LMFDB - Embedded newform 4600.2.e.h.4049.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, 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("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.e (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:	$36.7311849298$
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 920)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	4600.4049
Dual form			4600.2.e.h.4049.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +2.00000 q^{9} +2.00000 q^{11} +5.00000i q^{13} -4.00000i q^{17} +2.00000 q^{19} +1.00000i q^{23} +5.00000i q^{27} +3.00000 q^{29} +7.00000 q^{31} +2.00000i q^{33} -2.00000i q^{37} -5.00000 q^{39} -9.00000 q^{41} +4.00000i q^{43} -9.00000i q^{47} +7.00000 q^{49} +4.00000 q^{51} +6.00000i q^{53} +2.00000i q^{57} +2.00000 q^{61} -2.00000i q^{67} -1.00000 q^{69} -1.00000 q^{71} -1.00000i q^{73} +14.0000 q^{79} +1.00000 q^{81} +3.00000i q^{87} -16.0000 q^{89} +7.00000i q^{93} -4.00000i q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{9} + 4 q^{11} + 4 q^{19} + 6 q^{29} + 14 q^{31} - 10 q^{39} - 18 q^{41} + 14 q^{49} + 8 q^{51} + 4 q^{61} - 2 q^{69} - 2 q^{71} + 28 q^{79} + 2 q^{81} - 32 q^{89} + 8 q^{99}+O(q^{100}) $ 2 * q + 4 * q^9 + 4 * q^11 + 4 * q^19 + 6 * q^29 + 14 * q^31 - 10 * q^39 - 18 * q^41 + 14 * q^49 + 8 * q^51 + 4 * q^61 - 2 * q^69 - 2 * q^71 + 28 * q^79 + 2 * q^81 - 32 * q^89 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$.

$n$	$1151$	$1201$	$2301$	$2577$
$\chi(n)$	$1$	$1$	$1$	$-1$

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	0.957427	+	0.288675i	$0.0932147\pi$
$3$	1.00000i	0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	5.00000i	1.38675i	0.720577	+	0.693375i	$0.243877\pi$
$13$	5.00000i	1.38675i	−0.720577	+	0.693375i	$0.756123\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.00000i	− 0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$	− 4.00000i	− 0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000i	0.208514i
$23$	1.00000i	0.208514i
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.00000i	0.962250i
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	2.00000i	0.348155i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 2.00000i	− 0.328798i	−0.986394	−	0.164399i	$-0.947432\pi$
$37$	− 2.00000i	− 0.328798i	0.986394	−	0.164399i	$-0.0525685\pi$
$38$	0	0
$39$	−5.00000	−0.800641
$40$	0	0
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.00000i	− 1.31278i	−0.754420	−	0.656392i	$-0.772082\pi$
$47$	− 9.00000i	− 1.31278i	0.754420	−	0.656392i	$-0.227918\pi$
$48$	0	0
$49$	7.00000	1.00000
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.00000i	0.264906i
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.00000i	− 0.244339i	−0.992509	−	0.122169i	$-0.961015\pi$
$67$	− 2.00000i	− 0.244339i	0.992509	−	0.122169i	$-0.0389851\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−1.00000	−0.118678	−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000	−0.118678	−0.0593391	+	0.998238i	$0.518899\pi$
$72$	0	0
$73$	− 1.00000i	− 0.117041i	−0.998286	−	0.0585206i	$-0.981362\pi$
$73$	− 1.00000i	− 0.117041i	0.998286	−	0.0585206i	$-0.0186383\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0	0	1.00000			$0$
$83$	0	0	−1.00000			$\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.00000i	0.321634i
$88$	0	0
$89$	−16.0000	−1.69600	−0.847998	−	0.529999i	$-0.822192\pi$
$89$	−16.0000	−1.69600	−0.847998	+	0.529999i	$0.822192\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.00000i	0.725866i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.00000i	− 0.406138i	−0.979164	−	0.203069i	$-0.934908\pi$
$97$	− 4.00000i	− 0.406138i	0.979164	−	0.203069i	$-0.0650917\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	0	0	1.00000			$0$
$103$	0	0	−1.00000			$\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	10.0000i	0.966736i	0.875417	+	0.483368i	$0.160587\pi$
$107$	10.0000i	0.966736i	−0.875417	+	0.483368i	$0.839413\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	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
$117$	10.0000i	0.924500i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	− 9.00000i	− 0.811503i
$124$	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
$129$	−4.00000	−0.352180
$130$	0	0
$131$	15.0000	1.31056	0.655278	−	0.755388i	$-0.272551\pi$
$131$	15.0000	1.31056	0.655278	+	0.755388i	$0.272551\pi$
$132$	0	0
$133$	0	0
$134$	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
$139$	−9.00000	−0.763370	−0.381685	−	0.924292i	$-0.624656\pi$
$139$	−9.00000	−0.763370	−0.381685	+	0.924292i	$0.624656\pi$
$140$	0	0
$141$	9.00000	0.757937
$142$	0	0
$143$	10.0000i	0.836242i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.00000i	0.577350i
$148$	0	0
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	0	0
$151$	15.0000	1.22068	0.610341	−	0.792139i	$-0.291032\pi$
$151$	15.0000	1.22068	0.610341	+	0.792139i	$0.291032\pi$
$152$	0	0
$153$	− 8.00000i	− 0.646762i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 4.00000i	− 0.319235i	−0.987179	−	0.159617i	$-0.948974\pi$
$157$	− 4.00000i	− 0.319235i	0.987179	−	0.159617i	$-0.0510260\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	23.0000i	1.80150i	0.434339	+	0.900750i	$0.356982\pi$
$163$	23.0000i	1.80150i	−0.434339	+	0.900750i	$0.643018\pi$
$164$	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
$169$	−12.0000	−0.923077
$170$	0	0
$171$	4.00000	0.305888
$172$	0	0
$173$	− 14.0000i	− 1.06440i	−0.846619	−	0.532200i	$-0.821365\pi$
$173$	− 14.0000i	− 1.06440i	0.846619	−	0.532200i	$-0.178635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.00000	−0.672692	−0.336346	−	0.941739i	$-0.609191\pi$
$179$	−9.00000	−0.672692	−0.336346	+	0.941739i	$0.609191\pi$
$180$	0	0
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 8.00000i	− 0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.0000	1.01300	0.506502	−	0.862239i	$-0.330938\pi$
$191$	14.0000	1.01300	0.506502	+	0.862239i	$0.330938\pi$
$192$	0	0
$193$	− 5.00000i	− 0.359908i	−0.983675	−	0.179954i	$-0.942405\pi$
$193$	− 5.00000i	− 0.359908i	0.983675	−	0.179954i	$-0.0575949\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.0000i	1.78118i	0.454811	+	0.890588i	$0.349707\pi$
$197$	25.0000i	1.78118i	−0.454811	+	0.890588i	$0.650293\pi$
$198$	0	0
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	0	0
$201$	2.00000	0.141069
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	2.00000i	0.139010i
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	0	0
$213$	− 1.00000i	− 0.0685189i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1.00000	0.0675737
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	8.00000i	0.535720i	0.963458	+	0.267860i	$0.0863164\pi$
$223$	8.00000i	0.535720i	−0.963458	+	0.267860i	$0.913684\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 4.00000i	− 0.265489i	−0.991150	−	0.132745i	$-0.957621\pi$
$227$	− 4.00000i	− 0.265489i	0.991150	−	0.132745i	$-0.0423790\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	25.0000i	1.63780i	0.573933	+	0.818902i	$0.305417\pi$
$233$	25.0000i	1.63780i	−0.573933	+	0.818902i	$0.694583\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.0000i	0.909398i
$238$	0	0
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	16.0000i	1.02640i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.0000i	0.636285i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.00000	0.504956	0.252478	−	0.967603i	$-0.418755\pi$
$251$	8.00000	0.504956	0.252478	+	0.967603i	$0.418755\pi$
$252$	0	0
$253$	2.00000i	0.125739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.0000i	1.18519i	0.805502	+	0.592594i	$0.201896\pi$
$257$	19.0000i	1.18519i	−0.805502	+	0.592594i	$0.798104\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	− 14.0000i	− 0.863277i	−0.902047	−	0.431638i	$-0.857936\pi$
$263$	− 14.0000i	− 0.863277i	0.902047	−	0.431638i	$-0.142064\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 16.0000i	− 0.979184i
$268$	0	0
$269$	7.00000	0.426798	0.213399	−	0.976965i	$-0.431547\pi$
$269$	7.00000	0.426798	0.213399	+	0.976965i	$0.431547\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	13.0000i	0.781094i	0.920583	+	0.390547i	$0.127714\pi$
$277$	13.0000i	0.781094i	−0.920583	+	0.390547i	$0.872286\pi$
$278$	0	0
$279$	14.0000	0.838158
$280$	0	0
$281$	−32.0000	−1.90896	−0.954480	−	0.298275i	$-0.903589\pi$
$281$	−32.0000	−1.90896	−0.954480	+	0.298275i	$0.903589\pi$
$282$	0	0
$283$	22.0000i	1.30776i	0.756596	+	0.653882i	$0.226861\pi$
$283$	22.0000i	1.30776i	−0.756596	+	0.653882i	$0.773139\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	4.00000	0.234484
$292$	0	0
$293$	− 16.0000i	− 0.934730i	−0.884064	−	0.467365i	$-0.845203\pi$
$293$	− 16.0000i	− 0.934730i	0.884064	−	0.467365i	$-0.154797\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	10.0000i	0.580259i
$298$	0	0
$299$	−5.00000	−0.289157
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.0000i	0.574485i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.0000i	0.913168i	0.889680	+	0.456584i	$0.150927\pi$
$307$	16.0000i	0.913168i	−0.889680	+	0.456584i	$0.849073\pi$
$308$	0	0
$309$	0	0
$310$	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
$313$	− 8.00000i	− 0.452187i	−0.974106	−	0.226093i	$-0.927405\pi$
$313$	− 8.00000i	− 0.452187i	0.974106	−	0.226093i	$-0.0725954\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000i	0.112331i	0.998421	+	0.0561656i	$0.0178875\pi$
$317$	2.00000i	0.112331i	−0.998421	+	0.0561656i	$0.982113\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	−10.0000	−0.558146
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	4.00000i	0.221201i
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.0000	−1.04433	−0.522167	−	0.852843i	$-0.674876\pi$
$331$	−19.0000	−1.04433	−0.522167	+	0.852843i	$0.674876\pi$
$332$	0	0
$333$	− 4.00000i	− 0.219199i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.00000i	0.435788i	0.975972	+	0.217894i	$0.0699187\pi$
$337$	8.00000i	0.435788i	−0.975972	+	0.217894i	$0.930081\pi$
$338$	0	0
$339$	−16.0000	−0.869001
$340$	0	0
$341$	14.0000	0.758143
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 28.0000i	− 1.50312i	−0.659665	−	0.751559i	$-0.729302\pi$
$347$	− 28.0000i	− 1.50312i	0.659665	−	0.751559i	$-0.270698\pi$
$348$	0	0
$349$	−17.0000	−0.909989	−0.454995	−	0.890494i	$-0.650359\pi$
$349$	−17.0000	−0.909989	−0.454995	+	0.890494i	$0.650359\pi$
$350$	0	0
$351$	−25.0000	−1.33440
$352$	0	0
$353$	27.0000i	1.43706i	0.695493	+	0.718532i	$0.255186\pi$
$353$	27.0000i	1.43706i	−0.695493	+	0.718532i	$0.744814\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	− 7.00000i	− 0.367405i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 24.0000i	− 1.25279i	−0.779506	−	0.626395i	$-0.784530\pi$
$367$	− 24.0000i	− 1.25279i	0.779506	−	0.626395i	$-0.215470\pi$
$368$	0	0
$369$	−18.0000	−0.937043
$370$	0	0
$371$	0	0
$372$	0	0
$373$	− 14.0000i	− 0.724893i	−0.932005	−	0.362446i	$-0.881942\pi$
$373$	− 14.0000i	− 0.724893i	0.932005	−	0.362446i	$-0.118058\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.0000i	0.772539i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	7.00000	0.358621
$382$	0	0
$383$	− 16.0000i	− 0.817562i	−0.912633	−	0.408781i	$-0.865954\pi$
$383$	− 16.0000i	− 0.817562i	0.912633	−	0.408781i	$-0.134046\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000i	0.406663i
$388$	0	0
$389$	−8.00000	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−8.00000	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	0	0
$393$	15.0000i	0.756650i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3.00000i	0.150566i	0.997162	+	0.0752828i	$0.0239860\pi$
$397$	3.00000i	0.150566i	−0.997162	+	0.0752828i	$0.976014\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	35.0000i	1.74347i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 4.00000i	− 0.198273i
$408$	0	0
$409$	35.0000	1.73064	0.865319	−	0.501221i	$-0.167116\pi$
$409$	35.0000	1.73064	0.865319	+	0.501221i	$0.167116\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 9.00000i	− 0.440732i
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−8.00000	−0.389896	−0.194948	−	0.980814i	$-0.562454\pi$
$421$	−8.00000	−0.389896	−0.194948	+	0.980814i	$0.562454\pi$
$422$	0	0
$423$	− 18.0000i	− 0.875190i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−10.0000	−0.482805
$430$	0	0
$431$	28.0000	1.34871	0.674356	−	0.738406i	$-0.264421\pi$
$431$	28.0000	1.34871	0.674356	+	0.738406i	$0.264421\pi$
$432$	0	0
$433$	6.00000i	0.288342i	0.989553	+	0.144171i	$0.0460515\pi$
$433$	6.00000i	0.288342i	−0.989553	+	0.144171i	$0.953949\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00000i	0.0956730i
$438$	0	0
$439$	15.0000	0.715911	0.357955	−	0.933739i	$-0.383474\pi$
$439$	15.0000	0.715911	0.357955	+	0.933739i	$0.383474\pi$
$440$	0	0
$441$	14.0000	0.666667
$442$	0	0
$443$	25.0000i	1.18779i	0.804544	+	0.593893i	$0.202410\pi$
$443$	25.0000i	1.18779i	−0.804544	+	0.593893i	$0.797590\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 14.0000i	− 0.662177i
$448$	0	0
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	0	0
$453$	15.0000i	0.704761i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	32.0000i	1.49690i	0.663193	+	0.748448i	$0.269201\pi$
$457$	32.0000i	1.49690i	−0.663193	+	0.748448i	$0.730799\pi$
$458$	0	0
$459$	20.0000	0.933520
$460$	0	0
$461$	−7.00000	−0.326023	−0.163011	−	0.986624i	$-0.552121\pi$
$461$	−7.00000	−0.326023	−0.163011	+	0.986624i	$0.552121\pi$
$462$	0	0
$463$	24.0000i	1.11537i	0.830051	+	0.557687i	$0.188311\pi$
$463$	24.0000i	1.11537i	−0.830051	+	0.557687i	$0.811689\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 6.00000i	− 0.277647i	−0.990317	−	0.138823i	$-0.955668\pi$
$467$	− 6.00000i	− 0.277647i	0.990317	−	0.138823i	$-0.0443321\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	4.00000	0.184310
$472$	0	0
$473$	8.00000i	0.367840i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	12.0000i	0.549442i
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	10.0000	0.455961
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 23.0000i	− 1.04223i	−0.853487	−	0.521115i	$-0.825516\pi$
$487$	− 23.0000i	− 1.04223i	0.853487	−	0.521115i	$-0.174484\pi$
$488$	0	0
$489$	−23.0000	−1.04010
$490$	0	0
$491$	1.00000	0.0451294	0.0225647	−	0.999745i	$-0.492817\pi$
$491$	1.00000	0.0451294	0.0225647	+	0.999745i	$0.492817\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−3.00000	−0.134298	−0.0671492	−	0.997743i	$-0.521390\pi$
$499$	−3.00000	−0.134298	−0.0671492	+	0.997743i	$0.521390\pi$
$500$	0	0
$501$	−16.0000	−0.714827
$502$	0	0
$503$	− 22.0000i	− 0.980932i	−0.871460	−	0.490466i	$-0.836827\pi$
$503$	− 22.0000i	− 0.980932i	0.871460	−	0.490466i	$-0.163173\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 12.0000i	− 0.532939i
$508$	0	0
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	10.0000i	0.441511i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 18.0000i	− 0.791639i
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	− 28.0000i	− 1.22435i	−0.790721	−	0.612177i	$-0.790294\pi$
$523$	− 28.0000i	− 1.22435i	0.790721	−	0.612177i	$-0.209706\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 28.0000i	− 1.21970i
$528$	0	0
$529$	−1.00000	−0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 45.0000i	− 1.94917i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 9.00000i	− 0.388379i
$538$	0	0
$539$	14.0000	0.603023
$540$	0	0
$541$	−35.0000	−1.50477	−0.752384	−	0.658725i	$-0.771096\pi$
$541$	−35.0000	−1.50477	−0.752384	+	0.658725i	$0.771096\pi$
$542$	0	0
$543$	− 2.00000i	− 0.0858282i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 7.00000i	− 0.299298i	−0.988739	−	0.149649i	$-0.952186\pi$
$547$	− 7.00000i	− 0.299298i	0.988739	−	0.149649i	$-0.0478144\pi$
$548$	0	0
$549$	4.00000	0.170716
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 24.0000i	− 1.01691i	−0.861088	−	0.508456i	$-0.830216\pi$
$557$	− 24.0000i	− 1.01691i	0.861088	−	0.508456i	$-0.169784\pi$
$558$	0	0
$559$	−20.0000	−0.845910
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	− 36.0000i	− 1.51722i	−0.651546	−	0.758610i	$-0.725879\pi$
$563$	− 36.0000i	− 1.51722i	0.651546	−	0.758610i	$-0.274121\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2.00000	−0.0838444	−0.0419222	−	0.999121i	$-0.513348\pi$
$569$	−2.00000	−0.0838444	−0.0419222	+	0.999121i	$0.513348\pi$
$570$	0	0
$571$	4.00000	0.167395	0.0836974	−	0.996491i	$-0.473327\pi$
$571$	4.00000	0.167395	0.0836974	+	0.996491i	$0.473327\pi$
$572$	0	0
$573$	14.0000i	0.584858i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 11.0000i	− 0.457936i	−0.973434	−	0.228968i	$-0.926465\pi$
$577$	− 11.0000i	− 0.457936i	0.973434	−	0.228968i	$-0.0735351\pi$
$578$	0	0
$579$	5.00000	0.207793
$580$	0	0
$581$	0	0
$582$	0	0
$583$	12.0000i	0.496989i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 9.00000i	− 0.371470i	−0.982600	−	0.185735i	$-0.940533\pi$
$587$	− 9.00000i	− 0.371470i	0.982600	−	0.185735i	$-0.0594666\pi$
$588$	0	0
$589$	14.0000	0.576860
$590$	0	0
$591$	−25.0000	−1.02836
$592$	0	0
$593$	2.00000i	0.0821302i	0.999156	+	0.0410651i	$0.0130751\pi$
$593$	2.00000i	0.0821302i	−0.999156	+	0.0410651i	$0.986925\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	10.0000i	0.409273i
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	− 4.00000i	− 0.162893i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 20.0000i	− 0.811775i	−0.913923	−	0.405887i	$-0.866962\pi$
$607$	− 20.0000i	− 0.811775i	0.913923	−	0.405887i	$-0.133038\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	45.0000	1.82051
$612$	0	0
$613$	− 12.0000i	− 0.484675i	−0.970192	−	0.242338i	$-0.922086\pi$
$613$	− 12.0000i	− 0.484675i	0.970192	−	0.242338i	$-0.0779142\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 26.0000i	− 1.04672i	−0.852111	−	0.523360i	$-0.824678\pi$
$617$	− 26.0000i	− 1.04672i	0.852111	−	0.523360i	$-0.175322\pi$
$618$	0	0
$619$	48.0000	1.92928	0.964641	−	0.263566i	$-0.0848986\pi$
$619$	48.0000	1.92928	0.964641	+	0.263566i	$0.0848986\pi$
$620$	0	0
$621$	−5.00000	−0.200643
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	0	0
$633$	28.0000i	1.11290i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	35.0000i	1.38675i
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	−24.0000	−0.947943	−0.473972	−	0.880540i	$-0.657180\pi$
$641$	−24.0000	−0.947943	−0.473972	+	0.880540i	$0.657180\pi$
$642$	0	0
$643$	34.0000i	1.34083i	0.741987	+	0.670415i	$0.233884\pi$
$643$	34.0000i	1.34083i	−0.741987	+	0.670415i	$0.766116\pi$
$644$	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
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 19.0000i	− 0.743527i	−0.928327	−	0.371764i	$-0.878753\pi$
$653$	− 19.0000i	− 0.743527i	0.928327	−	0.371764i	$-0.121247\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 2.00000i	− 0.0780274i
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	0	0
$663$	20.0000i	0.776736i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.00000i	0.116160i
$668$	0	0
$669$	−8.00000	−0.309298
$670$	0	0
$671$	4.00000	0.154418
$672$	0	0
$673$	9.00000i	0.346925i	0.984841	+	0.173462i	$0.0554955\pi$
$673$	9.00000i	0.346925i	−0.984841	+	0.173462i	$0.944505\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 30.0000i	− 1.15299i	−0.817099	−	0.576497i	$-0.804419\pi$
$677$	− 30.0000i	− 1.15299i	0.817099	−	0.576497i	$-0.195581\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	4.00000	0.153280
$682$	0	0
$683$	− 5.00000i	− 0.191320i	−0.995414	−	0.0956598i	$-0.969504\pi$
$683$	− 5.00000i	− 0.191320i	0.995414	−	0.0956598i	$-0.0304961\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−30.0000	−1.14291
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.0000i	1.36360i
$698$	0	0
$699$	−25.0000	−0.945587
$700$	0	0
$701$	−16.0000	−0.604312	−0.302156	−	0.953259i	$-0.597706\pi$
$701$	−16.0000	−0.604312	−0.302156	+	0.953259i	$0.597706\pi$
$702$	0	0
$703$	− 4.00000i	− 0.150863i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.0000	0.450669	0.225335	−	0.974281i	$-0.427652\pi$
$709$	12.0000	0.450669	0.225335	+	0.974281i	$0.427652\pi$
$710$	0	0
$711$	28.0000	1.05008
$712$	0	0
$713$	7.00000i	0.262152i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	9.00000i	0.336111i
$718$	0	0
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 10.0000i	− 0.371904i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.0000i	1.70605i	0.521874	+	0.853023i	$0.325233\pi$
$727$	46.0000i	1.70605i	−0.521874	+	0.853023i	$0.674767\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	16.0000	0.591781
$732$	0	0
$733$	− 52.0000i	− 1.92066i	−0.278859	−	0.960332i	$-0.589956\pi$
$733$	− 52.0000i	− 1.92066i	0.278859	−	0.960332i	$-0.410044\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.00000i	− 0.147342i
$738$	0	0
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	0	0
$741$	−10.0000	−0.367359
$742$	0	0
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	0	0
$753$	8.00000i	0.291536i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	26.0000i	0.944986i	0.881334	+	0.472493i	$0.156646\pi$
$757$	26.0000i	0.944986i	−0.881334	+	0.472493i	$0.843354\pi$
$758$	0	0
$759$	−2.00000	−0.0725954
$760$	0	0
$761$	13.0000	0.471250	0.235625	−	0.971844i	$-0.424286\pi$
$761$	13.0000	0.471250	0.235625	+	0.971844i	$0.424286\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	0	0
$771$	−19.0000	−0.684268
$772$	0	0
$773$	− 24.0000i	− 0.863220i	−0.902060	−	0.431610i	$-0.857946\pi$
$773$	− 24.0000i	− 0.863220i	0.902060	−	0.431610i	$-0.142054\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−18.0000	−0.644917
$780$	0	0
$781$	−2.00000	−0.0715656
$782$	0	0
$783$	15.0000i	0.536056i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.00000i	− 0.142585i	−0.997455	−	0.0712923i	$-0.977288\pi$
$787$	− 4.00000i	− 0.142585i	0.997455	−	0.0712923i	$-0.0227123\pi$
$788$	0	0
$789$	14.0000	0.498413
$790$	0	0
$791$	0	0
$792$	0	0
$793$	10.0000i	0.355110i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000i	0.637593i	0.947823	+	0.318796i	$0.103279\pi$
$797$	18.0000i	0.637593i	−0.947823	+	0.318796i	$0.896721\pi$
$798$	0	0
$799$	−36.0000	−1.27359
$800$	0	0
$801$	−32.0000	−1.13066
$802$	0	0
$803$	− 2.00000i	− 0.0705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.00000i	0.246412i
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−29.0000	−1.01833	−0.509164	−	0.860670i	$-0.670045\pi$
$811$	−29.0000	−1.01833	−0.509164	+	0.860670i	$0.670045\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.00000i	0.279885i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	− 3.00000i	− 0.104573i	−0.998632	−	0.0522867i	$-0.983349\pi$
$823$	− 3.00000i	− 0.104573i	0.998632	−	0.0522867i	$-0.0166510\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 8.00000i	− 0.278187i	−0.990279	−	0.139094i	$-0.955581\pi$
$827$	− 8.00000i	− 0.278187i	0.990279	−	0.139094i	$-0.0444189\pi$
$828$	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
$831$	−13.0000	−0.450965
$832$	0	0
$833$	− 28.0000i	− 0.970143i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	35.0000i	1.20978i
$838$	0	0
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	− 32.0000i	− 1.10214i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−22.0000	−0.755038
$850$	0	0
$851$	2.00000	0.0685591
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 41.0000i	− 1.40053i	−0.713881	−	0.700267i	$-0.753064\pi$
$857$	− 41.0000i	− 1.40053i	0.713881	−	0.700267i	$-0.246936\pi$
$858$	0	0
$859$	−43.0000	−1.46714	−0.733571	−	0.679613i	$-0.762148\pi$
$859$	−43.0000	−1.46714	−0.733571	+	0.679613i	$0.762148\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 57.0000i	− 1.94030i	−0.242500	−	0.970151i	$-0.577968\pi$
$863$	− 57.0000i	− 1.94030i	0.242500	−	0.970151i	$-0.422032\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.00000i	0.0339618i
$868$	0	0
$869$	28.0000	0.949835
$870$	0	0
$871$	10.0000	0.338837
$872$	0	0
$873$	− 8.00000i	− 0.270759i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 50.0000i	− 1.68838i	−0.536044	−	0.844190i	$-0.680082\pi$
$877$	− 50.0000i	− 1.68838i	0.536044	−	0.844190i	$-0.319918\pi$
$878$	0	0
$879$	16.0000	0.539667
$880$	0	0
$881$	−28.0000	−0.943344	−0.471672	−	0.881774i	$-0.656349\pi$
$881$	−28.0000	−0.943344	−0.471672	+	0.881774i	$0.656349\pi$
$882$	0	0
$883$	− 52.0000i	− 1.74994i	−0.484178	−	0.874970i	$-0.660881\pi$
$883$	− 52.0000i	− 1.74994i	0.484178	−	0.874970i	$-0.339119\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 33.0000i	− 1.10803i	−0.832506	−	0.554016i	$-0.813095\pi$
$887$	− 33.0000i	− 1.10803i	0.832506	−	0.554016i	$-0.186905\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	− 18.0000i	− 0.602347i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 5.00000i	− 0.166945i
$898$	0	0
$899$	21.0000	0.700389
$900$	0	0
$901$	24.0000	0.799556
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 30.0000i	− 0.996134i	−0.867139	−	0.498067i	$-0.834043\pi$
$907$	− 30.0000i	− 0.996134i	0.867139	−	0.498067i	$-0.165957\pi$
$908$	0	0
$909$	20.0000	0.663358
$910$	0	0
$911$	50.0000	1.65657	0.828287	−	0.560304i	$-0.189316\pi$
$911$	50.0000	1.65657	0.828287	+	0.560304i	$0.189316\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−60.0000	−1.97922	−0.989609	−	0.143787i	$-0.954072\pi$
$919$	−60.0000	−1.97922	−0.989609	+	0.143787i	$0.954072\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	− 5.00000i	− 0.164577i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−23.0000	−0.754606	−0.377303	−	0.926090i	$-0.623148\pi$
$929$	−23.0000	−0.754606	−0.377303	+	0.926090i	$0.623148\pi$
$930$	0	0
$931$	14.0000	0.458831
$932$	0	0
$933$	− 9.00000i	− 0.294647i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 26.0000i	− 0.849383i	−0.905338	−	0.424691i	$-0.860383\pi$
$937$	− 26.0000i	− 0.849383i	0.905338	−	0.424691i	$-0.139617\pi$
$938$	0	0
$939$	8.00000	0.261070
$940$	0	0
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	0	0
$943$	− 9.00000i	− 0.293080i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.0000i	0.487435i	0.969846	+	0.243717i	$0.0783669\pi$
$947$	15.0000i	0.487435i	−0.969846	+	0.243717i	$0.921633\pi$
$948$	0	0
$949$	5.00000	0.162307
$950$	0	0
$951$	−2.00000	−0.0648544
$952$	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
$957$	6.00000i	0.193952i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	20.0000i	0.644491i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 3.00000i	− 0.0964735i	−0.998836	−	0.0482367i	$-0.984640\pi$
$967$	− 3.00000i	− 0.0964735i	0.998836	−	0.0482367i	$-0.0153602\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	22.0000	0.706014	0.353007	−	0.935621i	$-0.385159\pi$
$971$	22.0000	0.706014	0.353007	+	0.935621i	$0.385159\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 26.0000i	− 0.831814i	−0.909407	−	0.415907i	$-0.863464\pi$
$977$	− 26.0000i	− 0.831814i	0.909407	−	0.415907i	$-0.136536\pi$
$978$	0	0
$979$	−32.0000	−1.02272
$980$	0	0
$981$	8.00000	0.255420
$982$	0	0
$983$	− 50.0000i	− 1.59475i	−0.603483	−	0.797376i	$-0.706221\pi$
$983$	− 50.0000i	− 1.59475i	0.603483	−	0.797376i	$-0.293779\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	− 19.0000i	− 0.602947i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.e.h.4049.2		2
5.2	odd	4		4600.2.a.j.1.1		1
5.3	odd	4		920.2.a.b.1.1	✓	1
5.4	even	2	inner	4600.2.e.h.4049.1		2
15.8	even	4		8280.2.a.e.1.1		1
20.3	even	4		1840.2.a.g.1.1		1
20.7	even	4		9200.2.a.n.1.1		1
40.3	even	4		7360.2.a.g.1.1		1
40.13	odd	4		7360.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.b.1.1	✓	1	5.3	odd	4
1840.2.a.g.1.1		1	20.3	even	4
4600.2.a.j.1.1		1	5.2	odd	4
4600.2.e.h.4049.1		2	5.4	even	2	inner
4600.2.e.h.4049.2		2	1.1	even	1	trivial
7360.2.a.g.1.1		1	40.3	even	4
7360.2.a.t.1.1		1	40.13	odd	4
8280.2.a.e.1.1		1	15.8	even	4
9200.2.a.n.1.1		1	20.7	even	4

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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +2.00000 q^{9} +2.00000 q^{11} +5.00000i q^{13} -4.00000i q^{17} +2.00000 q^{19} +1.00000i q^{23} +5.00000i q^{27} +3.00000 q^{29} +7.00000 q^{31} +2.00000i q^{33} -2.00000i q^{37} -5.00000 q^{39} -9.00000 q^{41} +4.00000i q^{43} -9.00000i q^{47} +7.00000 q^{49} +4.00000 q^{51} +6.00000i q^{53} +2.00000i q^{57} +2.00000 q^{61} -2.00000i q^{67} -1.00000 q^{69} -1.00000 q^{71} -1.00000i q^{73} +14.0000 q^{79} +1.00000 q^{81} +3.00000i q^{87} -16.0000 q^{89} +7.00000i q^{93} -4.00000i q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{9} + 4 q^{11} + 4 q^{19} + 6 q^{29} + 14 q^{31} - 10 q^{39} - 18 q^{41} + 14 q^{49} + 8 q^{51} + 4 q^{61} - 2 q^{69} - 2 q^{71} + 28 q^{79} + 2 q^{81} - 32 q^{89} + 8 q^{99}+O(q^{100}) \) 2 * q + 4 * q^9 + 4 * q^11 + 4 * q^19 + 6 * q^29 + 14 * q^31 - 10 * q^39 - 18 * q^41 + 14 * q^49 + 8 * q^51 + 4 * q^61 - 2 * q^69 - 2 * q^71 + 28 * q^79 + 2 * q^81 - 32 * q^89 + 8 * q^99

Introduction

L-functions

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