LMFDB - Embedded newform 4014.2.a.o.1.2

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:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.713538$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.713538 q^{5} -1.77733 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.713538 q^{5} -1.77733 q^{7} +1.00000 q^{8} +0.713538 q^{10} -0.204402 q^{11} -2.71354 q^{13} -1.77733 q^{14} +1.00000 q^{16} +2.28646 q^{17} +4.47259 q^{19} +0.713538 q^{20} -0.204402 q^{22} -0.204402 q^{23} -4.49086 q^{25} -2.71354 q^{26} -1.77733 q^{28} +8.63148 q^{29} +4.34502 q^{31} +1.00000 q^{32} +2.28646 q^{34} -1.26819 q^{35} -0.204402 q^{37} +4.47259 q^{38} +0.713538 q^{40} +9.47259 q^{41} -3.47259 q^{43} -0.204402 q^{44} -0.204402 q^{46} +1.85939 q^{47} -3.84111 q^{49} -4.49086 q^{50} -2.71354 q^{52} -3.04551 q^{53} -0.145848 q^{55} -1.77733 q^{56} +8.63148 q^{58} +14.5311 q^{59} +7.33198 q^{61} +4.34502 q^{62} +1.00000 q^{64} -1.93621 q^{65} -10.5494 q^{67} +2.28646 q^{68} -1.26819 q^{70} +7.12758 q^{71} +1.87766 q^{73} -0.204402 q^{74} +4.47259 q^{76} +0.363288 q^{77} +15.8866 q^{79} +0.713538 q^{80} +9.47259 q^{82} +14.3085 q^{83} +1.63148 q^{85} -3.47259 q^{86} -0.204402 q^{88} -3.33198 q^{89} +4.82284 q^{91} -0.204402 q^{92} +1.85939 q^{94} +3.19136 q^{95} +3.45432 q^{97} -3.84111 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} + q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 + q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} + q^{7} + 3 q^{8} + q^{10} + 8 q^{11} - 7 q^{13} + q^{14} + 3 q^{16} + 8 q^{17} - 9 q^{19} + q^{20} + 8 q^{22} + 8 q^{23} - 6 q^{25} - 7 q^{26} + q^{28} + 15 q^{29} + q^{31} + 3 q^{32} + 8 q^{34} + 10 q^{35} + 8 q^{37} - 9 q^{38} + q^{40} + 6 q^{41} + 12 q^{43} + 8 q^{44} + 8 q^{46} + 9 q^{47} - 6 q^{50} - 7 q^{52} + 11 q^{53} - 5 q^{55} + q^{56} + 15 q^{58} + 8 q^{59} + 3 q^{61} + q^{62} + 3 q^{64} - 11 q^{65} - 11 q^{67} + 8 q^{68} + 10 q^{70} + 11 q^{71} + 24 q^{73} + 8 q^{74} - 9 q^{76} + 4 q^{77} + 16 q^{79} + q^{80} + 6 q^{82} + q^{83} - 6 q^{85} + 12 q^{86} + 8 q^{88} + 9 q^{89} - 12 q^{91} + 8 q^{92} + 9 q^{94} - 6 q^{95} - 27 q^{97}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 + q^7 + 3 * q^8 + q^10 + 8 * q^11 - 7 * q^13 + q^14 + 3 * q^16 + 8 * q^17 - 9 * q^19 + q^20 + 8 * q^22 + 8 * q^23 - 6 * q^25 - 7 * q^26 + q^28 + 15 * q^29 + q^31 + 3 * q^32 + 8 * q^34 + 10 * q^35 + 8 * q^37 - 9 * q^38 + q^40 + 6 * q^41 + 12 * q^43 + 8 * q^44 + 8 * q^46 + 9 * q^47 - 6 * q^50 - 7 * q^52 + 11 * q^53 - 5 * q^55 + q^56 + 15 * q^58 + 8 * q^59 + 3 * q^61 + q^62 + 3 * q^64 - 11 * q^65 - 11 * q^67 + 8 * q^68 + 10 * q^70 + 11 * q^71 + 24 * q^73 + 8 * q^74 - 9 * q^76 + 4 * q^77 + 16 * q^79 + q^80 + 6 * q^82 + q^83 - 6 * q^85 + 12 * q^86 + 8 * q^88 + 9 * q^89 - 12 * q^91 + 8 * q^92 + 9 * q^94 - 6 * q^95 - 27 * q^97

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
$5$	0.713538	0.319104	0.159552	−	0.987190i	$-0.448995\pi$
$5$	0.713538	0.319104	0.159552	+	0.987190i	$0.448995\pi$
$6$	0	0
$7$	−1.77733	−0.671766	−0.335883	−	0.941904i	$-0.609035\pi$
$7$	−1.77733	−0.671766	−0.335883	+	0.941904i	$0.609035\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.713538	0.225641
$11$	−0.204402	−0.0616294	−0.0308147	−	0.999525i	$-0.509810\pi$
$11$	−0.204402	−0.0616294	−0.0308147	+	0.999525i	$0.509810\pi$
$12$	0	0
$13$	−2.71354	−0.752600	−0.376300	−	0.926498i	$-0.622804\pi$
$13$	−2.71354	−0.752600	−0.376300	+	0.926498i	$0.622804\pi$
$14$	−1.77733	−0.475010
$15$	0	0
$16$	1.00000	0.250000
$17$	2.28646	0.554549	0.277274	−	0.960791i	$-0.410569\pi$
$17$	2.28646	0.554549	0.277274	+	0.960791i	$0.410569\pi$
$18$	0	0
$19$	4.47259	1.02608	0.513041	−	0.858364i	$-0.328519\pi$
$19$	4.47259	1.02608	0.513041	+	0.858364i	$0.328519\pi$
$20$	0.713538	0.159552
$21$	0	0
$22$	−0.204402	−0.0435786
$23$	−0.204402	−0.0426207	−0.0213103	−	0.999773i	$-0.506784\pi$
$23$	−0.204402	−0.0426207	−0.0213103	+	0.999773i	$0.506784\pi$
$24$	0	0
$25$	−4.49086	−0.898173
$26$	−2.71354	−0.532169
$27$	0	0
$28$	−1.77733	−0.335883
$29$	8.63148	1.60283	0.801413	−	0.598112i	$-0.204082\pi$
$29$	8.63148	1.60283	0.801413	+	0.598112i	$0.204082\pi$
$30$	0	0
$31$	4.34502	0.780388	0.390194	−	0.920733i	$-0.372408\pi$
$31$	4.34502	0.780388	0.390194	+	0.920733i	$0.372408\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.28646	0.392125
$35$	−1.26819	−0.214363
$36$	0	0
$37$	−0.204402	−0.0336034	−0.0168017	−	0.999859i	$-0.505348\pi$
$37$	−0.204402	−0.0336034	−0.0168017	+	0.999859i	$0.505348\pi$
$38$	4.47259	0.725550
$39$	0	0
$40$	0.713538	0.112820
$41$	9.47259	1.47937	0.739685	−	0.672953i	$-0.234974\pi$
$41$	9.47259	1.47937	0.739685	+	0.672953i	$0.234974\pi$
$42$	0	0
$43$	−3.47259	−0.529565	−0.264783	−	0.964308i	$-0.585300\pi$
$43$	−3.47259	−0.529565	−0.264783	+	0.964308i	$0.585300\pi$
$44$	−0.204402	−0.0308147
$45$	0	0
$46$	−0.204402	−0.0301374
$47$	1.85939	0.271219	0.135610	−	0.990762i	$-0.456701\pi$
$47$	1.85939	0.271219	0.135610	+	0.990762i	$0.456701\pi$
$48$	0	0
$49$	−3.84111	−0.548730
$50$	−4.49086	−0.635104
$51$	0	0
$52$	−2.71354	−0.376300
$53$	−3.04551	−0.418334	−0.209167	−	0.977880i	$-0.567075\pi$
$53$	−3.04551	−0.418334	−0.209167	+	0.977880i	$0.567075\pi$
$54$	0	0
$55$	−0.145848	−0.0196662
$56$	−1.77733	−0.237505
$57$	0	0
$58$	8.63148	1.13337
$59$	14.5311	1.89179	0.945897	−	0.324467i	$-0.105185\pi$
$59$	14.5311	1.89179	0.945897	+	0.324467i	$0.105185\pi$
$60$	0	0
$61$	7.33198	0.938763	0.469382	−	0.882995i	$-0.344477\pi$
$61$	7.33198	0.938763	0.469382	+	0.882995i	$0.344477\pi$
$62$	4.34502	0.551818
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.93621	−0.240158
$66$	0	0
$67$	−10.5494	−1.28882	−0.644408	−	0.764682i	$-0.722896\pi$
$67$	−10.5494	−1.28882	−0.644408	+	0.764682i	$0.722896\pi$
$68$	2.28646	0.277274
$69$	0	0
$70$	−1.26819	−0.151578
$71$	7.12758	0.845888	0.422944	−	0.906156i	$-0.360997\pi$
$71$	7.12758	0.845888	0.422944	+	0.906156i	$0.360997\pi$
$72$	0	0
$73$	1.87766	0.219763	0.109882	−	0.993945i	$-0.464953\pi$
$73$	1.87766	0.219763	0.109882	+	0.993945i	$0.464953\pi$
$74$	−0.204402	−0.0237612
$75$	0	0
$76$	4.47259	0.513041
$77$	0.363288	0.0414005
$78$	0	0
$79$	15.8866	1.78739	0.893693	−	0.448680i	$-0.148106\pi$
$79$	15.8866	1.78739	0.893693	+	0.448680i	$0.148106\pi$
$80$	0.713538	0.0797760
$81$	0	0
$82$	9.47259	1.04607
$83$	14.3085	1.57056	0.785279	−	0.619142i	$-0.212519\pi$
$83$	14.3085	1.57056	0.785279	+	0.619142i	$0.212519\pi$
$84$	0	0
$85$	1.63148	0.176959
$86$	−3.47259	−0.374459
$87$	0	0
$88$	−0.204402	−0.0217893
$89$	−3.33198	−0.353189	−0.176594	−	0.984284i	$-0.556508\pi$
$89$	−3.33198	−0.353189	−0.176594	+	0.984284i	$0.556508\pi$
$90$	0	0
$91$	4.82284	0.505571
$92$	−0.204402	−0.0213103
$93$	0	0
$94$	1.85939	0.191781
$95$	3.19136	0.327427
$96$	0	0
$97$	3.45432	0.350733	0.175366	−	0.984503i	$-0.443889\pi$
$97$	3.45432	0.350733	0.175366	+	0.984503i	$0.443889\pi$
$98$	−3.84111	−0.388011
$99$	0	0
$100$	−4.49086	−0.449086
$101$	−5.20440	−0.517857	−0.258929	−	0.965896i	$-0.583369\pi$
$101$	−5.20440	−0.517857	−0.258929	+	0.965896i	$0.583369\pi$
$102$	0	0
$103$	−13.1093	−1.29170	−0.645849	−	0.763465i	$-0.723496\pi$
$103$	−13.1093	−1.29170	−0.645849	+	0.763465i	$0.723496\pi$
$104$	−2.71354	−0.266084
$105$	0	0
$106$	−3.04551	−0.295807
$107$	−8.75382	−0.846264	−0.423132	−	0.906068i	$-0.639069\pi$
$107$	−8.75382	−0.846264	−0.423132	+	0.906068i	$0.639069\pi$
$108$	0	0
$109$	12.6367	1.21038	0.605189	−	0.796082i	$-0.293098\pi$
$109$	12.6367	1.21038	0.605189	+	0.796082i	$0.293098\pi$
$110$	−0.145848	−0.0139061
$111$	0	0
$112$	−1.77733	−0.167941
$113$	11.9049	1.11992	0.559959	−	0.828520i	$-0.310817\pi$
$113$	11.9049	1.11992	0.559959	+	0.828520i	$0.310817\pi$
$114$	0	0
$115$	−0.145848	−0.0136004
$116$	8.63148	0.801413
$117$	0	0
$118$	14.5311	1.33770
$119$	−4.06379	−0.372527
$120$	0	0
$121$	−10.9582	−0.996202
$122$	7.33198	0.663806
$123$	0	0
$124$	4.34502	0.390194
$125$	−6.77209	−0.605714
$126$	0	0
$127$	−9.02724	−0.801038	−0.400519	−	0.916288i	$-0.631170\pi$
$127$	−9.02724	−0.801038	−0.400519	+	0.916288i	$0.631170\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.93621	−0.169817
$131$	5.70050	0.498055	0.249027	−	0.968496i	$-0.419889\pi$
$131$	5.70050	0.498055	0.249027	+	0.968496i	$0.419889\pi$
$132$	0	0
$133$	−7.94925	−0.689287
$134$	−10.5494	−0.911331
$135$	0	0
$136$	2.28646	0.196063
$137$	9.62624	0.822425	0.411213	−	0.911539i	$-0.365105\pi$
$137$	9.62624	0.822425	0.411213	+	0.911539i	$0.365105\pi$
$138$	0	0
$139$	13.6770	1.16007	0.580034	−	0.814592i	$-0.303039\pi$
$139$	13.6770	1.16007	0.580034	+	0.814592i	$0.303039\pi$
$140$	−1.26819	−0.107182
$141$	0	0
$142$	7.12758	0.598133
$143$	0.554651	0.0463823
$144$	0	0
$145$	6.15889	0.511468
$146$	1.87766	0.155396
$147$	0	0
$148$	−0.204402	−0.0168017
$149$	−17.7355	−1.45295	−0.726476	−	0.687192i	$-0.758843\pi$
$149$	−17.7355	−1.45295	−0.726476	+	0.687192i	$0.758843\pi$
$150$	0	0
$151$	19.5819	1.59355	0.796776	−	0.604275i	$-0.206537\pi$
$151$	19.5819	1.59355	0.796776	+	0.604275i	$0.206537\pi$
$152$	4.47259	0.362775
$153$	0	0
$154$	0.363288	0.0292746
$155$	3.10033	0.249025
$156$	0	0
$157$	0.331977	0.0264947	0.0132473	−	0.999912i	$-0.495783\pi$
$157$	0.331977	0.0264947	0.0132473	+	0.999912i	$0.495783\pi$
$158$	15.8866	1.26387
$159$	0	0
$160$	0.713538	0.0564101
$161$	0.363288	0.0286311
$162$	0	0
$163$	−23.9269	−1.87410	−0.937050	−	0.349195i	$-0.886455\pi$
$163$	−23.9269	−1.87410	−0.937050	+	0.349195i	$0.886455\pi$
$164$	9.47259	0.739685
$165$	0	0
$166$	14.3085	1.11055
$167$	−12.1626	−0.941172	−0.470586	−	0.882354i	$-0.655957\pi$
$167$	−12.1626	−0.941172	−0.470586	+	0.882354i	$0.655957\pi$
$168$	0	0
$169$	−5.63671	−0.433593
$170$	1.63148	0.125129
$171$	0	0
$172$	−3.47259	−0.264783
$173$	12.9870	0.987380	0.493690	−	0.869638i	$-0.335648\pi$
$173$	12.9870	0.987380	0.493690	+	0.869638i	$0.335648\pi$
$174$	0	0
$175$	7.98173	0.603362
$176$	−0.204402	−0.0154073
$177$	0	0
$178$	−3.33198	−0.249742
$179$	13.8944	1.03852	0.519259	−	0.854617i	$-0.326208\pi$
$179$	13.8944	1.03852	0.519259	+	0.854617i	$0.326208\pi$
$180$	0	0
$181$	−11.3775	−0.845682	−0.422841	−	0.906204i	$-0.638967\pi$
$181$	−11.3775	−0.845682	−0.422841	+	0.906204i	$0.638967\pi$
$182$	4.82284	0.357493
$183$	0	0
$184$	−0.204402	−0.0150687
$185$	−0.145848	−0.0107230
$186$	0	0
$187$	−0.467356	−0.0341765
$188$	1.85939	0.135610
$189$	0	0
$190$	3.19136	0.231526
$191$	16.3085	1.18004	0.590020	−	0.807389i	$-0.299120\pi$
$191$	16.3085	1.18004	0.590020	+	0.807389i	$0.299120\pi$
$192$	0	0
$193$	−1.77733	−0.127935	−0.0639674	−	0.997952i	$-0.520375\pi$
$193$	−1.77733	−0.127935	−0.0639674	+	0.997952i	$0.520375\pi$
$194$	3.45432	0.248006
$195$	0	0
$196$	−3.84111	−0.274365
$197$	−7.39576	−0.526926	−0.263463	−	0.964669i	$-0.584865\pi$
$197$	−7.39576	−0.526926	−0.263463	+	0.964669i	$0.584865\pi$
$198$	0	0
$199$	−3.81387	−0.270358	−0.135179	−	0.990821i	$-0.543161\pi$
$199$	−3.81387	−0.270358	−0.135179	+	0.990821i	$0.543161\pi$
$200$	−4.49086	−0.317552
$201$	0	0
$202$	−5.20440	−0.366180
$203$	−15.3409	−1.07672
$204$	0	0
$205$	6.75905	0.472073
$206$	−13.1093	−0.913368
$207$	0	0
$208$	−2.71354	−0.188150
$209$	−0.914205	−0.0632368
$210$	0	0
$211$	−9.66395	−0.665294	−0.332647	−	0.943051i	$-0.607942\pi$
$211$	−9.66395	−0.665294	−0.332647	+	0.943051i	$0.607942\pi$
$212$	−3.04551	−0.209167
$213$	0	0
$214$	−8.75382	−0.598399
$215$	−2.47783	−0.168986
$216$	0	0
$217$	−7.72251	−0.524238
$218$	12.6367	0.855866
$219$	0	0
$220$	−0.145848	−0.00983309
$221$	−6.20440	−0.417353
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	−1.77733	−0.118753
$225$	0	0
$226$	11.9049	0.791902
$227$	−4.64975	−0.308615	−0.154307	−	0.988023i	$-0.549315\pi$
$227$	−4.64975	−0.308615	−0.154307	+	0.988023i	$0.549315\pi$
$228$	0	0
$229$	10.8448	0.716648	0.358324	−	0.933597i	$-0.383348\pi$
$229$	10.8448	0.716648	0.358324	+	0.933597i	$0.383348\pi$
$230$	−0.145848	−0.00961695
$231$	0	0
$232$	8.63148	0.566684
$233$	−0.994766	−0.0651693	−0.0325846	−	0.999469i	$-0.510374\pi$
$233$	−0.994766	−0.0651693	−0.0325846	+	0.999469i	$0.510374\pi$
$234$	0	0
$235$	1.32674	0.0865471
$236$	14.5311	0.945897
$237$	0	0
$238$	−4.06379	−0.263416
$239$	6.45432	0.417495	0.208748	−	0.977970i	$-0.433061\pi$
$239$	6.45432	0.417495	0.208748	+	0.977970i	$0.433061\pi$
$240$	0	0
$241$	12.4271	0.800498	0.400249	−	0.916406i	$-0.368924\pi$
$241$	12.4271	0.800498	0.400249	+	0.916406i	$0.368924\pi$
$242$	−10.9582	−0.704421
$243$	0	0
$244$	7.33198	0.469382
$245$	−2.74078	−0.175102
$246$	0	0
$247$	−12.1365	−0.772230
$248$	4.34502	0.275909
$249$	0	0
$250$	−6.77209	−0.428305
$251$	−3.39576	−0.214339	−0.107169	−	0.994241i	$-0.534179\pi$
$251$	−3.39576	−0.214339	−0.107169	+	0.994241i	$0.534179\pi$
$252$	0	0
$253$	0.0417800	0.00262669
$254$	−9.02724	−0.566420
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.9582	−1.05782	−0.528912	−	0.848676i	$-0.677400\pi$
$257$	−16.9582	−1.05782	−0.528912	+	0.848676i	$0.677400\pi$
$258$	0	0
$259$	0.363288	0.0225736
$260$	−1.93621	−0.120079
$261$	0	0
$262$	5.70050	0.352178
$263$	−12.8866	−0.794624	−0.397312	−	0.917684i	$-0.630057\pi$
$263$	−12.8866	−0.794624	−0.397312	+	0.917684i	$0.630057\pi$
$264$	0	0
$265$	−2.17309	−0.133492
$266$	−7.94925	−0.487400
$267$	0	0
$268$	−10.5494	−0.644408
$269$	−12.7721	−0.778728	−0.389364	−	0.921084i	$-0.627305\pi$
$269$	−12.7721	−0.778728	−0.389364	+	0.921084i	$0.627305\pi$
$270$	0	0
$271$	20.0910	1.22044	0.610222	−	0.792231i	$-0.291080\pi$
$271$	20.0910	1.22044	0.610222	+	0.792231i	$0.291080\pi$
$272$	2.28646	0.138637
$273$	0	0
$274$	9.62624	0.581543
$275$	0.917939	0.0553538
$276$	0	0
$277$	22.5547	1.35518	0.677589	−	0.735441i	$-0.263025\pi$
$277$	22.5547	1.35518	0.677589	+	0.735441i	$0.263025\pi$
$278$	13.6770	0.820291
$279$	0	0
$280$	−1.26819	−0.0757888
$281$	−18.1679	−1.08380	−0.541902	−	0.840442i	$-0.682296\pi$
$281$	−18.1679	−1.08380	−0.541902	+	0.840442i	$0.682296\pi$
$282$	0	0
$283$	3.92691	0.233431	0.116715	−	0.993165i	$-0.462764\pi$
$283$	3.92691	0.233431	0.116715	+	0.993165i	$0.462764\pi$
$284$	7.12758	0.422944
$285$	0	0
$286$	0.554651	0.0327972
$287$	−16.8359	−0.993791
$288$	0	0
$289$	−11.7721	−0.692476
$290$	6.15889	0.361662
$291$	0	0
$292$	1.87766	0.109882
$293$	−14.7135	−0.859574	−0.429787	−	0.902930i	$-0.641411\pi$
$293$	−14.7135	−0.859574	−0.429787	+	0.902930i	$0.641411\pi$
$294$	0	0
$295$	10.3685	0.603679
$296$	−0.204402	−0.0118806
$297$	0	0
$298$	−17.7355	−1.02739
$299$	0.554651	0.0320763
$300$	0	0
$301$	6.17192	0.355744
$302$	19.5819	1.12681
$303$	0	0
$304$	4.47259	0.256521
$305$	5.23164	0.299563
$306$	0	0
$307$	−10.0988	−0.576371	−0.288185	−	0.957575i	$-0.593052\pi$
$307$	−10.0988	−0.576371	−0.288185	+	0.957575i	$0.593052\pi$
$308$	0.363288	0.0207003
$309$	0	0
$310$	3.10033	0.176087
$311$	4.67699	0.265208	0.132604	−	0.991169i	$-0.457666\pi$
$311$	4.67699	0.265208	0.132604	+	0.991169i	$0.457666\pi$
$312$	0	0
$313$	16.6990	0.943883	0.471942	−	0.881630i	$-0.343553\pi$
$313$	16.6990	0.943883	0.471942	+	0.881630i	$0.343553\pi$
$314$	0.331977	0.0187345
$315$	0	0
$316$	15.8866	0.893693
$317$	−18.6498	−1.04747	−0.523737	−	0.851880i	$-0.675463\pi$
$317$	−18.6498	−1.04747	−0.523737	+	0.851880i	$0.675463\pi$
$318$	0	0
$319$	−1.76429	−0.0987811
$320$	0.713538	0.0398880
$321$	0	0
$322$	0.363288	0.0202453
$323$	10.2264	0.569013
$324$	0	0
$325$	12.1861	0.675965
$326$	−23.9269	−1.32519
$327$	0	0
$328$	9.47259	0.523036
$329$	−3.30473	−0.182196
$330$	0	0
$331$	14.8762	0.817668	0.408834	−	0.912609i	$-0.365936\pi$
$331$	14.8762	0.817668	0.408834	+	0.912609i	$0.365936\pi$
$332$	14.3085	0.785279
$333$	0	0
$334$	−12.1626	−0.665509
$335$	−7.52741	−0.411266
$336$	0	0
$337$	33.7393	1.83790	0.918948	−	0.394379i	$-0.129040\pi$
$337$	33.7393	1.83790	0.918948	+	0.394379i	$0.129040\pi$
$338$	−5.63671	−0.306597
$339$	0	0
$340$	1.63148	0.0884793
$341$	−0.888128	−0.0480948
$342$	0	0
$343$	19.2682	1.04038
$344$	−3.47259	−0.187230
$345$	0	0
$346$	12.9870	0.698183
$347$	18.3032	0.982569	0.491285	−	0.870999i	$-0.336528\pi$
$347$	18.3032	0.982569	0.491285	+	0.870999i	$0.336528\pi$
$348$	0	0
$349$	−11.1313	−0.595845	−0.297923	−	0.954590i	$-0.596294\pi$
$349$	−11.1313	−0.595845	−0.297923	+	0.954590i	$0.596294\pi$
$350$	7.98173	0.426641
$351$	0	0
$352$	−0.204402	−0.0108946
$353$	−14.1861	−0.755051	−0.377526	−	0.925999i	$-0.623225\pi$
$353$	−14.1861	−0.755051	−0.377526	+	0.925999i	$0.623225\pi$
$354$	0	0
$355$	5.08580	0.269926
$356$	−3.33198	−0.176594
$357$	0	0
$358$	13.8944	0.734344
$359$	−17.8724	−0.943270	−0.471635	−	0.881794i	$-0.656336\pi$
$359$	−17.8724	−0.943270	−0.471635	+	0.881794i	$0.656336\pi$
$360$	0	0
$361$	1.00407	0.0528457
$362$	−11.3775	−0.597988
$363$	0	0
$364$	4.82284	0.252786
$365$	1.33978	0.0701273
$366$	0	0
$367$	−3.33978	−0.174335	−0.0871676	−	0.996194i	$-0.527782\pi$
$367$	−3.33978	−0.174335	−0.0871676	+	0.996194i	$0.527782\pi$
$368$	−0.204402	−0.0106552
$369$	0	0
$370$	−0.145848	−0.00758229
$371$	5.41287	0.281022
$372$	0	0
$373$	−13.7643	−0.712688	−0.356344	−	0.934355i	$-0.615977\pi$
$373$	−13.7643	−0.712688	−0.356344	+	0.934355i	$0.615977\pi$
$374$	−0.467356	−0.0241664
$375$	0	0
$376$	1.85939	0.0958905
$377$	−23.4218	−1.20629
$378$	0	0
$379$	38.6300	1.98429	0.992144	−	0.125098i	$-0.0399247\pi$
$379$	38.6300	1.98429	0.992144	+	0.125098i	$0.0399247\pi$
$380$	3.19136	0.163713
$381$	0	0
$382$	16.3085	0.834414
$383$	33.2902	1.70105	0.850525	−	0.525935i	$-0.176284\pi$
$383$	33.2902	1.70105	0.850525	+	0.525935i	$0.176284\pi$
$384$	0	0
$385$	0.259220	0.0132111
$386$	−1.77733	−0.0904635
$387$	0	0
$388$	3.45432	0.175366
$389$	20.1313	1.02070	0.510349	−	0.859967i	$-0.329516\pi$
$389$	20.1313	1.02070	0.510349	+	0.859967i	$0.329516\pi$
$390$	0	0
$391$	−0.467356	−0.0236352
$392$	−3.84111	−0.194006
$393$	0	0
$394$	−7.39576	−0.372593
$395$	11.3357	0.570362
$396$	0	0
$397$	−20.1731	−1.01246	−0.506229	−	0.862399i	$-0.668961\pi$
$397$	−20.1731	−1.01246	−0.506229	+	0.862399i	$0.668961\pi$
$398$	−3.81387	−0.191172
$399$	0	0
$400$	−4.49086	−0.224543
$401$	−15.4140	−0.769740	−0.384870	−	0.922971i	$-0.625754\pi$
$401$	−15.4140	−0.769740	−0.384870	+	0.922971i	$0.625754\pi$
$402$	0	0
$403$	−11.7904	−0.587320
$404$	−5.20440	−0.258929
$405$	0	0
$406$	−15.3409	−0.761358
$407$	0.0417800	0.00207096
$408$	0	0
$409$	−20.1119	−0.994468	−0.497234	−	0.867616i	$-0.665651\pi$
$409$	−20.1119	−0.994468	−0.497234	+	0.867616i	$0.665651\pi$
$410$	6.75905	0.333806
$411$	0	0
$412$	−13.1093	−0.645849
$413$	−25.8266	−1.27084
$414$	0	0
$415$	10.2096	0.501171
$416$	−2.71354	−0.133042
$417$	0	0
$418$	−0.914205	−0.0447152
$419$	−37.4581	−1.82995	−0.914973	−	0.403515i	$-0.867788\pi$
$419$	−37.4581	−1.82995	−0.914973	+	0.403515i	$0.867788\pi$
$420$	0	0
$421$	−33.5621	−1.63572	−0.817859	−	0.575419i	$-0.804839\pi$
$421$	−33.5621	−1.63572	−0.817859	+	0.575419i	$0.804839\pi$
$422$	−9.66395	−0.470434
$423$	0	0
$424$	−3.04551	−0.147903
$425$	−10.2682	−0.498080
$426$	0	0
$427$	−13.0313	−0.630629
$428$	−8.75382	−0.423132
$429$	0	0
$430$	−2.47783	−0.119491
$431$	−4.27599	−0.205967	−0.102984	−	0.994683i	$-0.532839\pi$
$431$	−4.27599	−0.205967	−0.102984	+	0.994683i	$0.532839\pi$
$432$	0	0
$433$	−9.77733	−0.469868	−0.234934	−	0.972011i	$-0.575487\pi$
$433$	−9.77733	−0.469868	−0.234934	+	0.972011i	$0.575487\pi$
$434$	−7.72251	−0.370692
$435$	0	0
$436$	12.6367	0.605189
$437$	−0.914205	−0.0437323
$438$	0	0
$439$	−29.8579	−1.42504	−0.712520	−	0.701652i	$-0.752446\pi$
$439$	−29.8579	−1.42504	−0.712520	+	0.701652i	$0.752446\pi$
$440$	−0.145848	−0.00695304
$441$	0	0
$442$	−6.20440	−0.295113
$443$	4.57816	0.217515	0.108757	−	0.994068i	$-0.465313\pi$
$443$	4.57816	0.217515	0.108757	+	0.994068i	$0.465313\pi$
$444$	0	0
$445$	−2.37749	−0.112704
$446$	1.00000	0.0473514
$447$	0	0
$448$	−1.77733	−0.0839707
$449$	−36.2630	−1.71135	−0.855677	−	0.517510i	$-0.826859\pi$
$449$	−36.2630	−1.71135	−0.855677	+	0.517510i	$0.826859\pi$
$450$	0	0
$451$	−1.93621	−0.0911727
$452$	11.9049	0.559959
$453$	0	0
$454$	−4.64975	−0.218224
$455$	3.44128	0.161330
$456$	0	0
$457$	−3.64602	−0.170553	−0.0852767	−	0.996357i	$-0.527177\pi$
$457$	−3.64602	−0.170553	−0.0852767	+	0.996357i	$0.527177\pi$
$458$	10.8448	0.506746
$459$	0	0
$460$	−0.145848	−0.00680021
$461$	−1.67699	−0.0781053	−0.0390527	−	0.999237i	$-0.512434\pi$
$461$	−1.67699	−0.0781053	−0.0390527	+	0.999237i	$0.512434\pi$
$462$	0	0
$463$	0.421841	0.0196046	0.00980232	−	0.999952i	$-0.496880\pi$
$463$	0.421841	0.0196046	0.00980232	+	0.999952i	$0.496880\pi$
$464$	8.63148	0.400706
$465$	0	0
$466$	−0.994766	−0.0460816
$467$	7.73181	0.357786	0.178893	−	0.983869i	$-0.442748\pi$
$467$	7.73181	0.357786	0.178893	+	0.983869i	$0.442748\pi$
$468$	0	0
$469$	18.7497	0.865783
$470$	1.32674	0.0611981
$471$	0	0
$472$	14.5311	0.668850
$473$	0.709803	0.0326368
$474$	0	0
$475$	−20.0858	−0.921600
$476$	−4.06379	−0.186263
$477$	0	0
$478$	6.45432	0.295214
$479$	40.8448	1.86625	0.933124	−	0.359554i	$-0.117071\pi$
$479$	40.8448	1.86625	0.933124	+	0.359554i	$0.117071\pi$
$480$	0	0
$481$	0.554651	0.0252899
$482$	12.4271	0.566038
$483$	0	0
$484$	−10.9582	−0.498101
$485$	2.46479	0.111920
$486$	0	0
$487$	36.7027	1.66316	0.831580	−	0.555405i	$-0.187437\pi$
$487$	36.7027	1.66316	0.831580	+	0.555405i	$0.187437\pi$
$488$	7.33198	0.331903
$489$	0	0
$490$	−2.74078	−0.123816
$491$	7.00000	0.315906	0.157953	−	0.987447i	$-0.449511\pi$
$491$	7.00000	0.315906	0.157953	+	0.987447i	$0.449511\pi$
$492$	0	0
$493$	19.7355	0.888844
$494$	−12.1365	−0.546049
$495$	0	0
$496$	4.34502	0.195097
$497$	−12.6680	−0.568238
$498$	0	0
$499$	12.4360	0.556714	0.278357	−	0.960478i	$-0.410210\pi$
$499$	12.4360	0.556714	0.278357	+	0.960478i	$0.410210\pi$
$500$	−6.77209	−0.302857
$501$	0	0
$502$	−3.39576	−0.151560
$503$	6.27599	0.279833	0.139916	−	0.990163i	$-0.455317\pi$
$503$	6.27599	0.279833	0.139916	+	0.990163i	$0.455317\pi$
$504$	0	0
$505$	−3.71354	−0.165250
$506$	0.0417800	0.00185735
$507$	0	0
$508$	−9.02724	−0.400519
$509$	19.6352	0.870315	0.435158	−	0.900354i	$-0.356693\pi$
$509$	19.6352	0.870315	0.435158	+	0.900354i	$0.356693\pi$
$510$	0	0
$511$	−3.33721	−0.147630
$512$	1.00000	0.0441942
$513$	0	0
$514$	−16.9582	−0.747995
$515$	−9.35398	−0.412186
$516$	0	0
$517$	−0.380061	−0.0167151
$518$	0.363288	0.0159620
$519$	0	0
$520$	−1.93621	−0.0849085
$521$	−35.8448	−1.57039	−0.785196	−	0.619248i	$-0.787438\pi$
$521$	−35.8448	−1.57039	−0.785196	+	0.619248i	$0.787438\pi$
$522$	0	0
$523$	−5.41404	−0.236739	−0.118370	−	0.992970i	$-0.537767\pi$
$523$	−5.41404	−0.236739	−0.118370	+	0.992970i	$0.537767\pi$
$524$	5.70050	0.249027
$525$	0	0
$526$	−12.8866	−0.561884
$527$	9.93471	0.432763
$528$	0	0
$529$	−22.9582	−0.998183
$530$	−2.17309	−0.0943930
$531$	0	0
$532$	−7.94925	−0.344644
$533$	−25.7042	−1.11337
$534$	0	0
$535$	−6.24618	−0.270046
$536$	−10.5494	−0.455665
$537$	0	0
$538$	−12.7721	−0.550644
$539$	0.785130	0.0338179
$540$	0	0
$541$	36.6274	1.57474	0.787368	−	0.616484i	$-0.211443\pi$
$541$	36.6274	1.57474	0.787368	+	0.616484i	$0.211443\pi$
$542$	20.0910	0.862984
$543$	0	0
$544$	2.28646	0.0980313
$545$	9.01677	0.386236
$546$	0	0
$547$	−24.0780	−1.02950	−0.514750	−	0.857340i	$-0.672115\pi$
$547$	−24.0780	−1.02950	−0.514750	+	0.857340i	$0.672115\pi$
$548$	9.62624	0.411213
$549$	0	0
$550$	0.917939	0.0391411
$551$	38.6051	1.64463
$552$	0	0
$553$	−28.2357	−1.20070
$554$	22.5547	0.958255
$555$	0	0
$556$	13.6770	0.580034
$557$	−11.8489	−0.502055	−0.251027	−	0.967980i	$-0.580768\pi$
$557$	−11.8489	−0.502055	−0.251027	+	0.967980i	$0.580768\pi$
$558$	0	0
$559$	9.42301	0.398551
$560$	−1.26819	−0.0535908
$561$	0	0
$562$	−18.1679	−0.766365
$563$	−8.89036	−0.374684	−0.187342	−	0.982295i	$-0.559987\pi$
$563$	−8.89036	−0.374684	−0.187342	+	0.982295i	$0.559987\pi$
$564$	0	0
$565$	8.49460	0.357370
$566$	3.92691	0.165060
$567$	0	0
$568$	7.12758	0.299066
$569$	−9.04551	−0.379208	−0.189604	−	0.981861i	$-0.560720\pi$
$569$	−9.04551	−0.379208	−0.189604	+	0.981861i	$0.560720\pi$
$570$	0	0
$571$	−22.7303	−0.951234	−0.475617	−	0.879652i	$-0.657775\pi$
$571$	−22.7303	−0.951234	−0.475617	+	0.879652i	$0.657775\pi$
$572$	0.554651	0.0231911
$573$	0	0
$574$	−16.8359	−0.702716
$575$	0.917939	0.0382807
$576$	0	0
$577$	−23.7173	−0.987363	−0.493681	−	0.869643i	$-0.664349\pi$
$577$	−23.7173	−0.987363	−0.493681	+	0.869643i	$0.664349\pi$
$578$	−11.7721	−0.489654
$579$	0	0
$580$	6.15889	0.255734
$581$	−25.4308	−1.05505
$582$	0	0
$583$	0.622508	0.0257816
$584$	1.87766	0.0776981
$585$	0	0
$586$	−14.7135	−0.607811
$587$	−31.6677	−1.30707	−0.653533	−	0.756898i	$-0.726714\pi$
$587$	−31.6677	−1.30707	−0.653533	+	0.756898i	$0.726714\pi$
$588$	0	0
$589$	19.4335	0.800742
$590$	10.3685	0.426865
$591$	0	0
$592$	−0.204402	−0.00840085
$593$	38.4398	1.57853	0.789266	−	0.614051i	$-0.210461\pi$
$593$	38.4398	1.57853	0.789266	+	0.614051i	$0.210461\pi$
$594$	0	0
$595$	−2.89967	−0.118875
$596$	−17.7355	−0.726476
$597$	0	0
$598$	0.554651	0.0226814
$599$	−4.00897	−0.163802	−0.0819010	−	0.996640i	$-0.526099\pi$
$599$	−4.00897	−0.163802	−0.0819010	+	0.996640i	$0.526099\pi$
$600$	0	0
$601$	17.2589	0.704005	0.352002	−	0.935999i	$-0.385501\pi$
$601$	17.2589	0.704005	0.352002	+	0.935999i	$0.385501\pi$
$602$	6.17192	0.251549
$603$	0	0
$604$	19.5819	0.796776
$605$	−7.81911	−0.317892
$606$	0	0
$607$	−43.9306	−1.78309	−0.891545	−	0.452932i	$-0.850378\pi$
$607$	−43.9306	−1.78309	−0.891545	+	0.452932i	$0.850378\pi$
$608$	4.47259	0.181388
$609$	0	0
$610$	5.23164	0.211823
$611$	−5.04551	−0.204120
$612$	0	0
$613$	15.4271	0.623094	0.311547	−	0.950231i	$-0.399153\pi$
$613$	15.4271	0.623094	0.311547	+	0.950231i	$0.399153\pi$
$614$	−10.0988	−0.407556
$615$	0	0
$616$	0.363288	0.0146373
$617$	43.4674	1.74993	0.874965	−	0.484186i	$-0.160884\pi$
$617$	43.4674	1.74993	0.874965	+	0.484186i	$0.160884\pi$
$618$	0	0
$619$	−20.4178	−0.820660	−0.410330	−	0.911937i	$-0.634586\pi$
$619$	−20.4178	−0.820660	−0.410330	+	0.911937i	$0.634586\pi$
$620$	3.10033	0.124512
$621$	0	0
$622$	4.67699	0.187530
$623$	5.92201	0.237260
$624$	0	0
$625$	17.6222	0.704887
$626$	16.6990	0.667426
$627$	0	0
$628$	0.331977	0.0132473
$629$	−0.467356	−0.0186347
$630$	0	0
$631$	29.5181	1.17510	0.587549	−	0.809189i	$-0.300093\pi$
$631$	29.5181	1.17510	0.587549	+	0.809189i	$0.300093\pi$
$632$	15.8866	0.631936
$633$	0	0
$634$	−18.6498	−0.740676
$635$	−6.44128	−0.255614
$636$	0	0
$637$	10.4230	0.412975
$638$	−1.76429	−0.0698488
$639$	0	0
$640$	0.713538	0.0282051
$641$	−24.4398	−0.965313	−0.482657	−	0.875810i	$-0.660328\pi$
$641$	−24.4398	−0.965313	−0.482657	+	0.875810i	$0.660328\pi$
$642$	0	0
$643$	−15.6184	−0.615931	−0.307966	−	0.951398i	$-0.599648\pi$
$643$	−15.6184	−0.615931	−0.307966	+	0.951398i	$0.599648\pi$
$644$	0.363288	0.0143156
$645$	0	0
$646$	10.2264	0.402353
$647$	−15.6625	−0.615755	−0.307877	−	0.951426i	$-0.599619\pi$
$647$	−15.6625	−0.615755	−0.307877	+	0.951426i	$0.599619\pi$
$648$	0	0
$649$	−2.97019	−0.116590
$650$	12.1861	0.477979
$651$	0	0
$652$	−23.9269	−0.937050
$653$	32.4983	1.27176	0.635879	−	0.771789i	$-0.280638\pi$
$653$	32.4983	1.27176	0.635879	+	0.771789i	$0.280638\pi$
$654$	0	0
$655$	4.06752	0.158931
$656$	9.47259	0.369843
$657$	0	0
$658$	−3.30473	−0.128832
$659$	31.4726	1.22600	0.612999	−	0.790084i	$-0.289963\pi$
$659$	31.4726	1.22600	0.612999	+	0.790084i	$0.289963\pi$
$660$	0	0
$661$	−22.8956	−0.890536	−0.445268	−	0.895397i	$-0.646892\pi$
$661$	−22.8956	−0.890536	−0.445268	+	0.895397i	$0.646892\pi$
$662$	14.8762	0.578178
$663$	0	0
$664$	14.3085	0.555276
$665$	−5.67209	−0.219954
$666$	0	0
$667$	−1.76429	−0.0683135
$668$	−12.1626	−0.470586
$669$	0	0
$670$	−7.52741	−0.290809
$671$	−1.49867	−0.0578554
$672$	0	0
$673$	−32.3450	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$673$	−32.3450	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$674$	33.7393	1.29959
$675$	0	0
$676$	−5.63671	−0.216797
$677$	42.5674	1.63600	0.817998	−	0.575221i	$-0.195084\pi$
$677$	42.5674	1.63600	0.817998	+	0.575221i	$0.195084\pi$
$678$	0	0
$679$	−6.13945	−0.235610
$680$	1.63148	0.0625643
$681$	0	0
$682$	−0.888128	−0.0340082
$683$	20.6144	0.788787	0.394393	−	0.918942i	$-0.370955\pi$
$683$	20.6144	0.788787	0.394393	+	0.918942i	$0.370955\pi$
$684$	0	0
$685$	6.86869	0.262439
$686$	19.2682	0.735663
$687$	0	0
$688$	−3.47259	−0.132391
$689$	8.26412	0.314838
$690$	0	0
$691$	−0.705734	−0.0268474	−0.0134237	−	0.999910i	$-0.504273\pi$
$691$	−0.705734	−0.0268474	−0.0134237	+	0.999910i	$0.504273\pi$
$692$	12.9870	0.493690
$693$	0	0
$694$	18.3032	0.694781
$695$	9.75905	0.370182
$696$	0	0
$697$	21.6587	0.820383
$698$	−11.1313	−0.421326
$699$	0	0
$700$	7.98173	0.301681
$701$	−17.7956	−0.672130	−0.336065	−	0.941839i	$-0.609096\pi$
$701$	−17.7956	−0.672130	−0.336065	+	0.941839i	$0.609096\pi$
$702$	0	0
$703$	−0.914205	−0.0344799
$704$	−0.204402	−0.00770367
$705$	0	0
$706$	−14.1861	−0.533902
$707$	9.24992	0.347879
$708$	0	0
$709$	−8.12234	−0.305041	−0.152520	−	0.988300i	$-0.548739\pi$
$709$	−8.12234	−0.305041	−0.152520	+	0.988300i	$0.548739\pi$
$710$	5.08580	0.190867
$711$	0	0
$712$	−3.33198	−0.124871
$713$	−0.888128	−0.0332607
$714$	0	0
$715$	0.395765	0.0148008
$716$	13.8944	0.519259
$717$	0	0
$718$	−17.8724	−0.666993
$719$	−2.96719	−0.110657	−0.0553287	−	0.998468i	$-0.517621\pi$
$719$	−2.96719	−0.110657	−0.0553287	+	0.998468i	$0.517621\pi$
$720$	0	0
$721$	23.2995	0.867719
$722$	1.00407	0.0373676
$723$	0	0
$724$	−11.3775	−0.422841
$725$	−38.7628	−1.43961
$726$	0	0
$727$	10.7695	0.399419	0.199710	−	0.979855i	$-0.436000\pi$
$727$	10.7695	0.399419	0.199710	+	0.979855i	$0.436000\pi$
$728$	4.82284	0.178746
$729$	0	0
$730$	1.33978	0.0495875
$731$	−7.93995	−0.293670
$732$	0	0
$733$	−37.8684	−1.39870	−0.699350	−	0.714780i	$-0.746527\pi$
$733$	−37.8684	−1.39870	−0.699350	+	0.714780i	$0.746527\pi$
$734$	−3.33978	−0.123274
$735$	0	0
$736$	−0.204402	−0.00753434
$737$	2.15632	0.0794290
$738$	0	0
$739$	−22.9959	−0.845919	−0.422960	−	0.906149i	$-0.639009\pi$
$739$	−22.9959	−0.845919	−0.422960	+	0.906149i	$0.639009\pi$
$740$	−0.145848	−0.00536149
$741$	0	0
$742$	5.41287	0.198713
$743$	43.2861	1.58801	0.794007	−	0.607909i	$-0.207991\pi$
$743$	43.2861	1.58801	0.794007	+	0.607909i	$0.207991\pi$
$744$	0	0
$745$	−12.6550	−0.463643
$746$	−13.7643	−0.503946
$747$	0	0
$748$	−0.467356	−0.0170882
$749$	15.5584	0.568491
$750$	0	0
$751$	−35.4998	−1.29541	−0.647704	−	0.761892i	$-0.724271\pi$
$751$	−35.4998	−1.29541	−0.647704	+	0.761892i	$0.724271\pi$
$752$	1.85939	0.0678048
$753$	0	0
$754$	−23.4218	−0.852973
$755$	13.9724	0.508509
$756$	0	0
$757$	−1.16412	−0.0423107	−0.0211553	−	0.999776i	$-0.506734\pi$
$757$	−1.16412	−0.0423107	−0.0211553	+	0.999776i	$0.506734\pi$
$758$	38.6300	1.40310
$759$	0	0
$760$	3.19136	0.115763
$761$	−46.9907	−1.70341	−0.851706	−	0.524020i	$-0.824432\pi$
$761$	−46.9907	−1.70341	−0.851706	+	0.524020i	$0.824432\pi$
$762$	0	0
$763$	−22.4596	−0.813090
$764$	16.3085	0.590020
$765$	0	0
$766$	33.2902	1.20282
$767$	−39.4308	−1.42376
$768$	0	0
$769$	5.50390	0.198476	0.0992379	−	0.995064i	$-0.468360\pi$
$769$	5.50390	0.198476	0.0992379	+	0.995064i	$0.468360\pi$
$770$	0.259220	0.00934164
$771$	0	0
$772$	−1.77733	−0.0639674
$773$	−30.7591	−1.10633	−0.553163	−	0.833073i	$-0.686579\pi$
$773$	−30.7591	−1.10633	−0.553163	+	0.833073i	$0.686579\pi$
$774$	0	0
$775$	−19.5129	−0.700923
$776$	3.45432	0.124003
$777$	0	0
$778$	20.1313	0.721742
$779$	42.3670	1.51796
$780$	0	0
$781$	−1.45689	−0.0521315
$782$	−0.467356	−0.0167126
$783$	0	0
$784$	−3.84111	−0.137183
$785$	0.236878	0.00845455
$786$	0	0
$787$	5.61320	0.200089	0.100045	−	0.994983i	$-0.468101\pi$
$787$	5.61320	0.200089	0.100045	+	0.994983i	$0.468101\pi$
$788$	−7.39576	−0.263463
$789$	0	0
$790$	11.3357	0.403307
$791$	−21.1589	−0.752323
$792$	0	0
$793$	−19.8956	−0.706513
$794$	−20.1731	−0.715916
$795$	0	0
$796$	−3.81387	−0.135179
$797$	−38.7027	−1.37092	−0.685461	−	0.728110i	$-0.740399\pi$
$797$	−38.7027	−1.37092	−0.685461	+	0.728110i	$0.740399\pi$
$798$	0	0
$799$	4.25142	0.150404
$800$	−4.49086	−0.158776
$801$	0	0
$802$	−15.4140	−0.544289
$803$	−0.383796	−0.0135439
$804$	0	0
$805$	0.259220	0.00913630
$806$	−11.7904	−0.415298
$807$	0	0
$808$	−5.20440	−0.183090
$809$	−20.4987	−0.720695	−0.360347	−	0.932818i	$-0.617342\pi$
$809$	−20.4987	−0.720695	−0.360347	+	0.932818i	$0.617342\pi$
$810$	0	0
$811$	−49.8669	−1.75106	−0.875531	−	0.483162i	$-0.839488\pi$
$811$	−49.8669	−1.75106	−0.875531	+	0.483162i	$0.839488\pi$
$812$	−15.3409	−0.538362
$813$	0	0
$814$	0.0417800	0.00146439
$815$	−17.0728	−0.598033
$816$	0	0
$817$	−15.5315	−0.543378
$818$	−20.1119	−0.703195
$819$	0	0
$820$	6.75905	0.236036
$821$	30.4905	1.06413	0.532063	−	0.846705i	$-0.321417\pi$
$821$	30.4905	1.06413	0.532063	+	0.846705i	$0.321417\pi$
$822$	0	0
$823$	−40.1298	−1.39884	−0.699419	−	0.714712i	$-0.746558\pi$
$823$	−40.1298	−1.39884	−0.699419	+	0.714712i	$0.746558\pi$
$824$	−13.1093	−0.456684
$825$	0	0
$826$	−25.8266	−0.898621
$827$	−26.8773	−0.934616	−0.467308	−	0.884095i	$-0.654776\pi$
$827$	−26.8773	−0.934616	−0.467308	+	0.884095i	$0.654776\pi$
$828$	0	0
$829$	−30.0728	−1.04447	−0.522235	−	0.852802i	$-0.674902\pi$
$829$	−30.0728	−1.04447	−0.522235	+	0.852802i	$0.674902\pi$
$830$	10.2096	0.354382
$831$	0	0
$832$	−2.71354	−0.0940750
$833$	−8.78256	−0.304298
$834$	0	0
$835$	−8.67849	−0.300332
$836$	−0.914205	−0.0316184
$837$	0	0
$838$	−37.4581	−1.29397
$839$	−15.7941	−0.545273	−0.272636	−	0.962117i	$-0.587896\pi$
$839$	−15.7941	−0.545273	−0.272636	+	0.962117i	$0.587896\pi$
$840$	0	0
$841$	45.5024	1.56905
$842$	−33.5621	−1.15663
$843$	0	0
$844$	−9.66395	−0.332647
$845$	−4.02201	−0.138361
$846$	0	0
$847$	19.4763	0.669214
$848$	−3.04551	−0.104583
$849$	0	0
$850$	−10.2682	−0.352196
$851$	0.0417800	0.00143220
$852$	0	0
$853$	−14.6900	−0.502977	−0.251489	−	0.967860i	$-0.580920\pi$
$853$	−14.6900	−0.502977	−0.251489	+	0.967860i	$0.580920\pi$
$854$	−13.0313	−0.445922
$855$	0	0
$856$	−8.75382	−0.299199
$857$	−19.8606	−0.678424	−0.339212	−	0.940710i	$-0.610160\pi$
$857$	−19.8606	−0.678424	−0.339212	+	0.940710i	$0.610160\pi$
$858$	0	0
$859$	−47.6833	−1.62693	−0.813467	−	0.581612i	$-0.802422\pi$
$859$	−47.6833	−1.62693	−0.813467	+	0.581612i	$0.802422\pi$
$860$	−2.47783	−0.0844931
$861$	0	0
$862$	−4.27599	−0.145641
$863$	−23.4465	−0.798129	−0.399064	−	0.916923i	$-0.630665\pi$
$863$	−23.4465	−0.798129	−0.399064	+	0.916923i	$0.630665\pi$
$864$	0	0
$865$	9.26669	0.315077
$866$	−9.77733	−0.332247
$867$	0	0
$868$	−7.72251	−0.262119
$869$	−3.24725	−0.110155
$870$	0	0
$871$	28.6262	0.969963
$872$	12.6367	0.427933
$873$	0	0
$874$	−0.914205	−0.0309234
$875$	12.0362	0.406898
$876$	0	0
$877$	4.94892	0.167113	0.0835565	−	0.996503i	$-0.473372\pi$
$877$	4.94892	0.167113	0.0835565	+	0.996503i	$0.473372\pi$
$878$	−29.8579	−1.00765
$879$	0	0
$880$	−0.145848	−0.00491654
$881$	−19.8646	−0.669256	−0.334628	−	0.942350i	$-0.608611\pi$
$881$	−19.8646	−0.669256	−0.334628	+	0.942350i	$0.608611\pi$
$882$	0	0
$883$	7.11454	0.239423	0.119712	−	0.992809i	$-0.461803\pi$
$883$	7.11454	0.239423	0.119712	+	0.992809i	$0.461803\pi$
$884$	−6.20440	−0.208677
$885$	0	0
$886$	4.57816	0.153806
$887$	35.7445	1.20018	0.600092	−	0.799931i	$-0.295131\pi$
$887$	35.7445	1.20018	0.600092	+	0.799931i	$0.295131\pi$
$888$	0	0
$889$	16.0443	0.538110
$890$	−2.37749	−0.0796937
$891$	0	0
$892$	1.00000	0.0334825
$893$	8.31627	0.278293
$894$	0	0
$895$	9.91420	0.331395
$896$	−1.77733	−0.0593763
$897$	0	0
$898$	−36.2630	−1.21011
$899$	37.5039	1.25083
$900$	0	0
$901$	−6.96345	−0.231986
$902$	−1.93621	−0.0644688
$903$	0	0
$904$	11.9049	0.395951
$905$	−8.11827	−0.269860
$906$	0	0
$907$	27.2369	0.904386	0.452193	−	0.891920i	$-0.350642\pi$
$907$	27.2369	0.904386	0.452193	+	0.891920i	$0.350642\pi$
$908$	−4.64975	−0.154307
$909$	0	0
$910$	3.44128	0.114077
$911$	17.8046	0.589892	0.294946	−	0.955514i	$-0.404698\pi$
$911$	17.8046	0.589892	0.294946	+	0.955514i	$0.404698\pi$
$912$	0	0
$913$	−2.92467	−0.0967926
$914$	−3.64602	−0.120599
$915$	0	0
$916$	10.8448	0.358324
$917$	−10.1316	−0.334576
$918$	0	0
$919$	−20.7147	−0.683315	−0.341658	−	0.939825i	$-0.610988\pi$
$919$	−20.7147	−0.683315	−0.341658	+	0.939825i	$0.610988\pi$
$920$	−0.145848	−0.00480847
$921$	0	0
$922$	−1.67699	−0.0552288
$923$	−19.3409	−0.636615
$924$	0	0
$925$	0.917939	0.0301817
$926$	0.421841	0.0138626
$927$	0	0
$928$	8.63148	0.283342
$929$	18.5389	0.608243	0.304122	−	0.952633i	$-0.401637\pi$
$929$	18.5389	0.608243	0.304122	+	0.952633i	$0.401637\pi$
$930$	0	0
$931$	−17.1797	−0.563043
$932$	−0.994766	−0.0325846
$933$	0	0
$934$	7.73181	0.252993
$935$	−0.333477	−0.0109058
$936$	0	0
$937$	18.4868	0.603937	0.301969	−	0.953318i	$-0.402356\pi$
$937$	18.4868	0.603937	0.301969	+	0.953318i	$0.402356\pi$
$938$	18.7497	0.612201
$939$	0	0
$940$	1.32674	0.0432736
$941$	14.5584	0.474590	0.237295	−	0.971438i	$-0.423739\pi$
$941$	14.5584	0.474590	0.237295	+	0.971438i	$0.423739\pi$
$942$	0	0
$943$	−1.93621	−0.0630517
$944$	14.5311	0.472948
$945$	0	0
$946$	0.709803	0.0230777
$947$	51.0220	1.65799	0.828996	−	0.559254i	$-0.188912\pi$
$947$	51.0220	1.65799	0.828996	+	0.559254i	$0.188912\pi$
$948$	0	0
$949$	−5.09510	−0.165394
$950$	−20.0858	−0.651669
$951$	0	0
$952$	−4.06379	−0.131708
$953$	−33.6845	−1.09115	−0.545573	−	0.838063i	$-0.683688\pi$
$953$	−33.6845	−1.09115	−0.545573	+	0.838063i	$0.683688\pi$
$954$	0	0
$955$	11.6367	0.376555
$956$	6.45432	0.208748
$957$	0	0
$958$	40.8448	1.31964
$959$	−17.1090	−0.552477
$960$	0	0
$961$	−12.1208	−0.390995
$962$	0.554651	0.0178827
$963$	0	0
$964$	12.4271	0.400249
$965$	−1.26819	−0.0408245
$966$	0	0
$967$	16.9478	0.545003	0.272501	−	0.962155i	$-0.412149\pi$
$967$	16.9478	0.545003	0.272501	+	0.962155i	$0.412149\pi$
$968$	−10.9582	−0.352211
$969$	0	0
$970$	2.46479	0.0791395
$971$	−12.2656	−0.393622	−0.196811	−	0.980441i	$-0.563059\pi$
$971$	−12.2656	−0.393622	−0.196811	+	0.980441i	$0.563059\pi$
$972$	0	0
$973$	−24.3085	−0.779294
$974$	36.7027	1.17603
$975$	0	0
$976$	7.33198	0.234691
$977$	40.1623	1.28491	0.642453	−	0.766325i	$-0.277917\pi$
$977$	40.1623	1.28491	0.642453	+	0.766325i	$0.277917\pi$
$978$	0	0
$979$	0.681061	0.0217668
$980$	−2.74078	−0.0875510
$981$	0	0
$982$	7.00000	0.223379
$983$	40.2887	1.28501	0.642505	−	0.766282i	$-0.277895\pi$
$983$	40.2887	1.28501	0.642505	+	0.766282i	$0.277895\pi$
$984$	0	0
$985$	−5.27716	−0.168144
$986$	19.7355	0.628508
$987$	0	0
$988$	−12.1365	−0.386115
$989$	0.709803	0.0225704
$990$	0	0
$991$	21.0873	0.669860	0.334930	−	0.942243i	$-0.391287\pi$
$991$	21.0873	0.669860	0.334930	+	0.942243i	$0.391287\pi$
$992$	4.34502	0.137954
$993$	0	0
$994$	−12.6680	−0.401805
$995$	−2.72134	−0.0862723
$996$	0	0
$997$	21.6404	0.685360	0.342680	−	0.939452i	$-0.388665\pi$
$997$	21.6404	0.685360	0.342680	+	0.939452i	$0.388665\pi$
$998$	12.4360	0.393656
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.o.1.2		3
3.2	odd	2		1338.2.a.d.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.d.1.2	✓	3	3.2	odd	2
4014.2.a.o.1.2		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.713538 q^{5} -1.77733 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.713538 q^{5} -1.77733 q^{7} +1.00000 q^{8} +0.713538 q^{10} -0.204402 q^{11} -2.71354 q^{13} -1.77733 q^{14} +1.00000 q^{16} +2.28646 q^{17} +4.47259 q^{19} +0.713538 q^{20} -0.204402 q^{22} -0.204402 q^{23} -4.49086 q^{25} -2.71354 q^{26} -1.77733 q^{28} +8.63148 q^{29} +4.34502 q^{31} +1.00000 q^{32} +2.28646 q^{34} -1.26819 q^{35} -0.204402 q^{37} +4.47259 q^{38} +0.713538 q^{40} +9.47259 q^{41} -3.47259 q^{43} -0.204402 q^{44} -0.204402 q^{46} +1.85939 q^{47} -3.84111 q^{49} -4.49086 q^{50} -2.71354 q^{52} -3.04551 q^{53} -0.145848 q^{55} -1.77733 q^{56} +8.63148 q^{58} +14.5311 q^{59} +7.33198 q^{61} +4.34502 q^{62} +1.00000 q^{64} -1.93621 q^{65} -10.5494 q^{67} +2.28646 q^{68} -1.26819 q^{70} +7.12758 q^{71} +1.87766 q^{73} -0.204402 q^{74} +4.47259 q^{76} +0.363288 q^{77} +15.8866 q^{79} +0.713538 q^{80} +9.47259 q^{82} +14.3085 q^{83} +1.63148 q^{85} -3.47259 q^{86} -0.204402 q^{88} -3.33198 q^{89} +4.82284 q^{91} -0.204402 q^{92} +1.85939 q^{94} +3.19136 q^{95} +3.45432 q^{97} -3.84111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} + q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 + q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} + q^{7} + 3 q^{8} + q^{10} + 8 q^{11} - 7 q^{13} + q^{14} + 3 q^{16} + 8 q^{17} - 9 q^{19} + q^{20} + 8 q^{22} + 8 q^{23} - 6 q^{25} - 7 q^{26} + q^{28} + 15 q^{29} + q^{31} + 3 q^{32} + 8 q^{34} + 10 q^{35} + 8 q^{37} - 9 q^{38} + q^{40} + 6 q^{41} + 12 q^{43} + 8 q^{44} + 8 q^{46} + 9 q^{47} - 6 q^{50} - 7 q^{52} + 11 q^{53} - 5 q^{55} + q^{56} + 15 q^{58} + 8 q^{59} + 3 q^{61} + q^{62} + 3 q^{64} - 11 q^{65} - 11 q^{67} + 8 q^{68} + 10 q^{70} + 11 q^{71} + 24 q^{73} + 8 q^{74} - 9 q^{76} + 4 q^{77} + 16 q^{79} + q^{80} + 6 q^{82} + q^{83} - 6 q^{85} + 12 q^{86} + 8 q^{88} + 9 q^{89} - 12 q^{91} + 8 q^{92} + 9 q^{94} - 6 q^{95} - 27 q^{97}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 + q^7 + 3 * q^8 + q^10 + 8 * q^11 - 7 * q^13 + q^14 + 3 * q^16 + 8 * q^17 - 9 * q^19 + q^20 + 8 * q^22 + 8 * q^23 - 6 * q^25 - 7 * q^26 + q^28 + 15 * q^29 + q^31 + 3 * q^32 + 8 * q^34 + 10 * q^35 + 8 * q^37 - 9 * q^38 + q^40 + 6 * q^41 + 12 * q^43 + 8 * q^44 + 8 * q^46 + 9 * q^47 - 6 * q^50 - 7 * q^52 + 11 * q^53 - 5 * q^55 + q^56 + 15 * q^58 + 8 * q^59 + 3 * q^61 + q^62 + 3 * q^64 - 11 * q^65 - 11 * q^67 + 8 * q^68 + 10 * q^70 + 11 * q^71 + 24 * q^73 + 8 * q^74 - 9 * q^76 + 4 * q^77 + 16 * q^79 + q^80 + 6 * q^82 + q^83 - 6 * q^85 + 12 * q^86 + 8 * q^88 + 9 * q^89 - 12 * q^91 + 8 * q^92 + 9 * q^94 - 6 * q^95 - 27 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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