LMFDB - Embedded newform 1503.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.31801$ of defining polynomial
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.559296 q^{2} -1.68719 q^{4} +3.48658 q^{5} +3.13190 q^{7} +2.06223 q^{8} +O(q^{10})$	\(q-0.559296 q^{2} -1.68719 q^{4} +3.48658 q^{5} +3.13190 q^{7} +2.06223 q^{8} -1.95003 q^{10} +5.14945 q^{11} -2.47676 q^{13} -1.75166 q^{14} +2.22098 q^{16} +7.36613 q^{17} -1.17562 q^{19} -5.88251 q^{20} -2.88007 q^{22} -4.38769 q^{23} +7.15622 q^{25} +1.38524 q^{26} -5.28411 q^{28} +8.41200 q^{29} -4.26508 q^{31} -5.36664 q^{32} -4.11985 q^{34} +10.9196 q^{35} -6.00097 q^{37} +0.657519 q^{38} +7.19012 q^{40} -10.6359 q^{41} +1.62326 q^{43} -8.68809 q^{44} +2.45402 q^{46} -8.29637 q^{47} +2.80882 q^{49} -4.00244 q^{50} +4.17877 q^{52} +5.10653 q^{53} +17.9540 q^{55} +6.45870 q^{56} -4.70480 q^{58} -0.146919 q^{59} -7.55907 q^{61} +2.38544 q^{62} -1.44042 q^{64} -8.63543 q^{65} +12.8674 q^{67} -12.4280 q^{68} -6.10730 q^{70} -13.9141 q^{71} -2.23198 q^{73} +3.35632 q^{74} +1.98349 q^{76} +16.1276 q^{77} +3.03137 q^{79} +7.74362 q^{80} +5.94864 q^{82} +6.08068 q^{83} +25.6826 q^{85} -0.907881 q^{86} +10.6193 q^{88} +10.2591 q^{89} -7.75699 q^{91} +7.40286 q^{92} +4.64013 q^{94} -4.09889 q^{95} -7.33303 q^{97} -1.57096 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8	$ 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8} + 5 q^{10} + 15 q^{11} + 3 q^{14} + 10 q^{16} + 11 q^{17} - 16 q^{19} + 9 q^{20} + 7 q^{22} + 9 q^{23} + 6 q^{25} - 7 q^{26} + 9 q^{28} + q^{29} - 18 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 7 q^{37} - 20 q^{38} + 32 q^{40} + 10 q^{41} + 6 q^{43} + 5 q^{44} + 13 q^{46} + 7 q^{47} + 11 q^{49} - 3 q^{50} - 6 q^{52} + 9 q^{53} + 17 q^{55} + 15 q^{56} - 19 q^{58} + 37 q^{59} - 2 q^{61} - 18 q^{62} + 14 q^{64} + 16 q^{65} - 26 q^{68} + 7 q^{70} - 13 q^{71} - 6 q^{73} - 13 q^{74} - 50 q^{76} + 2 q^{77} + 14 q^{79} + 10 q^{80} + 39 q^{82} + 5 q^{83} + 29 q^{85} + 37 q^{86} + 48 q^{88} + 30 q^{89} - 33 q^{91} + 31 q^{92} + 34 q^{94} - 43 q^{95} - 9 q^{97} - 36 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8 + 5 * q^10 + 15 * q^11 + 3 * q^14 + 10 * q^16 + 11 * q^17 - 16 * q^19 + 9 * q^20 + 7 * q^22 + 9 * q^23 + 6 * q^25 - 7 * q^26 + 9 * q^28 + q^29 - 18 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 7 * q^37 - 20 * q^38 + 32 * q^40 + 10 * q^41 + 6 * q^43 + 5 * q^44 + 13 * q^46 + 7 * q^47 + 11 * q^49 - 3 * q^50 - 6 * q^52 + 9 * q^53 + 17 * q^55 + 15 * q^56 - 19 * q^58 + 37 * q^59 - 2 * q^61 - 18 * q^62 + 14 * q^64 + 16 * q^65 - 26 * q^68 + 7 * q^70 - 13 * q^71 - 6 * q^73 - 13 * q^74 - 50 * q^76 + 2 * q^77 + 14 * q^79 + 10 * q^80 + 39 * q^82 + 5 * q^83 + 29 * q^85 + 37 * q^86 + 48 * q^88 + 30 * q^89 - 33 * q^91 + 31 * q^92 + 34 * q^94 - 43 * q^95 - 9 * q^97 - 36 * 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 $

$2$	−0.559296	−0.395482	−0.197741	−	0.980254i	$-0.563360\pi$
$2$	−0.559296	−0.395482	−0.197741	+	0.980254i	$0.563360\pi$
$3$	0	0
$3$	0	0
$4$	−1.68719	−0.843594
$5$	3.48658	1.55924	0.779622	−	0.626250i	$-0.215411\pi$
$5$	3.48658	1.55924	0.779622	+	0.626250i	$0.215411\pi$
$6$	0	0
$7$	3.13190	1.18375	0.591874	−	0.806030i	$-0.298388\pi$
$7$	3.13190	1.18375	0.591874	+	0.806030i	$0.298388\pi$
$8$	2.06223	0.729108
$9$	0	0
$10$	−1.95003	−0.616653
$11$	5.14945	1.55262	0.776309	−	0.630353i	$-0.217090\pi$
$11$	5.14945	1.55262	0.776309	+	0.630353i	$0.217090\pi$
$12$	0	0
$13$	−2.47676	−0.686931	−0.343465	−	0.939165i	$-0.611601\pi$
$13$	−2.47676	−0.686931	−0.343465	+	0.939165i	$0.611601\pi$
$14$	−1.75166	−0.468151
$15$	0	0
$16$	2.22098	0.555245
$17$	7.36613	1.78655	0.893275	−	0.449512i	$-0.148402\pi$
$17$	7.36613	1.78655	0.893275	+	0.449512i	$0.148402\pi$
$18$	0	0
$19$	−1.17562	−0.269706	−0.134853	−	0.990866i	$-0.543056\pi$
$19$	−1.17562	−0.269706	−0.134853	+	0.990866i	$0.543056\pi$
$20$	−5.88251	−1.31537
$21$	0	0
$22$	−2.88007	−0.614032
$23$	−4.38769	−0.914896	−0.457448	−	0.889236i	$-0.651236\pi$
$23$	−4.38769	−0.914896	−0.457448	+	0.889236i	$0.651236\pi$
$24$	0	0
$25$	7.15622	1.43124
$26$	1.38524	0.271669
$27$	0	0
$28$	−5.28411	−0.998603
$29$	8.41200	1.56207	0.781035	−	0.624487i	$-0.214692\pi$
$29$	8.41200	1.56207	0.781035	+	0.624487i	$0.214692\pi$
$30$	0	0
$31$	−4.26508	−0.766032	−0.383016	−	0.923742i	$-0.625115\pi$
$31$	−4.26508	−0.766032	−0.383016	+	0.923742i	$0.625115\pi$
$32$	−5.36664	−0.948698
$33$	0	0
$34$	−4.11985	−0.706548
$35$	10.9196	1.84575
$36$	0	0
$37$	−6.00097	−0.986553	−0.493277	−	0.869873i	$-0.664201\pi$
$37$	−6.00097	−0.986553	−0.493277	+	0.869873i	$0.664201\pi$
$38$	0.657519	0.106664
$39$	0	0
$40$	7.19012	1.13686
$41$	−10.6359	−1.66105	−0.830527	−	0.556978i	$-0.811961\pi$
$41$	−10.6359	−1.66105	−0.830527	+	0.556978i	$0.811961\pi$
$42$	0	0
$43$	1.62326	0.247544	0.123772	−	0.992311i	$-0.460501\pi$
$43$	1.62326	0.247544	0.123772	+	0.992311i	$0.460501\pi$
$44$	−8.68809	−1.30978
$45$	0	0
$46$	2.45402	0.361825
$47$	−8.29637	−1.21015	−0.605075	−	0.796169i	$-0.706857\pi$
$47$	−8.29637	−1.21015	−0.605075	+	0.796169i	$0.706857\pi$
$48$	0	0
$49$	2.80882	0.401261
$50$	−4.00244	−0.566031
$51$	0	0
$52$	4.17877	0.579491
$53$	5.10653	0.701436	0.350718	−	0.936481i	$-0.385937\pi$
$53$	5.10653	0.701436	0.350718	+	0.936481i	$0.385937\pi$
$54$	0	0
$55$	17.9540	2.42091
$56$	6.45870	0.863081
$57$	0	0
$58$	−4.70480	−0.617770
$59$	−0.146919	−0.0191272	−0.00956362	−	0.999954i	$-0.503044\pi$
$59$	−0.146919	−0.0191272	−0.00956362	+	0.999954i	$0.503044\pi$
$60$	0	0
$61$	−7.55907	−0.967840	−0.483920	−	0.875112i	$-0.660787\pi$
$61$	−7.55907	−0.967840	−0.483920	+	0.875112i	$0.660787\pi$
$62$	2.38544	0.302952
$63$	0	0
$64$	−1.44042	−0.180052
$65$	−8.63543	−1.07109
$66$	0	0
$67$	12.8674	1.57200	0.786002	−	0.618224i	$-0.212147\pi$
$67$	12.8674	1.57200	0.786002	+	0.618224i	$0.212147\pi$
$68$	−12.4280	−1.50712
$69$	0	0
$70$	−6.10730	−0.729962
$71$	−13.9141	−1.65130	−0.825652	−	0.564180i	$-0.809193\pi$
$71$	−13.9141	−1.65130	−0.825652	+	0.564180i	$0.809193\pi$
$72$	0	0
$73$	−2.23198	−0.261234	−0.130617	−	0.991433i	$-0.541696\pi$
$73$	−2.23198	−0.261234	−0.130617	+	0.991433i	$0.541696\pi$
$74$	3.35632	0.390164
$75$	0	0
$76$	1.98349	0.227522
$77$	16.1276	1.83791
$78$	0	0
$79$	3.03137	0.341056	0.170528	−	0.985353i	$-0.445453\pi$
$79$	3.03137	0.341056	0.170528	+	0.985353i	$0.445453\pi$
$80$	7.74362	0.865763
$81$	0	0
$82$	5.94864	0.656917
$83$	6.08068	0.667441	0.333721	−	0.942672i	$-0.391696\pi$
$83$	6.08068	0.667441	0.333721	+	0.942672i	$0.391696\pi$
$84$	0	0
$85$	25.6826	2.78567
$86$	−0.907881	−0.0978993
$87$	0	0
$88$	10.6193	1.13203
$89$	10.2591	1.08746	0.543732	−	0.839259i	$-0.317011\pi$
$89$	10.2591	1.08746	0.543732	+	0.839259i	$0.317011\pi$
$90$	0	0
$91$	−7.75699	−0.813153
$92$	7.40286	0.771801
$93$	0	0
$94$	4.64013	0.478592
$95$	−4.09889	−0.420537
$96$	0	0
$97$	−7.33303	−0.744556	−0.372278	−	0.928121i	$-0.621423\pi$
$97$	−7.33303	−0.744556	−0.372278	+	0.928121i	$0.621423\pi$
$98$	−1.57096	−0.158691
$99$	0	0
$100$	−12.0739	−1.20739
$101$	−1.72959	−0.172101	−0.0860503	−	0.996291i	$-0.527425\pi$
$101$	−1.72959	−0.172101	−0.0860503	+	0.996291i	$0.527425\pi$
$102$	0	0
$103$	−9.84238	−0.969799	−0.484899	−	0.874570i	$-0.661144\pi$
$103$	−9.84238	−0.969799	−0.484899	+	0.874570i	$0.661144\pi$
$104$	−5.10765	−0.500847
$105$	0	0
$106$	−2.85606	−0.277405
$107$	−0.708156	−0.0684600	−0.0342300	−	0.999414i	$-0.510898\pi$
$107$	−0.708156	−0.0684600	−0.0342300	+	0.999414i	$0.510898\pi$
$108$	0	0
$109$	8.02849	0.768990	0.384495	−	0.923127i	$-0.374376\pi$
$109$	8.02849	0.768990	0.384495	+	0.923127i	$0.374376\pi$
$110$	−10.0416	−0.957426
$111$	0	0
$112$	6.95590	0.657270
$113$	8.65810	0.814485	0.407243	−	0.913320i	$-0.366490\pi$
$113$	8.65810	0.814485	0.407243	+	0.913320i	$0.366490\pi$
$114$	0	0
$115$	−15.2980	−1.42655
$116$	−14.1926	−1.31775
$117$	0	0
$118$	0.0821712	0.00756448
$119$	23.0700	2.11482
$120$	0	0
$121$	15.5168	1.41062
$122$	4.22776	0.382763
$123$	0	0
$124$	7.19600	0.646220
$125$	7.51782	0.672415
$126$	0	0
$127$	2.86053	0.253831	0.126915	−	0.991914i	$-0.459492\pi$
$127$	2.86053	0.253831	0.126915	+	0.991914i	$0.459492\pi$
$128$	11.5389	1.01990
$129$	0	0
$130$	4.82976	0.423598
$131$	6.98528	0.610306	0.305153	−	0.952303i	$-0.401292\pi$
$131$	6.98528	0.610306	0.305153	+	0.952303i	$0.401292\pi$
$132$	0	0
$133$	−3.68193	−0.319264
$134$	−7.19669	−0.621699
$135$	0	0
$136$	15.1907	1.30259
$137$	22.9270	1.95879	0.979395	−	0.201955i	$-0.0647296\pi$
$137$	22.9270	1.95879	0.979395	+	0.201955i	$0.0647296\pi$
$138$	0	0
$139$	9.08870	0.770894	0.385447	−	0.922730i	$-0.374047\pi$
$139$	9.08870	0.770894	0.385447	+	0.922730i	$0.374047\pi$
$140$	−18.4235	−1.55707
$141$	0	0
$142$	7.78212	0.653061
$143$	−12.7540	−1.06654
$144$	0	0
$145$	29.3291	2.43565
$146$	1.24834	0.103313
$147$	0	0
$148$	10.1248	0.832250
$149$	1.09136	0.0894074	0.0447037	−	0.999000i	$-0.485766\pi$
$149$	1.09136	0.0894074	0.0447037	+	0.999000i	$0.485766\pi$
$150$	0	0
$151$	1.31108	0.106694	0.0533471	−	0.998576i	$-0.483011\pi$
$151$	1.31108	0.106694	0.0533471	+	0.998576i	$0.483011\pi$
$152$	−2.42440	−0.196645
$153$	0	0
$154$	−9.02009	−0.726860
$155$	−14.8705	−1.19443
$156$	0	0
$157$	−16.8200	−1.34238	−0.671190	−	0.741285i	$-0.734217\pi$
$157$	−16.8200	−1.34238	−0.671190	+	0.741285i	$0.734217\pi$
$158$	−1.69543	−0.134881
$159$	0	0
$160$	−18.7112	−1.47925
$161$	−13.7418	−1.08301
$162$	0	0
$163$	−8.09630	−0.634151	−0.317076	−	0.948400i	$-0.602701\pi$
$163$	−8.09630	−0.634151	−0.317076	+	0.948400i	$0.602701\pi$
$164$	17.9448	1.40126
$165$	0	0
$166$	−3.40090	−0.263961
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−6.86564	−0.528126
$170$	−14.3642	−1.10168
$171$	0	0
$172$	−2.73874	−0.208827
$173$	−11.5024	−0.874513	−0.437257	−	0.899337i	$-0.644050\pi$
$173$	−11.5024	−0.874513	−0.437257	+	0.899337i	$0.644050\pi$
$174$	0	0
$175$	22.4126	1.69423
$176$	11.4368	0.862083
$177$	0	0
$178$	−5.73788	−0.430073
$179$	14.7851	1.10509	0.552545	−	0.833483i	$-0.313657\pi$
$179$	14.7851	1.10509	0.552545	+	0.833483i	$0.313657\pi$
$180$	0	0
$181$	−7.88729	−0.586257	−0.293129	−	0.956073i	$-0.594696\pi$
$181$	−7.88729	−0.586257	−0.293129	+	0.956073i	$0.594696\pi$
$182$	4.33845	0.321587
$183$	0	0
$184$	−9.04842	−0.667058
$185$	−20.9228	−1.53828
$186$	0	0
$187$	37.9315	2.77383
$188$	13.9975	1.02088
$189$	0	0
$190$	2.29249	0.166315
$191$	−22.6900	−1.64179	−0.820894	−	0.571080i	$-0.806525\pi$
$191$	−22.6900	−1.64179	−0.820894	+	0.571080i	$0.806525\pi$
$192$	0	0
$193$	−23.0432	−1.65869	−0.829343	−	0.558740i	$-0.811285\pi$
$193$	−23.0432	−1.65869	−0.829343	+	0.558740i	$0.811285\pi$
$194$	4.10133	0.294459
$195$	0	0
$196$	−4.73901	−0.338501
$197$	−1.54673	−0.110200	−0.0550998	−	0.998481i	$-0.517548\pi$
$197$	−1.54673	−0.110200	−0.0550998	+	0.998481i	$0.517548\pi$
$198$	0	0
$199$	−17.9970	−1.27577	−0.637887	−	0.770130i	$-0.720191\pi$
$199$	−17.9970	−1.27577	−0.637887	+	0.770130i	$0.720191\pi$
$200$	14.7578	1.04353
$201$	0	0
$202$	0.967353	0.0680627
$203$	26.3456	1.84910
$204$	0	0
$205$	−37.0830	−2.58999
$206$	5.50480	0.383538
$207$	0	0
$208$	−5.50084	−0.381415
$209$	−6.05379	−0.418750
$210$	0	0
$211$	18.2144	1.25393	0.626964	−	0.779048i	$-0.284297\pi$
$211$	18.2144	1.25393	0.626964	+	0.779048i	$0.284297\pi$
$212$	−8.61568	−0.591728
$213$	0	0
$214$	0.396069	0.0270747
$215$	5.65961	0.385982
$216$	0	0
$217$	−13.3578	−0.906789
$218$	−4.49030	−0.304122
$219$	0	0
$220$	−30.2917	−2.04227
$221$	−18.2442	−1.22724
$222$	0	0
$223$	−27.1693	−1.81939	−0.909695	−	0.415277i	$-0.863685\pi$
$223$	−27.1693	−1.81939	−0.909695	+	0.415277i	$0.863685\pi$
$224$	−16.8078	−1.12302
$225$	0	0
$226$	−4.84244	−0.322114
$227$	−2.40896	−0.159888	−0.0799441	−	0.996799i	$-0.525474\pi$
$227$	−2.40896	−0.159888	−0.0799441	+	0.996799i	$0.525474\pi$
$228$	0	0
$229$	19.0913	1.26159	0.630796	−	0.775949i	$-0.282729\pi$
$229$	19.0913	1.26159	0.630796	+	0.775949i	$0.282729\pi$
$230$	8.55612	0.564174
$231$	0	0
$232$	17.3475	1.13892
$233$	11.8734	0.777851	0.388925	−	0.921269i	$-0.372846\pi$
$233$	11.8734	0.777851	0.388925	+	0.921269i	$0.372846\pi$
$234$	0	0
$235$	−28.9259	−1.88692
$236$	0.247880	0.0161356
$237$	0	0
$238$	−12.9030	−0.836375
$239$	5.68491	0.367726	0.183863	−	0.982952i	$-0.441140\pi$
$239$	5.68491	0.367726	0.183863	+	0.982952i	$0.441140\pi$
$240$	0	0
$241$	2.95547	0.190379	0.0951894	−	0.995459i	$-0.469654\pi$
$241$	2.95547	0.190379	0.0951894	+	0.995459i	$0.469654\pi$
$242$	−8.67850	−0.557875
$243$	0	0
$244$	12.7536	0.816464
$245$	9.79318	0.625663
$246$	0	0
$247$	2.91173	0.185269
$248$	−8.79558	−0.558520
$249$	0	0
$250$	−4.20469	−0.265928
$251$	13.0979	0.826730	0.413365	−	0.910565i	$-0.364353\pi$
$251$	13.0979	0.826730	0.413365	+	0.910565i	$0.364353\pi$
$252$	0	0
$253$	−22.5942	−1.42048
$254$	−1.59988	−0.100385
$255$	0	0
$256$	−3.57283	−0.223302
$257$	−27.2262	−1.69832	−0.849161	−	0.528135i	$-0.822892\pi$
$257$	−27.2262	−1.69832	−0.849161	+	0.528135i	$0.822892\pi$
$258$	0	0
$259$	−18.7945	−1.16783
$260$	14.5696	0.903568
$261$	0	0
$262$	−3.90684	−0.241365
$263$	7.21748	0.445049	0.222524	−	0.974927i	$-0.428570\pi$
$263$	7.21748	0.445049	0.222524	+	0.974927i	$0.428570\pi$
$264$	0	0
$265$	17.8043	1.09371
$266$	2.05929	0.126263
$267$	0	0
$268$	−21.7097	−1.32613
$269$	0.488664	0.0297944	0.0148972	−	0.999889i	$-0.495258\pi$
$269$	0.488664	0.0297944	0.0148972	+	0.999889i	$0.495258\pi$
$270$	0	0
$271$	−7.69604	−0.467501	−0.233751	−	0.972297i	$-0.575100\pi$
$271$	−7.69604	−0.467501	−0.233751	+	0.972297i	$0.575100\pi$
$272$	16.3600	0.991972
$273$	0	0
$274$	−12.8230	−0.774666
$275$	36.8506	2.22217
$276$	0	0
$277$	20.7186	1.24486	0.622431	−	0.782675i	$-0.286145\pi$
$277$	20.7186	1.24486	0.622431	+	0.782675i	$0.286145\pi$
$278$	−5.08327	−0.304875
$279$	0	0
$280$	22.5188	1.34575
$281$	31.1451	1.85796	0.928982	−	0.370126i	$-0.120685\pi$
$281$	31.1451	1.85796	0.928982	+	0.370126i	$0.120685\pi$
$282$	0	0
$283$	−22.6017	−1.34353	−0.671765	−	0.740765i	$-0.734463\pi$
$283$	−22.6017	−1.34353	−0.671765	+	0.740765i	$0.734463\pi$
$284$	23.4758	1.39303
$285$	0	0
$286$	7.13324	0.421798
$287$	−33.3107	−1.96627
$288$	0	0
$289$	37.2599	2.19176
$290$	−16.4036	−0.963255
$291$	0	0
$292$	3.76577	0.220375
$293$	−11.1767	−0.652951	−0.326476	−	0.945206i	$-0.605861\pi$
$293$	−11.1767	−0.652951	−0.326476	+	0.945206i	$0.605861\pi$
$294$	0	0
$295$	−0.512245	−0.0298240
$296$	−12.3754	−0.719304
$297$	0	0
$298$	−0.610391	−0.0353590
$299$	10.8673	0.628470
$300$	0	0
$301$	5.08388	0.293030
$302$	−0.733282	−0.0421957
$303$	0	0
$304$	−2.61103	−0.149753
$305$	−26.3553	−1.50910
$306$	0	0
$307$	−20.6660	−1.17947	−0.589735	−	0.807597i	$-0.700768\pi$
$307$	−20.6660	−1.17947	−0.589735	+	0.807597i	$0.700768\pi$
$308$	−27.2103	−1.55045
$309$	0	0
$310$	8.31704	0.472376
$311$	−29.1465	−1.65274	−0.826372	−	0.563125i	$-0.809599\pi$
$311$	−29.1465	−1.65274	−0.826372	+	0.563125i	$0.809599\pi$
$312$	0	0
$313$	30.0446	1.69822	0.849110	−	0.528216i	$-0.177139\pi$
$313$	30.0446	1.69822	0.849110	+	0.528216i	$0.177139\pi$
$314$	9.40734	0.530887
$315$	0	0
$316$	−5.11449	−0.287713
$317$	−13.1392	−0.737970	−0.368985	−	0.929435i	$-0.620295\pi$
$317$	−13.1392	−0.737970	−0.368985	+	0.929435i	$0.620295\pi$
$318$	0	0
$319$	43.3172	2.42530
$320$	−5.02213	−0.280745
$321$	0	0
$322$	7.68574	0.428310
$323$	−8.65977	−0.481842
$324$	0	0
$325$	−17.7243	−0.983165
$326$	4.52823	0.250795
$327$	0	0
$328$	−21.9337	−1.21109
$329$	−25.9834	−1.43251
$330$	0	0
$331$	−9.37504	−0.515299	−0.257649	−	0.966238i	$-0.582948\pi$
$331$	−9.37504	−0.515299	−0.257649	+	0.966238i	$0.582948\pi$
$332$	−10.2592	−0.563049
$333$	0	0
$334$	0.559296	0.0306033
$335$	44.8632	2.45114
$336$	0	0
$337$	14.7899	0.805659	0.402830	−	0.915275i	$-0.368027\pi$
$337$	14.7899	0.805659	0.402830	+	0.915275i	$0.368027\pi$
$338$	3.83993	0.208864
$339$	0	0
$340$	−43.3313	−2.34997
$341$	−21.9628	−1.18935
$342$	0	0
$343$	−13.1264	−0.708757
$344$	3.34753	0.180487
$345$	0	0
$346$	6.43326	0.345854
$347$	−4.40910	−0.236693	−0.118347	−	0.992972i	$-0.537759\pi$
$347$	−4.40910	−0.236693	−0.118347	+	0.992972i	$0.537759\pi$
$348$	0	0
$349$	−13.3569	−0.714977	−0.357488	−	0.933918i	$-0.616367\pi$
$349$	−13.3569	−0.714977	−0.357488	+	0.933918i	$0.616367\pi$
$350$	−12.5353	−0.670039
$351$	0	0
$352$	−27.6353	−1.47296
$353$	20.4993	1.09107	0.545535	−	0.838088i	$-0.316327\pi$
$353$	20.4993	1.09107	0.545535	+	0.838088i	$0.316327\pi$
$354$	0	0
$355$	−48.5127	−2.57479
$356$	−17.3091	−0.917379
$357$	0	0
$358$	−8.26925	−0.437043
$359$	−0.219373	−0.0115780	−0.00578902	−	0.999983i	$-0.501843\pi$
$359$	−0.219373	−0.0115780	−0.00578902	+	0.999983i	$0.501843\pi$
$360$	0	0
$361$	−17.6179	−0.927259
$362$	4.41133	0.231854
$363$	0	0
$364$	13.0875	0.685971
$365$	−7.78198	−0.407327
$366$	0	0
$367$	21.8828	1.14227	0.571137	−	0.820855i	$-0.306503\pi$
$367$	21.8828	1.14227	0.571137	+	0.820855i	$0.306503\pi$
$368$	−9.74497	−0.507991
$369$	0	0
$370$	11.7021	0.608361
$371$	15.9932	0.830324
$372$	0	0
$373$	32.7568	1.69608	0.848041	−	0.529930i	$-0.177782\pi$
$373$	32.7568	1.69608	0.848041	+	0.529930i	$0.177782\pi$
$374$	−21.2149	−1.09700
$375$	0	0
$376$	−17.1090	−0.882330
$377$	−20.8345	−1.07303
$378$	0	0
$379$	−26.3586	−1.35395	−0.676974	−	0.736007i	$-0.736709\pi$
$379$	−26.3586	−1.35395	−0.676974	+	0.736007i	$0.736709\pi$
$380$	6.91560	0.354763
$381$	0	0
$382$	12.6904	0.649298
$383$	13.0910	0.668919	0.334460	−	0.942410i	$-0.391446\pi$
$383$	13.0910	0.668919	0.334460	+	0.942410i	$0.391446\pi$
$384$	0	0
$385$	56.2301	2.86575
$386$	12.8880	0.655980
$387$	0	0
$388$	12.3722	0.628103
$389$	−8.62641	−0.437377	−0.218688	−	0.975795i	$-0.570178\pi$
$389$	−8.62641	−0.437377	−0.218688	+	0.975795i	$0.570178\pi$
$390$	0	0
$391$	−32.3203	−1.63451
$392$	5.79244	0.292562
$393$	0	0
$394$	0.865077	0.0435820
$395$	10.5691	0.531790
$396$	0	0
$397$	−1.96271	−0.0985057	−0.0492529	−	0.998786i	$-0.515684\pi$
$397$	−1.96271	−0.0985057	−0.0492529	+	0.998786i	$0.515684\pi$
$398$	10.0657	0.504546
$399$	0	0
$400$	15.8938	0.794691
$401$	32.5639	1.62617	0.813083	−	0.582148i	$-0.197788\pi$
$401$	32.5639	1.62617	0.813083	+	0.582148i	$0.197788\pi$
$402$	0	0
$403$	10.5636	0.526211
$404$	2.91814	0.145183
$405$	0	0
$406$	−14.7350	−0.731285
$407$	−30.9017	−1.53174
$408$	0	0
$409$	−13.8736	−0.686007	−0.343003	−	0.939334i	$-0.611444\pi$
$409$	−13.8736	−0.686007	−0.343003	+	0.939334i	$0.611444\pi$
$410$	20.7404	1.02429
$411$	0	0
$412$	16.6059	0.818116
$413$	−0.460136	−0.0226418
$414$	0	0
$415$	21.2008	1.04070
$416$	13.2919	0.651689
$417$	0	0
$418$	3.38586	0.165608
$419$	−9.55415	−0.466751	−0.233375	−	0.972387i	$-0.574977\pi$
$419$	−9.55415	−0.466751	−0.233375	+	0.972387i	$0.574977\pi$
$420$	0	0
$421$	2.21491	0.107948	0.0539741	−	0.998542i	$-0.482811\pi$
$421$	2.21491	0.107948	0.0539741	+	0.998542i	$0.482811\pi$
$422$	−10.1872	−0.495906
$423$	0	0
$424$	10.5308	0.511423
$425$	52.7136	2.55699
$426$	0	0
$427$	−23.6743	−1.14568
$428$	1.19479	0.0577525
$429$	0	0
$430$	−3.16540	−0.152649
$431$	−22.7614	−1.09638	−0.548189	−	0.836355i	$-0.684682\pi$
$431$	−22.7614	−1.09638	−0.548189	+	0.836355i	$0.684682\pi$
$432$	0	0
$433$	−14.8738	−0.714791	−0.357396	−	0.933953i	$-0.616335\pi$
$433$	−14.8738	−0.714791	−0.357396	+	0.933953i	$0.616335\pi$
$434$	7.47098	0.358619
$435$	0	0
$436$	−13.5456	−0.648715
$437$	5.15825	0.246753
$438$	0	0
$439$	1.84773	0.0881872	0.0440936	−	0.999027i	$-0.485960\pi$
$439$	1.84773	0.0881872	0.0440936	+	0.999027i	$0.485960\pi$
$440$	37.0252	1.76511
$441$	0	0
$442$	10.2039	0.485349
$443$	−11.7435	−0.557950	−0.278975	−	0.960298i	$-0.589995\pi$
$443$	−11.7435	−0.557950	−0.278975	+	0.960298i	$0.589995\pi$
$444$	0	0
$445$	35.7692	1.69562
$446$	15.1957	0.719536
$447$	0	0
$448$	−4.51125	−0.213136
$449$	11.7259	0.553378	0.276689	−	0.960960i	$-0.410763\pi$
$449$	11.7259	0.553378	0.276689	+	0.960960i	$0.410763\pi$
$450$	0	0
$451$	−54.7692	−2.57898
$452$	−14.6078	−0.687095
$453$	0	0
$454$	1.34732	0.0632329
$455$	−27.0453	−1.26790
$456$	0	0
$457$	42.2015	1.97410	0.987051	−	0.160404i	$-0.0512798\pi$
$457$	42.2015	1.97410	0.987051	+	0.160404i	$0.0512798\pi$
$458$	−10.6777	−0.498937
$459$	0	0
$460$	25.8106	1.20343
$461$	−24.5098	−1.14153	−0.570767	−	0.821112i	$-0.693354\pi$
$461$	−24.5098	−1.14153	−0.570767	+	0.821112i	$0.693354\pi$
$462$	0	0
$463$	−3.45658	−0.160641	−0.0803205	−	0.996769i	$-0.525594\pi$
$463$	−3.45658	−0.160641	−0.0803205	+	0.996769i	$0.525594\pi$
$464$	18.6829	0.867331
$465$	0	0
$466$	−6.64073	−0.307626
$467$	−23.6624	−1.09496	−0.547482	−	0.836818i	$-0.684413\pi$
$467$	−23.6624	−1.09496	−0.547482	+	0.836818i	$0.684413\pi$
$468$	0	0
$469$	40.2995	1.86086
$470$	16.1782	0.746243
$471$	0	0
$472$	−0.302981	−0.0139458
$473$	8.35888	0.384342
$474$	0	0
$475$	−8.41299	−0.386015
$476$	−38.9235	−1.78405
$477$	0	0
$478$	−3.17955	−0.145429
$479$	30.2044	1.38007	0.690037	−	0.723774i	$-0.257594\pi$
$479$	30.2044	1.38007	0.690037	+	0.723774i	$0.257594\pi$
$480$	0	0
$481$	14.8630	0.677694
$482$	−1.65298	−0.0752914
$483$	0	0
$484$	−26.1798	−1.18999
$485$	−25.5672	−1.16095
$486$	0	0
$487$	8.65596	0.392239	0.196119	−	0.980580i	$-0.437166\pi$
$487$	8.65596	0.392239	0.196119	+	0.980580i	$0.437166\pi$
$488$	−15.5885	−0.705660
$489$	0	0
$490$	−5.47729	−0.247439
$491$	−28.6729	−1.29399	−0.646995	−	0.762495i	$-0.723974\pi$
$491$	−28.6729	−1.29399	−0.646995	+	0.762495i	$0.723974\pi$
$492$	0	0
$493$	61.9639	2.79071
$494$	−1.62852	−0.0732706
$495$	0	0
$496$	−9.47267	−0.425335
$497$	−43.5777	−1.95473
$498$	0	0
$499$	−17.4394	−0.780693	−0.390346	−	0.920668i	$-0.627645\pi$
$499$	−17.4394	−0.780693	−0.390346	+	0.920668i	$0.627645\pi$
$500$	−12.6840	−0.567245
$501$	0	0
$502$	−7.32558	−0.326957
$503$	25.1536	1.12154	0.560772	−	0.827970i	$-0.310504\pi$
$503$	25.1536	1.12154	0.560772	+	0.827970i	$0.310504\pi$
$504$	0	0
$505$	−6.03035	−0.268347
$506$	12.6368	0.561776
$507$	0	0
$508$	−4.82625	−0.214130
$509$	24.4858	1.08531	0.542657	−	0.839954i	$-0.317418\pi$
$509$	24.4858	1.08531	0.542657	+	0.839954i	$0.317418\pi$
$510$	0	0
$511$	−6.99035	−0.309235
$512$	−21.0795	−0.931593
$513$	0	0
$514$	15.2275	0.671655
$515$	−34.3162	−1.51215
$516$	0	0
$517$	−42.7217	−1.87890
$518$	10.5117	0.461856
$519$	0	0
$520$	−17.8082	−0.780943
$521$	−18.0239	−0.789639	−0.394820	−	0.918759i	$-0.629193\pi$
$521$	−18.0239	−0.789639	−0.394820	+	0.918759i	$0.629193\pi$
$522$	0	0
$523$	12.7597	0.557943	0.278971	−	0.960299i	$-0.410007\pi$
$523$	12.7597	0.557943	0.278971	+	0.960299i	$0.410007\pi$
$524$	−11.7855	−0.514851
$525$	0	0
$526$	−4.03671	−0.176009
$527$	−31.4172	−1.36855
$528$	0	0
$529$	−3.74819	−0.162965
$530$	−9.95789	−0.432543
$531$	0	0
$532$	6.21211	0.269329
$533$	26.3427	1.14103
$534$	0	0
$535$	−2.46904	−0.106746
$536$	26.5355	1.14616
$537$	0	0
$538$	−0.273308	−0.0117831
$539$	14.4639	0.623004
$540$	0	0
$541$	29.8187	1.28201	0.641003	−	0.767538i	$-0.278519\pi$
$541$	29.8187	1.28201	0.641003	+	0.767538i	$0.278519\pi$
$542$	4.30437	0.184888
$543$	0	0
$544$	−39.5314	−1.69489
$545$	27.9919	1.19904
$546$	0	0
$547$	−24.4152	−1.04392	−0.521958	−	0.852971i	$-0.674798\pi$
$547$	−24.4152	−1.04392	−0.521958	+	0.852971i	$0.674798\pi$
$548$	−38.6822	−1.65242
$549$	0	0
$550$	−20.6104	−0.878830
$551$	−9.88932	−0.421299
$552$	0	0
$553$	9.49396	0.403724
$554$	−11.5878	−0.492321
$555$	0	0
$556$	−15.3343	−0.650321
$557$	14.0494	0.595291	0.297645	−	0.954676i	$-0.403799\pi$
$557$	14.0494	0.595291	0.297645	+	0.954676i	$0.403799\pi$
$558$	0	0
$559$	−4.02042	−0.170046
$560$	24.2523	1.02485
$561$	0	0
$562$	−17.4194	−0.734791
$563$	31.8744	1.34335	0.671673	−	0.740848i	$-0.265576\pi$
$563$	31.8744	1.34335	0.671673	+	0.740848i	$0.265576\pi$
$564$	0	0
$565$	30.1871	1.26998
$566$	12.6410	0.531342
$567$	0	0
$568$	−28.6941	−1.20398
$569$	−35.9946	−1.50897	−0.754487	−	0.656315i	$-0.772114\pi$
$569$	−35.9946	−1.50897	−0.754487	+	0.656315i	$0.772114\pi$
$570$	0	0
$571$	0.742474	0.0310716	0.0155358	−	0.999879i	$-0.495055\pi$
$571$	0.742474	0.0310716	0.0155358	+	0.999879i	$0.495055\pi$
$572$	21.5183	0.899727
$573$	0	0
$574$	18.6306	0.777625
$575$	−31.3993	−1.30944
$576$	0	0
$577$	−24.8825	−1.03587	−0.517936	−	0.855419i	$-0.673300\pi$
$577$	−24.8825	−1.03587	−0.517936	+	0.855419i	$0.673300\pi$
$578$	−20.8393	−0.866801
$579$	0	0
$580$	−49.4837	−2.05470
$581$	19.0441	0.790082
$582$	0	0
$583$	26.2958	1.08906
$584$	−4.60286	−0.190468
$585$	0	0
$586$	6.25110	0.258230
$587$	−40.7068	−1.68015	−0.840075	−	0.542471i	$-0.817489\pi$
$587$	−40.7068	−1.68015	−0.840075	+	0.542471i	$0.817489\pi$
$588$	0	0
$589$	5.01412	0.206603
$590$	0.286496	0.0117949
$591$	0	0
$592$	−13.3280	−0.547779
$593$	15.9843	0.656398	0.328199	−	0.944609i	$-0.393558\pi$
$593$	15.9843	0.656398	0.328199	+	0.944609i	$0.393558\pi$
$594$	0	0
$595$	80.4354	3.29753
$596$	−1.84132	−0.0754236
$597$	0	0
$598$	−6.07802	−0.248549
$599$	−47.2868	−1.93209	−0.966043	−	0.258381i	$-0.916811\pi$
$599$	−47.2868	−1.93209	−0.966043	+	0.258381i	$0.916811\pi$
$600$	0	0
$601$	10.1748	0.415038	0.207519	−	0.978231i	$-0.433461\pi$
$601$	10.1748	0.415038	0.207519	+	0.978231i	$0.433461\pi$
$602$	−2.84340	−0.115888
$603$	0	0
$604$	−2.21204	−0.0900066
$605$	54.1006	2.19950
$606$	0	0
$607$	14.4349	0.585895	0.292947	−	0.956129i	$-0.405364\pi$
$607$	14.4349	0.585895	0.292947	+	0.956129i	$0.405364\pi$
$608$	6.30913	0.255869
$609$	0	0
$610$	14.7404	0.596822
$611$	20.5481	0.831289
$612$	0	0
$613$	31.5888	1.27586	0.637931	−	0.770094i	$-0.279791\pi$
$613$	31.5888	1.27586	0.637931	+	0.770094i	$0.279791\pi$
$614$	11.5584	0.466459
$615$	0	0
$616$	33.2588	1.34003
$617$	43.1763	1.73821	0.869106	−	0.494626i	$-0.164695\pi$
$617$	43.1763	1.73821	0.869106	+	0.494626i	$0.164695\pi$
$618$	0	0
$619$	−30.1013	−1.20987	−0.604937	−	0.796273i	$-0.706802\pi$
$619$	−30.1013	−1.20987	−0.604937	+	0.796273i	$0.706802\pi$
$620$	25.0894	1.00761
$621$	0	0
$622$	16.3015	0.653630
$623$	32.1306	1.28728
$624$	0	0
$625$	−9.56962	−0.382785
$626$	−16.8038	−0.671616
$627$	0	0
$628$	28.3785	1.13242
$629$	−44.2039	−1.76253
$630$	0	0
$631$	−14.7748	−0.588176	−0.294088	−	0.955778i	$-0.595016\pi$
$631$	−14.7748	−0.588176	−0.294088	+	0.955778i	$0.595016\pi$
$632$	6.25138	0.248667
$633$	0	0
$634$	7.34869	0.291854
$635$	9.97345	0.395784
$636$	0	0
$637$	−6.95679	−0.275638
$638$	−24.2271	−0.959161
$639$	0	0
$640$	40.2313	1.59028
$641$	−5.91157	−0.233493	−0.116747	−	0.993162i	$-0.537247\pi$
$641$	−5.91157	−0.233493	−0.116747	+	0.993162i	$0.537247\pi$
$642$	0	0
$643$	19.8852	0.784197	0.392098	−	0.919923i	$-0.371749\pi$
$643$	19.8852	0.784197	0.392098	+	0.919923i	$0.371749\pi$
$644$	23.1850	0.913618
$645$	0	0
$646$	4.84337	0.190560
$647$	40.7214	1.60092	0.800462	−	0.599383i	$-0.204587\pi$
$647$	40.7214	1.60092	0.800462	+	0.599383i	$0.204587\pi$
$648$	0	0
$649$	−0.756552	−0.0296973
$650$	9.91311	0.388824
$651$	0	0
$652$	13.6600	0.534966
$653$	−39.1398	−1.53166	−0.765829	−	0.643044i	$-0.777671\pi$
$653$	−39.1398	−1.53166	−0.765829	+	0.643044i	$0.777671\pi$
$654$	0	0
$655$	24.3547	0.951617
$656$	−23.6222	−0.922292
$657$	0	0
$658$	14.5324	0.566533
$659$	27.6703	1.07788	0.538941	−	0.842343i	$-0.318824\pi$
$659$	27.6703	1.07788	0.538941	+	0.842343i	$0.318824\pi$
$660$	0	0
$661$	3.89239	0.151396	0.0756982	−	0.997131i	$-0.475881\pi$
$661$	3.89239	0.151396	0.0756982	+	0.997131i	$0.475881\pi$
$662$	5.24342	0.203791
$663$	0	0
$664$	12.5398	0.486637
$665$	−12.8373	−0.497810
$666$	0	0
$667$	−36.9092	−1.42913
$668$	1.68719	0.0652793
$669$	0	0
$670$	−25.0918	−0.969381
$671$	−38.9251	−1.50269
$672$	0	0
$673$	−16.1127	−0.621097	−0.310549	−	0.950557i	$-0.600513\pi$
$673$	−16.1127	−0.621097	−0.310549	+	0.950557i	$0.600513\pi$
$674$	−8.27195	−0.318624
$675$	0	0
$676$	11.5836	0.445524
$677$	−4.73990	−0.182169	−0.0910845	−	0.995843i	$-0.529033\pi$
$677$	−4.73990	−0.182169	−0.0910845	+	0.995843i	$0.529033\pi$
$678$	0	0
$679$	−22.9663	−0.881367
$680$	52.9634	2.03105
$681$	0	0
$682$	12.2837	0.470368
$683$	−21.1490	−0.809243	−0.404622	−	0.914484i	$-0.632597\pi$
$683$	−21.1490	−0.809243	−0.404622	+	0.914484i	$0.632597\pi$
$684$	0	0
$685$	79.9369	3.05423
$686$	7.34152	0.280301
$687$	0	0
$688$	3.60522	0.137448
$689$	−12.6477	−0.481838
$690$	0	0
$691$	20.3791	0.775256	0.387628	−	0.921816i	$-0.373294\pi$
$691$	20.3791	0.775256	0.387628	+	0.921816i	$0.373294\pi$
$692$	19.4068	0.737734
$693$	0	0
$694$	2.46599	0.0936079
$695$	31.6885	1.20201
$696$	0	0
$697$	−78.3457	−2.96756
$698$	7.47044	0.282760
$699$	0	0
$700$	−37.8143	−1.42924
$701$	−0.365935	−0.0138212	−0.00691058	−	0.999976i	$-0.502200\pi$
$701$	−0.365935	−0.0138212	−0.00691058	+	0.999976i	$0.502200\pi$
$702$	0	0
$703$	7.05486	0.266079
$704$	−7.41736	−0.279552
$705$	0	0
$706$	−11.4652	−0.431498
$707$	−5.41691	−0.203724
$708$	0	0
$709$	5.88589	0.221049	0.110525	−	0.993873i	$-0.464747\pi$
$709$	5.88589	0.221049	0.110525	+	0.993873i	$0.464747\pi$
$710$	27.1330	1.01828
$711$	0	0
$712$	21.1567	0.792879
$713$	18.7139	0.700840
$714$	0	0
$715$	−44.4677	−1.66300
$716$	−24.9452	−0.932247
$717$	0	0
$718$	0.122694	0.00457891
$719$	−15.8978	−0.592886	−0.296443	−	0.955051i	$-0.595801\pi$
$719$	−15.8978	−0.592886	−0.296443	+	0.955051i	$0.595801\pi$
$720$	0	0
$721$	−30.8254	−1.14800
$722$	9.85363	0.366714
$723$	0	0
$724$	13.3073	0.494563
$725$	60.1981	2.23570
$726$	0	0
$727$	18.9730	0.703670	0.351835	−	0.936062i	$-0.385558\pi$
$727$	18.9730	0.703670	0.351835	+	0.936062i	$0.385558\pi$
$728$	−15.9967	−0.592877
$729$	0	0
$730$	4.35243	0.161091
$731$	11.9571	0.442250
$732$	0	0
$733$	5.90174	0.217986	0.108993	−	0.994043i	$-0.465237\pi$
$733$	5.90174	0.217986	0.108993	+	0.994043i	$0.465237\pi$
$734$	−12.2390	−0.451749
$735$	0	0
$736$	23.5472	0.867960
$737$	66.2601	2.44072
$738$	0	0
$739$	3.94346	0.145063	0.0725313	−	0.997366i	$-0.476892\pi$
$739$	3.94346	0.145063	0.0725313	+	0.997366i	$0.476892\pi$
$740$	35.3008	1.29768
$741$	0	0
$742$	−8.94492	−0.328378
$743$	−46.4212	−1.70303	−0.851514	−	0.524332i	$-0.824315\pi$
$743$	−46.4212	−1.70303	−0.851514	+	0.524332i	$0.824315\pi$
$744$	0	0
$745$	3.80510	0.139408
$746$	−18.3207	−0.670770
$747$	0	0
$748$	−63.9976	−2.33998
$749$	−2.21788	−0.0810395
$750$	0	0
$751$	12.3421	0.450370	0.225185	−	0.974316i	$-0.427701\pi$
$751$	12.3421	0.450370	0.225185	+	0.974316i	$0.427701\pi$
$752$	−18.4261	−0.671929
$753$	0	0
$754$	11.6527	0.424365
$755$	4.57119	0.166362
$756$	0	0
$757$	49.3731	1.79450	0.897248	−	0.441526i	$-0.145563\pi$
$757$	49.3731	1.79450	0.897248	+	0.441526i	$0.145563\pi$
$758$	14.7422	0.535462
$759$	0	0
$760$	−8.45285	−0.306617
$761$	23.7202	0.859856	0.429928	−	0.902863i	$-0.358539\pi$
$761$	23.7202	0.859856	0.429928	+	0.902863i	$0.358539\pi$
$762$	0	0
$763$	25.1445	0.910291
$764$	38.2823	1.38500
$765$	0	0
$766$	−7.32175	−0.264545
$767$	0.363884	0.0131391
$768$	0	0
$769$	−25.8993	−0.933952	−0.466976	−	0.884270i	$-0.654656\pi$
$769$	−25.8993	−0.933952	−0.466976	+	0.884270i	$0.654656\pi$
$770$	−31.4492	−1.13335
$771$	0	0
$772$	38.8782	1.39926
$773$	43.2023	1.55388	0.776940	−	0.629575i	$-0.216771\pi$
$773$	43.2023	1.55388	0.776940	+	0.629575i	$0.216771\pi$
$774$	0	0
$775$	−30.5219	−1.09638
$776$	−15.1224	−0.542862
$777$	0	0
$778$	4.82472	0.172975
$779$	12.5038	0.447996
$780$	0	0
$781$	−71.6502	−2.56384
$782$	18.0766	0.646418
$783$	0	0
$784$	6.23834	0.222798
$785$	−58.6441	−2.09310
$786$	0	0
$787$	−1.87347	−0.0667818	−0.0333909	−	0.999442i	$-0.510631\pi$
$787$	−1.87347	−0.0667818	−0.0333909	+	0.999442i	$0.510631\pi$
$788$	2.60962	0.0929638
$789$	0	0
$790$	−5.91126	−0.210313
$791$	27.1163	0.964146
$792$	0	0
$793$	18.7220	0.664839
$794$	1.09774	0.0389572
$795$	0	0
$796$	30.3643	1.07624
$797$	27.4642	0.972832	0.486416	−	0.873727i	$-0.338304\pi$
$797$	27.4642	0.972832	0.486416	+	0.873727i	$0.338304\pi$
$798$	0	0
$799$	−61.1121	−2.16199
$800$	−38.4049	−1.35782
$801$	0	0
$802$	−18.2129	−0.643119
$803$	−11.4935	−0.405596
$804$	0	0
$805$	−47.9119	−1.68867
$806$	−5.90818	−0.208107
$807$	0	0
$808$	−3.56681	−0.125480
$809$	22.1768	0.779695	0.389848	−	0.920879i	$-0.372528\pi$
$809$	22.1768	0.779695	0.389848	+	0.920879i	$0.372528\pi$
$810$	0	0
$811$	−17.7911	−0.624729	−0.312364	−	0.949962i	$-0.601121\pi$
$811$	−17.7911	−0.624729	−0.312364	+	0.949962i	$0.601121\pi$
$812$	−44.4500	−1.55989
$813$	0	0
$814$	17.2832	0.605775
$815$	−28.2284	−0.988797
$816$	0	0
$817$	−1.90833	−0.0667641
$818$	7.75946	0.271303
$819$	0	0
$820$	62.5660	2.18490
$821$	42.6849	1.48971	0.744857	−	0.667225i	$-0.232518\pi$
$821$	42.6849	1.48971	0.744857	+	0.667225i	$0.232518\pi$
$822$	0	0
$823$	11.7120	0.408254	0.204127	−	0.978944i	$-0.434564\pi$
$823$	11.7120	0.408254	0.204127	+	0.978944i	$0.434564\pi$
$824$	−20.2972	−0.707088
$825$	0	0
$826$	0.257352	0.00895444
$827$	−31.2241	−1.08577	−0.542884	−	0.839808i	$-0.682668\pi$
$827$	−31.2241	−1.08577	−0.542884	+	0.839808i	$0.682668\pi$
$828$	0	0
$829$	−6.77217	−0.235207	−0.117604	−	0.993061i	$-0.537521\pi$
$829$	−6.77217	−0.235207	−0.117604	+	0.993061i	$0.537521\pi$
$830$	−11.8575	−0.411580
$831$	0	0
$832$	3.56757	0.123683
$833$	20.6902	0.716872
$834$	0	0
$835$	−3.48658	−0.120658
$836$	10.2139	0.353255
$837$	0	0
$838$	5.34360	0.184592
$839$	25.6588	0.885840	0.442920	−	0.896561i	$-0.353943\pi$
$839$	25.6588	0.885840	0.442920	+	0.896561i	$0.353943\pi$
$840$	0	0
$841$	41.7618	1.44006
$842$	−1.23879	−0.0426916
$843$	0	0
$844$	−30.7311	−1.05781
$845$	−23.9376	−0.823478
$846$	0	0
$847$	48.5972	1.66982
$848$	11.3415	0.389469
$849$	0	0
$850$	−29.4825	−1.01124
$851$	26.3304	0.902594
$852$	0	0
$853$	−41.1575	−1.40921	−0.704603	−	0.709602i	$-0.748875\pi$
$853$	−41.1575	−1.40921	−0.704603	+	0.709602i	$0.748875\pi$
$854$	13.2409	0.453096
$855$	0	0
$856$	−1.46038	−0.0499148
$857$	−16.2850	−0.556284	−0.278142	−	0.960540i	$-0.589719\pi$
$857$	−16.2850	−0.556284	−0.278142	+	0.960540i	$0.589719\pi$
$858$	0	0
$859$	4.34497	0.148248	0.0741242	−	0.997249i	$-0.476384\pi$
$859$	4.34497	0.148248	0.0741242	+	0.997249i	$0.476384\pi$
$860$	−9.54882	−0.325612
$861$	0	0
$862$	12.7303	0.433597
$863$	−35.5748	−1.21098	−0.605490	−	0.795853i	$-0.707023\pi$
$863$	−35.5748	−1.21098	−0.605490	+	0.795853i	$0.707023\pi$
$864$	0	0
$865$	−40.1041	−1.36358
$866$	8.31888	0.282687
$867$	0	0
$868$	22.5372	0.764962
$869$	15.6099	0.529529
$870$	0	0
$871$	−31.8695	−1.07986
$872$	16.5566	0.560677
$873$	0	0
$874$	−2.88499	−0.0975862
$875$	23.5451	0.795970
$876$	0	0
$877$	34.3006	1.15825	0.579125	−	0.815239i	$-0.303394\pi$
$877$	34.3006	1.15825	0.579125	+	0.815239i	$0.303394\pi$
$878$	−1.03343	−0.0348765
$879$	0	0
$880$	39.8754	1.34420
$881$	11.3825	0.383485	0.191742	−	0.981445i	$-0.438586\pi$
$881$	11.3825	0.383485	0.191742	+	0.981445i	$0.438586\pi$
$882$	0	0
$883$	−3.47050	−0.116792	−0.0583959	−	0.998294i	$-0.518599\pi$
$883$	−3.47050	−0.116792	−0.0583959	+	0.998294i	$0.518599\pi$
$884$	30.7813	1.03529
$885$	0	0
$886$	6.56809	0.220659
$887$	44.3988	1.49077	0.745383	−	0.666636i	$-0.232266\pi$
$887$	44.3988	1.49077	0.745383	+	0.666636i	$0.232266\pi$
$888$	0	0
$889$	8.95889	0.300472
$890$	−20.0056	−0.670588
$891$	0	0
$892$	45.8397	1.53483
$893$	9.75337	0.326384
$894$	0	0
$895$	51.5494	1.72311
$896$	36.1388	1.20731
$897$	0	0
$898$	−6.55823	−0.218851
$899$	−35.8779	−1.19660
$900$	0	0
$901$	37.6154	1.25315
$902$	30.6322	1.01994
$903$	0	0
$904$	17.8550	0.593848
$905$	−27.4996	−0.914119
$906$	0	0
$907$	34.6801	1.15153	0.575767	−	0.817614i	$-0.304704\pi$
$907$	34.6801	1.15153	0.575767	+	0.817614i	$0.304704\pi$
$908$	4.06437	0.134881
$909$	0	0
$910$	15.1263	0.501433
$911$	−23.0692	−0.764316	−0.382158	−	0.924097i	$-0.624819\pi$
$911$	−23.0692	−0.764316	−0.382158	+	0.924097i	$0.624819\pi$
$912$	0	0
$913$	31.3121	1.03628
$914$	−23.6031	−0.780722
$915$	0	0
$916$	−32.2107	−1.06427
$917$	21.8772	0.722449
$918$	0	0
$919$	−50.5957	−1.66900	−0.834500	−	0.551008i	$-0.814243\pi$
$919$	−50.5957	−1.66900	−0.834500	+	0.551008i	$0.814243\pi$
$920$	−31.5480	−1.04011
$921$	0	0
$922$	13.7082	0.451456
$923$	34.4620	1.13433
$924$	0	0
$925$	−42.9442	−1.41200
$926$	1.93325	0.0635307
$927$	0	0
$928$	−45.1442	−1.48193
$929$	28.3578	0.930389	0.465195	−	0.885208i	$-0.345984\pi$
$929$	28.3578	0.930389	0.465195	+	0.885208i	$0.345984\pi$
$930$	0	0
$931$	−3.30211	−0.108222
$932$	−20.0326	−0.656190
$933$	0	0
$934$	13.2343	0.433038
$935$	132.251	4.32508
$936$	0	0
$937$	25.3720	0.828869	0.414434	−	0.910079i	$-0.363979\pi$
$937$	25.3720	0.828869	0.414434	+	0.910079i	$0.363979\pi$
$938$	−22.5393	−0.735936
$939$	0	0
$940$	48.8035	1.59179
$941$	20.3430	0.663163	0.331581	−	0.943427i	$-0.392418\pi$
$941$	20.3430	0.663163	0.331581	+	0.943427i	$0.392418\pi$
$942$	0	0
$943$	46.6672	1.51969
$944$	−0.326304	−0.0106203
$945$	0	0
$946$	−4.67509	−0.152000
$947$	36.4928	1.18586	0.592928	−	0.805255i	$-0.297972\pi$
$947$	36.4928	1.18586	0.592928	+	0.805255i	$0.297972\pi$
$948$	0	0
$949$	5.52809	0.179449
$950$	4.70535	0.152662
$951$	0	0
$952$	47.5757	1.54194
$953$	−38.5803	−1.24974	−0.624869	−	0.780730i	$-0.714848\pi$
$953$	−38.5803	−1.24974	−0.624869	+	0.780730i	$0.714848\pi$
$954$	0	0
$955$	−79.1103	−2.55995
$956$	−9.59151	−0.310212
$957$	0	0
$958$	−16.8932	−0.545794
$959$	71.8053	2.31871
$960$	0	0
$961$	−12.8091	−0.413195
$962$	−8.31280	−0.268016
$963$	0	0
$964$	−4.98644	−0.160602
$965$	−80.3419	−2.58630
$966$	0	0
$967$	−4.38572	−0.141035	−0.0705176	−	0.997511i	$-0.522465\pi$
$967$	−4.38572	−0.141035	−0.0705176	+	0.997511i	$0.522465\pi$
$968$	31.9993	1.02850
$969$	0	0
$970$	14.2996	0.459133
$971$	29.8005	0.956343	0.478172	−	0.878266i	$-0.341300\pi$
$971$	29.8005	0.956343	0.478172	+	0.878266i	$0.341300\pi$
$972$	0	0
$973$	28.4649	0.912544
$974$	−4.84124	−0.155123
$975$	0	0
$976$	−16.7886	−0.537388
$977$	−28.7521	−0.919862	−0.459931	−	0.887955i	$-0.652126\pi$
$977$	−28.7521	−0.919862	−0.459931	+	0.887955i	$0.652126\pi$
$978$	0	0
$979$	52.8288	1.68842
$980$	−16.5229	−0.527806
$981$	0	0
$982$	16.0366	0.511749
$983$	24.7559	0.789590	0.394795	−	0.918769i	$-0.370816\pi$
$983$	24.7559	0.789590	0.394795	+	0.918769i	$0.370816\pi$
$984$	0	0
$985$	−5.39278	−0.171828
$986$	−34.6562	−1.10368
$987$	0	0
$988$	−4.91264	−0.156292
$989$	−7.12234	−0.226477
$990$	0	0
$991$	−47.1674	−1.49832	−0.749161	−	0.662388i	$-0.769543\pi$
$991$	−47.1674	−1.49832	−0.749161	+	0.662388i	$0.769543\pi$
$992$	22.8892	0.726732
$993$	0	0
$994$	24.3729	0.773060
$995$	−62.7479	−1.98924
$996$	0	0
$997$	−32.9072	−1.04218	−0.521091	−	0.853501i	$-0.674475\pi$
$997$	−32.9072	−1.04218	−0.521091	+	0.853501i	$0.674475\pi$
$998$	9.75376	0.308750
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.a.d.1.2		5
3.2	odd	2		501.2.a.b.1.4	✓	5
12.11	even	2		8016.2.a.p.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.b.1.4	✓	5	3.2	odd	2
1503.2.a.d.1.2		5	1.1	even	1	trivial
8016.2.a.p.1.1		5	12.11	even	2

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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q-0.559296 q^{2} -1.68719 q^{4} +3.48658 q^{5} +3.13190 q^{7} +2.06223 q^{8} +O(q^{10})\)	\(q-0.559296 q^{2} -1.68719 q^{4} +3.48658 q^{5} +3.13190 q^{7} +2.06223 q^{8} -1.95003 q^{10} +5.14945 q^{11} -2.47676 q^{13} -1.75166 q^{14} +2.22098 q^{16} +7.36613 q^{17} -1.17562 q^{19} -5.88251 q^{20} -2.88007 q^{22} -4.38769 q^{23} +7.15622 q^{25} +1.38524 q^{26} -5.28411 q^{28} +8.41200 q^{29} -4.26508 q^{31} -5.36664 q^{32} -4.11985 q^{34} +10.9196 q^{35} -6.00097 q^{37} +0.657519 q^{38} +7.19012 q^{40} -10.6359 q^{41} +1.62326 q^{43} -8.68809 q^{44} +2.45402 q^{46} -8.29637 q^{47} +2.80882 q^{49} -4.00244 q^{50} +4.17877 q^{52} +5.10653 q^{53} +17.9540 q^{55} +6.45870 q^{56} -4.70480 q^{58} -0.146919 q^{59} -7.55907 q^{61} +2.38544 q^{62} -1.44042 q^{64} -8.63543 q^{65} +12.8674 q^{67} -12.4280 q^{68} -6.10730 q^{70} -13.9141 q^{71} -2.23198 q^{73} +3.35632 q^{74} +1.98349 q^{76} +16.1276 q^{77} +3.03137 q^{79} +7.74362 q^{80} +5.94864 q^{82} +6.08068 q^{83} +25.6826 q^{85} -0.907881 q^{86} +10.6193 q^{88} +10.2591 q^{89} -7.75699 q^{91} +7.40286 q^{92} +4.64013 q^{94} -4.09889 q^{95} -7.33303 q^{97} -1.57096 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8	\( 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8} + 5 q^{10} + 15 q^{11} + 3 q^{14} + 10 q^{16} + 11 q^{17} - 16 q^{19} + 9 q^{20} + 7 q^{22} + 9 q^{23} + 6 q^{25} - 7 q^{26} + 9 q^{28} + q^{29} - 18 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 7 q^{37} - 20 q^{38} + 32 q^{40} + 10 q^{41} + 6 q^{43} + 5 q^{44} + 13 q^{46} + 7 q^{47} + 11 q^{49} - 3 q^{50} - 6 q^{52} + 9 q^{53} + 17 q^{55} + 15 q^{56} - 19 q^{58} + 37 q^{59} - 2 q^{61} - 18 q^{62} + 14 q^{64} + 16 q^{65} - 26 q^{68} + 7 q^{70} - 13 q^{71} - 6 q^{73} - 13 q^{74} - 50 q^{76} + 2 q^{77} + 14 q^{79} + 10 q^{80} + 39 q^{82} + 5 q^{83} + 29 q^{85} + 37 q^{86} + 48 q^{88} + 30 q^{89} - 33 q^{91} + 31 q^{92} + 34 q^{94} - 43 q^{95} - 9 q^{97} - 36 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8 + 5 * q^10 + 15 * q^11 + 3 * q^14 + 10 * q^16 + 11 * q^17 - 16 * q^19 + 9 * q^20 + 7 * q^22 + 9 * q^23 + 6 * q^25 - 7 * q^26 + 9 * q^28 + q^29 - 18 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 7 * q^37 - 20 * q^38 + 32 * q^40 + 10 * q^41 + 6 * q^43 + 5 * q^44 + 13 * q^46 + 7 * q^47 + 11 * q^49 - 3 * q^50 - 6 * q^52 + 9 * q^53 + 17 * q^55 + 15 * q^56 - 19 * q^58 + 37 * q^59 - 2 * q^61 - 18 * q^62 + 14 * q^64 + 16 * q^65 - 26 * q^68 + 7 * q^70 - 13 * q^71 - 6 * q^73 - 13 * q^74 - 50 * q^76 + 2 * q^77 + 14 * q^79 + 10 * q^80 + 39 * q^82 + 5 * q^83 + 29 * q^85 + 37 * q^86 + 48 * q^88 + 30 * q^89 - 33 * q^91 + 31 * q^92 + 34 * q^94 - 43 * q^95 - 9 * q^97 - 36 * q^98

Introduction

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