LMFDB - Embedded newform 4140.2.f.b.829.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4140,2,Mod(829,4140)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4140, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4140.829");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4140.f (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:	$33.0580664368$
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:	no (minimal twist has level 460)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			829.2
Root			$-0.420790i$ of defining polynomial
Character	$\chi$	$=$	4140.829
Dual form			4140.2.f.b.829.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.77747 + 1.35668i) q^{5} +3.32224i q^{7} +O(q^{10})$	$q+(-1.77747 + 1.35668i) q^{5} +3.32224i q^{7} -5.77103 q^{11} -1.10197i q^{13} -0.893847i q^{17} -2.42839 q^{19} +1.00000i q^{23} +(1.31882 - 4.82294i) q^{25} +4.11268 q^{29} -9.54624 q^{31} +(-4.50722 - 5.90519i) q^{35} -7.69904i q^{37} -0.00418347 q^{41} +9.97045i q^{43} -10.0079i q^{47} -4.03726 q^{49} +6.25169i q^{53} +(10.2579 - 7.82946i) q^{55} -10.7764 q^{59} +10.5929 q^{61} +(1.49502 + 1.95872i) q^{65} +10.9529i q^{67} +12.9170 q^{71} -1.89943i q^{73} -19.1727i q^{77} +0.216085 q^{79} -5.38967i q^{83} +(1.21267 + 1.58879i) q^{85} +6.00657 q^{89} +3.66100 q^{91} +(4.31640 - 3.29456i) q^{95} +2.08104i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) $ 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times$.

$n$	$461$	$1657$	$2071$	$3961$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$		0			0
$3$		0			0
$4$		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			0
$10$		0			0
$11$	−5.77103			−1.74003			−0.870016	−	0.493024i	$-0.835891\pi$
$11$	−5.77103			−1.74003			−0.870016	+	0.493024i	$0.835891\pi$
$12$		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
$15$		0			0
$16$		0			0
$17$		−	0.893847i		−	0.216790i	−0.994108	−	0.108395i	$-0.965429\pi$
$17$		−	0.893847i		−	0.216790i	0.994108	−	0.108395i	$-0.0345711\pi$
$18$		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
$21$		0			0
$22$		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
$27$		0			0
$28$		0			0
$29$	4.11268			0.763705			0.381853	−	0.924223i	$-0.375286\pi$
$29$	4.11268			0.763705			0.381853	+	0.924223i	$0.375286\pi$
$30$		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
$33$		0			0
$34$		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
$39$		0			0
$40$		0			0
$41$	−0.00418347			−0.000653348			−0.000326674	−	1.00000i	$-0.500104\pi$
$41$	−0.00418347			−0.000653348			−0.000326674		1.00000i	$0.500104\pi$
$42$		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			0
$46$		0			0
$47$		−	10.0079i		−	1.45981i	−0.683551	−	0.729903i	$-0.739565\pi$
$47$		−	10.0079i		−	1.45981i	0.683551	−	0.729903i	$-0.260435\pi$
$48$		0			0
$49$	−4.03726			−0.576751
$50$		0			0
$51$		0			0
$52$		0			0
$53$			6.25169i			0.858735i	0.903130	+	0.429368i	$0.141264\pi$
$53$			6.25169i			0.858735i	−0.903130	+	0.429368i	$0.858736\pi$
$54$		0			0
$55$	10.2579	−	7.82946i	1.38317	−	1.05572i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.7764			−1.40297			−0.701486	−	0.712684i	$-0.747480\pi$
$59$	−10.7764			−1.40297			−0.701486	+	0.712684i	$0.747480\pi$
$60$		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			0
$64$		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
$69$		0			0
$70$		0			0
$71$	12.9170			1.53296			0.766481	−	0.642267i	$-0.222006\pi$
$71$	12.9170			1.53296			0.766481	+	0.642267i	$0.222006\pi$
$72$		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
$75$		0			0
$76$		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
$81$		0			0
$82$		0			0
$83$		−	5.38967i		−	0.591593i	−0.955251	−	0.295796i	$-0.904415\pi$
$83$		−	5.38967i		−	0.591593i	0.955251	−	0.295796i	$-0.0955850\pi$
$84$		0			0
$85$	1.21267	+	1.58879i	0.131532	+	0.172328i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.00657			0.636696			0.318348	−	0.947974i	$-0.396872\pi$
$89$	6.00657			0.636696			0.318348	+	0.947974i	$0.396872\pi$
$90$		0			0
$91$	3.66100			0.383777
$92$		0			0
$93$		0			0
$94$		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			0
$100$		0			0
$101$	13.1576			1.30923			0.654617	−	0.755961i	$-0.272830\pi$
$101$	13.1576			1.30923			0.654617	+	0.755961i	$0.272830\pi$
$102$		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
$105$		0			0
$106$		0			0
$107$		−	6.97263i		−	0.674069i	−0.941492	−	0.337035i	$-0.890576\pi$
$107$		−	6.97263i		−	0.674069i	0.941492	−	0.337035i	$-0.109424\pi$
$108$		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
$111$		0			0
$112$		0			0
$113$		−	5.53454i		−	0.520646i	−0.965522	−	0.260323i	$-0.916171\pi$
$113$		−	5.53454i		−	0.520646i	0.965522	−	0.260323i	$-0.0838290\pi$
$114$		0			0
$115$	−1.35668	−	1.77747i	−0.126511	−	0.165750i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.96957			0.272220
$120$		0			0
$121$	22.3048			2.02771
$122$		0			0
$123$		0			0
$124$		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
$129$		0			0
$130$		0			0
$131$	15.0421			1.31423			0.657117	−	0.753789i	$-0.271776\pi$
$131$	15.0421			1.31423			0.657117	+	0.753789i	$0.271776\pi$
$132$		0			0
$133$		−	8.06769i		−	0.699557i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2.21231i		−	0.189010i	−0.995524	−	0.0945050i	$-0.969873\pi$
$137$		−	2.21231i		−	0.189010i	0.995524	−	0.0945050i	$-0.0301268\pi$
$138$		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
$141$		0			0
$142$		0			0
$143$			6.35950i			0.531808i
$144$		0			0
$145$	−7.31017	+	5.57960i	−0.607077	+	0.463361i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−11.1844			−0.916263			−0.458131	−	0.888884i	$-0.651481\pi$
$149$	−11.1844			−0.916263			−0.458131	+	0.888884i	$0.651481\pi$
$150$		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			0
$154$		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
$159$		0			0
$160$		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
$165$		0			0
$166$		0			0
$167$		−	1.20759i		−	0.0934465i	−0.998908	−	0.0467232i	$-0.985122\pi$
$167$		−	1.20759i		−	0.0934465i	0.998908	−	0.0467232i	$-0.0148779\pi$
$168$		0			0
$169$	11.7857			0.906590
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	15.7392i		−	1.19663i	−0.801262	−	0.598313i	$-0.795838\pi$
$173$		−	15.7392i		−	1.19663i	0.801262	−	0.598313i	$-0.204162\pi$
$174$		0			0
$175$	16.0229	+	4.38143i	1.21122	+	0.331205i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.61963			0.494774			0.247387	−	0.968917i	$-0.420428\pi$
$179$	6.61963			0.494774			0.247387	+	0.968917i	$0.420428\pi$
$180$		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
$183$		0			0
$184$		0			0
$185$	10.4452	+	13.6848i	0.767944	+	1.00613i
$186$		0			0
$187$			5.15842i			0.377221i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.295590			−0.0213881			−0.0106941	−	0.999943i	$-0.503404\pi$
$191$	−0.295590			−0.0213881			−0.0106941	+	0.999943i	$0.503404\pi$
$192$		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
$195$		0			0
$196$		0			0
$197$			8.60107i			0.612801i	0.951903	+	0.306401i	$0.0991247\pi$
$197$			8.60107i			0.612801i	−0.951903	+	0.306401i	$0.900875\pi$
$198$		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
$201$		0			0
$202$		0			0
$203$			13.6633i			0.958975i
$204$		0			0
$205$	0.00743600	−	0.00567564i	0.000519353	−	0.000396404i
$206$		0			0
$207$		0			0
$208$		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
$213$		0			0
$214$		0			0
$215$	−13.5267	−	17.7222i	−0.922516	−	1.20864i
$216$		0			0
$217$		−	31.7149i		−	2.15295i
$218$		0			0
$219$		0			0
$220$		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			0
$226$		0			0
$227$			23.8021i			1.57980i	0.613236	+	0.789900i	$0.289867\pi$
$227$			23.8021i			1.57980i	−0.613236	+	0.789900i	$0.710133\pi$
$228$		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
$231$		0			0
$232$		0			0
$233$		−	20.8101i		−	1.36331i	−0.731672	−	0.681656i	$-0.761260\pi$
$233$		−	20.8101i		−	1.36331i	0.731672	−	0.681656i	$-0.238740\pi$
$234$		0			0
$235$	13.5776	+	17.7888i	0.885704	+	1.16041i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−25.2398			−1.63262			−0.816312	−	0.577611i	$-0.803985\pi$
$239$	−25.2398			−1.63262			−0.816312	+	0.577611i	$0.803985\pi$
$240$		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			0
$244$		0			0
$245$	7.17612	−	5.47728i	0.458466	−	0.349931i
$246$		0			0
$247$			2.67601i			0.170270i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	10.8439			0.684461			0.342230	−	0.939616i	$-0.388818\pi$
$251$	10.8439			0.684461			0.342230	+	0.939616i	$0.388818\pi$
$252$		0			0
$253$		−	5.77103i		−	0.362822i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	18.6238i		−	1.16172i	−0.814004	−	0.580859i	$-0.802717\pi$
$257$		−	18.6238i		−	1.16172i	0.814004	−	0.580859i	$-0.197283\pi$
$258$		0			0
$259$	25.5781			1.58934
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	19.4370i		−	1.19854i	−0.800548	−	0.599269i	$-0.795458\pi$
$263$		−	19.4370i		−	1.19854i	0.800548	−	0.599269i	$-0.204542\pi$
$264$		0			0
$265$	−8.48156	−	11.1122i	−0.521018	−	0.682617i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−3.80493			−0.231991			−0.115995	−	0.993250i	$-0.537006\pi$
$269$	−3.80493			−0.231991			−0.115995	+	0.993250i	$0.537006\pi$
$270$		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
$273$		0			0
$274$		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			0
$280$		0			0
$281$	6.29738			0.375670			0.187835	−	0.982201i	$-0.439853\pi$
$281$	6.29738			0.375670			0.187835	+	0.982201i	$0.439853\pi$
$282$		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
$285$		0			0
$286$		0			0
$287$		−	0.0138985i		−	0.000820401i
$288$		0			0
$289$	16.2010			0.953002
$290$		0			0
$291$		0			0
$292$		0			0
$293$			14.4907i			0.846556i	0.906000	+	0.423278i	$0.139121\pi$
$293$			14.4907i			0.846556i	−0.906000	+	0.423278i	$0.860879\pi$
$294$		0			0
$295$	19.1548	−	14.6202i	1.11524	−	0.851221i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.10197			0.0637285
$300$		0			0
$301$	−33.1242			−1.90925
$302$		0			0
$303$		0			0
$304$		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
$309$		0			0
$310$		0			0
$311$	−20.9031			−1.18531			−0.592654	−	0.805457i	$-0.701920\pi$
$311$	−20.9031			−1.18531			−0.592654	+	0.805457i	$0.701920\pi$
$312$		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			0
$316$		0			0
$317$		−	33.6939i		−	1.89244i	−0.323525	−	0.946220i	$-0.604868\pi$
$317$		−	33.6939i		−	1.89244i	0.323525	−	0.946220i	$-0.395132\pi$
$318$		0			0
$319$	−23.7344			−1.32887
$320$		0			0
$321$		0			0
$322$		0			0
$323$			2.17061i			0.120776i
$324$		0			0
$325$	−5.31473	−	1.45330i	−0.294808	−	0.0806146i
$326$		0			0
$327$		0			0
$328$		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			0
$334$		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
$339$		0			0
$340$		0			0
$341$	55.0917			2.98338
$342$		0			0
$343$			9.84292i			0.531468i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			9.15627i			0.491534i	0.969329	+	0.245767i	$0.0790399\pi$
$347$			9.15627i			0.491534i	−0.969329	+	0.245767i	$0.920960\pi$
$348$		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
$351$		0			0
$352$		0			0
$353$		−	2.40654i		−	0.128087i	−0.997947	−	0.0640436i	$-0.979600\pi$
$353$		−	2.40654i		−	0.128087i	0.997947	−	0.0640436i	$-0.0203997\pi$
$354$		0			0
$355$	−22.9596	+	17.5242i	−1.21857	+	0.930090i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	21.5236			1.13597			0.567985	−	0.823039i	$-0.307723\pi$
$359$	21.5236			1.13597			0.567985	+	0.823039i	$0.307723\pi$
$360$		0			0
$361$	−13.1029			−0.689628
$362$		0			0
$363$		0			0
$364$		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			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
$375$		0			0
$376$		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
$381$		0			0
$382$		0			0
$383$		−	19.4206i		−	0.992348i	−0.868223	−	0.496174i	$-0.834738\pi$
$383$		−	19.4206i		−	0.992348i	0.868223	−	0.496174i	$-0.165262\pi$
$384$		0			0
$385$	26.0113	+	34.0790i	1.32566	+	1.73683i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−26.1255			−1.32461			−0.662306	−	0.749233i	$-0.730422\pi$
$389$	−26.1255			−1.32461			−0.662306	+	0.749233i	$0.730422\pi$
$390$		0			0
$391$	0.893847			0.0452038
$392$		0			0
$393$		0			0
$394$		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
$399$		0			0
$400$		0			0
$401$	0.188549			0.00941568			0.00470784	−	0.999989i	$-0.498501\pi$
$401$	0.188549			0.00941568			0.00470784	+	0.999989i	$0.498501\pi$
$402$		0			0
$403$			10.5197i			0.524022i
$404$		0			0
$405$		0			0
$406$		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
$411$		0			0
$412$		0			0
$413$		−	35.8018i		−	1.76169i
$414$		0			0
$415$	7.31207	+	9.57999i	0.358935	+	0.470263i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.15604			−0.105329			−0.0526647	−	0.998612i	$-0.516771\pi$
$419$	−2.15604			−0.105329			−0.0526647	+	0.998612i	$0.516771\pi$
$420$		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			0
$424$		0			0
$425$	−4.31097	−	1.17882i	−0.209113	−	0.0571814i
$426$		0			0
$427$			35.1921i			1.70307i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	15.3343			0.738629			0.369315	−	0.929304i	$-0.379592\pi$
$431$	15.3343			0.738629			0.369315	+	0.929304i	$0.379592\pi$
$432$		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
$435$		0			0
$436$		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			0
$442$		0			0
$443$		−	29.2360i		−	1.38905i	−0.719471	−	0.694523i	$-0.755616\pi$
$443$		−	29.2360i		−	1.38905i	0.719471	−	0.694523i	$-0.244384\pi$
$444$		0			0
$445$	−10.6765	+	8.14902i	−0.506116	+	0.386301i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−21.2244			−1.00164			−0.500822	−	0.865550i	$-0.666969\pi$
$449$	−21.2244			−1.00164			−0.500822	+	0.865550i	$0.666969\pi$
$450$		0			0
$451$	0.0241429			0.00113685
$452$		0			0
$453$		0			0
$454$		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
$459$		0			0
$460$		0			0
$461$	−16.4858			−0.767820			−0.383910	−	0.923370i	$-0.625423\pi$
$461$	−16.4858			−0.767820			−0.383910	+	0.923370i	$0.625423\pi$
$462$		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
$465$		0			0
$466$		0			0
$467$			7.89645i			0.365404i	0.983168	+	0.182702i	$0.0584844\pi$
$467$			7.89645i			0.365404i	−0.983168	+	0.182702i	$0.941516\pi$
$468$		0			0
$469$	−36.3882			−1.68025
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	57.5398i		−	2.64568i
$474$		0			0
$475$	−3.20261	+	11.7120i	−0.146946	+	0.537382i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	39.5234			1.80587			0.902935	−	0.429778i	$-0.141408\pi$
$479$	39.5234			1.80587			0.902935	+	0.429778i	$0.141408\pi$
$480$		0			0
$481$	−8.48411			−0.386842
$482$		0			0
$483$		0			0
$484$		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
$489$		0			0
$490$		0			0
$491$	−34.8865			−1.57441			−0.787203	−	0.616694i	$-0.788472\pi$
$491$	−34.8865			−1.57441			−0.787203	+	0.616694i	$0.788472\pi$
$492$		0			0
$493$		−	3.67611i		−	0.165564i
$494$		0			0
$495$		0			0
$496$		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
$501$		0			0
$502$		0			0
$503$			7.05945i			0.314765i	0.987538	+	0.157383i	$0.0503056\pi$
$503$			7.05945i			0.314765i	−0.987538	+	0.157383i	$0.949694\pi$
$504$		0			0
$505$	−23.3874	+	17.8508i	−1.04072	+	0.794348i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	16.6812			0.739382			0.369691	−	0.929155i	$-0.379464\pi$
$509$	16.6812			0.739382			0.369691	+	0.929155i	$0.379464\pi$
$510$		0			0
$511$	6.31035			0.279154
$512$		0			0
$513$		0			0
$514$		0			0
$515$	24.5500	+	32.1645i	1.08180	+	1.41734i
$516$		0			0
$517$			57.7560i			2.54011i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.9941			1.09501			0.547507	−	0.836801i	$-0.315577\pi$
$521$	24.9941			1.09501			0.547507	+	0.836801i	$0.315577\pi$
$522$		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
$525$		0			0
$526$		0			0
$527$			8.53289i			0.371698i
$528$		0			0
$529$	−1.00000			−0.0434783
$530$		0			0
$531$		0			0
$532$		0			0
$533$			0.00461005i			0.000199683i
$534$		0			0
$535$	9.45965	+	12.3937i	0.408976	+	0.535824i
$536$		0			0
$537$		0			0
$538$		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
$543$		0			0
$544$		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			0
$550$		0			0
$551$	−9.98719			−0.425468
$552$		0			0
$553$			0.717884i			0.0305276i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			21.1923i			0.897945i	0.893546	+	0.448972i	$0.148210\pi$
$557$			21.1923i			0.897945i	−0.893546	+	0.448972i	$0.851790\pi$
$558$		0			0
$559$	10.9871			0.464706
$560$		0			0
$561$		0			0
$562$		0			0
$563$			23.6292i			0.995851i	0.867220	+	0.497925i	$0.165905\pi$
$563$			23.6292i			0.995851i	−0.867220	+	0.497925i	$0.834095\pi$
$564$		0			0
$565$	7.50862	+	9.83750i	0.315890	+	0.413867i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−33.7490			−1.41483			−0.707415	−	0.706798i	$-0.750139\pi$
$569$	−33.7490			−1.41483			−0.707415	+	0.706798i	$0.750139\pi$
$570$		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			0
$574$		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
$579$		0			0
$580$		0			0
$581$	17.9057			0.742856
$582$		0			0
$583$		−	36.0787i		−	1.49423i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			35.6609i			1.47188i	0.677045	+	0.735941i	$0.263260\pi$
$587$			35.6609i			1.47188i	−0.677045	+	0.735941i	$0.736740\pi$
$588$		0			0
$589$	23.1820			0.955198
$590$		0			0
$591$		0			0
$592$		0			0
$593$			20.8412i			0.855845i	0.903815	+	0.427922i	$0.140754\pi$
$593$			20.8412i			0.855845i	−0.903815	+	0.427922i	$0.859246\pi$
$594$		0			0
$595$	−5.27834	+	4.02877i	−0.216391	+	0.165163i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.9952			−0.449252			−0.224626	−	0.974445i	$-0.572116\pi$
$599$	−10.9952			−0.449252			−0.224626	+	0.974445i	$0.572116\pi$
$600$		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			0
$604$		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
$609$		0			0
$610$		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			0
$616$		0			0
$617$			29.9851i			1.20715i	0.797304	+	0.603577i	$0.206259\pi$
$617$			29.9851i			1.20715i	−0.797304	+	0.603577i	$0.793741\pi$
$618$		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
$621$		0			0
$622$		0			0
$623$			19.9553i			0.799491i
$624$		0			0
$625$	−21.5214	−	12.7212i	−0.860857	−	0.508847i
$626$		0			0
$627$		0			0
$628$		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
$633$		0			0
$634$		0			0
$635$	−15.1378	−	19.8330i	−0.600727	−	0.787049i
$636$		0			0
$637$			4.44894i			0.176273i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.27686			0.326916			0.163458	−	0.986550i	$-0.447735\pi$
$641$	8.27686			0.326916			0.163458	+	0.986550i	$0.447735\pi$
$642$		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
$645$		0			0
$646$		0			0
$647$		−	12.3508i		−	0.485561i	−0.970081	−	0.242781i	$-0.921940\pi$
$647$		−	12.3508i		−	0.485561i	0.970081	−	0.242781i	$-0.0780596\pi$
$648$		0			0
$649$	62.1911			2.44121
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	18.3211i		−	0.716959i	−0.933538	−	0.358480i	$-0.883295\pi$
$653$		−	18.3211i		−	0.716959i	0.933538	−	0.358480i	$-0.116705\pi$
$654$		0			0
$655$	−26.7369	+	20.4074i	−1.04470	+	0.797381i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−39.1896			−1.52661			−0.763306	−	0.646038i	$-0.776425\pi$
$659$	−39.1896			−1.52661			−0.763306	+	0.646038i	$0.776425\pi$
$660$		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
$663$		0			0
$664$		0			0
$665$	10.9453	+	14.3401i	0.424440	+	0.556085i
$666$		0			0
$667$			4.11268i			0.159244i
$668$		0			0
$669$		0			0
$670$		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
$675$		0			0
$676$		0			0
$677$			16.9461i			0.651293i	0.945492	+	0.325647i	$0.105582\pi$
$677$			16.9461i			0.651293i	−0.945492	+	0.325647i	$0.894418\pi$
$678$		0			0
$679$	−6.91370			−0.265324
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	0.111363i		−	0.00426119i	−0.999998	−	0.00213059i	$-0.999322\pi$
$683$		−	0.111363i		−	0.00426119i	0.999998	−	0.00213059i	$-0.000678190\pi$
$684$		0			0
$685$	3.00140	+	3.93231i	0.114678	+	0.150246i
$686$		0			0
$687$		0			0
$688$		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			0
$694$		0			0
$695$	−11.6199	+	8.86903i	−0.440766	+	0.336421i
$696$		0			0
$697$			0.00373938i			0.000141639i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−38.7698			−1.46431			−0.732157	−	0.681136i	$-0.761486\pi$
$701$	−38.7698			−1.46431			−0.732157	+	0.681136i	$0.761486\pi$
$702$		0			0
$703$			18.6963i			0.705144i
$704$		0			0
$705$		0			0
$706$		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			0
$712$		0			0
$713$		−	9.54624i		−	0.357510i
$714$		0			0
$715$	−8.62782	−	11.3038i	−0.322662	−	0.422739i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−17.7370			−0.661479			−0.330740	−	0.943722i	$-0.607298\pi$
$719$	−17.7370			−0.661479			−0.330740	+	0.943722i	$0.607298\pi$
$720$		0			0
$721$	60.1179			2.23891
$722$		0			0
$723$		0			0
$724$		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
$729$		0			0
$730$		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
$735$		0			0
$736$		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
$741$		0			0
$742$		0			0
$743$			38.9841i			1.43019i	0.699029	+	0.715094i	$0.253616\pi$
$743$			38.9841i			1.43019i	−0.699029	+	0.715094i	$0.746384\pi$
$744$		0			0
$745$	19.8800	−	15.1737i	0.728347	−	0.555922i
$746$		0			0
$747$		0			0
$748$		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
$753$		0			0
$754$		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
$759$		0			0
$760$		0			0
$761$	−5.62214			−0.203802			−0.101901	−	0.994795i	$-0.532493\pi$
$761$	−5.62214			−0.203802			−0.101901	+	0.994795i	$0.532493\pi$
$762$		0			0
$763$		−	34.1768i		−	1.23728i
$764$		0			0
$765$		0			0
$766$		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
$771$		0			0
$772$		0			0
$773$		−	46.8196i		−	1.68398i	−0.539491	−	0.841991i	$-0.681383\pi$
$773$		−	46.8196i		−	1.68398i	0.539491	−	0.841991i	$-0.318617\pi$
$774$		0			0
$775$	−12.5898	+	46.0409i	−0.452238	+	1.65384i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	0.0101591			0.000363987
$780$		0			0
$781$	−74.5442			−2.66740
$782$		0			0
$783$		0			0
$784$		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
$789$		0			0
$790$		0			0
$791$	18.3871			0.653769
$792$		0			0
$793$		−	11.6730i		−	0.414522i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	21.8566i		−	0.774200i	−0.922038	−	0.387100i	$-0.873477\pi$
$797$		−	21.8566i		−	0.774200i	0.922038	−	0.387100i	$-0.126523\pi$
$798$		0			0
$799$	−8.94556			−0.316471
$800$		0			0
$801$		0			0
$802$		0			0
$803$			10.9617i			0.386829i
$804$		0			0
$805$	5.90519	−	4.50722i	0.208130	−	0.158859i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.24801			−0.0790357			−0.0395178	−	0.999219i	$-0.512582\pi$
$809$	−2.24801			−0.0790357			−0.0395178	+	0.999219i	$0.512582\pi$
$810$		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
$813$		0			0
$814$		0			0
$815$	6.84128	+	8.96317i	0.239640	+	0.313966i
$816$		0			0
$817$		−	24.2121i		−	0.847076i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	29.1470			1.01724			0.508618	−	0.860992i	$-0.330157\pi$
$821$	29.1470			1.01724			0.508618	+	0.860992i	$0.330157\pi$
$822$		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
$825$		0			0
$826$		0			0
$827$		−	54.7619i		−	1.90426i	−0.305698	−	0.952129i	$-0.598890\pi$
$827$		−	54.7619i		−	1.90426i	0.305698	−	0.952129i	$-0.401110\pi$
$828$		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
$831$		0			0
$832$		0			0
$833$			3.60869i			0.125034i
$834$		0			0
$835$	1.63832	+	2.14647i	0.0566965	+	0.0742816i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8.60909			0.297219			0.148609	−	0.988896i	$-0.452520\pi$
$839$	8.60909			0.297219			0.148609	+	0.988896i	$0.452520\pi$
$840$		0			0
$841$	−12.0859			−0.416755
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−20.9487	+	15.9894i	−0.720657	+	0.550053i
$846$		0			0
$847$			74.1018i			2.54617i
$848$		0			0
$849$		0			0
$850$		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			0
$856$		0			0
$857$		−	13.3201i		−	0.455005i	−0.973778	−	0.227502i	$-0.926944\pi$
$857$		−	13.3201i		−	0.455005i	0.973778	−	0.227502i	$-0.0730560\pi$
$858$		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			0
$862$		0			0
$863$			21.8395i			0.743426i	0.928348	+	0.371713i	$0.121229\pi$
$863$			21.8395i			0.743426i	−0.928348	+	0.371713i	$0.878771\pi$
$864$		0			0
$865$	21.3531	+	27.9759i	0.726026	+	0.951211i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−1.24703			−0.0423026
$870$		0			0
$871$	12.0698			0.408968
$872$		0			0
$873$		0			0
$874$		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
$879$		0			0
$880$		0			0
$881$	44.7215			1.50671			0.753353	−	0.657616i	$-0.228435\pi$
$881$	44.7215			1.50671			0.753353	+	0.657616i	$0.228435\pi$
$882$		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
$885$		0			0
$886$		0			0
$887$			24.0317i			0.806906i	0.915000	+	0.403453i	$0.132190\pi$
$887$			24.0317i			0.806906i	−0.915000	+	0.403453i	$0.867810\pi$
$888$		0			0
$889$	−37.0694			−1.24327
$890$		0			0
$891$		0			0
$892$		0			0
$893$			24.3031i			0.813274i
$894$		0			0
$895$	−11.7662	+	8.98074i	−0.393301	+	0.300193i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−39.2606			−1.30942
$900$		0			0
$901$	5.58806			0.186165
$902$		0			0
$903$		0			0
$904$		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			0
$910$		0			0
$911$	−23.8128			−0.788952			−0.394476	−	0.918906i	$-0.629074\pi$
$911$	−23.8128			−0.788952			−0.394476	+	0.918906i	$0.629074\pi$
$912$		0			0
$913$			31.1039i			1.02939i
$914$		0			0
$915$		0			0
$916$		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
$921$		0			0
$922$		0			0
$923$		−	14.2341i		−	0.468521i
$924$		0			0
$925$	−37.1320	−	10.1537i	−1.22089	−	0.333850i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−5.19696			−0.170507			−0.0852534	−	0.996359i	$-0.527170\pi$
$929$	−5.19696			−0.170507			−0.0852534	+	0.996359i	$0.527170\pi$
$930$		0			0
$931$	9.80404			0.321315
$932$		0			0
$933$		0			0
$934$		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
$939$		0			0
$940$		0			0
$941$	−50.2138			−1.63692			−0.818461	−	0.574562i	$-0.805173\pi$
$941$	−50.2138			−1.63692			−0.818461	+	0.574562i	$0.805173\pi$
$942$		0			0
$943$		−	0.00418347i		−	0.000136232i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	18.3661i		−	0.596817i	−0.954438	−	0.298408i	$-0.903544\pi$
$947$		−	18.3661i		−	0.596817i	0.954438	−	0.298408i	$-0.0964558\pi$
$948$		0			0
$949$	−2.09311			−0.0679453
$950$		0			0
$951$		0			0
$952$		0			0
$953$			34.9542i			1.13228i	0.824310	+	0.566139i	$0.191563\pi$
$953$			34.9542i			1.13228i	−0.824310	+	0.566139i	$0.808437\pi$
$954$		0			0
$955$	0.525403	−	0.401022i	0.0170016	−	0.0129768i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.34980			0.237338
$960$		0			0
$961$	60.1308			1.93970
$962$		0			0
$963$		0			0
$964$		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
$969$		0			0
$970$		0			0
$971$	14.8815			0.477570			0.238785	−	0.971072i	$-0.423251\pi$
$971$	14.8815			0.477570			0.238785	+	0.971072i	$0.423251\pi$
$972$		0			0
$973$			21.7184i			0.696261i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	18.6833i		−	0.597730i	−0.954295	−	0.298865i	$-0.903392\pi$
$977$		−	18.6833i		−	0.597730i	0.954295	−	0.298865i	$-0.0966081\pi$
$978$		0			0
$979$	−34.6641			−1.10787
$980$		0			0
$981$		0			0
$982$		0			0
$983$			28.4191i			0.906428i	0.891402	+	0.453214i	$0.149723\pi$
$983$			28.4191i			0.906428i	−0.891402	+	0.453214i	$0.850277\pi$
$984$		0			0
$985$	−11.6689	−	15.2882i	−0.371803	−	0.487122i
$986$		0			0
$987$		0			0
$988$		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
$993$		0			0
$994$		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
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4140.2.f.b.829.2		12
3.2	odd	2		460.2.c.a.369.4	✓	12
5.4	even	2	inner	4140.2.f.b.829.1		12
12.11	even	2		1840.2.e.f.369.9		12
15.2	even	4		2300.2.a.n.1.3		6
15.8	even	4		2300.2.a.o.1.4		6
15.14	odd	2		460.2.c.a.369.9	yes	12
60.23	odd	4		9200.2.a.cx.1.3		6
60.47	odd	4		9200.2.a.cy.1.4		6
60.59	even	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	3.2	odd	2
460.2.c.a.369.9	yes	12	15.14	odd	2
1840.2.e.f.369.4		12	60.59	even	2
1840.2.e.f.369.9		12	12.11	even	2
2300.2.a.n.1.3		6	15.2	even	4
2300.2.a.o.1.4		6	15.8	even	4
4140.2.f.b.829.1		12	5.4	even	2	inner
4140.2.f.b.829.2		12	1.1	even	1	trivial
9200.2.a.cx.1.3		6	60.23	odd	4
9200.2.a.cy.1.4		6	60.47	odd	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:	\( N \)	\(=\)	\( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4140.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.77747 + 1.35668i) q^{5} +3.32224i q^{7} +O(q^{10})\)	\(q+(-1.77747 + 1.35668i) q^{5} +3.32224i q^{7} -5.77103 q^{11} -1.10197i q^{13} -0.893847i q^{17} -2.42839 q^{19} +1.00000i q^{23} +(1.31882 - 4.82294i) q^{25} +4.11268 q^{29} -9.54624 q^{31} +(-4.50722 - 5.90519i) q^{35} -7.69904i q^{37} -0.00418347 q^{41} +9.97045i q^{43} -10.0079i q^{47} -4.03726 q^{49} +6.25169i q^{53} +(10.2579 - 7.82946i) q^{55} -10.7764 q^{59} +10.5929 q^{61} +(1.49502 + 1.95872i) q^{65} +10.9529i q^{67} +12.9170 q^{71} -1.89943i q^{73} -19.1727i q^{77} +0.216085 q^{79} -5.38967i q^{83} +(1.21267 + 1.58879i) q^{85} +6.00657 q^{89} +3.66100 q^{91} +(4.31640 - 3.29456i) q^{95} +2.08104i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 4 q^{11} - 8 q^{19} + 8 q^{25} + 10 q^{29} + 18 q^{31} + 10 q^{35} + 2 q^{41} - 38 q^{49} + 16 q^{55} - 22 q^{59} - 8 q^{61} - 38 q^{65} + 34 q^{71} - 20 q^{79} + 6 q^{85} - 48 q^{89} - 8 q^{91} - 12 q^{95}+O(q^{100}) \) 12 * q - 4 * q^11 - 8 * q^19 + 8 * q^25 + 10 * q^29 + 18 * q^31 + 10 * q^35 + 2 * q^41 - 38 * q^49 + 16 * q^55 - 22 * q^59 - 8 * q^61 - 38 * q^65 + 34 * q^71 - 20 * q^79 + 6 * q^85 - 48 * q^89 - 8 * q^91 - 12 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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