LMFDB - Embedded newform 4012.2.a.i.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

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:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 8 x^{17} - 4 x^{16} + 178 x^{15} - 265 x^{14} - 1405 x^{13} + 3503 x^{12} + 4295 x^{11} + \cdots - 16 $ x^18 - 8x^17 - 4x^16 + 178x^15 - 265x^14 - 1405x^13 + 3503x^12 + 4295x^11 - 17374x^10 - 893x^9 + 38112x^8 - 18700x^7 - 32137x^6 + 26381x^5 + 3964x^4 - 5788x^3 - 108x^2 + 232*x - 16
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$2.60297$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.60297 q^{3} +0.183919 q^{5} +3.24534 q^{7} +3.77545 q^{9} +O(q^{10})$	$q+2.60297 q^{3} +0.183919 q^{5} +3.24534 q^{7} +3.77545 q^{9} +2.13693 q^{11} +1.69478 q^{13} +0.478736 q^{15} -1.00000 q^{17} +3.65056 q^{19} +8.44753 q^{21} +6.53848 q^{23} -4.96617 q^{25} +2.01847 q^{27} +3.31726 q^{29} -4.57757 q^{31} +5.56237 q^{33} +0.596880 q^{35} -3.89290 q^{37} +4.41146 q^{39} +7.20375 q^{41} -2.66187 q^{43} +0.694377 q^{45} -8.38769 q^{47} +3.53225 q^{49} -2.60297 q^{51} -8.06541 q^{53} +0.393023 q^{55} +9.50228 q^{57} -1.00000 q^{59} +3.08619 q^{61} +12.2526 q^{63} +0.311702 q^{65} +7.66017 q^{67} +17.0195 q^{69} -10.8657 q^{71} -5.91429 q^{73} -12.9268 q^{75} +6.93508 q^{77} +5.00108 q^{79} -6.07233 q^{81} -12.0316 q^{83} -0.183919 q^{85} +8.63472 q^{87} +7.08141 q^{89} +5.50014 q^{91} -11.9153 q^{93} +0.671407 q^{95} +17.4961 q^{97} +8.06788 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	$ 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) $ 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.60297	1.50282	0.751412	−	0.659833i	$-0.229373\pi$
$3$	2.60297	1.50282	0.751412	+	0.659833i	$0.229373\pi$
$4$	0	0
$5$	0.183919	0.0822511	0.0411255	−	0.999154i	$-0.486906\pi$
$5$	0.183919	0.0822511	0.0411255	+	0.999154i	$0.486906\pi$
$6$	0	0
$7$	3.24534	1.22662	0.613312	−	0.789841i	$-0.289837\pi$
$7$	3.24534	1.22662	0.613312	+	0.789841i	$0.289837\pi$
$8$	0	0
$9$	3.77545	1.25848
$10$	0	0
$11$	2.13693	0.644310	0.322155	−	0.946687i	$-0.395593\pi$
$11$	2.13693	0.644310	0.322155	+	0.946687i	$0.395593\pi$
$12$	0	0
$13$	1.69478	0.470047	0.235024	−	0.971990i	$-0.424483\pi$
$13$	1.69478	0.470047	0.235024	+	0.971990i	$0.424483\pi$
$14$	0	0
$15$	0.478736	0.123609
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	3.65056	0.837495	0.418747	−	0.908103i	$-0.362469\pi$
$19$	3.65056	0.837495	0.418747	+	0.908103i	$0.362469\pi$
$20$	0	0
$21$	8.44753	1.84340
$22$	0	0
$23$	6.53848	1.36337	0.681684	−	0.731647i	$-0.261248\pi$
$23$	6.53848	1.36337	0.681684	+	0.731647i	$0.261248\pi$
$24$	0	0
$25$	−4.96617	−0.993235
$26$	0	0
$27$	2.01847	0.388455
$28$	0	0
$29$	3.31726	0.615999	0.308000	−	0.951386i	$-0.400340\pi$
$29$	3.31726	0.615999	0.308000	+	0.951386i	$0.400340\pi$
$30$	0	0
$31$	−4.57757	−0.822156	−0.411078	−	0.911600i	$-0.634848\pi$
$31$	−4.57757	−0.822156	−0.411078	+	0.911600i	$0.634848\pi$
$32$	0	0
$33$	5.56237	0.968284
$34$	0	0
$35$	0.596880	0.100891
$36$	0	0
$37$	−3.89290	−0.639988	−0.319994	−	0.947420i	$-0.603681\pi$
$37$	−3.89290	−0.639988	−0.319994	+	0.947420i	$0.603681\pi$
$38$	0	0
$39$	4.41146	0.706399
$40$	0	0
$41$	7.20375	1.12504	0.562518	−	0.826785i	$-0.309833\pi$
$41$	7.20375	1.12504	0.562518	+	0.826785i	$0.309833\pi$
$42$	0	0
$43$	−2.66187	−0.405932	−0.202966	−	0.979186i	$-0.565058\pi$
$43$	−2.66187	−0.405932	−0.202966	+	0.979186i	$0.565058\pi$
$44$	0	0
$45$	0.694377	0.103512
$46$	0	0
$47$	−8.38769	−1.22347	−0.611735	−	0.791062i	$-0.709528\pi$
$47$	−8.38769	−1.22347	−0.611735	+	0.791062i	$0.709528\pi$
$48$	0	0
$49$	3.53225	0.504608
$50$	0	0
$51$	−2.60297	−0.364489
$52$	0	0
$53$	−8.06541	−1.10787	−0.553934	−	0.832560i	$-0.686874\pi$
$53$	−8.06541	−1.10787	−0.553934	+	0.832560i	$0.686874\pi$
$54$	0	0
$55$	0.393023	0.0529952
$56$	0	0
$57$	9.50228	1.25861
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	3.08619	0.395146	0.197573	−	0.980288i	$-0.436694\pi$
$61$	3.08619	0.395146	0.197573	+	0.980288i	$0.436694\pi$
$62$	0	0
$63$	12.2526	1.54369
$64$	0	0
$65$	0.311702	0.0386619
$66$	0	0
$67$	7.66017	0.935839	0.467919	−	0.883771i	$-0.345004\pi$
$67$	7.66017	0.935839	0.467919	+	0.883771i	$0.345004\pi$
$68$	0	0
$69$	17.0195	2.04890
$70$	0	0
$71$	−10.8657	−1.28953	−0.644763	−	0.764383i	$-0.723044\pi$
$71$	−10.8657	−1.28953	−0.644763	+	0.764383i	$0.723044\pi$
$72$	0	0
$73$	−5.91429	−0.692216	−0.346108	−	0.938195i	$-0.612497\pi$
$73$	−5.91429	−0.692216	−0.346108	+	0.938195i	$0.612497\pi$
$74$	0	0
$75$	−12.9268	−1.49266
$76$	0	0
$77$	6.93508	0.790326
$78$	0	0
$79$	5.00108	0.562665	0.281333	−	0.959610i	$-0.409224\pi$
$79$	5.00108	0.562665	0.281333	+	0.959610i	$0.409224\pi$
$80$	0	0
$81$	−6.07233	−0.674704
$82$	0	0
$83$	−12.0316	−1.32064	−0.660321	−	0.750984i	$-0.729580\pi$
$83$	−12.0316	−1.32064	−0.660321	+	0.750984i	$0.729580\pi$
$84$	0	0
$85$	−0.183919	−0.0199488
$86$	0	0
$87$	8.63472	0.925739
$88$	0	0
$89$	7.08141	0.750628	0.375314	−	0.926898i	$-0.377535\pi$
$89$	7.08141	0.750628	0.375314	+	0.926898i	$0.377535\pi$
$90$	0	0
$91$	5.50014	0.576571
$92$	0	0
$93$	−11.9153	−1.23556
$94$	0	0
$95$	0.671407	0.0688849
$96$	0	0
$97$	17.4961	1.77646	0.888228	−	0.459404i	$-0.151937\pi$
$97$	17.4961	1.77646	0.888228	+	0.459404i	$0.151937\pi$
$98$	0	0
$99$	8.06788	0.810853
$100$	0	0
$101$	−0.118123	−0.0117537	−0.00587683	−	0.999983i	$-0.501871\pi$
$101$	−0.118123	−0.0117537	−0.00587683	+	0.999983i	$0.501871\pi$
$102$	0	0
$103$	0.505459	0.0498043	0.0249022	−	0.999690i	$-0.492073\pi$
$103$	0.505459	0.0498043	0.0249022	+	0.999690i	$0.492073\pi$
$104$	0	0
$105$	1.55366	0.151622
$106$	0	0
$107$	9.11952	0.881617	0.440809	−	0.897601i	$-0.354692\pi$
$107$	9.11952	0.881617	0.440809	+	0.897601i	$0.354692\pi$
$108$	0	0
$109$	−0.365529	−0.0350113	−0.0175057	−	0.999847i	$-0.505573\pi$
$109$	−0.365529	−0.0350113	−0.0175057	+	0.999847i	$0.505573\pi$
$110$	0	0
$111$	−10.1331	−0.961790
$112$	0	0
$113$	7.03957	0.662227	0.331114	−	0.943591i	$-0.392576\pi$
$113$	7.03957	0.662227	0.331114	+	0.943591i	$0.392576\pi$
$114$	0	0
$115$	1.20255	0.112138
$116$	0	0
$117$	6.39855	0.591546
$118$	0	0
$119$	−3.24534	−0.297500
$120$	0	0
$121$	−6.43352	−0.584865
$122$	0	0
$123$	18.7511	1.69073
$124$	0	0
$125$	−1.83297	−0.163946
$126$	0	0
$127$	4.96968	0.440988	0.220494	−	0.975388i	$-0.429233\pi$
$127$	4.96968	0.440988	0.220494	+	0.975388i	$0.429233\pi$
$128$	0	0
$129$	−6.92878	−0.610045
$130$	0	0
$131$	−11.1572	−0.974806	−0.487403	−	0.873177i	$-0.662056\pi$
$131$	−11.1572	−0.974806	−0.487403	+	0.873177i	$0.662056\pi$
$132$	0	0
$133$	11.8473	1.02729
$134$	0	0
$135$	0.371235	0.0319508
$136$	0	0
$137$	−19.8885	−1.69919	−0.849594	−	0.527437i	$-0.823153\pi$
$137$	−19.8885	−1.69919	−0.849594	+	0.527437i	$0.823153\pi$
$138$	0	0
$139$	−18.4697	−1.56658	−0.783290	−	0.621656i	$-0.786460\pi$
$139$	−18.4697	−1.56658	−0.783290	+	0.621656i	$0.786460\pi$
$140$	0	0
$141$	−21.8329	−1.83866
$142$	0	0
$143$	3.62163	0.302856
$144$	0	0
$145$	0.610107	0.0506666
$146$	0	0
$147$	9.19435	0.758337
$148$	0	0
$149$	9.17504	0.751649	0.375824	−	0.926691i	$-0.377360\pi$
$149$	9.17504	0.751649	0.375824	+	0.926691i	$0.377360\pi$
$150$	0	0
$151$	0.0817592	0.00665347	0.00332673	−	0.999994i	$-0.498941\pi$
$151$	0.0817592	0.00665347	0.00332673	+	0.999994i	$0.498941\pi$
$152$	0	0
$153$	−3.77545	−0.305227
$154$	0	0
$155$	−0.841902	−0.0676232
$156$	0	0
$157$	−0.825742	−0.0659014	−0.0329507	−	0.999457i	$-0.510490\pi$
$157$	−0.825742	−0.0659014	−0.0329507	+	0.999457i	$0.510490\pi$
$158$	0	0
$159$	−20.9940	−1.66493
$160$	0	0
$161$	21.2196	1.67234
$162$	0	0
$163$	6.04508	0.473487	0.236744	−	0.971572i	$-0.423920\pi$
$163$	6.04508	0.473487	0.236744	+	0.971572i	$0.423920\pi$
$164$	0	0
$165$	1.02303	0.0796425
$166$	0	0
$167$	12.2713	0.949583	0.474792	−	0.880098i	$-0.342523\pi$
$167$	12.2713	0.949583	0.474792	+	0.880098i	$0.342523\pi$
$168$	0	0
$169$	−10.1277	−0.779056
$170$	0	0
$171$	13.7825	1.05397
$172$	0	0
$173$	15.7725	1.19916	0.599582	−	0.800313i	$-0.295333\pi$
$173$	15.7725	1.19916	0.599582	+	0.800313i	$0.295333\pi$
$174$	0	0
$175$	−16.1169	−1.21833
$176$	0	0
$177$	−2.60297	−0.195651
$178$	0	0
$179$	6.00318	0.448699	0.224349	−	0.974509i	$-0.427974\pi$
$179$	6.00318	0.448699	0.224349	+	0.974509i	$0.427974\pi$
$180$	0	0
$181$	−0.861323	−0.0640216	−0.0320108	−	0.999488i	$-0.510191\pi$
$181$	−0.861323	−0.0640216	−0.0320108	+	0.999488i	$0.510191\pi$
$182$	0	0
$183$	8.03325	0.593835
$184$	0	0
$185$	−0.715978	−0.0526397
$186$	0	0
$187$	−2.13693	−0.156268
$188$	0	0
$189$	6.55063	0.476488
$190$	0	0
$191$	1.42486	0.103099	0.0515497	−	0.998670i	$-0.483584\pi$
$191$	1.42486	0.103099	0.0515497	+	0.998670i	$0.483584\pi$
$192$	0	0
$193$	−6.91882	−0.498028	−0.249014	−	0.968500i	$-0.580106\pi$
$193$	−6.91882	−0.498028	−0.249014	+	0.968500i	$0.580106\pi$
$194$	0	0
$195$	0.811351	0.0581021
$196$	0	0
$197$	19.1858	1.36693	0.683465	−	0.729984i	$-0.260472\pi$
$197$	19.1858	1.36693	0.683465	+	0.729984i	$0.260472\pi$
$198$	0	0
$199$	−14.8550	−1.05305	−0.526523	−	0.850161i	$-0.676505\pi$
$199$	−14.8550	−1.05305	−0.526523	+	0.850161i	$0.676505\pi$
$200$	0	0
$201$	19.9392	1.40640
$202$	0	0
$203$	10.7656	0.755600
$204$	0	0
$205$	1.32491	0.0925355
$206$	0	0
$207$	24.6857	1.71577
$208$	0	0
$209$	7.80099	0.539606
$210$	0	0
$211$	−1.32963	−0.0915358	−0.0457679	−	0.998952i	$-0.514573\pi$
$211$	−1.32963	−0.0915358	−0.0457679	+	0.998952i	$0.514573\pi$
$212$	0	0
$213$	−28.2832	−1.93793
$214$	0	0
$215$	−0.489569	−0.0333883
$216$	0	0
$217$	−14.8558	−1.00848
$218$	0	0
$219$	−15.3947	−1.04028
$220$	0	0
$221$	−1.69478	−0.114003
$222$	0	0
$223$	−9.96669	−0.667419	−0.333709	−	0.942676i	$-0.608300\pi$
$223$	−9.96669	−0.667419	−0.333709	+	0.942676i	$0.608300\pi$
$224$	0	0
$225$	−18.7495	−1.24997
$226$	0	0
$227$	−5.96811	−0.396117	−0.198059	−	0.980190i	$-0.563464\pi$
$227$	−5.96811	−0.396117	−0.198059	+	0.980190i	$0.563464\pi$
$228$	0	0
$229$	2.72880	0.180324	0.0901622	−	0.995927i	$-0.471261\pi$
$229$	2.72880	0.180324	0.0901622	+	0.995927i	$0.471261\pi$
$230$	0	0
$231$	18.0518	1.18772
$232$	0	0
$233$	14.8885	0.975380	0.487690	−	0.873017i	$-0.337840\pi$
$233$	14.8885	0.975380	0.487690	+	0.873017i	$0.337840\pi$
$234$	0	0
$235$	−1.54266	−0.100632
$236$	0	0
$237$	13.0176	0.845587
$238$	0	0
$239$	−4.72236	−0.305464	−0.152732	−	0.988268i	$-0.548807\pi$
$239$	−4.72236	−0.305464	−0.152732	+	0.988268i	$0.548807\pi$
$240$	0	0
$241$	21.3523	1.37542	0.687712	−	0.725983i	$-0.258615\pi$
$241$	21.3523	1.37542	0.687712	+	0.725983i	$0.258615\pi$
$242$	0	0
$243$	−21.8615	−1.40242
$244$	0	0
$245$	0.649649	0.0415045
$246$	0	0
$247$	6.18689	0.393662
$248$	0	0
$249$	−31.3179	−1.98469
$250$	0	0
$251$	28.6463	1.80814	0.904069	−	0.427387i	$-0.140566\pi$
$251$	28.6463	1.80814	0.904069	+	0.427387i	$0.140566\pi$
$252$	0	0
$253$	13.9723	0.878431
$254$	0	0
$255$	−0.478736	−0.0299796
$256$	0	0
$257$	−12.3391	−0.769692	−0.384846	−	0.922981i	$-0.625745\pi$
$257$	−12.3391	−0.769692	−0.384846	+	0.922981i	$0.625745\pi$
$258$	0	0
$259$	−12.6338	−0.785025
$260$	0	0
$261$	12.5241	0.775225
$262$	0	0
$263$	−16.7665	−1.03387	−0.516934	−	0.856025i	$-0.672927\pi$
$263$	−16.7665	−1.03387	−0.516934	+	0.856025i	$0.672927\pi$
$264$	0	0
$265$	−1.48338	−0.0911234
$266$	0	0
$267$	18.4327	1.12806
$268$	0	0
$269$	5.54075	0.337825	0.168913	−	0.985631i	$-0.445974\pi$
$269$	5.54075	0.337825	0.168913	+	0.985631i	$0.445974\pi$
$270$	0	0
$271$	−10.0167	−0.608472	−0.304236	−	0.952597i	$-0.598401\pi$
$271$	−10.0167	−0.608472	−0.304236	+	0.952597i	$0.598401\pi$
$272$	0	0
$273$	14.3167	0.866486
$274$	0	0
$275$	−10.6124	−0.639951
$276$	0	0
$277$	−30.9671	−1.86063	−0.930317	−	0.366756i	$-0.880469\pi$
$277$	−30.9671	−1.86063	−0.930317	+	0.366756i	$0.880469\pi$
$278$	0	0
$279$	−17.2824	−1.03467
$280$	0	0
$281$	21.3375	1.27289	0.636443	−	0.771324i	$-0.280405\pi$
$281$	21.3375	1.27289	0.636443	+	0.771324i	$0.280405\pi$
$282$	0	0
$283$	9.10174	0.541042	0.270521	−	0.962714i	$-0.412804\pi$
$283$	9.10174	0.541042	0.270521	+	0.962714i	$0.412804\pi$
$284$	0	0
$285$	1.74765	0.103522
$286$	0	0
$287$	23.3786	1.38000
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	45.5417	2.66970
$292$	0	0
$293$	16.4989	0.963874	0.481937	−	0.876206i	$-0.339933\pi$
$293$	16.4989	0.963874	0.481937	+	0.876206i	$0.339933\pi$
$294$	0	0
$295$	−0.183919	−0.0107082
$296$	0	0
$297$	4.31333	0.250285
$298$	0	0
$299$	11.0813	0.640847
$300$	0	0
$301$	−8.63869	−0.497926
$302$	0	0
$303$	−0.307470	−0.0176637
$304$	0	0
$305$	0.567608	0.0325012
$306$	0	0
$307$	−3.92898	−0.224239	−0.112120	−	0.993695i	$-0.535764\pi$
$307$	−3.92898	−0.224239	−0.112120	+	0.993695i	$0.535764\pi$
$308$	0	0
$309$	1.31569	0.0748472
$310$	0	0
$311$	−12.3868	−0.702393	−0.351197	−	0.936302i	$-0.614225\pi$
$311$	−12.3868	−0.702393	−0.351197	+	0.936302i	$0.614225\pi$
$312$	0	0
$313$	5.35016	0.302409	0.151204	−	0.988503i	$-0.451685\pi$
$313$	5.35016	0.302409	0.151204	+	0.988503i	$0.451685\pi$
$314$	0	0
$315$	2.25349	0.126970
$316$	0	0
$317$	6.96467	0.391175	0.195587	−	0.980686i	$-0.437339\pi$
$317$	6.96467	0.391175	0.195587	+	0.980686i	$0.437339\pi$
$318$	0	0
$319$	7.08876	0.396894
$320$	0	0
$321$	23.7378	1.32492
$322$	0	0
$323$	−3.65056	−0.203122
$324$	0	0
$325$	−8.41657	−0.466867
$326$	0	0
$327$	−0.951461	−0.0526159
$328$	0	0
$329$	−27.2209	−1.50074
$330$	0	0
$331$	−22.9195	−1.25977	−0.629885	−	0.776688i	$-0.716898\pi$
$331$	−22.9195	−1.25977	−0.629885	+	0.776688i	$0.716898\pi$
$332$	0	0
$333$	−14.6974	−0.805414
$334$	0	0
$335$	1.40885	0.0769738
$336$	0	0
$337$	27.2584	1.48486	0.742429	−	0.669925i	$-0.233674\pi$
$337$	27.2584	1.48486	0.742429	+	0.669925i	$0.233674\pi$
$338$	0	0
$339$	18.3238	0.995211
$340$	0	0
$341$	−9.78196	−0.529723
$342$	0	0
$343$	−11.2540	−0.607660
$344$	0	0
$345$	3.13020	0.168524
$346$	0	0
$347$	32.1749	1.72724	0.863621	−	0.504142i	$-0.168191\pi$
$347$	32.1749	1.72724	0.863621	+	0.504142i	$0.168191\pi$
$348$	0	0
$349$	11.0679	0.592451	0.296226	−	0.955118i	$-0.404272\pi$
$349$	11.0679	0.592451	0.296226	+	0.955118i	$0.404272\pi$
$350$	0	0
$351$	3.42086	0.182592
$352$	0	0
$353$	−20.7585	−1.10486	−0.552431	−	0.833559i	$-0.686300\pi$
$353$	−20.7585	−1.10486	−0.552431	+	0.833559i	$0.686300\pi$
$354$	0	0
$355$	−1.99842	−0.106065
$356$	0	0
$357$	−8.44753	−0.447091
$358$	0	0
$359$	0.945197	0.0498856	0.0249428	−	0.999689i	$-0.492060\pi$
$359$	0.945197	0.0498856	0.0249428	+	0.999689i	$0.492060\pi$
$360$	0	0
$361$	−5.67344	−0.298602
$362$	0	0
$363$	−16.7462	−0.878950
$364$	0	0
$365$	−1.08775	−0.0569355
$366$	0	0
$367$	−31.4931	−1.64393	−0.821964	−	0.569540i	$-0.807121\pi$
$367$	−31.4931	−1.64393	−0.821964	+	0.569540i	$0.807121\pi$
$368$	0	0
$369$	27.1974	1.41584
$370$	0	0
$371$	−26.1750	−1.35894
$372$	0	0
$373$	−22.8434	−1.18279	−0.591394	−	0.806383i	$-0.701422\pi$
$373$	−22.8434	−1.18279	−0.591394	+	0.806383i	$0.701422\pi$
$374$	0	0
$375$	−4.77116	−0.246382
$376$	0	0
$377$	5.62202	0.289549
$378$	0	0
$379$	−25.5561	−1.31273	−0.656365	−	0.754444i	$-0.727907\pi$
$379$	−25.5561	−1.31273	−0.656365	+	0.754444i	$0.727907\pi$
$380$	0	0
$381$	12.9359	0.662727
$382$	0	0
$383$	−17.0633	−0.871896	−0.435948	−	0.899972i	$-0.643587\pi$
$383$	−17.0633	−0.871896	−0.435948	+	0.899972i	$0.643587\pi$
$384$	0	0
$385$	1.27549	0.0650052
$386$	0	0
$387$	−10.0498	−0.510858
$388$	0	0
$389$	−8.62659	−0.437385	−0.218693	−	0.975794i	$-0.570179\pi$
$389$	−8.62659	−0.437385	−0.218693	+	0.975794i	$0.570179\pi$
$390$	0	0
$391$	−6.53848	−0.330665
$392$	0	0
$393$	−29.0417	−1.46496
$394$	0	0
$395$	0.919793	0.0462798
$396$	0	0
$397$	−27.1729	−1.36377	−0.681885	−	0.731459i	$-0.738840\pi$
$397$	−27.1729	−1.36377	−0.681885	+	0.731459i	$0.738840\pi$
$398$	0	0
$399$	30.8382	1.54384
$400$	0	0
$401$	2.84509	0.142077	0.0710386	−	0.997474i	$-0.477369\pi$
$401$	2.84509	0.142077	0.0710386	+	0.997474i	$0.477369\pi$
$402$	0	0
$403$	−7.75797	−0.386452
$404$	0	0
$405$	−1.11682	−0.0554951
$406$	0	0
$407$	−8.31886	−0.412350
$408$	0	0
$409$	−29.3190	−1.44973	−0.724866	−	0.688890i	$-0.758098\pi$
$409$	−29.3190	−1.44973	−0.724866	+	0.688890i	$0.758098\pi$
$410$	0	0
$411$	−51.7691	−2.55358
$412$	0	0
$413$	−3.24534	−0.159693
$414$	0	0
$415$	−2.21284	−0.108624
$416$	0	0
$417$	−48.0761	−2.35430
$418$	0	0
$419$	−0.366603	−0.0179097	−0.00895487	−	0.999960i	$-0.502850\pi$
$419$	−0.366603	−0.0179097	−0.00895487	+	0.999960i	$0.502850\pi$
$420$	0	0
$421$	−13.2117	−0.643898	−0.321949	−	0.946757i	$-0.604338\pi$
$421$	−13.2117	−0.643898	−0.321949	+	0.946757i	$0.604338\pi$
$422$	0	0
$423$	−31.6673	−1.53972
$424$	0	0
$425$	4.96617	0.240895
$426$	0	0
$427$	10.0157	0.484695
$428$	0	0
$429$	9.42699	0.455139
$430$	0	0
$431$	32.7117	1.57567	0.787834	−	0.615888i	$-0.211203\pi$
$431$	32.7117	1.57567	0.787834	+	0.615888i	$0.211203\pi$
$432$	0	0
$433$	23.2253	1.11614	0.558069	−	0.829795i	$-0.311543\pi$
$433$	23.2253	1.11614	0.558069	+	0.829795i	$0.311543\pi$
$434$	0	0
$435$	1.58809	0.0761431
$436$	0	0
$437$	23.8691	1.14181
$438$	0	0
$439$	26.8785	1.28284	0.641421	−	0.767189i	$-0.278345\pi$
$439$	26.8785	1.28284	0.641421	+	0.767189i	$0.278345\pi$
$440$	0	0
$441$	13.3358	0.635040
$442$	0	0
$443$	−15.4738	−0.735181	−0.367591	−	0.929988i	$-0.619817\pi$
$443$	−15.4738	−0.735181	−0.367591	+	0.929988i	$0.619817\pi$
$444$	0	0
$445$	1.30241	0.0617400
$446$	0	0
$447$	23.8824	1.12960
$448$	0	0
$449$	10.4782	0.494498	0.247249	−	0.968952i	$-0.420473\pi$
$449$	10.4782	0.494498	0.247249	+	0.968952i	$0.420473\pi$
$450$	0	0
$451$	15.3939	0.724871
$452$	0	0
$453$	0.212817	0.00999900
$454$	0	0
$455$	1.01158	0.0474236
$456$	0	0
$457$	−4.36630	−0.204247	−0.102123	−	0.994772i	$-0.532564\pi$
$457$	−4.36630	−0.204247	−0.102123	+	0.994772i	$0.532564\pi$
$458$	0	0
$459$	−2.01847	−0.0942141
$460$	0	0
$461$	−17.5087	−0.815461	−0.407731	−	0.913102i	$-0.633680\pi$
$461$	−17.5087	−0.815461	−0.407731	+	0.913102i	$0.633680\pi$
$462$	0	0
$463$	11.9590	0.555780	0.277890	−	0.960613i	$-0.410365\pi$
$463$	11.9590	0.555780	0.277890	+	0.960613i	$0.410365\pi$
$464$	0	0
$465$	−2.19145	−0.101626
$466$	0	0
$467$	13.3561	0.618047	0.309024	−	0.951054i	$-0.399998\pi$
$467$	13.3561	0.618047	0.309024	+	0.951054i	$0.399998\pi$
$468$	0	0
$469$	24.8599	1.14792
$470$	0	0
$471$	−2.14938	−0.0990383
$472$	0	0
$473$	−5.68825	−0.261546
$474$	0	0
$475$	−18.1293	−0.831829
$476$	0	0
$477$	−30.4505	−1.39423
$478$	0	0
$479$	4.59012	0.209728	0.104864	−	0.994487i	$-0.466559\pi$
$479$	4.59012	0.209728	0.104864	+	0.994487i	$0.466559\pi$
$480$	0	0
$481$	−6.59760	−0.300825
$482$	0	0
$483$	55.2340	2.51323
$484$	0	0
$485$	3.21786	0.146115
$486$	0	0
$487$	−12.5186	−0.567270	−0.283635	−	0.958932i	$-0.591540\pi$
$487$	−12.5186	−0.567270	−0.283635	+	0.958932i	$0.591540\pi$
$488$	0	0
$489$	15.7352	0.711569
$490$	0	0
$491$	−6.78378	−0.306148	−0.153074	−	0.988215i	$-0.548917\pi$
$491$	−6.78378	−0.306148	−0.153074	+	0.988215i	$0.548917\pi$
$492$	0	0
$493$	−3.31726	−0.149402
$494$	0	0
$495$	1.48384	0.0666935
$496$	0	0
$497$	−35.2630	−1.58176
$498$	0	0
$499$	−22.3316	−0.999699	−0.499850	−	0.866112i	$-0.666611\pi$
$499$	−22.3316	−0.999699	−0.499850	+	0.866112i	$0.666611\pi$
$500$	0	0
$501$	31.9419	1.42706
$502$	0	0
$503$	−11.5501	−0.514995	−0.257497	−	0.966279i	$-0.582898\pi$
$503$	−11.5501	−0.514995	−0.257497	+	0.966279i	$0.582898\pi$
$504$	0	0
$505$	−0.0217251	−0.000966752 0
$506$	0	0
$507$	−26.3622	−1.17078
$508$	0	0
$509$	−36.5738	−1.62111	−0.810553	−	0.585665i	$-0.800833\pi$
$509$	−36.5738	−1.62111	−0.810553	+	0.585665i	$0.800833\pi$
$510$	0	0
$511$	−19.1939	−0.849089
$512$	0	0
$513$	7.36854	0.325329
$514$	0	0
$515$	0.0929635	0.00409646
$516$	0	0
$517$	−17.9239	−0.788294
$518$	0	0
$519$	41.0555	1.80213
$520$	0	0
$521$	26.2569	1.15034	0.575168	−	0.818035i	$-0.304937\pi$
$521$	26.2569	1.15034	0.575168	+	0.818035i	$0.304937\pi$
$522$	0	0
$523$	23.8328	1.04213	0.521067	−	0.853516i	$-0.325534\pi$
$523$	23.8328	1.04213	0.521067	+	0.853516i	$0.325534\pi$
$524$	0	0
$525$	−41.9519	−1.83093
$526$	0	0
$527$	4.57757	0.199402
$528$	0	0
$529$	19.7517	0.858771
$530$	0	0
$531$	−3.77545	−0.163841
$532$	0	0
$533$	12.2088	0.528820
$534$	0	0
$535$	1.67725	0.0725140
$536$	0	0
$537$	15.6261	0.674316
$538$	0	0
$539$	7.54819	0.325124
$540$	0	0
$541$	10.7617	0.462682	0.231341	−	0.972873i	$-0.425689\pi$
$541$	10.7617	0.462682	0.231341	+	0.972873i	$0.425689\pi$
$542$	0	0
$543$	−2.24200	−0.0962133
$544$	0	0
$545$	−0.0672277	−0.00287972
$546$	0	0
$547$	19.6808	0.841492	0.420746	−	0.907178i	$-0.361768\pi$
$547$	19.6808	0.841492	0.420746	+	0.907178i	$0.361768\pi$
$548$	0	0
$549$	11.6517	0.497284
$550$	0	0
$551$	12.1098	0.515896
$552$	0	0
$553$	16.2302	0.690179
$554$	0	0
$555$	−1.86367	−0.0791083
$556$	0	0
$557$	−16.4435	−0.696733	−0.348366	−	0.937358i	$-0.613264\pi$
$557$	−16.4435	−0.696733	−0.348366	+	0.937358i	$0.613264\pi$
$558$	0	0
$559$	−4.51129	−0.190807
$560$	0	0
$561$	−5.56237	−0.234843
$562$	0	0
$563$	13.0033	0.548024	0.274012	−	0.961726i	$-0.411649\pi$
$563$	13.0033	0.548024	0.274012	+	0.961726i	$0.411649\pi$
$564$	0	0
$565$	1.29471	0.0544689
$566$	0	0
$567$	−19.7068	−0.827608
$568$	0	0
$569$	−3.88723	−0.162961	−0.0814805	−	0.996675i	$-0.525965\pi$
$569$	−3.88723	−0.162961	−0.0814805	+	0.996675i	$0.525965\pi$
$570$	0	0
$571$	31.7365	1.32813	0.664065	−	0.747675i	$-0.268830\pi$
$571$	31.7365	1.32813	0.664065	+	0.747675i	$0.268830\pi$
$572$	0	0
$573$	3.70887	0.154940
$574$	0	0
$575$	−32.4712	−1.35414
$576$	0	0
$577$	41.8866	1.74376	0.871880	−	0.489719i	$-0.162901\pi$
$577$	41.8866	1.74376	0.871880	+	0.489719i	$0.162901\pi$
$578$	0	0
$579$	−18.0095	−0.748448
$580$	0	0
$581$	−39.0467	−1.61993
$582$	0	0
$583$	−17.2352	−0.713810
$584$	0	0
$585$	1.17682	0.0486553
$586$	0	0
$587$	17.2349	0.711362	0.355681	−	0.934607i	$-0.384249\pi$
$587$	17.2349	0.711362	0.355681	+	0.934607i	$0.384249\pi$
$588$	0	0
$589$	−16.7107	−0.688551
$590$	0	0
$591$	49.9400	2.05426
$592$	0	0
$593$	−30.7853	−1.26420	−0.632100	−	0.774887i	$-0.717807\pi$
$593$	−30.7853	−1.26420	−0.632100	+	0.774887i	$0.717807\pi$
$594$	0	0
$595$	−0.596880	−0.0244697
$596$	0	0
$597$	−38.6672	−1.58254
$598$	0	0
$599$	4.12331	0.168474	0.0842370	−	0.996446i	$-0.473155\pi$
$599$	4.12331	0.168474	0.0842370	+	0.996446i	$0.473155\pi$
$600$	0	0
$601$	−36.4662	−1.48749	−0.743743	−	0.668466i	$-0.766951\pi$
$601$	−36.4662	−1.48749	−0.743743	+	0.668466i	$0.766951\pi$
$602$	0	0
$603$	28.9206	1.17774
$604$	0	0
$605$	−1.18325	−0.0481058
$606$	0	0
$607$	35.2809	1.43201	0.716003	−	0.698097i	$-0.245969\pi$
$607$	35.2809	1.43201	0.716003	+	0.698097i	$0.245969\pi$
$608$	0	0
$609$	28.0226	1.13553
$610$	0	0
$611$	−14.2153	−0.575089
$612$	0	0
$613$	2.13748	0.0863321	0.0431661	−	0.999068i	$-0.486256\pi$
$613$	2.13748	0.0863321	0.0431661	+	0.999068i	$0.486256\pi$
$614$	0	0
$615$	3.44869	0.139065
$616$	0	0
$617$	7.20487	0.290057	0.145029	−	0.989427i	$-0.453673\pi$
$617$	7.20487	0.290057	0.145029	+	0.989427i	$0.453673\pi$
$618$	0	0
$619$	14.7759	0.593895	0.296947	−	0.954894i	$-0.404031\pi$
$619$	14.7759	0.593895	0.296947	+	0.954894i	$0.404031\pi$
$620$	0	0
$621$	13.1977	0.529606
$622$	0	0
$623$	22.9816	0.920739
$624$	0	0
$625$	24.4938	0.979750
$626$	0	0
$627$	20.3057	0.810933
$628$	0	0
$629$	3.89290	0.155220
$630$	0	0
$631$	−15.6050	−0.621224	−0.310612	−	0.950537i	$-0.600534\pi$
$631$	−15.6050	−0.621224	−0.310612	+	0.950537i	$0.600534\pi$
$632$	0	0
$633$	−3.46100	−0.137562
$634$	0	0
$635$	0.914019	0.0362717
$636$	0	0
$637$	5.98639	0.237190
$638$	0	0
$639$	−41.0230	−1.62285
$640$	0	0
$641$	38.5795	1.52380	0.761899	−	0.647696i	$-0.224267\pi$
$641$	38.5795	1.52380	0.761899	+	0.647696i	$0.224267\pi$
$642$	0	0
$643$	40.6467	1.60295	0.801475	−	0.598028i	$-0.204049\pi$
$643$	40.6467	1.60295	0.801475	+	0.598028i	$0.204049\pi$
$644$	0	0
$645$	−1.27433	−0.0501768
$646$	0	0
$647$	−33.8337	−1.33014	−0.665070	−	0.746781i	$-0.731598\pi$
$647$	−33.8337	−1.33014	−0.665070	+	0.746781i	$0.731598\pi$
$648$	0	0
$649$	−2.13693	−0.0838820
$650$	0	0
$651$	−38.6692	−1.51556
$652$	0	0
$653$	−43.9691	−1.72064	−0.860321	−	0.509752i	$-0.829737\pi$
$653$	−43.9691	−1.72064	−0.860321	+	0.509752i	$0.829737\pi$
$654$	0	0
$655$	−2.05201	−0.0801788
$656$	0	0
$657$	−22.3291	−0.871142
$658$	0	0
$659$	23.8087	0.927453	0.463727	−	0.885978i	$-0.346512\pi$
$659$	23.8087	0.927453	0.463727	+	0.885978i	$0.346512\pi$
$660$	0	0
$661$	−9.60095	−0.373434	−0.186717	−	0.982414i	$-0.559785\pi$
$661$	−9.60095	−0.373434	−0.186717	+	0.982414i	$0.559785\pi$
$662$	0	0
$663$	−4.41146	−0.171327
$664$	0	0
$665$	2.17895	0.0844959
$666$	0	0
$667$	21.6898	0.839833
$668$	0	0
$669$	−25.9430	−1.00301
$670$	0	0
$671$	6.59497	0.254596
$672$	0	0
$673$	11.3624	0.437988	0.218994	−	0.975726i	$-0.429722\pi$
$673$	11.3624	0.437988	0.218994	+	0.975726i	$0.429722\pi$
$674$	0	0
$675$	−10.0241	−0.385827
$676$	0	0
$677$	16.7465	0.643622	0.321811	−	0.946804i	$-0.395709\pi$
$677$	16.7465	0.643622	0.321811	+	0.946804i	$0.395709\pi$
$678$	0	0
$679$	56.7807	2.17904
$680$	0	0
$681$	−15.5348	−0.595295
$682$	0	0
$683$	38.9375	1.48990	0.744952	−	0.667119i	$-0.232473\pi$
$683$	38.9375	1.48990	0.744952	+	0.667119i	$0.232473\pi$
$684$	0	0
$685$	−3.65787	−0.139760
$686$	0	0
$687$	7.10299	0.270996
$688$	0	0
$689$	−13.6691	−0.520751
$690$	0	0
$691$	−25.1823	−0.957979	−0.478990	−	0.877821i	$-0.658997\pi$
$691$	−25.1823	−0.957979	−0.478990	+	0.877821i	$0.658997\pi$
$692$	0	0
$693$	26.1830	0.994612
$694$	0	0
$695$	−3.39693	−0.128853
$696$	0	0
$697$	−7.20375	−0.272861
$698$	0	0
$699$	38.7544	1.46583
$700$	0	0
$701$	−42.3224	−1.59849	−0.799247	−	0.601003i	$-0.794768\pi$
$701$	−42.3224	−1.59849	−0.799247	+	0.601003i	$0.794768\pi$
$702$	0	0
$703$	−14.2112	−0.535987
$704$	0	0
$705$	−4.01549	−0.151232
$706$	0	0
$707$	−0.383349	−0.0144173
$708$	0	0
$709$	−21.2136	−0.796692	−0.398346	−	0.917235i	$-0.630416\pi$
$709$	−21.2136	−0.796692	−0.398346	+	0.917235i	$0.630416\pi$
$710$	0	0
$711$	18.8813	0.708104
$712$	0	0
$713$	−29.9304	−1.12090
$714$	0	0
$715$	0.666087	0.0249102
$716$	0	0
$717$	−12.2922	−0.459059
$718$	0	0
$719$	14.4405	0.538539	0.269270	−	0.963065i	$-0.413218\pi$
$719$	14.4405	0.538539	0.269270	+	0.963065i	$0.413218\pi$
$720$	0	0
$721$	1.64039	0.0610912
$722$	0	0
$723$	55.5795	2.06702
$724$	0	0
$725$	−16.4741	−0.611832
$726$	0	0
$727$	−16.4292	−0.609327	−0.304663	−	0.952460i	$-0.598544\pi$
$727$	−16.4292	−0.609327	−0.304663	+	0.952460i	$0.598544\pi$
$728$	0	0
$729$	−38.6878	−1.43288
$730$	0	0
$731$	2.66187	0.0984530
$732$	0	0
$733$	0.953899	0.0352331	0.0176165	−	0.999845i	$-0.494392\pi$
$733$	0.953899	0.0352331	0.0176165	+	0.999845i	$0.494392\pi$
$734$	0	0
$735$	1.69102	0.0623741
$736$	0	0
$737$	16.3693	0.602970
$738$	0	0
$739$	−31.9730	−1.17615	−0.588073	−	0.808808i	$-0.700113\pi$
$739$	−31.9730	−1.17615	−0.588073	+	0.808808i	$0.700113\pi$
$740$	0	0
$741$	16.1043	0.591605
$742$	0	0
$743$	−31.6652	−1.16168	−0.580842	−	0.814016i	$-0.697277\pi$
$743$	−31.6652	−1.16168	−0.580842	+	0.814016i	$0.697277\pi$
$744$	0	0
$745$	1.68746	0.0618239
$746$	0	0
$747$	−45.4247	−1.66200
$748$	0	0
$749$	29.5960	1.08141
$750$	0	0
$751$	−44.9261	−1.63938	−0.819688	−	0.572811i	$-0.805853\pi$
$751$	−44.9261	−1.63938	−0.819688	+	0.572811i	$0.805853\pi$
$752$	0	0
$753$	74.5654	2.71731
$754$	0	0
$755$	0.0150371	0.000547255 0
$756$	0	0
$757$	−24.6888	−0.897331	−0.448665	−	0.893700i	$-0.648100\pi$
$757$	−24.6888	−0.897331	−0.448665	+	0.893700i	$0.648100\pi$
$758$	0	0
$759$	36.3695	1.32013
$760$	0	0
$761$	22.9158	0.830696	0.415348	−	0.909663i	$-0.363660\pi$
$761$	22.9158	0.830696	0.415348	+	0.909663i	$0.363660\pi$
$762$	0	0
$763$	−1.18627	−0.0429458
$764$	0	0
$765$	−0.694377	−0.0251053
$766$	0	0
$767$	−1.69478	−0.0611949
$768$	0	0
$769$	−12.3755	−0.446270	−0.223135	−	0.974788i	$-0.571629\pi$
$769$	−12.3755	−0.446270	−0.223135	+	0.974788i	$0.571629\pi$
$770$	0	0
$771$	−32.1183	−1.15671
$772$	0	0
$773$	18.8799	0.679062	0.339531	−	0.940595i	$-0.389732\pi$
$773$	18.8799	0.679062	0.339531	+	0.940595i	$0.389732\pi$
$774$	0	0
$775$	22.7330	0.816594
$776$	0	0
$777$	−32.8854	−1.17976
$778$	0	0
$779$	26.2977	0.942212
$780$	0	0
$781$	−23.2193	−0.830854
$782$	0	0
$783$	6.69578	0.239288
$784$	0	0
$785$	−0.151870	−0.00542046
$786$	0	0
$787$	−16.2831	−0.580428	−0.290214	−	0.956962i	$-0.593727\pi$
$787$	−16.2831	−0.580428	−0.290214	+	0.956962i	$0.593727\pi$
$788$	0	0
$789$	−43.6427	−1.55372
$790$	0	0
$791$	22.8458	0.812304
$792$	0	0
$793$	5.23040	0.185737
$794$	0	0
$795$	−3.86120	−0.136943
$796$	0	0
$797$	−18.2982	−0.648155	−0.324077	−	0.946031i	$-0.605054\pi$
$797$	−18.2982	−0.648155	−0.324077	+	0.946031i	$0.605054\pi$
$798$	0	0
$799$	8.38769	0.296735
$800$	0	0
$801$	26.7355	0.944652
$802$	0	0
$803$	−12.6385	−0.446001
$804$	0	0
$805$	3.90269	0.137552
$806$	0	0
$807$	14.4224	0.507692
$808$	0	0
$809$	7.56100	0.265831	0.132915	−	0.991127i	$-0.457566\pi$
$809$	7.56100	0.265831	0.132915	+	0.991127i	$0.457566\pi$
$810$	0	0
$811$	17.4716	0.613511	0.306755	−	0.951788i	$-0.400757\pi$
$811$	17.4716	0.613511	0.306755	+	0.951788i	$0.400757\pi$
$812$	0	0
$813$	−26.0732	−0.914428
$814$	0	0
$815$	1.11181	0.0389449
$816$	0	0
$817$	−9.71732	−0.339966
$818$	0	0
$819$	20.7655	0.725605
$820$	0	0
$821$	−40.8426	−1.42542	−0.712709	−	0.701460i	$-0.752532\pi$
$821$	−40.8426	−1.42542	−0.712709	+	0.701460i	$0.752532\pi$
$822$	0	0
$823$	−17.3304	−0.604100	−0.302050	−	0.953292i	$-0.597671\pi$
$823$	−17.3304	−0.604100	−0.302050	+	0.953292i	$0.597671\pi$
$824$	0	0
$825$	−27.6237	−0.961734
$826$	0	0
$827$	15.0880	0.524662	0.262331	−	0.964978i	$-0.415509\pi$
$827$	15.0880	0.524662	0.262331	+	0.964978i	$0.415509\pi$
$828$	0	0
$829$	−46.7604	−1.62406	−0.812029	−	0.583617i	$-0.801637\pi$
$829$	−46.7604	−1.62406	−0.812029	+	0.583617i	$0.801637\pi$
$830$	0	0
$831$	−80.6065	−2.79621
$832$	0	0
$833$	−3.53225	−0.122385
$834$	0	0
$835$	2.25693	0.0781043
$836$	0	0
$837$	−9.23969	−0.319370
$838$	0	0
$839$	10.4456	0.360621	0.180311	−	0.983610i	$-0.442290\pi$
$839$	10.4456	0.360621	0.180311	+	0.983610i	$0.442290\pi$
$840$	0	0
$841$	−17.9958	−0.620545
$842$	0	0
$843$	55.5408	1.91293
$844$	0	0
$845$	−1.86268	−0.0640782
$846$	0	0
$847$	−20.8790	−0.717410
$848$	0	0
$849$	23.6916	0.813092
$850$	0	0
$851$	−25.4536	−0.872539
$852$	0	0
$853$	−26.8169	−0.918195	−0.459097	−	0.888386i	$-0.651827\pi$
$853$	−26.8169	−0.918195	−0.459097	+	0.888386i	$0.651827\pi$
$854$	0	0
$855$	2.53486	0.0866904
$856$	0	0
$857$	27.6027	0.942891	0.471445	−	0.881895i	$-0.343732\pi$
$857$	27.6027	0.942891	0.471445	+	0.881895i	$0.343732\pi$
$858$	0	0
$859$	21.9599	0.749262	0.374631	−	0.927174i	$-0.377769\pi$
$859$	21.9599	0.749262	0.374631	+	0.927174i	$0.377769\pi$
$860$	0	0
$861$	60.8539	2.07389
$862$	0	0
$863$	−8.10215	−0.275801	−0.137900	−	0.990446i	$-0.544035\pi$
$863$	−8.10215	−0.275801	−0.137900	+	0.990446i	$0.544035\pi$
$864$	0	0
$865$	2.90087	0.0986326
$866$	0	0
$867$	2.60297	0.0884015
$868$	0	0
$869$	10.6870	0.362530
$870$	0	0
$871$	12.9823	0.439888
$872$	0	0
$873$	66.0555	2.23564
$874$	0	0
$875$	−5.94861	−0.201100
$876$	0	0
$877$	8.18009	0.276222	0.138111	−	0.990417i	$-0.455897\pi$
$877$	8.18009	0.276222	0.138111	+	0.990417i	$0.455897\pi$
$878$	0	0
$879$	42.9461	1.44853
$880$	0	0
$881$	40.3846	1.36059	0.680296	−	0.732937i	$-0.261851\pi$
$881$	40.3846	1.36059	0.680296	+	0.732937i	$0.261851\pi$
$882$	0	0
$883$	−39.8592	−1.34137	−0.670685	−	0.741743i	$-0.734000\pi$
$883$	−39.8592	−1.34137	−0.670685	+	0.741743i	$0.734000\pi$
$884$	0	0
$885$	−0.478736	−0.0160925
$886$	0	0
$887$	12.3243	0.413811	0.206905	−	0.978361i	$-0.433661\pi$
$887$	12.3243	0.413811	0.206905	+	0.978361i	$0.433661\pi$
$888$	0	0
$889$	16.1283	0.540926
$890$	0	0
$891$	−12.9762	−0.434718
$892$	0	0
$893$	−30.6197	−1.02465
$894$	0	0
$895$	1.10410	0.0369060
$896$	0	0
$897$	28.8442	0.963081
$898$	0	0
$899$	−15.1850	−0.506447
$900$	0	0
$901$	8.06541	0.268698
$902$	0	0
$903$	−22.4863	−0.748296
$904$	0	0
$905$	−0.158414	−0.00526585
$906$	0	0
$907$	29.0243	0.963735	0.481868	−	0.876244i	$-0.339959\pi$
$907$	29.0243	0.963735	0.481868	+	0.876244i	$0.339959\pi$
$908$	0	0
$909$	−0.445967	−0.0147918
$910$	0	0
$911$	27.5806	0.913785	0.456892	−	0.889522i	$-0.348962\pi$
$911$	27.5806	0.913785	0.456892	+	0.889522i	$0.348962\pi$
$912$	0	0
$913$	−25.7108	−0.850902
$914$	0	0
$915$	1.47747	0.0488435
$916$	0	0
$917$	−36.2088	−1.19572
$918$	0	0
$919$	−42.6436	−1.40668	−0.703341	−	0.710853i	$-0.748309\pi$
$919$	−42.6436	−1.40668	−0.703341	+	0.710853i	$0.748309\pi$
$920$	0	0
$921$	−10.2270	−0.336992
$922$	0	0
$923$	−18.4150	−0.606138
$924$	0	0
$925$	19.3328	0.635658
$926$	0	0
$927$	1.90833	0.0626779
$928$	0	0
$929$	9.44369	0.309837	0.154919	−	0.987927i	$-0.450488\pi$
$929$	9.44369	0.309837	0.154919	+	0.987927i	$0.450488\pi$
$930$	0	0
$931$	12.8947	0.422606
$932$	0	0
$933$	−32.2426	−1.05557
$934$	0	0
$935$	−0.393023	−0.0128532
$936$	0	0
$937$	−35.0831	−1.14612	−0.573058	−	0.819515i	$-0.694243\pi$
$937$	−35.0831	−1.14612	−0.573058	+	0.819515i	$0.694243\pi$
$938$	0	0
$939$	13.9263	0.454468
$940$	0	0
$941$	−31.0471	−1.01211	−0.506053	−	0.862502i	$-0.668896\pi$
$941$	−31.0471	−1.01211	−0.506053	+	0.862502i	$0.668896\pi$
$942$	0	0
$943$	47.1016	1.53384
$944$	0	0
$945$	1.20479	0.0391917
$946$	0	0
$947$	47.2669	1.53597	0.767983	−	0.640470i	$-0.221260\pi$
$947$	47.2669	1.53597	0.767983	+	0.640470i	$0.221260\pi$
$948$	0	0
$949$	−10.0234	−0.325374
$950$	0	0
$951$	18.1288	0.587867
$952$	0	0
$953$	−33.2110	−1.07581	−0.537904	−	0.843006i	$-0.680784\pi$
$953$	−33.2110	−1.07581	−0.537904	+	0.843006i	$0.680784\pi$
$954$	0	0
$955$	0.262059	0.00848004
$956$	0	0
$957$	18.4518	0.596463
$958$	0	0
$959$	−64.5450	−2.08427
$960$	0	0
$961$	−10.0459	−0.324060
$962$	0	0
$963$	34.4303	1.10950
$964$	0	0
$965$	−1.27250	−0.0409633
$966$	0	0
$967$	23.4762	0.754945	0.377473	−	0.926021i	$-0.376793\pi$
$967$	23.4762	0.754945	0.377473	+	0.926021i	$0.376793\pi$
$968$	0	0
$969$	−9.50228	−0.305257
$970$	0	0
$971$	23.2975	0.747651	0.373826	−	0.927499i	$-0.378046\pi$
$971$	23.2975	0.747651	0.373826	+	0.927499i	$0.378046\pi$
$972$	0	0
$973$	−59.9406	−1.92161
$974$	0	0
$975$	−21.9081	−0.701620
$976$	0	0
$977$	28.1370	0.900183	0.450092	−	0.892982i	$-0.351391\pi$
$977$	28.1370	0.900183	0.450092	+	0.892982i	$0.351391\pi$
$978$	0	0
$979$	15.1325	0.483637
$980$	0	0
$981$	−1.38004	−0.0440612
$982$	0	0
$983$	−4.62570	−0.147537	−0.0737684	−	0.997275i	$-0.523503\pi$
$983$	−4.62570	−0.147537	−0.0737684	+	0.997275i	$0.523503\pi$
$984$	0	0
$985$	3.52863	0.112431
$986$	0	0
$987$	−70.8553	−2.25535
$988$	0	0
$989$	−17.4046	−0.553434
$990$	0	0
$991$	−1.26892	−0.0403084	−0.0201542	−	0.999797i	$-0.506416\pi$
$991$	−1.26892	−0.0403084	−0.0201542	+	0.999797i	$0.506416\pi$
$992$	0	0
$993$	−59.6588	−1.89322
$994$	0	0
$995$	−2.73213	−0.0866142
$996$	0	0
$997$	−12.4826	−0.395329	−0.197665	−	0.980270i	$-0.563336\pi$
$997$	−12.4826	−0.395329	−0.197665	+	0.980270i	$0.563336\pi$
$998$	0	0
$999$	−7.85769	−0.248606

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.i.1.15	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.i.1.15	✓	18	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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+2.60297 q^{3} +0.183919 q^{5} +3.24534 q^{7} +3.77545 q^{9} +O(q^{10})\)	\(q+2.60297 q^{3} +0.183919 q^{5} +3.24534 q^{7} +3.77545 q^{9} +2.13693 q^{11} +1.69478 q^{13} +0.478736 q^{15} -1.00000 q^{17} +3.65056 q^{19} +8.44753 q^{21} +6.53848 q^{23} -4.96617 q^{25} +2.01847 q^{27} +3.31726 q^{29} -4.57757 q^{31} +5.56237 q^{33} +0.596880 q^{35} -3.89290 q^{37} +4.41146 q^{39} +7.20375 q^{41} -2.66187 q^{43} +0.694377 q^{45} -8.38769 q^{47} +3.53225 q^{49} -2.60297 q^{51} -8.06541 q^{53} +0.393023 q^{55} +9.50228 q^{57} -1.00000 q^{59} +3.08619 q^{61} +12.2526 q^{63} +0.311702 q^{65} +7.66017 q^{67} +17.0195 q^{69} -10.8657 q^{71} -5.91429 q^{73} -12.9268 q^{75} +6.93508 q^{77} +5.00108 q^{79} -6.07233 q^{81} -12.0316 q^{83} -0.183919 q^{85} +8.63472 q^{87} +7.08141 q^{89} +5.50014 q^{91} -11.9153 q^{93} +0.671407 q^{95} +17.4961 q^{97} +8.06788 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9}+O(q^{10}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9	\( 18 q + 8 q^{3} + 4 q^{5} + 2 q^{7} + 18 q^{9} + 12 q^{11} + 2 q^{13} - 18 q^{17} + 5 q^{19} - 3 q^{21} + 21 q^{23} + 16 q^{25} + 26 q^{27} + 14 q^{29} + 15 q^{31} + 19 q^{33} + 20 q^{35} + 2 q^{37} - 14 q^{39} + 34 q^{41} + 21 q^{43} + 49 q^{45} + 69 q^{47} + 28 q^{49} - 8 q^{51} - 4 q^{53} + 18 q^{55} + 5 q^{57} - 18 q^{59} + 11 q^{61} + 35 q^{63} + 27 q^{65} + 34 q^{67} - 4 q^{69} + 37 q^{71} + 18 q^{73} + 72 q^{75} + 11 q^{77} + 11 q^{79} + 30 q^{81} + 28 q^{83} - 4 q^{85} + 7 q^{87} + 44 q^{89} - 23 q^{91} - 3 q^{93} - 11 q^{95} + 11 q^{97} + 56 q^{99}+O(q^{100}) \) 18 * q + 8 * q^3 + 4 * q^5 + 2 * q^7 + 18 * q^9 + 12 * q^11 + 2 * q^13 - 18 * q^17 + 5 * q^19 - 3 * q^21 + 21 * q^23 + 16 * q^25 + 26 * q^27 + 14 * q^29 + 15 * q^31 + 19 * q^33 + 20 * q^35 + 2 * q^37 - 14 * q^39 + 34 * q^41 + 21 * q^43 + 49 * q^45 + 69 * q^47 + 28 * q^49 - 8 * q^51 - 4 * q^53 + 18 * q^55 + 5 * q^57 - 18 * q^59 + 11 * q^61 + 35 * q^63 + 27 * q^65 + 34 * q^67 - 4 * q^69 + 37 * q^71 + 18 * q^73 + 72 * q^75 + 11 * q^77 + 11 * q^79 + 30 * q^81 + 28 * q^83 - 4 * q^85 + 7 * q^87 + 44 * q^89 - 23 * q^91 - 3 * q^93 - 11 * q^95 + 11 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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