LMFDB - Embedded newform 4028.2.a.d.1.17

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			1.17
Root			$-2.53438$ 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)

$f(q)$	$=$	$q+2.53438 q^{3} -2.27936 q^{5} +1.27718 q^{7} +3.42306 q^{9} +O(q^{10})$	$q+2.53438 q^{3} -2.27936 q^{5} +1.27718 q^{7} +3.42306 q^{9} +0.683995 q^{11} -4.05395 q^{13} -5.77675 q^{15} +0.471295 q^{17} -1.00000 q^{19} +3.23685 q^{21} -4.28297 q^{23} +0.195468 q^{25} +1.07220 q^{27} -9.77368 q^{29} -9.37733 q^{31} +1.73350 q^{33} -2.91114 q^{35} +2.27596 q^{37} -10.2742 q^{39} +1.23060 q^{41} -0.166983 q^{43} -7.80238 q^{45} +5.60808 q^{47} -5.36882 q^{49} +1.19444 q^{51} +1.00000 q^{53} -1.55907 q^{55} -2.53438 q^{57} -7.90151 q^{59} +0.280568 q^{61} +4.37185 q^{63} +9.24039 q^{65} +2.83519 q^{67} -10.8547 q^{69} +9.88384 q^{71} +12.2089 q^{73} +0.495391 q^{75} +0.873582 q^{77} -9.28523 q^{79} -7.55184 q^{81} +12.4165 q^{83} -1.07425 q^{85} -24.7702 q^{87} -13.1886 q^{89} -5.17761 q^{91} -23.7657 q^{93} +2.27936 q^{95} -0.0580025 q^{97} +2.34136 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	2.53438	1.46322	0.731611	−	0.681722i	$-0.238769\pi$
$3$	2.53438	1.46322	0.731611	+	0.681722i	$0.238769\pi$
$4$	0	0
$5$	−2.27936	−1.01936	−0.509680	−	0.860364i	$-0.670236\pi$
$5$	−2.27936	−1.01936	−0.509680	+	0.860364i	$0.670236\pi$
$6$	0	0
$7$	1.27718	0.482728	0.241364	−	0.970435i	$-0.422405\pi$
$7$	1.27718	0.482728	0.241364	+	0.970435i	$0.422405\pi$
$8$	0	0
$9$	3.42306	1.14102
$10$	0	0
$11$	0.683995	0.206232	0.103116	−	0.994669i	$-0.467119\pi$
$11$	0.683995	0.206232	0.103116	+	0.994669i	$0.467119\pi$
$12$	0	0
$13$	−4.05395	−1.12436	−0.562181	−	0.827014i	$-0.690038\pi$
$13$	−4.05395	−1.12436	−0.562181	+	0.827014i	$0.690038\pi$
$14$	0	0
$15$	−5.77675	−1.49155
$16$	0	0
$17$	0.471295	0.114306	0.0571529	−	0.998365i	$-0.481798\pi$
$17$	0.471295	0.114306	0.0571529	+	0.998365i	$0.481798\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	3.23685	0.706338
$22$	0	0
$23$	−4.28297	−0.893061	−0.446531	−	0.894768i	$-0.647341\pi$
$23$	−4.28297	−0.893061	−0.446531	+	0.894768i	$0.647341\pi$
$24$	0	0
$25$	0.195468	0.0390937
$26$	0	0
$27$	1.07220	0.206344
$28$	0	0
$29$	−9.77368	−1.81493	−0.907464	−	0.420131i	$-0.861984\pi$
$29$	−9.77368	−1.81493	−0.907464	+	0.420131i	$0.861984\pi$
$30$	0	0
$31$	−9.37733	−1.68422	−0.842109	−	0.539308i	$-0.818686\pi$
$31$	−9.37733	−1.68422	−0.842109	+	0.539308i	$0.818686\pi$
$32$	0	0
$33$	1.73350	0.301764
$34$	0	0
$35$	−2.91114	−0.492073
$36$	0	0
$37$	2.27596	0.374166	0.187083	−	0.982344i	$-0.440097\pi$
$37$	2.27596	0.374166	0.187083	+	0.982344i	$0.440097\pi$
$38$	0	0
$39$	−10.2742	−1.64519
$40$	0	0
$41$	1.23060	0.192187	0.0960937	−	0.995372i	$-0.469365\pi$
$41$	1.23060	0.192187	0.0960937	+	0.995372i	$0.469365\pi$
$42$	0	0
$43$	−0.166983	−0.0254646	−0.0127323	−	0.999919i	$-0.504053\pi$
$43$	−0.166983	−0.0254646	−0.0127323	+	0.999919i	$0.504053\pi$
$44$	0	0
$45$	−7.80238	−1.16311
$46$	0	0
$47$	5.60808	0.818023	0.409011	−	0.912529i	$-0.365874\pi$
$47$	5.60808	0.818023	0.409011	+	0.912529i	$0.365874\pi$
$48$	0	0
$49$	−5.36882	−0.766974
$50$	0	0
$51$	1.19444	0.167255
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−1.55907	−0.210225
$56$	0	0
$57$	−2.53438	−0.335686
$58$	0	0
$59$	−7.90151	−1.02869	−0.514345	−	0.857584i	$-0.671965\pi$
$59$	−7.90151	−1.02869	−0.514345	+	0.857584i	$0.671965\pi$
$60$	0	0
$61$	0.280568	0.0359230	0.0179615	−	0.999839i	$-0.494282\pi$
$61$	0.280568	0.0359230	0.0179615	+	0.999839i	$0.494282\pi$
$62$	0	0
$63$	4.37185	0.550802
$64$	0	0
$65$	9.24039	1.14613
$66$	0	0
$67$	2.83519	0.346374	0.173187	−	0.984889i	$-0.444594\pi$
$67$	2.83519	0.346374	0.173187	+	0.984889i	$0.444594\pi$
$68$	0	0
$69$	−10.8547	−1.30675
$70$	0	0
$71$	9.88384	1.17300	0.586498	−	0.809951i	$-0.300506\pi$
$71$	9.88384	1.17300	0.586498	+	0.809951i	$0.300506\pi$
$72$	0	0
$73$	12.2089	1.42894	0.714471	−	0.699665i	$-0.246667\pi$
$73$	12.2089	1.42894	0.714471	+	0.699665i	$0.246667\pi$
$74$	0	0
$75$	0.495391	0.0572028
$76$	0	0
$77$	0.873582	0.0995539
$78$	0	0
$79$	−9.28523	−1.04467	−0.522335	−	0.852740i	$-0.674939\pi$
$79$	−9.28523	−1.04467	−0.522335	+	0.852740i	$0.674939\pi$
$80$	0	0
$81$	−7.55184	−0.839093
$82$	0	0
$83$	12.4165	1.36289	0.681444	−	0.731870i	$-0.261352\pi$
$83$	12.4165	1.36289	0.681444	+	0.731870i	$0.261352\pi$
$84$	0	0
$85$	−1.07425	−0.116519
$86$	0	0
$87$	−24.7702	−2.65564
$88$	0	0
$89$	−13.1886	−1.39799	−0.698995	−	0.715126i	$-0.746369\pi$
$89$	−13.1886	−1.39799	−0.698995	+	0.715126i	$0.746369\pi$
$90$	0	0
$91$	−5.17761	−0.542761
$92$	0	0
$93$	−23.7657	−2.46438
$94$	0	0
$95$	2.27936	0.233857
$96$	0	0
$97$	−0.0580025	−0.00588926	−0.00294463	−	0.999996i	$-0.500937\pi$
$97$	−0.0580025	−0.00588926	−0.00294463	+	0.999996i	$0.500937\pi$
$98$	0	0
$99$	2.34136	0.235315
$100$	0	0
$101$	−14.6990	−1.46260	−0.731302	−	0.682054i	$-0.761087\pi$
$101$	−14.6990	−1.46260	−0.731302	+	0.682054i	$0.761087\pi$
$102$	0	0
$103$	14.8976	1.46790	0.733952	−	0.679201i	$-0.237674\pi$
$103$	14.8976	1.46790	0.733952	+	0.679201i	$0.237674\pi$
$104$	0	0
$105$	−7.37793	−0.720012
$106$	0	0
$107$	18.2259	1.76196	0.880981	−	0.473152i	$-0.156884\pi$
$107$	18.2259	1.76196	0.880981	+	0.473152i	$0.156884\pi$
$108$	0	0
$109$	−8.25332	−0.790524	−0.395262	−	0.918568i	$-0.629346\pi$
$109$	−8.25332	−0.790524	−0.395262	+	0.918568i	$0.629346\pi$
$110$	0	0
$111$	5.76815	0.547488
$112$	0	0
$113$	11.7052	1.10113	0.550564	−	0.834793i	$-0.314413\pi$
$113$	11.7052	1.10113	0.550564	+	0.834793i	$0.314413\pi$
$114$	0	0
$115$	9.76242	0.910350
$116$	0	0
$117$	−13.8769	−1.28292
$118$	0	0
$119$	0.601927	0.0551786
$120$	0	0
$121$	−10.5322	−0.957468
$122$	0	0
$123$	3.11880	0.281213
$124$	0	0
$125$	10.9512	0.979509
$126$	0	0
$127$	−7.48112	−0.663842	−0.331921	−	0.943307i	$-0.607697\pi$
$127$	−7.48112	−0.663842	−0.331921	+	0.943307i	$0.607697\pi$
$128$	0	0
$129$	−0.423197	−0.0372604
$130$	0	0
$131$	−2.12916	−0.186026	−0.0930130	−	0.995665i	$-0.529650\pi$
$131$	−2.12916	−0.186026	−0.0930130	+	0.995665i	$0.529650\pi$
$132$	0	0
$133$	−1.27718	−0.110745
$134$	0	0
$135$	−2.44392	−0.210339
$136$	0	0
$137$	−1.92478	−0.164445	−0.0822224	−	0.996614i	$-0.526202\pi$
$137$	−1.92478	−0.164445	−0.0822224	+	0.996614i	$0.526202\pi$
$138$	0	0
$139$	−13.2021	−1.11979	−0.559894	−	0.828564i	$-0.689158\pi$
$139$	−13.2021	−1.11979	−0.559894	+	0.828564i	$0.689158\pi$
$140$	0	0
$141$	14.2130	1.19695
$142$	0	0
$143$	−2.77288	−0.231880
$144$	0	0
$145$	22.2777	1.85006
$146$	0	0
$147$	−13.6066	−1.12225
$148$	0	0
$149$	2.44056	0.199938	0.0999692	−	0.994991i	$-0.468126\pi$
$149$	2.44056	0.199938	0.0999692	+	0.994991i	$0.468126\pi$
$150$	0	0
$151$	−7.96295	−0.648016	−0.324008	−	0.946054i	$-0.605030\pi$
$151$	−7.96295	−0.648016	−0.324008	+	0.946054i	$0.605030\pi$
$152$	0	0
$153$	1.61327	0.130425
$154$	0	0
$155$	21.3743	1.71682
$156$	0	0
$157$	3.20936	0.256134	0.128067	−	0.991765i	$-0.459123\pi$
$157$	3.20936	0.256134	0.128067	+	0.991765i	$0.459123\pi$
$158$	0	0
$159$	2.53438	0.200989
$160$	0	0
$161$	−5.47011	−0.431105
$162$	0	0
$163$	−7.18403	−0.562697	−0.281348	−	0.959606i	$-0.590782\pi$
$163$	−7.18403	−0.562697	−0.281348	+	0.959606i	$0.590782\pi$
$164$	0	0
$165$	−3.95126	−0.307606
$166$	0	0
$167$	−6.29950	−0.487470	−0.243735	−	0.969842i	$-0.578373\pi$
$167$	−6.29950	−0.487470	−0.243735	+	0.969842i	$0.578373\pi$
$168$	0	0
$169$	3.43448	0.264191
$170$	0	0
$171$	−3.42306	−0.261768
$172$	0	0
$173$	−5.62900	−0.427965	−0.213983	−	0.976837i	$-0.568644\pi$
$173$	−5.62900	−0.427965	−0.213983	+	0.976837i	$0.568644\pi$
$174$	0	0
$175$	0.249648	0.0188716
$176$	0	0
$177$	−20.0254	−1.50520
$178$	0	0
$179$	−5.95898	−0.445395	−0.222698	−	0.974888i	$-0.571486\pi$
$179$	−5.95898	−0.445395	−0.222698	+	0.974888i	$0.571486\pi$
$180$	0	0
$181$	−12.4109	−0.922497	−0.461248	−	0.887271i	$-0.652598\pi$
$181$	−12.4109	−0.922497	−0.461248	+	0.887271i	$0.652598\pi$
$182$	0	0
$183$	0.711065	0.0525634
$184$	0	0
$185$	−5.18773	−0.381410
$186$	0	0
$187$	0.322363	0.0235735
$188$	0	0
$189$	1.36938	0.0996079
$190$	0	0
$191$	−14.3275	−1.03670	−0.518351	−	0.855168i	$-0.673454\pi$
$191$	−14.3275	−1.03670	−0.518351	+	0.855168i	$0.673454\pi$
$192$	0	0
$193$	−6.27032	−0.451347	−0.225674	−	0.974203i	$-0.572458\pi$
$193$	−6.27032	−0.451347	−0.225674	+	0.974203i	$0.572458\pi$
$194$	0	0
$195$	23.4186	1.67704
$196$	0	0
$197$	−17.3079	−1.23314	−0.616570	−	0.787300i	$-0.711478\pi$
$197$	−17.3079	−1.23314	−0.616570	+	0.787300i	$0.711478\pi$
$198$	0	0
$199$	13.6715	0.969146	0.484573	−	0.874751i	$-0.338975\pi$
$199$	13.6715	0.969146	0.484573	+	0.874751i	$0.338975\pi$
$200$	0	0
$201$	7.18544	0.506822
$202$	0	0
$203$	−12.4827	−0.876115
$204$	0	0
$205$	−2.80498	−0.195908
$206$	0	0
$207$	−14.6609	−1.01900
$208$	0	0
$209$	−0.683995	−0.0473129
$210$	0	0
$211$	18.0437	1.24218	0.621091	−	0.783739i	$-0.286690\pi$
$211$	18.0437	1.24218	0.621091	+	0.783739i	$0.286690\pi$
$212$	0	0
$213$	25.0494	1.71635
$214$	0	0
$215$	0.380614	0.0259576
$216$	0	0
$217$	−11.9765	−0.813018
$218$	0	0
$219$	30.9419	2.09086
$220$	0	0
$221$	−1.91061	−0.128521
$222$	0	0
$223$	5.67427	0.379977	0.189989	−	0.981786i	$-0.439155\pi$
$223$	5.67427	0.379977	0.189989	+	0.981786i	$0.439155\pi$
$224$	0	0
$225$	0.669101	0.0446067
$226$	0	0
$227$	28.4395	1.88760	0.943799	−	0.330521i	$-0.107224\pi$
$227$	28.4395	1.88760	0.943799	+	0.330521i	$0.107224\pi$
$228$	0	0
$229$	−23.0630	−1.52405	−0.762024	−	0.647548i	$-0.775794\pi$
$229$	−23.0630	−1.52405	−0.762024	+	0.647548i	$0.775794\pi$
$230$	0	0
$231$	2.21399	0.145670
$232$	0	0
$233$	−4.42250	−0.289727	−0.144864	−	0.989452i	$-0.546274\pi$
$233$	−4.42250	−0.289727	−0.144864	+	0.989452i	$0.546274\pi$
$234$	0	0
$235$	−12.7828	−0.833859
$236$	0	0
$237$	−23.5323	−1.52858
$238$	0	0
$239$	1.23936	0.0801675	0.0400837	−	0.999196i	$-0.487238\pi$
$239$	1.23936	0.0801675	0.0400837	+	0.999196i	$0.487238\pi$
$240$	0	0
$241$	−10.3316	−0.665518	−0.332759	−	0.943012i	$-0.607980\pi$
$241$	−10.3316	−0.665518	−0.332759	+	0.943012i	$0.607980\pi$
$242$	0	0
$243$	−22.3558	−1.43412
$244$	0	0
$245$	12.2375	0.781822
$246$	0	0
$247$	4.05395	0.257946
$248$	0	0
$249$	31.4681	1.99421
$250$	0	0
$251$	−3.88927	−0.245489	−0.122744	−	0.992438i	$-0.539170\pi$
$251$	−3.88927	−0.245489	−0.122744	+	0.992438i	$0.539170\pi$
$252$	0	0
$253$	−2.92953	−0.184178
$254$	0	0
$255$	−2.72255	−0.170493
$256$	0	0
$257$	7.70877	0.480860	0.240430	−	0.970667i	$-0.422712\pi$
$257$	7.70877	0.480860	0.240430	+	0.970667i	$0.422712\pi$
$258$	0	0
$259$	2.90681	0.180620
$260$	0	0
$261$	−33.4559	−2.07087
$262$	0	0
$263$	11.7299	0.723298	0.361649	−	0.932314i	$-0.382214\pi$
$263$	11.7299	0.723298	0.361649	+	0.932314i	$0.382214\pi$
$264$	0	0
$265$	−2.27936	−0.140020
$266$	0	0
$267$	−33.4249	−2.04557
$268$	0	0
$269$	−21.2401	−1.29503	−0.647516	−	0.762052i	$-0.724192\pi$
$269$	−21.2401	−1.29503	−0.647516	+	0.762052i	$0.724192\pi$
$270$	0	0
$271$	−10.4709	−0.636065	−0.318032	−	0.948080i	$-0.603022\pi$
$271$	−10.4709	−0.636065	−0.318032	+	0.948080i	$0.603022\pi$
$272$	0	0
$273$	−13.1220	−0.794180
$274$	0	0
$275$	0.133699	0.00806238
$276$	0	0
$277$	−18.1470	−1.09035	−0.545173	−	0.838323i	$-0.683536\pi$
$277$	−18.1470	−1.09035	−0.545173	+	0.838323i	$0.683536\pi$
$278$	0	0
$279$	−32.0992	−1.92173
$280$	0	0
$281$	−5.75924	−0.343568	−0.171784	−	0.985135i	$-0.554953\pi$
$281$	−5.75924	−0.343568	−0.171784	+	0.985135i	$0.554953\pi$
$282$	0	0
$283$	−13.4253	−0.798053	−0.399027	−	0.916939i	$-0.630652\pi$
$283$	−13.4253	−0.798053	−0.399027	+	0.916939i	$0.630652\pi$
$284$	0	0
$285$	5.77675	0.342185
$286$	0	0
$287$	1.57169	0.0927742
$288$	0	0
$289$	−16.7779	−0.986934
$290$	0	0
$291$	−0.147000	−0.00861730
$292$	0	0
$293$	2.18029	0.127374	0.0636868	−	0.997970i	$-0.479714\pi$
$293$	2.18029	0.127374	0.0636868	+	0.997970i	$0.479714\pi$
$294$	0	0
$295$	18.0104	1.04860
$296$	0	0
$297$	0.733376	0.0425548
$298$	0	0
$299$	17.3629	1.00412
$300$	0	0
$301$	−0.213267	−0.0122925
$302$	0	0
$303$	−37.2527	−2.14011
$304$	0	0
$305$	−0.639514	−0.0366185
$306$	0	0
$307$	15.4042	0.879165	0.439583	−	0.898202i	$-0.355126\pi$
$307$	15.4042	0.879165	0.439583	+	0.898202i	$0.355126\pi$
$308$	0	0
$309$	37.7561	2.14787
$310$	0	0
$311$	11.9536	0.677826	0.338913	−	0.940818i	$-0.389941\pi$
$311$	11.9536	0.677826	0.338913	+	0.940818i	$0.389941\pi$
$312$	0	0
$313$	0.866842	0.0489968	0.0244984	−	0.999700i	$-0.492201\pi$
$313$	0.866842	0.0489968	0.0244984	+	0.999700i	$0.492201\pi$
$314$	0	0
$315$	−9.96502	−0.561465
$316$	0	0
$317$	20.9663	1.17758	0.588792	−	0.808284i	$-0.299604\pi$
$317$	20.9663	1.17758	0.588792	+	0.808284i	$0.299604\pi$
$318$	0	0
$319$	−6.68515	−0.374296
$320$	0	0
$321$	46.1912	2.57814
$322$	0	0
$323$	−0.471295	−0.0262236
$324$	0	0
$325$	−0.792419	−0.0439555
$326$	0	0
$327$	−20.9170	−1.15671
$328$	0	0
$329$	7.16251	0.394882
$330$	0	0
$331$	27.0956	1.48931	0.744654	−	0.667451i	$-0.232615\pi$
$331$	27.0956	1.48931	0.744654	+	0.667451i	$0.232615\pi$
$332$	0	0
$333$	7.79076	0.426931
$334$	0	0
$335$	−6.46242	−0.353080
$336$	0	0
$337$	33.9085	1.84711	0.923556	−	0.383464i	$-0.125269\pi$
$337$	33.9085	1.84711	0.923556	+	0.383464i	$0.125269\pi$
$338$	0	0
$339$	29.6653	1.61120
$340$	0	0
$341$	−6.41404	−0.347340
$342$	0	0
$343$	−15.7972	−0.852967
$344$	0	0
$345$	24.7416	1.33205
$346$	0	0
$347$	−29.1422	−1.56444	−0.782219	−	0.623004i	$-0.785912\pi$
$347$	−29.1422	−1.56444	−0.782219	+	0.623004i	$0.785912\pi$
$348$	0	0
$349$	−11.2954	−0.604629	−0.302315	−	0.953208i	$-0.597759\pi$
$349$	−11.2954	−0.604629	−0.302315	+	0.953208i	$0.597759\pi$
$350$	0	0
$351$	−4.34662	−0.232005
$352$	0	0
$353$	22.3141	1.18766	0.593830	−	0.804591i	$-0.297615\pi$
$353$	22.3141	1.18766	0.593830	+	0.804591i	$0.297615\pi$
$354$	0	0
$355$	−22.5288	−1.19570
$356$	0	0
$357$	1.52551	0.0807385
$358$	0	0
$359$	32.3908	1.70952	0.854761	−	0.519021i	$-0.173704\pi$
$359$	32.3908	1.70952	0.854761	+	0.519021i	$0.173704\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−26.6924	−1.40099
$364$	0	0
$365$	−27.8284	−1.45661
$366$	0	0
$367$	21.8121	1.13858	0.569292	−	0.822135i	$-0.307217\pi$
$367$	21.8121	1.13858	0.569292	+	0.822135i	$0.307217\pi$
$368$	0	0
$369$	4.21242	0.219290
$370$	0	0
$371$	1.27718	0.0663077
$372$	0	0
$373$	9.02171	0.467127	0.233563	−	0.972342i	$-0.424961\pi$
$373$	9.02171	0.467127	0.233563	+	0.972342i	$0.424961\pi$
$374$	0	0
$375$	27.7546	1.43324
$376$	0	0
$377$	39.6220	2.04064
$378$	0	0
$379$	12.0094	0.616881	0.308440	−	0.951244i	$-0.400193\pi$
$379$	12.0094	0.616881	0.308440	+	0.951244i	$0.400193\pi$
$380$	0	0
$381$	−18.9600	−0.971349
$382$	0	0
$383$	−1.44318	−0.0737430	−0.0368715	−	0.999320i	$-0.511739\pi$
$383$	−1.44318	−0.0737430	−0.0368715	+	0.999320i	$0.511739\pi$
$384$	0	0
$385$	−1.99121	−0.101481
$386$	0	0
$387$	−0.571592	−0.0290557
$388$	0	0
$389$	32.6304	1.65443	0.827213	−	0.561889i	$-0.189925\pi$
$389$	32.6304	1.65443	0.827213	+	0.561889i	$0.189925\pi$
$390$	0	0
$391$	−2.01854	−0.102082
$392$	0	0
$393$	−5.39610	−0.272197
$394$	0	0
$395$	21.1644	1.06489
$396$	0	0
$397$	−19.5675	−0.982065	−0.491032	−	0.871141i	$-0.663380\pi$
$397$	−19.5675	−0.982065	−0.491032	+	0.871141i	$0.663380\pi$
$398$	0	0
$399$	−3.23685	−0.162045
$400$	0	0
$401$	−2.11783	−0.105759	−0.0528797	−	0.998601i	$-0.516840\pi$
$401$	−2.11783	−0.105759	−0.0528797	+	0.998601i	$0.516840\pi$
$402$	0	0
$403$	38.0152	1.89367
$404$	0	0
$405$	17.2133	0.855337
$406$	0	0
$407$	1.55675	0.0771651
$408$	0	0
$409$	11.6225	0.574694	0.287347	−	0.957827i	$-0.407227\pi$
$409$	11.6225	0.574694	0.287347	+	0.957827i	$0.407227\pi$
$410$	0	0
$411$	−4.87811	−0.240619
$412$	0	0
$413$	−10.0916	−0.496577
$414$	0	0
$415$	−28.3017	−1.38927
$416$	0	0
$417$	−33.4591	−1.63850
$418$	0	0
$419$	−10.4119	−0.508657	−0.254328	−	0.967118i	$-0.581854\pi$
$419$	−10.4119	−0.508657	−0.254328	+	0.967118i	$0.581854\pi$
$420$	0	0
$421$	8.67091	0.422594	0.211297	−	0.977422i	$-0.432231\pi$
$421$	8.67091	0.422594	0.211297	+	0.977422i	$0.432231\pi$
$422$	0	0
$423$	19.1968	0.933380
$424$	0	0
$425$	0.0921233	0.00446864
$426$	0	0
$427$	0.358335	0.0173410
$428$	0	0
$429$	−7.02752	−0.339292
$430$	0	0
$431$	−0.404087	−0.0194642	−0.00973210	−	0.999953i	$-0.503098\pi$
$431$	−0.404087	−0.0194642	−0.00973210	+	0.999953i	$0.503098\pi$
$432$	0	0
$433$	28.3610	1.36294	0.681472	−	0.731844i	$-0.261340\pi$
$433$	28.3610	1.36294	0.681472	+	0.731844i	$0.261340\pi$
$434$	0	0
$435$	56.4601	2.70705
$436$	0	0
$437$	4.28297	0.204882
$438$	0	0
$439$	36.2499	1.73011	0.865056	−	0.501676i	$-0.167283\pi$
$439$	36.2499	1.73011	0.865056	+	0.501676i	$0.167283\pi$
$440$	0	0
$441$	−18.3778	−0.875133
$442$	0	0
$443$	−6.10105	−0.289870	−0.144935	−	0.989441i	$-0.546297\pi$
$443$	−6.10105	−0.289870	−0.144935	+	0.989441i	$0.546297\pi$
$444$	0	0
$445$	30.0616	1.42505
$446$	0	0
$447$	6.18530	0.292554
$448$	0	0
$449$	8.12113	0.383260	0.191630	−	0.981467i	$-0.438623\pi$
$449$	8.12113	0.383260	0.191630	+	0.981467i	$0.438623\pi$
$450$	0	0
$451$	0.841724	0.0396352
$452$	0	0
$453$	−20.1811	−0.948191
$454$	0	0
$455$	11.8016	0.553268
$456$	0	0
$457$	−20.5365	−0.960657	−0.480329	−	0.877089i	$-0.659483\pi$
$457$	−20.5365	−0.960657	−0.480329	+	0.877089i	$0.659483\pi$
$458$	0	0
$459$	0.505320	0.0235863
$460$	0	0
$461$	2.96682	0.138178	0.0690892	−	0.997610i	$-0.477991\pi$
$461$	2.96682	0.138178	0.0690892	+	0.997610i	$0.477991\pi$
$462$	0	0
$463$	−23.4093	−1.08792	−0.543962	−	0.839110i	$-0.683076\pi$
$463$	−23.4093	−1.08792	−0.543962	+	0.839110i	$0.683076\pi$
$464$	0	0
$465$	54.1704	2.51209
$466$	0	0
$467$	27.8918	1.29068	0.645339	−	0.763896i	$-0.276716\pi$
$467$	27.8918	1.29068	0.645339	+	0.763896i	$0.276716\pi$
$468$	0	0
$469$	3.62104	0.167204
$470$	0	0
$471$	8.13371	0.374782
$472$	0	0
$473$	−0.114215	−0.00525163
$474$	0	0
$475$	−0.195468	−0.00896871
$476$	0	0
$477$	3.42306	0.156731
$478$	0	0
$479$	10.2425	0.467993	0.233996	−	0.972237i	$-0.424820\pi$
$479$	10.2425	0.467993	0.233996	+	0.972237i	$0.424820\pi$
$480$	0	0
$481$	−9.22663	−0.420698
$482$	0	0
$483$	−13.8633	−0.630803
$484$	0	0
$485$	0.132208	0.00600328
$486$	0	0
$487$	6.09948	0.276394	0.138197	−	0.990405i	$-0.455869\pi$
$487$	6.09948	0.276394	0.138197	+	0.990405i	$0.455869\pi$
$488$	0	0
$489$	−18.2070	−0.823350
$490$	0	0
$491$	23.8845	1.07789	0.538946	−	0.842340i	$-0.318823\pi$
$491$	23.8845	1.07789	0.538946	+	0.842340i	$0.318823\pi$
$492$	0	0
$493$	−4.60629	−0.207457
$494$	0	0
$495$	−5.33679	−0.239871
$496$	0	0
$497$	12.6234	0.566237
$498$	0	0
$499$	−35.4851	−1.58853	−0.794265	−	0.607572i	$-0.792144\pi$
$499$	−35.4851	−1.58853	−0.794265	+	0.607572i	$0.792144\pi$
$500$	0	0
$501$	−15.9653	−0.713277
$502$	0	0
$503$	21.8640	0.974867	0.487433	−	0.873160i	$-0.337933\pi$
$503$	21.8640	0.974867	0.487433	+	0.873160i	$0.337933\pi$
$504$	0	0
$505$	33.5042	1.49092
$506$	0	0
$507$	8.70427	0.386570
$508$	0	0
$509$	−28.8469	−1.27862	−0.639309	−	0.768950i	$-0.720779\pi$
$509$	−28.8469	−1.27862	−0.639309	+	0.768950i	$0.720779\pi$
$510$	0	0
$511$	15.5929	0.689790
$512$	0	0
$513$	−1.07220	−0.0473386
$514$	0	0
$515$	−33.9569	−1.49632
$516$	0	0
$517$	3.83590	0.168703
$518$	0	0
$519$	−14.2660	−0.626208
$520$	0	0
$521$	39.8171	1.74442	0.872209	−	0.489134i	$-0.162687\pi$
$521$	39.8171	1.74442	0.872209	+	0.489134i	$0.162687\pi$
$522$	0	0
$523$	−19.3763	−0.847265	−0.423633	−	0.905834i	$-0.639245\pi$
$523$	−19.3763	−0.847265	−0.423633	+	0.905834i	$0.639245\pi$
$524$	0	0
$525$	0.632701	0.0276134
$526$	0	0
$527$	−4.41949	−0.192516
$528$	0	0
$529$	−4.65616	−0.202442
$530$	0	0
$531$	−27.0474	−1.17376
$532$	0	0
$533$	−4.98879	−0.216088
$534$	0	0
$535$	−41.5433	−1.79607
$536$	0	0
$537$	−15.1023	−0.651712
$538$	0	0
$539$	−3.67224	−0.158175
$540$	0	0
$541$	17.7548	0.763341	0.381670	−	0.924299i	$-0.375349\pi$
$541$	17.7548	0.763341	0.381670	+	0.924299i	$0.375349\pi$
$542$	0	0
$543$	−31.4539	−1.34982
$544$	0	0
$545$	18.8123	0.805828
$546$	0	0
$547$	−25.5212	−1.09121	−0.545604	−	0.838043i	$-0.683700\pi$
$547$	−25.5212	−1.09121	−0.545604	+	0.838043i	$0.683700\pi$
$548$	0	0
$549$	0.960401	0.0409889
$550$	0	0
$551$	9.77368	0.416373
$552$	0	0
$553$	−11.8589	−0.504291
$554$	0	0
$555$	−13.1477	−0.558087
$556$	0	0
$557$	27.2435	1.15435	0.577173	−	0.816622i	$-0.304156\pi$
$557$	27.2435	1.15435	0.577173	+	0.816622i	$0.304156\pi$
$558$	0	0
$559$	0.676940	0.0286315
$560$	0	0
$561$	0.816990	0.0344933
$562$	0	0
$563$	17.7988	0.750130	0.375065	−	0.926999i	$-0.377620\pi$
$563$	17.7988	0.750130	0.375065	+	0.926999i	$0.377620\pi$
$564$	0	0
$565$	−26.6802	−1.12245
$566$	0	0
$567$	−9.64503	−0.405053
$568$	0	0
$569$	38.8181	1.62734	0.813671	−	0.581326i	$-0.197466\pi$
$569$	38.8181	1.62734	0.813671	+	0.581326i	$0.197466\pi$
$570$	0	0
$571$	14.7885	0.618880	0.309440	−	0.950919i	$-0.399858\pi$
$571$	14.7885	0.618880	0.309440	+	0.950919i	$0.399858\pi$
$572$	0	0
$573$	−36.3113	−1.51693
$574$	0	0
$575$	−0.837186	−0.0349131
$576$	0	0
$577$	−30.1701	−1.25600	−0.627999	−	0.778214i	$-0.716126\pi$
$577$	−30.1701	−1.25600	−0.627999	+	0.778214i	$0.716126\pi$
$578$	0	0
$579$	−15.8913	−0.660422
$580$	0	0
$581$	15.8581	0.657904
$582$	0	0
$583$	0.683995	0.0283282
$584$	0	0
$585$	31.6304	1.30776
$586$	0	0
$587$	−18.2448	−0.753044	−0.376522	−	0.926408i	$-0.622880\pi$
$587$	−18.2448	−0.753044	−0.376522	+	0.926408i	$0.622880\pi$
$588$	0	0
$589$	9.37733	0.386386
$590$	0	0
$591$	−43.8648	−1.80436
$592$	0	0
$593$	−26.0618	−1.07023	−0.535115	−	0.844779i	$-0.679732\pi$
$593$	−26.0618	−1.07023	−0.535115	+	0.844779i	$0.679732\pi$
$594$	0	0
$595$	−1.37201	−0.0562468
$596$	0	0
$597$	34.6487	1.41808
$598$	0	0
$599$	28.7331	1.17400	0.587002	−	0.809585i	$-0.300308\pi$
$599$	28.7331	1.17400	0.587002	+	0.809585i	$0.300308\pi$
$600$	0	0
$601$	−28.4158	−1.15910	−0.579552	−	0.814935i	$-0.696772\pi$
$601$	−28.4158	−1.15910	−0.579552	+	0.814935i	$0.696772\pi$
$602$	0	0
$603$	9.70504	0.395220
$604$	0	0
$605$	24.0065	0.976004
$606$	0	0
$607$	−30.7862	−1.24957	−0.624786	−	0.780796i	$-0.714814\pi$
$607$	−30.7862	−1.24957	−0.624786	+	0.780796i	$0.714814\pi$
$608$	0	0
$609$	−31.6359	−1.28195
$610$	0	0
$611$	−22.7349	−0.919754
$612$	0	0
$613$	25.5989	1.03393	0.516964	−	0.856007i	$-0.327062\pi$
$613$	25.5989	1.03393	0.516964	+	0.856007i	$0.327062\pi$
$614$	0	0
$615$	−7.10887	−0.286657
$616$	0	0
$617$	−39.7094	−1.59864	−0.799319	−	0.600907i	$-0.794806\pi$
$617$	−39.7094	−1.59864	−0.799319	+	0.600907i	$0.794806\pi$
$618$	0	0
$619$	−43.0046	−1.72850	−0.864251	−	0.503060i	$-0.832207\pi$
$619$	−43.0046	−1.72850	−0.864251	+	0.503060i	$0.832207\pi$
$620$	0	0
$621$	−4.59218	−0.184278
$622$	0	0
$623$	−16.8442	−0.674848
$624$	0	0
$625$	−25.9391	−1.03757
$626$	0	0
$627$	−1.73350	−0.0692293
$628$	0	0
$629$	1.07265	0.0427694
$630$	0	0
$631$	−39.0678	−1.55526	−0.777631	−	0.628720i	$-0.783579\pi$
$631$	−39.0678	−1.55526	−0.777631	+	0.628720i	$0.783579\pi$
$632$	0	0
$633$	45.7296	1.81759
$634$	0	0
$635$	17.0522	0.676694
$636$	0	0
$637$	21.7649	0.862357
$638$	0	0
$639$	33.8330	1.33841
$640$	0	0
$641$	36.3811	1.43697	0.718484	−	0.695543i	$-0.244836\pi$
$641$	36.3811	1.43697	0.718484	+	0.695543i	$0.244836\pi$
$642$	0	0
$643$	−31.4795	−1.24143	−0.620715	−	0.784037i	$-0.713157\pi$
$643$	−31.4795	−1.24143	−0.620715	+	0.784037i	$0.713157\pi$
$644$	0	0
$645$	0.964618	0.0379818
$646$	0	0
$647$	41.2170	1.62041	0.810203	−	0.586149i	$-0.199357\pi$
$647$	41.2170	1.62041	0.810203	+	0.586149i	$0.199357\pi$
$648$	0	0
$649$	−5.40459	−0.212149
$650$	0	0
$651$	−30.3530	−1.18963
$652$	0	0
$653$	−21.0470	−0.823631	−0.411816	−	0.911267i	$-0.635105\pi$
$653$	−21.0470	−0.823631	−0.411816	+	0.911267i	$0.635105\pi$
$654$	0	0
$655$	4.85312	0.189627
$656$	0	0
$657$	41.7918	1.63045
$658$	0	0
$659$	−47.7553	−1.86028	−0.930141	−	0.367203i	$-0.880315\pi$
$659$	−47.7553	−1.86028	−0.930141	+	0.367203i	$0.880315\pi$
$660$	0	0
$661$	−24.5466	−0.954752	−0.477376	−	0.878699i	$-0.658412\pi$
$661$	−24.5466	−0.954752	−0.477376	+	0.878699i	$0.658412\pi$
$662$	0	0
$663$	−4.84219	−0.188055
$664$	0	0
$665$	2.91114	0.112889
$666$	0	0
$667$	41.8604	1.62084
$668$	0	0
$669$	14.3807	0.555991
$670$	0	0
$671$	0.191907	0.00740849
$672$	0	0
$673$	−19.1448	−0.737979	−0.368989	−	0.929434i	$-0.620296\pi$
$673$	−19.1448	−0.737979	−0.368989	+	0.929434i	$0.620296\pi$
$674$	0	0
$675$	0.209580	0.00806675
$676$	0	0
$677$	−13.2507	−0.509264	−0.254632	−	0.967038i	$-0.581954\pi$
$677$	−13.2507	−0.509264	−0.254632	+	0.967038i	$0.581954\pi$
$678$	0	0
$679$	−0.0740795	−0.00284291
$680$	0	0
$681$	72.0764	2.76198
$682$	0	0
$683$	−43.8509	−1.67791	−0.838954	−	0.544203i	$-0.816832\pi$
$683$	−43.8509	−1.67791	−0.838954	+	0.544203i	$0.816832\pi$
$684$	0	0
$685$	4.38725	0.167628
$686$	0	0
$687$	−58.4504	−2.23002
$688$	0	0
$689$	−4.05395	−0.154443
$690$	0	0
$691$	39.5843	1.50586	0.752929	−	0.658102i	$-0.228641\pi$
$691$	39.5843	1.50586	0.752929	+	0.658102i	$0.228641\pi$
$692$	0	0
$693$	2.99033	0.113593
$694$	0	0
$695$	30.0923	1.14147
$696$	0	0
$697$	0.579976	0.0219682
$698$	0	0
$699$	−11.2083	−0.423936
$700$	0	0
$701$	−43.8680	−1.65687	−0.828435	−	0.560085i	$-0.810769\pi$
$701$	−43.8680	−1.65687	−0.828435	+	0.560085i	$0.810769\pi$
$702$	0	0
$703$	−2.27596	−0.0858396
$704$	0	0
$705$	−32.3965	−1.22012
$706$	0	0
$707$	−18.7732	−0.706039
$708$	0	0
$709$	−43.1969	−1.62229	−0.811147	−	0.584843i	$-0.801156\pi$
$709$	−43.1969	−1.62229	−0.811147	+	0.584843i	$0.801156\pi$
$710$	0	0
$711$	−31.7839	−1.19199
$712$	0	0
$713$	40.1628	1.50411
$714$	0	0
$715$	6.32038	0.236369
$716$	0	0
$717$	3.14100	0.117303
$718$	0	0
$719$	5.62546	0.209794	0.104897	−	0.994483i	$-0.466549\pi$
$719$	5.62546	0.209794	0.104897	+	0.994483i	$0.466549\pi$
$720$	0	0
$721$	19.0269	0.708598
$722$	0	0
$723$	−26.1842	−0.973800
$724$	0	0
$725$	−1.91045	−0.0709522
$726$	0	0
$727$	−4.63839	−0.172028	−0.0860141	−	0.996294i	$-0.527413\pi$
$727$	−4.63839	−0.172028	−0.0860141	+	0.996294i	$0.527413\pi$
$728$	0	0
$729$	−34.0024	−1.25935
$730$	0	0
$731$	−0.0786982	−0.00291076
$732$	0	0
$733$	−23.6696	−0.874257	−0.437128	−	0.899399i	$-0.644004\pi$
$733$	−23.6696	−0.874257	−0.437128	+	0.899399i	$0.644004\pi$
$734$	0	0
$735$	31.0143	1.14398
$736$	0	0
$737$	1.93926	0.0714334
$738$	0	0
$739$	−23.8493	−0.877311	−0.438656	−	0.898655i	$-0.644545\pi$
$739$	−23.8493	−0.877311	−0.438656	+	0.898655i	$0.644545\pi$
$740$	0	0
$741$	10.2742	0.377433
$742$	0	0
$743$	−4.10688	−0.150667	−0.0753334	−	0.997158i	$-0.524002\pi$
$743$	−4.10688	−0.150667	−0.0753334	+	0.997158i	$0.524002\pi$
$744$	0	0
$745$	−5.56291	−0.203809
$746$	0	0
$747$	42.5025	1.55508
$748$	0	0
$749$	23.2777	0.850548
$750$	0	0
$751$	39.5478	1.44312	0.721560	−	0.692352i	$-0.243425\pi$
$751$	39.5478	1.44312	0.721560	+	0.692352i	$0.243425\pi$
$752$	0	0
$753$	−9.85688	−0.359205
$754$	0	0
$755$	18.1504	0.660561
$756$	0	0
$757$	28.7611	1.04534	0.522670	−	0.852535i	$-0.324936\pi$
$757$	28.7611	1.04534	0.522670	+	0.852535i	$0.324936\pi$
$758$	0	0
$759$	−7.42453	−0.269493
$760$	0	0
$761$	−18.3747	−0.666083	−0.333042	−	0.942912i	$-0.608075\pi$
$761$	−18.3747	−0.666083	−0.333042	+	0.942912i	$0.608075\pi$
$762$	0	0
$763$	−10.5409	−0.381608
$764$	0	0
$765$	−3.67722	−0.132950
$766$	0	0
$767$	32.0323	1.15662
$768$	0	0
$769$	23.6592	0.853174	0.426587	−	0.904447i	$-0.359716\pi$
$769$	23.6592	0.853174	0.426587	+	0.904447i	$0.359716\pi$
$770$	0	0
$771$	19.5369	0.703605
$772$	0	0
$773$	−7.11589	−0.255941	−0.127970	−	0.991778i	$-0.540846\pi$
$773$	−7.11589	−0.255941	−0.127970	+	0.991778i	$0.540846\pi$
$774$	0	0
$775$	−1.83297	−0.0658423
$776$	0	0
$777$	7.36694	0.264288
$778$	0	0
$779$	−1.23060	−0.0440908
$780$	0	0
$781$	6.76049	0.241909
$782$	0	0
$783$	−10.4793	−0.374499
$784$	0	0
$785$	−7.31527	−0.261093
$786$	0	0
$787$	−18.0116	−0.642045	−0.321023	−	0.947072i	$-0.604027\pi$
$787$	−18.0116	−0.642045	−0.321023	+	0.947072i	$0.604027\pi$
$788$	0	0
$789$	29.7280	1.05835
$790$	0	0
$791$	14.9496	0.531545
$792$	0	0
$793$	−1.13741	−0.0403905
$794$	0	0
$795$	−5.77675	−0.204880
$796$	0	0
$797$	25.2309	0.893725	0.446863	−	0.894603i	$-0.352541\pi$
$797$	25.2309	0.893725	0.446863	+	0.894603i	$0.352541\pi$
$798$	0	0
$799$	2.64306	0.0935048
$800$	0	0
$801$	−45.1454	−1.59514
$802$	0	0
$803$	8.35082	0.294694
$804$	0	0
$805$	12.4683	0.439451
$806$	0	0
$807$	−53.8304	−1.89492
$808$	0	0
$809$	3.88600	0.136624	0.0683122	−	0.997664i	$-0.478239\pi$
$809$	3.88600	0.136624	0.0683122	+	0.997664i	$0.478239\pi$
$810$	0	0
$811$	−39.8142	−1.39807	−0.699033	−	0.715090i	$-0.746386\pi$
$811$	−39.8142	−1.39807	−0.699033	+	0.715090i	$0.746386\pi$
$812$	0	0
$813$	−26.5373	−0.930704
$814$	0	0
$815$	16.3750	0.573590
$816$	0	0
$817$	0.166983	0.00584199
$818$	0	0
$819$	−17.7233	−0.619301
$820$	0	0
$821$	16.6695	0.581769	0.290885	−	0.956758i	$-0.406050\pi$
$821$	16.6695	0.581769	0.290885	+	0.956758i	$0.406050\pi$
$822$	0	0
$823$	−17.4097	−0.606863	−0.303432	−	0.952853i	$-0.598132\pi$
$823$	−17.4097	−0.606863	−0.303432	+	0.952853i	$0.598132\pi$
$824$	0	0
$825$	0.338845	0.0117971
$826$	0	0
$827$	41.1975	1.43258	0.716288	−	0.697805i	$-0.245840\pi$
$827$	41.1975	1.43258	0.716288	+	0.697805i	$0.245840\pi$
$828$	0	0
$829$	20.2073	0.701830	0.350915	−	0.936407i	$-0.385871\pi$
$829$	20.2073	0.701830	0.350915	+	0.936407i	$0.385871\pi$
$830$	0	0
$831$	−45.9913	−1.59542
$832$	0	0
$833$	−2.53030	−0.0876696
$834$	0	0
$835$	14.3588	0.496907
$836$	0	0
$837$	−10.0543	−0.347528
$838$	0	0
$839$	−26.8630	−0.927412	−0.463706	−	0.885989i	$-0.653481\pi$
$839$	−26.8630	−0.927412	−0.463706	+	0.885989i	$0.653481\pi$
$840$	0	0
$841$	66.5249	2.29396
$842$	0	0
$843$	−14.5961	−0.502716
$844$	0	0
$845$	−7.82841	−0.269306
$846$	0	0
$847$	−13.4514	−0.462196
$848$	0	0
$849$	−34.0248	−1.16773
$850$	0	0
$851$	−9.74788	−0.334153
$852$	0	0
$853$	−11.3972	−0.390234	−0.195117	−	0.980780i	$-0.562509\pi$
$853$	−11.3972	−0.390234	−0.195117	+	0.980780i	$0.562509\pi$
$854$	0	0
$855$	7.80238	0.266836
$856$	0	0
$857$	36.1562	1.23507	0.617537	−	0.786542i	$-0.288131\pi$
$857$	36.1562	1.23507	0.617537	+	0.786542i	$0.288131\pi$
$858$	0	0
$859$	37.5140	1.27996	0.639981	−	0.768391i	$-0.278942\pi$
$859$	37.5140	1.27996	0.639981	+	0.768391i	$0.278942\pi$
$860$	0	0
$861$	3.98326	0.135749
$862$	0	0
$863$	−32.4883	−1.10591	−0.552957	−	0.833209i	$-0.686501\pi$
$863$	−32.4883	−1.10591	−0.552957	+	0.833209i	$0.686501\pi$
$864$	0	0
$865$	12.8305	0.436250
$866$	0	0
$867$	−42.5215	−1.44410
$868$	0	0
$869$	−6.35105	−0.215445
$870$	0	0
$871$	−11.4937	−0.389450
$872$	0	0
$873$	−0.198546	−0.00671977
$874$	0	0
$875$	13.9867	0.472836
$876$	0	0
$877$	18.9563	0.640109	0.320055	−	0.947399i	$-0.396299\pi$
$877$	18.9563	0.640109	0.320055	+	0.947399i	$0.396299\pi$
$878$	0	0
$879$	5.52567	0.186376
$880$	0	0
$881$	14.9684	0.504299	0.252149	−	0.967688i	$-0.418863\pi$
$881$	14.9684	0.504299	0.252149	+	0.967688i	$0.418863\pi$
$882$	0	0
$883$	42.9519	1.44545	0.722724	−	0.691137i	$-0.242890\pi$
$883$	42.9519	1.44545	0.722724	+	0.691137i	$0.242890\pi$
$884$	0	0
$885$	45.6450	1.53434
$886$	0	0
$887$	−33.7975	−1.13481	−0.567404	−	0.823440i	$-0.692052\pi$
$887$	−33.7975	−1.13481	−0.567404	+	0.823440i	$0.692052\pi$
$888$	0	0
$889$	−9.55472	−0.320455
$890$	0	0
$891$	−5.16542	−0.173048
$892$	0	0
$893$	−5.60808	−0.187667
$894$	0	0
$895$	13.5826	0.454018
$896$	0	0
$897$	44.0042	1.46926
$898$	0	0
$899$	91.6510	3.05673
$900$	0	0
$901$	0.471295	0.0157011
$902$	0	0
$903$	−0.540498	−0.0179866
$904$	0	0
$905$	28.2889	0.940356
$906$	0	0
$907$	−52.9182	−1.75712	−0.878560	−	0.477632i	$-0.841495\pi$
$907$	−52.9182	−1.75712	−0.878560	+	0.477632i	$0.841495\pi$
$908$	0	0
$909$	−50.3155	−1.66886
$910$	0	0
$911$	9.46241	0.313503	0.156752	−	0.987638i	$-0.449898\pi$
$911$	9.46241	0.313503	0.156752	+	0.987638i	$0.449898\pi$
$912$	0	0
$913$	8.49283	0.281071
$914$	0	0
$915$	−1.62077	−0.0535810
$916$	0	0
$917$	−2.71932	−0.0897998
$918$	0	0
$919$	−44.5335	−1.46902	−0.734512	−	0.678596i	$-0.762589\pi$
$919$	−44.5335	−1.46902	−0.734512	+	0.678596i	$0.762589\pi$
$920$	0	0
$921$	39.0401	1.28641
$922$	0	0
$923$	−40.0685	−1.31887
$924$	0	0
$925$	0.444879	0.0146275
$926$	0	0
$927$	50.9954	1.67491
$928$	0	0
$929$	58.7018	1.92594	0.962971	−	0.269604i	$-0.0868927\pi$
$929$	58.7018	1.92594	0.962971	+	0.269604i	$0.0868927\pi$
$930$	0	0
$931$	5.36882	0.175956
$932$	0	0
$933$	30.2949	0.991811
$934$	0	0
$935$	−0.734781	−0.0240299
$936$	0	0
$937$	−49.1190	−1.60465	−0.802323	−	0.596890i	$-0.796403\pi$
$937$	−49.1190	−1.60465	−0.802323	+	0.596890i	$0.796403\pi$
$938$	0	0
$939$	2.19690	0.0716932
$940$	0	0
$941$	0.186194	0.00606975	0.00303487	−	0.999995i	$-0.499034\pi$
$941$	0.186194	0.00606975	0.00303487	+	0.999995i	$0.499034\pi$
$942$	0	0
$943$	−5.27062	−0.171635
$944$	0	0
$945$	−3.12131	−0.101536
$946$	0	0
$947$	−26.5194	−0.861764	−0.430882	−	0.902408i	$-0.641797\pi$
$947$	−26.5194	−0.861764	−0.430882	+	0.902408i	$0.641797\pi$
$948$	0	0
$949$	−49.4942	−1.60665
$950$	0	0
$951$	53.1365	1.72307
$952$	0	0
$953$	−0.310683	−0.0100640	−0.00503200	−	0.999987i	$-0.501602\pi$
$953$	−0.310683	−0.0100640	−0.00503200	+	0.999987i	$0.501602\pi$
$954$	0	0
$955$	32.6575	1.05677
$956$	0	0
$957$	−16.9427	−0.547679
$958$	0	0
$959$	−2.45828	−0.0793820
$960$	0	0
$961$	56.9342	1.83659
$962$	0	0
$963$	62.3883	2.01043
$964$	0	0
$965$	14.2923	0.460085
$966$	0	0
$967$	−22.4435	−0.721734	−0.360867	−	0.932617i	$-0.617519\pi$
$967$	−22.4435	−0.721734	−0.360867	+	0.932617i	$0.617519\pi$
$968$	0	0
$969$	−1.19444	−0.0383709
$970$	0	0
$971$	3.77769	0.121232	0.0606159	−	0.998161i	$-0.480694\pi$
$971$	3.77769	0.121232	0.0606159	+	0.998161i	$0.480694\pi$
$972$	0	0
$973$	−16.8614	−0.540553
$974$	0	0
$975$	−2.00829	−0.0643167
$976$	0	0
$977$	−18.0410	−0.577184	−0.288592	−	0.957452i	$-0.593187\pi$
$977$	−18.0410	−0.577184	−0.288592	+	0.957452i	$0.593187\pi$
$978$	0	0
$979$	−9.02094	−0.288311
$980$	0	0
$981$	−28.2516	−0.902004
$982$	0	0
$983$	48.3170	1.54107	0.770537	−	0.637395i	$-0.219988\pi$
$983$	48.3170	1.54107	0.770537	+	0.637395i	$0.219988\pi$
$984$	0	0
$985$	39.4510	1.25701
$986$	0	0
$987$	18.1525	0.577800
$988$	0	0
$989$	0.715183	0.0227415
$990$	0	0
$991$	−23.2630	−0.738974	−0.369487	−	0.929236i	$-0.620467\pi$
$991$	−23.2630	−0.738974	−0.369487	+	0.929236i	$0.620467\pi$
$992$	0	0
$993$	68.6704	2.17919
$994$	0	0
$995$	−31.1622	−0.987908
$996$	0	0
$997$	15.2301	0.482343	0.241172	−	0.970482i	$-0.422468\pi$
$997$	15.2301	0.482343	0.241172	+	0.970482i	$0.422468\pi$
$998$	0	0
$999$	2.44028	0.0772069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.d.1.17	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.d.1.17	✓	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)

\(f(q)\)	\(=\)	\(q+2.53438 q^{3} -2.27936 q^{5} +1.27718 q^{7} +3.42306 q^{9} +O(q^{10})\)	\(q+2.53438 q^{3} -2.27936 q^{5} +1.27718 q^{7} +3.42306 q^{9} +0.683995 q^{11} -4.05395 q^{13} -5.77675 q^{15} +0.471295 q^{17} -1.00000 q^{19} +3.23685 q^{21} -4.28297 q^{23} +0.195468 q^{25} +1.07220 q^{27} -9.77368 q^{29} -9.37733 q^{31} +1.73350 q^{33} -2.91114 q^{35} +2.27596 q^{37} -10.2742 q^{39} +1.23060 q^{41} -0.166983 q^{43} -7.80238 q^{45} +5.60808 q^{47} -5.36882 q^{49} +1.19444 q^{51} +1.00000 q^{53} -1.55907 q^{55} -2.53438 q^{57} -7.90151 q^{59} +0.280568 q^{61} +4.37185 q^{63} +9.24039 q^{65} +2.83519 q^{67} -10.8547 q^{69} +9.88384 q^{71} +12.2089 q^{73} +0.495391 q^{75} +0.873582 q^{77} -9.28523 q^{79} -7.55184 q^{81} +12.4165 q^{83} -1.07425 q^{85} -24.7702 q^{87} -13.1886 q^{89} -5.17761 q^{91} -23.7657 q^{93} +2.27936 q^{95} -0.0580025 q^{97} +2.34136 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 4 q^{3} - 2 q^{5} - 11 q^{7} + 19 q^{9}+O(q^{10}) \) 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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