LMFDB - Embedded newform 6032.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 22 x^{11} + 96 x^{10} + 159 x^{9} - 827 x^{8} - 362 x^{7} + 3029 x^{6} - 142 x^{5} + \cdots + 16 $ x^13 - 4x^12 - 22x^11 + 96x^10 + 159x^9 - 827x^8 - 362x^7 + 3029x^6 - 142x^5 - 4316x^4 + 208x^3 + 2228x^2 + 376x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.69571$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69571 q^{3} +2.65571 q^{5} -3.33796 q^{7} +4.26683 q^{9} +O(q^{10})$	$q-2.69571 q^{3} +2.65571 q^{5} -3.33796 q^{7} +4.26683 q^{9} -0.287750 q^{11} -1.00000 q^{13} -7.15901 q^{15} +3.59227 q^{17} +6.39213 q^{19} +8.99816 q^{21} +5.37981 q^{23} +2.05280 q^{25} -3.41499 q^{27} +1.00000 q^{29} -4.23528 q^{31} +0.775688 q^{33} -8.86466 q^{35} +0.548563 q^{37} +2.69571 q^{39} +1.67323 q^{41} -10.4922 q^{43} +11.3315 q^{45} +2.86780 q^{47} +4.14199 q^{49} -9.68371 q^{51} -1.80297 q^{53} -0.764180 q^{55} -17.2313 q^{57} -3.83166 q^{59} -14.5972 q^{61} -14.2425 q^{63} -2.65571 q^{65} +12.4787 q^{67} -14.5024 q^{69} +0.702301 q^{71} +10.6421 q^{73} -5.53374 q^{75} +0.960498 q^{77} -4.03446 q^{79} -3.59468 q^{81} +16.7489 q^{83} +9.54004 q^{85} -2.69571 q^{87} +0.458322 q^{89} +3.33796 q^{91} +11.4171 q^{93} +16.9756 q^{95} +9.22728 q^{97} -1.22778 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	$ 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) $ 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * 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
$3$	−2.69571	−1.55637	−0.778183	−	0.628038i	$-0.783858\pi$
$3$	−2.69571	−1.55637	−0.778183	+	0.628038i	$0.783858\pi$
$4$	0	0
$5$	2.65571	1.18767	0.593835	−	0.804587i	$-0.297613\pi$
$5$	2.65571	1.18767	0.593835	+	0.804587i	$0.297613\pi$
$6$	0	0
$7$	−3.33796	−1.26163	−0.630815	−	0.775933i	$-0.717280\pi$
$7$	−3.33796	−1.26163	−0.630815	+	0.775933i	$0.717280\pi$
$8$	0	0
$9$	4.26683	1.42228
$10$	0	0
$11$	−0.287750	−0.0867598	−0.0433799	−	0.999059i	$-0.513813\pi$
$11$	−0.287750	−0.0867598	−0.0433799	+	0.999059i	$0.513813\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−7.15901	−1.84845
$16$	0	0
$17$	3.59227	0.871254	0.435627	−	0.900127i	$-0.356527\pi$
$17$	3.59227	0.871254	0.435627	+	0.900127i	$0.356527\pi$
$18$	0	0
$19$	6.39213	1.46645	0.733227	−	0.679984i	$-0.238013\pi$
$19$	6.39213	1.46645	0.733227	+	0.679984i	$0.238013\pi$
$20$	0	0
$21$	8.99816	1.96356
$22$	0	0
$23$	5.37981	1.12177	0.560883	−	0.827895i	$-0.310462\pi$
$23$	5.37981	1.12177	0.560883	+	0.827895i	$0.310462\pi$
$24$	0	0
$25$	2.05280	0.410560
$26$	0	0
$27$	−3.41499	−0.657215
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−4.23528	−0.760680	−0.380340	−	0.924847i	$-0.624193\pi$
$31$	−4.23528	−0.760680	−0.380340	+	0.924847i	$0.624193\pi$
$32$	0	0
$33$	0.775688	0.135030
$34$	0	0
$35$	−8.86466	−1.49840
$36$	0	0
$37$	0.548563	0.0901833	0.0450916	−	0.998983i	$-0.485642\pi$
$37$	0.548563	0.0901833	0.0450916	+	0.998983i	$0.485642\pi$
$38$	0	0
$39$	2.69571	0.431658
$40$	0	0
$41$	1.67323	0.261314	0.130657	−	0.991428i	$-0.458291\pi$
$41$	1.67323	0.261314	0.130657	+	0.991428i	$0.458291\pi$
$42$	0	0
$43$	−10.4922	−1.60005	−0.800024	−	0.599968i	$-0.795180\pi$
$43$	−10.4922	−1.60005	−0.800024	+	0.599968i	$0.795180\pi$
$44$	0	0
$45$	11.3315	1.68919
$46$	0	0
$47$	2.86780	0.418311	0.209156	−	0.977882i	$-0.432928\pi$
$47$	2.86780	0.418311	0.209156	+	0.977882i	$0.432928\pi$
$48$	0	0
$49$	4.14199	0.591713
$50$	0	0
$51$	−9.68371	−1.35599
$52$	0	0
$53$	−1.80297	−0.247656	−0.123828	−	0.992304i	$-0.539517\pi$
$53$	−1.80297	−0.247656	−0.123828	+	0.992304i	$0.539517\pi$
$54$	0	0
$55$	−0.764180	−0.103042
$56$	0	0
$57$	−17.2313	−2.28234
$58$	0	0
$59$	−3.83166	−0.498839	−0.249420	−	0.968395i	$-0.580240\pi$
$59$	−3.83166	−0.498839	−0.249420	+	0.968395i	$0.580240\pi$
$60$	0	0
$61$	−14.5972	−1.86898	−0.934492	−	0.355984i	$-0.884146\pi$
$61$	−14.5972	−1.86898	−0.934492	+	0.355984i	$0.884146\pi$
$62$	0	0
$63$	−14.2425	−1.79439
$64$	0	0
$65$	−2.65571	−0.329400
$66$	0	0
$67$	12.4787	1.52452	0.762260	−	0.647272i	$-0.224090\pi$
$67$	12.4787	1.52452	0.762260	+	0.647272i	$0.224090\pi$
$68$	0	0
$69$	−14.5024	−1.74588
$70$	0	0
$71$	0.702301	0.0833478	0.0416739	−	0.999131i	$-0.486731\pi$
$71$	0.702301	0.0833478	0.0416739	+	0.999131i	$0.486731\pi$
$72$	0	0
$73$	10.6421	1.24556	0.622780	−	0.782397i	$-0.286003\pi$
$73$	10.6421	1.24556	0.622780	+	0.782397i	$0.286003\pi$
$74$	0	0
$75$	−5.53374	−0.638982
$76$	0	0
$77$	0.960498	0.109459
$78$	0	0
$79$	−4.03446	−0.453912	−0.226956	−	0.973905i	$-0.572877\pi$
$79$	−4.03446	−0.453912	−0.226956	+	0.973905i	$0.572877\pi$
$80$	0	0
$81$	−3.59468	−0.399408
$82$	0	0
$83$	16.7489	1.83843	0.919216	−	0.393754i	$-0.128824\pi$
$83$	16.7489	1.83843	0.919216	+	0.393754i	$0.128824\pi$
$84$	0	0
$85$	9.54004	1.03476
$86$	0	0
$87$	−2.69571	−0.289010
$88$	0	0
$89$	0.458322	0.0485820	0.0242910	−	0.999705i	$-0.492267\pi$
$89$	0.458322	0.0485820	0.0242910	+	0.999705i	$0.492267\pi$
$90$	0	0
$91$	3.33796	0.349913
$92$	0	0
$93$	11.4171	1.18390
$94$	0	0
$95$	16.9756	1.74166
$96$	0	0
$97$	9.22728	0.936888	0.468444	−	0.883493i	$-0.344815\pi$
$97$	9.22728	0.936888	0.468444	+	0.883493i	$0.344815\pi$
$98$	0	0
$99$	−1.22778	−0.123396
$100$	0	0
$101$	4.95758	0.493298	0.246649	−	0.969105i	$-0.420671\pi$
$101$	4.95758	0.493298	0.246649	+	0.969105i	$0.420671\pi$
$102$	0	0
$103$	−18.2212	−1.79539	−0.897696	−	0.440615i	$-0.854760\pi$
$103$	−18.2212	−1.79539	−0.897696	+	0.440615i	$0.854760\pi$
$104$	0	0
$105$	23.8965	2.33206
$106$	0	0
$107$	0.695632	0.0672493	0.0336247	−	0.999435i	$-0.489295\pi$
$107$	0.695632	0.0672493	0.0336247	+	0.999435i	$0.489295\pi$
$108$	0	0
$109$	−9.85143	−0.943596	−0.471798	−	0.881707i	$-0.656395\pi$
$109$	−9.85143	−0.943596	−0.471798	+	0.881707i	$0.656395\pi$
$110$	0	0
$111$	−1.47876	−0.140358
$112$	0	0
$113$	−2.08223	−0.195880	−0.0979399	−	0.995192i	$-0.531225\pi$
$113$	−2.08223	−0.195880	−0.0979399	+	0.995192i	$0.531225\pi$
$114$	0	0
$115$	14.2872	1.33229
$116$	0	0
$117$	−4.26683	−0.394468
$118$	0	0
$119$	−11.9909	−1.09920
$120$	0	0
$121$	−10.9172	−0.992473
$122$	0	0
$123$	−4.51053	−0.406700
$124$	0	0
$125$	−7.82691	−0.700060
$126$	0	0
$127$	18.6952	1.65893	0.829467	−	0.558556i	$-0.188644\pi$
$127$	18.6952	1.65893	0.829467	+	0.558556i	$0.188644\pi$
$128$	0	0
$129$	28.2839	2.49026
$130$	0	0
$131$	13.6273	1.19062	0.595311	−	0.803495i	$-0.297029\pi$
$131$	13.6273	1.19062	0.595311	+	0.803495i	$0.297029\pi$
$132$	0	0
$133$	−21.3367	−1.85012
$134$	0	0
$135$	−9.06922	−0.780554
$136$	0	0
$137$	−3.45935	−0.295553	−0.147776	−	0.989021i	$-0.547212\pi$
$137$	−3.45935	−0.295553	−0.147776	+	0.989021i	$0.547212\pi$
$138$	0	0
$139$	−0.568889	−0.0482525	−0.0241263	−	0.999709i	$-0.507680\pi$
$139$	−0.568889	−0.0482525	−0.0241263	+	0.999709i	$0.507680\pi$
$140$	0	0
$141$	−7.73074	−0.651045
$142$	0	0
$143$	0.287750	0.0240628
$144$	0	0
$145$	2.65571	0.220545
$146$	0	0
$147$	−11.1656	−0.920922
$148$	0	0
$149$	12.3344	1.01047	0.505235	−	0.862982i	$-0.331406\pi$
$149$	12.3344	1.01047	0.505235	+	0.862982i	$0.331406\pi$
$150$	0	0
$151$	−11.5371	−0.938874	−0.469437	−	0.882966i	$-0.655543\pi$
$151$	−11.5371	−0.938874	−0.469437	+	0.882966i	$0.655543\pi$
$152$	0	0
$153$	15.3276	1.23916
$154$	0	0
$155$	−11.2477	−0.903436
$156$	0	0
$157$	−18.4555	−1.47291	−0.736453	−	0.676488i	$-0.763501\pi$
$157$	−18.4555	−1.47291	−0.736453	+	0.676488i	$0.763501\pi$
$158$	0	0
$159$	4.86027	0.385444
$160$	0	0
$161$	−17.9576	−1.41526
$162$	0	0
$163$	4.29465	0.336383	0.168191	−	0.985754i	$-0.446207\pi$
$163$	4.29465	0.336383	0.168191	+	0.985754i	$0.446207\pi$
$164$	0	0
$165$	2.06000	0.160371
$166$	0	0
$167$	11.8621	0.917917	0.458959	−	0.888458i	$-0.348223\pi$
$167$	11.8621	0.917917	0.458959	+	0.888458i	$0.348223\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	27.2741	2.08570
$172$	0	0
$173$	9.44518	0.718104	0.359052	−	0.933318i	$-0.383100\pi$
$173$	9.44518	0.718104	0.359052	+	0.933318i	$0.383100\pi$
$174$	0	0
$175$	−6.85217	−0.517975
$176$	0	0
$177$	10.3290	0.776377
$178$	0	0
$179$	−1.43668	−0.107383	−0.0536913	−	0.998558i	$-0.517099\pi$
$179$	−1.43668	−0.107383	−0.0536913	+	0.998558i	$0.517099\pi$
$180$	0	0
$181$	15.9935	1.18879	0.594395	−	0.804173i	$-0.297391\pi$
$181$	15.9935	1.18879	0.594395	+	0.804173i	$0.297391\pi$
$182$	0	0
$183$	39.3498	2.90882
$184$	0	0
$185$	1.45683	0.107108
$186$	0	0
$187$	−1.03368	−0.0755898
$188$	0	0
$189$	11.3991	0.829163
$190$	0	0
$191$	−16.0334	−1.16014	−0.580068	−	0.814568i	$-0.696974\pi$
$191$	−16.0334	−1.16014	−0.580068	+	0.814568i	$0.696974\pi$
$192$	0	0
$193$	8.87145	0.638581	0.319290	−	0.947657i	$-0.396555\pi$
$193$	8.87145	0.638581	0.319290	+	0.947657i	$0.396555\pi$
$194$	0	0
$195$	7.15901	0.512668
$196$	0	0
$197$	21.2970	1.51735	0.758674	−	0.651470i	$-0.225847\pi$
$197$	21.2970	1.51735	0.758674	+	0.651470i	$0.225847\pi$
$198$	0	0
$199$	−17.3485	−1.22980	−0.614900	−	0.788605i	$-0.710803\pi$
$199$	−17.3485	−1.22980	−0.614900	+	0.788605i	$0.710803\pi$
$200$	0	0
$201$	−33.6390	−2.37271
$202$	0	0
$203$	−3.33796	−0.234279
$204$	0	0
$205$	4.44361	0.310355
$206$	0	0
$207$	22.9547	1.59546
$208$	0	0
$209$	−1.83933	−0.127229
$210$	0	0
$211$	−4.42913	−0.304914	−0.152457	−	0.988310i	$-0.548718\pi$
$211$	−4.42913	−0.304914	−0.152457	+	0.988310i	$0.548718\pi$
$212$	0	0
$213$	−1.89320	−0.129720
$214$	0	0
$215$	−27.8643	−1.90033
$216$	0	0
$217$	14.1372	0.959697
$218$	0	0
$219$	−28.6879	−1.93855
$220$	0	0
$221$	−3.59227	−0.241642
$222$	0	0
$223$	20.2177	1.35388	0.676939	−	0.736039i	$-0.263306\pi$
$223$	20.2177	1.35388	0.676939	+	0.736039i	$0.263306\pi$
$224$	0	0
$225$	8.75894	0.583929
$226$	0	0
$227$	−6.56612	−0.435808	−0.217904	−	0.975970i	$-0.569922\pi$
$227$	−6.56612	−0.435808	−0.217904	+	0.975970i	$0.569922\pi$
$228$	0	0
$229$	1.36558	0.0902401	0.0451200	−	0.998982i	$-0.485633\pi$
$229$	1.36558	0.0902401	0.0451200	+	0.998982i	$0.485633\pi$
$230$	0	0
$231$	−2.58922	−0.170358
$232$	0	0
$233$	−1.79090	−0.117326	−0.0586628	−	0.998278i	$-0.518684\pi$
$233$	−1.79090	−0.117326	−0.0586628	+	0.998278i	$0.518684\pi$
$234$	0	0
$235$	7.61604	0.496816
$236$	0	0
$237$	10.8757	0.706453
$238$	0	0
$239$	28.1686	1.82208	0.911039	−	0.412321i	$-0.135282\pi$
$239$	28.1686	1.82208	0.911039	+	0.412321i	$0.135282\pi$
$240$	0	0
$241$	−24.6133	−1.58548	−0.792742	−	0.609558i	$-0.791347\pi$
$241$	−24.6133	−1.58548	−0.792742	+	0.609558i	$0.791347\pi$
$242$	0	0
$243$	19.9351	1.27884
$244$	0	0
$245$	10.9999	0.702760
$246$	0	0
$247$	−6.39213	−0.406721
$248$	0	0
$249$	−45.1501	−2.86127
$250$	0	0
$251$	27.4387	1.73192	0.865959	−	0.500116i	$-0.166709\pi$
$251$	27.4387	1.73192	0.865959	+	0.500116i	$0.166709\pi$
$252$	0	0
$253$	−1.54804	−0.0973243
$254$	0	0
$255$	−25.7171	−1.61047
$256$	0	0
$257$	20.4152	1.27347	0.636733	−	0.771084i	$-0.280285\pi$
$257$	20.4152	1.27347	0.636733	+	0.771084i	$0.280285\pi$
$258$	0	0
$259$	−1.83108	−0.113778
$260$	0	0
$261$	4.26683	0.264110
$262$	0	0
$263$	−14.8024	−0.912755	−0.456377	−	0.889786i	$-0.650853\pi$
$263$	−14.8024	−0.912755	−0.456377	+	0.889786i	$0.650853\pi$
$264$	0	0
$265$	−4.78816	−0.294134
$266$	0	0
$267$	−1.23550	−0.0756114
$268$	0	0
$269$	16.7957	1.02405	0.512025	−	0.858971i	$-0.328896\pi$
$269$	16.7957	1.02405	0.512025	+	0.858971i	$0.328896\pi$
$270$	0	0
$271$	−18.1966	−1.10536	−0.552681	−	0.833393i	$-0.686395\pi$
$271$	−18.1966	−1.10536	−0.552681	+	0.833393i	$0.686395\pi$
$272$	0	0
$273$	−8.99816	−0.544593
$274$	0	0
$275$	−0.590693	−0.0356201
$276$	0	0
$277$	−20.8930	−1.25534	−0.627670	−	0.778480i	$-0.715991\pi$
$277$	−20.8930	−1.25534	−0.627670	+	0.778480i	$0.715991\pi$
$278$	0	0
$279$	−18.0712	−1.08190
$280$	0	0
$281$	1.24731	0.0744082	0.0372041	−	0.999308i	$-0.488155\pi$
$281$	1.24731	0.0744082	0.0372041	+	0.999308i	$0.488155\pi$
$282$	0	0
$283$	−2.07229	−0.123185	−0.0615924	−	0.998101i	$-0.519618\pi$
$283$	−2.07229	−0.123185	−0.0615924	+	0.998101i	$0.519618\pi$
$284$	0	0
$285$	−45.7613	−2.71067
$286$	0	0
$287$	−5.58517	−0.329682
$288$	0	0
$289$	−4.09558	−0.240916
$290$	0	0
$291$	−24.8740	−1.45814
$292$	0	0
$293$	17.5676	1.02631	0.513155	−	0.858296i	$-0.328476\pi$
$293$	17.5676	1.02631	0.513155	+	0.858296i	$0.328476\pi$
$294$	0	0
$295$	−10.1758	−0.592457
$296$	0	0
$297$	0.982662	0.0570198
$298$	0	0
$299$	−5.37981	−0.311122
$300$	0	0
$301$	35.0226	2.01867
$302$	0	0
$303$	−13.3642	−0.767752
$304$	0	0
$305$	−38.7660	−2.21974
$306$	0	0
$307$	−7.83966	−0.447433	−0.223717	−	0.974654i	$-0.571819\pi$
$307$	−7.83966	−0.447433	−0.223717	+	0.974654i	$0.571819\pi$
$308$	0	0
$309$	49.1191	2.79429
$310$	0	0
$311$	16.3353	0.926289	0.463145	−	0.886283i	$-0.346721\pi$
$311$	16.3353	0.926289	0.463145	+	0.886283i	$0.346721\pi$
$312$	0	0
$313$	−6.44200	−0.364124	−0.182062	−	0.983287i	$-0.558277\pi$
$313$	−6.44200	−0.364124	−0.182062	+	0.983287i	$0.558277\pi$
$314$	0	0
$315$	−37.8240	−2.13114
$316$	0	0
$317$	20.3221	1.14140	0.570700	−	0.821158i	$-0.306672\pi$
$317$	20.3221	1.14140	0.570700	+	0.821158i	$0.306672\pi$
$318$	0	0
$319$	−0.287750	−0.0161109
$320$	0	0
$321$	−1.87522	−0.104665
$322$	0	0
$323$	22.9623	1.27765
$324$	0	0
$325$	−2.05280	−0.113869
$326$	0	0
$327$	26.5566	1.46858
$328$	0	0
$329$	−9.57260	−0.527754
$330$	0	0
$331$	10.8511	0.596430	0.298215	−	0.954499i	$-0.403609\pi$
$331$	10.8511	0.596430	0.298215	+	0.954499i	$0.403609\pi$
$332$	0	0
$333$	2.34062	0.128265
$334$	0	0
$335$	33.1399	1.81063
$336$	0	0
$337$	7.21177	0.392850	0.196425	−	0.980519i	$-0.437067\pi$
$337$	7.21177	0.392850	0.196425	+	0.980519i	$0.437067\pi$
$338$	0	0
$339$	5.61308	0.304861
$340$	0	0
$341$	1.21870	0.0659964
$342$	0	0
$343$	9.53993	0.515108
$344$	0	0
$345$	−38.5141	−2.07353
$346$	0	0
$347$	24.5154	1.31605	0.658027	−	0.752994i	$-0.271391\pi$
$347$	24.5154	1.31605	0.658027	+	0.752994i	$0.271391\pi$
$348$	0	0
$349$	16.0521	0.859247	0.429623	−	0.903008i	$-0.358646\pi$
$349$	16.0521	0.859247	0.429623	+	0.903008i	$0.358646\pi$
$350$	0	0
$351$	3.41499	0.182279
$352$	0	0
$353$	23.6886	1.26082	0.630409	−	0.776263i	$-0.282887\pi$
$353$	23.6886	1.26082	0.630409	+	0.776263i	$0.282887\pi$
$354$	0	0
$355$	1.86511	0.0989897
$356$	0	0
$357$	32.3238	1.71076
$358$	0	0
$359$	4.25928	0.224796	0.112398	−	0.993663i	$-0.464147\pi$
$359$	4.25928	0.224796	0.112398	+	0.993663i	$0.464147\pi$
$360$	0	0
$361$	21.8593	1.15049
$362$	0	0
$363$	29.4296	1.54465
$364$	0	0
$365$	28.2623	1.47931
$366$	0	0
$367$	31.4095	1.63956	0.819782	−	0.572676i	$-0.194095\pi$
$367$	31.4095	1.63956	0.819782	+	0.572676i	$0.194095\pi$
$368$	0	0
$369$	7.13937	0.371661
$370$	0	0
$371$	6.01823	0.312451
$372$	0	0
$373$	13.5420	0.701178	0.350589	−	0.936529i	$-0.385981\pi$
$373$	13.5420	0.701178	0.350589	+	0.936529i	$0.385981\pi$
$374$	0	0
$375$	21.0990	1.08955
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	28.2495	1.45108	0.725538	−	0.688182i	$-0.241591\pi$
$379$	28.2495	1.45108	0.725538	+	0.688182i	$0.241591\pi$
$380$	0	0
$381$	−50.3968	−2.58191
$382$	0	0
$383$	−33.9733	−1.73595	−0.867977	−	0.496604i	$-0.834580\pi$
$383$	−33.9733	−1.73595	−0.867977	+	0.496604i	$0.834580\pi$
$384$	0	0
$385$	2.55080	0.130001
$386$	0	0
$387$	−44.7684	−2.27571
$388$	0	0
$389$	28.5186	1.44595	0.722976	−	0.690873i	$-0.242774\pi$
$389$	28.5186	1.44595	0.722976	+	0.690873i	$0.242774\pi$
$390$	0	0
$391$	19.3257	0.977344
$392$	0	0
$393$	−36.7352	−1.85305
$394$	0	0
$395$	−10.7144	−0.539098
$396$	0	0
$397$	18.5304	0.930014	0.465007	−	0.885307i	$-0.346052\pi$
$397$	18.5304	0.930014	0.465007	+	0.885307i	$0.346052\pi$
$398$	0	0
$399$	57.5174	2.87947
$400$	0	0
$401$	15.1570	0.756907	0.378453	−	0.925620i	$-0.376456\pi$
$401$	15.1570	0.756907	0.378453	+	0.925620i	$0.376456\pi$
$402$	0	0
$403$	4.23528	0.210975
$404$	0	0
$405$	−9.54642	−0.474365
$406$	0	0
$407$	−0.157849	−0.00782428
$408$	0	0
$409$	−28.2033	−1.39456	−0.697281	−	0.716797i	$-0.745607\pi$
$409$	−28.2033	−1.39456	−0.697281	+	0.716797i	$0.745607\pi$
$410$	0	0
$411$	9.32540	0.459988
$412$	0	0
$413$	12.7899	0.629351
$414$	0	0
$415$	44.4803	2.18345
$416$	0	0
$417$	1.53356	0.0750986
$418$	0	0
$419$	−3.57958	−0.174874	−0.0874369	−	0.996170i	$-0.527868\pi$
$419$	−3.57958	−0.174874	−0.0874369	+	0.996170i	$0.527868\pi$
$420$	0	0
$421$	9.25338	0.450982	0.225491	−	0.974245i	$-0.427601\pi$
$421$	9.25338	0.450982	0.225491	+	0.974245i	$0.427601\pi$
$422$	0	0
$423$	12.2364	0.594954
$424$	0	0
$425$	7.37422	0.357702
$426$	0	0
$427$	48.7250	2.35797
$428$	0	0
$429$	−0.775688	−0.0374506
$430$	0	0
$431$	−28.5149	−1.37352	−0.686758	−	0.726886i	$-0.740967\pi$
$431$	−28.5149	−1.37352	−0.686758	+	0.726886i	$0.740967\pi$
$432$	0	0
$433$	23.8211	1.14477	0.572385	−	0.819985i	$-0.306018\pi$
$433$	23.8211	1.14477	0.572385	+	0.819985i	$0.306018\pi$
$434$	0	0
$435$	−7.15901	−0.343248
$436$	0	0
$437$	34.3884	1.64502
$438$	0	0
$439$	3.06727	0.146393	0.0731965	−	0.997318i	$-0.476680\pi$
$439$	3.06727	0.146393	0.0731965	+	0.997318i	$0.476680\pi$
$440$	0	0
$441$	17.6731	0.841578
$442$	0	0
$443$	−4.81995	−0.229003	−0.114501	−	0.993423i	$-0.536527\pi$
$443$	−4.81995	−0.229003	−0.114501	+	0.993423i	$0.536527\pi$
$444$	0	0
$445$	1.21717	0.0576994
$446$	0	0
$447$	−33.2498	−1.57266
$448$	0	0
$449$	8.69763	0.410466	0.205233	−	0.978713i	$-0.434205\pi$
$449$	8.69763	0.410466	0.205233	+	0.978713i	$0.434205\pi$
$450$	0	0
$451$	−0.481471	−0.0226716
$452$	0	0
$453$	31.1006	1.46123
$454$	0	0
$455$	8.86466	0.415582
$456$	0	0
$457$	−1.60496	−0.0750770	−0.0375385	−	0.999295i	$-0.511952\pi$
$457$	−1.60496	−0.0750770	−0.0375385	+	0.999295i	$0.511952\pi$
$458$	0	0
$459$	−12.2676	−0.572601
$460$	0	0
$461$	−7.36011	−0.342795	−0.171397	−	0.985202i	$-0.554828\pi$
$461$	−7.36011	−0.342795	−0.171397	+	0.985202i	$0.554828\pi$
$462$	0	0
$463$	−9.83878	−0.457247	−0.228623	−	0.973515i	$-0.573422\pi$
$463$	−9.83878	−0.457247	−0.228623	+	0.973515i	$0.573422\pi$
$464$	0	0
$465$	30.3205	1.40608
$466$	0	0
$467$	34.3722	1.59056	0.795279	−	0.606244i	$-0.207324\pi$
$467$	34.3722	1.59056	0.795279	+	0.606244i	$0.207324\pi$
$468$	0	0
$469$	−41.6535	−1.92338
$470$	0	0
$471$	49.7505	2.29238
$472$	0	0
$473$	3.01913	0.138820
$474$	0	0
$475$	13.1218	0.602068
$476$	0	0
$477$	−7.69294	−0.352236
$478$	0	0
$479$	16.4476	0.751510	0.375755	−	0.926719i	$-0.377383\pi$
$479$	16.4476	0.751510	0.375755	+	0.926719i	$0.377383\pi$
$480$	0	0
$481$	−0.548563	−0.0250123
$482$	0	0
$483$	48.4084	2.20266
$484$	0	0
$485$	24.5050	1.11271
$486$	0	0
$487$	−24.4733	−1.10899	−0.554496	−	0.832186i	$-0.687089\pi$
$487$	−24.4733	−1.10899	−0.554496	+	0.832186i	$0.687089\pi$
$488$	0	0
$489$	−11.5771	−0.523535
$490$	0	0
$491$	6.45110	0.291134	0.145567	−	0.989348i	$-0.453499\pi$
$491$	6.45110	0.291134	0.145567	+	0.989348i	$0.453499\pi$
$492$	0	0
$493$	3.59227	0.161788
$494$	0	0
$495$	−3.26062	−0.146554
$496$	0	0
$497$	−2.34426	−0.105154
$498$	0	0
$499$	−26.0596	−1.16659	−0.583295	−	0.812261i	$-0.698237\pi$
$499$	−26.0596	−1.16659	−0.583295	+	0.812261i	$0.698237\pi$
$500$	0	0
$501$	−31.9767	−1.42862
$502$	0	0
$503$	−10.4726	−0.466950	−0.233475	−	0.972363i	$-0.575010\pi$
$503$	−10.4726	−0.466950	−0.233475	+	0.972363i	$0.575010\pi$
$504$	0	0
$505$	13.1659	0.585875
$506$	0	0
$507$	−2.69571	−0.119720
$508$	0	0
$509$	8.48890	0.376264	0.188132	−	0.982144i	$-0.439757\pi$
$509$	8.48890	0.376264	0.188132	+	0.982144i	$0.439757\pi$
$510$	0	0
$511$	−35.5228	−1.57144
$512$	0	0
$513$	−21.8290	−0.963775
$514$	0	0
$515$	−48.3903	−2.13233
$516$	0	0
$517$	−0.825208	−0.0362926
$518$	0	0
$519$	−25.4614	−1.11763
$520$	0	0
$521$	−5.89788	−0.258391	−0.129196	−	0.991619i	$-0.541239\pi$
$521$	−5.89788	−0.258391	−0.129196	+	0.991619i	$0.541239\pi$
$522$	0	0
$523$	−3.95919	−0.173123	−0.0865617	−	0.996246i	$-0.527588\pi$
$523$	−3.95919	−0.173123	−0.0865617	+	0.996246i	$0.527588\pi$
$524$	0	0
$525$	18.4714	0.806159
$526$	0	0
$527$	−15.2143	−0.662745
$528$	0	0
$529$	5.94231	0.258361
$530$	0	0
$531$	−16.3490	−0.709487
$532$	0	0
$533$	−1.67323	−0.0724755
$534$	0	0
$535$	1.84740	0.0798700
$536$	0	0
$537$	3.87287	0.167127
$538$	0	0
$539$	−1.19186	−0.0513369
$540$	0	0
$541$	14.9470	0.642622	0.321311	−	0.946974i	$-0.395877\pi$
$541$	14.9470	0.642622	0.321311	+	0.946974i	$0.395877\pi$
$542$	0	0
$543$	−43.1139	−1.85019
$544$	0	0
$545$	−26.1626	−1.12068
$546$	0	0
$547$	38.6481	1.65247	0.826237	−	0.563322i	$-0.190477\pi$
$547$	38.6481	1.65247	0.826237	+	0.563322i	$0.190477\pi$
$548$	0	0
$549$	−62.2839	−2.65821
$550$	0	0
$551$	6.39213	0.272314
$552$	0	0
$553$	13.4669	0.572670
$554$	0	0
$555$	−3.92717	−0.166699
$556$	0	0
$557$	36.4756	1.54552	0.772760	−	0.634699i	$-0.218876\pi$
$557$	36.4756	1.54552	0.772760	+	0.634699i	$0.218876\pi$
$558$	0	0
$559$	10.4922	0.443773
$560$	0	0
$561$	2.78648	0.117645
$562$	0	0
$563$	21.4268	0.903033	0.451516	−	0.892263i	$-0.350883\pi$
$563$	21.4268	0.903033	0.451516	+	0.892263i	$0.350883\pi$
$564$	0	0
$565$	−5.52980	−0.232640
$566$	0	0
$567$	11.9989	0.503906
$568$	0	0
$569$	−15.0733	−0.631906	−0.315953	−	0.948775i	$-0.602324\pi$
$569$	−15.0733	−0.631906	−0.315953	+	0.948775i	$0.602324\pi$
$570$	0	0
$571$	−9.21298	−0.385551	−0.192776	−	0.981243i	$-0.561749\pi$
$571$	−9.21298	−0.385551	−0.192776	+	0.981243i	$0.561749\pi$
$572$	0	0
$573$	43.2213	1.80560
$574$	0	0
$575$	11.0437	0.460553
$576$	0	0
$577$	−22.2112	−0.924666	−0.462333	−	0.886706i	$-0.652988\pi$
$577$	−22.2112	−0.924666	−0.462333	+	0.886706i	$0.652988\pi$
$578$	0	0
$579$	−23.9148	−0.993865
$580$	0	0
$581$	−55.9072	−2.31942
$582$	0	0
$583$	0.518803	0.0214866
$584$	0	0
$585$	−11.3315	−0.468498
$586$	0	0
$587$	−8.16778	−0.337120	−0.168560	−	0.985691i	$-0.553912\pi$
$587$	−8.16778	−0.337120	−0.168560	+	0.985691i	$0.553912\pi$
$588$	0	0
$589$	−27.0725	−1.11550
$590$	0	0
$591$	−57.4104	−2.36155
$592$	0	0
$593$	24.0240	0.986547	0.493273	−	0.869874i	$-0.335800\pi$
$593$	24.0240	0.986547	0.493273	+	0.869874i	$0.335800\pi$
$594$	0	0
$595$	−31.8443	−1.30549
$596$	0	0
$597$	46.7663	1.91402
$598$	0	0
$599$	45.0656	1.84133	0.920665	−	0.390355i	$-0.127648\pi$
$599$	45.0656	1.84133	0.920665	+	0.390355i	$0.127648\pi$
$600$	0	0
$601$	−10.7181	−0.437201	−0.218600	−	0.975814i	$-0.570149\pi$
$601$	−10.7181	−0.437201	−0.218600	+	0.975814i	$0.570149\pi$
$602$	0	0
$603$	53.2445	2.16829
$604$	0	0
$605$	−28.9929	−1.17873
$606$	0	0
$607$	8.31627	0.337547	0.168774	−	0.985655i	$-0.446019\pi$
$607$	8.31627	0.337547	0.168774	+	0.985655i	$0.446019\pi$
$608$	0	0
$609$	8.99816	0.364624
$610$	0	0
$611$	−2.86780	−0.116019
$612$	0	0
$613$	−25.0630	−1.01228	−0.506142	−	0.862450i	$-0.668929\pi$
$613$	−25.0630	−1.01228	−0.506142	+	0.862450i	$0.668929\pi$
$614$	0	0
$615$	−11.9787	−0.483026
$616$	0	0
$617$	16.4208	0.661076	0.330538	−	0.943793i	$-0.392770\pi$
$617$	16.4208	0.661076	0.330538	+	0.943793i	$0.392770\pi$
$618$	0	0
$619$	8.16765	0.328286	0.164143	−	0.986437i	$-0.447514\pi$
$619$	8.16765	0.328286	0.164143	+	0.986437i	$0.447514\pi$
$620$	0	0
$621$	−18.3720	−0.737242
$622$	0	0
$623$	−1.52986	−0.0612926
$624$	0	0
$625$	−31.0500	−1.24200
$626$	0	0
$627$	4.95830	0.198015
$628$	0	0
$629$	1.97059	0.0785725
$630$	0	0
$631$	−27.5706	−1.09757	−0.548785	−	0.835964i	$-0.684909\pi$
$631$	−27.5706	−1.09757	−0.548785	+	0.835964i	$0.684909\pi$
$632$	0	0
$633$	11.9396	0.474557
$634$	0	0
$635$	49.6491	1.97027
$636$	0	0
$637$	−4.14199	−0.164112
$638$	0	0
$639$	2.99660	0.118544
$640$	0	0
$641$	10.0007	0.395003	0.197502	−	0.980303i	$-0.436717\pi$
$641$	10.0007	0.395003	0.197502	+	0.980303i	$0.436717\pi$
$642$	0	0
$643$	−27.0750	−1.06774	−0.533868	−	0.845568i	$-0.679262\pi$
$643$	−27.0750	−1.06774	−0.533868	+	0.845568i	$0.679262\pi$
$644$	0	0
$645$	75.1139	2.95761
$646$	0	0
$647$	−6.89768	−0.271176	−0.135588	−	0.990765i	$-0.543292\pi$
$647$	−6.89768	−0.271176	−0.135588	+	0.990765i	$0.543292\pi$
$648$	0	0
$649$	1.10256	0.0432792
$650$	0	0
$651$	−38.1098	−1.49364
$652$	0	0
$653$	16.5949	0.649410	0.324705	−	0.945815i	$-0.394735\pi$
$653$	16.5949	0.649410	0.324705	+	0.945815i	$0.394735\pi$
$654$	0	0
$655$	36.1902	1.41407
$656$	0	0
$657$	45.4079	1.77153
$658$	0	0
$659$	22.4437	0.874284	0.437142	−	0.899393i	$-0.355991\pi$
$659$	22.4437	0.874284	0.437142	+	0.899393i	$0.355991\pi$
$660$	0	0
$661$	31.5063	1.22545	0.612726	−	0.790296i	$-0.290073\pi$
$661$	31.5063	1.22545	0.612726	+	0.790296i	$0.290073\pi$
$662$	0	0
$663$	9.68371	0.376084
$664$	0	0
$665$	−56.6640	−2.19734
$666$	0	0
$667$	5.37981	0.208307
$668$	0	0
$669$	−54.5010	−2.10713
$670$	0	0
$671$	4.20035	0.162153
$672$	0	0
$673$	1.63617	0.0630697	0.0315348	−	0.999503i	$-0.489960\pi$
$673$	1.63617	0.0630697	0.0315348	+	0.999503i	$0.489960\pi$
$674$	0	0
$675$	−7.01029	−0.269826
$676$	0	0
$677$	−28.9765	−1.11366	−0.556828	−	0.830628i	$-0.687982\pi$
$677$	−28.9765	−1.11366	−0.556828	+	0.830628i	$0.687982\pi$
$678$	0	0
$679$	−30.8003	−1.18201
$680$	0	0
$681$	17.7003	0.678278
$682$	0	0
$683$	−9.16153	−0.350556	−0.175278	−	0.984519i	$-0.556082\pi$
$683$	−9.16153	−0.350556	−0.175278	+	0.984519i	$0.556082\pi$
$684$	0	0
$685$	−9.18705	−0.351019
$686$	0	0
$687$	−3.68120	−0.140447
$688$	0	0
$689$	1.80297	0.0686876
$690$	0	0
$691$	−31.6389	−1.20360	−0.601800	−	0.798647i	$-0.705550\pi$
$691$	−31.6389	−1.20360	−0.601800	+	0.798647i	$0.705550\pi$
$692$	0	0
$693$	4.09828	0.155681
$694$	0	0
$695$	−1.51080	−0.0573081
$696$	0	0
$697$	6.01069	0.227671
$698$	0	0
$699$	4.82773	0.182601
$700$	0	0
$701$	−26.1711	−0.988467	−0.494233	−	0.869329i	$-0.664551\pi$
$701$	−26.1711	−0.988467	−0.494233	+	0.869329i	$0.664551\pi$
$702$	0	0
$703$	3.50649	0.132250
$704$	0	0
$705$	−20.5306	−0.773227
$706$	0	0
$707$	−16.5482	−0.622360
$708$	0	0
$709$	−6.75988	−0.253872	−0.126936	−	0.991911i	$-0.540514\pi$
$709$	−6.75988	−0.253872	−0.126936	+	0.991911i	$0.540514\pi$
$710$	0	0
$711$	−17.2143	−0.645588
$712$	0	0
$713$	−22.7850	−0.853305
$714$	0	0
$715$	0.764180	0.0285787
$716$	0	0
$717$	−75.9343	−2.83582
$718$	0	0
$719$	23.5200	0.877149	0.438575	−	0.898695i	$-0.355484\pi$
$719$	23.5200	0.877149	0.438575	+	0.898695i	$0.355484\pi$
$720$	0	0
$721$	60.8218	2.26512
$722$	0	0
$723$	66.3503	2.46759
$724$	0	0
$725$	2.05280	0.0762391
$726$	0	0
$727$	22.8424	0.847179	0.423590	−	0.905854i	$-0.360770\pi$
$727$	22.8424	0.847179	0.423590	+	0.905854i	$0.360770\pi$
$728$	0	0
$729$	−42.9553	−1.59094
$730$	0	0
$731$	−37.6909	−1.39405
$732$	0	0
$733$	−4.45110	−0.164405	−0.0822026	−	0.996616i	$-0.526195\pi$
$733$	−4.45110	−0.164405	−0.0822026	+	0.996616i	$0.526195\pi$
$734$	0	0
$735$	−29.6526	−1.09375
$736$	0	0
$737$	−3.59075	−0.132267
$738$	0	0
$739$	23.0316	0.847233	0.423616	−	0.905842i	$-0.360760\pi$
$739$	23.0316	0.847233	0.423616	+	0.905842i	$0.360760\pi$
$740$	0	0
$741$	17.2313	0.633007
$742$	0	0
$743$	23.2176	0.851772	0.425886	−	0.904777i	$-0.359963\pi$
$743$	23.2176	0.851772	0.425886	+	0.904777i	$0.359963\pi$
$744$	0	0
$745$	32.7565	1.20010
$746$	0	0
$747$	71.4647	2.61476
$748$	0	0
$749$	−2.32199	−0.0848438
$750$	0	0
$751$	41.5906	1.51766	0.758832	−	0.651287i	$-0.225770\pi$
$751$	41.5906	1.51766	0.758832	+	0.651287i	$0.225770\pi$
$752$	0	0
$753$	−73.9667	−2.69550
$754$	0	0
$755$	−30.6392	−1.11507
$756$	0	0
$757$	−28.2033	−1.02507	−0.512534	−	0.858667i	$-0.671293\pi$
$757$	−28.2033	−1.02507	−0.512534	+	0.858667i	$0.671293\pi$
$758$	0	0
$759$	4.17305	0.151472
$760$	0	0
$761$	−0.699191	−0.0253457	−0.0126728	−	0.999920i	$-0.504034\pi$
$761$	−0.699191	−0.0253457	−0.0126728	+	0.999920i	$0.504034\pi$
$762$	0	0
$763$	32.8837	1.19047
$764$	0	0
$765$	40.7057	1.47172
$766$	0	0
$767$	3.83166	0.138353
$768$	0	0
$769$	−42.4592	−1.53112	−0.765560	−	0.643365i	$-0.777538\pi$
$769$	−42.4592	−1.53112	−0.765560	+	0.643365i	$0.777538\pi$
$770$	0	0
$771$	−55.0334	−1.98198
$772$	0	0
$773$	27.9014	1.00354	0.501772	−	0.865000i	$-0.332682\pi$
$773$	27.9014	1.00354	0.501772	+	0.865000i	$0.332682\pi$
$774$	0	0
$775$	−8.69419	−0.312305
$776$	0	0
$777$	4.93606	0.177080
$778$	0	0
$779$	10.6955	0.383205
$780$	0	0
$781$	−0.202087	−0.00723124
$782$	0	0
$783$	−3.41499	−0.122042
$784$	0	0
$785$	−49.0124	−1.74933
$786$	0	0
$787$	−7.02916	−0.250563	−0.125281	−	0.992121i	$-0.539983\pi$
$787$	−7.02916	−0.250563	−0.125281	+	0.992121i	$0.539983\pi$
$788$	0	0
$789$	39.9029	1.42058
$790$	0	0
$791$	6.95040	0.247128
$792$	0	0
$793$	14.5972	0.518363
$794$	0	0
$795$	12.9075	0.457780
$796$	0	0
$797$	0.00313844	0.000111169 0	5.55846e−5	−	1.00000i	$-0.499982\pi$
$797$	0.00313844	0.000111169 0	5.55846e−5		1.00000i	$0.499982\pi$
$798$	0	0
$799$	10.3019	0.364455
$800$	0	0
$801$	1.95558	0.0690970
$802$	0	0
$803$	−3.06225	−0.108065
$804$	0	0
$805$	−47.6902	−1.68086
$806$	0	0
$807$	−45.2762	−1.59380
$808$	0	0
$809$	−46.0975	−1.62070	−0.810351	−	0.585945i	$-0.800723\pi$
$809$	−46.0975	−1.62070	−0.810351	+	0.585945i	$0.800723\pi$
$810$	0	0
$811$	13.6631	0.479775	0.239887	−	0.970801i	$-0.422889\pi$
$811$	13.6631	0.479775	0.239887	+	0.970801i	$0.422889\pi$
$812$	0	0
$813$	49.0526	1.72035
$814$	0	0
$815$	11.4053	0.399512
$816$	0	0
$817$	−67.0675	−2.34640
$818$	0	0
$819$	14.2425	0.497673
$820$	0	0
$821$	−5.15570	−0.179935	−0.0899676	−	0.995945i	$-0.528676\pi$
$821$	−5.15570	−0.179935	−0.0899676	+	0.995945i	$0.528676\pi$
$822$	0	0
$823$	−25.4266	−0.886316	−0.443158	−	0.896444i	$-0.646142\pi$
$823$	−25.4266	−0.886316	−0.443158	+	0.896444i	$0.646142\pi$
$824$	0	0
$825$	1.59233	0.0554379
$826$	0	0
$827$	−24.2701	−0.843953	−0.421977	−	0.906607i	$-0.638664\pi$
$827$	−24.2701	−0.843953	−0.421977	+	0.906607i	$0.638664\pi$
$828$	0	0
$829$	−8.45854	−0.293777	−0.146889	−	0.989153i	$-0.546926\pi$
$829$	−8.45854	−0.293777	−0.146889	+	0.989153i	$0.546926\pi$
$830$	0	0
$831$	56.3214	1.95377
$832$	0	0
$833$	14.8792	0.515532
$834$	0	0
$835$	31.5023	1.09018
$836$	0	0
$837$	14.4634	0.499930
$838$	0	0
$839$	−3.25891	−0.112510	−0.0562550	−	0.998416i	$-0.517916\pi$
$839$	−3.25891	−0.112510	−0.0562550	+	0.998416i	$0.517916\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	−3.36238	−0.115806
$844$	0	0
$845$	2.65571	0.0913592
$846$	0	0
$847$	36.4412	1.25213
$848$	0	0
$849$	5.58628	0.191721
$850$	0	0
$851$	2.95116	0.101165
$852$	0	0
$853$	−39.8652	−1.36496	−0.682479	−	0.730905i	$-0.739098\pi$
$853$	−39.8652	−1.36496	−0.682479	+	0.730905i	$0.739098\pi$
$854$	0	0
$855$	72.4321	2.47713
$856$	0	0
$857$	−1.80737	−0.0617384	−0.0308692	−	0.999523i	$-0.509828\pi$
$857$	−1.80737	−0.0617384	−0.0308692	+	0.999523i	$0.509828\pi$
$858$	0	0
$859$	36.2791	1.23783	0.618914	−	0.785459i	$-0.287573\pi$
$859$	36.2791	1.23783	0.618914	+	0.785459i	$0.287573\pi$
$860$	0	0
$861$	15.0560	0.513106
$862$	0	0
$863$	17.8281	0.606874	0.303437	−	0.952851i	$-0.401866\pi$
$863$	17.8281	0.606874	0.303437	+	0.952851i	$0.401866\pi$
$864$	0	0
$865$	25.0837	0.852870
$866$	0	0
$867$	11.0405	0.374954
$868$	0	0
$869$	1.16091	0.0393813
$870$	0	0
$871$	−12.4787	−0.422826
$872$	0	0
$873$	39.3712	1.33251
$874$	0	0
$875$	26.1259	0.883218
$876$	0	0
$877$	0.597371	0.0201718	0.0100859	−	0.999949i	$-0.496790\pi$
$877$	0.597371	0.0201718	0.0100859	+	0.999949i	$0.496790\pi$
$878$	0	0
$879$	−47.3571	−1.59732
$880$	0	0
$881$	34.1314	1.14992	0.574959	−	0.818183i	$-0.305018\pi$
$881$	34.1314	1.14992	0.574959	+	0.818183i	$0.305018\pi$
$882$	0	0
$883$	14.5305	0.488989	0.244495	−	0.969651i	$-0.421378\pi$
$883$	14.5305	0.488989	0.244495	+	0.969651i	$0.421378\pi$
$884$	0	0
$885$	27.4309	0.922079
$886$	0	0
$887$	−27.3366	−0.917875	−0.458937	−	0.888469i	$-0.651770\pi$
$887$	−27.3366	−0.917875	−0.458937	+	0.888469i	$0.651770\pi$
$888$	0	0
$889$	−62.4040	−2.09296
$890$	0	0
$891$	1.03437	0.0346526
$892$	0	0
$893$	18.3313	0.613434
$894$	0	0
$895$	−3.81541	−0.127535
$896$	0	0
$897$	14.5024	0.484220
$898$	0	0
$899$	−4.23528	−0.141255
$900$	0	0
$901$	−6.47675	−0.215772
$902$	0	0
$903$	−94.4106	−3.14179
$904$	0	0
$905$	42.4742	1.41189
$906$	0	0
$907$	17.7259	0.588579	0.294289	−	0.955716i	$-0.404917\pi$
$907$	17.7259	0.588579	0.294289	+	0.955716i	$0.404917\pi$
$908$	0	0
$909$	21.1531	0.701605
$910$	0	0
$911$	10.1924	0.337689	0.168844	−	0.985643i	$-0.445996\pi$
$911$	10.1924	0.337689	0.168844	+	0.985643i	$0.445996\pi$
$912$	0	0
$913$	−4.81949	−0.159502
$914$	0	0
$915$	104.502	3.45472
$916$	0	0
$917$	−45.4874	−1.50213
$918$	0	0
$919$	−40.4533	−1.33443	−0.667216	−	0.744865i	$-0.732514\pi$
$919$	−40.4533	−1.33443	−0.667216	+	0.744865i	$0.732514\pi$
$920$	0	0
$921$	21.1334	0.696370
$922$	0	0
$923$	−0.702301	−0.0231165
$924$	0	0
$925$	1.12609	0.0370256
$926$	0	0
$927$	−77.7469	−2.55354
$928$	0	0
$929$	6.31413	0.207160	0.103580	−	0.994621i	$-0.466970\pi$
$929$	6.31413	0.207160	0.103580	+	0.994621i	$0.466970\pi$
$930$	0	0
$931$	26.4761	0.867720
$932$	0	0
$933$	−44.0351	−1.44164
$934$	0	0
$935$	−2.74514	−0.0897758
$936$	0	0
$937$	9.40575	0.307273	0.153636	−	0.988127i	$-0.450902\pi$
$937$	9.40575	0.307273	0.153636	+	0.988127i	$0.450902\pi$
$938$	0	0
$939$	17.3657	0.566709
$940$	0	0
$941$	4.39812	0.143375	0.0716873	−	0.997427i	$-0.477162\pi$
$941$	4.39812	0.143375	0.0716873	+	0.997427i	$0.477162\pi$
$942$	0	0
$943$	9.00164	0.293134
$944$	0	0
$945$	30.2727	0.984772
$946$	0	0
$947$	−30.0000	−0.974870	−0.487435	−	0.873159i	$-0.662067\pi$
$947$	−30.0000	−0.974870	−0.487435	+	0.873159i	$0.662067\pi$
$948$	0	0
$949$	−10.6421	−0.345456
$950$	0	0
$951$	−54.7823	−1.77644
$952$	0	0
$953$	52.3988	1.69736	0.848682	−	0.528903i	$-0.177397\pi$
$953$	52.3988	1.69736	0.848682	+	0.528903i	$0.177397\pi$
$954$	0	0
$955$	−42.5801	−1.37786
$956$	0	0
$957$	0.775688	0.0250744
$958$	0	0
$959$	11.5472	0.372878
$960$	0	0
$961$	−13.0624	−0.421367
$962$	0	0
$963$	2.96814	0.0956471
$964$	0	0
$965$	23.5600	0.758423
$966$	0	0
$967$	43.5005	1.39888	0.699441	−	0.714690i	$-0.253432\pi$
$967$	43.5005	1.39888	0.699441	+	0.714690i	$0.253432\pi$
$968$	0	0
$969$	−61.8995	−1.98850
$970$	0	0
$971$	44.2250	1.41925	0.709624	−	0.704580i	$-0.248865\pi$
$971$	44.2250	1.41925	0.709624	+	0.704580i	$0.248865\pi$
$972$	0	0
$973$	1.89893	0.0608769
$974$	0	0
$975$	5.53374	0.177222
$976$	0	0
$977$	−0.199639	−0.00638701	−0.00319351	−	0.999995i	$-0.501017\pi$
$977$	−0.199639	−0.00638701	−0.00319351	+	0.999995i	$0.501017\pi$
$978$	0	0
$979$	−0.131882	−0.00421497
$980$	0	0
$981$	−42.0344	−1.34205
$982$	0	0
$983$	−8.19282	−0.261310	−0.130655	−	0.991428i	$-0.541708\pi$
$983$	−8.19282	−0.261310	−0.130655	+	0.991428i	$0.541708\pi$
$984$	0	0
$985$	56.5587	1.80211
$986$	0	0
$987$	25.8049	0.821379
$988$	0	0
$989$	−56.4461	−1.79488
$990$	0	0
$991$	12.6299	0.401201	0.200600	−	0.979673i	$-0.435711\pi$
$991$	12.6299	0.401201	0.200600	+	0.979673i	$0.435711\pi$
$992$	0	0
$993$	−29.2514	−0.928264
$994$	0	0
$995$	−46.0725	−1.46060
$996$	0	0
$997$	60.3380	1.91092	0.955462	−	0.295113i	$-0.0953574\pi$
$997$	60.3380	1.91092	0.955462	+	0.295113i	$0.0953574\pi$
$998$	0	0
$999$	−1.87334	−0.0592698

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.be.1.2		13
4.3	odd	2		3016.2.a.k.1.12	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.k.1.12	✓	13	4.3	odd	2
6032.2.a.be.1.2		13	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69571 q^{3} +2.65571 q^{5} -3.33796 q^{7} +4.26683 q^{9} +O(q^{10})\)	\(q-2.69571 q^{3} +2.65571 q^{5} -3.33796 q^{7} +4.26683 q^{9} -0.287750 q^{11} -1.00000 q^{13} -7.15901 q^{15} +3.59227 q^{17} +6.39213 q^{19} +8.99816 q^{21} +5.37981 q^{23} +2.05280 q^{25} -3.41499 q^{27} +1.00000 q^{29} -4.23528 q^{31} +0.775688 q^{33} -8.86466 q^{35} +0.548563 q^{37} +2.69571 q^{39} +1.67323 q^{41} -10.4922 q^{43} +11.3315 q^{45} +2.86780 q^{47} +4.14199 q^{49} -9.68371 q^{51} -1.80297 q^{53} -0.764180 q^{55} -17.2313 q^{57} -3.83166 q^{59} -14.5972 q^{61} -14.2425 q^{63} -2.65571 q^{65} +12.4787 q^{67} -14.5024 q^{69} +0.702301 q^{71} +10.6421 q^{73} -5.53374 q^{75} +0.960498 q^{77} -4.03446 q^{79} -3.59468 q^{81} +16.7489 q^{83} +9.54004 q^{85} -2.69571 q^{87} +0.458322 q^{89} +3.33796 q^{91} +11.4171 q^{93} +16.9756 q^{95} +9.22728 q^{97} -1.22778 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9}+O(q^{10}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9	\( 13 q + 4 q^{3} + 5 q^{5} - 6 q^{7} + 21 q^{9} - 6 q^{11} - 13 q^{13} - 6 q^{15} + 4 q^{17} + 3 q^{19} + 5 q^{21} - 13 q^{23} + 24 q^{25} + 16 q^{27} + 13 q^{29} - 21 q^{31} + 21 q^{33} - 8 q^{35} + 23 q^{37} - 4 q^{39} + 4 q^{41} + 24 q^{43} + 9 q^{45} - 7 q^{47} + 41 q^{49} + q^{51} + 17 q^{53} + 8 q^{55} + 24 q^{57} - 11 q^{59} + 26 q^{61} - 17 q^{63} - 5 q^{65} + 35 q^{67} + 30 q^{69} - 30 q^{71} + 17 q^{73} + 43 q^{75} + 34 q^{77} - 3 q^{79} + 37 q^{81} + 18 q^{83} + 9 q^{85} + 4 q^{87} + 38 q^{89} + 6 q^{91} + 25 q^{93} + 9 q^{95} + 7 q^{97} + 40 q^{99}+O(q^{100}) \) 13 * q + 4 * q^3 + 5 * q^5 - 6 * q^7 + 21 * q^9 - 6 * q^11 - 13 * q^13 - 6 * q^15 + 4 * q^17 + 3 * q^19 + 5 * q^21 - 13 * q^23 + 24 * q^25 + 16 * q^27 + 13 * q^29 - 21 * q^31 + 21 * q^33 - 8 * q^35 + 23 * q^37 - 4 * q^39 + 4 * q^41 + 24 * q^43 + 9 * q^45 - 7 * q^47 + 41 * q^49 + q^51 + 17 * q^53 + 8 * q^55 + 24 * q^57 - 11 * q^59 + 26 * q^61 - 17 * q^63 - 5 * q^65 + 35 * q^67 + 30 * q^69 - 30 * q^71 + 17 * q^73 + 43 * q^75 + 34 * q^77 - 3 * q^79 + 37 * q^81 + 18 * q^83 + 9 * q^85 + 4 * q^87 + 38 * q^89 + 6 * q^91 + 25 * q^93 + 9 * q^95 + 7 * q^97 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

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