LMFDB - Embedded newform 1872.2.c.l.1585.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1872,2,Mod(1585,1872)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1872.1585"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1872 = 2^{4} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1872.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$14.9479952584$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 936)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1585.4
Root			$-0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1872.1585
Dual form			1872.2.c.l.1585.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.23607i q^{5} +1.74806i q^{7} +3.23607i q^{11} +(-2.23607 - 2.82843i) q^{13} -5.65685 q^{17} -7.40492i q^{19} +9.15298 q^{23} +3.47214 q^{25} +3.49613 q^{29} -1.74806i q^{31} +2.16073 q^{35} -5.65685i q^{37} -7.70820i q^{41} +8.94427 q^{43} +7.23607i q^{47} +3.94427 q^{49} -9.15298 q^{53} +4.00000 q^{55} +1.70820i q^{59} +8.47214 q^{61} +(-3.49613 + 2.76393i) q^{65} -3.90879i q^{67} +5.70820i q^{71} -5.65685 q^{77} +0.944272 q^{79} -11.2361i q^{83} +6.99226i q^{85} -10.7639i q^{89} +(4.94427 - 3.90879i) q^{91} -9.15298 q^{95} -18.3060i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{25} - 40 q^{49} + 32 q^{55} + 32 q^{61} - 64 q^{79} - 32 q^{91}+O(q^{100}) $ 8 * q - 8 * q^25 - 40 * q^49 + 32 * q^55 + 32 * q^61 - 64 * q^79 - 32 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$5$		−	1.23607i		−	0.552786i	−0.961045	−	0.276393i	$-0.910861\pi$
$5$		−	1.23607i		−	0.552786i	0.961045	−	0.276393i	$-0.0891392\pi$
$6$		0			0
$7$			1.74806i			0.660706i	0.943857	+	0.330353i	$0.107168\pi$
$7$			1.74806i			0.660706i	−0.943857	+	0.330353i	$0.892832\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.23607i			0.975711i	0.872924	+	0.487856i	$0.162221\pi$
$11$			3.23607i			0.975711i	−0.872924	+	0.487856i	$0.837779\pi$
$12$		0			0
$13$	−2.23607	−	2.82843i	−0.620174	−	0.784465i
$13$	−2.23607	−	2.82843i	−0.620174	−	0.784465i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.65685			−1.37199			−0.685994	−	0.727607i	$-0.740633\pi$
$17$	−5.65685			−1.37199			−0.685994	+	0.727607i	$0.740633\pi$
$18$		0			0
$19$		−	7.40492i		−	1.69880i	−0.527746	−	0.849402i	$-0.676963\pi$
$19$		−	7.40492i		−	1.69880i	0.527746	−	0.849402i	$-0.323037\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	9.15298			1.90853			0.954264	−	0.298964i	$-0.0966411\pi$
$23$	9.15298			1.90853			0.954264	+	0.298964i	$0.0966411\pi$
$24$		0			0
$25$	3.47214			0.694427
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.49613			0.649215			0.324607	−	0.945849i	$-0.394768\pi$
$29$	3.49613			0.649215			0.324607	+	0.945849i	$0.394768\pi$
$30$		0			0
$31$		−	1.74806i		−	0.313962i	−0.987602	−	0.156981i	$-0.949824\pi$
$31$		−	1.74806i		−	0.313962i	0.987602	−	0.156981i	$-0.0501761\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.16073			0.365229
$36$		0			0
$37$		−	5.65685i		−	0.929981i	−0.885316	−	0.464991i	$-0.846058\pi$
$37$		−	5.65685i		−	0.929981i	0.885316	−	0.464991i	$-0.153942\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.70820i		−	1.20382i	−0.798564	−	0.601910i	$-0.794407\pi$
$41$		−	7.70820i		−	1.20382i	0.798564	−	0.601910i	$-0.205593\pi$
$42$		0			0
$43$	8.94427			1.36399			0.681994	−	0.731357i	$-0.261113\pi$
$43$	8.94427			1.36399			0.681994	+	0.731357i	$0.261113\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.23607i			1.05549i	0.849403	+	0.527744i	$0.176962\pi$
$47$			7.23607i			1.05549i	−0.849403	+	0.527744i	$0.823038\pi$
$48$		0			0
$49$	3.94427			0.563467
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−9.15298			−1.25726			−0.628629	−	0.777705i	$-0.716384\pi$
$53$	−9.15298			−1.25726			−0.628629	+	0.777705i	$0.716384\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$			1.70820i			0.222389i	0.993799	+	0.111195i	$0.0354677\pi$
$59$			1.70820i			0.222389i	−0.993799	+	0.111195i	$0.964532\pi$
$60$		0			0
$61$	8.47214			1.08475			0.542373	−	0.840138i	$-0.317526\pi$
$61$	8.47214			1.08475			0.542373	+	0.840138i	$0.317526\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.49613	+	2.76393i	−0.433641	+	0.342824i
$66$		0			0
$67$		−	3.90879i		−	0.477535i	−0.971077	−	0.238767i	$-0.923257\pi$
$67$		−	3.90879i		−	0.477535i	0.971077	−	0.238767i	$-0.0767433\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			5.70820i			0.677439i	0.940887	+	0.338720i	$0.109994\pi$
$71$			5.70820i			0.677439i	−0.940887	+	0.338720i	$0.890006\pi$
$72$		0			0
$73$		0			0		1.00000			$0$
$73$		0			0		−1.00000			$\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−5.65685			−0.644658
$78$		0			0
$79$	0.944272			0.106239			0.0531194	−	0.998588i	$-0.483084\pi$
$79$	0.944272			0.106239			0.0531194	+	0.998588i	$0.483084\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	11.2361i		−	1.23332i	−0.787230	−	0.616659i	$-0.788486\pi$
$83$		−	11.2361i		−	1.23332i	0.787230	−	0.616659i	$-0.211514\pi$
$84$		0			0
$85$			6.99226i			0.758417i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	10.7639i		−	1.14097i	−0.821306	−	0.570487i	$-0.806754\pi$
$89$		−	10.7639i		−	1.14097i	0.821306	−	0.570487i	$-0.193246\pi$
$90$		0			0
$91$	4.94427	−	3.90879i	0.518301	−	0.409753i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−9.15298			−0.939076
$96$		0			0
$97$		−	18.3060i		−	1.85869i	−0.369214	−	0.929345i	$-0.620373\pi$
$97$		−	18.3060i		−	1.85869i	0.369214	−	0.929345i	$-0.379627\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	2.16073			0.215000			0.107500	−	0.994205i	$-0.465715\pi$
$101$	2.16073			0.215000			0.107500	+	0.994205i	$0.465715\pi$
$102$		0			0
$103$	−4.94427			−0.487174			−0.243587	−	0.969879i	$-0.578324\pi$
$103$	−4.94427			−0.487174			−0.243587	+	0.969879i	$0.578324\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.16073			0.208885			0.104443	−	0.994531i	$-0.466694\pi$
$107$	2.16073			0.208885			0.104443	+	0.994531i	$0.466694\pi$
$108$		0			0
$109$		−	12.6491i		−	1.21157i	−0.795630	−	0.605783i	$-0.792860\pi$
$109$		−	12.6491i		−	1.21157i	0.795630	−	0.605783i	$-0.207140\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	12.6491			1.18993			0.594964	−	0.803752i	$-0.297166\pi$
$113$	12.6491			1.18993			0.594964	+	0.803752i	$0.297166\pi$
$114$		0			0
$115$		−	11.3137i		−	1.05501i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	9.88854i		−	0.906481i
$120$		0			0
$121$	0.527864			0.0479876
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	10.4721i		−	0.936656i
$126$		0			0
$127$	−8.94427			−0.793676			−0.396838	−	0.917889i	$-0.629892\pi$
$127$	−8.94427			−0.793676			−0.396838	+	0.917889i	$0.629892\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−2.16073			−0.188784			−0.0943918	−	0.995535i	$-0.530091\pi$
$131$	−2.16073			−0.188784			−0.0943918	+	0.995535i	$0.530091\pi$
$132$		0			0
$133$	12.9443			1.12241
$134$		0			0
$135$		0			0
$136$		0			0
$137$			20.6525i			1.76446i	0.470819	+	0.882230i	$0.343959\pi$
$137$			20.6525i			1.76446i	−0.470819	+	0.882230i	$0.656041\pi$
$138$		0			0
$139$	16.9443			1.43719			0.718597	−	0.695427i	$-0.244785\pi$
$139$	16.9443			1.43719			0.718597	+	0.695427i	$0.244785\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.15298	−	7.23607i	0.765411	−	0.605110i
$144$		0			0
$145$		−	4.32145i		−	0.358877i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			14.1803i			1.16170i	0.814011	+	0.580849i	$0.197279\pi$
$149$			14.1803i			1.16170i	−0.814011	+	0.580849i	$0.802721\pi$
$150$		0			0
$151$		−	1.74806i		−	0.142255i	−0.997467	−	0.0711277i	$-0.977340\pi$
$151$		−	1.74806i		−	0.142255i	0.997467	−	0.0711277i	$-0.0226598\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.16073			−0.173554
$156$		0			0
$157$	−10.9443			−0.873448			−0.436724	−	0.899596i	$-0.643861\pi$
$157$	−10.9443			−0.873448			−0.436724	+	0.899596i	$0.643861\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			16.0000i			1.26098i
$162$		0			0
$163$		−	0.412662i		−	0.0323222i	−0.999869	−	0.0161611i	$-0.994856\pi$
$163$		−	0.412662i		−	0.0323222i	0.999869	−	0.0161611i	$-0.00514446\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			15.2361i			1.17900i	0.807768	+	0.589501i	$0.200676\pi$
$167$			15.2361i			1.17900i	−0.807768	+	0.589501i	$0.799324\pi$
$168$		0			0
$169$	−3.00000	+	12.6491i	−0.230769	+	0.973009i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	20.4667			1.55605			0.778027	−	0.628231i	$-0.216221\pi$
$173$	20.4667			1.55605			0.778027	+	0.628231i	$0.216221\pi$
$174$		0			0
$175$			6.06952i			0.458812i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−20.4667			−1.52975			−0.764876	−	0.644177i	$-0.777200\pi$
$179$	−20.4667			−1.52975			−0.764876	+	0.644177i	$0.777200\pi$
$180$		0			0
$181$	2.94427			0.218846			0.109423	−	0.993995i	$-0.465100\pi$
$181$	2.94427			0.218846			0.109423	+	0.993995i	$0.465100\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.99226			−0.514081
$186$		0			0
$187$		−	18.3060i		−	1.33866i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−13.4744			−0.974976			−0.487488	−	0.873130i	$-0.662087\pi$
$191$	−13.4744			−0.974976			−0.487488	+	0.873130i	$0.662087\pi$
$192$		0			0
$193$			11.3137i			0.814379i	0.913344	+	0.407189i	$0.133491\pi$
$193$			11.3137i			0.814379i	−0.913344	+	0.407189i	$0.866509\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	14.1803i		−	1.01031i	−0.863029	−	0.505154i	$-0.831436\pi$
$197$		−	14.1803i		−	1.01031i	0.863029	−	0.505154i	$-0.168564\pi$
$198$		0			0
$199$	−8.94427			−0.634043			−0.317021	−	0.948418i	$-0.602683\pi$
$199$	−8.94427			−0.634043			−0.317021	+	0.948418i	$0.602683\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			6.11146i			0.428940i
$204$		0			0
$205$	−9.52786			−0.665455
$206$		0			0
$207$		0			0
$208$		0			0
$209$	23.9628			1.65754
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	11.0557i		−	0.753994i
$216$		0			0
$217$	3.05573			0.207436
$218$		0			0
$219$		0			0
$220$		0			0
$221$	12.6491	+	16.0000i	0.850871	+	1.07628i
$222$		0			0
$223$			16.5579i			1.10880i	0.832251	+	0.554400i	$0.187052\pi$
$223$			16.5579i			1.10880i	−0.832251	+	0.554400i	$0.812948\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	9.70820i		−	0.644356i	−0.946679	−	0.322178i	$-0.895585\pi$
$227$		−	9.70820i		−	0.644356i	0.946679	−	0.322178i	$-0.104415\pi$
$228$		0			0
$229$		−	12.6491i		−	0.835877i	−0.908475	−	0.417938i	$-0.862753\pi$
$229$		−	12.6491i		−	0.835877i	0.908475	−	0.417938i	$-0.137247\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	11.3137			0.741186			0.370593	−	0.928795i	$-0.379155\pi$
$233$	11.3137			0.741186			0.370593	+	0.928795i	$0.379155\pi$
$234$		0			0
$235$	8.94427			0.583460
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	28.1803i		−	1.82283i	−0.411483	−	0.911417i	$-0.634989\pi$
$239$		−	28.1803i		−	1.82283i	0.411483	−	0.911417i	$-0.365011\pi$
$240$		0			0
$241$		−	18.3060i		−	1.17919i	−0.807699	−	0.589595i	$-0.799287\pi$
$241$		−	18.3060i		−	1.17919i	0.807699	−	0.589595i	$-0.200713\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		−	4.87539i		−	0.311477i
$246$		0			0
$247$	−20.9443	+	16.5579i	−1.33265	+	1.05355i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	18.3060			1.15546			0.577731	−	0.816227i	$-0.303938\pi$
$251$	18.3060			1.15546			0.577731	+	0.816227i	$0.303938\pi$
$252$		0			0
$253$			29.6197i			1.86217i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.3137			−0.705730			−0.352865	−	0.935674i	$-0.614792\pi$
$257$	−11.3137			−0.705730			−0.352865	+	0.935674i	$0.614792\pi$
$258$		0			0
$259$	9.88854			0.614444
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−22.6274			−1.39527			−0.697633	−	0.716455i	$-0.745763\pi$
$263$	−22.6274			−1.39527			−0.697633	+	0.716455i	$0.745763\pi$
$264$		0			0
$265$			11.3137i			0.694996i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−21.8021			−1.32930			−0.664649	−	0.747156i	$-0.731419\pi$
$269$	−21.8021			−1.32930			−0.664649	+	0.747156i	$0.731419\pi$
$270$		0			0
$271$			24.3755i			1.48071i	0.672219	+	0.740353i	$0.265341\pi$
$271$			24.3755i			1.48071i	−0.672219	+	0.740353i	$0.734659\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			11.2361i			0.677560i
$276$		0			0
$277$	10.0000			0.600842			0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000			0.600842			0.300421	+	0.953807i	$0.402873\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	8.29180i		−	0.494647i	−0.968933	−	0.247324i	$-0.920449\pi$
$281$		−	8.29180i		−	0.494647i	0.968933	−	0.247324i	$-0.0795511\pi$
$282$		0			0
$283$	−24.0000			−1.42665			−0.713326	−	0.700832i	$-0.752812\pi$
$283$	−24.0000			−1.42665			−0.713326	+	0.700832i	$0.752812\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	13.4744			0.795371
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$			11.7082i			0.684001i	0.939700	+	0.342000i	$0.111104\pi$
$293$			11.7082i			0.684001i	−0.939700	+	0.342000i	$0.888896\pi$
$294$		0			0
$295$	2.11146			0.122934
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−20.4667	−	25.8885i	−1.18362	−	1.49717i
$300$		0			0
$301$			15.6352i			0.901196i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	10.4721i		−	0.599633i
$306$		0			0
$307$		−	22.2148i		−	1.26786i	−0.773389	−	0.633932i	$-0.781440\pi$
$307$		−	22.2148i		−	1.26786i	0.773389	−	0.633932i	$-0.218560\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.83153			−0.273971			−0.136985	−	0.990573i	$-0.543741\pi$
$311$	−4.83153			−0.273971			−0.136985	+	0.990573i	$0.543741\pi$
$312$		0			0
$313$	−20.4721			−1.15715			−0.578577	−	0.815628i	$-0.696392\pi$
$313$	−20.4721			−1.15715			−0.578577	+	0.815628i	$0.696392\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			22.7639i			1.27855i	0.768978	+	0.639275i	$0.220765\pi$
$317$			22.7639i			1.27855i	−0.768978	+	0.639275i	$0.779235\pi$
$318$		0			0
$319$			11.3137i			0.633446i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			41.8885i			2.33074i
$324$		0			0
$325$	−7.76393	−	9.82068i	−0.430665	−	0.544754i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−12.6491			−0.697368
$330$		0			0
$331$		−	3.90879i		−	0.214847i	−0.994213	−	0.107423i	$-0.965740\pi$
$331$		−	3.90879i		−	0.214847i	0.994213	−	0.107423i	$-0.0342600\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.83153			−0.263975
$336$		0			0
$337$	−13.4164			−0.730838			−0.365419	−	0.930843i	$-0.619074\pi$
$337$	−13.4164			−0.730838			−0.365419	+	0.930843i	$0.619074\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.65685			0.306336
$342$		0			0
$343$			19.1313i			1.03299i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	18.3060			0.982716			0.491358	−	0.870958i	$-0.336501\pi$
$347$	18.3060			0.982716			0.491358	+	0.870958i	$0.336501\pi$
$348$		0			0
$349$			1.33540i			0.0714824i	0.999361	+	0.0357412i	$0.0113792\pi$
$349$			1.33540i			0.0714824i	−0.999361	+	0.0357412i	$0.988621\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	2.76393i		−	0.147109i	−0.997291	−	0.0735546i	$-0.976566\pi$
$353$		−	2.76393i		−	0.147109i	0.997291	−	0.0735546i	$-0.0234343\pi$
$354$		0			0
$355$	7.05573			0.374479
$356$		0			0
$357$		0			0
$358$		0			0
$359$			20.1803i			1.06508i	0.846406	+	0.532539i	$0.178762\pi$
$359$			20.1803i			1.06508i	−0.846406	+	0.532539i	$0.821238\pi$
$360$		0			0
$361$	−35.8328			−1.88594
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	3.05573			0.159508			0.0797539	−	0.996815i	$-0.474587\pi$
$367$	3.05573			0.159508			0.0797539	+	0.996815i	$0.474587\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	16.0000i		−	0.830679i
$372$		0			0
$373$	2.58359			0.133773			0.0668867	−	0.997761i	$-0.478693\pi$
$373$	2.58359			0.133773			0.0668867	+	0.997761i	$0.478693\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.81758	−	9.88854i	−0.402626	−	0.509286i
$378$		0			0
$379$		−	7.40492i		−	0.380365i	−0.981749	−	0.190183i	$-0.939092\pi$
$379$		−	7.40492i		−	0.380365i	0.981749	−	0.190183i	$-0.0609080\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	3.81966i		−	0.195176i	−0.995227	−	0.0975878i	$-0.968887\pi$
$383$		−	3.81966i		−	0.195176i	0.995227	−	0.0975878i	$-0.0311127\pi$
$384$		0			0
$385$			6.99226i			0.356358i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.81758			0.396367			0.198184	−	0.980165i	$-0.436496\pi$
$389$	7.81758			0.396367			0.198184	+	0.980165i	$0.436496\pi$
$390$		0			0
$391$	−51.7771			−2.61848
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	1.16718i		−	0.0587274i
$396$		0			0
$397$			30.9551i			1.55359i	0.629753	+	0.776795i	$0.283156\pi$
$397$			30.9551i			1.55359i	−0.629753	+	0.776795i	$0.716844\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	31.7082i		−	1.58343i	−0.610889	−	0.791716i	$-0.709188\pi$
$401$		−	31.7082i		−	1.58343i	0.610889	−	0.791716i	$-0.290812\pi$
$402$		0			0
$403$	−4.94427	+	3.90879i	−0.246292	+	0.194711i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.3060			0.907393
$408$		0			0
$409$			4.32145i			0.213682i	0.994276	+	0.106841i	$0.0340736\pi$
$409$			4.32145i			0.213682i	−0.994276	+	0.106841i	$0.965926\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2.98605			−0.146934
$414$		0			0
$415$	−13.8885			−0.681762
$416$		0			0
$417$		0			0
$418$		0			0
$419$	34.4512			1.68305			0.841526	−	0.540217i	$-0.181658\pi$
$419$	34.4512			1.68305			0.841526	+	0.540217i	$0.181658\pi$
$420$		0			0
$421$		−	5.65685i		−	0.275698i	−0.990453	−	0.137849i	$-0.955981\pi$
$421$		−	5.65685i		−	0.275698i	0.990453	−	0.137849i	$-0.0440189\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−19.6414			−0.952746
$426$		0			0
$427$			14.8098i			0.716698i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			18.2918i			0.881085i	0.897732	+	0.440542i	$0.145214\pi$
$431$			18.2918i			0.881085i	−0.897732	+	0.440542i	$0.854786\pi$
$432$		0			0
$433$	9.41641			0.452524			0.226262	−	0.974067i	$-0.427349\pi$
$433$	9.41641			0.452524			0.226262	+	0.974067i	$0.427349\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	67.7771i		−	3.24222i
$438$		0			0
$439$	−3.05573			−0.145842			−0.0729210	−	0.997338i	$-0.523232\pi$
$439$	−3.05573			−0.145842			−0.0729210	+	0.997338i	$0.523232\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.1452			−0.767083			−0.383542	−	0.923524i	$-0.625296\pi$
$443$	−16.1452			−0.767083			−0.383542	+	0.923524i	$0.625296\pi$
$444$		0			0
$445$	−13.3050			−0.630715
$446$		0			0
$447$		0			0
$448$		0			0
$449$			16.2918i			0.768857i	0.923155	+	0.384429i	$0.125602\pi$
$449$			16.2918i			0.768857i	−0.923155	+	0.384429i	$0.874398\pi$
$450$		0			0
$451$	24.9443			1.17458
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−4.83153	−	6.11146i	−0.226506	−	0.286509i
$456$		0			0
$457$			18.3060i			0.856317i	0.903704	+	0.428158i	$0.140837\pi$
$457$			18.3060i			0.856317i	−0.903704	+	0.428158i	$0.859163\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			6.76393i			0.315028i	0.987517	+	0.157514i	$0.0503479\pi$
$461$			6.76393i			0.315028i	−0.987517	+	0.157514i	$0.949652\pi$
$462$		0			0
$463$			35.6892i			1.65862i	0.558791	+	0.829309i	$0.311265\pi$
$463$			35.6892i			1.65862i	−0.558791	+	0.829309i	$0.688735\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	22.6274			1.04707			0.523536	−	0.852004i	$-0.324613\pi$
$467$	22.6274			1.04707			0.523536	+	0.852004i	$0.324613\pi$
$468$		0			0
$469$	6.83282			0.315510
$470$		0			0
$471$		0			0
$472$		0			0
$473$			28.9443i			1.33086i
$474$		0			0
$475$		−	25.7109i		−	1.17970i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			24.7639i			1.13149i	0.824579	+	0.565746i	$0.191412\pi$
$479$			24.7639i			1.13149i	−0.824579	+	0.565746i	$0.808588\pi$
$480$		0			0
$481$	−16.0000	+	12.6491i	−0.729537	+	0.576750i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−22.6274			−1.02746
$486$		0			0
$487$			23.5502i			1.06716i	0.845750	+	0.533580i	$0.179154\pi$
$487$			23.5502i			1.06716i	−0.845750	+	0.533580i	$0.820846\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.3060			0.826137			0.413068	−	0.910700i	$-0.364457\pi$
$491$	18.3060			0.826137			0.413068	+	0.910700i	$0.364457\pi$
$492$		0			0
$493$	−19.7771			−0.890715
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−9.97831			−0.447588
$498$		0			0
$499$			21.3894i			0.957522i	0.877945	+	0.478761i	$0.158914\pi$
$499$			21.3894i			0.957522i	−0.877945	+	0.478761i	$0.841086\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		−	2.67080i		−	0.118849i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			3.70820i			0.164363i	0.996617	+	0.0821816i	$0.0261888\pi$
$509$			3.70820i			0.164363i	−0.996617	+	0.0821816i	$0.973811\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6.11146i			0.269303i
$516$		0			0
$517$	−23.4164			−1.02985
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−28.2843			−1.23916			−0.619578	−	0.784935i	$-0.712696\pi$
$521$	−28.2843			−1.23916			−0.619578	+	0.784935i	$0.712696\pi$
$522$		0			0
$523$	24.9443			1.09074			0.545368	−	0.838196i	$-0.316390\pi$
$523$	24.9443			1.09074			0.545368	+	0.838196i	$0.316390\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			9.88854i			0.430752i
$528$		0			0
$529$	60.7771			2.64248
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−21.8021	+	17.2361i	−0.944353	+	0.746577i
$534$		0			0
$535$		−	2.67080i		−	0.115469i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			12.7639i			0.549781i
$540$		0			0
$541$			9.97831i			0.429001i	0.976724	+	0.214500i	$0.0688123\pi$
$541$			9.97831i			0.429001i	−0.976724	+	0.214500i	$0.931188\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.6352			−0.669737
$546$		0			0
$547$	−8.94427			−0.382429			−0.191215	−	0.981548i	$-0.561243\pi$
$547$	−8.94427			−0.382429			−0.191215	+	0.981548i	$0.561243\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	25.8885i		−	1.10289i
$552$		0			0
$553$			1.65065i			0.0701927i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	38.1803i		−	1.61775i	−0.587979	−	0.808876i	$-0.700076\pi$
$557$		−	38.1803i		−	1.61775i	0.587979	−	0.808876i	$-0.299924\pi$
$558$		0			0
$559$	−20.0000	−	25.2982i	−0.845910	−	1.07000i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−34.4512			−1.45195			−0.725973	−	0.687724i	$-0.758610\pi$
$563$	−34.4512			−1.45195			−0.725973	+	0.687724i	$0.758610\pi$
$564$		0			0
$565$		−	15.6352i		−	0.657776i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	40.9334			1.71602			0.858008	−	0.513636i	$-0.171702\pi$
$569$	40.9334			1.71602			0.858008	+	0.513636i	$0.171702\pi$
$570$		0			0
$571$	−24.0000			−1.00437			−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000			−1.00437			−0.502184	+	0.864761i	$0.667470\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	31.7804			1.32533
$576$		0			0
$577$			6.99226i			0.291091i	0.989352	+	0.145546i	$0.0464938\pi$
$577$			6.99226i			0.291091i	−0.989352	+	0.145546i	$0.953506\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	19.6414			0.814861
$582$		0			0
$583$		−	29.6197i		−	1.22672i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			37.4853i			1.54718i	0.633684	+	0.773592i	$0.281542\pi$
$587$			37.4853i			1.54718i	−0.633684	+	0.773592i	$0.718458\pi$
$588$		0			0
$589$	−12.9443			−0.533359
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	2.76393i		−	0.113501i	−0.998388	−	0.0567505i	$-0.981926\pi$
$593$		−	2.76393i		−	0.113501i	0.998388	−	0.0567505i	$-0.0180740\pi$
$594$		0			0
$595$	−12.2229			−0.501091
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−4.32145			−0.176570			−0.0882849	−	0.996095i	$-0.528139\pi$
$599$	−4.32145			−0.176570			−0.0882849	+	0.996095i	$0.528139\pi$
$600$		0			0
$601$	11.5279			0.470231			0.235116	−	0.971967i	$-0.424453\pi$
$601$	11.5279			0.470231			0.235116	+	0.971967i	$0.424453\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	0.652476i		−	0.0265269i
$606$		0			0
$607$	11.0557			0.448738			0.224369	−	0.974504i	$-0.427968\pi$
$607$	11.0557			0.448738			0.224369	+	0.974504i	$0.427968\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	20.4667	−	16.1803i	0.827994	−	0.654586i
$612$		0			0
$613$		−	42.2688i		−	1.70722i	−0.520913	−	0.853610i	$-0.674408\pi$
$613$		−	42.2688i		−	1.70722i	0.520913	−	0.853610i	$-0.325592\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	17.5967i		−	0.708418i	−0.935166	−	0.354209i	$-0.884750\pi$
$617$		−	17.5967i		−	0.708418i	0.935166	−	0.354209i	$-0.115250\pi$
$618$		0			0
$619$			21.3894i			0.859714i	0.902897	+	0.429857i	$0.141436\pi$
$619$			21.3894i			0.859714i	−0.902897	+	0.429857i	$0.858564\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	18.8160			0.753849
$624$		0			0
$625$	4.41641			0.176656
$626$		0			0
$627$		0			0
$628$		0			0
$629$			32.0000i			1.27592i
$630$		0			0
$631$			32.1931i			1.28159i	0.767714	+	0.640793i	$0.221394\pi$
$631$			32.1931i			1.28159i	−0.767714	+	0.640793i	$0.778606\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			11.0557i			0.438733i
$636$		0			0
$637$	−8.81966	−	11.1561i	−0.349448	−	0.442020i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.2843			1.11716			0.558581	−	0.829450i	$-0.311346\pi$
$641$	28.2843			1.11716			0.558581	+	0.829450i	$0.311346\pi$
$642$		0			0
$643$			0.412662i			0.0162738i	0.999967	+	0.00813690i	$0.00259008\pi$
$643$			0.412662i			0.0162738i	−0.999967	+	0.00813690i	$0.997410\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.3060			0.719682			0.359841	−	0.933014i	$-0.382831\pi$
$647$	18.3060			0.719682			0.359841	+	0.933014i	$0.382831\pi$
$648$		0			0
$649$	−5.52786			−0.216988
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−17.4806			−0.684070			−0.342035	−	0.939687i	$-0.611116\pi$
$653$	−17.4806			−0.684070			−0.342035	+	0.939687i	$0.611116\pi$
$654$		0			0
$655$			2.67080i			0.104357i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			19.6414i			0.763961i	0.924170	+	0.381980i	$0.124758\pi$
$661$			19.6414i			0.763961i	−0.924170	+	0.381980i	$0.875242\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	16.0000i		−	0.620453i
$666$		0			0
$667$	32.0000			1.23904
$668$		0			0
$669$		0			0
$670$		0			0
$671$			27.4164i			1.05840i
$672$		0			0
$673$	−21.0557			−0.811639			−0.405819	−	0.913953i	$-0.633014\pi$
$673$	−21.0557			−0.811639			−0.405819	+	0.913953i	$0.633014\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−2.16073			−0.0830434			−0.0415217	−	0.999138i	$-0.513221\pi$
$677$	−2.16073			−0.0830434			−0.0415217	+	0.999138i	$0.513221\pi$
$678$		0			0
$679$	32.0000			1.22805
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	43.5967i		−	1.66818i	−0.551626	−	0.834092i	$-0.685992\pi$
$683$		−	43.5967i		−	1.66818i	0.551626	−	0.834092i	$-0.314008\pi$
$684$		0			0
$685$	25.5279			0.975370
$686$		0			0
$687$		0			0
$688$		0			0
$689$	20.4667	+	25.8885i	0.779719	+	0.986275i
$690$		0			0
$691$		−	3.90879i		−	0.148697i	−0.997232	−	0.0743487i	$-0.976312\pi$
$691$		−	3.90879i		−	0.148697i	0.997232	−	0.0743487i	$-0.0236878\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	20.9443i		−	0.794462i
$696$		0			0
$697$			43.6042i			1.65163i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.81758			−0.295266			−0.147633	−	0.989042i	$-0.547165\pi$
$701$	−7.81758			−0.295266			−0.147633	+	0.989042i	$0.547165\pi$
$702$		0			0
$703$	−41.8885			−1.57986
$704$		0			0
$705$		0			0
$706$		0			0
$707$			3.77709i			0.142052i
$708$		0			0
$709$		−	28.2843i		−	1.06224i	−0.847297	−	0.531119i	$-0.821772\pi$
$709$		−	28.2843i		−	1.06224i	0.847297	−	0.531119i	$-0.178228\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	16.0000i		−	0.599205i
$714$		0			0
$715$	−8.94427	−	11.3137i	−0.334497	−	0.423109i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−31.7804			−1.18521			−0.592604	−	0.805494i	$-0.701900\pi$
$719$	−31.7804			−1.18521			−0.592604	+	0.805494i	$0.701900\pi$
$720$		0			0
$721$		−	8.64290i		−	0.321879i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	12.1390			0.450832
$726$		0			0
$727$	52.7214			1.95533			0.977663	−	0.210176i	$-0.0674038\pi$
$727$	52.7214			1.95533			0.977663	+	0.210176i	$0.0674038\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−50.5964			−1.87138
$732$		0			0
$733$			49.2610i			1.81950i	0.415159	+	0.909749i	$0.363726\pi$
$733$			49.2610i			1.81950i	−0.415159	+	0.909749i	$0.636274\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.6491			0.465936
$738$		0			0
$739$		−	40.5207i		−	1.49058i	−0.666741	−	0.745289i	$-0.732311\pi$
$739$		−	40.5207i		−	1.49058i	0.666741	−	0.745289i	$-0.267689\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	9.12461i		−	0.334750i	−0.985893	−	0.167375i	$-0.946471\pi$
$743$		−	9.12461i		−	0.334750i	0.985893	−	0.167375i	$-0.0535290\pi$
$744$		0			0
$745$	17.5279			0.642171
$746$		0			0
$747$		0			0
$748$		0			0
$749$			3.77709i			0.138012i
$750$		0			0
$751$	−12.9443			−0.472343			−0.236172	−	0.971711i	$-0.575893\pi$
$751$	−12.9443			−0.472343			−0.236172	+	0.971711i	$0.575893\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−2.16073			−0.0786369
$756$		0			0
$757$	41.4164			1.50530			0.752652	−	0.658418i	$-0.228774\pi$
$757$	41.4164			1.50530			0.752652	+	0.658418i	$0.228774\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			20.6525i			0.748652i	0.927297	+	0.374326i	$0.122126\pi$
$761$			20.6525i			0.748652i	−0.927297	+	0.374326i	$0.877874\pi$
$762$		0			0
$763$	22.1115			0.800488
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.83153	−	3.81966i	0.174456	−	0.137920i
$768$		0			0
$769$		−	11.3137i		−	0.407983i	−0.978973	−	0.203991i	$-0.934609\pi$
$769$		−	11.3137i		−	0.407983i	0.978973	−	0.203991i	$-0.0653915\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			11.1246i			0.400124i	0.979783	+	0.200062i	$0.0641144\pi$
$773$			11.1246i			0.400124i	−0.979783	+	0.200062i	$0.935886\pi$
$774$		0			0
$775$		−	6.06952i		−	0.218023i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−57.0786			−2.04505
$780$		0			0
$781$	−18.4721			−0.660985
$782$		0			0
$783$		0			0
$784$		0			0
$785$			13.5279i			0.482830i
$786$		0			0
$787$			11.7264i			0.418000i	0.977916	+	0.209000i	$0.0670209\pi$
$787$			11.7264i			0.418000i	−0.977916	+	0.209000i	$0.932979\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			22.1115i			0.786193i
$792$		0			0
$793$	−18.9443	−	23.9628i	−0.672731	−	0.850945i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−26.1235			−0.925343			−0.462672	−	0.886530i	$-0.653109\pi$
$797$	−26.1235			−0.925343			−0.462672	+	0.886530i	$0.653109\pi$
$798$		0			0
$799$		−	40.9334i		−	1.44812i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	19.7771			0.697051
$806$		0			0
$807$		0			0
$808$		0			0
$809$	42.2688			1.48609			0.743046	−	0.669241i	$-0.233381\pi$
$809$	42.2688			1.48609			0.743046	+	0.669241i	$0.233381\pi$
$810$		0			0
$811$		−	37.0246i		−	1.30011i	−0.759887	−	0.650055i	$-0.774746\pi$
$811$		−	37.0246i		−	1.30011i	0.759887	−	0.650055i	$-0.225254\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.510078			−0.0178673
$816$		0			0
$817$		−	66.2316i		−	2.31715i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			8.06888i			0.281606i	0.990038	+	0.140803i	$0.0449684\pi$
$821$			8.06888i			0.281606i	−0.990038	+	0.140803i	$0.955032\pi$
$822$		0			0
$823$	−12.9443			−0.451209			−0.225604	−	0.974219i	$-0.572436\pi$
$823$	−12.9443			−0.451209			−0.225604	+	0.974219i	$0.572436\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	1.70820i		−	0.0594001i	−0.999559	−	0.0297000i	$-0.990545\pi$
$827$		−	1.70820i		−	0.0594001i	0.999559	−	0.0297000i	$-0.00945521\pi$
$828$		0			0
$829$	−24.4721			−0.849952			−0.424976	−	0.905205i	$-0.639718\pi$
$829$	−24.4721			−0.849952			−0.424976	+	0.905205i	$0.639718\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−22.3122			−0.773071
$834$		0			0
$835$	18.8328			0.651736
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	35.0132i		−	1.20879i	−0.796685	−	0.604394i	$-0.793415\pi$
$839$		−	35.0132i		−	1.20879i	0.796685	−	0.604394i	$-0.206585\pi$
$840$		0			0
$841$	−16.7771			−0.578520
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15.6352	+	3.70820i	0.537866	+	0.127566i
$846$		0			0
$847$			0.922740i			0.0317057i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	51.7771i		−	1.77490i
$852$		0			0
$853$		−	16.9706i		−	0.581061i	−0.956866	−	0.290531i	$-0.906168\pi$
$853$		−	16.9706i		−	0.581061i	0.956866	−	0.290531i	$-0.0938318\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.99226			−0.238851			−0.119425	−	0.992843i	$-0.538105\pi$
$857$	−6.99226			−0.238851			−0.119425	+	0.992843i	$0.538105\pi$
$858$		0			0
$859$	25.8885			0.883306			0.441653	−	0.897186i	$-0.354392\pi$
$859$	25.8885			0.883306			0.441653	+	0.897186i	$0.354392\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	44.5410i		−	1.51619i	−0.652142	−	0.758097i	$-0.726129\pi$
$863$		−	44.5410i		−	1.51619i	0.652142	−	0.758097i	$-0.273871\pi$
$864$		0			0
$865$		−	25.2982i		−	0.860165i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			3.05573i			0.103658i
$870$		0			0
$871$	−11.0557	+	8.74032i	−0.374609	+	0.296154i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	18.3060			0.618855
$876$		0			0
$877$		−	12.6491i		−	0.427130i	−0.976929	−	0.213565i	$-0.931492\pi$
$877$		−	12.6491i		−	0.427130i	0.976929	−	0.213565i	$-0.0685075\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−50.5964			−1.70464			−0.852319	−	0.523023i	$-0.824804\pi$
$881$	−50.5964			−1.70464			−0.852319	+	0.523023i	$0.824804\pi$
$882$		0			0
$883$	−33.8885			−1.14044			−0.570220	−	0.821492i	$-0.693142\pi$
$883$	−33.8885			−1.14044			−0.570220	+	0.821492i	$0.693142\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−27.4589			−0.921981			−0.460991	−	0.887405i	$-0.652506\pi$
$887$	−27.4589			−0.921981			−0.460991	+	0.887405i	$0.652506\pi$
$888$		0			0
$889$		−	15.6352i		−	0.524386i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	53.5825			1.79307
$894$		0			0
$895$			25.2982i			0.845626i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	6.11146i		−	0.203828i
$900$		0			0
$901$	51.7771			1.72494
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	3.63932i		−	0.120975i
$906$		0			0
$907$	−57.8885			−1.92216			−0.961079	−	0.276274i	$-0.910900\pi$
$907$	−57.8885			−1.92216			−0.961079	+	0.276274i	$0.910900\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−13.4744			−0.446428			−0.223214	−	0.974769i	$-0.571655\pi$
$911$	−13.4744			−0.446428			−0.223214	+	0.974769i	$0.571655\pi$
$912$		0			0
$913$	36.3607			1.20336
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	3.77709i		−	0.124730i
$918$		0			0
$919$	52.7214			1.73912			0.869559	−	0.493830i	$-0.164403\pi$
$919$	52.7214			1.73912			0.869559	+	0.493830i	$0.164403\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16.1452	−	12.7639i	0.531427	−	0.420130i
$924$		0			0
$925$		−	19.6414i		−	0.645804i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			23.1246i			0.758694i	0.925255	+	0.379347i	$0.123851\pi$
$929$			23.1246i			0.758694i	−0.925255	+	0.379347i	$0.876149\pi$
$930$		0			0
$931$		−	29.2070i		−	0.957221i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−22.6274			−0.739996
$936$		0			0
$937$	−24.4721			−0.799470			−0.399735	−	0.916631i	$-0.630898\pi$
$937$	−24.4721			−0.799470			−0.399735	+	0.916631i	$0.630898\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	9.81966i		−	0.320112i	−0.987108	−	0.160056i	$-0.948833\pi$
$941$		−	9.81966i		−	0.320112i	0.987108	−	0.160056i	$-0.0511674\pi$
$942$		0			0
$943$		−	70.5531i		−	2.29752i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			9.70820i			0.315474i	0.987481	+	0.157737i	$0.0504199\pi$
$947$			9.70820i			0.315474i	−0.987481	+	0.157737i	$0.949580\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20.9768			0.679504			0.339752	−	0.940515i	$-0.389657\pi$
$953$	20.9768			0.679504			0.339752	+	0.940515i	$0.389657\pi$
$954$		0			0
$955$			16.6553i			0.538953i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−36.1019			−1.16579
$960$		0			0
$961$	27.9443			0.901428
$962$		0			0
$963$		0			0
$964$		0			0
$965$	13.9845			0.450177
$966$		0			0
$967$			16.5579i			0.532466i	0.963909	+	0.266233i	$0.0857791\pi$
$967$			16.5579i			0.532466i	−0.963909	+	0.266233i	$0.914221\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.6274			−0.726148			−0.363074	−	0.931760i	$-0.618273\pi$
$971$	−22.6274			−0.726148			−0.363074	+	0.931760i	$0.618273\pi$
$972$		0			0
$973$			29.6197i			0.949563i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			12.6525i			0.404789i	0.979304	+	0.202394i	$0.0648723\pi$
$977$			12.6525i			0.404789i	−0.979304	+	0.202394i	$0.935128\pi$
$978$		0			0
$979$	34.8328			1.11326
$980$		0			0
$981$		0			0
$982$		0			0
$983$			46.4296i			1.48087i	0.672126	+	0.740437i	$0.265381\pi$
$983$			46.4296i			1.48087i	−0.672126	+	0.740437i	$0.734619\pi$
$984$		0			0
$985$	−17.5279			−0.558484
$986$		0			0
$987$		0			0
$988$		0			0
$989$	81.8668			2.60321
$990$		0			0
$991$	35.0557			1.11358			0.556791	−	0.830653i	$-0.312032\pi$
$991$	35.0557			1.11358			0.556791	+	0.830653i	$0.312032\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			11.0557i			0.350490i
$996$		0			0
$997$	57.7771			1.82982			0.914909	−	0.403659i	$-0.132262\pi$
$997$	57.7771			1.82982			0.914909	+	0.403659i	$0.132262\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1872.2.c.l.1585.4		8
3.2	odd	2	inner	1872.2.c.l.1585.6		8
4.3	odd	2		936.2.c.e.649.3	✓	8
12.11	even	2		936.2.c.e.649.5	yes	8
13.12	even	2	inner	1872.2.c.l.1585.5		8
39.38	odd	2	inner	1872.2.c.l.1585.3		8
52.51	odd	2		936.2.c.e.649.6	yes	8
156.155	even	2		936.2.c.e.649.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.c.e.649.3	✓	8	4.3	odd	2
936.2.c.e.649.4	yes	8	156.155	even	2
936.2.c.e.649.5	yes	8	12.11	even	2
936.2.c.e.649.6	yes	8	52.51	odd	2
1872.2.c.l.1585.3		8	39.38	odd	2	inner
1872.2.c.l.1585.4		8	1.1	even	1	trivial
1872.2.c.l.1585.5		8	13.12	even	2	inner
1872.2.c.l.1585.6		8	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:	\( N \)	\(=\)	\( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1872.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.23607i q^{5} +1.74806i q^{7} +3.23607i q^{11} +(-2.23607 - 2.82843i) q^{13} -5.65685 q^{17} -7.40492i q^{19} +9.15298 q^{23} +3.47214 q^{25} +3.49613 q^{29} -1.74806i q^{31} +2.16073 q^{35} -5.65685i q^{37} -7.70820i q^{41} +8.94427 q^{43} +7.23607i q^{47} +3.94427 q^{49} -9.15298 q^{53} +4.00000 q^{55} +1.70820i q^{59} +8.47214 q^{61} +(-3.49613 + 2.76393i) q^{65} -3.90879i q^{67} +5.70820i q^{71} -5.65685 q^{77} +0.944272 q^{79} -11.2361i q^{83} +6.99226i q^{85} -10.7639i q^{89} +(4.94427 - 3.90879i) q^{91} -9.15298 q^{95} -18.3060i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{25} - 40 q^{49} + 32 q^{55} + 32 q^{61} - 64 q^{79} - 32 q^{91}+O(q^{100}) \) 8 * q - 8 * q^25 - 40 * q^49 + 32 * q^55 + 32 * q^61 - 64 * q^79 - 32 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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