LMFDB - Embedded newform 1341.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1341,2,Mod(1,1341)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1341, 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("1341.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1341 = 3^{2} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1341.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:	$10.7079389111$
Analytic rank:	$0$
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} - 3 x^{11} - 12 x^{10} + 38 x^{9} + 46 x^{8} - 162 x^{7} - 59 x^{6} + 280 x^{5} - 14 x^{4} + \cdots - 3 $ x^12 - 3x^11 - 12x^10 + 38x^9 + 46x^8 - 162x^7 - 59x^6 + 280x^5 - 14x^4 - 173x^3 + 49x^2 + 11*x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44716$ of defining polynomial
Character	$\chi$	$=$	1341.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44716 q^{2} +3.98860 q^{4} +3.45366 q^{5} -2.63647 q^{7} -4.86642 q^{8} +O(q^{10})$	\(q-2.44716 q^{2} +3.98860 q^{4} +3.45366 q^{5} -2.63647 q^{7} -4.86642 q^{8} -8.45165 q^{10} +2.70327 q^{11} -4.21903 q^{13} +6.45186 q^{14} +3.93172 q^{16} -0.343580 q^{17} -4.12343 q^{19} +13.7752 q^{20} -6.61535 q^{22} +1.12619 q^{23} +6.92774 q^{25} +10.3246 q^{26} -10.5158 q^{28} +5.09033 q^{29} +9.68696 q^{31} +0.111297 q^{32} +0.840796 q^{34} -9.10545 q^{35} +7.75136 q^{37} +10.0907 q^{38} -16.8069 q^{40} -6.04623 q^{41} +9.77697 q^{43} +10.7823 q^{44} -2.75598 q^{46} +6.97621 q^{47} -0.0490474 q^{49} -16.9533 q^{50} -16.8280 q^{52} -3.98024 q^{53} +9.33618 q^{55} +12.8301 q^{56} -12.4569 q^{58} +12.6004 q^{59} -2.49406 q^{61} -23.7055 q^{62} -8.13579 q^{64} -14.5711 q^{65} -14.1227 q^{67} -1.37040 q^{68} +22.2825 q^{70} +9.44746 q^{71} -9.50500 q^{73} -18.9688 q^{74} -16.4467 q^{76} -7.12709 q^{77} +16.3962 q^{79} +13.5788 q^{80} +14.7961 q^{82} +0.120163 q^{83} -1.18661 q^{85} -23.9258 q^{86} -13.1553 q^{88} +3.06897 q^{89} +11.1233 q^{91} +4.49193 q^{92} -17.0719 q^{94} -14.2409 q^{95} -8.77499 q^{97} +0.120027 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) $ 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8	$ 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8} + 22 q^{11} - 2 q^{13} + 12 q^{14} + 11 q^{16} + 8 q^{17} - 6 q^{19} + 24 q^{20} + 4 q^{22} + 12 q^{23} + 8 q^{25} + 26 q^{26} - 20 q^{28} + 16 q^{29} - 2 q^{31} + 21 q^{32} - 6 q^{34} + 32 q^{35} - 4 q^{37} + 15 q^{38} + 6 q^{40} + 24 q^{41} + 12 q^{43} + 37 q^{44} - 22 q^{46} + 26 q^{47} + 24 q^{49} - 5 q^{50} - 10 q^{52} + 20 q^{53} + 6 q^{55} - 13 q^{56} + 10 q^{58} + 72 q^{59} - 8 q^{61} - 2 q^{62} + q^{64} - 4 q^{65} - 14 q^{67} - 14 q^{68} + 18 q^{70} + 38 q^{71} - 27 q^{74} - 2 q^{76} + 6 q^{77} + 10 q^{79} + 56 q^{80} + 6 q^{82} + 42 q^{83} - 32 q^{85} - 14 q^{86} + 22 q^{88} + 44 q^{89} - 50 q^{92} + 2 q^{94} + 4 q^{95} - 22 q^{97} + 20 q^{98}+O(q^{100}) $ 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8 + 22 * q^11 - 2 * q^13 + 12 * q^14 + 11 * q^16 + 8 * q^17 - 6 * q^19 + 24 * q^20 + 4 * q^22 + 12 * q^23 + 8 * q^25 + 26 * q^26 - 20 * q^28 + 16 * q^29 - 2 * q^31 + 21 * q^32 - 6 * q^34 + 32 * q^35 - 4 * q^37 + 15 * q^38 + 6 * q^40 + 24 * q^41 + 12 * q^43 + 37 * q^44 - 22 * q^46 + 26 * q^47 + 24 * q^49 - 5 * q^50 - 10 * q^52 + 20 * q^53 + 6 * q^55 - 13 * q^56 + 10 * q^58 + 72 * q^59 - 8 * q^61 - 2 * q^62 + q^64 - 4 * q^65 - 14 * q^67 - 14 * q^68 + 18 * q^70 + 38 * q^71 - 27 * q^74 - 2 * q^76 + 6 * q^77 + 10 * q^79 + 56 * q^80 + 6 * q^82 + 42 * q^83 - 32 * q^85 - 14 * q^86 + 22 * q^88 + 44 * q^89 - 50 * q^92 + 2 * q^94 + 4 * q^95 - 22 * q^97 + 20 * 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 $

$2$	−2.44716	−1.73040	−0.865202	−	0.501423i	$-0.832810\pi$
$2$	−2.44716	−1.73040	−0.865202	+	0.501423i	$0.832810\pi$
$3$	0	0
$3$	0	0
$4$	3.98860	1.99430
$5$	3.45366	1.54452	0.772261	−	0.635305i	$-0.219126\pi$
$5$	3.45366	1.54452	0.772261	+	0.635305i	$0.219126\pi$
$6$	0	0
$7$	−2.63647	−0.996490	−0.498245	−	0.867036i	$-0.666022\pi$
$7$	−2.63647	−0.996490	−0.498245	+	0.867036i	$0.666022\pi$
$8$	−4.86642	−1.72054
$9$	0	0
$10$	−8.45165	−2.67265
$11$	2.70327	0.815068	0.407534	−	0.913190i	$-0.366389\pi$
$11$	2.70327	0.815068	0.407534	+	0.913190i	$0.366389\pi$
$12$	0	0
$13$	−4.21903	−1.17015	−0.585074	−	0.810980i	$-0.698934\pi$
$13$	−4.21903	−1.17015	−0.585074	+	0.810980i	$0.698934\pi$
$14$	6.45186	1.72433
$15$	0	0
$16$	3.93172	0.982929
$17$	−0.343580	−0.0833304	−0.0416652	−	0.999132i	$-0.513266\pi$
$17$	−0.343580	−0.0833304	−0.0416652	+	0.999132i	$0.513266\pi$
$18$	0	0
$19$	−4.12343	−0.945979	−0.472990	−	0.881068i	$-0.656825\pi$
$19$	−4.12343	−0.945979	−0.472990	+	0.881068i	$0.656825\pi$
$20$	13.7752	3.08024
$21$	0	0
$22$	−6.61535	−1.41040
$23$	1.12619	0.234828	0.117414	−	0.993083i	$-0.462540\pi$
$23$	1.12619	0.234828	0.117414	+	0.993083i	$0.462540\pi$
$24$	0	0
$25$	6.92774	1.38555
$26$	10.3246	2.02483
$27$	0	0
$28$	−10.5158	−1.98730
$29$	5.09033	0.945251	0.472626	−	0.881263i	$-0.343306\pi$
$29$	5.09033	0.945251	0.472626	+	0.881263i	$0.343306\pi$
$30$	0	0
$31$	9.68696	1.73983	0.869915	−	0.493202i	$-0.164174\pi$
$31$	9.68696	1.73983	0.869915	+	0.493202i	$0.164174\pi$
$32$	0.111297	0.0196747
$33$	0	0
$34$	0.840796	0.144195
$35$	−9.10545	−1.53910
$36$	0	0
$37$	7.75136	1.27431	0.637157	−	0.770734i	$-0.280110\pi$
$37$	7.75136	1.27431	0.637157	+	0.770734i	$0.280110\pi$
$38$	10.0907	1.63693
$39$	0	0
$40$	−16.8069	−2.65741
$41$	−6.04623	−0.944262	−0.472131	−	0.881528i	$-0.656515\pi$
$41$	−6.04623	−0.944262	−0.472131	+	0.881528i	$0.656515\pi$
$42$	0	0
$43$	9.77697	1.49097	0.745487	−	0.666520i	$-0.232217\pi$
$43$	9.77697	1.49097	0.745487	+	0.666520i	$0.232217\pi$
$44$	10.7823	1.62549
$45$	0	0
$46$	−2.75598	−0.406347
$47$	6.97621	1.01758	0.508792	−	0.860889i	$-0.330092\pi$
$47$	6.97621	1.01758	0.508792	+	0.860889i	$0.330092\pi$
$48$	0	0
$49$	−0.0490474	−0.00700677
$50$	−16.9533	−2.39756
$51$	0	0
$52$	−16.8280	−2.33362
$53$	−3.98024	−0.546729	−0.273364	−	0.961911i	$-0.588136\pi$
$53$	−3.98024	−0.546729	−0.273364	+	0.961911i	$0.588136\pi$
$54$	0	0
$55$	9.33618	1.25889
$56$	12.8301	1.71450
$57$	0	0
$58$	−12.4569	−1.63567
$59$	12.6004	1.64043	0.820215	−	0.572055i	$-0.193854\pi$
$59$	12.6004	1.64043	0.820215	+	0.572055i	$0.193854\pi$
$60$	0	0
$61$	−2.49406	−0.319332	−0.159666	−	0.987171i	$-0.551042\pi$
$61$	−2.49406	−0.319332	−0.159666	+	0.987171i	$0.551042\pi$
$62$	−23.7055	−3.01061
$63$	0	0
$64$	−8.13579	−1.01697
$65$	−14.5711	−1.80732
$66$	0	0
$67$	−14.1227	−1.72536	−0.862679	−	0.505751i	$-0.831215\pi$
$67$	−14.1227	−1.72536	−0.862679	+	0.505751i	$0.831215\pi$
$68$	−1.37040	−0.166186
$69$	0	0
$70$	22.2825	2.66327
$71$	9.44746	1.12121	0.560604	−	0.828084i	$-0.310569\pi$
$71$	9.44746	1.12121	0.560604	+	0.828084i	$0.310569\pi$
$72$	0	0
$73$	−9.50500	−1.11248	−0.556238	−	0.831023i	$-0.687756\pi$
$73$	−9.50500	−1.11248	−0.556238	+	0.831023i	$0.687756\pi$
$74$	−18.9688	−2.20508
$75$	0	0
$76$	−16.4467	−1.88657
$77$	−7.12709	−0.812207
$78$	0	0
$79$	16.3962	1.84472	0.922361	−	0.386330i	$-0.126257\pi$
$79$	16.3962	1.84472	0.922361	+	0.386330i	$0.126257\pi$
$80$	13.5788	1.51815
$81$	0	0
$82$	14.7961	1.63396
$83$	0.120163	0.0131896	0.00659481	−	0.999978i	$-0.497901\pi$
$83$	0.120163	0.0131896	0.00659481	+	0.999978i	$0.497901\pi$
$84$	0	0
$85$	−1.18661	−0.128706
$86$	−23.9258	−2.57999
$87$	0	0
$88$	−13.1553	−1.40236
$89$	3.06897	0.325310	0.162655	−	0.986683i	$-0.447994\pi$
$89$	3.06897	0.325310	0.162655	+	0.986683i	$0.447994\pi$
$90$	0	0
$91$	11.1233	1.16604
$92$	4.49193	0.468316
$93$	0	0
$94$	−17.0719	−1.76083
$95$	−14.2409	−1.46109
$96$	0	0
$97$	−8.77499	−0.890966	−0.445483	−	0.895290i	$-0.646968\pi$
$97$	−8.77499	−0.890966	−0.445483	+	0.895290i	$0.646968\pi$
$98$	0.120027	0.0121245
$99$	0	0
$100$	27.6320	2.76320
$101$	12.6361	1.25734	0.628668	−	0.777674i	$-0.283600\pi$
$101$	12.6361	1.25734	0.628668	+	0.777674i	$0.283600\pi$
$102$	0	0
$103$	3.65534	0.360172	0.180086	−	0.983651i	$-0.442362\pi$
$103$	3.65534	0.360172	0.180086	+	0.983651i	$0.442362\pi$
$104$	20.5315	2.01328
$105$	0	0
$106$	9.74030	0.946062
$107$	4.52947	0.437880	0.218940	−	0.975738i	$-0.429740\pi$
$107$	4.52947	0.437880	0.218940	+	0.975738i	$0.429740\pi$
$108$	0	0
$109$	8.86999	0.849591	0.424796	−	0.905289i	$-0.360346\pi$
$109$	8.86999	0.849591	0.424796	+	0.905289i	$0.360346\pi$
$110$	−22.8471	−2.17839
$111$	0	0
$112$	−10.3658	−0.979479
$113$	−2.09421	−0.197007	−0.0985034	−	0.995137i	$-0.531406\pi$
$113$	−2.09421	−0.197007	−0.0985034	+	0.995137i	$0.531406\pi$
$114$	0	0
$115$	3.88949	0.362696
$116$	20.3033	1.88511
$117$	0	0
$118$	−30.8352	−2.83861
$119$	0.905837	0.0830379
$120$	0	0
$121$	−3.69231	−0.335665
$122$	6.10337	0.552573
$123$	0	0
$124$	38.6374	3.46974
$125$	6.65775	0.595487
$126$	0	0
$127$	7.90365	0.701336	0.350668	−	0.936500i	$-0.385955\pi$
$127$	7.90365	0.701336	0.350668	+	0.936500i	$0.385955\pi$
$128$	19.6870	1.74010
$129$	0	0
$130$	35.6577	3.12739
$131$	11.0347	0.964109	0.482054	−	0.876141i	$-0.339891\pi$
$131$	11.0347	0.964109	0.482054	+	0.876141i	$0.339891\pi$
$132$	0	0
$133$	10.8713	0.942660
$134$	34.5604	2.98557
$135$	0	0
$136$	1.67200	0.143373
$137$	2.40113	0.205143	0.102571	−	0.994726i	$-0.467293\pi$
$137$	2.40113	0.205143	0.102571	+	0.994726i	$0.467293\pi$
$138$	0	0
$139$	−9.26412	−0.785772	−0.392886	−	0.919587i	$-0.628523\pi$
$139$	−9.26412	−0.785772	−0.392886	+	0.919587i	$0.628523\pi$
$140$	−36.3180	−3.06943
$141$	0	0
$142$	−23.1195	−1.94014
$143$	−11.4052	−0.953749
$144$	0	0
$145$	17.5803	1.45996
$146$	23.2603	1.92503
$147$	0	0
$148$	30.9170	2.54136
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	13.6420	1.11017	0.555084	−	0.831794i	$-0.312686\pi$
$151$	13.6420	1.11017	0.555084	+	0.831794i	$0.312686\pi$
$152$	20.0663	1.62759
$153$	0	0
$154$	17.4411	1.40545
$155$	33.4554	2.68720
$156$	0	0
$157$	13.0046	1.03788	0.518939	−	0.854811i	$-0.326327\pi$
$157$	13.0046	1.03788	0.518939	+	0.854811i	$0.326327\pi$
$158$	−40.1242	−3.19211
$159$	0	0
$160$	0.384381	0.0303880
$161$	−2.96917	−0.234003
$162$	0	0
$163$	3.95511	0.309788	0.154894	−	0.987931i	$-0.450496\pi$
$163$	3.95511	0.309788	0.154894	+	0.987931i	$0.450496\pi$
$164$	−24.1160	−1.88314
$165$	0	0
$166$	−0.294058	−0.0228234
$167$	−6.02763	−0.466432	−0.233216	−	0.972425i	$-0.574925\pi$
$167$	−6.02763	−0.466432	−0.233216	+	0.972425i	$0.574925\pi$
$168$	0	0
$169$	4.80018	0.369245
$170$	2.90382	0.222713
$171$	0	0
$172$	38.9964	2.97345
$173$	−0.700977	−0.0532943	−0.0266472	−	0.999645i	$-0.508483\pi$
$173$	−0.700977	−0.0532943	−0.0266472	+	0.999645i	$0.508483\pi$
$174$	0	0
$175$	−18.2647	−1.38069
$176$	10.6285	0.801153
$177$	0	0
$178$	−7.51026	−0.562918
$179$	4.54171	0.339464	0.169732	−	0.985490i	$-0.445710\pi$
$179$	4.54171	0.339464	0.169732	+	0.985490i	$0.445710\pi$
$180$	0	0
$181$	11.5090	0.855454	0.427727	−	0.903908i	$-0.359314\pi$
$181$	11.5090	0.855454	0.427727	+	0.903908i	$0.359314\pi$
$182$	−27.2206	−2.01772
$183$	0	0
$184$	−5.48053	−0.404030
$185$	26.7705	1.96821
$186$	0	0
$187$	−0.928791	−0.0679199
$188$	27.8253	2.02937
$189$	0	0
$190$	34.8498	2.52827
$191$	−15.6703	−1.13386	−0.566930	−	0.823766i	$-0.691869\pi$
$191$	−15.6703	−1.13386	−0.566930	+	0.823766i	$0.691869\pi$
$192$	0	0
$193$	−5.09005	−0.366390	−0.183195	−	0.983077i	$-0.558644\pi$
$193$	−5.09005	−0.366390	−0.183195	+	0.983077i	$0.558644\pi$
$194$	21.4738	1.54173
$195$	0	0
$196$	−0.195630	−0.0139736
$197$	−18.0835	−1.28839	−0.644197	−	0.764860i	$-0.722808\pi$
$197$	−18.0835	−1.28839	−0.644197	+	0.764860i	$0.722808\pi$
$198$	0	0
$199$	1.71487	0.121564	0.0607821	−	0.998151i	$-0.480641\pi$
$199$	1.71487	0.121564	0.0607821	+	0.998151i	$0.480641\pi$
$200$	−33.7133	−2.38389
$201$	0	0
$202$	−30.9225	−2.17570
$203$	−13.4205	−0.941934
$204$	0	0
$205$	−20.8816	−1.45843
$206$	−8.94522	−0.623243
$207$	0	0
$208$	−16.5880	−1.15017
$209$	−11.1468	−0.771037
$210$	0	0
$211$	−22.7473	−1.56599	−0.782995	−	0.622027i	$-0.786309\pi$
$211$	−22.7473	−1.56599	−0.782995	+	0.622027i	$0.786309\pi$
$212$	−15.8756	−1.09034
$213$	0	0
$214$	−11.0843	−0.757710
$215$	33.7663	2.30284
$216$	0	0
$217$	−25.5393	−1.73372
$218$	−21.7063	−1.47014
$219$	0	0
$220$	37.2382	2.51060
$221$	1.44957	0.0975088
$222$	0	0
$223$	−17.2418	−1.15460	−0.577300	−	0.816532i	$-0.695894\pi$
$223$	−17.2418	−1.15460	−0.577300	+	0.816532i	$0.695894\pi$
$224$	−0.293430	−0.0196056
$225$	0	0
$226$	5.12487	0.340902
$227$	28.4148	1.88595	0.942977	−	0.332857i	$-0.108013\pi$
$227$	28.4148	1.88595	0.942977	+	0.332857i	$0.108013\pi$
$228$	0	0
$229$	2.50669	0.165647	0.0828233	−	0.996564i	$-0.473606\pi$
$229$	2.50669	0.165647	0.0828233	+	0.996564i	$0.473606\pi$
$230$	−9.51820	−0.627611
$231$	0	0
$232$	−24.7717	−1.62634
$233$	−19.1832	−1.25673	−0.628365	−	0.777918i	$-0.716276\pi$
$233$	−19.1832	−1.25673	−0.628365	+	0.777918i	$0.716276\pi$
$234$	0	0
$235$	24.0934	1.57168
$236$	50.2579	3.27151
$237$	0	0
$238$	−2.21673	−0.143689
$239$	−7.47163	−0.483299	−0.241650	−	0.970364i	$-0.577688\pi$
$239$	−7.47163	−0.483299	−0.241650	+	0.970364i	$0.577688\pi$
$240$	0	0
$241$	15.6982	1.01121	0.505606	−	0.862764i	$-0.331269\pi$
$241$	15.6982	1.01121	0.505606	+	0.862764i	$0.331269\pi$
$242$	9.03569	0.580836
$243$	0	0
$244$	−9.94780	−0.636843
$245$	−0.169393	−0.0108221
$246$	0	0
$247$	17.3969	1.10694
$248$	−47.1408	−2.99344
$249$	0	0
$250$	−16.2926	−1.03043
$251$	−3.02040	−0.190646	−0.0953229	−	0.995446i	$-0.530388\pi$
$251$	−3.02040	−0.190646	−0.0953229	+	0.995446i	$0.530388\pi$
$252$	0	0
$253$	3.04441	0.191400
$254$	−19.3415	−1.21359
$255$	0	0
$256$	−31.9057	−1.99410
$257$	29.1677	1.81943	0.909715	−	0.415234i	$-0.136300\pi$
$257$	29.1677	1.81943	0.909715	+	0.415234i	$0.136300\pi$
$258$	0	0
$259$	−20.4362	−1.26984
$260$	−58.1181	−3.60433
$261$	0	0
$262$	−27.0038	−1.66830
$263$	−0.874373	−0.0539161	−0.0269581	−	0.999637i	$-0.508582\pi$
$263$	−0.874373	−0.0539161	−0.0269581	+	0.999637i	$0.508582\pi$
$264$	0	0
$265$	−13.7464	−0.844434
$266$	−26.6038	−1.63118
$267$	0	0
$268$	−56.3296	−3.44088
$269$	−24.3589	−1.48519	−0.742595	−	0.669741i	$-0.766405\pi$
$269$	−24.3589	−1.48519	−0.742595	+	0.669741i	$0.766405\pi$
$270$	0	0
$271$	−12.5113	−0.760004	−0.380002	−	0.924986i	$-0.624077\pi$
$271$	−12.5113	−0.760004	−0.380002	+	0.924986i	$0.624077\pi$
$272$	−1.35086	−0.0819078
$273$	0	0
$274$	−5.87596	−0.354980
$275$	18.7276	1.12932
$276$	0	0
$277$	−30.8460	−1.85336	−0.926678	−	0.375856i	$-0.877349\pi$
$277$	−30.8460	−1.85336	−0.926678	+	0.375856i	$0.877349\pi$
$278$	22.6708	1.35970
$279$	0	0
$280$	44.3109	2.64808
$281$	31.0781	1.85396	0.926982	−	0.375106i	$-0.122394\pi$
$281$	31.0781	1.85396	0.926982	+	0.375106i	$0.122394\pi$
$282$	0	0
$283$	−5.04101	−0.299657	−0.149828	−	0.988712i	$-0.547872\pi$
$283$	−5.04101	−0.299657	−0.149828	+	0.988712i	$0.547872\pi$
$284$	37.6821	2.23602
$285$	0	0
$286$	27.9103	1.65037
$287$	15.9407	0.940948
$288$	0	0
$289$	−16.8820	−0.993056
$290$	−43.0217	−2.52632
$291$	0	0
$292$	−37.9116	−2.21861
$293$	28.2854	1.65245	0.826226	−	0.563338i	$-0.190483\pi$
$293$	28.2854	1.65245	0.826226	+	0.563338i	$0.190483\pi$
$294$	0	0
$295$	43.5174	2.53368
$296$	−37.7213	−2.19251
$297$	0	0
$298$	−2.44716	−0.141760
$299$	−4.75144	−0.274783
$300$	0	0
$301$	−25.7766	−1.48574
$302$	−33.3841	−1.92104
$303$	0	0
$304$	−16.2121	−0.929830
$305$	−8.61362	−0.493215
$306$	0	0
$307$	8.97849	0.512430	0.256215	−	0.966620i	$-0.417525\pi$
$307$	8.97849	0.512430	0.256215	+	0.966620i	$0.417525\pi$
$308$	−28.4271	−1.61978
$309$	0	0
$310$	−81.8708	−4.64995
$311$	28.4658	1.61415	0.807073	−	0.590452i	$-0.201050\pi$
$311$	28.4658	1.61415	0.807073	+	0.590452i	$0.201050\pi$
$312$	0	0
$313$	29.8672	1.68819	0.844096	−	0.536193i	$-0.180138\pi$
$313$	29.8672	1.68819	0.844096	+	0.536193i	$0.180138\pi$
$314$	−31.8243	−1.79595
$315$	0	0
$316$	65.3980	3.67893
$317$	−12.4541	−0.699494	−0.349747	−	0.936844i	$-0.613732\pi$
$317$	−12.4541	−0.699494	−0.349747	+	0.936844i	$0.613732\pi$
$318$	0	0
$319$	13.7606	0.770444
$320$	−28.0982	−1.57074
$321$	0	0
$322$	7.26604	0.404921
$323$	1.41673	0.0788288
$324$	0	0
$325$	−29.2283	−1.62130
$326$	−9.67878	−0.536058
$327$	0	0
$328$	29.4235	1.62464
$329$	−18.3925	−1.01401
$330$	0	0
$331$	22.0252	1.21062	0.605308	−	0.795991i	$-0.293050\pi$
$331$	22.0252	1.21062	0.605308	+	0.795991i	$0.293050\pi$
$332$	0.479282	0.0263040
$333$	0	0
$334$	14.7506	0.807116
$335$	−48.7748	−2.66485
$336$	0	0
$337$	11.8884	0.647601	0.323800	−	0.946125i	$-0.395039\pi$
$337$	11.8884	0.647601	0.323800	+	0.946125i	$0.395039\pi$
$338$	−11.7468	−0.638943
$339$	0	0
$340$	−4.73290	−0.256677
$341$	26.1865	1.41808
$342$	0	0
$343$	18.5846	1.00347
$344$	−47.5788	−2.56528
$345$	0	0
$346$	1.71540	0.0922207
$347$	14.2435	0.764631	0.382315	−	0.924032i	$-0.375127\pi$
$347$	14.2435	0.764631	0.382315	+	0.924032i	$0.375127\pi$
$348$	0	0
$349$	−24.1912	−1.29493	−0.647463	−	0.762097i	$-0.724170\pi$
$349$	−24.1912	−1.29493	−0.647463	+	0.762097i	$0.724170\pi$
$350$	44.6968	2.38914
$351$	0	0
$352$	0.300865	0.0160362
$353$	−0.621749	−0.0330923	−0.0165462	−	0.999863i	$-0.505267\pi$
$353$	−0.621749	−0.0330923	−0.0165462	+	0.999863i	$0.505267\pi$
$354$	0	0
$355$	32.6283	1.73173
$356$	12.2409	0.648765
$357$	0	0
$358$	−11.1143	−0.587409
$359$	−11.6482	−0.614769	−0.307385	−	0.951585i	$-0.599454\pi$
$359$	−11.6482	−0.614769	−0.307385	+	0.951585i	$0.599454\pi$
$360$	0	0
$361$	−1.99733	−0.105123
$362$	−28.1643	−1.48028
$363$	0	0
$364$	44.3664	2.32543
$365$	−32.8270	−1.71824
$366$	0	0
$367$	−1.40583	−0.0733839	−0.0366919	−	0.999327i	$-0.511682\pi$
$367$	−1.40583	−0.0733839	−0.0366919	+	0.999327i	$0.511682\pi$
$368$	4.42787	0.230819
$369$	0	0
$370$	−65.5118	−3.40579
$371$	10.4938	0.544810
$372$	0	0
$373$	−13.7792	−0.713458	−0.356729	−	0.934208i	$-0.616108\pi$
$373$	−13.7792	−0.713458	−0.356729	+	0.934208i	$0.616108\pi$
$374$	2.27290	0.117529
$375$	0	0
$376$	−33.9491	−1.75079
$377$	−21.4762	−1.10608
$378$	0	0
$379$	−14.5274	−0.746221	−0.373110	−	0.927787i	$-0.621709\pi$
$379$	−14.5274	−0.746221	−0.373110	+	0.927787i	$0.621709\pi$
$380$	−56.8012	−2.91384
$381$	0	0
$382$	38.3476	1.96204
$383$	−21.1075	−1.07854	−0.539271	−	0.842132i	$-0.681300\pi$
$383$	−21.1075	−1.07854	−0.539271	+	0.842132i	$0.681300\pi$
$384$	0	0
$385$	−24.6145	−1.25447
$386$	12.4562	0.634002
$387$	0	0
$388$	−34.9999	−1.77685
$389$	−21.0460	−1.06707	−0.533536	−	0.845777i	$-0.679137\pi$
$389$	−21.0460	−1.06707	−0.533536	+	0.845777i	$0.679137\pi$
$390$	0	0
$391$	−0.386938	−0.0195683
$392$	0.238685	0.0120554
$393$	0	0
$394$	44.2532	2.22944
$395$	56.6270	2.84921
$396$	0	0
$397$	−9.82246	−0.492975	−0.246487	−	0.969146i	$-0.579276\pi$
$397$	−9.82246	−0.492975	−0.246487	+	0.969146i	$0.579276\pi$
$398$	−4.19657	−0.210355
$399$	0	0
$400$	27.2379	1.36189
$401$	38.3608	1.91565	0.957824	−	0.287355i	$-0.0927758\pi$
$401$	38.3608	1.91565	0.957824	+	0.287355i	$0.0927758\pi$
$402$	0	0
$403$	−40.8695	−2.03586
$404$	50.4002	2.50750
$405$	0	0
$406$	32.8421	1.62993
$407$	20.9540	1.03865
$408$	0	0
$409$	−3.84069	−0.189910	−0.0949548	−	0.995482i	$-0.530271\pi$
$409$	−3.84069	−0.189910	−0.0949548	+	0.995482i	$0.530271\pi$
$410$	51.1006	2.52368
$411$	0	0
$412$	14.5797	0.718290
$413$	−33.2205	−1.63467
$414$	0	0
$415$	0.415002	0.0203716
$416$	−0.469564	−0.0230223
$417$	0	0
$418$	27.2779	1.33421
$419$	6.23269	0.304487	0.152243	−	0.988343i	$-0.451350\pi$
$419$	6.23269	0.304487	0.152243	+	0.988343i	$0.451350\pi$
$420$	0	0
$421$	−32.5349	−1.58566	−0.792828	−	0.609445i	$-0.791392\pi$
$421$	−32.5349	−1.58566	−0.792828	+	0.609445i	$0.791392\pi$
$422$	55.6664	2.70980
$423$	0	0
$424$	19.3695	0.940668
$425$	−2.38023	−0.115458
$426$	0	0
$427$	6.57550	0.318211
$428$	18.0662	0.873264
$429$	0	0
$430$	−82.6315	−3.98485
$431$	−29.0409	−1.39885	−0.699426	−	0.714705i	$-0.746561\pi$
$431$	−29.0409	−1.39885	−0.699426	+	0.714705i	$0.746561\pi$
$432$	0	0
$433$	23.8576	1.14652	0.573262	−	0.819372i	$-0.305678\pi$
$433$	23.8576	1.14652	0.573262	+	0.819372i	$0.305678\pi$
$434$	62.4989	3.00004
$435$	0	0
$436$	35.3788	1.69434
$437$	−4.64378	−0.222142
$438$	0	0
$439$	−6.45717	−0.308184	−0.154092	−	0.988057i	$-0.549245\pi$
$439$	−6.45717	−0.308184	−0.154092	+	0.988057i	$0.549245\pi$
$440$	−45.4337	−2.16597
$441$	0	0
$442$	−3.54734	−0.168730
$443$	−35.3793	−1.68092	−0.840460	−	0.541874i	$-0.817715\pi$
$443$	−35.3793	−1.68092	−0.840460	+	0.541874i	$0.817715\pi$
$444$	0	0
$445$	10.5992	0.502449
$446$	42.1936	1.99792
$447$	0	0
$448$	21.4497	1.01340
$449$	−36.3318	−1.71460	−0.857302	−	0.514814i	$-0.827861\pi$
$449$	−36.3318	−1.71460	−0.857302	+	0.514814i	$0.827861\pi$
$450$	0	0
$451$	−16.3446	−0.769638
$452$	−8.35297	−0.392891
$453$	0	0
$454$	−69.5355	−3.26346
$455$	38.4161	1.80098
$456$	0	0
$457$	16.6430	0.778527	0.389263	−	0.921126i	$-0.372730\pi$
$457$	16.6430	0.778527	0.389263	+	0.921126i	$0.372730\pi$
$458$	−6.13427	−0.286636
$459$	0	0
$460$	15.5136	0.723325
$461$	19.4095	0.903990	0.451995	−	0.892021i	$-0.350713\pi$
$461$	19.4095	0.903990	0.451995	+	0.892021i	$0.350713\pi$
$462$	0	0
$463$	−15.6425	−0.726968	−0.363484	−	0.931600i	$-0.618413\pi$
$463$	−15.6425	−0.726968	−0.363484	+	0.931600i	$0.618413\pi$
$464$	20.0137	0.929115
$465$	0	0
$466$	46.9443	2.17465
$467$	20.7567	0.960505	0.480253	−	0.877130i	$-0.340545\pi$
$467$	20.7567	0.960505	0.480253	+	0.877130i	$0.340545\pi$
$468$	0	0
$469$	37.2339	1.71930
$470$	−58.9605	−2.71964
$471$	0	0
$472$	−61.3187	−2.82242
$473$	26.4298	1.21524
$474$	0	0
$475$	−28.5660	−1.31070
$476$	3.61302	0.165602
$477$	0	0
$478$	18.2843	0.836303
$479$	38.3604	1.75273	0.876366	−	0.481645i	$-0.159961\pi$
$479$	38.3604	1.75273	0.876366	+	0.481645i	$0.159961\pi$
$480$	0	0
$481$	−32.7032	−1.49114
$482$	−38.4161	−1.74981
$483$	0	0
$484$	−14.7272	−0.669416
$485$	−30.3058	−1.37612
$486$	0	0
$487$	−2.36747	−0.107280	−0.0536402	−	0.998560i	$-0.517082\pi$
$487$	−2.36747	−0.107280	−0.0536402	+	0.998560i	$0.517082\pi$
$488$	12.1371	0.549422
$489$	0	0
$490$	0.414531	0.0187266
$491$	−11.9257	−0.538199	−0.269100	−	0.963112i	$-0.586726\pi$
$491$	−11.9257	−0.538199	−0.269100	+	0.963112i	$0.586726\pi$
$492$	0	0
$493$	−1.74894	−0.0787681
$494$	−42.5729	−1.91545
$495$	0	0
$496$	38.0864	1.71013
$497$	−24.9079	−1.11727
$498$	0	0
$499$	−30.0105	−1.34346	−0.671728	−	0.740798i	$-0.734448\pi$
$499$	−30.0105	−1.34346	−0.671728	+	0.740798i	$0.734448\pi$
$500$	26.5551	1.18758
$501$	0	0
$502$	7.39140	0.329894
$503$	2.02796	0.0904224	0.0452112	−	0.998977i	$-0.485604\pi$
$503$	2.02796	0.0904224	0.0452112	+	0.998977i	$0.485604\pi$
$504$	0	0
$505$	43.6406	1.94198
$506$	−7.45016	−0.331200
$507$	0	0
$508$	31.5245	1.39867
$509$	3.47196	0.153892	0.0769460	−	0.997035i	$-0.475483\pi$
$509$	3.47196	0.153892	0.0769460	+	0.997035i	$0.475483\pi$
$510$	0	0
$511$	25.0596	1.10857
$512$	38.7043	1.71051
$513$	0	0
$514$	−71.3780	−3.14835
$515$	12.6243	0.556293
$516$	0	0
$517$	18.8586	0.829400
$518$	50.0106	2.19734
$519$	0	0
$520$	70.9089	3.10956
$521$	−39.8532	−1.74600	−0.873000	−	0.487720i	$-0.837829\pi$
$521$	−39.8532	−1.74600	−0.873000	+	0.487720i	$0.837829\pi$
$522$	0	0
$523$	15.8490	0.693026	0.346513	−	0.938045i	$-0.387366\pi$
$523$	15.8490	0.693026	0.346513	+	0.938045i	$0.387366\pi$
$524$	44.0131	1.92272
$525$	0	0
$526$	2.13973	0.0932967
$527$	−3.32825	−0.144981
$528$	0	0
$529$	−21.7317	−0.944856
$530$	33.6396	1.46121
$531$	0	0
$532$	43.3612	1.87994
$533$	25.5092	1.10493
$534$	0	0
$535$	15.6432	0.676316
$536$	68.7268	2.96855
$537$	0	0
$538$	59.6102	2.56998
$539$	−0.132588	−0.00571099
$540$	0	0
$541$	43.8774	1.88643	0.943217	−	0.332176i	$-0.107783\pi$
$541$	43.8774	1.88643	0.943217	+	0.332176i	$0.107783\pi$
$542$	30.6170	1.31511
$543$	0	0
$544$	−0.0382393	−0.00163950
$545$	30.6339	1.31221
$546$	0	0
$547$	3.40326	0.145513	0.0727565	−	0.997350i	$-0.476820\pi$
$547$	3.40326	0.145513	0.0727565	+	0.997350i	$0.476820\pi$
$548$	9.57715	0.409116
$549$	0	0
$550$	−45.8294	−1.95417
$551$	−20.9896	−0.894188
$552$	0	0
$553$	−43.2281	−1.83825
$554$	75.4851	3.20706
$555$	0	0
$556$	−36.9508	−1.56706
$557$	−31.8435	−1.34925	−0.674626	−	0.738160i	$-0.735695\pi$
$557$	−31.8435	−1.34925	−0.674626	+	0.738160i	$0.735695\pi$
$558$	0	0
$559$	−41.2493	−1.74466
$560$	−35.8000	−1.51283
$561$	0	0
$562$	−76.0531	−3.20811
$563$	−26.9329	−1.13509	−0.567544	−	0.823343i	$-0.692106\pi$
$563$	−26.9329	−1.13509	−0.567544	+	0.823343i	$0.692106\pi$
$564$	0	0
$565$	−7.23269	−0.304281
$566$	12.3362	0.518528
$567$	0	0
$568$	−45.9753	−1.92908
$569$	−8.88971	−0.372676	−0.186338	−	0.982486i	$-0.559662\pi$
$569$	−8.88971	−0.372676	−0.186338	+	0.982486i	$0.559662\pi$
$570$	0	0
$571$	−15.6281	−0.654017	−0.327008	−	0.945021i	$-0.606041\pi$
$571$	−15.6281	−0.654017	−0.327008	+	0.945021i	$0.606041\pi$
$572$	−45.4907	−1.90206
$573$	0	0
$574$	−39.0094	−1.62822
$575$	7.80198	0.325365
$576$	0	0
$577$	15.2772	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$577$	15.2772	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$578$	41.3129	1.71839
$579$	0	0
$580$	70.1206	2.91160
$581$	−0.316806	−0.0131433
$582$	0	0
$583$	−10.7597	−0.445621
$584$	46.2553	1.91406
$585$	0	0
$586$	−69.2190	−2.85941
$587$	40.3363	1.66486	0.832429	−	0.554131i	$-0.186949\pi$
$587$	40.3363	1.66486	0.832429	+	0.554131i	$0.186949\pi$
$588$	0	0
$589$	−39.9435	−1.64584
$590$	−106.494	−4.38429
$591$	0	0
$592$	30.4761	1.25256
$593$	−38.3255	−1.57384	−0.786921	−	0.617054i	$-0.788326\pi$
$593$	−38.3255	−1.57384	−0.786921	+	0.617054i	$0.788326\pi$
$594$	0	0
$595$	3.12845	0.128254
$596$	3.98860	0.163379
$597$	0	0
$598$	11.6275	0.475485
$599$	−3.79771	−0.155170	−0.0775850	−	0.996986i	$-0.524721\pi$
$599$	−3.79771	−0.155170	−0.0775850	+	0.996986i	$0.524721\pi$
$600$	0	0
$601$	40.4420	1.64967	0.824833	−	0.565377i	$-0.191269\pi$
$601$	40.4420	1.64967	0.824833	+	0.565377i	$0.191269\pi$
$602$	63.0796	2.57093
$603$	0	0
$604$	54.4124	2.21401
$605$	−12.7520	−0.518442
$606$	0	0
$607$	−0.560199	−0.0227378	−0.0113689	−	0.999935i	$-0.503619\pi$
$607$	−0.560199	−0.0227378	−0.0113689	+	0.999935i	$0.503619\pi$
$608$	−0.458924	−0.0186118
$609$	0	0
$610$	21.0789	0.853461
$611$	−29.4328	−1.19072
$612$	0	0
$613$	−17.5589	−0.709196	−0.354598	−	0.935019i	$-0.615382\pi$
$613$	−17.5589	−0.709196	−0.354598	+	0.935019i	$0.615382\pi$
$614$	−21.9718	−0.886710
$615$	0	0
$616$	34.6834	1.39743
$617$	39.4107	1.58661	0.793307	−	0.608822i	$-0.208358\pi$
$617$	39.4107	1.58661	0.793307	+	0.608822i	$0.208358\pi$
$618$	0	0
$619$	−6.65764	−0.267593	−0.133797	−	0.991009i	$-0.542717\pi$
$619$	−6.65764	−0.267593	−0.133797	+	0.991009i	$0.542717\pi$
$620$	133.440	5.35909
$621$	0	0
$622$	−69.6603	−2.79313
$623$	−8.09123	−0.324168
$624$	0	0
$625$	−11.6451	−0.465805
$626$	−73.0897	−2.92125
$627$	0	0
$628$	51.8700	2.06984
$629$	−2.66321	−0.106189
$630$	0	0
$631$	−35.7001	−1.42120	−0.710599	−	0.703598i	$-0.751576\pi$
$631$	−35.7001	−1.42120	−0.710599	+	0.703598i	$0.751576\pi$
$632$	−79.7910	−3.17391
$633$	0	0
$634$	30.4773	1.21041
$635$	27.2965	1.08323
$636$	0	0
$637$	0.206932	0.00819895
$638$	−33.6743	−1.33318
$639$	0	0
$640$	67.9921	2.68762
$641$	10.2386	0.404399	0.202200	−	0.979344i	$-0.435191\pi$
$641$	10.2386	0.404399	0.202200	+	0.979344i	$0.435191\pi$
$642$	0	0
$643$	19.0023	0.749376	0.374688	−	0.927151i	$-0.377750\pi$
$643$	19.0023	0.749376	0.374688	+	0.927151i	$0.377750\pi$
$644$	−11.8428	−0.466673
$645$	0	0
$646$	−3.46696	−0.136406
$647$	43.7451	1.71980	0.859900	−	0.510463i	$-0.170526\pi$
$647$	43.7451	1.71980	0.859900	+	0.510463i	$0.170526\pi$
$648$	0	0
$649$	34.0623	1.33706
$650$	71.5264	2.80550
$651$	0	0
$652$	15.7753	0.617809
$653$	−45.3321	−1.77398	−0.886992	−	0.461785i	$-0.847209\pi$
$653$	−45.3321	−1.77398	−0.886992	+	0.461785i	$0.847209\pi$
$654$	0	0
$655$	38.1102	1.48909
$656$	−23.7721	−0.928143
$657$	0	0
$658$	45.0095	1.75465
$659$	13.0894	0.509889	0.254945	−	0.966956i	$-0.417943\pi$
$659$	13.0894	0.509889	0.254945	+	0.966956i	$0.417943\pi$
$660$	0	0
$661$	17.6346	0.685907	0.342953	−	0.939352i	$-0.388573\pi$
$661$	17.6346	0.685907	0.342953	+	0.939352i	$0.388573\pi$
$662$	−53.8993	−2.09485
$663$	0	0
$664$	−0.584764	−0.0226932
$665$	37.5457	1.45596
$666$	0	0
$667$	5.73270	0.221971
$668$	−24.0418	−0.930205
$669$	0	0
$670$	119.360	4.61128
$671$	−6.74213	−0.260277
$672$	0	0
$673$	28.0904	1.08281	0.541403	−	0.840763i	$-0.317893\pi$
$673$	28.0904	1.08281	0.541403	+	0.840763i	$0.317893\pi$
$674$	−29.0928	−1.12061
$675$	0	0
$676$	19.1460	0.736385
$677$	−32.3097	−1.24176	−0.620882	−	0.783904i	$-0.713225\pi$
$677$	−32.3097	−1.24176	−0.620882	+	0.783904i	$0.713225\pi$
$678$	0	0
$679$	23.1350	0.887839
$680$	5.77453	0.221443
$681$	0	0
$682$	−64.0826	−2.45385
$683$	−13.3222	−0.509758	−0.254879	−	0.966973i	$-0.582036\pi$
$683$	−13.3222	−0.509758	−0.254879	+	0.966973i	$0.582036\pi$
$684$	0	0
$685$	8.29269	0.316847
$686$	−45.4794	−1.73641
$687$	0	0
$688$	38.4402	1.46552
$689$	16.7928	0.639753
$690$	0	0
$691$	27.2497	1.03663	0.518314	−	0.855190i	$-0.326560\pi$
$691$	27.2497	1.03663	0.518314	+	0.855190i	$0.326560\pi$
$692$	−2.79592	−0.106285
$693$	0	0
$694$	−34.8561	−1.32312
$695$	−31.9951	−1.21364
$696$	0	0
$697$	2.07736	0.0786858
$698$	59.1998	2.24074
$699$	0	0
$700$	−72.8507	−2.75350
$701$	45.5344	1.71981	0.859905	−	0.510455i	$-0.170523\pi$
$701$	45.5344	1.71981	0.859905	+	0.510455i	$0.170523\pi$
$702$	0	0
$703$	−31.9622	−1.20548
$704$	−21.9933	−0.828902
$705$	0	0
$706$	1.52152	0.0572631
$707$	−33.3146	−1.25292
$708$	0	0
$709$	−20.0255	−0.752074	−0.376037	−	0.926605i	$-0.622714\pi$
$709$	−20.0255	−0.752074	−0.376037	+	0.926605i	$0.622714\pi$
$710$	−79.8466	−2.99659
$711$	0	0
$712$	−14.9349	−0.559709
$713$	10.9094	0.408560
$714$	0	0
$715$	−39.3896	−1.47309
$716$	18.1151	0.676992
$717$	0	0
$718$	28.5051	1.06380
$719$	−9.65101	−0.359922	−0.179961	−	0.983674i	$-0.557597\pi$
$719$	−9.65101	−0.359922	−0.179961	+	0.983674i	$0.557597\pi$
$720$	0	0
$721$	−9.63719	−0.358908
$722$	4.88780	0.181905
$723$	0	0
$724$	45.9046	1.70603
$725$	35.2645	1.30969
$726$	0	0
$727$	−30.1381	−1.11776	−0.558880	−	0.829248i	$-0.688769\pi$
$727$	−30.1381	−1.11776	−0.558880	+	0.829248i	$0.688769\pi$
$728$	−54.1307	−2.00622
$729$	0	0
$730$	80.3329	2.97326
$731$	−3.35917	−0.124243
$732$	0	0
$733$	13.2331	0.488775	0.244387	−	0.969678i	$-0.421413\pi$
$733$	13.2331	0.488775	0.244387	+	0.969678i	$0.421413\pi$
$734$	3.44030	0.126984
$735$	0	0
$736$	0.125342	0.00462015
$737$	−38.1774	−1.40628
$738$	0	0
$739$	−36.0927	−1.32769	−0.663845	−	0.747870i	$-0.731077\pi$
$739$	−36.0927	−1.32769	−0.663845	+	0.747870i	$0.731077\pi$
$740$	106.777	3.92519
$741$	0	0
$742$	−25.6800	−0.942741
$743$	−14.3440	−0.526230	−0.263115	−	0.964764i	$-0.584750\pi$
$743$	−14.3440	−0.526230	−0.263115	+	0.964764i	$0.584750\pi$
$744$	0	0
$745$	3.45366	0.126532
$746$	33.7198	1.23457
$747$	0	0
$748$	−3.70457	−0.135453
$749$	−11.9418	−0.436344
$750$	0	0
$751$	−8.14048	−0.297050	−0.148525	−	0.988909i	$-0.547453\pi$
$751$	−8.14048	−0.297050	−0.148525	+	0.988909i	$0.547453\pi$
$752$	27.4285	1.00021
$753$	0	0
$754$	52.5558	1.91397
$755$	47.1147	1.71468
$756$	0	0
$757$	7.10300	0.258163	0.129081	−	0.991634i	$-0.458797\pi$
$757$	7.10300	0.258163	0.129081	+	0.991634i	$0.458797\pi$
$758$	35.5508	1.29126
$759$	0	0
$760$	69.3022	2.51386
$761$	−21.7186	−0.787297	−0.393649	−	0.919261i	$-0.628787\pi$
$761$	−21.7186	−0.787297	−0.393649	+	0.919261i	$0.628787\pi$
$762$	0	0
$763$	−23.3854	−0.846609
$764$	−62.5023	−2.26126
$765$	0	0
$766$	51.6534	1.86631
$767$	−53.1613	−1.91955
$768$	0	0
$769$	50.2926	1.81360	0.906799	−	0.421562i	$-0.138518\pi$
$769$	50.2926	1.81360	0.906799	+	0.421562i	$0.138518\pi$
$770$	60.2357	2.17074
$771$	0	0
$772$	−20.3022	−0.730691
$773$	8.02924	0.288792	0.144396	−	0.989520i	$-0.453876\pi$
$773$	8.02924	0.288792	0.144396	+	0.989520i	$0.453876\pi$
$774$	0	0
$775$	67.1087	2.41062
$776$	42.7028	1.53294
$777$	0	0
$778$	51.5028	1.84647
$779$	24.9312	0.893253
$780$	0	0
$781$	25.5391	0.913860
$782$	0.946899	0.0338610
$783$	0	0
$784$	−0.192840	−0.00688715
$785$	44.9133	1.60303
$786$	0	0
$787$	−41.2673	−1.47102	−0.735509	−	0.677515i	$-0.763057\pi$
$787$	−41.2673	−1.47102	−0.735509	+	0.677515i	$0.763057\pi$
$788$	−72.1277	−2.56944
$789$	0	0
$790$	−138.575	−4.93029
$791$	5.52132	0.196315
$792$	0	0
$793$	10.5225	0.373665
$794$	24.0371	0.853046
$795$	0	0
$796$	6.83994	0.242435
$797$	−41.5096	−1.47035	−0.735173	−	0.677879i	$-0.762899\pi$
$797$	−41.5096	−1.47035	−0.735173	+	0.677879i	$0.762899\pi$
$798$	0	0
$799$	−2.39689	−0.0847957
$800$	0.771035	0.0272602
$801$	0	0
$802$	−93.8751	−3.31485
$803$	−25.6946	−0.906743
$804$	0	0
$805$	−10.2545	−0.361423
$806$	100.014	3.52285
$807$	0	0
$808$	−61.4924	−2.16329
$809$	49.8696	1.75332	0.876661	−	0.481109i	$-0.159766\pi$
$809$	49.8696	1.75332	0.876661	+	0.481109i	$0.159766\pi$
$810$	0	0
$811$	39.9146	1.40159	0.700796	−	0.713361i	$-0.252828\pi$
$811$	39.9146	1.40159	0.700796	+	0.713361i	$0.252828\pi$
$812$	−53.5289	−1.87850
$813$	0	0
$814$	−51.2779	−1.79729
$815$	13.6596	0.478474
$816$	0	0
$817$	−40.3146	−1.41043
$818$	9.39878	0.328620
$819$	0	0
$820$	−83.2883	−2.90855
$821$	4.79372	0.167302	0.0836510	−	0.996495i	$-0.473342\pi$
$821$	4.79372	0.167302	0.0836510	+	0.996495i	$0.473342\pi$
$822$	0	0
$823$	−36.6692	−1.27821	−0.639104	−	0.769121i	$-0.720695\pi$
$823$	−36.6692	−1.27821	−0.639104	+	0.769121i	$0.720695\pi$
$824$	−17.7884	−0.619689
$825$	0	0
$826$	81.2959	2.82865
$827$	−17.2400	−0.599493	−0.299747	−	0.954019i	$-0.596902\pi$
$827$	−17.2400	−0.599493	−0.299747	+	0.954019i	$0.596902\pi$
$828$	0	0
$829$	−50.9717	−1.77032	−0.885161	−	0.465285i	$-0.845952\pi$
$829$	−50.9717	−1.77032	−0.885161	+	0.465285i	$0.845952\pi$
$830$	−1.01558	−0.0352512
$831$	0	0
$832$	34.3251	1.19001
$833$	0.0168517	0.000583877 0
$834$	0	0
$835$	−20.8174	−0.720414
$836$	−44.4599	−1.53768
$837$	0	0
$838$	−15.2524	−0.526885
$839$	11.8716	0.409853	0.204927	−	0.978777i	$-0.434304\pi$
$839$	11.8716	0.409853	0.204927	+	0.978777i	$0.434304\pi$
$840$	0	0
$841$	−3.08851	−0.106500
$842$	79.6182	2.74383
$843$	0	0
$844$	−90.7300	−3.12305
$845$	16.5782	0.570307
$846$	0	0
$847$	9.73466	0.334487
$848$	−15.6492	−0.537395
$849$	0	0
$850$	5.82481	0.199789
$851$	8.72953	0.299244
$852$	0	0
$853$	14.6530	0.501708	0.250854	−	0.968025i	$-0.419289\pi$
$853$	14.6530	0.501708	0.250854	+	0.968025i	$0.419289\pi$
$854$	−16.0913	−0.550634
$855$	0	0
$856$	−22.0423	−0.753390
$857$	−50.3159	−1.71876	−0.859380	−	0.511338i	$-0.829150\pi$
$857$	−50.3159	−1.71876	−0.859380	+	0.511338i	$0.829150\pi$
$858$	0	0
$859$	−34.2668	−1.16917	−0.584584	−	0.811333i	$-0.698742\pi$
$859$	−34.2668	−1.16917	−0.584584	+	0.811333i	$0.698742\pi$
$860$	134.680	4.59255
$861$	0	0
$862$	71.0678	2.42058
$863$	42.1015	1.43315	0.716576	−	0.697509i	$-0.245708\pi$
$863$	42.1015	1.43315	0.716576	+	0.697509i	$0.245708\pi$
$864$	0	0
$865$	−2.42093	−0.0823142
$866$	−58.3835	−1.98395
$867$	0	0
$868$	−101.866	−3.45756
$869$	44.3235	1.50357
$870$	0	0
$871$	59.5839	2.01892
$872$	−43.1651	−1.46175
$873$	0	0
$874$	11.3641	0.384396
$875$	−17.5529	−0.593397
$876$	0	0
$877$	−17.9601	−0.606471	−0.303235	−	0.952916i	$-0.598067\pi$
$877$	−17.9601	−0.606471	−0.303235	+	0.952916i	$0.598067\pi$
$878$	15.8017	0.533283
$879$	0	0
$880$	36.7072	1.23740
$881$	−25.6154	−0.863003	−0.431502	−	0.902112i	$-0.642016\pi$
$881$	−25.6154	−0.863003	−0.431502	+	0.902112i	$0.642016\pi$
$882$	0	0
$883$	14.2224	0.478622	0.239311	−	0.970943i	$-0.423078\pi$
$883$	14.2224	0.478622	0.239311	+	0.970943i	$0.423078\pi$
$884$	5.78176	0.194462
$885$	0	0
$886$	86.5788	2.90867
$887$	−22.8203	−0.766231	−0.383115	−	0.923701i	$-0.625149\pi$
$887$	−22.8203	−0.766231	−0.383115	+	0.923701i	$0.625149\pi$
$888$	0	0
$889$	−20.8377	−0.698874
$890$	−25.9379	−0.869439
$891$	0	0
$892$	−68.7708	−2.30262
$893$	−28.7659	−0.962614
$894$	0	0
$895$	15.6855	0.524309
$896$	−51.9041	−1.73399
$897$	0	0
$898$	88.9097	2.96696
$899$	49.3098	1.64458
$900$	0	0
$901$	1.36753	0.0455591
$902$	39.9979	1.33178
$903$	0	0
$904$	10.1913	0.338958
$905$	39.7480	1.32127
$906$	0	0
$907$	−5.79957	−0.192572	−0.0962858	−	0.995354i	$-0.530696\pi$
$907$	−5.79957	−0.192572	−0.0962858	+	0.995354i	$0.530696\pi$
$908$	113.335	3.76116
$909$	0	0
$910$	−94.0104	−3.11642
$911$	15.5383	0.514807	0.257403	−	0.966304i	$-0.417133\pi$
$911$	15.5383	0.514807	0.257403	+	0.966304i	$0.417133\pi$
$912$	0	0
$913$	0.324834	0.0107504
$914$	−40.7281	−1.34717
$915$	0	0
$916$	9.99817	0.330349
$917$	−29.0927	−0.960725
$918$	0	0
$919$	−41.2840	−1.36183	−0.680917	−	0.732360i	$-0.738419\pi$
$919$	−41.2840	−1.36183	−0.680917	+	0.732360i	$0.738419\pi$
$920$	−18.9279	−0.624033
$921$	0	0
$922$	−47.4981	−1.56427
$923$	−39.8591	−1.31198
$924$	0	0
$925$	53.6994	1.76562
$926$	38.2797	1.25795
$927$	0	0
$928$	0.566537	0.0185975
$929$	−13.0132	−0.426950	−0.213475	−	0.976949i	$-0.568478\pi$
$929$	−13.0132	−0.426950	−0.213475	+	0.976949i	$0.568478\pi$
$930$	0	0
$931$	0.202243	0.00662826
$932$	−76.5139	−2.50630
$933$	0	0
$934$	−50.7950	−1.66206
$935$	−3.20772	−0.104904
$936$	0	0
$937$	23.4237	0.765219	0.382609	−	0.923910i	$-0.375026\pi$
$937$	23.4237	0.765219	0.382609	+	0.923910i	$0.375026\pi$
$938$	−91.1174	−2.97509
$939$	0	0
$940$	96.0990	3.13440
$941$	7.19436	0.234529	0.117265	−	0.993101i	$-0.462587\pi$
$941$	7.19436	0.234529	0.117265	+	0.993101i	$0.462587\pi$
$942$	0	0
$943$	−6.80923	−0.221739
$944$	49.5411	1.61243
$945$	0	0
$946$	−64.6780	−2.10286
$947$	−14.5518	−0.472869	−0.236434	−	0.971647i	$-0.575979\pi$
$947$	−14.5518	−0.472869	−0.236434	+	0.971647i	$0.575979\pi$
$948$	0	0
$949$	40.1018	1.30176
$950$	69.9057	2.26804
$951$	0	0
$952$	−4.40818	−0.142870
$953$	−2.59161	−0.0839505	−0.0419753	−	0.999119i	$-0.513365\pi$
$953$	−2.59161	−0.0839505	−0.0419753	+	0.999119i	$0.513365\pi$
$954$	0	0
$955$	−54.1197	−1.75127
$956$	−29.8013	−0.963843
$957$	0	0
$958$	−93.8742	−3.03294
$959$	−6.33051	−0.204423
$960$	0	0
$961$	62.8372	2.02701
$962$	80.0299	2.58027
$963$	0	0
$964$	62.6140	2.01666
$965$	−17.5793	−0.565897
$966$	0	0
$967$	30.3462	0.975867	0.487934	−	0.872881i	$-0.337751\pi$
$967$	30.3462	0.975867	0.487934	+	0.872881i	$0.337751\pi$
$968$	17.9683	0.577524
$969$	0	0
$970$	74.1632	2.38124
$971$	25.3169	0.812459	0.406230	−	0.913771i	$-0.366843\pi$
$971$	25.3169	0.812459	0.406230	+	0.913771i	$0.366843\pi$
$972$	0	0
$973$	24.4245	0.783014
$974$	5.79358	0.185638
$975$	0	0
$976$	−9.80593	−0.313880
$977$	5.68782	0.181970	0.0909848	−	0.995852i	$-0.470999\pi$
$977$	5.68782	0.181970	0.0909848	+	0.995852i	$0.470999\pi$
$978$	0	0
$979$	8.29626	0.265150
$980$	−0.675640	−0.0215825
$981$	0	0
$982$	29.1841	0.931302
$983$	37.9448	1.21025	0.605125	−	0.796131i	$-0.293123\pi$
$983$	37.9448	1.21025	0.605125	+	0.796131i	$0.293123\pi$
$984$	0	0
$985$	−62.4541	−1.98995
$986$	4.27993	0.136301
$987$	0	0
$988$	69.3891	2.20756
$989$	11.0108	0.350122
$990$	0	0
$991$	−61.6868	−1.95954	−0.979772	−	0.200115i	$-0.935868\pi$
$991$	−61.6868	−1.95954	−0.979772	+	0.200115i	$0.935868\pi$
$992$	1.07813	0.0342306
$993$	0	0
$994$	60.9537	1.93333
$995$	5.92258	0.187758
$996$	0	0
$997$	−5.06637	−0.160454	−0.0802268	−	0.996777i	$-0.525564\pi$
$997$	−5.06637	−0.160454	−0.0802268	+	0.996777i	$0.525564\pi$
$998$	73.4406	2.32472
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.h.1.1	yes	12
3.2	odd	2		1341.2.a.g.1.12	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1341.2.a.g.1.12	✓	12	3.2	odd	2
1341.2.a.h.1.1	yes	12	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:	\( N \)	\(=\)	\( 1341 = 3^{2} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1341.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44716 q^{2} +3.98860 q^{4} +3.45366 q^{5} -2.63647 q^{7} -4.86642 q^{8} +O(q^{10})\)	\(q-2.44716 q^{2} +3.98860 q^{4} +3.45366 q^{5} -2.63647 q^{7} -4.86642 q^{8} -8.45165 q^{10} +2.70327 q^{11} -4.21903 q^{13} +6.45186 q^{14} +3.93172 q^{16} -0.343580 q^{17} -4.12343 q^{19} +13.7752 q^{20} -6.61535 q^{22} +1.12619 q^{23} +6.92774 q^{25} +10.3246 q^{26} -10.5158 q^{28} +5.09033 q^{29} +9.68696 q^{31} +0.111297 q^{32} +0.840796 q^{34} -9.10545 q^{35} +7.75136 q^{37} +10.0907 q^{38} -16.8069 q^{40} -6.04623 q^{41} +9.77697 q^{43} +10.7823 q^{44} -2.75598 q^{46} +6.97621 q^{47} -0.0490474 q^{49} -16.9533 q^{50} -16.8280 q^{52} -3.98024 q^{53} +9.33618 q^{55} +12.8301 q^{56} -12.4569 q^{58} +12.6004 q^{59} -2.49406 q^{61} -23.7055 q^{62} -8.13579 q^{64} -14.5711 q^{65} -14.1227 q^{67} -1.37040 q^{68} +22.2825 q^{70} +9.44746 q^{71} -9.50500 q^{73} -18.9688 q^{74} -16.4467 q^{76} -7.12709 q^{77} +16.3962 q^{79} +13.5788 q^{80} +14.7961 q^{82} +0.120163 q^{83} -1.18661 q^{85} -23.9258 q^{86} -13.1553 q^{88} +3.06897 q^{89} +11.1233 q^{91} +4.49193 q^{92} -17.0719 q^{94} -14.2409 q^{95} -8.77499 q^{97} +0.120027 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8}+O(q^{10}) \) 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8	\( 12 q + 3 q^{2} + 9 q^{4} + 8 q^{5} - 2 q^{7} + 9 q^{8} + 22 q^{11} - 2 q^{13} + 12 q^{14} + 11 q^{16} + 8 q^{17} - 6 q^{19} + 24 q^{20} + 4 q^{22} + 12 q^{23} + 8 q^{25} + 26 q^{26} - 20 q^{28} + 16 q^{29} - 2 q^{31} + 21 q^{32} - 6 q^{34} + 32 q^{35} - 4 q^{37} + 15 q^{38} + 6 q^{40} + 24 q^{41} + 12 q^{43} + 37 q^{44} - 22 q^{46} + 26 q^{47} + 24 q^{49} - 5 q^{50} - 10 q^{52} + 20 q^{53} + 6 q^{55} - 13 q^{56} + 10 q^{58} + 72 q^{59} - 8 q^{61} - 2 q^{62} + q^{64} - 4 q^{65} - 14 q^{67} - 14 q^{68} + 18 q^{70} + 38 q^{71} - 27 q^{74} - 2 q^{76} + 6 q^{77} + 10 q^{79} + 56 q^{80} + 6 q^{82} + 42 q^{83} - 32 q^{85} - 14 q^{86} + 22 q^{88} + 44 q^{89} - 50 q^{92} + 2 q^{94} + 4 q^{95} - 22 q^{97} + 20 q^{98}+O(q^{100}) \) 12 * q + 3 * q^2 + 9 * q^4 + 8 * q^5 - 2 * q^7 + 9 * q^8 + 22 * q^11 - 2 * q^13 + 12 * q^14 + 11 * q^16 + 8 * q^17 - 6 * q^19 + 24 * q^20 + 4 * q^22 + 12 * q^23 + 8 * q^25 + 26 * q^26 - 20 * q^28 + 16 * q^29 - 2 * q^31 + 21 * q^32 - 6 * q^34 + 32 * q^35 - 4 * q^37 + 15 * q^38 + 6 * q^40 + 24 * q^41 + 12 * q^43 + 37 * q^44 - 22 * q^46 + 26 * q^47 + 24 * q^49 - 5 * q^50 - 10 * q^52 + 20 * q^53 + 6 * q^55 - 13 * q^56 + 10 * q^58 + 72 * q^59 - 8 * q^61 - 2 * q^62 + q^64 - 4 * q^65 - 14 * q^67 - 14 * q^68 + 18 * q^70 + 38 * q^71 - 27 * q^74 - 2 * q^76 + 6 * q^77 + 10 * q^79 + 56 * q^80 + 6 * q^82 + 42 * q^83 - 32 * q^85 - 14 * q^86 + 22 * q^88 + 44 * q^89 - 50 * q^92 + 2 * q^94 + 4 * q^95 - 22 * q^97 + 20 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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