LMFDB - Embedded newform 169.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,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, 4, names="a")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$9.97132279097$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
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
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)

$f(q)$

$=$

$q+5.00000 q^{2} -7.00000 q^{3} +17.0000 q^{4} +7.00000 q^{5} -35.0000 q^{6} +13.0000 q^{7} +45.0000 q^{8} +22.0000 q^{9} +O(q^{10})$

\(q+5.00000 q^{2} -7.00000 q^{3} +17.0000 q^{4} +7.00000 q^{5} -35.0000 q^{6} +13.0000 q^{7} +45.0000 q^{8} +22.0000 q^{9} +35.0000 q^{10} +26.0000 q^{11} -119.000 q^{12} +65.0000 q^{14} -49.0000 q^{15} +89.0000 q^{16} +77.0000 q^{17} +110.000 q^{18} +126.000 q^{19} +119.000 q^{20} -91.0000 q^{21} +130.000 q^{22} -96.0000 q^{23} -315.000 q^{24} -76.0000 q^{25} +35.0000 q^{27} +221.000 q^{28} -82.0000 q^{29} -245.000 q^{30} -196.000 q^{31} +85.0000 q^{32} -182.000 q^{33} +385.000 q^{34} +91.0000 q^{35} +374.000 q^{36} +131.000 q^{37} +630.000 q^{38} +315.000 q^{40} -336.000 q^{41} -455.000 q^{42} -201.000 q^{43} +442.000 q^{44} +154.000 q^{45} -480.000 q^{46} +105.000 q^{47} -623.000 q^{48} -174.000 q^{49} -380.000 q^{50} -539.000 q^{51} -432.000 q^{53} +175.000 q^{54} +182.000 q^{55} +585.000 q^{56} -882.000 q^{57} -410.000 q^{58} +294.000 q^{59} -833.000 q^{60} -56.0000 q^{61} -980.000 q^{62} +286.000 q^{63} -287.000 q^{64} -910.000 q^{66} -478.000 q^{67} +1309.00 q^{68} +672.000 q^{69} +455.000 q^{70} -9.00000 q^{71} +990.000 q^{72} -98.0000 q^{73} +655.000 q^{74} +532.000 q^{75} +2142.00 q^{76} +338.000 q^{77} +1304.00 q^{79} +623.000 q^{80} -839.000 q^{81} -1680.00 q^{82} +308.000 q^{83} -1547.00 q^{84} +539.000 q^{85} -1005.00 q^{86} +574.000 q^{87} +1170.00 q^{88} +1190.00 q^{89} +770.000 q^{90} -1632.00 q^{92} +1372.00 q^{93} +525.000 q^{94} +882.000 q^{95} -595.000 q^{96} -70.0000 q^{97} -870.000 q^{98} +572.000 q^{99} +O(q^{100})\)

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 $

$2$	5.00000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	5.00000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	−7.00000	−1.34715	−0.673575	−	0.739119i	$-0.735242\pi$
$3$	−7.00000	−1.34715	−0.673575	+	0.739119i	$0.735242\pi$
$4$	17.0000	2.12500
$5$	7.00000	0.626099	0.313050	−	0.949737i	$-0.398649\pi$
$5$	7.00000	0.626099	0.313050	+	0.949737i	$0.398649\pi$
$6$	−35.0000	−2.38145
$7$	13.0000	0.701934	0.350967	−	0.936388i	$-0.385853\pi$
$7$	13.0000	0.701934	0.350967	+	0.936388i	$0.385853\pi$
$8$	45.0000	1.98874
$9$	22.0000	0.814815
$10$	35.0000	1.10680
$11$	26.0000	0.712663	0.356332	−	0.934360i	$-0.384027\pi$
$11$	26.0000	0.712663	0.356332	+	0.934360i	$0.384027\pi$
$12$	−119.000	−2.86270
$13$	0	0
$13$	0	0
$14$	65.0000	1.24086
$15$	−49.0000	−0.843450
$16$	89.0000	1.39062
$17$	77.0000	1.09854	0.549272	−	0.835644i	$-0.314905\pi$
$17$	77.0000	1.09854	0.549272	+	0.835644i	$0.314905\pi$
$18$	110.000	1.44040
$19$	126.000	1.52139	0.760694	−	0.649110i	$-0.224859\pi$
$19$	126.000	1.52139	0.760694	+	0.649110i	$0.224859\pi$
$20$	119.000	1.33046
$21$	−91.0000	−0.945611
$22$	130.000	1.25982
$23$	−96.0000	−0.870321	−0.435161	−	0.900353i	$-0.643308\pi$
$23$	−96.0000	−0.870321	−0.435161	+	0.900353i	$0.643308\pi$
$24$	−315.000	−2.67913
$25$	−76.0000	−0.608000
$26$	0	0
$27$	35.0000	0.249472
$28$	221.000	1.49161
$29$	−82.0000	−0.525070	−0.262535	−	0.964923i	$-0.584558\pi$
$29$	−82.0000	−0.525070	−0.262535	+	0.964923i	$0.584558\pi$
$30$	−245.000	−1.49102
$31$	−196.000	−1.13557	−0.567785	−	0.823177i	$-0.692199\pi$
$31$	−196.000	−1.13557	−0.567785	+	0.823177i	$0.692199\pi$
$32$	85.0000	0.469563
$33$	−182.000	−0.960065
$34$	385.000	1.94197
$35$	91.0000	0.439480
$36$	374.000	1.73148
$37$	131.000	0.582061	0.291031	−	0.956714i	$-0.406002\pi$
$37$	131.000	0.582061	0.291031	+	0.956714i	$0.406002\pi$
$38$	630.000	2.68946
$39$	0	0
$40$	315.000	1.24515
$41$	−336.000	−1.27986	−0.639932	−	0.768432i	$-0.721037\pi$
$41$	−336.000	−1.27986	−0.639932	+	0.768432i	$0.721037\pi$
$42$	−455.000	−1.67162
$43$	−201.000	−0.712842	−0.356421	−	0.934325i	$-0.616003\pi$
$43$	−201.000	−0.712842	−0.356421	+	0.934325i	$0.616003\pi$
$44$	442.000	1.51441
$45$	154.000	0.510155
$46$	−480.000	−1.53852
$47$	105.000	0.325869	0.162934	−	0.986637i	$-0.447904\pi$
$47$	105.000	0.325869	0.162934	+	0.986637i	$0.447904\pi$
$48$	−623.000	−1.87338
$49$	−174.000	−0.507289
$50$	−380.000	−1.07480
$51$	−539.000	−1.47990
$52$	0	0
$53$	−432.000	−1.11962	−0.559809	−	0.828622i	$-0.689126\pi$
$53$	−432.000	−1.11962	−0.559809	+	0.828622i	$0.689126\pi$
$54$	175.000	0.441009
$55$	182.000	0.446198
$56$	585.000	1.39596
$57$	−882.000	−2.04954
$58$	−410.000	−0.928201
$59$	294.000	0.648738	0.324369	−	0.945931i	$-0.394848\pi$
$59$	294.000	0.648738	0.324369	+	0.945931i	$0.394848\pi$
$60$	−833.000	−1.79233
$61$	−56.0000	−0.117542	−0.0587710	−	0.998271i	$-0.518718\pi$
$61$	−56.0000	−0.117542	−0.0587710	+	0.998271i	$0.518718\pi$
$62$	−980.000	−2.00742
$63$	286.000	0.571946
$64$	−287.000	−0.560547
$65$	0	0
$66$	−910.000	−1.69717
$67$	−478.000	−0.871597	−0.435798	−	0.900044i	$-0.643534\pi$
$67$	−478.000	−0.871597	−0.435798	+	0.900044i	$0.643534\pi$
$68$	1309.00	2.33441
$69$	672.000	1.17245
$70$	455.000	0.776899
$71$	−9.00000	−0.0150437	−0.00752186	−	0.999972i	$-0.502394\pi$
$71$	−9.00000	−0.0150437	−0.00752186	+	0.999972i	$0.502394\pi$
$72$	990.000	1.62045
$73$	−98.0000	−0.157124	−0.0785619	−	0.996909i	$-0.525033\pi$
$73$	−98.0000	−0.157124	−0.0785619	+	0.996909i	$0.525033\pi$
$74$	655.000	1.02895
$75$	532.000	0.819068
$76$	2142.00	3.23295
$77$	338.000	0.500243
$78$	0	0
$79$	1304.00	1.85711	0.928554	−	0.371198i	$-0.121053\pi$
$79$	1304.00	1.85711	0.928554	+	0.371198i	$0.121053\pi$
$80$	623.000	0.870669
$81$	−839.000	−1.15089
$82$	−1680.00	−2.26250
$83$	308.000	0.407318	0.203659	−	0.979042i	$-0.434717\pi$
$83$	308.000	0.407318	0.203659	+	0.979042i	$0.434717\pi$
$84$	−1547.00	−2.00942
$85$	539.000	0.687797
$86$	−1005.00	−1.26014
$87$	574.000	0.707348
$88$	1170.00	1.41730
$89$	1190.00	1.41730	0.708650	−	0.705560i	$-0.249304\pi$
$89$	1190.00	1.41730	0.708650	+	0.705560i	$0.249304\pi$
$90$	770.000	0.901835
$91$	0	0
$92$	−1632.00	−1.84943
$93$	1372.00	1.52978
$94$	525.000	0.576060
$95$	882.000	0.952540
$96$	−595.000	−0.632572
$97$	−70.0000	−0.0732724	−0.0366362	−	0.999329i	$-0.511664\pi$
$97$	−70.0000	−0.0732724	−0.0366362	+	0.999329i	$0.511664\pi$
$98$	−870.000	−0.896768
$99$	572.000	0.580689
$100$	−1292.00	−1.29200
$101$	420.000	0.413778	0.206889	−	0.978364i	$-0.433666\pi$
$101$	420.000	0.413778	0.206889	+	0.978364i	$0.433666\pi$
$102$	−2695.00	−2.61613
$103$	588.000	0.562499	0.281249	−	0.959635i	$-0.409251\pi$
$103$	588.000	0.562499	0.281249	+	0.959635i	$0.409251\pi$
$104$	0	0
$105$	−637.000	−0.592046
$106$	−2160.00	−1.97922
$107$	−684.000	−0.617989	−0.308994	−	0.951064i	$-0.599992\pi$
$107$	−684.000	−0.617989	−0.308994	+	0.951064i	$0.599992\pi$
$108$	595.000	0.530129
$109$	−373.000	−0.327770	−0.163885	−	0.986479i	$-0.552403\pi$
$109$	−373.000	−0.327770	−0.163885	+	0.986479i	$0.552403\pi$
$110$	910.000	0.788774
$111$	−917.000	−0.784124
$112$	1157.00	0.976127
$113$	−1734.00	−1.44355	−0.721774	−	0.692128i	$-0.756673\pi$
$113$	−1734.00	−1.44355	−0.721774	+	0.692128i	$0.756673\pi$
$114$	−4410.00	−3.62311
$115$	−672.000	−0.544907
$116$	−1394.00	−1.11577
$117$	0	0
$118$	1470.00	1.14682
$119$	1001.00	0.771105
$120$	−2205.00	−1.67740
$121$	−655.000	−0.492111
$122$	−280.000	−0.207787
$123$	2352.00	1.72417
$124$	−3332.00	−2.41308
$125$	−1407.00	−1.00677
$126$	1430.00	1.01107
$127$	1892.00	1.32195	0.660976	−	0.750407i	$-0.270143\pi$
$127$	1892.00	1.32195	0.660976	+	0.750407i	$0.270143\pi$
$128$	−2115.00	−1.46048
$129$	1407.00	0.960306
$130$	0	0
$131$	1435.00	0.957073	0.478536	−	0.878068i	$-0.341167\pi$
$131$	1435.00	0.957073	0.478536	+	0.878068i	$0.341167\pi$
$132$	−3094.00	−2.04014
$133$	1638.00	1.06791
$134$	−2390.00	−1.54078
$135$	245.000	0.156194
$136$	3465.00	2.18472
$137$	1776.00	1.10755	0.553773	−	0.832667i	$-0.313187\pi$
$137$	1776.00	1.10755	0.553773	+	0.832667i	$0.313187\pi$
$138$	3360.00	2.07262
$139$	−1869.00	−1.14048	−0.570239	−	0.821479i	$-0.693150\pi$
$139$	−1869.00	−1.14048	−0.570239	+	0.821479i	$0.693150\pi$
$140$	1547.00	0.933895
$141$	−735.000	−0.438994
$142$	−45.0000	−0.0265938
$143$	0	0
$144$	1958.00	1.13310
$145$	−574.000	−0.328746
$146$	−490.000	−0.277758
$147$	1218.00	0.683394
$148$	2227.00	1.23688
$149$	−2466.00	−1.35586	−0.677928	−	0.735128i	$-0.737122\pi$
$149$	−2466.00	−1.35586	−0.677928	+	0.735128i	$0.737122\pi$
$150$	2660.00	1.44792
$151$	3323.00	1.79087	0.895437	−	0.445189i	$-0.146863\pi$
$151$	3323.00	1.79087	0.895437	+	0.445189i	$0.146863\pi$
$152$	5670.00	3.02564
$153$	1694.00	0.895110
$154$	1690.00	0.884312
$155$	−1372.00	−0.710979
$156$	0	0
$157$	−2730.00	−1.38776	−0.693878	−	0.720092i	$-0.744099\pi$
$157$	−2730.00	−1.38776	−0.693878	+	0.720092i	$0.744099\pi$
$158$	6520.00	3.28293
$159$	3024.00	1.50829
$160$	595.000	0.293993
$161$	−1248.00	−0.610908
$162$	−4195.00	−2.03451
$163$	544.000	0.261407	0.130704	−	0.991421i	$-0.458276\pi$
$163$	544.000	0.261407	0.130704	+	0.991421i	$0.458276\pi$
$164$	−5712.00	−2.71971
$165$	−1274.00	−0.601096
$166$	1540.00	0.720043
$167$	−1624.00	−0.752508	−0.376254	−	0.926516i	$-0.622788\pi$
$167$	−1624.00	−0.752508	−0.376254	+	0.926516i	$0.622788\pi$
$168$	−4095.00	−1.88057
$169$	0	0
$170$	2695.00	1.21587
$171$	2772.00	1.23965
$172$	−3417.00	−1.51479
$173$	−336.000	−0.147662	−0.0738312	−	0.997271i	$-0.523523\pi$
$173$	−336.000	−0.147662	−0.0738312	+	0.997271i	$0.523523\pi$
$174$	2870.00	1.25043
$175$	−988.000	−0.426776
$176$	2314.00	0.991047
$177$	−2058.00	−0.873948
$178$	5950.00	2.50546
$179$	−3029.00	−1.26479	−0.632397	−	0.774645i	$-0.717929\pi$
$179$	−3029.00	−1.26479	−0.632397	+	0.774645i	$0.717929\pi$
$180$	2618.00	1.08408
$181$	−28.0000	−0.0114985	−0.00574924	−	0.999983i	$-0.501830\pi$
$181$	−28.0000	−0.0114985	−0.00574924	+	0.999983i	$0.501830\pi$
$182$	0	0
$183$	392.000	0.158347
$184$	−4320.00	−1.73084
$185$	917.000	0.364428
$186$	6860.00	2.70430
$187$	2002.00	0.782892
$188$	1785.00	0.692471
$189$	455.000	0.175113
$190$	4410.00	1.68387
$191$	422.000	0.159868	0.0799342	−	0.996800i	$-0.474529\pi$
$191$	422.000	0.159868	0.0799342	+	0.996800i	$0.474529\pi$
$192$	2009.00	0.755141
$193$	−492.000	−0.183497	−0.0917485	−	0.995782i	$-0.529246\pi$
$193$	−492.000	−0.183497	−0.0917485	+	0.995782i	$0.529246\pi$
$194$	−350.000	−0.129529
$195$	0	0
$196$	−2958.00	−1.07799
$197$	−2991.00	−1.08173	−0.540863	−	0.841111i	$-0.681902\pi$
$197$	−2991.00	−1.08173	−0.540863	+	0.841111i	$0.681902\pi$
$198$	2860.00	1.02652
$199$	−70.0000	−0.0249355	−0.0124678	−	0.999922i	$-0.503969\pi$
$199$	−70.0000	−0.0249355	−0.0124678	+	0.999922i	$0.503969\pi$
$200$	−3420.00	−1.20915
$201$	3346.00	1.17417
$202$	2100.00	0.731463
$203$	−1066.00	−0.368564
$204$	−9163.00	−3.14480
$205$	−2352.00	−0.801321
$206$	2940.00	0.994367
$207$	−2112.00	−0.709150
$208$	0	0
$209$	3276.00	1.08424
$210$	−3185.00	−1.04660
$211$	2851.00	0.930194	0.465097	−	0.885260i	$-0.346019\pi$
$211$	2851.00	0.930194	0.465097	+	0.885260i	$0.346019\pi$
$212$	−7344.00	−2.37919
$213$	63.0000	0.0202661
$214$	−3420.00	−1.09246
$215$	−1407.00	−0.446310
$216$	1575.00	0.496135
$217$	−2548.00	−0.797095
$218$	−1865.00	−0.579421
$219$	686.000	0.211669
$220$	3094.00	0.948170
$221$	0	0
$222$	−4585.00	−1.38615
$223$	−217.000	−0.0651632	−0.0325816	−	0.999469i	$-0.510373\pi$
$223$	−217.000	−0.0651632	−0.0325816	+	0.999469i	$0.510373\pi$
$224$	1105.00	0.329602
$225$	−1672.00	−0.495407
$226$	−8670.00	−2.55186
$227$	2576.00	0.753194	0.376597	−	0.926377i	$-0.377094\pi$
$227$	2576.00	0.753194	0.376597	+	0.926377i	$0.377094\pi$
$228$	−14994.0	−4.35527
$229$	−455.000	−0.131298	−0.0656490	−	0.997843i	$-0.520912\pi$
$229$	−455.000	−0.131298	−0.0656490	+	0.997843i	$0.520912\pi$
$230$	−3360.00	−0.963269
$231$	−2366.00	−0.673902
$232$	−3690.00	−1.04423
$233$	3061.00	0.860656	0.430328	−	0.902673i	$-0.358398\pi$
$233$	3061.00	0.860656	0.430328	+	0.902673i	$0.358398\pi$
$234$	0	0
$235$	735.000	0.204026
$236$	4998.00	1.37857
$237$	−9128.00	−2.50180
$238$	5005.00	1.36313
$239$	3477.00	0.941039	0.470520	−	0.882389i	$-0.344066\pi$
$239$	3477.00	0.941039	0.470520	+	0.882389i	$0.344066\pi$
$240$	−4361.00	−1.17292
$241$	1610.00	0.430329	0.215164	−	0.976578i	$-0.430971\pi$
$241$	1610.00	0.430329	0.215164	+	0.976578i	$0.430971\pi$
$242$	−3275.00	−0.869938
$243$	4928.00	1.30095
$244$	−952.000	−0.249777
$245$	−1218.00	−0.317613
$246$	11760.0	3.04793
$247$	0	0
$248$	−8820.00	−2.25835
$249$	−2156.00	−0.548719
$250$	−7035.00	−1.77973
$251$	1008.00	0.253484	0.126742	−	0.991936i	$-0.459548\pi$
$251$	1008.00	0.253484	0.126742	+	0.991936i	$0.459548\pi$
$252$	4862.00	1.21539
$253$	−2496.00	−0.620246
$254$	9460.00	2.33690
$255$	−3773.00	−0.926566
$256$	−8279.00	−2.02124
$257$	6041.00	1.46625	0.733127	−	0.680092i	$-0.238060\pi$
$257$	6041.00	1.46625	0.733127	+	0.680092i	$0.238060\pi$
$258$	7035.00	1.69760
$259$	1703.00	0.408569
$260$	0	0
$261$	−1804.00	−0.427834
$262$	7175.00	1.69188
$263$	−3708.00	−0.869373	−0.434686	−	0.900582i	$-0.643141\pi$
$263$	−3708.00	−0.869373	−0.434686	+	0.900582i	$0.643141\pi$
$264$	−8190.00	−1.90932
$265$	−3024.00	−0.700992
$266$	8190.00	1.88782
$267$	−8330.00	−1.90932
$268$	−8126.00	−1.85214
$269$	8344.00	1.89124	0.945618	−	0.325278i	$-0.105458\pi$
$269$	8344.00	1.89124	0.945618	+	0.325278i	$0.105458\pi$
$270$	1225.00	0.276115
$271$	1617.00	0.362457	0.181228	−	0.983441i	$-0.441993\pi$
$271$	1617.00	0.362457	0.181228	+	0.983441i	$0.441993\pi$
$272$	6853.00	1.52766
$273$	0	0
$274$	8880.00	1.95788
$275$	−1976.00	−0.433299
$276$	11424.0	2.49146
$277$	−3820.00	−0.828598	−0.414299	−	0.910141i	$-0.635973\pi$
$277$	−3820.00	−0.828598	−0.414299	+	0.910141i	$0.635973\pi$
$278$	−9345.00	−2.01610
$279$	−4312.00	−0.925278
$280$	4095.00	0.874011
$281$	6214.00	1.31920	0.659602	−	0.751615i	$-0.270725\pi$
$281$	6214.00	1.31920	0.659602	+	0.751615i	$0.270725\pi$
$282$	−3675.00	−0.776039
$283$	−5292.00	−1.11158	−0.555789	−	0.831323i	$-0.687584\pi$
$283$	−5292.00	−1.11158	−0.555789	+	0.831323i	$0.687584\pi$
$284$	−153.000	−0.0319679
$285$	−6174.00	−1.28321
$286$	0	0
$287$	−4368.00	−0.898379
$288$	1870.00	0.382607
$289$	1016.00	0.206798
$290$	−2870.00	−0.581146
$291$	490.000	0.0987090
$292$	−1666.00	−0.333888
$293$	903.000	0.180047	0.0900236	−	0.995940i	$-0.471306\pi$
$293$	903.000	0.180047	0.0900236	+	0.995940i	$0.471306\pi$
$294$	6090.00	1.20808
$295$	2058.00	0.406174
$296$	5895.00	1.15757
$297$	910.000	0.177790
$298$	−12330.0	−2.39684
$299$	0	0
$300$	9044.00	1.74052
$301$	−2613.00	−0.500368
$302$	16615.0	3.16585
$303$	−2940.00	−0.557421
$304$	11214.0	2.11568
$305$	−392.000	−0.0735930
$306$	8470.00	1.58235
$307$	−2114.00	−0.393004	−0.196502	−	0.980503i	$-0.562958\pi$
$307$	−2114.00	−0.393004	−0.196502	+	0.980503i	$0.562958\pi$
$308$	5746.00	1.06302
$309$	−4116.00	−0.757770
$310$	−6860.00	−1.25684
$311$	3402.00	0.620288	0.310144	−	0.950690i	$-0.399623\pi$
$311$	3402.00	0.620288	0.310144	+	0.950690i	$0.399623\pi$
$312$	0	0
$313$	−10689.0	−1.93028	−0.965141	−	0.261732i	$-0.915706\pi$
$313$	−10689.0	−1.93028	−0.965141	+	0.261732i	$0.915706\pi$
$314$	−13650.0	−2.45323
$315$	2002.00	0.358095
$316$	22168.0	3.94635
$317$	7054.00	1.24982	0.624909	−	0.780698i	$-0.285136\pi$
$317$	7054.00	1.24982	0.624909	+	0.780698i	$0.285136\pi$
$318$	15120.0	2.66631
$319$	−2132.00	−0.374198
$320$	−2009.00	−0.350958
$321$	4788.00	0.832524
$322$	−6240.00	−1.07994
$323$	9702.00	1.67131
$324$	−14263.0	−2.44564
$325$	0	0
$326$	2720.00	0.462107
$327$	2611.00	0.441555
$328$	−15120.0	−2.54531
$329$	1365.00	0.228738
$330$	−6370.00	−1.06260
$331$	−9704.00	−1.61142	−0.805710	−	0.592310i	$-0.798216\pi$
$331$	−9704.00	−1.61142	−0.805710	+	0.592310i	$0.798216\pi$
$332$	5236.00	0.865551
$333$	2882.00	0.474272
$334$	−8120.00	−1.33026
$335$	−3346.00	−0.545706
$336$	−8099.00	−1.31499
$337$	−10449.0	−1.68900	−0.844500	−	0.535555i	$-0.820103\pi$
$337$	−10449.0	−1.68900	−0.844500	+	0.535555i	$0.820103\pi$
$338$	0	0
$339$	12138.0	1.94468
$340$	9163.00	1.46157
$341$	−5096.00	−0.809278
$342$	13860.0	2.19141
$343$	−6721.00	−1.05802
$344$	−9045.00	−1.41766
$345$	4704.00	0.734072
$346$	−1680.00	−0.261033
$347$	−621.000	−0.0960721	−0.0480361	−	0.998846i	$-0.515296\pi$
$347$	−621.000	−0.0960721	−0.0480361	+	0.998846i	$0.515296\pi$
$348$	9758.00	1.50311
$349$	−12481.0	−1.91431	−0.957153	−	0.289584i	$-0.906483\pi$
$349$	−12481.0	−1.91431	−0.957153	+	0.289584i	$0.906483\pi$
$350$	−4940.00	−0.754440
$351$	0	0
$352$	2210.00	0.334640
$353$	1400.00	0.211089	0.105545	−	0.994415i	$-0.466341\pi$
$353$	1400.00	0.211089	0.105545	+	0.994415i	$0.466341\pi$
$354$	−10290.0	−1.54494
$355$	−63.0000	−0.00941885
$356$	20230.0	3.01176
$357$	−7007.00	−1.03879
$358$	−15145.0	−2.23586
$359$	4968.00	0.730365	0.365182	−	0.930936i	$-0.381007\pi$
$359$	4968.00	0.730365	0.365182	+	0.930936i	$0.381007\pi$
$360$	6930.00	1.01456
$361$	9017.00	1.31462
$362$	−140.000	−0.0203266
$363$	4585.00	0.662948
$364$	0	0
$365$	−686.000	−0.0983750
$366$	1960.00	0.279920
$367$	8722.00	1.24056	0.620279	−	0.784381i	$-0.287019\pi$
$367$	8722.00	1.24056	0.620279	+	0.784381i	$0.287019\pi$
$368$	−8544.00	−1.21029
$369$	−7392.00	−1.04285
$370$	4585.00	0.644224
$371$	−5616.00	−0.785898
$372$	23324.0	3.25079
$373$	10012.0	1.38982	0.694908	−	0.719098i	$-0.255445\pi$
$373$	10012.0	1.38982	0.694908	+	0.719098i	$0.255445\pi$
$374$	10010.0	1.38397
$375$	9849.00	1.35627
$376$	4725.00	0.648067
$377$	0	0
$378$	2275.00	0.309559
$379$	3372.00	0.457013	0.228507	−	0.973542i	$-0.426616\pi$
$379$	3372.00	0.457013	0.228507	+	0.973542i	$0.426616\pi$
$380$	14994.0	2.02415
$381$	−13244.0	−1.78087
$382$	2110.00	0.282610
$383$	847.000	0.113002	0.0565009	−	0.998403i	$-0.482006\pi$
$383$	847.000	0.113002	0.0565009	+	0.998403i	$0.482006\pi$
$384$	14805.0	1.96749
$385$	2366.00	0.313201
$386$	−2460.00	−0.324380
$387$	−4422.00	−0.580834
$388$	−1190.00	−0.155704
$389$	11314.0	1.47466	0.737330	−	0.675533i	$-0.236086\pi$
$389$	11314.0	1.47466	0.737330	+	0.675533i	$0.236086\pi$
$390$	0	0
$391$	−7392.00	−0.956086
$392$	−7830.00	−1.00886
$393$	−10045.0	−1.28932
$394$	−14955.0	−1.91224
$395$	9128.00	1.16273
$396$	9724.00	1.23396
$397$	−1862.00	−0.235393	−0.117697	−	0.993050i	$-0.537551\pi$
$397$	−1862.00	−0.235393	−0.117697	+	0.993050i	$0.537551\pi$
$398$	−350.000	−0.0440802
$399$	−11466.0	−1.43864
$400$	−6764.00	−0.845500
$401$	−6820.00	−0.849313	−0.424657	−	0.905355i	$-0.639605\pi$
$401$	−6820.00	−0.849313	−0.424657	+	0.905355i	$0.639605\pi$
$402$	16730.0	2.07566
$403$	0	0
$404$	7140.00	0.879278
$405$	−5873.00	−0.720572
$406$	−5330.00	−0.651536
$407$	3406.00	0.414814
$408$	−24255.0	−2.94314
$409$	12992.0	1.57069	0.785346	−	0.619057i	$-0.212485\pi$
$409$	12992.0	1.57069	0.785346	+	0.619057i	$0.212485\pi$
$410$	−11760.0	−1.41655
$411$	−12432.0	−1.49203
$412$	9996.00	1.19531
$413$	3822.00	0.455371
$414$	−10560.0	−1.25361
$415$	2156.00	0.255021
$416$	0	0
$417$	13083.0	1.53640
$418$	16380.0	1.91668
$419$	−7343.00	−0.856155	−0.428078	−	0.903742i	$-0.640809\pi$
$419$	−7343.00	−0.856155	−0.428078	+	0.903742i	$0.640809\pi$
$420$	−10829.0	−1.25810
$421$	5059.00	0.585655	0.292827	−	0.956165i	$-0.405404\pi$
$421$	5059.00	0.585655	0.292827	+	0.956165i	$0.405404\pi$
$422$	14255.0	1.64437
$423$	2310.00	0.265523
$424$	−19440.0	−2.22663
$425$	−5852.00	−0.667915
$426$	315.000	0.0358258
$427$	−728.000	−0.0825068
$428$	−11628.0	−1.31323
$429$	0	0
$430$	−7035.00	−0.788972
$431$	−3243.00	−0.362436	−0.181218	−	0.983443i	$-0.558004\pi$
$431$	−3243.00	−0.362436	−0.181218	+	0.983443i	$0.558004\pi$
$432$	3115.00	0.346922
$433$	11599.0	1.28733	0.643663	−	0.765309i	$-0.277414\pi$
$433$	11599.0	1.28733	0.643663	+	0.765309i	$0.277414\pi$
$434$	−12740.0	−1.40908
$435$	4018.00	0.442870
$436$	−6341.00	−0.696511
$437$	−12096.0	−1.32410
$438$	3430.00	0.374182
$439$	−17374.0	−1.88887	−0.944437	−	0.328692i	$-0.893392\pi$
$439$	−17374.0	−1.88887	−0.944437	+	0.328692i	$0.893392\pi$
$440$	8190.00	0.887370
$441$	−3828.00	−0.413346
$442$	0	0
$443$	989.000	0.106070	0.0530348	−	0.998593i	$-0.483111\pi$
$443$	989.000	0.106070	0.0530348	+	0.998593i	$0.483111\pi$
$444$	−15589.0	−1.66626
$445$	8330.00	0.887370
$446$	−1085.00	−0.115193
$447$	17262.0	1.82654
$448$	−3731.00	−0.393467
$449$	14474.0	1.52131	0.760657	−	0.649154i	$-0.224877\pi$
$449$	14474.0	1.52131	0.760657	+	0.649154i	$0.224877\pi$
$450$	−8360.00	−0.875765
$451$	−8736.00	−0.912111
$452$	−29478.0	−3.06754
$453$	−23261.0	−2.41258
$454$	12880.0	1.33147
$455$	0	0
$456$	−39690.0	−4.07600
$457$	1594.00	0.163160	0.0815801	−	0.996667i	$-0.474003\pi$
$457$	1594.00	0.163160	0.0815801	+	0.996667i	$0.474003\pi$
$458$	−2275.00	−0.232104
$459$	2695.00	0.274056
$460$	−11424.0	−1.15793
$461$	5915.00	0.597590	0.298795	−	0.954317i	$-0.403415\pi$
$461$	5915.00	0.597590	0.298795	+	0.954317i	$0.403415\pi$
$462$	−11830.0	−1.19130
$463$	11072.0	1.11136	0.555680	−	0.831396i	$-0.312458\pi$
$463$	11072.0	1.11136	0.555680	+	0.831396i	$0.312458\pi$
$464$	−7298.00	−0.730175
$465$	9604.00	0.957795
$466$	15305.0	1.52144
$467$	1260.00	0.124852	0.0624260	−	0.998050i	$-0.480116\pi$
$467$	1260.00	0.124852	0.0624260	+	0.998050i	$0.480116\pi$
$468$	0	0
$469$	−6214.00	−0.611804
$470$	3675.00	0.360670
$471$	19110.0	1.86952
$472$	13230.0	1.29017
$473$	−5226.00	−0.508016
$474$	−45640.0	−4.42260
$475$	−9576.00	−0.925004
$476$	17017.0	1.63860
$477$	−9504.00	−0.912281
$478$	17385.0	1.66354
$479$	12033.0	1.14781	0.573906	−	0.818921i	$-0.305428\pi$
$479$	12033.0	1.14781	0.573906	+	0.818921i	$0.305428\pi$
$480$	−4165.00	−0.396053
$481$	0	0
$482$	8050.00	0.760721
$483$	8736.00	0.822985
$484$	−11135.0	−1.04574
$485$	−490.000	−0.0458758
$486$	24640.0	2.29978
$487$	2280.00	0.212149	0.106075	−	0.994358i	$-0.466172\pi$
$487$	2280.00	0.212149	0.106075	+	0.994358i	$0.466172\pi$
$488$	−2520.00	−0.233760
$489$	−3808.00	−0.352155
$490$	−6090.00	−0.561466
$491$	16767.0	1.54111	0.770554	−	0.637375i	$-0.219980\pi$
$491$	16767.0	1.54111	0.770554	+	0.637375i	$0.219980\pi$
$492$	39984.0	3.66386
$493$	−6314.00	−0.576812
$494$	0	0
$495$	4004.00	0.363569
$496$	−17444.0	−1.57915
$497$	−117.000	−0.0105597
$498$	−10780.0	−0.970007
$499$	−12840.0	−1.15190	−0.575949	−	0.817485i	$-0.695367\pi$
$499$	−12840.0	−1.15190	−0.575949	+	0.817485i	$0.695367\pi$
$500$	−23919.0	−2.13938
$501$	11368.0	1.01374
$502$	5040.00	0.448100
$503$	−2198.00	−0.194839	−0.0974195	−	0.995243i	$-0.531059\pi$
$503$	−2198.00	−0.194839	−0.0974195	+	0.995243i	$0.531059\pi$
$504$	12870.0	1.13745
$505$	2940.00	0.259066
$506$	−12480.0	−1.09645
$507$	0	0
$508$	32164.0	2.80915
$509$	17066.0	1.48612	0.743062	−	0.669223i	$-0.233373\pi$
$509$	17066.0	1.48612	0.743062	+	0.669223i	$0.233373\pi$
$510$	−18865.0	−1.63795
$511$	−1274.00	−0.110290
$512$	−24475.0	−2.11260
$513$	4410.00	0.379544
$514$	30205.0	2.59200
$515$	4116.00	0.352180
$516$	23919.0	2.04065
$517$	2730.00	0.232235
$518$	8515.00	0.722254
$519$	2352.00	0.198924
$520$	0	0
$521$	2583.00	0.217204	0.108602	−	0.994085i	$-0.465363\pi$
$521$	2583.00	0.217204	0.108602	+	0.994085i	$0.465363\pi$
$522$	−9020.00	−0.756312
$523$	18620.0	1.55678	0.778390	−	0.627781i	$-0.216037\pi$
$523$	18620.0	1.55678	0.778390	+	0.627781i	$0.216037\pi$
$524$	24395.0	2.03378
$525$	6916.00	0.574931
$526$	−18540.0	−1.53685
$527$	−15092.0	−1.24747
$528$	−16198.0	−1.33509
$529$	−2951.00	−0.242541
$530$	−15120.0	−1.23919
$531$	6468.00	0.528601
$532$	27846.0	2.26932
$533$	0	0
$534$	−41650.0	−3.37523
$535$	−4788.00	−0.386922
$536$	−21510.0	−1.73338
$537$	21203.0	1.70387
$538$	41720.0	3.34327
$539$	−4524.00	−0.361526
$540$	4165.00	0.331913
$541$	16833.0	1.33772	0.668861	−	0.743388i	$-0.266782\pi$
$541$	16833.0	1.33772	0.668861	+	0.743388i	$0.266782\pi$
$542$	8085.00	0.640739
$543$	196.000	0.0154902
$544$	6545.00	0.515836
$545$	−2611.00	−0.205216
$546$	0	0
$547$	−8615.00	−0.673402	−0.336701	−	0.941612i	$-0.609311\pi$
$547$	−8615.00	−0.673402	−0.336701	+	0.941612i	$0.609311\pi$
$548$	30192.0	2.35354
$549$	−1232.00	−0.0957750
$550$	−9880.00	−0.765972
$551$	−10332.0	−0.798835
$552$	30240.0	2.33170
$553$	16952.0	1.30357
$554$	−19100.0	−1.46477
$555$	−6419.00	−0.490939
$556$	−31773.0	−2.42352
$557$	−8535.00	−0.649263	−0.324632	−	0.945841i	$-0.605240\pi$
$557$	−8535.00	−0.649263	−0.324632	+	0.945841i	$0.605240\pi$
$558$	−21560.0	−1.63568
$559$	0	0
$560$	8099.00	0.611152
$561$	−14014.0	−1.05467
$562$	31070.0	2.33204
$563$	−4641.00	−0.347415	−0.173708	−	0.984797i	$-0.555575\pi$
$563$	−4641.00	−0.347415	−0.173708	+	0.984797i	$0.555575\pi$
$564$	−12495.0	−0.932862
$565$	−12138.0	−0.903804
$566$	−26460.0	−1.96501
$567$	−10907.0	−0.807850
$568$	−405.000	−0.0299180
$569$	−4793.00	−0.353134	−0.176567	−	0.984289i	$-0.556499\pi$
$569$	−4793.00	−0.353134	−0.176567	+	0.984289i	$0.556499\pi$
$570$	−30870.0	−2.26842
$571$	−5563.00	−0.407713	−0.203857	−	0.979001i	$-0.565348\pi$
$571$	−5563.00	−0.407713	−0.203857	+	0.979001i	$0.565348\pi$
$572$	0	0
$573$	−2954.00	−0.215367
$574$	−21840.0	−1.58813
$575$	7296.00	0.529155
$576$	−6314.00	−0.456742
$577$	−24038.0	−1.73434	−0.867171	−	0.498011i	$-0.834064\pi$
$577$	−24038.0	−1.73434	−0.867171	+	0.498011i	$0.834064\pi$
$578$	5080.00	0.365571
$579$	3444.00	0.247198
$580$	−9758.00	−0.698584
$581$	4004.00	0.285910
$582$	2450.00	0.174494
$583$	−11232.0	−0.797911
$584$	−4410.00	−0.312478
$585$	0	0
$586$	4515.00	0.318281
$587$	21224.0	1.49235	0.746174	−	0.665751i	$-0.231889\pi$
$587$	21224.0	1.49235	0.746174	+	0.665751i	$0.231889\pi$
$588$	20706.0	1.45221
$589$	−24696.0	−1.72764
$590$	10290.0	0.718021
$591$	20937.0	1.45725
$592$	11659.0	0.809429
$593$	−4354.00	−0.301513	−0.150757	−	0.988571i	$-0.548171\pi$
$593$	−4354.00	−0.301513	−0.150757	+	0.988571i	$0.548171\pi$
$594$	4550.00	0.314291
$595$	7007.00	0.482788
$596$	−41922.0	−2.88119
$597$	490.000	0.0335919
$598$	0	0
$599$	7310.00	0.498629	0.249314	−	0.968423i	$-0.419795\pi$
$599$	7310.00	0.498629	0.249314	+	0.968423i	$0.419795\pi$
$600$	23940.0	1.62891
$601$	−7595.00	−0.515485	−0.257743	−	0.966214i	$-0.582979\pi$
$601$	−7595.00	−0.515485	−0.257743	+	0.966214i	$0.582979\pi$
$602$	−13065.0	−0.884534
$603$	−10516.0	−0.710190
$604$	56491.0	3.80561
$605$	−4585.00	−0.308110
$606$	−14700.0	−0.985391
$607$	−826.000	−0.0552328	−0.0276164	−	0.999619i	$-0.508792\pi$
$607$	−826.000	−0.0552328	−0.0276164	+	0.999619i	$0.508792\pi$
$608$	10710.0	0.714388
$609$	7462.00	0.496511
$610$	−1960.00	−0.130095
$611$	0	0
$612$	28798.0	1.90211
$613$	−14590.0	−0.961312	−0.480656	−	0.876909i	$-0.659602\pi$
$613$	−14590.0	−0.961312	−0.480656	+	0.876909i	$0.659602\pi$
$614$	−10570.0	−0.694740
$615$	16464.0	1.07950
$616$	15210.0	0.994851
$617$	−4888.00	−0.318936	−0.159468	−	0.987203i	$-0.550978\pi$
$617$	−4888.00	−0.318936	−0.159468	+	0.987203i	$0.550978\pi$
$618$	−20580.0	−1.33956
$619$	11004.0	0.714520	0.357260	−	0.934005i	$-0.383711\pi$
$619$	11004.0	0.714520	0.357260	+	0.934005i	$0.383711\pi$
$620$	−23324.0	−1.51083
$621$	−3360.00	−0.217121
$622$	17010.0	1.09653
$623$	15470.0	0.994851
$624$	0	0
$625$	−349.000	−0.0223360
$626$	−53445.0	−3.41229
$627$	−22932.0	−1.46063
$628$	−46410.0	−2.94898
$629$	10087.0	0.639420
$630$	10010.0	0.633028
$631$	4975.00	0.313869	0.156935	−	0.987609i	$-0.449839\pi$
$631$	4975.00	0.313869	0.156935	+	0.987609i	$0.449839\pi$
$632$	58680.0	3.69330
$633$	−19957.0	−1.25311
$634$	35270.0	2.20939
$635$	13244.0	0.827673
$636$	51408.0	3.20513
$637$	0	0
$638$	−10660.0	−0.661494
$639$	−198.000	−0.0122578
$640$	−14805.0	−0.914405
$641$	3950.00	0.243394	0.121697	−	0.992567i	$-0.461166\pi$
$641$	3950.00	0.243394	0.121697	+	0.992567i	$0.461166\pi$
$642$	23940.0	1.47171
$643$	3682.00	0.225823	0.112911	−	0.993605i	$-0.463982\pi$
$643$	3682.00	0.225823	0.112911	+	0.993605i	$0.463982\pi$
$644$	−21216.0	−1.29818
$645$	9849.00	0.601247
$646$	48510.0	2.95449
$647$	10402.0	0.632063	0.316032	−	0.948749i	$-0.397649\pi$
$647$	10402.0	0.632063	0.316032	+	0.948749i	$0.397649\pi$
$648$	−37755.0	−2.28882
$649$	7644.00	0.462332
$650$	0	0
$651$	17836.0	1.07381
$652$	9248.00	0.555490
$653$	−31680.0	−1.89852	−0.949260	−	0.314491i	$-0.898166\pi$
$653$	−31680.0	−1.89852	−0.949260	+	0.314491i	$0.898166\pi$
$654$	13055.0	0.780567
$655$	10045.0	0.599222
$656$	−29904.0	−1.77981
$657$	−2156.00	−0.128027
$658$	6825.00	0.404356
$659$	21940.0	1.29691	0.648453	−	0.761255i	$-0.275416\pi$
$659$	21940.0	1.29691	0.648453	+	0.761255i	$0.275416\pi$
$660$	−21658.0	−1.27733
$661$	31374.0	1.84615	0.923077	−	0.384616i	$-0.125666\pi$
$661$	31374.0	1.84615	0.923077	+	0.384616i	$0.125666\pi$
$662$	−48520.0	−2.84862
$663$	0	0
$664$	13860.0	0.810049
$665$	11466.0	0.668620
$666$	14410.0	0.838403
$667$	7872.00	0.456979
$668$	−27608.0	−1.59908
$669$	1519.00	0.0877847
$670$	−16730.0	−0.964681
$671$	−1456.00	−0.0837679
$672$	−7735.00	−0.444024
$673$	18013.0	1.03172	0.515862	−	0.856672i	$-0.327472\pi$
$673$	18013.0	1.03172	0.515862	+	0.856672i	$0.327472\pi$
$674$	−52245.0	−2.98576
$675$	−2660.00	−0.151679
$676$	0	0
$677$	−10640.0	−0.604030	−0.302015	−	0.953303i	$-0.597659\pi$
$677$	−10640.0	−0.604030	−0.302015	+	0.953303i	$0.597659\pi$
$678$	60690.0	3.43774
$679$	−910.000	−0.0514324
$680$	24255.0	1.36785
$681$	−18032.0	−1.01467
$682$	−25480.0	−1.43062
$683$	9336.00	0.523034	0.261517	−	0.965199i	$-0.415777\pi$
$683$	9336.00	0.523034	0.261517	+	0.965199i	$0.415777\pi$
$684$	47124.0	2.63426
$685$	12432.0	0.693434
$686$	−33605.0	−1.87033
$687$	3185.00	0.176878
$688$	−17889.0	−0.991296
$689$	0	0
$690$	23520.0	1.29767
$691$	−4200.00	−0.231224	−0.115612	−	0.993294i	$-0.536883\pi$
$691$	−4200.00	−0.231224	−0.115612	+	0.993294i	$0.536883\pi$
$692$	−5712.00	−0.313783
$693$	7436.00	0.407605
$694$	−3105.00	−0.169833
$695$	−13083.0	−0.714052
$696$	25830.0	1.40673
$697$	−25872.0	−1.40599
$698$	−62405.0	−3.38405
$699$	−21427.0	−1.15943
$700$	−16796.0	−0.906899
$701$	9872.00	0.531898	0.265949	−	0.963987i	$-0.414315\pi$
$701$	9872.00	0.531898	0.265949	+	0.963987i	$0.414315\pi$
$702$	0	0
$703$	16506.0	0.885541
$704$	−7462.00	−0.399481
$705$	−5145.00	−0.274854
$706$	7000.00	0.373156
$707$	5460.00	0.290445
$708$	−34986.0	−1.85714
$709$	−28450.0	−1.50700	−0.753499	−	0.657449i	$-0.771636\pi$
$709$	−28450.0	−1.50700	−0.753499	+	0.657449i	$0.771636\pi$
$710$	−315.000	−0.0166503
$711$	28688.0	1.51320
$712$	53550.0	2.81864
$713$	18816.0	0.988310
$714$	−35035.0	−1.83635
$715$	0	0
$716$	−51493.0	−2.68769
$717$	−24339.0	−1.26772
$718$	24840.0	1.29111
$719$	32718.0	1.69705	0.848523	−	0.529159i	$-0.177493\pi$
$719$	32718.0	1.69705	0.848523	+	0.529159i	$0.177493\pi$
$720$	13706.0	0.709434
$721$	7644.00	0.394837
$722$	45085.0	2.32395
$723$	−11270.0	−0.579718
$724$	−476.000	−0.0244343
$725$	6232.00	0.319242
$726$	22925.0	1.17194
$727$	−22834.0	−1.16488	−0.582439	−	0.812874i	$-0.697901\pi$
$727$	−22834.0	−1.16488	−0.582439	+	0.812874i	$0.697901\pi$
$728$	0	0
$729$	−11843.0	−0.601687
$730$	−3430.00	−0.173904
$731$	−15477.0	−0.783088
$732$	6664.00	0.336487
$733$	−7875.00	−0.396821	−0.198410	−	0.980119i	$-0.563578\pi$
$733$	−7875.00	−0.396821	−0.198410	+	0.980119i	$0.563578\pi$
$734$	43610.0	2.19302
$735$	8526.00	0.427872
$736$	−8160.00	−0.408671
$737$	−12428.0	−0.621155
$738$	−36960.0	−1.84352
$739$	2140.00	0.106524	0.0532620	−	0.998581i	$-0.483038\pi$
$739$	2140.00	0.106524	0.0532620	+	0.998581i	$0.483038\pi$
$740$	15589.0	0.774410
$741$	0	0
$742$	−28080.0	−1.38928
$743$	−31971.0	−1.57860	−0.789302	−	0.614006i	$-0.789557\pi$
$743$	−31971.0	−1.57860	−0.789302	+	0.614006i	$0.789557\pi$
$744$	61740.0	3.04234
$745$	−17262.0	−0.848900
$746$	50060.0	2.45687
$747$	6776.00	0.331889
$748$	34034.0	1.66364
$749$	−8892.00	−0.433787
$750$	49245.0	2.39756
$751$	−7432.00	−0.361115	−0.180558	−	0.983564i	$-0.557790\pi$
$751$	−7432.00	−0.361115	−0.180558	+	0.983564i	$0.557790\pi$
$752$	9345.00	0.453161
$753$	−7056.00	−0.341481
$754$	0	0
$755$	23261.0	1.12126
$756$	7735.00	0.372115
$757$	20176.0	0.968704	0.484352	−	0.874873i	$-0.339055\pi$
$757$	20176.0	0.968704	0.484352	+	0.874873i	$0.339055\pi$
$758$	16860.0	0.807893
$759$	17472.0	0.835564
$760$	39690.0	1.89435
$761$	9478.00	0.451481	0.225741	−	0.974187i	$-0.427520\pi$
$761$	9478.00	0.451481	0.225741	+	0.974187i	$0.427520\pi$
$762$	−66220.0	−3.14816
$763$	−4849.00	−0.230073
$764$	7174.00	0.339720
$765$	11858.0	0.560427
$766$	4235.00	0.199761
$767$	0	0
$768$	57953.0	2.72292
$769$	12096.0	0.567221	0.283610	−	0.958940i	$-0.408468\pi$
$769$	12096.0	0.567221	0.283610	+	0.958940i	$0.408468\pi$
$770$	11830.0	0.553667
$771$	−42287.0	−1.97526
$772$	−8364.00	−0.389931
$773$	−17941.0	−0.834790	−0.417395	−	0.908725i	$-0.637057\pi$
$773$	−17941.0	−0.834790	−0.417395	+	0.908725i	$0.637057\pi$
$774$	−22110.0	−1.02678
$775$	14896.0	0.690426
$776$	−3150.00	−0.145720
$777$	−11921.0	−0.550403
$778$	56570.0	2.60685
$779$	−42336.0	−1.94717
$780$	0	0
$781$	−234.000	−0.0107211
$782$	−36960.0	−1.69014
$783$	−2870.00	−0.130990
$784$	−15486.0	−0.705448
$785$	−19110.0	−0.868873
$786$	−50225.0	−2.27922
$787$	−6664.00	−0.301837	−0.150919	−	0.988546i	$-0.548223\pi$
$787$	−6664.00	−0.301837	−0.150919	+	0.988546i	$0.548223\pi$
$788$	−50847.0	−2.29867
$789$	25956.0	1.17118
$790$	45640.0	2.05544
$791$	−22542.0	−1.01328
$792$	25740.0	1.15484
$793$	0	0
$794$	−9310.00	−0.416120
$795$	21168.0	0.944342
$796$	−1190.00	−0.0529880
$797$	−1442.00	−0.0640882	−0.0320441	−	0.999486i	$-0.510202\pi$
$797$	−1442.00	−0.0640882	−0.0320441	+	0.999486i	$0.510202\pi$
$798$	−57330.0	−2.54318
$799$	8085.00	0.357981
$800$	−6460.00	−0.285494
$801$	26180.0	1.15484
$802$	−34100.0	−1.50139
$803$	−2548.00	−0.111976
$804$	56882.0	2.49512
$805$	−8736.00	−0.382489
$806$	0	0
$807$	−58408.0	−2.54778
$808$	18900.0	0.822896
$809$	30207.0	1.31276	0.656379	−	0.754431i	$-0.272087\pi$
$809$	30207.0	1.31276	0.656379	+	0.754431i	$0.272087\pi$
$810$	−29365.0	−1.27380
$811$	−21140.0	−0.915322	−0.457661	−	0.889127i	$-0.651313\pi$
$811$	−21140.0	−0.915322	−0.457661	+	0.889127i	$0.651313\pi$
$812$	−18122.0	−0.783199
$813$	−11319.0	−0.488284
$814$	17030.0	0.733294
$815$	3808.00	0.163667
$816$	−47971.0	−2.05799
$817$	−25326.0	−1.08451
$818$	64960.0	2.77662
$819$	0	0
$820$	−39984.0	−1.70281
$821$	−569.000	−0.0241879	−0.0120939	−	0.999927i	$-0.503850\pi$
$821$	−569.000	−0.0241879	−0.0120939	+	0.999927i	$0.503850\pi$
$822$	−62160.0	−2.63757
$823$	−8538.00	−0.361623	−0.180812	−	0.983518i	$-0.557872\pi$
$823$	−8538.00	−0.361623	−0.180812	+	0.983518i	$0.557872\pi$
$824$	26460.0	1.11866
$825$	13832.0	0.583719
$826$	19110.0	0.804990
$827$	32702.0	1.37504	0.687521	−	0.726164i	$-0.258699\pi$
$827$	32702.0	1.37504	0.687521	+	0.726164i	$0.258699\pi$
$828$	−35904.0	−1.50694
$829$	−21154.0	−0.886259	−0.443130	−	0.896458i	$-0.646132\pi$
$829$	−21154.0	−0.886259	−0.443130	+	0.896458i	$0.646132\pi$
$830$	10780.0	0.450818
$831$	26740.0	1.11625
$832$	0	0
$833$	−13398.0	−0.557279
$834$	65415.0	2.71599
$835$	−11368.0	−0.471145
$836$	55692.0	2.30400
$837$	−6860.00	−0.283293
$838$	−36715.0	−1.51348
$839$	2184.00	0.0898690	0.0449345	−	0.998990i	$-0.485692\pi$
$839$	2184.00	0.0898690	0.0449345	+	0.998990i	$0.485692\pi$
$840$	−28665.0	−1.17742
$841$	−17665.0	−0.724302
$842$	25295.0	1.03530
$843$	−43498.0	−1.77717
$844$	48467.0	1.97666
$845$	0	0
$846$	11550.0	0.469382
$847$	−8515.00	−0.345430
$848$	−38448.0	−1.55697
$849$	37044.0	1.49746
$850$	−29260.0	−1.18072
$851$	−12576.0	−0.506580
$852$	1071.00	0.0430656
$853$	−36687.0	−1.47261	−0.736307	−	0.676648i	$-0.763432\pi$
$853$	−36687.0	−1.47261	−0.736307	+	0.676648i	$0.763432\pi$
$854$	−3640.00	−0.145853
$855$	19404.0	0.776144
$856$	−30780.0	−1.22902
$857$	36806.0	1.46706	0.733529	−	0.679658i	$-0.237872\pi$
$857$	36806.0	1.46706	0.733529	+	0.679658i	$0.237872\pi$
$858$	0	0
$859$	4900.00	0.194628	0.0973142	−	0.995254i	$-0.468975\pi$
$859$	4900.00	0.194628	0.0973142	+	0.995254i	$0.468975\pi$
$860$	−23919.0	−0.948408
$861$	30576.0	1.21025
$862$	−16215.0	−0.640702
$863$	13697.0	0.540268	0.270134	−	0.962823i	$-0.412932\pi$
$863$	13697.0	0.540268	0.270134	+	0.962823i	$0.412932\pi$
$864$	2975.00	0.117143
$865$	−2352.00	−0.0924513
$866$	57995.0	2.27569
$867$	−7112.00	−0.278588
$868$	−43316.0	−1.69383
$869$	33904.0	1.32349
$870$	20090.0	0.782891
$871$	0	0
$872$	−16785.0	−0.651848
$873$	−1540.00	−0.0597034
$874$	−60480.0	−2.34069
$875$	−18291.0	−0.706684
$876$	11662.0	0.449797
$877$	−6239.00	−0.240224	−0.120112	−	0.992760i	$-0.538325\pi$
$877$	−6239.00	−0.240224	−0.120112	+	0.992760i	$0.538325\pi$
$878$	−86870.0	−3.33909
$879$	−6321.00	−0.242551
$880$	16198.0	0.620494
$881$	133.000	0.00508613	0.00254307	−	0.999997i	$-0.499191\pi$
$881$	133.000	0.00508613	0.00254307	+	0.999997i	$0.499191\pi$
$882$	−19140.0	−0.730700
$883$	−26003.0	−0.991020	−0.495510	−	0.868602i	$-0.665019\pi$
$883$	−26003.0	−0.991020	−0.495510	+	0.868602i	$0.665019\pi$
$884$	0	0
$885$	−14406.0	−0.547178
$886$	4945.00	0.187506
$887$	−31248.0	−1.18287	−0.591435	−	0.806353i	$-0.701438\pi$
$887$	−31248.0	−1.18287	−0.591435	+	0.806353i	$0.701438\pi$
$888$	−41265.0	−1.55942
$889$	24596.0	0.927923
$890$	41650.0	1.56866
$891$	−21814.0	−0.820198
$892$	−3689.00	−0.138472
$893$	13230.0	0.495773
$894$	86310.0	3.22890
$895$	−21203.0	−0.791886
$896$	−27495.0	−1.02516
$897$	0	0
$898$	72370.0	2.68933
$899$	16072.0	0.596253
$900$	−28424.0	−1.05274
$901$	−33264.0	−1.22995
$902$	−43680.0	−1.61240
$903$	18291.0	0.674071
$904$	−78030.0	−2.87084
$905$	−196.000	−0.00719918
$906$	−116305.	−4.26487
$907$	−38253.0	−1.40041	−0.700204	−	0.713943i	$-0.746908\pi$
$907$	−38253.0	−1.40041	−0.700204	+	0.713943i	$0.746908\pi$
$908$	43792.0	1.60054
$909$	9240.00	0.337152
$910$	0	0
$911$	36374.0	1.32286	0.661429	−	0.750007i	$-0.269950\pi$
$911$	36374.0	1.32286	0.661429	+	0.750007i	$0.269950\pi$
$912$	−78498.0	−2.85014
$913$	8008.00	0.290281
$914$	7970.00	0.288429
$915$	2744.00	0.0991408
$916$	−7735.00	−0.279008
$917$	18655.0	0.671802
$918$	13475.0	0.484468
$919$	−27648.0	−0.992408	−0.496204	−	0.868206i	$-0.665273\pi$
$919$	−27648.0	−0.992408	−0.496204	+	0.868206i	$0.665273\pi$
$920$	−30240.0	−1.08368
$921$	14798.0	0.529436
$922$	29575.0	1.05640
$923$	0	0
$924$	−40222.0	−1.43204
$925$	−9956.00	−0.353893
$926$	55360.0	1.96462
$927$	12936.0	0.458332
$928$	−6970.00	−0.246553
$929$	−756.000	−0.0266992	−0.0133496	−	0.999911i	$-0.504249\pi$
$929$	−756.000	−0.0266992	−0.0133496	+	0.999911i	$0.504249\pi$
$930$	48020.0	1.69316
$931$	−21924.0	−0.771783
$932$	52037.0	1.82889
$933$	−23814.0	−0.835622
$934$	6300.00	0.220709
$935$	14014.0	0.490168
$936$	0	0
$937$	20846.0	0.726797	0.363399	−	0.931634i	$-0.381616\pi$
$937$	20846.0	0.726797	0.363399	+	0.931634i	$0.381616\pi$
$938$	−31070.0	−1.08153
$939$	74823.0	2.60038
$940$	12495.0	0.433555
$941$	41321.0	1.43148	0.715742	−	0.698365i	$-0.246089\pi$
$941$	41321.0	1.43148	0.715742	+	0.698365i	$0.246089\pi$
$942$	95550.0	3.30487
$943$	32256.0	1.11389
$944$	26166.0	0.902151
$945$	3185.00	0.109638
$946$	−26130.0	−0.898055
$947$	−54966.0	−1.88612	−0.943060	−	0.332624i	$-0.892066\pi$
$947$	−54966.0	−1.88612	−0.943060	+	0.332624i	$0.892066\pi$
$948$	−155176.	−5.31633
$949$	0	0
$950$	−47880.0	−1.63519
$951$	−49378.0	−1.68369
$952$	45045.0	1.53353
$953$	−44553.0	−1.51439	−0.757195	−	0.653189i	$-0.773431\pi$
$953$	−44553.0	−1.51439	−0.757195	+	0.653189i	$0.773431\pi$
$954$	−47520.0	−1.61270
$955$	2954.00	0.100093
$956$	59109.0	1.99971
$957$	14924.0	0.504101
$958$	60165.0	2.02906
$959$	23088.0	0.777425
$960$	14063.0	0.472793
$961$	8625.00	0.289517
$962$	0	0
$963$	−15048.0	−0.503546
$964$	27370.0	0.914448
$965$	−3444.00	−0.114887
$966$	43680.0	1.45485
$967$	27907.0	0.928054	0.464027	−	0.885821i	$-0.346404\pi$
$967$	27907.0	0.928054	0.464027	+	0.885821i	$0.346404\pi$
$968$	−29475.0	−0.978680
$969$	−67914.0	−2.25151
$970$	−2450.00	−0.0810977
$971$	−16443.0	−0.543441	−0.271720	−	0.962376i	$-0.587593\pi$
$971$	−16443.0	−0.543441	−0.271720	+	0.962376i	$0.587593\pi$
$972$	83776.0	2.76452
$973$	−24297.0	−0.800541
$974$	11400.0	0.375030
$975$	0	0
$976$	−4984.00	−0.163457
$977$	45414.0	1.48713	0.743563	−	0.668666i	$-0.233134\pi$
$977$	45414.0	1.48713	0.743563	+	0.668666i	$0.233134\pi$
$978$	−19040.0	−0.622528
$979$	30940.0	1.01006
$980$	−20706.0	−0.674927
$981$	−8206.00	−0.267072
$982$	83835.0	2.72432
$983$	8981.00	0.291403	0.145702	−	0.989329i	$-0.453456\pi$
$983$	8981.00	0.291403	0.145702	+	0.989329i	$0.453456\pi$
$984$	105840.	3.42892
$985$	−20937.0	−0.677267
$986$	−31570.0	−1.01967
$987$	−9555.00	−0.308145
$988$	0	0
$989$	19296.0	0.620402
$990$	20020.0	0.642704
$991$	−17414.0	−0.558198	−0.279099	−	0.960262i	$-0.590036\pi$
$991$	−17414.0	−0.558198	−0.279099	+	0.960262i	$0.590036\pi$
$992$	−16660.0	−0.533221
$993$	67928.0	2.17083
$994$	−585.000	−0.0186671
$995$	−490.000	−0.0156121
$996$	−36652.0	−1.16603
$997$	−23702.0	−0.752909	−0.376454	−	0.926435i	$-0.622857\pi$
$997$	−23702.0	−0.752909	−0.376454	+	0.926435i	$0.622857\pi$
$998$	−64200.0	−2.03629
$999$	4585.00	0.145208

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.e.1.1		1
3.2	odd	2		1521.4.a.a.1.1		1
13.2	odd	12		169.4.e.e.147.2		4
13.3	even	3		169.4.c.a.22.1		2
13.4	even	6		169.4.c.e.146.1		2
13.5	odd	4		169.4.b.a.168.1		2
13.6	odd	12		169.4.e.e.23.1		4
13.7	odd	12		169.4.e.e.23.2		4
13.8	odd	4		169.4.b.a.168.2		2
13.9	even	3		169.4.c.a.146.1		2
13.10	even	6		169.4.c.e.22.1		2
13.11	odd	12		169.4.e.e.147.1		4
13.12	even	2		13.4.a.a.1.1	✓	1
39.38	odd	2		117.4.a.b.1.1		1
52.51	odd	2		208.4.a.g.1.1		1
65.12	odd	4		325.4.b.b.274.1		2
65.38	odd	4		325.4.b.b.274.2		2
65.64	even	2		325.4.a.d.1.1		1
91.90	odd	2		637.4.a.a.1.1		1
104.51	odd	2		832.4.a.a.1.1		1
104.77	even	2		832.4.a.r.1.1		1
143.142	odd	2		1573.4.a.a.1.1		1
156.155	even	2		1872.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.a.1.1	✓	1	13.12	even	2
117.4.a.b.1.1		1	39.38	odd	2
169.4.a.e.1.1		1	1.1	even	1	trivial
169.4.b.a.168.1		2	13.5	odd	4
169.4.b.a.168.2		2	13.8	odd	4
169.4.c.a.22.1		2	13.3	even	3
169.4.c.a.146.1		2	13.9	even	3
169.4.c.e.22.1		2	13.10	even	6
169.4.c.e.146.1		2	13.4	even	6
169.4.e.e.23.1		4	13.6	odd	12
169.4.e.e.23.2		4	13.7	odd	12
169.4.e.e.147.1		4	13.11	odd	12
169.4.e.e.147.2		4	13.2	odd	12
208.4.a.g.1.1		1	52.51	odd	2
325.4.a.d.1.1		1	65.64	even	2
325.4.b.b.274.1		2	65.12	odd	4
325.4.b.b.274.2		2	65.38	odd	4
637.4.a.a.1.1		1	91.90	odd	2
832.4.a.a.1.1		1	104.51	odd	2
832.4.a.r.1.1		1	104.77	even	2
1521.4.a.a.1.1		1	3.2	odd	2
1573.4.a.a.1.1		1	143.142	odd	2
1872.4.a.k.1.1		1	156.155	even	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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