LMFDB - Embedded newform 4028.2.a.e.1.3

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:	$0$
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} - 3 x^{18} - 35 x^{17} + 103 x^{16} + 501 x^{15} - 1437 x^{14} - 3775 x^{13} + 10450 x^{12} + \cdots - 4 $ x^19 - 3x^18 - 35x^17 + 103x^16 + 501x^15 - 1437x^14 - 3775x^13 + 10450x^12 + 16076x^11 - 42255x^10 - 38701x^9 + 93907x^8 + 49522x^7 - 106284x^6 - 26703x^5 + 50522x^4 - 446x^3 - 6265x^2 + 719x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.59429$ 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)

$f(q)$	$=$	$q-2.59429 q^{3} -4.28452 q^{5} +2.47228 q^{7} +3.73036 q^{9} +O(q^{10})$	$q-2.59429 q^{3} -4.28452 q^{5} +2.47228 q^{7} +3.73036 q^{9} +4.82881 q^{11} +2.84120 q^{13} +11.1153 q^{15} +0.195360 q^{17} +1.00000 q^{19} -6.41382 q^{21} -5.44622 q^{23} +13.3571 q^{25} -1.89477 q^{27} +6.56077 q^{29} -1.61368 q^{31} -12.5274 q^{33} -10.5925 q^{35} +2.49997 q^{37} -7.37091 q^{39} -3.46085 q^{41} +8.00519 q^{43} -15.9828 q^{45} -8.56512 q^{47} -0.887838 q^{49} -0.506821 q^{51} +1.00000 q^{53} -20.6891 q^{55} -2.59429 q^{57} -9.46258 q^{59} +3.49015 q^{61} +9.22249 q^{63} -12.1732 q^{65} +6.20630 q^{67} +14.1291 q^{69} +2.43322 q^{71} -2.35255 q^{73} -34.6522 q^{75} +11.9382 q^{77} +0.407637 q^{79} -6.27550 q^{81} +0.641217 q^{83} -0.837022 q^{85} -17.0206 q^{87} -7.32031 q^{89} +7.02424 q^{91} +4.18635 q^{93} -4.28452 q^{95} -3.69046 q^{97} +18.0132 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	$ 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) $ 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * 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
$3$	−2.59429	−1.49782	−0.748908	−	0.662674i	$-0.769422\pi$
$3$	−2.59429	−1.49782	−0.748908	+	0.662674i	$0.769422\pi$
$4$	0	0
$5$	−4.28452	−1.91609	−0.958047	−	0.286611i	$-0.907471\pi$
$5$	−4.28452	−1.91609	−0.958047	+	0.286611i	$0.907471\pi$
$6$	0	0
$7$	2.47228	0.934434	0.467217	−	0.884143i	$-0.345257\pi$
$7$	2.47228	0.934434	0.467217	+	0.884143i	$0.345257\pi$
$8$	0	0
$9$	3.73036	1.24345
$10$	0	0
$11$	4.82881	1.45594	0.727971	−	0.685608i	$-0.240464\pi$
$11$	4.82881	1.45594	0.727971	+	0.685608i	$0.240464\pi$
$12$	0	0
$13$	2.84120	0.788008	0.394004	−	0.919109i	$-0.371090\pi$
$13$	2.84120	0.788008	0.394004	+	0.919109i	$0.371090\pi$
$14$	0	0
$15$	11.1153	2.86996
$16$	0	0
$17$	0.195360	0.0473817	0.0236909	−	0.999719i	$-0.492458\pi$
$17$	0.195360	0.0473817	0.0236909	+	0.999719i	$0.492458\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−6.41382	−1.39961
$22$	0	0
$23$	−5.44622	−1.13562	−0.567808	−	0.823161i	$-0.692208\pi$
$23$	−5.44622	−1.13562	−0.567808	+	0.823161i	$0.692208\pi$
$24$	0	0
$25$	13.3571	2.67142
$26$	0	0
$27$	−1.89477	−0.364648
$28$	0	0
$29$	6.56077	1.21831	0.609153	−	0.793053i	$-0.291510\pi$
$29$	6.56077	1.21831	0.609153	+	0.793053i	$0.291510\pi$
$30$	0	0
$31$	−1.61368	−0.289825	−0.144912	−	0.989444i	$-0.546290\pi$
$31$	−1.61368	−0.289825	−0.144912	+	0.989444i	$0.546290\pi$
$32$	0	0
$33$	−12.5274	−2.18073
$34$	0	0
$35$	−10.5925	−1.79046
$36$	0	0
$37$	2.49997	0.410993	0.205496	−	0.978658i	$-0.434119\pi$
$37$	2.49997	0.410993	0.205496	+	0.978658i	$0.434119\pi$
$38$	0	0
$39$	−7.37091	−1.18029
$40$	0	0
$41$	−3.46085	−0.540494	−0.270247	−	0.962791i	$-0.587105\pi$
$41$	−3.46085	−0.540494	−0.270247	+	0.962791i	$0.587105\pi$
$42$	0	0
$43$	8.00519	1.22078	0.610390	−	0.792101i	$-0.291013\pi$
$43$	8.00519	1.22078	0.610390	+	0.792101i	$0.291013\pi$
$44$	0	0
$45$	−15.9828	−2.38257
$46$	0	0
$47$	−8.56512	−1.24935	−0.624676	−	0.780884i	$-0.714769\pi$
$47$	−8.56512	−1.24935	−0.624676	+	0.780884i	$0.714769\pi$
$48$	0	0
$49$	−0.887838	−0.126834
$50$	0	0
$51$	−0.506821	−0.0709691
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−20.6891	−2.78972
$56$	0	0
$57$	−2.59429	−0.343623
$58$	0	0
$59$	−9.46258	−1.23192	−0.615961	−	0.787776i	$-0.711232\pi$
$59$	−9.46258	−1.23192	−0.615961	+	0.787776i	$0.711232\pi$
$60$	0	0
$61$	3.49015	0.446868	0.223434	−	0.974719i	$-0.428273\pi$
$61$	3.49015	0.446868	0.223434	+	0.974719i	$0.428273\pi$
$62$	0	0
$63$	9.22249	1.16192
$64$	0	0
$65$	−12.1732	−1.50990
$66$	0	0
$67$	6.20630	0.758221	0.379110	−	0.925352i	$-0.376230\pi$
$67$	6.20630	0.758221	0.379110	+	0.925352i	$0.376230\pi$
$68$	0	0
$69$	14.1291	1.70094
$70$	0	0
$71$	2.43322	0.288770	0.144385	−	0.989522i	$-0.453880\pi$
$71$	2.43322	0.288770	0.144385	+	0.989522i	$0.453880\pi$
$72$	0	0
$73$	−2.35255	−0.275345	−0.137672	−	0.990478i	$-0.543962\pi$
$73$	−2.35255	−0.275345	−0.137672	+	0.990478i	$0.543962\pi$
$74$	0	0
$75$	−34.6522	−4.00129
$76$	0	0
$77$	11.9382	1.36048
$78$	0	0
$79$	0.407637	0.0458628	0.0229314	−	0.999737i	$-0.492700\pi$
$79$	0.407637	0.0458628	0.0229314	+	0.999737i	$0.492700\pi$
$80$	0	0
$81$	−6.27550	−0.697278
$82$	0	0
$83$	0.641217	0.0703827	0.0351914	−	0.999381i	$-0.488796\pi$
$83$	0.641217	0.0703827	0.0351914	+	0.999381i	$0.488796\pi$
$84$	0	0
$85$	−0.837022	−0.0907878
$86$	0	0
$87$	−17.0206	−1.82480
$88$	0	0
$89$	−7.32031	−0.775951	−0.387975	−	0.921670i	$-0.626825\pi$
$89$	−7.32031	−0.775951	−0.387975	+	0.921670i	$0.626825\pi$
$90$	0	0
$91$	7.02424	0.736341
$92$	0	0
$93$	4.18635	0.434104
$94$	0	0
$95$	−4.28452	−0.439582
$96$	0	0
$97$	−3.69046	−0.374710	−0.187355	−	0.982292i	$-0.559991\pi$
$97$	−3.69046	−0.374710	−0.187355	+	0.982292i	$0.559991\pi$
$98$	0	0
$99$	18.0132	1.81039
$100$	0	0
$101$	4.52120	0.449876	0.224938	−	0.974373i	$-0.427782\pi$
$101$	4.52120	0.449876	0.224938	+	0.974373i	$0.427782\pi$
$102$	0	0
$103$	9.05706	0.892418	0.446209	−	0.894929i	$-0.352774\pi$
$103$	9.05706	0.892418	0.446209	+	0.894929i	$0.352774\pi$
$104$	0	0
$105$	27.4801	2.68178
$106$	0	0
$107$	−6.75177	−0.652718	−0.326359	−	0.945246i	$-0.605822\pi$
$107$	−6.75177	−0.652718	−0.326359	+	0.945246i	$0.605822\pi$
$108$	0	0
$109$	−10.8996	−1.04399	−0.521995	−	0.852949i	$-0.674812\pi$
$109$	−10.8996	−1.04399	−0.521995	+	0.852949i	$0.674812\pi$
$110$	0	0
$111$	−6.48566	−0.615592
$112$	0	0
$113$	0.960989	0.0904022	0.0452011	−	0.998978i	$-0.485607\pi$
$113$	0.960989	0.0904022	0.0452011	+	0.998978i	$0.485607\pi$
$114$	0	0
$115$	23.3344	2.17595
$116$	0	0
$117$	10.5987	0.979850
$118$	0	0
$119$	0.482984	0.0442751
$120$	0	0
$121$	12.3174	1.11976
$122$	0	0
$123$	8.97846	0.809561
$124$	0	0
$125$	−35.8061	−3.20259
$126$	0	0
$127$	11.8797	1.05415	0.527077	−	0.849817i	$-0.323288\pi$
$127$	11.8797	1.05415	0.527077	+	0.849817i	$0.323288\pi$
$128$	0	0
$129$	−20.7678	−1.82850
$130$	0	0
$131$	8.31849	0.726790	0.363395	−	0.931635i	$-0.381618\pi$
$131$	8.31849	0.726790	0.363395	+	0.931635i	$0.381618\pi$
$132$	0	0
$133$	2.47228	0.214374
$134$	0	0
$135$	8.11816	0.698700
$136$	0	0
$137$	14.4264	1.23253	0.616264	−	0.787540i	$-0.288646\pi$
$137$	14.4264	1.23253	0.616264	+	0.787540i	$0.288646\pi$
$138$	0	0
$139$	−15.2597	−1.29431	−0.647157	−	0.762357i	$-0.724042\pi$
$139$	−15.2597	−1.29431	−0.647157	+	0.762357i	$0.724042\pi$
$140$	0	0
$141$	22.2204	1.87130
$142$	0	0
$143$	13.7196	1.14729
$144$	0	0
$145$	−28.1097	−2.33439
$146$	0	0
$147$	2.30331	0.189974
$148$	0	0
$149$	−8.45498	−0.692659	−0.346329	−	0.938113i	$-0.612572\pi$
$149$	−8.45498	−0.692659	−0.346329	+	0.938113i	$0.612572\pi$
$150$	0	0
$151$	−6.84884	−0.557351	−0.278675	−	0.960385i	$-0.589895\pi$
$151$	−6.84884	−0.557351	−0.278675	+	0.960385i	$0.589895\pi$
$152$	0	0
$153$	0.728762	0.0589169
$154$	0	0
$155$	6.91383	0.555332
$156$	0	0
$157$	24.7548	1.97565	0.987823	−	0.155581i	$-0.0497250\pi$
$157$	24.7548	1.97565	0.987823	+	0.155581i	$0.0497250\pi$
$158$	0	0
$159$	−2.59429	−0.205741
$160$	0	0
$161$	−13.4646	−1.06116
$162$	0	0
$163$	−4.40454	−0.344990	−0.172495	−	0.985010i	$-0.555183\pi$
$163$	−4.40454	−0.344990	−0.172495	+	0.985010i	$0.555183\pi$
$164$	0	0
$165$	53.6737	4.17849
$166$	0	0
$167$	15.9939	1.23764	0.618822	−	0.785532i	$-0.287610\pi$
$167$	15.9939	1.23764	0.618822	+	0.785532i	$0.287610\pi$
$168$	0	0
$169$	−4.92757	−0.379044
$170$	0	0
$171$	3.73036	0.285268
$172$	0	0
$173$	17.3677	1.32044	0.660222	−	0.751071i	$-0.270462\pi$
$173$	17.3677	1.32044	0.660222	+	0.751071i	$0.270462\pi$
$174$	0	0
$175$	33.0224	2.49626
$176$	0	0
$177$	24.5487	1.84519
$178$	0	0
$179$	−4.51356	−0.337359	−0.168680	−	0.985671i	$-0.553950\pi$
$179$	−4.51356	−0.337359	−0.168680	+	0.985671i	$0.553950\pi$
$180$	0	0
$181$	21.5845	1.60436	0.802181	−	0.597081i	$-0.203673\pi$
$181$	21.5845	1.60436	0.802181	+	0.597081i	$0.203673\pi$
$182$	0	0
$183$	−9.05448	−0.669327
$184$	0	0
$185$	−10.7112	−0.787501
$186$	0	0
$187$	0.943355	0.0689850
$188$	0	0
$189$	−4.68439	−0.340739
$190$	0	0
$191$	13.0035	0.940903	0.470451	−	0.882426i	$-0.344091\pi$
$191$	13.0035	0.940903	0.470451	+	0.882426i	$0.344091\pi$
$192$	0	0
$193$	10.4056	0.749014	0.374507	−	0.927224i	$-0.377812\pi$
$193$	10.4056	0.749014	0.374507	+	0.927224i	$0.377812\pi$
$194$	0	0
$195$	31.5808	2.26155
$196$	0	0
$197$	19.2662	1.37266	0.686331	−	0.727290i	$-0.259220\pi$
$197$	19.2662	1.37266	0.686331	+	0.727290i	$0.259220\pi$
$198$	0	0
$199$	17.9610	1.27322	0.636612	−	0.771185i	$-0.280335\pi$
$199$	17.9610	1.27322	0.636612	+	0.771185i	$0.280335\pi$
$200$	0	0
$201$	−16.1010	−1.13567
$202$	0	0
$203$	16.2201	1.13843
$204$	0	0
$205$	14.8281	1.03564
$206$	0	0
$207$	−20.3164	−1.41208
$208$	0	0
$209$	4.82881	0.334016
$210$	0	0
$211$	−5.22984	−0.360037	−0.180018	−	0.983663i	$-0.557616\pi$
$211$	−5.22984	−0.360037	−0.180018	+	0.983663i	$0.557616\pi$
$212$	0	0
$213$	−6.31249	−0.432524
$214$	0	0
$215$	−34.2984	−2.33913
$216$	0	0
$217$	−3.98946	−0.270822
$218$	0	0
$219$	6.10320	0.412416
$220$	0	0
$221$	0.555057	0.0373371
$222$	0	0
$223$	12.7009	0.850514	0.425257	−	0.905073i	$-0.360184\pi$
$223$	12.7009	0.850514	0.425257	+	0.905073i	$0.360184\pi$
$224$	0	0
$225$	49.8267	3.32178
$226$	0	0
$227$	21.1082	1.40100	0.700502	−	0.713651i	$-0.252960\pi$
$227$	21.1082	1.40100	0.700502	+	0.713651i	$0.252960\pi$
$228$	0	0
$229$	−19.6839	−1.30075	−0.650373	−	0.759615i	$-0.725387\pi$
$229$	−19.6839	−1.30075	−0.650373	+	0.759615i	$0.725387\pi$
$230$	0	0
$231$	−30.9711	−2.03775
$232$	0	0
$233$	−13.2924	−0.870813	−0.435406	−	0.900234i	$-0.643395\pi$
$233$	−13.2924	−0.870813	−0.435406	+	0.900234i	$0.643395\pi$
$234$	0	0
$235$	36.6974	2.39387
$236$	0	0
$237$	−1.05753	−0.0686940
$238$	0	0
$239$	29.2460	1.89177	0.945884	−	0.324505i	$-0.105198\pi$
$239$	29.2460	1.89177	0.945884	+	0.324505i	$0.105198\pi$
$240$	0	0
$241$	−11.6123	−0.748012	−0.374006	−	0.927426i	$-0.622016\pi$
$241$	−11.6123	−0.748012	−0.374006	+	0.927426i	$0.622016\pi$
$242$	0	0
$243$	21.9648	1.40904
$244$	0	0
$245$	3.80396	0.243026
$246$	0	0
$247$	2.84120	0.180781
$248$	0	0
$249$	−1.66351	−0.105420
$250$	0	0
$251$	9.24082	0.583275	0.291638	−	0.956529i	$-0.405800\pi$
$251$	9.24082	0.583275	0.291638	+	0.956529i	$0.405800\pi$
$252$	0	0
$253$	−26.2988	−1.65339
$254$	0	0
$255$	2.17148	0.135983
$256$	0	0
$257$	−29.1004	−1.81524	−0.907618	−	0.419797i	$-0.862101\pi$
$257$	−29.1004	−1.81524	−0.907618	+	0.419797i	$0.862101\pi$
$258$	0	0
$259$	6.18063	0.384045
$260$	0	0
$261$	24.4740	1.51491
$262$	0	0
$263$	−14.2691	−0.879873	−0.439937	−	0.898029i	$-0.644999\pi$
$263$	−14.2691	−0.879873	−0.439937	+	0.898029i	$0.644999\pi$
$264$	0	0
$265$	−4.28452	−0.263196
$266$	0	0
$267$	18.9910	1.16223
$268$	0	0
$269$	−23.9396	−1.45962	−0.729811	−	0.683649i	$-0.760392\pi$
$269$	−23.9396	−1.45962	−0.729811	+	0.683649i	$0.760392\pi$
$270$	0	0
$271$	−10.6770	−0.648580	−0.324290	−	0.945958i	$-0.605125\pi$
$271$	−10.6770	−0.648580	−0.324290	+	0.945958i	$0.605125\pi$
$272$	0	0
$273$	−18.2229	−1.10290
$274$	0	0
$275$	64.4988	3.88943
$276$	0	0
$277$	−26.9800	−1.62107	−0.810535	−	0.585690i	$-0.800824\pi$
$277$	−26.9800	−1.62107	−0.810535	+	0.585690i	$0.800824\pi$
$278$	0	0
$279$	−6.01959	−0.360384
$280$	0	0
$281$	20.2936	1.21061	0.605307	−	0.795992i	$-0.293050\pi$
$281$	20.2936	1.21061	0.605307	+	0.795992i	$0.293050\pi$
$282$	0	0
$283$	8.58766	0.510484	0.255242	−	0.966877i	$-0.417845\pi$
$283$	8.58766	0.510484	0.255242	+	0.966877i	$0.417845\pi$
$284$	0	0
$285$	11.1153	0.658413
$286$	0	0
$287$	−8.55619	−0.505056
$288$	0	0
$289$	−16.9618	−0.997755
$290$	0	0
$291$	9.57414	0.561246
$292$	0	0
$293$	15.0706	0.880437	0.440218	−	0.897891i	$-0.354901\pi$
$293$	15.0706	0.880437	0.440218	+	0.897891i	$0.354901\pi$
$294$	0	0
$295$	40.5426	2.36048
$296$	0	0
$297$	−9.14946	−0.530906
$298$	0	0
$299$	−15.4738	−0.894873
$300$	0	0
$301$	19.7911	1.14074
$302$	0	0
$303$	−11.7293	−0.673831
$304$	0	0
$305$	−14.9536	−0.856242
$306$	0	0
$307$	16.9632	0.968141	0.484070	−	0.875029i	$-0.339158\pi$
$307$	16.9632	0.968141	0.484070	+	0.875029i	$0.339158\pi$
$308$	0	0
$309$	−23.4967	−1.33668
$310$	0	0
$311$	−12.1658	−0.689858	−0.344929	−	0.938629i	$-0.612097\pi$
$311$	−12.1658	−0.689858	−0.344929	+	0.938629i	$0.612097\pi$
$312$	0	0
$313$	−33.3371	−1.88432	−0.942161	−	0.335160i	$-0.891210\pi$
$313$	−33.3371	−1.88432	−0.942161	+	0.335160i	$0.891210\pi$
$314$	0	0
$315$	−39.5139	−2.22636
$316$	0	0
$317$	1.74645	0.0980902	0.0490451	−	0.998797i	$-0.484382\pi$
$317$	1.74645	0.0980902	0.0490451	+	0.998797i	$0.484382\pi$
$318$	0	0
$319$	31.6807	1.77378
$320$	0	0
$321$	17.5161	0.977652
$322$	0	0
$323$	0.195360	0.0108701
$324$	0	0
$325$	37.9502	2.10510
$326$	0	0
$327$	28.2767	1.56371
$328$	0	0
$329$	−21.1754	−1.16744
$330$	0	0
$331$	21.4129	1.17696	0.588480	−	0.808512i	$-0.299726\pi$
$331$	21.4129	1.17696	0.588480	+	0.808512i	$0.299726\pi$
$332$	0	0
$333$	9.32579	0.511050
$334$	0	0
$335$	−26.5910	−1.45282
$336$	0	0
$337$	−28.5965	−1.55775	−0.778875	−	0.627179i	$-0.784210\pi$
$337$	−28.5965	−1.55775	−0.778875	+	0.627179i	$0.784210\pi$
$338$	0	0
$339$	−2.49309	−0.135406
$340$	0	0
$341$	−7.79214	−0.421968
$342$	0	0
$343$	−19.5009	−1.05295
$344$	0	0
$345$	−60.5363	−3.25917
$346$	0	0
$347$	−11.6489	−0.625346	−0.312673	−	0.949861i	$-0.601224\pi$
$347$	−11.6489	−0.625346	−0.312673	+	0.949861i	$0.601224\pi$
$348$	0	0
$349$	−21.3133	−1.14087	−0.570437	−	0.821341i	$-0.693226\pi$
$349$	−21.3133	−1.14087	−0.570437	+	0.821341i	$0.693226\pi$
$350$	0	0
$351$	−5.38341	−0.287345
$352$	0	0
$353$	−4.79810	−0.255377	−0.127689	−	0.991814i	$-0.540756\pi$
$353$	−4.79810	−0.255377	−0.127689	+	0.991814i	$0.540756\pi$
$354$	0	0
$355$	−10.4252	−0.553311
$356$	0	0
$357$	−1.25300	−0.0663159
$358$	0	0
$359$	18.5438	0.978704	0.489352	−	0.872086i	$-0.337233\pi$
$359$	18.5438	0.978704	0.489352	+	0.872086i	$0.337233\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−31.9550	−1.67720
$364$	0	0
$365$	10.0795	0.527587
$366$	0	0
$367$	−7.44506	−0.388629	−0.194314	−	0.980939i	$-0.562248\pi$
$367$	−7.44506	−0.388629	−0.194314	+	0.980939i	$0.562248\pi$
$368$	0	0
$369$	−12.9102	−0.672079
$370$	0	0
$371$	2.47228	0.128354
$372$	0	0
$373$	13.2592	0.686535	0.343268	−	0.939238i	$-0.388466\pi$
$373$	13.2592	0.686535	0.343268	+	0.939238i	$0.388466\pi$
$374$	0	0
$375$	92.8915	4.79690
$376$	0	0
$377$	18.6405	0.960034
$378$	0	0
$379$	−14.6051	−0.750214	−0.375107	−	0.926981i	$-0.622394\pi$
$379$	−14.6051	−0.750214	−0.375107	+	0.926981i	$0.622394\pi$
$380$	0	0
$381$	−30.8195	−1.57893
$382$	0	0
$383$	−27.2754	−1.39371	−0.696855	−	0.717213i	$-0.745418\pi$
$383$	−27.2754	−1.39371	−0.696855	+	0.717213i	$0.745418\pi$
$384$	0	0
$385$	−51.1493	−2.60681
$386$	0	0
$387$	29.8622	1.51798
$388$	0	0
$389$	7.08038	0.358989	0.179495	−	0.983759i	$-0.442554\pi$
$389$	7.08038	0.358989	0.179495	+	0.983759i	$0.442554\pi$
$390$	0	0
$391$	−1.06397	−0.0538074
$392$	0	0
$393$	−21.5806	−1.08860
$394$	0	0
$395$	−1.74653	−0.0878774
$396$	0	0
$397$	33.1017	1.66133	0.830664	−	0.556774i	$-0.187961\pi$
$397$	33.1017	1.66133	0.830664	+	0.556774i	$0.187961\pi$
$398$	0	0
$399$	−6.41382	−0.321092
$400$	0	0
$401$	−1.56535	−0.0781699	−0.0390850	−	0.999236i	$-0.512444\pi$
$401$	−1.56535	−0.0781699	−0.0390850	+	0.999236i	$0.512444\pi$
$402$	0	0
$403$	−4.58478	−0.228384
$404$	0	0
$405$	26.8875	1.33605
$406$	0	0
$407$	12.0719	0.598381
$408$	0	0
$409$	36.0927	1.78467	0.892334	−	0.451376i	$-0.149067\pi$
$409$	36.0927	1.78467	0.892334	+	0.451376i	$0.149067\pi$
$410$	0	0
$411$	−37.4262	−1.84610
$412$	0	0
$413$	−23.3941	−1.15115
$414$	0	0
$415$	−2.74731	−0.134860
$416$	0	0
$417$	39.5882	1.93864
$418$	0	0
$419$	21.5249	1.05156	0.525779	−	0.850621i	$-0.323774\pi$
$419$	21.5249	1.05156	0.525779	+	0.850621i	$0.323774\pi$
$420$	0	0
$421$	27.8331	1.35650	0.678252	−	0.734829i	$-0.262738\pi$
$421$	27.8331	1.35650	0.678252	+	0.734829i	$0.262738\pi$
$422$	0	0
$423$	−31.9510	−1.55351
$424$	0	0
$425$	2.60944	0.126576
$426$	0	0
$427$	8.62863	0.417569
$428$	0	0
$429$	−35.5927	−1.71843
$430$	0	0
$431$	12.8248	0.617749	0.308874	−	0.951103i	$-0.400048\pi$
$431$	12.8248	0.617749	0.308874	+	0.951103i	$0.400048\pi$
$432$	0	0
$433$	0.568966	0.0273427	0.0136714	−	0.999907i	$-0.495648\pi$
$433$	0.568966	0.0273427	0.0136714	+	0.999907i	$0.495648\pi$
$434$	0	0
$435$	72.9249	3.49648
$436$	0	0
$437$	−5.44622	−0.260528
$438$	0	0
$439$	14.0971	0.672820	0.336410	−	0.941716i	$-0.390787\pi$
$439$	14.0971	0.672820	0.336410	+	0.941716i	$0.390787\pi$
$440$	0	0
$441$	−3.31195	−0.157712
$442$	0	0
$443$	−4.74993	−0.225676	−0.112838	−	0.993613i	$-0.535994\pi$
$443$	−4.74993	−0.225676	−0.112838	+	0.993613i	$0.535994\pi$
$444$	0	0
$445$	31.3640	1.48680
$446$	0	0
$447$	21.9347	1.03748
$448$	0	0
$449$	−6.76133	−0.319087	−0.159543	−	0.987191i	$-0.551002\pi$
$449$	−6.76133	−0.319087	−0.159543	+	0.987191i	$0.551002\pi$
$450$	0	0
$451$	−16.7118	−0.786927
$452$	0	0
$453$	17.7679	0.834809
$454$	0	0
$455$	−30.0955	−1.41090
$456$	0	0
$457$	−35.5592	−1.66339	−0.831694	−	0.555235i	$-0.812628\pi$
$457$	−35.5592	−1.66339	−0.831694	+	0.555235i	$0.812628\pi$
$458$	0	0
$459$	−0.370161	−0.0172776
$460$	0	0
$461$	−22.8182	−1.06275	−0.531374	−	0.847137i	$-0.678324\pi$
$461$	−22.8182	−1.06275	−0.531374	+	0.847137i	$0.678324\pi$
$462$	0	0
$463$	40.3661	1.87597	0.937987	−	0.346670i	$-0.112688\pi$
$463$	40.3661	1.87597	0.937987	+	0.346670i	$0.112688\pi$
$464$	0	0
$465$	−17.9365	−0.831785
$466$	0	0
$467$	−26.9164	−1.24554	−0.622772	−	0.782404i	$-0.713993\pi$
$467$	−26.9164	−1.24554	−0.622772	+	0.782404i	$0.713993\pi$
$468$	0	0
$469$	15.3437	0.708507
$470$	0	0
$471$	−64.2212	−2.95915
$472$	0	0
$473$	38.6556	1.77738
$474$	0	0
$475$	13.3571	0.612865
$476$	0	0
$477$	3.73036	0.170801
$478$	0	0
$479$	−6.96332	−0.318162	−0.159081	−	0.987266i	$-0.550853\pi$
$479$	−6.96332	−0.318162	−0.159081	+	0.987266i	$0.550853\pi$
$480$	0	0
$481$	7.10292	0.323865
$482$	0	0
$483$	34.9311	1.58942
$484$	0	0
$485$	15.8118	0.717979
$486$	0	0
$487$	43.7318	1.98167	0.990837	−	0.135060i	$-0.0431226\pi$
$487$	43.7318	1.98167	0.990837	+	0.135060i	$0.0431226\pi$
$488$	0	0
$489$	11.4267	0.516732
$490$	0	0
$491$	0.898423	0.0405453	0.0202726	−	0.999794i	$-0.493547\pi$
$491$	0.898423	0.0405453	0.0202726	+	0.999794i	$0.493547\pi$
$492$	0	0
$493$	1.28171	0.0577254
$494$	0	0
$495$	−77.1778	−3.46889
$496$	0	0
$497$	6.01560	0.269836
$498$	0	0
$499$	17.0496	0.763244	0.381622	−	0.924319i	$-0.375366\pi$
$499$	17.0496	0.763244	0.381622	+	0.924319i	$0.375366\pi$
$500$	0	0
$501$	−41.4928	−1.85376
$502$	0	0
$503$	−18.8346	−0.839792	−0.419896	−	0.907572i	$-0.637933\pi$
$503$	−18.8346	−0.839792	−0.419896	+	0.907572i	$0.637933\pi$
$504$	0	0
$505$	−19.3711	−0.862005
$506$	0	0
$507$	12.7836	0.567738
$508$	0	0
$509$	−0.724754	−0.0321241	−0.0160621	−	0.999871i	$-0.505113\pi$
$509$	−0.724754	−0.0321241	−0.0160621	+	0.999871i	$0.505113\pi$
$510$	0	0
$511$	−5.81615	−0.257291
$512$	0	0
$513$	−1.89477	−0.0836560
$514$	0	0
$515$	−38.8051	−1.70996
$516$	0	0
$517$	−41.3593	−1.81898
$518$	0	0
$519$	−45.0570	−1.97778
$520$	0	0
$521$	−10.8926	−0.477216	−0.238608	−	0.971116i	$-0.576691\pi$
$521$	−10.8926	−0.477216	−0.238608	+	0.971116i	$0.576691\pi$
$522$	0	0
$523$	30.2714	1.32367	0.661837	−	0.749647i	$-0.269777\pi$
$523$	30.2714	1.32367	0.661837	+	0.749647i	$0.269777\pi$
$524$	0	0
$525$	−85.6699	−3.73894
$526$	0	0
$527$	−0.315248	−0.0137324
$528$	0	0
$529$	6.66131	0.289622
$530$	0	0
$531$	−35.2988	−1.53184
$532$	0	0
$533$	−9.83297	−0.425913
$534$	0	0
$535$	28.9281	1.25067
$536$	0	0
$537$	11.7095	0.505302
$538$	0	0
$539$	−4.28720	−0.184663
$540$	0	0
$541$	−1.27133	−0.0546588	−0.0273294	−	0.999626i	$-0.508700\pi$
$541$	−1.27133	−0.0546588	−0.0273294	+	0.999626i	$0.508700\pi$
$542$	0	0
$543$	−55.9965	−2.40304
$544$	0	0
$545$	46.6994	2.00038
$546$	0	0
$547$	12.0511	0.515268	0.257634	−	0.966243i	$-0.417057\pi$
$547$	12.0511	0.515268	0.257634	+	0.966243i	$0.417057\pi$
$548$	0	0
$549$	13.0195	0.555660
$550$	0	0
$551$	6.56077	0.279498
$552$	0	0
$553$	1.00779	0.0428557
$554$	0	0
$555$	27.7879	1.17953
$556$	0	0
$557$	−6.37141	−0.269965	−0.134983	−	0.990848i	$-0.543098\pi$
$557$	−6.37141	−0.269965	−0.134983	+	0.990848i	$0.543098\pi$
$558$	0	0
$559$	22.7444	0.961984
$560$	0	0
$561$	−2.44734	−0.103327
$562$	0	0
$563$	8.96446	0.377807	0.188903	−	0.981996i	$-0.439507\pi$
$563$	8.96446	0.377807	0.188903	+	0.981996i	$0.439507\pi$
$564$	0	0
$565$	−4.11737	−0.173219
$566$	0	0
$567$	−15.5148	−0.651560
$568$	0	0
$569$	−19.1128	−0.801249	−0.400625	−	0.916242i	$-0.631207\pi$
$569$	−19.1128	−0.801249	−0.400625	+	0.916242i	$0.631207\pi$
$570$	0	0
$571$	1.04117	0.0435716	0.0217858	−	0.999763i	$-0.493065\pi$
$571$	1.04117	0.0435716	0.0217858	+	0.999763i	$0.493065\pi$
$572$	0	0
$573$	−33.7350	−1.40930
$574$	0	0
$575$	−72.7456	−3.03370
$576$	0	0
$577$	−15.5166	−0.645963	−0.322981	−	0.946405i	$-0.604685\pi$
$577$	−15.5166	−0.645963	−0.322981	+	0.946405i	$0.604685\pi$
$578$	0	0
$579$	−26.9953	−1.12188
$580$	0	0
$581$	1.58527	0.0657680
$582$	0	0
$583$	4.82881	0.199989
$584$	0	0
$585$	−45.4103	−1.87749
$586$	0	0
$587$	17.2066	0.710193	0.355096	−	0.934830i	$-0.384448\pi$
$587$	17.2066	0.710193	0.355096	+	0.934830i	$0.384448\pi$
$588$	0	0
$589$	−1.61368	−0.0664904
$590$	0	0
$591$	−49.9822	−2.05599
$592$	0	0
$593$	17.3620	0.712971	0.356486	−	0.934301i	$-0.383975\pi$
$593$	17.3620	0.712971	0.356486	+	0.934301i	$0.383975\pi$
$594$	0	0
$595$	−2.06935	−0.0848352
$596$	0	0
$597$	−46.5962	−1.90705
$598$	0	0
$599$	−45.6512	−1.86526	−0.932629	−	0.360837i	$-0.882491\pi$
$599$	−45.6512	−1.86526	−0.932629	+	0.360837i	$0.882491\pi$
$600$	0	0
$601$	30.2452	1.23373	0.616864	−	0.787070i	$-0.288403\pi$
$601$	30.2452	1.23373	0.616864	+	0.787070i	$0.288403\pi$
$602$	0	0
$603$	23.1517	0.942812
$604$	0	0
$605$	−52.7741	−2.14557
$606$	0	0
$607$	5.15261	0.209138	0.104569	−	0.994518i	$-0.466654\pi$
$607$	5.15261	0.209138	0.104569	+	0.994518i	$0.466654\pi$
$608$	0	0
$609$	−42.0796	−1.70515
$610$	0	0
$611$	−24.3352	−0.984498
$612$	0	0
$613$	11.7857	0.476020	0.238010	−	0.971263i	$-0.423505\pi$
$613$	11.7857	0.476020	0.238010	+	0.971263i	$0.423505\pi$
$614$	0	0
$615$	−38.4684	−1.55119
$616$	0	0
$617$	−26.8787	−1.08210	−0.541048	−	0.840991i	$-0.681972\pi$
$617$	−26.8787	−1.08210	−0.541048	+	0.840991i	$0.681972\pi$
$618$	0	0
$619$	−11.4117	−0.458676	−0.229338	−	0.973347i	$-0.573656\pi$
$619$	−11.4117	−0.458676	−0.229338	+	0.973347i	$0.573656\pi$
$620$	0	0
$621$	10.3193	0.414100
$622$	0	0
$623$	−18.0978	−0.725075
$624$	0	0
$625$	86.6263	3.46505
$626$	0	0
$627$	−12.5274	−0.500294
$628$	0	0
$629$	0.488394	0.0194735
$630$	0	0
$631$	40.9103	1.62861	0.814306	−	0.580436i	$-0.197118\pi$
$631$	40.9103	1.62861	0.814306	+	0.580436i	$0.197118\pi$
$632$	0	0
$633$	13.5677	0.539269
$634$	0	0
$635$	−50.8989	−2.01986
$636$	0	0
$637$	−2.52253	−0.0999461
$638$	0	0
$639$	9.07678	0.359072
$640$	0	0
$641$	34.3277	1.35586	0.677931	−	0.735126i	$-0.262877\pi$
$641$	34.3277	1.35586	0.677931	+	0.735126i	$0.262877\pi$
$642$	0	0
$643$	16.8523	0.664588	0.332294	−	0.943176i	$-0.392177\pi$
$643$	16.8523	0.664588	0.332294	+	0.943176i	$0.392177\pi$
$644$	0	0
$645$	88.9801	3.50359
$646$	0	0
$647$	42.0276	1.65228	0.826138	−	0.563467i	$-0.190533\pi$
$647$	42.0276	1.65228	0.826138	+	0.563467i	$0.190533\pi$
$648$	0	0
$649$	−45.6930	−1.79361
$650$	0	0
$651$	10.3498	0.405642
$652$	0	0
$653$	48.8330	1.91098	0.955492	−	0.295016i	$-0.0953249\pi$
$653$	48.8330	1.91098	0.955492	+	0.295016i	$0.0953249\pi$
$654$	0	0
$655$	−35.6407	−1.39260
$656$	0	0
$657$	−8.77584	−0.342378
$658$	0	0
$659$	25.1785	0.980817	0.490408	−	0.871493i	$-0.336848\pi$
$659$	25.1785	0.980817	0.490408	+	0.871493i	$0.336848\pi$
$660$	0	0
$661$	9.98638	0.388425	0.194213	−	0.980959i	$-0.437785\pi$
$661$	9.98638	0.388425	0.194213	+	0.980959i	$0.437785\pi$
$662$	0	0
$663$	−1.43998	−0.0559242
$664$	0	0
$665$	−10.5925	−0.410760
$666$	0	0
$667$	−35.7314	−1.38353
$668$	0	0
$669$	−32.9498	−1.27391
$670$	0	0
$671$	16.8533	0.650614
$672$	0	0
$673$	6.86346	0.264567	0.132284	−	0.991212i	$-0.457769\pi$
$673$	6.86346	0.264567	0.132284	+	0.991212i	$0.457769\pi$
$674$	0	0
$675$	−25.3086	−0.974127
$676$	0	0
$677$	31.9805	1.22911	0.614556	−	0.788874i	$-0.289335\pi$
$677$	31.9805	1.22911	0.614556	+	0.788874i	$0.289335\pi$
$678$	0	0
$679$	−9.12385	−0.350141
$680$	0	0
$681$	−54.7610	−2.09845
$682$	0	0
$683$	−5.81046	−0.222331	−0.111166	−	0.993802i	$-0.535458\pi$
$683$	−5.81046	−0.222331	−0.111166	+	0.993802i	$0.535458\pi$
$684$	0	0
$685$	−61.8100	−2.36164
$686$	0	0
$687$	51.0657	1.94828
$688$	0	0
$689$	2.84120	0.108241
$690$	0	0
$691$	30.6697	1.16673	0.583366	−	0.812210i	$-0.301736\pi$
$691$	30.6697	1.16673	0.583366	+	0.812210i	$0.301736\pi$
$692$	0	0
$693$	44.5336	1.69169
$694$	0	0
$695$	65.3806	2.48003
$696$	0	0
$697$	−0.676111	−0.0256095
$698$	0	0
$699$	34.4843	1.30432
$700$	0	0
$701$	−14.4660	−0.546374	−0.273187	−	0.961961i	$-0.588078\pi$
$701$	−14.4660	−0.546374	−0.273187	+	0.961961i	$0.588078\pi$
$702$	0	0
$703$	2.49997	0.0942882
$704$	0	0
$705$	−95.2038	−3.58558
$706$	0	0
$707$	11.1777	0.420379
$708$	0	0
$709$	16.9396	0.636181	0.318090	−	0.948060i	$-0.396958\pi$
$709$	16.9396	0.636181	0.318090	+	0.948060i	$0.396958\pi$
$710$	0	0
$711$	1.52063	0.0570282
$712$	0	0
$713$	8.78844	0.329130
$714$	0	0
$715$	−58.7820	−2.19832
$716$	0	0
$717$	−75.8728	−2.83352
$718$	0	0
$719$	−3.35861	−0.125255	−0.0626274	−	0.998037i	$-0.519948\pi$
$719$	−3.35861	−0.125255	−0.0626274	+	0.998037i	$0.519948\pi$
$720$	0	0
$721$	22.3916	0.833905
$722$	0	0
$723$	30.1256	1.12038
$724$	0	0
$725$	87.6328	3.25460
$726$	0	0
$727$	23.6258	0.876233	0.438116	−	0.898918i	$-0.355646\pi$
$727$	23.6258	0.876233	0.438116	+	0.898918i	$0.355646\pi$
$728$	0	0
$729$	−38.1566	−1.41321
$730$	0	0
$731$	1.56389	0.0578427
$732$	0	0
$733$	−42.4968	−1.56966	−0.784828	−	0.619713i	$-0.787249\pi$
$733$	−42.4968	−1.56966	−0.784828	+	0.619713i	$0.787249\pi$
$734$	0	0
$735$	−9.86858	−0.364008
$736$	0	0
$737$	29.9691	1.10392
$738$	0	0
$739$	−0.203236	−0.00747617	−0.00373809	−	0.999993i	$-0.501190\pi$
$739$	−0.203236	−0.00747617	−0.00373809	+	0.999993i	$0.501190\pi$
$740$	0	0
$741$	−7.37091	−0.270777
$742$	0	0
$743$	43.6401	1.60100	0.800500	−	0.599333i	$-0.204568\pi$
$743$	43.6401	1.60100	0.800500	+	0.599333i	$0.204568\pi$
$744$	0	0
$745$	36.2255	1.32720
$746$	0	0
$747$	2.39197	0.0875176
$748$	0	0
$749$	−16.6923	−0.609922
$750$	0	0
$751$	−3.42440	−0.124958	−0.0624790	−	0.998046i	$-0.519901\pi$
$751$	−3.42440	−0.124958	−0.0624790	+	0.998046i	$0.519901\pi$
$752$	0	0
$753$	−23.9734	−0.873639
$754$	0	0
$755$	29.3440	1.06794
$756$	0	0
$757$	54.6396	1.98591	0.992955	−	0.118495i	$-0.0378070\pi$
$757$	54.6396	1.98591	0.992955	+	0.118495i	$0.0378070\pi$
$758$	0	0
$759$	68.2267	2.47647
$760$	0	0
$761$	−2.96015	−0.107305	−0.0536527	−	0.998560i	$-0.517086\pi$
$761$	−2.96015	−0.107305	−0.0536527	+	0.998560i	$0.517086\pi$
$762$	0	0
$763$	−26.9468	−0.975539
$764$	0	0
$765$	−3.12239	−0.112890
$766$	0	0
$767$	−26.8851	−0.970764
$768$	0	0
$769$	41.5735	1.49918	0.749590	−	0.661903i	$-0.230251\pi$
$769$	41.5735	1.49918	0.749590	+	0.661903i	$0.230251\pi$
$770$	0	0
$771$	75.4951	2.71889
$772$	0	0
$773$	−37.1035	−1.33452	−0.667260	−	0.744825i	$-0.732533\pi$
$773$	−37.1035	−1.33452	−0.667260	+	0.744825i	$0.732533\pi$
$774$	0	0
$775$	−21.5540	−0.774243
$776$	0	0
$777$	−16.0344	−0.575229
$778$	0	0
$779$	−3.46085	−0.123998
$780$	0	0
$781$	11.7496	0.420432
$782$	0	0
$783$	−12.4311	−0.444252
$784$	0	0
$785$	−106.062	−3.78552
$786$	0	0
$787$	46.6667	1.66349	0.831745	−	0.555158i	$-0.187342\pi$
$787$	46.6667	1.66349	0.831745	+	0.555158i	$0.187342\pi$
$788$	0	0
$789$	37.0184	1.31789
$790$	0	0
$791$	2.37583	0.0844749
$792$	0	0
$793$	9.91623	0.352136
$794$	0	0
$795$	11.1153	0.394219
$796$	0	0
$797$	27.4330	0.971727	0.485863	−	0.874035i	$-0.338505\pi$
$797$	27.4330	0.971727	0.485863	+	0.874035i	$0.338505\pi$
$798$	0	0
$799$	−1.67328	−0.0591964
$800$	0	0
$801$	−27.3074	−0.964859
$802$	0	0
$803$	−11.3600	−0.400886
$804$	0	0
$805$	57.6892	2.03328
$806$	0	0
$807$	62.1063	2.18625
$808$	0	0
$809$	23.4026	0.822791	0.411396	−	0.911457i	$-0.365041\pi$
$809$	23.4026	0.822791	0.411396	+	0.911457i	$0.365041\pi$
$810$	0	0
$811$	−22.3313	−0.784158	−0.392079	−	0.919931i	$-0.628244\pi$
$811$	−22.3313	−0.784158	−0.392079	+	0.919931i	$0.628244\pi$
$812$	0	0
$813$	27.6992	0.971453
$814$	0	0
$815$	18.8713	0.661034
$816$	0	0
$817$	8.00519	0.280066
$818$	0	0
$819$	26.2029	0.915605
$820$	0	0
$821$	17.7954	0.621063	0.310532	−	0.950563i	$-0.399493\pi$
$821$	17.7954	0.621063	0.310532	+	0.950563i	$0.399493\pi$
$822$	0	0
$823$	23.8914	0.832802	0.416401	−	0.909181i	$-0.363291\pi$
$823$	23.8914	0.832802	0.416401	+	0.909181i	$0.363291\pi$
$824$	0	0
$825$	−167.329	−5.82565
$826$	0	0
$827$	−51.6923	−1.79752	−0.898758	−	0.438445i	$-0.855529\pi$
$827$	−51.6923	−1.79752	−0.898758	+	0.438445i	$0.855529\pi$
$828$	0	0
$829$	−22.2083	−0.771326	−0.385663	−	0.922640i	$-0.626027\pi$
$829$	−22.2083	−0.771326	−0.385663	+	0.922640i	$0.626027\pi$
$830$	0	0
$831$	69.9940	2.42807
$832$	0	0
$833$	−0.173448	−0.00600961
$834$	0	0
$835$	−68.5260	−2.37144
$836$	0	0
$837$	3.05754	0.105684
$838$	0	0
$839$	−20.7948	−0.717916	−0.358958	−	0.933354i	$-0.616868\pi$
$839$	−20.7948	−0.717916	−0.358958	+	0.933354i	$0.616868\pi$
$840$	0	0
$841$	14.0438	0.484268
$842$	0	0
$843$	−52.6475	−1.81328
$844$	0	0
$845$	21.1123	0.726284
$846$	0	0
$847$	30.4521	1.04635
$848$	0	0
$849$	−22.2789	−0.764611
$850$	0	0
$851$	−13.6154	−0.466730
$852$	0	0
$853$	12.1192	0.414952	0.207476	−	0.978240i	$-0.433475\pi$
$853$	12.1192	0.414952	0.207476	+	0.978240i	$0.433475\pi$
$854$	0	0
$855$	−15.9828	−0.546600
$856$	0	0
$857$	4.99276	0.170549	0.0852747	−	0.996357i	$-0.472823\pi$
$857$	4.99276	0.170549	0.0852747	+	0.996357i	$0.472823\pi$
$858$	0	0
$859$	56.0485	1.91235	0.956175	−	0.292796i	$-0.0945858\pi$
$859$	56.0485	1.91235	0.956175	+	0.292796i	$0.0945858\pi$
$860$	0	0
$861$	22.1973	0.756480
$862$	0	0
$863$	54.6095	1.85893	0.929464	−	0.368912i	$-0.120270\pi$
$863$	54.6095	1.85893	0.929464	+	0.368912i	$0.120270\pi$
$864$	0	0
$865$	−74.4123	−2.53009
$866$	0	0
$867$	44.0040	1.49445
$868$	0	0
$869$	1.96840	0.0667735
$870$	0	0
$871$	17.6334	0.597484
$872$	0	0
$873$	−13.7667	−0.465934
$874$	0	0
$875$	−88.5226	−2.99261
$876$	0	0
$877$	55.3624	1.86946	0.934729	−	0.355363i	$-0.115643\pi$
$877$	55.3624	1.86946	0.934729	+	0.355363i	$0.115643\pi$
$878$	0	0
$879$	−39.0977	−1.31873
$880$	0	0
$881$	−28.4790	−0.959483	−0.479741	−	0.877410i	$-0.659269\pi$
$881$	−28.4790	−0.959483	−0.479741	+	0.877410i	$0.659269\pi$
$882$	0	0
$883$	−35.9834	−1.21094	−0.605470	−	0.795868i	$-0.707015\pi$
$883$	−35.9834	−1.21094	−0.605470	+	0.795868i	$0.707015\pi$
$884$	0	0
$885$	−105.179	−3.53556
$886$	0	0
$887$	−42.8502	−1.43877	−0.719384	−	0.694613i	$-0.755576\pi$
$887$	−42.8502	−1.43877	−0.719384	+	0.694613i	$0.755576\pi$
$888$	0	0
$889$	29.3700	0.985037
$890$	0	0
$891$	−30.3032	−1.01520
$892$	0	0
$893$	−8.56512	−0.286621
$894$	0	0
$895$	19.3384	0.646412
$896$	0	0
$897$	40.1436	1.34036
$898$	0	0
$899$	−10.5870	−0.353095
$900$	0	0
$901$	0.195360	0.00650838
$902$	0	0
$903$	−51.3438	−1.70862
$904$	0	0
$905$	−92.4791	−3.07411
$906$	0	0
$907$	22.2451	0.738636	0.369318	−	0.929303i	$-0.379591\pi$
$907$	22.2451	0.738636	0.369318	+	0.929303i	$0.379591\pi$
$908$	0	0
$909$	16.8657	0.559400
$910$	0	0
$911$	−33.3518	−1.10499	−0.552497	−	0.833515i	$-0.686325\pi$
$911$	−33.3518	−1.10499	−0.552497	+	0.833515i	$0.686325\pi$
$912$	0	0
$913$	3.09632	0.102473
$914$	0	0
$915$	38.7941	1.28249
$916$	0	0
$917$	20.5656	0.679137
$918$	0	0
$919$	1.06922	0.0352702	0.0176351	−	0.999844i	$-0.494386\pi$
$919$	1.06922	0.0352702	0.0176351	+	0.999844i	$0.494386\pi$
$920$	0	0
$921$	−44.0075	−1.45010
$922$	0	0
$923$	6.91327	0.227553
$924$	0	0
$925$	33.3923	1.09793
$926$	0	0
$927$	33.7861	1.10968
$928$	0	0
$929$	49.6270	1.62821	0.814105	−	0.580718i	$-0.197228\pi$
$929$	49.6270	1.62821	0.814105	+	0.580718i	$0.197228\pi$
$930$	0	0
$931$	−0.887838	−0.0290977
$932$	0	0
$933$	31.5616	1.03328
$934$	0	0
$935$	−4.04182	−0.132182
$936$	0	0
$937$	−53.2456	−1.73946	−0.869728	−	0.493531i	$-0.835706\pi$
$937$	−53.2456	−1.73946	−0.869728	+	0.493531i	$0.835706\pi$
$938$	0	0
$939$	86.4861	2.82237
$940$	0	0
$941$	11.4164	0.372162	0.186081	−	0.982534i	$-0.440421\pi$
$941$	11.4164	0.372162	0.186081	+	0.982534i	$0.440421\pi$
$942$	0	0
$943$	18.8485	0.613793
$944$	0	0
$945$	20.0703	0.652888
$946$	0	0
$947$	32.0734	1.04225	0.521123	−	0.853481i	$-0.325513\pi$
$947$	32.0734	1.04225	0.521123	+	0.853481i	$0.325513\pi$
$948$	0	0
$949$	−6.68406	−0.216974
$950$	0	0
$951$	−4.53079	−0.146921
$952$	0	0
$953$	21.2412	0.688071	0.344036	−	0.938957i	$-0.388206\pi$
$953$	21.2412	0.688071	0.344036	+	0.938957i	$0.388206\pi$
$954$	0	0
$955$	−55.7139	−1.80286
$956$	0	0
$957$	−82.1891	−2.65680
$958$	0	0
$959$	35.6660	1.15171
$960$	0	0
$961$	−28.3960	−0.916002
$962$	0	0
$963$	−25.1865	−0.811625
$964$	0	0
$965$	−44.5831	−1.43518
$966$	0	0
$967$	12.8981	0.414776	0.207388	−	0.978259i	$-0.433504\pi$
$967$	12.8981	0.414776	0.207388	+	0.978259i	$0.433504\pi$
$968$	0	0
$969$	−0.506821	−0.0162814
$970$	0	0
$971$	−29.6025	−0.949988	−0.474994	−	0.879989i	$-0.657550\pi$
$971$	−29.6025	−0.949988	−0.474994	+	0.879989i	$0.657550\pi$
$972$	0	0
$973$	−37.7263	−1.20945
$974$	0	0
$975$	−98.4539	−3.15305
$976$	0	0
$977$	27.4988	0.879765	0.439883	−	0.898055i	$-0.355020\pi$
$977$	27.4988	0.879765	0.439883	+	0.898055i	$0.355020\pi$
$978$	0	0
$979$	−35.3484	−1.12974
$980$	0	0
$981$	−40.6593	−1.29815
$982$	0	0
$983$	1.18306	0.0377338	0.0188669	−	0.999822i	$-0.493994\pi$
$983$	1.18306	0.0377338	0.0188669	+	0.999822i	$0.493994\pi$
$984$	0	0
$985$	−82.5465	−2.63015
$986$	0	0
$987$	54.9351	1.74860
$988$	0	0
$989$	−43.5980	−1.38634
$990$	0	0
$991$	−20.6861	−0.657117	−0.328558	−	0.944484i	$-0.606563\pi$
$991$	−20.6861	−0.657117	−0.328558	+	0.944484i	$0.606563\pi$
$992$	0	0
$993$	−55.5514	−1.76287
$994$	0	0
$995$	−76.9543	−2.43962
$996$	0	0
$997$	−40.4044	−1.27962	−0.639811	−	0.768533i	$-0.720987\pi$
$997$	−40.4044	−1.27962	−0.639811	+	0.768533i	$0.720987\pi$
$998$	0	0
$999$	−4.73686	−0.149868

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4028.2.a.e.1.3	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4028.2.a.e.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-2.59429 q^{3} -4.28452 q^{5} +2.47228 q^{7} +3.73036 q^{9} +O(q^{10})\)	\(q-2.59429 q^{3} -4.28452 q^{5} +2.47228 q^{7} +3.73036 q^{9} +4.82881 q^{11} +2.84120 q^{13} +11.1153 q^{15} +0.195360 q^{17} +1.00000 q^{19} -6.41382 q^{21} -5.44622 q^{23} +13.3571 q^{25} -1.89477 q^{27} +6.56077 q^{29} -1.61368 q^{31} -12.5274 q^{33} -10.5925 q^{35} +2.49997 q^{37} -7.37091 q^{39} -3.46085 q^{41} +8.00519 q^{43} -15.9828 q^{45} -8.56512 q^{47} -0.887838 q^{49} -0.506821 q^{51} +1.00000 q^{53} -20.6891 q^{55} -2.59429 q^{57} -9.46258 q^{59} +3.49015 q^{61} +9.22249 q^{63} -12.1732 q^{65} +6.20630 q^{67} +14.1291 q^{69} +2.43322 q^{71} -2.35255 q^{73} -34.6522 q^{75} +11.9382 q^{77} +0.407637 q^{79} -6.27550 q^{81} +0.641217 q^{83} -0.837022 q^{85} -17.0206 q^{87} -7.32031 q^{89} +7.02424 q^{91} +4.18635 q^{93} -4.28452 q^{95} -3.69046 q^{97} +18.0132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9}+O(q^{10}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9	\( 19 q + 3 q^{3} + 3 q^{5} + 6 q^{7} + 22 q^{9} + 5 q^{11} + 25 q^{13} + 20 q^{15} - 7 q^{17} + 19 q^{19} + 2 q^{21} + 18 q^{23} + 22 q^{25} + 15 q^{27} + 4 q^{29} + 12 q^{31} - 8 q^{33} + 11 q^{35} + 19 q^{37} + 9 q^{39} - 9 q^{41} + 31 q^{43} - 2 q^{45} - 2 q^{47} + 7 q^{49} + 5 q^{51} + 19 q^{53} + 11 q^{55} + 3 q^{57} + 2 q^{59} + 6 q^{61} + 52 q^{63} - 6 q^{65} + 50 q^{67} - 7 q^{69} + 25 q^{71} - 5 q^{73} + 22 q^{75} - 14 q^{77} + 36 q^{79} + 11 q^{81} + 20 q^{83} + 5 q^{85} + 18 q^{87} + 9 q^{89} + 61 q^{91} + q^{93} + 3 q^{95} + 7 q^{97} + 48 q^{99}+O(q^{100}) \) 19 * q + 3 * q^3 + 3 * q^5 + 6 * q^7 + 22 * q^9 + 5 * q^11 + 25 * q^13 + 20 * q^15 - 7 * q^17 + 19 * q^19 + 2 * q^21 + 18 * q^23 + 22 * q^25 + 15 * q^27 + 4 * q^29 + 12 * q^31 - 8 * q^33 + 11 * q^35 + 19 * q^37 + 9 * q^39 - 9 * q^41 + 31 * q^43 - 2 * q^45 - 2 * q^47 + 7 * q^49 + 5 * q^51 + 19 * q^53 + 11 * q^55 + 3 * q^57 + 2 * q^59 + 6 * q^61 + 52 * q^63 - 6 * q^65 + 50 * q^67 - 7 * q^69 + 25 * q^71 - 5 * q^73 + 22 * q^75 - 14 * q^77 + 36 * q^79 + 11 * q^81 + 20 * q^83 + 5 * q^85 + 18 * q^87 + 9 * q^89 + 61 * q^91 + q^93 + 3 * q^95 + 7 * q^97 + 48 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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