LMFDB - Embedded newform 1875.4.a.k.1.21

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,4,Mod(1,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, 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("1875.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1875.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:	$110.628581261$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.21
Character	$\chi$	$=$	1875.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.60577 q^{2} -3.00000 q^{3} +13.2131 q^{4} -13.8173 q^{6} -33.5484 q^{7} +24.0102 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.60577 q^{2} -3.00000 q^{3} +13.2131 q^{4} -13.8173 q^{6} -33.5484 q^{7} +24.0102 q^{8} +9.00000 q^{9} -43.8194 q^{11} -39.6392 q^{12} -7.37003 q^{13} -154.516 q^{14} +4.88082 q^{16} -22.6283 q^{17} +41.4519 q^{18} +128.610 q^{19} +100.645 q^{21} -201.822 q^{22} +165.615 q^{23} -72.0307 q^{24} -33.9446 q^{26} -27.0000 q^{27} -443.278 q^{28} -172.004 q^{29} -91.3035 q^{31} -169.602 q^{32} +131.458 q^{33} -104.221 q^{34} +118.918 q^{36} +235.949 q^{37} +592.349 q^{38} +22.1101 q^{39} +238.726 q^{41} +463.548 q^{42} -48.3382 q^{43} -578.989 q^{44} +762.783 q^{46} +289.512 q^{47} -14.6425 q^{48} +782.496 q^{49} +67.8850 q^{51} -97.3808 q^{52} -154.553 q^{53} -124.356 q^{54} -805.505 q^{56} -385.831 q^{57} -792.212 q^{58} +218.931 q^{59} +546.945 q^{61} -420.522 q^{62} -301.936 q^{63} -820.193 q^{64} +605.465 q^{66} -791.301 q^{67} -298.990 q^{68} -496.845 q^{69} +1142.61 q^{71} +216.092 q^{72} +251.151 q^{73} +1086.72 q^{74} +1699.34 q^{76} +1470.07 q^{77} +101.834 q^{78} -402.769 q^{79} +81.0000 q^{81} +1099.52 q^{82} -106.577 q^{83} +1329.83 q^{84} -222.634 q^{86} +516.013 q^{87} -1052.11 q^{88} +1608.81 q^{89} +247.253 q^{91} +2188.28 q^{92} +273.910 q^{93} +1333.42 q^{94} +508.806 q^{96} +984.943 q^{97} +3603.99 q^{98} -394.374 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9}+O(q^{10}) $ 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9	\( 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9} + 96 q^{11} - 399 q^{12} - 156 q^{13} + 92 q^{14} + 845 q^{16} + 46 q^{17} - 9 q^{18} + 182 q^{19} + 186 q^{21} - 158 q^{22} + 286 q^{23} + 81 q^{24} + 478 q^{26} - 648 q^{27} - 701 q^{28} + 1144 q^{29} + 64 q^{31} - 1212 q^{32} - 288 q^{33} + 961 q^{34} + 1197 q^{36} - 762 q^{37} - 474 q^{38} + 468 q^{39} + 1074 q^{41} - 276 q^{42} - 460 q^{43} + 319 q^{44} + 459 q^{46} + 960 q^{47} - 2535 q^{48} + 2680 q^{49} - 138 q^{51} - 2969 q^{52} - 914 q^{53} + 27 q^{54} + 1680 q^{56} - 546 q^{57} - 208 q^{58} + 208 q^{59} + 3520 q^{61} - 334 q^{62} - 558 q^{63} + 5747 q^{64} + 474 q^{66} - 154 q^{67} + 5727 q^{68} - 858 q^{69} - 252 q^{71} - 243 q^{72} - 4414 q^{73} + 5637 q^{74} + 627 q^{76} - 2344 q^{77} - 1434 q^{78} + 1110 q^{79} + 1944 q^{81} - 3714 q^{82} + 1488 q^{83} + 2103 q^{84} + 3036 q^{86} - 3432 q^{87} - 3947 q^{88} + 3402 q^{89} + 3504 q^{91} + 11163 q^{92} - 192 q^{93} + 3408 q^{94} + 3636 q^{96} - 534 q^{97} - 2244 q^{98} + 864 q^{99}+O(q^{100}) \) 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9 + 96 * q^11 - 399 * q^12 - 156 * q^13 + 92 * q^14 + 845 * q^16 + 46 * q^17 - 9 * q^18 + 182 * q^19 + 186 * q^21 - 158 * q^22 + 286 * q^23 + 81 * q^24 + 478 * q^26 - 648 * q^27 - 701 * q^28 + 1144 * q^29 + 64 * q^31 - 1212 * q^32 - 288 * q^33 + 961 * q^34 + 1197 * q^36 - 762 * q^37 - 474 * q^38 + 468 * q^39 + 1074 * q^41 - 276 * q^42 - 460 * q^43 + 319 * q^44 + 459 * q^46 + 960 * q^47 - 2535 * q^48 + 2680 * q^49 - 138 * q^51 - 2969 * q^52 - 914 * q^53 + 27 * q^54 + 1680 * q^56 - 546 * q^57 - 208 * q^58 + 208 * q^59 + 3520 * q^61 - 334 * q^62 - 558 * q^63 + 5747 * q^64 + 474 * q^66 - 154 * q^67 + 5727 * q^68 - 858 * q^69 - 252 * q^71 - 243 * q^72 - 4414 * q^73 + 5637 * q^74 + 627 * q^76 - 2344 * q^77 - 1434 * q^78 + 1110 * q^79 + 1944 * q^81 - 3714 * q^82 + 1488 * q^83 + 2103 * q^84 + 3036 * q^86 - 3432 * q^87 - 3947 * q^88 + 3402 * q^89 + 3504 * q^91 + 11163 * q^92 - 192 * q^93 + 3408 * q^94 + 3636 * q^96 - 534 * q^97 - 2244 * q^98 + 864 * q^99

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 $

$2$	4.60577	1.62838	0.814192	−	0.580596i	$-0.197180\pi$
$2$	4.60577	1.62838	0.814192	+	0.580596i	$0.197180\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	13.2131	1.65163
$5$	0	0
$5$	0	0
$6$	−13.8173	−0.940148
$7$	−33.5484	−1.81144	−0.905722	−	0.423872i	$-0.860671\pi$
$7$	−33.5484	−1.81144	−0.905722	+	0.423872i	$0.860671\pi$
$8$	24.0102	1.06111
$9$	9.00000	0.333333
$10$	0	0
$11$	−43.8194	−1.20109	−0.600547	−	0.799589i	$-0.705050\pi$
$11$	−43.8194	−1.20109	−0.600547	+	0.799589i	$0.705050\pi$
$12$	−39.6392	−0.953572
$13$	−7.37003	−0.157237	−0.0786184	−	0.996905i	$-0.525051\pi$
$13$	−7.37003	−0.157237	−0.0786184	+	0.996905i	$0.525051\pi$
$14$	−154.516	−2.94973
$15$	0	0
$16$	4.88082	0.0762628
$17$	−22.6283	−0.322834	−0.161417	−	0.986886i	$-0.551606\pi$
$17$	−22.6283	−0.322834	−0.161417	+	0.986886i	$0.551606\pi$
$18$	41.4519	0.542795
$19$	128.610	1.55291	0.776454	−	0.630174i	$-0.217017\pi$
$19$	128.610	1.55291	0.776454	+	0.630174i	$0.217017\pi$
$20$	0	0
$21$	100.645	1.04584
$22$	−201.822	−1.95584
$23$	165.615	1.50144	0.750719	−	0.660622i	$-0.229707\pi$
$23$	165.615	1.50144	0.750719	+	0.660622i	$0.229707\pi$
$24$	−72.0307	−0.612633
$25$	0	0
$26$	−33.9446	−0.256042
$27$	−27.0000	−0.192450
$28$	−443.278	−2.99184
$29$	−172.004	−1.10139	−0.550697	−	0.834705i	$-0.685638\pi$
$29$	−172.004	−1.10139	−0.550697	+	0.834705i	$0.685638\pi$
$30$	0	0
$31$	−91.3035	−0.528987	−0.264493	−	0.964388i	$-0.585205\pi$
$31$	−91.3035	−0.528987	−0.264493	+	0.964388i	$0.585205\pi$
$32$	−169.602	−0.936927
$33$	131.458	0.693452
$34$	−104.221	−0.525698
$35$	0	0
$36$	118.918	0.550545
$37$	235.949	1.04837	0.524186	−	0.851604i	$-0.324370\pi$
$37$	235.949	1.04837	0.524186	+	0.851604i	$0.324370\pi$
$38$	592.349	2.52873
$39$	22.1101	0.0907808
$40$	0	0
$41$	238.726	0.909335	0.454667	−	0.890661i	$-0.349758\pi$
$41$	238.726	0.909335	0.454667	+	0.890661i	$0.349758\pi$
$42$	463.548	1.70303
$43$	−48.3382	−0.171430	−0.0857152	−	0.996320i	$-0.527318\pi$
$43$	−48.3382	−0.171430	−0.0857152	+	0.996320i	$0.527318\pi$
$44$	−578.989	−1.98377
$45$	0	0
$46$	762.783	2.44492
$47$	289.512	0.898503	0.449252	−	0.893405i	$-0.351691\pi$
$47$	289.512	0.898503	0.449252	+	0.893405i	$0.351691\pi$
$48$	−14.6425	−0.0440304
$49$	782.496	2.28133
$50$	0	0
$51$	67.8850	0.186388
$52$	−97.3808	−0.259698
$53$	−154.553	−0.400556	−0.200278	−	0.979739i	$-0.564185\pi$
$53$	−154.553	−0.400556	−0.200278	+	0.979739i	$0.564185\pi$
$54$	−124.356	−0.313383
$55$	0	0
$56$	−805.505	−1.92214
$57$	−385.831	−0.896571
$58$	−792.212	−1.79349
$59$	218.931	0.483091	0.241545	−	0.970390i	$-0.422346\pi$
$59$	218.931	0.483091	0.241545	+	0.970390i	$0.422346\pi$
$60$	0	0
$61$	546.945	1.14802	0.574010	−	0.818849i	$-0.305387\pi$
$61$	546.945	1.14802	0.574010	+	0.818849i	$0.305387\pi$
$62$	−420.522	−0.861394
$63$	−301.936	−0.603815
$64$	−820.193	−1.60194
$65$	0	0
$66$	605.465	1.12921
$67$	−791.301	−1.44288	−0.721439	−	0.692478i	$-0.756519\pi$
$67$	−791.301	−1.44288	−0.721439	+	0.692478i	$0.756519\pi$
$68$	−298.990	−0.533204
$69$	−496.845	−0.866856
$70$	0	0
$71$	1142.61	1.90990	0.954951	−	0.296763i	$-0.0959070\pi$
$71$	1142.61	1.90990	0.954951	+	0.296763i	$0.0959070\pi$
$72$	216.092	0.353704
$73$	251.151	0.402672	0.201336	−	0.979522i	$-0.435472\pi$
$73$	251.151	0.402672	0.201336	+	0.979522i	$0.435472\pi$
$74$	1086.72	1.70715
$75$	0	0
$76$	1699.34	2.56484
$77$	1470.07	2.17571
$78$	101.834	0.147826
$79$	−402.769	−0.573609	−0.286804	−	0.957989i	$-0.592593\pi$
$79$	−402.769	−0.573609	−0.286804	+	0.957989i	$0.592593\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	1099.52	1.48075
$83$	−106.577	−0.140944	−0.0704720	−	0.997514i	$-0.522451\pi$
$83$	−106.577	−0.140944	−0.0704720	+	0.997514i	$0.522451\pi$
$84$	1329.83	1.72734
$85$	0	0
$86$	−222.634	−0.279154
$87$	516.013	0.635890
$88$	−1052.11	−1.27449
$89$	1608.81	1.91611	0.958053	−	0.286592i	$-0.0925223\pi$
$89$	1608.81	1.91611	0.958053	+	0.286592i	$0.0925223\pi$
$90$	0	0
$91$	247.253	0.284826
$92$	2188.28	2.47983
$93$	273.910	0.305411
$94$	1333.42	1.46311
$95$	0	0
$96$	508.806	0.540935
$97$	984.943	1.03099	0.515494	−	0.856893i	$-0.327608\pi$
$97$	984.943	1.03099	0.515494	+	0.856893i	$0.327608\pi$
$98$	3603.99	3.71488
$99$	−394.374	−0.400365
$100$	0	0
$101$	−805.442	−0.793509	−0.396755	−	0.917925i	$-0.629864\pi$
$101$	−805.442	−0.793509	−0.396755	+	0.917925i	$0.629864\pi$
$102$	312.662	0.303512
$103$	−1782.70	−1.70538	−0.852691	−	0.522416i	$-0.825031\pi$
$103$	−1782.70	−1.70538	−0.852691	+	0.522416i	$0.825031\pi$
$104$	−176.956	−0.166846
$105$	0	0
$106$	−711.834	−0.652259
$107$	953.536	0.861512	0.430756	−	0.902468i	$-0.358247\pi$
$107$	953.536	0.861512	0.430756	+	0.902468i	$0.358247\pi$
$108$	−356.753	−0.317857
$109$	2214.01	1.94554	0.972769	−	0.231777i	$-0.0744540\pi$
$109$	2214.01	1.94554	0.972769	+	0.231777i	$0.0744540\pi$
$110$	0	0
$111$	−707.846	−0.605277
$112$	−163.744	−0.138146
$113$	−1437.28	−1.19653	−0.598265	−	0.801298i	$-0.704143\pi$
$113$	−1437.28	−1.19653	−0.598265	+	0.801298i	$0.704143\pi$
$114$	−1777.05	−1.45996
$115$	0	0
$116$	−2272.71	−1.81910
$117$	−66.3303	−0.0524123
$118$	1008.34	0.786657
$119$	759.145	0.584796
$120$	0	0
$121$	589.136	0.442627
$122$	2519.10	1.86942
$123$	−716.178	−0.525005
$124$	−1206.40	−0.873693
$125$	0	0
$126$	−1390.65	−0.983242
$127$	713.871	0.498786	0.249393	−	0.968402i	$-0.419769\pi$
$127$	713.871	0.498786	0.249393	+	0.968402i	$0.419769\pi$
$128$	−2420.80	−1.67165
$129$	145.015	0.0989753
$130$	0	0
$131$	42.7900	0.0285388	0.0142694	−	0.999898i	$-0.495458\pi$
$131$	42.7900	0.0285388	0.0142694	+	0.999898i	$0.495458\pi$
$132$	1736.97	1.14533
$133$	−4314.67	−2.81300
$134$	−3644.55	−2.34956
$135$	0	0
$136$	−543.311	−0.342563
$137$	−2155.68	−1.34432	−0.672162	−	0.740404i	$-0.734634\pi$
$137$	−2155.68	−1.34432	−0.672162	+	0.740404i	$0.734634\pi$
$138$	−2288.35	−1.41157
$139$	−88.9660	−0.0542878	−0.0271439	−	0.999632i	$-0.508641\pi$
$139$	−88.9660	−0.0542878	−0.0271439	+	0.999632i	$0.508641\pi$
$140$	0	0
$141$	−868.536	−0.518751
$142$	5262.60	3.11006
$143$	322.950	0.188856
$144$	43.9274	0.0254209
$145$	0	0
$146$	1156.74	0.655705
$147$	−2347.49	−1.31713
$148$	3117.61	1.73153
$149$	2041.54	1.12248	0.561240	−	0.827653i	$-0.310324\pi$
$149$	2041.54	1.12248	0.561240	+	0.827653i	$0.310324\pi$
$150$	0	0
$151$	892.328	0.480905	0.240452	−	0.970661i	$-0.422704\pi$
$151$	892.328	0.480905	0.240452	+	0.970661i	$0.422704\pi$
$152$	3087.96	1.64781
$153$	−203.655	−0.107611
$154$	6770.80	3.54290
$155$	0	0
$156$	292.142	0.149937
$157$	1567.05	0.796585	0.398292	−	0.917258i	$-0.369603\pi$
$157$	1567.05	0.796585	0.398292	+	0.917258i	$0.369603\pi$
$158$	−1855.06	−0.934055
$159$	463.659	0.231261
$160$	0	0
$161$	−5556.11	−2.71977
$162$	373.067	0.180932
$163$	1325.13	0.636762	0.318381	−	0.947963i	$-0.396861\pi$
$163$	1325.13	0.636762	0.318381	+	0.947963i	$0.396861\pi$
$164$	3154.30	1.50189
$165$	0	0
$166$	−490.869	−0.229511
$167$	−156.572	−0.0725501	−0.0362751	−	0.999342i	$-0.511549\pi$
$167$	−156.572	−0.0725501	−0.0362751	+	0.999342i	$0.511549\pi$
$168$	2416.51	1.10975
$169$	−2142.68	−0.975277
$170$	0	0
$171$	1157.49	0.517636
$172$	−638.696	−0.283140
$173$	−1673.87	−0.735618	−0.367809	−	0.929901i	$-0.619892\pi$
$173$	−1673.87	−0.735618	−0.367809	+	0.929901i	$0.619892\pi$
$174$	2376.64	1.03547
$175$	0	0
$176$	−213.874	−0.0915988
$177$	−656.792	−0.278912
$178$	7409.80	3.12016
$179$	−2925.86	−1.22173	−0.610863	−	0.791736i	$-0.709177\pi$
$179$	−2925.86	−1.22173	−0.610863	+	0.791736i	$0.709177\pi$
$180$	0	0
$181$	−1613.12	−0.662443	−0.331222	−	0.943553i	$-0.607461\pi$
$181$	−1613.12	−0.662443	−0.331222	+	0.943553i	$0.607461\pi$
$182$	1138.79	0.463806
$183$	−1640.84	−0.662809
$184$	3976.45	1.59319
$185$	0	0
$186$	1261.57	0.497326
$187$	991.559	0.387754
$188$	3825.34	1.48400
$189$	905.807	0.348613
$190$	0	0
$191$	1599.27	0.605860	0.302930	−	0.953013i	$-0.402035\pi$
$191$	1599.27	0.605860	0.302930	+	0.953013i	$0.402035\pi$
$192$	2460.58	0.924880
$193$	2419.25	0.902288	0.451144	−	0.892451i	$-0.351016\pi$
$193$	2419.25	0.902288	0.451144	+	0.892451i	$0.351016\pi$
$194$	4536.42	1.67884
$195$	0	0
$196$	10339.2	3.76792
$197$	−1654.27	−0.598285	−0.299143	−	0.954208i	$-0.596701\pi$
$197$	−1654.27	−0.598285	−0.299143	+	0.954208i	$0.596701\pi$
$198$	−1816.40	−0.651947
$199$	−5102.28	−1.81754	−0.908771	−	0.417296i	$-0.862978\pi$
$199$	−5102.28	−1.81754	−0.908771	+	0.417296i	$0.862978\pi$
$200$	0	0
$201$	2373.90	0.833046
$202$	−3709.68	−1.29214
$203$	5770.47	1.99511
$204$	896.970	0.307845
$205$	0	0
$206$	−8210.68	−2.77702
$207$	1490.53	0.500479
$208$	−35.9718	−0.0119913
$209$	−5635.62	−1.86519
$210$	0	0
$211$	6033.10	1.96842	0.984208	−	0.177015i	$-0.0566441\pi$
$211$	6033.10	1.96842	0.984208	+	0.177015i	$0.0566441\pi$
$212$	−2042.12	−0.661572
$213$	−3427.84	−1.10268
$214$	4391.76	1.40287
$215$	0	0
$216$	−648.276	−0.204211
$217$	3063.09	0.958230
$218$	10197.2	3.16808
$219$	−753.454	−0.232483
$220$	0	0
$221$	166.772	0.0507614
$222$	−3260.17	−0.985624
$223$	−5350.14	−1.60660	−0.803301	−	0.595574i	$-0.796925\pi$
$223$	−5350.14	−1.60660	−0.803301	+	0.595574i	$0.796925\pi$
$224$	5689.87	1.69719
$225$	0	0
$226$	−6619.78	−1.94841
$227$	3065.13	0.896211	0.448106	−	0.893981i	$-0.352099\pi$
$227$	3065.13	0.896211	0.448106	+	0.893981i	$0.352099\pi$
$228$	−5098.02	−1.48081
$229$	5829.96	1.68234	0.841168	−	0.540774i	$-0.181869\pi$
$229$	5829.96	1.68234	0.841168	+	0.540774i	$0.181869\pi$
$230$	0	0
$231$	−4410.21	−1.25615
$232$	−4129.86	−1.16870
$233$	1555.52	0.437363	0.218681	−	0.975796i	$-0.429824\pi$
$233$	1555.52	0.437363	0.218681	+	0.975796i	$0.429824\pi$
$234$	−305.502	−0.0853473
$235$	0	0
$236$	2892.75	0.797889
$237$	1208.31	0.331173
$238$	3496.44	0.952272
$239$	−1588.23	−0.429849	−0.214924	−	0.976631i	$-0.568950\pi$
$239$	−1588.23	−0.429849	−0.214924	+	0.976631i	$0.568950\pi$
$240$	0	0
$241$	4225.08	1.12930	0.564651	−	0.825330i	$-0.309011\pi$
$241$	4225.08	1.12930	0.564651	+	0.825330i	$0.309011\pi$
$242$	2713.42	0.720766
$243$	−243.000	−0.0641500
$244$	7226.83	1.89611
$245$	0	0
$246$	−3298.55	−0.854909
$247$	−947.863	−0.244174
$248$	−2192.22	−0.561314
$249$	319.731	0.0813740
$250$	0	0
$251$	−1084.74	−0.272782	−0.136391	−	0.990655i	$-0.543550\pi$
$251$	−1084.74	−0.272782	−0.136391	+	0.990655i	$0.543550\pi$
$252$	−3989.50	−0.997281
$253$	−7257.14	−1.80337
$254$	3287.92	0.812214
$255$	0	0
$256$	−4588.10	−1.12014
$257$	3686.21	0.894705	0.447353	−	0.894358i	$-0.352367\pi$
$257$	3686.21	0.894705	0.447353	+	0.894358i	$0.352367\pi$
$258$	667.903	0.161170
$259$	−7915.71	−1.89907
$260$	0	0
$261$	−1548.04	−0.367131
$262$	197.081	0.0464721
$263$	2756.15	0.646204	0.323102	−	0.946364i	$-0.395274\pi$
$263$	2756.15	0.646204	0.323102	+	0.946364i	$0.395274\pi$
$264$	3156.34	0.735830
$265$	0	0
$266$	−19872.4	−4.58065
$267$	−4826.43	−1.10626
$268$	−10455.5	−2.38311
$269$	4914.24	1.11385	0.556927	−	0.830561i	$-0.311980\pi$
$269$	4914.24	1.11385	0.556927	+	0.830561i	$0.311980\pi$
$270$	0	0
$271$	−608.390	−0.136373	−0.0681865	−	0.997673i	$-0.521721\pi$
$271$	−608.390	−0.136373	−0.0681865	+	0.997673i	$0.521721\pi$
$272$	−110.445	−0.0246202
$273$	−741.759	−0.164444
$274$	−9928.58	−2.18908
$275$	0	0
$276$	−6564.85	−1.43173
$277$	3408.79	0.739402	0.369701	−	0.929151i	$-0.379460\pi$
$277$	3408.79	0.739402	0.369701	+	0.929151i	$0.379460\pi$
$278$	−409.757	−0.0884014
$279$	−821.731	−0.176329
$280$	0	0
$281$	−4436.65	−0.941881	−0.470940	−	0.882165i	$-0.656085\pi$
$281$	−4436.65	−0.941881	−0.470940	+	0.882165i	$0.656085\pi$
$282$	−4000.27	−0.844726
$283$	−3358.79	−0.705511	−0.352755	−	0.935716i	$-0.614755\pi$
$283$	−3358.79	−0.705511	−0.352755	+	0.935716i	$0.614755\pi$
$284$	15097.4	3.15446
$285$	0	0
$286$	1487.43	0.307531
$287$	−8008.88	−1.64721
$288$	−1526.42	−0.312309
$289$	−4400.96	−0.895778
$290$	0	0
$291$	−2954.83	−0.595241
$292$	3318.48	0.665067
$293$	2749.27	0.548171	0.274085	−	0.961705i	$-0.411625\pi$
$293$	2749.27	0.548171	0.274085	+	0.961705i	$0.411625\pi$
$294$	−10812.0	−2.14479
$295$	0	0
$296$	5665.18	1.11244
$297$	1183.12	0.231151
$298$	9402.86	1.82783
$299$	−1220.59	−0.236081
$300$	0	0
$301$	1621.67	0.310536
$302$	4109.85	0.783098
$303$	2416.32	0.458133
$304$	627.724	0.118429
$305$	0	0
$306$	−937.987	−0.175233
$307$	6981.55	1.29791	0.648954	−	0.760827i	$-0.275207\pi$
$307$	6981.55	1.29791	0.648954	+	0.760827i	$0.275207\pi$
$308$	19424.1	3.59349
$309$	5348.09	0.984603
$310$	0	0
$311$	7031.46	1.28205	0.641025	−	0.767520i	$-0.278509\pi$
$311$	7031.46	1.28205	0.641025	+	0.767520i	$0.278509\pi$
$312$	530.868	0.0963285
$313$	3182.12	0.574646	0.287323	−	0.957834i	$-0.407235\pi$
$313$	3182.12	0.574646	0.287323	+	0.957834i	$0.407235\pi$
$314$	7217.44	1.29715
$315$	0	0
$316$	−5321.82	−0.947392
$317$	−3964.31	−0.702390	−0.351195	−	0.936302i	$-0.614225\pi$
$317$	−3964.31	−0.702390	−0.351195	+	0.936302i	$0.614225\pi$
$318$	2135.50	0.376582
$319$	7537.12	1.32288
$320$	0	0
$321$	−2860.61	−0.497394
$322$	−25590.2	−4.42883
$323$	−2910.24	−0.501331
$324$	1070.26	0.183515
$325$	0	0
$326$	6103.24	1.03689
$327$	−6642.03	−1.12326
$328$	5731.86	0.964906
$329$	−9712.67	−1.62759
$330$	0	0
$331$	11219.5	1.86308	0.931541	−	0.363637i	$-0.118465\pi$
$331$	11219.5	1.86308	0.931541	+	0.363637i	$0.118465\pi$
$332$	−1408.21	−0.232788
$333$	2123.54	0.349457
$334$	−721.132	−0.118139
$335$	0	0
$336$	491.231	0.0797585
$337$	−2956.56	−0.477905	−0.238952	−	0.971031i	$-0.576804\pi$
$337$	−2956.56	−0.477905	−0.238952	+	0.971031i	$0.576804\pi$
$338$	−9868.69	−1.58812
$339$	4311.84	0.690817
$340$	0	0
$341$	4000.86	0.635363
$342$	5331.14	0.842910
$343$	−14744.4	−2.32106
$344$	−1160.61	−0.181907
$345$	0	0
$346$	−7709.45	−1.19787
$347$	−5466.64	−0.845720	−0.422860	−	0.906195i	$-0.638974\pi$
$347$	−5466.64	−0.845720	−0.422860	+	0.906195i	$0.638974\pi$
$348$	6818.12	1.05026
$349$	7548.63	1.15779	0.578895	−	0.815402i	$-0.303484\pi$
$349$	7548.63	1.15779	0.578895	+	0.815402i	$0.303484\pi$
$350$	0	0
$351$	198.991	0.0302603
$352$	7431.84	1.12534
$353$	−3170.61	−0.478058	−0.239029	−	0.971012i	$-0.576829\pi$
$353$	−3170.61	−0.478058	−0.239029	+	0.971012i	$0.576829\pi$
$354$	−3025.03	−0.454177
$355$	0	0
$356$	21257.3	3.16471
$357$	−2277.43	−0.337632
$358$	−13475.8	−1.98944
$359$	1784.51	0.262347	0.131174	−	0.991359i	$-0.458125\pi$
$359$	1784.51	0.262347	0.131174	+	0.991359i	$0.458125\pi$
$360$	0	0
$361$	9681.62	1.41152
$362$	−7429.65	−1.07871
$363$	−1767.41	−0.255551
$364$	3266.97	0.470428
$365$	0	0
$366$	−7557.31	−1.07931
$367$	−6511.59	−0.926164	−0.463082	−	0.886315i	$-0.653257\pi$
$367$	−6511.59	−0.926164	−0.463082	+	0.886315i	$0.653257\pi$
$368$	808.336	0.114504
$369$	2148.53	0.303112
$370$	0	0
$371$	5185.00	0.725585
$372$	3619.20	0.504427
$373$	−3936.95	−0.546509	−0.273254	−	0.961942i	$-0.588100\pi$
$373$	−3936.95	−0.546509	−0.273254	+	0.961942i	$0.588100\pi$
$374$	4566.89	0.631412
$375$	0	0
$376$	6951.25	0.953413
$377$	1267.68	0.173180
$378$	4171.94	0.567675
$379$	1134.62	0.153777	0.0768885	−	0.997040i	$-0.475501\pi$
$379$	1134.62	0.153777	0.0768885	+	0.997040i	$0.475501\pi$
$380$	0	0
$381$	−2141.61	−0.287974
$382$	7365.87	0.986572
$383$	7387.51	0.985599	0.492799	−	0.870143i	$-0.335974\pi$
$383$	7387.51	0.985599	0.492799	+	0.870143i	$0.335974\pi$
$384$	7262.41	0.965125
$385$	0	0
$386$	11142.5	1.46927
$387$	−435.044	−0.0571434
$388$	13014.1	1.70282
$389$	−2561.11	−0.333814	−0.166907	−	0.985973i	$-0.553378\pi$
$389$	−2561.11	−0.333814	−0.166907	+	0.985973i	$0.553378\pi$
$390$	0	0
$391$	−3747.59	−0.484715
$392$	18787.9	2.42075
$393$	−128.370	−0.0164769
$394$	−7619.20	−0.974238
$395$	0	0
$396$	−5210.90	−0.661256
$397$	−4075.29	−0.515196	−0.257598	−	0.966252i	$-0.582931\pi$
$397$	−4075.29	−0.515196	−0.257598	+	0.966252i	$0.582931\pi$
$398$	−23499.9	−2.95966
$399$	12944.0	1.62409
$400$	0	0
$401$	4491.11	0.559290	0.279645	−	0.960103i	$-0.409783\pi$
$401$	4491.11	0.559290	0.279645	+	0.960103i	$0.409783\pi$
$402$	10933.6	1.35652
$403$	672.910	0.0831762
$404$	−10642.4	−1.31059
$405$	0	0
$406$	26577.4	3.24881
$407$	−10339.1	−1.25919
$408$	1629.93	0.197779
$409$	−6325.03	−0.764677	−0.382338	−	0.924022i	$-0.624881\pi$
$409$	−6325.03	−0.764677	−0.382338	+	0.924022i	$0.624881\pi$
$410$	0	0
$411$	6467.05	0.776146
$412$	−23554.9	−2.81667
$413$	−7344.77	−0.875091
$414$	6865.05	0.814973
$415$	0	0
$416$	1249.97	0.147319
$417$	266.898	0.0313431
$418$	−25956.4	−3.03724
$419$	−1052.63	−0.122732	−0.0613658	−	0.998115i	$-0.519546\pi$
$419$	−1052.63	−0.122732	−0.0613658	+	0.998115i	$0.519546\pi$
$420$	0	0
$421$	10994.5	1.27277	0.636387	−	0.771370i	$-0.280428\pi$
$421$	10994.5	1.27277	0.636387	+	0.771370i	$0.280428\pi$
$422$	27787.1	3.20534
$423$	2605.61	0.299501
$424$	−3710.85	−0.425035
$425$	0	0
$426$	−15787.8	−1.79559
$427$	−18349.1	−2.07957
$428$	12599.1	1.42290
$429$	−968.850	−0.109036
$430$	0	0
$431$	17481.3	1.95370	0.976849	−	0.213931i	$-0.0686266\pi$
$431$	17481.3	1.95370	0.976849	+	0.213931i	$0.0686266\pi$
$432$	−131.782	−0.0146768
$433$	−3057.99	−0.339394	−0.169697	−	0.985496i	$-0.554279\pi$
$433$	−3057.99	−0.339394	−0.169697	+	0.985496i	$0.554279\pi$
$434$	14107.9	1.56037
$435$	0	0
$436$	29253.9	3.21332
$437$	21299.8	2.33159
$438$	−3470.23	−0.378571
$439$	−12594.8	−1.36929	−0.684646	−	0.728876i	$-0.740043\pi$
$439$	−12594.8	−1.36929	−0.684646	+	0.728876i	$0.740043\pi$
$440$	0	0
$441$	7042.46	0.760443
$442$	768.111	0.0826591
$443$	−6599.93	−0.707838	−0.353919	−	0.935276i	$-0.615151\pi$
$443$	−6599.93	−0.707838	−0.353919	+	0.935276i	$0.615151\pi$
$444$	−9352.83	−0.999697
$445$	0	0
$446$	−24641.5	−2.61616
$447$	−6124.62	−0.648064
$448$	27516.2	2.90182
$449$	12479.4	1.31167	0.655837	−	0.754903i	$-0.272316\pi$
$449$	12479.4	1.31167	0.655837	+	0.754903i	$0.272316\pi$
$450$	0	0
$451$	−10460.8	−1.09220
$452$	−18990.9	−1.97623
$453$	−2676.98	−0.277651
$454$	14117.3	1.45938
$455$	0	0
$456$	−9263.89	−0.951363
$457$	436.301	0.0446593	0.0223296	−	0.999751i	$-0.492892\pi$
$457$	436.301	0.0446593	0.0223296	+	0.999751i	$0.492892\pi$
$458$	26851.5	2.73949
$459$	610.965	0.0621294
$460$	0	0
$461$	−11749.4	−1.18704	−0.593521	−	0.804819i	$-0.702262\pi$
$461$	−11749.4	−1.18704	−0.593521	+	0.804819i	$0.702262\pi$
$462$	−20312.4	−2.04549
$463$	8031.27	0.806144	0.403072	−	0.915168i	$-0.367942\pi$
$463$	8031.27	0.806144	0.403072	+	0.915168i	$0.367942\pi$
$464$	−839.522	−0.0839954
$465$	0	0
$466$	7164.36	0.712194
$467$	−3981.50	−0.394522	−0.197261	−	0.980351i	$-0.563205\pi$
$467$	−3981.50	−0.394522	−0.197261	+	0.980351i	$0.563205\pi$
$468$	−876.427	−0.0865660
$469$	26546.9	2.61369
$470$	0	0
$471$	−4701.14	−0.459908
$472$	5256.57	0.512613
$473$	2118.15	0.205904
$474$	5565.18	0.539277
$475$	0	0
$476$	10030.6	0.965869
$477$	−1390.98	−0.133519
$478$	−7315.00	−0.699959
$479$	3852.79	0.367513	0.183756	−	0.982972i	$-0.441174\pi$
$479$	3852.79	0.367513	0.183756	+	0.982972i	$0.441174\pi$
$480$	0	0
$481$	−1738.95	−0.164843
$482$	19459.7	1.83894
$483$	16668.3	1.57026
$484$	7784.30	0.731057
$485$	0	0
$486$	−1119.20	−0.104461
$487$	8404.63	0.782033	0.391016	−	0.920384i	$-0.372124\pi$
$487$	8404.63	0.782033	0.391016	+	0.920384i	$0.372124\pi$
$488$	13132.3	1.21818
$489$	−3975.39	−0.367635
$490$	0	0
$491$	−16071.8	−1.47721	−0.738607	−	0.674136i	$-0.764516\pi$
$491$	−16071.8	−1.47721	−0.738607	+	0.674136i	$0.764516\pi$
$492$	−9462.91	−0.867116
$493$	3892.17	0.355567
$494$	−4365.63	−0.397610
$495$	0	0
$496$	−445.636	−0.0403420
$497$	−38332.8	−3.45968
$498$	1472.61	0.132508
$499$	2752.94	0.246971	0.123485	−	0.992346i	$-0.460593\pi$
$499$	2752.94	0.246971	0.123485	+	0.992346i	$0.460593\pi$
$500$	0	0
$501$	469.715	0.0418868
$502$	−4996.07	−0.444195
$503$	−21243.4	−1.88310	−0.941549	−	0.336875i	$-0.890630\pi$
$503$	−21243.4	−1.88310	−0.941549	+	0.336875i	$0.890630\pi$
$504$	−7249.54	−0.640715
$505$	0	0
$506$	−33424.7	−2.93658
$507$	6428.05	0.563076
$508$	9432.43	0.823812
$509$	3168.44	0.275911	0.137955	−	0.990438i	$-0.455947\pi$
$509$	3168.44	0.275911	0.137955	+	0.990438i	$0.455947\pi$
$510$	0	0
$511$	−8425.73	−0.729418
$512$	−1765.31	−0.152376
$513$	−3472.48	−0.298857
$514$	16977.8	1.45692
$515$	0	0
$516$	1916.09	0.163471
$517$	−12686.2	−1.07919
$518$	−36457.9	−3.09241
$519$	5021.61	0.424709
$520$	0	0
$521$	3701.03	0.311219	0.155609	−	0.987819i	$-0.450266\pi$
$521$	3701.03	0.311219	0.155609	+	0.987819i	$0.450266\pi$
$522$	−7129.91	−0.597830
$523$	−7547.89	−0.631064	−0.315532	−	0.948915i	$-0.602183\pi$
$523$	−7547.89	−0.631064	−0.315532	+	0.948915i	$0.602183\pi$
$524$	565.387	0.0471356
$525$	0	0
$526$	12694.2	1.05227
$527$	2066.05	0.170775
$528$	641.623	0.0528846
$529$	15261.3	1.25432
$530$	0	0
$531$	1970.38	0.161030
$532$	−57010.1	−4.64606
$533$	−1759.42	−0.142981
$534$	−22229.4	−1.80142
$535$	0	0
$536$	−18999.3	−1.53105
$537$	8777.58	0.705364
$538$	22633.9	1.81378
$539$	−34288.5	−2.74009
$540$	0	0
$541$	7164.72	0.569382	0.284691	−	0.958619i	$-0.408109\pi$
$541$	7164.72	0.569382	0.284691	+	0.958619i	$0.408109\pi$
$542$	−2802.10	−0.222068
$543$	4839.36	0.382462
$544$	3837.81	0.302472
$545$	0	0
$546$	−3416.37	−0.267778
$547$	17819.3	1.39287	0.696434	−	0.717621i	$-0.254769\pi$
$547$	17819.3	1.39287	0.696434	+	0.717621i	$0.254769\pi$
$548$	−28483.2	−2.22033
$549$	4922.51	0.382673
$550$	0	0
$551$	−22121.5	−1.71036
$552$	−11929.3	−0.919831
$553$	13512.3	1.03906
$554$	15700.1	1.20403
$555$	0	0
$556$	−1175.52	−0.0896636
$557$	9017.26	0.685949	0.342975	−	0.939345i	$-0.388566\pi$
$557$	9017.26	0.685949	0.342975	+	0.939345i	$0.388566\pi$
$558$	−3784.70	−0.287131
$559$	356.254	0.0269552
$560$	0	0
$561$	−2974.68	−0.223870
$562$	−20434.2	−1.53374
$563$	−12636.2	−0.945916	−0.472958	−	0.881085i	$-0.656814\pi$
$563$	−12636.2	−0.945916	−0.472958	+	0.881085i	$0.656814\pi$
$564$	−11476.0	−0.856787
$565$	0	0
$566$	−15469.8	−1.14884
$567$	−2717.42	−0.201272
$568$	27434.4	2.02662
$569$	−8645.88	−0.637002	−0.318501	−	0.947922i	$-0.603179\pi$
$569$	−8645.88	−0.637002	−0.318501	+	0.947922i	$0.603179\pi$
$570$	0	0
$571$	−12085.9	−0.885781	−0.442890	−	0.896576i	$-0.646047\pi$
$571$	−12085.9	−0.885781	−0.442890	+	0.896576i	$0.646047\pi$
$572$	4267.17	0.311922
$573$	−4797.81	−0.349793
$574$	−36887.0	−2.68229
$575$	0	0
$576$	−7381.74	−0.533980
$577$	−4644.56	−0.335105	−0.167553	−	0.985863i	$-0.553586\pi$
$577$	−4644.56	−0.335105	−0.167553	+	0.985863i	$0.553586\pi$
$578$	−20269.8	−1.45867
$579$	−7257.76	−0.520936
$580$	0	0
$581$	3575.49	0.255312
$582$	−13609.3	−0.969281
$583$	6772.41	0.481105
$584$	6030.20	0.427280
$585$	0	0
$586$	12662.5	0.892632
$587$	−7797.47	−0.548272	−0.274136	−	0.961691i	$-0.588392\pi$
$587$	−7797.47	−0.548272	−0.274136	+	0.961691i	$0.588392\pi$
$588$	−31017.5	−2.17541
$589$	−11742.6	−0.821467
$590$	0	0
$591$	4962.82	0.345420
$592$	1151.62	0.0799518
$593$	21835.1	1.51207	0.756037	−	0.654529i	$-0.227133\pi$
$593$	21835.1	1.51207	0.756037	+	0.654529i	$0.227133\pi$
$594$	5449.19	0.376402
$595$	0	0
$596$	26975.0	1.85393
$597$	15306.8	1.04936
$598$	−5621.74	−0.384431
$599$	9744.09	0.664662	0.332331	−	0.943163i	$-0.392165\pi$
$599$	9744.09	0.664662	0.332331	+	0.943163i	$0.392165\pi$
$600$	0	0
$601$	11576.7	0.785726	0.392863	−	0.919597i	$-0.371485\pi$
$601$	11576.7	0.785726	0.392863	+	0.919597i	$0.371485\pi$
$602$	7469.03	0.505673
$603$	−7121.71	−0.480959
$604$	11790.4	0.794279
$605$	0	0
$606$	11129.0	0.746016
$607$	−6719.64	−0.449327	−0.224664	−	0.974436i	$-0.572128\pi$
$607$	−6719.64	−0.449327	−0.224664	+	0.974436i	$0.572128\pi$
$608$	−21812.6	−1.45496
$609$	−17311.4	−1.15188
$610$	0	0
$611$	−2133.71	−0.141278
$612$	−2690.91	−0.177735
$613$	−3880.73	−0.255695	−0.127848	−	0.991794i	$-0.540807\pi$
$613$	−3880.73	−0.255695	−0.127848	+	0.991794i	$0.540807\pi$
$614$	32155.4	2.11349
$615$	0	0
$616$	35296.7	2.30868
$617$	22898.6	1.49410	0.747052	−	0.664766i	$-0.231469\pi$
$617$	22898.6	1.49410	0.747052	+	0.664766i	$0.231469\pi$
$618$	24632.1	1.60331
$619$	13421.8	0.871516	0.435758	−	0.900064i	$-0.356480\pi$
$619$	13421.8	0.871516	0.435758	+	0.900064i	$0.356480\pi$
$620$	0	0
$621$	−4471.60	−0.288952
$622$	32385.3	2.08767
$623$	−53973.0	−3.47092
$624$	107.915	0.00692320
$625$	0	0
$626$	14656.1	0.935745
$627$	16906.9	1.07687
$628$	20705.5	1.31567
$629$	−5339.13	−0.338450
$630$	0	0
$631$	−13089.3	−0.825797	−0.412898	−	0.910777i	$-0.635484\pi$
$631$	−13089.3	−0.825797	−0.412898	+	0.910777i	$0.635484\pi$
$632$	−9670.58	−0.608663
$633$	−18099.3	−1.13647
$634$	−18258.7	−1.14376
$635$	0	0
$636$	6126.36	0.381959
$637$	−5767.02	−0.358709
$638$	34714.2	2.15415
$639$	10283.5	0.636634
$640$	0	0
$641$	1160.65	0.0715179	0.0357590	−	0.999360i	$-0.488615\pi$
$641$	1160.65	0.0715179	0.0357590	+	0.999360i	$0.488615\pi$
$642$	−13175.3	−0.809949
$643$	−2318.80	−0.142215	−0.0711076	−	0.997469i	$-0.522653\pi$
$643$	−2318.80	−0.142215	−0.0711076	+	0.997469i	$0.522653\pi$
$644$	−73413.4	−4.49207
$645$	0	0
$646$	−13403.9	−0.816360
$647$	−26809.3	−1.62903	−0.814515	−	0.580143i	$-0.802997\pi$
$647$	−26809.3	−1.62903	−0.814515	+	0.580143i	$0.802997\pi$
$648$	1944.83	0.117901
$649$	−9593.40	−0.580237
$650$	0	0
$651$	−9189.26	−0.553234
$652$	17509.0	1.05170
$653$	−3318.86	−0.198893	−0.0994465	−	0.995043i	$-0.531707\pi$
$653$	−3318.86	−0.198893	−0.0994465	+	0.995043i	$0.531707\pi$
$654$	−30591.6	−1.82909
$655$	0	0
$656$	1165.18	0.0693485
$657$	2260.36	0.134224
$658$	−44734.3	−2.65034
$659$	−26163.5	−1.54656	−0.773281	−	0.634063i	$-0.781386\pi$
$659$	−26163.5	−1.54656	−0.773281	+	0.634063i	$0.781386\pi$
$660$	0	0
$661$	10785.7	0.634669	0.317335	−	0.948314i	$-0.397212\pi$
$661$	10785.7	0.634669	0.317335	+	0.948314i	$0.397212\pi$
$662$	51674.4	3.03381
$663$	−500.315	−0.0293071
$664$	−2558.94	−0.149557
$665$	0	0
$666$	9780.52	0.569050
$667$	−28486.5	−1.65367
$668$	−2068.79	−0.119826
$669$	16050.4	0.927572
$670$	0	0
$671$	−23966.8	−1.37888
$672$	−17069.6	−0.979873
$673$	12388.0	0.709541	0.354770	−	0.934953i	$-0.384559\pi$
$673$	12388.0	0.709541	0.354770	+	0.934953i	$0.384559\pi$
$674$	−13617.2	−0.778212
$675$	0	0
$676$	−28311.4	−1.61080
$677$	18741.0	1.06392	0.531960	−	0.846769i	$-0.321456\pi$
$677$	18741.0	1.06392	0.531960	+	0.846769i	$0.321456\pi$
$678$	19859.3	1.12492
$679$	−33043.3	−1.86758
$680$	0	0
$681$	−9195.39	−0.517428
$682$	18427.0	1.03461
$683$	−557.263	−0.0312198	−0.0156099	−	0.999878i	$-0.504969\pi$
$683$	−557.263	−0.0312198	−0.0156099	+	0.999878i	$0.504969\pi$
$684$	15294.0	0.854945
$685$	0	0
$686$	−67909.2	−3.77957
$687$	−17489.9	−0.971297
$688$	−235.930	−0.0130738
$689$	1139.06	0.0629822
$690$	0	0
$691$	−11777.2	−0.648374	−0.324187	−	0.945993i	$-0.605091\pi$
$691$	−11777.2	−0.648374	−0.324187	+	0.945993i	$0.605091\pi$
$692$	−22117.0	−1.21497
$693$	13230.6	0.725238
$694$	−25178.1	−1.37716
$695$	0	0
$696$	12389.6	0.674750
$697$	−5401.97	−0.293564
$698$	34767.2	1.88533
$699$	−4666.56	−0.252511
$700$	0	0
$701$	15025.5	0.809563	0.404782	−	0.914413i	$-0.367348\pi$
$701$	15025.5	0.809563	0.404782	+	0.914413i	$0.367348\pi$
$702$	916.505	0.0492753
$703$	30345.5	1.62802
$704$	35940.3	1.92408
$705$	0	0
$706$	−14603.1	−0.778462
$707$	27021.3	1.43740
$708$	−8678.24	−0.460662
$709$	−16819.9	−0.890950	−0.445475	−	0.895294i	$-0.646965\pi$
$709$	−16819.9	−0.890950	−0.445475	+	0.895294i	$0.646965\pi$
$710$	0	0
$711$	−3624.92	−0.191203
$712$	38627.9	2.03320
$713$	−15121.2	−0.794241
$714$	−10489.3	−0.549794
$715$	0	0
$716$	−38659.6	−2.01785
$717$	4764.68	0.248173
$718$	8219.02	0.427202
$719$	3172.56	0.164557	0.0822784	−	0.996609i	$-0.473780\pi$
$719$	3172.56	0.164557	0.0822784	+	0.996609i	$0.473780\pi$
$720$	0	0
$721$	59806.6	3.08920
$722$	44591.3	2.29850
$723$	−12675.3	−0.652002
$724$	−21314.3	−1.09411
$725$	0	0
$726$	−8140.27	−0.416134
$727$	−3030.09	−0.154580	−0.0772901	−	0.997009i	$-0.524627\pi$
$727$	−3030.09	−0.154580	−0.0772901	+	0.997009i	$0.524627\pi$
$728$	5936.60	0.302232
$729$	729.000	0.0370370
$730$	0	0
$731$	1093.81	0.0553435
$732$	−21680.5	−1.09472
$733$	−7784.47	−0.392259	−0.196130	−	0.980578i	$-0.562837\pi$
$733$	−7784.47	−0.392259	−0.196130	+	0.980578i	$0.562837\pi$
$734$	−29990.9	−1.50815
$735$	0	0
$736$	−28088.6	−1.40674
$737$	34674.3	1.73303
$738$	9895.64	0.493582
$739$	5925.54	0.294959	0.147479	−	0.989065i	$-0.452884\pi$
$739$	5925.54	0.294959	0.147479	+	0.989065i	$0.452884\pi$
$740$	0	0
$741$	2843.59	0.140974
$742$	23880.9	1.18153
$743$	−5801.80	−0.286470	−0.143235	−	0.989689i	$-0.545750\pi$
$743$	−5801.80	−0.286470	−0.143235	+	0.989689i	$0.545750\pi$
$744$	6576.65	0.324075
$745$	0	0
$746$	−18132.7	−0.889926
$747$	−959.193	−0.0469813
$748$	13101.5	0.640428
$749$	−31989.6	−1.56058
$750$	0	0
$751$	25182.5	1.22360	0.611799	−	0.791013i	$-0.290446\pi$
$751$	25182.5	1.22360	0.611799	+	0.791013i	$0.290446\pi$
$752$	1413.06	0.0685224
$753$	3254.23	0.157491
$754$	5838.63	0.282003
$755$	0	0
$756$	11968.5	0.575781
$757$	37908.7	1.82010	0.910050	−	0.414498i	$-0.136043\pi$
$757$	37908.7	1.82010	0.910050	+	0.414498i	$0.136043\pi$
$758$	5225.79	0.250408
$759$	21771.4	1.04118
$760$	0	0
$761$	−12693.8	−0.604666	−0.302333	−	0.953202i	$-0.597765\pi$
$761$	−12693.8	−0.604666	−0.302333	+	0.953202i	$0.597765\pi$
$762$	−9863.76	−0.468932
$763$	−74276.5	−3.52423
$764$	21131.3	1.00066
$765$	0	0
$766$	34025.2	1.60493
$767$	−1613.53	−0.0759596
$768$	13764.3	0.646714
$769$	19956.2	0.935811	0.467905	−	0.883779i	$-0.345009\pi$
$769$	19956.2	0.935811	0.467905	+	0.883779i	$0.345009\pi$
$770$	0	0
$771$	−11058.6	−0.516558
$772$	31965.8	1.49025
$773$	26532.8	1.23456	0.617282	−	0.786742i	$-0.288234\pi$
$773$	26532.8	1.23456	0.617282	+	0.786742i	$0.288234\pi$
$774$	−2003.71	−0.0930515
$775$	0	0
$776$	23648.7	1.09399
$777$	23747.1	1.09643
$778$	−11795.9	−0.543577
$779$	30702.6	1.41211
$780$	0	0
$781$	−50068.5	−2.29397
$782$	−17260.5	−0.789303
$783$	4644.12	0.211963
$784$	3819.22	0.173981
$785$	0	0
$786$	−591.242	−0.0268307
$787$	−22108.9	−1.00139	−0.500697	−	0.865622i	$-0.666923\pi$
$787$	−22108.9	−1.00139	−0.500697	+	0.865622i	$0.666923\pi$
$788$	−21858.1	−0.988149
$789$	−8268.46	−0.373086
$790$	0	0
$791$	48218.5	2.16745
$792$	−9469.01	−0.424832
$793$	−4031.00	−0.180511
$794$	−18769.8	−0.838937
$795$	0	0
$796$	−67416.8	−3.00191
$797$	39363.4	1.74946	0.874731	−	0.484608i	$-0.161038\pi$
$797$	39363.4	1.74946	0.874731	+	0.484608i	$0.161038\pi$
$798$	59617.1	2.64464
$799$	−6551.17	−0.290067
$800$	0	0
$801$	14479.3	0.638702
$802$	20685.0	0.910739
$803$	−11005.3	−0.483647
$804$	31366.6	1.37589
$805$	0	0
$806$	3099.26	0.135443
$807$	−14742.7	−0.643084
$808$	−19338.8	−0.842002
$809$	14593.9	0.634231	0.317116	−	0.948387i	$-0.397286\pi$
$809$	14593.9	0.634231	0.317116	+	0.948387i	$0.397286\pi$
$810$	0	0
$811$	6792.04	0.294082	0.147041	−	0.989130i	$-0.453025\pi$
$811$	6792.04	0.294082	0.147041	+	0.989130i	$0.453025\pi$
$812$	76245.7	3.29520
$813$	1825.17	0.0787350
$814$	−47619.6	−2.05045
$815$	0	0
$816$	331.335	0.0142145
$817$	−6216.79	−0.266215
$818$	−29131.6	−1.24519
$819$	2225.28	0.0949419
$820$	0	0
$821$	−13142.0	−0.558659	−0.279329	−	0.960195i	$-0.590112\pi$
$821$	−13142.0	−0.558659	−0.279329	+	0.960195i	$0.590112\pi$
$822$	29785.7	1.26386
$823$	2536.89	0.107449	0.0537245	−	0.998556i	$-0.482891\pi$
$823$	2536.89	0.107449	0.0537245	+	0.998556i	$0.482891\pi$
$824$	−42802.9	−1.80960
$825$	0	0
$826$	−33828.3	−1.42498
$827$	3502.79	0.147284	0.0736421	−	0.997285i	$-0.476538\pi$
$827$	3502.79	0.147284	0.0736421	+	0.997285i	$0.476538\pi$
$828$	19694.5	0.826609
$829$	−4725.94	−0.197996	−0.0989979	−	0.995088i	$-0.531564\pi$
$829$	−4725.94	−0.197996	−0.0989979	+	0.995088i	$0.531564\pi$
$830$	0	0
$831$	−10226.4	−0.426894
$832$	6044.85	0.251884
$833$	−17706.6	−0.736490
$834$	1229.27	0.0510385
$835$	0	0
$836$	−74463.9	−3.08061
$837$	2465.19	0.101804
$838$	−4848.19	−0.199854
$839$	−7199.51	−0.296251	−0.148126	−	0.988969i	$-0.547324\pi$
$839$	−7199.51	−0.296251	−0.148126	+	0.988969i	$0.547324\pi$
$840$	0	0
$841$	5196.49	0.213067
$842$	50638.0	2.07257
$843$	13310.0	0.543795
$844$	79715.8	3.25111
$845$	0	0
$846$	12000.8	0.487703
$847$	−19764.6	−0.801793
$848$	−754.345	−0.0305475
$849$	10076.4	0.407327
$850$	0	0
$851$	39076.6	1.57406
$852$	−45292.3	−1.82123
$853$	−42311.1	−1.69836	−0.849182	−	0.528100i	$-0.822904\pi$
$853$	−42311.1	−1.69836	−0.849182	+	0.528100i	$0.822904\pi$
$854$	−84511.9	−3.38634
$855$	0	0
$856$	22894.6	0.914161
$857$	28565.5	1.13860	0.569300	−	0.822130i	$-0.307214\pi$
$857$	28565.5	1.13860	0.569300	+	0.822130i	$0.307214\pi$
$858$	−4462.30	−0.177553
$859$	−11851.8	−0.470755	−0.235378	−	0.971904i	$-0.575633\pi$
$859$	−11851.8	−0.470755	−0.235378	+	0.971904i	$0.575633\pi$
$860$	0	0
$861$	24026.6	0.951017
$862$	80514.7	3.18137
$863$	27207.3	1.07317	0.536585	−	0.843846i	$-0.319714\pi$
$863$	27207.3	1.07317	0.536585	+	0.843846i	$0.319714\pi$
$864$	4579.25	0.180312
$865$	0	0
$866$	−14084.4	−0.552665
$867$	13202.9	0.517178
$868$	40472.8	1.58265
$869$	17649.1	0.688958
$870$	0	0
$871$	5831.91	0.226874
$872$	53158.8	2.06443
$873$	8864.49	0.343663
$874$	98101.8	3.79673
$875$	0	0
$876$	−9955.45	−0.383977
$877$	−13750.0	−0.529425	−0.264712	−	0.964327i	$-0.585277\pi$
$877$	−13750.0	−0.529425	−0.264712	+	0.964327i	$0.585277\pi$
$878$	−58008.9	−2.22973
$879$	−8247.81	−0.316486
$880$	0	0
$881$	−1662.82	−0.0635890	−0.0317945	−	0.999494i	$-0.510122\pi$
$881$	−1662.82	−0.0635890	−0.0317945	+	0.999494i	$0.510122\pi$
$882$	32435.9	1.23829
$883$	−4944.07	−0.188427	−0.0942136	−	0.995552i	$-0.530034\pi$
$883$	−4944.07	−0.188427	−0.0942136	+	0.995552i	$0.530034\pi$
$884$	2203.57	0.0838393
$885$	0	0
$886$	−30397.7	−1.15263
$887$	−23974.0	−0.907519	−0.453759	−	0.891124i	$-0.649917\pi$
$887$	−23974.0	−0.907519	−0.453759	+	0.891124i	$0.649917\pi$
$888$	−16995.5	−0.642267
$889$	−23949.2	−0.903522
$890$	0	0
$891$	−3549.37	−0.133455
$892$	−70691.9	−2.65352
$893$	37234.2	1.39529
$894$	−28208.6	−1.05530
$895$	0	0
$896$	81214.1	3.02809
$897$	3661.76	0.136302
$898$	57477.4	2.13591
$899$	15704.6	0.582622
$900$	0	0
$901$	3497.27	0.129313
$902$	−48180.1	−1.77852
$903$	−4865.01	−0.179288
$904$	−34509.4	−1.26965
$905$	0	0
$906$	−12329.6	−0.452122
$907$	40692.0	1.48970	0.744850	−	0.667232i	$-0.232521\pi$
$907$	40692.0	1.48970	0.744850	+	0.667232i	$0.232521\pi$
$908$	40499.8	1.48021
$909$	−7248.97	−0.264503
$910$	0	0
$911$	5532.73	0.201216	0.100608	−	0.994926i	$-0.467921\pi$
$911$	5532.73	0.201216	0.100608	+	0.994926i	$0.467921\pi$
$912$	−1883.17	−0.0683751
$913$	4670.14	0.169287
$914$	2009.50	0.0727225
$915$	0	0
$916$	77031.8	2.77860
$917$	−1435.54	−0.0516964
$918$	2813.96	0.101171
$919$	−29602.2	−1.06255	−0.531277	−	0.847198i	$-0.678288\pi$
$919$	−29602.2	−1.06255	−0.531277	+	0.847198i	$0.678288\pi$
$920$	0	0
$921$	−20944.6	−0.749348
$922$	−54115.2	−1.93296
$923$	−8421.09	−0.300307
$924$	−58272.4	−2.07470
$925$	0	0
$926$	36990.1	1.31271
$927$	−16044.3	−0.568461
$928$	29172.3	1.03192
$929$	−8456.51	−0.298654	−0.149327	−	0.988788i	$-0.547711\pi$
$929$	−8456.51	−0.298654	−0.149327	+	0.988788i	$0.547711\pi$
$930$	0	0
$931$	100637.	3.54269
$932$	20553.2	0.722363
$933$	−21094.4	−0.740192
$934$	−18337.9	−0.642434
$935$	0	0
$936$	−1592.61	−0.0556153
$937$	28862.3	1.00629	0.503143	−	0.864203i	$-0.332177\pi$
$937$	28862.3	1.00629	0.503143	+	0.864203i	$0.332177\pi$
$938$	122269.	4.25609
$939$	−9546.37	−0.331772
$940$	0	0
$941$	20891.0	0.723727	0.361863	−	0.932231i	$-0.382141\pi$
$941$	20891.0	0.723727	0.361863	+	0.932231i	$0.382141\pi$
$942$	−21652.3	−0.748908
$943$	39536.6	1.36531
$944$	1068.56	0.0368419
$945$	0	0
$946$	9755.69	0.335291
$947$	1974.72	0.0677610	0.0338805	−	0.999426i	$-0.489213\pi$
$947$	1974.72	0.0677610	0.0338805	+	0.999426i	$0.489213\pi$
$948$	15965.5	0.546977
$949$	−1850.99	−0.0633149
$950$	0	0
$951$	11892.9	0.405525
$952$	18227.2	0.620534
$953$	16786.4	0.570583	0.285292	−	0.958441i	$-0.407910\pi$
$953$	16786.4	0.570583	0.285292	+	0.958441i	$0.407910\pi$
$954$	−6406.51	−0.217420
$955$	0	0
$956$	−20985.4	−0.709953
$957$	−22611.4	−0.763763
$958$	17745.1	0.598452
$959$	72319.8	2.43517
$960$	0	0
$961$	−21454.7	−0.720173
$962$	−8009.20	−0.268427
$963$	8581.82	0.287171
$964$	55826.4	1.86519
$965$	0	0
$966$	76770.5	2.55699
$967$	−15006.2	−0.499036	−0.249518	−	0.968370i	$-0.580272\pi$
$967$	−15006.2	−0.499036	−0.249518	+	0.968370i	$0.580272\pi$
$968$	14145.3	0.469676
$969$	8730.71	0.289444
$970$	0	0
$971$	40317.1	1.33248	0.666240	−	0.745737i	$-0.267902\pi$
$971$	40317.1	1.33248	0.666240	+	0.745737i	$0.267902\pi$
$972$	−3210.78	−0.105952
$973$	2984.67	0.0983393
$974$	38709.7	1.27345
$975$	0	0
$976$	2669.54	0.0875512
$977$	3389.00	0.110976	0.0554881	−	0.998459i	$-0.482329\pi$
$977$	3389.00	0.110976	0.0554881	+	0.998459i	$0.482329\pi$
$978$	−18309.7	−0.598650
$979$	−70497.0	−2.30142
$980$	0	0
$981$	19926.1	0.648513
$982$	−74023.2	−2.40547
$983$	−57501.0	−1.86571	−0.932857	−	0.360246i	$-0.882693\pi$
$983$	−57501.0	−1.86571	−0.932857	+	0.360246i	$0.882693\pi$
$984$	−17195.6	−0.557089
$985$	0	0
$986$	17926.4	0.579000
$987$	29138.0	0.939689
$988$	−12524.2	−0.403287
$989$	−8005.52	−0.257392
$990$	0	0
$991$	1372.94	0.0440090	0.0220045	−	0.999758i	$-0.492995\pi$
$991$	1372.94	0.0440090	0.0220045	+	0.999758i	$0.492995\pi$
$992$	15485.2	0.495622
$993$	−33658.5	−1.07565
$994$	−176552.	−5.63369
$995$	0	0
$996$	4224.63	0.134400
$997$	18947.7	0.601884	0.300942	−	0.953642i	$-0.402699\pi$
$997$	18947.7	0.601884	0.300942	+	0.953642i	$0.402699\pi$
$998$	12679.4	0.402163
$999$	−6370.62	−0.201759

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.4.a.k.1.21	✓	24
5.4	even	2		1875.4.a.l.1.4	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.4.a.k.1.21	✓	24	1.1	even	1	trivial
1875.4.a.l.1.4	yes	24	5.4	even	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+4.60577 q^{2} -3.00000 q^{3} +13.2131 q^{4} -13.8173 q^{6} -33.5484 q^{7} +24.0102 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.60577 q^{2} -3.00000 q^{3} +13.2131 q^{4} -13.8173 q^{6} -33.5484 q^{7} +24.0102 q^{8} +9.00000 q^{9} -43.8194 q^{11} -39.6392 q^{12} -7.37003 q^{13} -154.516 q^{14} +4.88082 q^{16} -22.6283 q^{17} +41.4519 q^{18} +128.610 q^{19} +100.645 q^{21} -201.822 q^{22} +165.615 q^{23} -72.0307 q^{24} -33.9446 q^{26} -27.0000 q^{27} -443.278 q^{28} -172.004 q^{29} -91.3035 q^{31} -169.602 q^{32} +131.458 q^{33} -104.221 q^{34} +118.918 q^{36} +235.949 q^{37} +592.349 q^{38} +22.1101 q^{39} +238.726 q^{41} +463.548 q^{42} -48.3382 q^{43} -578.989 q^{44} +762.783 q^{46} +289.512 q^{47} -14.6425 q^{48} +782.496 q^{49} +67.8850 q^{51} -97.3808 q^{52} -154.553 q^{53} -124.356 q^{54} -805.505 q^{56} -385.831 q^{57} -792.212 q^{58} +218.931 q^{59} +546.945 q^{61} -420.522 q^{62} -301.936 q^{63} -820.193 q^{64} +605.465 q^{66} -791.301 q^{67} -298.990 q^{68} -496.845 q^{69} +1142.61 q^{71} +216.092 q^{72} +251.151 q^{73} +1086.72 q^{74} +1699.34 q^{76} +1470.07 q^{77} +101.834 q^{78} -402.769 q^{79} +81.0000 q^{81} +1099.52 q^{82} -106.577 q^{83} +1329.83 q^{84} -222.634 q^{86} +516.013 q^{87} -1052.11 q^{88} +1608.81 q^{89} +247.253 q^{91} +2188.28 q^{92} +273.910 q^{93} +1333.42 q^{94} +508.806 q^{96} +984.943 q^{97} +3603.99 q^{98} -394.374 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9}+O(q^{10}) \) 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9	\( 24 q - q^{2} - 72 q^{3} + 133 q^{4} + 3 q^{6} - 62 q^{7} - 27 q^{8} + 216 q^{9} + 96 q^{11} - 399 q^{12} - 156 q^{13} + 92 q^{14} + 845 q^{16} + 46 q^{17} - 9 q^{18} + 182 q^{19} + 186 q^{21} - 158 q^{22} + 286 q^{23} + 81 q^{24} + 478 q^{26} - 648 q^{27} - 701 q^{28} + 1144 q^{29} + 64 q^{31} - 1212 q^{32} - 288 q^{33} + 961 q^{34} + 1197 q^{36} - 762 q^{37} - 474 q^{38} + 468 q^{39} + 1074 q^{41} - 276 q^{42} - 460 q^{43} + 319 q^{44} + 459 q^{46} + 960 q^{47} - 2535 q^{48} + 2680 q^{49} - 138 q^{51} - 2969 q^{52} - 914 q^{53} + 27 q^{54} + 1680 q^{56} - 546 q^{57} - 208 q^{58} + 208 q^{59} + 3520 q^{61} - 334 q^{62} - 558 q^{63} + 5747 q^{64} + 474 q^{66} - 154 q^{67} + 5727 q^{68} - 858 q^{69} - 252 q^{71} - 243 q^{72} - 4414 q^{73} + 5637 q^{74} + 627 q^{76} - 2344 q^{77} - 1434 q^{78} + 1110 q^{79} + 1944 q^{81} - 3714 q^{82} + 1488 q^{83} + 2103 q^{84} + 3036 q^{86} - 3432 q^{87} - 3947 q^{88} + 3402 q^{89} + 3504 q^{91} + 11163 q^{92} - 192 q^{93} + 3408 q^{94} + 3636 q^{96} - 534 q^{97} - 2244 q^{98} + 864 q^{99}+O(q^{100}) \) 24 * q - q^2 - 72 * q^3 + 133 * q^4 + 3 * q^6 - 62 * q^7 - 27 * q^8 + 216 * q^9 + 96 * q^11 - 399 * q^12 - 156 * q^13 + 92 * q^14 + 845 * q^16 + 46 * q^17 - 9 * q^18 + 182 * q^19 + 186 * q^21 - 158 * q^22 + 286 * q^23 + 81 * q^24 + 478 * q^26 - 648 * q^27 - 701 * q^28 + 1144 * q^29 + 64 * q^31 - 1212 * q^32 - 288 * q^33 + 961 * q^34 + 1197 * q^36 - 762 * q^37 - 474 * q^38 + 468 * q^39 + 1074 * q^41 - 276 * q^42 - 460 * q^43 + 319 * q^44 + 459 * q^46 + 960 * q^47 - 2535 * q^48 + 2680 * q^49 - 138 * q^51 - 2969 * q^52 - 914 * q^53 + 27 * q^54 + 1680 * q^56 - 546 * q^57 - 208 * q^58 + 208 * q^59 + 3520 * q^61 - 334 * q^62 - 558 * q^63 + 5747 * q^64 + 474 * q^66 - 154 * q^67 + 5727 * q^68 - 858 * q^69 - 252 * q^71 - 243 * q^72 - 4414 * q^73 + 5637 * q^74 + 627 * q^76 - 2344 * q^77 - 1434 * q^78 + 1110 * q^79 + 1944 * q^81 - 3714 * q^82 + 1488 * q^83 + 2103 * q^84 + 3036 * q^86 - 3432 * q^87 - 3947 * q^88 + 3402 * q^89 + 3504 * q^91 + 11163 * q^92 - 192 * q^93 + 3408 * q^94 + 3636 * q^96 - 534 * q^97 - 2244 * q^98 + 864 * q^99

Introduction

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