LMFDB - Embedded newform 2368.2.g.m.961.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2368,2,Mod(961,2368)] 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("2368.961"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-6,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$18.9085751986$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.2992527616.3
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 13x^{6} + 52x^{4} + 61x^{2} + 1 $ x^8 + 13x^6 + 52x^4 + 61*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 1184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			961.4
Root			$2.48425i$ of defining polynomial
Character	$\chi$	$=$	2368.961
Dual form			2368.2.g.m.961.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91023 q^{3} +0.910233i q^{5} +3.26124 q^{7} +0.648991 q^{9} -2.79702 q^{11} -1.79702i q^{13} -1.73876i q^{15} +4.96849i q^{17} -2.96849i q^{19} -6.22973 q^{21} +1.35101i q^{23} +4.17148 q^{25} +4.49098 q^{27} +5.69396i q^{29} -6.05826i q^{31} +5.34295 q^{33} +2.96849i q^{35} +(-0.113218 + 6.08171i) q^{37} +3.43272i q^{39} -1.05826 q^{41} -0.851975i q^{43} +0.590733i q^{45} -2.91829 q^{47} +3.63570 q^{49} -9.49098i q^{51} -5.70725 q^{53} -2.54594i q^{55} +5.67051i q^{57} +0.342951i q^{59} +2.28469i q^{61} +2.11652 q^{63} +1.63570 q^{65} +3.35624 q^{67} -2.58074i q^{69} -9.19822 q^{71} +15.9289 q^{73} -7.96849 q^{75} -9.12175 q^{77} -9.06842i q^{79} -10.5258 q^{81} +6.73876 q^{83} -4.52249 q^{85} -10.8768i q^{87} +4.26647i q^{89} -5.86050i q^{91} +11.5727i q^{93} +2.70202 q^{95} +10.6390i q^{97} -1.81524 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} + 12 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{21} + 10 q^{25} - 36 q^{27} - 4 q^{33} - 12 q^{37} + 26 q^{41} - 56 q^{47} + 12 q^{49} - 16 q^{53} - 52 q^{63} - 4 q^{65} + 2 q^{67} + 28 q^{71} - 2 q^{73}+ \cdots - 16 q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 + 12 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^21 + 10 * q^25 - 36 * q^27 - 4 * q^33 - 12 * q^37 + 26 * q^41 - 56 * q^47 + 12 * q^49 - 16 * q^53 - 52 * q^63 - 4 * q^65 + 2 * q^67 + 28 * q^71 - 2 * q^73 - 20 * q^75 + 16 * q^77 + 24 * q^81 + 68 * q^83 - 8 * q^85 + 12 * q^95 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times$.

$n$	$705$	$1407$	$1925$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.91023			−1.10287			−0.551437	−	0.834217i	$-0.685920\pi$
$3$	−1.91023			−1.10287			−0.551437	+	0.834217i	$0.685920\pi$
$4$		0			0
$5$			0.910233i			0.407069i	0.979068	+	0.203534i	$0.0652428\pi$
$5$			0.910233i			0.407069i	−0.979068	+	0.203534i	$0.934757\pi$
$6$		0			0
$7$	3.26124			1.23263			0.616317	−	0.787498i	$-0.288624\pi$
$7$	3.26124			1.23263			0.616317	+	0.787498i	$0.288624\pi$
$8$		0			0
$9$	0.648991			0.216330
$10$		0			0
$11$	−2.79702			−0.843332			−0.421666	−	0.906751i	$-0.638554\pi$
$11$	−2.79702			−0.843332			−0.421666	+	0.906751i	$0.638554\pi$
$12$		0			0
$13$		−	1.79702i		−	0.498402i	−0.968452	−	0.249201i	$-0.919832\pi$
$13$		−	1.79702i		−	0.498402i	0.968452	−	0.249201i	$-0.0801680\pi$
$14$		0			0
$15$		−	1.73876i		−	0.448945i
$16$		0			0
$17$			4.96849i			1.20504i	0.798105	+	0.602518i	$0.205836\pi$
$17$			4.96849i			1.20504i	−0.798105	+	0.602518i	$0.794164\pi$
$18$		0			0
$19$		−	2.96849i		−	0.681019i	−0.940241	−	0.340509i	$-0.889401\pi$
$19$		−	2.96849i		−	0.681019i	0.940241	−	0.340509i	$-0.110599\pi$
$20$		0			0
$21$	−6.22973			−1.35944
$22$		0			0
$23$			1.35101i			0.281705i	0.990031	+	0.140852i	$0.0449843\pi$
$23$			1.35101i			0.281705i	−0.990031	+	0.140852i	$0.955016\pi$
$24$		0			0
$25$	4.17148			0.834295
$26$		0			0
$27$	4.49098			0.864289
$28$		0			0
$29$			5.69396i			1.05734i	0.848827	+	0.528671i	$0.177309\pi$
$29$			5.69396i			1.05734i	−0.848827	+	0.528671i	$0.822691\pi$
$30$		0			0
$31$		−	6.05826i		−	1.08810i	−0.839054	−	0.544048i	$-0.816891\pi$
$31$		−	6.05826i		−	1.08810i	0.839054	−	0.544048i	$-0.183109\pi$
$32$		0			0
$33$	5.34295			0.930088
$34$		0			0
$35$			2.96849i			0.501767i
$36$		0			0
$37$	−0.113218	+	6.08171i	−0.0186129	+	0.999827i
$37$	−0.113218	+	6.08171i	−0.0186129	+	0.999827i
$38$		0			0
$39$			3.43272i			0.549675i
$40$		0			0
$41$	−1.05826			−0.165272			−0.0826361	−	0.996580i	$-0.526334\pi$
$41$	−1.05826			−0.165272			−0.0826361	+	0.996580i	$0.526334\pi$
$42$		0			0
$43$		−	0.851975i		−	0.129925i	−0.997888	−	0.0649625i	$-0.979307\pi$
$43$		−	0.851975i		−	0.129925i	0.997888	−	0.0649625i	$-0.0206928\pi$
$44$		0			0
$45$			0.590733i			0.0880612i
$46$		0			0
$47$	−2.91829			−0.425677			−0.212838	−	0.977087i	$-0.568271\pi$
$47$	−2.91829			−0.425677			−0.212838	+	0.977087i	$0.568271\pi$
$48$		0			0
$49$	3.63570			0.519386
$50$		0			0
$51$		−	9.49098i		−	1.32900i
$52$		0			0
$53$	−5.70725			−0.783951			−0.391975	−	0.919976i	$-0.628208\pi$
$53$	−5.70725			−0.783951			−0.391975	+	0.919976i	$0.628208\pi$
$54$		0			0
$55$		−	2.54594i		−	0.343294i
$56$		0			0
$57$			5.67051i			0.751077i
$58$		0			0
$59$			0.342951i			0.0446485i	0.999751	+	0.0223242i	$0.00710661\pi$
$59$			0.342951i			0.0446485i	−0.999751	+	0.0223242i	$0.992893\pi$
$60$		0			0
$61$			2.28469i			0.292525i	0.989246	+	0.146263i	$0.0467244\pi$
$61$			2.28469i			0.292525i	−0.989246	+	0.146263i	$0.953276\pi$
$62$		0			0
$63$	2.11652			0.266656
$64$		0			0
$65$	1.63570			0.202884
$66$		0			0
$67$	3.35624			0.410030			0.205015	−	0.978759i	$-0.434276\pi$
$67$	3.35624			0.410030			0.205015	+	0.978759i	$0.434276\pi$
$68$		0			0
$69$		−	2.58074i		−	0.310685i
$70$		0			0
$71$	−9.19822			−1.09163			−0.545814	−	0.837906i	$-0.683780\pi$
$71$	−9.19822			−1.09163			−0.545814	+	0.837906i	$0.683780\pi$
$72$		0			0
$73$	15.9289			1.86434			0.932170	−	0.362021i	$-0.117913\pi$
$73$	15.9289			1.86434			0.932170	+	0.362021i	$0.117913\pi$
$74$		0			0
$75$	−7.96849			−0.920122
$76$		0			0
$77$	−9.12175			−1.03952
$78$		0			0
$79$		−	9.06842i		−	1.02028i	−0.860092	−	0.510139i	$-0.829594\pi$
$79$		−	9.06842i		−	1.02028i	0.860092	−	0.510139i	$-0.170406\pi$
$80$		0			0
$81$	−10.5258			−1.16953
$82$		0			0
$83$	6.73876			0.739675			0.369837	−	0.929097i	$-0.379413\pi$
$83$	6.73876			0.739675			0.369837	+	0.929097i	$0.379413\pi$
$84$		0			0
$85$	−4.52249			−0.490532
$86$		0			0
$87$		−	10.8768i		−	1.16611i
$88$		0			0
$89$			4.26647i			0.452245i	0.974099	+	0.226123i	$0.0726050\pi$
$89$			4.26647i			0.452245i	−0.974099	+	0.226123i	$0.927395\pi$
$90$		0			0
$91$		−	5.86050i		−	0.614348i
$92$		0			0
$93$			11.5727i			1.20003i
$94$		0			0
$95$	2.70202			0.277221
$96$		0			0
$97$			10.6390i			1.08023i	0.841592	+	0.540113i	$0.181619\pi$
$97$			10.6390i			1.08023i	−0.841592	+	0.540113i	$0.818381\pi$
$98$		0			0
$99$	−1.81524			−0.182438
$100$		0			0
$101$	2.45617			0.244398			0.122199	−	0.992506i	$-0.461005\pi$
$101$	2.45617			0.244398			0.122199	+	0.992506i	$0.461005\pi$
$102$		0			0
$103$			17.2532i			1.70001i	0.526777	+	0.850003i	$0.323400\pi$
$103$			17.2532i			1.70001i	−0.526777	+	0.850003i	$0.676600\pi$
$104$		0			0
$105$		−	5.67051i		−	0.553385i
$106$		0			0
$107$	10.7655			1.04074			0.520370	−	0.853941i	$-0.325794\pi$
$107$	10.7655			1.04074			0.520370	+	0.853941i	$0.325794\pi$
$108$		0			0
$109$			13.3114i			1.27500i	0.770448	+	0.637502i	$0.220032\pi$
$109$			13.3114i			1.27500i	−0.770448	+	0.637502i	$0.779968\pi$
$110$		0			0
$111$	0.216272	−	11.6175i	0.0205277	−	1.10268i
$112$		0			0
$113$			17.2350i			1.62133i	0.585511	+	0.810664i	$0.300894\pi$
$113$			17.2350i			1.62133i	−0.585511	+	0.810664i	$0.699106\pi$
$114$		0			0
$115$	−1.22973			−0.114673
$116$		0			0
$117$		−	1.16625i		−	0.107819i
$118$		0			0
$119$			16.2035i			1.48537i
$120$		0			0
$121$	−3.17671			−0.288791
$122$		0			0
$123$	2.02152			0.182274
$124$		0			0
$125$			8.34818i			0.746684i
$126$		0			0
$127$	−12.6757			−1.12479			−0.562395	−	0.826869i	$-0.690120\pi$
$127$	−12.6757			−1.12479			−0.562395	+	0.826869i	$0.690120\pi$
$128$		0			0
$129$			1.62747i			0.143291i
$130$		0			0
$131$			14.7421i			1.28802i	0.765017	+	0.644010i	$0.222730\pi$
$131$			14.7421i			1.28802i	−0.765017	+	0.644010i	$0.777270\pi$
$132$		0			0
$133$		−	9.68097i		−	0.839446i
$134$		0			0
$135$			4.08784i			0.351825i
$136$		0			0
$137$	−10.9237			−0.933274			−0.466637	−	0.884449i	$-0.654535\pi$
$137$	−10.9237			−0.933274			−0.466637	+	0.884449i	$0.654535\pi$
$138$		0			0
$139$	−1.10846			−0.0940181			−0.0470091	−	0.998894i	$-0.514969\pi$
$139$	−1.10846			−0.0940181			−0.0470091	+	0.998894i	$0.514969\pi$
$140$		0			0
$141$	5.57462			0.469467
$142$		0			0
$143$			5.02628i			0.420319i
$144$		0			0
$145$	−5.18283			−0.430411
$146$		0			0
$147$	−6.94504			−0.572817
$148$		0			0
$149$	3.93368			0.322260			0.161130	−	0.986933i	$-0.448486\pi$
$149$	3.93368			0.322260			0.161130	+	0.986933i	$0.448486\pi$
$150$		0			0
$151$	2.54906			0.207440			0.103720	−	0.994607i	$-0.466925\pi$
$151$	2.54906			0.207440			0.103720	+	0.994607i	$0.466925\pi$
$152$		0			0
$153$			3.22450i			0.260686i
$154$		0			0
$155$	5.51443			0.442929
$156$		0			0
$157$	−11.4647			−0.914982			−0.457491	−	0.889214i	$-0.651252\pi$
$157$	−11.4647			−0.914982			−0.457491	+	0.889214i	$0.651252\pi$
$158$		0			0
$159$	10.9022			0.864599
$160$		0			0
$161$			4.40597i			0.347239i
$162$		0			0
$163$			20.0239i			1.56839i	0.620512	+	0.784197i	$0.286925\pi$
$163$			20.0239i			1.56839i	−0.620512	+	0.784197i	$0.713075\pi$
$164$		0			0
$165$			4.86333i			0.378610i
$166$		0			0
$167$			9.80224i			0.758520i	0.925290	+	0.379260i	$0.123821\pi$
$167$			9.80224i			0.758520i	−0.925290	+	0.379260i	$0.876179\pi$
$168$		0			0
$169$	9.77074			0.751595
$170$		0			0
$171$		−	1.92652i		−	0.147325i
$172$		0			0
$173$	1.23167			0.0936418			0.0468209	−	0.998903i	$-0.485091\pi$
$173$	1.23167			0.0936418			0.0468209	+	0.998903i	$0.485091\pi$
$174$		0			0
$175$	13.6042			1.02838
$176$		0			0
$177$		−	0.655117i		−	0.0492416i
$178$		0			0
$179$			18.6390i			1.39314i	0.717487	+	0.696572i	$0.245292\pi$
$179$			18.6390i			1.39314i	−0.717487	+	0.696572i	$0.754708\pi$
$180$		0			0
$181$	0.588801			0.0437652			0.0218826	−	0.999761i	$-0.493034\pi$
$181$	0.588801			0.0437652			0.0218826	+	0.999761i	$0.493034\pi$
$182$		0			0
$183$		−	4.36430i		−	0.322618i
$184$		0			0
$185$	−5.53577	−	0.103055i	−0.406998	−	0.00757672i
$186$		0			0
$187$		−	13.8969i		−	1.01625i
$188$		0			0
$189$	14.6462			1.06535
$190$		0			0
$191$			14.9706i			1.08323i	0.840625	+	0.541617i	$0.182188\pi$
$191$			14.9706i			1.08323i	−0.840625	+	0.541617i	$0.817812\pi$
$192$		0			0
$193$		−	7.74399i		−	0.557424i	−0.960375	−	0.278712i	$-0.910092\pi$
$193$		−	7.74399i		−	0.557424i	0.960375	−	0.278712i	$-0.0899076\pi$
$194$		0			0
$195$	−3.12457			−0.223755
$196$		0			0
$197$	20.6892			1.47404			0.737022	−	0.675868i	$-0.236231\pi$
$197$	20.6892			1.47404			0.737022	+	0.675868i	$0.236231\pi$
$198$		0			0
$199$			14.1634i			1.00402i	0.864862	+	0.502009i	$0.167406\pi$
$199$			14.1634i			1.00402i	−0.864862	+	0.502009i	$0.832594\pi$
$200$		0			0
$201$	−6.41120			−0.452211
$202$		0			0
$203$			18.5694i			1.30332i
$204$		0			0
$205$		−	0.963261i		−	0.0672771i
$206$		0			0
$207$			0.876792i			0.0609413i
$208$		0			0
$209$			8.30291i			0.574325i
$210$		0			0
$211$	−5.77237			−0.397386			−0.198693	−	0.980062i	$-0.563670\pi$
$211$	−5.77237			−0.397386			−0.198693	+	0.980062i	$0.563670\pi$
$212$		0			0
$213$	17.5708			1.20393
$214$		0			0
$215$	0.775496			0.0528884
$216$		0			0
$217$		−	19.7574i		−	1.34122i
$218$		0			0
$219$	−30.4280			−2.05613
$220$		0			0
$221$	8.92845			0.600593
$222$		0			0
$223$	−21.6577			−1.45031			−0.725153	−	0.688588i	$-0.758231\pi$
$223$	−21.6577			−1.45031			−0.725153	+	0.688588i	$0.758231\pi$
$224$		0			0
$225$	2.70725			0.180483
$226$		0			0
$227$			6.79582i			0.451055i	0.974237	+	0.225527i	$0.0724105\pi$
$227$			6.79582i			0.451055i	−0.974237	+	0.225527i	$0.927590\pi$
$228$		0			0
$229$	−7.50114			−0.495689			−0.247845	−	0.968800i	$-0.579722\pi$
$229$	−7.50114			−0.495689			−0.247845	+	0.968800i	$0.579722\pi$
$230$		0			0
$231$	17.4247			1.14646
$232$		0			0
$233$	−12.0474			−0.789250			−0.394625	−	0.918842i	$-0.629125\pi$
$233$	−12.0474			−0.789250			−0.394625	+	0.918842i	$0.629125\pi$
$234$		0			0
$235$		−	2.65633i		−	0.173280i
$236$		0			0
$237$			17.3228i			1.12524i
$238$		0			0
$239$		−	5.86826i		−	0.379586i	−0.981824	−	0.189793i	$-0.939218\pi$
$239$		−	5.86826i		−	0.379586i	0.981824	−	0.189793i	$-0.0607818\pi$
$240$		0			0
$241$			22.4434i			1.44570i	0.691003	+	0.722852i	$0.257169\pi$
$241$			22.4434i			1.44570i	−0.691003	+	0.722852i	$0.742831\pi$
$242$		0			0
$243$	6.63377			0.425557
$244$		0			0
$245$			3.30934i			0.211426i
$246$		0			0
$247$	−5.33442			−0.339421
$248$		0			0
$249$	−12.8726			−0.815768
$250$		0			0
$251$		−	27.8178i		−	1.75584i	−0.478803	−	0.877922i	$-0.658929\pi$
$251$		−	27.8178i		−	1.75584i	0.478803	−	0.877922i	$-0.341071\pi$
$252$		0			0
$253$		−	3.77879i		−	0.237571i
$254$		0			0
$255$	8.63900			0.540995
$256$		0			0
$257$		−	0.0161162i		−	0.00100530i	−1.00000		0.000502650i	$-0.999840\pi$
$257$		−	0.0161162i		−	0.00100530i	1.00000		0.000502650i	$-0.000159998\pi$
$258$		0			0
$259$	−0.369231	+	19.8339i	−0.0229429	+	1.23242i
$260$		0			0
$261$			3.69533i			0.228735i
$262$		0			0
$263$	11.5573			0.712653			0.356327	−	0.934361i	$-0.384029\pi$
$263$	11.5573			0.712653			0.356327	+	0.934361i	$0.384029\pi$
$264$		0			0
$265$		−	5.19493i		−	0.319122i
$266$		0			0
$267$		−	8.14996i		−	0.498769i
$268$		0			0
$269$	17.0289			1.03827			0.519134	−	0.854693i	$-0.326255\pi$
$269$	17.0289			1.03827			0.519134	+	0.854693i	$0.326255\pi$
$270$		0			0
$271$	13.3309			0.809792			0.404896	−	0.914363i	$-0.367308\pi$
$271$	13.3309			0.809792			0.404896	+	0.914363i	$0.367308\pi$
$272$		0			0
$273$			11.1949i			0.677548i
$274$		0			0
$275$	−11.6677			−0.703588
$276$		0			0
$277$		−	12.1213i		−	0.728297i	−0.931341	−	0.364148i	$-0.881360\pi$
$277$		−	12.1213i		−	0.728297i	0.931341	−	0.364148i	$-0.118640\pi$
$278$		0			0
$279$		−	3.93175i		−	0.235388i
$280$		0			0
$281$		−	22.1700i		−	1.32255i	−0.750143	−	0.661276i	$-0.770015\pi$
$281$		−	22.1700i		−	1.32255i	0.750143	−	0.661276i	$-0.229985\pi$
$282$		0			0
$283$			1.80435i			0.107257i	0.998561	+	0.0536287i	$0.0170788\pi$
$283$			1.80435i			0.107257i	−0.998561	+	0.0536287i	$0.982921\pi$
$284$		0			0
$285$	−5.16149			−0.305740
$286$		0			0
$287$	−3.45124			−0.203720
$288$		0			0
$289$	−7.68590			−0.452112
$290$		0			0
$291$		−	20.3230i		−	1.19135i
$292$		0			0
$293$	−7.04497			−0.411572			−0.205786	−	0.978597i	$-0.565975\pi$
$293$	−7.04497			−0.411572			−0.205786	+	0.978597i	$0.565975\pi$
$294$		0			0
$295$	−0.312166			−0.0181750
$296$		0			0
$297$	−12.5613			−0.728882
$298$		0			0
$299$	2.42778			0.140402
$300$		0			0
$301$		−	2.77850i		−	0.160150i
$302$		0			0
$303$	−4.69186			−0.269540
$304$		0			0
$305$	−2.07960			−0.119078
$306$		0			0
$307$	28.5331			1.62847			0.814236	−	0.580535i	$-0.197156\pi$
$307$	28.5331			1.62847			0.814236	+	0.580535i	$0.197156\pi$
$308$		0			0
$309$		−	32.9576i		−	1.87489i
$310$		0			0
$311$		−	26.4668i		−	1.50079i	−0.660987	−	0.750397i	$-0.729862\pi$
$311$		−	26.4668i		−	1.50079i	0.660987	−	0.750397i	$-0.270138\pi$
$312$		0			0
$313$		−	12.6459i		−	0.714787i	−0.933954	−	0.357393i	$-0.883666\pi$
$313$		−	12.6459i		−	0.714787i	0.933954	−	0.357393i	$-0.116334\pi$
$314$		0			0
$315$			1.92652i			0.108547i
$316$		0			0
$317$	21.9209			1.23120			0.615599	−	0.788059i	$-0.288914\pi$
$317$	21.9209			1.23120			0.615599	+	0.788059i	$0.288914\pi$
$318$		0			0
$319$		−	15.9261i		−	0.891690i
$320$		0			0
$321$	−20.5646			−1.14781
$322$		0			0
$323$	14.7489			0.820652
$324$		0			0
$325$		−	7.49621i		−	0.415815i
$326$		0			0
$327$		−	25.4280i		−	1.40617i
$328$		0			0
$329$	−9.51726			−0.524703
$330$		0			0
$331$			6.71248i			0.368951i	0.982837	+	0.184476i	$0.0590587\pi$
$331$			6.71248i			0.368951i	−0.982837	+	0.184476i	$0.940941\pi$
$332$		0			0
$333$	−0.0734773	+	3.94697i	−0.00402653	+	0.216293i
$334$		0			0
$335$			3.05496i			0.166910i
$336$		0			0
$337$	5.30604			0.289038			0.144519	−	0.989502i	$-0.453836\pi$
$337$	5.30604			0.289038			0.144519	+	0.989502i	$0.453836\pi$
$338$		0			0
$339$		−	32.9228i		−	1.78812i
$340$		0			0
$341$			16.9450i			0.917625i
$342$		0			0
$343$	−10.9718			−0.592421
$344$		0			0
$345$	2.34908			0.126470
$346$		0			0
$347$		−	23.8205i		−	1.27875i	−0.768895	−	0.639375i	$-0.779193\pi$
$347$		−	23.8205i		−	1.27875i	0.768895	−	0.639375i	$-0.220807\pi$
$348$		0			0
$349$	−10.0000			−0.535288			−0.267644	−	0.963518i	$-0.586245\pi$
$349$	−10.0000			−0.535288			−0.267644	+	0.963518i	$0.586245\pi$
$350$		0			0
$351$		−	8.07035i		−	0.430764i
$352$		0			0
$353$		−	6.02392i		−	0.320621i	−0.987067	−	0.160310i	$-0.948750\pi$
$353$		−	6.02392i		−	0.320621i	0.987067	−	0.160310i	$-0.0512495\pi$
$354$		0			0
$355$		−	8.37253i		−	0.444368i
$356$		0			0
$357$		−	30.9524i		−	1.63817i
$358$		0			0
$359$	−28.1106			−1.48362			−0.741809	−	0.670611i	$-0.766032\pi$
$359$	−28.1106			−1.48362			−0.741809	+	0.670611i	$0.766032\pi$
$360$		0			0
$361$	10.1881			0.536214
$362$		0			0
$363$	6.06825			0.318500
$364$		0			0
$365$			14.4990i			0.758914i
$366$		0			0
$367$	−30.3430			−1.58389			−0.791945	−	0.610593i	$-0.790931\pi$
$367$	−30.3430			−1.58389			−0.791945	+	0.610593i	$0.790931\pi$
$368$		0			0
$369$	−0.686799			−0.0357533
$370$		0			0
$371$	−18.6127			−0.966324
$372$		0			0
$373$	−4.32354			−0.223864			−0.111932	−	0.993716i	$-0.535704\pi$
$373$	−4.32354			−0.223864			−0.111932	+	0.993716i	$0.535704\pi$
$374$		0			0
$375$		−	15.9470i		−	0.823498i
$376$		0			0
$377$	10.2321			0.526982
$378$		0			0
$379$	27.1025			1.39216			0.696081	−	0.717963i	$-0.254925\pi$
$379$	27.1025			1.39216			0.696081	+	0.717963i	$0.254925\pi$
$380$		0			0
$381$	24.2136			1.24050
$382$		0			0
$383$			27.1558i			1.38760i	0.720169	+	0.693799i	$0.244064\pi$
$383$			27.1558i			1.38760i	−0.720169	+	0.693799i	$0.755936\pi$
$384$		0			0
$385$		−	8.30291i		−	0.423156i
$386$		0			0
$387$		−	0.552924i		−	0.0281067i
$388$		0			0
$389$		−	8.21645i		−	0.416590i	−0.978066	−	0.208295i	$-0.933209\pi$
$389$		−	8.21645i		−	0.416590i	0.978066	−	0.208295i	$-0.0667915\pi$
$390$		0			0
$391$	−6.71248			−0.339465
$392$		0			0
$393$		−	28.1608i		−	1.42052i
$394$		0			0
$395$	8.25438			0.415323
$396$		0			0
$397$	−18.2297			−0.914924			−0.457462	−	0.889229i	$-0.651241\pi$
$397$	−18.2297			−0.914924			−0.457462	+	0.889229i	$0.651241\pi$
$398$		0			0
$399$			18.4929i			0.925803i
$400$		0			0
$401$			5.94385i			0.296822i	0.988926	+	0.148411i	$0.0474158\pi$
$401$			5.94385i			0.296822i	−0.988926	+	0.148411i	$0.952584\pi$
$402$		0			0
$403$	−10.8868			−0.542309
$404$		0			0
$405$		−	9.58092i		−	0.476080i
$406$		0			0
$407$	0.316672	−	17.0106i	0.0156968	−	0.843186i
$408$		0			0
$409$			4.13263i			0.204345i	0.994767	+	0.102173i	$0.0325794\pi$
$409$			4.13263i			0.204345i	−0.994767	+	0.102173i	$0.967421\pi$
$410$		0			0
$411$	20.8668			1.02928
$412$		0			0
$413$			1.11845i			0.0550352i
$414$		0			0
$415$			6.13384i			0.301098i
$416$		0			0
$417$	2.11741			0.103690
$418$		0			0
$419$	−7.00211			−0.342075			−0.171038	−	0.985265i	$-0.554712\pi$
$419$	−7.00211			−0.342075			−0.171038	+	0.985265i	$0.554712\pi$
$420$		0			0
$421$		−	32.9116i		−	1.60401i	−0.597315	−	0.802007i	$-0.703766\pi$
$421$		−	32.9116i		−	1.60401i	0.597315	−	0.802007i	$-0.296234\pi$
$422$		0			0
$423$	−1.89394			−0.0920867
$424$		0			0
$425$			20.7259i			1.00536i
$426$		0			0
$427$			7.45094i			0.360576i
$428$		0			0
$429$		−	9.60137i		−	0.463558i
$430$		0			0
$431$		−	23.1585i		−	1.11550i	−0.830007	−	0.557752i	$-0.811664\pi$
$431$		−	23.1585i		−	1.11550i	0.830007	−	0.557752i	$-0.188336\pi$
$432$		0			0
$433$	34.6096			1.66323			0.831616	−	0.555352i	$-0.187416\pi$
$433$	34.6096			1.66323			0.831616	+	0.555352i	$0.187416\pi$
$434$		0			0
$435$	9.90042			0.474689
$436$		0			0
$437$	4.01046			0.191846
$438$		0			0
$439$			3.46753i			0.165496i	0.996571	+	0.0827480i	$0.0263696\pi$
$439$			3.46753i			0.165496i	−0.996571	+	0.0827480i	$0.973630\pi$
$440$		0			0
$441$	2.35954			0.112359
$442$		0			0
$443$	−0.217169			−0.0103180			−0.00515901	−	0.999987i	$-0.501642\pi$
$443$	−0.217169			−0.0103180			−0.00515901	+	0.999987i	$0.501642\pi$
$444$		0			0
$445$	−3.88348			−0.184095
$446$		0			0
$447$	−7.51425			−0.355412
$448$		0			0
$449$		−	0.419430i		−	0.0197941i	−0.999951	−	0.00989707i	$-0.996850\pi$
$449$		−	0.419430i		−	0.0197941i	0.999951	−	0.00989707i	$-0.00315039\pi$
$450$		0			0
$451$	2.95996			0.139379
$452$		0			0
$453$	−4.86930			−0.228780
$454$		0			0
$455$	5.33442			0.250082
$456$		0			0
$457$		−	23.0528i		−	1.07836i	−0.842189	−	0.539182i	$-0.818734\pi$
$457$		−	23.0528i		−	1.07836i	0.842189	−	0.539182i	$-0.181266\pi$
$458$		0			0
$459$			22.3134i			1.04150i
$460$		0			0
$461$			36.4260i			1.69653i	0.529573	+	0.848265i	$0.322352\pi$
$461$			36.4260i			1.69653i	−0.529573	+	0.848265i	$0.677648\pi$
$462$		0			0
$463$			10.1104i			0.469870i	0.972011	+	0.234935i	$0.0754877\pi$
$463$			10.1104i			0.469870i	−0.972011	+	0.234935i	$0.924512\pi$
$464$		0			0
$465$	−10.5338			−0.488495
$466$		0			0
$467$			7.32877i			0.339135i	0.985519	+	0.169567i	$0.0542371\pi$
$467$			7.32877i			0.339135i	−0.985519	+	0.169567i	$0.945763\pi$
$468$		0			0
$469$	10.9455			0.505417
$470$		0			0
$471$	21.9002			1.00911
$472$		0			0
$473$			2.38299i			0.109570i
$474$		0			0
$475$		−	12.3830i		−	0.568170i
$476$		0			0
$477$	−3.70395			−0.169592
$478$		0			0
$479$			8.71860i			0.398363i	0.979963	+	0.199182i	$0.0638284\pi$
$479$			8.71860i			0.398363i	−0.979963	+	0.199182i	$0.936172\pi$
$480$		0			0
$481$	10.9289	+	0.203454i	0.498316	+	0.00927671i
$482$		0			0
$483$		−	8.41643i		−	0.382961i
$484$		0			0
$485$	−9.68397			−0.439726
$486$		0			0
$487$			16.8894i			0.765330i	0.923887	+	0.382665i	$0.124994\pi$
$487$			16.8894i			0.765330i	−0.923887	+	0.382665i	$0.875006\pi$
$488$		0			0
$489$		−	38.2504i		−	1.72974i
$490$		0			0
$491$	−29.2145			−1.31843			−0.659216	−	0.751953i	$-0.729112\pi$
$491$	−29.2145			−1.31843			−0.659216	+	0.751953i	$0.729112\pi$
$492$		0			0
$493$	−28.2904			−1.27414
$494$		0			0
$495$		−	1.65229i		−	0.0742648i
$496$		0			0
$497$	−29.9976			−1.34558
$498$		0			0
$499$			18.5168i			0.828927i	0.910066	+	0.414464i	$0.136031\pi$
$499$			18.5168i			0.828927i	−0.910066	+	0.414464i	$0.863969\pi$
$500$		0			0
$501$		−	18.7246i		−	0.836552i
$502$		0			0
$503$		−	10.4881i		−	0.467643i	−0.972279	−	0.233822i	$-0.924877\pi$
$503$		−	10.4881i		−	0.467643i	0.972279	−	0.233822i	$-0.0751232\pi$
$504$		0			0
$505$			2.23569i			0.0994867i
$506$		0			0
$507$	−18.6644			−0.828914
$508$		0			0
$509$	−32.9530			−1.46062			−0.730308	−	0.683118i	$-0.760624\pi$
$509$	−32.9530			−1.46062			−0.730308	+	0.683118i	$0.760624\pi$
$510$		0			0
$511$	51.9481			2.29805
$512$		0			0
$513$		−	13.3314i		−	0.588597i
$514$		0			0
$515$	−15.7044			−0.692019
$516$		0			0
$517$	8.16251			0.358987
$518$		0			0
$519$	−2.35277			−0.103275
$520$		0			0
$521$	4.56223			0.199875			0.0999374	−	0.994994i	$-0.468136\pi$
$521$	4.56223			0.199875			0.0999374	+	0.994994i	$0.468136\pi$
$522$		0			0
$523$			42.7259i			1.86827i	0.356914	+	0.934137i	$0.383829\pi$
$523$			42.7259i			1.86827i	−0.356914	+	0.934137i	$0.616171\pi$
$524$		0			0
$525$	−25.9872			−1.13417
$526$		0			0
$527$	30.1004			1.31119
$528$		0			0
$529$	21.1748			0.920642
$530$		0			0
$531$			0.222572i			0.00965881i
$532$		0			0
$533$			1.90171i			0.0823720i
$534$		0			0
$535$			9.79912i			0.423653i
$536$		0			0
$537$		−	35.6048i		−	1.53646i
$538$		0			0
$539$	−10.1691			−0.438015
$540$		0			0
$541$		−	0.654220i		−	0.0281271i	−0.999901	−	0.0140636i	$-0.995523\pi$
$541$		−	0.654220i		−	0.0281271i	0.999901	−	0.0140636i	$-0.00447672\pi$
$542$		0			0
$543$	−1.12475			−0.0482675
$544$		0			0
$545$	−12.1165			−0.519015
$546$		0			0
$547$		−	44.6068i		−	1.90725i	−0.301001	−	0.953624i	$-0.597321\pi$
$547$		−	44.6068i		−	1.90725i	0.301001	−	0.953624i	$-0.402679\pi$
$548$		0			0
$549$			1.48274i			0.0632820i
$550$		0			0
$551$	16.9025			0.720069
$552$		0			0
$553$		−	29.5743i		−	1.25763i
$554$		0			0
$555$	10.5746	+	0.196858i	0.448867	+	0.00835617i
$556$		0			0
$557$		−	33.9687i		−	1.43930i	−0.694338	−	0.719649i	$-0.744303\pi$
$557$		−	33.9687i		−	1.43930i	0.694338	−	0.719649i	$-0.255697\pi$
$558$		0			0
$559$	−1.53101			−0.0647549
$560$		0			0
$561$			26.5464i			1.12079i
$562$		0			0
$563$		−	34.8732i		−	1.46973i	−0.678212	−	0.734866i	$-0.737245\pi$
$563$		−	34.8732i		−	1.46973i	0.678212	−	0.734866i	$-0.262755\pi$
$564$		0			0
$565$	−15.6878			−0.659992
$566$		0			0
$567$	−34.3271			−1.44160
$568$		0			0
$569$			37.3088i			1.56407i	0.623237	+	0.782033i	$0.285817\pi$
$569$			37.3088i			1.56407i	−0.623237	+	0.782033i	$0.714183\pi$
$570$		0			0
$571$	−38.0372			−1.59181			−0.795904	−	0.605423i	$-0.793004\pi$
$571$	−38.0372			−1.59181			−0.795904	+	0.605423i	$0.793004\pi$
$572$		0			0
$573$		−	28.5973i		−	1.19467i
$574$		0			0
$575$			5.63570i			0.235025i
$576$		0			0
$577$		−	20.9993i		−	0.874211i	−0.899410	−	0.437106i	$-0.856004\pi$
$577$		−	20.9993i		−	0.874211i	0.899410	−	0.437106i	$-0.143996\pi$
$578$		0			0
$579$			14.7928i			0.614769i
$580$		0			0
$581$	21.9767			0.911748
$582$		0			0
$583$	15.9633			0.661131
$584$		0			0
$585$	1.06156			0.0438899
$586$		0			0
$587$			27.7345i			1.14472i	0.820001	+	0.572362i	$0.193973\pi$
$587$			27.7345i			1.14472i	−0.820001	+	0.572362i	$0.806027\pi$
$588$		0			0
$589$	−17.9839			−0.741013
$590$		0			0
$591$	−39.5212			−1.62568
$592$		0			0
$593$	39.7067			1.63056			0.815279	−	0.579068i	$-0.196583\pi$
$593$	39.7067			1.63056			0.815279	+	0.579068i	$0.196583\pi$
$594$		0			0
$595$	−14.7489			−0.604647
$596$		0			0
$597$		−	27.0554i		−	1.10731i
$598$		0			0
$599$	28.5431			1.16624			0.583120	−	0.812386i	$-0.301832\pi$
$599$	28.5431			1.16624			0.583120	+	0.812386i	$0.301832\pi$
$600$		0			0
$601$	16.5912			0.676769			0.338385	−	0.941008i	$-0.390120\pi$
$601$	16.5912			0.676769			0.338385	+	0.941008i	$0.390120\pi$
$602$		0			0
$603$	2.17817			0.0887018
$604$		0			0
$605$		−	2.89154i		−	0.117558i
$606$		0			0
$607$		−	4.78283i		−	0.194129i	−0.995278	−	0.0970646i	$-0.969055\pi$
$607$		−	4.78283i		−	0.194129i	0.995278	−	0.0970646i	$-0.0309453\pi$
$608$		0			0
$609$		−	35.4719i		−	1.43739i
$610$		0			0
$611$			5.24421i			0.212158i
$612$		0			0
$613$	−28.3666			−1.14572			−0.572858	−	0.819655i	$-0.694165\pi$
$613$	−28.3666			−1.14572			−0.572858	+	0.819655i	$0.694165\pi$
$614$		0			0
$615$			1.84005i			0.0741981i
$616$		0			0
$617$	13.9034			0.559729			0.279864	−	0.960040i	$-0.409711\pi$
$617$	13.9034			0.559729			0.279864	+	0.960040i	$0.409711\pi$
$618$		0			0
$619$	22.4541			0.902505			0.451253	−	0.892396i	$-0.350977\pi$
$619$	22.4541			0.902505			0.451253	+	0.892396i	$0.350977\pi$
$620$		0			0
$621$			6.06735i			0.243474i
$622$		0			0
$623$			13.9140i			0.557453i
$624$		0			0
$625$	13.2586			0.530343
$626$		0			0
$627$		−	15.8605i		−	0.633407i
$628$		0			0
$629$	−30.2169	−	0.562522i	−1.20483	−	0.0224292i
$630$		0			0
$631$			19.8239i			0.789178i	0.918858	+	0.394589i	$0.129113\pi$
$631$			19.8239i			0.789178i	−0.918858	+	0.394589i	$0.870887\pi$
$632$		0			0
$633$	11.0266			0.438267
$634$		0			0
$635$		−	11.5379i		−	0.457867i
$636$		0			0
$637$		−	6.53341i		−	0.258863i
$638$		0			0
$639$	−5.96956			−0.236152
$640$		0			0
$641$	−18.6096			−0.735035			−0.367517	−	0.930017i	$-0.619792\pi$
$641$	−18.6096			−0.735035			−0.367517	+	0.930017i	$0.619792\pi$
$642$		0			0
$643$			13.0142i			0.513229i	0.966514	+	0.256615i	$0.0826071\pi$
$643$			13.0142i			0.513229i	−0.966514	+	0.256615i	$0.917393\pi$
$644$		0			0
$645$	−1.48138			−0.0583292
$646$		0			0
$647$		−	45.0225i		−	1.77002i	−0.465576	−	0.885008i	$-0.654153\pi$
$647$		−	45.0225i		−	1.77002i	0.465576	−	0.885008i	$-0.345847\pi$
$648$		0			0
$649$		−	0.959240i		−	0.0376535i
$650$		0			0
$651$			37.7413i			1.47920i
$652$		0			0
$653$		−	6.43867i		−	0.251965i	−0.992032	−	0.125982i	$-0.959792\pi$
$653$		−	6.43867i		−	0.251965i	0.992032	−	0.125982i	$-0.0402083\pi$
$654$		0			0
$655$	−13.4187			−0.524312
$656$		0			0
$657$	10.3377			0.403313
$658$		0			0
$659$	47.4625			1.84888			0.924438	−	0.381331i	$-0.124534\pi$
$659$	47.4625			1.84888			0.924438	+	0.381331i	$0.124534\pi$
$660$		0			0
$661$			13.0845i			0.508930i	0.967082	+	0.254465i	$0.0818993\pi$
$661$			13.0845i			0.508930i	−0.967082	+	0.254465i	$0.918101\pi$
$662$		0			0
$663$	−17.0554			−0.662378
$664$		0			0
$665$	8.81194			0.341712
$666$		0			0
$667$	−7.69259			−0.297858
$668$		0			0
$669$	41.3712			1.59950
$670$		0			0
$671$		−	6.39032i		−	0.246696i
$672$		0			0
$673$	37.4652			1.44418			0.722088	−	0.691801i	$-0.243183\pi$
$673$	37.4652			1.44418			0.722088	+	0.691801i	$0.243183\pi$
$674$		0			0
$675$	18.7340			0.721072
$676$		0			0
$677$	27.3321			1.05046			0.525228	−	0.850961i	$-0.323980\pi$
$677$	27.3321			1.05046			0.525228	+	0.850961i	$0.323980\pi$
$678$		0			0
$679$			34.6964i			1.33152i
$680$		0			0
$681$		−	12.9816i		−	0.497456i
$682$		0			0
$683$		−	8.55627i		−	0.327397i	−0.986510	−	0.163698i	$-0.947658\pi$
$683$		−	8.55627i		−	0.327397i	0.986510	−	0.163698i	$-0.0523424\pi$
$684$		0			0
$685$		−	9.94311i		−	0.379907i
$686$		0			0
$687$	14.3289			0.546683
$688$		0			0
$689$			10.2560i			0.390723i
$690$		0			0
$691$	−48.6190			−1.84955			−0.924777	−	0.380509i	$-0.875749\pi$
$691$	−48.6190			−1.84955			−0.924777	+	0.380509i	$0.875749\pi$
$692$		0			0
$693$	−5.91993			−0.224879
$694$		0			0
$695$		−	1.00895i		−	0.0382718i
$696$		0			0
$697$		−	5.25794i		−	0.199159i
$698$		0			0
$699$	23.0133			0.870442
$700$		0			0
$701$			40.5485i			1.53150i	0.643141	+	0.765748i	$0.277631\pi$
$701$			40.5485i			1.53150i	−0.643141	+	0.765748i	$0.722369\pi$
$702$		0			0
$703$	18.0535	+	0.336086i	0.680901	+	0.0126757i
$704$		0			0
$705$			5.07420i			0.191105i
$706$		0			0
$707$	8.01016			0.301253
$708$		0			0
$709$		−	5.70185i		−	0.214137i	−0.994252	−	0.107069i	$-0.965854\pi$
$709$		−	5.70185i		−	0.214137i	0.994252	−	0.107069i	$-0.0341465\pi$
$710$		0			0
$711$		−	5.88532i		−	0.220717i
$712$		0			0
$713$	8.18476			0.306522
$714$		0			0
$715$	−4.57509			−0.171099
$716$		0			0
$717$			11.2098i			0.418636i
$718$		0			0
$719$	−18.2635			−0.681114			−0.340557	−	0.940224i	$-0.610616\pi$
$719$	−18.2635			−0.681114			−0.340557	+	0.940224i	$0.610616\pi$
$720$		0			0
$721$			56.2668i			2.09549i
$722$		0			0
$723$		−	42.8720i		−	1.59443i
$724$		0			0
$725$			23.7522i			0.882135i
$726$		0			0
$727$		−	17.9391i		−	0.665324i	−0.943046	−	0.332662i	$-0.892053\pi$
$727$		−	17.9391i		−	0.665324i	0.943046	−	0.332662i	$-0.107947\pi$
$728$		0			0
$729$	18.9053			0.700196
$730$		0			0
$731$	4.23303			0.156564
$732$		0			0
$733$	−42.1401			−1.55648			−0.778241	−	0.627966i	$-0.783888\pi$
$733$	−42.1401			−1.55648			−0.778241	+	0.627966i	$0.783888\pi$
$734$		0			0
$735$		−	6.32161i		−	0.233176i
$736$		0			0
$737$	−9.38745			−0.345791
$738$		0			0
$739$	10.0950			0.371350			0.185675	−	0.982611i	$-0.440553\pi$
$739$	10.0950			0.371350			0.185675	+	0.982611i	$0.440553\pi$
$740$		0			0
$741$	10.1900			0.374339
$742$		0			0
$743$	−21.0551			−0.772438			−0.386219	−	0.922407i	$-0.626219\pi$
$743$	−21.0551			−0.772438			−0.386219	+	0.922407i	$0.626219\pi$
$744$		0			0
$745$			3.58057i			0.131182i
$746$		0			0
$747$	4.37339			0.160014
$748$		0			0
$749$	35.1089			1.28285
$750$		0			0
$751$	28.6435			1.04522			0.522608	−	0.852573i	$-0.324959\pi$
$751$	28.6435			1.04522			0.522608	+	0.852573i	$0.324959\pi$
$752$		0			0
$753$			53.1385i			1.93647i
$754$		0			0
$755$			2.32024i			0.0844422i
$756$		0			0
$757$		−	32.8981i		−	1.19570i	−0.801608	−	0.597850i	$-0.796022\pi$
$757$		−	32.8981i		−	1.19570i	0.801608	−	0.597850i	$-0.203978\pi$
$758$		0			0
$759$			7.21838i			0.262011i
$760$		0			0
$761$	−28.2998			−1.02587			−0.512934	−	0.858428i	$-0.671441\pi$
$761$	−28.2998			−1.02587			−0.512934	+	0.858428i	$0.671441\pi$
$762$		0			0
$763$			43.4118i			1.57161i
$764$		0			0
$765$	−2.93505			−0.106117
$766$		0			0
$767$	0.616289			0.0222529
$768$		0			0
$769$		−	50.9189i		−	1.83618i	−0.396368	−	0.918092i	$-0.629729\pi$
$769$		−	50.9189i		−	1.83618i	0.396368	−	0.918092i	$-0.370271\pi$
$770$		0			0
$771$			0.0307857i			0.00110872i
$772$		0			0
$773$	−50.0407			−1.79984			−0.899919	−	0.436057i	$-0.856375\pi$
$773$	−50.0407			−1.79984			−0.899919	+	0.436057i	$0.856375\pi$
$774$		0			0
$775$		−	25.2719i		−	0.907793i
$776$		0			0
$777$	0.705317	−	37.8874i	0.0253031	−	1.35920i
$778$		0			0
$779$			3.14143i			0.112553i
$780$		0			0
$781$	25.7276			0.920605
$782$		0			0
$783$			25.5714i			0.913849i
$784$		0			0
$785$		−	10.4355i		−	0.372461i
$786$		0			0
$787$	29.2290			1.04190			0.520951	−	0.853587i	$-0.325577\pi$
$787$	29.2290			1.04190			0.520951	+	0.853587i	$0.325577\pi$
$788$		0			0
$789$	−22.0771			−0.785966
$790$		0			0
$791$			56.2074i			1.99850i
$792$		0			0
$793$	4.10563			0.145795
$794$		0			0
$795$			9.92352i			0.351951i
$796$		0			0
$797$		−	21.1819i		−	0.750303i	−0.926964	−	0.375151i	$-0.877591\pi$
$797$		−	21.1819i		−	0.750303i	0.926964	−	0.375151i	$-0.122409\pi$
$798$		0			0
$799$		−	14.4995i		−	0.512956i
$800$		0			0
$801$			2.76890i			0.0978343i
$802$		0			0
$803$	−44.5534			−1.57226
$804$		0			0
$805$	−4.01046			−0.141350
$806$		0			0
$807$	−32.5291			−1.14508
$808$		0			0
$809$		−	22.4021i		−	0.787616i	−0.919193	−	0.393808i	$-0.871157\pi$
$809$		−	22.4021i		−	0.787616i	0.919193	−	0.393808i	$-0.128843\pi$
$810$		0			0
$811$	−23.9788			−0.842011			−0.421005	−	0.907058i	$-0.638323\pi$
$811$	−23.9788			−0.842011			−0.421005	+	0.907058i	$0.638323\pi$
$812$		0			0
$813$	−25.4650			−0.893098
$814$		0			0
$815$	−18.2264			−0.638444
$816$		0			0
$817$	−2.52908			−0.0884813
$818$		0			0
$819$		−	3.80341i		−	0.132902i
$820$		0			0
$821$	30.8891			1.07804			0.539018	−	0.842294i	$-0.318796\pi$
$821$	30.8891			1.07804			0.539018	+	0.842294i	$0.318796\pi$
$822$		0			0
$823$	38.3576			1.33706			0.668531	−	0.743684i	$-0.266923\pi$
$823$	38.3576			1.33706			0.668531	+	0.743684i	$0.266923\pi$
$824$		0			0
$825$	22.2880			0.775968
$826$		0			0
$827$		−	42.8395i		−	1.48967i	−0.667247	−	0.744837i	$-0.732527\pi$
$827$		−	42.8395i		−	1.48967i	0.667247	−	0.744837i	$-0.267473\pi$
$828$		0			0
$829$			18.8390i			0.654305i	0.944972	+	0.327152i	$0.106089\pi$
$829$			18.8390i			0.654305i	−0.944972	+	0.327152i	$0.893911\pi$
$830$		0			0
$831$			23.1545i			0.803219i
$832$		0			0
$833$			18.0640i			0.625879i
$834$		0			0
$835$	−8.92233			−0.308770
$836$		0			0
$837$		−	27.2075i		−	0.940428i
$838$		0			0
$839$	−12.9754			−0.447959			−0.223980	−	0.974594i	$-0.571905\pi$
$839$	−12.9754			−0.447959			−0.223980	+	0.974594i	$0.571905\pi$
$840$		0			0
$841$	−3.42119			−0.117972
$842$		0			0
$843$			42.3499i			1.45861i
$844$		0			0
$845$			8.89365i			0.305951i
$846$		0			0
$847$	−10.3600			−0.355974
$848$		0			0
$849$		−	3.44673i		−	0.118291i
$850$		0			0
$851$	−8.21645	−	0.152958i	−0.281656	−	0.00524334i
$852$		0			0
$853$		−	39.3363i		−	1.34685i	−0.739257	−	0.673424i	$-0.764823\pi$
$853$		−	39.3363i		−	1.34685i	0.739257	−	0.673424i	$-0.235177\pi$
$854$		0			0
$855$	1.75358			0.0599713
$856$		0			0
$857$			31.2096i			1.06610i	0.846084	+	0.533050i	$0.178954\pi$
$857$			31.2096i			1.06610i	−0.846084	+	0.533050i	$0.821046\pi$
$858$		0			0
$859$			2.16221i			0.0737736i	0.999319	+	0.0368868i	$0.0117441\pi$
$859$			2.16221i			0.0737736i	−0.999319	+	0.0368868i	$0.988256\pi$
$860$		0			0
$861$	6.59266			0.224677
$862$		0			0
$863$	−26.3803			−0.897997			−0.448998	−	0.893533i	$-0.648219\pi$
$863$	−26.3803			−0.897997			−0.448998	+	0.893533i	$0.648219\pi$
$864$		0			0
$865$			1.12110i			0.0381186i
$866$		0			0
$867$	14.6819			0.498622
$868$		0			0
$869$			25.3645i			0.860432i
$870$		0			0
$871$		−	6.03121i		−	0.204360i
$872$		0			0
$873$			6.90461i			0.233686i
$874$		0			0
$875$			27.2254i			0.920388i
$876$		0			0
$877$	32.4230			1.09485			0.547424	−	0.836855i	$-0.315609\pi$
$877$	32.4230			1.09485			0.547424	+	0.836855i	$0.315609\pi$
$878$		0			0
$879$	13.4575			0.453911
$880$		0			0
$881$	−38.3448			−1.29187			−0.645934	−	0.763393i	$-0.723532\pi$
$881$	−38.3448			−1.29187			−0.645934	+	0.763393i	$0.723532\pi$
$882$		0			0
$883$			27.8047i			0.935703i	0.883807	+	0.467851i	$0.154972\pi$
$883$			27.8047i			0.935703i	−0.883807	+	0.467851i	$0.845028\pi$
$884$		0			0
$885$	0.596309			0.0200447
$886$		0			0
$887$	50.5700			1.69798			0.848988	−	0.528413i	$-0.177213\pi$
$887$	50.5700			1.69798			0.848988	+	0.528413i	$0.177213\pi$
$888$		0			0
$889$	−41.3387			−1.38645
$890$		0			0
$891$	29.4408			0.986303
$892$		0			0
$893$			8.66292i			0.289894i
$894$		0			0
$895$	−16.9658			−0.567105
$896$		0			0
$897$	−4.63763			−0.154846
$898$		0			0
$899$	34.4955			1.15049
$900$		0			0
$901$		−	28.3564i		−	0.944689i
$902$		0			0
$903$			5.30758i			0.176625i
$904$		0			0
$905$			0.535946i			0.0178155i
$906$		0			0
$907$		−	45.0193i		−	1.49484i	−0.664350	−	0.747421i	$-0.731292\pi$
$907$		−	45.0193i		−	1.49484i	0.664350	−	0.747421i	$-0.268708\pi$
$908$		0			0
$909$	1.59403			0.0528707
$910$		0			0
$911$		−	21.4683i		−	0.711275i	−0.934624	−	0.355638i	$-0.884264\pi$
$911$		−	21.4683i		−	0.711275i	0.934624	−	0.355638i	$-0.115736\pi$
$912$		0			0
$913$	−18.8484			−0.623791
$914$		0			0
$915$	3.97253			0.131328
$916$		0			0
$917$			48.0774i			1.58766i
$918$		0			0
$919$			36.2560i			1.19598i	0.801505	+	0.597988i	$0.204033\pi$
$919$			36.2560i			1.19598i	−0.801505	+	0.597988i	$0.795967\pi$
$920$		0			0
$921$	−54.5049			−1.79600
$922$		0			0
$923$			16.5294i			0.544070i
$924$		0			0
$925$	−0.472285	+	25.3697i	−0.0155286	+	0.834151i
$926$		0			0
$927$			11.1972i			0.367763i
$928$		0			0
$929$	36.9772			1.21318			0.606591	−	0.795014i	$-0.292537\pi$
$929$	36.9772			1.21318			0.606591	+	0.795014i	$0.292537\pi$
$930$		0			0
$931$		−	10.7926i		−	0.353712i
$932$		0			0
$933$			50.5578i			1.65519i
$934$		0			0
$935$	12.6495			0.413682
$936$		0			0
$937$	18.1098			0.591622			0.295811	−	0.955246i	$-0.404410\pi$
$937$	18.1098			0.591622			0.295811	+	0.955246i	$0.404410\pi$
$938$		0			0
$939$			24.1566i			0.788319i
$940$		0			0
$941$	16.7271			0.545289			0.272645	−	0.962115i	$-0.412102\pi$
$941$	16.7271			0.545289			0.272645	+	0.962115i	$0.412102\pi$
$942$		0			0
$943$		−	1.42972i		−	0.0465580i
$944$		0			0
$945$			13.3314i			0.433671i
$946$		0			0
$947$			30.1461i			0.979616i	0.871830	+	0.489808i	$0.162933\pi$
$947$			30.1461i			0.979616i	−0.871830	+	0.489808i	$0.837067\pi$
$948$		0			0
$949$		−	28.6245i		−	0.929191i
$950$		0			0
$951$	−41.8740			−1.35786
$952$		0			0
$953$	19.9223			0.645348			0.322674	−	0.946510i	$-0.395418\pi$
$953$	19.9223			0.645348			0.322674	+	0.946510i	$0.395418\pi$
$954$		0			0
$955$	−13.6267			−0.440951
$956$		0			0
$957$			30.4226i			0.983422i
$958$		0			0
$959$	−35.6248			−1.15039
$960$		0			0
$961$	−5.70249			−0.183951
$962$		0			0
$963$	6.98671			0.225144
$964$		0			0
$965$	7.04883			0.226910
$966$		0			0
$967$		−	46.4490i		−	1.49370i	−0.664992	−	0.746850i	$-0.731565\pi$
$967$		−	46.4490i		−	1.49370i	0.664992	−	0.746850i	$-0.268435\pi$
$968$		0			0
$969$	−28.1739			−0.905075
$970$		0			0
$971$	47.1179			1.51209			0.756043	−	0.654522i	$-0.227130\pi$
$971$	47.1179			1.51209			0.756043	+	0.654522i	$0.227130\pi$
$972$		0			0
$973$	−3.61495			−0.115890
$974$		0			0
$975$			14.3195i			0.458591i
$976$		0			0
$977$		−	35.3150i		−	1.12983i	−0.825150	−	0.564914i	$-0.808909\pi$
$977$		−	35.3150i		−	1.12983i	0.825150	−	0.564914i	$-0.191091\pi$
$978$		0			0
$979$		−	11.9334i		−	0.381393i
$980$		0			0
$981$			8.63900i			0.275822i
$982$		0			0
$983$	−20.2641			−0.646325			−0.323162	−	0.946344i	$-0.604746\pi$
$983$	−20.2641			−0.646325			−0.323162	+	0.946344i	$0.604746\pi$
$984$		0			0
$985$			18.8320i			0.600037i
$986$		0			0
$987$	18.1802			0.578681
$988$		0			0
$989$	1.15103			0.0366005
$990$		0			0
$991$			22.2138i			0.705644i	0.935690	+	0.352822i	$0.114778\pi$
$991$			22.2138i			0.705644i	−0.935690	+	0.352822i	$0.885222\pi$
$992$		0			0
$993$		−	12.8224i		−	0.406907i
$994$		0			0
$995$	−12.8920			−0.408704
$996$		0			0
$997$			38.5921i			1.22222i	0.791544	+	0.611112i	$0.209278\pi$
$997$			38.5921i			1.22222i	−0.791544	+	0.611112i	$0.790722\pi$
$998$		0			0
$999$	−0.508458	+	27.3128i	−0.0160869	+	0.864139i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.g.m.961.4		8
4.3	odd	2		2368.2.g.n.961.6		8
8.3	odd	2		1184.2.g.g.961.3	✓	8
8.5	even	2		1184.2.g.h.961.5	yes	8
37.36	even	2	inner	2368.2.g.m.961.3		8
148.147	odd	2		2368.2.g.n.961.5		8
296.147	odd	2		1184.2.g.g.961.4	yes	8
296.221	even	2		1184.2.g.h.961.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.g.g.961.3	✓	8	8.3	odd	2
1184.2.g.g.961.4	yes	8	296.147	odd	2
1184.2.g.h.961.5	yes	8	8.5	even	2
1184.2.g.h.961.6	yes	8	296.221	even	2
2368.2.g.m.961.3		8	37.36	even	2	inner
2368.2.g.m.961.4		8	1.1	even	1	trivial
2368.2.g.n.961.5		8	148.147	odd	2
2368.2.g.n.961.6		8	4.3	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:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.91023 q^{3} +0.910233i q^{5} +3.26124 q^{7} +0.648991 q^{9} -2.79702 q^{11} -1.79702i q^{13} -1.73876i q^{15} +4.96849i q^{17} -2.96849i q^{19} -6.22973 q^{21} +1.35101i q^{23} +4.17148 q^{25} +4.49098 q^{27} +5.69396i q^{29} -6.05826i q^{31} +5.34295 q^{33} +2.96849i q^{35} +(-0.113218 + 6.08171i) q^{37} +3.43272i q^{39} -1.05826 q^{41} -0.851975i q^{43} +0.590733i q^{45} -2.91829 q^{47} +3.63570 q^{49} -9.49098i q^{51} -5.70725 q^{53} -2.54594i q^{55} +5.67051i q^{57} +0.342951i q^{59} +2.28469i q^{61} +2.11652 q^{63} +1.63570 q^{65} +3.35624 q^{67} -2.58074i q^{69} -9.19822 q^{71} +15.9289 q^{73} -7.96849 q^{75} -9.12175 q^{77} -9.06842i q^{79} -10.5258 q^{81} +6.73876 q^{83} -4.52249 q^{85} -10.8768i q^{87} +4.26647i q^{89} -5.86050i q^{91} +11.5727i q^{93} +2.70202 q^{95} +10.6390i q^{97} -1.81524 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} + 12 q^{7} + 10 q^{9} - 2 q^{11} + 8 q^{21} + 10 q^{25} - 36 q^{27} - 4 q^{33} - 12 q^{37} + 26 q^{41} - 56 q^{47} + 12 q^{49} - 16 q^{53} - 52 q^{63} - 4 q^{65} + 2 q^{67} + 28 q^{71} - 2 q^{73}+ \cdots - 16 q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 + 12 * q^7 + 10 * q^9 - 2 * q^11 + 8 * q^21 + 10 * q^25 - 36 * q^27 - 4 * q^33 - 12 * q^37 + 26 * q^41 - 56 * q^47 + 12 * q^49 - 16 * q^53 - 52 * q^63 - 4 * q^65 + 2 * q^67 + 28 * q^71 - 2 * q^73 - 20 * q^75 + 16 * q^77 + 24 * q^81 + 68 * q^83 - 8 * q^85 + 12 * q^95 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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