LMFDB - Embedded newform 117.4.a.d.1.2

Error: no document with id 215702129 found in table mf_hecke_traces.

Error: table True does not exist

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,4,Mod(1,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

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

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{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$-1.56155$ 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)

$f(q)$	$=$	\(q+1.56155 q^{2} -5.56155 q^{4} +3.56155 q^{5} -27.1771 q^{7} -21.1771 q^{8} +5.56155 q^{10} -15.2614 q^{11} -13.0000 q^{13} -42.4384 q^{14} +11.4233 q^{16} -44.5464 q^{17} +23.9697 q^{19} -19.8078 q^{20} -23.8314 q^{22} -122.739 q^{23} -112.315 q^{25} -20.3002 q^{26} +151.147 q^{28} +219.909 q^{29} +27.0928 q^{31} +187.255 q^{32} -69.5616 q^{34} -96.7926 q^{35} +94.1922 q^{37} +37.4299 q^{38} -75.4233 q^{40} +160.354 q^{41} -151.302 q^{43} +84.8769 q^{44} -191.663 q^{46} -466.948 q^{47} +395.594 q^{49} -175.386 q^{50} +72.3002 q^{52} +120.847 q^{53} -54.3542 q^{55} +575.531 q^{56} +343.400 q^{58} +439.633 q^{59} -137.305 q^{61} +42.3068 q^{62} +201.022 q^{64} -46.3002 q^{65} +512.280 q^{67} +247.747 q^{68} -151.147 q^{70} -410.719 q^{71} -308.004 q^{73} +147.086 q^{74} -133.309 q^{76} +414.759 q^{77} -586.462 q^{79} +40.6847 q^{80} +250.401 q^{82} -1354.20 q^{83} -158.654 q^{85} -236.266 q^{86} +323.191 q^{88} -439.882 q^{89} +353.302 q^{91} +682.617 q^{92} -729.164 q^{94} +85.3693 q^{95} -1511.27 q^{97} +617.740 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - 7 q^{4} + 3 q^{5} - 9 q^{7} + 3 q^{8} + 7 q^{10} - 80 q^{11} - 26 q^{13} - 89 q^{14} - 39 q^{16} - 19 q^{17} - 84 q^{19} - 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} + 13 q^{26} + 125 q^{28}+ \cdots + 650 q^{98}+O(q^{100}) $ 2 * q - q^2 - 7 * q^4 + 3 * q^5 - 9 * q^7 + 3 * q^8 + 7 * q^10 - 80 * q^11 - 26 * q^13 - 89 * q^14 - 39 * q^16 - 19 * q^17 - 84 * q^19 - 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 + 13 * q^26 + 125 * q^28 + 44 * q^29 - 86 * q^31 + 123 * q^32 - 135 * q^34 - 107 * q^35 + 209 * q^37 + 314 * q^38 - 89 * q^40 + 230 * q^41 + 287 * q^43 + 178 * q^44 - 4 * q^46 - 435 * q^47 + 383 * q^49 + 144 * q^50 + 91 * q^52 + 118 * q^53 - 18 * q^55 + 1015 * q^56 + 794 * q^58 + 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 39 * q^65 + 68 * q^67 + 211 * q^68 - 125 * q^70 + 131 * q^71 + 456 * q^73 - 147 * q^74 + 22 * q^76 - 762 * q^77 - 1008 * q^79 + 69 * q^80 + 72 * q^82 - 1958 * q^83 - 173 * q^85 - 1359 * q^86 - 1242 * q^88 + 720 * q^89 + 117 * q^91 + 788 * q^92 - 811 * q^94 + 146 * q^95 - 928 * q^97 + 650 * 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 $

$2$	1.56155	0.552092	0.276046	−	0.961144i	$-0.410976\pi$
$2$	1.56155	0.552092	0.276046	+	0.961144i	$0.410976\pi$
$3$	0	0
$3$	0	0
$4$	−5.56155	−0.695194
$5$	3.56155	0.318555	0.159277	−	0.987234i	$-0.449084\pi$
$5$	3.56155	0.318555	0.159277	+	0.987234i	$0.449084\pi$
$6$	0	0
$7$	−27.1771	−1.46742	−0.733712	−	0.679460i	$-0.762214\pi$
$7$	−27.1771	−1.46742	−0.733712	+	0.679460i	$0.762214\pi$
$8$	−21.1771	−0.935904
$9$	0	0
$10$	5.56155	0.175872
$11$	−15.2614	−0.418316	−0.209158	−	0.977882i	$-0.567072\pi$
$11$	−15.2614	−0.418316	−0.209158	+	0.977882i	$0.567072\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−42.4384	−0.810154
$15$	0	0
$16$	11.4233	0.178489
$17$	−44.5464	−0.635535	−0.317767	−	0.948169i	$-0.602933\pi$
$17$	−44.5464	−0.635535	−0.317767	+	0.948169i	$0.602933\pi$
$18$	0	0
$19$	23.9697	0.289422	0.144711	−	0.989474i	$-0.453775\pi$
$19$	23.9697	0.289422	0.144711	+	0.989474i	$0.453775\pi$
$20$	−19.8078	−0.221458
$21$	0	0
$22$	−23.8314	−0.230949
$23$	−122.739	−1.11273	−0.556365	−	0.830938i	$-0.687804\pi$
$23$	−122.739	−1.11273	−0.556365	+	0.830938i	$0.687804\pi$
$24$	0	0
$25$	−112.315	−0.898523
$26$	−20.3002	−0.153123
$27$	0	0
$28$	151.147	1.02014
$29$	219.909	1.40814	0.704071	−	0.710130i	$-0.251364\pi$
$29$	219.909	1.40814	0.704071	+	0.710130i	$0.251364\pi$
$30$	0	0
$31$	27.0928	0.156968	0.0784840	−	0.996915i	$-0.474992\pi$
$31$	27.0928	0.156968	0.0784840	+	0.996915i	$0.474992\pi$
$32$	187.255	1.03445
$33$	0	0
$34$	−69.5616	−0.350874
$35$	−96.7926	−0.467455
$36$	0	0
$37$	94.1922	0.418516	0.209258	−	0.977860i	$-0.432895\pi$
$37$	94.1922	0.418516	0.209258	+	0.977860i	$0.432895\pi$
$38$	37.4299	0.159788
$39$	0	0
$40$	−75.4233	−0.298137
$41$	160.354	0.610808	0.305404	−	0.952223i	$-0.401209\pi$
$41$	160.354	0.610808	0.305404	+	0.952223i	$0.401209\pi$
$42$	0	0
$43$	−151.302	−0.536589	−0.268295	−	0.963337i	$-0.586460\pi$
$43$	−151.302	−0.536589	−0.268295	+	0.963337i	$0.586460\pi$
$44$	84.8769	0.290811
$45$	0	0
$46$	−191.663	−0.614329
$47$	−466.948	−1.44918	−0.724589	−	0.689181i	$-0.757970\pi$
$47$	−466.948	−1.44918	−0.724589	+	0.689181i	$0.757970\pi$
$48$	0	0
$49$	395.594	1.15333
$50$	−175.386	−0.496067
$51$	0	0
$52$	72.3002	0.192812
$53$	120.847	0.313199	0.156600	−	0.987662i	$-0.449947\pi$
$53$	120.847	0.313199	0.156600	+	0.987662i	$0.449947\pi$
$54$	0	0
$55$	−54.3542	−0.133257
$56$	575.531	1.37337
$57$	0	0
$58$	343.400	0.777424
$59$	439.633	0.970090	0.485045	−	0.874489i	$-0.338803\pi$
$59$	439.633	0.970090	0.485045	+	0.874489i	$0.338803\pi$
$60$	0	0
$61$	−137.305	−0.288198	−0.144099	−	0.989563i	$-0.546028\pi$
$61$	−137.305	−0.288198	−0.144099	+	0.989563i	$0.546028\pi$
$62$	42.3068	0.0866609
$63$	0	0
$64$	201.022	0.392621
$65$	−46.3002	−0.0883513
$66$	0	0
$67$	512.280	0.934104	0.467052	−	0.884230i	$-0.345316\pi$
$67$	512.280	0.934104	0.467052	+	0.884230i	$0.345316\pi$
$68$	247.747	0.441820
$69$	0	0
$70$	−151.147	−0.258078
$71$	−410.719	−0.686526	−0.343263	−	0.939239i	$-0.611532\pi$
$71$	−410.719	−0.686526	−0.343263	+	0.939239i	$0.611532\pi$
$72$	0	0
$73$	−308.004	−0.493823	−0.246912	−	0.969038i	$-0.579416\pi$
$73$	−308.004	−0.493823	−0.246912	+	0.969038i	$0.579416\pi$
$74$	147.086	0.231060
$75$	0	0
$76$	−133.309	−0.201205
$77$	414.759	0.613847
$78$	0	0
$79$	−586.462	−0.835217	−0.417608	−	0.908627i	$-0.637132\pi$
$79$	−586.462	−0.835217	−0.417608	+	0.908627i	$0.637132\pi$
$80$	40.6847	0.0568585
$81$	0	0
$82$	250.401	0.337222
$83$	−1354.20	−1.79088	−0.895440	−	0.445182i	$-0.853139\pi$
$83$	−1354.20	−1.79088	−0.895440	+	0.445182i	$0.853139\pi$
$84$	0	0
$85$	−158.654	−0.202453
$86$	−236.266	−0.296247
$87$	0	0
$88$	323.191	0.391503
$89$	−439.882	−0.523904	−0.261952	−	0.965081i	$-0.584366\pi$
$89$	−439.882	−0.523904	−0.261952	+	0.965081i	$0.584366\pi$
$90$	0	0
$91$	353.302	0.406990
$92$	682.617	0.773563
$93$	0	0
$94$	−729.164	−0.800080
$95$	85.3693	0.0921969
$96$	0	0
$97$	−1511.27	−1.58192	−0.790959	−	0.611869i	$-0.790418\pi$
$97$	−1511.27	−1.58192	−0.790959	+	0.611869i	$0.790418\pi$
$98$	617.740	0.636747
$99$	0	0
$100$	624.648	0.624648
$101$	−336.260	−0.331278	−0.165639	−	0.986186i	$-0.552969\pi$
$101$	−336.260	−0.331278	−0.165639	+	0.986186i	$0.552969\pi$
$102$	0	0
$103$	322.712	0.308716	0.154358	−	0.988015i	$-0.450669\pi$
$103$	322.712	0.308716	0.154358	+	0.988015i	$0.450669\pi$
$104$	275.302	0.259573
$105$	0	0
$106$	188.708	0.172915
$107$	−1434.62	−1.29617	−0.648083	−	0.761570i	$-0.724429\pi$
$107$	−1434.62	−1.29617	−0.648083	+	0.761570i	$0.724429\pi$
$108$	0	0
$109$	849.147	0.746179	0.373089	−	0.927795i	$-0.378298\pi$
$109$	849.147	0.746179	0.373089	+	0.927795i	$0.378298\pi$
$110$	−84.8769	−0.0735699
$111$	0	0
$112$	−310.452	−0.261919
$113$	−1614.53	−1.34409	−0.672044	−	0.740511i	$-0.734583\pi$
$113$	−1614.53	−1.34409	−0.672044	+	0.740511i	$0.734583\pi$
$114$	0	0
$115$	−437.140	−0.354465
$116$	−1223.04	−0.978931
$117$	0	0
$118$	686.509	0.535579
$119$	1210.64	0.932599
$120$	0	0
$121$	−1098.09	−0.825012
$122$	−214.409	−0.159112
$123$	0	0
$124$	−150.678	−0.109123
$125$	−845.211	−0.604784
$126$	0	0
$127$	865.174	0.604502	0.302251	−	0.953228i	$-0.402262\pi$
$127$	865.174	0.604502	0.302251	+	0.953228i	$0.402262\pi$
$128$	−1184.13	−0.817683
$129$	0	0
$130$	−72.3002	−0.0487780
$131$	281.400	0.187680	0.0938400	−	0.995587i	$-0.470086\pi$
$131$	281.400	0.187680	0.0938400	+	0.995587i	$0.470086\pi$
$132$	0	0
$133$	−651.426	−0.424705
$134$	799.953	0.515712
$135$	0	0
$136$	943.363	0.594799
$137$	2641.43	1.64725	0.823624	−	0.567137i	$-0.191949\pi$
$137$	2641.43	1.64725	0.823624	+	0.567137i	$0.191949\pi$
$138$	0	0
$139$	−1998.64	−1.21958	−0.609791	−	0.792562i	$-0.708747\pi$
$139$	−1998.64	−1.21958	−0.609791	+	0.792562i	$0.708747\pi$
$140$	538.317	0.324972
$141$	0	0
$142$	−641.359	−0.379026
$143$	198.398	0.116020
$144$	0	0
$145$	783.218	0.448570
$146$	−480.964	−0.272636
$147$	0	0
$148$	−523.855	−0.290950
$149$	1752.98	0.963824	0.481912	−	0.876220i	$-0.339942\pi$
$149$	1752.98	0.963824	0.481912	+	0.876220i	$0.339942\pi$
$150$	0	0
$151$	−2794.64	−1.50613	−0.753063	−	0.657949i	$-0.771424\pi$
$151$	−2794.64	−1.50613	−0.753063	+	0.657949i	$0.771424\pi$
$152$	−507.608	−0.270871
$153$	0	0
$154$	647.669	0.338900
$155$	96.4924	0.0500030
$156$	0	0
$157$	3244.87	1.64949	0.824743	−	0.565508i	$-0.191320\pi$
$157$	3244.87	1.64949	0.824743	+	0.565508i	$0.191320\pi$
$158$	−915.792	−0.461117
$159$	0	0
$160$	666.918	0.329528
$161$	3335.68	1.63285
$162$	0	0
$163$	3281.47	1.57684	0.788418	−	0.615139i	$-0.210900\pi$
$163$	3281.47	1.57684	0.788418	+	0.615139i	$0.210900\pi$
$164$	−891.818	−0.424630
$165$	0	0
$166$	−2114.66	−0.988731
$167$	3126.52	1.44873	0.724364	−	0.689418i	$-0.242134\pi$
$167$	3126.52	1.44873	0.724364	+	0.689418i	$0.242134\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−247.747	−0.111773
$171$	0	0
$172$	841.474	0.373034
$173$	−97.5698	−0.0428792	−0.0214396	−	0.999770i	$-0.506825\pi$
$173$	−97.5698	−0.0428792	−0.0214396	+	0.999770i	$0.506825\pi$
$174$	0	0
$175$	3052.40	1.31851
$176$	−174.335	−0.0746648
$177$	0	0
$178$	−686.900	−0.289243
$179$	34.7150	0.0144956	0.00724782	−	0.999974i	$-0.497693\pi$
$179$	34.7150	0.0144956	0.00724782	+	0.999974i	$0.497693\pi$
$180$	0	0
$181$	−1229.35	−0.504843	−0.252422	−	0.967617i	$-0.581227\pi$
$181$	−1229.35	−0.504843	−0.252422	+	0.967617i	$0.581227\pi$
$182$	551.700	0.224696
$183$	0	0
$184$	2599.25	1.04141
$185$	335.471	0.133320
$186$	0	0
$187$	679.839	0.265854
$188$	2596.96	1.00746
$189$	0	0
$190$	133.309	0.0509012
$191$	−4280.80	−1.62172	−0.810858	−	0.585243i	$-0.800999\pi$
$191$	−4280.80	−1.62172	−0.810858	+	0.585243i	$0.800999\pi$
$192$	0	0
$193$	472.320	0.176157	0.0880786	−	0.996114i	$-0.471927\pi$
$193$	472.320	0.176157	0.0880786	+	0.996114i	$0.471927\pi$
$194$	−2359.93	−0.873365
$195$	0	0
$196$	−2200.12	−0.801791
$197$	4484.37	1.62182	0.810908	−	0.585173i	$-0.198974\pi$
$197$	4484.37	1.62182	0.810908	+	0.585173i	$0.198974\pi$
$198$	0	0
$199$	−366.240	−0.130463	−0.0652314	−	0.997870i	$-0.520779\pi$
$199$	−366.240	−0.130463	−0.0652314	+	0.997870i	$0.520779\pi$
$200$	2378.51	0.840931
$201$	0	0
$202$	−525.087	−0.182896
$203$	−5976.49	−2.06634
$204$	0	0
$205$	571.110	0.194576
$206$	503.932	0.170440
$207$	0	0
$208$	−148.503	−0.0495039
$209$	−365.810	−0.121070
$210$	0	0
$211$	2122.55	0.692524	0.346262	−	0.938138i	$-0.387451\pi$
$211$	2122.55	0.692524	0.346262	+	0.938138i	$0.387451\pi$
$212$	−672.095	−0.217734
$213$	0	0
$214$	−2240.23	−0.715603
$215$	−538.870	−0.170933
$216$	0	0
$217$	−736.303	−0.230339
$218$	1325.99	0.411960
$219$	0	0
$220$	302.294	0.0926392
$221$	579.103	0.176266
$222$	0	0
$223$	−5926.42	−1.77965	−0.889826	−	0.456301i	$-0.849174\pi$
$223$	−5926.42	−1.77965	−0.889826	+	0.456301i	$0.849174\pi$
$224$	−5089.04	−1.51797
$225$	0	0
$226$	−2521.17	−0.742060
$227$	895.661	0.261881	0.130941	−	0.991390i	$-0.458200\pi$
$227$	895.661	0.261881	0.130941	+	0.991390i	$0.458200\pi$
$228$	0	0
$229$	627.717	0.181138	0.0905692	−	0.995890i	$-0.471131\pi$
$229$	627.717	0.181138	0.0905692	+	0.995890i	$0.471131\pi$
$230$	−682.617	−0.195698
$231$	0	0
$232$	−4657.03	−1.31788
$233$	−2303.72	−0.647734	−0.323867	−	0.946103i	$-0.604983\pi$
$233$	−2303.72	−0.647734	−0.323867	+	0.946103i	$0.604983\pi$
$234$	0	0
$235$	−1663.06	−0.461643
$236$	−2445.04	−0.674401
$237$	0	0
$238$	1890.48	0.514881
$239$	−544.622	−0.147400	−0.0737001	−	0.997280i	$-0.523481\pi$
$239$	−544.622	−0.147400	−0.0737001	+	0.997280i	$0.523481\pi$
$240$	0	0
$241$	5426.10	1.45031	0.725157	−	0.688584i	$-0.241767\pi$
$241$	5426.10	1.45031	0.725157	+	0.688584i	$0.241767\pi$
$242$	−1714.73	−0.455483
$243$	0	0
$244$	763.629	0.200354
$245$	1408.93	0.367400
$246$	0	0
$247$	−311.606	−0.0802713
$248$	−573.746	−0.146907
$249$	0	0
$250$	−1319.84	−0.333897
$251$	5221.22	1.31299	0.656494	−	0.754331i	$-0.272039\pi$
$251$	5221.22	1.31299	0.656494	+	0.754331i	$0.272039\pi$
$252$	0	0
$253$	1873.16	0.465472
$254$	1351.02	0.333741
$255$	0	0
$256$	−3457.26	−0.844057
$257$	−658.206	−0.159758	−0.0798789	−	0.996805i	$-0.525453\pi$
$257$	−658.206	−0.159758	−0.0798789	+	0.996805i	$0.525453\pi$
$258$	0	0
$259$	−2559.87	−0.614141
$260$	257.501	0.0614213
$261$	0	0
$262$	439.422	0.103617
$263$	−3246.45	−0.761160	−0.380580	−	0.924748i	$-0.624276\pi$
$263$	−3246.45	−0.761160	−0.380580	+	0.924748i	$0.624276\pi$
$264$	0	0
$265$	430.401	0.0997711
$266$	−1017.24	−0.234477
$267$	0	0
$268$	−2849.07	−0.649384
$269$	2585.80	0.586093	0.293047	−	0.956098i	$-0.405331\pi$
$269$	2585.80	0.586093	0.293047	+	0.956098i	$0.405331\pi$
$270$	0	0
$271$	988.933	0.221673	0.110836	−	0.993839i	$-0.464647\pi$
$271$	988.933	0.221673	0.110836	+	0.993839i	$0.464647\pi$
$272$	−508.867	−0.113436
$273$	0	0
$274$	4124.74	0.909433
$275$	1714.09	0.375866
$276$	0	0
$277$	8142.40	1.76617	0.883086	−	0.469211i	$-0.155462\pi$
$277$	8142.40	1.76617	0.883086	+	0.469211i	$0.155462\pi$
$278$	−3120.97	−0.673322
$279$	0	0
$280$	2049.78	0.437493
$281$	−1534.21	−0.325705	−0.162853	−	0.986650i	$-0.552070\pi$
$281$	−1534.21	−0.325705	−0.162853	+	0.986650i	$0.552070\pi$
$282$	0	0
$283$	−6965.00	−1.46299	−0.731495	−	0.681847i	$-0.761177\pi$
$283$	−6965.00	−1.46299	−0.731495	+	0.681847i	$0.761177\pi$
$284$	2284.23	0.477269
$285$	0	0
$286$	309.809	0.0640537
$287$	−4357.96	−0.896314
$288$	0	0
$289$	−2928.62	−0.596096
$290$	1223.04	0.247652
$291$	0	0
$292$	1712.98	0.343303
$293$	−640.029	−0.127614	−0.0638070	−	0.997962i	$-0.520324\pi$
$293$	−640.029	−0.127614	−0.0638070	+	0.997962i	$0.520324\pi$
$294$	0	0
$295$	1565.77	0.309027
$296$	−1994.72	−0.391691
$297$	0	0
$298$	2737.37	0.532120
$299$	1595.60	0.308616
$300$	0	0
$301$	4111.95	0.787404
$302$	−4363.99	−0.831520
$303$	0	0
$304$	273.813	0.0516587
$305$	−489.019	−0.0918070
$306$	0	0
$307$	−100.406	−0.0186660	−0.00933299	−	0.999956i	$-0.502971\pi$
$307$	−100.406	−0.0186660	−0.00933299	+	0.999956i	$0.502971\pi$
$308$	−2306.71	−0.426743
$309$	0	0
$310$	150.678	0.0276062
$311$	3878.92	0.707245	0.353623	−	0.935388i	$-0.384950\pi$
$311$	3878.92	0.707245	0.353623	+	0.935388i	$0.384950\pi$
$312$	0	0
$313$	−3789.39	−0.684311	−0.342155	−	0.939643i	$-0.611157\pi$
$313$	−3789.39	−0.684311	−0.342155	+	0.939643i	$0.611157\pi$
$314$	5067.04	0.910668
$315$	0	0
$316$	3261.64	0.580638
$317$	−4406.81	−0.780791	−0.390396	−	0.920647i	$-0.627662\pi$
$317$	−4406.81	−0.780791	−0.390396	+	0.920647i	$0.627662\pi$
$318$	0	0
$319$	−3356.11	−0.589048
$320$	715.950	0.125071
$321$	0	0
$322$	5208.84	0.901482
$323$	−1067.76	−0.183938
$324$	0	0
$325$	1460.10	0.249205
$326$	5124.19	0.870559
$327$	0	0
$328$	−3395.83	−0.571657
$329$	12690.3	2.12656
$330$	0	0
$331$	−4131.49	−0.686064	−0.343032	−	0.939324i	$-0.611454\pi$
$331$	−4131.49	−0.686064	−0.343032	+	0.939324i	$0.611454\pi$
$332$	7531.47	1.24501
$333$	0	0
$334$	4882.23	0.799831
$335$	1824.51	0.297564
$336$	0	0
$337$	−4560.82	−0.737221	−0.368611	−	0.929584i	$-0.620166\pi$
$337$	−4560.82	−0.737221	−0.368611	+	0.929584i	$0.620166\pi$
$338$	263.902	0.0424686
$339$	0	0
$340$	882.365	0.140744
$341$	−413.473	−0.0656622
$342$	0	0
$343$	−1429.34	−0.225007
$344$	3204.14	0.502196
$345$	0	0
$346$	−152.360	−0.0236733
$347$	−10069.4	−1.55779	−0.778896	−	0.627153i	$-0.784220\pi$
$347$	−10069.4	−1.55779	−0.778896	+	0.627153i	$0.784220\pi$
$348$	0	0
$349$	5879.32	0.901757	0.450878	−	0.892585i	$-0.351111\pi$
$349$	5879.32	0.901757	0.450878	+	0.892585i	$0.351111\pi$
$350$	4766.49	0.727942
$351$	0	0
$352$	−2857.76	−0.432725
$353$	9142.56	1.37850	0.689249	−	0.724525i	$-0.257941\pi$
$353$	9142.56	1.37850	0.689249	+	0.724525i	$0.257941\pi$
$354$	0	0
$355$	−1462.80	−0.218696
$356$	2446.43	0.364215
$357$	0	0
$358$	54.2093	0.00800293
$359$	2754.32	0.404924	0.202462	−	0.979290i	$-0.435106\pi$
$359$	2754.32	0.404924	0.202462	+	0.979290i	$0.435106\pi$
$360$	0	0
$361$	−6284.45	−0.916235
$362$	−1919.69	−0.278720
$363$	0	0
$364$	−1964.91	−0.282937
$365$	−1096.97	−0.157310
$366$	0	0
$367$	3040.19	0.432416	0.216208	−	0.976347i	$-0.430631\pi$
$367$	3040.19	0.432416	0.216208	+	0.976347i	$0.430631\pi$
$368$	−1402.08	−0.198610
$369$	0	0
$370$	523.855	0.0736052
$371$	−3284.26	−0.459596
$372$	0	0
$373$	−5384.72	−0.747481	−0.373740	−	0.927533i	$-0.621925\pi$
$373$	−5384.72	−0.747481	−0.373740	+	0.927533i	$0.621925\pi$
$374$	1061.60	0.146776
$375$	0	0
$376$	9888.59	1.35629
$377$	−2858.82	−0.390548
$378$	0	0
$379$	−3424.27	−0.464097	−0.232049	−	0.972704i	$-0.574543\pi$
$379$	−3424.27	−0.464097	−0.232049	+	0.972704i	$0.574543\pi$
$380$	−474.786	−0.0640948
$381$	0	0
$382$	−6684.69	−0.895336
$383$	382.985	0.0510956	0.0255478	−	0.999674i	$-0.491867\pi$
$383$	382.985	0.0510956	0.0255478	+	0.999674i	$0.491867\pi$
$384$	0	0
$385$	1477.19	0.195544
$386$	737.553	0.0972551
$387$	0	0
$388$	8405.00	1.09974
$389$	−8588.34	−1.11940	−0.559699	−	0.828696i	$-0.689083\pi$
$389$	−8588.34	−1.11940	−0.559699	+	0.828696i	$0.689083\pi$
$390$	0	0
$391$	5467.56	0.707178
$392$	−8377.52	−1.07941
$393$	0	0
$394$	7002.57	0.895392
$395$	−2088.72	−0.266063
$396$	0	0
$397$	−7239.16	−0.915171	−0.457586	−	0.889166i	$-0.651286\pi$
$397$	−7239.16	−0.915171	−0.457586	+	0.889166i	$0.651286\pi$
$398$	−571.904	−0.0720275
$399$	0	0
$400$	−1283.01	−0.160376
$401$	−4269.62	−0.531708	−0.265854	−	0.964013i	$-0.585654\pi$
$401$	−4269.62	−0.531708	−0.265854	+	0.964013i	$0.585654\pi$
$402$	0	0
$403$	−352.206	−0.0435351
$404$	1870.12	0.230302
$405$	0	0
$406$	−9332.60	−1.14081
$407$	−1437.50	−0.175072
$408$	0	0
$409$	13562.5	1.63967	0.819834	−	0.572602i	$-0.194066\pi$
$409$	13562.5	1.63967	0.819834	+	0.572602i	$0.194066\pi$
$410$	891.818	0.107424
$411$	0	0
$412$	−1794.78	−0.214618
$413$	−11947.9	−1.42353
$414$	0	0
$415$	−4823.06	−0.570494
$416$	−2434.31	−0.286904
$417$	0	0
$418$	−571.232	−0.0668418
$419$	14576.9	1.69959	0.849794	−	0.527114i	$-0.176726\pi$
$419$	14576.9	1.69959	0.849794	+	0.527114i	$0.176726\pi$
$420$	0	0
$421$	15848.4	1.83469	0.917343	−	0.398099i	$-0.130330\pi$
$421$	15848.4	1.83469	0.917343	+	0.398099i	$0.130330\pi$
$422$	3314.48	0.382337
$423$	0	0
$424$	−2559.18	−0.293124
$425$	5003.24	0.571042
$426$	0	0
$427$	3731.55	0.422909
$428$	7978.70	0.901087
$429$	0	0
$430$	−841.474	−0.0943709
$431$	−10694.7	−1.19524	−0.597618	−	0.801781i	$-0.703886\pi$
$431$	−10694.7	−1.19524	−0.597618	+	0.801781i	$0.703886\pi$
$432$	0	0
$433$	−16079.0	−1.78454	−0.892272	−	0.451498i	$-0.850890\pi$
$433$	−16079.0	−1.78454	−0.892272	+	0.451498i	$0.850890\pi$
$434$	−1149.78	−0.127168
$435$	0	0
$436$	−4722.57	−0.518739
$437$	−2942.01	−0.322049
$438$	0	0
$439$	6035.80	0.656203	0.328101	−	0.944643i	$-0.393591\pi$
$439$	6035.80	0.656203	0.328101	+	0.944643i	$0.393591\pi$
$440$	1151.06	0.124715
$441$	0	0
$442$	904.300	0.0973149
$443$	−10201.3	−1.09409	−0.547043	−	0.837105i	$-0.684247\pi$
$443$	−10201.3	−1.09409	−0.547043	+	0.837105i	$0.684247\pi$
$444$	0	0
$445$	−1566.66	−0.166892
$446$	−9254.41	−0.982532
$447$	0	0
$448$	−5463.19	−0.576141
$449$	5822.54	0.611988	0.305994	−	0.952033i	$-0.401011\pi$
$449$	5822.54	0.611988	0.305994	+	0.952033i	$0.401011\pi$
$450$	0	0
$451$	−2447.22	−0.255511
$452$	8979.27	0.934402
$453$	0	0
$454$	1398.62	0.144583
$455$	1258.30	0.129649
$456$	0	0
$457$	4621.60	0.473062	0.236531	−	0.971624i	$-0.423990\pi$
$457$	4621.60	0.473062	0.236531	+	0.971624i	$0.423990\pi$
$458$	980.213	0.100005
$459$	0	0
$460$	2431.18	0.246422
$461$	−5127.77	−0.518056	−0.259028	−	0.965870i	$-0.583402\pi$
$461$	−5127.77	−0.518056	−0.259028	+	0.965870i	$0.583402\pi$
$462$	0	0
$463$	6486.27	0.651064	0.325532	−	0.945531i	$-0.394457\pi$
$463$	6486.27	0.651064	0.325532	+	0.945531i	$0.394457\pi$
$464$	2512.09	0.251338
$465$	0	0
$466$	−3597.39	−0.357609
$467$	−12978.0	−1.28598	−0.642990	−	0.765875i	$-0.722306\pi$
$467$	−12978.0	−1.28598	−0.642990	+	0.765875i	$0.722306\pi$
$468$	0	0
$469$	−13922.3	−1.37073
$470$	−2596.96	−0.254869
$471$	0	0
$472$	−9310.13	−0.907910
$473$	2309.08	0.224464
$474$	0	0
$475$	−2692.16	−0.260053
$476$	−6733.04	−0.648337
$477$	0	0
$478$	−850.456	−0.0813786
$479$	5808.96	0.554109	0.277055	−	0.960854i	$-0.410642\pi$
$479$	5808.96	0.554109	0.277055	+	0.960854i	$0.410642\pi$
$480$	0	0
$481$	−1224.50	−0.116076
$482$	8473.14	0.800707
$483$	0	0
$484$	6107.09	0.573543
$485$	−5382.46	−0.503928
$486$	0	0
$487$	−5387.14	−0.501262	−0.250631	−	0.968083i	$-0.580638\pi$
$487$	−5387.14	−0.501262	−0.250631	+	0.968083i	$0.580638\pi$
$488$	2907.72	0.269726
$489$	0	0
$490$	2200.12	0.202839
$491$	−15259.1	−1.40251	−0.701255	−	0.712911i	$-0.747376\pi$
$491$	−15259.1	−1.40251	−0.701255	+	0.712911i	$0.747376\pi$
$492$	0	0
$493$	−9796.16	−0.894922
$494$	−486.589	−0.0443172
$495$	0	0
$496$	309.489	0.0280171
$497$	11162.1	1.00742
$498$	0	0
$499$	1856.04	0.166509	0.0832544	−	0.996528i	$-0.473469\pi$
$499$	1856.04	0.166509	0.0832544	+	0.996528i	$0.473469\pi$
$500$	4700.69	0.420442
$501$	0	0
$502$	8153.20	0.724891
$503$	−1049.46	−0.0930283	−0.0465142	−	0.998918i	$-0.514811\pi$
$503$	−1049.46	−0.0930283	−0.0465142	+	0.998918i	$0.514811\pi$
$504$	0	0
$505$	−1197.61	−0.105530
$506$	2925.04	0.256984
$507$	0	0
$508$	−4811.71	−0.420246
$509$	551.106	0.0479909	0.0239954	−	0.999712i	$-0.492361\pi$
$509$	551.106	0.0479909	0.0239954	+	0.999712i	$0.492361\pi$
$510$	0	0
$511$	8370.64	0.724649
$512$	4074.36	0.351686
$513$	0	0
$514$	−1027.82	−0.0882010
$515$	1149.36	0.0983431
$516$	0	0
$517$	7126.26	0.606214
$518$	−3997.37	−0.339063
$519$	0	0
$520$	980.503	0.0826883
$521$	8995.30	0.756413	0.378206	−	0.925721i	$-0.376541\pi$
$521$	8995.30	0.756413	0.378206	+	0.925721i	$0.376541\pi$
$522$	0	0
$523$	2663.91	0.222724	0.111362	−	0.993780i	$-0.464479\pi$
$523$	2663.91	0.222724	0.111362	+	0.993780i	$0.464479\pi$
$524$	−1565.02	−0.130474
$525$	0	0
$526$	−5069.51	−0.420230
$527$	−1206.89	−0.0997586
$528$	0	0
$529$	2897.77	0.238167
$530$	672.095	0.0550829
$531$	0	0
$532$	3622.94	0.295253
$533$	−2084.60	−0.169408
$534$	0	0
$535$	−5109.47	−0.412900
$536$	−10848.6	−0.874232
$537$	0	0
$538$	4037.86	0.323577
$539$	−6037.30	−0.482458
$540$	0	0
$541$	−6169.23	−0.490270	−0.245135	−	0.969489i	$-0.578832\pi$
$541$	−6169.23	−0.490270	−0.245135	+	0.969489i	$0.578832\pi$
$542$	1544.27	0.122384
$543$	0	0
$544$	−8341.52	−0.657426
$545$	3024.28	0.237699
$546$	0	0
$547$	5140.42	0.401807	0.200904	−	0.979611i	$-0.435612\pi$
$547$	5140.42	0.401807	0.200904	+	0.979611i	$0.435612\pi$
$548$	−14690.5	−1.14516
$549$	0	0
$550$	2676.64	0.207513
$551$	5271.15	0.407547
$552$	0	0
$553$	15938.3	1.22562
$554$	12714.8	0.975090
$555$	0	0
$556$	11115.5	0.847847
$557$	−2778.56	−0.211367	−0.105683	−	0.994400i	$-0.533703\pi$
$557$	−2778.56	−0.211367	−0.105683	+	0.994400i	$0.533703\pi$
$558$	0	0
$559$	1966.93	0.148823
$560$	−1105.69	−0.0834356
$561$	0	0
$562$	−2395.75	−0.179819
$563$	−4906.14	−0.367263	−0.183632	−	0.982995i	$-0.558785\pi$
$563$	−4906.14	−0.367263	−0.183632	+	0.982995i	$0.558785\pi$
$564$	0	0
$565$	−5750.22	−0.428166
$566$	−10876.2	−0.807706
$567$	0	0
$568$	8697.82	0.642522
$569$	9363.15	0.689849	0.344924	−	0.938631i	$-0.387905\pi$
$569$	9363.15	0.689849	0.344924	+	0.938631i	$0.387905\pi$
$570$	0	0
$571$	7199.32	0.527640	0.263820	−	0.964572i	$-0.415018\pi$
$571$	7199.32	0.527640	0.263820	+	0.964572i	$0.415018\pi$
$572$	−1103.40	−0.0806564
$573$	0	0
$574$	−6805.18	−0.494848
$575$	13785.4	0.999813
$576$	0	0
$577$	−11449.6	−0.826086	−0.413043	−	0.910711i	$-0.635534\pi$
$577$	−11449.6	−0.826086	−0.413043	+	0.910711i	$0.635534\pi$
$578$	−4573.19	−0.329100
$579$	0	0
$580$	−4355.91	−0.311843
$581$	36803.3	2.62798
$582$	0	0
$583$	−1844.28	−0.131016
$584$	6522.62	0.462171
$585$	0	0
$586$	−999.439	−0.0704547
$587$	5439.39	0.382466	0.191233	−	0.981545i	$-0.438751\pi$
$587$	5439.39	0.382466	0.191233	+	0.981545i	$0.438751\pi$
$588$	0	0
$589$	649.406	0.0454301
$590$	2445.04	0.170611
$591$	0	0
$592$	1075.99	0.0747006
$593$	28405.8	1.96709	0.983547	−	0.180651i	$-0.0578204\pi$
$593$	28405.8	1.96709	0.983547	+	0.180651i	$0.0578204\pi$
$594$	0	0
$595$	4311.76	0.297084
$596$	−9749.30	−0.670045
$597$	0	0
$598$	2491.62	0.170384
$599$	10482.3	0.715020	0.357510	−	0.933909i	$-0.383626\pi$
$599$	10482.3	0.715020	0.357510	+	0.933909i	$0.383626\pi$
$600$	0	0
$601$	3199.54	0.217158	0.108579	−	0.994088i	$-0.465370\pi$
$601$	3199.54	0.217158	0.108579	+	0.994088i	$0.465370\pi$
$602$	6421.02	0.434720
$603$	0	0
$604$	15542.6	1.04705
$605$	−3910.91	−0.262812
$606$	0	0
$607$	11342.8	0.758468	0.379234	−	0.925301i	$-0.376188\pi$
$607$	11342.8	0.758468	0.379234	+	0.925301i	$0.376188\pi$
$608$	4488.44	0.299392
$609$	0	0
$610$	−763.629	−0.0506859
$611$	6070.32	0.401930
$612$	0	0
$613$	14385.4	0.947831	0.473916	−	0.880570i	$-0.342840\pi$
$613$	14385.4	0.947831	0.473916	+	0.880570i	$0.342840\pi$
$614$	−156.789	−0.0103053
$615$	0	0
$616$	−8783.39	−0.574502
$617$	−22056.8	−1.43918	−0.719588	−	0.694401i	$-0.755669\pi$
$617$	−22056.8	−1.43918	−0.719588	+	0.694401i	$0.755669\pi$
$618$	0	0
$619$	13621.4	0.884477	0.442238	−	0.896898i	$-0.354185\pi$
$619$	13621.4	0.884477	0.442238	+	0.896898i	$0.354185\pi$
$620$	−536.648	−0.0347618
$621$	0	0
$622$	6057.14	0.390465
$623$	11954.7	0.768789
$624$	0	0
$625$	11029.2	0.705866
$626$	−5917.34	−0.377803
$627$	0	0
$628$	−18046.5	−1.14671
$629$	−4195.92	−0.265982
$630$	0	0
$631$	−18737.5	−1.18214	−0.591068	−	0.806622i	$-0.701293\pi$
$631$	−18737.5	−1.18214	−0.591068	+	0.806622i	$0.701293\pi$
$632$	12419.6	0.781683
$633$	0	0
$634$	−6881.46	−0.431069
$635$	3081.36	0.192567
$636$	0	0
$637$	−5142.72	−0.319877
$638$	−5240.75	−0.325209
$639$	0	0
$640$	−4217.35	−0.260477
$641$	−29798.7	−1.83616	−0.918081	−	0.396394i	$-0.870261\pi$
$641$	−29798.7	−1.83616	−0.918081	+	0.396394i	$0.870261\pi$
$642$	0	0
$643$	−22983.5	−1.40961	−0.704807	−	0.709399i	$-0.748966\pi$
$643$	−22983.5	−1.40961	−0.704807	+	0.709399i	$0.748966\pi$
$644$	−18551.5	−1.13515
$645$	0	0
$646$	−1667.37	−0.101551
$647$	24905.4	1.51334	0.756672	−	0.653794i	$-0.226824\pi$
$647$	24905.4	1.51334	0.756672	+	0.653794i	$0.226824\pi$
$648$	0	0
$649$	−6709.39	−0.405804
$650$	2280.02	0.137584
$651$	0	0
$652$	−18250.1	−1.09621
$653$	−10077.8	−0.603946	−0.301973	−	0.953316i	$-0.597645\pi$
$653$	−10077.8	−0.603946	−0.301973	+	0.953316i	$0.597645\pi$
$654$	0	0
$655$	1002.22	0.0597864
$656$	1831.77	0.109022
$657$	0	0
$658$	19816.5	1.17406
$659$	−12334.6	−0.729116	−0.364558	−	0.931181i	$-0.618780\pi$
$659$	−12334.6	−0.729116	−0.364558	+	0.931181i	$0.618780\pi$
$660$	0	0
$661$	−12749.1	−0.750202	−0.375101	−	0.926984i	$-0.622392\pi$
$661$	−12749.1	−0.750202	−0.375101	+	0.926984i	$0.622392\pi$
$662$	−6451.54	−0.378771
$663$	0	0
$664$	28678.1	1.67609
$665$	−2320.09	−0.135292
$666$	0	0
$667$	−26991.3	−1.56688
$668$	−17388.3	−1.00715
$669$	0	0
$670$	2849.07	0.164283
$671$	2095.46	0.120558
$672$	0	0
$673$	−13618.2	−0.780007	−0.390004	−	0.920813i	$-0.627526\pi$
$673$	−13618.2	−0.780007	−0.390004	+	0.920813i	$0.627526\pi$
$674$	−7121.96	−0.407014
$675$	0	0
$676$	−939.902	−0.0534765
$677$	−9655.67	−0.548150	−0.274075	−	0.961708i	$-0.588372\pi$
$677$	−9655.67	−0.548150	−0.274075	+	0.961708i	$0.588372\pi$
$678$	0	0
$679$	41071.9	2.32135
$680$	3359.84	0.189476
$681$	0	0
$682$	−645.660	−0.0362516
$683$	−16316.8	−0.914119	−0.457060	−	0.889436i	$-0.651097\pi$
$683$	−16316.8	−0.914119	−0.457060	+	0.889436i	$0.651097\pi$
$684$	0	0
$685$	9407.61	0.524739
$686$	−2232.00	−0.124225
$687$	0	0
$688$	−1728.37	−0.0957753
$689$	−1571.01	−0.0868658
$690$	0	0
$691$	2350.84	0.129421	0.0647106	−	0.997904i	$-0.479388\pi$
$691$	2350.84	0.129421	0.0647106	+	0.997904i	$0.479388\pi$
$692$	542.640	0.0298093
$693$	0	0
$694$	−15723.9	−0.860045
$695$	−7118.24	−0.388504
$696$	0	0
$697$	−7143.20	−0.388189
$698$	9180.88	0.497853
$699$	0	0
$700$	−16976.1	−0.916623
$701$	8076.90	0.435179	0.217589	−	0.976040i	$-0.430181\pi$
$701$	8076.90	0.435179	0.217589	+	0.976040i	$0.430181\pi$
$702$	0	0
$703$	2257.76	0.121128
$704$	−3067.87	−0.164239
$705$	0	0
$706$	14276.6	0.761058
$707$	9138.55	0.486125
$708$	0	0
$709$	−13624.9	−0.721712	−0.360856	−	0.932622i	$-0.617515\pi$
$709$	−13624.9	−0.721712	−0.360856	+	0.932622i	$0.617515\pi$
$710$	−2284.23	−0.120741
$711$	0	0
$712$	9315.43	0.490324
$713$	−3325.33	−0.174663
$714$	0	0
$715$	706.604	0.0369587
$716$	−193.069	−0.0100773
$717$	0	0
$718$	4301.02	0.223555
$719$	−16235.8	−0.842131	−0.421066	−	0.907030i	$-0.638344\pi$
$719$	−16235.8	−0.842131	−0.421066	+	0.907030i	$0.638344\pi$
$720$	0	0
$721$	−8770.37	−0.453018
$722$	−9813.51	−0.505846
$723$	0	0
$724$	6837.08	0.350964
$725$	−24699.2	−1.26525
$726$	0	0
$727$	24181.2	1.23361	0.616803	−	0.787118i	$-0.288428\pi$
$727$	24181.2	1.23361	0.616803	+	0.787118i	$0.288428\pi$
$728$	−7481.91	−0.380904
$729$	0	0
$730$	−1712.98	−0.0868496
$731$	6739.96	0.341021
$732$	0	0
$733$	3053.70	0.153876	0.0769379	−	0.997036i	$-0.475486\pi$
$733$	3053.70	0.153876	0.0769379	+	0.997036i	$0.475486\pi$
$734$	4747.41	0.238733
$735$	0	0
$736$	−22983.4	−1.15106
$737$	−7818.10	−0.390751
$738$	0	0
$739$	−8033.62	−0.399894	−0.199947	−	0.979807i	$-0.564077\pi$
$739$	−8033.62	−0.399894	−0.199947	+	0.979807i	$0.564077\pi$
$740$	−1865.74	−0.0926836
$741$	0	0
$742$	−5128.54	−0.253739
$743$	16139.6	0.796912	0.398456	−	0.917187i	$-0.369546\pi$
$743$	16139.6	0.796912	0.398456	+	0.917187i	$0.369546\pi$
$744$	0	0
$745$	6243.33	0.307031
$746$	−8408.53	−0.412678
$747$	0	0
$748$	−3780.96	−0.184820
$749$	38988.7	1.90202
$750$	0	0
$751$	−18491.1	−0.898469	−0.449235	−	0.893414i	$-0.648303\pi$
$751$	−18491.1	−0.898469	−0.449235	+	0.893414i	$0.648303\pi$
$752$	−5334.08	−0.258662
$753$	0	0
$754$	−4464.20	−0.215619
$755$	−9953.28	−0.479784
$756$	0	0
$757$	160.630	0.00771227	0.00385613	−	0.999993i	$-0.498773\pi$
$757$	160.630	0.00771227	0.00385613	+	0.999993i	$0.498773\pi$
$758$	−5347.17	−0.256224
$759$	0	0
$760$	−1807.87	−0.0862874
$761$	−26799.1	−1.27656	−0.638282	−	0.769803i	$-0.720355\pi$
$761$	−26799.1	−1.27656	−0.638282	+	0.769803i	$0.720355\pi$
$762$	0	0
$763$	−23077.3	−1.09496
$764$	23807.9	1.12741
$765$	0	0
$766$	598.052	0.0282095
$767$	−5715.22	−0.269054
$768$	0	0
$769$	−5145.82	−0.241304	−0.120652	−	0.992695i	$-0.538499\pi$
$769$	−5145.82	−0.241304	−0.120652	+	0.992695i	$0.538499\pi$
$770$	2306.71	0.107958
$771$	0	0
$772$	−2626.83	−0.122463
$773$	−12810.6	−0.596072	−0.298036	−	0.954555i	$-0.596332\pi$
$773$	−12810.6	−0.596072	−0.298036	+	0.954555i	$0.596332\pi$
$774$	0	0
$775$	−3042.94	−0.141039
$776$	32004.3	1.48052
$777$	0	0
$778$	−13411.1	−0.618011
$779$	3843.64	0.176781
$780$	0	0
$781$	6268.13	0.287185
$782$	8537.89	0.390428
$783$	0	0
$784$	4518.98	0.205857
$785$	11556.8	0.525452
$786$	0	0
$787$	28073.0	1.27153	0.635764	−	0.771883i	$-0.280685\pi$
$787$	28073.0	1.27153	0.635764	+	0.771883i	$0.280685\pi$
$788$	−24940.0	−1.12748
$789$	0	0
$790$	−3261.64	−0.146891
$791$	43878.1	1.97235
$792$	0	0
$793$	1784.96	0.0799318
$794$	−11304.3	−0.505259
$795$	0	0
$796$	2036.87	0.0906970
$797$	−30093.1	−1.33746	−0.668729	−	0.743507i	$-0.733161\pi$
$797$	−30093.1	−1.33746	−0.668729	+	0.743507i	$0.733161\pi$
$798$	0	0
$799$	20800.8	0.921003
$800$	−21031.6	−0.929473
$801$	0	0
$802$	−6667.24	−0.293552
$803$	4700.56	0.206574
$804$	0	0
$805$	11880.2	0.520151
$806$	−549.989	−0.0240354
$807$	0	0
$808$	7120.99	0.310044
$809$	24337.1	1.05766	0.528831	−	0.848727i	$-0.322631\pi$
$809$	24337.1	1.05766	0.528831	+	0.848727i	$0.322631\pi$
$810$	0	0
$811$	19078.7	0.826071	0.413035	−	0.910715i	$-0.364469\pi$
$811$	19078.7	0.826071	0.413035	+	0.910715i	$0.364469\pi$
$812$	33238.5	1.43651
$813$	0	0
$814$	−2244.74	−0.0966559
$815$	11687.1	0.502309
$816$	0	0
$817$	−3626.66	−0.155301
$818$	21178.6	0.905248
$819$	0	0
$820$	−3176.26	−0.135268
$821$	−2013.92	−0.0856104	−0.0428052	−	0.999083i	$-0.513629\pi$
$821$	−2013.92	−0.0856104	−0.0428052	+	0.999083i	$0.513629\pi$
$822$	0	0
$823$	−7692.10	−0.325795	−0.162898	−	0.986643i	$-0.552084\pi$
$823$	−7692.10	−0.325795	−0.162898	+	0.986643i	$0.552084\pi$
$824$	−6834.10	−0.288929
$825$	0	0
$826$	−18657.3	−0.785922
$827$	4762.76	0.200263	0.100131	−	0.994974i	$-0.468074\pi$
$827$	4762.76	0.200263	0.100131	+	0.994974i	$0.468074\pi$
$828$	0	0
$829$	19977.7	0.836976	0.418488	−	0.908222i	$-0.362560\pi$
$829$	19977.7	0.836976	0.418488	+	0.908222i	$0.362560\pi$
$830$	−7531.47	−0.314965
$831$	0	0
$832$	−2613.28	−0.108893
$833$	−17622.3	−0.732984
$834$	0	0
$835$	11135.3	0.461499
$836$	2034.47	0.0841671
$837$	0	0
$838$	22762.6	0.938330
$839$	30615.8	1.25980	0.629901	−	0.776676i	$-0.283095\pi$
$839$	30615.8	1.25980	0.629901	+	0.776676i	$0.283095\pi$
$840$	0	0
$841$	23971.0	0.982861
$842$	24748.1	1.01292
$843$	0	0
$844$	−11804.7	−0.481439
$845$	601.902	0.0245042
$846$	0	0
$847$	29842.9	1.21064
$848$	1380.47	0.0559026
$849$	0	0
$850$	7812.83	0.315268
$851$	−11561.0	−0.465696
$852$	0	0
$853$	5660.88	0.227227	0.113614	−	0.993525i	$-0.463757\pi$
$853$	5660.88	0.227227	0.113614	+	0.993525i	$0.463757\pi$
$854$	5827.01	0.233485
$855$	0	0
$856$	30381.0	1.21309
$857$	−41346.1	−1.64802	−0.824012	−	0.566572i	$-0.808269\pi$
$857$	−41346.1	−1.64802	−0.824012	+	0.566572i	$0.808269\pi$
$858$	0	0
$859$	−34810.5	−1.38268	−0.691339	−	0.722530i	$-0.742979\pi$
$859$	−34810.5	−1.38268	−0.691339	+	0.722530i	$0.742979\pi$
$860$	2996.96	0.118832
$861$	0	0
$862$	−16700.4	−0.659880
$863$	8360.51	0.329774	0.164887	−	0.986312i	$-0.447274\pi$
$863$	8360.51	0.329774	0.164887	+	0.986312i	$0.447274\pi$
$864$	0	0
$865$	−347.500	−0.0136594
$866$	−25108.2	−0.985233
$867$	0	0
$868$	4094.99	0.160130
$869$	8950.21	0.349385
$870$	0	0
$871$	−6659.64	−0.259074
$872$	−17982.4	−0.698352
$873$	0	0
$874$	−4594.10	−0.177801
$875$	22970.4	0.887475
$876$	0	0
$877$	−40579.3	−1.56245	−0.781223	−	0.624251i	$-0.785404\pi$
$877$	−40579.3	−1.56245	−0.781223	+	0.624251i	$0.785404\pi$
$878$	9425.22	0.362285
$879$	0	0
$880$	−620.903	−0.0237848
$881$	−10445.2	−0.399442	−0.199721	−	0.979853i	$-0.564004\pi$
$881$	−10445.2	−0.399442	−0.199721	+	0.979853i	$0.564004\pi$
$882$	0	0
$883$	18227.6	0.694685	0.347343	−	0.937738i	$-0.387084\pi$
$883$	18227.6	0.694685	0.347343	+	0.937738i	$0.387084\pi$
$884$	−3220.71	−0.122539
$885$	0	0
$886$	−15929.9	−0.604036
$887$	23517.7	0.890245	0.445122	−	0.895470i	$-0.353160\pi$
$887$	23517.7	0.890245	0.445122	+	0.895470i	$0.353160\pi$
$888$	0	0
$889$	−23512.9	−0.887061
$890$	−2446.43	−0.0921399
$891$	0	0
$892$	32960.1	1.23720
$893$	−11192.6	−0.419424
$894$	0	0
$895$	123.639	0.00461766
$896$	32181.2	1.19989
$897$	0	0
$898$	9092.21	0.337874
$899$	5957.95	0.221033
$900$	0	0
$901$	−5383.28	−0.199049
$902$	−3821.47	−0.141065
$903$	0	0
$904$	34191.0	1.25794
$905$	−4378.38	−0.160820
$906$	0	0
$907$	−30564.6	−1.11894	−0.559471	−	0.828850i	$-0.688996\pi$
$907$	−30564.6	−1.11894	−0.559471	+	0.828850i	$0.688996\pi$
$908$	−4981.26	−0.182058
$909$	0	0
$910$	1964.91	0.0715781
$911$	32766.5	1.19166	0.595831	−	0.803110i	$-0.296823\pi$
$911$	32766.5	1.19166	0.595831	+	0.803110i	$0.296823\pi$
$912$	0	0
$913$	20667.0	0.749154
$914$	7216.87	0.261174
$915$	0	0
$916$	−3491.08	−0.125926
$917$	−7647.64	−0.275406
$918$	0	0
$919$	20686.7	0.742538	0.371269	−	0.928525i	$-0.378923\pi$
$919$	20686.7	0.742538	0.371269	+	0.928525i	$0.378923\pi$
$920$	9257.35	0.331745
$921$	0	0
$922$	−8007.28	−0.286015
$923$	5339.34	0.190408
$924$	0	0
$925$	−10579.2	−0.376047
$926$	10128.6	0.359447
$927$	0	0
$928$	41179.0	1.45665
$929$	−45632.2	−1.61156	−0.805782	−	0.592212i	$-0.798255\pi$
$929$	−45632.2	−1.61156	−0.805782	+	0.592212i	$0.798255\pi$
$930$	0	0
$931$	9482.26	0.333801
$932$	12812.3	0.450301
$933$	0	0
$934$	−20265.9	−0.709979
$935$	2421.28	0.0846892
$936$	0	0
$937$	−17761.4	−0.619253	−0.309626	−	0.950858i	$-0.600204\pi$
$937$	−17761.4	−0.619253	−0.309626	+	0.950858i	$0.600204\pi$
$938$	−21740.4	−0.756768
$939$	0	0
$940$	9249.19	0.320931
$941$	44888.3	1.55507	0.777534	−	0.628841i	$-0.216470\pi$
$941$	44888.3	1.55507	0.777534	+	0.628841i	$0.216470\pi$
$942$	0	0
$943$	−19681.7	−0.679664
$944$	5022.05	0.173150
$945$	0	0
$946$	3605.74	0.123925
$947$	−16069.6	−0.551415	−0.275708	−	0.961242i	$-0.588912\pi$
$947$	−16069.6	−0.551415	−0.275708	+	0.961242i	$0.588912\pi$
$948$	0	0
$949$	4004.05	0.136962
$950$	−4203.96	−0.143573
$951$	0	0
$952$	−25637.8	−0.872823
$953$	−3512.03	−0.119377	−0.0596883	−	0.998217i	$-0.519011\pi$
$953$	−3512.03	−0.119377	−0.0596883	+	0.998217i	$0.519011\pi$
$954$	0	0
$955$	−15246.3	−0.516605
$956$	3028.94	0.102472
$957$	0	0
$958$	9071.00	0.305919
$959$	−71786.5	−2.41721
$960$	0	0
$961$	−29057.0	−0.975361
$962$	−1912.12	−0.0640844
$963$	0	0
$964$	−30177.5	−1.00825
$965$	1682.19	0.0561158
$966$	0	0
$967$	−37011.9	−1.23084	−0.615421	−	0.788199i	$-0.711014\pi$
$967$	−37011.9	−1.23084	−0.615421	+	0.788199i	$0.711014\pi$
$968$	23254.4	0.772132
$969$	0	0
$970$	−8405.00	−0.278215
$971$	−19532.3	−0.645542	−0.322771	−	0.946477i	$-0.604614\pi$
$971$	−19532.3	−0.645542	−0.322771	+	0.946477i	$0.604614\pi$
$972$	0	0
$973$	54317.1	1.78965
$974$	−8412.30	−0.276743
$975$	0	0
$976$	−1568.47	−0.0514402
$977$	−30201.2	−0.988970	−0.494485	−	0.869186i	$-0.664643\pi$
$977$	−30201.2	−0.988970	−0.494485	+	0.869186i	$0.664643\pi$
$978$	0	0
$979$	6713.21	0.219157
$980$	−7835.83	−0.255415
$981$	0	0
$982$	−23827.8	−0.774315
$983$	38774.9	1.25812	0.629058	−	0.777359i	$-0.283441\pi$
$983$	38774.9	1.25812	0.629058	+	0.777359i	$0.283441\pi$
$984$	0	0
$985$	15971.3	0.516638
$986$	−15297.2	−0.494080
$987$	0	0
$988$	1733.01	0.0558041
$989$	18570.6	0.597079
$990$	0	0
$991$	−27728.9	−0.888838	−0.444419	−	0.895819i	$-0.646590\pi$
$991$	−27728.9	−0.888838	−0.444419	+	0.895819i	$0.646590\pi$
$992$	5073.25	0.162375
$993$	0	0
$994$	17430.3	0.556192
$995$	−1304.38	−0.0415596
$996$	0	0
$997$	−48918.2	−1.55392	−0.776958	−	0.629552i	$-0.783239\pi$
$997$	−48918.2	−1.55392	−0.776958	+	0.629552i	$0.783239\pi$
$998$	2898.31	0.0919283
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.d.1.2		2
3.2	odd	2		13.4.a.b.1.1	✓	2
4.3	odd	2		1872.4.a.bb.1.2		2
12.11	even	2		208.4.a.h.1.1		2
13.12	even	2		1521.4.a.r.1.1		2
15.2	even	4		325.4.b.e.274.2		4
15.8	even	4		325.4.b.e.274.3		4
15.14	odd	2		325.4.a.f.1.2		2
21.20	even	2		637.4.a.b.1.1		2
24.5	odd	2		832.4.a.s.1.1		2
24.11	even	2		832.4.a.z.1.2		2
33.32	even	2		1573.4.a.b.1.2		2
39.2	even	12		169.4.e.f.147.2		8
39.5	even	4		169.4.b.f.168.3		4
39.8	even	4		169.4.b.f.168.2		4
39.11	even	12		169.4.e.f.147.3		8
39.17	odd	6		169.4.c.j.146.1		4
39.20	even	12		169.4.e.f.23.2		8
39.23	odd	6		169.4.c.j.22.1		4
39.29	odd	6		169.4.c.g.22.2		4
39.32	even	12		169.4.e.f.23.3		8
39.35	odd	6		169.4.c.g.146.2		4
39.38	odd	2		169.4.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.4.a.b.1.1	✓	2	3.2	odd	2
117.4.a.d.1.2		2	1.1	even	1	trivial
169.4.a.g.1.2		2	39.38	odd	2
169.4.b.f.168.2		4	39.8	even	4
169.4.b.f.168.3		4	39.5	even	4
169.4.c.g.22.2		4	39.29	odd	6
169.4.c.g.146.2		4	39.35	odd	6
169.4.c.j.22.1		4	39.23	odd	6
169.4.c.j.146.1		4	39.17	odd	6
169.4.e.f.23.2		8	39.20	even	12
169.4.e.f.23.3		8	39.32	even	12
169.4.e.f.147.2		8	39.2	even	12
169.4.e.f.147.3		8	39.11	even	12
208.4.a.h.1.1		2	12.11	even	2
325.4.a.f.1.2		2	15.14	odd	2
325.4.b.e.274.2		4	15.2	even	4
325.4.b.e.274.3		4	15.8	even	4
637.4.a.b.1.1		2	21.20	even	2
832.4.a.s.1.1		2	24.5	odd	2
832.4.a.z.1.2		2	24.11	even	2
1521.4.a.r.1.1		2	13.12	even	2
1573.4.a.b.1.2		2	33.32	even	2
1872.4.a.bb.1.2		2	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+1.56155 q^{2} -5.56155 q^{4} +3.56155 q^{5} -27.1771 q^{7} -21.1771 q^{8} +5.56155 q^{10} -15.2614 q^{11} -13.0000 q^{13} -42.4384 q^{14} +11.4233 q^{16} -44.5464 q^{17} +23.9697 q^{19} -19.8078 q^{20} -23.8314 q^{22} -122.739 q^{23} -112.315 q^{25} -20.3002 q^{26} +151.147 q^{28} +219.909 q^{29} +27.0928 q^{31} +187.255 q^{32} -69.5616 q^{34} -96.7926 q^{35} +94.1922 q^{37} +37.4299 q^{38} -75.4233 q^{40} +160.354 q^{41} -151.302 q^{43} +84.8769 q^{44} -191.663 q^{46} -466.948 q^{47} +395.594 q^{49} -175.386 q^{50} +72.3002 q^{52} +120.847 q^{53} -54.3542 q^{55} +575.531 q^{56} +343.400 q^{58} +439.633 q^{59} -137.305 q^{61} +42.3068 q^{62} +201.022 q^{64} -46.3002 q^{65} +512.280 q^{67} +247.747 q^{68} -151.147 q^{70} -410.719 q^{71} -308.004 q^{73} +147.086 q^{74} -133.309 q^{76} +414.759 q^{77} -586.462 q^{79} +40.6847 q^{80} +250.401 q^{82} -1354.20 q^{83} -158.654 q^{85} -236.266 q^{86} +323.191 q^{88} -439.882 q^{89} +353.302 q^{91} +682.617 q^{92} -729.164 q^{94} +85.3693 q^{95} -1511.27 q^{97} +617.740 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - 7 q^{4} + 3 q^{5} - 9 q^{7} + 3 q^{8} + 7 q^{10} - 80 q^{11} - 26 q^{13} - 89 q^{14} - 39 q^{16} - 19 q^{17} - 84 q^{19} - 19 q^{20} + 142 q^{22} - 196 q^{23} - 237 q^{25} + 13 q^{26} + 125 q^{28}+ \cdots + 650 q^{98}+O(q^{100}) \) 2 * q - q^2 - 7 * q^4 + 3 * q^5 - 9 * q^7 + 3 * q^8 + 7 * q^10 - 80 * q^11 - 26 * q^13 - 89 * q^14 - 39 * q^16 - 19 * q^17 - 84 * q^19 - 19 * q^20 + 142 * q^22 - 196 * q^23 - 237 * q^25 + 13 * q^26 + 125 * q^28 + 44 * q^29 - 86 * q^31 + 123 * q^32 - 135 * q^34 - 107 * q^35 + 209 * q^37 + 314 * q^38 - 89 * q^40 + 230 * q^41 + 287 * q^43 + 178 * q^44 - 4 * q^46 - 435 * q^47 + 383 * q^49 + 144 * q^50 + 91 * q^52 + 118 * q^53 - 18 * q^55 + 1015 * q^56 + 794 * q^58 + 368 * q^59 - 1058 * q^61 + 332 * q^62 + 769 * q^64 - 39 * q^65 + 68 * q^67 + 211 * q^68 - 125 * q^70 + 131 * q^71 + 456 * q^73 - 147 * q^74 + 22 * q^76 - 762 * q^77 - 1008 * q^79 + 69 * q^80 + 72 * q^82 - 1958 * q^83 - 173 * q^85 - 1359 * q^86 - 1242 * q^88 + 720 * q^89 + 117 * q^91 + 788 * q^92 - 811 * q^94 + 146 * q^95 - 928 * q^97 + 650 * q^98

Introduction

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