LMFDB - Embedded newform 3150.2.g.r.2899.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3150,2,Mod(2899,3150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3150, 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("3150.2899");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3150.g (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:	$25.1528766367$
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 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2899.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	3150.2899
Dual form			3150.2.g.r.2899.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} +6.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} -6.00000 q^{26} -1.00000i q^{28} -2.00000 q^{29} +1.00000i q^{32} -2.00000 q^{34} +10.0000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +8.00000 q^{46} -1.00000 q^{49} -6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{56} -2.00000i q^{58} +4.00000 q^{59} +6.00000 q^{61} -1.00000 q^{64} -4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{71} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} +4.00000i q^{77} +6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -4.00000i q^{88} -6.00000 q^{89} -6.00000 q^{91} +8.00000i q^{92} +14.0000i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} + 8 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} - 12 q^{26} - 4 q^{29} - 4 q^{34} + 12 q^{41} - 8 q^{44} + 16 q^{46} - 2 q^{49} + 2 q^{56} + 8 q^{59} + 12 q^{61} - 2 q^{64} - 16 q^{71} - 20 q^{74} - 8 q^{76} + 8 q^{86} - 12 q^{89} - 12 q^{91}+O(q^{100}) $ 2 * q - 2 * q^4 + 8 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^19 - 12 * q^26 - 4 * q^29 - 4 * q^34 + 12 * q^41 - 8 * q^44 + 16 * q^46 - 2 * q^49 + 2 * q^56 + 8 * q^59 + 12 * q^61 - 2 * q^64 - 16 * q^71 - 20 * q^74 - 8 * q^76 + 8 * q^86 - 12 * q^89 - 12 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$2801$
$\chi(n)$	$-1$	$1$	$1$

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
$2$	1.00000i	0.707107i
$3$	0	0
$3$	0	0
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000i	0.377964i
$7$	1.00000i	0.377964i
$8$	− 1.00000i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	6.00000i	1.66410i	0.554700	+	0.832050i	$0.312833\pi$
$13$	6.00000i	1.66410i	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000i	0.485071i	0.970143	+	0.242536i	$0.0779791\pi$
$17$	2.00000i	0.485071i	−0.970143	+	0.242536i	$0.922021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	4.00000i	0.852803i
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	−6.00000	−1.17670
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	−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
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	1.00000i	0.176777i
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	10.0000i	1.64399i	0.569495	+	0.821995i	$0.307139\pi$
$37$	10.0000i	1.64399i	−0.569495	+	0.821995i	$0.692861\pi$
$38$	4.00000i	0.648886i
$39$	0	0
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	− 4.00000i	− 0.609994i	−0.952353	−	0.304997i	$-0.901344\pi$
$43$	− 4.00000i	− 0.609994i	0.952353	−	0.304997i	$-0.0986555\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	− 6.00000i	− 0.832050i
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	1.00000	0.133631
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	−1.00000	−0.125000
$65$	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	$-0.0785709\pi$
$68$	− 2.00000i	− 0.242536i
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	−10.0000	−1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	4.00000i	0.455842i
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	0	0
$82$	6.00000i	0.662589i
$83$	4.00000i	0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$	4.00000i	0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$	0	0
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	− 4.00000i	− 0.426401i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	8.00000i	0.834058i
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.0000i	1.42148i	0.703452	+	0.710742i	$0.251641\pi$
$97$	14.0000i	1.42148i	−0.703452	+	0.710742i	$0.748359\pi$
$98$	− 1.00000i	− 0.101015i
$99$	0	0
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	8.00000i	0.788263i	0.919054	+	0.394132i	$0.128955\pi$
$103$	8.00000i	0.788263i	−0.919054	+	0.394132i	$0.871045\pi$
$104$	6.00000	0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	14.0000i	1.31701i	0.752577	+	0.658505i	$0.228811\pi$
$113$	14.0000i	1.31701i	−0.752577	+	0.658505i	$0.771189\pi$
$114$	0	0
$115$	0	0
$116$	2.00000	0.185695
$117$	0	0
$118$	4.00000i	0.368230i
$119$	−2.00000	−0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	6.00000i	0.543214i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	− 1.00000i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	4.00000	0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	− 8.00000i	− 0.671345i
$143$	24.0000i	2.00698i
$144$	0	0
$145$	0	0
$146$	−10.0000	−0.827606
$147$	0	0
$148$	− 10.0000i	− 0.821995i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	− 4.00000i	− 0.324443i
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	10.0000i	0.798087i	0.916932	+	0.399043i	$0.130658\pi$
$157$	10.0000i	0.798087i	−0.916932	+	0.399043i	$0.869342\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	20.0000i	1.56652i	0.621694	+	0.783260i	$0.286445\pi$
$163$	20.0000i	1.56652i	−0.621694	+	0.783260i	$0.713555\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	− 8.00000i	− 0.619059i	−0.950890	−	0.309529i	$-0.899829\pi$
$167$	− 8.00000i	− 0.619059i	0.950890	−	0.309529i	$-0.100171\pi$
$168$	0	0
$169$	−23.0000	−1.76923
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	− 22.0000i	− 1.67263i	−0.548250	−	0.836315i	$-0.684706\pi$
$173$	− 22.0000i	− 1.67263i	0.548250	−	0.836315i	$-0.315294\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	− 6.00000i	− 0.449719i
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	− 6.00000i	− 0.444750i
$183$	0	0
$184$	−8.00000	−0.589768
$185$	0	0
$186$	0	0
$187$	8.00000i	0.585018i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000i	0.143963i	0.997406	+	0.0719816i	$0.0229323\pi$
$193$	2.00000i	0.143963i	−0.997406	+	0.0719816i	$0.977068\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	− 10.0000i	− 0.712470i	−0.934396	−	0.356235i	$-0.884060\pi$
$197$	− 10.0000i	− 0.712470i	0.934396	−	0.356235i	$-0.115940\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	0	0
$202$	2.00000i	0.140720i
$203$	− 2.00000i	− 0.140372i
$204$	0	0
$205$	0	0
$206$	−8.00000	−0.557386
$207$	0	0
$208$	6.00000i	0.416025i
$209$	16.0000	1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	−12.0000	−0.820303
$215$	0	0
$216$	0	0
$217$	0	0
$218$	2.00000i	0.135457i
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	2.00000i	0.131306i
$233$	22.0000i	1.44127i	0.693316	+	0.720634i	$0.256149\pi$
$233$	22.0000i	1.44127i	−0.693316	+	0.720634i	$0.743851\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	− 2.00000i	− 0.129641i
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	−6.00000	−0.384111
$245$	0	0
$246$	0	0
$247$	24.0000i	1.52708i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	− 32.0000i	− 2.01182i
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	− 30.0000i	− 1.87135i	−0.352865	−	0.935674i	$-0.614792\pi$
$257$	− 30.0000i	− 1.87135i	0.352865	−	0.935674i	$-0.385208\pi$
$258$	0	0
$259$	−10.0000	−0.621370
$260$	0	0
$261$	0	0
$262$	20.0000i	1.23560i
$263$	24.0000i	1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$	24.0000i	1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	4.00000i	0.244339i
$269$	22.0000	1.34136	0.670682	−	0.741745i	$-0.266002\pi$
$269$	22.0000	1.34136	0.670682	+	0.741745i	$0.266002\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	2.00000i	0.121268i
$273$	0	0
$274$	−10.0000	−0.604122
$275$	0	0
$276$	0	0
$277$	10.0000i	0.600842i	0.953807	+	0.300421i	$0.0971271\pi$
$277$	10.0000i	0.600842i	−0.953807	+	0.300421i	$0.902873\pi$
$278$	− 4.00000i	− 0.239904i
$279$	0	0
$280$	0	0
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	4.00000i	0.237775i	0.992908	+	0.118888i	$0.0379328\pi$
$283$	4.00000i	0.237775i	−0.992908	+	0.118888i	$0.962067\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	6.00000i	0.354169i
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	0	0
$292$	− 10.0000i	− 0.585206i
$293$	− 30.0000i	− 1.75262i	−0.481749	−	0.876309i	$-0.659998\pi$
$293$	− 30.0000i	− 1.75262i	0.481749	−	0.876309i	$-0.340002\pi$
$294$	0	0
$295$	0	0
$296$	10.0000	0.581238
$297$	0	0
$298$	6.00000i	0.347571i
$299$	48.0000	2.77591
$300$	0	0
$301$	4.00000	0.230556
$302$	− 8.00000i	− 0.460348i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	− 28.0000i	− 1.59804i	−0.601302	−	0.799022i	$-0.705351\pi$
$307$	− 28.0000i	− 1.59804i	0.601302	−	0.799022i	$-0.294649\pi$
$308$	− 4.00000i	− 0.227921i
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	10.0000i	0.565233i	0.959233	+	0.282617i	$0.0912024\pi$
$313$	10.0000i	0.565233i	−0.959233	+	0.282617i	$0.908798\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	0	0
$317$	− 18.0000i	− 1.01098i	−0.862832	−	0.505490i	$-0.831312\pi$
$317$	− 18.0000i	− 1.01098i	0.862832	−	0.505490i	$-0.168688\pi$
$318$	0	0
$319$	−8.00000	−0.447914
$320$	0	0
$321$	0	0
$322$	8.00000i	0.445823i
$323$	8.00000i	0.445132i
$324$	0	0
$325$	0	0
$326$	−20.0000	−1.10770
$327$	0	0
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	− 4.00000i	− 0.219529i
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	0	0
$337$	− 18.0000i	− 0.980522i	−0.871576	−	0.490261i	$-0.836901\pi$
$337$	− 18.0000i	− 0.980522i	0.871576	−	0.490261i	$-0.163099\pi$
$338$	− 23.0000i	− 1.25104i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	− 1.00000i	− 0.0539949i
$344$	−4.00000	−0.215666
$345$	0	0
$346$	22.0000	1.18273
$347$	12.0000i	0.644194i	0.946707	+	0.322097i	$0.104388\pi$
$347$	12.0000i	0.644194i	−0.946707	+	0.322097i	$0.895612\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	4.00000i	0.213201i
$353$	30.0000i	1.59674i	0.602168	+	0.798369i	$0.294304\pi$
$353$	30.0000i	1.59674i	−0.602168	+	0.798369i	$0.705696\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	− 12.0000i	− 0.634220i
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 18.0000i	− 0.946059i
$363$	0	0
$364$	6.00000	0.314485
$365$	0	0
$366$	0	0
$367$	− 32.0000i	− 1.67039i	−0.549957	−	0.835193i	$-0.685356\pi$
$367$	− 32.0000i	− 1.67039i	0.549957	−	0.835193i	$-0.314644\pi$
$368$	− 8.00000i	− 0.417029i
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	22.0000i	1.13912i	0.821951	+	0.569558i	$0.192886\pi$
$373$	22.0000i	1.13912i	−0.821951	+	0.569558i	$0.807114\pi$
$374$	−8.00000	−0.413670
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	16.0000i	0.817562i	0.912633	+	0.408781i	$0.134046\pi$
$383$	16.0000i	0.817562i	−0.912633	+	0.408781i	$0.865954\pi$
$384$	0	0
$385$	0	0
$386$	−2.00000	−0.101797
$387$	0	0
$388$	− 14.0000i	− 0.710742i
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	1.00000i	0.0505076i
$393$	0	0
$394$	10.0000	0.503793
$395$	0	0
$396$	0	0
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	− 8.00000i	− 0.401004i
$399$	0	0
$400$	0	0
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	0	0
$406$	2.00000	0.0992583
$407$	40.0000i	1.98273i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	4.00000i	0.196827i
$414$	0	0
$415$	0	0
$416$	−6.00000	−0.294174
$417$	0	0
$418$	16.0000i	0.782586i
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	20.0000i	0.973585i
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	6.00000i	0.290360i
$428$	− 12.0000i	− 0.580042i
$429$	0	0
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	2.00000i	0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$	2.00000i	0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	− 32.0000i	− 1.53077i
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	4.00000i	0.190046i	0.995475	+	0.0950229i	$0.0302924\pi$
$443$	4.00000i	0.190046i	−0.995475	+	0.0950229i	$0.969708\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	− 1.00000i	− 0.0472456i
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	− 10.0000i	− 0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$	− 10.0000i	− 0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$	2.00000i	0.0934539i
$459$	0	0
$460$	0	0
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	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
$466$	−22.0000	−1.01913
$467$	28.0000i	1.29569i	0.761774	+	0.647843i	$0.224329\pi$
$467$	28.0000i	1.29569i	−0.761774	+	0.647843i	$0.775671\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	− 4.00000i	− 0.184115i
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	2.00000	0.0916698
$477$	0	0
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 6.00000i	− 0.271607i
$489$	0	0
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	− 4.00000i	− 0.180151i
$494$	−24.0000	−1.07981
$495$	0	0
$496$	0	0
$497$	− 8.00000i	− 0.358849i
$498$	0	0
$499$	44.0000	1.96971	0.984855	−	0.173379i	$-0.0554684\pi$
$499$	44.0000	1.96971	0.984855	+	0.173379i	$0.0554684\pi$
$500$	0	0
$501$	0	0
$502$	12.0000i	0.535586i
$503$	− 24.0000i	− 1.07011i	−0.844818	−	0.535054i	$-0.820291\pi$
$503$	− 24.0000i	− 1.07011i	0.844818	−	0.535054i	$-0.179709\pi$
$504$	0	0
$505$	0	0
$506$	32.0000	1.42257
$507$	0	0
$508$	0	0
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	1.00000i	0.0441942i
$513$	0	0
$514$	30.0000	1.32324
$515$	0	0
$516$	0	0
$517$	0	0
$518$	− 10.0000i	− 0.439375i
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	0	0
$532$	− 4.00000i	− 0.173422i
$533$	36.0000i	1.55933i
$534$	0	0
$535$	0	0
$536$	−4.00000	−0.172774
$537$	0	0
$538$	22.0000i	0.948487i
$539$	−4.00000	−0.172292
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	0	0
$546$	0	0
$547$	12.0000i	0.513083i	0.966533	+	0.256541i	$0.0825830\pi$
$547$	12.0000i	0.513083i	−0.966533	+	0.256541i	$0.917417\pi$
$548$	− 10.0000i	− 0.427179i
$549$	0	0
$550$	0	0
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	− 2.00000i	− 0.0847427i	−0.999102	−	0.0423714i	$-0.986509\pi$
$557$	− 2.00000i	− 0.0847427i	0.999102	−	0.0423714i	$-0.0134913\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	0	0
$562$	− 26.0000i	− 1.09674i
$563$	− 44.0000i	− 1.85438i	−0.374593	−	0.927189i	$-0.622217\pi$
$563$	− 44.0000i	− 1.85438i	0.374593	−	0.927189i	$-0.377783\pi$
$564$	0	0
$565$	0	0
$566$	−4.00000	−0.168133
$567$	0	0
$568$	8.00000i	0.335673i
$569$	−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
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	− 24.0000i	− 1.00349i
$573$	0	0
$574$	−6.00000	−0.250435
$575$	0	0
$576$	0	0
$577$	− 34.0000i	− 1.41544i	−0.706494	−	0.707719i	$-0.749724\pi$
$577$	− 34.0000i	− 1.41544i	0.706494	−	0.707719i	$-0.250276\pi$
$578$	13.0000i	0.540729i
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	10.0000	0.413803
$585$	0	0
$586$	30.0000	1.23929
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	10.0000i	0.410997i
$593$	− 18.0000i	− 0.739171i	−0.929197	−	0.369586i	$-0.879500\pi$
$593$	− 18.0000i	− 0.739171i	0.929197	−	0.369586i	$-0.120500\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	0	0
$598$	48.0000i	1.96287i
$599$	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
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	4.00000i	0.163028i
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	− 48.0000i	− 1.94826i	−0.225989	−	0.974130i	$-0.572561\pi$
$607$	− 48.0000i	− 1.94826i	0.225989	−	0.974130i	$-0.427439\pi$
$608$	4.00000i	0.162221i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	− 42.0000i	− 1.69636i	−0.529705	−	0.848182i	$-0.677697\pi$
$613$	− 42.0000i	− 1.69636i	0.529705	−	0.848182i	$-0.322303\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	4.00000	0.161165
$617$	− 22.0000i	− 0.885687i	−0.896599	−	0.442843i	$-0.853970\pi$
$617$	− 22.0000i	− 0.885687i	0.896599	−	0.442843i	$-0.146030\pi$
$618$	0	0
$619$	44.0000	1.76851	0.884255	−	0.467005i	$-0.154667\pi$
$619$	44.0000	1.76851	0.884255	+	0.467005i	$0.154667\pi$
$620$	0	0
$621$	0	0
$622$	8.00000i	0.320771i
$623$	− 6.00000i	− 0.240385i
$624$	0	0
$625$	0	0
$626$	−10.0000	−0.399680
$627$	0	0
$628$	− 10.0000i	− 0.399043i
$629$	−20.0000	−0.797452
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	18.0000	0.714871
$635$	0	0
$636$	0	0
$637$	− 6.00000i	− 0.237729i
$638$	− 8.00000i	− 0.316723i
$639$	0	0
$640$	0	0
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	− 4.00000i	− 0.157745i	−0.996885	−	0.0788723i	$-0.974868\pi$
$643$	− 4.00000i	− 0.157745i	0.996885	−	0.0788723i	$-0.0251319\pi$
$644$	−8.00000	−0.315244
$645$	0	0
$646$	−8.00000	−0.314756
$647$	24.0000i	0.943537i	0.881722	+	0.471769i	$0.156384\pi$
$647$	24.0000i	0.943537i	−0.881722	+	0.471769i	$0.843616\pi$
$648$	0	0
$649$	16.0000	0.628055
$650$	0	0
$651$	0	0
$652$	− 20.0000i	− 0.783260i
$653$	18.0000i	0.704394i	0.935926	+	0.352197i	$0.114565\pi$
$653$	18.0000i	0.704394i	−0.935926	+	0.352197i	$0.885435\pi$
$654$	0	0
$655$	0	0
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	−28.0000	−1.09073	−0.545363	−	0.838200i	$-0.683608\pi$
$659$	−28.0000	−1.09073	−0.545363	+	0.838200i	$0.683608\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	4.00000	0.155230
$665$	0	0
$666$	0	0
$667$	16.0000i	0.619522i
$668$	8.00000i	0.309529i
$669$	0	0
$670$	0	0
$671$	24.0000	0.926510
$672$	0	0
$673$	2.00000i	0.0770943i	0.999257	+	0.0385472i	$0.0122730\pi$
$673$	2.00000i	0.0770943i	−0.999257	+	0.0385472i	$0.987727\pi$
$674$	18.0000	0.693334
$675$	0	0
$676$	23.0000	0.884615
$677$	− 18.0000i	− 0.691796i	−0.938272	−	0.345898i	$-0.887574\pi$
$677$	− 18.0000i	− 0.691796i	0.938272	−	0.345898i	$-0.112426\pi$
$678$	0	0
$679$	−14.0000	−0.537271
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	1.00000	0.0381802
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	36.0000	1.37149
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	22.0000i	0.836315i
$693$	0	0
$694$	−12.0000	−0.455514
$695$	0	0
$696$	0	0
$697$	12.0000i	0.454532i
$698$	− 22.0000i	− 0.832712i
$699$	0	0
$700$	0	0
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	40.0000i	1.50863i
$704$	−4.00000	−0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	2.00000i	0.0752177i
$708$	0	0
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	0	0
$712$	6.00000i	0.224860i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	0	0
$718$	− 8.00000i	− 0.298557i
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	18.0000	0.668965
$725$	0	0
$726$	0	0
$727$	8.00000i	0.296704i	0.988935	+	0.148352i	$0.0473968\pi$
$727$	8.00000i	0.296704i	−0.988935	+	0.148352i	$0.952603\pi$
$728$	6.00000i	0.222375i
$729$	0	0
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	6.00000i	0.221615i	0.993842	+	0.110808i	$0.0353437\pi$
$733$	6.00000i	0.221615i	−0.993842	+	0.110808i	$0.964656\pi$
$734$	32.0000	1.18114
$735$	0	0
$736$	8.00000	0.294884
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	12.0000	0.441427	0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000	0.441427	0.220714	+	0.975339i	$0.429161\pi$
$740$	0	0
$741$	0	0
$742$	6.00000i	0.220267i
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	0	0
$745$	0	0
$746$	−22.0000	−0.805477
$747$	0	0
$748$	− 8.00000i	− 0.292509i
$749$	−12.0000	−0.438470
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	− 6.00000i	− 0.218074i	−0.994038	−	0.109037i	$-0.965223\pi$
$757$	− 6.00000i	− 0.218074i	0.994038	−	0.109037i	$-0.0347767\pi$
$758$	20.0000i	0.726433i
$759$	0	0
$760$	0	0
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	0	0
$766$	−16.0000	−0.578103
$767$	24.0000i	0.866590i
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	− 2.00000i	− 0.0719816i
$773$	2.00000i	0.0719350i	0.999353	+	0.0359675i	$0.0114513\pi$
$773$	2.00000i	0.0719350i	−0.999353	+	0.0359675i	$0.988549\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	− 26.0000i	− 0.932145i
$779$	24.0000	0.859889
$780$	0	0
$781$	−32.0000	−1.14505
$782$	16.0000i	0.572159i
$783$	0	0
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	0	0
$787$	36.0000i	1.28326i	0.767014	+	0.641631i	$0.221742\pi$
$787$	36.0000i	1.28326i	−0.767014	+	0.641631i	$0.778258\pi$
$788$	10.0000i	0.356235i
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	36.0000i	1.27840i
$794$	6.00000	0.212932
$795$	0	0
$796$	8.00000	0.283552
$797$	6.00000i	0.212531i	0.994338	+	0.106265i	$0.0338893\pi$
$797$	6.00000i	0.212531i	−0.994338	+	0.106265i	$0.966111\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	− 18.0000i	− 0.635602i
$803$	40.0000i	1.41157i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 2.00000i	− 0.0703598i
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	2.00000i	0.0701862i
$813$	0	0
$814$	−40.0000	−1.40200
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	22.0000i	0.769212i
$819$	0	0
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	− 56.0000i	− 1.95204i	−0.217687	−	0.976019i	$-0.569851\pi$
$823$	− 56.0000i	− 1.95204i	0.217687	−	0.976019i	$-0.430149\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	−4.00000	−0.139178
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	26.0000	0.903017	0.451509	−	0.892267i	$-0.350886\pi$
$829$	26.0000	0.903017	0.451509	+	0.892267i	$0.350886\pi$
$830$	0	0
$831$	0	0
$832$	− 6.00000i	− 0.208013i
$833$	− 2.00000i	− 0.0692959i
$834$	0	0
$835$	0	0
$836$	−16.0000	−0.553372
$837$	0	0
$838$	− 36.0000i	− 1.24360i
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	6.00000i	0.206774i
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	0	0
$847$	5.00000i	0.171802i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	80.0000	2.74236
$852$	0	0
$853$	14.0000i	0.479351i	0.970853	+	0.239675i	$0.0770410\pi$
$853$	14.0000i	0.479351i	−0.970853	+	0.239675i	$0.922959\pi$
$854$	−6.00000	−0.205316
$855$	0	0
$856$	12.0000	0.410152
$857$	42.0000i	1.43469i	0.696717	+	0.717346i	$0.254643\pi$
$857$	42.0000i	1.43469i	−0.696717	+	0.717346i	$0.745357\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	32.0000i	1.08929i	0.838666	+	0.544646i	$0.183336\pi$
$863$	32.0000i	1.08929i	−0.838666	+	0.544646i	$0.816664\pi$
$864$	0	0
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	− 2.00000i	− 0.0677285i
$873$	0	0
$874$	32.0000	1.08242
$875$	0	0
$876$	0	0
$877$	2.00000i	0.0675352i	0.999430	+	0.0337676i	$0.0107506\pi$
$877$	2.00000i	0.0675352i	−0.999430	+	0.0337676i	$0.989249\pi$
$878$	24.0000i	0.809961i
$879$	0	0
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	20.0000i	0.673054i	0.941674	+	0.336527i	$0.109252\pi$
$883$	20.0000i	0.673054i	−0.941674	+	0.336527i	$0.890748\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−4.00000	−0.134383
$887$	− 24.0000i	− 0.805841i	−0.915235	−	0.402921i	$-0.867995\pi$
$887$	− 24.0000i	− 0.805841i	0.915235	−	0.402921i	$-0.132005\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	16.0000i	0.535720i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	1.00000	0.0334077
$897$	0	0
$898$	34.0000i	1.13459i
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	24.0000i	0.799113i
$903$	0	0
$904$	14.0000	0.465633
$905$	0	0
$906$	0	0
$907$	− 12.0000i	− 0.398453i	−0.979953	−	0.199227i	$-0.936157\pi$
$907$	− 12.0000i	− 0.398453i	0.979953	−	0.199227i	$-0.0638430\pi$
$908$	− 12.0000i	− 0.398234i
$909$	0	0
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	16.0000i	0.529523i
$914$	10.0000	0.330771
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	20.0000i	0.660458i
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	0	0
$921$	0	0
$922$	− 22.0000i	− 0.724531i
$923$	− 48.0000i	− 1.57994i
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	0	0
$928$	− 2.00000i	− 0.0656532i
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	− 22.0000i	− 0.720634i
$933$	0	0
$934$	−28.0000	−0.916188
$935$	0	0
$936$	0	0
$937$	22.0000i	0.718709i	0.933201	+	0.359354i	$0.117003\pi$
$937$	22.0000i	0.718709i	−0.933201	+	0.359354i	$0.882997\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	0	0
$943$	− 48.0000i	− 1.56310i
$944$	4.00000	0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	0	0
$949$	−60.0000	−1.94768
$950$	0	0
$951$	0	0
$952$	2.00000i	0.0648204i
$953$	− 26.0000i	− 0.842223i	−0.907009	−	0.421111i	$-0.861640\pi$
$953$	− 26.0000i	− 0.842223i	0.907009	−	0.421111i	$-0.138360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	− 16.0000i	− 0.516937i
$959$	−10.0000	−0.322917
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 60.0000i	− 1.93448i
$963$	0	0
$964$	−2.00000	−0.0644157
$965$	0	0
$966$	0	0
$967$	− 8.00000i	− 0.257263i	−0.991692	−	0.128631i	$-0.958942\pi$
$967$	− 8.00000i	− 0.257263i	0.991692	−	0.128631i	$-0.0410584\pi$
$968$	− 5.00000i	− 0.160706i
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	− 4.00000i	− 0.128234i
$974$	8.00000	0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	18.0000i	0.575871i	0.957650	+	0.287936i	$0.0929689\pi$
$977$	18.0000i	0.575871i	−0.957650	+	0.287936i	$0.907031\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	0	0
$981$	0	0
$982$	− 12.0000i	− 0.382935i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	4.00000	0.127386
$987$	0	0
$988$	− 24.0000i	− 0.763542i
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	8.00000	0.253745
$995$	0	0
$996$	0	0
$997$	− 14.0000i	− 0.443384i	−0.975117	−	0.221692i	$-0.928842\pi$
$997$	− 14.0000i	− 0.443384i	0.975117	−	0.221692i	$-0.0711580\pi$
$998$	44.0000i	1.39280i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3150.2.g.r.2899.2		2
3.2	odd	2		1050.2.g.a.799.1		2
5.2	odd	4		126.2.a.a.1.1		1
5.3	odd	4		3150.2.a.bo.1.1		1
5.4	even	2	inner	3150.2.g.r.2899.1		2
15.2	even	4		42.2.a.a.1.1	✓	1
15.8	even	4		1050.2.a.i.1.1		1
15.14	odd	2		1050.2.g.a.799.2		2
20.7	even	4		1008.2.a.j.1.1		1
35.2	odd	12		882.2.g.h.361.1		2
35.12	even	12		882.2.g.j.361.1		2
35.17	even	12		882.2.g.j.667.1		2
35.27	even	4		882.2.a.b.1.1		1
35.32	odd	12		882.2.g.h.667.1		2
40.27	even	4		4032.2.a.m.1.1		1
40.37	odd	4		4032.2.a.e.1.1		1
45.2	even	12		1134.2.f.g.757.1		2
45.7	odd	12		1134.2.f.j.757.1		2
45.22	odd	12		1134.2.f.j.379.1		2
45.32	even	12		1134.2.f.g.379.1		2
60.23	odd	4		8400.2.a.k.1.1		1
60.47	odd	4		336.2.a.d.1.1		1
105.2	even	12		294.2.e.c.67.1		2
105.17	odd	12		294.2.e.a.79.1		2
105.32	even	12		294.2.e.c.79.1		2
105.47	odd	12		294.2.e.a.67.1		2
105.62	odd	4		294.2.a.g.1.1		1
105.83	odd	4		7350.2.a.f.1.1		1
120.77	even	4		1344.2.a.q.1.1		1
120.107	odd	4		1344.2.a.i.1.1		1
140.27	odd	4		7056.2.a.k.1.1		1
165.32	odd	4		5082.2.a.d.1.1		1
195.77	even	4		7098.2.a.f.1.1		1
240.77	even	4		5376.2.c.bc.2689.1		2
240.107	odd	4		5376.2.c.e.2689.1		2
240.197	even	4		5376.2.c.bc.2689.2		2
240.227	odd	4		5376.2.c.e.2689.2		2
420.47	even	12		2352.2.q.n.1537.1		2
420.107	odd	12		2352.2.q.i.1537.1		2
420.167	even	4		2352.2.a.l.1.1		1
420.227	even	12		2352.2.q.n.961.1		2
420.347	odd	12		2352.2.q.i.961.1		2
840.587	even	4		9408.2.a.bw.1.1		1
840.797	odd	4		9408.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	15.2	even	4
126.2.a.a.1.1		1	5.2	odd	4
294.2.a.g.1.1		1	105.62	odd	4
294.2.e.a.67.1		2	105.47	odd	12
294.2.e.a.79.1		2	105.17	odd	12
294.2.e.c.67.1		2	105.2	even	12
294.2.e.c.79.1		2	105.32	even	12
336.2.a.d.1.1		1	60.47	odd	4
882.2.a.b.1.1		1	35.27	even	4
882.2.g.h.361.1		2	35.2	odd	12
882.2.g.h.667.1		2	35.32	odd	12
882.2.g.j.361.1		2	35.12	even	12
882.2.g.j.667.1		2	35.17	even	12
1008.2.a.j.1.1		1	20.7	even	4
1050.2.a.i.1.1		1	15.8	even	4
1050.2.g.a.799.1		2	3.2	odd	2
1050.2.g.a.799.2		2	15.14	odd	2
1134.2.f.g.379.1		2	45.32	even	12
1134.2.f.g.757.1		2	45.2	even	12
1134.2.f.j.379.1		2	45.22	odd	12
1134.2.f.j.757.1		2	45.7	odd	12
1344.2.a.i.1.1		1	120.107	odd	4
1344.2.a.q.1.1		1	120.77	even	4
2352.2.a.l.1.1		1	420.167	even	4
2352.2.q.i.961.1		2	420.347	odd	12
2352.2.q.i.1537.1		2	420.107	odd	12
2352.2.q.n.961.1		2	420.227	even	12
2352.2.q.n.1537.1		2	420.47	even	12
3150.2.a.bo.1.1		1	5.3	odd	4
3150.2.g.r.2899.1		2	5.4	even	2	inner
3150.2.g.r.2899.2		2	1.1	even	1	trivial
4032.2.a.e.1.1		1	40.37	odd	4
4032.2.a.m.1.1		1	40.27	even	4
5082.2.a.d.1.1		1	165.32	odd	4
5376.2.c.e.2689.1		2	240.107	odd	4
5376.2.c.e.2689.2		2	240.227	odd	4
5376.2.c.bc.2689.1		2	240.77	even	4
5376.2.c.bc.2689.2		2	240.197	even	4
7056.2.a.k.1.1		1	140.27	odd	4
7098.2.a.f.1.1		1	195.77	even	4
7350.2.a.f.1.1		1	105.83	odd	4
8400.2.a.k.1.1		1	60.23	odd	4
9408.2.a.n.1.1		1	840.797	odd	4
9408.2.a.bw.1.1		1	840.587	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3150.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{7} -1.00000i q^{8} +4.00000 q^{11} +6.00000i q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.00000i q^{17} +4.00000 q^{19} +4.00000i q^{22} -8.00000i q^{23} -6.00000 q^{26} -1.00000i q^{28} -2.00000 q^{29} +1.00000i q^{32} -2.00000 q^{34} +10.0000i q^{37} +4.00000i q^{38} +6.00000 q^{41} -4.00000i q^{43} -4.00000 q^{44} +8.00000 q^{46} -1.00000 q^{49} -6.00000i q^{52} -6.00000i q^{53} +1.00000 q^{56} -2.00000i q^{58} +4.00000 q^{59} +6.00000 q^{61} -1.00000 q^{64} -4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{71} +10.0000i q^{73} -10.0000 q^{74} -4.00000 q^{76} +4.00000i q^{77} +6.00000i q^{82} +4.00000i q^{83} +4.00000 q^{86} -4.00000i q^{88} -6.00000 q^{89} -6.00000 q^{91} +8.00000i q^{92} +14.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} + 8 q^{11} - 2 q^{14} + 2 q^{16} + 8 q^{19} - 12 q^{26} - 4 q^{29} - 4 q^{34} + 12 q^{41} - 8 q^{44} + 16 q^{46} - 2 q^{49} + 2 q^{56} + 8 q^{59} + 12 q^{61} - 2 q^{64} - 16 q^{71} - 20 q^{74} - 8 q^{76} + 8 q^{86} - 12 q^{89} - 12 q^{91}+O(q^{100}) \) 2 * q - 2 * q^4 + 8 * q^11 - 2 * q^14 + 2 * q^16 + 8 * q^19 - 12 * q^26 - 4 * q^29 - 4 * q^34 + 12 * q^41 - 8 * q^44 + 16 * q^46 - 2 * q^49 + 2 * q^56 + 8 * q^59 + 12 * q^61 - 2 * q^64 - 16 * q^71 - 20 * q^74 - 8 * q^76 + 8 * q^86 - 12 * q^89 - 12 * q^91

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