LMFDB - Embedded newform 117.4.a.g.1.3

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:	$4$
Coefficient field:	4.4.1520092.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 40x^{2} - 9x + 81 $ x^4 - x^3 - 40x^2 - 9x + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.63814$ 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)

$f(q)$	$=$	$q+3.85588 q^{2} +6.86783 q^{4} +19.5342 q^{5} -6.26434 q^{7} -4.36551 q^{8} +O(q^{10})$	\(q+3.85588 q^{2} +6.86783 q^{4} +19.5342 q^{5} -6.26434 q^{7} -4.36551 q^{8} +75.3217 q^{10} +27.2460 q^{11} -13.0000 q^{13} -24.1546 q^{14} -71.7755 q^{16} -30.8471 q^{17} +127.850 q^{19} +134.158 q^{20} +105.057 q^{22} -84.3198 q^{23} +256.586 q^{25} -50.1265 q^{26} -43.0224 q^{28} -272.460 q^{29} -166.264 q^{31} -241.834 q^{32} -118.943 q^{34} -122.369 q^{35} -198.229 q^{37} +492.976 q^{38} -85.2769 q^{40} -160.385 q^{41} +158.943 q^{43} +187.121 q^{44} -325.127 q^{46} +305.889 q^{47} -303.758 q^{49} +989.366 q^{50} -89.2818 q^{52} -356.780 q^{53} +532.229 q^{55} +27.3470 q^{56} -1050.57 q^{58} +470.384 q^{59} -171.815 q^{61} -641.096 q^{62} -358.279 q^{64} -253.945 q^{65} +1022.49 q^{67} -211.852 q^{68} -471.840 q^{70} +188.616 q^{71} +959.975 q^{73} -764.349 q^{74} +878.055 q^{76} -170.678 q^{77} -1034.80 q^{79} -1402.08 q^{80} -618.424 q^{82} +105.383 q^{83} -602.574 q^{85} +612.864 q^{86} -118.943 q^{88} +649.860 q^{89} +81.4364 q^{91} -579.094 q^{92} +1179.47 q^{94} +2497.46 q^{95} +707.746 q^{97} -1171.26 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 58 q^{4} + 36 q^{7}+O(q^{10}) $ 4 * q + 58 * q^4 + 36 * q^7	$ 4 q + 58 q^{4} + 36 q^{7} - 4 q^{10} - 52 q^{13} + 354 q^{16} + 84 q^{19} + 176 q^{22} + 660 q^{25} + 988 q^{28} - 604 q^{31} - 720 q^{34} + 184 q^{37} - 2356 q^{40} + 880 q^{43} - 2888 q^{46} - 116 q^{49} - 754 q^{52} + 1152 q^{55} - 1760 q^{58} + 656 q^{61} + 3482 q^{64} + 3052 q^{67} - 4696 q^{70} - 312 q^{73} - 2044 q^{76} - 720 q^{79} + 396 q^{82} + 32 q^{85} - 720 q^{88} - 468 q^{91} + 4840 q^{94} - 344 q^{97}+O(q^{100}) $ 4 * q + 58 * q^4 + 36 * q^7 - 4 * q^10 - 52 * q^13 + 354 * q^16 + 84 * q^19 + 176 * q^22 + 660 * q^25 + 988 * q^28 - 604 * q^31 - 720 * q^34 + 184 * q^37 - 2356 * q^40 + 880 * q^43 - 2888 * q^46 - 116 * q^49 - 754 * q^52 + 1152 * q^55 - 1760 * q^58 + 656 * q^61 + 3482 * q^64 + 3052 * q^67 - 4696 * q^70 - 312 * q^73 - 2044 * q^76 - 720 * q^79 + 396 * q^82 + 32 * q^85 - 720 * q^88 - 468 * q^91 + 4840 * q^94 - 344 * q^97

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 $

$2$	3.85588	1.36326	0.681630	−	0.731697i	$-0.261271\pi$
$2$	3.85588	1.36326	0.681630	+	0.731697i	$0.261271\pi$
$3$	0	0
$3$	0	0
$4$	6.86783	0.858479
$5$	19.5342	1.74719	0.873597	−	0.486650i	$-0.161781\pi$
$5$	19.5342	1.74719	0.873597	+	0.486650i	$0.161781\pi$
$6$	0	0
$7$	−6.26434	−0.338242	−0.169121	−	0.985595i	$-0.554093\pi$
$7$	−6.26434	−0.338242	−0.169121	+	0.985595i	$0.554093\pi$
$8$	−4.36551	−0.192930
$9$	0	0
$10$	75.3217	2.38188
$11$	27.2460	0.746816	0.373408	−	0.927667i	$-0.378189\pi$
$11$	27.2460	0.746816	0.373408	+	0.927667i	$0.378189\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	−24.1546	−0.461113
$15$	0	0
$16$	−71.7755	−1.12149
$17$	−30.8471	−0.440089	−0.220044	−	0.975490i	$-0.570620\pi$
$17$	−30.8471	−0.440089	−0.220044	+	0.975490i	$0.570620\pi$
$18$	0	0
$19$	127.850	1.54373	0.771865	−	0.635786i	$-0.219324\pi$
$19$	127.850	1.54373	0.771865	+	0.635786i	$0.219324\pi$
$20$	134.158	1.49993
$21$	0	0
$22$	105.057	1.01810
$23$	−84.3198	−0.764430	−0.382215	−	0.924073i	$-0.624839\pi$
$23$	−84.3198	−0.764430	−0.382215	+	0.924073i	$0.624839\pi$
$24$	0	0
$25$	256.586	2.05269
$26$	−50.1265	−0.378100
$27$	0	0
$28$	−43.0224	−0.290374
$29$	−272.460	−1.74464	−0.872320	−	0.488936i	$-0.837385\pi$
$29$	−272.460	−1.74464	−0.872320	+	0.488936i	$0.837385\pi$
$30$	0	0
$31$	−166.264	−0.963289	−0.481644	−	0.876367i	$-0.659960\pi$
$31$	−166.264	−0.963289	−0.481644	+	0.876367i	$0.659960\pi$
$32$	−241.834	−1.33596
$33$	0	0
$34$	−118.943	−0.599956
$35$	−122.369	−0.590975
$36$	0	0
$37$	−198.229	−0.880776	−0.440388	−	0.897808i	$-0.645159\pi$
$37$	−198.229	−0.880776	−0.440388	+	0.897808i	$0.645159\pi$
$38$	492.976	2.10451
$39$	0	0
$40$	−85.2769	−0.337086
$41$	−160.385	−0.610923	−0.305462	−	0.952204i	$-0.598811\pi$
$41$	−160.385	−0.610923	−0.305462	+	0.952204i	$0.598811\pi$
$42$	0	0
$43$	158.943	0.563687	0.281843	−	0.959460i	$-0.409054\pi$
$43$	158.943	0.563687	0.281843	+	0.959460i	$0.409054\pi$
$44$	187.121	0.641126
$45$	0	0
$46$	−325.127	−1.04212
$47$	305.889	0.949329	0.474665	−	0.880167i	$-0.342569\pi$
$47$	305.889	0.949329	0.474665	+	0.880167i	$0.342569\pi$
$48$	0	0
$49$	−303.758	−0.885592
$50$	989.366	2.79835
$51$	0	0
$52$	−89.2818	−0.238099
$53$	−356.780	−0.924669	−0.462335	−	0.886706i	$-0.652988\pi$
$53$	−356.780	−0.924669	−0.462335	+	0.886706i	$0.652988\pi$
$54$	0	0
$55$	532.229	1.30483
$56$	27.3470	0.0652571
$57$	0	0
$58$	−1050.57	−2.37840
$59$	470.384	1.03795	0.518973	−	0.854791i	$-0.326315\pi$
$59$	470.384	1.03795	0.518973	+	0.854791i	$0.326315\pi$
$60$	0	0
$61$	−171.815	−0.360635	−0.180317	−	0.983608i	$-0.557712\pi$
$61$	−171.815	−0.360635	−0.180317	+	0.983608i	$0.557712\pi$
$62$	−641.096	−1.31321
$63$	0	0
$64$	−358.279	−0.699764
$65$	−253.945	−0.484585
$66$	0	0
$67$	1022.49	1.86444	0.932220	−	0.361892i	$-0.117869\pi$
$67$	1022.49	1.86444	0.932220	+	0.361892i	$0.117869\pi$
$68$	−211.852	−0.377807
$69$	0	0
$70$	−471.840	−0.805653
$71$	188.616	0.315276	0.157638	−	0.987497i	$-0.449612\pi$
$71$	188.616	0.315276	0.157638	+	0.987497i	$0.449612\pi$
$72$	0	0
$73$	959.975	1.53913	0.769566	−	0.638568i	$-0.220473\pi$
$73$	959.975	1.53913	0.769566	+	0.638568i	$0.220473\pi$
$74$	−764.349	−1.20073
$75$	0	0
$76$	878.055	1.32526
$77$	−170.678	−0.252605
$78$	0	0
$79$	−1034.80	−1.47373	−0.736863	−	0.676042i	$-0.763694\pi$
$79$	−1034.80	−1.47373	−0.736863	+	0.676042i	$0.763694\pi$
$80$	−1402.08	−1.95947
$81$	0	0
$82$	−618.424	−0.832847
$83$	105.383	0.139365	0.0696824	−	0.997569i	$-0.477801\pi$
$83$	105.383	0.139365	0.0696824	+	0.997569i	$0.477801\pi$
$84$	0	0
$85$	−602.574	−0.768921
$86$	612.864	0.768452
$87$	0	0
$88$	−118.943	−0.144083
$89$	649.860	0.773990	0.386995	−	0.922082i	$-0.373513\pi$
$89$	649.860	0.773990	0.386995	+	0.922082i	$0.373513\pi$
$90$	0	0
$91$	81.4364	0.0938116
$92$	−579.094	−0.656247
$93$	0	0
$94$	1179.47	1.29418
$95$	2497.46	2.69720
$96$	0	0
$97$	707.746	0.740832	0.370416	−	0.928866i	$-0.379215\pi$
$97$	707.746	0.740832	0.370416	+	0.928866i	$0.379215\pi$
$98$	−1171.26	−1.20729
$99$	0	0
$100$	1762.19	1.76219
$101$	−1409.59	−1.38871	−0.694353	−	0.719634i	$-0.744310\pi$
$101$	−1409.59	−1.38871	−0.694353	+	0.719634i	$0.744310\pi$
$102$	0	0
$103$	1519.63	1.45373	0.726863	−	0.686783i	$-0.240978\pi$
$103$	1519.63	1.45373	0.726863	+	0.686783i	$0.240978\pi$
$104$	56.7516	0.0535092
$105$	0	0
$106$	−1375.70	−1.26056
$107$	−1957.57	−1.76865	−0.884323	−	0.466875i	$-0.845380\pi$
$107$	−1957.57	−1.76865	−0.884323	+	0.466875i	$0.845380\pi$
$108$	0	0
$109$	1517.63	1.33360	0.666801	−	0.745236i	$-0.267663\pi$
$109$	1517.63	1.33360	0.666801	+	0.745236i	$0.267663\pi$
$110$	2052.21	1.77883
$111$	0	0
$112$	449.626	0.379336
$113$	2200.27	1.83171	0.915857	−	0.401505i	$-0.131513\pi$
$113$	2200.27	1.83171	0.915857	+	0.401505i	$0.131513\pi$
$114$	0	0
$115$	−1647.12	−1.33561
$116$	−1871.21	−1.49774
$117$	0	0
$118$	1813.75	1.41499
$119$	193.236	0.148857
$120$	0	0
$121$	−588.656	−0.442266
$122$	−662.500	−0.491639
$123$	0	0
$124$	−1141.88	−0.826963
$125$	2570.43	1.83925
$126$	0	0
$127$	1812.16	1.26617	0.633083	−	0.774084i	$-0.281789\pi$
$127$	1812.16	1.26617	0.633083	+	0.774084i	$0.281789\pi$
$128$	553.190	0.381996
$129$	0	0
$130$	−979.182	−0.660615
$131$	−1001.31	−0.667822	−0.333911	−	0.942605i	$-0.608369\pi$
$131$	−1001.31	−0.667822	−0.333911	+	0.942605i	$0.608369\pi$
$132$	0	0
$133$	−800.898	−0.522155
$134$	3942.62	2.54172
$135$	0	0
$136$	134.663	0.0849064
$137$	−1244.11	−0.775849	−0.387925	−	0.921691i	$-0.626808\pi$
$137$	−1244.11	−0.775849	−0.387925	+	0.921691i	$0.626808\pi$
$138$	0	0
$139$	1348.85	0.823077	0.411539	−	0.911392i	$-0.364991\pi$
$139$	1348.85	0.823077	0.411539	+	0.911392i	$0.364991\pi$
$140$	−840.410	−0.507340
$141$	0	0
$142$	727.281	0.429804
$143$	−354.198	−0.207129
$144$	0	0
$145$	−5322.29	−3.04822
$146$	3701.55	2.09824
$147$	0	0
$148$	−1361.41	−0.756128
$149$	1041.63	0.572710	0.286355	−	0.958124i	$-0.407556\pi$
$149$	1041.63	0.572710	0.286355	+	0.958124i	$0.407556\pi$
$150$	0	0
$151$	1361.90	0.733970	0.366985	−	0.930227i	$-0.380390\pi$
$151$	1361.90	0.733970	0.366985	+	0.930227i	$0.380390\pi$
$152$	−558.132	−0.297832
$153$	0	0
$154$	−658.115	−0.344366
$155$	−3247.85	−1.68305
$156$	0	0
$157$	86.8978	0.0441733	0.0220866	−	0.999756i	$-0.492969\pi$
$157$	86.8978	0.0441733	0.0220866	+	0.999756i	$0.492969\pi$
$158$	−3990.08	−2.00907
$159$	0	0
$160$	−4724.04	−2.33418
$161$	528.208	0.258563
$162$	0	0
$163$	−28.5386	−0.0137136	−0.00685679	−	0.999976i	$-0.502183\pi$
$163$	−28.5386	−0.0137136	−0.00685679	+	0.999976i	$0.502183\pi$
$164$	−1101.49	−0.524465
$165$	0	0
$166$	406.344	0.189990
$167$	−1289.87	−0.597683	−0.298842	−	0.954303i	$-0.596600\pi$
$167$	−1289.87	−0.597683	−0.298842	+	0.954303i	$0.596600\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	−2323.45	−1.04824
$171$	0	0
$172$	1091.59	0.483913
$173$	603.489	0.265216	0.132608	−	0.991169i	$-0.457665\pi$
$173$	603.489	0.265216	0.132608	+	0.991169i	$0.457665\pi$
$174$	0	0
$175$	−1607.34	−0.694306
$176$	−1955.60	−0.837549
$177$	0	0
$178$	2505.79	1.05515
$179$	−1985.29	−0.828980	−0.414490	−	0.910054i	$-0.636040\pi$
$179$	−1985.29	−0.828980	−0.414490	+	0.910054i	$0.636040\pi$
$180$	0	0
$181$	−3945.79	−1.62038	−0.810189	−	0.586169i	$-0.800635\pi$
$181$	−3945.79	−1.62038	−0.810189	+	0.586169i	$0.800635\pi$
$182$	314.009	0.127890
$183$	0	0
$184$	368.099	0.147482
$185$	−3872.26	−1.53889
$186$	0	0
$187$	−840.459	−0.328665
$188$	2100.79	0.814979
$189$	0	0
$190$	9629.91	3.67698
$191$	1446.48	0.547979	0.273989	−	0.961733i	$-0.411657\pi$
$191$	1446.48	0.547979	0.273989	+	0.961733i	$0.411657\pi$
$192$	0	0
$193$	784.523	0.292597	0.146299	−	0.989240i	$-0.453264\pi$
$193$	784.523	0.292597	0.146299	+	0.989240i	$0.453264\pi$
$194$	2728.98	1.00995
$195$	0	0
$196$	−2086.16	−0.760262
$197$	494.336	0.178782	0.0893909	−	0.995997i	$-0.471508\pi$
$197$	494.336	0.178782	0.0893909	+	0.995997i	$0.471508\pi$
$198$	0	0
$199$	−3022.59	−1.07671	−0.538357	−	0.842717i	$-0.680955\pi$
$199$	−3022.59	−1.07671	−0.538357	+	0.842717i	$0.680955\pi$
$200$	−1120.13	−0.396025
$201$	0	0
$202$	−5435.21	−1.89317
$203$	1706.78	0.590111
$204$	0	0
$205$	−3132.99	−1.06740
$206$	5859.52	1.98181
$207$	0	0
$208$	933.082	0.311046
$209$	3483.41	1.15288
$210$	0	0
$211$	3185.24	1.03925	0.519623	−	0.854396i	$-0.326073\pi$
$211$	3185.24	1.03925	0.519623	+	0.854396i	$0.326073\pi$
$212$	−2450.30	−0.793809
$213$	0	0
$214$	−7548.15	−2.41113
$215$	3104.82	0.984870
$216$	0	0
$217$	1041.54	0.325825
$218$	5851.81	1.81805
$219$	0	0
$220$	3655.26	1.12017
$221$	401.012	0.122059
$222$	0	0
$223$	997.686	0.299596	0.149798	−	0.988717i	$-0.452138\pi$
$223$	997.686	0.299596	0.149798	+	0.988717i	$0.452138\pi$
$224$	1514.93	0.451877
$225$	0	0
$226$	8483.97	2.49710
$227$	−6496.57	−1.89953	−0.949763	−	0.312969i	$-0.898676\pi$
$227$	−6496.57	−1.89953	−0.949763	+	0.312969i	$0.898676\pi$
$228$	0	0
$229$	708.434	0.204431	0.102215	−	0.994762i	$-0.467407\pi$
$229$	708.434	0.204431	0.102215	+	0.994762i	$0.467407\pi$
$230$	−6351.11	−1.82078
$231$	0	0
$232$	1189.43	0.336593
$233$	4919.02	1.38307	0.691536	−	0.722342i	$-0.256934\pi$
$233$	4919.02	1.38307	0.691536	+	0.722342i	$0.256934\pi$
$234$	0	0
$235$	5975.30	1.65866
$236$	3230.52	0.891054
$237$	0	0
$238$	745.097	0.202930
$239$	324.370	0.0877898	0.0438949	−	0.999036i	$-0.486023\pi$
$239$	324.370	0.0877898	0.0438949	+	0.999036i	$0.486023\pi$
$240$	0	0
$241$	−1940.43	−0.518649	−0.259324	−	0.965790i	$-0.583500\pi$
$241$	−1940.43	−0.518649	−0.259324	+	0.965790i	$0.583500\pi$
$242$	−2269.79	−0.602924
$243$	0	0
$244$	−1180.00	−0.309597
$245$	−5933.68	−1.54730
$246$	0	0
$247$	−1662.05	−0.428154
$248$	725.829	0.185847
$249$	0	0
$250$	9911.28	2.50738
$251$	−1958.45	−0.492495	−0.246248	−	0.969207i	$-0.579198\pi$
$251$	−1958.45	−0.492495	−0.246248	+	0.969207i	$0.579198\pi$
$252$	0	0
$253$	−2297.38	−0.570889
$254$	6987.47	1.72611
$255$	0	0
$256$	4999.27	1.22052
$257$	−4330.35	−1.05105	−0.525525	−	0.850778i	$-0.676131\pi$
$257$	−4330.35	−1.05105	−0.525525	+	0.850778i	$0.676131\pi$
$258$	0	0
$259$	1241.78	0.297916
$260$	−1744.05	−0.416006
$261$	0	0
$262$	−3860.93	−0.910415
$263$	−2933.60	−0.687807	−0.343904	−	0.939005i	$-0.611749\pi$
$263$	−2933.60	−0.687807	−0.343904	+	0.939005i	$0.611749\pi$
$264$	0	0
$265$	−6969.42	−1.61558
$266$	−3088.17	−0.711834
$267$	0	0
$268$	7022.31	1.60058
$269$	2458.25	0.557184	0.278592	−	0.960410i	$-0.410132\pi$
$269$	2458.25	0.557184	0.278592	+	0.960410i	$0.410132\pi$
$270$	0	0
$271$	2089.91	0.468462	0.234231	−	0.972181i	$-0.424743\pi$
$271$	2089.91	0.468462	0.234231	+	0.972181i	$0.424743\pi$
$272$	2214.06	0.493557
$273$	0	0
$274$	−4797.14	−1.05768
$275$	6990.94	1.53298
$276$	0	0
$277$	−2462.00	−0.534033	−0.267017	−	0.963692i	$-0.586038\pi$
$277$	−2462.00	−0.534033	−0.267017	+	0.963692i	$0.586038\pi$
$278$	5201.00	1.12207
$279$	0	0
$280$	534.203	0.114017
$281$	−5775.91	−1.22620	−0.613100	−	0.790005i	$-0.710078\pi$
$281$	−5775.91	−1.22620	−0.613100	+	0.790005i	$0.710078\pi$
$282$	0	0
$283$	5093.45	1.06987	0.534936	−	0.844892i	$-0.320336\pi$
$283$	5093.45	1.06987	0.534936	+	0.844892i	$0.320336\pi$
$284$	1295.38	0.270658
$285$	0	0
$286$	−1365.75	−0.282371
$287$	1004.70	0.206640
$288$	0	0
$289$	−3961.46	−0.806322
$290$	−20522.1	−4.15552
$291$	0	0
$292$	6592.95	1.32131
$293$	−1503.73	−0.299826	−0.149913	−	0.988699i	$-0.547899\pi$
$293$	−1503.73	−0.299826	−0.149913	+	0.988699i	$0.547899\pi$
$294$	0	0
$295$	9188.59	1.81349
$296$	865.372	0.169928
$297$	0	0
$298$	4016.41	0.780753
$299$	1096.16	0.212015
$300$	0	0
$301$	−995.670	−0.190663
$302$	5251.31	1.00059
$303$	0	0
$304$	−9176.53	−1.73128
$305$	−3356.28	−0.630099
$306$	0	0
$307$	−6567.59	−1.22095	−0.610476	−	0.792035i	$-0.709022\pi$
$307$	−6567.59	−1.22095	−0.610476	+	0.792035i	$0.709022\pi$
$308$	−1172.19	−0.216856
$309$	0	0
$310$	−12523.3	−2.29444
$311$	5714.66	1.04196	0.520978	−	0.853570i	$-0.325567\pi$
$311$	5714.66	1.04196	0.520978	+	0.853570i	$0.325567\pi$
$312$	0	0
$313$	−4953.59	−0.894548	−0.447274	−	0.894397i	$-0.647605\pi$
$313$	−4953.59	−0.894548	−0.447274	+	0.894397i	$0.647605\pi$
$314$	335.068	0.0602197
$315$	0	0
$316$	−7106.85	−1.26516
$317$	−2526.64	−0.447666	−0.223833	−	0.974628i	$-0.571857\pi$
$317$	−2526.64	−0.447666	−0.223833	+	0.974628i	$0.571857\pi$
$318$	0	0
$319$	−7423.44	−1.30292
$320$	−6998.71	−1.22262
$321$	0	0
$322$	2036.71	0.352488
$323$	−3943.81	−0.679379
$324$	0	0
$325$	−3335.62	−0.569313
$326$	−110.041	−0.0186952
$327$	0	0
$328$	700.160	0.117865
$329$	−1916.19	−0.321103
$330$	0	0
$331$	7436.06	1.23481	0.617406	−	0.786645i	$-0.288184\pi$
$331$	7436.06	1.23481	0.617406	+	0.786645i	$0.288184\pi$
$332$	723.752	0.119642
$333$	0	0
$334$	−4973.59	−0.814798
$335$	19973.6	3.25754
$336$	0	0
$337$	9467.96	1.53042	0.765212	−	0.643778i	$-0.222634\pi$
$337$	9467.96	1.53042	0.765212	+	0.643778i	$0.222634\pi$
$338$	651.644	0.104866
$339$	0	0
$340$	−4138.37	−0.660102
$341$	−4530.04	−0.719400
$342$	0	0
$343$	4051.51	0.637787
$344$	−693.866	−0.108752
$345$	0	0
$346$	2326.98	0.361559
$347$	2794.85	0.432379	0.216189	−	0.976351i	$-0.430637\pi$
$347$	2794.85	0.432379	0.216189	+	0.976351i	$0.430637\pi$
$348$	0	0
$349$	2602.91	0.399228	0.199614	−	0.979875i	$-0.436031\pi$
$349$	2602.91	0.399228	0.199614	+	0.979875i	$0.436031\pi$
$350$	−6197.72	−0.946520
$351$	0	0
$352$	−6589.01	−0.997714
$353$	8875.63	1.33825	0.669125	−	0.743150i	$-0.266669\pi$
$353$	8875.63	1.33825	0.669125	+	0.743150i	$0.266669\pi$
$354$	0	0
$355$	3684.47	0.550849
$356$	4463.13	0.664454
$357$	0	0
$358$	−7655.04	−1.13012
$359$	8946.74	1.31529	0.657647	−	0.753326i	$-0.271552\pi$
$359$	8946.74	1.31529	0.657647	+	0.753326i	$0.271552\pi$
$360$	0	0
$361$	9486.72	1.38310
$362$	−15214.5	−2.20900
$363$	0	0
$364$	559.291	0.0805353
$365$	18752.4	2.68916
$366$	0	0
$367$	−3965.30	−0.563997	−0.281998	−	0.959415i	$-0.590997\pi$
$367$	−3965.30	−0.563997	−0.281998	+	0.959415i	$0.590997\pi$
$368$	6052.10	0.857303
$369$	0	0
$370$	−14931.0	−2.09790
$371$	2234.99	0.312762
$372$	0	0
$373$	−1457.44	−0.202315	−0.101157	−	0.994870i	$-0.532255\pi$
$373$	−1457.44	−0.202315	−0.101157	+	0.994870i	$0.532255\pi$
$374$	−3240.71	−0.448057
$375$	0	0
$376$	−1335.36	−0.183154
$377$	3541.98	0.483876
$378$	0	0
$379$	1991.75	0.269945	0.134973	−	0.990849i	$-0.456905\pi$
$379$	1991.75	0.269945	0.134973	+	0.990849i	$0.456905\pi$
$380$	17152.1	2.31549
$381$	0	0
$382$	5577.48	0.747038
$383$	−2504.45	−0.334130	−0.167065	−	0.985946i	$-0.553429\pi$
$383$	−2504.45	−0.334130	−0.167065	+	0.985946i	$0.553429\pi$
$384$	0	0
$385$	−3334.06	−0.441350
$386$	3025.03	0.398886
$387$	0	0
$388$	4860.68	0.635988
$389$	1448.12	0.188747	0.0943735	−	0.995537i	$-0.469915\pi$
$389$	1448.12	0.188747	0.0943735	+	0.995537i	$0.469915\pi$
$390$	0	0
$391$	2601.02	0.336417
$392$	1326.06	0.170857
$393$	0	0
$394$	1906.10	0.243726
$395$	−20214.1	−2.57489
$396$	0	0
$397$	−5634.14	−0.712265	−0.356133	−	0.934435i	$-0.615905\pi$
$397$	−5634.14	−0.712265	−0.356133	+	0.934435i	$0.615905\pi$
$398$	−11654.8	−1.46784
$399$	0	0
$400$	−18416.6	−2.30208
$401$	−3648.07	−0.454304	−0.227152	−	0.973859i	$-0.572942\pi$
$401$	−3648.07	−0.454304	−0.227152	+	0.973859i	$0.572942\pi$
$402$	0	0
$403$	2161.44	0.267168
$404$	−9680.82	−1.19218
$405$	0	0
$406$	6581.15	0.804475
$407$	−5400.96	−0.657778
$408$	0	0
$409$	−4459.91	−0.539190	−0.269595	−	0.962974i	$-0.586890\pi$
$409$	−4459.91	−0.539190	−0.269595	+	0.962974i	$0.586890\pi$
$410$	−12080.4	−1.45515
$411$	0	0
$412$	10436.6	1.24799
$413$	−2946.64	−0.351077
$414$	0	0
$415$	2058.57	0.243497
$416$	3143.84	0.370528
$417$	0	0
$418$	13431.6	1.57168
$419$	4726.74	0.551114	0.275557	−	0.961285i	$-0.411138\pi$
$419$	4726.74	0.551114	0.275557	+	0.961285i	$0.411138\pi$
$420$	0	0
$421$	−3090.51	−0.357772	−0.178886	−	0.983870i	$-0.557249\pi$
$421$	−3090.51	−0.357772	−0.178886	+	0.983870i	$0.557249\pi$
$422$	12281.9	1.41676
$423$	0	0
$424$	1557.53	0.178397
$425$	−7914.92	−0.903365
$426$	0	0
$427$	1076.31	0.121982
$428$	−13444.2	−1.51835
$429$	0	0
$430$	11971.8	1.34263
$431$	−11131.9	−1.24409	−0.622045	−	0.782981i	$-0.713698\pi$
$431$	−11131.9	−1.24409	−0.622045	+	0.782981i	$0.713698\pi$
$432$	0	0
$433$	6773.33	0.751745	0.375872	−	0.926671i	$-0.377343\pi$
$433$	6773.33	0.751745	0.375872	+	0.926671i	$0.377343\pi$
$434$	4016.04	0.444185
$435$	0	0
$436$	10422.8	1.14487
$437$	−10780.3	−1.18007
$438$	0	0
$439$	−5384.95	−0.585443	−0.292722	−	0.956198i	$-0.594561\pi$
$439$	−5384.95	−0.585443	−0.292722	+	0.956198i	$0.594561\pi$
$440$	−2323.45	−0.251742
$441$	0	0
$442$	1546.25	0.166398
$443$	921.806	0.0988631	0.0494315	−	0.998778i	$-0.484259\pi$
$443$	921.806	0.0988631	0.0494315	+	0.998778i	$0.484259\pi$
$444$	0	0
$445$	12694.5	1.35231
$446$	3846.96	0.408428
$447$	0	0
$448$	2244.38	0.236690
$449$	−11902.6	−1.25104	−0.625521	−	0.780207i	$-0.715113\pi$
$449$	−11902.6	−1.25104	−0.625521	+	0.780207i	$0.715113\pi$
$450$	0	0
$451$	−4369.84	−0.456247
$452$	15111.1	1.57249
$453$	0	0
$454$	−25050.0	−2.58955
$455$	1590.80	0.163907
$456$	0	0
$457$	1420.66	0.145418	0.0727088	−	0.997353i	$-0.476836\pi$
$457$	1420.66	0.145418	0.0727088	+	0.997353i	$0.476836\pi$
$458$	2731.64	0.278692
$459$	0	0
$460$	−11312.2	−1.14659
$461$	−1768.23	−0.178644	−0.0893218	−	0.996003i	$-0.528470\pi$
$461$	−1768.23	−0.178644	−0.0893218	+	0.996003i	$0.528470\pi$
$462$	0	0
$463$	18637.4	1.87074	0.935370	−	0.353670i	$-0.115067\pi$
$463$	18637.4	1.87074	0.935370	+	0.353670i	$0.115067\pi$
$464$	19556.0	1.95660
$465$	0	0
$466$	18967.2	1.88549
$467$	−9042.01	−0.895962	−0.447981	−	0.894043i	$-0.647857\pi$
$467$	−9042.01	−0.895962	−0.447981	+	0.894043i	$0.647857\pi$
$468$	0	0
$469$	−6405.25	−0.630633
$470$	23040.1	2.26119
$471$	0	0
$472$	−2053.47	−0.200251
$473$	4330.55	0.420970
$474$	0	0
$475$	32804.6	3.16880
$476$	1327.12	0.127790
$477$	0	0
$478$	1250.73	0.119680
$479$	−9547.64	−0.910737	−0.455368	−	0.890303i	$-0.650492\pi$
$479$	−9547.64	−0.910737	−0.455368	+	0.890303i	$0.650492\pi$
$480$	0	0
$481$	2576.98	0.244283
$482$	−7482.08	−0.707053
$483$	0	0
$484$	−4042.79	−0.379676
$485$	13825.3	1.29438
$486$	0	0
$487$	3692.81	0.343608	0.171804	−	0.985131i	$-0.445040\pi$
$487$	3692.81	0.343608	0.171804	+	0.985131i	$0.445040\pi$
$488$	750.062	0.0695773
$489$	0	0
$490$	−22879.6	−2.10937
$491$	3487.77	0.320572	0.160286	−	0.987071i	$-0.448758\pi$
$491$	3487.77	0.320572	0.160286	+	0.987071i	$0.448758\pi$
$492$	0	0
$493$	8404.59	0.767796
$494$	−6408.69	−0.583685
$495$	0	0
$496$	11933.7	1.08032
$497$	−1181.55	−0.106640
$498$	0	0
$499$	15089.0	1.35366	0.676830	−	0.736139i	$-0.263353\pi$
$499$	15089.0	1.35366	0.676830	+	0.736139i	$0.263353\pi$
$500$	17653.3	1.57896
$501$	0	0
$502$	−7551.56	−0.671400
$503$	−8815.48	−0.781437	−0.390718	−	0.920510i	$-0.627773\pi$
$503$	−8815.48	−0.781437	−0.390718	+	0.920510i	$0.627773\pi$
$504$	0	0
$505$	−27535.2	−2.42634
$506$	−8858.41	−0.778270
$507$	0	0
$508$	12445.6	1.08698
$509$	−19491.2	−1.69731	−0.848655	−	0.528946i	$-0.822587\pi$
$509$	−19491.2	−1.69731	−0.848655	+	0.528946i	$0.822587\pi$
$510$	0	0
$511$	−6013.61	−0.520599
$512$	14851.1	1.28190
$513$	0	0
$514$	−16697.3	−1.43285
$515$	29684.8	2.53994
$516$	0	0
$517$	8334.24	0.708974
$518$	4788.14	0.406137
$519$	0	0
$520$	1108.60	0.0934909
$521$	6128.73	0.515364	0.257682	−	0.966230i	$-0.417041\pi$
$521$	6128.73	0.515364	0.257682	+	0.966230i	$0.417041\pi$
$522$	0	0
$523$	−22618.6	−1.89109	−0.945546	−	0.325489i	$-0.894471\pi$
$523$	−22618.6	−1.89109	−0.945546	+	0.325489i	$0.894471\pi$
$524$	−6876.82	−0.573311
$525$	0	0
$526$	−11311.6	−0.937660
$527$	5128.77	0.423933
$528$	0	0
$529$	−5057.17	−0.415647
$530$	−26873.3	−2.20245
$531$	0	0
$532$	−5500.43	−0.448259
$533$	2085.00	0.169440
$534$	0	0
$535$	−38239.6	−3.09017
$536$	−4463.71	−0.359707
$537$	0	0
$538$	9478.74	0.759587
$539$	−8276.19	−0.661374
$540$	0	0
$541$	−15950.5	−1.26759	−0.633796	−	0.773500i	$-0.718504\pi$
$541$	−15950.5	−1.26759	−0.633796	+	0.773500i	$0.718504\pi$
$542$	8058.47	0.638636
$543$	0	0
$544$	7459.87	0.587940
$545$	29645.7	2.33006
$546$	0	0
$547$	−1972.83	−0.154208	−0.0771042	−	0.997023i	$-0.524567\pi$
$547$	−1972.83	−0.154208	−0.0771042	+	0.997023i	$0.524567\pi$
$548$	−8544.33	−0.666050
$549$	0	0
$550$	26956.2	2.08985
$551$	−34834.1	−2.69325
$552$	0	0
$553$	6482.35	0.498477
$554$	−9493.18	−0.728027
$555$	0	0
$556$	9263.66	0.706595
$557$	19974.8	1.51950	0.759749	−	0.650217i	$-0.225322\pi$
$557$	19974.8	1.51950	0.759749	+	0.650217i	$0.225322\pi$
$558$	0	0
$559$	−2066.25	−0.156339
$560$	8783.10	0.662775
$561$	0	0
$562$	−22271.2	−1.67163
$563$	13153.4	0.984637	0.492319	−	0.870415i	$-0.336149\pi$
$563$	13153.4	0.984637	0.492319	+	0.870415i	$0.336149\pi$
$564$	0	0
$565$	42980.5	3.20036
$566$	19639.7	1.45851
$567$	0	0
$568$	−823.405	−0.0608263
$569$	−4133.86	−0.304570	−0.152285	−	0.988337i	$-0.548663\pi$
$569$	−4133.86	−0.304570	−0.152285	+	0.988337i	$0.548663\pi$
$570$	0	0
$571$	−3394.69	−0.248798	−0.124399	−	0.992232i	$-0.539700\pi$
$571$	−3394.69	−0.248798	−0.124399	+	0.992232i	$0.539700\pi$
$572$	−2432.57	−0.177816
$573$	0	0
$574$	3874.02	0.281704
$575$	−21635.3	−1.56914
$576$	0	0
$577$	4210.56	0.303792	0.151896	−	0.988396i	$-0.451462\pi$
$577$	4210.56	0.303792	0.151896	+	0.988396i	$0.451462\pi$
$578$	−15274.9	−1.09923
$579$	0	0
$580$	−36552.6	−2.61684
$581$	−660.154	−0.0471391
$582$	0	0
$583$	−9720.82	−0.690558
$584$	−4190.78	−0.296945
$585$	0	0
$586$	−5798.21	−0.408740
$587$	7252.27	0.509937	0.254969	−	0.966949i	$-0.417935\pi$
$587$	7252.27	0.509937	0.254969	+	0.966949i	$0.417935\pi$
$588$	0	0
$589$	−21257.0	−1.48706
$590$	35430.1	2.47226
$591$	0	0
$592$	14228.0	0.987784
$593$	13899.4	0.962531	0.481266	−	0.876575i	$-0.340177\pi$
$593$	13899.4	0.962531	0.481266	+	0.876575i	$0.340177\pi$
$594$	0	0
$595$	3774.72	0.260082
$596$	7153.75	0.491659
$597$	0	0
$598$	4226.65	0.289031
$599$	−18140.7	−1.23741	−0.618704	−	0.785624i	$-0.712342\pi$
$599$	−18140.7	−1.23741	−0.618704	+	0.785624i	$0.712342\pi$
$600$	0	0
$601$	−26808.3	−1.81952	−0.909761	−	0.415133i	$-0.863735\pi$
$601$	−26808.3	−1.81952	−0.909761	+	0.415133i	$0.863735\pi$
$602$	−3839.19	−0.259923
$603$	0	0
$604$	9353.27	0.630098
$605$	−11498.9	−0.772724
$606$	0	0
$607$	3769.98	0.252091	0.126045	−	0.992024i	$-0.459772\pi$
$607$	3769.98	0.252091	0.126045	+	0.992024i	$0.459772\pi$
$608$	−30918.6	−2.06236
$609$	0	0
$610$	−12941.4	−0.858989
$611$	−3976.55	−0.263297
$612$	0	0
$613$	2722.99	0.179413	0.0897067	−	0.995968i	$-0.471407\pi$
$613$	2722.99	0.179413	0.0897067	+	0.995968i	$0.471407\pi$
$614$	−25323.9	−1.66448
$615$	0	0
$616$	745.097	0.0487351
$617$	−11947.1	−0.779533	−0.389767	−	0.920914i	$-0.627444\pi$
$617$	−11947.1	−0.779533	−0.389767	+	0.920914i	$0.627444\pi$
$618$	0	0
$619$	−18386.8	−1.19391	−0.596953	−	0.802276i	$-0.703622\pi$
$619$	−18386.8	−1.19391	−0.596953	+	0.802276i	$0.703622\pi$
$620$	−22305.7	−1.44487
$621$	0	0
$622$	22035.1	1.42046
$623$	−4070.95	−0.261796
$624$	0	0
$625$	18138.1	1.16084
$626$	−19100.5	−1.21950
$627$	0	0
$628$	596.800	0.0379218
$629$	6114.79	0.387620
$630$	0	0
$631$	−25308.3	−1.59669	−0.798343	−	0.602202i	$-0.794290\pi$
$631$	−25308.3	−1.59669	−0.798343	+	0.602202i	$0.794290\pi$
$632$	4517.44	0.284326
$633$	0	0
$634$	−9742.42	−0.610285
$635$	35399.1	2.21224
$636$	0	0
$637$	3948.85	0.245619
$638$	−28623.9	−1.77623
$639$	0	0
$640$	10806.1	0.667422
$641$	9670.76	0.595901	0.297950	−	0.954581i	$-0.403697\pi$
$641$	9670.76	0.595901	0.297950	+	0.954581i	$0.403697\pi$
$642$	0	0
$643$	19673.0	1.20657	0.603286	−	0.797525i	$-0.293858\pi$
$643$	19673.0	1.20657	0.603286	+	0.797525i	$0.293858\pi$
$644$	3627.64	0.221971
$645$	0	0
$646$	−15206.9	−0.926170
$647$	13369.3	0.812367	0.406183	−	0.913792i	$-0.366859\pi$
$647$	13369.3	0.812367	0.406183	+	0.913792i	$0.366859\pi$
$648$	0	0
$649$	12816.1	0.775154
$650$	−12861.8	−0.776122
$651$	0	0
$652$	−195.998	−0.0117728
$653$	3088.52	0.185089	0.0925446	−	0.995709i	$-0.470500\pi$
$653$	3088.52	0.185089	0.0925446	+	0.995709i	$0.470500\pi$
$654$	0	0
$655$	−19559.8	−1.16682
$656$	11511.7	0.685146
$657$	0	0
$658$	−7388.61	−0.437748
$659$	−31326.8	−1.85177	−0.925886	−	0.377802i	$-0.876680\pi$
$659$	−31326.8	−1.85177	−0.925886	+	0.377802i	$0.876680\pi$
$660$	0	0
$661$	146.828	0.00863986	0.00431993	−	0.999991i	$-0.498625\pi$
$661$	146.828	0.00863986	0.00431993	+	0.999991i	$0.498625\pi$
$662$	28672.6	1.68337
$663$	0	0
$664$	−460.050	−0.0268877
$665$	−15644.9	−0.912307
$666$	0	0
$667$	22973.8	1.33365
$668$	−8858.61	−0.513098
$669$	0	0
$670$	77016.0	4.44087
$671$	−4681.28	−0.269328
$672$	0	0
$673$	−3463.75	−0.198392	−0.0991961	−	0.995068i	$-0.531627\pi$
$673$	−3463.75	−0.198392	−0.0991961	+	0.995068i	$0.531627\pi$
$674$	36507.4	2.08637
$675$	0	0
$676$	1160.66	0.0660368
$677$	534.786	0.0303597	0.0151798	−	0.999885i	$-0.495168\pi$
$677$	534.786	0.0303597	0.0151798	+	0.999885i	$0.495168\pi$
$678$	0	0
$679$	−4433.56	−0.250581
$680$	2630.54	0.148348
$681$	0	0
$682$	−17467.3	−0.980729
$683$	22369.0	1.25318	0.626592	−	0.779348i	$-0.284449\pi$
$683$	22369.0	1.25318	0.626592	+	0.779348i	$0.284449\pi$
$684$	0	0
$685$	−24302.7	−1.35556
$686$	15622.2	0.869470
$687$	0	0
$688$	−11408.2	−0.632171
$689$	4638.14	0.256457
$690$	0	0
$691$	25007.2	1.37673	0.688363	−	0.725366i	$-0.258329\pi$
$691$	25007.2	1.37673	0.688363	+	0.725366i	$0.258329\pi$
$692$	4144.66	0.227683
$693$	0	0
$694$	10776.6	0.589445
$695$	26348.7	1.43808
$696$	0	0
$697$	4947.39	0.268861
$698$	10036.5	0.544251
$699$	0	0
$700$	−11038.9	−0.596047
$701$	27977.1	1.50739	0.753696	−	0.657223i	$-0.228269\pi$
$701$	27977.1	1.50739	0.753696	+	0.657223i	$0.228269\pi$
$702$	0	0
$703$	−25343.7	−1.35968
$704$	−9761.67	−0.522595
$705$	0	0
$706$	34223.4	1.82438
$707$	8830.14	0.469720
$708$	0	0
$709$	6374.31	0.337648	0.168824	−	0.985646i	$-0.446003\pi$
$709$	6374.31	0.337648	0.168824	+	0.985646i	$0.446003\pi$
$710$	14206.9	0.750950
$711$	0	0
$712$	−2836.97	−0.149326
$713$	14019.4	0.736367
$714$	0	0
$715$	−6918.98	−0.361895
$716$	−13634.6	−0.711662
$717$	0	0
$718$	34497.6	1.79309
$719$	25433.3	1.31919	0.659597	−	0.751619i	$-0.270727\pi$
$719$	25433.3	1.31919	0.659597	+	0.751619i	$0.270727\pi$
$720$	0	0
$721$	−9519.48	−0.491711
$722$	36579.7	1.88553
$723$	0	0
$724$	−27099.0	−1.39106
$725$	−69909.4	−3.58120
$726$	0	0
$727$	−15847.8	−0.808476	−0.404238	−	0.914654i	$-0.632463\pi$
$727$	−15847.8	−0.808476	−0.404238	+	0.914654i	$0.632463\pi$
$728$	−355.511	−0.0180991
$729$	0	0
$730$	72306.9	3.66603
$731$	−4902.91	−0.248072
$732$	0	0
$733$	9378.34	0.472574	0.236287	−	0.971683i	$-0.424069\pi$
$733$	9378.34	0.472574	0.236287	+	0.971683i	$0.424069\pi$
$734$	−15289.7	−0.768874
$735$	0	0
$736$	20391.4	1.02125
$737$	27858.9	1.39239
$738$	0	0
$739$	12956.2	0.644928	0.322464	−	0.946582i	$-0.395489\pi$
$739$	12956.2	0.644928	0.322464	+	0.946582i	$0.395489\pi$
$740$	−26594.0	−1.32110
$741$	0	0
$742$	8617.85	0.426376
$743$	−16776.6	−0.828363	−0.414181	−	0.910194i	$-0.635932\pi$
$743$	−16776.6	−0.828363	−0.414181	+	0.910194i	$0.635932\pi$
$744$	0	0
$745$	20347.5	1.00064
$746$	−5619.72	−0.275808
$747$	0	0
$748$	−5772.13	−0.282152
$749$	12262.9	0.598231
$750$	0	0
$751$	11213.4	0.544849	0.272425	−	0.962177i	$-0.412174\pi$
$751$	11213.4	0.544849	0.272425	+	0.962177i	$0.412174\pi$
$752$	−21955.3	−1.06467
$753$	0	0
$754$	13657.5	0.659649
$755$	26603.6	1.28239
$756$	0	0
$757$	852.253	0.0409190	0.0204595	−	0.999791i	$-0.493487\pi$
$757$	852.253	0.0409190	0.0204595	+	0.999791i	$0.493487\pi$
$758$	7679.95	0.368006
$759$	0	0
$760$	−10902.7	−0.520371
$761$	−35706.1	−1.70085	−0.850425	−	0.526097i	$-0.823655\pi$
$761$	−35706.1	−1.70085	−0.850425	+	0.526097i	$0.823655\pi$
$762$	0	0
$763$	−9506.95	−0.451081
$764$	9934.21	0.470428
$765$	0	0
$766$	−9656.88	−0.455506
$767$	−6114.99	−0.287874
$768$	0	0
$769$	−3663.63	−0.171800	−0.0858999	−	0.996304i	$-0.527377\pi$
$769$	−3663.63	−0.171800	−0.0858999	+	0.996304i	$0.527377\pi$
$770$	−12855.8	−0.601675
$771$	0	0
$772$	5387.97	0.251188
$773$	−17985.6	−0.836865	−0.418432	−	0.908248i	$-0.637420\pi$
$773$	−17985.6	−0.836865	−0.418432	+	0.908248i	$0.637420\pi$
$774$	0	0
$775$	−42661.1	−1.97733
$776$	−3089.67	−0.142929
$777$	0	0
$778$	5583.78	0.257311
$779$	−20505.2	−0.943101
$780$	0	0
$781$	5139.03	0.235453
$782$	10029.2	0.458624
$783$	0	0
$784$	21802.4	0.993185
$785$	1697.48	0.0771793
$786$	0	0
$787$	−19322.0	−0.875167	−0.437584	−	0.899178i	$-0.644166\pi$
$787$	−19322.0	−0.875167	−0.437584	+	0.899178i	$0.644166\pi$
$788$	3395.02	0.153480
$789$	0	0
$790$	−77943.1	−3.51024
$791$	−13783.2	−0.619563
$792$	0	0
$793$	2233.60	0.100022
$794$	−21724.6	−0.971003
$795$	0	0
$796$	−20758.7	−0.924336
$797$	10689.5	0.475082	0.237541	−	0.971378i	$-0.423659\pi$
$797$	10689.5	0.475082	0.237541	+	0.971378i	$0.423659\pi$
$798$	0	0
$799$	−9435.77	−0.417789
$800$	−62051.2	−2.74230
$801$	0	0
$802$	−14066.5	−0.619335
$803$	26155.5	1.14945
$804$	0	0
$805$	10318.1	0.451759
$806$	8334.25	0.364220
$807$	0	0
$808$	6153.58	0.267923
$809$	−16756.4	−0.728213	−0.364107	−	0.931357i	$-0.618626\pi$
$809$	−16756.4	−0.728213	−0.364107	+	0.931357i	$0.618626\pi$
$810$	0	0
$811$	−33333.7	−1.44329	−0.721643	−	0.692266i	$-0.756613\pi$
$811$	−33333.7	−1.44329	−0.721643	+	0.692266i	$0.756613\pi$
$812$	11721.9	0.506598
$813$	0	0
$814$	−20825.5	−0.896722
$815$	−557.479	−0.0239603
$816$	0	0
$817$	20320.9	0.870180
$818$	−17196.9	−0.735056
$819$	0	0
$820$	−21516.8	−0.916342
$821$	−21242.1	−0.902988	−0.451494	−	0.892274i	$-0.649109\pi$
$821$	−21242.1	−0.902988	−0.451494	+	0.892274i	$0.649109\pi$
$822$	0	0
$823$	3994.06	0.169167	0.0845834	−	0.996416i	$-0.473044\pi$
$823$	3994.06	0.169167	0.0845834	+	0.996416i	$0.473044\pi$
$824$	−6633.96	−0.280467
$825$	0	0
$826$	−11361.9	−0.478610
$827$	16952.5	0.712814	0.356407	−	0.934331i	$-0.384002\pi$
$827$	16952.5	0.712814	0.356407	+	0.934331i	$0.384002\pi$
$828$	0	0
$829$	38917.5	1.63047	0.815237	−	0.579128i	$-0.196607\pi$
$829$	38917.5	1.63047	0.815237	+	0.579128i	$0.196607\pi$
$830$	7937.62	0.331950
$831$	0	0
$832$	4657.63	0.194080
$833$	9370.04	0.389739
$834$	0	0
$835$	−25196.6	−1.04427
$836$	23923.5	0.989726
$837$	0	0
$838$	18225.8	0.751311
$839$	25671.3	1.05634	0.528172	−	0.849138i	$-0.322878\pi$
$839$	25671.3	1.05634	0.528172	+	0.849138i	$0.322878\pi$
$840$	0	0
$841$	49845.4	2.04377
$842$	−11916.6	−0.487737
$843$	0	0
$844$	21875.7	0.892170
$845$	3301.28	0.134400
$846$	0	0
$847$	3687.54	0.149593
$848$	25608.1	1.03701
$849$	0	0
$850$	−30519.0	−1.23152
$851$	16714.7	0.673292
$852$	0	0
$853$	−37444.6	−1.50302	−0.751511	−	0.659720i	$-0.770675\pi$
$853$	−37444.6	−1.50302	−0.751511	+	0.659720i	$0.770675\pi$
$854$	4150.12	0.166293
$855$	0	0
$856$	8545.78	0.341225
$857$	−36610.6	−1.45927	−0.729636	−	0.683836i	$-0.760310\pi$
$857$	−36610.6	−1.45927	−0.729636	+	0.683836i	$0.760310\pi$
$858$	0	0
$859$	18043.5	0.716688	0.358344	−	0.933590i	$-0.383341\pi$
$859$	18043.5	0.716688	0.358344	+	0.933590i	$0.383341\pi$
$860$	21323.4	0.845490
$861$	0	0
$862$	−42923.1	−1.69602
$863$	−8697.94	−0.343084	−0.171542	−	0.985177i	$-0.554875\pi$
$863$	−8697.94	−0.343084	−0.171542	+	0.985177i	$0.554875\pi$
$864$	0	0
$865$	11788.7	0.463384
$866$	26117.2	1.02482
$867$	0	0
$868$	7153.09	0.279714
$869$	−28194.2	−1.10060
$870$	0	0
$871$	−13292.4	−0.517103
$872$	−6625.23	−0.257292
$873$	0	0
$874$	−41567.6	−1.60875
$875$	−16102.0	−0.622113
$876$	0	0
$877$	26657.4	1.02640	0.513202	−	0.858268i	$-0.328459\pi$
$877$	26657.4	1.02640	0.513202	+	0.858268i	$0.328459\pi$
$878$	−20763.7	−0.798111
$879$	0	0
$880$	−38201.1	−1.46336
$881$	13896.5	0.531424	0.265712	−	0.964053i	$-0.414393\pi$
$881$	13896.5	0.531424	0.265712	+	0.964053i	$0.414393\pi$
$882$	0	0
$883$	24343.3	0.927767	0.463884	−	0.885896i	$-0.346456\pi$
$883$	24343.3	0.927767	0.463884	+	0.885896i	$0.346456\pi$
$884$	2754.08	0.104785
$885$	0	0
$886$	3554.38	0.134776
$887$	4592.33	0.173839	0.0869196	−	0.996215i	$-0.472298\pi$
$887$	4592.33	0.173839	0.0869196	+	0.996215i	$0.472298\pi$
$888$	0	0
$889$	−11352.0	−0.428271
$890$	48948.6	1.84355
$891$	0	0
$892$	6851.94	0.257197
$893$	39108.0	1.46551
$894$	0	0
$895$	−38781.1	−1.44839
$896$	−3465.37	−0.129207
$897$	0	0
$898$	−45895.0	−1.70550
$899$	45300.4	1.68059
$900$	0	0
$901$	11005.6	0.406937
$902$	−16849.6	−0.621984
$903$	0	0
$904$	−9605.29	−0.353393
$905$	−77078.0	−2.83111
$906$	0	0
$907$	31406.7	1.14977	0.574885	−	0.818234i	$-0.305047\pi$
$907$	31406.7	1.14977	0.574885	+	0.818234i	$0.305047\pi$
$908$	−44617.4	−1.63070
$909$	0	0
$910$	6133.93	0.223448
$911$	−44825.0	−1.63021	−0.815103	−	0.579317i	$-0.803319\pi$
$911$	−44825.0	−1.63021	−0.815103	+	0.579317i	$0.803319\pi$
$912$	0	0
$913$	2871.26	0.104080
$914$	5477.91	0.198242
$915$	0	0
$916$	4865.40	0.175499
$917$	6272.53	0.225886
$918$	0	0
$919$	22241.1	0.798332	0.399166	−	0.916879i	$-0.369300\pi$
$919$	22241.1	0.798332	0.399166	+	0.916879i	$0.369300\pi$
$920$	7190.53	0.257679
$921$	0	0
$922$	−6818.09	−0.243538
$923$	−2452.01	−0.0874419
$924$	0	0
$925$	−50862.9	−1.80796
$926$	71863.6	2.55031
$927$	0	0
$928$	65890.1	2.33076
$929$	16292.5	0.575393	0.287697	−	0.957722i	$-0.407111\pi$
$929$	16292.5	0.575393	0.287697	+	0.957722i	$0.407111\pi$
$930$	0	0
$931$	−38835.6	−1.36712
$932$	33783.0	1.18734
$933$	0	0
$934$	−34864.9	−1.22143
$935$	−16417.7	−0.574242
$936$	0	0
$937$	24400.8	0.850734	0.425367	−	0.905021i	$-0.360145\pi$
$937$	24400.8	0.850734	0.425367	+	0.905021i	$0.360145\pi$
$938$	−24697.9	−0.859717
$939$	0	0
$940$	41037.4	1.42393
$941$	41529.9	1.43872	0.719360	−	0.694638i	$-0.244435\pi$
$941$	41529.9	1.43872	0.719360	+	0.694638i	$0.244435\pi$
$942$	0	0
$943$	13523.6	0.467008
$944$	−33762.1	−1.16405
$945$	0	0
$946$	16698.1	0.573892
$947$	−29981.8	−1.02880	−0.514402	−	0.857549i	$-0.671986\pi$
$947$	−29981.8	−1.02880	−0.514402	+	0.857549i	$0.671986\pi$
$948$	0	0
$949$	−12479.7	−0.426878
$950$	126491.	4.31990
$951$	0	0
$952$	−843.575	−0.0287189
$953$	−28594.4	−0.971943	−0.485972	−	0.873975i	$-0.661534\pi$
$953$	−28594.4	−0.971943	−0.485972	+	0.873975i	$0.661534\pi$
$954$	0	0
$955$	28256.0	0.957426
$956$	2227.72	0.0753657
$957$	0	0
$958$	−36814.6	−1.24157
$959$	7793.52	0.262425
$960$	0	0
$961$	−2147.17	−0.0720745
$962$	9936.54	0.333022
$963$	0	0
$964$	−13326.6	−0.445249
$965$	15325.1	0.511224
$966$	0	0
$967$	−28729.1	−0.955393	−0.477696	−	0.878525i	$-0.658528\pi$
$967$	−28729.1	−0.955393	−0.477696	+	0.878525i	$0.658528\pi$
$968$	2569.78	0.0853264
$969$	0	0
$970$	53308.6	1.76457
$971$	−27550.7	−0.910549	−0.455274	−	0.890351i	$-0.650459\pi$
$971$	−27550.7	−0.910549	−0.455274	+	0.890351i	$0.650459\pi$
$972$	0	0
$973$	−8449.64	−0.278400
$974$	14239.0	0.468427
$975$	0	0
$976$	12332.1	0.404449
$977$	23812.0	0.779746	0.389873	−	0.920869i	$-0.372519\pi$
$977$	23812.0	0.779746	0.389873	+	0.920869i	$0.372519\pi$
$978$	0	0
$979$	17706.1	0.578028
$980$	−40751.5	−1.32833
$981$	0	0
$982$	13448.4	0.437023
$983$	32668.3	1.05998	0.529988	−	0.848005i	$-0.322196\pi$
$983$	32668.3	1.05998	0.529988	+	0.848005i	$0.322196\pi$
$984$	0	0
$985$	9656.48	0.312367
$986$	32407.1	1.04671
$987$	0	0
$988$	−11414.7	−0.367561
$989$	−13402.0	−0.430899
$990$	0	0
$991$	−19654.5	−0.630015	−0.315008	−	0.949089i	$-0.602007\pi$
$991$	−19654.5	−0.630015	−0.315008	+	0.949089i	$0.602007\pi$
$992$	40208.4	1.28691
$993$	0	0
$994$	−4555.94	−0.145378
$995$	−59044.0	−1.88123
$996$	0	0
$997$	17666.5	0.561188	0.280594	−	0.959827i	$-0.409469\pi$
$997$	17666.5	0.561188	0.280594	+	0.959827i	$0.409469\pi$
$998$	58181.4	1.84539
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.4.a.g.1.3	yes	4
3.2	odd	2	inner	117.4.a.g.1.2	✓	4
4.3	odd	2		1872.4.a.bo.1.4		4
12.11	even	2		1872.4.a.bo.1.1		4
13.12	even	2		1521.4.a.ba.1.2		4
39.38	odd	2		1521.4.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.4.a.g.1.2	✓	4	3.2	odd	2	inner
117.4.a.g.1.3	yes	4	1.1	even	1	trivial
1521.4.a.ba.1.2		4	13.12	even	2
1521.4.a.ba.1.3		4	39.38	odd	2
1872.4.a.bo.1.1		4	12.11	even	2
1872.4.a.bo.1.4		4	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+3.85588 q^{2} +6.86783 q^{4} +19.5342 q^{5} -6.26434 q^{7} -4.36551 q^{8} +O(q^{10})\)	\(q+3.85588 q^{2} +6.86783 q^{4} +19.5342 q^{5} -6.26434 q^{7} -4.36551 q^{8} +75.3217 q^{10} +27.2460 q^{11} -13.0000 q^{13} -24.1546 q^{14} -71.7755 q^{16} -30.8471 q^{17} +127.850 q^{19} +134.158 q^{20} +105.057 q^{22} -84.3198 q^{23} +256.586 q^{25} -50.1265 q^{26} -43.0224 q^{28} -272.460 q^{29} -166.264 q^{31} -241.834 q^{32} -118.943 q^{34} -122.369 q^{35} -198.229 q^{37} +492.976 q^{38} -85.2769 q^{40} -160.385 q^{41} +158.943 q^{43} +187.121 q^{44} -325.127 q^{46} +305.889 q^{47} -303.758 q^{49} +989.366 q^{50} -89.2818 q^{52} -356.780 q^{53} +532.229 q^{55} +27.3470 q^{56} -1050.57 q^{58} +470.384 q^{59} -171.815 q^{61} -641.096 q^{62} -358.279 q^{64} -253.945 q^{65} +1022.49 q^{67} -211.852 q^{68} -471.840 q^{70} +188.616 q^{71} +959.975 q^{73} -764.349 q^{74} +878.055 q^{76} -170.678 q^{77} -1034.80 q^{79} -1402.08 q^{80} -618.424 q^{82} +105.383 q^{83} -602.574 q^{85} +612.864 q^{86} -118.943 q^{88} +649.860 q^{89} +81.4364 q^{91} -579.094 q^{92} +1179.47 q^{94} +2497.46 q^{95} +707.746 q^{97} -1171.26 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 58 q^{4} + 36 q^{7}+O(q^{10}) \) 4 * q + 58 * q^4 + 36 * q^7	\( 4 q + 58 q^{4} + 36 q^{7} - 4 q^{10} - 52 q^{13} + 354 q^{16} + 84 q^{19} + 176 q^{22} + 660 q^{25} + 988 q^{28} - 604 q^{31} - 720 q^{34} + 184 q^{37} - 2356 q^{40} + 880 q^{43} - 2888 q^{46} - 116 q^{49} - 754 q^{52} + 1152 q^{55} - 1760 q^{58} + 656 q^{61} + 3482 q^{64} + 3052 q^{67} - 4696 q^{70} - 312 q^{73} - 2044 q^{76} - 720 q^{79} + 396 q^{82} + 32 q^{85} - 720 q^{88} - 468 q^{91} + 4840 q^{94} - 344 q^{97}+O(q^{100}) \) 4 * q + 58 * q^4 + 36 * q^7 - 4 * q^10 - 52 * q^13 + 354 * q^16 + 84 * q^19 + 176 * q^22 + 660 * q^25 + 988 * q^28 - 604 * q^31 - 720 * q^34 + 184 * q^37 - 2356 * q^40 + 880 * q^43 - 2888 * q^46 - 116 * q^49 - 754 * q^52 + 1152 * q^55 - 1760 * q^58 + 656 * q^61 + 3482 * q^64 + 3052 * q^67 - 4696 * q^70 - 312 * q^73 - 2044 * q^76 - 720 * q^79 + 396 * q^82 + 32 * q^85 - 720 * q^88 - 468 * q^91 + 4840 * q^94 - 344 * q^97

Introduction

L-functions

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