LMFDB - Embedded newform 6036.2.a.h.1.9

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:	$24$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.497428 q^{5} -3.93416 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.497428 q^{5} -3.93416 q^{7} +1.00000 q^{9} -3.87914 q^{11} +5.59229 q^{13} -0.497428 q^{15} +4.95162 q^{17} -1.44420 q^{19} -3.93416 q^{21} -6.08724 q^{23} -4.75257 q^{25} +1.00000 q^{27} -1.24827 q^{29} +4.55700 q^{31} -3.87914 q^{33} +1.95696 q^{35} -4.23843 q^{37} +5.59229 q^{39} +8.76770 q^{41} -6.76790 q^{43} -0.497428 q^{45} -8.34030 q^{47} +8.47760 q^{49} +4.95162 q^{51} +12.1024 q^{53} +1.92959 q^{55} -1.44420 q^{57} +12.5123 q^{59} +14.3273 q^{61} -3.93416 q^{63} -2.78176 q^{65} +1.78162 q^{67} -6.08724 q^{69} -4.74350 q^{71} -12.1114 q^{73} -4.75257 q^{75} +15.2612 q^{77} +5.00178 q^{79} +1.00000 q^{81} +9.32299 q^{83} -2.46307 q^{85} -1.24827 q^{87} +0.499936 q^{89} -22.0009 q^{91} +4.55700 q^{93} +0.718387 q^{95} +2.87757 q^{97} -3.87914 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	$ 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) $ 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * 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
$5$	−0.497428	−0.222456	−0.111228	−	0.993795i	$-0.535478\pi$
$5$	−0.497428	−0.222456	−0.111228	+	0.993795i	$0.535478\pi$
$6$	0	0
$7$	−3.93416	−1.48697	−0.743486	−	0.668751i	$-0.766829\pi$
$7$	−3.93416	−1.48697	−0.743486	+	0.668751i	$0.766829\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.87914	−1.16961	−0.584803	−	0.811176i	$-0.698828\pi$
$11$	−3.87914	−1.16961	−0.584803	+	0.811176i	$0.698828\pi$
$12$	0	0
$13$	5.59229	1.55102	0.775511	−	0.631335i	$-0.217493\pi$
$13$	5.59229	1.55102	0.775511	+	0.631335i	$0.217493\pi$
$14$	0	0
$15$	−0.497428	−0.128435
$16$	0	0
$17$	4.95162	1.20095	0.600473	−	0.799645i	$-0.294979\pi$
$17$	4.95162	1.20095	0.600473	+	0.799645i	$0.294979\pi$
$18$	0	0
$19$	−1.44420	−0.331323	−0.165662	−	0.986183i	$-0.552976\pi$
$19$	−1.44420	−0.331323	−0.165662	+	0.986183i	$0.552976\pi$
$20$	0	0
$21$	−3.93416	−0.858504
$22$	0	0
$23$	−6.08724	−1.26928	−0.634639	−	0.772809i	$-0.718851\pi$
$23$	−6.08724	−1.26928	−0.634639	+	0.772809i	$0.718851\pi$
$24$	0	0
$25$	−4.75257	−0.950513
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.24827	−0.231798	−0.115899	−	0.993261i	$-0.536975\pi$
$29$	−1.24827	−0.231798	−0.115899	+	0.993261i	$0.536975\pi$
$30$	0	0
$31$	4.55700	0.818461	0.409230	−	0.912431i	$-0.365797\pi$
$31$	4.55700	0.818461	0.409230	+	0.912431i	$0.365797\pi$
$32$	0	0
$33$	−3.87914	−0.675272
$34$	0	0
$35$	1.95696	0.330786
$36$	0	0
$37$	−4.23843	−0.696794	−0.348397	−	0.937347i	$-0.613274\pi$
$37$	−4.23843	−0.696794	−0.348397	+	0.937347i	$0.613274\pi$
$38$	0	0
$39$	5.59229	0.895482
$40$	0	0
$41$	8.76770	1.36929	0.684643	−	0.728879i	$-0.259958\pi$
$41$	8.76770	1.36929	0.684643	+	0.728879i	$0.259958\pi$
$42$	0	0
$43$	−6.76790	−1.03210	−0.516048	−	0.856560i	$-0.672597\pi$
$43$	−6.76790	−1.03210	−0.516048	+	0.856560i	$0.672597\pi$
$44$	0	0
$45$	−0.497428	−0.0741521
$46$	0	0
$47$	−8.34030	−1.21656	−0.608279	−	0.793723i	$-0.708140\pi$
$47$	−8.34030	−1.21656	−0.608279	+	0.793723i	$0.708140\pi$
$48$	0	0
$49$	8.47760	1.21109
$50$	0	0
$51$	4.95162	0.693366
$52$	0	0
$53$	12.1024	1.66240	0.831199	−	0.555975i	$-0.187655\pi$
$53$	12.1024	1.66240	0.831199	+	0.555975i	$0.187655\pi$
$54$	0	0
$55$	1.92959	0.260186
$56$	0	0
$57$	−1.44420	−0.191289
$58$	0	0
$59$	12.5123	1.62897	0.814484	−	0.580186i	$-0.197020\pi$
$59$	12.5123	1.62897	0.814484	+	0.580186i	$0.197020\pi$
$60$	0	0
$61$	14.3273	1.83442	0.917209	−	0.398405i	$-0.130436\pi$
$61$	14.3273	1.83442	0.917209	+	0.398405i	$0.130436\pi$
$62$	0	0
$63$	−3.93416	−0.495657
$64$	0	0
$65$	−2.78176	−0.345035
$66$	0	0
$67$	1.78162	0.217659	0.108830	−	0.994060i	$-0.465290\pi$
$67$	1.78162	0.217659	0.108830	+	0.994060i	$0.465290\pi$
$68$	0	0
$69$	−6.08724	−0.732818
$70$	0	0
$71$	−4.74350	−0.562950	−0.281475	−	0.959569i	$-0.590824\pi$
$71$	−4.74350	−0.562950	−0.281475	+	0.959569i	$0.590824\pi$
$72$	0	0
$73$	−12.1114	−1.41754	−0.708768	−	0.705442i	$-0.750749\pi$
$73$	−12.1114	−1.41754	−0.708768	+	0.705442i	$0.750749\pi$
$74$	0	0
$75$	−4.75257	−0.548779
$76$	0	0
$77$	15.2612	1.73917
$78$	0	0
$79$	5.00178	0.562744	0.281372	−	0.959599i	$-0.409210\pi$
$79$	5.00178	0.562744	0.281372	+	0.959599i	$0.409210\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.32299	1.02333	0.511666	−	0.859185i	$-0.329029\pi$
$83$	9.32299	1.02333	0.511666	+	0.859185i	$0.329029\pi$
$84$	0	0
$85$	−2.46307	−0.267158
$86$	0	0
$87$	−1.24827	−0.133829
$88$	0	0
$89$	0.499936	0.0529932	0.0264966	−	0.999649i	$-0.491565\pi$
$89$	0.499936	0.0529932	0.0264966	+	0.999649i	$0.491565\pi$
$90$	0	0
$91$	−22.0009	−2.30633
$92$	0	0
$93$	4.55700	0.472539
$94$	0	0
$95$	0.718387	0.0737049
$96$	0	0
$97$	2.87757	0.292173	0.146087	−	0.989272i	$-0.453332\pi$
$97$	2.87757	0.292173	0.146087	+	0.989272i	$0.453332\pi$
$98$	0	0
$99$	−3.87914	−0.389868
$100$	0	0
$101$	2.89295	0.287859	0.143930	−	0.989588i	$-0.454026\pi$
$101$	2.89295	0.287859	0.143930	+	0.989588i	$0.454026\pi$
$102$	0	0
$103$	−5.45369	−0.537368	−0.268684	−	0.963228i	$-0.586589\pi$
$103$	−5.45369	−0.537368	−0.268684	+	0.963228i	$0.586589\pi$
$104$	0	0
$105$	1.95696	0.190980
$106$	0	0
$107$	−11.2751	−1.09000	−0.545002	−	0.838435i	$-0.683471\pi$
$107$	−11.2751	−1.09000	−0.545002	+	0.838435i	$0.683471\pi$
$108$	0	0
$109$	9.79791	0.938470	0.469235	−	0.883073i	$-0.344530\pi$
$109$	9.79791	0.938470	0.469235	+	0.883073i	$0.344530\pi$
$110$	0	0
$111$	−4.23843	−0.402294
$112$	0	0
$113$	1.58476	0.149082	0.0745410	−	0.997218i	$-0.476251\pi$
$113$	1.58476	0.149082	0.0745410	+	0.997218i	$0.476251\pi$
$114$	0	0
$115$	3.02796	0.282359
$116$	0	0
$117$	5.59229	0.517007
$118$	0	0
$119$	−19.4805	−1.78577
$120$	0	0
$121$	4.04774	0.367976
$122$	0	0
$123$	8.76770	0.790557
$124$	0	0
$125$	4.85120	0.433904
$126$	0	0
$127$	18.5456	1.64566	0.822829	−	0.568289i	$-0.192394\pi$
$127$	18.5456	1.64566	0.822829	+	0.568289i	$0.192394\pi$
$128$	0	0
$129$	−6.76790	−0.595880
$130$	0	0
$131$	13.0926	1.14391	0.571954	−	0.820286i	$-0.306186\pi$
$131$	13.0926	1.14391	0.571954	+	0.820286i	$0.306186\pi$
$132$	0	0
$133$	5.68173	0.492668
$134$	0	0
$135$	−0.497428	−0.0428118
$136$	0	0
$137$	−2.12742	−0.181758	−0.0908791	−	0.995862i	$-0.528968\pi$
$137$	−2.12742	−0.181758	−0.0908791	+	0.995862i	$0.528968\pi$
$138$	0	0
$139$	−5.70990	−0.484307	−0.242154	−	0.970238i	$-0.577854\pi$
$139$	−5.70990	−0.484307	−0.242154	+	0.970238i	$0.577854\pi$
$140$	0	0
$141$	−8.34030	−0.702380
$142$	0	0
$143$	−21.6933	−1.81408
$144$	0	0
$145$	0.620925	0.0515651
$146$	0	0
$147$	8.47760	0.699221
$148$	0	0
$149$	3.10697	0.254533	0.127267	−	0.991869i	$-0.459380\pi$
$149$	3.10697	0.254533	0.127267	+	0.991869i	$0.459380\pi$
$150$	0	0
$151$	2.67464	0.217659	0.108830	−	0.994060i	$-0.465290\pi$
$151$	2.67464	0.217659	0.108830	+	0.994060i	$0.465290\pi$
$152$	0	0
$153$	4.95162	0.400315
$154$	0	0
$155$	−2.26678	−0.182072
$156$	0	0
$157$	20.8669	1.66536	0.832679	−	0.553756i	$-0.186806\pi$
$157$	20.8669	1.66536	0.832679	+	0.553756i	$0.186806\pi$
$158$	0	0
$159$	12.1024	0.959786
$160$	0	0
$161$	23.9482	1.88738
$162$	0	0
$163$	−14.2481	−1.11600	−0.557999	−	0.829842i	$-0.688431\pi$
$163$	−14.2481	−1.11600	−0.557999	+	0.829842i	$0.688431\pi$
$164$	0	0
$165$	1.92959	0.150219
$166$	0	0
$167$	−0.0405751	−0.00313980	−0.00156990	−	0.999999i	$-0.500500\pi$
$167$	−0.0405751	−0.00313980	−0.00156990	+	0.999999i	$0.500500\pi$
$168$	0	0
$169$	18.2737	1.40567
$170$	0	0
$171$	−1.44420	−0.110441
$172$	0	0
$173$	1.14760	0.0872506	0.0436253	−	0.999048i	$-0.486109\pi$
$173$	1.14760	0.0872506	0.0436253	+	0.999048i	$0.486109\pi$
$174$	0	0
$175$	18.6973	1.41339
$176$	0	0
$177$	12.5123	0.940485
$178$	0	0
$179$	19.7463	1.47590	0.737952	−	0.674853i	$-0.235793\pi$
$179$	19.7463	1.47590	0.737952	+	0.674853i	$0.235793\pi$
$180$	0	0
$181$	−17.0060	−1.26404	−0.632022	−	0.774951i	$-0.717775\pi$
$181$	−17.0060	−1.26404	−0.632022	+	0.774951i	$0.717775\pi$
$182$	0	0
$183$	14.3273	1.05910
$184$	0	0
$185$	2.10831	0.155006
$186$	0	0
$187$	−19.2080	−1.40463
$188$	0	0
$189$	−3.93416	−0.286168
$190$	0	0
$191$	19.4108	1.40452	0.702258	−	0.711923i	$-0.252175\pi$
$191$	19.4108	1.40452	0.702258	+	0.711923i	$0.252175\pi$
$192$	0	0
$193$	17.3289	1.24736	0.623680	−	0.781680i	$-0.285637\pi$
$193$	17.3289	1.24736	0.623680	+	0.781680i	$0.285637\pi$
$194$	0	0
$195$	−2.78176	−0.199206
$196$	0	0
$197$	11.8102	0.841440	0.420720	−	0.907190i	$-0.361777\pi$
$197$	11.8102	0.841440	0.420720	+	0.907190i	$0.361777\pi$
$198$	0	0
$199$	−3.85397	−0.273200	−0.136600	−	0.990626i	$-0.543618\pi$
$199$	−3.85397	−0.273200	−0.136600	+	0.990626i	$0.543618\pi$
$200$	0	0
$201$	1.78162	0.125666
$202$	0	0
$203$	4.91090	0.344678
$204$	0	0
$205$	−4.36130	−0.304606
$206$	0	0
$207$	−6.08724	−0.423093
$208$	0	0
$209$	5.60227	0.387517
$210$	0	0
$211$	−3.71207	−0.255549	−0.127775	−	0.991803i	$-0.540783\pi$
$211$	−3.71207	−0.255549	−0.127775	+	0.991803i	$0.540783\pi$
$212$	0	0
$213$	−4.74350	−0.325019
$214$	0	0
$215$	3.36654	0.229596
$216$	0	0
$217$	−17.9280	−1.21703
$218$	0	0
$219$	−12.1114	−0.818414
$220$	0	0
$221$	27.6909	1.86269
$222$	0	0
$223$	6.19793	0.415044	0.207522	−	0.978230i	$-0.433460\pi$
$223$	6.19793	0.415044	0.207522	+	0.978230i	$0.433460\pi$
$224$	0	0
$225$	−4.75257	−0.316838
$226$	0	0
$227$	4.13264	0.274293	0.137146	−	0.990551i	$-0.456207\pi$
$227$	4.13264	0.274293	0.137146	+	0.990551i	$0.456207\pi$
$228$	0	0
$229$	2.46880	0.163143	0.0815714	−	0.996667i	$-0.474006\pi$
$229$	2.46880	0.163143	0.0815714	+	0.996667i	$0.474006\pi$
$230$	0	0
$231$	15.2612	1.00411
$232$	0	0
$233$	−3.80807	−0.249475	−0.124737	−	0.992190i	$-0.539809\pi$
$233$	−3.80807	−0.249475	−0.124737	+	0.992190i	$0.539809\pi$
$234$	0	0
$235$	4.14870	0.270631
$236$	0	0
$237$	5.00178	0.324901
$238$	0	0
$239$	11.5208	0.745221	0.372610	−	0.927988i	$-0.378463\pi$
$239$	11.5208	0.745221	0.372610	+	0.927988i	$0.378463\pi$
$240$	0	0
$241$	25.2538	1.62674	0.813370	−	0.581746i	$-0.197630\pi$
$241$	25.2538	1.62674	0.813370	+	0.581746i	$0.197630\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.21699	−0.269414
$246$	0	0
$247$	−8.07640	−0.513889
$248$	0	0
$249$	9.32299	0.590821
$250$	0	0
$251$	−11.1481	−0.703660	−0.351830	−	0.936064i	$-0.614440\pi$
$251$	−11.1481	−0.703660	−0.351830	+	0.936064i	$0.614440\pi$
$252$	0	0
$253$	23.6133	1.48455
$254$	0	0
$255$	−2.46307	−0.154244
$256$	0	0
$257$	−12.4605	−0.777266	−0.388633	−	0.921393i	$-0.627053\pi$
$257$	−12.4605	−0.777266	−0.388633	+	0.921393i	$0.627053\pi$
$258$	0	0
$259$	16.6747	1.03611
$260$	0	0
$261$	−1.24827	−0.0772662
$262$	0	0
$263$	19.2879	1.18934	0.594672	−	0.803968i	$-0.297282\pi$
$263$	19.2879	1.18934	0.594672	+	0.803968i	$0.297282\pi$
$264$	0	0
$265$	−6.02009	−0.369811
$266$	0	0
$267$	0.499936	0.0305956
$268$	0	0
$269$	−19.4006	−1.18288	−0.591439	−	0.806350i	$-0.701440\pi$
$269$	−19.4006	−1.18288	−0.591439	+	0.806350i	$0.701440\pi$
$270$	0	0
$271$	25.6124	1.55584	0.777922	−	0.628361i	$-0.216274\pi$
$271$	25.6124	1.55584	0.777922	+	0.628361i	$0.216274\pi$
$272$	0	0
$273$	−22.0009	−1.33156
$274$	0	0
$275$	18.4359	1.11173
$276$	0	0
$277$	21.2245	1.27525	0.637627	−	0.770345i	$-0.279916\pi$
$277$	21.2245	1.27525	0.637627	+	0.770345i	$0.279916\pi$
$278$	0	0
$279$	4.55700	0.272820
$280$	0	0
$281$	11.4608	0.683693	0.341846	−	0.939756i	$-0.388948\pi$
$281$	11.4608	0.683693	0.341846	+	0.939756i	$0.388948\pi$
$282$	0	0
$283$	−4.71339	−0.280182	−0.140091	−	0.990139i	$-0.544739\pi$
$283$	−4.71339	−0.280182	−0.140091	+	0.990139i	$0.544739\pi$
$284$	0	0
$285$	0.718387	0.0425536
$286$	0	0
$287$	−34.4935	−2.03609
$288$	0	0
$289$	7.51858	0.442269
$290$	0	0
$291$	2.87757	0.168686
$292$	0	0
$293$	−12.1146	−0.707743	−0.353872	−	0.935294i	$-0.615135\pi$
$293$	−12.1146	−0.707743	−0.353872	+	0.935294i	$0.615135\pi$
$294$	0	0
$295$	−6.22398	−0.362374
$296$	0	0
$297$	−3.87914	−0.225091
$298$	0	0
$299$	−34.0416	−1.96868
$300$	0	0
$301$	26.6260	1.53470
$302$	0	0
$303$	2.89295	0.166196
$304$	0	0
$305$	−7.12678	−0.408078
$306$	0	0
$307$	−14.6263	−0.834766	−0.417383	−	0.908731i	$-0.637053\pi$
$307$	−14.6263	−0.834766	−0.417383	+	0.908731i	$0.637053\pi$
$308$	0	0
$309$	−5.45369	−0.310250
$310$	0	0
$311$	25.6780	1.45607	0.728034	−	0.685541i	$-0.240434\pi$
$311$	25.6780	1.45607	0.728034	+	0.685541i	$0.240434\pi$
$312$	0	0
$313$	−8.07696	−0.456537	−0.228268	−	0.973598i	$-0.573306\pi$
$313$	−8.07696	−0.456537	−0.228268	+	0.973598i	$0.573306\pi$
$314$	0	0
$315$	1.95696	0.110262
$316$	0	0
$317$	10.6564	0.598525	0.299262	−	0.954171i	$-0.403259\pi$
$317$	10.6564	0.598525	0.299262	+	0.954171i	$0.403259\pi$
$318$	0	0
$319$	4.84223	0.271113
$320$	0	0
$321$	−11.2751	−0.629314
$322$	0	0
$323$	−7.15115	−0.397901
$324$	0	0
$325$	−26.5777	−1.47427
$326$	0	0
$327$	9.79791	0.541826
$328$	0	0
$329$	32.8121	1.80899
$330$	0	0
$331$	18.6801	1.02675	0.513375	−	0.858165i	$-0.328395\pi$
$331$	18.6801	1.02675	0.513375	+	0.858165i	$0.328395\pi$
$332$	0	0
$333$	−4.23843	−0.232265
$334$	0	0
$335$	−0.886226	−0.0484197
$336$	0	0
$337$	−16.5324	−0.900577	−0.450289	−	0.892883i	$-0.648679\pi$
$337$	−16.5324	−0.900577	−0.450289	+	0.892883i	$0.648679\pi$
$338$	0	0
$339$	1.58476	0.0860726
$340$	0	0
$341$	−17.6772	−0.957276
$342$	0	0
$343$	−5.81312	−0.313879
$344$	0	0
$345$	3.02796	0.163020
$346$	0	0
$347$	31.0294	1.66575	0.832873	−	0.553464i	$-0.186694\pi$
$347$	31.0294	1.66575	0.832873	+	0.553464i	$0.186694\pi$
$348$	0	0
$349$	−18.6293	−0.997206	−0.498603	−	0.866830i	$-0.666153\pi$
$349$	−18.6293	−0.997206	−0.498603	+	0.866830i	$0.666153\pi$
$350$	0	0
$351$	5.59229	0.298494
$352$	0	0
$353$	14.2492	0.758408	0.379204	−	0.925313i	$-0.376198\pi$
$353$	14.2492	0.758408	0.379204	+	0.925313i	$0.376198\pi$
$354$	0	0
$355$	2.35955	0.125232
$356$	0	0
$357$	−19.4805	−1.03102
$358$	0	0
$359$	16.7717	0.885175	0.442587	−	0.896725i	$-0.354061\pi$
$359$	16.7717	0.885175	0.442587	+	0.896725i	$0.354061\pi$
$360$	0	0
$361$	−16.9143	−0.890225
$362$	0	0
$363$	4.04774	0.212451
$364$	0	0
$365$	6.02456	0.315340
$366$	0	0
$367$	−12.4003	−0.647290	−0.323645	−	0.946179i	$-0.604908\pi$
$367$	−12.4003	−0.647290	−0.323645	+	0.946179i	$0.604908\pi$
$368$	0	0
$369$	8.76770	0.456428
$370$	0	0
$371$	−47.6129	−2.47194
$372$	0	0
$373$	0.145087	0.00751232	0.00375616	−	0.999993i	$-0.498804\pi$
$373$	0.145087	0.00751232	0.00375616	+	0.999993i	$0.498804\pi$
$374$	0	0
$375$	4.85120	0.250515
$376$	0	0
$377$	−6.98070	−0.359524
$378$	0	0
$379$	−8.63854	−0.443732	−0.221866	−	0.975077i	$-0.571215\pi$
$379$	−8.63854	−0.443732	−0.221866	+	0.975077i	$0.571215\pi$
$380$	0	0
$381$	18.5456	0.950121
$382$	0	0
$383$	7.26191	0.371066	0.185533	−	0.982638i	$-0.440599\pi$
$383$	7.26191	0.371066	0.185533	+	0.982638i	$0.440599\pi$
$384$	0	0
$385$	−7.59132	−0.386890
$386$	0	0
$387$	−6.76790	−0.344032
$388$	0	0
$389$	−26.5993	−1.34864	−0.674319	−	0.738440i	$-0.735563\pi$
$389$	−26.5993	−1.34864	−0.674319	+	0.738440i	$0.735563\pi$
$390$	0	0
$391$	−30.1417	−1.52433
$392$	0	0
$393$	13.0926	0.660435
$394$	0	0
$395$	−2.48802	−0.125186
$396$	0	0
$397$	15.7057	0.788248	0.394124	−	0.919057i	$-0.371048\pi$
$397$	15.7057	0.788248	0.394124	+	0.919057i	$0.371048\pi$
$398$	0	0
$399$	5.68173	0.284442
$400$	0	0
$401$	−34.7848	−1.73707	−0.868536	−	0.495626i	$-0.834939\pi$
$401$	−34.7848	−1.73707	−0.868536	+	0.495626i	$0.834939\pi$
$402$	0	0
$403$	25.4840	1.26945
$404$	0	0
$405$	−0.497428	−0.0247174
$406$	0	0
$407$	16.4415	0.814974
$408$	0	0
$409$	−25.7242	−1.27198	−0.635989	−	0.771698i	$-0.719408\pi$
$409$	−25.7242	−1.27198	−0.635989	+	0.771698i	$0.719408\pi$
$410$	0	0
$411$	−2.12742	−0.104938
$412$	0	0
$413$	−49.2255	−2.42223
$414$	0	0
$415$	−4.63751	−0.227647
$416$	0	0
$417$	−5.70990	−0.279615
$418$	0	0
$419$	1.93718	0.0946375	0.0473187	−	0.998880i	$-0.484932\pi$
$419$	1.93718	0.0946375	0.0473187	+	0.998880i	$0.484932\pi$
$420$	0	0
$421$	21.9795	1.07122	0.535609	−	0.844466i	$-0.320082\pi$
$421$	21.9795	1.07122	0.535609	+	0.844466i	$0.320082\pi$
$422$	0	0
$423$	−8.34030	−0.405519
$424$	0	0
$425$	−23.5329	−1.14151
$426$	0	0
$427$	−56.3657	−2.72773
$428$	0	0
$429$	−21.6933	−1.04736
$430$	0	0
$431$	0.650134	0.0313159	0.0156579	−	0.999877i	$-0.495016\pi$
$431$	0.650134	0.0313159	0.0156579	+	0.999877i	$0.495016\pi$
$432$	0	0
$433$	−26.9706	−1.29612	−0.648062	−	0.761588i	$-0.724420\pi$
$433$	−26.9706	−1.29612	−0.648062	+	0.761588i	$0.724420\pi$
$434$	0	0
$435$	0.620925	0.0297711
$436$	0	0
$437$	8.79122	0.420541
$438$	0	0
$439$	4.69426	0.224045	0.112022	−	0.993706i	$-0.464267\pi$
$439$	4.69426	0.224045	0.112022	+	0.993706i	$0.464267\pi$
$440$	0	0
$441$	8.47760	0.403695
$442$	0	0
$443$	26.6928	1.26821	0.634106	−	0.773246i	$-0.281368\pi$
$443$	26.6928	1.26821	0.634106	+	0.773246i	$0.281368\pi$
$444$	0	0
$445$	−0.248682	−0.0117887
$446$	0	0
$447$	3.10697	0.146955
$448$	0	0
$449$	−11.7236	−0.553269	−0.276635	−	0.960975i	$-0.589219\pi$
$449$	−11.7236	−0.553269	−0.276635	+	0.960975i	$0.589219\pi$
$450$	0	0
$451$	−34.0112	−1.60152
$452$	0	0
$453$	2.67464	0.125666
$454$	0	0
$455$	10.9439	0.513057
$456$	0	0
$457$	−26.7576	−1.25167	−0.625834	−	0.779956i	$-0.715241\pi$
$457$	−26.7576	−1.25167	−0.625834	+	0.779956i	$0.715241\pi$
$458$	0	0
$459$	4.95162	0.231122
$460$	0	0
$461$	−19.8359	−0.923852	−0.461926	−	0.886918i	$-0.652841\pi$
$461$	−19.8359	−0.923852	−0.461926	+	0.886918i	$0.652841\pi$
$462$	0	0
$463$	−30.9670	−1.43916	−0.719579	−	0.694410i	$-0.755665\pi$
$463$	−30.9670	−1.43916	−0.719579	+	0.694410i	$0.755665\pi$
$464$	0	0
$465$	−2.26678	−0.105119
$466$	0	0
$467$	38.9764	1.80361	0.901807	−	0.432138i	$-0.142241\pi$
$467$	38.9764	1.80361	0.901807	+	0.432138i	$0.142241\pi$
$468$	0	0
$469$	−7.00917	−0.323653
$470$	0	0
$471$	20.8669	0.961495
$472$	0	0
$473$	26.2536	1.20714
$474$	0	0
$475$	6.86367	0.314927
$476$	0	0
$477$	12.1024	0.554132
$478$	0	0
$479$	10.6385	0.486086	0.243043	−	0.970016i	$-0.421854\pi$
$479$	10.6385	0.486086	0.243043	+	0.970016i	$0.421854\pi$
$480$	0	0
$481$	−23.7025	−1.08074
$482$	0	0
$483$	23.9482	1.08968
$484$	0	0
$485$	−1.43139	−0.0649959
$486$	0	0
$487$	−26.2943	−1.19151	−0.595754	−	0.803167i	$-0.703147\pi$
$487$	−26.2943	−1.19151	−0.595754	+	0.803167i	$0.703147\pi$
$488$	0	0
$489$	−14.2481	−0.644322
$490$	0	0
$491$	35.0438	1.58151	0.790753	−	0.612135i	$-0.209689\pi$
$491$	35.0438	1.58151	0.790753	+	0.612135i	$0.209689\pi$
$492$	0	0
$493$	−6.18098	−0.278377
$494$	0	0
$495$	1.92959	0.0867287
$496$	0	0
$497$	18.6617	0.837091
$498$	0	0
$499$	−21.3332	−0.955007	−0.477504	−	0.878630i	$-0.658458\pi$
$499$	−21.3332	−0.955007	−0.477504	+	0.878630i	$0.658458\pi$
$500$	0	0
$501$	−0.0405751	−0.00181276
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−1.43903	−0.0640361
$506$	0	0
$507$	18.2737	0.811562
$508$	0	0
$509$	−7.76973	−0.344387	−0.172194	−	0.985063i	$-0.555085\pi$
$509$	−7.76973	−0.344387	−0.172194	+	0.985063i	$0.555085\pi$
$510$	0	0
$511$	47.6483	2.10784
$512$	0	0
$513$	−1.44420	−0.0637631
$514$	0	0
$515$	2.71282	0.119541
$516$	0	0
$517$	32.3532	1.42289
$518$	0	0
$519$	1.14760	0.0503742
$520$	0	0
$521$	−33.8416	−1.48263	−0.741314	−	0.671159i	$-0.765797\pi$
$521$	−33.8416	−1.48263	−0.741314	+	0.671159i	$0.765797\pi$
$522$	0	0
$523$	42.4242	1.85508	0.927540	−	0.373725i	$-0.121920\pi$
$523$	42.4242	1.85508	0.927540	+	0.373725i	$0.121920\pi$
$524$	0	0
$525$	18.6973	0.816019
$526$	0	0
$527$	22.5645	0.982927
$528$	0	0
$529$	14.0545	0.611066
$530$	0	0
$531$	12.5123	0.542989
$532$	0	0
$533$	49.0315	2.12379
$534$	0	0
$535$	5.60854	0.242478
$536$	0	0
$537$	19.7463	0.852114
$538$	0	0
$539$	−32.8858	−1.41649
$540$	0	0
$541$	−18.5897	−0.799234	−0.399617	−	0.916682i	$-0.630857\pi$
$541$	−18.5897	−0.799234	−0.399617	+	0.916682i	$0.630857\pi$
$542$	0	0
$543$	−17.0060	−0.729796
$544$	0	0
$545$	−4.87375	−0.208769
$546$	0	0
$547$	17.6437	0.754389	0.377194	−	0.926134i	$-0.376889\pi$
$547$	17.6437	0.754389	0.377194	+	0.926134i	$0.376889\pi$
$548$	0	0
$549$	14.3273	0.611473
$550$	0	0
$551$	1.80276	0.0768002
$552$	0	0
$553$	−19.6778	−0.836785
$554$	0	0
$555$	2.10831	0.0894929
$556$	0	0
$557$	−12.1014	−0.512751	−0.256376	−	0.966577i	$-0.582528\pi$
$557$	−12.1014	−0.512751	−0.256376	+	0.966577i	$0.582528\pi$
$558$	0	0
$559$	−37.8480	−1.60080
$560$	0	0
$561$	−19.2080	−0.810964
$562$	0	0
$563$	19.5138	0.822409	0.411204	−	0.911543i	$-0.365108\pi$
$563$	19.5138	0.822409	0.411204	+	0.911543i	$0.365108\pi$
$564$	0	0
$565$	−0.788305	−0.0331643
$566$	0	0
$567$	−3.93416	−0.165219
$568$	0	0
$569$	−35.6871	−1.49608	−0.748041	−	0.663653i	$-0.769005\pi$
$569$	−35.6871	−1.49608	−0.748041	+	0.663653i	$0.769005\pi$
$570$	0	0
$571$	40.1313	1.67944	0.839722	−	0.543016i	$-0.182718\pi$
$571$	40.1313	1.67944	0.839722	+	0.543016i	$0.182718\pi$
$572$	0	0
$573$	19.4108	0.810897
$574$	0	0
$575$	28.9300	1.20647
$576$	0	0
$577$	47.8414	1.99166	0.995832	−	0.0912075i	$-0.0290727\pi$
$577$	47.8414	1.99166	0.995832	+	0.0912075i	$0.0290727\pi$
$578$	0	0
$579$	17.3289	0.720163
$580$	0	0
$581$	−36.6781	−1.52167
$582$	0	0
$583$	−46.9471	−1.94435
$584$	0	0
$585$	−2.78176	−0.115012
$586$	0	0
$587$	35.9460	1.48365	0.741825	−	0.670594i	$-0.233961\pi$
$587$	35.9460	1.48365	0.741825	+	0.670594i	$0.233961\pi$
$588$	0	0
$589$	−6.58123	−0.271175
$590$	0	0
$591$	11.8102	0.485806
$592$	0	0
$593$	29.3259	1.20427	0.602134	−	0.798395i	$-0.294317\pi$
$593$	29.3259	1.20427	0.602134	+	0.798395i	$0.294317\pi$
$594$	0	0
$595$	9.69013	0.397256
$596$	0	0
$597$	−3.85397	−0.157732
$598$	0	0
$599$	25.0672	1.02422	0.512109	−	0.858921i	$-0.328864\pi$
$599$	25.0672	1.02422	0.512109	+	0.858921i	$0.328864\pi$
$600$	0	0
$601$	22.3595	0.912064	0.456032	−	0.889963i	$-0.349270\pi$
$601$	22.3595	0.912064	0.456032	+	0.889963i	$0.349270\pi$
$602$	0	0
$603$	1.78162	0.0725531
$604$	0	0
$605$	−2.01346	−0.0818587
$606$	0	0
$607$	−38.6674	−1.56946	−0.784730	−	0.619837i	$-0.787199\pi$
$607$	−38.6674	−1.56946	−0.784730	+	0.619837i	$0.787199\pi$
$608$	0	0
$609$	4.91090	0.199000
$610$	0	0
$611$	−46.6414	−1.88691
$612$	0	0
$613$	−20.9148	−0.844741	−0.422371	−	0.906423i	$-0.638802\pi$
$613$	−20.9148	−0.844741	−0.422371	+	0.906423i	$0.638802\pi$
$614$	0	0
$615$	−4.36130	−0.175865
$616$	0	0
$617$	−1.60794	−0.0647333	−0.0323667	−	0.999476i	$-0.510304\pi$
$617$	−1.60794	−0.0647333	−0.0323667	+	0.999476i	$0.510304\pi$
$618$	0	0
$619$	−10.8366	−0.435561	−0.217781	−	0.975998i	$-0.569882\pi$
$619$	−10.8366	−0.435561	−0.217781	+	0.975998i	$0.569882\pi$
$620$	0	0
$621$	−6.08724	−0.244273
$622$	0	0
$623$	−1.96683	−0.0787993
$624$	0	0
$625$	21.3497	0.853988
$626$	0	0
$627$	5.60227	0.223733
$628$	0	0
$629$	−20.9871	−0.836811
$630$	0	0
$631$	−29.5958	−1.17819	−0.589096	−	0.808063i	$-0.700516\pi$
$631$	−29.5958	−1.17819	−0.589096	+	0.808063i	$0.700516\pi$
$632$	0	0
$633$	−3.71207	−0.147541
$634$	0	0
$635$	−9.22511	−0.366087
$636$	0	0
$637$	47.4092	1.87842
$638$	0	0
$639$	−4.74350	−0.187650
$640$	0	0
$641$	24.8705	0.982327	0.491164	−	0.871067i	$-0.336572\pi$
$641$	24.8705	0.982327	0.491164	+	0.871067i	$0.336572\pi$
$642$	0	0
$643$	−24.3517	−0.960336	−0.480168	−	0.877177i	$-0.659424\pi$
$643$	−24.3517	−0.960336	−0.480168	+	0.877177i	$0.659424\pi$
$644$	0	0
$645$	3.36654	0.132557
$646$	0	0
$647$	−3.17866	−0.124966	−0.0624831	−	0.998046i	$-0.519902\pi$
$647$	−3.17866	−0.124966	−0.0624831	+	0.998046i	$0.519902\pi$
$648$	0	0
$649$	−48.5371	−1.90525
$650$	0	0
$651$	−17.9280	−0.702652
$652$	0	0
$653$	29.5934	1.15808	0.579039	−	0.815300i	$-0.303428\pi$
$653$	29.5934	1.15808	0.579039	+	0.815300i	$0.303428\pi$
$654$	0	0
$655$	−6.51263	−0.254470
$656$	0	0
$657$	−12.1114	−0.472512
$658$	0	0
$659$	10.1095	0.393812	0.196906	−	0.980422i	$-0.436911\pi$
$659$	10.1095	0.393812	0.196906	+	0.980422i	$0.436911\pi$
$660$	0	0
$661$	18.6770	0.726450	0.363225	−	0.931701i	$-0.381676\pi$
$661$	18.6770	0.726450	0.363225	+	0.931701i	$0.381676\pi$
$662$	0	0
$663$	27.6909	1.07543
$664$	0	0
$665$	−2.82625	−0.109597
$666$	0	0
$667$	7.59854	0.294217
$668$	0	0
$669$	6.19793	0.239626
$670$	0	0
$671$	−55.5775	−2.14555
$672$	0	0
$673$	−18.3364	−0.706817	−0.353408	−	0.935469i	$-0.614977\pi$
$673$	−18.3364	−0.706817	−0.353408	+	0.935469i	$0.614977\pi$
$674$	0	0
$675$	−4.75257	−0.182926
$676$	0	0
$677$	−30.0504	−1.15493	−0.577466	−	0.816415i	$-0.695958\pi$
$677$	−30.0504	−1.15493	−0.577466	+	0.816415i	$0.695958\pi$
$678$	0	0
$679$	−11.3208	−0.434454
$680$	0	0
$681$	4.13264	0.158363
$682$	0	0
$683$	−16.0140	−0.612758	−0.306379	−	0.951910i	$-0.599118\pi$
$683$	−16.0140	−0.612758	−0.306379	+	0.951910i	$0.599118\pi$
$684$	0	0
$685$	1.05824	0.0404333
$686$	0	0
$687$	2.46880	0.0941906
$688$	0	0
$689$	67.6803	2.57841
$690$	0	0
$691$	−4.24542	−0.161503	−0.0807517	−	0.996734i	$-0.525732\pi$
$691$	−4.24542	−0.161503	−0.0807517	+	0.996734i	$0.525732\pi$
$692$	0	0
$693$	15.2612	0.579723
$694$	0	0
$695$	2.84026	0.107737
$696$	0	0
$697$	43.4144	1.64444
$698$	0	0
$699$	−3.80807	−0.144034
$700$	0	0
$701$	37.6344	1.42143	0.710715	−	0.703480i	$-0.248371\pi$
$701$	37.6344	1.42143	0.710715	+	0.703480i	$0.248371\pi$
$702$	0	0
$703$	6.12116	0.230864
$704$	0	0
$705$	4.14870	0.156249
$706$	0	0
$707$	−11.3813	−0.428039
$708$	0	0
$709$	−19.6809	−0.739132	−0.369566	−	0.929205i	$-0.620494\pi$
$709$	−19.6809	−0.739132	−0.369566	+	0.929205i	$0.620494\pi$
$710$	0	0
$711$	5.00178	0.187581
$712$	0	0
$713$	−27.7396	−1.03885
$714$	0	0
$715$	10.7908	0.403554
$716$	0	0
$717$	11.5208	0.430253
$718$	0	0
$719$	−30.0618	−1.12112	−0.560558	−	0.828115i	$-0.689413\pi$
$719$	−30.0618	−1.12112	−0.560558	+	0.828115i	$0.689413\pi$
$720$	0	0
$721$	21.4557	0.799052
$722$	0	0
$723$	25.2538	0.939199
$724$	0	0
$725$	5.93250	0.220327
$726$	0	0
$727$	−39.0689	−1.44899	−0.724493	−	0.689282i	$-0.757926\pi$
$727$	−39.0689	−1.44899	−0.724493	+	0.689282i	$0.757926\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.5121	−1.23949
$732$	0	0
$733$	−33.0526	−1.22083	−0.610414	−	0.792083i	$-0.708997\pi$
$733$	−33.0526	−1.22083	−0.610414	+	0.792083i	$0.708997\pi$
$734$	0	0
$735$	−4.21699	−0.155546
$736$	0	0
$737$	−6.91115	−0.254575
$738$	0	0
$739$	−44.2909	−1.62927	−0.814634	−	0.579975i	$-0.803062\pi$
$739$	−44.2909	−1.62927	−0.814634	+	0.579975i	$0.803062\pi$
$740$	0	0
$741$	−8.07640	−0.296694
$742$	0	0
$743$	−1.00447	−0.0368503	−0.0184252	−	0.999830i	$-0.505865\pi$
$743$	−1.00447	−0.0368503	−0.0184252	+	0.999830i	$0.505865\pi$
$744$	0	0
$745$	−1.54549	−0.0566225
$746$	0	0
$747$	9.32299	0.341110
$748$	0	0
$749$	44.3580	1.62080
$750$	0	0
$751$	26.0562	0.950805	0.475402	−	0.879768i	$-0.342302\pi$
$751$	26.0562	0.950805	0.475402	+	0.879768i	$0.342302\pi$
$752$	0	0
$753$	−11.1481	−0.406258
$754$	0	0
$755$	−1.33044	−0.0484197
$756$	0	0
$757$	−6.16775	−0.224171	−0.112085	−	0.993699i	$-0.535753\pi$
$757$	−6.16775	−0.224171	−0.112085	+	0.993699i	$0.535753\pi$
$758$	0	0
$759$	23.6133	0.857108
$760$	0	0
$761$	−11.9985	−0.434944	−0.217472	−	0.976067i	$-0.569781\pi$
$761$	−11.9985	−0.434944	−0.217472	+	0.976067i	$0.569781\pi$
$762$	0	0
$763$	−38.5465	−1.39548
$764$	0	0
$765$	−2.46307	−0.0890526
$766$	0	0
$767$	69.9726	2.52656
$768$	0	0
$769$	−40.7697	−1.47019	−0.735097	−	0.677962i	$-0.762863\pi$
$769$	−40.7697	−1.47019	−0.735097	+	0.677962i	$0.762863\pi$
$770$	0	0
$771$	−12.4605	−0.448754
$772$	0	0
$773$	36.2246	1.30291	0.651454	−	0.758688i	$-0.274159\pi$
$773$	36.2246	1.30291	0.651454	+	0.758688i	$0.274159\pi$
$774$	0	0
$775$	−21.6574	−0.777958
$776$	0	0
$777$	16.6747	0.598200
$778$	0	0
$779$	−12.6623	−0.453676
$780$	0	0
$781$	18.4007	0.658429
$782$	0	0
$783$	−1.24827	−0.0446096
$784$	0	0
$785$	−10.3798	−0.370469
$786$	0	0
$787$	−36.7118	−1.30863	−0.654317	−	0.756220i	$-0.727044\pi$
$787$	−36.7118	−1.30863	−0.654317	+	0.756220i	$0.727044\pi$
$788$	0	0
$789$	19.2879	0.686668
$790$	0	0
$791$	−6.23471	−0.221681
$792$	0	0
$793$	80.1222	2.84522
$794$	0	0
$795$	−6.02009	−0.213510
$796$	0	0
$797$	11.6323	0.412038	0.206019	−	0.978548i	$-0.433949\pi$
$797$	11.6323	0.412038	0.206019	+	0.978548i	$0.433949\pi$
$798$	0	0
$799$	−41.2980	−1.46102
$800$	0	0
$801$	0.499936	0.0176644
$802$	0	0
$803$	46.9819	1.65796
$804$	0	0
$805$	−11.9125	−0.419860
$806$	0	0
$807$	−19.4006	−0.682935
$808$	0	0
$809$	4.72439	0.166101	0.0830504	−	0.996545i	$-0.473534\pi$
$809$	4.72439	0.166101	0.0830504	+	0.996545i	$0.473534\pi$
$810$	0	0
$811$	−55.8136	−1.95988	−0.979941	−	0.199287i	$-0.936137\pi$
$811$	−55.8136	−1.95988	−0.979941	+	0.199287i	$0.936137\pi$
$812$	0	0
$813$	25.6124	0.898267
$814$	0	0
$815$	7.08740	0.248261
$816$	0	0
$817$	9.77423	0.341957
$818$	0	0
$819$	−22.0009	−0.768775
$820$	0	0
$821$	14.9459	0.521617	0.260808	−	0.965391i	$-0.416011\pi$
$821$	14.9459	0.521617	0.260808	+	0.965391i	$0.416011\pi$
$822$	0	0
$823$	−15.6344	−0.544983	−0.272491	−	0.962158i	$-0.587848\pi$
$823$	−15.6344	−0.544983	−0.272491	+	0.962158i	$0.587848\pi$
$824$	0	0
$825$	18.4359	0.641855
$826$	0	0
$827$	−28.2500	−0.982348	−0.491174	−	0.871061i	$-0.663432\pi$
$827$	−28.2500	−0.982348	−0.491174	+	0.871061i	$0.663432\pi$
$828$	0	0
$829$	22.6583	0.786955	0.393477	−	0.919334i	$-0.371272\pi$
$829$	22.6583	0.786955	0.393477	+	0.919334i	$0.371272\pi$
$830$	0	0
$831$	21.2245	0.736268
$832$	0	0
$833$	41.9779	1.45445
$834$	0	0
$835$	0.0201832	0.000698468 0
$836$	0	0
$837$	4.55700	0.157513
$838$	0	0
$839$	5.88181	0.203063	0.101531	−	0.994832i	$-0.467626\pi$
$839$	5.88181	0.203063	0.101531	+	0.994832i	$0.467626\pi$
$840$	0	0
$841$	−27.4418	−0.946269
$842$	0	0
$843$	11.4608	0.394730
$844$	0	0
$845$	−9.08982	−0.312699
$846$	0	0
$847$	−15.9244	−0.547170
$848$	0	0
$849$	−4.71339	−0.161763
$850$	0	0
$851$	25.8004	0.884425
$852$	0	0
$853$	−21.8576	−0.748390	−0.374195	−	0.927350i	$-0.622081\pi$
$853$	−21.8576	−0.748390	−0.374195	+	0.927350i	$0.622081\pi$
$854$	0	0
$855$	0.718387	0.0245683
$856$	0	0
$857$	13.2090	0.451211	0.225606	−	0.974219i	$-0.427564\pi$
$857$	13.2090	0.451211	0.225606	+	0.974219i	$0.427564\pi$
$858$	0	0
$859$	5.87933	0.200600	0.100300	−	0.994957i	$-0.468020\pi$
$859$	5.87933	0.200600	0.100300	+	0.994957i	$0.468020\pi$
$860$	0	0
$861$	−34.4935	−1.17554
$862$	0	0
$863$	23.1326	0.787444	0.393722	−	0.919230i	$-0.371187\pi$
$863$	23.1326	0.787444	0.393722	+	0.919230i	$0.371187\pi$
$864$	0	0
$865$	−0.570849	−0.0194095
$866$	0	0
$867$	7.51858	0.255344
$868$	0	0
$869$	−19.4026	−0.658189
$870$	0	0
$871$	9.96332	0.337594
$872$	0	0
$873$	2.87757	0.0973911
$874$	0	0
$875$	−19.0854	−0.645203
$876$	0	0
$877$	47.5937	1.60713	0.803563	−	0.595220i	$-0.202935\pi$
$877$	47.5937	1.60713	0.803563	+	0.595220i	$0.202935\pi$
$878$	0	0
$879$	−12.1146	−0.408616
$880$	0	0
$881$	8.25554	0.278136	0.139068	−	0.990283i	$-0.455589\pi$
$881$	8.25554	0.278136	0.139068	+	0.990283i	$0.455589\pi$
$882$	0	0
$883$	−2.25496	−0.0758856	−0.0379428	−	0.999280i	$-0.512080\pi$
$883$	−2.25496	−0.0758856	−0.0379428	+	0.999280i	$0.512080\pi$
$884$	0	0
$885$	−6.22398	−0.209217
$886$	0	0
$887$	−9.80681	−0.329280	−0.164640	−	0.986354i	$-0.552646\pi$
$887$	−9.80681	−0.329280	−0.164640	+	0.986354i	$0.552646\pi$
$888$	0	0
$889$	−72.9614	−2.44705
$890$	0	0
$891$	−3.87914	−0.129956
$892$	0	0
$893$	12.0451	0.403074
$894$	0	0
$895$	−9.82234	−0.328325
$896$	0	0
$897$	−34.0416	−1.13662
$898$	0	0
$899$	−5.68838	−0.189718
$900$	0	0
$901$	59.9267	1.99645
$902$	0	0
$903$	26.6260	0.886057
$904$	0	0
$905$	8.45924	0.281195
$906$	0	0
$907$	−53.4398	−1.77444	−0.887220	−	0.461346i	$-0.847367\pi$
$907$	−53.4398	−1.77444	−0.887220	+	0.461346i	$0.847367\pi$
$908$	0	0
$909$	2.89295	0.0959530
$910$	0	0
$911$	14.6800	0.486372	0.243186	−	0.969980i	$-0.421808\pi$
$911$	14.6800	0.486372	0.243186	+	0.969980i	$0.421808\pi$
$912$	0	0
$913$	−36.1652	−1.19689
$914$	0	0
$915$	−7.12678	−0.235604
$916$	0	0
$917$	−51.5085	−1.70096
$918$	0	0
$919$	10.9287	0.360504	0.180252	−	0.983620i	$-0.442309\pi$
$919$	10.9287	0.360504	0.180252	+	0.983620i	$0.442309\pi$
$920$	0	0
$921$	−14.6263	−0.481952
$922$	0	0
$923$	−26.5270	−0.873147
$924$	0	0
$925$	20.1434	0.662312
$926$	0	0
$927$	−5.45369	−0.179123
$928$	0	0
$929$	−45.3152	−1.48674	−0.743371	−	0.668879i	$-0.766774\pi$
$929$	−45.3152	−1.48674	−0.743371	+	0.668879i	$0.766774\pi$
$930$	0	0
$931$	−12.2434	−0.401261
$932$	0	0
$933$	25.6780	0.840661
$934$	0	0
$935$	9.55461	0.312469
$936$	0	0
$937$	−35.1840	−1.14941	−0.574706	−	0.818360i	$-0.694883\pi$
$937$	−35.1840	−1.14941	−0.574706	+	0.818360i	$0.694883\pi$
$938$	0	0
$939$	−8.07696	−0.263582
$940$	0	0
$941$	−47.2624	−1.54071	−0.770356	−	0.637614i	$-0.779921\pi$
$941$	−47.2624	−1.54071	−0.770356	+	0.637614i	$0.779921\pi$
$942$	0	0
$943$	−53.3711	−1.73800
$944$	0	0
$945$	1.95696	0.0636599
$946$	0	0
$947$	−2.23723	−0.0727002	−0.0363501	−	0.999339i	$-0.511573\pi$
$947$	−2.23723	−0.0727002	−0.0363501	+	0.999339i	$0.511573\pi$
$948$	0	0
$949$	−67.7306	−2.19863
$950$	0	0
$951$	10.6564	0.345558
$952$	0	0
$953$	2.85769	0.0925697	0.0462849	−	0.998928i	$-0.485262\pi$
$953$	2.85769	0.0925697	0.0462849	+	0.998928i	$0.485262\pi$
$954$	0	0
$955$	−9.65546	−0.312444
$956$	0	0
$957$	4.84223	0.156527
$958$	0	0
$959$	8.36962	0.270269
$960$	0	0
$961$	−10.2338	−0.330122
$962$	0	0
$963$	−11.2751	−0.363334
$964$	0	0
$965$	−8.61986	−0.277483
$966$	0	0
$967$	15.5514	0.500100	0.250050	−	0.968233i	$-0.419553\pi$
$967$	15.5514	0.500100	0.250050	+	0.968233i	$0.419553\pi$
$968$	0	0
$969$	−7.15115	−0.229728
$970$	0	0
$971$	51.6072	1.65615	0.828077	−	0.560614i	$-0.189435\pi$
$971$	51.6072	1.65615	0.828077	+	0.560614i	$0.189435\pi$
$972$	0	0
$973$	22.4637	0.720152
$974$	0	0
$975$	−26.5777	−0.851168
$976$	0	0
$977$	−10.6951	−0.342165	−0.171082	−	0.985257i	$-0.554726\pi$
$977$	−10.6951	−0.342165	−0.171082	+	0.985257i	$0.554726\pi$
$978$	0	0
$979$	−1.93932	−0.0619811
$980$	0	0
$981$	9.79791	0.312823
$982$	0	0
$983$	31.6063	1.00808	0.504042	−	0.863679i	$-0.331846\pi$
$983$	31.6063	1.00808	0.504042	+	0.863679i	$0.331846\pi$
$984$	0	0
$985$	−5.87471	−0.187184
$986$	0	0
$987$	32.8121	1.04442
$988$	0	0
$989$	41.1978	1.31002
$990$	0	0
$991$	47.0985	1.49613	0.748067	−	0.663623i	$-0.230982\pi$
$991$	47.0985	1.49613	0.748067	+	0.663623i	$0.230982\pi$
$992$	0	0
$993$	18.6801	0.592794
$994$	0	0
$995$	1.91707	0.0607752
$996$	0	0
$997$	29.8979	0.946877	0.473439	−	0.880827i	$-0.343013\pi$
$997$	29.8979	0.946877	0.473439	+	0.880827i	$0.343013\pi$
$998$	0	0
$999$	−4.23843	−0.134098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.h.1.9	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.h.1.9	✓	24	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.497428 q^{5} -3.93416 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.497428 q^{5} -3.93416 q^{7} +1.00000 q^{9} -3.87914 q^{11} +5.59229 q^{13} -0.497428 q^{15} +4.95162 q^{17} -1.44420 q^{19} -3.93416 q^{21} -6.08724 q^{23} -4.75257 q^{25} +1.00000 q^{27} -1.24827 q^{29} +4.55700 q^{31} -3.87914 q^{33} +1.95696 q^{35} -4.23843 q^{37} +5.59229 q^{39} +8.76770 q^{41} -6.76790 q^{43} -0.497428 q^{45} -8.34030 q^{47} +8.47760 q^{49} +4.95162 q^{51} +12.1024 q^{53} +1.92959 q^{55} -1.44420 q^{57} +12.5123 q^{59} +14.3273 q^{61} -3.93416 q^{63} -2.78176 q^{65} +1.78162 q^{67} -6.08724 q^{69} -4.74350 q^{71} -12.1114 q^{73} -4.75257 q^{75} +15.2612 q^{77} +5.00178 q^{79} +1.00000 q^{81} +9.32299 q^{83} -2.46307 q^{85} -1.24827 q^{87} +0.499936 q^{89} -22.0009 q^{91} +4.55700 q^{93} +0.718387 q^{95} +2.87757 q^{97} -3.87914 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9}+O(q^{10}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9	\( 24 q + 24 q^{3} + 18 q^{5} + 9 q^{7} + 24 q^{9} + 9 q^{11} + 7 q^{13} + 18 q^{15} + 12 q^{17} - q^{19} + 9 q^{21} + 22 q^{23} + 48 q^{25} + 24 q^{27} + 44 q^{29} + 15 q^{31} + 9 q^{33} + 3 q^{35} + 18 q^{37} + 7 q^{39} + 37 q^{41} - 8 q^{43} + 18 q^{45} + 24 q^{47} + 63 q^{49} + 12 q^{51} + 59 q^{53} + 20 q^{55} - q^{57} + 44 q^{59} + 33 q^{61} + 9 q^{63} + 15 q^{65} + 13 q^{67} + 22 q^{69} + 47 q^{71} + 28 q^{73} + 48 q^{75} + 19 q^{77} + 12 q^{79} + 24 q^{81} + 17 q^{83} + 31 q^{85} + 44 q^{87} + 41 q^{89} - q^{91} + 15 q^{93} + 58 q^{95} + 51 q^{97} + 9 q^{99}+O(q^{100}) \) 24 * q + 24 * q^3 + 18 * q^5 + 9 * q^7 + 24 * q^9 + 9 * q^11 + 7 * q^13 + 18 * q^15 + 12 * q^17 - q^19 + 9 * q^21 + 22 * q^23 + 48 * q^25 + 24 * q^27 + 44 * q^29 + 15 * q^31 + 9 * q^33 + 3 * q^35 + 18 * q^37 + 7 * q^39 + 37 * q^41 - 8 * q^43 + 18 * q^45 + 24 * q^47 + 63 * q^49 + 12 * q^51 + 59 * q^53 + 20 * q^55 - q^57 + 44 * q^59 + 33 * q^61 + 9 * q^63 + 15 * q^65 + 13 * q^67 + 22 * q^69 + 47 * q^71 + 28 * q^73 + 48 * q^75 + 19 * q^77 + 12 * q^79 + 24 * q^81 + 17 * q^83 + 31 * q^85 + 44 * q^87 + 41 * q^89 - q^91 + 15 * q^93 + 58 * q^95 + 51 * q^97 + 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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