LMFDB - Embedded newform 4900.2.a.bb.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4900,2,Mod(1,4900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4900, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4900.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4900.a (trivial)

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:	yes
Analytic conductor:	$39.1266969904$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 700)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.91223$ of defining polynomial
Character	$\chi$	$=$	4900.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56885 q^{3} +3.59899 q^{9} -1.56885 q^{11} -5.56885 q^{13} -7.16784 q^{17} -3.16784 q^{19} +5.73669 q^{23} +1.53871 q^{27} +1.96986 q^{29} -0.969861 q^{31} -4.03014 q^{33} -6.70655 q^{37} -14.3055 q^{39} +8.87439 q^{41} -4.59899 q^{43} -0.401012 q^{47} -18.4131 q^{51} +9.53871 q^{53} -8.13770 q^{57} +4.56885 q^{59} -15.3055 q^{61} +0.862301 q^{67} +14.7367 q^{69} -12.1377 q^{71} -4.00000 q^{73} -12.8744 q^{79} -6.84425 q^{81} -17.1076 q^{83} +5.06028 q^{87} +5.59899 q^{89} -2.49143 q^{93} -0.233174 q^{97} -5.64627 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{3} + 8 q^{9} + 4 q^{11} - 8 q^{13} - 10 q^{17} + 2 q^{19} - 3 q^{23} - 10 q^{27} + 3 q^{31} - 18 q^{33} + 6 q^{37} - 14 q^{39} - 11 q^{41} - 11 q^{43} - 4 q^{47} - 3 q^{51} + 14 q^{53} - 7 q^{57}+ \cdots + 21 q^{99}+O(q^{100}) $ 3 * q - q^3 + 8 * q^9 + 4 * q^11 - 8 * q^13 - 10 * q^17 + 2 * q^19 - 3 * q^23 - 10 * q^27 + 3 * q^31 - 18 * q^33 + 6 * q^37 - 14 * q^39 - 11 * q^41 - 11 * q^43 - 4 * q^47 - 3 * q^51 + 14 * q^53 - 7 * q^57 + 5 * q^59 - 17 * q^61 + 20 * q^67 + 24 * q^69 - 19 * q^71 - 12 * q^73 - q^79 + 23 * q^81 - 28 * q^83 + 27 * q^87 + 14 * q^89 - 28 * q^93 - 15 * q^97 + 21 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.56885	1.48313	0.741563	−	0.670883i	$-0.234085\pi$
$3$	2.56885	1.48313	0.741563	+	0.670883i	$0.234085\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.59899	1.19966
$10$	0	0
$11$	−1.56885	−0.473026	−0.236513	−	0.971628i	$-0.576005\pi$
$11$	−1.56885	−0.473026	−0.236513	+	0.971628i	$0.576005\pi$
$12$	0	0
$13$	−5.56885	−1.54452	−0.772260	−	0.635306i	$-0.780874\pi$
$13$	−5.56885	−1.54452	−0.772260	+	0.635306i	$0.780874\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.16784	−1.73846	−0.869228	−	0.494411i	$-0.835384\pi$
$17$	−7.16784	−1.73846	−0.869228	+	0.494411i	$0.835384\pi$
$18$	0	0
$19$	−3.16784	−0.726752	−0.363376	−	0.931643i	$-0.618376\pi$
$19$	−3.16784	−0.726752	−0.363376	+	0.931643i	$0.618376\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.73669	1.19618	0.598091	−	0.801428i	$-0.295926\pi$
$23$	5.73669	1.19618	0.598091	+	0.801428i	$0.295926\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.53871	0.296125
$28$	0	0
$29$	1.96986	0.365794	0.182897	−	0.983132i	$-0.441453\pi$
$29$	1.96986	0.365794	0.182897	+	0.983132i	$0.441453\pi$
$30$	0	0
$31$	−0.969861	−0.174192	−0.0870961	−	0.996200i	$-0.527759\pi$
$31$	−0.969861	−0.174192	−0.0870961	+	0.996200i	$0.527759\pi$
$32$	0	0
$33$	−4.03014	−0.701557
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.70655	−1.10255	−0.551275	−	0.834324i	$-0.685858\pi$
$37$	−6.70655	−1.10255	−0.551275	+	0.834324i	$0.685858\pi$
$38$	0	0
$39$	−14.3055	−2.29072
$40$	0	0
$41$	8.87439	1.38595	0.692973	−	0.720963i	$-0.256300\pi$
$41$	8.87439	1.38595	0.692973	+	0.720963i	$0.256300\pi$
$42$	0	0
$43$	−4.59899	−0.701339	−0.350670	−	0.936499i	$-0.614046\pi$
$43$	−4.59899	−0.701339	−0.350670	+	0.936499i	$0.614046\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.401012	−0.0584936	−0.0292468	−	0.999572i	$-0.509311\pi$
$47$	−0.401012	−0.0584936	−0.0292468	+	0.999572i	$0.509311\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−18.4131	−2.57835
$52$	0	0
$53$	9.53871	1.31024	0.655121	−	0.755524i	$-0.272617\pi$
$53$	9.53871	1.31024	0.655121	+	0.755524i	$0.272617\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−8.13770	−1.07786
$58$	0	0
$59$	4.56885	0.594814	0.297407	−	0.954751i	$-0.403878\pi$
$59$	4.56885	0.594814	0.297407	+	0.954751i	$0.403878\pi$
$60$	0	0
$61$	−15.3055	−1.95967	−0.979837	−	0.199800i	$-0.935971\pi$
$61$	−15.3055	−1.95967	−0.979837	+	0.199800i	$0.935971\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	0.862301	0.105347	0.0526734	−	0.998612i	$-0.483226\pi$
$67$	0.862301	0.105347	0.0526734	+	0.998612i	$0.483226\pi$
$68$	0	0
$69$	14.7367	1.77409
$70$	0	0
$71$	−12.1377	−1.44048	−0.720240	−	0.693725i	$-0.755968\pi$
$71$	−12.1377	−1.44048	−0.720240	+	0.693725i	$0.755968\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.8744	−1.44848	−0.724241	−	0.689547i	$-0.757810\pi$
$79$	−12.8744	−1.44848	−0.724241	+	0.689547i	$0.757810\pi$
$80$	0	0
$81$	−6.84425	−0.760472
$82$	0	0
$83$	−17.1076	−1.87780	−0.938899	−	0.344192i	$-0.888153\pi$
$83$	−17.1076	−1.87780	−0.938899	+	0.344192i	$0.888153\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.06028	0.542519
$88$	0	0
$89$	5.59899	0.593492	0.296746	−	0.954957i	$-0.404099\pi$
$89$	5.59899	0.593492	0.296746	+	0.954957i	$0.404099\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.49143	−0.258349
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−0.233174	−0.0236752	−0.0118376	−	0.999930i	$-0.503768\pi$
$97$	−0.233174	−0.0236752	−0.0118376	+	0.999930i	$0.503768\pi$
$98$	0	0
$99$	−5.64627	−0.567472
$100$	0	0
$101$	−12.5035	−1.24415	−0.622073	−	0.782959i	$-0.713709\pi$
$101$	−12.5035	−1.24415	−0.622073	+	0.782959i	$0.713709\pi$
$102$	0	0
$103$	−3.37087	−0.332142	−0.166071	−	0.986114i	$-0.553108\pi$
$103$	−3.37087	−0.332142	−0.166071	+	0.986114i	$0.553108\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.7367	−1.71467	−0.857335	−	0.514759i	$-0.827882\pi$
$107$	−17.7367	−1.71467	−0.857335	+	0.514759i	$0.827882\pi$
$108$	0	0
$109$	5.90453	0.565551	0.282775	−	0.959186i	$-0.408745\pi$
$109$	5.90453	0.565551	0.282775	+	0.959186i	$0.408745\pi$
$110$	0	0
$111$	−17.2281	−1.63522
$112$	0	0
$113$	15.6412	1.47140	0.735701	−	0.677307i	$-0.236853\pi$
$113$	15.6412	1.47140	0.735701	+	0.677307i	$0.236853\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−20.0422	−1.85290
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.53871	−0.776246
$122$	0	0
$123$	22.7970	2.05553
$124$	0	0
$125$	0	0
$126$	0	0
$127$	12.8744	1.14242	0.571209	−	0.820805i	$-0.306475\pi$
$127$	12.8744	1.14242	0.571209	+	0.820805i	$0.306475\pi$
$128$	0	0
$129$	−11.8141	−1.04017
$130$	0	0
$131$	0.228115	0.0199305	0.00996526	−	0.999950i	$-0.496828\pi$
$131$	0.228115	0.0199305	0.00996526	+	0.999950i	$0.496828\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−9.67641	−0.826712	−0.413356	−	0.910570i	$-0.635643\pi$
$137$	−9.67641	−0.826712	−0.413356	+	0.910570i	$0.635643\pi$
$138$	0	0
$139$	12.3709	1.04928	0.524642	−	0.851323i	$-0.324199\pi$
$139$	12.3709	1.04928	0.524642	+	0.851323i	$0.324199\pi$
$140$	0	0
$141$	−1.03014	−0.0867533
$142$	0	0
$143$	8.73669	0.730599
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.9096	0.975671	0.487836	−	0.872936i	$-0.337787\pi$
$149$	11.9096	0.975671	0.487836	+	0.872936i	$0.337787\pi$
$150$	0	0
$151$	9.96986	0.811336	0.405668	−	0.914021i	$-0.367039\pi$
$151$	9.96986	0.811336	0.405668	+	0.914021i	$0.367039\pi$
$152$	0	0
$153$	−25.7970	−2.08556
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0422	0.961074	0.480537	−	0.876974i	$-0.340442\pi$
$157$	12.0422	0.961074	0.480537	+	0.876974i	$0.340442\pi$
$158$	0	0
$159$	24.5035	1.94326
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.7367	−1.31092	−0.655459	−	0.755231i	$-0.727525\pi$
$163$	−16.7367	−1.31092	−0.655459	+	0.755231i	$0.727525\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.76683	0.523633	0.261816	−	0.965118i	$-0.415679\pi$
$167$	6.76683	0.523633	0.261816	+	0.965118i	$0.415679\pi$
$168$	0	0
$169$	18.0121	1.38555
$170$	0	0
$171$	−11.4010	−0.871857
$172$	0	0
$173$	−4.69446	−0.356913	−0.178457	−	0.983948i	$-0.557110\pi$
$173$	−4.69446	−0.356913	−0.178457	+	0.983948i	$0.557110\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	11.7367	0.882183
$178$	0	0
$179$	11.4734	0.857560	0.428780	−	0.903409i	$-0.358943\pi$
$179$	11.4734	0.857560	0.428780	+	0.903409i	$0.358943\pi$
$180$	0	0
$181$	−0.832162	−0.0618541	−0.0309271	−	0.999522i	$-0.509846\pi$
$181$	−0.832162	−0.0618541	−0.0309271	+	0.999522i	$0.509846\pi$
$182$	0	0
$183$	−39.3176	−2.90644
$184$	0	0
$185$	0	0
$186$	0	0
$187$	11.2453	0.822335
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.93972	0.502141	0.251070	−	0.967969i	$-0.419217\pi$
$191$	6.93972	0.502141	0.251070	+	0.967969i	$0.419217\pi$
$192$	0	0
$193$	19.6764	1.41634	0.708169	−	0.706042i	$-0.249521\pi$
$193$	19.6764	1.41634	0.708169	+	0.706042i	$0.249521\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−16.0422	−1.14296	−0.571481	−	0.820616i	$-0.693631\pi$
$197$	−16.0422	−1.14296	−0.571481	+	0.820616i	$0.693631\pi$
$198$	0	0
$199$	10.6764	0.756831	0.378415	−	0.925636i	$-0.376469\pi$
$199$	10.6764	0.756831	0.378415	+	0.925636i	$0.376469\pi$
$200$	0	0
$201$	2.21512	0.156243
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	20.6463	1.43502
$208$	0	0
$209$	4.96986	0.343772
$210$	0	0
$211$	11.3658	0.782455	0.391227	−	0.920294i	$-0.372051\pi$
$211$	11.3658	0.782455	0.391227	+	0.920294i	$0.372051\pi$
$212$	0	0
$213$	−31.1799	−2.13641
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−10.2754	−0.694347
$220$	0	0
$221$	39.9166	2.68508
$222$	0	0
$223$	6.04222	0.404617	0.202309	−	0.979322i	$-0.435156\pi$
$223$	6.04222	0.404617	0.202309	+	0.979322i	$0.435156\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−21.6412	−1.43638	−0.718189	−	0.695848i	$-0.755029\pi$
$227$	−21.6412	−1.43638	−0.718189	+	0.695848i	$0.755029\pi$
$228$	0	0
$229$	5.43115	0.358901	0.179450	−	0.983767i	$-0.442568\pi$
$229$	5.43115	0.358901	0.179450	+	0.983767i	$0.442568\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.93972	−0.454636	−0.227318	−	0.973821i	$-0.572996\pi$
$233$	−6.93972	−0.454636	−0.227318	+	0.973821i	$0.572996\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−33.0724	−2.14828
$238$	0	0
$239$	1.69446	0.109606	0.0548028	−	0.998497i	$-0.482547\pi$
$239$	1.69446	0.109606	0.0548028	+	0.998497i	$0.482547\pi$
$240$	0	0
$241$	14.9347	0.962026	0.481013	−	0.876713i	$-0.340269\pi$
$241$	14.9347	0.962026	0.481013	+	0.876713i	$0.340269\pi$
$242$	0	0
$243$	−22.1980	−1.42400
$244$	0	0
$245$	0	0
$246$	0	0
$247$	17.6412	1.12248
$248$	0	0
$249$	−43.9468	−2.78501
$250$	0	0
$251$	−8.90958	−0.562368	−0.281184	−	0.959654i	$-0.590727\pi$
$251$	−8.90958	−0.562368	−0.281184	+	0.959654i	$0.590727\pi$
$252$	0	0
$253$	−9.00000	−0.565825
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.66432	−0.602844	−0.301422	−	0.953491i	$-0.597461\pi$
$257$	−9.66432	−0.602844	−0.301422	+	0.953491i	$0.597461\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	7.08951	0.438830
$262$	0	0
$263$	−6.10250	−0.376296	−0.188148	−	0.982141i	$-0.560249\pi$
$263$	−6.10250	−0.376296	−0.188148	+	0.982141i	$0.560249\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	14.3830	0.880223
$268$	0	0
$269$	−22.5337	−1.37390	−0.686951	−	0.726704i	$-0.741051\pi$
$269$	−22.5337	−1.37390	−0.686951	+	0.726704i	$0.741051\pi$
$270$	0	0
$271$	−4.46129	−0.271004	−0.135502	−	0.990777i	$-0.543265\pi$
$271$	−4.46129	−0.271004	−0.135502	+	0.990777i	$0.543265\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.50857	0.391062	0.195531	−	0.980698i	$-0.437357\pi$
$277$	6.50857	0.391062	0.195531	+	0.980698i	$0.437357\pi$
$278$	0	0
$279$	−3.49052	−0.208972
$280$	0	0
$281$	−11.1980	−0.668015	−0.334008	−	0.942570i	$-0.608401\pi$
$281$	−11.1980	−0.668015	−0.334008	+	0.942570i	$0.608401\pi$
$282$	0	0
$283$	7.64627	0.454524	0.227262	−	0.973834i	$-0.427023\pi$
$283$	7.64627	0.454524	0.227262	+	0.973834i	$0.427023\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	34.3779	2.02223
$290$	0	0
$291$	−0.598988	−0.0351133
$292$	0	0
$293$	−6.12561	−0.357862	−0.178931	−	0.983862i	$-0.557264\pi$
$293$	−6.12561	−0.357862	−0.178931	+	0.983862i	$0.557264\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−2.41401	−0.140075
$298$	0	0
$299$	−31.9468	−1.84753
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−32.1196	−1.84523
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−26.4080	−1.50719	−0.753593	−	0.657341i	$-0.771681\pi$
$307$	−26.4080	−1.50719	−0.753593	+	0.657341i	$0.771681\pi$
$308$	0	0
$309$	−8.65927	−0.492608
$310$	0	0
$311$	19.3528	1.09740	0.548699	−	0.836020i	$-0.315123\pi$
$311$	19.3528	1.09740	0.548699	+	0.836020i	$0.315123\pi$
$312$	0	0
$313$	−8.70655	−0.492123	−0.246062	−	0.969254i	$-0.579137\pi$
$313$	−8.70655	−0.492123	−0.246062	+	0.969254i	$0.579137\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.2754	0.857952	0.428976	−	0.903316i	$-0.358875\pi$
$317$	15.2754	0.857952	0.428976	+	0.903316i	$0.358875\pi$
$318$	0	0
$319$	−3.09042	−0.173030
$320$	0	0
$321$	−45.5629	−2.54307
$322$	0	0
$323$	22.7065	1.26343
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.1678	0.838783
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.7488	0.920596	0.460298	−	0.887765i	$-0.347743\pi$
$331$	16.7488	0.920596	0.460298	+	0.887765i	$0.347743\pi$
$332$	0	0
$333$	−24.1368	−1.32269
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−31.2453	−1.70204	−0.851019	−	0.525135i	$-0.824015\pi$
$337$	−31.2453	−1.70204	−0.851019	+	0.525135i	$0.824015\pi$
$338$	0	0
$339$	40.1799	2.18227
$340$	0	0
$341$	1.52157	0.0823974
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.1025	0.649696	0.324848	−	0.945766i	$-0.394687\pi$
$347$	12.1025	0.649696	0.324848	+	0.945766i	$0.394687\pi$
$348$	0	0
$349$	3.87439	0.207391	0.103696	−	0.994609i	$-0.466933\pi$
$349$	3.87439	0.207391	0.103696	+	0.994609i	$0.466933\pi$
$350$	0	0
$351$	−8.56885	−0.457371
$352$	0	0
$353$	2.56379	0.136457	0.0682284	−	0.997670i	$-0.478265\pi$
$353$	2.56379	0.136457	0.0682284	+	0.997670i	$0.478265\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.8392	1.09985	0.549925	−	0.835214i	$-0.314656\pi$
$359$	20.8392	1.09985	0.549925	+	0.835214i	$0.314656\pi$
$360$	0	0
$361$	−8.96480	−0.471832
$362$	0	0
$363$	−21.9347	−1.15127
$364$	0	0
$365$	0	0
$366$	0	0
$367$	15.8322	0.826432	0.413216	−	0.910633i	$-0.364405\pi$
$367$	15.8322	0.826432	0.413216	+	0.910633i	$0.364405\pi$
$368$	0	0
$369$	31.9388	1.66267
$370$	0	0
$371$	0	0
$372$	0	0
$373$	22.5035	1.16519	0.582594	−	0.812763i	$-0.302038\pi$
$373$	22.5035	1.16519	0.582594	+	0.812763i	$0.302038\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.9699	−0.564977
$378$	0	0
$379$	−7.36581	−0.378356	−0.189178	−	0.981943i	$-0.560582\pi$
$379$	−7.36581	−0.378356	−0.189178	+	0.981943i	$0.560582\pi$
$380$	0	0
$381$	33.0724	1.69435
$382$	0	0
$383$	−33.6533	−1.71960	−0.859802	−	0.510628i	$-0.829413\pi$
$383$	−33.6533	−1.71960	−0.859802	+	0.510628i	$0.829413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−16.5517	−0.841370
$388$	0	0
$389$	17.4855	0.886548	0.443274	−	0.896386i	$-0.353817\pi$
$389$	17.4855	0.886548	0.443274	+	0.896386i	$0.353817\pi$
$390$	0	0
$391$	−41.1196	−2.07951
$392$	0	0
$393$	0.585994	0.0295595
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.59899	0.381382	0.190691	−	0.981650i	$-0.438927\pi$
$397$	7.59899	0.381382	0.190691	+	0.981650i	$0.438927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.8442	0.541536	0.270768	−	0.962645i	$-0.412722\pi$
$401$	10.8442	0.541536	0.270768	+	0.962645i	$0.412722\pi$
$402$	0	0
$403$	5.40101	0.269044
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.5216	0.521535
$408$	0	0
$409$	16.8693	0.834135	0.417067	−	0.908876i	$-0.363058\pi$
$409$	16.8693	0.834135	0.417067	+	0.908876i	$0.363058\pi$
$410$	0	0
$411$	−24.8572	−1.22612
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	31.7789	1.55622
$418$	0	0
$419$	4.03014	0.196885	0.0984426	−	0.995143i	$-0.468614\pi$
$419$	4.03014	0.196885	0.0984426	+	0.995143i	$0.468614\pi$
$420$	0	0
$421$	25.1196	1.22426	0.612128	−	0.790758i	$-0.290314\pi$
$421$	25.1196	1.22426	0.612128	+	0.790758i	$0.290314\pi$
$422$	0	0
$423$	−1.44324	−0.0701726
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	22.4432	1.08357
$430$	0	0
$431$	30.2402	1.45662	0.728310	−	0.685248i	$-0.240306\pi$
$431$	30.2402	1.45662	0.728310	+	0.685248i	$0.240306\pi$
$432$	0	0
$433$	22.9769	1.10420	0.552099	−	0.833778i	$-0.313827\pi$
$433$	22.9769	1.10420	0.552099	+	0.833778i	$0.313827\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−18.1729	−0.869328
$438$	0	0
$439$	−11.9518	−0.570429	−0.285214	−	0.958464i	$-0.592065\pi$
$439$	−11.9518	−0.570429	−0.285214	+	0.958464i	$0.592065\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−14.1076	−0.670270	−0.335135	−	0.942170i	$-0.608782\pi$
$443$	−14.1076	−0.670270	−0.335135	+	0.942170i	$0.608782\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	30.5939	1.44704
$448$	0	0
$449$	37.9468	1.79082	0.895409	−	0.445245i	$-0.146883\pi$
$449$	37.9468	1.79082	0.895409	+	0.445245i	$0.146883\pi$
$450$	0	0
$451$	−13.9226	−0.655589
$452$	0	0
$453$	25.6111	1.20331
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.5457	−0.867533	−0.433767	−	0.901025i	$-0.642816\pi$
$457$	−18.5457	−0.867533	−0.433767	+	0.901025i	$0.642816\pi$
$458$	0	0
$459$	−11.0292	−0.514800
$460$	0	0
$461$	23.2453	1.08264	0.541320	−	0.840817i	$-0.317925\pi$
$461$	23.2453	1.08264	0.541320	+	0.840817i	$0.317925\pi$
$462$	0	0
$463$	7.59393	0.352920	0.176460	−	0.984308i	$-0.443535\pi$
$463$	7.59393	0.352920	0.176460	+	0.984308i	$0.443535\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−28.1678	−1.30345	−0.651726	−	0.758454i	$-0.725955\pi$
$467$	−28.1678	−1.30345	−0.651726	+	0.758454i	$0.725955\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	30.9347	1.42539
$472$	0	0
$473$	7.21512	0.331752
$474$	0	0
$475$	0	0
$476$	0	0
$477$	34.3297	1.57185
$478$	0	0
$479$	34.1799	1.56172	0.780860	−	0.624706i	$-0.214781\pi$
$479$	34.1799	1.56172	0.780860	+	0.624706i	$0.214781\pi$
$480$	0	0
$481$	37.3478	1.70291
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−11.5035	−0.521274	−0.260637	−	0.965437i	$-0.583933\pi$
$487$	−11.5035	−0.521274	−0.260637	+	0.965437i	$0.583933\pi$
$488$	0	0
$489$	−42.9940	−1.94426
$490$	0	0
$491$	−25.5337	−1.15232	−0.576159	−	0.817338i	$-0.695449\pi$
$491$	−25.5337	−1.15232	−0.576159	+	0.817338i	$0.695449\pi$
$492$	0	0
$493$	−14.1196	−0.635917
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−11.3960	−0.510153	−0.255076	−	0.966921i	$-0.582101\pi$
$499$	−11.3960	−0.510153	−0.255076	+	0.966921i	$0.582101\pi$
$500$	0	0
$501$	17.3830	0.776613
$502$	0	0
$503$	2.49649	0.111313	0.0556564	−	0.998450i	$-0.482275\pi$
$503$	2.49649	0.111313	0.0556564	+	0.998450i	$0.482275\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	46.2703	2.05494
$508$	0	0
$509$	−11.5438	−0.511669	−0.255834	−	0.966721i	$-0.582350\pi$
$509$	−11.5438	−0.511669	−0.255834	+	0.966721i	$0.582350\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.87439	−0.215209
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.629127	0.0276690
$518$	0	0
$519$	−12.0594	−0.529348
$520$	0	0
$521$	−23.7488	−1.04045	−0.520226	−	0.854028i	$-0.674152\pi$
$521$	−23.7488	−1.04045	−0.520226	+	0.854028i	$0.674152\pi$
$522$	0	0
$523$	−16.1377	−0.705652	−0.352826	−	0.935689i	$-0.614779\pi$
$523$	−16.1377	−0.705652	−0.352826	+	0.935689i	$0.614779\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.95181	0.302826
$528$	0	0
$529$	9.90958	0.430851
$530$	0	0
$531$	16.4432	0.713576
$532$	0	0
$533$	−49.4201	−2.14062
$534$	0	0
$535$	0	0
$536$	0	0
$537$	29.4734	1.27187
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.53871	−0.0661543	−0.0330772	−	0.999453i	$-0.510531\pi$
$541$	−1.53871	−0.0661543	−0.0330772	+	0.999453i	$0.510531\pi$
$542$	0	0
$543$	−2.13770	−0.0917375
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−14.3779	−0.614755	−0.307377	−	0.951588i	$-0.599451\pi$
$547$	−14.3779	−0.614755	−0.307377	+	0.951588i	$0.599451\pi$
$548$	0	0
$549$	−55.0844	−2.35095
$550$	0	0
$551$	−6.24020	−0.265842
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−16.9347	−0.717545	−0.358772	−	0.933425i	$-0.616805\pi$
$557$	−16.9347	−0.717545	−0.358772	+	0.933425i	$0.616805\pi$
$558$	0	0
$559$	25.6111	1.08323
$560$	0	0
$561$	28.8874	1.21963
$562$	0	0
$563$	−0.664324	−0.0279979	−0.0139990	−	0.999902i	$-0.504456\pi$
$563$	−0.664324	−0.0279979	−0.0139990	+	0.999902i	$0.504456\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.0372	−1.72037	−0.860184	−	0.509984i	$-0.829651\pi$
$569$	−41.0372	−1.72037	−0.860184	+	0.509984i	$0.829651\pi$
$570$	0	0
$571$	−28.6291	−1.19809	−0.599046	−	0.800715i	$-0.704453\pi$
$571$	−28.6291	−1.19809	−0.599046	+	0.800715i	$0.704453\pi$
$572$	0	0
$573$	17.8271	0.744738
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−31.4604	−1.30971	−0.654856	−	0.755753i	$-0.727271\pi$
$577$	−31.4604	−1.30971	−0.654856	+	0.755753i	$0.727271\pi$
$578$	0	0
$579$	50.5457	2.10061
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−14.9648	−0.619779
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.53871	0.393705	0.196852	−	0.980433i	$-0.436928\pi$
$587$	9.53871	0.393705	0.196852	+	0.980433i	$0.436928\pi$
$588$	0	0
$589$	3.07236	0.126595
$590$	0	0
$591$	−41.2101	−1.69516
$592$	0	0
$593$	−9.93972	−0.408175	−0.204088	−	0.978953i	$-0.565423\pi$
$593$	−9.93972	−0.408175	−0.204088	+	0.978953i	$0.565423\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	27.4261	1.12248
$598$	0	0
$599$	9.41310	0.384609	0.192304	−	0.981335i	$-0.438404\pi$
$599$	9.41310	0.384609	0.192304	+	0.981335i	$0.438404\pi$
$600$	0	0
$601$	3.00506	0.122579	0.0612894	−	0.998120i	$-0.480479\pi$
$601$	3.00506	0.122579	0.0612894	+	0.998120i	$0.480479\pi$
$602$	0	0
$603$	3.10341	0.126381
$604$	0	0
$605$	0	0
$606$	0	0
$607$	18.8322	0.764374	0.382187	−	0.924085i	$-0.375171\pi$
$607$	18.8322	0.764374	0.382187	+	0.924085i	$0.375171\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.23317	0.0903445
$612$	0	0
$613$	15.8392	0.639739	0.319869	−	0.947462i	$-0.396361\pi$
$613$	15.8392	0.639739	0.319869	+	0.947462i	$0.396361\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−0.778912	−0.0313578	−0.0156789	−	0.999877i	$-0.504991\pi$
$617$	−0.778912	−0.0313578	−0.0156789	+	0.999877i	$0.504991\pi$
$618$	0	0
$619$	−9.94675	−0.399794	−0.199897	−	0.979817i	$-0.564061\pi$
$619$	−9.94675	−0.399794	−0.199897	+	0.979817i	$0.564061\pi$
$620$	0	0
$621$	8.82710	0.354219
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	12.7668	0.509858
$628$	0	0
$629$	48.0715	1.91673
$630$	0	0
$631$	−12.5638	−0.500157	−0.250078	−	0.968226i	$-0.580456\pi$
$631$	−12.5638	−0.500157	−0.250078	+	0.968226i	$0.580456\pi$
$632$	0	0
$633$	29.1971	1.16048
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−43.6834	−1.72809
$640$	0	0
$641$	−11.5207	−0.455039	−0.227519	−	0.973774i	$-0.573061\pi$
$641$	−11.5207	−0.455039	−0.227519	+	0.973774i	$0.573061\pi$
$642$	0	0
$643$	9.74175	0.384177	0.192088	−	0.981378i	$-0.438474\pi$
$643$	9.74175	0.384177	0.192088	+	0.981378i	$0.438474\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25.7538	1.01249	0.506244	−	0.862390i	$-0.331034\pi$
$647$	25.7538	1.01249	0.506244	+	0.862390i	$0.331034\pi$
$648$	0	0
$649$	−7.16784	−0.281362
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−15.1929	−0.594545	−0.297272	−	0.954793i	$-0.596077\pi$
$653$	−15.1929	−0.594545	−0.297272	+	0.954793i	$0.596077\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−14.3960	−0.561640
$658$	0	0
$659$	−14.6894	−0.572218	−0.286109	−	0.958197i	$-0.592362\pi$
$659$	−14.6894	−0.572218	−0.286109	+	0.958197i	$0.592362\pi$
$660$	0	0
$661$	−4.33568	−0.168638	−0.0843191	−	0.996439i	$-0.526872\pi$
$661$	−4.33568	−0.168638	−0.0843191	+	0.996439i	$0.526872\pi$
$662$	0	0
$663$	102.540	3.98231
$664$	0	0
$665$	0	0
$666$	0	0
$667$	11.3005	0.437556
$668$	0	0
$669$	15.5216	0.600098
$670$	0	0
$671$	24.0121	0.926976
$672$	0	0
$673$	34.1971	1.31820	0.659100	−	0.752055i	$-0.270937\pi$
$673$	34.1971	1.31820	0.659100	+	0.752055i	$0.270937\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0904	−0.695271	−0.347636	−	0.937630i	$-0.613015\pi$
$677$	−18.0904	−0.695271	−0.347636	+	0.937630i	$0.613015\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−55.5930	−2.13033
$682$	0	0
$683$	−24.1256	−0.923141	−0.461570	−	0.887104i	$-0.652714\pi$
$683$	−24.1256	−0.923141	−0.461570	+	0.887104i	$0.652714\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	13.9518	0.532295
$688$	0	0
$689$	−53.1196	−2.02370
$690$	0	0
$691$	−17.4483	−0.663764	−0.331882	−	0.943321i	$-0.607684\pi$
$691$	−17.4483	−0.663764	−0.331882	+	0.943321i	$0.607684\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−63.6102	−2.40941
$698$	0	0
$699$	−17.8271	−0.674283
$700$	0	0
$701$	20.7015	0.781885	0.390942	−	0.920415i	$-0.372149\pi$
$701$	20.7015	0.781885	0.390942	+	0.920415i	$0.372149\pi$
$702$	0	0
$703$	21.2453	0.801280
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	7.09547	0.266476	0.133238	−	0.991084i	$-0.457463\pi$
$709$	7.09547	0.266476	0.133238	+	0.991084i	$0.457463\pi$
$710$	0	0
$711$	−46.3348	−1.73769
$712$	0	0
$713$	−5.56379	−0.208366
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.35282	0.162559
$718$	0	0
$719$	−52.4432	−1.95580	−0.977901	−	0.209067i	$-0.932957\pi$
$719$	−52.4432	−1.95580	−0.977901	+	0.209067i	$0.932957\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	38.3649	1.42681
$724$	0	0
$725$	0	0
$726$	0	0
$727$	29.9045	1.10910	0.554549	−	0.832151i	$-0.312891\pi$
$727$	29.9045	1.10910	0.554549	+	0.832151i	$0.312891\pi$
$728$	0	0
$729$	−36.4905	−1.35150
$730$	0	0
$731$	32.9648	1.21925
$732$	0	0
$733$	−5.58094	−0.206137	−0.103068	−	0.994674i	$-0.532866\pi$
$733$	−5.58094	−0.206137	−0.103068	+	0.994674i	$0.532866\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.35282	−0.0498318
$738$	0	0
$739$	40.7488	1.49897	0.749484	−	0.662023i	$-0.230302\pi$
$739$	40.7488	1.49897	0.749484	+	0.662023i	$0.230302\pi$
$740$	0	0
$741$	45.3176	1.66478
$742$	0	0
$743$	50.5327	1.85387	0.926933	−	0.375226i	$-0.122435\pi$
$743$	50.5327	1.85387	0.926933	+	0.375226i	$0.122435\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−61.5699	−2.25273
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−30.7136	−1.12075	−0.560377	−	0.828238i	$-0.689344\pi$
$751$	−30.7136	−1.12075	−0.560377	+	0.828238i	$0.689344\pi$
$752$	0	0
$753$	−22.8874	−0.834063
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−20.7116	−0.752776	−0.376388	−	0.926462i	$-0.622834\pi$
$757$	−20.7116	−0.752776	−0.376388	+	0.926462i	$0.622834\pi$
$758$	0	0
$759$	−23.1196	−0.839190
$760$	0	0
$761$	34.3528	1.24529	0.622644	−	0.782505i	$-0.286058\pi$
$761$	34.3528	1.24529	0.622644	+	0.782505i	$0.286058\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−25.4432	−0.918702
$768$	0	0
$769$	−46.8392	−1.68906	−0.844532	−	0.535505i	$-0.820121\pi$
$769$	−46.8392	−1.68906	−0.844532	+	0.535505i	$0.820121\pi$
$770$	0	0
$771$	−24.8262	−0.894094
$772$	0	0
$773$	−33.3306	−1.19882	−0.599409	−	0.800443i	$-0.704598\pi$
$773$	−33.3306	−1.19882	−0.599409	+	0.800443i	$0.704598\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−28.1126	−1.00724
$780$	0	0
$781$	19.0422	0.681384
$782$	0	0
$783$	3.03105	0.108321
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−8.40804	−0.299714	−0.149857	−	0.988708i	$-0.547881\pi$
$787$	−8.40804	−0.299714	−0.149857	+	0.988708i	$0.547881\pi$
$788$	0	0
$789$	−15.6764	−0.558095
$790$	0	0
$791$	0	0
$792$	0	0
$793$	85.2342	3.02676
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16.2935	−0.577144	−0.288572	−	0.957458i	$-0.593180\pi$
$797$	−16.2935	−0.577144	−0.288572	+	0.957458i	$0.593180\pi$
$798$	0	0
$799$	2.87439	0.101688
$800$	0	0
$801$	20.1507	0.711990
$802$	0	0
$803$	6.27540	0.221454
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−57.8856	−2.03767
$808$	0	0
$809$	−15.5035	−0.545075	−0.272537	−	0.962145i	$-0.587863\pi$
$809$	−15.5035	−0.545075	−0.272537	+	0.962145i	$0.587863\pi$
$810$	0	0
$811$	36.6232	1.28601	0.643007	−	0.765861i	$-0.277687\pi$
$811$	36.6232	1.28601	0.643007	+	0.765861i	$0.277687\pi$
$812$	0	0
$813$	−11.4604	−0.401933
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.5688	0.509700
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−43.3297	−1.51222	−0.756109	−	0.654446i	$-0.772902\pi$
$821$	−43.3297	−1.51222	−0.756109	+	0.654446i	$0.772902\pi$
$822$	0	0
$823$	41.0543	1.43106	0.715532	−	0.698580i	$-0.246185\pi$
$823$	41.0543	1.43106	0.715532	+	0.698580i	$0.246185\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.0473	1.04485	0.522423	−	0.852686i	$-0.325028\pi$
$827$	30.0473	1.04485	0.522423	+	0.852686i	$0.325028\pi$
$828$	0	0
$829$	42.0292	1.45974	0.729868	−	0.683588i	$-0.239582\pi$
$829$	42.0292	1.45974	0.729868	+	0.683588i	$0.239582\pi$
$830$	0	0
$831$	16.7195	0.579995
$832$	0	0
$833$	0	0
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.49234	−0.0515827
$838$	0	0
$839$	−3.38893	−0.116999	−0.0584994	−	0.998287i	$-0.518632\pi$
$839$	−3.38893	−0.116999	−0.0584994	+	0.998287i	$0.518632\pi$
$840$	0	0
$841$	−25.1196	−0.866195
$842$	0	0
$843$	−28.7659	−0.990751
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	19.6421	0.674116
$850$	0	0
$851$	−38.4734	−1.31885
$852$	0	0
$853$	−25.1267	−0.860321	−0.430160	−	0.902752i	$-0.641543\pi$
$853$	−25.1267	−0.860321	−0.430160	+	0.902752i	$0.641543\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.7367	−1.63065	−0.815327	−	0.579001i	$-0.803443\pi$
$857$	−47.7367	−1.63065	−0.815327	+	0.579001i	$0.803443\pi$
$858$	0	0
$859$	41.0422	1.40034	0.700171	−	0.713975i	$-0.253107\pi$
$859$	41.0422	1.40034	0.700171	+	0.713975i	$0.253107\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.1326	1.05977	0.529884	−	0.848070i	$-0.322236\pi$
$863$	31.1326	1.05977	0.529884	+	0.848070i	$0.322236\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	88.3117	2.99922
$868$	0	0
$869$	20.1980	0.685169
$870$	0	0
$871$	−4.80202	−0.162710
$872$	0	0
$873$	−0.839190	−0.0284023
$874$	0	0
$875$	0	0
$876$	0	0
$877$	7.93972	0.268105	0.134053	−	0.990974i	$-0.457201\pi$
$877$	7.93972	0.268105	0.134053	+	0.990974i	$0.457201\pi$
$878$	0	0
$879$	−15.7358	−0.530755
$880$	0	0
$881$	29.3709	0.989530	0.494765	−	0.869027i	$-0.335254\pi$
$881$	29.3709	0.989530	0.494765	+	0.869027i	$0.335254\pi$
$882$	0	0
$883$	−26.9718	−0.907674	−0.453837	−	0.891085i	$-0.649945\pi$
$883$	−26.9718	−0.907674	−0.453837	+	0.891085i	$0.649945\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	23.4855	0.788565	0.394282	−	0.918989i	$-0.370993\pi$
$887$	23.4855	0.788565	0.394282	+	0.918989i	$0.370993\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	10.7376	0.359723
$892$	0	0
$893$	1.27034	0.0425103
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−82.0664	−2.74012
$898$	0	0
$899$	−1.91049	−0.0637185
$900$	0	0
$901$	−68.3719	−2.27780
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	9.58690	0.318328	0.159164	−	0.987252i	$-0.449120\pi$
$907$	9.58690	0.318328	0.159164	+	0.987252i	$0.449120\pi$
$908$	0	0
$909$	−45.0000	−1.49256
$910$	0	0
$911$	−3.78994	−0.125566	−0.0627831	−	0.998027i	$-0.519998\pi$
$911$	−3.78994	−0.125566	−0.0627831	+	0.998027i	$0.519998\pi$
$912$	0	0
$913$	26.8392	0.888248
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	23.3900	0.771564	0.385782	−	0.922590i	$-0.373932\pi$
$919$	23.3900	0.771564	0.385782	+	0.922590i	$0.373932\pi$
$920$	0	0
$921$	−67.8383	−2.23535
$922$	0	0
$923$	67.5930	2.22485
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−12.1317	−0.398458
$928$	0	0
$929$	−50.2573	−1.64889	−0.824445	−	0.565942i	$-0.808513\pi$
$929$	−50.2573	−1.64889	−0.824445	+	0.565942i	$0.808513\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	49.7145	1.62758
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.43621	0.0795875	0.0397937	−	0.999208i	$-0.487330\pi$
$937$	2.43621	0.0795875	0.0397937	+	0.999208i	$0.487330\pi$
$938$	0	0
$939$	−22.3658	−0.729881
$940$	0	0
$941$	23.9648	0.781230	0.390615	−	0.920554i	$-0.372262\pi$
$941$	23.9648	0.781230	0.390615	+	0.920554i	$0.372262\pi$
$942$	0	0
$943$	50.9096	1.65784
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−51.1619	−1.66254	−0.831269	−	0.555871i	$-0.812385\pi$
$947$	−51.1619	−1.66254	−0.831269	+	0.555871i	$0.812385\pi$
$948$	0	0
$949$	22.2754	0.723090
$950$	0	0
$951$	39.2402	1.27245
$952$	0	0
$953$	33.3306	1.07968	0.539842	−	0.841766i	$-0.318484\pi$
$953$	33.3306	1.07968	0.539842	+	0.841766i	$0.318484\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−7.93881	−0.256625
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−30.0594	−0.969657
$962$	0	0
$963$	−63.8341	−2.05703
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−31.8392	−1.02388	−0.511940	−	0.859021i	$-0.671073\pi$
$967$	−31.8392	−1.02388	−0.511940	+	0.859021i	$0.671073\pi$
$968$	0	0
$969$	58.3297	1.87382
$970$	0	0
$971$	−1.94675	−0.0624742	−0.0312371	−	0.999512i	$-0.509945\pi$
$971$	−1.94675	−0.0624742	−0.0312371	+	0.999512i	$0.509945\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.1196	−1.60347	−0.801735	−	0.597680i	$-0.796089\pi$
$977$	−50.1196	−1.60347	−0.801735	+	0.597680i	$0.796089\pi$
$978$	0	0
$979$	−8.78397	−0.280737
$980$	0	0
$981$	21.2503	0.678470
$982$	0	0
$983$	47.9217	1.52846	0.764232	−	0.644941i	$-0.223118\pi$
$983$	47.9217	1.52846	0.764232	+	0.644941i	$0.223118\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−26.3830	−0.838929
$990$	0	0
$991$	8.86933	0.281743	0.140872	−	0.990028i	$-0.455009\pi$
$991$	8.86933	0.281743	0.140872	+	0.990028i	$0.455009\pi$
$992$	0	0
$993$	43.0251	1.36536
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−12.2101	−0.386697	−0.193348	−	0.981130i	$-0.561935\pi$
$997$	−12.2101	−0.386697	−0.193348	+	0.981130i	$0.561935\pi$
$998$	0	0
$999$	−10.3194	−0.326493

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4900.2.a.bb.1.3		3
5.2	odd	4		4900.2.e.s.2549.2		6
5.3	odd	4		4900.2.e.s.2549.5		6
5.4	even	2		4900.2.a.bd.1.1		3
7.3	odd	6		700.2.i.d.401.3	✓	6
7.5	odd	6		700.2.i.d.501.3	yes	6
7.6	odd	2		4900.2.a.bc.1.1		3
35.3	even	12		700.2.r.d.149.2		12
35.12	even	12		700.2.r.d.249.2		12
35.13	even	4		4900.2.e.t.2549.2		6
35.17	even	12		700.2.r.d.149.5		12
35.19	odd	6		700.2.i.e.501.1	yes	6
35.24	odd	6		700.2.i.e.401.1	yes	6
35.27	even	4		4900.2.e.t.2549.5		6
35.33	even	12		700.2.r.d.249.5		12
35.34	odd	2		4900.2.a.ba.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
700.2.i.d.401.3	✓	6	7.3	odd	6
700.2.i.d.501.3	yes	6	7.5	odd	6
700.2.i.e.401.1	yes	6	35.24	odd	6
700.2.i.e.501.1	yes	6	35.19	odd	6
700.2.r.d.149.2		12	35.3	even	12
700.2.r.d.149.5		12	35.17	even	12
700.2.r.d.249.2		12	35.12	even	12
700.2.r.d.249.5		12	35.33	even	12
4900.2.a.ba.1.3		3	35.34	odd	2
4900.2.a.bb.1.3		3	1.1	even	1	trivial
4900.2.a.bc.1.1		3	7.6	odd	2
4900.2.a.bd.1.1		3	5.4	even	2
4900.2.e.s.2549.2		6	5.2	odd	4
4900.2.e.s.2549.5		6	5.3	odd	4
4900.2.e.t.2549.2		6	35.13	even	4
4900.2.e.t.2549.5		6	35.27	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:	\( N \)	\(=\)	\( 4900 = 2^{2} \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4900.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56885 q^{3} +3.59899 q^{9} -1.56885 q^{11} -5.56885 q^{13} -7.16784 q^{17} -3.16784 q^{19} +5.73669 q^{23} +1.53871 q^{27} +1.96986 q^{29} -0.969861 q^{31} -4.03014 q^{33} -6.70655 q^{37} -14.3055 q^{39} +8.87439 q^{41} -4.59899 q^{43} -0.401012 q^{47} -18.4131 q^{51} +9.53871 q^{53} -8.13770 q^{57} +4.56885 q^{59} -15.3055 q^{61} +0.862301 q^{67} +14.7367 q^{69} -12.1377 q^{71} -4.00000 q^{73} -12.8744 q^{79} -6.84425 q^{81} -17.1076 q^{83} +5.06028 q^{87} +5.59899 q^{89} -2.49143 q^{93} -0.233174 q^{97} -5.64627 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{3} + 8 q^{9} + 4 q^{11} - 8 q^{13} - 10 q^{17} + 2 q^{19} - 3 q^{23} - 10 q^{27} + 3 q^{31} - 18 q^{33} + 6 q^{37} - 14 q^{39} - 11 q^{41} - 11 q^{43} - 4 q^{47} - 3 q^{51} + 14 q^{53} - 7 q^{57}+ \cdots + 21 q^{99}+O(q^{100}) \) 3 * q - q^3 + 8 * q^9 + 4 * q^11 - 8 * q^13 - 10 * q^17 + 2 * q^19 - 3 * q^23 - 10 * q^27 + 3 * q^31 - 18 * q^33 + 6 * q^37 - 14 * q^39 - 11 * q^41 - 11 * q^43 - 4 * q^47 - 3 * q^51 + 14 * q^53 - 7 * q^57 + 5 * q^59 - 17 * q^61 + 20 * q^67 + 24 * q^69 - 19 * q^71 - 12 * q^73 - q^79 + 23 * q^81 - 28 * q^83 + 27 * q^87 + 14 * q^89 - 28 * q^93 - 15 * q^97 + 21 * q^99

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