LMFDB - Embedded newform 2151.4.a.g.1.49

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, 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("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.49
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.99709 q^{2} +7.97675 q^{4} +4.04840 q^{5} -1.10463 q^{7} -0.0929295 q^{8} +O(q^{10})$	\(q+3.99709 q^{2} +7.97675 q^{4} +4.04840 q^{5} -1.10463 q^{7} -0.0929295 q^{8} +16.1818 q^{10} -47.9991 q^{11} +50.4246 q^{13} -4.41533 q^{14} -64.1855 q^{16} +51.1190 q^{17} -87.9833 q^{19} +32.2931 q^{20} -191.857 q^{22} +141.821 q^{23} -108.610 q^{25} +201.552 q^{26} -8.81139 q^{28} -88.7250 q^{29} +242.373 q^{31} -255.812 q^{32} +204.328 q^{34} -4.47200 q^{35} +93.5246 q^{37} -351.677 q^{38} -0.376216 q^{40} -437.766 q^{41} +173.062 q^{43} -382.877 q^{44} +566.870 q^{46} -147.320 q^{47} -341.780 q^{49} -434.126 q^{50} +402.224 q^{52} -118.071 q^{53} -194.320 q^{55} +0.102653 q^{56} -354.642 q^{58} -674.993 q^{59} +122.013 q^{61} +968.787 q^{62} -509.020 q^{64} +204.139 q^{65} -711.424 q^{67} +407.764 q^{68} -17.8750 q^{70} -1026.79 q^{71} -1224.17 q^{73} +373.827 q^{74} -701.821 q^{76} +53.0214 q^{77} -13.8124 q^{79} -259.849 q^{80} -1749.79 q^{82} +113.011 q^{83} +206.950 q^{85} +691.743 q^{86} +4.46053 q^{88} +1140.93 q^{89} -55.7007 q^{91} +1131.27 q^{92} -588.853 q^{94} -356.192 q^{95} -476.494 q^{97} -1366.13 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	3.99709	1.41319	0.706593	−	0.707620i	$-0.250231\pi$
$2$	3.99709	1.41319	0.706593	+	0.707620i	$0.250231\pi$
$3$	0	0
$3$	0	0
$4$	7.97675	0.997094
$5$	4.04840	0.362100	0.181050	−	0.983474i	$-0.442050\pi$
$5$	4.04840	0.362100	0.181050	+	0.983474i	$0.442050\pi$
$6$	0	0
$7$	−1.10463	−0.0596446	−0.0298223	−	0.999555i	$-0.509494\pi$
$7$	−1.10463	−0.0596446	−0.0298223	+	0.999555i	$0.509494\pi$
$8$	−0.0929295	−0.00410694
$9$	0	0
$10$	16.1818	0.511715
$11$	−47.9991	−1.31566	−0.657830	−	0.753166i	$-0.728526\pi$
$11$	−47.9991	−1.31566	−0.657830	+	0.753166i	$0.728526\pi$
$12$	0	0
$13$	50.4246	1.07579	0.537894	−	0.843012i	$-0.319220\pi$
$13$	50.4246	1.07579	0.537894	+	0.843012i	$0.319220\pi$
$14$	−4.41533	−0.0842890
$15$	0	0
$16$	−64.1855	−1.00290
$17$	51.1190	0.729305	0.364653	−	0.931144i	$-0.381188\pi$
$17$	51.1190	0.729305	0.364653	+	0.931144i	$0.381188\pi$
$18$	0	0
$19$	−87.9833	−1.06235	−0.531177	−	0.847261i	$-0.678250\pi$
$19$	−87.9833	−1.06235	−0.531177	+	0.847261i	$0.678250\pi$
$20$	32.2931	0.361048
$21$	0	0
$22$	−191.857	−1.85927
$23$	141.821	1.28572	0.642862	−	0.765982i	$-0.277747\pi$
$23$	141.821	1.28572	0.642862	+	0.765982i	$0.277747\pi$
$24$	0	0
$25$	−108.610	−0.868884
$26$	201.552	1.52029
$27$	0	0
$28$	−8.81139	−0.0594713
$29$	−88.7250	−0.568132	−0.284066	−	0.958805i	$-0.591683\pi$
$29$	−88.7250	−0.568132	−0.284066	+	0.958805i	$0.591683\pi$
$30$	0	0
$31$	242.373	1.40424	0.702120	−	0.712058i	$-0.252237\pi$
$31$	242.373	1.40424	0.702120	+	0.712058i	$0.252237\pi$
$32$	−255.812	−1.41317
$33$	0	0
$34$	204.328	1.03064
$35$	−4.47200	−0.0215973
$36$	0	0
$37$	93.5246	0.415550	0.207775	−	0.978177i	$-0.433378\pi$
$37$	93.5246	0.415550	0.207775	+	0.978177i	$0.433378\pi$
$38$	−351.677	−1.50130
$39$	0	0
$40$	−0.376216	−0.00148712
$41$	−437.766	−1.66750	−0.833750	−	0.552142i	$-0.813811\pi$
$41$	−437.766	−1.66750	−0.833750	+	0.552142i	$0.813811\pi$
$42$	0	0
$43$	173.062	0.613759	0.306880	−	0.951748i	$-0.400715\pi$
$43$	173.062	0.613759	0.306880	+	0.951748i	$0.400715\pi$
$44$	−382.877	−1.31184
$45$	0	0
$46$	566.870	1.81697
$47$	−147.320	−0.457210	−0.228605	−	0.973519i	$-0.573416\pi$
$47$	−147.320	−0.457210	−0.228605	+	0.973519i	$0.573416\pi$
$48$	0	0
$49$	−341.780	−0.996443
$50$	−434.126	−1.22789
$51$	0	0
$52$	402.224	1.07266
$53$	−118.071	−0.306006	−0.153003	−	0.988226i	$-0.548894\pi$
$53$	−118.071	−0.306006	−0.153003	+	0.988226i	$0.548894\pi$
$54$	0	0
$55$	−194.320	−0.476401
$56$	0.102653	0.000244957 0
$57$	0	0
$58$	−354.642	−0.802875
$59$	−674.993	−1.48943	−0.744717	−	0.667381i	$-0.767415\pi$
$59$	−674.993	−1.48943	−0.744717	+	0.667381i	$0.767415\pi$
$60$	0	0
$61$	122.013	0.256101	0.128050	−	0.991768i	$-0.459128\pi$
$61$	122.013	0.256101	0.128050	+	0.991768i	$0.459128\pi$
$62$	968.787	1.98445
$63$	0	0
$64$	−509.020	−0.994179
$65$	204.139	0.389543
$66$	0	0
$67$	−711.424	−1.29723	−0.648614	−	0.761117i	$-0.724651\pi$
$67$	−711.424	−1.29723	−0.648614	+	0.761117i	$0.724651\pi$
$68$	407.764	0.727186
$69$	0	0
$70$	−17.8750	−0.0305210
$71$	−1026.79	−1.71631	−0.858155	−	0.513391i	$-0.828389\pi$
$71$	−1026.79	−1.71631	−0.858155	+	0.513391i	$0.828389\pi$
$72$	0	0
$73$	−1224.17	−1.96272	−0.981358	−	0.192190i	$-0.938441\pi$
$73$	−1224.17	−1.96272	−0.981358	+	0.192190i	$0.938441\pi$
$74$	373.827	0.587250
$75$	0	0
$76$	−701.821	−1.05927
$77$	53.0214	0.0784721
$78$	0	0
$79$	−13.8124	−0.0196711	−0.00983556	−	0.999952i	$-0.503131\pi$
$79$	−13.8124	−0.0196711	−0.00983556	+	0.999952i	$0.503131\pi$
$80$	−259.849	−0.363149
$81$	0	0
$82$	−1749.79	−2.35649
$83$	113.011	0.149452	0.0747262	−	0.997204i	$-0.476192\pi$
$83$	113.011	0.149452	0.0747262	+	0.997204i	$0.476192\pi$
$84$	0	0
$85$	206.950	0.264081
$86$	691.743	0.867356
$87$	0	0
$88$	4.46053	0.00540334
$89$	1140.93	1.35886	0.679431	−	0.733739i	$-0.262227\pi$
$89$	1140.93	1.35886	0.679431	+	0.733739i	$0.262227\pi$
$90$	0	0
$91$	−55.7007	−0.0641651
$92$	1131.27	1.28199
$93$	0	0
$94$	−588.853	−0.646122
$95$	−356.192	−0.384679
$96$	0	0
$97$	−476.494	−0.498770	−0.249385	−	0.968404i	$-0.580228\pi$
$97$	−476.494	−0.498770	−0.249385	+	0.968404i	$0.580228\pi$
$98$	−1366.13	−1.40816
$99$	0	0
$100$	−866.358	−0.866358
$101$	−27.6630	−0.0272532	−0.0136266	−	0.999907i	$-0.504338\pi$
$101$	−27.6630	−0.0272532	−0.0136266	+	0.999907i	$0.504338\pi$
$102$	0	0
$103$	535.926	0.512683	0.256341	−	0.966586i	$-0.417483\pi$
$103$	535.926	0.512683	0.256341	+	0.966586i	$0.417483\pi$
$104$	−4.68593	−0.00441820
$105$	0	0
$106$	−471.941	−0.432443
$107$	−507.713	−0.458715	−0.229357	−	0.973342i	$-0.573662\pi$
$107$	−507.713	−0.458715	−0.229357	+	0.973342i	$0.573662\pi$
$108$	0	0
$109$	−468.604	−0.411781	−0.205890	−	0.978575i	$-0.566009\pi$
$109$	−468.604	−0.411781	−0.205890	+	0.978575i	$0.566009\pi$
$110$	−776.713	−0.673243
$111$	0	0
$112$	70.9015	0.0598175
$113$	275.943	0.229722	0.114861	−	0.993382i	$-0.463358\pi$
$113$	275.943	0.229722	0.114861	+	0.993382i	$0.463358\pi$
$114$	0	0
$115$	574.147	0.465561
$116$	−707.737	−0.566480
$117$	0	0
$118$	−2698.01	−2.10485
$119$	−56.4678	−0.0434991
$120$	0	0
$121$	972.912	0.730963
$122$	487.697	0.361918
$123$	0	0
$124$	1933.35	1.40016
$125$	−945.749	−0.676723
$126$	0	0
$127$	1511.06	1.05578	0.527892	−	0.849311i	$-0.322982\pi$
$127$	1511.06	1.05578	0.527892	+	0.849311i	$0.322982\pi$
$128$	11.8949	0.00821385
$129$	0	0
$130$	815.962	0.550497
$131$	1892.47	1.26218	0.631091	−	0.775709i	$-0.282607\pi$
$131$	1892.47	1.26218	0.631091	+	0.775709i	$0.282607\pi$
$132$	0	0
$133$	97.1893	0.0633638
$134$	−2843.63	−1.83322
$135$	0	0
$136$	−4.75046	−0.00299521
$137$	−426.762	−0.266137	−0.133069	−	0.991107i	$-0.542483\pi$
$137$	−426.762	−0.266137	−0.133069	+	0.991107i	$0.542483\pi$
$138$	0	0
$139$	−1346.23	−0.821479	−0.410739	−	0.911753i	$-0.634729\pi$
$139$	−1346.23	−0.821479	−0.410739	+	0.911753i	$0.634729\pi$
$140$	−35.6721	−0.0215346
$141$	0	0
$142$	−4104.19	−2.42546
$143$	−2420.33	−1.41537
$144$	0	0
$145$	−359.194	−0.205720
$146$	−4893.12	−2.77368
$147$	0	0
$148$	746.023	0.414343
$149$	−2819.32	−1.55012	−0.775060	−	0.631887i	$-0.782281\pi$
$149$	−2819.32	−1.55012	−0.775060	+	0.631887i	$0.782281\pi$
$150$	0	0
$151$	−1597.84	−0.861129	−0.430564	−	0.902560i	$-0.641685\pi$
$151$	−1597.84	−0.861129	−0.430564	+	0.902560i	$0.641685\pi$
$152$	8.17624	0.00436303
$153$	0	0
$154$	211.932	0.110896
$155$	981.223	0.508476
$156$	0	0
$157$	1731.97	0.880422	0.440211	−	0.897894i	$-0.354904\pi$
$157$	1731.97	0.880422	0.440211	+	0.897894i	$0.354904\pi$
$158$	−55.2095	−0.0277990
$159$	0	0
$160$	−1035.63	−0.511710
$161$	−156.660	−0.0766865
$162$	0	0
$163$	−2310.95	−1.11048	−0.555238	−	0.831692i	$-0.687373\pi$
$163$	−2310.95	−1.11048	−0.555238	+	0.831692i	$0.687373\pi$
$164$	−3491.95	−1.66265
$165$	0	0
$166$	451.715	0.211204
$167$	1880.73	0.871469	0.435734	−	0.900075i	$-0.356489\pi$
$167$	1880.73	0.871469	0.435734	+	0.900075i	$0.356489\pi$
$168$	0	0
$169$	345.636	0.157322
$170$	827.200	0.373196
$171$	0	0
$172$	1380.47	0.611976
$173$	−419.954	−0.184558	−0.0922789	−	0.995733i	$-0.529415\pi$
$173$	−419.954	−0.184558	−0.0922789	+	0.995733i	$0.529415\pi$
$174$	0	0
$175$	119.975	0.0518243
$176$	3080.84	1.31947
$177$	0	0
$178$	4560.42	1.92032
$179$	1980.03	0.826783	0.413391	−	0.910553i	$-0.364344\pi$
$179$	1980.03	0.826783	0.413391	+	0.910553i	$0.364344\pi$
$180$	0	0
$181$	1301.87	0.534627	0.267313	−	0.963610i	$-0.413864\pi$
$181$	1301.87	0.534627	0.267313	+	0.963610i	$0.413864\pi$
$182$	−222.641	−0.0906771
$183$	0	0
$184$	−13.1793	−0.00528039
$185$	378.625	0.150471
$186$	0	0
$187$	−2453.67	−0.959518
$188$	−1175.14	−0.455881
$189$	0	0
$190$	−1423.73	−0.543623
$191$	−2560.05	−0.969836	−0.484918	−	0.874560i	$-0.661151\pi$
$191$	−2560.05	−0.969836	−0.484918	+	0.874560i	$0.661151\pi$
$192$	0	0
$193$	−974.394	−0.363411	−0.181706	−	0.983353i	$-0.558162\pi$
$193$	−974.394	−0.363411	−0.181706	+	0.983353i	$0.558162\pi$
$194$	−1904.59	−0.704854
$195$	0	0
$196$	−2726.29	−0.993547
$197$	−757.309	−0.273889	−0.136944	−	0.990579i	$-0.543728\pi$
$197$	−757.309	−0.273889	−0.136944	+	0.990579i	$0.543728\pi$
$198$	0	0
$199$	−1792.51	−0.638533	−0.319266	−	0.947665i	$-0.603436\pi$
$199$	−1792.51	−0.638533	−0.319266	+	0.947665i	$0.603436\pi$
$200$	10.0931	0.00356845
$201$	0	0
$202$	−110.572	−0.0385138
$203$	98.0087	0.0338860
$204$	0	0
$205$	−1772.25	−0.603802
$206$	2142.15	0.724516
$207$	0	0
$208$	−3236.52	−1.07891
$209$	4223.12	1.39770
$210$	0	0
$211$	−2277.10	−0.742947	−0.371474	−	0.928444i	$-0.621147\pi$
$211$	−2277.10	−0.742947	−0.371474	+	0.928444i	$0.621147\pi$
$212$	−941.823	−0.305116
$213$	0	0
$214$	−2029.38	−0.648249
$215$	700.623	0.222242
$216$	0	0
$217$	−267.733	−0.0837554
$218$	−1873.05	−0.581923
$219$	0	0
$220$	−1550.04	−0.475016
$221$	2577.65	0.784578
$222$	0	0
$223$	2554.41	0.767066	0.383533	−	0.923527i	$-0.374707\pi$
$223$	2554.41	0.767066	0.383533	+	0.923527i	$0.374707\pi$
$224$	282.578	0.0842883
$225$	0	0
$226$	1102.97	0.324639
$227$	2642.83	0.772735	0.386367	−	0.922345i	$-0.373730\pi$
$227$	2642.83	0.772735	0.386367	+	0.922345i	$0.373730\pi$
$228$	0	0
$229$	4665.27	1.34624	0.673122	−	0.739531i	$-0.264953\pi$
$229$	4665.27	1.34624	0.673122	+	0.739531i	$0.264953\pi$
$230$	2294.92	0.657924
$231$	0	0
$232$	8.24516	0.00233328
$233$	−949.475	−0.266962	−0.133481	−	0.991051i	$-0.542616\pi$
$233$	−949.475	−0.266962	−0.133481	+	0.991051i	$0.542616\pi$
$234$	0	0
$235$	−596.411	−0.165556
$236$	−5384.25	−1.48511
$237$	0	0
$238$	−225.707	−0.0614724
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	5411.45	1.44640	0.723200	−	0.690639i	$-0.242671\pi$
$241$	5411.45	1.44640	0.723200	+	0.690639i	$0.242671\pi$
$242$	3888.82	1.03299
$243$	0	0
$244$	973.267	0.255357
$245$	−1383.66	−0.360812
$246$	0	0
$247$	−4436.52	−1.14287
$248$	−22.5236	−0.00576713
$249$	0	0
$250$	−3780.25	−0.956335
$251$	−6171.16	−1.55187	−0.775936	−	0.630811i	$-0.782722\pi$
$251$	−6171.16	−1.55187	−0.775936	+	0.630811i	$0.782722\pi$
$252$	0	0
$253$	−6807.26	−1.69158
$254$	6039.84	1.49202
$255$	0	0
$256$	4119.70	1.00579
$257$	1152.39	0.279704	0.139852	−	0.990172i	$-0.455337\pi$
$257$	1152.39	0.279704	0.139852	+	0.990172i	$0.455337\pi$
$258$	0	0
$259$	−103.311	−0.0247853
$260$	1628.36	0.388411
$261$	0	0
$262$	7564.38	1.78370
$263$	−3770.83	−0.884103	−0.442052	−	0.896990i	$-0.645749\pi$
$263$	−3770.83	−0.884103	−0.442052	+	0.896990i	$0.645749\pi$
$264$	0	0
$265$	−477.999	−0.110805
$266$	388.475	0.0895448
$267$	0	0
$268$	−5674.85	−1.29346
$269$	−4716.65	−1.06907	−0.534534	−	0.845147i	$-0.679513\pi$
$269$	−4716.65	−1.06907	−0.534534	+	0.845147i	$0.679513\pi$
$270$	0	0
$271$	10.3194	0.00231313	0.00115656	−	0.999999i	$-0.499632\pi$
$271$	10.3194	0.00231313	0.00115656	+	0.999999i	$0.499632\pi$
$272$	−3281.10	−0.731418
$273$	0	0
$274$	−1705.81	−0.376101
$275$	5213.20	1.14316
$276$	0	0
$277$	7671.93	1.66412	0.832060	−	0.554685i	$-0.187161\pi$
$277$	7671.93	1.66412	0.832060	+	0.554685i	$0.187161\pi$
$278$	−5381.00	−1.16090
$279$	0	0
$280$	0.415581	8.86990e−5 0
$281$	−62.9370	−0.0133612	−0.00668061	−	0.999978i	$-0.502127\pi$
$281$	−62.9370	−0.0133612	−0.00668061	+	0.999978i	$0.502127\pi$
$282$	0	0
$283$	7377.75	1.54969	0.774844	−	0.632152i	$-0.217828\pi$
$283$	7377.75	1.54969	0.774844	+	0.632152i	$0.217828\pi$
$284$	−8190.48	−1.71132
$285$	0	0
$286$	−9674.29	−2.00019
$287$	483.571	0.0994575
$288$	0	0
$289$	−2299.84	−0.468114
$290$	−1435.73	−0.290721
$291$	0	0
$292$	−9764.90	−1.95701
$293$	−2769.63	−0.552231	−0.276115	−	0.961125i	$-0.589047\pi$
$293$	−2769.63	−0.552231	−0.276115	+	0.961125i	$0.589047\pi$
$294$	0	0
$295$	−2732.64	−0.539324
$296$	−8.69119	−0.00170664
$297$	0	0
$298$	−11269.1	−2.19061
$299$	7151.24	1.38317
$300$	0	0
$301$	−191.170	−0.0366075
$302$	−6386.72	−1.21693
$303$	0	0
$304$	5647.25	1.06543
$305$	493.957	0.0927342
$306$	0	0
$307$	9496.98	1.76554	0.882771	−	0.469804i	$-0.155675\pi$
$307$	9496.98	1.76554	0.882771	+	0.469804i	$0.155675\pi$
$308$	422.939	0.0782441
$309$	0	0
$310$	3922.04	0.718570
$311$	6132.04	1.11806	0.559029	−	0.829148i	$-0.311174\pi$
$311$	6132.04	1.11806	0.559029	+	0.829148i	$0.311174\pi$
$312$	0	0
$313$	−7597.42	−1.37199	−0.685993	−	0.727608i	$-0.740632\pi$
$313$	−7597.42	−1.37199	−0.685993	+	0.727608i	$0.740632\pi$
$314$	6922.84	1.24420
$315$	0	0
$316$	−110.178	−0.0196140
$317$	3010.40	0.533378	0.266689	−	0.963783i	$-0.414070\pi$
$317$	3010.40	0.533378	0.266689	+	0.963783i	$0.414070\pi$
$318$	0	0
$319$	4258.72	0.747468
$320$	−2060.72	−0.359992
$321$	0	0
$322$	−626.184	−0.108372
$323$	−4497.62	−0.774781
$324$	0	0
$325$	−5476.63	−0.934735
$326$	−9237.08	−1.56931
$327$	0	0
$328$	40.6813	0.00684833
$329$	162.735	0.0272701
$330$	0	0
$331$	3742.19	0.621418	0.310709	−	0.950505i	$-0.399434\pi$
$331$	3742.19	0.621418	0.310709	+	0.950505i	$0.399434\pi$
$332$	901.460	0.149018
$333$	0	0
$334$	7517.45	1.23155
$335$	−2880.13	−0.469726
$336$	0	0
$337$	−2950.76	−0.476968	−0.238484	−	0.971146i	$-0.576651\pi$
$337$	−2950.76	−0.476968	−0.238484	+	0.971146i	$0.576651\pi$
$338$	1381.54	0.222325
$339$	0	0
$340$	1650.79	0.263314
$341$	−11633.7	−1.84750
$342$	0	0
$343$	756.431	0.119077
$344$	−16.0825	−0.00252067
$345$	0	0
$346$	−1678.59	−0.260814
$347$	−8511.49	−1.31677	−0.658387	−	0.752679i	$-0.728761\pi$
$347$	−8511.49	−1.31677	−0.658387	+	0.752679i	$0.728761\pi$
$348$	0	0
$349$	−8812.19	−1.35159	−0.675796	−	0.737088i	$-0.736200\pi$
$349$	−8812.19	−1.35159	−0.675796	+	0.737088i	$0.736200\pi$
$350$	479.551	0.0732373
$351$	0	0
$352$	12278.7	1.85926
$353$	8301.38	1.25167	0.625833	−	0.779957i	$-0.284759\pi$
$353$	8301.38	1.25167	0.625833	+	0.779957i	$0.284759\pi$
$354$	0	0
$355$	−4156.87	−0.621476
$356$	9100.94	1.35491
$357$	0	0
$358$	7914.35	1.16840
$359$	5375.68	0.790300	0.395150	−	0.918617i	$-0.370693\pi$
$359$	5375.68	0.790300	0.395150	+	0.918617i	$0.370693\pi$
$360$	0	0
$361$	882.055	0.128598
$362$	5203.71	0.755527
$363$	0	0
$364$	−444.311	−0.0639786
$365$	−4955.93	−0.710699
$366$	0	0
$367$	7875.44	1.12015	0.560075	−	0.828442i	$-0.310772\pi$
$367$	7875.44	1.12015	0.560075	+	0.828442i	$0.310772\pi$
$368$	−9102.82	−1.28945
$369$	0	0
$370$	1513.40	0.212643
$371$	130.425	0.0182516
$372$	0	0
$373$	−523.441	−0.0726615	−0.0363308	−	0.999340i	$-0.511567\pi$
$373$	−523.441	−0.0726615	−0.0363308	+	0.999340i	$0.511567\pi$
$374$	−9807.53	−1.35598
$375$	0	0
$376$	13.6904	0.00187773
$377$	−4473.92	−0.611190
$378$	0	0
$379$	854.837	0.115858	0.0579288	−	0.998321i	$-0.481550\pi$
$379$	854.837	0.115858	0.0579288	+	0.998321i	$0.481550\pi$
$380$	−2841.25	−0.383561
$381$	0	0
$382$	−10232.8	−1.37056
$383$	2973.40	0.396694	0.198347	−	0.980132i	$-0.436443\pi$
$383$	2973.40	0.396694	0.198347	+	0.980132i	$0.436443\pi$
$384$	0	0
$385$	214.652	0.0284148
$386$	−3894.74	−0.513568
$387$	0	0
$388$	−3800.88	−0.497320
$389$	−8861.43	−1.15499	−0.577497	−	0.816393i	$-0.695970\pi$
$389$	−8861.43	−1.15499	−0.577497	+	0.816393i	$0.695970\pi$
$390$	0	0
$391$	7249.73	0.937685
$392$	31.7614	0.00409233
$393$	0	0
$394$	−3027.04	−0.387055
$395$	−55.9182	−0.00712292
$396$	0	0
$397$	−373.265	−0.0471879	−0.0235940	−	0.999722i	$-0.507511\pi$
$397$	−373.265	−0.0471879	−0.0235940	+	0.999722i	$0.507511\pi$
$398$	−7164.85	−0.902365
$399$	0	0
$400$	6971.21	0.871401
$401$	844.656	0.105187	0.0525936	−	0.998616i	$-0.483251\pi$
$401$	844.656	0.105187	0.0525936	+	0.998616i	$0.483251\pi$
$402$	0	0
$403$	12221.5	1.51067
$404$	−220.661	−0.0271740
$405$	0	0
$406$	391.750	0.0478872
$407$	−4489.10	−0.546723
$408$	0	0
$409$	−2089.86	−0.252657	−0.126328	−	0.991988i	$-0.540319\pi$
$409$	−2089.86	−0.252657	−0.126328	+	0.991988i	$0.540319\pi$
$410$	−7083.86	−0.853285
$411$	0	0
$412$	4274.95	0.511193
$413$	745.620	0.0888367
$414$	0	0
$415$	457.513	0.0541168
$416$	−12899.2	−1.52028
$417$	0	0
$418$	16880.2	1.97521
$419$	11396.1	1.32873	0.664363	−	0.747410i	$-0.268703\pi$
$419$	11396.1	1.32873	0.664363	+	0.747410i	$0.268703\pi$
$420$	0	0
$421$	−4878.55	−0.564765	−0.282383	−	0.959302i	$-0.591125\pi$
$421$	−4878.55	−0.564765	−0.282383	+	0.959302i	$0.591125\pi$
$422$	−9101.77	−1.04992
$423$	0	0
$424$	10.9723	0.00125675
$425$	−5552.06	−0.633681
$426$	0	0
$427$	−134.780	−0.0152750
$428$	−4049.90	−0.457382
$429$	0	0
$430$	2800.46	0.314070
$431$	−13782.1	−1.54028	−0.770139	−	0.637876i	$-0.779813\pi$
$431$	−13782.1	−1.54028	−0.770139	+	0.637876i	$0.779813\pi$
$432$	0	0
$433$	2146.21	0.238199	0.119099	−	0.992882i	$-0.461999\pi$
$433$	2146.21	0.238199	0.119099	+	0.992882i	$0.461999\pi$
$434$	−1070.16	−0.118362
$435$	0	0
$436$	−3737.94	−0.410584
$437$	−12477.8	−1.36590
$438$	0	0
$439$	8338.78	0.906579	0.453289	−	0.891363i	$-0.350250\pi$
$439$	8338.78	0.906579	0.453289	+	0.891363i	$0.350250\pi$
$440$	18.0580	0.00195655
$441$	0	0
$442$	10303.1	1.10875
$443$	12614.7	1.35292	0.676461	−	0.736479i	$-0.263513\pi$
$443$	12614.7	1.35292	0.676461	+	0.736479i	$0.263513\pi$
$444$	0	0
$445$	4618.96	0.492044
$446$	10210.2	1.08401
$447$	0	0
$448$	562.281	0.0592975
$449$	−7675.63	−0.806761	−0.403380	−	0.915032i	$-0.632165\pi$
$449$	−7675.63	−0.806761	−0.403380	+	0.915032i	$0.632165\pi$
$450$	0	0
$451$	21012.4	2.19387
$452$	2201.13	0.229054
$453$	0	0
$454$	10563.6	1.09202
$455$	−225.499	−0.0232342
$456$	0	0
$457$	−320.495	−0.0328055	−0.0164028	−	0.999865i	$-0.505221\pi$
$457$	−320.495	−0.0328055	−0.0164028	+	0.999865i	$0.505221\pi$
$458$	18647.5	1.90249
$459$	0	0
$460$	4579.83	0.464208
$461$	−11008.9	−1.11223	−0.556113	−	0.831107i	$-0.687708\pi$
$461$	−11008.9	−1.11223	−0.556113	+	0.831107i	$0.687708\pi$
$462$	0	0
$463$	−6268.55	−0.629210	−0.314605	−	0.949223i	$-0.601872\pi$
$463$	−6268.55	−0.629210	−0.314605	+	0.949223i	$0.601872\pi$
$464$	5694.85	0.569778
$465$	0	0
$466$	−3795.14	−0.377267
$467$	−5453.97	−0.540427	−0.270214	−	0.962800i	$-0.587094\pi$
$467$	−5453.97	−0.540427	−0.270214	+	0.962800i	$0.587094\pi$
$468$	0	0
$469$	785.863	0.0773727
$470$	−2383.91	−0.233961
$471$	0	0
$472$	62.7267	0.00611702
$473$	−8306.80	−0.807499
$474$	0	0
$475$	9555.90	0.923063
$476$	−450.430	−0.0433727
$477$	0	0
$478$	955.305	0.0914114
$479$	15898.3	1.51652	0.758260	−	0.651952i	$-0.226050\pi$
$479$	15898.3	1.51652	0.758260	+	0.651952i	$0.226050\pi$
$480$	0	0
$481$	4715.94	0.447044
$482$	21630.1	2.04403
$483$	0	0
$484$	7760.68	0.728839
$485$	−1929.04	−0.180605
$486$	0	0
$487$	−15727.9	−1.46345	−0.731726	−	0.681599i	$-0.761285\pi$
$487$	−15727.9	−1.46345	−0.731726	+	0.681599i	$0.761285\pi$
$488$	−11.3386	−0.00105179
$489$	0	0
$490$	−5530.63	−0.509894
$491$	−10116.1	−0.929802	−0.464901	−	0.885363i	$-0.653910\pi$
$491$	−10116.1	−0.929802	−0.464901	+	0.885363i	$0.653910\pi$
$492$	0	0
$493$	−4535.54	−0.414341
$494$	−17733.2	−1.61509
$495$	0	0
$496$	−15556.8	−1.40831
$497$	1134.23	0.102369
$498$	0	0
$499$	−1710.86	−0.153484	−0.0767420	−	0.997051i	$-0.524452\pi$
$499$	−1710.86	−0.153484	−0.0767420	+	0.997051i	$0.524452\pi$
$500$	−7544.00	−0.674756
$501$	0	0
$502$	−24666.7	−2.19308
$503$	−14432.0	−1.27930	−0.639652	−	0.768664i	$-0.720922\pi$
$503$	−14432.0	−1.27930	−0.639652	+	0.768664i	$0.720922\pi$
$504$	0	0
$505$	−111.991	−0.00986838
$506$	−27209.3	−2.39051
$507$	0	0
$508$	12053.3	1.05272
$509$	3625.68	0.315728	0.157864	−	0.987461i	$-0.449539\pi$
$509$	3625.68	0.315728	0.157864	+	0.987461i	$0.449539\pi$
$510$	0	0
$511$	1352.26	0.117065
$512$	16371.7	1.41315
$513$	0	0
$514$	4606.20	0.395274
$515$	2169.64	0.185643
$516$	0	0
$517$	7071.24	0.601533
$518$	−412.942	−0.0350263
$519$	0	0
$520$	−18.9705	−0.00159983
$521$	6058.73	0.509478	0.254739	−	0.967010i	$-0.418011\pi$
$521$	6058.73	0.509478	0.254739	+	0.967010i	$0.418011\pi$
$522$	0	0
$523$	17176.8	1.43612	0.718058	−	0.695983i	$-0.245031\pi$
$523$	17176.8	1.43612	0.718058	+	0.695983i	$0.245031\pi$
$524$	15095.8	1.25851
$525$	0	0
$526$	−15072.3	−1.24940
$527$	12389.9	1.02412
$528$	0	0
$529$	7946.09	0.653086
$530$	−1910.61	−0.156588
$531$	0	0
$532$	775.255	0.0631796
$533$	−22074.1	−1.79388
$534$	0	0
$535$	−2055.43	−0.166101
$536$	66.1122	0.00532764
$537$	0	0
$538$	−18852.9	−1.51079
$539$	16405.1	1.31098
$540$	0	0
$541$	−3891.14	−0.309230	−0.154615	−	0.987975i	$-0.549414\pi$
$541$	−3891.14	−0.309230	−0.154615	+	0.987975i	$0.549414\pi$
$542$	41.2475	0.00326888
$543$	0	0
$544$	−13076.9	−1.03063
$545$	−1897.10	−0.149106
$546$	0	0
$547$	13449.8	1.05132	0.525659	−	0.850695i	$-0.323819\pi$
$547$	13449.8	1.05132	0.525659	+	0.850695i	$0.323819\pi$
$548$	−3404.18	−0.265364
$549$	0	0
$550$	20837.7	1.61549
$551$	7806.31	0.603557
$552$	0	0
$553$	15.2577	0.00117328
$554$	30665.4	2.35171
$555$	0	0
$556$	−10738.5	−0.819091
$557$	−18563.3	−1.41212	−0.706062	−	0.708150i	$-0.749530\pi$
$557$	−18563.3	−1.41212	−0.706062	+	0.708150i	$0.749530\pi$
$558$	0	0
$559$	8726.56	0.660275
$560$	287.038	0.0216599
$561$	0	0
$562$	−251.565	−0.0188819
$563$	−1625.00	−0.121644	−0.0608219	−	0.998149i	$-0.519372\pi$
$563$	−1625.00	−0.121644	−0.0608219	+	0.998149i	$0.519372\pi$
$564$	0	0
$565$	1117.13	0.0831822
$566$	29489.6	2.19000
$567$	0	0
$568$	95.4194	0.00704878
$569$	−3009.34	−0.221719	−0.110859	−	0.993836i	$-0.535360\pi$
$569$	−3009.34	−0.221719	−0.110859	+	0.993836i	$0.535360\pi$
$570$	0	0
$571$	16739.6	1.22685	0.613424	−	0.789754i	$-0.289792\pi$
$571$	16739.6	1.22685	0.613424	+	0.789754i	$0.289792\pi$
$572$	−19306.4	−1.41126
$573$	0	0
$574$	1932.88	0.140552
$575$	−15403.2	−1.11714
$576$	0	0
$577$	−11635.1	−0.839471	−0.419735	−	0.907647i	$-0.637877\pi$
$577$	−11635.1	−0.839471	−0.419735	+	0.907647i	$0.637877\pi$
$578$	−9192.69	−0.661532
$579$	0	0
$580$	−2865.20	−0.205123
$581$	−124.836	−0.00891404
$582$	0	0
$583$	5667.30	0.402599
$584$	113.761	0.00806076
$585$	0	0
$586$	−11070.5	−0.780405
$587$	−20826.5	−1.46440	−0.732198	−	0.681092i	$-0.761505\pi$
$587$	−20826.5	−1.46440	−0.732198	+	0.681092i	$0.761505\pi$
$588$	0	0
$589$	−21324.8	−1.49180
$590$	−10922.6	−0.762165
$591$	0	0
$592$	−6002.92	−0.416754
$593$	−2889.49	−0.200097	−0.100048	−	0.994983i	$-0.531900\pi$
$593$	−2889.49	−0.200097	−0.100048	+	0.994983i	$0.531900\pi$
$594$	0	0
$595$	−228.605	−0.0157510
$596$	−22489.0	−1.54562
$597$	0	0
$598$	28584.2	1.95467
$599$	−10424.7	−0.711085	−0.355543	−	0.934660i	$-0.615704\pi$
$599$	−10424.7	−0.711085	−0.355543	+	0.934660i	$0.615704\pi$
$600$	0	0
$601$	10091.1	0.684899	0.342449	−	0.939536i	$-0.388743\pi$
$601$	10091.1	0.684899	0.342449	+	0.939536i	$0.388743\pi$
$602$	−764.123	−0.0517331
$603$	0	0
$604$	−12745.6	−0.858626
$605$	3938.74	0.264682
$606$	0	0
$607$	6410.41	0.428650	0.214325	−	0.976762i	$-0.431245\pi$
$607$	6410.41	0.428650	0.214325	+	0.976762i	$0.431245\pi$
$608$	22507.2	1.50129
$609$	0	0
$610$	1974.39	0.131051
$611$	−7428.56	−0.491861
$612$	0	0
$613$	10238.7	0.674611	0.337306	−	0.941395i	$-0.390484\pi$
$613$	10238.7	0.674611	0.337306	+	0.941395i	$0.390484\pi$
$614$	37960.3	2.49504
$615$	0	0
$616$	−4.92725	−0.000322280 0
$617$	−5867.50	−0.382847	−0.191424	−	0.981508i	$-0.561310\pi$
$617$	−5867.50	−0.382847	−0.191424	+	0.981508i	$0.561310\pi$
$618$	0	0
$619$	15855.6	1.02955	0.514775	−	0.857325i	$-0.327875\pi$
$619$	15855.6	1.02955	0.514775	+	0.857325i	$0.327875\pi$
$620$	7826.97	0.506998
$621$	0	0
$622$	24510.3	1.58002
$623$	−1260.31	−0.0810488
$624$	0	0
$625$	9747.53	0.623842
$626$	−30367.6	−1.93887
$627$	0	0
$628$	13815.5	0.877863
$629$	4780.89	0.303063
$630$	0	0
$631$	31300.6	1.97473	0.987367	−	0.158451i	$-0.0506499\pi$
$631$	31300.6	1.97473	0.987367	+	0.158451i	$0.0506499\pi$
$632$	1.28358	8.07882e−5 0
$633$	0	0
$634$	12032.8	0.753762
$635$	6117.37	0.382300
$636$	0	0
$637$	−17234.1	−1.07196
$638$	17022.5	1.05631
$639$	0	0
$640$	48.1554	0.00297423
$641$	4920.23	0.303178	0.151589	−	0.988444i	$-0.451561\pi$
$641$	4920.23	0.303178	0.151589	+	0.988444i	$0.451561\pi$
$642$	0	0
$643$	−26573.3	−1.62978	−0.814889	−	0.579617i	$-0.803202\pi$
$643$	−26573.3	−1.62978	−0.814889	+	0.579617i	$0.803202\pi$
$644$	−1249.64	−0.0764637
$645$	0	0
$646$	−17977.4	−1.09491
$647$	14348.5	0.871870	0.435935	−	0.899978i	$-0.356418\pi$
$647$	14348.5	0.871870	0.435935	+	0.899978i	$0.356418\pi$
$648$	0	0
$649$	32399.0	1.95959
$650$	−21890.6	−1.32095
$651$	0	0
$652$	−18433.9	−1.10725
$653$	22712.9	1.36114	0.680569	−	0.732684i	$-0.261732\pi$
$653$	22712.9	1.36114	0.680569	+	0.732684i	$0.261732\pi$
$654$	0	0
$655$	7661.48	0.457036
$656$	28098.2	1.67233
$657$	0	0
$658$	650.467	0.0385377
$659$	5250.90	0.310389	0.155194	−	0.987884i	$-0.450400\pi$
$659$	5250.90	0.310389	0.155194	+	0.987884i	$0.450400\pi$
$660$	0	0
$661$	30600.4	1.80063	0.900315	−	0.435238i	$-0.143336\pi$
$661$	30600.4	1.80063	0.900315	+	0.435238i	$0.143336\pi$
$662$	14957.9	0.878178
$663$	0	0
$664$	−10.5020	−0.000613792 0
$665$	393.461	0.0229440
$666$	0	0
$667$	−12583.0	−0.730460
$668$	15002.1	0.868936
$669$	0	0
$670$	−11512.1	−0.663811
$671$	−5856.51	−0.336942
$672$	0	0
$673$	4580.09	0.262332	0.131166	−	0.991360i	$-0.458128\pi$
$673$	4580.09	0.262332	0.131166	+	0.991360i	$0.458128\pi$
$674$	−11794.5	−0.674045
$675$	0	0
$676$	2757.05	0.156865
$677$	−11317.7	−0.642505	−0.321252	−	0.946994i	$-0.604104\pi$
$677$	−11317.7	−0.642505	−0.321252	+	0.946994i	$0.604104\pi$
$678$	0	0
$679$	526.352	0.0297489
$680$	−19.2318	−0.00108457
$681$	0	0
$682$	−46500.9	−2.61087
$683$	6562.83	0.367671	0.183836	−	0.982957i	$-0.441149\pi$
$683$	6562.83	0.367671	0.183836	+	0.982957i	$0.441149\pi$
$684$	0	0
$685$	−1727.71	−0.0963682
$686$	3023.53	0.168278
$687$	0	0
$688$	−11108.0	−0.615538
$689$	−5953.68	−0.329197
$690$	0	0
$691$	30308.7	1.66859	0.834295	−	0.551318i	$-0.185875\pi$
$691$	30308.7	1.66859	0.834295	+	0.551318i	$0.185875\pi$
$692$	−3349.87	−0.184021
$693$	0	0
$694$	−34021.2	−1.86085
$695$	−5450.07	−0.297458
$696$	0	0
$697$	−22378.2	−1.21612
$698$	−35223.1	−1.91005
$699$	0	0
$700$	957.009	0.0516736
$701$	−8394.73	−0.452303	−0.226152	−	0.974092i	$-0.572615\pi$
$701$	−8394.73	−0.452303	−0.226152	+	0.974092i	$0.572615\pi$
$702$	0	0
$703$	−8228.60	−0.441462
$704$	24432.5	1.30800
$705$	0	0
$706$	33181.4	1.76884
$707$	30.5575	0.00162551
$708$	0	0
$709$	24500.7	1.29780	0.648902	−	0.760872i	$-0.275229\pi$
$709$	24500.7	1.29780	0.648902	+	0.760872i	$0.275229\pi$
$710$	−16615.4	−0.878261
$711$	0	0
$712$	−106.026	−0.00558077
$713$	34373.5	1.80547
$714$	0	0
$715$	−9798.48	−0.512507
$716$	15794.2	0.824380
$717$	0	0
$718$	21487.1	1.11684
$719$	−2682.97	−0.139163	−0.0695814	−	0.997576i	$-0.522166\pi$
$719$	−2682.97	−0.139163	−0.0695814	+	0.997576i	$0.522166\pi$
$720$	0	0
$721$	−592.002	−0.0305788
$722$	3525.65	0.181733
$723$	0	0
$724$	10384.7	0.533073
$725$	9636.46	0.493640
$726$	0	0
$727$	15504.9	0.790981	0.395491	−	0.918470i	$-0.370575\pi$
$727$	15504.9	0.790981	0.395491	+	0.918470i	$0.370575\pi$
$728$	5.17624	0.000263522 0
$729$	0	0
$730$	−19809.3	−1.00435
$731$	8846.74	0.447618
$732$	0	0
$733$	−31040.9	−1.56415	−0.782074	−	0.623185i	$-0.785838\pi$
$733$	−31040.9	−1.56415	−0.782074	+	0.623185i	$0.785838\pi$
$734$	31478.9	1.58298
$735$	0	0
$736$	−36279.4	−1.81695
$737$	34147.7	1.70671
$738$	0	0
$739$	34245.2	1.70464	0.852320	−	0.523021i	$-0.175195\pi$
$739$	34245.2	1.70464	0.852320	+	0.523021i	$0.175195\pi$
$740$	3020.20	0.150033
$741$	0	0
$742$	521.322	0.0257929
$743$	−26315.7	−1.29937	−0.649684	−	0.760204i	$-0.725099\pi$
$743$	−26315.7	−1.29937	−0.649684	+	0.760204i	$0.725099\pi$
$744$	0	0
$745$	−11413.8	−0.561299
$746$	−2092.24	−0.102684
$747$	0	0
$748$	−19572.3	−0.956730
$749$	560.837	0.0273599
$750$	0	0
$751$	−13876.1	−0.674229	−0.337114	−	0.941464i	$-0.609451\pi$
$751$	−13876.1	−0.674229	−0.337114	+	0.941464i	$0.609451\pi$
$752$	9455.81	0.458535
$753$	0	0
$754$	−17882.7	−0.863724
$755$	−6468.70	−0.311815
$756$	0	0
$757$	5912.49	0.283875	0.141937	−	0.989876i	$-0.454667\pi$
$757$	5912.49	0.283875	0.141937	+	0.989876i	$0.454667\pi$
$758$	3416.86	0.163728
$759$	0	0
$760$	33.1007	0.00157985
$761$	19059.5	0.907891	0.453945	−	0.891030i	$-0.350016\pi$
$761$	19059.5	0.907891	0.453945	+	0.891030i	$0.350016\pi$
$762$	0	0
$763$	517.636	0.0245605
$764$	−20420.9	−0.967018
$765$	0	0
$766$	11885.0	0.560602
$767$	−34036.2	−1.60232
$768$	0	0
$769$	25233.6	1.18329	0.591644	−	0.806200i	$-0.298479\pi$
$769$	25233.6	1.18329	0.591644	+	0.806200i	$0.298479\pi$
$770$	857.984	0.0401553
$771$	0	0
$772$	−7772.50	−0.362355
$773$	−919.765	−0.0427965	−0.0213982	−	0.999771i	$-0.506812\pi$
$773$	−919.765	−0.0427965	−0.0213982	+	0.999771i	$0.506812\pi$
$774$	0	0
$775$	−26324.2	−1.22012
$776$	44.2804	0.00204842
$777$	0	0
$778$	−35420.0	−1.63222
$779$	38516.1	1.77148
$780$	0	0
$781$	49285.2	2.25808
$782$	28977.9	1.32512
$783$	0	0
$784$	21937.3	0.999330
$785$	7011.71	0.318801
$786$	0	0
$787$	−27714.4	−1.25529	−0.627643	−	0.778501i	$-0.715980\pi$
$787$	−27714.4	−1.25529	−0.627643	+	0.778501i	$0.715980\pi$
$788$	−6040.87	−0.273093
$789$	0	0
$790$	−223.510	−0.0100660
$791$	−304.816	−0.0137017
$792$	0	0
$793$	6152.45	0.275511
$794$	−1491.97	−0.0666853
$795$	0	0
$796$	−14298.4	−0.636677
$797$	−17889.7	−0.795091	−0.397545	−	0.917583i	$-0.630138\pi$
$797$	−17889.7	−0.795091	−0.397545	+	0.917583i	$0.630138\pi$
$798$	0	0
$799$	−7530.87	−0.333445
$800$	27783.8	1.22788
$801$	0	0
$802$	3376.17	0.148649
$803$	58759.0	2.58227
$804$	0	0
$805$	−634.222	−0.0277682
$806$	48850.6	2.13485
$807$	0	0
$808$	2.57071	0.000111927 0
$809$	−41008.0	−1.78216	−0.891078	−	0.453849i	$-0.850050\pi$
$809$	−41008.0	−1.78216	−0.891078	+	0.453849i	$0.850050\pi$
$810$	0	0
$811$	37353.0	1.61732	0.808658	−	0.588279i	$-0.200195\pi$
$811$	37353.0	1.61732	0.808658	+	0.588279i	$0.200195\pi$
$812$	781.791	0.0337875
$813$	0	0
$814$	−17943.3	−0.772621
$815$	−9355.65	−0.402103
$816$	0	0
$817$	−15226.5	−0.652030
$818$	−8353.35	−0.357051
$819$	0	0
$820$	−14136.8	−0.602047
$821$	10694.8	0.454628	0.227314	−	0.973821i	$-0.427006\pi$
$821$	10694.8	0.454628	0.227314	+	0.973821i	$0.427006\pi$
$822$	0	0
$823$	−9279.81	−0.393042	−0.196521	−	0.980500i	$-0.562964\pi$
$823$	−9279.81	−0.393042	−0.196521	+	0.980500i	$0.562964\pi$
$824$	−49.8033	−0.00210556
$825$	0	0
$826$	2980.31	0.125543
$827$	−3262.63	−0.137186	−0.0685930	−	0.997645i	$-0.521851\pi$
$827$	−3262.63	−0.137186	−0.0685930	+	0.997645i	$0.521851\pi$
$828$	0	0
$829$	31780.5	1.33146	0.665732	−	0.746191i	$-0.268119\pi$
$829$	31780.5	1.33146	0.665732	+	0.746191i	$0.268119\pi$
$830$	1828.72	0.0764770
$831$	0	0
$832$	−25667.1	−1.06953
$833$	−17471.5	−0.726711
$834$	0	0
$835$	7613.95	0.315559
$836$	33686.7	1.39364
$837$	0	0
$838$	45551.3	1.87774
$839$	32049.1	1.31878	0.659390	−	0.751801i	$-0.270814\pi$
$839$	32049.1	1.31878	0.659390	+	0.751801i	$0.270814\pi$
$840$	0	0
$841$	−16516.9	−0.677227
$842$	−19500.0	−0.798118
$843$	0	0
$844$	−18163.8	−0.740788
$845$	1399.27	0.0569662
$846$	0	0
$847$	−1074.71	−0.0435980
$848$	7578.44	0.306892
$849$	0	0
$850$	−22192.1	−0.895509
$851$	13263.7	0.534283
$852$	0	0
$853$	−29521.1	−1.18497	−0.592487	−	0.805580i	$-0.701854\pi$
$853$	−29521.1	−1.18497	−0.592487	+	0.805580i	$0.701854\pi$
$854$	−538.727	−0.0215865
$855$	0	0
$856$	47.1815	0.00188391
$857$	17125.3	0.682602	0.341301	−	0.939954i	$-0.389132\pi$
$857$	17125.3	0.682602	0.341301	+	0.939954i	$0.389132\pi$
$858$	0	0
$859$	−8707.51	−0.345863	−0.172932	−	0.984934i	$-0.555324\pi$
$859$	−8707.51	−0.345863	−0.172932	+	0.984934i	$0.555324\pi$
$860$	5588.70	0.221596
$861$	0	0
$862$	−55088.3	−2.17670
$863$	−21710.8	−0.856366	−0.428183	−	0.903692i	$-0.640846\pi$
$863$	−21710.8	−0.856366	−0.428183	+	0.903692i	$0.640846\pi$
$864$	0	0
$865$	−1700.14	−0.0668284
$866$	8578.58	0.336619
$867$	0	0
$868$	−2135.64	−0.0835120
$869$	662.984	0.0258805
$870$	0	0
$871$	−35873.2	−1.39554
$872$	43.5471	0.00169116
$873$	0	0
$874$	−49875.1	−1.93026
$875$	1044.71	0.0403629
$876$	0	0
$877$	8646.67	0.332927	0.166464	−	0.986048i	$-0.446765\pi$
$877$	8646.67	0.332927	0.166464	+	0.986048i	$0.446765\pi$
$878$	33330.9	1.28116
$879$	0	0
$880$	12472.5	0.477781
$881$	−14872.3	−0.568740	−0.284370	−	0.958715i	$-0.591784\pi$
$881$	−14872.3	−0.568740	−0.284370	+	0.958715i	$0.591784\pi$
$882$	0	0
$883$	30906.4	1.17790	0.588949	−	0.808170i	$-0.299542\pi$
$883$	30906.4	1.17790	0.588949	+	0.808170i	$0.299542\pi$
$884$	20561.3	0.782298
$885$	0	0
$886$	50422.3	1.91193
$887$	−16725.2	−0.633122	−0.316561	−	0.948572i	$-0.602528\pi$
$887$	−16725.2	−0.633122	−0.316561	+	0.948572i	$0.602528\pi$
$888$	0	0
$889$	−1669.17	−0.0629719
$890$	18462.4	0.695350
$891$	0	0
$892$	20375.9	0.764837
$893$	12961.7	0.485719
$894$	0	0
$895$	8015.94	0.299378
$896$	−13.1395	−0.000489912 0
$897$	0	0
$898$	−30680.2	−1.14010
$899$	−21504.5	−0.797793
$900$	0	0
$901$	−6035.67	−0.223171
$902$	83988.3	3.10034
$903$	0	0
$904$	−25.6433	−0.000943453 0
$905$	5270.50	0.193588
$906$	0	0
$907$	−8883.30	−0.325210	−0.162605	−	0.986691i	$-0.551990\pi$
$907$	−8883.30	−0.325210	−0.162605	+	0.986691i	$0.551990\pi$
$908$	21081.2	0.770489
$909$	0	0
$910$	−901.340	−0.0328342
$911$	−5440.31	−0.197855	−0.0989274	−	0.995095i	$-0.531541\pi$
$911$	−5440.31	−0.197855	−0.0989274	+	0.995095i	$0.531541\pi$
$912$	0	0
$913$	−5424.42	−0.196629
$914$	−1281.05	−0.0463603
$915$	0	0
$916$	37213.7	1.34233
$917$	−2090.49	−0.0752824
$918$	0	0
$919$	29325.4	1.05262	0.526309	−	0.850294i	$-0.323576\pi$
$919$	29325.4	1.05262	0.526309	+	0.850294i	$0.323576\pi$
$920$	−53.3552	−0.00191203
$921$	0	0
$922$	−44003.6	−1.57178
$923$	−51775.6	−1.84639
$924$	0	0
$925$	−10157.8	−0.361065
$926$	−25056.0	−0.889191
$927$	0	0
$928$	22696.9	0.802869
$929$	−50671.0	−1.78952	−0.894759	−	0.446549i	$-0.852653\pi$
$929$	−50671.0	−1.78952	−0.894759	+	0.446549i	$0.852653\pi$
$930$	0	0
$931$	30070.9	1.05858
$932$	−7573.73	−0.266186
$933$	0	0
$934$	−21800.0	−0.763724
$935$	−9933.43	−0.347442
$936$	0	0
$937$	−13209.3	−0.460543	−0.230271	−	0.973126i	$-0.573961\pi$
$937$	−13209.3	−0.460543	−0.230271	+	0.973126i	$0.573961\pi$
$938$	3141.17	0.109342
$939$	0	0
$940$	−4757.43	−0.165075
$941$	45678.9	1.58246	0.791228	−	0.611521i	$-0.209442\pi$
$941$	45678.9	1.58246	0.791228	+	0.611521i	$0.209442\pi$
$942$	0	0
$943$	−62084.2	−2.14395
$944$	43324.7	1.49375
$945$	0	0
$946$	−33203.0	−1.14115
$947$	−42368.9	−1.45386	−0.726930	−	0.686712i	$-0.759053\pi$
$947$	−42368.9	−1.45386	−0.726930	+	0.686712i	$0.759053\pi$
$948$	0	0
$949$	−61728.2	−2.11147
$950$	38195.8	1.30446
$951$	0	0
$952$	5.24753	0.000178648 0
$953$	32335.5	1.09911	0.549553	−	0.835459i	$-0.314798\pi$
$953$	32335.5	1.09911	0.549553	+	0.835459i	$0.314798\pi$
$954$	0	0
$955$	−10364.1	−0.351178
$956$	1906.44	0.0644966
$957$	0	0
$958$	63547.1	2.14312
$959$	471.416	0.0158736
$960$	0	0
$961$	28953.6	0.971891
$962$	18850.0	0.631757
$963$	0	0
$964$	43165.8	1.44220
$965$	−3944.74	−0.131591
$966$	0	0
$967$	−35204.6	−1.17074	−0.585368	−	0.810768i	$-0.699050\pi$
$967$	−35204.6	−1.17074	−0.585368	+	0.810768i	$0.699050\pi$
$968$	−90.4122	−0.00300202
$969$	0	0
$970$	−7710.55	−0.255228
$971$	18320.8	0.605501	0.302751	−	0.953070i	$-0.402095\pi$
$971$	18320.8	0.605501	0.302751	+	0.953070i	$0.402095\pi$
$972$	0	0
$973$	1487.09	0.0489968
$974$	−62866.0	−2.06813
$975$	0	0
$976$	−7831.46	−0.256843
$977$	8044.38	0.263421	0.131711	−	0.991288i	$-0.457953\pi$
$977$	8044.38	0.263421	0.131711	+	0.991288i	$0.457953\pi$
$978$	0	0
$979$	−54763.8	−1.78780
$980$	−11037.1	−0.359763
$981$	0	0
$982$	−40435.0	−1.31398
$983$	−2326.32	−0.0754812	−0.0377406	−	0.999288i	$-0.512016\pi$
$983$	−2326.32	−0.0754812	−0.0377406	+	0.999288i	$0.512016\pi$
$984$	0	0
$985$	−3065.89	−0.0991751
$986$	−18129.0	−0.585541
$987$	0	0
$988$	−35389.0	−1.13955
$989$	24543.7	0.789125
$990$	0	0
$991$	−40978.0	−1.31353	−0.656766	−	0.754094i	$-0.728076\pi$
$991$	−40978.0	−1.31353	−0.656766	+	0.754094i	$0.728076\pi$
$992$	−62001.8	−1.98444
$993$	0	0
$994$	4533.63	0.144666
$995$	−7256.82	−0.231213
$996$	0	0
$997$	−10403.3	−0.330467	−0.165233	−	0.986255i	$-0.552838\pi$
$997$	−10403.3	−0.330467	−0.165233	+	0.986255i	$0.552838\pi$
$998$	−6838.46	−0.216901
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.49	✓	59
3.2	odd	2		2151.4.a.h.1.11	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.49	✓	59	1.1	even	1	trivial
2151.4.a.h.1.11	yes	59	3.2	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q+3.99709 q^{2} +7.97675 q^{4} +4.04840 q^{5} -1.10463 q^{7} -0.0929295 q^{8} +O(q^{10})\)	\(q+3.99709 q^{2} +7.97675 q^{4} +4.04840 q^{5} -1.10463 q^{7} -0.0929295 q^{8} +16.1818 q^{10} -47.9991 q^{11} +50.4246 q^{13} -4.41533 q^{14} -64.1855 q^{16} +51.1190 q^{17} -87.9833 q^{19} +32.2931 q^{20} -191.857 q^{22} +141.821 q^{23} -108.610 q^{25} +201.552 q^{26} -8.81139 q^{28} -88.7250 q^{29} +242.373 q^{31} -255.812 q^{32} +204.328 q^{34} -4.47200 q^{35} +93.5246 q^{37} -351.677 q^{38} -0.376216 q^{40} -437.766 q^{41} +173.062 q^{43} -382.877 q^{44} +566.870 q^{46} -147.320 q^{47} -341.780 q^{49} -434.126 q^{50} +402.224 q^{52} -118.071 q^{53} -194.320 q^{55} +0.102653 q^{56} -354.642 q^{58} -674.993 q^{59} +122.013 q^{61} +968.787 q^{62} -509.020 q^{64} +204.139 q^{65} -711.424 q^{67} +407.764 q^{68} -17.8750 q^{70} -1026.79 q^{71} -1224.17 q^{73} +373.827 q^{74} -701.821 q^{76} +53.0214 q^{77} -13.8124 q^{79} -259.849 q^{80} -1749.79 q^{82} +113.011 q^{83} +206.950 q^{85} +691.743 q^{86} +4.46053 q^{88} +1140.93 q^{89} -55.7007 q^{91} +1131.27 q^{92} -588.853 q^{94} -356.192 q^{95} -476.494 q^{97} -1366.13 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

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