LMFDB - Embedded newform 8036.2.a.p.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$0.729469$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.729469 q^{3} -3.02160 q^{5} -2.46788 q^{9} +O(q^{10})$	$q+0.729469 q^{3} -3.02160 q^{5} -2.46788 q^{9} -2.63548 q^{11} +0.419531 q^{13} -2.20416 q^{15} -4.48814 q^{17} +4.82614 q^{19} +0.672994 q^{23} +4.13008 q^{25} -3.98864 q^{27} -7.40739 q^{29} -3.91525 q^{31} -1.92250 q^{33} -2.78938 q^{37} +0.306035 q^{39} +1.00000 q^{41} +9.85531 q^{43} +7.45694 q^{45} -12.2186 q^{47} -3.27396 q^{51} -12.6009 q^{53} +7.96337 q^{55} +3.52052 q^{57} +8.17437 q^{59} -5.69538 q^{61} -1.26766 q^{65} -4.13105 q^{67} +0.490928 q^{69} +16.7449 q^{71} +11.9924 q^{73} +3.01276 q^{75} -5.02996 q^{79} +4.49403 q^{81} +4.57801 q^{83} +13.5614 q^{85} -5.40346 q^{87} -12.8020 q^{89} -2.85605 q^{93} -14.5827 q^{95} +4.57826 q^{97} +6.50404 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	$ 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * 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
$3$	0.729469	0.421159	0.210580	−	0.977577i	$-0.432465\pi$
$3$	0.729469	0.421159	0.210580	+	0.977577i	$0.432465\pi$
$4$	0	0
$5$	−3.02160	−1.35130	−0.675651	−	0.737222i	$-0.736137\pi$
$5$	−3.02160	−1.35130	−0.675651	+	0.737222i	$0.736137\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.46788	−0.822625
$10$	0	0
$11$	−2.63548	−0.794627	−0.397314	−	0.917683i	$-0.630057\pi$
$11$	−2.63548	−0.794627	−0.397314	+	0.917683i	$0.630057\pi$
$12$	0	0
$13$	0.419531	0.116357	0.0581785	−	0.998306i	$-0.481471\pi$
$13$	0.419531	0.116357	0.0581785	+	0.998306i	$0.481471\pi$
$14$	0	0
$15$	−2.20416	−0.569113
$16$	0	0
$17$	−4.48814	−1.08853	−0.544267	−	0.838912i	$-0.683192\pi$
$17$	−4.48814	−1.08853	−0.544267	+	0.838912i	$0.683192\pi$
$18$	0	0
$19$	4.82614	1.10719	0.553596	−	0.832785i	$-0.313255\pi$
$19$	4.82614	1.10719	0.553596	+	0.832785i	$0.313255\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.672994	0.140329	0.0701644	−	0.997535i	$-0.477648\pi$
$23$	0.672994	0.140329	0.0701644	+	0.997535i	$0.477648\pi$
$24$	0	0
$25$	4.13008	0.826016
$26$	0	0
$27$	−3.98864	−0.767615
$28$	0	0
$29$	−7.40739	−1.37552	−0.687759	−	0.725939i	$-0.741405\pi$
$29$	−7.40739	−1.37552	−0.687759	+	0.725939i	$0.741405\pi$
$30$	0	0
$31$	−3.91525	−0.703200	−0.351600	−	0.936150i	$-0.614362\pi$
$31$	−3.91525	−0.703200	−0.351600	+	0.936150i	$0.614362\pi$
$32$	0	0
$33$	−1.92250	−0.334664
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.78938	−0.458572	−0.229286	−	0.973359i	$-0.573639\pi$
$37$	−2.78938	−0.458572	−0.229286	+	0.973359i	$0.573639\pi$
$38$	0	0
$39$	0.306035	0.0490048
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	9.85531	1.50292	0.751460	−	0.659779i	$-0.229350\pi$
$43$	9.85531	1.50292	0.751460	+	0.659779i	$0.229350\pi$
$44$	0	0
$45$	7.45694	1.11161
$46$	0	0
$47$	−12.2186	−1.78227	−0.891134	−	0.453741i	$-0.850089\pi$
$47$	−12.2186	−1.78227	−0.891134	+	0.453741i	$0.850089\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.27396	−0.458446
$52$	0	0
$53$	−12.6009	−1.73087	−0.865434	−	0.501023i	$-0.832957\pi$
$53$	−12.6009	−1.73087	−0.865434	+	0.501023i	$0.832957\pi$
$54$	0	0
$55$	7.96337	1.07378
$56$	0	0
$57$	3.52052	0.466304
$58$	0	0
$59$	8.17437	1.06421	0.532106	−	0.846678i	$-0.321401\pi$
$59$	8.17437	1.06421	0.532106	+	0.846678i	$0.321401\pi$
$60$	0	0
$61$	−5.69538	−0.729218	−0.364609	−	0.931161i	$-0.618797\pi$
$61$	−5.69538	−0.729218	−0.364609	+	0.931161i	$0.618797\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.26766	−0.157233
$66$	0	0
$67$	−4.13105	−0.504688	−0.252344	−	0.967638i	$-0.581201\pi$
$67$	−4.13105	−0.504688	−0.252344	+	0.967638i	$0.581201\pi$
$68$	0	0
$69$	0.490928	0.0591008
$70$	0	0
$71$	16.7449	1.98726	0.993629	−	0.112696i	$-0.0359487\pi$
$71$	16.7449	1.98726	0.993629	+	0.112696i	$0.0359487\pi$
$72$	0	0
$73$	11.9924	1.40360	0.701800	−	0.712374i	$-0.252380\pi$
$73$	11.9924	1.40360	0.701800	+	0.712374i	$0.252380\pi$
$74$	0	0
$75$	3.01276	0.347884
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.02996	−0.565915	−0.282957	−	0.959132i	$-0.591316\pi$
$79$	−5.02996	−0.565915	−0.282957	+	0.959132i	$0.591316\pi$
$80$	0	0
$81$	4.49403	0.499337
$82$	0	0
$83$	4.57801	0.502502	0.251251	−	0.967922i	$-0.419158\pi$
$83$	4.57801	0.502502	0.251251	+	0.967922i	$0.419158\pi$
$84$	0	0
$85$	13.5614	1.47094
$86$	0	0
$87$	−5.40346	−0.579312
$88$	0	0
$89$	−12.8020	−1.35701	−0.678504	−	0.734596i	$-0.737372\pi$
$89$	−12.8020	−1.35701	−0.678504	+	0.734596i	$0.737372\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2.85605	−0.296159
$94$	0	0
$95$	−14.5827	−1.49615
$96$	0	0
$97$	4.57826	0.464852	0.232426	−	0.972614i	$-0.425334\pi$
$97$	4.57826	0.464852	0.232426	+	0.972614i	$0.425334\pi$
$98$	0	0
$99$	6.50404	0.653680
$100$	0	0
$101$	7.02429	0.698943	0.349472	−	0.936947i	$-0.386361\pi$
$101$	7.02429	0.698943	0.349472	+	0.936947i	$0.386361\pi$
$102$	0	0
$103$	5.94473	0.585752	0.292876	−	0.956150i	$-0.405388\pi$
$103$	5.94473	0.585752	0.292876	+	0.956150i	$0.405388\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.94944	−0.671828	−0.335914	−	0.941893i	$-0.609045\pi$
$107$	−6.94944	−0.671828	−0.335914	+	0.941893i	$0.609045\pi$
$108$	0	0
$109$	−18.6652	−1.78780	−0.893900	−	0.448267i	$-0.852042\pi$
$109$	−18.6652	−1.78780	−0.893900	+	0.448267i	$0.852042\pi$
$110$	0	0
$111$	−2.03477	−0.193132
$112$	0	0
$113$	−15.0598	−1.41671	−0.708355	−	0.705857i	$-0.750562\pi$
$113$	−15.0598	−1.41671	−0.708355	+	0.705857i	$0.750562\pi$
$114$	0	0
$115$	−2.03352	−0.189627
$116$	0	0
$117$	−1.03535	−0.0957181
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.05424	−0.368568
$122$	0	0
$123$	0.729469	0.0657740
$124$	0	0
$125$	2.62855	0.235105
$126$	0	0
$127$	3.79505	0.336756	0.168378	−	0.985722i	$-0.446147\pi$
$127$	3.79505	0.336756	0.168378	+	0.985722i	$0.446147\pi$
$128$	0	0
$129$	7.18914	0.632968
$130$	0	0
$131$	−0.172315	−0.0150552	−0.00752761	−	0.999972i	$-0.502396\pi$
$131$	−0.172315	−0.0150552	−0.00752761	+	0.999972i	$0.502396\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	12.0521	1.03728
$136$	0	0
$137$	1.94598	0.166256	0.0831282	−	0.996539i	$-0.473509\pi$
$137$	1.94598	0.166256	0.0831282	+	0.996539i	$0.473509\pi$
$138$	0	0
$139$	16.5097	1.40033	0.700166	−	0.713980i	$-0.253109\pi$
$139$	16.5097	1.40033	0.700166	+	0.713980i	$0.253109\pi$
$140$	0	0
$141$	−8.91310	−0.750618
$142$	0	0
$143$	−1.10567	−0.0924604
$144$	0	0
$145$	22.3822	1.85874
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.14372	−0.175620	−0.0878101	−	0.996137i	$-0.527987\pi$
$149$	−2.14372	−0.175620	−0.0878101	+	0.996137i	$0.527987\pi$
$150$	0	0
$151$	13.6070	1.10732	0.553659	−	0.832743i	$-0.313231\pi$
$151$	13.6070	1.10732	0.553659	+	0.832743i	$0.313231\pi$
$152$	0	0
$153$	11.0762	0.895455
$154$	0	0
$155$	11.8303	0.950235
$156$	0	0
$157$	16.6320	1.32738	0.663690	−	0.748008i	$-0.268990\pi$
$157$	16.6320	1.32738	0.663690	+	0.748008i	$0.268990\pi$
$158$	0	0
$159$	−9.19197	−0.728971
$160$	0	0
$161$	0	0
$162$	0	0
$163$	2.94527	0.230691	0.115346	−	0.993325i	$-0.463202\pi$
$163$	2.94527	0.230691	0.115346	+	0.993325i	$0.463202\pi$
$164$	0	0
$165$	5.80903	0.452233
$166$	0	0
$167$	16.6880	1.29136	0.645678	−	0.763610i	$-0.276575\pi$
$167$	16.6880	1.29136	0.645678	+	0.763610i	$0.276575\pi$
$168$	0	0
$169$	−12.8240	−0.986461
$170$	0	0
$171$	−11.9103	−0.910804
$172$	0	0
$173$	20.1557	1.53241	0.766203	−	0.642598i	$-0.222144\pi$
$173$	20.1557	1.53241	0.766203	+	0.642598i	$0.222144\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.96295	0.448203
$178$	0	0
$179$	−19.3960	−1.44972	−0.724861	−	0.688895i	$-0.758096\pi$
$179$	−19.3960	−1.44972	−0.724861	+	0.688895i	$0.758096\pi$
$180$	0	0
$181$	1.65214	0.122802	0.0614012	−	0.998113i	$-0.480443\pi$
$181$	1.65214	0.122802	0.0614012	+	0.998113i	$0.480443\pi$
$182$	0	0
$183$	−4.15460	−0.307117
$184$	0	0
$185$	8.42841	0.619669
$186$	0	0
$187$	11.8284	0.864979
$188$	0	0
$189$	0	0
$190$	0	0
$191$	23.4468	1.69655	0.848274	−	0.529557i	$-0.177642\pi$
$191$	23.4468	1.69655	0.848274	+	0.529557i	$0.177642\pi$
$192$	0	0
$193$	16.2357	1.16867	0.584334	−	0.811513i	$-0.301356\pi$
$193$	16.2357	1.16867	0.584334	+	0.811513i	$0.301356\pi$
$194$	0	0
$195$	−0.924715	−0.0662202
$196$	0	0
$197$	−15.0901	−1.07512	−0.537561	−	0.843225i	$-0.680654\pi$
$197$	−15.0901	−1.07512	−0.537561	+	0.843225i	$0.680654\pi$
$198$	0	0
$199$	10.9859	0.778772	0.389386	−	0.921075i	$-0.372687\pi$
$199$	10.9859	0.778772	0.389386	+	0.921075i	$0.372687\pi$
$200$	0	0
$201$	−3.01347	−0.212554
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.02160	−0.211038
$206$	0	0
$207$	−1.66086	−0.115438
$208$	0	0
$209$	−12.7192	−0.879805
$210$	0	0
$211$	23.7238	1.63321	0.816606	−	0.577196i	$-0.195853\pi$
$211$	23.7238	1.63321	0.816606	+	0.577196i	$0.195853\pi$
$212$	0	0
$213$	12.2149	0.836952
$214$	0	0
$215$	−29.7788	−2.03090
$216$	0	0
$217$	0	0
$218$	0	0
$219$	8.74805	0.591139
$220$	0	0
$221$	−1.88291	−0.126658
$222$	0	0
$223$	−21.5235	−1.44132	−0.720659	−	0.693290i	$-0.756161\pi$
$223$	−21.5235	−1.44132	−0.720659	+	0.693290i	$0.756161\pi$
$224$	0	0
$225$	−10.1925	−0.679501
$226$	0	0
$227$	4.57776	0.303837	0.151918	−	0.988393i	$-0.451455\pi$
$227$	4.57776	0.303837	0.151918	+	0.988393i	$0.451455\pi$
$228$	0	0
$229$	−4.92402	−0.325389	−0.162694	−	0.986677i	$-0.552018\pi$
$229$	−4.92402	−0.325389	−0.162694	+	0.986677i	$0.552018\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.6354	−1.15534	−0.577668	−	0.816272i	$-0.696037\pi$
$233$	−17.6354	−1.15534	−0.577668	+	0.816272i	$0.696037\pi$
$234$	0	0
$235$	36.9198	2.40838
$236$	0	0
$237$	−3.66920	−0.238340
$238$	0	0
$239$	0.532642	0.0344538	0.0172269	−	0.999852i	$-0.494516\pi$
$239$	0.532642	0.0344538	0.0172269	+	0.999852i	$0.494516\pi$
$240$	0	0
$241$	10.4604	0.673814	0.336907	−	0.941538i	$-0.390619\pi$
$241$	10.4604	0.673814	0.336907	+	0.941538i	$0.390619\pi$
$242$	0	0
$243$	15.2442	0.977915
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.02471	0.128830
$248$	0	0
$249$	3.33952	0.211633
$250$	0	0
$251$	24.4434	1.54286	0.771428	−	0.636317i	$-0.219543\pi$
$251$	24.4434	1.54286	0.771428	+	0.636317i	$0.219543\pi$
$252$	0	0
$253$	−1.77366	−0.111509
$254$	0	0
$255$	9.89260	0.619499
$256$	0	0
$257$	−13.1025	−0.817309	−0.408655	−	0.912689i	$-0.634002\pi$
$257$	−13.1025	−0.817309	−0.408655	+	0.912689i	$0.634002\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	18.2805	1.13153
$262$	0	0
$263$	4.37037	0.269489	0.134744	−	0.990880i	$-0.456979\pi$
$263$	4.37037	0.269489	0.134744	+	0.990880i	$0.456979\pi$
$264$	0	0
$265$	38.0749	2.33892
$266$	0	0
$267$	−9.33866	−0.571517
$268$	0	0
$269$	21.3721	1.30308	0.651540	−	0.758615i	$-0.274123\pi$
$269$	21.3721	1.30308	0.651540	+	0.758615i	$0.274123\pi$
$270$	0	0
$271$	−16.3061	−0.990526	−0.495263	−	0.868743i	$-0.664928\pi$
$271$	−16.3061	−0.990526	−0.495263	+	0.868743i	$0.664928\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.8847	−0.656375
$276$	0	0
$277$	−19.3362	−1.16180	−0.580901	−	0.813974i	$-0.697300\pi$
$277$	−19.3362	−1.16180	−0.580901	+	0.813974i	$0.697300\pi$
$278$	0	0
$279$	9.66235	0.578470
$280$	0	0
$281$	24.9338	1.48743	0.743714	−	0.668498i	$-0.233062\pi$
$281$	24.9338	1.48743	0.743714	+	0.668498i	$0.233062\pi$
$282$	0	0
$283$	3.46955	0.206243	0.103122	−	0.994669i	$-0.467117\pi$
$283$	3.46955	0.206243	0.103122	+	0.994669i	$0.467117\pi$
$284$	0	0
$285$	−10.6376	−0.630117
$286$	0	0
$287$	0	0
$288$	0	0
$289$	3.14340	0.184906
$290$	0	0
$291$	3.33970	0.195777
$292$	0	0
$293$	−17.5746	−1.02672	−0.513359	−	0.858174i	$-0.671599\pi$
$293$	−17.5746	−1.02672	−0.513359	+	0.858174i	$0.671599\pi$
$294$	0	0
$295$	−24.6997	−1.43807
$296$	0	0
$297$	10.5120	0.609968
$298$	0	0
$299$	0.282342	0.0163282
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.12400	0.294366
$304$	0	0
$305$	17.2092	0.985393
$306$	0	0
$307$	18.7869	1.07222	0.536112	−	0.844147i	$-0.319892\pi$
$307$	18.7869	1.07222	0.536112	+	0.844147i	$0.319892\pi$
$308$	0	0
$309$	4.33650	0.246695
$310$	0	0
$311$	24.4618	1.38710	0.693550	−	0.720408i	$-0.256046\pi$
$311$	24.4618	1.38710	0.693550	+	0.720408i	$0.256046\pi$
$312$	0	0
$313$	18.0320	1.01923	0.509613	−	0.860404i	$-0.329789\pi$
$313$	18.0320	1.01923	0.509613	+	0.860404i	$0.329789\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.26355	0.183299	0.0916496	−	0.995791i	$-0.470786\pi$
$317$	3.26355	0.183299	0.0916496	+	0.995791i	$0.470786\pi$
$318$	0	0
$319$	19.5220	1.09302
$320$	0	0
$321$	−5.06940	−0.282946
$322$	0	0
$323$	−21.6604	−1.20522
$324$	0	0
$325$	1.73270	0.0961127
$326$	0	0
$327$	−13.6157	−0.752948
$328$	0	0
$329$	0	0
$330$	0	0
$331$	10.1594	0.558411	0.279206	−	0.960231i	$-0.409929\pi$
$331$	10.1594	0.558411	0.279206	+	0.960231i	$0.409929\pi$
$332$	0	0
$333$	6.88385	0.377233
$334$	0	0
$335$	12.4824	0.681985
$336$	0	0
$337$	13.3482	0.727124	0.363562	−	0.931570i	$-0.381561\pi$
$337$	13.3482	0.727124	0.363562	+	0.931570i	$0.381561\pi$
$338$	0	0
$339$	−10.9857	−0.596660
$340$	0	0
$341$	10.3186	0.558782
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1.48339	−0.0798630
$346$	0	0
$347$	−6.32015	−0.339283	−0.169642	−	0.985506i	$-0.554261\pi$
$347$	−6.32015	−0.339283	−0.169642	+	0.985506i	$0.554261\pi$
$348$	0	0
$349$	−8.29144	−0.443831	−0.221915	−	0.975066i	$-0.571231\pi$
$349$	−8.29144	−0.443831	−0.221915	+	0.975066i	$0.571231\pi$
$350$	0	0
$351$	−1.67336	−0.0893173
$352$	0	0
$353$	−30.3657	−1.61620	−0.808102	−	0.589043i	$-0.799505\pi$
$353$	−30.3657	−1.61620	−0.808102	+	0.589043i	$0.799505\pi$
$354$	0	0
$355$	−50.5966	−2.68539
$356$	0	0
$357$	0	0
$358$	0	0
$359$	32.3876	1.70935	0.854675	−	0.519163i	$-0.173756\pi$
$359$	32.3876	1.70935	0.854675	+	0.519163i	$0.173756\pi$
$360$	0	0
$361$	4.29162	0.225875
$362$	0	0
$363$	−2.95744	−0.155226
$364$	0	0
$365$	−36.2361	−1.89669
$366$	0	0
$367$	16.0161	0.836031	0.418016	−	0.908440i	$-0.362726\pi$
$367$	16.0161	0.836031	0.418016	+	0.908440i	$0.362726\pi$
$368$	0	0
$369$	−2.46788	−0.128472
$370$	0	0
$371$	0	0
$372$	0	0
$373$	9.50112	0.491949	0.245975	−	0.969276i	$-0.420892\pi$
$373$	9.50112	0.491949	0.245975	+	0.969276i	$0.420892\pi$
$374$	0	0
$375$	1.91745	0.0990166
$376$	0	0
$377$	−3.10763	−0.160051
$378$	0	0
$379$	8.18315	0.420340	0.210170	−	0.977665i	$-0.432598\pi$
$379$	8.18315	0.420340	0.210170	+	0.977665i	$0.432598\pi$
$380$	0	0
$381$	2.76837	0.141828
$382$	0	0
$383$	17.5295	0.895713	0.447857	−	0.894105i	$-0.352188\pi$
$383$	17.5295	0.895713	0.447857	+	0.894105i	$0.352188\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−24.3217	−1.23634
$388$	0	0
$389$	−17.9349	−0.909336	−0.454668	−	0.890661i	$-0.650242\pi$
$389$	−17.9349	−0.909336	−0.454668	+	0.890661i	$0.650242\pi$
$390$	0	0
$391$	−3.02049	−0.152753
$392$	0	0
$393$	−0.125698	−0.00634064
$394$	0	0
$395$	15.1985	0.764722
$396$	0	0
$397$	−25.4109	−1.27534	−0.637668	−	0.770311i	$-0.720101\pi$
$397$	−25.4109	−1.27534	−0.637668	+	0.770311i	$0.720101\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.64011	0.131841	0.0659204	−	0.997825i	$-0.479002\pi$
$401$	2.64011	0.131841	0.0659204	+	0.997825i	$0.479002\pi$
$402$	0	0
$403$	−1.64257	−0.0818222
$404$	0	0
$405$	−13.5792	−0.674755
$406$	0	0
$407$	7.35137	0.364394
$408$	0	0
$409$	−26.1419	−1.29263	−0.646317	−	0.763069i	$-0.723692\pi$
$409$	−26.1419	−1.29263	−0.646317	+	0.763069i	$0.723692\pi$
$410$	0	0
$411$	1.41953	0.0700204
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−13.8329	−0.679032
$416$	0	0
$417$	12.0433	0.589763
$418$	0	0
$419$	2.11337	0.103245	0.0516224	−	0.998667i	$-0.483561\pi$
$419$	2.11337	0.103245	0.0516224	+	0.998667i	$0.483561\pi$
$420$	0	0
$421$	13.5523	0.660500	0.330250	−	0.943894i	$-0.392867\pi$
$421$	13.5523	0.660500	0.330250	+	0.943894i	$0.392867\pi$
$422$	0	0
$423$	30.1540	1.46614
$424$	0	0
$425$	−18.5364	−0.899146
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.806549	−0.0389405
$430$	0	0
$431$	−13.8099	−0.665200	−0.332600	−	0.943068i	$-0.607926\pi$
$431$	−13.8099	−0.665200	−0.332600	+	0.943068i	$0.607926\pi$
$432$	0	0
$433$	2.72356	0.130886	0.0654429	−	0.997856i	$-0.479154\pi$
$433$	2.72356	0.130886	0.0654429	+	0.997856i	$0.479154\pi$
$434$	0	0
$435$	16.3271	0.782825
$436$	0	0
$437$	3.24796	0.155371
$438$	0	0
$439$	12.3395	0.588931	0.294465	−	0.955662i	$-0.404858\pi$
$439$	12.3395	0.588931	0.294465	+	0.955662i	$0.404858\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.66457	−0.269132	−0.134566	−	0.990905i	$-0.542964\pi$
$443$	−5.66457	−0.269132	−0.134566	+	0.990905i	$0.542964\pi$
$444$	0	0
$445$	38.6825	1.83373
$446$	0	0
$447$	−1.56378	−0.0739640
$448$	0	0
$449$	−23.6028	−1.11388	−0.556942	−	0.830552i	$-0.688025\pi$
$449$	−23.6028	−1.11388	−0.556942	+	0.830552i	$0.688025\pi$
$450$	0	0
$451$	−2.63548	−0.124100
$452$	0	0
$453$	9.92585	0.466357
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.8068	0.645854	0.322927	−	0.946424i	$-0.395333\pi$
$457$	13.8068	0.645854	0.322927	+	0.946424i	$0.395333\pi$
$458$	0	0
$459$	17.9016	0.835575
$460$	0	0
$461$	7.33393	0.341575	0.170788	−	0.985308i	$-0.445369\pi$
$461$	7.33393	0.341575	0.170788	+	0.985308i	$0.445369\pi$
$462$	0	0
$463$	−32.3928	−1.50542	−0.752710	−	0.658352i	$-0.771254\pi$
$463$	−32.3928	−1.50542	−0.752710	+	0.658352i	$0.771254\pi$
$464$	0	0
$465$	8.62986	0.400200
$466$	0	0
$467$	−0.471234	−0.0218061	−0.0109030	−	0.999941i	$-0.503471\pi$
$467$	−0.471234	−0.0218061	−0.0109030	+	0.999941i	$0.503471\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	12.1325	0.559038
$472$	0	0
$473$	−25.9735	−1.19426
$474$	0	0
$475$	19.9323	0.914558
$476$	0	0
$477$	31.0975	1.42386
$478$	0	0
$479$	−37.2215	−1.70069	−0.850346	−	0.526224i	$-0.823607\pi$
$479$	−37.2215	−1.70069	−0.850346	+	0.526224i	$0.823607\pi$
$480$	0	0
$481$	−1.17023	−0.0533580
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.8337	−0.628155
$486$	0	0
$487$	−38.3423	−1.73745	−0.868727	−	0.495292i	$-0.835061\pi$
$487$	−38.3423	−1.73745	−0.868727	+	0.495292i	$0.835061\pi$
$488$	0	0
$489$	2.14848	0.0971578
$490$	0	0
$491$	0.220444	0.00994850	0.00497425	−	0.999988i	$-0.498417\pi$
$491$	0.220444	0.00994850	0.00497425	+	0.999988i	$0.498417\pi$
$492$	0	0
$493$	33.2454	1.49730
$494$	0	0
$495$	−19.6526	−0.883319
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−1.31343	−0.0587971	−0.0293985	−	0.999568i	$-0.509359\pi$
$499$	−1.31343	−0.0587971	−0.0293985	+	0.999568i	$0.509359\pi$
$500$	0	0
$501$	12.1734	0.543866
$502$	0	0
$503$	24.5426	1.09430	0.547151	−	0.837034i	$-0.315712\pi$
$503$	24.5426	1.09430	0.547151	+	0.837034i	$0.315712\pi$
$504$	0	0
$505$	−21.2246	−0.944483
$506$	0	0
$507$	−9.35470	−0.415457
$508$	0	0
$509$	−13.1233	−0.581679	−0.290839	−	0.956772i	$-0.593934\pi$
$509$	−13.1233	−0.581679	−0.290839	+	0.956772i	$0.593934\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−19.2498	−0.849897
$514$	0	0
$515$	−17.9626	−0.791528
$516$	0	0
$517$	32.2019	1.41624
$518$	0	0
$519$	14.7029	0.645387
$520$	0	0
$521$	31.6139	1.38503	0.692515	−	0.721403i	$-0.256503\pi$
$521$	31.6139	1.38503	0.692515	+	0.721403i	$0.256503\pi$
$522$	0	0
$523$	−2.75542	−0.120486	−0.0602431	−	0.998184i	$-0.519188\pi$
$523$	−2.75542	−0.120486	−0.0602431	+	0.998184i	$0.519188\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17.5722	0.765457
$528$	0	0
$529$	−22.5471	−0.980308
$530$	0	0
$531$	−20.1733	−0.875448
$532$	0	0
$533$	0.419531	0.0181719
$534$	0	0
$535$	20.9984	0.907842
$536$	0	0
$537$	−14.1487	−0.610564
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−7.02547	−0.302049	−0.151024	−	0.988530i	$-0.548257\pi$
$541$	−7.02547	−0.302049	−0.151024	+	0.988530i	$0.548257\pi$
$542$	0	0
$543$	1.20518	0.0517193
$544$	0	0
$545$	56.3987	2.41586
$546$	0	0
$547$	13.3015	0.568732	0.284366	−	0.958716i	$-0.408217\pi$
$547$	13.3015	0.568732	0.284366	+	0.958716i	$0.408217\pi$
$548$	0	0
$549$	14.0555	0.599873
$550$	0	0
$551$	−35.7491	−1.52296
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6.14826	0.260979
$556$	0	0
$557$	38.9873	1.65194	0.825972	−	0.563711i	$-0.190627\pi$
$557$	38.9873	1.65194	0.825972	+	0.563711i	$0.190627\pi$
$558$	0	0
$559$	4.13461	0.174875
$560$	0	0
$561$	8.62845	0.364294
$562$	0	0
$563$	−5.28994	−0.222945	−0.111472	−	0.993768i	$-0.535557\pi$
$563$	−5.28994	−0.222945	−0.111472	+	0.993768i	$0.535557\pi$
$564$	0	0
$565$	45.5048	1.91440
$566$	0	0
$567$	0	0
$568$	0	0
$569$	39.3456	1.64945	0.824726	−	0.565533i	$-0.191329\pi$
$569$	39.3456	1.64945	0.824726	+	0.565533i	$0.191329\pi$
$570$	0	0
$571$	15.8773	0.664446	0.332223	−	0.943201i	$-0.392201\pi$
$571$	15.8773	0.664446	0.332223	+	0.943201i	$0.392201\pi$
$572$	0	0
$573$	17.1037	0.714517
$574$	0	0
$575$	2.77952	0.115914
$576$	0	0
$577$	8.01458	0.333651	0.166826	−	0.985986i	$-0.446648\pi$
$577$	8.01458	0.333651	0.166826	+	0.985986i	$0.446648\pi$
$578$	0	0
$579$	11.8434	0.492195
$580$	0	0
$581$	0	0
$582$	0	0
$583$	33.2094	1.37539
$584$	0	0
$585$	3.12842	0.129344
$586$	0	0
$587$	6.48213	0.267546	0.133773	−	0.991012i	$-0.457291\pi$
$587$	6.48213	0.267546	0.133773	+	0.991012i	$0.457291\pi$
$588$	0	0
$589$	−18.8955	−0.778577
$590$	0	0
$591$	−11.0077	−0.452798
$592$	0	0
$593$	7.02097	0.288317	0.144158	−	0.989555i	$-0.453953\pi$
$593$	7.02097	0.288317	0.144158	+	0.989555i	$0.453953\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.01390	0.327987
$598$	0	0
$599$	−14.5721	−0.595399	−0.297699	−	0.954660i	$-0.596219\pi$
$599$	−14.5721	−0.595399	−0.297699	+	0.954660i	$0.596219\pi$
$600$	0	0
$601$	−7.94938	−0.324262	−0.162131	−	0.986769i	$-0.551837\pi$
$601$	−7.94938	−0.324262	−0.162131	+	0.986769i	$0.551837\pi$
$602$	0	0
$603$	10.1949	0.415169
$604$	0	0
$605$	12.2503	0.498046
$606$	0	0
$607$	−5.44728	−0.221098	−0.110549	−	0.993871i	$-0.535261\pi$
$607$	−5.44728	−0.221098	−0.110549	+	0.993871i	$0.535261\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−5.12609	−0.207379
$612$	0	0
$613$	30.9589	1.25042	0.625209	−	0.780458i	$-0.285014\pi$
$613$	30.9589	1.25042	0.625209	+	0.780458i	$0.285014\pi$
$614$	0	0
$615$	−2.20416	−0.0888805
$616$	0	0
$617$	−27.7514	−1.11723	−0.558615	−	0.829427i	$-0.688667\pi$
$617$	−27.7514	−1.11723	−0.558615	+	0.829427i	$0.688667\pi$
$618$	0	0
$619$	13.2653	0.533176	0.266588	−	0.963811i	$-0.414104\pi$
$619$	13.2653	0.533176	0.266588	+	0.963811i	$0.414104\pi$
$620$	0	0
$621$	−2.68433	−0.107719
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.5928	−1.14371
$626$	0	0
$627$	−9.27826	−0.370538
$628$	0	0
$629$	12.5191	0.499171
$630$	0	0
$631$	−25.6466	−1.02097	−0.510487	−	0.859885i	$-0.670535\pi$
$631$	−25.6466	−1.02097	−0.510487	+	0.859885i	$0.670535\pi$
$632$	0	0
$633$	17.3057	0.687842
$634$	0	0
$635$	−11.4671	−0.455059
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−41.3244	−1.63477
$640$	0	0
$641$	−1.47574	−0.0582884	−0.0291442	−	0.999575i	$-0.509278\pi$
$641$	−1.47574	−0.0582884	−0.0291442	+	0.999575i	$0.509278\pi$
$642$	0	0
$643$	13.4628	0.530922	0.265461	−	0.964122i	$-0.414476\pi$
$643$	13.4628	0.530922	0.265461	+	0.964122i	$0.414476\pi$
$644$	0	0
$645$	−21.7227	−0.855331
$646$	0	0
$647$	43.7466	1.71986	0.859929	−	0.510414i	$-0.170508\pi$
$647$	43.7466	1.71986	0.859929	+	0.510414i	$0.170508\pi$
$648$	0	0
$649$	−21.5434	−0.845652
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−42.6172	−1.66774	−0.833869	−	0.551962i	$-0.813879\pi$
$653$	−42.6172	−1.66774	−0.833869	+	0.551962i	$0.813879\pi$
$654$	0	0
$655$	0.520667	0.0203441
$656$	0	0
$657$	−29.5956	−1.15464
$658$	0	0
$659$	13.4790	0.525068	0.262534	−	0.964923i	$-0.415442\pi$
$659$	13.4790	0.525068	0.262534	+	0.964923i	$0.415442\pi$
$660$	0	0
$661$	−26.6145	−1.03518	−0.517592	−	0.855628i	$-0.673171\pi$
$661$	−26.6145	−1.03518	−0.517592	+	0.855628i	$0.673171\pi$
$662$	0	0
$663$	−1.37353	−0.0533434
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.98512	−0.193025
$668$	0	0
$669$	−15.7007	−0.607024
$670$	0	0
$671$	15.0100	0.579456
$672$	0	0
$673$	34.8109	1.34186	0.670930	−	0.741521i	$-0.265895\pi$
$673$	34.8109	1.34186	0.670930	+	0.741521i	$0.265895\pi$
$674$	0	0
$675$	−16.4734	−0.634062
$676$	0	0
$677$	14.2444	0.547455	0.273728	−	0.961807i	$-0.411743\pi$
$677$	14.2444	0.547455	0.273728	+	0.961807i	$0.411743\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.33933	0.127964
$682$	0	0
$683$	29.2620	1.11968	0.559839	−	0.828601i	$-0.310863\pi$
$683$	29.2620	1.11968	0.559839	+	0.828601i	$0.310863\pi$
$684$	0	0
$685$	−5.87998	−0.224663
$686$	0	0
$687$	−3.59192	−0.137040
$688$	0	0
$689$	−5.28647	−0.201398
$690$	0	0
$691$	−8.59879	−0.327113	−0.163557	−	0.986534i	$-0.552297\pi$
$691$	−8.59879	−0.327113	−0.163557	+	0.986534i	$0.552297\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−49.8857	−1.89227
$696$	0	0
$697$	−4.48814	−0.170000
$698$	0	0
$699$	−12.8645	−0.486580
$700$	0	0
$701$	−15.1495	−0.572188	−0.286094	−	0.958201i	$-0.592357\pi$
$701$	−15.1495	−0.572188	−0.286094	+	0.958201i	$0.592357\pi$
$702$	0	0
$703$	−13.4620	−0.507727
$704$	0	0
$705$	26.9318	1.01431
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−15.4893	−0.581713	−0.290857	−	0.956767i	$-0.593940\pi$
$709$	−15.4893	−0.581713	−0.290857	+	0.956767i	$0.593940\pi$
$710$	0	0
$711$	12.4133	0.465536
$712$	0	0
$713$	−2.63494	−0.0986792
$714$	0	0
$715$	3.34088	0.124942
$716$	0	0
$717$	0.388546	0.0145105
$718$	0	0
$719$	44.8745	1.67353	0.836767	−	0.547559i	$-0.184443\pi$
$719$	44.8745	1.67353	0.836767	+	0.547559i	$0.184443\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	7.63054	0.283783
$724$	0	0
$725$	−30.5931	−1.13620
$726$	0	0
$727$	19.9602	0.740284	0.370142	−	0.928975i	$-0.379309\pi$
$727$	19.9602	0.740284	0.370142	+	0.928975i	$0.379309\pi$
$728$	0	0
$729$	−2.36194	−0.0874791
$730$	0	0
$731$	−44.2320	−1.63598
$732$	0	0
$733$	−10.0255	−0.370299	−0.185150	−	0.982710i	$-0.559277\pi$
$733$	−10.0255	−0.370299	−0.185150	+	0.982710i	$0.559277\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.8873	0.401039
$738$	0	0
$739$	14.4879	0.532947	0.266474	−	0.963842i	$-0.414141\pi$
$739$	14.4879	0.532947	0.266474	+	0.963842i	$0.414141\pi$
$740$	0	0
$741$	1.47697	0.0542577
$742$	0	0
$743$	−19.2464	−0.706081	−0.353041	−	0.935608i	$-0.614852\pi$
$743$	−19.2464	−0.706081	−0.353041	+	0.935608i	$0.614852\pi$
$744$	0	0
$745$	6.47746	0.237316
$746$	0	0
$747$	−11.2980	−0.413371
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−17.3680	−0.633766	−0.316883	−	0.948465i	$-0.602636\pi$
$751$	−17.3680	−0.633766	−0.316883	+	0.948465i	$0.602636\pi$
$752$	0	0
$753$	17.8307	0.649787
$754$	0	0
$755$	−41.1148	−1.49632
$756$	0	0
$757$	−8.55997	−0.311117	−0.155559	−	0.987827i	$-0.549718\pi$
$757$	−8.55997	−0.311117	−0.155559	+	0.987827i	$0.549718\pi$
$758$	0	0
$759$	−1.29383	−0.0469631
$760$	0	0
$761$	22.8820	0.829474	0.414737	−	0.909941i	$-0.363874\pi$
$761$	22.8820	0.829474	0.414737	+	0.909941i	$0.363874\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−33.4678	−1.21003
$766$	0	0
$767$	3.42940	0.123829
$768$	0	0
$769$	−42.5966	−1.53607	−0.768036	−	0.640407i	$-0.778766\pi$
$769$	−42.5966	−1.53607	−0.768036	+	0.640407i	$0.778766\pi$
$770$	0	0
$771$	−9.55784	−0.344217
$772$	0	0
$773$	−39.0856	−1.40581	−0.702906	−	0.711282i	$-0.748115\pi$
$773$	−39.0856	−1.40581	−0.702906	+	0.711282i	$0.748115\pi$
$774$	0	0
$775$	−16.1703	−0.580854
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.82614	0.172914
$780$	0	0
$781$	−44.1310	−1.57913
$782$	0	0
$783$	29.5454	1.05587
$784$	0	0
$785$	−50.2553	−1.79369
$786$	0	0
$787$	5.56896	0.198512	0.0992561	−	0.995062i	$-0.468354\pi$
$787$	5.56896	0.198512	0.0992561	+	0.995062i	$0.468354\pi$
$788$	0	0
$789$	3.18805	0.113498
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.38939	−0.0848496
$794$	0	0
$795$	27.7745	0.985059
$796$	0	0
$797$	−22.0942	−0.782616	−0.391308	−	0.920260i	$-0.627977\pi$
$797$	−22.0942	−0.782616	−0.391308	+	0.920260i	$0.627977\pi$
$798$	0	0
$799$	54.8388	1.94006
$800$	0	0
$801$	31.5937	1.11631
$802$	0	0
$803$	−31.6056	−1.11534
$804$	0	0
$805$	0	0
$806$	0	0
$807$	15.5903	0.548804
$808$	0	0
$809$	−15.2610	−0.536548	−0.268274	−	0.963343i	$-0.586453\pi$
$809$	−15.2610	−0.536548	−0.268274	+	0.963343i	$0.586453\pi$
$810$	0	0
$811$	−25.3582	−0.890446	−0.445223	−	0.895420i	$-0.646876\pi$
$811$	−25.3582	−0.890446	−0.445223	+	0.895420i	$0.646876\pi$
$812$	0	0
$813$	−11.8948	−0.417169
$814$	0	0
$815$	−8.89944	−0.311734
$816$	0	0
$817$	47.5631	1.66402
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.2829	0.603178	0.301589	−	0.953438i	$-0.402483\pi$
$821$	17.2829	0.603178	0.301589	+	0.953438i	$0.402483\pi$
$822$	0	0
$823$	−40.8641	−1.42443	−0.712216	−	0.701960i	$-0.752308\pi$
$823$	−40.8641	−1.42443	−0.712216	+	0.701960i	$0.752308\pi$
$824$	0	0
$825$	−7.94008	−0.276438
$826$	0	0
$827$	13.5665	0.471755	0.235877	−	0.971783i	$-0.424204\pi$
$827$	13.5665	0.471755	0.235877	+	0.971783i	$0.424204\pi$
$828$	0	0
$829$	−10.0661	−0.349609	−0.174805	−	0.984603i	$-0.555929\pi$
$829$	−10.0661	−0.349609	−0.174805	+	0.984603i	$0.555929\pi$
$830$	0	0
$831$	−14.1052	−0.489303
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−50.4245	−1.74501
$836$	0	0
$837$	15.6165	0.539787
$838$	0	0
$839$	−44.4731	−1.53538	−0.767691	−	0.640820i	$-0.778595\pi$
$839$	−44.4731	−1.53538	−0.767691	+	0.640820i	$0.778595\pi$
$840$	0	0
$841$	25.8694	0.892048
$842$	0	0
$843$	18.1885	0.626444
$844$	0	0
$845$	38.7490	1.33301
$846$	0	0
$847$	0	0
$848$	0	0
$849$	2.53093	0.0868612
$850$	0	0
$851$	−1.87724	−0.0643509
$852$	0	0
$853$	41.7118	1.42818	0.714092	−	0.700051i	$-0.246840\pi$
$853$	41.7118	1.42818	0.714092	+	0.700051i	$0.246840\pi$
$854$	0	0
$855$	35.9882	1.23077
$856$	0	0
$857$	32.3777	1.10600	0.553000	−	0.833181i	$-0.313483\pi$
$857$	32.3777	1.10600	0.553000	+	0.833181i	$0.313483\pi$
$858$	0	0
$859$	10.3027	0.351525	0.175762	−	0.984433i	$-0.443761\pi$
$859$	10.3027	0.351525	0.175762	+	0.984433i	$0.443761\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−11.4435	−0.389543	−0.194771	−	0.980849i	$-0.562396\pi$
$863$	−11.4435	−0.389543	−0.194771	+	0.980849i	$0.562396\pi$
$864$	0	0
$865$	−60.9024	−2.07074
$866$	0	0
$867$	2.29301	0.0778748
$868$	0	0
$869$	13.2564	0.449691
$870$	0	0
$871$	−1.73310	−0.0587239
$872$	0	0
$873$	−11.2986	−0.382399
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−41.8720	−1.41392	−0.706959	−	0.707255i	$-0.749933\pi$
$877$	−41.8720	−1.41392	−0.706959	+	0.707255i	$0.749933\pi$
$878$	0	0
$879$	−12.8201	−0.432411
$880$	0	0
$881$	−10.7494	−0.362158	−0.181079	−	0.983469i	$-0.557959\pi$
$881$	−10.7494	−0.362158	−0.181079	+	0.983469i	$0.557959\pi$
$882$	0	0
$883$	16.8468	0.566940	0.283470	−	0.958981i	$-0.408514\pi$
$883$	16.8468	0.566940	0.283470	+	0.958981i	$0.408514\pi$
$884$	0	0
$885$	−18.0177	−0.605657
$886$	0	0
$887$	−29.6898	−0.996887	−0.498444	−	0.866922i	$-0.666095\pi$
$887$	−29.6898	−0.996887	−0.498444	+	0.866922i	$0.666095\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−11.8439	−0.396787
$892$	0	0
$893$	−58.9687	−1.97331
$894$	0	0
$895$	58.6069	1.95901
$896$	0	0
$897$	0.205959	0.00687679
$898$	0	0
$899$	29.0018	0.967263
$900$	0	0
$901$	56.5546	1.88411
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.99210	−0.165943
$906$	0	0
$907$	−18.9752	−0.630060	−0.315030	−	0.949082i	$-0.602015\pi$
$907$	−18.9752	−0.630060	−0.315030	+	0.949082i	$0.602015\pi$
$908$	0	0
$909$	−17.3351	−0.574968
$910$	0	0
$911$	−29.1631	−0.966216	−0.483108	−	0.875561i	$-0.660492\pi$
$911$	−29.1631	−0.966216	−0.483108	+	0.875561i	$0.660492\pi$
$912$	0	0
$913$	−12.0653	−0.399302
$914$	0	0
$915$	12.5535	0.415007
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−14.2264	−0.469286	−0.234643	−	0.972082i	$-0.575392\pi$
$919$	−14.2264	−0.469286	−0.234643	+	0.972082i	$0.575392\pi$
$920$	0	0
$921$	13.7045	0.451577
$922$	0	0
$923$	7.02502	0.231231
$924$	0	0
$925$	−11.5204	−0.378788
$926$	0	0
$927$	−14.6709	−0.481854
$928$	0	0
$929$	9.29313	0.304898	0.152449	−	0.988311i	$-0.451284\pi$
$929$	9.29313	0.304898	0.152449	+	0.988311i	$0.451284\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	17.8441	0.584190
$934$	0	0
$935$	−35.7407	−1.16885
$936$	0	0
$937$	40.4919	1.32281	0.661406	−	0.750028i	$-0.269960\pi$
$937$	40.4919	1.32281	0.661406	+	0.750028i	$0.269960\pi$
$938$	0	0
$939$	13.1537	0.429256
$940$	0	0
$941$	40.7076	1.32703	0.663515	−	0.748163i	$-0.269064\pi$
$941$	40.7076	1.32703	0.663515	+	0.748163i	$0.269064\pi$
$942$	0	0
$943$	0.672994	0.0219157
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.0839	0.880107	0.440054	−	0.897972i	$-0.354959\pi$
$947$	27.0839	0.880107	0.440054	+	0.897972i	$0.354959\pi$
$948$	0	0
$949$	5.03117	0.163319
$950$	0	0
$951$	2.38066	0.0771981
$952$	0	0
$953$	54.2939	1.75875	0.879376	−	0.476128i	$-0.157960\pi$
$953$	54.2939	1.75875	0.879376	+	0.476128i	$0.157960\pi$
$954$	0	0
$955$	−70.8468	−2.29255
$956$	0	0
$957$	14.2407	0.460337
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.6708	−0.505510
$962$	0	0
$963$	17.1503	0.552662
$964$	0	0
$965$	−49.0577	−1.57922
$966$	0	0
$967$	−11.2843	−0.362878	−0.181439	−	0.983402i	$-0.558075\pi$
$967$	−11.2843	−0.362878	−0.181439	+	0.983402i	$0.558075\pi$
$968$	0	0
$969$	−15.8006	−0.507588
$970$	0	0
$971$	−54.7201	−1.75605	−0.878025	−	0.478614i	$-0.841139\pi$
$971$	−54.7201	−1.75605	−0.878025	+	0.478614i	$0.841139\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	1.26395	0.0404787
$976$	0	0
$977$	−45.2859	−1.44882	−0.724412	−	0.689368i	$-0.757889\pi$
$977$	−45.2859	−1.44882	−0.724412	+	0.689368i	$0.757889\pi$
$978$	0	0
$979$	33.7394	1.07832
$980$	0	0
$981$	46.0633	1.47069
$982$	0	0
$983$	24.0163	0.766000	0.383000	−	0.923748i	$-0.374891\pi$
$983$	24.0163	0.766000	0.383000	+	0.923748i	$0.374891\pi$
$984$	0	0
$985$	45.5962	1.45282
$986$	0	0
$987$	0	0
$988$	0	0
$989$	6.63256	0.210903
$990$	0	0
$991$	19.1437	0.608119	0.304060	−	0.952653i	$-0.401658\pi$
$991$	19.1437	0.608119	0.304060	+	0.952653i	$0.401658\pi$
$992$	0	0
$993$	7.41097	0.235180
$994$	0	0
$995$	−33.1951	−1.05236
$996$	0	0
$997$	1.60423	0.0508066	0.0254033	−	0.999677i	$-0.491913\pi$
$997$	1.60423	0.0508066	0.0254033	+	0.999677i	$0.491913\pi$
$998$	0	0
$999$	11.1259	0.352007

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.p.1.6	yes	10
7.6	odd	2		8036.2.a.o.1.5	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.5	✓	10	7.6	odd	2
8036.2.a.p.1.6	yes	10	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+0.729469 q^{3} -3.02160 q^{5} -2.46788 q^{9} +O(q^{10})\)	\(q+0.729469 q^{3} -3.02160 q^{5} -2.46788 q^{9} -2.63548 q^{11} +0.419531 q^{13} -2.20416 q^{15} -4.48814 q^{17} +4.82614 q^{19} +0.672994 q^{23} +4.13008 q^{25} -3.98864 q^{27} -7.40739 q^{29} -3.91525 q^{31} -1.92250 q^{33} -2.78938 q^{37} +0.306035 q^{39} +1.00000 q^{41} +9.85531 q^{43} +7.45694 q^{45} -12.2186 q^{47} -3.27396 q^{51} -12.6009 q^{53} +7.96337 q^{55} +3.52052 q^{57} +8.17437 q^{59} -5.69538 q^{61} -1.26766 q^{65} -4.13105 q^{67} +0.490928 q^{69} +16.7449 q^{71} +11.9924 q^{73} +3.01276 q^{75} -5.02996 q^{79} +4.49403 q^{81} +4.57801 q^{83} +13.5614 q^{85} -5.40346 q^{87} -12.8020 q^{89} -2.85605 q^{93} -14.5827 q^{95} +4.57826 q^{97} +6.50404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9	\( 10 q + 2 q^{3} + 4 q^{5} + 10 q^{9} + 6 q^{13} - 4 q^{15} + 12 q^{17} + 8 q^{19} + 4 q^{23} + 16 q^{25} + 8 q^{27} + 2 q^{29} + 8 q^{31} - 6 q^{33} - 2 q^{37} - 2 q^{39} + 10 q^{41} - 2 q^{43} + 44 q^{45} - 14 q^{47} + 14 q^{51} + 8 q^{53} + 8 q^{55} - 10 q^{57} + 24 q^{59} + 14 q^{61} + 2 q^{65} - 8 q^{67} + 16 q^{69} + 10 q^{71} + 44 q^{73} - 50 q^{75} + 10 q^{79} - 14 q^{81} + 20 q^{83} + 8 q^{85} + 20 q^{87} + 6 q^{89} + 8 q^{93} + 4 q^{95} + 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q + 2 * q^3 + 4 * q^5 + 10 * q^9 + 6 * q^13 - 4 * q^15 + 12 * q^17 + 8 * q^19 + 4 * q^23 + 16 * q^25 + 8 * q^27 + 2 * q^29 + 8 * q^31 - 6 * q^33 - 2 * q^37 - 2 * q^39 + 10 * q^41 - 2 * q^43 + 44 * q^45 - 14 * q^47 + 14 * q^51 + 8 * q^53 + 8 * q^55 - 10 * q^57 + 24 * q^59 + 14 * q^61 + 2 * q^65 - 8 * q^67 + 16 * q^69 + 10 * q^71 + 44 * q^73 - 50 * q^75 + 10 * q^79 - 14 * q^81 + 20 * q^83 + 8 * q^85 + 20 * q^87 + 6 * q^89 + 8 * q^93 + 4 * q^95 + 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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