LMFDB - Embedded newform 9248.2.a.bl.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9248,2,Mod(1,9248)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9248.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9248, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9248 = 2^{5} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9248.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-18,0,0,0,0, 0,0,0,-24,0,0,0,0,0,0,0,0,0,-24,0,0,0,-24,0,30,0,0,0,0,0,-24,0,0,0,0,0, 0,0,0,0,0,0,-24,0,0,0,0,0,0,0,0,0,0,0,0,0,-18,0,-48,0,0,0,-24,0,-24,0, 0,0,48,0,0,0,0,0,0,0,48,0,-24,0,0,0,0,0,0,0,-48,0,0,0,24,0,-48,0,0,0,6, 0,-48,0,0,0,24,0,0,0,0,0,0,0,24,0,-24,0,0,0,0,0,0,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(145)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.8456517893$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.3359232.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 36x^{2} - 8 $ x^6 - 12x^4 + 36x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.65785$ of defining polynomial
Character	$\chi$	$=$	9248.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.65785 q^{3} +1.41421 q^{5} -1.67555 q^{7} +4.06418 q^{9} -5.48628 q^{11} -3.06418 q^{13} +3.75877 q^{15} +3.75877 q^{19} -4.45336 q^{21} +4.50398 q^{23} -3.00000 q^{25} +2.82843 q^{27} -5.74762 q^{29} +4.16283 q^{31} -14.5817 q^{33} -2.36959 q^{35} -3.37882 q^{37} -8.14413 q^{39} -0.550391 q^{41} -1.63041 q^{43} +5.74762 q^{45} -10.1284 q^{47} -4.19253 q^{49} +13.6459 q^{53} -7.75877 q^{55} +9.99026 q^{57} -9.88713 q^{59} +10.0810 q^{61} -6.80973 q^{63} -4.33340 q^{65} -4.00000 q^{67} +11.9709 q^{69} -3.64015 q^{71} -9.21719 q^{73} -7.97356 q^{75} +9.19253 q^{77} -4.84513 q^{79} -4.67499 q^{81} -13.1480 q^{83} -15.2763 q^{87} -15.9709 q^{89} +5.13418 q^{91} +11.0642 q^{93} +5.31570 q^{95} -2.05537 q^{97} -22.2972 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{9} - 18 q^{25} - 24 q^{33} - 24 q^{43} - 24 q^{47} + 30 q^{49} - 24 q^{55} - 24 q^{67} - 18 q^{81} - 48 q^{83} - 24 q^{87} - 24 q^{89} + 48 q^{93}+O(q^{100}) $ 6 * q + 6 * q^9 - 18 * q^25 - 24 * q^33 - 24 * q^43 - 24 * q^47 + 30 * q^49 - 24 * q^55 - 24 * q^67 - 18 * q^81 - 48 * q^83 - 24 * q^87 - 24 * q^89 + 48 * q^93

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
$3$	2.65785	1.53451	0.767256	−	0.641341i	$-0.221622\pi$
$3$	2.65785	1.53451	0.767256	+	0.641341i	$0.221622\pi$
$4$	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
$7$	−1.67555	−0.633298	−0.316649	−	0.948543i	$-0.602558\pi$
$7$	−1.67555	−0.633298	−0.316649	+	0.948543i	$0.602558\pi$
$8$	0	0
$9$	4.06418	1.35473
$10$	0	0
$11$	−5.48628	−1.65418	−0.827088	−	0.562073i	$-0.810004\pi$
$11$	−5.48628	−1.65418	−0.827088	+	0.562073i	$0.810004\pi$
$12$	0	0
$13$	−3.06418	−0.849850	−0.424925	−	0.905229i	$-0.639700\pi$
$13$	−3.06418	−0.849850	−0.424925	+	0.905229i	$0.639700\pi$
$14$	0	0
$15$	3.75877	0.970510
$16$	0	0
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.75877	0.862321	0.431161	−	0.902275i	$-0.358104\pi$
$19$	3.75877	0.862321	0.431161	+	0.902275i	$0.358104\pi$
$20$	0	0
$21$	−4.45336	−0.971804
$22$	0	0
$23$	4.50398	0.939144	0.469572	−	0.882894i	$-0.344408\pi$
$23$	4.50398	0.939144	0.469572	+	0.882894i	$0.344408\pi$
$24$	0	0
$25$	−3.00000	−0.600000
$26$	0	0
$27$	2.82843	0.544331
$28$	0	0
$29$	−5.74762	−1.06731	−0.533653	−	0.845704i	$-0.679181\pi$
$29$	−5.74762	−1.06731	−0.533653	+	0.845704i	$0.679181\pi$
$30$	0	0
$31$	4.16283	0.747666	0.373833	−	0.927496i	$-0.378043\pi$
$31$	4.16283	0.747666	0.373833	+	0.927496i	$0.378043\pi$
$32$	0	0
$33$	−14.5817	−2.53835
$34$	0	0
$35$	−2.36959	−0.400533
$36$	0	0
$37$	−3.37882	−0.555474	−0.277737	−	0.960657i	$-0.589584\pi$
$37$	−3.37882	−0.555474	−0.277737	+	0.960657i	$0.589584\pi$
$38$	0	0
$39$	−8.14413	−1.30410
$40$	0	0
$41$	−0.550391	−0.0859567	−0.0429783	−	0.999076i	$-0.513685\pi$
$41$	−0.550391	−0.0859567	−0.0429783	+	0.999076i	$0.513685\pi$
$42$	0	0
$43$	−1.63041	−0.248636	−0.124318	−	0.992242i	$-0.539674\pi$
$43$	−1.63041	−0.248636	−0.124318	+	0.992242i	$0.539674\pi$
$44$	0	0
$45$	5.74762	0.856804
$46$	0	0
$47$	−10.1284	−1.47737	−0.738686	−	0.674049i	$-0.764553\pi$
$47$	−10.1284	−1.47737	−0.738686	+	0.674049i	$0.764553\pi$
$48$	0	0
$49$	−4.19253	−0.598933
$50$	0	0
$51$	0	0
$52$	0	0
$53$	13.6459	1.87441	0.937204	−	0.348782i	$-0.113405\pi$
$53$	13.6459	1.87441	0.937204	+	0.348782i	$0.113405\pi$
$54$	0	0
$55$	−7.75877	−1.04619
$56$	0	0
$57$	9.99026	1.32324
$58$	0	0
$59$	−9.88713	−1.28719	−0.643597	−	0.765364i	$-0.722559\pi$
$59$	−9.88713	−1.28719	−0.643597	+	0.765364i	$0.722559\pi$
$60$	0	0
$61$	10.0810	1.29074	0.645371	−	0.763869i	$-0.276703\pi$
$61$	10.0810	1.29074	0.645371	+	0.763869i	$0.276703\pi$
$62$	0	0
$63$	−6.80973	−0.857946
$64$	0	0
$65$	−4.33340	−0.537492
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	11.9709	1.44113
$70$	0	0
$71$	−3.64015	−0.432007	−0.216003	−	0.976393i	$-0.569302\pi$
$71$	−3.64015	−0.432007	−0.216003	+	0.976393i	$0.569302\pi$
$72$	0	0
$73$	−9.21719	−1.07879	−0.539396	−	0.842053i	$-0.681347\pi$
$73$	−9.21719	−1.07879	−0.539396	+	0.842053i	$0.681347\pi$
$74$	0	0
$75$	−7.97356	−0.920707
$76$	0	0
$77$	9.19253	1.04759
$78$	0	0
$79$	−4.84513	−0.545119	−0.272560	−	0.962139i	$-0.587870\pi$
$79$	−4.84513	−0.545119	−0.272560	+	0.962139i	$0.587870\pi$
$80$	0	0
$81$	−4.67499	−0.519444
$82$	0	0
$83$	−13.1480	−1.44318	−0.721588	−	0.692323i	$-0.756587\pi$
$83$	−13.1480	−1.44318	−0.721588	+	0.692323i	$0.756587\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−15.2763	−1.63779
$88$	0	0
$89$	−15.9709	−1.69291	−0.846456	−	0.532458i	$-0.821268\pi$
$89$	−15.9709	−1.69291	−0.846456	+	0.532458i	$0.821268\pi$
$90$	0	0
$91$	5.13418	0.538209
$92$	0	0
$93$	11.0642	1.14730
$94$	0	0
$95$	5.31570	0.545380
$96$	0	0
$97$	−2.05537	−0.208691	−0.104345	−	0.994541i	$-0.533275\pi$
$97$	−2.05537	−0.208691	−0.104345	+	0.994541i	$0.533275\pi$
$98$	0	0
$99$	−22.2972	−2.24095
$100$	0	0
$101$	11.0642	1.10093	0.550463	−	0.834859i	$-0.314451\pi$
$101$	11.0642	1.10093	0.550463	+	0.834859i	$0.314451\pi$
$102$	0	0
$103$	−6.77837	−0.667893	−0.333946	−	0.942592i	$-0.608380\pi$
$103$	−6.77837	−0.667893	−0.333946	+	0.942592i	$0.608380\pi$
$104$	0	0
$105$	−6.29801	−0.614623
$106$	0	0
$107$	9.93816	0.960758	0.480379	−	0.877061i	$-0.340499\pi$
$107$	9.93816	0.960758	0.480379	+	0.877061i	$0.340499\pi$
$108$	0	0
$109$	−19.8486	−1.90115	−0.950576	−	0.310493i	$-0.899506\pi$
$109$	−19.8486	−1.90115	−0.950576	+	0.310493i	$0.899506\pi$
$110$	0	0
$111$	−8.98040	−0.852382
$112$	0	0
$113$	16.3790	1.54081	0.770404	−	0.637555i	$-0.220054\pi$
$113$	16.3790	1.54081	0.770404	+	0.637555i	$0.220054\pi$
$114$	0	0
$115$	6.36959	0.593967
$116$	0	0
$117$	−12.4534	−1.15131
$118$	0	0
$119$	0	0
$120$	0	0
$121$	19.0993	1.73630
$122$	0	0
$123$	−1.46286	−0.131902
$124$	0	0
$125$	−11.3137	−1.01193
$126$	0	0
$127$	21.4047	1.89936	0.949679	−	0.313226i	$-0.101410\pi$
$127$	21.4047	1.89936	0.949679	+	0.313226i	$0.101410\pi$
$128$	0	0
$129$	−4.33340	−0.381535
$130$	0	0
$131$	−3.34015	−0.291830	−0.145915	−	0.989297i	$-0.546613\pi$
$131$	−3.34015	−0.291830	−0.145915	+	0.989297i	$0.546613\pi$
$132$	0	0
$133$	−6.29801	−0.546106
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	7.97090	0.681000	0.340500	−	0.940244i	$-0.389404\pi$
$137$	7.97090	0.681000	0.340500	+	0.940244i	$0.389404\pi$
$138$	0	0
$139$	2.47633	0.210040	0.105020	−	0.994470i	$-0.466509\pi$
$139$	2.47633	0.210040	0.105020	+	0.994470i	$0.466509\pi$
$140$	0	0
$141$	−26.9197	−2.26705
$142$	0	0
$143$	16.8109	1.40580
$144$	0	0
$145$	−8.12836	−0.675023
$146$	0	0
$147$	−11.1431	−0.919070
$148$	0	0
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	0	0
$151$	−10.3696	−0.843865	−0.421932	−	0.906627i	$-0.638648\pi$
$151$	−10.3696	−0.843865	−0.421932	+	0.906627i	$0.638648\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	5.88713	0.472865
$156$	0	0
$157$	2.48246	0.198122	0.0990609	−	0.995081i	$-0.468416\pi$
$157$	2.48246	0.198122	0.0990609	+	0.995081i	$0.468416\pi$
$158$	0	0
$159$	36.2688	2.87630
$160$	0	0
$161$	−7.54664	−0.594758
$162$	0	0
$163$	−10.1197	−0.792635	−0.396317	−	0.918114i	$-0.629712\pi$
$163$	−10.1197	−0.792635	−0.396317	+	0.918114i	$0.629712\pi$
$164$	0	0
$165$	−20.6217	−1.60539
$166$	0	0
$167$	−4.32245	−0.334482	−0.167241	−	0.985916i	$-0.553486\pi$
$167$	−4.32245	−0.334482	−0.167241	+	0.985916i	$0.553486\pi$
$168$	0	0
$169$	−3.61081	−0.277755
$170$	0	0
$171$	15.2763	1.16821
$172$	0	0
$173$	4.70227	0.357507	0.178754	−	0.983894i	$-0.442794\pi$
$173$	4.70227	0.357507	0.178754	+	0.983894i	$0.442794\pi$
$174$	0	0
$175$	5.02665	0.379979
$176$	0	0
$177$	−26.2785	−1.97521
$178$	0	0
$179$	−7.75877	−0.579918	−0.289959	−	0.957039i	$-0.593642\pi$
$179$	−7.75877	−0.579918	−0.289959	+	0.957039i	$0.593642\pi$
$180$	0	0
$181$	−5.10646	−0.379560	−0.189780	−	0.981827i	$-0.560778\pi$
$181$	−5.10646	−0.379560	−0.189780	+	0.981827i	$0.560778\pi$
$182$	0	0
$183$	26.7939	1.98066
$184$	0	0
$185$	−4.77837	−0.351313
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−4.73917	−0.344724
$190$	0	0
$191$	−19.6851	−1.42436	−0.712182	−	0.701995i	$-0.752293\pi$
$191$	−19.6851	−1.42436	−0.712182	+	0.701995i	$0.752293\pi$
$192$	0	0
$193$	−4.42416	−0.318458	−0.159229	−	0.987242i	$-0.550901\pi$
$193$	−4.42416	−0.318458	−0.159229	+	0.987242i	$0.550901\pi$
$194$	0	0
$195$	−11.5175	−0.824788
$196$	0	0
$197$	−9.43986	−0.672563	−0.336281	−	0.941762i	$-0.609169\pi$
$197$	−9.43986	−0.672563	−0.336281	+	0.941762i	$0.609169\pi$
$198$	0	0
$199$	−21.1334	−1.49811	−0.749053	−	0.662510i	$-0.769491\pi$
$199$	−21.1334	−1.49811	−0.749053	+	0.662510i	$0.769491\pi$
$200$	0	0
$201$	−10.6314	−0.749882
$202$	0	0
$203$	9.63041	0.675923
$204$	0	0
$205$	−0.778371	−0.0543638
$206$	0	0
$207$	18.3050	1.27228
$208$	0	0
$209$	−20.6217	−1.42643
$210$	0	0
$211$	15.0169	1.03381	0.516903	−	0.856044i	$-0.327085\pi$
$211$	15.0169	1.03381	0.516903	+	0.856044i	$0.327085\pi$
$212$	0	0
$213$	−9.67499	−0.662920
$214$	0	0
$215$	−2.30575	−0.157251
$216$	0	0
$217$	−6.97502	−0.473495
$218$	0	0
$219$	−24.4979	−1.65542
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−20.4979	−1.37264	−0.686322	−	0.727298i	$-0.740776\pi$
$223$	−20.4979	−1.37264	−0.686322	+	0.727298i	$0.740776\pi$
$224$	0	0
$225$	−12.1925	−0.812836
$226$	0	0
$227$	−24.1022	−1.59972	−0.799859	−	0.600188i	$-0.795093\pi$
$227$	−24.1022	−1.59972	−0.799859	+	0.600188i	$0.795093\pi$
$228$	0	0
$229$	29.9709	1.98053	0.990267	−	0.139184i	$-0.0444479\pi$
$229$	29.9709	1.98053	0.990267	+	0.139184i	$0.0444479\pi$
$230$	0	0
$231$	24.4324	1.60753
$232$	0	0
$233$	16.1975	1.06113	0.530567	−	0.847643i	$-0.321979\pi$
$233$	16.1975	1.06113	0.530567	+	0.847643i	$0.321979\pi$
$234$	0	0
$235$	−14.3237	−0.934372
$236$	0	0
$237$	−12.8776	−0.836492
$238$	0	0
$239$	16.9067	1.09361	0.546803	−	0.837262i	$-0.315845\pi$
$239$	16.9067	1.09361	0.546803	+	0.837262i	$0.315845\pi$
$240$	0	0
$241$	−10.0810	−0.649375	−0.324688	−	0.945821i	$-0.605259\pi$
$241$	−10.0810	−0.649375	−0.324688	+	0.945821i	$0.605259\pi$
$242$	0	0
$243$	−20.9107	−1.34142
$244$	0	0
$245$	−5.92914	−0.378799
$246$	0	0
$247$	−11.5175	−0.732844
$248$	0	0
$249$	−34.9453	−2.21457
$250$	0	0
$251$	−17.8135	−1.12438	−0.562188	−	0.827010i	$-0.690040\pi$
$251$	−17.8135	−1.12438	−0.562188	+	0.827010i	$0.690040\pi$
$252$	0	0
$253$	−24.7101	−1.55351
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.28581	−0.392098	−0.196049	−	0.980594i	$-0.562811\pi$
$257$	−6.28581	−0.392098	−0.196049	+	0.980594i	$0.562811\pi$
$258$	0	0
$259$	5.66138	0.351781
$260$	0	0
$261$	−23.3593	−1.44591
$262$	0	0
$263$	1.40467	0.0866155	0.0433077	−	0.999062i	$-0.486210\pi$
$263$	1.40467	0.0866155	0.0433077	+	0.999062i	$0.486210\pi$
$264$	0	0
$265$	19.2982	1.18548
$266$	0	0
$267$	−42.4483	−2.59779
$268$	0	0
$269$	9.21719	0.561982	0.280991	−	0.959710i	$-0.409337\pi$
$269$	9.21719	0.561982	0.280991	+	0.959710i	$0.409337\pi$
$270$	0	0
$271$	−0.906726	−0.0550797	−0.0275399	−	0.999621i	$-0.508767\pi$
$271$	−0.906726	−0.0550797	−0.0275399	+	0.999621i	$0.508767\pi$
$272$	0	0
$273$	13.6459	0.825887
$274$	0	0
$275$	16.4588	0.992505
$276$	0	0
$277$	17.2428	1.03602	0.518011	−	0.855374i	$-0.326673\pi$
$277$	17.2428	1.03602	0.518011	+	0.855374i	$0.326673\pi$
$278$	0	0
$279$	16.9185	1.01288
$280$	0	0
$281$	18.7784	1.12022	0.560112	−	0.828417i	$-0.310758\pi$
$281$	18.7784	1.12022	0.560112	+	0.828417i	$0.310758\pi$
$282$	0	0
$283$	11.8473	0.704251	0.352125	−	0.935953i	$-0.385459\pi$
$283$	11.8473	0.704251	0.352125	+	0.935953i	$0.385459\pi$
$284$	0	0
$285$	14.1284	0.836892
$286$	0	0
$287$	0.922208	0.0544362
$288$	0	0
$289$	0	0
$290$	0	0
$291$	−5.46286	−0.320238
$292$	0	0
$293$	−20.3851	−1.19091	−0.595454	−	0.803389i	$-0.703028\pi$
$293$	−20.3851	−1.19091	−0.595454	+	0.803389i	$0.703028\pi$
$294$	0	0
$295$	−13.9825	−0.814093
$296$	0	0
$297$	−15.5175	−0.900419
$298$	0	0
$299$	−13.8010	−0.798132
$300$	0	0
$301$	2.73184	0.157461
$302$	0	0
$303$	29.4069	1.68939
$304$	0	0
$305$	14.2567	0.816337
$306$	0	0
$307$	−14.1284	−0.806348	−0.403174	−	0.915123i	$-0.632093\pi$
$307$	−14.1284	−0.806348	−0.403174	+	0.915123i	$0.632093\pi$
$308$	0	0
$309$	−18.0159	−1.02489
$310$	0	0
$311$	−10.0012	−0.567116	−0.283558	−	0.958955i	$-0.591515\pi$
$311$	−10.0012	−0.567116	−0.283558	+	0.958955i	$0.591515\pi$
$312$	0	0
$313$	−8.98023	−0.507593	−0.253796	−	0.967258i	$-0.581679\pi$
$313$	−8.98023	−0.507593	−0.253796	+	0.967258i	$0.581679\pi$
$314$	0	0
$315$	−9.63041	−0.542612
$316$	0	0
$317$	30.3396	1.70404	0.852022	−	0.523506i	$-0.175376\pi$
$317$	30.3396	1.70404	0.852022	+	0.523506i	$0.175376\pi$
$318$	0	0
$319$	31.5330	1.76551
$320$	0	0
$321$	26.4142	1.47429
$322$	0	0
$323$	0	0
$324$	0	0
$325$	9.19253	0.509910
$326$	0	0
$327$	−52.7547	−2.91734
$328$	0	0
$329$	16.9706	0.935617
$330$	0	0
$331$	−19.4439	−1.06873	−0.534366	−	0.845253i	$-0.679450\pi$
$331$	−19.4439	−1.06873	−0.534366	+	0.845253i	$0.679450\pi$
$332$	0	0
$333$	−13.7321	−0.752515
$334$	0	0
$335$	−5.65685	−0.309067
$336$	0	0
$337$	21.5762	1.17533	0.587667	−	0.809103i	$-0.300047\pi$
$337$	21.5762	1.17533	0.587667	+	0.809103i	$0.300047\pi$
$338$	0	0
$339$	43.5330	2.36439
$340$	0	0
$341$	−22.8384	−1.23677
$342$	0	0
$343$	18.7536	1.01260
$344$	0	0
$345$	16.9294	0.911449
$346$	0	0
$347$	−9.88272	−0.530532	−0.265266	−	0.964175i	$-0.585460\pi$
$347$	−9.88272	−0.530532	−0.265266	+	0.964175i	$0.585460\pi$
$348$	0	0
$349$	−18.8675	−1.00996	−0.504978	−	0.863132i	$-0.668499\pi$
$349$	−18.8675	−1.00996	−0.504978	+	0.863132i	$0.668499\pi$
$350$	0	0
$351$	−8.66680	−0.462600
$352$	0	0
$353$	−12.1284	−0.645527	−0.322764	−	0.946480i	$-0.604612\pi$
$353$	−12.1284	−0.645527	−0.322764	+	0.946480i	$0.604612\pi$
$354$	0	0
$355$	−5.14796	−0.273225
$356$	0	0
$357$	0	0
$358$	0	0
$359$	17.2371	0.909740	0.454870	−	0.890558i	$-0.349686\pi$
$359$	17.2371	0.909740	0.454870	+	0.890558i	$0.349686\pi$
$360$	0	0
$361$	−4.87164	−0.256402
$362$	0	0
$363$	50.7630	2.66437
$364$	0	0
$365$	−13.0351	−0.682287
$366$	0	0
$367$	30.9010	1.61302	0.806509	−	0.591222i	$-0.201354\pi$
$367$	30.9010	1.61302	0.806509	+	0.591222i	$0.201354\pi$
$368$	0	0
$369$	−2.23689	−0.116448
$370$	0	0
$371$	−22.8644	−1.18706
$372$	0	0
$373$	−14.4142	−0.746337	−0.373169	−	0.927764i	$-0.621729\pi$
$373$	−14.4142	−0.746337	−0.373169	+	0.927764i	$0.621729\pi$
$374$	0	0
$375$	−30.0702	−1.55282
$376$	0	0
$377$	17.6117	0.907049
$378$	0	0
$379$	34.8932	1.79234	0.896172	−	0.443706i	$-0.146337\pi$
$379$	34.8932	1.79234	0.896172	+	0.443706i	$0.146337\pi$
$380$	0	0
$381$	56.8904	2.91459
$382$	0	0
$383$	6.62630	0.338588	0.169294	−	0.985566i	$-0.445851\pi$
$383$	6.62630	0.338588	0.169294	+	0.985566i	$0.445851\pi$
$384$	0	0
$385$	13.0002	0.662552
$386$	0	0
$387$	−6.62630	−0.336834
$388$	0	0
$389$	13.3209	0.675396	0.337698	−	0.941255i	$-0.390352\pi$
$389$	13.3209	0.675396	0.337698	+	0.941255i	$0.390352\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−8.87763	−0.447817
$394$	0	0
$395$	−6.85204	−0.344764
$396$	0	0
$397$	27.4701	1.37868	0.689341	−	0.724437i	$-0.257900\pi$
$397$	27.4701	1.37868	0.689341	+	0.724437i	$0.257900\pi$
$398$	0	0
$399$	−16.7392	−0.838007
$400$	0	0
$401$	−7.25259	−0.362177	−0.181089	−	0.983467i	$-0.557962\pi$
$401$	−7.25259	−0.362177	−0.181089	+	0.983467i	$0.557962\pi$
$402$	0	0
$403$	−12.7556	−0.635404
$404$	0	0
$405$	−6.61144	−0.328525
$406$	0	0
$407$	18.5371	0.918852
$408$	0	0
$409$	−26.4243	−1.30660	−0.653298	−	0.757101i	$-0.726615\pi$
$409$	−26.4243	−1.30660	−0.653298	+	0.757101i	$0.726615\pi$
$410$	0	0
$411$	21.1855	1.04500
$412$	0	0
$413$	16.5664	0.815178
$414$	0	0
$415$	−18.5940	−0.912744
$416$	0	0
$417$	6.58172	0.322308
$418$	0	0
$419$	−34.3151	−1.67640	−0.838202	−	0.545361i	$-0.816393\pi$
$419$	−34.3151	−1.67640	−0.838202	+	0.545361i	$0.816393\pi$
$420$	0	0
$421$	22.4142	1.09240	0.546200	−	0.837655i	$-0.316074\pi$
$421$	22.4142	1.09240	0.546200	+	0.837655i	$0.316074\pi$
$422$	0	0
$423$	−41.1634	−2.00143
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−16.8912	−0.817425
$428$	0	0
$429$	44.6810	2.15722
$430$	0	0
$431$	36.0352	1.73575	0.867876	−	0.496780i	$-0.165485\pi$
$431$	36.0352	1.73575	0.867876	+	0.496780i	$0.165485\pi$
$432$	0	0
$433$	−14.9358	−0.717770	−0.358885	−	0.933382i	$-0.616843\pi$
$433$	−14.9358	−0.717770	−0.358885	+	0.933382i	$0.616843\pi$
$434$	0	0
$435$	−21.6040	−1.03583
$436$	0	0
$437$	16.9294	0.809844
$438$	0	0
$439$	−5.36780	−0.256191	−0.128096	−	0.991762i	$-0.540886\pi$
$439$	−5.36780	−0.256191	−0.128096	+	0.991762i	$0.540886\pi$
$440$	0	0
$441$	−17.0392	−0.811391
$442$	0	0
$443$	13.4783	0.640375	0.320188	−	0.947354i	$-0.396254\pi$
$443$	13.4783	0.640375	0.320188	+	0.947354i	$0.396254\pi$
$444$	0	0
$445$	−22.5863	−1.07069
$446$	0	0
$447$	26.5785	1.25712
$448$	0	0
$449$	−9.07682	−0.428362	−0.214181	−	0.976794i	$-0.568708\pi$
$449$	−9.07682	−0.428362	−0.214181	+	0.976794i	$0.568708\pi$
$450$	0	0
$451$	3.01960	0.142187
$452$	0	0
$453$	−27.5608	−1.29492
$454$	0	0
$455$	7.26083	0.340393
$456$	0	0
$457$	31.7844	1.48681	0.743405	−	0.668842i	$-0.233210\pi$
$457$	31.7844	1.48681	0.743405	+	0.668842i	$0.233210\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−10.8675	−0.506151	−0.253076	−	0.967447i	$-0.581442\pi$
$461$	−10.8675	−0.506151	−0.253076	+	0.967447i	$0.581442\pi$
$462$	0	0
$463$	−20.0000	−0.929479	−0.464739	−	0.885448i	$-0.653852\pi$
$463$	−20.0000	−0.929479	−0.464739	+	0.885448i	$0.653852\pi$
$464$	0	0
$465$	15.6471	0.725617
$466$	0	0
$467$	−20.9804	−0.970857	−0.485429	−	0.874276i	$-0.661336\pi$
$467$	−20.9804	−0.970857	−0.485429	+	0.874276i	$0.661336\pi$
$468$	0	0
$469$	6.70220	0.309479
$470$	0	0
$471$	6.59801	0.304020
$472$	0	0
$473$	8.94491	0.411287
$474$	0	0
$475$	−11.2763	−0.517393
$476$	0	0
$477$	55.4593	2.53931
$478$	0	0
$479$	−2.09403	−0.0956788	−0.0478394	−	0.998855i	$-0.515234\pi$
$479$	−2.09403	−0.0956788	−0.0478394	+	0.998855i	$0.515234\pi$
$480$	0	0
$481$	10.3533	0.472070
$482$	0	0
$483$	−20.0578	−0.912664
$484$	0	0
$485$	−2.90673	−0.131988
$486$	0	0
$487$	6.65010	0.301345	0.150672	−	0.988584i	$-0.451856\pi$
$487$	6.65010	0.301345	0.150672	+	0.988584i	$0.451856\pi$
$488$	0	0
$489$	−26.8966	−1.21631
$490$	0	0
$491$	0.0154816	0.000698674 0	0.000349337	−	1.00000i	$-0.499889\pi$
$491$	0.0154816	0.000698674 0	0.000349337		1.00000i	$0.499889\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−31.5330	−1.41730
$496$	0	0
$497$	6.09926	0.273589
$498$	0	0
$499$	15.3581	0.687521	0.343760	−	0.939057i	$-0.388299\pi$
$499$	15.3581	0.687521	0.343760	+	0.939057i	$0.388299\pi$
$500$	0	0
$501$	−11.4884	−0.513266
$502$	0	0
$503$	−0.630205	−0.0280995	−0.0140497	−	0.999901i	$-0.504472\pi$
$503$	−0.630205	−0.0280995	−0.0140497	+	0.999901i	$0.504472\pi$
$504$	0	0
$505$	15.6471	0.696287
$506$	0	0
$507$	−9.59701	−0.426218
$508$	0	0
$509$	40.9769	1.81627	0.908134	−	0.418679i	$-0.137507\pi$
$509$	40.9769	1.81627	0.908134	+	0.418679i	$0.137507\pi$
$510$	0	0
$511$	15.4439	0.683196
$512$	0	0
$513$	10.6314	0.469388
$514$	0	0
$515$	−9.58606	−0.422412
$516$	0	0
$517$	55.5670	2.44383
$518$	0	0
$519$	12.4979	0.548599
$520$	0	0
$521$	−0.131909	−0.00577903	−0.00288951	−	0.999996i	$-0.500920\pi$
$521$	−0.131909	−0.00577903	−0.00288951	+	0.999996i	$0.500920\pi$
$522$	0	0
$523$	−29.7351	−1.30022	−0.650112	−	0.759839i	$-0.725278\pi$
$523$	−29.7351	−1.30022	−0.650112	+	0.759839i	$0.725278\pi$
$524$	0	0
$525$	13.3601	0.583082
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−2.71419	−0.118008
$530$	0	0
$531$	−40.1830	−1.74380
$532$	0	0
$533$	1.68650	0.0730503
$534$	0	0
$535$	14.0547	0.607637
$536$	0	0
$537$	−20.6217	−0.889891
$538$	0	0
$539$	23.0014	0.990741
$540$	0	0
$541$	−36.6376	−1.57518	−0.787588	−	0.616203i	$-0.788670\pi$
$541$	−36.6376	−1.57518	−0.787588	+	0.616203i	$0.788670\pi$
$542$	0	0
$543$	−13.5722	−0.582440
$544$	0	0
$545$	−28.0702	−1.20239
$546$	0	0
$547$	26.3306	1.12582	0.562908	−	0.826520i	$-0.309683\pi$
$547$	26.3306	1.12582	0.562908	+	0.826520i	$0.309683\pi$
$548$	0	0
$549$	40.9710	1.74860
$550$	0	0
$551$	−21.6040	−0.920360
$552$	0	0
$553$	8.11825	0.345223
$554$	0	0
$555$	−12.7002	−0.539094
$556$	0	0
$557$	−4.54252	−0.192473	−0.0962363	−	0.995359i	$-0.530680\pi$
$557$	−4.54252	−0.192473	−0.0962363	+	0.995359i	$0.530680\pi$
$558$	0	0
$559$	4.99588	0.211303
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−22.8830	−0.964404	−0.482202	−	0.876060i	$-0.660163\pi$
$563$	−22.8830	−0.964404	−0.482202	+	0.876060i	$0.660163\pi$
$564$	0	0
$565$	23.1634	0.974493
$566$	0	0
$567$	7.83318	0.328963
$568$	0	0
$569$	3.77425	0.158225	0.0791124	−	0.996866i	$-0.474791\pi$
$569$	3.77425	0.158225	0.0791124	+	0.996866i	$0.474791\pi$
$570$	0	0
$571$	23.3207	0.975939	0.487970	−	0.872861i	$-0.337738\pi$
$571$	23.3207	0.975939	0.487970	+	0.872861i	$0.337738\pi$
$572$	0	0
$573$	−52.3201	−2.18570
$574$	0	0
$575$	−13.5119	−0.563486
$576$	0	0
$577$	11.7142	0.487668	0.243834	−	0.969817i	$-0.421595\pi$
$577$	11.7142	0.487668	0.243834	+	0.969817i	$0.421595\pi$
$578$	0	0
$579$	−11.7588	−0.488678
$580$	0	0
$581$	22.0301	0.913961
$582$	0	0
$583$	−74.8652	−3.10060
$584$	0	0
$585$	−17.6117	−0.728155
$586$	0	0
$587$	−14.4587	−0.596776	−0.298388	−	0.954445i	$-0.596449\pi$
$587$	−14.4587	−0.596776	−0.298388	+	0.954445i	$0.596449\pi$
$588$	0	0
$589$	15.6471	0.644728
$590$	0	0
$591$	−25.0898	−1.03205
$592$	0	0
$593$	6.03920	0.248000	0.124000	−	0.992282i	$-0.460428\pi$
$593$	6.03920	0.248000	0.124000	+	0.992282i	$0.460428\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−56.1694	−2.29886
$598$	0	0
$599$	6.77837	0.276957	0.138478	−	0.990365i	$-0.455779\pi$
$599$	6.77837	0.276957	0.138478	+	0.990365i	$0.455779\pi$
$600$	0	0
$601$	−7.07107	−0.288435	−0.144217	−	0.989546i	$-0.546066\pi$
$601$	−7.07107	−0.288435	−0.144217	+	0.989546i	$0.546066\pi$
$602$	0	0
$603$	−16.2567	−0.662024
$604$	0	0
$605$	27.0104	1.09813
$606$	0	0
$607$	7.51393	0.304981	0.152490	−	0.988305i	$-0.451271\pi$
$607$	7.51393	0.304981	0.152490	+	0.988305i	$0.451271\pi$
$608$	0	0
$609$	25.5962	1.03721
$610$	0	0
$611$	31.0351	1.25555
$612$	0	0
$613$	0.443258	0.0179030	0.00895152	−	0.999960i	$-0.497151\pi$
$613$	0.443258	0.0179030	0.00895152	+	0.999960i	$0.497151\pi$
$614$	0	0
$615$	−2.06879	−0.0834218
$616$	0	0
$617$	21.5351	0.866970	0.433485	−	0.901161i	$-0.357284\pi$
$617$	21.5351	0.866970	0.433485	+	0.901161i	$0.357284\pi$
$618$	0	0
$619$	14.3900	0.578385	0.289192	−	0.957271i	$-0.406613\pi$
$619$	14.3900	0.578385	0.289192	+	0.957271i	$0.406613\pi$
$620$	0	0
$621$	12.7392	0.511205
$622$	0	0
$623$	26.7600	1.07212
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	−54.8093	−2.18887
$628$	0	0
$629$	0	0
$630$	0	0
$631$	−31.0196	−1.23487	−0.617435	−	0.786622i	$-0.711828\pi$
$631$	−31.0196	−1.23487	−0.617435	+	0.786622i	$0.711828\pi$
$632$	0	0
$633$	39.9127	1.58639
$634$	0	0
$635$	30.2708	1.20126
$636$	0	0
$637$	12.8467	0.509003
$638$	0	0
$639$	−14.7942	−0.585251
$640$	0	0
$641$	33.7681	1.33376	0.666879	−	0.745166i	$-0.267630\pi$
$641$	33.7681	1.33376	0.666879	+	0.745166i	$0.267630\pi$
$642$	0	0
$643$	5.66780	0.223516	0.111758	−	0.993735i	$-0.464352\pi$
$643$	5.66780	0.223516	0.111758	+	0.993735i	$0.464352\pi$
$644$	0	0
$645$	−6.12836	−0.241304
$646$	0	0
$647$	−3.34998	−0.131701	−0.0658507	−	0.997829i	$-0.520976\pi$
$647$	−3.34998	−0.131701	−0.0658507	+	0.997829i	$0.520976\pi$
$648$	0	0
$649$	54.2435	2.12924
$650$	0	0
$651$	−18.5386	−0.726584
$652$	0	0
$653$	32.4858	1.27127	0.635633	−	0.771991i	$-0.280739\pi$
$653$	32.4858	1.27127	0.635633	+	0.771991i	$0.280739\pi$
$654$	0	0
$655$	−4.72369	−0.184570
$656$	0	0
$657$	−37.4603	−1.46147
$658$	0	0
$659$	37.4201	1.45768	0.728841	−	0.684683i	$-0.240059\pi$
$659$	37.4201	1.45768	0.728841	+	0.684683i	$0.240059\pi$
$660$	0	0
$661$	3.68510	0.143334	0.0716668	−	0.997429i	$-0.477168\pi$
$661$	3.68510	0.143334	0.0716668	+	0.997429i	$0.477168\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−8.90673	−0.345388
$666$	0	0
$667$	−25.8871	−1.00235
$668$	0	0
$669$	−54.4805	−2.10634
$670$	0	0
$671$	−55.3073	−2.13511
$672$	0	0
$673$	16.3236	0.629228	0.314614	−	0.949220i	$-0.398125\pi$
$673$	16.3236	0.629228	0.314614	+	0.949220i	$0.398125\pi$
$674$	0	0
$675$	−8.48528	−0.326599
$676$	0	0
$677$	19.7082	0.757449	0.378724	−	0.925510i	$-0.376363\pi$
$677$	19.7082	0.757449	0.378724	+	0.925510i	$0.376363\pi$
$678$	0	0
$679$	3.44387	0.132164
$680$	0	0
$681$	−64.0601	−2.45479
$682$	0	0
$683$	−8.78194	−0.336032	−0.168016	−	0.985784i	$-0.553736\pi$
$683$	−8.78194	−0.336032	−0.168016	+	0.985784i	$0.553736\pi$
$684$	0	0
$685$	11.2726	0.430702
$686$	0	0
$687$	79.6582	3.03915
$688$	0	0
$689$	−41.8135	−1.59297
$690$	0	0
$691$	−8.73319	−0.332226	−0.166113	−	0.986107i	$-0.553122\pi$
$691$	−8.73319	−0.332226	−0.166113	+	0.986107i	$0.553122\pi$
$692$	0	0
$693$	37.3601	1.41919
$694$	0	0
$695$	3.50206	0.132841
$696$	0	0
$697$	0	0
$698$	0	0
$699$	43.0506	1.62832
$700$	0	0
$701$	26.6709	1.00734	0.503672	−	0.863895i	$-0.331982\pi$
$701$	26.6709	1.00734	0.503672	+	0.863895i	$0.331982\pi$
$702$	0	0
$703$	−12.7002	−0.478997
$704$	0	0
$705$	−38.0702	−1.43381
$706$	0	0
$707$	−18.5386	−0.697215
$708$	0	0
$709$	32.1665	1.20804	0.604019	−	0.796970i	$-0.293565\pi$
$709$	32.1665	1.20804	0.604019	+	0.796970i	$0.293565\pi$
$710$	0	0
$711$	−19.6915	−0.738487
$712$	0	0
$713$	18.7493	0.702166
$714$	0	0
$715$	23.7743	0.889107
$716$	0	0
$717$	44.9356	1.67815
$718$	0	0
$719$	21.6561	0.807635	0.403817	−	0.914840i	$-0.367683\pi$
$719$	21.6561	0.807635	0.403817	+	0.914840i	$0.367683\pi$
$720$	0	0
$721$	11.3575	0.422975
$722$	0	0
$723$	−26.7939	−0.996474
$724$	0	0
$725$	17.2428	0.640383
$726$	0	0
$727$	−16.6500	−0.617515	−0.308757	−	0.951141i	$-0.599913\pi$
$727$	−16.6500	−0.617515	−0.308757	+	0.951141i	$0.599913\pi$
$728$	0	0
$729$	−41.5526	−1.53899
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2.20676	−0.0815084	−0.0407542	−	0.999169i	$-0.512976\pi$
$733$	−2.20676	−0.0815084	−0.0407542	+	0.999169i	$0.512976\pi$
$734$	0	0
$735$	−15.7588	−0.581271
$736$	0	0
$737$	21.9451	0.808359
$738$	0	0
$739$	35.5330	1.30710	0.653552	−	0.756882i	$-0.273278\pi$
$739$	35.5330	1.30710	0.653552	+	0.756882i	$0.273278\pi$
$740$	0	0
$741$	−30.6119	−1.12456
$742$	0	0
$743$	−43.0785	−1.58040	−0.790199	−	0.612851i	$-0.790023\pi$
$743$	−43.0785	−1.58040	−0.790199	+	0.612851i	$0.790023\pi$
$744$	0	0
$745$	14.1421	0.518128
$746$	0	0
$747$	−53.4356	−1.95511
$748$	0	0
$749$	−16.6519	−0.608447
$750$	0	0
$751$	15.2950	0.558123	0.279061	−	0.960273i	$-0.409977\pi$
$751$	15.2950	0.558123	0.279061	+	0.960273i	$0.409977\pi$
$752$	0	0
$753$	−47.3455	−1.72537
$754$	0	0
$755$	−14.6648	−0.533707
$756$	0	0
$757$	−19.8425	−0.721190	−0.360595	−	0.932723i	$-0.617426\pi$
$757$	−19.8425	−0.721190	−0.360595	+	0.932723i	$0.617426\pi$
$758$	0	0
$759$	−65.6757	−2.38388
$760$	0	0
$761$	−3.37908	−0.122492	−0.0612458	−	0.998123i	$-0.519507\pi$
$761$	−3.37908	−0.122492	−0.0612458	+	0.998123i	$0.519507\pi$
$762$	0	0
$763$	33.2573	1.20400
$764$	0	0
$765$	0	0
$766$	0	0
$767$	30.2959	1.09392
$768$	0	0
$769$	11.3209	0.408242	0.204121	−	0.978946i	$-0.434566\pi$
$769$	11.3209	0.408242	0.204121	+	0.978946i	$0.434566\pi$
$770$	0	0
$771$	−16.7067	−0.601679
$772$	0	0
$773$	49.5776	1.78318	0.891591	−	0.452841i	$-0.149590\pi$
$773$	49.5776	1.78318	0.891591	+	0.452841i	$0.149590\pi$
$774$	0	0
$775$	−12.4885	−0.448599
$776$	0	0
$777$	15.0471	0.539812
$778$	0	0
$779$	−2.06879	−0.0741223
$780$	0	0
$781$	19.9709	0.714615
$782$	0	0
$783$	−16.2567	−0.580967
$784$	0	0
$785$	3.51073	0.125303
$786$	0	0
$787$	18.1865	0.648278	0.324139	−	0.946009i	$-0.394925\pi$
$787$	18.1865	0.648278	0.324139	+	0.946009i	$0.394925\pi$
$788$	0	0
$789$	3.73340	0.132912
$790$	0	0
$791$	−27.4439	−0.975792
$792$	0	0
$793$	−30.8900	−1.09694
$794$	0	0
$795$	51.2918	1.81913
$796$	0	0
$797$	18.3851	0.651232	0.325616	−	0.945502i	$-0.394428\pi$
$797$	18.3851	0.651232	0.325616	+	0.945502i	$0.394428\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	−64.9086	−2.29343
$802$	0	0
$803$	50.5681	1.78451
$804$	0	0
$805$	−10.6726	−0.376158
$806$	0	0
$807$	24.4979	0.862368
$808$	0	0
$809$	−25.0458	−0.880564	−0.440282	−	0.897859i	$-0.645122\pi$
$809$	−25.0458	−0.880564	−0.440282	+	0.897859i	$0.645122\pi$
$810$	0	0
$811$	−8.25927	−0.290022	−0.145011	−	0.989430i	$-0.546322\pi$
$811$	−8.25927	−0.290022	−0.145011	+	0.989430i	$0.546322\pi$
$812$	0	0
$813$	−2.40994	−0.0845204
$814$	0	0
$815$	−14.3114	−0.501306
$816$	0	0
$817$	−6.12836	−0.214404
$818$	0	0
$819$	20.8662	0.729125
$820$	0	0
$821$	10.3037	0.359601	0.179801	−	0.983703i	$-0.442455\pi$
$821$	10.3037	0.359601	0.179801	+	0.983703i	$0.442455\pi$
$822$	0	0
$823$	−23.1199	−0.805909	−0.402954	−	0.915220i	$-0.632017\pi$
$823$	−23.1199	−0.805909	−0.402954	+	0.915220i	$0.632017\pi$
$824$	0	0
$825$	43.7452	1.52301
$826$	0	0
$827$	0.715141	0.0248679	0.0124340	−	0.999923i	$-0.496042\pi$
$827$	0.715141	0.0248679	0.0124340	+	0.999923i	$0.496042\pi$
$828$	0	0
$829$	−20.5526	−0.713822	−0.356911	−	0.934138i	$-0.616170\pi$
$829$	−20.5526	−0.713822	−0.356911	+	0.934138i	$0.616170\pi$
$830$	0	0
$831$	45.8289	1.58979
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−6.11287	−0.211545
$836$	0	0
$837$	11.7743	0.406978
$838$	0	0
$839$	0.393245	0.0135763	0.00678816	−	0.999977i	$-0.497839\pi$
$839$	0.393245	0.0135763	0.00678816	+	0.999977i	$0.497839\pi$
$840$	0	0
$841$	4.03508	0.139141
$842$	0	0
$843$	49.9101	1.71900
$844$	0	0
$845$	−5.10646	−0.175668
$846$	0	0
$847$	−32.0018	−1.09959
$848$	0	0
$849$	31.4884	1.08068
$850$	0	0
$851$	−15.2181	−0.521670
$852$	0	0
$853$	−9.48101	−0.324624	−0.162312	−	0.986740i	$-0.551895\pi$
$853$	−9.48101	−0.324624	−0.162312	+	0.986740i	$0.551895\pi$
$854$	0	0
$855$	21.6040	0.738840
$856$	0	0
$857$	53.6520	1.83272	0.916359	−	0.400358i	$-0.131114\pi$
$857$	53.6520	1.83272	0.916359	+	0.400358i	$0.131114\pi$
$858$	0	0
$859$	18.8830	0.644280	0.322140	−	0.946692i	$-0.395598\pi$
$859$	18.8830	0.644280	0.322140	+	0.946692i	$0.395598\pi$
$860$	0	0
$861$	2.45109	0.0835330
$862$	0	0
$863$	5.81345	0.197892	0.0989461	−	0.995093i	$-0.468453\pi$
$863$	5.81345	0.197892	0.0989461	+	0.995093i	$0.468453\pi$
$864$	0	0
$865$	6.65002	0.226107
$866$	0	0
$867$	0	0
$868$	0	0
$869$	26.5817	0.901723
$870$	0	0
$871$	12.2567	0.415303
$872$	0	0
$873$	−8.35337	−0.282719
$874$	0	0
$875$	18.9567	0.640853
$876$	0	0
$877$	20.1124	0.679148	0.339574	−	0.940579i	$-0.389717\pi$
$877$	20.1124	0.679148	0.339574	+	0.940579i	$0.389717\pi$
$878$	0	0
$879$	−54.1805	−1.82746
$880$	0	0
$881$	−50.9613	−1.71693	−0.858465	−	0.512872i	$-0.828581\pi$
$881$	−50.9613	−1.71693	−0.858465	+	0.512872i	$0.828581\pi$
$882$	0	0
$883$	−12.7082	−0.427665	−0.213833	−	0.976870i	$-0.568595\pi$
$883$	−12.7082	−0.427665	−0.213833	+	0.976870i	$0.568595\pi$
$884$	0	0
$885$	−37.1634	−1.24924
$886$	0	0
$887$	1.51592	0.0508997	0.0254498	−	0.999676i	$-0.491898\pi$
$887$	1.51592	0.0508997	0.0254498	+	0.999676i	$0.491898\pi$
$888$	0	0
$889$	−35.8646	−1.20286
$890$	0	0
$891$	25.6483	0.859251
$892$	0	0
$893$	−38.0702	−1.27397
$894$	0	0
$895$	−10.9726	−0.366772
$896$	0	0
$897$	−36.6810	−1.22474
$898$	0	0
$899$	−23.9263	−0.797988
$900$	0	0
$901$	0	0
$902$	0	0
$903$	7.26083	0.241625
$904$	0	0
$905$	−7.22163	−0.240055
$906$	0	0
$907$	−9.86083	−0.327423	−0.163712	−	0.986508i	$-0.552347\pi$
$907$	−9.86083	−0.327423	−0.163712	+	0.986508i	$0.552347\pi$
$908$	0	0
$909$	44.9668	1.49145
$910$	0	0
$911$	−8.01470	−0.265539	−0.132769	−	0.991147i	$-0.542387\pi$
$911$	−8.01470	−0.265539	−0.132769	+	0.991147i	$0.542387\pi$
$912$	0	0
$913$	72.1334	2.38727
$914$	0	0
$915$	37.8922	1.25268
$916$	0	0
$917$	5.59659	0.184816
$918$	0	0
$919$	−16.6500	−0.549233	−0.274617	−	0.961554i	$-0.588551\pi$
$919$	−16.6500	−0.549233	−0.274617	+	0.961554i	$0.588551\pi$
$920$	0	0
$921$	−37.5511	−1.23735
$922$	0	0
$923$	11.1541	0.367141
$924$	0	0
$925$	10.1365	0.333285
$926$	0	0
$927$	−27.5485	−0.904812
$928$	0	0
$929$	−42.6575	−1.39955	−0.699774	−	0.714364i	$-0.746716\pi$
$929$	−42.6575	−1.39955	−0.699774	+	0.714364i	$0.746716\pi$
$930$	0	0
$931$	−15.7588	−0.516473
$932$	0	0
$933$	−26.5817	−0.870246
$934$	0	0
$935$	0	0
$936$	0	0
$937$	41.3310	1.35022	0.675112	−	0.737715i	$-0.264095\pi$
$937$	41.3310	1.35022	0.675112	+	0.737715i	$0.264095\pi$
$938$	0	0
$939$	−23.8681	−0.778907
$940$	0	0
$941$	−13.1876	−0.429902	−0.214951	−	0.976625i	$-0.568959\pi$
$941$	−13.1876	−0.429902	−0.214951	+	0.976625i	$0.568959\pi$
$942$	0	0
$943$	−2.47895	−0.0807257
$944$	0	0
$945$	−6.70220	−0.218023
$946$	0	0
$947$	−10.4608	−0.339931	−0.169966	−	0.985450i	$-0.554366\pi$
$947$	−10.4608	−0.339931	−0.169966	+	0.985450i	$0.554366\pi$
$948$	0	0
$949$	28.2431	0.916811
$950$	0	0
$951$	80.6383	2.61488
$952$	0	0
$953$	−20.6209	−0.667977	−0.333989	−	0.942577i	$-0.608395\pi$
$953$	−20.6209	−0.667977	−0.333989	+	0.942577i	$0.608395\pi$
$954$	0	0
$955$	−27.8389	−0.900847
$956$	0	0
$957$	83.8101	2.70920
$958$	0	0
$959$	−13.3556	−0.431276
$960$	0	0
$961$	−13.6709	−0.440996
$962$	0	0
$963$	40.3905	1.30156
$964$	0	0
$965$	−6.25671	−0.201411
$966$	0	0
$967$	−52.1248	−1.67622	−0.838111	−	0.545500i	$-0.816340\pi$
$967$	−52.1248	−1.67622	−0.838111	+	0.545500i	$0.816340\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−4.43975	−0.142478	−0.0712392	−	0.997459i	$-0.522695\pi$
$971$	−4.43975	−0.142478	−0.0712392	+	0.997459i	$0.522695\pi$
$972$	0	0
$973$	−4.14921	−0.133018
$974$	0	0
$975$	24.4324	0.782463
$976$	0	0
$977$	−19.7743	−0.632634	−0.316317	−	0.948653i	$-0.602446\pi$
$977$	−19.7743	−0.632634	−0.316317	+	0.948653i	$0.602446\pi$
$978$	0	0
$979$	87.6208	2.80037
$980$	0	0
$981$	−80.6683	−2.57554
$982$	0	0
$983$	2.01670	0.0643227	0.0321614	−	0.999483i	$-0.489761\pi$
$983$	2.01670	0.0643227	0.0321614	+	0.999483i	$0.489761\pi$
$984$	0	0
$985$	−13.3500	−0.425366
$986$	0	0
$987$	45.1052	1.43572
$988$	0	0
$989$	−7.34335	−0.233505
$990$	0	0
$991$	14.6346	0.464884	0.232442	−	0.972610i	$-0.425328\pi$
$991$	14.6346	0.464884	0.232442	+	0.972610i	$0.425328\pi$
$992$	0	0
$993$	−51.6789	−1.63998
$994$	0	0
$995$	−29.8871	−0.947486
$996$	0	0
$997$	5.30228	0.167925	0.0839624	−	0.996469i	$-0.473242\pi$
$997$	5.30228	0.167925	0.0839624	+	0.996469i	$0.473242\pi$
$998$	0	0
$999$	−9.55674	−0.302362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9248.2.a.bl.1.6		6
4.3	odd	2		9248.2.a.bm.1.1		6
17.8	even	8		544.2.o.h.353.1	yes	6
17.15	even	8		544.2.o.h.225.1	yes	6
17.16	even	2	inner	9248.2.a.bl.1.1		6
68.15	odd	8		544.2.o.g.225.3	✓	6
68.59	odd	8		544.2.o.g.353.3	yes	6
68.67	odd	2		9248.2.a.bm.1.6		6
136.59	odd	8		1088.2.o.v.897.1		6
136.83	odd	8		1088.2.o.v.769.1		6
136.93	even	8		1088.2.o.u.897.3		6
136.117	even	8		1088.2.o.u.769.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.o.g.225.3	✓	6	68.15	odd	8
544.2.o.g.353.3	yes	6	68.59	odd	8
544.2.o.h.225.1	yes	6	17.15	even	8
544.2.o.h.353.1	yes	6	17.8	even	8
1088.2.o.u.769.3		6	136.117	even	8
1088.2.o.u.897.3		6	136.93	even	8
1088.2.o.v.769.1		6	136.83	odd	8
1088.2.o.v.897.1		6	136.59	odd	8
9248.2.a.bl.1.1		6	17.16	even	2	inner
9248.2.a.bl.1.6		6	1.1	even	1	trivial
9248.2.a.bm.1.1		6	4.3	odd	2
9248.2.a.bm.1.6		6	68.67	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9248 = 2^{5} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9248.a (trivial)

\(f(q)\)	\(=\)	\(q+2.65785 q^{3} +1.41421 q^{5} -1.67555 q^{7} +4.06418 q^{9} -5.48628 q^{11} -3.06418 q^{13} +3.75877 q^{15} +3.75877 q^{19} -4.45336 q^{21} +4.50398 q^{23} -3.00000 q^{25} +2.82843 q^{27} -5.74762 q^{29} +4.16283 q^{31} -14.5817 q^{33} -2.36959 q^{35} -3.37882 q^{37} -8.14413 q^{39} -0.550391 q^{41} -1.63041 q^{43} +5.74762 q^{45} -10.1284 q^{47} -4.19253 q^{49} +13.6459 q^{53} -7.75877 q^{55} +9.99026 q^{57} -9.88713 q^{59} +10.0810 q^{61} -6.80973 q^{63} -4.33340 q^{65} -4.00000 q^{67} +11.9709 q^{69} -3.64015 q^{71} -9.21719 q^{73} -7.97356 q^{75} +9.19253 q^{77} -4.84513 q^{79} -4.67499 q^{81} -13.1480 q^{83} -15.2763 q^{87} -15.9709 q^{89} +5.13418 q^{91} +11.0642 q^{93} +5.31570 q^{95} -2.05537 q^{97} -22.2972 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{9} - 18 q^{25} - 24 q^{33} - 24 q^{43} - 24 q^{47} + 30 q^{49} - 24 q^{55} - 24 q^{67} - 18 q^{81} - 48 q^{83} - 24 q^{87} - 24 q^{89} + 48 q^{93}+O(q^{100}) \) 6 * q + 6 * q^9 - 18 * q^25 - 24 * q^33 - 24 * q^43 - 24 * q^47 + 30 * q^49 - 24 * q^55 - 24 * q^67 - 18 * q^81 - 48 * q^83 - 24 * q^87 - 24 * q^89 + 48 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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