LMFDB - Embedded newform 4014.2.a.x.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 14x^{6} + 28x^{5} + 43x^{4} - 90x^{3} - 23x^{2} + 82x - 28 $ x^8 - 2x^7 - 14x^6 + 28x^5 + 43x^4 - 90x^3 - 23x^2 + 82*x - 28
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.541762$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +3.88234 q^{5} -1.74188 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +3.88234 q^{5} -1.74188 q^{7} -1.00000 q^{8} -3.88234 q^{10} -0.798915 q^{11} +0.679382 q^{13} +1.74188 q^{14} +1.00000 q^{16} -5.58768 q^{17} +6.11294 q^{19} +3.88234 q^{20} +0.798915 q^{22} +8.82640 q^{23} +10.0726 q^{25} -0.679382 q^{26} -1.74188 q^{28} -4.65831 q^{29} -8.02989 q^{31} -1.00000 q^{32} +5.58768 q^{34} -6.76256 q^{35} +4.24863 q^{37} -6.11294 q^{38} -3.88234 q^{40} +4.20624 q^{41} -7.19007 q^{43} -0.798915 q^{44} -8.82640 q^{46} +8.05077 q^{47} -3.96587 q^{49} -10.0726 q^{50} +0.679382 q^{52} +3.31622 q^{53} -3.10166 q^{55} +1.74188 q^{56} +4.65831 q^{58} +2.34603 q^{59} +11.7944 q^{61} +8.02989 q^{62} +1.00000 q^{64} +2.63759 q^{65} -2.36537 q^{67} -5.58768 q^{68} +6.76256 q^{70} +8.93943 q^{71} +2.95045 q^{73} -4.24863 q^{74} +6.11294 q^{76} +1.39161 q^{77} -3.88168 q^{79} +3.88234 q^{80} -4.20624 q^{82} +13.3424 q^{83} -21.6933 q^{85} +7.19007 q^{86} +0.798915 q^{88} +16.9614 q^{89} -1.18340 q^{91} +8.82640 q^{92} -8.05077 q^{94} +23.7325 q^{95} +16.6802 q^{97} +3.96587 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8}+O(q^{10}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8	$ 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8} - 6 q^{10} + 11 q^{11} - q^{13} + 6 q^{14} + 8 q^{16} + 16 q^{17} - q^{19} + 6 q^{20} - 11 q^{22} + 14 q^{23} + 10 q^{25} + q^{26} - 6 q^{28} + 21 q^{29} - 6 q^{31} - 8 q^{32} - 16 q^{34} + 8 q^{35} - 14 q^{37} + q^{38} - 6 q^{40} + 16 q^{41} - 29 q^{43} + 11 q^{44} - 14 q^{46} + 9 q^{47} - 2 q^{49} - 10 q^{50} - q^{52} + 11 q^{53} - 22 q^{55} + 6 q^{56} - 21 q^{58} + 21 q^{59} + 3 q^{61} + 6 q^{62} + 8 q^{64} + 24 q^{65} - 20 q^{67} + 16 q^{68} - 8 q^{70} + 32 q^{71} + 13 q^{73} + 14 q^{74} - q^{76} + 4 q^{77} + 21 q^{79} + 6 q^{80} - 16 q^{82} + 28 q^{83} - 14 q^{85} + 29 q^{86} - 11 q^{88} + 54 q^{89} - 36 q^{91} + 14 q^{92} - 9 q^{94} + 30 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8 - 6 * q^10 + 11 * q^11 - q^13 + 6 * q^14 + 8 * q^16 + 16 * q^17 - q^19 + 6 * q^20 - 11 * q^22 + 14 * q^23 + 10 * q^25 + q^26 - 6 * q^28 + 21 * q^29 - 6 * q^31 - 8 * q^32 - 16 * q^34 + 8 * q^35 - 14 * q^37 + q^38 - 6 * q^40 + 16 * q^41 - 29 * q^43 + 11 * q^44 - 14 * q^46 + 9 * q^47 - 2 * q^49 - 10 * q^50 - q^52 + 11 * q^53 - 22 * q^55 + 6 * q^56 - 21 * q^58 + 21 * q^59 + 3 * q^61 + 6 * q^62 + 8 * q^64 + 24 * q^65 - 20 * q^67 + 16 * q^68 - 8 * q^70 + 32 * q^71 + 13 * q^73 + 14 * q^74 - q^76 + 4 * q^77 + 21 * q^79 + 6 * q^80 - 16 * q^82 + 28 * q^83 - 14 * q^85 + 29 * q^86 - 11 * q^88 + 54 * q^89 - 36 * q^91 + 14 * q^92 - 9 * q^94 + 30 * q^95 + 10 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.88234	1.73624	0.868118	−	0.496357i	$-0.165329\pi$
$5$	3.88234	1.73624	0.868118	+	0.496357i	$0.165329\pi$
$6$	0	0
$7$	−1.74188	−0.658367	−0.329184	−	0.944266i	$-0.606774\pi$
$7$	−1.74188	−0.658367	−0.329184	+	0.944266i	$0.606774\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.88234	−1.22770
$11$	−0.798915	−0.240882	−0.120441	−	0.992720i	$-0.538431\pi$
$11$	−0.798915	−0.240882	−0.120441	+	0.992720i	$0.538431\pi$
$12$	0	0
$13$	0.679382	0.188427	0.0942133	−	0.995552i	$-0.469966\pi$
$13$	0.679382	0.188427	0.0942133	+	0.995552i	$0.469966\pi$
$14$	1.74188	0.465536
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.58768	−1.35521	−0.677606	−	0.735425i	$-0.736983\pi$
$17$	−5.58768	−1.35521	−0.677606	+	0.735425i	$0.736983\pi$
$18$	0	0
$19$	6.11294	1.40241	0.701203	−	0.712962i	$-0.252647\pi$
$19$	6.11294	1.40241	0.701203	+	0.712962i	$0.252647\pi$
$20$	3.88234	0.868118
$21$	0	0
$22$	0.798915	0.170329
$23$	8.82640	1.84043	0.920216	−	0.391411i	$-0.128013\pi$
$23$	8.82640	1.84043	0.920216	+	0.391411i	$0.128013\pi$
$24$	0	0
$25$	10.0726	2.01452
$26$	−0.679382	−0.133238
$27$	0	0
$28$	−1.74188	−0.329184
$29$	−4.65831	−0.865027	−0.432513	−	0.901628i	$-0.642373\pi$
$29$	−4.65831	−0.865027	−0.432513	+	0.901628i	$0.642373\pi$
$30$	0	0
$31$	−8.02989	−1.44221	−0.721105	−	0.692825i	$-0.756366\pi$
$31$	−8.02989	−1.44221	−0.721105	+	0.692825i	$0.756366\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.58768	0.958280
$35$	−6.76256	−1.14308
$36$	0	0
$37$	4.24863	0.698470	0.349235	−	0.937035i	$-0.386441\pi$
$37$	4.24863	0.698470	0.349235	+	0.937035i	$0.386441\pi$
$38$	−6.11294	−0.991651
$39$	0	0
$40$	−3.88234	−0.613852
$41$	4.20624	0.656905	0.328452	−	0.944521i	$-0.393473\pi$
$41$	4.20624	0.656905	0.328452	+	0.944521i	$0.393473\pi$
$42$	0	0
$43$	−7.19007	−1.09647	−0.548237	−	0.836323i	$-0.684701\pi$
$43$	−7.19007	−1.09647	−0.548237	+	0.836323i	$0.684701\pi$
$44$	−0.798915	−0.120441
$45$	0	0
$46$	−8.82640	−1.30138
$47$	8.05077	1.17433	0.587163	−	0.809469i	$-0.300245\pi$
$47$	8.05077	1.17433	0.587163	+	0.809469i	$0.300245\pi$
$48$	0	0
$49$	−3.96587	−0.566553
$50$	−10.0726	−1.42448
$51$	0	0
$52$	0.679382	0.0942133
$53$	3.31622	0.455518	0.227759	−	0.973718i	$-0.426860\pi$
$53$	3.31622	0.455518	0.227759	+	0.973718i	$0.426860\pi$
$54$	0	0
$55$	−3.10166	−0.418228
$56$	1.74188	0.232768
$57$	0	0
$58$	4.65831	0.611666
$59$	2.34603	0.305428	0.152714	−	0.988270i	$-0.451199\pi$
$59$	2.34603	0.305428	0.152714	+	0.988270i	$0.451199\pi$
$60$	0	0
$61$	11.7944	1.51012	0.755059	−	0.655657i	$-0.227608\pi$
$61$	11.7944	1.51012	0.755059	+	0.655657i	$0.227608\pi$
$62$	8.02989	1.01980
$63$	0	0
$64$	1.00000	0.125000
$65$	2.63759	0.327153
$66$	0	0
$67$	−2.36537	−0.288976	−0.144488	−	0.989507i	$-0.546154\pi$
$67$	−2.36537	−0.288976	−0.144488	+	0.989507i	$0.546154\pi$
$68$	−5.58768	−0.677606
$69$	0	0
$70$	6.76256	0.808281
$71$	8.93943	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$71$	8.93943	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$72$	0	0
$73$	2.95045	0.345324	0.172662	−	0.984981i	$-0.444763\pi$
$73$	2.95045	0.345324	0.172662	+	0.984981i	$0.444763\pi$
$74$	−4.24863	−0.493893
$75$	0	0
$76$	6.11294	0.701203
$77$	1.39161	0.158589
$78$	0	0
$79$	−3.88168	−0.436723	−0.218361	−	0.975868i	$-0.570071\pi$
$79$	−3.88168	−0.436723	−0.218361	+	0.975868i	$0.570071\pi$
$80$	3.88234	0.434059
$81$	0	0
$82$	−4.20624	−0.464502
$83$	13.3424	1.46451	0.732257	−	0.681028i	$-0.238467\pi$
$83$	13.3424	1.46451	0.732257	+	0.681028i	$0.238467\pi$
$84$	0	0
$85$	−21.6933	−2.35297
$86$	7.19007	0.775325
$87$	0	0
$88$	0.798915	0.0851646
$89$	16.9614	1.79790	0.898950	−	0.438051i	$-0.144331\pi$
$89$	16.9614	1.79790	0.898950	+	0.438051i	$0.144331\pi$
$90$	0	0
$91$	−1.18340	−0.124054
$92$	8.82640	0.920216
$93$	0	0
$94$	−8.05077	−0.830374
$95$	23.7325	2.43491
$96$	0	0
$97$	16.6802	1.69361	0.846806	−	0.531901i	$-0.178522\pi$
$97$	16.6802	1.69361	0.846806	+	0.531901i	$0.178522\pi$
$98$	3.96587	0.400613
$99$	0	0
$100$	10.0726	1.00726
$101$	10.0727	1.00227	0.501137	−	0.865368i	$-0.332915\pi$
$101$	10.0727	1.00227	0.501137	+	0.865368i	$0.332915\pi$
$102$	0	0
$103$	−11.8705	−1.16963	−0.584816	−	0.811166i	$-0.698833\pi$
$103$	−11.8705	−1.16963	−0.584816	+	0.811166i	$0.698833\pi$
$104$	−0.679382	−0.0666189
$105$	0	0
$106$	−3.31622	−0.322100
$107$	−13.2478	−1.28071	−0.640354	−	0.768079i	$-0.721213\pi$
$107$	−13.2478	−1.28071	−0.640354	+	0.768079i	$0.721213\pi$
$108$	0	0
$109$	1.87668	0.179754	0.0898768	−	0.995953i	$-0.471353\pi$
$109$	1.87668	0.179754	0.0898768	+	0.995953i	$0.471353\pi$
$110$	3.10166	0.295732
$111$	0	0
$112$	−1.74188	−0.164592
$113$	6.28762	0.591489	0.295745	−	0.955267i	$-0.404432\pi$
$113$	6.28762	0.591489	0.295745	+	0.955267i	$0.404432\pi$
$114$	0	0
$115$	34.2671	3.19542
$116$	−4.65831	−0.432513
$117$	0	0
$118$	−2.34603	−0.215970
$119$	9.73305	0.892228
$120$	0	0
$121$	−10.3617	−0.941976
$122$	−11.7944	−1.06781
$123$	0	0
$124$	−8.02989	−0.721105
$125$	19.6935	1.76144
$126$	0	0
$127$	−7.72443	−0.685432	−0.342716	−	0.939439i	$-0.611347\pi$
$127$	−7.72443	−0.685432	−0.342716	+	0.939439i	$0.611347\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.63759	−0.231332
$131$	−11.4950	−1.00433	−0.502163	−	0.864773i	$-0.667462\pi$
$131$	−11.4950	−1.00433	−0.502163	+	0.864773i	$0.667462\pi$
$132$	0	0
$133$	−10.6480	−0.923298
$134$	2.36537	0.204337
$135$	0	0
$136$	5.58768	0.479140
$137$	−0.670656	−0.0572980	−0.0286490	−	0.999590i	$-0.509121\pi$
$137$	−0.670656	−0.0572980	−0.0286490	+	0.999590i	$0.509121\pi$
$138$	0	0
$139$	−11.8477	−1.00491	−0.502453	−	0.864605i	$-0.667569\pi$
$139$	−11.8477	−1.00491	−0.502453	+	0.864605i	$0.667569\pi$
$140$	−6.76256	−0.571541
$141$	0	0
$142$	−8.93943	−0.750180
$143$	−0.542768	−0.0453886
$144$	0	0
$145$	−18.0852	−1.50189
$146$	−2.95045	−0.244181
$147$	0	0
$148$	4.24863	0.349235
$149$	11.3799	0.932279	0.466140	−	0.884711i	$-0.345644\pi$
$149$	11.3799	0.932279	0.466140	+	0.884711i	$0.345644\pi$
$150$	0	0
$151$	12.0911	0.983959	0.491979	−	0.870607i	$-0.336274\pi$
$151$	12.0911	0.983959	0.491979	+	0.870607i	$0.336274\pi$
$152$	−6.11294	−0.495825
$153$	0	0
$154$	−1.39161	−0.112139
$155$	−31.1748	−2.50402
$156$	0	0
$157$	1.66707	0.133047	0.0665235	−	0.997785i	$-0.478809\pi$
$157$	1.66707	0.133047	0.0665235	+	0.997785i	$0.478809\pi$
$158$	3.88168	0.308810
$159$	0	0
$160$	−3.88234	−0.306926
$161$	−15.3745	−1.21168
$162$	0	0
$163$	−3.89698	−0.305235	−0.152617	−	0.988285i	$-0.548770\pi$
$163$	−3.89698	−0.305235	−0.152617	+	0.988285i	$0.548770\pi$
$164$	4.20624	0.328452
$165$	0	0
$166$	−13.3424	−1.03557
$167$	−11.2913	−0.873748	−0.436874	−	0.899523i	$-0.643914\pi$
$167$	−11.2913	−0.873748	−0.436874	+	0.899523i	$0.643914\pi$
$168$	0	0
$169$	−12.5384	−0.964495
$170$	21.6933	1.66380
$171$	0	0
$172$	−7.19007	−0.548237
$173$	14.5097	1.10315	0.551577	−	0.834124i	$-0.314026\pi$
$173$	14.5097	1.10315	0.551577	+	0.834124i	$0.314026\pi$
$174$	0	0
$175$	−17.5452	−1.32629
$176$	−0.798915	−0.0602205
$177$	0	0
$178$	−16.9614	−1.27131
$179$	21.4466	1.60299	0.801496	−	0.598000i	$-0.204038\pi$
$179$	21.4466	1.60299	0.801496	+	0.598000i	$0.204038\pi$
$180$	0	0
$181$	−8.65603	−0.643398	−0.321699	−	0.946842i	$-0.604254\pi$
$181$	−8.65603	−0.643398	−0.321699	+	0.946842i	$0.604254\pi$
$182$	1.18340	0.0877194
$183$	0	0
$184$	−8.82640	−0.650691
$185$	16.4946	1.21271
$186$	0	0
$187$	4.46408	0.326446
$188$	8.05077	0.587163
$189$	0	0
$190$	−23.7325	−1.72174
$191$	16.1675	1.16984	0.584918	−	0.811092i	$-0.301127\pi$
$191$	16.1675	1.16984	0.584918	+	0.811092i	$0.301127\pi$
$192$	0	0
$193$	13.6739	0.984271	0.492135	−	0.870519i	$-0.336217\pi$
$193$	13.6739	0.984271	0.492135	+	0.870519i	$0.336217\pi$
$194$	−16.6802	−1.19756
$195$	0	0
$196$	−3.96587	−0.283276
$197$	−17.1970	−1.22524	−0.612619	−	0.790378i	$-0.709884\pi$
$197$	−17.1970	−1.22524	−0.612619	+	0.790378i	$0.709884\pi$
$198$	0	0
$199$	13.9264	0.987220	0.493610	−	0.869683i	$-0.335677\pi$
$199$	13.9264	0.987220	0.493610	+	0.869683i	$0.335677\pi$
$200$	−10.0726	−0.712240
$201$	0	0
$202$	−10.0727	−0.708715
$203$	8.11420	0.569505
$204$	0	0
$205$	16.3301	1.14054
$206$	11.8705	0.827055
$207$	0	0
$208$	0.679382	0.0471067
$209$	−4.88372	−0.337814
$210$	0	0
$211$	−19.3934	−1.33510	−0.667550	−	0.744565i	$-0.732657\pi$
$211$	−19.3934	−1.33510	−0.667550	+	0.744565i	$0.732657\pi$
$212$	3.31622	0.227759
$213$	0	0
$214$	13.2478	0.905598
$215$	−27.9143	−1.90374
$216$	0	0
$217$	13.9871	0.949504
$218$	−1.87668	−0.127105
$219$	0	0
$220$	−3.10166	−0.209114
$221$	−3.79617	−0.255358
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	1.74188	0.116384
$225$	0	0
$226$	−6.28762	−0.418246
$227$	1.94508	0.129099	0.0645497	−	0.997914i	$-0.479439\pi$
$227$	1.94508	0.129099	0.0645497	+	0.997914i	$0.479439\pi$
$228$	0	0
$229$	19.5340	1.29084	0.645422	−	0.763826i	$-0.276681\pi$
$229$	19.5340	1.29084	0.645422	+	0.763826i	$0.276681\pi$
$230$	−34.2671	−2.25951
$231$	0	0
$232$	4.65831	0.305833
$233$	−6.71844	−0.440140	−0.220070	−	0.975484i	$-0.570629\pi$
$233$	−6.71844	−0.440140	−0.220070	+	0.975484i	$0.570629\pi$
$234$	0	0
$235$	31.2559	2.03891
$236$	2.34603	0.152714
$237$	0	0
$238$	−9.73305	−0.630900
$239$	−1.51874	−0.0982394	−0.0491197	−	0.998793i	$-0.515642\pi$
$239$	−1.51874	−0.0982394	−0.0491197	+	0.998793i	$0.515642\pi$
$240$	0	0
$241$	14.3078	0.921647	0.460823	−	0.887492i	$-0.347554\pi$
$241$	14.3078	0.921647	0.460823	+	0.887492i	$0.347554\pi$
$242$	10.3617	0.666078
$243$	0	0
$244$	11.7944	0.755059
$245$	−15.3969	−0.983669
$246$	0	0
$247$	4.15302	0.264251
$248$	8.02989	0.509899
$249$	0	0
$250$	−19.6935	−1.24553
$251$	−5.90106	−0.372472	−0.186236	−	0.982505i	$-0.559629\pi$
$251$	−5.90106	−0.372472	−0.186236	+	0.982505i	$0.559629\pi$
$252$	0	0
$253$	−7.05154	−0.443327
$254$	7.72443	0.484674
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.2480	−1.01352	−0.506761	−	0.862087i	$-0.669157\pi$
$257$	−16.2480	−1.01352	−0.506761	+	0.862087i	$0.669157\pi$
$258$	0	0
$259$	−7.40059	−0.459850
$260$	2.63759	0.163577
$261$	0	0
$262$	11.4950	0.710166
$263$	−6.60909	−0.407534	−0.203767	−	0.979019i	$-0.565318\pi$
$263$	−6.60909	−0.407534	−0.203767	+	0.979019i	$0.565318\pi$
$264$	0	0
$265$	12.8747	0.790887
$266$	10.6480	0.652870
$267$	0	0
$268$	−2.36537	−0.144488
$269$	27.5461	1.67952	0.839759	−	0.542960i	$-0.182696\pi$
$269$	27.5461	1.67952	0.839759	+	0.542960i	$0.182696\pi$
$270$	0	0
$271$	−18.4748	−1.12227	−0.561133	−	0.827726i	$-0.689634\pi$
$271$	−18.4748	−1.12227	−0.561133	+	0.827726i	$0.689634\pi$
$272$	−5.58768	−0.338803
$273$	0	0
$274$	0.670656	0.0405158
$275$	−8.04714	−0.485261
$276$	0	0
$277$	−24.3355	−1.46218	−0.731088	−	0.682283i	$-0.760987\pi$
$277$	−24.3355	−1.46218	−0.731088	+	0.682283i	$0.760987\pi$
$278$	11.8477	0.710575
$279$	0	0
$280$	6.76256	0.404140
$281$	18.7636	1.11934	0.559672	−	0.828714i	$-0.310927\pi$
$281$	18.7636	1.11934	0.559672	+	0.828714i	$0.310927\pi$
$282$	0	0
$283$	−6.62499	−0.393815	−0.196907	−	0.980422i	$-0.563090\pi$
$283$	−6.62499	−0.393815	−0.196907	+	0.980422i	$0.563090\pi$
$284$	8.93943	0.530458
$285$	0	0
$286$	0.542768	0.0320946
$287$	−7.32675	−0.432484
$288$	0	0
$289$	14.2222	0.836601
$290$	18.0852	1.06200
$291$	0	0
$292$	2.95045	0.172662
$293$	6.44277	0.376391	0.188195	−	0.982132i	$-0.439736\pi$
$293$	6.44277	0.376391	0.188195	+	0.982132i	$0.439736\pi$
$294$	0	0
$295$	9.10811	0.530295
$296$	−4.24863	−0.246947
$297$	0	0
$298$	−11.3799	−0.659221
$299$	5.99650	0.346786
$300$	0	0
$301$	12.5242	0.721883
$302$	−12.0911	−0.695764
$303$	0	0
$304$	6.11294	0.350601
$305$	45.7899	2.62192
$306$	0	0
$307$	29.1353	1.66284	0.831420	−	0.555645i	$-0.187529\pi$
$307$	29.1353	1.66284	0.831420	+	0.555645i	$0.187529\pi$
$308$	1.39161	0.0792944
$309$	0	0
$310$	31.1748	1.77061
$311$	−28.5391	−1.61831	−0.809153	−	0.587598i	$-0.800074\pi$
$311$	−28.5391	−1.61831	−0.809153	+	0.587598i	$0.800074\pi$
$312$	0	0
$313$	23.1151	1.30654	0.653271	−	0.757125i	$-0.273396\pi$
$313$	23.1151	1.30654	0.653271	+	0.757125i	$0.273396\pi$
$314$	−1.66707	−0.0940785
$315$	0	0
$316$	−3.88168	−0.218361
$317$	19.5188	1.09629	0.548143	−	0.836385i	$-0.315335\pi$
$317$	19.5188	1.09629	0.548143	+	0.836385i	$0.315335\pi$
$318$	0	0
$319$	3.72159	0.208369
$320$	3.88234	0.217030
$321$	0	0
$322$	15.3745	0.856787
$323$	−34.1572	−1.90056
$324$	0	0
$325$	6.84314	0.379589
$326$	3.89698	0.215834
$327$	0	0
$328$	−4.20624	−0.232251
$329$	−14.0235	−0.773138
$330$	0	0
$331$	−26.0372	−1.43113	−0.715566	−	0.698545i	$-0.753831\pi$
$331$	−26.0372	−1.43113	−0.715566	+	0.698545i	$0.753831\pi$
$332$	13.3424	0.732257
$333$	0	0
$334$	11.2913	0.617833
$335$	−9.18318	−0.501731
$336$	0	0
$337$	−32.6296	−1.77745	−0.888723	−	0.458445i	$-0.848407\pi$
$337$	−32.6296	−1.77745	−0.888723	+	0.458445i	$0.848407\pi$
$338$	12.5384	0.682001
$339$	0	0
$340$	−21.6933	−1.17648
$341$	6.41520	0.347402
$342$	0	0
$343$	19.1012	1.03137
$344$	7.19007	0.387662
$345$	0	0
$346$	−14.5097	−0.780048
$347$	−29.4651	−1.58177	−0.790885	−	0.611965i	$-0.790379\pi$
$347$	−29.4651	−1.58177	−0.790885	+	0.611965i	$0.790379\pi$
$348$	0	0
$349$	14.8025	0.792359	0.396180	−	0.918173i	$-0.370336\pi$
$349$	14.8025	0.792359	0.396180	+	0.918173i	$0.370336\pi$
$350$	17.5452	0.937830
$351$	0	0
$352$	0.798915	0.0425823
$353$	14.4302	0.768041	0.384021	−	0.923324i	$-0.374539\pi$
$353$	14.4302	0.768041	0.384021	+	0.923324i	$0.374539\pi$
$354$	0	0
$355$	34.7059	1.84200
$356$	16.9614	0.898950
$357$	0	0
$358$	−21.4466	−1.13349
$359$	11.8789	0.626946	0.313473	−	0.949597i	$-0.398508\pi$
$359$	11.8789	0.626946	0.313473	+	0.949597i	$0.398508\pi$
$360$	0	0
$361$	18.3681	0.966742
$362$	8.65603	0.454951
$363$	0	0
$364$	−1.18340	−0.0620270
$365$	11.4547	0.599565
$366$	0	0
$367$	−19.4404	−1.01478	−0.507390	−	0.861716i	$-0.669390\pi$
$367$	−19.4404	−1.01478	−0.507390	+	0.861716i	$0.669390\pi$
$368$	8.82640	0.460108
$369$	0	0
$370$	−16.4946	−0.857515
$371$	−5.77645	−0.299898
$372$	0	0
$373$	7.97069	0.412706	0.206353	−	0.978478i	$-0.433840\pi$
$373$	7.97069	0.412706	0.206353	+	0.978478i	$0.433840\pi$
$374$	−4.46408	−0.230832
$375$	0	0
$376$	−8.05077	−0.415187
$377$	−3.16477	−0.162994
$378$	0	0
$379$	0.161014	0.00827075	0.00413537	−	0.999991i	$-0.498684\pi$
$379$	0.161014	0.00827075	0.00413537	+	0.999991i	$0.498684\pi$
$380$	23.7325	1.21745
$381$	0	0
$382$	−16.1675	−0.827199
$383$	−31.3010	−1.59940	−0.799702	−	0.600397i	$-0.795009\pi$
$383$	−31.3010	−1.59940	−0.799702	+	0.600397i	$0.795009\pi$
$384$	0	0
$385$	5.40271	0.275348
$386$	−13.6739	−0.695984
$387$	0	0
$388$	16.6802	0.846806
$389$	19.2985	0.978474	0.489237	−	0.872151i	$-0.337275\pi$
$389$	19.2985	0.978474	0.489237	+	0.872151i	$0.337275\pi$
$390$	0	0
$391$	−49.3191	−2.49418
$392$	3.96587	0.200307
$393$	0	0
$394$	17.1970	0.866374
$395$	−15.0700	−0.758254
$396$	0	0
$397$	−5.21217	−0.261591	−0.130796	−	0.991409i	$-0.541753\pi$
$397$	−5.21217	−0.261591	−0.130796	+	0.991409i	$0.541753\pi$
$398$	−13.9264	−0.698070
$399$	0	0
$400$	10.0726	0.503629
$401$	−37.5946	−1.87738	−0.938691	−	0.344759i	$-0.887961\pi$
$401$	−37.5946	−1.87738	−0.938691	+	0.344759i	$0.887961\pi$
$402$	0	0
$403$	−5.45536	−0.271751
$404$	10.0727	0.501137
$405$	0	0
$406$	−8.11420	−0.402701
$407$	−3.39429	−0.168249
$408$	0	0
$409$	34.3815	1.70006	0.850029	−	0.526737i	$-0.176585\pi$
$409$	34.3815	1.70006	0.850029	+	0.526737i	$0.176585\pi$
$410$	−16.3301	−0.806485
$411$	0	0
$412$	−11.8705	−0.584816
$413$	−4.08650	−0.201084
$414$	0	0
$415$	51.7996	2.54274
$416$	−0.679382	−0.0333094
$417$	0	0
$418$	4.88372	0.238871
$419$	15.7143	0.767692	0.383846	−	0.923397i	$-0.374599\pi$
$419$	15.7143	0.767692	0.383846	+	0.923397i	$0.374599\pi$
$420$	0	0
$421$	15.8955	0.774700	0.387350	−	0.921933i	$-0.373391\pi$
$421$	15.8955	0.774700	0.387350	+	0.921933i	$0.373391\pi$
$422$	19.3934	0.944058
$423$	0	0
$424$	−3.31622	−0.161050
$425$	−56.2825	−2.73010
$426$	0	0
$427$	−20.5444	−0.994212
$428$	−13.2478	−0.640354
$429$	0	0
$430$	27.9143	1.34615
$431$	31.2884	1.50711	0.753555	−	0.657384i	$-0.228337\pi$
$431$	31.2884	1.50711	0.753555	+	0.657384i	$0.228337\pi$
$432$	0	0
$433$	−17.0963	−0.821598	−0.410799	−	0.911726i	$-0.634750\pi$
$433$	−17.0963	−0.821598	−0.410799	+	0.911726i	$0.634750\pi$
$434$	−13.9871	−0.671401
$435$	0	0
$436$	1.87668	0.0898768
$437$	53.9553	2.58103
$438$	0	0
$439$	−5.12879	−0.244784	−0.122392	−	0.992482i	$-0.539057\pi$
$439$	−5.12879	−0.244784	−0.122392	+	0.992482i	$0.539057\pi$
$440$	3.10166	0.147866
$441$	0	0
$442$	3.79617	0.180566
$443$	13.5828	0.645337	0.322669	−	0.946512i	$-0.395420\pi$
$443$	13.5828	0.645337	0.322669	+	0.946512i	$0.395420\pi$
$444$	0	0
$445$	65.8498	3.12158
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−1.74188	−0.0822959
$449$	−3.24053	−0.152930	−0.0764650	−	0.997072i	$-0.524363\pi$
$449$	−3.24053	−0.152930	−0.0764650	+	0.997072i	$0.524363\pi$
$450$	0	0
$451$	−3.36043	−0.158236
$452$	6.28762	0.295745
$453$	0	0
$454$	−1.94508	−0.0912870
$455$	−4.59436	−0.215387
$456$	0	0
$457$	4.06131	0.189980	0.0949900	−	0.995478i	$-0.469718\pi$
$457$	4.06131	0.189980	0.0949900	+	0.995478i	$0.469718\pi$
$458$	−19.5340	−0.912765
$459$	0	0
$460$	34.2671	1.59771
$461$	30.9868	1.44320	0.721599	−	0.692312i	$-0.243408\pi$
$461$	30.9868	1.44320	0.721599	+	0.692312i	$0.243408\pi$
$462$	0	0
$463$	−11.2425	−0.522481	−0.261241	−	0.965274i	$-0.584132\pi$
$463$	−11.2425	−0.522481	−0.261241	+	0.965274i	$0.584132\pi$
$464$	−4.65831	−0.216257
$465$	0	0
$466$	6.71844	0.311226
$467$	−31.3967	−1.45287	−0.726434	−	0.687236i	$-0.758824\pi$
$467$	−31.3967	−1.45287	−0.726434	+	0.687236i	$0.758824\pi$
$468$	0	0
$469$	4.12018	0.190252
$470$	−31.2559	−1.44173
$471$	0	0
$472$	−2.34603	−0.107985
$473$	5.74425	0.264121
$474$	0	0
$475$	61.5732	2.82517
$476$	9.73305	0.446114
$477$	0	0
$478$	1.51874	0.0694658
$479$	1.78988	0.0817818	0.0408909	−	0.999164i	$-0.486980\pi$
$479$	1.78988	0.0817818	0.0408909	+	0.999164i	$0.486980\pi$
$480$	0	0
$481$	2.88644	0.131610
$482$	−14.3078	−0.651703
$483$	0	0
$484$	−10.3617	−0.470988
$485$	64.7581	2.94051
$486$	0	0
$487$	5.72076	0.259232	0.129616	−	0.991564i	$-0.458625\pi$
$487$	5.72076	0.259232	0.129616	+	0.991564i	$0.458625\pi$
$488$	−11.7944	−0.533907
$489$	0	0
$490$	15.3969	0.695559
$491$	37.2724	1.68208	0.841041	−	0.540971i	$-0.181943\pi$
$491$	37.2724	1.68208	0.841041	+	0.540971i	$0.181943\pi$
$492$	0	0
$493$	26.0292	1.17230
$494$	−4.15302	−0.186853
$495$	0	0
$496$	−8.02989	−0.360553
$497$	−15.5714	−0.698472
$498$	0	0
$499$	10.0464	0.449738	0.224869	−	0.974389i	$-0.427805\pi$
$499$	10.0464	0.449738	0.224869	+	0.974389i	$0.427805\pi$
$500$	19.6935	0.880721
$501$	0	0
$502$	5.90106	0.263377
$503$	−26.9403	−1.20121	−0.600604	−	0.799547i	$-0.705073\pi$
$503$	−26.9403	−1.20121	−0.600604	+	0.799547i	$0.705073\pi$
$504$	0	0
$505$	39.1058	1.74019
$506$	7.05154	0.313479
$507$	0	0
$508$	−7.72443	−0.342716
$509$	15.0434	0.666789	0.333394	−	0.942787i	$-0.391806\pi$
$509$	15.0434	0.666789	0.333394	+	0.942787i	$0.391806\pi$
$510$	0	0
$511$	−5.13932	−0.227350
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	16.2480	0.716668
$515$	−46.0852	−2.03076
$516$	0	0
$517$	−6.43188	−0.282874
$518$	7.40059	0.325163
$519$	0	0
$520$	−2.63759	−0.115666
$521$	−8.24146	−0.361065	−0.180533	−	0.983569i	$-0.557782\pi$
$521$	−8.24146	−0.361065	−0.180533	+	0.983569i	$0.557782\pi$
$522$	0	0
$523$	−11.9092	−0.520754	−0.260377	−	0.965507i	$-0.583847\pi$
$523$	−11.9092	−0.520754	−0.260377	+	0.965507i	$0.583847\pi$
$524$	−11.4950	−0.502163
$525$	0	0
$526$	6.60909	0.288170
$527$	44.8685	1.95450
$528$	0	0
$529$	54.9053	2.38719
$530$	−12.8747	−0.559242
$531$	0	0
$532$	−10.6480	−0.461649
$533$	2.85764	0.123778
$534$	0	0
$535$	−51.4323	−2.22361
$536$	2.36537	0.102168
$537$	0	0
$538$	−27.5461	−1.18760
$539$	3.16839	0.136472
$540$	0	0
$541$	−20.1515	−0.866382	−0.433191	−	0.901302i	$-0.642612\pi$
$541$	−20.1515	−0.866382	−0.433191	+	0.901302i	$0.642612\pi$
$542$	18.4748	0.793562
$543$	0	0
$544$	5.58768	0.239570
$545$	7.28593	0.312095
$546$	0	0
$547$	11.5032	0.491843	0.245921	−	0.969290i	$-0.420910\pi$
$547$	11.5032	0.491843	0.245921	+	0.969290i	$0.420910\pi$
$548$	−0.670656	−0.0286490
$549$	0	0
$550$	8.04714	0.343131
$551$	−28.4760	−1.21312
$552$	0	0
$553$	6.76140	0.287524
$554$	24.3355	1.03391
$555$	0	0
$556$	−11.8477	−0.502453
$557$	−31.1881	−1.32148	−0.660741	−	0.750614i	$-0.729758\pi$
$557$	−31.1881	−1.32148	−0.660741	+	0.750614i	$0.729758\pi$
$558$	0	0
$559$	−4.88480	−0.206605
$560$	−6.76256	−0.285770
$561$	0	0
$562$	−18.7636	−0.791495
$563$	2.46349	0.103824	0.0519118	−	0.998652i	$-0.483469\pi$
$563$	2.46349	0.103824	0.0519118	+	0.998652i	$0.483469\pi$
$564$	0	0
$565$	24.4107	1.02697
$566$	6.62499	0.278469
$567$	0	0
$568$	−8.93943	−0.375090
$569$	0.890193	0.0373188	0.0186594	−	0.999826i	$-0.494060\pi$
$569$	0.890193	0.0373188	0.0186594	+	0.999826i	$0.494060\pi$
$570$	0	0
$571$	8.04094	0.336503	0.168251	−	0.985744i	$-0.446188\pi$
$571$	8.04094	0.336503	0.168251	+	0.985744i	$0.446188\pi$
$572$	−0.542768	−0.0226943
$573$	0	0
$574$	7.32675	0.305813
$575$	88.9047	3.70758
$576$	0	0
$577$	−18.2990	−0.761796	−0.380898	−	0.924617i	$-0.624385\pi$
$577$	−18.2990	−0.761796	−0.380898	+	0.924617i	$0.624385\pi$
$578$	−14.2222	−0.591566
$579$	0	0
$580$	−18.0852	−0.750946
$581$	−23.2407	−0.964188
$582$	0	0
$583$	−2.64938	−0.109726
$584$	−2.95045	−0.122091
$585$	0	0
$586$	−6.44277	−0.266148
$587$	28.2837	1.16739	0.583696	−	0.811972i	$-0.301606\pi$
$587$	28.2837	1.16739	0.583696	+	0.811972i	$0.301606\pi$
$588$	0	0
$589$	−49.0863	−2.02256
$590$	−9.10811	−0.374975
$591$	0	0
$592$	4.24863	0.174618
$593$	−1.77004	−0.0726869	−0.0363435	−	0.999339i	$-0.511571\pi$
$593$	−1.77004	−0.0726869	−0.0363435	+	0.999339i	$0.511571\pi$
$594$	0	0
$595$	37.7871	1.54912
$596$	11.3799	0.466140
$597$	0	0
$598$	−5.99650	−0.245215
$599$	25.7050	1.05028	0.525139	−	0.851016i	$-0.324013\pi$
$599$	25.7050	1.05028	0.525139	+	0.851016i	$0.324013\pi$
$600$	0	0
$601$	39.0637	1.59344	0.796720	−	0.604349i	$-0.206567\pi$
$601$	39.0637	1.59344	0.796720	+	0.604349i	$0.206567\pi$
$602$	−12.5242	−0.510448
$603$	0	0
$604$	12.0911	0.491979
$605$	−40.2278	−1.63549
$606$	0	0
$607$	6.53075	0.265075	0.132537	−	0.991178i	$-0.457688\pi$
$607$	6.53075	0.265075	0.132537	+	0.991178i	$0.457688\pi$
$608$	−6.11294	−0.247913
$609$	0	0
$610$	−45.7899	−1.85398
$611$	5.46955	0.221274
$612$	0	0
$613$	2.80923	0.113464	0.0567318	−	0.998389i	$-0.481932\pi$
$613$	2.80923	0.113464	0.0567318	+	0.998389i	$0.481932\pi$
$614$	−29.1353	−1.17580
$615$	0	0
$616$	−1.39161	−0.0560696
$617$	−17.2583	−0.694791	−0.347396	−	0.937719i	$-0.612934\pi$
$617$	−17.2583	−0.694791	−0.347396	+	0.937719i	$0.612934\pi$
$618$	0	0
$619$	−27.2134	−1.09380	−0.546900	−	0.837198i	$-0.684192\pi$
$619$	−27.2134	−1.09380	−0.546900	+	0.837198i	$0.684192\pi$
$620$	−31.1748	−1.25201
$621$	0	0
$622$	28.5391	1.14432
$623$	−29.5446	−1.18368
$624$	0	0
$625$	26.0941	1.04376
$626$	−23.1151	−0.923864
$627$	0	0
$628$	1.66707	0.0665235
$629$	−23.7400	−0.946576
$630$	0	0
$631$	3.40827	0.135681	0.0678405	−	0.997696i	$-0.478389\pi$
$631$	3.40827	0.135681	0.0678405	+	0.997696i	$0.478389\pi$
$632$	3.88168	0.154405
$633$	0	0
$634$	−19.5188	−0.775191
$635$	−29.9889	−1.19007
$636$	0	0
$637$	−2.69434	−0.106754
$638$	−3.72159	−0.147339
$639$	0	0
$640$	−3.88234	−0.153463
$641$	27.6635	1.09264	0.546321	−	0.837576i	$-0.316028\pi$
$641$	27.6635	1.09264	0.546321	+	0.837576i	$0.316028\pi$
$642$	0	0
$643$	−36.0538	−1.42182	−0.710912	−	0.703281i	$-0.751718\pi$
$643$	−36.0538	−1.42182	−0.710912	+	0.703281i	$0.751718\pi$
$644$	−15.3745	−0.605840
$645$	0	0
$646$	34.1572	1.34390
$647$	−39.0201	−1.53404	−0.767020	−	0.641623i	$-0.778261\pi$
$647$	−39.0201	−1.53404	−0.767020	+	0.641623i	$0.778261\pi$
$648$	0	0
$649$	−1.87428	−0.0735720
$650$	−6.84314	−0.268410
$651$	0	0
$652$	−3.89698	−0.152617
$653$	1.36955	0.0535947	0.0267973	−	0.999641i	$-0.491469\pi$
$653$	1.36955	0.0535947	0.0267973	+	0.999641i	$0.491469\pi$
$654$	0	0
$655$	−44.6277	−1.74375
$656$	4.20624	0.164226
$657$	0	0
$658$	14.0235	0.546691
$659$	−38.1895	−1.48765	−0.743827	−	0.668373i	$-0.766991\pi$
$659$	−38.1895	−1.48765	−0.743827	+	0.668373i	$0.766991\pi$
$660$	0	0
$661$	−27.3135	−1.06237	−0.531186	−	0.847255i	$-0.678254\pi$
$661$	−27.3135	−1.06237	−0.531186	+	0.847255i	$0.678254\pi$
$662$	26.0372	1.01196
$663$	0	0
$664$	−13.3424	−0.517784
$665$	−41.3392	−1.60306
$666$	0	0
$667$	−41.1161	−1.59202
$668$	−11.2913	−0.436874
$669$	0	0
$670$	9.18318	0.354777
$671$	−9.42272	−0.363760
$672$	0	0
$673$	−43.5564	−1.67898	−0.839488	−	0.543378i	$-0.817145\pi$
$673$	−43.5564	−1.67898	−0.839488	+	0.543378i	$0.817145\pi$
$674$	32.6296	1.25684
$675$	0	0
$676$	−12.5384	−0.482248
$677$	10.4932	0.403286	0.201643	−	0.979459i	$-0.435372\pi$
$677$	10.4932	0.403286	0.201643	+	0.979459i	$0.435372\pi$
$678$	0	0
$679$	−29.0548	−1.11502
$680$	21.6933	0.831900
$681$	0	0
$682$	−6.41520	−0.245651
$683$	−1.15990	−0.0443823	−0.0221912	−	0.999754i	$-0.507064\pi$
$683$	−1.15990	−0.0443823	−0.0221912	+	0.999754i	$0.507064\pi$
$684$	0	0
$685$	−2.60372	−0.0994829
$686$	−19.1012	−0.729287
$687$	0	0
$688$	−7.19007	−0.274119
$689$	2.25298	0.0858317
$690$	0	0
$691$	25.6232	0.974752	0.487376	−	0.873192i	$-0.337954\pi$
$691$	25.6232	0.974752	0.487376	+	0.873192i	$0.337954\pi$
$692$	14.5097	0.551577
$693$	0	0
$694$	29.4651	1.11848
$695$	−45.9967	−1.74475
$696$	0	0
$697$	−23.5032	−0.890245
$698$	−14.8025	−0.560283
$699$	0	0
$700$	−17.5452	−0.663146
$701$	22.9465	0.866677	0.433338	−	0.901231i	$-0.357336\pi$
$701$	22.9465	0.866677	0.433338	+	0.901231i	$0.357336\pi$
$702$	0	0
$703$	25.9716	0.979539
$704$	−0.798915	−0.0301102
$705$	0	0
$706$	−14.4302	−0.543087
$707$	−17.5455	−0.659865
$708$	0	0
$709$	−30.9792	−1.16345	−0.581724	−	0.813387i	$-0.697621\pi$
$709$	−30.9792	−1.16345	−0.581724	+	0.813387i	$0.697621\pi$
$710$	−34.7059	−1.30249
$711$	0	0
$712$	−16.9614	−0.635654
$713$	−70.8750	−2.65429
$714$	0	0
$715$	−2.10721	−0.0788053
$716$	21.4466	0.801496
$717$	0	0
$718$	−11.8789	−0.443318
$719$	26.2722	0.979787	0.489894	−	0.871782i	$-0.337036\pi$
$719$	26.2722	0.979787	0.489894	+	0.871782i	$0.337036\pi$
$720$	0	0
$721$	20.6769	0.770048
$722$	−18.3681	−0.683590
$723$	0	0
$724$	−8.65603	−0.321699
$725$	−46.9213	−1.74261
$726$	0	0
$727$	−17.0543	−0.632509	−0.316254	−	0.948674i	$-0.602425\pi$
$727$	−17.0543	−0.632509	−0.316254	+	0.948674i	$0.602425\pi$
$728$	1.18340	0.0438597
$729$	0	0
$730$	−11.4547	−0.423956
$731$	40.1758	1.48596
$732$	0	0
$733$	−35.1098	−1.29681	−0.648405	−	0.761295i	$-0.724564\pi$
$733$	−35.1098	−1.29681	−0.648405	+	0.761295i	$0.724564\pi$
$734$	19.4404	0.717558
$735$	0	0
$736$	−8.82640	−0.325345
$737$	1.88973	0.0696091
$738$	0	0
$739$	10.8061	0.397507	0.198754	−	0.980049i	$-0.436311\pi$
$739$	10.8061	0.397507	0.198754	+	0.980049i	$0.436311\pi$
$740$	16.4946	0.606355
$741$	0	0
$742$	5.77645	0.212060
$743$	24.1142	0.884663	0.442331	−	0.896852i	$-0.354151\pi$
$743$	24.1142	0.884663	0.442331	+	0.896852i	$0.354151\pi$
$744$	0	0
$745$	44.1807	1.61866
$746$	−7.97069	−0.291828
$747$	0	0
$748$	4.46408	0.163223
$749$	23.0759	0.843177
$750$	0	0
$751$	7.72415	0.281858	0.140929	−	0.990020i	$-0.454991\pi$
$751$	7.72415	0.281858	0.140929	+	0.990020i	$0.454991\pi$
$752$	8.05077	0.293582
$753$	0	0
$754$	3.16477	0.115254
$755$	46.9417	1.70839
$756$	0	0
$757$	−15.2467	−0.554150	−0.277075	−	0.960848i	$-0.589365\pi$
$757$	−15.2467	−0.554150	−0.277075	+	0.960848i	$0.589365\pi$
$758$	−0.161014	−0.00584830
$759$	0	0
$760$	−23.7325	−0.860870
$761$	−44.0657	−1.59738	−0.798690	−	0.601742i	$-0.794473\pi$
$761$	−44.0657	−1.59738	−0.798690	+	0.601742i	$0.794473\pi$
$762$	0	0
$763$	−3.26895	−0.118344
$764$	16.1675	0.584918
$765$	0	0
$766$	31.3010	1.13095
$767$	1.59385	0.0575507
$768$	0	0
$769$	−5.97069	−0.215309	−0.107654	−	0.994188i	$-0.534334\pi$
$769$	−5.97069	−0.215309	−0.107654	+	0.994188i	$0.534334\pi$
$770$	−5.40271	−0.194700
$771$	0	0
$772$	13.6739	0.492135
$773$	−12.4187	−0.446669	−0.223334	−	0.974742i	$-0.571694\pi$
$773$	−12.4187	−0.446669	−0.223334	+	0.974742i	$0.571694\pi$
$774$	0	0
$775$	−80.8818	−2.90536
$776$	−16.6802	−0.598782
$777$	0	0
$778$	−19.2985	−0.691886
$779$	25.7125	0.921247
$780$	0	0
$781$	−7.14184	−0.255555
$782$	49.3191	1.76365
$783$	0	0
$784$	−3.96587	−0.141638
$785$	6.47216	0.231001
$786$	0	0
$787$	40.4562	1.44211	0.721054	−	0.692879i	$-0.243658\pi$
$787$	40.4562	1.44211	0.721054	+	0.692879i	$0.243658\pi$
$788$	−17.1970	−0.612619
$789$	0	0
$790$	15.0700	0.536167
$791$	−10.9522	−0.389417
$792$	0	0
$793$	8.01290	0.284546
$794$	5.21217	0.184973
$795$	0	0
$796$	13.9264	0.493610
$797$	30.3892	1.07644	0.538221	−	0.842804i	$-0.319097\pi$
$797$	30.3892	1.07644	0.538221	+	0.842804i	$0.319097\pi$
$798$	0	0
$799$	−44.9852	−1.59146
$800$	−10.0726	−0.356120
$801$	0	0
$802$	37.5946	1.32751
$803$	−2.35716	−0.0831824
$804$	0	0
$805$	−59.6891	−2.10376
$806$	5.45536	0.192157
$807$	0	0
$808$	−10.0727	−0.354357
$809$	−20.2721	−0.712731	−0.356365	−	0.934347i	$-0.615984\pi$
$809$	−20.2721	−0.712731	−0.356365	+	0.934347i	$0.615984\pi$
$810$	0	0
$811$	−46.1582	−1.62083	−0.810416	−	0.585854i	$-0.800759\pi$
$811$	−46.1582	−1.62083	−0.810416	+	0.585854i	$0.800759\pi$
$812$	8.11420	0.284753
$813$	0	0
$814$	3.39429	0.118970
$815$	−15.1294	−0.529960
$816$	0	0
$817$	−43.9525	−1.53770
$818$	−34.3815	−1.20212
$819$	0	0
$820$	16.3301	0.570271
$821$	−41.3653	−1.44366	−0.721829	−	0.692071i	$-0.756698\pi$
$821$	−41.3653	−1.44366	−0.721829	+	0.692071i	$0.756698\pi$
$822$	0	0
$823$	0.125934	0.00438979	0.00219490	−	0.999998i	$-0.499301\pi$
$823$	0.125934	0.00438979	0.00219490	+	0.999998i	$0.499301\pi$
$824$	11.8705	0.413528
$825$	0	0
$826$	4.08650	0.142188
$827$	35.2232	1.22483	0.612416	−	0.790536i	$-0.290198\pi$
$827$	35.2232	1.22483	0.612416	+	0.790536i	$0.290198\pi$
$828$	0	0
$829$	−37.7837	−1.31228	−0.656140	−	0.754639i	$-0.727812\pi$
$829$	−37.7837	−1.31228	−0.656140	+	0.754639i	$0.727812\pi$
$830$	−51.7996	−1.79799
$831$	0	0
$832$	0.679382	0.0235533
$833$	22.1600	0.767799
$834$	0	0
$835$	−43.8368	−1.51703
$836$	−4.88372	−0.168907
$837$	0	0
$838$	−15.7143	−0.542840
$839$	−1.93155	−0.0666846	−0.0333423	−	0.999444i	$-0.510615\pi$
$839$	−1.93155	−0.0666846	−0.0333423	+	0.999444i	$0.510615\pi$
$840$	0	0
$841$	−7.30013	−0.251729
$842$	−15.8955	−0.547795
$843$	0	0
$844$	−19.3934	−0.667550
$845$	−48.6785	−1.67459
$846$	0	0
$847$	18.0489	0.620166
$848$	3.31622	0.113879
$849$	0	0
$850$	56.2825	1.93047
$851$	37.5001	1.28549
$852$	0	0
$853$	56.6605	1.94002	0.970009	−	0.243071i	$-0.0781547\pi$
$853$	56.6605	1.94002	0.970009	+	0.243071i	$0.0781547\pi$
$854$	20.5444	0.703014
$855$	0	0
$856$	13.2478	0.452799
$857$	8.82296	0.301387	0.150693	−	0.988581i	$-0.451849\pi$
$857$	8.82296	0.301387	0.150693	+	0.988581i	$0.451849\pi$
$858$	0	0
$859$	23.2234	0.792374	0.396187	−	0.918170i	$-0.370333\pi$
$859$	23.2234	0.792374	0.396187	+	0.918170i	$0.370333\pi$
$860$	−27.9143	−0.951870
$861$	0	0
$862$	−31.2884	−1.06569
$863$	−39.8393	−1.35615	−0.678073	−	0.734995i	$-0.737185\pi$
$863$	−39.8393	−1.35615	−0.678073	+	0.734995i	$0.737185\pi$
$864$	0	0
$865$	56.3317	1.91534
$866$	17.0963	0.580958
$867$	0	0
$868$	13.9871	0.474752
$869$	3.10113	0.105199
$870$	0	0
$871$	−1.60699	−0.0544508
$872$	−1.87668	−0.0635525
$873$	0	0
$874$	−53.9553	−1.82507
$875$	−34.3037	−1.15968
$876$	0	0
$877$	35.7875	1.20846	0.604229	−	0.796811i	$-0.293481\pi$
$877$	35.7875	1.20846	0.604229	+	0.796811i	$0.293481\pi$
$878$	5.12879	0.173088
$879$	0	0
$880$	−3.10166	−0.104557
$881$	−25.4941	−0.858919	−0.429460	−	0.903086i	$-0.641296\pi$
$881$	−25.4941	−0.858919	−0.429460	+	0.903086i	$0.641296\pi$
$882$	0	0
$883$	10.4183	0.350605	0.175302	−	0.984515i	$-0.443910\pi$
$883$	10.4183	0.350605	0.175302	+	0.984515i	$0.443910\pi$
$884$	−3.79617	−0.127679
$885$	0	0
$886$	−13.5828	−0.456322
$887$	7.69652	0.258424	0.129212	−	0.991617i	$-0.458755\pi$
$887$	7.69652	0.258424	0.129212	+	0.991617i	$0.458755\pi$
$888$	0	0
$889$	13.4550	0.451266
$890$	−65.8498	−2.20729
$891$	0	0
$892$	1.00000	0.0334825
$893$	49.2139	1.64688
$894$	0	0
$895$	83.2630	2.78317
$896$	1.74188	0.0581920
$897$	0	0
$898$	3.24053	0.108138
$899$	37.4057	1.24755
$900$	0	0
$901$	−18.5300	−0.617324
$902$	3.36043	0.111890
$903$	0	0
$904$	−6.28762	−0.209123
$905$	−33.6057	−1.11709
$906$	0	0
$907$	−47.6925	−1.58360	−0.791802	−	0.610778i	$-0.790857\pi$
$907$	−47.6925	−1.58360	−0.791802	+	0.610778i	$0.790857\pi$
$908$	1.94508	0.0645497
$909$	0	0
$910$	4.59436	0.152302
$911$	−20.8393	−0.690436	−0.345218	−	0.938523i	$-0.612195\pi$
$911$	−20.8393	−0.690436	−0.345218	+	0.938523i	$0.612195\pi$
$912$	0	0
$913$	−10.6594	−0.352775
$914$	−4.06131	−0.134336
$915$	0	0
$916$	19.5340	0.645422
$917$	20.0229	0.661215
$918$	0	0
$919$	24.3359	0.802767	0.401383	−	0.915910i	$-0.368530\pi$
$919$	24.3359	0.802767	0.401383	+	0.915910i	$0.368530\pi$
$920$	−34.2671	−1.12975
$921$	0	0
$922$	−30.9868	−1.02049
$923$	6.07329	0.199905
$924$	0	0
$925$	42.7947	1.40708
$926$	11.2425	0.369450
$927$	0	0
$928$	4.65831	0.152917
$929$	−37.7379	−1.23814	−0.619070	−	0.785336i	$-0.712490\pi$
$929$	−37.7379	−1.23814	−0.619070	+	0.785336i	$0.712490\pi$
$930$	0	0
$931$	−24.2431	−0.794537
$932$	−6.71844	−0.220070
$933$	0	0
$934$	31.3967	1.02733
$935$	17.3311	0.566788
$936$	0	0
$937$	−49.5370	−1.61830	−0.809151	−	0.587601i	$-0.800073\pi$
$937$	−49.5370	−1.61830	−0.809151	+	0.587601i	$0.800073\pi$
$938$	−4.12018	−0.134529
$939$	0	0
$940$	31.2559	1.01945
$941$	39.2210	1.27857	0.639285	−	0.768970i	$-0.279231\pi$
$941$	39.2210	1.27857	0.639285	+	0.768970i	$0.279231\pi$
$942$	0	0
$943$	37.1260	1.20899
$944$	2.34603	0.0763569
$945$	0	0
$946$	−5.74425	−0.186762
$947$	9.25620	0.300786	0.150393	−	0.988626i	$-0.451946\pi$
$947$	9.25620	0.300786	0.150393	+	0.988626i	$0.451946\pi$
$948$	0	0
$949$	2.00448	0.0650683
$950$	−61.5732	−1.99770
$951$	0	0
$952$	−9.73305	−0.315450
$953$	−31.9429	−1.03473	−0.517366	−	0.855764i	$-0.673088\pi$
$953$	−31.9429	−1.03473	−0.517366	+	0.855764i	$0.673088\pi$
$954$	0	0
$955$	62.7676	2.03111
$956$	−1.51874	−0.0491197
$957$	0	0
$958$	−1.78988	−0.0578285
$959$	1.16820	0.0377231
$960$	0	0
$961$	33.4791	1.07997
$962$	−2.88644	−0.0930626
$963$	0	0
$964$	14.3078	0.460823
$965$	53.0869	1.70893
$966$	0	0
$967$	32.0405	1.03035	0.515177	−	0.857084i	$-0.327726\pi$
$967$	32.0405	1.03035	0.515177	+	0.857084i	$0.327726\pi$
$968$	10.3617	0.333039
$969$	0	0
$970$	−64.7581	−2.07926
$971$	31.3349	1.00558	0.502792	−	0.864407i	$-0.332306\pi$
$971$	31.3349	1.00558	0.502792	+	0.864407i	$0.332306\pi$
$972$	0	0
$973$	20.6372	0.661597
$974$	−5.72076	−0.183305
$975$	0	0
$976$	11.7944	0.377529
$977$	−22.1243	−0.707819	−0.353909	−	0.935280i	$-0.615148\pi$
$977$	−22.1243	−0.707819	−0.353909	+	0.935280i	$0.615148\pi$
$978$	0	0
$979$	−13.5507	−0.433082
$980$	−15.3969	−0.491835
$981$	0	0
$982$	−37.2724	−1.18941
$983$	−36.7467	−1.17204	−0.586019	−	0.810298i	$-0.699305\pi$
$983$	−36.7467	−1.17204	−0.586019	+	0.810298i	$0.699305\pi$
$984$	0	0
$985$	−66.7648	−2.12730
$986$	−26.0292	−0.828938
$987$	0	0
$988$	4.15302	0.132125
$989$	−63.4624	−2.01799
$990$	0	0
$991$	−16.0627	−0.510248	−0.255124	−	0.966908i	$-0.582116\pi$
$991$	−16.0627	−0.510248	−0.255124	+	0.966908i	$0.582116\pi$
$992$	8.02989	0.254949
$993$	0	0
$994$	15.5714	0.493894
$995$	54.0672	1.71405
$996$	0	0
$997$	2.36320	0.0748434	0.0374217	−	0.999300i	$-0.488086\pi$
$997$	2.36320	0.0748434	0.0374217	+	0.999300i	$0.488086\pi$
$998$	−10.0464	−0.318013
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.x.1.8	✓	8
3.2	odd	2		4014.2.a.y.1.1	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.x.1.8	✓	8	1.1	even	1	trivial
4014.2.a.y.1.1	yes	8	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.88234 q^{5} -1.74188 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +3.88234 q^{5} -1.74188 q^{7} -1.00000 q^{8} -3.88234 q^{10} -0.798915 q^{11} +0.679382 q^{13} +1.74188 q^{14} +1.00000 q^{16} -5.58768 q^{17} +6.11294 q^{19} +3.88234 q^{20} +0.798915 q^{22} +8.82640 q^{23} +10.0726 q^{25} -0.679382 q^{26} -1.74188 q^{28} -4.65831 q^{29} -8.02989 q^{31} -1.00000 q^{32} +5.58768 q^{34} -6.76256 q^{35} +4.24863 q^{37} -6.11294 q^{38} -3.88234 q^{40} +4.20624 q^{41} -7.19007 q^{43} -0.798915 q^{44} -8.82640 q^{46} +8.05077 q^{47} -3.96587 q^{49} -10.0726 q^{50} +0.679382 q^{52} +3.31622 q^{53} -3.10166 q^{55} +1.74188 q^{56} +4.65831 q^{58} +2.34603 q^{59} +11.7944 q^{61} +8.02989 q^{62} +1.00000 q^{64} +2.63759 q^{65} -2.36537 q^{67} -5.58768 q^{68} +6.76256 q^{70} +8.93943 q^{71} +2.95045 q^{73} -4.24863 q^{74} +6.11294 q^{76} +1.39161 q^{77} -3.88168 q^{79} +3.88234 q^{80} -4.20624 q^{82} +13.3424 q^{83} -21.6933 q^{85} +7.19007 q^{86} +0.798915 q^{88} +16.9614 q^{89} -1.18340 q^{91} +8.82640 q^{92} -8.05077 q^{94} +23.7325 q^{95} +16.6802 q^{97} +3.96587 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8}+O(q^{10}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8	\( 8 q - 8 q^{2} + 8 q^{4} + 6 q^{5} - 6 q^{7} - 8 q^{8} - 6 q^{10} + 11 q^{11} - q^{13} + 6 q^{14} + 8 q^{16} + 16 q^{17} - q^{19} + 6 q^{20} - 11 q^{22} + 14 q^{23} + 10 q^{25} + q^{26} - 6 q^{28} + 21 q^{29} - 6 q^{31} - 8 q^{32} - 16 q^{34} + 8 q^{35} - 14 q^{37} + q^{38} - 6 q^{40} + 16 q^{41} - 29 q^{43} + 11 q^{44} - 14 q^{46} + 9 q^{47} - 2 q^{49} - 10 q^{50} - q^{52} + 11 q^{53} - 22 q^{55} + 6 q^{56} - 21 q^{58} + 21 q^{59} + 3 q^{61} + 6 q^{62} + 8 q^{64} + 24 q^{65} - 20 q^{67} + 16 q^{68} - 8 q^{70} + 32 q^{71} + 13 q^{73} + 14 q^{74} - q^{76} + 4 q^{77} + 21 q^{79} + 6 q^{80} - 16 q^{82} + 28 q^{83} - 14 q^{85} + 29 q^{86} - 11 q^{88} + 54 q^{89} - 36 q^{91} + 14 q^{92} - 9 q^{94} + 30 q^{95} + 10 q^{97} + 2 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 8 * q^4 + 6 * q^5 - 6 * q^7 - 8 * q^8 - 6 * q^10 + 11 * q^11 - q^13 + 6 * q^14 + 8 * q^16 + 16 * q^17 - q^19 + 6 * q^20 - 11 * q^22 + 14 * q^23 + 10 * q^25 + q^26 - 6 * q^28 + 21 * q^29 - 6 * q^31 - 8 * q^32 - 16 * q^34 + 8 * q^35 - 14 * q^37 + q^38 - 6 * q^40 + 16 * q^41 - 29 * q^43 + 11 * q^44 - 14 * q^46 + 9 * q^47 - 2 * q^49 - 10 * q^50 - q^52 + 11 * q^53 - 22 * q^55 + 6 * q^56 - 21 * q^58 + 21 * q^59 + 3 * q^61 + 6 * q^62 + 8 * q^64 + 24 * q^65 - 20 * q^67 + 16 * q^68 - 8 * q^70 + 32 * q^71 + 13 * q^73 + 14 * q^74 - q^76 + 4 * q^77 + 21 * q^79 + 6 * q^80 - 16 * q^82 + 28 * q^83 - 14 * q^85 + 29 * q^86 - 11 * q^88 + 54 * q^89 - 36 * q^91 + 14 * q^92 - 9 * q^94 + 30 * q^95 + 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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