LMFDB - Embedded newform 4624.2.a.bq.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4624,2,Mod(1,4624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4624, 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("4624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4624 = 2^{4} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4624.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:	$36.9228258946$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{13})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 9x^{2} + 10x - 1 $ x^4 - 2x^3 - 9x^2 + 10*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 68)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-2.71699$ of defining polynomial
Character	$\chi$	$=$	4624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.25662 q^{3} +1.41421 q^{5} +3.25662 q^{7} +7.60555 q^{9} +O(q^{10})$	$q+3.25662 q^{3} +1.41421 q^{5} +3.25662 q^{7} +7.60555 q^{9} +0.428189 q^{11} -2.60555 q^{13} +4.60555 q^{15} -0.605551 q^{19} +10.6056 q^{21} +6.08504 q^{23} -3.00000 q^{25} +14.9985 q^{27} +2.27059 q^{29} -6.08504 q^{31} +1.39445 q^{33} +4.60555 q^{35} +4.24264 q^{37} -8.48528 q^{39} -1.41421 q^{41} -3.39445 q^{43} +10.7559 q^{45} -4.00000 q^{47} +3.60555 q^{49} -5.21110 q^{53} +0.605551 q^{55} -1.97205 q^{57} -8.60555 q^{59} +8.78383 q^{61} +24.7684 q^{63} -3.68481 q^{65} -9.21110 q^{67} +19.8167 q^{69} -4.11300 q^{71} +9.89949 q^{73} -9.76985 q^{75} +1.39445 q^{77} +0.428189 q^{79} +26.0278 q^{81} -17.8167 q^{83} +7.39445 q^{87} -7.81665 q^{89} -8.48528 q^{91} -19.8167 q^{93} -0.856379 q^{95} +10.7559 q^{97} +3.25662 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{9}+O(q^{10}) $ 4 * q + 16 * q^9	$ 4 q + 16 q^{9} + 4 q^{13} + 4 q^{15} + 12 q^{19} + 28 q^{21} - 12 q^{25} + 20 q^{33} + 4 q^{35} - 28 q^{43} - 16 q^{47} + 8 q^{53} - 12 q^{55} - 20 q^{59} - 8 q^{67} + 36 q^{69} + 20 q^{77} + 32 q^{81} - 28 q^{83} + 44 q^{87} + 12 q^{89} - 36 q^{93}+O(q^{100}) $ 4 * q + 16 * q^9 + 4 * q^13 + 4 * q^15 + 12 * q^19 + 28 * q^21 - 12 * q^25 + 20 * q^33 + 4 * q^35 - 28 * q^43 - 16 * q^47 + 8 * q^53 - 12 * q^55 - 20 * q^59 - 8 * q^67 + 36 * q^69 + 20 * q^77 + 32 * q^81 - 28 * q^83 + 44 * q^87 + 12 * q^89 - 36 * q^93

Display 100 coefficients 10 coefficients

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
$3$	3.25662	1.88021	0.940104	−	0.340887i	$-0.110727\pi$
$3$	3.25662	1.88021	0.940104	+	0.340887i	$0.110727\pi$
$4$	0	0
$5$	1.41421	0.632456	0.316228	−	0.948683i	$-0.397584\pi$
$5$	1.41421	0.632456	0.316228	+	0.948683i	$0.397584\pi$
$6$	0	0
$7$	3.25662	1.23089	0.615443	−	0.788182i	$-0.288977\pi$
$7$	3.25662	1.23089	0.615443	+	0.788182i	$0.288977\pi$
$8$	0	0
$9$	7.60555	2.53518
$10$	0	0
$11$	0.428189	0.129104	0.0645520	−	0.997914i	$-0.479438\pi$
$11$	0.428189	0.129104	0.0645520	+	0.997914i	$0.479438\pi$
$12$	0	0
$13$	−2.60555	−0.722650	−0.361325	−	0.932440i	$-0.617675\pi$
$13$	−2.60555	−0.722650	−0.361325	+	0.932440i	$0.617675\pi$
$14$	0	0
$15$	4.60555	1.18915
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	−0.605551	−0.138923	−0.0694615	−	0.997585i	$-0.522128\pi$
$19$	−0.605551	−0.138923	−0.0694615	+	0.997585i	$0.522128\pi$
$20$	0	0
$21$	10.6056	2.31432
$22$	0	0
$23$	6.08504	1.26882	0.634410	−	0.772997i	$-0.281243\pi$
$23$	6.08504	1.26882	0.634410	+	0.772997i	$0.281243\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	14.9985	2.88647
$28$	0	0
$29$	2.27059	0.421638	0.210819	−	0.977525i	$-0.432387\pi$
$29$	2.27059	0.421638	0.210819	+	0.977525i	$0.432387\pi$
$30$	0	0
$31$	−6.08504	−1.09291	−0.546453	−	0.837490i	$-0.684022\pi$
$31$	−6.08504	−1.09291	−0.546453	+	0.837490i	$0.684022\pi$
$32$	0	0
$33$	1.39445	0.242742
$34$	0	0
$35$	4.60555	0.778480
$36$	0	0
$37$	4.24264	0.697486	0.348743	−	0.937218i	$-0.386609\pi$
$37$	4.24264	0.697486	0.348743	+	0.937218i	$0.386609\pi$
$38$	0	0
$39$	−8.48528	−1.35873
$40$	0	0
$41$	−1.41421	−0.220863	−0.110432	−	0.993884i	$-0.535223\pi$
$41$	−1.41421	−0.220863	−0.110432	+	0.993884i	$0.535223\pi$
$42$	0	0
$43$	−3.39445	−0.517649	−0.258824	−	0.965924i	$-0.583335\pi$
$43$	−3.39445	−0.517649	−0.258824	+	0.965924i	$0.583335\pi$
$44$	0	0
$45$	10.7559	1.60339
$46$	0	0
$47$	−4.00000	−0.583460	−0.291730	−	0.956501i	$-0.594231\pi$
$47$	−4.00000	−0.583460	−0.291730	+	0.956501i	$0.594231\pi$
$48$	0	0
$49$	3.60555	0.515079
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−5.21110	−0.715800	−0.357900	−	0.933760i	$-0.616507\pi$
$53$	−5.21110	−0.715800	−0.357900	+	0.933760i	$0.616507\pi$
$54$	0	0
$55$	0.605551	0.0816525
$56$	0	0
$57$	−1.97205	−0.261204
$58$	0	0
$59$	−8.60555	−1.12035	−0.560174	−	0.828375i	$-0.689266\pi$
$59$	−8.60555	−1.12035	−0.560174	+	0.828375i	$0.689266\pi$
$60$	0	0
$61$	8.78383	1.12465	0.562327	−	0.826915i	$-0.309906\pi$
$61$	8.78383	1.12465	0.562327	+	0.826915i	$0.309906\pi$
$62$	0	0
$63$	24.7684	3.12052
$64$	0	0
$65$	−3.68481	−0.457044
$66$	0	0
$67$	−9.21110	−1.12532	−0.562658	−	0.826690i	$-0.690221\pi$
$67$	−9.21110	−1.12532	−0.562658	+	0.826690i	$0.690221\pi$
$68$	0	0
$69$	19.8167	2.38564
$70$	0	0
$71$	−4.11300	−0.488123	−0.244061	−	0.969760i	$-0.578480\pi$
$71$	−4.11300	−0.488123	−0.244061	+	0.969760i	$0.578480\pi$
$72$	0	0
$73$	9.89949	1.15865	0.579324	−	0.815097i	$-0.303317\pi$
$73$	9.89949	1.15865	0.579324	+	0.815097i	$0.303317\pi$
$74$	0	0
$75$	−9.76985	−1.12813
$76$	0	0
$77$	1.39445	0.158912
$78$	0	0
$79$	0.428189	0.0481751	0.0240875	−	0.999710i	$-0.492332\pi$
$79$	0.428189	0.0481751	0.0240875	+	0.999710i	$0.492332\pi$
$80$	0	0
$81$	26.0278	2.89197
$82$	0	0
$83$	−17.8167	−1.95563	−0.977816	−	0.209466i	$-0.932827\pi$
$83$	−17.8167	−1.95563	−0.977816	+	0.209466i	$0.932827\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	7.39445	0.792768
$88$	0	0
$89$	−7.81665	−0.828564	−0.414282	−	0.910149i	$-0.635967\pi$
$89$	−7.81665	−0.828564	−0.414282	+	0.910149i	$0.635967\pi$
$90$	0	0
$91$	−8.48528	−0.889499
$92$	0	0
$93$	−19.8167	−2.05489
$94$	0	0
$95$	−0.856379	−0.0878626
$96$	0	0
$97$	10.7559	1.09209	0.546047	−	0.837755i	$-0.316132\pi$
$97$	10.7559	1.09209	0.546047	+	0.837755i	$0.316132\pi$
$98$	0	0
$99$	3.25662	0.327302
$100$	0	0
$101$	10.6056	1.05529	0.527646	−	0.849464i	$-0.323075\pi$
$101$	10.6056	1.05529	0.527646	+	0.849464i	$0.323075\pi$
$102$	0	0
$103$	13.2111	1.30173	0.650864	−	0.759194i	$-0.274407\pi$
$103$	13.2111	1.30173	0.650864	+	0.759194i	$0.274407\pi$
$104$	0	0
$105$	14.9985	1.46371
$106$	0	0
$107$	8.05709	0.778908	0.389454	−	0.921046i	$-0.372664\pi$
$107$	8.05709	0.778908	0.389454	+	0.921046i	$0.372664\pi$
$108$	0	0
$109$	8.78383	0.841338	0.420669	−	0.907214i	$-0.361795\pi$
$109$	8.78383	0.841338	0.420669	+	0.907214i	$0.361795\pi$
$110$	0	0
$111$	13.8167	1.31142
$112$	0	0
$113$	13.5843	1.27790	0.638952	−	0.769247i	$-0.279368\pi$
$113$	13.5843	1.27790	0.638952	+	0.769247i	$0.279368\pi$
$114$	0	0
$115$	8.60555	0.802472
$116$	0	0
$117$	−19.8167	−1.83205
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.8167	−0.983332
$122$	0	0
$123$	−4.60555	−0.415269
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	−0.605551	−0.0537340	−0.0268670	−	0.999639i	$-0.508553\pi$
$127$	−0.605551	−0.0537340	−0.0268670	+	0.999639i	$0.508553\pi$
$128$	0	0
$129$	−11.0544	−0.973287
$130$	0	0
$131$	−8.91347	−0.778774	−0.389387	−	0.921074i	$-0.627313\pi$
$131$	−8.91347	−0.778774	−0.389387	+	0.921074i	$0.627313\pi$
$132$	0	0
$133$	−1.97205	−0.170998
$134$	0	0
$135$	21.2111	1.82556
$136$	0	0
$137$	−2.60555	−0.222607	−0.111304	−	0.993786i	$-0.535503\pi$
$137$	−2.60555	−0.222607	−0.111304	+	0.993786i	$0.535503\pi$
$138$	0	0
$139$	−0.428189	−0.0363186	−0.0181593	−	0.999835i	$-0.505781\pi$
$139$	−0.428189	−0.0363186	−0.0181593	+	0.999835i	$0.505781\pi$
$140$	0	0
$141$	−13.0265	−1.09703
$142$	0	0
$143$	−1.11567	−0.0932970
$144$	0	0
$145$	3.21110	0.266668
$146$	0	0
$147$	11.7419	0.968455
$148$	0	0
$149$	−8.42221	−0.689974	−0.344987	−	0.938607i	$-0.612117\pi$
$149$	−8.42221	−0.689974	−0.344987	+	0.938607i	$0.612117\pi$
$150$	0	0
$151$	−4.60555	−0.374794	−0.187397	−	0.982284i	$-0.560005\pi$
$151$	−4.60555	−0.374794	−0.187397	+	0.982284i	$0.560005\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.60555	−0.691215
$156$	0	0
$157$	8.42221	0.672165	0.336083	−	0.941833i	$-0.390898\pi$
$157$	8.42221	0.672165	0.336083	+	0.941833i	$0.390898\pi$
$158$	0	0
$159$	−16.9706	−1.34585
$160$	0	0
$161$	19.8167	1.56177
$162$	0	0
$163$	−6.08504	−0.476617	−0.238309	−	0.971189i	$-0.576593\pi$
$163$	−6.08504	−0.476617	−0.238309	+	0.971189i	$0.576593\pi$
$164$	0	0
$165$	1.97205	0.153524
$166$	0	0
$167$	−21.0836	−1.63149	−0.815747	−	0.578408i	$-0.803674\pi$
$167$	−21.0836	−1.63149	−0.815747	+	0.578408i	$0.803674\pi$
$168$	0	0
$169$	−6.21110	−0.477777
$170$	0	0
$171$	−4.60555	−0.352195
$172$	0	0
$173$	−20.9539	−1.59310	−0.796548	−	0.604575i	$-0.793343\pi$
$173$	−20.9539	−1.59310	−0.796548	+	0.604575i	$0.793343\pi$
$174$	0	0
$175$	−9.76985	−0.738531
$176$	0	0
$177$	−28.0250	−2.10649
$178$	0	0
$179$	−15.0278	−1.12323	−0.561614	−	0.827400i	$-0.689819\pi$
$179$	−15.0278	−1.12323	−0.561614	+	0.827400i	$0.689819\pi$
$180$	0	0
$181$	−18.3848	−1.36653	−0.683265	−	0.730171i	$-0.739441\pi$
$181$	−18.3848	−1.36653	−0.683265	+	0.730171i	$0.739441\pi$
$182$	0	0
$183$	28.6056	2.11458
$184$	0	0
$185$	6.00000	0.441129
$186$	0	0
$187$	0	0
$188$	0	0
$189$	48.8444	3.55291
$190$	0	0
$191$	6.78890	0.491227	0.245614	−	0.969368i	$-0.421011\pi$
$191$	6.78890	0.491227	0.245614	+	0.969368i	$0.421011\pi$
$192$	0	0
$193$	−11.6123	−0.835868	−0.417934	−	0.908477i	$-0.637246\pi$
$193$	−11.6123	−0.835868	−0.417934	+	0.908477i	$0.637246\pi$
$194$	0	0
$195$	−12.0000	−0.859338
$196$	0	0
$197$	24.8980	1.77391	0.886955	−	0.461856i	$-0.152816\pi$
$197$	24.8980	1.77391	0.886955	+	0.461856i	$0.152816\pi$
$198$	0	0
$199$	−14.3110	−1.01448	−0.507241	−	0.861804i	$-0.669335\pi$
$199$	−14.3110	−1.01448	−0.507241	+	0.861804i	$0.669335\pi$
$200$	0	0
$201$	−29.9970	−2.11583
$202$	0	0
$203$	7.39445	0.518989
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	46.2801	3.21669
$208$	0	0
$209$	−0.259291	−0.0179355
$210$	0	0
$211$	21.9399	1.51041	0.755204	−	0.655490i	$-0.227538\pi$
$211$	21.9399	1.51041	0.755204	+	0.655490i	$0.227538\pi$
$212$	0	0
$213$	−13.3944	−0.917773
$214$	0	0
$215$	−4.80048	−0.327390
$216$	0	0
$217$	−19.8167	−1.34524
$218$	0	0
$219$	32.2389	2.17850
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.8167	1.19309	0.596546	−	0.802579i	$-0.296539\pi$
$223$	17.8167	1.19309	0.596546	+	0.802579i	$0.296539\pi$
$224$	0	0
$225$	−22.8167	−1.52111
$226$	0	0
$227$	0.428189	0.0284199	0.0142100	−	0.999899i	$-0.495477\pi$
$227$	0.428189	0.0284199	0.0142100	+	0.999899i	$0.495477\pi$
$228$	0	0
$229$	−4.60555	−0.304343	−0.152172	−	0.988354i	$-0.548627\pi$
$229$	−4.60555	−0.304343	−0.152172	+	0.988354i	$0.548627\pi$
$230$	0	0
$231$	4.54118	0.298788
$232$	0	0
$233$	−7.92745	−0.519344	−0.259672	−	0.965697i	$-0.583614\pi$
$233$	−7.92745	−0.519344	−0.259672	+	0.965697i	$0.583614\pi$
$234$	0	0
$235$	−5.65685	−0.369012
$236$	0	0
$237$	1.39445	0.0905792
$238$	0	0
$239$	14.7889	0.956614	0.478307	−	0.878193i	$-0.341251\pi$
$239$	14.7889	0.956614	0.478307	+	0.878193i	$0.341251\pi$
$240$	0	0
$241$	12.7279	0.819878	0.409939	−	0.912113i	$-0.365550\pi$
$241$	12.7279	0.819878	0.409939	+	0.912113i	$0.365550\pi$
$242$	0	0
$243$	39.7669	2.55105
$244$	0	0
$245$	5.09902	0.325764
$246$	0	0
$247$	1.57779	0.100393
$248$	0	0
$249$	−58.0220	−3.67700
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	2.60555	0.163810
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.3944	1.20979	0.604896	−	0.796304i	$-0.293215\pi$
$257$	19.3944	1.20979	0.604896	+	0.796304i	$0.293215\pi$
$258$	0	0
$259$	13.8167	0.858525
$260$	0	0
$261$	17.2691	1.06893
$262$	0	0
$263$	23.0278	1.41995	0.709976	−	0.704226i	$-0.248706\pi$
$263$	23.0278	1.41995	0.709976	+	0.704226i	$0.248706\pi$
$264$	0	0
$265$	−7.36961	−0.452712
$266$	0	0
$267$	−25.4558	−1.55787
$268$	0	0
$269$	24.6387	1.50225	0.751125	−	0.660160i	$-0.229512\pi$
$269$	24.6387	1.50225	0.751125	+	0.660160i	$0.229512\pi$
$270$	0	0
$271$	−1.21110	−0.0735692	−0.0367846	−	0.999323i	$-0.511712\pi$
$271$	−1.21110	−0.0735692	−0.0367846	+	0.999323i	$0.511712\pi$
$272$	0	0
$273$	−27.6333	−1.67244
$274$	0	0
$275$	−1.28457	−0.0774624
$276$	0	0
$277$	−5.09902	−0.306370	−0.153185	−	0.988197i	$-0.548953\pi$
$277$	−5.09902	−0.306370	−0.153185	+	0.988197i	$0.548953\pi$
$278$	0	0
$279$	−46.2801	−2.77072
$280$	0	0
$281$	−18.4222	−1.09898	−0.549488	−	0.835501i	$-0.685177\pi$
$281$	−18.4222	−1.09898	−0.549488	+	0.835501i	$0.685177\pi$
$282$	0	0
$283$	−2.99733	−0.178173	−0.0890863	−	0.996024i	$-0.528395\pi$
$283$	−2.99733	−0.178173	−0.0890863	+	0.996024i	$0.528395\pi$
$284$	0	0
$285$	−2.78890	−0.165200
$286$	0	0
$287$	−4.60555	−0.271857
$288$	0	0
$289$	0	0
$290$	0	0
$291$	35.0278	2.05336
$292$	0	0
$293$	21.6333	1.26383	0.631916	−	0.775037i	$-0.282269\pi$
$293$	21.6333	1.26383	0.631916	+	0.775037i	$0.282269\pi$
$294$	0	0
$295$	−12.1701	−0.708570
$296$	0	0
$297$	6.42221	0.372654
$298$	0	0
$299$	−15.8549	−0.916912
$300$	0	0
$301$	−11.0544	−0.637166
$302$	0	0
$303$	34.5382	1.98417
$304$	0	0
$305$	12.4222	0.711293
$306$	0	0
$307$	−21.2111	−1.21058	−0.605291	−	0.796004i	$-0.706943\pi$
$307$	−21.2111	−1.21058	−0.605291	+	0.796004i	$0.706943\pi$
$308$	0	0
$309$	43.0235	2.44752
$310$	0	0
$311$	26.7404	1.51631	0.758155	−	0.652075i	$-0.226101\pi$
$311$	26.7404	1.51631	0.758155	+	0.652075i	$0.226101\pi$
$312$	0	0
$313$	−8.78383	−0.496491	−0.248246	−	0.968697i	$-0.579854\pi$
$313$	−8.78383	−0.496491	−0.248246	+	0.968697i	$0.579854\pi$
$314$	0	0
$315$	35.0278	1.97359
$316$	0	0
$317$	27.7264	1.55727	0.778636	−	0.627476i	$-0.215912\pi$
$317$	27.7264	1.55727	0.778636	+	0.627476i	$0.215912\pi$
$318$	0	0
$319$	0.972244	0.0544352
$320$	0	0
$321$	26.2389	1.46451
$322$	0	0
$323$	0	0
$324$	0	0
$325$	7.81665	0.433590
$326$	0	0
$327$	28.6056	1.58189
$328$	0	0
$329$	−13.0265	−0.718172
$330$	0	0
$331$	−3.39445	−0.186576	−0.0932879	−	0.995639i	$-0.529738\pi$
$331$	−3.39445	−0.186576	−0.0932879	+	0.995639i	$0.529738\pi$
$332$	0	0
$333$	32.2676	1.76825
$334$	0	0
$335$	−13.0265	−0.711712
$336$	0	0
$337$	−22.9260	−1.24886	−0.624428	−	0.781082i	$-0.714668\pi$
$337$	−22.9260	−1.24886	−0.624428	+	0.781082i	$0.714668\pi$
$338$	0	0
$339$	44.2389	2.40273
$340$	0	0
$341$	−2.60555	−0.141099
$342$	0	0
$343$	−11.0544	−0.596882
$344$	0	0
$345$	28.0250	1.50881
$346$	0	0
$347$	−7.79780	−0.418608	−0.209304	−	0.977851i	$-0.567120\pi$
$347$	−7.79780	−0.418608	−0.209304	+	0.977851i	$0.567120\pi$
$348$	0	0
$349$	−23.6333	−1.26506	−0.632531	−	0.774535i	$-0.717984\pi$
$349$	−23.6333	−1.26506	−0.632531	+	0.774535i	$0.717984\pi$
$350$	0	0
$351$	−39.0794	−2.08590
$352$	0	0
$353$	31.2111	1.66120	0.830600	−	0.556870i	$-0.187998\pi$
$353$	31.2111	1.66120	0.830600	+	0.556870i	$0.187998\pi$
$354$	0	0
$355$	−5.81665	−0.308716
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−16.6056	−0.876407	−0.438204	−	0.898876i	$-0.644385\pi$
$359$	−16.6056	−0.876407	−0.438204	+	0.898876i	$0.644385\pi$
$360$	0	0
$361$	−18.6333	−0.980700
$362$	0	0
$363$	−35.2257	−1.84887
$364$	0	0
$365$	14.0000	0.732793
$366$	0	0
$367$	−9.76985	−0.509982	−0.254991	−	0.966943i	$-0.582073\pi$
$367$	−9.76985	−0.509982	−0.254991	+	0.966943i	$0.582073\pi$
$368$	0	0
$369$	−10.7559	−0.559928
$370$	0	0
$371$	−16.9706	−0.881068
$372$	0	0
$373$	21.0278	1.08878	0.544388	−	0.838834i	$-0.316762\pi$
$373$	21.0278	1.08878	0.544388	+	0.838834i	$0.316762\pi$
$374$	0	0
$375$	−36.8444	−1.90264
$376$	0	0
$377$	−5.91614	−0.304697
$378$	0	0
$379$	−29.3095	−1.50553	−0.752765	−	0.658289i	$-0.771280\pi$
$379$	−29.3095	−1.50553	−0.752765	+	0.658289i	$0.771280\pi$
$380$	0	0
$381$	−1.97205	−0.101031
$382$	0	0
$383$	−19.3944	−0.991010	−0.495505	−	0.868605i	$-0.665017\pi$
$383$	−19.3944	−0.991010	−0.495505	+	0.868605i	$0.665017\pi$
$384$	0	0
$385$	1.97205	0.100505
$386$	0	0
$387$	−25.8167	−1.31233
$388$	0	0
$389$	10.1833	0.516316	0.258158	−	0.966103i	$-0.416884\pi$
$389$	10.1833	0.516316	0.258158	+	0.966103i	$0.416884\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−29.0278	−1.46426
$394$	0	0
$395$	0.605551	0.0304686
$396$	0	0
$397$	−12.7279	−0.638796	−0.319398	−	0.947621i	$-0.603481\pi$
$397$	−12.7279	−0.638796	−0.319398	+	0.947621i	$0.603481\pi$
$398$	0	0
$399$	−6.42221	−0.321512
$400$	0	0
$401$	9.89949	0.494357	0.247179	−	0.968970i	$-0.420497\pi$
$401$	9.89949	0.494357	0.247179	+	0.968970i	$0.420497\pi$
$402$	0	0
$403$	15.8549	0.789788
$404$	0	0
$405$	36.8088	1.82904
$406$	0	0
$407$	1.81665	0.0900482
$408$	0	0
$409$	23.2111	1.14772	0.573858	−	0.818955i	$-0.305446\pi$
$409$	23.2111	1.14772	0.573858	+	0.818955i	$0.305446\pi$
$410$	0	0
$411$	−8.48528	−0.418548
$412$	0	0
$413$	−28.0250	−1.37902
$414$	0	0
$415$	−25.1966	−1.23685
$416$	0	0
$417$	−1.39445	−0.0682864
$418$	0	0
$419$	−3.25662	−0.159096	−0.0795481	−	0.996831i	$-0.525348\pi$
$419$	−3.25662	−0.159096	−0.0795481	+	0.996831i	$0.525348\pi$
$420$	0	0
$421$	−18.6056	−0.906779	−0.453390	−	0.891312i	$-0.649785\pi$
$421$	−18.6056	−0.906779	−0.453390	+	0.891312i	$0.649785\pi$
$422$	0	0
$423$	−30.4222	−1.47918
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.6056	1.38432
$428$	0	0
$429$	−3.63331	−0.175418
$430$	0	0
$431$	2.65953	0.128105	0.0640525	−	0.997947i	$-0.479597\pi$
$431$	2.65953	0.128105	0.0640525	+	0.997947i	$0.479597\pi$
$432$	0	0
$433$	15.0278	0.722188	0.361094	−	0.932529i	$-0.382403\pi$
$433$	15.0278	0.722188	0.361094	+	0.932529i	$0.382403\pi$
$434$	0	0
$435$	10.4573	0.501391
$436$	0	0
$437$	−3.68481	−0.176268
$438$	0	0
$439$	37.1977	1.77535	0.887676	−	0.460469i	$-0.152319\pi$
$439$	37.1977	1.77535	0.887676	+	0.460469i	$0.152319\pi$
$440$	0	0
$441$	27.4222	1.30582
$442$	0	0
$443$	17.2111	0.817724	0.408862	−	0.912596i	$-0.365926\pi$
$443$	17.2111	0.817724	0.408862	+	0.912596i	$0.365926\pi$
$444$	0	0
$445$	−11.0544	−0.524030
$446$	0	0
$447$	−27.4279	−1.29729
$448$	0	0
$449$	23.7823	1.12236	0.561179	−	0.827694i	$-0.310348\pi$
$449$	23.7823	1.12236	0.561179	+	0.827694i	$0.310348\pi$
$450$	0	0
$451$	−0.605551	−0.0285143
$452$	0	0
$453$	−14.9985	−0.704692
$454$	0	0
$455$	−12.0000	−0.562569
$456$	0	0
$457$	−36.6056	−1.71234	−0.856168	−	0.516698i	$-0.827161\pi$
$457$	−36.6056	−1.71234	−0.856168	+	0.516698i	$0.827161\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.2111	1.36050	0.680248	−	0.732982i	$-0.261872\pi$
$461$	29.2111	1.36050	0.680248	+	0.732982i	$0.261872\pi$
$462$	0	0
$463$	−39.6333	−1.84192	−0.920958	−	0.389662i	$-0.872592\pi$
$463$	−39.6333	−1.84192	−0.920958	+	0.389662i	$0.872592\pi$
$464$	0	0
$465$	−28.0250	−1.29963
$466$	0	0
$467$	16.2389	0.751445	0.375722	−	0.926732i	$-0.377395\pi$
$467$	16.2389	0.751445	0.375722	+	0.926732i	$0.377395\pi$
$468$	0	0
$469$	−29.9970	−1.38513
$470$	0	0
$471$	27.4279	1.26381
$472$	0	0
$473$	−1.45347	−0.0668305
$474$	0	0
$475$	1.81665	0.0833538
$476$	0	0
$477$	−39.6333	−1.81468
$478$	0	0
$479$	−17.3988	−0.794969	−0.397485	−	0.917609i	$-0.630117\pi$
$479$	−17.3988	−0.794969	−0.397485	+	0.917609i	$0.630117\pi$
$480$	0	0
$481$	−11.0544	−0.504038
$482$	0	0
$483$	64.5352	2.93646
$484$	0	0
$485$	15.2111	0.690701
$486$	0	0
$487$	24.5091	1.11061	0.555306	−	0.831646i	$-0.312601\pi$
$487$	24.5091	1.11061	0.555306	+	0.831646i	$0.312601\pi$
$488$	0	0
$489$	−19.8167	−0.896140
$490$	0	0
$491$	9.81665	0.443019	0.221510	−	0.975158i	$-0.428902\pi$
$491$	9.81665	0.443019	0.221510	+	0.975158i	$0.428902\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	4.60555	0.207004
$496$	0	0
$497$	−13.3944	−0.600823
$498$	0	0
$499$	−14.5703	−0.652257	−0.326129	−	0.945325i	$-0.605744\pi$
$499$	−14.5703	−0.652257	−0.326129	+	0.945325i	$0.605744\pi$
$500$	0	0
$501$	−68.6611	−3.06755
$502$	0	0
$503$	38.9105	1.73493	0.867467	−	0.497495i	$-0.165747\pi$
$503$	38.9105	1.73493	0.867467	+	0.497495i	$0.165747\pi$
$504$	0	0
$505$	14.9985	0.667425
$506$	0	0
$507$	−20.2272	−0.898321
$508$	0	0
$509$	−3.21110	−0.142330	−0.0711648	−	0.997465i	$-0.522672\pi$
$509$	−3.21110	−0.142330	−0.0711648	+	0.997465i	$0.522672\pi$
$510$	0	0
$511$	32.2389	1.42616
$512$	0	0
$513$	−9.08237	−0.400996
$514$	0	0
$515$	18.6833	0.823285
$516$	0	0
$517$	−1.71276	−0.0753270
$518$	0	0
$519$	−68.2389	−2.99535
$520$	0	0
$521$	18.3848	0.805452	0.402726	−	0.915321i	$-0.368063\pi$
$521$	18.3848	0.805452	0.402726	+	0.915321i	$0.368063\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	−31.8167	−1.38859
$526$	0	0
$527$	0	0
$528$	0	0
$529$	14.0278	0.609902
$530$	0	0
$531$	−65.4500	−2.84029
$532$	0	0
$533$	3.68481	0.159607
$534$	0	0
$535$	11.3944	0.492625
$536$	0	0
$537$	−48.9396	−2.11190
$538$	0	0
$539$	1.54386	0.0664987
$540$	0	0
$541$	4.24264	0.182405	0.0912027	−	0.995832i	$-0.470929\pi$
$541$	4.24264	0.182405	0.0912027	+	0.995832i	$0.470929\pi$
$542$	0	0
$543$	−59.8722	−2.56936
$544$	0	0
$545$	12.4222	0.532109
$546$	0	0
$547$	16.2831	0.696214	0.348107	−	0.937455i	$-0.386825\pi$
$547$	16.2831	0.696214	0.348107	+	0.937455i	$0.386825\pi$
$548$	0	0
$549$	66.8058	2.85120
$550$	0	0
$551$	−1.37496	−0.0585753
$552$	0	0
$553$	1.39445	0.0592980
$554$	0	0
$555$	19.5397	0.829414
$556$	0	0
$557$	21.3944	0.906512	0.453256	−	0.891380i	$-0.350262\pi$
$557$	21.3944	0.906512	0.453256	+	0.891380i	$0.350262\pi$
$558$	0	0
$559$	8.84441	0.374079
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−20.6056	−0.868420	−0.434210	−	0.900812i	$-0.642972\pi$
$563$	−20.6056	−0.868420	−0.434210	+	0.900812i	$0.642972\pi$
$564$	0	0
$565$	19.2111	0.808217
$566$	0	0
$567$	84.7624	3.55969
$568$	0	0
$569$	30.4222	1.27537	0.637683	−	0.770299i	$-0.279893\pi$
$569$	30.4222	1.27537	0.637683	+	0.770299i	$0.279893\pi$
$570$	0	0
$571$	3.25662	0.136285	0.0681426	−	0.997676i	$-0.478293\pi$
$571$	3.25662	0.136285	0.0681426	+	0.997676i	$0.478293\pi$
$572$	0	0
$573$	22.1088	0.923610
$574$	0	0
$575$	−18.2551	−0.761292
$576$	0	0
$577$	−31.4500	−1.30928	−0.654640	−	0.755941i	$-0.727180\pi$
$577$	−31.4500	−1.30928	−0.654640	+	0.755941i	$0.727180\pi$
$578$	0	0
$579$	−37.8167	−1.57161
$580$	0	0
$581$	−58.0220	−2.40716
$582$	0	0
$583$	−2.23134	−0.0924126
$584$	0	0
$585$	−28.0250	−1.15869
$586$	0	0
$587$	−4.60555	−0.190091	−0.0950457	−	0.995473i	$-0.530300\pi$
$587$	−4.60555	−0.190091	−0.0950457	+	0.995473i	$0.530300\pi$
$588$	0	0
$589$	3.68481	0.151830
$590$	0	0
$591$	81.0833	3.33532
$592$	0	0
$593$	1.57779	0.0647923	0.0323961	−	0.999475i	$-0.489686\pi$
$593$	1.57779	0.0647923	0.0323961	+	0.999475i	$0.489686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−46.6056	−1.90744
$598$	0	0
$599$	26.0555	1.06460	0.532300	−	0.846556i	$-0.321328\pi$
$599$	26.0555	1.06460	0.532300	+	0.846556i	$0.321328\pi$
$600$	0	0
$601$	−5.95540	−0.242926	−0.121463	−	0.992596i	$-0.538759\pi$
$601$	−5.95540	−0.242926	−0.121463	+	0.992596i	$0.538759\pi$
$602$	0	0
$603$	−70.0555	−2.85288
$604$	0	0
$605$	−15.2971	−0.621914
$606$	0	0
$607$	13.4547	0.546108	0.273054	−	0.961999i	$-0.411966\pi$
$607$	13.4547	0.546108	0.273054	+	0.961999i	$0.411966\pi$
$608$	0	0
$609$	24.0809	0.975807
$610$	0	0
$611$	10.4222	0.421637
$612$	0	0
$613$	40.4222	1.63264	0.816319	−	0.577602i	$-0.196011\pi$
$613$	40.4222	1.63264	0.816319	+	0.577602i	$0.196011\pi$
$614$	0	0
$615$	−6.51323	−0.262639
$616$	0	0
$617$	−2.27059	−0.0914106	−0.0457053	−	0.998955i	$-0.514554\pi$
$617$	−2.27059	−0.0914106	−0.0457053	+	0.998955i	$0.514554\pi$
$618$	0	0
$619$	20.4865	0.823421	0.411710	−	0.911315i	$-0.364932\pi$
$619$	20.4865	0.823421	0.411710	+	0.911315i	$0.364932\pi$
$620$	0	0
$621$	91.2666	3.66240
$622$	0	0
$623$	−25.4558	−1.01987
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	−0.844410	−0.0337225
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−24.2389	−0.964934	−0.482467	−	0.875914i	$-0.660259\pi$
$631$	−24.2389	−0.964934	−0.482467	+	0.875914i	$0.660259\pi$
$632$	0	0
$633$	71.4500	2.83988
$634$	0	0
$635$	−0.856379	−0.0339844
$636$	0	0
$637$	−9.39445	−0.372222
$638$	0	0
$639$	−31.2816	−1.23748
$640$	0	0
$641$	−43.5813	−1.72136	−0.860680	−	0.509147i	$-0.829961\pi$
$641$	−43.5813	−1.72136	−0.860680	+	0.509147i	$0.829961\pi$
$642$	0	0
$643$	−5.82575	−0.229745	−0.114873	−	0.993380i	$-0.536646\pi$
$643$	−5.82575	−0.229745	−0.114873	+	0.993380i	$0.536646\pi$
$644$	0	0
$645$	−15.6333	−0.615561
$646$	0	0
$647$	17.2111	0.676638	0.338319	−	0.941031i	$-0.390142\pi$
$647$	17.2111	0.676638	0.338319	+	0.941031i	$0.390142\pi$
$648$	0	0
$649$	−3.68481	−0.144641
$650$	0	0
$651$	−64.5352	−2.52934
$652$	0	0
$653$	−35.0960	−1.37341	−0.686707	−	0.726934i	$-0.740945\pi$
$653$	−35.0960	−1.37341	−0.686707	+	0.726934i	$0.740945\pi$
$654$	0	0
$655$	−12.6056	−0.492540
$656$	0	0
$657$	75.2911	2.93739
$658$	0	0
$659$	−32.0000	−1.24654	−0.623272	−	0.782006i	$-0.714197\pi$
$659$	−32.0000	−1.24654	−0.623272	+	0.782006i	$0.714197\pi$
$660$	0	0
$661$	−1.21110	−0.0471064	−0.0235532	−	0.999723i	$-0.507498\pi$
$661$	−1.21110	−0.0471064	−0.0235532	+	0.999723i	$0.507498\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.78890	−0.108149
$666$	0	0
$667$	13.8167	0.534983
$668$	0	0
$669$	58.0220	2.24326
$670$	0	0
$671$	3.76114	0.145197
$672$	0	0
$673$	−7.92745	−0.305581	−0.152790	−	0.988259i	$-0.548826\pi$
$673$	−7.92745	−0.305581	−0.152790	+	0.988259i	$0.548826\pi$
$674$	0	0
$675$	−44.9955	−1.73188
$676$	0	0
$677$	−22.6667	−0.871151	−0.435575	−	0.900152i	$-0.643455\pi$
$677$	−22.6667	−0.871151	−0.435575	+	0.900152i	$0.643455\pi$
$678$	0	0
$679$	35.0278	1.34424
$680$	0	0
$681$	1.39445	0.0534354
$682$	0	0
$683$	1.02528	0.0392312	0.0196156	−	0.999808i	$-0.493756\pi$
$683$	1.02528	0.0392312	0.0196156	+	0.999808i	$0.493756\pi$
$684$	0	0
$685$	−3.68481	−0.140789
$686$	0	0
$687$	−14.9985	−0.572229
$688$	0	0
$689$	13.5778	0.517273
$690$	0	0
$691$	7.79780	0.296642	0.148321	−	0.988939i	$-0.452613\pi$
$691$	7.79780	0.296642	0.148321	+	0.988939i	$0.452613\pi$
$692$	0	0
$693$	10.6056	0.402872
$694$	0	0
$695$	−0.605551	−0.0229699
$696$	0	0
$697$	0	0
$698$	0	0
$699$	−25.8167	−0.976476
$700$	0	0
$701$	7.81665	0.295231	0.147615	−	0.989045i	$-0.452840\pi$
$701$	7.81665	0.295231	0.147615	+	0.989045i	$0.452840\pi$
$702$	0	0
$703$	−2.56914	−0.0968968
$704$	0	0
$705$	−18.4222	−0.693820
$706$	0	0
$707$	34.5382	1.29894
$708$	0	0
$709$	−20.3568	−0.764517	−0.382258	−	0.924056i	$-0.624854\pi$
$709$	−20.3568	−0.764517	−0.382258	+	0.924056i	$0.624854\pi$
$710$	0	0
$711$	3.25662	0.122133
$712$	0	0
$713$	−37.0278	−1.38670
$714$	0	0
$715$	−1.57779	−0.0590062
$716$	0	0
$717$	48.1618	1.79863
$718$	0	0
$719$	−6.68213	−0.249201	−0.124601	−	0.992207i	$-0.539765\pi$
$719$	−6.68213	−0.249201	−0.124601	+	0.992207i	$0.539765\pi$
$720$	0	0
$721$	43.0235	1.60228
$722$	0	0
$723$	41.4500	1.54154
$724$	0	0
$725$	−6.81178	−0.252983
$726$	0	0
$727$	−6.42221	−0.238186	−0.119093	−	0.992883i	$-0.537999\pi$
$727$	−6.42221	−0.238186	−0.119093	+	0.992883i	$0.537999\pi$
$728$	0	0
$729$	51.4222	1.90453
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−46.0555	−1.70110	−0.850550	−	0.525895i	$-0.823731\pi$
$733$	−46.0555	−1.70110	−0.850550	+	0.525895i	$0.823731\pi$
$734$	0	0
$735$	16.6056	0.612505
$736$	0	0
$737$	−3.94410	−0.145283
$738$	0	0
$739$	34.6611	1.27503	0.637514	−	0.770439i	$-0.279963\pi$
$739$	34.6611	1.27503	0.637514	+	0.770439i	$0.279963\pi$
$740$	0	0
$741$	5.13827	0.188759
$742$	0	0
$743$	20.8243	0.763968	0.381984	−	0.924169i	$-0.375241\pi$
$743$	20.8243	0.763968	0.381984	+	0.924169i	$0.375241\pi$
$744$	0	0
$745$	−11.9108	−0.436378
$746$	0	0
$747$	−135.505	−4.95789
$748$	0	0
$749$	26.2389	0.958747
$750$	0	0
$751$	−2.40024	−0.0875859	−0.0437930	−	0.999041i	$-0.513944\pi$
$751$	−2.40024	−0.0875859	−0.0437930	+	0.999041i	$0.513944\pi$
$752$	0	0
$753$	−39.0794	−1.42413
$754$	0	0
$755$	−6.51323	−0.237041
$756$	0	0
$757$	−39.0278	−1.41849	−0.709244	−	0.704963i	$-0.750964\pi$
$757$	−39.0278	−1.41849	−0.709244	+	0.704963i	$0.750964\pi$
$758$	0	0
$759$	8.48528	0.307996
$760$	0	0
$761$	31.4500	1.14006	0.570030	−	0.821624i	$-0.306932\pi$
$761$	31.4500	1.14006	0.570030	+	0.821624i	$0.306932\pi$
$762$	0	0
$763$	28.6056	1.03559
$764$	0	0
$765$	0	0
$766$	0	0
$767$	22.4222	0.809619
$768$	0	0
$769$	18.6056	0.670933	0.335467	−	0.942052i	$-0.391106\pi$
$769$	18.6056	0.670933	0.335467	+	0.942052i	$0.391106\pi$
$770$	0	0
$771$	63.1603	2.27466
$772$	0	0
$773$	28.6056	1.02887	0.514435	−	0.857529i	$-0.328002\pi$
$773$	28.6056	1.02887	0.514435	+	0.857529i	$0.328002\pi$
$774$	0	0
$775$	18.2551	0.655744
$776$	0	0
$777$	44.9955	1.61421
$778$	0	0
$779$	0.856379	0.0306830
$780$	0	0
$781$	−1.76114	−0.0630186
$782$	0	0
$783$	34.0555	1.21704
$784$	0	0
$785$	11.9108	0.425115
$786$	0	0
$787$	20.8243	0.742305	0.371152	−	0.928572i	$-0.378963\pi$
$787$	20.8243	0.742305	0.371152	+	0.928572i	$0.378963\pi$
$788$	0	0
$789$	74.9926	2.66981
$790$	0	0
$791$	44.2389	1.57295
$792$	0	0
$793$	−22.8867	−0.812731
$794$	0	0
$795$	−24.0000	−0.851192
$796$	0	0
$797$	3.63331	0.128698	0.0643492	−	0.997927i	$-0.479503\pi$
$797$	3.63331	0.128698	0.0643492	+	0.997927i	$0.479503\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−59.4500	−2.10056
$802$	0	0
$803$	4.23886	0.149586
$804$	0	0
$805$	28.0250	0.987751
$806$	0	0
$807$	80.2389	2.82454
$808$	0	0
$809$	29.4392	1.03503	0.517513	−	0.855675i	$-0.326858\pi$
$809$	29.4392	1.03503	0.517513	+	0.855675i	$0.326858\pi$
$810$	0	0
$811$	15.1674	0.532600	0.266300	−	0.963890i	$-0.414199\pi$
$811$	15.1674	0.532600	0.266300	+	0.963890i	$0.414199\pi$
$812$	0	0
$813$	−3.94410	−0.138326
$814$	0	0
$815$	−8.60555	−0.301439
$816$	0	0
$817$	2.05551	0.0719133
$818$	0	0
$819$	−64.5352	−2.25504
$820$	0	0
$821$	−18.1255	−0.632584	−0.316292	−	0.948662i	$-0.602438\pi$
$821$	−18.1255	−0.632584	−0.316292	+	0.948662i	$0.602438\pi$
$822$	0	0
$823$	18.2551	0.636334	0.318167	−	0.948035i	$-0.396933\pi$
$823$	18.2551	0.636334	0.318167	+	0.948035i	$0.396933\pi$
$824$	0	0
$825$	−4.18335	−0.145645
$826$	0	0
$827$	−8.91347	−0.309952	−0.154976	−	0.987918i	$-0.549530\pi$
$827$	−8.91347	−0.309952	−0.154976	+	0.987918i	$0.549530\pi$
$828$	0	0
$829$	18.8444	0.654493	0.327247	−	0.944939i	$-0.393879\pi$
$829$	18.8444	0.654493	0.327247	+	0.944939i	$0.393879\pi$
$830$	0	0
$831$	−16.6056	−0.576040
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−29.8167	−1.03185
$836$	0	0
$837$	−91.2666	−3.15464
$838$	0	0
$839$	21.0836	0.727885	0.363943	−	0.931421i	$-0.381430\pi$
$839$	21.0836	0.727885	0.363943	+	0.931421i	$0.381430\pi$
$840$	0	0
$841$	−23.8444	−0.822221
$842$	0	0
$843$	−59.9941	−2.06631
$844$	0	0
$845$	−8.78383	−0.302173
$846$	0	0
$847$	−35.2257	−1.21037
$848$	0	0
$849$	−9.76114	−0.335001
$850$	0	0
$851$	25.8167	0.884983
$852$	0	0
$853$	41.6093	1.42467	0.712337	−	0.701837i	$-0.247637\pi$
$853$	41.6093	1.42467	0.712337	+	0.701837i	$0.247637\pi$
$854$	0	0
$855$	−6.51323	−0.222748
$856$	0	0
$857$	26.3515	0.900149	0.450075	−	0.892991i	$-0.351397\pi$
$857$	26.3515	0.900149	0.450075	+	0.892991i	$0.351397\pi$
$858$	0	0
$859$	1.81665	0.0619834	0.0309917	−	0.999520i	$-0.490133\pi$
$859$	1.81665	0.0619834	0.0309917	+	0.999520i	$0.490133\pi$
$860$	0	0
$861$	−14.9985	−0.511148
$862$	0	0
$863$	−52.4777	−1.78636	−0.893181	−	0.449697i	$-0.851532\pi$
$863$	−52.4777	−1.78636	−0.893181	+	0.449697i	$0.851532\pi$
$864$	0	0
$865$	−29.6333	−1.00756
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.183346	0.00621959
$870$	0	0
$871$	24.0000	0.813209
$872$	0	0
$873$	81.8043	2.76866
$874$	0	0
$875$	−36.8444	−1.24557
$876$	0	0
$877$	−2.52988	−0.0854281	−0.0427140	−	0.999087i	$-0.513600\pi$
$877$	−2.52988	−0.0854281	−0.0427140	+	0.999087i	$0.513600\pi$
$878$	0	0
$879$	70.4514	2.37627
$880$	0	0
$881$	−34.2397	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$881$	−34.2397	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$882$	0	0
$883$	−25.5778	−0.860761	−0.430381	−	0.902647i	$-0.641621\pi$
$883$	−25.5778	−0.860761	−0.430381	+	0.902647i	$0.641621\pi$
$884$	0	0
$885$	−39.6333	−1.33226
$886$	0	0
$887$	16.5424	0.555439	0.277719	−	0.960662i	$-0.410421\pi$
$887$	16.5424	0.555439	0.277719	+	0.960662i	$0.410421\pi$
$888$	0	0
$889$	−1.97205	−0.0661404
$890$	0	0
$891$	11.1448	0.373365
$892$	0	0
$893$	2.42221	0.0810560
$894$	0	0
$895$	−21.2525	−0.710391
$896$	0	0
$897$	−51.6333	−1.72399
$898$	0	0
$899$	−13.8167	−0.460811
$900$	0	0
$901$	0	0
$902$	0	0
$903$	−36.0000	−1.19800
$904$	0	0
$905$	−26.0000	−0.864269
$906$	0	0
$907$	−34.7071	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$907$	−34.7071	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$908$	0	0
$909$	80.6611	2.67536
$910$	0	0
$911$	6.68213	0.221389	0.110694	−	0.993854i	$-0.464693\pi$
$911$	6.68213	0.221389	0.110694	+	0.993854i	$0.464693\pi$
$912$	0	0
$913$	−7.62890	−0.252480
$914$	0	0
$915$	40.4544	1.33738
$916$	0	0
$917$	−29.0278	−0.958581
$918$	0	0
$919$	−40.8444	−1.34733	−0.673666	−	0.739036i	$-0.735282\pi$
$919$	−40.8444	−1.34733	−0.673666	+	0.739036i	$0.735282\pi$
$920$	0	0
$921$	−69.0764	−2.27615
$922$	0	0
$923$	10.7166	0.352742
$924$	0	0
$925$	−12.7279	−0.418491
$926$	0	0
$927$	100.478	3.30012
$928$	0	0
$929$	1.41421	0.0463988	0.0231994	−	0.999731i	$-0.492615\pi$
$929$	1.41421	0.0463988	0.0231994	+	0.999731i	$0.492615\pi$
$930$	0	0
$931$	−2.18335	−0.0715563
$932$	0	0
$933$	87.0833	2.85098
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.42221	0.209804	0.104902	−	0.994483i	$-0.466547\pi$
$937$	6.42221	0.209804	0.104902	+	0.994483i	$0.466547\pi$
$938$	0	0
$939$	−28.6056	−0.933507
$940$	0	0
$941$	−22.6667	−0.738912	−0.369456	−	0.929248i	$-0.620456\pi$
$941$	−22.6667	−0.738912	−0.369456	+	0.929248i	$0.620456\pi$
$942$	0	0
$943$	−8.60555	−0.280235
$944$	0	0
$945$	69.0764	2.24706
$946$	0	0
$947$	−13.1169	−0.426241	−0.213120	−	0.977026i	$-0.568363\pi$
$947$	−13.1169	−0.426241	−0.213120	+	0.977026i	$0.568363\pi$
$948$	0	0
$949$	−25.7936	−0.837297
$950$	0	0
$951$	90.2944	2.92800
$952$	0	0
$953$	50.2389	1.62740	0.813698	−	0.581288i	$-0.197451\pi$
$953$	50.2389	1.62740	0.813698	+	0.581288i	$0.197451\pi$
$954$	0	0
$955$	9.60095	0.310679
$956$	0	0
$957$	3.16622	0.102350
$958$	0	0
$959$	−8.48528	−0.274004
$960$	0	0
$961$	6.02776	0.194444
$962$	0	0
$963$	61.2786	1.97468
$964$	0	0
$965$	−16.4222	−0.528649
$966$	0	0
$967$	−21.8167	−0.701576	−0.350788	−	0.936455i	$-0.614086\pi$
$967$	−21.8167	−0.701576	−0.350788	+	0.936455i	$0.614086\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−55.0278	−1.76592	−0.882962	−	0.469444i	$-0.844454\pi$
$971$	−55.0278	−1.76592	−0.882962	+	0.469444i	$0.844454\pi$
$972$	0	0
$973$	−1.39445	−0.0447040
$974$	0	0
$975$	25.4558	0.815239
$976$	0	0
$977$	−16.8444	−0.538900	−0.269450	−	0.963014i	$-0.586842\pi$
$977$	−16.8444	−0.538900	−0.269450	+	0.963014i	$0.586842\pi$
$978$	0	0
$979$	−3.34701	−0.106971
$980$	0	0
$981$	66.8058	2.13295
$982$	0	0
$983$	−11.4826	−0.366238	−0.183119	−	0.983091i	$-0.558619\pi$
$983$	−11.4826	−0.366238	−0.183119	+	0.983091i	$0.558619\pi$
$984$	0	0
$985$	35.2111	1.12192
$986$	0	0
$987$	−42.4222	−1.35031
$988$	0	0
$989$	−20.6554	−0.656803
$990$	0	0
$991$	−33.2536	−1.05634	−0.528168	−	0.849140i	$-0.677121\pi$
$991$	−33.2536	−1.05634	−0.528168	+	0.849140i	$0.677121\pi$
$992$	0	0
$993$	−11.0544	−0.350801
$994$	0	0
$995$	−20.2389	−0.641615
$996$	0	0
$997$	55.4921	1.75745	0.878727	−	0.477325i	$-0.158394\pi$
$997$	55.4921	1.75745	0.878727	+	0.477325i	$0.158394\pi$
$998$	0	0
$999$	63.6333	2.01327

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4624.2.a.bq.1.4		4
4.3	odd	2		1156.2.a.h.1.1		4
17.2	even	8		272.2.o.g.225.2		4
17.9	even	8		272.2.o.g.81.2		4
17.16	even	2	inner	4624.2.a.bq.1.1		4
51.2	odd	8		2448.2.be.u.1585.1		4
51.26	odd	8		2448.2.be.u.1441.1		4
68.3	even	16		1156.2.h.e.757.1		16
68.7	even	16		1156.2.h.e.1001.4		16
68.11	even	16		1156.2.h.e.733.4		16
68.15	odd	8		1156.2.e.c.905.2		4
68.19	odd	8		68.2.e.a.21.1	yes	4
68.23	even	16		1156.2.h.e.733.1		16
68.27	even	16		1156.2.h.e.1001.1		16
68.31	even	16		1156.2.h.e.757.4		16
68.39	even	16		1156.2.h.e.977.4		16
68.43	odd	8		68.2.e.a.13.1	✓	4
68.47	odd	4		1156.2.b.a.577.1		4
68.55	odd	4		1156.2.b.a.577.4		4
68.59	odd	8		1156.2.e.c.829.2		4
68.63	even	16		1156.2.h.e.977.1		16
68.67	odd	2		1156.2.a.h.1.4		4
136.19	odd	8		1088.2.o.t.769.2		4
136.43	odd	8		1088.2.o.t.897.2		4
136.53	even	8		1088.2.o.s.769.1		4
136.77	even	8		1088.2.o.s.897.1		4
204.155	even	8		612.2.k.e.361.2		4
204.179	even	8		612.2.k.e.217.2		4
340.19	odd	8		1700.2.o.c.701.2		4
340.43	even	8		1700.2.m.b.149.2		4
340.87	even	8		1700.2.m.b.1449.2		4
340.179	odd	8		1700.2.o.c.1101.2		4
340.223	even	8		1700.2.m.a.1449.1		4
340.247	even	8		1700.2.m.a.149.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.e.a.13.1	✓	4	68.43	odd	8
68.2.e.a.21.1	yes	4	68.19	odd	8
272.2.o.g.81.2		4	17.9	even	8
272.2.o.g.225.2		4	17.2	even	8
612.2.k.e.217.2		4	204.179	even	8
612.2.k.e.361.2		4	204.155	even	8
1088.2.o.s.769.1		4	136.53	even	8
1088.2.o.s.897.1		4	136.77	even	8
1088.2.o.t.769.2		4	136.19	odd	8
1088.2.o.t.897.2		4	136.43	odd	8
1156.2.a.h.1.1		4	4.3	odd	2
1156.2.a.h.1.4		4	68.67	odd	2
1156.2.b.a.577.1		4	68.47	odd	4
1156.2.b.a.577.4		4	68.55	odd	4
1156.2.e.c.829.2		4	68.59	odd	8
1156.2.e.c.905.2		4	68.15	odd	8
1156.2.h.e.733.1		16	68.23	even	16
1156.2.h.e.733.4		16	68.11	even	16
1156.2.h.e.757.1		16	68.3	even	16
1156.2.h.e.757.4		16	68.31	even	16
1156.2.h.e.977.1		16	68.63	even	16
1156.2.h.e.977.4		16	68.39	even	16
1156.2.h.e.1001.1		16	68.27	even	16
1156.2.h.e.1001.4		16	68.7	even	16
1700.2.m.a.149.1		4	340.247	even	8
1700.2.m.a.1449.1		4	340.223	even	8
1700.2.m.b.149.2		4	340.43	even	8
1700.2.m.b.1449.2		4	340.87	even	8
1700.2.o.c.701.2		4	340.19	odd	8
1700.2.o.c.1101.2		4	340.179	odd	8
2448.2.be.u.1441.1		4	51.26	odd	8
2448.2.be.u.1585.1		4	51.2	odd	8
4624.2.a.bq.1.1		4	17.16	even	2	inner
4624.2.a.bq.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4624 = 2^{4} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4624.a (trivial)

\(f(q)\)	\(=\)	\(q+3.25662 q^{3} +1.41421 q^{5} +3.25662 q^{7} +7.60555 q^{9} +O(q^{10})\)	\(q+3.25662 q^{3} +1.41421 q^{5} +3.25662 q^{7} +7.60555 q^{9} +0.428189 q^{11} -2.60555 q^{13} +4.60555 q^{15} -0.605551 q^{19} +10.6056 q^{21} +6.08504 q^{23} -3.00000 q^{25} +14.9985 q^{27} +2.27059 q^{29} -6.08504 q^{31} +1.39445 q^{33} +4.60555 q^{35} +4.24264 q^{37} -8.48528 q^{39} -1.41421 q^{41} -3.39445 q^{43} +10.7559 q^{45} -4.00000 q^{47} +3.60555 q^{49} -5.21110 q^{53} +0.605551 q^{55} -1.97205 q^{57} -8.60555 q^{59} +8.78383 q^{61} +24.7684 q^{63} -3.68481 q^{65} -9.21110 q^{67} +19.8167 q^{69} -4.11300 q^{71} +9.89949 q^{73} -9.76985 q^{75} +1.39445 q^{77} +0.428189 q^{79} +26.0278 q^{81} -17.8167 q^{83} +7.39445 q^{87} -7.81665 q^{89} -8.48528 q^{91} -19.8167 q^{93} -0.856379 q^{95} +10.7559 q^{97} +3.25662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{9}+O(q^{10}) \) 4 * q + 16 * q^9	\( 4 q + 16 q^{9} + 4 q^{13} + 4 q^{15} + 12 q^{19} + 28 q^{21} - 12 q^{25} + 20 q^{33} + 4 q^{35} - 28 q^{43} - 16 q^{47} + 8 q^{53} - 12 q^{55} - 20 q^{59} - 8 q^{67} + 36 q^{69} + 20 q^{77} + 32 q^{81} - 28 q^{83} + 44 q^{87} + 12 q^{89} - 36 q^{93}+O(q^{100}) \) 4 * q + 16 * q^9 + 4 * q^13 + 4 * q^15 + 12 * q^19 + 28 * q^21 - 12 * q^25 + 20 * q^33 + 4 * q^35 - 28 * q^43 - 16 * q^47 + 8 * q^53 - 12 * q^55 - 20 * q^59 - 8 * q^67 + 36 * q^69 + 20 * q^77 + 32 * q^81 - 28 * q^83 + 44 * q^87 + 12 * q^89 - 36 * q^93

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