LMFDB - Embedded newform 8046.2.a.i.1.5

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:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 23 x^{10} + 142 x^{9} + 104 x^{8} - 1302 x^{7} + 607 x^{6} + 4323 x^{5} - 4461 x^{4} + \cdots - 553 $ x^12 - 5x^11 - 23x^10 + 142x^9 + 104x^8 - 1302x^7 + 607x^6 + 4323x^5 - 4461x^4 - 3333x^3 + 4805x^2 - 224*x - 553
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.33540$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.33540 q^{5} +2.26685 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -1.33540 q^{5} +2.26685 q^{7} -1.00000 q^{8} +1.33540 q^{10} -5.80942 q^{11} +2.07946 q^{13} -2.26685 q^{14} +1.00000 q^{16} +5.11591 q^{17} +8.67108 q^{19} -1.33540 q^{20} +5.80942 q^{22} -3.21691 q^{23} -3.21671 q^{25} -2.07946 q^{26} +2.26685 q^{28} -3.55075 q^{29} -4.87939 q^{31} -1.00000 q^{32} -5.11591 q^{34} -3.02716 q^{35} +4.63624 q^{37} -8.67108 q^{38} +1.33540 q^{40} -8.09935 q^{41} -7.14821 q^{43} -5.80942 q^{44} +3.21691 q^{46} +4.49388 q^{47} -1.86137 q^{49} +3.21671 q^{50} +2.07946 q^{52} -1.64247 q^{53} +7.75790 q^{55} -2.26685 q^{56} +3.55075 q^{58} -1.45742 q^{59} +2.95663 q^{61} +4.87939 q^{62} +1.00000 q^{64} -2.77691 q^{65} -11.2036 q^{67} +5.11591 q^{68} +3.02716 q^{70} +6.47410 q^{71} -3.65611 q^{73} -4.63624 q^{74} +8.67108 q^{76} -13.1691 q^{77} +4.22676 q^{79} -1.33540 q^{80} +8.09935 q^{82} -0.157895 q^{83} -6.83179 q^{85} +7.14821 q^{86} +5.80942 q^{88} +4.44985 q^{89} +4.71382 q^{91} -3.21691 q^{92} -4.49388 q^{94} -11.5794 q^{95} +8.30166 q^{97} +1.86137 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 6 q^{11} + 3 q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} + 8 q^{19} - 5 q^{20} + 6 q^{22} - 11 q^{23} + 11 q^{25} - 3 q^{26} + 6 q^{28} - 29 q^{29} + 2 q^{31} - 12 q^{32} + 6 q^{34} - 4 q^{35} + 5 q^{37} - 8 q^{38} + 5 q^{40} - 22 q^{41} + 9 q^{43} - 6 q^{44} + 11 q^{46} - 15 q^{47} + 14 q^{49} - 11 q^{50} + 3 q^{52} - 12 q^{53} + 13 q^{55} - 6 q^{56} + 29 q^{58} - 34 q^{59} - 4 q^{61} - 2 q^{62} + 12 q^{64} - 12 q^{65} + q^{67} - 6 q^{68} + 4 q^{70} - 21 q^{71} - 2 q^{73} - 5 q^{74} + 8 q^{76} - 34 q^{77} + 9 q^{79} - 5 q^{80} + 22 q^{82} - 10 q^{83} + 5 q^{85} - 9 q^{86} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 11 q^{92} + 15 q^{94} - 69 q^{95} - 13 q^{97} - 14 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 6 * q^11 + 3 * q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 + 8 * q^19 - 5 * q^20 + 6 * q^22 - 11 * q^23 + 11 * q^25 - 3 * q^26 + 6 * q^28 - 29 * q^29 + 2 * q^31 - 12 * q^32 + 6 * q^34 - 4 * q^35 + 5 * q^37 - 8 * q^38 + 5 * q^40 - 22 * q^41 + 9 * q^43 - 6 * q^44 + 11 * q^46 - 15 * q^47 + 14 * q^49 - 11 * q^50 + 3 * q^52 - 12 * q^53 + 13 * q^55 - 6 * q^56 + 29 * q^58 - 34 * q^59 - 4 * q^61 - 2 * q^62 + 12 * q^64 - 12 * q^65 + q^67 - 6 * q^68 + 4 * q^70 - 21 * q^71 - 2 * q^73 - 5 * q^74 + 8 * q^76 - 34 * q^77 + 9 * q^79 - 5 * q^80 + 22 * q^82 - 10 * q^83 + 5 * q^85 - 9 * q^86 + 6 * q^88 + 2 * q^89 + 17 * q^91 - 11 * q^92 + 15 * q^94 - 69 * q^95 - 13 * q^97 - 14 * 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
$5$	−1.33540	−0.597209	−0.298605	−	0.954377i	$-0.596521\pi$
$5$	−1.33540	−0.597209	−0.298605	+	0.954377i	$0.596521\pi$
$6$	0	0
$7$	2.26685	0.856790	0.428395	−	0.903591i	$-0.359079\pi$
$7$	2.26685	0.856790	0.428395	+	0.903591i	$0.359079\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.33540	0.422291
$11$	−5.80942	−1.75161	−0.875803	−	0.482668i	$-0.839668\pi$
$11$	−5.80942	−1.75161	−0.875803	+	0.482668i	$0.839668\pi$
$12$	0	0
$13$	2.07946	0.576737	0.288369	−	0.957519i	$-0.406887\pi$
$13$	2.07946	0.576737	0.288369	+	0.957519i	$0.406887\pi$
$14$	−2.26685	−0.605842
$15$	0	0
$16$	1.00000	0.250000
$17$	5.11591	1.24079	0.620395	−	0.784289i	$-0.286972\pi$
$17$	5.11591	1.24079	0.620395	+	0.784289i	$0.286972\pi$
$18$	0	0
$19$	8.67108	1.98928	0.994641	−	0.103393i	$-0.0329699\pi$
$19$	8.67108	1.98928	0.994641	+	0.103393i	$0.0329699\pi$
$20$	−1.33540	−0.298605
$21$	0	0
$22$	5.80942	1.23857
$23$	−3.21691	−0.670771	−0.335386	−	0.942081i	$-0.608867\pi$
$23$	−3.21691	−0.670771	−0.335386	+	0.942081i	$0.608867\pi$
$24$	0	0
$25$	−3.21671	−0.643341
$26$	−2.07946	−0.407815
$27$	0	0
$28$	2.26685	0.428395
$29$	−3.55075	−0.659357	−0.329679	−	0.944093i	$-0.606940\pi$
$29$	−3.55075	−0.659357	−0.329679	+	0.944093i	$0.606940\pi$
$30$	0	0
$31$	−4.87939	−0.876364	−0.438182	−	0.898886i	$-0.644378\pi$
$31$	−4.87939	−0.876364	−0.438182	+	0.898886i	$0.644378\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.11591	−0.877372
$35$	−3.02716	−0.511683
$36$	0	0
$37$	4.63624	0.762193	0.381096	−	0.924535i	$-0.375547\pi$
$37$	4.63624	0.762193	0.381096	+	0.924535i	$0.375547\pi$
$38$	−8.67108	−1.40663
$39$	0	0
$40$	1.33540	0.211145
$41$	−8.09935	−1.26491	−0.632453	−	0.774599i	$-0.717952\pi$
$41$	−8.09935	−1.26491	−0.632453	+	0.774599i	$0.717952\pi$
$42$	0	0
$43$	−7.14821	−1.09009	−0.545046	−	0.838406i	$-0.683488\pi$
$43$	−7.14821	−1.09009	−0.545046	+	0.838406i	$0.683488\pi$
$44$	−5.80942	−0.875803
$45$	0	0
$46$	3.21691	0.474307
$47$	4.49388	0.655500	0.327750	−	0.944764i	$-0.393710\pi$
$47$	4.49388	0.655500	0.327750	+	0.944764i	$0.393710\pi$
$48$	0	0
$49$	−1.86137	−0.265910
$50$	3.21671	0.454911
$51$	0	0
$52$	2.07946	0.288369
$53$	−1.64247	−0.225611	−0.112805	−	0.993617i	$-0.535984\pi$
$53$	−1.64247	−0.225611	−0.112805	+	0.993617i	$0.535984\pi$
$54$	0	0
$55$	7.75790	1.04608
$56$	−2.26685	−0.302921
$57$	0	0
$58$	3.55075	0.466236
$59$	−1.45742	−0.189740	−0.0948700	−	0.995490i	$-0.530244\pi$
$59$	−1.45742	−0.189740	−0.0948700	+	0.995490i	$0.530244\pi$
$60$	0	0
$61$	2.95663	0.378557	0.189279	−	0.981923i	$-0.439385\pi$
$61$	2.95663	0.378557	0.189279	+	0.981923i	$0.439385\pi$
$62$	4.87939	0.619683
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.77691	−0.344433
$66$	0	0
$67$	−11.2036	−1.36874	−0.684371	−	0.729134i	$-0.739923\pi$
$67$	−11.2036	−1.36874	−0.684371	+	0.729134i	$0.739923\pi$
$68$	5.11591	0.620395
$69$	0	0
$70$	3.02716	0.361814
$71$	6.47410	0.768334	0.384167	−	0.923264i	$-0.374489\pi$
$71$	6.47410	0.768334	0.384167	+	0.923264i	$0.374489\pi$
$72$	0	0
$73$	−3.65611	−0.427915	−0.213958	−	0.976843i	$-0.568635\pi$
$73$	−3.65611	−0.427915	−0.213958	+	0.976843i	$0.568635\pi$
$74$	−4.63624	−0.538952
$75$	0	0
$76$	8.67108	0.994641
$77$	−13.1691	−1.50076
$78$	0	0
$79$	4.22676	0.475547	0.237774	−	0.971321i	$-0.423582\pi$
$79$	4.22676	0.475547	0.237774	+	0.971321i	$0.423582\pi$
$80$	−1.33540	−0.149302
$81$	0	0
$82$	8.09935	0.894423
$83$	−0.157895	−0.0173312	−0.00866560	−	0.999962i	$-0.502758\pi$
$83$	−0.157895	−0.0173312	−0.00866560	+	0.999962i	$0.502758\pi$
$84$	0	0
$85$	−6.83179	−0.741012
$86$	7.14821	0.770811
$87$	0	0
$88$	5.80942	0.619286
$89$	4.44985	0.471683	0.235842	−	0.971792i	$-0.424215\pi$
$89$	4.44985	0.471683	0.235842	+	0.971792i	$0.424215\pi$
$90$	0	0
$91$	4.71382	0.494143
$92$	−3.21691	−0.335386
$93$	0	0
$94$	−4.49388	−0.463509
$95$	−11.5794	−1.18802
$96$	0	0
$97$	8.30166	0.842906	0.421453	−	0.906850i	$-0.361520\pi$
$97$	8.30166	0.842906	0.421453	+	0.906850i	$0.361520\pi$
$98$	1.86137	0.188027
$99$	0	0
$100$	−3.21671	−0.321671
$101$	−1.68201	−0.167366	−0.0836831	−	0.996492i	$-0.526668\pi$
$101$	−1.68201	−0.167366	−0.0836831	+	0.996492i	$0.526668\pi$
$102$	0	0
$103$	−3.89775	−0.384057	−0.192028	−	0.981389i	$-0.561507\pi$
$103$	−3.89775	−0.384057	−0.192028	+	0.981389i	$0.561507\pi$
$104$	−2.07946	−0.203907
$105$	0	0
$106$	1.64247	0.159531
$107$	14.6552	1.41677	0.708385	−	0.705827i	$-0.249424\pi$
$107$	14.6552	1.41677	0.708385	+	0.705827i	$0.249424\pi$
$108$	0	0
$109$	−10.4977	−1.00549	−0.502747	−	0.864434i	$-0.667677\pi$
$109$	−10.4977	−1.00549	−0.502747	+	0.864434i	$0.667677\pi$
$110$	−7.75790	−0.739687
$111$	0	0
$112$	2.26685	0.214198
$113$	13.8839	1.30609	0.653043	−	0.757321i	$-0.273492\pi$
$113$	13.8839	1.30609	0.653043	+	0.757321i	$0.273492\pi$
$114$	0	0
$115$	4.29586	0.400591
$116$	−3.55075	−0.329679
$117$	0	0
$118$	1.45742	0.134166
$119$	11.5970	1.06310
$120$	0	0
$121$	22.7494	2.06813
$122$	−2.95663	−0.267681
$123$	0	0
$124$	−4.87939	−0.438182
$125$	10.9726	0.981418
$126$	0	0
$127$	−18.9253	−1.67935	−0.839676	−	0.543088i	$-0.817255\pi$
$127$	−18.9253	−1.67935	−0.839676	+	0.543088i	$0.817255\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.77691	0.243551
$131$	7.87984	0.688465	0.344233	−	0.938884i	$-0.388139\pi$
$131$	7.87984	0.688465	0.344233	+	0.938884i	$0.388139\pi$
$132$	0	0
$133$	19.6561	1.70440
$134$	11.2036	0.967847
$135$	0	0
$136$	−5.11591	−0.438686
$137$	−16.3169	−1.39405	−0.697025	−	0.717047i	$-0.745493\pi$
$137$	−16.3169	−1.39405	−0.697025	+	0.717047i	$0.745493\pi$
$138$	0	0
$139$	22.4436	1.90364	0.951821	−	0.306654i	$-0.0992094\pi$
$139$	22.4436	1.90364	0.951821	+	0.306654i	$0.0992094\pi$
$140$	−3.02716	−0.255841
$141$	0	0
$142$	−6.47410	−0.543294
$143$	−12.0804	−1.01022
$144$	0	0
$145$	4.74167	0.393774
$146$	3.65611	0.302582
$147$	0	0
$148$	4.63624	0.381096
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−1.03910	−0.0845608	−0.0422804	−	0.999106i	$-0.513462\pi$
$151$	−1.03910	−0.0845608	−0.0422804	+	0.999106i	$0.513462\pi$
$152$	−8.67108	−0.703317
$153$	0	0
$154$	13.1691	1.06120
$155$	6.51594	0.523373
$156$	0	0
$157$	−11.3346	−0.904600	−0.452300	−	0.891866i	$-0.649396\pi$
$157$	−11.3346	−0.904600	−0.452300	+	0.891866i	$0.649396\pi$
$158$	−4.22676	−0.336263
$159$	0	0
$160$	1.33540	0.105573
$161$	−7.29226	−0.574711
$162$	0	0
$163$	−12.4023	−0.971420	−0.485710	−	0.874120i	$-0.661439\pi$
$163$	−12.4023	−0.971420	−0.485710	+	0.874120i	$0.661439\pi$
$164$	−8.09935	−0.632453
$165$	0	0
$166$	0.157895	0.0122550
$167$	−7.40538	−0.573046	−0.286523	−	0.958073i	$-0.592499\pi$
$167$	−7.40538	−0.573046	−0.286523	+	0.958073i	$0.592499\pi$
$168$	0	0
$169$	−8.67586	−0.667374
$170$	6.83179	0.523974
$171$	0	0
$172$	−7.14821	−0.545046
$173$	−4.62449	−0.351594	−0.175797	−	0.984426i	$-0.556250\pi$
$173$	−4.62449	−0.351594	−0.175797	+	0.984426i	$0.556250\pi$
$174$	0	0
$175$	−7.29181	−0.551209
$176$	−5.80942	−0.437902
$177$	0	0
$178$	−4.44985	−0.333530
$179$	−11.2805	−0.843141	−0.421570	−	0.906796i	$-0.638521\pi$
$179$	−11.2805	−0.843141	−0.421570	+	0.906796i	$0.638521\pi$
$180$	0	0
$181$	−4.51927	−0.335915	−0.167957	−	0.985794i	$-0.553717\pi$
$181$	−4.51927	−0.335915	−0.167957	+	0.985794i	$0.553717\pi$
$182$	−4.71382	−0.349412
$183$	0	0
$184$	3.21691	0.237154
$185$	−6.19123	−0.455188
$186$	0	0
$187$	−29.7205	−2.17338
$188$	4.49388	0.327750
$189$	0	0
$190$	11.5794	0.840055
$191$	5.82375	0.421392	0.210696	−	0.977552i	$-0.432427\pi$
$191$	5.82375	0.421392	0.210696	+	0.977552i	$0.432427\pi$
$192$	0	0
$193$	−7.80698	−0.561959	−0.280979	−	0.959714i	$-0.590659\pi$
$193$	−7.80698	−0.561959	−0.280979	+	0.959714i	$0.590659\pi$
$194$	−8.30166	−0.596025
$195$	0	0
$196$	−1.86137	−0.132955
$197$	5.58618	0.397999	0.198999	−	0.980000i	$-0.436231\pi$
$197$	5.58618	0.397999	0.198999	+	0.980000i	$0.436231\pi$
$198$	0	0
$199$	−10.7298	−0.760613	−0.380306	−	0.924861i	$-0.624181\pi$
$199$	−10.7298	−0.760613	−0.380306	+	0.924861i	$0.624181\pi$
$200$	3.21671	0.227456
$201$	0	0
$202$	1.68201	0.118346
$203$	−8.04902	−0.564931
$204$	0	0
$205$	10.8159	0.755413
$206$	3.89775	0.271569
$207$	0	0
$208$	2.07946	0.144184
$209$	−50.3739	−3.48444
$210$	0	0
$211$	4.18033	0.287786	0.143893	−	0.989593i	$-0.454038\pi$
$211$	4.18033	0.287786	0.143893	+	0.989593i	$0.454038\pi$
$212$	−1.64247	−0.112805
$213$	0	0
$214$	−14.6552	−1.00181
$215$	9.54572	0.651013
$216$	0	0
$217$	−11.0609	−0.750861
$218$	10.4977	0.710991
$219$	0	0
$220$	7.75790	0.523038
$221$	10.6383	0.715610
$222$	0	0
$223$	12.2349	0.819312	0.409656	−	0.912240i	$-0.365649\pi$
$223$	12.2349	0.819312	0.409656	+	0.912240i	$0.365649\pi$
$224$	−2.26685	−0.151461
$225$	0	0
$226$	−13.8839	−0.923542
$227$	10.8799	0.722124	0.361062	−	0.932542i	$-0.382414\pi$
$227$	10.8799	0.722124	0.361062	+	0.932542i	$0.382414\pi$
$228$	0	0
$229$	−25.2384	−1.66780	−0.833900	−	0.551916i	$-0.813897\pi$
$229$	−25.2384	−1.66780	−0.833900	+	0.551916i	$0.813897\pi$
$230$	−4.29586	−0.283260
$231$	0	0
$232$	3.55075	0.233118
$233$	−28.3597	−1.85790	−0.928952	−	0.370201i	$-0.879289\pi$
$233$	−28.3597	−1.85790	−0.928952	+	0.370201i	$0.879289\pi$
$234$	0	0
$235$	−6.00113	−0.391471
$236$	−1.45742	−0.0948700
$237$	0	0
$238$	−11.5970	−0.751724
$239$	−7.12040	−0.460580	−0.230290	−	0.973122i	$-0.573968\pi$
$239$	−7.12040	−0.460580	−0.230290	+	0.973122i	$0.573968\pi$
$240$	0	0
$241$	−4.08678	−0.263253	−0.131626	−	0.991299i	$-0.542020\pi$
$241$	−4.08678	−0.263253	−0.131626	+	0.991299i	$0.542020\pi$
$242$	−22.7494	−1.46239
$243$	0	0
$244$	2.95663	0.189279
$245$	2.48568	0.158804
$246$	0	0
$247$	18.0311	1.14729
$248$	4.87939	0.309842
$249$	0	0
$250$	−10.9726	−0.693968
$251$	13.6479	0.861447	0.430724	−	0.902484i	$-0.358258\pi$
$251$	13.6479	0.861447	0.430724	+	0.902484i	$0.358258\pi$
$252$	0	0
$253$	18.6884	1.17493
$254$	18.9253	1.18748
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.5964	−1.03525	−0.517627	−	0.855607i	$-0.673184\pi$
$257$	−16.5964	−1.03525	−0.517627	+	0.855607i	$0.673184\pi$
$258$	0	0
$259$	10.5097	0.653039
$260$	−2.77691	−0.172216
$261$	0	0
$262$	−7.87984	−0.486818
$263$	12.5008	0.770835	0.385417	−	0.922742i	$-0.374057\pi$
$263$	12.5008	0.770835	0.385417	+	0.922742i	$0.374057\pi$
$264$	0	0
$265$	2.19336	0.134737
$266$	−19.6561	−1.20519
$267$	0	0
$268$	−11.2036	−0.684371
$269$	−20.6728	−1.26044	−0.630222	−	0.776415i	$-0.717036\pi$
$269$	−20.6728	−1.26044	−0.630222	+	0.776415i	$0.717036\pi$
$270$	0	0
$271$	30.2724	1.83891	0.919457	−	0.393190i	$-0.128628\pi$
$271$	30.2724	1.83891	0.919457	+	0.393190i	$0.128628\pi$
$272$	5.11591	0.310198
$273$	0	0
$274$	16.3169	0.985742
$275$	18.6872	1.12688
$276$	0	0
$277$	15.0407	0.903708	0.451854	−	0.892092i	$-0.350763\pi$
$277$	15.0407	0.903708	0.451854	+	0.892092i	$0.350763\pi$
$278$	−22.4436	−1.34608
$279$	0	0
$280$	3.02716	0.180907
$281$	13.3323	0.795337	0.397668	−	0.917529i	$-0.369819\pi$
$281$	13.3323	0.795337	0.397668	+	0.917529i	$0.369819\pi$
$282$	0	0
$283$	−15.7754	−0.937750	−0.468875	−	0.883265i	$-0.655340\pi$
$283$	−15.7754	−0.937750	−0.468875	+	0.883265i	$0.655340\pi$
$284$	6.47410	0.384167
$285$	0	0
$286$	12.0804	0.714331
$287$	−18.3600	−1.08376
$288$	0	0
$289$	9.17256	0.539562
$290$	−4.74167	−0.278440
$291$	0	0
$292$	−3.65611	−0.213958
$293$	−4.83921	−0.282710	−0.141355	−	0.989959i	$-0.545146\pi$
$293$	−4.83921	−0.282710	−0.141355	+	0.989959i	$0.545146\pi$
$294$	0	0
$295$	1.94624	0.113314
$296$	−4.63624	−0.269476
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−6.68942	−0.386859
$300$	0	0
$301$	−16.2039	−0.933980
$302$	1.03910	0.0597935
$303$	0	0
$304$	8.67108	0.497320
$305$	−3.94828	−0.226078
$306$	0	0
$307$	−18.6584	−1.06489	−0.532446	−	0.846464i	$-0.678727\pi$
$307$	−18.6584	−1.06489	−0.532446	+	0.846464i	$0.678727\pi$
$308$	−13.1691	−0.750380
$309$	0	0
$310$	−6.51594	−0.370080
$311$	−34.2651	−1.94300	−0.971498	−	0.237048i	$-0.923820\pi$
$311$	−34.2651	−1.94300	−0.971498	+	0.237048i	$0.923820\pi$
$312$	0	0
$313$	0.723019	0.0408674	0.0204337	−	0.999791i	$-0.493495\pi$
$313$	0.723019	0.0408674	0.0204337	+	0.999791i	$0.493495\pi$
$314$	11.3346	0.639649
$315$	0	0
$316$	4.22676	0.237774
$317$	−9.57951	−0.538039	−0.269019	−	0.963135i	$-0.586700\pi$
$317$	−9.57951	−0.538039	−0.269019	+	0.963135i	$0.586700\pi$
$318$	0	0
$319$	20.6278	1.15493
$320$	−1.33540	−0.0746511
$321$	0	0
$322$	7.29226	0.406382
$323$	44.3605	2.46828
$324$	0	0
$325$	−6.68900	−0.371039
$326$	12.4023	0.686897
$327$	0	0
$328$	8.09935	0.447212
$329$	10.1870	0.561626
$330$	0	0
$331$	15.6251	0.858831	0.429416	−	0.903107i	$-0.358720\pi$
$331$	15.6251	0.858831	0.429416	+	0.903107i	$0.358720\pi$
$332$	−0.157895	−0.00866560
$333$	0	0
$334$	7.40538	0.405205
$335$	14.9613	0.817426
$336$	0	0
$337$	−8.12119	−0.442390	−0.221195	−	0.975230i	$-0.570996\pi$
$337$	−8.12119	−0.442390	−0.221195	+	0.975230i	$0.570996\pi$
$338$	8.67586	0.471905
$339$	0	0
$340$	−6.83179	−0.370506
$341$	28.3464	1.53505
$342$	0	0
$343$	−20.0874	−1.08462
$344$	7.14821	0.385406
$345$	0	0
$346$	4.62449	0.248614
$347$	−6.52384	−0.350218	−0.175109	−	0.984549i	$-0.556028\pi$
$347$	−6.52384	−0.350218	−0.175109	+	0.984549i	$0.556028\pi$
$348$	0	0
$349$	−19.1926	−1.02736	−0.513678	−	0.857983i	$-0.671717\pi$
$349$	−19.1926	−1.02736	−0.513678	+	0.857983i	$0.671717\pi$
$350$	7.29181	0.389763
$351$	0	0
$352$	5.80942	0.309643
$353$	−9.72044	−0.517367	−0.258683	−	0.965962i	$-0.583289\pi$
$353$	−9.72044	−0.517367	−0.258683	+	0.965962i	$0.583289\pi$
$354$	0	0
$355$	−8.64551	−0.458856
$356$	4.44985	0.235842
$357$	0	0
$358$	11.2805	0.596191
$359$	−14.8797	−0.785323	−0.392661	−	0.919683i	$-0.628446\pi$
$359$	−14.8797	−0.785323	−0.392661	+	0.919683i	$0.628446\pi$
$360$	0	0
$361$	56.1876	2.95724
$362$	4.51927	0.237528
$363$	0	0
$364$	4.71382	0.247071
$365$	4.88237	0.255555
$366$	0	0
$367$	−35.7769	−1.86754	−0.933769	−	0.357877i	$-0.883501\pi$
$367$	−35.7769	−1.86754	−0.933769	+	0.357877i	$0.883501\pi$
$368$	−3.21691	−0.167693
$369$	0	0
$370$	6.19123	0.321867
$371$	−3.72325	−0.193301
$372$	0	0
$373$	33.2864	1.72351	0.861753	−	0.507327i	$-0.169367\pi$
$373$	33.2864	1.72351	0.861753	+	0.507327i	$0.169367\pi$
$374$	29.7205	1.53681
$375$	0	0
$376$	−4.49388	−0.231754
$377$	−7.38362	−0.380276
$378$	0	0
$379$	−2.08326	−0.107010	−0.0535048	−	0.998568i	$-0.517039\pi$
$379$	−2.08326	−0.107010	−0.0535048	+	0.998568i	$0.517039\pi$
$380$	−11.5794	−0.594008
$381$	0	0
$382$	−5.82375	−0.297969
$383$	18.2511	0.932590	0.466295	−	0.884629i	$-0.345589\pi$
$383$	18.2511	0.932590	0.466295	+	0.884629i	$0.345589\pi$
$384$	0	0
$385$	17.5860	0.896267
$386$	7.80698	0.397365
$387$	0	0
$388$	8.30166	0.421453
$389$	11.6927	0.592843	0.296422	−	0.955057i	$-0.404207\pi$
$389$	11.6927	0.592843	0.296422	+	0.955057i	$0.404207\pi$
$390$	0	0
$391$	−16.4574	−0.832287
$392$	1.86137	0.0940134
$393$	0	0
$394$	−5.58618	−0.281428
$395$	−5.64441	−0.284001
$396$	0	0
$397$	0.213966	0.0107386	0.00536932	−	0.999986i	$-0.498291\pi$
$397$	0.213966	0.0107386	0.00536932	+	0.999986i	$0.498291\pi$
$398$	10.7298	0.537835
$399$	0	0
$400$	−3.21671	−0.160835
$401$	−10.3209	−0.515403	−0.257701	−	0.966225i	$-0.582965\pi$
$401$	−10.3209	−0.515403	−0.257701	+	0.966225i	$0.582965\pi$
$402$	0	0
$403$	−10.1465	−0.505432
$404$	−1.68201	−0.0836831
$405$	0	0
$406$	8.04902	0.399466
$407$	−26.9339	−1.33506
$408$	0	0
$409$	0.211630	0.0104644	0.00523222	−	0.999986i	$-0.498335\pi$
$409$	0.211630	0.0104644	0.00523222	+	0.999986i	$0.498335\pi$
$410$	−10.8159	−0.534158
$411$	0	0
$412$	−3.89775	−0.192028
$413$	−3.30376	−0.162567
$414$	0	0
$415$	0.210853	0.0103503
$416$	−2.07946	−0.101954
$417$	0	0
$418$	50.3739	2.46387
$419$	−3.39484	−0.165849	−0.0829243	−	0.996556i	$-0.526426\pi$
$419$	−3.39484	−0.165849	−0.0829243	+	0.996556i	$0.526426\pi$
$420$	0	0
$421$	5.07013	0.247103	0.123552	−	0.992338i	$-0.460572\pi$
$421$	5.07013	0.247103	0.123552	+	0.992338i	$0.460572\pi$
$422$	−4.18033	−0.203495
$423$	0	0
$424$	1.64247	0.0797655
$425$	−16.4564	−0.798252
$426$	0	0
$427$	6.70225	0.324344
$428$	14.6552	0.708385
$429$	0	0
$430$	−9.54572	−0.460335
$431$	1.42121	0.0684573	0.0342287	−	0.999414i	$-0.489103\pi$
$431$	1.42121	0.0684573	0.0342287	+	0.999414i	$0.489103\pi$
$432$	0	0
$433$	−8.16167	−0.392225	−0.196112	−	0.980581i	$-0.562832\pi$
$433$	−8.16167	−0.392225	−0.196112	+	0.980581i	$0.562832\pi$
$434$	11.0609	0.530939
$435$	0	0
$436$	−10.4977	−0.502747
$437$	−27.8940	−1.33435
$438$	0	0
$439$	−24.2959	−1.15958	−0.579791	−	0.814765i	$-0.696866\pi$
$439$	−24.2959	−1.15958	−0.579791	+	0.814765i	$0.696866\pi$
$440$	−7.75790	−0.369843
$441$	0	0
$442$	−10.6383	−0.506013
$443$	−21.8590	−1.03855	−0.519277	−	0.854606i	$-0.673799\pi$
$443$	−21.8590	−1.03855	−0.519277	+	0.854606i	$0.673799\pi$
$444$	0	0
$445$	−5.94233	−0.281693
$446$	−12.2349	−0.579341
$447$	0	0
$448$	2.26685	0.107099
$449$	−31.3083	−1.47753	−0.738765	−	0.673964i	$-0.764590\pi$
$449$	−31.3083	−1.47753	−0.738765	+	0.673964i	$0.764590\pi$
$450$	0	0
$451$	47.0525	2.21562
$452$	13.8839	0.653043
$453$	0	0
$454$	−10.8799	−0.510618
$455$	−6.29484	−0.295107
$456$	0	0
$457$	−22.4776	−1.05146	−0.525729	−	0.850652i	$-0.676207\pi$
$457$	−22.4776	−1.05146	−0.525729	+	0.850652i	$0.676207\pi$
$458$	25.2384	1.17931
$459$	0	0
$460$	4.29586	0.200295
$461$	24.6409	1.14764	0.573820	−	0.818982i	$-0.305461\pi$
$461$	24.6409	1.14764	0.573820	+	0.818982i	$0.305461\pi$
$462$	0	0
$463$	−23.3886	−1.08696	−0.543479	−	0.839423i	$-0.682893\pi$
$463$	−23.3886	−1.08696	−0.543479	+	0.839423i	$0.682893\pi$
$464$	−3.55075	−0.164839
$465$	0	0
$466$	28.3597	1.31374
$467$	−30.4777	−1.41034	−0.705170	−	0.709038i	$-0.749129\pi$
$467$	−30.4777	−1.41034	−0.705170	+	0.709038i	$0.749129\pi$
$468$	0	0
$469$	−25.3970	−1.17273
$470$	6.00113	0.276812
$471$	0	0
$472$	1.45742	0.0670832
$473$	41.5270	1.90941
$474$	0	0
$475$	−27.8923	−1.27979
$476$	11.5970	0.531549
$477$	0	0
$478$	7.12040	0.325680
$479$	−7.57443	−0.346084	−0.173042	−	0.984914i	$-0.555360\pi$
$479$	−7.57443	−0.346084	−0.173042	+	0.984914i	$0.555360\pi$
$480$	0	0
$481$	9.64085	0.439585
$482$	4.08678	0.186148
$483$	0	0
$484$	22.7494	1.03406
$485$	−11.0860	−0.503391
$486$	0	0
$487$	40.2731	1.82495	0.912474	−	0.409134i	$-0.134169\pi$
$487$	40.2731	1.82495	0.912474	+	0.409134i	$0.134169\pi$
$488$	−2.95663	−0.133840
$489$	0	0
$490$	−2.48568	−0.112291
$491$	−15.3506	−0.692765	−0.346383	−	0.938093i	$-0.612590\pi$
$491$	−15.3506	−0.692765	−0.346383	+	0.938093i	$0.612590\pi$
$492$	0	0
$493$	−18.1653	−0.818124
$494$	−18.0311	−0.811258
$495$	0	0
$496$	−4.87939	−0.219091
$497$	14.6758	0.658302
$498$	0	0
$499$	41.3580	1.85144	0.925719	−	0.378212i	$-0.123461\pi$
$499$	41.3580	1.85144	0.925719	+	0.378212i	$0.123461\pi$
$500$	10.9726	0.490709
$501$	0	0
$502$	−13.6479	−0.609135
$503$	5.57208	0.248447	0.124223	−	0.992254i	$-0.460356\pi$
$503$	5.57208	0.248447	0.124223	+	0.992254i	$0.460356\pi$
$504$	0	0
$505$	2.24616	0.0999526
$506$	−18.6884	−0.830799
$507$	0	0
$508$	−18.9253	−0.839676
$509$	−29.8590	−1.32348	−0.661740	−	0.749734i	$-0.730182\pi$
$509$	−29.8590	−1.32348	−0.661740	+	0.749734i	$0.730182\pi$
$510$	0	0
$511$	−8.28787	−0.366634
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	16.5964	0.732034
$515$	5.20506	0.229362
$516$	0	0
$517$	−26.1069	−1.14818
$518$	−10.5097	−0.461769
$519$	0	0
$520$	2.77691	0.121775
$521$	8.13727	0.356500	0.178250	−	0.983985i	$-0.442956\pi$
$521$	8.13727	0.356500	0.178250	+	0.983985i	$0.442956\pi$
$522$	0	0
$523$	−7.53980	−0.329692	−0.164846	−	0.986319i	$-0.552713\pi$
$523$	−7.53980	−0.329692	−0.164846	+	0.986319i	$0.552713\pi$
$524$	7.87984	0.344233
$525$	0	0
$526$	−12.5008	−0.545062
$527$	−24.9625	−1.08738
$528$	0	0
$529$	−12.6515	−0.550066
$530$	−2.19336	−0.0952734
$531$	0	0
$532$	19.6561	0.852199
$533$	−16.8422	−0.729518
$534$	0	0
$535$	−19.5705	−0.846107
$536$	11.2036	0.483924
$537$	0	0
$538$	20.6728	0.891268
$539$	10.8135	0.465770
$540$	0	0
$541$	−25.3168	−1.08845	−0.544227	−	0.838938i	$-0.683177\pi$
$541$	−25.3168	−1.08845	−0.544227	+	0.838938i	$0.683177\pi$
$542$	−30.2724	−1.30031
$543$	0	0
$544$	−5.11591	−0.219343
$545$	14.0186	0.600490
$546$	0	0
$547$	9.87966	0.422424	0.211212	−	0.977440i	$-0.432259\pi$
$547$	9.87966	0.422424	0.211212	+	0.977440i	$0.432259\pi$
$548$	−16.3169	−0.697025
$549$	0	0
$550$	−18.6872	−0.796825
$551$	−30.7888	−1.31165
$552$	0	0
$553$	9.58144	0.407444
$554$	−15.0407	−0.639018
$555$	0	0
$556$	22.4436	0.951821
$557$	23.0656	0.977320	0.488660	−	0.872474i	$-0.337486\pi$
$557$	23.0656	0.977320	0.488660	+	0.872474i	$0.337486\pi$
$558$	0	0
$559$	−14.8644	−0.628696
$560$	−3.02716	−0.127921
$561$	0	0
$562$	−13.3323	−0.562388
$563$	−7.90993	−0.333364	−0.166682	−	0.986011i	$-0.553305\pi$
$563$	−7.90993	−0.333364	−0.166682	+	0.986011i	$0.553305\pi$
$564$	0	0
$565$	−18.5405	−0.780006
$566$	15.7754	0.663089
$567$	0	0
$568$	−6.47410	−0.271647
$569$	2.89792	0.121487	0.0607435	−	0.998153i	$-0.480653\pi$
$569$	2.89792	0.121487	0.0607435	+	0.998153i	$0.480653\pi$
$570$	0	0
$571$	32.6023	1.36436	0.682182	−	0.731183i	$-0.261031\pi$
$571$	32.6023	1.36436	0.682182	+	0.731183i	$0.261031\pi$
$572$	−12.0804	−0.505108
$573$	0	0
$574$	18.3600	0.766333
$575$	10.3478	0.431535
$576$	0	0
$577$	3.55957	0.148187	0.0740935	−	0.997251i	$-0.476394\pi$
$577$	3.55957	0.148187	0.0740935	+	0.997251i	$0.476394\pi$
$578$	−9.17256	−0.381528
$579$	0	0
$580$	4.74167	0.196887
$581$	−0.357924	−0.0148492
$582$	0	0
$583$	9.54182	0.395182
$584$	3.65611	0.151291
$585$	0	0
$586$	4.83921	0.199906
$587$	−18.3174	−0.756039	−0.378019	−	0.925798i	$-0.623395\pi$
$587$	−18.3174	−0.756039	−0.378019	+	0.925798i	$0.623395\pi$
$588$	0	0
$589$	−42.3096	−1.74333
$590$	−1.94624	−0.0801254
$591$	0	0
$592$	4.63624	0.190548
$593$	6.58426	0.270383	0.135192	−	0.990819i	$-0.456835\pi$
$593$	6.58426	0.270383	0.135192	+	0.990819i	$0.456835\pi$
$594$	0	0
$595$	−15.4867	−0.634892
$596$	1.00000	0.0409616
$597$	0	0
$598$	6.68942	0.273551
$599$	−1.77773	−0.0726360	−0.0363180	−	0.999340i	$-0.511563\pi$
$599$	−1.77773	−0.0726360	−0.0363180	+	0.999340i	$0.511563\pi$
$600$	0	0
$601$	11.8709	0.484222	0.242111	−	0.970249i	$-0.422160\pi$
$601$	11.8709	0.484222	0.242111	+	0.970249i	$0.422160\pi$
$602$	16.2039	0.660424
$603$	0	0
$604$	−1.03910	−0.0422804
$605$	−30.3795	−1.23510
$606$	0	0
$607$	−5.55085	−0.225302	−0.112651	−	0.993635i	$-0.535934\pi$
$607$	−5.55085	−0.225302	−0.112651	+	0.993635i	$0.535934\pi$
$608$	−8.67108	−0.351659
$609$	0	0
$610$	3.94828	0.159861
$611$	9.34483	0.378051
$612$	0	0
$613$	16.5635	0.668993	0.334496	−	0.942397i	$-0.391434\pi$
$613$	16.5635	0.668993	0.334496	+	0.942397i	$0.391434\pi$
$614$	18.6584	0.752993
$615$	0	0
$616$	13.1691	0.530599
$617$	−0.583822	−0.0235038	−0.0117519	−	0.999931i	$-0.503741\pi$
$617$	−0.583822	−0.0235038	−0.0117519	+	0.999931i	$0.503741\pi$
$618$	0	0
$619$	−21.5344	−0.865539	−0.432769	−	0.901505i	$-0.642464\pi$
$619$	−21.5344	−0.865539	−0.432769	+	0.901505i	$0.642464\pi$
$620$	6.51594	0.261686
$621$	0	0
$622$	34.2651	1.37391
$623$	10.0872	0.404134
$624$	0	0
$625$	1.43074	0.0572295
$626$	−0.723019	−0.0288976
$627$	0	0
$628$	−11.3346	−0.452300
$629$	23.7186	0.945722
$630$	0	0
$631$	0.665901	0.0265091	0.0132546	−	0.999912i	$-0.495781\pi$
$631$	0.665901	0.0265091	0.0132546	+	0.999912i	$0.495781\pi$
$632$	−4.22676	−0.168131
$633$	0	0
$634$	9.57951	0.380451
$635$	25.2729	1.00292
$636$	0	0
$637$	−3.87064	−0.153360
$638$	−20.6278	−0.816662
$639$	0	0
$640$	1.33540	0.0527863
$641$	−49.8567	−1.96922	−0.984611	−	0.174758i	$-0.944086\pi$
$641$	−49.8567	−1.96922	−0.984611	+	0.174758i	$0.944086\pi$
$642$	0	0
$643$	−37.0304	−1.46034	−0.730169	−	0.683267i	$-0.760559\pi$
$643$	−37.0304	−1.46034	−0.730169	+	0.683267i	$0.760559\pi$
$644$	−7.29226	−0.287355
$645$	0	0
$646$	−44.3605	−1.74534
$647$	−45.7407	−1.79825	−0.899126	−	0.437691i	$-0.855797\pi$
$647$	−45.7407	−1.79825	−0.899126	+	0.437691i	$0.855797\pi$
$648$	0	0
$649$	8.46677	0.332350
$650$	6.68900	0.262364
$651$	0	0
$652$	−12.4023	−0.485710
$653$	−16.6391	−0.651138	−0.325569	−	0.945518i	$-0.605556\pi$
$653$	−16.6391	−0.651138	−0.325569	+	0.945518i	$0.605556\pi$
$654$	0	0
$655$	−10.5227	−0.411158
$656$	−8.09935	−0.316226
$657$	0	0
$658$	−10.1870	−0.397130
$659$	32.1205	1.25124	0.625619	−	0.780129i	$-0.284846\pi$
$659$	32.1205	1.25124	0.625619	+	0.780129i	$0.284846\pi$
$660$	0	0
$661$	−15.5899	−0.606377	−0.303188	−	0.952931i	$-0.598051\pi$
$661$	−15.5899	−0.606377	−0.303188	+	0.952931i	$0.598051\pi$
$662$	−15.6251	−0.607285
$663$	0	0
$664$	0.157895	0.00612750
$665$	−26.2487	−1.01788
$666$	0	0
$667$	11.4224	0.442278
$668$	−7.40538	−0.286523
$669$	0	0
$670$	−14.9613	−0.578007
$671$	−17.1763	−0.663084
$672$	0	0
$673$	−38.3356	−1.47773	−0.738865	−	0.673853i	$-0.764638\pi$
$673$	−38.3356	−1.47773	−0.738865	+	0.673853i	$0.764638\pi$
$674$	8.12119	0.312817
$675$	0	0
$676$	−8.67586	−0.333687
$677$	34.3362	1.31965	0.659823	−	0.751421i	$-0.270631\pi$
$677$	34.3362	1.31965	0.659823	+	0.751421i	$0.270631\pi$
$678$	0	0
$679$	18.8187	0.722194
$680$	6.83179	0.261987
$681$	0	0
$682$	−28.3464	−1.08544
$683$	−8.44528	−0.323150	−0.161575	−	0.986860i	$-0.551657\pi$
$683$	−8.44528	−0.323150	−0.161575	+	0.986860i	$0.551657\pi$
$684$	0	0
$685$	21.7896	0.832539
$686$	20.0874	0.766942
$687$	0	0
$688$	−7.14821	−0.272523
$689$	−3.41545	−0.130118
$690$	0	0
$691$	−29.3854	−1.11787	−0.558937	−	0.829210i	$-0.688790\pi$
$691$	−29.3854	−1.11787	−0.558937	+	0.829210i	$0.688790\pi$
$692$	−4.62449	−0.175797
$693$	0	0
$694$	6.52384	0.247641
$695$	−29.9712	−1.13687
$696$	0	0
$697$	−41.4355	−1.56948
$698$	19.1926	0.726450
$699$	0	0
$700$	−7.29181	−0.275604
$701$	−40.5189	−1.53038	−0.765189	−	0.643806i	$-0.777355\pi$
$701$	−40.5189	−1.53038	−0.765189	+	0.643806i	$0.777355\pi$
$702$	0	0
$703$	40.2012	1.51622
$704$	−5.80942	−0.218951
$705$	0	0
$706$	9.72044	0.365834
$707$	−3.81287	−0.143398
$708$	0	0
$709$	−29.3299	−1.10151	−0.550753	−	0.834668i	$-0.685659\pi$
$709$	−29.3299	−1.10151	−0.550753	+	0.834668i	$0.685659\pi$
$710$	8.64551	0.324460
$711$	0	0
$712$	−4.44985	−0.166765
$713$	15.6965	0.587840
$714$	0	0
$715$	16.1322	0.603311
$716$	−11.2805	−0.421570
$717$	0	0
$718$	14.8797	0.555307
$719$	−23.3369	−0.870320	−0.435160	−	0.900353i	$-0.643308\pi$
$719$	−23.3369	−0.870320	−0.435160	+	0.900353i	$0.643308\pi$
$720$	0	0
$721$	−8.83563	−0.329056
$722$	−56.1876	−2.09108
$723$	0	0
$724$	−4.51927	−0.167957
$725$	11.4217	0.424192
$726$	0	0
$727$	−19.8362	−0.735682	−0.367841	−	0.929889i	$-0.619903\pi$
$727$	−19.8362	−0.735682	−0.367841	+	0.929889i	$0.619903\pi$
$728$	−4.71382	−0.174706
$729$	0	0
$730$	−4.88237	−0.180705
$731$	−36.5696	−1.35258
$732$	0	0
$733$	0.000758868 0	2.80294e−5 0	1.40147e−5	−	1.00000i	$-0.499996\pi$
$733$	0.000758868 0	2.80294e−5 0	1.40147e−5		1.00000i	$0.499996\pi$
$734$	35.7769	1.32055
$735$	0	0
$736$	3.21691	0.118577
$737$	65.0867	2.39750
$738$	0	0
$739$	23.4783	0.863665	0.431832	−	0.901954i	$-0.357867\pi$
$739$	23.4783	0.863665	0.431832	+	0.901954i	$0.357867\pi$
$740$	−6.19123	−0.227594
$741$	0	0
$742$	3.72325	0.136685
$743$	−25.1755	−0.923601	−0.461801	−	0.886984i	$-0.652796\pi$
$743$	−25.1755	−0.923601	−0.461801	+	0.886984i	$0.652796\pi$
$744$	0	0
$745$	−1.33540	−0.0489253
$746$	−33.2864	−1.21870
$747$	0	0
$748$	−29.7205	−1.08669
$749$	33.2211	1.21387
$750$	0	0
$751$	25.2907	0.922870	0.461435	−	0.887174i	$-0.347335\pi$
$751$	25.2907	0.922870	0.461435	+	0.887174i	$0.347335\pi$
$752$	4.49388	0.163875
$753$	0	0
$754$	7.38362	0.268896
$755$	1.38761	0.0505005
$756$	0	0
$757$	−10.1046	−0.367257	−0.183628	−	0.982996i	$-0.558784\pi$
$757$	−10.1046	−0.367257	−0.183628	+	0.982996i	$0.558784\pi$
$758$	2.08326	0.0756672
$759$	0	0
$760$	11.5794	0.420027
$761$	38.6287	1.40029	0.700145	−	0.714001i	$-0.253119\pi$
$761$	38.6287	1.40029	0.700145	+	0.714001i	$0.253119\pi$
$762$	0	0
$763$	−23.7967	−0.861497
$764$	5.82375	0.210696
$765$	0	0
$766$	−18.2511	−0.659440
$767$	−3.03064	−0.109430
$768$	0	0
$769$	−8.70976	−0.314082	−0.157041	−	0.987592i	$-0.550195\pi$
$769$	−8.70976	−0.314082	−0.157041	+	0.987592i	$0.550195\pi$
$770$	−17.5860	−0.633757
$771$	0	0
$772$	−7.80698	−0.280979
$773$	7.97472	0.286831	0.143415	−	0.989663i	$-0.454192\pi$
$773$	7.97472	0.286831	0.143415	+	0.989663i	$0.454192\pi$
$774$	0	0
$775$	15.6956	0.563801
$776$	−8.30166	−0.298012
$777$	0	0
$778$	−11.6927	−0.419204
$779$	−70.2300	−2.51625
$780$	0	0
$781$	−37.6108	−1.34582
$782$	16.4574	0.588516
$783$	0	0
$784$	−1.86137	−0.0664775
$785$	15.1362	0.540235
$786$	0	0
$787$	−51.5425	−1.83729	−0.918646	−	0.395082i	$-0.870716\pi$
$787$	−51.5425	−1.83729	−0.918646	+	0.395082i	$0.870716\pi$
$788$	5.58618	0.198999
$789$	0	0
$790$	5.64441	0.200819
$791$	31.4727	1.11904
$792$	0	0
$793$	6.14818	0.218328
$794$	−0.213966	−0.00759336
$795$	0	0
$796$	−10.7298	−0.380306
$797$	51.1112	1.81045	0.905225	−	0.424933i	$-0.139702\pi$
$797$	51.1112	1.81045	0.905225	+	0.424933i	$0.139702\pi$
$798$	0	0
$799$	22.9903	0.813339
$800$	3.21671	0.113728
$801$	0	0
$802$	10.3209	0.364445
$803$	21.2399	0.749539
$804$	0	0
$805$	9.73808	0.343222
$806$	10.1465	0.357394
$807$	0	0
$808$	1.68201	0.0591729
$809$	7.09891	0.249584	0.124792	−	0.992183i	$-0.460174\pi$
$809$	7.09891	0.249584	0.124792	+	0.992183i	$0.460174\pi$
$810$	0	0
$811$	−41.1182	−1.44386	−0.721928	−	0.691968i	$-0.756744\pi$
$811$	−41.1182	−1.44386	−0.721928	+	0.691968i	$0.756744\pi$
$812$	−8.04902	−0.282465
$813$	0	0
$814$	26.9339	0.944031
$815$	16.5620	0.580140
$816$	0	0
$817$	−61.9827	−2.16850
$818$	−0.211630	−0.00739948
$819$	0	0
$820$	10.8159	0.377706
$821$	−2.66641	−0.0930585	−0.0465293	−	0.998917i	$-0.514816\pi$
$821$	−2.66641	−0.0930585	−0.0465293	+	0.998917i	$0.514816\pi$
$822$	0	0
$823$	29.8142	1.03926	0.519630	−	0.854392i	$-0.326070\pi$
$823$	29.8142	1.03926	0.519630	+	0.854392i	$0.326070\pi$
$824$	3.89775	0.135785
$825$	0	0
$826$	3.30376	0.114953
$827$	47.2228	1.64210	0.821049	−	0.570857i	$-0.193389\pi$
$827$	47.2228	1.64210	0.821049	+	0.570857i	$0.193389\pi$
$828$	0	0
$829$	−27.3167	−0.948747	−0.474373	−	0.880324i	$-0.657325\pi$
$829$	−27.3167	−0.948747	−0.474373	+	0.880324i	$0.657325\pi$
$830$	−0.210853	−0.00731880
$831$	0	0
$832$	2.07946	0.0720922
$833$	−9.52261	−0.329939
$834$	0	0
$835$	9.88915	0.342228
$836$	−50.3739	−1.74222
$837$	0	0
$838$	3.39484	0.117273
$839$	−7.28988	−0.251675	−0.125837	−	0.992051i	$-0.540162\pi$
$839$	−7.28988	−0.251675	−0.125837	+	0.992051i	$0.540162\pi$
$840$	0	0
$841$	−16.3922	−0.565248
$842$	−5.07013	−0.174728
$843$	0	0
$844$	4.18033	0.143893
$845$	11.5857	0.398562
$846$	0	0
$847$	51.5695	1.77195
$848$	−1.64247	−0.0564027
$849$	0	0
$850$	16.4564	0.564450
$851$	−14.9143	−0.511257
$852$	0	0
$853$	26.7076	0.914452	0.457226	−	0.889351i	$-0.348843\pi$
$853$	26.7076	0.914452	0.457226	+	0.889351i	$0.348843\pi$
$854$	−6.70225	−0.229346
$855$	0	0
$856$	−14.6552	−0.500904
$857$	21.2306	0.725224	0.362612	−	0.931940i	$-0.381885\pi$
$857$	21.2306	0.725224	0.362612	+	0.931940i	$0.381885\pi$
$858$	0	0
$859$	15.7136	0.536141	0.268071	−	0.963399i	$-0.413614\pi$
$859$	15.7136	0.536141	0.268071	+	0.963399i	$0.413614\pi$
$860$	9.54572	0.325506
$861$	0	0
$862$	−1.42121	−0.0484066
$863$	55.9282	1.90382	0.951910	−	0.306378i	$-0.0991172\pi$
$863$	55.9282	1.90382	0.951910	+	0.306378i	$0.0991172\pi$
$864$	0	0
$865$	6.17555	0.209975
$866$	8.16167	0.277345
$867$	0	0
$868$	−11.0609	−0.375430
$869$	−24.5550	−0.832972
$870$	0	0
$871$	−23.2975	−0.789405
$872$	10.4977	0.355496
$873$	0	0
$874$	27.8940	0.943530
$875$	24.8733	0.840870
$876$	0	0
$877$	−48.2782	−1.63024	−0.815120	−	0.579292i	$-0.803329\pi$
$877$	−48.2782	−1.63024	−0.815120	+	0.579292i	$0.803329\pi$
$878$	24.2959	0.819948
$879$	0	0
$880$	7.75790	0.261519
$881$	11.0099	0.370934	0.185467	−	0.982651i	$-0.440620\pi$
$881$	11.0099	0.370934	0.185467	+	0.982651i	$0.440620\pi$
$882$	0	0
$883$	19.0571	0.641321	0.320661	−	0.947194i	$-0.396095\pi$
$883$	19.0571	0.641321	0.320661	+	0.947194i	$0.396095\pi$
$884$	10.6383	0.357805
$885$	0	0
$886$	21.8590	0.734368
$887$	−31.8567	−1.06964	−0.534822	−	0.844965i	$-0.679621\pi$
$887$	−31.8567	−1.06964	−0.534822	+	0.844965i	$0.679621\pi$
$888$	0	0
$889$	−42.9010	−1.43885
$890$	5.94233	0.199187
$891$	0	0
$892$	12.2349	0.409656
$893$	38.9668	1.30397
$894$	0	0
$895$	15.0639	0.503531
$896$	−2.26685	−0.0757303
$897$	0	0
$898$	31.3083	1.04477
$899$	17.3255	0.577837
$900$	0	0
$901$	−8.40275	−0.279936
$902$	−47.0525	−1.56668
$903$	0	0
$904$	−13.8839	−0.461771
$905$	6.03503	0.200611
$906$	0	0
$907$	−27.0250	−0.897352	−0.448676	−	0.893695i	$-0.648104\pi$
$907$	−27.0250	−0.897352	−0.448676	+	0.893695i	$0.648104\pi$
$908$	10.8799	0.361062
$909$	0	0
$910$	6.29484	0.208672
$911$	−21.8821	−0.724987	−0.362493	−	0.931986i	$-0.618074\pi$
$911$	−21.8821	−0.724987	−0.362493	+	0.931986i	$0.618074\pi$
$912$	0	0
$913$	0.917277	0.0303574
$914$	22.4776	0.743493
$915$	0	0
$916$	−25.2384	−0.833900
$917$	17.8625	0.589870
$918$	0	0
$919$	30.5127	1.00652	0.503261	−	0.864134i	$-0.332133\pi$
$919$	30.5127	1.00652	0.503261	+	0.864134i	$0.332133\pi$
$920$	−4.29586	−0.141630
$921$	0	0
$922$	−24.6409	−0.811504
$923$	13.4626	0.443127
$924$	0	0
$925$	−14.9134	−0.490350
$926$	23.3886	0.768595
$927$	0	0
$928$	3.55075	0.116559
$929$	29.4298	0.965562	0.482781	−	0.875741i	$-0.339627\pi$
$929$	29.4298	0.965562	0.482781	+	0.875741i	$0.339627\pi$
$930$	0	0
$931$	−16.1401	−0.528970
$932$	−28.3597	−0.928952
$933$	0	0
$934$	30.4777	0.997261
$935$	39.6887	1.29796
$936$	0	0
$937$	47.9101	1.56515	0.782577	−	0.622554i	$-0.213905\pi$
$937$	47.9101	1.56515	0.782577	+	0.622554i	$0.213905\pi$
$938$	25.3970	0.829242
$939$	0	0
$940$	−6.00113	−0.195735
$941$	−7.06453	−0.230297	−0.115149	−	0.993348i	$-0.536734\pi$
$941$	−7.06453	−0.230297	−0.115149	+	0.993348i	$0.536734\pi$
$942$	0	0
$943$	26.0548	0.848463
$944$	−1.45742	−0.0474350
$945$	0	0
$946$	−41.5270	−1.35016
$947$	42.3321	1.37561	0.687804	−	0.725896i	$-0.258575\pi$
$947$	42.3321	1.37561	0.687804	+	0.725896i	$0.258575\pi$
$948$	0	0
$949$	−7.60272	−0.246795
$950$	27.8923	0.904946
$951$	0	0
$952$	−11.5970	−0.375862
$953$	−46.5984	−1.50947	−0.754735	−	0.656029i	$-0.772235\pi$
$953$	−46.5984	−1.50947	−0.754735	+	0.656029i	$0.772235\pi$
$954$	0	0
$955$	−7.77704	−0.251659
$956$	−7.12040	−0.230290
$957$	0	0
$958$	7.57443	0.244719
$959$	−36.9881	−1.19441
$960$	0	0
$961$	−7.19156	−0.231986
$962$	−9.64085	−0.310833
$963$	0	0
$964$	−4.08678	−0.131626
$965$	10.4254	0.335607
$966$	0	0
$967$	36.9542	1.18837	0.594184	−	0.804329i	$-0.297475\pi$
$967$	36.9542	1.18837	0.594184	+	0.804329i	$0.297475\pi$
$968$	−22.7494	−0.731193
$969$	0	0
$970$	11.0860	0.355951
$971$	−13.4887	−0.432872	−0.216436	−	0.976297i	$-0.569443\pi$
$971$	−13.4887	−0.432872	−0.216436	+	0.976297i	$0.569443\pi$
$972$	0	0
$973$	50.8764	1.63102
$974$	−40.2731	−1.29043
$975$	0	0
$976$	2.95663	0.0946394
$977$	−1.91216	−0.0611755	−0.0305878	−	0.999532i	$-0.509738\pi$
$977$	−1.91216	−0.0611755	−0.0305878	+	0.999532i	$0.509738\pi$
$978$	0	0
$979$	−25.8511	−0.826203
$980$	2.48568	0.0794020
$981$	0	0
$982$	15.3506	0.489859
$983$	−53.5309	−1.70737	−0.853686	−	0.520788i	$-0.825638\pi$
$983$	−53.5309	−1.70737	−0.853686	+	0.520788i	$0.825638\pi$
$984$	0	0
$985$	−7.45979	−0.237689
$986$	18.1653	0.578501
$987$	0	0
$988$	18.0311	0.573646
$989$	22.9951	0.731202
$990$	0	0
$991$	−17.4059	−0.552915	−0.276458	−	0.961026i	$-0.589161\pi$
$991$	−17.4059	−0.552915	−0.276458	+	0.961026i	$0.589161\pi$
$992$	4.87939	0.154921
$993$	0	0
$994$	−14.6758	−0.465490
$995$	14.3285	0.454245
$996$	0	0
$997$	59.0850	1.87124	0.935620	−	0.353010i	$-0.114842\pi$
$997$	59.0850	1.87124	0.935620	+	0.353010i	$0.114842\pi$
$998$	−41.3580	−1.30916
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.i.1.5	✓	12
3.2	odd	2		8046.2.a.p.1.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.i.1.5	✓	12	1.1	even	1	trivial
8046.2.a.p.1.8	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.33540 q^{5} +2.26685 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.33540 q^{5} +2.26685 q^{7} -1.00000 q^{8} +1.33540 q^{10} -5.80942 q^{11} +2.07946 q^{13} -2.26685 q^{14} +1.00000 q^{16} +5.11591 q^{17} +8.67108 q^{19} -1.33540 q^{20} +5.80942 q^{22} -3.21691 q^{23} -3.21671 q^{25} -2.07946 q^{26} +2.26685 q^{28} -3.55075 q^{29} -4.87939 q^{31} -1.00000 q^{32} -5.11591 q^{34} -3.02716 q^{35} +4.63624 q^{37} -8.67108 q^{38} +1.33540 q^{40} -8.09935 q^{41} -7.14821 q^{43} -5.80942 q^{44} +3.21691 q^{46} +4.49388 q^{47} -1.86137 q^{49} +3.21671 q^{50} +2.07946 q^{52} -1.64247 q^{53} +7.75790 q^{55} -2.26685 q^{56} +3.55075 q^{58} -1.45742 q^{59} +2.95663 q^{61} +4.87939 q^{62} +1.00000 q^{64} -2.77691 q^{65} -11.2036 q^{67} +5.11591 q^{68} +3.02716 q^{70} +6.47410 q^{71} -3.65611 q^{73} -4.63624 q^{74} +8.67108 q^{76} -13.1691 q^{77} +4.22676 q^{79} -1.33540 q^{80} +8.09935 q^{82} -0.157895 q^{83} -6.83179 q^{85} +7.14821 q^{86} +5.80942 q^{88} +4.44985 q^{89} +4.71382 q^{91} -3.21691 q^{92} -4.49388 q^{94} -11.5794 q^{95} +8.30166 q^{97} +1.86137 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} - 5 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 6 q^{11} + 3 q^{13} - 6 q^{14} + 12 q^{16} - 6 q^{17} + 8 q^{19} - 5 q^{20} + 6 q^{22} - 11 q^{23} + 11 q^{25} - 3 q^{26} + 6 q^{28} - 29 q^{29} + 2 q^{31} - 12 q^{32} + 6 q^{34} - 4 q^{35} + 5 q^{37} - 8 q^{38} + 5 q^{40} - 22 q^{41} + 9 q^{43} - 6 q^{44} + 11 q^{46} - 15 q^{47} + 14 q^{49} - 11 q^{50} + 3 q^{52} - 12 q^{53} + 13 q^{55} - 6 q^{56} + 29 q^{58} - 34 q^{59} - 4 q^{61} - 2 q^{62} + 12 q^{64} - 12 q^{65} + q^{67} - 6 q^{68} + 4 q^{70} - 21 q^{71} - 2 q^{73} - 5 q^{74} + 8 q^{76} - 34 q^{77} + 9 q^{79} - 5 q^{80} + 22 q^{82} - 10 q^{83} + 5 q^{85} - 9 q^{86} + 6 q^{88} + 2 q^{89} + 17 q^{91} - 11 q^{92} + 15 q^{94} - 69 q^{95} - 13 q^{97} - 14 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 - 5 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 6 * q^11 + 3 * q^13 - 6 * q^14 + 12 * q^16 - 6 * q^17 + 8 * q^19 - 5 * q^20 + 6 * q^22 - 11 * q^23 + 11 * q^25 - 3 * q^26 + 6 * q^28 - 29 * q^29 + 2 * q^31 - 12 * q^32 + 6 * q^34 - 4 * q^35 + 5 * q^37 - 8 * q^38 + 5 * q^40 - 22 * q^41 + 9 * q^43 - 6 * q^44 + 11 * q^46 - 15 * q^47 + 14 * q^49 - 11 * q^50 + 3 * q^52 - 12 * q^53 + 13 * q^55 - 6 * q^56 + 29 * q^58 - 34 * q^59 - 4 * q^61 - 2 * q^62 + 12 * q^64 - 12 * q^65 + q^67 - 6 * q^68 + 4 * q^70 - 21 * q^71 - 2 * q^73 - 5 * q^74 + 8 * q^76 - 34 * q^77 + 9 * q^79 - 5 * q^80 + 22 * q^82 - 10 * q^83 + 5 * q^85 - 9 * q^86 + 6 * q^88 + 2 * q^89 + 17 * q^91 - 11 * q^92 + 15 * q^94 - 69 * q^95 - 13 * q^97 - 14 * q^98

Introduction

L-functions

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