LMFDB - Embedded newform 315.2.bl.f.146.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,2,Mod(41,315)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([5, 0, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("315.41");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	315.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.51528766367$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			146.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	315.146
Dual form			315.2.bl.f.41.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})$	\(q+(-1.50000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-3.00000 - 1.73205i) q^{11} +(3.00000 + 1.73205i) q^{12} +(-4.50000 + 2.59808i) q^{13} +(-1.50000 - 0.866025i) q^{15} +(-2.00000 + 3.46410i) q^{16} -3.00000 q^{17} +3.46410i q^{19} +(1.00000 - 1.73205i) q^{20} +(-3.00000 - 3.46410i) q^{21} +(-6.00000 + 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.19615i q^{27} +(4.00000 - 3.46410i) q^{28} +(1.50000 + 0.866025i) q^{29} +(-3.00000 + 1.73205i) q^{31} +6.00000 q^{33} +(-2.00000 + 1.73205i) q^{35} -6.00000 q^{36} +10.0000 q^{37} +(4.50000 - 7.79423i) q^{39} +(-6.00000 - 10.3923i) q^{41} +(4.00000 - 6.92820i) q^{43} +6.92820i q^{44} +3.00000 q^{45} -6.92820i q^{48} +(-6.50000 + 2.59808i) q^{49} +(4.50000 - 2.59808i) q^{51} +(9.00000 + 5.19615i) q^{52} -3.46410i q^{53} -3.46410i q^{55} +(-3.00000 - 5.19615i) q^{57} +3.46410i q^{60} +(6.00000 + 3.46410i) q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +(-4.50000 - 2.59808i) q^{65} +(5.00000 + 8.66025i) q^{67} +(3.00000 + 5.19615i) q^{68} +(6.00000 - 10.3923i) q^{69} +12.1244i q^{71} +1.73205i q^{73} -1.73205i q^{75} +(6.00000 - 3.46410i) q^{76} +(3.00000 - 8.66025i) q^{77} +(-4.00000 + 6.92820i) q^{79} -4.00000 q^{80} +(-4.50000 - 7.79423i) q^{81} +(-4.50000 + 7.79423i) q^{83} +(-3.00000 + 8.66025i) q^{84} +(-1.50000 - 2.59808i) q^{85} -3.00000 q^{87} +6.00000 q^{89} +(-9.00000 - 10.3923i) q^{91} +(12.0000 + 6.92820i) q^{92} +(3.00000 - 5.19615i) q^{93} +(-3.00000 + 1.73205i) q^{95} +(-6.00000 - 3.46410i) q^{97} +(-9.00000 + 5.19615i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - 2 q^{4} + q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - 3 * q^3 - 2 * q^4 + q^5 + q^7 + 3 * q^9	$ 2 q - 3 q^{3} - 2 q^{4} + q^{5} + q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{12} - 9 q^{13} - 3 q^{15} - 4 q^{16} - 6 q^{17} + 2 q^{20} - 6 q^{21} - 12 q^{23} - q^{25} + 8 q^{28} + 3 q^{29} - 6 q^{31} + 12 q^{33} - 4 q^{35} - 12 q^{36} + 20 q^{37} + 9 q^{39} - 12 q^{41} + 8 q^{43} + 6 q^{45} - 13 q^{49} + 9 q^{51} + 18 q^{52} - 6 q^{57} + 12 q^{61} + 15 q^{63} + 16 q^{64} - 9 q^{65} + 10 q^{67} + 6 q^{68} + 12 q^{69} + 12 q^{76} + 6 q^{77} - 8 q^{79} - 8 q^{80} - 9 q^{81} - 9 q^{83} - 6 q^{84} - 3 q^{85} - 6 q^{87} + 12 q^{89} - 18 q^{91} + 24 q^{92} + 6 q^{93} - 6 q^{95} - 12 q^{97} - 18 q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - 2 * q^4 + q^5 + q^7 + 3 * q^9 - 6 * q^11 + 6 * q^12 - 9 * q^13 - 3 * q^15 - 4 * q^16 - 6 * q^17 + 2 * q^20 - 6 * q^21 - 12 * q^23 - q^25 + 8 * q^28 + 3 * q^29 - 6 * q^31 + 12 * q^33 - 4 * q^35 - 12 * q^36 + 20 * q^37 + 9 * q^39 - 12 * q^41 + 8 * q^43 + 6 * q^45 - 13 * q^49 + 9 * q^51 + 18 * q^52 - 6 * q^57 + 12 * q^61 + 15 * q^63 + 16 * q^64 - 9 * q^65 + 10 * q^67 + 6 * q^68 + 12 * q^69 + 12 * q^76 + 6 * q^77 - 8 * q^79 - 8 * q^80 - 9 * q^81 - 9 * q^83 - 6 * q^84 - 3 * q^85 - 6 * q^87 + 12 * q^89 - 18 * q^91 + 24 * q^92 + 6 * q^93 - 6 * q^95 - 12 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.50000	+	0.866025i	−0.866025	+	0.500000i
$3$	−1.50000	+	0.866025i	−0.866025	+	0.500000i
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$7$	0.500000	+	2.59808i	0.188982	+	0.981981i
$8$		0			0
$9$	1.50000	−	2.59808i	0.500000	−	0.866025i
$10$		0			0
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.878668	+	0.477432i	$0.841568\pi$
$12$	3.00000	+	1.73205i	0.866025	+	0.500000i
$13$	−4.50000	+	2.59808i	−1.24808	+	0.720577i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−4.50000	+	2.59808i	−1.24808	+	0.720577i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$	−1.50000	−	0.866025i	−0.387298	−	0.223607i
$16$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$17$	−3.00000			−0.727607			−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000			−0.727607			−0.363803	+	0.931476i	$0.618522\pi$
$18$		0			0
$19$			3.46410i			0.794719i	0.917663	+	0.397360i	$0.130073\pi$
$19$			3.46410i			0.794719i	−0.917663	+	0.397360i	$0.869927\pi$
$20$	1.00000	−	1.73205i	0.223607	−	0.387298i
$21$	−3.00000	−	3.46410i	−0.654654	−	0.755929i
$22$		0			0
$23$	−6.00000	+	3.46410i	−1.25109	+	0.722315i	−0.971325	−	0.237754i	$-0.923589\pi$
$23$	−6.00000	+	3.46410i	−1.25109	+	0.722315i	−0.279761	+	0.960070i	$0.590255\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$			5.19615i			1.00000i
$28$	4.00000	−	3.46410i	0.755929	−	0.654654i
$29$	1.50000	+	0.866025i	0.278543	+	0.160817i	0.632764	−	0.774345i	$-0.281920\pi$
$29$	1.50000	+	0.866025i	0.278543	+	0.160817i	−0.354221	+	0.935162i	$0.615254\pi$
$30$		0			0
$31$	−3.00000	+	1.73205i	−0.538816	+	0.311086i	−0.744599	−	0.667512i	$-0.767359\pi$
$31$	−3.00000	+	1.73205i	−0.538816	+	0.311086i	0.205783	+	0.978598i	$0.434026\pi$
$32$		0			0
$33$	6.00000			1.04447
$34$		0			0
$35$	−2.00000	+	1.73205i	−0.338062	+	0.292770i
$36$	−6.00000			−1.00000
$37$	10.0000			1.64399			0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000			1.64399			0.821995	+	0.569495i	$0.192861\pi$
$38$		0			0
$39$	4.50000	−	7.79423i	0.720577	−	1.24808i
$40$		0			0
$41$	−6.00000	−	10.3923i	−0.937043	−	1.62301i	−0.770950	−	0.636895i	$-0.780218\pi$
$41$	−6.00000	−	10.3923i	−0.937043	−	1.62301i	−0.166092	−	0.986110i	$-0.553115\pi$
$42$		0			0
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	4.00000	−	6.92820i	0.609994	−	1.05654i	0.991241	−	0.132068i	$-0.0421616\pi$
$44$			6.92820i			1.04447i
$45$	3.00000			0.447214
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		−	6.92820i		−	1.00000i
$49$	−6.50000	+	2.59808i	−0.928571	+	0.371154i
$50$		0			0
$51$	4.50000	−	2.59808i	0.630126	−	0.363803i
$52$	9.00000	+	5.19615i	1.24808	+	0.720577i
$53$		−	3.46410i		−	0.475831i	−0.971286	−	0.237915i	$-0.923536\pi$
$53$		−	3.46410i		−	0.475831i	0.971286	−	0.237915i	$-0.0764641\pi$
$54$		0			0
$55$		−	3.46410i		−	0.467099i
$56$		0			0
$57$	−3.00000	−	5.19615i	−0.397360	−	0.688247i
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$			3.46410i			0.447214i
$61$	6.00000	+	3.46410i	0.768221	+	0.443533i	0.832240	−	0.554416i	$-0.187058\pi$
$61$	6.00000	+	3.46410i	0.768221	+	0.443533i	−0.0640184	+	0.997949i	$0.520392\pi$
$62$		0			0
$63$	7.50000	+	2.59808i	0.944911	+	0.327327i
$64$	8.00000			1.00000
$65$	−4.50000	−	2.59808i	−0.558156	−	0.322252i
$66$		0			0
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	0.991098	+	0.133135i	$0.0425044\pi$
$67$	5.00000	+	8.66025i	0.610847	+	1.05802i	−0.380251	+	0.924883i	$0.624162\pi$
$68$	3.00000	+	5.19615i	0.363803	+	0.630126i
$69$	6.00000	−	10.3923i	0.722315	−	1.25109i
$70$		0			0
$71$			12.1244i			1.43890i	0.694546	+	0.719448i	$0.255605\pi$
$71$			12.1244i			1.43890i	−0.694546	+	0.719448i	$0.744395\pi$
$72$		0			0
$73$			1.73205i			0.202721i	0.994850	+	0.101361i	$0.0323196\pi$
$73$			1.73205i			0.202721i	−0.994850	+	0.101361i	$0.967680\pi$
$74$		0			0
$75$		−	1.73205i		−	0.200000i
$76$	6.00000	−	3.46410i	0.688247	−	0.397360i
$77$	3.00000	−	8.66025i	0.341882	−	0.986928i
$78$		0			0
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	−0.998388	−	0.0567635i	$-0.981922\pi$
$79$	−4.00000	+	6.92820i	−0.450035	+	0.779484i	0.548352	+	0.836247i	$0.315255\pi$
$80$	−4.00000			−0.447214
$81$	−4.50000	−	7.79423i	−0.500000	−	0.866025i
$82$		0			0
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	−0.999976	−	0.00698436i	$-0.997777\pi$
$83$	−4.50000	+	7.79423i	−0.493939	+	0.855528i	0.506036	+	0.862512i	$0.331110\pi$
$84$	−3.00000	+	8.66025i	−0.327327	+	0.944911i
$85$	−1.50000	−	2.59808i	−0.162698	−	0.281801i
$86$		0			0
$87$	−3.00000			−0.321634
$88$		0			0
$89$	6.00000			0.635999			0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000			0.635999			0.317999	+	0.948091i	$0.396989\pi$
$90$		0			0
$91$	−9.00000	−	10.3923i	−0.943456	−	1.08941i
$92$	12.0000	+	6.92820i	1.25109	+	0.722315i
$93$	3.00000	−	5.19615i	0.311086	−	0.538816i
$94$		0			0
$95$	−3.00000	+	1.73205i	−0.307794	+	0.177705i
$96$		0			0
$97$	−6.00000	−	3.46410i	−0.609208	−	0.351726i	0.163448	−	0.986552i	$-0.447739\pi$
$97$	−6.00000	−	3.46410i	−0.609208	−	0.351726i	−0.772655	+	0.634826i	$0.781072\pi$
$98$		0			0
$99$	−9.00000	+	5.19615i	−0.904534	+	0.522233i
$100$	2.00000			0.200000
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.0623905	−	0.998052i	$-0.480128\pi$
$101$	9.00000	−	15.5885i	0.895533	−	1.55111i	0.833143	−	0.553058i	$-0.186539\pi$
$102$		0			0
$103$	−7.50000	+	4.33013i	−0.738997	+	0.426660i	−0.821705	−	0.569914i	$-0.806977\pi$
$103$	−7.50000	+	4.33013i	−0.738997	+	0.426660i	0.0827075	+	0.996574i	$0.473643\pi$
$104$		0			0
$105$	1.50000	−	4.33013i	0.146385	−	0.422577i
$106$		0			0
$107$		−	17.3205i		−	1.67444i	−0.546869	−	0.837218i	$-0.684180\pi$
$107$		−	17.3205i		−	1.67444i	0.546869	−	0.837218i	$-0.315820\pi$
$108$	9.00000	−	5.19615i	0.866025	−	0.500000i
$109$	−2.00000			−0.191565			−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000			−0.191565			−0.0957826	+	0.995402i	$0.530535\pi$
$110$		0			0
$111$	−15.0000	+	8.66025i	−1.42374	+	0.821995i
$112$	−10.0000	−	3.46410i	−0.944911	−	0.327327i
$113$	−9.00000	+	5.19615i	−0.846649	+	0.488813i	−0.859519	−	0.511104i	$-0.829237\pi$
$113$	−9.00000	+	5.19615i	−0.846649	+	0.488813i	0.0128699	+	0.999917i	$0.495903\pi$
$114$		0			0
$115$	−6.00000	−	3.46410i	−0.559503	−	0.323029i
$116$		−	3.46410i		−	0.321634i
$117$			15.5885i			1.44115i
$118$		0			0
$119$	−1.50000	−	7.79423i	−0.137505	−	0.714496i
$120$		0			0
$121$	0.500000	+	0.866025i	0.0454545	+	0.0787296i
$122$		0			0
$123$	18.0000	+	10.3923i	1.62301	+	0.937043i
$124$	6.00000	+	3.46410i	0.538816	+	0.311086i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$			13.8564i			1.21999i
$130$		0			0
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	0.966803	−	0.255524i	$-0.0822479\pi$
$131$	3.00000	+	5.19615i	0.262111	+	0.453990i	−0.704692	+	0.709514i	$0.748915\pi$
$132$	−6.00000	−	10.3923i	−0.522233	−	0.904534i
$133$	−9.00000	+	1.73205i	−0.780399	+	0.150188i
$134$		0			0
$135$	−4.50000	+	2.59808i	−0.387298	+	0.223607i
$136$		0			0
$137$	−3.00000	−	1.73205i	−0.256307	−	0.147979i	0.366342	−	0.930480i	$-0.380610\pi$
$137$	−3.00000	−	1.73205i	−0.256307	−	0.147979i	−0.622649	+	0.782501i	$0.713943\pi$
$138$		0			0
$139$	6.00000	−	3.46410i	0.508913	−	0.293821i	−0.223474	−	0.974710i	$-0.571740\pi$
$139$	6.00000	−	3.46410i	0.508913	−	0.293821i	0.732387	+	0.680889i	$0.238406\pi$
$140$	5.00000	+	1.73205i	0.422577	+	0.146385i
$141$		0			0
$142$		0			0
$143$	18.0000			1.50524
$144$	6.00000	+	10.3923i	0.500000	+	0.866025i
$145$			1.73205i			0.143839i
$146$		0			0
$147$	7.50000	−	9.52628i	0.618590	−	0.785714i
$148$	−10.0000	−	17.3205i	−0.821995	−	1.42374i
$149$	4.50000	−	2.59808i	0.368654	−	0.212843i	−0.304216	−	0.952603i	$-0.598394\pi$
$149$	4.50000	−	2.59808i	0.368654	−	0.212843i	0.672870	+	0.739760i	$0.265061\pi$
$150$		0			0
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	0.162758	+	0.986666i	$0.447961\pi$
$151$	−9.50000	+	16.4545i	−0.773099	+	1.33905i	−0.935857	+	0.352381i	$0.885372\pi$
$152$		0			0
$153$	−4.50000	+	7.79423i	−0.363803	+	0.630126i
$154$		0			0
$155$	−3.00000	−	1.73205i	−0.240966	−	0.139122i
$156$	−18.0000			−1.44115
$157$	7.50000	−	4.33013i	0.598565	−	0.345582i	−0.169912	−	0.985459i	$-0.554348\pi$
$157$	7.50000	−	4.33013i	0.598565	−	0.345582i	0.768477	+	0.639878i	$0.221015\pi$
$158$		0			0
$159$	3.00000	+	5.19615i	0.237915	+	0.412082i
$160$		0			0
$161$	−12.0000	−	13.8564i	−0.945732	−	1.09204i
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	−12.0000	+	20.7846i	−0.937043	+	1.62301i
$165$	3.00000	+	5.19615i	0.233550	+	0.404520i
$166$		0			0
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	0.985933	−	0.167139i	$-0.0534527\pi$
$167$	4.50000	+	7.79423i	0.348220	+	0.603136i	−0.637713	+	0.770274i	$0.720119\pi$
$168$		0			0
$169$	7.00000	−	12.1244i	0.538462	−	0.932643i
$170$		0			0
$171$	9.00000	+	5.19615i	0.688247	+	0.397360i
$172$	−16.0000			−1.21999
$173$	−10.5000	+	18.1865i	−0.798300	+	1.38270i	0.122422	+	0.992478i	$0.460934\pi$
$173$	−10.5000	+	18.1865i	−0.798300	+	1.38270i	−0.920722	+	0.390218i	$0.872399\pi$
$174$		0			0
$175$	−2.50000	−	0.866025i	−0.188982	−	0.0654654i
$176$	12.0000	−	6.92820i	0.904534	−	0.522233i
$177$		0			0
$178$		0			0
$179$		−	15.5885i		−	1.16514i	−0.812782	−	0.582568i	$-0.802048\pi$
$179$		−	15.5885i		−	1.16514i	0.812782	−	0.582568i	$-0.197952\pi$
$180$	−3.00000	−	5.19615i	−0.223607	−	0.387298i
$181$			10.3923i			0.772454i	0.922404	+	0.386227i	$0.126222\pi$
$181$			10.3923i			0.772454i	−0.922404	+	0.386227i	$0.873778\pi$
$182$		0			0
$183$	−12.0000			−0.887066
$184$		0			0
$185$	5.00000	+	8.66025i	0.367607	+	0.636715i
$186$		0			0
$187$	9.00000	+	5.19615i	0.658145	+	0.379980i
$188$		0			0
$189$	−13.5000	+	2.59808i	−0.981981	+	0.188982i
$190$		0			0
$191$	10.5000	+	6.06218i	0.759753	+	0.438644i	0.829207	−	0.558941i	$-0.188792\pi$
$191$	10.5000	+	6.06218i	0.759753	+	0.438644i	−0.0694538	+	0.997585i	$0.522126\pi$
$192$	−12.0000	+	6.92820i	−0.866025	+	0.500000i
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	0.785022	−	0.619467i	$-0.212651\pi$
$193$	−2.00000	−	3.46410i	−0.143963	−	0.249351i	−0.928986	+	0.370116i	$0.879318\pi$
$194$		0			0
$195$	9.00000			0.644503
$196$	11.0000	+	8.66025i	0.785714	+	0.618590i
$197$			24.2487i			1.72765i	0.503793	+	0.863825i	$0.331938\pi$
$197$			24.2487i			1.72765i	−0.503793	+	0.863825i	$0.668062\pi$
$198$		0			0
$199$		−	10.3923i		−	0.736691i	−0.929689	−	0.368345i	$-0.879924\pi$
$199$		−	10.3923i		−	0.736691i	0.929689	−	0.368345i	$-0.120076\pi$
$200$		0			0
$201$	−15.0000	−	8.66025i	−1.05802	−	0.610847i
$202$		0			0
$203$	−1.50000	+	4.33013i	−0.105279	+	0.303915i
$204$	−9.00000	−	5.19615i	−0.630126	−	0.363803i
$205$	6.00000	−	10.3923i	0.419058	−	0.725830i
$206$		0			0
$207$			20.7846i			1.44463i
$208$		−	20.7846i		−	1.44115i
$209$	6.00000	−	10.3923i	0.415029	−	0.718851i
$210$		0			0
$211$	4.00000	+	6.92820i	0.275371	+	0.476957i	0.970229	−	0.242190i	$-0.0778659\pi$
$211$	4.00000	+	6.92820i	0.275371	+	0.476957i	−0.694857	+	0.719148i	$0.744533\pi$
$212$	−6.00000	+	3.46410i	−0.412082	+	0.237915i
$213$	−10.5000	−	18.1865i	−0.719448	−	1.24612i
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	−6.00000	−	6.92820i	−0.407307	−	0.470317i
$218$		0			0
$219$	−1.50000	−	2.59808i	−0.101361	−	0.175562i
$220$	−6.00000	+	3.46410i	−0.404520	+	0.233550i
$221$	13.5000	−	7.79423i	0.908108	−	0.524297i
$222$		0			0
$223$	−16.5000	−	9.52628i	−1.10492	−	0.637927i	−0.167412	−	0.985887i	$-0.553541\pi$
$223$	−16.5000	−	9.52628i	−1.10492	−	0.637927i	−0.937509	+	0.347960i	$0.886874\pi$
$224$		0			0
$225$	1.50000	+	2.59808i	0.100000	+	0.173205i
$226$		0			0
$227$	−1.50000	+	2.59808i	−0.0995585	+	0.172440i	−0.911502	−	0.411296i	$-0.865076\pi$
$227$	−1.50000	+	2.59808i	−0.0995585	+	0.172440i	0.811943	+	0.583736i	$0.198410\pi$
$228$	−6.00000	+	10.3923i	−0.397360	+	0.688247i
$229$	3.00000	−	1.73205i	0.198246	−	0.114457i	−0.397591	−	0.917563i	$-0.630154\pi$
$229$	3.00000	−	1.73205i	0.198246	−	0.114457i	0.595837	+	0.803105i	$0.296820\pi$
$230$		0			0
$231$	3.00000	+	15.5885i	0.197386	+	1.02565i
$232$		0			0
$233$			6.92820i			0.453882i	0.973909	+	0.226941i	$0.0728724\pi$
$233$			6.92820i			0.453882i	−0.973909	+	0.226941i	$0.927128\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	13.8564i		−	0.900070i
$238$		0			0
$239$	1.50000	−	0.866025i	0.0970269	−	0.0560185i	−0.450701	−	0.892675i	$-0.648826\pi$
$239$	1.50000	−	0.866025i	0.0970269	−	0.0560185i	0.547728	+	0.836656i	$0.315493\pi$
$240$	6.00000	−	3.46410i	0.387298	−	0.223607i
$241$	9.00000	+	5.19615i	0.579741	+	0.334714i	0.761030	−	0.648716i	$-0.224694\pi$
$241$	9.00000	+	5.19615i	0.579741	+	0.334714i	−0.181289	+	0.983430i	$0.558027\pi$
$242$		0			0
$243$	13.5000	+	7.79423i	0.866025	+	0.500000i
$244$		−	13.8564i		−	0.887066i
$245$	−5.50000	−	4.33013i	−0.351382	−	0.276642i
$246$		0			0
$247$	−9.00000	−	15.5885i	−0.572656	−	0.991870i
$248$		0			0
$249$		−	15.5885i		−	0.987878i
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−3.00000	−	15.5885i	−0.188982	−	0.981981i
$253$	24.0000			1.50887
$254$		0			0
$255$	4.50000	+	2.59808i	0.281801	+	0.162698i
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−7.50000	−	12.9904i	−0.467837	−	0.810318i	0.531487	−	0.847066i	$-0.321633\pi$
$257$	−7.50000	−	12.9904i	−0.467837	−	0.810318i	−0.999325	+	0.0367485i	$0.988300\pi$
$258$		0			0
$259$	5.00000	+	25.9808i	0.310685	+	1.61437i
$260$			10.3923i			0.644503i
$261$	4.50000	−	2.59808i	0.278543	−	0.160817i
$262$		0			0
$263$	−3.00000	−	1.73205i	−0.184988	−	0.106803i	0.404646	−	0.914473i	$-0.367395\pi$
$263$	−3.00000	−	1.73205i	−0.184988	−	0.106803i	−0.589634	+	0.807671i	$0.700728\pi$
$264$		0			0
$265$	3.00000	−	1.73205i	0.184289	−	0.106399i
$266$		0			0
$267$	−9.00000	+	5.19615i	−0.550791	+	0.317999i
$268$	10.0000	−	17.3205i	0.610847	−	1.05802i
$269$	−24.0000			−1.46331			−0.731653	−	0.681677i	$-0.761251\pi$
$269$	−24.0000			−1.46331			−0.731653	+	0.681677i	$0.761251\pi$
$270$		0			0
$271$			3.46410i			0.210429i	0.994450	+	0.105215i	$0.0335529\pi$
$271$			3.46410i			0.210429i	−0.994450	+	0.105215i	$0.966447\pi$
$272$	6.00000	−	10.3923i	0.363803	−	0.630126i
$273$	22.5000	+	7.79423i	1.36176	+	0.471728i
$274$		0			0
$275$	3.00000	−	1.73205i	0.180907	−	0.104447i
$276$	−24.0000			−1.44463
$277$	−13.0000	+	22.5167i	−0.781094	+	1.35290i	0.150210	+	0.988654i	$0.452005\pi$
$277$	−13.0000	+	22.5167i	−0.781094	+	1.35290i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$			10.3923i			0.622171i
$280$		0			0
$281$	12.0000	+	6.92820i	0.715860	+	0.413302i	0.813227	−	0.581947i	$-0.197709\pi$
$281$	12.0000	+	6.92820i	0.715860	+	0.413302i	−0.0973670	+	0.995249i	$0.531042\pi$
$282$		0			0
$283$	−3.00000	+	1.73205i	−0.178331	+	0.102960i	−0.586509	−	0.809943i	$-0.699498\pi$
$283$	−3.00000	+	1.73205i	−0.178331	+	0.102960i	0.408177	+	0.912903i	$0.366165\pi$
$284$	21.0000	−	12.1244i	1.24612	−	0.719448i
$285$	3.00000	−	5.19615i	0.177705	−	0.307794i
$286$		0			0
$287$	24.0000	−	20.7846i	1.41668	−	1.22688i
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$	12.0000			0.703452
$292$	3.00000	−	1.73205i	0.175562	−	0.101361i
$293$	4.50000	+	7.79423i	0.262893	+	0.455344i	0.967009	−	0.254741i	$-0.0819901\pi$
$293$	4.50000	+	7.79423i	0.262893	+	0.455344i	−0.704117	+	0.710084i	$0.748657\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	9.00000	−	15.5885i	0.522233	−	0.904534i
$298$		0			0
$299$	18.0000	−	31.1769i	1.04097	−	1.80301i
$300$	−3.00000	+	1.73205i	−0.173205	+	0.100000i
$301$	20.0000	+	6.92820i	1.15278	+	0.399335i
$302$		0			0
$303$			31.1769i			1.79107i
$304$	−12.0000	−	6.92820i	−0.688247	−	0.397360i
$305$			6.92820i			0.396708i
$306$		0			0
$307$			22.5167i			1.28509i	0.766246	+	0.642547i	$0.222122\pi$
$307$			22.5167i			1.28509i	−0.766246	+	0.642547i	$0.777878\pi$
$308$	−18.0000	+	3.46410i	−1.02565	+	0.197386i
$309$	7.50000	−	12.9904i	0.426660	−	0.738997i
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−4.50000	−	2.59808i	−0.254355	−	0.146852i	0.367402	−	0.930062i	$-0.380247\pi$
$313$	−4.50000	−	2.59808i	−0.254355	−	0.146852i	−0.621757	+	0.783210i	$0.713581\pi$
$314$		0			0
$315$	1.50000	+	7.79423i	0.0845154	+	0.439155i
$316$	16.0000			0.900070
$317$	21.0000	+	12.1244i	1.17948	+	0.680972i	0.955894	−	0.293713i	$-0.0948910\pi$
$317$	21.0000	+	12.1244i	1.17948	+	0.680972i	0.223584	+	0.974685i	$0.428224\pi$
$318$		0			0
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$	4.00000	+	6.92820i	0.223607	+	0.387298i
$321$	15.0000	+	25.9808i	0.837218	+	1.45010i
$322$		0			0
$323$		−	10.3923i		−	0.578243i
$324$	−9.00000	+	15.5885i	−0.500000	+	0.866025i
$325$		−	5.19615i		−	0.288231i
$326$		0			0
$327$	3.00000	−	1.73205i	0.165900	−	0.0957826i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.500000	+	0.866025i	−0.0274825	+	0.0476011i	−0.879440	−	0.476011i	$-0.842082\pi$
$331$	−0.500000	+	0.866025i	−0.0274825	+	0.0476011i	0.851957	+	0.523612i	$0.175416\pi$
$332$	18.0000			0.987878
$333$	15.0000	−	25.9808i	0.821995	−	1.42374i
$334$		0			0
$335$	−5.00000	+	8.66025i	−0.273179	+	0.473160i
$336$	18.0000	−	3.46410i	0.981981	−	0.188982i
$337$	4.00000	+	6.92820i	0.217894	+	0.377403i	0.954164	−	0.299285i	$-0.0967480\pi$
$337$	4.00000	+	6.92820i	0.217894	+	0.377403i	−0.736270	+	0.676688i	$0.763415\pi$
$338$		0			0
$339$	9.00000	−	15.5885i	0.488813	−	0.846649i
$340$	−3.00000	+	5.19615i	−0.162698	+	0.281801i
$341$	12.0000			0.649836
$342$		0			0
$343$	−10.0000	−	15.5885i	−0.539949	−	0.841698i
$344$		0			0
$345$	12.0000			0.646058
$346$		0			0
$347$	−21.0000	+	12.1244i	−1.12734	+	0.650870i	−0.943264	−	0.332043i	$-0.892262\pi$
$347$	−21.0000	+	12.1244i	−1.12734	+	0.650870i	−0.184075	+	0.982912i	$0.558929\pi$
$348$	3.00000	+	5.19615i	0.160817	+	0.278543i
$349$	−6.00000	−	3.46410i	−0.321173	−	0.185429i	0.330743	−	0.943721i	$-0.392701\pi$
$349$	−6.00000	−	3.46410i	−0.321173	−	0.185429i	−0.651915	+	0.758292i	$0.726034\pi$
$350$		0			0
$351$	−13.5000	−	23.3827i	−0.720577	−	1.24808i
$352$		0			0
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	−0.903179	−	0.429263i	$-0.858773\pi$
$353$	−1.50000	+	2.59808i	−0.0798369	+	0.138282i	0.823343	+	0.567545i	$0.192107\pi$
$354$		0			0
$355$	−10.5000	+	6.06218i	−0.557282	+	0.321747i
$356$	−6.00000	−	10.3923i	−0.317999	−	0.550791i
$357$	9.00000	+	10.3923i	0.476331	+	0.550019i
$358$		0			0
$359$			31.1769i			1.64545i	0.568436	+	0.822727i	$0.307549\pi$
$359$			31.1769i			1.64545i	−0.568436	+	0.822727i	$0.692451\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$	−1.50000	−	0.866025i	−0.0787296	−	0.0454545i
$364$	−9.00000	+	25.9808i	−0.471728	+	1.36176i
$365$	−1.50000	+	0.866025i	−0.0785136	+	0.0453298i
$366$		0			0
$367$	−22.5000	−	12.9904i	−1.17449	−	0.678092i	−0.219757	−	0.975555i	$-0.570526\pi$
$367$	−22.5000	−	12.9904i	−1.17449	−	0.678092i	−0.954734	+	0.297462i	$0.903860\pi$
$368$		−	27.7128i		−	1.44463i
$369$	−36.0000			−1.87409
$370$		0			0
$371$	9.00000	−	1.73205i	0.467257	−	0.0899236i
$372$	−12.0000			−0.622171
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	0.913147	−	0.407630i	$-0.133645\pi$
$373$	2.00000	+	3.46410i	0.103556	+	0.179364i	−0.809591	+	0.586994i	$0.800311\pi$
$374$		0			0
$375$	1.50000	−	0.866025i	0.0774597	−	0.0447214i
$376$		0			0
$377$	−9.00000			−0.463524
$378$		0			0
$379$	−5.00000			−0.256833			−0.128416	−	0.991720i	$-0.540989\pi$
$379$	−5.00000			−0.256833			−0.128416	+	0.991720i	$0.540989\pi$
$380$	6.00000	+	3.46410i	0.307794	+	0.177705i
$381$	−3.00000	+	1.73205i	−0.153695	+	0.0887357i
$382$		0			0
$383$	−7.50000	−	12.9904i	−0.383232	−	0.663777i	0.608290	−	0.793715i	$-0.291856\pi$
$383$	−7.50000	−	12.9904i	−0.383232	−	0.663777i	−0.991522	+	0.129937i	$0.958522\pi$
$384$		0			0
$385$	9.00000	−	1.73205i	0.458682	−	0.0882735i
$386$		0			0
$387$	−12.0000	−	20.7846i	−0.609994	−	1.05654i
$388$			13.8564i			0.703452i
$389$	−24.0000	−	13.8564i	−1.21685	−	0.702548i	−0.252606	−	0.967569i	$-0.581288\pi$
$389$	−24.0000	−	13.8564i	−1.21685	−	0.702548i	−0.964242	+	0.265022i	$0.914621\pi$
$390$		0			0
$391$	18.0000	−	10.3923i	0.910299	−	0.525561i
$392$		0			0
$393$	−9.00000	−	5.19615i	−0.453990	−	0.262111i
$394$		0			0
$395$	−8.00000			−0.402524
$396$	18.0000	+	10.3923i	0.904534	+	0.522233i
$397$		−	32.9090i		−	1.65165i	−0.563924	−	0.825827i	$-0.690709\pi$
$397$		−	32.9090i		−	1.65165i	0.563924	−	0.825827i	$-0.309291\pi$
$398$		0			0
$399$	12.0000	−	10.3923i	0.600751	−	0.520266i
$400$	−2.00000	−	3.46410i	−0.100000	−	0.173205i
$401$	4.50000	−	2.59808i	0.224719	−	0.129742i	−0.383414	−	0.923576i	$-0.625252\pi$
$401$	4.50000	−	2.59808i	0.224719	−	0.129742i	0.608134	+	0.793835i	$0.291919\pi$
$402$		0			0
$403$	9.00000	−	15.5885i	0.448322	−	0.776516i
$404$	−36.0000			−1.79107
$405$	4.50000	−	7.79423i	0.223607	−	0.387298i
$406$		0			0
$407$	−30.0000	−	17.3205i	−1.48704	−	0.858546i
$408$		0			0
$409$	18.0000	−	10.3923i	0.890043	−	0.513866i	0.0160862	−	0.999871i	$-0.494879\pi$
$409$	18.0000	−	10.3923i	0.890043	−	0.513866i	0.873956	+	0.486004i	$0.161546\pi$
$410$		0			0
$411$	6.00000			0.295958
$412$	15.0000	+	8.66025i	0.738997	+	0.426660i
$413$		0			0
$414$		0			0
$415$	−9.00000			−0.441793
$416$		0			0
$417$	−6.00000	+	10.3923i	−0.293821	+	0.508913i
$418$		0			0
$419$	15.0000	+	25.9808i	0.732798	+	1.26924i	0.955683	+	0.294398i	$0.0951193\pi$
$419$	15.0000	+	25.9808i	0.732798	+	1.26924i	−0.222885	+	0.974845i	$0.571547\pi$
$420$	−9.00000	+	1.73205i	−0.439155	+	0.0845154i
$421$	11.0000	−	19.0526i	0.536107	−	0.928565i	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	11.0000	−	19.0526i	0.536107	−	0.928565i	0.999109	−	0.0422075i	$-0.0134391\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.50000	−	2.59808i	0.0727607	−	0.126025i
$426$		0			0
$427$	−6.00000	+	17.3205i	−0.290360	+	0.838198i
$428$	−30.0000	+	17.3205i	−1.45010	+	0.837218i
$429$	−27.0000	+	15.5885i	−1.30357	+	0.752618i
$430$		0			0
$431$		−	15.5885i		−	0.750870i	−0.926849	−	0.375435i	$-0.877493\pi$
$431$		−	15.5885i		−	0.750870i	0.926849	−	0.375435i	$-0.122507\pi$
$432$	−18.0000	−	10.3923i	−0.866025	−	0.500000i
$433$			34.6410i			1.66474i	0.554220	+	0.832370i	$0.313017\pi$
$433$			34.6410i			1.66474i	−0.554220	+	0.832370i	$0.686983\pi$
$434$		0			0
$435$	−1.50000	−	2.59808i	−0.0719195	−	0.124568i
$436$	2.00000	+	3.46410i	0.0957826	+	0.165900i
$437$	−12.0000	−	20.7846i	−0.574038	−	0.994263i
$438$		0			0
$439$	6.00000	+	3.46410i	0.286364	+	0.165333i	0.636301	−	0.771441i	$-0.280464\pi$
$439$	6.00000	+	3.46410i	0.286364	+	0.165333i	−0.349937	+	0.936773i	$0.613797\pi$
$440$		0			0
$441$	−3.00000	+	20.7846i	−0.142857	+	0.989743i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	30.0000	+	17.3205i	1.42374	+	0.821995i
$445$	3.00000	+	5.19615i	0.142214	+	0.246321i
$446$		0			0
$447$	−4.50000	+	7.79423i	−0.212843	+	0.368654i
$448$	4.00000	+	20.7846i	0.188982	+	0.981981i
$449$			36.3731i			1.71655i	0.513189	+	0.858276i	$0.328464\pi$
$449$			36.3731i			1.71655i	−0.513189	+	0.858276i	$0.671536\pi$
$450$		0			0
$451$			41.5692i			1.95742i
$452$	18.0000	+	10.3923i	0.846649	+	0.488813i
$453$		−	32.9090i		−	1.54620i
$454$		0			0
$455$	4.50000	−	12.9904i	0.210963	−	0.608998i
$456$		0			0
$457$	10.0000	−	17.3205i	0.467780	−	0.810219i	−0.531542	−	0.847032i	$-0.678387\pi$
$457$	10.0000	−	17.3205i	0.467780	−	0.810219i	0.999322	+	0.0368128i	$0.0117205\pi$
$458$		0			0
$459$		−	15.5885i		−	0.727607i
$460$			13.8564i			0.646058i
$461$	−15.0000	+	25.9808i	−0.698620	+	1.21004i	0.270326	+	0.962769i	$0.412869\pi$
$461$	−15.0000	+	25.9808i	−0.698620	+	1.21004i	−0.968945	+	0.247276i	$0.920465\pi$
$462$		0			0
$463$	11.0000	+	19.0526i	0.511213	+	0.885448i	0.999916	+	0.0129968i	$0.00413714\pi$
$463$	11.0000	+	19.0526i	0.511213	+	0.885448i	−0.488702	+	0.872451i	$0.662530\pi$
$464$	−6.00000	+	3.46410i	−0.278543	+	0.160817i
$465$	6.00000			0.278243
$466$		0			0
$467$	−36.0000			−1.66588			−0.832941	−	0.553362i	$-0.813345\pi$
$467$	−36.0000			−1.66588			−0.832941	+	0.553362i	$0.813345\pi$
$468$	27.0000	−	15.5885i	1.24808	−	0.720577i
$469$	−20.0000	+	17.3205i	−0.923514	+	0.799787i
$470$		0			0
$471$	−7.50000	+	12.9904i	−0.345582	+	0.598565i
$472$		0			0
$473$	−24.0000	+	13.8564i	−1.10352	+	0.637118i
$474$		0			0
$475$	−3.00000	−	1.73205i	−0.137649	−	0.0794719i
$476$	−12.0000	+	10.3923i	−0.550019	+	0.476331i
$477$	−9.00000	−	5.19615i	−0.412082	−	0.237915i
$478$		0			0
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	−0.583803	−	0.811895i	$-0.698436\pi$
$479$	9.00000	−	15.5885i	0.411220	−	0.712255i	0.995023	+	0.0996406i	$0.0317693\pi$
$480$		0			0
$481$	−45.0000	+	25.9808i	−2.05182	+	1.18462i
$482$		0			0
$483$	30.0000	+	10.3923i	1.36505	+	0.472866i
$484$	1.00000	−	1.73205i	0.0454545	−	0.0787296i
$485$		−	6.92820i		−	0.314594i
$486$		0			0
$487$	−2.00000			−0.0906287			−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000			−0.0906287			−0.0453143	+	0.998973i	$0.514429\pi$
$488$		0			0
$489$	6.00000	−	3.46410i	0.271329	−	0.156652i
$490$		0			0
$491$	−19.5000	+	11.2583i	−0.880023	+	0.508081i	−0.870666	−	0.491875i	$-0.836312\pi$
$491$	−19.5000	+	11.2583i	−0.880023	+	0.508081i	−0.00935679	+	0.999956i	$0.502978\pi$
$492$		−	41.5692i		−	1.87409i
$493$	−4.50000	−	2.59808i	−0.202670	−	0.117011i
$494$		0			0
$495$	−9.00000	−	5.19615i	−0.404520	−	0.233550i
$496$		−	13.8564i		−	0.622171i
$497$	−31.5000	+	6.06218i	−1.41297	+	0.271926i
$498$		0			0
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	0.962472	−	0.271380i	$-0.0874801\pi$
$499$	5.50000	+	9.52628i	0.246214	+	0.426455i	−0.716258	+	0.697835i	$0.754147\pi$
$500$	1.00000	+	1.73205i	0.0447214	+	0.0774597i
$501$	−13.5000	−	7.79423i	−0.603136	−	0.348220i
$502$		0			0
$503$	21.0000			0.936344			0.468172	−	0.883637i	$-0.344913\pi$
$503$	21.0000			0.936344			0.468172	+	0.883637i	$0.344913\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$		0			0
$507$			24.2487i			1.07692i
$508$	−2.00000	−	3.46410i	−0.0887357	−	0.153695i
$509$	3.00000	+	5.19615i	0.132973	+	0.230315i	0.924821	−	0.380402i	$-0.124214\pi$
$509$	3.00000	+	5.19615i	0.132973	+	0.230315i	−0.791849	+	0.610718i	$0.790881\pi$
$510$		0			0
$511$	−4.50000	+	0.866025i	−0.199068	+	0.0383107i
$512$		0			0
$513$	−18.0000			−0.794719
$514$		0			0
$515$	−7.50000	−	4.33013i	−0.330489	−	0.190808i
$516$	24.0000	−	13.8564i	1.05654	−	0.609994i
$517$		0			0
$518$		0			0
$519$		−	36.3731i		−	1.59660i
$520$		0			0
$521$	18.0000			0.788594			0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000			0.788594			0.394297	+	0.918983i	$0.370988\pi$
$522$		0			0
$523$		−	8.66025i		−	0.378686i	−0.981911	−	0.189343i	$-0.939364\pi$
$523$		−	8.66025i		−	0.378686i	0.981911	−	0.189343i	$-0.0606359\pi$
$524$	6.00000	−	10.3923i	0.262111	−	0.453990i
$525$	4.50000	−	0.866025i	0.196396	−	0.0377964i
$526$		0			0
$527$	9.00000	−	5.19615i	0.392046	−	0.226348i
$528$	−12.0000	+	20.7846i	−0.522233	+	0.904534i
$529$	12.5000	−	21.6506i	0.543478	−	0.941332i
$530$		0			0
$531$		0			0
$532$	12.0000	+	13.8564i	0.520266	+	0.600751i
$533$	54.0000	+	31.1769i	2.33900	+	1.35042i
$534$		0			0
$535$	15.0000	−	8.66025i	0.648507	−	0.374415i
$536$		0			0
$537$	13.5000	+	23.3827i	0.582568	+	1.00904i
$538$		0			0
$539$	24.0000	+	3.46410i	1.03375	+	0.149209i
$540$	9.00000	+	5.19615i	0.387298	+	0.223607i
$541$	17.0000			0.730887			0.365444	−	0.930834i	$-0.380917\pi$
$541$	17.0000			0.730887			0.365444	+	0.930834i	$0.380917\pi$
$542$		0			0
$543$	−9.00000	−	15.5885i	−0.386227	−	0.668965i
$544$		0			0
$545$	−1.00000	−	1.73205i	−0.0428353	−	0.0741929i
$546$		0			0
$547$	11.0000	−	19.0526i	0.470326	−	0.814629i	−0.529098	−	0.848561i	$-0.677470\pi$
$547$	11.0000	−	19.0526i	0.470326	−	0.814629i	0.999424	+	0.0339321i	$0.0108030\pi$
$548$			6.92820i			0.295958i
$549$	18.0000	−	10.3923i	0.768221	−	0.443533i
$550$		0			0
$551$	−3.00000	+	5.19615i	−0.127804	+	0.221364i
$552$		0			0
$553$	−20.0000	−	6.92820i	−0.850487	−	0.294617i
$554$		0			0
$555$	−15.0000	−	8.66025i	−0.636715	−	0.367607i
$556$	−12.0000	−	6.92820i	−0.508913	−	0.293821i
$557$			20.7846i			0.880672i	0.897833	+	0.440336i	$0.145141\pi$
$557$			20.7846i			0.880672i	−0.897833	+	0.440336i	$0.854859\pi$
$558$		0			0
$559$			41.5692i			1.75819i
$560$	−2.00000	−	10.3923i	−0.0845154	−	0.439155i
$561$	−18.0000			−0.759961
$562$		0			0
$563$	−10.5000	−	18.1865i	−0.442522	−	0.766471i	0.555354	−	0.831614i	$-0.312583\pi$
$563$	−10.5000	−	18.1865i	−0.442522	−	0.766471i	−0.997876	+	0.0651433i	$0.979250\pi$
$564$		0			0
$565$	−9.00000	−	5.19615i	−0.378633	−	0.218604i
$566$		0			0
$567$	18.0000	−	15.5885i	0.755929	−	0.654654i
$568$		0			0
$569$	−31.5000	−	18.1865i	−1.32055	−	0.762419i	−0.336733	−	0.941600i	$-0.609322\pi$
$569$	−31.5000	−	18.1865i	−1.32055	−	0.762419i	−0.983816	+	0.179181i	$0.942655\pi$
$570$		0			0
$571$	15.5000	+	26.8468i	0.648655	+	1.12350i	0.983444	+	0.181210i	$0.0580014\pi$
$571$	15.5000	+	26.8468i	0.648655	+	1.12350i	−0.334790	+	0.942293i	$0.608665\pi$
$572$	−18.0000	−	31.1769i	−0.752618	−	1.30357i
$573$	−21.0000			−0.877288
$574$		0			0
$575$		−	6.92820i		−	0.288926i
$576$	12.0000	−	20.7846i	0.500000	−	0.866025i
$577$			13.8564i			0.576850i	0.957503	+	0.288425i	$0.0931316\pi$
$577$			13.8564i			0.576850i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		0			0
$579$	6.00000	+	3.46410i	0.249351	+	0.143963i
$580$	3.00000	−	1.73205i	0.124568	−	0.0719195i
$581$	−22.5000	−	7.79423i	−0.933457	−	0.323359i
$582$		0			0
$583$	−6.00000	+	10.3923i	−0.248495	+	0.430405i
$584$		0			0
$585$	−13.5000	+	7.79423i	−0.558156	+	0.322252i
$586$		0			0
$587$	6.00000	−	10.3923i	0.247647	−	0.428936i	−0.715226	−	0.698893i	$-0.753676\pi$
$587$	6.00000	−	10.3923i	0.247647	−	0.428936i	0.962872	+	0.269957i	$0.0870095\pi$
$588$	−24.0000	−	3.46410i	−0.989743	−	0.142857i
$589$	−6.00000	−	10.3923i	−0.247226	−	0.428207i
$590$		0			0
$591$	−21.0000	−	36.3731i	−0.863825	−	1.49619i
$592$	−20.0000	+	34.6410i	−0.821995	+	1.42374i
$593$	−15.0000			−0.615976			−0.307988	−	0.951390i	$-0.599656\pi$
$593$	−15.0000			−0.615976			−0.307988	+	0.951390i	$0.599656\pi$
$594$		0			0
$595$	6.00000	−	5.19615i	0.245976	−	0.213021i
$596$	−9.00000	−	5.19615i	−0.368654	−	0.212843i
$597$	9.00000	+	15.5885i	0.368345	+	0.637993i
$598$		0			0
$599$	31.5000	−	18.1865i	1.28706	−	0.743082i	0.308927	−	0.951086i	$-0.400030\pi$
$599$	31.5000	−	18.1865i	1.28706	−	0.743082i	0.978128	+	0.208004i	$0.0666967\pi$
$600$		0			0
$601$	15.0000	+	8.66025i	0.611863	+	0.353259i	0.773694	−	0.633559i	$-0.218407\pi$
$601$	15.0000	+	8.66025i	0.611863	+	0.353259i	−0.161831	+	0.986818i	$0.551740\pi$
$602$		0			0
$603$	30.0000			1.22169
$604$	38.0000			1.54620
$605$	−0.500000	+	0.866025i	−0.0203279	+	0.0352089i
$606$		0			0
$607$	34.5000	−	19.9186i	1.40031	−	0.808470i	0.405887	−	0.913923i	$-0.366962\pi$
$607$	34.5000	−	19.9186i	1.40031	−	0.808470i	0.994424	+	0.105453i	$0.0336291\pi$
$608$		0			0
$609$	−1.50000	−	7.79423i	−0.0607831	−	0.315838i
$610$		0			0
$611$		0			0
$612$	18.0000			0.727607
$613$	−16.0000			−0.646234			−0.323117	−	0.946359i	$-0.604731\pi$
$613$	−16.0000			−0.646234			−0.323117	+	0.946359i	$0.604731\pi$
$614$		0			0
$615$			20.7846i			0.838116i
$616$		0			0
$617$	12.0000	−	6.92820i	0.483102	−	0.278919i	−0.238606	−	0.971116i	$-0.576691\pi$
$617$	12.0000	−	6.92820i	0.483102	−	0.278919i	0.721708	+	0.692197i	$0.243357\pi$
$618$		0			0
$619$	−33.0000	−	19.0526i	−1.32638	−	0.765787i	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−33.0000	−	19.0526i	−1.32638	−	0.765787i	−0.984738	+	0.174042i	$0.944317\pi$
$620$			6.92820i			0.278243i
$621$	−18.0000	−	31.1769i	−0.722315	−	1.25109i
$622$		0			0
$623$	3.00000	+	15.5885i	0.120192	+	0.624538i
$624$	18.0000	+	31.1769i	0.720577	+	1.24808i
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$			20.7846i			0.830057i
$628$	−15.0000	−	8.66025i	−0.598565	−	0.345582i
$629$	−30.0000			−1.19618
$630$		0			0
$631$	−40.0000			−1.59237			−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000			−1.59237			−0.796187	+	0.605050i	$0.793153\pi$
$632$		0			0
$633$	−12.0000	−	6.92820i	−0.476957	−	0.275371i
$634$		0			0
$635$	1.00000	+	1.73205i	0.0396838	+	0.0687343i
$636$	6.00000	−	10.3923i	0.237915	−	0.412082i
$637$	22.5000	−	28.5788i	0.891482	−	1.13233i
$638$		0			0
$639$	31.5000	+	18.1865i	1.24612	+	0.719448i
$640$		0			0
$641$	18.0000	+	10.3923i	0.710957	+	0.410471i	0.811415	−	0.584470i	$-0.198698\pi$
$641$	18.0000	+	10.3923i	0.710957	+	0.410471i	−0.100458	+	0.994941i	$0.532031\pi$
$642$		0			0
$643$	27.0000	−	15.5885i	1.06478	−	0.614749i	0.138027	−	0.990429i	$-0.455924\pi$
$643$	27.0000	−	15.5885i	1.06478	−	0.614749i	0.926750	+	0.375680i	$0.122591\pi$
$644$	−12.0000	+	34.6410i	−0.472866	+	1.36505i
$645$	−12.0000	+	6.92820i	−0.472500	+	0.272798i
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	15.0000	+	5.19615i	0.587896	+	0.203653i
$652$	4.00000	+	6.92820i	0.156652	+	0.271329i
$653$	−6.00000	+	3.46410i	−0.234798	+	0.135561i	−0.612784	−	0.790251i	$-0.709950\pi$
$653$	−6.00000	+	3.46410i	−0.234798	+	0.135561i	0.377985	+	0.925812i	$0.376617\pi$
$654$		0			0
$655$	−3.00000	+	5.19615i	−0.117220	+	0.203030i
$656$	48.0000			1.87409
$657$	4.50000	+	2.59808i	0.175562	+	0.101361i
$658$		0			0
$659$	1.50000	+	0.866025i	0.0584317	+	0.0337356i	0.528931	−	0.848665i	$-0.322593\pi$
$659$	1.50000	+	0.866025i	0.0584317	+	0.0337356i	−0.470500	+	0.882400i	$0.655926\pi$
$660$	6.00000	−	10.3923i	0.233550	−	0.404520i
$661$	39.0000	−	22.5167i	1.51692	−	0.875797i	0.517122	−	0.855912i	$-0.327003\pi$
$661$	39.0000	−	22.5167i	1.51692	−	0.875797i	0.999802	−	0.0198848i	$-0.00632996\pi$
$662$		0			0
$663$	−13.5000	+	23.3827i	−0.524297	+	0.908108i
$664$		0			0
$665$	−6.00000	−	6.92820i	−0.232670	−	0.268664i
$666$		0			0
$667$	−12.0000			−0.464642
$668$	9.00000	−	15.5885i	0.348220	−	0.603136i
$669$	33.0000			1.27585
$670$		0			0
$671$	−12.0000	−	20.7846i	−0.463255	−	0.802381i
$672$		0			0
$673$	4.00000	−	6.92820i	0.154189	−	0.267063i	−0.778575	−	0.627552i	$-0.784057\pi$
$673$	4.00000	−	6.92820i	0.154189	−	0.267063i	0.932763	+	0.360489i	$0.117390\pi$
$674$		0			0
$675$	−4.50000	−	2.59808i	−0.173205	−	0.100000i
$676$	−28.0000			−1.07692
$677$	−10.5000	+	18.1865i	−0.403548	+	0.698965i	−0.994151	−	0.107997i	$-0.965556\pi$
$677$	−10.5000	+	18.1865i	−0.403548	+	0.698965i	0.590603	+	0.806962i	$0.298890\pi$
$678$		0			0
$679$	6.00000	−	17.3205i	0.230259	−	0.664700i
$680$		0			0
$681$		−	5.19615i		−	0.199117i
$682$		0			0
$683$		−	3.46410i		−	0.132550i	−0.997801	−	0.0662751i	$-0.978889\pi$
$683$		−	3.46410i		−	0.132550i	0.997801	−	0.0662751i	$-0.0211115\pi$
$684$		−	20.7846i		−	0.794719i
$685$		−	3.46410i		−	0.132357i
$686$		0			0
$687$	−3.00000	+	5.19615i	−0.114457	+	0.198246i
$688$	16.0000	+	27.7128i	0.609994	+	1.05654i
$689$	9.00000	+	15.5885i	0.342873	+	0.593873i
$690$		0			0
$691$	−36.0000	−	20.7846i	−1.36950	−	0.790684i	−0.378640	−	0.925544i	$-0.623608\pi$
$691$	−36.0000	−	20.7846i	−1.36950	−	0.790684i	−0.990865	+	0.134860i	$0.956941\pi$
$692$	42.0000			1.59660
$693$	−18.0000	−	20.7846i	−0.683763	−	0.789542i
$694$		0			0
$695$	6.00000	+	3.46410i	0.227593	+	0.131401i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$		0			0
$699$	−6.00000	−	10.3923i	−0.226941	−	0.393073i
$700$	1.00000	+	5.19615i	0.0377964	+	0.196396i
$701$		−	12.1244i		−	0.457931i	−0.973435	−	0.228965i	$-0.926466\pi$
$701$		−	12.1244i		−	0.457931i	0.973435	−	0.228965i	$-0.0735342\pi$
$702$		0			0
$703$			34.6410i			1.30651i
$704$	−24.0000	−	13.8564i	−0.904534	−	0.522233i
$705$		0			0
$706$		0			0
$707$	45.0000	+	15.5885i	1.69240	+	0.586264i
$708$		0			0
$709$	−0.500000	+	0.866025i	−0.0187779	+	0.0325243i	−0.875262	−	0.483650i	$-0.839311\pi$
$709$	−0.500000	+	0.866025i	−0.0187779	+	0.0325243i	0.856484	+	0.516174i	$0.172644\pi$
$710$		0			0
$711$	12.0000	+	20.7846i	0.450035	+	0.779484i
$712$		0			0
$713$	12.0000	−	20.7846i	0.449404	−	0.778390i
$714$		0			0
$715$	9.00000	+	15.5885i	0.336581	+	0.582975i
$716$	−27.0000	+	15.5885i	−1.00904	+	0.582568i
$717$	−1.50000	+	2.59808i	−0.0560185	+	0.0970269i
$718$		0			0
$719$	6.00000			0.223762			0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000			0.223762			0.111881	+	0.993722i	$0.464312\pi$
$720$	−6.00000	+	10.3923i	−0.223607	+	0.387298i
$721$	−15.0000	−	17.3205i	−0.558629	−	0.645049i
$722$		0			0
$723$	−18.0000			−0.669427
$724$	18.0000	−	10.3923i	0.668965	−	0.386227i
$725$	−1.50000	+	0.866025i	−0.0557086	+	0.0321634i
$726$		0			0
$727$	−37.5000	−	21.6506i	−1.39080	−	0.802978i	−0.397394	−	0.917648i	$-0.630086\pi$
$727$	−37.5000	−	21.6506i	−1.39080	−	0.802978i	−0.993404	+	0.114670i	$0.963419\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−12.0000	+	20.7846i	−0.443836	+	0.768747i
$732$	12.0000	+	20.7846i	0.443533	+	0.768221i
$733$	13.5000	−	7.79423i	0.498634	−	0.287886i	−0.229515	−	0.973305i	$-0.573714\pi$
$733$	13.5000	−	7.79423i	0.498634	−	0.287886i	0.728149	+	0.685419i	$0.240381\pi$
$734$		0			0
$735$	12.0000	+	1.73205i	0.442627	+	0.0638877i
$736$		0			0
$737$		−	34.6410i		−	1.27602i
$738$		0			0
$739$	5.00000			0.183928			0.0919640	−	0.995762i	$-0.470686\pi$
$739$	5.00000			0.183928			0.0919640	+	0.995762i	$0.470686\pi$
$740$	10.0000	−	17.3205i	0.367607	−	0.636715i
$741$	27.0000	+	15.5885i	0.991870	+	0.572656i
$742$		0			0
$743$	12.0000	−	6.92820i	0.440237	−	0.254171i	−0.263461	−	0.964670i	$-0.584864\pi$
$743$	12.0000	−	6.92820i	0.440237	−	0.254171i	0.703698	+	0.710499i	$0.251531\pi$
$744$		0			0
$745$	4.50000	+	2.59808i	0.164867	+	0.0951861i
$746$		0			0
$747$	13.5000	+	23.3827i	0.493939	+	0.855528i
$748$		−	20.7846i		−	0.759961i
$749$	45.0000	−	8.66025i	1.64426	−	0.316439i
$750$		0			0
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	−0.996993	−	0.0774878i	$-0.975310\pi$
$751$	−15.5000	−	26.8468i	−0.565603	−	0.979653i	0.431390	−	0.902165i	$-0.358023\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19.0000			−0.691481
$756$	18.0000	+	20.7846i	0.654654	+	0.755929i
$757$	16.0000			0.581530			0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000			0.581530			0.290765	+	0.956795i	$0.406090\pi$
$758$		0			0
$759$	−36.0000	+	20.7846i	−1.30672	+	0.754434i
$760$		0			0
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	0.915264	−	0.402854i	$-0.131982\pi$
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	−0.806514	+	0.591215i	$0.798649\pi$
$762$		0			0
$763$	−1.00000	−	5.19615i	−0.0362024	−	0.188113i
$764$		−	24.2487i		−	0.877288i
$765$	−9.00000			−0.325396
$766$		0			0
$767$		0			0
$768$	24.0000	+	13.8564i	0.866025	+	0.500000i
$769$	−36.0000	+	20.7846i	−1.29819	+	0.749512i	−0.980092	−	0.198545i	$-0.936379\pi$
$769$	−36.0000	+	20.7846i	−1.29819	+	0.749512i	−0.318101	+	0.948057i	$0.603045\pi$
$770$		0			0
$771$	22.5000	+	12.9904i	0.810318	+	0.467837i
$772$	−4.00000	+	6.92820i	−0.143963	+	0.249351i
$773$	18.0000			0.647415			0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000			0.647415			0.323708	+	0.946157i	$0.395071\pi$
$774$		0			0
$775$		−	3.46410i		−	0.124434i
$776$		0			0
$777$	−30.0000	−	34.6410i	−1.07624	−	1.24274i
$778$		0			0
$779$	36.0000	−	20.7846i	1.28983	−	0.744686i
$780$	−9.00000	−	15.5885i	−0.322252	−	0.558156i
$781$	21.0000	−	36.3731i	0.751439	−	1.30153i
$782$		0			0
$783$	−4.50000	+	7.79423i	−0.160817	+	0.278543i
$784$	4.00000	−	27.7128i	0.142857	−	0.989743i
$785$	7.50000	+	4.33013i	0.267686	+	0.154549i
$786$		0			0
$787$	28.5000	−	16.4545i	1.01592	−	0.586539i	0.102997	−	0.994682i	$-0.467157\pi$
$787$	28.5000	−	16.4545i	1.01592	−	0.586539i	0.912918	+	0.408143i	$0.133823\pi$
$788$	42.0000	−	24.2487i	1.49619	−	0.863825i
$789$	6.00000			0.213606
$790$		0			0
$791$	−18.0000	−	20.7846i	−0.640006	−	0.739016i
$792$		0			0
$793$	−36.0000			−1.27840
$794$		0			0
$795$	−3.00000	+	5.19615i	−0.106399	+	0.184289i
$796$	−18.0000	+	10.3923i	−0.637993	+	0.368345i
$797$	25.5000	+	44.1673i	0.903256	+	1.56449i	0.823241	+	0.567692i	$0.192164\pi$
$797$	25.5000	+	44.1673i	0.903256	+	1.56449i	0.0800155	+	0.996794i	$0.474503\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	9.00000	−	15.5885i	0.317999	−	0.550791i
$802$		0			0
$803$	3.00000	−	5.19615i	0.105868	−	0.183368i
$804$			34.6410i			1.22169i
$805$	6.00000	−	17.3205i	0.211472	−	0.610468i
$806$		0			0
$807$	36.0000	−	20.7846i	1.26726	−	0.731653i
$808$		0			0
$809$			32.9090i			1.15702i	0.815676	+	0.578509i	$0.196365\pi$
$809$			32.9090i			1.15702i	−0.815676	+	0.578509i	$0.803635\pi$
$810$		0			0
$811$		−	38.1051i		−	1.33805i	−0.743239	−	0.669026i	$-0.766712\pi$
$811$		−	38.1051i		−	1.33805i	0.743239	−	0.669026i	$-0.233288\pi$
$812$	9.00000	−	1.73205i	0.315838	−	0.0607831i
$813$	−3.00000	−	5.19615i	−0.105215	−	0.182237i
$814$		0			0
$815$	−2.00000	−	3.46410i	−0.0700569	−	0.121342i
$816$			20.7846i			0.727607i
$817$	24.0000	+	13.8564i	0.839654	+	0.484774i
$818$		0			0
$819$	−40.5000	+	7.79423i	−1.41518	+	0.272352i
$820$	−24.0000			−0.838116
$821$	−10.5000	−	6.06218i	−0.366453	−	0.211571i	0.305455	−	0.952207i	$-0.401191\pi$
$821$	−10.5000	−	6.06218i	−0.366453	−	0.211571i	−0.671908	+	0.740635i	$0.734525\pi$
$822$		0			0
$823$	5.00000	+	8.66025i	0.174289	+	0.301877i	0.939915	−	0.341409i	$-0.110904\pi$
$823$	5.00000	+	8.66025i	0.174289	+	0.301877i	−0.765626	+	0.643286i	$0.777571\pi$
$824$		0			0
$825$	−3.00000	+	5.19615i	−0.104447	+	0.180907i
$826$		0			0
$827$			24.2487i			0.843210i	0.906780	+	0.421605i	$0.138533\pi$
$827$			24.2487i			0.843210i	−0.906780	+	0.421605i	$0.861467\pi$
$828$	36.0000	−	20.7846i	1.25109	−	0.722315i
$829$			13.8564i			0.481253i	0.970618	+	0.240626i	$0.0773529\pi$
$829$			13.8564i			0.481253i	−0.970618	+	0.240626i	$0.922647\pi$
$830$		0			0
$831$		−	45.0333i		−	1.56219i
$832$	−36.0000	+	20.7846i	−1.24808	+	0.720577i
$833$	19.5000	−	7.79423i	0.675635	−	0.270054i
$834$		0			0
$835$	−4.50000	+	7.79423i	−0.155729	+	0.269730i
$836$	−24.0000			−0.830057
$837$	−9.00000	−	15.5885i	−0.311086	−	0.538816i
$838$		0			0
$839$	−15.0000	+	25.9808i	−0.517858	+	0.896956i	0.481927	+	0.876211i	$0.339937\pi$
$839$	−15.0000	+	25.9808i	−0.517858	+	0.896956i	−0.999785	+	0.0207443i	$0.993396\pi$
$840$		0			0
$841$	−13.0000	−	22.5167i	−0.448276	−	0.776437i
$842$		0			0
$843$	−24.0000			−0.826604
$844$	8.00000	−	13.8564i	0.275371	−	0.476957i
$845$	14.0000			0.481615
$846$		0			0
$847$	−2.00000	+	1.73205i	−0.0687208	+	0.0595140i
$848$	12.0000	+	6.92820i	0.412082	+	0.237915i
$849$	3.00000	−	5.19615i	0.102960	−	0.178331i
$850$		0			0
$851$	−60.0000	+	34.6410i	−2.05677	+	1.18748i
$852$	−21.0000	+	36.3731i	−0.719448	+	1.24612i
$853$	36.0000	+	20.7846i	1.23262	+	0.711651i	0.967575	−	0.252585i	$-0.0812809\pi$
$853$	36.0000	+	20.7846i	1.23262	+	0.711651i	0.265042	+	0.964237i	$0.414614\pi$
$854$		0			0
$855$			10.3923i			0.355409i
$856$		0			0
$857$	−1.50000	+	2.59808i	−0.0512390	+	0.0887486i	−0.890507	−	0.454969i	$-0.849650\pi$
$857$	−1.50000	+	2.59808i	−0.0512390	+	0.0887486i	0.839268	+	0.543718i	$0.182984\pi$
$858$		0			0
$859$	3.00000	−	1.73205i	0.102359	−	0.0590968i	−0.447947	−	0.894060i	$-0.647845\pi$
$859$	3.00000	−	1.73205i	0.102359	−	0.0590968i	0.550305	+	0.834963i	$0.314511\pi$
$860$	−8.00000	−	13.8564i	−0.272798	−	0.472500i
$861$	−18.0000	+	51.9615i	−0.613438	+	1.77084i
$862$		0			0
$863$		−	17.3205i		−	0.589597i	−0.955559	−	0.294798i	$-0.904747\pi$
$863$		−	17.3205i		−	0.589597i	0.955559	−	0.294798i	$-0.0952525\pi$
$864$		0			0
$865$	−21.0000			−0.714021
$866$		0			0
$867$	12.0000	−	6.92820i	0.407541	−	0.235294i
$868$	−6.00000	+	17.3205i	−0.203653	+	0.587896i
$869$	24.0000	−	13.8564i	0.814144	−	0.470046i
$870$		0			0
$871$	−45.0000	−	25.9808i	−1.52477	−	0.880325i
$872$		0			0
$873$	−18.0000	+	10.3923i	−0.609208	+	0.351726i
$874$		0			0
$875$	−0.500000	−	2.59808i	−0.0169031	−	0.0878310i
$876$	−3.00000	+	5.19615i	−0.101361	+	0.175562i
$877$	2.00000	+	3.46410i	0.0675352	+	0.116974i	0.897816	−	0.440371i	$-0.145153\pi$
$877$	2.00000	+	3.46410i	0.0675352	+	0.116974i	−0.830281	+	0.557346i	$0.811820\pi$
$878$		0			0
$879$	−13.5000	−	7.79423i	−0.455344	−	0.262893i
$880$	12.0000	+	6.92820i	0.404520	+	0.233550i
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−26.0000			−0.874970			−0.437485	−	0.899226i	$-0.644131\pi$
$883$	−26.0000			−0.874970			−0.437485	+	0.899226i	$0.644131\pi$
$884$	−27.0000	−	15.5885i	−0.908108	−	0.524297i
$885$		0			0
$886$		0			0
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	0.915722	+	0.401813i	$0.131620\pi$
$887$	24.0000	+	41.5692i	0.805841	+	1.39576i	−0.109881	+	0.993945i	$0.535047\pi$
$888$		0			0
$889$	1.00000	+	5.19615i	0.0335389	+	0.174273i
$890$		0			0
$891$			31.1769i			1.04447i
$892$			38.1051i			1.27585i
$893$		0			0
$894$		0			0
$895$	13.5000	−	7.79423i	0.451255	−	0.260532i
$896$		0			0
$897$			62.3538i			2.08193i
$898$		0			0
$899$	−6.00000			−0.200111
$900$	3.00000	−	5.19615i	0.100000	−	0.173205i
$901$			10.3923i			0.346218i
$902$		0			0
$903$	−36.0000	+	6.92820i	−1.19800	+	0.230556i
$904$		0			0
$905$	−9.00000	+	5.19615i	−0.299170	+	0.172726i
$906$		0			0
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	−0.226173	−	0.974087i	$-0.572621\pi$
$907$	22.0000	−	38.1051i	0.730498	−	1.26526i	0.956671	−	0.291172i	$-0.0940453\pi$
$908$	6.00000			0.199117
$909$	−27.0000	−	46.7654i	−0.895533	−	1.55111i
$910$		0			0
$911$	−19.5000	−	11.2583i	−0.646064	−	0.373005i	0.140883	−	0.990026i	$-0.455006\pi$
$911$	−19.5000	−	11.2583i	−0.646064	−	0.373005i	−0.786947	+	0.617021i	$0.788339\pi$
$912$	24.0000			0.794719
$913$	27.0000	−	15.5885i	0.893570	−	0.515903i
$914$		0			0
$915$	−6.00000	−	10.3923i	−0.198354	−	0.343559i
$916$	−6.00000	−	3.46410i	−0.198246	−	0.114457i
$917$	−12.0000	+	10.3923i	−0.396275	+	0.343184i
$918$		0			0
$919$	−53.0000			−1.74831			−0.874154	−	0.485648i	$-0.838584\pi$
$919$	−53.0000			−1.74831			−0.874154	+	0.485648i	$0.838584\pi$
$920$		0			0
$921$	−19.5000	−	33.7750i	−0.642547	−	1.11292i
$922$		0			0
$923$	−31.5000	−	54.5596i	−1.03684	−	1.79585i
$924$	24.0000	−	20.7846i	0.789542	−	0.683763i
$925$	−5.00000	+	8.66025i	−0.164399	+	0.284747i
$926$		0			0
$927$			25.9808i			0.853320i
$928$		0			0
$929$	−21.0000	+	36.3731i	−0.688988	+	1.19336i	0.283178	+	0.959067i	$0.408611\pi$
$929$	−21.0000	+	36.3731i	−0.688988	+	1.19336i	−0.972166	+	0.234294i	$0.924722\pi$
$930$		0			0
$931$	−9.00000	−	22.5167i	−0.294963	−	0.737954i
$932$	12.0000	−	6.92820i	0.393073	−	0.226941i
$933$		0			0
$934$		0			0
$935$			10.3923i			0.339865i
$936$		0			0
$937$		−	1.73205i		−	0.0565836i	−0.999600	−	0.0282918i	$-0.990993\pi$
$937$		−	1.73205i		−	0.0565836i	0.999600	−	0.0282918i	$-0.00900677\pi$
$938$		0			0
$939$	9.00000			0.293704
$940$		0			0
$941$	21.0000	+	36.3731i	0.684580	+	1.18573i	0.973568	+	0.228395i	$0.0733479\pi$
$941$	21.0000	+	36.3731i	0.684580	+	1.18573i	−0.288988	+	0.957333i	$0.593319\pi$
$942$		0			0
$943$	72.0000	+	41.5692i	2.34464	+	1.35368i
$944$		0			0
$945$	−9.00000	−	10.3923i	−0.292770	−	0.338062i
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−24.0000	+	13.8564i	−0.779484	+	0.450035i
$949$	−4.50000	−	7.79423i	−0.146076	−	0.253011i
$950$		0			0
$951$	−42.0000			−1.36194
$952$		0			0
$953$		−	48.4974i		−	1.57099i	−0.618871	−	0.785493i	$-0.712410\pi$
$953$		−	48.4974i		−	1.57099i	0.618871	−	0.785493i	$-0.287590\pi$
$954$		0			0
$955$			12.1244i			0.392335i
$956$	−3.00000	−	1.73205i	−0.0970269	−	0.0560185i
$957$	9.00000	+	5.19615i	0.290929	+	0.167968i
$958$		0			0
$959$	3.00000	−	8.66025i	0.0968751	−	0.279654i
$960$	−12.0000	−	6.92820i	−0.387298	−	0.223607i
$961$	−9.50000	+	16.4545i	−0.306452	+	0.530790i
$962$		0			0
$963$	−45.0000	−	25.9808i	−1.45010	−	0.837218i
$964$		−	20.7846i		−	0.669427i
$965$	2.00000	−	3.46410i	0.0643823	−	0.111513i
$966$		0			0
$967$	−10.0000	−	17.3205i	−0.321578	−	0.556990i	0.659236	−	0.751936i	$-0.270880\pi$
$967$	−10.0000	−	17.3205i	−0.321578	−	0.556990i	−0.980814	+	0.194946i	$0.937547\pi$
$968$		0			0
$969$	9.00000	+	15.5885i	0.289122	+	0.500773i
$970$		0			0
$971$	48.0000			1.54039			0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000			1.54039			0.770197	+	0.637806i	$0.220158\pi$
$972$		−	31.1769i		−	1.00000i
$973$	12.0000	+	13.8564i	0.384702	+	0.444216i
$974$		0			0
$975$	4.50000	+	7.79423i	0.144115	+	0.249615i
$976$	−24.0000	+	13.8564i	−0.768221	+	0.443533i
$977$	−21.0000	+	12.1244i	−0.671850	+	0.387893i	−0.796777	−	0.604273i	$-0.793463\pi$
$977$	−21.0000	+	12.1244i	−0.671850	+	0.387893i	0.124928	+	0.992166i	$0.460130\pi$
$978$		0			0
$979$	−18.0000	−	10.3923i	−0.575282	−	0.332140i
$980$	−2.00000	+	13.8564i	−0.0638877	+	0.442627i
$981$	−3.00000	+	5.19615i	−0.0957826	+	0.165900i
$982$		0			0
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	−0.991453	−	0.130465i	$-0.958353\pi$
$983$	−12.0000	+	20.7846i	−0.382741	+	0.662926i	0.608712	+	0.793391i	$0.291686\pi$
$984$		0			0
$985$	−21.0000	+	12.1244i	−0.669116	+	0.386314i
$986$		0			0
$987$		0			0
$988$	−18.0000	+	31.1769i	−0.572656	+	0.991870i
$989$			55.4256i			1.76243i
$990$		0			0
$991$	19.0000			0.603555			0.301777	−	0.953378i	$-0.402420\pi$
$991$	19.0000			0.603555			0.301777	+	0.953378i	$0.402420\pi$
$992$		0			0
$993$		−	1.73205i		−	0.0549650i
$994$		0			0
$995$	9.00000	−	5.19615i	0.285319	−	0.164729i
$996$	−27.0000	+	15.5885i	−0.855528	+	0.493939i
$997$	−22.5000	−	12.9904i	−0.712582	−	0.411409i	0.0994342	−	0.995044i	$-0.468297\pi$
$997$	−22.5000	−	12.9904i	−0.712582	−	0.411409i	−0.812016	+	0.583635i	$0.801630\pi$
$998$		0			0
$999$			51.9615i			1.64399i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.2.bl.f.146.1	yes	2
3.2	odd	2		945.2.bl.b.251.1		2
7.6	odd	2		315.2.bl.g.146.1	yes	2
9.4	even	3		945.2.bl.d.881.1		2
9.5	odd	6		315.2.bl.g.41.1	yes	2
21.20	even	2		945.2.bl.d.251.1		2
63.13	odd	6		945.2.bl.b.881.1		2
63.41	even	6	inner	315.2.bl.f.41.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.2.bl.f.41.1	✓	2	63.41	even	6	inner
315.2.bl.f.146.1	yes	2	1.1	even	1	trivial
315.2.bl.g.41.1	yes	2	9.5	odd	6
315.2.bl.g.146.1	yes	2	7.6	odd	2
945.2.bl.b.251.1		2	3.2	odd	2
945.2.bl.b.881.1		2	63.13	odd	6
945.2.bl.d.251.1		2	21.20	even	2
945.2.bl.d.881.1		2	9.4	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	315.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +O(q^{10})\)	\(q+(-1.50000 + 0.866025i) q^{3} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +(0.500000 + 2.59808i) q^{7} +(1.50000 - 2.59808i) q^{9} +(-3.00000 - 1.73205i) q^{11} +(3.00000 + 1.73205i) q^{12} +(-4.50000 + 2.59808i) q^{13} +(-1.50000 - 0.866025i) q^{15} +(-2.00000 + 3.46410i) q^{16} -3.00000 q^{17} +3.46410i q^{19} +(1.00000 - 1.73205i) q^{20} +(-3.00000 - 3.46410i) q^{21} +(-6.00000 + 3.46410i) q^{23} +(-0.500000 + 0.866025i) q^{25} +5.19615i q^{27} +(4.00000 - 3.46410i) q^{28} +(1.50000 + 0.866025i) q^{29} +(-3.00000 + 1.73205i) q^{31} +6.00000 q^{33} +(-2.00000 + 1.73205i) q^{35} -6.00000 q^{36} +10.0000 q^{37} +(4.50000 - 7.79423i) q^{39} +(-6.00000 - 10.3923i) q^{41} +(4.00000 - 6.92820i) q^{43} +6.92820i q^{44} +3.00000 q^{45} -6.92820i q^{48} +(-6.50000 + 2.59808i) q^{49} +(4.50000 - 2.59808i) q^{51} +(9.00000 + 5.19615i) q^{52} -3.46410i q^{53} -3.46410i q^{55} +(-3.00000 - 5.19615i) q^{57} +3.46410i q^{60} +(6.00000 + 3.46410i) q^{61} +(7.50000 + 2.59808i) q^{63} +8.00000 q^{64} +(-4.50000 - 2.59808i) q^{65} +(5.00000 + 8.66025i) q^{67} +(3.00000 + 5.19615i) q^{68} +(6.00000 - 10.3923i) q^{69} +12.1244i q^{71} +1.73205i q^{73} -1.73205i q^{75} +(6.00000 - 3.46410i) q^{76} +(3.00000 - 8.66025i) q^{77} +(-4.00000 + 6.92820i) q^{79} -4.00000 q^{80} +(-4.50000 - 7.79423i) q^{81} +(-4.50000 + 7.79423i) q^{83} +(-3.00000 + 8.66025i) q^{84} +(-1.50000 - 2.59808i) q^{85} -3.00000 q^{87} +6.00000 q^{89} +(-9.00000 - 10.3923i) q^{91} +(12.0000 + 6.92820i) q^{92} +(3.00000 - 5.19615i) q^{93} +(-3.00000 + 1.73205i) q^{95} +(-6.00000 - 3.46410i) q^{97} +(-9.00000 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 2 q^{4} + q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - 3 * q^3 - 2 * q^4 + q^5 + q^7 + 3 * q^9	\( 2 q - 3 q^{3} - 2 q^{4} + q^{5} + q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{12} - 9 q^{13} - 3 q^{15} - 4 q^{16} - 6 q^{17} + 2 q^{20} - 6 q^{21} - 12 q^{23} - q^{25} + 8 q^{28} + 3 q^{29} - 6 q^{31} + 12 q^{33} - 4 q^{35} - 12 q^{36} + 20 q^{37} + 9 q^{39} - 12 q^{41} + 8 q^{43} + 6 q^{45} - 13 q^{49} + 9 q^{51} + 18 q^{52} - 6 q^{57} + 12 q^{61} + 15 q^{63} + 16 q^{64} - 9 q^{65} + 10 q^{67} + 6 q^{68} + 12 q^{69} + 12 q^{76} + 6 q^{77} - 8 q^{79} - 8 q^{80} - 9 q^{81} - 9 q^{83} - 6 q^{84} - 3 q^{85} - 6 q^{87} + 12 q^{89} - 18 q^{91} + 24 q^{92} + 6 q^{93} - 6 q^{95} - 12 q^{97} - 18 q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - 2 * q^4 + q^5 + q^7 + 3 * q^9 - 6 * q^11 + 6 * q^12 - 9 * q^13 - 3 * q^15 - 4 * q^16 - 6 * q^17 + 2 * q^20 - 6 * q^21 - 12 * q^23 - q^25 + 8 * q^28 + 3 * q^29 - 6 * q^31 + 12 * q^33 - 4 * q^35 - 12 * q^36 + 20 * q^37 + 9 * q^39 - 12 * q^41 + 8 * q^43 + 6 * q^45 - 13 * q^49 + 9 * q^51 + 18 * q^52 - 6 * q^57 + 12 * q^61 + 15 * q^63 + 16 * q^64 - 9 * q^65 + 10 * q^67 + 6 * q^68 + 12 * q^69 + 12 * q^76 + 6 * q^77 - 8 * q^79 - 8 * q^80 - 9 * q^81 - 9 * q^83 - 6 * q^84 - 3 * q^85 - 6 * q^87 + 12 * q^89 - 18 * q^91 + 24 * q^92 + 6 * q^93 - 6 * q^95 - 12 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more