LMFDB - Embedded newform 117.4.a.c.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{14}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 14 $ x^2 - 14
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.74166$ of defining polynomial
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)

$f(q)$	$=$	$q-4.74166 q^{2} +14.4833 q^{4} -4.51669 q^{5} -7.48331 q^{7} -30.7417 q^{8} +O(q^{10})$	\(q-4.74166 q^{2} +14.4833 q^{4} -4.51669 q^{5} -7.48331 q^{7} -30.7417 q^{8} +21.4166 q^{10} +66.8999 q^{11} -13.0000 q^{13} +35.4833 q^{14} +29.8999 q^{16} -96.9666 q^{17} +31.4833 q^{19} -65.4166 q^{20} -317.216 q^{22} -183.600 q^{23} -104.600 q^{25} +61.6415 q^{26} -108.383 q^{28} -112.200 q^{29} -77.2831 q^{31} +104.158 q^{32} +459.783 q^{34} +33.7998 q^{35} +54.7664 q^{37} -149.283 q^{38} +138.850 q^{40} -451.716 q^{41} -113.434 q^{43} +968.932 q^{44} +870.566 q^{46} +42.2670 q^{47} -287.000 q^{49} +495.975 q^{50} -188.283 q^{52} +530.999 q^{53} -302.166 q^{55} +230.049 q^{56} +532.015 q^{58} -219.666 q^{59} +822.865 q^{61} +366.450 q^{62} -733.082 q^{64} +58.7169 q^{65} -872.082 q^{67} -1404.40 q^{68} -160.267 q^{70} +100.299 q^{71} -165.634 q^{73} -259.684 q^{74} +455.983 q^{76} -500.633 q^{77} -545.266 q^{79} -135.048 q^{80} +2141.88 q^{82} +454.534 q^{83} +437.968 q^{85} +537.864 q^{86} -2056.61 q^{88} +230.915 q^{89} +97.2831 q^{91} -2659.13 q^{92} -200.415 q^{94} -142.200 q^{95} -1089.16 q^{97} +1360.86 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8	$ 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28} - 404 q^{29} + 40 q^{31} + 126 q^{32} + 276 q^{34} - 112 q^{35} - 100 q^{37} - 104 q^{38} + 592 q^{40} - 200 q^{41} - 616 q^{43} + 980 q^{44} + 1352 q^{46} + 324 q^{47} - 574 q^{49} + 1194 q^{50} - 182 q^{52} + 164 q^{53} + 144 q^{55} + 56 q^{56} - 268 q^{58} - 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 312 q^{65} - 472 q^{67} - 1372 q^{68} - 560 q^{70} - 428 q^{71} - 900 q^{73} - 684 q^{74} + 448 q^{76} - 672 q^{77} - 432 q^{79} + 1032 q^{80} + 2832 q^{82} + 1388 q^{83} + 1744 q^{85} - 840 q^{86} - 1524 q^{88} - 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} - 532 q^{97} + 574 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 - 32 * q^10 + 44 * q^11 - 26 * q^13 + 56 * q^14 - 30 * q^16 - 164 * q^17 + 48 * q^19 - 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 26 * q^26 - 112 * q^28 - 404 * q^29 + 40 * q^31 + 126 * q^32 + 276 * q^34 - 112 * q^35 - 100 * q^37 - 104 * q^38 + 592 * q^40 - 200 * q^41 - 616 * q^43 + 980 * q^44 + 1352 * q^46 + 324 * q^47 - 574 * q^49 + 1194 * q^50 - 182 * q^52 + 164 * q^53 + 144 * q^55 + 56 * q^56 - 268 * q^58 - 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 312 * q^65 - 472 * q^67 - 1372 * q^68 - 560 * q^70 - 428 * q^71 - 900 * q^73 - 684 * q^74 + 448 * q^76 - 672 * q^77 - 432 * q^79 + 1032 * q^80 + 2832 * q^82 + 1388 * q^83 + 1744 * q^85 - 840 * q^86 - 1524 * q^88 - 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 - 532 * q^97 + 574 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−4.74166	−1.67643	−0.838215	−	0.545341i	$-0.816400\pi$
$2$	−4.74166	−1.67643	−0.838215	+	0.545341i	$0.816400\pi$
$3$	0	0
$3$	0	0
$4$	14.4833	1.81041
$5$	−4.51669	−0.403985	−0.201992	−	0.979387i	$-0.564742\pi$
$5$	−4.51669	−0.403985	−0.201992	+	0.979387i	$0.564742\pi$
$6$	0	0
$7$	−7.48331	−0.404061	−0.202031	−	0.979379i	$-0.564754\pi$
$7$	−7.48331	−0.404061	−0.202031	+	0.979379i	$0.564754\pi$
$8$	−30.7417	−1.35860
$9$	0	0
$10$	21.4166	0.677252
$11$	66.8999	1.83373	0.916867	−	0.399193i	$-0.130710\pi$
$11$	66.8999	1.83373	0.916867	+	0.399193i	$0.130710\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	35.4833	0.677380
$15$	0	0
$16$	29.8999	0.467186
$17$	−96.9666	−1.38340	−0.691702	−	0.722183i	$-0.743139\pi$
$17$	−96.9666	−1.38340	−0.691702	+	0.722183i	$0.743139\pi$
$18$	0	0
$19$	31.4833	0.380146	0.190073	−	0.981770i	$-0.439128\pi$
$19$	31.4833	0.380146	0.190073	+	0.981770i	$0.439128\pi$
$20$	−65.4166	−0.731380
$21$	0	0
$22$	−317.216	−3.07413
$23$	−183.600	−1.66448	−0.832242	−	0.554412i	$-0.812943\pi$
$23$	−183.600	−1.66448	−0.832242	+	0.554412i	$0.812943\pi$
$24$	0	0
$25$	−104.600	−0.836796
$26$	61.6415	0.464958
$27$	0	0
$28$	−108.383	−0.731518
$29$	−112.200	−0.718450	−0.359225	−	0.933251i	$-0.616959\pi$
$29$	−112.200	−0.718450	−0.359225	+	0.933251i	$0.616959\pi$
$30$	0	0
$31$	−77.2831	−0.447757	−0.223878	−	0.974617i	$-0.571872\pi$
$31$	−77.2831	−0.447757	−0.223878	+	0.974617i	$0.571872\pi$
$32$	104.158	0.575398
$33$	0	0
$34$	459.783	2.31918
$35$	33.7998	0.163234
$36$	0	0
$37$	54.7664	0.243339	0.121669	−	0.992571i	$-0.461175\pi$
$37$	54.7664	0.243339	0.121669	+	0.992571i	$0.461175\pi$
$38$	−149.283	−0.637287
$39$	0	0
$40$	138.850	0.548854
$41$	−451.716	−1.72064	−0.860319	−	0.509756i	$-0.829736\pi$
$41$	−451.716	−1.72064	−0.860319	+	0.509756i	$0.829736\pi$
$42$	0	0
$43$	−113.434	−0.402291	−0.201145	−	0.979561i	$-0.564466\pi$
$43$	−113.434	−0.402291	−0.201145	+	0.979561i	$0.564466\pi$
$44$	968.932	3.31982
$45$	0	0
$46$	870.566	2.79039
$47$	42.2670	0.131176	0.0655880	−	0.997847i	$-0.479108\pi$
$47$	42.2670	0.131176	0.0655880	+	0.997847i	$0.479108\pi$
$48$	0	0
$49$	−287.000	−0.836735
$50$	495.975	1.40283
$51$	0	0
$52$	−188.283	−0.502119
$53$	530.999	1.37619	0.688097	−	0.725618i	$-0.258446\pi$
$53$	530.999	1.37619	0.688097	+	0.725618i	$0.258446\pi$
$54$	0	0
$55$	−302.166	−0.740800
$56$	230.049	0.548958
$57$	0	0
$58$	532.015	1.20443
$59$	−219.666	−0.484714	−0.242357	−	0.970187i	$-0.577920\pi$
$59$	−219.666	−0.484714	−0.242357	+	0.970187i	$0.577920\pi$
$60$	0	0
$61$	822.865	1.72717	0.863583	−	0.504207i	$-0.168215\pi$
$61$	822.865	1.72717	0.863583	+	0.504207i	$0.168215\pi$
$62$	366.450	0.750632
$63$	0	0
$64$	−733.082	−1.43180
$65$	58.7169	0.112045
$66$	0	0
$67$	−872.082	−1.59018	−0.795088	−	0.606495i	$-0.792575\pi$
$67$	−872.082	−1.59018	−0.795088	+	0.606495i	$0.792575\pi$
$68$	−1404.40	−2.50453
$69$	0	0
$70$	−160.267	−0.273651
$71$	100.299	0.167653	0.0838263	−	0.996480i	$-0.473286\pi$
$71$	100.299	0.167653	0.0838263	+	0.996480i	$0.473286\pi$
$72$	0	0
$73$	−165.634	−0.265562	−0.132781	−	0.991145i	$-0.542391\pi$
$73$	−165.634	−0.265562	−0.132781	+	0.991145i	$0.542391\pi$
$74$	−259.684	−0.407941
$75$	0	0
$76$	455.983	0.688221
$77$	−500.633	−0.740940
$78$	0	0
$79$	−545.266	−0.776547	−0.388273	−	0.921544i	$-0.626928\pi$
$79$	−545.266	−0.776547	−0.388273	+	0.921544i	$0.626928\pi$
$80$	−135.048	−0.188736
$81$	0	0
$82$	2141.88	2.88453
$83$	454.534	0.601103	0.300552	−	0.953766i	$-0.402829\pi$
$83$	454.534	0.601103	0.300552	+	0.953766i	$0.402829\pi$
$84$	0	0
$85$	437.968	0.558874
$86$	537.864	0.674412
$87$	0	0
$88$	−2056.61	−2.49132
$89$	230.915	0.275022	0.137511	−	0.990500i	$-0.456090\pi$
$89$	230.915	0.275022	0.137511	+	0.990500i	$0.456090\pi$
$90$	0	0
$91$	97.2831	0.112066
$92$	−2659.13	−3.01341
$93$	0	0
$94$	−200.415	−0.219907
$95$	−142.200	−0.153573
$96$	0	0
$97$	−1089.16	−1.14008	−0.570041	−	0.821616i	$-0.693073\pi$
$97$	−1089.16	−1.14008	−0.570041	+	0.821616i	$0.693073\pi$
$98$	1360.86	1.40273
$99$	0	0
$100$	−1514.95	−1.51495
$101$	77.2336	0.0760894	0.0380447	−	0.999276i	$-0.487887\pi$
$101$	77.2336	0.0760894	0.0380447	+	0.999276i	$0.487887\pi$
$102$	0	0
$103$	1351.36	1.29275	0.646377	−	0.763018i	$-0.276283\pi$
$103$	1351.36	1.29275	0.646377	+	0.763018i	$0.276283\pi$
$104$	399.642	0.376808
$105$	0	0
$106$	−2517.81	−2.30709
$107$	−1133.67	−1.02426	−0.512129	−	0.858908i	$-0.671143\pi$
$107$	−1133.67	−1.02426	−0.512129	+	0.858908i	$0.671143\pi$
$108$	0	0
$109$	1017.66	0.894262	0.447131	−	0.894469i	$-0.352446\pi$
$109$	1017.66	0.894262	0.447131	+	0.894469i	$0.352446\pi$
$110$	1432.77	1.24190
$111$	0	0
$112$	−223.750	−0.188772
$113$	−1570.06	−1.30707	−0.653536	−	0.756895i	$-0.726715\pi$
$113$	−1570.06	−1.30707	−0.653536	+	0.756895i	$0.726715\pi$
$114$	0	0
$115$	829.261	0.672426
$116$	−1625.03	−1.30069
$117$	0	0
$118$	1041.58	0.812588
$119$	725.632	0.558979
$120$	0	0
$121$	3144.60	2.36258
$122$	−3901.75	−2.89547
$123$	0	0
$124$	−1119.32	−0.810625
$125$	1037.03	0.742037
$126$	0	0
$127$	−1248.16	−0.872099	−0.436050	−	0.899923i	$-0.643623\pi$
$127$	−1248.16	−0.872099	−0.436050	+	0.899923i	$0.643623\pi$
$128$	2642.76	1.82491
$129$	0	0
$130$	−278.415	−0.187836
$131$	1274.80	0.850227	0.425113	−	0.905140i	$-0.360234\pi$
$131$	1274.80	0.850227	0.425113	+	0.905140i	$0.360234\pi$
$132$	0	0
$133$	−235.600	−0.153602
$134$	4135.11	2.66582
$135$	0	0
$136$	2980.91	1.87950
$137$	−874.915	−0.545613	−0.272807	−	0.962069i	$-0.587952\pi$
$137$	−874.915	−0.545613	−0.272807	+	0.962069i	$0.587952\pi$
$138$	0	0
$139$	310.334	0.189368	0.0946840	−	0.995507i	$-0.469816\pi$
$139$	310.334	0.189368	0.0946840	+	0.995507i	$0.469816\pi$
$140$	489.533	0.295522
$141$	0	0
$142$	−475.585	−0.281058
$143$	−869.699	−0.508586
$144$	0	0
$145$	506.773	0.290243
$146$	785.380	0.445195
$147$	0	0
$148$	793.199	0.440544
$149$	−5.08064	−0.00279344	−0.00139672	−	0.999999i	$-0.500445\pi$
$149$	−5.08064	−0.00279344	−0.00139672	+	0.999999i	$0.500445\pi$
$150$	0	0
$151$	6.54894	0.00352944	0.00176472	−	0.999998i	$-0.499438\pi$
$151$	6.54894	0.00352944	0.00176472	+	0.999998i	$0.499438\pi$
$152$	−967.849	−0.516467
$153$	0	0
$154$	2373.83	1.24213
$155$	349.063	0.180887
$156$	0	0
$157$	−2297.60	−1.16795	−0.583975	−	0.811772i	$-0.698503\pi$
$157$	−2297.60	−1.16795	−0.583975	+	0.811772i	$0.698503\pi$
$158$	2585.46	1.30183
$159$	0	0
$160$	−470.450	−0.232452
$161$	1373.93	0.672553
$162$	0	0
$163$	−1085.49	−0.521606	−0.260803	−	0.965392i	$-0.583987\pi$
$163$	−1085.49	−0.521606	−0.260803	+	0.965392i	$0.583987\pi$
$164$	−6542.34	−3.11507
$165$	0	0
$166$	−2155.24	−1.00771
$167$	109.066	0.0505374	0.0252687	−	0.999681i	$-0.491956\pi$
$167$	109.066	0.0505374	0.0252687	+	0.999681i	$0.491956\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−2076.69	−0.936912
$171$	0	0
$172$	−1642.90	−0.728313
$173$	1889.17	0.830236	0.415118	−	0.909768i	$-0.363740\pi$
$173$	1889.17	0.830236	0.415118	+	0.909768i	$0.363740\pi$
$174$	0	0
$175$	782.751	0.338117
$176$	2000.30	0.856694
$177$	0	0
$178$	−1094.92	−0.461054
$179$	−3427.86	−1.43134	−0.715672	−	0.698437i	$-0.753879\pi$
$179$	−3427.86	−1.43134	−0.715672	+	0.698437i	$0.753879\pi$
$180$	0	0
$181$	208.403	0.0855826	0.0427913	−	0.999084i	$-0.486375\pi$
$181$	208.403	0.0855826	0.0427913	+	0.999084i	$0.486375\pi$
$182$	−461.283	−0.187871
$183$	0	0
$184$	5644.15	2.26137
$185$	−247.363	−0.0983052
$186$	0	0
$187$	−6487.06	−2.53679
$188$	612.166	0.237483
$189$	0	0
$190$	674.265	0.257454
$191$	−957.735	−0.362824	−0.181412	−	0.983407i	$-0.558067\pi$
$191$	−957.735	−0.362824	−0.181412	+	0.983407i	$0.558067\pi$
$192$	0	0
$193$	−512.730	−0.191228	−0.0956142	−	0.995418i	$-0.530482\pi$
$193$	−512.730	−0.191228	−0.0956142	+	0.995418i	$0.530482\pi$
$194$	5164.45	1.91127
$195$	0	0
$196$	−4156.71	−1.51484
$197$	3870.35	1.39975	0.699876	−	0.714265i	$-0.253239\pi$
$197$	3870.35	1.39975	0.699876	+	0.714265i	$0.253239\pi$
$198$	0	0
$199$	2305.83	0.821388	0.410694	−	0.911773i	$-0.365286\pi$
$199$	2305.83	0.821388	0.410694	+	0.911773i	$0.365286\pi$
$200$	3215.56	1.13687
$201$	0	0
$202$	−366.215	−0.127558
$203$	839.630	0.290298
$204$	0	0
$205$	2040.26	0.695111
$206$	−6407.70	−2.16721
$207$	0	0
$208$	−388.699	−0.129574
$209$	2106.23	0.697086
$210$	0	0
$211$	−3672.40	−1.19819	−0.599096	−	0.800677i	$-0.704473\pi$
$211$	−3672.40	−1.19819	−0.599096	+	0.800677i	$0.704473\pi$
$212$	7690.62	2.49148
$213$	0	0
$214$	5375.46	1.71710
$215$	512.345	0.162519
$216$	0	0
$217$	578.334	0.180921
$218$	−4825.41	−1.49917
$219$	0	0
$220$	−4376.36	−1.34116
$221$	1260.57	0.383687
$222$	0	0
$223$	5087.05	1.52760	0.763798	−	0.645455i	$-0.223332\pi$
$223$	5087.05	1.52760	0.763798	+	0.645455i	$0.223332\pi$
$224$	−779.449	−0.232496
$225$	0	0
$226$	7444.71	2.19122
$227$	2625.83	0.767763	0.383882	−	0.923382i	$-0.374587\pi$
$227$	2625.83	0.767763	0.383882	+	0.923382i	$0.374587\pi$
$228$	0	0
$229$	1678.73	0.484425	0.242213	−	0.970223i	$-0.422127\pi$
$229$	1678.73	0.484425	0.242213	+	0.970223i	$0.422127\pi$
$230$	−3932.07	−1.12727
$231$	0	0
$232$	3449.22	0.976088
$233$	−648.506	−0.182339	−0.0911696	−	0.995835i	$-0.529061\pi$
$233$	−648.506	−0.182339	−0.0911696	+	0.995835i	$0.529061\pi$
$234$	0	0
$235$	−190.907	−0.0529931
$236$	−3181.50	−0.877533
$237$	0	0
$238$	−3440.70	−0.937089
$239$	−5219.69	−1.41269	−0.706347	−	0.707866i	$-0.749658\pi$
$239$	−5219.69	−1.41269	−0.706347	+	0.707866i	$0.749658\pi$
$240$	0	0
$241$	6103.56	1.63139	0.815695	−	0.578483i	$-0.196355\pi$
$241$	6103.56	1.63139	0.815695	+	0.578483i	$0.196355\pi$
$242$	−14910.6	−3.96070
$243$	0	0
$244$	11917.8	3.12689
$245$	1296.29	0.338028
$246$	0	0
$247$	−409.283	−0.105433
$248$	2375.81	0.608323
$249$	0	0
$250$	−4917.24	−1.24397
$251$	−6423.40	−1.61530	−0.807652	−	0.589660i	$-0.799262\pi$
$251$	−6423.40	−1.61530	−0.807652	+	0.589660i	$0.799262\pi$
$252$	0	0
$253$	−12282.8	−3.05222
$254$	5918.36	1.46201
$255$	0	0
$256$	−6666.39	−1.62754
$257$	1230.23	0.298597	0.149299	−	0.988792i	$-0.452298\pi$
$257$	1230.23	0.298597	0.149299	+	0.988792i	$0.452298\pi$
$258$	0	0
$259$	−409.834	−0.0983238
$260$	850.415	0.202848
$261$	0	0
$262$	−6044.66	−1.42534
$263$	514.992	0.120744	0.0603722	−	0.998176i	$-0.480771\pi$
$263$	514.992	0.120744	0.0603722	+	0.998176i	$0.480771\pi$
$264$	0	0
$265$	−2398.35	−0.555961
$266$	1117.13	0.257503
$267$	0	0
$268$	−12630.6	−2.87888
$269$	5132.60	1.16335	0.581673	−	0.813423i	$-0.302398\pi$
$269$	5132.60	1.16335	0.581673	+	0.813423i	$0.302398\pi$
$270$	0	0
$271$	−4300.00	−0.963862	−0.481931	−	0.876209i	$-0.660064\pi$
$271$	−4300.00	−0.963862	−0.481931	+	0.876209i	$0.660064\pi$
$272$	−2899.29	−0.646306
$273$	0	0
$274$	4148.55	0.914682
$275$	−6997.70	−1.53446
$276$	0	0
$277$	−1812.80	−0.393215	−0.196607	−	0.980482i	$-0.562992\pi$
$277$	−1812.80	−0.393215	−0.196607	+	0.980482i	$0.562992\pi$
$278$	−1471.50	−0.317462
$279$	0	0
$280$	−1039.06	−0.221771
$281$	4073.08	0.864696	0.432348	−	0.901707i	$-0.357685\pi$
$281$	4073.08	0.864696	0.432348	+	0.901707i	$0.357685\pi$
$282$	0	0
$283$	6346.29	1.33303	0.666516	−	0.745491i	$-0.267785\pi$
$283$	6346.29	1.33303	0.666516	+	0.745491i	$0.267785\pi$
$284$	1452.67	0.303520
$285$	0	0
$286$	4123.81	0.852609
$287$	3380.33	0.695243
$288$	0	0
$289$	4489.53	0.913806
$290$	−2402.94	−0.486572
$291$	0	0
$292$	−2398.93	−0.480777
$293$	8390.97	1.67306	0.836529	−	0.547923i	$-0.184581\pi$
$293$	8390.97	1.67306	0.836529	+	0.547923i	$0.184581\pi$
$294$	0	0
$295$	992.164	0.195817
$296$	−1683.61	−0.330601
$297$	0	0
$298$	24.0907	0.00468300
$299$	2386.79	0.461645
$300$	0	0
$301$	848.861	0.162550
$302$	−31.0528	−0.00591685
$303$	0	0
$304$	941.348	0.177599
$305$	−3716.62	−0.697748
$306$	0	0
$307$	4005.27	0.744603	0.372301	−	0.928112i	$-0.378569\pi$
$307$	4005.27	0.744603	0.372301	+	0.928112i	$0.378569\pi$
$308$	−7250.82	−1.34141
$309$	0	0
$310$	−1655.14	−0.303244
$311$	−5836.53	−1.06418	−0.532088	−	0.846689i	$-0.678593\pi$
$311$	−5836.53	−1.06418	−0.532088	+	0.846689i	$0.678593\pi$
$312$	0	0
$313$	1763.19	0.318407	0.159204	−	0.987246i	$-0.449107\pi$
$313$	1763.19	0.318407	0.159204	+	0.987246i	$0.449107\pi$
$314$	10894.4	1.95798
$315$	0	0
$316$	−7897.26	−1.40587
$317$	6106.35	1.08191	0.540957	−	0.841050i	$-0.318062\pi$
$317$	6106.35	1.08191	0.540957	+	0.841050i	$0.318062\pi$
$318$	0	0
$319$	−7506.18	−1.31745
$320$	3311.10	0.578425
$321$	0	0
$322$	−6514.72	−1.12749
$323$	−3052.83	−0.525895
$324$	0	0
$325$	1359.79	0.232086
$326$	5147.00	0.874436
$327$	0	0
$328$	13886.5	2.33766
$329$	−316.297	−0.0530031
$330$	0	0
$331$	7490.38	1.24383	0.621916	−	0.783084i	$-0.286354\pi$
$331$	7490.38	1.24383	0.621916	+	0.783084i	$0.286354\pi$
$332$	6583.16	1.08825
$333$	0	0
$334$	−517.152	−0.0847224
$335$	3938.92	0.642406
$336$	0	0
$337$	9462.46	1.52953	0.764767	−	0.644307i	$-0.222854\pi$
$337$	9462.46	1.52953	0.764767	+	0.644307i	$0.222854\pi$
$338$	−801.340	−0.128956
$339$	0	0
$340$	6343.22	1.01179
$341$	−5170.23	−0.821066
$342$	0	0
$343$	4714.49	0.742153
$344$	3487.14	0.546553
$345$	0	0
$346$	−8957.79	−1.39183
$347$	−11460.3	−1.77297	−0.886487	−	0.462753i	$-0.846862\pi$
$347$	−11460.3	−1.77297	−0.886487	+	0.462753i	$0.846862\pi$
$348$	0	0
$349$	−3673.96	−0.563503	−0.281751	−	0.959487i	$-0.590915\pi$
$349$	−3673.96	−0.563503	−0.281751	+	0.959487i	$0.590915\pi$
$350$	−3711.54	−0.566829
$351$	0	0
$352$	6968.17	1.05513
$353$	−3388.65	−0.510935	−0.255467	−	0.966818i	$-0.582229\pi$
$353$	−3388.65	−0.510935	−0.255467	+	0.966818i	$0.582229\pi$
$354$	0	0
$355$	−453.020	−0.0677290
$356$	3344.41	0.497903
$357$	0	0
$358$	16253.7	2.39955
$359$	9673.98	1.42221	0.711105	−	0.703086i	$-0.248195\pi$
$359$	9673.98	1.42221	0.711105	+	0.703086i	$0.248195\pi$
$360$	0	0
$361$	−5867.80	−0.855489
$362$	−988.174	−0.143473
$363$	0	0
$364$	1408.98	0.202887
$365$	748.117	0.107283
$366$	0	0
$367$	−8715.98	−1.23970	−0.619851	−	0.784720i	$-0.712807\pi$
$367$	−8715.98	−1.23970	−0.619851	+	0.784720i	$0.712807\pi$
$368$	−5489.61	−0.777624
$369$	0	0
$370$	1172.91	0.164802
$371$	−3973.63	−0.556067
$372$	0	0
$373$	4667.99	0.647987	0.323994	−	0.946059i	$-0.394974\pi$
$373$	4667.99	0.647987	0.323994	+	0.946059i	$0.394974\pi$
$374$	30759.4	4.25276
$375$	0	0
$376$	−1299.36	−0.178216
$377$	1458.60	0.199262
$378$	0	0
$379$	10862.7	1.47225	0.736123	−	0.676848i	$-0.236655\pi$
$379$	10862.7	1.47225	0.736123	+	0.676848i	$0.236655\pi$
$380$	−2059.53	−0.278031
$381$	0	0
$382$	4541.25	0.608248
$383$	−10054.2	−1.34137	−0.670686	−	0.741742i	$-0.734000\pi$
$383$	−10054.2	−1.34137	−0.670686	+	0.741742i	$0.734000\pi$
$384$	0	0
$385$	2261.20	0.299329
$386$	2431.19	0.320581
$387$	0	0
$388$	−15774.7	−2.06402
$389$	−6418.50	−0.836584	−0.418292	−	0.908313i	$-0.637371\pi$
$389$	−6418.50	−0.836584	−0.418292	+	0.908313i	$0.637371\pi$
$390$	0	0
$391$	17803.0	2.30265
$392$	8822.86	1.13679
$393$	0	0
$394$	−18351.9	−2.34658
$395$	2462.79	0.313713
$396$	0	0
$397$	−12019.9	−1.51955	−0.759775	−	0.650186i	$-0.774691\pi$
$397$	−12019.9	−1.51955	−0.759775	+	0.650186i	$0.774691\pi$
$398$	−10933.5	−1.37700
$399$	0	0
$400$	−3127.52	−0.390939
$401$	−3599.80	−0.448293	−0.224147	−	0.974555i	$-0.571959\pi$
$401$	−3599.80	−0.448293	−0.224147	+	0.974555i	$0.571959\pi$
$402$	0	0
$403$	1004.68	0.124185
$404$	1118.60	0.137753
$405$	0	0
$406$	−3981.24	−0.486664
$407$	3663.87	0.446219
$408$	0	0
$409$	−48.0968	−0.00581475	−0.00290737	−	0.999996i	$-0.500925\pi$
$409$	−48.0968	−0.00581475	−0.00290737	+	0.999996i	$0.500925\pi$
$410$	−9674.20	−1.16530
$411$	0	0
$412$	19572.2	2.34042
$413$	1643.83	0.195854
$414$	0	0
$415$	−2052.99	−0.242837
$416$	−1354.06	−0.159587
$417$	0	0
$418$	−9987.02	−1.16862
$419$	−723.462	−0.0843518	−0.0421759	−	0.999110i	$-0.513429\pi$
$419$	−723.462	−0.0843518	−0.0421759	+	0.999110i	$0.513429\pi$
$420$	0	0
$421$	−14845.5	−1.71859	−0.859295	−	0.511481i	$-0.829097\pi$
$421$	−14845.5	−1.71859	−0.859295	+	0.511481i	$0.829097\pi$
$422$	17413.3	2.00868
$423$	0	0
$424$	−16323.8	−1.86970
$425$	10142.7	1.15763
$426$	0	0
$427$	−6157.76	−0.697880
$428$	−16419.2	−1.85433
$429$	0	0
$430$	−2429.36	−0.272452
$431$	−1103.11	−0.123283	−0.0616417	−	0.998098i	$-0.519634\pi$
$431$	−1103.11	−0.123283	−0.0616417	+	0.998098i	$0.519634\pi$
$432$	0	0
$433$	8893.53	0.987057	0.493528	−	0.869730i	$-0.335707\pi$
$433$	8893.53	0.987057	0.493528	+	0.869730i	$0.335707\pi$
$434$	−2742.26	−0.303301
$435$	0	0
$436$	14739.1	1.61898
$437$	−5780.32	−0.632747
$438$	0	0
$439$	−10901.7	−1.18521	−0.592607	−	0.805492i	$-0.701901\pi$
$439$	−10901.7	−1.18521	−0.592607	+	0.805492i	$0.701901\pi$
$440$	9289.08	1.00645
$441$	0	0
$442$	−5977.17	−0.643224
$443$	3781.37	0.405550	0.202775	−	0.979225i	$-0.435004\pi$
$443$	3781.37	0.405550	0.202775	+	0.979225i	$0.435004\pi$
$444$	0	0
$445$	−1042.97	−0.111105
$446$	−24121.0	−2.56091
$447$	0	0
$448$	5485.88	0.578535
$449$	−106.834	−0.0112289	−0.00561447	−	0.999984i	$-0.501787\pi$
$449$	−106.834	−0.0112289	−0.00561447	+	0.999984i	$0.501787\pi$
$450$	0	0
$451$	−30219.7	−3.15519
$452$	−22739.7	−2.36634
$453$	0	0
$454$	−12450.8	−1.28710
$455$	−439.397	−0.0452731
$456$	0	0
$457$	−1237.64	−0.126684	−0.0633419	−	0.997992i	$-0.520176\pi$
$457$	−1237.64	−0.126684	−0.0633419	+	0.997992i	$0.520176\pi$
$458$	−7959.95	−0.812105
$459$	0	0
$460$	12010.5	1.21737
$461$	−8790.90	−0.888141	−0.444071	−	0.895992i	$-0.646466\pi$
$461$	−8790.90	−0.888141	−0.444071	+	0.895992i	$0.646466\pi$
$462$	0	0
$463$	−3861.55	−0.387606	−0.193803	−	0.981040i	$-0.562082\pi$
$463$	−3861.55	−0.387606	−0.193803	+	0.981040i	$0.562082\pi$
$464$	−3354.77	−0.335650
$465$	0	0
$466$	3074.99	0.305679
$467$	8991.38	0.890945	0.445473	−	0.895296i	$-0.353036\pi$
$467$	8991.38	0.890945	0.445473	+	0.895296i	$0.353036\pi$
$468$	0	0
$469$	6526.06	0.642528
$470$	905.214	0.0888391
$471$	0	0
$472$	6752.91	0.658533
$473$	−7588.71	−0.737694
$474$	0	0
$475$	−3293.14	−0.318105
$476$	10509.6	1.01198
$477$	0	0
$478$	24750.0	2.36828
$479$	−4179.82	−0.398707	−0.199354	−	0.979928i	$-0.563884\pi$
$479$	−4179.82	−0.398707	−0.199354	+	0.979928i	$0.563884\pi$
$480$	0	0
$481$	−711.963	−0.0674901
$482$	−28941.0	−2.73491
$483$	0	0
$484$	45544.2	4.27725
$485$	4919.41	0.460575
$486$	0	0
$487$	−18443.8	−1.71616	−0.858078	−	0.513519i	$-0.828342\pi$
$487$	−18443.8	−1.71616	−0.858078	+	0.513519i	$0.828342\pi$
$488$	−25296.2	−2.34653
$489$	0	0
$490$	−6146.56	−0.566680
$491$	−8093.26	−0.743877	−0.371939	−	0.928257i	$-0.621307\pi$
$491$	−8093.26	−0.743877	−0.371939	+	0.928257i	$0.621307\pi$
$492$	0	0
$493$	10879.7	0.993907
$494$	1940.68	0.176752
$495$	0	0
$496$	−2310.76	−0.209185
$497$	−750.571	−0.0677418
$498$	0	0
$499$	−10941.6	−0.981591	−0.490796	−	0.871275i	$-0.663294\pi$
$499$	−10941.6	−0.981591	−0.490796	+	0.871275i	$0.663294\pi$
$500$	15019.6	1.34340
$501$	0	0
$502$	30457.5	2.70794
$503$	9260.11	0.820851	0.410425	−	0.911894i	$-0.365380\pi$
$503$	9260.11	0.820851	0.410425	+	0.911894i	$0.365380\pi$
$504$	0	0
$505$	−348.840	−0.0307389
$506$	58240.8	5.11684
$507$	0	0
$508$	−18077.5	−1.57886
$509$	9996.40	0.870497	0.435248	−	0.900310i	$-0.356661\pi$
$509$	9996.40	0.870497	0.435248	+	0.900310i	$0.356661\pi$
$510$	0	0
$511$	1239.49	0.107303
$512$	10467.7	0.903538
$513$	0	0
$514$	−5833.32	−0.500577
$515$	−6103.68	−0.522253
$516$	0	0
$517$	2827.66	0.240542
$518$	1943.29	0.164833
$519$	0	0
$520$	−1805.06	−0.152225
$521$	−11427.9	−0.960973	−0.480486	−	0.877002i	$-0.659540\pi$
$521$	−11427.9	−0.960973	−0.480486	+	0.877002i	$0.659540\pi$
$522$	0	0
$523$	−4810.47	−0.402193	−0.201097	−	0.979571i	$-0.564451\pi$
$523$	−4810.47	−0.402193	−0.201097	+	0.979571i	$0.564451\pi$
$524$	18463.3	1.53926
$525$	0	0
$526$	−2441.92	−0.202419
$527$	7493.88	0.619428
$528$	0	0
$529$	21541.8	1.77051
$530$	11372.2	0.932030
$531$	0	0
$532$	−3412.26	−0.278083
$533$	5872.31	0.477219
$534$	0	0
$535$	5120.41	0.413785
$536$	26809.2	2.16042
$537$	0	0
$538$	−24337.0	−1.95027
$539$	−19200.3	−1.53435
$540$	0	0
$541$	2411.99	0.191681	0.0958406	−	0.995397i	$-0.469446\pi$
$541$	2411.99	0.191681	0.0958406	+	0.995397i	$0.469446\pi$
$542$	20389.1	1.61585
$543$	0	0
$544$	−10099.9	−0.796008
$545$	−4596.47	−0.361268
$546$	0	0
$547$	−4396.34	−0.343646	−0.171823	−	0.985128i	$-0.554966\pi$
$547$	−4396.34	−0.343646	−0.171823	+	0.985128i	$0.554966\pi$
$548$	−12671.7	−0.987786
$549$	0	0
$550$	33180.7	2.57242
$551$	−3532.43	−0.273116
$552$	0	0
$553$	4080.40	0.313772
$554$	8595.67	0.659197
$555$	0	0
$556$	4494.66	0.342835
$557$	17488.0	1.33032	0.665160	−	0.746701i	$-0.268363\pi$
$557$	17488.0	1.33032	0.665160	+	0.746701i	$0.268363\pi$
$558$	0	0
$559$	1474.64	0.111575
$560$	1010.61	0.0762608
$561$	0	0
$562$	−19313.2	−1.44960
$563$	6881.77	0.515154	0.257577	−	0.966258i	$-0.417076\pi$
$563$	6881.77	0.515154	0.257577	+	0.966258i	$0.417076\pi$
$564$	0	0
$565$	7091.49	0.528037
$566$	−30092.0	−2.23473
$567$	0	0
$568$	−3083.36	−0.227773
$569$	−14733.5	−1.08552	−0.542758	−	0.839889i	$-0.682620\pi$
$569$	−14733.5	−1.08552	−0.542758	+	0.839889i	$0.682620\pi$
$570$	0	0
$571$	4488.51	0.328964	0.164482	−	0.986380i	$-0.447405\pi$
$571$	4488.51	0.328964	0.164482	+	0.986380i	$0.447405\pi$
$572$	−12596.1	−0.920752
$573$	0	0
$574$	−16028.4	−1.16553
$575$	19204.4	1.39284
$576$	0	0
$577$	−10552.2	−0.761338	−0.380669	−	0.924711i	$-0.624306\pi$
$577$	−10552.2	−0.761338	−0.380669	+	0.924711i	$0.624306\pi$
$578$	−21287.8	−1.53193
$579$	0	0
$580$	7339.75	0.525460
$581$	−3401.42	−0.242882
$582$	0	0
$583$	35523.8	2.52357
$584$	5091.86	0.360793
$585$	0	0
$586$	−39787.1	−2.80476
$587$	1637.20	0.115118	0.0575591	−	0.998342i	$-0.481668\pi$
$587$	1637.20	0.115118	0.0575591	+	0.998342i	$0.481668\pi$
$588$	0	0
$589$	−2433.13	−0.170213
$590$	−4704.50	−0.328273
$591$	0	0
$592$	1637.51	0.113685
$593$	−14024.0	−0.971155	−0.485577	−	0.874194i	$-0.661391\pi$
$593$	−14024.0	−0.971155	−0.485577	+	0.874194i	$0.661391\pi$
$594$	0	0
$595$	−3277.45	−0.225819
$596$	−73.5845	−0.00505728
$597$	0	0
$598$	−11317.4	−0.773915
$599$	−365.860	−0.0249560	−0.0124780	−	0.999922i	$-0.503972\pi$
$599$	−365.860	−0.0249560	−0.0124780	+	0.999922i	$0.503972\pi$
$600$	0	0
$601$	−10128.9	−0.687465	−0.343733	−	0.939068i	$-0.611691\pi$
$601$	−10128.9	−0.687465	−0.343733	+	0.939068i	$0.611691\pi$
$602$	−4025.01	−0.272503
$603$	0	0
$604$	94.8504	0.00638975
$605$	−14203.1	−0.954446
$606$	0	0
$607$	2324.97	0.155465	0.0777327	−	0.996974i	$-0.475232\pi$
$607$	2324.97	0.155465	0.0777327	+	0.996974i	$0.475232\pi$
$608$	3279.25	0.218735
$609$	0	0
$610$	17623.0	1.16973
$611$	−549.471	−0.0363817
$612$	0	0
$613$	633.133	0.0417162	0.0208581	−	0.999782i	$-0.493360\pi$
$613$	633.133	0.0417162	0.0208581	+	0.999782i	$0.493360\pi$
$614$	−18991.6	−1.24827
$615$	0	0
$616$	15390.3	1.00664
$617$	2981.85	0.194562	0.0972810	−	0.995257i	$-0.468985\pi$
$617$	2981.85	0.194562	0.0972810	+	0.995257i	$0.468985\pi$
$618$	0	0
$619$	15158.6	0.984292	0.492146	−	0.870513i	$-0.336213\pi$
$619$	15158.6	0.984292	0.492146	+	0.870513i	$0.336213\pi$
$620$	5055.60	0.327480
$621$	0	0
$622$	27674.8	1.78402
$623$	−1728.01	−0.111126
$624$	0	0
$625$	8391.01	0.537025
$626$	−8360.45	−0.533787
$627$	0	0
$628$	−33276.8	−2.11447
$629$	−5310.51	−0.336636
$630$	0	0
$631$	5562.30	0.350922	0.175461	−	0.984486i	$-0.443858\pi$
$631$	5562.30	0.350922	0.175461	+	0.984486i	$0.443858\pi$
$632$	16762.4	1.05502
$633$	0	0
$634$	−28954.2	−1.81375
$635$	5637.56	0.352315
$636$	0	0
$637$	3731.00	0.232068
$638$	35591.7	2.20861
$639$	0	0
$640$	−11936.5	−0.737237
$641$	24140.7	1.48752	0.743761	−	0.668446i	$-0.233040\pi$
$641$	24140.7	1.48752	0.743761	+	0.668446i	$0.233040\pi$
$642$	0	0
$643$	−1749.69	−0.107311	−0.0536555	−	0.998560i	$-0.517087\pi$
$643$	−1749.69	−0.107311	−0.0536555	+	0.998560i	$0.517087\pi$
$644$	19899.1	1.21760
$645$	0	0
$646$	14475.5	0.881626
$647$	−1489.51	−0.0905083	−0.0452542	−	0.998976i	$-0.514410\pi$
$647$	−1489.51	−0.0905083	−0.0452542	+	0.998976i	$0.514410\pi$
$648$	0	0
$649$	−14695.7	−0.888836
$650$	−6447.68	−0.389075
$651$	0	0
$652$	−15721.4	−0.944323
$653$	−10668.2	−0.639327	−0.319663	−	0.947531i	$-0.603570\pi$
$653$	−10668.2	−0.639327	−0.319663	+	0.947531i	$0.603570\pi$
$654$	0	0
$655$	−5757.86	−0.343478
$656$	−13506.3	−0.803858
$657$	0	0
$658$	1499.77	0.0888559
$659$	13933.1	0.823608	0.411804	−	0.911272i	$-0.364899\pi$
$659$	13933.1	0.823608	0.411804	+	0.911272i	$0.364899\pi$
$660$	0	0
$661$	−2349.69	−0.138264	−0.0691320	−	0.997608i	$-0.522023\pi$
$661$	−2349.69	−0.138264	−0.0691320	+	0.997608i	$0.522023\pi$
$662$	−35516.8	−2.08520
$663$	0	0
$664$	−13973.1	−0.816660
$665$	1064.13	0.0620529
$666$	0	0
$667$	20599.9	1.19585
$668$	1579.63	0.0914937
$669$	0	0
$670$	−18677.0	−1.07695
$671$	55049.6	3.16716
$672$	0	0
$673$	−32421.0	−1.85697	−0.928483	−	0.371374i	$-0.878887\pi$
$673$	−32421.0	−1.85697	−0.928483	+	0.371374i	$0.878887\pi$
$674$	−44867.7	−2.56415
$675$	0	0
$676$	2447.68	0.139263
$677$	1071.74	0.0608421	0.0304211	−	0.999537i	$-0.490315\pi$
$677$	1071.74	0.0608421	0.0304211	+	0.999537i	$0.490315\pi$
$678$	0	0
$679$	8150.56	0.460663
$680$	−13463.9	−0.759287
$681$	0	0
$682$	24515.5	1.37646
$683$	305.487	0.0171144	0.00855721	−	0.999963i	$-0.497276\pi$
$683$	305.487	0.0171144	0.00855721	+	0.999963i	$0.497276\pi$
$684$	0	0
$685$	3951.72	0.220419
$686$	−22354.5	−1.24417
$687$	0	0
$688$	−3391.66	−0.187944
$689$	−6902.99	−0.381688
$690$	0	0
$691$	2180.81	0.120061	0.0600303	−	0.998197i	$-0.480880\pi$
$691$	2180.81	0.120061	0.0600303	+	0.998197i	$0.480880\pi$
$692$	27361.4	1.50307
$693$	0	0
$694$	54340.9	2.97227
$695$	−1401.68	−0.0765018
$696$	0	0
$697$	43801.4	2.38034
$698$	17420.6	0.944672
$699$	0	0
$700$	11336.8	0.612132
$701$	15168.3	0.817259	0.408629	−	0.912700i	$-0.366007\pi$
$701$	15168.3	0.817259	0.408629	+	0.912700i	$0.366007\pi$
$702$	0	0
$703$	1724.23	0.0925043
$704$	−49043.1	−2.62554
$705$	0	0
$706$	16067.8	0.856545
$707$	−577.963	−0.0307448
$708$	0	0
$709$	−8988.17	−0.476104	−0.238052	−	0.971252i	$-0.576509\pi$
$709$	−8988.17	−0.476104	−0.238052	+	0.971252i	$0.576509\pi$
$710$	2148.07	0.113543
$711$	0	0
$712$	−7098.71	−0.373645
$713$	14189.1	0.745284
$714$	0	0
$715$	3928.15	0.205461
$716$	−49646.8	−2.59132
$717$	0	0
$718$	−45870.7	−2.38423
$719$	−8448.18	−0.438198	−0.219099	−	0.975703i	$-0.570312\pi$
$719$	−8448.18	−0.438198	−0.219099	+	0.975703i	$0.570312\pi$
$720$	0	0
$721$	−10112.7	−0.522352
$722$	27823.1	1.43417
$723$	0	0
$724$	3018.36	0.154940
$725$	11736.1	0.601197
$726$	0	0
$727$	−8624.18	−0.439963	−0.219982	−	0.975504i	$-0.570600\pi$
$727$	−8624.18	−0.439963	−0.219982	+	0.975504i	$0.570600\pi$
$728$	−2990.64	−0.152254
$729$	0	0
$730$	−3547.31	−0.179852
$731$	10999.3	0.556530
$732$	0	0
$733$	31124.2	1.56835	0.784174	−	0.620541i	$-0.213087\pi$
$733$	31124.2	1.56835	0.784174	+	0.620541i	$0.213087\pi$
$734$	41328.2	2.07827
$735$	0	0
$736$	−19123.4	−0.957742
$737$	−58342.2	−2.91596
$738$	0	0
$739$	−17671.1	−0.879626	−0.439813	−	0.898089i	$-0.644955\pi$
$739$	−17671.1	−0.879626	−0.439813	+	0.898089i	$0.644955\pi$
$740$	−3582.63	−0.177973
$741$	0	0
$742$	18841.6	0.932206
$743$	21331.1	1.05325	0.526623	−	0.850099i	$-0.323458\pi$
$743$	21331.1	1.05325	0.526623	+	0.850099i	$0.323458\pi$
$744$	0	0
$745$	22.9477	0.00112851
$746$	−22134.0	−1.08630
$747$	0	0
$748$	−93954.1	−4.59265
$749$	8483.58	0.413863
$750$	0	0
$751$	11712.9	0.569122	0.284561	−	0.958658i	$-0.408152\pi$
$751$	11712.9	0.569122	0.284561	+	0.958658i	$0.408152\pi$
$752$	1263.78	0.0612835
$753$	0	0
$754$	−6916.20	−0.334049
$755$	−29.5795	−0.00142584
$756$	0	0
$757$	−16610.9	−0.797537	−0.398768	−	0.917052i	$-0.630562\pi$
$757$	−16610.9	−0.797537	−0.398768	+	0.917052i	$0.630562\pi$
$758$	−51507.4	−2.46812
$759$	0	0
$760$	4371.47	0.208645
$761$	29365.5	1.39882	0.699408	−	0.714723i	$-0.253447\pi$
$761$	29365.5	1.39882	0.699408	+	0.714723i	$0.253447\pi$
$762$	0	0
$763$	−7615.50	−0.361336
$764$	−13871.2	−0.656861
$765$	0	0
$766$	47673.5	2.24871
$767$	2855.66	0.134435
$768$	0	0
$769$	−28599.9	−1.34114	−0.670572	−	0.741844i	$-0.733951\pi$
$769$	−28599.9	−1.34114	−0.670572	+	0.741844i	$0.733951\pi$
$770$	−10721.8	−0.501803
$771$	0	0
$772$	−7426.03	−0.346203
$773$	−13491.8	−0.627772	−0.313886	−	0.949461i	$-0.601631\pi$
$773$	−13491.8	−0.627772	−0.313886	+	0.949461i	$0.601631\pi$
$774$	0	0
$775$	8083.78	0.374681
$776$	33482.7	1.54892
$777$	0	0
$778$	30434.3	1.40247
$779$	−14221.5	−0.654093
$780$	0	0
$781$	6710.01	0.307430
$782$	−84415.9	−3.86024
$783$	0	0
$784$	−8581.27	−0.390911
$785$	10377.5	0.471834
$786$	0	0
$787$	25876.7	1.17205	0.586025	−	0.810293i	$-0.300692\pi$
$787$	25876.7	1.17205	0.586025	+	0.810293i	$0.300692\pi$
$788$	56055.5	2.53413
$789$	0	0
$790$	−11677.7	−0.525918
$791$	11749.3	0.528137
$792$	0	0
$793$	−10697.3	−0.479030
$794$	56994.2	2.54742
$795$	0	0
$796$	33396.1	1.48705
$797$	−21936.4	−0.974938	−0.487469	−	0.873140i	$-0.662080\pi$
$797$	−21936.4	−0.974938	−0.487469	+	0.873140i	$0.662080\pi$
$798$	0	0
$799$	−4098.49	−0.181469
$800$	−10894.9	−0.481491
$801$	0	0
$802$	17069.0	0.751531
$803$	−11080.9	−0.486969
$804$	0	0
$805$	−6205.62	−0.271701
$806$	−4763.85	−0.208188
$807$	0	0
$808$	−2374.29	−0.103375
$809$	−5583.23	−0.242640	−0.121320	−	0.992613i	$-0.538713\pi$
$809$	−5583.23	−0.242640	−0.121320	+	0.992613i	$0.538713\pi$
$810$	0	0
$811$	12925.4	0.559647	0.279823	−	0.960052i	$-0.409724\pi$
$811$	12925.4	0.559647	0.279823	+	0.960052i	$0.409724\pi$
$812$	12160.6	0.525559
$813$	0	0
$814$	−17372.8	−0.748054
$815$	4902.80	0.210721
$816$	0	0
$817$	−3571.27	−0.152929
$818$	228.058	0.00974801
$819$	0	0
$820$	29549.7	1.25844
$821$	10153.2	0.431608	0.215804	−	0.976437i	$-0.430763\pi$
$821$	10153.2	0.431608	0.215804	+	0.976437i	$0.430763\pi$
$822$	0	0
$823$	3282.93	0.139047	0.0695235	−	0.997580i	$-0.477852\pi$
$823$	3282.93	0.139047	0.0695235	+	0.997580i	$0.477852\pi$
$824$	−41543.1	−1.75634
$825$	0	0
$826$	−7794.49	−0.328335
$827$	17689.3	0.743795	0.371897	−	0.928274i	$-0.378707\pi$
$827$	17689.3	0.743795	0.371897	+	0.928274i	$0.378707\pi$
$828$	0	0
$829$	38181.5	1.59964	0.799818	−	0.600243i	$-0.204930\pi$
$829$	38181.5	1.59964	0.799818	+	0.600243i	$0.204930\pi$
$830$	9734.56	0.407098
$831$	0	0
$832$	9530.06	0.397110
$833$	27829.4	1.15754
$834$	0	0
$835$	−492.615	−0.0204163
$836$	30505.2	1.26201
$837$	0	0
$838$	3430.41	0.141410
$839$	−43895.2	−1.80623	−0.903117	−	0.429395i	$-0.858727\pi$
$839$	−43895.2	−1.80623	−0.903117	+	0.429395i	$0.858727\pi$
$840$	0	0
$841$	−11800.1	−0.483829
$842$	70392.4	2.88109
$843$	0	0
$844$	−53188.5	−2.16922
$845$	−763.320	−0.0310757
$846$	0	0
$847$	−23532.0	−0.954627
$848$	15876.8	0.642938
$849$	0	0
$850$	−48093.0	−1.94068
$851$	−10055.1	−0.405034
$852$	0	0
$853$	−19955.2	−0.800998	−0.400499	−	0.916297i	$-0.631163\pi$
$853$	−19955.2	−0.800998	−0.400499	+	0.916297i	$0.631163\pi$
$854$	29198.0	1.16995
$855$	0	0
$856$	34850.8	1.39156
$857$	−26030.4	−1.03755	−0.518776	−	0.854910i	$-0.673612\pi$
$857$	−26030.4	−1.03755	−0.518776	+	0.854910i	$0.673612\pi$
$858$	0	0
$859$	−45617.4	−1.81193	−0.905964	−	0.423354i	$-0.860853\pi$
$859$	−45617.4	−1.81193	−0.905964	+	0.423354i	$0.860853\pi$
$860$	7420.45	0.294227
$861$	0	0
$862$	5230.59	0.206676
$863$	2010.93	0.0793195	0.0396597	−	0.999213i	$-0.487373\pi$
$863$	2010.93	0.0793195	0.0396597	+	0.999213i	$0.487373\pi$
$864$	0	0
$865$	−8532.78	−0.335403
$866$	−42170.1	−1.65473
$867$	0	0
$868$	8376.19	0.327542
$869$	−36478.2	−1.42398
$870$	0	0
$871$	11337.1	0.441035
$872$	−31284.7	−1.21495
$873$	0	0
$874$	27408.3	1.06076
$875$	−7760.41	−0.299828
$876$	0	0
$877$	−36767.3	−1.41567	−0.707836	−	0.706376i	$-0.750329\pi$
$877$	−36767.3	−1.41567	−0.707836	+	0.706376i	$0.750329\pi$
$878$	51692.0	1.98693
$879$	0	0
$880$	−9034.72	−0.346091
$881$	−35401.2	−1.35380	−0.676899	−	0.736076i	$-0.736677\pi$
$881$	−35401.2	−1.35380	−0.676899	+	0.736076i	$0.736677\pi$
$882$	0	0
$883$	−11928.0	−0.454596	−0.227298	−	0.973825i	$-0.572989\pi$
$883$	−11928.0	−0.454596	−0.227298	+	0.973825i	$0.572989\pi$
$884$	18257.2	0.694633
$885$	0	0
$886$	−17930.0	−0.679875
$887$	32939.3	1.24689	0.623447	−	0.781866i	$-0.285732\pi$
$887$	32939.3	1.24689	0.623447	+	0.781866i	$0.285732\pi$
$888$	0	0
$889$	9340.40	0.352381
$890$	4945.41	0.186259
$891$	0	0
$892$	73677.3	2.76558
$893$	1330.70	0.0498660
$894$	0	0
$895$	15482.6	0.578241
$896$	−19776.6	−0.737376
$897$	0	0
$898$	506.569	0.0188245
$899$	8671.18	0.321691
$900$	0	0
$901$	−51489.2	−1.90383
$902$	143292.	5.28946
$903$	0	0
$904$	48266.4	1.77579
$905$	−941.289	−0.0345740
$906$	0	0
$907$	1500.89	0.0549461	0.0274731	−	0.999623i	$-0.491254\pi$
$907$	1500.89	0.0549461	0.0274731	+	0.999623i	$0.491254\pi$
$908$	38030.7	1.38997
$909$	0	0
$910$	2083.47	0.0758971
$911$	−19728.2	−0.717481	−0.358740	−	0.933437i	$-0.616794\pi$
$911$	−19728.2	−0.717481	−0.358740	+	0.933437i	$0.616794\pi$
$912$	0	0
$913$	30408.3	1.10226
$914$	5868.48	0.212376
$915$	0	0
$916$	24313.5	0.877011
$917$	−9539.72	−0.343543
$918$	0	0
$919$	19992.0	0.717602	0.358801	−	0.933414i	$-0.383186\pi$
$919$	19992.0	0.717602	0.358801	+	0.933414i	$0.383186\pi$
$920$	−25492.9	−0.913560
$921$	0	0
$922$	41683.4	1.48891
$923$	−1303.89	−0.0464984
$924$	0	0
$925$	−5728.54	−0.203625
$926$	18310.2	0.649794
$927$	0	0
$928$	−11686.6	−0.413395
$929$	1923.62	0.0679354	0.0339677	−	0.999423i	$-0.489186\pi$
$929$	1923.62	0.0679354	0.0339677	+	0.999423i	$0.489186\pi$
$930$	0	0
$931$	−9035.71	−0.318081
$932$	−9392.52	−0.330110
$933$	0	0
$934$	−42634.0	−1.49361
$935$	29300.0	1.02483
$936$	0	0
$937$	−10252.9	−0.357468	−0.178734	−	0.983897i	$-0.557200\pi$
$937$	−10252.9	−0.357468	−0.178734	+	0.983897i	$0.557200\pi$
$938$	−30944.4	−1.07715
$939$	0	0
$940$	−2764.96	−0.0959394
$941$	−11043.0	−0.382563	−0.191282	−	0.981535i	$-0.561264\pi$
$941$	−11043.0	−0.382563	−0.191282	+	0.981535i	$0.561264\pi$
$942$	0	0
$943$	82934.8	2.86398
$944$	−6568.00	−0.226451
$945$	0	0
$946$	35983.1	1.23669
$947$	−32105.2	−1.10167	−0.550833	−	0.834615i	$-0.685690\pi$
$947$	−32105.2	−1.10167	−0.550833	+	0.834615i	$0.685690\pi$
$948$	0	0
$949$	2153.24	0.0736535
$950$	15614.9	0.533280
$951$	0	0
$952$	−22307.1	−0.759431
$953$	9473.37	0.322007	0.161003	−	0.986954i	$-0.448527\pi$
$953$	9473.37	0.322007	0.161003	+	0.986954i	$0.448527\pi$
$954$	0	0
$955$	4325.79	0.146575
$956$	−75598.4	−2.55756
$957$	0	0
$958$	19819.3	0.668404
$959$	6547.26	0.220461
$960$	0	0
$961$	−23818.3	−0.799514
$962$	3375.89	0.113142
$963$	0	0
$964$	88399.8	2.95349
$965$	2315.84	0.0772534
$966$	0	0
$967$	−5310.75	−0.176610	−0.0883052	−	0.996093i	$-0.528145\pi$
$967$	−5310.75	−0.176610	−0.0883052	+	0.996093i	$0.528145\pi$
$968$	−96670.1	−3.20981
$969$	0	0
$970$	−23326.2	−0.772122
$971$	−24271.2	−0.802164	−0.401082	−	0.916042i	$-0.631366\pi$
$971$	−24271.2	−0.802164	−0.401082	+	0.916042i	$0.631366\pi$
$972$	0	0
$973$	−2322.32	−0.0765163
$974$	87454.2	2.87702
$975$	0	0
$976$	24603.6	0.806907
$977$	49602.5	1.62428	0.812142	−	0.583460i	$-0.198301\pi$
$977$	49602.5	1.62428	0.812142	+	0.583460i	$0.198301\pi$
$978$	0	0
$979$	15448.2	0.504317
$980$	18774.6	0.611971
$981$	0	0
$982$	38375.5	1.24706
$983$	−47385.7	−1.53751	−0.768753	−	0.639545i	$-0.779123\pi$
$983$	−47385.7	−1.53751	−0.768753	+	0.639545i	$0.779123\pi$
$984$	0	0
$985$	−17481.2	−0.565478
$986$	−51587.7	−1.66621
$987$	0	0
$988$	−5927.78	−0.190878
$989$	20826.4	0.669607
$990$	0	0
$991$	−8947.33	−0.286802	−0.143401	−	0.989665i	$-0.545804\pi$
$991$	−8947.33	−0.286802	−0.143401	+	0.989665i	$0.545804\pi$
$992$	−8049.67	−0.257638
$993$	0	0
$994$	3558.95	0.113564
$995$	−10414.7	−0.331828
$996$	0	0
$997$	−14908.6	−0.473582	−0.236791	−	0.971561i	$-0.576096\pi$
$997$	−14908.6	−0.473582	−0.236791	+	0.971561i	$0.576096\pi$
$998$	51881.4	1.64557
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.c.1.1		2
3.2	odd	2		39.4.a.b.1.2	✓	2
4.3	odd	2		1872.4.a.t.1.2		2
12.11	even	2		624.4.a.r.1.1		2
13.12	even	2		1521.4.a.s.1.2		2
15.14	odd	2		975.4.a.j.1.1		2
21.20	even	2		1911.4.a.h.1.2		2
24.5	odd	2		2496.4.a.bc.1.2		2
24.11	even	2		2496.4.a.s.1.2		2
39.5	even	4		507.4.b.f.337.1		4
39.8	even	4		507.4.b.f.337.4		4
39.38	odd	2		507.4.a.f.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.b.1.2	✓	2	3.2	odd	2
117.4.a.c.1.1		2	1.1	even	1	trivial
507.4.a.f.1.1		2	39.38	odd	2
507.4.b.f.337.1		4	39.5	even	4
507.4.b.f.337.4		4	39.8	even	4
624.4.a.r.1.1		2	12.11	even	2
975.4.a.j.1.1		2	15.14	odd	2
1521.4.a.s.1.2		2	13.12	even	2
1872.4.a.t.1.2		2	4.3	odd	2
1911.4.a.h.1.2		2	21.20	even	2
2496.4.a.s.1.2		2	24.11	even	2
2496.4.a.bc.1.2		2	24.5	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-4.74166 q^{2} +14.4833 q^{4} -4.51669 q^{5} -7.48331 q^{7} -30.7417 q^{8} +O(q^{10})\)	\(q-4.74166 q^{2} +14.4833 q^{4} -4.51669 q^{5} -7.48331 q^{7} -30.7417 q^{8} +21.4166 q^{10} +66.8999 q^{11} -13.0000 q^{13} +35.4833 q^{14} +29.8999 q^{16} -96.9666 q^{17} +31.4833 q^{19} -65.4166 q^{20} -317.216 q^{22} -183.600 q^{23} -104.600 q^{25} +61.6415 q^{26} -108.383 q^{28} -112.200 q^{29} -77.2831 q^{31} +104.158 q^{32} +459.783 q^{34} +33.7998 q^{35} +54.7664 q^{37} -149.283 q^{38} +138.850 q^{40} -451.716 q^{41} -113.434 q^{43} +968.932 q^{44} +870.566 q^{46} +42.2670 q^{47} -287.000 q^{49} +495.975 q^{50} -188.283 q^{52} +530.999 q^{53} -302.166 q^{55} +230.049 q^{56} +532.015 q^{58} -219.666 q^{59} +822.865 q^{61} +366.450 q^{62} -733.082 q^{64} +58.7169 q^{65} -872.082 q^{67} -1404.40 q^{68} -160.267 q^{70} +100.299 q^{71} -165.634 q^{73} -259.684 q^{74} +455.983 q^{76} -500.633 q^{77} -545.266 q^{79} -135.048 q^{80} +2141.88 q^{82} +454.534 q^{83} +437.968 q^{85} +537.864 q^{86} -2056.61 q^{88} +230.915 q^{89} +97.2831 q^{91} -2659.13 q^{92} -200.415 q^{94} -142.200 q^{95} -1089.16 q^{97} +1360.86 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8	\( 2 q - 2 q^{2} + 14 q^{4} - 24 q^{5} - 54 q^{8} - 32 q^{10} + 44 q^{11} - 26 q^{13} + 56 q^{14} - 30 q^{16} - 164 q^{17} + 48 q^{19} - 56 q^{20} - 380 q^{22} - 8 q^{23} + 150 q^{25} + 26 q^{26} - 112 q^{28} - 404 q^{29} + 40 q^{31} + 126 q^{32} + 276 q^{34} - 112 q^{35} - 100 q^{37} - 104 q^{38} + 592 q^{40} - 200 q^{41} - 616 q^{43} + 980 q^{44} + 1352 q^{46} + 324 q^{47} - 574 q^{49} + 1194 q^{50} - 182 q^{52} + 164 q^{53} + 144 q^{55} + 56 q^{56} - 268 q^{58} - 140 q^{59} + 628 q^{61} + 688 q^{62} - 194 q^{64} + 312 q^{65} - 472 q^{67} - 1372 q^{68} - 560 q^{70} - 428 q^{71} - 900 q^{73} - 684 q^{74} + 448 q^{76} - 672 q^{77} - 432 q^{79} + 1032 q^{80} + 2832 q^{82} + 1388 q^{83} + 1744 q^{85} - 840 q^{86} - 1524 q^{88} - 960 q^{89} - 2744 q^{92} + 572 q^{94} - 464 q^{95} - 532 q^{97} + 574 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 14 * q^4 - 24 * q^5 - 54 * q^8 - 32 * q^10 + 44 * q^11 - 26 * q^13 + 56 * q^14 - 30 * q^16 - 164 * q^17 + 48 * q^19 - 56 * q^20 - 380 * q^22 - 8 * q^23 + 150 * q^25 + 26 * q^26 - 112 * q^28 - 404 * q^29 + 40 * q^31 + 126 * q^32 + 276 * q^34 - 112 * q^35 - 100 * q^37 - 104 * q^38 + 592 * q^40 - 200 * q^41 - 616 * q^43 + 980 * q^44 + 1352 * q^46 + 324 * q^47 - 574 * q^49 + 1194 * q^50 - 182 * q^52 + 164 * q^53 + 144 * q^55 + 56 * q^56 - 268 * q^58 - 140 * q^59 + 628 * q^61 + 688 * q^62 - 194 * q^64 + 312 * q^65 - 472 * q^67 - 1372 * q^68 - 560 * q^70 - 428 * q^71 - 900 * q^73 - 684 * q^74 + 448 * q^76 - 672 * q^77 - 432 * q^79 + 1032 * q^80 + 2832 * q^82 + 1388 * q^83 + 1744 * q^85 - 840 * q^86 - 1524 * q^88 - 960 * q^89 - 2744 * q^92 + 572 * q^94 - 464 * q^95 - 532 * q^97 + 574 * q^98

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