LMFDB - Embedded newform 6036.2.a.i.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.62125 q^{5} +5.08723 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.62125 q^{5} +5.08723 q^{7} +1.00000 q^{9} -4.78702 q^{11} +3.75038 q^{13} -1.62125 q^{15} +8.05516 q^{17} +2.54548 q^{19} -5.08723 q^{21} -6.47688 q^{23} -2.37155 q^{25} -1.00000 q^{27} +1.84482 q^{29} +11.0732 q^{31} +4.78702 q^{33} +8.24767 q^{35} -2.26721 q^{37} -3.75038 q^{39} +3.95793 q^{41} -9.87920 q^{43} +1.62125 q^{45} -1.11380 q^{47} +18.8800 q^{49} -8.05516 q^{51} +5.65830 q^{53} -7.76096 q^{55} -2.54548 q^{57} -9.39418 q^{59} +4.19568 q^{61} +5.08723 q^{63} +6.08030 q^{65} +0.706924 q^{67} +6.47688 q^{69} +14.2100 q^{71} +16.1734 q^{73} +2.37155 q^{75} -24.3527 q^{77} +2.99763 q^{79} +1.00000 q^{81} -11.2872 q^{83} +13.0594 q^{85} -1.84482 q^{87} -5.82489 q^{89} +19.0790 q^{91} -11.0732 q^{93} +4.12685 q^{95} -8.69438 q^{97} -4.78702 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.62125	0.725045	0.362522	−	0.931975i	$-0.381916\pi$
$5$	1.62125	0.725045	0.362522	+	0.931975i	$0.381916\pi$
$6$	0	0
$7$	5.08723	1.92279	0.961397	−	0.275165i	$-0.0887326\pi$
$7$	5.08723	1.92279	0.961397	+	0.275165i	$0.0887326\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.78702	−1.44334	−0.721671	−	0.692236i	$-0.756626\pi$
$11$	−4.78702	−1.44334	−0.721671	+	0.692236i	$0.756626\pi$
$12$	0	0
$13$	3.75038	1.04017	0.520084	−	0.854115i	$-0.325901\pi$
$13$	3.75038	1.04017	0.520084	+	0.854115i	$0.325901\pi$
$14$	0	0
$15$	−1.62125	−0.418605
$16$	0	0
$17$	8.05516	1.95366	0.976832	−	0.214009i	$-0.0686520\pi$
$17$	8.05516	1.95366	0.976832	+	0.214009i	$0.0686520\pi$
$18$	0	0
$19$	2.54548	0.583972	0.291986	−	0.956423i	$-0.405684\pi$
$19$	2.54548	0.583972	0.291986	+	0.956423i	$0.405684\pi$
$20$	0	0
$21$	−5.08723	−1.11013
$22$	0	0
$23$	−6.47688	−1.35052	−0.675261	−	0.737579i	$-0.735969\pi$
$23$	−6.47688	−1.35052	−0.675261	+	0.737579i	$0.735969\pi$
$24$	0	0
$25$	−2.37155	−0.474310
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.84482	0.342574	0.171287	−	0.985221i	$-0.445207\pi$
$29$	1.84482	0.342574	0.171287	+	0.985221i	$0.445207\pi$
$30$	0	0
$31$	11.0732	1.98881	0.994403	−	0.105657i	$-0.0336945\pi$
$31$	11.0732	1.98881	0.994403	+	0.105657i	$0.0336945\pi$
$32$	0	0
$33$	4.78702	0.833314
$34$	0	0
$35$	8.24767	1.39411
$36$	0	0
$37$	−2.26721	−0.372727	−0.186364	−	0.982481i	$-0.559670\pi$
$37$	−2.26721	−0.372727	−0.186364	+	0.982481i	$0.559670\pi$
$38$	0	0
$39$	−3.75038	−0.600541
$40$	0	0
$41$	3.95793	0.618126	0.309063	−	0.951042i	$-0.399985\pi$
$41$	3.95793	0.618126	0.309063	+	0.951042i	$0.399985\pi$
$42$	0	0
$43$	−9.87920	−1.50656	−0.753282	−	0.657698i	$-0.771530\pi$
$43$	−9.87920	−1.50656	−0.753282	+	0.657698i	$0.771530\pi$
$44$	0	0
$45$	1.62125	0.241682
$46$	0	0
$47$	−1.11380	−0.162464	−0.0812320	−	0.996695i	$-0.525885\pi$
$47$	−1.11380	−0.162464	−0.0812320	+	0.996695i	$0.525885\pi$
$48$	0	0
$49$	18.8800	2.69714
$50$	0	0
$51$	−8.05516	−1.12795
$52$	0	0
$53$	5.65830	0.777228	0.388614	−	0.921401i	$-0.372954\pi$
$53$	5.65830	0.777228	0.388614	+	0.921401i	$0.372954\pi$
$54$	0	0
$55$	−7.76096	−1.04649
$56$	0	0
$57$	−2.54548	−0.337156
$58$	0	0
$59$	−9.39418	−1.22302	−0.611509	−	0.791237i	$-0.709437\pi$
$59$	−9.39418	−1.22302	−0.611509	+	0.791237i	$0.709437\pi$
$60$	0	0
$61$	4.19568	0.537201	0.268601	−	0.963252i	$-0.413439\pi$
$61$	4.19568	0.537201	0.268601	+	0.963252i	$0.413439\pi$
$62$	0	0
$63$	5.08723	0.640931
$64$	0	0
$65$	6.08030	0.754168
$66$	0	0
$67$	0.706924	0.0863645	0.0431822	−	0.999067i	$-0.486250\pi$
$67$	0.706924	0.0863645	0.0431822	+	0.999067i	$0.486250\pi$
$68$	0	0
$69$	6.47688	0.779724
$70$	0	0
$71$	14.2100	1.68642	0.843210	−	0.537585i	$-0.180663\pi$
$71$	14.2100	1.68642	0.843210	+	0.537585i	$0.180663\pi$
$72$	0	0
$73$	16.1734	1.89295	0.946476	−	0.322775i	$-0.104616\pi$
$73$	16.1734	1.89295	0.946476	+	0.322775i	$0.104616\pi$
$74$	0	0
$75$	2.37155	0.273843
$76$	0	0
$77$	−24.3527	−2.77525
$78$	0	0
$79$	2.99763	0.337260	0.168630	−	0.985679i	$-0.446066\pi$
$79$	2.99763	0.337260	0.168630	+	0.985679i	$0.446066\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.2872	−1.23893	−0.619465	−	0.785024i	$-0.712650\pi$
$83$	−11.2872	−1.23893	−0.619465	+	0.785024i	$0.712650\pi$
$84$	0	0
$85$	13.0594	1.41649
$86$	0	0
$87$	−1.84482	−0.197785
$88$	0	0
$89$	−5.82489	−0.617437	−0.308719	−	0.951153i	$-0.599900\pi$
$89$	−5.82489	−0.617437	−0.308719	+	0.951153i	$0.599900\pi$
$90$	0	0
$91$	19.0790	2.00003
$92$	0	0
$93$	−11.0732	−1.14824
$94$	0	0
$95$	4.12685	0.423406
$96$	0	0
$97$	−8.69438	−0.882781	−0.441391	−	0.897315i	$-0.645515\pi$
$97$	−8.69438	−0.882781	−0.441391	+	0.897315i	$0.645515\pi$
$98$	0	0
$99$	−4.78702	−0.481114
$100$	0	0
$101$	−7.46387	−0.742683	−0.371342	−	0.928496i	$-0.621102\pi$
$101$	−7.46387	−0.742683	−0.371342	+	0.928496i	$0.621102\pi$
$102$	0	0
$103$	−4.05463	−0.399515	−0.199757	−	0.979845i	$-0.564015\pi$
$103$	−4.05463	−0.399515	−0.199757	+	0.979845i	$0.564015\pi$
$104$	0	0
$105$	−8.24767	−0.804891
$106$	0	0
$107$	8.22324	0.794971	0.397485	−	0.917608i	$-0.369883\pi$
$107$	8.22324	0.794971	0.397485	+	0.917608i	$0.369883\pi$
$108$	0	0
$109$	−4.91908	−0.471162	−0.235581	−	0.971855i	$-0.575699\pi$
$109$	−4.91908	−0.471162	−0.235581	+	0.971855i	$0.575699\pi$
$110$	0	0
$111$	2.26721	0.215194
$112$	0	0
$113$	−10.3271	−0.971495	−0.485747	−	0.874099i	$-0.661453\pi$
$113$	−10.3271	−0.971495	−0.485747	+	0.874099i	$0.661453\pi$
$114$	0	0
$115$	−10.5006	−0.979189
$116$	0	0
$117$	3.75038	0.346723
$118$	0	0
$119$	40.9785	3.75649
$120$	0	0
$121$	11.9156	1.08324
$122$	0	0
$123$	−3.95793	−0.356875
$124$	0	0
$125$	−11.9511	−1.06894
$126$	0	0
$127$	12.1345	1.07676	0.538379	−	0.842703i	$-0.319037\pi$
$127$	12.1345	1.07676	0.538379	+	0.842703i	$0.319037\pi$
$128$	0	0
$129$	9.87920	0.869815
$130$	0	0
$131$	−8.79470	−0.768396	−0.384198	−	0.923251i	$-0.625522\pi$
$131$	−8.79470	−0.768396	−0.384198	+	0.923251i	$0.625522\pi$
$132$	0	0
$133$	12.9494	1.12286
$134$	0	0
$135$	−1.62125	−0.139535
$136$	0	0
$137$	3.83330	0.327501	0.163750	−	0.986502i	$-0.447641\pi$
$137$	3.83330	0.327501	0.163750	+	0.986502i	$0.447641\pi$
$138$	0	0
$139$	1.81069	0.153580	0.0767902	−	0.997047i	$-0.475533\pi$
$139$	1.81069	0.153580	0.0767902	+	0.997047i	$0.475533\pi$
$140$	0	0
$141$	1.11380	0.0937986
$142$	0	0
$143$	−17.9531	−1.50132
$144$	0	0
$145$	2.99091	0.248382
$146$	0	0
$147$	−18.8800	−1.55719
$148$	0	0
$149$	−19.9656	−1.63564	−0.817822	−	0.575472i	$-0.804819\pi$
$149$	−19.9656	−1.63564	−0.817822	+	0.575472i	$0.804819\pi$
$150$	0	0
$151$	−12.5577	−1.02193	−0.510966	−	0.859601i	$-0.670712\pi$
$151$	−12.5577	−1.02193	−0.510966	+	0.859601i	$0.670712\pi$
$152$	0	0
$153$	8.05516	0.651221
$154$	0	0
$155$	17.9524	1.44197
$156$	0	0
$157$	24.4197	1.94890	0.974452	−	0.224597i	$-0.0721066\pi$
$157$	24.4197	1.94890	0.974452	+	0.224597i	$0.0721066\pi$
$158$	0	0
$159$	−5.65830	−0.448733
$160$	0	0
$161$	−32.9494	−2.59678
$162$	0	0
$163$	1.07379	0.0841057	0.0420528	−	0.999115i	$-0.486610\pi$
$163$	1.07379	0.0841057	0.0420528	+	0.999115i	$0.486610\pi$
$164$	0	0
$165$	7.76096	0.604190
$166$	0	0
$167$	−7.25268	−0.561230	−0.280615	−	0.959820i	$-0.590538\pi$
$167$	−7.25268	−0.561230	−0.280615	+	0.959820i	$0.590538\pi$
$168$	0	0
$169$	1.06533	0.0819485
$170$	0	0
$171$	2.54548	0.194657
$172$	0	0
$173$	10.7486	0.817197	0.408599	−	0.912714i	$-0.366018\pi$
$173$	10.7486	0.817197	0.408599	+	0.912714i	$0.366018\pi$
$174$	0	0
$175$	−12.0646	−0.912001
$176$	0	0
$177$	9.39418	0.706110
$178$	0	0
$179$	−5.28959	−0.395363	−0.197681	−	0.980266i	$-0.563341\pi$
$179$	−5.28959	−0.395363	−0.197681	+	0.980266i	$0.563341\pi$
$180$	0	0
$181$	−7.10431	−0.528059	−0.264030	−	0.964515i	$-0.585052\pi$
$181$	−7.10431	−0.528059	−0.264030	+	0.964515i	$0.585052\pi$
$182$	0	0
$183$	−4.19568	−0.310153
$184$	0	0
$185$	−3.67571	−0.270244
$186$	0	0
$187$	−38.5603	−2.81981
$188$	0	0
$189$	−5.08723	−0.370042
$190$	0	0
$191$	16.2208	1.17369	0.586846	−	0.809698i	$-0.300369\pi$
$191$	16.2208	1.17369	0.586846	+	0.809698i	$0.300369\pi$
$192$	0	0
$193$	−15.9161	−1.14567	−0.572834	−	0.819671i	$-0.694156\pi$
$193$	−15.9161	−1.14567	−0.572834	+	0.819671i	$0.694156\pi$
$194$	0	0
$195$	−6.08030	−0.435419
$196$	0	0
$197$	6.48555	0.462077	0.231038	−	0.972945i	$-0.425788\pi$
$197$	6.48555	0.462077	0.231038	+	0.972945i	$0.425788\pi$
$198$	0	0
$199$	14.1090	1.00016	0.500081	−	0.865979i	$-0.333304\pi$
$199$	14.1090	1.00016	0.500081	+	0.865979i	$0.333304\pi$
$200$	0	0
$201$	−0.706924	−0.0498625
$202$	0	0
$203$	9.38503	0.658700
$204$	0	0
$205$	6.41680	0.448169
$206$	0	0
$207$	−6.47688	−0.450174
$208$	0	0
$209$	−12.1853	−0.842872
$210$	0	0
$211$	3.02938	0.208551	0.104276	−	0.994548i	$-0.466748\pi$
$211$	3.02938	0.208551	0.104276	+	0.994548i	$0.466748\pi$
$212$	0	0
$213$	−14.2100	−0.973655
$214$	0	0
$215$	−16.0166	−1.09233
$216$	0	0
$217$	56.3320	3.82406
$218$	0	0
$219$	−16.1734	−1.09290
$220$	0	0
$221$	30.2099	2.03214
$222$	0	0
$223$	−18.1230	−1.21360	−0.606801	−	0.794853i	$-0.707548\pi$
$223$	−18.1230	−1.21360	−0.606801	+	0.794853i	$0.707548\pi$
$224$	0	0
$225$	−2.37155	−0.158103
$226$	0	0
$227$	7.36207	0.488638	0.244319	−	0.969695i	$-0.421436\pi$
$227$	7.36207	0.488638	0.244319	+	0.969695i	$0.421436\pi$
$228$	0	0
$229$	−28.8730	−1.90798	−0.953990	−	0.299839i	$-0.903067\pi$
$229$	−28.8730	−1.90798	−0.953990	+	0.299839i	$0.903067\pi$
$230$	0	0
$231$	24.3527	1.60229
$232$	0	0
$233$	22.1642	1.45202	0.726011	−	0.687683i	$-0.241372\pi$
$233$	22.1642	1.45202	0.726011	+	0.687683i	$0.241372\pi$
$234$	0	0
$235$	−1.80574	−0.117794
$236$	0	0
$237$	−2.99763	−0.194717
$238$	0	0
$239$	−8.28176	−0.535702	−0.267851	−	0.963460i	$-0.586314\pi$
$239$	−8.28176	−0.535702	−0.267851	+	0.963460i	$0.586314\pi$
$240$	0	0
$241$	1.60328	0.103277	0.0516383	−	0.998666i	$-0.483556\pi$
$241$	1.60328	0.103277	0.0516383	+	0.998666i	$0.483556\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	30.6091	1.95554
$246$	0	0
$247$	9.54649	0.607429
$248$	0	0
$249$	11.2872	0.715296
$250$	0	0
$251$	28.7613	1.81540	0.907698	−	0.419623i	$-0.137838\pi$
$251$	28.7613	1.81540	0.907698	+	0.419623i	$0.137838\pi$
$252$	0	0
$253$	31.0050	1.94927
$254$	0	0
$255$	−13.0594	−0.817813
$256$	0	0
$257$	14.8466	0.926104	0.463052	−	0.886331i	$-0.346754\pi$
$257$	14.8466	0.926104	0.463052	+	0.886331i	$0.346754\pi$
$258$	0	0
$259$	−11.5338	−0.716677
$260$	0	0
$261$	1.84482	0.114191
$262$	0	0
$263$	−12.6492	−0.779982	−0.389991	−	0.920819i	$-0.627522\pi$
$263$	−12.6492	−0.779982	−0.389991	+	0.920819i	$0.627522\pi$
$264$	0	0
$265$	9.17352	0.563525
$266$	0	0
$267$	5.82489	0.356478
$268$	0	0
$269$	12.5501	0.765194	0.382597	−	0.923915i	$-0.375030\pi$
$269$	12.5501	0.765194	0.382597	+	0.923915i	$0.375030\pi$
$270$	0	0
$271$	−23.0080	−1.39764	−0.698818	−	0.715299i	$-0.746290\pi$
$271$	−23.0080	−1.39764	−0.698818	+	0.715299i	$0.746290\pi$
$272$	0	0
$273$	−19.0790	−1.15472
$274$	0	0
$275$	11.3527	0.684592
$276$	0	0
$277$	20.6993	1.24370	0.621849	−	0.783137i	$-0.286382\pi$
$277$	20.6993	1.24370	0.621849	+	0.783137i	$0.286382\pi$
$278$	0	0
$279$	11.0732	0.662935
$280$	0	0
$281$	−30.9147	−1.84422	−0.922110	−	0.386929i	$-0.873536\pi$
$281$	−30.9147	−1.84422	−0.922110	+	0.386929i	$0.873536\pi$
$282$	0	0
$283$	−15.7644	−0.937097	−0.468548	−	0.883438i	$-0.655223\pi$
$283$	−15.7644	−0.937097	−0.468548	+	0.883438i	$0.655223\pi$
$284$	0	0
$285$	−4.12685	−0.244453
$286$	0	0
$287$	20.1349	1.18853
$288$	0	0
$289$	47.8856	2.81680
$290$	0	0
$291$	8.69438	0.509674
$292$	0	0
$293$	−9.59637	−0.560626	−0.280313	−	0.959909i	$-0.590438\pi$
$293$	−9.59637	−0.560626	−0.280313	+	0.959909i	$0.590438\pi$
$294$	0	0
$295$	−15.2303	−0.886743
$296$	0	0
$297$	4.78702	0.277771
$298$	0	0
$299$	−24.2907	−1.40477
$300$	0	0
$301$	−50.2578	−2.89681
$302$	0	0
$303$	7.46387	0.428788
$304$	0	0
$305$	6.80224	0.389495
$306$	0	0
$307$	20.3194	1.15969	0.579844	−	0.814727i	$-0.303113\pi$
$307$	20.3194	1.15969	0.579844	+	0.814727i	$0.303113\pi$
$308$	0	0
$309$	4.05463	0.230660
$310$	0	0
$311$	−21.7343	−1.23244	−0.616218	−	0.787575i	$-0.711336\pi$
$311$	−21.7343	−1.23244	−0.616218	+	0.787575i	$0.711336\pi$
$312$	0	0
$313$	−10.0035	−0.565432	−0.282716	−	0.959204i	$-0.591235\pi$
$313$	−10.0035	−0.565432	−0.282716	+	0.959204i	$0.591235\pi$
$314$	0	0
$315$	8.24767	0.464704
$316$	0	0
$317$	4.74953	0.266760	0.133380	−	0.991065i	$-0.457417\pi$
$317$	4.74953	0.266760	0.133380	+	0.991065i	$0.457417\pi$
$318$	0	0
$319$	−8.83120	−0.494452
$320$	0	0
$321$	−8.22324	−0.458977
$322$	0	0
$323$	20.5042	1.14088
$324$	0	0
$325$	−8.89421	−0.493362
$326$	0	0
$327$	4.91908	0.272026
$328$	0	0
$329$	−5.66614	−0.312385
$330$	0	0
$331$	−5.77266	−0.317294	−0.158647	−	0.987335i	$-0.550713\pi$
$331$	−5.77266	−0.317294	−0.158647	+	0.987335i	$0.550713\pi$
$332$	0	0
$333$	−2.26721	−0.124242
$334$	0	0
$335$	1.14610	0.0626181
$336$	0	0
$337$	18.3300	0.998496	0.499248	−	0.866459i	$-0.333610\pi$
$337$	18.3300	0.998496	0.499248	+	0.866459i	$0.333610\pi$
$338$	0	0
$339$	10.3271	0.560893
$340$	0	0
$341$	−53.0077	−2.87053
$342$	0	0
$343$	60.4361	3.26324
$344$	0	0
$345$	10.5006	0.565335
$346$	0	0
$347$	−25.3602	−1.36140	−0.680702	−	0.732560i	$-0.738325\pi$
$347$	−25.3602	−1.36140	−0.680702	+	0.732560i	$0.738325\pi$
$348$	0	0
$349$	−7.44859	−0.398714	−0.199357	−	0.979927i	$-0.563885\pi$
$349$	−7.44859	−0.398714	−0.199357	+	0.979927i	$0.563885\pi$
$350$	0	0
$351$	−3.75038	−0.200180
$352$	0	0
$353$	−20.2269	−1.07657	−0.538284	−	0.842764i	$-0.680927\pi$
$353$	−20.2269	−1.07657	−0.538284	+	0.842764i	$0.680927\pi$
$354$	0	0
$355$	23.0380	1.22273
$356$	0	0
$357$	−40.9785	−2.16881
$358$	0	0
$359$	4.20284	0.221818	0.110909	−	0.993831i	$-0.464624\pi$
$359$	4.20284	0.221818	0.110909	+	0.993831i	$0.464624\pi$
$360$	0	0
$361$	−12.5206	−0.658977
$362$	0	0
$363$	−11.9156	−0.625407
$364$	0	0
$365$	26.2211	1.37247
$366$	0	0
$367$	13.7248	0.716430	0.358215	−	0.933639i	$-0.383386\pi$
$367$	13.7248	0.716430	0.358215	+	0.933639i	$0.383386\pi$
$368$	0	0
$369$	3.95793	0.206042
$370$	0	0
$371$	28.7851	1.49445
$372$	0	0
$373$	36.3665	1.88299	0.941493	−	0.337032i	$-0.109423\pi$
$373$	36.3665	1.88299	0.941493	+	0.337032i	$0.109423\pi$
$374$	0	0
$375$	11.9511	0.617153
$376$	0	0
$377$	6.91877	0.356335
$378$	0	0
$379$	16.9142	0.868822	0.434411	−	0.900715i	$-0.356957\pi$
$379$	16.9142	0.868822	0.434411	+	0.900715i	$0.356957\pi$
$380$	0	0
$381$	−12.1345	−0.621667
$382$	0	0
$383$	9.23564	0.471919	0.235960	−	0.971763i	$-0.424177\pi$
$383$	9.23564	0.471919	0.235960	+	0.971763i	$0.424177\pi$
$384$	0	0
$385$	−39.4818	−2.01218
$386$	0	0
$387$	−9.87920	−0.502188
$388$	0	0
$389$	20.0808	1.01814	0.509069	−	0.860726i	$-0.329990\pi$
$389$	20.0808	1.01814	0.509069	+	0.860726i	$0.329990\pi$
$390$	0	0
$391$	−52.1723	−2.63847
$392$	0	0
$393$	8.79470	0.443634
$394$	0	0
$395$	4.85991	0.244529
$396$	0	0
$397$	8.03548	0.403289	0.201645	−	0.979459i	$-0.435371\pi$
$397$	8.03548	0.403289	0.201645	+	0.979459i	$0.435371\pi$
$398$	0	0
$399$	−12.9494	−0.648282
$400$	0	0
$401$	−20.6983	−1.03362	−0.516812	−	0.856099i	$-0.672881\pi$
$401$	−20.6983	−1.03362	−0.516812	+	0.856099i	$0.672881\pi$
$402$	0	0
$403$	41.5287	2.06869
$404$	0	0
$405$	1.62125	0.0805605
$406$	0	0
$407$	10.8532	0.537973
$408$	0	0
$409$	24.7004	1.22136	0.610679	−	0.791878i	$-0.290897\pi$
$409$	24.7004	1.22136	0.610679	+	0.791878i	$0.290897\pi$
$410$	0	0
$411$	−3.83330	−0.189083
$412$	0	0
$413$	−47.7904	−2.35161
$414$	0	0
$415$	−18.2993	−0.898279
$416$	0	0
$417$	−1.81069	−0.0886697
$418$	0	0
$419$	25.5548	1.24843	0.624217	−	0.781251i	$-0.285418\pi$
$419$	25.5548	1.24843	0.624217	+	0.781251i	$0.285418\pi$
$420$	0	0
$421$	−12.2638	−0.597701	−0.298850	−	0.954300i	$-0.596603\pi$
$421$	−12.2638	−0.597701	−0.298850	+	0.954300i	$0.596603\pi$
$422$	0	0
$423$	−1.11380	−0.0541546
$424$	0	0
$425$	−19.1032	−0.926643
$426$	0	0
$427$	21.3444	1.03293
$428$	0	0
$429$	17.9531	0.866786
$430$	0	0
$431$	16.4921	0.794398	0.397199	−	0.917733i	$-0.369982\pi$
$431$	16.4921	0.794398	0.397199	+	0.917733i	$0.369982\pi$
$432$	0	0
$433$	−20.3288	−0.976939	−0.488469	−	0.872581i	$-0.662445\pi$
$433$	−20.3288	−0.976939	−0.488469	+	0.872581i	$0.662445\pi$
$434$	0	0
$435$	−2.99091	−0.143403
$436$	0	0
$437$	−16.4867	−0.788667
$438$	0	0
$439$	23.8416	1.13790	0.568949	−	0.822373i	$-0.307350\pi$
$439$	23.8416	1.13790	0.568949	+	0.822373i	$0.307350\pi$
$440$	0	0
$441$	18.8800	0.899045
$442$	0	0
$443$	−17.8300	−0.847131	−0.423565	−	0.905865i	$-0.639222\pi$
$443$	−17.8300	−0.847131	−0.423565	+	0.905865i	$0.639222\pi$
$444$	0	0
$445$	−9.44360	−0.447670
$446$	0	0
$447$	19.9656	0.944339
$448$	0	0
$449$	19.1086	0.901788	0.450894	−	0.892577i	$-0.351105\pi$
$449$	19.1086	0.901788	0.450894	+	0.892577i	$0.351105\pi$
$450$	0	0
$451$	−18.9467	−0.892167
$452$	0	0
$453$	12.5577	0.590012
$454$	0	0
$455$	30.9319	1.45011
$456$	0	0
$457$	27.9632	1.30806	0.654031	−	0.756467i	$-0.273076\pi$
$457$	27.9632	1.30806	0.654031	+	0.756467i	$0.273076\pi$
$458$	0	0
$459$	−8.05516	−0.375983
$460$	0	0
$461$	6.83764	0.318461	0.159230	−	0.987241i	$-0.449099\pi$
$461$	6.83764	0.318461	0.159230	+	0.987241i	$0.449099\pi$
$462$	0	0
$463$	−29.5091	−1.37140	−0.685702	−	0.727882i	$-0.740505\pi$
$463$	−29.5091	−1.37140	−0.685702	+	0.727882i	$0.740505\pi$
$464$	0	0
$465$	−17.9524	−0.832523
$466$	0	0
$467$	24.3776	1.12806	0.564030	−	0.825754i	$-0.309250\pi$
$467$	24.3776	1.12806	0.564030	+	0.825754i	$0.309250\pi$
$468$	0	0
$469$	3.59629	0.166061
$470$	0	0
$471$	−24.4197	−1.12520
$472$	0	0
$473$	47.2920	2.17449
$474$	0	0
$475$	−6.03673	−0.276984
$476$	0	0
$477$	5.65830	0.259076
$478$	0	0
$479$	15.5820	0.711960	0.355980	−	0.934494i	$-0.384147\pi$
$479$	15.5820	0.711960	0.355980	+	0.934494i	$0.384147\pi$
$480$	0	0
$481$	−8.50289	−0.387699
$482$	0	0
$483$	32.9494	1.49925
$484$	0	0
$485$	−14.0958	−0.640056
$486$	0	0
$487$	−29.1356	−1.32026	−0.660131	−	0.751151i	$-0.729499\pi$
$487$	−29.1356	−1.32026	−0.660131	+	0.751151i	$0.729499\pi$
$488$	0	0
$489$	−1.07379	−0.0485584
$490$	0	0
$491$	15.3303	0.691849	0.345924	−	0.938262i	$-0.387565\pi$
$491$	15.3303	0.691849	0.345924	+	0.938262i	$0.387565\pi$
$492$	0	0
$493$	14.8603	0.669275
$494$	0	0
$495$	−7.76096	−0.348829
$496$	0	0
$497$	72.2897	3.24264
$498$	0	0
$499$	−20.9521	−0.937947	−0.468974	−	0.883212i	$-0.655376\pi$
$499$	−20.9521	−0.937947	−0.468974	+	0.883212i	$0.655376\pi$
$500$	0	0
$501$	7.25268	0.324026
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−12.1008	−0.538478
$506$	0	0
$507$	−1.06533	−0.0473130
$508$	0	0
$509$	21.6036	0.957564	0.478782	−	0.877934i	$-0.341078\pi$
$509$	21.6036	0.957564	0.478782	+	0.877934i	$0.341078\pi$
$510$	0	0
$511$	82.2778	3.63975
$512$	0	0
$513$	−2.54548	−0.112385
$514$	0	0
$515$	−6.57357	−0.289666
$516$	0	0
$517$	5.33177	0.234491
$518$	0	0
$519$	−10.7486	−0.471809
$520$	0	0
$521$	34.5773	1.51486	0.757430	−	0.652916i	$-0.226455\pi$
$521$	34.5773	1.51486	0.757430	+	0.652916i	$0.226455\pi$
$522$	0	0
$523$	16.6517	0.728130	0.364065	−	0.931374i	$-0.381389\pi$
$523$	16.6517	0.728130	0.364065	+	0.931374i	$0.381389\pi$
$524$	0	0
$525$	12.0646	0.526544
$526$	0	0
$527$	89.1964	3.88546
$528$	0	0
$529$	18.9499	0.823910
$530$	0	0
$531$	−9.39418	−0.407673
$532$	0	0
$533$	14.8437	0.642954
$534$	0	0
$535$	13.3319	0.576389
$536$	0	0
$537$	5.28959	0.228263
$538$	0	0
$539$	−90.3788	−3.89289
$540$	0	0
$541$	−12.2237	−0.525536	−0.262768	−	0.964859i	$-0.584635\pi$
$541$	−12.2237	−0.525536	−0.262768	+	0.964859i	$0.584635\pi$
$542$	0	0
$543$	7.10431	0.304875
$544$	0	0
$545$	−7.97505	−0.341614
$546$	0	0
$547$	42.3294	1.80988	0.904938	−	0.425544i	$-0.139917\pi$
$547$	42.3294	1.80988	0.904938	+	0.425544i	$0.139917\pi$
$548$	0	0
$549$	4.19568	0.179067
$550$	0	0
$551$	4.69594	0.200054
$552$	0	0
$553$	15.2497	0.648482
$554$	0	0
$555$	3.67571	0.156025
$556$	0	0
$557$	17.4486	0.739321	0.369661	−	0.929167i	$-0.379474\pi$
$557$	17.4486	0.739321	0.369661	+	0.929167i	$0.379474\pi$
$558$	0	0
$559$	−37.0507	−1.56708
$560$	0	0
$561$	38.5603	1.62802
$562$	0	0
$563$	−36.4161	−1.53475	−0.767377	−	0.641196i	$-0.778438\pi$
$563$	−36.4161	−1.53475	−0.767377	+	0.641196i	$0.778438\pi$
$564$	0	0
$565$	−16.7429	−0.704377
$566$	0	0
$567$	5.08723	0.213644
$568$	0	0
$569$	30.2694	1.26896	0.634481	−	0.772939i	$-0.281214\pi$
$569$	30.2694	1.26896	0.634481	+	0.772939i	$0.281214\pi$
$570$	0	0
$571$	18.3774	0.769068	0.384534	−	0.923111i	$-0.374362\pi$
$571$	18.3774	0.769068	0.384534	+	0.923111i	$0.374362\pi$
$572$	0	0
$573$	−16.2208	−0.677632
$574$	0	0
$575$	15.3602	0.640567
$576$	0	0
$577$	−11.9921	−0.499236	−0.249618	−	0.968344i	$-0.580305\pi$
$577$	−11.9921	−0.499236	−0.249618	+	0.968344i	$0.580305\pi$
$578$	0	0
$579$	15.9161	0.661452
$580$	0	0
$581$	−57.4206	−2.38221
$582$	0	0
$583$	−27.0864	−1.12181
$584$	0	0
$585$	6.08030	0.251389
$586$	0	0
$587$	−0.765014	−0.0315755	−0.0157877	−	0.999875i	$-0.505026\pi$
$587$	−0.765014	−0.0315755	−0.0157877	+	0.999875i	$0.505026\pi$
$588$	0	0
$589$	28.1866	1.16141
$590$	0	0
$591$	−6.48555	−0.266780
$592$	0	0
$593$	−19.5386	−0.802353	−0.401176	−	0.916001i	$-0.631399\pi$
$593$	−19.5386	−0.802353	−0.401176	+	0.916001i	$0.631399\pi$
$594$	0	0
$595$	66.4363	2.72362
$596$	0	0
$597$	−14.1090	−0.577443
$598$	0	0
$599$	−12.3808	−0.505864	−0.252932	−	0.967484i	$-0.581395\pi$
$599$	−12.3808	−0.505864	−0.252932	+	0.967484i	$0.581395\pi$
$600$	0	0
$601$	39.1993	1.59897	0.799487	−	0.600683i	$-0.205105\pi$
$601$	39.1993	1.59897	0.799487	+	0.600683i	$0.205105\pi$
$602$	0	0
$603$	0.706924	0.0287882
$604$	0	0
$605$	19.3182	0.785395
$606$	0	0
$607$	4.83993	0.196447	0.0982234	−	0.995164i	$-0.468684\pi$
$607$	4.83993	0.196447	0.0982234	+	0.995164i	$0.468684\pi$
$608$	0	0
$609$	−9.38503	−0.380301
$610$	0	0
$611$	−4.17716	−0.168990
$612$	0	0
$613$	−6.04924	−0.244327	−0.122163	−	0.992510i	$-0.538983\pi$
$613$	−6.04924	−0.244327	−0.122163	+	0.992510i	$0.538983\pi$
$614$	0	0
$615$	−6.41680	−0.258750
$616$	0	0
$617$	32.0507	1.29031	0.645156	−	0.764051i	$-0.276792\pi$
$617$	32.0507	1.29031	0.645156	+	0.764051i	$0.276792\pi$
$618$	0	0
$619$	−10.8170	−0.434772	−0.217386	−	0.976086i	$-0.569753\pi$
$619$	−10.8170	−0.434772	−0.217386	+	0.976086i	$0.569753\pi$
$620$	0	0
$621$	6.47688	0.259908
$622$	0	0
$623$	−29.6326	−1.18720
$624$	0	0
$625$	−7.51799	−0.300720
$626$	0	0
$627$	12.1853	0.486632
$628$	0	0
$629$	−18.2627	−0.728183
$630$	0	0
$631$	−5.54692	−0.220819	−0.110410	−	0.993886i	$-0.535216\pi$
$631$	−5.54692	−0.220819	−0.110410	+	0.993886i	$0.535216\pi$
$632$	0	0
$633$	−3.02938	−0.120407
$634$	0	0
$635$	19.6730	0.780698
$636$	0	0
$637$	70.8069	2.80547
$638$	0	0
$639$	14.2100	0.562140
$640$	0	0
$641$	14.4980	0.572635	0.286318	−	0.958135i	$-0.407569\pi$
$641$	14.4980	0.572635	0.286318	+	0.958135i	$0.407569\pi$
$642$	0	0
$643$	34.5051	1.36075	0.680373	−	0.732866i	$-0.261818\pi$
$643$	34.5051	1.36075	0.680373	+	0.732866i	$0.261818\pi$
$644$	0	0
$645$	16.0166	0.630655
$646$	0	0
$647$	0.260969	0.0102597	0.00512987	−	0.999987i	$-0.498367\pi$
$647$	0.260969	0.0102597	0.00512987	+	0.999987i	$0.498367\pi$
$648$	0	0
$649$	44.9702	1.76523
$650$	0	0
$651$	−56.3320	−2.20782
$652$	0	0
$653$	−6.80054	−0.266126	−0.133063	−	0.991108i	$-0.542481\pi$
$653$	−6.80054	−0.266126	−0.133063	+	0.991108i	$0.542481\pi$
$654$	0	0
$655$	−14.2584	−0.557121
$656$	0	0
$657$	16.1734	0.630984
$658$	0	0
$659$	−32.0742	−1.24943	−0.624717	−	0.780852i	$-0.714786\pi$
$659$	−32.0742	−1.24943	−0.624717	+	0.780852i	$0.714786\pi$
$660$	0	0
$661$	−2.65863	−0.103409	−0.0517043	−	0.998662i	$-0.516465\pi$
$661$	−2.65863	−0.103409	−0.0517043	+	0.998662i	$0.516465\pi$
$662$	0	0
$663$	−30.2099	−1.17326
$664$	0	0
$665$	20.9943	0.814122
$666$	0	0
$667$	−11.9487	−0.462654
$668$	0	0
$669$	18.1230	0.700674
$670$	0	0
$671$	−20.0848	−0.775365
$672$	0	0
$673$	−12.8856	−0.496704	−0.248352	−	0.968670i	$-0.579889\pi$
$673$	−12.8856	−0.496704	−0.248352	+	0.968670i	$0.579889\pi$
$674$	0	0
$675$	2.37155	0.0912811
$676$	0	0
$677$	−8.88973	−0.341660	−0.170830	−	0.985301i	$-0.554645\pi$
$677$	−8.88973	−0.341660	−0.170830	+	0.985301i	$0.554645\pi$
$678$	0	0
$679$	−44.2304	−1.69741
$680$	0	0
$681$	−7.36207	−0.282115
$682$	0	0
$683$	34.8155	1.33218	0.666089	−	0.745872i	$-0.267967\pi$
$683$	34.8155	1.33218	0.666089	+	0.745872i	$0.267967\pi$
$684$	0	0
$685$	6.21473	0.237453
$686$	0	0
$687$	28.8730	1.10157
$688$	0	0
$689$	21.2208	0.808447
$690$	0	0
$691$	1.97469	0.0751209	0.0375605	−	0.999294i	$-0.488041\pi$
$691$	1.97469	0.0751209	0.0375605	+	0.999294i	$0.488041\pi$
$692$	0	0
$693$	−24.3527	−0.925083
$694$	0	0
$695$	2.93557	0.111353
$696$	0	0
$697$	31.8818	1.20761
$698$	0	0
$699$	−22.1642	−0.838326
$700$	0	0
$701$	13.5969	0.513549	0.256774	−	0.966471i	$-0.417340\pi$
$701$	13.5969	0.513549	0.256774	+	0.966471i	$0.417340\pi$
$702$	0	0
$703$	−5.77113	−0.217662
$704$	0	0
$705$	1.80574	0.0680082
$706$	0	0
$707$	−37.9705	−1.42803
$708$	0	0
$709$	−36.6935	−1.37805	−0.689027	−	0.724735i	$-0.741962\pi$
$709$	−36.6935	−1.37805	−0.689027	+	0.724735i	$0.741962\pi$
$710$	0	0
$711$	2.99763	0.112420
$712$	0	0
$713$	−71.7198	−2.68593
$714$	0	0
$715$	−29.1065	−1.08852
$716$	0	0
$717$	8.28176	0.309288
$718$	0	0
$719$	−12.7127	−0.474104	−0.237052	−	0.971497i	$-0.576181\pi$
$719$	−12.7127	−0.474104	−0.237052	+	0.971497i	$0.576181\pi$
$720$	0	0
$721$	−20.6269	−0.768184
$722$	0	0
$723$	−1.60328	−0.0596267
$724$	0	0
$725$	−4.37508	−0.162487
$726$	0	0
$727$	−21.6489	−0.802914	−0.401457	−	0.915878i	$-0.631496\pi$
$727$	−21.6489	−0.802914	−0.401457	+	0.915878i	$0.631496\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−79.5785	−2.94332
$732$	0	0
$733$	3.39863	0.125531	0.0627657	−	0.998028i	$-0.480008\pi$
$733$	3.39863	0.125531	0.0627657	+	0.998028i	$0.480008\pi$
$734$	0	0
$735$	−30.6091	−1.12903
$736$	0	0
$737$	−3.38406	−0.124653
$738$	0	0
$739$	−18.8891	−0.694848	−0.347424	−	0.937708i	$-0.612944\pi$
$739$	−18.8891	−0.694848	−0.347424	+	0.937708i	$0.612944\pi$
$740$	0	0
$741$	−9.54649	−0.350699
$742$	0	0
$743$	−31.3785	−1.15116	−0.575582	−	0.817744i	$-0.695224\pi$
$743$	−31.3785	−1.15116	−0.575582	+	0.817744i	$0.695224\pi$
$744$	0	0
$745$	−32.3692	−1.18591
$746$	0	0
$747$	−11.2872	−0.412977
$748$	0	0
$749$	41.8336	1.52857
$750$	0	0
$751$	−12.2331	−0.446393	−0.223197	−	0.974773i	$-0.571649\pi$
$751$	−12.2331	−0.446393	−0.223197	+	0.974773i	$0.571649\pi$
$752$	0	0
$753$	−28.7613	−1.04812
$754$	0	0
$755$	−20.3592	−0.740946
$756$	0	0
$757$	−43.3208	−1.57452	−0.787261	−	0.616620i	$-0.788502\pi$
$757$	−43.3208	−1.57452	−0.787261	+	0.616620i	$0.788502\pi$
$758$	0	0
$759$	−31.0050	−1.12541
$760$	0	0
$761$	−12.9918	−0.470951	−0.235475	−	0.971880i	$-0.575665\pi$
$761$	−12.9918	−0.470951	−0.235475	+	0.971880i	$0.575665\pi$
$762$	0	0
$763$	−25.0245	−0.905948
$764$	0	0
$765$	13.0594	0.472164
$766$	0	0
$767$	−35.2317	−1.27214
$768$	0	0
$769$	−10.8066	−0.389697	−0.194848	−	0.980833i	$-0.562421\pi$
$769$	−10.8066	−0.389697	−0.194848	+	0.980833i	$0.562421\pi$
$770$	0	0
$771$	−14.8466	−0.534687
$772$	0	0
$773$	−42.1613	−1.51643	−0.758217	−	0.652002i	$-0.773929\pi$
$773$	−42.1613	−1.51643	−0.758217	+	0.652002i	$0.773929\pi$
$774$	0	0
$775$	−26.2607	−0.943311
$776$	0	0
$777$	11.5338	0.413774
$778$	0	0
$779$	10.0748	0.360968
$780$	0	0
$781$	−68.0237	−2.43408
$782$	0	0
$783$	−1.84482	−0.0659285
$784$	0	0
$785$	39.5904	1.41304
$786$	0	0
$787$	42.5411	1.51643	0.758214	−	0.652006i	$-0.226072\pi$
$787$	42.5411	1.51643	0.758214	+	0.652006i	$0.226072\pi$
$788$	0	0
$789$	12.6492	0.450323
$790$	0	0
$791$	−52.5365	−1.86798
$792$	0	0
$793$	15.7354	0.558779
$794$	0	0
$795$	−9.17352	−0.325351
$796$	0	0
$797$	−19.3105	−0.684011	−0.342006	−	0.939698i	$-0.611106\pi$
$797$	−19.3105	−0.684011	−0.342006	+	0.939698i	$0.611106\pi$
$798$	0	0
$799$	−8.97181	−0.317400
$800$	0	0
$801$	−5.82489	−0.205812
$802$	0	0
$803$	−77.4224	−2.73218
$804$	0	0
$805$	−53.4192	−1.88278
$806$	0	0
$807$	−12.5501	−0.441785
$808$	0	0
$809$	10.3817	0.365003	0.182501	−	0.983206i	$-0.441581\pi$
$809$	10.3817	0.365003	0.182501	+	0.983206i	$0.441581\pi$
$810$	0	0
$811$	−3.30087	−0.115909	−0.0579546	−	0.998319i	$-0.518458\pi$
$811$	−3.30087	−0.115909	−0.0579546	+	0.998319i	$0.518458\pi$
$812$	0	0
$813$	23.0080	0.806926
$814$	0	0
$815$	1.74088	0.0609804
$816$	0	0
$817$	−25.1473	−0.879791
$818$	0	0
$819$	19.0790	0.666676
$820$	0	0
$821$	23.1377	0.807511	0.403755	−	0.914867i	$-0.367705\pi$
$821$	23.1377	0.807511	0.403755	+	0.914867i	$0.367705\pi$
$822$	0	0
$823$	−30.8167	−1.07420	−0.537102	−	0.843517i	$-0.680481\pi$
$823$	−30.8167	−1.07420	−0.537102	+	0.843517i	$0.680481\pi$
$824$	0	0
$825$	−11.3527	−0.395249
$826$	0	0
$827$	−42.7231	−1.48563	−0.742815	−	0.669497i	$-0.766510\pi$
$827$	−42.7231	−1.48563	−0.742815	+	0.669497i	$0.766510\pi$
$828$	0	0
$829$	11.6954	0.406198	0.203099	−	0.979158i	$-0.434899\pi$
$829$	11.6954	0.406198	0.203099	+	0.979158i	$0.434899\pi$
$830$	0	0
$831$	−20.6993	−0.718050
$832$	0	0
$833$	152.081	5.26930
$834$	0	0
$835$	−11.7584	−0.406916
$836$	0	0
$837$	−11.0732	−0.382746
$838$	0	0
$839$	−31.7883	−1.09745	−0.548727	−	0.836002i	$-0.684887\pi$
$839$	−31.7883	−1.09745	−0.548727	+	0.836002i	$0.684887\pi$
$840$	0	0
$841$	−25.5966	−0.882643
$842$	0	0
$843$	30.9147	1.06476
$844$	0	0
$845$	1.72717	0.0594163
$846$	0	0
$847$	60.6175	2.08284
$848$	0	0
$849$	15.7644	0.541033
$850$	0	0
$851$	14.6844	0.503376
$852$	0	0
$853$	1.86112	0.0637234	0.0318617	−	0.999492i	$-0.489856\pi$
$853$	1.86112	0.0637234	0.0318617	+	0.999492i	$0.489856\pi$
$854$	0	0
$855$	4.12685	0.141135
$856$	0	0
$857$	−35.0269	−1.19650	−0.598249	−	0.801311i	$-0.704136\pi$
$857$	−35.0269	−1.19650	−0.598249	+	0.801311i	$0.704136\pi$
$858$	0	0
$859$	−47.9439	−1.63582	−0.817912	−	0.575344i	$-0.804868\pi$
$859$	−47.9439	−1.63582	−0.817912	+	0.575344i	$0.804868\pi$
$860$	0	0
$861$	−20.1349	−0.686197
$862$	0	0
$863$	−22.5634	−0.768068	−0.384034	−	0.923319i	$-0.625466\pi$
$863$	−22.5634	−0.768068	−0.384034	+	0.923319i	$0.625466\pi$
$864$	0	0
$865$	17.4261	0.592505
$866$	0	0
$867$	−47.8856	−1.62628
$868$	0	0
$869$	−14.3498	−0.486782
$870$	0	0
$871$	2.65123	0.0898335
$872$	0	0
$873$	−8.69438	−0.294260
$874$	0	0
$875$	−60.7982	−2.05535
$876$	0	0
$877$	2.97906	0.100596	0.0502979	−	0.998734i	$-0.483983\pi$
$877$	2.97906	0.100596	0.0502979	+	0.998734i	$0.483983\pi$
$878$	0	0
$879$	9.59637	0.323678
$880$	0	0
$881$	−24.7033	−0.832275	−0.416138	−	0.909302i	$-0.636617\pi$
$881$	−24.7033	−0.832275	−0.416138	+	0.909302i	$0.636617\pi$
$882$	0	0
$883$	22.0936	0.743510	0.371755	−	0.928331i	$-0.378756\pi$
$883$	22.0936	0.743510	0.371755	+	0.928331i	$0.378756\pi$
$884$	0	0
$885$	15.2303	0.511961
$886$	0	0
$887$	−7.00169	−0.235094	−0.117547	−	0.993067i	$-0.537503\pi$
$887$	−7.00169	−0.235094	−0.117547	+	0.993067i	$0.537503\pi$
$888$	0	0
$889$	61.7308	2.07038
$890$	0	0
$891$	−4.78702	−0.160371
$892$	0	0
$893$	−2.83514	−0.0948744
$894$	0	0
$895$	−8.57574	−0.286655
$896$	0	0
$897$	24.2907	0.811044
$898$	0	0
$899$	20.4281	0.681314
$900$	0	0
$901$	45.5785	1.51844
$902$	0	0
$903$	50.2578	1.67247
$904$	0	0
$905$	−11.5179	−0.382867
$906$	0	0
$907$	16.0503	0.532943	0.266472	−	0.963843i	$-0.414142\pi$
$907$	16.0503	0.532943	0.266472	+	0.963843i	$0.414142\pi$
$908$	0	0
$909$	−7.46387	−0.247561
$910$	0	0
$911$	8.69646	0.288127	0.144063	−	0.989568i	$-0.453983\pi$
$911$	8.69646	0.288127	0.144063	+	0.989568i	$0.453983\pi$
$912$	0	0
$913$	54.0320	1.78820
$914$	0	0
$915$	−6.80224	−0.224875
$916$	0	0
$917$	−44.7407	−1.47747
$918$	0	0
$919$	24.0545	0.793484	0.396742	−	0.917930i	$-0.370141\pi$
$919$	24.0545	0.793484	0.396742	+	0.917930i	$0.370141\pi$
$920$	0	0
$921$	−20.3194	−0.669547
$922$	0	0
$923$	53.2930	1.75416
$924$	0	0
$925$	5.37681	0.176788
$926$	0	0
$927$	−4.05463	−0.133172
$928$	0	0
$929$	−2.25584	−0.0740118	−0.0370059	−	0.999315i	$-0.511782\pi$
$929$	−2.25584	−0.0740118	−0.0370059	+	0.999315i	$0.511782\pi$
$930$	0	0
$931$	48.0585	1.57505
$932$	0	0
$933$	21.7343	0.711548
$934$	0	0
$935$	−62.5158	−2.04448
$936$	0	0
$937$	5.56080	0.181663	0.0908317	−	0.995866i	$-0.471047\pi$
$937$	5.56080	0.181663	0.0908317	+	0.995866i	$0.471047\pi$
$938$	0	0
$939$	10.0035	0.326452
$940$	0	0
$941$	−42.0092	−1.36946	−0.684731	−	0.728796i	$-0.740080\pi$
$941$	−42.0092	−1.36946	−0.684731	+	0.728796i	$0.740080\pi$
$942$	0	0
$943$	−25.6351	−0.834792
$944$	0	0
$945$	−8.24767	−0.268297
$946$	0	0
$947$	−43.9687	−1.42879	−0.714395	−	0.699743i	$-0.753298\pi$
$947$	−43.9687	−1.42879	−0.714395	+	0.699743i	$0.753298\pi$
$948$	0	0
$949$	60.6563	1.96899
$950$	0	0
$951$	−4.74953	−0.154014
$952$	0	0
$953$	−16.8158	−0.544718	−0.272359	−	0.962196i	$-0.587804\pi$
$953$	−16.8158	−0.544718	−0.272359	+	0.962196i	$0.587804\pi$
$954$	0	0
$955$	26.2979	0.850980
$956$	0	0
$957$	8.83120	0.285472
$958$	0	0
$959$	19.5009	0.629717
$960$	0	0
$961$	91.6157	2.95535
$962$	0	0
$963$	8.22324	0.264990
$964$	0	0
$965$	−25.8040	−0.830661
$966$	0	0
$967$	−12.9469	−0.416344	−0.208172	−	0.978092i	$-0.566751\pi$
$967$	−12.9469	−0.416344	−0.208172	+	0.978092i	$0.566751\pi$
$968$	0	0
$969$	−20.5042	−0.658690
$970$	0	0
$971$	22.0433	0.707403	0.353702	−	0.935358i	$-0.384923\pi$
$971$	22.0433	0.707403	0.353702	+	0.935358i	$0.384923\pi$
$972$	0	0
$973$	9.21139	0.295304
$974$	0	0
$975$	8.89421	0.284843
$976$	0	0
$977$	−9.91543	−0.317223	−0.158611	−	0.987341i	$-0.550702\pi$
$977$	−9.91543	−0.317223	−0.158611	+	0.987341i	$0.550702\pi$
$978$	0	0
$979$	27.8839	0.891173
$980$	0	0
$981$	−4.91908	−0.157054
$982$	0	0
$983$	32.2011	1.02705	0.513527	−	0.858073i	$-0.328338\pi$
$983$	32.2011	1.02705	0.513527	+	0.858073i	$0.328338\pi$
$984$	0	0
$985$	10.5147	0.335026
$986$	0	0
$987$	5.66614	0.180355
$988$	0	0
$989$	63.9863	2.03465
$990$	0	0
$991$	20.1212	0.639169	0.319585	−	0.947558i	$-0.396457\pi$
$991$	20.1212	0.639169	0.319585	+	0.947558i	$0.396457\pi$
$992$	0	0
$993$	5.77266	0.183190
$994$	0	0
$995$	22.8742	0.725161
$996$	0	0
$997$	−48.8248	−1.54630	−0.773148	−	0.634226i	$-0.781319\pi$
$997$	−48.8248	−1.54630	−0.773148	+	0.634226i	$0.781319\pi$
$998$	0	0
$999$	2.26721	0.0717314

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.16	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.16	✓	26	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:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.62125 q^{5} +5.08723 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.62125 q^{5} +5.08723 q^{7} +1.00000 q^{9} -4.78702 q^{11} +3.75038 q^{13} -1.62125 q^{15} +8.05516 q^{17} +2.54548 q^{19} -5.08723 q^{21} -6.47688 q^{23} -2.37155 q^{25} -1.00000 q^{27} +1.84482 q^{29} +11.0732 q^{31} +4.78702 q^{33} +8.24767 q^{35} -2.26721 q^{37} -3.75038 q^{39} +3.95793 q^{41} -9.87920 q^{43} +1.62125 q^{45} -1.11380 q^{47} +18.8800 q^{49} -8.05516 q^{51} +5.65830 q^{53} -7.76096 q^{55} -2.54548 q^{57} -9.39418 q^{59} +4.19568 q^{61} +5.08723 q^{63} +6.08030 q^{65} +0.706924 q^{67} +6.47688 q^{69} +14.2100 q^{71} +16.1734 q^{73} +2.37155 q^{75} -24.3527 q^{77} +2.99763 q^{79} +1.00000 q^{81} -11.2872 q^{83} +13.0594 q^{85} -1.84482 q^{87} -5.82489 q^{89} +19.0790 q^{91} -11.0732 q^{93} +4.12685 q^{95} -8.69438 q^{97} -4.78702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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