LMFDB - Embedded newform 4016.2.a.i.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, 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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
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} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-1.20604$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.20604 q^{3} -1.79940 q^{5} +2.78727 q^{7} -1.54547 q^{9} +O(q^{10})$	$q+1.20604 q^{3} -1.79940 q^{5} +2.78727 q^{7} -1.54547 q^{9} -0.183550 q^{11} -4.03963 q^{13} -2.17015 q^{15} +1.33145 q^{17} +0.976694 q^{19} +3.36156 q^{21} +6.76113 q^{23} -1.76215 q^{25} -5.48202 q^{27} -3.70176 q^{29} -5.56037 q^{31} -0.221369 q^{33} -5.01541 q^{35} +2.65718 q^{37} -4.87195 q^{39} -10.1556 q^{41} +2.03219 q^{43} +2.78092 q^{45} -11.3649 q^{47} +0.768858 q^{49} +1.60578 q^{51} +12.2932 q^{53} +0.330281 q^{55} +1.17793 q^{57} +10.3582 q^{59} +2.65032 q^{61} -4.30763 q^{63} +7.26891 q^{65} -1.94822 q^{67} +8.15420 q^{69} -5.71480 q^{71} -8.31463 q^{73} -2.12523 q^{75} -0.511603 q^{77} -15.2426 q^{79} -1.97513 q^{81} -8.57230 q^{83} -2.39581 q^{85} -4.46448 q^{87} +0.496905 q^{89} -11.2595 q^{91} -6.70603 q^{93} -1.75746 q^{95} -0.552787 q^{97} +0.283671 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	1.20604	0.696308	0.348154	−	0.937437i	$-0.386809\pi$
$3$	1.20604	0.696308	0.348154	+	0.937437i	$0.386809\pi$
$4$	0	0
$5$	−1.79940	−0.804717	−0.402358	−	0.915482i	$-0.631809\pi$
$5$	−1.79940	−0.804717	−0.402358	+	0.915482i	$0.631809\pi$
$6$	0	0
$7$	2.78727	1.05349	0.526744	−	0.850024i	$-0.323413\pi$
$7$	2.78727	1.05349	0.526744	+	0.850024i	$0.323413\pi$
$8$	0	0
$9$	−1.54547	−0.515156
$10$	0	0
$11$	−0.183550	−0.0553425	−0.0276712	−	0.999617i	$-0.508809\pi$
$11$	−0.183550	−0.0553425	−0.0276712	+	0.999617i	$0.508809\pi$
$12$	0	0
$13$	−4.03963	−1.12039	−0.560195	−	0.828361i	$-0.689274\pi$
$13$	−4.03963	−1.12039	−0.560195	+	0.828361i	$0.689274\pi$
$14$	0	0
$15$	−2.17015	−0.560331
$16$	0	0
$17$	1.33145	0.322924	0.161462	−	0.986879i	$-0.448379\pi$
$17$	1.33145	0.322924	0.161462	+	0.986879i	$0.448379\pi$
$18$	0	0
$19$	0.976694	0.224069	0.112034	−	0.993704i	$-0.464263\pi$
$19$	0.976694	0.224069	0.112034	+	0.993704i	$0.464263\pi$
$20$	0	0
$21$	3.36156	0.733552
$22$	0	0
$23$	6.76113	1.40979	0.704897	−	0.709310i	$-0.250993\pi$
$23$	6.76113	1.40979	0.704897	+	0.709310i	$0.250993\pi$
$24$	0	0
$25$	−1.76215	−0.352431
$26$	0	0
$27$	−5.48202	−1.05501
$28$	0	0
$29$	−3.70176	−0.687400	−0.343700	−	0.939079i	$-0.611680\pi$
$29$	−3.70176	−0.687400	−0.343700	+	0.939079i	$0.611680\pi$
$30$	0	0
$31$	−5.56037	−0.998672	−0.499336	−	0.866408i	$-0.666423\pi$
$31$	−5.56037	−0.998672	−0.499336	+	0.866408i	$0.666423\pi$
$32$	0	0
$33$	−0.221369	−0.0385354
$34$	0	0
$35$	−5.01541	−0.847760
$36$	0	0
$37$	2.65718	0.436837	0.218419	−	0.975855i	$-0.429910\pi$
$37$	2.65718	0.436837	0.218419	+	0.975855i	$0.429910\pi$
$38$	0	0
$39$	−4.87195	−0.780137
$40$	0	0
$41$	−10.1556	−1.58603	−0.793017	−	0.609200i	$-0.791491\pi$
$41$	−10.1556	−1.58603	−0.793017	+	0.609200i	$0.791491\pi$
$42$	0	0
$43$	2.03219	0.309906	0.154953	−	0.987922i	$-0.450477\pi$
$43$	2.03219	0.309906	0.154953	+	0.987922i	$0.450477\pi$
$44$	0	0
$45$	2.78092	0.414554
$46$	0	0
$47$	−11.3649	−1.65774	−0.828872	−	0.559438i	$-0.811017\pi$
$47$	−11.3649	−1.65774	−0.828872	+	0.559438i	$0.811017\pi$
$48$	0	0
$49$	0.768858	0.109837
$50$	0	0
$51$	1.60578	0.224854
$52$	0	0
$53$	12.2932	1.68860	0.844298	−	0.535874i	$-0.180018\pi$
$53$	12.2932	1.68860	0.844298	+	0.535874i	$0.180018\pi$
$54$	0	0
$55$	0.330281	0.0445350
$56$	0	0
$57$	1.17793	0.156021
$58$	0	0
$59$	10.3582	1.34853	0.674263	−	0.738491i	$-0.264461\pi$
$59$	10.3582	1.34853	0.674263	+	0.738491i	$0.264461\pi$
$60$	0	0
$61$	2.65032	0.339338	0.169669	−	0.985501i	$-0.445730\pi$
$61$	2.65032	0.339338	0.169669	+	0.985501i	$0.445730\pi$
$62$	0	0
$63$	−4.30763	−0.542710
$64$	0	0
$65$	7.26891	0.901597
$66$	0	0
$67$	−1.94822	−0.238013	−0.119007	−	0.992893i	$-0.537971\pi$
$67$	−1.94822	−0.238013	−0.119007	+	0.992893i	$0.537971\pi$
$68$	0	0
$69$	8.15420	0.981650
$70$	0	0
$71$	−5.71480	−0.678222	−0.339111	−	0.940746i	$-0.610126\pi$
$71$	−5.71480	−0.678222	−0.339111	+	0.940746i	$0.610126\pi$
$72$	0	0
$73$	−8.31463	−0.973153	−0.486577	−	0.873638i	$-0.661755\pi$
$73$	−8.31463	−0.973153	−0.486577	+	0.873638i	$0.661755\pi$
$74$	0	0
$75$	−2.12523	−0.245400
$76$	0	0
$77$	−0.511603	−0.0583026
$78$	0	0
$79$	−15.2426	−1.71493	−0.857463	−	0.514546i	$-0.827960\pi$
$79$	−15.2426	−1.71493	−0.857463	+	0.514546i	$0.827960\pi$
$80$	0	0
$81$	−1.97513	−0.219459
$82$	0	0
$83$	−8.57230	−0.940932	−0.470466	−	0.882418i	$-0.655914\pi$
$83$	−8.57230	−0.940932	−0.470466	+	0.882418i	$0.655914\pi$
$84$	0	0
$85$	−2.39581	−0.259862
$86$	0	0
$87$	−4.46448	−0.478642
$88$	0	0
$89$	0.496905	0.0526719	0.0263359	−	0.999653i	$-0.491616\pi$
$89$	0.496905	0.0526719	0.0263359	+	0.999653i	$0.491616\pi$
$90$	0	0
$91$	−11.2595	−1.18032
$92$	0	0
$93$	−6.70603	−0.695383
$94$	0	0
$95$	−1.75746	−0.180312
$96$	0	0
$97$	−0.552787	−0.0561271	−0.0280635	−	0.999606i	$-0.508934\pi$
$97$	−0.552787	−0.0561271	−0.0280635	+	0.999606i	$0.508934\pi$
$98$	0	0
$99$	0.283671	0.0285100
$100$	0	0
$101$	−3.21382	−0.319787	−0.159893	−	0.987134i	$-0.551115\pi$
$101$	−3.21382	−0.319787	−0.159893	+	0.987134i	$0.551115\pi$
$102$	0	0
$103$	0.284523	0.0280349	0.0140175	−	0.999902i	$-0.495538\pi$
$103$	0.284523	0.0280349	0.0140175	+	0.999902i	$0.495538\pi$
$104$	0	0
$105$	−6.04879	−0.590302
$106$	0	0
$107$	−13.4908	−1.30421	−0.652103	−	0.758131i	$-0.726113\pi$
$107$	−13.4908	−1.30421	−0.652103	+	0.758131i	$0.726113\pi$
$108$	0	0
$109$	4.41187	0.422581	0.211290	−	0.977423i	$-0.432233\pi$
$109$	4.41187	0.422581	0.211290	+	0.977423i	$0.432233\pi$
$110$	0	0
$111$	3.20466	0.304173
$112$	0	0
$113$	−16.2654	−1.53012	−0.765059	−	0.643961i	$-0.777290\pi$
$113$	−16.2654	−1.53012	−0.765059	+	0.643961i	$0.777290\pi$
$114$	0	0
$115$	−12.1660	−1.13449
$116$	0	0
$117$	6.24311	0.577176
$118$	0	0
$119$	3.71110	0.340196
$120$	0	0
$121$	−10.9663	−0.996937
$122$	0	0
$123$	−12.2480	−1.10437
$124$	0	0
$125$	12.1678	1.08832
$126$	0	0
$127$	−9.03591	−0.801808	−0.400904	−	0.916120i	$-0.631304\pi$
$127$	−9.03591	−0.801808	−0.400904	+	0.916120i	$0.631304\pi$
$128$	0	0
$129$	2.45090	0.215790
$130$	0	0
$131$	8.69192	0.759417	0.379708	−	0.925106i	$-0.376024\pi$
$131$	8.69192	0.759417	0.379708	+	0.925106i	$0.376024\pi$
$132$	0	0
$133$	2.72231	0.236054
$134$	0	0
$135$	9.86435	0.848988
$136$	0	0
$137$	−0.486879	−0.0415969	−0.0207984	−	0.999784i	$-0.506621\pi$
$137$	−0.486879	−0.0415969	−0.0207984	+	0.999784i	$0.506621\pi$
$138$	0	0
$139$	−1.14836	−0.0974023	−0.0487012	−	0.998813i	$-0.515508\pi$
$139$	−1.14836	−0.0974023	−0.0487012	+	0.998813i	$0.515508\pi$
$140$	0	0
$141$	−13.7066	−1.15430
$142$	0	0
$143$	0.741474	0.0620052
$144$	0	0
$145$	6.66096	0.553163
$146$	0	0
$147$	0.927274	0.0764802
$148$	0	0
$149$	4.75478	0.389527	0.194763	−	0.980850i	$-0.437606\pi$
$149$	4.75478	0.389527	0.194763	+	0.980850i	$0.437606\pi$
$150$	0	0
$151$	16.9121	1.37629	0.688144	−	0.725574i	$-0.258426\pi$
$151$	16.9121	1.37629	0.688144	+	0.725574i	$0.258426\pi$
$152$	0	0
$153$	−2.05771	−0.166356
$154$	0	0
$155$	10.0053	0.803648
$156$	0	0
$157$	13.4286	1.07172	0.535859	−	0.844308i	$-0.319988\pi$
$157$	13.4286	1.07172	0.535859	+	0.844308i	$0.319988\pi$
$158$	0	0
$159$	14.8261	1.17578
$160$	0	0
$161$	18.8451	1.48520
$162$	0	0
$163$	−8.82873	−0.691520	−0.345760	−	0.938323i	$-0.612379\pi$
$163$	−8.82873	−0.691520	−0.345760	+	0.938323i	$0.612379\pi$
$164$	0	0
$165$	0.398332	0.0310101
$166$	0	0
$167$	−11.6374	−0.900525	−0.450263	−	0.892896i	$-0.648670\pi$
$167$	−11.6374	−0.900525	−0.450263	+	0.892896i	$0.648670\pi$
$168$	0	0
$169$	3.31858	0.255275
$170$	0	0
$171$	−1.50945	−0.115430
$172$	0	0
$173$	−5.77854	−0.439334	−0.219667	−	0.975575i	$-0.570497\pi$
$173$	−5.77854	−0.439334	−0.219667	+	0.975575i	$0.570497\pi$
$174$	0	0
$175$	−4.91159	−0.371281
$176$	0	0
$177$	12.4924	0.938990
$178$	0	0
$179$	−14.2233	−1.06310	−0.531550	−	0.847027i	$-0.678390\pi$
$179$	−14.2233	−1.06310	−0.531550	+	0.847027i	$0.678390\pi$
$180$	0	0
$181$	3.02526	0.224865	0.112433	−	0.993659i	$-0.464136\pi$
$181$	3.02526	0.224865	0.112433	+	0.993659i	$0.464136\pi$
$182$	0	0
$183$	3.19639	0.236284
$184$	0	0
$185$	−4.78133	−0.351530
$186$	0	0
$187$	−0.244388	−0.0178714
$188$	0	0
$189$	−15.2798	−1.11145
$190$	0	0
$191$	−1.55755	−0.112700	−0.0563501	−	0.998411i	$-0.517946\pi$
$191$	−1.55755	−0.112700	−0.0563501	+	0.998411i	$0.517946\pi$
$192$	0	0
$193$	−9.00603	−0.648268	−0.324134	−	0.946011i	$-0.605073\pi$
$193$	−9.00603	−0.648268	−0.324134	+	0.946011i	$0.605073\pi$
$194$	0	0
$195$	8.76660	0.627789
$196$	0	0
$197$	4.29641	0.306106	0.153053	−	0.988218i	$-0.451089\pi$
$197$	4.29641	0.306106	0.153053	+	0.988218i	$0.451089\pi$
$198$	0	0
$199$	10.0736	0.714097	0.357049	−	0.934086i	$-0.383783\pi$
$199$	10.0736	0.714097	0.357049	+	0.934086i	$0.383783\pi$
$200$	0	0
$201$	−2.34964	−0.165730
$202$	0	0
$203$	−10.3178	−0.724168
$204$	0	0
$205$	18.2739	1.27631
$206$	0	0
$207$	−10.4491	−0.726263
$208$	0	0
$209$	−0.179272	−0.0124005
$210$	0	0
$211$	−12.5342	−0.862893	−0.431446	−	0.902139i	$-0.641997\pi$
$211$	−12.5342	−0.862893	−0.431446	+	0.902139i	$0.641997\pi$
$212$	0	0
$213$	−6.89228	−0.472251
$214$	0	0
$215$	−3.65672	−0.249387
$216$	0	0
$217$	−15.4982	−1.05209
$218$	0	0
$219$	−10.0278	−0.677614
$220$	0	0
$221$	−5.37855	−0.361801
$222$	0	0
$223$	−9.11216	−0.610195	−0.305098	−	0.952321i	$-0.598689\pi$
$223$	−9.11216	−0.610195	−0.305098	+	0.952321i	$0.598689\pi$
$224$	0	0
$225$	2.72335	0.181557
$226$	0	0
$227$	13.9957	0.928927	0.464464	−	0.885592i	$-0.346247\pi$
$227$	13.9957	0.928927	0.464464	+	0.885592i	$0.346247\pi$
$228$	0	0
$229$	−8.15121	−0.538647	−0.269323	−	0.963050i	$-0.586800\pi$
$229$	−8.15121	−0.538647	−0.269323	+	0.963050i	$0.586800\pi$
$230$	0	0
$231$	−0.617014	−0.0405966
$232$	0	0
$233$	3.16347	0.207246	0.103623	−	0.994617i	$-0.466956\pi$
$233$	3.16347	0.207246	0.103623	+	0.994617i	$0.466956\pi$
$234$	0	0
$235$	20.4501	1.33402
$236$	0	0
$237$	−18.3832	−1.19412
$238$	0	0
$239$	14.2776	0.923540	0.461770	−	0.887000i	$-0.347215\pi$
$239$	14.2776	0.923540	0.461770	+	0.887000i	$0.347215\pi$
$240$	0	0
$241$	−14.5848	−0.939489	−0.469745	−	0.882802i	$-0.655654\pi$
$241$	−14.5848	−0.939489	−0.469745	+	0.882802i	$0.655654\pi$
$242$	0	0
$243$	14.0640	0.902203
$244$	0	0
$245$	−1.38348	−0.0883876
$246$	0	0
$247$	−3.94548	−0.251045
$248$	0	0
$249$	−10.3385	−0.655178
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−1.24101	−0.0780215
$254$	0	0
$255$	−2.88944	−0.180944
$256$	0	0
$257$	26.1664	1.63222	0.816108	−	0.577899i	$-0.196127\pi$
$257$	26.1664	1.63222	0.816108	+	0.577899i	$0.196127\pi$
$258$	0	0
$259$	7.40626	0.460203
$260$	0	0
$261$	5.72095	0.354118
$262$	0	0
$263$	−30.5839	−1.88588	−0.942942	−	0.332958i	$-0.891953\pi$
$263$	−30.5839	−1.88588	−0.942942	+	0.332958i	$0.891953\pi$
$264$	0	0
$265$	−22.1203	−1.35884
$266$	0	0
$267$	0.599288	0.0366758
$268$	0	0
$269$	18.9482	1.15529	0.577647	−	0.816287i	$-0.303971\pi$
$269$	18.9482	1.15529	0.577647	+	0.816287i	$0.303971\pi$
$270$	0	0
$271$	25.5917	1.55458	0.777291	−	0.629141i	$-0.216593\pi$
$271$	25.5917	1.55458	0.777291	+	0.629141i	$0.216593\pi$
$272$	0	0
$273$	−13.5794	−0.821865
$274$	0	0
$275$	0.323444	0.0195044
$276$	0	0
$277$	−27.2776	−1.63895	−0.819475	−	0.573115i	$-0.805735\pi$
$277$	−27.2776	−1.63895	−0.819475	+	0.573115i	$0.805735\pi$
$278$	0	0
$279$	8.59337	0.514471
$280$	0	0
$281$	−21.2169	−1.26570	−0.632848	−	0.774276i	$-0.718114\pi$
$281$	−21.2169	−1.26570	−0.632848	+	0.774276i	$0.718114\pi$
$282$	0	0
$283$	−3.40273	−0.202272	−0.101136	−	0.994873i	$-0.532248\pi$
$283$	−3.40273	−0.202272	−0.101136	+	0.994873i	$0.532248\pi$
$284$	0	0
$285$	−2.11957	−0.125553
$286$	0	0
$287$	−28.3063	−1.67087
$288$	0	0
$289$	−15.2272	−0.895720
$290$	0	0
$291$	−0.666684	−0.0390817
$292$	0	0
$293$	14.3568	0.838735	0.419367	−	0.907817i	$-0.362252\pi$
$293$	14.3568	0.838735	0.419367	+	0.907817i	$0.362252\pi$
$294$	0	0
$295$	−18.6386	−1.08518
$296$	0	0
$297$	1.00623	0.0583871
$298$	0	0
$299$	−27.3125	−1.57952
$300$	0	0
$301$	5.66425	0.326482
$302$	0	0
$303$	−3.87599	−0.222670
$304$	0	0
$305$	−4.76898	−0.273071
$306$	0	0
$307$	−7.53649	−0.430130	−0.215065	−	0.976600i	$-0.568996\pi$
$307$	−7.53649	−0.430130	−0.215065	+	0.976600i	$0.568996\pi$
$308$	0	0
$309$	0.343147	0.0195209
$310$	0	0
$311$	26.7244	1.51540	0.757701	−	0.652602i	$-0.226323\pi$
$311$	26.7244	1.51540	0.757701	+	0.652602i	$0.226323\pi$
$312$	0	0
$313$	13.6125	0.769423	0.384712	−	0.923037i	$-0.374301\pi$
$313$	13.6125	0.769423	0.384712	+	0.923037i	$0.374301\pi$
$314$	0	0
$315$	7.75115	0.436728
$316$	0	0
$317$	30.4593	1.71076	0.855381	−	0.517999i	$-0.173323\pi$
$317$	30.4593	1.71076	0.855381	+	0.517999i	$0.173323\pi$
$318$	0	0
$319$	0.679460	0.0380424
$320$	0	0
$321$	−16.2705	−0.908128
$322$	0	0
$323$	1.30042	0.0723572
$324$	0	0
$325$	7.11844	0.394860
$326$	0	0
$327$	5.32090	0.294246
$328$	0	0
$329$	−31.6771	−1.74641
$330$	0	0
$331$	−11.5398	−0.634283	−0.317142	−	0.948378i	$-0.602723\pi$
$331$	−11.5398	−0.634283	−0.317142	+	0.948378i	$0.602723\pi$
$332$	0	0
$333$	−4.10658	−0.225039
$334$	0	0
$335$	3.50564	0.191533
$336$	0	0
$337$	−14.4178	−0.785390	−0.392695	−	0.919669i	$-0.628457\pi$
$337$	−14.4178	−0.785390	−0.392695	+	0.919669i	$0.628457\pi$
$338$	0	0
$339$	−19.6167	−1.06543
$340$	0	0
$341$	1.02061	0.0552690
$342$	0	0
$343$	−17.3679	−0.937776
$344$	0	0
$345$	−14.6727	−0.789951
$346$	0	0
$347$	16.6499	0.893815	0.446907	−	0.894580i	$-0.352525\pi$
$347$	16.6499	0.893815	0.446907	+	0.894580i	$0.352525\pi$
$348$	0	0
$349$	−20.0107	−1.07115	−0.535575	−	0.844488i	$-0.679905\pi$
$349$	−20.0107	−1.07115	−0.535575	+	0.844488i	$0.679905\pi$
$350$	0	0
$351$	22.1453	1.18203
$352$	0	0
$353$	23.3999	1.24545	0.622725	−	0.782441i	$-0.286026\pi$
$353$	23.3999	1.24545	0.622725	+	0.782441i	$0.286026\pi$
$354$	0	0
$355$	10.2832	0.545777
$356$	0	0
$357$	4.47574	0.236881
$358$	0	0
$359$	−7.74852	−0.408951	−0.204476	−	0.978872i	$-0.565549\pi$
$359$	−7.74852	−0.408951	−0.204476	+	0.978872i	$0.565549\pi$
$360$	0	0
$361$	−18.0461	−0.949793
$362$	0	0
$363$	−13.2258	−0.694175
$364$	0	0
$365$	14.9614	0.783113
$366$	0	0
$367$	16.3802	0.855041	0.427520	−	0.904006i	$-0.359387\pi$
$367$	16.3802	0.855041	0.427520	+	0.904006i	$0.359387\pi$
$368$	0	0
$369$	15.6951	0.817054
$370$	0	0
$371$	34.2643	1.77892
$372$	0	0
$373$	−33.8771	−1.75409	−0.877044	−	0.480410i	$-0.840488\pi$
$373$	−33.8771	−1.75409	−0.877044	+	0.480410i	$0.840488\pi$
$374$	0	0
$375$	14.6749	0.757808
$376$	0	0
$377$	14.9537	0.770157
$378$	0	0
$379$	9.89644	0.508346	0.254173	−	0.967159i	$-0.418197\pi$
$379$	9.89644	0.508346	0.254173	+	0.967159i	$0.418197\pi$
$380$	0	0
$381$	−10.8977	−0.558305
$382$	0	0
$383$	−6.52469	−0.333396	−0.166698	−	0.986008i	$-0.553310\pi$
$383$	−6.52469	−0.333396	−0.166698	+	0.986008i	$0.553310\pi$
$384$	0	0
$385$	0.920580	0.0469171
$386$	0	0
$387$	−3.14068	−0.159650
$388$	0	0
$389$	−9.26110	−0.469557	−0.234778	−	0.972049i	$-0.575436\pi$
$389$	−9.26110	−0.469557	−0.234778	+	0.972049i	$0.575436\pi$
$390$	0	0
$391$	9.00210	0.455256
$392$	0	0
$393$	10.4828	0.528788
$394$	0	0
$395$	27.4275	1.38003
$396$	0	0
$397$	−9.39549	−0.471546	−0.235773	−	0.971808i	$-0.575762\pi$
$397$	−9.39549	−0.471546	−0.235773	+	0.971808i	$0.575762\pi$
$398$	0	0
$399$	3.28321	0.164366
$400$	0	0
$401$	30.6780	1.53199	0.765993	−	0.642849i	$-0.222248\pi$
$401$	30.6780	1.53199	0.765993	+	0.642849i	$0.222248\pi$
$402$	0	0
$403$	22.4618	1.11890
$404$	0	0
$405$	3.55406	0.176603
$406$	0	0
$407$	−0.487725	−0.0241756
$408$	0	0
$409$	14.2928	0.706732	0.353366	−	0.935485i	$-0.385037\pi$
$409$	14.2928	0.706732	0.353366	+	0.935485i	$0.385037\pi$
$410$	0	0
$411$	−0.587195	−0.0289642
$412$	0	0
$413$	28.8712	1.42066
$414$	0	0
$415$	15.4250	0.757184
$416$	0	0
$417$	−1.38496	−0.0678220
$418$	0	0
$419$	−22.2612	−1.08753	−0.543767	−	0.839237i	$-0.683002\pi$
$419$	−22.2612	−1.08753	−0.543767	+	0.839237i	$0.683002\pi$
$420$	0	0
$421$	12.5518	0.611737	0.305868	−	0.952074i	$-0.401053\pi$
$421$	12.5518	0.611737	0.305868	+	0.952074i	$0.401053\pi$
$422$	0	0
$423$	17.5641	0.853996
$424$	0	0
$425$	−2.34622	−0.113808
$426$	0	0
$427$	7.38714	0.357489
$428$	0	0
$429$	0.894248	0.0431747
$430$	0	0
$431$	18.5641	0.894199	0.447100	−	0.894484i	$-0.352457\pi$
$431$	18.5641	0.894199	0.447100	+	0.894484i	$0.352457\pi$
$432$	0	0
$433$	−38.2672	−1.83900	−0.919502	−	0.393086i	$-0.871407\pi$
$433$	−38.2672	−1.83900	−0.919502	+	0.393086i	$0.871407\pi$
$434$	0	0
$435$	8.03339	0.385171
$436$	0	0
$437$	6.60356	0.315891
$438$	0	0
$439$	−7.13433	−0.340503	−0.170251	−	0.985401i	$-0.554458\pi$
$439$	−7.13433	−0.340503	−0.170251	+	0.985401i	$0.554458\pi$
$440$	0	0
$441$	−1.18824	−0.0565831
$442$	0	0
$443$	−27.0350	−1.28447	−0.642237	−	0.766506i	$-0.721993\pi$
$443$	−27.0350	−1.28447	−0.642237	+	0.766506i	$0.721993\pi$
$444$	0	0
$445$	−0.894132	−0.0423859
$446$	0	0
$447$	5.73445	0.271230
$448$	0	0
$449$	−25.6697	−1.21143	−0.605715	−	0.795682i	$-0.707113\pi$
$449$	−25.6697	−1.21143	−0.605715	+	0.795682i	$0.707113\pi$
$450$	0	0
$451$	1.86406	0.0877750
$452$	0	0
$453$	20.3967	0.958320
$454$	0	0
$455$	20.2604	0.949822
$456$	0	0
$457$	−6.47932	−0.303090	−0.151545	−	0.988450i	$-0.548425\pi$
$457$	−6.47932	−0.303090	−0.151545	+	0.988450i	$0.548425\pi$
$458$	0	0
$459$	−7.29902	−0.340689
$460$	0	0
$461$	20.4117	0.950667	0.475334	−	0.879806i	$-0.342327\pi$
$461$	20.4117	0.950667	0.475334	+	0.879806i	$0.342327\pi$
$462$	0	0
$463$	−0.590184	−0.0274282	−0.0137141	−	0.999906i	$-0.504365\pi$
$463$	−0.590184	−0.0274282	−0.0137141	+	0.999906i	$0.504365\pi$
$464$	0	0
$465$	12.0668	0.559586
$466$	0	0
$467$	40.9116	1.89317	0.946583	−	0.322462i	$-0.104510\pi$
$467$	40.9116	1.89317	0.946583	+	0.322462i	$0.104510\pi$
$468$	0	0
$469$	−5.43022	−0.250744
$470$	0	0
$471$	16.1954	0.746245
$472$	0	0
$473$	−0.373009	−0.0171510
$474$	0	0
$475$	−1.72108	−0.0789687
$476$	0	0
$477$	−18.9987	−0.869890
$478$	0	0
$479$	34.9473	1.59678	0.798392	−	0.602138i	$-0.205684\pi$
$479$	34.9473	1.59678	0.798392	+	0.602138i	$0.205684\pi$
$480$	0	0
$481$	−10.7340	−0.489428
$482$	0	0
$483$	22.7279	1.03416
$484$	0	0
$485$	0.994687	0.0451664
$486$	0	0
$487$	2.84644	0.128984	0.0644922	−	0.997918i	$-0.479457\pi$
$487$	2.84644	0.128984	0.0644922	+	0.997918i	$0.479457\pi$
$488$	0	0
$489$	−10.6478	−0.481510
$490$	0	0
$491$	5.68782	0.256688	0.128344	−	0.991730i	$-0.459034\pi$
$491$	5.68782	0.256688	0.128344	+	0.991730i	$0.459034\pi$
$492$	0	0
$493$	−4.92871	−0.221978
$494$	0	0
$495$	−0.510438	−0.0229425
$496$	0	0
$497$	−15.9287	−0.714499
$498$	0	0
$499$	−29.9905	−1.34256	−0.671280	−	0.741204i	$-0.734255\pi$
$499$	−29.9905	−1.34256	−0.671280	+	0.741204i	$0.734255\pi$
$500$	0	0
$501$	−14.0351	−0.627043
$502$	0	0
$503$	24.8131	1.10636	0.553181	−	0.833061i	$-0.313414\pi$
$503$	24.8131	1.10636	0.553181	+	0.833061i	$0.313414\pi$
$504$	0	0
$505$	5.78294	0.257338
$506$	0	0
$507$	4.00234	0.177750
$508$	0	0
$509$	4.33617	0.192198	0.0960988	−	0.995372i	$-0.469364\pi$
$509$	4.33617	0.192198	0.0960988	+	0.995372i	$0.469364\pi$
$510$	0	0
$511$	−23.1751	−1.02521
$512$	0	0
$513$	−5.35425	−0.236396
$514$	0	0
$515$	−0.511972	−0.0225602
$516$	0	0
$517$	2.08603	0.0917437
$518$	0	0
$519$	−6.96915	−0.305912
$520$	0	0
$521$	33.2101	1.45496	0.727481	−	0.686128i	$-0.240691\pi$
$521$	33.2101	1.45496	0.727481	+	0.686128i	$0.240691\pi$
$522$	0	0
$523$	−9.93913	−0.434608	−0.217304	−	0.976104i	$-0.569726\pi$
$523$	−9.93913	−0.434608	−0.217304	+	0.976104i	$0.569726\pi$
$524$	0	0
$525$	−5.92358	−0.258526
$526$	0	0
$527$	−7.40335	−0.322495
$528$	0	0
$529$	22.7129	0.987519
$530$	0	0
$531$	−16.0083	−0.694701
$532$	0	0
$533$	41.0247	1.77698
$534$	0	0
$535$	24.2754	1.04952
$536$	0	0
$537$	−17.1539	−0.740245
$538$	0	0
$539$	−0.141124	−0.00607864
$540$	0	0
$541$	5.55952	0.239022	0.119511	−	0.992833i	$-0.461867\pi$
$541$	5.55952	0.239022	0.119511	+	0.992833i	$0.461867\pi$
$542$	0	0
$543$	3.64858	0.156576
$544$	0	0
$545$	−7.93873	−0.340058
$546$	0	0
$547$	3.19094	0.136435	0.0682174	−	0.997670i	$-0.478269\pi$
$547$	3.19094	0.136435	0.0682174	+	0.997670i	$0.478269\pi$
$548$	0	0
$549$	−4.09598	−0.174812
$550$	0	0
$551$	−3.61549	−0.154025
$552$	0	0
$553$	−42.4852	−1.80665
$554$	0	0
$555$	−5.76647	−0.244773
$556$	0	0
$557$	−2.57675	−0.109181	−0.0545903	−	0.998509i	$-0.517385\pi$
$557$	−2.57675	−0.109181	−0.0545903	+	0.998509i	$0.517385\pi$
$558$	0	0
$559$	−8.20928	−0.347216
$560$	0	0
$561$	−0.294741	−0.0124440
$562$	0	0
$563$	−16.9980	−0.716380	−0.358190	−	0.933649i	$-0.616606\pi$
$563$	−16.9980	−0.716380	−0.358190	+	0.933649i	$0.616606\pi$
$564$	0	0
$565$	29.2679	1.23131
$566$	0	0
$567$	−5.50522	−0.231198
$568$	0	0
$569$	3.19324	0.133868	0.0669338	−	0.997757i	$-0.478678\pi$
$569$	3.19324	0.133868	0.0669338	+	0.997757i	$0.478678\pi$
$570$	0	0
$571$	−45.1426	−1.88916	−0.944579	−	0.328285i	$-0.893529\pi$
$571$	−45.1426	−1.88916	−0.944579	+	0.328285i	$0.893529\pi$
$572$	0	0
$573$	−1.87847	−0.0784741
$574$	0	0
$575$	−11.9142	−0.496855
$576$	0	0
$577$	−40.5176	−1.68677	−0.843386	−	0.537309i	$-0.819441\pi$
$577$	−40.5176	−1.68677	−0.843386	+	0.537309i	$0.819441\pi$
$578$	0	0
$579$	−10.8616	−0.451394
$580$	0	0
$581$	−23.8933	−0.991261
$582$	0	0
$583$	−2.25641	−0.0934511
$584$	0	0
$585$	−11.2339	−0.464463
$586$	0	0
$587$	−33.5623	−1.38527	−0.692633	−	0.721290i	$-0.743549\pi$
$587$	−33.5623	−1.38527	−0.692633	+	0.721290i	$0.743549\pi$
$588$	0	0
$589$	−5.43078	−0.223771
$590$	0	0
$591$	5.18164	0.213144
$592$	0	0
$593$	12.4771	0.512372	0.256186	−	0.966627i	$-0.417534\pi$
$593$	12.4771	0.512372	0.256186	+	0.966627i	$0.417534\pi$
$594$	0	0
$595$	−6.67777	−0.273762
$596$	0	0
$597$	12.1491	0.497232
$598$	0	0
$599$	8.28604	0.338559	0.169279	−	0.985568i	$-0.445856\pi$
$599$	8.28604	0.338559	0.169279	+	0.985568i	$0.445856\pi$
$600$	0	0
$601$	32.4190	1.32240	0.661199	−	0.750211i	$-0.270048\pi$
$601$	32.4190	1.32240	0.661199	+	0.750211i	$0.270048\pi$
$602$	0	0
$603$	3.01091	0.122614
$604$	0	0
$605$	19.7328	0.802252
$606$	0	0
$607$	29.8166	1.21022	0.605109	−	0.796142i	$-0.293129\pi$
$607$	29.8166	1.21022	0.605109	+	0.796142i	$0.293129\pi$
$608$	0	0
$609$	−12.4437	−0.504244
$610$	0	0
$611$	45.9100	1.85732
$612$	0	0
$613$	26.7504	1.08044	0.540220	−	0.841524i	$-0.318341\pi$
$613$	26.7504	1.08044	0.540220	+	0.841524i	$0.318341\pi$
$614$	0	0
$615$	22.0391	0.888703
$616$	0	0
$617$	44.4238	1.78843	0.894217	−	0.447634i	$-0.147733\pi$
$617$	44.4238	1.78843	0.894217	+	0.447634i	$0.147733\pi$
$618$	0	0
$619$	−7.37193	−0.296303	−0.148151	−	0.988965i	$-0.547332\pi$
$619$	−7.37193	−0.296303	−0.148151	+	0.988965i	$0.547332\pi$
$620$	0	0
$621$	−37.0646	−1.48735
$622$	0	0
$623$	1.38501	0.0554892
$624$	0	0
$625$	−13.0841	−0.523362
$626$	0	0
$627$	−0.216210	−0.00863458
$628$	0	0
$629$	3.53789	0.141065
$630$	0	0
$631$	−32.5436	−1.29554	−0.647770	−	0.761836i	$-0.724298\pi$
$631$	−32.5436	−1.29554	−0.647770	+	0.761836i	$0.724298\pi$
$632$	0	0
$633$	−15.1168	−0.600839
$634$	0	0
$635$	16.2592	0.645228
$636$	0	0
$637$	−3.10590	−0.123060
$638$	0	0
$639$	8.83204	0.349390
$640$	0	0
$641$	29.5920	1.16881	0.584407	−	0.811461i	$-0.301327\pi$
$641$	29.5920	1.16881	0.584407	+	0.811461i	$0.301327\pi$
$642$	0	0
$643$	26.5393	1.04661	0.523303	−	0.852146i	$-0.324699\pi$
$643$	26.5393	1.04661	0.523303	+	0.852146i	$0.324699\pi$
$644$	0	0
$645$	−4.41016	−0.173650
$646$	0	0
$647$	31.4942	1.23817	0.619083	−	0.785326i	$-0.287505\pi$
$647$	31.4942	1.23817	0.619083	+	0.785326i	$0.287505\pi$
$648$	0	0
$649$	−1.90126	−0.0746308
$650$	0	0
$651$	−18.6915	−0.732578
$652$	0	0
$653$	20.0351	0.784035	0.392018	−	0.919958i	$-0.371777\pi$
$653$	20.0351	0.784035	0.392018	+	0.919958i	$0.371777\pi$
$654$	0	0
$655$	−15.6403	−0.611116
$656$	0	0
$657$	12.8500	0.501325
$658$	0	0
$659$	−33.0775	−1.28852	−0.644258	−	0.764809i	$-0.722834\pi$
$659$	−33.0775	−1.28852	−0.644258	+	0.764809i	$0.722834\pi$
$660$	0	0
$661$	−22.2500	−0.865426	−0.432713	−	0.901532i	$-0.642444\pi$
$661$	−22.2500	−0.865426	−0.432713	+	0.901532i	$0.642444\pi$
$662$	0	0
$663$	−6.48675	−0.251925
$664$	0	0
$665$	−4.89852	−0.189957
$666$	0	0
$667$	−25.0281	−0.969093
$668$	0	0
$669$	−10.9896	−0.424884
$670$	0	0
$671$	−0.486466	−0.0187798
$672$	0	0
$673$	12.6130	0.486196	0.243098	−	0.970002i	$-0.421836\pi$
$673$	12.6130	0.486196	0.243098	+	0.970002i	$0.421836\pi$
$674$	0	0
$675$	9.66015	0.371819
$676$	0	0
$677$	−28.5013	−1.09540	−0.547698	−	0.836676i	$-0.684496\pi$
$677$	−28.5013	−1.09540	−0.547698	+	0.836676i	$0.684496\pi$
$678$	0	0
$679$	−1.54077	−0.0591292
$680$	0	0
$681$	16.8794	0.646819
$682$	0	0
$683$	10.2874	0.393638	0.196819	−	0.980440i	$-0.436939\pi$
$683$	10.2874	0.393638	0.196819	+	0.980440i	$0.436939\pi$
$684$	0	0
$685$	0.876091	0.0334737
$686$	0	0
$687$	−9.83068	−0.375064
$688$	0	0
$689$	−49.6598	−1.89189
$690$	0	0
$691$	38.7854	1.47547	0.737733	−	0.675092i	$-0.235896\pi$
$691$	38.7854	1.47547	0.737733	+	0.675092i	$0.235896\pi$
$692$	0	0
$693$	0.790666	0.0300349
$694$	0	0
$695$	2.06635	0.0783813
$696$	0	0
$697$	−13.5216	−0.512168
$698$	0	0
$699$	3.81527	0.144307
$700$	0	0
$701$	44.2404	1.67094	0.835468	−	0.549539i	$-0.185197\pi$
$701$	44.2404	1.67094	0.835468	+	0.549539i	$0.185197\pi$
$702$	0	0
$703$	2.59525	0.0978816
$704$	0	0
$705$	24.6636	0.928885
$706$	0	0
$707$	−8.95776	−0.336891
$708$	0	0
$709$	48.9555	1.83856	0.919281	−	0.393602i	$-0.128771\pi$
$709$	48.9555	1.83856	0.919281	+	0.393602i	$0.128771\pi$
$710$	0	0
$711$	23.5569	0.883453
$712$	0	0
$713$	−37.5944	−1.40792
$714$	0	0
$715$	−1.33421	−0.0498966
$716$	0	0
$717$	17.2193	0.643068
$718$	0	0
$719$	2.21557	0.0826268	0.0413134	−	0.999146i	$-0.486846\pi$
$719$	2.21557	0.0826268	0.0413134	+	0.999146i	$0.486846\pi$
$720$	0	0
$721$	0.793043	0.0295345
$722$	0	0
$723$	−17.5898	−0.654173
$724$	0	0
$725$	6.52307	0.242261
$726$	0	0
$727$	−27.5393	−1.02138	−0.510688	−	0.859766i	$-0.670609\pi$
$727$	−27.5393	−1.02138	−0.510688	+	0.859766i	$0.670609\pi$
$728$	0	0
$729$	22.8871	0.847670
$730$	0	0
$731$	2.70576	0.100076
$732$	0	0
$733$	18.6031	0.687121	0.343560	−	0.939131i	$-0.388367\pi$
$733$	18.6031	0.687121	0.343560	+	0.939131i	$0.388367\pi$
$734$	0	0
$735$	−1.66854	−0.0615449
$736$	0	0
$737$	0.357597	0.0131722
$738$	0	0
$739$	15.5002	0.570184	0.285092	−	0.958500i	$-0.407976\pi$
$739$	15.5002	0.570184	0.285092	+	0.958500i	$0.407976\pi$
$740$	0	0
$741$	−4.75840	−0.174804
$742$	0	0
$743$	−34.1494	−1.25282	−0.626410	−	0.779494i	$-0.715476\pi$
$743$	−34.1494	−1.25282	−0.626410	+	0.779494i	$0.715476\pi$
$744$	0	0
$745$	−8.55576	−0.313459
$746$	0	0
$747$	13.2482	0.484727
$748$	0	0
$749$	−37.6025	−1.37396
$750$	0	0
$751$	34.2413	1.24948	0.624741	−	0.780832i	$-0.285204\pi$
$751$	34.2413	1.24948	0.624741	+	0.780832i	$0.285204\pi$
$752$	0	0
$753$	−1.20604	−0.0439506
$754$	0	0
$755$	−30.4317	−1.10752
$756$	0	0
$757$	48.5530	1.76469	0.882344	−	0.470605i	$-0.155964\pi$
$757$	48.5530	1.76469	0.882344	+	0.470605i	$0.155964\pi$
$758$	0	0
$759$	−1.49671	−0.0543270
$760$	0	0
$761$	−20.1270	−0.729604	−0.364802	−	0.931085i	$-0.618863\pi$
$761$	−20.1270	−0.729604	−0.364802	+	0.931085i	$0.618863\pi$
$762$	0	0
$763$	12.2971	0.445184
$764$	0	0
$765$	3.70265	0.133869
$766$	0	0
$767$	−41.8434	−1.51088
$768$	0	0
$769$	9.97987	0.359883	0.179942	−	0.983677i	$-0.442409\pi$
$769$	9.97987	0.359883	0.179942	+	0.983677i	$0.442409\pi$
$770$	0	0
$771$	31.5578	1.13653
$772$	0	0
$773$	−16.0640	−0.577783	−0.288892	−	0.957362i	$-0.593287\pi$
$773$	−16.0640	−0.577783	−0.288892	+	0.957362i	$0.593287\pi$
$774$	0	0
$775$	9.79822	0.351963
$776$	0	0
$777$	8.93225	0.320443
$778$	0	0
$779$	−9.91888	−0.355381
$780$	0	0
$781$	1.04895	0.0375345
$782$	0	0
$783$	20.2931	0.725217
$784$	0	0
$785$	−24.1634	−0.862429
$786$	0	0
$787$	28.6547	1.02143	0.510714	−	0.859751i	$-0.329381\pi$
$787$	28.6547	1.02143	0.510714	+	0.859751i	$0.329381\pi$
$788$	0	0
$789$	−36.8854	−1.31315
$790$	0	0
$791$	−45.3359	−1.61196
$792$	0	0
$793$	−10.7063	−0.380191
$794$	0	0
$795$	−26.6780	−0.946172
$796$	0	0
$797$	51.6224	1.82856	0.914280	−	0.405083i	$-0.132757\pi$
$797$	51.6224	1.82856	0.914280	+	0.405083i	$0.132757\pi$
$798$	0	0
$799$	−15.1318	−0.535325
$800$	0	0
$801$	−0.767951	−0.0271342
$802$	0	0
$803$	1.52615	0.0538567
$804$	0	0
$805$	−33.9099	−1.19517
$806$	0	0
$807$	22.8523	0.804440
$808$	0	0
$809$	13.9246	0.489561	0.244781	−	0.969578i	$-0.421284\pi$
$809$	13.9246	0.489561	0.244781	+	0.969578i	$0.421284\pi$
$810$	0	0
$811$	15.1811	0.533080	0.266540	−	0.963824i	$-0.414120\pi$
$811$	15.1811	0.533080	0.266540	+	0.963824i	$0.414120\pi$
$812$	0	0
$813$	30.8646	1.08247
$814$	0	0
$815$	15.8864	0.556477
$816$	0	0
$817$	1.98483	0.0694403
$818$	0	0
$819$	17.4012	0.608047
$820$	0	0
$821$	3.58153	0.124996	0.0624981	−	0.998045i	$-0.480093\pi$
$821$	3.58153	0.124996	0.0624981	+	0.998045i	$0.480093\pi$
$822$	0	0
$823$	−14.5308	−0.506511	−0.253256	−	0.967399i	$-0.581501\pi$
$823$	−14.5308	−0.506511	−0.253256	+	0.967399i	$0.581501\pi$
$824$	0	0
$825$	0.390086	0.0135811
$826$	0	0
$827$	−40.1211	−1.39515	−0.697573	−	0.716514i	$-0.745737\pi$
$827$	−40.1211	−1.39515	−0.697573	+	0.716514i	$0.745737\pi$
$828$	0	0
$829$	29.3305	1.01869	0.509345	−	0.860563i	$-0.329888\pi$
$829$	29.3305	1.01869	0.509345	+	0.860563i	$0.329888\pi$
$830$	0	0
$831$	−32.8978	−1.14121
$832$	0	0
$833$	1.02369	0.0354689
$834$	0	0
$835$	20.9403	0.724668
$836$	0	0
$837$	30.4820	1.05361
$838$	0	0
$839$	−34.5367	−1.19234	−0.596170	−	0.802859i	$-0.703311\pi$
$839$	−34.5367	−1.19234	−0.596170	+	0.802859i	$0.703311\pi$
$840$	0	0
$841$	−15.2969	−0.527481
$842$	0	0
$843$	−25.5885	−0.881314
$844$	0	0
$845$	−5.97146	−0.205424
$846$	0	0
$847$	−30.5660	−1.05026
$848$	0	0
$849$	−4.10383	−0.140843
$850$	0	0
$851$	17.9655	0.615850
$852$	0	0
$853$	−36.0079	−1.23289	−0.616443	−	0.787399i	$-0.711427\pi$
$853$	−36.0079	−1.23289	−0.616443	+	0.787399i	$0.711427\pi$
$854$	0	0
$855$	2.71610	0.0928887
$856$	0	0
$857$	−31.0898	−1.06201	−0.531004	−	0.847369i	$-0.678185\pi$
$857$	−31.0898	−1.06201	−0.531004	+	0.847369i	$0.678185\pi$
$858$	0	0
$859$	6.13714	0.209396	0.104698	−	0.994504i	$-0.466612\pi$
$859$	6.13714	0.209396	0.104698	+	0.994504i	$0.466612\pi$
$860$	0	0
$861$	−34.1385	−1.16344
$862$	0	0
$863$	−0.556674	−0.0189494	−0.00947471	−	0.999955i	$-0.503016\pi$
$863$	−0.556674	−0.0189494	−0.00947471	+	0.999955i	$0.503016\pi$
$864$	0	0
$865$	10.3979	0.353540
$866$	0	0
$867$	−18.3647	−0.623697
$868$	0	0
$869$	2.79778	0.0949082
$870$	0	0
$871$	7.87009	0.266668
$872$	0	0
$873$	0.854315	0.0289142
$874$	0	0
$875$	33.9150	1.14654
$876$	0	0
$877$	26.8634	0.907112	0.453556	−	0.891228i	$-0.350155\pi$
$877$	26.8634	0.907112	0.453556	+	0.891228i	$0.350155\pi$
$878$	0	0
$879$	17.3149	0.584018
$880$	0	0
$881$	27.0591	0.911643	0.455821	−	0.890071i	$-0.349346\pi$
$881$	27.0591	0.911643	0.455821	+	0.890071i	$0.349346\pi$
$882$	0	0
$883$	14.7927	0.497815	0.248907	−	0.968527i	$-0.419928\pi$
$883$	14.7927	0.497815	0.248907	+	0.968527i	$0.419928\pi$
$884$	0	0
$885$	−22.4789	−0.755621
$886$	0	0
$887$	−7.62226	−0.255930	−0.127965	−	0.991779i	$-0.540845\pi$
$887$	−7.62226	−0.255930	−0.127965	+	0.991779i	$0.540845\pi$
$888$	0	0
$889$	−25.1855	−0.844695
$890$	0	0
$891$	0.362536	0.0121454
$892$	0	0
$893$	−11.1000	−0.371449
$894$	0	0
$895$	25.5934	0.855495
$896$	0	0
$897$	−32.9399	−1.09983
$898$	0	0
$899$	20.5832	0.686487
$900$	0	0
$901$	16.3677	0.545288
$902$	0	0
$903$	6.83132	0.227332
$904$	0	0
$905$	−5.44365	−0.180953
$906$	0	0
$907$	−27.9241	−0.927205	−0.463602	−	0.886043i	$-0.653443\pi$
$907$	−27.9241	−0.927205	−0.463602	+	0.886043i	$0.653443\pi$
$908$	0	0
$909$	4.96684	0.164740
$910$	0	0
$911$	−0.477331	−0.0158147	−0.00790734	−	0.999969i	$-0.502517\pi$
$911$	−0.477331	−0.0158147	−0.00790734	+	0.999969i	$0.502517\pi$
$912$	0	0
$913$	1.57345	0.0520735
$914$	0	0
$915$	−5.75159	−0.190142
$916$	0	0
$917$	24.2267	0.800036
$918$	0	0
$919$	−9.50053	−0.313394	−0.156697	−	0.987647i	$-0.550085\pi$
$919$	−9.50053	−0.313394	−0.156697	+	0.987647i	$0.550085\pi$
$920$	0	0
$921$	−9.08931	−0.299503
$922$	0	0
$923$	23.0857	0.759874
$924$	0	0
$925$	−4.68235	−0.153955
$926$	0	0
$927$	−0.439722	−0.0144423
$928$	0	0
$929$	−16.8176	−0.551767	−0.275884	−	0.961191i	$-0.588970\pi$
$929$	−16.8176	−0.551767	−0.275884	+	0.961191i	$0.588970\pi$
$930$	0	0
$931$	0.750939	0.0246110
$932$	0	0
$933$	32.2307	1.05519
$934$	0	0
$935$	0.439752	0.0143814
$936$	0	0
$937$	48.3645	1.58000	0.790000	−	0.613107i	$-0.210081\pi$
$937$	48.3645	1.58000	0.790000	+	0.613107i	$0.210081\pi$
$938$	0	0
$939$	16.4172	0.535756
$940$	0	0
$941$	19.6763	0.641429	0.320715	−	0.947176i	$-0.396077\pi$
$941$	19.6763	0.641429	0.320715	+	0.947176i	$0.396077\pi$
$942$	0	0
$943$	−68.6632	−2.23598
$944$	0	0
$945$	27.4946	0.894399
$946$	0	0
$947$	−30.3309	−0.985623	−0.492812	−	0.870136i	$-0.664031\pi$
$947$	−30.3309	−0.985623	−0.492812	+	0.870136i	$0.664031\pi$
$948$	0	0
$949$	33.5880	1.09031
$950$	0	0
$951$	36.7351	1.19122
$952$	0	0
$953$	16.9306	0.548435	0.274218	−	0.961668i	$-0.411581\pi$
$953$	16.9306	0.548435	0.274218	+	0.961668i	$0.411581\pi$
$954$	0	0
$955$	2.80266	0.0906918
$956$	0	0
$957$	0.819456	0.0264892
$958$	0	0
$959$	−1.35706	−0.0438218
$960$	0	0
$961$	−0.0822835	−0.00265431
$962$	0	0
$963$	20.8496	0.671868
$964$	0	0
$965$	16.2055	0.521673
$966$	0	0
$967$	−19.4258	−0.624691	−0.312345	−	0.949969i	$-0.601115\pi$
$967$	−19.4258	−0.624691	−0.312345	+	0.949969i	$0.601115\pi$
$968$	0	0
$969$	1.56836	0.0503828
$970$	0	0
$971$	6.15915	0.197656	0.0988282	−	0.995105i	$-0.468491\pi$
$971$	6.15915	0.197656	0.0988282	+	0.995105i	$0.468491\pi$
$972$	0	0
$973$	−3.20078	−0.102612
$974$	0	0
$975$	8.58513	0.274944
$976$	0	0
$977$	44.8892	1.43613	0.718067	−	0.695974i	$-0.245027\pi$
$977$	44.8892	1.43613	0.718067	+	0.695974i	$0.245027\pi$
$978$	0	0
$979$	−0.0912071	−0.00291499
$980$	0	0
$981$	−6.81840	−0.217695
$982$	0	0
$983$	−58.7210	−1.87291	−0.936454	−	0.350789i	$-0.885913\pi$
$983$	−58.7210	−1.87291	−0.936454	+	0.350789i	$0.885913\pi$
$984$	0	0
$985$	−7.73097	−0.246329
$986$	0	0
$987$	−38.2038	−1.21604
$988$	0	0
$989$	13.7399	0.436904
$990$	0	0
$991$	5.12912	0.162932	0.0814660	−	0.996676i	$-0.474040\pi$
$991$	5.12912	0.162932	0.0814660	+	0.996676i	$0.474040\pi$
$992$	0	0
$993$	−13.9174	−0.441656
$994$	0	0
$995$	−18.1264	−0.574646
$996$	0	0
$997$	−27.4395	−0.869018	−0.434509	−	0.900667i	$-0.643078\pi$
$997$	−27.4395	−0.869018	−0.434509	+	0.900667i	$0.643078\pi$
$998$	0	0
$999$	−14.5667	−0.460869

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.10		12
4.3	odd	2		2008.2.a.b.1.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.3	✓	12	4.3	odd	2
4016.2.a.i.1.10		12	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.20604 q^{3} -1.79940 q^{5} +2.78727 q^{7} -1.54547 q^{9} +O(q^{10})\)	\(q+1.20604 q^{3} -1.79940 q^{5} +2.78727 q^{7} -1.54547 q^{9} -0.183550 q^{11} -4.03963 q^{13} -2.17015 q^{15} +1.33145 q^{17} +0.976694 q^{19} +3.36156 q^{21} +6.76113 q^{23} -1.76215 q^{25} -5.48202 q^{27} -3.70176 q^{29} -5.56037 q^{31} -0.221369 q^{33} -5.01541 q^{35} +2.65718 q^{37} -4.87195 q^{39} -10.1556 q^{41} +2.03219 q^{43} +2.78092 q^{45} -11.3649 q^{47} +0.768858 q^{49} +1.60578 q^{51} +12.2932 q^{53} +0.330281 q^{55} +1.17793 q^{57} +10.3582 q^{59} +2.65032 q^{61} -4.30763 q^{63} +7.26891 q^{65} -1.94822 q^{67} +8.15420 q^{69} -5.71480 q^{71} -8.31463 q^{73} -2.12523 q^{75} -0.511603 q^{77} -15.2426 q^{79} -1.97513 q^{81} -8.57230 q^{83} -2.39581 q^{85} -4.46448 q^{87} +0.496905 q^{89} -11.2595 q^{91} -6.70603 q^{93} -1.75746 q^{95} -0.552787 q^{97} +0.283671 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

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