LMFDB - Embedded newform 169.4.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,4,Mod(1,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$169 = 13^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	169.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$9.97132279097$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 13)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ 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)

$f(q)$	$=$	$q-3.46410 q^{2} -7.00000 q^{3} +4.00000 q^{4} -13.8564 q^{5} +24.2487 q^{6} +22.5167 q^{7} +13.8564 q^{8} +22.0000 q^{9} +48.0000 q^{10} +22.5167 q^{11} -28.0000 q^{12} -78.0000 q^{14} +96.9948 q^{15} -80.0000 q^{16} -27.0000 q^{17} -76.2102 q^{18} +88.3346 q^{19} -55.4256 q^{20} -157.617 q^{21} -78.0000 q^{22} -57.0000 q^{23} -96.9948 q^{24} +67.0000 q^{25} +35.0000 q^{27} +90.0666 q^{28} -69.0000 q^{29} -336.000 q^{30} +72.7461 q^{31} +166.277 q^{32} -157.617 q^{33} +93.5307 q^{34} -312.000 q^{35} +88.0000 q^{36} +39.8372 q^{37} -306.000 q^{38} -192.000 q^{40} -393.176 q^{41} +546.000 q^{42} +85.0000 q^{43} +90.0666 q^{44} -304.841 q^{45} +197.454 q^{46} -342.946 q^{47} +560.000 q^{48} +164.000 q^{49} -232.095 q^{50} +189.000 q^{51} +426.000 q^{53} -121.244 q^{54} -312.000 q^{55} +312.000 q^{56} -618.342 q^{57} +239.023 q^{58} -19.0526 q^{59} +387.979 q^{60} -17.0000 q^{61} -252.000 q^{62} +495.367 q^{63} +64.0000 q^{64} +546.000 q^{66} +164.545 q^{67} -108.000 q^{68} +399.000 q^{69} +1080.80 q^{70} -583.701 q^{71} +304.841 q^{72} +1004.59 q^{73} -138.000 q^{74} -469.000 q^{75} +353.338 q^{76} +507.000 q^{77} -1244.00 q^{79} +1108.51 q^{80} -839.000 q^{81} +1362.00 q^{82} +426.084 q^{83} -630.466 q^{84} +374.123 q^{85} -294.449 q^{86} +483.000 q^{87} +312.000 q^{88} -306.573 q^{89} +1056.00 q^{90} -228.000 q^{92} -509.223 q^{93} +1188.00 q^{94} -1224.00 q^{95} -1163.94 q^{96} -1234.95 q^{97} -568.113 q^{98} +495.367 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 14 q^{3} + 8 q^{4} + 44 q^{9} + 96 q^{10} - 56 q^{12} - 156 q^{14} - 160 q^{16} - 54 q^{17} - 156 q^{22} - 114 q^{23} + 134 q^{25} + 70 q^{27} - 138 q^{29} - 672 q^{30} - 624 q^{35} + 176 q^{36} - 612 q^{38}+ \cdots - 2448 q^{95}+O(q^{100})$ 2 * q - 14 * q^3 + 8 * q^4 + 44 * q^9 + 96 * q^10 - 56 * q^12 - 156 * q^14 - 160 * q^16 - 54 * q^17 - 156 * q^22 - 114 * q^23 + 134 * q^25 + 70 * q^27 - 138 * q^29 - 672 * q^30 - 624 * q^35 + 176 * q^36 - 612 * q^38 - 384 * q^40 + 1092 * q^42 + 170 * q^43 + 1120 * q^48 + 328 * q^49 + 378 * q^51 + 852 * q^53 - 624 * q^55 + 624 * q^56 - 34 * q^61 - 504 * q^62 + 128 * q^64 + 1092 * q^66 - 216 * q^68 + 798 * q^69 - 276 * q^74 - 938 * q^75 + 1014 * q^77 - 2488 * q^79 - 1678 * q^81 + 2724 * q^82 + 966 * q^87 + 624 * q^88 + 2112 * q^90 - 456 * q^92 + 2376 * q^94 - 2448 * q^95

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

)

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

$2$	−3.46410	−1.22474	−0.612372	−	0.790569i	$-0.709785\pi$
$2$	−3.46410	−1.22474	−0.612372	+	0.790569i	$0.709785\pi$
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	4.00000	0.500000
$5$	−13.8564	−1.23935	−0.619677	−	0.784857i	$-0.712737\pi$
$5$	−13.8564	−1.23935	−0.619677	+	0.784857i	$0.712737\pi$
$6$	24.2487	1.64992
$7$	22.5167	1.21579	0.607893	−	0.794019i	$-0.292015\pi$
$7$	22.5167	1.21579	0.607893	+	0.794019i	$0.292015\pi$
$8$	13.8564	0.612372
$9$	22.0000	0.814815
$10$	48.0000	1.51789
$11$	22.5167	0.617184	0.308592	−	0.951194i	$-0.400142\pi$
$11$	22.5167	0.617184	0.308592	+	0.951194i	$0.400142\pi$
$12$	−28.0000	−0.673575
$13$	0	0
$13$	0	0
$14$	−78.0000	−1.48903
$15$	96.9948	1.66960
$16$	−80.0000	−1.25000
$17$	−27.0000	−0.385204	−0.192602	−	0.981277i	$-0.561693\pi$
$17$	−27.0000	−0.385204	−0.192602	+	0.981277i	$0.561693\pi$
$18$	−76.2102	−0.997940
$19$	88.3346	1.06660	0.533299	−	0.845927i	$-0.320952\pi$
$19$	88.3346	1.06660	0.533299	+	0.845927i	$0.320952\pi$
$20$	−55.4256	−0.619677
$21$	−157.617	−1.63785
$22$	−78.0000	−0.755893
$23$	−57.0000	−0.516753	−0.258377	−	0.966044i	$-0.583188\pi$
$23$	−57.0000	−0.516753	−0.258377	+	0.966044i	$0.583188\pi$
$24$	−96.9948	−0.824958
$25$	67.0000	0.536000
$26$	0	0
$27$	35.0000	0.249472
$28$	90.0666	0.607893
$29$	−69.0000	−0.441827	−0.220913	−	0.975293i	$-0.570904\pi$
$29$	−69.0000	−0.441827	−0.220913	+	0.975293i	$0.570904\pi$
$30$	−336.000	−2.04483
$31$	72.7461	0.421471	0.210735	−	0.977543i	$-0.432414\pi$
$31$	72.7461	0.421471	0.210735	+	0.977543i	$0.432414\pi$
$32$	166.277	0.918559
$33$	−157.617	−0.831440
$34$	93.5307	0.471776
$35$	−312.000	−1.50679
$36$	88.0000	0.407407
$37$	39.8372	0.177005	0.0885026	−	0.996076i	$-0.471792\pi$
$37$	39.8372	0.177005	0.0885026	+	0.996076i	$0.471792\pi$
$38$	−306.000	−1.30631
$39$	0	0
$40$	−192.000	−0.758947
$41$	−393.176	−1.49765	−0.748826	−	0.662767i	$-0.769382\pi$
$41$	−393.176	−1.49765	−0.748826	+	0.662767i	$0.769382\pi$
$42$	546.000	2.00594
$43$	85.0000	0.301451	0.150725	−	0.988576i	$-0.451839\pi$
$43$	85.0000	0.301451	0.150725	+	0.988576i	$0.451839\pi$
$44$	90.0666	0.308592
$45$	−304.841	−1.00984
$46$	197.454	0.632891
$47$	−342.946	−1.06434	−0.532168	−	0.846639i	$-0.678623\pi$
$47$	−342.946	−1.06434	−0.532168	+	0.846639i	$0.678623\pi$
$48$	560.000	1.68394
$49$	164.000	0.478134
$50$	−232.095	−0.656463
$51$	189.000	0.518927
$52$	0	0
$53$	426.000	1.10407	0.552034	−	0.833822i	$-0.313852\pi$
$53$	426.000	1.10407	0.552034	+	0.833822i	$0.313852\pi$
$54$	−121.244	−0.305540
$55$	−312.000	−0.764910
$56$	312.000	0.744513
$57$	−618.342	−1.43687
$58$	239.023	0.541125
$59$	−19.0526	−0.0420412	−0.0210206	−	0.999779i	$-0.506692\pi$
$59$	−19.0526	−0.0420412	−0.0210206	+	0.999779i	$0.506692\pi$
$60$	387.979	0.834799
$61$	−17.0000	−0.0356824	−0.0178412	−	0.999841i	$-0.505679\pi$
$61$	−17.0000	−0.0356824	−0.0178412	+	0.999841i	$0.505679\pi$
$62$	−252.000	−0.516194
$63$	495.367	0.990640
$64$	64.0000	0.125000
$65$	0	0
$66$	546.000	1.01830
$67$	164.545	0.300035	0.150018	−	0.988683i	$-0.452067\pi$
$67$	164.545	0.300035	0.150018	+	0.988683i	$0.452067\pi$
$68$	−108.000	−0.192602
$69$	399.000	0.696144
$70$	1080.80	1.84543
$71$	−583.701	−0.975670	−0.487835	−	0.872936i	$-0.662213\pi$
$71$	−583.701	−0.975670	−0.487835	+	0.872936i	$0.662213\pi$
$72$	304.841	0.498970
$73$	1004.59	1.61066	0.805331	−	0.592826i	$-0.201988\pi$
$73$	1004.59	1.61066	0.805331	+	0.592826i	$0.201988\pi$
$74$	−138.000	−0.216786
$75$	−469.000	−0.722073
$76$	353.338	0.533299
$77$	507.000	0.750364
$78$	0	0
$79$	−1244.00	−1.77166	−0.885829	−	0.464012i	$-0.846409\pi$
$79$	−1244.00	−1.77166	−0.885829	+	0.464012i	$0.846409\pi$
$80$	1108.51	1.54919
$81$	−839.000	−1.15089
$82$	1362.00	1.83424
$83$	426.084	0.563480	0.281740	−	0.959491i	$-0.409088\pi$
$83$	426.084	0.563480	0.281740	+	0.959491i	$0.409088\pi$
$84$	−630.466	−0.818923
$85$	374.123	0.477404
$86$	−294.449	−0.369200
$87$	483.000	0.595207
$88$	312.000	0.377947
$89$	−306.573	−0.365131	−0.182566	−	0.983194i	$-0.558440\pi$
$89$	−306.573	−0.365131	−0.182566	+	0.983194i	$0.558440\pi$
$90$	1056.00	1.23680
$91$	0	0
$92$	−228.000	−0.258377
$93$	−509.223	−0.567785
$94$	1188.00	1.30354
$95$	−1224.00	−1.32189
$96$	−1163.94	−1.23744
$97$	−1234.95	−1.29268	−0.646342	−	0.763048i	$-0.723702\pi$
$97$	−1234.95	−1.29268	−0.646342	+	0.763048i	$0.723702\pi$
$98$	−568.113	−0.585592
$99$	495.367	0.502891
$100$	268.000	0.268000
$101$	−1959.00	−1.92998	−0.964989	−	0.262290i	$-0.915522\pi$
$101$	−1959.00	−1.92998	−0.964989	+	0.262290i	$0.915522\pi$
$102$	−654.715	−0.635554
$103$	−1856.00	−1.77551	−0.887753	−	0.460320i	$-0.847735\pi$
$103$	−1856.00	−1.77551	−0.887753	+	0.460320i	$0.847735\pi$
$104$	0	0
$105$	2184.00	2.02987
$106$	−1475.71	−1.35220
$107$	−255.000	−0.230390	−0.115195	−	0.993343i	$-0.536749\pi$
$107$	−255.000	−0.230390	−0.115195	+	0.993343i	$0.536749\pi$
$108$	140.000	0.124736
$109$	609.682	0.535752	0.267876	−	0.963453i	$-0.413678\pi$
$109$	609.682	0.535752	0.267876	+	0.963453i	$0.413678\pi$
$110$	1080.80	0.936820
$111$	−278.860	−0.238453
$112$	−1801.33	−1.51973
$113$	411.000	0.342156	0.171078	−	0.985257i	$-0.445275\pi$
$113$	411.000	0.342156	0.171078	+	0.985257i	$0.445275\pi$
$114$	2142.00	1.75980
$115$	789.815	0.640440
$116$	−276.000	−0.220913
$117$	0	0
$118$	66.0000	0.0514898
$119$	−607.950	−0.468325
$120$	1344.00	1.02242
$121$	−824.000	−0.619083
$122$	58.8897	0.0437018
$123$	2752.23	2.01756
$124$	290.985	0.210735
$125$	803.672	0.575061
$126$	−1716.00	−1.21328
$127$	2243.00	1.56720	0.783599	−	0.621267i	$-0.213382\pi$
$127$	2243.00	1.56720	0.783599	+	0.621267i	$0.213382\pi$
$128$	−1551.92	−1.07165
$129$	−595.000	−0.406099
$130$	0	0
$131$	−372.000	−0.248105	−0.124053	−	0.992276i	$-0.539589\pi$
$131$	−372.000	−0.248105	−0.124053	+	0.992276i	$0.539589\pi$
$132$	−630.466	−0.415720
$133$	1989.00	1.29675
$134$	−570.000	−0.367466
$135$	−484.974	−0.309185
$136$	−374.123	−0.235888
$137$	−1189.92	−0.742056	−0.371028	−	0.928622i	$-0.620995\pi$
$137$	−1189.92	−0.742056	−0.371028	+	0.928622i	$0.620995\pi$
$138$	−1382.18	−0.852599
$139$	−2545.00	−1.55298	−0.776490	−	0.630130i	$-0.783002\pi$
$139$	−2545.00	−1.55298	−0.776490	+	0.630130i	$0.783002\pi$
$140$	−1248.00	−0.753395
$141$	2400.62	1.43382
$142$	2022.00	1.19495
$143$	0	0
$144$	−1760.00	−1.01852
$145$	956.092	0.547580
$146$	−3480.00	−1.97265
$147$	−1148.00	−0.644119
$148$	159.349	0.0885026
$149$	−1304.23	−0.717094	−0.358547	−	0.933512i	$-0.616728\pi$
$149$	−1304.23	−0.717094	−0.358547	+	0.933512i	$0.616728\pi$
$150$	1624.66	0.884355
$151$	−86.6025	−0.0466729	−0.0233365	−	0.999728i	$-0.507429\pi$
$151$	−86.6025	−0.0466729	−0.0233365	+	0.999728i	$0.507429\pi$
$152$	1224.00	0.653155
$153$	−594.000	−0.313870
$154$	−1756.30	−0.919004
$155$	−1008.00	−0.522352
$156$	0	0
$157$	−1534.00	−0.779787	−0.389893	−	0.920860i	$-0.627488\pi$
$157$	−1534.00	−0.779787	−0.389893	+	0.920860i	$0.627488\pi$
$158$	4309.34	2.16983
$159$	−2982.00	−1.48735
$160$	−2304.00	−1.13842
$161$	−1283.45	−0.628261
$162$	2906.38	1.40955
$163$	−1633.32	−0.784858	−0.392429	−	0.919782i	$-0.628365\pi$
$163$	−1633.32	−0.784858	−0.392429	+	0.919782i	$0.628365\pi$
$164$	−1572.70	−0.748826
$165$	2184.00	1.03045
$166$	−1476.00	−0.690119
$167$	−1626.40	−0.753618	−0.376809	−	0.926291i	$-0.622979\pi$
$167$	−1626.40	−0.753618	−0.376809	+	0.926291i	$0.622979\pi$
$168$	−2184.00	−1.00297
$169$	0	0
$170$	−1296.00	−0.584698
$171$	1943.36	0.869079
$172$	340.000	0.150725
$173$	873.000	0.383659	0.191829	−	0.981428i	$-0.438558\pi$
$173$	873.000	0.383659	0.191829	+	0.981428i	$0.438558\pi$
$174$	−1673.16	−0.728977
$175$	1508.62	0.651661
$176$	−1801.33	−0.771481
$177$	133.368	0.0566359
$178$	1062.00	0.447193
$179$	1287.00	0.537402	0.268701	−	0.963224i	$-0.413406\pi$
$179$	1287.00	0.537402	0.268701	+	0.963224i	$0.413406\pi$
$180$	−1219.36	−0.504922
$181$	−2.00000	−0.000821319 0	−0.000410660	−	1.00000i	$-0.500131\pi$
$181$	−2.00000	−0.000821319 0	−0.000410660		1.00000i	$0.500131\pi$
$182$	0	0
$183$	119.000	0.0480696
$184$	−789.815	−0.316445
$185$	−552.000	−0.219372
$186$	1764.00	0.695391
$187$	−607.950	−0.237742
$188$	−1371.78	−0.532168
$189$	788.083	0.303305
$190$	4240.06	1.61898
$191$	−2841.00	−1.07627	−0.538135	−	0.842859i	$-0.680871\pi$
$191$	−2841.00	−1.07627	−0.538135	+	0.842859i	$0.680871\pi$
$192$	−448.000	−0.168394
$193$	4245.26	1.58332	0.791659	−	0.610964i	$-0.209218\pi$
$193$	4245.26	1.58332	0.791659	+	0.610964i	$0.209218\pi$
$194$	4278.00	1.58321
$195$	0	0
$196$	656.000	0.239067
$197$	2752.23	0.995371	0.497686	−	0.867357i	$-0.334183\pi$
$197$	2752.23	0.995371	0.497686	+	0.867357i	$0.334183\pi$
$198$	−1716.00	−0.615913
$199$	1685.00	0.600234	0.300117	−	0.953902i	$-0.402974\pi$
$199$	1685.00	0.600234	0.300117	+	0.953902i	$0.402974\pi$
$200$	928.379	0.328232
$201$	−1151.81	−0.404192
$202$	6786.18	2.36373
$203$	−1553.65	−0.537167
$204$	756.000	0.259464
$205$	5448.00	1.85612
$206$	6429.37	2.17454
$207$	−1254.00	−0.421058
$208$	0	0
$209$	1989.00	0.658287
$210$	−7565.60	−2.48608
$211$	1681.00	0.548459	0.274229	−	0.961664i	$-0.411577\pi$
$211$	1681.00	0.548459	0.274229	+	0.961664i	$0.411577\pi$
$212$	1704.00	0.552034
$213$	4085.91	1.31437
$214$	883.346	0.282170
$215$	−1177.79	−0.373604
$216$	484.974	0.152770
$217$	1638.00	0.512418
$218$	−2112.00	−0.656159
$219$	−7032.13	−2.16980
$220$	−1248.00	−0.382455
$221$	0	0
$222$	966.000	0.292044
$223$	−4096.30	−1.23008	−0.615042	−	0.788495i	$-0.710861\pi$
$223$	−4096.30	−1.23008	−0.615042	+	0.788495i	$0.710861\pi$
$224$	3744.00	1.11677
$225$	1474.00	0.436741
$226$	−1423.75	−0.419054
$227$	−438.209	−0.128128	−0.0640638	−	0.997946i	$-0.520406\pi$
$227$	−438.209	−0.128128	−0.0640638	+	0.997946i	$0.520406\pi$
$228$	−2473.37	−0.718433
$229$	180.133	0.0519805	0.0259903	−	0.999662i	$-0.491726\pi$
$229$	180.133	0.0519805	0.0259903	+	0.999662i	$0.491726\pi$
$230$	−2736.00	−0.784376
$231$	−3549.00	−1.01085
$232$	−956.092	−0.270563
$233$	5778.00	1.62459	0.812295	−	0.583247i	$-0.198218\pi$
$233$	5778.00	1.62459	0.812295	+	0.583247i	$0.198218\pi$
$234$	0	0
$235$	4752.00	1.31909
$236$	−76.2102	−0.0210206
$237$	8708.00	2.38669
$238$	2106.00	0.573579
$239$	1860.22	0.503464	0.251732	−	0.967797i	$-0.419000\pi$
$239$	1860.22	0.503464	0.251732	+	0.967797i	$0.419000\pi$
$240$	−7759.59	−2.08700
$241$	2059.41	0.550449	0.275224	−	0.961380i	$-0.411248\pi$
$241$	2059.41	0.550449	0.275224	+	0.961380i	$0.411248\pi$
$242$	2854.42	0.758219
$243$	4928.00	1.30095
$244$	−68.0000	−0.0178412
$245$	−2272.45	−0.592578
$246$	−9534.00	−2.47100
$247$	0	0
$248$	1008.00	0.258097
$249$	−2982.59	−0.759093
$250$	−2784.00	−0.704302
$251$	−4491.00	−1.12936	−0.564680	−	0.825310i	$-0.691000\pi$
$251$	−4491.00	−1.12936	−0.564680	+	0.825310i	$0.691000\pi$
$252$	1981.47	0.495320
$253$	−1283.45	−0.318932
$254$	−7769.98	−1.91942
$255$	−2618.86	−0.643135
$256$	4864.00	1.18750
$257$	−5451.00	−1.32305	−0.661525	−	0.749923i	$-0.730091\pi$
$257$	−5451.00	−1.32305	−0.661525	+	0.749923i	$0.730091\pi$
$258$	2061.14	0.497368
$259$	897.000	0.215200
$260$	0	0
$261$	−1518.00	−0.360007
$262$	1288.65	0.303866
$263$	−783.000	−0.183581	−0.0917906	−	0.995778i	$-0.529259\pi$
$263$	−783.000	−0.183581	−0.0917906	+	0.995778i	$0.529259\pi$
$264$	−2184.00	−0.509151
$265$	−5902.83	−1.36833
$266$	−6890.10	−1.58819
$267$	2146.01	0.491887
$268$	658.179	0.150018
$269$	−5085.00	−1.15256	−0.576279	−	0.817253i	$-0.695496\pi$
$269$	−5085.00	−1.15256	−0.576279	+	0.817253i	$0.695496\pi$
$270$	1680.00	0.378672
$271$	1325.02	0.297008	0.148504	−	0.988912i	$-0.452554\pi$
$271$	1325.02	0.297008	0.148504	+	0.988912i	$0.452554\pi$
$272$	2160.00	0.481505
$273$	0	0
$274$	4122.00	0.908829
$275$	1508.62	0.330811
$276$	1596.00	0.348072
$277$	3421.00	0.742050	0.371025	−	0.928623i	$-0.379006\pi$
$277$	3421.00	0.742050	0.371025	+	0.928623i	$0.379006\pi$
$278$	8816.14	1.90200
$279$	1600.41	0.343421
$280$	−4323.20	−0.922716
$281$	810.600	0.172087	0.0860433	−	0.996291i	$-0.472578\pi$
$281$	810.600	0.172087	0.0860433	+	0.996291i	$0.472578\pi$
$282$	−8316.00	−1.75607
$283$	−7177.00	−1.50752	−0.753760	−	0.657149i	$-0.771762\pi$
$283$	−7177.00	−1.50752	−0.753760	+	0.657149i	$0.771762\pi$
$284$	−2334.80	−0.487835
$285$	8568.00	1.78079
$286$	0	0
$287$	−8853.00	−1.82082
$288$	3658.09	0.748455
$289$	−4184.00	−0.851618
$290$	−3312.00	−0.670646
$291$	8644.67	1.74144
$292$	4018.36	0.805331
$293$	9313.24	1.85695	0.928473	−	0.371400i	$-0.121122\pi$
$293$	9313.24	1.85695	0.928473	+	0.371400i	$0.121122\pi$
$294$	3976.79	0.788881
$295$	264.000	0.0521040
$296$	552.000	0.108393
$297$	788.083	0.153970
$298$	4518.00	0.878257
$299$	0	0
$300$	−1876.00	−0.361036
$301$	1913.92	0.366499
$302$	300.000	0.0571625
$303$	13713.0	2.59997
$304$	−7066.77	−1.33325
$305$	235.559	0.0442232
$306$	2057.68	0.384410
$307$	−4777.00	−0.888070	−0.444035	−	0.896009i	$-0.646453\pi$
$307$	−4777.00	−0.888070	−0.444035	+	0.896009i	$0.646453\pi$
$308$	2028.00	0.375182
$309$	12992.0	2.39187
$310$	3491.81	0.639748
$311$	−6192.00	−1.12899	−0.564495	−	0.825436i	$-0.690929\pi$
$311$	−6192.00	−1.12899	−0.564495	+	0.825436i	$0.690929\pi$
$312$	0	0
$313$	−770.000	−0.139051	−0.0695255	−	0.997580i	$-0.522149\pi$
$313$	−770.000	−0.139051	−0.0695255	+	0.997580i	$0.522149\pi$
$314$	5313.93	0.955040
$315$	−6864.00	−1.22775
$316$	−4976.00	−0.885829
$317$	−8057.50	−1.42762	−0.713808	−	0.700341i	$-0.753031\pi$
$317$	−8057.50	−1.42762	−0.713808	+	0.700341i	$0.753031\pi$
$318$	10330.0	1.82162
$319$	−1553.65	−0.272689
$320$	−886.810	−0.154919
$321$	1785.00	0.310371
$322$	4446.00	0.769459
$323$	−2385.03	−0.410857
$324$	−3356.00	−0.575446
$325$	0	0
$326$	5658.00	0.961250
$327$	−4267.77	−0.721738
$328$	−5448.00	−0.917120
$329$	−7722.00	−1.29400
$330$	−7565.60	−1.26204
$331$	5277.56	0.876377	0.438189	−	0.898883i	$-0.355620\pi$
$331$	5277.56	0.876377	0.438189	+	0.898883i	$0.355620\pi$
$332$	1704.34	0.281740
$333$	876.418	0.144226
$334$	5634.00	0.922990
$335$	−2280.00	−0.371850
$336$	12609.3	2.04731
$337$	−8278.00	−1.33808	−0.669038	−	0.743228i	$-0.733294\pi$
$337$	−8278.00	−1.33808	−0.669038	+	0.743228i	$0.733294\pi$
$338$	0	0
$339$	−2877.00	−0.460936
$340$	1496.49	0.238702
$341$	1638.00	0.260125
$342$	−6732.00	−1.06440
$343$	−4030.48	−0.634477
$344$	1177.79	0.184600
$345$	−5528.71	−0.862770
$346$	−3024.16	−0.469884
$347$	6867.00	1.06236	0.531181	−	0.847258i	$-0.321748\pi$
$347$	6867.00	1.06236	0.531181	+	0.847258i	$0.321748\pi$
$348$	1932.00	0.297604
$349$	−12153.8	−1.86412	−0.932060	−	0.362303i	$-0.881990\pi$
$349$	−12153.8	−1.86412	−0.932060	+	0.362303i	$0.881990\pi$
$350$	−5226.00	−0.798118
$351$	0	0
$352$	3744.00	0.566920
$353$	5807.57	0.875653	0.437827	−	0.899059i	$-0.355748\pi$
$353$	5807.57	0.875653	0.437827	+	0.899059i	$0.355748\pi$
$354$	−462.000	−0.0693645
$355$	8088.00	1.20920
$356$	−1226.29	−0.182566
$357$	4255.65	0.630904
$358$	−4458.30	−0.658180
$359$	1340.61	0.197088	0.0985439	−	0.995133i	$-0.468581\pi$
$359$	1340.61	0.197088	0.0985439	+	0.995133i	$0.468581\pi$
$360$	−4224.00	−0.618401
$361$	944.000	0.137629
$362$	6.92820	0.00100591
$363$	5768.00	0.833999
$364$	0	0
$365$	−13920.0	−1.99618
$366$	−412.228	−0.0588730
$367$	3665.00	0.521285	0.260642	−	0.965435i	$-0.416066\pi$
$367$	3665.00	0.521285	0.260642	+	0.965435i	$0.416066\pi$
$368$	4560.00	0.645941
$369$	−8649.86	−1.22031
$370$	1912.18	0.268675
$371$	9592.10	1.34231
$372$	−2036.89	−0.283892
$373$	5371.00	0.745576	0.372788	−	0.927917i	$-0.378402\pi$
$373$	5371.00	0.745576	0.372788	+	0.927917i	$0.378402\pi$
$374$	2106.00	0.291173
$375$	−5625.70	−0.774693
$376$	−4752.00	−0.651770
$377$	0	0
$378$	−2730.00	−0.371471
$379$	−11509.5	−1.55990	−0.779950	−	0.625842i	$-0.784756\pi$
$379$	−11509.5	−1.55990	−0.779950	+	0.625842i	$0.784756\pi$
$380$	−4896.00	−0.660946
$381$	−15701.0	−2.11125
$382$	9841.51	1.31816
$383$	−2419.67	−0.322819	−0.161409	−	0.986888i	$-0.551604\pi$
$383$	−2419.67	−0.322819	−0.161409	+	0.986888i	$0.551604\pi$
$384$	10863.4	1.44368
$385$	−7025.20	−0.929967
$386$	−14706.0	−1.93916
$387$	1870.00	0.245626
$388$	−4939.81	−0.646342
$389$	9858.00	1.28489	0.642443	−	0.766334i	$-0.277921\pi$
$389$	9858.00	1.28489	0.642443	+	0.766334i	$0.277921\pi$
$390$	0	0
$391$	1539.00	0.199055
$392$	2272.45	0.292796
$393$	2604.00	0.334235
$394$	−9534.00	−1.21908
$395$	17237.4	2.19571
$396$	1981.47	0.251446
$397$	−8720.88	−1.10249	−0.551245	−	0.834344i	$-0.685847\pi$
$397$	−8720.88	−1.10249	−0.551245	+	0.834344i	$0.685847\pi$
$398$	−5837.01	−0.735133
$399$	−13923.0	−1.74692
$400$	−5360.00	−0.670000
$401$	7584.65	0.944537	0.472269	−	0.881455i	$-0.343435\pi$
$401$	7584.65	0.944537	0.472269	+	0.881455i	$0.343435\pi$
$402$	3990.00	0.495033
$403$	0	0
$404$	−7836.00	−0.964989
$405$	11625.5	1.42636
$406$	5382.00	0.657892
$407$	897.000	0.109245
$408$	2618.86	0.317777
$409$	−4304.15	−0.520358	−0.260179	−	0.965560i	$-0.583782\pi$
$409$	−4304.15	−0.520358	−0.260179	+	0.965560i	$0.583782\pi$
$410$	−18872.4	−2.27327
$411$	8329.43	0.999661
$412$	−7424.00	−0.887753
$413$	−429.000	−0.0511131
$414$	4343.98	0.515689
$415$	−5904.00	−0.698352
$416$	0	0
$417$	17815.0	2.09210
$418$	−6890.10	−0.806234
$419$	5397.00	0.629262	0.314631	−	0.949214i	$-0.398119\pi$
$419$	5397.00	0.629262	0.314631	+	0.949214i	$0.398119\pi$
$420$	8736.00	1.01494
$421$	−7260.76	−0.840541	−0.420270	−	0.907399i	$-0.638065\pi$
$421$	−7260.76	−0.840541	−0.420270	+	0.907399i	$0.638065\pi$
$422$	−5823.15	−0.671722
$423$	−7544.81	−0.867237
$424$	5902.83	0.676101
$425$	−1809.00	−0.206469
$426$	−14154.0	−1.60977
$427$	−382.783	−0.0433822
$428$	−1020.00	−0.115195
$429$	0	0
$430$	4080.00	0.457570
$431$	−486.706	−0.0543940	−0.0271970	−	0.999630i	$-0.508658\pi$
$431$	−486.706	−0.0543940	−0.0271970	+	0.999630i	$0.508658\pi$
$432$	−2800.00	−0.311840
$433$	−12139.0	−1.34726	−0.673629	−	0.739069i	$-0.735266\pi$
$433$	−12139.0	−1.34726	−0.673629	+	0.739069i	$0.735266\pi$
$434$	−5674.20	−0.627581
$435$	−6692.64	−0.737673
$436$	2438.73	0.267876
$437$	−5035.07	−0.551167
$438$	24360.0	2.65746
$439$	−461.000	−0.0501192	−0.0250596	−	0.999686i	$-0.507978\pi$
$439$	−461.000	−0.0501192	−0.0250596	+	0.999686i	$0.507978\pi$
$440$	−4323.20	−0.468410
$441$	3608.00	0.389591
$442$	0	0
$443$	12156.0	1.30372	0.651861	−	0.758338i	$-0.273988\pi$
$443$	12156.0	1.30372	0.651861	+	0.758338i	$0.273988\pi$
$444$	−1115.44	−0.119226
$445$	4248.00	0.452527
$446$	14190.0	1.50654
$447$	9129.64	0.966034
$448$	1441.07	0.151973
$449$	−296.181	−0.0311306	−0.0155653	−	0.999879i	$-0.504955\pi$
$449$	−296.181	−0.0311306	−0.0155653	+	0.999879i	$0.504955\pi$
$450$	−5106.09	−0.534896
$451$	−8853.00	−0.924327
$452$	1644.00	0.171078
$453$	606.218	0.0628755
$454$	1518.00	0.156924
$455$	0	0
$456$	−8568.00	−0.879898
$457$	611.414	0.0625837	0.0312918	−	0.999510i	$-0.490038\pi$
$457$	611.414	0.0625837	0.0312918	+	0.999510i	$0.490038\pi$
$458$	−624.000	−0.0636629
$459$	−945.000	−0.0960977
$460$	3159.26	0.320220
$461$	13127.2	1.32624	0.663119	−	0.748514i	$-0.269233\pi$
$461$	13127.2	1.32624	0.663119	+	0.748514i	$0.269233\pi$
$462$	12294.1	1.23804
$463$	−834.848	−0.0837985	−0.0418992	−	0.999122i	$-0.513341\pi$
$463$	−834.848	−0.0837985	−0.0418992	+	0.999122i	$0.513341\pi$
$464$	5520.00	0.552284
$465$	7056.00	0.703686
$466$	−20015.6	−1.98971
$467$	−14496.0	−1.43639	−0.718196	−	0.695841i	$-0.755032\pi$
$467$	−14496.0	−1.43639	−0.718196	+	0.695841i	$0.755032\pi$
$468$	0	0
$469$	3705.00	0.364778
$470$	−16461.4	−1.61555
$471$	10738.0	1.05049
$472$	−264.000	−0.0257449
$473$	1913.92	0.186051
$474$	−30165.4	−2.92309
$475$	5918.42	0.571696
$476$	−2431.80	−0.234162
$477$	9372.00	0.899611
$478$	−6444.00	−0.616614
$479$	8897.54	0.848725	0.424362	−	0.905492i	$-0.360498\pi$
$479$	8897.54	0.848725	0.424362	+	0.905492i	$0.360498\pi$
$480$	16128.0	1.53362
$481$	0	0
$482$	−7134.00	−0.674159
$483$	8984.15	0.846362
$484$	−3296.00	−0.309542
$485$	17112.0	1.60209
$486$	−17071.1	−1.59333
$487$	−4754.48	−0.442394	−0.221197	−	0.975229i	$-0.570996\pi$
$487$	−4754.48	−0.442394	−0.221197	+	0.975229i	$0.570996\pi$
$488$	−235.559	−0.0218509
$489$	11433.3	1.05732
$490$	7872.00	0.725757
$491$	1635.00	0.150278	0.0751390	−	0.997173i	$-0.476060\pi$
$491$	1635.00	0.150278	0.0751390	+	0.997173i	$0.476060\pi$
$492$	11008.9	1.00878
$493$	1863.00	0.170193
$494$	0	0
$495$	−6864.00	−0.623260
$496$	−5819.69	−0.526838
$497$	−13143.0	−1.18621
$498$	10332.0	0.929695
$499$	14434.9	1.29498	0.647490	−	0.762074i	$-0.275819\pi$
$499$	14434.9	1.29498	0.647490	+	0.762074i	$0.275819\pi$
$500$	3214.69	0.287530
$501$	11384.8	1.01524
$502$	15557.3	1.38318
$503$	12687.0	1.12462	0.562312	−	0.826925i	$-0.309912\pi$
$503$	12687.0	1.12462	0.562312	+	0.826925i	$0.309912\pi$
$504$	6864.00	0.606641
$505$	27144.7	2.39193
$506$	4446.00	0.390610
$507$	0	0
$508$	8972.00	0.783599
$509$	−5748.68	−0.500600	−0.250300	−	0.968168i	$-0.580529\pi$
$509$	−5748.68	−0.500600	−0.250300	+	0.968168i	$0.580529\pi$
$510$	9072.00	0.787676
$511$	22620.0	1.95822
$512$	−4434.05	−0.382733
$513$	3091.71	0.266086
$514$	18882.8	1.62040
$515$	25717.5	2.20048
$516$	−2380.00	−0.203050
$517$	−7722.00	−0.656892
$518$	−3107.30	−0.263565
$519$	−6111.00	−0.516846
$520$	0	0
$521$	6054.00	0.509080	0.254540	−	0.967062i	$-0.418076\pi$
$521$	6054.00	0.509080	0.254540	+	0.967062i	$0.418076\pi$
$522$	5258.51	0.440917
$523$	−14803.0	−1.23765	−0.618824	−	0.785530i	$-0.712391\pi$
$523$	−14803.0	−1.23765	−0.618824	+	0.785530i	$0.712391\pi$
$524$	−1488.00	−0.124053
$525$	−10560.3	−0.877885
$526$	2712.39	0.224840
$527$	−1964.15	−0.162352
$528$	12609.3	1.03930
$529$	−8918.00	−0.732966
$530$	20448.0	1.67586
$531$	−419.156	−0.0342558
$532$	7956.00	0.648377
$533$	0	0
$534$	−7434.00	−0.602436
$535$	3533.38	0.285536
$536$	2280.00	0.183733
$537$	−9009.00	−0.723961
$538$	17615.0	1.41159
$539$	3692.73	0.295097
$540$	−1939.90	−0.154592
$541$	−21470.5	−1.70626	−0.853132	−	0.521695i	$-0.825300\pi$
$541$	−21470.5	−1.70626	−0.853132	+	0.521695i	$0.825300\pi$
$542$	−4590.00	−0.363759
$543$	14.0000	0.00110644
$544$	−4489.48	−0.353832
$545$	−8448.00	−0.663986
$546$	0	0
$547$	−13516.0	−1.05649	−0.528247	−	0.849091i	$-0.677151\pi$
$547$	−13516.0	−1.05649	−0.528247	+	0.849091i	$0.677151\pi$
$548$	−4759.68	−0.371028
$549$	−374.000	−0.0290746
$550$	−5226.00	−0.405159
$551$	−6095.09	−0.471251
$552$	5528.71	0.426300
$553$	−28010.7	−2.15396
$554$	−11850.7	−0.908822
$555$	3864.00	0.295527
$556$	−10180.0	−0.776490
$557$	−2890.79	−0.219905	−0.109952	−	0.993937i	$-0.535070\pi$
$557$	−2890.79	−0.219905	−0.109952	+	0.993937i	$0.535070\pi$
$558$	−5544.00	−0.420603
$559$	0	0
$560$	24960.0	1.88349
$561$	4255.65	0.320274
$562$	−2808.00	−0.210762
$563$	−11583.0	−0.867079	−0.433539	−	0.901135i	$-0.642735\pi$
$563$	−11583.0	−0.867079	−0.433539	+	0.901135i	$0.642735\pi$
$564$	9602.49	0.716911
$565$	−5694.98	−0.424053
$566$	24861.9	1.84633
$567$	−18891.5	−1.39924
$568$	−8088.00	−0.597473
$569$	−12879.0	−0.948885	−0.474443	−	0.880286i	$-0.657350\pi$
$569$	−12879.0	−0.948885	−0.474443	+	0.880286i	$0.657350\pi$
$570$	−29680.4	−2.18101
$571$	11636.0	0.852805	0.426402	−	0.904534i	$-0.359781\pi$
$571$	11636.0	0.852805	0.426402	+	0.904534i	$0.359781\pi$
$572$	0	0
$573$	19887.0	1.44990
$574$	30667.7	2.23004
$575$	−3819.00	−0.276980
$576$	1408.00	0.101852
$577$	−12311.4	−0.888269	−0.444134	−	0.895960i	$-0.646489\pi$
$577$	−12311.4	−0.888269	−0.444134	+	0.895960i	$0.646489\pi$
$578$	14493.8	1.04301
$579$	−29716.8	−2.13297
$580$	3824.37	0.273790
$581$	9594.00	0.685071
$582$	−29946.0	−2.13282
$583$	9592.10	0.681414
$584$	13920.0	0.986325
$585$	0	0
$586$	−32262.0	−2.27428
$587$	−15645.6	−1.10011	−0.550054	−	0.835129i	$-0.685393\pi$
$587$	−15645.6	−1.10011	−0.550054	+	0.835129i	$0.685393\pi$
$588$	−4592.00	−0.322059
$589$	6426.00	0.449539
$590$	−914.523	−0.0638141
$591$	−19265.6	−1.34092
$592$	−3186.97	−0.221256
$593$	25821.4	1.78813	0.894063	−	0.447942i	$-0.147843\pi$
$593$	25821.4	1.78813	0.894063	+	0.447942i	$0.147843\pi$
$594$	−2730.00	−0.188575
$595$	8424.00	0.580421
$596$	−5216.94	−0.358547
$597$	−11795.0	−0.808605
$598$	0	0
$599$	1668.00	0.113777	0.0568887	−	0.998381i	$-0.481882\pi$
$599$	1668.00	0.113777	0.0568887	+	0.998381i	$0.481882\pi$
$600$	−6498.65	−0.442177
$601$	13699.0	0.929773	0.464887	−	0.885370i	$-0.346095\pi$
$601$	13699.0	0.929773	0.464887	+	0.885370i	$0.346095\pi$
$602$	−6630.00	−0.448868
$603$	3619.99	0.244473
$604$	−346.410	−0.0233365
$605$	11417.7	0.767264
$606$	−47503.2	−3.18430
$607$	−23173.0	−1.54953	−0.774764	−	0.632251i	$-0.782131\pi$
$607$	−23173.0	−1.54953	−0.774764	+	0.632251i	$0.782131\pi$
$608$	14688.0	0.979732
$609$	10875.5	0.723644
$610$	−816.000	−0.0541621
$611$	0	0
$612$	−2376.00	−0.156935
$613$	16615.6	1.09477	0.547387	−	0.836880i	$-0.315623\pi$
$613$	16615.6	1.09477	0.547387	+	0.836880i	$0.315623\pi$
$614$	16548.0	1.08766
$615$	−38136.0	−2.50047
$616$	7025.20	0.459502
$617$	−28393.5	−1.85264	−0.926321	−	0.376736i	$-0.877046\pi$
$617$	−28393.5	−1.85264	−0.926321	+	0.376736i	$0.877046\pi$
$618$	−45005.6	−2.92944
$619$	−6245.78	−0.405556	−0.202778	−	0.979225i	$-0.564997\pi$
$619$	−6245.78	−0.405556	−0.202778	+	0.979225i	$0.564997\pi$
$620$	−4032.00	−0.261176
$621$	−1995.00	−0.128916
$622$	21449.7	1.38273
$623$	−6903.00	−0.443921
$624$	0	0
$625$	−19511.0	−1.24870
$626$	2667.36	0.170302
$627$	−13923.0	−0.886812
$628$	−6136.00	−0.389893
$629$	−1075.60	−0.0681830
$630$	23777.6	1.50369
$631$	22379.8	1.41193	0.705964	−	0.708247i	$-0.250514\pi$
$631$	22379.8	1.41193	0.705964	+	0.708247i	$0.250514\pi$
$632$	−17237.4	−1.08491
$633$	−11767.0	−0.738857
$634$	27912.0	1.74847
$635$	−31079.9	−1.94231
$636$	−11928.0	−0.743673
$637$	0	0
$638$	5382.00	0.333974
$639$	−12841.4	−0.794990
$640$	21504.0	1.32816
$641$	−19827.0	−1.22172	−0.610858	−	0.791740i	$-0.709175\pi$
$641$	−19827.0	−1.22172	−0.610858	+	0.791740i	$0.709175\pi$
$642$	−6183.42	−0.380125
$643$	8450.68	0.518293	0.259146	−	0.965838i	$-0.416559\pi$
$643$	8450.68	0.518293	0.259146	+	0.965838i	$0.416559\pi$
$644$	−5133.80	−0.314130
$645$	8244.56	0.503301
$646$	8262.00	0.503195
$647$	−2949.00	−0.179192	−0.0895959	−	0.995978i	$-0.528558\pi$
$647$	−2949.00	−0.179192	−0.0895959	+	0.995978i	$0.528558\pi$
$648$	−11625.5	−0.704774
$649$	−429.000	−0.0259472
$650$	0	0
$651$	−11466.0	−0.690304
$652$	−6533.30	−0.392429
$653$	12039.0	0.721474	0.360737	−	0.932668i	$-0.382525\pi$
$653$	12039.0	0.721474	0.360737	+	0.932668i	$0.382525\pi$
$654$	14784.0	0.883945
$655$	5154.58	0.307490
$656$	31454.0	1.87206
$657$	22101.0	1.31239
$658$	26749.8	1.58483
$659$	3363.00	0.198792	0.0993960	−	0.995048i	$-0.468309\pi$
$659$	3363.00	0.198792	0.0993960	+	0.995048i	$0.468309\pi$
$660$	8736.00	0.515225
$661$	−10158.5	−0.597759	−0.298880	−	0.954291i	$-0.596613\pi$
$661$	−10158.5	−0.597759	−0.298880	+	0.954291i	$0.596613\pi$
$662$	−18282.0	−1.07334
$663$	0	0
$664$	5904.00	0.345060
$665$	−27560.4	−1.60714
$666$	−3036.00	−0.176641
$667$	3933.00	0.228315
$668$	−6505.58	−0.376809
$669$	28674.1	1.65711
$670$	7898.15	0.455421
$671$	−382.783	−0.0220226
$672$	−26208.0	−1.50446
$673$	18169.0	1.04066	0.520329	−	0.853966i	$-0.325809\pi$
$673$	18169.0	1.04066	0.520329	+	0.853966i	$0.325809\pi$
$674$	28675.8	1.63880
$675$	2345.00	0.133717
$676$	0	0
$677$	9042.00	0.513312	0.256656	−	0.966503i	$-0.417379\pi$
$677$	9042.00	0.513312	0.256656	+	0.966503i	$0.417379\pi$
$678$	9966.22	0.564529
$679$	−27807.0	−1.57163
$680$	5184.00	0.292349
$681$	3067.46	0.172607
$682$	−5674.20	−0.318587
$683$	−12462.1	−0.698169	−0.349084	−	0.937091i	$-0.613507\pi$
$683$	−12462.1	−0.698169	−0.349084	+	0.937091i	$0.613507\pi$
$684$	7773.44	0.434540
$685$	16488.0	0.919670
$686$	13962.0	0.777072
$687$	−1260.93	−0.0700256
$688$	−6800.00	−0.376813
$689$	0	0
$690$	19152.0	1.05667
$691$	−4318.00	−0.237720	−0.118860	−	0.992911i	$-0.537924\pi$
$691$	−4318.00	−0.237720	−0.118860	+	0.992911i	$0.537924\pi$
$692$	3492.00	0.191829
$693$	11154.0	0.611408
$694$	−23788.0	−1.30112
$695$	35264.6	1.92469
$696$	6692.64	0.364489
$697$	10615.7	0.576901
$698$	42102.0	2.28307
$699$	−40446.0	−2.18857
$700$	6034.47	0.325830
$701$	18270.0	0.984377	0.492189	−	0.870489i	$-0.336197\pi$
$701$	18270.0	0.984377	0.492189	+	0.870489i	$0.336197\pi$
$702$	0	0
$703$	3519.00	0.188793
$704$	1441.07	0.0771481
$705$	−33264.0	−1.77701
$706$	−20118.0	−1.07245
$707$	−44110.1	−2.34644
$708$	533.472	0.0283179
$709$	1629.86	0.0863338	0.0431669	−	0.999068i	$-0.486255\pi$
$709$	1629.86	0.0863338	0.0431669	+	0.999068i	$0.486255\pi$
$710$	−28017.7	−1.48096
$711$	−27368.0	−1.44357
$712$	−4248.00	−0.223596
$713$	−4146.53	−0.217796
$714$	−14742.0	−0.772697
$715$	0	0
$716$	5148.00	0.268701
$717$	−13021.6	−0.678241
$718$	−4644.00	−0.241382
$719$	−9831.00	−0.509923	−0.254961	−	0.966951i	$-0.582063\pi$
$719$	−9831.00	−0.509923	−0.254961	+	0.966951i	$0.582063\pi$
$720$	24387.3	1.26231
$721$	−41790.9	−2.15863
$722$	−3270.11	−0.168561
$723$	−14415.9	−0.741537
$724$	−8.00000	−0.000410660 0
$725$	−4623.00	−0.236819
$726$	−19980.9	−1.02144
$727$	15464.0	0.788897	0.394448	−	0.918918i	$-0.370936\pi$
$727$	15464.0	0.788897	0.394448	+	0.918918i	$0.370936\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	48220.3	2.44481
$731$	−2295.00	−0.116120
$732$	476.000	0.0240348
$733$	12616.3	0.635733	0.317866	−	0.948136i	$-0.397034\pi$
$733$	12616.3	0.635733	0.317866	+	0.948136i	$0.397034\pi$
$734$	−12695.9	−0.638441
$735$	15907.2	0.798291
$736$	−9477.78	−0.474668
$737$	3705.00	0.185177
$738$	29964.0	1.49457
$739$	16283.0	0.810528	0.405264	−	0.914200i	$-0.367180\pi$
$739$	16283.0	0.810528	0.405264	+	0.914200i	$0.367180\pi$
$740$	−2208.00	−0.109686
$741$	0	0
$742$	−33228.0	−1.64399
$743$	10806.3	0.533571	0.266786	−	0.963756i	$-0.414038\pi$
$743$	10806.3	0.533571	0.266786	+	0.963756i	$0.414038\pi$
$744$	−7056.00	−0.347696
$745$	18072.0	0.888734
$746$	−18605.7	−0.913140
$747$	9373.86	0.459132
$748$	−2431.80	−0.118871
$749$	−5741.75	−0.280105
$750$	19488.0	0.948802
$751$	13615.0	0.661542	0.330771	−	0.943711i	$-0.392691\pi$
$751$	13615.0	0.661542	0.330771	+	0.943711i	$0.392691\pi$
$752$	27435.7	1.33042
$753$	31437.0	1.52142
$754$	0	0
$755$	1200.00	0.0578443
$756$	3152.33	0.151652
$757$	5551.00	0.266519	0.133259	−	0.991081i	$-0.457456\pi$
$757$	5551.00	0.266519	0.133259	+	0.991081i	$0.457456\pi$
$758$	39870.0	1.91048
$759$	8984.15	0.429649
$760$	−16960.2	−0.809490
$761$	10082.3	0.480265	0.240133	−	0.970740i	$-0.422809\pi$
$761$	10082.3	0.480265	0.240133	+	0.970740i	$0.422809\pi$
$762$	54389.9	2.58574
$763$	13728.0	0.651359
$764$	−11364.0	−0.538135
$765$	8230.71	0.388996
$766$	8382.00	0.395371
$767$	0	0
$768$	−34048.0	−1.59974
$769$	29758.4	1.39547	0.697733	−	0.716357i	$-0.254192\pi$
$769$	29758.4	1.39547	0.697733	+	0.716357i	$0.254192\pi$
$770$	24336.0	1.13897
$771$	38157.0	1.78235
$772$	16981.0	0.791659
$773$	−27735.3	−1.29052	−0.645259	−	0.763964i	$-0.723251\pi$
$773$	−27735.3	−1.29052	−0.645259	+	0.763964i	$0.723251\pi$
$774$	−6477.87	−0.300830
$775$	4873.99	0.225908
$776$	−17112.0	−0.791604
$777$	−6279.00	−0.289907
$778$	−34149.1	−1.57366
$779$	−34731.0	−1.59739
$780$	0	0
$781$	−13143.0	−0.602168
$782$	−5331.25	−0.243792
$783$	−2415.00	−0.110224
$784$	−13120.0	−0.597668
$785$	21255.7	0.966432
$786$	−9020.52	−0.409353
$787$	−31549.3	−1.42899	−0.714493	−	0.699643i	$-0.753342\pi$
$787$	−31549.3	−1.42899	−0.714493	+	0.699643i	$0.753342\pi$
$788$	11008.9	0.497686
$789$	5481.00	0.247311
$790$	−59712.0	−2.68919
$791$	9254.35	0.415988
$792$	6864.00	0.307957
$793$	0	0
$794$	30210.0	1.35027
$795$	41319.8	1.84335
$796$	6740.00	0.300117
$797$	−1455.00	−0.0646659	−0.0323330	−	0.999477i	$-0.510294\pi$
$797$	−1455.00	−0.0646659	−0.0323330	+	0.999477i	$0.510294\pi$
$798$	48230.7	2.13953
$799$	9259.54	0.409986
$800$	11140.6	0.492347
$801$	−6744.61	−0.297514
$802$	−26274.0	−1.15682
$803$	22620.0	0.994075
$804$	−4607.26	−0.202096
$805$	17784.0	0.778638
$806$	0	0
$807$	35595.0	1.55267
$808$	−27144.7	−1.18187
$809$	1659.00	0.0720981	0.0360490	−	0.999350i	$-0.488523\pi$
$809$	1659.00	0.0720981	0.0360490	+	0.999350i	$0.488523\pi$
$810$	−40272.0	−1.74693
$811$	−4402.87	−0.190636	−0.0953180	−	0.995447i	$-0.530387\pi$
$811$	−4402.87	−0.190636	−0.0953180	+	0.995447i	$0.530387\pi$
$812$	−6214.60	−0.268583
$813$	−9275.13	−0.400114
$814$	−3107.30	−0.133797
$815$	22632.0	0.972717
$816$	−15120.0	−0.648659
$817$	7508.44	0.321526
$818$	14910.0	0.637306
$819$	0	0
$820$	21792.0	0.928061
$821$	−28701.8	−1.22010	−0.610049	−	0.792364i	$-0.708850\pi$
$821$	−28701.8	−1.22010	−0.610049	+	0.792364i	$0.708850\pi$
$822$	−28854.0	−1.22433
$823$	−15779.0	−0.668313	−0.334156	−	0.942518i	$-0.608451\pi$
$823$	−15779.0	−0.668313	−0.334156	+	0.942518i	$0.608451\pi$
$824$	−25717.5	−1.08727
$825$	−10560.3	−0.445652
$826$	1486.10	0.0626005
$827$	−7354.29	−0.309231	−0.154615	−	0.987975i	$-0.549414\pi$
$827$	−7354.29	−0.309231	−0.154615	+	0.987975i	$0.549414\pi$
$828$	−5016.00	−0.210529
$829$	−17371.0	−0.727768	−0.363884	−	0.931444i	$-0.618550\pi$
$829$	−17371.0	−0.727768	−0.363884	+	0.931444i	$0.618550\pi$
$830$	20452.1	0.855303
$831$	−23947.0	−0.999654
$832$	0	0
$833$	−4428.00	−0.184179
$834$	−61713.0	−2.56228
$835$	22536.0	0.934001
$836$	7956.00	0.329144
$837$	2546.11	0.105145
$838$	−18695.8	−0.770685
$839$	29474.3	1.21283	0.606416	−	0.795148i	$-0.292607\pi$
$839$	29474.3	1.21283	0.606416	+	0.795148i	$0.292607\pi$
$840$	30262.4	1.24304
$841$	−19628.0	−0.804789
$842$	25152.0	1.02945
$843$	−5674.20	−0.231827
$844$	6724.00	0.274229
$845$	0	0
$846$	26136.0	1.06214
$847$	−18553.7	−0.752673
$848$	−34080.0	−1.38008
$849$	50239.0	2.03086
$850$	6266.56	0.252872
$851$	−2270.72	−0.0914680
$852$	16343.6	0.657187
$853$	2909.85	0.116801	0.0584005	−	0.998293i	$-0.481400\pi$
$853$	2909.85	0.116801	0.0584005	+	0.998293i	$0.481400\pi$
$854$	1326.00	0.0531321
$855$	−26928.0	−1.07710
$856$	−3533.38	−0.141085
$857$	5346.00	0.213087	0.106544	−	0.994308i	$-0.466022\pi$
$857$	5346.00	0.213087	0.106544	+	0.994308i	$0.466022\pi$
$858$	0	0
$859$	24244.0	0.962974	0.481487	−	0.876453i	$-0.340097\pi$
$859$	24244.0	0.962974	0.481487	+	0.876453i	$0.340097\pi$
$860$	−4711.18	−0.186802
$861$	61971.0	2.45292
$862$	1686.00	0.0666188
$863$	−32780.8	−1.29301	−0.646507	−	0.762908i	$-0.723771\pi$
$863$	−32780.8	−1.29301	−0.646507	+	0.762908i	$0.723771\pi$
$864$	5819.69	0.229155
$865$	−12096.6	−0.475489
$866$	42050.7	1.65005
$867$	29288.0	1.14726
$868$	6552.00	0.256209
$869$	−28010.7	−1.09344
$870$	23184.0	0.903461
$871$	0	0
$872$	8448.00	0.328080
$873$	−27168.9	−1.05330
$874$	17442.0	0.675039
$875$	18096.0	0.699150
$876$	−28128.5	−1.08490
$877$	4543.17	0.174928	0.0874640	−	0.996168i	$-0.472124\pi$
$877$	4543.17	0.174928	0.0874640	+	0.996168i	$0.472124\pi$
$878$	1596.95	0.0613832
$879$	−65192.7	−2.50159
$880$	24960.0	0.956138
$881$	20517.0	0.784603	0.392302	−	0.919837i	$-0.371679\pi$
$881$	20517.0	0.784603	0.392302	+	0.919837i	$0.371679\pi$
$882$	−12498.5	−0.477149
$883$	23852.0	0.909042	0.454521	−	0.890736i	$-0.349811\pi$
$883$	23852.0	0.909042	0.454521	+	0.890736i	$0.349811\pi$
$884$	0	0
$885$	−1848.00	−0.0701919
$886$	−42109.6	−1.59673
$887$	38757.0	1.46712	0.733558	−	0.679626i	$-0.237858\pi$
$887$	38757.0	1.46712	0.733558	+	0.679626i	$0.237858\pi$
$888$	−3864.00	−0.146022
$889$	50504.9	1.90538
$890$	−14715.5	−0.554230
$891$	−18891.5	−0.710312
$892$	−16385.2	−0.615042
$893$	−30294.0	−1.13522
$894$	−31626.0	−1.18315
$895$	−17833.2	−0.666031
$896$	−34944.0	−1.30290
$897$	0	0
$898$	1026.00	0.0381270
$899$	−5019.48	−0.186217
$900$	5896.00	0.218370
$901$	−11502.0	−0.425291
$902$	30667.7	1.13206
$903$	−13397.4	−0.493730
$904$	5694.98	0.209527
$905$	27.7128	0.00101791
$906$	−2100.00	−0.0770064
$907$	39071.0	1.43035	0.715177	−	0.698943i	$-0.246346\pi$
$907$	39071.0	1.43035	0.715177	+	0.698943i	$0.246346\pi$
$908$	−1752.84	−0.0640638
$909$	−43098.0	−1.57257
$910$	0	0
$911$	−53040.0	−1.92897	−0.964486	−	0.264134i	$-0.914914\pi$
$911$	−53040.0	−1.92897	−0.964486	+	0.264134i	$0.914914\pi$
$912$	49467.4	1.79608
$913$	9594.00	0.347771
$914$	−2118.00	−0.0766490
$915$	−1648.91	−0.0595753
$916$	720.533	0.0259903
$917$	−8376.20	−0.301643
$918$	3273.58	0.117695
$919$	367.000	0.0131732	0.00658662	−	0.999978i	$-0.497903\pi$
$919$	367.000	0.0131732	0.00658662	+	0.999978i	$0.497903\pi$
$920$	10944.0	0.392188
$921$	33439.0	1.19636
$922$	−45474.0	−1.62430
$923$	0	0
$924$	−14196.0	−0.505427
$925$	2669.09	0.0948748
$926$	2892.00	0.102632
$927$	−40832.0	−1.44671
$928$	−11473.1	−0.405844
$929$	29935.0	1.05720	0.528599	−	0.848872i	$-0.322718\pi$
$929$	29935.0	1.05720	0.528599	+	0.848872i	$0.322718\pi$
$930$	−24442.7	−0.861836
$931$	14486.9	0.509976
$932$	23112.0	0.812295
$933$	43344.0	1.52092
$934$	50215.6	1.75921
$935$	8424.00	0.294646
$936$	0	0
$937$	42166.0	1.47012	0.735060	−	0.678002i	$-0.237154\pi$
$937$	42166.0	1.47012	0.735060	+	0.678002i	$0.237154\pi$
$938$	−12834.5	−0.446760
$939$	5390.00	0.187323
$940$	19008.0	0.659545
$941$	35022.1	1.21327	0.606635	−	0.794981i	$-0.292519\pi$
$941$	35022.1	1.21327	0.606635	+	0.794981i	$0.292519\pi$
$942$	−37197.5	−1.28658
$943$	22411.0	0.773916
$944$	1524.20	0.0525515
$945$	−10920.0	−0.375902
$946$	−6630.00	−0.227865
$947$	2599.81	0.0892106	0.0446053	−	0.999005i	$-0.485797\pi$
$947$	2599.81	0.0892106	0.0446053	+	0.999005i	$0.485797\pi$
$948$	34832.0	1.19334
$949$	0	0
$950$	−20502.0	−0.700182
$951$	56402.5	1.92321
$952$	−8424.00	−0.286789
$953$	−10623.0	−0.361084	−0.180542	−	0.983567i	$-0.557785\pi$
$953$	−10623.0	−0.361084	−0.180542	+	0.983567i	$0.557785\pi$
$954$	−32465.6	−1.10179
$955$	39366.1	1.33388
$956$	7440.89	0.251732
$957$	10875.5	0.367353
$958$	−30822.0	−1.03947
$959$	−26793.0	−0.902180
$960$	6207.67	0.208700
$961$	−24499.0	−0.822362
$962$	0	0
$963$	−5610.00	−0.187726
$964$	8237.63	0.275224
$965$	−58824.0	−1.96229
$966$	−31122.0	−1.03658
$967$	−20199.2	−0.671729	−0.335864	−	0.941910i	$-0.609028\pi$
$967$	−20199.2	−0.671729	−0.335864	+	0.941910i	$0.609028\pi$
$968$	−11417.7	−0.379110
$969$	16695.2	0.553486
$970$	−59277.7	−1.96216
$971$	−2325.00	−0.0768412	−0.0384206	−	0.999262i	$-0.512233\pi$
$971$	−2325.00	−0.0768412	−0.0384206	+	0.999262i	$0.512233\pi$
$972$	19712.0	0.650476
$973$	−57304.9	−1.88809
$974$	16470.0	0.541820
$975$	0	0
$976$	1360.00	0.0446030
$977$	32938.4	1.07860	0.539300	−	0.842113i	$-0.318689\pi$
$977$	32938.4	1.07860	0.539300	+	0.842113i	$0.318689\pi$
$978$	−39606.0	−1.29495
$979$	−6903.00	−0.225353
$980$	−9089.80	−0.296289
$981$	13413.0	0.436538
$982$	−5663.81	−0.184052
$983$	42702.0	1.38554	0.692768	−	0.721161i	$-0.256391\pi$
$983$	42702.0	1.38554	0.692768	+	0.721161i	$0.256391\pi$
$984$	38136.0	1.23550
$985$	−38136.0	−1.23362
$986$	−6453.62	−0.208443
$987$	54054.0	1.74322
$988$	0	0
$989$	−4845.00	−0.155776
$990$	23777.6	0.763335
$991$	−4843.00	−0.155240	−0.0776201	−	0.996983i	$-0.524732\pi$
$991$	−4843.00	−0.155240	−0.0776201	+	0.996983i	$0.524732\pi$
$992$	12096.0	0.387146
$993$	−36942.9	−1.18061
$994$	45528.7	1.45280
$995$	−23348.0	−0.743902
$996$	−11930.4	−0.379546
$997$	10943.0	0.347611	0.173806	−	0.984780i	$-0.444394\pi$
$997$	10943.0	0.347611	0.173806	+	0.984780i	$0.444394\pi$
$998$	−50004.0	−1.58602
$999$	1394.30	0.0441579

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	169.4.a.h.1.1		2
3.2	odd	2		1521.4.a.q.1.2		2
13.2	odd	12		13.4.e.a.4.1	✓	2
13.3	even	3		169.4.c.i.22.2		4
13.4	even	6		169.4.c.i.146.1		4
13.5	odd	4		169.4.b.b.168.2		2
13.6	odd	12		169.4.e.b.23.1		2
13.7	odd	12		13.4.e.a.10.1	yes	2
13.8	odd	4		169.4.b.b.168.1		2
13.9	even	3		169.4.c.i.146.2		4
13.10	even	6		169.4.c.i.22.1		4
13.11	odd	12		169.4.e.b.147.1		2
13.12	even	2	inner	169.4.a.h.1.2		2
39.2	even	12		117.4.q.c.82.1		2
39.20	even	12		117.4.q.c.10.1		2
39.38	odd	2		1521.4.a.q.1.1		2
52.7	even	12		208.4.w.a.49.1		2
52.15	even	12		208.4.w.a.17.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.e.a.4.1	✓	2	13.2	odd	12
13.4.e.a.10.1	yes	2	13.7	odd	12
117.4.q.c.10.1		2	39.20	even	12
117.4.q.c.82.1		2	39.2	even	12
169.4.a.h.1.1		2	1.1	even	1	trivial
169.4.a.h.1.2		2	13.12	even	2	inner
169.4.b.b.168.1		2	13.8	odd	4
169.4.b.b.168.2		2	13.5	odd	4
169.4.c.i.22.1		4	13.10	even	6
169.4.c.i.22.2		4	13.3	even	3
169.4.c.i.146.1		4	13.4	even	6
169.4.c.i.146.2		4	13.9	even	3
169.4.e.b.23.1		2	13.6	odd	12
169.4.e.b.147.1		2	13.11	odd	12
208.4.w.a.17.1		2	52.15	even	12
208.4.w.a.49.1		2	52.7	even	12
1521.4.a.q.1.1		2	39.38	odd	2
1521.4.a.q.1.2		2	3.2	odd	2

Overview	Random
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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

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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	169.4.a.h.1.1

Level	$169$
Weight	$4$
Character	169.1
Self dual	yes
Analytic conductor	$9.971$
Analytic rank	$1$
Dimension	$2$
Inner twists	$2$