LMFDB - Embedded newform 936.2.c.e.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [936,2,Mod(649,936)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(936, 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("936.649"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.c (of order $2$, degree $1$, 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:	$7.47399762919$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	936.649
Dual form			936.2.c.e.649.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.23607i q^{5} -4.57649i q^{7} -1.23607i q^{11} +(2.23607 + 2.82843i) q^{13} -5.65685 q^{17} -1.08036i q^{19} +3.49613 q^{23} -5.47214 q^{25} -9.15298 q^{29} +4.57649i q^{31} -14.8098 q^{35} +5.65685i q^{37} -5.70820i q^{41} +8.94427 q^{43} +2.76393i q^{47} -13.9443 q^{49} +3.49613 q^{53} -4.00000 q^{55} -11.7082i q^{59} -0.472136 q^{61} +(9.15298 - 7.23607i) q^{65} -10.2333i q^{67} -7.70820i q^{71} -5.65685 q^{77} +16.9443 q^{79} -6.76393i q^{83} +18.3060i q^{85} +15.2361i q^{89} +(12.9443 - 10.2333i) q^{91} -3.49613 q^{95} -6.99226i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

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

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
$5$		−	3.23607i		−	1.44721i	−0.690212	−	0.723607i	$-0.742483\pi$
$5$		−	3.23607i		−	1.44721i	0.690212	−	0.723607i	$-0.257517\pi$
$6$		0			0
$7$		−	4.57649i		−	1.72975i	−0.501986	−	0.864876i	$-0.667397\pi$
$7$		−	4.57649i		−	1.72975i	0.501986	−	0.864876i	$-0.332603\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	1.23607i		−	0.372689i	−0.982485	−	0.186344i	$-0.940336\pi$
$11$		−	1.23607i		−	0.372689i	0.982485	−	0.186344i	$-0.0596640\pi$
$12$		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
$19$		−	1.08036i		−	0.247852i	−0.992291	−	0.123926i	$-0.960451\pi$
$19$		−	1.08036i		−	0.247852i	0.992291	−	0.123926i	$-0.0395486\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	3.49613			0.728993			0.364497	−	0.931205i	$-0.381241\pi$
$23$	3.49613			0.728993			0.364497	+	0.931205i	$0.381241\pi$
$24$		0			0
$25$	−5.47214			−1.09443
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−9.15298			−1.69967			−0.849833	−	0.527052i	$-0.823297\pi$
$29$	−9.15298			−1.69967			−0.849833	+	0.527052i	$0.823297\pi$
$30$		0			0
$31$			4.57649i			0.821962i	0.911644	+	0.410981i	$0.134814\pi$
$31$			4.57649i			0.821962i	−0.911644	+	0.410981i	$0.865186\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−14.8098			−2.50332
$36$		0			0
$37$			5.65685i			0.929981i	0.885316	+	0.464991i	$0.153942\pi$
$37$			5.65685i			0.929981i	−0.885316	+	0.464991i	$0.846058\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.70820i		−	0.891472i	−0.895165	−	0.445736i	$-0.852942\pi$
$41$		−	5.70820i		−	0.891472i	0.895165	−	0.445736i	$-0.147058\pi$
$42$		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
$47$			2.76393i			0.403161i	0.979472	+	0.201580i	$0.0646078\pi$
$47$			2.76393i			0.403161i	−0.979472	+	0.201580i	$0.935392\pi$
$48$		0			0
$49$	−13.9443			−1.99204
$50$		0			0
$51$		0			0
$52$		0			0
$53$	3.49613			0.480230			0.240115	−	0.970744i	$-0.422815\pi$
$53$	3.49613			0.480230			0.240115	+	0.970744i	$0.422815\pi$
$54$		0			0
$55$	−4.00000			−0.539360
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	11.7082i		−	1.52428i	−0.647413	−	0.762139i	$-0.724149\pi$
$59$		−	11.7082i		−	1.52428i	0.647413	−	0.762139i	$-0.275851\pi$
$60$		0			0
$61$	−0.472136			−0.0604508			−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136			−0.0604508			−0.0302254	+	0.999543i	$0.509623\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	9.15298	−	7.23607i	1.13529	−	0.897524i
$66$		0			0
$67$		−	10.2333i		−	1.25020i	−0.780544	−	0.625101i	$-0.785058\pi$
$67$		−	10.2333i		−	1.25020i	0.780544	−	0.625101i	$-0.214942\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.70820i		−	0.914796i	−0.889262	−	0.457398i	$-0.848782\pi$
$71$		−	7.70820i		−	0.914796i	0.889262	−	0.457398i	$-0.151218\pi$
$72$		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
$79$	16.9443			1.90638			0.953190	−	0.302373i	$-0.0977787\pi$
$79$	16.9443			1.90638			0.953190	+	0.302373i	$0.0977787\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	6.76393i		−	0.742438i	−0.928545	−	0.371219i	$-0.878940\pi$
$83$		−	6.76393i		−	0.742438i	0.928545	−	0.371219i	$-0.121060\pi$
$84$		0			0
$85$			18.3060i			1.98556i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			15.2361i			1.61502i	0.589854	+	0.807510i	$0.299185\pi$
$89$			15.2361i			1.61502i	−0.589854	+	0.807510i	$0.700815\pi$
$90$		0			0
$91$	12.9443	−	10.2333i	1.35693	−	1.07275i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.49613			−0.358695
$96$		0			0
$97$		−	6.99226i		−	0.709956i	−0.934875	−	0.354978i	$-0.884488\pi$
$97$		−	6.99226i		−	0.709956i	0.934875	−	0.354978i	$-0.115512\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	14.8098			1.47363			0.736817	−	0.676092i	$-0.236328\pi$
$101$	14.8098			1.47363			0.736817	+	0.676092i	$0.236328\pi$
$102$		0			0
$103$	−12.9443			−1.27544			−0.637719	−	0.770270i	$-0.720122\pi$
$103$	−12.9443			−1.27544			−0.637719	+	0.770270i	$0.720122\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.8098			−1.43172			−0.715860	−	0.698243i	$-0.753965\pi$
$107$	−14.8098			−1.43172			−0.715860	+	0.698243i	$0.753965\pi$
$108$		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	$-0.702834\pi$
$113$	−12.6491			−1.18993			−0.594964	+	0.803752i	$0.702834\pi$
$114$		0			0
$115$		−	11.3137i		−	1.05501i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			25.8885i			2.37320i
$120$		0			0
$121$	9.47214			0.861103
$122$		0			0
$123$		0			0
$124$		0			0
$125$			1.52786i			0.136656i
$126$		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
$131$	14.8098			1.29394			0.646971	−	0.762515i	$-0.276036\pi$
$131$	14.8098			1.29394			0.646971	+	0.762515i	$0.276036\pi$
$132$		0			0
$133$	−4.94427			−0.428723
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.6525i			0.910102i	0.890465	+	0.455051i	$0.150379\pi$
$137$			10.6525i			0.910102i	−0.890465	+	0.455051i	$0.849621\pi$
$138$		0			0
$139$	0.944272			0.0800921			0.0400460	−	0.999198i	$-0.487250\pi$
$139$	0.944272			0.0800921			0.0400460	+	0.999198i	$0.487250\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.49613	−	2.76393i	0.292361	−	0.231132i
$144$		0			0
$145$			29.6197i			2.45978i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			8.18034i			0.670160i	0.942190	+	0.335080i	$0.108763\pi$
$149$			8.18034i			0.670160i	−0.942190	+	0.335080i	$0.891237\pi$
$150$		0			0
$151$			4.57649i			0.372430i	0.982509	+	0.186215i	$0.0596220\pi$
$151$			4.57649i			0.372430i	−0.982509	+	0.186215i	$0.940378\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.8098			1.18955
$156$		0			0
$157$	6.94427			0.554213			0.277107	−	0.960839i	$-0.410624\pi$
$157$	6.94427			0.554213			0.277107	+	0.960839i	$0.410624\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	16.0000i		−	1.26098i
$162$		0			0
$163$		−	19.3863i		−	1.51845i	−0.650826	−	0.759227i	$-0.725577\pi$
$163$		−	19.3863i		−	1.51845i	0.650826	−	0.759227i	$-0.274423\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			10.7639i			0.832938i	0.909150	+	0.416469i	$0.136733\pi$
$167$			10.7639i			0.832938i	−0.909150	+	0.416469i	$0.863267\pi$
$168$		0			0
$169$	−3.00000	+	12.6491i	−0.230769	+	0.973009i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.81758			0.594360			0.297180	−	0.954822i	$-0.403954\pi$
$173$	7.81758			0.594360			0.297180	+	0.954822i	$0.403954\pi$
$174$		0			0
$175$			25.0432i			1.89309i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.81758			0.584313			0.292157	−	0.956370i	$-0.405627\pi$
$179$	7.81758			0.584313			0.292157	+	0.956370i	$0.405627\pi$
$180$		0			0
$181$	−14.9443			−1.11080			−0.555399	−	0.831584i	$-0.687435\pi$
$181$	−14.9443			−1.11080			−0.555399	+	0.831584i	$0.687435\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	18.3060			1.34588
$186$		0			0
$187$			6.99226i			0.511324i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	26.1235			1.89023			0.945117	−	0.326733i	$-0.105948\pi$
$191$	26.1235			1.89023			0.945117	+	0.326733i	$0.105948\pi$
$192$		0			0
$193$		−	11.3137i		−	0.814379i	−0.913344	−	0.407189i	$-0.866509\pi$
$193$		−	11.3137i		−	0.814379i	0.913344	−	0.407189i	$-0.133491\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	8.18034i		−	0.582825i	−0.956597	−	0.291413i	$-0.905875\pi$
$197$		−	8.18034i		−	0.582825i	0.956597	−	0.291413i	$-0.0941252\pi$
$198$		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
$203$			41.8885i			2.94000i
$204$		0			0
$205$	−18.4721			−1.29015
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1.33540			−0.0923717
$210$		0			0
$211$	−8.00000			−0.550743			−0.275371	−	0.961338i	$-0.588801\pi$
$211$	−8.00000			−0.550743			−0.275371	+	0.961338i	$0.588801\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	28.9443i		−	1.97398i
$216$		0			0
$217$	20.9443			1.42179
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−12.6491	−	16.0000i	−0.850871	−	1.07628i
$222$		0			0
$223$		−	2.41577i		−	0.161772i	−0.996723	−	0.0808858i	$-0.974225\pi$
$223$		−	2.41577i		−	0.161772i	0.996723	−	0.0808858i	$-0.0257749\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			3.70820i			0.246122i	0.992399	+	0.123061i	$0.0392711\pi$
$227$			3.70820i			0.246122i	−0.992399	+	0.123061i	$0.960729\pi$
$228$		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
$239$		−	5.81966i		−	0.376443i	−0.982127	−	0.188221i	$-0.939728\pi$
$239$		−	5.81966i		−	0.376443i	0.982127	−	0.188221i	$-0.0602722\pi$
$240$		0			0
$241$		−	6.99226i		−	0.450411i	−0.974311	−	0.225205i	$-0.927695\pi$
$241$		−	6.99226i		−	0.450411i	0.974311	−	0.225205i	$-0.0723053\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			45.1246i			2.88291i
$246$		0			0
$247$	3.05573	−	2.41577i	0.194431	−	0.153711i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.99226			0.441347			0.220674	−	0.975348i	$-0.429174\pi$
$251$	6.99226			0.441347			0.220674	+	0.975348i	$0.429174\pi$
$252$		0			0
$253$		−	4.32145i		−	0.271687i
$254$		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
$259$	25.8885			1.60864
$260$		0			0
$261$		0			0
$262$		0			0
$263$	22.6274			1.39527			0.697633	−	0.716455i	$-0.254237\pi$
$263$	22.6274			1.39527			0.697633	+	0.716455i	$0.254237\pi$
$264$		0			0
$265$		−	11.3137i		−	0.694996i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	16.1452			0.984393			0.492196	−	0.870484i	$-0.336194\pi$
$269$	16.1452			0.984393			0.492196	+	0.870484i	$0.336194\pi$
$270$		0			0
$271$			18.0509i			1.09652i	0.836309	+	0.548258i	$0.184709\pi$
$271$			18.0509i			1.09652i	−0.836309	+	0.548258i	$0.815291\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			6.76393i			0.407880i
$276$		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
$281$			21.7082i			1.29500i	0.762064	+	0.647501i	$0.224186\pi$
$281$			21.7082i			1.29500i	−0.762064	+	0.647501i	$0.775814\pi$
$282$		0			0
$283$	24.0000			1.42665			0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000			1.42665			0.713326	+	0.700832i	$0.247188\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−26.1235			−1.54202
$288$		0			0
$289$	15.0000			0.882353
$290$		0			0
$291$		0			0
$292$		0			0
$293$			1.70820i			0.0997943i	0.998754	+	0.0498972i	$0.0158894\pi$
$293$			1.70820i			0.0997943i	−0.998754	+	0.0498972i	$0.984111\pi$
$294$		0			0
$295$	−37.8885			−2.20596
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.81758	+	9.88854i	0.452102	+	0.571869i
$300$		0			0
$301$		−	40.9334i		−	2.35936i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.52786i			0.0874852i
$306$		0			0
$307$		−	3.24109i		−	0.184979i	−0.995714	−	0.0924894i	$-0.970518\pi$
$307$		−	3.24109i		−	0.184979i	0.995714	−	0.0924894i	$-0.0294824\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−33.1158			−1.87782			−0.938912	−	0.344156i	$-0.888165\pi$
$311$	−33.1158			−1.87782			−0.938912	+	0.344156i	$0.888165\pi$
$312$		0			0
$313$	−11.5279			−0.651593			−0.325797	−	0.945440i	$-0.605632\pi$
$313$	−11.5279			−0.651593			−0.325797	+	0.945440i	$0.605632\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	27.2361i		−	1.52973i	−0.644191	−	0.764865i	$-0.722806\pi$
$317$		−	27.2361i		−	1.52973i	0.644191	−	0.764865i	$-0.277194\pi$
$318$		0			0
$319$			11.3137i			0.633446i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			6.11146i			0.340051i
$324$		0			0
$325$	−12.2361	−	15.4775i	−0.678735	−	0.858539i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	12.6491			0.697368
$330$		0			0
$331$		−	10.2333i		−	0.562476i	−0.959638	−	0.281238i	$-0.909255\pi$
$331$		−	10.2333i		−	0.562476i	0.959638	−	0.281238i	$-0.0907450\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−33.1158			−1.80931
$336$		0			0
$337$	13.4164			0.730838			0.365419	−	0.930843i	$-0.380926\pi$
$337$	13.4164			0.730838			0.365419	+	0.930843i	$0.380926\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5.65685			0.306336
$342$		0			0
$343$			31.7804i			1.71598i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.99226			0.375364			0.187682	−	0.982230i	$-0.439903\pi$
$347$	6.99226			0.375364			0.187682	+	0.982230i	$0.439903\pi$
$348$		0			0
$349$			23.9628i			1.28270i	0.767248	+	0.641350i	$0.221625\pi$
$349$			23.9628i			1.28270i	−0.767248	+	0.641350i	$0.778375\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			7.23607i			0.385137i	0.981284	+	0.192569i	$0.0616818\pi$
$353$			7.23607i			0.385137i	−0.981284	+	0.192569i	$0.938318\pi$
$354$		0			0
$355$	−24.9443			−1.32390
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	2.18034i		−	0.115074i	−0.998343	−	0.0575370i	$-0.981675\pi$
$359$		−	2.18034i		−	0.115074i	0.998343	−	0.0575370i	$-0.0183247\pi$
$360$		0			0
$361$	17.8328			0.938569
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−20.9443			−1.09328			−0.546641	−	0.837367i	$-0.684094\pi$
$367$	−20.9443			−1.09328			−0.546641	+	0.837367i	$0.684094\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	16.0000i		−	0.830679i
$372$		0			0
$373$	29.4164			1.52312			0.761562	−	0.648092i	$-0.224433\pi$
$373$	29.4164			1.52312			0.761562	+	0.648092i	$0.224433\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−20.4667	−	25.8885i	−1.05409	−	1.33333i
$378$		0			0
$379$		−	1.08036i		−	0.0554945i	−0.999615	−	0.0277473i	$-0.991167\pi$
$379$		−	1.08036i		−	0.0554945i	0.999615	−	0.0277473i	$-0.00883336\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	26.1803i		−	1.33775i	−0.743374	−	0.668876i	$-0.766776\pi$
$383$		−	26.1803i		−	1.33775i	0.743374	−	0.668876i	$-0.233224\pi$
$384$		0			0
$385$			18.3060i			0.932958i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	20.4667			1.03770			0.518851	−	0.854865i	$-0.326360\pi$
$389$	20.4667			1.03770			0.518851	+	0.854865i	$0.326360\pi$
$390$		0			0
$391$	−19.7771			−1.00017
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	54.8328i		−	2.75894i
$396$		0			0
$397$			19.6414i			0.985772i	0.870094	+	0.492886i	$0.164058\pi$
$397$			19.6414i			0.985772i	−0.870094	+	0.492886i	$0.835942\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.2918i			0.913449i	0.889608	+	0.456724i	$0.150977\pi$
$401$			18.2918i			0.913449i	−0.889608	+	0.456724i	$0.849023\pi$
$402$		0			0
$403$	−12.9443	+	10.2333i	−0.644800	+	0.509759i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.99226			0.346593
$408$		0			0
$409$		−	29.6197i		−	1.46460i	−0.680983	−	0.732299i	$-0.738447\pi$
$409$		−	29.6197i		−	1.46460i	0.680983	−	0.732299i	$-0.261553\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−53.5825			−2.63662
$414$		0			0
$415$	−21.8885			−1.07447
$416$		0			0
$417$		0			0
$418$		0			0
$419$	28.7943			1.40670			0.703348	−	0.710846i	$-0.251688\pi$
$419$	28.7943			1.40670			0.703348	+	0.710846i	$0.251688\pi$
$420$		0			0
$421$			5.65685i			0.275698i	0.990453	+	0.137849i	$0.0440189\pi$
$421$			5.65685i			0.275698i	−0.990453	+	0.137849i	$0.955981\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	30.9551			1.50154
$426$		0			0
$427$			2.16073i			0.104565i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			31.7082i			1.52733i	0.645612	+	0.763665i	$0.276602\pi$
$431$			31.7082i			1.52733i	−0.645612	+	0.763665i	$0.723398\pi$
$432$		0			0
$433$	−17.4164			−0.836979			−0.418490	−	0.908222i	$-0.637440\pi$
$433$	−17.4164			−0.836979			−0.418490	+	0.908222i	$0.637440\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	3.77709i		−	0.180683i
$438$		0			0
$439$	20.9443			0.999616			0.499808	−	0.866136i	$-0.333404\pi$
$439$	20.9443			0.999616			0.499808	+	0.866136i	$0.333404\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−21.8021			−1.03585			−0.517924	−	0.855426i	$-0.673295\pi$
$443$	−21.8021			−1.03585			−0.517924	+	0.855426i	$0.673295\pi$
$444$		0			0
$445$	49.3050			2.33728
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	29.7082i		−	1.40202i	−0.713153	−	0.701008i	$-0.752734\pi$
$449$		−	29.7082i		−	1.40202i	0.713153	−	0.701008i	$-0.247266\pi$
$450$		0			0
$451$	−7.05573			−0.332241
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−33.1158	−	41.8885i	−1.55249	−	1.96377i
$456$		0			0
$457$			6.99226i			0.327084i	0.986536	+	0.163542i	$0.0522919\pi$
$457$			6.99226i			0.327084i	−0.986536	+	0.163542i	$0.947708\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	11.2361i		−	0.523316i	−0.965161	−	0.261658i	$-0.915731\pi$
$461$		−	11.2361i		−	0.523316i	0.965161	−	0.261658i	$-0.0842692\pi$
$462$		0			0
$463$			29.3646i			1.36469i	0.731030	+	0.682345i	$0.239040\pi$
$463$			29.3646i			1.36469i	−0.731030	+	0.682345i	$0.760960\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.6274			−1.04707			−0.523536	−	0.852004i	$-0.675387\pi$
$467$	−22.6274			−1.04707			−0.523536	+	0.852004i	$0.675387\pi$
$468$		0			0
$469$	−46.8328			−2.16254
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	11.0557i		−	0.508343i
$474$		0			0
$475$			5.91189i			0.271256i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			29.2361i			1.33583i	0.744238	+	0.667915i	$0.232813\pi$
$479$			29.2361i			1.33583i	−0.744238	+	0.667915i	$0.767187\pi$
$480$		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
$487$		−	20.7217i		−	0.938991i	−0.882935	−	0.469496i	$-0.844436\pi$
$487$		−	20.7217i		−	0.938991i	0.882935	−	0.469496i	$-0.155564\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.99226			0.315556			0.157778	−	0.987475i	$-0.449567\pi$
$491$	6.99226			0.315556			0.157778	+	0.987475i	$0.449567\pi$
$492$		0			0
$493$	51.7771			2.33192
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−35.2765			−1.58237
$498$		0			0
$499$		−	35.5316i		−	1.59061i	−0.606209	−	0.795306i	$-0.707310\pi$
$499$		−	35.5316i		−	1.59061i	0.606209	−	0.795306i	$-0.292690\pi$
$500$		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
$505$		−	47.9256i		−	2.13266i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			9.70820i			0.430309i	0.976580	+	0.215154i	$0.0690254\pi$
$509$			9.70820i			0.430309i	−0.976580	+	0.215154i	$0.930975\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			41.8885i			1.84583i
$516$		0			0
$517$	3.41641			0.150253
$518$		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
$523$	−7.05573			−0.308525			−0.154263	−	0.988030i	$-0.549300\pi$
$523$	−7.05573			−0.308525			−0.154263	+	0.988030i	$0.549300\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	25.8885i		−	1.12772i
$528$		0			0
$529$	−10.7771			−0.468569
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16.1452	−	12.7639i	0.699328	−	0.552867i
$534$		0			0
$535$			47.9256i			2.07201i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			17.2361i			0.742410i
$540$		0			0
$541$		−	35.2765i		−	1.51666i	−0.651873	−	0.758328i	$-0.726016\pi$
$541$		−	35.2765i		−	1.51666i	0.651873	−	0.758328i	$-0.273984\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−40.9334			−1.75339
$546$		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
$551$			9.88854i			0.421266i
$552$		0			0
$553$		−	77.5453i		−	3.29756i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			15.8197i			0.670301i	0.942165	+	0.335150i	$0.108787\pi$
$557$			15.8197i			0.670301i	−0.942165	+	0.335150i	$0.891213\pi$
$558$		0			0
$559$	20.0000	+	25.2982i	0.845910	+	1.07000i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−28.7943			−1.21354			−0.606769	−	0.794879i	$-0.707535\pi$
$563$	−28.7943			−1.21354			−0.606769	+	0.794879i	$0.707535\pi$
$564$		0			0
$565$			40.9334i			1.72208i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15.6352			0.655460			0.327730	−	0.944771i	$-0.393716\pi$
$569$	15.6352			0.655460			0.327730	+	0.944771i	$0.393716\pi$
$570$		0			0
$571$	24.0000			1.00437			0.502184	−	0.864761i	$-0.332530\pi$
$571$	24.0000			1.00437			0.502184	+	0.864761i	$0.332530\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−19.1313			−0.797830
$576$		0			0
$577$			18.3060i			0.762087i	0.924557	+	0.381044i	$0.124435\pi$
$577$			18.3060i			0.762087i	−0.924557	+	0.381044i	$0.875565\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−30.9551			−1.28423
$582$		0			0
$583$		−	4.32145i		−	0.178976i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	47.4853i		−	1.95993i	−0.199175	−	0.979964i	$-0.563826\pi$
$587$		−	47.4853i		−	1.95993i	0.199175	−	0.979964i	$-0.436174\pi$
$588$		0			0
$589$	4.94427			0.203725
$590$		0			0
$591$		0			0
$592$		0			0
$593$			7.23607i			0.297150i	0.988901	+	0.148575i	$0.0474686\pi$
$593$			7.23607i			0.297150i	−0.988901	+	0.148575i	$0.952531\pi$
$594$		0			0
$595$	83.7771			3.43453
$596$		0			0
$597$		0			0
$598$		0			0
$599$	29.6197			1.21023			0.605114	−	0.796139i	$-0.293128\pi$
$599$	29.6197			1.21023			0.605114	+	0.796139i	$0.293128\pi$
$600$		0			0
$601$	20.4721			0.835076			0.417538	−	0.908659i	$-0.362893\pi$
$601$	20.4721			0.835076			0.417538	+	0.908659i	$0.362893\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	30.6525i		−	1.24620i
$606$		0			0
$607$	−28.9443			−1.17481			−0.587406	−	0.809292i	$-0.699851\pi$
$607$	−28.9443			−1.17481			−0.587406	+	0.809292i	$0.699851\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−7.81758	+	6.18034i	−0.316265	+	0.250030i
$612$		0			0
$613$		−	8.32766i		−	0.336351i	−0.985757	−	0.168175i	$-0.946212\pi$
$613$		−	8.32766i		−	0.336351i	0.985757	−	0.168175i	$-0.0537875\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	31.5967i		−	1.27204i	−0.771674	−	0.636019i	$-0.780580\pi$
$617$		−	31.5967i		−	1.27204i	0.771674	−	0.636019i	$-0.219420\pi$
$618$		0			0
$619$		−	35.5316i		−	1.42813i	−0.700077	−	0.714067i	$-0.746851\pi$
$619$		−	35.5316i		−	1.42813i	0.700077	−	0.714067i	$-0.253149\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	69.7277			2.79358
$624$		0			0
$625$	−22.4164			−0.896656
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	32.0000i		−	1.27592i
$630$		0			0
$631$			38.5176i			1.53336i	0.642028	+	0.766681i	$0.278093\pi$
$631$			38.5176i			1.53336i	−0.642028	+	0.766681i	$0.721907\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			28.9443i			1.14862i
$636$		0			0
$637$	−31.1803	−	39.4404i	−1.23541	−	1.56268i
$638$		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
$643$			19.3863i			0.764522i	0.924054	+	0.382261i	$0.124854\pi$
$643$			19.3863i			0.764522i	−0.924054	+	0.382261i	$0.875146\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	6.99226			0.274894			0.137447	−	0.990509i	$-0.456110\pi$
$647$	6.99226			0.274894			0.137447	+	0.990509i	$0.456110\pi$
$648$		0			0
$649$	−14.4721			−0.568081
$650$		0			0
$651$		0			0
$652$		0			0
$653$	45.7649			1.79092			0.895460	−	0.445143i	$-0.146847\pi$
$653$	45.7649			1.79092			0.895460	+	0.445143i	$0.146847\pi$
$654$		0			0
$655$		−	47.9256i		−	1.87261i
$656$		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
$661$			30.9551i			1.20401i	0.798491	+	0.602006i	$0.205632\pi$
$661$			30.9551i			1.20401i	−0.798491	+	0.602006i	$0.794368\pi$
$662$		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
$671$			0.583592i			0.0225293i
$672$		0			0
$673$	−38.9443			−1.50119			−0.750596	−	0.660762i	$-0.770233\pi$
$673$	−38.9443			−1.50119			−0.750596	+	0.660762i	$0.770233\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−14.8098			−0.569188			−0.284594	−	0.958648i	$-0.591859\pi$
$677$	−14.8098			−0.569188			−0.284594	+	0.958648i	$0.591859\pi$
$678$		0			0
$679$	−32.0000			−1.22805
$680$		0			0
$681$		0			0
$682$		0			0
$683$			5.59675i			0.214154i	0.994251	+	0.107077i	$0.0341491\pi$
$683$			5.59675i			0.214154i	−0.994251	+	0.107077i	$0.965851\pi$
$684$		0			0
$685$	34.4721			1.31711
$686$		0			0
$687$		0			0
$688$		0			0
$689$	7.81758	+	9.88854i	0.297826	+	0.376724i
$690$		0			0
$691$		−	10.2333i		−	0.389295i	−0.980873	−	0.194647i	$-0.937644\pi$
$691$		−	10.2333i		−	0.389295i	0.980873	−	0.194647i	$-0.0623563\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	3.05573i		−	0.115910i
$696$		0			0
$697$			32.2905i			1.22309i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−20.4667			−0.773016			−0.386508	−	0.922286i	$-0.626319\pi$
$701$	−20.4667			−0.773016			−0.386508	+	0.922286i	$0.626319\pi$
$702$		0			0
$703$	6.11146			0.230498
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	67.7771i		−	2.54902i
$708$		0			0
$709$			28.2843i			1.06224i	0.847297	+	0.531119i	$0.178228\pi$
$709$			28.2843i			1.06224i	−0.847297	+	0.531119i	$0.821772\pi$
$710$		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
$719$	19.1313			0.713477			0.356738	−	0.934204i	$-0.383889\pi$
$719$	19.1313			0.713477			0.356738	+	0.934204i	$0.383889\pi$
$720$		0			0
$721$			59.2393i			2.20619i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	50.0864			1.86016
$726$		0			0
$727$	36.7214			1.36192			0.680960	−	0.732321i	$-0.261563\pi$
$727$	36.7214			1.36192			0.680960	+	0.732321i	$0.261563\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−50.5964			−1.87138
$732$		0			0
$733$			26.6336i			0.983735i	0.870670	+	0.491868i	$0.163686\pi$
$733$			26.6336i			0.983735i	−0.870670	+	0.491868i	$0.836314\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.6491			−0.465936
$738$		0			0
$739$			3.75117i			0.137989i	0.997617	+	0.0689945i	$0.0219791\pi$
$739$			3.75117i			0.137989i	−0.997617	+	0.0689945i	$0.978021\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			31.1246i			1.14185i	0.821002	+	0.570926i	$0.193416\pi$
$743$			31.1246i			1.14185i	−0.821002	+	0.570926i	$0.806584\pi$
$744$		0			0
$745$	26.4721			0.969864
$746$		0			0
$747$		0			0
$748$		0			0
$749$			67.7771i			2.47652i
$750$		0			0
$751$	−4.94427			−0.180419			−0.0902095	−	0.995923i	$-0.528754\pi$
$751$	−4.94427			−0.180419			−0.0902095	+	0.995923i	$0.528754\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	14.8098			0.538985
$756$		0			0
$757$	14.5836			0.530050			0.265025	−	0.964242i	$-0.414620\pi$
$757$	14.5836			0.530050			0.265025	+	0.964242i	$0.414620\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			10.6525i			0.386152i	0.981184	+	0.193076i	$0.0618464\pi$
$761$			10.6525i			0.386152i	−0.981184	+	0.193076i	$0.938154\pi$
$762$		0			0
$763$	−57.8885			−2.09571
$764$		0			0
$765$		0			0
$766$		0			0
$767$	33.1158	−	26.1803i	1.19574	−	0.945317i
$768$		0			0
$769$			11.3137i			0.407983i	0.978973	+	0.203991i	$0.0653915\pi$
$769$			11.3137i			0.407983i	−0.978973	+	0.203991i	$0.934609\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			29.1246i			1.04754i	0.851860	+	0.523770i	$0.175475\pi$
$773$			29.1246i			1.04754i	−0.851860	+	0.523770i	$0.824525\pi$
$774$		0			0
$775$		−	25.0432i		−	0.899578i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−6.16693			−0.220953
$780$		0			0
$781$	−9.52786			−0.340934
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	22.4721i		−	0.802065i
$786$		0			0
$787$			30.7000i			1.09434i	0.837022	+	0.547169i	$0.184295\pi$
$787$			30.7000i			1.09434i	−0.837022	+	0.547169i	$0.815705\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			57.8885i			2.05828i
$792$		0			0
$793$	−1.05573	−	1.33540i	−0.0374900	−	0.0474215i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−13.4744			−0.477289			−0.238644	−	0.971107i	$-0.576703\pi$
$797$	−13.4744			−0.477289			−0.238644	+	0.971107i	$0.576703\pi$
$798$		0			0
$799$		−	15.6352i		−	0.553132i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$	−51.7771			−1.82490
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−8.32766			−0.292785			−0.146392	−	0.989227i	$-0.546766\pi$
$809$	−8.32766			−0.292785			−0.146392	+	0.989227i	$0.546766\pi$
$810$		0			0
$811$		−	5.40182i		−	0.189683i	−0.995492	−	0.0948417i	$-0.969766\pi$
$811$		−	5.40182i		−	0.189683i	0.995492	−	0.0948417i	$-0.0302345\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−62.7355			−2.19753
$816$		0			0
$817$		−	9.66306i		−	0.338068i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			50.0689i			1.74742i	0.486451	+	0.873708i	$0.338291\pi$
$821$			50.0689i			1.74742i	−0.486451	+	0.873708i	$0.661709\pi$
$822$		0			0
$823$	−4.94427			−0.172346			−0.0861732	−	0.996280i	$-0.527464\pi$
$823$	−4.94427			−0.172346			−0.0861732	+	0.996280i	$0.527464\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			11.7082i			0.407134i	0.979061	+	0.203567i	$0.0652535\pi$
$827$			11.7082i			0.407134i	−0.979061	+	0.203567i	$0.934747\pi$
$828$		0			0
$829$	−15.5279			−0.539305			−0.269653	−	0.962958i	$-0.586909\pi$
$829$	−15.5279			−0.539305			−0.269653	+	0.962958i	$0.586909\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	78.8807			2.73305
$834$		0			0
$835$	34.8328			1.20544
$836$		0			0
$837$		0			0
$838$		0			0
$839$			41.0132i			1.41593i	0.706247	+	0.707966i	$0.250387\pi$
$839$			41.0132i			1.41593i	−0.706247	+	0.707966i	$0.749613\pi$
$840$		0			0
$841$	54.7771			1.88887
$842$		0			0
$843$		0			0
$844$		0			0
$845$	40.9334	+	9.70820i	1.40815	+	0.333972i
$846$		0			0
$847$		−	43.3491i		−	1.48949i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			19.7771i			0.677950i
$852$		0			0
$853$			16.9706i			0.581061i	0.956866	+	0.290531i	$0.0938318\pi$
$853$			16.9706i			0.581061i	−0.956866	+	0.290531i	$0.906168\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.3060			0.625320			0.312660	−	0.949865i	$-0.398780\pi$
$857$	18.3060			0.625320			0.312660	+	0.949865i	$0.398780\pi$
$858$		0			0
$859$	9.88854			0.337393			0.168696	−	0.985668i	$-0.446044\pi$
$859$	9.88854			0.337393			0.168696	+	0.985668i	$0.446044\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			22.5410i			0.767305i	0.923477	+	0.383653i	$0.125334\pi$
$863$			22.5410i			0.767305i	−0.923477	+	0.383653i	$0.874666\pi$
$864$		0			0
$865$		−	25.2982i		−	0.860165i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	20.9443i		−	0.710486i
$870$		0			0
$871$	28.9443	−	22.8825i	0.980739	−	0.775342i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	6.99226			0.236381
$876$		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	$-0.175196\pi$
$881$	50.5964			1.70464			0.852319	+	0.523023i	$0.175196\pi$
$882$		0			0
$883$	−1.88854			−0.0635546			−0.0317773	−	0.999495i	$-0.510117\pi$
$883$	−1.88854			−0.0635546			−0.0317773	+	0.999495i	$0.510117\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.4884			−0.352165			−0.176083	−	0.984375i	$-0.556343\pi$
$887$	−10.4884			−0.352165			−0.176083	+	0.984375i	$0.556343\pi$
$888$		0			0
$889$			40.9334i			1.37286i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2.98605			0.0999244
$894$		0			0
$895$		−	25.2982i		−	0.845626i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	41.8885i		−	1.39706i
$900$		0			0
$901$	−19.7771			−0.658870
$902$		0			0
$903$		0			0
$904$		0			0
$905$			48.3607i			1.60756i
$906$		0			0
$907$	22.1115			0.734199			0.367099	−	0.930182i	$-0.380351\pi$
$907$	22.1115			0.734199			0.367099	+	0.930182i	$0.380351\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	26.1235			0.865512			0.432756	−	0.901511i	$-0.357541\pi$
$911$	26.1235			0.865512			0.432756	+	0.901511i	$0.357541\pi$
$912$		0			0
$913$	−8.36068			−0.276698
$914$		0			0
$915$		0			0
$916$		0			0
$917$		−	67.7771i		−	2.23820i
$918$		0			0
$919$	36.7214			1.21133			0.605663	−	0.795721i	$-0.292908\pi$
$919$	36.7214			1.21133			0.605663	+	0.795721i	$0.292908\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	21.8021	−	17.2361i	0.717625	−	0.567332i
$924$		0			0
$925$		−	30.9551i		−	1.01780i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			17.1246i			0.561840i	0.959731	+	0.280920i	$0.0906396\pi$
$929$			17.1246i			0.561840i	−0.959731	+	0.280920i	$0.909360\pi$
$930$		0			0
$931$			15.0649i			0.493731i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	22.6274			0.739996
$936$		0			0
$937$	−15.5279			−0.507273			−0.253637	−	0.967300i	$-0.581627\pi$
$937$	−15.5279			−0.507273			−0.253637	+	0.967300i	$0.581627\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			32.1803i			1.04905i	0.851395	+	0.524525i	$0.175757\pi$
$941$			32.1803i			1.04905i	−0.851395	+	0.524525i	$0.824243\pi$
$942$		0			0
$943$		−	19.9566i		−	0.649877i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	3.70820i		−	0.120500i	−0.998183	−	0.0602502i	$-0.980810\pi$
$947$		−	3.70820i		−	0.120500i	0.998183	−	0.0602502i	$-0.0191899\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−54.9179			−1.77897			−0.889483	−	0.456969i	$-0.848935\pi$
$953$	−54.9179			−1.77897			−0.889483	+	0.456969i	$0.848935\pi$
$954$		0			0
$955$		−	84.5376i		−	2.73557i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	48.7510			1.57425
$960$		0			0
$961$	10.0557			0.324378
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−36.6119			−1.17858
$966$		0			0
$967$		−	2.41577i		−	0.0776858i	−0.999245	−	0.0388429i	$-0.987633\pi$
$967$		−	2.41577i		−	0.0776858i	0.999245	−	0.0388429i	$-0.0123672\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22.6274			0.726148			0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274			0.726148			0.363074	+	0.931760i	$0.381727\pi$
$972$		0			0
$973$		−	4.32145i		−	0.138539i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			18.6525i			0.596746i	0.954449	+	0.298373i	$0.0964439\pi$
$977$			18.6525i			0.596746i	−0.954449	+	0.298373i	$0.903556\pi$
$978$		0			0
$979$	18.8328			0.601899
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	56.4296i		−	1.79982i	−0.436072	−	0.899912i	$-0.643631\pi$
$983$		−	56.4296i		−	1.79982i	0.436072	−	0.899912i	$-0.356369\pi$
$984$		0			0
$985$	−26.4721			−0.843472
$986$		0			0
$987$		0			0
$988$		0			0
$989$	31.2703			0.994338
$990$		0			0
$991$	−52.9443			−1.68183			−0.840915	−	0.541167i	$-0.817983\pi$
$991$	−52.9443			−1.68183			−0.840915	+	0.541167i	$0.817983\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			28.9443i			0.917595i
$996$		0			0
$997$	−13.7771			−0.436325			−0.218162	−	0.975912i	$-0.570006\pi$
$997$	−13.7771			−0.436325			−0.218162	+	0.975912i	$0.570006\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	936.2.c.e.649.1	✓	8
3.2	odd	2	inner	936.2.c.e.649.7	yes	8
4.3	odd	2		1872.2.c.l.1585.2		8
12.11	even	2		1872.2.c.l.1585.8		8
13.12	even	2	inner	936.2.c.e.649.8	yes	8
39.38	odd	2	inner	936.2.c.e.649.2	yes	8
52.51	odd	2		1872.2.c.l.1585.7		8
156.155	even	2		1872.2.c.l.1585.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
936.2.c.e.649.1	✓	8	1.1	even	1	trivial
936.2.c.e.649.2	yes	8	39.38	odd	2	inner
936.2.c.e.649.7	yes	8	3.2	odd	2	inner
936.2.c.e.649.8	yes	8	13.12	even	2	inner
1872.2.c.l.1585.1		8	156.155	even	2
1872.2.c.l.1585.2		8	4.3	odd	2
1872.2.c.l.1585.7		8	52.51	odd	2
1872.2.c.l.1585.8		8	12.11	even	2

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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.23607i q^{5} -4.57649i q^{7} -1.23607i q^{11} +(2.23607 + 2.82843i) q^{13} -5.65685 q^{17} -1.08036i q^{19} +3.49613 q^{23} -5.47214 q^{25} -9.15298 q^{29} +4.57649i q^{31} -14.8098 q^{35} +5.65685i q^{37} -5.70820i q^{41} +8.94427 q^{43} +2.76393i q^{47} -13.9443 q^{49} +3.49613 q^{53} -4.00000 q^{55} -11.7082i q^{59} -0.472136 q^{61} +(9.15298 - 7.23607i) q^{65} -10.2333i q^{67} -7.70820i q^{71} -5.65685 q^{77} +16.9443 q^{79} -6.76393i q^{83} +18.3060i q^{85} +15.2361i q^{89} +(12.9443 - 10.2333i) q^{91} -3.49613 q^{95} -6.99226i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{25} - 40 q^{49} - 32 q^{55} + 32 q^{61} + 64 q^{79} + 32 q^{91}+O(q^{100}) \) 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

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