LMFDB - Embedded newform 4028.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4028,2,Mod(1,4028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4028, 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("4028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4028.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:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 4 x^{18} - 30 x^{17} + 132 x^{16} + 332 x^{15} - 1714 x^{14} - 1598 x^{13} + 11179 x^{12} + \cdots - 49 $ x^19 - 4x^18 - 30x^17 + 132x^16 + 332x^15 - 1714x^14 - 1598x^13 + 11179x^12 + 2544x^11 - 38897x^10 + 3416x^9 + 71354x^8 - 10941x^7 - 64854x^6 + 219x^5 + 26515x^4 + 5796x^3 - 2375x^2 - 826x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.40459$ of defining polynomial
Character	$\chi$	$=$	4028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.40459 q^{3} -0.489610 q^{5} +4.26722 q^{7} +2.78204 q^{9} +O(q^{10})$	$q-2.40459 q^{3} -0.489610 q^{5} +4.26722 q^{7} +2.78204 q^{9} -0.593113 q^{11} -1.15631 q^{13} +1.17731 q^{15} -0.567069 q^{17} -1.00000 q^{19} -10.2609 q^{21} -0.283453 q^{23} -4.76028 q^{25} +0.524106 q^{27} +7.84950 q^{29} -1.45614 q^{31} +1.42619 q^{33} -2.08928 q^{35} -10.2044 q^{37} +2.78046 q^{39} -1.80512 q^{41} -10.9526 q^{43} -1.36212 q^{45} -0.268056 q^{47} +11.2092 q^{49} +1.36357 q^{51} +1.00000 q^{53} +0.290395 q^{55} +2.40459 q^{57} -7.12660 q^{59} -3.60308 q^{61} +11.8716 q^{63} +0.566143 q^{65} +13.0044 q^{67} +0.681587 q^{69} +5.30469 q^{71} +6.82454 q^{73} +11.4465 q^{75} -2.53095 q^{77} +6.03777 q^{79} -9.60638 q^{81} -18.1576 q^{83} +0.277643 q^{85} -18.8748 q^{87} +11.2560 q^{89} -4.93425 q^{91} +3.50142 q^{93} +0.489610 q^{95} +16.7957 q^{97} -1.65006 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	$ 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) $ 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + 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
$3$	−2.40459	−1.38829	−0.694144	−	0.719836i	$-0.744217\pi$
$3$	−2.40459	−1.38829	−0.694144	+	0.719836i	$0.744217\pi$
$4$	0	0
$5$	−0.489610	−0.218960	−0.109480	−	0.993989i	$-0.534919\pi$
$5$	−0.489610	−0.218960	−0.109480	+	0.993989i	$0.534919\pi$
$6$	0	0
$7$	4.26722	1.61286	0.806429	−	0.591331i	$-0.201397\pi$
$7$	4.26722	1.61286	0.806429	+	0.591331i	$0.201397\pi$
$8$	0	0
$9$	2.78204	0.927346
$10$	0	0
$11$	−0.593113	−0.178830	−0.0894152	−	0.995994i	$-0.528500\pi$
$11$	−0.593113	−0.178830	−0.0894152	+	0.995994i	$0.528500\pi$
$12$	0	0
$13$	−1.15631	−0.320704	−0.160352	−	0.987060i	$-0.551263\pi$
$13$	−1.15631	−0.320704	−0.160352	+	0.987060i	$0.551263\pi$
$14$	0	0
$15$	1.17731	0.303980
$16$	0	0
$17$	−0.567069	−0.137534	−0.0687672	−	0.997633i	$-0.521907\pi$
$17$	−0.567069	−0.137534	−0.0687672	+	0.997633i	$0.521907\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−10.2609	−2.23911
$22$	0	0
$23$	−0.283453	−0.0591040	−0.0295520	−	0.999563i	$-0.509408\pi$
$23$	−0.283453	−0.0591040	−0.0295520	+	0.999563i	$0.509408\pi$
$24$	0	0
$25$	−4.76028	−0.952056
$26$	0	0
$27$	0.524106	0.100864
$28$	0	0
$29$	7.84950	1.45762	0.728808	−	0.684718i	$-0.240075\pi$
$29$	7.84950	1.45762	0.728808	+	0.684718i	$0.240075\pi$
$30$	0	0
$31$	−1.45614	−0.261531	−0.130765	−	0.991413i	$-0.541743\pi$
$31$	−1.45614	−0.261531	−0.130765	+	0.991413i	$0.541743\pi$
$32$	0	0
$33$	1.42619	0.248268
$34$	0	0
$35$	−2.08928	−0.353152
$36$	0	0
$37$	−10.2044	−1.67759	−0.838796	−	0.544446i	$-0.816740\pi$
$37$	−10.2044	−1.67759	−0.838796	+	0.544446i	$0.816740\pi$
$38$	0	0
$39$	2.78046	0.445229
$40$	0	0
$41$	−1.80512	−0.281912	−0.140956	−	0.990016i	$-0.545018\pi$
$41$	−1.80512	−0.281912	−0.140956	+	0.990016i	$0.545018\pi$
$42$	0	0
$43$	−10.9526	−1.67025	−0.835126	−	0.550059i	$-0.814605\pi$
$43$	−10.9526	−1.67025	−0.835126	+	0.550059i	$0.814605\pi$
$44$	0	0
$45$	−1.36212	−0.203052
$46$	0	0
$47$	−0.268056	−0.0391000	−0.0195500	−	0.999809i	$-0.506223\pi$
$47$	−0.268056	−0.0391000	−0.0195500	+	0.999809i	$0.506223\pi$
$48$	0	0
$49$	11.2092	1.60131
$50$	0	0
$51$	1.36357	0.190937
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	0.290395	0.0391568
$56$	0	0
$57$	2.40459	0.318495
$58$	0	0
$59$	−7.12660	−0.927804	−0.463902	−	0.885887i	$-0.653551\pi$
$59$	−7.12660	−0.927804	−0.463902	+	0.885887i	$0.653551\pi$
$60$	0	0
$61$	−3.60308	−0.461327	−0.230664	−	0.973034i	$-0.574090\pi$
$61$	−3.60308	−0.461327	−0.230664	+	0.973034i	$0.574090\pi$
$62$	0	0
$63$	11.8716	1.49568
$64$	0	0
$65$	0.566143	0.0702214
$66$	0	0
$67$	13.0044	1.58874	0.794369	−	0.607436i	$-0.207802\pi$
$67$	13.0044	1.58874	0.794369	+	0.607436i	$0.207802\pi$
$68$	0	0
$69$	0.681587	0.0820534
$70$	0	0
$71$	5.30469	0.629551	0.314776	−	0.949166i	$-0.398071\pi$
$71$	5.30469	0.629551	0.314776	+	0.949166i	$0.398071\pi$
$72$	0	0
$73$	6.82454	0.798752	0.399376	−	0.916787i	$-0.369227\pi$
$73$	6.82454	0.798752	0.399376	+	0.916787i	$0.369227\pi$
$74$	0	0
$75$	11.4465	1.32173
$76$	0	0
$77$	−2.53095	−0.288428
$78$	0	0
$79$	6.03777	0.679302	0.339651	−	0.940552i	$-0.389691\pi$
$79$	6.03777	0.679302	0.339651	+	0.940552i	$0.389691\pi$
$80$	0	0
$81$	−9.60638	−1.06738
$82$	0	0
$83$	−18.1576	−1.99305	−0.996526	−	0.0832803i	$-0.973460\pi$
$83$	−18.1576	−1.99305	−0.996526	+	0.0832803i	$0.973460\pi$
$84$	0	0
$85$	0.277643	0.0301146
$86$	0	0
$87$	−18.8748	−2.02359
$88$	0	0
$89$	11.2560	1.19314	0.596568	−	0.802563i	$-0.296531\pi$
$89$	11.2560	1.19314	0.596568	+	0.802563i	$0.296531\pi$
$90$	0	0
$91$	−4.93425	−0.517250
$92$	0	0
$93$	3.50142	0.363080
$94$	0	0
$95$	0.489610	0.0502330
$96$	0	0
$97$	16.7957	1.70534	0.852671	−	0.522448i	$-0.174981\pi$
$97$	16.7957	1.70534	0.852671	+	0.522448i	$0.174981\pi$
$98$	0	0
$99$	−1.65006	−0.165838
$100$	0	0
$101$	−6.59077	−0.655807	−0.327903	−	0.944711i	$-0.606342\pi$
$101$	−6.59077	−0.655807	−0.327903	+	0.944711i	$0.606342\pi$
$102$	0	0
$103$	3.34526	0.329618	0.164809	−	0.986325i	$-0.447299\pi$
$103$	3.34526	0.329618	0.164809	+	0.986325i	$0.447299\pi$
$104$	0	0
$105$	5.02385	0.490277
$106$	0	0
$107$	−9.55246	−0.923471	−0.461736	−	0.887018i	$-0.652773\pi$
$107$	−9.55246	−0.923471	−0.461736	+	0.887018i	$0.652773\pi$
$108$	0	0
$109$	0.195040	0.0186815	0.00934074	−	0.999956i	$-0.497027\pi$
$109$	0.195040	0.0186815	0.00934074	+	0.999956i	$0.497027\pi$
$110$	0	0
$111$	24.5373	2.32898
$112$	0	0
$113$	−13.8267	−1.30070	−0.650352	−	0.759633i	$-0.725379\pi$
$113$	−13.8267	−1.30070	−0.650352	+	0.759633i	$0.725379\pi$
$114$	0	0
$115$	0.138781	0.0129414
$116$	0	0
$117$	−3.21691	−0.297403
$118$	0	0
$119$	−2.41981	−0.221823
$120$	0	0
$121$	−10.6482	−0.968020
$122$	0	0
$123$	4.34057	0.391376
$124$	0	0
$125$	4.77874	0.427423
$126$	0	0
$127$	13.5583	1.20311	0.601553	−	0.798833i	$-0.294549\pi$
$127$	13.5583	1.20311	0.601553	+	0.798833i	$0.294549\pi$
$128$	0	0
$129$	26.3364	2.31879
$130$	0	0
$131$	11.3534	0.991953	0.495977	−	0.868336i	$-0.334810\pi$
$131$	11.3534	0.991953	0.495977	+	0.868336i	$0.334810\pi$
$132$	0	0
$133$	−4.26722	−0.370015
$134$	0	0
$135$	−0.256608	−0.0220853
$136$	0	0
$137$	10.2091	0.872223	0.436112	−	0.899893i	$-0.356355\pi$
$137$	10.2091	0.872223	0.436112	+	0.899893i	$0.356355\pi$
$138$	0	0
$139$	0.848744	0.0719895	0.0359947	−	0.999352i	$-0.488540\pi$
$139$	0.848744	0.0719895	0.0359947	+	0.999352i	$0.488540\pi$
$140$	0	0
$141$	0.644564	0.0542821
$142$	0	0
$143$	0.685825	0.0573516
$144$	0	0
$145$	−3.84320	−0.319160
$146$	0	0
$147$	−26.9534	−2.22308
$148$	0	0
$149$	19.7506	1.61803	0.809015	−	0.587788i	$-0.200001\pi$
$149$	19.7506	1.61803	0.809015	+	0.587788i	$0.200001\pi$
$150$	0	0
$151$	−17.9990	−1.46474	−0.732368	−	0.680909i	$-0.761585\pi$
$151$	−17.9990	−1.46474	−0.732368	+	0.680909i	$0.761585\pi$
$152$	0	0
$153$	−1.57761	−0.127542
$154$	0	0
$155$	0.712942	0.0572649
$156$	0	0
$157$	−5.29269	−0.422402	−0.211201	−	0.977443i	$-0.567738\pi$
$157$	−5.29269	−0.422402	−0.211201	+	0.977443i	$0.567738\pi$
$158$	0	0
$159$	−2.40459	−0.190696
$160$	0	0
$161$	−1.20956	−0.0953263
$162$	0	0
$163$	−22.9884	−1.80059	−0.900296	−	0.435278i	$-0.856650\pi$
$163$	−22.9884	−1.80059	−0.900296	+	0.435278i	$0.856650\pi$
$164$	0	0
$165$	−0.698279	−0.0543609
$166$	0	0
$167$	−18.0882	−1.39971	−0.699855	−	0.714285i	$-0.746752\pi$
$167$	−18.0882	−1.39971	−0.699855	+	0.714285i	$0.746752\pi$
$168$	0	0
$169$	−11.6629	−0.897149
$170$	0	0
$171$	−2.78204	−0.212748
$172$	0	0
$173$	10.3205	0.784651	0.392325	−	0.919827i	$-0.371671\pi$
$173$	10.3205	0.784651	0.392325	+	0.919827i	$0.371671\pi$
$174$	0	0
$175$	−20.3132	−1.53553
$176$	0	0
$177$	17.1365	1.28806
$178$	0	0
$179$	−11.7825	−0.880666	−0.440333	−	0.897834i	$-0.645140\pi$
$179$	−11.7825	−0.880666	−0.440333	+	0.897834i	$0.645140\pi$
$180$	0	0
$181$	−10.7511	−0.799125	−0.399562	−	0.916706i	$-0.630838\pi$
$181$	−10.7511	−0.799125	−0.399562	+	0.916706i	$0.630838\pi$
$182$	0	0
$183$	8.66392	0.640456
$184$	0	0
$185$	4.99618	0.367326
$186$	0	0
$187$	0.336336	0.0245953
$188$	0	0
$189$	2.23648	0.162680
$190$	0	0
$191$	−17.9185	−1.29654	−0.648268	−	0.761412i	$-0.724506\pi$
$191$	−17.9185	−1.29654	−0.648268	+	0.761412i	$0.724506\pi$
$192$	0	0
$193$	−14.7314	−1.06039	−0.530195	−	0.847876i	$-0.677881\pi$
$193$	−14.7314	−1.06039	−0.530195	+	0.847876i	$0.677881\pi$
$194$	0	0
$195$	−1.36134	−0.0974876
$196$	0	0
$197$	8.16837	0.581973	0.290986	−	0.956727i	$-0.406017\pi$
$197$	8.16837	0.581973	0.290986	+	0.956727i	$0.406017\pi$
$198$	0	0
$199$	−26.2015	−1.85738	−0.928688	−	0.370861i	$-0.879062\pi$
$199$	−26.2015	−1.85738	−0.928688	+	0.370861i	$0.879062\pi$
$200$	0	0
$201$	−31.2702	−2.20563
$202$	0	0
$203$	33.4956	2.35093
$204$	0	0
$205$	0.883805	0.0617276
$206$	0	0
$207$	−0.788577	−0.0548099
$208$	0	0
$209$	0.593113	0.0410265
$210$	0	0
$211$	−24.1902	−1.66532	−0.832661	−	0.553782i	$-0.813184\pi$
$211$	−24.1902	−1.66532	−0.832661	+	0.553782i	$0.813184\pi$
$212$	0	0
$213$	−12.7556	−0.873999
$214$	0	0
$215$	5.36249	0.365719
$216$	0	0
$217$	−6.21368	−0.421812
$218$	0	0
$219$	−16.4102	−1.10890
$220$	0	0
$221$	0.655709	0.0441078
$222$	0	0
$223$	−3.32596	−0.222723	−0.111361	−	0.993780i	$-0.535521\pi$
$223$	−3.32596	−0.222723	−0.111361	+	0.993780i	$0.535521\pi$
$224$	0	0
$225$	−13.2433	−0.882886
$226$	0	0
$227$	−13.7421	−0.912096	−0.456048	−	0.889955i	$-0.650735\pi$
$227$	−13.7421	−0.912096	−0.456048	+	0.889955i	$0.650735\pi$
$228$	0	0
$229$	−28.4253	−1.87840	−0.939200	−	0.343371i	$-0.888431\pi$
$229$	−28.4253	−1.87840	−0.939200	+	0.343371i	$0.888431\pi$
$230$	0	0
$231$	6.08588	0.400422
$232$	0	0
$233$	−25.2345	−1.65316	−0.826582	−	0.562816i	$-0.809718\pi$
$233$	−25.2345	−1.65316	−0.826582	+	0.562816i	$0.809718\pi$
$234$	0	0
$235$	0.131243	0.00856136
$236$	0	0
$237$	−14.5183	−0.943067
$238$	0	0
$239$	19.4760	1.25980	0.629900	−	0.776676i	$-0.283096\pi$
$239$	19.4760	1.25980	0.629900	+	0.776676i	$0.283096\pi$
$240$	0	0
$241$	6.36949	0.410295	0.205147	−	0.978731i	$-0.434233\pi$
$241$	6.36949	0.410295	0.205147	+	0.978731i	$0.434233\pi$
$242$	0	0
$243$	21.5270	1.38096
$244$	0	0
$245$	−5.48813	−0.350624
$246$	0	0
$247$	1.15631	0.0735745
$248$	0	0
$249$	43.6615	2.76693
$250$	0	0
$251$	−14.0762	−0.888479	−0.444240	−	0.895908i	$-0.646526\pi$
$251$	−14.0762	−0.888479	−0.444240	+	0.895908i	$0.646526\pi$
$252$	0	0
$253$	0.168120	0.0105696
$254$	0	0
$255$	−0.667616	−0.0418077
$256$	0	0
$257$	23.6236	1.47360	0.736801	−	0.676110i	$-0.236336\pi$
$257$	23.6236	1.47360	0.736801	+	0.676110i	$0.236336\pi$
$258$	0	0
$259$	−43.5444	−2.70572
$260$	0	0
$261$	21.8376	1.35171
$262$	0	0
$263$	−25.3321	−1.56204	−0.781022	−	0.624503i	$-0.785301\pi$
$263$	−25.3321	−1.56204	−0.781022	+	0.624503i	$0.785301\pi$
$264$	0	0
$265$	−0.489610	−0.0300765
$266$	0	0
$267$	−27.0661	−1.65642
$268$	0	0
$269$	10.0059	0.610072	0.305036	−	0.952341i	$-0.401331\pi$
$269$	10.0059	0.610072	0.305036	+	0.952341i	$0.401331\pi$
$270$	0	0
$271$	−8.45285	−0.513474	−0.256737	−	0.966481i	$-0.582647\pi$
$271$	−8.45285	−0.513474	−0.256737	+	0.966481i	$0.582647\pi$
$272$	0	0
$273$	11.8648	0.718092
$274$	0	0
$275$	2.82339	0.170257
$276$	0	0
$277$	13.6319	0.819063	0.409531	−	0.912296i	$-0.365692\pi$
$277$	13.6319	0.819063	0.409531	+	0.912296i	$0.365692\pi$
$278$	0	0
$279$	−4.05104	−0.242530
$280$	0	0
$281$	8.21933	0.490324	0.245162	−	0.969482i	$-0.421159\pi$
$281$	8.21933	0.490324	0.245162	+	0.969482i	$0.421159\pi$
$282$	0	0
$283$	−17.9266	−1.06562	−0.532812	−	0.846234i	$-0.678865\pi$
$283$	−17.9266	−1.06562	−0.532812	+	0.846234i	$0.678865\pi$
$284$	0	0
$285$	−1.17731	−0.0697379
$286$	0	0
$287$	−7.70284	−0.454684
$288$	0	0
$289$	−16.6784	−0.981084
$290$	0	0
$291$	−40.3867	−2.36751
$292$	0	0
$293$	25.3915	1.48339	0.741694	−	0.670739i	$-0.234023\pi$
$293$	25.3915	1.48339	0.741694	+	0.670739i	$0.234023\pi$
$294$	0	0
$295$	3.48926	0.203152
$296$	0	0
$297$	−0.310854	−0.0180376
$298$	0	0
$299$	0.327760	0.0189549
$300$	0	0
$301$	−46.7370	−2.69388
$302$	0	0
$303$	15.8481	0.910449
$304$	0	0
$305$	1.76411	0.101012
$306$	0	0
$307$	−6.58449	−0.375797	−0.187898	−	0.982188i	$-0.560168\pi$
$307$	−6.58449	−0.375797	−0.187898	+	0.982188i	$0.560168\pi$
$308$	0	0
$309$	−8.04397	−0.457605
$310$	0	0
$311$	3.50525	0.198765	0.0993823	−	0.995049i	$-0.468313\pi$
$311$	3.50525	0.198765	0.0993823	+	0.995049i	$0.468313\pi$
$312$	0	0
$313$	11.0456	0.624337	0.312168	−	0.950027i	$-0.398945\pi$
$313$	11.0456	0.624337	0.312168	+	0.950027i	$0.398945\pi$
$314$	0	0
$315$	−5.81245	−0.327494
$316$	0	0
$317$	−8.51919	−0.478486	−0.239243	−	0.970960i	$-0.576899\pi$
$317$	−8.51919	−0.478486	−0.239243	+	0.970960i	$0.576899\pi$
$318$	0	0
$319$	−4.65565	−0.260666
$320$	0	0
$321$	22.9697	1.28204
$322$	0	0
$323$	0.567069	0.0315525
$324$	0	0
$325$	5.50438	0.305328
$326$	0	0
$327$	−0.468992	−0.0259353
$328$	0	0
$329$	−1.14386	−0.0630628
$330$	0	0
$331$	15.8010	0.868503	0.434252	−	0.900792i	$-0.357013\pi$
$331$	15.8010	0.868503	0.434252	+	0.900792i	$0.357013\pi$
$332$	0	0
$333$	−28.3890	−1.55571
$334$	0	0
$335$	−6.36708	−0.347871
$336$	0	0
$337$	−5.91933	−0.322446	−0.161223	−	0.986918i	$-0.551544\pi$
$337$	−5.91933	−0.322446	−0.161223	+	0.986918i	$0.551544\pi$
$338$	0	0
$339$	33.2474	1.80575
$340$	0	0
$341$	0.863657	0.0467697
$342$	0	0
$343$	17.9615	0.969829
$344$	0	0
$345$	−0.333712	−0.0179665
$346$	0	0
$347$	−33.3477	−1.79020	−0.895100	−	0.445865i	$-0.852896\pi$
$347$	−33.3477	−1.79020	−0.895100	+	0.445865i	$0.852896\pi$
$348$	0	0
$349$	16.4860	0.882475	0.441238	−	0.897390i	$-0.354540\pi$
$349$	16.4860	0.882475	0.441238	+	0.897390i	$0.354540\pi$
$350$	0	0
$351$	−0.606031	−0.0323475
$352$	0	0
$353$	−26.5765	−1.41453	−0.707263	−	0.706950i	$-0.750071\pi$
$353$	−26.5765	−1.41453	−0.707263	+	0.706950i	$0.750071\pi$
$354$	0	0
$355$	−2.59723	−0.137847
$356$	0	0
$357$	5.81864	0.307955
$358$	0	0
$359$	26.3425	1.39030	0.695152	−	0.718863i	$-0.255337\pi$
$359$	26.3425	1.39030	0.695152	+	0.718863i	$0.255337\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	25.6046	1.34389
$364$	0	0
$365$	−3.34137	−0.174895
$366$	0	0
$367$	7.93086	0.413987	0.206994	−	0.978342i	$-0.433632\pi$
$367$	7.93086	0.413987	0.206994	+	0.978342i	$0.433632\pi$
$368$	0	0
$369$	−5.02191	−0.261430
$370$	0	0
$371$	4.26722	0.221543
$372$	0	0
$373$	−20.1490	−1.04328	−0.521638	−	0.853167i	$-0.674679\pi$
$373$	−20.1490	−1.04328	−0.521638	+	0.853167i	$0.674679\pi$
$374$	0	0
$375$	−11.4909	−0.593387
$376$	0	0
$377$	−9.07649	−0.467463
$378$	0	0
$379$	2.74775	0.141142	0.0705712	−	0.997507i	$-0.477518\pi$
$379$	2.74775	0.141142	0.0705712	+	0.997507i	$0.477518\pi$
$380$	0	0
$381$	−32.6022	−1.67026
$382$	0	0
$383$	18.5199	0.946321	0.473160	−	0.880976i	$-0.343113\pi$
$383$	18.5199	0.946321	0.473160	+	0.880976i	$0.343113\pi$
$384$	0	0
$385$	1.23918	0.0631543
$386$	0	0
$387$	−30.4705	−1.54890
$388$	0	0
$389$	16.0162	0.812052	0.406026	−	0.913861i	$-0.366914\pi$
$389$	16.0162	0.812052	0.406026	+	0.913861i	$0.366914\pi$
$390$	0	0
$391$	0.160737	0.00812883
$392$	0	0
$393$	−27.3003	−1.37712
$394$	0	0
$395$	−2.95615	−0.148740
$396$	0	0
$397$	−26.7822	−1.34416	−0.672080	−	0.740479i	$-0.734599\pi$
$397$	−26.7822	−1.34416	−0.672080	+	0.740479i	$0.734599\pi$
$398$	0	0
$399$	10.2609	0.513688
$400$	0	0
$401$	11.0341	0.551018	0.275509	−	0.961299i	$-0.411154\pi$
$401$	11.0341	0.551018	0.275509	+	0.961299i	$0.411154\pi$
$402$	0	0
$403$	1.68376	0.0838739
$404$	0	0
$405$	4.70338	0.233713
$406$	0	0
$407$	6.05236	0.300004
$408$	0	0
$409$	−5.49748	−0.271833	−0.135916	−	0.990720i	$-0.543398\pi$
$409$	−5.49748	−0.271833	−0.135916	+	0.990720i	$0.543398\pi$
$410$	0	0
$411$	−24.5487	−1.21090
$412$	0	0
$413$	−30.4108	−1.49642
$414$	0	0
$415$	8.89014	0.436400
$416$	0	0
$417$	−2.04088	−0.0999422
$418$	0	0
$419$	−18.5023	−0.903897	−0.451949	−	0.892044i	$-0.649271\pi$
$419$	−18.5023	−0.903897	−0.451949	+	0.892044i	$0.649271\pi$
$420$	0	0
$421$	−28.2312	−1.37591	−0.687953	−	0.725755i	$-0.741490\pi$
$421$	−28.2312	−1.37591	−0.687953	+	0.725755i	$0.741490\pi$
$422$	0	0
$423$	−0.745743	−0.0362593
$424$	0	0
$425$	2.69941	0.130940
$426$	0	0
$427$	−15.3751	−0.744055
$428$	0	0
$429$	−1.64913	−0.0796206
$430$	0	0
$431$	−2.69590	−0.129857	−0.0649285	−	0.997890i	$-0.520682\pi$
$431$	−2.69590	−0.129857	−0.0649285	+	0.997890i	$0.520682\pi$
$432$	0	0
$433$	19.3997	0.932291	0.466145	−	0.884708i	$-0.345642\pi$
$433$	19.3997	0.932291	0.466145	+	0.884708i	$0.345642\pi$
$434$	0	0
$435$	9.24130	0.443087
$436$	0	0
$437$	0.283453	0.0135594
$438$	0	0
$439$	−36.3458	−1.73469	−0.867344	−	0.497708i	$-0.834175\pi$
$439$	−36.3458	−1.73469	−0.867344	+	0.497708i	$0.834175\pi$
$440$	0	0
$441$	31.1844	1.48497
$442$	0	0
$443$	−19.9977	−0.950120	−0.475060	−	0.879953i	$-0.657574\pi$
$443$	−19.9977	−0.950120	−0.475060	+	0.879953i	$0.657574\pi$
$444$	0	0
$445$	−5.51106	−0.261249
$446$	0	0
$447$	−47.4920	−2.24629
$448$	0	0
$449$	−9.85329	−0.465006	−0.232503	−	0.972596i	$-0.574691\pi$
$449$	−9.85329	−0.465006	−0.232503	+	0.972596i	$0.574691\pi$
$450$	0	0
$451$	1.07064	0.0504145
$452$	0	0
$453$	43.2801	2.03348
$454$	0	0
$455$	2.41586	0.113257
$456$	0	0
$457$	24.0250	1.12384	0.561922	−	0.827190i	$-0.310062\pi$
$457$	24.0250	1.12384	0.561922	+	0.827190i	$0.310062\pi$
$458$	0	0
$459$	−0.297204	−0.0138723
$460$	0	0
$461$	26.0400	1.21280	0.606401	−	0.795159i	$-0.292613\pi$
$461$	26.0400	1.21280	0.606401	+	0.795159i	$0.292613\pi$
$462$	0	0
$463$	12.1893	0.566483	0.283241	−	0.959049i	$-0.408590\pi$
$463$	12.1893	0.566483	0.283241	+	0.959049i	$0.408590\pi$
$464$	0	0
$465$	−1.71433	−0.0795002
$466$	0	0
$467$	11.9943	0.555030	0.277515	−	0.960721i	$-0.410489\pi$
$467$	11.9943	0.555030	0.277515	+	0.960721i	$0.410489\pi$
$468$	0	0
$469$	55.4925	2.56241
$470$	0	0
$471$	12.7267	0.586416
$472$	0	0
$473$	6.49612	0.298692
$474$	0	0
$475$	4.76028	0.218417
$476$	0	0
$477$	2.78204	0.127381
$478$	0	0
$479$	37.4019	1.70894	0.854469	−	0.519503i	$-0.173883\pi$
$479$	37.4019	1.70894	0.854469	+	0.519503i	$0.173883\pi$
$480$	0	0
$481$	11.7995	0.538010
$482$	0	0
$483$	2.90848	0.132341
$484$	0	0
$485$	−8.22334	−0.373403
$486$	0	0
$487$	−5.58556	−0.253106	−0.126553	−	0.991960i	$-0.540391\pi$
$487$	−5.58556	−0.253106	−0.126553	+	0.991960i	$0.540391\pi$
$488$	0	0
$489$	55.2777	2.49974
$490$	0	0
$491$	−0.376459	−0.0169893	−0.00849467	−	0.999964i	$-0.502704\pi$
$491$	−0.376459	−0.0169893	−0.00849467	+	0.999964i	$0.502704\pi$
$492$	0	0
$493$	−4.45121	−0.200472
$494$	0	0
$495$	0.807889	0.0363119
$496$	0	0
$497$	22.6363	1.01538
$498$	0	0
$499$	4.55394	0.203862	0.101931	−	0.994791i	$-0.467498\pi$
$499$	4.55394	0.203862	0.101931	+	0.994791i	$0.467498\pi$
$500$	0	0
$501$	43.4948	1.94320
$502$	0	0
$503$	21.1855	0.944616	0.472308	−	0.881433i	$-0.343421\pi$
$503$	21.1855	0.944616	0.472308	+	0.881433i	$0.343421\pi$
$504$	0	0
$505$	3.22691	0.143596
$506$	0	0
$507$	28.0446	1.24550
$508$	0	0
$509$	23.8635	1.05773	0.528865	−	0.848706i	$-0.322618\pi$
$509$	23.8635	1.05773	0.528865	+	0.848706i	$0.322618\pi$
$510$	0	0
$511$	29.1218	1.28827
$512$	0	0
$513$	−0.524106	−0.0231398
$514$	0	0
$515$	−1.63787	−0.0721733
$516$	0	0
$517$	0.158988	0.00699227
$518$	0	0
$519$	−24.8165	−1.08932
$520$	0	0
$521$	32.2098	1.41114	0.705568	−	0.708642i	$-0.250692\pi$
$521$	32.2098	1.41114	0.705568	+	0.708642i	$0.250692\pi$
$522$	0	0
$523$	−2.73250	−0.119484	−0.0597420	−	0.998214i	$-0.519028\pi$
$523$	−2.73250	−0.119484	−0.0597420	+	0.998214i	$0.519028\pi$
$524$	0	0
$525$	48.8448	2.13176
$526$	0	0
$527$	0.825732	0.0359695
$528$	0	0
$529$	−22.9197	−0.996507
$530$	0	0
$531$	−19.8265	−0.860396
$532$	0	0
$533$	2.08728	0.0904103
$534$	0	0
$535$	4.67698	0.202204
$536$	0	0
$537$	28.3321	1.22262
$538$	0	0
$539$	−6.64831	−0.286363
$540$	0	0
$541$	−21.6113	−0.929143	−0.464572	−	0.885536i	$-0.653792\pi$
$541$	−21.6113	−0.929143	−0.464572	+	0.885536i	$0.653792\pi$
$542$	0	0
$543$	25.8520	1.10942
$544$	0	0
$545$	−0.0954939	−0.00409051
$546$	0	0
$547$	−14.0368	−0.600168	−0.300084	−	0.953913i	$-0.597015\pi$
$547$	−14.0368	−0.600168	−0.300084	+	0.953913i	$0.597015\pi$
$548$	0	0
$549$	−10.0239	−0.427810
$550$	0	0
$551$	−7.84950	−0.334400
$552$	0	0
$553$	25.7645	1.09562
$554$	0	0
$555$	−12.0137	−0.509955
$556$	0	0
$557$	17.4768	0.740517	0.370258	−	0.928929i	$-0.379269\pi$
$557$	17.4768	0.740517	0.370258	+	0.928929i	$0.379269\pi$
$558$	0	0
$559$	12.6646	0.535656
$560$	0	0
$561$	−0.808749	−0.0341454
$562$	0	0
$563$	5.49829	0.231725	0.115863	−	0.993265i	$-0.463037\pi$
$563$	5.49829	0.231725	0.115863	+	0.993265i	$0.463037\pi$
$564$	0	0
$565$	6.76968	0.284803
$566$	0	0
$567$	−40.9925	−1.72152
$568$	0	0
$569$	−0.659955	−0.0276668	−0.0138334	−	0.999904i	$-0.504403\pi$
$569$	−0.659955	−0.0276668	−0.0138334	+	0.999904i	$0.504403\pi$
$570$	0	0
$571$	40.4557	1.69302	0.846509	−	0.532374i	$-0.178700\pi$
$571$	40.4557	1.69302	0.846509	+	0.532374i	$0.178700\pi$
$572$	0	0
$573$	43.0866	1.79997
$574$	0	0
$575$	1.34931	0.0562703
$576$	0	0
$577$	−10.5897	−0.440855	−0.220428	−	0.975403i	$-0.570745\pi$
$577$	−10.5897	−0.440855	−0.220428	+	0.975403i	$0.570745\pi$
$578$	0	0
$579$	35.4230	1.47213
$580$	0	0
$581$	−77.4824	−3.21451
$582$	0	0
$583$	−0.593113	−0.0245642
$584$	0	0
$585$	1.57503	0.0651196
$586$	0	0
$587$	−26.2330	−1.08275	−0.541376	−	0.840780i	$-0.682097\pi$
$587$	−26.2330	−1.08275	−0.541376	+	0.840780i	$0.682097\pi$
$588$	0	0
$589$	1.45614	0.0599993
$590$	0	0
$591$	−19.6416	−0.807946
$592$	0	0
$593$	42.9710	1.76461	0.882304	−	0.470680i	$-0.155991\pi$
$593$	42.9710	1.76461	0.882304	+	0.470680i	$0.155991\pi$
$594$	0	0
$595$	1.18476	0.0485705
$596$	0	0
$597$	63.0038	2.57858
$598$	0	0
$599$	−20.4694	−0.836355	−0.418178	−	0.908365i	$-0.637331\pi$
$599$	−20.4694	−0.836355	−0.418178	+	0.908365i	$0.637331\pi$
$600$	0	0
$601$	8.64710	0.352723	0.176361	−	0.984325i	$-0.443567\pi$
$601$	8.64710	0.352723	0.176361	+	0.984325i	$0.443567\pi$
$602$	0	0
$603$	36.1787	1.47331
$604$	0	0
$605$	5.21348	0.211958
$606$	0	0
$607$	−29.5194	−1.19815	−0.599077	−	0.800691i	$-0.704466\pi$
$607$	−29.5194	−1.19815	−0.599077	+	0.800691i	$0.704466\pi$
$608$	0	0
$609$	−80.5430	−3.26377
$610$	0	0
$611$	0.309957	0.0125395
$612$	0	0
$613$	−14.6360	−0.591144	−0.295572	−	0.955320i	$-0.595510\pi$
$613$	−14.6360	−0.591144	−0.295572	+	0.955320i	$0.595510\pi$
$614$	0	0
$615$	−2.12519	−0.0856958
$616$	0	0
$617$	−5.60037	−0.225462	−0.112731	−	0.993626i	$-0.535960\pi$
$617$	−5.60037	−0.225462	−0.112731	+	0.993626i	$0.535960\pi$
$618$	0	0
$619$	−10.0552	−0.404151	−0.202075	−	0.979370i	$-0.564769\pi$
$619$	−10.0552	−0.404151	−0.202075	+	0.979370i	$0.564769\pi$
$620$	0	0
$621$	−0.148559	−0.00596148
$622$	0	0
$623$	48.0319	1.92436
$624$	0	0
$625$	21.4617	0.858468
$626$	0	0
$627$	−1.42619	−0.0569567
$628$	0	0
$629$	5.78659	0.230726
$630$	0	0
$631$	42.9043	1.70799	0.853996	−	0.520280i	$-0.174172\pi$
$631$	42.9043	1.70799	0.853996	+	0.520280i	$0.174172\pi$
$632$	0	0
$633$	58.1675	2.31195
$634$	0	0
$635$	−6.63829	−0.263433
$636$	0	0
$637$	−12.9613	−0.513546
$638$	0	0
$639$	14.7579	0.583812
$640$	0	0
$641$	−13.7730	−0.544003	−0.272001	−	0.962297i	$-0.587686\pi$
$641$	−13.7730	−0.544003	−0.272001	+	0.962297i	$0.587686\pi$
$642$	0	0
$643$	−33.2396	−1.31084	−0.655421	−	0.755263i	$-0.727509\pi$
$643$	−33.2396	−1.31084	−0.655421	+	0.755263i	$0.727509\pi$
$644$	0	0
$645$	−12.8946	−0.507724
$646$	0	0
$647$	−37.8612	−1.48848	−0.744238	−	0.667914i	$-0.767187\pi$
$647$	−37.8612	−1.48848	−0.744238	+	0.667914i	$0.767187\pi$
$648$	0	0
$649$	4.22688	0.165920
$650$	0	0
$651$	14.9413	0.585597
$652$	0	0
$653$	24.1029	0.943219	0.471610	−	0.881807i	$-0.343673\pi$
$653$	24.1029	0.943219	0.471610	+	0.881807i	$0.343673\pi$
$654$	0	0
$655$	−5.55875	−0.217199
$656$	0	0
$657$	18.9861	0.740720
$658$	0	0
$659$	−46.9261	−1.82798	−0.913991	−	0.405735i	$-0.867016\pi$
$659$	−46.9261	−1.82798	−0.913991	+	0.405735i	$0.867016\pi$
$660$	0	0
$661$	11.8119	0.459431	0.229715	−	0.973258i	$-0.426220\pi$
$661$	11.8119	0.459431	0.229715	+	0.973258i	$0.426220\pi$
$662$	0	0
$663$	−1.57671	−0.0612343
$664$	0	0
$665$	2.08928	0.0810186
$666$	0	0
$667$	−2.22496	−0.0861509
$668$	0	0
$669$	7.99757	0.309204
$670$	0	0
$671$	2.13704	0.0824994
$672$	0	0
$673$	−27.1703	−1.04734	−0.523668	−	0.851922i	$-0.675437\pi$
$673$	−27.1703	−1.04734	−0.523668	+	0.851922i	$0.675437\pi$
$674$	0	0
$675$	−2.49489	−0.0960285
$676$	0	0
$677$	−24.4682	−0.940390	−0.470195	−	0.882563i	$-0.655816\pi$
$677$	−24.4682	−0.940390	−0.470195	+	0.882563i	$0.655816\pi$
$678$	0	0
$679$	71.6709	2.75048
$680$	0	0
$681$	33.0441	1.26625
$682$	0	0
$683$	−28.7130	−1.09867	−0.549337	−	0.835601i	$-0.685120\pi$
$683$	−28.7130	−1.09867	−0.549337	+	0.835601i	$0.685120\pi$
$684$	0	0
$685$	−4.99849	−0.190982
$686$	0	0
$687$	68.3512	2.60776
$688$	0	0
$689$	−1.15631	−0.0440520
$690$	0	0
$691$	−23.2097	−0.882938	−0.441469	−	0.897276i	$-0.645543\pi$
$691$	−23.2097	−0.882938	−0.441469	+	0.897276i	$0.645543\pi$
$692$	0	0
$693$	−7.04119	−0.267473
$694$	0	0
$695$	−0.415554	−0.0157629
$696$	0	0
$697$	1.02363	0.0387726
$698$	0	0
$699$	60.6785	2.29507
$700$	0	0
$701$	14.2814	0.539402	0.269701	−	0.962944i	$-0.413075\pi$
$701$	14.2814	0.539402	0.269701	+	0.962944i	$0.413075\pi$
$702$	0	0
$703$	10.2044	0.384866
$704$	0	0
$705$	−0.315585	−0.0118856
$706$	0	0
$707$	−28.1243	−1.05772
$708$	0	0
$709$	−14.5116	−0.544993	−0.272497	−	0.962157i	$-0.587849\pi$
$709$	−14.5116	−0.544993	−0.272497	+	0.962157i	$0.587849\pi$
$710$	0	0
$711$	16.7973	0.629948
$712$	0	0
$713$	0.412747	0.0154575
$714$	0	0
$715$	−0.335787	−0.0125577
$716$	0	0
$717$	−46.8318	−1.74897
$718$	0	0
$719$	49.2609	1.83712	0.918561	−	0.395280i	$-0.129353\pi$
$719$	49.2609	1.83712	0.918561	+	0.395280i	$0.129353\pi$
$720$	0	0
$721$	14.2750	0.531627
$722$	0	0
$723$	−15.3160	−0.569608
$724$	0	0
$725$	−37.3658	−1.38773
$726$	0	0
$727$	−17.9463	−0.665593	−0.332796	−	0.942999i	$-0.607992\pi$
$727$	−17.9463	−0.665593	−0.332796	+	0.942999i	$0.607992\pi$
$728$	0	0
$729$	−22.9445	−0.849798
$730$	0	0
$731$	6.21086	0.229717
$732$	0	0
$733$	−14.7118	−0.543393	−0.271697	−	0.962383i	$-0.587585\pi$
$733$	−14.7118	−0.543393	−0.271697	+	0.962383i	$0.587585\pi$
$734$	0	0
$735$	13.1967	0.486767
$736$	0	0
$737$	−7.71307	−0.284115
$738$	0	0
$739$	−13.6777	−0.503141	−0.251570	−	0.967839i	$-0.580947\pi$
$739$	−13.6777	−0.503141	−0.251570	+	0.967839i	$0.580947\pi$
$740$	0	0
$741$	−2.78046	−0.102143
$742$	0	0
$743$	33.9122	1.24412	0.622059	−	0.782971i	$-0.286296\pi$
$743$	33.9122	1.24412	0.622059	+	0.782971i	$0.286296\pi$
$744$	0	0
$745$	−9.67008	−0.354284
$746$	0	0
$747$	−50.5151	−1.84825
$748$	0	0
$749$	−40.7625	−1.48943
$750$	0	0
$751$	20.4999	0.748050	0.374025	−	0.927419i	$-0.377977\pi$
$751$	20.4999	0.748050	0.374025	+	0.927419i	$0.377977\pi$
$752$	0	0
$753$	33.8473	1.23347
$754$	0	0
$755$	8.81249	0.320719
$756$	0	0
$757$	−31.6654	−1.15090	−0.575450	−	0.817837i	$-0.695173\pi$
$757$	−31.6654	−1.15090	−0.575450	+	0.817837i	$0.695173\pi$
$758$	0	0
$759$	−0.404258	−0.0146736
$760$	0	0
$761$	−12.6218	−0.457540	−0.228770	−	0.973480i	$-0.573470\pi$
$761$	−12.6218	−0.457540	−0.228770	+	0.973480i	$0.573470\pi$
$762$	0	0
$763$	0.832281	0.0301306
$764$	0	0
$765$	0.772413	0.0279266
$766$	0	0
$767$	8.24058	0.297550
$768$	0	0
$769$	−47.2496	−1.70386	−0.851932	−	0.523652i	$-0.824569\pi$
$769$	−47.2496	−1.70386	−0.851932	+	0.523652i	$0.824569\pi$
$770$	0	0
$771$	−56.8051	−2.04579
$772$	0	0
$773$	39.3017	1.41358	0.706791	−	0.707422i	$-0.250142\pi$
$773$	39.3017	1.41358	0.706791	+	0.707422i	$0.250142\pi$
$774$	0	0
$775$	6.93165	0.248992
$776$	0	0
$777$	104.706	3.75632
$778$	0	0
$779$	1.80512	0.0646751
$780$	0	0
$781$	−3.14628	−0.112583
$782$	0	0
$783$	4.11397	0.147021
$784$	0	0
$785$	2.59135	0.0924894
$786$	0	0
$787$	16.7525	0.597163	0.298581	−	0.954384i	$-0.403487\pi$
$787$	16.7525	0.597163	0.298581	+	0.954384i	$0.403487\pi$
$788$	0	0
$789$	60.9133	2.16857
$790$	0	0
$791$	−59.0015	−2.09785
$792$	0	0
$793$	4.16629	0.147949
$794$	0	0
$795$	1.17731	0.0417549
$796$	0	0
$797$	43.4218	1.53808	0.769039	−	0.639202i	$-0.220735\pi$
$797$	43.4218	1.53808	0.769039	+	0.639202i	$0.220735\pi$
$798$	0	0
$799$	0.152006	0.00537759
$800$	0	0
$801$	31.3147	1.10645
$802$	0	0
$803$	−4.04773	−0.142841
$804$	0	0
$805$	0.592211	0.0208727
$806$	0	0
$807$	−24.0601	−0.846956
$808$	0	0
$809$	−42.1872	−1.48322	−0.741611	−	0.670831i	$-0.765938\pi$
$809$	−42.1872	−1.48322	−0.741611	+	0.670831i	$0.765938\pi$
$810$	0	0
$811$	5.39281	0.189367	0.0946837	−	0.995507i	$-0.469816\pi$
$811$	5.39281	0.189367	0.0946837	+	0.995507i	$0.469816\pi$
$812$	0	0
$813$	20.3256	0.712850
$814$	0	0
$815$	11.2554	0.394258
$816$	0	0
$817$	10.9526	0.383182
$818$	0	0
$819$	−13.7273	−0.479669
$820$	0	0
$821$	37.9182	1.32335	0.661677	−	0.749789i	$-0.269845\pi$
$821$	37.9182	1.32335	0.661677	+	0.749789i	$0.269845\pi$
$822$	0	0
$823$	−42.2072	−1.47125	−0.735625	−	0.677389i	$-0.763111\pi$
$823$	−42.2072	−1.47125	−0.735625	+	0.677389i	$0.763111\pi$
$824$	0	0
$825$	−6.78908	−0.236365
$826$	0	0
$827$	9.53128	0.331435	0.165718	−	0.986173i	$-0.447006\pi$
$827$	9.53128	0.331435	0.165718	+	0.986173i	$0.447006\pi$
$828$	0	0
$829$	−7.77654	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$829$	−7.77654	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$830$	0	0
$831$	−32.7791	−1.13710
$832$	0	0
$833$	−6.35637	−0.220235
$834$	0	0
$835$	8.85619	0.306481
$836$	0	0
$837$	−0.763173	−0.0263791
$838$	0	0
$839$	−9.60525	−0.331610	−0.165805	−	0.986159i	$-0.553022\pi$
$839$	−9.60525	−0.331610	−0.165805	+	0.986159i	$0.553022\pi$
$840$	0	0
$841$	32.6147	1.12464
$842$	0	0
$843$	−19.7641	−0.680712
$844$	0	0
$845$	5.71030	0.196440
$846$	0	0
$847$	−45.4383	−1.56128
$848$	0	0
$849$	43.1060	1.47939
$850$	0	0
$851$	2.89246	0.0991523
$852$	0	0
$853$	−3.76954	−0.129066	−0.0645332	−	0.997916i	$-0.520556\pi$
$853$	−3.76954	−0.129066	−0.0645332	+	0.997916i	$0.520556\pi$
$854$	0	0
$855$	1.36212	0.0465834
$856$	0	0
$857$	−31.8511	−1.08801	−0.544007	−	0.839081i	$-0.683094\pi$
$857$	−31.8511	−1.08801	−0.544007	+	0.839081i	$0.683094\pi$
$858$	0	0
$859$	−45.7453	−1.56081	−0.780404	−	0.625275i	$-0.784987\pi$
$859$	−45.7453	−1.56081	−0.780404	+	0.625275i	$0.784987\pi$
$860$	0	0
$861$	18.5222	0.631233
$862$	0	0
$863$	14.2650	0.485586	0.242793	−	0.970078i	$-0.421936\pi$
$863$	14.2650	0.485586	0.242793	+	0.970078i	$0.421936\pi$
$864$	0	0
$865$	−5.05301	−0.171807
$866$	0	0
$867$	40.1047	1.36203
$868$	0	0
$869$	−3.58108	−0.121480
$870$	0	0
$871$	−15.0371	−0.509514
$872$	0	0
$873$	46.7262	1.58144
$874$	0	0
$875$	20.3919	0.689373
$876$	0	0
$877$	4.98368	0.168287	0.0841434	−	0.996454i	$-0.473185\pi$
$877$	4.98368	0.168287	0.0841434	+	0.996454i	$0.473185\pi$
$878$	0	0
$879$	−61.0561	−2.05937
$880$	0	0
$881$	−35.2448	−1.18743	−0.593713	−	0.804677i	$-0.702339\pi$
$881$	−35.2448	−1.18743	−0.593713	+	0.804677i	$0.702339\pi$
$882$	0	0
$883$	17.1327	0.576562	0.288281	−	0.957546i	$-0.406916\pi$
$883$	17.1327	0.576562	0.288281	+	0.957546i	$0.406916\pi$
$884$	0	0
$885$	−8.39022	−0.282034
$886$	0	0
$887$	−12.7821	−0.429180	−0.214590	−	0.976704i	$-0.568841\pi$
$887$	−12.7821	−0.429180	−0.214590	+	0.976704i	$0.568841\pi$
$888$	0	0
$889$	57.8563	1.94044
$890$	0	0
$891$	5.69767	0.190879
$892$	0	0
$893$	0.268056	0.00897016
$894$	0	0
$895$	5.76884	0.192831
$896$	0	0
$897$	−0.788128	−0.0263148
$898$	0	0
$899$	−11.4300	−0.381211
$900$	0	0
$901$	−0.567069	−0.0188918
$902$	0	0
$903$	112.383	3.73988
$904$	0	0
$905$	5.26386	0.174977
$906$	0	0
$907$	−40.2799	−1.33747	−0.668735	−	0.743501i	$-0.733164\pi$
$907$	−40.2799	−1.33747	−0.668735	+	0.743501i	$0.733164\pi$
$908$	0	0
$909$	−18.3358	−0.608160
$910$	0	0
$911$	41.3166	1.36888	0.684440	−	0.729069i	$-0.260047\pi$
$911$	41.3166	1.36888	0.684440	+	0.729069i	$0.260047\pi$
$912$	0	0
$913$	10.7695	0.356418
$914$	0	0
$915$	−4.24195	−0.140234
$916$	0	0
$917$	48.4476	1.59988
$918$	0	0
$919$	56.6644	1.86919	0.934593	−	0.355720i	$-0.115764\pi$
$919$	56.6644	1.86919	0.934593	+	0.355720i	$0.115764\pi$
$920$	0	0
$921$	15.8330	0.521714
$922$	0	0
$923$	−6.13389	−0.201899
$924$	0	0
$925$	48.5758	1.59716
$926$	0	0
$927$	9.30664	0.305670
$928$	0	0
$929$	41.3730	1.35740	0.678702	−	0.734414i	$-0.262543\pi$
$929$	41.3730	1.35740	0.678702	+	0.734414i	$0.262543\pi$
$930$	0	0
$931$	−11.2092	−0.367366
$932$	0	0
$933$	−8.42868	−0.275943
$934$	0	0
$935$	−0.164674	−0.00538540
$936$	0	0
$937$	4.98784	0.162946	0.0814728	−	0.996676i	$-0.474038\pi$
$937$	4.98784	0.162946	0.0814728	+	0.996676i	$0.474038\pi$
$938$	0	0
$939$	−26.5602	−0.866760
$940$	0	0
$941$	−17.1052	−0.557612	−0.278806	−	0.960347i	$-0.589939\pi$
$941$	−17.1052	−0.557612	−0.278806	+	0.960347i	$0.589939\pi$
$942$	0	0
$943$	0.511666	0.0166621
$944$	0	0
$945$	−1.09500	−0.0356204
$946$	0	0
$947$	−0.775986	−0.0252162	−0.0126081	−	0.999921i	$-0.504013\pi$
$947$	−0.775986	−0.0252162	−0.0126081	+	0.999921i	$0.504013\pi$
$948$	0	0
$949$	−7.89131	−0.256163
$950$	0	0
$951$	20.4851	0.664276
$952$	0	0
$953$	−24.0180	−0.778018	−0.389009	−	0.921234i	$-0.627182\pi$
$953$	−24.0180	−0.778018	−0.389009	+	0.921234i	$0.627182\pi$
$954$	0	0
$955$	8.77308	0.283890
$956$	0	0
$957$	11.1949	0.361880
$958$	0	0
$959$	43.5645	1.40677
$960$	0	0
$961$	−28.8797	−0.931602
$962$	0	0
$963$	−26.5753	−0.856378
$964$	0	0
$965$	7.21265	0.232183
$966$	0	0
$967$	7.13796	0.229541	0.114771	−	0.993392i	$-0.463387\pi$
$967$	7.13796	0.229541	0.114771	+	0.993392i	$0.463387\pi$
$968$	0	0
$969$	−1.36357	−0.0438040
$970$	0	0
$971$	−14.3344	−0.460014	−0.230007	−	0.973189i	$-0.573875\pi$
$971$	−14.3344	−0.460014	−0.230007	+	0.973189i	$0.573875\pi$
$972$	0	0
$973$	3.62178	0.116109
$974$	0	0
$975$	−13.2358	−0.423884
$976$	0	0
$977$	−38.5586	−1.23360	−0.616800	−	0.787120i	$-0.711571\pi$
$977$	−38.5586	−1.23360	−0.616800	+	0.787120i	$0.711571\pi$
$978$	0	0
$979$	−6.67610	−0.213369
$980$	0	0
$981$	0.542610	0.0173242
$982$	0	0
$983$	−17.3991	−0.554944	−0.277472	−	0.960734i	$-0.589497\pi$
$983$	−17.3991	−0.554944	−0.277472	+	0.960734i	$0.589497\pi$
$984$	0	0
$985$	−3.99932	−0.127429
$986$	0	0
$987$	2.75050	0.0875493
$988$	0	0
$989$	3.10454	0.0987185
$990$	0	0
$991$	47.6058	1.51225	0.756123	−	0.654429i	$-0.227091\pi$
$991$	47.6058	1.51225	0.756123	+	0.654429i	$0.227091\pi$
$992$	0	0
$993$	−37.9950	−1.20573
$994$	0	0
$995$	12.8285	0.406692
$996$	0	0
$997$	−50.9178	−1.61258	−0.806292	−	0.591518i	$-0.798529\pi$
$997$	−50.9178	−1.61258	−0.806292	+	0.591518i	$0.798529\pi$
$998$	0	0
$999$	−5.34818	−0.169209

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.4	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.4	✓	19	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.40459 q^{3} -0.489610 q^{5} +4.26722 q^{7} +2.78204 q^{9} +O(q^{10})\)	\(q-2.40459 q^{3} -0.489610 q^{5} +4.26722 q^{7} +2.78204 q^{9} -0.593113 q^{11} -1.15631 q^{13} +1.17731 q^{15} -0.567069 q^{17} -1.00000 q^{19} -10.2609 q^{21} -0.283453 q^{23} -4.76028 q^{25} +0.524106 q^{27} +7.84950 q^{29} -1.45614 q^{31} +1.42619 q^{33} -2.08928 q^{35} -10.2044 q^{37} +2.78046 q^{39} -1.80512 q^{41} -10.9526 q^{43} -1.36212 q^{45} -0.268056 q^{47} +11.2092 q^{49} +1.36357 q^{51} +1.00000 q^{53} +0.290395 q^{55} +2.40459 q^{57} -7.12660 q^{59} -3.60308 q^{61} +11.8716 q^{63} +0.566143 q^{65} +13.0044 q^{67} +0.681587 q^{69} +5.30469 q^{71} +6.82454 q^{73} +11.4465 q^{75} -2.53095 q^{77} +6.03777 q^{79} -9.60638 q^{81} -18.1576 q^{83} +0.277643 q^{85} -18.8748 q^{87} +11.2560 q^{89} -4.93425 q^{91} +3.50142 q^{93} +0.489610 q^{95} +16.7957 q^{97} -1.65006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9} - 11 q^{11} + q^{13} - 20 q^{15} - q^{17} - 19 q^{19} - 10 q^{21} - 16 q^{23} + 21 q^{25} - 4 q^{27} - 9 q^{31} + 7 q^{33} - 25 q^{37} - 25 q^{39} + q^{41} - 41 q^{43} - 27 q^{45} - 29 q^{47} + 14 q^{49} - 24 q^{51} + 19 q^{53} - 28 q^{55} + 4 q^{57} - 42 q^{59} + q^{61} - 41 q^{63} + 2 q^{65} - 41 q^{67} - 25 q^{69} - 20 q^{73} + 11 q^{75} - 19 q^{77} - 38 q^{79} + 23 q^{81} - 36 q^{83} - 58 q^{85} - 30 q^{87} - 25 q^{89} - 55 q^{91} - 38 q^{93} + 2 q^{95} - 13 q^{97} + q^{99}+O(q^{100}) \) 19 * q - 4 * q^3 - 2 * q^5 - 11 * q^7 + 19 * q^9 - 11 * q^11 + q^13 - 20 * q^15 - q^17 - 19 * q^19 - 10 * q^21 - 16 * q^23 + 21 * q^25 - 4 * q^27 - 9 * q^31 + 7 * q^33 - 25 * q^37 - 25 * q^39 + q^41 - 41 * q^43 - 27 * q^45 - 29 * q^47 + 14 * q^49 - 24 * q^51 + 19 * q^53 - 28 * q^55 + 4 * q^57 - 42 * q^59 + q^61 - 41 * q^63 + 2 * q^65 - 41 * q^67 - 25 * q^69 - 20 * q^73 + 11 * q^75 - 19 * q^77 - 38 * q^79 + 23 * q^81 - 36 * q^83 - 58 * q^85 - 30 * q^87 - 25 * q^89 - 55 * q^91 - 38 * q^93 + 2 * q^95 - 13 * q^97 + q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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