LMFDB - Embedded newform 966.2.f.a.461.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,2,Mod(461,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

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

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

chi := DirichletCharacter("966.461");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	966.f (of order $2$, degree $1$, 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:	$7.71354883526$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			461.3
Root			$0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	966.461
Dual form			966.2.f.a.461.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.73205i q^{3} -1.00000 q^{4} -1.73205 q^{5} +1.73205 q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})$	\(q+1.00000i q^{2} -1.73205i q^{3} -1.00000 q^{4} -1.73205 q^{5} +1.73205 q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} -3.00000 q^{9} -1.73205i q^{10} +1.73205i q^{12} -1.73205i q^{13} +(-1.73205 + 2.00000i) q^{14} +3.00000i q^{15} +1.00000 q^{16} -3.46410 q^{17} -3.00000i q^{18} -6.92820i q^{19} +1.73205 q^{20} +(3.00000 - 3.46410i) q^{21} -1.00000i q^{23} -1.73205 q^{24} -2.00000 q^{25} +1.73205 q^{26} +5.19615i q^{27} +(-2.00000 - 1.73205i) q^{28} +9.00000i q^{29} -3.00000 q^{30} -10.3923i q^{31} +1.00000i q^{32} -3.46410i q^{34} +(-3.46410 - 3.00000i) q^{35} +3.00000 q^{36} -5.00000 q^{37} +6.92820 q^{38} -3.00000 q^{39} +1.73205i q^{40} -12.1244 q^{41} +(3.46410 + 3.00000i) q^{42} +1.00000 q^{43} +5.19615 q^{45} +1.00000 q^{46} -8.66025 q^{47} -1.73205i q^{48} +(1.00000 + 6.92820i) q^{49} -2.00000i q^{50} +6.00000i q^{51} +1.73205i q^{52} -12.0000i q^{53} -5.19615 q^{54} +(1.73205 - 2.00000i) q^{56} -12.0000 q^{57} -9.00000 q^{58} +3.46410 q^{59} -3.00000i q^{60} +10.3923 q^{62} +(-6.00000 - 5.19615i) q^{63} -1.00000 q^{64} +3.00000i q^{65} -4.00000 q^{67} +3.46410 q^{68} -1.73205 q^{69} +(3.00000 - 3.46410i) q^{70} -12.0000i q^{71} +3.00000i q^{72} -6.92820i q^{73} -5.00000i q^{74} +3.46410i q^{75} +6.92820i q^{76} -3.00000i q^{78} -10.0000 q^{79} -1.73205 q^{80} +9.00000 q^{81} -12.1244i q^{82} -3.46410 q^{83} +(-3.00000 + 3.46410i) q^{84} +6.00000 q^{85} +1.00000i q^{86} +15.5885 q^{87} +3.46410 q^{89} +5.19615i q^{90} +(3.00000 - 3.46410i) q^{91} +1.00000i q^{92} -18.0000 q^{93} -8.66025i q^{94} +12.0000i q^{95} +1.73205 q^{96} +12.1244i q^{97} +(-6.92820 + 1.00000i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^9	$ 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9} + 4 q^{16} + 12 q^{21} - 8 q^{25} - 8 q^{28} - 12 q^{30} + 12 q^{36} - 20 q^{37} - 12 q^{39} + 4 q^{43} + 4 q^{46} + 4 q^{49} - 48 q^{57} - 36 q^{58} - 24 q^{63} - 4 q^{64} - 16 q^{67} + 12 q^{70} - 40 q^{79} + 36 q^{81} - 12 q^{84} + 24 q^{85} + 12 q^{91} - 72 q^{93}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^9 + 4 * q^16 + 12 * q^21 - 8 * q^25 - 8 * q^28 - 12 * q^30 + 12 * q^36 - 20 * q^37 - 12 * q^39 + 4 * q^43 + 4 * q^46 + 4 * q^49 - 48 * q^57 - 36 * q^58 - 24 * q^63 - 4 * q^64 - 16 * q^67 + 12 * q^70 - 40 * q^79 + 36 * q^81 - 12 * q^84 + 24 * q^85 + 12 * q^91 - 72 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$.

$n$	$323$	$829$	$925$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$			1.00000i			0.707107i
$2$			1.00000i			0.707107i
$3$		−	1.73205i		−	1.00000i
$3$		−	1.73205i		−	1.00000i
$4$	−1.00000			−0.500000
$5$	−1.73205			−0.774597			−0.387298	−	0.921954i	$-0.626592\pi$
$5$	−1.73205			−0.774597			−0.387298	+	0.921954i	$0.626592\pi$
$6$	1.73205			0.707107
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$7$	2.00000	+	1.73205i	0.755929	+	0.654654i
$8$		−	1.00000i		−	0.353553i
$9$	−3.00000			−1.00000
$10$		−	1.73205i		−	0.547723i
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			1.73205i			0.500000i
$13$		−	1.73205i		−	0.480384i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$		−	1.73205i		−	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$14$	−1.73205	+	2.00000i	−0.462910	+	0.534522i
$15$			3.00000i			0.774597i
$16$	1.00000			0.250000
$17$	−3.46410			−0.840168			−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410			−0.840168			−0.420084	+	0.907485i	$0.637999\pi$
$18$		−	3.00000i		−	0.707107i
$19$		−	6.92820i		−	1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$		−	6.92820i		−	1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	1.73205			0.387298
$21$	3.00000	−	3.46410i	0.654654	−	0.755929i
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$	−1.73205			−0.353553
$25$	−2.00000			−0.400000
$26$	1.73205			0.339683
$27$			5.19615i			1.00000i
$28$	−2.00000	−	1.73205i	−0.377964	−	0.327327i
$29$			9.00000i			1.67126i	0.549294	+	0.835629i	$0.314897\pi$
$29$			9.00000i			1.67126i	−0.549294	+	0.835629i	$0.685103\pi$
$30$	−3.00000			−0.547723
$31$		−	10.3923i		−	1.86651i	−0.359211	−	0.933257i	$-0.616954\pi$
$31$		−	10.3923i		−	1.86651i	0.359211	−	0.933257i	$-0.383046\pi$
$32$			1.00000i			0.176777i
$33$		0			0
$34$		−	3.46410i		−	0.594089i
$35$	−3.46410	−	3.00000i	−0.585540	−	0.507093i
$36$	3.00000			0.500000
$37$	−5.00000			−0.821995			−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000			−0.821995			−0.410997	+	0.911636i	$0.634819\pi$
$38$	6.92820			1.12390
$39$	−3.00000			−0.480384
$40$			1.73205i			0.273861i
$41$	−12.1244			−1.89351			−0.946753	−	0.321960i	$-0.895658\pi$
$41$	−12.1244			−1.89351			−0.946753	+	0.321960i	$0.895658\pi$
$42$	3.46410	+	3.00000i	0.534522	+	0.462910i
$43$	1.00000			0.152499			0.0762493	−	0.997089i	$-0.475706\pi$
$43$	1.00000			0.152499			0.0762493	+	0.997089i	$0.475706\pi$
$44$		0			0
$45$	5.19615			0.774597
$46$	1.00000			0.147442
$47$	−8.66025			−1.26323			−0.631614	−	0.775283i	$-0.717607\pi$
$47$	−8.66025			−1.26323			−0.631614	+	0.775283i	$0.717607\pi$
$48$		−	1.73205i		−	0.250000i
$49$	1.00000	+	6.92820i	0.142857	+	0.989743i
$50$		−	2.00000i		−	0.282843i
$51$			6.00000i			0.840168i
$52$			1.73205i			0.240192i
$53$		−	12.0000i		−	1.64833i	−0.566352	−	0.824163i	$-0.691646\pi$
$53$		−	12.0000i		−	1.64833i	0.566352	−	0.824163i	$-0.308354\pi$
$54$	−5.19615			−0.707107
$55$		0			0
$56$	1.73205	−	2.00000i	0.231455	−	0.267261i
$57$	−12.0000			−1.58944
$58$	−9.00000			−1.18176
$59$	3.46410			0.450988			0.225494	−	0.974245i	$-0.427600\pi$
$59$	3.46410			0.450988			0.225494	+	0.974245i	$0.427600\pi$
$60$		−	3.00000i		−	0.387298i
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$	10.3923			1.31982
$63$	−6.00000	−	5.19615i	−0.755929	−	0.654654i
$64$	−1.00000			−0.125000
$65$			3.00000i			0.372104i
$66$		0			0
$67$	−4.00000			−0.488678			−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000			−0.488678			−0.244339	+	0.969690i	$0.578571\pi$
$68$	3.46410			0.420084
$69$	−1.73205			−0.208514
$70$	3.00000	−	3.46410i	0.358569	−	0.414039i
$71$		−	12.0000i		−	1.42414i	−0.702109	−	0.712069i	$-0.747758\pi$
$71$		−	12.0000i		−	1.42414i	0.702109	−	0.712069i	$-0.252242\pi$
$72$			3.00000i			0.353553i
$73$		−	6.92820i		−	0.810885i	−0.914121	−	0.405442i	$-0.867117\pi$
$73$		−	6.92820i		−	0.810885i	0.914121	−	0.405442i	$-0.132883\pi$
$74$		−	5.00000i		−	0.581238i
$75$			3.46410i			0.400000i
$76$			6.92820i			0.794719i
$77$		0			0
$78$		−	3.00000i		−	0.339683i
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$	−1.73205			−0.193649
$81$	9.00000			1.00000
$82$		−	12.1244i		−	1.33891i
$83$	−3.46410			−0.380235			−0.190117	−	0.981761i	$-0.560887\pi$
$83$	−3.46410			−0.380235			−0.190117	+	0.981761i	$0.560887\pi$
$84$	−3.00000	+	3.46410i	−0.327327	+	0.377964i
$85$	6.00000			0.650791
$86$			1.00000i			0.107833i
$87$	15.5885			1.67126
$88$		0			0
$89$	3.46410			0.367194			0.183597	−	0.983002i	$-0.441226\pi$
$89$	3.46410			0.367194			0.183597	+	0.983002i	$0.441226\pi$
$90$			5.19615i			0.547723i
$91$	3.00000	−	3.46410i	0.314485	−	0.363137i
$92$			1.00000i			0.104257i
$93$	−18.0000			−1.86651
$94$		−	8.66025i		−	0.893237i
$95$			12.0000i			1.23117i
$96$	1.73205			0.176777
$97$			12.1244i			1.23104i	0.788121	+	0.615521i	$0.211054\pi$
$97$			12.1244i			1.23104i	−0.788121	+	0.615521i	$0.788946\pi$
$98$	−6.92820	+	1.00000i	−0.699854	+	0.101015i
$99$		0			0
$100$	2.00000			0.200000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−6.00000			−0.594089
$103$			5.19615i			0.511992i	0.966678	+	0.255996i	$0.0824034\pi$
$103$			5.19615i			0.511992i	−0.966678	+	0.255996i	$0.917597\pi$
$104$	−1.73205			−0.169842
$105$	−5.19615	+	6.00000i	−0.507093	+	0.585540i
$106$	12.0000			1.16554
$107$			18.0000i			1.74013i	0.492941	+	0.870063i	$0.335922\pi$
$107$			18.0000i			1.74013i	−0.492941	+	0.870063i	$0.664078\pi$
$108$		−	5.19615i		−	0.500000i
$109$	−11.0000			−1.05361			−0.526804	−	0.849987i	$-0.676610\pi$
$109$	−11.0000			−1.05361			−0.526804	+	0.849987i	$0.676610\pi$
$110$		0			0
$111$			8.66025i			0.821995i
$112$	2.00000	+	1.73205i	0.188982	+	0.163663i
$113$		−	9.00000i		−	0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$		−	9.00000i		−	0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$		−	12.0000i		−	1.12390i
$115$			1.73205i			0.161515i
$116$		−	9.00000i		−	0.835629i
$117$			5.19615i			0.480384i
$118$			3.46410i			0.318896i
$119$	−6.92820	−	6.00000i	−0.635107	−	0.550019i
$120$	3.00000			0.273861
$121$	11.0000			1.00000
$122$		0			0
$123$			21.0000i			1.89351i
$124$			10.3923i			0.933257i
$125$	12.1244			1.08444
$126$	5.19615	−	6.00000i	0.462910	−	0.534522i
$127$	17.0000			1.50851			0.754253	−	0.656584i	$-0.227999\pi$
$127$	17.0000			1.50851			0.754253	+	0.656584i	$0.227999\pi$
$128$		−	1.00000i		−	0.0883883i
$129$		−	1.73205i		−	0.152499i
$130$	−3.00000			−0.263117
$131$	13.8564			1.21064			0.605320	−	0.795982i	$-0.293045\pi$
$131$	13.8564			1.21064			0.605320	+	0.795982i	$0.293045\pi$
$132$		0			0
$133$	12.0000	−	13.8564i	1.04053	−	1.20150i
$134$		−	4.00000i		−	0.345547i
$135$		−	9.00000i		−	0.774597i
$136$			3.46410i			0.297044i
$137$		−	15.0000i		−	1.28154i	−0.767734	−	0.640768i	$-0.778616\pi$
$137$		−	15.0000i		−	1.28154i	0.767734	−	0.640768i	$-0.221384\pi$
$138$		−	1.73205i		−	0.147442i
$139$			5.19615i			0.440732i	0.975417	+	0.220366i	$0.0707252\pi$
$139$			5.19615i			0.440732i	−0.975417	+	0.220366i	$0.929275\pi$
$140$	3.46410	+	3.00000i	0.292770	+	0.253546i
$141$			15.0000i			1.26323i
$142$	12.0000			1.00702
$143$		0			0
$144$	−3.00000			−0.250000
$145$		−	15.5885i		−	1.29455i
$146$	6.92820			0.573382
$147$	12.0000	−	1.73205i	0.989743	−	0.142857i
$148$	5.00000			0.410997
$149$		−	18.0000i		−	1.47462i	−0.675556	−	0.737309i	$-0.736096\pi$
$149$		−	18.0000i		−	1.47462i	0.675556	−	0.737309i	$-0.263904\pi$
$150$	−3.46410			−0.282843
$151$	19.0000			1.54620			0.773099	−	0.634285i	$-0.218706\pi$
$151$	19.0000			1.54620			0.773099	+	0.634285i	$0.218706\pi$
$152$	−6.92820			−0.561951
$153$	10.3923			0.840168
$154$		0			0
$155$			18.0000i			1.44579i
$156$	3.00000			0.240192
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		−	10.0000i		−	0.795557i
$159$	−20.7846			−1.64833
$160$		−	1.73205i		−	0.136931i
$161$	1.73205	−	2.00000i	0.136505	−	0.157622i
$162$			9.00000i			0.707107i
$163$	14.0000			1.09656			0.548282	−	0.836293i	$-0.315282\pi$
$163$	14.0000			1.09656			0.548282	+	0.836293i	$0.315282\pi$
$164$	12.1244			0.946753
$165$		0			0
$166$		−	3.46410i		−	0.268866i
$167$	−17.3205			−1.34030			−0.670151	−	0.742225i	$-0.733770\pi$
$167$	−17.3205			−1.34030			−0.670151	+	0.742225i	$0.733770\pi$
$168$	−3.46410	−	3.00000i	−0.267261	−	0.231455i
$169$	10.0000			0.769231
$170$			6.00000i			0.460179i
$171$			20.7846i			1.58944i
$172$	−1.00000			−0.0762493
$173$	3.46410			0.263371			0.131685	−	0.991292i	$-0.457961\pi$
$173$	3.46410			0.263371			0.131685	+	0.991292i	$0.457961\pi$
$174$			15.5885i			1.18176i
$175$	−4.00000	−	3.46410i	−0.302372	−	0.261861i
$176$		0			0
$177$		−	6.00000i		−	0.450988i
$178$			3.46410i			0.259645i
$179$			21.0000i			1.56961i	0.619740	+	0.784807i	$0.287238\pi$
$179$			21.0000i			1.56961i	−0.619740	+	0.784807i	$0.712762\pi$
$180$	−5.19615			−0.387298
$181$			17.3205i			1.28742i	0.765268	+	0.643712i	$0.222606\pi$
$181$			17.3205i			1.28742i	−0.765268	+	0.643712i	$0.777394\pi$
$182$	3.46410	+	3.00000i	0.256776	+	0.222375i
$183$		0			0
$184$	−1.00000			−0.0737210
$185$	8.66025			0.636715
$186$		−	18.0000i		−	1.31982i
$187$		0			0
$188$	8.66025			0.631614
$189$	−9.00000	+	10.3923i	−0.654654	+	0.755929i
$190$	−12.0000			−0.870572
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			1.73205i			0.125000i
$193$	−5.00000			−0.359908			−0.179954	−	0.983675i	$-0.557595\pi$
$193$	−5.00000			−0.359908			−0.179954	+	0.983675i	$0.557595\pi$
$194$	−12.1244			−0.870478
$195$	5.19615			0.372104
$196$	−1.00000	−	6.92820i	−0.0714286	−	0.494872i
$197$		−	15.0000i		−	1.06871i	−0.845262	−	0.534353i	$-0.820555\pi$
$197$		−	15.0000i		−	1.06871i	0.845262	−	0.534353i	$-0.179445\pi$
$198$		0			0
$199$		−	12.1244i		−	0.859473i	−0.902954	−	0.429736i	$-0.858606\pi$
$199$		−	12.1244i		−	0.859473i	0.902954	−	0.429736i	$-0.141394\pi$
$200$			2.00000i			0.141421i
$201$			6.92820i			0.488678i
$202$		0			0
$203$	−15.5885	+	18.0000i	−1.09410	+	1.26335i
$204$		−	6.00000i		−	0.420084i
$205$	21.0000			1.46670
$206$	−5.19615			−0.362033
$207$			3.00000i			0.208514i
$208$		−	1.73205i		−	0.120096i
$209$		0			0
$210$	−6.00000	−	5.19615i	−0.414039	−	0.358569i
$211$	−26.0000			−1.78991			−0.894957	−	0.446153i	$-0.852794\pi$
$211$	−26.0000			−1.78991			−0.894957	+	0.446153i	$0.852794\pi$
$212$			12.0000i			0.824163i
$213$	−20.7846			−1.42414
$214$	−18.0000			−1.23045
$215$	−1.73205			−0.118125
$216$	5.19615			0.353553
$217$	18.0000	−	20.7846i	1.22192	−	1.41095i
$218$		−	11.0000i		−	0.745014i
$219$	−12.0000			−0.810885
$220$		0			0
$221$			6.00000i			0.403604i
$222$	−8.66025			−0.581238
$223$		−	6.92820i		−	0.463947i	−0.972722	−	0.231973i	$-0.925482\pi$
$223$		−	6.92820i		−	0.463947i	0.972722	−	0.231973i	$-0.0745182\pi$
$224$	−1.73205	+	2.00000i	−0.115728	+	0.133631i
$225$	6.00000			0.400000
$226$	9.00000			0.598671
$227$	15.5885			1.03464			0.517321	−	0.855791i	$-0.326929\pi$
$227$	15.5885			1.03464			0.517321	+	0.855791i	$0.326929\pi$
$228$	12.0000			0.794719
$229$			24.2487i			1.60240i	0.598397	+	0.801200i	$0.295805\pi$
$229$			24.2487i			1.60240i	−0.598397	+	0.801200i	$0.704195\pi$
$230$	−1.73205			−0.114208
$231$		0			0
$232$	9.00000			0.590879
$233$		−	12.0000i		−	0.786146i	−0.919507	−	0.393073i	$-0.871412\pi$
$233$		−	12.0000i		−	0.786146i	0.919507	−	0.393073i	$-0.128588\pi$
$234$	−5.19615			−0.339683
$235$	15.0000			0.978492
$236$	−3.46410			−0.225494
$237$			17.3205i			1.12509i
$238$	6.00000	−	6.92820i	0.388922	−	0.449089i
$239$		−	6.00000i		−	0.388108i	−0.980991	−	0.194054i	$-0.937836\pi$
$239$		−	6.00000i		−	0.388108i	0.980991	−	0.194054i	$-0.0621637\pi$
$240$			3.00000i			0.193649i
$241$		−	15.5885i		−	1.00414i	−0.864827	−	0.502070i	$-0.832572\pi$
$241$		−	15.5885i		−	1.00414i	0.864827	−	0.502070i	$-0.167428\pi$
$242$			11.0000i			0.707107i
$243$		−	15.5885i		−	1.00000i
$244$		0			0
$245$	−1.73205	−	12.0000i	−0.110657	−	0.766652i
$246$	−21.0000			−1.33891
$247$	−12.0000			−0.763542
$248$	−10.3923			−0.659912
$249$			6.00000i			0.380235i
$250$			12.1244i			0.766812i
$251$	−22.5167			−1.42124			−0.710620	−	0.703577i	$-0.751585\pi$
$251$	−22.5167			−1.42124			−0.710620	+	0.703577i	$0.751585\pi$
$252$	6.00000	+	5.19615i	0.377964	+	0.327327i
$253$		0			0
$254$			17.0000i			1.06667i
$255$		−	10.3923i		−	0.650791i
$256$	1.00000			0.0625000
$257$	6.92820			0.432169			0.216085	−	0.976375i	$-0.430671\pi$
$257$	6.92820			0.432169			0.216085	+	0.976375i	$0.430671\pi$
$258$	1.73205			0.107833
$259$	−10.0000	−	8.66025i	−0.621370	−	0.538122i
$260$		−	3.00000i		−	0.186052i
$261$		−	27.0000i		−	1.67126i
$262$			13.8564i			0.856052i
$263$			27.0000i			1.66489i	0.554107	+	0.832446i	$0.313060\pi$
$263$			27.0000i			1.66489i	−0.554107	+	0.832446i	$0.686940\pi$
$264$		0			0
$265$			20.7846i			1.27679i
$266$	13.8564	+	12.0000i	0.849591	+	0.735767i
$267$		−	6.00000i		−	0.367194i
$268$	4.00000			0.244339
$269$	13.8564			0.844840			0.422420	−	0.906400i	$-0.361181\pi$
$269$	13.8564			0.844840			0.422420	+	0.906400i	$0.361181\pi$
$270$	9.00000			0.547723
$271$			20.7846i			1.26258i	0.775549	+	0.631288i	$0.217473\pi$
$271$			20.7846i			1.26258i	−0.775549	+	0.631288i	$0.782527\pi$
$272$	−3.46410			−0.210042
$273$	−6.00000	−	5.19615i	−0.363137	−	0.314485i
$274$	15.0000			0.906183
$275$		0			0
$276$	1.73205			0.104257
$277$	−10.0000			−0.600842			−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000			−0.600842			−0.300421	+	0.953807i	$0.597127\pi$
$278$	−5.19615			−0.311645
$279$			31.1769i			1.86651i
$280$	−3.00000	+	3.46410i	−0.179284	+	0.207020i
$281$		−	9.00000i		−	0.536895i	−0.963294	−	0.268447i	$-0.913489\pi$
$281$		−	9.00000i		−	0.536895i	0.963294	−	0.268447i	$-0.0865106\pi$
$282$	−15.0000			−0.893237
$283$		−	3.46410i		−	0.205919i	−0.994686	−	0.102960i	$-0.967169\pi$
$283$		−	3.46410i		−	0.205919i	0.994686	−	0.102960i	$-0.0328313\pi$
$284$			12.0000i			0.712069i
$285$	20.7846			1.23117
$286$		0			0
$287$	−24.2487	−	21.0000i	−1.43136	−	1.23959i
$288$		−	3.00000i		−	0.176777i
$289$	−5.00000			−0.294118
$290$	15.5885			0.915386
$291$	21.0000			1.23104
$292$			6.92820i			0.405442i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	1.73205	+	12.0000i	0.101015	+	0.699854i
$295$	−6.00000			−0.349334
$296$			5.00000i			0.290619i
$297$		0			0
$298$	18.0000			1.04271
$299$	−1.73205			−0.100167
$300$		−	3.46410i		−	0.200000i
$301$	2.00000	+	1.73205i	0.115278	+	0.0998337i
$302$			19.0000i			1.09333i
$303$		0			0
$304$		−	6.92820i		−	0.397360i
$305$		0			0
$306$			10.3923i			0.594089i
$307$			1.73205i			0.0988534i	0.998778	+	0.0494267i	$0.0157394\pi$
$307$			1.73205i			0.0988534i	−0.998778	+	0.0494267i	$0.984261\pi$
$308$		0			0
$309$	9.00000			0.511992
$310$	−18.0000			−1.02233
$311$	−10.3923			−0.589294			−0.294647	−	0.955606i	$-0.595202\pi$
$311$	−10.3923			−0.589294			−0.294647	+	0.955606i	$0.595202\pi$
$312$			3.00000i			0.169842i
$313$		0			0		1.00000			$0$
$313$		0			0		−1.00000			$\pi$
$314$		0			0
$315$	10.3923	+	9.00000i	0.585540	+	0.507093i
$316$	10.0000			0.562544
$317$			3.00000i			0.168497i	0.996445	+	0.0842484i	$0.0268489\pi$
$317$			3.00000i			0.168497i	−0.996445	+	0.0842484i	$0.973151\pi$
$318$		−	20.7846i		−	1.16554i
$319$		0			0
$320$	1.73205			0.0968246
$321$	31.1769			1.74013
$322$	2.00000	+	1.73205i	0.111456	+	0.0965234i
$323$			24.0000i			1.33540i
$324$	−9.00000			−0.500000
$325$			3.46410i			0.192154i
$326$			14.0000i			0.775388i
$327$			19.0526i			1.05361i
$328$			12.1244i			0.669456i
$329$	−17.3205	−	15.0000i	−0.954911	−	0.826977i
$330$		0			0
$331$	10.0000			0.549650			0.274825	−	0.961494i	$-0.411380\pi$
$331$	10.0000			0.549650			0.274825	+	0.961494i	$0.411380\pi$
$332$	3.46410			0.190117
$333$	15.0000			0.821995
$334$		−	17.3205i		−	0.947736i
$335$	6.92820			0.378528
$336$	3.00000	−	3.46410i	0.163663	−	0.188982i
$337$	4.00000			0.217894			0.108947	−	0.994048i	$-0.465252\pi$
$337$	4.00000			0.217894			0.108947	+	0.994048i	$0.465252\pi$
$338$			10.0000i			0.543928i
$339$	−15.5885			−0.846649
$340$	−6.00000			−0.325396
$341$		0			0
$342$	−20.7846			−1.12390
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		−	1.00000i		−	0.0539164i
$345$	3.00000			0.161515
$346$			3.46410i			0.186231i
$347$		−	9.00000i		−	0.483145i	−0.970383	−	0.241573i	$-0.922337\pi$
$347$		−	9.00000i		−	0.483145i	0.970383	−	0.241573i	$-0.0776632\pi$
$348$	−15.5885			−0.835629
$349$		−	13.8564i		−	0.741716i	−0.928689	−	0.370858i	$-0.879064\pi$
$349$		−	13.8564i		−	0.741716i	0.928689	−	0.370858i	$-0.120936\pi$
$350$	3.46410	−	4.00000i	0.185164	−	0.213809i
$351$	9.00000			0.480384
$352$		0			0
$353$	−5.19615			−0.276563			−0.138282	−	0.990393i	$-0.544158\pi$
$353$	−5.19615			−0.276563			−0.138282	+	0.990393i	$0.544158\pi$
$354$	6.00000			0.318896
$355$			20.7846i			1.10313i
$356$	−3.46410			−0.183597
$357$	−10.3923	+	12.0000i	−0.550019	+	0.635107i
$358$	−21.0000			−1.10988
$359$		−	15.0000i		−	0.791670i	−0.918322	−	0.395835i	$-0.870455\pi$
$359$		−	15.0000i		−	0.791670i	0.918322	−	0.395835i	$-0.129545\pi$
$360$		−	5.19615i		−	0.273861i
$361$	−29.0000			−1.52632
$362$	−17.3205			−0.910346
$363$		−	19.0526i		−	1.00000i
$364$	−3.00000	+	3.46410i	−0.157243	+	0.181568i
$365$			12.0000i			0.628109i
$366$		0			0
$367$			1.73205i			0.0904123i	0.998978	+	0.0452062i	$0.0143945\pi$
$367$			1.73205i			0.0904123i	−0.998978	+	0.0452062i	$0.985606\pi$
$368$		−	1.00000i		−	0.0521286i
$369$	36.3731			1.89351
$370$			8.66025i			0.450225i
$371$	20.7846	−	24.0000i	1.07908	−	1.24602i
$372$	18.0000			0.933257
$373$	−10.0000			−0.517780			−0.258890	−	0.965907i	$-0.583357\pi$
$373$	−10.0000			−0.517780			−0.258890	+	0.965907i	$0.583357\pi$
$374$		0			0
$375$		−	21.0000i		−	1.08444i
$376$			8.66025i			0.446619i
$377$	15.5885			0.802846
$378$	−10.3923	−	9.00000i	−0.534522	−	0.462910i
$379$	−11.0000			−0.565032			−0.282516	−	0.959263i	$-0.591169\pi$
$379$	−11.0000			−0.565032			−0.282516	+	0.959263i	$0.591169\pi$
$380$		−	12.0000i		−	0.615587i
$381$		−	29.4449i		−	1.50851i
$382$		0			0
$383$	27.7128			1.41606			0.708029	−	0.706183i	$-0.249584\pi$
$383$	27.7128			1.41606			0.708029	+	0.706183i	$0.249584\pi$
$384$	−1.73205			−0.0883883
$385$		0			0
$386$		−	5.00000i		−	0.254493i
$387$	−3.00000			−0.152499
$388$		−	12.1244i		−	0.615521i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$			5.19615i			0.263117i
$391$			3.46410i			0.175187i
$392$	6.92820	−	1.00000i	0.349927	−	0.0505076i
$393$		−	24.0000i		−	1.21064i
$394$	15.0000			0.755689
$395$	17.3205			0.871489
$396$		0			0
$397$			13.8564i			0.695433i	0.937600	+	0.347717i	$0.113043\pi$
$397$			13.8564i			0.695433i	−0.937600	+	0.347717i	$0.886957\pi$
$398$	12.1244			0.607739
$399$	−24.0000	−	20.7846i	−1.20150	−	1.04053i
$400$	−2.00000			−0.100000
$401$			18.0000i			0.898877i	0.893311	+	0.449439i	$0.148376\pi$
$401$			18.0000i			0.898877i	−0.893311	+	0.449439i	$0.851624\pi$
$402$	−6.92820			−0.345547
$403$	−18.0000			−0.896644
$404$		0			0
$405$	−15.5885			−0.774597
$406$	−18.0000	−	15.5885i	−0.893325	−	0.773642i
$407$		0			0
$408$	6.00000			0.297044
$409$		−	13.8564i		−	0.685155i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$		−	13.8564i		−	0.685155i	0.939490	−	0.342578i	$-0.111300\pi$
$410$			21.0000i			1.03712i
$411$	−25.9808			−1.28154
$412$		−	5.19615i		−	0.255996i
$413$	6.92820	+	6.00000i	0.340915	+	0.295241i
$414$	−3.00000			−0.147442
$415$	6.00000			0.294528
$416$	1.73205			0.0849208
$417$	9.00000			0.440732
$418$		0			0
$419$	−3.46410			−0.169232			−0.0846162	−	0.996414i	$-0.526966\pi$
$419$	−3.46410			−0.169232			−0.0846162	+	0.996414i	$0.526966\pi$
$420$	5.19615	−	6.00000i	0.253546	−	0.292770i
$421$	−19.0000			−0.926003			−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000			−0.926003			−0.463002	+	0.886357i	$0.653228\pi$
$422$		−	26.0000i		−	1.26566i
$423$	25.9808			1.26323
$424$	−12.0000			−0.582772
$425$	6.92820			0.336067
$426$		−	20.7846i		−	1.00702i
$427$		0			0
$428$		−	18.0000i		−	0.870063i
$429$		0			0
$430$		−	1.73205i		−	0.0835269i
$431$		−	27.0000i		−	1.30054i	−0.759701	−	0.650272i	$-0.774655\pi$
$431$		−	27.0000i		−	1.30054i	0.759701	−	0.650272i	$-0.225345\pi$
$432$			5.19615i			0.250000i
$433$			19.0526i			0.915608i	0.889053	+	0.457804i	$0.151364\pi$
$433$			19.0526i			0.915608i	−0.889053	+	0.457804i	$0.848636\pi$
$434$	20.7846	+	18.0000i	0.997693	+	0.864028i
$435$	−27.0000			−1.29455
$436$	11.0000			0.526804
$437$	−6.92820			−0.331421
$438$		−	12.0000i		−	0.573382i
$439$		−	27.7128i		−	1.32266i	−0.750095	−	0.661330i	$-0.769992\pi$
$439$		−	27.7128i		−	1.32266i	0.750095	−	0.661330i	$-0.230008\pi$
$440$		0			0
$441$	−3.00000	−	20.7846i	−0.142857	−	0.989743i
$442$	−6.00000			−0.285391
$443$			9.00000i			0.427603i	0.976877	+	0.213801i	$0.0685846\pi$
$443$			9.00000i			0.427603i	−0.976877	+	0.213801i	$0.931415\pi$
$444$		−	8.66025i		−	0.410997i
$445$	−6.00000			−0.284427
$446$	6.92820			0.328060
$447$	−31.1769			−1.47462
$448$	−2.00000	−	1.73205i	−0.0944911	−	0.0818317i
$449$		−	24.0000i		−	1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$		−	24.0000i		−	1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$			6.00000i			0.282843i
$451$		0			0
$452$			9.00000i			0.423324i
$453$		−	32.9090i		−	1.54620i
$454$			15.5885i			0.731603i
$455$	−5.19615	+	6.00000i	−0.243599	+	0.281284i
$456$			12.0000i			0.561951i
$457$	22.0000			1.02912			0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000			1.02912			0.514558	+	0.857455i	$0.327956\pi$
$458$	−24.2487			−1.13307
$459$		−	18.0000i		−	0.840168i
$460$		−	1.73205i		−	0.0807573i
$461$	−31.1769			−1.45205			−0.726027	−	0.687666i	$-0.758635\pi$
$461$	−31.1769			−1.45205			−0.726027	+	0.687666i	$0.758635\pi$
$462$		0			0
$463$	17.0000			0.790057			0.395029	−	0.918669i	$-0.370735\pi$
$463$	17.0000			0.790057			0.395029	+	0.918669i	$0.370735\pi$
$464$			9.00000i			0.417815i
$465$	31.1769			1.44579
$466$	12.0000			0.555889
$467$	−19.0526			−0.881647			−0.440824	−	0.897594i	$-0.645314\pi$
$467$	−19.0526			−0.881647			−0.440824	+	0.897594i	$0.645314\pi$
$468$		−	5.19615i		−	0.240192i
$469$	−8.00000	−	6.92820i	−0.369406	−	0.319915i
$470$			15.0000i			0.691898i
$471$		0			0
$472$		−	3.46410i		−	0.159448i
$473$		0			0
$474$	−17.3205			−0.795557
$475$			13.8564i			0.635776i
$476$	6.92820	+	6.00000i	0.317554	+	0.275010i
$477$			36.0000i			1.64833i
$478$	6.00000			0.274434
$479$	−27.7128			−1.26623			−0.633115	−	0.774057i	$-0.718224\pi$
$479$	−27.7128			−1.26623			−0.633115	+	0.774057i	$0.718224\pi$
$480$	−3.00000			−0.136931
$481$			8.66025i			0.394874i
$482$	15.5885			0.710035
$483$	−3.46410	−	3.00000i	−0.157622	−	0.136505i
$484$	−11.0000			−0.500000
$485$		−	21.0000i		−	0.953561i
$486$	15.5885			0.707107
$487$	−29.0000			−1.31412			−0.657058	−	0.753840i	$-0.728199\pi$
$487$	−29.0000			−1.31412			−0.657058	+	0.753840i	$0.728199\pi$
$488$		0			0
$489$		−	24.2487i		−	1.09656i
$490$	12.0000	−	1.73205i	0.542105	−	0.0782461i
$491$		−	36.0000i		−	1.62466i	−0.583200	−	0.812329i	$-0.698200\pi$
$491$		−	36.0000i		−	1.62466i	0.583200	−	0.812329i	$-0.301800\pi$
$492$		−	21.0000i		−	0.946753i
$493$		−	31.1769i		−	1.40414i
$494$		−	12.0000i		−	0.539906i
$495$		0			0
$496$		−	10.3923i		−	0.466628i
$497$	20.7846	−	24.0000i	0.932317	−	1.07655i
$498$	−6.00000			−0.268866
$499$	4.00000			0.179065			0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000			0.179065			0.0895323	+	0.995984i	$0.471463\pi$
$500$	−12.1244			−0.542218
$501$			30.0000i			1.34030i
$502$		−	22.5167i		−	1.00497i
$503$	−20.7846			−0.926740			−0.463370	−	0.886165i	$-0.653360\pi$
$503$	−20.7846			−0.926740			−0.463370	+	0.886165i	$0.653360\pi$
$504$	−5.19615	+	6.00000i	−0.231455	+	0.267261i
$505$		0			0
$506$		0			0
$507$		−	17.3205i		−	0.769231i
$508$	−17.0000			−0.754253
$509$	24.2487			1.07481			0.537403	−	0.843326i	$-0.319406\pi$
$509$	24.2487			1.07481			0.537403	+	0.843326i	$0.319406\pi$
$510$	10.3923			0.460179
$511$	12.0000	−	13.8564i	0.530849	−	0.612971i
$512$			1.00000i			0.0441942i
$513$	36.0000			1.58944
$514$			6.92820i			0.305590i
$515$		−	9.00000i		−	0.396587i
$516$			1.73205i			0.0762493i
$517$		0			0
$518$	8.66025	−	10.0000i	0.380510	−	0.439375i
$519$		−	6.00000i		−	0.263371i
$520$	3.00000			0.131559
$521$	10.3923			0.455295			0.227648	−	0.973744i	$-0.426897\pi$
$521$	10.3923			0.455295			0.227648	+	0.973744i	$0.426897\pi$
$522$	27.0000			1.18176
$523$		−	3.46410i		−	0.151475i	−0.997128	−	0.0757373i	$-0.975869\pi$
$523$		−	3.46410i		−	0.151475i	0.997128	−	0.0757373i	$-0.0241310\pi$
$524$	−13.8564			−0.605320
$525$	−6.00000	+	6.92820i	−0.261861	+	0.302372i
$526$	−27.0000			−1.17726
$527$			36.0000i			1.56818i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$	−20.7846			−0.902826
$531$	−10.3923			−0.450988
$532$	−12.0000	+	13.8564i	−0.520266	+	0.600751i
$533$			21.0000i			0.909611i
$534$	6.00000			0.259645
$535$		−	31.1769i		−	1.34790i
$536$			4.00000i			0.172774i
$537$	36.3731			1.56961
$538$			13.8564i			0.597392i
$539$		0			0
$540$			9.00000i			0.387298i
$541$	38.0000			1.63375			0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000			1.63375			0.816874	+	0.576816i	$0.195705\pi$
$542$	−20.7846			−0.892775
$543$	30.0000			1.28742
$544$		−	3.46410i		−	0.148522i
$545$	19.0526			0.816122
$546$	5.19615	−	6.00000i	0.222375	−	0.256776i
$547$	26.0000			1.11168			0.555840	−	0.831289i	$-0.312397\pi$
$547$	26.0000			1.11168			0.555840	+	0.831289i	$0.312397\pi$
$548$			15.0000i			0.640768i
$549$		0			0
$550$		0			0
$551$	62.3538			2.65636
$552$			1.73205i			0.0737210i
$553$	−20.0000	−	17.3205i	−0.850487	−	0.736543i
$554$		−	10.0000i		−	0.424859i
$555$		−	15.0000i		−	0.636715i
$556$		−	5.19615i		−	0.220366i
$557$			6.00000i			0.254228i	0.991888	+	0.127114i	$0.0405714\pi$
$557$			6.00000i			0.254228i	−0.991888	+	0.127114i	$0.959429\pi$
$558$	−31.1769			−1.31982
$559$		−	1.73205i		−	0.0732579i
$560$	−3.46410	−	3.00000i	−0.146385	−	0.126773i
$561$		0			0
$562$	9.00000			0.379642
$563$	32.9090			1.38695			0.693474	−	0.720482i	$-0.256079\pi$
$563$	32.9090			1.38695			0.693474	+	0.720482i	$0.256079\pi$
$564$		−	15.0000i		−	0.631614i
$565$			15.5885i			0.655811i
$566$	3.46410			0.145607
$567$	18.0000	+	15.5885i	0.755929	+	0.654654i
$568$	−12.0000			−0.503509
$569$			9.00000i			0.377300i	0.982044	+	0.188650i	$0.0604111\pi$
$569$			9.00000i			0.377300i	−0.982044	+	0.188650i	$0.939589\pi$
$570$			20.7846i			0.870572i
$571$	−28.0000			−1.17176			−0.585882	−	0.810397i	$-0.699252\pi$
$571$	−28.0000			−1.17176			−0.585882	+	0.810397i	$0.699252\pi$
$572$		0			0
$573$		0			0
$574$	21.0000	−	24.2487i	0.876523	−	1.01212i
$575$			2.00000i			0.0834058i
$576$	3.00000			0.125000
$577$			13.8564i			0.576850i	0.957503	+	0.288425i	$0.0931316\pi$
$577$			13.8564i			0.576850i	−0.957503	+	0.288425i	$0.906868\pi$
$578$		−	5.00000i		−	0.207973i
$579$			8.66025i			0.359908i
$580$			15.5885i			0.647275i
$581$	−6.92820	−	6.00000i	−0.287430	−	0.248922i
$582$			21.0000i			0.870478i
$583$		0			0
$584$	−6.92820			−0.286691
$585$		−	9.00000i		−	0.372104i
$586$		0			0
$587$	38.1051			1.57277			0.786383	−	0.617739i	$-0.211951\pi$
$587$	38.1051			1.57277			0.786383	+	0.617739i	$0.211951\pi$
$588$	−12.0000	+	1.73205i	−0.494872	+	0.0714286i
$589$	−72.0000			−2.96671
$590$		−	6.00000i		−	0.247016i
$591$	−25.9808			−1.06871
$592$	−5.00000			−0.205499
$593$	−5.19615			−0.213380			−0.106690	−	0.994292i	$-0.534025\pi$
$593$	−5.19615			−0.213380			−0.106690	+	0.994292i	$0.534025\pi$
$594$		0			0
$595$	12.0000	+	10.3923i	0.491952	+	0.426043i
$596$			18.0000i			0.737309i
$597$	−21.0000			−0.859473
$598$		−	1.73205i		−	0.0708288i
$599$		−	18.0000i		−	0.735460i	−0.929933	−	0.367730i	$-0.880135\pi$
$599$		−	18.0000i		−	0.735460i	0.929933	−	0.367730i	$-0.119865\pi$
$600$	3.46410			0.141421
$601$		−	24.2487i		−	0.989126i	−0.869142	−	0.494563i	$-0.835328\pi$
$601$		−	24.2487i		−	0.989126i	0.869142	−	0.494563i	$-0.164672\pi$
$602$	−1.73205	+	2.00000i	−0.0705931	+	0.0815139i
$603$	12.0000			0.488678
$604$	−19.0000			−0.773099
$605$	−19.0526			−0.774597
$606$		0			0
$607$		−	17.3205i		−	0.703018i	−0.936185	−	0.351509i	$-0.885669\pi$
$607$		−	17.3205i		−	0.703018i	0.936185	−	0.351509i	$-0.114331\pi$
$608$	6.92820			0.280976
$609$	31.1769	+	27.0000i	1.26335	+	1.09410i
$610$		0			0
$611$			15.0000i			0.606835i
$612$	−10.3923			−0.420084
$613$	−31.0000			−1.25208			−0.626039	−	0.779792i	$-0.715325\pi$
$613$	−31.0000			−1.25208			−0.626039	+	0.779792i	$0.715325\pi$
$614$	−1.73205			−0.0698999
$615$		−	36.3731i		−	1.46670i
$616$		0			0
$617$			18.0000i			0.724653i	0.932051	+	0.362326i	$0.118017\pi$
$617$			18.0000i			0.724653i	−0.932051	+	0.362326i	$0.881983\pi$
$618$			9.00000i			0.362033i
$619$		−	6.92820i		−	0.278468i	−0.990260	−	0.139234i	$-0.955536\pi$
$619$		−	6.92820i		−	0.278468i	0.990260	−	0.139234i	$-0.0444640\pi$
$620$		−	18.0000i		−	0.722897i
$621$	5.19615			0.208514
$622$		−	10.3923i		−	0.416693i
$623$	6.92820	+	6.00000i	0.277573	+	0.240385i
$624$	−3.00000			−0.120096
$625$	−11.0000			−0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	17.3205			0.690614
$630$	−9.00000	+	10.3923i	−0.358569	+	0.414039i
$631$	−28.0000			−1.11466			−0.557331	−	0.830290i	$-0.688175\pi$
$631$	−28.0000			−1.11466			−0.557331	+	0.830290i	$0.688175\pi$
$632$			10.0000i			0.397779i
$633$			45.0333i			1.78991i
$634$	−3.00000			−0.119145
$635$	−29.4449			−1.16848
$636$	20.7846			0.824163
$637$	12.0000	−	1.73205i	0.475457	−	0.0686264i
$638$		0			0
$639$			36.0000i			1.42414i
$640$			1.73205i			0.0684653i
$641$		−	27.0000i		−	1.06644i	−0.845978	−	0.533218i	$-0.820983\pi$
$641$		−	27.0000i		−	1.06644i	0.845978	−	0.533218i	$-0.179017\pi$
$642$			31.1769i			1.23045i
$643$			38.1051i			1.50272i	0.659893	+	0.751360i	$0.270602\pi$
$643$			38.1051i			1.50272i	−0.659893	+	0.751360i	$0.729398\pi$
$644$	−1.73205	+	2.00000i	−0.0682524	+	0.0788110i
$645$			3.00000i			0.118125i
$646$	−24.0000			−0.944267
$647$	−17.3205			−0.680939			−0.340470	−	0.940255i	$-0.610586\pi$
$647$	−17.3205			−0.680939			−0.340470	+	0.940255i	$0.610586\pi$
$648$		−	9.00000i		−	0.353553i
$649$		0			0
$650$	−3.46410			−0.135873
$651$	−36.0000	−	31.1769i	−1.41095	−	1.22192i
$652$	−14.0000			−0.548282
$653$		−	21.0000i		−	0.821794i	−0.911682	−	0.410897i	$-0.865216\pi$
$653$		−	21.0000i		−	0.821794i	0.911682	−	0.410897i	$-0.134784\pi$
$654$	−19.0526			−0.745014
$655$	−24.0000			−0.937758
$656$	−12.1244			−0.473377
$657$			20.7846i			0.810885i
$658$	15.0000	−	17.3205i	0.584761	−	0.675224i
$659$			6.00000i			0.233727i	0.993148	+	0.116863i	$0.0372840\pi$
$659$			6.00000i			0.233727i	−0.993148	+	0.116863i	$0.962716\pi$
$660$		0			0
$661$			6.92820i			0.269476i	0.990881	+	0.134738i	$0.0430193\pi$
$661$			6.92820i			0.269476i	−0.990881	+	0.134738i	$0.956981\pi$
$662$			10.0000i			0.388661i
$663$	10.3923			0.403604
$664$			3.46410i			0.134433i
$665$	−20.7846	+	24.0000i	−0.805993	+	0.930680i
$666$			15.0000i			0.581238i
$667$	9.00000			0.348481
$668$	17.3205			0.670151
$669$	−12.0000			−0.463947
$670$			6.92820i			0.267660i
$671$		0			0
$672$	3.46410	+	3.00000i	0.133631	+	0.115728i
$673$	25.0000			0.963679			0.481840	−	0.876259i	$-0.339969\pi$
$673$	25.0000			0.963679			0.481840	+	0.876259i	$0.339969\pi$
$674$			4.00000i			0.154074i
$675$		−	10.3923i		−	0.400000i
$676$	−10.0000			−0.384615
$677$	−27.7128			−1.06509			−0.532545	−	0.846402i	$-0.678764\pi$
$677$	−27.7128			−1.06509			−0.532545	+	0.846402i	$0.678764\pi$
$678$		−	15.5885i		−	0.598671i
$679$	−21.0000	+	24.2487i	−0.805906	+	0.930580i
$680$		−	6.00000i		−	0.230089i
$681$		−	27.0000i		−	1.03464i
$682$		0			0
$683$			12.0000i			0.459167i	0.973289	+	0.229584i	$0.0737364\pi$
$683$			12.0000i			0.459167i	−0.973289	+	0.229584i	$0.926264\pi$
$684$		−	20.7846i		−	0.794719i
$685$			25.9808i			0.992674i
$686$	−15.5885	−	10.0000i	−0.595170	−	0.381802i
$687$	42.0000			1.60240
$688$	1.00000			0.0381246
$689$	−20.7846			−0.791831
$690$			3.00000i			0.114208i
$691$		−	43.3013i		−	1.64726i	−0.567129	−	0.823629i	$-0.691946\pi$
$691$		−	43.3013i		−	1.64726i	0.567129	−	0.823629i	$-0.308054\pi$
$692$	−3.46410			−0.131685
$693$		0			0
$694$	9.00000			0.341635
$695$		−	9.00000i		−	0.341389i
$696$		−	15.5885i		−	0.590879i
$697$	42.0000			1.59086
$698$	13.8564			0.524473
$699$	−20.7846			−0.786146
$700$	4.00000	+	3.46410i	0.151186	+	0.130931i
$701$			6.00000i			0.226617i	0.993560	+	0.113308i	$0.0361448\pi$
$701$			6.00000i			0.226617i	−0.993560	+	0.113308i	$0.963855\pi$
$702$			9.00000i			0.339683i
$703$			34.6410i			1.30651i
$704$		0			0
$705$		−	25.9808i		−	0.978492i
$706$		−	5.19615i		−	0.195560i
$707$		0			0
$708$			6.00000i			0.225494i
$709$	22.0000			0.826227			0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000			0.826227			0.413114	+	0.910679i	$0.364441\pi$
$710$	−20.7846			−0.780033
$711$	30.0000			1.12509
$712$		−	3.46410i		−	0.129823i
$713$	−10.3923			−0.389195
$714$	−12.0000	−	10.3923i	−0.449089	−	0.388922i
$715$		0			0
$716$		−	21.0000i		−	0.784807i
$717$	−10.3923			−0.388108
$718$	15.0000			0.559795
$719$	12.1244			0.452162			0.226081	−	0.974108i	$-0.427409\pi$
$719$	12.1244			0.452162			0.226081	+	0.974108i	$0.427409\pi$
$720$	5.19615			0.193649
$721$	−9.00000	+	10.3923i	−0.335178	+	0.387030i
$722$		−	29.0000i		−	1.07927i
$723$	−27.0000			−1.00414
$724$		−	17.3205i		−	0.643712i
$725$		−	18.0000i		−	0.668503i
$726$	19.0526			0.707107
$727$			31.1769i			1.15629i	0.815935	+	0.578144i	$0.196223\pi$
$727$			31.1769i			1.15629i	−0.815935	+	0.578144i	$0.803777\pi$
$728$	−3.46410	−	3.00000i	−0.128388	−	0.111187i
$729$	−27.0000			−1.00000
$730$	−12.0000			−0.444140
$731$	−3.46410			−0.128124
$732$		0			0
$733$			3.46410i			0.127950i	0.997952	+	0.0639748i	$0.0203777\pi$
$733$			3.46410i			0.127950i	−0.997952	+	0.0639748i	$0.979622\pi$
$734$	−1.73205			−0.0639312
$735$	−20.7846	+	3.00000i	−0.766652	+	0.110657i
$736$	1.00000			0.0368605
$737$		0			0
$738$			36.3731i			1.33891i
$739$	20.0000			0.735712			0.367856	−	0.929883i	$-0.380092\pi$
$739$	20.0000			0.735712			0.367856	+	0.929883i	$0.380092\pi$
$740$	−8.66025			−0.318357
$741$			20.7846i			0.763542i
$742$	24.0000	+	20.7846i	0.881068	+	0.763027i
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$			18.0000i			0.659912i
$745$			31.1769i			1.14223i
$746$		−	10.0000i		−	0.366126i
$747$	10.3923			0.380235
$748$		0			0
$749$	−31.1769	+	36.0000i	−1.13918	+	1.31541i
$750$	21.0000			0.766812
$751$	−14.0000			−0.510867			−0.255434	−	0.966827i	$-0.582218\pi$
$751$	−14.0000			−0.510867			−0.255434	+	0.966827i	$0.582218\pi$
$752$	−8.66025			−0.315807
$753$			39.0000i			1.42124i
$754$			15.5885i			0.567698i
$755$	−32.9090			−1.19768
$756$	9.00000	−	10.3923i	0.327327	−	0.377964i
$757$	10.0000			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$757$	10.0000			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$758$		−	11.0000i		−	0.399538i
$759$		0			0
$760$	12.0000			0.435286
$761$	27.7128			1.00459			0.502294	−	0.864697i	$-0.332489\pi$
$761$	27.7128			1.00459			0.502294	+	0.864697i	$0.332489\pi$
$762$	29.4449			1.06667
$763$	−22.0000	−	19.0526i	−0.796453	−	0.689749i
$764$		0			0
$765$	−18.0000			−0.650791
$766$			27.7128i			1.00130i
$767$		−	6.00000i		−	0.216647i
$768$		−	1.73205i		−	0.0625000i
$769$		−	8.66025i		−	0.312297i	−0.987734	−	0.156148i	$-0.950092\pi$
$769$		−	8.66025i		−	0.312297i	0.987734	−	0.156148i	$-0.0499078\pi$
$770$		0			0
$771$		−	12.0000i		−	0.432169i
$772$	5.00000			0.179954
$773$	−1.73205			−0.0622975			−0.0311488	−	0.999515i	$-0.509917\pi$
$773$	−1.73205			−0.0622975			−0.0311488	+	0.999515i	$0.509917\pi$
$774$		−	3.00000i		−	0.107833i
$775$			20.7846i			0.746605i
$776$	12.1244			0.435239
$777$	−15.0000	+	17.3205i	−0.538122	+	0.621370i
$778$		0			0
$779$			84.0000i			3.00961i
$780$	−5.19615			−0.186052
$781$		0			0
$782$	−3.46410			−0.123876
$783$	−46.7654			−1.67126
$784$	1.00000	+	6.92820i	0.0357143	+	0.247436i
$785$		0			0
$786$	24.0000			0.856052
$787$			17.3205i			0.617409i	0.951158	+	0.308705i	$0.0998955\pi$
$787$			17.3205i			0.617409i	−0.951158	+	0.308705i	$0.900105\pi$
$788$			15.0000i			0.534353i
$789$	46.7654			1.66489
$790$			17.3205i			0.616236i
$791$	15.5885	−	18.0000i	0.554262	−	0.640006i
$792$		0			0
$793$		0			0
$794$	−13.8564			−0.491745
$795$	36.0000			1.27679
$796$			12.1244i			0.429736i
$797$	15.5885			0.552171			0.276086	−	0.961133i	$-0.410963\pi$
$797$	15.5885			0.552171			0.276086	+	0.961133i	$0.410963\pi$
$798$	20.7846	−	24.0000i	0.735767	−	0.849591i
$799$	30.0000			1.06132
$800$		−	2.00000i		−	0.0707107i
$801$	−10.3923			−0.367194
$802$	−18.0000			−0.635602
$803$		0			0
$804$		−	6.92820i		−	0.244339i
$805$	−3.00000	+	3.46410i	−0.105736	+	0.122094i
$806$		−	18.0000i		−	0.634023i
$807$		−	24.0000i		−	0.844840i
$808$		0			0
$809$		−	48.0000i		−	1.68759i	−0.536666	−	0.843795i	$-0.680316\pi$
$809$		−	48.0000i		−	1.68759i	0.536666	−	0.843795i	$-0.319684\pi$
$810$		−	15.5885i		−	0.547723i
$811$			1.73205i			0.0608205i	0.999538	+	0.0304103i	$0.00968138\pi$
$811$			1.73205i			0.0608205i	−0.999538	+	0.0304103i	$0.990319\pi$
$812$	15.5885	−	18.0000i	0.547048	−	0.631676i
$813$	36.0000			1.26258
$814$		0			0
$815$	−24.2487			−0.849395
$816$			6.00000i			0.210042i
$817$		−	6.92820i		−	0.242387i
$818$	13.8564			0.484478
$819$	−9.00000	+	10.3923i	−0.314485	+	0.363137i
$820$	−21.0000			−0.733352
$821$		−	6.00000i		−	0.209401i	−0.994504	−	0.104701i	$-0.966612\pi$
$821$		−	6.00000i		−	0.209401i	0.994504	−	0.104701i	$-0.0333885\pi$
$822$		−	25.9808i		−	0.906183i
$823$	−31.0000			−1.08059			−0.540296	−	0.841475i	$-0.681688\pi$
$823$	−31.0000			−1.08059			−0.540296	+	0.841475i	$0.681688\pi$
$824$	5.19615			0.181017
$825$		0			0
$826$	−6.00000	+	6.92820i	−0.208767	+	0.241063i
$827$		−	30.0000i		−	1.04320i	−0.853189	−	0.521601i	$-0.825335\pi$
$827$		−	30.0000i		−	1.04320i	0.853189	−	0.521601i	$-0.174665\pi$
$828$		−	3.00000i		−	0.104257i
$829$			6.92820i			0.240626i	0.992736	+	0.120313i	$0.0383899\pi$
$829$			6.92820i			0.240626i	−0.992736	+	0.120313i	$0.961610\pi$
$830$			6.00000i			0.208263i
$831$			17.3205i			0.600842i
$832$			1.73205i			0.0600481i
$833$	−3.46410	−	24.0000i	−0.120024	−	0.831551i
$834$			9.00000i			0.311645i
$835$	30.0000			1.03819
$836$		0			0
$837$	54.0000			1.86651
$838$		−	3.46410i		−	0.119665i
$839$	−45.0333			−1.55472			−0.777361	−	0.629054i	$-0.783442\pi$
$839$	−45.0333			−1.55472			−0.777361	+	0.629054i	$0.783442\pi$
$840$	6.00000	+	5.19615i	0.207020	+	0.179284i
$841$	−52.0000			−1.79310
$842$		−	19.0000i		−	0.654783i
$843$	−15.5885			−0.536895
$844$	26.0000			0.894957
$845$	−17.3205			−0.595844
$846$			25.9808i			0.893237i
$847$	22.0000	+	19.0526i	0.755929	+	0.654654i
$848$		−	12.0000i		−	0.412082i
$849$	−6.00000			−0.205919
$850$			6.92820i			0.237635i
$851$			5.00000i			0.171398i
$852$	20.7846			0.712069
$853$		−	22.5167i		−	0.770956i	−0.922717	−	0.385478i	$-0.874037\pi$
$853$		−	22.5167i		−	0.770956i	0.922717	−	0.385478i	$-0.125963\pi$
$854$		0			0
$855$		−	36.0000i		−	1.23117i
$856$	18.0000			0.615227
$857$	−50.2295			−1.71581			−0.857903	−	0.513812i	$-0.828233\pi$
$857$	−50.2295			−1.71581			−0.857903	+	0.513812i	$0.828233\pi$
$858$		0			0
$859$			8.66025i			0.295484i	0.989026	+	0.147742i	$0.0472005\pi$
$859$			8.66025i			0.295484i	−0.989026	+	0.147742i	$0.952799\pi$
$860$	1.73205			0.0590624
$861$	−36.3731	+	42.0000i	−1.23959	+	1.43136i
$862$	27.0000			0.919624
$863$			48.0000i			1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$			48.0000i			1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	−5.19615			−0.176777
$865$	−6.00000			−0.204006
$866$	−19.0526			−0.647432
$867$			8.66025i			0.294118i
$868$	−18.0000	+	20.7846i	−0.610960	+	0.705476i
$869$		0			0
$870$		−	27.0000i		−	0.915386i
$871$			6.92820i			0.234753i
$872$			11.0000i			0.372507i
$873$		−	36.3731i		−	1.23104i
$874$		−	6.92820i		−	0.234350i
$875$	24.2487	+	21.0000i	0.819756	+	0.709930i
$876$	12.0000			0.405442
$877$	32.0000			1.08056			0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000			1.08056			0.540282	+	0.841484i	$0.318318\pi$
$878$	27.7128			0.935262
$879$		0			0
$880$		0			0
$881$	3.46410			0.116709			0.0583543	−	0.998296i	$-0.481415\pi$
$881$	3.46410			0.116709			0.0583543	+	0.998296i	$0.481415\pi$
$882$	20.7846	−	3.00000i	0.699854	−	0.101015i
$883$	10.0000			0.336527			0.168263	−	0.985742i	$-0.446184\pi$
$883$	10.0000			0.336527			0.168263	+	0.985742i	$0.446184\pi$
$884$		−	6.00000i		−	0.201802i
$885$			10.3923i			0.349334i
$886$	−9.00000			−0.302361
$887$	−51.9615			−1.74470			−0.872349	−	0.488884i	$-0.837404\pi$
$887$	−51.9615			−1.74470			−0.872349	+	0.488884i	$0.837404\pi$
$888$	8.66025			0.290619
$889$	34.0000	+	29.4449i	1.14032	+	0.987549i
$890$		−	6.00000i		−	0.201120i
$891$		0			0
$892$			6.92820i			0.231973i
$893$			60.0000i			2.00782i
$894$		−	31.1769i		−	1.04271i
$895$		−	36.3731i		−	1.21582i
$896$	1.73205	−	2.00000i	0.0578638	−	0.0668153i
$897$			3.00000i			0.100167i
$898$	24.0000			0.800890
$899$	93.5307			3.11942
$900$	−6.00000			−0.200000
$901$			41.5692i			1.38487i
$902$		0			0
$903$	3.00000	−	3.46410i	0.0998337	−	0.115278i
$904$	−9.00000			−0.299336
$905$		−	30.0000i		−	0.997234i
$906$	32.9090			1.09333
$907$	37.0000			1.22856			0.614282	−	0.789086i	$-0.289446\pi$
$907$	37.0000			1.22856			0.614282	+	0.789086i	$0.289446\pi$
$908$	−15.5885			−0.517321
$909$		0			0
$910$	−6.00000	−	5.19615i	−0.198898	−	0.172251i
$911$			9.00000i			0.298183i	0.988823	+	0.149092i	$0.0476349\pi$
$911$			9.00000i			0.298183i	−0.988823	+	0.149092i	$0.952365\pi$
$912$	−12.0000			−0.397360
$913$		0			0
$914$			22.0000i			0.727695i
$915$		0			0
$916$		−	24.2487i		−	0.801200i
$917$	27.7128	+	24.0000i	0.915158	+	0.792550i
$918$	18.0000			0.594089
$919$	−26.0000			−0.857661			−0.428830	−	0.903385i	$-0.641074\pi$
$919$	−26.0000			−0.857661			−0.428830	+	0.903385i	$0.641074\pi$
$920$	1.73205			0.0571040
$921$	3.00000			0.0988534
$922$		−	31.1769i		−	1.02676i
$923$	−20.7846			−0.684134
$924$		0			0
$925$	10.0000			0.328798
$926$			17.0000i			0.558655i
$927$		−	15.5885i		−	0.511992i
$928$	−9.00000			−0.295439
$929$	50.2295			1.64798			0.823988	−	0.566608i	$-0.191744\pi$
$929$	50.2295			1.64798			0.823988	+	0.566608i	$0.191744\pi$
$930$			31.1769i			1.02233i
$931$	48.0000	−	6.92820i	1.57314	−	0.227063i
$932$			12.0000i			0.393073i
$933$			18.0000i			0.589294i
$934$		−	19.0526i		−	0.623419i
$935$		0			0
$936$	5.19615			0.169842
$937$		−	5.19615i		−	0.169751i	−0.996392	−	0.0848755i	$-0.972951\pi$
$937$		−	5.19615i		−	0.169751i	0.996392	−	0.0848755i	$-0.0270492\pi$
$938$	6.92820	−	8.00000i	0.226214	−	0.261209i
$939$		0			0
$940$	−15.0000			−0.489246
$941$	−39.8372			−1.29865			−0.649327	−	0.760509i	$-0.724949\pi$
$941$	−39.8372			−1.29865			−0.649327	+	0.760509i	$0.724949\pi$
$942$		0			0
$943$			12.1244i			0.394823i
$944$	3.46410			0.112747
$945$	15.5885	−	18.0000i	0.507093	−	0.585540i
$946$		0			0
$947$			27.0000i			0.877382i	0.898638	+	0.438691i	$0.144558\pi$
$947$			27.0000i			0.877382i	−0.898638	+	0.438691i	$0.855442\pi$
$948$		−	17.3205i		−	0.562544i
$949$	−12.0000			−0.389536
$950$	−13.8564			−0.449561
$951$	5.19615			0.168497
$952$	−6.00000	+	6.92820i	−0.194461	+	0.224544i
$953$			42.0000i			1.36051i	0.732974	+	0.680257i	$0.238132\pi$
$953$			42.0000i			1.36051i	−0.732974	+	0.680257i	$0.761868\pi$
$954$	−36.0000			−1.16554
$955$		0			0
$956$			6.00000i			0.194054i
$957$		0			0
$958$		−	27.7128i		−	0.895360i
$959$	25.9808	−	30.0000i	0.838963	−	0.968751i
$960$		−	3.00000i		−	0.0968246i
$961$	−77.0000			−2.48387
$962$	−8.66025			−0.279218
$963$		−	54.0000i		−	1.74013i
$964$			15.5885i			0.502070i
$965$	8.66025			0.278783
$966$	3.00000	−	3.46410i	0.0965234	−	0.111456i
$967$	8.00000			0.257263			0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000			0.257263			0.128631	+	0.991692i	$0.458942\pi$
$968$		−	11.0000i		−	0.353553i
$969$	41.5692			1.33540
$970$	21.0000			0.674269
$971$	10.3923			0.333505			0.166752	−	0.985999i	$-0.446672\pi$
$971$	10.3923			0.333505			0.166752	+	0.985999i	$0.446672\pi$
$972$			15.5885i			0.500000i
$973$	−9.00000	+	10.3923i	−0.288527	+	0.333162i
$974$		−	29.0000i		−	0.929220i
$975$	6.00000			0.192154
$976$		0			0
$977$		−	57.0000i		−	1.82359i	−0.410644	−	0.911796i	$-0.634696\pi$
$977$		−	57.0000i		−	1.82359i	0.410644	−	0.911796i	$-0.365304\pi$
$978$	24.2487			0.775388
$979$		0			0
$980$	1.73205	+	12.0000i	0.0553283	+	0.383326i
$981$	33.0000			1.05361
$982$	36.0000			1.14881
$983$	−17.3205			−0.552438			−0.276219	−	0.961095i	$-0.589082\pi$
$983$	−17.3205			−0.552438			−0.276219	+	0.961095i	$0.589082\pi$
$984$	21.0000			0.669456
$985$			25.9808i			0.827816i
$986$	31.1769			0.992875
$987$	−25.9808	+	30.0000i	−0.826977	+	0.954911i
$988$	12.0000			0.381771
$989$		−	1.00000i		−	0.0317982i
$990$		0			0
$991$	−8.00000			−0.254128			−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000			−0.254128			−0.127064	+	0.991894i	$0.540555\pi$
$992$	10.3923			0.329956
$993$		−	17.3205i		−	0.549650i
$994$	24.0000	+	20.7846i	0.761234	+	0.659248i
$995$			21.0000i			0.665745i
$996$		−	6.00000i		−	0.190117i
$997$			6.92820i			0.219418i	0.993964	+	0.109709i	$0.0349920\pi$
$997$			6.92820i			0.219418i	−0.993964	+	0.109709i	$0.965008\pi$
$998$			4.00000i			0.126618i
$999$		−	25.9808i		−	0.821995i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.2.f.a.461.3	yes	4
3.2	odd	2	inner	966.2.f.a.461.1	✓	4
7.6	odd	2	inner	966.2.f.a.461.4	yes	4
21.20	even	2	inner	966.2.f.a.461.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.f.a.461.1	✓	4	3.2	odd	2	inner
966.2.f.a.461.2	yes	4	21.20	even	2	inner
966.2.f.a.461.3	yes	4	1.1	even	1	trivial
966.2.f.a.461.4	yes	4	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	966.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.73205i q^{3} -1.00000 q^{4} -1.73205 q^{5} +1.73205 q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} -3.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.73205i q^{3} -1.00000 q^{4} -1.73205 q^{5} +1.73205 q^{6} +(2.00000 + 1.73205i) q^{7} -1.00000i q^{8} -3.00000 q^{9} -1.73205i q^{10} +1.73205i q^{12} -1.73205i q^{13} +(-1.73205 + 2.00000i) q^{14} +3.00000i q^{15} +1.00000 q^{16} -3.46410 q^{17} -3.00000i q^{18} -6.92820i q^{19} +1.73205 q^{20} +(3.00000 - 3.46410i) q^{21} -1.00000i q^{23} -1.73205 q^{24} -2.00000 q^{25} +1.73205 q^{26} +5.19615i q^{27} +(-2.00000 - 1.73205i) q^{28} +9.00000i q^{29} -3.00000 q^{30} -10.3923i q^{31} +1.00000i q^{32} -3.46410i q^{34} +(-3.46410 - 3.00000i) q^{35} +3.00000 q^{36} -5.00000 q^{37} +6.92820 q^{38} -3.00000 q^{39} +1.73205i q^{40} -12.1244 q^{41} +(3.46410 + 3.00000i) q^{42} +1.00000 q^{43} +5.19615 q^{45} +1.00000 q^{46} -8.66025 q^{47} -1.73205i q^{48} +(1.00000 + 6.92820i) q^{49} -2.00000i q^{50} +6.00000i q^{51} +1.73205i q^{52} -12.0000i q^{53} -5.19615 q^{54} +(1.73205 - 2.00000i) q^{56} -12.0000 q^{57} -9.00000 q^{58} +3.46410 q^{59} -3.00000i q^{60} +10.3923 q^{62} +(-6.00000 - 5.19615i) q^{63} -1.00000 q^{64} +3.00000i q^{65} -4.00000 q^{67} +3.46410 q^{68} -1.73205 q^{69} +(3.00000 - 3.46410i) q^{70} -12.0000i q^{71} +3.00000i q^{72} -6.92820i q^{73} -5.00000i q^{74} +3.46410i q^{75} +6.92820i q^{76} -3.00000i q^{78} -10.0000 q^{79} -1.73205 q^{80} +9.00000 q^{81} -12.1244i q^{82} -3.46410 q^{83} +(-3.00000 + 3.46410i) q^{84} +6.00000 q^{85} +1.00000i q^{86} +15.5885 q^{87} +3.46410 q^{89} +5.19615i q^{90} +(3.00000 - 3.46410i) q^{91} +1.00000i q^{92} -18.0000 q^{93} -8.66025i q^{94} +12.0000i q^{95} +1.73205 q^{96} +12.1244i q^{97} +(-6.92820 + 1.00000i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^9	\( 4 q - 4 q^{4} + 8 q^{7} - 12 q^{9} + 4 q^{16} + 12 q^{21} - 8 q^{25} - 8 q^{28} - 12 q^{30} + 12 q^{36} - 20 q^{37} - 12 q^{39} + 4 q^{43} + 4 q^{46} + 4 q^{49} - 48 q^{57} - 36 q^{58} - 24 q^{63} - 4 q^{64} - 16 q^{67} + 12 q^{70} - 40 q^{79} + 36 q^{81} - 12 q^{84} + 24 q^{85} + 12 q^{91} - 72 q^{93}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^7 - 12 * q^9 + 4 * q^16 + 12 * q^21 - 8 * q^25 - 8 * q^28 - 12 * q^30 + 12 * q^36 - 20 * q^37 - 12 * q^39 + 4 * q^43 + 4 * q^46 + 4 * q^49 - 48 * q^57 - 36 * q^58 - 24 * q^63 - 4 * q^64 - 16 * q^67 + 12 * q^70 - 40 * q^79 + 36 * q^81 - 12 * q^84 + 24 * q^85 + 12 * q^91 - 72 * q^93

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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