LMFDB - Embedded newform 3420.2.t.b.3241.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3420,2,Mod(1261,3420)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3420, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("3420.1261");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.t (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:	$27.3088374913$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 380)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			3241.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3420.3241
Dual form			3420.2.t.b.1261.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +O(q^{10})$	$q+(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +3.00000 q^{11} +(-3.00000 - 5.19615i) q^{13} +(1.00000 - 1.73205i) q^{17} +(3.50000 + 2.59808i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(0.500000 + 0.866025i) q^{29} -5.00000 q^{31} +(2.00000 - 3.46410i) q^{35} -4.00000 q^{37} +(1.00000 - 1.73205i) q^{41} +(3.00000 + 5.19615i) q^{47} +9.00000 q^{49} +(-3.00000 - 5.19615i) q^{53} +(-1.50000 + 2.59808i) q^{55} +(-0.500000 + 0.866025i) q^{59} +(3.50000 + 6.06218i) q^{61} +6.00000 q^{65} +(7.00000 + 12.1244i) q^{67} +(7.50000 - 12.9904i) q^{71} +(-6.00000 + 10.3923i) q^{73} -12.0000 q^{77} +(0.500000 - 0.866025i) q^{79} -16.0000 q^{83} +(1.00000 + 1.73205i) q^{85} +(8.50000 + 14.7224i) q^{89} +(12.0000 + 20.7846i) q^{91} +(-4.00000 + 1.73205i) q^{95} +(-6.00000 + 10.3923i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{5} - 8 q^{7}+O(q^{10}) $ 2 * q - q^5 - 8 * q^7	$ 2 q - q^{5} - 8 q^{7} + 6 q^{11} - 6 q^{13} + 2 q^{17} + 7 q^{19} + 4 q^{23} - q^{25} + q^{29} - 10 q^{31} + 4 q^{35} - 8 q^{37} + 2 q^{41} + 6 q^{47} + 18 q^{49} - 6 q^{53} - 3 q^{55} - q^{59} + 7 q^{61} + 12 q^{65} + 14 q^{67} + 15 q^{71} - 12 q^{73} - 24 q^{77} + q^{79} - 32 q^{83} + 2 q^{85} + 17 q^{89} + 24 q^{91} - 8 q^{95} - 12 q^{97}+O(q^{100}) $ 2 * q - q^5 - 8 * q^7 + 6 * q^11 - 6 * q^13 + 2 * q^17 + 7 * q^19 + 4 * q^23 - q^25 + q^29 - 10 * q^31 + 4 * q^35 - 8 * q^37 + 2 * q^41 + 6 * q^47 + 18 * q^49 - 6 * q^53 - 3 * q^55 - q^59 + 7 * q^61 + 12 * q^65 + 14 * q^67 + 15 * q^71 - 12 * q^73 - 24 * q^77 + q^79 - 32 * q^83 + 2 * q^85 + 17 * q^89 + 24 * q^91 - 8 * q^95 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	−4.00000			−1.51186			−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000			−1.51186			−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000			0.904534			0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000			0.904534			0.452267	+	0.891883i	$0.350615\pi$
$12$		0			0
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	−0.896410	−	0.443227i	$-0.853834\pi$
$13$	−3.00000	−	5.19615i	−0.832050	−	1.44115i	0.0643593	−	0.997927i	$-0.479500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	−0.718900	−	0.695113i	$-0.755354\pi$
$17$	1.00000	−	1.73205i	0.242536	−	0.420084i	0.961436	+	0.275029i	$0.0886875\pi$
$18$		0			0
$19$	3.50000	+	2.59808i	0.802955	+	0.596040i
$19$	3.50000	+	2.59808i	0.802955	+	0.596040i
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	0.995639	−	0.0932891i	$-0.0297381\pi$
$23$	2.00000	+	3.46410i	0.417029	+	0.722315i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	0.500000	+	0.866025i	0.0928477	+	0.160817i	0.908708	−	0.417432i	$-0.137070\pi$
$29$	0.500000	+	0.866025i	0.0928477	+	0.160817i	−0.815861	+	0.578249i	$0.803736\pi$
$30$		0			0
$31$	−5.00000			−0.898027			−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000			−0.898027			−0.449013	+	0.893525i	$0.648224\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.00000	−	3.46410i	0.338062	−	0.585540i
$36$		0			0
$37$	−4.00000			−0.657596			−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000			−0.657596			−0.328798	+	0.944400i	$0.606644\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	1.00000	−	1.73205i	0.156174	−	0.270501i	−0.777312	−	0.629115i	$-0.783417\pi$
$41$	1.00000	−	1.73205i	0.156174	−	0.270501i	0.933486	+	0.358614i	$0.116751\pi$
$42$		0			0
$43$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$43$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	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
$49$	9.00000			1.28571
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	0.583036	−	0.812447i	$-0.301865\pi$
$53$	−3.00000	−	5.19615i	−0.412082	−	0.713746i	−0.995117	+	0.0987002i	$0.968532\pi$
$54$		0			0
$55$	−1.50000	+	2.59808i	−0.202260	+	0.350325i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.500000	+	0.866025i	−0.0650945	+	0.112747i	−0.896736	−	0.442566i	$-0.854068\pi$
$59$	−0.500000	+	0.866025i	−0.0650945	+	0.112747i	0.831641	+	0.555313i	$0.187402\pi$
$60$		0			0
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	0.998264	−	0.0588933i	$-0.0187572\pi$
$61$	3.50000	+	6.06218i	0.448129	+	0.776182i	−0.550135	+	0.835076i	$0.685424\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.00000			0.744208
$66$		0			0
$67$	7.00000	+	12.1244i	0.855186	+	1.48123i	0.876472	+	0.481452i	$0.159891\pi$
$67$	7.00000	+	12.1244i	0.855186	+	1.48123i	−0.0212861	+	0.999773i	$0.506776\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	7.50000	−	12.9904i	0.890086	−	1.54167i	0.0503155	−	0.998733i	$-0.483977\pi$
$71$	7.50000	−	12.9904i	0.890086	−	1.54167i	0.839771	−	0.542941i	$-0.182689\pi$
$72$		0			0
$73$	−6.00000	+	10.3923i	−0.702247	+	1.21633i	0.265429	+	0.964130i	$0.414486\pi$
$73$	−6.00000	+	10.3923i	−0.702247	+	1.21633i	−0.967676	+	0.252197i	$0.918847\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.0000			−1.36753
$78$		0			0
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	−0.836527	−	0.547926i	$-0.815418\pi$
$79$	0.500000	−	0.866025i	0.0562544	−	0.0974355i	0.892781	+	0.450490i	$0.148751\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−16.0000			−1.75623			−0.878114	−	0.478451i	$-0.841198\pi$
$83$	−16.0000			−1.75623			−0.878114	+	0.478451i	$0.841198\pi$
$84$		0			0
$85$	1.00000	+	1.73205i	0.108465	+	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.50000	+	14.7224i	0.900998	+	1.56057i	0.826201	+	0.563376i	$0.190498\pi$
$89$	8.50000	+	14.7224i	0.900998	+	1.56057i	0.0747975	+	0.997199i	$0.476169\pi$
$90$		0			0
$91$	12.0000	+	20.7846i	1.25794	+	2.17882i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−4.00000	+	1.73205i	−0.410391	+	0.177705i
$96$		0			0
$97$	−6.00000	+	10.3923i	−0.609208	+	1.05518i	0.382164	+	0.924095i	$0.375179\pi$
$97$	−6.00000	+	10.3923i	−0.609208	+	1.05518i	−0.991371	+	0.131084i	$0.958154\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.50000	+	9.52628i	0.547270	+	0.947900i	0.998460	+	0.0554722i	$0.0176664\pi$
$101$	5.50000	+	9.52628i	0.547270	+	0.947900i	−0.451190	+	0.892428i	$0.649000\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	−0.983531	−	0.180741i	$-0.942150\pi$
$109$	−3.50000	+	6.06218i	−0.335239	+	0.580651i	0.648292	+	0.761392i	$0.275484\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−14.0000			−1.31701			−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000			−1.31701			−0.658505	+	0.752577i	$0.728811\pi$
$114$		0			0
$115$	−4.00000			−0.373002
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.00000	+	6.92820i	−0.366679	+	0.635107i
$120$		0			0
$121$	−2.00000			−0.181818
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−9.00000	−	15.5885i	−0.798621	−	1.38325i	−0.920514	−	0.390709i	$-0.872230\pi$
$127$	−9.00000	−	15.5885i	−0.798621	−	1.38325i	0.121894	−	0.992543i	$-0.461103\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−10.0000	+	17.3205i	−0.873704	+	1.51330i	−0.0155672	+	0.999879i	$0.504955\pi$
$131$	−10.0000	+	17.3205i	−0.873704	+	1.51330i	−0.858137	+	0.513421i	$0.828378\pi$
$132$		0			0
$133$	−14.0000	−	10.3923i	−1.21395	−	0.901127i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	0.999893	+	0.0146279i	$0.00465636\pi$
$137$	6.00000	+	10.3923i	0.512615	+	0.887875i	−0.487278	+	0.873247i	$0.662010\pi$
$138$		0			0
$139$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$139$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−9.00000	−	15.5885i	−0.752618	−	1.30357i
$144$		0			0
$145$	−1.00000			−0.0830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	0.273377	+	0.961907i	$0.411859\pi$
$149$	−8.50000	+	14.7224i	−0.696347	+	1.20611i	−0.969724	+	0.244202i	$0.921474\pi$
$150$		0			0
$151$	3.00000			0.244137			0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000			0.244137			0.122068	+	0.992522i	$0.461047\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.50000	−	4.33013i	0.200805	−	0.347804i
$156$		0			0
$157$	−9.00000	+	15.5885i	−0.718278	+	1.24409i	0.243403	+	0.969925i	$0.421736\pi$
$157$	−9.00000	+	15.5885i	−0.718278	+	1.24409i	−0.961681	+	0.274169i	$0.911597\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−8.00000	−	13.8564i	−0.630488	−	1.09204i
$162$		0			0
$163$	18.0000			1.40987			0.704934	−	0.709273i	$-0.250976\pi$
$163$	18.0000			1.40987			0.704934	+	0.709273i	$0.250976\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.00000	+	6.92820i	0.309529	+	0.536120i	0.978259	−	0.207385i	$-0.0664952\pi$
$167$	4.00000	+	6.92820i	0.309529	+	0.536120i	−0.668730	+	0.743505i	$0.733162\pi$
$168$		0			0
$169$	−11.5000	+	19.9186i	−0.884615	+	1.53220i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	2.00000	+	3.46410i	0.151186	+	0.261861i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$		0			0
$181$	−1.00000	−	1.73205i	−0.0743294	−	0.128742i	0.826465	−	0.562988i	$-0.190348\pi$
$181$	−1.00000	−	1.73205i	−0.0743294	−	0.128742i	−0.900794	+	0.434246i	$0.857015\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.00000	−	3.46410i	0.147043	−	0.254686i
$186$		0			0
$187$	3.00000	−	5.19615i	0.219382	−	0.379980i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	13.0000			0.940647			0.470323	−	0.882494i	$-0.344137\pi$
$191$	13.0000			0.940647			0.470323	+	0.882494i	$0.344137\pi$
$192$		0			0
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	0.420096	+	0.907480i	$0.361996\pi$
$193$	−8.00000	+	13.8564i	−0.575853	+	0.997406i	−0.995948	+	0.0899262i	$0.971337\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	−11.5000	−	19.9186i	−0.815213	−	1.41199i	−0.909175	−	0.416415i	$-0.863286\pi$
$199$	−11.5000	−	19.9186i	−0.815213	−	1.41199i	0.0939612	−	0.995576i	$-0.470047\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2.00000	−	3.46410i	−0.140372	−	0.243132i
$204$		0			0
$205$	1.00000	+	1.73205i	0.0698430	+	0.120972i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.5000	+	7.79423i	0.726300	+	0.539138i
$210$		0			0
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	−0.848301	−	0.529514i	$-0.822374\pi$
$211$	0.500000	−	0.866025i	0.0344214	−	0.0596196i	0.882723	+	0.469894i	$0.155708\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	20.0000			1.35769
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.0000			−0.807207
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.00000			0.265489			0.132745	−	0.991150i	$-0.457621\pi$
$227$	4.00000			0.265489			0.132745	+	0.991150i	$0.457621\pi$
$228$		0			0
$229$	21.0000			1.38772			0.693860	−	0.720110i	$-0.255909\pi$
$229$	21.0000			1.38772			0.693860	+	0.720110i	$0.255909\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.00000	+	8.66025i	−0.327561	+	0.567352i	−0.982027	−	0.188739i	$-0.939560\pi$
$233$	−5.00000	+	8.66025i	−0.327561	+	0.567352i	0.654466	+	0.756091i	$0.272893\pi$
$234$		0			0
$235$	−6.00000			−0.391397
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.0000			−0.970269			−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000			−0.970269			−0.485135	+	0.874439i	$0.661229\pi$
$240$		0			0
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	0.774202	−	0.632938i	$-0.218151\pi$
$241$	−2.50000	−	4.33013i	−0.161039	−	0.278928i	−0.935242	+	0.354010i	$0.884818\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−4.50000	+	7.79423i	−0.287494	+	0.497955i
$246$		0			0
$247$	3.00000	−	25.9808i	0.190885	−	1.65312i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.5000	+	26.8468i	0.978351	+	1.69455i	0.668400	+	0.743802i	$0.266979\pi$
$251$	15.5000	+	26.8468i	0.978351	+	1.69455i	0.309951	+	0.950753i	$0.399687\pi$
$252$		0			0
$253$	6.00000	+	10.3923i	0.377217	+	0.653359i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	0.997374	+	0.0724199i	$0.0230722\pi$
$257$	9.00000	+	15.5885i	0.561405	+	0.972381i	−0.435970	+	0.899961i	$0.643595\pi$
$258$		0			0
$259$	16.0000			0.994192
$260$		0			0
$261$		0			0
$262$		0			0
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	−0.669680	−	0.742650i	$-0.733569\pi$
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	0.977993	+	0.208635i	$0.0669022\pi$
$264$		0			0
$265$	6.00000			0.368577
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.500000	+	0.866025i	−0.0304855	+	0.0528025i	−0.880866	−	0.473366i	$-0.843039\pi$
$269$	−0.500000	+	0.866025i	−0.0304855	+	0.0528025i	0.850380	+	0.526169i	$0.176372\pi$
$270$		0			0
$271$	−8.50000	+	14.7224i	−0.516338	+	0.894324i	0.483482	+	0.875354i	$0.339372\pi$
$271$	−8.50000	+	14.7224i	−0.516338	+	0.894324i	−0.999820	+	0.0189696i	$0.993961\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.50000	−	2.59808i	−0.0904534	−	0.156670i
$276$		0			0
$277$	16.0000			0.961347			0.480673	−	0.876900i	$-0.340392\pi$
$277$	16.0000			0.961347			0.480673	+	0.876900i	$0.340392\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−13.0000	−	22.5167i	−0.775515	−	1.34323i	−0.934505	−	0.355951i	$-0.884157\pi$
$281$	−13.0000	−	22.5167i	−0.775515	−	1.34323i	0.158990	−	0.987280i	$-0.449176\pi$
$282$		0			0
$283$	13.0000	−	22.5167i	0.772770	−	1.33848i	−0.163270	−	0.986581i	$-0.552204\pi$
$283$	13.0000	−	22.5167i	0.772770	−	1.33848i	0.936039	−	0.351895i	$-0.114463\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.00000	+	6.92820i	−0.236113	+	0.408959i
$288$		0			0
$289$	6.50000	+	11.2583i	0.382353	+	0.662255i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−16.0000			−0.934730			−0.467365	−	0.884064i	$-0.654797\pi$
$293$	−16.0000			−0.934730			−0.467365	+	0.884064i	$0.654797\pi$
$294$		0			0
$295$	−0.500000	−	0.866025i	−0.0291111	−	0.0504219i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	−	20.7846i	0.693978	−	1.20201i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−7.00000			−0.400819
$306$		0			0
$307$	7.00000	−	12.1244i	0.399511	−	0.691974i	−0.594154	−	0.804351i	$-0.702513\pi$
$307$	7.00000	−	12.1244i	0.399511	−	0.691974i	0.993666	+	0.112377i	$0.0358466\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−17.0000	−	29.4449i	−0.960897	−	1.66432i	−0.720257	−	0.693708i	$-0.755976\pi$
$313$	−17.0000	−	29.4449i	−0.960897	−	1.66432i	−0.240640	−	0.970614i	$-0.577357\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	0.646872	−	0.762598i	$-0.276077\pi$
$317$	−6.00000	−	10.3923i	−0.336994	−	0.583690i	−0.983866	+	0.178908i	$0.942743\pi$
$318$		0			0
$319$	1.50000	+	2.59808i	0.0839839	+	0.145464i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	8.00000	−	3.46410i	0.445132	−	0.192748i
$324$		0			0
$325$	−3.00000	+	5.19615i	−0.166410	+	0.288231i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.0000	−	20.7846i	−0.661581	−	1.14589i
$330$		0			0
$331$	−20.0000			−1.09930			−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000			−1.09930			−0.549650	+	0.835395i	$0.685239\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−14.0000			−0.764902
$336$		0			0
$337$	−2.00000	+	3.46410i	−0.108947	+	0.188702i	−0.915344	−	0.402673i	$-0.868081\pi$
$337$	−2.00000	+	3.46410i	−0.108947	+	0.188702i	0.806397	+	0.591375i	$0.201415\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−15.0000			−0.812296
$342$		0			0
$343$	−8.00000			−0.431959
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	−0.658824	−	0.752297i	$-0.728946\pi$
$347$	6.00000	−	10.3923i	0.322097	−	0.557888i	0.980921	+	0.194409i	$0.0622790\pi$
$348$		0			0
$349$	−2.00000			−0.107058			−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000			−0.107058			−0.0535288	+	0.998566i	$0.517047\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	36.0000			1.91609			0.958043	−	0.286623i	$-0.0925328\pi$
$353$	36.0000			1.91609			0.958043	+	0.286623i	$0.0925328\pi$
$354$		0			0
$355$	7.50000	+	12.9904i	0.398059	+	0.689458i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$359$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$360$		0			0
$361$	5.50000	+	18.1865i	0.289474	+	0.957186i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−6.00000	−	10.3923i	−0.314054	−	0.543958i
$366$		0			0
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	0.956544	+	0.291587i	$0.0941834\pi$
$367$	14.0000	+	24.2487i	0.730794	+	1.26577i	−0.225750	+	0.974185i	$0.572483\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12.0000	+	20.7846i	0.623009	+	1.07908i
$372$		0			0
$373$	−12.0000			−0.621336			−0.310668	−	0.950518i	$-0.600553\pi$
$373$	−12.0000			−0.621336			−0.310668	+	0.950518i	$0.600553\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.00000	−	5.19615i	0.154508	−	0.267615i
$378$		0			0
$379$	−19.0000			−0.975964			−0.487982	−	0.872854i	$-0.662267\pi$
$379$	−19.0000			−0.975964			−0.487982	+	0.872854i	$0.662267\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	−0.998954	−	0.0457244i	$-0.985440\pi$
$383$	−9.00000	+	15.5885i	−0.459879	+	0.796533i	0.539076	+	0.842257i	$0.318774\pi$
$384$		0			0
$385$	6.00000	−	10.3923i	0.305788	−	0.529641i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−10.5000	−	18.1865i	−0.532371	−	0.922094i	−0.999286	−	0.0377914i	$-0.987968\pi$
$389$	−10.5000	−	18.1865i	−0.532371	−	0.922094i	0.466915	−	0.884302i	$-0.345366\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$		0			0
$393$		0			0
$394$		0			0
$395$	0.500000	+	0.866025i	0.0251577	+	0.0435745i
$396$		0			0
$397$	4.00000	−	6.92820i	0.200754	−	0.347717i	−0.748017	−	0.663679i	$-0.768994\pi$
$397$	4.00000	−	6.92820i	0.200754	−	0.347717i	0.948772	+	0.315963i	$0.102327\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.50000	−	12.9904i	0.374532	−	0.648709i	−0.615725	−	0.787961i	$-0.711137\pi$
$401$	7.50000	−	12.9904i	0.374532	−	0.648709i	0.990257	+	0.139253i	$0.0444700\pi$
$402$		0			0
$403$	15.0000	+	25.9808i	0.747203	+	1.29419i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−12.0000			−0.594818
$408$		0			0
$409$	0.500000	+	0.866025i	0.0247234	+	0.0428222i	0.878122	−	0.478436i	$-0.158796\pi$
$409$	0.500000	+	0.866025i	0.0247234	+	0.0428222i	−0.853399	+	0.521258i	$0.825463\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.00000	−	3.46410i	0.0984136	−	0.170457i
$414$		0			0
$415$	8.00000	−	13.8564i	0.392705	−	0.680184i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.00000			0.244266			0.122133	−	0.992514i	$-0.461027\pi$
$419$	5.00000			0.244266			0.122133	+	0.992514i	$0.461027\pi$
$420$		0			0
$421$	−12.5000	+	21.6506i	−0.609213	+	1.05519i	0.382158	+	0.924097i	$0.375181\pi$
$421$	−12.5000	+	21.6506i	−0.609213	+	1.05519i	−0.991370	+	0.131090i	$0.958152\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.00000			−0.0970143
$426$		0			0
$427$	−14.0000	−	24.2487i	−0.677507	−	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.5000	+	23.3827i	0.650272	+	1.12630i	0.983057	+	0.183301i	$0.0586785\pi$
$431$	13.5000	+	23.3827i	0.650272	+	1.12630i	−0.332785	+	0.943003i	$0.607988\pi$
$432$		0			0
$433$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$433$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.00000	+	17.3205i	−0.0956730	+	0.828552i
$438$		0			0
$439$	−0.500000	+	0.866025i	−0.0238637	+	0.0413331i	−0.877711	−	0.479191i	$-0.840930\pi$
$439$	−0.500000	+	0.866025i	−0.0238637	+	0.0413331i	0.853847	+	0.520524i	$0.174263\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	0.687557	−	0.726130i	$-0.258683\pi$
$443$	−6.00000	−	10.3923i	−0.285069	−	0.493753i	−0.972626	+	0.232377i	$0.925350\pi$
$444$		0			0
$445$	−17.0000			−0.805877
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.00000			0.0471929			0.0235965	−	0.999722i	$-0.492488\pi$
$449$	1.00000			0.0471929			0.0235965	+	0.999722i	$0.492488\pi$
$450$		0			0
$451$	3.00000	−	5.19615i	0.141264	−	0.244677i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−24.0000			−1.12514
$456$		0			0
$457$	−2.00000			−0.0935561			−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000			−0.0935561			−0.0467780	+	0.998905i	$0.514895\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	17.5000	−	30.3109i	0.815056	−	1.41172i	−0.0942312	−	0.995550i	$-0.530039\pi$
$461$	17.5000	−	30.3109i	0.815056	−	1.41172i	0.909288	−	0.416169i	$-0.136627\pi$
$462$		0			0
$463$	−8.00000			−0.371792			−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000			−0.371792			−0.185896	+	0.982569i	$0.559519\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	20.0000			0.925490			0.462745	−	0.886492i	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		0			0
$469$	−28.0000	−	48.4974i	−1.29292	−	2.23940i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.500000	−	4.33013i	0.0229416	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.50000	−	2.59808i	−0.0685367	−	0.118709i	0.829721	−	0.558179i	$-0.188500\pi$
$479$	−1.50000	−	2.59808i	−0.0685367	−	0.118709i	−0.898257	+	0.439470i	$0.855166\pi$
$480$		0			0
$481$	12.0000	+	20.7846i	0.547153	+	0.947697i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.00000	−	10.3923i	−0.272446	−	0.471890i
$486$		0			0
$487$	34.0000			1.54069			0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000			1.54069			0.770344	+	0.637629i	$0.220085\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−12.5000	+	21.6506i	−0.564117	+	0.977079i	0.433014	+	0.901387i	$0.357450\pi$
$491$	−12.5000	+	21.6506i	−0.564117	+	0.977079i	−0.997131	+	0.0756923i	$0.975883\pi$
$492$		0			0
$493$	2.00000			0.0900755
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−30.0000	+	51.9615i	−1.34568	+	2.33079i
$498$		0			0
$499$	−18.0000	+	31.1769i	−0.805791	+	1.39567i	0.109965	+	0.993935i	$0.464926\pi$
$499$	−18.0000	+	31.1769i	−0.805791	+	1.39567i	−0.915756	+	0.401735i	$0.868407\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	11.0000	+	19.0526i	0.490466	+	0.849512i	0.999940	−	0.0109744i	$-0.00349334\pi$
$503$	11.0000	+	19.0526i	0.490466	+	0.849512i	−0.509474	+	0.860486i	$0.670160\pi$
$504$		0			0
$505$	−11.0000			−0.489494
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.0000	+	25.9808i	0.664863	+	1.15158i	0.979322	+	0.202306i	$0.0648436\pi$
$509$	15.0000	+	25.9808i	0.664863	+	1.15158i	−0.314459	+	0.949271i	$0.601823\pi$
$510$		0			0
$511$	24.0000	−	41.5692i	1.06170	−	1.83891i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$	9.00000	+	15.5885i	0.395820	+	0.685580i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	45.0000			1.97149			0.985743	−	0.168259i	$-0.0538144\pi$
$521$	45.0000			1.97149			0.985743	+	0.168259i	$0.0538144\pi$
$522$		0			0
$523$	5.00000	+	8.66025i	0.218635	+	0.378686i	0.954391	−	0.298560i	$-0.0965063\pi$
$523$	5.00000	+	8.66025i	0.218635	+	0.378686i	−0.735756	+	0.677247i	$0.763173\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5.00000	+	8.66025i	−0.217803	+	0.377247i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−12.0000			−0.519778
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	27.0000			1.16297
$540$		0			0
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	0.999042	+	0.0437584i	$0.0139332\pi$
$541$	12.5000	+	21.6506i	0.537417	+	0.930834i	−0.461625	+	0.887075i	$0.652733\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.50000	−	6.06218i	−0.149924	−	0.259675i
$546$		0			0
$547$	4.00000	+	6.92820i	0.171028	+	0.296229i	0.938779	−	0.344519i	$-0.111958\pi$
$547$	4.00000	+	6.92820i	0.171028	+	0.296229i	−0.767752	+	0.640747i	$0.778625\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−0.500000	+	4.33013i	−0.0213007	+	0.184470i
$552$		0			0
$553$	−2.00000	+	3.46410i	−0.0850487	+	0.147309i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−20.0000	−	34.6410i	−0.847427	−	1.46779i	−0.883497	−	0.468438i	$-0.844817\pi$
$557$	−20.0000	−	34.6410i	−0.847427	−	1.46779i	0.0360693	−	0.999349i	$-0.488516\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.0000			−0.421450			−0.210725	−	0.977545i	$-0.567582\pi$
$563$	−10.0000			−0.421450			−0.210725	+	0.977545i	$0.567582\pi$
$564$		0			0
$565$	7.00000	−	12.1244i	0.294492	−	0.510075i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	17.0000			0.712677			0.356339	−	0.934357i	$-0.384025\pi$
$569$	17.0000			0.712677			0.356339	+	0.934357i	$0.384025\pi$
$570$		0			0
$571$	−17.0000			−0.711428			−0.355714	−	0.934595i	$-0.615762\pi$
$571$	−17.0000			−0.711428			−0.355714	+	0.934595i	$0.615762\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.00000	−	3.46410i	0.0834058	−	0.144463i
$576$		0			0
$577$	−20.0000			−0.832611			−0.416305	−	0.909225i	$-0.636675\pi$
$577$	−20.0000			−0.832611			−0.416305	+	0.909225i	$0.636675\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	64.0000			2.65517
$582$		0			0
$583$	−9.00000	−	15.5885i	−0.372742	−	0.645608i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−6.00000	+	10.3923i	−0.247647	+	0.428936i	−0.962872	−	0.269957i	$-0.912990\pi$
$587$	−6.00000	+	10.3923i	−0.247647	+	0.428936i	0.715226	+	0.698893i	$0.246324\pi$
$588$		0			0
$589$	−17.5000	−	12.9904i	−0.721075	−	0.535259i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−2.00000	−	3.46410i	−0.0821302	−	0.142254i	0.822035	−	0.569438i	$-0.192839\pi$
$593$	−2.00000	−	3.46410i	−0.0821302	−	0.142254i	−0.904165	+	0.427184i	$0.859506\pi$
$594$		0			0
$595$	−4.00000	−	6.92820i	−0.163984	−	0.284029i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	0.509631	−	0.860393i	$-0.329782\pi$
$599$	−12.0000	−	20.7846i	−0.490307	−	0.849236i	−0.999938	+	0.0111569i	$0.996449\pi$
$600$		0			0
$601$	17.0000			0.693444			0.346722	−	0.937968i	$-0.387295\pi$
$601$	17.0000			0.693444			0.346722	+	0.937968i	$0.387295\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.00000	−	1.73205i	0.0406558	−	0.0704179i
$606$		0			0
$607$	−14.0000			−0.568242			−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000			−0.568242			−0.284121	+	0.958788i	$0.591702\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	18.0000	−	31.1769i	0.728202	−	1.26128i
$612$		0			0
$613$	11.0000	−	19.0526i	0.444286	−	0.769526i	−0.553716	−	0.832705i	$-0.686791\pi$
$613$	11.0000	−	19.0526i	0.444286	−	0.769526i	0.998002	+	0.0631797i	$0.0201241\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−4.00000	−	6.92820i	−0.161034	−	0.278919i	0.774206	−	0.632934i	$-0.218150\pi$
$617$	−4.00000	−	6.92820i	−0.161034	−	0.278919i	−0.935240	+	0.354015i	$0.884816\pi$
$618$		0			0
$619$	−20.0000			−0.803868			−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000			−0.803868			−0.401934	+	0.915669i	$0.631662\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−34.0000	−	58.8897i	−1.36218	−	2.35937i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.00000	+	6.92820i	−0.159490	+	0.276246i
$630$		0			0
$631$	16.5000	+	28.5788i	0.656855	+	1.13771i	0.981425	+	0.191844i	$0.0614468\pi$
$631$	16.5000	+	28.5788i	0.656855	+	1.13771i	−0.324571	+	0.945861i	$0.605220\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.0000			0.714308
$636$		0			0
$637$	−27.0000	−	46.7654i	−1.06978	−	1.85291i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	−0.763367	−	0.645966i	$-0.776455\pi$
$641$	4.50000	−	7.79423i	0.177739	−	0.307854i	0.941106	+	0.338112i	$0.109788\pi$
$642$		0			0
$643$	1.00000	−	1.73205i	0.0394362	−	0.0683054i	−0.845634	−	0.533764i	$-0.820777\pi$
$643$	1.00000	−	1.73205i	0.0394362	−	0.0683054i	0.885070	+	0.465458i	$0.154110\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	30.0000			1.17942			0.589711	−	0.807614i	$-0.299242\pi$
$647$	30.0000			1.17942			0.589711	+	0.807614i	$0.299242\pi$
$648$		0			0
$649$	−1.50000	+	2.59808i	−0.0588802	+	0.101983i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−12.0000			−0.469596			−0.234798	−	0.972044i	$-0.575443\pi$
$653$	−12.0000			−0.469596			−0.234798	+	0.972044i	$0.575443\pi$
$654$		0			0
$655$	−10.0000	−	17.3205i	−0.390732	−	0.676768i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	−0.968052	−	0.250748i	$-0.919323\pi$
$659$	−18.0000	−	31.1769i	−0.701180	−	1.21448i	0.266872	−	0.963732i	$-0.414010\pi$
$660$		0			0
$661$	−20.5000	−	35.5070i	−0.797358	−	1.38106i	−0.921331	−	0.388778i	$-0.872897\pi$
$661$	−20.5000	−	35.5070i	−0.797358	−	1.38106i	0.123974	−	0.992286i	$-0.460436\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	16.0000	−	6.92820i	0.620453	−	0.268664i
$666$		0			0
$667$	−2.00000	+	3.46410i	−0.0774403	+	0.134131i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	10.5000	+	18.1865i	0.405348	+	0.702083i
$672$		0			0
$673$	−28.0000			−1.07932			−0.539660	−	0.841883i	$-0.681447\pi$
$673$	−28.0000			−1.07932			−0.539660	+	0.841883i	$0.681447\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	32.0000			1.22986			0.614930	−	0.788582i	$-0.289184\pi$
$677$	32.0000			1.22986			0.614930	+	0.788582i	$0.289184\pi$
$678$		0			0
$679$	24.0000	−	41.5692i	0.921035	−	1.59528i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−6.00000			−0.229584			−0.114792	−	0.993390i	$-0.536620\pi$
$683$	−6.00000			−0.229584			−0.114792	+	0.993390i	$0.536620\pi$
$684$		0			0
$685$	−12.0000			−0.458496
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.0000	+	31.1769i	−0.685745	+	1.18775i
$690$		0			0
$691$	37.0000			1.40755			0.703773	−	0.710425i	$-0.251497\pi$
$691$	37.0000			1.40755			0.703773	+	0.710425i	$0.251497\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−2.00000	−	3.46410i	−0.0757554	−	0.131212i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−21.0000	+	36.3731i	−0.793159	+	1.37379i	0.130843	+	0.991403i	$0.458232\pi$
$701$	−21.0000	+	36.3731i	−0.793159	+	1.37379i	−0.924002	+	0.382389i	$0.875102\pi$
$702$		0			0
$703$	−14.0000	−	10.3923i	−0.528020	−	0.391953i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.0000	−	38.1051i	−0.827395	−	1.43309i
$708$		0			0
$709$	−2.50000	−	4.33013i	−0.0938895	−	0.162621i	0.815255	−	0.579102i	$-0.196597\pi$
$709$	−2.50000	−	4.33013i	−0.0938895	−	0.162621i	−0.909145	+	0.416481i	$0.863263\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−10.0000	−	17.3205i	−0.374503	−	0.648658i
$714$		0			0
$715$	18.0000			0.673162
$716$		0			0
$717$		0			0
$718$		0			0
$719$	14.5000	−	25.1147i	0.540759	−	0.936622i	−0.458102	−	0.888900i	$-0.651471\pi$
$719$	14.5000	−	25.1147i	0.540759	−	0.936622i	0.998861	−	0.0477220i	$-0.0151961\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	0.500000	−	0.866025i	0.0185695	−	0.0321634i
$726$		0			0
$727$	−13.0000	+	22.5167i	−0.482143	+	0.835097i	−0.999790	−	0.0204978i	$-0.993475\pi$
$727$	−13.0000	+	22.5167i	−0.482143	+	0.835097i	0.517647	+	0.855595i	$0.326808\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−12.0000			−0.443230			−0.221615	−	0.975134i	$-0.571133\pi$
$733$	−12.0000			−0.443230			−0.221615	+	0.975134i	$0.571133\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	21.0000	+	36.3731i	0.773545	+	1.33982i
$738$		0			0
$739$	9.50000	−	16.4545i	0.349463	−	0.605288i	−0.636691	−	0.771119i	$-0.719697\pi$
$739$	9.50000	−	16.4545i	0.349463	−	0.605288i	0.986154	+	0.165831i	$0.0530307\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.0000	−	45.0333i	0.953847	−	1.65211i	0.216864	−	0.976202i	$-0.430417\pi$
$743$	26.0000	−	45.0333i	0.953847	−	1.65211i	0.736984	−	0.675910i	$-0.236249\pi$
$744$		0			0
$745$	−8.50000	−	14.7224i	−0.311416	−	0.539388i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−48.0000			−1.75388
$750$		0			0
$751$	−9.50000	−	16.4545i	−0.346660	−	0.600433i	0.638994	−	0.769212i	$-0.279351\pi$
$751$	−9.50000	−	16.4545i	−0.346660	−	0.600433i	−0.985654	+	0.168779i	$0.946018\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.50000	+	2.59808i	−0.0545906	+	0.0945537i
$756$		0			0
$757$	7.00000	−	12.1244i	0.254419	−	0.440667i	−0.710318	−	0.703881i	$-0.751449\pi$
$757$	7.00000	−	12.1244i	0.254419	−	0.440667i	0.964738	+	0.263213i	$0.0847823\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−54.0000			−1.95750			−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000			−1.95750			−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$	14.0000	−	24.2487i	0.506834	−	0.877862i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6.00000			0.216647
$768$		0			0
$769$	−6.50000	−	11.2583i	−0.234396	−	0.405986i	0.724701	−	0.689063i	$-0.241978\pi$
$769$	−6.50000	−	11.2583i	−0.234396	−	0.405986i	−0.959097	+	0.283078i	$0.908645\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27.0000	−	46.7654i	−0.971123	−	1.68203i	−0.692179	−	0.721726i	$-0.743349\pi$
$773$	−27.0000	−	46.7654i	−0.971123	−	1.68203i	−0.278944	−	0.960307i	$-0.589984\pi$
$774$		0			0
$775$	2.50000	+	4.33013i	0.0898027	+	0.155543i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.00000	−	3.46410i	0.286630	−	0.124114i
$780$		0			0
$781$	22.5000	−	38.9711i	0.805113	−	1.39450i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−9.00000	−	15.5885i	−0.321224	−	0.556376i
$786$		0			0
$787$	38.0000			1.35455			0.677277	−	0.735728i	$-0.263160\pi$
$787$	38.0000			1.35455			0.677277	+	0.735728i	$0.263160\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	56.0000			1.99113
$792$		0			0
$793$	21.0000	−	36.3731i	0.745732	−	1.29165i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22.0000			0.779280			0.389640	−	0.920967i	$-0.372599\pi$
$797$	22.0000			0.779280			0.389640	+	0.920967i	$0.372599\pi$
$798$		0			0
$799$	12.0000			0.424529
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−18.0000	+	31.1769i	−0.635206	+	1.10021i
$804$		0			0
$805$	16.0000			0.563926
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−45.0000			−1.58212			−0.791058	−	0.611741i	$-0.790469\pi$
$809$	−45.0000			−1.58212			−0.791058	+	0.611741i	$0.790469\pi$
$810$		0			0
$811$	6.50000	+	11.2583i	0.228246	+	0.395333i	0.957288	−	0.289135i	$-0.0933677\pi$
$811$	6.50000	+	11.2583i	0.228246	+	0.395333i	−0.729042	+	0.684468i	$0.760034\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−9.00000	+	15.5885i	−0.315256	+	0.546040i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	12.5000	+	21.6506i	0.436253	+	0.755612i	0.997397	−	0.0721058i	$-0.0229719\pi$
$821$	12.5000	+	21.6506i	0.436253	+	0.755612i	−0.561144	+	0.827718i	$0.689639\pi$
$822$		0			0
$823$	−13.0000	−	22.5167i	−0.453152	−	0.784881i	0.545428	−	0.838157i	$-0.316367\pi$
$823$	−13.0000	−	22.5167i	−0.453152	−	0.784881i	−0.998580	+	0.0532760i	$0.983034\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−9.00000	−	15.5885i	−0.312961	−	0.542064i	0.666041	−	0.745915i	$-0.267987\pi$
$827$	−9.00000	−	15.5885i	−0.312961	−	0.542064i	−0.979002	+	0.203851i	$0.934654\pi$
$828$		0			0
$829$	−30.0000			−1.04194			−0.520972	−	0.853574i	$-0.674430\pi$
$829$	−30.0000			−1.04194			−0.520972	+	0.853574i	$0.674430\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9.00000	−	15.5885i	0.311832	−	0.540108i
$834$		0			0
$835$	−8.00000			−0.276851
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$839$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$840$		0			0
$841$	14.0000	−	24.2487i	0.482759	−	0.836162i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−11.5000	−	19.9186i	−0.395612	−	0.685220i
$846$		0			0
$847$	8.00000			0.274883
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−8.00000	−	13.8564i	−0.274236	−	0.474991i
$852$		0			0
$853$	−19.0000	+	32.9090i	−0.650548	+	1.12678i	0.332443	+	0.943123i	$0.392127\pi$
$853$	−19.0000	+	32.9090i	−0.650548	+	1.12678i	−0.982990	+	0.183658i	$0.941206\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	0.500000	+	0.866025i	0.0170598	+	0.0295484i	0.874429	−	0.485153i	$-0.161236\pi$
$859$	0.500000	+	0.866025i	0.0170598	+	0.0295484i	−0.857369	+	0.514701i	$0.827903\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18.0000			0.612727			0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000			0.612727			0.306364	+	0.951915i	$0.400888\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.50000	−	2.59808i	0.0508840	−	0.0881337i
$870$		0			0
$871$	42.0000	−	72.7461i	1.42312	−	2.46491i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−4.00000			−0.135225
$876$		0			0
$877$	−20.0000	+	34.6410i	−0.675352	+	1.16974i	0.301014	+	0.953620i	$0.402675\pi$
$877$	−20.0000	+	34.6410i	−0.675352	+	1.16974i	−0.976366	+	0.216124i	$0.930658\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	7.00000			0.235836			0.117918	−	0.993023i	$-0.462378\pi$
$881$	7.00000			0.235836			0.117918	+	0.993023i	$0.462378\pi$
$882$		0			0
$883$	−4.00000	−	6.92820i	−0.134611	−	0.233153i	0.790838	−	0.612026i	$-0.209645\pi$
$883$	−4.00000	−	6.92820i	−0.134611	−	0.233153i	−0.925449	+	0.378873i	$0.876312\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.0000	+	24.2487i	0.470074	+	0.814192i	0.999414	−	0.0342175i	$-0.0108939\pi$
$887$	14.0000	+	24.2487i	0.470074	+	0.814192i	−0.529340	+	0.848410i	$0.677561\pi$
$888$		0			0
$889$	36.0000	+	62.3538i	1.20740	+	2.09128i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−3.00000	+	25.9808i	−0.100391	+	0.869413i
$894$		0			0
$895$	−7.50000	+	12.9904i	−0.250697	+	0.434221i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−2.50000	−	4.33013i	−0.0833797	−	0.144418i
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.00000			0.0664822
$906$		0			0
$907$	−16.0000	+	27.7128i	−0.531271	+	0.920189i	0.468063	+	0.883695i	$0.344952\pi$
$907$	−16.0000	+	27.7128i	−0.531271	+	0.920189i	−0.999334	+	0.0364935i	$0.988381\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.00000			−0.231920			−0.115960	−	0.993254i	$-0.536994\pi$
$911$	−7.00000			−0.231920			−0.115960	+	0.993254i	$0.536994\pi$
$912$		0			0
$913$	−48.0000			−1.58857
$914$		0			0
$915$		0			0
$916$		0			0
$917$	40.0000	−	69.2820i	1.32092	−	2.28789i
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−90.0000			−2.96239
$924$		0			0
$925$	2.00000	+	3.46410i	0.0657596	+	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−10.5000	+	18.1865i	−0.344494	+	0.596681i	−0.985262	−	0.171054i	$-0.945283\pi$
$929$	−10.5000	+	18.1865i	−0.344494	+	0.596681i	0.640768	+	0.767735i	$0.278616\pi$
$930$		0			0
$931$	31.5000	+	23.3827i	1.03237	+	0.766337i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.00000	+	5.19615i	0.0981105	+	0.169932i
$936$		0			0
$937$	−1.00000	−	1.73205i	−0.0326686	−	0.0565836i	0.849229	−	0.528025i	$-0.177067\pi$
$937$	−1.00000	−	1.73205i	−0.0326686	−	0.0565836i	−0.881897	+	0.471441i	$0.843734\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	17.5000	+	30.3109i	0.570484	+	0.988107i	0.996516	+	0.0833989i	$0.0265776\pi$
$941$	17.5000	+	30.3109i	0.570484	+	0.988107i	−0.426033	+	0.904708i	$0.640089\pi$
$942$		0			0
$943$	8.00000			0.260516
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−5.00000	+	8.66025i	−0.162478	+	0.281420i	−0.935757	−	0.352646i	$-0.885282\pi$
$947$	−5.00000	+	8.66025i	−0.162478	+	0.281420i	0.773279	+	0.634066i	$0.218615\pi$
$948$		0			0
$949$	72.0000			2.33722
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−14.0000	+	24.2487i	−0.453504	+	0.785493i	−0.998601	−	0.0528806i	$-0.983160\pi$
$953$	−14.0000	+	24.2487i	−0.453504	+	0.785493i	0.545096	+	0.838373i	$0.316493\pi$
$954$		0			0
$955$	−6.50000	+	11.2583i	−0.210335	+	0.364311i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−24.0000	−	41.5692i	−0.775000	−	1.34234i
$960$		0			0
$961$	−6.00000			−0.193548
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−8.00000	−	13.8564i	−0.257529	−	0.446054i
$966$		0			0
$967$	6.00000	−	10.3923i	0.192947	−	0.334194i	−0.753279	−	0.657702i	$-0.771529\pi$
$967$	6.00000	−	10.3923i	0.192947	−	0.334194i	0.946226	+	0.323508i	$0.104862\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	30.0000	−	51.9615i	0.962746	−	1.66752i	0.247193	−	0.968966i	$-0.420492\pi$
$971$	30.0000	−	51.9615i	0.962746	−	1.66752i	0.715553	−	0.698558i	$-0.246175\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−12.0000			−0.383914			−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000			−0.383914			−0.191957	+	0.981403i	$0.561483\pi$
$978$		0			0
$979$	25.5000	+	44.1673i	0.814984	+	1.41159i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	0.422027	+	0.906583i	$0.361319\pi$
$983$	−18.0000	+	31.1769i	−0.574111	+	0.994389i	−0.996138	+	0.0878058i	$0.972015\pi$
$984$		0			0
$985$	1.00000	−	1.73205i	0.0318626	−	0.0551877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	0.491698	+	0.870766i	$0.336377\pi$
$991$	−16.0000	+	27.7128i	−0.508257	+	0.880327i	−0.999954	+	0.00956046i	$0.996957\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	23.0000			0.729149
$996$		0			0
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	0.934274	−	0.356555i	$-0.116049\pi$
$997$	5.00000	+	8.66025i	0.158352	+	0.274273i	−0.775923	+	0.630828i	$0.782715\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.t.b.3241.1		2
3.2	odd	2		380.2.i.a.201.1	yes	2
12.11	even	2		1520.2.q.g.961.1		2
15.2	even	4		1900.2.s.b.49.2		4
15.8	even	4		1900.2.s.b.49.1		4
15.14	odd	2		1900.2.i.b.201.1		2
19.7	even	3	inner	3420.2.t.b.1261.1		2
57.8	even	6		7220.2.a.a.1.1		1
57.11	odd	6		7220.2.a.e.1.1		1
57.26	odd	6		380.2.i.a.121.1	✓	2
228.83	even	6		1520.2.q.g.881.1		2
285.83	even	12		1900.2.s.b.349.2		4
285.197	even	12		1900.2.s.b.349.1		4
285.254	odd	6		1900.2.i.b.501.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
380.2.i.a.121.1	✓	2	57.26	odd	6
380.2.i.a.201.1	yes	2	3.2	odd	2
1520.2.q.g.881.1		2	228.83	even	6
1520.2.q.g.961.1		2	12.11	even	2
1900.2.i.b.201.1		2	15.14	odd	2
1900.2.i.b.501.1		2	285.254	odd	6
1900.2.s.b.49.1		4	15.8	even	4
1900.2.s.b.49.2		4	15.2	even	4
1900.2.s.b.349.1		4	285.197	even	12
1900.2.s.b.349.2		4	285.83	even	12
3420.2.t.b.1261.1		2	19.7	even	3	inner
3420.2.t.b.3241.1		2	1.1	even	1	trivial
7220.2.a.a.1.1		1	57.8	even	6
7220.2.a.e.1.1		1	57.11	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:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.t (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +O(q^{10})\)	\(q+(-0.500000 + 0.866025i) q^{5} -4.00000 q^{7} +3.00000 q^{11} +(-3.00000 - 5.19615i) q^{13} +(1.00000 - 1.73205i) q^{17} +(3.50000 + 2.59808i) q^{19} +(2.00000 + 3.46410i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(0.500000 + 0.866025i) q^{29} -5.00000 q^{31} +(2.00000 - 3.46410i) q^{35} -4.00000 q^{37} +(1.00000 - 1.73205i) q^{41} +(3.00000 + 5.19615i) q^{47} +9.00000 q^{49} +(-3.00000 - 5.19615i) q^{53} +(-1.50000 + 2.59808i) q^{55} +(-0.500000 + 0.866025i) q^{59} +(3.50000 + 6.06218i) q^{61} +6.00000 q^{65} +(7.00000 + 12.1244i) q^{67} +(7.50000 - 12.9904i) q^{71} +(-6.00000 + 10.3923i) q^{73} -12.0000 q^{77} +(0.500000 - 0.866025i) q^{79} -16.0000 q^{83} +(1.00000 + 1.73205i) q^{85} +(8.50000 + 14.7224i) q^{89} +(12.0000 + 20.7846i) q^{91} +(-4.00000 + 1.73205i) q^{95} +(-6.00000 + 10.3923i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{5} - 8 q^{7}+O(q^{10}) \) 2 * q - q^5 - 8 * q^7	\( 2 q - q^{5} - 8 q^{7} + 6 q^{11} - 6 q^{13} + 2 q^{17} + 7 q^{19} + 4 q^{23} - q^{25} + q^{29} - 10 q^{31} + 4 q^{35} - 8 q^{37} + 2 q^{41} + 6 q^{47} + 18 q^{49} - 6 q^{53} - 3 q^{55} - q^{59} + 7 q^{61} + 12 q^{65} + 14 q^{67} + 15 q^{71} - 12 q^{73} - 24 q^{77} + q^{79} - 32 q^{83} + 2 q^{85} + 17 q^{89} + 24 q^{91} - 8 q^{95} - 12 q^{97}+O(q^{100}) \) 2 * q - q^5 - 8 * q^7 + 6 * q^11 - 6 * q^13 + 2 * q^17 + 7 * q^19 + 4 * q^23 - q^25 + q^29 - 10 * q^31 + 4 * q^35 - 8 * q^37 + 2 * q^41 + 6 * q^47 + 18 * q^49 - 6 * q^53 - 3 * q^55 - q^59 + 7 * q^61 + 12 * q^65 + 14 * q^67 + 15 * q^71 - 12 * q^73 - 24 * q^77 + q^79 - 32 * q^83 + 2 * q^85 + 17 * q^89 + 24 * q^91 - 8 * q^95 - 12 * q^97

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