LMFDB - Embedded newform 4012.2.a.h.1.11

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:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 5 x^{14} - 16 x^{13} + 123 x^{12} - 76 x^{11} - 630 x^{10} + 937 x^{9} + 1083 x^{8} - 2602 x^{7} + \cdots - 2 $ x^15 - 5x^14 - 16x^13 + 123x^12 - 76x^11 - 630x^10 + 937x^9 + 1083x^8 - 2602x^7 - 249x^6 + 2736x^5 - 801x^4 - 900x^3 + 429x^2 - 36x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$3.62663$ 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)

$f(q)$	$=$	$q+1.57548 q^{3} +3.44709 q^{5} -4.23195 q^{7} -0.517858 q^{9} +O(q^{10})$	$q+1.57548 q^{3} +3.44709 q^{5} -4.23195 q^{7} -0.517858 q^{9} -5.87827 q^{11} +5.16420 q^{13} +5.43082 q^{15} +1.00000 q^{17} -5.38694 q^{19} -6.66735 q^{21} -2.25478 q^{23} +6.88240 q^{25} -5.54232 q^{27} -9.66616 q^{29} +5.16881 q^{31} -9.26111 q^{33} -14.5879 q^{35} -2.63472 q^{37} +8.13611 q^{39} -4.74011 q^{41} +11.3464 q^{43} -1.78510 q^{45} -7.12549 q^{47} +10.9094 q^{49} +1.57548 q^{51} -5.14259 q^{53} -20.2629 q^{55} -8.48703 q^{57} -1.00000 q^{59} +7.57215 q^{61} +2.19155 q^{63} +17.8015 q^{65} +0.377854 q^{67} -3.55237 q^{69} -11.3156 q^{71} +6.63450 q^{73} +10.8431 q^{75} +24.8765 q^{77} -14.7362 q^{79} -7.17825 q^{81} -9.16094 q^{83} +3.44709 q^{85} -15.2289 q^{87} -16.1826 q^{89} -21.8546 q^{91} +8.14336 q^{93} -18.5693 q^{95} -5.97258 q^{97} +3.04411 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	$ 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) $ 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * 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
$3$	1.57548	0.909605	0.454802	−	0.890592i	$-0.349710\pi$
$3$	1.57548	0.909605	0.454802	+	0.890592i	$0.349710\pi$
$4$	0	0
$5$	3.44709	1.54158	0.770792	−	0.637087i	$-0.219861\pi$
$5$	3.44709	1.54158	0.770792	+	0.637087i	$0.219861\pi$
$6$	0	0
$7$	−4.23195	−1.59952	−0.799762	−	0.600317i	$-0.795041\pi$
$7$	−4.23195	−1.59952	−0.799762	+	0.600317i	$0.795041\pi$
$8$	0	0
$9$	−0.517858	−0.172619
$10$	0	0
$11$	−5.87827	−1.77237	−0.886183	−	0.463336i	$-0.846652\pi$
$11$	−5.87827	−1.77237	−0.886183	+	0.463336i	$0.846652\pi$
$12$	0	0
$13$	5.16420	1.43229	0.716146	−	0.697950i	$-0.245904\pi$
$13$	5.16420	1.43229	0.716146	+	0.697950i	$0.245904\pi$
$14$	0	0
$15$	5.43082	1.40223
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−5.38694	−1.23585	−0.617925	−	0.786237i	$-0.712026\pi$
$19$	−5.38694	−1.23585	−0.617925	+	0.786237i	$0.712026\pi$
$20$	0	0
$21$	−6.66735	−1.45494
$22$	0	0
$23$	−2.25478	−0.470155	−0.235077	−	0.971977i	$-0.575534\pi$
$23$	−2.25478	−0.470155	−0.235077	+	0.971977i	$0.575534\pi$
$24$	0	0
$25$	6.88240	1.37648
$26$	0	0
$27$	−5.54232	−1.06662
$28$	0	0
$29$	−9.66616	−1.79496	−0.897481	−	0.441054i	$-0.854605\pi$
$29$	−9.66616	−1.79496	−0.897481	+	0.441054i	$0.854605\pi$
$30$	0	0
$31$	5.16881	0.928346	0.464173	−	0.885745i	$-0.346352\pi$
$31$	5.16881	0.928346	0.464173	+	0.885745i	$0.346352\pi$
$32$	0	0
$33$	−9.26111	−1.61215
$34$	0	0
$35$	−14.5879	−2.46580
$36$	0	0
$37$	−2.63472	−0.433146	−0.216573	−	0.976266i	$-0.569488\pi$
$37$	−2.63472	−0.433146	−0.216573	+	0.976266i	$0.569488\pi$
$38$	0	0
$39$	8.13611	1.30282
$40$	0	0
$41$	−4.74011	−0.740281	−0.370141	−	0.928976i	$-0.620691\pi$
$41$	−4.74011	−0.740281	−0.370141	+	0.928976i	$0.620691\pi$
$42$	0	0
$43$	11.3464	1.73031	0.865156	−	0.501503i	$-0.167219\pi$
$43$	11.3464	1.73031	0.865156	+	0.501503i	$0.167219\pi$
$44$	0	0
$45$	−1.78510	−0.266107
$46$	0	0
$47$	−7.12549	−1.03936	−0.519680	−	0.854361i	$-0.673949\pi$
$47$	−7.12549	−1.03936	−0.519680	+	0.854361i	$0.673949\pi$
$48$	0	0
$49$	10.9094	1.55848
$50$	0	0
$51$	1.57548	0.220612
$52$	0	0
$53$	−5.14259	−0.706390	−0.353195	−	0.935550i	$-0.614905\pi$
$53$	−5.14259	−0.706390	−0.353195	+	0.935550i	$0.614905\pi$
$54$	0	0
$55$	−20.2629	−2.73225
$56$	0	0
$57$	−8.48703	−1.12413
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	7.57215	0.969514	0.484757	−	0.874649i	$-0.338908\pi$
$61$	7.57215	0.969514	0.484757	+	0.874649i	$0.338908\pi$
$62$	0	0
$63$	2.19155	0.276109
$64$	0	0
$65$	17.8015	2.20800
$66$	0	0
$67$	0.377854	0.0461623	0.0230811	−	0.999734i	$-0.492652\pi$
$67$	0.377854	0.0461623	0.0230811	+	0.999734i	$0.492652\pi$
$68$	0	0
$69$	−3.55237	−0.427655
$70$	0	0
$71$	−11.3156	−1.34292	−0.671460	−	0.741041i	$-0.734333\pi$
$71$	−11.3156	−1.34292	−0.671460	+	0.741041i	$0.734333\pi$
$72$	0	0
$73$	6.63450	0.776509	0.388254	−	0.921552i	$-0.373078\pi$
$73$	6.63450	0.776509	0.388254	+	0.921552i	$0.373078\pi$
$74$	0	0
$75$	10.8431	1.25205
$76$	0	0
$77$	24.8765	2.83494
$78$	0	0
$79$	−14.7362	−1.65795	−0.828975	−	0.559285i	$-0.811076\pi$
$79$	−14.7362	−1.65795	−0.828975	+	0.559285i	$0.811076\pi$
$80$	0	0
$81$	−7.17825	−0.797583
$82$	0	0
$83$	−9.16094	−1.00554	−0.502772	−	0.864419i	$-0.667686\pi$
$83$	−9.16094	−1.00554	−0.502772	+	0.864419i	$0.667686\pi$
$84$	0	0
$85$	3.44709	0.373889
$86$	0	0
$87$	−15.2289	−1.63271
$88$	0	0
$89$	−16.1826	−1.71536	−0.857678	−	0.514187i	$-0.828094\pi$
$89$	−16.1826	−1.71536	−0.857678	+	0.514187i	$0.828094\pi$
$90$	0	0
$91$	−21.8546	−2.29099
$92$	0	0
$93$	8.14336	0.844427
$94$	0	0
$95$	−18.5693	−1.90517
$96$	0	0
$97$	−5.97258	−0.606424	−0.303212	−	0.952923i	$-0.598059\pi$
$97$	−5.97258	−0.606424	−0.303212	+	0.952923i	$0.598059\pi$
$98$	0	0
$99$	3.04411	0.305945
$100$	0	0
$101$	3.85064	0.383153	0.191576	−	0.981478i	$-0.438640\pi$
$101$	3.85064	0.383153	0.191576	+	0.981478i	$0.438640\pi$
$102$	0	0
$103$	10.0195	0.987249	0.493625	−	0.869675i	$-0.335672\pi$
$103$	10.0195	0.987249	0.493625	+	0.869675i	$0.335672\pi$
$104$	0	0
$105$	−22.9829	−2.24290
$106$	0	0
$107$	19.7726	1.91149	0.955746	−	0.294193i	$-0.0950509\pi$
$107$	19.7726	1.91149	0.955746	+	0.294193i	$0.0950509\pi$
$108$	0	0
$109$	−11.8591	−1.13589	−0.567947	−	0.823065i	$-0.692262\pi$
$109$	−11.8591	−1.13589	−0.567947	+	0.823065i	$0.692262\pi$
$110$	0	0
$111$	−4.15096	−0.393992
$112$	0	0
$113$	2.49545	0.234752	0.117376	−	0.993088i	$-0.462552\pi$
$113$	2.49545	0.234752	0.117376	+	0.993088i	$0.462552\pi$
$114$	0	0
$115$	−7.77243	−0.724783
$116$	0	0
$117$	−2.67433	−0.247241
$118$	0	0
$119$	−4.23195	−0.387942
$120$	0	0
$121$	23.5541	2.14128
$122$	0	0
$123$	−7.46796	−0.673363
$124$	0	0
$125$	6.48878	0.580374
$126$	0	0
$127$	−9.48580	−0.841729	−0.420864	−	0.907124i	$-0.638273\pi$
$127$	−9.48580	−0.841729	−0.420864	+	0.907124i	$0.638273\pi$
$128$	0	0
$129$	17.8761	1.57390
$130$	0	0
$131$	−20.9381	−1.82937	−0.914685	−	0.404168i	$-0.867561\pi$
$131$	−20.9381	−1.82937	−0.914685	+	0.404168i	$0.867561\pi$
$132$	0	0
$133$	22.7973	1.97677
$134$	0	0
$135$	−19.1048	−1.64428
$136$	0	0
$137$	−13.1496	−1.12344	−0.561722	−	0.827326i	$-0.689861\pi$
$137$	−13.1496	−1.12344	−0.561722	+	0.827326i	$0.689861\pi$
$138$	0	0
$139$	7.72218	0.654987	0.327493	−	0.944853i	$-0.393796\pi$
$139$	7.72218	0.654987	0.327493	+	0.944853i	$0.393796\pi$
$140$	0	0
$141$	−11.2261	−0.945406
$142$	0	0
$143$	−30.3566	−2.53855
$144$	0	0
$145$	−33.3201	−2.76708
$146$	0	0
$147$	17.1875	1.41760
$148$	0	0
$149$	6.45799	0.529059	0.264530	−	0.964378i	$-0.414783\pi$
$149$	6.45799	0.529059	0.264530	+	0.964378i	$0.414783\pi$
$150$	0	0
$151$	13.5912	1.10603	0.553017	−	0.833170i	$-0.313476\pi$
$151$	13.5912	1.10603	0.553017	+	0.833170i	$0.313476\pi$
$152$	0	0
$153$	−0.517858	−0.0418664
$154$	0	0
$155$	17.8173	1.43112
$156$	0	0
$157$	9.29223	0.741601	0.370800	−	0.928713i	$-0.379083\pi$
$157$	9.29223	0.741601	0.370800	+	0.928713i	$0.379083\pi$
$158$	0	0
$159$	−8.10206	−0.642535
$160$	0	0
$161$	9.54212	0.752024
$162$	0	0
$163$	−5.40898	−0.423664	−0.211832	−	0.977306i	$-0.567943\pi$
$163$	−5.40898	−0.423664	−0.211832	+	0.977306i	$0.567943\pi$
$164$	0	0
$165$	−31.9238	−2.48527
$166$	0	0
$167$	−13.8062	−1.06836	−0.534178	−	0.845372i	$-0.679379\pi$
$167$	−13.8062	−1.06836	−0.534178	+	0.845372i	$0.679379\pi$
$168$	0	0
$169$	13.6690	1.05146
$170$	0	0
$171$	2.78967	0.213332
$172$	0	0
$173$	12.5605	0.954957	0.477479	−	0.878643i	$-0.341551\pi$
$173$	12.5605	0.954957	0.477479	+	0.878643i	$0.341551\pi$
$174$	0	0
$175$	−29.1259	−2.20171
$176$	0	0
$177$	−1.57548	−0.118420
$178$	0	0
$179$	−16.0520	−1.19979	−0.599893	−	0.800080i	$-0.704790\pi$
$179$	−16.0520	−1.19979	−0.599893	+	0.800080i	$0.704790\pi$
$180$	0	0
$181$	−15.3408	−1.14027	−0.570134	−	0.821551i	$-0.693109\pi$
$181$	−15.3408	−1.14027	−0.570134	+	0.821551i	$0.693109\pi$
$182$	0	0
$183$	11.9298	0.881875
$184$	0	0
$185$	−9.08212	−0.667730
$186$	0	0
$187$	−5.87827	−0.429862
$188$	0	0
$189$	23.4548	1.70609
$190$	0	0
$191$	11.2468	0.813793	0.406897	−	0.913474i	$-0.366611\pi$
$191$	11.2468	0.813793	0.406897	+	0.913474i	$0.366611\pi$
$192$	0	0
$193$	9.17892	0.660713	0.330356	−	0.943856i	$-0.392831\pi$
$193$	9.17892	0.660713	0.330356	+	0.943856i	$0.392831\pi$
$194$	0	0
$195$	28.0459	2.00841
$196$	0	0
$197$	4.13784	0.294809	0.147404	−	0.989076i	$-0.452908\pi$
$197$	4.13784	0.294809	0.147404	+	0.989076i	$0.452908\pi$
$198$	0	0
$199$	−1.59412	−0.113004	−0.0565019	−	0.998402i	$-0.517995\pi$
$199$	−1.59412	−0.113004	−0.0565019	+	0.998402i	$0.517995\pi$
$200$	0	0
$201$	0.595303	0.0419894
$202$	0	0
$203$	40.9067	2.87109
$204$	0	0
$205$	−16.3396	−1.14121
$206$	0	0
$207$	1.16766	0.0811578
$208$	0	0
$209$	31.6659	2.19038
$210$	0	0
$211$	8.45799	0.582272	0.291136	−	0.956682i	$-0.405967\pi$
$211$	8.45799	0.582272	0.291136	+	0.956682i	$0.405967\pi$
$212$	0	0
$213$	−17.8276	−1.22153
$214$	0	0
$215$	39.1121	2.66742
$216$	0	0
$217$	−21.8741	−1.48491
$218$	0	0
$219$	10.4525	0.706316
$220$	0	0
$221$	5.16420	0.347382
$222$	0	0
$223$	22.5298	1.50871	0.754354	−	0.656468i	$-0.227950\pi$
$223$	22.5298	1.50871	0.754354	+	0.656468i	$0.227950\pi$
$224$	0	0
$225$	−3.56410	−0.237607
$226$	0	0
$227$	−1.28963	−0.0855956	−0.0427978	−	0.999084i	$-0.513627\pi$
$227$	−1.28963	−0.0855956	−0.0427978	+	0.999084i	$0.513627\pi$
$228$	0	0
$229$	−2.94642	−0.194705	−0.0973523	−	0.995250i	$-0.531037\pi$
$229$	−2.94642	−0.194705	−0.0973523	+	0.995250i	$0.531037\pi$
$230$	0	0
$231$	39.1925	2.57868
$232$	0	0
$233$	16.1570	1.05848	0.529240	−	0.848472i	$-0.322477\pi$
$233$	16.1570	1.05848	0.529240	+	0.848472i	$0.322477\pi$
$234$	0	0
$235$	−24.5622	−1.60226
$236$	0	0
$237$	−23.2166	−1.50808
$238$	0	0
$239$	11.6238	0.751880	0.375940	−	0.926644i	$-0.377320\pi$
$239$	11.6238	0.751880	0.375940	+	0.926644i	$0.377320\pi$
$240$	0	0
$241$	−17.5868	−1.13287	−0.566434	−	0.824107i	$-0.691677\pi$
$241$	−17.5868	−1.13287	−0.566434	+	0.824107i	$0.691677\pi$
$242$	0	0
$243$	5.31776	0.341135
$244$	0	0
$245$	37.6055	2.40253
$246$	0	0
$247$	−27.8193	−1.77010
$248$	0	0
$249$	−14.4329	−0.914647
$250$	0	0
$251$	27.3663	1.72735	0.863673	−	0.504053i	$-0.168158\pi$
$251$	27.3663	1.72735	0.863673	+	0.504053i	$0.168158\pi$
$252$	0	0
$253$	13.2542	0.833286
$254$	0	0
$255$	5.43082	0.340091
$256$	0	0
$257$	24.1406	1.50585	0.752926	−	0.658105i	$-0.228642\pi$
$257$	24.1406	1.50585	0.752926	+	0.658105i	$0.228642\pi$
$258$	0	0
$259$	11.1500	0.692828
$260$	0	0
$261$	5.00570	0.309845
$262$	0	0
$263$	4.04841	0.249636	0.124818	−	0.992180i	$-0.460165\pi$
$263$	4.04841	0.249636	0.124818	+	0.992180i	$0.460165\pi$
$264$	0	0
$265$	−17.7270	−1.08896
$266$	0	0
$267$	−25.4954	−1.56030
$268$	0	0
$269$	−26.0957	−1.59108	−0.795542	−	0.605898i	$-0.792814\pi$
$269$	−26.0957	−1.59108	−0.795542	+	0.605898i	$0.792814\pi$
$270$	0	0
$271$	−30.0945	−1.82811	−0.914055	−	0.405591i	$-0.867066\pi$
$271$	−30.0945	−1.82811	−0.914055	+	0.405591i	$0.867066\pi$
$272$	0	0
$273$	−34.4316	−2.08389
$274$	0	0
$275$	−40.4566	−2.43962
$276$	0	0
$277$	9.86888	0.592963	0.296482	−	0.955039i	$-0.404187\pi$
$277$	9.86888	0.592963	0.296482	+	0.955039i	$0.404187\pi$
$278$	0	0
$279$	−2.67671	−0.160250
$280$	0	0
$281$	21.8448	1.30315	0.651576	−	0.758584i	$-0.274108\pi$
$281$	21.8448	1.30315	0.651576	+	0.758584i	$0.274108\pi$
$282$	0	0
$283$	12.9114	0.767505	0.383752	−	0.923436i	$-0.374632\pi$
$283$	12.9114	0.767505	0.383752	+	0.923436i	$0.374632\pi$
$284$	0	0
$285$	−29.2555	−1.73295
$286$	0	0
$287$	20.0599	1.18410
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−9.40969	−0.551606
$292$	0	0
$293$	11.7543	0.686696	0.343348	−	0.939208i	$-0.388439\pi$
$293$	11.7543	0.686696	0.343348	+	0.939208i	$0.388439\pi$
$294$	0	0
$295$	−3.44709	−0.200697
$296$	0	0
$297$	32.5793	1.89044
$298$	0	0
$299$	−11.6442	−0.673399
$300$	0	0
$301$	−48.0174	−2.76768
$302$	0	0
$303$	6.06661	0.348517
$304$	0	0
$305$	26.1018	1.49459
$306$	0	0
$307$	22.9299	1.30868	0.654340	−	0.756200i	$-0.272947\pi$
$307$	22.9299	1.30868	0.654340	+	0.756200i	$0.272947\pi$
$308$	0	0
$309$	15.7855	0.898007
$310$	0	0
$311$	−7.65598	−0.434131	−0.217065	−	0.976157i	$-0.569648\pi$
$311$	−7.65598	−0.434131	−0.217065	+	0.976157i	$0.569648\pi$
$312$	0	0
$313$	4.83338	0.273199	0.136599	−	0.990626i	$-0.456383\pi$
$313$	4.83338	0.273199	0.136599	+	0.990626i	$0.456383\pi$
$314$	0	0
$315$	7.55445	0.425645
$316$	0	0
$317$	−25.0904	−1.40922	−0.704609	−	0.709596i	$-0.748878\pi$
$317$	−25.0904	−1.40922	−0.704609	+	0.709596i	$0.748878\pi$
$318$	0	0
$319$	56.8203	3.18133
$320$	0	0
$321$	31.1514	1.73870
$322$	0	0
$323$	−5.38694	−0.299738
$324$	0	0
$325$	35.5421	1.97152
$326$	0	0
$327$	−18.6838	−1.03321
$328$	0	0
$329$	30.1547	1.66248
$330$	0	0
$331$	1.08862	0.0598361	0.0299180	−	0.999552i	$-0.490475\pi$
$331$	1.08862	0.0598361	0.0299180	+	0.999552i	$0.490475\pi$
$332$	0	0
$333$	1.36441	0.0747694
$334$	0	0
$335$	1.30250	0.0711630
$336$	0	0
$337$	25.4901	1.38854	0.694268	−	0.719717i	$-0.255728\pi$
$337$	25.4901	1.38854	0.694268	+	0.719717i	$0.255728\pi$
$338$	0	0
$339$	3.93153	0.213531
$340$	0	0
$341$	−30.3837	−1.64537
$342$	0	0
$343$	−16.5442	−0.893303
$344$	0	0
$345$	−12.2453	−0.659266
$346$	0	0
$347$	19.9133	1.06900	0.534500	−	0.845168i	$-0.320500\pi$
$347$	19.9133	1.06900	0.534500	+	0.845168i	$0.320500\pi$
$348$	0	0
$349$	−6.96886	−0.373035	−0.186517	−	0.982452i	$-0.559720\pi$
$349$	−6.96886	−0.373035	−0.186517	+	0.982452i	$0.559720\pi$
$350$	0	0
$351$	−28.6217	−1.52771
$352$	0	0
$353$	−31.0887	−1.65469	−0.827343	−	0.561698i	$-0.810148\pi$
$353$	−31.0887	−1.65469	−0.827343	+	0.561698i	$0.810148\pi$
$354$	0	0
$355$	−39.0060	−2.07022
$356$	0	0
$357$	−6.66735	−0.352874
$358$	0	0
$359$	−19.2140	−1.01408	−0.507039	−	0.861923i	$-0.669260\pi$
$359$	−19.2140	−1.01408	−0.507039	+	0.861923i	$0.669260\pi$
$360$	0	0
$361$	10.0192	0.527325
$362$	0	0
$363$	37.1090	1.94772
$364$	0	0
$365$	22.8697	1.19705
$366$	0	0
$367$	0.647712	0.0338103	0.0169051	−	0.999857i	$-0.494619\pi$
$367$	0.647712	0.0338103	0.0169051	+	0.999857i	$0.494619\pi$
$368$	0	0
$369$	2.45471	0.127787
$370$	0	0
$371$	21.7632	1.12989
$372$	0	0
$373$	17.8226	0.922820	0.461410	−	0.887187i	$-0.347344\pi$
$373$	17.8226	0.922820	0.461410	+	0.887187i	$0.347344\pi$
$374$	0	0
$375$	10.2229	0.527911
$376$	0	0
$377$	−49.9180	−2.57091
$378$	0	0
$379$	−33.6058	−1.72621	−0.863106	−	0.505023i	$-0.831484\pi$
$379$	−33.6058	−1.72621	−0.863106	+	0.505023i	$0.831484\pi$
$380$	0	0
$381$	−14.9447	−0.765640
$382$	0	0
$383$	13.6682	0.698414	0.349207	−	0.937046i	$-0.386451\pi$
$383$	13.6682	0.698414	0.349207	+	0.937046i	$0.386451\pi$
$384$	0	0
$385$	85.7515	4.37030
$386$	0	0
$387$	−5.87583	−0.298685
$388$	0	0
$389$	−15.2026	−0.770801	−0.385401	−	0.922749i	$-0.625937\pi$
$389$	−15.2026	−0.770801	−0.385401	+	0.922749i	$0.625937\pi$
$390$	0	0
$391$	−2.25478	−0.114029
$392$	0	0
$393$	−32.9876	−1.66400
$394$	0	0
$395$	−50.7969	−2.55587
$396$	0	0
$397$	−21.4298	−1.07553	−0.537766	−	0.843094i	$-0.680732\pi$
$397$	−21.4298	−1.07553	−0.537766	+	0.843094i	$0.680732\pi$
$398$	0	0
$399$	35.9167	1.79808
$400$	0	0
$401$	1.27530	0.0636856	0.0318428	−	0.999493i	$-0.489862\pi$
$401$	1.27530	0.0636856	0.0318428	+	0.999493i	$0.489862\pi$
$402$	0	0
$403$	26.6928	1.32966
$404$	0	0
$405$	−24.7440	−1.22954
$406$	0	0
$407$	15.4876	0.767693
$408$	0	0
$409$	−17.6163	−0.871070	−0.435535	−	0.900172i	$-0.643441\pi$
$409$	−17.6163	−0.871070	−0.435535	+	0.900172i	$0.643441\pi$
$410$	0	0
$411$	−20.7169	−1.02189
$412$	0	0
$413$	4.23195	0.208240
$414$	0	0
$415$	−31.5785	−1.55013
$416$	0	0
$417$	12.1662	0.595779
$418$	0	0
$419$	−28.0308	−1.36939	−0.684696	−	0.728829i	$-0.740065\pi$
$419$	−28.0308	−1.36939	−0.684696	+	0.728829i	$0.740065\pi$
$420$	0	0
$421$	−22.2109	−1.08249	−0.541245	−	0.840865i	$-0.682047\pi$
$421$	−22.2109	−1.08249	−0.541245	+	0.840865i	$0.682047\pi$
$422$	0	0
$423$	3.68999	0.179414
$424$	0	0
$425$	6.88240	0.333845
$426$	0	0
$427$	−32.0449	−1.55076
$428$	0	0
$429$	−47.8262	−2.30907
$430$	0	0
$431$	9.58659	0.461770	0.230885	−	0.972981i	$-0.425838\pi$
$431$	9.58659	0.461770	0.230885	+	0.972981i	$0.425838\pi$
$432$	0	0
$433$	−2.50507	−0.120386	−0.0601930	−	0.998187i	$-0.519172\pi$
$433$	−2.50507	−0.120386	−0.0601930	+	0.998187i	$0.519172\pi$
$434$	0	0
$435$	−52.4952	−2.51695
$436$	0	0
$437$	12.1464	0.581041
$438$	0	0
$439$	−17.7041	−0.844969	−0.422485	−	0.906370i	$-0.638842\pi$
$439$	−17.7041	−0.844969	−0.422485	+	0.906370i	$0.638842\pi$
$440$	0	0
$441$	−5.64950	−0.269024
$442$	0	0
$443$	1.43854	0.0683471	0.0341736	−	0.999416i	$-0.489120\pi$
$443$	1.43854	0.0683471	0.0341736	+	0.999416i	$0.489120\pi$
$444$	0	0
$445$	−55.7829	−2.64436
$446$	0	0
$447$	10.1744	0.481235
$448$	0	0
$449$	−25.4709	−1.20205	−0.601023	−	0.799232i	$-0.705240\pi$
$449$	−25.4709	−1.20205	−0.601023	+	0.799232i	$0.705240\pi$
$450$	0	0
$451$	27.8637	1.31205
$452$	0	0
$453$	21.4127	1.00605
$454$	0	0
$455$	−75.3348	−3.53175
$456$	0	0
$457$	−8.50285	−0.397747	−0.198873	−	0.980025i	$-0.563728\pi$
$457$	−8.50285	−0.397747	−0.198873	+	0.980025i	$0.563728\pi$
$458$	0	0
$459$	−5.54232	−0.258693
$460$	0	0
$461$	9.12771	0.425120	0.212560	−	0.977148i	$-0.431820\pi$
$461$	9.12771	0.425120	0.212560	+	0.977148i	$0.431820\pi$
$462$	0	0
$463$	25.8520	1.20144	0.600722	−	0.799458i	$-0.294880\pi$
$463$	25.8520	1.20144	0.600722	+	0.799458i	$0.294880\pi$
$464$	0	0
$465$	28.0709	1.30176
$466$	0	0
$467$	−8.20674	−0.379763	−0.189881	−	0.981807i	$-0.560810\pi$
$467$	−8.20674	−0.379763	−0.189881	+	0.981807i	$0.560810\pi$
$468$	0	0
$469$	−1.59906	−0.0738377
$470$	0	0
$471$	14.6397	0.674564
$472$	0	0
$473$	−66.6973	−3.06675
$474$	0	0
$475$	−37.0751	−1.70112
$476$	0	0
$477$	2.66313	0.121937
$478$	0	0
$479$	4.47341	0.204396	0.102198	−	0.994764i	$-0.467413\pi$
$479$	4.47341	0.204396	0.102198	+	0.994764i	$0.467413\pi$
$480$	0	0
$481$	−13.6063	−0.620392
$482$	0	0
$483$	15.0334	0.684045
$484$	0	0
$485$	−20.5880	−0.934852
$486$	0	0
$487$	30.9168	1.40097	0.700486	−	0.713666i	$-0.252967\pi$
$487$	30.9168	1.40097	0.700486	+	0.713666i	$0.252967\pi$
$488$	0	0
$489$	−8.52175	−0.385367
$490$	0	0
$491$	−19.1068	−0.862276	−0.431138	−	0.902286i	$-0.641888\pi$
$491$	−19.1068	−0.862276	−0.431138	+	0.902286i	$0.641888\pi$
$492$	0	0
$493$	−9.66616	−0.435342
$494$	0	0
$495$	10.4933	0.471639
$496$	0	0
$497$	47.8872	2.14803
$498$	0	0
$499$	−0.128241	−0.00574085	−0.00287043	−	0.999996i	$-0.500914\pi$
$499$	−0.128241	−0.00574085	−0.00287043	+	0.999996i	$0.500914\pi$
$500$	0	0
$501$	−21.7514	−0.971782
$502$	0	0
$503$	32.5827	1.45279	0.726395	−	0.687277i	$-0.241194\pi$
$503$	32.5827	1.45279	0.726395	+	0.687277i	$0.241194\pi$
$504$	0	0
$505$	13.2735	0.590662
$506$	0	0
$507$	21.5353	0.956415
$508$	0	0
$509$	15.1279	0.670532	0.335266	−	0.942124i	$-0.391174\pi$
$509$	15.1279	0.670532	0.335266	+	0.942124i	$0.391174\pi$
$510$	0	0
$511$	−28.0768	−1.24205
$512$	0	0
$513$	29.8562	1.31818
$514$	0	0
$515$	34.5380	1.52193
$516$	0	0
$517$	41.8856	1.84213
$518$	0	0
$519$	19.7888	0.868634
$520$	0	0
$521$	29.8544	1.30795	0.653973	−	0.756518i	$-0.273101\pi$
$521$	29.8544	1.30795	0.653973	+	0.756518i	$0.273101\pi$
$522$	0	0
$523$	19.7050	0.861638	0.430819	−	0.902438i	$-0.358225\pi$
$523$	19.7050	0.861638	0.430819	+	0.902438i	$0.358225\pi$
$524$	0	0
$525$	−45.8873	−2.00269
$526$	0	0
$527$	5.16881	0.225157
$528$	0	0
$529$	−17.9160	−0.778954
$530$	0	0
$531$	0.517858	0.0224731
$532$	0	0
$533$	−24.4789	−1.06030
$534$	0	0
$535$	68.1579	2.94672
$536$	0	0
$537$	−25.2897	−1.09133
$538$	0	0
$539$	−64.1282	−2.76220
$540$	0	0
$541$	−29.2189	−1.25622	−0.628110	−	0.778124i	$-0.716171\pi$
$541$	−29.2189	−1.25622	−0.628110	+	0.778124i	$0.716171\pi$
$542$	0	0
$543$	−24.1691	−1.03719
$544$	0	0
$545$	−40.8793	−1.75108
$546$	0	0
$547$	−33.0214	−1.41189	−0.705946	−	0.708265i	$-0.749478\pi$
$547$	−33.0214	−1.41189	−0.705946	+	0.708265i	$0.749478\pi$
$548$	0	0
$549$	−3.92130	−0.167357
$550$	0	0
$551$	52.0711	2.21830
$552$	0	0
$553$	62.3627	2.65193
$554$	0	0
$555$	−14.3087	−0.607371
$556$	0	0
$557$	31.2342	1.32343	0.661717	−	0.749753i	$-0.269828\pi$
$557$	31.2342	1.32343	0.661717	+	0.749753i	$0.269828\pi$
$558$	0	0
$559$	58.5952	2.47831
$560$	0	0
$561$	−9.26111	−0.391004
$562$	0	0
$563$	−5.52616	−0.232900	−0.116450	−	0.993197i	$-0.537151\pi$
$563$	−5.52616	−0.232900	−0.116450	+	0.993197i	$0.537151\pi$
$564$	0	0
$565$	8.60201	0.361889
$566$	0	0
$567$	30.3780	1.27575
$568$	0	0
$569$	28.1198	1.17884	0.589421	−	0.807826i	$-0.299356\pi$
$569$	28.1198	1.17884	0.589421	+	0.807826i	$0.299356\pi$
$570$	0	0
$571$	−14.4858	−0.606212	−0.303106	−	0.952957i	$-0.598024\pi$
$571$	−14.4858	−0.606212	−0.303106	+	0.952957i	$0.598024\pi$
$572$	0	0
$573$	17.7192	0.740230
$574$	0	0
$575$	−15.5183	−0.647158
$576$	0	0
$577$	−9.01021	−0.375100	−0.187550	−	0.982255i	$-0.560055\pi$
$577$	−9.01021	−0.375100	−0.187550	+	0.982255i	$0.560055\pi$
$578$	0	0
$579$	14.4612	0.600987
$580$	0	0
$581$	38.7686	1.60839
$582$	0	0
$583$	30.2296	1.25198
$584$	0	0
$585$	−9.21863	−0.381143
$586$	0	0
$587$	−35.2400	−1.45451	−0.727255	−	0.686368i	$-0.759204\pi$
$587$	−35.2400	−1.45451	−0.727255	+	0.686368i	$0.759204\pi$
$588$	0	0
$589$	−27.8441	−1.14730
$590$	0	0
$591$	6.51909	0.268159
$592$	0	0
$593$	−16.3059	−0.669602	−0.334801	−	0.942289i	$-0.608669\pi$
$593$	−16.3059	−0.669602	−0.334801	+	0.942289i	$0.608669\pi$
$594$	0	0
$595$	−14.5879	−0.598045
$596$	0	0
$597$	−2.51150	−0.102789
$598$	0	0
$599$	29.5305	1.20658	0.603291	−	0.797521i	$-0.293856\pi$
$599$	29.5305	1.20658	0.603291	+	0.797521i	$0.293856\pi$
$600$	0	0
$601$	−22.2172	−0.906256	−0.453128	−	0.891445i	$-0.649692\pi$
$601$	−22.2172	−0.906256	−0.453128	+	0.891445i	$0.649692\pi$
$602$	0	0
$603$	−0.195675	−0.00796850
$604$	0	0
$605$	81.1929	3.30096
$606$	0	0
$607$	6.94665	0.281956	0.140978	−	0.990013i	$-0.454975\pi$
$607$	6.94665	0.281956	0.140978	+	0.990013i	$0.454975\pi$
$608$	0	0
$609$	64.4477	2.61155
$610$	0	0
$611$	−36.7975	−1.48867
$612$	0	0
$613$	21.0517	0.850272	0.425136	−	0.905130i	$-0.360226\pi$
$613$	21.0517	0.850272	0.425136	+	0.905130i	$0.360226\pi$
$614$	0	0
$615$	−25.7427	−1.03805
$616$	0	0
$617$	−13.1645	−0.529981	−0.264991	−	0.964251i	$-0.585369\pi$
$617$	−13.1645	−0.529981	−0.264991	+	0.964251i	$0.585369\pi$
$618$	0	0
$619$	17.1341	0.688676	0.344338	−	0.938846i	$-0.388103\pi$
$619$	17.1341	0.688676	0.344338	+	0.938846i	$0.388103\pi$
$620$	0	0
$621$	12.4967	0.501477
$622$	0	0
$623$	68.4840	2.74376
$624$	0	0
$625$	−12.0446	−0.481784
$626$	0	0
$627$	49.8891	1.99238
$628$	0	0
$629$	−2.63472	−0.105053
$630$	0	0
$631$	5.01244	0.199542	0.0997710	−	0.995010i	$-0.468189\pi$
$631$	5.01244	0.199542	0.0997710	+	0.995010i	$0.468189\pi$
$632$	0	0
$633$	13.3254	0.529637
$634$	0	0
$635$	−32.6984	−1.29759
$636$	0	0
$637$	56.3382	2.23220
$638$	0	0
$639$	5.85990	0.231814
$640$	0	0
$641$	−12.4038	−0.489920	−0.244960	−	0.969533i	$-0.578775\pi$
$641$	−12.4038	−0.489920	−0.244960	+	0.969533i	$0.578775\pi$
$642$	0	0
$643$	48.7051	1.92074	0.960370	−	0.278727i	$-0.0899126\pi$
$643$	48.7051	1.92074	0.960370	+	0.278727i	$0.0899126\pi$
$644$	0	0
$645$	61.6203	2.42630
$646$	0	0
$647$	−21.7790	−0.856219	−0.428110	−	0.903727i	$-0.640820\pi$
$647$	−21.7790	−0.856219	−0.428110	+	0.903727i	$0.640820\pi$
$648$	0	0
$649$	5.87827	0.230742
$650$	0	0
$651$	−34.4623	−1.35068
$652$	0	0
$653$	−21.4710	−0.840224	−0.420112	−	0.907472i	$-0.638009\pi$
$653$	−21.4710	−0.840224	−0.420112	+	0.907472i	$0.638009\pi$
$654$	0	0
$655$	−72.1754	−2.82013
$656$	0	0
$657$	−3.43573	−0.134040
$658$	0	0
$659$	−35.8336	−1.39588	−0.697939	−	0.716158i	$-0.745899\pi$
$659$	−35.8336	−1.39588	−0.697939	+	0.716158i	$0.745899\pi$
$660$	0	0
$661$	13.8612	0.539139	0.269570	−	0.962981i	$-0.413119\pi$
$661$	13.8612	0.539139	0.269570	+	0.962981i	$0.413119\pi$
$662$	0	0
$663$	8.13611	0.315980
$664$	0	0
$665$	78.5841	3.04736
$666$	0	0
$667$	21.7951	0.843910
$668$	0	0
$669$	35.4953	1.37233
$670$	0	0
$671$	−44.5111	−1.71833
$672$	0	0
$673$	−14.7367	−0.568059	−0.284030	−	0.958816i	$-0.591671\pi$
$673$	−14.7367	−0.568059	−0.284030	+	0.958816i	$0.591671\pi$
$674$	0	0
$675$	−38.1444	−1.46818
$676$	0	0
$677$	−36.3703	−1.39782	−0.698911	−	0.715208i	$-0.746332\pi$
$677$	−36.3703	−1.39782	−0.698911	+	0.715208i	$0.746332\pi$
$678$	0	0
$679$	25.2756	0.969990
$680$	0	0
$681$	−2.03179	−0.0778582
$682$	0	0
$683$	32.3920	1.23945	0.619723	−	0.784821i	$-0.287245\pi$
$683$	32.3920	1.23945	0.619723	+	0.784821i	$0.287245\pi$
$684$	0	0
$685$	−45.3277	−1.73188
$686$	0	0
$687$	−4.64202	−0.177104
$688$	0	0
$689$	−26.5574	−1.01176
$690$	0	0
$691$	−5.70724	−0.217114	−0.108557	−	0.994090i	$-0.534623\pi$
$691$	−5.70724	−0.217114	−0.108557	+	0.994090i	$0.534623\pi$
$692$	0	0
$693$	−12.8825	−0.489366
$694$	0	0
$695$	26.6190	1.00972
$696$	0	0
$697$	−4.74011	−0.179545
$698$	0	0
$699$	25.4550	0.962798
$700$	0	0
$701$	1.78793	0.0675292	0.0337646	−	0.999430i	$-0.489250\pi$
$701$	1.78793	0.0675292	0.0337646	+	0.999430i	$0.489250\pi$
$702$	0	0
$703$	14.1931	0.535303
$704$	0	0
$705$	−38.6973	−1.45742
$706$	0	0
$707$	−16.2957	−0.612862
$708$	0	0
$709$	40.4253	1.51820	0.759102	−	0.650971i	$-0.225638\pi$
$709$	40.4253	1.51820	0.759102	+	0.650971i	$0.225638\pi$
$710$	0	0
$711$	7.63126	0.286194
$712$	0	0
$713$	−11.6545	−0.436466
$714$	0	0
$715$	−104.642	−3.91338
$716$	0	0
$717$	18.3131	0.683913
$718$	0	0
$719$	37.7534	1.40796	0.703982	−	0.710218i	$-0.251404\pi$
$719$	37.7534	1.40796	0.703982	+	0.710218i	$0.251404\pi$
$720$	0	0
$721$	−42.4019	−1.57913
$722$	0	0
$723$	−27.7077	−1.03046
$724$	0	0
$725$	−66.5264	−2.47073
$726$	0	0
$727$	−14.6561	−0.543566	−0.271783	−	0.962359i	$-0.587613\pi$
$727$	−14.6561	−0.543566	−0.271783	+	0.962359i	$0.587613\pi$
$728$	0	0
$729$	29.9128	1.10788
$730$	0	0
$731$	11.3464	0.419662
$732$	0	0
$733$	−34.8082	−1.28567	−0.642835	−	0.766005i	$-0.722242\pi$
$733$	−34.8082	−1.28567	−0.642835	+	0.766005i	$0.722242\pi$
$734$	0	0
$735$	59.2468	2.18535
$736$	0	0
$737$	−2.22113	−0.0818164
$738$	0	0
$739$	−14.3544	−0.528035	−0.264017	−	0.964518i	$-0.585048\pi$
$739$	−14.3544	−0.528035	−0.264017	+	0.964518i	$0.585048\pi$
$740$	0	0
$741$	−43.8288	−1.61009
$742$	0	0
$743$	23.1839	0.850536	0.425268	−	0.905068i	$-0.360180\pi$
$743$	23.1839	0.850536	0.425268	+	0.905068i	$0.360180\pi$
$744$	0	0
$745$	22.2612	0.815589
$746$	0	0
$747$	4.74407	0.173576
$748$	0	0
$749$	−83.6767	−3.05748
$750$	0	0
$751$	−36.8698	−1.34540	−0.672699	−	0.739916i	$-0.734865\pi$
$751$	−36.8698	−1.34540	−0.672699	+	0.739916i	$0.734865\pi$
$752$	0	0
$753$	43.1151	1.57120
$754$	0	0
$755$	46.8500	1.70504
$756$	0	0
$757$	5.86701	0.213240	0.106620	−	0.994300i	$-0.465997\pi$
$757$	5.86701	0.213240	0.106620	+	0.994300i	$0.465997\pi$
$758$	0	0
$759$	20.8818	0.757961
$760$	0	0
$761$	14.5000	0.525626	0.262813	−	0.964847i	$-0.415350\pi$
$761$	14.5000	0.525626	0.262813	+	0.964847i	$0.415350\pi$
$762$	0	0
$763$	50.1870	1.81689
$764$	0	0
$765$	−1.78510	−0.0645405
$766$	0	0
$767$	−5.16420	−0.186469
$768$	0	0
$769$	−19.1731	−0.691399	−0.345699	−	0.938345i	$-0.612358\pi$
$769$	−19.1731	−0.691399	−0.345699	+	0.938345i	$0.612358\pi$
$770$	0	0
$771$	38.0331	1.36973
$772$	0	0
$773$	23.9339	0.860844	0.430422	−	0.902628i	$-0.358365\pi$
$773$	23.9339	0.860844	0.430422	+	0.902628i	$0.358365\pi$
$774$	0	0
$775$	35.5738	1.27785
$776$	0	0
$777$	17.5666	0.630199
$778$	0	0
$779$	25.5347	0.914876
$780$	0	0
$781$	66.5164	2.38015
$782$	0	0
$783$	53.5730	1.91454
$784$	0	0
$785$	32.0311	1.14324
$786$	0	0
$787$	37.4553	1.33514	0.667569	−	0.744548i	$-0.267335\pi$
$787$	37.4553	1.33514	0.667569	+	0.744548i	$0.267335\pi$
$788$	0	0
$789$	6.37820	0.227070
$790$	0	0
$791$	−10.5606	−0.375491
$792$	0	0
$793$	39.1041	1.38863
$794$	0	0
$795$	−27.9285	−0.990522
$796$	0	0
$797$	−15.1630	−0.537103	−0.268551	−	0.963265i	$-0.586545\pi$
$797$	−15.1630	−0.537103	−0.268551	+	0.963265i	$0.586545\pi$
$798$	0	0
$799$	−7.12549	−0.252082
$800$	0	0
$801$	8.38031	0.296104
$802$	0	0
$803$	−38.9994	−1.37626
$804$	0	0
$805$	32.8925	1.15931
$806$	0	0
$807$	−41.1133	−1.44726
$808$	0	0
$809$	−37.6661	−1.32427	−0.662135	−	0.749384i	$-0.730350\pi$
$809$	−37.6661	−1.32427	−0.662135	+	0.749384i	$0.730350\pi$
$810$	0	0
$811$	34.4522	1.20978	0.604890	−	0.796309i	$-0.293217\pi$
$811$	34.4522	1.20978	0.604890	+	0.796309i	$0.293217\pi$
$812$	0	0
$813$	−47.4133	−1.66286
$814$	0	0
$815$	−18.6452	−0.653113
$816$	0	0
$817$	−61.1225	−2.13841
$818$	0	0
$819$	11.3176	0.395469
$820$	0	0
$821$	6.44994	0.225105	0.112552	−	0.993646i	$-0.464097\pi$
$821$	6.44994	0.225105	0.112552	+	0.993646i	$0.464097\pi$
$822$	0	0
$823$	−33.8079	−1.17847	−0.589235	−	0.807962i	$-0.700571\pi$
$823$	−33.8079	−1.17847	−0.589235	+	0.807962i	$0.700571\pi$
$824$	0	0
$825$	−63.7386	−2.21909
$826$	0	0
$827$	−2.64538	−0.0919890	−0.0459945	−	0.998942i	$-0.514646\pi$
$827$	−2.64538	−0.0919890	−0.0459945	+	0.998942i	$0.514646\pi$
$828$	0	0
$829$	−9.83827	−0.341697	−0.170849	−	0.985297i	$-0.554651\pi$
$829$	−9.83827	−0.341697	−0.170849	+	0.985297i	$0.554651\pi$
$830$	0	0
$831$	15.5482	0.539362
$832$	0	0
$833$	10.9094	0.377987
$834$	0	0
$835$	−47.5912	−1.64696
$836$	0	0
$837$	−28.6472	−0.990192
$838$	0	0
$839$	−16.6616	−0.575221	−0.287610	−	0.957747i	$-0.592861\pi$
$839$	−16.6616	−0.575221	−0.287610	+	0.957747i	$0.592861\pi$
$840$	0	0
$841$	64.4347	2.22189
$842$	0	0
$843$	34.4161	1.18535
$844$	0	0
$845$	47.1182	1.62092
$846$	0	0
$847$	−99.6795	−3.42503
$848$	0	0
$849$	20.3417	0.698126
$850$	0	0
$851$	5.94073	0.203646
$852$	0	0
$853$	−6.33938	−0.217056	−0.108528	−	0.994093i	$-0.534614\pi$
$853$	−6.33938	−0.217056	−0.108528	+	0.994093i	$0.534614\pi$
$854$	0	0
$855$	9.61624	0.328868
$856$	0	0
$857$	−27.7025	−0.946298	−0.473149	−	0.880982i	$-0.656883\pi$
$857$	−27.7025	−0.946298	−0.473149	+	0.880982i	$0.656883\pi$
$858$	0	0
$859$	−26.8824	−0.917217	−0.458608	−	0.888638i	$-0.651652\pi$
$859$	−26.8824	−0.917217	−0.458608	+	0.888638i	$0.651652\pi$
$860$	0	0
$861$	31.6040	1.07706
$862$	0	0
$863$	19.9192	0.678058	0.339029	−	0.940776i	$-0.389902\pi$
$863$	19.9192	0.678058	0.339029	+	0.940776i	$0.389902\pi$
$864$	0	0
$865$	43.2971	1.47215
$866$	0	0
$867$	1.57548	0.0535062
$868$	0	0
$869$	86.6233	2.93849
$870$	0	0
$871$	1.95132	0.0661179
$872$	0	0
$873$	3.09295	0.104680
$874$	0	0
$875$	−27.4602	−0.928323
$876$	0	0
$877$	−27.6210	−0.932696	−0.466348	−	0.884601i	$-0.654430\pi$
$877$	−27.6210	−0.932696	−0.466348	+	0.884601i	$0.654430\pi$
$878$	0	0
$879$	18.5187	0.624622
$880$	0	0
$881$	13.7158	0.462098	0.231049	−	0.972942i	$-0.425784\pi$
$881$	13.7158	0.462098	0.231049	+	0.972942i	$0.425784\pi$
$882$	0	0
$883$	−3.39120	−0.114123	−0.0570614	−	0.998371i	$-0.518173\pi$
$883$	−3.39120	−0.114123	−0.0570614	+	0.998371i	$0.518173\pi$
$884$	0	0
$885$	−5.43082	−0.182555
$886$	0	0
$887$	1.86338	0.0625661	0.0312831	−	0.999511i	$-0.490041\pi$
$887$	1.86338	0.0625661	0.0312831	+	0.999511i	$0.490041\pi$
$888$	0	0
$889$	40.1434	1.34637
$890$	0	0
$891$	42.1957	1.41361
$892$	0	0
$893$	38.3846	1.28449
$894$	0	0
$895$	−55.3328	−1.84957
$896$	0	0
$897$	−18.3452	−0.612527
$898$	0	0
$899$	−49.9626	−1.66634
$900$	0	0
$901$	−5.14259	−0.171325
$902$	0	0
$903$	−75.6505	−2.51749
$904$	0	0
$905$	−52.8809	−1.75782
$906$	0	0
$907$	9.11163	0.302547	0.151273	−	0.988492i	$-0.451663\pi$
$907$	9.11163	0.302547	0.151273	+	0.988492i	$0.451663\pi$
$908$	0	0
$909$	−1.99408	−0.0661396
$910$	0	0
$911$	−18.8415	−0.624246	−0.312123	−	0.950042i	$-0.601040\pi$
$911$	−18.8415	−0.624246	−0.312123	+	0.950042i	$0.601040\pi$
$912$	0	0
$913$	53.8505	1.78219
$914$	0	0
$915$	41.1230	1.35948
$916$	0	0
$917$	88.6089	2.92612
$918$	0	0
$919$	0.190478	0.00628328	0.00314164	−	0.999995i	$-0.499000\pi$
$919$	0.190478	0.00628328	0.00314164	+	0.999995i	$0.499000\pi$
$920$	0	0
$921$	36.1257	1.19038
$922$	0	0
$923$	−58.4363	−1.92345
$924$	0	0
$925$	−18.1332	−0.596216
$926$	0	0
$927$	−5.18867	−0.170418
$928$	0	0
$929$	−17.8425	−0.585392	−0.292696	−	0.956206i	$-0.594552\pi$
$929$	−17.8425	−0.585392	−0.292696	+	0.956206i	$0.594552\pi$
$930$	0	0
$931$	−58.7681	−1.92605
$932$	0	0
$933$	−12.0619	−0.394887
$934$	0	0
$935$	−20.2629	−0.662668
$936$	0	0
$937$	1.06183	0.0346886	0.0173443	−	0.999850i	$-0.494479\pi$
$937$	1.06183	0.0346886	0.0173443	+	0.999850i	$0.494479\pi$
$938$	0	0
$939$	7.61490	0.248503
$940$	0	0
$941$	57.3762	1.87041	0.935206	−	0.354105i	$-0.115215\pi$
$941$	57.3762	1.87041	0.935206	+	0.354105i	$0.115215\pi$
$942$	0	0
$943$	10.6879	0.348047
$944$	0	0
$945$	80.8507	2.63007
$946$	0	0
$947$	−51.0069	−1.65750	−0.828750	−	0.559619i	$-0.810947\pi$
$947$	−51.0069	−1.65750	−0.828750	+	0.559619i	$0.810947\pi$
$948$	0	0
$949$	34.2619	1.11219
$950$	0	0
$951$	−39.5295	−1.28183
$952$	0	0
$953$	−14.7579	−0.478054	−0.239027	−	0.971013i	$-0.576828\pi$
$953$	−14.7579	−0.478054	−0.239027	+	0.971013i	$0.576828\pi$
$954$	0	0
$955$	38.7688	1.25453
$956$	0	0
$957$	89.5194	2.89375
$958$	0	0
$959$	55.6482	1.79698
$960$	0	0
$961$	−4.28341	−0.138175
$962$	0	0
$963$	−10.2394	−0.329961
$964$	0	0
$965$	31.6405	1.01854
$966$	0	0
$967$	−24.5207	−0.788532	−0.394266	−	0.918996i	$-0.629001\pi$
$967$	−24.5207	−0.788532	−0.394266	+	0.918996i	$0.629001\pi$
$968$	0	0
$969$	−8.48703	−0.272643
$970$	0	0
$971$	−48.3283	−1.55093	−0.775465	−	0.631391i	$-0.782484\pi$
$971$	−48.3283	−1.55093	−0.775465	+	0.631391i	$0.782484\pi$
$972$	0	0
$973$	−32.6798	−1.04767
$974$	0	0
$975$	55.9959	1.79330
$976$	0	0
$977$	8.08764	0.258746	0.129373	−	0.991596i	$-0.458703\pi$
$977$	8.08764	0.258746	0.129373	+	0.991596i	$0.458703\pi$
$978$	0	0
$979$	95.1259	3.04024
$980$	0	0
$981$	6.14132	0.196077
$982$	0	0
$983$	−6.52400	−0.208083	−0.104042	−	0.994573i	$-0.533178\pi$
$983$	−6.52400	−0.208083	−0.104042	+	0.994573i	$0.533178\pi$
$984$	0	0
$985$	14.2635	0.454472
$986$	0	0
$987$	47.5082	1.51220
$988$	0	0
$989$	−25.5837	−0.813515
$990$	0	0
$991$	−50.2997	−1.59782	−0.798911	−	0.601449i	$-0.794590\pi$
$991$	−50.2997	−1.59782	−0.798911	+	0.601449i	$0.794590\pi$
$992$	0	0
$993$	1.71510	0.0544272
$994$	0	0
$995$	−5.49505	−0.174205
$996$	0	0
$997$	28.8940	0.915082	0.457541	−	0.889189i	$-0.348730\pi$
$997$	28.8940	0.915082	0.457541	+	0.889189i	$0.348730\pi$
$998$	0	0
$999$	14.6025	0.462002

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.h.1.11	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.h.1.11	✓	15	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.57548 q^{3} +3.44709 q^{5} -4.23195 q^{7} -0.517858 q^{9} +O(q^{10})\)	\(q+1.57548 q^{3} +3.44709 q^{5} -4.23195 q^{7} -0.517858 q^{9} -5.87827 q^{11} +5.16420 q^{13} +5.43082 q^{15} +1.00000 q^{17} -5.38694 q^{19} -6.66735 q^{21} -2.25478 q^{23} +6.88240 q^{25} -5.54232 q^{27} -9.66616 q^{29} +5.16881 q^{31} -9.26111 q^{33} -14.5879 q^{35} -2.63472 q^{37} +8.13611 q^{39} -4.74011 q^{41} +11.3464 q^{43} -1.78510 q^{45} -7.12549 q^{47} +10.9094 q^{49} +1.57548 q^{51} -5.14259 q^{53} -20.2629 q^{55} -8.48703 q^{57} -1.00000 q^{59} +7.57215 q^{61} +2.19155 q^{63} +17.8015 q^{65} +0.377854 q^{67} -3.55237 q^{69} -11.3156 q^{71} +6.63450 q^{73} +10.8431 q^{75} +24.8765 q^{77} -14.7362 q^{79} -7.17825 q^{81} -9.16094 q^{83} +3.44709 q^{85} -15.2289 q^{87} -16.1826 q^{89} -21.8546 q^{91} +8.14336 q^{93} -18.5693 q^{95} -5.97258 q^{97} +3.04411 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9}+O(q^{10}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9	\( 15 q - q^{3} + q^{5} - 11 q^{7} + 16 q^{9} - 12 q^{11} - 10 q^{13} - 13 q^{15} + 15 q^{17} - 4 q^{21} - 21 q^{23} + 4 q^{25} - 37 q^{27} + 23 q^{29} - 31 q^{31} - 11 q^{33} - 35 q^{35} - 10 q^{37} - 2 q^{39} - 15 q^{41} - 3 q^{43} + 15 q^{45} - 47 q^{47} + 18 q^{49} - q^{51} - 7 q^{53} - 20 q^{55} - 30 q^{57} - 15 q^{59} + q^{61} + 19 q^{63} + 11 q^{65} - 20 q^{67} - 24 q^{69} - 13 q^{71} - 24 q^{73} - 10 q^{75} + 21 q^{77} - 34 q^{79} - 21 q^{81} - 40 q^{83} + q^{85} - 60 q^{87} - 46 q^{89} - 19 q^{91} + 33 q^{93} - 4 q^{95} - 5 q^{97} - 74 q^{99}+O(q^{100}) \) 15 * q - q^3 + q^5 - 11 * q^7 + 16 * q^9 - 12 * q^11 - 10 * q^13 - 13 * q^15 + 15 * q^17 - 4 * q^21 - 21 * q^23 + 4 * q^25 - 37 * q^27 + 23 * q^29 - 31 * q^31 - 11 * q^33 - 35 * q^35 - 10 * q^37 - 2 * q^39 - 15 * q^41 - 3 * q^43 + 15 * q^45 - 47 * q^47 + 18 * q^49 - q^51 - 7 * q^53 - 20 * q^55 - 30 * q^57 - 15 * q^59 + q^61 + 19 * q^63 + 11 * q^65 - 20 * q^67 - 24 * q^69 - 13 * q^71 - 24 * q^73 - 10 * q^75 + 21 * q^77 - 34 * q^79 - 21 * q^81 - 40 * q^83 + q^85 - 60 * q^87 - 46 * q^89 - 19 * q^91 + 33 * q^93 - 4 * q^95 - 5 * q^97 - 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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