LMFDB - Embedded newform 2151.4.a.c.1.17

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:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
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)

$f(q)$	$=$	$q+1.64745 q^{2} -5.28592 q^{4} -5.57976 q^{5} +35.6078 q^{7} -21.8878 q^{8} +O(q^{10})$	\(q+1.64745 q^{2} -5.28592 q^{4} -5.57976 q^{5} +35.6078 q^{7} -21.8878 q^{8} -9.19235 q^{10} +54.2616 q^{11} -19.0551 q^{13} +58.6620 q^{14} +6.22837 q^{16} +90.4987 q^{17} -78.5936 q^{19} +29.4942 q^{20} +89.3930 q^{22} +172.386 q^{23} -93.8663 q^{25} -31.3922 q^{26} -188.220 q^{28} -79.9221 q^{29} +48.5690 q^{31} +185.364 q^{32} +149.092 q^{34} -198.683 q^{35} +437.671 q^{37} -129.479 q^{38} +122.129 q^{40} -64.7499 q^{41} -440.739 q^{43} -286.823 q^{44} +283.997 q^{46} +210.972 q^{47} +924.917 q^{49} -154.640 q^{50} +100.724 q^{52} -19.8918 q^{53} -302.766 q^{55} -779.378 q^{56} -131.667 q^{58} +163.118 q^{59} -439.784 q^{61} +80.0148 q^{62} +255.549 q^{64} +106.323 q^{65} -889.752 q^{67} -478.369 q^{68} -327.319 q^{70} +5.50954 q^{71} -601.222 q^{73} +721.039 q^{74} +415.440 q^{76} +1932.14 q^{77} -93.0720 q^{79} -34.7528 q^{80} -106.672 q^{82} -212.225 q^{83} -504.961 q^{85} -726.094 q^{86} -1187.67 q^{88} +461.248 q^{89} -678.509 q^{91} -911.220 q^{92} +347.565 q^{94} +438.533 q^{95} +885.983 q^{97} +1523.75 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	$ 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) $ 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * 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 $

$2$	1.64745	0.582460	0.291230	−	0.956653i	$-0.405936\pi$
$2$	1.64745	0.582460	0.291230	+	0.956653i	$0.405936\pi$
$3$	0	0
$3$	0	0
$4$	−5.28592	−0.660740
$5$	−5.57976	−0.499069	−0.249534	−	0.968366i	$-0.580278\pi$
$5$	−5.57976	−0.499069	−0.249534	+	0.968366i	$0.580278\pi$
$6$	0	0
$7$	35.6078	1.92264	0.961321	−	0.275431i	$-0.0888204\pi$
$7$	35.6078	1.92264	0.961321	+	0.275431i	$0.0888204\pi$
$8$	−21.8878	−0.967315
$9$	0	0
$10$	−9.19235	−0.290687
$11$	54.2616	1.48732	0.743658	−	0.668560i	$-0.233089\pi$
$11$	54.2616	1.48732	0.743658	+	0.668560i	$0.233089\pi$
$12$	0	0
$13$	−19.0551	−0.406533	−0.203266	−	0.979123i	$-0.565156\pi$
$13$	−19.0551	−0.406533	−0.203266	+	0.979123i	$0.565156\pi$
$14$	58.6620	1.11986
$15$	0	0
$16$	6.22837	0.0973183
$17$	90.4987	1.29113	0.645564	−	0.763706i	$-0.276622\pi$
$17$	90.4987	1.29113	0.645564	+	0.763706i	$0.276622\pi$
$18$	0	0
$19$	−78.5936	−0.948979	−0.474490	−	0.880261i	$-0.657367\pi$
$19$	−78.5936	−0.948979	−0.474490	+	0.880261i	$0.657367\pi$
$20$	29.4942	0.329755
$21$	0	0
$22$	89.3930	0.866302
$23$	172.386	1.56283	0.781413	−	0.624014i	$-0.214499\pi$
$23$	172.386	1.56283	0.781413	+	0.624014i	$0.214499\pi$
$24$	0	0
$25$	−93.8663	−0.750930
$26$	−31.3922	−0.236789
$27$	0	0
$28$	−188.220	−1.27037
$29$	−79.9221	−0.511764	−0.255882	−	0.966708i	$-0.582366\pi$
$29$	−79.9221	−0.511764	−0.255882	+	0.966708i	$0.582366\pi$
$30$	0	0
$31$	48.5690	0.281395	0.140698	−	0.990053i	$-0.455065\pi$
$31$	48.5690	0.281395	0.140698	+	0.990053i	$0.455065\pi$
$32$	185.364	1.02400
$33$	0	0
$34$	149.092	0.752030
$35$	−198.683	−0.959530
$36$	0	0
$37$	437.671	1.94467	0.972333	−	0.233599i	$-0.0750503\pi$
$37$	437.671	1.94467	0.972333	+	0.233599i	$0.0750503\pi$
$38$	−129.479	−0.552742
$39$	0	0
$40$	122.129	0.482756
$41$	−64.7499	−0.246640	−0.123320	−	0.992367i	$-0.539354\pi$
$41$	−64.7499	−0.246640	−0.123320	+	0.992367i	$0.539354\pi$
$42$	0	0
$43$	−440.739	−1.56307	−0.781536	−	0.623860i	$-0.785564\pi$
$43$	−440.739	−1.56307	−0.781536	+	0.623860i	$0.785564\pi$
$44$	−286.823	−0.982730
$45$	0	0
$46$	283.997	0.910283
$47$	210.972	0.654755	0.327377	−	0.944894i	$-0.393835\pi$
$47$	210.972	0.654755	0.327377	+	0.944894i	$0.393835\pi$
$48$	0	0
$49$	924.917	2.69655
$50$	−154.640	−0.437387
$51$	0	0
$52$	100.724	0.268613
$53$	−19.8918	−0.0515538	−0.0257769	−	0.999668i	$-0.508206\pi$
$53$	−19.8918	−0.0515538	−0.0257769	+	0.999668i	$0.508206\pi$
$54$	0	0
$55$	−302.766	−0.742273
$56$	−779.378	−1.85980
$57$	0	0
$58$	−131.667	−0.298082
$59$	163.118	0.359935	0.179967	−	0.983673i	$-0.442401\pi$
$59$	163.118	0.359935	0.179967	+	0.983673i	$0.442401\pi$
$60$	0	0
$61$	−439.784	−0.923091	−0.461545	−	0.887117i	$-0.652705\pi$
$61$	−439.784	−0.923091	−0.461545	+	0.887117i	$0.652705\pi$
$62$	80.0148	0.163902
$63$	0	0
$64$	255.549	0.499120
$65$	106.323	0.202888
$66$	0	0
$67$	−889.752	−1.62240	−0.811198	−	0.584772i	$-0.801184\pi$
$67$	−889.752	−1.62240	−0.811198	+	0.584772i	$0.801184\pi$
$68$	−478.369	−0.853100
$69$	0	0
$70$	−327.319	−0.558888
$71$	5.50954	0.00920933	0.00460466	−	0.999989i	$-0.498534\pi$
$71$	5.50954	0.00920933	0.00460466	+	0.999989i	$0.498534\pi$
$72$	0	0
$73$	−601.222	−0.963941	−0.481970	−	0.876188i	$-0.660079\pi$
$73$	−601.222	−0.963941	−0.481970	+	0.876188i	$0.660079\pi$
$74$	721.039	1.13269
$75$	0	0
$76$	415.440	0.627029
$77$	1932.14	2.85958
$78$	0	0
$79$	−93.0720	−0.132550	−0.0662748	−	0.997801i	$-0.521111\pi$
$79$	−93.0720	−0.132550	−0.0662748	+	0.997801i	$0.521111\pi$
$80$	−34.7528	−0.0485685
$81$	0	0
$82$	−106.672	−0.143658
$83$	−212.225	−0.280659	−0.140330	−	0.990105i	$-0.544816\pi$
$83$	−212.225	−0.280659	−0.140330	+	0.990105i	$0.544816\pi$
$84$	0	0
$85$	−504.961	−0.644361
$86$	−726.094	−0.910427
$87$	0	0
$88$	−1187.67	−1.43870
$89$	461.248	0.549351	0.274675	−	0.961537i	$-0.411430\pi$
$89$	461.248	0.549351	0.274675	+	0.961537i	$0.411430\pi$
$90$	0	0
$91$	−678.509	−0.781617
$92$	−911.220	−1.03262
$93$	0	0
$94$	347.565	0.381369
$95$	438.533	0.473606
$96$	0	0
$97$	885.983	0.927401	0.463701	−	0.885992i	$-0.346521\pi$
$97$	885.983	0.927401	0.463701	+	0.885992i	$0.346521\pi$
$98$	1523.75	1.57063
$99$	0	0
$100$	496.170	0.496170
$101$	265.340	0.261409	0.130705	−	0.991421i	$-0.458276\pi$
$101$	265.340	0.261409	0.130705	+	0.991421i	$0.458276\pi$
$102$	0	0
$103$	−498.373	−0.476759	−0.238380	−	0.971172i	$-0.576616\pi$
$103$	−498.373	−0.476759	−0.238380	+	0.971172i	$0.576616\pi$
$104$	417.074	0.393245
$105$	0	0
$106$	−32.7707	−0.0300281
$107$	−330.252	−0.298380	−0.149190	−	0.988809i	$-0.547667\pi$
$107$	−330.252	−0.298380	−0.149190	+	0.988809i	$0.547667\pi$
$108$	0	0
$109$	179.639	0.157856	0.0789282	−	0.996880i	$-0.474850\pi$
$109$	179.639	0.157856	0.0789282	+	0.996880i	$0.474850\pi$
$110$	−498.791	−0.432344
$111$	0	0
$112$	221.779	0.187108
$113$	1527.27	1.27145	0.635725	−	0.771915i	$-0.280701\pi$
$113$	1527.27	1.27145	0.635725	+	0.771915i	$0.280701\pi$
$114$	0	0
$115$	−961.873	−0.779957
$116$	422.462	0.338143
$117$	0	0
$118$	268.728	0.209648
$119$	3222.46	2.48238
$120$	0	0
$121$	1613.32	1.21211
$122$	−724.520	−0.537663
$123$	0	0
$124$	−256.732	−0.185929
$125$	1221.22	0.873834
$126$	0	0
$127$	1374.93	0.960670	0.480335	−	0.877085i	$-0.340515\pi$
$127$	1374.93	0.960670	0.480335	+	0.877085i	$0.340515\pi$
$128$	−1061.90	−0.733281
$129$	0	0
$130$	175.161	0.118174
$131$	−1046.00	−0.697627	−0.348813	−	0.937192i	$-0.613415\pi$
$131$	−1046.00	−0.697627	−0.348813	+	0.937192i	$0.613415\pi$
$132$	0	0
$133$	−2798.55	−1.82455
$134$	−1465.82	−0.944981
$135$	0	0
$136$	−1980.82	−1.24893
$137$	−956.425	−0.596445	−0.298222	−	0.954496i	$-0.596394\pi$
$137$	−956.425	−0.596445	−0.298222	+	0.954496i	$0.596394\pi$
$138$	0	0
$139$	1779.97	1.08615	0.543075	−	0.839684i	$-0.317260\pi$
$139$	1779.97	1.08615	0.543075	+	0.839684i	$0.317260\pi$
$140$	1050.22	0.634000
$141$	0	0
$142$	9.07667	0.00536406
$143$	−1033.96	−0.604643
$144$	0	0
$145$	445.946	0.255405
$146$	−990.480	−0.561457
$147$	0	0
$148$	−2313.49	−1.28492
$149$	−2389.19	−1.31363	−0.656813	−	0.754054i	$-0.728096\pi$
$149$	−2389.19	−1.31363	−0.656813	+	0.754054i	$0.728096\pi$
$150$	0	0
$151$	2983.32	1.60781	0.803904	−	0.594759i	$-0.202752\pi$
$151$	2983.32	1.60781	0.803904	+	0.594759i	$0.202752\pi$
$152$	1720.24	0.917962
$153$	0	0
$154$	3183.09	1.66559
$155$	−271.003	−0.140436
$156$	0	0
$157$	1895.07	0.963330	0.481665	−	0.876355i	$-0.340032\pi$
$157$	1895.07	0.963330	0.481665	+	0.876355i	$0.340032\pi$
$158$	−153.331	−0.0772048
$159$	0	0
$160$	−1034.28	−0.511046
$161$	6138.30	3.00475
$162$	0	0
$163$	−874.514	−0.420228	−0.210114	−	0.977677i	$-0.567384\pi$
$163$	−874.514	−0.420228	−0.210114	+	0.977677i	$0.567384\pi$
$164$	342.263	0.162965
$165$	0	0
$166$	−349.629	−0.163473
$167$	4191.06	1.94200	0.971001	−	0.239076i	$-0.0768444\pi$
$167$	4191.06	1.94200	0.971001	+	0.239076i	$0.0768444\pi$
$168$	0	0
$169$	−1833.90	−0.834731
$170$	−831.896	−0.375315
$171$	0	0
$172$	2329.71	1.03279
$173$	1048.67	0.460861	0.230431	−	0.973089i	$-0.425987\pi$
$173$	1048.67	0.460861	0.230431	+	0.973089i	$0.425987\pi$
$174$	0	0
$175$	−3342.38	−1.44377
$176$	337.961	0.144743
$177$	0	0
$178$	759.882	0.319975
$179$	4139.45	1.72848	0.864238	−	0.503083i	$-0.167801\pi$
$179$	4139.45	1.72848	0.864238	+	0.503083i	$0.167801\pi$
$180$	0	0
$181$	−3662.97	−1.50423	−0.752117	−	0.659030i	$-0.770967\pi$
$181$	−3662.97	−1.50423	−0.752117	+	0.659030i	$0.770967\pi$
$182$	−1117.81	−0.455260
$183$	0	0
$184$	−3773.16	−1.51174
$185$	−2442.10	−0.970522
$186$	0	0
$187$	4910.60	1.92032
$188$	−1115.18	−0.432623
$189$	0	0
$190$	722.459	0.275856
$191$	2845.69	1.07805	0.539023	−	0.842291i	$-0.318794\pi$
$191$	2845.69	1.07805	0.539023	+	0.842291i	$0.318794\pi$
$192$	0	0
$193$	4714.14	1.75819	0.879096	−	0.476645i	$-0.158147\pi$
$193$	4714.14	1.75819	0.879096	+	0.476645i	$0.158147\pi$
$194$	1459.61	0.540174
$195$	0	0
$196$	−4889.04	−1.78172
$197$	4032.32	1.45833	0.729165	−	0.684337i	$-0.239908\pi$
$197$	4032.32	1.45833	0.729165	+	0.684337i	$0.239908\pi$
$198$	0	0
$199$	−933.417	−0.332503	−0.166252	−	0.986083i	$-0.553166\pi$
$199$	−933.417	−0.332503	−0.166252	+	0.986083i	$0.553166\pi$
$200$	2054.53	0.726386
$201$	0	0
$202$	437.133	0.152260
$203$	−2845.85	−0.983939
$204$	0	0
$205$	361.289	0.123090
$206$	−821.043	−0.277693
$207$	0	0
$208$	−118.682	−0.0395631
$209$	−4264.61	−1.41143
$210$	0	0
$211$	−1813.40	−0.591658	−0.295829	−	0.955241i	$-0.595596\pi$
$211$	−1813.40	−0.591658	−0.295829	+	0.955241i	$0.595596\pi$
$212$	105.147	0.0340637
$213$	0	0
$214$	−544.072	−0.173794
$215$	2459.22	0.780081
$216$	0	0
$217$	1729.44	0.541022
$218$	295.946	0.0919450
$219$	0	0
$220$	1600.40	0.490450
$221$	−1724.46	−0.524885
$222$	0	0
$223$	2352.07	0.706306	0.353153	−	0.935566i	$-0.385110\pi$
$223$	2352.07	0.706306	0.353153	+	0.935566i	$0.385110\pi$
$224$	6600.39	1.96878
$225$	0	0
$226$	2516.10	0.740569
$227$	3432.14	1.00352	0.501761	−	0.865006i	$-0.332686\pi$
$227$	3432.14	1.00352	0.501761	+	0.865006i	$0.332686\pi$
$228$	0	0
$229$	4121.07	1.18920	0.594602	−	0.804020i	$-0.297310\pi$
$229$	4121.07	1.18920	0.594602	+	0.804020i	$0.297310\pi$
$230$	−1584.63	−0.454294
$231$	0	0
$232$	1749.32	0.495037
$233$	1117.49	0.314202	0.157101	−	0.987583i	$-0.449785\pi$
$233$	1117.49	0.314202	0.157101	+	0.987583i	$0.449785\pi$
$234$	0	0
$235$	−1177.17	−0.326768
$236$	−862.229	−0.237823
$237$	0	0
$238$	5308.83	1.44588
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−4803.11	−1.28380	−0.641899	−	0.766789i	$-0.721853\pi$
$241$	−4803.11	−1.28380	−0.641899	+	0.766789i	$0.721853\pi$
$242$	2657.86	0.706006
$243$	0	0
$244$	2324.66	0.609923
$245$	−5160.81	−1.34576
$246$	0	0
$247$	1497.61	0.385791
$248$	−1063.07	−0.272198
$249$	0	0
$250$	2011.89	0.508974
$251$	−3912.29	−0.983831	−0.491915	−	0.870643i	$-0.663703\pi$
$251$	−3912.29	−0.983831	−0.491915	+	0.870643i	$0.663703\pi$
$252$	0	0
$253$	9353.94	2.32442
$254$	2265.12	0.559552
$255$	0	0
$256$	−3793.83	−0.926227
$257$	−6174.12	−1.49856	−0.749282	−	0.662251i	$-0.769601\pi$
$257$	−6174.12	−1.49856	−0.749282	+	0.662251i	$0.769601\pi$
$258$	0	0
$259$	15584.5	3.73890
$260$	−562.013	−0.134056
$261$	0	0
$262$	−1723.22	−0.406340
$263$	6934.32	1.62581	0.812906	−	0.582395i	$-0.197884\pi$
$263$	6934.32	1.62581	0.812906	+	0.582395i	$0.197884\pi$
$264$	0	0
$265$	110.992	0.0257289
$266$	−4610.45	−1.06273
$267$	0	0
$268$	4703.16	1.07198
$269$	5537.65	1.25515	0.627577	−	0.778555i	$-0.284047\pi$
$269$	5537.65	1.25515	0.627577	+	0.778555i	$0.284047\pi$
$270$	0	0
$271$	7680.70	1.72166	0.860829	−	0.508894i	$-0.169945\pi$
$271$	7680.70	1.72166	0.860829	+	0.508894i	$0.169945\pi$
$272$	563.660	0.125650
$273$	0	0
$274$	−1575.66	−0.347405
$275$	−5093.33	−1.11687
$276$	0	0
$277$	−4927.67	−1.06886	−0.534432	−	0.845212i	$-0.679474\pi$
$277$	−4927.67	−1.06886	−0.534432	+	0.845212i	$0.679474\pi$
$278$	2932.40	0.632639
$279$	0	0
$280$	4348.74	0.928168
$281$	2240.98	0.475751	0.237875	−	0.971296i	$-0.423549\pi$
$281$	2240.98	0.475751	0.237875	+	0.971296i	$0.423549\pi$
$282$	0	0
$283$	4151.98	0.872119	0.436059	−	0.899918i	$-0.356374\pi$
$283$	4151.98	0.872119	0.436059	+	0.899918i	$0.356374\pi$
$284$	−29.1230	−0.00608497
$285$	0	0
$286$	−1703.39	−0.352180
$287$	−2305.60	−0.474200
$288$	0	0
$289$	3277.02	0.667011
$290$	734.671	0.148763
$291$	0	0
$292$	3178.01	0.636914
$293$	−3487.66	−0.695396	−0.347698	−	0.937607i	$-0.613037\pi$
$293$	−3487.66	−0.695396	−0.347698	+	0.937607i	$0.613037\pi$
$294$	0	0
$295$	−910.159	−0.179632
$296$	−9579.67	−1.88110
$297$	0	0
$298$	−3936.06	−0.765134
$299$	−3284.83	−0.635340
$300$	0	0
$301$	−15693.8	−3.00523
$302$	4914.86	0.936484
$303$	0	0
$304$	−489.510	−0.0923531
$305$	2453.89	0.460686
$306$	0	0
$307$	−1383.54	−0.257209	−0.128604	−	0.991696i	$-0.541050\pi$
$307$	−1383.54	−0.257209	−0.128604	+	0.991696i	$0.541050\pi$
$308$	−10213.1	−1.88944
$309$	0	0
$310$	−446.463	−0.0817981
$311$	6117.12	1.11534	0.557669	−	0.830063i	$-0.311696\pi$
$311$	6117.12	1.11534	0.557669	+	0.830063i	$0.311696\pi$
$312$	0	0
$313$	−2006.29	−0.362308	−0.181154	−	0.983455i	$-0.557983\pi$
$313$	−2006.29	−0.362308	−0.181154	+	0.983455i	$0.557983\pi$
$314$	3122.02	0.561101
$315$	0	0
$316$	491.972	0.0875809
$317$	−5133.38	−0.909525	−0.454763	−	0.890613i	$-0.650276\pi$
$317$	−5133.38	−0.909525	−0.454763	+	0.890613i	$0.650276\pi$
$318$	0	0
$319$	−4336.70	−0.761155
$320$	−1425.90	−0.249095
$321$	0	0
$322$	10112.5	1.75015
$323$	−7112.62	−1.22525
$324$	0	0
$325$	1788.63	0.305278
$326$	−1440.71	−0.244766
$327$	0	0
$328$	1417.24	0.238578
$329$	7512.27	1.25886
$330$	0	0
$331$	1703.75	0.282920	0.141460	−	0.989944i	$-0.454820\pi$
$331$	1703.75	0.282920	0.141460	+	0.989944i	$0.454820\pi$
$332$	1121.81	0.185443
$333$	0	0
$334$	6904.55	1.13114
$335$	4964.60	0.809687
$336$	0	0
$337$	5307.52	0.857920	0.428960	−	0.903324i	$-0.358880\pi$
$337$	5307.52	0.857920	0.428960	+	0.903324i	$0.358880\pi$
$338$	−3021.26	−0.486198
$339$	0	0
$340$	2669.19	0.425756
$341$	2635.43	0.418524
$342$	0	0
$343$	20720.8	3.26186
$344$	9646.83	1.51198
$345$	0	0
$346$	1727.63	0.268433
$347$	−11128.3	−1.72161	−0.860804	−	0.508937i	$-0.830039\pi$
$347$	−11128.3	−1.72161	−0.860804	+	0.508937i	$0.830039\pi$
$348$	0	0
$349$	498.178	0.0764093	0.0382047	−	0.999270i	$-0.487836\pi$
$349$	498.178	0.0764093	0.0382047	+	0.999270i	$0.487836\pi$
$350$	−5506.38	−0.840938
$351$	0	0
$352$	10058.1	1.52301
$353$	4016.63	0.605619	0.302809	−	0.953051i	$-0.402075\pi$
$353$	4016.63	0.605619	0.302809	+	0.953051i	$0.402075\pi$
$354$	0	0
$355$	−30.7419	−0.00459609
$356$	−2438.12	−0.362978
$357$	0	0
$358$	6819.52	1.00677
$359$	−864.748	−0.127130	−0.0635650	−	0.997978i	$-0.520247\pi$
$359$	−864.748	−0.127130	−0.0635650	+	0.997978i	$0.520247\pi$
$360$	0	0
$361$	−682.047	−0.0994383
$362$	−6034.54	−0.876156
$363$	0	0
$364$	3586.55	0.516446
$365$	3354.67	0.481073
$366$	0	0
$367$	−2722.63	−0.387249	−0.193624	−	0.981076i	$-0.562024\pi$
$367$	−2722.63	−0.387249	−0.193624	+	0.981076i	$0.562024\pi$
$368$	1073.68	0.152092
$369$	0	0
$370$	−4023.22	−0.565290
$371$	−708.305	−0.0991196
$372$	0	0
$373$	−8534.44	−1.18471	−0.592354	−	0.805677i	$-0.701801\pi$
$373$	−8534.44	−1.18471	−0.592354	+	0.805677i	$0.701801\pi$
$374$	8089.95	1.11851
$375$	0	0
$376$	−4617.73	−0.633354
$377$	1522.92	0.208049
$378$	0	0
$379$	−3521.92	−0.477331	−0.238666	−	0.971102i	$-0.576710\pi$
$379$	−3521.92	−0.477331	−0.238666	+	0.971102i	$0.576710\pi$
$380$	−2318.05	−0.312930
$381$	0	0
$382$	4688.11	0.627918
$383$	3611.09	0.481770	0.240885	−	0.970554i	$-0.422562\pi$
$383$	3611.09	0.481770	0.240885	+	0.970554i	$0.422562\pi$
$384$	0	0
$385$	−10780.9	−1.42713
$386$	7766.28	1.02408
$387$	0	0
$388$	−4683.24	−0.612771
$389$	6408.88	0.835329	0.417665	−	0.908601i	$-0.362849\pi$
$389$	6408.88	0.835329	0.417665	+	0.908601i	$0.362849\pi$
$390$	0	0
$391$	15600.7	2.01781
$392$	−20244.4	−2.60841
$393$	0	0
$394$	6643.03	0.849419
$395$	519.319	0.0661514
$396$	0	0
$397$	9621.06	1.21629	0.608145	−	0.793826i	$-0.291914\pi$
$397$	9621.06	1.21629	0.608145	+	0.793826i	$0.291914\pi$
$398$	−1537.75	−0.193670
$399$	0	0
$400$	−584.634	−0.0730793
$401$	−12078.8	−1.50421	−0.752105	−	0.659043i	$-0.770962\pi$
$401$	−12078.8	−1.50421	−0.752105	+	0.659043i	$0.770962\pi$
$402$	0	0
$403$	−925.486	−0.114396
$404$	−1402.57	−0.172724
$405$	0	0
$406$	−4688.38	−0.573105
$407$	23748.7	2.89233
$408$	0	0
$409$	−9252.85	−1.11864	−0.559320	−	0.828952i	$-0.688938\pi$
$409$	−9252.85	−1.11864	−0.559320	+	0.828952i	$0.688938\pi$
$410$	595.204	0.0716951
$411$	0	0
$412$	2634.36	0.315014
$413$	5808.28	0.692026
$414$	0	0
$415$	1184.16	0.140068
$416$	−3532.11	−0.416289
$417$	0	0
$418$	−7025.72	−0.822103
$419$	−3089.02	−0.360164	−0.180082	−	0.983652i	$-0.557636\pi$
$419$	−3089.02	−0.360164	−0.180082	+	0.983652i	$0.557636\pi$
$420$	0	0
$421$	1044.46	0.120912	0.0604558	−	0.998171i	$-0.480745\pi$
$421$	1044.46	0.120912	0.0604558	+	0.998171i	$0.480745\pi$
$422$	−2987.48	−0.344617
$423$	0	0
$424$	435.389	0.0498688
$425$	−8494.78	−0.969547
$426$	0	0
$427$	−15659.7	−1.77477
$428$	1745.68	0.197152
$429$	0	0
$430$	4051.43	0.454366
$431$	−2532.27	−0.283005	−0.141502	−	0.989938i	$-0.545193\pi$
$431$	−2532.27	−0.283005	−0.141502	+	0.989938i	$0.545193\pi$
$432$	0	0
$433$	17903.2	1.98700	0.993501	−	0.113827i	$-0.0363109\pi$
$433$	17903.2	1.98700	0.993501	+	0.113827i	$0.0363109\pi$
$434$	2849.15	0.315124
$435$	0	0
$436$	−949.560	−0.104302
$437$	−13548.4	−1.48309
$438$	0	0
$439$	−13212.4	−1.43643	−0.718214	−	0.695822i	$-0.755040\pi$
$439$	−13212.4	−1.43643	−0.718214	+	0.695822i	$0.755040\pi$
$440$	6626.90	0.718012
$441$	0	0
$442$	−2840.95	−0.305725
$443$	−4677.28	−0.501635	−0.250818	−	0.968034i	$-0.580699\pi$
$443$	−4677.28	−0.501635	−0.250818	+	0.968034i	$0.580699\pi$
$444$	0	0
$445$	−2573.65	−0.274164
$446$	3874.91	0.411395
$447$	0	0
$448$	9099.56	0.959629
$449$	−12810.1	−1.34643	−0.673216	−	0.739446i	$-0.735088\pi$
$449$	−12810.1	−1.34643	−0.673216	+	0.739446i	$0.735088\pi$
$450$	0	0
$451$	−3513.43	−0.366832
$452$	−8073.06	−0.840099
$453$	0	0
$454$	5654.27	0.584511
$455$	3785.92	0.390080
$456$	0	0
$457$	−19402.9	−1.98606	−0.993032	−	0.117846i	$-0.962401\pi$
$457$	−19402.9	−1.98606	−0.993032	+	0.117846i	$0.962401\pi$
$458$	6789.24	0.692664
$459$	0	0
$460$	5084.39	0.515349
$461$	13983.8	1.41278	0.706388	−	0.707825i	$-0.250323\pi$
$461$	13983.8	1.41278	0.706388	+	0.707825i	$0.250323\pi$
$462$	0	0
$463$	−7800.22	−0.782953	−0.391476	−	0.920188i	$-0.628036\pi$
$463$	−7800.22	−0.782953	−0.391476	+	0.920188i	$0.628036\pi$
$464$	−497.784	−0.0498040
$465$	0	0
$466$	1841.00	0.183010
$467$	13713.3	1.35883	0.679415	−	0.733754i	$-0.262234\pi$
$467$	13713.3	1.35883	0.679415	+	0.733754i	$0.262234\pi$
$468$	0	0
$469$	−31682.1	−3.11929
$470$	−1939.33	−0.190329
$471$	0	0
$472$	−3570.30	−0.348170
$473$	−23915.2	−2.32478
$474$	0	0
$475$	7377.29	0.712618
$476$	−17033.7	−1.64021
$477$	0	0
$478$	393.739	0.0376762
$479$	−15954.9	−1.52192	−0.760958	−	0.648801i	$-0.775271\pi$
$479$	−15954.9	−1.52192	−0.760958	+	0.648801i	$0.775271\pi$
$480$	0	0
$481$	−8339.85	−0.790570
$482$	−7912.86	−0.747761
$483$	0	0
$484$	−8527.88	−0.800890
$485$	−4943.57	−0.462837
$486$	0	0
$487$	5449.51	0.507065	0.253533	−	0.967327i	$-0.418408\pi$
$487$	5449.51	0.507065	0.253533	+	0.967327i	$0.418408\pi$
$488$	9625.91	0.892919
$489$	0	0
$490$	−8502.16	−0.783854
$491$	−7614.50	−0.699873	−0.349936	−	0.936773i	$-0.613797\pi$
$491$	−7614.50	−0.699873	−0.349936	+	0.936773i	$0.613797\pi$
$492$	0	0
$493$	−7232.85	−0.660753
$494$	2467.22	0.224708
$495$	0	0
$496$	302.506	0.0273849
$497$	196.183	0.0177062
$498$	0	0
$499$	3547.26	0.318231	0.159115	−	0.987260i	$-0.449136\pi$
$499$	3547.26	0.318231	0.159115	+	0.987260i	$0.449136\pi$
$500$	−6455.28	−0.577378
$501$	0	0
$502$	−6445.28	−0.573042
$503$	−15506.5	−1.37455	−0.687276	−	0.726397i	$-0.741194\pi$
$503$	−15506.5	−1.37455	−0.687276	+	0.726397i	$0.741194\pi$
$504$	0	0
$505$	−1480.53	−0.130461
$506$	15410.1	1.35388
$507$	0	0
$508$	−7267.76	−0.634753
$509$	−18965.4	−1.65153	−0.825763	−	0.564018i	$-0.809255\pi$
$509$	−18965.4	−1.65153	−0.825763	+	0.564018i	$0.809255\pi$
$510$	0	0
$511$	−21408.2	−1.85331
$512$	2245.12	0.193791
$513$	0	0
$514$	−10171.5	−0.872854
$515$	2780.80	0.237936
$516$	0	0
$517$	11447.7	0.973828
$518$	25674.6	2.17776
$519$	0	0
$520$	−2327.17	−0.196256
$521$	4545.27	0.382211	0.191105	−	0.981570i	$-0.438793\pi$
$521$	4545.27	0.382211	0.191105	+	0.981570i	$0.438793\pi$
$522$	0	0
$523$	19500.8	1.63042	0.815210	−	0.579165i	$-0.196621\pi$
$523$	19500.8	1.63042	0.815210	+	0.579165i	$0.196621\pi$
$524$	5529.06	0.460950
$525$	0	0
$526$	11423.9	0.946971
$527$	4395.44	0.363317
$528$	0	0
$529$	17550.0	1.44242
$530$	182.853	0.0149861
$531$	0	0
$532$	14792.9	1.20555
$533$	1233.81	0.100267
$534$	0	0
$535$	1842.72	0.148912
$536$	19474.7	1.56937
$537$	0	0
$538$	9122.97	0.731077
$539$	50187.5	4.01063
$540$	0	0
$541$	−14008.5	−1.11325	−0.556627	−	0.830762i	$-0.687905\pi$
$541$	−14008.5	−1.11325	−0.556627	+	0.830762i	$0.687905\pi$
$542$	12653.5	1.00280
$543$	0	0
$544$	16775.2	1.32211
$545$	−1002.34	−0.0787811
$546$	0	0
$547$	−12009.3	−0.938720	−0.469360	−	0.883007i	$-0.655515\pi$
$547$	−12009.3	−0.938720	−0.469360	+	0.883007i	$0.655515\pi$
$548$	5055.59	0.394095
$549$	0	0
$550$	−8390.99	−0.650533
$551$	6281.36	0.485653
$552$	0	0
$553$	−3314.09	−0.254845
$554$	−8118.07	−0.622570
$555$	0	0
$556$	−9408.76	−0.717663
$557$	13403.1	1.01958	0.509790	−	0.860299i	$-0.329723\pi$
$557$	13403.1	1.01958	0.509790	+	0.860299i	$0.329723\pi$
$558$	0	0
$559$	8398.32	0.635440
$560$	−1237.47	−0.0933799
$561$	0	0
$562$	3691.90	0.277106
$563$	13027.5	0.975210	0.487605	−	0.873064i	$-0.337871\pi$
$563$	13027.5	0.975210	0.487605	+	0.873064i	$0.337871\pi$
$564$	0	0
$565$	−8521.82	−0.634541
$566$	6840.16	0.507974
$567$	0	0
$568$	−120.592	−0.00890832
$569$	−7195.64	−0.530153	−0.265076	−	0.964227i	$-0.585397\pi$
$569$	−7195.64	−0.530153	−0.265076	+	0.964227i	$0.585397\pi$
$570$	0	0
$571$	−13680.8	−1.00267	−0.501333	−	0.865255i	$-0.667157\pi$
$571$	−13680.8	−1.00267	−0.501333	+	0.865255i	$0.667157\pi$
$572$	5465.42	0.399512
$573$	0	0
$574$	−3798.36	−0.276203
$575$	−16181.2	−1.17357
$576$	0	0
$577$	19096.2	1.37779	0.688897	−	0.724859i	$-0.258095\pi$
$577$	19096.2	1.37779	0.688897	+	0.724859i	$0.258095\pi$
$578$	5398.72	0.388507
$579$	0	0
$580$	−2357.23	−0.168757
$581$	−7556.87	−0.539607
$582$	0	0
$583$	−1079.36	−0.0766769
$584$	13159.4	0.932434
$585$	0	0
$586$	−5745.73	−0.405041
$587$	−16405.3	−1.15353	−0.576763	−	0.816911i	$-0.695685\pi$
$587$	−16405.3	−1.15353	−0.576763	+	0.816911i	$0.695685\pi$
$588$	0	0
$589$	−3817.21	−0.267038
$590$	−1499.44	−0.104629
$591$	0	0
$592$	2725.98	0.189252
$593$	−19025.6	−1.31751	−0.658757	−	0.752356i	$-0.728917\pi$
$593$	−19025.6	−1.31751	−0.658757	+	0.752356i	$0.728917\pi$
$594$	0	0
$595$	−17980.6	−1.23888
$596$	12629.1	0.867965
$597$	0	0
$598$	−5411.58	−0.370060
$599$	3751.43	0.255892	0.127946	−	0.991781i	$-0.459162\pi$
$599$	3751.43	0.255892	0.127946	+	0.991781i	$0.459162\pi$
$600$	0	0
$601$	−12654.5	−0.858881	−0.429440	−	0.903095i	$-0.641289\pi$
$601$	−12654.5	−0.858881	−0.429440	+	0.903095i	$0.641289\pi$
$602$	−25854.6	−1.75043
$603$	0	0
$604$	−15769.6	−1.06234
$605$	−9001.93	−0.604926
$606$	0	0
$607$	5552.85	0.371307	0.185653	−	0.982615i	$-0.440560\pi$
$607$	5552.85	0.371307	0.185653	+	0.982615i	$0.440560\pi$
$608$	−14568.4	−0.971754
$609$	0	0
$610$	4042.64	0.268331
$611$	−4020.09	−0.266179
$612$	0	0
$613$	19445.2	1.28122	0.640609	−	0.767868i	$-0.278682\pi$
$613$	19445.2	1.28122	0.640609	+	0.767868i	$0.278682\pi$
$614$	−2279.31	−0.149814
$615$	0	0
$616$	−42290.3	−2.76611
$617$	7681.31	0.501196	0.250598	−	0.968091i	$-0.419373\pi$
$617$	7681.31	0.501196	0.250598	+	0.968091i	$0.419373\pi$
$618$	0	0
$619$	23937.0	1.55430	0.777149	−	0.629317i	$-0.216665\pi$
$619$	23937.0	1.55430	0.777149	+	0.629317i	$0.216665\pi$
$620$	1432.50	0.0927915
$621$	0	0
$622$	10077.6	0.649640
$623$	16424.1	1.05621
$624$	0	0
$625$	4919.17	0.314827
$626$	−3305.26	−0.211030
$627$	0	0
$628$	−10017.2	−0.636511
$629$	39608.7	2.51081
$630$	0	0
$631$	11248.1	0.709632	0.354816	−	0.934936i	$-0.384543\pi$
$631$	11248.1	0.709632	0.354816	+	0.934936i	$0.384543\pi$
$632$	2037.14	0.128217
$633$	0	0
$634$	−8456.97	−0.529762
$635$	−7671.76	−0.479440
$636$	0	0
$637$	−17624.4	−1.09624
$638$	−7144.47	−0.443342
$639$	0	0
$640$	5925.17	0.365958
$641$	−17640.5	−1.08699	−0.543493	−	0.839414i	$-0.682899\pi$
$641$	−17640.5	−1.08699	−0.543493	+	0.839414i	$0.682899\pi$
$642$	0	0
$643$	−13020.3	−0.798555	−0.399278	−	0.916830i	$-0.630739\pi$
$643$	−13020.3	−0.798555	−0.399278	+	0.916830i	$0.630739\pi$
$644$	−32446.6	−1.98536
$645$	0	0
$646$	−11717.7	−0.713661
$647$	2580.88	0.156824	0.0784119	−	0.996921i	$-0.475015\pi$
$647$	2580.88	0.156824	0.0784119	+	0.996921i	$0.475015\pi$
$648$	0	0
$649$	8851.04	0.535337
$650$	2946.67	0.177812
$651$	0	0
$652$	4622.61	0.277662
$653$	9853.27	0.590487	0.295244	−	0.955422i	$-0.404599\pi$
$653$	9853.27	0.590487	0.295244	+	0.955422i	$0.404599\pi$
$654$	0	0
$655$	5836.41	0.348164
$656$	−403.286	−0.0240026
$657$	0	0
$658$	12376.1	0.733235
$659$	3950.82	0.233539	0.116769	−	0.993159i	$-0.462746\pi$
$659$	3950.82	0.233539	0.116769	+	0.993159i	$0.462746\pi$
$660$	0	0
$661$	23360.3	1.37460	0.687299	−	0.726374i	$-0.258796\pi$
$661$	23360.3	1.37460	0.687299	+	0.726374i	$0.258796\pi$
$662$	2806.83	0.164789
$663$	0	0
$664$	4645.15	0.271486
$665$	15615.2	0.910574
$666$	0	0
$667$	−13777.5	−0.799798
$668$	−22153.6	−1.28316
$669$	0	0
$670$	8178.91	0.471610
$671$	−23863.4	−1.37293
$672$	0	0
$673$	20404.8	1.16872	0.584360	−	0.811495i	$-0.301346\pi$
$673$	20404.8	1.16872	0.584360	+	0.811495i	$0.301346\pi$
$674$	8743.85	0.499704
$675$	0	0
$676$	9693.88	0.551541
$677$	17244.1	0.978941	0.489470	−	0.872020i	$-0.337190\pi$
$677$	17244.1	0.978941	0.489470	+	0.872020i	$0.337190\pi$
$678$	0	0
$679$	31547.9	1.78306
$680$	11052.5	0.623300
$681$	0	0
$682$	4341.73	0.243773
$683$	22468.3	1.25875	0.629374	−	0.777102i	$-0.283311\pi$
$683$	22468.3	1.25875	0.629374	+	0.777102i	$0.283311\pi$
$684$	0	0
$685$	5336.62	0.297667
$686$	34136.4	1.89990
$687$	0	0
$688$	−2745.09	−0.152116
$689$	379.040	0.0209583
$690$	0	0
$691$	−1335.99	−0.0735508	−0.0367754	−	0.999324i	$-0.511709\pi$
$691$	−1335.99	−0.0735508	−0.0367754	+	0.999324i	$0.511709\pi$
$692$	−5543.19	−0.304510
$693$	0	0
$694$	−18333.3	−1.00277
$695$	−9931.78	−0.542063
$696$	0	0
$697$	−5859.79	−0.318444
$698$	820.721	0.0445054
$699$	0	0
$700$	17667.5	0.953957
$701$	−14909.4	−0.803308	−0.401654	−	0.915792i	$-0.631565\pi$
$701$	−14909.4	−0.803308	−0.401654	+	0.915792i	$0.631565\pi$
$702$	0	0
$703$	−34398.1	−1.84545
$704$	13866.5	0.742349
$705$	0	0
$706$	6617.17	0.352749
$707$	9448.18	0.502596
$708$	0	0
$709$	543.498	0.0287891	0.0143946	−	0.999896i	$-0.495418\pi$
$709$	543.498	0.0287891	0.0143946	+	0.999896i	$0.495418\pi$
$710$	−50.6456	−0.00267704
$711$	0	0
$712$	−10095.7	−0.531395
$713$	8372.63	0.439772
$714$	0	0
$715$	5769.23	0.301758
$716$	−21880.8	−1.14207
$717$	0	0
$718$	−1424.63	−0.0740481
$719$	25212.4	1.30774	0.653870	−	0.756607i	$-0.273144\pi$
$719$	25212.4	1.30774	0.653870	+	0.756607i	$0.273144\pi$
$720$	0	0
$721$	−17746.0	−0.916637
$722$	−1123.64	−0.0579188
$723$	0	0
$724$	19362.2	0.993908
$725$	7501.99	0.384299
$726$	0	0
$727$	17816.1	0.908890	0.454445	−	0.890775i	$-0.349838\pi$
$727$	17816.1	0.908890	0.454445	+	0.890775i	$0.349838\pi$
$728$	14851.1	0.756069
$729$	0	0
$730$	5526.64	0.280205
$731$	−39886.4	−2.01813
$732$	0	0
$733$	−27700.5	−1.39583	−0.697914	−	0.716181i	$-0.745888\pi$
$733$	−27700.5	−1.39583	−0.697914	+	0.716181i	$0.745888\pi$
$734$	−4485.39	−0.225557
$735$	0	0
$736$	31954.1	1.60033
$737$	−48279.4	−2.41302
$738$	0	0
$739$	177.348	0.00882794	0.00441397	−	0.999990i	$-0.498595\pi$
$739$	177.348	0.00882794	0.00441397	+	0.999990i	$0.498595\pi$
$740$	12908.7	0.641263
$741$	0	0
$742$	−1166.89	−0.0577332
$743$	17677.9	0.872865	0.436432	−	0.899737i	$-0.356242\pi$
$743$	17677.9	0.872865	0.436432	+	0.899737i	$0.356242\pi$
$744$	0	0
$745$	13331.1	0.655589
$746$	−14060.0	−0.690046
$747$	0	0
$748$	−25957.1	−1.26883
$749$	−11759.5	−0.573677
$750$	0	0
$751$	27506.4	1.33652	0.668258	−	0.743930i	$-0.267040\pi$
$751$	27506.4	1.33652	0.668258	+	0.743930i	$0.267040\pi$
$752$	1314.01	0.0637196
$753$	0	0
$754$	2508.93	0.121180
$755$	−16646.2	−0.802407
$756$	0	0
$757$	−5969.17	−0.286596	−0.143298	−	0.989680i	$-0.545771\pi$
$757$	−5969.17	−0.286596	−0.143298	+	0.989680i	$0.545771\pi$
$758$	−5802.16	−0.278026
$759$	0	0
$760$	−9598.54	−0.458126
$761$	13976.6	0.665772	0.332886	−	0.942967i	$-0.391978\pi$
$761$	13976.6	0.665772	0.332886	+	0.942967i	$0.391978\pi$
$762$	0	0
$763$	6396.57	0.303501
$764$	−15042.1	−0.712308
$765$	0	0
$766$	5949.07	0.280612
$767$	−3108.22	−0.146325
$768$	0	0
$769$	21818.2	1.02313	0.511564	−	0.859245i	$-0.329066\pi$
$769$	21818.2	1.02313	0.511564	+	0.859245i	$0.329066\pi$
$770$	−17760.9	−0.831243
$771$	0	0
$772$	−24918.6	−1.16171
$773$	−2656.81	−0.123621	−0.0618104	−	0.998088i	$-0.519687\pi$
$773$	−2656.81	−0.123621	−0.0618104	+	0.998088i	$0.519687\pi$
$774$	0	0
$775$	−4559.00	−0.211308
$776$	−19392.2	−0.897089
$777$	0	0
$778$	10558.3	0.486546
$779$	5088.93	0.234056
$780$	0	0
$781$	298.956	0.0136972
$782$	25701.3	1.17529
$783$	0	0
$784$	5760.73	0.262424
$785$	−10574.0	−0.480768
$786$	0	0
$787$	−19700.1	−0.892292	−0.446146	−	0.894960i	$-0.647204\pi$
$787$	−19700.1	−0.892292	−0.446146	+	0.894960i	$0.647204\pi$
$788$	−21314.6	−0.963578
$789$	0	0
$790$	855.550	0.0385305
$791$	54382.9	2.44454
$792$	0	0
$793$	8380.11	0.375267
$794$	15850.2	0.708440
$795$	0	0
$796$	4933.97	0.219698
$797$	18458.3	0.820362	0.410181	−	0.912004i	$-0.365466\pi$
$797$	18458.3	0.820362	0.410181	+	0.912004i	$0.365466\pi$
$798$	0	0
$799$	19092.7	0.845372
$800$	−17399.4	−0.768952
$801$	0	0
$802$	−19899.2	−0.876143
$803$	−32623.2	−1.43368
$804$	0	0
$805$	−34250.2	−1.49958
$806$	−1524.69	−0.0666313
$807$	0	0
$808$	−5807.72	−0.252865
$809$	−17976.7	−0.781246	−0.390623	−	0.920551i	$-0.627740\pi$
$809$	−17976.7	−0.781246	−0.390623	+	0.920551i	$0.627740\pi$
$810$	0	0
$811$	−28715.6	−1.24333	−0.621665	−	0.783283i	$-0.713544\pi$
$811$	−28715.6	−1.24333	−0.621665	+	0.783283i	$0.713544\pi$
$812$	15042.9	0.650128
$813$	0	0
$814$	39124.7	1.68467
$815$	4879.57	0.209723
$816$	0	0
$817$	34639.3	1.48332
$818$	−15243.6	−0.651563
$819$	0	0
$820$	−1909.74	−0.0813307
$821$	2245.35	0.0954487	0.0477243	−	0.998861i	$-0.484803\pi$
$821$	2245.35	0.0954487	0.0477243	+	0.998861i	$0.484803\pi$
$822$	0	0
$823$	−6607.49	−0.279857	−0.139929	−	0.990162i	$-0.544687\pi$
$823$	−6607.49	−0.279857	−0.139929	+	0.990162i	$0.544687\pi$
$824$	10908.3	0.461176
$825$	0	0
$826$	9568.82	0.403077
$827$	6354.94	0.267210	0.133605	−	0.991035i	$-0.457345\pi$
$827$	6354.94	0.267210	0.133605	+	0.991035i	$0.457345\pi$
$828$	0	0
$829$	−16373.2	−0.685963	−0.342982	−	0.939342i	$-0.611437\pi$
$829$	−16373.2	−0.685963	−0.342982	+	0.939342i	$0.611437\pi$
$830$	1950.85	0.0815842
$831$	0	0
$832$	−4869.51	−0.202909
$833$	83703.9	3.48159
$834$	0	0
$835$	−23385.1	−0.969192
$836$	22542.4	0.932591
$837$	0	0
$838$	−5089.00	−0.209781
$839$	−17839.6	−0.734076	−0.367038	−	0.930206i	$-0.619628\pi$
$839$	−17839.6	−0.734076	−0.367038	+	0.930206i	$0.619628\pi$
$840$	0	0
$841$	−18001.5	−0.738098
$842$	1720.69	0.0704262
$843$	0	0
$844$	9585.51	0.390932
$845$	10232.7	0.416588
$846$	0	0
$847$	57446.8	2.33045
$848$	−123.894	−0.00501713
$849$	0	0
$850$	−13994.7	−0.564722
$851$	75448.4	3.03917
$852$	0	0
$853$	−9279.28	−0.372469	−0.186235	−	0.982505i	$-0.559628\pi$
$853$	−9279.28	−0.372469	−0.186235	+	0.982505i	$0.559628\pi$
$854$	−25798.6	−1.03373
$855$	0	0
$856$	7228.49	0.288627
$857$	−9631.39	−0.383900	−0.191950	−	0.981405i	$-0.561481\pi$
$857$	−9631.39	−0.383900	−0.191950	+	0.981405i	$0.561481\pi$
$858$	0	0
$859$	−4086.40	−0.162312	−0.0811560	−	0.996701i	$-0.525861\pi$
$859$	−4086.40	−0.162312	−0.0811560	+	0.996701i	$0.525861\pi$
$860$	−12999.2	−0.515431
$861$	0	0
$862$	−4171.77	−0.164839
$863$	−34804.9	−1.37285	−0.686426	−	0.727200i	$-0.740821\pi$
$863$	−34804.9	−1.37285	−0.686426	+	0.727200i	$0.740821\pi$
$864$	0	0
$865$	−5851.33	−0.230001
$866$	29494.5	1.15735
$867$	0	0
$868$	−9141.67	−0.357475
$869$	−5050.23	−0.197143
$870$	0	0
$871$	16954.3	0.659557
$872$	−3931.92	−0.152697
$873$	0	0
$874$	−22320.3	−0.863840
$875$	43485.0	1.68007
$876$	0	0
$877$	−16077.5	−0.619041	−0.309521	−	0.950893i	$-0.600169\pi$
$877$	−16077.5	−0.619041	−0.309521	+	0.950893i	$0.600169\pi$
$878$	−21766.7	−0.836662
$879$	0	0
$880$	−1885.74	−0.0722368
$881$	29258.8	1.11890	0.559452	−	0.828863i	$-0.311012\pi$
$881$	29258.8	1.11890	0.559452	+	0.828863i	$0.311012\pi$
$882$	0	0
$883$	−9180.31	−0.349878	−0.174939	−	0.984579i	$-0.555973\pi$
$883$	−9180.31	−0.349878	−0.174939	+	0.984579i	$0.555973\pi$
$884$	9115.36	0.346813
$885$	0	0
$886$	−7705.57	−0.292182
$887$	27405.2	1.03740	0.518701	−	0.854956i	$-0.326416\pi$
$887$	27405.2	1.03740	0.518701	+	0.854956i	$0.326416\pi$
$888$	0	0
$889$	48958.2	1.84702
$890$	−4239.95	−0.159689
$891$	0	0
$892$	−12432.9	−0.466685
$893$	−16581.1	−0.621349
$894$	0	0
$895$	−23097.1	−0.862628
$896$	−37812.1	−1.40984
$897$	0	0
$898$	−21104.0	−0.784243
$899$	−3881.74	−0.144008
$900$	0	0
$901$	−1800.19	−0.0665626
$902$	−5788.19	−0.213665
$903$	0	0
$904$	−33428.7	−1.22989
$905$	20438.5	0.750716
$906$	0	0
$907$	−23293.7	−0.852760	−0.426380	−	0.904544i	$-0.640211\pi$
$907$	−23293.7	−0.852760	−0.426380	+	0.904544i	$0.640211\pi$
$908$	−18142.0	−0.663067
$909$	0	0
$910$	6237.09	0.227206
$911$	37781.1	1.37403	0.687017	−	0.726642i	$-0.258920\pi$
$911$	37781.1	1.37403	0.687017	+	0.726642i	$0.258920\pi$
$912$	0	0
$913$	−11515.7	−0.417429
$914$	−31965.3	−1.15680
$915$	0	0
$916$	−21783.7	−0.785756
$917$	−37245.7	−1.34129
$918$	0	0
$919$	32968.9	1.18340	0.591699	−	0.806159i	$-0.298457\pi$
$919$	32968.9	1.18340	0.591699	+	0.806159i	$0.298457\pi$
$920$	21053.3	0.754464
$921$	0	0
$922$	23037.5	0.822886
$923$	−104.985	−0.00374389
$924$	0	0
$925$	−41082.5	−1.46031
$926$	−12850.4	−0.456039
$927$	0	0
$928$	−14814.6	−0.524046
$929$	−27745.8	−0.979881	−0.489940	−	0.871756i	$-0.662981\pi$
$929$	−27745.8	−0.979881	−0.489940	+	0.871756i	$0.662981\pi$
$930$	0	0
$931$	−72692.6	−2.55897
$932$	−5906.96	−0.207606
$933$	0	0
$934$	22591.8	0.791464
$935$	−27400.0	−0.958369
$936$	0	0
$937$	−3424.81	−0.119406	−0.0597032	−	0.998216i	$-0.519015\pi$
$937$	−3424.81	−0.119406	−0.0597032	+	0.998216i	$0.519015\pi$
$938$	−52194.6	−1.81686
$939$	0	0
$940$	6222.45	0.215909
$941$	−29835.8	−1.03360	−0.516800	−	0.856106i	$-0.672877\pi$
$941$	−29835.8	−1.03360	−0.516800	+	0.856106i	$0.672877\pi$
$942$	0	0
$943$	−11162.0	−0.385455
$944$	1015.96	0.0350282
$945$	0	0
$946$	−39399.0	−1.35409
$947$	−7223.68	−0.247875	−0.123938	−	0.992290i	$-0.539552\pi$
$947$	−7223.68	−0.247875	−0.123938	+	0.992290i	$0.539552\pi$
$948$	0	0
$949$	11456.3	0.391873
$950$	12153.7	0.415071
$951$	0	0
$952$	−70532.8	−2.40124
$953$	−50384.7	−1.71261	−0.856306	−	0.516469i	$-0.827246\pi$
$953$	−50384.7	−1.71261	−0.856306	+	0.516469i	$0.827246\pi$
$954$	0	0
$955$	−15878.2	−0.538018
$956$	−1263.34	−0.0427397
$957$	0	0
$958$	−26284.8	−0.886455
$959$	−34056.2	−1.14675
$960$	0	0
$961$	−27432.0	−0.920817
$962$	−13739.4	−0.460475
$963$	0	0
$964$	25388.9	0.848257
$965$	−26303.7	−0.877458
$966$	0	0
$967$	17397.6	0.578562	0.289281	−	0.957244i	$-0.406584\pi$
$967$	17397.6	0.578562	0.289281	+	0.957244i	$0.406584\pi$
$968$	−35312.1	−1.17249
$969$	0	0
$970$	−8144.26	−0.269584
$971$	−712.563	−0.0235502	−0.0117751	−	0.999931i	$-0.503748\pi$
$971$	−712.563	−0.0235502	−0.0117751	+	0.999931i	$0.503748\pi$
$972$	0	0
$973$	63380.7	2.08828
$974$	8977.76	0.295345
$975$	0	0
$976$	−2739.14	−0.0898336
$977$	39634.3	1.29786	0.648932	−	0.760847i	$-0.275216\pi$
$977$	39634.3	1.29786	0.648932	+	0.760847i	$0.275216\pi$
$978$	0	0
$979$	25028.1	0.817059
$980$	27279.7	0.889201
$981$	0	0
$982$	−12544.5	−0.407648
$983$	37996.8	1.23287	0.616434	−	0.787406i	$-0.288577\pi$
$983$	37996.8	1.23287	0.616434	+	0.787406i	$0.288577\pi$
$984$	0	0
$985$	−22499.4	−0.727807
$986$	−11915.7	−0.384862
$987$	0	0
$988$	−7916.23	−0.254908
$989$	−75977.4	−2.44281
$990$	0	0
$991$	−26751.3	−0.857502	−0.428751	−	0.903423i	$-0.641046\pi$
$991$	−26751.3	−0.857502	−0.428751	+	0.903423i	$0.641046\pi$
$992$	9002.93	0.288148
$993$	0	0
$994$	323.200	0.0103132
$995$	5208.24	0.165942
$996$	0	0
$997$	−28884.7	−0.917541	−0.458770	−	0.888555i	$-0.651710\pi$
$997$	−28884.7	−0.917541	−0.458770	+	0.888555i	$0.651710\pi$
$998$	5843.91	0.185357
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.c.1.17		28
3.2	odd	2		717.4.a.a.1.12	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.a.1.12	✓	28	3.2	odd	2
2151.4.a.c.1.17		28	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.64745 q^{2} -5.28592 q^{4} -5.57976 q^{5} +35.6078 q^{7} -21.8878 q^{8} +O(q^{10})\)	\(q+1.64745 q^{2} -5.28592 q^{4} -5.57976 q^{5} +35.6078 q^{7} -21.8878 q^{8} -9.19235 q^{10} +54.2616 q^{11} -19.0551 q^{13} +58.6620 q^{14} +6.22837 q^{16} +90.4987 q^{17} -78.5936 q^{19} +29.4942 q^{20} +89.3930 q^{22} +172.386 q^{23} -93.8663 q^{25} -31.3922 q^{26} -188.220 q^{28} -79.9221 q^{29} +48.5690 q^{31} +185.364 q^{32} +149.092 q^{34} -198.683 q^{35} +437.671 q^{37} -129.479 q^{38} +122.129 q^{40} -64.7499 q^{41} -440.739 q^{43} -286.823 q^{44} +283.997 q^{46} +210.972 q^{47} +924.917 q^{49} -154.640 q^{50} +100.724 q^{52} -19.8918 q^{53} -302.766 q^{55} -779.378 q^{56} -131.667 q^{58} +163.118 q^{59} -439.784 q^{61} +80.0148 q^{62} +255.549 q^{64} +106.323 q^{65} -889.752 q^{67} -478.369 q^{68} -327.319 q^{70} +5.50954 q^{71} -601.222 q^{73} +721.039 q^{74} +415.440 q^{76} +1932.14 q^{77} -93.0720 q^{79} -34.7528 q^{80} -106.672 q^{82} -212.225 q^{83} -504.961 q^{85} -726.094 q^{86} -1187.67 q^{88} +461.248 q^{89} -678.509 q^{91} -911.220 q^{92} +347.565 q^{94} +438.533 q^{95} +885.983 q^{97} +1523.75 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8}+O(q^{10}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8	\( 28 q + 13 q^{2} + 99 q^{4} + 74 q^{5} - 82 q^{7} + 135 q^{8} - 68 q^{10} + 258 q^{11} - 134 q^{13} + 292 q^{14} + 327 q^{16} + 364 q^{17} - 278 q^{19} + 986 q^{20} - 179 q^{22} + 668 q^{23} + 490 q^{25} + 760 q^{26} - 802 q^{28} + 714 q^{29} - 608 q^{31} + 918 q^{32} - 228 q^{34} + 934 q^{35} - 1080 q^{37} + 1395 q^{38} - 563 q^{40} + 1796 q^{41} - 1934 q^{43} + 3157 q^{44} - 940 q^{46} + 2032 q^{47} + 762 q^{49} + 1754 q^{50} - 2328 q^{52} + 1790 q^{53} - 478 q^{55} + 3557 q^{56} - 2626 q^{58} + 3622 q^{59} + 324 q^{61} + 796 q^{62} + 2023 q^{64} + 2200 q^{65} - 2444 q^{67} - 357 q^{68} + 4305 q^{70} + 1298 q^{71} - 1368 q^{73} - 813 q^{74} + 1390 q^{76} + 1408 q^{77} - 1378 q^{79} + 7684 q^{80} + 9001 q^{82} + 3524 q^{83} + 60 q^{85} + 2543 q^{86} + 1749 q^{88} + 7854 q^{89} + 850 q^{91} + 496 q^{92} + 6634 q^{94} + 3696 q^{95} - 1746 q^{97} + 4632 q^{98}+O(q^{100}) \) 28 * q + 13 * q^2 + 99 * q^4 + 74 * q^5 - 82 * q^7 + 135 * q^8 - 68 * q^10 + 258 * q^11 - 134 * q^13 + 292 * q^14 + 327 * q^16 + 364 * q^17 - 278 * q^19 + 986 * q^20 - 179 * q^22 + 668 * q^23 + 490 * q^25 + 760 * q^26 - 802 * q^28 + 714 * q^29 - 608 * q^31 + 918 * q^32 - 228 * q^34 + 934 * q^35 - 1080 * q^37 + 1395 * q^38 - 563 * q^40 + 1796 * q^41 - 1934 * q^43 + 3157 * q^44 - 940 * q^46 + 2032 * q^47 + 762 * q^49 + 1754 * q^50 - 2328 * q^52 + 1790 * q^53 - 478 * q^55 + 3557 * q^56 - 2626 * q^58 + 3622 * q^59 + 324 * q^61 + 796 * q^62 + 2023 * q^64 + 2200 * q^65 - 2444 * q^67 - 357 * q^68 + 4305 * q^70 + 1298 * q^71 - 1368 * q^73 - 813 * q^74 + 1390 * q^76 + 1408 * q^77 - 1378 * q^79 + 7684 * q^80 + 9001 * q^82 + 3524 * q^83 + 60 * q^85 + 2543 * q^86 + 1749 * q^88 + 7854 * q^89 + 850 * q^91 + 496 * q^92 + 6634 * q^94 + 3696 * q^95 - 1746 * q^97 + 4632 * q^98

Introduction

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