LMFDB - Embedded newform 8024.2.a.v.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 9 x^{17} - 2 x^{16} + 212 x^{15} - 289 x^{14} - 2094 x^{13} + 3933 x^{12} + 11326 x^{11} + \cdots + 1136 $ x^18 - 9x^17 - 2x^16 + 212x^15 - 289x^14 - 2094x^13 + 3933x^12 + 11326x^11 - 23166x^10 - 36429x^9 + 72042x^8 + 69272x^7 - 119982x^6 - 69890x^5 + 98783x^4 + 26872x^3 - 30932x^2 + 1600*x + 1136
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Root			$-1.86517$ of defining polynomial
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.86517 q^{3} -3.00205 q^{5} +3.89738 q^{7} +5.20917 q^{9} +O(q^{10})$	$q+2.86517 q^{3} -3.00205 q^{5} +3.89738 q^{7} +5.20917 q^{9} +2.75032 q^{11} +7.07696 q^{13} -8.60138 q^{15} +1.00000 q^{17} +3.96217 q^{19} +11.1666 q^{21} +2.50107 q^{23} +4.01233 q^{25} +6.32964 q^{27} -6.77067 q^{29} +10.7312 q^{31} +7.88013 q^{33} -11.7001 q^{35} +9.11084 q^{37} +20.2767 q^{39} -5.93402 q^{41} -2.16380 q^{43} -15.6382 q^{45} +2.01633 q^{47} +8.18955 q^{49} +2.86517 q^{51} -12.1100 q^{53} -8.25662 q^{55} +11.3523 q^{57} -1.00000 q^{59} -14.1037 q^{61} +20.3021 q^{63} -21.2454 q^{65} -9.34532 q^{67} +7.16598 q^{69} -7.15597 q^{71} -1.43113 q^{73} +11.4960 q^{75} +10.7190 q^{77} -16.1282 q^{79} +2.50795 q^{81} +2.91492 q^{83} -3.00205 q^{85} -19.3991 q^{87} -4.82557 q^{89} +27.5816 q^{91} +30.7467 q^{93} -11.8946 q^{95} +7.50368 q^{97} +14.3269 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) $ 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9	$ 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) $ 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9 - q^11 + 15 * q^13 + 3 * q^15 + 18 * q^17 + 26 * q^19 - 4 * q^21 + 22 * q^23 + 42 * q^25 + 45 * q^27 + 6 * q^29 + 13 * q^31 - 5 * q^33 + 4 * q^35 + 4 * q^37 + 36 * q^39 - 15 * q^41 + 12 * q^43 + 14 * q^45 + 8 * q^47 - 13 * q^49 + 9 * q^51 - 11 * q^53 + 55 * q^55 - 20 * q^57 - 18 * q^59 + 53 * q^61 + 29 * q^63 - 26 * q^65 + 2 * q^67 + 32 * q^69 + 8 * q^71 - 42 * q^73 + 72 * q^75 + 6 * q^77 - 9 * q^79 + 42 * q^81 - 4 * q^83 + 2 * q^85 + 36 * q^87 + 13 * q^89 + 68 * q^91 + q^93 + 3 * q^95 - 56 * q^97 - 13 * 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
$3$	2.86517	1.65420	0.827102	−	0.562052i	$-0.189988\pi$
$3$	2.86517	1.65420	0.827102	+	0.562052i	$0.189988\pi$
$4$	0	0
$5$	−3.00205	−1.34256	−0.671280	−	0.741204i	$-0.734255\pi$
$5$	−3.00205	−1.34256	−0.671280	+	0.741204i	$0.734255\pi$
$6$	0	0
$7$	3.89738	1.47307	0.736535	−	0.676399i	$-0.236461\pi$
$7$	3.89738	1.47307	0.736535	+	0.676399i	$0.236461\pi$
$8$	0	0
$9$	5.20917	1.73639
$10$	0	0
$11$	2.75032	0.829253	0.414627	−	0.909992i	$-0.363912\pi$
$11$	2.75032	0.829253	0.414627	+	0.909992i	$0.363912\pi$
$12$	0	0
$13$	7.07696	1.96280	0.981398	−	0.191983i	$-0.0614919\pi$
$13$	7.07696	1.96280	0.981398	+	0.191983i	$0.0614919\pi$
$14$	0	0
$15$	−8.60138	−2.22087
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.96217	0.908984	0.454492	−	0.890751i	$-0.349821\pi$
$19$	3.96217	0.908984	0.454492	+	0.890751i	$0.349821\pi$
$20$	0	0
$21$	11.1666	2.43676
$22$	0	0
$23$	2.50107	0.521509	0.260755	−	0.965405i	$-0.416029\pi$
$23$	2.50107	0.521509	0.260755	+	0.965405i	$0.416029\pi$
$24$	0	0
$25$	4.01233	0.802466
$26$	0	0
$27$	6.32964	1.21814
$28$	0	0
$29$	−6.77067	−1.25728	−0.628641	−	0.777696i	$-0.716389\pi$
$29$	−6.77067	−1.25728	−0.628641	+	0.777696i	$0.716389\pi$
$30$	0	0
$31$	10.7312	1.92738	0.963690	−	0.267023i	$-0.0860399\pi$
$31$	10.7312	1.92738	0.963690	+	0.267023i	$0.0860399\pi$
$32$	0	0
$33$	7.88013	1.37175
$34$	0	0
$35$	−11.7001	−1.97768
$36$	0	0
$37$	9.11084	1.49781	0.748906	−	0.662676i	$-0.230579\pi$
$37$	9.11084	1.49781	0.748906	+	0.662676i	$0.230579\pi$
$38$	0	0
$39$	20.2767	3.24687
$40$	0	0
$41$	−5.93402	−0.926738	−0.463369	−	0.886165i	$-0.653360\pi$
$41$	−5.93402	−0.926738	−0.463369	+	0.886165i	$0.653360\pi$
$42$	0	0
$43$	−2.16380	−0.329976	−0.164988	−	0.986296i	$-0.552759\pi$
$43$	−2.16380	−0.329976	−0.164988	+	0.986296i	$0.552759\pi$
$44$	0	0
$45$	−15.6382	−2.33121
$46$	0	0
$47$	2.01633	0.294112	0.147056	−	0.989128i	$-0.453020\pi$
$47$	2.01633	0.294112	0.147056	+	0.989128i	$0.453020\pi$
$48$	0	0
$49$	8.18955	1.16994
$50$	0	0
$51$	2.86517	0.401203
$52$	0	0
$53$	−12.1100	−1.66344	−0.831719	−	0.555197i	$-0.812643\pi$
$53$	−12.1100	−1.66344	−0.831719	+	0.555197i	$0.812643\pi$
$54$	0	0
$55$	−8.25662	−1.11332
$56$	0	0
$57$	11.3523	1.50364
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−14.1037	−1.80580	−0.902899	−	0.429852i	$-0.858566\pi$
$61$	−14.1037	−1.80580	−0.902899	+	0.429852i	$0.858566\pi$
$62$	0	0
$63$	20.3021	2.55782
$64$	0	0
$65$	−21.2454	−2.63517
$66$	0	0
$67$	−9.34532	−1.14171	−0.570856	−	0.821050i	$-0.693389\pi$
$67$	−9.34532	−1.14171	−0.570856	+	0.821050i	$0.693389\pi$
$68$	0	0
$69$	7.16598	0.862683
$70$	0	0
$71$	−7.15597	−0.849257	−0.424629	−	0.905368i	$-0.639595\pi$
$71$	−7.15597	−0.849257	−0.424629	+	0.905368i	$0.639595\pi$
$72$	0	0
$73$	−1.43113	−0.167501	−0.0837505	−	0.996487i	$-0.526690\pi$
$73$	−1.43113	−0.167501	−0.0837505	+	0.996487i	$0.526690\pi$
$74$	0	0
$75$	11.4960	1.32744
$76$	0	0
$77$	10.7190	1.22155
$78$	0	0
$79$	−16.1282	−1.81457	−0.907283	−	0.420521i	$-0.861847\pi$
$79$	−16.1282	−1.81457	−0.907283	+	0.420521i	$0.861847\pi$
$80$	0	0
$81$	2.50795	0.278662
$82$	0	0
$83$	2.91492	0.319954	0.159977	−	0.987121i	$-0.448858\pi$
$83$	2.91492	0.319954	0.159977	+	0.987121i	$0.448858\pi$
$84$	0	0
$85$	−3.00205	−0.325619
$86$	0	0
$87$	−19.3991	−2.07980
$88$	0	0
$89$	−4.82557	−0.511510	−0.255755	−	0.966742i	$-0.582324\pi$
$89$	−4.82557	−0.511510	−0.255755	+	0.966742i	$0.582324\pi$
$90$	0	0
$91$	27.5816	2.89134
$92$	0	0
$93$	30.7467	3.18828
$94$	0	0
$95$	−11.8946	−1.22036
$96$	0	0
$97$	7.50368	0.761883	0.380942	−	0.924599i	$-0.375600\pi$
$97$	7.50368	0.761883	0.380942	+	0.924599i	$0.375600\pi$
$98$	0	0
$99$	14.3269	1.43991
$100$	0	0
$101$	2.57809	0.256530	0.128265	−	0.991740i	$-0.459059\pi$
$101$	2.57809	0.256530	0.128265	+	0.991740i	$0.459059\pi$
$102$	0	0
$103$	−10.9252	−1.07650	−0.538248	−	0.842786i	$-0.680914\pi$
$103$	−10.9252	−1.07650	−0.538248	+	0.842786i	$0.680914\pi$
$104$	0	0
$105$	−33.5228	−3.27149
$106$	0	0
$107$	−10.5249	−1.01748	−0.508738	−	0.860921i	$-0.669888\pi$
$107$	−10.5249	−1.01748	−0.508738	+	0.860921i	$0.669888\pi$
$108$	0	0
$109$	8.89432	0.851921	0.425961	−	0.904742i	$-0.359936\pi$
$109$	8.89432	0.851921	0.425961	+	0.904742i	$0.359936\pi$
$110$	0	0
$111$	26.1040	2.47769
$112$	0	0
$113$	−12.7230	−1.19688	−0.598439	−	0.801168i	$-0.704212\pi$
$113$	−12.7230	−1.19688	−0.598439	+	0.801168i	$0.704212\pi$
$114$	0	0
$115$	−7.50835	−0.700157
$116$	0	0
$117$	36.8651	3.40818
$118$	0	0
$119$	3.89738	0.357272
$120$	0	0
$121$	−3.43573	−0.312339
$122$	0	0
$123$	−17.0019	−1.53301
$124$	0	0
$125$	2.96504	0.265201
$126$	0	0
$127$	5.46190	0.484665	0.242333	−	0.970193i	$-0.422087\pi$
$127$	5.46190	0.484665	0.242333	+	0.970193i	$0.422087\pi$
$128$	0	0
$129$	−6.19964	−0.545848
$130$	0	0
$131$	17.3959	1.51989	0.759945	−	0.649987i	$-0.225226\pi$
$131$	17.3959	1.51989	0.759945	+	0.649987i	$0.225226\pi$
$132$	0	0
$133$	15.4421	1.33900
$134$	0	0
$135$	−19.0019	−1.63543
$136$	0	0
$137$	−14.3692	−1.22764	−0.613820	−	0.789446i	$-0.710368\pi$
$137$	−14.3692	−1.22764	−0.613820	+	0.789446i	$0.710368\pi$
$138$	0	0
$139$	−11.0693	−0.938885	−0.469442	−	0.882963i	$-0.655545\pi$
$139$	−11.0693	−0.938885	−0.469442	+	0.882963i	$0.655545\pi$
$140$	0	0
$141$	5.77712	0.486522
$142$	0	0
$143$	19.4639	1.62766
$144$	0	0
$145$	20.3259	1.68798
$146$	0	0
$147$	23.4644	1.93531
$148$	0	0
$149$	14.1879	1.16232	0.581158	−	0.813791i	$-0.302600\pi$
$149$	14.1879	1.16232	0.581158	+	0.813791i	$0.302600\pi$
$150$	0	0
$151$	10.8801	0.885411	0.442705	−	0.896667i	$-0.354019\pi$
$151$	10.8801	0.885411	0.442705	+	0.896667i	$0.354019\pi$
$152$	0	0
$153$	5.20917	0.421137
$154$	0	0
$155$	−32.2156	−2.58762
$156$	0	0
$157$	12.5086	0.998293	0.499146	−	0.866518i	$-0.333647\pi$
$157$	12.5086	0.998293	0.499146	+	0.866518i	$0.333647\pi$
$158$	0	0
$159$	−34.6972	−2.75167
$160$	0	0
$161$	9.74761	0.768220
$162$	0	0
$163$	16.3713	1.28230	0.641149	−	0.767416i	$-0.278458\pi$
$163$	16.3713	1.28230	0.641149	+	0.767416i	$0.278458\pi$
$164$	0	0
$165$	−23.6566	−1.84166
$166$	0	0
$167$	−7.45470	−0.576862	−0.288431	−	0.957501i	$-0.593134\pi$
$167$	−7.45470	−0.576862	−0.288431	+	0.957501i	$0.593134\pi$
$168$	0	0
$169$	37.0834	2.85257
$170$	0	0
$171$	20.6396	1.57835
$172$	0	0
$173$	13.0992	0.995916	0.497958	−	0.867201i	$-0.334083\pi$
$173$	13.0992	0.995916	0.497958	+	0.867201i	$0.334083\pi$
$174$	0	0
$175$	15.6376	1.18209
$176$	0	0
$177$	−2.86517	−0.215359
$178$	0	0
$179$	−4.29643	−0.321131	−0.160565	−	0.987025i	$-0.551332\pi$
$179$	−4.29643	−0.321131	−0.160565	+	0.987025i	$0.551332\pi$
$180$	0	0
$181$	0.846722	0.0629364	0.0314682	−	0.999505i	$-0.489982\pi$
$181$	0.846722	0.0629364	0.0314682	+	0.999505i	$0.489982\pi$
$182$	0	0
$183$	−40.4095	−2.98716
$184$	0	0
$185$	−27.3512	−2.01090
$186$	0	0
$187$	2.75032	0.201123
$188$	0	0
$189$	24.6690	1.79441
$190$	0	0
$191$	−5.71273	−0.413359	−0.206679	−	0.978409i	$-0.566266\pi$
$191$	−5.71273	−0.413359	−0.206679	+	0.978409i	$0.566266\pi$
$192$	0	0
$193$	16.4017	1.18062	0.590309	−	0.807177i	$-0.299006\pi$
$193$	16.4017	1.18062	0.590309	+	0.807177i	$0.299006\pi$
$194$	0	0
$195$	−60.8717	−4.35911
$196$	0	0
$197$	25.7910	1.83753	0.918767	−	0.394801i	$-0.129186\pi$
$197$	25.7910	1.83753	0.918767	+	0.394801i	$0.129186\pi$
$198$	0	0
$199$	−15.7016	−1.11305	−0.556527	−	0.830829i	$-0.687867\pi$
$199$	−15.7016	−1.11305	−0.556527	+	0.830829i	$0.687867\pi$
$200$	0	0
$201$	−26.7759	−1.88862
$202$	0	0
$203$	−26.3879	−1.85206
$204$	0	0
$205$	17.8142	1.24420
$206$	0	0
$207$	13.0285	0.905544
$208$	0	0
$209$	10.8972	0.753778
$210$	0	0
$211$	−8.42712	−0.580147	−0.290073	−	0.957004i	$-0.593680\pi$
$211$	−8.42712	−0.580147	−0.290073	+	0.957004i	$0.593680\pi$
$212$	0	0
$213$	−20.5030	−1.40484
$214$	0	0
$215$	6.49584	0.443013
$216$	0	0
$217$	41.8235	2.83917
$218$	0	0
$219$	−4.10042	−0.277081
$220$	0	0
$221$	7.07696	0.476048
$222$	0	0
$223$	6.68609	0.447734	0.223867	−	0.974620i	$-0.428132\pi$
$223$	6.68609	0.447734	0.223867	+	0.974620i	$0.428132\pi$
$224$	0	0
$225$	20.9009	1.39339
$226$	0	0
$227$	−21.7472	−1.44341	−0.721705	−	0.692200i	$-0.756641\pi$
$227$	−21.7472	−1.44341	−0.721705	+	0.692200i	$0.756641\pi$
$228$	0	0
$229$	−6.52413	−0.431127	−0.215563	−	0.976490i	$-0.569159\pi$
$229$	−6.52413	−0.431127	−0.215563	+	0.976490i	$0.569159\pi$
$230$	0	0
$231$	30.7118	2.02069
$232$	0	0
$233$	9.21658	0.603798	0.301899	−	0.953340i	$-0.402379\pi$
$233$	9.21658	0.603798	0.301899	+	0.953340i	$0.402379\pi$
$234$	0	0
$235$	−6.05314	−0.394863
$236$	0	0
$237$	−46.2100	−3.00166
$238$	0	0
$239$	16.4645	1.06500	0.532501	−	0.846429i	$-0.321252\pi$
$239$	16.4645	1.06500	0.532501	+	0.846429i	$0.321252\pi$
$240$	0	0
$241$	−26.2069	−1.68814	−0.844068	−	0.536237i	$-0.819845\pi$
$241$	−26.2069	−1.68814	−0.844068	+	0.536237i	$0.819845\pi$
$242$	0	0
$243$	−11.8032	−0.757177
$244$	0	0
$245$	−24.5855	−1.57071
$246$	0	0
$247$	28.0401	1.78415
$248$	0	0
$249$	8.35173	0.529269
$250$	0	0
$251$	5.15234	0.325213	0.162606	−	0.986691i	$-0.448010\pi$
$251$	5.15234	0.325213	0.162606	+	0.986691i	$0.448010\pi$
$252$	0	0
$253$	6.87875	0.432463
$254$	0	0
$255$	−8.60138	−0.538639
$256$	0	0
$257$	−3.93280	−0.245321	−0.122661	−	0.992449i	$-0.539143\pi$
$257$	−3.93280	−0.245321	−0.122661	+	0.992449i	$0.539143\pi$
$258$	0	0
$259$	35.5084	2.20638
$260$	0	0
$261$	−35.2696	−2.18313
$262$	0	0
$263$	−8.05180	−0.496495	−0.248248	−	0.968697i	$-0.579855\pi$
$263$	−8.05180	−0.496495	−0.248248	+	0.968697i	$0.579855\pi$
$264$	0	0
$265$	36.3549	2.23326
$266$	0	0
$267$	−13.8261	−0.846142
$268$	0	0
$269$	−10.9366	−0.666815	−0.333408	−	0.942783i	$-0.608199\pi$
$269$	−10.9366	−0.666815	−0.333408	+	0.942783i	$0.608199\pi$
$270$	0	0
$271$	−27.6286	−1.67832	−0.839158	−	0.543888i	$-0.816952\pi$
$271$	−27.6286	−1.67832	−0.839158	+	0.543888i	$0.816952\pi$
$272$	0	0
$273$	79.0258	4.78286
$274$	0	0
$275$	11.0352	0.665448
$276$	0	0
$277$	0.310312	0.0186449	0.00932243	−	0.999957i	$-0.497033\pi$
$277$	0.310312	0.0186449	0.00932243	+	0.999957i	$0.497033\pi$
$278$	0	0
$279$	55.9007	3.34669
$280$	0	0
$281$	0.932275	0.0556149	0.0278074	−	0.999613i	$-0.491147\pi$
$281$	0.932275	0.0556149	0.0278074	+	0.999613i	$0.491147\pi$
$282$	0	0
$283$	−19.7817	−1.17590	−0.587949	−	0.808898i	$-0.700065\pi$
$283$	−19.7817	−1.17590	−0.587949	+	0.808898i	$0.700065\pi$
$284$	0	0
$285$	−34.0801	−2.01873
$286$	0	0
$287$	−23.1271	−1.36515
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	21.4993	1.26031
$292$	0	0
$293$	−11.9304	−0.696981	−0.348491	−	0.937312i	$-0.613306\pi$
$293$	−11.9304	−0.696981	−0.348491	+	0.937312i	$0.613306\pi$
$294$	0	0
$295$	3.00205	0.174786
$296$	0	0
$297$	17.4086	1.01015
$298$	0	0
$299$	17.7000	1.02362
$300$	0	0
$301$	−8.43314	−0.486078
$302$	0	0
$303$	7.38666	0.424353
$304$	0	0
$305$	42.3402	2.42439
$306$	0	0
$307$	−8.53773	−0.487274	−0.243637	−	0.969866i	$-0.578341\pi$
$307$	−8.53773	−0.487274	−0.243637	+	0.969866i	$0.578341\pi$
$308$	0	0
$309$	−31.3026	−1.78075
$310$	0	0
$311$	−20.3241	−1.15247	−0.576236	−	0.817283i	$-0.695479\pi$
$311$	−20.3241	−1.15247	−0.576236	+	0.817283i	$0.695479\pi$
$312$	0	0
$313$	−12.1364	−0.685991	−0.342995	−	0.939337i	$-0.611441\pi$
$313$	−12.1364	−0.685991	−0.342995	+	0.939337i	$0.611441\pi$
$314$	0	0
$315$	−60.9480	−3.43403
$316$	0	0
$317$	14.0320	0.788117	0.394058	−	0.919085i	$-0.371071\pi$
$317$	14.0320	0.788117	0.394058	+	0.919085i	$0.371071\pi$
$318$	0	0
$319$	−18.6215	−1.04261
$320$	0	0
$321$	−30.1554	−1.68311
$322$	0	0
$323$	3.96217	0.220461
$324$	0	0
$325$	28.3951	1.57508
$326$	0	0
$327$	25.4837	1.40925
$328$	0	0
$329$	7.85841	0.433248
$330$	0	0
$331$	−30.2516	−1.66278	−0.831389	−	0.555691i	$-0.812454\pi$
$331$	−30.2516	−1.66278	−0.831389	+	0.555691i	$0.812454\pi$
$332$	0	0
$333$	47.4599	2.60079
$334$	0	0
$335$	28.0552	1.53282
$336$	0	0
$337$	−10.2579	−0.558781	−0.279391	−	0.960178i	$-0.590132\pi$
$337$	−10.2579	−0.558781	−0.279391	+	0.960178i	$0.590132\pi$
$338$	0	0
$339$	−36.4535	−1.97988
$340$	0	0
$341$	29.5143	1.59829
$342$	0	0
$343$	4.63611	0.250326
$344$	0	0
$345$	−21.5127	−1.15820
$346$	0	0
$347$	12.1808	0.653900	0.326950	−	0.945042i	$-0.393979\pi$
$347$	12.1808	0.653900	0.326950	+	0.945042i	$0.393979\pi$
$348$	0	0
$349$	31.4969	1.68599	0.842996	−	0.537920i	$-0.180790\pi$
$349$	31.4969	1.68599	0.842996	+	0.537920i	$0.180790\pi$
$350$	0	0
$351$	44.7946	2.39096
$352$	0	0
$353$	−16.9934	−0.904466	−0.452233	−	0.891900i	$-0.649372\pi$
$353$	−16.9934	−0.904466	−0.452233	+	0.891900i	$0.649372\pi$
$354$	0	0
$355$	21.4826	1.14018
$356$	0	0
$357$	11.1666	0.591001
$358$	0	0
$359$	27.3651	1.44428	0.722138	−	0.691749i	$-0.243160\pi$
$359$	27.3651	1.44428	0.722138	+	0.691749i	$0.243160\pi$
$360$	0	0
$361$	−3.30122	−0.173748
$362$	0	0
$363$	−9.84393	−0.516672
$364$	0	0
$365$	4.29633	0.224880
$366$	0	0
$367$	24.7595	1.29244	0.646218	−	0.763153i	$-0.276350\pi$
$367$	24.7595	1.29244	0.646218	+	0.763153i	$0.276350\pi$
$368$	0	0
$369$	−30.9113	−1.60918
$370$	0	0
$371$	−47.1973	−2.45036
$372$	0	0
$373$	−20.0433	−1.03780	−0.518901	−	0.854834i	$-0.673659\pi$
$373$	−20.0433	−1.03780	−0.518901	+	0.854834i	$0.673659\pi$
$374$	0	0
$375$	8.49532	0.438696
$376$	0	0
$377$	−47.9158	−2.46779
$378$	0	0
$379$	−19.2980	−0.991274	−0.495637	−	0.868530i	$-0.665065\pi$
$379$	−19.2980	−0.991274	−0.495637	+	0.868530i	$0.665065\pi$
$380$	0	0
$381$	15.6492	0.801735
$382$	0	0
$383$	20.5628	1.05071	0.525355	−	0.850883i	$-0.323933\pi$
$383$	20.5628	1.05071	0.525355	+	0.850883i	$0.323933\pi$
$384$	0	0
$385$	−32.1791	−1.64000
$386$	0	0
$387$	−11.2716	−0.572968
$388$	0	0
$389$	−4.11101	−0.208437	−0.104218	−	0.994554i	$-0.533234\pi$
$389$	−4.11101	−0.208437	−0.104218	+	0.994554i	$0.533234\pi$
$390$	0	0
$391$	2.50107	0.126485
$392$	0	0
$393$	49.8422	2.51421
$394$	0	0
$395$	48.4178	2.43616
$396$	0	0
$397$	5.45873	0.273966	0.136983	−	0.990573i	$-0.456259\pi$
$397$	5.45873	0.273966	0.136983	+	0.990573i	$0.456259\pi$
$398$	0	0
$399$	44.2441	2.21497
$400$	0	0
$401$	25.2107	1.25896	0.629481	−	0.777016i	$-0.283268\pi$
$401$	25.2107	1.25896	0.629481	+	0.777016i	$0.283268\pi$
$402$	0	0
$403$	75.9443	3.78306
$404$	0	0
$405$	−7.52901	−0.374120
$406$	0	0
$407$	25.0577	1.24207
$408$	0	0
$409$	−9.98970	−0.493959	−0.246979	−	0.969021i	$-0.579438\pi$
$409$	−9.98970	−0.493959	−0.246979	+	0.969021i	$0.579438\pi$
$410$	0	0
$411$	−41.1700	−2.03077
$412$	0	0
$413$	−3.89738	−0.191777
$414$	0	0
$415$	−8.75075	−0.429557
$416$	0	0
$417$	−31.7153	−1.55311
$418$	0	0
$419$	−11.4209	−0.557947	−0.278974	−	0.960299i	$-0.589994\pi$
$419$	−11.4209	−0.557947	−0.278974	+	0.960299i	$0.589994\pi$
$420$	0	0
$421$	8.78881	0.428340	0.214170	−	0.976796i	$-0.431295\pi$
$421$	8.78881	0.428340	0.214170	+	0.976796i	$0.431295\pi$
$422$	0	0
$423$	10.5034	0.510694
$424$	0	0
$425$	4.01233	0.194627
$426$	0	0
$427$	−54.9676	−2.66007
$428$	0	0
$429$	55.7674	2.69247
$430$	0	0
$431$	9.21329	0.443789	0.221894	−	0.975071i	$-0.428776\pi$
$431$	9.21329	0.443789	0.221894	+	0.975071i	$0.428776\pi$
$432$	0	0
$433$	−18.3201	−0.880410	−0.440205	−	0.897897i	$-0.645094\pi$
$433$	−18.3201	−0.880410	−0.440205	+	0.897897i	$0.645094\pi$
$434$	0	0
$435$	58.2371	2.79226
$436$	0	0
$437$	9.90966	0.474044
$438$	0	0
$439$	−17.7684	−0.848037	−0.424019	−	0.905653i	$-0.639381\pi$
$439$	−17.7684	−0.848037	−0.424019	+	0.905653i	$0.639381\pi$
$440$	0	0
$441$	42.6607	2.03146
$442$	0	0
$443$	−7.02085	−0.333571	−0.166785	−	0.985993i	$-0.553339\pi$
$443$	−7.02085	−0.333571	−0.166785	+	0.985993i	$0.553339\pi$
$444$	0	0
$445$	14.4866	0.686732
$446$	0	0
$447$	40.6506	1.92271
$448$	0	0
$449$	−12.1617	−0.573948	−0.286974	−	0.957938i	$-0.592649\pi$
$449$	−12.1617	−0.573948	−0.286974	+	0.957938i	$0.592649\pi$
$450$	0	0
$451$	−16.3205	−0.768501
$452$	0	0
$453$	31.1733	1.46465
$454$	0	0
$455$	−82.8014	−3.88179
$456$	0	0
$457$	12.9890	0.607601	0.303801	−	0.952736i	$-0.401744\pi$
$457$	12.9890	0.607601	0.303801	+	0.952736i	$0.401744\pi$
$458$	0	0
$459$	6.32964	0.295442
$460$	0	0
$461$	15.6756	0.730085	0.365043	−	0.930991i	$-0.381054\pi$
$461$	15.6756	0.730085	0.365043	+	0.930991i	$0.381054\pi$
$462$	0	0
$463$	18.1369	0.842896	0.421448	−	0.906853i	$-0.361522\pi$
$463$	18.1369	0.842896	0.421448	+	0.906853i	$0.361522\pi$
$464$	0	0
$465$	−92.3032	−4.28046
$466$	0	0
$467$	−23.4426	−1.08479	−0.542397	−	0.840122i	$-0.682483\pi$
$467$	−23.4426	−1.08479	−0.542397	+	0.840122i	$0.682483\pi$
$468$	0	0
$469$	−36.4222	−1.68182
$470$	0	0
$471$	35.8391	1.65138
$472$	0	0
$473$	−5.95115	−0.273634
$474$	0	0
$475$	15.8975	0.729429
$476$	0	0
$477$	−63.0831	−2.88838
$478$	0	0
$479$	8.62514	0.394093	0.197046	−	0.980394i	$-0.436865\pi$
$479$	8.62514	0.394093	0.197046	+	0.980394i	$0.436865\pi$
$480$	0	0
$481$	64.4770	2.93990
$482$	0	0
$483$	27.9285	1.27079
$484$	0	0
$485$	−22.5265	−1.02287
$486$	0	0
$487$	6.84293	0.310083	0.155041	−	0.987908i	$-0.450449\pi$
$487$	6.84293	0.310083	0.155041	+	0.987908i	$0.450449\pi$
$488$	0	0
$489$	46.9064	2.12118
$490$	0	0
$491$	−7.34304	−0.331387	−0.165693	−	0.986177i	$-0.552986\pi$
$491$	−7.34304	−0.331387	−0.165693	+	0.986177i	$0.552986\pi$
$492$	0	0
$493$	−6.77067	−0.304936
$494$	0	0
$495$	−43.0101	−1.93316
$496$	0	0
$497$	−27.8895	−1.25102
$498$	0	0
$499$	−15.3592	−0.687571	−0.343785	−	0.939048i	$-0.611709\pi$
$499$	−15.3592	−0.687571	−0.343785	+	0.939048i	$0.611709\pi$
$500$	0	0
$501$	−21.3589	−0.954247
$502$	0	0
$503$	30.0883	1.34157	0.670786	−	0.741651i	$-0.265957\pi$
$503$	30.0883	1.34157	0.670786	+	0.741651i	$0.265957\pi$
$504$	0	0
$505$	−7.73958	−0.344407
$506$	0	0
$507$	106.250	4.71873
$508$	0	0
$509$	37.8425	1.67734	0.838669	−	0.544642i	$-0.183334\pi$
$509$	37.8425	1.67734	0.838669	+	0.544642i	$0.183334\pi$
$510$	0	0
$511$	−5.57765	−0.246741
$512$	0	0
$513$	25.0791	1.10727
$514$	0	0
$515$	32.7982	1.44526
$516$	0	0
$517$	5.54556	0.243894
$518$	0	0
$519$	37.5314	1.64745
$520$	0	0
$521$	35.3226	1.54751	0.773755	−	0.633485i	$-0.218376\pi$
$521$	35.3226	1.54751	0.773755	+	0.633485i	$0.218376\pi$
$522$	0	0
$523$	25.8845	1.13185	0.565925	−	0.824457i	$-0.308519\pi$
$523$	25.8845	1.13185	0.565925	+	0.824457i	$0.308519\pi$
$524$	0	0
$525$	44.8042	1.95542
$526$	0	0
$527$	10.7312	0.467458
$528$	0	0
$529$	−16.7446	−0.728028
$530$	0	0
$531$	−5.20917	−0.226059
$532$	0	0
$533$	−41.9948	−1.81900
$534$	0	0
$535$	31.5962	1.36602
$536$	0	0
$537$	−12.3100	−0.531215
$538$	0	0
$539$	22.5239	0.970173
$540$	0	0
$541$	−16.9773	−0.729912	−0.364956	−	0.931025i	$-0.618916\pi$
$541$	−16.9773	−0.729912	−0.364956	+	0.931025i	$0.618916\pi$
$542$	0	0
$543$	2.42600	0.104110
$544$	0	0
$545$	−26.7012	−1.14376
$546$	0	0
$547$	4.66392	0.199415	0.0997075	−	0.995017i	$-0.468209\pi$
$547$	4.66392	0.199415	0.0997075	+	0.995017i	$0.468209\pi$
$548$	0	0
$549$	−73.4688	−3.13557
$550$	0	0
$551$	−26.8265	−1.14285
$552$	0	0
$553$	−62.8577	−2.67298
$554$	0	0
$555$	−78.3658	−3.32644
$556$	0	0
$557$	40.9052	1.73321	0.866604	−	0.498997i	$-0.166298\pi$
$557$	40.9052	1.73321	0.866604	+	0.498997i	$0.166298\pi$
$558$	0	0
$559$	−15.3131	−0.647676
$560$	0	0
$561$	7.88013	0.332699
$562$	0	0
$563$	−30.8059	−1.29831	−0.649157	−	0.760655i	$-0.724878\pi$
$563$	−30.8059	−1.29831	−0.649157	+	0.760655i	$0.724878\pi$
$564$	0	0
$565$	38.1951	1.60688
$566$	0	0
$567$	9.77444	0.410488
$568$	0	0
$569$	−14.4858	−0.607276	−0.303638	−	0.952787i	$-0.598201\pi$
$569$	−14.4858	−0.607276	−0.303638	+	0.952787i	$0.598201\pi$
$570$	0	0
$571$	26.1975	1.09633	0.548166	−	0.836370i	$-0.315326\pi$
$571$	26.1975	1.09633	0.548166	+	0.836370i	$0.315326\pi$
$572$	0	0
$573$	−16.3679	−0.683780
$574$	0	0
$575$	10.0351	0.418494
$576$	0	0
$577$	−46.4980	−1.93574	−0.967869	−	0.251456i	$-0.919091\pi$
$577$	−46.4980	−1.93574	−0.967869	+	0.251456i	$0.919091\pi$
$578$	0	0
$579$	46.9935	1.95298
$580$	0	0
$581$	11.3605	0.471315
$582$	0	0
$583$	−33.3064	−1.37941
$584$	0	0
$585$	−110.671	−4.57569
$586$	0	0
$587$	43.1891	1.78261	0.891303	−	0.453408i	$-0.149792\pi$
$587$	43.1891	1.78261	0.891303	+	0.453408i	$0.149792\pi$
$588$	0	0
$589$	42.5188	1.75196
$590$	0	0
$591$	73.8955	3.03966
$592$	0	0
$593$	−41.1478	−1.68974	−0.844869	−	0.534974i	$-0.820322\pi$
$593$	−41.1478	−1.68974	−0.844869	+	0.534974i	$0.820322\pi$
$594$	0	0
$595$	−11.7001	−0.479659
$596$	0	0
$597$	−44.9876	−1.84122
$598$	0	0
$599$	7.22078	0.295033	0.147516	−	0.989060i	$-0.452872\pi$
$599$	7.22078	0.295033	0.147516	+	0.989060i	$0.452872\pi$
$600$	0	0
$601$	43.4435	1.77210	0.886048	−	0.463593i	$-0.153440\pi$
$601$	43.4435	1.77210	0.886048	+	0.463593i	$0.153440\pi$
$602$	0	0
$603$	−48.6814	−1.98246
$604$	0	0
$605$	10.3142	0.419333
$606$	0	0
$607$	36.4025	1.47753	0.738767	−	0.673961i	$-0.235408\pi$
$607$	36.4025	1.47753	0.738767	+	0.673961i	$0.235408\pi$
$608$	0	0
$609$	−75.6056	−3.06369
$610$	0	0
$611$	14.2695	0.577282
$612$	0	0
$613$	23.0342	0.930344	0.465172	−	0.885220i	$-0.345992\pi$
$613$	23.0342	0.930344	0.465172	+	0.885220i	$0.345992\pi$
$614$	0	0
$615$	51.0408	2.05816
$616$	0	0
$617$	29.6171	1.19234	0.596170	−	0.802858i	$-0.296688\pi$
$617$	29.6171	1.19234	0.596170	+	0.802858i	$0.296688\pi$
$618$	0	0
$619$	17.1598	0.689712	0.344856	−	0.938656i	$-0.387928\pi$
$619$	17.1598	0.689712	0.344856	+	0.938656i	$0.387928\pi$
$620$	0	0
$621$	15.8309	0.635271
$622$	0	0
$623$	−18.8071	−0.753490
$624$	0	0
$625$	−28.9629	−1.15851
$626$	0	0
$627$	31.2224	1.24690
$628$	0	0
$629$	9.11084	0.363273
$630$	0	0
$631$	−16.1145	−0.641507	−0.320753	−	0.947163i	$-0.603936\pi$
$631$	−16.1145	−0.641507	−0.320753	+	0.947163i	$0.603936\pi$
$632$	0	0
$633$	−24.1451	−0.959681
$634$	0	0
$635$	−16.3969	−0.650692
$636$	0	0
$637$	57.9571	2.29634
$638$	0	0
$639$	−37.2767	−1.47464
$640$	0	0
$641$	34.2470	1.35267	0.676337	−	0.736592i	$-0.263566\pi$
$641$	34.2470	1.35267	0.676337	+	0.736592i	$0.263566\pi$
$642$	0	0
$643$	−18.3975	−0.725526	−0.362763	−	0.931881i	$-0.618167\pi$
$643$	−18.3975	−0.725526	−0.362763	+	0.931881i	$0.618167\pi$
$644$	0	0
$645$	18.6117	0.732834
$646$	0	0
$647$	7.28086	0.286240	0.143120	−	0.989705i	$-0.454287\pi$
$647$	7.28086	0.286240	0.143120	+	0.989705i	$0.454287\pi$
$648$	0	0
$649$	−2.75032	−0.107960
$650$	0	0
$651$	119.831	4.69656
$652$	0	0
$653$	15.4693	0.605361	0.302681	−	0.953092i	$-0.402118\pi$
$653$	15.4693	0.605361	0.302681	+	0.953092i	$0.402118\pi$
$654$	0	0
$655$	−52.2236	−2.04054
$656$	0	0
$657$	−7.45500	−0.290847
$658$	0	0
$659$	15.1871	0.591606	0.295803	−	0.955249i	$-0.404413\pi$
$659$	15.1871	0.591606	0.295803	+	0.955249i	$0.404413\pi$
$660$	0	0
$661$	−42.4641	−1.65166	−0.825832	−	0.563917i	$-0.809294\pi$
$661$	−42.4641	−1.65166	−0.825832	+	0.563917i	$0.809294\pi$
$662$	0	0
$663$	20.2767	0.787481
$664$	0	0
$665$	−46.3579	−1.79768
$666$	0	0
$667$	−16.9339	−0.655684
$668$	0	0
$669$	19.1568	0.740643
$670$	0	0
$671$	−38.7898	−1.49746
$672$	0	0
$673$	−5.92570	−0.228419	−0.114210	−	0.993457i	$-0.536434\pi$
$673$	−5.92570	−0.228419	−0.114210	+	0.993457i	$0.536434\pi$
$674$	0	0
$675$	25.3966	0.977516
$676$	0	0
$677$	−44.1752	−1.69779	−0.848896	−	0.528560i	$-0.822732\pi$
$677$	−44.1752	−1.69779	−0.848896	+	0.528560i	$0.822732\pi$
$678$	0	0
$679$	29.2447	1.12231
$680$	0	0
$681$	−62.3093	−2.38770
$682$	0	0
$683$	3.10583	0.118841	0.0594207	−	0.998233i	$-0.481075\pi$
$683$	3.10583	0.118841	0.0594207	+	0.998233i	$0.481075\pi$
$684$	0	0
$685$	43.1370	1.64818
$686$	0	0
$687$	−18.6927	−0.713171
$688$	0	0
$689$	−85.7021	−3.26499
$690$	0	0
$691$	−6.71176	−0.255327	−0.127664	−	0.991818i	$-0.540748\pi$
$691$	−6.71176	−0.255327	−0.127664	+	0.991818i	$0.540748\pi$
$692$	0	0
$693$	55.8373	2.12108
$694$	0	0
$695$	33.2306	1.26051
$696$	0	0
$697$	−5.93402	−0.224767
$698$	0	0
$699$	26.4070	0.998805
$700$	0	0
$701$	−15.7444	−0.594658	−0.297329	−	0.954775i	$-0.596096\pi$
$701$	−15.7444	−0.594658	−0.297329	+	0.954775i	$0.596096\pi$
$702$	0	0
$703$	36.0987	1.36149
$704$	0	0
$705$	−17.3432	−0.653184
$706$	0	0
$707$	10.0478	0.377886
$708$	0	0
$709$	13.4324	0.504465	0.252232	−	0.967667i	$-0.418835\pi$
$709$	13.4324	0.504465	0.252232	+	0.967667i	$0.418835\pi$
$710$	0	0
$711$	−84.0146	−3.15079
$712$	0	0
$713$	26.8395	1.00515
$714$	0	0
$715$	−58.4318	−2.18522
$716$	0	0
$717$	47.1736	1.76173
$718$	0	0
$719$	12.4736	0.465187	0.232594	−	0.972574i	$-0.425279\pi$
$719$	12.4736	0.465187	0.232594	+	0.972574i	$0.425279\pi$
$720$	0	0
$721$	−42.5798	−1.58576
$722$	0	0
$723$	−75.0871	−2.79252
$724$	0	0
$725$	−27.1662	−1.00893
$726$	0	0
$727$	−41.3088	−1.53206	−0.766029	−	0.642806i	$-0.777770\pi$
$727$	−41.3088	−1.53206	−0.766029	+	0.642806i	$0.777770\pi$
$728$	0	0
$729$	−41.3420	−1.53119
$730$	0	0
$731$	−2.16380	−0.0800310
$732$	0	0
$733$	−31.1810	−1.15170	−0.575849	−	0.817556i	$-0.695328\pi$
$733$	−31.1810	−1.15170	−0.575849	+	0.817556i	$0.695328\pi$
$734$	0	0
$735$	−70.4414	−2.59827
$736$	0	0
$737$	−25.7026	−0.946769
$738$	0	0
$739$	4.26427	0.156864	0.0784319	−	0.996919i	$-0.475009\pi$
$739$	4.26427	0.156864	0.0784319	+	0.996919i	$0.475009\pi$
$740$	0	0
$741$	80.3396	2.95135
$742$	0	0
$743$	−14.5710	−0.534557	−0.267279	−	0.963619i	$-0.586124\pi$
$743$	−14.5710	−0.534557	−0.267279	+	0.963619i	$0.586124\pi$
$744$	0	0
$745$	−42.5928	−1.56048
$746$	0	0
$747$	15.1843	0.555565
$748$	0	0
$749$	−41.0193	−1.49881
$750$	0	0
$751$	−25.7693	−0.940336	−0.470168	−	0.882577i	$-0.655807\pi$
$751$	−25.7693	−0.940336	−0.470168	+	0.882577i	$0.655807\pi$
$752$	0	0
$753$	14.7623	0.537968
$754$	0	0
$755$	−32.6627	−1.18872
$756$	0	0
$757$	39.0222	1.41829	0.709144	−	0.705064i	$-0.249082\pi$
$757$	39.0222	1.41829	0.709144	+	0.705064i	$0.249082\pi$
$758$	0	0
$759$	19.7088	0.715383
$760$	0	0
$761$	−18.1255	−0.657047	−0.328524	−	0.944496i	$-0.606551\pi$
$761$	−18.1255	−0.657047	−0.328524	+	0.944496i	$0.606551\pi$
$762$	0	0
$763$	34.6645	1.25494
$764$	0	0
$765$	−15.6382	−0.565401
$766$	0	0
$767$	−7.07696	−0.255534
$768$	0	0
$769$	−15.8675	−0.572198	−0.286099	−	0.958200i	$-0.592359\pi$
$769$	−15.8675	−0.572198	−0.286099	+	0.958200i	$0.592359\pi$
$770$	0	0
$771$	−11.2681	−0.405812
$772$	0	0
$773$	21.4588	0.771820	0.385910	−	0.922536i	$-0.373887\pi$
$773$	21.4588	0.771820	0.385910	+	0.922536i	$0.373887\pi$
$774$	0	0
$775$	43.0571	1.54666
$776$	0	0
$777$	101.737	3.64981
$778$	0	0
$779$	−23.5116	−0.842390
$780$	0	0
$781$	−19.6812	−0.704249
$782$	0	0
$783$	−42.8559	−1.53155
$784$	0	0
$785$	−37.5514	−1.34027
$786$	0	0
$787$	−27.4662	−0.979064	−0.489532	−	0.871985i	$-0.662832\pi$
$787$	−27.4662	−0.979064	−0.489532	+	0.871985i	$0.662832\pi$
$788$	0	0
$789$	−23.0697	−0.821304
$790$	0	0
$791$	−49.5863	−1.76308
$792$	0	0
$793$	−99.8117	−3.54442
$794$	0	0
$795$	104.163	3.69427
$796$	0	0
$797$	6.41081	0.227083	0.113541	−	0.993533i	$-0.463781\pi$
$797$	6.41081	0.227083	0.113541	+	0.993533i	$0.463781\pi$
$798$	0	0
$799$	2.01633	0.0713327
$800$	0	0
$801$	−25.1372	−0.888181
$802$	0	0
$803$	−3.93607	−0.138901
$804$	0	0
$805$	−29.2629	−1.03138
$806$	0	0
$807$	−31.3351	−1.10305
$808$	0	0
$809$	14.5932	0.513069	0.256535	−	0.966535i	$-0.417419\pi$
$809$	14.5932	0.513069	0.256535	+	0.966535i	$0.417419\pi$
$810$	0	0
$811$	10.8886	0.382351	0.191175	−	0.981556i	$-0.438770\pi$
$811$	10.8886	0.382351	0.191175	+	0.981556i	$0.438770\pi$
$812$	0	0
$813$	−79.1604	−2.77628
$814$	0	0
$815$	−49.1475	−1.72156
$816$	0	0
$817$	−8.57334	−0.299943
$818$	0	0
$819$	143.677	5.02049
$820$	0	0
$821$	−35.0300	−1.22255	−0.611277	−	0.791417i	$-0.709344\pi$
$821$	−35.0300	−1.22255	−0.611277	+	0.791417i	$0.709344\pi$
$822$	0	0
$823$	30.9738	1.07968	0.539839	−	0.841768i	$-0.318485\pi$
$823$	30.9738	1.07968	0.539839	+	0.841768i	$0.318485\pi$
$824$	0	0
$825$	31.6177	1.10079
$826$	0	0
$827$	7.77266	0.270282	0.135141	−	0.990826i	$-0.456851\pi$
$827$	7.77266	0.270282	0.135141	+	0.990826i	$0.456851\pi$
$828$	0	0
$829$	15.4077	0.535132	0.267566	−	0.963540i	$-0.413781\pi$
$829$	15.4077	0.535132	0.267566	+	0.963540i	$0.413781\pi$
$830$	0	0
$831$	0.889096	0.0308424
$832$	0	0
$833$	8.18955	0.283751
$834$	0	0
$835$	22.3794	0.774471
$836$	0	0
$837$	67.9247	2.34782
$838$	0	0
$839$	−12.7980	−0.441835	−0.220917	−	0.975293i	$-0.570905\pi$
$839$	−12.7980	−0.441835	−0.220917	+	0.975293i	$0.570905\pi$
$840$	0	0
$841$	16.8420	0.580758
$842$	0	0
$843$	2.67112	0.0919983
$844$	0	0
$845$	−111.326	−3.82975
$846$	0	0
$847$	−13.3903	−0.460097
$848$	0	0
$849$	−56.6778	−1.94518
$850$	0	0
$851$	22.7868	0.781123
$852$	0	0
$853$	−31.6417	−1.08339	−0.541695	−	0.840575i	$-0.682217\pi$
$853$	−31.6417	−1.08339	−0.541695	+	0.840575i	$0.682217\pi$
$854$	0	0
$855$	−61.9613	−2.11903
$856$	0	0
$857$	−32.2406	−1.10132	−0.550659	−	0.834730i	$-0.685624\pi$
$857$	−32.2406	−1.10132	−0.550659	+	0.834730i	$0.685624\pi$
$858$	0	0
$859$	−5.13799	−0.175306	−0.0876529	−	0.996151i	$-0.527937\pi$
$859$	−5.13799	−0.175306	−0.0876529	+	0.996151i	$0.527937\pi$
$860$	0	0
$861$	−66.2630	−2.25824
$862$	0	0
$863$	−16.2937	−0.554646	−0.277323	−	0.960777i	$-0.589447\pi$
$863$	−16.2937	−0.554646	−0.277323	+	0.960777i	$0.589447\pi$
$864$	0	0
$865$	−39.3246	−1.33708
$866$	0	0
$867$	2.86517	0.0973061
$868$	0	0
$869$	−44.3578	−1.50473
$870$	0	0
$871$	−66.1365	−2.24095
$872$	0	0
$873$	39.0880	1.32293
$874$	0	0
$875$	11.5559	0.390660
$876$	0	0
$877$	8.70573	0.293972	0.146986	−	0.989139i	$-0.453043\pi$
$877$	8.70573	0.293972	0.146986	+	0.989139i	$0.453043\pi$
$878$	0	0
$879$	−34.1826	−1.15295
$880$	0	0
$881$	3.61467	0.121781	0.0608907	−	0.998144i	$-0.480606\pi$
$881$	3.61467	0.121781	0.0608907	+	0.998144i	$0.480606\pi$
$882$	0	0
$883$	−19.5698	−0.658578	−0.329289	−	0.944229i	$-0.606809\pi$
$883$	−19.5698	−0.658578	−0.329289	+	0.944229i	$0.606809\pi$
$884$	0	0
$885$	8.60138	0.289132
$886$	0	0
$887$	−36.4334	−1.22331	−0.611656	−	0.791124i	$-0.709496\pi$
$887$	−36.4334	−1.22331	−0.611656	+	0.791124i	$0.709496\pi$
$888$	0	0
$889$	21.2871	0.713946
$890$	0	0
$891$	6.89768	0.231081
$892$	0	0
$893$	7.98905	0.267343
$894$	0	0
$895$	12.8981	0.431137
$896$	0	0
$897$	50.7134	1.69327
$898$	0	0
$899$	−72.6574	−2.42326
$900$	0	0
$901$	−12.1100	−0.403443
$902$	0	0
$903$	−24.1623	−0.804072
$904$	0	0
$905$	−2.54191	−0.0844958
$906$	0	0
$907$	47.0554	1.56245	0.781224	−	0.624251i	$-0.214596\pi$
$907$	47.0554	1.56245	0.781224	+	0.624251i	$0.214596\pi$
$908$	0	0
$909$	13.4297	0.445436
$910$	0	0
$911$	34.0705	1.12881	0.564403	−	0.825500i	$-0.309107\pi$
$911$	34.0705	1.12881	0.564403	+	0.825500i	$0.309107\pi$
$912$	0	0
$913$	8.01697	0.265323
$914$	0	0
$915$	121.312	4.01044
$916$	0	0
$917$	67.7985	2.23890
$918$	0	0
$919$	9.52865	0.314321	0.157161	−	0.987573i	$-0.449766\pi$
$919$	9.52865	0.314321	0.157161	+	0.987573i	$0.449766\pi$
$920$	0	0
$921$	−24.4620	−0.806051
$922$	0	0
$923$	−50.6425	−1.66692
$924$	0	0
$925$	36.5557	1.20194
$926$	0	0
$927$	−56.9115	−1.86922
$928$	0	0
$929$	55.9731	1.83642	0.918208	−	0.396098i	$-0.129636\pi$
$929$	55.9731	1.83642	0.918208	+	0.396098i	$0.129636\pi$
$930$	0	0
$931$	32.4484	1.06345
$932$	0	0
$933$	−58.2318	−1.90642
$934$	0	0
$935$	−8.25662	−0.270020
$936$	0	0
$937$	41.0074	1.33965	0.669826	−	0.742518i	$-0.266369\pi$
$937$	41.0074	1.33965	0.669826	+	0.742518i	$0.266369\pi$
$938$	0	0
$939$	−34.7728	−1.13477
$940$	0	0
$941$	−40.8341	−1.33116	−0.665578	−	0.746329i	$-0.731815\pi$
$941$	−40.8341	−1.33116	−0.665578	+	0.746329i	$0.731815\pi$
$942$	0	0
$943$	−14.8414	−0.483302
$944$	0	0
$945$	−74.0577	−2.40910
$946$	0	0
$947$	28.6950	0.932461	0.466231	−	0.884663i	$-0.345612\pi$
$947$	28.6950	0.932461	0.466231	+	0.884663i	$0.345612\pi$
$948$	0	0
$949$	−10.1280	−0.328770
$950$	0	0
$951$	40.2040	1.30371
$952$	0	0
$953$	49.1053	1.59068	0.795338	−	0.606166i	$-0.207293\pi$
$953$	49.1053	1.59068	0.795338	+	0.606166i	$0.207293\pi$
$954$	0	0
$955$	17.1499	0.554959
$956$	0	0
$957$	−53.3538	−1.72468
$958$	0	0
$959$	−56.0020	−1.80840
$960$	0	0
$961$	84.1587	2.71480
$962$	0	0
$963$	−54.8258	−1.76674
$964$	0	0
$965$	−49.2387	−1.58505
$966$	0	0
$967$	−1.36597	−0.0439265	−0.0219633	−	0.999759i	$-0.506992\pi$
$967$	−1.36597	−0.0439265	−0.0219633	+	0.999759i	$0.506992\pi$
$968$	0	0
$969$	11.3523	0.364687
$970$	0	0
$971$	−44.8761	−1.44014	−0.720071	−	0.693901i	$-0.755891\pi$
$971$	−44.8761	−1.44014	−0.720071	+	0.693901i	$0.755891\pi$
$972$	0	0
$973$	−43.1412	−1.38304
$974$	0	0
$975$	81.3567	2.60550
$976$	0	0
$977$	−18.3070	−0.585694	−0.292847	−	0.956159i	$-0.594603\pi$
$977$	−18.3070	−0.585694	−0.292847	+	0.956159i	$0.594603\pi$
$978$	0	0
$979$	−13.2719	−0.424171
$980$	0	0
$981$	46.3320	1.47927
$982$	0	0
$983$	40.4845	1.29126	0.645628	−	0.763652i	$-0.276596\pi$
$983$	40.4845	1.29126	0.645628	+	0.763652i	$0.276596\pi$
$984$	0	0
$985$	−77.4260	−2.46700
$986$	0	0
$987$	22.5156	0.716680
$988$	0	0
$989$	−5.41182	−0.172086
$990$	0	0
$991$	−48.2583	−1.53298	−0.766488	−	0.642259i	$-0.777998\pi$
$991$	−48.2583	−1.53298	−0.766488	+	0.642259i	$0.777998\pi$
$992$	0	0
$993$	−86.6758	−2.75057
$994$	0	0
$995$	47.1370	1.49434
$996$	0	0
$997$	36.2939	1.14944	0.574719	−	0.818351i	$-0.305111\pi$
$997$	36.2939	1.14944	0.574719	+	0.818351i	$0.305111\pi$
$998$	0	0
$999$	57.6683	1.82454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.v.1.15	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.v.1.15	✓	18	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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+2.86517 q^{3} -3.00205 q^{5} +3.89738 q^{7} +5.20917 q^{9} +O(q^{10})\)	\(q+2.86517 q^{3} -3.00205 q^{5} +3.89738 q^{7} +5.20917 q^{9} +2.75032 q^{11} +7.07696 q^{13} -8.60138 q^{15} +1.00000 q^{17} +3.96217 q^{19} +11.1666 q^{21} +2.50107 q^{23} +4.01233 q^{25} +6.32964 q^{27} -6.77067 q^{29} +10.7312 q^{31} +7.88013 q^{33} -11.7001 q^{35} +9.11084 q^{37} +20.2767 q^{39} -5.93402 q^{41} -2.16380 q^{43} -15.6382 q^{45} +2.01633 q^{47} +8.18955 q^{49} +2.86517 q^{51} -12.1100 q^{53} -8.25662 q^{55} +11.3523 q^{57} -1.00000 q^{59} -14.1037 q^{61} +20.3021 q^{63} -21.2454 q^{65} -9.34532 q^{67} +7.16598 q^{69} -7.15597 q^{71} -1.43113 q^{73} +11.4960 q^{75} +10.7190 q^{77} -16.1282 q^{79} +2.50795 q^{81} +2.91492 q^{83} -3.00205 q^{85} -19.3991 q^{87} -4.82557 q^{89} +27.5816 q^{91} +30.7467 q^{93} -11.8946 q^{95} +7.50368 q^{97} +14.3269 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9}+O(q^{10}) \) 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9	\( 18 q + 9 q^{3} + 2 q^{5} + 11 q^{7} + 31 q^{9} - q^{11} + 15 q^{13} + 3 q^{15} + 18 q^{17} + 26 q^{19} - 4 q^{21} + 22 q^{23} + 42 q^{25} + 45 q^{27} + 6 q^{29} + 13 q^{31} - 5 q^{33} + 4 q^{35} + 4 q^{37} + 36 q^{39} - 15 q^{41} + 12 q^{43} + 14 q^{45} + 8 q^{47} - 13 q^{49} + 9 q^{51} - 11 q^{53} + 55 q^{55} - 20 q^{57} - 18 q^{59} + 53 q^{61} + 29 q^{63} - 26 q^{65} + 2 q^{67} + 32 q^{69} + 8 q^{71} - 42 q^{73} + 72 q^{75} + 6 q^{77} - 9 q^{79} + 42 q^{81} - 4 q^{83} + 2 q^{85} + 36 q^{87} + 13 q^{89} + 68 q^{91} + q^{93} + 3 q^{95} - 56 q^{97} - 13 q^{99}+O(q^{100}) \) 18 * q + 9 * q^3 + 2 * q^5 + 11 * q^7 + 31 * q^9 - q^11 + 15 * q^13 + 3 * q^15 + 18 * q^17 + 26 * q^19 - 4 * q^21 + 22 * q^23 + 42 * q^25 + 45 * q^27 + 6 * q^29 + 13 * q^31 - 5 * q^33 + 4 * q^35 + 4 * q^37 + 36 * q^39 - 15 * q^41 + 12 * q^43 + 14 * q^45 + 8 * q^47 - 13 * q^49 + 9 * q^51 - 11 * q^53 + 55 * q^55 - 20 * q^57 - 18 * q^59 + 53 * q^61 + 29 * q^63 - 26 * q^65 + 2 * q^67 + 32 * q^69 + 8 * q^71 - 42 * q^73 + 72 * q^75 + 6 * q^77 - 9 * q^79 + 42 * q^81 - 4 * q^83 + 2 * q^85 + 36 * q^87 + 13 * q^89 + 68 * q^91 + q^93 + 3 * q^95 - 56 * q^97 - 13 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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