LMFDB - Embedded newform 464.3.h.c.463.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [464,3,Mod(463,464)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(464, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("464.463");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 464 = 2^{4} \cdot 29 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	464.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.6430842663$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 58 x^{18} + 2256 x^{16} + 48540 x^{14} + 757617 x^{12} + 7081908 x^{10} + 46200312 x^{8} + \cdots + 529984 $ x^20 + 58x^18 + 2256x^16 + 48540x^14 + 757617x^12 + 7081908x^10 + 46200312x^8 + 108855714x^6 + 191707065x^4 + 10229128*x^2 + 529984
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{39}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			463.14
Root			$0.798572 - 1.38317i$ of defining polynomial
Character	$\chi$	$=$	464.463
Dual form			464.3.h.c.463.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.59714 q^{3} -4.84550 q^{5} +1.34200i q^{7} -6.44913 q^{9} +O(q^{10})$	$q+1.59714 q^{3} -4.84550 q^{5} +1.34200i q^{7} -6.44913 q^{9} +7.78881 q^{11} +11.8362 q^{13} -7.73896 q^{15} +14.3222i q^{17} -33.3031 q^{19} +2.14337i q^{21} +36.0872i q^{23} -1.52109 q^{25} -24.6745 q^{27} +(-10.3151 + 27.1035i) q^{29} -34.0106 q^{31} +12.4398 q^{33} -6.50267i q^{35} +70.2553i q^{37} +18.9041 q^{39} -63.3016i q^{41} -20.7303 q^{43} +31.2493 q^{45} -42.5246 q^{47} +47.1990 q^{49} +22.8745i q^{51} +62.6841 q^{53} -37.7407 q^{55} -53.1898 q^{57} -56.7687i q^{59} +25.2148i q^{61} -8.65474i q^{63} -57.3524 q^{65} +97.3696i q^{67} +57.6364i q^{69} -15.3545i q^{71} -6.08230i q^{73} -2.42940 q^{75} +10.4526i q^{77} +28.1831 q^{79} +18.6335 q^{81} -86.8954i q^{83} -69.3981i q^{85} +(-16.4747 + 43.2881i) q^{87} -156.628i q^{89} +15.8842i q^{91} -54.3198 q^{93} +161.370 q^{95} +105.767i q^{97} -50.2311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 8 q^{5} + 52 q^{9}+O(q^{10}) $ 20 * q - 8 * q^5 + 52 * q^9	$ 20 q - 8 q^{5} + 52 q^{9} - 16 q^{13} + 12 q^{25} + 4 q^{29} - 96 q^{33} - 112 q^{45} - 76 q^{49} - 112 q^{53} + 120 q^{57} + 136 q^{65} - 52 q^{81}+O(q^{100}) $ 20 * q - 8 * q^5 + 52 * q^9 - 16 * q^13 + 12 * q^25 + 4 * q^29 - 96 * q^33 - 112 * q^45 - 76 * q^49 - 112 * q^53 + 120 * q^57 + 136 * q^65 - 52 * q^81

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/464\mathbb{Z}\right)^\times$.

$n$	$117$	$175$	$321$
$\chi(n)$	$1$	$-1$	$-1$

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
$3$	1.59714			0.532381			0.266191	−	0.963920i	$-0.414235\pi$
$3$	1.59714			0.532381			0.266191	+	0.963920i	$0.414235\pi$
$4$		0			0
$5$	−4.84550			−0.969101			−0.484550	−	0.874763i	$-0.661017\pi$
$5$	−4.84550			−0.969101			−0.484550	+	0.874763i	$0.661017\pi$
$6$		0			0
$7$			1.34200i			0.191714i	0.995395	+	0.0958571i	$0.0305592\pi$
$7$			1.34200i			0.191714i	−0.995395	+	0.0958571i	$0.969441\pi$
$8$		0			0
$9$	−6.44913			−0.716570
$10$		0			0
$11$	7.78881			0.708074			0.354037	−	0.935231i	$-0.384809\pi$
$11$	7.78881			0.708074			0.354037	+	0.935231i	$0.384809\pi$
$12$		0			0
$13$	11.8362			0.910478			0.455239	−	0.890369i	$-0.349554\pi$
$13$	11.8362			0.910478			0.455239	+	0.890369i	$0.349554\pi$
$14$		0			0
$15$	−7.73896			−0.515931
$16$		0			0
$17$			14.3222i			0.842480i	0.906949	+	0.421240i	$0.138405\pi$
$17$			14.3222i			0.842480i	−0.906949	+	0.421240i	$0.861595\pi$
$18$		0			0
$19$	−33.3031			−1.75279			−0.876396	−	0.481591i	$-0.840059\pi$
$19$	−33.3031			−1.75279			−0.876396	+	0.481591i	$0.840059\pi$
$20$		0			0
$21$			2.14337i			0.102065i
$22$		0			0
$23$			36.0872i			1.56901i	0.620123	+	0.784504i	$0.287083\pi$
$23$			36.0872i			1.56901i	−0.620123	+	0.784504i	$0.712917\pi$
$24$		0			0
$25$	−1.52109			−0.0608437
$26$		0			0
$27$	−24.6745			−0.913870
$28$		0			0
$29$	−10.3151	+	27.1035i	−0.355694	+	0.934602i
$29$	−10.3151	+	27.1035i	−0.355694	+	0.934602i
$30$		0			0
$31$	−34.0106			−1.09712			−0.548558	−	0.836113i	$-0.684823\pi$
$31$	−34.0106			−1.09712			−0.548558	+	0.836113i	$0.684823\pi$
$32$		0			0
$33$	12.4398			0.376965
$34$		0			0
$35$		−	6.50267i		−	0.185790i
$36$		0			0
$37$			70.2553i			1.89879i	0.314081	+	0.949396i	$0.398304\pi$
$37$			70.2553i			1.89879i	−0.314081	+	0.949396i	$0.601696\pi$
$38$		0			0
$39$	18.9041			0.484721
$40$		0			0
$41$		−	63.3016i		−	1.54394i	−0.635658	−	0.771971i	$-0.719271\pi$
$41$		−	63.3016i		−	1.54394i	0.635658	−	0.771971i	$-0.280729\pi$
$42$		0			0
$43$	−20.7303			−0.482100			−0.241050	−	0.970513i	$-0.577492\pi$
$43$	−20.7303			−0.482100			−0.241050	+	0.970513i	$0.577492\pi$
$44$		0			0
$45$	31.2493			0.694429
$46$		0			0
$47$	−42.5246			−0.904780			−0.452390	−	0.891820i	$-0.649428\pi$
$47$	−42.5246			−0.904780			−0.452390	+	0.891820i	$0.649428\pi$
$48$		0			0
$49$	47.1990			0.963246
$50$		0			0
$51$			22.8745i			0.448520i
$52$		0			0
$53$	62.6841			1.18272			0.591359	−	0.806408i	$-0.298592\pi$
$53$	62.6841			1.18272			0.591359	+	0.806408i	$0.298592\pi$
$54$		0			0
$55$	−37.7407			−0.686195
$56$		0			0
$57$	−53.1898			−0.933154
$58$		0			0
$59$		−	56.7687i		−	0.962181i	−0.876671	−	0.481090i	$-0.840241\pi$
$59$		−	56.7687i		−	0.962181i	0.876671	−	0.481090i	$-0.159759\pi$
$60$		0			0
$61$			25.2148i			0.413358i	0.978409	+	0.206679i	$0.0662655\pi$
$61$			25.2148i			0.413358i	−0.978409	+	0.206679i	$0.933734\pi$
$62$		0			0
$63$		−	8.65474i		−	0.137377i
$64$		0			0
$65$	−57.3524			−0.882345
$66$		0			0
$67$			97.3696i			1.45328i	0.687020	+	0.726638i	$0.258918\pi$
$67$			97.3696i			1.45328i	−0.687020	+	0.726638i	$0.741082\pi$
$68$		0			0
$69$			57.6364i			0.835311i
$70$		0			0
$71$		−	15.3545i		−	0.216260i	−0.994137	−	0.108130i	$-0.965514\pi$
$71$		−	15.3545i		−	0.216260i	0.994137	−	0.108130i	$-0.0344863\pi$
$72$		0			0
$73$		−	6.08230i		−	0.0833191i	−0.999132	−	0.0416596i	$-0.986736\pi$
$73$		−	6.08230i		−	0.0833191i	0.999132	−	0.0416596i	$-0.0132645\pi$
$74$		0			0
$75$	−2.42940			−0.0323920
$76$		0			0
$77$			10.4526i			0.135748i
$78$		0			0
$79$	28.1831			0.356748			0.178374	−	0.983963i	$-0.442916\pi$
$79$	28.1831			0.356748			0.178374	+	0.983963i	$0.442916\pi$
$80$		0			0
$81$	18.6335			0.230043
$82$		0			0
$83$		−	86.8954i		−	1.04693i	−0.852046	−	0.523466i	$-0.824639\pi$
$83$		−	86.8954i		−	1.04693i	0.852046	−	0.523466i	$-0.175361\pi$
$84$		0			0
$85$		−	69.3981i		−	0.816448i
$86$		0			0
$87$	−16.4747	+	43.2881i	−0.189365	+	0.497565i
$88$		0			0
$89$		−	156.628i		−	1.75987i	−0.475095	−	0.879935i	$-0.657586\pi$
$89$		−	156.628i		−	1.75987i	0.475095	−	0.879935i	$-0.342414\pi$
$90$		0			0
$91$			15.8842i			0.174552i
$92$		0			0
$93$	−54.3198			−0.584084
$94$		0			0
$95$	161.370			1.69863
$96$		0			0
$97$			105.767i			1.09039i	0.838311	+	0.545193i	$0.183543\pi$
$97$			105.767i			1.09039i	−0.838311	+	0.545193i	$0.816457\pi$
$98$		0			0
$99$	−50.2311			−0.507385
$100$		0			0
$101$			64.1730i			0.635376i	0.948195	+	0.317688i	$0.102907\pi$
$101$			64.1730i			0.635376i	−0.948195	+	0.317688i	$0.897093\pi$
$102$		0			0
$103$		−	3.70360i		−	0.0359573i	−0.999838	−	0.0179787i	$-0.994277\pi$
$103$		−	3.70360i		−	0.0359573i	0.999838	−	0.0179787i	$-0.00572309\pi$
$104$		0			0
$105$		−	10.3857i		−	0.0989113i
$106$		0			0
$107$			68.1096i			0.636538i	0.948000	+	0.318269i	$0.103102\pi$
$107$			68.1096i			0.636538i	−0.948000	+	0.318269i	$0.896898\pi$
$108$		0			0
$109$	0.162432			0.00149021			0.000745103	−	1.00000i	$-0.499763\pi$
$109$	0.162432			0.00149021			0.000745103		1.00000i	$0.499763\pi$
$110$		0			0
$111$			112.208i			1.01088i
$112$		0			0
$113$			54.2274i			0.479889i	0.970787	+	0.239944i	$0.0771292\pi$
$113$			54.2274i			0.479889i	−0.970787	+	0.239944i	$0.922871\pi$
$114$		0			0
$115$		−	174.861i		−	1.52053i
$116$		0			0
$117$	−76.3333			−0.652422
$118$		0			0
$119$	−19.2203			−0.161515
$120$		0			0
$121$	−60.3344			−0.498632
$122$		0			0
$123$		−	101.102i		−	0.821965i
$124$		0			0
$125$	128.508			1.02806
$126$		0			0
$127$	−109.721			−0.863949			−0.431974	−	0.901886i	$-0.642183\pi$
$127$	−109.721			−0.863949			−0.431974	+	0.901886i	$0.642183\pi$
$128$		0			0
$129$	−33.1093			−0.256661
$130$		0			0
$131$	116.657			0.890514			0.445257	−	0.895403i	$-0.353112\pi$
$131$	116.657			0.890514			0.445257	+	0.895403i	$0.353112\pi$
$132$		0			0
$133$		−	44.6927i		−	0.336035i
$134$		0			0
$135$	119.560			0.885632
$136$		0			0
$137$		−	106.618i		−	0.778236i	−0.921188	−	0.389118i	$-0.872780\pi$
$137$		−	106.618i		−	0.778236i	0.921188	−	0.389118i	$-0.127220\pi$
$138$		0			0
$139$			215.142i			1.54779i	0.633316	+	0.773894i	$0.281694\pi$
$139$			215.142i			1.54779i	−0.633316	+	0.773894i	$0.718306\pi$
$140$		0			0
$141$	−67.9179			−0.481688
$142$		0			0
$143$	92.1901			0.644686
$144$		0			0
$145$	49.9820	−	131.330i	0.344703	−	0.905724i
$146$		0			0
$147$	75.3836			0.512814
$148$		0			0
$149$	−60.4169			−0.405483			−0.202741	−	0.979232i	$-0.564985\pi$
$149$	−60.4169			−0.405483			−0.202741	+	0.979232i	$0.564985\pi$
$150$		0			0
$151$		−	269.240i		−	1.78304i	−0.452978	−	0.891522i	$-0.649638\pi$
$151$		−	269.240i		−	1.78304i	0.452978	−	0.891522i	$-0.350362\pi$
$152$		0			0
$153$		−	92.3655i		−	0.603696i
$154$		0			0
$155$	164.798			1.06322
$156$		0			0
$157$		−	8.24613i		−	0.0525231i	−0.999655	−	0.0262616i	$-0.991640\pi$
$157$		−	8.24613i		−	0.0525231i	0.999655	−	0.0262616i	$-0.00836027\pi$
$158$		0			0
$159$	100.115			0.629657
$160$		0			0
$161$	−48.4290			−0.300801
$162$		0			0
$163$	128.172			0.786332			0.393166	−	0.919467i	$-0.371380\pi$
$163$	128.172			0.786332			0.393166	+	0.919467i	$0.371380\pi$
$164$		0			0
$165$	−60.2773			−0.365317
$166$		0			0
$167$		−	195.533i		−	1.17086i	−0.810725	−	0.585428i	$-0.800927\pi$
$167$		−	195.533i		−	1.17086i	0.810725	−	0.585428i	$-0.199073\pi$
$168$		0			0
$169$	−28.9040			−0.171029
$170$		0			0
$171$	214.776			1.25600
$172$		0			0
$173$	−11.3326			−0.0655064			−0.0327532	−	0.999463i	$-0.510428\pi$
$173$	−11.3326			−0.0655064			−0.0327532	+	0.999463i	$0.510428\pi$
$174$		0			0
$175$		−	2.04131i		−	0.0116646i
$176$		0			0
$177$		−	90.6677i		−	0.512247i
$178$		0			0
$179$		−	161.557i		−	0.902552i	−0.892385	−	0.451276i	$-0.850969\pi$
$179$		−	161.557i		−	0.902552i	0.892385	−	0.451276i	$-0.149031\pi$
$180$		0			0
$181$	235.014			1.29842			0.649211	−	0.760608i	$-0.275099\pi$
$181$	235.014			1.29842			0.649211	+	0.760608i	$0.275099\pi$
$182$		0			0
$183$			40.2717i			0.220064i
$184$		0			0
$185$		−	340.422i		−	1.84012i
$186$		0			0
$187$			111.553i			0.596538i
$188$		0			0
$189$		−	33.1131i		−	0.175202i
$190$		0			0
$191$	−10.5793			−0.0553889			−0.0276944	−	0.999616i	$-0.508817\pi$
$191$	−10.5793			−0.0553889			−0.0276944	+	0.999616i	$0.508817\pi$
$192$		0			0
$193$		−	146.743i		−	0.760328i	−0.924919	−	0.380164i	$-0.875867\pi$
$193$		−	146.743i		−	0.760328i	0.924919	−	0.380164i	$-0.124133\pi$
$194$		0			0
$195$	−91.6001			−0.469744
$196$		0			0
$197$	275.301			1.39747			0.698734	−	0.715381i	$-0.253747\pi$
$197$	275.301			1.39747			0.698734	+	0.715381i	$0.253747\pi$
$198$		0			0
$199$		−	58.1890i		−	0.292407i	−0.989255	−	0.146203i	$-0.953295\pi$
$199$		−	58.1890i		−	0.292407i	0.989255	−	0.146203i	$-0.0467054\pi$
$200$		0			0
$201$			155.513i			0.773697i
$202$		0			0
$203$	−36.3729	−	13.8429i	−0.179177	−	0.0681916i
$204$		0			0
$205$			306.728i			1.49623i
$206$		0			0
$207$		−	232.731i		−	1.12431i
$208$		0			0
$209$	−259.391			−1.24111
$210$		0			0
$211$	−387.928			−1.83852			−0.919262	−	0.393647i	$-0.871213\pi$
$211$	−387.928			−1.83852			−0.919262	+	0.393647i	$0.871213\pi$
$212$		0			0
$213$		−	24.5233i		−	0.115133i
$214$		0			0
$215$	100.449			0.467204
$216$		0			0
$217$		−	45.6422i		−	0.210333i
$218$		0			0
$219$		−	9.71430i		−	0.0443575i
$220$		0			0
$221$			169.520i			0.767060i
$222$		0			0
$223$			340.110i			1.52516i	0.646896	+	0.762578i	$0.276067\pi$
$223$			340.110i			1.52516i	−0.646896	+	0.762578i	$0.723933\pi$
$224$		0			0
$225$	9.80973			0.0435988
$226$		0			0
$227$			298.044i			1.31297i	0.754340	+	0.656484i	$0.227957\pi$
$227$			298.044i			1.31297i	−0.754340	+	0.656484i	$0.772043\pi$
$228$		0			0
$229$			268.024i			1.17041i	0.810886	+	0.585205i	$0.198986\pi$
$229$			268.024i			1.17041i	−0.810886	+	0.585205i	$0.801014\pi$
$230$		0			0
$231$			16.6943i			0.0722696i
$232$		0			0
$233$	−116.035			−0.498006			−0.249003	−	0.968503i	$-0.580103\pi$
$233$	−116.035			−0.498006			−0.249003	+	0.968503i	$0.580103\pi$
$234$		0			0
$235$	206.053			0.876823
$236$		0			0
$237$	45.0124			0.189926
$238$		0			0
$239$			140.616i			0.588349i	0.955752	+	0.294175i	$0.0950448\pi$
$239$			140.616i			0.588349i	−0.955752	+	0.294175i	$0.904955\pi$
$240$		0			0
$241$	222.022			0.921255			0.460628	−	0.887594i	$-0.347624\pi$
$241$	222.022			0.921255			0.460628	+	0.887594i	$0.347624\pi$
$242$		0			0
$243$	251.831			1.03634
$244$		0			0
$245$	−228.703			−0.933482
$246$		0			0
$247$	−394.182			−1.59588
$248$		0			0
$249$		−	138.784i		−	0.557367i
$250$		0			0
$251$	−380.154			−1.51456			−0.757279	−	0.653091i	$-0.773472\pi$
$251$	−380.154			−1.51456			−0.757279	+	0.653091i	$0.773472\pi$
$252$		0			0
$253$			281.076i			1.11097i
$254$		0			0
$255$		−	110.839i		−	0.434662i
$256$		0			0
$257$	64.0288			0.249139			0.124570	−	0.992211i	$-0.460245\pi$
$257$	64.0288			0.249139			0.124570	+	0.992211i	$0.460245\pi$
$258$		0			0
$259$	−94.2826			−0.364026
$260$		0			0
$261$	66.5236	−	174.794i	0.254880	−	0.669708i
$262$		0			0
$263$	−91.5624			−0.348146			−0.174073	−	0.984733i	$-0.555693\pi$
$263$	−91.5624			−0.348146			−0.174073	+	0.984733i	$0.555693\pi$
$264$		0			0
$265$	−303.736			−1.14617
$266$		0			0
$267$		−	250.158i		−	0.936921i
$268$		0			0
$269$			36.7635i			0.136667i	0.997663	+	0.0683336i	$0.0217682\pi$
$269$			36.7635i			0.136667i	−0.997663	+	0.0683336i	$0.978232\pi$
$270$		0			0
$271$	−391.235			−1.44367			−0.721835	−	0.692065i	$-0.756701\pi$
$271$	−391.235			−1.44367			−0.721835	+	0.692065i	$0.756701\pi$
$272$		0			0
$273$			25.3693i			0.0929280i
$274$		0			0
$275$	−11.8475			−0.0430818
$276$		0			0
$277$	−187.485			−0.676840			−0.338420	−	0.940995i	$-0.609893\pi$
$277$	−187.485			−0.676840			−0.338420	+	0.940995i	$0.609893\pi$
$278$		0			0
$279$	219.339			0.786161
$280$		0			0
$281$	402.820			1.43352			0.716761	−	0.697319i	$-0.245624\pi$
$281$	402.820			1.43352			0.716761	+	0.697319i	$0.245624\pi$
$282$		0			0
$283$		−	123.183i		−	0.435276i	−0.976030	−	0.217638i	$-0.930165\pi$
$283$		−	123.183i		−	0.435276i	0.976030	−	0.217638i	$-0.0698352\pi$
$284$		0			0
$285$	257.731			0.904320
$286$		0			0
$287$	84.9507			0.295996
$288$		0			0
$289$	83.8757			0.290227
$290$		0			0
$291$			168.926i			0.580500i
$292$		0			0
$293$			362.655i			1.23773i	0.785498	+	0.618865i	$0.212407\pi$
$293$			362.655i			1.23773i	−0.785498	+	0.618865i	$0.787593\pi$
$294$		0			0
$295$			275.073i			0.932450i
$296$		0			0
$297$	−192.185			−0.647087
$298$		0			0
$299$			427.136i			1.42855i
$300$		0			0
$301$		−	27.8201i		−	0.0924255i
$302$		0			0
$303$			102.494i			0.338262i
$304$		0			0
$305$		−	122.179i		−	0.400585i
$306$		0			0
$307$	−145.491			−0.473911			−0.236955	−	0.971521i	$-0.576150\pi$
$307$	−145.491			−0.473911			−0.236955	+	0.971521i	$0.576150\pi$
$308$		0			0
$309$		−	5.91519i		−	0.0191430i
$310$		0			0
$311$	149.442			0.480521			0.240261	−	0.970708i	$-0.422767\pi$
$311$	149.442			0.480521			0.240261	+	0.970708i	$0.422767\pi$
$312$		0			0
$313$	−413.125			−1.31989			−0.659944	−	0.751314i	$-0.729420\pi$
$313$	−413.125			−1.31989			−0.659944	+	0.751314i	$0.729420\pi$
$314$		0			0
$315$			41.9366i			0.133132i
$316$		0			0
$317$			345.773i			1.09077i	0.838187	+	0.545383i	$0.183616\pi$
$317$			345.773i			1.09077i	−0.838187	+	0.545383i	$0.816384\pi$
$318$		0			0
$319$	−80.3426	+	211.104i	−0.251858	+	0.661767i
$320$		0			0
$321$			108.781i			0.338881i
$322$		0			0
$323$		−	476.972i		−	1.47669i
$324$		0			0
$325$	−18.0040			−0.0553968
$326$		0			0
$327$	0.259428			0.000793357
$328$		0			0
$329$		−	57.0681i		−	0.173459i
$330$		0			0
$331$	−406.466			−1.22799			−0.613997	−	0.789308i	$-0.710439\pi$
$331$	−406.466			−1.22799			−0.613997	+	0.789308i	$0.710439\pi$
$332$		0			0
$333$		−	453.086i		−	1.36062i
$334$		0			0
$335$		−	471.805i		−	1.40837i
$336$		0			0
$337$			470.818i			1.39709i	0.715568	+	0.698543i	$0.246168\pi$
$337$			470.818i			1.39709i	−0.715568	+	0.698543i	$0.753832\pi$
$338$		0			0
$339$			86.6089i			0.255484i
$340$		0			0
$341$	−264.902			−0.776839
$342$		0			0
$343$			129.099i			0.376382i
$344$		0			0
$345$		−	279.278i		−	0.809500i
$346$		0			0
$347$			234.768i			0.676565i	0.941045	+	0.338282i	$0.109846\pi$
$347$			234.768i			0.676565i	−0.941045	+	0.338282i	$0.890154\pi$
$348$		0			0
$349$	581.492			1.66617			0.833084	−	0.553147i	$-0.186573\pi$
$349$	581.492			1.66617			0.833084	+	0.553147i	$0.186573\pi$
$350$		0			0
$351$	−292.053			−0.832058
$352$		0			0
$353$	−70.6501			−0.200142			−0.100071	−	0.994980i	$-0.531907\pi$
$353$	−70.6501			−0.200142			−0.100071	+	0.994980i	$0.531907\pi$
$354$		0			0
$355$			74.4002i			0.209578i
$356$		0			0
$357$	−30.6976			−0.0859878
$358$		0			0
$359$	−629.986			−1.75484			−0.877418	−	0.479727i	$-0.840736\pi$
$359$	−629.986			−1.75484			−0.877418	+	0.479727i	$0.840736\pi$
$360$		0			0
$361$	748.093			2.07228
$362$		0			0
$363$	−96.3627			−0.265462
$364$		0			0
$365$			29.4718i			0.0807446i
$366$		0			0
$367$	539.173			1.46914			0.734569	−	0.678534i	$-0.237385\pi$
$367$	539.173			1.46914			0.734569	+	0.678534i	$0.237385\pi$
$368$		0			0
$369$			408.240i			1.10634i
$370$		0			0
$371$			84.1220i			0.226744i
$372$		0			0
$373$	118.182			0.316842			0.158421	−	0.987372i	$-0.449360\pi$
$373$	118.182			0.316842			0.158421	+	0.987372i	$0.449360\pi$
$374$		0			0
$375$	205.246			0.547322
$376$		0			0
$377$	−122.092	+	320.803i	−0.323852	+	0.850935i
$378$		0			0
$379$	−208.655			−0.550540			−0.275270	−	0.961367i	$-0.588767\pi$
$379$	−208.655			−0.550540			−0.275270	+	0.961367i	$0.588767\pi$
$380$		0			0
$381$	−175.241			−0.459950
$382$		0			0
$383$		−	385.597i		−	1.00678i	−0.864059	−	0.503390i	$-0.832086\pi$
$383$		−	385.597i		−	1.00678i	0.864059	−	0.503390i	$-0.167914\pi$
$384$		0			0
$385$		−	50.6480i		−	0.131553i
$386$		0			0
$387$	133.693			0.345459
$388$		0			0
$389$		−	75.8778i		−	0.195059i	−0.995233	−	0.0975294i	$-0.968906\pi$
$389$		−	75.8778i		−	0.195059i	0.995233	−	0.0975294i	$-0.0310940\pi$
$390$		0			0
$391$	−516.847			−1.32186
$392$		0			0
$393$	186.319			0.474093
$394$		0			0
$395$	−136.561			−0.345724
$396$		0			0
$397$	−178.443			−0.449478			−0.224739	−	0.974419i	$-0.572153\pi$
$397$	−178.443			−0.449478			−0.224739	+	0.974419i	$0.572153\pi$
$398$		0			0
$399$		−	71.3806i		−	0.178899i
$400$		0			0
$401$	297.700			0.742393			0.371197	−	0.928554i	$-0.378948\pi$
$401$	297.700			0.742393			0.371197	+	0.928554i	$0.378948\pi$
$402$		0			0
$403$	−402.557			−0.998900
$404$		0			0
$405$	−90.2888			−0.222935
$406$		0			0
$407$			547.205i			1.34448i
$408$		0			0
$409$			274.381i			0.670859i	0.942065	+	0.335430i	$0.108881\pi$
$409$			274.381i			0.670859i	−0.942065	+	0.335430i	$0.891119\pi$
$410$		0			0
$411$		−	170.285i		−	0.414318i
$412$		0			0
$413$	76.1835			0.184464
$414$		0			0
$415$			421.052i			1.01458i
$416$		0			0
$417$			343.613i			0.824013i
$418$		0			0
$419$		−	131.319i		−	0.313410i	−0.987645	−	0.156705i	$-0.949913\pi$
$419$		−	131.319i		−	0.313410i	0.987645	−	0.156705i	$-0.0500872\pi$
$420$		0			0
$421$		−	489.174i		−	1.16193i	−0.813927	−	0.580967i	$-0.802675\pi$
$421$		−	489.174i		−	1.16193i	0.813927	−	0.580967i	$-0.197325\pi$
$422$		0			0
$423$	274.247			0.648338
$424$		0			0
$425$		−	21.7853i		−	0.0512596i
$426$		0			0
$427$	−33.8383			−0.0792466
$428$		0			0
$429$	147.241			0.343218
$430$		0			0
$431$			546.445i			1.26785i	0.773393	+	0.633927i	$0.218558\pi$
$431$			546.445i			1.26785i	−0.773393	+	0.633927i	$0.781442\pi$
$432$		0			0
$433$		−	163.240i		−	0.376997i	−0.982073	−	0.188499i	$-0.939638\pi$
$433$		−	163.240i		−	0.376997i	0.982073	−	0.188499i	$-0.0603621\pi$
$434$		0			0
$435$	79.8284	−	209.753i	0.183514	−	0.482190i
$436$		0			0
$437$		−	1201.81i		−	2.75015i
$438$		0			0
$439$		−	609.676i		−	1.38878i	−0.719597	−	0.694392i	$-0.755674\pi$
$439$		−	609.676i		−	1.38878i	0.719597	−	0.694392i	$-0.244326\pi$
$440$		0			0
$441$	−304.393			−0.690233
$442$		0			0
$443$	593.828			1.34047			0.670235	−	0.742149i	$-0.266193\pi$
$443$	593.828			1.34047			0.670235	+	0.742149i	$0.266193\pi$
$444$		0			0
$445$			758.943i			1.70549i
$446$		0			0
$447$	−96.4945			−0.215871
$448$		0			0
$449$			45.4861i			0.101305i	0.998716	+	0.0506526i	$0.0161301\pi$
$449$			45.4861i			0.101305i	−0.998716	+	0.0506526i	$0.983870\pi$
$450$		0			0
$451$		−	493.044i		−	1.09322i
$452$		0			0
$453$		−	430.014i		−	0.949259i
$454$		0			0
$455$		−	76.9670i		−	0.169158i
$456$		0			0
$457$	−90.4775			−0.197981			−0.0989907	−	0.995088i	$-0.531561\pi$
$457$	−90.4775			−0.197981			−0.0989907	+	0.995088i	$0.531561\pi$
$458$		0			0
$459$		−	353.392i		−	0.769917i
$460$		0			0
$461$			35.4157i			0.0768236i	0.999262	+	0.0384118i	$0.0122299\pi$
$461$			35.4157i			0.0768236i	−0.999262	+	0.0384118i	$0.987770\pi$
$462$		0			0
$463$			76.0579i			0.164272i	0.996621	+	0.0821360i	$0.0261742\pi$
$463$			76.0579i			0.164272i	−0.996621	+	0.0821360i	$0.973826\pi$
$464$		0			0
$465$	263.207			0.566036
$466$		0			0
$467$	538.988			1.15415			0.577075	−	0.816691i	$-0.304194\pi$
$467$	538.988			1.15415			0.577075	+	0.816691i	$0.304194\pi$
$468$		0			0
$469$	−130.670			−0.278614
$470$		0			0
$471$		−	13.1702i		−	0.0279623i
$472$		0			0
$473$	−161.465			−0.341363
$474$		0			0
$475$	50.6570			0.106646
$476$		0			0
$477$	−404.258			−0.847501
$478$		0			0
$479$	268.940			0.561461			0.280731	−	0.959787i	$-0.409423\pi$
$479$	268.940			0.561461			0.280731	+	0.959787i	$0.409423\pi$
$480$		0			0
$481$			831.557i			1.72881i
$482$		0			0
$483$	−77.3481			−0.160141
$484$		0			0
$485$		−	512.496i		−	1.05669i
$486$		0			0
$487$			359.511i			0.738215i	0.929387	+	0.369108i	$0.120337\pi$
$487$			359.511i			0.738215i	−0.929387	+	0.369108i	$0.879663\pi$
$488$		0			0
$489$	204.709			0.418628
$490$		0			0
$491$	337.256			0.686875			0.343438	−	0.939175i	$-0.388409\pi$
$491$	337.256			0.686875			0.343438	+	0.939175i	$0.388409\pi$
$492$		0			0
$493$	−388.180	−	147.735i	−0.787384	−	0.299665i
$494$		0			0
$495$	243.395			0.491707
$496$		0			0
$497$	20.6057			0.0414602
$498$		0			0
$499$		−	58.1753i		−	0.116584i	−0.998300	−	0.0582918i	$-0.981435\pi$
$499$		−	58.1753i		−	0.116584i	0.998300	−	0.0582918i	$-0.0185654\pi$
$500$		0			0
$501$		−	312.294i		−	0.623341i
$502$		0			0
$503$	872.403			1.73440			0.867200	−	0.497960i	$-0.165917\pi$
$503$	872.403			1.73440			0.867200	+	0.497960i	$0.165917\pi$
$504$		0			0
$505$		−	310.951i		−	0.615744i
$506$		0			0
$507$	−46.1638			−0.0910528
$508$		0			0
$509$	−518.840			−1.01933			−0.509666	−	0.860373i	$-0.670231\pi$
$509$	−518.840			−1.01933			−0.509666	+	0.860373i	$0.670231\pi$
$510$		0			0
$511$	8.16244			0.0159735
$512$		0			0
$513$	821.736			1.60182
$514$		0			0
$515$			17.9458i			0.0348463i
$516$		0			0
$517$	−331.216			−0.640651
$518$		0			0
$519$	−18.0998			−0.0348744
$520$		0			0
$521$	−265.518			−0.509631			−0.254815	−	0.966990i	$-0.582015\pi$
$521$	−265.518			−0.509631			−0.254815	+	0.966990i	$0.582015\pi$
$522$		0			0
$523$			709.399i			1.35640i	0.734876	+	0.678201i	$0.237240\pi$
$523$			709.399i			1.35640i	−0.734876	+	0.678201i	$0.762760\pi$
$524$		0			0
$525$		−	3.26026i		−	0.00621001i
$526$		0			0
$527$		−	487.105i		−	0.924298i
$528$		0			0
$529$	−773.286			−1.46179
$530$		0			0
$531$			366.109i			0.689470i
$532$		0			0
$533$		−	749.251i		−	1.40572i
$534$		0			0
$535$		−	330.025i		−	0.616870i
$536$		0			0
$537$		−	258.029i		−	0.480501i
$538$		0			0
$539$	367.624			0.682049
$540$		0			0
$541$			699.685i			1.29332i	0.762779	+	0.646659i	$0.223834\pi$
$541$			699.685i			1.29332i	−0.762779	+	0.646659i	$0.776166\pi$
$542$		0			0
$543$	375.352			0.691256
$544$		0			0
$545$	−0.787067			−0.00144416
$546$		0			0
$547$			325.389i			0.594861i	0.954743	+	0.297430i	$0.0961297\pi$
$547$			325.389i			0.594861i	−0.954743	+	0.297430i	$0.903870\pi$
$548$		0			0
$549$		−	162.614i		−	0.296200i
$550$		0			0
$551$	343.525	−	902.628i	0.623458	−	1.63816i
$552$		0			0
$553$			37.8217i			0.0683936i
$554$		0			0
$555$		−	543.703i		−	0.979646i
$556$		0			0
$557$	−12.5552			−0.0225408			−0.0112704	−	0.999936i	$-0.503588\pi$
$557$	−12.5552			−0.0225408			−0.0112704	+	0.999936i	$0.503588\pi$
$558$		0			0
$559$	−245.369			−0.438942
$560$		0			0
$561$			178.165i			0.317586i
$562$		0			0
$563$	814.191			1.44617			0.723083	−	0.690761i	$-0.242724\pi$
$563$	814.191			1.44617			0.723083	+	0.690761i	$0.242724\pi$
$564$		0			0
$565$		−	262.759i		−	0.465060i
$566$		0			0
$567$			25.0062i			0.0441026i
$568$		0			0
$569$		−	352.953i		−	0.620304i	−0.950687	−	0.310152i	$-0.899620\pi$
$569$		−	352.953i		−	0.620304i	0.950687	−	0.310152i	$-0.100380\pi$
$570$		0			0
$571$			522.188i			0.914515i	0.889334	+	0.457258i	$0.151168\pi$
$571$			522.188i			0.914515i	−0.889334	+	0.457258i	$0.848832\pi$
$572$		0			0
$573$	−16.8966			−0.0294880
$574$		0			0
$575$		−	54.8920i		−	0.0954643i
$576$		0			0
$577$			17.2753i			0.0299398i	0.999888	+	0.0149699i	$0.00476525\pi$
$577$			17.2753i			0.0299398i	−0.999888	+	0.0149699i	$0.995235\pi$
$578$		0			0
$579$		−	234.370i		−	0.404784i
$580$		0			0
$581$	116.614			0.200712
$582$		0			0
$583$	488.234			0.837452
$584$		0			0
$585$	369.874			0.632262
$586$		0			0
$587$		−	717.528i		−	1.22236i	−0.791490	−	0.611182i	$-0.790694\pi$
$587$		−	717.528i		−	1.22236i	0.791490	−	0.611182i	$-0.209306\pi$
$588$		0			0
$589$	1132.66			1.92302
$590$		0			0
$591$	439.696			0.743986
$592$		0			0
$593$	−105.763			−0.178352			−0.0891761	−	0.996016i	$-0.528423\pi$
$593$	−105.763			−0.178352			−0.0891761	+	0.996016i	$0.528423\pi$
$594$		0			0
$595$	93.1322			0.156525
$596$		0			0
$597$		−	92.9361i		−	0.155672i
$598$		0			0
$599$	−298.126			−0.497706			−0.248853	−	0.968541i	$-0.580054\pi$
$599$	−298.126			−0.497706			−0.248853	+	0.968541i	$0.580054\pi$
$600$		0			0
$601$		−	427.045i		−	0.710557i	−0.934761	−	0.355278i	$-0.884386\pi$
$601$		−	427.045i		−	0.710557i	0.934761	−	0.355278i	$-0.115614\pi$
$602$		0			0
$603$		−	627.949i		−	1.04138i
$604$		0			0
$605$	292.351			0.483224
$606$		0			0
$607$	−657.752			−1.08361			−0.541806	−	0.840504i	$-0.682259\pi$
$607$	−657.752			−1.08361			−0.541806	+	0.840504i	$0.682259\pi$
$608$		0			0
$609$	−58.0927	−	22.1091i	−0.0953902	−	0.0363039i
$610$		0			0
$611$	−503.331			−0.823782
$612$		0			0
$613$	−968.156			−1.57937			−0.789687	−	0.613510i	$-0.789757\pi$
$613$	−968.156			−1.57937			−0.789687	+	0.613510i	$0.789757\pi$
$614$		0			0
$615$			489.889i			0.796567i
$616$		0			0
$617$			819.667i			1.32847i	0.747523	+	0.664236i	$0.231243\pi$
$617$			819.667i			1.32847i	−0.747523	+	0.664236i	$0.768757\pi$
$618$		0			0
$619$	390.644			0.631089			0.315544	−	0.948911i	$-0.397813\pi$
$619$	390.644			0.631089			0.315544	+	0.948911i	$0.397813\pi$
$620$		0			0
$621$		−	890.433i		−	1.43387i
$622$		0			0
$623$	210.195			0.337392
$624$		0			0
$625$	−584.659			−0.935454
$626$		0			0
$627$	−414.285			−0.660741
$628$		0			0
$629$	−1006.21			−1.59969
$630$		0			0
$631$			316.296i			0.501261i	0.968083	+	0.250630i	$0.0806379\pi$
$631$			316.296i			0.501261i	−0.968083	+	0.250630i	$0.919362\pi$
$632$		0			0
$633$	−619.577			−0.978795
$634$		0			0
$635$	531.656			0.837253
$636$		0			0
$637$	558.658			0.877014
$638$		0			0
$639$			99.0230i			0.154966i
$640$		0			0
$641$			692.756i			1.08074i	0.841427	+	0.540371i	$0.181716\pi$
$641$			692.756i			1.08074i	−0.841427	+	0.540371i	$0.818284\pi$
$642$		0			0
$643$			1061.06i			1.65018i	0.565004	+	0.825088i	$0.308875\pi$
$643$			1061.06i			1.65018i	−0.565004	+	0.825088i	$0.691125\pi$
$644$		0			0
$645$	160.431			0.248731
$646$		0			0
$647$			729.612i			1.12768i	0.825883	+	0.563842i	$0.190677\pi$
$647$			729.612i			1.12768i	−0.825883	+	0.563842i	$0.809323\pi$
$648$		0			0
$649$		−	442.160i		−	0.681295i
$650$		0			0
$651$		−	72.8971i		−	0.111977i
$652$		0			0
$653$			216.126i			0.330974i	0.986212	+	0.165487i	$0.0529196\pi$
$653$			216.126i			0.330974i	−0.986212	+	0.165487i	$0.947080\pi$
$654$		0			0
$655$	−565.264			−0.862998
$656$		0			0
$657$			39.2255i			0.0597040i
$658$		0			0
$659$	−261.677			−0.397082			−0.198541	−	0.980093i	$-0.563620\pi$
$659$	−261.677			−0.397082			−0.198541	+	0.980093i	$0.563620\pi$
$660$		0			0
$661$	1023.21			1.54797			0.773984	−	0.633205i	$-0.218261\pi$
$661$	1023.21			1.54797			0.773984	+	0.633205i	$0.218261\pi$
$662$		0			0
$663$			270.748i			0.408368i
$664$		0			0
$665$			216.559i			0.325652i
$666$		0			0
$667$	−978.089	−	372.244i	−1.46640	−	0.558087i
$668$		0			0
$669$			543.204i			0.811964i
$670$		0			0
$671$			196.393i			0.292688i
$672$		0			0
$673$	762.216			1.13257			0.566283	−	0.824211i	$-0.308381\pi$
$673$	762.216			1.13257			0.566283	+	0.824211i	$0.308381\pi$
$674$		0			0
$675$	37.5322			0.0556032
$676$		0			0
$677$		−	734.544i		−	1.08500i	−0.840056	−	0.542499i	$-0.817478\pi$
$677$		−	734.544i		−	1.08500i	0.840056	−	0.542499i	$-0.182522\pi$
$678$		0			0
$679$	−141.940			−0.209042
$680$		0			0
$681$			476.019i			0.699000i
$682$		0			0
$683$		−	812.445i		−	1.18952i	−0.803902	−	0.594762i	$-0.797246\pi$
$683$		−	812.445i		−	1.18952i	0.803902	−	0.594762i	$-0.202754\pi$
$684$		0			0
$685$			516.620i			0.754189i
$686$		0			0
$687$			428.072i			0.623104i
$688$		0			0
$689$	741.942			1.07684
$690$		0			0
$691$			103.579i			0.149897i	0.997187	+	0.0749484i	$0.0238792\pi$
$691$			103.579i			0.149897i	−0.997187	+	0.0749484i	$0.976121\pi$
$692$		0			0
$693$		−	67.4101i		−	0.0972729i
$694$		0			0
$695$		−	1042.47i		−	1.49996i
$696$		0			0
$697$	906.615			1.30074
$698$		0			0
$699$	−185.325			−0.265129
$700$		0			0
$701$	216.745			0.309194			0.154597	−	0.987978i	$-0.450592\pi$
$701$	216.745			0.309194			0.154597	+	0.987978i	$0.450592\pi$
$702$		0			0
$703$		−	2339.72i		−	3.32819i
$704$		0			0
$705$	329.097			0.466804
$706$		0			0
$707$	−86.1202			−0.121811
$708$		0			0
$709$	−902.308			−1.27265			−0.636324	−	0.771422i	$-0.719546\pi$
$709$	−902.308			−1.27265			−0.636324	+	0.771422i	$0.719546\pi$
$710$		0			0
$711$	−181.756			−0.255635
$712$		0			0
$713$		−	1227.35i		−	1.72138i
$714$		0			0
$715$	−446.707			−0.624765
$716$		0			0
$717$			224.583i			0.313226i
$718$		0			0
$719$			1066.68i			1.48357i	0.670640	+	0.741783i	$0.266019\pi$
$719$			1066.68i			1.48357i	−0.670640	+	0.741783i	$0.733981\pi$
$720$		0			0
$721$	4.97024			0.00689353
$722$		0			0
$723$	354.602			0.490459
$724$		0			0
$725$	15.6903	−	41.2269i	0.0216417	−	0.0568647i
$726$		0			0
$727$	−13.3901			−0.0184183			−0.00920914	−	0.999958i	$-0.502931\pi$
$727$	−13.3901			−0.0184183			−0.00920914	+	0.999958i	$0.502931\pi$
$728$		0			0
$729$	234.508			0.321685
$730$		0			0
$731$		−	296.903i		−	0.406160i
$732$		0			0
$733$		−	121.763i		−	0.166115i	−0.996545	−	0.0830577i	$-0.973531\pi$
$733$		−	121.763i		−	0.166115i	0.996545	−	0.0830577i	$-0.0264686\pi$
$734$		0			0
$735$	−365.272			−0.496968
$736$		0			0
$737$			758.393i			1.02903i
$738$		0			0
$739$	1088.55			1.47300			0.736499	−	0.676439i	$-0.236478\pi$
$739$	1088.55			1.47300			0.736499	+	0.676439i	$0.236478\pi$
$740$		0			0
$741$	−629.565			−0.849616
$742$		0			0
$743$	413.457			0.556470			0.278235	−	0.960513i	$-0.410251\pi$
$743$	413.457			0.556470			0.278235	+	0.960513i	$0.410251\pi$
$744$		0			0
$745$	292.750			0.392954
$746$		0			0
$747$			560.400i			0.750201i
$748$		0			0
$749$	−91.4031			−0.122033
$750$		0			0
$751$	−177.768			−0.236709			−0.118354	−	0.992971i	$-0.537762\pi$
$751$	−177.768			−0.236709			−0.118354	+	0.992971i	$0.537762\pi$
$752$		0			0
$753$	−607.161			−0.806322
$754$		0			0
$755$			1304.60i			1.72795i
$756$		0			0
$757$		−	496.506i		−	0.655887i	−0.944697	−	0.327943i	$-0.893645\pi$
$757$		−	496.506i		−	0.655887i	0.944697	−	0.327943i	$-0.106355\pi$
$758$		0			0
$759$			448.919i			0.591462i
$760$		0			0
$761$	−595.426			−0.782426			−0.391213	−	0.920300i	$-0.627944\pi$
$761$	−595.426			−0.782426			−0.391213	+	0.920300i	$0.627944\pi$
$762$		0			0
$763$			0.217984i			0.000285694i
$764$		0			0
$765$			447.557i			0.585042i
$766$		0			0
$767$		−	671.926i		−	0.876045i
$768$		0			0
$769$			320.137i			0.416303i	0.978097	+	0.208152i	$0.0667448\pi$
$769$			320.137i			0.416303i	−0.978097	+	0.208152i	$0.933255\pi$
$770$		0			0
$771$	102.263			0.132637
$772$		0			0
$773$		−	1518.25i		−	1.96410i	−0.188633	−	0.982048i	$-0.560406\pi$
$773$		−	1518.25i		−	1.96410i	0.188633	−	0.982048i	$-0.439594\pi$
$774$		0			0
$775$	51.7332			0.0667526
$776$		0			0
$777$	−150.583			−0.193800
$778$		0			0
$779$			2108.14i			2.70621i
$780$		0			0
$781$		−	119.593i		−	0.153128i
$782$		0			0
$783$	254.520	−	668.764i	0.325058	−	0.854105i
$784$		0			0
$785$			39.9566i			0.0509002i
$786$		0			0
$787$		−	457.252i		−	0.581006i	−0.956874	−	0.290503i	$-0.906177\pi$
$787$		−	457.252i		−	0.581006i	0.956874	−	0.290503i	$-0.0938226\pi$
$788$		0			0
$789$	−146.238			−0.185346
$790$		0			0
$791$	−72.7732			−0.0920015
$792$		0			0
$793$			298.448i			0.376353i
$794$		0			0
$795$	−485.110			−0.610201
$796$		0			0
$797$			948.085i			1.18957i	0.803886	+	0.594784i	$0.202762\pi$
$797$			948.085i			1.18957i	−0.803886	+	0.594784i	$0.797238\pi$
$798$		0			0
$799$		−	609.045i		−	0.762259i
$800$		0			0
$801$			1010.12i			1.26107i
$802$		0			0
$803$		−	47.3738i		−	0.0589961i
$804$		0			0
$805$	234.663			0.291507
$806$		0			0
$807$			58.7166i			0.0727591i
$808$		0			0
$809$			1509.79i			1.86624i	0.359569	+	0.933119i	$0.382924\pi$
$809$			1509.79i			1.86624i	−0.359569	+	0.933119i	$0.617076\pi$
$810$		0			0
$811$			152.245i			0.187725i	0.995585	+	0.0938627i	$0.0299215\pi$
$811$			152.245i			0.187725i	−0.995585	+	0.0938627i	$0.970079\pi$
$812$		0			0
$813$	−624.858			−0.768583
$814$		0			0
$815$	−621.059			−0.762035
$816$		0			0
$817$	690.383			0.845022
$818$		0			0
$819$		−	102.439i		−	0.125079i
$820$		0			0
$821$	202.358			0.246478			0.123239	−	0.992377i	$-0.460672\pi$
$821$	202.358			0.246478			0.123239	+	0.992377i	$0.460672\pi$
$822$		0			0
$823$	−841.841			−1.02289			−0.511447	−	0.859315i	$-0.670890\pi$
$823$	−841.841			−1.02289			−0.511447	+	0.859315i	$0.670890\pi$
$824$		0			0
$825$	−18.9222			−0.0229359
$826$		0			0
$827$	−1456.78			−1.76152			−0.880761	−	0.473562i	$-0.842968\pi$
$827$	−1456.78			−1.76152			−0.880761	+	0.473562i	$0.842968\pi$
$828$		0			0
$829$		−	424.434i		−	0.511983i	−0.966679	−	0.255992i	$-0.917598\pi$
$829$		−	424.434i		−	0.511983i	0.966679	−	0.255992i	$-0.0824020\pi$
$830$		0			0
$831$	−299.440			−0.360337
$832$		0			0
$833$			675.992i			0.811515i
$834$		0			0
$835$			947.455i			1.13468i
$836$		0			0
$837$	839.194			1.00262
$838$		0			0
$839$	−12.7492			−0.0151957			−0.00759784	−	0.999971i	$-0.502418\pi$
$839$	−12.7492			−0.0151957			−0.00759784	+	0.999971i	$0.502418\pi$
$840$		0			0
$841$	−628.196	−	559.151i	−0.746964	−	0.664865i
$842$		0			0
$843$	643.361			0.763180
$844$		0			0
$845$	140.054			0.165745
$846$		0			0
$847$		−	80.9688i		−	0.0955948i
$848$		0			0
$849$		−	196.741i		−	0.231733i
$850$		0			0
$851$	−2535.32			−2.97922
$852$		0			0
$853$			1246.14i			1.46089i	0.682971	+	0.730445i	$0.260687\pi$
$853$			1246.14i			1.46089i	−0.682971	+	0.730445i	$0.739313\pi$
$854$		0			0
$855$	−1040.70			−1.21719
$856$		0			0
$857$	−42.0728			−0.0490932			−0.0245466	−	0.999699i	$-0.507814\pi$
$857$	−42.0728			−0.0490932			−0.0245466	+	0.999699i	$0.507814\pi$
$858$		0			0
$859$	548.010			0.637963			0.318981	−	0.947761i	$-0.396659\pi$
$859$	548.010			0.637963			0.318981	+	0.947761i	$0.396659\pi$
$860$		0			0
$861$	135.678			0.157582
$862$		0			0
$863$			947.501i			1.09791i	0.835850	+	0.548957i	$0.184975\pi$
$863$			947.501i			1.09791i	−0.835850	+	0.548957i	$0.815025\pi$
$864$		0			0
$865$	54.9122			0.0634823
$866$		0			0
$867$	133.962			0.154512
$868$		0			0
$869$	219.513			0.252604
$870$		0			0
$871$			1152.49i			1.32318i
$872$		0			0
$873$		−	682.108i		−	0.781338i
$874$		0			0
$875$			172.458i			0.197095i
$876$		0			0
$877$	−1035.81			−1.18109			−0.590543	−	0.807006i	$-0.701086\pi$
$877$	−1035.81			−1.18109			−0.590543	+	0.807006i	$0.701086\pi$
$878$		0			0
$879$			579.211i			0.658944i
$880$		0			0
$881$			492.392i			0.558902i	0.960160	+	0.279451i	$0.0901524\pi$
$881$			492.392i			0.558902i	−0.960160	+	0.279451i	$0.909848\pi$
$882$		0			0
$883$		−	1169.14i		−	1.32406i	−0.749479	−	0.662028i	$-0.769696\pi$
$883$		−	1169.14i		−	1.32406i	0.749479	−	0.662028i	$-0.230304\pi$
$884$		0			0
$885$			439.331i			0.496419i
$886$		0			0
$887$	−1320.63			−1.48887			−0.744437	−	0.667693i	$-0.767282\pi$
$887$	−1320.63			−1.48887			−0.744437	+	0.667693i	$0.767282\pi$
$888$		0			0
$889$		−	147.246i		−	0.165631i
$890$		0			0
$891$	145.133			0.162888
$892$		0			0
$893$	1416.20			1.58589
$894$		0			0
$895$			782.824i			0.874664i
$896$		0			0
$897$			682.197i			0.760532i
$898$		0			0
$899$	350.823	−	921.805i	0.390237	−	1.02537i
$900$		0			0
$901$			897.771i			0.996417i
$902$		0			0
$903$		−	44.4327i		−	0.0492056i
$904$		0			0
$905$	−1138.76			−1.25830
$906$		0			0
$907$	247.660			0.273054			0.136527	−	0.990636i	$-0.456406\pi$
$907$	247.660			0.273054			0.136527	+	0.990636i	$0.456406\pi$
$908$		0			0
$909$		−	413.860i		−	0.455292i
$910$		0			0
$911$	−1064.10			−1.16806			−0.584029	−	0.811733i	$-0.698525\pi$
$911$	−1064.10			−1.16806			−0.584029	+	0.811733i	$0.698525\pi$
$912$		0			0
$913$		−	676.812i		−	0.741305i
$914$		0			0
$915$		−	195.137i		−	0.213264i
$916$		0			0
$917$			156.554i			0.170724i
$918$		0			0
$919$			811.345i			0.882857i	0.897296	+	0.441428i	$0.145528\pi$
$919$			811.345i			0.882857i	−0.897296	+	0.441428i	$0.854472\pi$
$920$		0			0
$921$	−232.369			−0.252301
$922$		0			0
$923$		−	181.739i		−	0.196900i
$924$		0			0
$925$		−	106.865i		−	0.115530i
$926$		0			0
$927$			23.8850i			0.0257660i
$928$		0			0
$929$	1204.95			1.29704			0.648520	−	0.761198i	$-0.275388\pi$
$929$	1204.95			1.29704			0.648520	+	0.761198i	$0.275388\pi$
$930$		0			0
$931$	−1571.87			−1.68837
$932$		0			0
$933$	238.681			0.255821
$934$		0			0
$935$		−	540.528i		−	0.578105i
$936$		0			0
$937$	−667.281			−0.712147			−0.356073	−	0.934458i	$-0.615885\pi$
$937$	−667.281			−0.712147			−0.356073	+	0.934458i	$0.615885\pi$
$938$		0			0
$939$	−659.820			−0.702684
$940$		0			0
$941$	−899.678			−0.956087			−0.478043	−	0.878336i	$-0.658654\pi$
$941$	−899.678			−0.956087			−0.478043	+	0.878336i	$0.658654\pi$
$942$		0			0
$943$	2284.38			2.42246
$944$		0			0
$945$			160.450i			0.169788i
$946$		0			0
$947$	1501.95			1.58601			0.793005	−	0.609215i	$-0.208516\pi$
$947$	1501.95			1.58601			0.793005	+	0.609215i	$0.208516\pi$
$948$		0			0
$949$		−	71.9914i		−	0.0758602i
$950$		0			0
$951$			552.249i			0.580704i
$952$		0			0
$953$	1520.32			1.59530			0.797648	−	0.603123i	$-0.206077\pi$
$953$	1520.32			1.59530			0.797648	+	0.603123i	$0.206077\pi$
$954$		0			0
$955$	51.2619			0.0536774
$956$		0			0
$957$	−128.319	+	337.163i	−0.134084	+	0.352312i
$958$		0			0
$959$	143.082			0.149199
$960$		0			0
$961$	195.720			0.203663
$962$		0			0
$963$		−	439.248i		−	0.456125i
$964$		0			0
$965$			711.046i			0.736835i
$966$		0			0
$967$	1160.13			1.19972			0.599861	−	0.800104i	$-0.295223\pi$
$967$	1160.13			1.19972			0.599861	+	0.800104i	$0.295223\pi$
$968$		0			0
$969$		−	761.792i		−	0.786163i
$970$		0			0
$971$	−260.015			−0.267780			−0.133890	−	0.990996i	$-0.542747\pi$
$971$	−260.015			−0.267780			−0.133890	+	0.990996i	$0.542747\pi$
$972$		0			0
$973$	−288.721			−0.296733
$974$		0			0
$975$	−28.7549			−0.0294922
$976$		0			0
$977$	1653.17			1.69209			0.846045	−	0.533111i	$-0.178977\pi$
$977$	1653.17			1.69209			0.846045	+	0.533111i	$0.178977\pi$
$978$		0			0
$979$		−	1219.95i		−	1.24612i
$980$		0			0
$981$	−1.04755			−0.00106784
$982$		0			0
$983$	915.473			0.931305			0.465652	−	0.884968i	$-0.345820\pi$
$983$	915.473			0.931305			0.465652	+	0.884968i	$0.345820\pi$
$984$		0			0
$985$	−1333.97			−1.35429
$986$		0			0
$987$		−	91.1459i		−	0.0923464i
$988$		0			0
$989$		−	748.099i		−	0.756420i
$990$		0			0
$991$			1754.47i			1.77040i	0.465208	+	0.885201i	$0.345979\pi$
$991$			1754.47i			1.77040i	−0.465208	+	0.885201i	$0.654021\pi$
$992$		0			0
$993$	−649.185			−0.653761
$994$		0			0
$995$			281.955i			0.283372i
$996$		0			0
$997$			44.8688i			0.0450038i	0.999747	+	0.0225019i	$0.00716318\pi$
$997$			44.8688i			0.0450038i	−0.999747	+	0.0225019i	$0.992837\pi$
$998$		0			0
$999$		−	1733.51i		−	1.73525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	464.3.h.c.463.14	yes	20
4.3	odd	2	inner	464.3.h.c.463.7	✓	20
29.28	even	2	inner	464.3.h.c.463.8	yes	20
116.115	odd	2	inner	464.3.h.c.463.13	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
464.3.h.c.463.7	✓	20	4.3	odd	2	inner
464.3.h.c.463.8	yes	20	29.28	even	2	inner
464.3.h.c.463.13	yes	20	116.115	odd	2	inner
464.3.h.c.463.14	yes	20	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:	\( N \)	\(=\)	\( 464 = 2^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	464.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.59714 q^{3} -4.84550 q^{5} +1.34200i q^{7} -6.44913 q^{9} +O(q^{10})\)	\(q+1.59714 q^{3} -4.84550 q^{5} +1.34200i q^{7} -6.44913 q^{9} +7.78881 q^{11} +11.8362 q^{13} -7.73896 q^{15} +14.3222i q^{17} -33.3031 q^{19} +2.14337i q^{21} +36.0872i q^{23} -1.52109 q^{25} -24.6745 q^{27} +(-10.3151 + 27.1035i) q^{29} -34.0106 q^{31} +12.4398 q^{33} -6.50267i q^{35} +70.2553i q^{37} +18.9041 q^{39} -63.3016i q^{41} -20.7303 q^{43} +31.2493 q^{45} -42.5246 q^{47} +47.1990 q^{49} +22.8745i q^{51} +62.6841 q^{53} -37.7407 q^{55} -53.1898 q^{57} -56.7687i q^{59} +25.2148i q^{61} -8.65474i q^{63} -57.3524 q^{65} +97.3696i q^{67} +57.6364i q^{69} -15.3545i q^{71} -6.08230i q^{73} -2.42940 q^{75} +10.4526i q^{77} +28.1831 q^{79} +18.6335 q^{81} -86.8954i q^{83} -69.3981i q^{85} +(-16.4747 + 43.2881i) q^{87} -156.628i q^{89} +15.8842i q^{91} -54.3198 q^{93} +161.370 q^{95} +105.767i q^{97} -50.2311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 8 q^{5} + 52 q^{9}+O(q^{10}) \) 20 * q - 8 * q^5 + 52 * q^9	\( 20 q - 8 q^{5} + 52 q^{9} - 16 q^{13} + 12 q^{25} + 4 q^{29} - 96 q^{33} - 112 q^{45} - 76 q^{49} - 112 q^{53} + 120 q^{57} + 136 q^{65} - 52 q^{81}+O(q^{100}) \) 20 * q - 8 * q^5 + 52 * q^9 - 16 * q^13 + 12 * q^25 + 4 * q^29 - 96 * q^33 - 112 * q^45 - 76 * q^49 - 112 * q^53 + 120 * q^57 + 136 * q^65 - 52 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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