LMFDB - Embedded newform 1425.2.c.g.799.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1425,2,Mod(799,1425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1425, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1425.799");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1425 = 3 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1425.c (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.3786822880$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			799.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1425.799
Dual form			1425.2.c.g.799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} -1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -4.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +5.00000i q^{32} +6.00000 q^{34} -1.00000 q^{36} -10.0000i q^{37} +1.00000i q^{38} -6.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} +4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -1.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -7.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} -10.0000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -6.00000i q^{78} +1.00000 q^{81} -2.00000i q^{82} -16.0000i q^{83} -4.00000 q^{86} +2.00000i q^{87} +2.00000 q^{89} -4.00000i q^{92} -8.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9	$ 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{16} + 2 q^{19} + 6 q^{24} + 12 q^{26} - 4 q^{29} + 16 q^{31} + 12 q^{34} - 2 q^{36} - 12 q^{39} - 4 q^{41} + 8 q^{46} + 14 q^{49} - 12 q^{51} - 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{69} + 20 q^{74} + 2 q^{76} + 2 q^{81} - 8 q^{86} + 4 q^{89} - 24 q^{94} + 10 q^{96}+O(q^{100}) $ 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^16 + 2 * q^19 + 6 * q^24 + 12 * q^26 - 4 * q^29 + 16 * q^31 + 12 * q^34 - 2 * q^36 - 12 * q^39 - 4 * q^41 + 8 * q^46 + 14 * q^49 - 12 * q^51 - 2 * q^54 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^69 + 20 * q^74 + 2 * q^76 + 2 * q^81 - 8 * q^86 + 4 * q^89 - 24 * q^94 + 10 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1425\mathbb{Z}\right)^\times$.

$n$	$476$	$1027$	$1351$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	− 1.00000i	− 0.577350i
$3$	− 1.00000i	− 0.577350i
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	1.00000	0.408248
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	3.00000i	1.06066i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	− 1.00000i	− 0.288675i
$13$	− 6.00000i	− 1.66410i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	− 6.00000i	− 1.66410i	0.554700	−	0.832050i	$-0.312833\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	− 1.00000i	− 0.235702i
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 4.00000i	− 0.834058i	−0.908893	−	0.417029i	$-0.863071\pi$
$23$	− 4.00000i	− 0.834058i	0.908893	−	0.417029i	$-0.136929\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	6.00000	1.17670
$27$	1.00000i	0.192450i
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	8.00000	1.43684	0.718421	−	0.695608i	$-0.244865\pi$
$31$	8.00000	1.43684	0.718421	+	0.695608i	$0.244865\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	6.00000	1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	− 10.0000i	− 1.64399i	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	− 10.0000i	− 1.64399i	0.569495	−	0.821995i	$-0.307139\pi$
$38$	1.00000i	0.162221i
$39$	−6.00000	−0.960769
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	12.0000i	1.75038i	0.483779	+	0.875190i	$0.339264\pi$
$47$	12.0000i	1.75038i	−0.483779	+	0.875190i	$0.660736\pi$
$48$	1.00000i	0.144338i
$49$	7.00000	1.00000
$50$	0	0
$51$	−6.00000	−0.840168
$52$	− 6.00000i	− 0.832050i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	− 1.00000i	− 0.132453i
$58$	− 2.00000i	− 0.262613i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	8.00000i	1.01600i
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	− 4.00000i	− 0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$	− 4.00000i	− 0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$	− 6.00000i	− 0.727607i
$69$	−4.00000	−0.481543
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	− 3.00000i	− 0.353553i
$73$	− 10.0000i	− 1.17041i	−0.810885	−	0.585206i	$-0.801014\pi$
$73$	− 10.0000i	− 1.17041i	0.810885	−	0.585206i	$-0.198986\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	1.00000	0.114708
$77$	0	0
$78$	− 6.00000i	− 0.679366i
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 2.00000i	− 0.220863i
$83$	− 16.0000i	− 1.75623i	−0.478451	−	0.878114i	$-0.658802\pi$
$83$	− 16.0000i	− 1.75623i	0.478451	−	0.878114i	$-0.341198\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	2.00000i	0.214423i
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	− 4.00000i	− 0.417029i
$93$	− 8.00000i	− 0.829561i
$94$	−12.0000	−1.23771
$95$	0	0
$96$	5.00000	0.510310
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	7.00000i	0.707107i
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	− 6.00000i	− 0.594089i
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	18.0000	1.76505
$105$	0	0
$106$	−6.00000	−0.582772
$107$	4.00000i	0.386695i	0.981130	+	0.193347i	$0.0619344\pi$
$107$	4.00000i	0.386695i	−0.981130	+	0.193347i	$0.938066\pi$
$108$	1.00000i	0.0962250i
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	−10.0000	−0.949158
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	1.00000	0.0936586
$115$	0	0
$116$	−2.00000	−0.185695
$117$	6.00000i	0.554700i
$118$	12.0000i	1.10469i
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	− 2.00000i	− 0.181071i
$123$	2.00000i	0.180334i
$124$	8.00000	0.718421
$125$	0	0
$126$	0	0
$127$	− 8.00000i	− 0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$	− 8.00000i	− 0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$	3.00000i	0.265165i
$129$	4.00000	0.352180
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	0	0
$134$	4.00000	0.345547
$135$	0	0
$136$	18.0000	1.54349
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	− 4.00000i	− 0.340503i
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.0000	0.827606
$147$	− 7.00000i	− 0.577350i
$148$	− 10.0000i	− 0.821995i
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	3.00000i	0.243332i
$153$	6.00000i	0.485071i
$154$	0	0
$155$	0	0
$156$	−6.00000	−0.480384
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000i	0.0785674i
$163$	4.00000i	0.313304i	0.987654	+	0.156652i	$0.0500701\pi$
$163$	4.00000i	0.313304i	−0.987654	+	0.156652i	$0.949930\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	16.0000	1.24184
$167$	24.0000i	1.85718i	0.371113	+	0.928588i	$0.378976\pi$
$167$	24.0000i	1.85718i	−0.371113	+	0.928588i	$0.621024\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	4.00000i	0.304997i
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	−2.00000	−0.151620
$175$	0	0
$176$	0	0
$177$	− 12.0000i	− 0.901975i
$178$	2.00000i	0.149906i
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	2.00000i	0.147844i
$184$	12.0000	0.884652
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	12.0000i	0.875190i
$189$	0	0
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	7.00000i	0.505181i
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	−10.0000	−0.717958
$195$	0	0
$196$	7.00000	0.500000
$197$	− 2.00000i	− 0.142494i	−0.997459	−	0.0712470i	$-0.977302\pi$
$197$	− 2.00000i	− 0.142494i	0.997459	−	0.0712470i	$-0.0226979\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	− 10.0000i	− 0.703598i
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	8.00000	0.557386
$207$	4.00000i	0.278019i
$208$	6.00000i	0.416025i
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−4.00000	−0.273434
$215$	0	0
$216$	−3.00000	−0.204124
$217$	0	0
$218$	10.0000i	0.677285i
$219$	−10.0000	−0.675737
$220$	0	0
$221$	−36.0000	−2.42162
$222$	− 10.0000i	− 0.671156i
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	0	0
$225$	0	0
$226$	6.00000	0.399114
$227$	− 12.0000i	− 0.796468i	−0.917284	−	0.398234i	$-0.869623\pi$
$227$	− 12.0000i	− 0.796468i	0.917284	−	0.398234i	$-0.130377\pi$
$228$	− 1.00000i	− 0.0662266i
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 10.0000i	− 0.655122i	−0.944830	−	0.327561i	$-0.893773\pi$
$233$	− 10.0000i	− 0.655122i	0.944830	−	0.327561i	$-0.106227\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	− 11.0000i	− 0.707107i
$243$	− 1.00000i	− 0.0641500i
$244$	−2.00000	−0.128037
$245$	0	0
$246$	−2.00000	−0.127515
$247$	− 6.00000i	− 0.381771i
$248$	24.0000i	1.52400i
$249$	−16.0000	−1.01396
$250$	0	0
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	14.0000i	0.873296i	0.899632	+	0.436648i	$0.143834\pi$
$257$	14.0000i	0.873296i	−0.899632	+	0.436648i	$0.856166\pi$
$258$	4.00000i	0.249029i
$259$	0	0
$260$	0	0
$261$	2.00000	0.123797
$262$	8.00000i	0.494242i
$263$	− 12.0000i	− 0.739952i	−0.929041	−	0.369976i	$-0.879366\pi$
$263$	− 12.0000i	− 0.739952i	0.929041	−	0.369976i	$-0.120634\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 2.00000i	− 0.122398i
$268$	− 4.00000i	− 0.244339i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	6.00000i	0.363803i
$273$	0	0
$274$	−18.0000	−1.08742
$275$	0	0
$276$	−4.00000	−0.240772
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 4.00000i	− 0.239904i
$279$	−8.00000	−0.478947
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	12.0000i	0.714590i
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	− 5.00000i	− 0.294628i
$289$	−19.0000	−1.11765
$290$	0	0
$291$	10.0000	0.586210
$292$	− 10.0000i	− 0.585206i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	30.0000	1.74371
$297$	0	0
$298$	− 6.00000i	− 0.347571i
$299$	−24.0000	−1.38796
$300$	0	0
$301$	0	0
$302$	− 8.00000i	− 0.460348i
$303$	10.0000i	0.574485i
$304$	−1.00000	−0.0573539
$305$	0	0
$306$	−6.00000	−0.342997
$307$	− 12.0000i	− 0.684876i	−0.939540	−	0.342438i	$-0.888747\pi$
$307$	− 12.0000i	− 0.684876i	0.939540	−	0.342438i	$-0.111253\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	4.00000	0.226819	0.113410	−	0.993548i	$-0.463823\pi$
$311$	4.00000	0.226819	0.113410	+	0.993548i	$0.463823\pi$
$312$	− 18.0000i	− 1.01905i
$313$	22.0000i	1.24351i	0.783210	+	0.621757i	$0.213581\pi$
$313$	22.0000i	1.24351i	−0.783210	+	0.621757i	$0.786419\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	0	0
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	6.00000i	0.336463i
$319$	0	0
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	− 6.00000i	− 0.333849i
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	− 10.0000i	− 0.553001i
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	− 16.0000i	− 0.878114i
$333$	10.0000i	0.547997i
$334$	−24.0000	−1.31322
$335$	0	0
$336$	0	0
$337$	− 22.0000i	− 1.19842i	−0.800593	−	0.599208i	$-0.795482\pi$
$337$	− 22.0000i	− 1.19842i	0.800593	−	0.599208i	$-0.204518\pi$
$338$	− 23.0000i	− 1.25104i
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	− 1.00000i	− 0.0540738i
$343$	0	0
$344$	−12.0000	−0.646997
$345$	0	0
$346$	−22.0000	−1.18273
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	2.00000i	0.107211i
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	0	0
$353$	22.0000i	1.17094i	0.810693	+	0.585471i	$0.199090\pi$
$353$	22.0000i	1.17094i	−0.810693	+	0.585471i	$0.800910\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	2.00000	0.106000
$357$	0	0
$358$	4.00000i	0.211407i
$359$	20.0000	1.05556	0.527780	−	0.849381i	$-0.323025\pi$
$359$	20.0000	1.05556	0.527780	+	0.849381i	$0.323025\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	14.0000i	0.735824i
$363$	11.0000i	0.577350i
$364$	0	0
$365$	0	0
$366$	−2.00000	−0.104542
$367$	32.0000i	1.67039i	0.549957	+	0.835193i	$0.314644\pi$
$367$	32.0000i	1.67039i	−0.549957	+	0.835193i	$0.685356\pi$
$368$	4.00000i	0.208514i
$369$	2.00000	0.104116
$370$	0	0
$371$	0	0
$372$	− 8.00000i	− 0.414781i
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	0	0
$375$	0	0
$376$	−36.0000	−1.85656
$377$	12.0000i	0.618031i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	− 12.0000i	− 0.613973i
$383$	− 8.00000i	− 0.408781i	−0.978889	−	0.204390i	$-0.934479\pi$
$383$	− 8.00000i	− 0.408781i	0.978889	−	0.204390i	$-0.0655212\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	−14.0000	−0.712581
$387$	− 4.00000i	− 0.203331i
$388$	10.0000i	0.507673i
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	−24.0000	−1.21373
$392$	21.0000i	1.06066i
$393$	− 8.00000i	− 0.403547i
$394$	2.00000	0.100759
$395$	0	0
$396$	0	0
$397$	14.0000i	0.702640i	0.936255	+	0.351320i	$0.114267\pi$
$397$	14.0000i	0.702640i	−0.936255	+	0.351320i	$0.885733\pi$
$398$	8.00000i	0.401004i
$399$	0	0
$400$	0	0
$401$	38.0000	1.89763	0.948815	−	0.315833i	$-0.102284\pi$
$401$	38.0000	1.89763	0.948815	+	0.315833i	$0.102284\pi$
$402$	− 4.00000i	− 0.199502i
$403$	− 48.0000i	− 2.39105i
$404$	−10.0000	−0.497519
$405$	0	0
$406$	0	0
$407$	0	0
$408$	− 18.0000i	− 0.891133i
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	− 8.00000i	− 0.394132i
$413$	0	0
$414$	−4.00000	−0.196589
$415$	0	0
$416$	30.0000	1.47087
$417$	4.00000i	0.195881i
$418$	0	0
$419$	−8.00000	−0.390826	−0.195413	−	0.980721i	$-0.562605\pi$
$419$	−8.00000	−0.390826	−0.195413	+	0.980721i	$0.562605\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	− 4.00000i	− 0.194717i
$423$	− 12.0000i	− 0.583460i
$424$	−18.0000	−0.874157
$425$	0	0
$426$	0	0
$427$	0	0
$428$	4.00000i	0.193347i
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	− 1.00000i	− 0.0481125i
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	10.0000	0.478913
$437$	− 4.00000i	− 0.191346i
$438$	− 10.0000i	− 0.477818i
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	− 36.0000i	− 1.71235i
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	−10.0000	−0.474579
$445$	0	0
$446$	16.0000	0.757622
$447$	6.00000i	0.283790i
$448$	0	0
$449$	2.00000	0.0943858	0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000	0.0943858	0.0471929	+	0.998886i	$0.484972\pi$
$450$	0	0
$451$	0	0
$452$	− 6.00000i	− 0.282216i
$453$	8.00000i	0.375873i
$454$	12.0000	0.563188
$455$	0	0
$456$	3.00000	0.140488
$457$	− 6.00000i	− 0.280668i	−0.990104	−	0.140334i	$-0.955182\pi$
$457$	− 6.00000i	− 0.280668i	0.990104	−	0.140334i	$-0.0448177\pi$
$458$	− 6.00000i	− 0.280362i
$459$	6.00000	0.280056
$460$	0	0
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	0	0
$463$	− 32.0000i	− 1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$	− 32.0000i	− 1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	10.0000	0.463241
$467$	− 32.0000i	− 1.48078i	−0.672176	−	0.740392i	$-0.734640\pi$
$467$	− 32.0000i	− 1.48078i	0.672176	−	0.740392i	$-0.265360\pi$
$468$	6.00000i	0.277350i
$469$	0	0
$470$	0	0
$471$	−2.00000	−0.0921551
$472$	36.0000i	1.65703i
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 6.00000i	− 0.274721i
$478$	12.0000i	0.548867i
$479$	−20.0000	−0.913823	−0.456912	−	0.889512i	$-0.651044\pi$
$479$	−20.0000	−0.913823	−0.456912	+	0.889512i	$0.651044\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	− 6.00000i	− 0.273293i
$483$	0	0
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	− 6.00000i	− 0.271607i
$489$	4.00000	0.180886
$490$	0	0
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	2.00000i	0.0901670i
$493$	12.0000i	0.540453i
$494$	6.00000	0.269953
$495$	0	0
$496$	−8.00000	−0.359211
$497$	0	0
$498$	− 16.0000i	− 0.716977i
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	0	0
$501$	24.0000	1.07224
$502$	− 24.0000i	− 1.07117i
$503$	− 12.0000i	− 0.535054i	−0.963550	−	0.267527i	$-0.913794\pi$
$503$	− 12.0000i	− 0.535054i	0.963550	−	0.267527i	$-0.0862064\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	23.0000i	1.02147i
$508$	− 8.00000i	− 0.354943i
$509$	22.0000	0.975133	0.487566	−	0.873086i	$-0.337885\pi$
$509$	22.0000	0.975133	0.487566	+	0.873086i	$0.337885\pi$
$510$	0	0
$511$	0	0
$512$	− 11.0000i	− 0.486136i
$513$	1.00000i	0.0441511i
$514$	−14.0000	−0.617514
$515$	0	0
$516$	4.00000	0.176090
$517$	0	0
$518$	0	0
$519$	22.0000	0.965693
$520$	0	0
$521$	14.0000	0.613351	0.306676	−	0.951814i	$-0.400783\pi$
$521$	14.0000	0.613351	0.306676	+	0.951814i	$0.400783\pi$
$522$	2.00000i	0.0875376i
$523$	28.0000i	1.22435i	0.790721	+	0.612177i	$0.209706\pi$
$523$	28.0000i	1.22435i	−0.790721	+	0.612177i	$0.790294\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	12.0000	0.523225
$527$	− 48.0000i	− 2.09091i
$528$	0	0
$529$	7.00000	0.304348
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	12.0000i	0.519778i
$534$	2.00000	0.0865485
$535$	0	0
$536$	12.0000	0.518321
$537$	− 4.00000i	− 0.172613i
$538$	6.00000i	0.258678i
$539$	0	0
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	− 14.0000i	− 0.600798i
$544$	30.0000	1.28624
$545$	0	0
$546$	0	0
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	18.0000i	0.768922i
$549$	2.00000	0.0853579
$550$	0	0
$551$	−2.00000	−0.0852029
$552$	− 12.0000i	− 0.510754i
$553$	0	0
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−4.00000	−0.169638
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	− 8.00000i	− 0.338667i
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	− 10.0000i	− 0.421825i
$563$	− 20.0000i	− 0.842900i	−0.906852	−	0.421450i	$-0.861521\pi$
$563$	− 20.0000i	− 0.842900i	0.906852	−	0.421450i	$-0.138479\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	0	0
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	44.0000	1.84134	0.920671	−	0.390339i	$-0.127642\pi$
$571$	44.0000	1.84134	0.920671	+	0.390339i	$0.127642\pi$
$572$	0	0
$573$	12.0000i	0.501307i
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	− 19.0000i	− 0.790296i
$579$	14.0000	0.581820
$580$	0	0
$581$	0	0
$582$	10.0000i	0.414513i
$583$	0	0
$584$	30.0000	1.24141
$585$	0	0
$586$	−14.0000	−0.578335
$587$	8.00000i	0.330195i	0.986277	+	0.165098i	$0.0527939\pi$
$587$	8.00000i	0.330195i	−0.986277	+	0.165098i	$0.947206\pi$
$588$	− 7.00000i	− 0.288675i
$589$	8.00000	0.329634
$590$	0	0
$591$	−2.00000	−0.0822690
$592$	10.0000i	0.410997i
$593$	30.0000i	1.23195i	0.787765	+	0.615976i	$0.211238\pi$
$593$	30.0000i	1.23195i	−0.787765	+	0.615976i	$0.788762\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	− 8.00000i	− 0.327418i
$598$	− 24.0000i	− 0.981433i
$599$	24.0000	0.980613	0.490307	−	0.871550i	$-0.336885\pi$
$599$	24.0000	0.980613	0.490307	+	0.871550i	$0.336885\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	4.00000i	0.162893i
$604$	−8.00000	−0.325515
$605$	0	0
$606$	−10.0000	−0.406222
$607$	24.0000i	0.974130i	0.873366	+	0.487065i	$0.161933\pi$
$607$	24.0000i	0.974130i	−0.873366	+	0.487065i	$0.838067\pi$
$608$	5.00000i	0.202777i
$609$	0	0
$610$	0	0
$611$	72.0000	2.91281
$612$	6.00000i	0.242536i
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	0	0
$617$	2.00000i	0.0805170i	0.999189	+	0.0402585i	$0.0128181\pi$
$617$	2.00000i	0.0805170i	−0.999189	+	0.0402585i	$0.987182\pi$
$618$	− 8.00000i	− 0.321807i
$619$	28.0000	1.12542	0.562708	−	0.826656i	$-0.309760\pi$
$619$	28.0000	1.12542	0.562708	+	0.826656i	$0.309760\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	4.00000i	0.160385i
$623$	0	0
$624$	6.00000	0.240192
$625$	0	0
$626$	−22.0000	−0.879297
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	−60.0000	−2.39236
$630$	0	0
$631$	−32.0000	−1.27390	−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000	−1.27390	−0.636950	+	0.770905i	$0.719804\pi$
$632$	0	0
$633$	4.00000i	0.158986i
$634$	6.00000	0.238290
$635$	0	0
$636$	6.00000	0.237915
$637$	− 42.0000i	− 1.66410i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	38.0000	1.50091	0.750455	−	0.660922i	$-0.229834\pi$
$641$	38.0000	1.50091	0.750455	+	0.660922i	$0.229834\pi$
$642$	4.00000i	0.157867i
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	6.00000	0.236067
$647$	36.0000i	1.41531i	0.706560	+	0.707653i	$0.250246\pi$
$647$	36.0000i	1.41531i	−0.706560	+	0.707653i	$0.749754\pi$
$648$	3.00000i	0.117851i
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000i	0.156652i
$653$	− 14.0000i	− 0.547862i	−0.961749	−	0.273931i	$-0.911676\pi$
$653$	− 14.0000i	− 0.547862i	0.961749	−	0.273931i	$-0.0883240\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	2.00000	0.0780869
$657$	10.0000i	0.390137i
$658$	0	0
$659$	44.0000	1.71400	0.856998	−	0.515319i	$-0.172327\pi$
$659$	44.0000	1.71400	0.856998	+	0.515319i	$0.172327\pi$
$660$	0	0
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	12.0000i	0.466393i
$663$	36.0000i	1.39812i
$664$	48.0000	1.86276
$665$	0	0
$666$	−10.0000	−0.387492
$667$	8.00000i	0.309761i
$668$	24.0000i	0.928588i
$669$	−16.0000	−0.618596
$670$	0	0
$671$	0	0
$672$	0	0
$673$	46.0000i	1.77317i	0.462566	+	0.886585i	$0.346929\pi$
$673$	46.0000i	1.77317i	−0.462566	+	0.886585i	$0.653071\pi$
$674$	22.0000	0.847408
$675$	0	0
$676$	−23.0000	−0.884615
$677$	− 22.0000i	− 0.845529i	−0.906240	−	0.422764i	$-0.861060\pi$
$677$	− 22.0000i	− 0.845529i	0.906240	−	0.422764i	$-0.138940\pi$
$678$	− 6.00000i	− 0.230429i
$679$	0	0
$680$	0	0
$681$	−12.0000	−0.459841
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	−1.00000	−0.0382360
$685$	0	0
$686$	0	0
$687$	6.00000i	0.228914i
$688$	− 4.00000i	− 0.152499i
$689$	36.0000	1.37149
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	22.0000i	0.836315i
$693$	0	0
$694$	0	0
$695$	0	0
$696$	−6.00000	−0.227429
$697$	12.0000i	0.454532i
$698$	2.00000i	0.0757011i
$699$	−10.0000	−0.378235
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	6.00000i	0.226455i
$703$	− 10.0000i	− 0.377157i
$704$	0	0
$705$	0	0
$706$	−22.0000	−0.827981
$707$	0	0
$708$	− 12.0000i	− 0.450988i
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	− 32.0000i	− 1.19841i
$714$	0	0
$715$	0	0
$716$	4.00000	0.149487
$717$	− 12.0000i	− 0.448148i
$718$	20.0000i	0.746393i
$719$	20.0000	0.745874	0.372937	−	0.927857i	$-0.378351\pi$
$719$	20.0000	0.745874	0.372937	+	0.927857i	$0.378351\pi$
$720$	0	0
$721$	0	0
$722$	1.00000i	0.0372161i
$723$	6.00000i	0.223142i
$724$	14.0000	0.520306
$725$	0	0
$726$	−11.0000	−0.408248
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	24.0000	0.887672
$732$	2.00000i	0.0739221i
$733$	− 46.0000i	− 1.69905i	−0.527549	−	0.849524i	$-0.676889\pi$
$733$	− 46.0000i	− 1.69905i	0.527549	−	0.849524i	$-0.323111\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	20.0000	0.737210
$737$	0	0
$738$	2.00000i	0.0736210i
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	−6.00000	−0.220416
$742$	0	0
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	24.0000	0.879883
$745$	0	0
$746$	−10.0000	−0.366126
$747$	16.0000i	0.585409i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	16.0000	0.583848	0.291924	−	0.956441i	$-0.405705\pi$
$751$	16.0000	0.583848	0.291924	+	0.956441i	$0.405705\pi$
$752$	− 12.0000i	− 0.437595i
$753$	24.0000i	0.874609i
$754$	−12.0000	−0.437014
$755$	0	0
$756$	0	0
$757$	38.0000i	1.38113i	0.723269	+	0.690567i	$0.242639\pi$
$757$	38.0000i	1.38113i	−0.723269	+	0.690567i	$0.757361\pi$
$758$	− 12.0000i	− 0.435860i
$759$	0	0
$760$	0	0
$761$	50.0000	1.81250	0.906249	−	0.422744i	$-0.138933\pi$
$761$	50.0000	1.81250	0.906249	+	0.422744i	$0.138933\pi$
$762$	− 8.00000i	− 0.289809i
$763$	0	0
$764$	−12.0000	−0.434145
$765$	0	0
$766$	8.00000	0.289052
$767$	− 72.0000i	− 2.59977i
$768$	17.0000i	0.613435i
$769$	−18.0000	−0.649097	−0.324548	−	0.945869i	$-0.605212\pi$
$769$	−18.0000	−0.649097	−0.324548	+	0.945869i	$0.605212\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	14.0000i	0.503871i
$773$	− 18.0000i	− 0.647415i	−0.946157	−	0.323708i	$-0.895071\pi$
$773$	− 18.0000i	− 0.647415i	0.946157	−	0.323708i	$-0.104929\pi$
$774$	4.00000	0.143777
$775$	0	0
$776$	−30.0000	−1.07694
$777$	0	0
$778$	− 30.0000i	− 1.07555i
$779$	−2.00000	−0.0716574
$780$	0	0
$781$	0	0
$782$	− 24.0000i	− 0.858238i
$783$	− 2.00000i	− 0.0714742i
$784$	−7.00000	−0.250000
$785$	0	0
$786$	8.00000	0.285351
$787$	44.0000i	1.56843i	0.620489	+	0.784215i	$0.286934\pi$
$787$	44.0000i	1.56843i	−0.620489	+	0.784215i	$0.713066\pi$
$788$	− 2.00000i	− 0.0712470i
$789$	−12.0000	−0.427211
$790$	0	0
$791$	0	0
$792$	0	0
$793$	12.0000i	0.426132i
$794$	−14.0000	−0.496841
$795$	0	0
$796$	8.00000	0.283552
$797$	− 6.00000i	− 0.212531i	−0.994338	−	0.106265i	$-0.966111\pi$
$797$	− 6.00000i	− 0.212531i	0.994338	−	0.106265i	$-0.0338893\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	38.0000i	1.34183i
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	48.0000	1.69073
$807$	− 6.00000i	− 0.211210i
$808$	− 30.0000i	− 1.05540i
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	6.00000	0.210042
$817$	4.00000i	0.139942i
$818$	14.0000i	0.489499i
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	18.0000i	0.627822i
$823$	32.0000i	1.11545i	0.830026	+	0.557725i	$0.188326\pi$
$823$	32.0000i	1.11545i	−0.830026	+	0.557725i	$0.811674\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	0	0
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	4.00000i	0.139010i
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	0	0
$831$	22.0000	0.763172
$832$	42.0000i	1.45609i
$833$	− 42.0000i	− 1.45521i
$834$	−4.00000	−0.138509
$835$	0	0
$836$	0	0
$837$	8.00000i	0.276520i
$838$	− 8.00000i	− 0.276355i
$839$	0	0		−	1.00000i	$-0.5\pi$
$839$	0	0			1.00000i	$0.5\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	14.0000i	0.482472i
$843$	10.0000i	0.344418i
$844$	−4.00000	−0.137686
$845$	0	0
$846$	12.0000	0.412568
$847$	0	0
$848$	− 6.00000i	− 0.206041i
$849$	20.0000	0.686398
$850$	0	0
$851$	−40.0000	−1.37118
$852$	0	0
$853$	− 22.0000i	− 0.753266i	−0.926363	−	0.376633i	$-0.877082\pi$
$853$	− 22.0000i	− 0.753266i	0.926363	−	0.376633i	$-0.122918\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	30.0000i	1.02478i	0.858753	+	0.512390i	$0.171240\pi$
$857$	30.0000i	1.02478i	−0.858753	+	0.512390i	$0.828760\pi$
$858$	0	0
$859$	28.0000	0.955348	0.477674	−	0.878537i	$-0.341480\pi$
$859$	28.0000	0.955348	0.477674	+	0.878537i	$0.341480\pi$
$860$	0	0
$861$	0	0
$862$	− 24.0000i	− 0.817443i
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−14.0000	−0.475739
$867$	19.0000i	0.645274i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−24.0000	−0.813209
$872$	30.0000i	1.01593i
$873$	− 10.0000i	− 0.338449i
$874$	4.00000	0.135302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	− 34.0000i	− 1.14810i	−0.818821	−	0.574049i	$-0.805372\pi$
$877$	− 34.0000i	− 1.14810i	0.818821	−	0.574049i	$-0.194628\pi$
$878$	8.00000i	0.269987i
$879$	14.0000	0.472208
$880$	0	0
$881$	−38.0000	−1.28025	−0.640126	−	0.768270i	$-0.721118\pi$
$881$	−38.0000	−1.28025	−0.640126	+	0.768270i	$0.721118\pi$
$882$	− 7.00000i	− 0.235702i
$883$	36.0000i	1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$	36.0000i	1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$	−36.0000	−1.21081
$885$	0	0
$886$	0	0
$887$	− 40.0000i	− 1.34307i	−0.740973	−	0.671534i	$-0.765636\pi$
$887$	− 40.0000i	− 1.34307i	0.740973	−	0.671534i	$-0.234364\pi$
$888$	− 30.0000i	− 1.00673i
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	12.0000i	0.401565i
$894$	−6.00000	−0.200670
$895$	0	0
$896$	0	0
$897$	24.0000i	0.801337i
$898$	2.00000i	0.0667409i
$899$	−16.0000	−0.533630
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	−8.00000	−0.265782
$907$	4.00000i	0.132818i	0.997792	+	0.0664089i	$0.0211542\pi$
$907$	4.00000i	0.132818i	−0.997792	+	0.0664089i	$0.978846\pi$
$908$	− 12.0000i	− 0.398234i
$909$	10.0000	0.331679
$910$	0	0
$911$	−16.0000	−0.530104	−0.265052	−	0.964234i	$-0.585389\pi$
$911$	−16.0000	−0.530104	−0.265052	+	0.964234i	$0.585389\pi$
$912$	1.00000i	0.0331133i
$913$	0	0
$914$	6.00000	0.198462
$915$	0	0
$916$	−6.00000	−0.198246
$917$	0	0
$918$	6.00000i	0.198030i
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	−12.0000	−0.395413
$922$	− 18.0000i	− 0.592798i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	8.00000i	0.262754i
$928$	− 10.0000i	− 0.328266i
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	7.00000	0.229416
$932$	− 10.0000i	− 0.327561i
$933$	− 4.00000i	− 0.130954i
$934$	32.0000	1.04707
$935$	0	0
$936$	−18.0000	−0.588348
$937$	− 22.0000i	− 0.718709i	−0.933201	−	0.359354i	$-0.882997\pi$
$937$	− 22.0000i	− 0.718709i	0.933201	−	0.359354i	$-0.117003\pi$
$938$	0	0
$939$	22.0000	0.717943
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	− 2.00000i	− 0.0651635i
$943$	8.00000i	0.260516i
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	− 8.00000i	− 0.259965i	−0.991516	−	0.129983i	$-0.958508\pi$
$947$	− 8.00000i	− 0.259965i	0.991516	−	0.129983i	$-0.0414921\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	− 38.0000i	− 1.23094i	−0.788160	−	0.615470i	$-0.788966\pi$
$953$	− 38.0000i	− 1.23094i	0.788160	−	0.615470i	$-0.211034\pi$
$954$	6.00000	0.194257
$955$	0	0
$956$	12.0000	0.388108
$957$	0	0
$958$	− 20.0000i	− 0.646171i
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	− 60.0000i	− 1.93448i
$963$	− 4.00000i	− 0.128898i
$964$	−6.00000	−0.193247
$965$	0	0
$966$	0	0
$967$	− 32.0000i	− 1.02905i	−0.857475	−	0.514525i	$-0.827968\pi$
$967$	− 32.0000i	− 1.02905i	0.857475	−	0.514525i	$-0.172032\pi$
$968$	− 33.0000i	− 1.06066i
$969$	−6.00000	−0.192748
$970$	0	0
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	− 1.00000i	− 0.0320750i
$973$	0	0
$974$	−32.0000	−1.02535
$975$	0	0
$976$	2.00000	0.0640184
$977$	− 42.0000i	− 1.34370i	−0.740688	−	0.671850i	$-0.765500\pi$
$977$	− 42.0000i	− 1.34370i	0.740688	−	0.671850i	$-0.234500\pi$
$978$	4.00000i	0.127906i
$979$	0	0
$980$	0	0
$981$	−10.0000	−0.319275
$982$	− 32.0000i	− 1.02116i
$983$	8.00000i	0.255160i	0.991828	+	0.127580i	$0.0407210\pi$
$983$	8.00000i	0.255160i	−0.991828	+	0.127580i	$0.959279\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	−12.0000	−0.382158
$987$	0	0
$988$	− 6.00000i	− 0.190885i
$989$	16.0000	0.508770
$990$	0	0
$991$	40.0000	1.27064	0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000	1.27064	0.635321	+	0.772248i	$0.280868\pi$
$992$	40.0000i	1.27000i
$993$	− 12.0000i	− 0.380808i
$994$	0	0
$995$	0	0
$996$	−16.0000	−0.506979
$997$	− 58.0000i	− 1.83688i	−0.395562	−	0.918439i	$-0.629450\pi$
$997$	− 58.0000i	− 1.83688i	0.395562	−	0.918439i	$-0.370550\pi$
$998$	− 28.0000i	− 0.886325i
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1425.2.c.g.799.2		2
5.2	odd	4		1425.2.a.a.1.1		1
5.3	odd	4		57.2.a.c.1.1	✓	1
5.4	even	2	inner	1425.2.c.g.799.1		2
15.2	even	4		4275.2.a.m.1.1		1
15.8	even	4		171.2.a.a.1.1		1
20.3	even	4		912.2.a.b.1.1		1
35.13	even	4		2793.2.a.i.1.1		1
40.3	even	4		3648.2.a.bf.1.1		1
40.13	odd	4		3648.2.a.o.1.1		1
55.43	even	4		6897.2.a.a.1.1		1
60.23	odd	4		2736.2.a.s.1.1		1
65.38	odd	4		9633.2.a.h.1.1		1
95.18	even	4		1083.2.a.a.1.1		1
105.83	odd	4		8379.2.a.e.1.1		1
285.113	odd	4		3249.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.a.c.1.1	✓	1	5.3	odd	4
171.2.a.a.1.1		1	15.8	even	4
912.2.a.b.1.1		1	20.3	even	4
1083.2.a.a.1.1		1	95.18	even	4
1425.2.a.a.1.1		1	5.2	odd	4
1425.2.c.g.799.1		2	5.4	even	2	inner
1425.2.c.g.799.2		2	1.1	even	1	trivial
2736.2.a.s.1.1		1	60.23	odd	4
2793.2.a.i.1.1		1	35.13	even	4
3249.2.a.g.1.1		1	285.113	odd	4
3648.2.a.o.1.1		1	40.13	odd	4
3648.2.a.bf.1.1		1	40.3	even	4
4275.2.a.m.1.1		1	15.2	even	4
6897.2.a.a.1.1		1	55.43	even	4
8379.2.a.e.1.1		1	105.83	odd	4
9633.2.a.h.1.1		1	65.38	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1425 = 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1425.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000i q^{3} +1.00000 q^{4} +1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} -1.00000i q^{12} -6.00000i q^{13} -1.00000 q^{16} -6.00000i q^{17} -1.00000i q^{18} +1.00000 q^{19} -4.00000i q^{23} +3.00000 q^{24} +6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} +8.00000 q^{31} +5.00000i q^{32} +6.00000 q^{34} -1.00000 q^{36} -10.0000i q^{37} +1.00000i q^{38} -6.00000 q^{39} -2.00000 q^{41} +4.00000i q^{43} +4.00000 q^{46} +12.0000i q^{47} +1.00000i q^{48} +7.00000 q^{49} -6.00000 q^{51} -6.00000i q^{52} +6.00000i q^{53} -1.00000 q^{54} -1.00000i q^{57} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} +8.00000i q^{62} -7.00000 q^{64} -4.00000i q^{67} -6.00000i q^{68} -4.00000 q^{69} -3.00000i q^{72} -10.0000i q^{73} +10.0000 q^{74} +1.00000 q^{76} -6.00000i q^{78} +1.00000 q^{81} -2.00000i q^{82} -16.0000i q^{83} -4.00000 q^{86} +2.00000i q^{87} +2.00000 q^{89} -4.00000i q^{92} -8.00000i q^{93} -12.0000 q^{94} +5.00000 q^{96} +10.0000i q^{97} +7.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9	\( 2 q + 2 q^{4} + 2 q^{6} - 2 q^{9} - 2 q^{16} + 2 q^{19} + 6 q^{24} + 12 q^{26} - 4 q^{29} + 16 q^{31} + 12 q^{34} - 2 q^{36} - 12 q^{39} - 4 q^{41} + 8 q^{46} + 14 q^{49} - 12 q^{51} - 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{69} + 20 q^{74} + 2 q^{76} + 2 q^{81} - 8 q^{86} + 4 q^{89} - 24 q^{94} + 10 q^{96}+O(q^{100}) \) 2 * q + 2 * q^4 + 2 * q^6 - 2 * q^9 - 2 * q^16 + 2 * q^19 + 6 * q^24 + 12 * q^26 - 4 * q^29 + 16 * q^31 + 12 * q^34 - 2 * q^36 - 12 * q^39 - 4 * q^41 + 8 * q^46 + 14 * q^49 - 12 * q^51 - 2 * q^54 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 8 * q^69 + 20 * q^74 + 2 * q^76 + 2 * q^81 - 8 * q^86 + 4 * q^89 - 24 * q^94 + 10 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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