LMFDB - Embedded newform 117.4.a.f.1.2

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:	$3$
Coefficient field:	3.3.3144.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 16x - 8 $ x^3 - x^2 - 16*x - 8
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.526440$ 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)

$f(q)$	$=$	\(q-1.52644 q^{2} -5.66998 q^{4} -19.3400 q^{5} +4.84136 q^{7} +20.8664 q^{8} +29.5213 q^{10} +61.0728 q^{11} +13.0000 q^{13} -7.39005 q^{14} +13.5085 q^{16} +41.7885 q^{17} -107.561 q^{19} +109.657 q^{20} -93.2239 q^{22} -28.5138 q^{23} +249.034 q^{25} -19.8437 q^{26} -27.4504 q^{28} +89.8886 q^{29} +183.108 q^{31} -187.551 q^{32} -63.7876 q^{34} -93.6318 q^{35} +418.029 q^{37} +164.185 q^{38} -403.555 q^{40} +142.674 q^{41} -71.0935 q^{43} -346.281 q^{44} +43.5246 q^{46} -323.711 q^{47} -319.561 q^{49} -380.136 q^{50} -73.7098 q^{52} +25.1047 q^{53} -1181.15 q^{55} +101.022 q^{56} -137.210 q^{58} +684.508 q^{59} +308.125 q^{61} -279.503 q^{62} +178.217 q^{64} -251.420 q^{65} +672.808 q^{67} -236.940 q^{68} +142.923 q^{70} +326.837 q^{71} +24.3058 q^{73} -638.095 q^{74} +609.869 q^{76} +295.675 q^{77} +166.810 q^{79} -261.255 q^{80} -217.783 q^{82} +201.093 q^{83} -808.188 q^{85} +108.520 q^{86} +1274.37 q^{88} -108.834 q^{89} +62.9377 q^{91} +161.673 q^{92} +494.126 q^{94} +2080.23 q^{95} +1157.95 q^{97} +487.791 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8} - 4 q^{10} + 16 q^{11} + 39 q^{13} + 176 q^{14} - 110 q^{16} + 146 q^{17} + 94 q^{19} + 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 26 q^{26}+ \cdots + 5070 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8 - 4 * q^10 + 16 * q^11 + 39 * q^13 + 176 * q^14 - 110 * q^16 + 146 * q^17 + 94 * q^19 + 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 26 * q^26 + 80 * q^28 + 2 * q^29 + 302 * q^31 - 154 * q^32 + 164 * q^34 - 80 * q^35 + 374 * q^37 - 312 * q^38 - 516 * q^40 - 480 * q^41 - 260 * q^43 - 712 * q^44 - 1104 * q^46 + 24 * q^47 + 447 * q^49 - 814 * q^50 + 130 * q^52 + 678 * q^53 - 1552 * q^55 - 96 * q^56 - 628 * q^58 + 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 52 * q^65 + 74 * q^67 + 460 * q^68 + 1216 * q^70 + 948 * q^71 - 222 * q^73 - 1724 * q^74 + 2392 * q^76 - 112 * q^77 - 24 * q^79 - 1100 * q^80 + 564 * q^82 + 796 * q^83 - 248 * q^85 - 1800 * q^86 + 1608 * q^88 - 1436 * q^89 + 390 * q^91 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 + 3242 * q^97 + 5070 * q^98

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 $

$2$	−1.52644	−0.539678	−0.269839	−	0.962905i	$-0.586970\pi$
$2$	−1.52644	−0.539678	−0.269839	+	0.962905i	$0.586970\pi$
$3$	0	0
$3$	0	0
$4$	−5.66998	−0.708748
$5$	−19.3400	−1.72982	−0.864909	−	0.501928i	$-0.832624\pi$
$5$	−19.3400	−1.72982	−0.864909	+	0.501928i	$0.832624\pi$
$6$	0	0
$7$	4.84136	0.261409	0.130704	−	0.991421i	$-0.458276\pi$
$7$	4.84136	0.261409	0.130704	+	0.991421i	$0.458276\pi$
$8$	20.8664	0.922173
$9$	0	0
$10$	29.5213	0.933545
$11$	61.0728	1.67401	0.837006	−	0.547194i	$-0.184304\pi$
$11$	61.0728	1.67401	0.837006	+	0.547194i	$0.184304\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	−7.39005	−0.141077
$15$	0	0
$16$	13.5085	0.211071
$17$	41.7885	0.596188	0.298094	−	0.954537i	$-0.403649\pi$
$17$	41.7885	0.596188	0.298094	+	0.954537i	$0.403649\pi$
$18$	0	0
$19$	−107.561	−1.29875	−0.649374	−	0.760469i	$-0.724969\pi$
$19$	−107.561	−1.29875	−0.649374	+	0.760469i	$0.724969\pi$
$20$	109.657	1.22601
$21$	0	0
$22$	−93.2239	−0.903427
$23$	−28.5138	−0.258502	−0.129251	−	0.991612i	$-0.541257\pi$
$23$	−28.5138	−0.258502	−0.129251	+	0.991612i	$0.541257\pi$
$24$	0	0
$25$	249.034	1.99227
$26$	−19.8437	−0.149680
$27$	0	0
$28$	−27.4504	−0.185273
$29$	89.8886	0.575583	0.287791	−	0.957693i	$-0.407079\pi$
$29$	89.8886	0.575583	0.287791	+	0.957693i	$0.407079\pi$
$30$	0	0
$31$	183.108	1.06087	0.530437	−	0.847724i	$-0.322028\pi$
$31$	183.108	1.06087	0.530437	+	0.847724i	$0.322028\pi$
$32$	−187.551	−1.03608
$33$	0	0
$34$	−63.7876	−0.321750
$35$	−93.6318	−0.452190
$36$	0	0
$37$	418.029	1.85739	0.928696	−	0.370843i	$-0.120931\pi$
$37$	418.029	1.85739	0.928696	+	0.370843i	$0.120931\pi$
$38$	164.185	0.700905
$39$	0	0
$40$	−403.555	−1.59519
$41$	142.674	0.543460	0.271730	−	0.962373i	$-0.412404\pi$
$41$	142.674	0.543460	0.271730	+	0.962373i	$0.412404\pi$
$42$	0	0
$43$	−71.0935	−0.252132	−0.126066	−	0.992022i	$-0.540235\pi$
$43$	−71.0935	−0.252132	−0.126066	+	0.992022i	$0.540235\pi$
$44$	−346.281	−1.18645
$45$	0	0
$46$	43.5246	0.139508
$47$	−323.711	−1.00464	−0.502321	−	0.864681i	$-0.667520\pi$
$47$	−323.711	−1.00464	−0.502321	+	0.864681i	$0.667520\pi$
$48$	0	0
$49$	−319.561	−0.931665
$50$	−380.136	−1.07519
$51$	0	0
$52$	−73.7098	−0.196571
$53$	25.1047	0.0650641	0.0325321	−	0.999471i	$-0.489643\pi$
$53$	25.1047	0.0650641	0.0325321	+	0.999471i	$0.489643\pi$
$54$	0	0
$55$	−1181.15	−2.89574
$56$	101.022	0.241064
$57$	0	0
$58$	−137.210	−0.310629
$59$	684.508	1.51043	0.755215	−	0.655477i	$-0.227533\pi$
$59$	684.508	1.51043	0.755215	+	0.655477i	$0.227533\pi$
$60$	0	0
$61$	308.125	0.646744	0.323372	−	0.946272i	$-0.395184\pi$
$61$	308.125	0.646744	0.323372	+	0.946272i	$0.395184\pi$
$62$	−279.503	−0.572531
$63$	0	0
$64$	178.217	0.348081
$65$	−251.420	−0.479765
$66$	0	0
$67$	672.808	1.22681	0.613407	−	0.789767i	$-0.289798\pi$
$67$	672.808	1.22681	0.613407	+	0.789767i	$0.289798\pi$
$68$	−236.940	−0.422547
$69$	0	0
$70$	142.923	0.244037
$71$	326.837	0.546315	0.273158	−	0.961969i	$-0.411932\pi$
$71$	326.837	0.546315	0.273158	+	0.961969i	$0.411932\pi$
$72$	0	0
$73$	24.3058	0.0389695	0.0194847	−	0.999810i	$-0.493797\pi$
$73$	24.3058	0.0389695	0.0194847	+	0.999810i	$0.493797\pi$
$74$	−638.095	−1.00239
$75$	0	0
$76$	609.869	0.920484
$77$	295.675	0.437602
$78$	0	0
$79$	166.810	0.237565	0.118783	−	0.992920i	$-0.462101\pi$
$79$	166.810	0.237565	0.118783	+	0.992920i	$0.462101\pi$
$80$	−261.255	−0.365115
$81$	0	0
$82$	−217.783	−0.293294
$83$	201.093	0.265938	0.132969	−	0.991120i	$-0.457549\pi$
$83$	201.093	0.265938	0.132969	+	0.991120i	$0.457549\pi$
$84$	0	0
$85$	−808.188	−1.03130
$86$	108.520	0.136070
$87$	0	0
$88$	1274.37	1.54373
$89$	−108.834	−0.129622	−0.0648109	−	0.997898i	$-0.520644\pi$
$89$	−108.834	−0.129622	−0.0648109	+	0.997898i	$0.520644\pi$
$90$	0	0
$91$	62.9377	0.0725018
$92$	161.673	0.183212
$93$	0	0
$94$	494.126	0.542183
$95$	2080.23	2.24660
$96$	0	0
$97$	1157.95	1.21208	0.606041	−	0.795434i	$-0.292757\pi$
$97$	1157.95	1.21208	0.606041	+	0.795434i	$0.292757\pi$
$98$	487.791	0.502799
$99$	0	0
$100$	−1412.02	−1.41202
$101$	−1702.75	−1.67752	−0.838761	−	0.544500i	$-0.816719\pi$
$101$	−1702.75	−1.67752	−0.838761	+	0.544500i	$0.816719\pi$
$102$	0	0
$103$	−1455.14	−1.39203	−0.696015	−	0.718027i	$-0.745045\pi$
$103$	−1455.14	−1.39203	−0.696015	+	0.718027i	$0.745045\pi$
$104$	271.263	0.255765
$105$	0	0
$106$	−38.3209	−0.0351137
$107$	822.762	0.743359	0.371679	−	0.928361i	$-0.378782\pi$
$107$	822.762	0.743359	0.371679	+	0.928361i	$0.378782\pi$
$108$	0	0
$109$	457.264	0.401816	0.200908	−	0.979610i	$-0.435611\pi$
$109$	457.264	0.401816	0.200908	+	0.979610i	$0.435611\pi$
$110$	1802.95	1.56277
$111$	0	0
$112$	65.3998	0.0551759
$113$	381.693	0.317758	0.158879	−	0.987298i	$-0.449212\pi$
$113$	381.693	0.317758	0.158879	+	0.987298i	$0.449212\pi$
$114$	0	0
$115$	551.456	0.447161
$116$	−509.667	−0.407943
$117$	0	0
$118$	−1044.86	−0.815146
$119$	202.313	0.155849
$120$	0	0
$121$	2398.88	1.80232
$122$	−470.334	−0.349033
$123$	0	0
$124$	−1038.22	−0.751892
$125$	−2398.82	−1.71645
$126$	0	0
$127$	−1129.09	−0.788905	−0.394452	−	0.918916i	$-0.629066\pi$
$127$	−1129.09	−0.788905	−0.394452	+	0.918916i	$0.629066\pi$
$128$	1228.37	0.848232
$129$	0	0
$130$	383.777	0.258919
$131$	852.761	0.568749	0.284374	−	0.958713i	$-0.408214\pi$
$131$	852.761	0.568749	0.284374	+	0.958713i	$0.408214\pi$
$132$	0	0
$133$	−520.742	−0.339504
$134$	−1027.00	−0.662085
$135$	0	0
$136$	871.975	0.549789
$137$	488.903	0.304889	0.152445	−	0.988312i	$-0.451285\pi$
$137$	488.903	0.304889	0.152445	+	0.988312i	$0.451285\pi$
$138$	0	0
$139$	407.123	0.248430	0.124215	−	0.992255i	$-0.460359\pi$
$139$	407.123	0.248430	0.124215	+	0.992255i	$0.460359\pi$
$140$	530.890	0.320489
$141$	0	0
$142$	−498.897	−0.294834
$143$	793.946	0.464287
$144$	0	0
$145$	−1738.44	−0.995654
$146$	−37.1013	−0.0210310
$147$	0	0
$148$	−2370.21	−1.31642
$149$	−1717.63	−0.944388	−0.472194	−	0.881495i	$-0.656538\pi$
$149$	−1717.63	−0.944388	−0.472194	+	0.881495i	$0.656538\pi$
$150$	0	0
$151$	1341.79	0.723133	0.361567	−	0.932346i	$-0.382242\pi$
$151$	1341.79	0.723133	0.361567	+	0.932346i	$0.382242\pi$
$152$	−2244.41	−1.19767
$153$	0	0
$154$	−451.331	−0.236164
$155$	−3541.30	−1.83512
$156$	0	0
$157$	−760.546	−0.386612	−0.193306	−	0.981138i	$-0.561921\pi$
$157$	−760.546	−0.386612	−0.193306	+	0.981138i	$0.561921\pi$
$158$	−254.626	−0.128209
$159$	0	0
$160$	3627.23	1.79224
$161$	−138.046	−0.0675746
$162$	0	0
$163$	2712.09	1.30323	0.651616	−	0.758549i	$-0.274091\pi$
$163$	2712.09	1.30323	0.651616	+	0.758549i	$0.274091\pi$
$164$	−808.957	−0.385176
$165$	0	0
$166$	−306.957	−0.143521
$167$	−1551.69	−0.719004	−0.359502	−	0.933144i	$-0.617053\pi$
$167$	−1551.69	−0.719004	−0.359502	+	0.933144i	$0.617053\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	1233.65	0.556568
$171$	0	0
$172$	403.099	0.178698
$173$	3970.26	1.74482	0.872409	−	0.488777i	$-0.162557\pi$
$173$	3970.26	1.74482	0.872409	+	0.488777i	$0.162557\pi$
$174$	0	0
$175$	1205.66	0.520798
$176$	825.004	0.353335
$177$	0	0
$178$	166.128	0.0699540
$179$	2690.95	1.12364	0.561818	−	0.827261i	$-0.310102\pi$
$179$	2690.95	1.12364	0.561818	+	0.827261i	$0.310102\pi$
$180$	0	0
$181$	−4371.10	−1.79503	−0.897517	−	0.440980i	$-0.854631\pi$
$181$	−4371.10	−1.79503	−0.897517	+	0.440980i	$0.854631\pi$
$182$	−96.0706	−0.0391276
$183$	0	0
$184$	−594.980	−0.238383
$185$	−8084.66	−3.21295
$186$	0	0
$187$	2552.14	0.998026
$188$	1835.44	0.712038
$189$	0	0
$190$	−3175.34	−1.21244
$191$	−1408.47	−0.533578	−0.266789	−	0.963755i	$-0.585963\pi$
$191$	−1408.47	−0.533578	−0.266789	+	0.963755i	$0.585963\pi$
$192$	0	0
$193$	−4131.69	−1.54096	−0.770481	−	0.637463i	$-0.779984\pi$
$193$	−4131.69	−1.54096	−0.770481	+	0.637463i	$0.779984\pi$
$194$	−1767.54	−0.654134
$195$	0	0
$196$	1811.91	0.660316
$197$	3401.23	1.23009	0.615045	−	0.788492i	$-0.289138\pi$
$197$	3401.23	1.23009	0.615045	+	0.788492i	$0.289138\pi$
$198$	0	0
$199$	−3520.74	−1.25416	−0.627081	−	0.778954i	$-0.715751\pi$
$199$	−3520.74	−1.25416	−0.627081	+	0.778954i	$0.715751\pi$
$200$	5196.45	1.83722
$201$	0	0
$202$	2599.14	0.905321
$203$	435.183	0.150463
$204$	0	0
$205$	−2759.30	−0.940088
$206$	2221.18	0.751248
$207$	0	0
$208$	175.611	0.0585406
$209$	−6569.05	−2.17412
$210$	0	0
$211$	−2245.22	−0.732545	−0.366272	−	0.930508i	$-0.619366\pi$
$211$	−2245.22	−0.732545	−0.366272	+	0.930508i	$0.619366\pi$
$212$	−142.343	−0.0461141
$213$	0	0
$214$	−1255.90	−0.401174
$215$	1374.95	0.436142
$216$	0	0
$217$	886.490	0.277322
$218$	−697.986	−0.216851
$219$	0	0
$220$	6697.07	2.05235
$221$	543.250	0.165353
$222$	0	0
$223$	3431.26	1.03038	0.515188	−	0.857077i	$-0.327722\pi$
$223$	3431.26	1.03038	0.515188	+	0.857077i	$0.327722\pi$
$224$	−908.003	−0.270842
$225$	0	0
$226$	−582.631	−0.171487
$227$	4757.91	1.39116	0.695581	−	0.718448i	$-0.255147\pi$
$227$	4757.91	1.39116	0.695581	+	0.718448i	$0.255147\pi$
$228$	0	0
$229$	−4368.93	−1.26073	−0.630364	−	0.776300i	$-0.717094\pi$
$229$	−4368.93	−1.26073	−0.630364	+	0.776300i	$0.717094\pi$
$230$	−841.764	−0.241323
$231$	0	0
$232$	1875.65	0.530787
$233$	3642.00	1.02401	0.512007	−	0.858981i	$-0.328902\pi$
$233$	3642.00	1.02401	0.512007	+	0.858981i	$0.328902\pi$
$234$	0	0
$235$	6260.57	1.73785
$236$	−3881.15	−1.07051
$237$	0	0
$238$	−308.819	−0.0841082
$239$	−2236.17	−0.605213	−0.302606	−	0.953116i	$-0.597857\pi$
$239$	−2236.17	−0.605213	−0.302606	+	0.953116i	$0.597857\pi$
$240$	0	0
$241$	6538.78	1.74772	0.873858	−	0.486181i	$-0.161610\pi$
$241$	6538.78	1.74772	0.873858	+	0.486181i	$0.161610\pi$
$242$	−3661.75	−0.972670
$243$	0	0
$244$	−1747.06	−0.458378
$245$	6180.30	1.61161
$246$	0	0
$247$	−1398.29	−0.360208
$248$	3820.80	0.978310
$249$	0	0
$250$	3661.65	0.926332
$251$	−2507.12	−0.630470	−0.315235	−	0.949014i	$-0.602083\pi$
$251$	−2507.12	−0.630470	−0.315235	+	0.949014i	$0.602083\pi$
$252$	0	0
$253$	−1741.42	−0.432735
$254$	1723.49	0.425755
$255$	0	0
$256$	−3300.77	−0.805853
$257$	808.131	0.196147	0.0980735	−	0.995179i	$-0.468732\pi$
$257$	808.131	0.196147	0.0980735	+	0.995179i	$0.468732\pi$
$258$	0	0
$259$	2023.83	0.485539
$260$	1425.54	0.340033
$261$	0	0
$262$	−1301.69	−0.306941
$263$	−2940.70	−0.689472	−0.344736	−	0.938700i	$-0.612032\pi$
$263$	−2940.70	−0.689472	−0.344736	+	0.938700i	$0.612032\pi$
$264$	0	0
$265$	−485.525	−0.112549
$266$	794.881	0.183223
$267$	0	0
$268$	−3814.81	−0.869502
$269$	−7111.50	−1.61188	−0.805940	−	0.591997i	$-0.798340\pi$
$269$	−7111.50	−1.61188	−0.805940	+	0.591997i	$0.798340\pi$
$270$	0	0
$271$	2034.96	0.456145	0.228072	−	0.973644i	$-0.426758\pi$
$271$	2034.96	0.456145	0.228072	+	0.973644i	$0.426758\pi$
$272$	564.502	0.125838
$273$	0	0
$274$	−746.281	−0.164542
$275$	15209.2	3.33509
$276$	0	0
$277$	2723.20	0.590689	0.295345	−	0.955391i	$-0.404566\pi$
$277$	2723.20	0.590689	0.295345	+	0.955391i	$0.404566\pi$
$278$	−621.449	−0.134072
$279$	0	0
$280$	−1953.76	−0.416998
$281$	−3265.56	−0.693263	−0.346632	−	0.938001i	$-0.612675\pi$
$281$	−3265.56	−0.693263	−0.346632	+	0.938001i	$0.612675\pi$
$282$	0	0
$283$	1144.02	0.240299	0.120150	−	0.992756i	$-0.461663\pi$
$283$	1144.02	0.240299	0.120150	+	0.992756i	$0.461663\pi$
$284$	−1853.16	−0.387200
$285$	0	0
$286$	−1211.91	−0.250566
$287$	690.735	0.142065
$288$	0	0
$289$	−3166.72	−0.644560
$290$	2653.63	0.537333
$291$	0	0
$292$	−137.813	−0.0276195
$293$	1677.35	0.334444	0.167222	−	0.985919i	$-0.446520\pi$
$293$	1677.35	0.334444	0.167222	+	0.985919i	$0.446520\pi$
$294$	0	0
$295$	−13238.4	−2.61277
$296$	8722.75	1.71284
$297$	0	0
$298$	2621.86	0.509666
$299$	−370.679	−0.0716954
$300$	0	0
$301$	−344.190	−0.0659095
$302$	−2048.16	−0.390259
$303$	0	0
$304$	−1452.99	−0.274128
$305$	−5959.13	−1.11875
$306$	0	0
$307$	7207.70	1.33995	0.669975	−	0.742383i	$-0.266305\pi$
$307$	7207.70	1.33995	0.669975	+	0.742383i	$0.266305\pi$
$308$	−1676.47	−0.310149
$309$	0	0
$310$	5405.57	0.990374
$311$	−412.963	−0.0752958	−0.0376479	−	0.999291i	$-0.511987\pi$
$311$	−412.963	−0.0752958	−0.0376479	+	0.999291i	$0.511987\pi$
$312$	0	0
$313$	2936.39	0.530270	0.265135	−	0.964211i	$-0.414583\pi$
$313$	2936.39	0.530270	0.265135	+	0.964211i	$0.414583\pi$
$314$	1160.93	0.208646
$315$	0	0
$316$	−945.812	−0.168374
$317$	−377.956	−0.0669657	−0.0334828	−	0.999439i	$-0.510660\pi$
$317$	−377.956	−0.0669657	−0.0334828	+	0.999439i	$0.510660\pi$
$318$	0	0
$319$	5489.75	0.963533
$320$	−3446.71	−0.602116
$321$	0	0
$322$	210.718	0.0364685
$323$	−4494.81	−0.774298
$324$	0	0
$325$	3237.44	0.552557
$326$	−4139.83	−0.703326
$327$	0	0
$328$	2977.09	0.501165
$329$	−1567.20	−0.262622
$330$	0	0
$331$	−4428.17	−0.735330	−0.367665	−	0.929958i	$-0.619843\pi$
$331$	−4428.17	−0.735330	−0.367665	+	0.929958i	$0.619843\pi$
$332$	−1140.20	−0.188483
$333$	0	0
$334$	2368.57	0.388031
$335$	−13012.1	−2.12217
$336$	0	0
$337$	−1768.76	−0.285907	−0.142953	−	0.989729i	$-0.545660\pi$
$337$	−1768.76	−0.285907	−0.142953	+	0.989729i	$0.545660\pi$
$338$	−257.968	−0.0415137
$339$	0	0
$340$	4582.41	0.730930
$341$	11182.9	1.77592
$342$	0	0
$343$	−3207.70	−0.504955
$344$	−1483.47	−0.232509
$345$	0	0
$346$	−6060.36	−0.941639
$347$	−2412.97	−0.373300	−0.186650	−	0.982426i	$-0.559763\pi$
$347$	−2412.97	−0.373300	−0.186650	+	0.982426i	$0.559763\pi$
$348$	0	0
$349$	−9967.45	−1.52878	−0.764392	−	0.644752i	$-0.776961\pi$
$349$	−9967.45	−1.52878	−0.764392	+	0.644752i	$0.776961\pi$
$350$	−1840.37	−0.281063
$351$	0	0
$352$	−11454.3	−1.73442
$353$	−4516.30	−0.680959	−0.340479	−	0.940252i	$-0.610589\pi$
$353$	−4516.30	−0.680959	−0.340479	+	0.940252i	$0.610589\pi$
$354$	0	0
$355$	−6321.01	−0.945027
$356$	617.084	0.0918691
$357$	0	0
$358$	−4107.57	−0.606401
$359$	−12159.8	−1.78767	−0.893833	−	0.448400i	$-0.851994\pi$
$359$	−12159.8	−1.78767	−0.893833	+	0.448400i	$0.851994\pi$
$360$	0	0
$361$	4710.38	0.686745
$362$	6672.22	0.968740
$363$	0	0
$364$	−356.856	−0.0513855
$365$	−470.072	−0.0674102
$366$	0	0
$367$	−2674.25	−0.380367	−0.190183	−	0.981749i	$-0.560908\pi$
$367$	−2674.25	−0.380367	−0.190183	+	0.981749i	$0.560908\pi$
$368$	−385.180	−0.0545622
$369$	0	0
$370$	12340.7	1.73396
$371$	121.541	0.0170083
$372$	0	0
$373$	9601.74	1.33287	0.666433	−	0.745564i	$-0.267820\pi$
$373$	9601.74	1.33287	0.666433	+	0.745564i	$0.267820\pi$
$374$	−3895.69	−0.538613
$375$	0	0
$376$	−6754.69	−0.926454
$377$	1168.55	0.159638
$378$	0	0
$379$	9019.65	1.22245	0.611225	−	0.791457i	$-0.290677\pi$
$379$	9019.65	1.22245	0.611225	+	0.791457i	$0.290677\pi$
$380$	−11794.9	−1.59227
$381$	0	0
$382$	2149.95	0.287960
$383$	4015.34	0.535703	0.267852	−	0.963460i	$-0.413686\pi$
$383$	4015.34	0.535703	0.267852	+	0.963460i	$0.413686\pi$
$384$	0	0
$385$	−5718.35	−0.756972
$386$	6306.78	0.831623
$387$	0	0
$388$	−6565.55	−0.859060
$389$	2725.35	0.355221	0.177610	−	0.984101i	$-0.443163\pi$
$389$	2725.35	0.355221	0.177610	+	0.984101i	$0.443163\pi$
$390$	0	0
$391$	−1191.55	−0.154115
$392$	−6668.09	−0.859157
$393$	0	0
$394$	−5191.78	−0.663853
$395$	−3226.11	−0.410945
$396$	0	0
$397$	−4391.59	−0.555182	−0.277591	−	0.960699i	$-0.589536\pi$
$397$	−4391.59	−0.555182	−0.277591	+	0.960699i	$0.589536\pi$
$398$	5374.19	0.676844
$399$	0	0
$400$	3364.09	0.420511
$401$	−3762.48	−0.468552	−0.234276	−	0.972170i	$-0.575272\pi$
$401$	−3762.48	−0.468552	−0.234276	+	0.972170i	$0.575272\pi$
$402$	0	0
$403$	2380.40	0.294234
$404$	9654.55	1.18894
$405$	0	0
$406$	−664.281	−0.0812013
$407$	25530.2	3.10930
$408$	0	0
$409$	−6797.81	−0.821833	−0.410917	−	0.911673i	$-0.634791\pi$
$409$	−6797.81	−0.821833	−0.410917	+	0.911673i	$0.634791\pi$
$410$	4211.91	0.507345
$411$	0	0
$412$	8250.61	0.986599
$413$	3313.95	0.394840
$414$	0	0
$415$	−3889.14	−0.460025
$416$	−2438.17	−0.287358
$417$	0	0
$418$	10027.3	1.17332
$419$	12594.2	1.46841	0.734207	−	0.678925i	$-0.237554\pi$
$419$	12594.2	1.46841	0.734207	+	0.678925i	$0.237554\pi$
$420$	0	0
$421$	6888.04	0.797393	0.398697	−	0.917083i	$-0.369463\pi$
$421$	6888.04	0.797393	0.398697	+	0.917083i	$0.369463\pi$
$422$	3427.19	0.395338
$423$	0	0
$424$	523.845	0.0600004
$425$	10406.8	1.18777
$426$	0	0
$427$	1491.74	0.169065
$428$	−4665.04	−0.526854
$429$	0	0
$430$	−2098.77	−0.235376
$431$	7384.53	0.825291	0.412645	−	0.910892i	$-0.364605\pi$
$431$	7384.53	0.825291	0.412645	+	0.910892i	$0.364605\pi$
$432$	0	0
$433$	9068.33	1.00646	0.503229	−	0.864153i	$-0.332145\pi$
$433$	9068.33	1.00646	0.503229	+	0.864153i	$0.332145\pi$
$434$	−1353.17	−0.149665
$435$	0	0
$436$	−2592.68	−0.284786
$437$	3066.97	0.335728
$438$	0	0
$439$	16875.4	1.83466	0.917331	−	0.398125i	$-0.130339\pi$
$439$	16875.4	1.83466	0.917331	+	0.398125i	$0.130339\pi$
$440$	−24646.3	−2.67037
$441$	0	0
$442$	−829.239	−0.0892373
$443$	6766.18	0.725668	0.362834	−	0.931854i	$-0.381809\pi$
$443$	6766.18	0.725668	0.362834	+	0.931854i	$0.381809\pi$
$444$	0	0
$445$	2104.84	0.224222
$446$	−5237.61	−0.556072
$447$	0	0
$448$	862.814	0.0909914
$449$	−140.944	−0.0148141	−0.00740706	−	0.999973i	$-0.502358\pi$
$449$	−140.944	−0.0148141	−0.00740706	+	0.999973i	$0.502358\pi$
$450$	0	0
$451$	8713.47	0.909759
$452$	−2164.19	−0.225210
$453$	0	0
$454$	−7262.67	−0.750779
$455$	−1217.21	−0.125415
$456$	0	0
$457$	−17733.1	−1.81514	−0.907571	−	0.419898i	$-0.862066\pi$
$457$	−17733.1	−1.81514	−0.907571	+	0.419898i	$0.862066\pi$
$458$	6668.90	0.680387
$459$	0	0
$460$	−3126.74	−0.316924
$461$	−2293.37	−0.231699	−0.115849	−	0.993267i	$-0.536959\pi$
$461$	−2293.37	−0.231699	−0.115849	+	0.993267i	$0.536959\pi$
$462$	0	0
$463$	13770.9	1.38226	0.691129	−	0.722731i	$-0.257113\pi$
$463$	13770.9	1.38226	0.691129	+	0.722731i	$0.257113\pi$
$464$	1214.27	0.121489
$465$	0	0
$466$	−5559.29	−0.552638
$467$	3477.37	0.344568	0.172284	−	0.985047i	$-0.444885\pi$
$467$	3477.37	0.344568	0.172284	+	0.985047i	$0.444885\pi$
$468$	0	0
$469$	3257.31	0.320700
$470$	−9556.38	−0.937879
$471$	0	0
$472$	14283.2	1.39288
$473$	−4341.88	−0.422072
$474$	0	0
$475$	−26786.4	−2.58746
$476$	−1147.11	−0.110458
$477$	0	0
$478$	3413.38	0.326620
$479$	3137.39	0.299271	0.149636	−	0.988741i	$-0.452190\pi$
$479$	3137.39	0.299271	0.149636	+	0.988741i	$0.452190\pi$
$480$	0	0
$481$	5434.37	0.515148
$482$	−9981.05	−0.943204
$483$	0	0
$484$	−13601.6	−1.27739
$485$	−22394.7	−2.09668
$486$	0	0
$487$	5996.52	0.557964	0.278982	−	0.960296i	$-0.410003\pi$
$487$	5996.52	0.557964	0.278982	+	0.960296i	$0.410003\pi$
$488$	6429.46	0.596410
$489$	0	0
$490$	−9433.86	−0.869752
$491$	9401.49	0.864121	0.432060	−	0.901845i	$-0.357787\pi$
$491$	9401.49	0.864121	0.432060	+	0.901845i	$0.357787\pi$
$492$	0	0
$493$	3756.31	0.343156
$494$	2134.41	0.194396
$495$	0	0
$496$	2473.52	0.223920
$497$	1582.34	0.142812
$498$	0	0
$499$	−5052.33	−0.453253	−0.226626	−	0.973982i	$-0.572770\pi$
$499$	−5052.33	−0.453253	−0.226626	+	0.973982i	$0.572770\pi$
$500$	13601.2	1.21653
$501$	0	0
$502$	3826.96	0.340251
$503$	−8184.02	−0.725462	−0.362731	−	0.931894i	$-0.618156\pi$
$503$	−8184.02	−0.725462	−0.362731	+	0.931894i	$0.618156\pi$
$504$	0	0
$505$	32931.1	2.90181
$506$	2658.17	0.233537
$507$	0	0
$508$	6401.94	0.559134
$509$	−6039.12	−0.525892	−0.262946	−	0.964811i	$-0.584694\pi$
$509$	−6039.12	−0.525892	−0.262946	+	0.964811i	$0.584694\pi$
$510$	0	0
$511$	117.673	0.0101870
$512$	−4788.54	−0.413331
$513$	0	0
$514$	−1233.56	−0.105856
$515$	28142.3	2.40796
$516$	0	0
$517$	−19770.0	−1.68178
$518$	−3089.25	−0.262035
$519$	0	0
$520$	−5246.22	−0.442427
$521$	14602.5	1.22792	0.613960	−	0.789337i	$-0.289576\pi$
$521$	14602.5	1.22792	0.613960	+	0.789337i	$0.289576\pi$
$522$	0	0
$523$	8910.70	0.745005	0.372502	−	0.928031i	$-0.378500\pi$
$523$	8910.70	0.745005	0.372502	+	0.928031i	$0.378500\pi$
$524$	−4835.14	−0.403099
$525$	0	0
$526$	4488.80	0.372093
$527$	7651.79	0.632481
$528$	0	0
$529$	−11354.0	−0.933177
$530$	741.124	0.0607403
$531$	0	0
$532$	2952.60	0.240623
$533$	1854.76	0.150729
$534$	0	0
$535$	−15912.2	−1.28588
$536$	14039.1	1.13134
$537$	0	0
$538$	10855.3	0.869897
$539$	−19516.5	−1.55962
$540$	0	0
$541$	−13313.6	−1.05803	−0.529017	−	0.848611i	$-0.677439\pi$
$541$	−13313.6	−1.05803	−0.529017	+	0.848611i	$0.677439\pi$
$542$	−3106.25	−0.246171
$543$	0	0
$544$	−7837.48	−0.617701
$545$	−8843.47	−0.695069
$546$	0	0
$547$	−4116.94	−0.321806	−0.160903	−	0.986970i	$-0.551441\pi$
$547$	−4116.94	−0.321806	−0.160903	+	0.986970i	$0.551441\pi$
$548$	−2772.07	−0.216090
$549$	0	0
$550$	−23215.9	−1.79987
$551$	−9668.52	−0.747537
$552$	0	0
$553$	807.590	0.0621016
$554$	−4156.79	−0.318782
$555$	0	0
$556$	−2308.38	−0.176074
$557$	6888.37	0.524003	0.262002	−	0.965067i	$-0.415617\pi$
$557$	6888.37	0.524003	0.262002	+	0.965067i	$0.415617\pi$
$558$	0	0
$559$	−924.216	−0.0699288
$560$	−1264.83	−0.0954443
$561$	0	0
$562$	4984.68	0.374139
$563$	−10537.1	−0.788782	−0.394391	−	0.918943i	$-0.629045\pi$
$563$	−10537.1	−0.788782	−0.394391	+	0.918943i	$0.629045\pi$
$564$	0	0
$565$	−7381.92	−0.549664
$566$	−1746.27	−0.129684
$567$	0	0
$568$	6819.91	0.503798
$569$	−26930.1	−1.98413	−0.992065	−	0.125722i	$-0.959875\pi$
$569$	−26930.1	−1.98413	−0.992065	+	0.125722i	$0.959875\pi$
$570$	0	0
$571$	−3125.60	−0.229076	−0.114538	−	0.993419i	$-0.536539\pi$
$571$	−3125.60	−0.229076	−0.114538	+	0.993419i	$0.536539\pi$
$572$	−4501.66	−0.329063
$573$	0	0
$574$	−1054.36	−0.0766696
$575$	−7100.91	−0.515006
$576$	0	0
$577$	4787.13	0.345391	0.172696	−	0.984975i	$-0.444752\pi$
$577$	4787.13	0.345391	0.172696	+	0.984975i	$0.444752\pi$
$578$	4833.81	0.347855
$579$	0	0
$580$	9856.94	0.705668
$581$	973.566	0.0695186
$582$	0	0
$583$	1533.22	0.108918
$584$	507.174	0.0359366
$585$	0	0
$586$	−2560.38	−0.180492
$587$	−18380.8	−1.29243	−0.646214	−	0.763156i	$-0.723649\pi$
$587$	−18380.8	−1.29243	−0.646214	+	0.763156i	$0.723649\pi$
$588$	0	0
$589$	−19695.3	−1.37781
$590$	20207.6	1.41005
$591$	0	0
$592$	5646.96	0.392042
$593$	13831.7	0.957843	0.478922	−	0.877858i	$-0.341028\pi$
$593$	13831.7	0.957843	0.478922	+	0.877858i	$0.341028\pi$
$594$	0	0
$595$	−3912.73	−0.269590
$596$	9738.94	0.669333
$597$	0	0
$598$	565.819	0.0386924
$599$	−12248.6	−0.835502	−0.417751	−	0.908562i	$-0.637182\pi$
$599$	−12248.6	−0.835502	−0.417751	+	0.908562i	$0.637182\pi$
$600$	0	0
$601$	9719.56	0.659682	0.329841	−	0.944036i	$-0.393005\pi$
$601$	9719.56	0.659682	0.329841	+	0.944036i	$0.393005\pi$
$602$	525.385	0.0355699
$603$	0	0
$604$	−7607.91	−0.512519
$605$	−46394.3	−3.11768
$606$	0	0
$607$	1607.83	0.107512	0.0537560	−	0.998554i	$-0.482881\pi$
$607$	1607.83	0.107512	0.0537560	+	0.998554i	$0.482881\pi$
$608$	20173.2	1.34561
$609$	0	0
$610$	9096.25	0.603764
$611$	−4208.25	−0.278638
$612$	0	0
$613$	14731.1	0.970610	0.485305	−	0.874345i	$-0.338709\pi$
$613$	14731.1	0.970610	0.485305	+	0.874345i	$0.338709\pi$
$614$	−11002.1	−0.723142
$615$	0	0
$616$	6169.68	0.403545
$617$	−27951.8	−1.82382	−0.911909	−	0.410392i	$-0.865392\pi$
$617$	−27951.8	−1.82382	−0.911909	+	0.410392i	$0.865392\pi$
$618$	0	0
$619$	16200.2	1.05192	0.525961	−	0.850509i	$-0.323706\pi$
$619$	16200.2	1.05192	0.525961	+	0.850509i	$0.323706\pi$
$620$	20079.1	1.30064
$621$	0	0
$622$	630.364	0.0406355
$623$	−526.903	−0.0338843
$624$	0	0
$625$	15263.8	0.976880
$626$	−4482.22	−0.286175
$627$	0	0
$628$	4312.28	0.274011
$629$	17468.8	1.10735
$630$	0	0
$631$	12731.8	0.803239	0.401619	−	0.915807i	$-0.368447\pi$
$631$	12731.8	0.803239	0.401619	+	0.915807i	$0.368447\pi$
$632$	3480.73	0.219076
$633$	0	0
$634$	576.927	0.0361399
$635$	21836.6	1.36466
$636$	0	0
$637$	−4154.30	−0.258397
$638$	−8379.77	−0.519997
$639$	0	0
$640$	−23756.7	−1.46729
$641$	11556.9	0.712119	0.356059	−	0.934463i	$-0.384120\pi$
$641$	11556.9	0.712119	0.356059	+	0.934463i	$0.384120\pi$
$642$	0	0
$643$	−9181.25	−0.563100	−0.281550	−	0.959547i	$-0.590848\pi$
$643$	−9181.25	−0.563100	−0.281550	+	0.959547i	$0.590848\pi$
$644$	782.716	0.0478934
$645$	0	0
$646$	6861.06	0.417871
$647$	−5244.11	−0.318651	−0.159326	−	0.987226i	$-0.550932\pi$
$647$	−5244.11	−0.318651	−0.159326	+	0.987226i	$0.550932\pi$
$648$	0	0
$649$	41804.8	2.52848
$650$	−4941.76	−0.298203
$651$	0	0
$652$	−15377.5	−0.923663
$653$	16421.4	0.984106	0.492053	−	0.870565i	$-0.336247\pi$
$653$	16421.4	0.984106	0.492053	+	0.870565i	$0.336247\pi$
$654$	0	0
$655$	−16492.4	−0.983832
$656$	1927.31	0.114709
$657$	0	0
$658$	2392.24	0.141732
$659$	−1838.11	−0.108653	−0.0543266	−	0.998523i	$-0.517301\pi$
$659$	−1838.11	−0.108653	−0.0543266	+	0.998523i	$0.517301\pi$
$660$	0	0
$661$	−5500.93	−0.323694	−0.161847	−	0.986816i	$-0.551745\pi$
$661$	−5500.93	−0.323694	−0.161847	+	0.986816i	$0.551745\pi$
$662$	6759.34	0.396841
$663$	0	0
$664$	4196.10	0.245241
$665$	10071.1	0.587281
$666$	0	0
$667$	−2563.07	−0.148789
$668$	8798.08	0.509593
$669$	0	0
$670$	19862.2	1.14529
$671$	18818.0	1.08266
$672$	0	0
$673$	−25986.7	−1.48843	−0.744216	−	0.667939i	$-0.767177\pi$
$673$	−25986.7	−1.48843	−0.744216	+	0.667939i	$0.767177\pi$
$674$	2699.91	0.154298
$675$	0	0
$676$	−958.227	−0.0545191
$677$	11691.3	0.663714	0.331857	−	0.943330i	$-0.392325\pi$
$677$	11691.3	0.663714	0.331857	+	0.943330i	$0.392325\pi$
$678$	0	0
$679$	5606.05	0.316849
$680$	−16864.0	−0.951035
$681$	0	0
$682$	−17070.0	−0.958423
$683$	11111.5	0.622501	0.311251	−	0.950328i	$-0.399252\pi$
$683$	11111.5	0.622501	0.311251	+	0.950328i	$0.399252\pi$
$684$	0	0
$685$	−9455.37	−0.527403
$686$	4896.36	0.272513
$687$	0	0
$688$	−960.370	−0.0532177
$689$	326.361	0.0180455
$690$	0	0
$691$	−7542.55	−0.415242	−0.207621	−	0.978209i	$-0.566572\pi$
$691$	−7542.55	−0.415242	−0.207621	+	0.978209i	$0.566572\pi$
$692$	−22511.3	−1.23664
$693$	0	0
$694$	3683.26	0.201462
$695$	−7873.75	−0.429738
$696$	0	0
$697$	5962.11	0.324005
$698$	15214.7	0.825051
$699$	0	0
$700$	−6836.10	−0.369114
$701$	8231.17	0.443491	0.221745	−	0.975105i	$-0.428825\pi$
$701$	8231.17	0.443491	0.221745	+	0.975105i	$0.428825\pi$
$702$	0	0
$703$	−44963.6	−2.41228
$704$	10884.2	0.582691
$705$	0	0
$706$	6893.86	0.367498
$707$	−8243.61	−0.438519
$708$	0	0
$709$	28044.6	1.48553	0.742764	−	0.669554i	$-0.233515\pi$
$709$	28044.6	1.48553	0.742764	+	0.669554i	$0.233515\pi$
$710$	9648.64	0.510010
$711$	0	0
$712$	−2270.96	−0.119534
$713$	−5221.09	−0.274238
$714$	0	0
$715$	−15354.9	−0.803133
$716$	−15257.6	−0.796374
$717$	0	0
$718$	18561.3	0.964764
$719$	−29686.4	−1.53980	−0.769901	−	0.638163i	$-0.779694\pi$
$719$	−29686.4	−1.53980	−0.769901	+	0.638163i	$0.779694\pi$
$720$	0	0
$721$	−7044.86	−0.363889
$722$	−7190.12	−0.370621
$723$	0	0
$724$	24784.0	1.27223
$725$	22385.3	1.14672
$726$	0	0
$727$	−27654.5	−1.41080	−0.705398	−	0.708812i	$-0.749232\pi$
$727$	−27654.5	−1.41080	−0.705398	+	0.708812i	$0.749232\pi$
$728$	1313.28	0.0668592
$729$	0	0
$730$	717.537	0.0363798
$731$	−2970.89	−0.150318
$732$	0	0
$733$	−13077.5	−0.658975	−0.329488	−	0.944160i	$-0.606876\pi$
$733$	−13077.5	−0.658975	−0.329488	+	0.944160i	$0.606876\pi$
$734$	4082.08	0.205276
$735$	0	0
$736$	5347.79	0.267829
$737$	41090.3	2.05370
$738$	0	0
$739$	−4218.33	−0.209978	−0.104989	−	0.994473i	$-0.533481\pi$
$739$	−4218.33	−0.209978	−0.104989	+	0.994473i	$0.533481\pi$
$740$	45839.9	2.27717
$741$	0	0
$742$	−185.525	−0.00917903
$743$	−7725.54	−0.381457	−0.190728	−	0.981643i	$-0.561085\pi$
$743$	−7725.54	−0.381457	−0.190728	+	0.981643i	$0.561085\pi$
$744$	0	0
$745$	33218.9	1.63362
$746$	−14656.5	−0.719319
$747$	0	0
$748$	−14470.6	−0.707349
$749$	3983.29	0.194321
$750$	0	0
$751$	7506.12	0.364717	0.182358	−	0.983232i	$-0.441627\pi$
$751$	7506.12	0.364717	0.182358	+	0.983232i	$0.441627\pi$
$752$	−4372.87	−0.212051
$753$	0	0
$754$	−1783.72	−0.0861531
$755$	−25950.1	−1.25089
$756$	0	0
$757$	−2741.62	−0.131632	−0.0658162	−	0.997832i	$-0.520965\pi$
$757$	−2741.62	−0.131632	−0.0658162	+	0.997832i	$0.520965\pi$
$758$	−13768.0	−0.659729
$759$	0	0
$760$	43406.9	2.07175
$761$	−29740.0	−1.41666	−0.708328	−	0.705883i	$-0.750550\pi$
$761$	−29740.0	−1.41666	−0.708328	+	0.705883i	$0.750550\pi$
$762$	0	0
$763$	2213.78	0.105038
$764$	7986.01	0.378172
$765$	0	0
$766$	−6129.18	−0.289107
$767$	8898.60	0.418918
$768$	0	0
$769$	−19896.3	−0.933004	−0.466502	−	0.884520i	$-0.654486\pi$
$769$	−19896.3	−0.933004	−0.466502	+	0.884520i	$0.654486\pi$
$770$	8728.72	0.408521
$771$	0	0
$772$	23426.6	1.09215
$773$	13601.3	0.632866	0.316433	−	0.948615i	$-0.397515\pi$
$773$	13601.3	0.632866	0.316433	+	0.948615i	$0.397515\pi$
$774$	0	0
$775$	45600.1	2.11355
$776$	24162.2	1.11775
$777$	0	0
$778$	−4160.09	−0.191705
$779$	−15346.1	−0.705818
$780$	0	0
$781$	19960.8	0.914539
$782$	1818.83	0.0831727
$783$	0	0
$784$	−4316.81	−0.196648
$785$	14708.9	0.668770
$786$	0	0
$787$	−1498.29	−0.0678631	−0.0339315	−	0.999424i	$-0.510803\pi$
$787$	−1498.29	−0.0678631	−0.0339315	+	0.999424i	$0.510803\pi$
$788$	−19284.9	−0.871824
$789$	0	0
$790$	4924.46	0.221778
$791$	1847.91	0.0830647
$792$	0	0
$793$	4005.62	0.179374
$794$	6703.49	0.299620
$795$	0	0
$796$	19962.5	0.888885
$797$	3713.30	0.165034	0.0825168	−	0.996590i	$-0.473704\pi$
$797$	3713.30	0.165034	0.0825168	+	0.996590i	$0.473704\pi$
$798$	0	0
$799$	−13527.4	−0.598955
$800$	−46706.7	−2.06416
$801$	0	0
$802$	5743.20	0.252867
$803$	1484.42	0.0652354
$804$	0	0
$805$	2669.80	0.116892
$806$	−3633.54	−0.158791
$807$	0	0
$808$	−35530.2	−1.54697
$809$	34527.6	1.50053	0.750263	−	0.661139i	$-0.229927\pi$
$809$	34527.6	1.50053	0.750263	+	0.661139i	$0.229927\pi$
$810$	0	0
$811$	−37279.2	−1.61412	−0.807059	−	0.590471i	$-0.798942\pi$
$811$	−37279.2	−1.61412	−0.807059	+	0.590471i	$0.798942\pi$
$812$	−2467.48	−0.106640
$813$	0	0
$814$	−38970.3	−1.67802
$815$	−52451.6	−2.25436
$816$	0	0
$817$	7646.90	0.327455
$818$	10376.4	0.443525
$819$	0	0
$820$	15645.2	0.666285
$821$	−13877.9	−0.589943	−0.294972	−	0.955506i	$-0.595310\pi$
$821$	−13877.9	−0.589943	−0.294972	+	0.955506i	$0.595310\pi$
$822$	0	0
$823$	18945.1	0.802410	0.401205	−	0.915988i	$-0.368592\pi$
$823$	18945.1	0.802410	0.401205	+	0.915988i	$0.368592\pi$
$824$	−30363.5	−1.28369
$825$	0	0
$826$	−5058.55	−0.213086
$827$	7804.75	0.328171	0.164086	−	0.986446i	$-0.447533\pi$
$827$	7804.75	0.328171	0.164086	+	0.986446i	$0.447533\pi$
$828$	0	0
$829$	5784.85	0.242360	0.121180	−	0.992631i	$-0.461332\pi$
$829$	5784.85	0.242360	0.121180	+	0.992631i	$0.461332\pi$
$830$	5936.54	0.248265
$831$	0	0
$832$	2316.82	0.0965402
$833$	−13354.0	−0.555448
$834$	0	0
$835$	30009.7	1.24375
$836$	37246.4	1.54090
$837$	0	0
$838$	−19224.3	−0.792471
$839$	5011.42	0.206214	0.103107	−	0.994670i	$-0.467122\pi$
$839$	5011.42	0.206214	0.103107	+	0.994670i	$0.467122\pi$
$840$	0	0
$841$	−16309.0	−0.668704
$842$	−10514.2	−0.430336
$843$	0	0
$844$	12730.3	0.519190
$845$	−3268.45	−0.133063
$846$	0	0
$847$	11613.9	0.471142
$848$	339.128	0.0137332
$849$	0	0
$850$	−15885.3	−0.641013
$851$	−11919.6	−0.480138
$852$	0	0
$853$	−22059.0	−0.885446	−0.442723	−	0.896659i	$-0.645987\pi$
$853$	−22059.0	−0.885446	−0.442723	+	0.896659i	$0.645987\pi$
$854$	−2277.06	−0.0912404
$855$	0	0
$856$	17168.1	0.685506
$857$	−13956.2	−0.556283	−0.278141	−	0.960540i	$-0.589718\pi$
$857$	−13956.2	−0.556283	−0.278141	+	0.960540i	$0.589718\pi$
$858$	0	0
$859$	12498.5	0.496442	0.248221	−	0.968703i	$-0.420154\pi$
$859$	12498.5	0.496442	0.248221	+	0.968703i	$0.420154\pi$
$860$	−7795.92	−0.309115
$861$	0	0
$862$	−11272.0	−0.445391
$863$	38631.2	1.52378	0.761890	−	0.647707i	$-0.224272\pi$
$863$	38631.2	1.52378	0.761890	+	0.647707i	$0.224272\pi$
$864$	0	0
$865$	−76784.7	−3.01822
$866$	−13842.3	−0.543163
$867$	0	0
$868$	−5026.38	−0.196551
$869$	10187.6	0.397687
$870$	0	0
$871$	8746.51	0.340257
$872$	9541.46	0.370544
$873$	0	0
$874$	−4681.55	−0.181185
$875$	−11613.5	−0.448696
$876$	0	0
$877$	−856.756	−0.0329881	−0.0164941	−	0.999864i	$-0.505250\pi$
$877$	−856.756	−0.0329881	−0.0164941	+	0.999864i	$0.505250\pi$
$878$	−25759.2	−0.990127
$879$	0	0
$880$	−15955.6	−0.611206
$881$	−33638.6	−1.28640	−0.643198	−	0.765700i	$-0.722393\pi$
$881$	−33638.6	−1.28640	−0.643198	+	0.765700i	$0.722393\pi$
$882$	0	0
$883$	−31109.1	−1.18562	−0.592811	−	0.805342i	$-0.701982\pi$
$883$	−31109.1	−1.18562	−0.592811	+	0.805342i	$0.701982\pi$
$884$	−3080.22	−0.117193
$885$	0	0
$886$	−10328.2	−0.391627
$887$	−26080.3	−0.987248	−0.493624	−	0.869675i	$-0.664328\pi$
$887$	−26080.3	−0.987248	−0.493624	+	0.869675i	$0.664328\pi$
$888$	0	0
$889$	−5466.35	−0.206227
$890$	−3212.91	−0.121008
$891$	0	0
$892$	−19455.2	−0.730277
$893$	34818.8	1.30478
$894$	0	0
$895$	−52042.8	−1.94369
$896$	5946.99	0.221736
$897$	0	0
$898$	215.142	0.00799486
$899$	16459.3	0.610621
$900$	0	0
$901$	1049.09	0.0387905
$902$	−13300.6	−0.490977
$903$	0	0
$904$	7964.56	0.293028
$905$	84536.9	3.10508
$906$	0	0
$907$	20169.0	0.738369	0.369184	−	0.929356i	$-0.379637\pi$
$907$	20169.0	0.738369	0.369184	+	0.929356i	$0.379637\pi$
$908$	−26977.3	−0.985983
$909$	0	0
$910$	1858.00	0.0676837
$911$	−19982.2	−0.726716	−0.363358	−	0.931650i	$-0.618370\pi$
$911$	−19982.2	−0.726716	−0.363358	+	0.931650i	$0.618370\pi$
$912$	0	0
$913$	12281.3	0.445184
$914$	27068.5	0.979593
$915$	0	0
$916$	24771.7	0.893538
$917$	4128.52	0.148676
$918$	0	0
$919$	38513.1	1.38241	0.691203	−	0.722661i	$-0.257081\pi$
$919$	38513.1	1.38241	0.691203	+	0.722661i	$0.257081\pi$
$920$	11506.9	0.412360
$921$	0	0
$922$	3500.70	0.125043
$923$	4248.88	0.151521
$924$	0	0
$925$	104103.	3.70043
$926$	−21020.4	−0.745975
$927$	0	0
$928$	−16858.7	−0.596352
$929$	−23218.9	−0.820009	−0.410005	−	0.912083i	$-0.634473\pi$
$929$	−23218.9	−0.820009	−0.410005	+	0.912083i	$0.634473\pi$
$930$	0	0
$931$	34372.3	1.21000
$932$	−20650.1	−0.725768
$933$	0	0
$934$	−5307.99	−0.185956
$935$	−49358.3	−1.72640
$936$	0	0
$937$	−11112.9	−0.387452	−0.193726	−	0.981056i	$-0.562057\pi$
$937$	−11112.9	−0.387452	−0.193726	+	0.981056i	$0.562057\pi$
$938$	−4972.08	−0.173075
$939$	0	0
$940$	−35497.3	−1.23170
$941$	−45570.4	−1.57869	−0.789347	−	0.613947i	$-0.789581\pi$
$941$	−45570.4	−1.57869	−0.789347	+	0.613947i	$0.789581\pi$
$942$	0	0
$943$	−4068.16	−0.140485
$944$	9246.71	0.318808
$945$	0	0
$946$	6627.62	0.227783
$947$	−34903.9	−1.19770	−0.598852	−	0.800860i	$-0.704376\pi$
$947$	−34903.9	−1.19770	−0.598852	+	0.800860i	$0.704376\pi$
$948$	0	0
$949$	315.975	0.0108082
$950$	40887.8	1.39640
$951$	0	0
$952$	4221.55	0.143720
$953$	2886.52	0.0981151	0.0490575	−	0.998796i	$-0.484378\pi$
$953$	2886.52	0.0981151	0.0490575	+	0.998796i	$0.484378\pi$
$954$	0	0
$955$	27239.8	0.922994
$956$	12679.0	0.428943
$957$	0	0
$958$	−4789.03	−0.161510
$959$	2366.96	0.0797008
$960$	0	0
$961$	3737.42	0.125455
$962$	−8295.24	−0.278014
$963$	0	0
$964$	−37074.8	−1.23869
$965$	79906.8	2.66559
$966$	0	0
$967$	11593.8	0.385556	0.192778	−	0.981242i	$-0.438250\pi$
$967$	11593.8	0.385556	0.192778	+	0.981242i	$0.438250\pi$
$968$	50056.1	1.66205
$969$	0	0
$970$	34184.1	1.13153
$971$	4952.12	0.163667	0.0818337	−	0.996646i	$-0.473922\pi$
$971$	4952.12	0.163667	0.0818337	+	0.996646i	$0.473922\pi$
$972$	0	0
$973$	1971.03	0.0649418
$974$	−9153.33	−0.301121
$975$	0	0
$976$	4162.32	0.136509
$977$	19650.1	0.643462	0.321731	−	0.946831i	$-0.395735\pi$
$977$	19650.1	0.643462	0.321731	+	0.946831i	$0.395735\pi$
$978$	0	0
$979$	−6646.77	−0.216988
$980$	−35042.2	−1.14223
$981$	0	0
$982$	−14350.8	−0.466347
$983$	−56818.4	−1.84357	−0.921783	−	0.387707i	$-0.873267\pi$
$983$	−56818.4	−1.84357	−0.921783	+	0.387707i	$0.873267\pi$
$984$	0	0
$985$	−65779.7	−2.12783
$986$	−5733.78	−0.185193
$987$	0	0
$988$	7928.30	0.255296
$989$	2027.15	0.0651764
$990$	0	0
$991$	−19120.4	−0.612897	−0.306448	−	0.951887i	$-0.599141\pi$
$991$	−19120.4	−0.612897	−0.306448	+	0.951887i	$0.599141\pi$
$992$	−34342.1	−1.09915
$993$	0	0
$994$	−2415.34	−0.0770723
$995$	68090.9	2.16947
$996$	0	0
$997$	38887.9	1.23530	0.617650	−	0.786453i	$-0.288085\pi$
$997$	38887.9	1.23530	0.617650	+	0.786453i	$0.288085\pi$
$998$	7712.07	0.244611
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.f.1.2		3
3.2	odd	2		39.4.a.c.1.2	✓	3
4.3	odd	2		1872.4.a.bk.1.1		3
12.11	even	2		624.4.a.t.1.3		3
13.12	even	2		1521.4.a.u.1.2		3
15.14	odd	2		975.4.a.l.1.2		3
21.20	even	2		1911.4.a.k.1.2		3
24.5	odd	2		2496.4.a.bl.1.1		3
24.11	even	2		2496.4.a.bp.1.1		3
39.5	even	4		507.4.b.g.337.3		6
39.8	even	4		507.4.b.g.337.4		6
39.38	odd	2		507.4.a.h.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.c.1.2	✓	3	3.2	odd	2
117.4.a.f.1.2		3	1.1	even	1	trivial
507.4.a.h.1.2		3	39.38	odd	2
507.4.b.g.337.3		6	39.5	even	4
507.4.b.g.337.4		6	39.8	even	4
624.4.a.t.1.3		3	12.11	even	2
975.4.a.l.1.2		3	15.14	odd	2
1521.4.a.u.1.2		3	13.12	even	2
1872.4.a.bk.1.1		3	4.3	odd	2
1911.4.a.k.1.2		3	21.20	even	2
2496.4.a.bl.1.1		3	24.5	odd	2
2496.4.a.bp.1.1		3	24.11	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:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	117.a (trivial)

\(f(q)\)	\(=\)	\(q-1.52644 q^{2} -5.66998 q^{4} -19.3400 q^{5} +4.84136 q^{7} +20.8664 q^{8} +29.5213 q^{10} +61.0728 q^{11} +13.0000 q^{13} -7.39005 q^{14} +13.5085 q^{16} +41.7885 q^{17} -107.561 q^{19} +109.657 q^{20} -93.2239 q^{22} -28.5138 q^{23} +249.034 q^{25} -19.8437 q^{26} -27.4504 q^{28} +89.8886 q^{29} +183.108 q^{31} -187.551 q^{32} -63.7876 q^{34} -93.6318 q^{35} +418.029 q^{37} +164.185 q^{38} -403.555 q^{40} +142.674 q^{41} -71.0935 q^{43} -346.281 q^{44} +43.5246 q^{46} -323.711 q^{47} -319.561 q^{49} -380.136 q^{50} -73.7098 q^{52} +25.1047 q^{53} -1181.15 q^{55} +101.022 q^{56} -137.210 q^{58} +684.508 q^{59} +308.125 q^{61} -279.503 q^{62} +178.217 q^{64} -251.420 q^{65} +672.808 q^{67} -236.940 q^{68} +142.923 q^{70} +326.837 q^{71} +24.3058 q^{73} -638.095 q^{74} +609.869 q^{76} +295.675 q^{77} +166.810 q^{79} -261.255 q^{80} -217.783 q^{82} +201.093 q^{83} -808.188 q^{85} +108.520 q^{86} +1274.37 q^{88} -108.834 q^{89} +62.9377 q^{91} +161.673 q^{92} +494.126 q^{94} +2080.23 q^{95} +1157.95 q^{97} +487.791 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 10 q^{4} - 4 q^{5} + 30 q^{7} + 6 q^{8} - 4 q^{10} + 16 q^{11} + 39 q^{13} + 176 q^{14} - 110 q^{16} + 146 q^{17} + 94 q^{19} + 244 q^{20} - 56 q^{22} + 48 q^{23} + 145 q^{25} - 26 q^{26}+ \cdots + 5070 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 10 * q^4 - 4 * q^5 + 30 * q^7 + 6 * q^8 - 4 * q^10 + 16 * q^11 + 39 * q^13 + 176 * q^14 - 110 * q^16 + 146 * q^17 + 94 * q^19 + 244 * q^20 - 56 * q^22 + 48 * q^23 + 145 * q^25 - 26 * q^26 + 80 * q^28 + 2 * q^29 + 302 * q^31 - 154 * q^32 + 164 * q^34 - 80 * q^35 + 374 * q^37 - 312 * q^38 - 516 * q^40 - 480 * q^41 - 260 * q^43 - 712 * q^44 - 1104 * q^46 + 24 * q^47 + 447 * q^49 - 814 * q^50 + 130 * q^52 + 678 * q^53 - 1552 * q^55 - 96 * q^56 - 628 * q^58 + 1788 * q^59 + 230 * q^61 - 1952 * q^62 - 750 * q^64 - 52 * q^65 + 74 * q^67 + 460 * q^68 + 1216 * q^70 + 948 * q^71 - 222 * q^73 - 1724 * q^74 + 2392 * q^76 - 112 * q^77 - 24 * q^79 - 1100 * q^80 + 564 * q^82 + 796 * q^83 - 248 * q^85 - 1800 * q^86 + 1608 * q^88 - 1436 * q^89 + 390 * q^91 + 1296 * q^92 - 1920 * q^94 + 4032 * q^95 + 3242 * q^97 + 5070 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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