LMFDB - Embedded newform 460.2.c.a.369.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [460,2,Mod(369,460)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(460, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("460.369");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 460 = 2^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	460.c (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:	$3.67311849298$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 24x^{10} + 188x^{8} + 530x^{6} + 508x^{4} + 80x^{2} + 1 $ x^12 + 24x^10 + 188x^8 + 530x^6 + 508x^4 + 80*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			369.4
Root			$0.420790i$ of defining polynomial
Character	$\chi$	$=$	460.369
Dual form			460.2.c.a.369.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73961i q^{3} +(1.77747 - 1.35668i) q^{5} +3.32224i q^{7} -0.0262434 q^{9} +O(q^{10})$	$q-1.73961i q^{3} +(1.77747 - 1.35668i) q^{5} +3.32224i q^{7} -0.0262434 q^{9} +5.77103 q^{11} -1.10197i q^{13} +(-2.36010 - 3.09211i) q^{15} +0.893847i q^{17} -2.42839 q^{19} +5.77940 q^{21} -1.00000i q^{23} +(1.31882 - 4.82294i) q^{25} -5.17318i q^{27} -4.11268 q^{29} -9.54624 q^{31} -10.0393i q^{33} +(4.50722 + 5.90519i) q^{35} -7.69904i q^{37} -1.91700 q^{39} +0.00418347 q^{41} +9.97045i q^{43} +(-0.0466469 + 0.0356040i) q^{45} +10.0079i q^{47} -4.03726 q^{49} +1.55495 q^{51} -6.25169i q^{53} +(10.2579 - 7.82946i) q^{55} +4.22445i q^{57} +10.7764 q^{59} +10.5929 q^{61} -0.0871868i q^{63} +(-1.49502 - 1.95872i) q^{65} +10.9529i q^{67} -1.73961 q^{69} -12.9170 q^{71} -1.89943i q^{73} +(-8.39003 - 2.29423i) q^{75} +19.1727i q^{77} +0.216085 q^{79} -9.07804 q^{81} +5.38967i q^{83} +(1.21267 + 1.58879i) q^{85} +7.15446i q^{87} -6.00657 q^{89} +3.66100 q^{91} +16.6067i q^{93} +(-4.31640 + 3.29456i) q^{95} +2.08104i q^{97} -0.151451 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 20 q^{9}+O(q^{10}) $ 12 * q - 20 * q^9	$ 12 q - 20 q^{9} + 4 q^{11} + 2 q^{15} - 8 q^{19} + 8 q^{25} - 10 q^{29} + 18 q^{31} - 10 q^{35} + 16 q^{39} - 2 q^{41} + 2 q^{45} - 38 q^{49} - 24 q^{51} + 16 q^{55} + 22 q^{59} - 8 q^{61} + 38 q^{65} - 8 q^{69} - 34 q^{71} + 16 q^{75} - 20 q^{79} + 28 q^{81} + 6 q^{85} + 48 q^{89} - 8 q^{91} + 12 q^{95} + 32 q^{99}+O(q^{100}) $ 12 * q - 20 * q^9 + 4 * q^11 + 2 * q^15 - 8 * q^19 + 8 * q^25 - 10 * q^29 + 18 * q^31 - 10 * q^35 + 16 * q^39 - 2 * q^41 + 2 * q^45 - 38 * q^49 - 24 * q^51 + 16 * q^55 + 22 * q^59 - 8 * q^61 + 38 * q^65 - 8 * q^69 - 34 * q^71 + 16 * q^75 - 20 * q^79 + 28 * q^81 + 6 * q^85 + 48 * q^89 - 8 * q^91 + 12 * q^95 + 32 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/460\mathbb{Z}\right)^\times$.

$n$	$231$	$277$	$281$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	1.73961i		−	1.00436i	−0.864762	−	0.502182i	$-0.832531\pi$
$3$		−	1.73961i		−	1.00436i	0.864762	−	0.502182i	$-0.167469\pi$
$4$		0			0
$5$	1.77747	−	1.35668i	0.794910	−	0.606727i
$5$	1.77747	−	1.35668i	0.794910	−	0.606727i
$6$		0			0
$7$			3.32224i			1.25569i	0.778339	+	0.627844i	$0.216062\pi$
$7$			3.32224i			1.25569i	−0.778339	+	0.627844i	$0.783938\pi$
$8$		0			0
$9$	−0.0262434			−0.00874780
$10$		0			0
$11$	5.77103			1.74003			0.870016	−	0.493024i	$-0.164109\pi$
$11$	5.77103			1.74003			0.870016	+	0.493024i	$0.164109\pi$
$12$		0			0
$13$		−	1.10197i		−	0.305631i	−0.988255	−	0.152816i	$-0.951166\pi$
$13$		−	1.10197i		−	0.305631i	0.988255	−	0.152816i	$-0.0488341\pi$
$14$		0			0
$15$	−2.36010	−	3.09211i	−0.609375	−	0.798379i
$16$		0			0
$17$			0.893847i			0.216790i	0.994108	+	0.108395i	$0.0345711\pi$
$17$			0.893847i			0.216790i	−0.994108	+	0.108395i	$0.965429\pi$
$18$		0			0
$19$	−2.42839			−0.557111			−0.278555	−	0.960420i	$-0.589856\pi$
$19$	−2.42839			−0.557111			−0.278555	+	0.960420i	$0.589856\pi$
$20$		0			0
$21$	5.77940			1.26117
$22$		0			0
$23$		−	1.00000i		−	0.208514i
$23$		−	1.00000i		−	0.208514i
$24$		0			0
$25$	1.31882	−	4.82294i	0.263764	−	0.964587i
$26$		0			0
$27$		−	5.17318i		−	0.995578i
$28$		0			0
$29$	−4.11268			−0.763705			−0.381853	−	0.924223i	$-0.624714\pi$
$29$	−4.11268			−0.763705			−0.381853	+	0.924223i	$0.624714\pi$
$30$		0			0
$31$	−9.54624			−1.71456			−0.857278	−	0.514854i	$-0.827846\pi$
$31$	−9.54624			−1.71456			−0.857278	+	0.514854i	$0.827846\pi$
$32$		0			0
$33$		−	10.0393i		−	1.74763i
$34$		0			0
$35$	4.50722	+	5.90519i	0.761860	+	0.998159i
$36$		0			0
$37$		−	7.69904i		−	1.26572i	−0.774268	−	0.632858i	$-0.781882\pi$
$37$		−	7.69904i		−	1.26572i	0.774268	−	0.632858i	$-0.218118\pi$
$38$		0			0
$39$	−1.91700			−0.306965
$40$		0			0
$41$	0.00418347			0.000653348			0.000326674	−	1.00000i	$-0.499896\pi$
$41$	0.00418347			0.000653348			0.000326674		1.00000i	$0.499896\pi$
$42$		0			0
$43$			9.97045i			1.52048i	0.649643	+	0.760240i	$0.274919\pi$
$43$			9.97045i			1.52048i	−0.649643	+	0.760240i	$0.725081\pi$
$44$		0			0
$45$	−0.0466469	+	0.0356040i	−0.00695371	+	0.00530753i
$46$		0			0
$47$			10.0079i			1.45981i	0.683551	+	0.729903i	$0.260435\pi$
$47$			10.0079i			1.45981i	−0.683551	+	0.729903i	$0.739565\pi$
$48$		0			0
$49$	−4.03726			−0.576751
$50$		0			0
$51$	1.55495			0.217736
$52$		0			0
$53$		−	6.25169i		−	0.858735i	−0.903130	−	0.429368i	$-0.858736\pi$
$53$		−	6.25169i		−	0.858735i	0.903130	−	0.429368i	$-0.141264\pi$
$54$		0			0
$55$	10.2579	−	7.82946i	1.38317	−	1.05572i
$56$		0			0
$57$			4.22445i			0.559542i
$58$		0			0
$59$	10.7764			1.40297			0.701486	−	0.712684i	$-0.252520\pi$
$59$	10.7764			1.40297			0.701486	+	0.712684i	$0.252520\pi$
$60$		0			0
$61$	10.5929			1.35628			0.678140	−	0.734932i	$-0.262786\pi$
$61$	10.5929			1.35628			0.678140	+	0.734932i	$0.262786\pi$
$62$		0			0
$63$		−	0.0871868i		−	0.0109845i
$64$		0			0
$65$	−1.49502	−	1.95872i	−0.185435	−	0.242949i
$66$		0			0
$67$			10.9529i			1.33811i	0.743213	+	0.669055i	$0.233301\pi$
$67$			10.9529i			1.33811i	−0.743213	+	0.669055i	$0.766699\pi$
$68$		0			0
$69$	−1.73961			−0.209424
$70$		0			0
$71$	−12.9170			−1.53296			−0.766481	−	0.642267i	$-0.777994\pi$
$71$	−12.9170			−1.53296			−0.766481	+	0.642267i	$0.777994\pi$
$72$		0			0
$73$		−	1.89943i		−	0.222311i	−0.993803	−	0.111156i	$-0.964545\pi$
$73$		−	1.89943i		−	0.222311i	0.993803	−	0.111156i	$-0.0354552\pi$
$74$		0			0
$75$	−8.39003	−	2.29423i	−0.968797	−	0.264915i
$76$		0			0
$77$			19.1727i			2.18494i
$78$		0			0
$79$	0.216085			0.0243114			0.0121557	−	0.999926i	$-0.496131\pi$
$79$	0.216085			0.0243114			0.0121557	+	0.999926i	$0.496131\pi$
$80$		0			0
$81$	−9.07804			−1.00867
$82$		0			0
$83$			5.38967i			0.591593i	0.955251	+	0.295796i	$0.0955850\pi$
$83$			5.38967i			0.591593i	−0.955251	+	0.295796i	$0.904415\pi$
$84$		0			0
$85$	1.21267	+	1.58879i	0.131532	+	0.172328i
$86$		0			0
$87$			7.15446i			0.767038i
$88$		0			0
$89$	−6.00657			−0.636696			−0.318348	−	0.947974i	$-0.603128\pi$
$89$	−6.00657			−0.636696			−0.318348	+	0.947974i	$0.603128\pi$
$90$		0			0
$91$	3.66100			0.383777
$92$		0			0
$93$			16.6067i			1.72204i
$94$		0			0
$95$	−4.31640	+	3.29456i	−0.442853	+	0.338014i
$96$		0			0
$97$			2.08104i			0.211297i	0.994404	+	0.105649i	$0.0336919\pi$
$97$			2.08104i			0.211297i	−0.994404	+	0.105649i	$0.966308\pi$
$98$		0			0
$99$	−0.151451			−0.0152214
$100$		0			0
$101$	−13.1576			−1.30923			−0.654617	−	0.755961i	$-0.727170\pi$
$101$	−13.1576			−1.30923			−0.654617	+	0.755961i	$0.727170\pi$
$102$		0			0
$103$		−	18.0956i		−	1.78301i	−0.453008	−	0.891507i	$-0.649649\pi$
$103$		−	18.0956i		−	1.78301i	0.453008	−	0.891507i	$-0.350351\pi$
$104$		0			0
$105$	10.2727	−	7.84081i	1.00252	−	0.765185i
$106$		0			0
$107$			6.97263i			0.674069i	0.941492	+	0.337035i	$0.109424\pi$
$107$			6.97263i			0.674069i	−0.941492	+	0.337035i	$0.890576\pi$
$108$		0			0
$109$	−10.2873			−0.985344			−0.492672	−	0.870215i	$-0.663980\pi$
$109$	−10.2873			−0.985344			−0.492672	+	0.870215i	$0.663980\pi$
$110$		0			0
$111$	−13.3933			−1.27124
$112$		0			0
$113$			5.53454i			0.520646i	0.965522	+	0.260323i	$0.0838290\pi$
$113$			5.53454i			0.520646i	−0.965522	+	0.260323i	$0.916171\pi$
$114$		0			0
$115$	−1.35668	−	1.77747i	−0.126511	−	0.165750i
$116$		0			0
$117$			0.0289194i			0.00267360i
$118$		0			0
$119$	−2.96957			−0.272220
$120$		0			0
$121$	22.3048			2.02771
$122$		0			0
$123$		−	0.00727760i		−	0.000656199i
$124$		0			0
$125$	−4.19903	−	10.3619i	−0.375573	−	0.926793i
$126$		0			0
$127$			11.1580i			0.990110i	0.868862	+	0.495055i	$0.164852\pi$
$127$			11.1580i			0.990110i	−0.868862	+	0.495055i	$0.835148\pi$
$128$		0			0
$129$	17.3447			1.52712
$130$		0			0
$131$	−15.0421			−1.31423			−0.657117	−	0.753789i	$-0.728224\pi$
$131$	−15.0421			−1.31423			−0.657117	+	0.753789i	$0.728224\pi$
$132$		0			0
$133$		−	8.06769i		−	0.699557i
$134$		0			0
$135$	−7.01836	−	9.19518i	−0.604045	−	0.791395i
$136$		0			0
$137$			2.21231i			0.189010i	0.995524	+	0.0945050i	$0.0301268\pi$
$137$			2.21231i			0.189010i	−0.995524	+	0.0945050i	$0.969873\pi$
$138$		0			0
$139$	6.53729			0.554486			0.277243	−	0.960800i	$-0.410579\pi$
$139$	6.53729			0.554486			0.277243	+	0.960800i	$0.410579\pi$
$140$		0			0
$141$	17.4099			1.46618
$142$		0			0
$143$		−	6.35950i		−	0.531808i
$144$		0			0
$145$	−7.31017	+	5.57960i	−0.607077	+	0.463361i
$146$		0			0
$147$			7.02326i			0.579269i
$148$		0			0
$149$	11.1844			0.916263			0.458131	−	0.888884i	$-0.348519\pi$
$149$	11.1844			0.916263			0.458131	+	0.888884i	$0.348519\pi$
$150$		0			0
$151$	−1.29176			−0.105122			−0.0525610	−	0.998618i	$-0.516738\pi$
$151$	−1.29176			−0.105122			−0.0525610	+	0.998618i	$0.516738\pi$
$152$		0			0
$153$		−	0.0234576i		−	0.00189643i
$154$		0			0
$155$	−16.9682	+	12.9512i	−1.36292	+	1.04027i
$156$		0			0
$157$			11.0465i			0.881609i	0.897603	+	0.440805i	$0.145307\pi$
$157$			11.0465i			0.881609i	−0.897603	+	0.440805i	$0.854693\pi$
$158$		0			0
$159$	−10.8755			−0.862483
$160$		0			0
$161$	3.32224			0.261829
$162$		0			0
$163$		−	5.04265i		−	0.394971i	−0.980306	−	0.197485i	$-0.936723\pi$
$163$		−	5.04265i		−	0.394971i	0.980306	−	0.197485i	$-0.0632775\pi$
$164$		0			0
$165$	−13.6202	−	17.8447i	−1.06033	−	1.38921i
$166$		0			0
$167$			1.20759i			0.0934465i	0.998908	+	0.0467232i	$0.0148779\pi$
$167$			1.20759i			0.0934465i	−0.998908	+	0.0467232i	$0.985122\pi$
$168$		0			0
$169$	11.7857			0.906590
$170$		0			0
$171$	0.0637292			0.00487349
$172$		0			0
$173$			15.7392i			1.19663i	0.801262	+	0.598313i	$0.204162\pi$
$173$			15.7392i			1.19663i	−0.801262	+	0.598313i	$0.795838\pi$
$174$		0			0
$175$	16.0229	+	4.38143i	1.21122	+	0.331205i
$176$		0			0
$177$		−	18.7468i		−	1.40909i
$178$		0			0
$179$	−6.61963			−0.494774			−0.247387	−	0.968917i	$-0.579572\pi$
$179$	−6.61963			−0.494774			−0.247387	+	0.968917i	$0.579572\pi$
$180$		0			0
$181$	−16.0433			−1.19249			−0.596245	−	0.802803i	$-0.703341\pi$
$181$	−16.0433			−1.19249			−0.596245	+	0.802803i	$0.703341\pi$
$182$		0			0
$183$		−	18.4275i		−	1.36220i
$184$		0			0
$185$	−10.4452	−	13.6848i	−0.767944	−	1.00613i
$186$		0			0
$187$			5.15842i			0.377221i
$188$		0			0
$189$	17.1865			1.25014
$190$		0			0
$191$	0.295590			0.0213881			0.0106941	−	0.999943i	$-0.496596\pi$
$191$	0.295590			0.0213881			0.0106941	+	0.999943i	$0.496596\pi$
$192$		0			0
$193$			1.42564i			0.102620i	0.998683	+	0.0513101i	$0.0163397\pi$
$193$			1.42564i			0.102620i	−0.998683	+	0.0513101i	$0.983660\pi$
$194$		0			0
$195$	−3.40741	+	2.60076i	−0.244010	+	0.186244i
$196$		0			0
$197$		−	8.60107i		−	0.612801i	−0.951903	−	0.306401i	$-0.900875\pi$
$197$		−	8.60107i		−	0.612801i	0.951903	−	0.306401i	$-0.0991247\pi$
$198$		0			0
$199$	−3.81515			−0.270449			−0.135224	−	0.990815i	$-0.543176\pi$
$199$	−3.81515			−0.270449			−0.135224	+	0.990815i	$0.543176\pi$
$200$		0			0
$201$	19.0538			1.34395
$202$		0			0
$203$		−	13.6633i		−	0.958975i
$204$		0			0
$205$	0.00743600	−	0.00567564i	0.000519353	−	0.000396404i
$206$		0			0
$207$			0.0262434i			0.00182404i
$208$		0			0
$209$	−14.0143			−0.969390
$210$		0			0
$211$	1.32370			0.0911273			0.0455637	−	0.998961i	$-0.485492\pi$
$211$	1.32370			0.0911273			0.0455637	+	0.998961i	$0.485492\pi$
$212$		0			0
$213$			22.4705i			1.53965i
$214$		0			0
$215$	13.5267	+	17.7222i	0.922516	+	1.20864i
$216$		0			0
$217$		−	31.7149i		−	2.15295i
$218$		0			0
$219$	−3.30427			−0.223282
$220$		0			0
$221$	0.984992			0.0662577
$222$		0			0
$223$		−	25.1286i		−	1.68274i	−0.540462	−	0.841368i	$-0.681751\pi$
$223$		−	25.1286i		−	1.68274i	0.540462	−	0.841368i	$-0.318249\pi$
$224$		0			0
$225$	−0.0346103	+	0.126570i	−0.00230736	+	0.00843802i
$226$		0			0
$227$		−	23.8021i		−	1.57980i	−0.613236	−	0.789900i	$-0.710133\pi$
$227$		−	23.8021i		−	1.57980i	0.613236	−	0.789900i	$-0.289867\pi$
$228$		0			0
$229$	0.789351			0.0521618			0.0260809	−	0.999660i	$-0.491697\pi$
$229$	0.789351			0.0521618			0.0260809	+	0.999660i	$0.491697\pi$
$230$		0			0
$231$	33.3531			2.19447
$232$		0			0
$233$			20.8101i			1.36331i	0.731672	+	0.681656i	$0.238740\pi$
$233$			20.8101i			1.36331i	−0.731672	+	0.681656i	$0.761260\pi$
$234$		0			0
$235$	13.5776	+	17.7888i	0.885704	+	1.16041i
$236$		0			0
$237$		−	0.375903i		−	0.0244175i
$238$		0			0
$239$	25.2398			1.63262			0.816312	−	0.577611i	$-0.196015\pi$
$239$	25.2398			1.63262			0.816312	+	0.577611i	$0.196015\pi$
$240$		0			0
$241$	5.27710			0.339928			0.169964	−	0.985450i	$-0.445635\pi$
$241$	5.27710			0.339928			0.169964	+	0.985450i	$0.445635\pi$
$242$		0			0
$243$			0.272722i			0.0174951i
$244$		0			0
$245$	−7.17612	+	5.47728i	−0.458466	+	0.349931i
$246$		0			0
$247$			2.67601i			0.170270i
$248$		0			0
$249$	9.37592			0.594175
$250$		0			0
$251$	−10.8439			−0.684461			−0.342230	−	0.939616i	$-0.611182\pi$
$251$	−10.8439			−0.684461			−0.342230	+	0.939616i	$0.611182\pi$
$252$		0			0
$253$		−	5.77103i		−	0.362822i
$254$		0			0
$255$	2.76387	−	2.10957i	0.173081	−	0.132106i
$256$		0			0
$257$			18.6238i			1.16172i	0.814004	+	0.580859i	$0.197283\pi$
$257$			18.6238i			1.16172i	−0.814004	+	0.580859i	$0.802717\pi$
$258$		0			0
$259$	25.5781			1.58934
$260$		0			0
$261$	0.107931			0.00668074
$262$		0			0
$263$			19.4370i			1.19854i	0.800548	+	0.599269i	$0.204542\pi$
$263$			19.4370i			1.19854i	−0.800548	+	0.599269i	$0.795458\pi$
$264$		0			0
$265$	−8.48156	−	11.1122i	−0.521018	−	0.682617i
$266$		0			0
$267$			10.4491i			0.639474i
$268$		0			0
$269$	3.80493			0.231991			0.115995	−	0.993250i	$-0.462994\pi$
$269$	3.80493			0.231991			0.115995	+	0.993250i	$0.462994\pi$
$270$		0			0
$271$	−18.2100			−1.10618			−0.553089	−	0.833122i	$-0.686551\pi$
$271$	−18.2100			−1.10618			−0.553089	+	0.833122i	$0.686551\pi$
$272$		0			0
$273$		−	6.36872i		−	0.385452i
$274$		0			0
$275$	7.61095	−	27.8333i	0.458958	−	1.67841i
$276$		0			0
$277$			14.7624i			0.886988i	0.896277	+	0.443494i	$0.146261\pi$
$277$			14.7624i			0.886988i	−0.896277	+	0.443494i	$0.853739\pi$
$278$		0			0
$279$	0.250526			0.0149986
$280$		0			0
$281$	−6.29738			−0.375670			−0.187835	−	0.982201i	$-0.560147\pi$
$281$	−6.29738			−0.375670			−0.187835	+	0.982201i	$0.560147\pi$
$282$		0			0
$283$		−	28.4443i		−	1.69084i	−0.534105	−	0.845418i	$-0.679351\pi$
$283$		−	28.4443i		−	1.69084i	0.534105	−	0.845418i	$-0.320649\pi$
$284$		0			0
$285$	5.73124	+	7.50885i	0.339490	+	0.444786i
$286$		0			0
$287$			0.0138985i			0.000820401i
$288$		0			0
$289$	16.2010			0.953002
$290$		0			0
$291$	3.62020			0.212220
$292$		0			0
$293$		−	14.4907i		−	0.846556i	−0.906000	−	0.423278i	$-0.860879\pi$
$293$		−	14.4907i		−	0.846556i	0.906000	−	0.423278i	$-0.139121\pi$
$294$		0			0
$295$	19.1548	−	14.6202i	1.11524	−	0.851221i
$296$		0			0
$297$		−	29.8546i		−	1.73234i
$298$		0			0
$299$	−1.10197			−0.0637285
$300$		0			0
$301$	−33.1242			−1.90925
$302$		0			0
$303$			22.8892i			1.31495i
$304$		0			0
$305$	18.8286	−	14.3712i	1.07812	−	0.822893i
$306$		0			0
$307$		−	30.4707i		−	1.73905i	−0.493885	−	0.869527i	$-0.664424\pi$
$307$		−	30.4707i		−	1.73905i	0.493885	−	0.869527i	$-0.335576\pi$
$308$		0			0
$309$	−31.4793			−1.79080
$310$		0			0
$311$	20.9031			1.18531			0.592654	−	0.805457i	$-0.298080\pi$
$311$	20.9031			1.18531			0.592654	+	0.805457i	$0.298080\pi$
$312$		0			0
$313$		−	14.3614i		−	0.811757i	−0.913927	−	0.405878i	$-0.866966\pi$
$313$		−	14.3614i		−	0.811757i	0.913927	−	0.405878i	$-0.133034\pi$
$314$		0			0
$315$	−0.118285	−	0.154972i	−0.00666460	−	0.00873169i
$316$		0			0
$317$			33.6939i			1.89244i	0.323525	+	0.946220i	$0.395132\pi$
$317$			33.6939i			1.89244i	−0.323525	+	0.946220i	$0.604868\pi$
$318$		0			0
$319$	−23.7344			−1.32887
$320$		0			0
$321$	12.1297			0.677011
$322$		0			0
$323$		−	2.17061i		−	0.120776i
$324$		0			0
$325$	−5.31473	−	1.45330i	−0.294808	−	0.0806146i
$326$		0			0
$327$			17.8959i			0.989644i
$328$		0			0
$329$	−33.2487			−1.83306
$330$		0			0
$331$	9.82292			0.539917			0.269958	−	0.962872i	$-0.412990\pi$
$331$	9.82292			0.539917			0.269958	+	0.962872i	$0.412990\pi$
$332$		0			0
$333$			0.202049i			0.0110722i
$334$		0			0
$335$	14.8596	+	19.4685i	0.811868	+	1.06368i
$336$		0			0
$337$		−	13.2410i		−	0.721286i	−0.932704	−	0.360643i	$-0.882557\pi$
$337$		−	13.2410i		−	0.721286i	0.932704	−	0.360643i	$-0.117443\pi$
$338$		0			0
$339$	9.62795			0.522918
$340$		0			0
$341$	−55.0917			−2.98338
$342$		0			0
$343$			9.84292i			0.531468i
$344$		0			0
$345$	−3.09211	+	2.36010i	−0.166474	+	0.127064i
$346$		0			0
$347$		−	9.15627i		−	0.491534i	−0.969329	−	0.245767i	$-0.920960\pi$
$347$		−	9.15627i		−	0.491534i	0.969329	−	0.245767i	$-0.0790399\pi$
$348$		0			0
$349$	−25.7744			−1.37967			−0.689836	−	0.723966i	$-0.742317\pi$
$349$	−25.7744			−1.37967			−0.689836	+	0.723966i	$0.742317\pi$
$350$		0			0
$351$	−5.70068			−0.304280
$352$		0			0
$353$			2.40654i			0.128087i	0.997947	+	0.0640436i	$0.0203997\pi$
$353$			2.40654i			0.128087i	−0.997947	+	0.0640436i	$0.979600\pi$
$354$		0			0
$355$	−22.9596	+	17.5242i	−1.21857	+	0.930090i
$356$		0			0
$357$			5.16590i			0.273408i
$358$		0			0
$359$	−21.5236			−1.13597			−0.567985	−	0.823039i	$-0.692277\pi$
$359$	−21.5236			−1.13597			−0.567985	+	0.823039i	$0.692277\pi$
$360$		0			0
$361$	−13.1029			−0.689628
$362$		0			0
$363$		−	38.8016i		−	2.03656i
$364$		0			0
$365$	−2.57692	−	3.37618i	−0.134882	−	0.176717i
$366$		0			0
$367$		−	6.47203i		−	0.337837i	−0.985630	−	0.168919i	$-0.945972\pi$
$367$		−	6.47203i		−	0.337837i	0.985630	−	0.168919i	$-0.0540275\pi$
$368$		0			0
$369$	−0.000109788	0		−5.71536e−6	0
$370$		0			0
$371$	20.7696			1.07830
$372$		0			0
$373$			27.7866i			1.43874i	0.694629	+	0.719368i	$0.255569\pi$
$373$			27.7866i			1.43874i	−0.694629	+	0.719368i	$0.744431\pi$
$374$		0			0
$375$	−18.0256	+	7.30467i	−0.930838	+	0.377212i
$376$		0			0
$377$			4.53204i			0.233412i
$378$		0			0
$379$	−7.63904			−0.392391			−0.196196	−	0.980565i	$-0.562859\pi$
$379$	−7.63904			−0.392391			−0.196196	+	0.980565i	$0.562859\pi$
$380$		0			0
$381$	19.4105			0.994432
$382$		0			0
$383$			19.4206i			0.992348i	0.868223	+	0.496174i	$0.165262\pi$
$383$			19.4206i			0.992348i	−0.868223	+	0.496174i	$0.834738\pi$
$384$		0			0
$385$	26.0113	+	34.0790i	1.32566	+	1.73683i
$386$		0			0
$387$		−	0.261659i		−	0.0133009i
$388$		0			0
$389$	26.1255			1.32461			0.662306	−	0.749233i	$-0.269578\pi$
$389$	26.1255			1.32461			0.662306	+	0.749233i	$0.269578\pi$
$390$		0			0
$391$	0.893847			0.0452038
$392$		0			0
$393$			26.1674i			1.31997i
$394$		0			0
$395$	0.384085	−	0.293158i	0.0193254	−	0.0147504i
$396$		0			0
$397$		−	11.5388i		−	0.579117i	−0.957160	−	0.289558i	$-0.906492\pi$
$397$		−	11.5388i		−	0.579117i	0.957160	−	0.289558i	$-0.0935085\pi$
$398$		0			0
$399$	−14.0346			−0.702610
$400$		0			0
$401$	−0.188549			−0.00941568			−0.00470784	−	0.999989i	$-0.501499\pi$
$401$	−0.188549			−0.00941568			−0.00470784	+	0.999989i	$0.501499\pi$
$402$		0			0
$403$			10.5197i			0.524022i
$404$		0			0
$405$	−16.1360	+	12.3160i	−0.801803	+	0.611988i
$406$		0			0
$407$		−	44.4314i		−	2.20238i
$408$		0			0
$409$	8.48327			0.419471			0.209735	−	0.977758i	$-0.432740\pi$
$409$	8.48327			0.419471			0.209735	+	0.977758i	$0.432740\pi$
$410$		0			0
$411$	3.84855			0.189835
$412$		0			0
$413$			35.8018i			1.76169i
$414$		0			0
$415$	7.31207	+	9.57999i	0.358935	+	0.470263i
$416$		0			0
$417$		−	11.3723i		−	0.556906i
$418$		0			0
$419$	2.15604			0.105329			0.0526647	−	0.998612i	$-0.483229\pi$
$419$	2.15604			0.105329			0.0526647	+	0.998612i	$0.483229\pi$
$420$		0			0
$421$	8.01842			0.390794			0.195397	−	0.980724i	$-0.437400\pi$
$421$	8.01842			0.390794			0.195397	+	0.980724i	$0.437400\pi$
$422$		0			0
$423$		−	0.262642i		−	0.0127701i
$424$		0			0
$425$	4.31097	+	1.17882i	0.209113	+	0.0571814i
$426$		0			0
$427$			35.1921i			1.70307i
$428$		0			0
$429$	−11.0630			−0.534129
$430$		0			0
$431$	−15.3343			−0.738629			−0.369315	−	0.929304i	$-0.620408\pi$
$431$	−15.3343			−0.738629			−0.369315	+	0.929304i	$0.620408\pi$
$432$		0			0
$433$		−	16.3093i		−	0.783777i	−0.920013	−	0.391888i	$-0.871822\pi$
$433$		−	16.3093i		−	0.783777i	0.920013	−	0.391888i	$-0.128178\pi$
$434$		0			0
$435$	9.70633	+	12.7169i	0.465383	+	0.609726i
$436$		0			0
$437$			2.42839i			0.116166i
$438$		0			0
$439$	23.9534			1.14323			0.571617	−	0.820520i	$-0.306316\pi$
$439$	23.9534			1.14323			0.571617	+	0.820520i	$0.306316\pi$
$440$		0			0
$441$	0.105951			0.00504531
$442$		0			0
$443$			29.2360i			1.38905i	0.719471	+	0.694523i	$0.244384\pi$
$443$			29.2360i			1.38905i	−0.719471	+	0.694523i	$0.755616\pi$
$444$		0			0
$445$	−10.6765	+	8.14902i	−0.506116	+	0.386301i
$446$		0			0
$447$		−	19.4565i		−	0.920262i
$448$		0			0
$449$	21.2244			1.00164			0.500822	−	0.865550i	$-0.333031\pi$
$449$	21.2244			1.00164			0.500822	+	0.865550i	$0.333031\pi$
$450$		0			0
$451$	0.0241429			0.00113685
$452$		0			0
$453$			2.24716i			0.105581i
$454$		0			0
$455$	6.50733	−	4.96682i	0.305069	−	0.232848i
$456$		0			0
$457$		−	31.5959i		−	1.47799i	−0.673710	−	0.738996i	$-0.735300\pi$
$457$		−	31.5959i		−	1.47799i	0.673710	−	0.738996i	$-0.264700\pi$
$458$		0			0
$459$	4.62403			0.215831
$460$		0			0
$461$	16.4858			0.767820			0.383910	−	0.923370i	$-0.374577\pi$
$461$	16.4858			0.767820			0.383910	+	0.923370i	$0.374577\pi$
$462$		0			0
$463$		−	30.1555i		−	1.40144i	−0.713435	−	0.700722i	$-0.752861\pi$
$463$		−	30.1555i		−	1.40144i	0.713435	−	0.700722i	$-0.247139\pi$
$464$		0			0
$465$	22.5301	+	29.5180i	1.04481	+	1.36887i
$466$		0			0
$467$		−	7.89645i		−	0.365404i	−0.983168	−	0.182702i	$-0.941516\pi$
$467$		−	7.89645i		−	0.365404i	0.983168	−	0.182702i	$-0.0584844\pi$
$468$		0			0
$469$	−36.3882			−1.68025
$470$		0			0
$471$	19.2167			0.885457
$472$		0			0
$473$			57.5398i			2.64568i
$474$		0			0
$475$	−3.20261	+	11.7120i	−0.146946	+	0.537382i
$476$		0			0
$477$			0.164066i			0.00751205i
$478$		0			0
$479$	−39.5234			−1.80587			−0.902935	−	0.429778i	$-0.858592\pi$
$479$	−39.5234			−1.80587			−0.902935	+	0.429778i	$0.858592\pi$
$480$		0			0
$481$	−8.48411			−0.386842
$482$		0			0
$483$		−	5.77940i		−	0.262972i
$484$		0			0
$485$	2.82331	+	3.69899i	0.128200	+	0.167962i
$486$		0			0
$487$			3.20444i			0.145207i	0.997361	+	0.0726035i	$0.0231308\pi$
$487$			3.20444i			0.145207i	−0.997361	+	0.0726035i	$0.976869\pi$
$488$		0			0
$489$	−8.77224			−0.396695
$490$		0			0
$491$	34.8865			1.57441			0.787203	−	0.616694i	$-0.211528\pi$
$491$	34.8865			1.57441			0.787203	+	0.616694i	$0.211528\pi$
$492$		0			0
$493$		−	3.67611i		−	0.165564i
$494$		0			0
$495$	−0.269201	+	0.205472i	−0.0120997	+	0.00923527i
$496$		0			0
$497$		−	42.9132i		−	1.92492i
$498$		0			0
$499$	−33.1594			−1.48442			−0.742210	−	0.670167i	$-0.766222\pi$
$499$	−33.1594			−1.48442			−0.742210	+	0.670167i	$0.766222\pi$
$500$		0			0
$501$	2.10074			0.0938543
$502$		0			0
$503$		−	7.05945i		−	0.314765i	−0.987538	−	0.157383i	$-0.949694\pi$
$503$		−	7.05945i		−	0.314765i	0.987538	−	0.157383i	$-0.0503056\pi$
$504$		0			0
$505$	−23.3874	+	17.8508i	−1.04072	+	0.794348i
$506$		0			0
$507$		−	20.5025i		−	0.910546i
$508$		0			0
$509$	−16.6812			−0.739382			−0.369691	−	0.929155i	$-0.620536\pi$
$509$	−16.6812			−0.739382			−0.369691	+	0.929155i	$0.620536\pi$
$510$		0			0
$511$	6.31035			0.279154
$512$		0			0
$513$			12.5625i			0.554648i
$514$		0			0
$515$	−24.5500	−	32.1645i	−1.08180	−	1.41734i
$516$		0			0
$517$			57.7560i			2.54011i
$518$		0			0
$519$	27.3800			1.20185
$520$		0			0
$521$	−24.9941			−1.09501			−0.547507	−	0.836801i	$-0.684423\pi$
$521$	−24.9941			−1.09501			−0.547507	+	0.836801i	$0.684423\pi$
$522$		0			0
$523$			14.3283i			0.626535i	0.949665	+	0.313267i	$0.101424\pi$
$523$			14.3283i			0.626535i	−0.949665	+	0.313267i	$0.898576\pi$
$524$		0			0
$525$	7.62199	−	27.8737i	0.332651	−	1.21651i
$526$		0			0
$527$		−	8.53289i		−	0.371698i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$	−0.282810			−0.0122729
$532$		0			0
$533$		−	0.00461005i		−	0.000199683i
$534$		0			0
$535$	9.45965	+	12.3937i	0.408976	+	0.535824i
$536$		0			0
$537$			11.5156i			0.496934i
$538$		0			0
$539$	−23.2992			−1.00357
$540$		0			0
$541$	22.0323			0.947244			0.473622	−	0.880728i	$-0.342946\pi$
$541$	22.0323			0.947244			0.473622	+	0.880728i	$0.342946\pi$
$542$		0			0
$543$			27.9091i			1.19769i
$544$		0			0
$545$	−18.2854	+	13.9566i	−0.783260	+	0.597835i
$546$		0			0
$547$			21.5194i			0.920104i	0.887892	+	0.460052i	$0.152169\pi$
$547$			21.5194i			0.920104i	−0.887892	+	0.460052i	$0.847831\pi$
$548$		0			0
$549$	−0.277993			−0.0118645
$550$		0			0
$551$	9.98719			0.425468
$552$		0			0
$553$			0.717884i			0.0305276i
$554$		0			0
$555$	−23.8063	+	18.1705i	−1.01052	+	0.771295i
$556$		0			0
$557$		−	21.1923i		−	0.897945i	−0.893546	−	0.448972i	$-0.851790\pi$
$557$		−	21.1923i		−	0.897945i	0.893546	−	0.448972i	$-0.148210\pi$
$558$		0			0
$559$	10.9871			0.464706
$560$		0			0
$561$	8.97364			0.378867
$562$		0			0
$563$		−	23.6292i		−	0.995851i	−0.867220	−	0.497925i	$-0.834095\pi$
$563$		−	23.6292i		−	0.995851i	0.867220	−	0.497925i	$-0.165905\pi$
$564$		0			0
$565$	7.50862	+	9.83750i	0.315890	+	0.413867i
$566$		0			0
$567$		−	30.1594i		−	1.26658i
$568$		0			0
$569$	33.7490			1.41483			0.707415	−	0.706798i	$-0.249861\pi$
$569$	33.7490			1.41483			0.707415	+	0.706798i	$0.249861\pi$
$570$		0			0
$571$	−25.4667			−1.06575			−0.532874	−	0.846194i	$-0.678888\pi$
$571$	−25.4667			−1.06575			−0.532874	+	0.846194i	$0.678888\pi$
$572$		0			0
$573$		−	0.514211i		−	0.0214815i
$574$		0			0
$575$	−4.82294	−	1.31882i	−0.201130	−	0.0549986i
$576$		0			0
$577$		−	22.7293i		−	0.946234i	−0.881000	−	0.473117i	$-0.843129\pi$
$577$		−	22.7293i		−	0.946234i	0.881000	−	0.473117i	$-0.156871\pi$
$578$		0			0
$579$	2.48006			0.103068
$580$		0			0
$581$	−17.9057			−0.742856
$582$		0			0
$583$		−	36.0787i		−	1.49423i
$584$		0			0
$585$	0.0392345	+	0.0514035i	0.00162215	+	0.00212527i
$586$		0			0
$587$		−	35.6609i		−	1.47188i	−0.677045	−	0.735941i	$-0.736740\pi$
$587$		−	35.6609i		−	1.47188i	0.677045	−	0.735941i	$-0.263260\pi$
$588$		0			0
$589$	23.1820			0.955198
$590$		0			0
$591$	−14.9625			−0.615476
$592$		0			0
$593$		−	20.8412i		−	0.855845i	−0.903815	−	0.427922i	$-0.859246\pi$
$593$		−	20.8412i		−	0.855845i	0.903815	−	0.427922i	$-0.140754\pi$
$594$		0			0
$595$	−5.27834	+	4.02877i	−0.216391	+	0.165163i
$596$		0			0
$597$			6.63687i			0.271629i
$598$		0			0
$599$	10.9952			0.449252			0.224626	−	0.974445i	$-0.427884\pi$
$599$	10.9952			0.449252			0.224626	+	0.974445i	$0.427884\pi$
$600$		0			0
$601$	2.35032			0.0958714			0.0479357	−	0.998850i	$-0.484736\pi$
$601$	2.35032			0.0958714			0.0479357	+	0.998850i	$0.484736\pi$
$602$		0			0
$603$		−	0.287442i		−	0.0117055i
$604$		0			0
$605$	39.6462	−	30.2605i	1.61185	−	1.23027i
$606$		0			0
$607$			31.8412i			1.29239i	0.763170	+	0.646197i	$0.223642\pi$
$607$			31.8412i			1.29239i	−0.763170	+	0.646197i	$0.776358\pi$
$608$		0			0
$609$	−23.7688			−0.963160
$610$		0			0
$611$	11.0284			0.446162
$612$		0			0
$613$		−	29.0597i		−	1.17371i	−0.809692	−	0.586856i	$-0.800366\pi$
$613$		−	29.0597i		−	1.17371i	0.809692	−	0.586856i	$-0.199634\pi$
$614$		0			0
$615$	−0.00987340	−	0.0129357i	−0.000398134	−	0.000521619i
$616$		0			0
$617$		−	29.9851i		−	1.20715i	−0.797304	−	0.603577i	$-0.793741\pi$
$617$		−	29.9851i		−	1.20715i	0.797304	−	0.603577i	$-0.206259\pi$
$618$		0			0
$619$	16.9204			0.680089			0.340045	−	0.940409i	$-0.389558\pi$
$619$	16.9204			0.680089			0.340045	+	0.940409i	$0.389558\pi$
$620$		0			0
$621$	−5.17318			−0.207592
$622$		0			0
$623$		−	19.9553i		−	0.799491i
$624$		0			0
$625$	−21.5214	−	12.7212i	−0.860857	−	0.508847i
$626$		0			0
$627$			24.3794i			0.973621i
$628$		0			0
$629$	6.88177			0.274394
$630$		0			0
$631$	−20.4522			−0.814188			−0.407094	−	0.913386i	$-0.633458\pi$
$631$	−20.4522			−0.814188			−0.407094	+	0.913386i	$0.633458\pi$
$632$		0			0
$633$		−	2.30272i		−	0.0915250i
$634$		0			0
$635$	15.1378	+	19.8330i	0.600727	+	0.787049i
$636$		0			0
$637$			4.44894i			0.176273i
$638$		0			0
$639$	0.338985			0.0134100
$640$		0			0
$641$	−8.27686			−0.326916			−0.163458	−	0.986550i	$-0.552265\pi$
$641$	−8.27686			−0.326916			−0.163458	+	0.986550i	$0.552265\pi$
$642$		0			0
$643$		−	0.627685i		−	0.0247535i	−0.999923	−	0.0123767i	$-0.996060\pi$
$643$		−	0.627685i		−	0.0247535i	0.999923	−	0.0123767i	$-0.00393974\pi$
$644$		0			0
$645$	30.8297	−	23.5313i	1.21392	−	0.926543i
$646$		0			0
$647$			12.3508i			0.485561i	0.970081	+	0.242781i	$0.0780596\pi$
$647$			12.3508i			0.485561i	−0.970081	+	0.242781i	$0.921940\pi$
$648$		0			0
$649$	62.1911			2.44121
$650$		0			0
$651$	−55.1715			−2.16234
$652$		0			0
$653$			18.3211i			0.716959i	0.933538	+	0.358480i	$0.116705\pi$
$653$			18.3211i			0.716959i	−0.933538	+	0.358480i	$0.883295\pi$
$654$		0			0
$655$	−26.7369	+	20.4074i	−1.04470	+	0.797381i
$656$		0			0
$657$			0.0498475i			0.00194473i
$658$		0			0
$659$	39.1896			1.52661			0.763306	−	0.646038i	$-0.223575\pi$
$659$	39.1896			1.52661			0.763306	+	0.646038i	$0.223575\pi$
$660$		0			0
$661$	9.38565			0.365060			0.182530	−	0.983200i	$-0.441571\pi$
$661$	9.38565			0.365060			0.182530	+	0.983200i	$0.441571\pi$
$662$		0			0
$663$		−	1.71350i		−	0.0665469i
$664$		0			0
$665$	−10.9453	−	14.3401i	−0.424440	−	0.556085i
$666$		0			0
$667$			4.11268i			0.159244i
$668$		0			0
$669$	−43.7140			−1.69008
$670$		0			0
$671$	61.1319			2.35997
$672$		0			0
$673$		−	8.05600i		−	0.310536i	−0.987872	−	0.155268i	$-0.950376\pi$
$673$		−	8.05600i		−	0.310536i	0.987872	−	0.155268i	$-0.0496241\pi$
$674$		0			0
$675$	−24.9499	−	6.82249i	−0.960322	−	0.262598i
$676$		0			0
$677$		−	16.9461i		−	0.651293i	−0.945492	−	0.325647i	$-0.894418\pi$
$677$		−	16.9461i		−	0.651293i	0.945492	−	0.325647i	$-0.105582\pi$
$678$		0			0
$679$	−6.91370			−0.265324
$680$		0			0
$681$	−41.4063			−1.58669
$682$		0			0
$683$			0.111363i			0.00426119i	0.999998	+	0.00213059i	$0.000678190\pi$
$683$			0.111363i			0.00426119i	−0.999998	+	0.00213059i	$0.999322\pi$
$684$		0			0
$685$	3.00140	+	3.93231i	0.114678	+	0.150246i
$686$		0			0
$687$		−	1.37316i		−	0.0523894i
$688$		0			0
$689$	−6.88917			−0.262456
$690$		0			0
$691$	−38.1718			−1.45212			−0.726062	−	0.687629i	$-0.758652\pi$
$691$	−38.1718			−1.45212			−0.726062	+	0.687629i	$0.758652\pi$
$692$		0			0
$693$		−	0.503158i		−	0.0191134i
$694$		0			0
$695$	11.6199	−	8.86903i	0.440766	−	0.336421i
$696$		0			0
$697$			0.00373938i			0.000141639i
$698$		0			0
$699$	36.2014			1.36926
$700$		0			0
$701$	38.7698			1.46431			0.732157	−	0.681136i	$-0.238514\pi$
$701$	38.7698			1.46431			0.732157	+	0.681136i	$0.238514\pi$
$702$		0			0
$703$			18.6963i			0.705144i
$704$		0			0
$705$	30.9456	−	23.6197i	1.16548	−	0.889569i
$706$		0			0
$707$		−	43.7128i		−	1.64399i
$708$		0			0
$709$	47.4028			1.78025			0.890126	−	0.455715i	$-0.150617\pi$
$709$	47.4028			1.78025			0.890126	+	0.455715i	$0.150617\pi$
$710$		0			0
$711$	−0.00567080			−0.000212671
$712$		0			0
$713$			9.54624i			0.357510i
$714$		0			0
$715$	−8.62782	−	11.3038i	−0.322662	−	0.422739i
$716$		0			0
$717$		−	43.9073i		−	1.63975i
$718$		0			0
$719$	17.7370			0.661479			0.330740	−	0.943722i	$-0.392702\pi$
$719$	17.7370			0.661479			0.330740	+	0.943722i	$0.392702\pi$
$720$		0			0
$721$	60.1179			2.23891
$722$		0			0
$723$		−	9.18010i		−	0.341412i
$724$		0			0
$725$	−5.42388	+	19.8352i	−0.201438	+	0.736660i
$726$		0			0
$727$		−	50.7341i		−	1.88163i	−0.338927	−	0.940813i	$-0.610064\pi$
$727$		−	50.7341i		−	1.88163i	0.338927	−	0.940813i	$-0.389936\pi$
$728$		0			0
$729$	−26.7597			−0.991100
$730$		0			0
$731$	−8.91206			−0.329625
$732$		0			0
$733$			18.9970i			0.701670i	0.936437	+	0.350835i	$0.114102\pi$
$733$			18.9970i			0.701670i	−0.936437	+	0.350835i	$0.885898\pi$
$734$		0			0
$735$	9.52834	+	12.4837i	0.351458	+	0.460466i
$736$		0			0
$737$			63.2096i			2.32835i
$738$		0			0
$739$	44.6999			1.64431			0.822156	−	0.569263i	$-0.192771\pi$
$739$	44.6999			1.64431			0.822156	+	0.569263i	$0.192771\pi$
$740$		0			0
$741$	4.65522			0.171014
$742$		0			0
$743$		−	38.9841i		−	1.43019i	−0.699029	−	0.715094i	$-0.746384\pi$
$743$		−	38.9841i		−	1.43019i	0.699029	−	0.715094i	$-0.253616\pi$
$744$		0			0
$745$	19.8800	−	15.1737i	0.728347	−	0.555922i
$746$		0			0
$747$		−	0.141443i		−	0.00517513i
$748$		0			0
$749$	−23.1647			−0.846421
$750$		0			0
$751$	3.97027			0.144877			0.0724386	−	0.997373i	$-0.476922\pi$
$751$	3.97027			0.144877			0.0724386	+	0.997373i	$0.476922\pi$
$752$		0			0
$753$			18.8641i			0.687448i
$754$		0			0
$755$	−2.29607	+	1.75251i	−0.0835625	+	0.0637804i
$756$		0			0
$757$		−	23.2885i		−	0.846436i	−0.906028	−	0.423218i	$-0.860900\pi$
$757$		−	23.2885i		−	0.846436i	0.906028	−	0.423218i	$-0.139100\pi$
$758$		0			0
$759$	−10.0393			−0.364405
$760$		0			0
$761$	5.62214			0.203802			0.101901	−	0.994795i	$-0.467507\pi$
$761$	5.62214			0.203802			0.101901	+	0.994795i	$0.467507\pi$
$762$		0			0
$763$		−	34.1768i		−	1.23728i
$764$		0			0
$765$	−0.0318245	−	0.0416952i	−0.00115062	−	0.00150749i
$766$		0			0
$767$		−	11.8753i		−	0.428792i
$768$		0			0
$769$	33.0457			1.19166			0.595828	−	0.803112i	$-0.296824\pi$
$769$	33.0457			1.19166			0.595828	+	0.803112i	$0.296824\pi$
$770$		0			0
$771$	32.3981			1.16679
$772$		0			0
$773$			46.8196i			1.68398i	0.539491	+	0.841991i	$0.318617\pi$
$773$			46.8196i			1.68398i	−0.539491	+	0.841991i	$0.681383\pi$
$774$		0			0
$775$	−12.5898	+	46.0409i	−0.452238	+	1.65384i
$776$		0			0
$777$		−	44.4958i		−	1.59628i
$778$		0			0
$779$	−0.0101591			−0.000363987
$780$		0			0
$781$	−74.5442			−2.66740
$782$		0			0
$783$			21.2756i			0.760328i
$784$		0			0
$785$	14.9866	+	19.6349i	0.534896	+	0.700800i
$786$		0			0
$787$			35.1470i			1.25286i	0.779479	+	0.626428i	$0.215484\pi$
$787$			35.1470i			1.25286i	−0.779479	+	0.626428i	$0.784516\pi$
$788$		0			0
$789$	33.8128			1.20377
$790$		0			0
$791$	−18.3871			−0.653769
$792$		0			0
$793$		−	11.6730i		−	0.414522i
$794$		0			0
$795$	−19.3309	+	14.7546i	−0.685597	+	0.523292i
$796$		0			0
$797$			21.8566i			0.774200i	0.922038	+	0.387100i	$0.126523\pi$
$797$			21.8566i			0.774200i	−0.922038	+	0.387100i	$0.873477\pi$
$798$		0			0
$799$	−8.94556			−0.316471
$800$		0			0
$801$	0.157633			0.00556969
$802$		0			0
$803$		−	10.9617i		−	0.386829i
$804$		0			0
$805$	5.90519	−	4.50722i	0.208130	−	0.158859i
$806$		0			0
$807$		−	6.61909i		−	0.233003i
$808$		0			0
$809$	2.24801			0.0790357			0.0395178	−	0.999219i	$-0.487418\pi$
$809$	2.24801			0.0790357			0.0395178	+	0.999219i	$0.487418\pi$
$810$		0			0
$811$	29.5155			1.03643			0.518215	−	0.855250i	$-0.326597\pi$
$811$	29.5155			1.03643			0.518215	+	0.855250i	$0.326597\pi$
$812$		0			0
$813$			31.6783i			1.11101i
$814$		0			0
$815$	−6.84128	−	8.96317i	−0.239640	−	0.313966i
$816$		0			0
$817$		−	24.2121i		−	0.847076i
$818$		0			0
$819$	−0.0960772			−0.00335721
$820$		0			0
$821$	−29.1470			−1.01724			−0.508618	−	0.860992i	$-0.669843\pi$
$821$	−29.1470			−1.01724			−0.508618	+	0.860992i	$0.669843\pi$
$822$		0			0
$823$			13.1354i			0.457873i	0.973441	+	0.228936i	$0.0735248\pi$
$823$			13.1354i			0.457873i	−0.973441	+	0.228936i	$0.926475\pi$
$824$		0			0
$825$	−48.4191	−	13.2401i	−1.68574	−	0.460961i
$826$		0			0
$827$			54.7619i			1.90426i	0.305698	+	0.952129i	$0.401110\pi$
$827$			54.7619i			1.90426i	−0.305698	+	0.952129i	$0.598890\pi$
$828$		0			0
$829$	41.3545			1.43630			0.718151	−	0.695887i	$-0.244989\pi$
$829$	41.3545			1.43630			0.718151	+	0.695887i	$0.244989\pi$
$830$		0			0
$831$	25.6809			0.890859
$832$		0			0
$833$		−	3.60869i		−	0.125034i
$834$		0			0
$835$	1.63832	+	2.14647i	0.0566965	+	0.0742816i
$836$		0			0
$837$			49.3844i			1.70698i
$838$		0			0
$839$	−8.60909			−0.297219			−0.148609	−	0.988896i	$-0.547480\pi$
$839$	−8.60909			−0.297219			−0.148609	+	0.988896i	$0.547480\pi$
$840$		0			0
$841$	−12.0859			−0.416755
$842$		0			0
$843$			10.9550i			0.377310i
$844$		0			0
$845$	20.9487	−	15.9894i	0.720657	−	0.550053i
$846$		0			0
$847$			74.1018i			2.54617i
$848$		0			0
$849$	−49.4819			−1.69822
$850$		0			0
$851$	−7.69904			−0.263920
$852$		0			0
$853$		−	53.2697i		−	1.82392i	−0.410280	−	0.911960i	$-0.634569\pi$
$853$		−	53.2697i		−	1.82392i	0.410280	−	0.911960i	$-0.365431\pi$
$854$		0			0
$855$	0.113277	−	0.0864604i	0.00387399	−	0.00295688i
$856$		0			0
$857$			13.3201i			0.455005i	0.973778	+	0.227502i	$0.0730560\pi$
$857$			13.3201i			0.455005i	−0.973778	+	0.227502i	$0.926944\pi$
$858$		0			0
$859$	−23.3425			−0.796434			−0.398217	−	0.917291i	$-0.630371\pi$
$859$	−23.3425			−0.796434			−0.398217	+	0.917291i	$0.630371\pi$
$860$		0			0
$861$	0.0241779			0.000823981
$862$		0			0
$863$		−	21.8395i		−	0.743426i	−0.928348	−	0.371713i	$-0.878771\pi$
$863$		−	21.8395i		−	0.743426i	0.928348	−	0.371713i	$-0.121229\pi$
$864$		0			0
$865$	21.3531	+	27.9759i	0.726026	+	0.951211i
$866$		0			0
$867$		−	28.1835i		−	0.957161i
$868$		0			0
$869$	1.24703			0.0423026
$870$		0			0
$871$	12.0698			0.408968
$872$		0			0
$873$		−	0.0546135i		−	0.00184839i
$874$		0			0
$875$	34.4246	−	13.9502i	1.16376	−	0.471602i
$876$		0			0
$877$		−	29.0903i		−	0.982309i	−0.871072	−	0.491155i	$-0.836575\pi$
$877$		−	29.0903i		−	0.982309i	0.871072	−	0.491155i	$-0.163425\pi$
$878$		0			0
$879$	−25.2082			−0.850251
$880$		0			0
$881$	−44.7215			−1.50671			−0.753353	−	0.657616i	$-0.771565\pi$
$881$	−44.7215			−1.50671			−0.753353	+	0.657616i	$0.771565\pi$
$882$		0			0
$883$		−	19.4889i		−	0.655855i	−0.944703	−	0.327927i	$-0.893650\pi$
$883$		−	19.4889i		−	0.655855i	0.944703	−	0.327927i	$-0.106350\pi$
$884$		0			0
$885$	−25.4334	−	33.3219i	−0.854936	−	1.12010i
$886$		0			0
$887$		−	24.0317i		−	0.806906i	−0.915000	−	0.403453i	$-0.867810\pi$
$887$		−	24.0317i		−	0.806906i	0.915000	−	0.403453i	$-0.132190\pi$
$888$		0			0
$889$	−37.0694			−1.24327
$890$		0			0
$891$	−52.3897			−1.75512
$892$		0			0
$893$		−	24.3031i		−	0.813274i
$894$		0			0
$895$	−11.7662	+	8.98074i	−0.393301	+	0.300193i
$896$		0			0
$897$			1.91700i			0.0640067i
$898$		0			0
$899$	39.2606			1.30942
$900$		0			0
$901$	5.58806			0.186165
$902$		0			0
$903$			57.6232i			1.91758i
$904$		0			0
$905$	−28.5165	+	21.7657i	−0.947922	+	0.723516i
$906$		0			0
$907$			16.2777i			0.540491i	0.962792	+	0.270245i	$0.0871048\pi$
$907$			16.2777i			0.540491i	−0.962792	+	0.270245i	$0.912895\pi$
$908$		0			0
$909$	0.345301			0.0114529
$910$		0			0
$911$	23.8128			0.788952			0.394476	−	0.918906i	$-0.370926\pi$
$911$	23.8128			0.788952			0.394476	+	0.918906i	$0.370926\pi$
$912$		0			0
$913$			31.1039i			1.02939i
$914$		0			0
$915$	−25.0003	−	32.7544i	−0.826484	−	1.08283i
$916$		0			0
$917$		−	49.9734i		−	1.65027i
$918$		0			0
$919$	−22.0433			−0.727141			−0.363571	−	0.931567i	$-0.618442\pi$
$919$	−22.0433			−0.727141			−0.363571	+	0.931567i	$0.618442\pi$
$920$		0			0
$921$	−53.0071			−1.74664
$922$		0			0
$923$			14.2341i			0.468521i
$924$		0			0
$925$	−37.1320	−	10.1537i	−1.22089	−	0.333850i
$926$		0			0
$927$			0.474890i			0.0155974i
$928$		0			0
$929$	5.19696			0.170507			0.0852534	−	0.996359i	$-0.472830\pi$
$929$	5.19696			0.170507			0.0852534	+	0.996359i	$0.472830\pi$
$930$		0			0
$931$	9.80404			0.321315
$932$		0			0
$933$		−	36.3633i		−	1.19048i
$934$		0			0
$935$	6.99834	+	9.16895i	0.228870	+	0.299857i
$936$		0			0
$937$		−	29.0029i		−	0.947482i	−0.880664	−	0.473741i	$-0.842903\pi$
$937$		−	29.0029i		−	0.947482i	0.880664	−	0.473741i	$-0.157097\pi$
$938$		0			0
$939$	−24.9833			−0.815300
$940$		0			0
$941$	50.2138			1.63692			0.818461	−	0.574562i	$-0.194827\pi$
$941$	50.2138			1.63692			0.818461	+	0.574562i	$0.194827\pi$
$942$		0			0
$943$		−	0.00418347i		−	0.000136232i
$944$		0			0
$945$	30.5486	−	23.3167i	0.993745	−	0.758491i
$946$		0			0
$947$			18.3661i			0.596817i	0.954438	+	0.298408i	$0.0964558\pi$
$947$			18.3661i			0.596817i	−0.954438	+	0.298408i	$0.903544\pi$
$948$		0			0
$949$	−2.09311			−0.0679453
$950$		0			0
$951$	58.6143			1.90070
$952$		0			0
$953$		−	34.9542i		−	1.13228i	−0.824310	−	0.566139i	$-0.808437\pi$
$953$		−	34.9542i		−	1.13228i	0.824310	−	0.566139i	$-0.191563\pi$
$954$		0			0
$955$	0.525403	−	0.401022i	0.0170016	−	0.0129768i
$956$		0			0
$957$			41.2886i			1.33467i
$958$		0			0
$959$	−7.34980			−0.237338
$960$		0			0
$961$	60.1308			1.93970
$962$		0			0
$963$		−	0.182985i		−	0.00589662i
$964$		0			0
$965$	1.93415	+	2.53404i	0.0622624	+	0.0815738i
$966$		0			0
$967$			48.3107i			1.55357i	0.629767	+	0.776784i	$0.283150\pi$
$967$			48.3107i			1.55357i	−0.629767	+	0.776784i	$0.716850\pi$
$968$		0			0
$969$	−3.77602			−0.121303
$970$		0			0
$971$	−14.8815			−0.477570			−0.238785	−	0.971072i	$-0.576749\pi$
$971$	−14.8815			−0.477570			−0.238785	+	0.971072i	$0.576749\pi$
$972$		0			0
$973$			21.7184i			0.696261i
$974$		0			0
$975$	−2.52817	+	9.24555i	−0.0809664	+	0.296095i
$976$		0			0
$977$			18.6833i			0.597730i	0.954295	+	0.298865i	$0.0966081\pi$
$977$			18.6833i			0.597730i	−0.954295	+	0.298865i	$0.903392\pi$
$978$		0			0
$979$	−34.6641			−1.10787
$980$		0			0
$981$	0.269974			0.00861959
$982$		0			0
$983$		−	28.4191i		−	0.906428i	−0.891402	−	0.453214i	$-0.850277\pi$
$983$		−	28.4191i		−	0.906428i	0.891402	−	0.453214i	$-0.149723\pi$
$984$		0			0
$985$	−11.6689	−	15.2882i	−0.371803	−	0.487122i
$986$		0			0
$987$			57.8398i			1.84106i
$988$		0			0
$989$	9.97045			0.317042
$990$		0			0
$991$	−28.4942			−0.905148			−0.452574	−	0.891727i	$-0.649494\pi$
$991$	−28.4942			−0.905148			−0.452574	+	0.891727i	$0.649494\pi$
$992$		0			0
$993$		−	17.0881i		−	0.542273i
$994$		0			0
$995$	−6.78133	+	5.17595i	−0.214983	+	0.164089i
$996$		0			0
$997$			26.5435i			0.840642i	0.907375	+	0.420321i	$0.138083\pi$
$997$			26.5435i			0.840642i	−0.907375	+	0.420321i	$0.861917\pi$
$998$		0			0
$999$	−39.8285			−1.26012

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	460.2.c.a.369.4	✓	12
3.2	odd	2		4140.2.f.b.829.2		12
4.3	odd	2		1840.2.e.f.369.9		12
5.2	odd	4		2300.2.a.n.1.3		6
5.3	odd	4		2300.2.a.o.1.4		6
5.4	even	2	inner	460.2.c.a.369.9	yes	12
15.14	odd	2		4140.2.f.b.829.1		12
20.3	even	4		9200.2.a.cx.1.3		6
20.7	even	4		9200.2.a.cy.1.4		6
20.19	odd	2		1840.2.e.f.369.4		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.2.c.a.369.4	✓	12	1.1	even	1	trivial
460.2.c.a.369.9	yes	12	5.4	even	2	inner
1840.2.e.f.369.4		12	20.19	odd	2
1840.2.e.f.369.9		12	4.3	odd	2
2300.2.a.n.1.3		6	5.2	odd	4
2300.2.a.o.1.4		6	5.3	odd	4
4140.2.f.b.829.1		12	15.14	odd	2
4140.2.f.b.829.2		12	3.2	odd	2
9200.2.a.cx.1.3		6	20.3	even	4
9200.2.a.cy.1.4		6	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 460 = 2^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	460.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73961i q^{3} +(1.77747 - 1.35668i) q^{5} +3.32224i q^{7} -0.0262434 q^{9} +O(q^{10})\)	\(q-1.73961i q^{3} +(1.77747 - 1.35668i) q^{5} +3.32224i q^{7} -0.0262434 q^{9} +5.77103 q^{11} -1.10197i q^{13} +(-2.36010 - 3.09211i) q^{15} +0.893847i q^{17} -2.42839 q^{19} +5.77940 q^{21} -1.00000i q^{23} +(1.31882 - 4.82294i) q^{25} -5.17318i q^{27} -4.11268 q^{29} -9.54624 q^{31} -10.0393i q^{33} +(4.50722 + 5.90519i) q^{35} -7.69904i q^{37} -1.91700 q^{39} +0.00418347 q^{41} +9.97045i q^{43} +(-0.0466469 + 0.0356040i) q^{45} +10.0079i q^{47} -4.03726 q^{49} +1.55495 q^{51} -6.25169i q^{53} +(10.2579 - 7.82946i) q^{55} +4.22445i q^{57} +10.7764 q^{59} +10.5929 q^{61} -0.0871868i q^{63} +(-1.49502 - 1.95872i) q^{65} +10.9529i q^{67} -1.73961 q^{69} -12.9170 q^{71} -1.89943i q^{73} +(-8.39003 - 2.29423i) q^{75} +19.1727i q^{77} +0.216085 q^{79} -9.07804 q^{81} +5.38967i q^{83} +(1.21267 + 1.58879i) q^{85} +7.15446i q^{87} -6.00657 q^{89} +3.66100 q^{91} +16.6067i q^{93} +(-4.31640 + 3.29456i) q^{95} +2.08104i q^{97} -0.151451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 20 q^{9}+O(q^{10}) \) 12 * q - 20 * q^9	\( 12 q - 20 q^{9} + 4 q^{11} + 2 q^{15} - 8 q^{19} + 8 q^{25} - 10 q^{29} + 18 q^{31} - 10 q^{35} + 16 q^{39} - 2 q^{41} + 2 q^{45} - 38 q^{49} - 24 q^{51} + 16 q^{55} + 22 q^{59} - 8 q^{61} + 38 q^{65} - 8 q^{69} - 34 q^{71} + 16 q^{75} - 20 q^{79} + 28 q^{81} + 6 q^{85} + 48 q^{89} - 8 q^{91} + 12 q^{95} + 32 q^{99}+O(q^{100}) \) 12 * q - 20 * q^9 + 4 * q^11 + 2 * q^15 - 8 * q^19 + 8 * q^25 - 10 * q^29 + 18 * q^31 - 10 * q^35 + 16 * q^39 - 2 * q^41 + 2 * q^45 - 38 * q^49 - 24 * q^51 + 16 * q^55 + 22 * q^59 - 8 * q^61 + 38 * q^65 - 8 * q^69 - 34 * q^71 + 16 * q^75 - 20 * q^79 + 28 * q^81 + 6 * q^85 + 48 * q^89 - 8 * q^91 + 12 * q^95 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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