LMFDB - Embedded newform 117.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,4,Mod(1,117)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(117, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("117.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	117.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:	$6.90322347067$
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	$\chi$	$=$	117.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+5.00000 q^{2} +17.0000 q^{4} +7.00000 q^{5} -13.0000 q^{7} +45.0000 q^{8} +35.0000 q^{10} +26.0000 q^{11} +13.0000 q^{13} -65.0000 q^{14} +89.0000 q^{16} -77.0000 q^{17} -126.000 q^{19} +119.000 q^{20} +130.000 q^{22} +96.0000 q^{23} -76.0000 q^{25} +65.0000 q^{26} -221.000 q^{28} +82.0000 q^{29} +196.000 q^{31} +85.0000 q^{32} -385.000 q^{34} -91.0000 q^{35} -131.000 q^{37} -630.000 q^{38} +315.000 q^{40} -336.000 q^{41} -201.000 q^{43} +442.000 q^{44} +480.000 q^{46} +105.000 q^{47} -174.000 q^{49} -380.000 q^{50} +221.000 q^{52} +432.000 q^{53} +182.000 q^{55} -585.000 q^{56} +410.000 q^{58} +294.000 q^{59} -56.0000 q^{61} +980.000 q^{62} -287.000 q^{64} +91.0000 q^{65} +478.000 q^{67} -1309.00 q^{68} -455.000 q^{70} -9.00000 q^{71} +98.0000 q^{73} -655.000 q^{74} -2142.00 q^{76} -338.000 q^{77} +1304.00 q^{79} +623.000 q^{80} -1680.00 q^{82} +308.000 q^{83} -539.000 q^{85} -1005.00 q^{86} +1170.00 q^{88} +1190.00 q^{89} -169.000 q^{91} +1632.00 q^{92} +525.000 q^{94} -882.000 q^{95} +70.0000 q^{97} -870.000 q^{98} +O(q^{100})\)

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	$0.154920\pi$
$3$	0	0
$3$	0	0
$4$	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	$0.398649\pi$
$6$	0	0
$7$	−13.0000	−0.701934	−0.350967	−	0.936388i	$-0.614147\pi$
$7$	−13.0000	−0.701934	−0.350967	+	0.936388i	$0.614147\pi$
$8$	45.0000	1.98874
$9$	0	0
$10$	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	$0.384027\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−65.0000	−1.24086
$15$	0	0
$16$	89.0000	1.39062
$17$	−77.0000	−1.09854	−0.549272	−	0.835644i	$-0.685095\pi$
$17$	−77.0000	−1.09854	−0.549272	+	0.835644i	$0.685095\pi$
$18$	0	0
$19$	−126.000	−1.52139	−0.760694	−	0.649110i	$-0.775141\pi$
$19$	−126.000	−1.52139	−0.760694	+	0.649110i	$0.775141\pi$
$20$	119.000	1.33046
$21$	0	0
$22$	130.000	1.25982
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	65.0000	0.490290
$27$	0	0
$28$	−221.000	−1.49161
$29$	82.0000	0.525070	0.262535	−	0.964923i	$-0.415442\pi$
$29$	82.0000	0.525070	0.262535	+	0.964923i	$0.415442\pi$
$30$	0	0
$31$	196.000	1.13557	0.567785	−	0.823177i	$-0.307801\pi$
$31$	196.000	1.13557	0.567785	+	0.823177i	$0.307801\pi$
$32$	85.0000	0.469563
$33$	0	0
$34$	−385.000	−1.94197
$35$	−91.0000	−0.439480
$36$	0	0
$37$	−131.000	−0.582061	−0.291031	−	0.956714i	$-0.593998\pi$
$37$	−131.000	−0.582061	−0.291031	+	0.956714i	$0.593998\pi$
$38$	−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	$0.721037\pi$
$42$	0	0
$43$	−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
$45$	0	0
$46$	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	$0.447904\pi$
$48$	0	0
$49$	−174.000	−0.507289
$50$	−380.000	−1.07480
$51$	0	0
$52$	221.000	0.589369
$53$	432.000	1.11962	0.559809	−	0.828622i	$-0.310874\pi$
$53$	432.000	1.11962	0.559809	+	0.828622i	$0.310874\pi$
$54$	0	0
$55$	182.000	0.446198
$56$	−585.000	−1.39596
$57$	0	0
$58$	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	$0.394848\pi$
$60$	0	0
$61$	−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
$63$	0	0
$64$	−287.000	−0.560547
$65$	91.0000	0.173649
$66$	0	0
$67$	478.000	0.871597	0.435798	−	0.900044i	$-0.356466\pi$
$67$	478.000	0.871597	0.435798	+	0.900044i	$0.356466\pi$
$68$	−1309.00	−2.33441
$69$	0	0
$70$	−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	$0.502394\pi$
$72$	0	0
$73$	98.0000	0.157124	0.0785619	−	0.996909i	$-0.474967\pi$
$73$	98.0000	0.157124	0.0785619	+	0.996909i	$0.474967\pi$
$74$	−655.000	−1.02895
$75$	0	0
$76$	−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
$81$	0	0
$82$	−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	$0.434717\pi$
$84$	0	0
$85$	−539.000	−0.687797
$86$	−1005.00	−1.26014
$87$	0	0
$88$	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	$0.249304\pi$
$90$	0	0
$91$	−169.000	−0.194681
$92$	1632.00	1.84943
$93$	0	0
$94$	525.000	0.576060
$95$	−882.000	−0.952540
$96$	0	0
$97$	70.0000	0.0732724	0.0366362	−	0.999329i	$-0.488336\pi$
$97$	70.0000	0.0732724	0.0366362	+	0.999329i	$0.488336\pi$
$98$	−870.000	−0.896768
$99$	0	0
$100$	−1292.00	−1.29200
$101$	−420.000	−0.413778	−0.206889	−	0.978364i	$-0.566334\pi$
$101$	−420.000	−0.413778	−0.206889	+	0.978364i	$0.566334\pi$
$102$	0	0
$103$	588.000	0.562499	0.281249	−	0.959635i	$-0.409251\pi$
$103$	588.000	0.562499	0.281249	+	0.959635i	$0.409251\pi$
$104$	585.000	0.551577
$105$	0	0
$106$	2160.00	1.97922
$107$	684.000	0.617989	0.308994	−	0.951064i	$-0.400008\pi$
$107$	684.000	0.617989	0.308994	+	0.951064i	$0.400008\pi$
$108$	0	0
$109$	373.000	0.327770	0.163885	−	0.986479i	$-0.447597\pi$
$109$	373.000	0.327770	0.163885	+	0.986479i	$0.447597\pi$
$110$	910.000	0.788774
$111$	0	0
$112$	−1157.00	−0.976127
$113$	1734.00	1.44355	0.721774	−	0.692128i	$-0.243327\pi$
$113$	1734.00	1.44355	0.721774	+	0.692128i	$0.243327\pi$
$114$	0	0
$115$	672.000	0.544907
$116$	1394.00	1.11577
$117$	0	0
$118$	1470.00	1.14682
$119$	1001.00	0.771105
$120$	0	0
$121$	−655.000	−0.492111
$122$	−280.000	−0.207787
$123$	0	0
$124$	3332.00	2.41308
$125$	−1407.00	−1.00677
$126$	0	0
$127$	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
$129$	0	0
$130$	455.000	0.306970
$131$	−1435.00	−0.957073	−0.478536	−	0.878068i	$-0.658833\pi$
$131$	−1435.00	−0.957073	−0.478536	+	0.878068i	$0.658833\pi$
$132$	0	0
$133$	1638.00	1.06791
$134$	2390.00	1.54078
$135$	0	0
$136$	−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	$0.313187\pi$
$138$	0	0
$139$	−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
$141$	0	0
$142$	−45.0000	−0.0265938
$143$	338.000	0.197657
$144$	0	0
$145$	574.000	0.328746
$146$	490.000	0.277758
$147$	0	0
$148$	−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	$0.737122\pi$
$150$	0	0
$151$	−3323.00	−1.79087	−0.895437	−	0.445189i	$-0.853137\pi$
$151$	−3323.00	−1.79087	−0.895437	+	0.445189i	$0.853137\pi$
$152$	−5670.00	−3.02564
$153$	0	0
$154$	−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
$159$	0	0
$160$	595.000	0.293993
$161$	−1248.00	−0.610908
$162$	0	0
$163$	−544.000	−0.261407	−0.130704	−	0.991421i	$-0.541724\pi$
$163$	−544.000	−0.261407	−0.130704	+	0.991421i	$0.541724\pi$
$164$	−5712.00	−2.71971
$165$	0	0
$166$	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	$0.622788\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−2695.00	−1.21587
$171$	0	0
$172$	−3417.00	−1.51479
$173$	336.000	0.147662	0.0738312	−	0.997271i	$-0.476477\pi$
$173$	336.000	0.147662	0.0738312	+	0.997271i	$0.476477\pi$
$174$	0	0
$175$	988.000	0.426776
$176$	2314.00	0.991047
$177$	0	0
$178$	5950.00	2.50546
$179$	3029.00	1.26479	0.632397	−	0.774645i	$-0.282071\pi$
$179$	3029.00	1.26479	0.632397	+	0.774645i	$0.282071\pi$
$180$	0	0
$181$	−28.0000	−0.0114985	−0.00574924	−	0.999983i	$-0.501830\pi$
$181$	−28.0000	−0.0114985	−0.00574924	+	0.999983i	$0.501830\pi$
$182$	−845.000	−0.344151
$183$	0	0
$184$	4320.00	1.73084
$185$	−917.000	−0.364428
$186$	0	0
$187$	−2002.00	−0.782892
$188$	1785.00	0.692471
$189$	0	0
$190$	−4410.00	−1.68387
$191$	−422.000	−0.159868	−0.0799342	−	0.996800i	$-0.525471\pi$
$191$	−422.000	−0.159868	−0.0799342	+	0.996800i	$0.525471\pi$
$192$	0	0
$193$	492.000	0.183497	0.0917485	−	0.995782i	$-0.470754\pi$
$193$	492.000	0.183497	0.0917485	+	0.995782i	$0.470754\pi$
$194$	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	$0.681902\pi$
$198$	0	0
$199$	−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
$201$	0	0
$202$	−2100.00	−0.731463
$203$	−1066.00	−0.368564
$204$	0	0
$205$	−2352.00	−0.801321
$206$	2940.00	0.994367
$207$	0	0
$208$	1157.00	0.385690
$209$	−3276.00	−1.08424
$210$	0	0
$211$	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
$213$	0	0
$214$	3420.00	1.09246
$215$	−1407.00	−0.446310
$216$	0	0
$217$	−2548.00	−0.797095
$218$	1865.00	0.579421
$219$	0	0
$220$	3094.00	0.948170
$221$	−1001.00	−0.304681
$222$	0	0
$223$	217.000	0.0651632	0.0325816	−	0.999469i	$-0.489627\pi$
$223$	217.000	0.0651632	0.0325816	+	0.999469i	$0.489627\pi$
$224$	−1105.00	−0.329602
$225$	0	0
$226$	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	$0.377094\pi$
$228$	0	0
$229$	455.000	0.131298	0.0656490	−	0.997843i	$-0.479088\pi$
$229$	455.000	0.131298	0.0656490	+	0.997843i	$0.479088\pi$
$230$	3360.00	0.963269
$231$	0	0
$232$	3690.00	1.04423
$233$	−3061.00	−0.860656	−0.430328	−	0.902673i	$-0.641602\pi$
$233$	−3061.00	−0.860656	−0.430328	+	0.902673i	$0.641602\pi$
$234$	0	0
$235$	735.000	0.204026
$236$	4998.00	1.37857
$237$	0	0
$238$	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	$0.344066\pi$
$240$	0	0
$241$	−1610.00	−0.430329	−0.215164	−	0.976578i	$-0.569029\pi$
$241$	−1610.00	−0.430329	−0.215164	+	0.976578i	$0.569029\pi$
$242$	−3275.00	−0.869938
$243$	0	0
$244$	−952.000	−0.249777
$245$	−1218.00	−0.317613
$246$	0	0
$247$	−1638.00	−0.421957
$248$	8820.00	2.25835
$249$	0	0
$250$	−7035.00	−1.77973
$251$	−1008.00	−0.253484	−0.126742	−	0.991936i	$-0.540452\pi$
$251$	−1008.00	−0.253484	−0.126742	+	0.991936i	$0.540452\pi$
$252$	0	0
$253$	2496.00	0.620246
$254$	9460.00	2.33690
$255$	0	0
$256$	−8279.00	−2.02124
$257$	−6041.00	−1.46625	−0.733127	−	0.680092i	$-0.761940\pi$
$257$	−6041.00	−1.46625	−0.733127	+	0.680092i	$0.761940\pi$
$258$	0	0
$259$	1703.00	0.408569
$260$	1547.00	0.369003
$261$	0	0
$262$	−7175.00	−1.69188
$263$	3708.00	0.869373	0.434686	−	0.900582i	$-0.356859\pi$
$263$	3708.00	0.869373	0.434686	+	0.900582i	$0.356859\pi$
$264$	0	0
$265$	3024.00	0.700992
$266$	8190.00	1.88782
$267$	0	0
$268$	8126.00	1.85214
$269$	−8344.00	−1.89124	−0.945618	−	0.325278i	$-0.894542\pi$
$269$	−8344.00	−1.89124	−0.945618	+	0.325278i	$0.894542\pi$
$270$	0	0
$271$	−1617.00	−0.362457	−0.181228	−	0.983441i	$-0.558007\pi$
$271$	−1617.00	−0.362457	−0.181228	+	0.983441i	$0.558007\pi$
$272$	−6853.00	−1.52766
$273$	0	0
$274$	8880.00	1.95788
$275$	−1976.00	−0.433299
$276$	0	0
$277$	−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
$279$	0	0
$280$	−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	$0.270725\pi$
$282$	0	0
$283$	−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
$285$	0	0
$286$	1690.00	0.349412
$287$	4368.00	0.898379
$288$	0	0
$289$	1016.00	0.206798
$290$	2870.00	0.581146
$291$	0	0
$292$	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	$0.471306\pi$
$294$	0	0
$295$	2058.00	0.406174
$296$	−5895.00	−1.15757
$297$	0	0
$298$	−12330.0	−2.39684
$299$	1248.00	0.241384
$300$	0	0
$301$	2613.00	0.500368
$302$	−16615.0	−3.16585
$303$	0	0
$304$	−11214.0	−2.11568
$305$	−392.000	−0.0735930
$306$	0	0
$307$	2114.00	0.393004	0.196502	−	0.980503i	$-0.437042\pi$
$307$	2114.00	0.393004	0.196502	+	0.980503i	$0.437042\pi$
$308$	−5746.00	−1.06302
$309$	0	0
$310$	6860.00	1.25684
$311$	−3402.00	−0.620288	−0.310144	−	0.950690i	$-0.600377\pi$
$311$	−3402.00	−0.620288	−0.310144	+	0.950690i	$0.600377\pi$
$312$	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
$315$	0	0
$316$	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	$0.285136\pi$
$318$	0	0
$319$	2132.00	0.374198
$320$	−2009.00	−0.350958
$321$	0	0
$322$	−6240.00	−1.07994
$323$	9702.00	1.67131
$324$	0	0
$325$	−988.000	−0.168629
$326$	−2720.00	−0.462107
$327$	0	0
$328$	−15120.0	−2.54531
$329$	−1365.00	−0.228738
$330$	0	0
$331$	9704.00	1.61142	0.805710	−	0.592310i	$-0.201784\pi$
$331$	9704.00	1.61142	0.805710	+	0.592310i	$0.201784\pi$
$332$	5236.00	0.865551
$333$	0	0
$334$	−8120.00	−1.33026
$335$	3346.00	0.545706
$336$	0	0
$337$	−10449.0	−1.68900	−0.844500	−	0.535555i	$-0.820103\pi$
$337$	−10449.0	−1.68900	−0.844500	+	0.535555i	$0.820103\pi$
$338$	845.000	0.135982
$339$	0	0
$340$	−9163.00	−1.46157
$341$	5096.00	0.809278
$342$	0	0
$343$	6721.00	1.05802
$344$	−9045.00	−1.41766
$345$	0	0
$346$	1680.00	0.261033
$347$	621.000	0.0960721	0.0480361	−	0.998846i	$-0.484704\pi$
$347$	621.000	0.0960721	0.0480361	+	0.998846i	$0.484704\pi$
$348$	0	0
$349$	12481.0	1.91431	0.957153	−	0.289584i	$-0.0935168\pi$
$349$	12481.0	1.91431	0.957153	+	0.289584i	$0.0935168\pi$
$350$	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	$0.466341\pi$
$354$	0	0
$355$	−63.0000	−0.00941885
$356$	20230.0	3.01176
$357$	0	0
$358$	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	$0.381007\pi$
$360$	0	0
$361$	9017.00	1.31462
$362$	−140.000	−0.0203266
$363$	0	0
$364$	−2873.00	−0.413698
$365$	686.000	0.0983750
$366$	0	0
$367$	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
$369$	0	0
$370$	−4585.00	−0.644224
$371$	−5616.00	−0.785898
$372$	0	0
$373$	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
$375$	0	0
$376$	4725.00	0.648067
$377$	1066.00	0.145628
$378$	0	0
$379$	−3372.00	−0.457013	−0.228507	−	0.973542i	$-0.573384\pi$
$379$	−3372.00	−0.457013	−0.228507	+	0.973542i	$0.573384\pi$
$380$	−14994.0	−2.02415
$381$	0	0
$382$	−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	$0.482006\pi$
$384$	0	0
$385$	−2366.00	−0.313201
$386$	2460.00	0.324380
$387$	0	0
$388$	1190.00	0.155704
$389$	−11314.0	−1.47466	−0.737330	−	0.675533i	$-0.763914\pi$
$389$	−11314.0	−1.47466	−0.737330	+	0.675533i	$0.763914\pi$
$390$	0	0
$391$	−7392.00	−0.956086
$392$	−7830.00	−1.00886
$393$	0	0
$394$	−14955.0	−1.91224
$395$	9128.00	1.16273
$396$	0	0
$397$	1862.00	0.235393	0.117697	−	0.993050i	$-0.462449\pi$
$397$	1862.00	0.235393	0.117697	+	0.993050i	$0.462449\pi$
$398$	−350.000	−0.0440802
$399$	0	0
$400$	−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	$0.639605\pi$
$402$	0	0
$403$	2548.00	0.314950
$404$	−7140.00	−0.879278
$405$	0	0
$406$	−5330.00	−0.651536
$407$	−3406.00	−0.414814
$408$	0	0
$409$	−12992.0	−1.57069	−0.785346	−	0.619057i	$-0.787515\pi$
$409$	−12992.0	−1.57069	−0.785346	+	0.619057i	$0.787515\pi$
$410$	−11760.0	−1.41655
$411$	0	0
$412$	9996.00	1.19531
$413$	−3822.00	−0.455371
$414$	0	0
$415$	2156.00	0.255021
$416$	1105.00	0.130233
$417$	0	0
$418$	−16380.0	−1.91668
$419$	7343.00	0.856155	0.428078	−	0.903742i	$-0.359191\pi$
$419$	7343.00	0.856155	0.428078	+	0.903742i	$0.359191\pi$
$420$	0	0
$421$	−5059.00	−0.585655	−0.292827	−	0.956165i	$-0.594596\pi$
$421$	−5059.00	−0.585655	−0.292827	+	0.956165i	$0.594596\pi$
$422$	14255.0	1.64437
$423$	0	0
$424$	19440.0	2.22663
$425$	5852.00	0.667915
$426$	0	0
$427$	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	$0.558004\pi$
$432$	0	0
$433$	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
$435$	0	0
$436$	6341.00	0.696511
$437$	−12096.0	−1.32410
$438$	0	0
$439$	−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
$441$	0	0
$442$	−5005.00	−0.538605
$443$	−989.000	−0.106070	−0.0530348	−	0.998593i	$-0.516889\pi$
$443$	−989.000	−0.106070	−0.0530348	+	0.998593i	$0.516889\pi$
$444$	0	0
$445$	8330.00	0.887370
$446$	1085.00	0.115193
$447$	0	0
$448$	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	$0.224877\pi$
$450$	0	0
$451$	−8736.00	−0.912111
$452$	29478.0	3.06754
$453$	0	0
$454$	12880.0	1.33147
$455$	−1183.00	−0.121890
$456$	0	0
$457$	−1594.00	−0.163160	−0.0815801	−	0.996667i	$-0.525997\pi$
$457$	−1594.00	−0.163160	−0.0815801	+	0.996667i	$0.525997\pi$
$458$	2275.00	0.232104
$459$	0	0
$460$	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	$0.403415\pi$
$462$	0	0
$463$	−11072.0	−1.11136	−0.555680	−	0.831396i	$-0.687542\pi$
$463$	−11072.0	−1.11136	−0.555680	+	0.831396i	$0.687542\pi$
$464$	7298.00	0.730175
$465$	0	0
$466$	−15305.0	−1.52144
$467$	−1260.00	−0.124852	−0.0624260	−	0.998050i	$-0.519884\pi$
$467$	−1260.00	−0.124852	−0.0624260	+	0.998050i	$0.519884\pi$
$468$	0	0
$469$	−6214.00	−0.611804
$470$	3675.00	0.360670
$471$	0	0
$472$	13230.0	1.29017
$473$	−5226.00	−0.508016
$474$	0	0
$475$	9576.00	0.925004
$476$	17017.0	1.63860
$477$	0	0
$478$	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	$0.305428\pi$
$480$	0	0
$481$	−1703.00	−0.161435
$482$	−8050.00	−0.760721
$483$	0	0
$484$	−11135.0	−1.04574
$485$	490.000	0.0458758
$486$	0	0
$487$	−2280.00	−0.212149	−0.106075	−	0.994358i	$-0.533828\pi$
$487$	−2280.00	−0.212149	−0.106075	+	0.994358i	$0.533828\pi$
$488$	−2520.00	−0.233760
$489$	0	0
$490$	−6090.00	−0.561466
$491$	−16767.0	−1.54111	−0.770554	−	0.637375i	$-0.780020\pi$
$491$	−16767.0	−1.54111	−0.770554	+	0.637375i	$0.780020\pi$
$492$	0	0
$493$	−6314.00	−0.576812
$494$	−8190.00	−0.745922
$495$	0	0
$496$	17444.0	1.57915
$497$	117.000	0.0105597
$498$	0	0
$499$	12840.0	1.15190	0.575949	−	0.817485i	$-0.304633\pi$
$499$	12840.0	1.15190	0.575949	+	0.817485i	$0.304633\pi$
$500$	−23919.0	−2.13938
$501$	0	0
$502$	−5040.00	−0.448100
$503$	2198.00	0.194839	0.0974195	−	0.995243i	$-0.468941\pi$
$503$	2198.00	0.194839	0.0974195	+	0.995243i	$0.468941\pi$
$504$	0	0
$505$	−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	$0.233373\pi$
$510$	0	0
$511$	−1274.00	−0.110290
$512$	−24475.0	−2.11260
$513$	0	0
$514$	−30205.0	−2.59200
$515$	4116.00	0.352180
$516$	0	0
$517$	2730.00	0.232235
$518$	8515.00	0.722254
$519$	0	0
$520$	4095.00	0.345342
$521$	−2583.00	−0.217204	−0.108602	−	0.994085i	$-0.534637\pi$
$521$	−2583.00	−0.217204	−0.108602	+	0.994085i	$0.534637\pi$
$522$	0	0
$523$	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
$525$	0	0
$526$	18540.0	1.53685
$527$	−15092.0	−1.24747
$528$	0	0
$529$	−2951.00	−0.242541
$530$	15120.0	1.23919
$531$	0	0
$532$	27846.0	2.26932
$533$	−4368.00	−0.354970
$534$	0	0
$535$	4788.00	0.386922
$536$	21510.0	1.73338
$537$	0	0
$538$	−41720.0	−3.34327
$539$	−4524.00	−0.361526
$540$	0	0
$541$	−16833.0	−1.33772	−0.668861	−	0.743388i	$-0.733218\pi$
$541$	−16833.0	−1.33772	−0.668861	+	0.743388i	$0.733218\pi$
$542$	−8085.00	−0.640739
$543$	0	0
$544$	−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
$549$	0	0
$550$	−9880.00	−0.765972
$551$	−10332.0	−0.798835
$552$	0	0
$553$	−16952.0	−1.30357
$554$	−19100.0	−1.46477
$555$	0	0
$556$	−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	$0.605240\pi$
$558$	0	0
$559$	−2613.00	−0.197707
$560$	−8099.00	−0.611152
$561$	0	0
$562$	31070.0	2.33204
$563$	4641.00	0.347415	0.173708	−	0.984797i	$-0.444425\pi$
$563$	4641.00	0.347415	0.173708	+	0.984797i	$0.444425\pi$
$564$	0	0
$565$	12138.0	0.903804
$566$	−26460.0	−1.96501
$567$	0	0
$568$	−405.000	−0.0299180
$569$	4793.00	0.353134	0.176567	−	0.984289i	$-0.443501\pi$
$569$	4793.00	0.353134	0.176567	+	0.984289i	$0.443501\pi$
$570$	0	0
$571$	−5563.00	−0.407713	−0.203857	−	0.979001i	$-0.565348\pi$
$571$	−5563.00	−0.407713	−0.203857	+	0.979001i	$0.565348\pi$
$572$	5746.00	0.420022
$573$	0	0
$574$	21840.0	1.58813
$575$	−7296.00	−0.529155
$576$	0	0
$577$	24038.0	1.73434	0.867171	−	0.498011i	$-0.165936\pi$
$577$	24038.0	1.73434	0.867171	+	0.498011i	$0.165936\pi$
$578$	5080.00	0.365571
$579$	0	0
$580$	9758.00	0.698584
$581$	−4004.00	−0.285910
$582$	0	0
$583$	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	$0.231889\pi$
$588$	0	0
$589$	−24696.0	−1.72764
$590$	10290.0	0.718021
$591$	0	0
$592$	−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	$0.548171\pi$
$594$	0	0
$595$	7007.00	0.482788
$596$	−41922.0	−2.88119
$597$	0	0
$598$	6240.00	0.426710
$599$	−7310.00	−0.498629	−0.249314	−	0.968423i	$-0.580205\pi$
$599$	−7310.00	−0.498629	−0.249314	+	0.968423i	$0.580205\pi$
$600$	0	0
$601$	−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
$603$	0	0
$604$	−56491.0	−3.80561
$605$	−4585.00	−0.308110
$606$	0	0
$607$	−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
$609$	0	0
$610$	−1960.00	−0.130095
$611$	1365.00	0.0903797
$612$	0	0
$613$	14590.0	0.961312	0.480656	−	0.876909i	$-0.340398\pi$
$613$	14590.0	0.961312	0.480656	+	0.876909i	$0.340398\pi$
$614$	10570.0	0.694740
$615$	0	0
$616$	−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	$0.550978\pi$
$618$	0	0
$619$	−11004.0	−0.714520	−0.357260	−	0.934005i	$-0.616289\pi$
$619$	−11004.0	−0.714520	−0.357260	+	0.934005i	$0.616289\pi$
$620$	23324.0	1.51083
$621$	0	0
$622$	−17010.0	−1.09653
$623$	−15470.0	−0.994851
$624$	0	0
$625$	−349.000	−0.0223360
$626$	−53445.0	−3.41229
$627$	0	0
$628$	−46410.0	−2.94898
$629$	10087.0	0.639420
$630$	0	0
$631$	−4975.00	−0.313869	−0.156935	−	0.987609i	$-0.550161\pi$
$631$	−4975.00	−0.313869	−0.156935	+	0.987609i	$0.550161\pi$
$632$	58680.0	3.69330
$633$	0	0
$634$	35270.0	2.20939
$635$	13244.0	0.827673
$636$	0	0
$637$	−2262.00	−0.140697
$638$	10660.0	0.661494
$639$	0	0
$640$	−14805.0	−0.914405
$641$	−3950.00	−0.243394	−0.121697	−	0.992567i	$-0.538834\pi$
$641$	−3950.00	−0.243394	−0.121697	+	0.992567i	$0.538834\pi$
$642$	0	0
$643$	−3682.00	−0.225823	−0.112911	−	0.993605i	$-0.536018\pi$
$643$	−3682.00	−0.225823	−0.112911	+	0.993605i	$0.536018\pi$
$644$	−21216.0	−1.29818
$645$	0	0
$646$	48510.0	2.95449
$647$	−10402.0	−0.632063	−0.316032	−	0.948749i	$-0.602351\pi$
$647$	−10402.0	−0.632063	−0.316032	+	0.948749i	$0.602351\pi$
$648$	0	0
$649$	7644.00	0.462332
$650$	−4940.00	−0.298097
$651$	0	0
$652$	−9248.00	−0.555490
$653$	31680.0	1.89852	0.949260	−	0.314491i	$-0.101834\pi$
$653$	31680.0	1.89852	0.949260	+	0.314491i	$0.101834\pi$
$654$	0	0
$655$	−10045.0	−0.599222
$656$	−29904.0	−1.77981
$657$	0	0
$658$	−6825.00	−0.404356
$659$	−21940.0	−1.29691	−0.648453	−	0.761255i	$-0.724584\pi$
$659$	−21940.0	−1.29691	−0.648453	+	0.761255i	$0.724584\pi$
$660$	0	0
$661$	−31374.0	−1.84615	−0.923077	−	0.384616i	$-0.874334\pi$
$661$	−31374.0	−1.84615	−0.923077	+	0.384616i	$0.874334\pi$
$662$	48520.0	2.84862
$663$	0	0
$664$	13860.0	0.810049
$665$	11466.0	0.668620
$666$	0	0
$667$	7872.00	0.456979
$668$	−27608.0	−1.59908
$669$	0	0
$670$	16730.0	0.964681
$671$	−1456.00	−0.0837679
$672$	0	0
$673$	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
$675$	0	0
$676$	2873.00	0.163462
$677$	10640.0	0.604030	0.302015	−	0.953303i	$-0.402341\pi$
$677$	10640.0	0.604030	0.302015	+	0.953303i	$0.402341\pi$
$678$	0	0
$679$	−910.000	−0.0514324
$680$	−24255.0	−1.36785
$681$	0	0
$682$	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	$0.415777\pi$
$684$	0	0
$685$	12432.0	0.693434
$686$	33605.0	1.87033
$687$	0	0
$688$	−17889.0	−0.991296
$689$	5616.00	0.310526
$690$	0	0
$691$	4200.00	0.231224	0.115612	−	0.993294i	$-0.463117\pi$
$691$	4200.00	0.231224	0.115612	+	0.993294i	$0.463117\pi$
$692$	5712.00	0.313783
$693$	0	0
$694$	3105.00	0.169833
$695$	−13083.0	−0.714052
$696$	0	0
$697$	25872.0	1.40599
$698$	62405.0	3.38405
$699$	0	0
$700$	16796.0	0.906899
$701$	−9872.00	−0.531898	−0.265949	−	0.963987i	$-0.585685\pi$
$701$	−9872.00	−0.531898	−0.265949	+	0.963987i	$0.585685\pi$
$702$	0	0
$703$	16506.0	0.885541
$704$	−7462.00	−0.399481
$705$	0	0
$706$	7000.00	0.373156
$707$	5460.00	0.290445
$708$	0	0
$709$	28450.0	1.50700	0.753499	−	0.657449i	$-0.228364\pi$
$709$	28450.0	1.50700	0.753499	+	0.657449i	$0.228364\pi$
$710$	−315.000	−0.0166503
$711$	0	0
$712$	53550.0	2.81864
$713$	18816.0	0.988310
$714$	0	0
$715$	2366.00	0.123753
$716$	51493.0	2.68769
$717$	0	0
$718$	24840.0	1.29111
$719$	−32718.0	−1.69705	−0.848523	−	0.529159i	$-0.822507\pi$
$719$	−32718.0	−1.69705	−0.848523	+	0.529159i	$0.822507\pi$
$720$	0	0
$721$	−7644.00	−0.394837
$722$	45085.0	2.32395
$723$	0	0
$724$	−476.000	−0.0244343
$725$	−6232.00	−0.319242
$726$	0	0
$727$	−22834.0	−1.16488	−0.582439	−	0.812874i	$-0.697901\pi$
$727$	−22834.0	−1.16488	−0.582439	+	0.812874i	$0.697901\pi$
$728$	−7605.00	−0.387170
$729$	0	0
$730$	3430.00	0.173904
$731$	15477.0	0.783088
$732$	0	0
$733$	7875.00	0.396821	0.198410	−	0.980119i	$-0.436422\pi$
$733$	7875.00	0.396821	0.198410	+	0.980119i	$0.436422\pi$
$734$	43610.0	2.19302
$735$	0	0
$736$	8160.00	0.408671
$737$	12428.0	0.621155
$738$	0	0
$739$	−2140.00	−0.106524	−0.0532620	−	0.998581i	$-0.516962\pi$
$739$	−2140.00	−0.106524	−0.0532620	+	0.998581i	$0.516962\pi$
$740$	−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	$0.789557\pi$
$744$	0	0
$745$	−17262.0	−0.848900
$746$	50060.0	2.45687
$747$	0	0
$748$	−34034.0	−1.66364
$749$	−8892.00	−0.433787
$750$	0	0
$751$	−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
$753$	0	0
$754$	5330.00	0.257437
$755$	−23261.0	−1.12126
$756$	0	0
$757$	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
$759$	0	0
$760$	−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	$0.427520\pi$
$762$	0	0
$763$	−4849.00	−0.230073
$764$	−7174.00	−0.339720
$765$	0	0
$766$	4235.00	0.199761
$767$	3822.00	0.179928
$768$	0	0
$769$	−12096.0	−0.567221	−0.283610	−	0.958940i	$-0.591532\pi$
$769$	−12096.0	−0.567221	−0.283610	+	0.958940i	$0.591532\pi$
$770$	−11830.0	−0.553667
$771$	0	0
$772$	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	$0.637057\pi$
$774$	0	0
$775$	−14896.0	−0.690426
$776$	3150.00	0.145720
$777$	0	0
$778$	−56570.0	−2.60685
$779$	42336.0	1.94717
$780$	0	0
$781$	−234.000	−0.0107211
$782$	−36960.0	−1.69014
$783$	0	0
$784$	−15486.0	−0.705448
$785$	−19110.0	−0.868873
$786$	0	0
$787$	6664.00	0.301837	0.150919	−	0.988546i	$-0.451777\pi$
$787$	6664.00	0.301837	0.150919	+	0.988546i	$0.451777\pi$
$788$	−50847.0	−2.29867
$789$	0	0
$790$	45640.0	2.05544
$791$	−22542.0	−1.01328
$792$	0	0
$793$	−728.000	−0.0326003
$794$	9310.00	0.416120
$795$	0	0
$796$	−1190.00	−0.0529880
$797$	1442.00	0.0640882	0.0320441	−	0.999486i	$-0.489798\pi$
$797$	1442.00	0.0640882	0.0320441	+	0.999486i	$0.489798\pi$
$798$	0	0
$799$	−8085.00	−0.357981
$800$	−6460.00	−0.285494
$801$	0	0
$802$	−34100.0	−1.50139
$803$	2548.00	0.111976
$804$	0	0
$805$	−8736.00	−0.382489
$806$	12740.0	0.556759
$807$	0	0
$808$	−18900.0	−0.822896
$809$	−30207.0	−1.31276	−0.656379	−	0.754431i	$-0.727913\pi$
$809$	−30207.0	−1.31276	−0.656379	+	0.754431i	$0.727913\pi$
$810$	0	0
$811$	21140.0	0.915322	0.457661	−	0.889127i	$-0.348687\pi$
$811$	21140.0	0.915322	0.457661	+	0.889127i	$0.348687\pi$
$812$	−18122.0	−0.783199
$813$	0	0
$814$	−17030.0	−0.733294
$815$	−3808.00	−0.163667
$816$	0	0
$817$	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	$0.503850\pi$
$822$	0	0
$823$	−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
$825$	0	0
$826$	−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	$0.258699\pi$
$828$	0	0
$829$	−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
$831$	0	0
$832$	−3731.00	−0.155468
$833$	13398.0	0.557279
$834$	0	0
$835$	−11368.0	−0.471145
$836$	−55692.0	−2.30400
$837$	0	0
$838$	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	$0.485692\pi$
$840$	0	0
$841$	−17665.0	−0.724302
$842$	−25295.0	−1.03530
$843$	0	0
$844$	48467.0	1.97666
$845$	1183.00	0.0481615
$846$	0	0
$847$	8515.00	0.345430
$848$	38448.0	1.55697
$849$	0	0
$850$	29260.0	1.18072
$851$	−12576.0	−0.506580
$852$	0	0
$853$	36687.0	1.47261	0.736307	−	0.676648i	$-0.236568\pi$
$853$	36687.0	1.47261	0.736307	+	0.676648i	$0.236568\pi$
$854$	3640.00	0.145853
$855$	0	0
$856$	30780.0	1.22902
$857$	−36806.0	−1.46706	−0.733529	−	0.679658i	$-0.762128\pi$
$857$	−36806.0	−1.46706	−0.733529	+	0.679658i	$0.762128\pi$
$858$	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
$861$	0	0
$862$	−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	$0.412932\pi$
$864$	0	0
$865$	2352.00	0.0924513
$866$	57995.0	2.27569
$867$	0	0
$868$	−43316.0	−1.69383
$869$	33904.0	1.32349
$870$	0	0
$871$	6214.00	0.241737
$872$	16785.0	0.651848
$873$	0	0
$874$	−60480.0	−2.34069
$875$	18291.0	0.706684
$876$	0	0
$877$	6239.00	0.240224	0.120112	−	0.992760i	$-0.461675\pi$
$877$	6239.00	0.240224	0.120112	+	0.992760i	$0.461675\pi$
$878$	−86870.0	−3.33909
$879$	0	0
$880$	16198.0	0.620494
$881$	−133.000	−0.00508613	−0.00254307	−	0.999997i	$-0.500809\pi$
$881$	−133.000	−0.00508613	−0.00254307	+	0.999997i	$0.500809\pi$
$882$	0	0
$883$	−26003.0	−0.991020	−0.495510	−	0.868602i	$-0.665019\pi$
$883$	−26003.0	−0.991020	−0.495510	+	0.868602i	$0.665019\pi$
$884$	−17017.0	−0.647448
$885$	0	0
$886$	−4945.00	−0.187506
$887$	31248.0	1.18287	0.591435	−	0.806353i	$-0.298562\pi$
$887$	31248.0	1.18287	0.591435	+	0.806353i	$0.298562\pi$
$888$	0	0
$889$	−24596.0	−0.927923
$890$	41650.0	1.56866
$891$	0	0
$892$	3689.00	0.138472
$893$	−13230.0	−0.495773
$894$	0	0
$895$	21203.0	0.791886
$896$	27495.0	1.02516
$897$	0	0
$898$	72370.0	2.68933
$899$	16072.0	0.596253
$900$	0	0
$901$	−33264.0	−1.22995
$902$	−43680.0	−1.61240
$903$	0	0
$904$	78030.0	2.87084
$905$	−196.000	−0.00719918
$906$	0	0
$907$	−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
$909$	0	0
$910$	−5915.00	−0.215473
$911$	−36374.0	−1.32286	−0.661429	−	0.750007i	$-0.730050\pi$
$911$	−36374.0	−1.32286	−0.661429	+	0.750007i	$0.730050\pi$
$912$	0	0
$913$	8008.00	0.290281
$914$	−7970.00	−0.288429
$915$	0	0
$916$	7735.00	0.279008
$917$	18655.0	0.671802
$918$	0	0
$919$	−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
$921$	0	0
$922$	29575.0	1.05640
$923$	−117.000	−0.00417237
$924$	0	0
$925$	9956.00	0.353893
$926$	−55360.0	−1.96462
$927$	0	0
$928$	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	$0.504249\pi$
$930$	0	0
$931$	21924.0	0.771783
$932$	−52037.0	−1.82889
$933$	0	0
$934$	−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
$939$	0	0
$940$	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	$0.246089\pi$
$942$	0	0
$943$	−32256.0	−1.11389
$944$	26166.0	0.902151
$945$	0	0
$946$	−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	$0.892066\pi$
$948$	0	0
$949$	1274.00	0.0435783
$950$	47880.0	1.63519
$951$	0	0
$952$	45045.0	1.53353
$953$	44553.0	1.51439	0.757195	−	0.653189i	$-0.226569\pi$
$953$	44553.0	1.51439	0.757195	+	0.653189i	$0.226569\pi$
$954$	0	0
$955$	−2954.00	−0.100093
$956$	59109.0	1.99971
$957$	0	0
$958$	60165.0	2.02906
$959$	−23088.0	−0.777425
$960$	0	0
$961$	8625.00	0.289517
$962$	−8515.00	−0.285379
$963$	0	0
$964$	−27370.0	−0.914448
$965$	3444.00	0.114887
$966$	0	0
$967$	−27907.0	−0.928054	−0.464027	−	0.885821i	$-0.653596\pi$
$967$	−27907.0	−0.928054	−0.464027	+	0.885821i	$0.653596\pi$
$968$	−29475.0	−0.978680
$969$	0	0
$970$	2450.00	0.0810977
$971$	16443.0	0.543441	0.271720	−	0.962376i	$-0.412407\pi$
$971$	16443.0	0.543441	0.271720	+	0.962376i	$0.412407\pi$
$972$	0	0
$973$	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	$0.233134\pi$
$978$	0	0
$979$	30940.0	1.01006
$980$	−20706.0	−0.674927
$981$	0	0
$982$	−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	$0.453456\pi$
$984$	0	0
$985$	−20937.0	−0.677267
$986$	−31570.0	−1.01967
$987$	0	0
$988$	−27846.0	−0.896659
$989$	−19296.0	−0.620402
$990$	0	0
$991$	−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
$993$	0	0
$994$	585.000	0.0186671
$995$	−490.000	−0.0156121
$996$	0	0
$997$	−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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.b.1.1		1
3.2	odd	2		13.4.a.a.1.1	✓	1
4.3	odd	2		1872.4.a.k.1.1		1
12.11	even	2		208.4.a.g.1.1		1
13.12	even	2		1521.4.a.a.1.1		1
15.2	even	4		325.4.b.b.274.1		2
15.8	even	4		325.4.b.b.274.2		2
15.14	odd	2		325.4.a.d.1.1		1
21.20	even	2		637.4.a.a.1.1		1
24.5	odd	2		832.4.a.r.1.1		1
24.11	even	2		832.4.a.a.1.1		1
33.32	even	2		1573.4.a.a.1.1		1
39.2	even	12		169.4.e.e.147.1		4
39.5	even	4		169.4.b.a.168.2		2
39.8	even	4		169.4.b.a.168.1		2
39.11	even	12		169.4.e.e.147.2		4
39.17	odd	6		169.4.c.a.146.1		2
39.20	even	12		169.4.e.e.23.1		4
39.23	odd	6		169.4.c.a.22.1		2
39.29	odd	6		169.4.c.e.22.1		2
39.32	even	12		169.4.e.e.23.2		4
39.35	odd	6		169.4.c.e.146.1		2
39.38	odd	2		169.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.a.1.1	✓	1	3.2	odd	2
117.4.a.b.1.1		1	1.1	even	1	trivial
169.4.a.e.1.1		1	39.38	odd	2
169.4.b.a.168.1		2	39.8	even	4
169.4.b.a.168.2		2	39.5	even	4
169.4.c.a.22.1		2	39.23	odd	6
169.4.c.a.146.1		2	39.17	odd	6
169.4.c.e.22.1		2	39.29	odd	6
169.4.c.e.146.1		2	39.35	odd	6
169.4.e.e.23.1		4	39.20	even	12
169.4.e.e.23.2		4	39.32	even	12
169.4.e.e.147.1		4	39.2	even	12
169.4.e.e.147.2		4	39.11	even	12
208.4.a.g.1.1		1	12.11	even	2
325.4.a.d.1.1		1	15.14	odd	2
325.4.b.b.274.1		2	15.2	even	4
325.4.b.b.274.2		2	15.8	even	4
637.4.a.a.1.1		1	21.20	even	2
832.4.a.a.1.1		1	24.11	even	2
832.4.a.r.1.1		1	24.5	odd	2
1521.4.a.a.1.1		1	13.12	even	2
1573.4.a.a.1.1		1	33.32	even	2
1872.4.a.k.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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