LMFDB - Embedded newform 8037.2.a.m.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8037,2,Mod(1,8037)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8037, 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("8037.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8037 = 3^{2} \cdot 19 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8037.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:	$64.1757681045$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} - 15 x^{10} + 14 x^{9} + 84 x^{8} - 76 x^{7} - 213 x^{6} + 196 x^{5} + 225 x^{4} + \cdots - 17 $ x^12 - x^11 - 15x^10 + 14x^9 + 84x^8 - 76x^7 - 213x^6 + 196x^5 + 225x^4 - 229x^3 - 49x^2 + 83x - 17
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 893)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.821767$ of defining polynomial
Character	$\chi$	$=$	8037.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.821767 q^{2} -1.32470 q^{4} +0.409207 q^{5} -1.21348 q^{7} -2.73213 q^{8} +O(q^{10})$	\(q+0.821767 q^{2} -1.32470 q^{4} +0.409207 q^{5} -1.21348 q^{7} -2.73213 q^{8} +0.336273 q^{10} -2.14165 q^{11} +2.77986 q^{13} -0.997196 q^{14} +0.404222 q^{16} -6.55590 q^{17} +1.00000 q^{19} -0.542076 q^{20} -1.75994 q^{22} +9.08375 q^{23} -4.83255 q^{25} +2.28440 q^{26} +1.60749 q^{28} +9.54358 q^{29} -6.57178 q^{31} +5.79643 q^{32} -5.38743 q^{34} -0.496563 q^{35} +11.6932 q^{37} +0.821767 q^{38} -1.11801 q^{40} +11.4538 q^{41} -1.71020 q^{43} +2.83704 q^{44} +7.46473 q^{46} -1.00000 q^{47} -5.52747 q^{49} -3.97123 q^{50} -3.68247 q^{52} -3.41879 q^{53} -0.876378 q^{55} +3.31538 q^{56} +7.84260 q^{58} -1.27077 q^{59} -1.06140 q^{61} -5.40048 q^{62} +3.95488 q^{64} +1.13754 q^{65} -11.7419 q^{67} +8.68459 q^{68} -0.408059 q^{70} -0.0620319 q^{71} -5.73749 q^{73} +9.60910 q^{74} -1.32470 q^{76} +2.59884 q^{77} -6.77808 q^{79} +0.165410 q^{80} +9.41237 q^{82} +3.31568 q^{83} -2.68272 q^{85} -1.40539 q^{86} +5.85127 q^{88} -3.80089 q^{89} -3.37329 q^{91} -12.0332 q^{92} -0.821767 q^{94} +0.409207 q^{95} -7.05338 q^{97} -4.54230 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	$ 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) $ 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

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 $

$2$	0.821767	0.581077	0.290539	−	0.956863i	$-0.406166\pi$
$2$	0.821767	0.581077	0.290539	+	0.956863i	$0.406166\pi$
$3$	0	0
$3$	0	0
$4$	−1.32470	−0.662349
$5$	0.409207	0.183003	0.0915014	−	0.995805i	$-0.470833\pi$
$5$	0.409207	0.183003	0.0915014	+	0.995805i	$0.470833\pi$
$6$	0	0
$7$	−1.21348	−0.458651	−0.229326	−	0.973350i	$-0.573652\pi$
$7$	−1.21348	−0.458651	−0.229326	+	0.973350i	$0.573652\pi$
$8$	−2.73213	−0.965953
$9$	0	0
$10$	0.336273	0.106339
$11$	−2.14165	−0.645732	−0.322866	−	0.946445i	$-0.604646\pi$
$11$	−2.14165	−0.645732	−0.322866	+	0.946445i	$0.604646\pi$
$12$	0	0
$13$	2.77986	0.770993	0.385497	−	0.922709i	$-0.374030\pi$
$13$	2.77986	0.770993	0.385497	+	0.922709i	$0.374030\pi$
$14$	−0.997196	−0.266512
$15$	0	0
$16$	0.404222	0.101055
$17$	−6.55590	−1.59004	−0.795020	−	0.606583i	$-0.792540\pi$
$17$	−6.55590	−1.59004	−0.795020	+	0.606583i	$0.792540\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.542076	−0.121212
$21$	0	0
$22$	−1.75994	−0.375220
$23$	9.08375	1.89409	0.947047	−	0.321095i	$-0.104051\pi$
$23$	9.08375	1.89409	0.947047	+	0.321095i	$0.104051\pi$
$24$	0	0
$25$	−4.83255	−0.966510
$26$	2.28440	0.448007
$27$	0	0
$28$	1.60749	0.303787
$29$	9.54358	1.77220	0.886099	−	0.463496i	$-0.153405\pi$
$29$	9.54358	1.77220	0.886099	+	0.463496i	$0.153405\pi$
$30$	0	0
$31$	−6.57178	−1.18033	−0.590164	−	0.807284i	$-0.700937\pi$
$31$	−6.57178	−1.18033	−0.590164	+	0.807284i	$0.700937\pi$
$32$	5.79643	1.02467
$33$	0	0
$34$	−5.38743	−0.923936
$35$	−0.496563	−0.0839345
$36$	0	0
$37$	11.6932	1.92235	0.961176	−	0.275935i	$-0.0889874\pi$
$37$	11.6932	1.92235	0.961176	+	0.275935i	$0.0889874\pi$
$38$	0.821767	0.133308
$39$	0	0
$40$	−1.11801	−0.176772
$41$	11.4538	1.78878	0.894392	−	0.447283i	$-0.147608\pi$
$41$	11.4538	1.78878	0.894392	+	0.447283i	$0.147608\pi$
$42$	0	0
$43$	−1.71020	−0.260804	−0.130402	−	0.991461i	$-0.541627\pi$
$43$	−1.71020	−0.260804	−0.130402	+	0.991461i	$0.541627\pi$
$44$	2.83704	0.427700
$45$	0	0
$46$	7.46473	1.10061
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	0	0
$49$	−5.52747	−0.789639
$50$	−3.97123	−0.561617
$51$	0	0
$52$	−3.68247	−0.510667
$53$	−3.41879	−0.469608	−0.234804	−	0.972043i	$-0.575445\pi$
$53$	−3.41879	−0.469608	−0.234804	+	0.972043i	$0.575445\pi$
$54$	0	0
$55$	−0.876378	−0.118171
$56$	3.31538	0.443036
$57$	0	0
$58$	7.84260	1.02978
$59$	−1.27077	−0.165440	−0.0827198	−	0.996573i	$-0.526361\pi$
$59$	−1.27077	−0.165440	−0.0827198	+	0.996573i	$0.526361\pi$
$60$	0	0
$61$	−1.06140	−0.135899	−0.0679495	−	0.997689i	$-0.521646\pi$
$61$	−1.06140	−0.135899	−0.0679495	+	0.997689i	$0.521646\pi$
$62$	−5.40048	−0.685861
$63$	0	0
$64$	3.95488	0.494360
$65$	1.13754	0.141094
$66$	0	0
$67$	−11.7419	−1.43450	−0.717250	−	0.696816i	$-0.754599\pi$
$67$	−11.7419	−1.43450	−0.717250	+	0.696816i	$0.754599\pi$
$68$	8.68459	1.05316
$69$	0	0
$70$	−0.408059	−0.0487724
$71$	−0.0620319	−0.00736183	−0.00368092	−	0.999993i	$-0.501172\pi$
$71$	−0.0620319	−0.00736183	−0.00368092	+	0.999993i	$0.501172\pi$
$72$	0	0
$73$	−5.73749	−0.671522	−0.335761	−	0.941947i	$-0.608993\pi$
$73$	−5.73749	−0.671522	−0.335761	+	0.941947i	$0.608993\pi$
$74$	9.60910	1.11704
$75$	0	0
$76$	−1.32470	−0.151953
$77$	2.59884	0.296166
$78$	0	0
$79$	−6.77808	−0.762594	−0.381297	−	0.924453i	$-0.624522\pi$
$79$	−6.77808	−0.762594	−0.381297	+	0.924453i	$0.624522\pi$
$80$	0.165410	0.0184934
$81$	0	0
$82$	9.41237	1.03942
$83$	3.31568	0.363944	0.181972	−	0.983304i	$-0.441752\pi$
$83$	3.31568	0.363944	0.181972	+	0.983304i	$0.441752\pi$
$84$	0	0
$85$	−2.68272	−0.290982
$86$	−1.40539	−0.151547
$87$	0	0
$88$	5.85127	0.623747
$89$	−3.80089	−0.402893	−0.201447	−	0.979499i	$-0.564564\pi$
$89$	−3.80089	−0.402893	−0.201447	+	0.979499i	$0.564564\pi$
$90$	0	0
$91$	−3.37329	−0.353617
$92$	−12.0332	−1.25455
$93$	0	0
$94$	−0.821767	−0.0847588
$95$	0.409207	0.0419837
$96$	0	0
$97$	−7.05338	−0.716163	−0.358081	−	0.933690i	$-0.616569\pi$
$97$	−7.05338	−0.716163	−0.358081	+	0.933690i	$0.616569\pi$
$98$	−4.54230	−0.458841
$99$	0	0
$100$	6.40167	0.640167
$101$	−17.5031	−1.74163	−0.870814	−	0.491612i	$-0.836408\pi$
$101$	−17.5031	−1.74163	−0.870814	+	0.491612i	$0.836408\pi$
$102$	0	0
$103$	−0.135279	−0.0133294	−0.00666471	−	0.999978i	$-0.502121\pi$
$103$	−0.135279	−0.0133294	−0.00666471	+	0.999978i	$0.502121\pi$
$104$	−7.59493	−0.744744
$105$	0	0
$106$	−2.80945	−0.272878
$107$	−10.0562	−0.972174	−0.486087	−	0.873910i	$-0.661576\pi$
$107$	−10.0562	−0.972174	−0.486087	+	0.873910i	$0.661576\pi$
$108$	0	0
$109$	−5.96392	−0.571240	−0.285620	−	0.958343i	$-0.592200\pi$
$109$	−5.96392	−0.571240	−0.285620	+	0.958343i	$0.592200\pi$
$110$	−0.720179	−0.0686664
$111$	0	0
$112$	−0.490514	−0.0463492
$113$	19.2550	1.81135	0.905677	−	0.423969i	$-0.139363\pi$
$113$	19.2550	1.81135	0.905677	+	0.423969i	$0.139363\pi$
$114$	0	0
$115$	3.71713	0.346625
$116$	−12.6424	−1.17381
$117$	0	0
$118$	−1.04427	−0.0961332
$119$	7.95544	0.729274
$120$	0	0
$121$	−6.41333	−0.583030
$122$	−0.872228	−0.0789678
$123$	0	0
$124$	8.70563	0.781789
$125$	−4.02355	−0.359877
$126$	0	0
$127$	−3.86414	−0.342887	−0.171443	−	0.985194i	$-0.554843\pi$
$127$	−3.86414	−0.342887	−0.171443	+	0.985194i	$0.554843\pi$
$128$	−8.34288	−0.737413
$129$	0	0
$130$	0.934790	0.0819865
$131$	−14.1359	−1.23506	−0.617531	−	0.786546i	$-0.711867\pi$
$131$	−14.1359	−1.23506	−0.617531	+	0.786546i	$0.711867\pi$
$132$	0	0
$133$	−1.21348	−0.105222
$134$	−9.64910	−0.833556
$135$	0	0
$136$	17.9116	1.53590
$137$	3.52759	0.301382	0.150691	−	0.988581i	$-0.451850\pi$
$137$	3.52759	0.301382	0.150691	+	0.988581i	$0.451850\pi$
$138$	0	0
$139$	−9.87201	−0.837333	−0.418666	−	0.908140i	$-0.637502\pi$
$139$	−9.87201	−0.837333	−0.418666	+	0.908140i	$0.637502\pi$
$140$	0.657796	0.0555939
$141$	0	0
$142$	−0.0509758	−0.00427780
$143$	−5.95348	−0.497855
$144$	0	0
$145$	3.90530	0.324317
$146$	−4.71488	−0.390206
$147$	0	0
$148$	−15.4900	−1.27327
$149$	12.1982	0.999314	0.499657	−	0.866223i	$-0.333459\pi$
$149$	12.1982	0.999314	0.499657	+	0.866223i	$0.333459\pi$
$150$	0	0
$151$	−17.2426	−1.40318	−0.701591	−	0.712579i	$-0.747527\pi$
$151$	−17.2426	−1.40318	−0.701591	+	0.712579i	$0.747527\pi$
$152$	−2.73213	−0.221605
$153$	0	0
$154$	2.13565	0.172095
$155$	−2.68922	−0.216003
$156$	0	0
$157$	19.4830	1.55491	0.777456	−	0.628937i	$-0.216510\pi$
$157$	19.4830	1.55491	0.777456	+	0.628937i	$0.216510\pi$
$158$	−5.57001	−0.443126
$159$	0	0
$160$	2.37194	0.187518
$161$	−11.0229	−0.868728
$162$	0	0
$163$	−2.88511	−0.225979	−0.112990	−	0.993596i	$-0.536043\pi$
$163$	−2.88511	−0.225979	−0.112990	+	0.993596i	$0.536043\pi$
$164$	−15.1728	−1.18480
$165$	0	0
$166$	2.72472	0.211479
$167$	−15.6681	−1.21244	−0.606218	−	0.795298i	$-0.707314\pi$
$167$	−15.6681	−1.21244	−0.606218	+	0.795298i	$0.707314\pi$
$168$	0	0
$169$	−5.27240	−0.405569
$170$	−2.20457	−0.169083
$171$	0	0
$172$	2.26550	0.172743
$173$	−12.8846	−0.979602	−0.489801	−	0.871834i	$-0.662930\pi$
$173$	−12.8846	−0.979602	−0.489801	+	0.871834i	$0.662930\pi$
$174$	0	0
$175$	5.86419	0.443291
$176$	−0.865702	−0.0652547
$177$	0	0
$178$	−3.12345	−0.234112
$179$	16.5879	1.23983	0.619917	−	0.784667i	$-0.287166\pi$
$179$	16.5879	1.23983	0.619917	+	0.784667i	$0.287166\pi$
$180$	0	0
$181$	−4.45954	−0.331475	−0.165738	−	0.986170i	$-0.553000\pi$
$181$	−4.45954	−0.331475	−0.165738	+	0.986170i	$0.553000\pi$
$182$	−2.77206	−0.205479
$183$	0	0
$184$	−24.8180	−1.82961
$185$	4.78494	0.351796
$186$	0	0
$187$	14.0405	1.02674
$188$	1.32470	0.0966135
$189$	0	0
$190$	0.336273	0.0243958
$191$	−18.6908	−1.35242	−0.676210	−	0.736709i	$-0.736379\pi$
$191$	−18.6908	−1.35242	−0.676210	+	0.736709i	$0.736379\pi$
$192$	0	0
$193$	3.71585	0.267473	0.133737	−	0.991017i	$-0.457302\pi$
$193$	3.71585	0.267473	0.133737	+	0.991017i	$0.457302\pi$
$194$	−5.79624	−0.416146
$195$	0	0
$196$	7.32223	0.523017
$197$	0.754726	0.0537720	0.0268860	−	0.999639i	$-0.491441\pi$
$197$	0.754726	0.0537720	0.0268860	+	0.999639i	$0.491441\pi$
$198$	0	0
$199$	−1.43253	−0.101549	−0.0507746	−	0.998710i	$-0.516169\pi$
$199$	−1.43253	−0.101549	−0.0507746	+	0.998710i	$0.516169\pi$
$200$	13.2031	0.933604
$201$	0	0
$202$	−14.3835	−1.01202
$203$	−11.5809	−0.812821
$204$	0	0
$205$	4.68698	0.327353
$206$	−0.111168	−0.00774542
$207$	0	0
$208$	1.12368	0.0779131
$209$	−2.14165	−0.148141
$210$	0	0
$211$	15.4825	1.06586	0.532930	−	0.846159i	$-0.321091\pi$
$211$	15.4825	1.06586	0.532930	+	0.846159i	$0.321091\pi$
$212$	4.52887	0.311044
$213$	0	0
$214$	−8.26390	−0.564909
$215$	−0.699827	−0.0477278
$216$	0	0
$217$	7.97471	0.541359
$218$	−4.90096	−0.331935
$219$	0	0
$220$	1.16094	0.0782703
$221$	−18.2245	−1.22591
$222$	0	0
$223$	−3.09263	−0.207098	−0.103549	−	0.994624i	$-0.533020\pi$
$223$	−3.09263	−0.207098	−0.103549	+	0.994624i	$0.533020\pi$
$224$	−7.03384	−0.469968
$225$	0	0
$226$	15.8231	1.05254
$227$	−2.22879	−0.147930	−0.0739648	−	0.997261i	$-0.523565\pi$
$227$	−2.22879	−0.147930	−0.0739648	+	0.997261i	$0.523565\pi$
$228$	0	0
$229$	19.3066	1.27582	0.637908	−	0.770113i	$-0.279800\pi$
$229$	19.3066	1.27582	0.637908	+	0.770113i	$0.279800\pi$
$230$	3.05462	0.201416
$231$	0	0
$232$	−26.0743	−1.71186
$233$	17.2159	1.12785	0.563927	−	0.825825i	$-0.309290\pi$
$233$	17.2159	1.12785	0.563927	+	0.825825i	$0.309290\pi$
$234$	0	0
$235$	−0.409207	−0.0266937
$236$	1.68338	0.109579
$237$	0	0
$238$	6.53752	0.423765
$239$	26.5296	1.71606	0.858030	−	0.513599i	$-0.171688\pi$
$239$	26.5296	1.71606	0.858030	+	0.513599i	$0.171688\pi$
$240$	0	0
$241$	−3.17946	−0.204807	−0.102404	−	0.994743i	$-0.532653\pi$
$241$	−3.17946	−0.204807	−0.102404	+	0.994743i	$0.532653\pi$
$242$	−5.27027	−0.338786
$243$	0	0
$244$	1.40604	0.0900125
$245$	−2.26188	−0.144506
$246$	0	0
$247$	2.77986	0.176878
$248$	17.9550	1.14014
$249$	0	0
$250$	−3.30642	−0.209116
$251$	16.0672	1.01415	0.507077	−	0.861901i	$-0.330726\pi$
$251$	16.0672	1.01415	0.507077	+	0.861901i	$0.330726\pi$
$252$	0	0
$253$	−19.4542	−1.22308
$254$	−3.17542	−0.199244
$255$	0	0
$256$	−14.7657	−0.922854
$257$	−25.7290	−1.60493	−0.802465	−	0.596699i	$-0.796479\pi$
$257$	−25.7290	−1.60493	−0.802465	+	0.596699i	$0.796479\pi$
$258$	0	0
$259$	−14.1894	−0.881689
$260$	−1.50689	−0.0934535
$261$	0	0
$262$	−11.6164	−0.717667
$263$	10.3371	0.637414	0.318707	−	0.947853i	$-0.396751\pi$
$263$	10.3371	0.637414	0.318707	+	0.947853i	$0.396751\pi$
$264$	0	0
$265$	−1.39899	−0.0859395
$266$	−0.997196	−0.0611420
$267$	0	0
$268$	15.5545	0.950140
$269$	−26.3307	−1.60541	−0.802704	−	0.596377i	$-0.796606\pi$
$269$	−26.3307	−1.60541	−0.802704	+	0.596377i	$0.796606\pi$
$270$	0	0
$271$	−15.4606	−0.939164	−0.469582	−	0.882889i	$-0.655595\pi$
$271$	−15.4606	−0.939164	−0.469582	+	0.882889i	$0.655595\pi$
$272$	−2.65004	−0.160682
$273$	0	0
$274$	2.89886	0.175126
$275$	10.3496	0.624106
$276$	0	0
$277$	−0.672372	−0.0403989	−0.0201994	−	0.999796i	$-0.506430\pi$
$277$	−0.672372	−0.0403989	−0.0201994	+	0.999796i	$0.506430\pi$
$278$	−8.11250	−0.486555
$279$	0	0
$280$	1.35667	0.0810768
$281$	6.46513	0.385678	0.192839	−	0.981230i	$-0.438231\pi$
$281$	6.46513	0.385678	0.192839	+	0.981230i	$0.438231\pi$
$282$	0	0
$283$	−4.15301	−0.246871	−0.123435	−	0.992353i	$-0.539391\pi$
$283$	−4.15301	−0.246871	−0.123435	+	0.992353i	$0.539391\pi$
$284$	0.0821736	0.00487610
$285$	0	0
$286$	−4.89238	−0.289292
$287$	−13.8989	−0.820428
$288$	0	0
$289$	25.9799	1.52823
$290$	3.20925	0.188453
$291$	0	0
$292$	7.60044	0.444782
$293$	−19.0946	−1.11552	−0.557760	−	0.830003i	$-0.688339\pi$
$293$	−19.0946	−1.11552	−0.557760	+	0.830003i	$0.688339\pi$
$294$	0	0
$295$	−0.520006	−0.0302759
$296$	−31.9474	−1.85690
$297$	0	0
$298$	10.0241	0.580679
$299$	25.2515	1.46033
$300$	0	0
$301$	2.07529	0.119618
$302$	−14.1694	−0.815358
$303$	0	0
$304$	0.404222	0.0231837
$305$	−0.434334	−0.0248699
$306$	0	0
$307$	−17.3816	−0.992022	−0.496011	−	0.868316i	$-0.665202\pi$
$307$	−17.3816	−0.992022	−0.496011	+	0.868316i	$0.665202\pi$
$308$	−3.44268	−0.196165
$309$	0	0
$310$	−2.20991	−0.125515
$311$	−23.4195	−1.32800	−0.663998	−	0.747734i	$-0.731142\pi$
$311$	−23.4195	−1.32800	−0.663998	+	0.747734i	$0.731142\pi$
$312$	0	0
$313$	−5.36015	−0.302974	−0.151487	−	0.988459i	$-0.548406\pi$
$313$	−5.36015	−0.302974	−0.151487	+	0.988459i	$0.548406\pi$
$314$	16.0105	0.903525
$315$	0	0
$316$	8.97892	0.505103
$317$	−7.26085	−0.407810	−0.203905	−	0.978991i	$-0.565363\pi$
$317$	−7.26085	−0.407810	−0.203905	+	0.978991i	$0.565363\pi$
$318$	0	0
$319$	−20.4390	−1.14437
$320$	1.61836	0.0904693
$321$	0	0
$322$	−9.05828	−0.504798
$323$	−6.55590	−0.364780
$324$	0	0
$325$	−13.4338	−0.745173
$326$	−2.37089	−0.131312
$327$	0	0
$328$	−31.2933	−1.72788
$329$	1.21348	0.0669012
$330$	0	0
$331$	12.1810	0.669527	0.334763	−	0.942302i	$-0.391344\pi$
$331$	12.1810	0.669527	0.334763	+	0.942302i	$0.391344\pi$
$332$	−4.39228	−0.241058
$333$	0	0
$334$	−12.8756	−0.704519
$335$	−4.80486	−0.262518
$336$	0	0
$337$	−11.7505	−0.640090	−0.320045	−	0.947402i	$-0.603698\pi$
$337$	−11.7505	−0.640090	−0.320045	+	0.947402i	$0.603698\pi$
$338$	−4.33268	−0.235667
$339$	0	0
$340$	3.55380	0.192732
$341$	14.0745	0.762175
$342$	0	0
$343$	15.2018	0.820820
$344$	4.67250	0.251924
$345$	0	0
$346$	−10.5882	−0.569224
$347$	23.3226	1.25202	0.626011	−	0.779814i	$-0.284686\pi$
$347$	23.3226	1.25202	0.626011	+	0.779814i	$0.284686\pi$
$348$	0	0
$349$	−27.7914	−1.48764	−0.743820	−	0.668380i	$-0.766988\pi$
$349$	−27.7914	−1.48764	−0.743820	+	0.668380i	$0.766988\pi$
$350$	4.81900	0.257586
$351$	0	0
$352$	−12.4139	−0.661665
$353$	−4.16495	−0.221678	−0.110839	−	0.993838i	$-0.535354\pi$
$353$	−4.16495	−0.221678	−0.110839	+	0.993838i	$0.535354\pi$
$354$	0	0
$355$	−0.0253839	−0.00134724
$356$	5.03503	0.266856
$357$	0	0
$358$	13.6314	0.720440
$359$	13.0179	0.687060	0.343530	−	0.939142i	$-0.388377\pi$
$359$	13.0179	0.687060	0.343530	+	0.939142i	$0.388377\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−3.66471	−0.192613
$363$	0	0
$364$	4.46859	0.234218
$365$	−2.34782	−0.122890
$366$	0	0
$367$	−21.4588	−1.12014	−0.560070	−	0.828446i	$-0.689226\pi$
$367$	−21.4588	−1.12014	−0.560070	+	0.828446i	$0.689226\pi$
$368$	3.67185	0.191408
$369$	0	0
$370$	3.93211	0.204421
$371$	4.14863	0.215386
$372$	0	0
$373$	2.65784	0.137618	0.0688088	−	0.997630i	$-0.478080\pi$
$373$	2.65784	0.137618	0.0688088	+	0.997630i	$0.478080\pi$
$374$	11.5380	0.596615
$375$	0	0
$376$	2.73213	0.140899
$377$	26.5298	1.36635
$378$	0	0
$379$	6.51123	0.334459	0.167230	−	0.985918i	$-0.446518\pi$
$379$	6.51123	0.334459	0.167230	+	0.985918i	$0.446518\pi$
$380$	−0.542076	−0.0278079
$381$	0	0
$382$	−15.3595	−0.785860
$383$	−22.7933	−1.16468	−0.582341	−	0.812944i	$-0.697863\pi$
$383$	−22.7933	−1.16468	−0.582341	+	0.812944i	$0.697863\pi$
$384$	0	0
$385$	1.06346	0.0541992
$386$	3.05357	0.155423
$387$	0	0
$388$	9.34361	0.474350
$389$	−15.0334	−0.762224	−0.381112	−	0.924529i	$-0.624459\pi$
$389$	−15.0334	−0.762224	−0.381112	+	0.924529i	$0.624459\pi$
$390$	0	0
$391$	−59.5522	−3.01169
$392$	15.1018	0.762755
$393$	0	0
$394$	0.620209	0.0312457
$395$	−2.77364	−0.139557
$396$	0	0
$397$	28.3355	1.42212	0.711060	−	0.703131i	$-0.248215\pi$
$397$	28.3355	1.42212	0.711060	+	0.703131i	$0.248215\pi$
$398$	−1.17720	−0.0590079
$399$	0	0
$400$	−1.95342	−0.0976711
$401$	−17.4693	−0.872375	−0.436187	−	0.899856i	$-0.643671\pi$
$401$	−17.4693	−0.872375	−0.436187	+	0.899856i	$0.643671\pi$
$402$	0	0
$403$	−18.2686	−0.910025
$404$	23.1864	1.15357
$405$	0	0
$406$	−9.51682	−0.472312
$407$	−25.0428	−1.24132
$408$	0	0
$409$	9.50869	0.470174	0.235087	−	0.971974i	$-0.424462\pi$
$409$	9.50869	0.470174	0.235087	+	0.971974i	$0.424462\pi$
$410$	3.85161	0.190217
$411$	0	0
$412$	0.179204	0.00882873
$413$	1.54205	0.0758791
$414$	0	0
$415$	1.35680	0.0666027
$416$	16.1133	0.790017
$417$	0	0
$418$	−1.75994	−0.0860814
$419$	−22.8121	−1.11444	−0.557222	−	0.830363i	$-0.688133\pi$
$419$	−22.8121	−1.11444	−0.557222	+	0.830363i	$0.688133\pi$
$420$	0	0
$421$	14.6221	0.712639	0.356319	−	0.934364i	$-0.384032\pi$
$421$	14.6221	0.712639	0.356319	+	0.934364i	$0.384032\pi$
$422$	12.7230	0.619347
$423$	0	0
$424$	9.34059	0.453619
$425$	31.6817	1.53679
$426$	0	0
$427$	1.28799	0.0623302
$428$	13.3215	0.643919
$429$	0	0
$430$	−0.575095	−0.0277336
$431$	27.3845	1.31906	0.659532	−	0.751676i	$-0.270754\pi$
$431$	27.3845	1.31906	0.659532	+	0.751676i	$0.270754\pi$
$432$	0	0
$433$	−13.7244	−0.659551	−0.329775	−	0.944059i	$-0.606973\pi$
$433$	−13.7244	−0.659551	−0.329775	+	0.944059i	$0.606973\pi$
$434$	6.55336	0.314571
$435$	0	0
$436$	7.90040	0.378360
$437$	9.08375	0.434535
$438$	0	0
$439$	7.14990	0.341246	0.170623	−	0.985336i	$-0.445422\pi$
$439$	7.14990	0.341246	0.170623	+	0.985336i	$0.445422\pi$
$440$	2.39438	0.114148
$441$	0	0
$442$	−14.9763	−0.712349
$443$	17.8971	0.850317	0.425159	−	0.905119i	$-0.360218\pi$
$443$	17.8971	0.850317	0.425159	+	0.905119i	$0.360218\pi$
$444$	0	0
$445$	−1.55535	−0.0737306
$446$	−2.54143	−0.120340
$447$	0	0
$448$	−4.79915	−0.226739
$449$	−34.2325	−1.61553	−0.807765	−	0.589504i	$-0.799323\pi$
$449$	−34.2325	−1.61553	−0.807765	+	0.589504i	$0.799323\pi$
$450$	0	0
$451$	−24.5301	−1.15508
$452$	−25.5070	−1.19975
$453$	0	0
$454$	−1.83154	−0.0859586
$455$	−1.38037	−0.0647129
$456$	0	0
$457$	−22.4109	−1.04834	−0.524168	−	0.851615i	$-0.675623\pi$
$457$	−22.4109	−1.04834	−0.524168	+	0.851615i	$0.675623\pi$
$458$	15.8655	0.741347
$459$	0	0
$460$	−4.92408	−0.229586
$461$	−9.20541	−0.428739	−0.214369	−	0.976753i	$-0.568770\pi$
$461$	−9.20541	−0.428739	−0.214369	+	0.976753i	$0.568770\pi$
$462$	0	0
$463$	−9.16002	−0.425702	−0.212851	−	0.977085i	$-0.568275\pi$
$463$	−9.16002	−0.425702	−0.212851	+	0.977085i	$0.568275\pi$
$464$	3.85772	0.179090
$465$	0	0
$466$	14.1475	0.655371
$467$	33.1209	1.53265	0.766327	−	0.642451i	$-0.222082\pi$
$467$	33.1209	1.53265	0.766327	+	0.642451i	$0.222082\pi$
$468$	0	0
$469$	14.2485	0.657935
$470$	−0.336273	−0.0155111
$471$	0	0
$472$	3.47190	0.159807
$473$	3.66266	0.168409
$474$	0	0
$475$	−4.83255	−0.221733
$476$	−10.5386	−0.483034
$477$	0	0
$478$	21.8012	0.997164
$479$	−14.3619	−0.656214	−0.328107	−	0.944641i	$-0.606411\pi$
$479$	−14.3619	−0.656214	−0.328107	+	0.944641i	$0.606411\pi$
$480$	0	0
$481$	32.5055	1.48212
$482$	−2.61278	−0.119009
$483$	0	0
$484$	8.49573	0.386169
$485$	−2.88629	−0.131060
$486$	0	0
$487$	−40.9513	−1.85568	−0.927840	−	0.372979i	$-0.878336\pi$
$487$	−40.9513	−1.85568	−0.927840	+	0.372979i	$0.878336\pi$
$488$	2.89989	0.131272
$489$	0	0
$490$	−1.85874	−0.0839693
$491$	−19.8733	−0.896868	−0.448434	−	0.893816i	$-0.648018\pi$
$491$	−19.8733	−0.896868	−0.448434	+	0.893816i	$0.648018\pi$
$492$	0	0
$493$	−62.5668	−2.81787
$494$	2.28440	0.102780
$495$	0	0
$496$	−2.65646	−0.119278
$497$	0.0752743	0.00337651
$498$	0	0
$499$	−24.0897	−1.07841	−0.539203	−	0.842176i	$-0.681274\pi$
$499$	−24.0897	−1.07841	−0.539203	+	0.842176i	$0.681274\pi$
$500$	5.32999	0.238364
$501$	0	0
$502$	13.2035	0.589302
$503$	−15.9184	−0.709765	−0.354883	−	0.934911i	$-0.615479\pi$
$503$	−15.9184	−0.709765	−0.354883	+	0.934911i	$0.615479\pi$
$504$	0	0
$505$	−7.16241	−0.318723
$506$	−15.9869	−0.710702
$507$	0	0
$508$	5.11882	0.227111
$509$	−2.52612	−0.111968	−0.0559842	−	0.998432i	$-0.517830\pi$
$509$	−2.52612	−0.111968	−0.0559842	+	0.998432i	$0.517830\pi$
$510$	0	0
$511$	6.96231	0.307994
$512$	4.55182	0.201164
$513$	0	0
$514$	−21.1432	−0.932588
$515$	−0.0553570	−0.00243932
$516$	0	0
$517$	2.14165	0.0941897
$518$	−11.6604	−0.512330
$519$	0	0
$520$	−3.10790	−0.136290
$521$	−24.0819	−1.05505	−0.527524	−	0.849540i	$-0.676879\pi$
$521$	−24.0819	−1.05505	−0.527524	+	0.849540i	$0.676879\pi$
$522$	0	0
$523$	−32.9333	−1.44007	−0.720036	−	0.693937i	$-0.755875\pi$
$523$	−32.9333	−1.44007	−0.720036	+	0.693937i	$0.755875\pi$
$524$	18.7258	0.818042
$525$	0	0
$526$	8.49471	0.370387
$527$	43.0840	1.87677
$528$	0	0
$529$	59.5146	2.58759
$530$	−1.14965	−0.0499375
$531$	0	0
$532$	1.60749	0.0696936
$533$	31.8399	1.37914
$534$	0	0
$535$	−4.11509	−0.177911
$536$	32.0804	1.38566
$537$	0	0
$538$	−21.6377	−0.932867
$539$	11.8379	0.509895
$540$	0	0
$541$	−24.6563	−1.06006	−0.530029	−	0.847979i	$-0.677819\pi$
$541$	−24.6563	−1.06006	−0.530029	+	0.847979i	$0.677819\pi$
$542$	−12.7050	−0.545727
$543$	0	0
$544$	−38.0009	−1.62927
$545$	−2.44048	−0.104539
$546$	0	0
$547$	30.5012	1.30414	0.652068	−	0.758160i	$-0.273902\pi$
$547$	30.5012	1.30414	0.652068	+	0.758160i	$0.273902\pi$
$548$	−4.67299	−0.199620
$549$	0	0
$550$	8.50499	0.362654
$551$	9.54358	0.406570
$552$	0	0
$553$	8.22505	0.349765
$554$	−0.552533	−0.0234749
$555$	0	0
$556$	13.0774	0.554607
$557$	−3.65243	−0.154758	−0.0773792	−	0.997002i	$-0.524655\pi$
$557$	−3.65243	−0.154758	−0.0773792	+	0.997002i	$0.524655\pi$
$558$	0	0
$559$	−4.75412	−0.201078
$560$	−0.200722	−0.00848203
$561$	0	0
$562$	5.31284	0.224108
$563$	−10.5396	−0.444190	−0.222095	−	0.975025i	$-0.571290\pi$
$563$	−10.5396	−0.444190	−0.222095	+	0.975025i	$0.571290\pi$
$564$	0	0
$565$	7.87926	0.331483
$566$	−3.41281	−0.143451
$567$	0	0
$568$	0.169479	0.00711119
$569$	−18.4134	−0.771932	−0.385966	−	0.922513i	$-0.626132\pi$
$569$	−18.4134	−0.771932	−0.385966	+	0.922513i	$0.626132\pi$
$570$	0	0
$571$	28.9656	1.21217	0.606086	−	0.795399i	$-0.292739\pi$
$571$	28.9656	1.21217	0.606086	+	0.795399i	$0.292739\pi$
$572$	7.88657	0.329754
$573$	0	0
$574$	−11.4217	−0.476732
$575$	−43.8977	−1.83066
$576$	0	0
$577$	−32.2625	−1.34311	−0.671553	−	0.740956i	$-0.734373\pi$
$577$	−32.2625	−1.34311	−0.671553	+	0.740956i	$0.734373\pi$
$578$	21.3494	0.888019
$579$	0	0
$580$	−5.17334	−0.214811
$581$	−4.02351	−0.166923
$582$	0	0
$583$	7.32186	0.303241
$584$	15.6756	0.648659
$585$	0	0
$586$	−15.6913	−0.648203
$587$	14.5010	0.598520	0.299260	−	0.954172i	$-0.403260\pi$
$587$	14.5010	0.598520	0.299260	+	0.954172i	$0.403260\pi$
$588$	0	0
$589$	−6.57178	−0.270786
$590$	−0.427324	−0.0175927
$591$	0	0
$592$	4.72665	0.194264
$593$	34.3870	1.41210	0.706051	−	0.708161i	$-0.250475\pi$
$593$	34.3870	1.41210	0.706051	+	0.708161i	$0.250475\pi$
$594$	0	0
$595$	3.25542	0.133459
$596$	−16.1589	−0.661895
$597$	0	0
$598$	20.7509	0.848567
$599$	−4.72237	−0.192951	−0.0964753	−	0.995335i	$-0.530757\pi$
$599$	−4.72237	−0.192951	−0.0964753	+	0.995335i	$0.530757\pi$
$600$	0	0
$601$	−5.34061	−0.217848	−0.108924	−	0.994050i	$-0.534741\pi$
$601$	−5.34061	−0.217848	−0.108924	+	0.994050i	$0.534741\pi$
$602$	1.70541	0.0695072
$603$	0	0
$604$	22.8412	0.929397
$605$	−2.62438	−0.106696
$606$	0	0
$607$	30.2826	1.22913	0.614567	−	0.788865i	$-0.289331\pi$
$607$	30.2826	1.22913	0.614567	+	0.788865i	$0.289331\pi$
$608$	5.79643	0.235076
$609$	0	0
$610$	−0.356922	−0.0144513
$611$	−2.77986	−0.112461
$612$	0	0
$613$	20.1496	0.813836	0.406918	−	0.913465i	$-0.366603\pi$
$613$	20.1496	0.813836	0.406918	+	0.913465i	$0.366603\pi$
$614$	−14.2837	−0.576441
$615$	0	0
$616$	−7.10038	−0.286082
$617$	−36.8742	−1.48450	−0.742250	−	0.670123i	$-0.766241\pi$
$617$	−36.8742	−1.48450	−0.742250	+	0.670123i	$0.766241\pi$
$618$	0	0
$619$	16.7072	0.671518	0.335759	−	0.941948i	$-0.391007\pi$
$619$	16.7072	0.671518	0.335759	+	0.941948i	$0.391007\pi$
$620$	3.56240	0.143070
$621$	0	0
$622$	−19.2454	−0.771669
$623$	4.61229	0.184787
$624$	0	0
$625$	22.5163	0.900651
$626$	−4.40480	−0.176051
$627$	0	0
$628$	−25.8091	−1.02990
$629$	−76.6596	−3.05662
$630$	0	0
$631$	−35.3548	−1.40745	−0.703726	−	0.710471i	$-0.748482\pi$
$631$	−35.3548	−1.40745	−0.703726	+	0.710471i	$0.748482\pi$
$632$	18.5186	0.736630
$633$	0	0
$634$	−5.96673	−0.236969
$635$	−1.58123	−0.0627493
$636$	0	0
$637$	−15.3656	−0.608807
$638$	−16.7961	−0.664965
$639$	0	0
$640$	−3.41396	−0.134949
$641$	25.0193	0.988203	0.494101	−	0.869404i	$-0.335497\pi$
$641$	25.0193	0.988203	0.494101	+	0.869404i	$0.335497\pi$
$642$	0	0
$643$	−14.4346	−0.569245	−0.284622	−	0.958640i	$-0.591868\pi$
$643$	−14.4346	−0.569245	−0.284622	+	0.958640i	$0.591868\pi$
$644$	14.6021	0.575401
$645$	0	0
$646$	−5.38743	−0.211966
$647$	44.5052	1.74968	0.874841	−	0.484411i	$-0.160966\pi$
$647$	44.5052	1.74968	0.874841	+	0.484411i	$0.160966\pi$
$648$	0	0
$649$	2.72154	0.106830
$650$	−11.0395	−0.433003
$651$	0	0
$652$	3.82190	0.149677
$653$	42.4252	1.66023	0.830113	−	0.557595i	$-0.188276\pi$
$653$	42.4252	1.66023	0.830113	+	0.557595i	$0.188276\pi$
$654$	0	0
$655$	−5.78452	−0.226020
$656$	4.62988	0.180766
$657$	0	0
$658$	0.997196	0.0388747
$659$	20.1247	0.783946	0.391973	−	0.919977i	$-0.371793\pi$
$659$	20.1247	0.783946	0.391973	+	0.919977i	$0.371793\pi$
$660$	0	0
$661$	11.4631	0.445862	0.222931	−	0.974834i	$-0.428438\pi$
$661$	11.4631	0.445862	0.222931	+	0.974834i	$0.428438\pi$
$662$	10.0099	0.389047
$663$	0	0
$664$	−9.05888	−0.351553
$665$	−0.496563	−0.0192559
$666$	0	0
$667$	86.6915	3.35671
$668$	20.7555	0.803056
$669$	0	0
$670$	−3.94848	−0.152543
$671$	2.27316	0.0877543
$672$	0	0
$673$	31.5548	1.21635	0.608175	−	0.793803i	$-0.291902\pi$
$673$	31.5548	1.21635	0.608175	+	0.793803i	$0.291902\pi$
$674$	−9.65617	−0.371942
$675$	0	0
$676$	6.98434	0.268628
$677$	1.52175	0.0584856	0.0292428	−	0.999572i	$-0.490690\pi$
$677$	1.52175	0.0584856	0.0292428	+	0.999572i	$0.490690\pi$
$678$	0	0
$679$	8.55912	0.328469
$680$	7.32954	0.281075
$681$	0	0
$682$	11.5659	0.442883
$683$	−27.7363	−1.06130	−0.530649	−	0.847591i	$-0.678052\pi$
$683$	−27.7363	−1.06130	−0.530649	+	0.847591i	$0.678052\pi$
$684$	0	0
$685$	1.44351	0.0551538
$686$	12.4923	0.476960
$687$	0	0
$688$	−0.691301	−0.0263556
$689$	−9.50376	−0.362064
$690$	0	0
$691$	43.3650	1.64968	0.824841	−	0.565365i	$-0.191265\pi$
$691$	43.3650	1.64968	0.824841	+	0.565365i	$0.191265\pi$
$692$	17.0683	0.648838
$693$	0	0
$694$	19.1658	0.727522
$695$	−4.03969	−0.153234
$696$	0	0
$697$	−75.0901	−2.84424
$698$	−22.8381	−0.864434
$699$	0	0
$700$	−7.76828	−0.293613
$701$	−12.6082	−0.476203	−0.238102	−	0.971240i	$-0.576525\pi$
$701$	−12.6082	−0.476203	−0.238102	+	0.971240i	$0.576525\pi$
$702$	0	0
$703$	11.6932	0.441018
$704$	−8.46997	−0.319224
$705$	0	0
$706$	−3.42262	−0.128812
$707$	21.2397	0.798800
$708$	0	0
$709$	27.3787	1.02823	0.514114	−	0.857722i	$-0.328121\pi$
$709$	27.3787	1.02823	0.514114	+	0.857722i	$0.328121\pi$
$710$	−0.0208597	−0.000782849 0
$711$	0	0
$712$	10.3845	0.389176
$713$	−59.6965	−2.23565
$714$	0	0
$715$	−2.43621	−0.0911089
$716$	−21.9739	−0.821204
$717$	0	0
$718$	10.6977	0.399235
$719$	−5.93611	−0.221380	−0.110690	−	0.993855i	$-0.535306\pi$
$719$	−5.93611	−0.221380	−0.110690	+	0.993855i	$0.535306\pi$
$720$	0	0
$721$	0.164158	0.00611355
$722$	0.821767	0.0305830
$723$	0	0
$724$	5.90755	0.219552
$725$	−46.1198	−1.71285
$726$	0	0
$727$	−33.5443	−1.24409	−0.622044	−	0.782982i	$-0.713698\pi$
$727$	−33.5443	−1.24409	−0.622044	+	0.782982i	$0.713698\pi$
$728$	9.21627	0.341578
$729$	0	0
$730$	−1.92936	−0.0714089
$731$	11.2119	0.414688
$732$	0	0
$733$	18.2587	0.674402	0.337201	−	0.941433i	$-0.390520\pi$
$733$	18.2587	0.674402	0.337201	+	0.941433i	$0.390520\pi$
$734$	−17.6341	−0.650887
$735$	0	0
$736$	52.6534	1.94083
$737$	25.1470	0.926303
$738$	0	0
$739$	−18.3693	−0.675727	−0.337863	−	0.941195i	$-0.609704\pi$
$739$	−18.3693	−0.675727	−0.337863	+	0.941195i	$0.609704\pi$
$740$	−6.33861	−0.233012
$741$	0	0
$742$	3.40921	0.125156
$743$	−1.04585	−0.0383684	−0.0191842	−	0.999816i	$-0.506107\pi$
$743$	−1.04585	−0.0383684	−0.0191842	+	0.999816i	$0.506107\pi$
$744$	0	0
$745$	4.99158	0.182877
$746$	2.18412	0.0799665
$747$	0	0
$748$	−18.5994	−0.680060
$749$	12.2030	0.445889
$750$	0	0
$751$	−33.0446	−1.20582	−0.602908	−	0.797811i	$-0.705991\pi$
$751$	−33.0446	−1.20582	−0.602908	+	0.797811i	$0.705991\pi$
$752$	−0.404222	−0.0147404
$753$	0	0
$754$	21.8013	0.793957
$755$	−7.05579	−0.256787
$756$	0	0
$757$	0.960854	0.0349228	0.0174614	−	0.999848i	$-0.494442\pi$
$757$	0.960854	0.0349228	0.0174614	+	0.999848i	$0.494442\pi$
$758$	5.35071	0.194347
$759$	0	0
$760$	−1.11801	−0.0405543
$761$	25.6788	0.930857	0.465429	−	0.885085i	$-0.345900\pi$
$761$	25.6788	0.930857	0.465429	+	0.885085i	$0.345900\pi$
$762$	0	0
$763$	7.23708	0.262000
$764$	24.7597	0.895774
$765$	0	0
$766$	−18.7308	−0.676771
$767$	−3.53255	−0.127553
$768$	0	0
$769$	3.83553	0.138313	0.0691564	−	0.997606i	$-0.477969\pi$
$769$	3.83553	0.138313	0.0691564	+	0.997606i	$0.477969\pi$
$770$	0.873921	0.0314939
$771$	0	0
$772$	−4.92239	−0.177161
$773$	−25.3278	−0.910977	−0.455488	−	0.890242i	$-0.650535\pi$
$773$	−25.3278	−0.910977	−0.455488	+	0.890242i	$0.650535\pi$
$774$	0	0
$775$	31.7585	1.14080
$776$	19.2708	0.691780
$777$	0	0
$778$	−12.3540	−0.442911
$779$	11.4538	0.410375
$780$	0	0
$781$	0.132851	0.00475377
$782$	−48.9381	−1.75002
$783$	0	0
$784$	−2.23432	−0.0797973
$785$	7.97258	0.284554
$786$	0	0
$787$	35.5152	1.26598	0.632989	−	0.774160i	$-0.281828\pi$
$787$	35.5152	1.26598	0.632989	+	0.774160i	$0.281828\pi$
$788$	−0.999784	−0.0356158
$789$	0	0
$790$	−2.27929	−0.0810934
$791$	−23.3654	−0.830779
$792$	0	0
$793$	−2.95055	−0.104777
$794$	23.2852	0.826362
$795$	0	0
$796$	1.89767	0.0672610
$797$	38.3841	1.35963	0.679817	−	0.733382i	$-0.262059\pi$
$797$	38.3841	1.35963	0.679817	+	0.733382i	$0.262059\pi$
$798$	0	0
$799$	6.55590	0.231931
$800$	−28.0116	−0.990358
$801$	0	0
$802$	−14.3557	−0.506917
$803$	12.2877	0.433623
$804$	0	0
$805$	−4.51066	−0.158980
$806$	−15.0126	−0.528795
$807$	0	0
$808$	47.8209	1.68233
$809$	8.01559	0.281813	0.140907	−	0.990023i	$-0.454998\pi$
$809$	8.01559	0.281813	0.140907	+	0.990023i	$0.454998\pi$
$810$	0	0
$811$	24.8032	0.870958	0.435479	−	0.900199i	$-0.356579\pi$
$811$	24.8032	0.870958	0.435479	+	0.900199i	$0.356579\pi$
$812$	15.3412	0.538371
$813$	0	0
$814$	−20.5793	−0.721306
$815$	−1.18061	−0.0413549
$816$	0	0
$817$	−1.71020	−0.0598325
$818$	7.81393	0.273208
$819$	0	0
$820$	−6.20883	−0.216822
$821$	−27.2624	−0.951464	−0.475732	−	0.879590i	$-0.657817\pi$
$821$	−27.2624	−0.951464	−0.475732	+	0.879590i	$0.657817\pi$
$822$	0	0
$823$	−2.42972	−0.0846948	−0.0423474	−	0.999103i	$-0.513484\pi$
$823$	−2.42972	−0.0846948	−0.0423474	+	0.999103i	$0.513484\pi$
$824$	0.369599	0.0128756
$825$	0	0
$826$	1.26720	0.0440916
$827$	0.0245168	0.000852534 0	0.000426267	−	1.00000i	$-0.499864\pi$
$827$	0.0245168	0.000852534 0	0.000426267		1.00000i	$0.499864\pi$
$828$	0	0
$829$	26.0306	0.904079	0.452040	−	0.891998i	$-0.350697\pi$
$829$	26.0306	0.904079	0.452040	+	0.891998i	$0.350697\pi$
$830$	1.11497	0.0387013
$831$	0	0
$832$	10.9940	0.381148
$833$	36.2376	1.25556
$834$	0	0
$835$	−6.41151	−0.221879
$836$	2.83704	0.0981211
$837$	0	0
$838$	−18.7462	−0.647578
$839$	−23.0573	−0.796027	−0.398013	−	0.917380i	$-0.630300\pi$
$839$	−23.0573	−0.796027	−0.398013	+	0.917380i	$0.630300\pi$
$840$	0	0
$841$	62.0799	2.14069
$842$	12.0160	0.414098
$843$	0	0
$844$	−20.5097	−0.705972
$845$	−2.15750	−0.0742203
$846$	0	0
$847$	7.78243	0.267407
$848$	−1.38195	−0.0474564
$849$	0	0
$850$	26.0350	0.892994
$851$	106.218	3.64112
$852$	0	0
$853$	−8.49702	−0.290933	−0.145466	−	0.989363i	$-0.546468\pi$
$853$	−8.49702	−0.290933	−0.145466	+	0.989363i	$0.546468\pi$
$854$	1.05843	0.0362187
$855$	0	0
$856$	27.4750	0.939075
$857$	−14.8407	−0.506949	−0.253474	−	0.967342i	$-0.581573\pi$
$857$	−14.8407	−0.506949	−0.253474	+	0.967342i	$0.581573\pi$
$858$	0	0
$859$	−37.5272	−1.28041	−0.640206	−	0.768203i	$-0.721151\pi$
$859$	−37.5272	−1.28041	−0.640206	+	0.768203i	$0.721151\pi$
$860$	0.927060	0.0316125
$861$	0	0
$862$	22.5037	0.766478
$863$	−13.3818	−0.455522	−0.227761	−	0.973717i	$-0.573141\pi$
$863$	−13.3818	−0.455522	−0.227761	+	0.973717i	$0.573141\pi$
$864$	0	0
$865$	−5.27249	−0.179270
$866$	−11.2782	−0.383250
$867$	0	0
$868$	−10.5641	−0.358568
$869$	14.5163	0.492431
$870$	0	0
$871$	−32.6408	−1.10599
$872$	16.2942	0.551791
$873$	0	0
$874$	7.46473	0.252498
$875$	4.88248	0.165058
$876$	0	0
$877$	−17.0398	−0.575392	−0.287696	−	0.957722i	$-0.592889\pi$
$877$	−17.0398	−0.575392	−0.287696	+	0.957722i	$0.592889\pi$
$878$	5.87555	0.198290
$879$	0	0
$880$	−0.354251	−0.0119418
$881$	−51.5560	−1.73697	−0.868483	−	0.495719i	$-0.834904\pi$
$881$	−51.5560	−1.73697	−0.868483	+	0.495719i	$0.834904\pi$
$882$	0	0
$883$	−8.09273	−0.272342	−0.136171	−	0.990685i	$-0.543480\pi$
$883$	−8.09273	−0.272342	−0.136171	+	0.990685i	$0.543480\pi$
$884$	24.1419	0.811981
$885$	0	0
$886$	14.7073	0.494100
$887$	−24.5597	−0.824633	−0.412316	−	0.911041i	$-0.635280\pi$
$887$	−24.5597	−0.824633	−0.412316	+	0.911041i	$0.635280\pi$
$888$	0	0
$889$	4.68904	0.157265
$890$	−1.27814	−0.0428432
$891$	0	0
$892$	4.09681	0.137171
$893$	−1.00000	−0.0334637
$894$	0	0
$895$	6.78787	0.226893
$896$	10.1239	0.338215
$897$	0	0
$898$	−28.1311	−0.938748
$899$	−62.7183	−2.09177
$900$	0	0
$901$	22.4133	0.746695
$902$	−20.1580	−0.671188
$903$	0	0
$904$	−52.6070	−1.74968
$905$	−1.82488	−0.0606609
$906$	0	0
$907$	−55.2963	−1.83609	−0.918043	−	0.396482i	$-0.870231\pi$
$907$	−55.2963	−1.83609	−0.918043	+	0.396482i	$0.870231\pi$
$908$	2.95247	0.0979811
$909$	0	0
$910$	−1.13435	−0.0376032
$911$	20.9491	0.694074	0.347037	−	0.937852i	$-0.387188\pi$
$911$	20.9491	0.694074	0.347037	+	0.937852i	$0.387188\pi$
$912$	0	0
$913$	−7.10104	−0.235010
$914$	−18.4165	−0.609164
$915$	0	0
$916$	−25.5754	−0.845035
$917$	17.1536	0.566463
$918$	0	0
$919$	−29.8962	−0.986183	−0.493092	−	0.869977i	$-0.664133\pi$
$919$	−29.8962	−0.986183	−0.493092	+	0.869977i	$0.664133\pi$
$920$	−10.1557	−0.334823
$921$	0	0
$922$	−7.56471	−0.249130
$923$	−0.172440	−0.00567593
$924$	0	0
$925$	−56.5080	−1.85797
$926$	−7.52741	−0.247366
$927$	0	0
$928$	55.3187	1.81593
$929$	16.0710	0.527274	0.263637	−	0.964622i	$-0.415078\pi$
$929$	16.0710	0.527274	0.263637	+	0.964622i	$0.415078\pi$
$930$	0	0
$931$	−5.52747	−0.181156
$932$	−22.8059	−0.747033
$933$	0	0
$934$	27.2177	0.890591
$935$	5.74545	0.187896
$936$	0	0
$937$	−40.9734	−1.33854	−0.669271	−	0.743018i	$-0.733394\pi$
$937$	−40.9734	−1.33854	−0.669271	+	0.743018i	$0.733394\pi$
$938$	11.7090	0.382311
$939$	0	0
$940$	0.542076	0.0176806
$941$	28.8380	0.940092	0.470046	−	0.882642i	$-0.344237\pi$
$941$	28.8380	0.940092	0.470046	+	0.882642i	$0.344237\pi$
$942$	0	0
$943$	104.044	3.38813
$944$	−0.513671	−0.0167186
$945$	0	0
$946$	3.00985	0.0978588
$947$	3.37397	0.109639	0.0548197	−	0.998496i	$-0.482542\pi$
$947$	3.37397	0.109639	0.0548197	+	0.998496i	$0.482542\pi$
$948$	0	0
$949$	−15.9494	−0.517739
$950$	−3.97123	−0.128844
$951$	0	0
$952$	−21.7353	−0.704445
$953$	7.42143	0.240404	0.120202	−	0.992749i	$-0.461646\pi$
$953$	7.42143	0.240404	0.120202	+	0.992749i	$0.461646\pi$
$954$	0	0
$955$	−7.64841	−0.247497
$956$	−35.1438	−1.13663
$957$	0	0
$958$	−11.8022	−0.381311
$959$	−4.28065	−0.138229
$960$	0	0
$961$	12.1883	0.393172
$962$	26.7119	0.861227
$963$	0	0
$964$	4.21183	0.135654
$965$	1.52055	0.0489483
$966$	0	0
$967$	42.1262	1.35469	0.677344	−	0.735666i	$-0.263131\pi$
$967$	42.1262	1.35469	0.677344	+	0.735666i	$0.263131\pi$
$968$	17.5220	0.563180
$969$	0	0
$970$	−2.37186	−0.0761559
$971$	−53.0085	−1.70112	−0.850561	−	0.525876i	$-0.823738\pi$
$971$	−53.0085	−1.70112	−0.850561	+	0.525876i	$0.823738\pi$
$972$	0	0
$973$	11.9795	0.384044
$974$	−33.6524	−1.07829
$975$	0	0
$976$	−0.429043	−0.0137333
$977$	−19.0714	−0.610148	−0.305074	−	0.952329i	$-0.598681\pi$
$977$	−19.0714	−0.610148	−0.305074	+	0.952329i	$0.598681\pi$
$978$	0	0
$979$	8.14018	0.260161
$980$	2.99631	0.0957136
$981$	0	0
$982$	−16.3312	−0.521150
$983$	16.4657	0.525176	0.262588	−	0.964908i	$-0.415424\pi$
$983$	16.4657	0.525176	0.262588	+	0.964908i	$0.415424\pi$
$984$	0	0
$985$	0.308839	0.00984043
$986$	−51.4154	−1.63740
$987$	0	0
$988$	−3.68247	−0.117155
$989$	−15.5351	−0.493986
$990$	0	0
$991$	−43.8196	−1.39197	−0.695987	−	0.718054i	$-0.745033\pi$
$991$	−43.8196	−1.39197	−0.695987	+	0.718054i	$0.745033\pi$
$992$	−38.0929	−1.20945
$993$	0	0
$994$	0.0618580	0.00196202
$995$	−0.586200	−0.0185838
$996$	0	0
$997$	−3.06269	−0.0969962	−0.0484981	−	0.998823i	$-0.515443\pi$
$997$	−3.06269	−0.0969962	−0.0484981	+	0.998823i	$0.515443\pi$
$998$	−19.7962	−0.626637
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8037.2.a.m.1.8		12
3.2	odd	2		893.2.a.a.1.5	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
893.2.a.a.1.5	✓	12	3.2	odd	2
8037.2.a.m.1.8		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8037.a (trivial)

\(f(q)\)	\(=\)	\(q+0.821767 q^{2} -1.32470 q^{4} +0.409207 q^{5} -1.21348 q^{7} -2.73213 q^{8} +O(q^{10})\)	\(q+0.821767 q^{2} -1.32470 q^{4} +0.409207 q^{5} -1.21348 q^{7} -2.73213 q^{8} +0.336273 q^{10} -2.14165 q^{11} +2.77986 q^{13} -0.997196 q^{14} +0.404222 q^{16} -6.55590 q^{17} +1.00000 q^{19} -0.542076 q^{20} -1.75994 q^{22} +9.08375 q^{23} -4.83255 q^{25} +2.28440 q^{26} +1.60749 q^{28} +9.54358 q^{29} -6.57178 q^{31} +5.79643 q^{32} -5.38743 q^{34} -0.496563 q^{35} +11.6932 q^{37} +0.821767 q^{38} -1.11801 q^{40} +11.4538 q^{41} -1.71020 q^{43} +2.83704 q^{44} +7.46473 q^{46} -1.00000 q^{47} -5.52747 q^{49} -3.97123 q^{50} -3.68247 q^{52} -3.41879 q^{53} -0.876378 q^{55} +3.31538 q^{56} +7.84260 q^{58} -1.27077 q^{59} -1.06140 q^{61} -5.40048 q^{62} +3.95488 q^{64} +1.13754 q^{65} -11.7419 q^{67} +8.68459 q^{68} -0.408059 q^{70} -0.0620319 q^{71} -5.73749 q^{73} +9.60910 q^{74} -1.32470 q^{76} +2.59884 q^{77} -6.77808 q^{79} +0.165410 q^{80} +9.41237 q^{82} +3.31568 q^{83} -2.68272 q^{85} -1.40539 q^{86} +5.85127 q^{88} -3.80089 q^{89} -3.37329 q^{91} -12.0332 q^{92} -0.821767 q^{94} +0.409207 q^{95} -7.05338 q^{97} -4.54230 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7}+O(q^{10}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7	\( 12 q + q^{2} + 7 q^{4} + 7 q^{5} - 13 q^{7} + q^{10} + 4 q^{11} - 17 q^{13} - 3 q^{14} - 19 q^{16} + 6 q^{17} + 12 q^{19} - 5 q^{20} - 8 q^{22} + 13 q^{23} - 7 q^{25} + 19 q^{26} - 29 q^{28} + 2 q^{29} - 14 q^{31} + 21 q^{32} - 6 q^{34} + 3 q^{35} - 2 q^{37} + q^{38} + 8 q^{40} - 8 q^{41} - 42 q^{43} - 24 q^{44} - 9 q^{46} - 12 q^{47} - 5 q^{49} + 33 q^{50} - 26 q^{52} - 3 q^{53} - 12 q^{55} - 7 q^{56} - 16 q^{58} - 8 q^{59} - 6 q^{61} + 24 q^{62} - 22 q^{64} - 22 q^{65} - 29 q^{67} + 30 q^{68} - 34 q^{70} + 7 q^{71} - 48 q^{73} - 25 q^{74} + 7 q^{76} + 18 q^{77} - 11 q^{79} - 3 q^{80} + 28 q^{82} + 57 q^{83} - 7 q^{85} - 9 q^{86} - 11 q^{88} + 2 q^{89} - 4 q^{91} + 13 q^{92} - q^{94} + 7 q^{95} - 14 q^{97} - 58 q^{98}+O(q^{100}) \) 12 * q + q^2 + 7 * q^4 + 7 * q^5 - 13 * q^7 + q^10 + 4 * q^11 - 17 * q^13 - 3 * q^14 - 19 * q^16 + 6 * q^17 + 12 * q^19 - 5 * q^20 - 8 * q^22 + 13 * q^23 - 7 * q^25 + 19 * q^26 - 29 * q^28 + 2 * q^29 - 14 * q^31 + 21 * q^32 - 6 * q^34 + 3 * q^35 - 2 * q^37 + q^38 + 8 * q^40 - 8 * q^41 - 42 * q^43 - 24 * q^44 - 9 * q^46 - 12 * q^47 - 5 * q^49 + 33 * q^50 - 26 * q^52 - 3 * q^53 - 12 * q^55 - 7 * q^56 - 16 * q^58 - 8 * q^59 - 6 * q^61 + 24 * q^62 - 22 * q^64 - 22 * q^65 - 29 * q^67 + 30 * q^68 - 34 * q^70 + 7 * q^71 - 48 * q^73 - 25 * q^74 + 7 * q^76 + 18 * q^77 - 11 * q^79 - 3 * q^80 + 28 * q^82 + 57 * q^83 - 7 * q^85 - 9 * q^86 - 11 * q^88 + 2 * q^89 - 4 * q^91 + 13 * q^92 - q^94 + 7 * q^95 - 14 * q^97 - 58 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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