LMFDB - Embedded newform 8046.2.a.t.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
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			1.16
Root			$4.36260$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +4.36260 q^{5} +2.57952 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +4.36260 q^{5} +2.57952 q^{7} +1.00000 q^{8} +4.36260 q^{10} -6.22907 q^{11} +0.768400 q^{13} +2.57952 q^{14} +1.00000 q^{16} +2.14011 q^{17} +3.60185 q^{19} +4.36260 q^{20} -6.22907 q^{22} -0.549412 q^{23} +14.0323 q^{25} +0.768400 q^{26} +2.57952 q^{28} -9.37769 q^{29} +3.83934 q^{31} +1.00000 q^{32} +2.14011 q^{34} +11.2534 q^{35} +3.34752 q^{37} +3.60185 q^{38} +4.36260 q^{40} -1.84301 q^{41} +11.6658 q^{43} -6.22907 q^{44} -0.549412 q^{46} -1.30340 q^{47} -0.346089 q^{49} +14.0323 q^{50} +0.768400 q^{52} -7.86564 q^{53} -27.1749 q^{55} +2.57952 q^{56} -9.37769 q^{58} +11.9672 q^{59} -1.43378 q^{61} +3.83934 q^{62} +1.00000 q^{64} +3.35222 q^{65} +1.90317 q^{67} +2.14011 q^{68} +11.2534 q^{70} +11.9673 q^{71} -6.31479 q^{73} +3.34752 q^{74} +3.60185 q^{76} -16.0680 q^{77} -5.58864 q^{79} +4.36260 q^{80} -1.84301 q^{82} -6.77087 q^{83} +9.33643 q^{85} +11.6658 q^{86} -6.22907 q^{88} -1.81991 q^{89} +1.98210 q^{91} -0.549412 q^{92} -1.30340 q^{94} +15.7134 q^{95} +2.30485 q^{97} -0.346089 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	$ 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) $ 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * 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
$5$	4.36260	1.95101	0.975506	−	0.219971i	$-0.0705963\pi$
$5$	4.36260	1.95101	0.975506	+	0.219971i	$0.0705963\pi$
$6$	0	0
$7$	2.57952	0.974966	0.487483	−	0.873132i	$-0.337915\pi$
$7$	2.57952	0.974966	0.487483	+	0.873132i	$0.337915\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	4.36260	1.37957
$11$	−6.22907	−1.87814	−0.939068	−	0.343732i	$-0.888309\pi$
$11$	−6.22907	−1.87814	−0.939068	+	0.343732i	$0.888309\pi$
$12$	0	0
$13$	0.768400	0.213116	0.106558	−	0.994306i	$-0.466017\pi$
$13$	0.768400	0.213116	0.106558	+	0.994306i	$0.466017\pi$
$14$	2.57952	0.689405
$15$	0	0
$16$	1.00000	0.250000
$17$	2.14011	0.519053	0.259526	−	0.965736i	$-0.416434\pi$
$17$	2.14011	0.519053	0.259526	+	0.965736i	$0.416434\pi$
$18$	0	0
$19$	3.60185	0.826320	0.413160	−	0.910658i	$-0.364425\pi$
$19$	3.60185	0.826320	0.413160	+	0.910658i	$0.364425\pi$
$20$	4.36260	0.975506
$21$	0	0
$22$	−6.22907	−1.32804
$23$	−0.549412	−0.114560	−0.0572802	−	0.998358i	$-0.518243\pi$
$23$	−0.549412	−0.114560	−0.0572802	+	0.998358i	$0.518243\pi$
$24$	0	0
$25$	14.0323	2.80645
$26$	0.768400	0.150696
$27$	0	0
$28$	2.57952	0.487483
$29$	−9.37769	−1.74139	−0.870696	−	0.491821i	$-0.836331\pi$
$29$	−9.37769	−1.74139	−0.870696	+	0.491821i	$0.836331\pi$
$30$	0	0
$31$	3.83934	0.689566	0.344783	−	0.938682i	$-0.387952\pi$
$31$	3.83934	0.689566	0.344783	+	0.938682i	$0.387952\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.14011	0.367026
$35$	11.2534	1.90217
$36$	0	0
$37$	3.34752	0.550330	0.275165	−	0.961397i	$-0.411268\pi$
$37$	3.34752	0.550330	0.275165	+	0.961397i	$0.411268\pi$
$38$	3.60185	0.584297
$39$	0	0
$40$	4.36260	0.689787
$41$	−1.84301	−0.287830	−0.143915	−	0.989590i	$-0.545969\pi$
$41$	−1.84301	−0.287830	−0.143915	+	0.989590i	$0.545969\pi$
$42$	0	0
$43$	11.6658	1.77902	0.889508	−	0.456919i	$-0.151047\pi$
$43$	11.6658	1.77902	0.889508	+	0.456919i	$0.151047\pi$
$44$	−6.22907	−0.939068
$45$	0	0
$46$	−0.549412	−0.0810064
$47$	−1.30340	−0.190120	−0.0950601	−	0.995472i	$-0.530304\pi$
$47$	−1.30340	−0.190120	−0.0950601	+	0.995472i	$0.530304\pi$
$48$	0	0
$49$	−0.346089	−0.0494413
$50$	14.0323	1.98446
$51$	0	0
$52$	0.768400	0.106558
$53$	−7.86564	−1.08043	−0.540215	−	0.841527i	$-0.681657\pi$
$53$	−7.86564	−1.08043	−0.540215	+	0.841527i	$0.681657\pi$
$54$	0	0
$55$	−27.1749	−3.66427
$56$	2.57952	0.344703
$57$	0	0
$58$	−9.37769	−1.23135
$59$	11.9672	1.55799	0.778996	−	0.627029i	$-0.215729\pi$
$59$	11.9672	1.55799	0.778996	+	0.627029i	$0.215729\pi$
$60$	0	0
$61$	−1.43378	−0.183577	−0.0917885	−	0.995779i	$-0.529258\pi$
$61$	−1.43378	−0.183577	−0.0917885	+	0.995779i	$0.529258\pi$
$62$	3.83934	0.487597
$63$	0	0
$64$	1.00000	0.125000
$65$	3.35222	0.415792
$66$	0	0
$67$	1.90317	0.232509	0.116254	−	0.993219i	$-0.462911\pi$
$67$	1.90317	0.232509	0.116254	+	0.993219i	$0.462911\pi$
$68$	2.14011	0.259526
$69$	0	0
$70$	11.2534	1.34504
$71$	11.9673	1.42026	0.710129	−	0.704072i	$-0.248637\pi$
$71$	11.9673	1.42026	0.710129	+	0.704072i	$0.248637\pi$
$72$	0	0
$73$	−6.31479	−0.739091	−0.369545	−	0.929213i	$-0.620487\pi$
$73$	−6.31479	−0.739091	−0.369545	+	0.929213i	$0.620487\pi$
$74$	3.34752	0.389142
$75$	0	0
$76$	3.60185	0.413160
$77$	−16.0680	−1.83112
$78$	0	0
$79$	−5.58864	−0.628771	−0.314386	−	0.949295i	$-0.601799\pi$
$79$	−5.58864	−0.628771	−0.314386	+	0.949295i	$0.601799\pi$
$80$	4.36260	0.487753
$81$	0	0
$82$	−1.84301	−0.203527
$83$	−6.77087	−0.743200	−0.371600	−	0.928393i	$-0.621191\pi$
$83$	−6.77087	−0.743200	−0.371600	+	0.928393i	$0.621191\pi$
$84$	0	0
$85$	9.33643	1.01268
$86$	11.6658	1.25795
$87$	0	0
$88$	−6.22907	−0.664021
$89$	−1.81991	−0.192911	−0.0964553	−	0.995337i	$-0.530750\pi$
$89$	−1.81991	−0.192911	−0.0964553	+	0.995337i	$0.530750\pi$
$90$	0	0
$91$	1.98210	0.207781
$92$	−0.549412	−0.0572802
$93$	0	0
$94$	−1.30340	−0.134435
$95$	15.7134	1.61216
$96$	0	0
$97$	2.30485	0.234022	0.117011	−	0.993131i	$-0.462669\pi$
$97$	2.30485	0.234022	0.117011	+	0.993131i	$0.462669\pi$
$98$	−0.346089	−0.0349603
$99$	0	0
$100$	14.0323	1.40323
$101$	12.3563	1.22950	0.614750	−	0.788722i	$-0.289257\pi$
$101$	12.3563	1.22950	0.614750	+	0.788722i	$0.289257\pi$
$102$	0	0
$103$	11.3992	1.12320	0.561598	−	0.827410i	$-0.310187\pi$
$103$	11.3992	1.12320	0.561598	+	0.827410i	$0.310187\pi$
$104$	0.768400	0.0753478
$105$	0	0
$106$	−7.86564	−0.763979
$107$	−2.63616	−0.254847	−0.127424	−	0.991848i	$-0.540671\pi$
$107$	−2.63616	−0.254847	−0.127424	+	0.991848i	$0.540671\pi$
$108$	0	0
$109$	4.99948	0.478863	0.239432	−	0.970913i	$-0.423039\pi$
$109$	4.99948	0.478863	0.239432	+	0.970913i	$0.423039\pi$
$110$	−27.1749	−2.59103
$111$	0	0
$112$	2.57952	0.243741
$113$	12.6376	1.18885	0.594423	−	0.804153i	$-0.297381\pi$
$113$	12.6376	1.18885	0.594423	+	0.804153i	$0.297381\pi$
$114$	0	0
$115$	−2.39687	−0.223509
$116$	−9.37769	−0.870696
$117$	0	0
$118$	11.9672	1.10167
$119$	5.52045	0.506059
$120$	0	0
$121$	27.8013	2.52739
$122$	−1.43378	−0.129809
$123$	0	0
$124$	3.83934	0.344783
$125$	39.4041	3.52441
$126$	0	0
$127$	1.79523	0.159301	0.0796507	−	0.996823i	$-0.474620\pi$
$127$	1.79523	0.159301	0.0796507	+	0.996823i	$0.474620\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	3.35222	0.294009
$131$	−13.2575	−1.15831	−0.579156	−	0.815217i	$-0.696618\pi$
$131$	−13.2575	−1.15831	−0.579156	+	0.815217i	$0.696618\pi$
$132$	0	0
$133$	9.29103	0.805634
$134$	1.90317	0.164409
$135$	0	0
$136$	2.14011	0.183513
$137$	−14.1812	−1.21158	−0.605792	−	0.795623i	$-0.707144\pi$
$137$	−14.1812	−1.21158	−0.605792	+	0.795623i	$0.707144\pi$
$138$	0	0
$139$	18.1403	1.53864	0.769320	−	0.638864i	$-0.220595\pi$
$139$	18.1403	1.53864	0.769320	+	0.638864i	$0.220595\pi$
$140$	11.2534	0.951086
$141$	0	0
$142$	11.9673	1.00427
$143$	−4.78642	−0.400260
$144$	0	0
$145$	−40.9111	−3.39748
$146$	−6.31479	−0.522616
$147$	0	0
$148$	3.34752	0.275165
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	0.137426	0.0111835	0.00559177	−	0.999984i	$-0.498220\pi$
$151$	0.137426	0.0111835	0.00559177	+	0.999984i	$0.498220\pi$
$152$	3.60185	0.292148
$153$	0	0
$154$	−16.0680	−1.29480
$155$	16.7495	1.34535
$156$	0	0
$157$	−22.5932	−1.80313	−0.901567	−	0.432639i	$-0.857583\pi$
$157$	−22.5932	−1.80313	−0.901567	+	0.432639i	$0.857583\pi$
$158$	−5.58864	−0.444609
$159$	0	0
$160$	4.36260	0.344894
$161$	−1.41722	−0.111693
$162$	0	0
$163$	5.47799	0.429070	0.214535	−	0.976716i	$-0.431176\pi$
$163$	5.47799	0.429070	0.214535	+	0.976716i	$0.431176\pi$
$164$	−1.84301	−0.143915
$165$	0	0
$166$	−6.77087	−0.525522
$167$	10.3890	0.803927	0.401964	−	0.915656i	$-0.368328\pi$
$167$	10.3890	0.803927	0.401964	+	0.915656i	$0.368328\pi$
$168$	0	0
$169$	−12.4096	−0.954582
$170$	9.33643	0.716072
$171$	0	0
$172$	11.6658	0.889508
$173$	21.0179	1.59796	0.798980	−	0.601357i	$-0.205373\pi$
$173$	21.0179	1.59796	0.798980	+	0.601357i	$0.205373\pi$
$174$	0	0
$175$	36.1964	2.73619
$176$	−6.22907	−0.469534
$177$	0	0
$178$	−1.81991	−0.136408
$179$	0.338792	0.0253225	0.0126613	−	0.999920i	$-0.495970\pi$
$179$	0.338792	0.0253225	0.0126613	+	0.999920i	$0.495970\pi$
$180$	0	0
$181$	−12.6109	−0.937363	−0.468681	−	0.883367i	$-0.655271\pi$
$181$	−12.6109	−0.937363	−0.468681	+	0.883367i	$0.655271\pi$
$182$	1.98210	0.146923
$183$	0	0
$184$	−0.549412	−0.0405032
$185$	14.6039	1.07370
$186$	0	0
$187$	−13.3309	−0.974851
$188$	−1.30340	−0.0950601
$189$	0	0
$190$	15.7134	1.13997
$191$	−27.4287	−1.98467	−0.992335	−	0.123576i	$-0.960564\pi$
$191$	−27.4287	−1.98467	−0.992335	+	0.123576i	$0.960564\pi$
$192$	0	0
$193$	−19.8628	−1.42975	−0.714877	−	0.699251i	$-0.753517\pi$
$193$	−19.8628	−1.42975	−0.714877	+	0.699251i	$0.753517\pi$
$194$	2.30485	0.165479
$195$	0	0
$196$	−0.346089	−0.0247207
$197$	2.39019	0.170294	0.0851469	−	0.996368i	$-0.472864\pi$
$197$	2.39019	0.170294	0.0851469	+	0.996368i	$0.472864\pi$
$198$	0	0
$199$	18.0680	1.28081	0.640403	−	0.768039i	$-0.278767\pi$
$199$	18.0680	1.28081	0.640403	+	0.768039i	$0.278767\pi$
$200$	14.0323	0.992230
$201$	0	0
$202$	12.3563	0.869387
$203$	−24.1899	−1.69780
$204$	0	0
$205$	−8.04032	−0.561561
$206$	11.3992	0.794220
$207$	0	0
$208$	0.768400	0.0532790
$209$	−22.4362	−1.55194
$210$	0	0
$211$	−7.18315	−0.494508	−0.247254	−	0.968951i	$-0.579528\pi$
$211$	−7.18315	−0.494508	−0.247254	+	0.968951i	$0.579528\pi$
$212$	−7.86564	−0.540215
$213$	0	0
$214$	−2.63616	−0.180204
$215$	50.8932	3.47088
$216$	0	0
$217$	9.90365	0.672303
$218$	4.99948	0.338607
$219$	0	0
$220$	−27.1749	−1.83213
$221$	1.64446	0.110618
$222$	0	0
$223$	−19.1171	−1.28018	−0.640088	−	0.768302i	$-0.721102\pi$
$223$	−19.1171	−1.28018	−0.640088	+	0.768302i	$0.721102\pi$
$224$	2.57952	0.172351
$225$	0	0
$226$	12.6376	0.840640
$227$	−15.4575	−1.02595	−0.512975	−	0.858403i	$-0.671457\pi$
$227$	−15.4575	−1.02595	−0.512975	+	0.858403i	$0.671457\pi$
$228$	0	0
$229$	−18.6958	−1.23546	−0.617728	−	0.786392i	$-0.711947\pi$
$229$	−18.6958	−1.23546	−0.617728	+	0.786392i	$0.711947\pi$
$230$	−2.39687	−0.158045
$231$	0	0
$232$	−9.37769	−0.615675
$233$	−23.5054	−1.53989	−0.769944	−	0.638111i	$-0.779716\pi$
$233$	−23.5054	−1.53989	−0.769944	+	0.638111i	$0.779716\pi$
$234$	0	0
$235$	−5.68620	−0.370927
$236$	11.9672	0.778996
$237$	0	0
$238$	5.52045	0.357837
$239$	−0.553298	−0.0357899	−0.0178949	−	0.999840i	$-0.505696\pi$
$239$	−0.553298	−0.0357899	−0.0178949	+	0.999840i	$0.505696\pi$
$240$	0	0
$241$	−17.6976	−1.14000	−0.570002	−	0.821643i	$-0.693058\pi$
$241$	−17.6976	−1.14000	−0.570002	+	0.821643i	$0.693058\pi$
$242$	27.8013	1.78714
$243$	0	0
$244$	−1.43378	−0.0917885
$245$	−1.50985	−0.0964606
$246$	0	0
$247$	2.76766	0.176102
$248$	3.83934	0.243798
$249$	0	0
$250$	39.4041	2.49213
$251$	−18.2527	−1.15210	−0.576049	−	0.817415i	$-0.695406\pi$
$251$	−18.2527	−1.15210	−0.576049	+	0.817415i	$0.695406\pi$
$252$	0	0
$253$	3.42233	0.215160
$254$	1.79523	0.112643
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.6423	1.10050	0.550248	−	0.835001i	$-0.314533\pi$
$257$	17.6423	1.10050	0.550248	+	0.835001i	$0.314533\pi$
$258$	0	0
$259$	8.63500	0.536553
$260$	3.35222	0.207896
$261$	0	0
$262$	−13.2575	−0.819050
$263$	13.9081	0.857611	0.428806	−	0.903397i	$-0.358934\pi$
$263$	13.9081	0.857611	0.428806	+	0.903397i	$0.358934\pi$
$264$	0	0
$265$	−34.3146	−2.10793
$266$	9.29103	0.569669
$267$	0	0
$268$	1.90317	0.116254
$269$	−8.56765	−0.522379	−0.261189	−	0.965288i	$-0.584115\pi$
$269$	−8.56765	−0.522379	−0.261189	+	0.965288i	$0.584115\pi$
$270$	0	0
$271$	23.5969	1.43341	0.716706	−	0.697376i	$-0.245649\pi$
$271$	23.5969	1.43341	0.716706	+	0.697376i	$0.245649\pi$
$272$	2.14011	0.129763
$273$	0	0
$274$	−14.1812	−0.856719
$275$	−87.4079	−5.27089
$276$	0	0
$277$	−25.6480	−1.54104	−0.770518	−	0.637418i	$-0.780002\pi$
$277$	−25.6480	−1.54104	−0.770518	+	0.637418i	$0.780002\pi$
$278$	18.1403	1.08798
$279$	0	0
$280$	11.2534	0.672519
$281$	0.200523	0.0119622	0.00598111	−	0.999982i	$-0.498096\pi$
$281$	0.200523	0.0119622	0.00598111	+	0.999982i	$0.498096\pi$
$282$	0	0
$283$	−4.87942	−0.290051	−0.145026	−	0.989428i	$-0.546326\pi$
$283$	−4.87942	−0.290051	−0.145026	+	0.989428i	$0.546326\pi$
$284$	11.9673	0.710129
$285$	0	0
$286$	−4.78642	−0.283027
$287$	−4.75409	−0.280625
$288$	0	0
$289$	−12.4199	−0.730584
$290$	−40.9111	−2.40238
$291$	0	0
$292$	−6.31479	−0.369545
$293$	4.82024	0.281601	0.140801	−	0.990038i	$-0.455032\pi$
$293$	4.82024	0.281601	0.140801	+	0.990038i	$0.455032\pi$
$294$	0	0
$295$	52.2079	3.03966
$296$	3.34752	0.194571
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−0.422169	−0.0244146
$300$	0	0
$301$	30.0921	1.73448
$302$	0.137426	0.00790796
$303$	0	0
$304$	3.60185	0.206580
$305$	−6.25501	−0.358161
$306$	0	0
$307$	20.0016	1.14155	0.570777	−	0.821105i	$-0.306642\pi$
$307$	20.0016	1.14155	0.570777	+	0.821105i	$0.306642\pi$
$308$	−16.0680	−0.915559
$309$	0	0
$310$	16.7495	0.951308
$311$	11.7001	0.663453	0.331726	−	0.943376i	$-0.392369\pi$
$311$	11.7001	0.663453	0.331726	+	0.943376i	$0.392369\pi$
$312$	0	0
$313$	−6.49795	−0.367286	−0.183643	−	0.982993i	$-0.558789\pi$
$313$	−6.49795	−0.367286	−0.183643	+	0.982993i	$0.558789\pi$
$314$	−22.5932	−1.27501
$315$	0	0
$316$	−5.58864	−0.314386
$317$	−11.6737	−0.655660	−0.327830	−	0.944737i	$-0.606317\pi$
$317$	−11.6737	−0.655660	−0.327830	+	0.944737i	$0.606317\pi$
$318$	0	0
$319$	58.4143	3.27057
$320$	4.36260	0.243877
$321$	0	0
$322$	−1.41722	−0.0789785
$323$	7.70834	0.428904
$324$	0	0
$325$	10.7824	0.598099
$326$	5.47799	0.303398
$327$	0	0
$328$	−1.84301	−0.101763
$329$	−3.36214	−0.185361
$330$	0	0
$331$	−6.38419	−0.350907	−0.175453	−	0.984488i	$-0.556139\pi$
$331$	−6.38419	−0.350907	−0.175453	+	0.984488i	$0.556139\pi$
$332$	−6.77087	−0.371600
$333$	0	0
$334$	10.3890	0.568463
$335$	8.30276	0.453628
$336$	0	0
$337$	7.00440	0.381554	0.190777	−	0.981633i	$-0.438899\pi$
$337$	7.00440	0.381554	0.190777	+	0.981633i	$0.438899\pi$
$338$	−12.4096	−0.674991
$339$	0	0
$340$	9.33643	0.506339
$341$	−23.9155	−1.29510
$342$	0	0
$343$	−18.9494	−1.02317
$344$	11.6658	0.628977
$345$	0	0
$346$	21.0179	1.12993
$347$	33.6399	1.80588	0.902942	−	0.429762i	$-0.141402\pi$
$347$	33.6399	1.80588	0.902942	+	0.429762i	$0.141402\pi$
$348$	0	0
$349$	−21.7389	−1.16366	−0.581829	−	0.813311i	$-0.697663\pi$
$349$	−21.7389	−1.16366	−0.581829	+	0.813311i	$0.697663\pi$
$350$	36.1964	1.93478
$351$	0	0
$352$	−6.22907	−0.332011
$353$	−5.74264	−0.305650	−0.152825	−	0.988253i	$-0.548837\pi$
$353$	−5.74264	−0.305650	−0.152825	+	0.988253i	$0.548837\pi$
$354$	0	0
$355$	52.2085	2.77094
$356$	−1.81991	−0.0964553
$357$	0	0
$358$	0.338792	0.0179057
$359$	−2.72337	−0.143734	−0.0718670	−	0.997414i	$-0.522896\pi$
$359$	−2.72337	−0.143734	−0.0718670	+	0.997414i	$0.522896\pi$
$360$	0	0
$361$	−6.02670	−0.317195
$362$	−12.6109	−0.662816
$363$	0	0
$364$	1.98210	0.103890
$365$	−27.5489	−1.44198
$366$	0	0
$367$	17.4084	0.908709	0.454355	−	0.890821i	$-0.349870\pi$
$367$	17.4084	0.908709	0.454355	+	0.890821i	$0.349870\pi$
$368$	−0.549412	−0.0286401
$369$	0	0
$370$	14.6039	0.759221
$371$	−20.2896	−1.05338
$372$	0	0
$373$	21.7954	1.12853	0.564263	−	0.825595i	$-0.309161\pi$
$373$	21.7954	1.12853	0.564263	+	0.825595i	$0.309161\pi$
$374$	−13.3309	−0.689324
$375$	0	0
$376$	−1.30340	−0.0672176
$377$	−7.20582	−0.371118
$378$	0	0
$379$	−25.2733	−1.29820	−0.649101	−	0.760702i	$-0.724855\pi$
$379$	−25.2733	−1.29820	−0.649101	+	0.760702i	$0.724855\pi$
$380$	15.7134	0.806081
$381$	0	0
$382$	−27.4287	−1.40337
$383$	−11.3936	−0.582188	−0.291094	−	0.956694i	$-0.594019\pi$
$383$	−11.3936	−0.582188	−0.291094	+	0.956694i	$0.594019\pi$
$384$	0	0
$385$	−70.0982	−3.57253
$386$	−19.8628	−1.01099
$387$	0	0
$388$	2.30485	0.117011
$389$	−21.7032	−1.10040	−0.550199	−	0.835034i	$-0.685448\pi$
$389$	−21.7032	−1.10040	−0.550199	+	0.835034i	$0.685448\pi$
$390$	0	0
$391$	−1.17580	−0.0594629
$392$	−0.346089	−0.0174801
$393$	0	0
$394$	2.39019	0.120416
$395$	−24.3810	−1.22674
$396$	0	0
$397$	7.28845	0.365797	0.182898	−	0.983132i	$-0.441452\pi$
$397$	7.28845	0.365797	0.182898	+	0.983132i	$0.441452\pi$
$398$	18.0680	0.905667
$399$	0	0
$400$	14.0323	0.701613
$401$	−37.0109	−1.84824	−0.924119	−	0.382105i	$-0.875199\pi$
$401$	−37.0109	−1.84824	−0.924119	+	0.382105i	$0.875199\pi$
$402$	0	0
$403$	2.95015	0.146957
$404$	12.3563	0.614750
$405$	0	0
$406$	−24.1899	−1.20052
$407$	−20.8520	−1.03359
$408$	0	0
$409$	−15.9935	−0.790827	−0.395413	−	0.918503i	$-0.629399\pi$
$409$	−15.9935	−0.790827	−0.395413	+	0.918503i	$0.629399\pi$
$410$	−8.04032	−0.397083
$411$	0	0
$412$	11.3992	0.561598
$413$	30.8695	1.51899
$414$	0	0
$415$	−29.5386	−1.44999
$416$	0.768400	0.0376739
$417$	0	0
$418$	−22.4362	−1.09739
$419$	7.62172	0.372346	0.186173	−	0.982517i	$-0.440392\pi$
$419$	7.62172	0.372346	0.186173	+	0.982517i	$0.440392\pi$
$420$	0	0
$421$	25.9985	1.26709	0.633545	−	0.773706i	$-0.281599\pi$
$421$	25.9985	1.26709	0.633545	+	0.773706i	$0.281599\pi$
$422$	−7.18315	−0.349670
$423$	0	0
$424$	−7.86564	−0.381989
$425$	30.0305	1.45670
$426$	0	0
$427$	−3.69847	−0.178981
$428$	−2.63616	−0.127424
$429$	0	0
$430$	50.8932	2.45429
$431$	−28.6985	−1.38236	−0.691178	−	0.722684i	$-0.742908\pi$
$431$	−28.6985	−1.38236	−0.691178	+	0.722684i	$0.742908\pi$
$432$	0	0
$433$	33.5827	1.61388	0.806941	−	0.590632i	$-0.201121\pi$
$433$	33.5827	1.61388	0.806941	+	0.590632i	$0.201121\pi$
$434$	9.90365	0.475390
$435$	0	0
$436$	4.99948	0.239432
$437$	−1.97890	−0.0946636
$438$	0	0
$439$	19.4715	0.929323	0.464661	−	0.885488i	$-0.346176\pi$
$439$	19.4715	0.929323	0.464661	+	0.885488i	$0.346176\pi$
$440$	−27.1749	−1.29551
$441$	0	0
$442$	1.64446	0.0782190
$443$	31.4442	1.49396	0.746979	−	0.664847i	$-0.231503\pi$
$443$	31.4442	1.49396	0.746979	+	0.664847i	$0.231503\pi$
$444$	0	0
$445$	−7.93955	−0.376371
$446$	−19.1171	−0.905221
$447$	0	0
$448$	2.57952	0.121871
$449$	−26.2096	−1.23691	−0.618453	−	0.785822i	$-0.712240\pi$
$449$	−26.2096	−1.23691	−0.618453	+	0.785822i	$0.712240\pi$
$450$	0	0
$451$	11.4803	0.540584
$452$	12.6376	0.594423
$453$	0	0
$454$	−15.4575	−0.725456
$455$	8.64711	0.405383
$456$	0	0
$457$	−1.18648	−0.0555012	−0.0277506	−	0.999615i	$-0.508834\pi$
$457$	−1.18648	−0.0555012	−0.0277506	+	0.999615i	$0.508834\pi$
$458$	−18.6958	−0.873600
$459$	0	0
$460$	−2.39687	−0.111754
$461$	−1.52992	−0.0712557	−0.0356278	−	0.999365i	$-0.511343\pi$
$461$	−1.52992	−0.0712557	−0.0356278	+	0.999365i	$0.511343\pi$
$462$	0	0
$463$	−13.3040	−0.618290	−0.309145	−	0.951015i	$-0.600043\pi$
$463$	−13.3040	−0.618290	−0.309145	+	0.951015i	$0.600043\pi$
$464$	−9.37769	−0.435348
$465$	0	0
$466$	−23.5054	−1.08887
$467$	17.4121	0.805736	0.402868	−	0.915258i	$-0.368013\pi$
$467$	17.4121	0.805736	0.402868	+	0.915258i	$0.368013\pi$
$468$	0	0
$469$	4.90926	0.226688
$470$	−5.68620	−0.262285
$471$	0	0
$472$	11.9672	0.550833
$473$	−72.6671	−3.34123
$474$	0	0
$475$	50.5420	2.31903
$476$	5.52045	0.253029
$477$	0	0
$478$	−0.553298	−0.0253072
$479$	−38.5162	−1.75985	−0.879926	−	0.475111i	$-0.842408\pi$
$479$	−38.5162	−1.75985	−0.879926	+	0.475111i	$0.842408\pi$
$480$	0	0
$481$	2.57224	0.117284
$482$	−17.6976	−0.806105
$483$	0	0
$484$	27.8013	1.26370
$485$	10.0551	0.456580
$486$	0	0
$487$	−18.8092	−0.852326	−0.426163	−	0.904646i	$-0.640135\pi$
$487$	−18.8092	−0.852326	−0.426163	+	0.904646i	$0.640135\pi$
$488$	−1.43378	−0.0649043
$489$	0	0
$490$	−1.50985	−0.0682080
$491$	−14.2549	−0.643313	−0.321656	−	0.946856i	$-0.604240\pi$
$491$	−14.2549	−0.643313	−0.321656	+	0.946856i	$0.604240\pi$
$492$	0	0
$493$	−20.0693	−0.903874
$494$	2.76766	0.124523
$495$	0	0
$496$	3.83934	0.172392
$497$	30.8699	1.38470
$498$	0	0
$499$	20.3226	0.909765	0.454883	−	0.890551i	$-0.349681\pi$
$499$	20.3226	0.909765	0.454883	+	0.890551i	$0.349681\pi$
$500$	39.4041	1.76220
$501$	0	0
$502$	−18.2527	−0.814657
$503$	−31.8995	−1.42233	−0.711164	−	0.703026i	$-0.751832\pi$
$503$	−31.8995	−1.42233	−0.711164	+	0.703026i	$0.751832\pi$
$504$	0	0
$505$	53.9056	2.39877
$506$	3.42233	0.152141
$507$	0	0
$508$	1.79523	0.0796507
$509$	−0.776500	−0.0344177	−0.0172089	−	0.999852i	$-0.505478\pi$
$509$	−0.776500	−0.0344177	−0.0172089	+	0.999852i	$0.505478\pi$
$510$	0	0
$511$	−16.2891	−0.720588
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.6423	0.778169
$515$	49.7301	2.19137
$516$	0	0
$517$	8.11896	0.357071
$518$	8.63500	0.379400
$519$	0	0
$520$	3.35222	0.147005
$521$	−32.7620	−1.43533	−0.717665	−	0.696388i	$-0.754789\pi$
$521$	−32.7620	−1.43533	−0.717665	+	0.696388i	$0.754789\pi$
$522$	0	0
$523$	16.2967	0.712605	0.356302	−	0.934371i	$-0.384037\pi$
$523$	16.2967	0.712605	0.356302	+	0.934371i	$0.384037\pi$
$524$	−13.2575	−0.579156
$525$	0	0
$526$	13.9081	0.606423
$527$	8.21661	0.357921
$528$	0	0
$529$	−22.6981	−0.986876
$530$	−34.3146	−1.49053
$531$	0	0
$532$	9.29103	0.402817
$533$	−1.41617	−0.0613412
$534$	0	0
$535$	−11.5005	−0.497211
$536$	1.90317	0.0822043
$537$	0	0
$538$	−8.56765	−0.369377
$539$	2.15581	0.0928575
$540$	0	0
$541$	14.2174	0.611255	0.305627	−	0.952151i	$-0.401134\pi$
$541$	14.2174	0.611255	0.305627	+	0.952151i	$0.401134\pi$
$542$	23.5969	1.01358
$543$	0	0
$544$	2.14011	0.0917564
$545$	21.8107	0.934268
$546$	0	0
$547$	24.5546	1.04988	0.524940	−	0.851139i	$-0.324088\pi$
$547$	24.5546	1.04988	0.524940	+	0.851139i	$0.324088\pi$
$548$	−14.1812	−0.605792
$549$	0	0
$550$	−87.4079	−3.72709
$551$	−33.7770	−1.43895
$552$	0	0
$553$	−14.4160	−0.613031
$554$	−25.6480	−1.08968
$555$	0	0
$556$	18.1403	0.769320
$557$	24.9133	1.05561	0.527805	−	0.849365i	$-0.323015\pi$
$557$	24.9133	1.05561	0.527805	+	0.849365i	$0.323015\pi$
$558$	0	0
$559$	8.96400	0.379137
$560$	11.2534	0.475543
$561$	0	0
$562$	0.200523	0.00845857
$563$	−14.8658	−0.626518	−0.313259	−	0.949668i	$-0.601421\pi$
$563$	−14.8658	−0.626518	−0.313259	+	0.949668i	$0.601421\pi$
$564$	0	0
$565$	55.1327	2.31945
$566$	−4.87942	−0.205097
$567$	0	0
$568$	11.9673	0.502137
$569$	−20.4958	−0.859227	−0.429613	−	0.903013i	$-0.641350\pi$
$569$	−20.4958	−0.859227	−0.429613	+	0.903013i	$0.641350\pi$
$570$	0	0
$571$	20.0126	0.837503	0.418751	−	0.908101i	$-0.362468\pi$
$571$	20.0126	0.837503	0.418751	+	0.908101i	$0.362468\pi$
$572$	−4.78642	−0.200130
$573$	0	0
$574$	−4.75409	−0.198432
$575$	−7.70950	−0.321508
$576$	0	0
$577$	11.0986	0.462040	0.231020	−	0.972949i	$-0.425794\pi$
$577$	11.0986	0.462040	0.231020	+	0.972949i	$0.425794\pi$
$578$	−12.4199	−0.516601
$579$	0	0
$580$	−40.9111	−1.69874
$581$	−17.4656	−0.724595
$582$	0	0
$583$	48.9957	2.02919
$584$	−6.31479	−0.261308
$585$	0	0
$586$	4.82024	0.199122
$587$	4.03860	0.166691	0.0833455	−	0.996521i	$-0.473439\pi$
$587$	4.03860	0.166691	0.0833455	+	0.996521i	$0.473439\pi$
$588$	0	0
$589$	13.8287	0.569802
$590$	52.2079	2.14937
$591$	0	0
$592$	3.34752	0.137582
$593$	19.5768	0.803923	0.401962	−	0.915656i	$-0.368328\pi$
$593$	19.5768	0.803923	0.401962	+	0.915656i	$0.368328\pi$
$594$	0	0
$595$	24.0835	0.987327
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	−0.422169	−0.0172638
$599$	−29.8591	−1.22001	−0.610006	−	0.792397i	$-0.708833\pi$
$599$	−29.8591	−1.22001	−0.610006	+	0.792397i	$0.708833\pi$
$600$	0	0
$601$	28.9432	1.18062	0.590310	−	0.807177i	$-0.299006\pi$
$601$	28.9432	1.18062	0.590310	+	0.807177i	$0.299006\pi$
$602$	30.0921	1.22646
$603$	0	0
$604$	0.137426	0.00559177
$605$	121.286	4.93098
$606$	0	0
$607$	7.53944	0.306016	0.153008	−	0.988225i	$-0.451104\pi$
$607$	7.53944	0.306016	0.153008	+	0.988225i	$0.451104\pi$
$608$	3.60185	0.146074
$609$	0	0
$610$	−6.25501	−0.253258
$611$	−1.00153	−0.0405176
$612$	0	0
$613$	−30.2438	−1.22154	−0.610768	−	0.791810i	$-0.709139\pi$
$613$	−30.2438	−1.22154	−0.610768	+	0.791810i	$0.709139\pi$
$614$	20.0016	0.807201
$615$	0	0
$616$	−16.0680	−0.647398
$617$	24.5517	0.988414	0.494207	−	0.869344i	$-0.335459\pi$
$617$	24.5517	0.988414	0.494207	+	0.869344i	$0.335459\pi$
$618$	0	0
$619$	−0.721154	−0.0289856	−0.0144928	−	0.999895i	$-0.504613\pi$
$619$	−0.721154	−0.0289856	−0.0144928	+	0.999895i	$0.504613\pi$
$620$	16.7495	0.672676
$621$	0	0
$622$	11.7001	0.469132
$623$	−4.69450	−0.188081
$624$	0	0
$625$	101.743	4.06972
$626$	−6.49795	−0.259710
$627$	0	0
$628$	−22.5932	−0.901567
$629$	7.16407	0.285650
$630$	0	0
$631$	−22.6928	−0.903388	−0.451694	−	0.892173i	$-0.649180\pi$
$631$	−22.6928	−0.903388	−0.451694	+	0.892173i	$0.649180\pi$
$632$	−5.58864	−0.222304
$633$	0	0
$634$	−11.6737	−0.463622
$635$	7.83189	0.310799
$636$	0	0
$637$	−0.265935	−0.0105367
$638$	58.4143	2.31264
$639$	0	0
$640$	4.36260	0.172447
$641$	−10.1229	−0.399830	−0.199915	−	0.979813i	$-0.564067\pi$
$641$	−10.1229	−0.399830	−0.199915	+	0.979813i	$0.564067\pi$
$642$	0	0
$643$	14.7704	0.582487	0.291243	−	0.956649i	$-0.405931\pi$
$643$	14.7704	0.582487	0.291243	+	0.956649i	$0.405931\pi$
$644$	−1.41722	−0.0558463
$645$	0	0
$646$	7.70834	0.303281
$647$	46.8709	1.84268	0.921342	−	0.388753i	$-0.127094\pi$
$647$	46.8709	1.84268	0.921342	+	0.388753i	$0.127094\pi$
$648$	0	0
$649$	−74.5443	−2.92612
$650$	10.7824	0.422920
$651$	0	0
$652$	5.47799	0.214535
$653$	6.54800	0.256243	0.128122	−	0.991758i	$-0.459105\pi$
$653$	6.54800	0.256243	0.128122	+	0.991758i	$0.459105\pi$
$654$	0	0
$655$	−57.8370	−2.25988
$656$	−1.84301	−0.0719576
$657$	0	0
$658$	−3.36214	−0.131070
$659$	−29.7899	−1.16045	−0.580225	−	0.814456i	$-0.697036\pi$
$659$	−29.7899	−1.16045	−0.580225	+	0.814456i	$0.697036\pi$
$660$	0	0
$661$	−15.2677	−0.593844	−0.296922	−	0.954902i	$-0.595960\pi$
$661$	−15.2677	−0.593844	−0.296922	+	0.954902i	$0.595960\pi$
$662$	−6.38419	−0.248129
$663$	0	0
$664$	−6.77087	−0.262761
$665$	40.5330	1.57180
$666$	0	0
$667$	5.15222	0.199495
$668$	10.3890	0.401964
$669$	0	0
$670$	8.30276	0.320763
$671$	8.93113	0.344782
$672$	0	0
$673$	−27.2802	−1.05158	−0.525788	−	0.850616i	$-0.676229\pi$
$673$	−27.2802	−1.05158	−0.525788	+	0.850616i	$0.676229\pi$
$674$	7.00440	0.269799
$675$	0	0
$676$	−12.4096	−0.477291
$677$	31.8870	1.22552	0.612759	−	0.790270i	$-0.290060\pi$
$677$	31.8870	1.22552	0.612759	+	0.790270i	$0.290060\pi$
$678$	0	0
$679$	5.94540	0.228164
$680$	9.33643	0.358036
$681$	0	0
$682$	−23.9155	−0.915773
$683$	−2.77228	−0.106079	−0.0530393	−	0.998592i	$-0.516891\pi$
$683$	−2.77228	−0.106079	−0.0530393	+	0.998592i	$0.516891\pi$
$684$	0	0
$685$	−61.8670	−2.36382
$686$	−18.9494	−0.723490
$687$	0	0
$688$	11.6658	0.444754
$689$	−6.04396	−0.230257
$690$	0	0
$691$	−5.69332	−0.216584	−0.108292	−	0.994119i	$-0.534538\pi$
$691$	−5.69332	−0.216584	−0.108292	+	0.994119i	$0.534538\pi$
$692$	21.0179	0.798980
$693$	0	0
$694$	33.6399	1.27695
$695$	79.1388	3.00191
$696$	0	0
$697$	−3.94425	−0.149399
$698$	−21.7389	−0.822831
$699$	0	0
$700$	36.1964	1.36810
$701$	38.7619	1.46402	0.732009	−	0.681295i	$-0.238583\pi$
$701$	38.7619	1.46402	0.732009	+	0.681295i	$0.238583\pi$
$702$	0	0
$703$	12.0573	0.454749
$704$	−6.22907	−0.234767
$705$	0	0
$706$	−5.74264	−0.216127
$707$	31.8733	1.19872
$708$	0	0
$709$	1.63024	0.0612251	0.0306126	−	0.999531i	$-0.490254\pi$
$709$	1.63024	0.0612251	0.0306126	+	0.999531i	$0.490254\pi$
$710$	52.2085	1.95935
$711$	0	0
$712$	−1.81991	−0.0682042
$713$	−2.10938	−0.0789970
$714$	0	0
$715$	−20.8812	−0.780913
$716$	0.338792	0.0126613
$717$	0	0
$718$	−2.72337	−0.101635
$719$	−10.8629	−0.405119	−0.202559	−	0.979270i	$-0.564926\pi$
$719$	−10.8629	−0.405119	−0.202559	+	0.979270i	$0.564926\pi$
$720$	0	0
$721$	29.4044	1.09508
$722$	−6.02670	−0.224291
$723$	0	0
$724$	−12.6109	−0.468681
$725$	−131.590	−4.88713
$726$	0	0
$727$	14.6404	0.542984	0.271492	−	0.962441i	$-0.412483\pi$
$727$	14.6404	0.542984	0.271492	+	0.962441i	$0.412483\pi$
$728$	1.98210	0.0734616
$729$	0	0
$730$	−27.5489	−1.01963
$731$	24.9661	0.923403
$732$	0	0
$733$	−14.1094	−0.521141	−0.260570	−	0.965455i	$-0.583911\pi$
$733$	−14.1094	−0.521141	−0.260570	+	0.965455i	$0.583911\pi$
$734$	17.4084	0.642555
$735$	0	0
$736$	−0.549412	−0.0202516
$737$	−11.8550	−0.436683
$738$	0	0
$739$	−33.0669	−1.21639	−0.608193	−	0.793789i	$-0.708105\pi$
$739$	−33.0669	−1.21639	−0.608193	+	0.793789i	$0.708105\pi$
$740$	14.6039	0.536850
$741$	0	0
$742$	−20.2896	−0.744854
$743$	−51.5895	−1.89263	−0.946317	−	0.323240i	$-0.895228\pi$
$743$	−51.5895	−1.89263	−0.946317	+	0.323240i	$0.895228\pi$
$744$	0	0
$745$	−4.36260	−0.159833
$746$	21.7954	0.797988
$747$	0	0
$748$	−13.3309	−0.487425
$749$	−6.80003	−0.248468
$750$	0	0
$751$	−0.745493	−0.0272034	−0.0136017	−	0.999907i	$-0.504330\pi$
$751$	−0.745493	−0.0272034	−0.0136017	+	0.999907i	$0.504330\pi$
$752$	−1.30340	−0.0475300
$753$	0	0
$754$	−7.20582	−0.262420
$755$	0.599533	0.0218192
$756$	0	0
$757$	−51.3170	−1.86515	−0.932574	−	0.360980i	$-0.882442\pi$
$757$	−51.3170	−1.86515	−0.932574	+	0.360980i	$0.882442\pi$
$758$	−25.2733	−0.917967
$759$	0	0
$760$	15.7134	0.569985
$761$	36.8724	1.33662	0.668312	−	0.743881i	$-0.267017\pi$
$761$	36.8724	1.33662	0.668312	+	0.743881i	$0.267017\pi$
$762$	0	0
$763$	12.8962	0.466875
$764$	−27.4287	−0.992335
$765$	0	0
$766$	−11.3936	−0.411669
$767$	9.19557	0.332033
$768$	0	0
$769$	16.3455	0.589433	0.294717	−	0.955585i	$-0.404775\pi$
$769$	16.3455	0.589433	0.294717	+	0.955585i	$0.404775\pi$
$770$	−70.0982	−2.52616
$771$	0	0
$772$	−19.8628	−0.714877
$773$	52.4497	1.88649	0.943243	−	0.332103i	$-0.107758\pi$
$773$	52.4497	1.88649	0.943243	+	0.332103i	$0.107758\pi$
$774$	0	0
$775$	53.8746	1.93523
$776$	2.30485	0.0827393
$777$	0	0
$778$	−21.7032	−0.778098
$779$	−6.63825	−0.237840
$780$	0	0
$781$	−74.5452	−2.66744
$782$	−1.17580	−0.0420466
$783$	0	0
$784$	−0.346089	−0.0123603
$785$	−98.5651	−3.51794
$786$	0	0
$787$	−13.7279	−0.489345	−0.244673	−	0.969606i	$-0.578680\pi$
$787$	−13.7279	−0.489345	−0.244673	+	0.969606i	$0.578680\pi$
$788$	2.39019	0.0851469
$789$	0	0
$790$	−24.3810	−0.867437
$791$	32.5989	1.15908
$792$	0	0
$793$	−1.10172	−0.0391232
$794$	7.28845	0.258657
$795$	0	0
$796$	18.0680	0.640403
$797$	−15.7478	−0.557815	−0.278907	−	0.960318i	$-0.589972\pi$
$797$	−15.7478	−0.557815	−0.278907	+	0.960318i	$0.589972\pi$
$798$	0	0
$799$	−2.78941	−0.0986824
$800$	14.0323	0.496115
$801$	0	0
$802$	−37.0109	−1.30690
$803$	39.3353	1.38811
$804$	0	0
$805$	−6.18276	−0.217914
$806$	2.95015	0.103915
$807$	0	0
$808$	12.3563	0.434694
$809$	−30.9434	−1.08791	−0.543956	−	0.839114i	$-0.683074\pi$
$809$	−30.9434	−1.08791	−0.543956	+	0.839114i	$0.683074\pi$
$810$	0	0
$811$	3.69831	0.129865	0.0649326	−	0.997890i	$-0.479317\pi$
$811$	3.69831	0.129865	0.0649326	+	0.997890i	$0.479317\pi$
$812$	−24.1899	−0.848899
$813$	0	0
$814$	−20.8520	−0.730861
$815$	23.8983	0.837120
$816$	0	0
$817$	42.0184	1.47004
$818$	−15.9935	−0.559199
$819$	0	0
$820$	−8.04032	−0.280780
$821$	45.3937	1.58425	0.792126	−	0.610358i	$-0.208974\pi$
$821$	45.3937	1.58425	0.792126	+	0.610358i	$0.208974\pi$
$822$	0	0
$823$	25.1131	0.875386	0.437693	−	0.899124i	$-0.355796\pi$
$823$	25.1131	0.875386	0.437693	+	0.899124i	$0.355796\pi$
$824$	11.3992	0.397110
$825$	0	0
$826$	30.8695	1.07409
$827$	−7.80963	−0.271567	−0.135784	−	0.990739i	$-0.543355\pi$
$827$	−7.80963	−0.271567	−0.135784	+	0.990739i	$0.543355\pi$
$828$	0	0
$829$	−20.4816	−0.711357	−0.355678	−	0.934608i	$-0.615750\pi$
$829$	−20.4816	−0.711357	−0.355678	+	0.934608i	$0.615750\pi$
$830$	−29.5386	−1.02530
$831$	0	0
$832$	0.768400	0.0266395
$833$	−0.740668	−0.0256626
$834$	0	0
$835$	45.3232	1.56847
$836$	−22.4362	−0.775971
$837$	0	0
$838$	7.62172	0.263288
$839$	−30.3182	−1.04670	−0.523349	−	0.852118i	$-0.675318\pi$
$839$	−30.3182	−1.04670	−0.523349	+	0.852118i	$0.675318\pi$
$840$	0	0
$841$	58.9410	2.03245
$842$	25.9985	0.895967
$843$	0	0
$844$	−7.18315	−0.247254
$845$	−54.1379	−1.86240
$846$	0	0
$847$	71.7140	2.46412
$848$	−7.86564	−0.270107
$849$	0	0
$850$	30.0305	1.03004
$851$	−1.83917	−0.0630460
$852$	0	0
$853$	24.0398	0.823106	0.411553	−	0.911386i	$-0.364987\pi$
$853$	24.0398	0.823106	0.411553	+	0.911386i	$0.364987\pi$
$854$	−3.69847	−0.126559
$855$	0	0
$856$	−2.63616	−0.0901022
$857$	51.2042	1.74910	0.874551	−	0.484933i	$-0.161156\pi$
$857$	51.2042	1.74910	0.874551	+	0.484933i	$0.161156\pi$
$858$	0	0
$859$	42.0494	1.43471	0.717353	−	0.696709i	$-0.245353\pi$
$859$	42.0494	1.43471	0.717353	+	0.696709i	$0.245353\pi$
$860$	50.8932	1.73544
$861$	0	0
$862$	−28.6985	−0.977474
$863$	−49.3782	−1.68085	−0.840426	−	0.541926i	$-0.817695\pi$
$863$	−49.3782	−1.68085	−0.840426	+	0.541926i	$0.817695\pi$
$864$	0	0
$865$	91.6926	3.11764
$866$	33.5827	1.14119
$867$	0	0
$868$	9.90365	0.336152
$869$	34.8120	1.18092
$870$	0	0
$871$	1.46239	0.0495514
$872$	4.99948	0.169304
$873$	0	0
$874$	−1.97890	−0.0669373
$875$	101.644	3.43618
$876$	0	0
$877$	55.8873	1.88718	0.943589	−	0.331118i	$-0.107426\pi$
$877$	55.8873	1.88718	0.943589	+	0.331118i	$0.107426\pi$
$878$	19.4715	0.657131
$879$	0	0
$880$	−27.1749	−0.916067
$881$	−28.0144	−0.943830	−0.471915	−	0.881644i	$-0.656437\pi$
$881$	−28.0144	−0.943830	−0.471915	+	0.881644i	$0.656437\pi$
$882$	0	0
$883$	−38.3410	−1.29028	−0.645139	−	0.764065i	$-0.723201\pi$
$883$	−38.3410	−1.29028	−0.645139	+	0.764065i	$0.723201\pi$
$884$	1.64446	0.0553092
$885$	0	0
$886$	31.4442	1.05639
$887$	23.4599	0.787707	0.393853	−	0.919173i	$-0.371142\pi$
$887$	23.4599	0.787707	0.393853	+	0.919173i	$0.371142\pi$
$888$	0	0
$889$	4.63084	0.155313
$890$	−7.93955	−0.266134
$891$	0	0
$892$	−19.1171	−0.640088
$893$	−4.69464	−0.157100
$894$	0	0
$895$	1.47802	0.0494046
$896$	2.57952	0.0861756
$897$	0	0
$898$	−26.2096	−0.874624
$899$	−36.0041	−1.20081
$900$	0	0
$901$	−16.8333	−0.560800
$902$	11.4803	0.382251
$903$	0	0
$904$	12.6376	0.420320
$905$	−55.0164	−1.82881
$906$	0	0
$907$	−33.1096	−1.09939	−0.549694	−	0.835366i	$-0.685256\pi$
$907$	−33.1096	−1.09939	−0.549694	+	0.835366i	$0.685256\pi$
$908$	−15.4575	−0.512975
$909$	0	0
$910$	8.64711	0.286649
$911$	−2.23405	−0.0740173	−0.0370086	−	0.999315i	$-0.511783\pi$
$911$	−2.23405	−0.0740173	−0.0370086	+	0.999315i	$0.511783\pi$
$912$	0	0
$913$	42.1763	1.39583
$914$	−1.18648	−0.0392453
$915$	0	0
$916$	−18.6958	−0.617728
$917$	−34.1979	−1.12931
$918$	0	0
$919$	33.1692	1.09415	0.547075	−	0.837083i	$-0.315741\pi$
$919$	33.1692	1.09415	0.547075	+	0.837083i	$0.315741\pi$
$920$	−2.39687	−0.0790223
$921$	0	0
$922$	−1.52992	−0.0503854
$923$	9.19568	0.302680
$924$	0	0
$925$	46.9733	1.54447
$926$	−13.3040	−0.437197
$927$	0	0
$928$	−9.37769	−0.307838
$929$	18.9732	0.622490	0.311245	−	0.950330i	$-0.399254\pi$
$929$	18.9732	0.622490	0.311245	+	0.950330i	$0.399254\pi$
$930$	0	0
$931$	−1.24656	−0.0408544
$932$	−23.5054	−0.769944
$933$	0	0
$934$	17.4121	0.569741
$935$	−58.1573	−1.90195
$936$	0	0
$937$	1.96335	0.0641398	0.0320699	−	0.999486i	$-0.489790\pi$
$937$	1.96335	0.0641398	0.0320699	+	0.999486i	$0.489790\pi$
$938$	4.90926	0.160293
$939$	0	0
$940$	−5.68620	−0.185463
$941$	1.03177	0.0336347	0.0168174	−	0.999859i	$-0.494647\pi$
$941$	1.03177	0.0336347	0.0168174	+	0.999859i	$0.494647\pi$
$942$	0	0
$943$	1.01257	0.0329740
$944$	11.9672	0.389498
$945$	0	0
$946$	−72.6671	−2.36261
$947$	39.8807	1.29595	0.647974	−	0.761663i	$-0.275617\pi$
$947$	39.8807	1.29595	0.647974	+	0.761663i	$0.275617\pi$
$948$	0	0
$949$	−4.85229	−0.157512
$950$	50.5420	1.63980
$951$	0	0
$952$	5.52045	0.178919
$953$	−30.2546	−0.980044	−0.490022	−	0.871710i	$-0.663011\pi$
$953$	−30.2546	−0.980044	−0.490022	+	0.871710i	$0.663011\pi$
$954$	0	0
$955$	−119.660	−3.87212
$956$	−0.553298	−0.0178949
$957$	0	0
$958$	−38.5162	−1.24440
$959$	−36.5807	−1.18125
$960$	0	0
$961$	−16.2595	−0.524499
$962$	2.57224	0.0829323
$963$	0	0
$964$	−17.6976	−0.570002
$965$	−86.6533	−2.78947
$966$	0	0
$967$	−31.8802	−1.02520	−0.512598	−	0.858628i	$-0.671317\pi$
$967$	−31.8802	−1.02520	−0.512598	+	0.858628i	$0.671317\pi$
$968$	27.8013	0.893568
$969$	0	0
$970$	10.0551	0.322851
$971$	−55.1264	−1.76909	−0.884545	−	0.466456i	$-0.845531\pi$
$971$	−55.1264	−1.76909	−0.884545	+	0.466456i	$0.845531\pi$
$972$	0	0
$973$	46.7932	1.50012
$974$	−18.8092	−0.602685
$975$	0	0
$976$	−1.43378	−0.0458942
$977$	−39.4265	−1.26137	−0.630683	−	0.776041i	$-0.717225\pi$
$977$	−39.4265	−1.26137	−0.630683	+	0.776041i	$0.717225\pi$
$978$	0	0
$979$	11.3364	0.362312
$980$	−1.50985	−0.0482303
$981$	0	0
$982$	−14.2549	−0.454891
$983$	13.6964	0.436846	0.218423	−	0.975854i	$-0.429909\pi$
$983$	13.6964	0.436846	0.218423	+	0.975854i	$0.429909\pi$
$984$	0	0
$985$	10.4274	0.332245
$986$	−20.0693	−0.639136
$987$	0	0
$988$	2.76766	0.0880510
$989$	−6.40933	−0.203805
$990$	0	0
$991$	31.6484	1.00535	0.502673	−	0.864477i	$-0.332350\pi$
$991$	31.6484	1.00535	0.502673	+	0.864477i	$0.332350\pi$
$992$	3.83934	0.121899
$993$	0	0
$994$	30.8699	0.979133
$995$	78.8234	2.49887
$996$	0	0
$997$	−4.19629	−0.132898	−0.0664489	−	0.997790i	$-0.521167\pi$
$997$	−4.19629	−0.132898	−0.0664489	+	0.997790i	$0.521167\pi$
$998$	20.3226	0.643301
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.t.1.16	yes	16
3.2	odd	2		8046.2.a.s.1.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.1	✓	16	3.2	odd	2
8046.2.a.t.1.16	yes	16	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.36260 q^{5} +2.57952 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +4.36260 q^{5} +2.57952 q^{7} +1.00000 q^{8} +4.36260 q^{10} -6.22907 q^{11} +0.768400 q^{13} +2.57952 q^{14} +1.00000 q^{16} +2.14011 q^{17} +3.60185 q^{19} +4.36260 q^{20} -6.22907 q^{22} -0.549412 q^{23} +14.0323 q^{25} +0.768400 q^{26} +2.57952 q^{28} -9.37769 q^{29} +3.83934 q^{31} +1.00000 q^{32} +2.14011 q^{34} +11.2534 q^{35} +3.34752 q^{37} +3.60185 q^{38} +4.36260 q^{40} -1.84301 q^{41} +11.6658 q^{43} -6.22907 q^{44} -0.549412 q^{46} -1.30340 q^{47} -0.346089 q^{49} +14.0323 q^{50} +0.768400 q^{52} -7.86564 q^{53} -27.1749 q^{55} +2.57952 q^{56} -9.37769 q^{58} +11.9672 q^{59} -1.43378 q^{61} +3.83934 q^{62} +1.00000 q^{64} +3.35222 q^{65} +1.90317 q^{67} +2.14011 q^{68} +11.2534 q^{70} +11.9673 q^{71} -6.31479 q^{73} +3.34752 q^{74} +3.60185 q^{76} -16.0680 q^{77} -5.58864 q^{79} +4.36260 q^{80} -1.84301 q^{82} -6.77087 q^{83} +9.33643 q^{85} +11.6658 q^{86} -6.22907 q^{88} -1.81991 q^{89} +1.98210 q^{91} -0.549412 q^{92} -1.30340 q^{94} +15.7134 q^{95} +2.30485 q^{97} -0.346089 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8}+O(q^{10}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8	\( 16 q + 16 q^{2} + 16 q^{4} + 4 q^{5} + 6 q^{7} + 16 q^{8} + 4 q^{10} + 6 q^{11} + 6 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 10 q^{19} + 4 q^{20} + 6 q^{22} + 10 q^{23} + 28 q^{25} + 6 q^{26} + 6 q^{28} + 6 q^{29} + 21 q^{31} + 16 q^{32} + q^{34} + 16 q^{35} + 17 q^{37} + 10 q^{38} + 4 q^{40} - 4 q^{41} + 16 q^{43} + 6 q^{44} + 10 q^{46} + 25 q^{47} + 36 q^{49} + 28 q^{50} + 6 q^{52} + 14 q^{53} + 19 q^{55} + 6 q^{56} + 6 q^{58} + 6 q^{59} + 23 q^{61} + 21 q^{62} + 16 q^{64} + 20 q^{65} + 22 q^{67} + q^{68} + 16 q^{70} + 10 q^{71} + 16 q^{73} + 17 q^{74} + 10 q^{76} - 2 q^{77} + 37 q^{79} + 4 q^{80} - 4 q^{82} + 33 q^{83} + 43 q^{85} + 16 q^{86} + 6 q^{88} - 3 q^{89} + 28 q^{91} + 10 q^{92} + 25 q^{94} + 14 q^{95} - 3 q^{97} + 36 q^{98}+O(q^{100}) \) 16 * q + 16 * q^2 + 16 * q^4 + 4 * q^5 + 6 * q^7 + 16 * q^8 + 4 * q^10 + 6 * q^11 + 6 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 10 * q^19 + 4 * q^20 + 6 * q^22 + 10 * q^23 + 28 * q^25 + 6 * q^26 + 6 * q^28 + 6 * q^29 + 21 * q^31 + 16 * q^32 + q^34 + 16 * q^35 + 17 * q^37 + 10 * q^38 + 4 * q^40 - 4 * q^41 + 16 * q^43 + 6 * q^44 + 10 * q^46 + 25 * q^47 + 36 * q^49 + 28 * q^50 + 6 * q^52 + 14 * q^53 + 19 * q^55 + 6 * q^56 + 6 * q^58 + 6 * q^59 + 23 * q^61 + 21 * q^62 + 16 * q^64 + 20 * q^65 + 22 * q^67 + q^68 + 16 * q^70 + 10 * q^71 + 16 * q^73 + 17 * q^74 + 10 * q^76 - 2 * q^77 + 37 * q^79 + 4 * q^80 - 4 * q^82 + 33 * q^83 + 43 * q^85 + 16 * q^86 + 6 * q^88 - 3 * q^89 + 28 * q^91 + 10 * q^92 + 25 * q^94 + 14 * q^95 - 3 * q^97 + 36 * q^98

Introduction

L-functions

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Database

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