LMFDB - Embedded newform 378.2.g.e.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(109,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

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

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

chi := DirichletCharacter("378.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.g (of order $3$, 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:	$3.01834519640$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	378.109
Dual form			378.2.g.e.163.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +O(q^{10})$	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-3.00000 + 5.19615i) q^{11} +5.00000 q^{13} +(-2.50000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.00000 + 5.19615i) q^{17} +(2.00000 + 3.46410i) q^{19} -6.00000 q^{22} +(-3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} +(2.50000 + 4.33013i) q^{26} +(-0.500000 - 2.59808i) q^{28} +6.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(0.500000 - 0.866025i) q^{32} -6.00000 q^{34} +(0.500000 + 0.866025i) q^{37} +(-2.00000 + 3.46410i) q^{38} -6.00000 q^{41} -1.00000 q^{43} +(-3.00000 - 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{46} +(3.00000 + 5.19615i) q^{47} +(1.00000 - 6.92820i) q^{49} +5.00000 q^{50} +(-2.50000 + 4.33013i) q^{52} +(3.00000 - 5.19615i) q^{53} +(2.00000 - 1.73205i) q^{56} +(3.00000 + 5.19615i) q^{58} +(3.00000 - 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +1.00000 q^{62} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{67} +(-3.00000 - 5.19615i) q^{68} +12.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-0.500000 + 0.866025i) q^{74} -4.00000 q^{76} +(-3.00000 - 15.5885i) q^{77} +(0.500000 + 0.866025i) q^{79} +(-3.00000 - 5.19615i) q^{82} +6.00000 q^{83} +(-0.500000 - 0.866025i) q^{86} +(3.00000 - 5.19615i) q^{88} +(-10.0000 + 8.66025i) q^{91} +6.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +17.0000 q^{97} +(6.50000 - 2.59808i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8	$ 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8} - 6 q^{11} + 10 q^{13} - 5 q^{14} - q^{16} - 6 q^{17} + 4 q^{19} - 12 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - q^{28} + 12 q^{29} + q^{31} + q^{32} - 12 q^{34} + q^{37} - 4 q^{38} - 12 q^{41} - 2 q^{43} - 6 q^{44} + 6 q^{46} + 6 q^{47} + 2 q^{49} + 10 q^{50} - 5 q^{52} + 6 q^{53} + 4 q^{56} + 6 q^{58} + 6 q^{59} + q^{61} + 2 q^{62} + 2 q^{64} + q^{67} - 6 q^{68} + 24 q^{71} - 2 q^{73} - q^{74} - 8 q^{76} - 6 q^{77} + q^{79} - 6 q^{82} + 12 q^{83} - q^{86} + 6 q^{88} - 20 q^{91} + 12 q^{92} - 6 q^{94} + 34 q^{97} + 13 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8 - 6 * q^11 + 10 * q^13 - 5 * q^14 - q^16 - 6 * q^17 + 4 * q^19 - 12 * q^22 - 6 * q^23 + 5 * q^25 + 5 * q^26 - q^28 + 12 * q^29 + q^31 + q^32 - 12 * q^34 + q^37 - 4 * q^38 - 12 * q^41 - 2 * q^43 - 6 * q^44 + 6 * q^46 + 6 * q^47 + 2 * q^49 + 10 * q^50 - 5 * q^52 + 6 * q^53 + 4 * q^56 + 6 * q^58 + 6 * q^59 + q^61 + 2 * q^62 + 2 * q^64 + q^67 - 6 * q^68 + 24 * q^71 - 2 * q^73 - q^74 - 8 * q^76 - 6 * q^77 + q^79 - 6 * q^82 + 12 * q^83 - q^86 + 6 * q^88 - 20 * q^91 + 12 * q^92 - 6 * q^94 + 34 * q^97 + 13 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$2$	0.500000	+	0.866025i	0.353553	+	0.612372i
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.250000	+	0.433013i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$7$	−2.00000	+	1.73205i	−0.755929	+	0.654654i
$8$	−1.00000			−0.353553
$9$		0			0
$10$		0			0
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.0829925	+	0.996550i	$0.526448\pi$
$11$	−3.00000	+	5.19615i	−0.904534	+	1.56670i	−0.821541	+	0.570149i	$0.806886\pi$
$12$		0			0
$13$	5.00000			1.38675			0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000			1.38675			0.693375	+	0.720577i	$0.256123\pi$
$14$	−2.50000	−	0.866025i	−0.668153	−	0.231455i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	0.230285	+	0.973123i	$0.426034\pi$
$17$	−3.00000	+	5.19615i	−0.727607	+	1.26025i	−0.957892	+	0.287129i	$0.907299\pi$
$18$		0			0
$19$	2.00000	+	3.46410i	0.458831	+	0.794719i	0.998899	−	0.0469020i	$-0.0149348\pi$
$19$	2.00000	+	3.46410i	0.458831	+	0.794719i	−0.540068	+	0.841621i	$0.681602\pi$
$20$		0			0
$21$		0			0
$22$	−6.00000			−1.27920
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	−0.988436	−	0.151642i	$-0.951544\pi$
$23$	−3.00000	−	5.19615i	−0.625543	−	1.08347i	0.362892	−	0.931831i	$-0.381789\pi$
$24$		0			0
$25$	2.50000	−	4.33013i	0.500000	−	0.866025i
$26$	2.50000	+	4.33013i	0.490290	+	0.849208i
$27$		0			0
$28$	−0.500000	−	2.59808i	−0.0944911	−	0.490990i
$29$	6.00000			1.11417			0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000			1.11417			0.557086	+	0.830455i	$0.311919\pi$
$30$		0			0
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	−0.817625	−	0.575751i	$-0.804710\pi$
$31$	0.500000	−	0.866025i	0.0898027	−	0.155543i	0.907428	+	0.420208i	$0.138043\pi$
$32$	0.500000	−	0.866025i	0.0883883	−	0.153093i
$33$		0			0
$34$	−6.00000			−1.02899
$35$		0			0
$36$		0			0
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	0.904194	−	0.427121i	$-0.140472\pi$
$37$	0.500000	+	0.866025i	0.0821995	+	0.142374i	−0.821995	+	0.569495i	$0.807139\pi$
$38$	−2.00000	+	3.46410i	−0.324443	+	0.561951i
$39$		0			0
$40$		0			0
$41$	−6.00000			−0.937043			−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000			−0.937043			−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−	5.19615i	−0.452267	−	0.783349i
$45$		0			0
$46$	3.00000	−	5.19615i	0.442326	−	0.766131i
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	0.997503	−	0.0706177i	$-0.0224970\pi$
$47$	3.00000	+	5.19615i	0.437595	+	0.757937i	−0.559908	+	0.828554i	$0.689164\pi$
$48$		0			0
$49$	1.00000	−	6.92820i	0.142857	−	0.989743i
$50$	5.00000			0.707107
$51$		0			0
$52$	−2.50000	+	4.33013i	−0.346688	+	0.600481i
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	−0.583036	−	0.812447i	$-0.698135\pi$
$53$	3.00000	−	5.19615i	0.412082	−	0.713746i	0.995117	+	0.0987002i	$0.0314685\pi$
$54$		0			0
$55$		0			0
$56$	2.00000	−	1.73205i	0.267261	−	0.231455i
$57$		0			0
$58$	3.00000	+	5.19615i	0.393919	+	0.682288i
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	−0.601958	−	0.798528i	$-0.705612\pi$
$59$	3.00000	−	5.19615i	0.390567	−	0.676481i	0.992524	+	0.122047i	$0.0389457\pi$
$60$		0			0
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	0.896258	−	0.443533i	$-0.146275\pi$
$61$	0.500000	+	0.866025i	0.0640184	+	0.110883i	−0.832240	+	0.554416i	$0.812942\pi$
$62$	1.00000			0.127000
$63$		0			0
$64$	1.00000			0.125000
$65$		0			0
$66$		0			0
$67$	0.500000	−	0.866025i	0.0610847	−	0.105802i	−0.833866	−	0.551967i	$-0.813877\pi$
$67$	0.500000	−	0.866025i	0.0610847	−	0.105802i	0.894951	+	0.446165i	$0.147211\pi$
$68$	−3.00000	−	5.19615i	−0.363803	−	0.630126i
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	−0.918594	−	0.395203i	$-0.870674\pi$
$73$	−1.00000	+	1.73205i	−0.117041	+	0.202721i	0.801553	+	0.597924i	$0.204008\pi$
$74$	−0.500000	+	0.866025i	−0.0581238	+	0.100673i
$75$		0			0
$76$	−4.00000			−0.458831
$77$	−3.00000	−	15.5885i	−0.341882	−	1.77647i
$78$		0			0
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	0.892781	−	0.450490i	$-0.148751\pi$
$79$	0.500000	+	0.866025i	0.0562544	+	0.0974355i	−0.836527	+	0.547926i	$0.815418\pi$
$80$		0			0
$81$		0			0
$82$	−3.00000	−	5.19615i	−0.331295	−	0.573819i
$83$	6.00000			0.658586			0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000			0.658586			0.329293	+	0.944228i	$0.393190\pi$
$84$		0			0
$85$		0			0
$86$	−0.500000	−	0.866025i	−0.0539164	−	0.0933859i
$87$		0			0
$88$	3.00000	−	5.19615i	0.319801	−	0.553912i
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−10.0000	+	8.66025i	−1.04828	+	0.907841i
$92$	6.00000			0.625543
$93$		0			0
$94$	−3.00000	+	5.19615i	−0.309426	+	0.535942i
$95$		0			0
$96$		0			0
$97$	17.0000			1.72609			0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000			1.72609			0.863044	+	0.505128i	$0.168555\pi$
$98$	6.50000	−	2.59808i	0.656599	−	0.262445i
$99$		0			0
$100$	2.50000	+	4.33013i	0.250000	+	0.433013i
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	0.396236	+	0.918149i	$0.370316\pi$
$101$	−6.00000	+	10.3923i	−0.597022	+	1.03407i	−0.993258	+	0.115924i	$0.963017\pi$
$102$		0			0
$103$	9.50000	+	16.4545i	0.936063	+	1.62131i	0.772728	+	0.634738i	$0.218892\pi$
$103$	9.50000	+	16.4545i	0.936063	+	1.62131i	0.163335	+	0.986571i	$0.447775\pi$
$104$	−5.00000			−0.490290
$105$		0			0
$106$	6.00000			0.582772
$107$	−9.00000	−	15.5885i	−0.870063	−	1.50699i	−0.861931	−	0.507026i	$-0.830745\pi$
$107$	−9.00000	−	15.5885i	−0.870063	−	1.50699i	−0.00813215	−	0.999967i	$-0.502589\pi$
$108$		0			0
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	−0.960558	−	0.278078i	$-0.910303\pi$
$109$	−2.50000	+	4.33013i	−0.239457	+	0.414751i	0.721102	+	0.692829i	$0.243636\pi$
$110$		0			0
$111$		0			0
$112$	2.50000	+	0.866025i	0.236228	+	0.0818317i
$113$	12.0000			1.12887			0.564433	−	0.825479i	$-0.309095\pi$
$113$	12.0000			1.12887			0.564433	+	0.825479i	$0.309095\pi$
$114$		0			0
$115$		0			0
$116$	−3.00000	+	5.19615i	−0.278543	+	0.482451i
$117$		0			0
$118$	6.00000			0.552345
$119$	−3.00000	−	15.5885i	−0.275010	−	1.42899i
$120$		0			0
$121$	−12.5000	−	21.6506i	−1.13636	−	1.96824i
$122$	−0.500000	+	0.866025i	−0.0452679	+	0.0784063i
$123$		0			0
$124$	0.500000	+	0.866025i	0.0449013	+	0.0777714i
$125$		0			0
$126$		0			0
$127$	−13.0000			−1.15356			−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000			−1.15356			−0.576782	+	0.816898i	$0.695692\pi$
$128$	0.500000	+	0.866025i	0.0441942	+	0.0765466i
$129$		0			0
$130$		0			0
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$	−10.0000	−	3.46410i	−0.867110	−	0.300376i
$134$	1.00000			0.0863868
$135$		0			0
$136$	3.00000	−	5.19615i	0.257248	−	0.445566i
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	−7.00000			−0.593732			−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000			−0.593732			−0.296866	+	0.954919i	$0.595942\pi$
$140$		0			0
$141$		0			0
$142$	6.00000	+	10.3923i	0.503509	+	0.872103i
$143$	−15.0000	+	25.9808i	−1.25436	+	2.17262i
$144$		0			0
$145$		0			0
$146$	−2.00000			−0.165521
$147$		0			0
$148$	−1.00000			−0.0821995
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	0.999953	−	0.00974235i	$-0.00310113\pi$
$149$	6.00000	+	10.3923i	0.491539	+	0.851371i	−0.508413	+	0.861113i	$0.669768\pi$
$150$		0			0
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	0.279554	+	0.960130i	$0.409814\pi$
$151$	−8.50000	+	14.7224i	−0.691720	+	1.19809i	−0.971274	+	0.237964i	$0.923520\pi$
$152$	−2.00000	−	3.46410i	−0.162221	−	0.280976i
$153$		0			0
$154$	12.0000	−	10.3923i	0.966988	−	0.837436i
$155$		0			0
$156$		0			0
$157$	11.0000	−	19.0526i	0.877896	−	1.52056i	0.0242497	−	0.999706i	$-0.492280\pi$
$157$	11.0000	−	19.0526i	0.877896	−	1.52056i	0.853646	−	0.520854i	$-0.174386\pi$
$158$	−0.500000	+	0.866025i	−0.0397779	+	0.0688973i
$159$		0			0
$160$		0			0
$161$	15.0000	+	5.19615i	1.18217	+	0.409514i
$162$		0			0
$163$	−5.50000	−	9.52628i	−0.430793	−	0.746156i	0.566149	−	0.824303i	$-0.308433\pi$
$163$	−5.50000	−	9.52628i	−0.430793	−	0.746156i	−0.996942	+	0.0781474i	$0.975100\pi$
$164$	3.00000	−	5.19615i	0.234261	−	0.405751i
$165$		0			0
$166$	3.00000	+	5.19615i	0.232845	+	0.403300i
$167$	−18.0000			−1.39288			−0.696441	−	0.717614i	$-0.745234\pi$
$167$	−18.0000			−1.39288			−0.696441	+	0.717614i	$0.745234\pi$
$168$		0			0
$169$	12.0000			0.923077
$170$		0			0
$171$		0			0
$172$	0.500000	−	0.866025i	0.0381246	−	0.0660338i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	2.50000	+	12.9904i	0.188982	+	0.981981i
$176$	6.00000			0.452267
$177$		0			0
$178$		0			0
$179$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$179$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$	−12.5000	−	4.33013i	−0.926562	−	0.320970i
$183$		0			0
$184$	3.00000	+	5.19615i	0.221163	+	0.383065i
$185$		0			0
$186$		0			0
$187$	−18.0000	−	31.1769i	−1.31629	−	2.27988i
$188$	−6.00000			−0.437595
$189$		0			0
$190$		0			0
$191$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$191$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$192$		0			0
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	−0.847469	−	0.530845i	$-0.821875\pi$
$193$	0.500000	−	0.866025i	0.0359908	−	0.0623379i	0.883460	+	0.468507i	$0.155208\pi$
$194$	8.50000	+	14.7224i	0.610264	+	1.05701i
$195$		0			0
$196$	5.50000	+	4.33013i	0.392857	+	0.309295i
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	−5.50000	+	9.52628i	−0.389885	+	0.675300i	−0.992434	−	0.122782i	$-0.960818\pi$
$199$	−5.50000	+	9.52628i	−0.389885	+	0.675300i	0.602549	+	0.798082i	$0.294152\pi$
$200$	−2.50000	+	4.33013i	−0.176777	+	0.306186i
$201$		0			0
$202$	−12.0000			−0.844317
$203$	−12.0000	+	10.3923i	−0.842235	+	0.729397i
$204$		0			0
$205$		0			0
$206$	−9.50000	+	16.4545i	−0.661896	+	1.14644i
$207$		0			0
$208$	−2.50000	−	4.33013i	−0.173344	−	0.300240i
$209$	−24.0000			−1.66011
$210$		0			0
$211$	−13.0000			−0.894957			−0.447478	−	0.894295i	$-0.647678\pi$
$211$	−13.0000			−0.894957			−0.447478	+	0.894295i	$0.647678\pi$
$212$	3.00000	+	5.19615i	0.206041	+	0.356873i
$213$		0			0
$214$	9.00000	−	15.5885i	0.615227	−	1.06561i
$215$		0			0
$216$		0			0
$217$	0.500000	+	2.59808i	0.0339422	+	0.176369i
$218$	−5.00000			−0.338643
$219$		0			0
$220$		0			0
$221$	−15.0000	+	25.9808i	−1.00901	+	1.74766i
$222$		0			0
$223$	8.00000			0.535720			0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000			0.535720			0.267860	+	0.963458i	$0.413684\pi$
$224$	0.500000	+	2.59808i	0.0334077	+	0.173591i
$225$		0			0
$226$	6.00000	+	10.3923i	0.399114	+	0.691286i
$227$	3.00000	−	5.19615i	0.199117	−	0.344881i	−0.749125	−	0.662428i	$-0.769526\pi$
$227$	3.00000	−	5.19615i	0.199117	−	0.344881i	0.948242	+	0.317547i	$0.102859\pi$
$228$		0			0
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	0.771523	−	0.636201i	$-0.219495\pi$
$229$	−2.50000	−	4.33013i	−0.165205	−	0.286143i	−0.936728	+	0.350058i	$0.886162\pi$
$230$		0			0
$231$		0			0
$232$	−6.00000			−0.393919
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	−0.928312	−	0.371802i	$-0.878740\pi$
$233$	−12.0000	−	20.7846i	−0.786146	−	1.36165i	0.142166	−	0.989843i	$-0.454593\pi$
$234$		0			0
$235$		0			0
$236$	3.00000	+	5.19615i	0.195283	+	0.338241i
$237$		0			0
$238$	12.0000	−	10.3923i	0.777844	−	0.673633i
$239$	24.0000			1.55243			0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000			1.55243			0.776215	+	0.630468i	$0.217137\pi$
$240$		0			0
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$	12.5000	−	21.6506i	0.803530	−	1.39176i
$243$		0			0
$244$	−1.00000			−0.0640184
$245$		0			0
$246$		0			0
$247$	10.0000	+	17.3205i	0.636285	+	1.10208i
$248$	−0.500000	+	0.866025i	−0.0317500	+	0.0549927i
$249$		0			0
$250$		0			0
$251$	18.0000			1.13615			0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000			1.13615			0.568075	+	0.822977i	$0.307688\pi$
$252$		0			0
$253$	36.0000			2.26330
$254$	−6.50000	−	11.2583i	−0.407846	−	0.706410i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	0.944294	−	0.329104i	$-0.106747\pi$
$257$	3.00000	+	5.19615i	0.187135	+	0.324127i	−0.757159	+	0.653231i	$0.773413\pi$
$258$		0			0
$259$	−2.50000	−	0.866025i	−0.155342	−	0.0538122i
$260$		0			0
$261$		0			0
$262$	−6.00000	+	10.3923i	−0.370681	+	0.642039i
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	−0.943572	−	0.331166i	$-0.892558\pi$
$263$	−3.00000	+	5.19615i	−0.184988	+	0.320408i	0.758585	+	0.651575i	$0.225891\pi$
$264$		0			0
$265$		0			0
$266$	−2.00000	−	10.3923i	−0.122628	−	0.637193i
$267$		0			0
$268$	0.500000	+	0.866025i	0.0305424	+	0.0529009i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	3.50000	+	6.06218i	0.212610	+	0.368251i	0.952531	−	0.304443i	$-0.0984703\pi$
$271$	3.50000	+	6.06218i	0.212610	+	0.368251i	−0.739921	+	0.672694i	$0.765137\pi$
$272$	6.00000			0.363803
$273$		0			0
$274$	6.00000			0.362473
$275$	15.0000	+	25.9808i	0.904534	+	1.56670i
$276$		0			0
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	0.489207	+	0.872167i	$0.337286\pi$
$277$	−8.50000	+	14.7224i	−0.510716	+	0.884585i	−0.999923	+	0.0124177i	$0.996047\pi$
$278$	−3.50000	−	6.06218i	−0.209916	−	0.363585i
$279$		0			0
$280$		0			0
$281$	12.0000			0.715860			0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000			0.715860			0.357930	+	0.933748i	$0.383483\pi$
$282$		0			0
$283$	6.50000	−	11.2583i	0.386385	−	0.669238i	−0.605575	−	0.795788i	$-0.707057\pi$
$283$	6.50000	−	11.2583i	0.386385	−	0.669238i	0.991960	+	0.126550i	$0.0403903\pi$
$284$	−6.00000	+	10.3923i	−0.356034	+	0.616670i
$285$		0			0
$286$	−30.0000			−1.77394
$287$	12.0000	−	10.3923i	0.708338	−	0.613438i
$288$		0			0
$289$	−9.50000	−	16.4545i	−0.558824	−	0.967911i
$290$		0			0
$291$		0			0
$292$	−1.00000	−	1.73205i	−0.0585206	−	0.101361i
$293$	18.0000			1.05157			0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000			1.05157			0.525786	+	0.850617i	$0.323771\pi$
$294$		0			0
$295$		0			0
$296$	−0.500000	−	0.866025i	−0.0290619	−	0.0503367i
$297$		0			0
$298$	−6.00000	+	10.3923i	−0.347571	+	0.602010i
$299$	−15.0000	−	25.9808i	−0.867472	−	1.50251i
$300$		0			0
$301$	2.00000	−	1.73205i	0.115278	−	0.0998337i
$302$	−17.0000			−0.978240
$303$		0			0
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$		0			0
$306$		0			0
$307$	17.0000			0.970241			0.485121	−	0.874447i	$-0.338776\pi$
$307$	17.0000			0.970241			0.485121	+	0.874447i	$0.338776\pi$
$308$	15.0000	+	5.19615i	0.854704	+	0.296078i
$309$		0			0
$310$		0			0
$311$	6.00000	−	10.3923i	0.340229	−	0.589294i	−0.644246	−	0.764818i	$-0.722829\pi$
$311$	6.00000	−	10.3923i	0.340229	−	0.589294i	0.984475	+	0.175525i	$0.0561621\pi$
$312$		0			0
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	0.989158	+	0.146852i	$0.0469141\pi$
$313$	11.0000	+	19.0526i	0.621757	+	1.07691i	−0.367402	+	0.930062i	$0.619753\pi$
$314$	22.0000			1.24153
$315$		0			0
$316$	−1.00000			−0.0562544
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	−0.999980	−	0.00635137i	$-0.997978\pi$
$317$	−9.00000	−	15.5885i	−0.505490	−	0.875535i	0.494489	−	0.869184i	$-0.335355\pi$
$318$		0			0
$319$	−18.0000	+	31.1769i	−1.00781	+	1.74557i
$320$		0			0
$321$		0			0
$322$	3.00000	+	15.5885i	0.167183	+	0.868711i
$323$	−24.0000			−1.33540
$324$		0			0
$325$	12.5000	−	21.6506i	0.693375	−	1.20096i
$326$	5.50000	−	9.52628i	0.304617	−	0.527612i
$327$		0			0
$328$	6.00000			0.331295
$329$	−15.0000	−	5.19615i	−0.826977	−	0.286473i
$330$		0			0
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	−0.998298	−	0.0583130i	$-0.981428\pi$
$331$	−10.0000	−	17.3205i	−0.549650	−	0.952021i	0.448649	−	0.893708i	$-0.351905\pi$
$332$	−3.00000	+	5.19615i	−0.164646	+	0.285176i
$333$		0			0
$334$	−9.00000	−	15.5885i	−0.492458	−	0.852962i
$335$		0			0
$336$		0			0
$337$	2.00000			0.108947			0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000			0.108947			0.0544735	+	0.998515i	$0.482652\pi$
$338$	6.00000	+	10.3923i	0.326357	+	0.565267i
$339$		0			0
$340$		0			0
$341$	3.00000	+	5.19615i	0.162459	+	0.281387i
$342$		0			0
$343$	10.0000	+	15.5885i	0.539949	+	0.841698i
$344$	1.00000			0.0539164
$345$		0			0
$346$		0			0
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	−0.340293	−	0.940319i	$-0.610526\pi$
$347$	12.0000	−	20.7846i	0.644194	−	1.11578i	0.984487	−	0.175457i	$-0.0561403\pi$
$348$		0			0
$349$	−1.00000			−0.0535288			−0.0267644	−	0.999642i	$-0.508520\pi$
$349$	−1.00000			−0.0535288			−0.0267644	+	0.999642i	$0.508520\pi$
$350$	−10.0000	+	8.66025i	−0.534522	+	0.462910i
$351$		0			0
$352$	3.00000	+	5.19615i	0.159901	+	0.276956i
$353$	18.0000	−	31.1769i	0.958043	−	1.65938i	0.230799	−	0.973002i	$-0.425866\pi$
$353$	18.0000	−	31.1769i	0.958043	−	1.65938i	0.727245	−	0.686378i	$-0.240800\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	0.775934	−	0.630814i	$-0.217279\pi$
$359$	−3.00000	−	5.19615i	−0.158334	−	0.274242i	−0.934268	+	0.356572i	$0.883946\pi$
$360$		0			0
$361$	1.50000	−	2.59808i	0.0789474	−	0.136741i
$362$	−5.00000	−	8.66025i	−0.262794	−	0.455173i
$363$		0			0
$364$	−2.50000	−	12.9904i	−0.131036	−	0.680881i
$365$		0			0
$366$		0			0
$367$	14.0000	−	24.2487i	0.730794	−	1.26577i	−0.225750	−	0.974185i	$-0.572483\pi$
$367$	14.0000	−	24.2487i	0.730794	−	1.26577i	0.956544	−	0.291587i	$-0.0941834\pi$
$368$	−3.00000	+	5.19615i	−0.156386	+	0.270868i
$369$		0			0
$370$		0			0
$371$	3.00000	+	15.5885i	0.155752	+	0.809312i
$372$		0			0
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	−0.977016	−	0.213165i	$-0.931623\pi$
$373$	−13.0000	−	22.5167i	−0.673114	−	1.16587i	0.303902	−	0.952703i	$-0.401711\pi$
$374$	18.0000	−	31.1769i	0.930758	−	1.61212i
$375$		0			0
$376$	−3.00000	−	5.19615i	−0.154713	−	0.267971i
$377$	30.0000			1.54508
$378$		0			0
$379$	−1.00000			−0.0513665			−0.0256833	−	0.999670i	$-0.508176\pi$
$379$	−1.00000			−0.0513665			−0.0256833	+	0.999670i	$0.508176\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	0.779143	−	0.626846i	$-0.215654\pi$
$383$	−3.00000	−	5.19615i	−0.153293	−	0.265511i	−0.932436	+	0.361335i	$0.882321\pi$
$384$		0			0
$385$		0			0
$386$	1.00000			0.0508987
$387$		0			0
$388$	−8.50000	+	14.7224i	−0.431522	+	0.747418i
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	−0.672874	−	0.739758i	$-0.734940\pi$
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	0.977086	+	0.212847i	$0.0682735\pi$
$390$		0			0
$391$	36.0000			1.82060
$392$	−1.00000	+	6.92820i	−0.0505076	+	0.349927i
$393$		0			0
$394$	−6.00000	−	10.3923i	−0.302276	−	0.523557i
$395$		0			0
$396$		0			0
$397$	6.50000	+	11.2583i	0.326226	+	0.565039i	0.981760	−	0.190126i	$-0.0608897\pi$
$397$	6.50000	+	11.2583i	0.326226	+	0.565039i	−0.655534	+	0.755166i	$0.727556\pi$
$398$	−11.0000			−0.551380
$399$		0			0
$400$	−5.00000			−0.250000
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	0.931158	−	0.364615i	$-0.118800\pi$
$401$	3.00000	+	5.19615i	0.149813	+	0.259483i	−0.781345	+	0.624099i	$0.785466\pi$
$402$		0			0
$403$	2.50000	−	4.33013i	0.124534	−	0.215699i
$404$	−6.00000	−	10.3923i	−0.298511	−	0.517036i
$405$		0			0
$406$	−15.0000	−	5.19615i	−0.744438	−	0.257881i
$407$	−6.00000			−0.297409
$408$		0			0
$409$	−14.5000	+	25.1147i	−0.716979	+	1.24184i	0.245212	+	0.969469i	$0.421142\pi$
$409$	−14.5000	+	25.1147i	−0.716979	+	1.24184i	−0.962191	+	0.272374i	$0.912191\pi$
$410$		0			0
$411$		0			0
$412$	−19.0000			−0.936063
$413$	3.00000	+	15.5885i	0.147620	+	0.767058i
$414$		0			0
$415$		0			0
$416$	2.50000	−	4.33013i	0.122573	−	0.212302i
$417$		0			0
$418$	−12.0000	−	20.7846i	−0.586939	−	1.01661i
$419$	−24.0000			−1.17248			−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000			−1.17248			−0.586238	+	0.810139i	$0.699392\pi$
$420$		0			0
$421$	14.0000			0.682318			0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000			0.682318			0.341159	+	0.940006i	$0.389181\pi$
$422$	−6.50000	−	11.2583i	−0.316415	−	0.548047i
$423$		0			0
$424$	−3.00000	+	5.19615i	−0.145693	+	0.252347i
$425$	15.0000	+	25.9808i	0.727607	+	1.26025i
$426$		0			0
$427$	−2.50000	−	0.866025i	−0.120983	−	0.0419099i
$428$	18.0000			0.870063
$429$		0			0
$430$		0			0
$431$	9.00000	−	15.5885i	0.433515	−	0.750870i	−0.563658	−	0.826008i	$-0.690607\pi$
$431$	9.00000	−	15.5885i	0.433515	−	0.750870i	0.997173	+	0.0751385i	$0.0239399\pi$
$432$		0			0
$433$	−25.0000			−1.20142			−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000			−1.20142			−0.600712	+	0.799466i	$0.705116\pi$
$434$	−2.00000	+	1.73205i	−0.0960031	+	0.0831411i
$435$		0			0
$436$	−2.50000	−	4.33013i	−0.119728	−	0.207375i
$437$	12.0000	−	20.7846i	0.574038	−	0.994263i
$438$		0			0
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	0.754642	−	0.656136i	$-0.227810\pi$
$439$	−4.00000	−	6.92820i	−0.190910	−	0.330665i	−0.945552	+	0.325471i	$0.894477\pi$
$440$		0			0
$441$		0			0
$442$	−30.0000			−1.42695
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	0.996551	+	0.0829786i	$0.0264433\pi$
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	−0.426414	+	0.904528i	$0.640223\pi$
$444$		0			0
$445$		0			0
$446$	4.00000	+	6.92820i	0.189405	+	0.328060i
$447$		0			0
$448$	−2.00000	+	1.73205i	−0.0944911	+	0.0818317i
$449$	−24.0000			−1.13263			−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000			−1.13263			−0.566315	+	0.824189i	$0.691631\pi$
$450$		0			0
$451$	18.0000	−	31.1769i	0.847587	−	1.46806i
$452$	−6.00000	+	10.3923i	−0.282216	+	0.488813i
$453$		0			0
$454$	6.00000			0.281594
$455$		0			0
$456$		0			0
$457$	9.50000	+	16.4545i	0.444391	+	0.769708i	0.998010	−	0.0630623i	$-0.0200867\pi$
$457$	9.50000	+	16.4545i	0.444391	+	0.769708i	−0.553618	+	0.832771i	$0.686753\pi$
$458$	2.50000	−	4.33013i	0.116817	−	0.202334i
$459$		0			0
$460$		0			0
$461$	18.0000			0.838344			0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000			0.838344			0.419172	+	0.907907i	$0.362320\pi$
$462$		0			0
$463$	8.00000			0.371792			0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000			0.371792			0.185896	+	0.982569i	$0.440481\pi$
$464$	−3.00000	−	5.19615i	−0.139272	−	0.241225i
$465$		0			0
$466$	12.0000	−	20.7846i	0.555889	−	0.962828i
$467$	−6.00000	−	10.3923i	−0.277647	−	0.480899i	0.693153	−	0.720791i	$-0.256221\pi$
$467$	−6.00000	−	10.3923i	−0.277647	−	0.480899i	−0.970799	+	0.239892i	$0.922888\pi$
$468$		0			0
$469$	0.500000	+	2.59808i	0.0230879	+	0.119968i
$470$		0			0
$471$		0			0
$472$	−3.00000	+	5.19615i	−0.138086	+	0.239172i
$473$	3.00000	−	5.19615i	0.137940	−	0.238919i
$474$		0			0
$475$	20.0000			0.917663
$476$	15.0000	+	5.19615i	0.687524	+	0.238165i
$477$		0			0
$478$	12.0000	+	20.7846i	0.548867	+	0.950666i
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	−0.969920	−	0.243426i	$-0.921729\pi$
$479$	−6.00000	+	10.3923i	−0.274147	+	0.474837i	0.695773	+	0.718262i	$0.255062\pi$
$480$		0			0
$481$	2.50000	+	4.33013i	0.113990	+	0.197437i
$482$	−17.0000			−0.774329
$483$		0			0
$484$	25.0000			1.13636
$485$		0			0
$486$		0			0
$487$	20.0000	−	34.6410i	0.906287	−	1.56973i	0.0871056	−	0.996199i	$-0.472238\pi$
$487$	20.0000	−	34.6410i	0.906287	−	1.56973i	0.819181	−	0.573535i	$-0.194428\pi$
$488$	−0.500000	−	0.866025i	−0.0226339	−	0.0392031i
$489$		0			0
$490$		0			0
$491$	−30.0000			−1.35388			−0.676941	−	0.736038i	$-0.736695\pi$
$491$	−30.0000			−1.35388			−0.676941	+	0.736038i	$0.736695\pi$
$492$		0			0
$493$	−18.0000	+	31.1769i	−0.810679	+	1.40414i
$494$	−10.0000	+	17.3205i	−0.449921	+	0.779287i
$495$		0			0
$496$	−1.00000			−0.0449013
$497$	−24.0000	+	20.7846i	−1.07655	+	0.932317i
$498$		0			0
$499$	12.5000	+	21.6506i	0.559577	+	0.969216i	0.997532	+	0.0702185i	$0.0223697\pi$
$499$	12.5000	+	21.6506i	0.559577	+	0.969216i	−0.437955	+	0.898997i	$0.644297\pi$
$500$		0			0
$501$		0			0
$502$	9.00000	+	15.5885i	0.401690	+	0.695747i
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$		0			0
$506$	18.0000	+	31.1769i	0.800198	+	1.38598i
$507$		0			0
$508$	6.50000	−	11.2583i	0.288391	−	0.499508i
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	−0.979322	−	0.202306i	$-0.935156\pi$
$509$	−15.0000	−	25.9808i	−0.664863	−	1.15158i	0.314459	−	0.949271i	$-0.398177\pi$
$510$		0			0
$511$	−1.00000	−	5.19615i	−0.0442374	−	0.229864i
$512$	−1.00000			−0.0441942
$513$		0			0
$514$	−3.00000	+	5.19615i	−0.132324	+	0.229192i
$515$		0			0
$516$		0			0
$517$	−36.0000			−1.58328
$518$	−0.500000	−	2.59808i	−0.0219687	−	0.114153i
$519$		0			0
$520$		0			0
$521$	−21.0000	+	36.3731i	−0.920027	+	1.59353i	−0.120656	+	0.992694i	$0.538500\pi$
$521$	−21.0000	+	36.3731i	−0.920027	+	1.59353i	−0.799370	+	0.600839i	$0.794833\pi$
$522$		0			0
$523$	−11.5000	−	19.9186i	−0.502860	−	0.870979i	−0.999995	−	0.00330547i	$-0.998948\pi$
$523$	−11.5000	−	19.9186i	−0.502860	−	0.870979i	0.497135	−	0.867673i	$-0.334385\pi$
$524$	−12.0000			−0.524222
$525$		0			0
$526$	−6.00000			−0.261612
$527$	3.00000	+	5.19615i	0.130682	+	0.226348i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$		0			0
$531$		0			0
$532$	8.00000	−	6.92820i	0.346844	−	0.300376i
$533$	−30.0000			−1.29944
$534$		0			0
$535$		0			0
$536$	−0.500000	+	0.866025i	−0.0215967	+	0.0374066i
$537$		0			0
$538$		0			0
$539$	33.0000	+	25.9808i	1.42141	+	1.11907i
$540$		0			0
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	0.953262	−	0.302144i	$-0.0977023\pi$
$541$	5.00000	+	8.66025i	0.214967	+	0.372333i	−0.738296	+	0.674477i	$0.764369\pi$
$542$	−3.50000	+	6.06218i	−0.150338	+	0.260393i
$543$		0			0
$544$	3.00000	+	5.19615i	0.128624	+	0.222783i
$545$		0			0
$546$		0			0
$547$	−37.0000			−1.58201			−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000			−1.58201			−0.791003	+	0.611812i	$0.790441\pi$
$548$	3.00000	+	5.19615i	0.128154	+	0.221969i
$549$		0			0
$550$	−15.0000	+	25.9808i	−0.639602	+	1.10782i
$551$	12.0000	+	20.7846i	0.511217	+	0.885454i
$552$		0			0
$553$	−2.50000	−	0.866025i	−0.106311	−	0.0368271i
$554$	−17.0000			−0.722261
$555$		0			0
$556$	3.50000	−	6.06218i	0.148433	−	0.257094i
$557$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$557$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$558$		0			0
$559$	−5.00000			−0.211477
$560$		0			0
$561$		0			0
$562$	6.00000	+	10.3923i	0.253095	+	0.438373i
$563$	21.0000	−	36.3731i	0.885044	−	1.53294i	0.0393818	−	0.999224i	$-0.487461\pi$
$563$	21.0000	−	36.3731i	0.885044	−	1.53294i	0.845663	−	0.533718i	$-0.179206\pi$
$564$		0			0
$565$		0			0
$566$	13.0000			0.546431
$567$		0			0
$568$	−12.0000			−0.503509
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	0.945573	+	0.325409i	$0.105502\pi$
$569$	18.0000	+	31.1769i	0.754599	+	1.30700i	−0.190974	+	0.981595i	$0.561165\pi$
$570$		0			0
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	8.00000	−	13.8564i	0.334790	−	0.579873i	0.983444	+	0.181210i	$0.0580014\pi$
$572$	−15.0000	−	25.9808i	−0.627182	−	1.08631i
$573$		0			0
$574$	15.0000	+	5.19615i	0.626088	+	0.216883i
$575$	−30.0000			−1.25109
$576$		0			0
$577$	−11.5000	+	19.9186i	−0.478751	+	0.829222i	−0.999703	−	0.0243645i	$-0.992244\pi$
$577$	−11.5000	+	19.9186i	−0.478751	+	0.829222i	0.520952	+	0.853586i	$0.325577\pi$
$578$	9.50000	−	16.4545i	0.395148	−	0.684416i
$579$		0			0
$580$		0			0
$581$	−12.0000	+	10.3923i	−0.497844	+	0.431145i
$582$		0			0
$583$	18.0000	+	31.1769i	0.745484	+	1.29122i
$584$	1.00000	−	1.73205i	0.0413803	−	0.0716728i
$585$		0			0
$586$	9.00000	+	15.5885i	0.371787	+	0.643953i
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$		0			0
$589$	4.00000			0.164817
$590$		0			0
$591$		0			0
$592$	0.500000	−	0.866025i	0.0205499	−	0.0355934i
$593$	−18.0000	−	31.1769i	−0.739171	−	1.28028i	−0.952869	−	0.303383i	$-0.901884\pi$
$593$	−18.0000	−	31.1769i	−0.739171	−	1.28028i	0.213697	−	0.976900i	$-0.431449\pi$
$594$		0			0
$595$		0			0
$596$	−12.0000			−0.491539
$597$		0			0
$598$	15.0000	−	25.9808i	0.613396	−	1.06243i
$599$	−18.0000	+	31.1769i	−0.735460	+	1.27385i	0.219061	+	0.975711i	$0.429701\pi$
$599$	−18.0000	+	31.1769i	−0.735460	+	1.27385i	−0.954521	+	0.298143i	$0.903633\pi$
$600$		0			0
$601$	47.0000			1.91717			0.958585	−	0.284807i	$-0.0919294\pi$
$601$	47.0000			1.91717			0.958585	+	0.284807i	$0.0919294\pi$
$602$	2.50000	+	0.866025i	0.101892	+	0.0352966i
$603$		0			0
$604$	−8.50000	−	14.7224i	−0.345860	−	0.599047i
$605$		0			0
$606$		0			0
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	0.981454	−	0.191700i	$-0.0614000\pi$
$607$	8.00000	+	13.8564i	0.324710	+	0.562414i	−0.656744	+	0.754114i	$0.728067\pi$
$608$	4.00000			0.162221
$609$		0			0
$610$		0			0
$611$	15.0000	+	25.9808i	0.606835	+	1.05107i
$612$		0			0
$613$	−5.50000	+	9.52628i	−0.222143	+	0.384763i	−0.955458	−	0.295126i	$-0.904638\pi$
$613$	−5.50000	+	9.52628i	−0.222143	+	0.384763i	0.733316	+	0.679888i	$0.237972\pi$
$614$	8.50000	+	14.7224i	0.343032	+	0.594149i
$615$		0			0
$616$	3.00000	+	15.5885i	0.120873	+	0.628077i
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		0			0
$619$	12.5000	−	21.6506i	0.502417	−	0.870212i	−0.497579	−	0.867419i	$-0.665777\pi$
$619$	12.5000	−	21.6506i	0.502417	−	0.870212i	0.999996	−	0.00279365i	$-0.000889247\pi$
$620$		0			0
$621$		0			0
$622$	12.0000			0.481156
$623$		0			0
$624$		0			0
$625$	−12.5000	−	21.6506i	−0.500000	−	0.866025i
$626$	−11.0000	+	19.0526i	−0.439648	+	0.761493i
$627$		0			0
$628$	11.0000	+	19.0526i	0.438948	+	0.760280i
$629$	−6.00000			−0.239236
$630$		0			0
$631$	−7.00000			−0.278666			−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000			−0.278666			−0.139333	+	0.990246i	$0.544496\pi$
$632$	−0.500000	−	0.866025i	−0.0198889	−	0.0344486i
$633$		0			0
$634$	9.00000	−	15.5885i	0.357436	−	0.619097i
$635$		0			0
$636$		0			0
$637$	5.00000	−	34.6410i	0.198107	−	1.37253i
$638$	−36.0000			−1.42525
$639$		0			0
$640$		0			0
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	−0.987200	−	0.159489i	$-0.949015\pi$
$641$	−9.00000	+	15.5885i	−0.355479	+	0.615707i	0.631721	+	0.775196i	$0.282349\pi$
$642$		0			0
$643$	−25.0000			−0.985904			−0.492952	−	0.870057i	$-0.664082\pi$
$643$	−25.0000			−0.985904			−0.492952	+	0.870057i	$0.664082\pi$
$644$	−12.0000	+	10.3923i	−0.472866	+	0.409514i
$645$		0			0
$646$	−12.0000	−	20.7846i	−0.472134	−	0.817760i
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	−0.723645	−	0.690172i	$-0.757535\pi$
$647$	6.00000	−	10.3923i	0.235884	−	0.408564i	0.959529	+	0.281609i	$0.0908680\pi$
$648$		0			0
$649$	18.0000	+	31.1769i	0.706562	+	1.22380i
$650$	25.0000			0.980581
$651$		0			0
$652$	11.0000			0.430793
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	0.986634	−	0.162951i	$-0.0521013\pi$
$653$	9.00000	+	15.5885i	0.352197	+	0.610023i	−0.634437	+	0.772975i	$0.718768\pi$
$654$		0			0
$655$		0			0
$656$	3.00000	+	5.19615i	0.117130	+	0.202876i
$657$		0			0
$658$	−3.00000	−	15.5885i	−0.116952	−	0.607701i
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$	−25.0000	+	43.3013i	−0.972387	+	1.68422i	−0.284087	+	0.958799i	$0.591690\pi$
$661$	−25.0000	+	43.3013i	−0.972387	+	1.68422i	−0.688301	+	0.725426i	$0.741643\pi$
$662$	10.0000	−	17.3205i	0.388661	−	0.673181i
$663$		0			0
$664$	−6.00000			−0.232845
$665$		0			0
$666$		0			0
$667$	−18.0000	−	31.1769i	−0.696963	−	1.20717i
$668$	9.00000	−	15.5885i	0.348220	−	0.603136i
$669$		0			0
$670$		0			0
$671$	−6.00000			−0.231627
$672$		0			0
$673$	−34.0000			−1.31060			−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000			−1.31060			−0.655302	+	0.755367i	$0.727459\pi$
$674$	1.00000	+	1.73205i	0.0385186	+	0.0667161i
$675$		0			0
$676$	−6.00000	+	10.3923i	−0.230769	+	0.399704i
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	0.727386	−	0.686229i	$-0.240735\pi$
$677$	−6.00000	−	10.3923i	−0.230599	−	0.399409i	−0.957984	+	0.286820i	$0.907402\pi$
$678$		0			0
$679$	−34.0000	+	29.4449i	−1.30480	+	1.12999i
$680$		0			0
$681$		0			0
$682$	−3.00000	+	5.19615i	−0.114876	+	0.198971i
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	−0.957685	−	0.287819i	$-0.907070\pi$
$683$	−6.00000	+	10.3923i	−0.229584	+	0.397650i	0.728101	+	0.685470i	$0.240403\pi$
$684$		0			0
$685$		0			0
$686$	−8.50000	+	16.4545i	−0.324532	+	0.628235i
$687$		0			0
$688$	0.500000	+	0.866025i	0.0190623	+	0.0330169i
$689$	15.0000	−	25.9808i	0.571454	−	0.989788i
$690$		0			0
$691$	−8.50000	−	14.7224i	−0.323355	−	0.560068i	0.657823	−	0.753173i	$-0.271478\pi$
$691$	−8.50000	−	14.7224i	−0.323355	−	0.560068i	−0.981178	+	0.193105i	$0.938144\pi$
$692$		0			0
$693$		0			0
$694$	24.0000			0.911028
$695$		0			0
$696$		0			0
$697$	18.0000	−	31.1769i	0.681799	−	1.18091i
$698$	−0.500000	−	0.866025i	−0.0189253	−	0.0327795i
$699$		0			0
$700$	−12.5000	−	4.33013i	−0.472456	−	0.163663i
$701$	48.0000			1.81293			0.906467	−	0.422276i	$-0.138769\pi$
$701$	48.0000			1.81293			0.906467	+	0.422276i	$0.138769\pi$
$702$		0			0
$703$	−2.00000	+	3.46410i	−0.0754314	+	0.130651i
$704$	−3.00000	+	5.19615i	−0.113067	+	0.195837i
$705$		0			0
$706$	36.0000			1.35488
$707$	−6.00000	−	31.1769i	−0.225653	−	1.17253i
$708$		0			0
$709$	−8.50000	−	14.7224i	−0.319224	−	0.552913i	0.661102	−	0.750296i	$-0.270089\pi$
$709$	−8.50000	−	14.7224i	−0.319224	−	0.552913i	−0.980326	+	0.197383i	$0.936756\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.00000			−0.224702
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	3.00000	−	5.19615i	0.111959	−	0.193919i
$719$	−6.00000	−	10.3923i	−0.223762	−	0.387568i	0.732185	−	0.681106i	$-0.238501\pi$
$719$	−6.00000	−	10.3923i	−0.223762	−	0.387568i	−0.955947	+	0.293538i	$0.905167\pi$
$720$		0			0
$721$	−47.5000	−	16.4545i	−1.76899	−	0.612797i
$722$	3.00000			0.111648
$723$		0			0
$724$	5.00000	−	8.66025i	0.185824	−	0.321856i
$725$	15.0000	−	25.9808i	0.557086	−	0.964901i
$726$		0			0
$727$	5.00000			0.185440			0.0927199	−	0.995692i	$-0.470444\pi$
$727$	5.00000			0.185440			0.0927199	+	0.995692i	$0.470444\pi$
$728$	10.0000	−	8.66025i	0.370625	−	0.320970i
$729$		0			0
$730$		0			0
$731$	3.00000	−	5.19615i	0.110959	−	0.192187i
$732$		0			0
$733$	−8.50000	−	14.7224i	−0.313955	−	0.543785i	0.665260	−	0.746612i	$-0.268321\pi$
$733$	−8.50000	−	14.7224i	−0.313955	−	0.543785i	−0.979215	+	0.202826i	$0.934987\pi$
$734$	28.0000			1.03350
$735$		0			0
$736$	−6.00000			−0.221163
$737$	3.00000	+	5.19615i	0.110506	+	0.191403i
$738$		0			0
$739$	24.5000	−	42.4352i	0.901247	−	1.56101i	0.0753699	−	0.997156i	$-0.475986\pi$
$739$	24.5000	−	42.4352i	0.901247	−	1.56101i	0.825877	−	0.563850i	$-0.190680\pi$
$740$		0			0
$741$		0			0
$742$	−12.0000	+	10.3923i	−0.440534	+	0.381514i
$743$	18.0000			0.660356			0.330178	−	0.943919i	$-0.392891\pi$
$743$	18.0000			0.660356			0.330178	+	0.943919i	$0.392891\pi$
$744$		0			0
$745$		0			0
$746$	13.0000	−	22.5167i	0.475964	−	0.824394i
$747$		0			0
$748$	36.0000			1.31629
$749$	45.0000	+	15.5885i	1.64426	+	0.569590i
$750$		0			0
$751$	8.00000	+	13.8564i	0.291924	+	0.505627i	0.974265	−	0.225407i	$-0.0723712\pi$
$751$	8.00000	+	13.8564i	0.291924	+	0.505627i	−0.682341	+	0.731034i	$0.739038\pi$
$752$	3.00000	−	5.19615i	0.109399	−	0.189484i
$753$		0			0
$754$	15.0000	+	25.9808i	0.546268	+	0.946164i
$755$		0			0
$756$		0			0
$757$	29.0000			1.05402			0.527011	−	0.849858i	$-0.323312\pi$
$757$	29.0000			1.05402			0.527011	+	0.849858i	$0.323312\pi$
$758$	−0.500000	−	0.866025i	−0.0181608	−	0.0314555i
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−2.50000	−	12.9904i	−0.0905061	−	0.470283i
$764$		0			0
$765$		0			0
$766$	3.00000	−	5.19615i	0.108394	−	0.187745i
$767$	15.0000	−	25.9808i	0.541619	−	0.938111i
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$		0			0
$772$	0.500000	+	0.866025i	0.0179954	+	0.0311689i
$773$	12.0000	−	20.7846i	0.431610	−	0.747570i	−0.565402	−	0.824815i	$-0.691279\pi$
$773$	12.0000	−	20.7846i	0.431610	−	0.747570i	0.997012	+	0.0772449i	$0.0246123\pi$
$774$		0			0
$775$	−2.50000	−	4.33013i	−0.0898027	−	0.155543i
$776$	−17.0000			−0.610264
$777$		0			0
$778$	12.0000			0.430221
$779$	−12.0000	−	20.7846i	−0.429945	−	0.744686i
$780$		0			0
$781$	−36.0000	+	62.3538i	−1.28818	+	2.23120i
$782$	18.0000	+	31.1769i	0.643679	+	1.11488i
$783$		0			0
$784$	−6.50000	+	2.59808i	−0.232143	+	0.0927884i
$785$		0			0
$786$		0			0
$787$	15.5000	−	26.8468i	0.552515	−	0.956985i	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	15.5000	−	26.8468i	0.552515	−	0.956985i	0.998092	−	0.0617409i	$-0.0196653\pi$
$788$	6.00000	−	10.3923i	0.213741	−	0.370211i
$789$		0			0
$790$		0			0
$791$	−24.0000	+	20.7846i	−0.853342	+	0.739016i
$792$		0			0
$793$	2.50000	+	4.33013i	0.0887776	+	0.153767i
$794$	−6.50000	+	11.2583i	−0.230676	+	0.399543i
$795$		0			0
$796$	−5.50000	−	9.52628i	−0.194942	−	0.337650i
$797$	−30.0000			−1.06265			−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000			−1.06265			−0.531327	+	0.847167i	$0.678307\pi$
$798$		0			0
$799$	−36.0000			−1.27359
$800$	−2.50000	−	4.33013i	−0.0883883	−	0.153093i
$801$		0			0
$802$	−3.00000	+	5.19615i	−0.105934	+	0.183483i
$803$	−6.00000	−	10.3923i	−0.211735	−	0.366736i
$804$		0			0
$805$		0			0
$806$	5.00000			0.176117
$807$		0			0
$808$	6.00000	−	10.3923i	0.211079	−	0.365600i
$809$	15.0000	−	25.9808i	0.527372	−	0.913435i	−0.472119	−	0.881535i	$-0.656511\pi$
$809$	15.0000	−	25.9808i	0.527372	−	0.913435i	0.999491	−	0.0319002i	$-0.0101559\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$	−3.00000	−	15.5885i	−0.105279	−	0.547048i
$813$		0			0
$814$	−3.00000	−	5.19615i	−0.105150	−	0.182125i
$815$		0			0
$816$		0			0
$817$	−2.00000	−	3.46410i	−0.0699711	−	0.121194i
$818$	−29.0000			−1.01396
$819$		0			0
$820$		0			0
$821$	−3.00000	−	5.19615i	−0.104701	−	0.181347i	0.808915	−	0.587925i	$-0.200055\pi$
$821$	−3.00000	−	5.19615i	−0.104701	−	0.181347i	−0.913616	+	0.406578i	$0.866722\pi$
$822$		0			0
$823$	15.5000	−	26.8468i	0.540296	−	0.935820i	−0.458591	−	0.888648i	$-0.651646\pi$
$823$	15.5000	−	26.8468i	0.540296	−	0.935820i	0.998887	−	0.0471726i	$-0.0150211\pi$
$824$	−9.50000	−	16.4545i	−0.330948	−	0.573219i
$825$		0			0
$826$	−12.0000	+	10.3923i	−0.417533	+	0.361595i
$827$	−12.0000			−0.417281			−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000			−0.417281			−0.208640	+	0.977992i	$0.566904\pi$
$828$		0			0
$829$	−7.00000	+	12.1244i	−0.243120	+	0.421096i	−0.961601	−	0.274450i	$-0.911504\pi$
$829$	−7.00000	+	12.1244i	−0.243120	+	0.421096i	0.718481	+	0.695546i	$0.244838\pi$
$830$		0			0
$831$		0			0
$832$	5.00000			0.173344
$833$	33.0000	+	25.9808i	1.14338	+	0.900180i
$834$		0			0
$835$		0			0
$836$	12.0000	−	20.7846i	0.415029	−	0.718851i
$837$		0			0
$838$	−12.0000	−	20.7846i	−0.414533	−	0.717992i
$839$	18.0000			0.621429			0.310715	−	0.950503i	$-0.399432\pi$
$839$	18.0000			0.621429			0.310715	+	0.950503i	$0.399432\pi$
$840$		0			0
$841$	7.00000			0.241379
$842$	7.00000	+	12.1244i	0.241236	+	0.417833i
$843$		0			0
$844$	6.50000	−	11.2583i	0.223739	−	0.387528i
$845$		0			0
$846$		0			0
$847$	62.5000	+	21.6506i	2.14753	+	0.743925i
$848$	−6.00000			−0.206041
$849$		0			0
$850$	−15.0000	+	25.9808i	−0.514496	+	0.891133i
$851$	3.00000	−	5.19615i	0.102839	−	0.178122i
$852$		0			0
$853$	−46.0000			−1.57501			−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000			−1.57501			−0.787505	+	0.616308i	$0.788628\pi$
$854$	−0.500000	−	2.59808i	−0.0171096	−	0.0889043i
$855$		0			0
$856$	9.00000	+	15.5885i	0.307614	+	0.532803i
$857$	6.00000	−	10.3923i	0.204956	−	0.354994i	−0.745163	−	0.666883i	$-0.767628\pi$
$857$	6.00000	−	10.3923i	0.204956	−	0.354994i	0.950119	+	0.311888i	$0.100962\pi$
$858$		0			0
$859$	−11.5000	−	19.9186i	−0.392375	−	0.679613i	0.600387	−	0.799709i	$-0.295013\pi$
$859$	−11.5000	−	19.9186i	−0.392375	−	0.679613i	−0.992762	+	0.120096i	$0.961680\pi$
$860$		0			0
$861$		0			0
$862$	18.0000			0.613082
$863$	−18.0000	−	31.1769i	−0.612727	−	1.06127i	−0.990779	−	0.135490i	$-0.956739\pi$
$863$	−18.0000	−	31.1769i	−0.612727	−	1.06127i	0.378052	−	0.925785i	$-0.376594\pi$
$864$		0			0
$865$		0			0
$866$	−12.5000	−	21.6506i	−0.424767	−	0.735719i
$867$		0			0
$868$	−2.50000	−	0.866025i	−0.0848555	−	0.0293948i
$869$	−6.00000			−0.203536
$870$		0			0
$871$	2.50000	−	4.33013i	0.0847093	−	0.146721i
$872$	2.50000	−	4.33013i	0.0846607	−	0.146637i
$873$		0			0
$874$	24.0000			0.811812
$875$		0			0
$876$		0			0
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	0.954652	−	0.297724i	$-0.0962275\pi$
$877$	6.50000	+	11.2583i	0.219489	+	0.380167i	−0.735163	+	0.677891i	$0.762894\pi$
$878$	4.00000	−	6.92820i	0.134993	−	0.233816i
$879$		0			0
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−40.0000			−1.34611			−0.673054	−	0.739594i	$-0.735018\pi$
$883$	−40.0000			−1.34611			−0.673054	+	0.739594i	$0.735018\pi$
$884$	−15.0000	−	25.9808i	−0.504505	−	0.873828i
$885$		0			0
$886$	−12.0000	+	20.7846i	−0.403148	+	0.698273i
$887$	9.00000	+	15.5885i	0.302190	+	0.523409i	0.976632	−	0.214919i	$-0.0689488\pi$
$887$	9.00000	+	15.5885i	0.302190	+	0.523409i	−0.674441	+	0.738328i	$0.735615\pi$
$888$		0			0
$889$	26.0000	−	22.5167i	0.872012	−	0.755185i
$890$		0			0
$891$		0			0
$892$	−4.00000	+	6.92820i	−0.133930	+	0.231973i
$893$	−12.0000	+	20.7846i	−0.401565	+	0.695530i
$894$		0			0
$895$		0			0
$896$	−2.50000	−	0.866025i	−0.0835191	−	0.0289319i
$897$		0			0
$898$	−12.0000	−	20.7846i	−0.400445	−	0.693591i
$899$	3.00000	−	5.19615i	0.100056	−	0.173301i
$900$		0			0
$901$	18.0000	+	31.1769i	0.599667	+	1.03865i
$902$	36.0000			1.19867
$903$		0			0
$904$	−12.0000			−0.399114
$905$		0			0
$906$		0			0
$907$	18.5000	−	32.0429i	0.614282	−	1.06397i	−0.376228	−	0.926527i	$-0.622779\pi$
$907$	18.5000	−	32.0429i	0.614282	−	1.06397i	0.990510	−	0.137441i	$-0.0438878\pi$
$908$	3.00000	+	5.19615i	0.0995585	+	0.172440i
$909$		0			0
$910$		0			0
$911$	−6.00000			−0.198789			−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000			−0.198789			−0.0993944	+	0.995048i	$0.531691\pi$
$912$		0			0
$913$	−18.0000	+	31.1769i	−0.595713	+	1.03181i
$914$	−9.50000	+	16.4545i	−0.314232	+	0.544266i
$915$		0			0
$916$	5.00000			0.165205
$917$	−30.0000	−	10.3923i	−0.990687	−	0.343184i
$918$		0			0
$919$	27.5000	+	47.6314i	0.907141	+	1.57121i	0.818017	+	0.575194i	$0.195074\pi$
$919$	27.5000	+	47.6314i	0.907141	+	1.57121i	0.0891245	+	0.996020i	$0.471593\pi$
$920$		0			0
$921$		0			0
$922$	9.00000	+	15.5885i	0.296399	+	0.513378i
$923$	60.0000			1.97492
$924$		0			0
$925$	5.00000			0.164399
$926$	4.00000	+	6.92820i	0.131448	+	0.227675i
$927$		0			0
$928$	3.00000	−	5.19615i	0.0984798	−	0.170572i
$929$	−24.0000	−	41.5692i	−0.787414	−	1.36384i	−0.927546	−	0.373709i	$-0.878086\pi$
$929$	−24.0000	−	41.5692i	−0.787414	−	1.36384i	0.140132	−	0.990133i	$-0.455247\pi$
$930$		0			0
$931$	26.0000	−	10.3923i	0.852116	−	0.340594i
$932$	24.0000			0.786146
$933$		0			0
$934$	6.00000	−	10.3923i	0.196326	−	0.340047i
$935$		0			0
$936$		0			0
$937$	−37.0000			−1.20874			−0.604369	−	0.796705i	$-0.706575\pi$
$937$	−37.0000			−1.20874			−0.604369	+	0.796705i	$0.706575\pi$
$938$	−2.00000	+	1.73205i	−0.0653023	+	0.0565535i
$939$		0			0
$940$		0			0
$941$	−9.00000	+	15.5885i	−0.293392	+	0.508169i	−0.974609	−	0.223912i	$-0.928117\pi$
$941$	−9.00000	+	15.5885i	−0.293392	+	0.508169i	0.681218	+	0.732081i	$0.261451\pi$
$942$		0			0
$943$	18.0000	+	31.1769i	0.586161	+	1.01526i
$944$	−6.00000			−0.195283
$945$		0			0
$946$	6.00000			0.195077
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	0.681930	−	0.731417i	$-0.261141\pi$
$947$	−9.00000	−	15.5885i	−0.292461	−	0.506557i	−0.974391	+	0.224860i	$0.927807\pi$
$948$		0			0
$949$	−5.00000	+	8.66025i	−0.162307	+	0.281124i
$950$	10.0000	+	17.3205i	0.324443	+	0.561951i
$951$		0			0
$952$	3.00000	+	15.5885i	0.0972306	+	0.505225i
$953$	−6.00000			−0.194359			−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000			−0.194359			−0.0971795	+	0.995267i	$0.530982\pi$
$954$		0			0
$955$		0			0
$956$	−12.0000	+	20.7846i	−0.388108	+	0.672222i
$957$		0			0
$958$	−12.0000			−0.387702
$959$	3.00000	+	15.5885i	0.0968751	+	0.503378i
$960$		0			0
$961$	15.0000	+	25.9808i	0.483871	+	0.838089i
$962$	−2.50000	+	4.33013i	−0.0806032	+	0.139609i
$963$		0			0
$964$	−8.50000	−	14.7224i	−0.273767	−	0.474178i
$965$		0			0
$966$		0			0
$967$	35.0000			1.12552			0.562762	−	0.826619i	$-0.309739\pi$
$967$	35.0000			1.12552			0.562762	+	0.826619i	$0.309739\pi$
$968$	12.5000	+	21.6506i	0.401765	+	0.695878i
$969$		0			0
$970$		0			0
$971$	18.0000	+	31.1769i	0.577647	+	1.00051i	0.995748	+	0.0921142i	$0.0293625\pi$
$971$	18.0000	+	31.1769i	0.577647	+	1.00051i	−0.418101	+	0.908401i	$0.637304\pi$
$972$		0			0
$973$	14.0000	−	12.1244i	0.448819	−	0.388689i
$974$	40.0000			1.28168
$975$		0			0
$976$	0.500000	−	0.866025i	0.0160046	−	0.0277208i
$977$	−3.00000	+	5.19615i	−0.0959785	+	0.166240i	−0.910017	−	0.414572i	$-0.863931\pi$
$977$	−3.00000	+	5.19615i	−0.0959785	+	0.166240i	0.814038	+	0.580812i	$0.197265\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$	−15.0000	−	25.9808i	−0.478669	−	0.829079i
$983$	12.0000	−	20.7846i	0.382741	−	0.662926i	−0.608712	−	0.793391i	$-0.708314\pi$
$983$	12.0000	−	20.7846i	0.382741	−	0.662926i	0.991453	+	0.130465i	$0.0416470\pi$
$984$		0			0
$985$		0			0
$986$	−36.0000			−1.14647
$987$		0			0
$988$	−20.0000			−0.636285
$989$	3.00000	+	5.19615i	0.0953945	+	0.165228i
$990$		0			0
$991$	−2.50000	+	4.33013i	−0.0794151	+	0.137551i	−0.902998	−	0.429645i	$-0.858639\pi$
$991$	−2.50000	+	4.33013i	−0.0794151	+	0.137551i	0.823583	+	0.567196i	$0.191972\pi$
$992$	−0.500000	−	0.866025i	−0.0158750	−	0.0274963i
$993$		0			0
$994$	−30.0000	−	10.3923i	−0.951542	−	0.329624i
$995$		0			0
$996$		0			0
$997$	−11.5000	+	19.9186i	−0.364209	+	0.630828i	−0.988649	−	0.150245i	$-0.951994\pi$
$997$	−11.5000	+	19.9186i	−0.364209	+	0.630828i	0.624440	+	0.781073i	$0.285327\pi$
$998$	−12.5000	+	21.6506i	−0.395681	+	0.685339i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.g.e.109.1	yes	2
3.2	odd	2		378.2.g.b.109.1	✓	2
7.2	even	3	inner	378.2.g.e.163.1	yes	2
7.3	odd	6		2646.2.a.g.1.1		1
7.4	even	3		2646.2.a.h.1.1		1
9.2	odd	6		1134.2.h.d.109.1		2
9.4	even	3		1134.2.e.c.865.1		2
9.5	odd	6		1134.2.e.m.865.1		2
9.7	even	3		1134.2.h.n.109.1		2
21.2	odd	6		378.2.g.b.163.1	yes	2
21.11	odd	6		2646.2.a.x.1.1		1
21.17	even	6		2646.2.a.w.1.1		1
63.2	odd	6		1134.2.e.m.919.1		2
63.16	even	3		1134.2.e.c.919.1		2
63.23	odd	6		1134.2.h.d.541.1		2
63.58	even	3		1134.2.h.n.541.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.g.b.109.1	✓	2	3.2	odd	2
378.2.g.b.163.1	yes	2	21.2	odd	6
378.2.g.e.109.1	yes	2	1.1	even	1	trivial
378.2.g.e.163.1	yes	2	7.2	even	3	inner
1134.2.e.c.865.1		2	9.4	even	3
1134.2.e.c.919.1		2	63.16	even	3
1134.2.e.m.865.1		2	9.5	odd	6
1134.2.e.m.919.1		2	63.2	odd	6
1134.2.h.d.109.1		2	9.2	odd	6
1134.2.h.d.541.1		2	63.23	odd	6
1134.2.h.n.109.1		2	9.7	even	3
1134.2.h.n.541.1		2	63.58	even	3
2646.2.a.g.1.1		1	7.3	odd	6
2646.2.a.h.1.1		1	7.4	even	3
2646.2.a.w.1.1		1	21.17	even	6
2646.2.a.x.1.1		1	21.11	odd	6

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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	378.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 + 0.866025i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(-2.00000 + 1.73205i) q^{7} -1.00000 q^{8} +(-3.00000 + 5.19615i) q^{11} +5.00000 q^{13} +(-2.50000 - 0.866025i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(-3.00000 + 5.19615i) q^{17} +(2.00000 + 3.46410i) q^{19} -6.00000 q^{22} +(-3.00000 - 5.19615i) q^{23} +(2.50000 - 4.33013i) q^{25} +(2.50000 + 4.33013i) q^{26} +(-0.500000 - 2.59808i) q^{28} +6.00000 q^{29} +(0.500000 - 0.866025i) q^{31} +(0.500000 - 0.866025i) q^{32} -6.00000 q^{34} +(0.500000 + 0.866025i) q^{37} +(-2.00000 + 3.46410i) q^{38} -6.00000 q^{41} -1.00000 q^{43} +(-3.00000 - 5.19615i) q^{44} +(3.00000 - 5.19615i) q^{46} +(3.00000 + 5.19615i) q^{47} +(1.00000 - 6.92820i) q^{49} +5.00000 q^{50} +(-2.50000 + 4.33013i) q^{52} +(3.00000 - 5.19615i) q^{53} +(2.00000 - 1.73205i) q^{56} +(3.00000 + 5.19615i) q^{58} +(3.00000 - 5.19615i) q^{59} +(0.500000 + 0.866025i) q^{61} +1.00000 q^{62} +1.00000 q^{64} +(0.500000 - 0.866025i) q^{67} +(-3.00000 - 5.19615i) q^{68} +12.0000 q^{71} +(-1.00000 + 1.73205i) q^{73} +(-0.500000 + 0.866025i) q^{74} -4.00000 q^{76} +(-3.00000 - 15.5885i) q^{77} +(0.500000 + 0.866025i) q^{79} +(-3.00000 - 5.19615i) q^{82} +6.00000 q^{83} +(-0.500000 - 0.866025i) q^{86} +(3.00000 - 5.19615i) q^{88} +(-10.0000 + 8.66025i) q^{91} +6.00000 q^{92} +(-3.00000 + 5.19615i) q^{94} +17.0000 q^{97} +(6.50000 - 2.59808i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8	\( 2 q + q^{2} - q^{4} - 4 q^{7} - 2 q^{8} - 6 q^{11} + 10 q^{13} - 5 q^{14} - q^{16} - 6 q^{17} + 4 q^{19} - 12 q^{22} - 6 q^{23} + 5 q^{25} + 5 q^{26} - q^{28} + 12 q^{29} + q^{31} + q^{32} - 12 q^{34} + q^{37} - 4 q^{38} - 12 q^{41} - 2 q^{43} - 6 q^{44} + 6 q^{46} + 6 q^{47} + 2 q^{49} + 10 q^{50} - 5 q^{52} + 6 q^{53} + 4 q^{56} + 6 q^{58} + 6 q^{59} + q^{61} + 2 q^{62} + 2 q^{64} + q^{67} - 6 q^{68} + 24 q^{71} - 2 q^{73} - q^{74} - 8 q^{76} - 6 q^{77} + q^{79} - 6 q^{82} + 12 q^{83} - q^{86} + 6 q^{88} - 20 q^{91} + 12 q^{92} - 6 q^{94} + 34 q^{97} + 13 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - 4 * q^7 - 2 * q^8 - 6 * q^11 + 10 * q^13 - 5 * q^14 - q^16 - 6 * q^17 + 4 * q^19 - 12 * q^22 - 6 * q^23 + 5 * q^25 + 5 * q^26 - q^28 + 12 * q^29 + q^31 + q^32 - 12 * q^34 + q^37 - 4 * q^38 - 12 * q^41 - 2 * q^43 - 6 * q^44 + 6 * q^46 + 6 * q^47 + 2 * q^49 + 10 * q^50 - 5 * q^52 + 6 * q^53 + 4 * q^56 + 6 * q^58 + 6 * q^59 + q^61 + 2 * q^62 + 2 * q^64 + q^67 - 6 * q^68 + 24 * q^71 - 2 * q^73 - q^74 - 8 * q^76 - 6 * q^77 + q^79 - 6 * q^82 + 12 * q^83 - q^86 + 6 * q^88 - 20 * q^91 + 12 * q^92 - 6 * q^94 + 34 * q^97 + 13 * q^98

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