LMFDB - Embedded newform 1341.2.a.b.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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 149)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.24698$ 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)

$f(q)$	$=$	$q-1.24698 q^{2} -0.445042 q^{4} +2.69202 q^{5} -0.198062 q^{7} +3.04892 q^{8} +O(q^{10})$	\(q-1.24698 q^{2} -0.445042 q^{4} +2.69202 q^{5} -0.198062 q^{7} +3.04892 q^{8} -3.35690 q^{10} -0.137063 q^{11} -2.35690 q^{13} +0.246980 q^{14} -2.91185 q^{16} -5.96077 q^{17} -7.35690 q^{19} -1.19806 q^{20} +0.170915 q^{22} -1.19806 q^{23} +2.24698 q^{25} +2.93900 q^{26} +0.0881460 q^{28} +4.96077 q^{29} -6.33513 q^{31} -2.46681 q^{32} +7.43296 q^{34} -0.533188 q^{35} +10.4819 q^{37} +9.17390 q^{38} +8.20775 q^{40} +5.78986 q^{41} -9.57002 q^{43} +0.0609989 q^{44} +1.49396 q^{46} +4.62565 q^{47} -6.96077 q^{49} -2.80194 q^{50} +1.04892 q^{52} -10.3448 q^{53} -0.368977 q^{55} -0.603875 q^{56} -6.18598 q^{58} -1.46681 q^{59} -5.43296 q^{61} +7.89977 q^{62} +8.89977 q^{64} -6.34481 q^{65} -6.19806 q^{67} +2.65279 q^{68} +0.664874 q^{70} -8.61356 q^{71} +14.4644 q^{73} -13.0707 q^{74} +3.27413 q^{76} +0.0271471 q^{77} +3.09783 q^{79} -7.83877 q^{80} -7.21983 q^{82} -0.554958 q^{83} -16.0465 q^{85} +11.9336 q^{86} -0.417895 q^{88} +9.76809 q^{89} +0.466812 q^{91} +0.533188 q^{92} -5.76809 q^{94} -19.8049 q^{95} -12.1903 q^{97} +8.67994 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - q^{4} + 3 q^{5} - 5 q^{7}+O(q^{10}) $ 3 * q + q^2 - q^4 + 3 * q^5 - 5 * q^7	$ 3 q + q^{2} - q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{10} + 5 q^{11} - 3 q^{13} - 4 q^{14} - 5 q^{16} - 5 q^{17} - 18 q^{19} - 8 q^{20} + 11 q^{22} - 8 q^{23} + 2 q^{25} - q^{26} + 4 q^{28} + 2 q^{29} - 18 q^{31} - 4 q^{32} + 3 q^{34} - 5 q^{35} + 3 q^{37} - 6 q^{38} + 7 q^{40} - 6 q^{41} - 4 q^{43} + 10 q^{44} - 5 q^{46} + 2 q^{47} - 8 q^{49} - 4 q^{50} - 6 q^{52} - 8 q^{53} - 16 q^{55} + 7 q^{56} - 4 q^{58} - q^{59} + 3 q^{61} + q^{62} + 4 q^{64} + 4 q^{65} - 23 q^{67} - 10 q^{68} + 3 q^{70} + 5 q^{71} - q^{73} - 27 q^{74} - q^{76} - 6 q^{77} - 9 q^{79} + 9 q^{80} - 23 q^{82} - 2 q^{83} + 2 q^{85} + 29 q^{86} - 7 q^{88} + 9 q^{89} - 2 q^{91} + 5 q^{92} + 3 q^{94} - 11 q^{95} - 3 q^{97} + 2 q^{98}+O(q^{100}) $ 3 * q + q^2 - q^4 + 3 * q^5 - 5 * q^7 - 6 * q^10 + 5 * q^11 - 3 * q^13 - 4 * q^14 - 5 * q^16 - 5 * q^17 - 18 * q^19 - 8 * q^20 + 11 * q^22 - 8 * q^23 + 2 * q^25 - q^26 + 4 * q^28 + 2 * q^29 - 18 * q^31 - 4 * q^32 + 3 * q^34 - 5 * q^35 + 3 * q^37 - 6 * q^38 + 7 * q^40 - 6 * q^41 - 4 * q^43 + 10 * q^44 - 5 * q^46 + 2 * q^47 - 8 * q^49 - 4 * q^50 - 6 * q^52 - 8 * q^53 - 16 * q^55 + 7 * q^56 - 4 * q^58 - q^59 + 3 * q^61 + q^62 + 4 * q^64 + 4 * q^65 - 23 * q^67 - 10 * q^68 + 3 * q^70 + 5 * q^71 - q^73 - 27 * q^74 - q^76 - 6 * q^77 - 9 * q^79 + 9 * q^80 - 23 * q^82 - 2 * q^83 + 2 * q^85 + 29 * q^86 - 7 * q^88 + 9 * q^89 - 2 * q^91 + 5 * q^92 + 3 * q^94 - 11 * q^95 - 3 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.24698	−0.881748	−0.440874	−	0.897569i	$-0.645331\pi$
$2$	−1.24698	−0.881748	−0.440874	+	0.897569i	$0.645331\pi$
$3$	0	0
$3$	0	0
$4$	−0.445042	−0.222521
$5$	2.69202	1.20391	0.601954	−	0.798531i	$-0.294389\pi$
$5$	2.69202	1.20391	0.601954	+	0.798531i	$0.294389\pi$
$6$	0	0
$7$	−0.198062	−0.0748605	−0.0374302	−	0.999299i	$-0.511917\pi$
$7$	−0.198062	−0.0748605	−0.0374302	+	0.999299i	$0.511917\pi$
$8$	3.04892	1.07796
$9$	0	0
$10$	−3.35690	−1.06154
$11$	−0.137063	−0.0413262	−0.0206631	−	0.999786i	$-0.506578\pi$
$11$	−0.137063	−0.0413262	−0.0206631	+	0.999786i	$0.506578\pi$
$12$	0	0
$13$	−2.35690	−0.653685	−0.326843	−	0.945079i	$-0.605985\pi$
$13$	−2.35690	−0.653685	−0.326843	+	0.945079i	$0.605985\pi$
$14$	0.246980	0.0660081
$15$	0	0
$16$	−2.91185	−0.727963
$17$	−5.96077	−1.44570	−0.722850	−	0.691005i	$-0.757168\pi$
$17$	−5.96077	−1.44570	−0.722850	+	0.691005i	$0.757168\pi$
$18$	0	0
$19$	−7.35690	−1.68779	−0.843894	−	0.536510i	$-0.819742\pi$
$19$	−7.35690	−1.68779	−0.843894	+	0.536510i	$0.819742\pi$
$20$	−1.19806	−0.267895
$21$	0	0
$22$	0.170915	0.0364392
$23$	−1.19806	−0.249813	−0.124907	−	0.992169i	$-0.539863\pi$
$23$	−1.19806	−0.249813	−0.124907	+	0.992169i	$0.539863\pi$
$24$	0	0
$25$	2.24698	0.449396
$26$	2.93900	0.576386
$27$	0	0
$28$	0.0881460	0.0166580
$29$	4.96077	0.921192	0.460596	−	0.887610i	$-0.347636\pi$
$29$	4.96077	0.921192	0.460596	+	0.887610i	$0.347636\pi$
$30$	0	0
$31$	−6.33513	−1.13782	−0.568911	−	0.822399i	$-0.692635\pi$
$31$	−6.33513	−1.13782	−0.568911	+	0.822399i	$0.692635\pi$
$32$	−2.46681	−0.436075
$33$	0	0
$34$	7.43296	1.27474
$35$	−0.533188	−0.0901252
$36$	0	0
$37$	10.4819	1.72321	0.861605	−	0.507579i	$-0.169460\pi$
$37$	10.4819	1.72321	0.861605	+	0.507579i	$0.169460\pi$
$38$	9.17390	1.48820
$39$	0	0
$40$	8.20775	1.29776
$41$	5.78986	0.904224	0.452112	−	0.891961i	$-0.350671\pi$
$41$	5.78986	0.904224	0.452112	+	0.891961i	$0.350671\pi$
$42$	0	0
$43$	−9.57002	−1.45941	−0.729707	−	0.683759i	$-0.760344\pi$
$43$	−9.57002	−1.45941	−0.729707	+	0.683759i	$0.760344\pi$
$44$	0.0609989	0.00919593
$45$	0	0
$46$	1.49396	0.220272
$47$	4.62565	0.674720	0.337360	−	0.941376i	$-0.390466\pi$
$47$	4.62565	0.674720	0.337360	+	0.941376i	$0.390466\pi$
$48$	0	0
$49$	−6.96077	−0.994396
$50$	−2.80194	−0.396254
$51$	0	0
$52$	1.04892	0.145459
$53$	−10.3448	−1.42097	−0.710485	−	0.703713i	$-0.751524\pi$
$53$	−10.3448	−1.42097	−0.710485	+	0.703713i	$0.751524\pi$
$54$	0	0
$55$	−0.368977	−0.0497529
$56$	−0.603875	−0.0806963
$57$	0	0
$58$	−6.18598	−0.812259
$59$	−1.46681	−0.190963	−0.0954813	−	0.995431i	$-0.530439\pi$
$59$	−1.46681	−0.190963	−0.0954813	+	0.995431i	$0.530439\pi$
$60$	0	0
$61$	−5.43296	−0.695619	−0.347810	−	0.937565i	$-0.613074\pi$
$61$	−5.43296	−0.695619	−0.347810	+	0.937565i	$0.613074\pi$
$62$	7.89977	1.00327
$63$	0	0
$64$	8.89977	1.11247
$65$	−6.34481	−0.786977
$66$	0	0
$67$	−6.19806	−0.757214	−0.378607	−	0.925558i	$-0.623597\pi$
$67$	−6.19806	−0.757214	−0.378607	+	0.925558i	$0.623597\pi$
$68$	2.65279	0.321698
$69$	0	0
$70$	0.664874	0.0794677
$71$	−8.61356	−1.02224	−0.511121	−	0.859509i	$-0.670770\pi$
$71$	−8.61356	−1.02224	−0.511121	+	0.859509i	$0.670770\pi$
$72$	0	0
$73$	14.4644	1.69293	0.846466	−	0.532443i	$-0.178726\pi$
$73$	14.4644	1.69293	0.846466	+	0.532443i	$0.178726\pi$
$74$	−13.0707	−1.51944
$75$	0	0
$76$	3.27413	0.375568
$77$	0.0271471	0.00309370
$78$	0	0
$79$	3.09783	0.348534	0.174267	−	0.984698i	$-0.444244\pi$
$79$	3.09783	0.348534	0.174267	+	0.984698i	$0.444244\pi$
$80$	−7.83877	−0.876402
$81$	0	0
$82$	−7.21983	−0.797297
$83$	−0.554958	−0.0609146	−0.0304573	−	0.999536i	$-0.509696\pi$
$83$	−0.554958	−0.0609146	−0.0304573	+	0.999536i	$0.509696\pi$
$84$	0	0
$85$	−16.0465	−1.74049
$86$	11.9336	1.28684
$87$	0	0
$88$	−0.417895	−0.0445477
$89$	9.76809	1.03542	0.517708	−	0.855558i	$-0.326786\pi$
$89$	9.76809	1.03542	0.517708	+	0.855558i	$0.326786\pi$
$90$	0	0
$91$	0.466812	0.0489352
$92$	0.533188	0.0555887
$93$	0	0
$94$	−5.76809	−0.594933
$95$	−19.8049	−2.03194
$96$	0	0
$97$	−12.1903	−1.23774	−0.618868	−	0.785495i	$-0.712409\pi$
$97$	−12.1903	−1.23774	−0.618868	+	0.785495i	$0.712409\pi$
$98$	8.67994	0.876806
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−16.0858	−1.60059	−0.800296	−	0.599605i	$-0.795324\pi$
$101$	−16.0858	−1.60059	−0.800296	+	0.599605i	$0.795324\pi$
$102$	0	0
$103$	−16.5187	−1.62764	−0.813819	−	0.581119i	$-0.802615\pi$
$103$	−16.5187	−1.62764	−0.813819	+	0.581119i	$0.802615\pi$
$104$	−7.18598	−0.704643
$105$	0	0
$106$	12.8998	1.25294
$107$	15.1196	1.46167	0.730834	−	0.682556i	$-0.239131\pi$
$107$	15.1196	1.46167	0.730834	+	0.682556i	$0.239131\pi$
$108$	0	0
$109$	4.28621	0.410544	0.205272	−	0.978705i	$-0.434192\pi$
$109$	4.28621	0.410544	0.205272	+	0.978705i	$0.434192\pi$
$110$	0.460107	0.0438695
$111$	0	0
$112$	0.576728	0.0544957
$113$	−4.86831	−0.457972	−0.228986	−	0.973430i	$-0.573541\pi$
$113$	−4.86831	−0.457972	−0.228986	+	0.973430i	$0.573541\pi$
$114$	0	0
$115$	−3.22521	−0.300752
$116$	−2.20775	−0.204985
$117$	0	0
$118$	1.82908	0.168381
$119$	1.18060	0.108226
$120$	0	0
$121$	−10.9812	−0.998292
$122$	6.77479	0.613361
$123$	0	0
$124$	2.81940	0.253189
$125$	−7.41119	−0.662877
$126$	0	0
$127$	4.72587	0.419353	0.209677	−	0.977771i	$-0.432759\pi$
$127$	4.72587	0.419353	0.209677	+	0.977771i	$0.432759\pi$
$128$	−6.16421	−0.544844
$129$	0	0
$130$	7.91185	0.693915
$131$	−1.78448	−0.155911	−0.0779553	−	0.996957i	$-0.524839\pi$
$131$	−1.78448	−0.155911	−0.0779553	+	0.996957i	$0.524839\pi$
$132$	0	0
$133$	1.45712	0.126349
$134$	7.72886	0.667672
$135$	0	0
$136$	−18.1739	−1.55840
$137$	2.00969	0.171699	0.0858496	−	0.996308i	$-0.472640\pi$
$137$	2.00969	0.171699	0.0858496	+	0.996308i	$0.472640\pi$
$138$	0	0
$139$	16.2543	1.37867	0.689335	−	0.724443i	$-0.257903\pi$
$139$	16.2543	1.37867	0.689335	+	0.724443i	$0.257903\pi$
$140$	0.237291	0.0200547
$141$	0	0
$142$	10.7409	0.901360
$143$	0.323044	0.0270143
$144$	0	0
$145$	13.3545	1.10903
$146$	−18.0368	−1.49274
$147$	0	0
$148$	−4.66487	−0.383450
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−4.95539	−0.403264	−0.201632	−	0.979461i	$-0.564625\pi$
$151$	−4.95539	−0.403264	−0.201632	+	0.979461i	$0.564625\pi$
$152$	−22.4306	−1.81936
$153$	0	0
$154$	−0.0338518	−0.00272786
$155$	−17.0543	−1.36983
$156$	0	0
$157$	2.00538	0.160046	0.0800232	−	0.996793i	$-0.474501\pi$
$157$	2.00538	0.160046	0.0800232	+	0.996793i	$0.474501\pi$
$158$	−3.86294	−0.307319
$159$	0	0
$160$	−6.64071	−0.524994
$161$	0.237291	0.0187011
$162$	0	0
$163$	4.29052	0.336059	0.168030	−	0.985782i	$-0.446260\pi$
$163$	4.29052	0.336059	0.168030	+	0.985782i	$0.446260\pi$
$164$	−2.57673	−0.201209
$165$	0	0
$166$	0.692021	0.0537113
$167$	−15.6896	−1.21410	−0.607050	−	0.794664i	$-0.707647\pi$
$167$	−15.6896	−1.21410	−0.607050	+	0.794664i	$0.707647\pi$
$168$	0	0
$169$	−7.44504	−0.572696
$170$	20.0097	1.53467
$171$	0	0
$172$	4.25906	0.324750
$173$	−8.60925	−0.654549	−0.327275	−	0.944929i	$-0.606130\pi$
$173$	−8.60925	−0.654549	−0.327275	+	0.944929i	$0.606130\pi$
$174$	0	0
$175$	−0.445042	−0.0336420
$176$	0.399108	0.0300839
$177$	0	0
$178$	−12.1806	−0.912975
$179$	5.28083	0.394708	0.197354	−	0.980332i	$-0.436765\pi$
$179$	5.28083	0.394708	0.197354	+	0.980332i	$0.436765\pi$
$180$	0	0
$181$	−23.1685	−1.72210	−0.861051	−	0.508518i	$-0.830194\pi$
$181$	−23.1685	−1.72210	−0.861051	+	0.508518i	$0.830194\pi$
$182$	−0.582105	−0.0431485
$183$	0	0
$184$	−3.65279	−0.269287
$185$	28.2174	2.07459
$186$	0	0
$187$	0.817003	0.0597452
$188$	−2.05861	−0.150139
$189$	0	0
$190$	24.6963	1.79166
$191$	3.77479	0.273134	0.136567	−	0.990631i	$-0.456393\pi$
$191$	3.77479	0.273134	0.136567	+	0.990631i	$0.456393\pi$
$192$	0	0
$193$	−11.7017	−0.842308	−0.421154	−	0.906989i	$-0.638375\pi$
$193$	−11.7017	−0.842308	−0.421154	+	0.906989i	$0.638375\pi$
$194$	15.2010	1.09137
$195$	0	0
$196$	3.09783	0.221274
$197$	17.1642	1.22290	0.611450	−	0.791283i	$-0.290587\pi$
$197$	17.1642	1.22290	0.611450	+	0.791283i	$0.290587\pi$
$198$	0	0
$199$	4.59717	0.325885	0.162942	−	0.986636i	$-0.447902\pi$
$199$	4.59717	0.325885	0.162942	+	0.986636i	$0.447902\pi$
$200$	6.85086	0.484429
$201$	0	0
$202$	20.0586	1.41132
$203$	−0.982542	−0.0689609
$204$	0	0
$205$	15.5864	1.08860
$206$	20.5985	1.43517
$207$	0	0
$208$	6.86294	0.475859
$209$	1.00836	0.0697498
$210$	0	0
$211$	2.20344	0.151691	0.0758455	−	0.997120i	$-0.475834\pi$
$211$	2.20344	0.151691	0.0758455	+	0.997120i	$0.475834\pi$
$212$	4.60388	0.316195
$213$	0	0
$214$	−18.8538	−1.28882
$215$	−25.7627	−1.75700
$216$	0	0
$217$	1.25475	0.0851779
$218$	−5.34481	−0.361997
$219$	0	0
$220$	0.164210	0.0110711
$221$	14.0489	0.945032
$222$	0	0
$223$	−2.48188	−0.166199	−0.0830994	−	0.996541i	$-0.526482\pi$
$223$	−2.48188	−0.166199	−0.0830994	+	0.996541i	$0.526482\pi$
$224$	0.488582	0.0326448
$225$	0	0
$226$	6.07069	0.403816
$227$	14.3230	0.950654	0.475327	−	0.879809i	$-0.342330\pi$
$227$	14.3230	0.950654	0.475327	+	0.879809i	$0.342330\pi$
$228$	0	0
$229$	−15.8780	−1.04925	−0.524624	−	0.851334i	$-0.675794\pi$
$229$	−15.8780	−1.04925	−0.524624	+	0.851334i	$0.675794\pi$
$230$	4.02177	0.265188
$231$	0	0
$232$	15.1250	0.993004
$233$	23.1323	1.51545	0.757723	−	0.652576i	$-0.226312\pi$
$233$	23.1323	1.51545	0.757723	+	0.652576i	$0.226312\pi$
$234$	0	0
$235$	12.4523	0.812301
$236$	0.652793	0.0424932
$237$	0	0
$238$	−1.47219	−0.0954278
$239$	18.2892	1.18303	0.591515	−	0.806294i	$-0.298530\pi$
$239$	18.2892	1.18303	0.591515	+	0.806294i	$0.298530\pi$
$240$	0	0
$241$	16.7168	1.07682	0.538411	−	0.842682i	$-0.319025\pi$
$241$	16.7168	1.07682	0.538411	+	0.842682i	$0.319025\pi$
$242$	13.6933	0.880242
$243$	0	0
$244$	2.41789	0.154790
$245$	−18.7385	−1.19716
$246$	0	0
$247$	17.3394	1.10328
$248$	−19.3153	−1.22652
$249$	0	0
$250$	9.24160	0.584490
$251$	−21.1511	−1.33504	−0.667522	−	0.744590i	$-0.732645\pi$
$251$	−21.1511	−1.33504	−0.667522	+	0.744590i	$0.732645\pi$
$252$	0	0
$253$	0.164210	0.0103238
$254$	−5.89307	−0.369764
$255$	0	0
$256$	−10.1129	−0.632056
$257$	−9.27173	−0.578355	−0.289177	−	0.957276i	$-0.593382\pi$
$257$	−9.27173	−0.578355	−0.289177	+	0.957276i	$0.593382\pi$
$258$	0	0
$259$	−2.07606	−0.129000
$260$	2.82371	0.175119
$261$	0	0
$262$	2.22521	0.137474
$263$	−17.6039	−1.08550	−0.542751	−	0.839894i	$-0.682617\pi$
$263$	−17.6039	−1.08550	−0.542751	+	0.839894i	$0.682617\pi$
$264$	0	0
$265$	−27.8485	−1.71072
$266$	−1.81700	−0.111408
$267$	0	0
$268$	2.75840	0.168496
$269$	13.3351	0.813057	0.406528	−	0.913638i	$-0.366739\pi$
$269$	13.3351	0.813057	0.406528	+	0.913638i	$0.366739\pi$
$270$	0	0
$271$	−14.4601	−0.878389	−0.439194	−	0.898392i	$-0.644736\pi$
$271$	−14.4601	−0.878389	−0.439194	+	0.898392i	$0.644736\pi$
$272$	17.3569	1.05242
$273$	0	0
$274$	−2.50604	−0.151395
$275$	−0.307979	−0.0185718
$276$	0	0
$277$	−2.91723	−0.175279	−0.0876397	−	0.996152i	$-0.527932\pi$
$277$	−2.91723	−0.175279	−0.0876397	+	0.996152i	$0.527932\pi$
$278$	−20.2687	−1.21564
$279$	0	0
$280$	−1.62565	−0.0971509
$281$	26.1812	1.56184	0.780920	−	0.624632i	$-0.214751\pi$
$281$	26.1812	1.56184	0.780920	+	0.624632i	$0.214751\pi$
$282$	0	0
$283$	−4.56465	−0.271340	−0.135670	−	0.990754i	$-0.543319\pi$
$283$	−4.56465	−0.271340	−0.135670	+	0.990754i	$0.543319\pi$
$284$	3.83340	0.227470
$285$	0	0
$286$	−0.402829	−0.0238198
$287$	−1.14675	−0.0676906
$288$	0	0
$289$	18.5308	1.09005
$290$	−16.6528	−0.977886
$291$	0	0
$292$	−6.43727	−0.376713
$293$	32.1739	1.87962	0.939810	−	0.341698i	$-0.111002\pi$
$293$	32.1739	1.87962	0.939810	+	0.341698i	$0.111002\pi$
$294$	0	0
$295$	−3.94869	−0.229902
$296$	31.9584	1.85754
$297$	0	0
$298$	−1.24698	−0.0722356
$299$	2.82371	0.163299
$300$	0	0
$301$	1.89546	0.109253
$302$	6.17928	0.355577
$303$	0	0
$304$	21.4222	1.22865
$305$	−14.6256	−0.837462
$306$	0	0
$307$	−1.37329	−0.0783778	−0.0391889	−	0.999232i	$-0.512477\pi$
$307$	−1.37329	−0.0783778	−0.0391889	+	0.999232i	$0.512477\pi$
$308$	−0.0120816	−0.000688412 0
$309$	0	0
$310$	21.2664	1.20785
$311$	−33.3056	−1.88859	−0.944293	−	0.329105i	$-0.893253\pi$
$311$	−33.3056	−1.88859	−0.944293	+	0.329105i	$0.893253\pi$
$312$	0	0
$313$	5.27413	0.298111	0.149056	−	0.988829i	$-0.452377\pi$
$313$	5.27413	0.298111	0.149056	+	0.988829i	$0.452377\pi$
$314$	−2.50066	−0.141121
$315$	0	0
$316$	−1.37867	−0.0775560
$317$	13.5144	0.759044	0.379522	−	0.925183i	$-0.376088\pi$
$317$	13.5144	0.759044	0.379522	+	0.925183i	$0.376088\pi$
$318$	0	0
$319$	−0.679940	−0.0380693
$320$	23.9584	1.33931
$321$	0	0
$322$	−0.295897	−0.0164897
$323$	43.8528	2.44003
$324$	0	0
$325$	−5.29590	−0.293764
$326$	−5.35019	−0.296320
$327$	0	0
$328$	17.6528	0.974712
$329$	−0.916166	−0.0505099
$330$	0	0
$331$	−9.55735	−0.525320	−0.262660	−	0.964889i	$-0.584600\pi$
$331$	−9.55735	−0.525320	−0.262660	+	0.964889i	$0.584600\pi$
$332$	0.246980	0.0135548
$333$	0	0
$334$	19.5646	1.07053
$335$	−16.6853	−0.911616
$336$	0	0
$337$	12.6703	0.690193	0.345096	−	0.938567i	$-0.387846\pi$
$337$	12.6703	0.690193	0.345096	+	0.938567i	$0.387846\pi$
$338$	9.28382	0.504973
$339$	0	0
$340$	7.14138	0.387295
$341$	0.868313	0.0470218
$342$	0	0
$343$	2.76510	0.149301
$344$	−29.1782	−1.57318
$345$	0	0
$346$	10.7356	0.577147
$347$	−25.3569	−1.36123	−0.680615	−	0.732642i	$-0.738287\pi$
$347$	−25.3569	−1.36123	−0.680615	+	0.732642i	$0.738287\pi$
$348$	0	0
$349$	10.6722	0.571268	0.285634	−	0.958339i	$-0.407796\pi$
$349$	10.6722	0.571268	0.285634	+	0.958339i	$0.407796\pi$
$350$	0.554958	0.0296638
$351$	0	0
$352$	0.338110	0.0180213
$353$	8.77048	0.466805	0.233403	−	0.972380i	$-0.425014\pi$
$353$	8.77048	0.466805	0.233403	+	0.972380i	$0.425014\pi$
$354$	0	0
$355$	−23.1879	−1.23069
$356$	−4.34721	−0.230402
$357$	0	0
$358$	−6.58509	−0.348033
$359$	9.24459	0.487911	0.243955	−	0.969786i	$-0.421555\pi$
$359$	9.24459	0.487911	0.243955	+	0.969786i	$0.421555\pi$
$360$	0	0
$361$	35.1239	1.84863
$362$	28.8907	1.51846
$363$	0	0
$364$	−0.207751	−0.0108891
$365$	38.9385	2.03814
$366$	0	0
$367$	−18.8984	−0.986491	−0.493245	−	0.869890i	$-0.664190\pi$
$367$	−18.8984	−0.986491	−0.493245	+	0.869890i	$0.664190\pi$
$368$	3.48858	0.181855
$369$	0	0
$370$	−35.1866	−1.82926
$371$	2.04892	0.106374
$372$	0	0
$373$	15.1196	0.782863	0.391432	−	0.920207i	$-0.371980\pi$
$373$	15.1196	0.782863	0.391432	+	0.920207i	$0.371980\pi$
$374$	−1.01879	−0.0526802
$375$	0	0
$376$	14.1032	0.727318
$377$	−11.6920	−0.602170
$378$	0	0
$379$	−23.2905	−1.19635	−0.598177	−	0.801364i	$-0.704108\pi$
$379$	−23.2905	−1.19635	−0.598177	+	0.801364i	$0.704108\pi$
$380$	8.81402	0.452150
$381$	0	0
$382$	−4.70709	−0.240836
$383$	−8.70469	−0.444789	−0.222395	−	0.974957i	$-0.571387\pi$
$383$	−8.70469	−0.444789	−0.222395	+	0.974957i	$0.571387\pi$
$384$	0	0
$385$	0.0730805	0.00372453
$386$	14.5918	0.742703
$387$	0	0
$388$	5.42519	0.275422
$389$	−12.4286	−0.630157	−0.315079	−	0.949066i	$-0.602031\pi$
$389$	−12.4286	−0.630157	−0.315079	+	0.949066i	$0.602031\pi$
$390$	0	0
$391$	7.14138	0.361155
$392$	−21.2228	−1.07191
$393$	0	0
$394$	−21.4034	−1.07829
$395$	8.33944	0.419603
$396$	0	0
$397$	36.2597	1.81982	0.909910	−	0.414806i	$-0.136151\pi$
$397$	36.2597	1.81982	0.909910	+	0.414806i	$0.136151\pi$
$398$	−5.73258	−0.287348
$399$	0	0
$400$	−6.54288	−0.327144
$401$	−17.6612	−0.881956	−0.440978	−	0.897518i	$-0.645368\pi$
$401$	−17.6612	−0.881956	−0.440978	+	0.897518i	$0.645368\pi$
$402$	0	0
$403$	14.9312	0.743778
$404$	7.15883	0.356165
$405$	0	0
$406$	1.22521	0.0608061
$407$	−1.43668	−0.0712136
$408$	0	0
$409$	3.04652	0.150641	0.0753205	−	0.997159i	$-0.476002\pi$
$409$	3.04652	0.150641	0.0753205	+	0.997159i	$0.476002\pi$
$410$	−19.4359	−0.959873
$411$	0	0
$412$	7.35152	0.362183
$413$	0.290520	0.0142956
$414$	0	0
$415$	−1.49396	−0.0733356
$416$	5.81402	0.285056
$417$	0	0
$418$	−1.25741	−0.0615017
$419$	4.52111	0.220870	0.110435	−	0.993883i	$-0.464776\pi$
$419$	4.52111	0.220870	0.110435	+	0.993883i	$0.464776\pi$
$420$	0	0
$421$	30.8243	1.50228	0.751142	−	0.660140i	$-0.229503\pi$
$421$	30.8243	1.50228	0.751142	+	0.660140i	$0.229503\pi$
$422$	−2.74764	−0.133753
$423$	0	0
$424$	−31.5405	−1.53174
$425$	−13.3937	−0.649691
$426$	0	0
$427$	1.07606	0.0520744
$428$	−6.72886	−0.325252
$429$	0	0
$430$	32.1256	1.54923
$431$	0.646088	0.0311210	0.0155605	−	0.999879i	$-0.495047\pi$
$431$	0.646088	0.0311210	0.0155605	+	0.999879i	$0.495047\pi$
$432$	0	0
$433$	−5.14244	−0.247130	−0.123565	−	0.992336i	$-0.539433\pi$
$433$	−5.14244	−0.247130	−0.123565	+	0.992336i	$0.539433\pi$
$434$	−1.56465	−0.0751055
$435$	0	0
$436$	−1.90754	−0.0913547
$437$	8.81402	0.421632
$438$	0	0
$439$	20.3870	0.973020	0.486510	−	0.873675i	$-0.338270\pi$
$439$	20.3870	0.973020	0.486510	+	0.873675i	$0.338270\pi$
$440$	−1.12498	−0.0536314
$441$	0	0
$442$	−17.5187	−0.833280
$443$	−2.36658	−0.112440	−0.0562199	−	0.998418i	$-0.517905\pi$
$443$	−2.36658	−0.112440	−0.0562199	+	0.998418i	$0.517905\pi$
$444$	0	0
$445$	26.2959	1.24655
$446$	3.09485	0.146545
$447$	0	0
$448$	−1.76271	−0.0832802
$449$	25.9119	1.22286	0.611428	−	0.791300i	$-0.290595\pi$
$449$	25.9119	1.22286	0.611428	+	0.791300i	$0.290595\pi$
$450$	0	0
$451$	−0.793577	−0.0373681
$452$	2.16660	0.101908
$453$	0	0
$454$	−17.8605	−0.838237
$455$	1.25667	0.0589135
$456$	0	0
$457$	28.0877	1.31389	0.656943	−	0.753940i	$-0.271849\pi$
$457$	28.0877	1.31389	0.656943	+	0.753940i	$0.271849\pi$
$458$	19.7995	0.925172
$459$	0	0
$460$	1.43535	0.0669237
$461$	−1.77346	−0.0825984	−0.0412992	−	0.999147i	$-0.513150\pi$
$461$	−1.77346	−0.0825984	−0.0412992	+	0.999147i	$0.513150\pi$
$462$	0	0
$463$	−5.40581	−0.251229	−0.125615	−	0.992079i	$-0.540090\pi$
$463$	−5.40581	−0.251229	−0.125615	+	0.992079i	$0.540090\pi$
$464$	−14.4450	−0.670594
$465$	0	0
$466$	−28.8455	−1.33624
$467$	−15.0465	−0.696270	−0.348135	−	0.937444i	$-0.613185\pi$
$467$	−15.0465	−0.696270	−0.348135	+	0.937444i	$0.613185\pi$
$468$	0	0
$469$	1.22760	0.0566854
$470$	−15.5278	−0.716245
$471$	0	0
$472$	−4.47219	−0.205849
$473$	1.31170	0.0603120
$474$	0	0
$475$	−16.5308	−0.758485
$476$	−0.525418	−0.0240825
$477$	0	0
$478$	−22.8062	−1.04313
$479$	−32.7942	−1.49840	−0.749202	−	0.662342i	$-0.769563\pi$
$479$	−32.7942	−1.49840	−0.749202	+	0.662342i	$0.769563\pi$
$480$	0	0
$481$	−24.7047	−1.12644
$482$	−20.8455	−0.949486
$483$	0	0
$484$	4.88710	0.222141
$485$	−32.8165	−1.49012
$486$	0	0
$487$	20.7090	0.938415	0.469207	−	0.883088i	$-0.344540\pi$
$487$	20.7090	0.938415	0.469207	+	0.883088i	$0.344540\pi$
$488$	−16.5646	−0.749846
$489$	0	0
$490$	23.3666	1.05559
$491$	−11.2731	−0.508746	−0.254373	−	0.967106i	$-0.581869\pi$
$491$	−11.2731	−0.508746	−0.254373	+	0.967106i	$0.581869\pi$
$492$	0	0
$493$	−29.5700	−1.33177
$494$	−21.6219	−0.972816
$495$	0	0
$496$	18.4470	0.828293
$497$	1.70602	0.0765255
$498$	0	0
$499$	−27.4838	−1.23034	−0.615172	−	0.788393i	$-0.710913\pi$
$499$	−27.4838	−1.23034	−0.615172	+	0.788393i	$0.710913\pi$
$500$	3.29829	0.147504
$501$	0	0
$502$	26.3749	1.17717
$503$	−5.96077	−0.265778	−0.132889	−	0.991131i	$-0.542425\pi$
$503$	−5.96077	−0.265778	−0.132889	+	0.991131i	$0.542425\pi$
$504$	0	0
$505$	−43.3032	−1.92697
$506$	−0.204767	−0.00910301
$507$	0	0
$508$	−2.10321	−0.0933149
$509$	−4.58642	−0.203289	−0.101645	−	0.994821i	$-0.532410\pi$
$509$	−4.58642	−0.203289	−0.101645	+	0.994821i	$0.532410\pi$
$510$	0	0
$511$	−2.86486	−0.126734
$512$	24.9390	1.10216
$513$	0	0
$514$	11.5617	0.509963
$515$	−44.4687	−1.95953
$516$	0	0
$517$	−0.634006	−0.0278836
$518$	2.58881	0.113746
$519$	0	0
$520$	−19.3448	−0.848326
$521$	0.650400	0.0284945	0.0142473	−	0.999899i	$-0.495465\pi$
$521$	0.650400	0.0284945	0.0142473	+	0.999899i	$0.495465\pi$
$522$	0	0
$523$	40.6045	1.77551	0.887755	−	0.460317i	$-0.152264\pi$
$523$	40.6045	1.77551	0.887755	+	0.460317i	$0.152264\pi$
$524$	0.794168	0.0346934
$525$	0	0
$526$	21.9517	0.957138
$527$	37.7622	1.64495
$528$	0	0
$529$	−21.5646	−0.937593
$530$	34.7265	1.50842
$531$	0	0
$532$	−0.648481	−0.0281152
$533$	−13.6461	−0.591078
$534$	0	0
$535$	40.7023	1.75971
$536$	−18.8974	−0.816242
$537$	0	0
$538$	−16.6286	−0.716911
$539$	0.954067	0.0410946
$540$	0	0
$541$	−1.88530	−0.0810553	−0.0405276	−	0.999178i	$-0.512904\pi$
$541$	−1.88530	−0.0810553	−0.0405276	+	0.999178i	$0.512904\pi$
$542$	18.0315	0.774517
$543$	0	0
$544$	14.7041	0.630433
$545$	11.5386	0.494258
$546$	0	0
$547$	−26.1051	−1.11617	−0.558087	−	0.829782i	$-0.688465\pi$
$547$	−26.1051	−1.11617	−0.558087	+	0.829782i	$0.688465\pi$
$548$	−0.894396	−0.0382067
$549$	0	0
$550$	0.384043	0.0163756
$551$	−36.4959	−1.55478
$552$	0	0
$553$	−0.613564	−0.0260914
$554$	3.63773	0.154552
$555$	0	0
$556$	−7.23383	−0.306783
$557$	−4.87071	−0.206378	−0.103189	−	0.994662i	$-0.532905\pi$
$557$	−4.87071	−0.206378	−0.103189	+	0.994662i	$0.532905\pi$
$558$	0	0
$559$	22.5555	0.953998
$560$	1.55257	0.0656079
$561$	0	0
$562$	−32.6474	−1.37715
$563$	44.5840	1.87899	0.939496	−	0.342559i	$-0.111294\pi$
$563$	44.5840	1.87899	0.939496	+	0.342559i	$0.111294\pi$
$564$	0	0
$565$	−13.1056	−0.551357
$566$	5.69202	0.239254
$567$	0	0
$568$	−26.2620	−1.10193
$569$	5.94092	0.249056	0.124528	−	0.992216i	$-0.460258\pi$
$569$	5.94092	0.249056	0.124528	+	0.992216i	$0.460258\pi$
$570$	0	0
$571$	−11.5308	−0.482549	−0.241274	−	0.970457i	$-0.577565\pi$
$571$	−11.5308	−0.482549	−0.241274	+	0.970457i	$0.577565\pi$
$572$	−0.143768	−0.00601125
$573$	0	0
$574$	1.42998	0.0596861
$575$	−2.69202	−0.112265
$576$	0	0
$577$	15.0519	0.626619	0.313309	−	0.949651i	$-0.398562\pi$
$577$	15.0519	0.626619	0.313309	+	0.949651i	$0.398562\pi$
$578$	−23.1075	−0.961146
$579$	0	0
$580$	−5.94331	−0.246783
$581$	0.109916	0.00456010
$582$	0	0
$583$	1.41789	0.0587232
$584$	44.1008	1.82490
$585$	0	0
$586$	−40.1202	−1.65735
$587$	22.0030	0.908160	0.454080	−	0.890961i	$-0.349968\pi$
$587$	22.0030	0.908160	0.454080	+	0.890961i	$0.349968\pi$
$588$	0	0
$589$	46.6069	1.92040
$590$	4.92394	0.202715
$591$	0	0
$592$	−30.5217	−1.25443
$593$	7.12200	0.292465	0.146233	−	0.989250i	$-0.453285\pi$
$593$	7.12200	0.292465	0.146233	+	0.989250i	$0.453285\pi$
$594$	0	0
$595$	3.17821	0.130294
$596$	−0.445042	−0.0182296
$597$	0	0
$598$	−3.52111	−0.143989
$599$	8.51871	0.348065	0.174033	−	0.984740i	$-0.444320\pi$
$599$	8.51871	0.348065	0.174033	+	0.984740i	$0.444320\pi$
$600$	0	0
$601$	−41.3594	−1.68709	−0.843543	−	0.537062i	$-0.819534\pi$
$601$	−41.3594	−1.68709	−0.843543	+	0.537062i	$0.819534\pi$
$602$	−2.36360	−0.0963332
$603$	0	0
$604$	2.20536	0.0897347
$605$	−29.5617	−1.20185
$606$	0	0
$607$	33.5013	1.35977	0.679887	−	0.733317i	$-0.262029\pi$
$607$	33.5013	1.35977	0.679887	+	0.733317i	$0.262029\pi$
$608$	18.1481	0.736002
$609$	0	0
$610$	18.2379	0.738430
$611$	−10.9022	−0.441054
$612$	0	0
$613$	−1.85995	−0.0751228	−0.0375614	−	0.999294i	$-0.511959\pi$
$613$	−1.85995	−0.0751228	−0.0375614	+	0.999294i	$0.511959\pi$
$614$	1.71246	0.0691094
$615$	0	0
$616$	0.0827692	0.00333487
$617$	5.95108	0.239582	0.119791	−	0.992799i	$-0.461778\pi$
$617$	5.95108	0.239582	0.119791	+	0.992799i	$0.461778\pi$
$618$	0	0
$619$	−34.4510	−1.38470	−0.692351	−	0.721560i	$-0.743425\pi$
$619$	−34.4510	−1.38470	−0.692351	+	0.721560i	$0.743425\pi$
$620$	7.58987	0.304817
$621$	0	0
$622$	41.5314	1.66526
$623$	−1.93469	−0.0775117
$624$	0	0
$625$	−31.1860	−1.24744
$626$	−6.57673	−0.262859
$627$	0	0
$628$	−0.892477	−0.0356137
$629$	−62.4801	−2.49124
$630$	0	0
$631$	−2.43727	−0.0970263	−0.0485131	−	0.998823i	$-0.515448\pi$
$631$	−2.43727	−0.0970263	−0.0485131	+	0.998823i	$0.515448\pi$
$632$	9.44504	0.375704
$633$	0	0
$634$	−16.8522	−0.669286
$635$	12.7222	0.504863
$636$	0	0
$637$	16.4058	0.650022
$638$	0.847871	0.0335675
$639$	0	0
$640$	−16.5942	−0.655943
$641$	10.1709	0.401727	0.200863	−	0.979619i	$-0.435625\pi$
$641$	10.1709	0.401727	0.200863	+	0.979619i	$0.435625\pi$
$642$	0	0
$643$	29.6644	1.16985	0.584925	−	0.811087i	$-0.301124\pi$
$643$	29.6644	1.16985	0.584925	+	0.811087i	$0.301124\pi$
$644$	−0.105604	−0.00416140
$645$	0	0
$646$	−54.6835	−2.15149
$647$	5.43967	0.213855	0.106928	−	0.994267i	$-0.465899\pi$
$647$	5.43967	0.213855	0.106928	+	0.994267i	$0.465899\pi$
$648$	0	0
$649$	0.201046	0.00789175
$650$	6.60388	0.259025
$651$	0	0
$652$	−1.90946	−0.0747803
$653$	1.53856	0.0602087	0.0301043	−	0.999547i	$-0.490416\pi$
$653$	1.53856	0.0602087	0.0301043	+	0.999547i	$0.490416\pi$
$654$	0	0
$655$	−4.80386	−0.187702
$656$	−16.8592	−0.658242
$657$	0	0
$658$	1.14244	0.0445370
$659$	−21.0325	−0.819311	−0.409655	−	0.912240i	$-0.634351\pi$
$659$	−21.0325	−0.819311	−0.409655	+	0.912240i	$0.634351\pi$
$660$	0	0
$661$	−14.4179	−0.560791	−0.280396	−	0.959885i	$-0.590466\pi$
$661$	−14.4179	−0.560791	−0.280396	+	0.959885i	$0.590466\pi$
$662$	11.9178	0.463199
$663$	0	0
$664$	−1.69202	−0.0656632
$665$	3.92261	0.152112
$666$	0	0
$667$	−5.94331	−0.230126
$668$	6.98254	0.270163
$669$	0	0
$670$	20.8062	0.803816
$671$	0.744660	0.0287473
$672$	0	0
$673$	−35.2054	−1.35707	−0.678533	−	0.734570i	$-0.737384\pi$
$673$	−35.2054	−1.35707	−0.678533	+	0.734570i	$0.737384\pi$
$674$	−15.7995	−0.608576
$675$	0	0
$676$	3.31336	0.127437
$677$	−10.1239	−0.389094	−0.194547	−	0.980893i	$-0.562324\pi$
$677$	−10.1239	−0.389094	−0.194547	+	0.980893i	$0.562324\pi$
$678$	0	0
$679$	2.41444	0.0926576
$680$	−48.9245	−1.87617
$681$	0	0
$682$	−1.08277	−0.0414614
$683$	−39.6364	−1.51664	−0.758322	−	0.651880i	$-0.773981\pi$
$683$	−39.6364	−1.51664	−0.758322	+	0.651880i	$0.773981\pi$
$684$	0	0
$685$	5.41013	0.206710
$686$	−3.44803	−0.131646
$687$	0	0
$688$	27.8665	1.06240
$689$	24.3817	0.928867
$690$	0	0
$691$	−22.0664	−0.839444	−0.419722	−	0.907653i	$-0.637873\pi$
$691$	−22.0664	−0.839444	−0.419722	+	0.907653i	$0.637873\pi$
$692$	3.83148	0.145651
$693$	0	0
$694$	31.6195	1.20026
$695$	43.7569	1.65979
$696$	0	0
$697$	−34.5120	−1.30724
$698$	−13.3080	−0.503714
$699$	0	0
$700$	0.198062	0.00748605
$701$	9.13946	0.345192	0.172596	−	0.984993i	$-0.444784\pi$
$701$	9.13946	0.345192	0.172596	+	0.984993i	$0.444784\pi$
$702$	0	0
$703$	−77.1141	−2.90841
$704$	−1.21983	−0.0459742
$705$	0	0
$706$	−10.9366	−0.411605
$707$	3.18598	0.119821
$708$	0	0
$709$	12.9815	0.487530	0.243765	−	0.969834i	$-0.421618\pi$
$709$	12.9815	0.487530	0.243765	+	0.969834i	$0.421618\pi$
$710$	28.9148	1.08515
$711$	0	0
$712$	29.7821	1.11613
$713$	7.58987	0.284243
$714$	0	0
$715$	0.869641	0.0325227
$716$	−2.35019	−0.0878308
$717$	0	0
$718$	−11.5278	−0.430214
$719$	6.84787	0.255383	0.127691	−	0.991814i	$-0.459243\pi$
$719$	6.84787	0.255383	0.127691	+	0.991814i	$0.459243\pi$
$720$	0	0
$721$	3.27173	0.121846
$722$	−43.7988	−1.63002
$723$	0	0
$724$	10.3110	0.383204
$725$	11.1468	0.413980
$726$	0	0
$727$	−1.76271	−0.0653753	−0.0326876	−	0.999466i	$-0.510407\pi$
$727$	−1.76271	−0.0653753	−0.0326876	+	0.999466i	$0.510407\pi$
$728$	1.42327	0.0527500
$729$	0	0
$730$	−48.5555	−1.79712
$731$	57.0447	2.10988
$732$	0	0
$733$	−44.5851	−1.64679	−0.823394	−	0.567470i	$-0.807922\pi$
$733$	−44.5851	−1.64679	−0.823394	+	0.567470i	$0.807922\pi$
$734$	23.5660	0.869836
$735$	0	0
$736$	2.95539	0.108937
$737$	0.849527	0.0312927
$738$	0	0
$739$	−5.46250	−0.200941	−0.100471	−	0.994940i	$-0.532035\pi$
$739$	−5.46250	−0.200941	−0.100471	+	0.994940i	$0.532035\pi$
$740$	−12.5579	−0.461639
$741$	0	0
$742$	−2.55496	−0.0937955
$743$	−14.3351	−0.525905	−0.262952	−	0.964809i	$-0.584696\pi$
$743$	−14.3351	−0.525905	−0.262952	+	0.964809i	$0.584696\pi$
$744$	0	0
$745$	2.69202	0.0986280
$746$	−18.8538	−0.690288
$747$	0	0
$748$	−0.363601	−0.0132946
$749$	−2.99462	−0.109421
$750$	0	0
$751$	25.4136	0.927355	0.463677	−	0.886004i	$-0.346530\pi$
$751$	25.4136	0.927355	0.463677	+	0.886004i	$0.346530\pi$
$752$	−13.4692	−0.491171
$753$	0	0
$754$	14.5797	0.530962
$755$	−13.3400	−0.485493
$756$	0	0
$757$	−37.4704	−1.36188	−0.680942	−	0.732337i	$-0.738430\pi$
$757$	−37.4704	−1.36188	−0.680942	+	0.732337i	$0.738430\pi$
$758$	29.0428	1.05488
$759$	0	0
$760$	−60.3836	−2.19034
$761$	−1.13514	−0.0411490	−0.0205745	−	0.999788i	$-0.506550\pi$
$761$	−1.13514	−0.0411490	−0.0205745	+	0.999788i	$0.506550\pi$
$762$	0	0
$763$	−0.848936	−0.0307336
$764$	−1.67994	−0.0607781
$765$	0	0
$766$	10.8546	0.392192
$767$	3.45712	0.124829
$768$	0	0
$769$	9.50796	0.342866	0.171433	−	0.985196i	$-0.445160\pi$
$769$	9.50796	0.342866	0.171433	+	0.985196i	$0.445160\pi$
$770$	−0.0911299	−0.00328409
$771$	0	0
$772$	5.20775	0.187431
$773$	39.0374	1.40408	0.702039	−	0.712138i	$-0.252273\pi$
$773$	39.0374	1.40408	0.702039	+	0.712138i	$0.252273\pi$
$774$	0	0
$775$	−14.2349	−0.511333
$776$	−37.1672	−1.33422
$777$	0	0
$778$	15.4983	0.555640
$779$	−42.5954	−1.52614
$780$	0	0
$781$	1.18060	0.0422453
$782$	−8.90515	−0.318447
$783$	0	0
$784$	20.2687	0.723884
$785$	5.39852	0.192681
$786$	0	0
$787$	3.91590	0.139587	0.0697934	−	0.997561i	$-0.477766\pi$
$787$	3.91590	0.139587	0.0697934	+	0.997561i	$0.477766\pi$
$788$	−7.63879	−0.272121
$789$	0	0
$790$	−10.3991	−0.369984
$791$	0.964229	0.0342840
$792$	0	0
$793$	12.8049	0.454716
$794$	−45.2150	−1.60462
$795$	0	0
$796$	−2.04593	−0.0725162
$797$	−16.8616	−0.597269	−0.298634	−	0.954368i	$-0.596531\pi$
$797$	−16.8616	−0.597269	−0.298634	+	0.954368i	$0.596531\pi$
$798$	0	0
$799$	−27.5724	−0.975442
$800$	−5.54288	−0.195970
$801$	0	0
$802$	22.0231	0.777663
$803$	−1.98254	−0.0699624
$804$	0	0
$805$	0.638792	0.0225145
$806$	−18.6189	−0.655824
$807$	0	0
$808$	−49.0441	−1.72537
$809$	−12.5211	−0.440219	−0.220109	−	0.975475i	$-0.570641\pi$
$809$	−12.5211	−0.440219	−0.220109	+	0.975475i	$0.570641\pi$
$810$	0	0
$811$	−52.5086	−1.84382	−0.921912	−	0.387399i	$-0.873374\pi$
$811$	−52.5086	−1.84382	−0.921912	+	0.387399i	$0.873374\pi$
$812$	0.437272	0.0153452
$813$	0	0
$814$	1.79151	0.0627925
$815$	11.5502	0.404585
$816$	0	0
$817$	70.4057	2.46318
$818$	−3.79895	−0.132827
$819$	0	0
$820$	−6.93661	−0.242237
$821$	−16.3217	−0.569632	−0.284816	−	0.958582i	$-0.591932\pi$
$821$	−16.3217	−0.569632	−0.284816	+	0.958582i	$0.591932\pi$
$822$	0	0
$823$	−5.38942	−0.187863	−0.0939317	−	0.995579i	$-0.529944\pi$
$823$	−5.38942	−0.187863	−0.0939317	+	0.995579i	$0.529944\pi$
$824$	−50.3642	−1.75452
$825$	0	0
$826$	−0.362273	−0.0126051
$827$	48.4292	1.68405	0.842025	−	0.539439i	$-0.181364\pi$
$827$	48.4292	1.68405	0.842025	+	0.539439i	$0.181364\pi$
$828$	0	0
$829$	29.2567	1.01613	0.508063	−	0.861320i	$-0.330362\pi$
$829$	29.2567	1.01613	0.508063	+	0.861320i	$0.330362\pi$
$830$	1.86294	0.0646635
$831$	0	0
$832$	−20.9758	−0.727206
$833$	41.4916	1.43760
$834$	0	0
$835$	−42.2368	−1.46167
$836$	−0.448763	−0.0155208
$837$	0	0
$838$	−5.63773	−0.194752
$839$	−6.27891	−0.216772	−0.108386	−	0.994109i	$-0.534568\pi$
$839$	−6.27891	−0.216772	−0.108386	+	0.994109i	$0.534568\pi$
$840$	0	0
$841$	−4.39075	−0.151405
$842$	−38.4373	−1.32464
$843$	0	0
$844$	−0.980623	−0.0337544
$845$	−20.0422	−0.689473
$846$	0	0
$847$	2.17496	0.0747326
$848$	30.1226	1.03441
$849$	0	0
$850$	16.7017	0.572864
$851$	−12.5579	−0.430481
$852$	0	0
$853$	0.702775	0.0240626	0.0120313	−	0.999928i	$-0.496170\pi$
$853$	0.702775	0.0240626	0.0120313	+	0.999928i	$0.496170\pi$
$854$	−1.34183	−0.0459165
$855$	0	0
$856$	46.0984	1.57561
$857$	14.4789	0.494590	0.247295	−	0.968940i	$-0.420458\pi$
$857$	14.4789	0.494590	0.247295	+	0.968940i	$0.420458\pi$
$858$	0	0
$859$	−27.8062	−0.948737	−0.474368	−	0.880326i	$-0.657324\pi$
$859$	−27.8062	−0.948737	−0.474368	+	0.880326i	$0.657324\pi$
$860$	11.4655	0.390970
$861$	0	0
$862$	−0.805659	−0.0274408
$863$	39.1269	1.33190	0.665948	−	0.745999i	$-0.268027\pi$
$863$	39.1269	1.33190	0.665948	+	0.745999i	$0.268027\pi$
$864$	0	0
$865$	−23.1763	−0.788018
$866$	6.41252	0.217906
$867$	0	0
$868$	−0.558416	−0.0189539
$869$	−0.424600	−0.0144036
$870$	0	0
$871$	14.6082	0.494980
$872$	13.0683	0.442548
$873$	0	0
$874$	−10.9909	−0.371773
$875$	1.46788	0.0496233
$876$	0	0
$877$	17.4257	0.588423	0.294211	−	0.955740i	$-0.404943\pi$
$877$	17.4257	0.588423	0.294211	+	0.955740i	$0.404943\pi$
$878$	−25.4222	−0.857958
$879$	0	0
$880$	1.07441	0.0362183
$881$	12.5405	0.422500	0.211250	−	0.977432i	$-0.432247\pi$
$881$	12.5405	0.422500	0.211250	+	0.977432i	$0.432247\pi$
$882$	0	0
$883$	38.8678	1.30801	0.654003	−	0.756492i	$-0.273088\pi$
$883$	38.8678	1.30801	0.654003	+	0.756492i	$0.273088\pi$
$884$	−6.25236	−0.210290
$885$	0	0
$886$	2.95108	0.0991436
$887$	6.49694	0.218146	0.109073	−	0.994034i	$-0.465212\pi$
$887$	6.49694	0.218146	0.109073	+	0.994034i	$0.465212\pi$
$888$	0	0
$889$	−0.936017	−0.0313930
$890$	−32.7904	−1.09914
$891$	0	0
$892$	1.10454	0.0369827
$893$	−34.0304	−1.13878
$894$	0	0
$895$	14.2161	0.475192
$896$	1.22090	0.0407873
$897$	0	0
$898$	−32.3116	−1.07825
$899$	−31.4271	−1.04815
$900$	0	0
$901$	61.6631	2.05429
$902$	0.989574	0.0329492
$903$	0	0
$904$	−14.8431	−0.493674
$905$	−62.3702	−2.07325
$906$	0	0
$907$	41.5754	1.38049	0.690244	−	0.723576i	$-0.257503\pi$
$907$	41.5754	1.38049	0.690244	+	0.723576i	$0.257503\pi$
$908$	−6.37435	−0.211540
$909$	0	0
$910$	−1.56704	−0.0519469
$911$	12.3757	0.410025	0.205012	−	0.978759i	$-0.434277\pi$
$911$	12.3757	0.410025	0.205012	+	0.978759i	$0.434277\pi$
$912$	0	0
$913$	0.0760644	0.00251736
$914$	−35.0248	−1.15852
$915$	0	0
$916$	7.06638	0.233480
$917$	0.353438	0.0116716
$918$	0	0
$919$	−6.23357	−0.205627	−0.102813	−	0.994701i	$-0.532784\pi$
$919$	−6.23357	−0.205627	−0.102813	+	0.994701i	$0.532784\pi$
$920$	−9.83340	−0.324197
$921$	0	0
$922$	2.21147	0.0728309
$923$	20.3013	0.668225
$924$	0	0
$925$	23.5526	0.774404
$926$	6.74094	0.221521
$927$	0	0
$928$	−12.2373	−0.401709
$929$	−8.72886	−0.286385	−0.143192	−	0.989695i	$-0.545737\pi$
$929$	−8.72886	−0.286385	−0.143192	+	0.989695i	$0.545737\pi$
$930$	0	0
$931$	51.2097	1.67833
$932$	−10.2948	−0.337218
$933$	0	0
$934$	18.7627	0.613935
$935$	2.19939	0.0719278
$936$	0	0
$937$	13.7584	0.449467	0.224734	−	0.974420i	$-0.427849\pi$
$937$	13.7584	0.449467	0.224734	+	0.974420i	$0.427849\pi$
$938$	−1.53079	−0.0499822
$939$	0	0
$940$	−5.54181	−0.180754
$941$	21.5883	0.703760	0.351880	−	0.936045i	$-0.385543\pi$
$941$	21.5883	0.703760	0.351880	+	0.936045i	$0.385543\pi$
$942$	0	0
$943$	−6.93661	−0.225887
$944$	4.27114	0.139014
$945$	0	0
$946$	−1.63566	−0.0531800
$947$	50.7348	1.64866	0.824330	−	0.566109i	$-0.191552\pi$
$947$	50.7348	1.64866	0.824330	+	0.566109i	$0.191552\pi$
$948$	0	0
$949$	−34.0911	−1.10664
$950$	20.6136	0.668792
$951$	0	0
$952$	3.59956	0.116663
$953$	11.3327	0.367103	0.183552	−	0.983010i	$-0.441241\pi$
$953$	11.3327	0.367103	0.183552	+	0.983010i	$0.441241\pi$
$954$	0	0
$955$	10.1618	0.328829
$956$	−8.13946	−0.263249
$957$	0	0
$958$	40.8937	1.32121
$959$	−0.398043	−0.0128535
$960$	0	0
$961$	9.13382	0.294639
$962$	30.8062	0.993233
$963$	0	0
$964$	−7.43967	−0.239615
$965$	−31.5013	−1.01406
$966$	0	0
$967$	−4.83041	−0.155336	−0.0776678	−	0.996979i	$-0.524747\pi$
$967$	−4.83041	−0.155336	−0.0776678	+	0.996979i	$0.524747\pi$
$968$	−33.4808	−1.07611
$969$	0	0
$970$	40.9215	1.31391
$971$	−26.5961	−0.853510	−0.426755	−	0.904367i	$-0.640343\pi$
$971$	−26.5961	−0.853510	−0.426755	+	0.904367i	$0.640343\pi$
$972$	0	0
$973$	−3.21936	−0.103208
$974$	−25.8237	−0.827445
$975$	0	0
$976$	15.8200	0.506385
$977$	−3.26577	−0.104481	−0.0522406	−	0.998635i	$-0.516636\pi$
$977$	−3.26577	−0.104481	−0.0522406	+	0.998635i	$0.516636\pi$
$978$	0	0
$979$	−1.33885	−0.0427897
$980$	8.33944	0.266394
$981$	0	0
$982$	14.0573	0.448586
$983$	−19.7928	−0.631294	−0.315647	−	0.948877i	$-0.602221\pi$
$983$	−19.7928	−0.631294	−0.315647	+	0.948877i	$0.602221\pi$
$984$	0	0
$985$	46.2064	1.47226
$986$	36.8732	1.17428
$987$	0	0
$988$	−7.71678	−0.245503
$989$	11.4655	0.364581
$990$	0	0
$991$	−33.6937	−1.07031	−0.535157	−	0.844752i	$-0.679748\pi$
$991$	−33.6937	−1.07031	−0.535157	+	0.844752i	$0.679748\pi$
$992$	15.6276	0.496176
$993$	0	0
$994$	−2.12737	−0.0674762
$995$	12.3757	0.392336
$996$	0	0
$997$	39.9729	1.26595	0.632976	−	0.774171i	$-0.281833\pi$
$997$	39.9729	1.26595	0.632976	+	0.774171i	$0.281833\pi$
$998$	34.2717	1.08485
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1341.2.a.b.1.1		3
3.2	odd	2		149.2.a.a.1.3	✓	3
12.11	even	2		2384.2.a.e.1.3		3
15.14	odd	2		3725.2.a.b.1.1		3
21.20	even	2		7301.2.a.f.1.3		3
24.5	odd	2		9536.2.a.m.1.3		3
24.11	even	2		9536.2.a.j.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
149.2.a.a.1.3	✓	3	3.2	odd	2
1341.2.a.b.1.1		3	1.1	even	1	trivial
2384.2.a.e.1.3		3	12.11	even	2
3725.2.a.b.1.1		3	15.14	odd	2
7301.2.a.f.1.3		3	21.20	even	2
9536.2.a.j.1.1		3	24.11	even	2
9536.2.a.m.1.3		3	24.5	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:	\( N \)	\(=\)	\( 1341 = 3^{2} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1341.a (trivial)

\(f(q)\)	\(=\)	\(q-1.24698 q^{2} -0.445042 q^{4} +2.69202 q^{5} -0.198062 q^{7} +3.04892 q^{8} +O(q^{10})\)	\(q-1.24698 q^{2} -0.445042 q^{4} +2.69202 q^{5} -0.198062 q^{7} +3.04892 q^{8} -3.35690 q^{10} -0.137063 q^{11} -2.35690 q^{13} +0.246980 q^{14} -2.91185 q^{16} -5.96077 q^{17} -7.35690 q^{19} -1.19806 q^{20} +0.170915 q^{22} -1.19806 q^{23} +2.24698 q^{25} +2.93900 q^{26} +0.0881460 q^{28} +4.96077 q^{29} -6.33513 q^{31} -2.46681 q^{32} +7.43296 q^{34} -0.533188 q^{35} +10.4819 q^{37} +9.17390 q^{38} +8.20775 q^{40} +5.78986 q^{41} -9.57002 q^{43} +0.0609989 q^{44} +1.49396 q^{46} +4.62565 q^{47} -6.96077 q^{49} -2.80194 q^{50} +1.04892 q^{52} -10.3448 q^{53} -0.368977 q^{55} -0.603875 q^{56} -6.18598 q^{58} -1.46681 q^{59} -5.43296 q^{61} +7.89977 q^{62} +8.89977 q^{64} -6.34481 q^{65} -6.19806 q^{67} +2.65279 q^{68} +0.664874 q^{70} -8.61356 q^{71} +14.4644 q^{73} -13.0707 q^{74} +3.27413 q^{76} +0.0271471 q^{77} +3.09783 q^{79} -7.83877 q^{80} -7.21983 q^{82} -0.554958 q^{83} -16.0465 q^{85} +11.9336 q^{86} -0.417895 q^{88} +9.76809 q^{89} +0.466812 q^{91} +0.533188 q^{92} -5.76809 q^{94} -19.8049 q^{95} -12.1903 q^{97} +8.67994 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - q^{4} + 3 q^{5} - 5 q^{7}+O(q^{10}) \) 3 * q + q^2 - q^4 + 3 * q^5 - 5 * q^7	\( 3 q + q^{2} - q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{10} + 5 q^{11} - 3 q^{13} - 4 q^{14} - 5 q^{16} - 5 q^{17} - 18 q^{19} - 8 q^{20} + 11 q^{22} - 8 q^{23} + 2 q^{25} - q^{26} + 4 q^{28} + 2 q^{29} - 18 q^{31} - 4 q^{32} + 3 q^{34} - 5 q^{35} + 3 q^{37} - 6 q^{38} + 7 q^{40} - 6 q^{41} - 4 q^{43} + 10 q^{44} - 5 q^{46} + 2 q^{47} - 8 q^{49} - 4 q^{50} - 6 q^{52} - 8 q^{53} - 16 q^{55} + 7 q^{56} - 4 q^{58} - q^{59} + 3 q^{61} + q^{62} + 4 q^{64} + 4 q^{65} - 23 q^{67} - 10 q^{68} + 3 q^{70} + 5 q^{71} - q^{73} - 27 q^{74} - q^{76} - 6 q^{77} - 9 q^{79} + 9 q^{80} - 23 q^{82} - 2 q^{83} + 2 q^{85} + 29 q^{86} - 7 q^{88} + 9 q^{89} - 2 q^{91} + 5 q^{92} + 3 q^{94} - 11 q^{95} - 3 q^{97} + 2 q^{98}+O(q^{100}) \) 3 * q + q^2 - q^4 + 3 * q^5 - 5 * q^7 - 6 * q^10 + 5 * q^11 - 3 * q^13 - 4 * q^14 - 5 * q^16 - 5 * q^17 - 18 * q^19 - 8 * q^20 + 11 * q^22 - 8 * q^23 + 2 * q^25 - q^26 + 4 * q^28 + 2 * q^29 - 18 * q^31 - 4 * q^32 + 3 * q^34 - 5 * q^35 + 3 * q^37 - 6 * q^38 + 7 * q^40 - 6 * q^41 - 4 * q^43 + 10 * q^44 - 5 * q^46 + 2 * q^47 - 8 * q^49 - 4 * q^50 - 6 * q^52 - 8 * q^53 - 16 * q^55 + 7 * q^56 - 4 * q^58 - q^59 + 3 * q^61 + q^62 + 4 * q^64 + 4 * q^65 - 23 * q^67 - 10 * q^68 + 3 * q^70 + 5 * q^71 - q^73 - 27 * q^74 - q^76 - 6 * q^77 - 9 * q^79 + 9 * q^80 - 23 * q^82 - 2 * q^83 + 2 * q^85 + 29 * q^86 - 7 * q^88 + 9 * q^89 - 2 * q^91 + 5 * q^92 + 3 * q^94 - 11 * q^95 - 3 * q^97 + 2 * q^98

Introduction

L-functions

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