LMFDB - Embedded newform 169.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,2,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$169 = 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	169.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:	$1.34947179416$
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:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.554958 q^{2} +0.801938 q^{3} -1.69202 q^{4} -2.80194 q^{5} -0.445042 q^{6} -2.69202 q^{7} +2.04892 q^{8} -2.35690 q^{9} +1.55496 q^{10} -1.19806 q^{11} -1.35690 q^{12} +1.49396 q^{14} -2.24698 q^{15} +2.24698 q^{16} +1.13706 q^{17} +1.30798 q^{18} +1.93900 q^{19} +4.74094 q^{20} -2.15883 q^{21} +0.664874 q^{22} -4.60388 q^{23} +1.64310 q^{24} +2.85086 q^{25} -4.29590 q^{27} +4.55496 q^{28} -7.89977 q^{29} +1.24698 q^{30} +5.89977 q^{31} -5.34481 q^{32} -0.960771 q^{33} -0.631023 q^{34} +7.54288 q^{35} +3.98792 q^{36} +0.951083 q^{37} -1.07606 q^{38} -5.74094 q^{40} +3.31767 q^{41} +1.19806 q^{42} +7.15883 q^{43} +2.02715 q^{44} +6.60388 q^{45} +2.55496 q^{46} -7.69202 q^{47} +1.80194 q^{48} +0.246980 q^{49} -1.58211 q^{50} +0.911854 q^{51} +5.87263 q^{53} +2.38404 q^{54} +3.35690 q^{55} -5.51573 q^{56} +1.55496 q^{57} +4.38404 q^{58} -0.0120816 q^{59} +3.80194 q^{60} -8.03684 q^{61} -3.27413 q^{62} +6.34481 q^{63} -1.52781 q^{64} +0.533188 q^{66} -9.25667 q^{67} -1.92394 q^{68} -3.69202 q^{69} -4.18598 q^{70} -13.7409 q^{71} -4.82908 q^{72} +12.8170 q^{73} -0.527811 q^{74} +2.28621 q^{75} -3.28083 q^{76} +3.22521 q^{77} +0.807315 q^{79} -6.29590 q^{80} +3.62565 q^{81} -1.84117 q^{82} -16.3327 q^{83} +3.65279 q^{84} -3.18598 q^{85} -3.97285 q^{86} -6.33513 q^{87} -2.45473 q^{88} -14.7289 q^{89} -3.66487 q^{90} +7.78986 q^{92} +4.73125 q^{93} +4.26875 q^{94} -5.43296 q^{95} -4.28621 q^{96} +3.13169 q^{97} -0.137063 q^{98} +2.82371 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$3 q - 2 q^{2} - 2 q^{3} - 4 q^{5} - q^{6} - 3 q^{7} - 3 q^{8} - 3 q^{9} + 5 q^{10} - 8 q^{11} - 5 q^{14} - 2 q^{15} + 2 q^{16} - 2 q^{17} + 9 q^{18} - 4 q^{19} + 2 q^{21} + 3 q^{22} - 5 q^{23} + 9 q^{24}+ \cdots + q^{99}+O(q^{100})$ 3 * q - 2 * q^2 - 2 * q^3 - 4 * q^5 - q^6 - 3 * q^7 - 3 * q^8 - 3 * q^9 + 5 * q^10 - 8 * q^11 - 5 * q^14 - 2 * q^15 + 2 * q^16 - 2 * q^17 + 9 * q^18 - 4 * q^19 + 2 * q^21 + 3 * q^22 - 5 * q^23 + 9 * q^24 - 5 * q^25 + q^27 + 14 * q^28 - q^29 - q^30 - 5 * q^31 + 7 * q^32 + 10 * q^33 + 13 * q^34 + 4 * q^35 - 7 * q^36 + 12 * q^37 + 12 * q^38 - 3 * q^40 - 7 * q^41 + 8 * q^42 + 13 * q^43 + 11 * q^45 + 8 * q^46 - 18 * q^47 + q^48 - 4 * q^49 + q^50 - q^51 + q^53 - 3 * q^54 + 6 * q^55 - 4 * q^56 + 5 * q^57 + 3 * q^58 - 19 * q^59 + 7 * q^60 + 4 * q^61 + q^62 - 4 * q^63 - 11 * q^64 + 5 * q^66 - q^67 - 21 * q^68 - 6 * q^69 + 2 * q^70 - 27 * q^71 - 4 * q^72 + 9 * q^73 - 8 * q^74 + 15 * q^75 - 21 * q^76 + 8 * q^77 - 5 * q^79 - 5 * q^80 - q^81 - 14 * q^82 - 7 * q^83 - 7 * q^84 + 5 * q^85 - 18 * q^86 - 18 * q^87 + 15 * q^88 - 11 * q^89 - 12 * q^90 + 22 * q^93 + 5 * q^94 + 3 * q^95 - 21 * q^96 + 7 * q^97 + 5 * q^98 + q^99

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

)

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

$2$	−0.554958	−0.392415	−0.196207	−	0.980562i	$-0.562863\pi$
$2$	−0.554958	−0.392415	−0.196207	+	0.980562i	$0.562863\pi$
$3$	0.801938	0.462999	0.231499	−	0.972835i	$-0.425637\pi$
$3$	0.801938	0.462999	0.231499	+	0.972835i	$0.425637\pi$
$4$	−1.69202	−0.846011
$5$	−2.80194	−1.25306	−0.626532	−	0.779395i	$-0.715526\pi$
$5$	−2.80194	−1.25306	−0.626532	+	0.779395i	$0.715526\pi$
$6$	−0.445042	−0.181688
$7$	−2.69202	−1.01749	−0.508744	−	0.860918i	$-0.669890\pi$
$7$	−2.69202	−1.01749	−0.508744	+	0.860918i	$0.669890\pi$
$8$	2.04892	0.724402
$9$	−2.35690	−0.785632
$10$	1.55496	0.491721
$11$	−1.19806	−0.361229	−0.180615	−	0.983554i	$-0.557809\pi$
$11$	−1.19806	−0.361229	−0.180615	+	0.983554i	$0.557809\pi$
$12$	−1.35690	−0.391702
$13$	0	0
$13$	0	0
$14$	1.49396	0.399277
$15$	−2.24698	−0.580168
$16$	2.24698	0.561745
$17$	1.13706	0.275778	0.137889	−	0.990448i	$-0.455968\pi$
$17$	1.13706	0.275778	0.137889	+	0.990448i	$0.455968\pi$
$18$	1.30798	0.308293
$19$	1.93900	0.444837	0.222419	−	0.974951i	$-0.428605\pi$
$19$	1.93900	0.444837	0.222419	+	0.974951i	$0.428605\pi$
$20$	4.74094	1.06011
$21$	−2.15883	−0.471096
$22$	0.664874	0.141752
$23$	−4.60388	−0.959974	−0.479987	−	0.877275i	$-0.659359\pi$
$23$	−4.60388	−0.959974	−0.479987	+	0.877275i	$0.659359\pi$
$24$	1.64310	0.335397
$25$	2.85086	0.570171
$26$	0	0
$27$	−4.29590	−0.826746
$28$	4.55496	0.860806
$29$	−7.89977	−1.46695	−0.733475	−	0.679716i	$-0.762103\pi$
$29$	−7.89977	−1.46695	−0.733475	+	0.679716i	$0.762103\pi$
$30$	1.24698	0.227666
$31$	5.89977	1.05963	0.529815	−	0.848113i	$-0.322261\pi$
$31$	5.89977	1.05963	0.529815	+	0.848113i	$0.322261\pi$
$32$	−5.34481	−0.944839
$33$	−0.960771	−0.167249
$34$	−0.631023	−0.108219
$35$	7.54288	1.27498
$36$	3.98792	0.664653
$37$	0.951083	0.156357	0.0781785	−	0.996939i	$-0.475090\pi$
$37$	0.951083	0.156357	0.0781785	+	0.996939i	$0.475090\pi$
$38$	−1.07606	−0.174561
$39$	0	0
$40$	−5.74094	−0.907722
$41$	3.31767	0.518133	0.259066	−	0.965860i	$-0.416585\pi$
$41$	3.31767	0.518133	0.259066	+	0.965860i	$0.416585\pi$
$42$	1.19806	0.184865
$43$	7.15883	1.09171	0.545856	−	0.837879i	$-0.316205\pi$
$43$	7.15883	1.09171	0.545856	+	0.837879i	$0.316205\pi$
$44$	2.02715	0.305604
$45$	6.60388	0.984448
$46$	2.55496	0.376708
$47$	−7.69202	−1.12200	−0.560998	−	0.827817i	$-0.689583\pi$
$47$	−7.69202	−1.12200	−0.560998	+	0.827817i	$0.689583\pi$
$48$	1.80194	0.260087
$49$	0.246980	0.0352828
$50$	−1.58211	−0.223743
$51$	0.911854	0.127685
$52$	0	0
$53$	5.87263	0.806667	0.403334	−	0.915053i	$-0.367851\pi$
$53$	5.87263	0.806667	0.403334	+	0.915053i	$0.367851\pi$
$54$	2.38404	0.324427
$55$	3.35690	0.452644
$56$	−5.51573	−0.737070
$57$	1.55496	0.205959
$58$	4.38404	0.575653
$59$	−0.0120816	−0.00157289	−0.000786444	−	1.00000i	$-0.500250\pi$
$59$	−0.0120816	−0.00157289	−0.000786444		1.00000i	$0.500250\pi$
$60$	3.80194	0.490828
$61$	−8.03684	−1.02901	−0.514506	−	0.857487i	$-0.672025\pi$
$61$	−8.03684	−1.02901	−0.514506	+	0.857487i	$0.672025\pi$
$62$	−3.27413	−0.415815
$63$	6.34481	0.799371
$64$	−1.52781	−0.190976
$65$	0	0
$66$	0.533188	0.0656309
$67$	−9.25667	−1.13088	−0.565441	−	0.824789i	$-0.691294\pi$
$67$	−9.25667	−1.13088	−0.565441	+	0.824789i	$0.691294\pi$
$68$	−1.92394	−0.233311
$69$	−3.69202	−0.444467
$70$	−4.18598	−0.500320
$71$	−13.7409	−1.63075	−0.815375	−	0.578934i	$-0.803469\pi$
$71$	−13.7409	−1.63075	−0.815375	+	0.578934i	$0.803469\pi$
$72$	−4.82908	−0.569113
$73$	12.8170	1.50012	0.750058	−	0.661372i	$-0.230025\pi$
$73$	12.8170	1.50012	0.750058	+	0.661372i	$0.230025\pi$
$74$	−0.527811	−0.0613568
$75$	2.28621	0.263989
$76$	−3.28083	−0.376337
$77$	3.22521	0.367547
$78$	0	0
$79$	0.807315	0.0908300	0.0454150	−	0.998968i	$-0.485539\pi$
$79$	0.807315	0.0908300	0.0454150	+	0.998968i	$0.485539\pi$
$80$	−6.29590	−0.703903
$81$	3.62565	0.402850
$82$	−1.84117	−0.203323
$83$	−16.3327	−1.79275	−0.896375	−	0.443296i	$-0.853809\pi$
$83$	−16.3327	−1.79275	−0.896375	+	0.443296i	$0.853809\pi$
$84$	3.65279	0.398552
$85$	−3.18598	−0.345568
$86$	−3.97285	−0.428404
$87$	−6.33513	−0.679197
$88$	−2.45473	−0.261675
$89$	−14.7289	−1.56126	−0.780628	−	0.624996i	$-0.785101\pi$
$89$	−14.7289	−1.56126	−0.780628	+	0.624996i	$0.785101\pi$
$90$	−3.66487	−0.386312
$91$	0	0
$92$	7.78986	0.812149
$93$	4.73125	0.490608
$94$	4.26875	0.440288
$95$	−5.43296	−0.557410
$96$	−4.28621	−0.437459
$97$	3.13169	0.317975	0.158987	−	0.987281i	$-0.449177\pi$
$97$	3.13169	0.317975	0.158987	+	0.987281i	$0.449177\pi$
$98$	−0.137063	−0.0138455
$99$	2.82371	0.283793
$100$	−4.82371	−0.482371
$101$	5.29052	0.526426	0.263213	−	0.964738i	$-0.415218\pi$
$101$	5.29052	0.526426	0.263213	+	0.964738i	$0.415218\pi$
$102$	−0.506041	−0.0501055
$103$	−13.5308	−1.33323	−0.666614	−	0.745403i	$-0.732257\pi$
$103$	−13.5308	−1.33323	−0.666614	+	0.745403i	$0.732257\pi$
$104$	0	0
$105$	6.04892	0.590314
$106$	−3.25906	−0.316548
$107$	5.63102	0.544371	0.272186	−	0.962245i	$-0.412253\pi$
$107$	5.63102	0.544371	0.272186	+	0.962245i	$0.412253\pi$
$108$	7.26875	0.699436
$109$	4.17629	0.400016	0.200008	−	0.979794i	$-0.435903\pi$
$109$	4.17629	0.400016	0.200008	+	0.979794i	$0.435903\pi$
$110$	−1.86294	−0.177624
$111$	0.762709	0.0723931
$112$	−6.04892	−0.571569
$113$	7.64310	0.719003	0.359501	−	0.933145i	$-0.382947\pi$
$113$	7.64310	0.719003	0.359501	+	0.933145i	$0.382947\pi$
$114$	−0.862937	−0.0808214
$115$	12.8998	1.20291
$116$	13.3666	1.24106
$117$	0	0
$118$	0.00670477	0.000617224 0
$119$	−3.06100	−0.280601
$120$	−4.60388	−0.420274
$121$	−9.56465	−0.869513
$122$	4.46011	0.403799
$123$	2.66056	0.239895
$124$	−9.98254	−0.896459
$125$	6.02177	0.538604
$126$	−3.52111	−0.313685
$127$	6.77777	0.601430	0.300715	−	0.953714i	$-0.402775\pi$
$127$	6.77777	0.601430	0.300715	+	0.953714i	$0.402775\pi$
$128$	11.5375	1.01978
$129$	5.74094	0.505461
$130$	0	0
$131$	−13.6799	−1.19522	−0.597611	−	0.801786i	$-0.703883\pi$
$131$	−13.6799	−1.19522	−0.597611	+	0.801786i	$0.703883\pi$
$132$	1.62565	0.141494
$133$	−5.21983	−0.452617
$134$	5.13706	0.443775
$135$	12.0368	1.03597
$136$	2.32975	0.199774
$137$	12.9879	1.10963	0.554816	−	0.831973i	$-0.312789\pi$
$137$	12.9879	1.10963	0.554816	+	0.831973i	$0.312789\pi$
$138$	2.04892	0.174415
$139$	12.0465	1.02177	0.510886	−	0.859648i	$-0.329317\pi$
$139$	12.0465	1.02177	0.510886	+	0.859648i	$0.329317\pi$
$140$	−12.7627	−1.07865
$141$	−6.16852	−0.519483
$142$	7.62565	0.639930
$143$	0	0
$144$	−5.29590	−0.441325
$145$	22.1347	1.83818
$146$	−7.11290	−0.588668
$147$	0.198062	0.0163359
$148$	−1.60925	−0.132280
$149$	−0.740939	−0.0607001	−0.0303500	−	0.999539i	$-0.509662\pi$
$149$	−0.740939	−0.0607001	−0.0303500	+	0.999539i	$0.509662\pi$
$150$	−1.26875	−0.103593
$151$	−19.0737	−1.55219	−0.776097	−	0.630614i	$-0.782803\pi$
$151$	−19.0737	−1.55219	−0.776097	+	0.630614i	$0.782803\pi$
$152$	3.97285	0.322241
$153$	−2.67994	−0.216660
$154$	−1.78986	−0.144231
$155$	−16.5308	−1.32779
$156$	0	0
$157$	−4.02177	−0.320972	−0.160486	−	0.987038i	$-0.551306\pi$
$157$	−4.02177	−0.320972	−0.160486	+	0.987038i	$0.551306\pi$
$158$	−0.448026	−0.0356430
$159$	4.70948	0.373486
$160$	14.9758	1.18394
$161$	12.3937	0.976763
$162$	−2.01208	−0.158084
$163$	15.1371	1.18563	0.592813	−	0.805340i	$-0.298017\pi$
$163$	15.1371	1.18563	0.592813	+	0.805340i	$0.298017\pi$
$164$	−5.61356	−0.438346
$165$	2.69202	0.209574
$166$	9.06398	0.703502
$167$	−6.26337	−0.484674	−0.242337	−	0.970192i	$-0.577914\pi$
$167$	−6.26337	−0.484674	−0.242337	+	0.970192i	$0.577914\pi$
$168$	−4.42327	−0.341263
$169$	0	0
$170$	1.76809	0.135606
$171$	−4.57002	−0.349478
$172$	−12.1129	−0.923600
$173$	16.3913	1.24621	0.623105	−	0.782138i	$-0.285871\pi$
$173$	16.3913	1.24621	0.623105	+	0.782138i	$0.285871\pi$
$174$	3.51573	0.266527
$175$	−7.67456	−0.580142
$176$	−2.69202	−0.202919
$177$	−0.00968868	−0.000728246 0
$178$	8.17390	0.612660
$179$	−2.45473	−0.183475	−0.0917376	−	0.995783i	$-0.529242\pi$
$179$	−2.45473	−0.183475	−0.0917376	+	0.995783i	$0.529242\pi$
$180$	−11.1739	−0.832853
$181$	11.8073	0.877631	0.438815	−	0.898577i	$-0.355398\pi$
$181$	11.8073	0.877631	0.438815	+	0.898577i	$0.355398\pi$
$182$	0	0
$183$	−6.44504	−0.476431
$184$	−9.43296	−0.695407
$185$	−2.66487	−0.195925
$186$	−2.62565	−0.192522
$187$	−1.36227	−0.0996192
$188$	13.0151	0.949221
$189$	11.5646	0.841204
$190$	3.01507	0.218736
$191$	−8.99330	−0.650732	−0.325366	−	0.945588i	$-0.605488\pi$
$191$	−8.99330	−0.650732	−0.325366	+	0.945588i	$0.605488\pi$
$192$	−1.22521	−0.0884219
$193$	13.5254	0.973581	0.486790	−	0.873519i	$-0.338168\pi$
$193$	13.5254	0.973581	0.486790	+	0.873519i	$0.338168\pi$
$194$	−1.73795	−0.124778
$195$	0	0
$196$	−0.417895	−0.0298496
$197$	12.9758	0.924490	0.462245	−	0.886752i	$-0.347044\pi$
$197$	12.9758	0.924490	0.462245	+	0.886752i	$0.347044\pi$
$198$	−1.56704	−0.111365
$199$	−13.5864	−0.963116	−0.481558	−	0.876414i	$-0.659929\pi$
$199$	−13.5864	−0.963116	−0.481558	+	0.876414i	$0.659929\pi$
$200$	5.84117	0.413033
$201$	−7.42327	−0.523597
$202$	−2.93602	−0.206577
$203$	21.2664	1.49261
$204$	−1.54288	−0.108023
$205$	−9.29590	−0.649254
$206$	7.50902	0.523179
$207$	10.8509	0.754187
$208$	0	0
$209$	−2.32304	−0.160688
$210$	−3.35690	−0.231648
$211$	10.4601	0.720103	0.360052	−	0.932932i	$-0.382759\pi$
$211$	10.4601	0.720103	0.360052	+	0.932932i	$0.382759\pi$
$212$	−9.93661	−0.682449
$213$	−11.0194	−0.755035
$214$	−3.12498	−0.213619
$215$	−20.0586	−1.36799
$216$	−8.80194	−0.598896
$217$	−15.8823	−1.07816
$218$	−2.31767	−0.156972
$219$	10.2784	0.694553
$220$	−5.67994	−0.382941
$221$	0	0
$222$	−0.423272	−0.0284081
$223$	−11.4058	−0.763790	−0.381895	−	0.924206i	$-0.624728\pi$
$223$	−11.4058	−0.763790	−0.381895	+	0.924206i	$0.624728\pi$
$224$	14.3884	0.961362
$225$	−6.71917	−0.447945
$226$	−4.24160	−0.282147
$227$	10.6407	0.706249	0.353124	−	0.935576i	$-0.385119\pi$
$227$	10.6407	0.706249	0.353124	+	0.935576i	$0.385119\pi$
$228$	−2.63102	−0.174244
$229$	−1.13946	−0.0752974	−0.0376487	−	0.999291i	$-0.511987\pi$
$229$	−1.13946	−0.0752974	−0.0376487	+	0.999291i	$0.511987\pi$
$230$	−7.15883	−0.472040
$231$	2.58642	0.170174
$232$	−16.1860	−1.06266
$233$	−10.8509	−0.710863	−0.355432	−	0.934702i	$-0.615666\pi$
$233$	−10.8509	−0.710863	−0.355432	+	0.934702i	$0.615666\pi$
$234$	0	0
$235$	21.5526	1.40593
$236$	0.0204423	0.00133068
$237$	0.647416	0.0420542
$238$	1.69873	0.110112
$239$	−11.9293	−0.771643	−0.385822	−	0.922573i	$-0.626082\pi$
$239$	−11.9293	−0.771643	−0.385822	+	0.922573i	$0.626082\pi$
$240$	−5.04892	−0.325906
$241$	−3.64848	−0.235019	−0.117510	−	0.993072i	$-0.537491\pi$
$241$	−3.64848	−0.235019	−0.117510	+	0.993072i	$0.537491\pi$
$242$	5.30798	0.341210
$243$	15.7952	1.01326
$244$	13.5985	0.870555
$245$	−0.692021	−0.0442116
$246$	−1.47650	−0.0941383
$247$	0	0
$248$	12.0881	0.767598
$249$	−13.0978	−0.830042
$250$	−3.34183	−0.211356
$251$	1.37329	0.0866813	0.0433406	−	0.999060i	$-0.486200\pi$
$251$	1.37329	0.0866813	0.0433406	+	0.999060i	$0.486200\pi$
$252$	−10.7356	−0.676277
$253$	5.51573	0.346771
$254$	−3.76138	−0.236010
$255$	−2.55496	−0.159998
$256$	−3.34721	−0.209200
$257$	29.4359	1.83616	0.918082	−	0.396391i	$-0.129737\pi$
$257$	29.4359	1.83616	0.918082	+	0.396391i	$0.129737\pi$
$258$	−3.18598	−0.198350
$259$	−2.56033	−0.159091
$260$	0	0
$261$	18.6189	1.15248
$262$	7.59179	0.469023
$263$	10.6963	0.659564	0.329782	−	0.944057i	$-0.393025\pi$
$263$	10.6963	0.659564	0.329782	+	0.944057i	$0.393025\pi$
$264$	−1.96854	−0.121155
$265$	−16.4547	−1.01081
$266$	2.89679	0.177613
$267$	−11.8116	−0.722860
$268$	15.6625	0.956738
$269$	−10.1860	−0.621050	−0.310525	−	0.950565i	$-0.600505\pi$
$269$	−10.1860	−0.621050	−0.310525	+	0.950565i	$0.600505\pi$
$270$	−6.67994	−0.406528
$271$	−29.4523	−1.78910	−0.894551	−	0.446966i	$-0.852505\pi$
$271$	−29.4523	−1.78910	−0.894551	+	0.446966i	$0.852505\pi$
$272$	2.55496	0.154917
$273$	0	0
$274$	−7.20775	−0.435436
$275$	−3.41550	−0.205963
$276$	6.24698	0.376024
$277$	−10.2446	−0.615538	−0.307769	−	0.951461i	$-0.599582\pi$
$277$	−10.2446	−0.615538	−0.307769	+	0.951461i	$0.599582\pi$
$278$	−6.68532	−0.400959
$279$	−13.9051	−0.832480
$280$	15.4547	0.923597
$281$	−11.5646	−0.689889	−0.344944	−	0.938623i	$-0.612102\pi$
$281$	−11.5646	−0.689889	−0.344944	+	0.938623i	$0.612102\pi$
$282$	3.42327	0.203853
$283$	−30.7090	−1.82546	−0.912730	−	0.408562i	$-0.866030\pi$
$283$	−30.7090	−1.82546	−0.912730	+	0.408562i	$0.866030\pi$
$284$	23.2500	1.37963
$285$	−4.35690	−0.258080
$286$	0	0
$287$	−8.93123	−0.527194
$288$	12.5972	0.742295
$289$	−15.7071	−0.923946
$290$	−12.2838	−0.721330
$291$	2.51142	0.147222
$292$	−21.6866	−1.26911
$293$	−18.6082	−1.08710	−0.543551	−	0.839376i	$-0.682921\pi$
$293$	−18.6082	−1.08710	−0.543551	+	0.839376i	$0.682921\pi$
$294$	−0.109916	−0.00641045
$295$	0.0338518	0.00197093
$296$	1.94869	0.113265
$297$	5.14675	0.298645
$298$	0.411190	0.0238196
$299$	0	0
$300$	−3.86831	−0.223337
$301$	−19.2717	−1.11080
$302$	10.5851	0.609103
$303$	4.24267	0.243735
$304$	4.35690	0.249885
$305$	22.5187	1.28942
$306$	1.48725	0.0850207
$307$	8.94438	0.510483	0.255241	−	0.966877i	$-0.417845\pi$
$307$	8.94438	0.510483	0.255241	+	0.966877i	$0.417845\pi$
$308$	−5.45712	−0.310948
$309$	−10.8509	−0.617284
$310$	9.17390	0.521042
$311$	21.0398	1.19306	0.596529	−	0.802591i	$-0.296546\pi$
$311$	21.0398	1.19306	0.596529	+	0.802591i	$0.296546\pi$
$312$	0	0
$313$	−7.12737	−0.402863	−0.201432	−	0.979503i	$-0.564559\pi$
$313$	−7.12737	−0.402863	−0.201432	+	0.979503i	$0.564559\pi$
$314$	2.23191	0.125954
$315$	−17.7778	−1.00166
$316$	−1.36599	−0.0768431
$317$	23.9651	1.34601	0.673007	−	0.739636i	$-0.265003\pi$
$317$	23.9651	1.34601	0.673007	+	0.739636i	$0.265003\pi$
$318$	−2.61356	−0.146561
$319$	9.46442	0.529906
$320$	4.28083	0.239306
$321$	4.51573	0.252043
$322$	−6.87800	−0.383296
$323$	2.20477	0.122677
$324$	−6.13467	−0.340815
$325$	0	0
$326$	−8.40044	−0.465257
$327$	3.34913	0.185207
$328$	6.79763	0.375336
$329$	20.7071	1.14162
$330$	−1.49396	−0.0822397
$331$	2.89546	0.159149	0.0795745	−	0.996829i	$-0.474644\pi$
$331$	2.89546	0.159149	0.0795745	+	0.996829i	$0.474644\pi$
$332$	27.6353	1.51669
$333$	−2.24160	−0.122839
$334$	3.47591	0.190193
$335$	25.9366	1.41707
$336$	−4.85086	−0.264636
$337$	−3.10560	−0.169173	−0.0845865	−	0.996416i	$-0.526957\pi$
$337$	−3.10560	−0.169173	−0.0845865	+	0.996416i	$0.526957\pi$
$338$	0	0
$339$	6.12929	0.332898
$340$	5.39075	0.292354
$341$	−7.06829	−0.382770
$342$	2.53617	0.137140
$343$	18.1793	0.981589
$344$	14.6679	0.790838
$345$	10.3448	0.556946
$346$	−9.09651	−0.489031
$347$	−11.3787	−0.610839	−0.305419	−	0.952218i	$-0.598797\pi$
$347$	−11.3787	−0.610839	−0.305419	+	0.952218i	$0.598797\pi$
$348$	10.7192	0.574608
$349$	−3.34721	−0.179172	−0.0895859	−	0.995979i	$-0.528554\pi$
$349$	−3.34721	−0.179172	−0.0895859	+	0.995979i	$0.528554\pi$
$350$	4.25906	0.227656
$351$	0	0
$352$	6.40342	0.341303
$353$	0.637727	0.0339428	0.0169714	−	0.999856i	$-0.494598\pi$
$353$	0.637727	0.0339428	0.0169714	+	0.999856i	$0.494598\pi$
$354$	0.00537681	0.000285774 0
$355$	38.5013	2.04343
$356$	24.9215	1.32084
$357$	−2.45473	−0.129918
$358$	1.36227	0.0719983
$359$	−21.4590	−1.13256	−0.566282	−	0.824211i	$-0.691619\pi$
$359$	−21.4590	−1.13256	−0.566282	+	0.824211i	$0.691619\pi$
$360$	13.5308	0.713136
$361$	−15.2403	−0.802120
$362$	−6.55257	−0.344395
$363$	−7.67025	−0.402584
$364$	0	0
$365$	−35.9124	−1.87974
$366$	3.57673	0.186959
$367$	−9.38703	−0.489999	−0.244999	−	0.969523i	$-0.578788\pi$
$367$	−9.38703	−0.489999	−0.244999	+	0.969523i	$0.578788\pi$
$368$	−10.3448	−0.539261
$369$	−7.81940	−0.407062
$370$	1.47889	0.0768840
$371$	−15.8092	−0.820775
$372$	−8.00538	−0.415059
$373$	27.7265	1.43562	0.717811	−	0.696238i	$-0.245144\pi$
$373$	27.7265	1.43562	0.717811	+	0.696238i	$0.245144\pi$
$374$	0.756004	0.0390921
$375$	4.82908	0.249373
$376$	−15.7603	−0.812776
$377$	0	0
$378$	−6.41789	−0.330101
$379$	35.8702	1.84253	0.921265	−	0.388935i	$-0.127157\pi$
$379$	35.8702	1.84253	0.921265	+	0.388935i	$0.127157\pi$
$380$	9.19269	0.471575
$381$	5.43535	0.278462
$382$	4.99090	0.255357
$383$	4.85517	0.248087	0.124044	−	0.992277i	$-0.460414\pi$
$383$	4.85517	0.248087	0.124044	+	0.992277i	$0.460414\pi$
$384$	9.25236	0.472157
$385$	−9.03684	−0.460560
$386$	−7.50604	−0.382047
$387$	−16.8726	−0.857684
$388$	−5.29888	−0.269010
$389$	2.38537	0.120943	0.0604716	−	0.998170i	$-0.480740\pi$
$389$	2.38537	0.120943	0.0604716	+	0.998170i	$0.480740\pi$
$390$	0	0
$391$	−5.23490	−0.264740
$392$	0.506041	0.0255589
$393$	−10.9705	−0.553387
$394$	−7.20105	−0.362783
$395$	−2.26205	−0.113816
$396$	−4.77777	−0.240092
$397$	−15.2664	−0.766196	−0.383098	−	0.923708i	$-0.625143\pi$
$397$	−15.2664	−0.766196	−0.383098	+	0.923708i	$0.625143\pi$
$398$	7.53989	0.377941
$399$	−4.18598	−0.209561
$400$	6.40581	0.320291
$401$	12.7584	0.637124	0.318562	−	0.947902i	$-0.396800\pi$
$401$	12.7584	0.637124	0.318562	+	0.947902i	$0.396800\pi$
$402$	4.11960	0.205467
$403$	0	0
$404$	−8.95167	−0.445362
$405$	−10.1588	−0.504797
$406$	−11.8019	−0.585720
$407$	−1.13946	−0.0564807
$408$	1.86831	0.0924953
$409$	−25.3588	−1.25391	−0.626956	−	0.779054i	$-0.715700\pi$
$409$	−25.3588	−1.25391	−0.626956	+	0.779054i	$0.715700\pi$
$410$	5.15883	0.254777
$411$	10.4155	0.513759
$412$	22.8944	1.12793
$413$	0.0325239	0.00160040
$414$	−6.02177	−0.295954
$415$	45.7633	2.24643
$416$	0	0
$417$	9.66056	0.473080
$418$	1.28919	0.0630565
$419$	−11.6673	−0.569983	−0.284992	−	0.958530i	$-0.591991\pi$
$419$	−11.6673	−0.569983	−0.284992	+	0.958530i	$0.591991\pi$
$420$	−10.2349	−0.499412
$421$	−8.29291	−0.404172	−0.202086	−	0.979368i	$-0.564772\pi$
$421$	−8.29291	−0.404172	−0.202086	+	0.979368i	$0.564772\pi$
$422$	−5.80492	−0.282579
$423$	18.1293	0.881476
$424$	12.0325	0.584351
$425$	3.24160	0.157241
$426$	6.11529	0.296287
$427$	21.6353	1.04701
$428$	−9.52781	−0.460544
$429$	0	0
$430$	11.1317	0.536818
$431$	0.932296	0.0449071	0.0224536	−	0.999748i	$-0.492852\pi$
$431$	0.932296	0.0449071	0.0224536	+	0.999748i	$0.492852\pi$
$432$	−9.65279	−0.464420
$433$	−13.3502	−0.641569	−0.320785	−	0.947152i	$-0.603947\pi$
$433$	−13.3502	−0.641569	−0.320785	+	0.947152i	$0.603947\pi$
$434$	8.81402	0.423086
$435$	17.7506	0.851077
$436$	−7.06638	−0.338418
$437$	−8.92692	−0.427032
$438$	−5.70410	−0.272553
$439$	13.9922	0.667813	0.333906	−	0.942606i	$-0.391633\pi$
$439$	13.9922	0.667813	0.333906	+	0.942606i	$0.391633\pi$
$440$	6.87800	0.327896
$441$	−0.582105	−0.0277193
$442$	0	0
$443$	23.7017	1.12610	0.563051	−	0.826422i	$-0.309627\pi$
$443$	23.7017	1.12610	0.563051	+	0.826422i	$0.309627\pi$
$444$	−1.29052	−0.0612454
$445$	41.2693	1.95635
$446$	6.32975	0.299722
$447$	−0.594187	−0.0281041
$448$	4.11290	0.194316
$449$	−12.5864	−0.593990	−0.296995	−	0.954879i	$-0.595984\pi$
$449$	−12.5864	−0.593990	−0.296995	+	0.954879i	$0.595984\pi$
$450$	3.72886	0.175780
$451$	−3.97477	−0.187165
$452$	−12.9323	−0.608284
$453$	−15.2959	−0.718664
$454$	−5.90515	−0.277142
$455$	0	0
$456$	3.18598	0.149197
$457$	−33.6383	−1.57353	−0.786767	−	0.617250i	$-0.788247\pi$
$457$	−33.6383	−1.57353	−0.786767	+	0.617250i	$0.788247\pi$
$458$	0.632351	0.0295478
$459$	−4.88471	−0.227999
$460$	−21.8267	−1.01767
$461$	−1.40283	−0.0653363	−0.0326681	−	0.999466i	$-0.510400\pi$
$461$	−1.40283	−0.0653363	−0.0326681	+	0.999466i	$0.510400\pi$
$462$	−1.43535	−0.0667787
$463$	−15.2010	−0.706453	−0.353226	−	0.935538i	$-0.614915\pi$
$463$	−15.2010	−0.706453	−0.353226	+	0.935538i	$0.614915\pi$
$464$	−17.7506	−0.824052
$465$	−13.2567	−0.614763
$466$	6.02177	0.278953
$467$	−39.3414	−1.82050	−0.910250	−	0.414058i	$-0.864111\pi$
$467$	−39.3414	−1.82050	−0.910250	+	0.414058i	$0.864111\pi$
$468$	0	0
$469$	24.9191	1.15066
$470$	−11.9608	−0.551709
$471$	−3.22521	−0.148610
$472$	−0.0247542	−0.00113940
$473$	−8.57673	−0.394358
$474$	−0.359289	−0.0165027
$475$	5.52781	0.253633
$476$	5.17928	0.237392
$477$	−13.8412	−0.633743
$478$	6.62027	0.302804
$479$	−22.3690	−1.02206	−0.511032	−	0.859561i	$-0.670737\pi$
$479$	−22.3690	−1.02206	−0.511032	+	0.859561i	$0.670737\pi$
$480$	12.0097	0.548165
$481$	0	0
$482$	2.02475	0.0922250
$483$	9.93900	0.452240
$484$	16.1836	0.735618
$485$	−8.77479	−0.398443
$486$	−8.76569	−0.397620
$487$	22.9205	1.03863	0.519313	−	0.854584i	$-0.326188\pi$
$487$	22.9205	1.03863	0.519313	+	0.854584i	$0.326188\pi$
$488$	−16.4668	−0.745418
$489$	12.1390	0.548944
$490$	0.384043	0.0173493
$491$	1.84356	0.0831987	0.0415993	−	0.999134i	$-0.486755\pi$
$491$	1.84356	0.0831987	0.0415993	+	0.999134i	$0.486755\pi$
$492$	−4.50173	−0.202954
$493$	−8.98254	−0.404553
$494$	0	0
$495$	−7.91185	−0.355611
$496$	13.2567	0.595242
$497$	36.9909	1.65927
$498$	7.26875	0.325720
$499$	−12.0344	−0.538736	−0.269368	−	0.963037i	$-0.586815\pi$
$499$	−12.0344	−0.538736	−0.269368	+	0.963037i	$0.586815\pi$
$500$	−10.1890	−0.455664
$501$	−5.02284	−0.224404
$502$	−0.762118	−0.0340150
$503$	−30.5056	−1.36018	−0.680088	−	0.733130i	$-0.738058\pi$
$503$	−30.5056	−1.36018	−0.680088	+	0.733130i	$0.738058\pi$
$504$	13.0000	0.579066
$505$	−14.8237	−0.659646
$506$	−3.06100	−0.136078
$507$	0	0
$508$	−11.4681	−0.508816
$509$	−1.51142	−0.0669924	−0.0334962	−	0.999439i	$-0.510664\pi$
$509$	−1.51142	−0.0669924	−0.0334962	+	0.999439i	$0.510664\pi$
$510$	1.41789	0.0627854
$511$	−34.5036	−1.52635
$512$	−21.2174	−0.937687
$513$	−8.32975	−0.367767
$514$	−16.3357	−0.720538
$515$	37.9124	1.67062
$516$	−9.71379	−0.427626
$517$	9.21552	0.405298
$518$	1.42088	0.0624298
$519$	13.1448	0.576994
$520$	0	0
$521$	−5.64012	−0.247098	−0.123549	−	0.992338i	$-0.539428\pi$
$521$	−5.64012	−0.247098	−0.123549	+	0.992338i	$0.539428\pi$
$522$	−10.3327	−0.452251
$523$	−31.7506	−1.38836	−0.694179	−	0.719802i	$-0.744232\pi$
$523$	−31.7506	−1.38836	−0.694179	+	0.719802i	$0.744232\pi$
$524$	23.1468	1.01117
$525$	−6.15452	−0.268605
$526$	−5.93602	−0.258823
$527$	6.70841	0.292223
$528$	−2.15883	−0.0939512
$529$	−1.80433	−0.0784492
$530$	9.13169	0.396655
$531$	0.0284750	0.00123571
$532$	8.83207	0.382919
$533$	0	0
$534$	6.55496	0.283661
$535$	−15.7778	−0.682133
$536$	−18.9661	−0.819213
$537$	−1.96854	−0.0849488
$538$	5.65279	0.243709
$539$	−0.295897	−0.0127452
$540$	−20.3666	−0.876438
$541$	24.3297	1.04602	0.523009	−	0.852327i	$-0.324809\pi$
$541$	24.3297	1.04602	0.523009	+	0.852327i	$0.324809\pi$
$542$	16.3448	0.702070
$543$	9.46873	0.406342
$544$	−6.07739	−0.260566
$545$	−11.7017	−0.501246
$546$	0	0
$547$	−8.18896	−0.350135	−0.175067	−	0.984556i	$-0.556014\pi$
$547$	−8.18896	−0.350135	−0.175067	+	0.984556i	$0.556014\pi$
$548$	−21.9758	−0.938761
$549$	18.9420	0.808424
$550$	1.89546	0.0808227
$551$	−15.3177	−0.652555
$552$	−7.56465	−0.321973
$553$	−2.17331	−0.0924185
$554$	5.68532	0.241546
$555$	−2.13706	−0.0907133
$556$	−20.3830	−0.864431
$557$	25.3327	1.07338	0.536691	−	0.843779i	$-0.319674\pi$
$557$	25.3327	1.07338	0.536691	+	0.843779i	$0.319674\pi$
$558$	7.71678	0.326677
$559$	0	0
$560$	16.9487	0.716213
$561$	−1.09246	−0.0461236
$562$	6.41789	0.270723
$563$	−25.3937	−1.07022	−0.535109	−	0.844783i	$-0.679730\pi$
$563$	−25.3937	−1.07022	−0.535109	+	0.844783i	$0.679730\pi$
$564$	10.4373	0.439488
$565$	−21.4155	−0.900957
$566$	17.0422	0.716338
$567$	−9.76032	−0.409895
$568$	−28.1540	−1.18132
$569$	−31.1347	−1.30523	−0.652617	−	0.757688i	$-0.726329\pi$
$569$	−31.1347	−1.30523	−0.652617	+	0.757688i	$0.726329\pi$
$570$	2.41789	0.101274
$571$	−20.5090	−0.858276	−0.429138	−	0.903239i	$-0.641183\pi$
$571$	−20.5090	−0.858276	−0.429138	+	0.903239i	$0.641183\pi$
$572$	0	0
$573$	−7.21206	−0.301288
$574$	4.95646	0.206879
$575$	−13.1250	−0.547350
$576$	3.60089	0.150037
$577$	15.6890	0.653143	0.326572	−	0.945172i	$-0.394107\pi$
$577$	15.6890	0.653143	0.326572	+	0.945172i	$0.394107\pi$
$578$	8.71678	0.362570
$579$	10.8465	0.450767
$580$	−37.4523	−1.55512
$581$	43.9681	1.82410
$582$	−1.39373	−0.0577720
$583$	−7.03577	−0.291392
$584$	26.2610	1.08669
$585$	0	0
$586$	10.3268	0.426595
$587$	30.5687	1.26171	0.630853	−	0.775903i	$-0.282705\pi$
$587$	30.5687	1.26171	0.630853	+	0.775903i	$0.282705\pi$
$588$	−0.335126	−0.0138203
$589$	11.4397	0.471363
$590$	−0.0187864	−0.000773422 0
$591$	10.4058	0.428038
$592$	2.13706	0.0878328
$593$	−29.6883	−1.21915	−0.609576	−	0.792727i	$-0.708660\pi$
$593$	−29.6883	−1.21915	−0.609576	+	0.792727i	$0.708660\pi$
$594$	−2.85623	−0.117193
$595$	8.57673	0.351612
$596$	1.25368	0.0513529
$597$	−10.8955	−0.445922
$598$	0	0
$599$	24.2325	0.990113	0.495057	−	0.868861i	$-0.335147\pi$
$599$	24.2325	0.990113	0.495057	+	0.868861i	$0.335147\pi$
$600$	4.68425	0.191234
$601$	16.4819	0.672310	0.336155	−	0.941807i	$-0.390873\pi$
$601$	16.4819	0.672310	0.336155	+	0.941807i	$0.390873\pi$
$602$	10.6950	0.435896
$603$	21.8170	0.888457
$604$	32.2731	1.31317
$605$	26.7995	1.08956
$606$	−2.35450	−0.0956451
$607$	1.43190	0.0581188	0.0290594	−	0.999578i	$-0.490749\pi$
$607$	1.43190	0.0581188	0.0290594	+	0.999578i	$0.490749\pi$
$608$	−10.3636	−0.420300
$609$	17.0543	0.691075
$610$	−12.4969	−0.505986
$611$	0	0
$612$	4.53452	0.183297
$613$	3.84846	0.155438	0.0777190	−	0.996975i	$-0.475236\pi$
$613$	3.84846	0.155438	0.0777190	+	0.996975i	$0.475236\pi$
$614$	−4.96376	−0.200321
$615$	−7.45473	−0.300604
$616$	6.60819	0.266251
$617$	−15.0388	−0.605437	−0.302719	−	0.953080i	$-0.597894\pi$
$617$	−15.0388	−0.605437	−0.302719	+	0.953080i	$0.597894\pi$
$618$	6.02177	0.242231
$619$	12.8170	0.515159	0.257579	−	0.966257i	$-0.417075\pi$
$619$	12.8170	0.515159	0.257579	+	0.966257i	$0.417075\pi$
$620$	27.9705	1.12332
$621$	19.7778	0.793655
$622$	−11.6762	−0.468174
$623$	39.6504	1.58856
$624$	0	0
$625$	−31.1269	−1.24508
$626$	3.95539	0.158089
$627$	−1.86294	−0.0743985
$628$	6.80492	0.271546
$629$	1.08144	0.0431199
$630$	9.86592	0.393068
$631$	25.7517	1.02516	0.512579	−	0.858640i	$-0.328690\pi$
$631$	25.7517	1.02516	0.512579	+	0.858640i	$0.328690\pi$
$632$	1.65412	0.0657974
$633$	8.38835	0.333407
$634$	−13.2996	−0.528195
$635$	−18.9909	−0.753631
$636$	−7.96854	−0.315973
$637$	0	0
$638$	−5.25236	−0.207943
$639$	32.3860	1.28117
$640$	−32.3274	−1.27785
$641$	24.4571	0.965998	0.482999	−	0.875621i	$-0.339547\pi$
$641$	24.4571	0.965998	0.482999	+	0.875621i	$0.339547\pi$
$642$	−2.50604	−0.0989055
$643$	−9.97344	−0.393314	−0.196657	−	0.980472i	$-0.563009\pi$
$643$	−9.97344	−0.393314	−0.196657	+	0.980472i	$0.563009\pi$
$644$	−20.9705	−0.826352
$645$	−16.0858	−0.633376
$646$	−1.22355	−0.0481401
$647$	11.8431	0.465600	0.232800	−	0.972525i	$-0.425211\pi$
$647$	11.8431	0.465600	0.232800	+	0.972525i	$0.425211\pi$
$648$	7.42865	0.291825
$649$	0.0144745	0.000568173 0
$650$	0	0
$651$	−12.7366	−0.499188
$652$	−25.6122	−1.00305
$653$	−7.47411	−0.292484	−0.146242	−	0.989249i	$-0.546718\pi$
$653$	−7.47411	−0.292484	−0.146242	+	0.989249i	$0.546718\pi$
$654$	−1.85862	−0.0726780
$655$	38.3303	1.49769
$656$	7.45473	0.291058
$657$	−30.2083	−1.17854
$658$	−11.4916	−0.447988
$659$	34.1739	1.33123	0.665613	−	0.746297i	$-0.268170\pi$
$659$	34.1739	1.33123	0.665613	+	0.746297i	$0.268170\pi$
$660$	−4.55496	−0.177302
$661$	33.6088	1.30723	0.653615	−	0.756827i	$-0.273252\pi$
$661$	33.6088	1.30723	0.653615	+	0.756827i	$0.273252\pi$
$662$	−1.60686	−0.0624524
$663$	0	0
$664$	−33.4644	−1.29867
$665$	14.6256	0.567158
$666$	1.24400	0.0482039
$667$	36.3696	1.40824
$668$	10.5978	0.410040
$669$	−9.14675	−0.353634
$670$	−14.3937	−0.556078
$671$	9.62863	0.371709
$672$	11.5386	0.445110
$673$	48.0320	1.85150	0.925750	−	0.378137i	$-0.123435\pi$
$673$	48.0320	1.85150	0.925750	+	0.378137i	$0.123435\pi$
$674$	1.72348	0.0663860
$675$	−12.2470	−0.471386
$676$	0	0
$677$	−33.6582	−1.29359	−0.646794	−	0.762665i	$-0.723891\pi$
$677$	−33.6582	−1.29359	−0.646794	+	0.762665i	$0.723891\pi$
$678$	−3.40150	−0.130634
$679$	−8.43057	−0.323535
$680$	−6.52781	−0.250330
$681$	8.53319	0.326992
$682$	3.92261	0.150204
$683$	15.9041	0.608553	0.304276	−	0.952584i	$-0.401585\pi$
$683$	15.9041	0.608553	0.304276	+	0.952584i	$0.401585\pi$
$684$	7.73258	0.295663
$685$	−36.3913	−1.39044
$686$	−10.0887	−0.385190
$687$	−0.913773	−0.0348626
$688$	16.0858	0.613264
$689$	0	0
$690$	−5.74094	−0.218554
$691$	33.1903	1.26262	0.631309	−	0.775531i	$-0.282518\pi$
$691$	33.1903	1.26262	0.631309	+	0.775531i	$0.282518\pi$
$692$	−27.7345	−1.05431
$693$	−7.60148	−0.288756
$694$	6.31468	0.239702
$695$	−33.7536	−1.28035
$696$	−12.9801	−0.492011
$697$	3.77240	0.142890
$698$	1.85756	0.0703097
$699$	−8.70171	−0.329129
$700$	12.9855	0.490807
$701$	−14.9129	−0.563253	−0.281627	−	0.959524i	$-0.590874\pi$
$701$	−14.9129	−0.563253	−0.281627	+	0.959524i	$0.590874\pi$
$702$	0	0
$703$	1.84415	0.0695534
$704$	1.83041	0.0689863
$705$	17.2838	0.650946
$706$	−0.353912	−0.0133197
$707$	−14.2422	−0.535633
$708$	0.0163935	0.000616104 0
$709$	38.4312	1.44331	0.721656	−	0.692252i	$-0.243381\pi$
$709$	38.4312	1.44331	0.721656	+	0.692252i	$0.243381\pi$
$710$	−21.3666	−0.801874
$711$	−1.90276	−0.0713589
$712$	−30.1782	−1.13098
$713$	−27.1618	−1.01722
$714$	1.36227	0.0509818
$715$	0	0
$716$	4.15346	0.155222
$717$	−9.56657	−0.357270
$718$	11.9089	0.444435
$719$	−11.4373	−0.426538	−0.213269	−	0.976993i	$-0.568411\pi$
$719$	−11.4373	−0.426538	−0.213269	+	0.976993i	$0.568411\pi$
$720$	14.8388	0.553008
$721$	36.4252	1.35654
$722$	8.45771	0.314764
$723$	−2.92585	−0.108814
$724$	−19.9782	−0.742485
$725$	−22.5211	−0.836413
$726$	4.25667	0.157980
$727$	3.63640	0.134867	0.0674333	−	0.997724i	$-0.478519\pi$
$727$	3.63640	0.134867	0.0674333	+	0.997724i	$0.478519\pi$
$728$	0	0
$729$	1.78986	0.0662910
$730$	19.9299	0.737639
$731$	8.14005	0.301071
$732$	10.9051	0.403066
$733$	−3.52217	−0.130094	−0.0650472	−	0.997882i	$-0.520720\pi$
$733$	−3.52217	−0.130094	−0.0650472	+	0.997882i	$0.520720\pi$
$734$	5.20941	0.192283
$735$	−0.554958	−0.0204699
$736$	24.6069	0.907021
$737$	11.0901	0.408508
$738$	4.33944	0.159737
$739$	0.420288	0.0154605	0.00773027	−	0.999970i	$-0.497539\pi$
$739$	0.420288	0.0154605	0.00773027	+	0.999970i	$0.497539\pi$
$740$	4.50902	0.165755
$741$	0	0
$742$	8.77346	0.322084
$743$	−25.3623	−0.930452	−0.465226	−	0.885192i	$-0.654027\pi$
$743$	−25.3623	−0.930452	−0.465226	+	0.885192i	$0.654027\pi$
$744$	9.69394	0.355397
$745$	2.07606	0.0760611
$746$	−15.3870	−0.563359
$747$	38.4946	1.40844
$748$	2.30499	0.0842790
$749$	−15.1588	−0.553892
$750$	−2.67994	−0.0978576
$751$	0.650874	0.0237507	0.0118754	−	0.999929i	$-0.496220\pi$
$751$	0.650874	0.0237507	0.0118754	+	0.999929i	$0.496220\pi$
$752$	−17.2838	−0.630276
$753$	1.10129	0.0401333
$754$	0	0
$755$	53.4432	1.94500
$756$	−19.5676	−0.711668
$757$	−16.7909	−0.610276	−0.305138	−	0.952308i	$-0.598703\pi$
$757$	−16.7909	−0.610276	−0.305138	+	0.952308i	$0.598703\pi$
$758$	−19.9065	−0.723036
$759$	4.42327	0.160555
$760$	−11.1317	−0.403789
$761$	30.9221	1.12093	0.560463	−	0.828179i	$-0.310623\pi$
$761$	30.9221	1.12093	0.560463	+	0.828179i	$0.310623\pi$
$762$	−3.01639	−0.109272
$763$	−11.2427	−0.407012
$764$	15.2168	0.550526
$765$	7.50902	0.271489
$766$	−2.69441	−0.0973531
$767$	0	0
$768$	−2.68425	−0.0968596
$769$	−43.7689	−1.57835	−0.789174	−	0.614169i	$-0.789491\pi$
$769$	−43.7689	−1.57835	−0.789174	+	0.614169i	$0.789491\pi$
$770$	5.01507	0.180730
$771$	23.6058	0.850142
$772$	−22.8853	−0.823660
$773$	−42.4209	−1.52577	−0.762886	−	0.646532i	$-0.776218\pi$
$773$	−42.4209	−1.52577	−0.762886	+	0.646532i	$0.776218\pi$
$774$	9.36360	0.336568
$775$	16.8194	0.604171
$776$	6.41657	0.230341
$777$	−2.05323	−0.0736592
$778$	−1.32378	−0.0474598
$779$	6.43296	0.230485
$780$	0	0
$781$	16.4625	0.589075
$782$	2.90515	0.103888
$783$	33.9366	1.21280
$784$	0.554958	0.0198199
$785$	11.2687	0.402199
$786$	6.08815	0.217157
$787$	−36.0116	−1.28368	−0.641838	−	0.766841i	$-0.721828\pi$
$787$	−36.0116	−1.28368	−0.641838	+	0.766841i	$0.721828\pi$
$788$	−21.9554	−0.782129
$789$	8.57779	0.305378
$790$	1.25534	0.0446630
$791$	−20.5754	−0.731577
$792$	5.78554	0.205580
$793$	0	0
$794$	8.47219	0.300667
$795$	−13.1957	−0.468002
$796$	22.9885	0.814806
$797$	−31.7101	−1.12323	−0.561614	−	0.827399i	$-0.689819\pi$
$797$	−31.7101	−1.12323	−0.561614	+	0.827399i	$0.689819\pi$
$798$	2.32304	0.0822349
$799$	−8.74632	−0.309422
$800$	−15.2373	−0.538720
$801$	34.7144	1.22657
$802$	−7.08038	−0.250017
$803$	−15.3556	−0.541886
$804$	12.5603	0.442969
$805$	−34.7265	−1.22395
$806$	0	0
$807$	−8.16852	−0.287546
$808$	10.8398	0.381344
$809$	−45.2814	−1.59201	−0.796005	−	0.605290i	$-0.793057\pi$
$809$	−45.2814	−1.59201	−0.796005	+	0.605290i	$0.793057\pi$
$810$	5.63773	0.198090
$811$	−42.8635	−1.50514	−0.752571	−	0.658511i	$-0.771187\pi$
$811$	−42.8635	−1.50514	−0.752571	+	0.658511i	$0.771187\pi$
$812$	−35.9831	−1.26276
$813$	−23.6189	−0.828352
$814$	0.632351	0.0221639
$815$	−42.4131	−1.48567
$816$	2.04892	0.0717265
$817$	13.8810	0.485634
$818$	14.0731	0.492054
$819$	0	0
$820$	15.7289	0.549276
$821$	−7.82776	−0.273191	−0.136595	−	0.990627i	$-0.543616\pi$
$821$	−7.82776	−0.273191	−0.136595	+	0.990627i	$0.543616\pi$
$822$	−5.78017	−0.201606
$823$	−36.7754	−1.28191	−0.640955	−	0.767579i	$-0.721461\pi$
$823$	−36.7754	−1.28191	−0.640955	+	0.767579i	$0.721461\pi$
$824$	−27.7235	−0.965793
$825$	−2.73902	−0.0953604
$826$	−0.0180494	−0.000628019 0
$827$	47.3293	1.64580	0.822900	−	0.568186i	$-0.192355\pi$
$827$	47.3293	1.64580	0.822900	+	0.568186i	$0.192355\pi$
$828$	−18.3599	−0.638050
$829$	25.2687	0.877620	0.438810	−	0.898580i	$-0.355400\pi$
$829$	25.2687	0.877620	0.438810	+	0.898580i	$0.355400\pi$
$830$	−25.3967	−0.881533
$831$	−8.21552	−0.284993
$832$	0	0
$833$	0.280831	0.00973023
$834$	−5.36121	−0.185643
$835$	17.5496	0.607328
$836$	3.93064	0.135944
$837$	−25.3448	−0.876045
$838$	6.47484	0.223670
$839$	−37.6883	−1.30114	−0.650572	−	0.759444i	$-0.725471\pi$
$839$	−37.6883	−1.30114	−0.650572	+	0.759444i	$0.725471\pi$
$840$	12.3937	0.427624
$841$	33.4064	1.15194
$842$	4.60222	0.158603
$843$	−9.27413	−0.319418
$844$	−17.6987	−0.609215
$845$	0	0
$846$	−10.0610	−0.345904
$847$	25.7482	0.884720
$848$	13.1957	0.453141
$849$	−24.6267	−0.845187
$850$	−1.79895	−0.0617036
$851$	−4.37867	−0.150099
$852$	18.6450	0.638768
$853$	−31.0121	−1.06183	−0.530917	−	0.847424i	$-0.678152\pi$
$853$	−31.0121	−1.06183	−0.530917	+	0.847424i	$0.678152\pi$
$854$	−12.0067	−0.410861
$855$	12.8049	0.437919
$856$	11.5375	0.394344
$857$	12.4692	0.425940	0.212970	−	0.977059i	$-0.431686\pi$
$857$	12.4692	0.425940	0.212970	+	0.977059i	$0.431686\pi$
$858$	0	0
$859$	−17.3163	−0.590826	−0.295413	−	0.955370i	$-0.595457\pi$
$859$	−17.3163	−0.590826	−0.295413	+	0.955370i	$0.595457\pi$
$860$	33.9396	1.15733
$861$	−7.16229	−0.244090
$862$	−0.517385	−0.0176222
$863$	−3.46383	−0.117910	−0.0589550	−	0.998261i	$-0.518777\pi$
$863$	−3.46383	−0.117910	−0.0589550	+	0.998261i	$0.518777\pi$
$864$	22.9608	0.781141
$865$	−45.9275	−1.56158
$866$	7.40880	0.251761
$867$	−12.5961	−0.427786
$868$	26.8732	0.912136
$869$	−0.967213	−0.0328105
$870$	−9.85086	−0.333975
$871$	0	0
$872$	8.55688	0.289772
$873$	−7.38106	−0.249811
$874$	4.95407	0.167574
$875$	−16.2107	−0.548023
$876$	−17.3913	−0.587599
$877$	57.2549	1.93336	0.966680	−	0.255989i	$-0.0824010\pi$
$877$	57.2549	1.93336	0.966680	+	0.255989i	$0.0824010\pi$
$878$	−7.76510	−0.262059
$879$	−14.9226	−0.503327
$880$	7.54288	0.254270
$881$	−43.1782	−1.45471	−0.727355	−	0.686261i	$-0.759251\pi$
$881$	−43.1782	−1.45471	−0.727355	+	0.686261i	$0.759251\pi$
$882$	0.323044	0.0108775
$883$	49.9560	1.68115	0.840576	−	0.541693i	$-0.182216\pi$
$883$	49.9560	1.68115	0.840576	+	0.541693i	$0.182216\pi$
$884$	0	0
$885$	0.0271471	0.000912539 0
$886$	−13.1535	−0.441899
$887$	17.6746	0.593454	0.296727	−	0.954962i	$-0.404105\pi$
$887$	17.6746	0.593454	0.296727	+	0.954962i	$0.404105\pi$
$888$	1.56273	0.0524417
$889$	−18.2459	−0.611948
$890$	−22.9028	−0.767702
$891$	−4.34375	−0.145521
$892$	19.2989	0.646174
$893$	−14.9148	−0.499106
$894$	0.329749	0.0110284
$895$	6.87800	0.229906
$896$	−31.0592	−1.03761
$897$	0	0
$898$	6.98493	0.233090
$899$	−46.6069	−1.55443
$900$	11.3690	0.378966
$901$	6.67755	0.222461
$902$	2.20583	0.0734462
$903$	−15.4547	−0.514301
$904$	15.6601	0.520847
$905$	−33.0834	−1.09973
$906$	8.48858	0.282014
$907$	7.73423	0.256811	0.128406	−	0.991722i	$-0.459014\pi$
$907$	7.73423	0.256811	0.128406	+	0.991722i	$0.459014\pi$
$908$	−18.0043	−0.597494
$909$	−12.4692	−0.413577
$910$	0	0
$911$	39.6179	1.31260	0.656299	−	0.754501i	$-0.272121\pi$
$911$	39.6179	1.31260	0.656299	+	0.754501i	$0.272121\pi$
$912$	3.49396	0.115697
$913$	19.5676	0.647594
$914$	18.6679	0.617478
$915$	18.0586	0.596999
$916$	1.92798	0.0637024
$917$	36.8267	1.21612
$918$	2.71081	0.0894700
$919$	14.6213	0.482313	0.241157	−	0.970486i	$-0.422473\pi$
$919$	14.6213	0.482313	0.241157	+	0.970486i	$0.422473\pi$
$920$	26.4306	0.871390
$921$	7.17283	0.236353
$922$	0.778512	0.0256389
$923$	0	0
$924$	−4.37627	−0.143969
$925$	2.71140	0.0891502
$926$	8.43594	0.277222
$927$	31.8907	1.04743
$928$	42.2228	1.38603
$929$	3.55735	0.116713	0.0583565	−	0.998296i	$-0.481414\pi$
$929$	3.55735	0.116713	0.0583565	+	0.998296i	$0.481414\pi$
$930$	7.35690	0.241242
$931$	0.478894	0.0156951
$932$	18.3599	0.601398
$933$	16.8726	0.552385
$934$	21.8328	0.714391
$935$	3.81700	0.124829
$936$	0	0
$937$	34.5526	1.12878	0.564392	−	0.825507i	$-0.309111\pi$
$937$	34.5526	1.12878	0.564392	+	0.825507i	$0.309111\pi$
$938$	−13.8291	−0.451536
$939$	−5.71571	−0.186525
$940$	−36.4674	−1.18944
$941$	20.6233	0.672299	0.336149	−	0.941809i	$-0.390875\pi$
$941$	20.6233	0.672299	0.336149	+	0.941809i	$0.390875\pi$
$942$	1.78986	0.0583167
$943$	−15.2741	−0.497394
$944$	−0.0271471	−0.000883562 0
$945$	−32.4034	−1.05408
$946$	4.75973	0.154752
$947$	29.4999	0.958619	0.479309	−	0.877646i	$-0.340887\pi$
$947$	29.4999	0.958619	0.479309	+	0.877646i	$0.340887\pi$
$948$	−1.09544	−0.0355783
$949$	0	0
$950$	−3.06770	−0.0995295
$951$	19.2185	0.623203
$952$	−6.27173	−0.203268
$953$	26.2389	0.849963	0.424981	−	0.905202i	$-0.360281\pi$
$953$	26.2389	0.849963	0.424981	+	0.905202i	$0.360281\pi$
$954$	7.68127	0.248690
$955$	25.1987	0.815409
$956$	20.1847	0.652818
$957$	7.58987	0.245346
$958$	12.4138	0.401073
$959$	−34.9638	−1.12904
$960$	3.43296	0.110798
$961$	3.80731	0.122817
$962$	0	0
$963$	−13.2717	−0.427676
$964$	6.17331	0.198829
$965$	−37.8974	−1.21996
$966$	−5.51573	−0.177466
$967$	17.5176	0.563330	0.281665	−	0.959513i	$-0.409113\pi$
$967$	17.5176	0.563330	0.281665	+	0.959513i	$0.409113\pi$
$968$	−19.5972	−0.629877
$969$	1.76809	0.0567991
$970$	4.86964	0.156355
$971$	20.5120	0.658262	0.329131	−	0.944284i	$-0.393244\pi$
$971$	20.5120	0.658262	0.329131	+	0.944284i	$0.393244\pi$
$972$	−26.7259	−0.857233
$973$	−32.4295	−1.03964
$974$	−12.7199	−0.407572
$975$	0	0
$976$	−18.0586	−0.578042
$977$	25.4450	0.814059	0.407030	−	0.913415i	$-0.366565\pi$
$977$	25.4450	0.814059	0.407030	+	0.913415i	$0.366565\pi$
$978$	−6.73663	−0.215414
$979$	17.6461	0.563971
$980$	1.17092	0.0374035
$981$	−9.84309	−0.314266
$982$	−1.02310	−0.0326484
$983$	−39.5244	−1.26063	−0.630316	−	0.776339i	$-0.717074\pi$
$983$	−39.5244	−1.26063	−0.630316	+	0.776339i	$0.717074\pi$
$984$	5.45127	0.173780
$985$	−36.3575	−1.15845
$986$	4.98493	0.158753
$987$	16.6058	0.528568
$988$	0	0
$989$	−32.9584	−1.04802
$990$	4.39075	0.139547
$991$	−29.8377	−0.947826	−0.473913	−	0.880572i	$-0.657159\pi$
$991$	−29.8377	−0.947826	−0.473913	+	0.880572i	$0.657159\pi$
$992$	−31.5332	−1.00118
$993$	2.32198	0.0736858
$994$	−20.5284	−0.651121
$995$	38.0683	1.20685
$996$	22.1618	0.702224
$997$	−4.93123	−0.156174	−0.0780868	−	0.996947i	$-0.524881\pi$
$997$	−4.93123	−0.156174	−0.0780868	+	0.996947i	$0.524881\pi$
$998$	6.67861	0.211408
$999$	−4.08575	−0.129268

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.2.a.b.1.2	✓	3
3.2	odd	2		1521.2.a.r.1.2		3
4.3	odd	2		2704.2.a.z.1.1		3
5.4	even	2		4225.2.a.bg.1.2		3
7.6	odd	2		8281.2.a.bf.1.2		3
13.2	odd	12		169.2.e.b.147.3		12
13.3	even	3		169.2.c.c.22.2		6
13.4	even	6		169.2.c.b.146.2		6
13.5	odd	4		169.2.b.b.168.4		6
13.6	odd	12		169.2.e.b.23.4		12
13.7	odd	12		169.2.e.b.23.3		12
13.8	odd	4		169.2.b.b.168.3		6
13.9	even	3		169.2.c.c.146.2		6
13.10	even	6		169.2.c.b.22.2		6
13.11	odd	12		169.2.e.b.147.4		12
13.12	even	2		169.2.a.c.1.2	yes	3
39.5	even	4		1521.2.b.l.1351.3		6
39.8	even	4		1521.2.b.l.1351.4		6
39.38	odd	2		1521.2.a.o.1.2		3
52.31	even	4		2704.2.f.o.337.2		6
52.47	even	4		2704.2.f.o.337.1		6
52.51	odd	2		2704.2.a.ba.1.1		3
65.64	even	2		4225.2.a.bb.1.2		3
91.90	odd	2		8281.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.2.a.b.1.2	✓	3	1.1	even	1	trivial
169.2.a.c.1.2	yes	3	13.12	even	2
169.2.b.b.168.3		6	13.8	odd	4
169.2.b.b.168.4		6	13.5	odd	4
169.2.c.b.22.2		6	13.10	even	6
169.2.c.b.146.2		6	13.4	even	6
169.2.c.c.22.2		6	13.3	even	3
169.2.c.c.146.2		6	13.9	even	3
169.2.e.b.23.3		12	13.7	odd	12
169.2.e.b.23.4		12	13.6	odd	12
169.2.e.b.147.3		12	13.2	odd	12
169.2.e.b.147.4		12	13.11	odd	12
1521.2.a.o.1.2		3	39.38	odd	2
1521.2.a.r.1.2		3	3.2	odd	2
1521.2.b.l.1351.3		6	39.5	even	4
1521.2.b.l.1351.4		6	39.8	even	4
2704.2.a.z.1.1		3	4.3	odd	2
2704.2.a.ba.1.1		3	52.51	odd	2
2704.2.f.o.337.1		6	52.47	even	4
2704.2.f.o.337.2		6	52.31	even	4
4225.2.a.bb.1.2		3	65.64	even	2
4225.2.a.bg.1.2		3	5.4	even	2
8281.2.a.bf.1.2		3	7.6	odd	2
8281.2.a.bj.1.2		3	91.90	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	169.2.a.b.1.2

Level	$169$
Weight	$2$
Character	169.1
Self dual	yes
Analytic conductor	$1.349$
Analytic rank	$1$
Dimension	$3$
CM	no
Inner twists	$1$