LMFDB - Embedded newform 8700.2.g.k.349.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8700,2,Mod(349,8700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8700, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8700.349");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8700.g (of order $2$, degree $1$, not 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:	$69.4698497585$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1740)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			349.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	8700.349
Dual form			8700.2.g.k.349.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} -2.00000i q^{13} +4.00000i q^{17} -6.00000 q^{19} +2.00000 q^{21} -1.00000i q^{23} +1.00000i q^{27} -1.00000 q^{29} +4.00000 q^{31} -3.00000i q^{33} -3.00000i q^{37} -2.00000 q^{39} +3.00000 q^{41} +1.00000i q^{43} -2.00000i q^{47} +3.00000 q^{49} +4.00000 q^{51} -1.00000i q^{53} +6.00000i q^{57} -4.00000 q^{59} +4.00000 q^{61} -2.00000i q^{63} -10.0000i q^{67} -1.00000 q^{69} +6.00000 q^{71} +1.00000i q^{73} +6.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +17.0000i q^{83} +1.00000i q^{87} -6.00000 q^{89} +4.00000 q^{91} -4.00000i q^{93} +13.0000i q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^9	$ 2 q - 2 q^{9} + 6 q^{11} - 12 q^{19} + 4 q^{21} - 2 q^{29} + 8 q^{31} - 4 q^{39} + 6 q^{41} + 6 q^{49} + 8 q^{51} - 8 q^{59} + 8 q^{61} - 2 q^{69} + 12 q^{71} + 8 q^{79} + 2 q^{81} - 12 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^9 + 6 * q^11 - 12 * q^19 + 4 * q^21 - 2 * q^29 + 8 * q^31 - 4 * q^39 + 6 * q^41 + 6 * q^49 + 8 * q^51 - 8 * q^59 + 8 * q^61 - 2 * q^69 + 12 * q^71 + 8 * q^79 + 2 * q^81 - 12 * q^89 + 8 * q^91 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8700\mathbb{Z}\right)^\times$.

$n$	$901$	$4177$	$4351$	$5801$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.00000i	0.755929i	0.925820	+	0.377964i	$0.123376\pi$
$7$	2.00000i	0.755929i	−0.925820	+	0.377964i	$0.876624\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	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
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.00000i	0.970143i	0.874475	+	0.485071i	$0.161206\pi$
$17$	4.00000i	0.970143i	−0.874475	+	0.485071i	$0.838794\pi$
$18$	0	0
$19$	−6.00000	−1.37649	−0.688247	−	0.725476i	$-0.741620\pi$
$19$	−6.00000	−1.37649	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	− 1.00000i	− 0.208514i	−0.994550	−	0.104257i	$-0.966753\pi$
$23$	− 1.00000i	− 0.208514i	0.994550	−	0.104257i	$-0.0332465\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	− 3.00000i	− 0.522233i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 3.00000i	− 0.493197i	−0.969118	−	0.246598i	$-0.920687\pi$
$37$	− 3.00000i	− 0.493197i	0.969118	−	0.246598i	$-0.0793129\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2.00000i	− 0.291730i	−0.989305	−	0.145865i	$-0.953403\pi$
$47$	− 2.00000i	− 0.291730i	0.989305	−	0.145865i	$-0.0465965\pi$
$48$	0	0
$49$	3.00000	0.428571
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	− 1.00000i	− 0.137361i	−0.997639	−	0.0686803i	$-0.978121\pi$
$53$	− 1.00000i	− 0.137361i	0.997639	−	0.0686803i	$-0.0218788\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.00000i	0.794719i
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	− 2.00000i	− 0.251976i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 10.0000i	− 1.22169i	−0.791748	−	0.610847i	$-0.790829\pi$
$67$	− 10.0000i	− 1.22169i	0.791748	−	0.610847i	$-0.209171\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	1.00000i	0.117041i	0.998286	+	0.0585206i	$0.0186383\pi$
$73$	1.00000i	0.117041i	−0.998286	+	0.0585206i	$0.981362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	6.00000i	0.683763i
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	17.0000i	1.86599i	0.359886	+	0.932996i	$0.382816\pi$
$83$	17.0000i	1.86599i	−0.359886	+	0.932996i	$0.617184\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.00000i	0.107211i
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	− 4.00000i	− 0.414781i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	13.0000i	1.31995i	0.751288	+	0.659975i	$0.229433\pi$
$97$	13.0000i	1.31995i	−0.751288	+	0.659975i	$0.770567\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	0	0
$103$	14.0000i	1.37946i	0.724066	+	0.689730i	$0.242271\pi$
$103$	14.0000i	1.37946i	−0.724066	+	0.689730i	$0.757729\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	0	0
$113$	16.0000i	1.50515i	0.658505	+	0.752577i	$0.271189\pi$
$113$	16.0000i	1.50515i	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	2.00000i	0.184900i
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	− 3.00000i	− 0.270501i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 7.00000i	− 0.621150i	−0.950549	−	0.310575i	$-0.899478\pi$
$127$	− 7.00000i	− 0.621150i	0.950549	−	0.310575i	$-0.100522\pi$
$128$	0	0
$129$	1.00000	0.0880451
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	− 12.0000i	− 1.04053i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 12.0000i	− 1.02523i	−0.858619	−	0.512615i	$-0.828677\pi$
$137$	− 12.0000i	− 1.02523i	0.858619	−	0.512615i	$-0.171323\pi$
$138$	0	0
$139$	5.00000	0.424094	0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000	0.424094	0.212047	+	0.977259i	$0.431987\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	0	0
$143$	− 6.00000i	− 0.501745i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 3.00000i	− 0.247436i
$148$	0	0
$149$	−2.00000	−0.163846	−0.0819232	−	0.996639i	$-0.526106\pi$
$149$	−2.00000	−0.163846	−0.0819232	+	0.996639i	$0.526106\pi$
$150$	0	0
$151$	−9.00000	−0.732410	−0.366205	−	0.930534i	$-0.619343\pi$
$151$	−9.00000	−0.732410	−0.366205	+	0.930534i	$0.619343\pi$
$152$	0	0
$153$	− 4.00000i	− 0.323381i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.0000i	1.11732i	0.829396	+	0.558661i	$0.188685\pi$
$157$	14.0000i	1.11732i	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	−1.00000	−0.0793052
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	11.0000i	0.861586i	0.902451	+	0.430793i	$0.141766\pi$
$163$	11.0000i	0.861586i	−0.902451	+	0.430793i	$0.858234\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	16.0000i	1.23812i	0.785345	+	0.619059i	$0.212486\pi$
$167$	16.0000i	1.23812i	−0.785345	+	0.619059i	$0.787514\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	6.00000	0.458831
$172$	0	0
$173$	7.00000i	0.532200i	0.963945	+	0.266100i	$0.0857352\pi$
$173$	7.00000i	0.532200i	−0.963945	+	0.266100i	$0.914265\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	4.00000i	0.300658i
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	− 4.00000i	− 0.295689i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	12.0000i	0.877527i
$188$	0	0
$189$	−2.00000	−0.145479
$190$	0	0
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	6.00000i	0.431889i	0.976406	+	0.215945i	$0.0692831\pi$
$193$	6.00000i	0.431889i	−0.976406	+	0.215945i	$0.930717\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.00000i	0.0712470i	0.999365	+	0.0356235i	$0.0113417\pi$
$197$	1.00000i	0.0712470i	−0.999365	+	0.0356235i	$0.988658\pi$
$198$	0	0
$199$	−13.0000	−0.921546	−0.460773	−	0.887518i	$-0.652428\pi$
$199$	−13.0000	−0.921546	−0.460773	+	0.887518i	$0.652428\pi$
$200$	0	0
$201$	−10.0000	−0.705346
$202$	0	0
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.00000i	0.0695048i
$208$	0	0
$209$	−18.0000	−1.24509
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	− 6.00000i	− 0.411113i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.00000i	0.543075i
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	8.00000	0.538138
$222$	0	0
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.00000i	− 0.199117i	−0.995032	−	0.0995585i	$-0.968257\pi$
$227$	− 3.00000i	− 0.199117i	0.995032	−	0.0995585i	$-0.0317430\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	6.00000	0.394771
$232$	0	0
$233$	1.00000i	0.0655122i	0.999463	+	0.0327561i	$0.0104285\pi$
$233$	1.00000i	0.0655122i	−0.999463	+	0.0327561i	$0.989572\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.00000i	− 0.259828i
$238$	0	0
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	17.0000	1.09507	0.547533	−	0.836784i	$-0.315567\pi$
$241$	17.0000	1.09507	0.547533	+	0.836784i	$0.315567\pi$
$242$	0	0
$243$	− 1.00000i	− 0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	12.0000i	0.763542i
$248$	0	0
$249$	17.0000	1.07733
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	− 3.00000i	− 0.188608i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 13.0000i	− 0.810918i	−0.914113	−	0.405459i	$-0.867112\pi$
$257$	− 13.0000i	− 0.810918i	0.914113	−	0.405459i	$-0.132888\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	1.00000	0.0618984
$262$	0	0
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	6.00000i	0.367194i
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	10.0000	0.607457	0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000	0.607457	0.303728	+	0.952759i	$0.401768\pi$
$272$	0	0
$273$	− 4.00000i	− 0.242091i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 4.00000i	− 0.240337i	−0.992754	−	0.120168i	$-0.961657\pi$
$277$	− 4.00000i	− 0.240337i	0.992754	−	0.120168i	$-0.0383434\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	6.00000i	0.356663i	0.983970	+	0.178331i	$0.0570699\pi$
$283$	6.00000i	0.356663i	−0.983970	+	0.178331i	$0.942930\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.00000i	0.354169i
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.0000	0.762073
$292$	0	0
$293$	− 12.0000i	− 0.701047i	−0.936554	−	0.350524i	$-0.886004\pi$
$293$	− 12.0000i	− 0.701047i	0.936554	−	0.350524i	$-0.113996\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3.00000i	0.174078i
$298$	0	0
$299$	−2.00000	−0.115663
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	− 5.00000i	− 0.287242i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	33.0000i	1.88341i	0.336440	+	0.941705i	$0.390777\pi$
$307$	33.0000i	1.88341i	−0.336440	+	0.941705i	$0.609223\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	−9.00000	−0.510343	−0.255172	−	0.966896i	$-0.582132\pi$
$311$	−9.00000	−0.510343	−0.255172	+	0.966896i	$0.582132\pi$
$312$	0	0
$313$	12.0000i	0.678280i	0.940736	+	0.339140i	$0.110136\pi$
$313$	12.0000i	0.678280i	−0.940736	+	0.339140i	$0.889864\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.0000i	1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$	32.0000i	1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$	0	0
$319$	−3.00000	−0.167968
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 9.00000i	− 0.497701i
$328$	0	0
$329$	4.00000	0.220527
$330$	0	0
$331$	−36.0000	−1.97874	−0.989369	−	0.145424i	$-0.953545\pi$
$331$	−36.0000	−1.97874	−0.989369	+	0.145424i	$0.953545\pi$
$332$	0	0
$333$	3.00000i	0.164399i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	30.0000i	1.63420i	0.576493	+	0.817102i	$0.304421\pi$
$337$	30.0000i	1.63420i	−0.576493	+	0.817102i	$0.695579\pi$
$338$	0	0
$339$	16.0000	0.869001
$340$	0	0
$341$	12.0000	0.649836
$342$	0	0
$343$	20.0000i	1.07990i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.00000i	0.268414i	0.990953	+	0.134207i	$0.0428487\pi$
$347$	5.00000i	0.268414i	−0.990953	+	0.134207i	$0.957151\pi$
$348$	0	0
$349$	29.0000	1.55233	0.776167	−	0.630527i	$-0.217161\pi$
$349$	29.0000	1.55233	0.776167	+	0.630527i	$0.217161\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	0	0
$353$	− 14.0000i	− 0.745145i	−0.928003	−	0.372572i	$-0.878476\pi$
$353$	− 14.0000i	− 0.745145i	0.928003	−	0.372572i	$-0.121524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.00000i	0.423405i
$358$	0	0
$359$	25.0000	1.31945	0.659725	−	0.751507i	$-0.270673\pi$
$359$	25.0000	1.31945	0.659725	+	0.751507i	$0.270673\pi$
$360$	0	0
$361$	17.0000	0.894737
$362$	0	0
$363$	2.00000i	0.104973i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.0000i	0.887393i	0.896177	+	0.443696i	$0.146333\pi$
$367$	17.0000i	0.887393i	−0.896177	+	0.443696i	$0.853667\pi$
$368$	0	0
$369$	−3.00000	−0.156174
$370$	0	0
$371$	2.00000	0.103835
$372$	0	0
$373$	− 12.0000i	− 0.621336i	−0.950518	−	0.310668i	$-0.899447\pi$
$373$	− 12.0000i	− 0.621336i	0.950518	−	0.310668i	$-0.100553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000i	0.103005i
$378$	0	0
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	0	0
$381$	−7.00000	−0.358621
$382$	0	0
$383$	15.0000i	0.766464i	0.923652	+	0.383232i	$0.125189\pi$
$383$	15.0000i	0.766464i	−0.923652	+	0.383232i	$0.874811\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 1.00000i	− 0.0508329i
$388$	0	0
$389$	−11.0000	−0.557722	−0.278861	−	0.960331i	$-0.589957\pi$
$389$	−11.0000	−0.557722	−0.278861	+	0.960331i	$0.589957\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	− 20.0000i	− 1.00887i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000i	1.10415i	0.833795	+	0.552074i	$0.186163\pi$
$397$	22.0000i	1.10415i	−0.833795	+	0.552074i	$0.813837\pi$
$398$	0	0
$399$	−12.0000	−0.600751
$400$	0	0
$401$	12.0000	0.599251	0.299626	−	0.954057i	$-0.403138\pi$
$401$	12.0000	0.599251	0.299626	+	0.954057i	$0.403138\pi$
$402$	0	0
$403$	− 8.00000i	− 0.398508i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 9.00000i	− 0.446113i
$408$	0	0
$409$	24.0000	1.18672	0.593362	−	0.804936i	$-0.297800\pi$
$409$	24.0000	1.18672	0.593362	+	0.804936i	$0.297800\pi$
$410$	0	0
$411$	−12.0000	−0.591916
$412$	0	0
$413$	− 8.00000i	− 0.393654i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 5.00000i	− 0.244851i
$418$	0	0
$419$	6.00000	0.293119	0.146560	−	0.989202i	$-0.453180\pi$
$419$	6.00000	0.293119	0.146560	+	0.989202i	$0.453180\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	2.00000i	0.0972433i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.00000i	0.387147i
$428$	0	0
$429$	−6.00000	−0.289683
$430$	0	0
$431$	34.0000	1.63772	0.818861	−	0.573992i	$-0.194606\pi$
$431$	34.0000	1.63772	0.818861	+	0.573992i	$0.194606\pi$
$432$	0	0
$433$	29.0000i	1.39365i	0.717241	+	0.696826i	$0.245405\pi$
$433$	29.0000i	1.39365i	−0.717241	+	0.696826i	$0.754595\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000i	0.287019i
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	− 16.0000i	− 0.760183i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$	− 16.0000i	− 0.760183i	0.924949	−	0.380091i	$-0.124107\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	2.00000i	0.0945968i
$448$	0	0
$449$	−21.0000	−0.991051	−0.495526	−	0.868593i	$-0.665025\pi$
$449$	−21.0000	−0.991051	−0.495526	+	0.868593i	$0.665025\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	9.00000i	0.422857i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	12.0000i	0.561336i	0.959805	+	0.280668i	$0.0905560\pi$
$457$	12.0000i	0.561336i	−0.959805	+	0.280668i	$0.909444\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	27.0000	1.25752	0.628758	−	0.777601i	$-0.283564\pi$
$461$	27.0000	1.25752	0.628758	+	0.777601i	$0.283564\pi$
$462$	0	0
$463$	14.0000i	0.650635i	0.945605	+	0.325318i	$0.105471\pi$
$463$	14.0000i	0.650635i	−0.945605	+	0.325318i	$0.894529\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 18.0000i	− 0.832941i	−0.909149	−	0.416470i	$-0.863267\pi$
$467$	− 18.0000i	− 0.832941i	0.909149	−	0.416470i	$-0.136733\pi$
$468$	0	0
$469$	20.0000	0.923514
$470$	0	0
$471$	14.0000	0.645086
$472$	0	0
$473$	3.00000i	0.137940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.00000i	0.0457869i
$478$	0	0
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	0	0
$483$	− 2.00000i	− 0.0910032i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	12.0000i	0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$	12.0000i	0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$	0	0
$489$	11.0000	0.497437
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	12.0000i	0.538274i
$498$	0	0
$499$	−24.0000	−1.07439	−0.537194	−	0.843459i	$-0.680516\pi$
$499$	−24.0000	−1.07439	−0.537194	+	0.843459i	$0.680516\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	0	0
$503$	− 10.0000i	− 0.445878i	−0.974832	−	0.222939i	$-0.928435\pi$
$503$	− 10.0000i	− 0.445878i	0.974832	−	0.222939i	$-0.0715651\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 9.00000i	− 0.399704i
$508$	0	0
$509$	42.0000	1.86162	0.930809	−	0.365507i	$-0.119104\pi$
$509$	42.0000	1.86162	0.930809	+	0.365507i	$0.119104\pi$
$510$	0	0
$511$	−2.00000	−0.0884748
$512$	0	0
$513$	− 6.00000i	− 0.264906i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 6.00000i	− 0.263880i
$518$	0	0
$519$	7.00000	0.307266
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.0000i	0.696971i
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	− 6.00000i	− 0.259889i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 6.00000i	− 0.258919i
$538$	0	0
$539$	9.00000	0.387657
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	− 5.00000i	− 0.214571i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 32.0000i	− 1.36822i	−0.729378	−	0.684111i	$-0.760191\pi$
$547$	− 32.0000i	− 1.36822i	0.729378	−	0.684111i	$-0.239809\pi$
$548$	0	0
$549$	−4.00000	−0.170716
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	8.00000i	0.340195i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 23.0000i	− 0.974541i	−0.873251	−	0.487271i	$-0.837993\pi$
$557$	− 23.0000i	− 0.974541i	0.873251	−	0.487271i	$-0.162007\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	0	0
$561$	12.0000	0.506640
$562$	0	0
$563$	− 6.00000i	− 0.252870i	−0.991975	−	0.126435i	$-0.959647\pi$
$563$	− 6.00000i	− 0.252870i	0.991975	−	0.126435i	$-0.0403535\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.00000i	0.0839921i
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−21.0000	−0.878823	−0.439411	−	0.898286i	$-0.644813\pi$
$571$	−21.0000	−0.878823	−0.439411	+	0.898286i	$0.644813\pi$
$572$	0	0
$573$	− 3.00000i	− 0.125327i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 30.0000i	− 1.24892i	−0.781058	−	0.624458i	$-0.785320\pi$
$577$	− 30.0000i	− 1.24892i	0.781058	−	0.624458i	$-0.214680\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	0	0
$581$	−34.0000	−1.41056
$582$	0	0
$583$	− 3.00000i	− 0.124247i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	0	0
$591$	1.00000	0.0411345
$592$	0	0
$593$	− 2.00000i	− 0.0821302i	−0.999156	−	0.0410651i	$-0.986925\pi$
$593$	− 2.00000i	− 0.0821302i	0.999156	−	0.0410651i	$-0.0130751\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.0000i	0.532055i
$598$	0	0
$599$	4.00000	0.163436	0.0817178	−	0.996656i	$-0.473959\pi$
$599$	4.00000	0.163436	0.0817178	+	0.996656i	$0.473959\pi$
$600$	0	0
$601$	0	0		−	1.00000i	$-0.5\pi$
$601$	0	0			1.00000i	$0.5\pi$
$602$	0	0
$603$	10.0000i	0.407231i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 8.00000i	− 0.324710i	−0.986732	−	0.162355i	$-0.948091\pi$
$607$	− 8.00000i	− 0.324710i	0.986732	−	0.162355i	$-0.0519090\pi$
$608$	0	0
$609$	−2.00000	−0.0810441
$610$	0	0
$611$	−4.00000	−0.161823
$612$	0	0
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	−6.00000	−0.241160	−0.120580	−	0.992704i	$-0.538475\pi$
$619$	−6.00000	−0.241160	−0.120580	+	0.992704i	$0.538475\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	− 12.0000i	− 0.480770i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	18.0000i	0.718851i
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	0	0
$633$	10.0000i	0.397464i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	0	0
$639$	−6.00000	−0.237356
$640$	0	0
$641$	13.0000	0.513469	0.256735	−	0.966482i	$-0.417353\pi$
$641$	13.0000	0.513469	0.256735	+	0.966482i	$0.417353\pi$
$642$	0	0
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	7.00000i	0.275198i	0.990488	+	0.137599i	$0.0439386\pi$
$647$	7.00000i	0.275198i	−0.990488	+	0.137599i	$0.956061\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	8.00000	0.313545
$652$	0	0
$653$	6.00000i	0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$	6.00000i	0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.00000i	− 0.0390137i
$658$	0	0
$659$	13.0000	0.506408	0.253204	−	0.967413i	$-0.418516\pi$
$659$	13.0000	0.506408	0.253204	+	0.967413i	$0.418516\pi$
$660$	0	0
$661$	3.00000	0.116686	0.0583432	−	0.998297i	$-0.481418\pi$
$661$	3.00000	0.116686	0.0583432	+	0.998297i	$0.481418\pi$
$662$	0	0
$663$	− 8.00000i	− 0.310694i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.00000i	0.0387202i
$668$	0	0
$669$	−4.00000	−0.154649
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	26.0000i	1.00223i	0.865382	+	0.501113i	$0.167076\pi$
$673$	26.0000i	1.00223i	−0.865382	+	0.501113i	$0.832924\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.00000i	0.307465i	0.988113	+	0.153732i	$0.0491294\pi$
$677$	8.00000i	0.307465i	−0.988113	+	0.153732i	$0.950871\pi$
$678$	0	0
$679$	−26.0000	−0.997788
$680$	0	0
$681$	−3.00000	−0.114960
$682$	0	0
$683$	− 33.0000i	− 1.26271i	−0.775494	−	0.631355i	$-0.782499\pi$
$683$	− 33.0000i	− 1.26271i	0.775494	−	0.631355i	$-0.217501\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 2.00000i	− 0.0763048i
$688$	0	0
$689$	−2.00000	−0.0761939
$690$	0	0
$691$	−12.0000	−0.456502	−0.228251	−	0.973602i	$-0.573301\pi$
$691$	−12.0000	−0.456502	−0.228251	+	0.973602i	$0.573301\pi$
$692$	0	0
$693$	− 6.00000i	− 0.227921i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	0	0
$699$	1.00000	0.0378235
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	18.0000i	0.678883i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.0000i	0.376089i
$708$	0	0
$709$	21.0000	0.788672	0.394336	−	0.918966i	$-0.370975\pi$
$709$	21.0000	0.788672	0.394336	+	0.918966i	$0.370975\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	0	0
$713$	− 4.00000i	− 0.149801i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 6.00000i	− 0.224074i
$718$	0	0
$719$	−40.0000	−1.49175	−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000	−1.49175	−0.745874	+	0.666087i	$0.767968\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	0	0
$723$	− 17.0000i	− 0.632237i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	20.0000i	0.741759i	0.928681	+	0.370879i	$0.120944\pi$
$727$	20.0000i	0.741759i	−0.928681	+	0.370879i	$0.879056\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−4.00000	−0.147945
$732$	0	0
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 30.0000i	− 1.10506i
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	0	0
$743$	− 54.0000i	− 1.98107i	−0.137268	−	0.990534i	$-0.543832\pi$
$743$	− 54.0000i	− 1.98107i	0.137268	−	0.990534i	$-0.456168\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 17.0000i	− 0.621997i
$748$	0	0
$749$	8.00000	0.292314
$750$	0	0
$751$	−20.0000	−0.729810	−0.364905	−	0.931045i	$-0.618899\pi$
$751$	−20.0000	−0.729810	−0.364905	+	0.931045i	$0.618899\pi$
$752$	0	0
$753$	− 12.0000i	− 0.437304i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 33.0000i	− 1.19941i	−0.800223	−	0.599703i	$-0.795286\pi$
$757$	− 33.0000i	− 1.19941i	0.800223	−	0.599703i	$-0.204714\pi$
$758$	0	0
$759$	−3.00000	−0.108893
$760$	0	0
$761$	36.0000	1.30500	0.652499	−	0.757789i	$-0.273720\pi$
$761$	36.0000	1.30500	0.652499	+	0.757789i	$0.273720\pi$
$762$	0	0
$763$	18.0000i	0.651644i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8.00000i	0.288863i
$768$	0	0
$769$	20.0000	0.721218	0.360609	−	0.932717i	$-0.382569\pi$
$769$	20.0000	0.721218	0.360609	+	0.932717i	$0.382569\pi$
$770$	0	0
$771$	−13.0000	−0.468184
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 6.00000i	− 0.215249i
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	18.0000	0.644091
$782$	0	0
$783$	− 1.00000i	− 0.0357371i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 8.00000i	− 0.285169i	−0.989783	−	0.142585i	$-0.954459\pi$
$787$	− 8.00000i	− 0.285169i	0.989783	−	0.142585i	$-0.0455413\pi$
$788$	0	0
$789$	−12.0000	−0.427211
$790$	0	0
$791$	−32.0000	−1.13779
$792$	0	0
$793$	− 8.00000i	− 0.284088i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 42.0000i	− 1.48772i	−0.668338	−	0.743858i	$-0.732994\pi$
$797$	− 42.0000i	− 1.48772i	0.668338	−	0.743858i	$-0.267006\pi$
$798$	0	0
$799$	8.00000	0.283020
$800$	0	0
$801$	6.00000	0.212000
$802$	0	0
$803$	3.00000i	0.105868i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	2.00000i	0.0704033i
$808$	0	0
$809$	21.0000	0.738321	0.369160	−	0.929366i	$-0.379645\pi$
$809$	21.0000	0.738321	0.369160	+	0.929366i	$0.379645\pi$
$810$	0	0
$811$	33.0000	1.15879	0.579393	−	0.815048i	$-0.303290\pi$
$811$	33.0000	1.15879	0.579393	+	0.815048i	$0.303290\pi$
$812$	0	0
$813$	− 10.0000i	− 0.350715i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 6.00000i	− 0.209913i
$818$	0	0
$819$	−4.00000	−0.139771
$820$	0	0
$821$	0	0		−	1.00000i	$-0.5\pi$
$821$	0	0			1.00000i	$0.5\pi$
$822$	0	0
$823$	40.0000i	1.39431i	0.716919	+	0.697156i	$0.245552\pi$
$823$	40.0000i	1.39431i	−0.716919	+	0.697156i	$0.754448\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 18.0000i	− 0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$	− 18.0000i	− 0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$	0	0
$829$	46.0000	1.59765	0.798823	−	0.601566i	$-0.205456\pi$
$829$	46.0000	1.59765	0.798823	+	0.601566i	$0.205456\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	0	0
$833$	12.0000i	0.415775i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.00000i	0.138260i
$838$	0	0
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	16.0000i	0.551069i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 4.00000i	− 0.137442i
$848$	0	0
$849$	6.00000	0.205919
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	− 27.0000i	− 0.924462i	−0.886759	−	0.462231i	$-0.847049\pi$
$853$	− 27.0000i	− 0.924462i	0.886759	−	0.462231i	$-0.152951\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.00000i	0.307434i	0.988115	+	0.153717i	$0.0491244\pi$
$857$	9.00000i	0.307434i	−0.988115	+	0.153717i	$0.950876\pi$
$858$	0	0
$859$	−22.0000	−0.750630	−0.375315	−	0.926897i	$-0.622466\pi$
$859$	−22.0000	−0.750630	−0.375315	+	0.926897i	$0.622466\pi$
$860$	0	0
$861$	6.00000	0.204479
$862$	0	0
$863$	48.0000i	1.63394i	0.576681	+	0.816970i	$0.304348\pi$
$863$	48.0000i	1.63394i	−0.576681	+	0.816970i	$0.695652\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.00000i	− 0.0339618i
$868$	0	0
$869$	12.0000	0.407072
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	− 13.0000i	− 0.439983i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 4.00000i	− 0.135070i	−0.997717	−	0.0675352i	$-0.978487\pi$
$877$	− 4.00000i	− 0.135070i	0.997717	−	0.0675352i	$-0.0215135\pi$
$878$	0	0
$879$	−12.0000	−0.404750
$880$	0	0
$881$	45.0000	1.51609	0.758044	−	0.652203i	$-0.226155\pi$
$881$	45.0000	1.51609	0.758044	+	0.652203i	$0.226155\pi$
$882$	0	0
$883$	16.0000i	0.538443i	0.963078	+	0.269221i	$0.0867663\pi$
$883$	16.0000i	0.538443i	−0.963078	+	0.269221i	$0.913234\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.0000i	0.335767i	0.985807	+	0.167884i	$0.0536933\pi$
$887$	10.0000i	0.335767i	−0.985807	+	0.167884i	$0.946307\pi$
$888$	0	0
$889$	14.0000	0.469545
$890$	0	0
$891$	3.00000	0.100504
$892$	0	0
$893$	12.0000i	0.401565i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	2.00000i	0.0667781i
$898$	0	0
$899$	−4.00000	−0.133407
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	2.00000i	0.0665558i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 3.00000i	− 0.0996134i	−0.998759	−	0.0498067i	$-0.984139\pi$
$907$	− 3.00000i	− 0.0996134i	0.998759	−	0.0498067i	$-0.0158605\pi$
$908$	0	0
$909$	−5.00000	−0.165840
$910$	0	0
$911$	−27.0000	−0.894550	−0.447275	−	0.894397i	$-0.647605\pi$
$911$	−27.0000	−0.894550	−0.447275	+	0.894397i	$0.647605\pi$
$912$	0	0
$913$	51.0000i	1.68785i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.0000i	1.32092i
$918$	0	0
$919$	−44.0000	−1.45143	−0.725713	−	0.687998i	$-0.758490\pi$
$919$	−44.0000	−1.45143	−0.725713	+	0.687998i	$0.758490\pi$
$920$	0	0
$921$	33.0000	1.08739
$922$	0	0
$923$	− 12.0000i	− 0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 14.0000i	− 0.459820i
$928$	0	0
$929$	8.00000	0.262471	0.131236	−	0.991351i	$-0.458106\pi$
$929$	8.00000	0.262471	0.131236	+	0.991351i	$0.458106\pi$
$930$	0	0
$931$	−18.0000	−0.589926
$932$	0	0
$933$	9.00000i	0.294647i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 28.0000i	− 0.914720i	−0.889282	−	0.457360i	$-0.848795\pi$
$937$	− 28.0000i	− 0.914720i	0.889282	−	0.457360i	$-0.151205\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	50.0000	1.62995	0.814977	−	0.579494i	$-0.196750\pi$
$941$	50.0000	1.62995	0.814977	+	0.579494i	$0.196750\pi$
$942$	0	0
$943$	− 3.00000i	− 0.0976934i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	14.0000i	0.454939i	0.973785	+	0.227469i	$0.0730452\pi$
$947$	14.0000i	0.454939i	−0.973785	+	0.227469i	$0.926955\pi$
$948$	0	0
$949$	2.00000	0.0649227
$950$	0	0
$951$	32.0000	1.03767
$952$	0	0
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.00000i	0.0969762i
$958$	0	0
$959$	24.0000	0.775000
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	4.00000i	0.128898i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.0000i	0.932577i	0.884633	+	0.466289i	$0.154409\pi$
$967$	29.0000i	0.932577i	−0.884633	+	0.466289i	$0.845591\pi$
$968$	0	0
$969$	−24.0000	−0.770991
$970$	0	0
$971$	−9.00000	−0.288824	−0.144412	−	0.989518i	$-0.546129\pi$
$971$	−9.00000	−0.288824	−0.144412	+	0.989518i	$0.546129\pi$
$972$	0	0
$973$	10.0000i	0.320585i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 23.0000i	− 0.735835i	−0.929858	−	0.367918i	$-0.880071\pi$
$977$	− 23.0000i	− 0.735835i	0.929858	−	0.367918i	$-0.119929\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	−9.00000	−0.287348
$982$	0	0
$983$	10.0000i	0.318950i	0.987202	+	0.159475i	$0.0509802\pi$
$983$	10.0000i	0.318950i	−0.987202	+	0.159475i	$0.949020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 4.00000i	− 0.127321i
$988$	0	0
$989$	1.00000	0.0317982
$990$	0	0
$991$	−51.0000	−1.62007	−0.810034	−	0.586383i	$-0.800552\pi$
$991$	−51.0000	−1.62007	−0.810034	+	0.586383i	$0.800552\pi$
$992$	0	0
$993$	36.0000i	1.14243i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 7.00000i	− 0.221692i	−0.993838	−	0.110846i	$-0.964644\pi$
$997$	− 7.00000i	− 0.221692i	0.993838	−	0.110846i	$-0.0353561\pi$
$998$	0	0
$999$	3.00000	0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8700.2.g.k.349.1		2
5.2	odd	4		1740.2.a.b.1.1	✓	1
5.3	odd	4		8700.2.a.r.1.1		1
5.4	even	2	inner	8700.2.g.k.349.2		2
15.2	even	4		5220.2.a.c.1.1		1
20.7	even	4		6960.2.a.bl.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1740.2.a.b.1.1	✓	1	5.2	odd	4
5220.2.a.c.1.1		1	15.2	even	4
6960.2.a.bl.1.1		1	20.7	even	4
8700.2.a.r.1.1		1	5.3	odd	4
8700.2.g.k.349.1		2	1.1	even	1	trivial
8700.2.g.k.349.2		2	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8700 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8700.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.00000i q^{7} -1.00000 q^{9} +3.00000 q^{11} -2.00000i q^{13} +4.00000i q^{17} -6.00000 q^{19} +2.00000 q^{21} -1.00000i q^{23} +1.00000i q^{27} -1.00000 q^{29} +4.00000 q^{31} -3.00000i q^{33} -3.00000i q^{37} -2.00000 q^{39} +3.00000 q^{41} +1.00000i q^{43} -2.00000i q^{47} +3.00000 q^{49} +4.00000 q^{51} -1.00000i q^{53} +6.00000i q^{57} -4.00000 q^{59} +4.00000 q^{61} -2.00000i q^{63} -10.0000i q^{67} -1.00000 q^{69} +6.00000 q^{71} +1.00000i q^{73} +6.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} +17.0000i q^{83} +1.00000i q^{87} -6.00000 q^{89} +4.00000 q^{91} -4.00000i q^{93} +13.0000i q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^9	\( 2 q - 2 q^{9} + 6 q^{11} - 12 q^{19} + 4 q^{21} - 2 q^{29} + 8 q^{31} - 4 q^{39} + 6 q^{41} + 6 q^{49} + 8 q^{51} - 8 q^{59} + 8 q^{61} - 2 q^{69} + 12 q^{71} + 8 q^{79} + 2 q^{81} - 12 q^{89} + 8 q^{91} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^9 + 6 * q^11 - 12 * q^19 + 4 * q^21 - 2 * q^29 + 8 * q^31 - 4 * q^39 + 6 * q^41 + 6 * q^49 + 8 * q^51 - 8 * q^59 + 8 * q^61 - 2 * q^69 + 12 * q^71 + 8 * q^79 + 2 * q^81 - 12 * q^89 + 8 * q^91 - 6 * q^99

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