LMFDB - Embedded newform 1584.3.j.i.1297.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(1297,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.1297"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1584.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,84] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{46})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 46x^{2} + 2116 $ x^4 + 46*x^2 + 2116
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 99)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.2
Root			$-3.39116 - 5.87367i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.i.1297.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.78233 q^{5} +11.7473i q^{7} +(-6.78233 + 8.66025i) q^{11} +11.7473i q^{13} +10.3923i q^{17} -33.9116 q^{23} +21.0000 q^{25} +34.6410i q^{29} +10.0000 q^{31} -79.6743i q^{35} +50.0000 q^{37} -34.6410i q^{41} +46.9894i q^{43} -33.9116 q^{47} -89.0000 q^{49} +33.9116 q^{53} +(46.0000 - 58.7367i) q^{55} +67.8233 q^{59} +58.7367i q^{61} -79.6743i q^{65} +10.0000 q^{67} -33.9116 q^{71} -70.4840i q^{73} +(-101.735 - 79.6743i) q^{77} -58.7367i q^{79} -76.2102i q^{83} -70.4840i q^{85} +13.5647 q^{89} -138.000 q^{91} -40.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 84 q^{25} + 40 q^{31} + 200 q^{37} - 356 q^{49} + 184 q^{55} + 40 q^{67} - 552 q^{91} - 160 q^{97}+O(q^{100}) $ 4 * q + 84 * q^25 + 40 * q^31 + 200 * q^37 - 356 * q^49 + 184 * q^55 + 40 * q^67 - 552 * q^91 - 160 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−6.78233			−1.35647			−0.678233	−	0.734847i	$-0.737254\pi$
$5$	−6.78233			−1.35647			−0.678233	+	0.734847i	$0.737254\pi$
$6$		0			0
$7$			11.7473i			1.67819i	0.543984	+	0.839096i	$0.316915\pi$
$7$			11.7473i			1.67819i	−0.543984	+	0.839096i	$0.683085\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.78233	+	8.66025i	−0.616575	+	0.787296i
$11$	−6.78233	+	8.66025i	−0.616575	+	0.787296i
$12$		0			0
$13$			11.7473i			0.903642i	0.892109	+	0.451821i	$0.149225\pi$
$13$			11.7473i			0.903642i	−0.892109	+	0.451821i	$0.850775\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			10.3923i			0.611312i	0.952142	+	0.305656i	$0.0988758\pi$
$17$			10.3923i			0.611312i	−0.952142	+	0.305656i	$0.901124\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−33.9116			−1.47442			−0.737210	−	0.675664i	$-0.763857\pi$
$23$	−33.9116			−1.47442			−0.737210	+	0.675664i	$0.763857\pi$
$24$		0			0
$25$	21.0000			0.840000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			34.6410i			1.19452i	0.802049	+	0.597259i	$0.203744\pi$
$29$			34.6410i			1.19452i	−0.802049	+	0.597259i	$0.796256\pi$
$30$		0			0
$31$	10.0000			0.322581			0.161290	−	0.986907i	$-0.448434\pi$
$31$	10.0000			0.322581			0.161290	+	0.986907i	$0.448434\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	79.6743i		−	2.27641i
$36$		0			0
$37$	50.0000			1.35135			0.675676	−	0.737199i	$-0.263852\pi$
$37$	50.0000			1.35135			0.675676	+	0.737199i	$0.263852\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	34.6410i		−	0.844903i	−0.906386	−	0.422451i	$-0.861170\pi$
$41$		−	34.6410i		−	0.844903i	0.906386	−	0.422451i	$-0.138830\pi$
$42$		0			0
$43$			46.9894i			1.09278i	0.837532	+	0.546388i	$0.183998\pi$
$43$			46.9894i			1.09278i	−0.837532	+	0.546388i	$0.816002\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−33.9116			−0.721524			−0.360762	−	0.932658i	$-0.617483\pi$
$47$	−33.9116			−0.721524			−0.360762	+	0.932658i	$0.617483\pi$
$48$		0			0
$49$	−89.0000			−1.81633
$50$		0			0
$51$		0			0
$52$		0			0
$53$	33.9116			0.639842			0.319921	−	0.947444i	$-0.396344\pi$
$53$	33.9116			0.639842			0.319921	+	0.947444i	$0.396344\pi$
$54$		0			0
$55$	46.0000	−	58.7367i	0.836364	−	1.06794i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	67.8233			1.14955			0.574774	−	0.818312i	$-0.305090\pi$
$59$	67.8233			1.14955			0.574774	+	0.818312i	$0.305090\pi$
$60$		0			0
$61$			58.7367i			0.962897i	0.876474	+	0.481448i	$0.159889\pi$
$61$			58.7367i			0.962897i	−0.876474	+	0.481448i	$0.840111\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	79.6743i		−	1.22576i
$66$		0			0
$67$	10.0000			0.149254			0.0746269	−	0.997212i	$-0.476223\pi$
$67$	10.0000			0.149254			0.0746269	+	0.997212i	$0.476223\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−33.9116			−0.477629			−0.238814	−	0.971065i	$-0.576759\pi$
$71$	−33.9116			−0.477629			−0.238814	+	0.971065i	$0.576759\pi$
$72$		0			0
$73$		−	70.4840i		−	0.965535i	−0.875749	−	0.482767i	$-0.839632\pi$
$73$		−	70.4840i		−	0.965535i	0.875749	−	0.482767i	$-0.160368\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−101.735	−	79.6743i	−1.32123	−	1.03473i
$78$		0			0
$79$		−	58.7367i		−	0.743503i	−0.928332	−	0.371751i	$-0.878757\pi$
$79$		−	58.7367i		−	0.743503i	0.928332	−	0.371751i	$-0.121243\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	76.2102i		−	0.918196i	−0.888386	−	0.459098i	$-0.848173\pi$
$83$		−	76.2102i		−	0.918196i	0.888386	−	0.459098i	$-0.151827\pi$
$84$		0			0
$85$		−	70.4840i		−	0.829224i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	13.5647			0.152412			0.0762060	−	0.997092i	$-0.475719\pi$
$89$	13.5647			0.152412			0.0762060	+	0.997092i	$0.475719\pi$
$90$		0			0
$91$	−138.000			−1.51648
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−40.0000			−0.412371			−0.206186	−	0.978513i	$-0.566105\pi$
$97$	−40.0000			−0.412371			−0.206186	+	0.978513i	$0.566105\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	69.2820i		−	0.685961i	−0.939343	−	0.342980i	$-0.888564\pi$
$101$		−	69.2820i		−	0.685961i	0.939343	−	0.342980i	$-0.111436\pi$
$102$		0			0
$103$	−170.000			−1.65049			−0.825243	−	0.564778i	$-0.808962\pi$
$103$	−170.000			−1.65049			−0.825243	+	0.564778i	$0.808962\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	93.5307i		−	0.874119i	−0.899433	−	0.437060i	$-0.856020\pi$
$107$		−	93.5307i		−	0.874119i	0.899433	−	0.437060i	$-0.143980\pi$
$108$		0			0
$109$			176.210i			1.61661i	0.588766	+	0.808303i	$0.299614\pi$
$109$			176.210i			1.61661i	−0.588766	+	0.808303i	$0.700386\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−67.8233			−0.600206			−0.300103	−	0.953907i	$-0.597021\pi$
$113$	−67.8233			−0.600206			−0.300103	+	0.953907i	$0.597021\pi$
$114$		0			0
$115$	230.000			2.00000
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−122.082			−1.02590
$120$		0			0
$121$	−29.0000	−	117.473i	−0.239669	−	0.970855i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	27.1293			0.217035
$126$		0			0
$127$			105.726i			0.832489i	0.909253	+	0.416244i	$0.136654\pi$
$127$			105.726i			0.832489i	−0.909253	+	0.416244i	$0.863346\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			34.6410i			0.264435i	0.991221	+	0.132218i	$0.0422098\pi$
$131$			34.6410i			0.264435i	−0.991221	+	0.132218i	$0.957790\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−271.293			−1.98024			−0.990121	−	0.140214i	$-0.955221\pi$
$137$	−271.293			−1.98024			−0.990121	+	0.140214i	$0.955221\pi$
$138$		0			0
$139$		−	117.473i		−	0.845132i	−0.906332	−	0.422566i	$-0.861129\pi$
$139$		−	117.473i		−	0.845132i	0.906332	−	0.422566i	$-0.138871\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−101.735	−	79.6743i	−0.711433	−	0.557163i
$144$		0			0
$145$		−	234.947i		−	1.62032i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	138.564i		−	0.929960i	−0.885321	−	0.464980i	$-0.846061\pi$
$149$		−	138.564i		−	0.929960i	0.885321	−	0.464980i	$-0.153939\pi$
$150$		0			0
$151$		−	58.7367i		−	0.388985i	−0.980904	−	0.194492i	$-0.937694\pi$
$151$		−	58.7367i		−	0.388985i	0.980904	−	0.194492i	$-0.0623060\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−67.8233			−0.437570
$156$		0			0
$157$	170.000			1.08280			0.541401	−	0.840764i	$-0.317894\pi$
$157$	170.000			1.08280			0.541401	+	0.840764i	$0.317894\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	398.372i		−	2.47436i
$162$		0			0
$163$	−140.000			−0.858896			−0.429448	−	0.903092i	$-0.641292\pi$
$163$	−140.000			−0.858896			−0.429448	+	0.903092i	$0.641292\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			200.918i			1.20310i	0.798835	+	0.601551i	$0.205450\pi$
$167$			200.918i			1.20310i	−0.798835	+	0.601551i	$0.794550\pi$
$168$		0			0
$169$	31.0000			0.183432
$170$		0			0
$171$		0			0
$172$		0			0
$173$			304.841i			1.76209i	0.473036	+	0.881043i	$0.343158\pi$
$173$			304.841i			1.76209i	−0.473036	+	0.881043i	$0.656842\pi$
$174$		0			0
$175$			246.694i			1.40968i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	149.211			0.833582			0.416791	−	0.909002i	$-0.363155\pi$
$179$	149.211			0.833582			0.416791	+	0.909002i	$0.363155\pi$
$180$		0			0
$181$	−130.000			−0.718232			−0.359116	−	0.933293i	$-0.616922\pi$
$181$	−130.000			−0.718232			−0.359116	+	0.933293i	$0.616922\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−339.116			−1.83306
$186$		0			0
$187$	−90.0000	−	70.4840i	−0.481283	−	0.376920i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	88.1703			0.461625			0.230812	−	0.972998i	$-0.425862\pi$
$191$	88.1703			0.461625			0.230812	+	0.972998i	$0.425862\pi$
$192$		0			0
$193$			187.957i			0.973873i	0.873437	+	0.486936i	$0.161886\pi$
$193$			187.957i			0.973873i	−0.873437	+	0.486936i	$0.838114\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	41.5692i		−	0.211011i	−0.994419	−	0.105506i	$-0.966354\pi$
$197$		−	41.5692i		−	0.211011i	0.994419	−	0.105506i	$-0.0336461\pi$
$198$		0			0
$199$	202.000			1.01508			0.507538	−	0.861630i	$-0.330556\pi$
$199$	202.000			1.01508			0.507538	+	0.861630i	$0.330556\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−406.940			−2.00463
$204$		0			0
$205$			234.947i			1.14608i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	117.473i		−	0.556746i	−0.960473	−	0.278373i	$-0.910205\pi$
$211$		−	117.473i		−	0.556746i	0.960473	−	0.278373i	$-0.0897951\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	318.697i		−	1.48231i
$216$		0			0
$217$			117.473i			0.541352i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−122.082			−0.552407
$222$		0			0
$223$	250.000			1.12108			0.560538	−	0.828129i	$-0.310594\pi$
$223$	250.000			1.12108			0.560538	+	0.828129i	$0.310594\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	45.0333i		−	0.198385i	−0.995068	−	0.0991923i	$-0.968374\pi$
$227$		−	45.0333i		−	0.198385i	0.995068	−	0.0991923i	$-0.0316259\pi$
$228$		0			0
$229$	158.000			0.689956			0.344978	−	0.938611i	$-0.387886\pi$
$229$	158.000			0.689956			0.344978	+	0.938611i	$0.387886\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	62.3538i		−	0.267613i	−0.991007	−	0.133807i	$-0.957280\pi$
$233$		−	62.3538i		−	0.267613i	0.991007	−	0.133807i	$-0.0427201\pi$
$234$		0			0
$235$	230.000			0.978723
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	381.051i		−	1.59436i	−0.603744	−	0.797178i	$-0.706325\pi$
$239$		−	381.051i		−	1.59436i	0.603744	−	0.797178i	$-0.293675\pi$
$240$		0			0
$241$			234.947i			0.974883i	0.873156	+	0.487441i	$0.162070\pi$
$241$			234.947i			0.974883i	−0.873156	+	0.487441i	$0.837930\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	603.627			2.46379
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−135.647			−0.540425			−0.270212	−	0.962801i	$-0.587094\pi$
$251$	−135.647			−0.540425			−0.270212	+	0.962801i	$0.587094\pi$
$252$		0			0
$253$	230.000	−	293.684i	0.909091	−	1.16080i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	339.116			1.31952			0.659760	−	0.751477i	$-0.270658\pi$
$257$	339.116			1.31952			0.659760	+	0.751477i	$0.270658\pi$
$258$		0			0
$259$			587.367i			2.26783i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	96.9948i		−	0.368802i	−0.982851	−	0.184401i	$-0.940966\pi$
$263$		−	96.9948i		−	0.368802i	0.982851	−	0.184401i	$-0.0590345\pi$
$264$		0			0
$265$	−230.000			−0.867925
$266$		0			0
$267$		0			0
$268$		0			0
$269$	33.9116			0.126066			0.0630328	−	0.998011i	$-0.479923\pi$
$269$	33.9116			0.126066			0.0630328	+	0.998011i	$0.479923\pi$
$270$		0			0
$271$			176.210i			0.650222i	0.945676	+	0.325111i	$0.105402\pi$
$271$			176.210i			0.650222i	−0.945676	+	0.325111i	$0.894598\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−142.429	+	181.865i	−0.517923	+	0.661328i
$276$		0			0
$277$		−	11.7473i		−	0.0424092i	−0.999775	−	0.0212046i	$-0.993250\pi$
$277$		−	11.7473i		−	0.0424092i	0.999775	−	0.0212046i	$-0.00675013\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			190.526i			0.678027i	0.940782	+	0.339014i	$0.110093\pi$
$281$			190.526i			0.678027i	−0.940782	+	0.339014i	$0.889907\pi$
$282$		0			0
$283$		−	399.410i		−	1.41134i	−0.708540	−	0.705671i	$-0.750646\pi$
$283$		−	399.410i		−	1.41134i	0.708540	−	0.705671i	$-0.249354\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	406.940			1.41791
$288$		0			0
$289$	181.000			0.626298
$290$		0			0
$291$		0			0
$292$		0			0
$293$			422.620i			1.44239i	0.692732	+	0.721195i	$0.256407\pi$
$293$			422.620i			1.44239i	−0.692732	+	0.721195i	$0.743593\pi$
$294$		0			0
$295$	−460.000			−1.55932
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	398.372i		−	1.33235i
$300$		0			0
$301$	−552.000			−1.83389
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	398.372i		−	1.30614i
$306$		0			0
$307$			70.4840i			0.229590i	0.993389	+	0.114795i	$0.0366211\pi$
$307$			70.4840i			0.229590i	−0.993389	+	0.114795i	$0.963379\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	373.028			1.19945			0.599724	−	0.800207i	$-0.295277\pi$
$311$	373.028			1.19945			0.599724	+	0.800207i	$0.295277\pi$
$312$		0			0
$313$	−70.0000			−0.223642			−0.111821	−	0.993728i	$-0.535668\pi$
$313$	−70.0000			−0.223642			−0.111821	+	0.993728i	$0.535668\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−169.558			−0.534884			−0.267442	−	0.963574i	$-0.586178\pi$
$317$	−169.558			−0.534884			−0.267442	+	0.963574i	$0.586178\pi$
$318$		0			0
$319$	−300.000	−	234.947i	−0.940439	−	0.736510i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$			246.694i			0.759059i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	398.372i		−	1.21086i
$330$		0			0
$331$	40.0000			0.120846			0.0604230	−	0.998173i	$-0.480755\pi$
$331$	40.0000			0.120846			0.0604230	+	0.998173i	$0.480755\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−67.8233			−0.202458
$336$		0			0
$337$		−	164.463i		−	0.488020i	−0.969773	−	0.244010i	$-0.921537\pi$
$337$		−	164.463i		−	0.488020i	0.969773	−	0.244010i	$-0.0784630\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−67.8233	+	86.6025i	−0.198895	+	0.253966i
$342$		0			0
$343$		−	469.894i		−	1.36995i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			491.902i			1.41759i	0.705416	+	0.708793i	$0.250760\pi$
$347$			491.902i			1.41759i	−0.705416	+	0.708793i	$0.749240\pi$
$348$		0			0
$349$		−	646.104i		−	1.85130i	−0.378381	−	0.925650i	$-0.623519\pi$
$349$		−	646.104i		−	1.85130i	0.378381	−	0.925650i	$-0.376481\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	135.647			0.384268			0.192134	−	0.981369i	$-0.438459\pi$
$353$	135.647			0.384268			0.192134	+	0.981369i	$0.438459\pi$
$354$		0			0
$355$	230.000			0.647887
$356$		0			0
$357$		0			0
$358$		0			0
$359$			103.923i			0.289479i	0.989470	+	0.144740i	$0.0462345\pi$
$359$			103.923i			0.289479i	−0.989470	+	0.144740i	$0.953766\pi$
$360$		0			0
$361$	361.000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$			478.046i			1.30972i
$366$		0			0
$367$	130.000			0.354223			0.177112	−	0.984191i	$-0.443325\pi$
$367$	130.000			0.354223			0.177112	+	0.984191i	$0.443325\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			398.372i			1.07378i
$372$		0			0
$373$		−	481.641i		−	1.29126i	−0.763649	−	0.645631i	$-0.776594\pi$
$373$		−	481.641i		−	1.29126i	0.763649	−	0.645631i	$-0.223406\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−406.940			−1.07942
$378$		0			0
$379$	−308.000			−0.812665			−0.406332	−	0.913725i	$-0.633193\pi$
$379$	−308.000			−0.812665			−0.406332	+	0.913725i	$0.633193\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−237.382			−0.619795			−0.309898	−	0.950770i	$-0.600295\pi$
$383$	−237.382			−0.619795			−0.309898	+	0.950770i	$0.600295\pi$
$384$		0			0
$385$	690.000	+	540.378i	1.79221	+	1.40358i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	644.321			1.65635			0.828177	−	0.560467i	$-0.189378\pi$
$389$	644.321			1.65635			0.828177	+	0.560467i	$0.189378\pi$
$390$		0			0
$391$		−	352.420i		−	0.901330i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			398.372i			1.00854i
$396$		0			0
$397$	−310.000			−0.780856			−0.390428	−	0.920633i	$-0.627673\pi$
$397$	−310.000			−0.780856			−0.390428	+	0.920633i	$0.627673\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−556.151			−1.38691			−0.693455	−	0.720500i	$-0.743912\pi$
$401$	−556.151			−1.38691			−0.693455	+	0.720500i	$0.743912\pi$
$402$		0			0
$403$			117.473i			0.291497i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−339.116	+	433.013i	−0.833210	+	1.06391i
$408$		0			0
$409$		−	587.367i		−	1.43611i	−0.695989	−	0.718053i	$-0.745034\pi$
$409$		−	587.367i		−	1.43611i	0.695989	−	0.718053i	$-0.254966\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			796.743i			1.92916i
$414$		0			0
$415$			516.883i			1.24550i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−54.2586			−0.129496			−0.0647478	−	0.997902i	$-0.520624\pi$
$419$	−54.2586			−0.129496			−0.0647478	+	0.997902i	$0.520624\pi$
$420$		0			0
$421$	−310.000			−0.736342			−0.368171	−	0.929758i	$-0.620016\pi$
$421$	−310.000			−0.736342			−0.368171	+	0.929758i	$0.620016\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			218.238i			0.513502i
$426$		0			0
$427$	−690.000			−1.61593
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	103.923i		−	0.241121i	−0.992706	−	0.120560i	$-0.961531\pi$
$431$		−	103.923i		−	0.241121i	0.992706	−	0.120560i	$-0.0384691\pi$
$432$		0			0
$433$	−610.000			−1.40878			−0.704388	−	0.709815i	$-0.748778\pi$
$433$	−610.000			−1.40878			−0.704388	+	0.709815i	$0.748778\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	411.157i		−	0.936576i	−0.883576	−	0.468288i	$-0.844871\pi$
$439$		−	411.157i		−	0.936576i	0.883576	−	0.468288i	$-0.155129\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	271.293			0.612400			0.306200	−	0.951967i	$-0.400942\pi$
$443$	271.293			0.612400			0.306200	+	0.951967i	$0.400942\pi$
$444$		0			0
$445$	−92.0000			−0.206742
$446$		0			0
$447$		0			0
$448$		0			0
$449$	217.035			0.483373			0.241687	−	0.970354i	$-0.422299\pi$
$449$	217.035			0.483373			0.241687	+	0.970354i	$0.422299\pi$
$450$		0			0
$451$	300.000	+	234.947i	0.665188	+	0.520946i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	935.962			2.05706
$456$		0			0
$457$			164.463i			0.359875i	0.983678	+	0.179937i	$0.0575895\pi$
$457$			164.463i			0.359875i	−0.983678	+	0.179937i	$0.942410\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			658.179i			1.42772i	0.700288	+	0.713860i	$0.253055\pi$
$461$			658.179i			1.42772i	−0.700288	+	0.713860i	$0.746945\pi$
$462$		0			0
$463$	670.000			1.44708			0.723542	−	0.690280i	$-0.242513\pi$
$463$	670.000			1.44708			0.723542	+	0.690280i	$0.242513\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−339.116			−0.726160			−0.363080	−	0.931758i	$-0.618275\pi$
$467$	−339.116			−0.726160			−0.363080	+	0.931758i	$0.618275\pi$
$468$		0			0
$469$			117.473i			0.250476i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−406.940	−	318.697i	−0.860338	−	0.673779i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$			484.974i			1.01247i	0.862395	+	0.506236i	$0.168964\pi$
$479$			484.974i			1.01247i	−0.862395	+	0.506236i	$0.831036\pi$
$480$		0			0
$481$			587.367i			1.22114i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	271.293			0.559367
$486$		0			0
$487$	−230.000			−0.472279			−0.236140	−	0.971719i	$-0.575882\pi$
$487$	−230.000			−0.472279			−0.236140	+	0.971719i	$0.575882\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	17.3205i		−	0.0352760i	−0.999844	−	0.0176380i	$-0.994385\pi$
$491$		−	17.3205i		−	0.0352760i	0.999844	−	0.0176380i	$-0.00561464\pi$
$492$		0			0
$493$	−360.000			−0.730223
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	398.372i		−	0.801553i
$498$		0			0
$499$	−152.000			−0.304609			−0.152305	−	0.988334i	$-0.548669\pi$
$499$	−152.000			−0.304609			−0.152305	+	0.988334i	$0.548669\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			769.031i			1.52889i	0.644690	+	0.764444i	$0.276986\pi$
$503$			769.031i			1.52889i	−0.644690	+	0.764444i	$0.723014\pi$
$504$		0			0
$505$			469.894i			0.930482i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	644.321			1.26586			0.632929	−	0.774210i	$-0.281853\pi$
$509$	644.321			1.26586			0.632929	+	0.774210i	$0.281853\pi$
$510$		0			0
$511$	828.000			1.62035
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1153.00			2.23883
$516$		0			0
$517$	230.000	−	293.684i	0.444874	−	0.568053i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−149.211			−0.286394			−0.143197	−	0.989694i	$-0.545738\pi$
$521$	−149.211			−0.286394			−0.143197	+	0.989694i	$0.545738\pi$
$522$		0			0
$523$		−	281.936i		−	0.539075i	−0.962990	−	0.269537i	$-0.913129\pi$
$523$		−	281.936i		−	0.539075i	0.962990	−	0.269537i	$-0.0868708\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			103.923i			0.197197i
$528$		0			0
$529$	621.000			1.17391
$530$		0			0
$531$		0			0
$532$		0			0
$533$	406.940			0.763489
$534$		0			0
$535$			634.356i			1.18571i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	603.627	−	770.763i	1.11990	−	1.42999i
$540$		0			0
$541$			881.051i			1.62856i	0.580473	+	0.814280i	$0.302868\pi$
$541$			881.051i			1.62856i	−0.580473	+	0.814280i	$0.697132\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	1195.12i		−	2.19287i
$546$		0			0
$547$			187.957i			0.343615i	0.985131	+	0.171808i	$0.0549607\pi$
$547$			187.957i			0.343615i	−0.985131	+	0.171808i	$0.945039\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	690.000			1.24774
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	478.046i		−	0.858251i	−0.903245	−	0.429126i	$-0.858822\pi$
$557$		−	478.046i		−	0.858251i	0.903245	−	0.429126i	$-0.141178\pi$
$558$		0			0
$559$	−552.000			−0.987478
$560$		0			0
$561$		0			0
$562$		0			0
$563$			1084.26i			1.92587i	0.269737	+	0.962934i	$0.413063\pi$
$563$			1084.26i			1.92587i	−0.269737	+	0.962934i	$0.586937\pi$
$564$		0			0
$565$	460.000			0.814159
$566$		0			0
$567$		0			0
$568$		0			0
$569$			606.218i			1.06541i	0.846301	+	0.532705i	$0.178824\pi$
$569$			606.218i			1.06541i	−0.846301	+	0.532705i	$0.821176\pi$
$570$		0			0
$571$		−	117.473i		−	0.205733i	−0.994695	−	0.102866i	$-0.967199\pi$
$571$		−	117.473i		−	0.205733i	0.994695	−	0.102866i	$-0.0328014\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−712.145			−1.23851
$576$		0			0
$577$	−400.000			−0.693241			−0.346620	−	0.938005i	$-0.612671\pi$
$577$	−400.000			−0.693241			−0.346620	+	0.938005i	$0.612671\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	895.268			1.54091
$582$		0			0
$583$	−230.000	+	293.684i	−0.394511	+	0.503745i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	881.703			1.50205			0.751025	−	0.660274i	$-0.229560\pi$
$587$	881.703			1.50205			0.751025	+	0.660274i	$0.229560\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			613.146i			1.03397i	0.855994	+	0.516986i	$0.172946\pi$
$593$			613.146i			1.03397i	−0.855994	+	0.516986i	$0.827054\pi$
$594$		0			0
$595$	828.000			1.39160
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1064.83			1.77767			0.888836	−	0.458225i	$-0.151515\pi$
$599$	1064.83			1.77767			0.888836	+	0.458225i	$0.151515\pi$
$600$		0			0
$601$		−	117.473i		−	0.195463i	−0.995213	−	0.0977316i	$-0.968841\pi$
$601$		−	117.473i		−	0.195463i	0.995213	−	0.0977316i	$-0.0311587\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	196.688	+	796.743i	0.325103	+	1.31693i
$606$		0			0
$607$			693.093i			1.14183i	0.821008	+	0.570917i	$0.193412\pi$
$607$			693.093i			1.14183i	−0.821008	+	0.570917i	$0.806588\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	398.372i		−	0.651999i
$612$		0			0
$613$		−	105.726i		−	0.172473i	−0.996275	−	0.0862366i	$-0.972516\pi$
$613$		−	105.726i		−	0.172473i	0.996275	−	0.0862366i	$-0.0274841\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−67.8233			−0.109924			−0.0549622	−	0.998488i	$-0.517504\pi$
$617$	−67.8233			−0.109924			−0.0549622	+	0.998488i	$0.517504\pi$
$618$		0			0
$619$	490.000			0.791599			0.395800	−	0.918337i	$-0.370467\pi$
$619$	490.000			0.791599			0.395800	+	0.918337i	$0.370467\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$			159.349i			0.255776i
$624$		0			0
$625$	−709.000			−1.13440
$626$		0			0
$627$		0			0
$628$		0			0
$629$			519.615i			0.826097i
$630$		0			0
$631$	−338.000			−0.535658			−0.267829	−	0.963467i	$-0.586306\pi$
$631$	−338.000			−0.535658			−0.267829	+	0.963467i	$0.586306\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	717.069i		−	1.12924i
$636$		0			0
$637$		−	1045.51i		−	1.64131i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	257.729			0.402073			0.201036	−	0.979584i	$-0.435569\pi$
$641$	257.729			0.402073			0.201036	+	0.979584i	$0.435569\pi$
$642$		0			0
$643$	130.000			0.202177			0.101089	−	0.994877i	$-0.467767\pi$
$643$	130.000			0.202177			0.101089	+	0.994877i	$0.467767\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1051.26			−1.62482			−0.812412	−	0.583084i	$-0.801846\pi$
$647$	−1051.26			−1.62482			−0.812412	+	0.583084i	$0.801846\pi$
$648$		0			0
$649$	−460.000	+	587.367i	−0.708783	+	0.905034i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	237.382			0.363525			0.181762	−	0.983342i	$-0.441820\pi$
$653$	237.382			0.363525			0.181762	+	0.983342i	$0.441820\pi$
$654$		0			0
$655$		−	234.947i		−	0.358697i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	398.372i		−	0.604509i	−0.953227	−	0.302255i	$-0.902261\pi$
$659$		−	398.372i		−	0.604509i	0.953227	−	0.302255i	$-0.0977393\pi$
$660$		0			0
$661$	278.000			0.420575			0.210287	−	0.977640i	$-0.432560\pi$
$661$	278.000			0.420575			0.210287	+	0.977640i	$0.432560\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	1174.73i		−	1.76122i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−508.675	−	398.372i	−0.758085	−	0.593698i
$672$		0			0
$673$		−	540.378i		−	0.802939i	−0.915872	−	0.401469i	$-0.868500\pi$
$673$		−	540.378i		−	0.802939i	0.915872	−	0.401469i	$-0.131500\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			491.902i			0.726591i	0.931674	+	0.363296i	$0.118349\pi$
$677$			491.902i			0.726591i	−0.931674	+	0.363296i	$0.881651\pi$
$678$		0			0
$679$		−	469.894i		−	0.692038i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−949.526			−1.39023			−0.695114	−	0.718899i	$-0.744646\pi$
$683$	−949.526			−1.39023			−0.695114	+	0.718899i	$0.744646\pi$
$684$		0			0
$685$	1840.00			2.68613
$686$		0			0
$687$		0			0
$688$		0			0
$689$			398.372i			0.578188i
$690$		0			0
$691$	−482.000			−0.697540			−0.348770	−	0.937208i	$-0.613401\pi$
$691$	−482.000			−0.697540			−0.348770	+	0.937208i	$0.613401\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			796.743i			1.14639i
$696$		0			0
$697$	360.000			0.516499
$698$		0			0
$699$		0			0
$700$		0			0
$701$			207.846i			0.296499i	0.988950	+	0.148250i	$0.0473639\pi$
$701$			207.846i			0.296499i	−0.988950	+	0.148250i	$0.952636\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	813.880			1.15117
$708$		0			0
$709$	−1210.00			−1.70663			−0.853315	−	0.521397i	$-0.825411\pi$
$709$	−1210.00			−1.70663			−0.853315	+	0.521397i	$0.825411\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−339.116			−0.475619
$714$		0			0
$715$	690.000	+	540.378i	0.965035	+	0.755773i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1132.65			−1.57531			−0.787656	−	0.616115i	$-0.788705\pi$
$719$	−1132.65			−1.57531			−0.787656	+	0.616115i	$0.788705\pi$
$720$		0			0
$721$		−	1997.05i		−	2.76983i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			727.461i			1.00339i
$726$		0			0
$727$	−1130.00			−1.55433			−0.777166	−	0.629295i	$-0.783344\pi$
$727$	−1130.00			−1.55433			−0.777166	+	0.629295i	$0.783344\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−488.328			−0.668027
$732$		0			0
$733$			928.040i			1.26608i	0.774117	+	0.633042i	$0.218194\pi$
$733$			928.040i			1.26608i	−0.774117	+	0.633042i	$0.781806\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−67.8233	+	86.6025i	−0.0920262	+	0.117507i
$738$		0			0
$739$			1057.26i			1.43066i	0.698785	+	0.715332i	$0.253724\pi$
$739$			1057.26i			1.43066i	−0.698785	+	0.715332i	$0.746276\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	131.636i		−	0.177168i	−0.996069	−	0.0885840i	$-0.971766\pi$
$743$		−	131.636i		−	0.177168i	0.996069	−	0.0885840i	$-0.0282342\pi$
$744$		0			0
$745$			939.787i			1.26146i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1098.74			1.46694
$750$		0			0
$751$	298.000			0.396804			0.198402	−	0.980121i	$-0.436425\pi$
$751$	298.000			0.396804			0.198402	+	0.980121i	$0.436425\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			398.372i			0.527645i
$756$		0			0
$757$	−490.000			−0.647292			−0.323646	−	0.946178i	$-0.604909\pi$
$757$	−490.000			−0.647292			−0.323646	+	0.946178i	$0.604909\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	502.295i		−	0.660046i	−0.943973	−	0.330023i	$-0.892944\pi$
$761$		−	502.295i		−	0.660046i	0.943973	−	0.330023i	$-0.107056\pi$
$762$		0			0
$763$	−2070.00			−2.71298
$764$		0			0
$765$		0			0
$766$		0			0
$767$			796.743i			1.03878i
$768$		0			0
$769$		−	234.947i		−	0.305522i	−0.988263	−	0.152761i	$-0.951183\pi$
$769$		−	234.947i		−	0.305522i	0.988263	−	0.152761i	$-0.0488166\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−1390.38			−1.79868			−0.899339	−	0.437253i	$-0.855952\pi$
$773$	−1390.38			−1.79868			−0.899339	+	0.437253i	$0.855952\pi$
$774$		0			0
$775$	210.000			0.270968
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	230.000	−	293.684i	0.294494	−	0.376035i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1153.00			−1.46878
$786$		0			0
$787$		−	187.957i		−	0.238828i	−0.992845	−	0.119414i	$-0.961898\pi$
$787$		−	187.957i		−	0.238828i	0.992845	−	0.119414i	$-0.0381015\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	796.743i		−	1.00726i
$792$		0			0
$793$	−690.000			−0.870113
$794$		0			0
$795$		0			0
$796$		0			0
$797$	237.382			0.297844			0.148922	−	0.988849i	$-0.452420\pi$
$797$	237.382			0.297844			0.148922	+	0.988849i	$0.452420\pi$
$798$		0			0
$799$		−	352.420i		−	0.441077i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	610.410	+	478.046i	0.760162	+	0.595325i
$804$		0			0
$805$			2701.89i			3.35638i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	1143.15i		−	1.41305i	−0.707691	−	0.706523i	$-0.750263\pi$
$809$		−	1143.15i		−	1.41305i	0.707691	−	0.706523i	$-0.249737\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	949.526			1.16506
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$			969.948i			1.18142i	0.806883	+	0.590712i	$0.201153\pi$
$821$			969.948i			1.18142i	−0.806883	+	0.590712i	$0.798847\pi$
$822$		0			0
$823$	310.000			0.376671			0.188335	−	0.982105i	$-0.439691\pi$
$823$	310.000			0.376671			0.188335	+	0.982105i	$0.439691\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	893.738i		−	1.08070i	−0.841441	−	0.540350i	$-0.818292\pi$
$827$		−	893.738i		−	1.08070i	0.841441	−	0.540350i	$-0.181708\pi$
$828$		0			0
$829$	−850.000			−1.02533			−0.512666	−	0.858588i	$-0.671342\pi$
$829$	−850.000			−1.02533			−0.512666	+	0.858588i	$0.671342\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	924.915i		−	1.11034i
$834$		0			0
$835$		−	1362.69i		−	1.63197i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−440.851			−0.525449			−0.262724	−	0.964871i	$-0.584621\pi$
$839$	−440.851			−0.525449			−0.262724	+	0.964871i	$0.584621\pi$
$840$		0			0
$841$	−359.000			−0.426873
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−210.252			−0.248819
$846$		0			0
$847$	1380.00	−	340.673i	1.62928	−	0.402211i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1695.58			−1.99246
$852$		0			0
$853$		−	1069.01i		−	1.25323i	−0.779328	−	0.626617i	$-0.784439\pi$
$853$		−	1069.01i		−	1.25323i	0.779328	−	0.626617i	$-0.215561\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	963.020i		−	1.12371i	−0.827235	−	0.561855i	$-0.810088\pi$
$857$		−	963.020i		−	1.12371i	0.827235	−	0.561855i	$-0.189912\pi$
$858$		0			0
$859$	820.000			0.954598			0.477299	−	0.878741i	$-0.341616\pi$
$859$	820.000			0.954598			0.477299	+	0.878741i	$0.341616\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	779.968			0.903787			0.451893	−	0.892072i	$-0.350749\pi$
$863$	779.968			0.903787			0.451893	+	0.892072i	$0.350749\pi$
$864$		0			0
$865$		−	2067.53i		−	2.39021i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	508.675	+	398.372i	0.585356	+	0.458425i
$870$		0			0
$871$			117.473i			0.134872i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			318.697i			0.364226i
$876$		0			0
$877$		−	693.093i		−	0.790300i	−0.918617	−	0.395150i	$-0.870693\pi$
$877$		−	693.093i		−	0.790300i	0.918617	−	0.395150i	$-0.129307\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1492.11			−1.69366			−0.846829	−	0.531865i	$-0.821491\pi$
$881$	−1492.11			−1.69366			−0.846829	+	0.531865i	$0.821491\pi$
$882$		0			0
$883$	−230.000			−0.260476			−0.130238	−	0.991483i	$-0.541574\pi$
$883$	−230.000			−0.260476			−0.130238	+	0.991483i	$0.541574\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	547.328i		−	0.617055i	−0.951215	−	0.308528i	$-0.900164\pi$
$887$		−	547.328i		−	0.617055i	0.951215	−	0.308528i	$-0.0998362\pi$
$888$		0			0
$889$	−1242.00			−1.39708
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−1012.00			−1.13073
$896$		0			0
$897$		0			0
$898$		0			0
$899$			346.410i			0.385328i
$900$		0			0
$901$			352.420i			0.391143i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	881.703			0.974257
$906$		0			0
$907$	430.000			0.474090			0.237045	−	0.971499i	$-0.423821\pi$
$907$	430.000			0.474090			0.237045	+	0.971499i	$0.423821\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−155.994			−0.171233			−0.0856167	−	0.996328i	$-0.527286\pi$
$911$	−155.994			−0.171233			−0.0856167	+	0.996328i	$0.527286\pi$
$912$		0			0
$913$	660.000	+	516.883i	0.722892	+	0.566137i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−406.940			−0.443773
$918$		0			0
$919$		−	1233.47i		−	1.34219i	−0.741372	−	0.671094i	$-0.765825\pi$
$919$		−	1233.47i		−	1.34219i	0.741372	−	0.671094i	$-0.234175\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	398.372i		−	0.431605i
$924$		0			0
$925$	1050.00			1.13514
$926$		0			0
$927$		0			0
$928$		0			0
$929$	949.526			1.02209			0.511047	−	0.859552i	$-0.329258\pi$
$929$	949.526			1.02209			0.511047	+	0.859552i	$0.329258\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	610.410	+	478.046i	0.652845	+	0.511279i
$936$		0			0
$937$		−	986.777i		−	1.05312i	−0.850137	−	0.526562i	$-0.823481\pi$
$937$		−	986.777i		−	1.05312i	0.850137	−	0.526562i	$-0.176519\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	34.6410i		−	0.0368130i	−0.999831	−	0.0184065i	$-0.994141\pi$
$941$		−	34.6410i		−	0.0368130i	0.999831	−	0.0184065i	$-0.00585930\pi$
$942$		0			0
$943$			1174.73i			1.24574i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	271.293			0.286476			0.143238	−	0.989688i	$-0.454248\pi$
$947$	271.293			0.286476			0.143238	+	0.989688i	$0.454248\pi$
$948$		0			0
$949$	828.000			0.872497
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	145.492i		−	0.152668i	−0.997082	−	0.0763338i	$-0.975679\pi$
$953$		−	145.492i		−	0.152668i	0.997082	−	0.0763338i	$-0.0243215\pi$
$954$		0			0
$955$	−598.000			−0.626178
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	3186.97i		−	3.32323i
$960$		0			0
$961$	−861.000			−0.895942
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	1274.79i		−	1.32103i
$966$		0			0
$967$			693.093i			0.716746i	0.933579	+	0.358373i	$0.116668\pi$
$967$			693.093i			0.716746i	−0.933579	+	0.358373i	$0.883332\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−868.138			−0.894066			−0.447033	−	0.894517i	$-0.647519\pi$
$971$	−868.138			−0.894066			−0.447033	+	0.894517i	$0.647519\pi$
$972$		0			0
$973$	1380.00			1.41829
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−881.703			−0.902459			−0.451230	−	0.892408i	$-0.649014\pi$
$977$	−881.703			−0.902459			−0.451230	+	0.892408i	$0.649014\pi$
$978$		0			0
$979$	−92.0000	+	117.473i	−0.0939734	+	0.119993i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−33.9116			−0.0344981			−0.0172491	−	0.999851i	$-0.505491\pi$
$983$	−33.9116			−0.0344981			−0.0172491	+	0.999851i	$0.505491\pi$
$984$		0			0
$985$			281.936i			0.286230i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1593.49i		−	1.61121i
$990$		0			0
$991$	−830.000			−0.837538			−0.418769	−	0.908093i	$-0.637538\pi$
$991$	−830.000			−0.837538			−0.418769	+	0.908093i	$0.637538\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1370.03			−1.37692
$996$		0			0
$997$			1186.48i			1.19005i	0.803707	+	0.595026i	$0.202858\pi$
$997$			1186.48i			1.19005i	−0.803707	+	0.595026i	$0.797142\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.i.1297.2		4
3.2	odd	2	inner	1584.3.j.i.1297.4		4
4.3	odd	2		99.3.c.c.10.1	✓	4
11.10	odd	2	inner	1584.3.j.i.1297.1		4
12.11	even	2		99.3.c.c.10.4	yes	4
33.32	even	2	inner	1584.3.j.i.1297.3		4
44.43	even	2		99.3.c.c.10.3	yes	4
132.131	odd	2		99.3.c.c.10.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.3.c.c.10.1	✓	4	4.3	odd	2
99.3.c.c.10.2	yes	4	132.131	odd	2
99.3.c.c.10.3	yes	4	44.43	even	2
99.3.c.c.10.4	yes	4	12.11	even	2
1584.3.j.i.1297.1		4	11.10	odd	2	inner
1584.3.j.i.1297.2		4	1.1	even	1	trivial
1584.3.j.i.1297.3		4	33.32	even	2	inner
1584.3.j.i.1297.4		4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1584.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-6.78233 q^{5} +11.7473i q^{7} +(-6.78233 + 8.66025i) q^{11} +11.7473i q^{13} +10.3923i q^{17} -33.9116 q^{23} +21.0000 q^{25} +34.6410i q^{29} +10.0000 q^{31} -79.6743i q^{35} +50.0000 q^{37} -34.6410i q^{41} +46.9894i q^{43} -33.9116 q^{47} -89.0000 q^{49} +33.9116 q^{53} +(46.0000 - 58.7367i) q^{55} +67.8233 q^{59} +58.7367i q^{61} -79.6743i q^{65} +10.0000 q^{67} -33.9116 q^{71} -70.4840i q^{73} +(-101.735 - 79.6743i) q^{77} -58.7367i q^{79} -76.2102i q^{83} -70.4840i q^{85} +13.5647 q^{89} -138.000 q^{91} -40.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 84 q^{25} + 40 q^{31} + 200 q^{37} - 356 q^{49} + 184 q^{55} + 40 q^{67} - 552 q^{91} - 160 q^{97}+O(q^{100}) \) 4 * q + 84 * q^25 + 40 * q^31 + 200 * q^37 - 356 * q^49 + 184 * q^55 + 40 * q^67 - 552 * q^91 - 160 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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