LMFDB - Embedded newform 1875.4.a.l.1.23

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			1.23
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)

$f(q)$	$=$	$q+5.42895 q^{2} +3.00000 q^{3} +21.4736 q^{4} +16.2869 q^{6} +15.6883 q^{7} +73.1473 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+5.42895 q^{2} +3.00000 q^{3} +21.4736 q^{4} +16.2869 q^{6} +15.6883 q^{7} +73.1473 q^{8} +9.00000 q^{9} -11.2476 q^{11} +64.4207 q^{12} +70.4992 q^{13} +85.1709 q^{14} +225.325 q^{16} -117.926 q^{17} +48.8606 q^{18} +133.131 q^{19} +47.0648 q^{21} -61.0629 q^{22} +4.39598 q^{23} +219.442 q^{24} +382.737 q^{26} +27.0000 q^{27} +336.883 q^{28} +17.8263 q^{29} -179.188 q^{31} +638.101 q^{32} -33.7429 q^{33} -640.217 q^{34} +193.262 q^{36} +86.0592 q^{37} +722.762 q^{38} +211.498 q^{39} -375.112 q^{41} +255.513 q^{42} -400.245 q^{43} -241.527 q^{44} +23.8656 q^{46} -456.367 q^{47} +675.975 q^{48} -96.8781 q^{49} -353.779 q^{51} +1513.87 q^{52} +124.140 q^{53} +146.582 q^{54} +1147.55 q^{56} +399.393 q^{57} +96.7780 q^{58} -275.226 q^{59} +499.511 q^{61} -972.802 q^{62} +141.194 q^{63} +1661.62 q^{64} -183.189 q^{66} +635.801 q^{67} -2532.30 q^{68} +13.1879 q^{69} +812.581 q^{71} +658.326 q^{72} +1121.42 q^{73} +467.212 q^{74} +2858.80 q^{76} -176.456 q^{77} +1148.21 q^{78} -479.934 q^{79} +81.0000 q^{81} -2036.47 q^{82} -942.918 q^{83} +1010.65 q^{84} -2172.91 q^{86} +53.4788 q^{87} -822.734 q^{88} +461.792 q^{89} +1106.01 q^{91} +94.3972 q^{92} -537.563 q^{93} -2477.60 q^{94} +1914.30 q^{96} +1277.82 q^{97} -525.947 q^{98} -101.229 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	5.42895	1.91943	0.959713	−	0.280983i	$-0.0906605\pi$
$2$	5.42895	1.91943	0.959713	+	0.280983i	$0.0906605\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	21.4736	2.68419
$5$	0	0
$5$	0	0
$6$	16.2869	1.10818
$7$	15.6883	0.847087	0.423544	−	0.905876i	$-0.360786\pi$
$7$	15.6883	0.847087	0.423544	+	0.905876i	$0.360786\pi$
$8$	73.1473	3.23268
$9$	9.00000	0.333333
$10$	0	0
$11$	−11.2476	−0.308299	−0.154150	−	0.988048i	$-0.549264\pi$
$11$	−11.2476	−0.308299	−0.154150	+	0.988048i	$0.549264\pi$
$12$	64.4207	1.54972
$13$	70.4992	1.50407	0.752037	−	0.659121i	$-0.229072\pi$
$13$	70.4992	1.50407	0.752037	+	0.659121i	$0.229072\pi$
$14$	85.1709	1.62592
$15$	0	0
$16$	225.325	3.52070
$17$	−117.926	−1.68243	−0.841216	−	0.540700i	$-0.818159\pi$
$17$	−117.926	−1.68243	−0.841216	+	0.540700i	$0.818159\pi$
$18$	48.8606	0.639808
$19$	133.131	1.60749	0.803746	−	0.594973i	$-0.202837\pi$
$19$	133.131	1.60749	0.803746	+	0.594973i	$0.202837\pi$
$20$	0	0
$21$	47.0648	0.489066
$22$	−61.0629	−0.591757
$23$	4.39598	0.0398532	0.0199266	−	0.999801i	$-0.493657\pi$
$23$	4.39598	0.0398532	0.0199266	+	0.999801i	$0.493657\pi$
$24$	219.442	1.86639
$25$	0	0
$26$	382.737	2.88696
$27$	27.0000	0.192450
$28$	336.883	2.27375
$29$	17.8263	0.114147	0.0570734	−	0.998370i	$-0.481823\pi$
$29$	17.8263	0.114147	0.0570734	+	0.998370i	$0.481823\pi$
$30$	0	0
$31$	−179.188	−1.03816	−0.519082	−	0.854725i	$-0.673726\pi$
$31$	−179.188	−1.03816	−0.519082	+	0.854725i	$0.673726\pi$
$32$	638.101	3.52504
$33$	−33.7429	−0.177997
$34$	−640.217	−3.22930
$35$	0	0
$36$	193.262	0.894731
$37$	86.0592	0.382380	0.191190	−	0.981553i	$-0.438765\pi$
$37$	86.0592	0.382380	0.191190	+	0.981553i	$0.438765\pi$
$38$	722.762	3.08546
$39$	211.498	0.868377
$40$	0	0
$41$	−375.112	−1.42885	−0.714423	−	0.699714i	$-0.753311\pi$
$41$	−375.112	−1.42885	−0.714423	+	0.699714i	$0.753311\pi$
$42$	255.513	0.938726
$43$	−400.245	−1.41946	−0.709731	−	0.704473i	$-0.751183\pi$
$43$	−400.245	−1.41946	−0.709731	+	0.704473i	$0.751183\pi$
$44$	−241.527	−0.827535
$45$	0	0
$46$	23.8656	0.0764953
$47$	−456.367	−1.41634	−0.708170	−	0.706042i	$-0.750479\pi$
$47$	−456.367	−1.41634	−0.708170	+	0.706042i	$0.750479\pi$
$48$	675.975	2.03268
$49$	−96.8781	−0.282444
$50$	0	0
$51$	−353.779	−0.971352
$52$	1513.87	4.03722
$53$	124.140	0.321733	0.160867	−	0.986976i	$-0.448571\pi$
$53$	124.140	0.321733	0.160867	+	0.986976i	$0.448571\pi$
$54$	146.582	0.369394
$55$	0	0
$56$	1147.55	2.73836
$57$	399.393	0.928086
$58$	96.7780	0.219096
$59$	−275.226	−0.607311	−0.303655	−	0.952782i	$-0.598207\pi$
$59$	−275.226	−0.607311	−0.303655	+	0.952782i	$0.598207\pi$
$60$	0	0
$61$	499.511	1.04846	0.524228	−	0.851578i	$-0.324354\pi$
$61$	499.511	1.04846	0.524228	+	0.851578i	$0.324354\pi$
$62$	−972.802	−1.99268
$63$	141.194	0.282362
$64$	1661.62	3.24535
$65$	0	0
$66$	−183.189	−0.341651
$67$	635.801	1.15933	0.579667	−	0.814853i	$-0.303183\pi$
$67$	635.801	1.15933	0.579667	+	0.814853i	$0.303183\pi$
$68$	−2532.30	−4.51597
$69$	13.1879	0.0230093
$70$	0	0
$71$	812.581	1.35825	0.679124	−	0.734023i	$-0.262360\pi$
$71$	812.581	1.35825	0.679124	+	0.734023i	$0.262360\pi$
$72$	658.326	1.07756
$73$	1121.42	1.79798	0.898992	−	0.437966i	$-0.144301\pi$
$73$	1121.42	1.79798	0.898992	+	0.437966i	$0.144301\pi$
$74$	467.212	0.733949
$75$	0	0
$76$	2858.80	4.31482
$77$	−176.456	−0.261156
$78$	1148.21	1.66679
$79$	−479.934	−0.683503	−0.341752	−	0.939790i	$-0.611020\pi$
$79$	−479.934	−0.683503	−0.341752	+	0.939790i	$0.611020\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	−2036.47	−2.74256
$83$	−942.918	−1.24697	−0.623486	−	0.781834i	$-0.714284\pi$
$83$	−942.918	−1.24697	−0.623486	+	0.781834i	$0.714284\pi$
$84$	1010.65	1.31275
$85$	0	0
$86$	−2172.91	−2.72455
$87$	53.4788	0.0659026
$88$	−822.734	−0.996634
$89$	461.792	0.549999	0.274999	−	0.961444i	$-0.411322\pi$
$89$	461.792	0.549999	0.274999	+	0.961444i	$0.411322\pi$
$90$	0	0
$91$	1106.01	1.27408
$92$	94.3972	0.106974
$93$	−537.563	−0.599384
$94$	−2477.60	−2.71856
$95$	0	0
$96$	1914.30	2.03518
$97$	1277.82	1.33756	0.668780	−	0.743460i	$-0.266817\pi$
$97$	1277.82	1.33756	0.668780	+	0.743460i	$0.266817\pi$
$98$	−525.947	−0.542129
$99$	−101.229	−0.102766
$100$	0	0
$101$	−237.701	−0.234179	−0.117090	−	0.993121i	$-0.537357\pi$
$101$	−237.701	−0.234179	−0.117090	+	0.993121i	$0.537357\pi$
$102$	−1920.65	−1.86444
$103$	−165.940	−0.158743	−0.0793715	−	0.996845i	$-0.525291\pi$
$103$	−165.940	−0.158743	−0.0793715	+	0.996845i	$0.525291\pi$
$104$	5156.82	4.86219
$105$	0	0
$106$	673.948	0.617543
$107$	1078.63	0.974536	0.487268	−	0.873253i	$-0.337994\pi$
$107$	1078.63	0.974536	0.487268	+	0.873253i	$0.337994\pi$
$108$	579.786	0.516573
$109$	−756.864	−0.665086	−0.332543	−	0.943088i	$-0.607907\pi$
$109$	−756.864	−0.665086	−0.332543	+	0.943088i	$0.607907\pi$
$110$	0	0
$111$	258.178	0.220767
$112$	3534.96	2.98234
$113$	−1689.34	−1.40637	−0.703186	−	0.711006i	$-0.748240\pi$
$113$	−1689.34	−1.40637	−0.703186	+	0.711006i	$0.748240\pi$
$114$	2168.29	1.78139
$115$	0	0
$116$	382.793	0.306392
$117$	634.493	0.501358
$118$	−1494.19	−1.16569
$119$	−1850.06	−1.42517
$120$	0	0
$121$	−1204.49	−0.904952
$122$	2711.82	2.01243
$123$	−1125.34	−0.824945
$124$	−3847.80	−2.78663
$125$	0	0
$126$	766.538	0.541973
$127$	1896.28	1.32495	0.662473	−	0.749086i	$-0.269507\pi$
$127$	1896.28	1.32495	0.662473	+	0.749086i	$0.269507\pi$
$128$	3916.05	2.70417
$129$	−1200.74	−0.819527
$130$	0	0
$131$	−255.266	−0.170250	−0.0851249	−	0.996370i	$-0.527129\pi$
$131$	−255.266	−0.170250	−0.0851249	+	0.996370i	$0.527129\pi$
$132$	−724.580	−0.477777
$133$	2088.60	1.36169
$134$	3451.73	2.22526
$135$	0	0
$136$	−8625.99	−5.43877
$137$	−451.992	−0.281871	−0.140935	−	0.990019i	$-0.545011\pi$
$137$	−451.992	−0.281871	−0.140935	+	0.990019i	$0.545011\pi$
$138$	71.5967	0.0441646
$139$	−887.038	−0.541277	−0.270639	−	0.962681i	$-0.587235\pi$
$139$	−887.038	−0.541277	−0.270639	+	0.962681i	$0.587235\pi$
$140$	0	0
$141$	−1369.10	−0.817724
$142$	4411.47	2.60706
$143$	−792.949	−0.463704
$144$	2027.92	1.17357
$145$	0	0
$146$	6088.16	3.45110
$147$	−290.634	−0.163069
$148$	1848.00	1.02638
$149$	1919.96	1.05563	0.527815	−	0.849359i	$-0.323011\pi$
$149$	1919.96	1.05563	0.527815	+	0.849359i	$0.323011\pi$
$150$	0	0
$151$	566.825	0.305481	0.152740	−	0.988266i	$-0.451190\pi$
$151$	566.825	0.305481	0.152740	+	0.988266i	$0.451190\pi$
$152$	9738.17	5.19651
$153$	−1061.34	−0.560810
$154$	−957.972	−0.501270
$155$	0	0
$156$	4541.60	2.33089
$157$	−1682.76	−0.855404	−0.427702	−	0.903920i	$-0.640677\pi$
$157$	−1682.76	−0.855404	−0.427702	+	0.903920i	$0.640677\pi$
$158$	−2605.54	−1.31193
$159$	372.419	0.185753
$160$	0	0
$161$	68.9653	0.0337592
$162$	439.745	0.213269
$163$	549.241	0.263926	0.131963	−	0.991255i	$-0.457872\pi$
$163$	549.241	0.263926	0.131963	+	0.991255i	$0.457872\pi$
$164$	−8054.99	−3.83530
$165$	0	0
$166$	−5119.06	−2.39347
$167$	−3225.55	−1.49461	−0.747306	−	0.664480i	$-0.768653\pi$
$167$	−3225.55	−1.49461	−0.747306	+	0.664480i	$0.768653\pi$
$168$	3442.66	1.58100
$169$	2773.13	1.26224
$170$	0	0
$171$	1198.18	0.535831
$172$	−8594.69	−3.81011
$173$	−110.360	−0.0485003	−0.0242501	−	0.999706i	$-0.507720\pi$
$173$	−110.360	−0.0485003	−0.0242501	+	0.999706i	$0.507720\pi$
$174$	290.334	0.126495
$175$	0	0
$176$	−2534.37	−1.08543
$177$	−825.677	−0.350631
$178$	2507.05	1.05568
$179$	−3298.04	−1.37713	−0.688567	−	0.725173i	$-0.741760\pi$
$179$	−3298.04	−1.37713	−0.688567	+	0.725173i	$0.741760\pi$
$180$	0	0
$181$	−2255.79	−0.926361	−0.463181	−	0.886264i	$-0.653292\pi$
$181$	−2255.79	−0.926361	−0.463181	+	0.886264i	$0.653292\pi$
$182$	6004.48	2.44550
$183$	1498.53	0.605327
$184$	321.554	0.128833
$185$	0	0
$186$	−2918.41	−1.15047
$187$	1326.39	0.518692
$188$	−9799.82	−3.80173
$189$	423.583	0.163022
$190$	0	0
$191$	−3344.41	−1.26698	−0.633489	−	0.773751i	$-0.718378\pi$
$191$	−3344.41	−1.26698	−0.633489	+	0.773751i	$0.718378\pi$
$192$	4984.86	1.87370
$193$	1097.44	0.409304	0.204652	−	0.978835i	$-0.434394\pi$
$193$	1097.44	0.409304	0.204652	+	0.978835i	$0.434394\pi$
$194$	6937.25	2.56735
$195$	0	0
$196$	−2080.32	−0.758133
$197$	392.522	0.141960	0.0709798	−	0.997478i	$-0.477387\pi$
$197$	392.522	0.141960	0.0709798	+	0.997478i	$0.477387\pi$
$198$	−549.566	−0.197252
$199$	4007.33	1.42750	0.713750	−	0.700401i	$-0.246995\pi$
$199$	4007.33	1.42750	0.713750	+	0.700401i	$0.246995\pi$
$200$	0	0
$201$	1907.40	0.669342
$202$	−1290.47	−0.449490
$203$	279.663	0.0966922
$204$	−7596.89	−2.60730
$205$	0	0
$206$	−900.879	−0.304695
$207$	39.5638	0.0132844
$208$	15885.2	5.29539
$209$	−1497.41	−0.495588
$210$	0	0
$211$	−2003.50	−0.653682	−0.326841	−	0.945079i	$-0.605984\pi$
$211$	−2003.50	−0.653682	−0.326841	+	0.945079i	$0.605984\pi$
$212$	2665.72	0.863595
$213$	2437.74	0.784185
$214$	5855.85	1.87055
$215$	0	0
$216$	1974.98	0.622130
$217$	−2811.15	−0.879415
$218$	−4108.98	−1.27658
$219$	3364.27	1.03807
$220$	0	0
$221$	−8313.71	−2.53050
$222$	1401.63	0.423746
$223$	974.853	0.292740	0.146370	−	0.989230i	$-0.453241\pi$
$223$	974.853	0.292740	0.146370	+	0.989230i	$0.453241\pi$
$224$	10010.7	2.98602
$225$	0	0
$226$	−9171.37	−2.69943
$227$	4993.14	1.45994	0.729970	−	0.683479i	$-0.239534\pi$
$227$	4993.14	1.45994	0.729970	+	0.683479i	$0.239534\pi$
$228$	8576.39	2.49116
$229$	−1489.99	−0.429963	−0.214982	−	0.976618i	$-0.568969\pi$
$229$	−1489.99	−0.429963	−0.214982	+	0.976618i	$0.568969\pi$
$230$	0	0
$231$	−529.368	−0.150779
$232$	1303.94	0.369000
$233$	996.732	0.280249	0.140125	−	0.990134i	$-0.455250\pi$
$233$	996.732	0.280249	0.140125	+	0.990134i	$0.455250\pi$
$234$	3444.63	0.962319
$235$	0	0
$236$	−5910.07	−1.63014
$237$	−1439.80	−0.394621
$238$	−10043.9	−2.73550
$239$	−3952.97	−1.06986	−0.534930	−	0.844896i	$-0.679662\pi$
$239$	−3952.97	−1.06986	−0.534930	+	0.844896i	$0.679662\pi$
$240$	0	0
$241$	−79.7102	−0.0213053	−0.0106527	−	0.999943i	$-0.503391\pi$
$241$	−79.7102	−0.0213053	−0.0106527	+	0.999943i	$0.503391\pi$
$242$	−6539.13	−1.73699
$243$	243.000	0.0641500
$244$	10726.3	2.81426
$245$	0	0
$246$	−6109.40	−1.58342
$247$	9385.63	2.41779
$248$	−13107.1	−3.35605
$249$	−2828.75	−0.719940
$250$	0	0
$251$	2211.04	0.556015	0.278008	−	0.960579i	$-0.410326\pi$
$251$	2211.04	0.556015	0.278008	+	0.960579i	$0.410326\pi$
$252$	3031.95	0.757915
$253$	−49.4443	−0.0122867
$254$	10294.8	2.54313
$255$	0	0
$256$	7967.12	1.94510
$257$	−2567.10	−0.623079	−0.311539	−	0.950233i	$-0.600845\pi$
$257$	−2567.10	−0.623079	−0.311539	+	0.950233i	$0.600845\pi$
$258$	−6518.74	−1.57302
$259$	1350.12	0.323909
$260$	0	0
$261$	160.436	0.0380489
$262$	−1385.83	−0.326782
$263$	2879.64	0.675158	0.337579	−	0.941297i	$-0.390392\pi$
$263$	2879.64	0.675158	0.337579	+	0.941297i	$0.390392\pi$
$264$	−2468.20	−0.575407
$265$	0	0
$266$	11338.9	2.61365
$267$	1385.38	0.317542
$268$	13652.9	3.11188
$269$	1882.45	0.426673	0.213336	−	0.976979i	$-0.431567\pi$
$269$	1882.45	0.426673	0.213336	+	0.976979i	$0.431567\pi$
$270$	0	0
$271$	901.970	0.202180	0.101090	−	0.994877i	$-0.467767\pi$
$271$	901.970	0.202180	0.101090	+	0.994877i	$0.467767\pi$
$272$	−26571.7	−5.92334
$273$	3318.03	0.735591
$274$	−2453.84	−0.541030
$275$	0	0
$276$	283.192	0.0617613
$277$	−5023.27	−1.08960	−0.544800	−	0.838566i	$-0.683394\pi$
$277$	−5023.27	−1.08960	−0.544800	+	0.838566i	$0.683394\pi$
$278$	−4815.69	−1.03894
$279$	−1612.69	−0.346055
$280$	0	0
$281$	3160.01	0.670856	0.335428	−	0.942066i	$-0.391119\pi$
$281$	3160.01	0.670856	0.335428	+	0.942066i	$0.391119\pi$
$282$	−7432.79	−1.56956
$283$	−4118.93	−0.865176	−0.432588	−	0.901592i	$-0.642400\pi$
$283$	−4118.93	−0.865176	−0.432588	+	0.901592i	$0.642400\pi$
$284$	17449.0	3.64580
$285$	0	0
$286$	−4304.89	−0.890046
$287$	−5884.86	−1.21036
$288$	5742.91	1.17501
$289$	8993.61	1.83057
$290$	0	0
$291$	3833.47	0.772241
$292$	24081.0	4.82614
$293$	−5457.00	−1.08806	−0.544030	−	0.839066i	$-0.683102\pi$
$293$	−5457.00	−1.08806	−0.544030	+	0.839066i	$0.683102\pi$
$294$	−1577.84	−0.312999
$295$	0	0
$296$	6295.00	1.23611
$297$	−303.686	−0.0593322
$298$	10423.4	2.02620
$299$	309.913	0.0599422
$300$	0	0
$301$	−6279.16	−1.20241
$302$	3077.27	0.586347
$303$	−713.103	−0.135204
$304$	29997.7	5.65950
$305$	0	0
$306$	−5761.95	−1.07643
$307$	8828.49	1.64127	0.820633	−	0.571456i	$-0.193621\pi$
$307$	8828.49	1.64127	0.820633	+	0.571456i	$0.193621\pi$
$308$	−3789.14	−0.700994
$309$	−497.819	−0.0916503
$310$	0	0
$311$	415.573	0.0757717	0.0378859	−	0.999282i	$-0.487938\pi$
$311$	415.573	0.0757717	0.0378859	+	0.999282i	$0.487938\pi$
$312$	15470.5	2.80719
$313$	−6398.38	−1.15546	−0.577728	−	0.816229i	$-0.696061\pi$
$313$	−6398.38	−1.15546	−0.577728	+	0.816229i	$0.696061\pi$
$314$	−9135.60	−1.64188
$315$	0	0
$316$	−10305.9	−1.83466
$317$	3855.59	0.683128	0.341564	−	0.939859i	$-0.389043\pi$
$317$	3855.59	0.683128	0.341564	+	0.939859i	$0.389043\pi$
$318$	2021.84	0.356539
$319$	−200.503	−0.0351913
$320$	0	0
$321$	3235.90	0.562648
$322$	374.409	0.0647982
$323$	−15699.6	−2.70449
$324$	1739.36	0.298244
$325$	0	0
$326$	2981.81	0.506586
$327$	−2270.59	−0.383988
$328$	−27438.5	−4.61901
$329$	−7159.61	−1.19976
$330$	0	0
$331$	−1220.60	−0.202690	−0.101345	−	0.994851i	$-0.532315\pi$
$331$	−1220.60	−0.202690	−0.101345	+	0.994851i	$0.532315\pi$
$332$	−20247.8	−3.34712
$333$	774.533	0.127460
$334$	−17511.3	−2.86880
$335$	0	0
$336$	10604.9	1.72186
$337$	−7072.73	−1.14325	−0.571627	−	0.820514i	$-0.693687\pi$
$337$	−7072.73	−1.14325	−0.571627	+	0.820514i	$0.693687\pi$
$338$	15055.2	2.42277
$339$	−5068.03	−0.811969
$340$	0	0
$341$	2015.44	0.320065
$342$	6504.86	1.02849
$343$	−6900.93	−1.08634
$344$	−29276.9	−4.58867
$345$	0	0
$346$	−599.141	−0.0930926
$347$	6696.03	1.03591	0.517957	−	0.855407i	$-0.326693\pi$
$347$	6696.03	1.03591	0.517957	+	0.855407i	$0.326693\pi$
$348$	1148.38	0.176895
$349$	−7983.78	−1.22453	−0.612266	−	0.790652i	$-0.709742\pi$
$349$	−7983.78	−1.22453	−0.612266	+	0.790652i	$0.709742\pi$
$350$	0	0
$351$	1903.48	0.289459
$352$	−7177.12	−1.08677
$353$	−7844.48	−1.18278	−0.591388	−	0.806387i	$-0.701420\pi$
$353$	−7844.48	−1.18278	−0.591388	+	0.806387i	$0.701420\pi$
$354$	−4482.56	−0.673010
$355$	0	0
$356$	9916.32	1.47630
$357$	−5550.18	−0.822820
$358$	−17904.9	−2.64331
$359$	−5325.01	−0.782850	−0.391425	−	0.920210i	$-0.628018\pi$
$359$	−5325.01	−0.782850	−0.391425	+	0.920210i	$0.628018\pi$
$360$	0	0
$361$	10864.9	1.58403
$362$	−12246.6	−1.77808
$363$	−3613.47	−0.522474
$364$	23750.0	3.41988
$365$	0	0
$366$	8135.47	1.16188
$367$	−2845.11	−0.404670	−0.202335	−	0.979316i	$-0.564853\pi$
$367$	−2845.11	−0.404670	−0.202335	+	0.979316i	$0.564853\pi$
$368$	990.523	0.140311
$369$	−3376.01	−0.476282
$370$	0	0
$371$	1947.53	0.272536
$372$	−11543.4	−1.60886
$373$	−3843.73	−0.533567	−0.266784	−	0.963756i	$-0.585961\pi$
$373$	−3843.73	−0.533567	−0.266784	+	0.963756i	$0.585961\pi$
$374$	7200.92	0.995591
$375$	0	0
$376$	−33382.0	−4.57858
$377$	1256.74	0.171685
$378$	2299.61	0.312909
$379$	−5030.53	−0.681797	−0.340898	−	0.940100i	$-0.610731\pi$
$379$	−5030.53	−0.681797	−0.340898	+	0.940100i	$0.610731\pi$
$380$	0	0
$381$	5688.85	0.764957
$382$	−18156.6	−2.43187
$383$	−147.734	−0.0197098	−0.00985489	−	0.999951i	$-0.503137\pi$
$383$	−147.734	−0.0197098	−0.00985489	+	0.999951i	$0.503137\pi$
$384$	11748.2	1.56125
$385$	0	0
$386$	5957.97	0.785629
$387$	−3602.21	−0.473154
$388$	27439.4	3.59027
$389$	11758.0	1.53254	0.766268	−	0.642522i	$-0.222112\pi$
$389$	11758.0	1.53254	0.766268	+	0.642522i	$0.222112\pi$
$390$	0	0
$391$	−518.401	−0.0670503
$392$	−7086.37	−0.913051
$393$	−765.799	−0.0982938
$394$	2130.99	0.272481
$395$	0	0
$396$	−2173.74	−0.275845
$397$	−1125.22	−0.142249	−0.0711246	−	0.997467i	$-0.522659\pi$
$397$	−1125.22	−0.142249	−0.0711246	+	0.997467i	$0.522659\pi$
$398$	21755.6	2.73998
$399$	6265.79	0.786170
$400$	0	0
$401$	3636.86	0.452908	0.226454	−	0.974022i	$-0.427287\pi$
$401$	3636.86	0.452908	0.226454	+	0.974022i	$0.427287\pi$
$402$	10355.2	1.28475
$403$	−12632.6	−1.56147
$404$	−5104.28	−0.628583
$405$	0	0
$406$	1518.28	0.185593
$407$	−967.963	−0.117887
$408$	−25878.0	−3.14007
$409$	3405.29	0.411689	0.205845	−	0.978585i	$-0.434006\pi$
$409$	3405.29	0.411689	0.205845	+	0.978585i	$0.434006\pi$
$410$	0	0
$411$	−1355.98	−0.162738
$412$	−3563.31	−0.426097
$413$	−4317.81	−0.514445
$414$	214.790	0.0254984
$415$	0	0
$416$	44985.6	5.30192
$417$	−2661.11	−0.312507
$418$	−8129.37	−0.951245
$419$	−5055.23	−0.589414	−0.294707	−	0.955588i	$-0.595222\pi$
$419$	−5055.23	−0.589414	−0.294707	+	0.955588i	$0.595222\pi$
$420$	0	0
$421$	8185.46	0.947589	0.473794	−	0.880636i	$-0.342884\pi$
$421$	8185.46	0.947589	0.473794	+	0.880636i	$0.342884\pi$
$422$	−10876.9	−1.25469
$423$	−4107.30	−0.472113
$424$	9080.47	1.04006
$425$	0	0
$426$	13234.4	1.50518
$427$	7836.46	0.888134
$428$	23162.1	2.61584
$429$	−2378.85	−0.267720
$430$	0	0
$431$	1721.19	0.192359	0.0961795	−	0.995364i	$-0.469338\pi$
$431$	1721.19	0.192359	0.0961795	+	0.995364i	$0.469338\pi$
$432$	6083.77	0.677559
$433$	7678.69	0.852227	0.426114	−	0.904670i	$-0.359882\pi$
$433$	7678.69	0.852227	0.426114	+	0.904670i	$0.359882\pi$
$434$	−15261.6	−1.68797
$435$	0	0
$436$	−16252.5	−1.78522
$437$	585.241	0.0640637
$438$	18264.5	1.99249
$439$	16002.6	1.73977	0.869887	−	0.493251i	$-0.164192\pi$
$439$	16002.6	1.73977	0.869887	+	0.493251i	$0.164192\pi$
$440$	0	0
$441$	−871.903	−0.0941478
$442$	−45134.7	−4.85711
$443$	4053.97	0.434785	0.217392	−	0.976084i	$-0.430245\pi$
$443$	4053.97	0.434785	0.217392	+	0.976084i	$0.430245\pi$
$444$	5543.99	0.592581
$445$	0	0
$446$	5292.43	0.561892
$447$	5759.87	0.609468
$448$	26067.9	2.74909
$449$	2140.40	0.224971	0.112485	−	0.993653i	$-0.464119\pi$
$449$	2140.40	0.224971	0.112485	+	0.993653i	$0.464119\pi$
$450$	0	0
$451$	4219.13	0.440512
$452$	−36276.2	−3.77497
$453$	1700.47	0.176369
$454$	27107.5	2.80225
$455$	0	0
$456$	29214.5	3.00021
$457$	−11149.0	−1.14120	−0.570600	−	0.821228i	$-0.693289\pi$
$457$	−11149.0	−1.14120	−0.570600	+	0.821228i	$0.693289\pi$
$458$	−8089.11	−0.825282
$459$	−3184.01	−0.323784
$460$	0	0
$461$	14564.3	1.47142	0.735711	−	0.677296i	$-0.236848\pi$
$461$	14564.3	1.47142	0.735711	+	0.677296i	$0.236848\pi$
$462$	−2873.91	−0.289408
$463$	−10825.0	−1.08657	−0.543283	−	0.839549i	$-0.682819\pi$
$463$	−10825.0	−1.08657	−0.543283	+	0.839549i	$0.682819\pi$
$464$	4016.70	0.401877
$465$	0	0
$466$	5411.21	0.537918
$467$	−541.915	−0.0536977	−0.0268489	−	0.999640i	$-0.508547\pi$
$467$	−541.915	−0.0536977	−0.0268489	+	0.999640i	$0.508547\pi$
$468$	13624.8	1.34574
$469$	9974.61	0.982057
$470$	0	0
$471$	−5048.27	−0.493868
$472$	−20132.0	−1.96324
$473$	4501.82	0.437619
$474$	−7816.61	−0.757445
$475$	0	0
$476$	−39727.4	−3.82542
$477$	1117.26	0.107244
$478$	−21460.5	−2.05352
$479$	−9361.52	−0.892983	−0.446491	−	0.894788i	$-0.647327\pi$
$479$	−9361.52	−0.892983	−0.446491	+	0.894788i	$0.647327\pi$
$480$	0	0
$481$	6067.10	0.575127
$482$	−432.743	−0.0408940
$483$	206.896	0.0194909
$484$	−25864.7	−2.42907
$485$	0	0
$486$	1319.24	0.123131
$487$	−2176.19	−0.202489	−0.101245	−	0.994862i	$-0.532283\pi$
$487$	−2176.19	−0.202489	−0.101245	+	0.994862i	$0.532283\pi$
$488$	36537.9	3.38933
$489$	1647.72	0.152378
$490$	0	0
$491$	12044.9	1.10708	0.553541	−	0.832822i	$-0.313276\pi$
$491$	12044.9	1.10708	0.553541	+	0.832822i	$0.313276\pi$
$492$	−24165.0	−2.21431
$493$	−2102.19	−0.192044
$494$	50954.1	4.64076
$495$	0	0
$496$	−40375.5	−3.65506
$497$	12748.0	1.15055
$498$	−15357.2	−1.38187
$499$	3682.01	0.330319	0.165160	−	0.986267i	$-0.447186\pi$
$499$	3682.01	0.330319	0.165160	+	0.986267i	$0.447186\pi$
$500$	0	0
$501$	−9676.64	−0.862915
$502$	12003.7	1.06723
$503$	4569.29	0.405039	0.202519	−	0.979278i	$-0.435087\pi$
$503$	4569.29	0.405039	0.202519	+	0.979278i	$0.435087\pi$
$504$	10328.0	0.912788
$505$	0	0
$506$	−268.431	−0.0235834
$507$	8319.40	0.728753
$508$	40720.0	3.55641
$509$	−14758.6	−1.28520	−0.642598	−	0.766203i	$-0.722143\pi$
$509$	−14758.6	−1.28520	−0.642598	+	0.766203i	$0.722143\pi$
$510$	0	0
$511$	17593.2	1.52305
$512$	11924.7	1.02930
$513$	3594.54	0.309362
$514$	−13936.7	−1.19595
$515$	0	0
$516$	−25784.1	−2.19977
$517$	5133.05	0.436656
$518$	7329.74	0.621719
$519$	−331.081	−0.0280016
$520$	0	0
$521$	773.872	0.0650748	0.0325374	−	0.999471i	$-0.489641\pi$
$521$	773.872	0.0650748	0.0325374	+	0.999471i	$0.489641\pi$
$522$	871.002	0.0730320
$523$	−2741.98	−0.229251	−0.114626	−	0.993409i	$-0.536567\pi$
$523$	−2741.98	−0.229251	−0.114626	+	0.993409i	$0.536567\pi$
$524$	−5481.48	−0.456984
$525$	0	0
$526$	15633.5	1.29591
$527$	21130.9	1.74664
$528$	−7603.12	−0.626673
$529$	−12147.7	−0.998412
$530$	0	0
$531$	−2477.03	−0.202437
$532$	44849.6	3.65503
$533$	−26445.1	−2.14909
$534$	7521.15	0.609498
$535$	0	0
$536$	46507.1	3.74776
$537$	−9894.11	−0.795089
$538$	10219.7	0.818966
$539$	1089.65	0.0870771
$540$	0	0
$541$	−6688.61	−0.531545	−0.265773	−	0.964036i	$-0.585627\pi$
$541$	−6688.61	−0.531545	−0.265773	+	0.964036i	$0.585627\pi$
$542$	4896.75	0.388069
$543$	−6767.36	−0.534835
$544$	−75248.8	−5.93064
$545$	0	0
$546$	18013.4	1.41191
$547$	−9080.97	−0.709825	−0.354912	−	0.934900i	$-0.615489\pi$
$547$	−9080.97	−0.709825	−0.354912	+	0.934900i	$0.615489\pi$
$548$	−9705.87	−0.756596
$549$	4495.60	0.349485
$550$	0	0
$551$	2373.23	0.183490
$552$	964.661	0.0743817
$553$	−7529.33	−0.578987
$554$	−27271.1	−2.09140
$555$	0	0
$556$	−19047.8	−1.45289
$557$	−3271.53	−0.248867	−0.124434	−	0.992228i	$-0.539711\pi$
$557$	−3271.53	−0.248867	−0.124434	+	0.992228i	$0.539711\pi$
$558$	−8755.22	−0.664226
$559$	−28217.0	−2.13497
$560$	0	0
$561$	3979.18	0.299467
$562$	17155.6	1.28766
$563$	−18770.8	−1.40514	−0.702569	−	0.711615i	$-0.747964\pi$
$563$	−18770.8	−1.40514	−0.702569	+	0.711615i	$0.747964\pi$
$564$	−29399.5	−2.19493
$565$	0	0
$566$	−22361.5	−1.66064
$567$	1270.75	0.0941208
$568$	59438.1	4.39079
$569$	9692.66	0.714125	0.357063	−	0.934080i	$-0.383778\pi$
$569$	9692.66	0.714125	0.357063	+	0.934080i	$0.383778\pi$
$570$	0	0
$571$	733.711	0.0537738	0.0268869	−	0.999638i	$-0.491441\pi$
$571$	733.711	0.0537738	0.0268869	+	0.999638i	$0.491441\pi$
$572$	−17027.4	−1.24467
$573$	−10033.2	−0.731491
$574$	−31948.7	−2.32319
$575$	0	0
$576$	14954.6	1.08178
$577$	23838.8	1.71997	0.859983	−	0.510322i	$-0.170474\pi$
$577$	23838.8	1.71997	0.859983	+	0.510322i	$0.170474\pi$
$578$	48825.9	3.51365
$579$	3292.33	0.236312
$580$	0	0
$581$	−14792.8	−1.05629
$582$	20811.7	1.48226
$583$	−1396.28	−0.0991901
$584$	82029.2	5.81231
$585$	0	0
$586$	−29625.8	−2.08845
$587$	−8210.64	−0.577325	−0.288662	−	0.957431i	$-0.593210\pi$
$587$	−8210.64	−0.577325	−0.288662	+	0.957431i	$0.593210\pi$
$588$	−6240.95	−0.437708
$589$	−23855.4	−1.66884
$590$	0	0
$591$	1177.57	0.0819604
$592$	19391.3	1.34624
$593$	24272.6	1.68087	0.840435	−	0.541912i	$-0.182300\pi$
$593$	24272.6	1.68087	0.840435	+	0.541912i	$0.182300\pi$
$594$	−1648.70	−0.113884
$595$	0	0
$596$	41228.3	2.83352
$597$	12022.0	0.824167
$598$	1682.50	0.115055
$599$	−19682.1	−1.34255	−0.671275	−	0.741208i	$-0.734253\pi$
$599$	−19682.1	−1.34255	−0.671275	+	0.741208i	$0.734253\pi$
$600$	0	0
$601$	−6347.72	−0.430830	−0.215415	−	0.976523i	$-0.569110\pi$
$601$	−6347.72	−0.430830	−0.215415	+	0.976523i	$0.569110\pi$
$602$	−34089.3	−2.30793
$603$	5722.20	0.386445
$604$	12171.7	0.819969
$605$	0	0
$606$	−3871.40	−0.259513
$607$	−28233.8	−1.88793	−0.943967	−	0.330041i	$-0.892938\pi$
$607$	−28233.8	−1.88793	−0.943967	+	0.330041i	$0.892938\pi$
$608$	84951.0	5.66648
$609$	838.990	0.0558253
$610$	0	0
$611$	−32173.5	−2.13028
$612$	−22790.7	−1.50532
$613$	−8573.54	−0.564897	−0.282449	−	0.959282i	$-0.591147\pi$
$613$	−8573.54	−0.564897	−0.282449	+	0.959282i	$0.591147\pi$
$614$	47929.5	3.15029
$615$	0	0
$616$	−12907.3	−0.844235
$617$	28070.1	1.83154	0.915770	−	0.401702i	$-0.131581\pi$
$617$	28070.1	1.83154	0.915770	+	0.401702i	$0.131581\pi$
$618$	−2702.64	−0.175916
$619$	4929.32	0.320075	0.160037	−	0.987111i	$-0.448839\pi$
$619$	4929.32	0.320075	0.160037	+	0.987111i	$0.448839\pi$
$620$	0	0
$621$	118.691	0.00766976
$622$	2256.13	0.145438
$623$	7244.72	0.465897
$624$	47655.7	3.05730
$625$	0	0
$626$	−34736.5	−2.21781
$627$	−4492.23	−0.286128
$628$	−36134.7	−2.29607
$629$	−10148.6	−0.643327
$630$	0	0
$631$	−2096.09	−0.132241	−0.0661205	−	0.997812i	$-0.521062\pi$
$631$	−2096.09	−0.132241	−0.0661205	+	0.997812i	$0.521062\pi$
$632$	−35105.9	−2.20955
$633$	−6010.51	−0.377403
$634$	20931.8	1.31121
$635$	0	0
$636$	7997.15	0.498597
$637$	−6829.83	−0.424816
$638$	−1088.52	−0.0675471
$639$	7313.23	0.452749
$640$	0	0
$641$	22890.4	1.41048	0.705239	−	0.708970i	$-0.250840\pi$
$641$	22890.4	1.41048	0.705239	+	0.708970i	$0.250840\pi$
$642$	17567.5	1.07996
$643$	−14092.3	−0.864301	−0.432151	−	0.901801i	$-0.642245\pi$
$643$	−14092.3	−0.864301	−0.432151	+	0.901801i	$0.642245\pi$
$644$	1480.93	0.0906161
$645$	0	0
$646$	−85232.7	−5.19108
$647$	12030.2	0.730998	0.365499	−	0.930812i	$-0.380898\pi$
$647$	12030.2	0.730998	0.365499	+	0.930812i	$0.380898\pi$
$648$	5924.93	0.359187
$649$	3095.64	0.187233
$650$	0	0
$651$	−8433.44	−0.507730
$652$	11794.2	0.708428
$653$	−32432.6	−1.94362	−0.971812	−	0.235757i	$-0.924243\pi$
$653$	−32432.6	−1.94362	−0.971812	+	0.235757i	$0.924243\pi$
$654$	−12326.9	−0.737036
$655$	0	0
$656$	−84522.2	−5.03054
$657$	10092.8	0.599328
$658$	−38869.2	−2.30286
$659$	17057.4	1.00829	0.504145	−	0.863619i	$-0.331808\pi$
$659$	17057.4	1.00829	0.504145	+	0.863619i	$0.331808\pi$
$660$	0	0
$661$	−3278.12	−0.192895	−0.0964477	−	0.995338i	$-0.530748\pi$
$661$	−3278.12	−0.192895	−0.0964477	+	0.995338i	$0.530748\pi$
$662$	−6626.59	−0.389048
$663$	−24941.1	−1.46098
$664$	−68971.9	−4.03107
$665$	0	0
$666$	4204.90	0.244650
$667$	78.3638	0.00454911
$668$	−69263.9	−4.01183
$669$	2924.56	0.169013
$670$	0	0
$671$	−5618.32	−0.323238
$672$	30032.1	1.72398
$673$	−32212.6	−1.84503	−0.922516	−	0.385959i	$-0.873870\pi$
$673$	−32212.6	−1.84503	−0.922516	+	0.385959i	$0.873870\pi$
$674$	−38397.5	−2.19439
$675$	0	0
$676$	59549.0	3.38809
$677$	−28728.9	−1.63093	−0.815465	−	0.578806i	$-0.803519\pi$
$677$	−28728.9	−1.63093	−0.815465	+	0.578806i	$0.803519\pi$
$678$	−27514.1	−1.55851
$679$	20046.8	1.13303
$680$	0	0
$681$	14979.4	0.842897
$682$	10941.7	0.614341
$683$	32284.2	1.80867	0.904334	−	0.426825i	$-0.140368\pi$
$683$	32284.2	1.80867	0.904334	+	0.426825i	$0.140368\pi$
$684$	25729.2	1.43827
$685$	0	0
$686$	−37464.8	−2.08515
$687$	−4469.98	−0.248239
$688$	−90185.3	−4.99750
$689$	8751.73	0.483911
$690$	0	0
$691$	−11416.7	−0.628527	−0.314263	−	0.949336i	$-0.601758\pi$
$691$	−11416.7	−0.628527	−0.314263	+	0.949336i	$0.601758\pi$
$692$	−2369.83	−0.130184
$693$	−1588.10	−0.0870521
$694$	36352.5	1.98836
$695$	0	0
$696$	3911.83	0.213042
$697$	44235.6	2.40394
$698$	−43343.6	−2.35040
$699$	2990.20	0.161802
$700$	0	0
$701$	23396.2	1.26057	0.630287	−	0.776363i	$-0.282937\pi$
$701$	23396.2	1.26057	0.630287	+	0.776363i	$0.282937\pi$
$702$	10333.9	0.555595
$703$	11457.1	0.614672
$704$	−18689.3	−1.00054
$705$	0	0
$706$	−42587.3	−2.27025
$707$	−3729.12	−0.198370
$708$	−17730.2	−0.941161
$709$	22720.8	1.20352	0.601762	−	0.798675i	$-0.294465\pi$
$709$	22720.8	1.20352	0.601762	+	0.798675i	$0.294465\pi$
$710$	0	0
$711$	−4319.40	−0.227834
$712$	33778.9	1.77797
$713$	−787.705	−0.0413742
$714$	−30131.7	−1.57934
$715$	0	0
$716$	−70820.6	−3.69649
$717$	−11858.9	−0.617684
$718$	−28909.2	−1.50262
$719$	24374.7	1.26429	0.632144	−	0.774851i	$-0.282175\pi$
$719$	24374.7	1.26429	0.632144	+	0.774851i	$0.282175\pi$
$720$	0	0
$721$	−2603.31	−0.134469
$722$	58984.9	3.04043
$723$	−239.130	−0.0123006
$724$	−48439.8	−2.48653
$725$	0	0
$726$	−19617.4	−1.00285
$727$	12465.6	0.635936	0.317968	−	0.948102i	$-0.397000\pi$
$727$	12465.6	0.635936	0.317968	+	0.948102i	$0.397000\pi$
$728$	80901.7	4.11870
$729$	729.000	0.0370370
$730$	0	0
$731$	47199.5	2.38815
$732$	32178.8	1.62481
$733$	22491.5	1.13335	0.566673	−	0.823943i	$-0.308230\pi$
$733$	22491.5	1.13335	0.566673	+	0.823943i	$0.308230\pi$
$734$	−15446.0	−0.776733
$735$	0	0
$736$	2805.07	0.140484
$737$	−7151.25	−0.357422
$738$	−18328.2	−0.914188
$739$	27917.5	1.38966	0.694832	−	0.719172i	$-0.255479\pi$
$739$	27917.5	1.38966	0.694832	+	0.719172i	$0.255479\pi$
$740$	0	0
$741$	28156.9	1.39591
$742$	10573.1	0.523113
$743$	939.627	0.0463951	0.0231976	−	0.999731i	$-0.492615\pi$
$743$	939.627	0.0463951	0.0231976	+	0.999731i	$0.492615\pi$
$744$	−39321.3	−1.93762
$745$	0	0
$746$	−20867.4	−1.02414
$747$	−8486.26	−0.415657
$748$	28482.4	1.39227
$749$	16921.9	0.825517
$750$	0	0
$751$	31745.1	1.54247	0.771235	−	0.636550i	$-0.219639\pi$
$751$	31745.1	1.54247	0.771235	+	0.636550i	$0.219639\pi$
$752$	−102831.	−4.98651
$753$	6633.13	0.321016
$754$	6822.77	0.329537
$755$	0	0
$756$	9095.84	0.437583
$757$	−34845.4	−1.67302	−0.836510	−	0.547952i	$-0.815408\pi$
$757$	−34845.4	−1.67302	−0.836510	+	0.547952i	$0.815408\pi$
$758$	−27310.5	−1.30866
$759$	−148.333	−0.00709374
$760$	0	0
$761$	9176.18	0.437104	0.218552	−	0.975825i	$-0.429867\pi$
$761$	9176.18	0.437104	0.218552	+	0.975825i	$0.429867\pi$
$762$	30884.5	1.46828
$763$	−11873.9	−0.563386
$764$	−71816.3	−3.40082
$765$	0	0
$766$	−802.041	−0.0378315
$767$	−19403.2	−0.913440
$768$	23901.4	1.12300
$769$	−8778.48	−0.411652	−0.205826	−	0.978589i	$-0.565988\pi$
$769$	−8778.48	−0.411652	−0.205826	+	0.978589i	$0.565988\pi$
$770$	0	0
$771$	−7701.30	−0.359735
$772$	23566.0	1.09865
$773$	−19813.9	−0.921936	−0.460968	−	0.887417i	$-0.652498\pi$
$773$	−19813.9	−0.921936	−0.460968	+	0.887417i	$0.652498\pi$
$774$	−19556.2	−0.908184
$775$	0	0
$776$	93469.3	4.32391
$777$	4050.36	0.187009
$778$	63833.9	2.94159
$779$	−49939.1	−2.29686
$780$	0	0
$781$	−9139.62	−0.418747
$782$	−2814.38	−0.128698
$783$	481.309	0.0219675
$784$	−21829.1	−0.994400
$785$	0	0
$786$	−4157.49	−0.188668
$787$	−22904.2	−1.03741	−0.518707	−	0.854952i	$-0.673587\pi$
$787$	−22904.2	−1.03741	−0.518707	+	0.854952i	$0.673587\pi$
$788$	8428.85	0.381047
$789$	8638.93	0.389802
$790$	0	0
$791$	−26502.9	−1.19132
$792$	−7404.61	−0.332211
$793$	35215.1	1.57696
$794$	−6108.75	−0.273037
$795$	0	0
$796$	86051.7	3.83168
$797$	−21697.8	−0.964336	−0.482168	−	0.876079i	$-0.660151\pi$
$797$	−21697.8	−0.964336	−0.482168	+	0.876079i	$0.660151\pi$
$798$	34016.7	1.50899
$799$	53817.7	2.38289
$800$	0	0
$801$	4156.13	0.183333
$802$	19744.4	0.869323
$803$	−12613.4	−0.554317
$804$	40958.7	1.79664
$805$	0	0
$806$	−68581.7	−2.99713
$807$	5647.35	0.246340
$808$	−17387.2	−0.757028
$809$	10414.1	0.452585	0.226292	−	0.974059i	$-0.427340\pi$
$809$	10414.1	0.452585	0.226292	+	0.974059i	$0.427340\pi$
$810$	0	0
$811$	4970.93	0.215232	0.107616	−	0.994193i	$-0.465678\pi$
$811$	4970.93	0.215232	0.107616	+	0.994193i	$0.465678\pi$
$812$	6005.36	0.259541
$813$	2705.91	0.116729
$814$	−5255.03	−0.226276
$815$	0	0
$816$	−79715.2	−3.41984
$817$	−53285.1	−2.28177
$818$	18487.2	0.790207
$819$	9954.09	0.424694
$820$	0	0
$821$	28799.2	1.22424	0.612119	−	0.790766i	$-0.290317\pi$
$821$	28799.2	1.22424	0.612119	+	0.790766i	$0.290317\pi$
$822$	−7361.53	−0.312364
$823$	627.388	0.0265728	0.0132864	−	0.999912i	$-0.495771\pi$
$823$	627.388	0.0265728	0.0132864	+	0.999912i	$0.495771\pi$
$824$	−12138.0	−0.513166
$825$	0	0
$826$	−23441.2	−0.987439
$827$	11942.7	0.502164	0.251082	−	0.967966i	$-0.419214\pi$
$827$	11942.7	0.502164	0.251082	+	0.967966i	$0.419214\pi$
$828$	849.575	0.0356579
$829$	−19617.2	−0.821872	−0.410936	−	0.911664i	$-0.634798\pi$
$829$	−19617.2	−0.821872	−0.410936	+	0.911664i	$0.634798\pi$
$830$	0	0
$831$	−15069.8	−0.629080
$832$	117143.	4.88125
$833$	11424.5	0.475192
$834$	−14447.1	−0.599833
$835$	0	0
$836$	−32154.7	−1.33026
$837$	−4838.07	−0.199795
$838$	−27444.6	−1.13134
$839$	35418.8	1.45744	0.728720	−	0.684812i	$-0.240116\pi$
$839$	35418.8	1.45744	0.728720	+	0.684812i	$0.240116\pi$
$840$	0	0
$841$	−24071.2	−0.986971
$842$	44438.5	1.81883
$843$	9480.04	0.387319
$844$	−43022.3	−1.75461
$845$	0	0
$846$	−22298.4	−0.906186
$847$	−18896.4	−0.766573
$848$	27971.7	1.13273
$849$	−12356.8	−0.499510
$850$	0	0
$851$	378.314	0.0152391
$852$	52347.0	2.10490
$853$	−20327.4	−0.815941	−0.407971	−	0.912995i	$-0.633763\pi$
$853$	−20327.4	−0.815941	−0.407971	+	0.912995i	$0.633763\pi$
$854$	42543.8	1.70471
$855$	0	0
$856$	78899.0	3.15037
$857$	24549.3	0.978516	0.489258	−	0.872139i	$-0.337268\pi$
$857$	24549.3	0.978516	0.489258	+	0.872139i	$0.337268\pi$
$858$	−12914.7	−0.513868
$859$	−41870.0	−1.66308	−0.831540	−	0.555465i	$-0.812540\pi$
$859$	−41870.0	−1.66308	−0.831540	+	0.555465i	$0.812540\pi$
$860$	0	0
$861$	−17654.6	−0.698800
$862$	9344.25	0.369219
$863$	−15174.4	−0.598542	−0.299271	−	0.954168i	$-0.596743\pi$
$863$	−15174.4	−0.598542	−0.299271	+	0.954168i	$0.596743\pi$
$864$	17228.7	0.678395
$865$	0	0
$866$	41687.3	1.63579
$867$	26980.8	1.05688
$868$	−60365.3	−2.36052
$869$	5398.12	0.210723
$870$	0	0
$871$	44823.4	1.74372
$872$	−55362.5	−2.15001
$873$	11500.4	0.445853
$874$	3177.24	0.122966
$875$	0	0
$876$	72242.9	2.78637
$877$	11441.0	0.440518	0.220259	−	0.975441i	$-0.429310\pi$
$877$	11441.0	0.440518	0.220259	+	0.975441i	$0.429310\pi$
$878$	86877.2	3.33937
$879$	−16371.0	−0.628192
$880$	0	0
$881$	36680.4	1.40272	0.701359	−	0.712808i	$-0.252577\pi$
$881$	36680.4	1.40272	0.701359	+	0.712808i	$0.252577\pi$
$882$	−4733.52	−0.180710
$883$	21092.1	0.803855	0.401928	−	0.915671i	$-0.368340\pi$
$883$	21092.1	0.803855	0.401928	+	0.915671i	$0.368340\pi$
$884$	−178525.	−6.79235
$885$	0	0
$886$	22008.8	0.834537
$887$	38009.0	1.43880	0.719401	−	0.694595i	$-0.244416\pi$
$887$	38009.0	1.43880	0.719401	+	0.694595i	$0.244416\pi$
$888$	18885.0	0.713670
$889$	29749.4	1.12234
$890$	0	0
$891$	−911.059	−0.0342555
$892$	20933.6	0.785771
$893$	−60756.6	−2.27675
$894$	31270.1	1.16983
$895$	0	0
$896$	61436.1	2.29067
$897$	929.738	0.0346076
$898$	11620.2	0.431815
$899$	−3194.25	−0.118503
$900$	0	0
$901$	−14639.3	−0.541294
$902$	22905.5	0.845530
$903$	−18837.5	−0.694210
$904$	−123571.	−4.54636
$905$	0	0
$906$	9231.80	0.338528
$907$	9111.57	0.333566	0.166783	−	0.985994i	$-0.446662\pi$
$907$	9111.57	0.333566	0.166783	+	0.985994i	$0.446662\pi$
$908$	107220.	3.91876
$909$	−2139.31	−0.0780598
$910$	0	0
$911$	27910.8	1.01507	0.507534	−	0.861632i	$-0.330557\pi$
$911$	27910.8	1.01507	0.507534	+	0.861632i	$0.330557\pi$
$912$	89993.2	3.26751
$913$	10605.6	0.384440
$914$	−60527.5	−2.19045
$915$	0	0
$916$	−31995.4	−1.15410
$917$	−4004.69	−0.144216
$918$	−17285.8	−0.621479
$919$	16689.7	0.599068	0.299534	−	0.954086i	$-0.403169\pi$
$919$	16689.7	0.599068	0.299534	+	0.954086i	$0.403169\pi$
$920$	0	0
$921$	26485.5	0.947585
$922$	79068.7	2.82428
$923$	57286.3	2.04291
$924$	−11367.4	−0.404719
$925$	0	0
$926$	−58768.4	−2.08558
$927$	−1493.46	−0.0529143
$928$	11375.0	0.402372
$929$	16039.5	0.566459	0.283229	−	0.959052i	$-0.408594\pi$
$929$	16039.5	0.566459	0.283229	+	0.959052i	$0.408594\pi$
$930$	0	0
$931$	−12897.5	−0.454026
$932$	21403.4	0.752244
$933$	1246.72	0.0437468
$934$	−2942.03	−0.103069
$935$	0	0
$936$	46411.4	1.62073
$937$	15543.0	0.541909	0.270955	−	0.962592i	$-0.412661\pi$
$937$	15543.0	0.541909	0.270955	+	0.962592i	$0.412661\pi$
$938$	54151.7	1.88499
$939$	−19195.1	−0.667103
$940$	0	0
$941$	44607.8	1.54535	0.772675	−	0.634802i	$-0.218918\pi$
$941$	44607.8	1.54535	0.772675	+	0.634802i	$0.218918\pi$
$942$	−27406.8	−0.947943
$943$	−1648.98	−0.0569441
$944$	−62015.2	−2.13816
$945$	0	0
$946$	24440.2	0.839977
$947$	19516.5	0.669696	0.334848	−	0.942272i	$-0.391315\pi$
$947$	19516.5	0.669696	0.334848	+	0.942272i	$0.391315\pi$
$948$	−30917.6	−1.05924
$949$	79059.5	2.70430
$950$	0	0
$951$	11566.8	0.394404
$952$	−135327.	−4.60711
$953$	−22443.0	−0.762853	−0.381426	−	0.924399i	$-0.624567\pi$
$953$	−22443.0	−0.762853	−0.381426	+	0.924399i	$0.624567\pi$
$954$	6065.53	0.205848
$955$	0	0
$956$	−84884.3	−2.87171
$957$	−601.510	−0.0203177
$958$	−50823.3	−1.71401
$959$	−7090.97	−0.238769
$960$	0	0
$961$	2317.25	0.0777835
$962$	32938.0	1.10391
$963$	9707.69	0.324845
$964$	−1711.66	−0.0571876
$965$	0	0
$966$	1123.23	0.0374112
$967$	5732.52	0.190636	0.0953181	−	0.995447i	$-0.469613\pi$
$967$	5732.52	0.190636	0.0953181	+	0.995447i	$0.469613\pi$
$968$	−88105.2	−2.92542
$969$	−47098.9	−1.56144
$970$	0	0
$971$	−32738.3	−1.08200	−0.540999	−	0.841023i	$-0.681954\pi$
$971$	−32738.3	−1.08200	−0.540999	+	0.841023i	$0.681954\pi$
$972$	5218.07	0.172191
$973$	−13916.1	−0.458509
$974$	−11814.4	−0.388663
$975$	0	0
$976$	112552.	3.69130
$977$	53785.2	1.76125	0.880626	−	0.473813i	$-0.157123\pi$
$977$	53785.2	1.76125	0.880626	+	0.473813i	$0.157123\pi$
$978$	8945.42	0.292477
$979$	−5194.07	−0.169564
$980$	0	0
$981$	−6811.77	−0.221695
$982$	65391.1	2.12496
$983$	−21847.2	−0.708868	−0.354434	−	0.935081i	$-0.615326\pi$
$983$	−21847.2	−0.708868	−0.354434	+	0.935081i	$0.615326\pi$
$984$	−82315.4	−2.66679
$985$	0	0
$986$	−11412.7	−0.368614
$987$	−21478.8	−0.692683
$988$	201543.	6.48981
$989$	−1759.47	−0.0565701
$990$	0	0
$991$	−3705.24	−0.118770	−0.0593848	−	0.998235i	$-0.518914\pi$
$991$	−3705.24	−0.118770	−0.0593848	+	0.998235i	$0.518914\pi$
$992$	−114340.	−3.65957
$993$	−3661.80	−0.117023
$994$	69208.3	2.20840
$995$	0	0
$996$	−60743.4	−1.93246
$997$	−22989.7	−0.730282	−0.365141	−	0.930952i	$-0.618979\pi$
$997$	−22989.7	−0.730282	−0.365141	+	0.930952i	$0.618979\pi$
$998$	19989.5	0.634023
$999$	2323.60	0.0735890

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.4.a.l.1.23	yes	24
5.4	even	2		1875.4.a.k.1.2	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.4.a.k.1.2	✓	24	5.4	even	2
1875.4.a.l.1.23	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1875.a (trivial)

\(f(q)\)	\(=\)	\(q+5.42895 q^{2} +3.00000 q^{3} +21.4736 q^{4} +16.2869 q^{6} +15.6883 q^{7} +73.1473 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+5.42895 q^{2} +3.00000 q^{3} +21.4736 q^{4} +16.2869 q^{6} +15.6883 q^{7} +73.1473 q^{8} +9.00000 q^{9} -11.2476 q^{11} +64.4207 q^{12} +70.4992 q^{13} +85.1709 q^{14} +225.325 q^{16} -117.926 q^{17} +48.8606 q^{18} +133.131 q^{19} +47.0648 q^{21} -61.0629 q^{22} +4.39598 q^{23} +219.442 q^{24} +382.737 q^{26} +27.0000 q^{27} +336.883 q^{28} +17.8263 q^{29} -179.188 q^{31} +638.101 q^{32} -33.7429 q^{33} -640.217 q^{34} +193.262 q^{36} +86.0592 q^{37} +722.762 q^{38} +211.498 q^{39} -375.112 q^{41} +255.513 q^{42} -400.245 q^{43} -241.527 q^{44} +23.8656 q^{46} -456.367 q^{47} +675.975 q^{48} -96.8781 q^{49} -353.779 q^{51} +1513.87 q^{52} +124.140 q^{53} +146.582 q^{54} +1147.55 q^{56} +399.393 q^{57} +96.7780 q^{58} -275.226 q^{59} +499.511 q^{61} -972.802 q^{62} +141.194 q^{63} +1661.62 q^{64} -183.189 q^{66} +635.801 q^{67} -2532.30 q^{68} +13.1879 q^{69} +812.581 q^{71} +658.326 q^{72} +1121.42 q^{73} +467.212 q^{74} +2858.80 q^{76} -176.456 q^{77} +1148.21 q^{78} -479.934 q^{79} +81.0000 q^{81} -2036.47 q^{82} -942.918 q^{83} +1010.65 q^{84} -2172.91 q^{86} +53.4788 q^{87} -822.734 q^{88} +461.792 q^{89} +1106.01 q^{91} +94.3972 q^{92} -537.563 q^{93} -2477.60 q^{94} +1914.30 q^{96} +1277.82 q^{97} -525.947 q^{98} -101.229 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 72 q^{3} + 133 q^{4} + 3 q^{6} + 62 q^{7} + 27 q^{8} + 216 q^{9}+O(q^{10}) \) 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

L-functions

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