LMFDB - Embedded newform 2057.4.a.s.1.16

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2057,4,Mod(1,2057)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2057.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2057, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2057 = 11^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2057.a (trivial)

Newform invariants

comment:select newform

sage:traces = [44,-1,-28,139,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$121.366928882$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	no (minimal twist has level 187)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	2057.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.63323 q^{2} -6.45520 q^{3} -5.33256 q^{4} +3.96216 q^{5} +10.5428 q^{6} +20.2168 q^{7} +21.7751 q^{8} +14.6696 q^{9} -6.47113 q^{10} +34.4227 q^{12} +14.7304 q^{13} -33.0188 q^{14} -25.5766 q^{15} +7.09659 q^{16} -17.0000 q^{17} -23.9589 q^{18} +62.5498 q^{19} -21.1285 q^{20} -130.504 q^{21} +140.512 q^{23} -140.563 q^{24} -109.301 q^{25} -24.0582 q^{26} +79.5951 q^{27} -107.807 q^{28} -149.067 q^{29} +41.7725 q^{30} -184.056 q^{31} -185.792 q^{32} +27.7649 q^{34} +80.1025 q^{35} -78.2265 q^{36} +168.441 q^{37} -102.158 q^{38} -95.0877 q^{39} +86.2767 q^{40} -301.281 q^{41} +213.143 q^{42} -200.110 q^{43} +58.1234 q^{45} -229.489 q^{46} +75.4054 q^{47} -45.8099 q^{48} +65.7209 q^{49} +178.514 q^{50} +109.738 q^{51} -78.5507 q^{52} -452.999 q^{53} -129.997 q^{54} +440.225 q^{56} -403.772 q^{57} +243.460 q^{58} -20.0854 q^{59} +136.388 q^{60} -36.5652 q^{61} +300.606 q^{62} +296.573 q^{63} +246.668 q^{64} +58.3643 q^{65} +239.105 q^{67} +90.6534 q^{68} -907.035 q^{69} -130.826 q^{70} +857.657 q^{71} +319.433 q^{72} -53.5367 q^{73} -275.103 q^{74} +705.561 q^{75} -333.550 q^{76} +155.300 q^{78} -259.291 q^{79} +28.1178 q^{80} -909.882 q^{81} +492.061 q^{82} +830.750 q^{83} +695.919 q^{84} -67.3568 q^{85} +326.826 q^{86} +962.255 q^{87} -50.0168 q^{89} -94.9290 q^{90} +297.802 q^{91} -749.289 q^{92} +1188.12 q^{93} -123.155 q^{94} +247.833 q^{95} +1199.32 q^{96} -522.527 q^{97} -107.337 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) $ 44 * q - q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 + 24 * q^6 + 2 * q^7 + 63 * q^8 + 356 * q^9 + 65 * q^10 - 337 * q^12 - 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 - 748 * q^17 + 55 * q^18 + 36 * q^19 - 209 * q^20 + 240 * q^21 - 656 * q^23 + 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 - 63 * q^28 + 698 * q^29 - 304 * q^30 - 1418 * q^31 + 83 * q^32 + 17 * q^34 - 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 - 114 * q^39 + 281 * q^40 - 54 * q^41 - 1943 * q^42 - 382 * q^43 - 1226 * q^45 - 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 - 1307 * q^50 + 476 * q^51 - 3112 * q^52 - 2492 * q^53 - 2400 * q^54 + 27 * q^56 - 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 + 964 * q^61 + 3256 * q^62 - 2322 * q^63 - 3653 * q^64 + 152 * q^65 - 3442 * q^67 - 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 + 1281 * q^72 + 2240 * q^73 - 3494 * q^74 - 1892 * q^75 - 1261 * q^76 + 1120 * q^78 + 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 + 1656 * q^83 + 5401 * q^84 + 408 * q^85 - 1373 * q^86 - 1968 * q^87 - 2992 * q^89 - 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 + 694 * q^94 + 8280 * q^95 - 2203 * q^96 - 4746 * q^97 - 2302 * q^98

Coefficient data

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

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

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

$2$	−1.63323	−0.577434	−0.288717	−	0.957414i	$-0.593229\pi$
$2$	−1.63323	−0.577434	−0.288717	+	0.957414i	$0.593229\pi$
$3$	−6.45520	−1.24230	−0.621152	−	0.783690i	$-0.713335\pi$
$3$	−6.45520	−1.24230	−0.621152	+	0.783690i	$0.713335\pi$
$4$	−5.33256	−0.666569
$5$	3.96216	0.354387	0.177193	−	0.984176i	$-0.443298\pi$
$5$	3.96216	0.354387	0.177193	+	0.984176i	$0.443298\pi$
$6$	10.5428	0.717349
$7$	20.2168	1.09161	0.545804	−	0.837913i	$-0.316224\pi$
$7$	20.2168	1.09161	0.545804	+	0.837913i	$0.316224\pi$
$8$	21.7751	0.962335
$9$	14.6696	0.543319
$10$	−6.47113	−0.204635
$11$	0	0
$11$	0	0
$12$	34.4227	0.828082
$13$	14.7304	0.314268	0.157134	−	0.987577i	$-0.449775\pi$
$13$	14.7304	0.314268	0.157134	+	0.987577i	$0.449775\pi$
$14$	−33.0188	−0.630332
$15$	−25.5766	−0.440256
$16$	7.09659	0.110884
$17$	−17.0000	−0.242536
$17$	−17.0000	−0.242536
$18$	−23.9589	−0.313731
$19$	62.5498	0.755259	0.377629	−	0.925957i	$-0.376739\pi$
$19$	62.5498	0.755259	0.377629	+	0.925957i	$0.376739\pi$
$20$	−21.1285	−0.236223
$21$	−130.504	−1.35611
$22$	0	0
$23$	140.512	1.27386	0.636931	−	0.770921i	$-0.280204\pi$
$23$	140.512	1.27386	0.636931	+	0.770921i	$0.280204\pi$
$24$	−140.563	−1.19551
$25$	−109.301	−0.874410
$26$	−24.0582	−0.181469
$27$	79.5951	0.567337
$28$	−107.807	−0.727632
$29$	−149.067	−0.954516	−0.477258	−	0.878763i	$-0.658369\pi$
$29$	−149.067	−0.954516	−0.477258	+	0.878763i	$0.658369\pi$
$30$	41.7725	0.254219
$31$	−184.056	−1.06637	−0.533184	−	0.845999i	$-0.679005\pi$
$31$	−184.056	−1.06637	−0.533184	+	0.845999i	$0.679005\pi$
$32$	−185.792	−1.02636
$33$	0	0
$34$	27.7649	0.140048
$35$	80.1025	0.386851
$36$	−78.2265	−0.362160
$37$	168.441	0.748420	0.374210	−	0.927344i	$-0.377914\pi$
$37$	168.441	0.748420	0.374210	+	0.927344i	$0.377914\pi$
$38$	−102.158	−0.436112
$39$	−95.0877	−0.390416
$40$	86.2767	0.341039
$41$	−301.281	−1.14761	−0.573807	−	0.818991i	$-0.694534\pi$
$41$	−301.281	−1.14761	−0.573807	+	0.818991i	$0.694534\pi$
$42$	213.143	0.783063
$43$	−200.110	−0.709685	−0.354843	−	0.934926i	$-0.615466\pi$
$43$	−200.110	−0.709685	−0.354843	+	0.934926i	$0.615466\pi$
$44$	0	0
$45$	58.1234	0.192545
$46$	−229.489	−0.735572
$47$	75.4054	0.234022	0.117011	−	0.993131i	$-0.462669\pi$
$47$	75.4054	0.234022	0.117011	+	0.993131i	$0.462669\pi$
$48$	−45.8099	−0.137752
$49$	65.7209	0.191606
$50$	178.514	0.504914
$51$	109.738	0.301303
$52$	−78.5507	−0.209481
$53$	−452.999	−1.17404	−0.587021	−	0.809572i	$-0.699700\pi$
$53$	−452.999	−1.17404	−0.587021	+	0.809572i	$0.699700\pi$
$54$	−129.997	−0.327600
$55$	0	0
$56$	440.225	1.05049
$57$	−403.772	−0.938261
$58$	243.460	0.551171
$59$	−20.0854	−0.0443203	−0.0221601	−	0.999754i	$-0.507054\pi$
$59$	−20.0854	−0.0443203	−0.0221601	+	0.999754i	$0.507054\pi$
$60$	136.388	0.293461
$61$	−36.5652	−0.0767491	−0.0383746	−	0.999263i	$-0.512218\pi$
$61$	−36.5652	−0.0767491	−0.0383746	+	0.999263i	$0.512218\pi$
$62$	300.606	0.615757
$63$	296.573	0.593091
$64$	246.668	0.481773
$65$	58.3643	0.111372
$66$	0	0
$67$	239.105	0.435990	0.217995	−	0.975950i	$-0.430048\pi$
$67$	239.105	0.435990	0.217995	+	0.975950i	$0.430048\pi$
$68$	90.6534	0.161667
$69$	−907.035	−1.58252
$70$	−130.826	−0.223381
$71$	857.657	1.43359	0.716797	−	0.697282i	$-0.245608\pi$
$71$	857.657	1.43359	0.716797	+	0.697282i	$0.245608\pi$
$72$	319.433	0.522855
$73$	−53.5367	−0.0858355	−0.0429178	−	0.999079i	$-0.513665\pi$
$73$	−53.5367	−0.0858355	−0.0429178	+	0.999079i	$0.513665\pi$
$74$	−275.103	−0.432163
$75$	705.561	1.08628
$76$	−333.550	−0.503432
$77$	0	0
$78$	155.300	0.225440
$79$	−259.291	−0.369272	−0.184636	−	0.982807i	$-0.559111\pi$
$79$	−259.291	−0.369272	−0.184636	+	0.982807i	$0.559111\pi$
$80$	28.1178	0.0392959
$81$	−909.882	−1.24812
$82$	492.061	0.662672
$83$	830.750	1.09864	0.549318	−	0.835614i	$-0.314888\pi$
$83$	830.750	1.09864	0.549318	+	0.835614i	$0.314888\pi$
$84$	695.919	0.903940
$85$	−67.3568	−0.0859514
$86$	326.826	0.409797
$87$	962.255	1.18580
$88$	0	0
$89$	−50.0168	−0.0595704	−0.0297852	−	0.999556i	$-0.509482\pi$
$89$	−50.0168	−0.0595704	−0.0297852	+	0.999556i	$0.509482\pi$
$90$	−94.9290	−0.111182
$91$	297.802	0.343057
$92$	−749.289	−0.849117
$93$	1188.12	1.32475
$94$	−123.155	−0.135132
$95$	247.833	0.267654
$96$	1199.32	1.27505
$97$	−522.527	−0.546955	−0.273477	−	0.961878i	$-0.588174\pi$
$97$	−522.527	−0.546955	−0.273477	+	0.961878i	$0.588174\pi$
$98$	−107.337	−0.110640
$99$	0	0
$100$	582.855	0.582855
$101$	−1546.56	−1.52365	−0.761825	−	0.647782i	$-0.775697\pi$
$101$	−1546.56	−1.52365	−0.761825	+	0.647782i	$0.775697\pi$
$102$	−179.228	−0.173983
$103$	1692.32	1.61893	0.809465	−	0.587169i	$-0.199757\pi$
$103$	1692.32	1.61893	0.809465	+	0.587169i	$0.199757\pi$
$104$	320.757	0.302431
$105$	−517.078	−0.480587
$106$	739.853	0.677933
$107$	682.580	0.616706	0.308353	−	0.951272i	$-0.400222\pi$
$107$	682.580	0.616706	0.308353	+	0.951272i	$0.400222\pi$
$108$	−424.445	−0.378169
$109$	−313.666	−0.275631	−0.137815	−	0.990458i	$-0.544008\pi$
$109$	−313.666	−0.275631	−0.137815	+	0.990458i	$0.544008\pi$
$110$	0	0
$111$	−1087.32	−0.929765
$112$	143.471	0.121042
$113$	658.111	0.547875	0.273937	−	0.961748i	$-0.411674\pi$
$113$	658.111	0.547875	0.273937	+	0.961748i	$0.411674\pi$
$114$	659.453	0.541784
$115$	556.733	0.451440
$116$	794.906	0.636251
$117$	216.089	0.170748
$118$	32.8041	0.0255921
$119$	−343.686	−0.264754
$120$	−556.934	−0.423674
$121$	0	0
$122$	59.7195	0.0443176
$123$	1944.83	1.42568
$124$	981.488	0.710808
$125$	−928.340	−0.664266
$126$	−484.373	−0.342471
$127$	1952.31	1.36409	0.682046	−	0.731309i	$-0.261090\pi$
$127$	1952.31	1.36409	0.682046	+	0.731309i	$0.261090\pi$
$128$	1083.47	0.748171
$129$	1291.75	0.881645
$130$	−95.3224	−0.0643102
$131$	−984.972	−0.656927	−0.328463	−	0.944517i	$-0.606531\pi$
$131$	−984.972	−0.656927	−0.328463	+	0.944517i	$0.606531\pi$
$132$	0	0
$133$	1264.56	0.824446
$134$	−390.513	−0.251755
$135$	315.369	0.201057
$136$	−370.178	−0.233400
$137$	1558.12	0.971671	0.485836	−	0.874050i	$-0.338515\pi$
$137$	1558.12	0.971671	0.485836	+	0.874050i	$0.338515\pi$
$138$	1481.40	0.913804
$139$	−2572.86	−1.56998	−0.784990	−	0.619509i	$-0.787332\pi$
$139$	−2572.86	−1.56998	−0.784990	+	0.619509i	$0.787332\pi$
$140$	−427.151	−0.257863
$141$	−486.757	−0.290726
$142$	−1400.75	−0.827806
$143$	0	0
$144$	104.104	0.0602455
$145$	−590.626	−0.338268
$146$	87.4378	0.0495644
$147$	−424.241	−0.238033
$148$	−898.221	−0.498874
$149$	2210.62	1.21544	0.607721	−	0.794150i	$-0.292084\pi$
$149$	2210.62	1.21544	0.607721	+	0.794150i	$0.292084\pi$
$150$	−1152.35	−0.627257
$151$	1091.89	0.588455	0.294227	−	0.955735i	$-0.404938\pi$
$151$	1091.89	0.588455	0.294227	+	0.955735i	$0.404938\pi$
$152$	1362.03	0.726812
$153$	−249.383	−0.131774
$154$	0	0
$155$	−729.259	−0.377907
$156$	507.060	0.260239
$157$	652.687	0.331784	0.165892	−	0.986144i	$-0.446950\pi$
$157$	652.687	0.331784	0.165892	+	0.986144i	$0.446950\pi$
$158$	423.482	0.213230
$159$	2924.20	1.45852
$160$	−736.137	−0.363729
$161$	2840.71	1.39056
$162$	1486.05	0.720710
$163$	−3305.43	−1.58835	−0.794175	−	0.607690i	$-0.792096\pi$
$163$	−3305.43	−1.58835	−0.794175	+	0.607690i	$0.792096\pi$
$164$	1606.60	0.764964
$165$	0	0
$166$	−1356.81	−0.634390
$167$	2908.14	1.34754	0.673768	−	0.738943i	$-0.264675\pi$
$167$	2908.14	1.34754	0.673768	+	0.738943i	$0.264675\pi$
$168$	−2841.74	−1.30503
$169$	−1980.02	−0.901236
$170$	110.009	0.0496313
$171$	917.582	0.410346
$172$	1067.10	0.473054
$173$	2862.82	1.25813	0.629064	−	0.777354i	$-0.283438\pi$
$173$	2862.82	1.25813	0.629064	+	0.777354i	$0.283438\pi$
$174$	−1571.58	−0.684721
$175$	−2209.73	−0.954512
$176$	0	0
$177$	129.655	0.0550593
$178$	81.6890	0.0343980
$179$	−4354.95	−1.81846	−0.909229	−	0.416295i	$-0.863328\pi$
$179$	−4354.95	−1.81846	−0.909229	+	0.416295i	$0.863328\pi$
$180$	−309.946	−0.128345
$181$	3612.74	1.48361	0.741803	−	0.670617i	$-0.233971\pi$
$181$	3612.74	1.48361	0.741803	+	0.670617i	$0.233971\pi$
$182$	−486.380	−0.198093
$183$	236.036	0.0953458
$184$	3059.67	1.22588
$185$	667.391	0.265230
$186$	−1940.47	−0.764958
$187$	0	0
$188$	−402.104	−0.155992
$189$	1609.16	0.619309
$190$	−404.768	−0.154552
$191$	−3637.41	−1.37798	−0.688989	−	0.724772i	$-0.741945\pi$
$191$	−3637.41	−1.37798	−0.688989	+	0.724772i	$0.741945\pi$
$192$	−1592.29	−0.598509
$193$	−2249.32	−0.838911	−0.419455	−	0.907776i	$-0.637779\pi$
$193$	−2249.32	−0.838911	−0.419455	+	0.907776i	$0.637779\pi$
$194$	853.408	0.315831
$195$	−376.753	−0.138358
$196$	−350.460	−0.127719
$197$	−3735.32	−1.35092	−0.675459	−	0.737398i	$-0.736054\pi$
$197$	−3735.32	−1.35092	−0.675459	+	0.737398i	$0.736054\pi$
$198$	0	0
$199$	−4554.27	−1.62233	−0.811166	−	0.584817i	$-0.801166\pi$
$199$	−4554.27	−1.62233	−0.811166	+	0.584817i	$0.801166\pi$
$200$	−2380.05	−0.841475
$201$	−1543.47	−0.541631
$202$	2525.89	0.879809
$203$	−3013.66	−1.04196
$204$	−585.186	−0.200839
$205$	−1193.72	−0.406699
$206$	−2763.96	−0.934826
$207$	2061.26	0.692113
$208$	104.536	0.0348473
$209$	0	0
$210$	844.507	0.277507
$211$	4006.02	1.30704	0.653521	−	0.756909i	$-0.273291\pi$
$211$	4006.02	1.30704	0.653521	+	0.756909i	$0.273291\pi$
$212$	2415.64	0.782581
$213$	−5536.35	−1.78096
$214$	−1114.81	−0.356107
$215$	−792.868	−0.251503
$216$	1733.20	0.545968
$217$	−3721.03	−1.16405
$218$	512.289	0.159159
$219$	345.590	0.106634
$220$	0	0
$221$	−250.417	−0.0762211
$222$	1775.85	0.536878
$223$	−4099.86	−1.23115	−0.615576	−	0.788077i	$-0.711077\pi$
$223$	−4099.86	−1.23115	−0.615576	+	0.788077i	$0.711077\pi$
$224$	−3756.12	−1.12039
$225$	−1603.41	−0.475084
$226$	−1074.85	−0.316362
$227$	1406.95	0.411376	0.205688	−	0.978618i	$-0.434057\pi$
$227$	1406.95	0.411376	0.205688	+	0.978618i	$0.434057\pi$
$228$	2153.13	0.625416
$229$	4448.47	1.28368	0.641842	−	0.766837i	$-0.278171\pi$
$229$	4448.47	1.28368	0.641842	+	0.766837i	$0.278171\pi$
$230$	−909.273	−0.260677
$231$	0	0
$232$	−3245.95	−0.918564
$233$	−2878.32	−0.809293	−0.404646	−	0.914473i	$-0.632605\pi$
$233$	−2878.32	−0.809293	−0.404646	+	0.914473i	$0.632605\pi$
$234$	−352.924	−0.0985955
$235$	298.769	0.0829341
$236$	107.107	0.0295425
$237$	1673.77	0.458748
$238$	561.319	0.152878
$239$	−6694.22	−1.81177	−0.905886	−	0.423522i	$-0.860794\pi$
$239$	−6694.22	−1.81177	−0.905886	+	0.423522i	$0.860794\pi$
$240$	−181.506	−0.0488174
$241$	3014.76	0.805801	0.402900	−	0.915244i	$-0.368002\pi$
$241$	3014.76	0.805801	0.402900	+	0.915244i	$0.368002\pi$
$242$	0	0
$243$	3724.40	0.983212
$244$	194.986	0.0511586
$245$	260.397	0.0679026
$246$	−3176.35	−0.823240
$247$	921.384	0.237353
$248$	−4007.84	−1.02620
$249$	−5362.66	−1.36484
$250$	1516.19	0.383570
$251$	2551.37	0.641597	0.320799	−	0.947147i	$-0.396049\pi$
$251$	2551.37	0.641597	0.320799	+	0.947147i	$0.396049\pi$
$252$	−1581.49	−0.395336
$253$	0	0
$254$	−3188.58	−0.787674
$255$	434.802	0.106778
$256$	−3742.89	−0.913793
$257$	−64.1421	−0.0155684	−0.00778419	−	0.999970i	$-0.502478\pi$
$257$	−64.1421	−0.0155684	−0.00778419	+	0.999970i	$0.502478\pi$
$258$	−2109.72	−0.509092
$259$	3405.35	0.816980
$260$	−311.231	−0.0742373
$261$	−2186.75	−0.518607
$262$	1608.69	0.379332
$263$	1854.53	0.434810	0.217405	−	0.976081i	$-0.430241\pi$
$263$	1854.53	0.434810	0.217405	+	0.976081i	$0.430241\pi$
$264$	0	0
$265$	−1794.86	−0.416065
$266$	−2065.32	−0.476063
$267$	322.868	0.0740046
$268$	−1275.04	−0.290617
$269$	−4874.36	−1.10481	−0.552407	−	0.833575i	$-0.686290\pi$
$269$	−4874.36	−1.10481	−0.552407	+	0.833575i	$0.686290\pi$
$270$	−515.070	−0.116097
$271$	−4465.51	−1.00096	−0.500481	−	0.865748i	$-0.666843\pi$
$271$	−4465.51	−1.00096	−0.500481	+	0.865748i	$0.666843\pi$
$272$	−120.642	−0.0268934
$273$	−1922.37	−0.426181
$274$	−2544.77	−0.561077
$275$	0	0
$276$	4836.81	1.05486
$277$	−2572.24	−0.557945	−0.278973	−	0.960299i	$-0.589994\pi$
$277$	−2572.24	−0.557945	−0.278973	+	0.960299i	$0.589994\pi$
$278$	4202.08	0.906560
$279$	−2700.03	−0.579378
$280$	1744.24	0.372280
$281$	−1628.09	−0.345635	−0.172818	−	0.984954i	$-0.555287\pi$
$281$	−1628.09	−0.345635	−0.172818	+	0.984954i	$0.555287\pi$
$282$	794.987	0.167875
$283$	765.047	0.160697	0.0803487	−	0.996767i	$-0.474397\pi$
$283$	765.047	0.160697	0.0803487	+	0.996767i	$0.474397\pi$
$284$	−4573.50	−0.955589
$285$	−1599.81	−0.332507
$286$	0	0
$287$	−6090.95	−1.25274
$288$	−2725.49	−0.557643
$289$	289.000	0.0588235
$290$	964.630	0.195328
$291$	3373.02	0.679484
$292$	285.487	0.0572153
$293$	−2402.10	−0.478950	−0.239475	−	0.970903i	$-0.576975\pi$
$293$	−2402.10	−0.478950	−0.239475	+	0.970903i	$0.576975\pi$
$294$	692.884	0.137448
$295$	−79.5817	−0.0157065
$296$	3667.83	0.720230
$297$	0	0
$298$	−3610.45	−0.701838
$299$	2069.80	0.400334
$300$	−3762.45	−0.724083
$301$	−4045.59	−0.774697
$302$	−1783.31	−0.339794
$303$	9983.37	1.89284
$304$	443.890	0.0837462
$305$	−144.877	−0.0271989
$306$	407.301	0.0760910
$307$	−3321.67	−0.617516	−0.308758	−	0.951141i	$-0.599913\pi$
$307$	−3321.67	−0.617516	−0.308758	+	0.951141i	$0.599913\pi$
$308$	0	0
$309$	−10924.3	−2.01120
$310$	1191.05	0.218216
$311$	−4927.94	−0.898514	−0.449257	−	0.893403i	$-0.648311\pi$
$311$	−4927.94	−0.898514	−0.449257	+	0.893403i	$0.648311\pi$
$312$	−2070.55	−0.375711
$313$	1299.21	0.234619	0.117309	−	0.993095i	$-0.462573\pi$
$313$	1299.21	0.234619	0.117309	+	0.993095i	$0.462573\pi$
$314$	−1065.99	−0.191584
$315$	1175.07	0.210184
$316$	1382.68	0.246145
$317$	6071.82	1.07580	0.537898	−	0.843010i	$-0.319219\pi$
$317$	6071.82	1.07580	0.537898	+	0.843010i	$0.319219\pi$
$318$	−4775.90	−0.842198
$319$	0	0
$320$	977.339	0.170734
$321$	−4406.19	−0.766136
$322$	−4639.54	−0.802955
$323$	−1063.35	−0.183177
$324$	4852.00	0.831961
$325$	−1610.05	−0.274799
$326$	5398.52	0.917168
$327$	2024.78	0.342417
$328$	−6560.43	−1.10439
$329$	1524.46	0.255460
$330$	0	0
$331$	−4153.62	−0.689739	−0.344870	−	0.938651i	$-0.612077\pi$
$331$	−4153.62	−0.689739	−0.344870	+	0.938651i	$0.612077\pi$
$332$	−4430.02	−0.732316
$333$	2470.96	0.406631
$334$	−4749.66	−0.778114
$335$	947.373	0.154509
$336$	−926.132	−0.150371
$337$	−5608.52	−0.906574	−0.453287	−	0.891365i	$-0.649749\pi$
$337$	−5608.52	−0.906574	−0.453287	+	0.891365i	$0.649749\pi$
$338$	3233.82	0.520405
$339$	−4248.24	−0.680627
$340$	359.184	0.0572926
$341$	0	0
$342$	−1498.62	−0.236948
$343$	−5605.71	−0.882449
$344$	−4357.42	−0.682955
$345$	−3593.82	−0.560826
$346$	−4675.64	−0.726486
$347$	11018.2	1.70457	0.852287	−	0.523075i	$-0.175215\pi$
$347$	11018.2	1.70457	0.852287	+	0.523075i	$0.175215\pi$
$348$	−5131.28	−0.790418
$349$	501.129	0.0768620	0.0384310	−	0.999261i	$-0.487764\pi$
$349$	501.129	0.0768620	0.0384310	+	0.999261i	$0.487764\pi$
$350$	3608.99	0.551168
$351$	1172.47	0.178295
$352$	0	0
$353$	773.750	0.116664	0.0583322	−	0.998297i	$-0.481422\pi$
$353$	773.750	0.116664	0.0583322	+	0.998297i	$0.481422\pi$
$354$	−211.757	−0.0317931
$355$	3398.18	0.508047
$356$	266.717	0.0397078
$357$	2218.56	0.328904
$358$	7112.64	1.05004
$359$	4961.29	0.729379	0.364689	−	0.931129i	$-0.381175\pi$
$359$	4961.29	0.729379	0.364689	+	0.931129i	$0.381175\pi$
$360$	1265.65	0.185293
$361$	−2946.52	−0.429584
$362$	−5900.44	−0.856686
$363$	0	0
$364$	−1588.05	−0.228671
$365$	−212.121	−0.0304190
$366$	−385.501	−0.0550559
$367$	2866.12	0.407657	0.203828	−	0.979007i	$-0.434662\pi$
$367$	2866.12	0.407657	0.203828	+	0.979007i	$0.434662\pi$
$368$	997.157	0.141251
$369$	−4419.67	−0.623520
$370$	−1090.00	−0.153153
$371$	−9158.22	−1.28159
$372$	−6335.70	−0.883040
$373$	7998.80	1.11035	0.555177	−	0.831732i	$-0.312650\pi$
$373$	7998.80	1.11035	0.555177	+	0.831732i	$0.312650\pi$
$374$	0	0
$375$	5992.62	0.825220
$376$	1641.96	0.225207
$377$	−2195.81	−0.299974
$378$	−2628.13	−0.357610
$379$	−10507.9	−1.42415	−0.712077	−	0.702101i	$-0.752245\pi$
$379$	−10507.9	−1.42415	−0.712077	+	0.702101i	$0.752245\pi$
$380$	−1321.58	−0.178410
$381$	−12602.6	−1.69462
$382$	5940.74	0.795692
$383$	4419.84	0.589669	0.294834	−	0.955548i	$-0.404736\pi$
$383$	4419.84	0.589669	0.294834	+	0.955548i	$0.404736\pi$
$384$	−6993.99	−0.929455
$385$	0	0
$386$	3673.66	0.484416
$387$	−2935.53	−0.385585
$388$	2786.41	0.364583
$389$	−3633.81	−0.473628	−0.236814	−	0.971555i	$-0.576103\pi$
$389$	−3633.81	−0.473628	−0.236814	+	0.971555i	$0.576103\pi$
$390$	615.325	0.0798928
$391$	−2388.71	−0.308957
$392$	1431.08	0.184389
$393$	6358.19	0.816102
$394$	6100.65	0.780066
$395$	−1027.35	−0.130865
$396$	0	0
$397$	−8527.26	−1.07801	−0.539006	−	0.842302i	$-0.681200\pi$
$397$	−8527.26	−1.07801	−0.539006	+	0.842302i	$0.681200\pi$
$398$	7438.18	0.936790
$399$	−8162.99	−1.02421
$400$	−775.666	−0.0969582
$401$	−3113.23	−0.387699	−0.193850	−	0.981031i	$-0.562097\pi$
$401$	−3113.23	−0.387699	−0.193850	+	0.981031i	$0.562097\pi$
$402$	2520.84	0.312757
$403$	−2711.22	−0.335125
$404$	8247.13	1.01562
$405$	−3605.10	−0.442318
$406$	4922.00	0.601662
$407$	0	0
$408$	2389.57	0.289954
$409$	−2211.85	−0.267406	−0.133703	−	0.991021i	$-0.542687\pi$
$409$	−2211.85	−0.267406	−0.133703	+	0.991021i	$0.542687\pi$
$410$	1949.63	0.234842
$411$	−10058.0	−1.20711
$412$	−9024.42	−1.07913
$413$	−406.064	−0.0483803
$414$	−3366.51	−0.399650
$415$	3291.57	0.389342
$416$	−2736.78	−0.322553
$417$	16608.3	1.95039
$418$	0	0
$419$	−2522.36	−0.294094	−0.147047	−	0.989129i	$-0.546977\pi$
$419$	−2522.36	−0.294094	−0.147047	+	0.989129i	$0.546977\pi$
$420$	2757.34	0.320344
$421$	2247.17	0.260143	0.130072	−	0.991505i	$-0.458479\pi$
$421$	2247.17	0.260143	0.130072	+	0.991505i	$0.458479\pi$
$422$	−6542.76	−0.754731
$423$	1106.17	0.127148
$424$	−9864.13	−1.12982
$425$	1858.12	0.212076
$426$	9042.13	1.02839
$427$	−739.234	−0.0837799
$428$	−3639.90	−0.411077
$429$	0	0
$430$	1294.94	0.145227
$431$	6710.81	0.749996	0.374998	−	0.927026i	$-0.377643\pi$
$431$	6710.81	0.749996	0.374998	+	0.927026i	$0.377643\pi$
$432$	564.854	0.0629086
$433$	12697.2	1.40921	0.704603	−	0.709601i	$-0.251125\pi$
$433$	12697.2	1.40921	0.704603	+	0.709601i	$0.251125\pi$
$434$	6077.30	0.672165
$435$	3812.61	0.420232
$436$	1672.64	0.183727
$437$	8789.02	0.962095
$438$	−564.428	−0.0615740
$439$	10145.5	1.10300	0.551502	−	0.834174i	$-0.314055\pi$
$439$	10145.5	1.10300	0.551502	+	0.834174i	$0.314055\pi$
$440$	0	0
$441$	964.100	0.104103
$442$	408.989	0.0440127
$443$	11378.8	1.22037	0.610186	−	0.792258i	$-0.291095\pi$
$443$	11378.8	1.22037	0.610186	+	0.792258i	$0.291095\pi$
$444$	5798.20	0.619753
$445$	−198.175	−0.0211110
$446$	6696.02	0.710910
$447$	−14270.0	−1.50995
$448$	4986.85	0.525907
$449$	−3367.48	−0.353945	−0.176972	−	0.984216i	$-0.556630\pi$
$449$	−3367.48	−0.353945	−0.176972	+	0.984216i	$0.556630\pi$
$450$	2618.73	0.274330
$451$	0	0
$452$	−3509.41	−0.365196
$453$	−7048.36	−0.731040
$454$	−2297.87	−0.237543
$455$	1179.94	0.121575
$456$	−8792.19	−0.902921
$457$	11422.8	1.16923	0.584615	−	0.811311i	$-0.301246\pi$
$457$	11422.8	1.16923	0.584615	+	0.811311i	$0.301246\pi$
$458$	−7265.39	−0.741243
$459$	−1353.12	−0.137599
$460$	−2968.81	−0.300916
$461$	−19405.4	−1.96052	−0.980262	−	0.197705i	$-0.936651\pi$
$461$	−19405.4	−1.96052	−0.980262	+	0.197705i	$0.936651\pi$
$462$	0	0
$463$	5450.90	0.547138	0.273569	−	0.961852i	$-0.411796\pi$
$463$	5450.90	0.547138	0.273569	+	0.961852i	$0.411796\pi$
$464$	−1057.86	−0.105841
$465$	4707.52	0.469475
$466$	4700.97	0.467313
$467$	−16269.6	−1.61213	−0.806066	−	0.591826i	$-0.798407\pi$
$467$	−16269.6	−1.61213	−0.806066	+	0.591826i	$0.798407\pi$
$468$	−1152.31	−0.113815
$469$	4833.94	0.475929
$470$	−487.959	−0.0478890
$471$	−4213.23	−0.412177
$472$	−437.363	−0.0426509
$473$	0	0
$474$	−2733.66	−0.264897
$475$	−6836.77	−0.660406
$476$	1832.73	0.176477
$477$	−6645.33	−0.637880
$478$	10933.2	1.04618
$479$	−1924.61	−0.183586	−0.0917932	−	0.995778i	$-0.529260\pi$
$479$	−1924.61	−0.183586	−0.0917932	+	0.995778i	$0.529260\pi$
$480$	4751.91	0.451863
$481$	2481.20	0.235204
$482$	−4923.80	−0.465297
$483$	−18337.4	−1.72749
$484$	0	0
$485$	−2070.34	−0.193834
$486$	−6082.81	−0.567741
$487$	14878.5	1.38442	0.692208	−	0.721698i	$-0.256638\pi$
$487$	14878.5	1.38442	0.692208	+	0.721698i	$0.256638\pi$
$488$	−796.213	−0.0738584
$489$	21337.2	1.97321
$490$	−425.288	−0.0392093
$491$	4998.60	0.459437	0.229718	−	0.973257i	$-0.426219\pi$
$491$	4998.60	0.459437	0.229718	+	0.973257i	$0.426219\pi$
$492$	−10370.9	−0.950318
$493$	2534.13	0.231504
$494$	−1504.83	−0.137056
$495$	0	0
$496$	−1306.17	−0.118243
$497$	17339.1	1.56492
$498$	8758.47	0.788105
$499$	12802.9	1.14857	0.574285	−	0.818655i	$-0.305280\pi$
$499$	12802.9	1.14857	0.574285	+	0.818655i	$0.305280\pi$
$500$	4950.43	0.442779
$501$	−18772.6	−1.67405
$502$	−4166.97	−0.370480
$503$	6743.55	0.597773	0.298887	−	0.954289i	$-0.403385\pi$
$503$	6743.55	0.597773	0.298887	+	0.954289i	$0.403385\pi$
$504$	6457.93	0.570752
$505$	−6127.74	−0.539962
$506$	0	0
$507$	12781.4	1.11961
$508$	−10410.8	−0.909262
$509$	−18949.4	−1.65014	−0.825068	−	0.565033i	$-0.808863\pi$
$509$	−18949.4	−1.65014	−0.825068	+	0.565033i	$0.808863\pi$
$510$	−710.132	−0.0616572
$511$	−1082.34	−0.0936987
$512$	−2554.72	−0.220515
$513$	4978.66	0.428486
$514$	104.759	0.00898972
$515$	6705.27	0.573727
$516$	−6888.32	−0.587677
$517$	0	0
$518$	−5561.72	−0.471752
$519$	−18480.1	−1.56298
$520$	1270.89	0.107177
$521$	−3002.35	−0.252467	−0.126234	−	0.992001i	$-0.540289\pi$
$521$	−3002.35	−0.252467	−0.126234	+	0.992001i	$0.540289\pi$
$522$	3571.47	0.299462
$523$	4701.84	0.393111	0.196556	−	0.980493i	$-0.437024\pi$
$523$	4701.84	0.393111	0.196556	+	0.980493i	$0.437024\pi$
$524$	5252.42	0.437887
$525$	14264.2	1.18579
$526$	−3028.87	−0.251074
$527$	3128.95	0.258632
$528$	0	0
$529$	7576.69	0.622724
$530$	2931.42	0.240250
$531$	−294.645	−0.0240801
$532$	−6743.34	−0.549550
$533$	−4437.99	−0.360658
$534$	−527.319	−0.0427328
$535$	2704.49	0.218552
$536$	5206.54	0.419568
$537$	28112.1	2.25908
$538$	7960.95	0.637957
$539$	0	0
$540$	−1681.72	−0.134018
$541$	−3668.08	−0.291503	−0.145751	−	0.989321i	$-0.546560\pi$
$541$	−3668.08	−0.291503	−0.145751	+	0.989321i	$0.546560\pi$
$542$	7293.22	0.577990
$543$	−23321.0	−1.84309
$544$	3158.46	0.248930
$545$	−1242.80	−0.0976799
$546$	3139.68	0.246091
$547$	−11481.5	−0.897463	−0.448731	−	0.893667i	$-0.648124\pi$
$547$	−11481.5	−0.897463	−0.448731	+	0.893667i	$0.648124\pi$
$548$	−8308.75	−0.647686
$549$	−536.398	−0.0416993
$550$	0	0
$551$	−9324.09	−0.720907
$552$	−19750.8	−1.52292
$553$	−5242.04	−0.403100
$554$	4201.06	0.322177
$555$	−4308.14	−0.329496
$556$	13719.9	1.04650
$557$	13622.3	1.03626	0.518130	−	0.855302i	$-0.326628\pi$
$557$	13622.3	1.03626	0.518130	+	0.855302i	$0.326628\pi$
$558$	4409.77	0.334553
$559$	−2947.70	−0.223031
$560$	568.454	0.0428957
$561$	0	0
$562$	2659.04	0.199582
$563$	−2660.51	−0.199160	−0.0995799	−	0.995030i	$-0.531750\pi$
$563$	−2660.51	−0.199160	−0.0995799	+	0.995030i	$0.531750\pi$
$564$	2595.66	0.193789
$565$	2607.54	0.194160
$566$	−1249.50	−0.0927922
$567$	−18394.9	−1.36246
$568$	18675.6	1.37960
$569$	−23294.5	−1.71627	−0.858134	−	0.513427i	$-0.828376\pi$
$569$	−23294.5	−1.71627	−0.858134	+	0.513427i	$0.828376\pi$
$570$	2612.86	0.192001
$571$	−15661.2	−1.14781	−0.573907	−	0.818920i	$-0.694573\pi$
$571$	−15661.2	−1.14781	−0.573907	+	0.818920i	$0.694573\pi$
$572$	0	0
$573$	23480.2	1.71187
$574$	9947.93	0.723377
$575$	−15358.2	−1.11388
$576$	3618.52	0.261757
$577$	5242.38	0.378238	0.189119	−	0.981954i	$-0.439437\pi$
$577$	5242.38	0.378238	0.189119	+	0.981954i	$0.439437\pi$
$578$	−472.004	−0.0339667
$579$	14519.8	1.04218
$580$	3149.55	0.225479
$581$	16795.2	1.19928
$582$	−5508.92	−0.392358
$583$	0	0
$584$	−1165.77	−0.0826025
$585$	856.181	0.0605107
$586$	3923.19	0.276562
$587$	16225.8	1.14090	0.570451	−	0.821332i	$-0.306768\pi$
$587$	16225.8	1.14090	0.570451	+	0.821332i	$0.306768\pi$
$588$	2262.29	0.158665
$589$	−11512.7	−0.805383
$590$	129.975	0.00906949
$591$	24112.3	1.67825
$592$	1195.36	0.0829879
$593$	3155.64	0.218527	0.109263	−	0.994013i	$-0.465151\pi$
$593$	3155.64	0.218527	0.109263	+	0.994013i	$0.465151\pi$
$594$	0	0
$595$	−1361.74	−0.0938252
$596$	−11788.2	−0.810177
$597$	29398.7	2.01543
$598$	−3380.46	−0.231166
$599$	15082.2	1.02879	0.514393	−	0.857554i	$-0.328017\pi$
$599$	15082.2	1.02879	0.514393	+	0.857554i	$0.328017\pi$
$600$	15363.7	1.04537
$601$	−17491.9	−1.18720	−0.593601	−	0.804760i	$-0.702294\pi$
$601$	−17491.9	−1.18720	−0.593601	+	0.804760i	$0.702294\pi$
$602$	6607.38	0.447337
$603$	3507.57	0.236881
$604$	−5822.56	−0.392246
$605$	0	0
$606$	−16305.2	−1.09299
$607$	−29157.1	−1.94967	−0.974835	−	0.222926i	$-0.928439\pi$
$607$	−29157.1	−1.94967	−0.974835	+	0.222926i	$0.928439\pi$
$608$	−11621.2	−0.775170
$609$	19453.8	1.29443
$610$	236.618	0.0157056
$611$	1110.75	0.0735454
$612$	1329.85	0.0878367
$613$	−29441.8	−1.93987	−0.969937	−	0.243357i	$-0.921751\pi$
$613$	−29441.8	−1.93987	−0.969937	+	0.243357i	$0.921751\pi$
$614$	5425.05	0.356575
$615$	7705.73	0.505244
$616$	0	0
$617$	−2096.27	−0.136779	−0.0683894	−	0.997659i	$-0.521786\pi$
$617$	−2096.27	−0.136779	−0.0683894	+	0.997659i	$0.521786\pi$
$618$	17841.9	1.16134
$619$	−16773.6	−1.08916	−0.544579	−	0.838709i	$-0.683311\pi$
$619$	−16773.6	−1.08916	−0.544579	+	0.838709i	$0.683311\pi$
$620$	3888.82	0.251901
$621$	11184.1	0.722709
$622$	8048.46	0.518833
$623$	−1011.18	−0.0650275
$624$	−674.798	−0.0432909
$625$	9984.42	0.639003
$626$	−2121.91	−0.135477
$627$	0	0
$628$	−3480.49	−0.221157
$629$	−2863.50	−0.181518
$630$	−1919.17	−0.121367
$631$	17285.8	1.09055	0.545275	−	0.838257i	$-0.316425\pi$
$631$	17285.8	1.09055	0.545275	+	0.838257i	$0.316425\pi$
$632$	−5646.10	−0.355363
$633$	−25859.7	−1.62374
$634$	−9916.68	−0.621202
$635$	7735.38	0.483416
$636$	−15593.5	−0.972203
$637$	968.095	0.0602155
$638$	0	0
$639$	12581.5	0.778899
$640$	4292.87	0.265142
$641$	−23271.6	−1.43397	−0.716984	−	0.697089i	$-0.754478\pi$
$641$	−23271.6	−1.43397	−0.716984	+	0.697089i	$0.754478\pi$
$642$	7196.33	0.442393
$643$	−3987.59	−0.244565	−0.122282	−	0.992495i	$-0.539021\pi$
$643$	−3987.59	−0.244565	−0.122282	+	0.992495i	$0.539021\pi$
$644$	−15148.3	−0.926903
$645$	5118.12	0.312443
$646$	1736.69	0.105773
$647$	−26371.7	−1.60244	−0.801219	−	0.598371i	$-0.795815\pi$
$647$	−26371.7	−1.60244	−0.801219	+	0.598371i	$0.795815\pi$
$648$	−19812.8	−1.20111
$649$	0	0
$650$	2629.59	0.158678
$651$	24020.0	1.44611
$652$	17626.4	1.05874
$653$	5739.92	0.343982	0.171991	−	0.985098i	$-0.444980\pi$
$653$	5739.92	0.343982	0.171991	+	0.985098i	$0.444980\pi$
$654$	−3306.93	−0.197724
$655$	−3902.62	−0.232806
$656$	−2138.07	−0.127252
$657$	−785.362	−0.0466361
$658$	−2489.80	−0.147511
$659$	7895.02	0.466687	0.233343	−	0.972394i	$-0.425033\pi$
$659$	7895.02	0.466687	0.233343	+	0.972394i	$0.425033\pi$
$660$	0	0
$661$	8605.00	0.506347	0.253174	−	0.967421i	$-0.418526\pi$
$661$	8605.00	0.506347	0.253174	+	0.967421i	$0.418526\pi$
$662$	6783.83	0.398279
$663$	1616.49	0.0946897
$664$	18089.7	1.05725
$665$	5010.40	0.292173
$666$	−4035.66	−0.234803
$667$	−20945.7	−1.21592
$668$	−15507.8	−0.898226
$669$	26465.4	1.52947
$670$	−1547.28	−0.0892188
$671$	0	0
$672$	24246.5	1.39186
$673$	488.042	0.0279534	0.0139767	−	0.999902i	$-0.495551\pi$
$673$	488.042	0.0279534	0.0139767	+	0.999902i	$0.495551\pi$
$674$	9160.01	0.523487
$675$	−8699.85	−0.496085
$676$	10558.5	0.600736
$677$	−2967.47	−0.168462	−0.0842312	−	0.996446i	$-0.526843\pi$
$677$	−2967.47	−0.168462	−0.0842312	+	0.996446i	$0.526843\pi$
$678$	6938.35	0.393017
$679$	−10563.9	−0.597060
$680$	−1466.70	−0.0827140
$681$	−9082.12	−0.511054
$682$	0	0
$683$	−29110.8	−1.63088	−0.815442	−	0.578838i	$-0.803506\pi$
$683$	−29110.8	−1.63088	−0.815442	+	0.578838i	$0.803506\pi$
$684$	−4893.06	−0.273524
$685$	6173.52	0.344347
$686$	9155.42	0.509556
$687$	−28715.8	−1.59472
$688$	−1420.10	−0.0786929
$689$	−6672.86	−0.368963
$690$	5869.54	0.323840
$691$	−32097.0	−1.76705	−0.883523	−	0.468388i	$-0.844835\pi$
$691$	−32097.0	−1.76705	−0.883523	+	0.468388i	$0.844835\pi$
$692$	−15266.1	−0.838629
$693$	0	0
$694$	−17995.2	−0.984280
$695$	−10194.1	−0.556380
$696$	20953.2	1.14114
$697$	5121.77	0.278337
$698$	−818.460	−0.0443828
$699$	18580.1	1.00539
$700$	11783.5	0.636249
$701$	−23483.6	−1.26528	−0.632642	−	0.774444i	$-0.718030\pi$
$701$	−23483.6	−1.26528	−0.632642	+	0.774444i	$0.718030\pi$
$702$	−1914.91	−0.102954
$703$	10536.0	0.565250
$704$	0	0
$705$	−1928.61	−0.103029
$706$	−1263.71	−0.0673661
$707$	−31266.6	−1.66323
$708$	−691.394	−0.0367008
$709$	32605.5	1.72711	0.863557	−	0.504251i	$-0.168231\pi$
$709$	32605.5	1.72711	0.863557	+	0.504251i	$0.168231\pi$
$710$	−5550.01	−0.293364
$711$	−3803.70	−0.200633
$712$	−1089.12	−0.0573267
$713$	−25862.1	−1.35841
$714$	−3623.43	−0.189921
$715$	0	0
$716$	23223.0	1.21213
$717$	43212.6	2.25077
$718$	−8102.94	−0.421168
$719$	3240.34	0.168073	0.0840363	−	0.996463i	$-0.473219\pi$
$719$	3240.34	0.168073	0.0840363	+	0.996463i	$0.473219\pi$
$720$	412.478	0.0213502
$721$	34213.5	1.76723
$722$	4812.35	0.248057
$723$	−19460.9	−1.00105
$724$	−19265.1	−0.988927
$725$	16293.2	0.834639
$726$	0	0
$727$	−1573.82	−0.0802883	−0.0401441	−	0.999194i	$-0.512782\pi$
$727$	−1573.82	−0.0802883	−0.0401441	+	0.999194i	$0.512782\pi$
$728$	6484.69	0.330135
$729$	525.048	0.0266752
$730$	346.443	0.0175650
$731$	3401.87	0.172124
$732$	−1258.67	−0.0635546
$733$	15795.1	0.795916	0.397958	−	0.917404i	$-0.369719\pi$
$733$	15795.1	0.795916	0.397958	+	0.917404i	$0.369719\pi$
$734$	−4681.03	−0.235395
$735$	−1680.91	−0.0843557
$736$	−26106.0	−1.30744
$737$	0	0
$738$	7218.35	0.360042
$739$	−4117.06	−0.204937	−0.102469	−	0.994736i	$-0.532674\pi$
$739$	−4117.06	−0.204937	−0.102469	+	0.994736i	$0.532674\pi$
$740$	−3558.90	−0.176794
$741$	−5947.72	−0.294865
$742$	14957.5	0.740036
$743$	18382.1	0.907635	0.453818	−	0.891095i	$-0.350062\pi$
$743$	18382.1	0.907635	0.453818	+	0.891095i	$0.350062\pi$
$744$	25871.4	1.27486
$745$	8758.83	0.430737
$746$	−13063.9	−0.641157
$747$	12186.8	0.596909
$748$	0	0
$749$	13799.6	0.673200
$750$	−9787.34	−0.476511
$751$	−17310.0	−0.841078	−0.420539	−	0.907275i	$-0.638159\pi$
$751$	−17310.0	−0.841078	−0.420539	+	0.907275i	$0.638159\pi$
$752$	535.121	0.0259493
$753$	−16469.6	−0.797059
$754$	3586.27	0.173215
$755$	4326.24	0.208541
$756$	−8580.95	−0.412812
$757$	22413.0	1.07611	0.538053	−	0.842911i	$-0.319160\pi$
$757$	22413.0	1.07611	0.538053	+	0.842911i	$0.319160\pi$
$758$	17161.8	0.822356
$759$	0	0
$760$	5396.59	0.257572
$761$	−886.688	−0.0422371	−0.0211185	−	0.999777i	$-0.506723\pi$
$761$	−886.688	−0.0422371	−0.0211185	+	0.999777i	$0.506723\pi$
$762$	20582.9	0.978530
$763$	−6341.34	−0.300881
$764$	19396.7	0.918518
$765$	−988.098	−0.0466990
$766$	−7218.62	−0.340495
$767$	−295.866	−0.0139284
$768$	24161.1	1.13521
$769$	23045.9	1.08070	0.540350	−	0.841440i	$-0.318292\pi$
$769$	23045.9	1.08070	0.540350	+	0.841440i	$0.318292\pi$
$770$	0	0
$771$	414.050	0.0193407
$772$	11994.6	0.559192
$773$	7737.97	0.360046	0.180023	−	0.983662i	$-0.442383\pi$
$773$	7737.97	0.360046	0.180023	+	0.983662i	$0.442383\pi$
$774$	4794.41	0.222650
$775$	20117.5	0.932442
$776$	−11378.1	−0.526354
$777$	−21982.2	−1.01494
$778$	5934.85	0.273489
$779$	−18845.1	−0.866745
$780$	2009.06	0.0922253
$781$	0	0
$782$	3901.31	0.178402
$783$	−11865.0	−0.541532
$784$	466.394	0.0212461
$785$	2586.05	0.117580
$786$	−10384.4	−0.471246
$787$	−13686.0	−0.619890	−0.309945	−	0.950754i	$-0.600311\pi$
$787$	−13686.0	−0.619890	−0.309945	+	0.950754i	$0.600311\pi$
$788$	19918.8	0.900480
$789$	−11971.3	−0.540166
$790$	1677.90	0.0755660
$791$	13304.9	0.598064
$792$	0	0
$793$	−538.620	−0.0241198
$794$	13927.0	0.622482
$795$	11586.2	0.516879
$796$	24285.9	1.08140
$797$	−19058.8	−0.847049	−0.423525	−	0.905885i	$-0.639207\pi$
$797$	−19058.8	−0.847049	−0.423525	+	0.905885i	$0.639207\pi$
$798$	13332.1	0.591415
$799$	−1281.89	−0.0567586
$800$	20307.2	0.897462
$801$	−733.727	−0.0323657
$802$	5084.63	0.223871
$803$	0	0
$804$	8230.64	0.361035
$805$	11255.4	0.492795
$806$	4428.04	0.193513
$807$	31464.9	1.37251
$808$	−33676.6	−1.46626
$809$	−27381.8	−1.18998	−0.594989	−	0.803734i	$-0.702844\pi$
$809$	−27381.8	−1.18998	−0.594989	+	0.803734i	$0.702844\pi$
$810$	5887.97	0.255410
$811$	−37341.9	−1.61683	−0.808416	−	0.588611i	$-0.799675\pi$
$811$	−37341.9	−1.61683	−0.808416	+	0.588611i	$0.799675\pi$
$812$	16070.5	0.694537
$813$	28825.8	1.24350
$814$	0	0
$815$	−13096.6	−0.562890
$816$	778.768	0.0334097
$817$	−12516.8	−0.535996
$818$	3612.46	0.154409
$819$	4368.64	0.186389
$820$	6365.60	0.271093
$821$	2414.41	0.102635	0.0513176	−	0.998682i	$-0.483658\pi$
$821$	2414.41	0.102635	0.0513176	+	0.998682i	$0.483658\pi$
$822$	16427.0	0.697028
$823$	−7042.05	−0.298263	−0.149131	−	0.988817i	$-0.547648\pi$
$823$	−7042.05	−0.298263	−0.149131	+	0.988817i	$0.547648\pi$
$824$	36850.6	1.55795
$825$	0	0
$826$	663.196	0.0279365
$827$	34455.2	1.44876	0.724380	−	0.689401i	$-0.242126\pi$
$827$	34455.2	1.44876	0.724380	+	0.689401i	$0.242126\pi$
$828$	−10991.8	−0.461342
$829$	−36349.3	−1.52288	−0.761438	−	0.648238i	$-0.775506\pi$
$829$	−36349.3	−1.52288	−0.761438	+	0.648238i	$0.775506\pi$
$830$	−5375.90	−0.224819
$831$	16604.3	0.693138
$832$	3633.52	0.151406
$833$	−1117.25	−0.0464713
$834$	−27125.2	−1.12622
$835$	11522.5	0.477549
$836$	0	0
$837$	−14649.9	−0.604989
$838$	4119.60	0.169820
$839$	22196.0	0.913340	0.456670	−	0.889636i	$-0.349042\pi$
$839$	22196.0	0.913340	0.456670	+	0.889636i	$0.349042\pi$
$840$	−11259.4	−0.462485
$841$	−2168.15	−0.0888985
$842$	−3670.15	−0.150216
$843$	10509.6	0.429384
$844$	−21362.3	−0.871234
$845$	−7845.15	−0.319386
$846$	−1806.63	−0.0734199
$847$	0	0
$848$	−3214.75	−0.130183
$849$	−4938.53	−0.199635
$850$	−3034.74	−0.122460
$851$	23668.0	0.953383
$852$	29522.9	1.18713
$853$	1277.35	0.0512727	0.0256363	−	0.999671i	$-0.491839\pi$
$853$	1277.35	0.0512727	0.0256363	+	0.999671i	$0.491839\pi$
$854$	1207.34	0.0483774
$855$	3635.61	0.145421
$856$	14863.3	0.593477
$857$	17995.1	0.717271	0.358635	−	0.933478i	$-0.383242\pi$
$857$	17995.1	0.717271	0.358635	+	0.933478i	$0.383242\pi$
$858$	0	0
$859$	−11261.4	−0.447303	−0.223652	−	0.974669i	$-0.571798\pi$
$859$	−11261.4	−0.447303	−0.223652	+	0.974669i	$0.571798\pi$
$860$	4228.01	0.167644
$861$	39318.3	1.55629
$862$	−10960.3	−0.433074
$863$	−29202.3	−1.15186	−0.575932	−	0.817497i	$-0.695361\pi$
$863$	−29202.3	−1.15186	−0.575932	+	0.817497i	$0.695361\pi$
$864$	−14788.1	−0.582293
$865$	11343.0	0.445864
$866$	−20737.4	−0.813725
$867$	−1865.55	−0.0730767
$868$	19842.6	0.775923
$869$	0	0
$870$	−6226.88	−0.242656
$871$	3522.11	0.137017
$872$	−6830.13	−0.265249
$873$	−7665.28	−0.297171
$874$	−14354.5	−0.555547
$875$	−18768.1	−0.725118
$876$	−1842.88	−0.0710788
$877$	3815.52	0.146911	0.0734554	−	0.997299i	$-0.476597\pi$
$877$	3815.52	0.146911	0.0734554	+	0.997299i	$0.476597\pi$
$878$	−16570.0	−0.636913
$879$	15506.1	0.595001
$880$	0	0
$881$	47895.5	1.83160	0.915800	−	0.401636i	$-0.131558\pi$
$881$	47895.5	1.83160	0.915800	+	0.401636i	$0.131558\pi$
$882$	−1574.60	−0.0601128
$883$	−31981.4	−1.21887	−0.609433	−	0.792838i	$-0.708603\pi$
$883$	−31981.4	−1.21887	−0.609433	+	0.792838i	$0.708603\pi$
$884$	1335.36	0.0508066
$885$	513.716	0.0195123
$886$	−18584.3	−0.704685
$887$	−50162.7	−1.89887	−0.949436	−	0.313961i	$-0.898344\pi$
$887$	−50162.7	−1.89887	−0.949436	+	0.313961i	$0.898344\pi$
$888$	−23676.6	−0.894745
$889$	39469.6	1.48905
$890$	323.665	0.0121902
$891$	0	0
$892$	21862.7	0.820648
$893$	4716.60	0.176747
$894$	23306.2	0.871896
$895$	−17255.0	−0.644438
$896$	21904.3	0.816708
$897$	−13361.0	−0.497336
$898$	5499.87	0.204380
$899$	27436.6	1.01787
$900$	8550.26	0.316676
$901$	7700.99	0.284747
$902$	0	0
$903$	26115.1	0.962410
$904$	14330.5	0.527239
$905$	14314.3	0.525771
$906$	11511.6	0.422128
$907$	−22728.3	−0.832063	−0.416032	−	0.909350i	$-0.636579\pi$
$907$	−22728.3	−0.832063	−0.416032	+	0.909350i	$0.636579\pi$
$908$	−7502.62	−0.274210
$909$	−22687.5	−0.827829
$910$	−1927.12	−0.0702015
$911$	−27347.1	−0.994566	−0.497283	−	0.867588i	$-0.665669\pi$
$911$	−27347.1	−0.994566	−0.497283	+	0.867588i	$0.665669\pi$
$912$	−2865.40	−0.104038
$913$	0	0
$914$	−18656.1	−0.675153
$915$	935.213	0.0337893
$916$	−23721.7	−0.855664
$917$	−19913.0	−0.717106
$918$	2209.95	0.0794546
$919$	4106.32	0.147394	0.0736969	−	0.997281i	$-0.476520\pi$
$919$	4106.32	0.147394	0.0736969	+	0.997281i	$0.476520\pi$
$920$	12122.9	0.434436
$921$	21442.0	0.767143
$922$	31693.6	1.13207
$923$	12633.6	0.450532
$924$	0	0
$925$	−18410.8	−0.654426
$926$	−8902.58	−0.315936
$927$	24825.8	0.879595
$928$	27695.3	0.979680
$929$	−13485.7	−0.476265	−0.238132	−	0.971233i	$-0.576535\pi$
$929$	−13485.7	−0.476265	−0.238132	+	0.971233i	$0.576535\pi$
$930$	−7688.46	−0.271091
$931$	4110.83	0.144712
$932$	15348.8	0.539450
$933$	31810.8	1.11623
$934$	26572.0	0.930900
$935$	0	0
$936$	4705.38	0.164316
$937$	3169.10	0.110491	0.0552455	−	0.998473i	$-0.482406\pi$
$937$	3169.10	0.110491	0.0552455	+	0.998473i	$0.482406\pi$
$938$	−7894.95	−0.274818
$939$	−8386.66	−0.291468
$940$	−1593.20	−0.0552814
$941$	7635.79	0.264527	0.132263	−	0.991215i	$-0.457776\pi$
$941$	7635.79	0.264527	0.132263	+	0.991215i	$0.457776\pi$
$942$	6881.18	0.238005
$943$	−42333.6	−1.46190
$944$	−142.538	−0.00491442
$945$	6375.77	0.219475
$946$	0	0
$947$	15567.9	0.534201	0.267101	−	0.963669i	$-0.413934\pi$
$947$	15567.9	0.534201	0.267101	+	0.963669i	$0.413934\pi$
$948$	−8925.49	−0.305787
$949$	−788.617	−0.0269753
$950$	11166.0	0.381341
$951$	−39194.8	−1.33647
$952$	−7483.82	−0.254782
$953$	4639.24	0.157691	0.0788456	−	0.996887i	$-0.474877\pi$
$953$	4639.24	0.157691	0.0788456	+	0.996887i	$0.474877\pi$
$954$	10853.4	0.368334
$955$	−14412.0	−0.488337
$956$	35697.3	1.20767
$957$	0	0
$958$	3143.34	0.106009
$959$	31500.2	1.06068
$960$	−6308.92	−0.212104
$961$	4085.53	0.137140
$962$	−4052.38	−0.135815
$963$	10013.2	0.335068
$964$	−16076.4	−0.537122
$965$	−8912.19	−0.297299
$966$	29949.2	0.997515
$967$	−19550.7	−0.650165	−0.325082	−	0.945686i	$-0.605392\pi$
$967$	−19550.7	−0.650165	−0.325082	+	0.945686i	$0.605392\pi$
$968$	0	0
$969$	6864.12	0.227562
$970$	3381.34	0.111926
$971$	52636.2	1.73963	0.869813	−	0.493381i	$-0.164239\pi$
$971$	52636.2	1.73963	0.869813	+	0.493381i	$0.164239\pi$
$972$	−19860.6	−0.655379
$973$	−52015.1	−1.71380
$974$	−24300.1	−0.799410
$975$	10393.2	0.341383
$976$	−259.488	−0.00851027
$977$	−31333.4	−1.02604	−0.513022	−	0.858376i	$-0.671474\pi$
$977$	−31333.4	−1.02604	−0.513022	+	0.858376i	$0.671474\pi$
$978$	−34848.6	−1.13940
$979$	0	0
$980$	−1388.58	−0.0452618
$981$	−4601.36	−0.149755
$982$	−8163.87	−0.265295
$983$	−18830.2	−0.610977	−0.305488	−	0.952196i	$-0.598820\pi$
$983$	−18830.2	−0.610977	−0.305488	+	0.952196i	$0.598820\pi$
$984$	42348.9	1.37199
$985$	−14800.0	−0.478747
$986$	−4138.82	−0.133679
$987$	−9840.70	−0.317358
$988$	−4913.33	−0.158212
$989$	−28117.9	−0.904041
$990$	0	0
$991$	−28111.1	−0.901089	−0.450544	−	0.892754i	$-0.648770\pi$
$991$	−28111.1	−0.901089	−0.450544	+	0.892754i	$0.648770\pi$
$992$	34196.0	1.09448
$993$	26812.5	0.856866
$994$	−28318.8	−0.903639
$995$	−18044.8	−0.574933
$996$	28596.7	0.909760
$997$	−5443.48	−0.172915	−0.0864577	−	0.996256i	$-0.527555\pi$
$997$	−5443.48	−0.172915	−0.0864577	+	0.996256i	$0.527555\pi$
$998$	−20910.1	−0.663224
$999$	13407.1	0.424606

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2057.4.a.s.1.16		44
11.2	odd	10		187.4.g.a.103.7	yes	88
11.6	odd	10		187.4.g.a.69.7	✓	88
11.10	odd	2		2057.4.a.t.1.29		44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.4.g.a.69.7	✓	88	11.6	odd	10
187.4.g.a.103.7	yes	88	11.2	odd	10
2057.4.a.s.1.16		44	1.1	even	1	trivial
2057.4.a.t.1.29		44	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2057 = 11^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2057.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63323 q^{2} -6.45520 q^{3} -5.33256 q^{4} +3.96216 q^{5} +10.5428 q^{6} +20.2168 q^{7} +21.7751 q^{8} +14.6696 q^{9} -6.47113 q^{10} +34.4227 q^{12} +14.7304 q^{13} -33.0188 q^{14} -25.5766 q^{15} +7.09659 q^{16} -17.0000 q^{17} -23.9589 q^{18} +62.5498 q^{19} -21.1285 q^{20} -130.504 q^{21} +140.512 q^{23} -140.563 q^{24} -109.301 q^{25} -24.0582 q^{26} +79.5951 q^{27} -107.807 q^{28} -149.067 q^{29} +41.7725 q^{30} -184.056 q^{31} -185.792 q^{32} +27.7649 q^{34} +80.1025 q^{35} -78.2265 q^{36} +168.441 q^{37} -102.158 q^{38} -95.0877 q^{39} +86.2767 q^{40} -301.281 q^{41} +213.143 q^{42} -200.110 q^{43} +58.1234 q^{45} -229.489 q^{46} +75.4054 q^{47} -45.8099 q^{48} +65.7209 q^{49} +178.514 q^{50} +109.738 q^{51} -78.5507 q^{52} -452.999 q^{53} -129.997 q^{54} +440.225 q^{56} -403.772 q^{57} +243.460 q^{58} -20.0854 q^{59} +136.388 q^{60} -36.5652 q^{61} +300.606 q^{62} +296.573 q^{63} +246.668 q^{64} +58.3643 q^{65} +239.105 q^{67} +90.6534 q^{68} -907.035 q^{69} -130.826 q^{70} +857.657 q^{71} +319.433 q^{72} -53.5367 q^{73} -275.103 q^{74} +705.561 q^{75} -333.550 q^{76} +155.300 q^{78} -259.291 q^{79} +28.1178 q^{80} -909.882 q^{81} +492.061 q^{82} +830.750 q^{83} +695.919 q^{84} -67.3568 q^{85} +326.826 q^{86} +962.255 q^{87} -50.0168 q^{89} -94.9290 q^{90} +297.802 q^{91} -749.289 q^{92} +1188.12 q^{93} -123.155 q^{94} +247.833 q^{95} +1199.32 q^{96} -522.527 q^{97} -107.337 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - q^{2} - 28 q^{3} + 139 q^{4} - 24 q^{5} + 24 q^{6} + 2 q^{7} + 63 q^{8} + 356 q^{9} + 65 q^{10} - 337 q^{12} - 12 q^{13} - 151 q^{14} - 320 q^{15} + 311 q^{16} - 748 q^{17} + 55 q^{18} + 36 q^{19}+ \cdots - 2302 q^{98}+O(q^{100}) \) 44 * q - q^2 - 28 * q^3 + 139 * q^4 - 24 * q^5 + 24 * q^6 + 2 * q^7 + 63 * q^8 + 356 * q^9 + 65 * q^10 - 337 * q^12 - 12 * q^13 - 151 * q^14 - 320 * q^15 + 311 * q^16 - 748 * q^17 + 55 * q^18 + 36 * q^19 - 209 * q^20 + 240 * q^21 - 656 * q^23 + 104 * q^24 + 458 * q^25 - 455 * q^26 - 538 * q^27 - 63 * q^28 + 698 * q^29 - 304 * q^30 - 1418 * q^31 + 83 * q^32 + 17 * q^34 - 102 * q^35 - 85 * q^36 - 1358 * q^37 - 568 * q^38 - 114 * q^39 + 281 * q^40 - 54 * q^41 - 1943 * q^42 - 382 * q^43 - 1226 * q^45 - 232 * q^46 - 544 * q^47 - 2743 * q^48 + 864 * q^49 - 1307 * q^50 + 476 * q^51 - 3112 * q^52 - 2492 * q^53 - 2400 * q^54 + 27 * q^56 - 56 * q^57 - 3746 * q^58 - 662 * q^59 - 1392 * q^60 + 964 * q^61 + 3256 * q^62 - 2322 * q^63 - 3653 * q^64 + 152 * q^65 - 3442 * q^67 - 2363 * q^68 - 1664 * q^69 - 3130 * q^70 - 2936 * q^71 + 1281 * q^72 + 2240 * q^73 - 3494 * q^74 - 1892 * q^75 - 1261 * q^76 + 1120 * q^78 + 1016 * q^79 - 3280 * q^80 + 2140 * q^81 - 3985 * q^82 + 1656 * q^83 + 5401 * q^84 + 408 * q^85 - 1373 * q^86 - 1968 * q^87 - 2992 * q^89 - 1287 * q^90 - 8310 * q^91 - 2442 * q^92 - 4058 * q^93 + 694 * q^94 + 8280 * q^95 - 2203 * q^96 - 4746 * q^97 - 2302 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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