LMFDB - Embedded newform 4050.2.c.ba.649.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4050,2,Mod(649,4050)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4050, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("4050.649"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4050 = 2 \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4050.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-4,0,0,0,0,0,0,6,0,0,-2,0,4,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$32.3394128186$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{33})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 17x^{2} + 64 $ x^4 + 17*x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.4
Root			$3.37228i$ of defining polynomial
Character	$\chi$	$=$	4050.649
Dual form			4050.2.c.ba.649.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +3.37228i q^{7} -1.00000i q^{8} +4.37228 q^{11} +6.74456i q^{13} -3.37228 q^{14} +1.00000 q^{16} +1.62772i q^{17} +2.37228 q^{19} +4.37228i q^{22} +1.37228i q^{23} -6.74456 q^{26} -3.37228i q^{28} +1.37228 q^{29} +4.74456 q^{31} +1.00000i q^{32} -1.62772 q^{34} -4.00000i q^{37} +2.37228i q^{38} +3.00000 q^{41} +5.62772i q^{43} -4.37228 q^{44} -1.37228 q^{46} +7.37228i q^{47} -4.37228 q^{49} -6.74456i q^{52} -11.4891i q^{53} +3.37228 q^{56} +1.37228i q^{58} +4.37228 q^{59} -8.11684 q^{61} +4.74456i q^{62} -1.00000 q^{64} -7.00000i q^{67} -1.62772i q^{68} +6.00000 q^{71} -3.11684i q^{73} +4.00000 q^{74} -2.37228 q^{76} +14.7446i q^{77} -2.00000 q^{79} +3.00000i q^{82} +7.37228i q^{83} -5.62772 q^{86} -4.37228i q^{88} +16.1168 q^{89} -22.7446 q^{91} -1.37228i q^{92} -7.37228 q^{94} -8.37228i q^{97} -4.37228i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 6 q^{11} - 2 q^{14} + 4 q^{16} - 2 q^{19} - 4 q^{26} - 6 q^{29} - 4 q^{31} - 18 q^{34} + 12 q^{41} - 6 q^{44} + 6 q^{46} - 6 q^{49} + 2 q^{56} + 6 q^{59} + 2 q^{61} - 4 q^{64} + 24 q^{71}+ \cdots - 18 q^{94}+O(q^{100}) $ 4 * q - 4 * q^4 + 6 * q^11 - 2 * q^14 + 4 * q^16 - 2 * q^19 - 4 * q^26 - 6 * q^29 - 4 * q^31 - 18 * q^34 + 12 * q^41 - 6 * q^44 + 6 * q^46 - 6 * q^49 + 2 * q^56 + 6 * q^59 + 2 * q^61 - 4 * q^64 + 24 * q^71 + 16 * q^74 + 2 * q^76 - 8 * q^79 - 34 * q^86 + 30 * q^89 - 68 * q^91 - 18 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4050\mathbb{Z}\right)^\times$.

$n$	$2351$	$3727$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.00000i	0.707107i
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.37228i	1.27460i	0.770615	+	0.637301i	$0.219949\pi$
$7$	3.37228i	1.27460i	−0.770615	+	0.637301i	$0.780051\pi$
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	4.37228	1.31829	0.659146	−	0.752015i	$-0.270918\pi$
$11$	4.37228	1.31829	0.659146	+	0.752015i	$0.270918\pi$
$12$	0	0
$13$	6.74456i	1.87061i	0.353849	+	0.935303i	$0.384873\pi$
$13$	6.74456i	1.87061i	−0.353849	+	0.935303i	$0.615127\pi$
$14$	−3.37228	−0.901280
$15$	0	0
$16$	1.00000	0.250000
$17$	1.62772i	0.394780i	0.980325	+	0.197390i	$0.0632465\pi$
$17$	1.62772i	0.394780i	−0.980325	+	0.197390i	$0.936754\pi$
$18$	0	0
$19$	2.37228	0.544239	0.272119	−	0.962264i	$-0.412275\pi$
$19$	2.37228	0.544239	0.272119	+	0.962264i	$0.412275\pi$
$20$	0	0
$21$	0	0
$22$	4.37228i	0.932174i
$23$	1.37228i	0.286140i	0.989713	+	0.143070i	$0.0456975\pi$
$23$	1.37228i	0.286140i	−0.989713	+	0.143070i	$0.954303\pi$
$24$	0	0
$25$	0	0
$26$	−6.74456	−1.32272
$27$	0	0
$28$	− 3.37228i	− 0.637301i
$29$	1.37228	0.254826	0.127413	−	0.991850i	$-0.459333\pi$
$29$	1.37228	0.254826	0.127413	+	0.991850i	$0.459333\pi$
$30$	0	0
$31$	4.74456	0.852149	0.426074	−	0.904688i	$-0.359896\pi$
$31$	4.74456	0.852149	0.426074	+	0.904688i	$0.359896\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−1.62772	−0.279151
$35$	0	0
$36$	0	0
$37$	− 4.00000i	− 0.657596i	−0.944400	−	0.328798i	$-0.893356\pi$
$37$	− 4.00000i	− 0.657596i	0.944400	−	0.328798i	$-0.106644\pi$
$38$	2.37228i	0.384835i
$39$	0	0
$40$	0	0
$41$	3.00000	0.468521	0.234261	−	0.972174i	$-0.424733\pi$
$41$	3.00000	0.468521	0.234261	+	0.972174i	$0.424733\pi$
$42$	0	0
$43$	5.62772i	0.858219i	0.903252	+	0.429110i	$0.141173\pi$
$43$	5.62772i	0.858219i	−0.903252	+	0.429110i	$0.858827\pi$
$44$	−4.37228	−0.659146
$45$	0	0
$46$	−1.37228	−0.202332
$47$	7.37228i	1.07536i	0.843150	+	0.537679i	$0.180699\pi$
$47$	7.37228i	1.07536i	−0.843150	+	0.537679i	$0.819301\pi$
$48$	0	0
$49$	−4.37228	−0.624612
$50$	0	0
$51$	0	0
$52$	− 6.74456i	− 0.935303i
$53$	− 11.4891i	− 1.57815i	−0.614295	−	0.789076i	$-0.710560\pi$
$53$	− 11.4891i	− 1.57815i	0.614295	−	0.789076i	$-0.289440\pi$
$54$	0	0
$55$	0	0
$56$	3.37228	0.450640
$57$	0	0
$58$	1.37228i	0.180189i
$59$	4.37228	0.569223	0.284611	−	0.958643i	$-0.408135\pi$
$59$	4.37228	0.569223	0.284611	+	0.958643i	$0.408135\pi$
$60$	0	0
$61$	−8.11684	−1.03926	−0.519628	−	0.854393i	$-0.673929\pi$
$61$	−8.11684	−1.03926	−0.519628	+	0.854393i	$0.673929\pi$
$62$	4.74456i	0.602560i
$63$	0	0
$64$	−1.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	− 1.62772i	− 0.197390i
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	− 3.11684i	− 0.364799i	−0.983225	−	0.182399i	$-0.941614\pi$
$73$	− 3.11684i	− 0.364799i	0.983225	−	0.182399i	$-0.0583864\pi$
$74$	4.00000	0.464991
$75$	0	0
$76$	−2.37228	−0.272119
$77$	14.7446i	1.68030i
$78$	0	0
$79$	−2.00000	−0.225018	−0.112509	−	0.993651i	$-0.535889\pi$
$79$	−2.00000	−0.225018	−0.112509	+	0.993651i	$0.535889\pi$
$80$	0	0
$81$	0	0
$82$	3.00000i	0.331295i
$83$	7.37228i	0.809213i	0.914491	+	0.404607i	$0.132592\pi$
$83$	7.37228i	0.809213i	−0.914491	+	0.404607i	$0.867408\pi$
$84$	0	0
$85$	0	0
$86$	−5.62772	−0.606853
$87$	0	0
$88$	− 4.37228i	− 0.466087i
$89$	16.1168	1.70838	0.854191	−	0.519959i	$-0.174053\pi$
$89$	16.1168	1.70838	0.854191	+	0.519959i	$0.174053\pi$
$90$	0	0
$91$	−22.7446	−2.38428
$92$	− 1.37228i	− 0.143070i
$93$	0	0
$94$	−7.37228	−0.760393
$95$	0	0
$96$	0	0
$97$	− 8.37228i	− 0.850076i	−0.905175	−	0.425038i	$-0.860261\pi$
$97$	− 8.37228i	− 0.850076i	0.905175	−	0.425038i	$-0.139739\pi$
$98$	− 4.37228i	− 0.441667i
$99$	0	0
$100$	0	0
$101$	−2.74456	−0.273094	−0.136547	−	0.990634i	$-0.543601\pi$
$101$	−2.74456	−0.273094	−0.136547	+	0.990634i	$0.543601\pi$
$102$	0	0
$103$	16.0000i	1.57653i	0.615338	+	0.788263i	$0.289020\pi$
$103$	16.0000i	1.57653i	−0.615338	+	0.788263i	$0.710980\pi$
$104$	6.74456	0.661359
$105$	0	0
$106$	11.4891	1.11592
$107$	− 8.48913i	− 0.820675i	−0.911934	−	0.410337i	$-0.865411\pi$
$107$	− 8.48913i	− 0.820675i	0.911934	−	0.410337i	$-0.134589\pi$
$108$	0	0
$109$	−15.3723	−1.47240	−0.736199	−	0.676765i	$-0.763381\pi$
$109$	−15.3723	−1.47240	−0.736199	+	0.676765i	$0.763381\pi$
$110$	0	0
$111$	0	0
$112$	3.37228i	0.318651i
$113$	3.25544i	0.306246i	0.988207	+	0.153123i	$0.0489330\pi$
$113$	3.25544i	0.306246i	−0.988207	+	0.153123i	$0.951067\pi$
$114$	0	0
$115$	0	0
$116$	−1.37228	−0.127413
$117$	0	0
$118$	4.37228i	0.402501i
$119$	−5.48913	−0.503187
$120$	0	0
$121$	8.11684	0.737895
$122$	− 8.11684i	− 0.734865i
$123$	0	0
$124$	−4.74456	−0.426074
$125$	0	0
$126$	0	0
$127$	− 8.11684i	− 0.720253i	−0.932903	−	0.360127i	$-0.882733\pi$
$127$	− 8.11684i	− 0.720253i	0.932903	−	0.360127i	$-0.117267\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	2.74456	0.239794	0.119897	−	0.992786i	$-0.461744\pi$
$131$	2.74456	0.239794	0.119897	+	0.992786i	$0.461744\pi$
$132$	0	0
$133$	8.00000i	0.693688i
$134$	7.00000	0.604708
$135$	0	0
$136$	1.62772	0.139576
$137$	19.1168i	1.63326i	0.577160	+	0.816631i	$0.304161\pi$
$137$	19.1168i	1.63326i	−0.577160	+	0.816631i	$0.695839\pi$
$138$	0	0
$139$	−0.883156	−0.0749083	−0.0374542	−	0.999298i	$-0.511925\pi$
$139$	−0.883156	−0.0749083	−0.0374542	+	0.999298i	$0.511925\pi$
$140$	0	0
$141$	0	0
$142$	6.00000i	0.503509i
$143$	29.4891i	2.46600i
$144$	0	0
$145$	0	0
$146$	3.11684	0.257952
$147$	0	0
$148$	4.00000i	0.328798i
$149$	1.88316	0.154274	0.0771371	−	0.997020i	$-0.475422\pi$
$149$	1.88316	0.154274	0.0771371	+	0.997020i	$0.475422\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	− 2.37228i	− 0.192417i
$153$	0	0
$154$	−14.7446	−1.18815
$155$	0	0
$156$	0	0
$157$	− 6.74456i	− 0.538275i	−0.963102	−	0.269137i	$-0.913261\pi$
$157$	− 6.74456i	− 0.538275i	0.963102	−	0.269137i	$-0.0867386\pi$
$158$	− 2.00000i	− 0.159111i
$159$	0	0
$160$	0	0
$161$	−4.62772	−0.364715
$162$	0	0
$163$	21.4891i	1.68316i	0.540134	+	0.841579i	$0.318374\pi$
$163$	21.4891i	1.68316i	−0.540134	+	0.841579i	$0.681626\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	−7.37228	−0.572200
$167$	− 13.3723i	− 1.03478i	−0.855750	−	0.517389i	$-0.826904\pi$
$167$	− 13.3723i	− 1.03478i	0.855750	−	0.517389i	$-0.173096\pi$
$168$	0	0
$169$	−32.4891	−2.49916
$170$	0	0
$171$	0	0
$172$	− 5.62772i	− 0.429110i
$173$	20.7446i	1.57718i	0.614919	+	0.788590i	$0.289189\pi$
$173$	20.7446i	1.57718i	−0.614919	+	0.788590i	$0.710811\pi$
$174$	0	0
$175$	0	0
$176$	4.37228	0.329573
$177$	0	0
$178$	16.1168i	1.20801i
$179$	−14.7446	−1.10206	−0.551030	−	0.834485i	$-0.685765\pi$
$179$	−14.7446	−1.10206	−0.551030	+	0.834485i	$0.685765\pi$
$180$	0	0
$181$	20.8614	1.55062	0.775308	−	0.631583i	$-0.217595\pi$
$181$	20.8614	1.55062	0.775308	+	0.631583i	$0.217595\pi$
$182$	− 22.7446i	− 1.68594i
$183$	0	0
$184$	1.37228	0.101166
$185$	0	0
$186$	0	0
$187$	7.11684i	0.520435i
$188$	− 7.37228i	− 0.537679i
$189$	0	0
$190$	0	0
$191$	−17.4891	−1.26547	−0.632734	−	0.774369i	$-0.718067\pi$
$191$	−17.4891	−1.26547	−0.632734	+	0.774369i	$0.718067\pi$
$192$	0	0
$193$	− 21.1168i	− 1.52002i	−0.649909	−	0.760012i	$-0.725193\pi$
$193$	− 21.1168i	− 1.52002i	0.649909	−	0.760012i	$-0.274807\pi$
$194$	8.37228	0.601095
$195$	0	0
$196$	4.37228	0.312306
$197$	5.48913i	0.391084i	0.980695	+	0.195542i	$0.0626466\pi$
$197$	5.48913i	0.391084i	−0.980695	+	0.195542i	$0.937353\pi$
$198$	0	0
$199$	−13.4891	−0.956219	−0.478109	−	0.878300i	$-0.658678\pi$
$199$	−13.4891	−0.956219	−0.478109	+	0.878300i	$0.658678\pi$
$200$	0	0
$201$	0	0
$202$	− 2.74456i	− 0.193107i
$203$	4.62772i	0.324802i
$204$	0	0
$205$	0	0
$206$	−16.0000	−1.11477
$207$	0	0
$208$	6.74456i	0.467651i
$209$	10.3723	0.717466
$210$	0	0
$211$	−18.7446	−1.29043	−0.645214	−	0.764002i	$-0.723232\pi$
$211$	−18.7446	−1.29043	−0.645214	+	0.764002i	$0.723232\pi$
$212$	11.4891i	0.789076i
$213$	0	0
$214$	8.48913	0.580305
$215$	0	0
$216$	0	0
$217$	16.0000i	1.08615i
$218$	− 15.3723i	− 1.04114i
$219$	0	0
$220$	0	0
$221$	−10.9783	−0.738477
$222$	0	0
$223$	− 6.62772i	− 0.443825i	−0.975067	−	0.221912i	$-0.928770\pi$
$223$	− 6.62772i	− 0.443825i	0.975067	−	0.221912i	$-0.0712299\pi$
$224$	−3.37228	−0.225320
$225$	0	0
$226$	−3.25544	−0.216548
$227$	− 19.1168i	− 1.26883i	−0.772993	−	0.634415i	$-0.781241\pi$
$227$	− 19.1168i	− 1.26883i	0.772993	−	0.634415i	$-0.218759\pi$
$228$	0	0
$229$	−12.6277	−0.834463	−0.417232	−	0.908800i	$-0.637000\pi$
$229$	−12.6277	−0.834463	−0.417232	+	0.908800i	$0.637000\pi$
$230$	0	0
$231$	0	0
$232$	− 1.37228i	− 0.0900947i
$233$	7.11684i	0.466240i	0.972448	+	0.233120i	$0.0748935\pi$
$233$	7.11684i	0.466240i	−0.972448	+	0.233120i	$0.925107\pi$
$234$	0	0
$235$	0	0
$236$	−4.37228	−0.284611
$237$	0	0
$238$	− 5.48913i	− 0.355807i
$239$	−3.25544	−0.210577	−0.105288	−	0.994442i	$-0.533577\pi$
$239$	−3.25544	−0.210577	−0.105288	+	0.994442i	$0.533577\pi$
$240$	0	0
$241$	−12.4891	−0.804495	−0.402248	−	0.915531i	$-0.631771\pi$
$241$	−12.4891	−0.804495	−0.402248	+	0.915531i	$0.631771\pi$
$242$	8.11684i	0.521770i
$243$	0	0
$244$	8.11684	0.519628
$245$	0	0
$246$	0	0
$247$	16.0000i	1.01806i
$248$	− 4.74456i	− 0.301280i
$249$	0	0
$250$	0	0
$251$	24.6060	1.55312	0.776558	−	0.630046i	$-0.216964\pi$
$251$	24.6060	1.55312	0.776558	+	0.630046i	$0.216964\pi$
$252$	0	0
$253$	6.00000i	0.377217i
$254$	8.11684	0.509296
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 4.37228i	− 0.272735i	−0.990658	−	0.136368i	$-0.956457\pi$
$257$	− 4.37228i	− 0.272735i	0.990658	−	0.136368i	$-0.0435429\pi$
$258$	0	0
$259$	13.4891	0.838173
$260$	0	0
$261$	0	0
$262$	2.74456i	0.169560i
$263$	− 17.4891i	− 1.07843i	−0.842170	−	0.539213i	$-0.818722\pi$
$263$	− 17.4891i	− 1.07843i	0.842170	−	0.539213i	$-0.181278\pi$
$264$	0	0
$265$	0	0
$266$	−8.00000	−0.490511
$267$	0	0
$268$	7.00000i	0.427593i
$269$	−1.37228	−0.0836695	−0.0418347	−	0.999125i	$-0.513320\pi$
$269$	−1.37228	−0.0836695	−0.0418347	+	0.999125i	$0.513320\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	1.62772i	0.0986949i
$273$	0	0
$274$	−19.1168	−1.15489
$275$	0	0
$276$	0	0
$277$	16.7446i	1.00608i	0.864262	+	0.503042i	$0.167786\pi$
$277$	16.7446i	1.00608i	−0.864262	+	0.503042i	$0.832214\pi$
$278$	− 0.883156i	− 0.0529682i
$279$	0	0
$280$	0	0
$281$	1.37228	0.0818634	0.0409317	−	0.999162i	$-0.486967\pi$
$281$	1.37228	0.0818634	0.0409317	+	0.999162i	$0.486967\pi$
$282$	0	0
$283$	3.13859i	0.186570i	0.995639	+	0.0932850i	$0.0297368\pi$
$283$	3.13859i	0.186570i	−0.995639	+	0.0932850i	$0.970263\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	−29.4891	−1.74373
$287$	10.1168i	0.597178i
$288$	0	0
$289$	14.3505	0.844149
$290$	0	0
$291$	0	0
$292$	3.11684i	0.182399i
$293$	26.2337i	1.53259i	0.642490	+	0.766294i	$0.277901\pi$
$293$	26.2337i	1.53259i	−0.642490	+	0.766294i	$0.722099\pi$
$294$	0	0
$295$	0	0
$296$	−4.00000	−0.232495
$297$	0	0
$298$	1.88316i	0.109088i
$299$	−9.25544	−0.535256
$300$	0	0
$301$	−18.9783	−1.09389
$302$	− 10.0000i	− 0.575435i
$303$	0	0
$304$	2.37228	0.136060
$305$	0	0
$306$	0	0
$307$	1.23369i	0.0704103i	0.999380	+	0.0352051i	$0.0112085\pi$
$307$	1.23369i	0.0704103i	−0.999380	+	0.0352051i	$0.988792\pi$
$308$	− 14.7446i	− 0.840149i
$309$	0	0
$310$	0	0
$311$	20.7446	1.17632	0.588158	−	0.808746i	$-0.299853\pi$
$311$	20.7446	1.17632	0.588158	+	0.808746i	$0.299853\pi$
$312$	0	0
$313$	− 3.62772i	− 0.205051i	−0.994730	−	0.102525i	$-0.967308\pi$
$313$	− 3.62772i	− 0.205051i	0.994730	−	0.102525i	$-0.0326923\pi$
$314$	6.74456	0.380618
$315$	0	0
$316$	2.00000	0.112509
$317$	− 2.74456i	− 0.154150i	−0.997025	−	0.0770750i	$-0.975442\pi$
$317$	− 2.74456i	− 0.154150i	0.997025	−	0.0770750i	$-0.0245581\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	− 4.62772i	− 0.257893i
$323$	3.86141i	0.214854i
$324$	0	0
$325$	0	0
$326$	−21.4891	−1.19017
$327$	0	0
$328$	− 3.00000i	− 0.165647i
$329$	−24.8614	−1.37065
$330$	0	0
$331$	16.2337	0.892284	0.446142	−	0.894962i	$-0.352798\pi$
$331$	16.2337	0.892284	0.446142	+	0.894962i	$0.352798\pi$
$332$	− 7.37228i	− 0.404607i
$333$	0	0
$334$	13.3723	0.731699
$335$	0	0
$336$	0	0
$337$	− 2.37228i	− 0.129226i	−0.997910	−	0.0646132i	$-0.979419\pi$
$337$	− 2.37228i	− 0.129226i	0.997910	−	0.0646132i	$-0.0205814\pi$
$338$	− 32.4891i	− 1.76718i
$339$	0	0
$340$	0	0
$341$	20.7446	1.12338
$342$	0	0
$343$	8.86141i	0.478471i
$344$	5.62772	0.303426
$345$	0	0
$346$	−20.7446	−1.11523
$347$	4.88316i	0.262142i	0.991373	+	0.131071i	$0.0418415\pi$
$347$	4.88316i	0.262142i	−0.991373	+	0.131071i	$0.958158\pi$
$348$	0	0
$349$	−18.1168	−0.969772	−0.484886	−	0.874577i	$-0.661139\pi$
$349$	−18.1168	−0.969772	−0.484886	+	0.874577i	$0.661139\pi$
$350$	0	0
$351$	0	0
$352$	4.37228i	0.233043i
$353$	− 21.3505i	− 1.13637i	−0.822899	−	0.568187i	$-0.807645\pi$
$353$	− 21.3505i	− 1.13637i	0.822899	−	0.568187i	$-0.192355\pi$
$354$	0	0
$355$	0	0
$356$	−16.1168	−0.854191
$357$	0	0
$358$	− 14.7446i	− 0.779274i
$359$	−17.4891	−0.923041	−0.461520	−	0.887130i	$-0.652696\pi$
$359$	−17.4891	−0.923041	−0.461520	+	0.887130i	$0.652696\pi$
$360$	0	0
$361$	−13.3723	−0.703804
$362$	20.8614i	1.09645i
$363$	0	0
$364$	22.7446	1.19214
$365$	0	0
$366$	0	0
$367$	− 16.0000i	− 0.835193i	−0.908633	−	0.417597i	$-0.862873\pi$
$367$	− 16.0000i	− 0.835193i	0.908633	−	0.417597i	$-0.137127\pi$
$368$	1.37228i	0.0715351i
$369$	0	0
$370$	0	0
$371$	38.7446	2.01152
$372$	0	0
$373$	15.4891i	0.801997i	0.916079	+	0.400998i	$0.131337\pi$
$373$	15.4891i	0.801997i	−0.916079	+	0.400998i	$0.868663\pi$
$374$	−7.11684	−0.368003
$375$	0	0
$376$	7.37228	0.380196
$377$	9.25544i	0.476679i
$378$	0	0
$379$	−17.8614	−0.917479	−0.458739	−	0.888571i	$-0.651699\pi$
$379$	−17.8614	−0.917479	−0.458739	+	0.888571i	$0.651699\pi$
$380$	0	0
$381$	0	0
$382$	− 17.4891i	− 0.894821i
$383$	− 22.9783i	− 1.17413i	−0.809538	−	0.587067i	$-0.800283\pi$
$383$	− 22.9783i	− 1.17413i	0.809538	−	0.587067i	$-0.199717\pi$
$384$	0	0
$385$	0	0
$386$	21.1168	1.07482
$387$	0	0
$388$	8.37228i	0.425038i
$389$	−4.62772	−0.234635	−0.117317	−	0.993094i	$-0.537429\pi$
$389$	−4.62772	−0.234635	−0.117317	+	0.993094i	$0.537429\pi$
$390$	0	0
$391$	−2.23369	−0.112962
$392$	4.37228i	0.220834i
$393$	0	0
$394$	−5.48913	−0.276538
$395$	0	0
$396$	0	0
$397$	22.7446i	1.14152i	0.821118	+	0.570758i	$0.193351\pi$
$397$	22.7446i	1.14152i	−0.821118	+	0.570758i	$0.806649\pi$
$398$	− 13.4891i	− 0.676149i
$399$	0	0
$400$	0	0
$401$	1.11684	0.0557725	0.0278863	−	0.999611i	$-0.491122\pi$
$401$	1.11684	0.0557725	0.0278863	+	0.999611i	$0.491122\pi$
$402$	0	0
$403$	32.0000i	1.59403i
$404$	2.74456	0.136547
$405$	0	0
$406$	−4.62772	−0.229670
$407$	− 17.4891i	− 0.866904i
$408$	0	0
$409$	−17.8614	−0.883190	−0.441595	−	0.897215i	$-0.645587\pi$
$409$	−17.8614	−0.883190	−0.441595	+	0.897215i	$0.645587\pi$
$410$	0	0
$411$	0	0
$412$	− 16.0000i	− 0.788263i
$413$	14.7446i	0.725532i
$414$	0	0
$415$	0	0
$416$	−6.74456	−0.330679
$417$	0	0
$418$	10.3723i	0.507325i
$419$	25.7228	1.25664	0.628321	−	0.777954i	$-0.283743\pi$
$419$	25.7228	1.25664	0.628321	+	0.777954i	$0.283743\pi$
$420$	0	0
$421$	30.4674	1.48489	0.742445	−	0.669908i	$-0.233666\pi$
$421$	30.4674	1.48489	0.742445	+	0.669908i	$0.233666\pi$
$422$	− 18.7446i	− 0.912471i
$423$	0	0
$424$	−11.4891	−0.557961
$425$	0	0
$426$	0	0
$427$	− 27.3723i	− 1.32464i
$428$	8.48913i	0.410337i
$429$	0	0
$430$	0	0
$431$	−8.23369	−0.396603	−0.198301	−	0.980141i	$-0.563542\pi$
$431$	−8.23369	−0.396603	−0.198301	+	0.980141i	$0.563542\pi$
$432$	0	0
$433$	− 6.37228i	− 0.306232i	−0.988208	−	0.153116i	$-0.951069\pi$
$433$	− 6.37228i	− 0.306232i	0.988208	−	0.153116i	$-0.0489309\pi$
$434$	−16.0000	−0.768025
$435$	0	0
$436$	15.3723	0.736199
$437$	3.25544i	0.155729i
$438$	0	0
$439$	18.2337	0.870246	0.435123	−	0.900371i	$-0.356705\pi$
$439$	18.2337	0.870246	0.435123	+	0.900371i	$0.356705\pi$
$440$	0	0
$441$	0	0
$442$	− 10.9783i	− 0.522182i
$443$	− 3.51087i	− 0.166807i	−0.996516	−	0.0834033i	$-0.973421\pi$
$443$	− 3.51087i	− 0.166807i	0.996516	−	0.0834033i	$-0.0265790\pi$
$444$	0	0
$445$	0	0
$446$	6.62772	0.313832
$447$	0	0
$448$	− 3.37228i	− 0.159325i
$449$	9.86141	0.465389	0.232694	−	0.972550i	$-0.425246\pi$
$449$	9.86141	0.465389	0.232694	+	0.972550i	$0.425246\pi$
$450$	0	0
$451$	13.1168	0.617648
$452$	− 3.25544i	− 0.153123i
$453$	0	0
$454$	19.1168	0.897198
$455$	0	0
$456$	0	0
$457$	12.8832i	0.602649i	0.953522	+	0.301324i	$0.0974287\pi$
$457$	12.8832i	0.602649i	−0.953522	+	0.301324i	$0.902571\pi$
$458$	− 12.6277i	− 0.590055i
$459$	0	0
$460$	0	0
$461$	−1.88316	−0.0877073	−0.0438537	−	0.999038i	$-0.513964\pi$
$461$	−1.88316	−0.0877073	−0.0438537	+	0.999038i	$0.513964\pi$
$462$	0	0
$463$	− 20.0000i	− 0.929479i	−0.885448	−	0.464739i	$-0.846148\pi$
$463$	− 20.0000i	− 0.929479i	0.885448	−	0.464739i	$-0.153852\pi$
$464$	1.37228	0.0637066
$465$	0	0
$466$	−7.11684	−0.329681
$467$	− 43.1168i	− 1.99521i	−0.0691713	−	0.997605i	$-0.522036\pi$
$467$	− 43.1168i	− 1.99521i	0.0691713	−	0.997605i	$-0.477964\pi$
$468$	0	0
$469$	23.6060	1.09002
$470$	0	0
$471$	0	0
$472$	− 4.37228i	− 0.201251i
$473$	24.6060i	1.13138i
$474$	0	0
$475$	0	0
$476$	5.48913	0.251594
$477$	0	0
$478$	− 3.25544i	− 0.148900i
$479$	−0.510875	−0.0233425	−0.0116712	−	0.999932i	$-0.503715\pi$
$479$	−0.510875	−0.0233425	−0.0116712	+	0.999932i	$0.503715\pi$
$480$	0	0
$481$	26.9783	1.23010
$482$	− 12.4891i	− 0.568864i
$483$	0	0
$484$	−8.11684	−0.368947
$485$	0	0
$486$	0	0
$487$	− 12.7446i	− 0.577511i	−0.957403	−	0.288756i	$-0.906758\pi$
$487$	− 12.7446i	− 0.577511i	0.957403	−	0.288756i	$-0.0932415\pi$
$488$	8.11684i	0.367432i
$489$	0	0
$490$	0	0
$491$	36.6060	1.65200	0.826002	−	0.563667i	$-0.190610\pi$
$491$	36.6060	1.65200	0.826002	+	0.563667i	$0.190610\pi$
$492$	0	0
$493$	2.23369i	0.100600i
$494$	−16.0000	−0.719874
$495$	0	0
$496$	4.74456	0.213037
$497$	20.2337i	0.907605i
$498$	0	0
$499$	−15.1168	−0.676723	−0.338361	−	0.941016i	$-0.609873\pi$
$499$	−15.1168	−0.676723	−0.338361	+	0.941016i	$0.609873\pi$
$500$	0	0
$501$	0	0
$502$	24.6060i	1.09822i
$503$	− 6.86141i	− 0.305935i	−0.988231	−	0.152967i	$-0.951117\pi$
$503$	− 6.86141i	− 0.305935i	0.988231	−	0.152967i	$-0.0488830\pi$
$504$	0	0
$505$	0	0
$506$	−6.00000	−0.266733
$507$	0	0
$508$	8.11684i	0.360127i
$509$	−42.3505	−1.87715	−0.938577	−	0.345069i	$-0.887855\pi$
$509$	−42.3505	−1.87715	−0.938577	+	0.345069i	$0.887855\pi$
$510$	0	0
$511$	10.5109	0.464974
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	4.37228	0.192853
$515$	0	0
$516$	0	0
$517$	32.2337i	1.41764i
$518$	13.4891i	0.592678i
$519$	0	0
$520$	0	0
$521$	6.76631	0.296438	0.148219	−	0.988955i	$-0.452646\pi$
$521$	6.76631	0.296438	0.148219	+	0.988955i	$0.452646\pi$
$522$	0	0
$523$	− 6.11684i	− 0.267471i	−0.991017	−	0.133735i	$-0.957303\pi$
$523$	− 6.11684i	− 0.267471i	0.991017	−	0.133735i	$-0.0426972\pi$
$524$	−2.74456	−0.119897
$525$	0	0
$526$	17.4891	0.762562
$527$	7.72281i	0.336411i
$528$	0	0
$529$	21.1168	0.918124
$530$	0	0
$531$	0	0
$532$	− 8.00000i	− 0.346844i
$533$	20.2337i	0.876418i
$534$	0	0
$535$	0	0
$536$	−7.00000	−0.302354
$537$	0	0
$538$	− 1.37228i	− 0.0591632i
$539$	−19.1168	−0.823421
$540$	0	0
$541$	27.3723	1.17683	0.588413	−	0.808560i	$-0.299753\pi$
$541$	27.3723	1.17683	0.588413	+	0.808560i	$0.299753\pi$
$542$	8.00000i	0.343629i
$543$	0	0
$544$	−1.62772	−0.0697879
$545$	0	0
$546$	0	0
$547$	− 29.4674i	− 1.25993i	−0.776622	−	0.629967i	$-0.783068\pi$
$547$	− 29.4674i	− 1.25993i	0.776622	−	0.629967i	$-0.216932\pi$
$548$	− 19.1168i	− 0.816631i
$549$	0	0
$550$	0	0
$551$	3.25544	0.138686
$552$	0	0
$553$	− 6.74456i	− 0.286808i
$554$	−16.7446	−0.711408
$555$	0	0
$556$	0.883156	0.0374542
$557$	44.2337i	1.87424i	0.349005	+	0.937121i	$0.386520\pi$
$557$	44.2337i	1.87424i	−0.349005	+	0.937121i	$0.613480\pi$
$558$	0	0
$559$	−37.9565	−1.60539
$560$	0	0
$561$	0	0
$562$	1.37228i	0.0578862i
$563$	− 40.7228i	− 1.71626i	−0.513431	−	0.858131i	$-0.671626\pi$
$563$	− 40.7228i	− 1.71626i	0.513431	−	0.858131i	$-0.328374\pi$
$564$	0	0
$565$	0	0
$566$	−3.13859	−0.131925
$567$	0	0
$568$	− 6.00000i	− 0.251754i
$569$	−30.6060	−1.28307	−0.641534	−	0.767094i	$-0.721702\pi$
$569$	−30.6060	−1.28307	−0.641534	+	0.767094i	$0.721702\pi$
$570$	0	0
$571$	8.60597	0.360149	0.180074	−	0.983653i	$-0.442366\pi$
$571$	8.60597	0.360149	0.180074	+	0.983653i	$0.442366\pi$
$572$	− 29.4891i	− 1.23300i
$573$	0	0
$574$	−10.1168	−0.422269
$575$	0	0
$576$	0	0
$577$	− 41.1168i	− 1.71172i	−0.517210	−	0.855858i	$-0.673030\pi$
$577$	− 41.1168i	− 1.71172i	0.517210	−	0.855858i	$-0.326970\pi$
$578$	14.3505i	0.596903i
$579$	0	0
$580$	0	0
$581$	−24.8614	−1.03142
$582$	0	0
$583$	− 50.2337i	− 2.08047i
$584$	−3.11684	−0.128976
$585$	0	0
$586$	−26.2337	−1.08370
$587$	− 27.0000i	− 1.11441i	−0.830375	−	0.557205i	$-0.811874\pi$
$587$	− 27.0000i	− 1.11441i	0.830375	−	0.557205i	$-0.188126\pi$
$588$	0	0
$589$	11.2554	0.463772
$590$	0	0
$591$	0	0
$592$	− 4.00000i	− 0.164399i
$593$	− 19.7228i	− 0.809919i	−0.914335	−	0.404959i	$-0.867286\pi$
$593$	− 19.7228i	− 0.809919i	0.914335	−	0.404959i	$-0.132714\pi$
$594$	0	0
$595$	0	0
$596$	−1.88316	−0.0771371
$597$	0	0
$598$	− 9.25544i	− 0.378483i
$599$	−3.76631	−0.153887	−0.0769437	−	0.997035i	$-0.524516\pi$
$599$	−3.76631	−0.153887	−0.0769437	+	0.997035i	$0.524516\pi$
$600$	0	0
$601$	−1.86141	−0.0759284	−0.0379642	−	0.999279i	$-0.512087\pi$
$601$	−1.86141	−0.0759284	−0.0379642	+	0.999279i	$0.512087\pi$
$602$	− 18.9783i	− 0.773496i
$603$	0	0
$604$	10.0000	0.406894
$605$	0	0
$606$	0	0
$607$	18.1168i	0.735340i	0.929956	+	0.367670i	$0.119844\pi$
$607$	18.1168i	0.735340i	−0.929956	+	0.367670i	$0.880156\pi$
$608$	2.37228i	0.0962087i
$609$	0	0
$610$	0	0
$611$	−49.7228	−2.01157
$612$	0	0
$613$	− 34.2337i	− 1.38269i	−0.722527	−	0.691343i	$-0.757019\pi$
$613$	− 34.2337i	− 1.38269i	0.722527	−	0.691343i	$-0.242981\pi$
$614$	−1.23369	−0.0497876
$615$	0	0
$616$	14.7446	0.594075
$617$	4.88316i	0.196588i	0.995157	+	0.0982942i	$0.0313386\pi$
$617$	4.88316i	0.196588i	−0.995157	+	0.0982942i	$0.968661\pi$
$618$	0	0
$619$	20.8832	0.839365	0.419682	−	0.907671i	$-0.362141\pi$
$619$	20.8832	0.839365	0.419682	+	0.907671i	$0.362141\pi$
$620$	0	0
$621$	0	0
$622$	20.7446i	0.831781i
$623$	54.3505i	2.17751i
$624$	0	0
$625$	0	0
$626$	3.62772	0.144993
$627$	0	0
$628$	6.74456i	0.269137i
$629$	6.51087	0.259606
$630$	0	0
$631$	−23.7228	−0.944390	−0.472195	−	0.881494i	$-0.656538\pi$
$631$	−23.7228	−0.944390	−0.472195	+	0.881494i	$0.656538\pi$
$632$	2.00000i	0.0795557i
$633$	0	0
$634$	2.74456	0.109001
$635$	0	0
$636$	0	0
$637$	− 29.4891i	− 1.16840i
$638$	6.00000i	0.237542i
$639$	0	0
$640$	0	0
$641$	39.0000	1.54041	0.770204	−	0.637798i	$-0.220155\pi$
$641$	39.0000	1.54041	0.770204	+	0.637798i	$0.220155\pi$
$642$	0	0
$643$	− 11.0000i	− 0.433798i	−0.976194	−	0.216899i	$-0.930406\pi$
$643$	− 11.0000i	− 0.433798i	0.976194	−	0.216899i	$-0.0695942\pi$
$644$	4.62772	0.182358
$645$	0	0
$646$	−3.86141	−0.151925
$647$	39.0951i	1.53699i	0.639858	+	0.768493i	$0.278993\pi$
$647$	39.0951i	1.53699i	−0.639858	+	0.768493i	$0.721007\pi$
$648$	0	0
$649$	19.1168	0.750402
$650$	0	0
$651$	0	0
$652$	− 21.4891i	− 0.841579i
$653$	19.7228i	0.771813i	0.922538	+	0.385907i	$0.126111\pi$
$653$	19.7228i	0.771813i	−0.922538	+	0.385907i	$0.873889\pi$
$654$	0	0
$655$	0	0
$656$	3.00000	0.117130
$657$	0	0
$658$	− 24.8614i	− 0.969199i
$659$	17.4891	0.681280	0.340640	−	0.940194i	$-0.389356\pi$
$659$	17.4891	0.681280	0.340640	+	0.940194i	$0.389356\pi$
$660$	0	0
$661$	−12.2337	−0.475835	−0.237918	−	0.971285i	$-0.576465\pi$
$661$	−12.2337	−0.475835	−0.237918	+	0.971285i	$0.576465\pi$
$662$	16.2337i	0.630940i
$663$	0	0
$664$	7.37228	0.286100
$665$	0	0
$666$	0	0
$667$	1.88316i	0.0729161i
$668$	13.3723i	0.517389i
$669$	0	0
$670$	0	0
$671$	−35.4891	−1.37004
$672$	0	0
$673$	10.0000i	0.385472i	0.981251	+	0.192736i	$0.0617360\pi$
$673$	10.0000i	0.385472i	−0.981251	+	0.192736i	$0.938264\pi$
$674$	2.37228	0.0913769
$675$	0	0
$676$	32.4891	1.24958
$677$	− 13.7228i	− 0.527411i	−0.964603	−	0.263705i	$-0.915055\pi$
$677$	− 13.7228i	− 0.527411i	0.964603	−	0.263705i	$-0.0849446\pi$
$678$	0	0
$679$	28.2337	1.08351
$680$	0	0
$681$	0	0
$682$	20.7446i	0.794350i
$683$	− 30.0951i	− 1.15156i	−0.817606	−	0.575778i	$-0.804699\pi$
$683$	− 30.0951i	− 1.15156i	0.817606	−	0.575778i	$-0.195301\pi$
$684$	0	0
$685$	0	0
$686$	−8.86141	−0.338330
$687$	0	0
$688$	5.62772i	0.214555i
$689$	77.4891	2.95210
$690$	0	0
$691$	−36.2337	−1.37839	−0.689197	−	0.724574i	$-0.742037\pi$
$691$	−36.2337	−1.37839	−0.689197	+	0.724574i	$0.742037\pi$
$692$	− 20.7446i	− 0.788590i
$693$	0	0
$694$	−4.88316	−0.185362
$695$	0	0
$696$	0	0
$697$	4.88316i	0.184963i
$698$	− 18.1168i	− 0.685733i
$699$	0	0
$700$	0	0
$701$	−42.8614	−1.61885	−0.809426	−	0.587221i	$-0.800222\pi$
$701$	−42.8614	−1.61885	−0.809426	+	0.587221i	$0.800222\pi$
$702$	0	0
$703$	− 9.48913i	− 0.357889i
$704$	−4.37228	−0.164787
$705$	0	0
$706$	21.3505	0.803538
$707$	− 9.25544i	− 0.348087i
$708$	0	0
$709$	−2.86141	−0.107462	−0.0537312	−	0.998555i	$-0.517111\pi$
$709$	−2.86141	−0.107462	−0.0537312	+	0.998555i	$0.517111\pi$
$710$	0	0
$711$	0	0
$712$	− 16.1168i	− 0.604004i
$713$	6.51087i	0.243834i
$714$	0	0
$715$	0	0
$716$	14.7446	0.551030
$717$	0	0
$718$	− 17.4891i	− 0.652688i
$719$	−3.76631	−0.140460	−0.0702299	−	0.997531i	$-0.522373\pi$
$719$	−3.76631	−0.140460	−0.0702299	+	0.997531i	$0.522373\pi$
$720$	0	0
$721$	−53.9565	−2.00945
$722$	− 13.3723i	− 0.497665i
$723$	0	0
$724$	−20.8614	−0.775308
$725$	0	0
$726$	0	0
$727$	18.1168i	0.671917i	0.941877	+	0.335958i	$0.109060\pi$
$727$	18.1168i	0.671917i	−0.941877	+	0.335958i	$0.890940\pi$
$728$	22.7446i	0.842970i
$729$	0	0
$730$	0	0
$731$	−9.16034	−0.338808
$732$	0	0
$733$	0.233688i	0.00863146i	0.999991	+	0.00431573i	$0.00137374\pi$
$733$	0.233688i	0.00863146i	−0.999991	+	0.00431573i	$0.998626\pi$
$734$	16.0000	0.590571
$735$	0	0
$736$	−1.37228	−0.0505830
$737$	− 30.6060i	− 1.12739i
$738$	0	0
$739$	41.1168	1.51251	0.756254	−	0.654278i	$-0.227028\pi$
$739$	41.1168	1.51251	0.756254	+	0.654278i	$0.227028\pi$
$740$	0	0
$741$	0	0
$742$	38.7446i	1.42236i
$743$	7.88316i	0.289205i	0.989490	+	0.144602i	$0.0461903\pi$
$743$	7.88316i	0.289205i	−0.989490	+	0.144602i	$0.953810\pi$
$744$	0	0
$745$	0	0
$746$	−15.4891	−0.567097
$747$	0	0
$748$	− 7.11684i	− 0.260218i
$749$	28.6277	1.04603
$750$	0	0
$751$	16.2337	0.592376	0.296188	−	0.955130i	$-0.404285\pi$
$751$	16.2337	0.592376	0.296188	+	0.955130i	$0.404285\pi$
$752$	7.37228i	0.268839i
$753$	0	0
$754$	−9.25544	−0.337063
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	− 17.8614i	− 0.648756i
$759$	0	0
$760$	0	0
$761$	51.0951	1.85220	0.926098	−	0.377283i	$-0.123142\pi$
$761$	51.0951	1.85220	0.926098	+	0.377283i	$0.123142\pi$
$762$	0	0
$763$	− 51.8397i	− 1.87672i
$764$	17.4891	0.632734
$765$	0	0
$766$	22.9783	0.830238
$767$	29.4891i	1.06479i
$768$	0	0
$769$	46.8614	1.68987	0.844933	−	0.534873i	$-0.179640\pi$
$769$	46.8614	1.68987	0.844933	+	0.534873i	$0.179640\pi$
$770$	0	0
$771$	0	0
$772$	21.1168i	0.760012i
$773$	3.25544i	0.117090i	0.998285	+	0.0585450i	$0.0186461\pi$
$773$	3.25544i	0.117090i	−0.998285	+	0.0585450i	$0.981354\pi$
$774$	0	0
$775$	0	0
$776$	−8.37228	−0.300547
$777$	0	0
$778$	− 4.62772i	− 0.165912i
$779$	7.11684	0.254987
$780$	0	0
$781$	26.2337	0.938715
$782$	− 2.23369i	− 0.0798765i
$783$	0	0
$784$	−4.37228	−0.156153
$785$	0	0
$786$	0	0
$787$	− 28.0000i	− 0.998092i	−0.866575	−	0.499046i	$-0.833684\pi$
$787$	− 28.0000i	− 0.998092i	0.866575	−	0.499046i	$-0.166316\pi$
$788$	− 5.48913i	− 0.195542i
$789$	0	0
$790$	0	0
$791$	−10.9783	−0.390342
$792$	0	0
$793$	− 54.7446i	− 1.94404i
$794$	−22.7446	−0.807174
$795$	0	0
$796$	13.4891	0.478109
$797$	− 14.7446i	− 0.522279i	−0.965301	−	0.261140i	$-0.915902\pi$
$797$	− 14.7446i	− 0.522279i	0.965301	−	0.261140i	$-0.0840983\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	0	0
$802$	1.11684i	0.0394371i
$803$	− 13.6277i	− 0.480912i
$804$	0	0
$805$	0	0
$806$	−32.0000	−1.12715
$807$	0	0
$808$	2.74456i	0.0965534i
$809$	39.3505	1.38349	0.691746	−	0.722141i	$-0.256842\pi$
$809$	39.3505	1.38349	0.691746	+	0.722141i	$0.256842\pi$
$810$	0	0
$811$	3.62772	0.127386	0.0636932	−	0.997970i	$-0.479712\pi$
$811$	3.62772	0.127386	0.0636932	+	0.997970i	$0.479712\pi$
$812$	− 4.62772i	− 0.162401i
$813$	0	0
$814$	17.4891	0.612994
$815$	0	0
$816$	0	0
$817$	13.3505i	0.467076i
$818$	− 17.8614i	− 0.624509i
$819$	0	0
$820$	0	0
$821$	23.8397	0.832010	0.416005	−	0.909362i	$-0.363430\pi$
$821$	23.8397	0.832010	0.416005	+	0.909362i	$0.363430\pi$
$822$	0	0
$823$	25.0951i	0.874760i	0.899277	+	0.437380i	$0.144094\pi$
$823$	25.0951i	0.874760i	−0.899277	+	0.437380i	$0.855906\pi$
$824$	16.0000	0.557386
$825$	0	0
$826$	−14.7446	−0.513029
$827$	− 36.8614i	− 1.28180i	−0.767626	−	0.640898i	$-0.778562\pi$
$827$	− 36.8614i	− 1.28180i	0.767626	−	0.640898i	$-0.221438\pi$
$828$	0	0
$829$	50.1168	1.74063	0.870315	−	0.492496i	$-0.163915\pi$
$829$	50.1168	1.74063	0.870315	+	0.492496i	$0.163915\pi$
$830$	0	0
$831$	0	0
$832$	− 6.74456i	− 0.233826i
$833$	− 7.11684i	− 0.246584i
$834$	0	0
$835$	0	0
$836$	−10.3723	−0.358733
$837$	0	0
$838$	25.7228i	0.888580i
$839$	9.76631	0.337171	0.168585	−	0.985687i	$-0.446080\pi$
$839$	9.76631	0.337171	0.168585	+	0.985687i	$0.446080\pi$
$840$	0	0
$841$	−27.1168	−0.935064
$842$	30.4674i	1.04998i
$843$	0	0
$844$	18.7446	0.645214
$845$	0	0
$846$	0	0
$847$	27.3723i	0.940523i
$848$	− 11.4891i	− 0.394538i
$849$	0	0
$850$	0	0
$851$	5.48913	0.188165
$852$	0	0
$853$	36.2337i	1.24062i	0.784358	+	0.620309i	$0.212993\pi$
$853$	36.2337i	1.24062i	−0.784358	+	0.620309i	$0.787007\pi$
$854$	27.3723	0.936660
$855$	0	0
$856$	−8.48913	−0.290152
$857$	− 0.510875i	− 0.0174511i	−0.999962	−	0.00872557i	$-0.997223\pi$
$857$	− 0.510875i	− 0.0174511i	0.999962	−	0.00872557i	$-0.00277747\pi$
$858$	0	0
$859$	17.1168	0.584019	0.292010	−	0.956415i	$-0.405676\pi$
$859$	17.1168	0.584019	0.292010	+	0.956415i	$0.405676\pi$
$860$	0	0
$861$	0	0
$862$	− 8.23369i	− 0.280441i
$863$	− 2.39403i	− 0.0814938i	−0.999170	−	0.0407469i	$-0.987026\pi$
$863$	− 2.39403i	− 0.0814938i	0.999170	−	0.0407469i	$-0.0129737\pi$
$864$	0	0
$865$	0	0
$866$	6.37228	0.216539
$867$	0	0
$868$	− 16.0000i	− 0.543075i
$869$	−8.74456	−0.296639
$870$	0	0
$871$	47.2119	1.59972
$872$	15.3723i	0.520571i
$873$	0	0
$874$	−3.25544	−0.110117
$875$	0	0
$876$	0	0
$877$	35.9565i	1.21416i	0.794639	+	0.607082i	$0.207660\pi$
$877$	35.9565i	1.21416i	−0.794639	+	0.607082i	$0.792340\pi$
$878$	18.2337i	0.615357i
$879$	0	0
$880$	0	0
$881$	24.3505	0.820390	0.410195	−	0.911998i	$-0.365461\pi$
$881$	24.3505	0.820390	0.410195	+	0.911998i	$0.365461\pi$
$882$	0	0
$883$	44.7228i	1.50504i	0.658568	+	0.752521i	$0.271163\pi$
$883$	44.7228i	1.50504i	−0.658568	+	0.752521i	$0.728837\pi$
$884$	10.9783	0.369239
$885$	0	0
$886$	3.51087	0.117950
$887$	− 19.7228i	− 0.662227i	−0.943591	−	0.331114i	$-0.892576\pi$
$887$	− 19.7228i	− 0.662227i	0.943591	−	0.331114i	$-0.107424\pi$
$888$	0	0
$889$	27.3723	0.918037
$890$	0	0
$891$	0	0
$892$	6.62772i	0.221912i
$893$	17.4891i	0.585251i
$894$	0	0
$895$	0	0
$896$	3.37228	0.112660
$897$	0	0
$898$	9.86141i	0.329079i
$899$	6.51087	0.217150
$900$	0	0
$901$	18.7011	0.623023
$902$	13.1168i	0.436743i
$903$	0	0
$904$	3.25544	0.108274
$905$	0	0
$906$	0	0
$907$	− 7.00000i	− 0.232431i	−0.993224	−	0.116216i	$-0.962924\pi$
$907$	− 7.00000i	− 0.232431i	0.993224	−	0.116216i	$-0.0370764\pi$
$908$	19.1168i	0.634415i
$909$	0	0
$910$	0	0
$911$	42.0000	1.39152	0.695761	−	0.718273i	$-0.255067\pi$
$911$	42.0000	1.39152	0.695761	+	0.718273i	$0.255067\pi$
$912$	0	0
$913$	32.2337i	1.06678i
$914$	−12.8832	−0.426137
$915$	0	0
$916$	12.6277	0.417232
$917$	9.25544i	0.305641i
$918$	0	0
$919$	26.4674	0.873078	0.436539	−	0.899685i	$-0.356204\pi$
$919$	26.4674	0.873078	0.436539	+	0.899685i	$0.356204\pi$
$920$	0	0
$921$	0	0
$922$	− 1.88316i	− 0.0620184i
$923$	40.4674i	1.33200i
$924$	0	0
$925$	0	0
$926$	20.0000	0.657241
$927$	0	0
$928$	1.37228i	0.0450473i
$929$	51.9565	1.70464	0.852319	−	0.523023i	$-0.175196\pi$
$929$	51.9565	1.70464	0.852319	+	0.523023i	$0.175196\pi$
$930$	0	0
$931$	−10.3723	−0.339938
$932$	− 7.11684i	− 0.233120i
$933$	0	0
$934$	43.1168	1.41083
$935$	0	0
$936$	0	0
$937$	39.7228i	1.29769i	0.760922	+	0.648844i	$0.224747\pi$
$937$	39.7228i	1.29769i	−0.760922	+	0.648844i	$0.775253\pi$
$938$	23.6060i	0.770762i
$939$	0	0
$940$	0	0
$941$	−43.3723	−1.41390	−0.706948	−	0.707266i	$-0.749929\pi$
$941$	−43.3723	−1.41390	−0.706948	+	0.707266i	$0.749929\pi$
$942$	0	0
$943$	4.11684i	0.134063i
$944$	4.37228	0.142306
$945$	0	0
$946$	−24.6060	−0.800009
$947$	34.7228i	1.12834i	0.825659	+	0.564170i	$0.190804\pi$
$947$	34.7228i	1.12834i	−0.825659	+	0.564170i	$0.809196\pi$
$948$	0	0
$949$	21.0217	0.682395
$950$	0	0
$951$	0	0
$952$	5.48913i	0.177904i
$953$	2.13859i	0.0692758i	0.999400	+	0.0346379i	$0.0110278\pi$
$953$	2.13859i	0.0692758i	−0.999400	+	0.0346379i	$0.988972\pi$
$954$	0	0
$955$	0	0
$956$	3.25544	0.105288
$957$	0	0
$958$	− 0.510875i	− 0.0165056i
$959$	−64.4674	−2.08176
$960$	0	0
$961$	−8.48913	−0.273843
$962$	26.9783i	0.869814i
$963$	0	0
$964$	12.4891	0.402248
$965$	0	0
$966$	0	0
$967$	36.1168i	1.16144i	0.814104	+	0.580720i	$0.197229\pi$
$967$	36.1168i	1.16144i	−0.814104	+	0.580720i	$0.802771\pi$
$968$	− 8.11684i	− 0.260885i
$969$	0	0
$970$	0	0
$971$	22.9783	0.737407	0.368704	−	0.929547i	$-0.379802\pi$
$971$	22.9783	0.737407	0.368704	+	0.929547i	$0.379802\pi$
$972$	0	0
$973$	− 2.97825i	− 0.0954783i
$974$	12.7446	0.408362
$975$	0	0
$976$	−8.11684	−0.259814
$977$	22.8832i	0.732097i	0.930596	+	0.366049i	$0.119290\pi$
$977$	22.8832i	0.732097i	−0.930596	+	0.366049i	$0.880710\pi$
$978$	0	0
$979$	70.4674	2.25215
$980$	0	0
$981$	0	0
$982$	36.6060i	1.16814i
$983$	− 12.8614i	− 0.410215i	−0.978739	−	0.205108i	$-0.934246\pi$
$983$	− 12.8614i	− 0.410215i	0.978739	−	0.205108i	$-0.0657544\pi$
$984$	0	0
$985$	0	0
$986$	−2.23369	−0.0711351
$987$	0	0
$988$	− 16.0000i	− 0.509028i
$989$	−7.72281	−0.245571
$990$	0	0
$991$	16.2337	0.515680	0.257840	−	0.966188i	$-0.416989\pi$
$991$	16.2337	0.515680	0.257840	+	0.966188i	$0.416989\pi$
$992$	4.74456i	0.150640i
$993$	0	0
$994$	−20.2337	−0.641774
$995$	0	0
$996$	0	0
$997$	14.0000i	0.443384i	0.975117	+	0.221692i	$0.0711580\pi$
$997$	14.0000i	0.443384i	−0.975117	+	0.221692i	$0.928842\pi$
$998$	− 15.1168i	− 0.478515i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4050.2.c.ba.649.4		4
3.2	odd	2		4050.2.c.v.649.2		4
5.2	odd	4		4050.2.a.bo.1.1		2
5.3	odd	4		810.2.a.k.1.2		2
5.4	even	2	inner	4050.2.c.ba.649.1		4
9.2	odd	6		450.2.j.g.49.2		8
9.4	even	3		1350.2.j.f.1099.1		8
9.5	odd	6		450.2.j.g.349.3		8
9.7	even	3		1350.2.j.f.199.4		8
15.2	even	4		4050.2.a.bw.1.1		2
15.8	even	4		810.2.a.i.1.2		2
15.14	odd	2		4050.2.c.v.649.3		4
20.3	even	4		6480.2.a.bn.1.1		2
45.2	even	12		450.2.e.j.301.1		4
45.4	even	6		1350.2.j.f.1099.4		8
45.7	odd	12		1350.2.e.l.901.2		4
45.13	odd	12		270.2.e.c.181.1		4
45.14	odd	6		450.2.j.g.349.2		8
45.22	odd	12		1350.2.e.l.451.2		4
45.23	even	12		90.2.e.c.61.1	yes	4
45.29	odd	6		450.2.j.g.49.3		8
45.32	even	12		450.2.e.j.151.2		4
45.34	even	6		1350.2.j.f.199.1		8
45.38	even	12		90.2.e.c.31.2	✓	4
45.43	odd	12		270.2.e.c.91.1		4
60.23	odd	4		6480.2.a.be.1.1		2
180.23	odd	12		720.2.q.f.241.2		4
180.43	even	12		2160.2.q.f.1441.2		4
180.83	odd	12		720.2.q.f.481.1		4
180.103	even	12		2160.2.q.f.721.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.2.e.c.31.2	✓	4	45.38	even	12
90.2.e.c.61.1	yes	4	45.23	even	12
270.2.e.c.91.1		4	45.43	odd	12
270.2.e.c.181.1		4	45.13	odd	12
450.2.e.j.151.2		4	45.32	even	12
450.2.e.j.301.1		4	45.2	even	12
450.2.j.g.49.2		8	9.2	odd	6
450.2.j.g.49.3		8	45.29	odd	6
450.2.j.g.349.2		8	45.14	odd	6
450.2.j.g.349.3		8	9.5	odd	6
720.2.q.f.241.2		4	180.23	odd	12
720.2.q.f.481.1		4	180.83	odd	12
810.2.a.i.1.2		2	15.8	even	4
810.2.a.k.1.2		2	5.3	odd	4
1350.2.e.l.451.2		4	45.22	odd	12
1350.2.e.l.901.2		4	45.7	odd	12
1350.2.j.f.199.1		8	45.34	even	6
1350.2.j.f.199.4		8	9.7	even	3
1350.2.j.f.1099.1		8	9.4	even	3
1350.2.j.f.1099.4		8	45.4	even	6
2160.2.q.f.721.2		4	180.103	even	12
2160.2.q.f.1441.2		4	180.43	even	12
4050.2.a.bo.1.1		2	5.2	odd	4
4050.2.a.bw.1.1		2	15.2	even	4
4050.2.c.v.649.2		4	3.2	odd	2
4050.2.c.v.649.3		4	15.14	odd	2
4050.2.c.ba.649.1		4	5.4	even	2	inner
4050.2.c.ba.649.4		4	1.1	even	1	trivial
6480.2.a.be.1.1		2	60.23	odd	4
6480.2.a.bn.1.1		2	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4050 = 2 \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4050.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +3.37228i q^{7} -1.00000i q^{8} +4.37228 q^{11} +6.74456i q^{13} -3.37228 q^{14} +1.00000 q^{16} +1.62772i q^{17} +2.37228 q^{19} +4.37228i q^{22} +1.37228i q^{23} -6.74456 q^{26} -3.37228i q^{28} +1.37228 q^{29} +4.74456 q^{31} +1.00000i q^{32} -1.62772 q^{34} -4.00000i q^{37} +2.37228i q^{38} +3.00000 q^{41} +5.62772i q^{43} -4.37228 q^{44} -1.37228 q^{46} +7.37228i q^{47} -4.37228 q^{49} -6.74456i q^{52} -11.4891i q^{53} +3.37228 q^{56} +1.37228i q^{58} +4.37228 q^{59} -8.11684 q^{61} +4.74456i q^{62} -1.00000 q^{64} -7.00000i q^{67} -1.62772i q^{68} +6.00000 q^{71} -3.11684i q^{73} +4.00000 q^{74} -2.37228 q^{76} +14.7446i q^{77} -2.00000 q^{79} +3.00000i q^{82} +7.37228i q^{83} -5.62772 q^{86} -4.37228i q^{88} +16.1168 q^{89} -22.7446 q^{91} -1.37228i q^{92} -7.37228 q^{94} -8.37228i q^{97} -4.37228i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 6 q^{11} - 2 q^{14} + 4 q^{16} - 2 q^{19} - 4 q^{26} - 6 q^{29} - 4 q^{31} - 18 q^{34} + 12 q^{41} - 6 q^{44} + 6 q^{46} - 6 q^{49} + 2 q^{56} + 6 q^{59} + 2 q^{61} - 4 q^{64} + 24 q^{71}+ \cdots - 18 q^{94}+O(q^{100}) \) 4 * q - 4 * q^4 + 6 * q^11 - 2 * q^14 + 4 * q^16 - 2 * q^19 - 4 * q^26 - 6 * q^29 - 4 * q^31 - 18 * q^34 + 12 * q^41 - 6 * q^44 + 6 * q^46 - 6 * q^49 + 2 * q^56 + 6 * q^59 + 2 * q^61 - 4 * q^64 + 24 * q^71 + 16 * q^74 + 2 * q^76 - 8 * q^79 - 34 * q^86 + 30 * q^89 - 68 * q^91 - 18 * q^94

Introduction

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