LMFDB - Embedded newform 1148.4.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,4,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1148.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1148.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:	$67.7341926866$
Analytic rank:	$1$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 247 x^{13} - 6 x^{12} + 23870 x^{11} + 940 x^{10} - 1147074 x^{9} - 8966 x^{8} + \cdots + 1720288256 $ x^15 - 247x^13 - 6x^12 + 23870x^11 + 940x^10 - 1147074x^9 - 8966x^8 + 28798759x^7 - 3139244x^6 - 363586916x^5 + 114207044x^4 + 2060613491x^3 - 892545712x^2 - 4098298668*x + 1720288256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.72571$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-7.72571 q^{3} +13.6642 q^{5} -7.00000 q^{7} +32.6866 q^{9} +O(q^{10})$	$q-7.72571 q^{3} +13.6642 q^{5} -7.00000 q^{7} +32.6866 q^{9} -23.9071 q^{11} +58.0340 q^{13} -105.566 q^{15} +25.7656 q^{17} -140.516 q^{19} +54.0800 q^{21} +31.0246 q^{23} +61.7100 q^{25} -43.9329 q^{27} -81.6885 q^{29} -55.8686 q^{31} +184.699 q^{33} -95.6493 q^{35} -5.44062 q^{37} -448.354 q^{39} -41.0000 q^{41} +376.067 q^{43} +446.636 q^{45} -165.330 q^{47} +49.0000 q^{49} -199.058 q^{51} +372.351 q^{53} -326.671 q^{55} +1085.59 q^{57} -327.285 q^{59} -239.240 q^{61} -228.806 q^{63} +792.988 q^{65} +145.876 q^{67} -239.687 q^{69} +962.911 q^{71} +761.370 q^{73} -476.754 q^{75} +167.349 q^{77} -708.752 q^{79} -543.125 q^{81} -384.847 q^{83} +352.067 q^{85} +631.102 q^{87} -58.8876 q^{89} -406.238 q^{91} +431.625 q^{93} -1920.04 q^{95} +66.9217 q^{97} -781.440 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	$ 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) $ 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * 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 $

$2$	0	0
$2$	0	0
$3$	−7.72571	−1.48681	−0.743407	−	0.668840i	$-0.766791\pi$
$3$	−7.72571	−1.48681	−0.743407	+	0.668840i	$0.766791\pi$
$4$	0	0
$5$	13.6642	1.22216	0.611081	−	0.791568i	$-0.290735\pi$
$5$	13.6642	1.22216	0.611081	+	0.791568i	$0.290735\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	32.6866	1.21061
$10$	0	0
$11$	−23.9071	−0.655295	−0.327648	−	0.944800i	$-0.606256\pi$
$11$	−23.9071	−0.655295	−0.327648	+	0.944800i	$0.606256\pi$
$12$	0	0
$13$	58.0340	1.23813	0.619067	−	0.785338i	$-0.287511\pi$
$13$	58.0340	1.23813	0.619067	+	0.785338i	$0.287511\pi$
$14$	0	0
$15$	−105.566	−1.81713
$16$	0	0
$17$	25.7656	0.367593	0.183797	−	0.982964i	$-0.441161\pi$
$17$	25.7656	0.367593	0.183797	+	0.982964i	$0.441161\pi$
$18$	0	0
$19$	−140.516	−1.69667	−0.848333	−	0.529463i	$-0.822393\pi$
$19$	−140.516	−1.69667	−0.848333	+	0.529463i	$0.822393\pi$
$20$	0	0
$21$	54.0800	0.561963
$22$	0	0
$23$	31.0246	0.281265	0.140632	−	0.990062i	$-0.455086\pi$
$23$	31.0246	0.281265	0.140632	+	0.990062i	$0.455086\pi$
$24$	0	0
$25$	61.7100	0.493680
$26$	0	0
$27$	−43.9329	−0.313144
$28$	0	0
$29$	−81.6885	−0.523075	−0.261538	−	0.965193i	$-0.584230\pi$
$29$	−81.6885	−0.523075	−0.261538	+	0.965193i	$0.584230\pi$
$30$	0	0
$31$	−55.8686	−0.323687	−0.161844	−	0.986816i	$-0.551744\pi$
$31$	−55.8686	−0.323687	−0.161844	+	0.986816i	$0.551744\pi$
$32$	0	0
$33$	184.699	0.974302
$34$	0	0
$35$	−95.6493	−0.461934
$36$	0	0
$37$	−5.44062	−0.0241738	−0.0120869	−	0.999927i	$-0.503847\pi$
$37$	−5.44062	−0.0241738	−0.0120869	+	0.999927i	$0.503847\pi$
$38$	0	0
$39$	−448.354	−1.84087
$40$	0	0
$41$	−41.0000	−0.156174
$41$	−41.0000	−0.156174
$42$	0	0
$43$	376.067	1.33371	0.666857	−	0.745185i	$-0.267639\pi$
$43$	376.067	1.33371	0.666857	+	0.745185i	$0.267639\pi$
$44$	0	0
$45$	446.636	1.47957
$46$	0	0
$47$	−165.330	−0.513103	−0.256551	−	0.966531i	$-0.582586\pi$
$47$	−165.330	−0.513103	−0.256551	+	0.966531i	$0.582586\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−199.058	−0.546543
$52$	0	0
$53$	372.351	0.965027	0.482513	−	0.875889i	$-0.339724\pi$
$53$	372.351	0.965027	0.482513	+	0.875889i	$0.339724\pi$
$54$	0	0
$55$	−326.671	−0.800877
$56$	0	0
$57$	1085.59	2.52263
$58$	0	0
$59$	−327.285	−0.722185	−0.361093	−	0.932530i	$-0.617596\pi$
$59$	−327.285	−0.722185	−0.361093	+	0.932530i	$0.617596\pi$
$60$	0	0
$61$	−239.240	−0.502156	−0.251078	−	0.967967i	$-0.580785\pi$
$61$	−239.240	−0.502156	−0.251078	+	0.967967i	$0.580785\pi$
$62$	0	0
$63$	−228.806	−0.457569
$64$	0	0
$65$	792.988	1.51320
$66$	0	0
$67$	145.876	0.265993	0.132997	−	0.991116i	$-0.457540\pi$
$67$	145.876	0.265993	0.132997	+	0.991116i	$0.457540\pi$
$68$	0	0
$69$	−239.687	−0.418188
$70$	0	0
$71$	962.911	1.60953	0.804764	−	0.593594i	$-0.202292\pi$
$71$	962.911	1.60953	0.804764	+	0.593594i	$0.202292\pi$
$72$	0	0
$73$	761.370	1.22071	0.610354	−	0.792129i	$-0.291027\pi$
$73$	761.370	1.22071	0.610354	+	0.792129i	$0.291027\pi$
$74$	0	0
$75$	−476.754	−0.734010
$76$	0	0
$77$	167.349	0.247678
$78$	0	0
$79$	−708.752	−1.00938	−0.504688	−	0.863302i	$-0.668393\pi$
$79$	−708.752	−1.00938	−0.504688	+	0.863302i	$0.668393\pi$
$80$	0	0
$81$	−543.125	−0.745028
$82$	0	0
$83$	−384.847	−0.508945	−0.254472	−	0.967080i	$-0.581902\pi$
$83$	−384.847	−0.508945	−0.254472	+	0.967080i	$0.581902\pi$
$84$	0	0
$85$	352.067	0.449259
$86$	0	0
$87$	631.102	0.777715
$88$	0	0
$89$	−58.8876	−0.0701357	−0.0350678	−	0.999385i	$-0.511165\pi$
$89$	−58.8876	−0.0701357	−0.0350678	+	0.999385i	$0.511165\pi$
$90$	0	0
$91$	−406.238	−0.467971
$92$	0	0
$93$	431.625	0.481262
$94$	0	0
$95$	−1920.04	−2.07360
$96$	0	0
$97$	66.9217	0.0700502	0.0350251	−	0.999386i	$-0.488849\pi$
$97$	66.9217	0.0700502	0.0350251	+	0.999386i	$0.488849\pi$
$98$	0	0
$99$	−781.440	−0.793310
$100$	0	0
$101$	−128.543	−0.126639	−0.0633195	−	0.997993i	$-0.520169\pi$
$101$	−128.543	−0.126639	−0.0633195	+	0.997993i	$0.520169\pi$
$102$	0	0
$103$	−641.141	−0.613335	−0.306667	−	0.951817i	$-0.599214\pi$
$103$	−641.141	−0.613335	−0.306667	+	0.951817i	$0.599214\pi$
$104$	0	0
$105$	738.959	0.686809
$106$	0	0
$107$	−538.749	−0.486755	−0.243378	−	0.969932i	$-0.578255\pi$
$107$	−538.749	−0.486755	−0.243378	+	0.969932i	$0.578255\pi$
$108$	0	0
$109$	−624.899	−0.549124	−0.274562	−	0.961569i	$-0.588533\pi$
$109$	−624.899	−0.549124	−0.274562	+	0.961569i	$0.588533\pi$
$110$	0	0
$111$	42.0326	0.0359420
$112$	0	0
$113$	−1009.75	−0.840616	−0.420308	−	0.907382i	$-0.638078\pi$
$113$	−1009.75	−0.840616	−0.420308	+	0.907382i	$0.638078\pi$
$114$	0	0
$115$	423.927	0.343751
$116$	0	0
$117$	1896.93	1.49890
$118$	0	0
$119$	−180.359	−0.138937
$120$	0	0
$121$	−759.452	−0.570588
$122$	0	0
$123$	316.754	0.232201
$124$	0	0
$125$	−864.806	−0.618805
$126$	0	0
$127$	896.212	0.626189	0.313094	−	0.949722i	$-0.398634\pi$
$127$	896.212	0.626189	0.313094	+	0.949722i	$0.398634\pi$
$128$	0	0
$129$	−2905.39	−1.98298
$130$	0	0
$131$	941.175	0.627716	0.313858	−	0.949470i	$-0.398378\pi$
$131$	941.175	0.627716	0.313858	+	0.949470i	$0.398378\pi$
$132$	0	0
$133$	983.614	0.641280
$134$	0	0
$135$	−600.307	−0.382712
$136$	0	0
$137$	−1687.08	−1.05210	−0.526049	−	0.850454i	$-0.676327\pi$
$137$	−1687.08	−1.05210	−0.526049	+	0.850454i	$0.676327\pi$
$138$	0	0
$139$	1148.08	0.700566	0.350283	−	0.936644i	$-0.386085\pi$
$139$	1148.08	0.700566	0.350283	+	0.936644i	$0.386085\pi$
$140$	0	0
$141$	1277.29	0.762888
$142$	0	0
$143$	−1387.42	−0.811344
$144$	0	0
$145$	−1116.21	−0.639283
$146$	0	0
$147$	−378.560	−0.212402
$148$	0	0
$149$	−709.796	−0.390260	−0.195130	−	0.980777i	$-0.562513\pi$
$149$	−709.796	−0.390260	−0.195130	+	0.980777i	$0.562513\pi$
$150$	0	0
$151$	−1024.95	−0.552377	−0.276189	−	0.961103i	$-0.589071\pi$
$151$	−1024.95	−0.552377	−0.276189	+	0.961103i	$0.589071\pi$
$152$	0	0
$153$	842.191	0.445014
$154$	0	0
$155$	−763.399	−0.395598
$156$	0	0
$157$	−2641.92	−1.34298	−0.671491	−	0.741013i	$-0.734346\pi$
$157$	−2641.92	−1.34298	−0.671491	+	0.741013i	$0.734346\pi$
$158$	0	0
$159$	−2876.68	−1.43481
$160$	0	0
$161$	−217.172	−0.106308
$162$	0	0
$163$	−2163.97	−1.03985	−0.519924	−	0.854212i	$-0.674040\pi$
$163$	−2163.97	−1.03985	−0.519924	+	0.854212i	$0.674040\pi$
$164$	0	0
$165$	2523.76	1.19076
$166$	0	0
$167$	1691.25	0.783672	0.391836	−	0.920035i	$-0.371840\pi$
$167$	1691.25	0.783672	0.391836	+	0.920035i	$0.371840\pi$
$168$	0	0
$169$	1170.95	0.532977
$170$	0	0
$171$	−4593.00	−2.05401
$172$	0	0
$173$	−2453.36	−1.07818	−0.539091	−	0.842247i	$-0.681232\pi$
$173$	−2453.36	−1.07818	−0.539091	+	0.842247i	$0.681232\pi$
$174$	0	0
$175$	−431.970	−0.186594
$176$	0	0
$177$	2528.51	1.07375
$178$	0	0
$179$	−3413.65	−1.42541	−0.712704	−	0.701465i	$-0.752530\pi$
$179$	−3413.65	−1.42541	−0.712704	+	0.701465i	$0.752530\pi$
$180$	0	0
$181$	1412.67	0.580128	0.290064	−	0.957007i	$-0.406323\pi$
$181$	1412.67	0.580128	0.290064	+	0.957007i	$0.406323\pi$
$182$	0	0
$183$	1848.30	0.746612
$184$	0	0
$185$	−74.3416	−0.0295444
$186$	0	0
$187$	−615.981	−0.240882
$188$	0	0
$189$	307.530	0.118357
$190$	0	0
$191$	−1316.49	−0.498733	−0.249367	−	0.968409i	$-0.580222\pi$
$191$	−1316.49	−0.498733	−0.249367	+	0.968409i	$0.580222\pi$
$192$	0	0
$193$	−4571.06	−1.70483	−0.852414	−	0.522868i	$-0.824862\pi$
$193$	−4571.06	−1.70483	−0.852414	+	0.522868i	$0.824862\pi$
$194$	0	0
$195$	−6126.39	−2.24985
$196$	0	0
$197$	−3111.51	−1.12531	−0.562654	−	0.826693i	$-0.690220\pi$
$197$	−3111.51	−1.12531	−0.562654	+	0.826693i	$0.690220\pi$
$198$	0	0
$199$	−4115.22	−1.46593	−0.732965	−	0.680267i	$-0.761864\pi$
$199$	−4115.22	−1.46593	−0.732965	+	0.680267i	$0.761864\pi$
$200$	0	0
$201$	−1126.99	−0.395482
$202$	0	0
$203$	571.820	0.197704
$204$	0	0
$205$	−560.232	−0.190870
$206$	0	0
$207$	1014.09	0.340503
$208$	0	0
$209$	3359.33	1.11182
$210$	0	0
$211$	−1243.50	−0.405715	−0.202857	−	0.979208i	$-0.565023\pi$
$211$	−1243.50	−0.405715	−0.202857	+	0.979208i	$0.565023\pi$
$212$	0	0
$213$	−7439.17	−2.39307
$214$	0	0
$215$	5138.65	1.63002
$216$	0	0
$217$	391.080	0.122342
$218$	0	0
$219$	−5882.12	−1.81496
$220$	0	0
$221$	1495.28	0.455130
$222$	0	0
$223$	2374.97	0.713183	0.356592	−	0.934260i	$-0.383939\pi$
$223$	2374.97	0.713183	0.356592	+	0.934260i	$0.383939\pi$
$224$	0	0
$225$	2017.09	0.597656
$226$	0	0
$227$	−5714.87	−1.67097	−0.835484	−	0.549515i	$-0.814812\pi$
$227$	−5714.87	−1.67097	−0.835484	+	0.549515i	$0.814812\pi$
$228$	0	0
$229$	−1847.01	−0.532986	−0.266493	−	0.963837i	$-0.585865\pi$
$229$	−1847.01	−0.532986	−0.266493	+	0.963837i	$0.585865\pi$
$230$	0	0
$231$	−1292.89	−0.368252
$232$	0	0
$233$	−3103.75	−0.872675	−0.436338	−	0.899783i	$-0.643725\pi$
$233$	−3103.75	−0.872675	−0.436338	+	0.899783i	$0.643725\pi$
$234$	0	0
$235$	−2259.10	−0.627095
$236$	0	0
$237$	5475.61	1.50076
$238$	0	0
$239$	1324.69	0.358524	0.179262	−	0.983801i	$-0.442629\pi$
$239$	1324.69	0.358524	0.179262	+	0.983801i	$0.442629\pi$
$240$	0	0
$241$	787.741	0.210551	0.105276	−	0.994443i	$-0.466428\pi$
$241$	787.741	0.210551	0.105276	+	0.994443i	$0.466428\pi$
$242$	0	0
$243$	5382.21	1.42086
$244$	0	0
$245$	669.545	0.174595
$246$	0	0
$247$	−8154.73	−2.10070
$248$	0	0
$249$	2973.21	0.756706
$250$	0	0
$251$	5004.74	1.25855	0.629275	−	0.777183i	$-0.283352\pi$
$251$	5004.74	1.25855	0.629275	+	0.777183i	$0.283352\pi$
$252$	0	0
$253$	−741.708	−0.184311
$254$	0	0
$255$	−2719.96	−0.667964
$256$	0	0
$257$	−2035.71	−0.494101	−0.247050	−	0.969003i	$-0.579461\pi$
$257$	−2035.71	−0.494101	−0.247050	+	0.969003i	$0.579461\pi$
$258$	0	0
$259$	38.0843	0.00913685
$260$	0	0
$261$	−2670.12	−0.633242
$262$	0	0
$263$	−5294.15	−1.24126	−0.620629	−	0.784104i	$-0.713123\pi$
$263$	−5294.15	−1.24126	−0.620629	+	0.784104i	$0.713123\pi$
$264$	0	0
$265$	5087.88	1.17942
$266$	0	0
$267$	454.949	0.104279
$268$	0	0
$269$	3640.10	0.825060	0.412530	−	0.910944i	$-0.364645\pi$
$269$	3640.10	0.825060	0.412530	+	0.910944i	$0.364645\pi$
$270$	0	0
$271$	3193.76	0.715894	0.357947	−	0.933742i	$-0.383477\pi$
$271$	3193.76	0.715894	0.357947	+	0.933742i	$0.383477\pi$
$272$	0	0
$273$	3138.48	0.695785
$274$	0	0
$275$	−1475.31	−0.323506
$276$	0	0
$277$	4633.57	1.00507	0.502535	−	0.864557i	$-0.332401\pi$
$277$	4633.57	1.00507	0.502535	+	0.864557i	$0.332401\pi$
$278$	0	0
$279$	−1826.15	−0.391860
$280$	0	0
$281$	−1758.06	−0.373229	−0.186614	−	0.982433i	$-0.559751\pi$
$281$	−1758.06	−0.373229	−0.186614	+	0.982433i	$0.559751\pi$
$282$	0	0
$283$	2902.44	0.609655	0.304828	−	0.952408i	$-0.401401\pi$
$283$	2902.44	0.609655	0.304828	+	0.952408i	$0.401401\pi$
$284$	0	0
$285$	14833.7	3.08306
$286$	0	0
$287$	287.000	0.0590281
$288$	0	0
$289$	−4249.13	−0.864875
$290$	0	0
$291$	−517.018	−0.104152
$292$	0	0
$293$	−2679.13	−0.534185	−0.267092	−	0.963671i	$-0.586063\pi$
$293$	−2679.13	−0.534185	−0.267092	+	0.963671i	$0.586063\pi$
$294$	0	0
$295$	−4472.09	−0.882627
$296$	0	0
$297$	1050.31	0.205202
$298$	0	0
$299$	1800.49	0.348243
$300$	0	0
$301$	−2632.47	−0.504097
$302$	0	0
$303$	993.089	0.188289
$304$	0	0
$305$	−3269.02	−0.613716
$306$	0	0
$307$	−2864.55	−0.532535	−0.266268	−	0.963899i	$-0.585791\pi$
$307$	−2864.55	−0.532535	−0.266268	+	0.963899i	$0.585791\pi$
$308$	0	0
$309$	4953.27	0.911914
$310$	0	0
$311$	−6953.03	−1.26775	−0.633874	−	0.773436i	$-0.718536\pi$
$311$	−6953.03	−1.26775	−0.633874	+	0.773436i	$0.718536\pi$
$312$	0	0
$313$	6153.48	1.11123	0.555615	−	0.831440i	$-0.312483\pi$
$313$	6153.48	1.11123	0.555615	+	0.831440i	$0.312483\pi$
$314$	0	0
$315$	−3126.45	−0.559224
$316$	0	0
$317$	−4008.37	−0.710197	−0.355098	−	0.934829i	$-0.615553\pi$
$317$	−4008.37	−0.710197	−0.355098	+	0.934829i	$0.615553\pi$
$318$	0	0
$319$	1952.93	0.342769
$320$	0	0
$321$	4162.22	0.723714
$322$	0	0
$323$	−3620.49	−0.623683
$324$	0	0
$325$	3581.28	0.611243
$326$	0	0
$327$	4827.79	0.816445
$328$	0	0
$329$	1157.31	0.193935
$330$	0	0
$331$	368.177	0.0611385	0.0305692	−	0.999533i	$-0.490268\pi$
$331$	368.177	0.0611385	0.0305692	+	0.999533i	$0.490268\pi$
$332$	0	0
$333$	−177.835	−0.0292652
$334$	0	0
$335$	1993.27	0.325087
$336$	0	0
$337$	−442.903	−0.0715919	−0.0357959	−	0.999359i	$-0.511397\pi$
$337$	−442.903	−0.0715919	−0.0357959	+	0.999359i	$0.511397\pi$
$338$	0	0
$339$	7801.06	1.24984
$340$	0	0
$341$	1335.65	0.212111
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3275.13	−0.511093
$346$	0	0
$347$	3488.51	0.539692	0.269846	−	0.962903i	$-0.413027\pi$
$347$	3488.51	0.539692	0.269846	+	0.962903i	$0.413027\pi$
$348$	0	0
$349$	7869.72	1.20704	0.603520	−	0.797348i	$-0.293764\pi$
$349$	7869.72	1.20704	0.603520	+	0.797348i	$0.293764\pi$
$350$	0	0
$351$	−2549.60	−0.387714
$352$	0	0
$353$	2657.85	0.400746	0.200373	−	0.979720i	$-0.435785\pi$
$353$	2657.85	0.400746	0.200373	+	0.979720i	$0.435785\pi$
$354$	0	0
$355$	13157.4	1.96711
$356$	0	0
$357$	1393.40	0.206574
$358$	0	0
$359$	12413.1	1.82489	0.912446	−	0.409197i	$-0.134191\pi$
$359$	12413.1	1.82489	0.912446	+	0.409197i	$0.134191\pi$
$360$	0	0
$361$	12885.8	1.87868
$362$	0	0
$363$	5867.31	0.848358
$364$	0	0
$365$	10403.5	1.49190
$366$	0	0
$367$	5144.18	0.731673	0.365836	−	0.930679i	$-0.380783\pi$
$367$	5144.18	0.731673	0.365836	+	0.930679i	$0.380783\pi$
$368$	0	0
$369$	−1340.15	−0.189066
$370$	0	0
$371$	−2606.46	−0.364746
$372$	0	0
$373$	12609.6	1.75041	0.875204	−	0.483754i	$-0.160727\pi$
$373$	12609.6	1.75041	0.875204	+	0.483754i	$0.160727\pi$
$374$	0	0
$375$	6681.24	0.920047
$376$	0	0
$377$	−4740.71	−0.647637
$378$	0	0
$379$	−5012.99	−0.679420	−0.339710	−	0.940530i	$-0.610329\pi$
$379$	−5012.99	−0.679420	−0.339710	+	0.940530i	$0.610329\pi$
$380$	0	0
$381$	−6923.87	−0.931026
$382$	0	0
$383$	−1926.74	−0.257055	−0.128527	−	0.991706i	$-0.541025\pi$
$383$	−1926.74	−0.257055	−0.128527	+	0.991706i	$0.541025\pi$
$384$	0	0
$385$	2286.69	0.302703
$386$	0	0
$387$	12292.4	1.61461
$388$	0	0
$389$	−1882.60	−0.245377	−0.122688	−	0.992445i	$-0.539152\pi$
$389$	−1882.60	−0.245377	−0.122688	+	0.992445i	$0.539152\pi$
$390$	0	0
$391$	799.370	0.103391
$392$	0	0
$393$	−7271.24	−0.933297
$394$	0	0
$395$	−9684.51	−1.23362
$396$	0	0
$397$	181.997	0.0230080	0.0115040	−	0.999934i	$-0.496338\pi$
$397$	181.997	0.0230080	0.0115040	+	0.999934i	$0.496338\pi$
$398$	0	0
$399$	−7599.12	−0.953463
$400$	0	0
$401$	−11199.4	−1.39469	−0.697347	−	0.716734i	$-0.745636\pi$
$401$	−11199.4	−1.39469	−0.697347	+	0.716734i	$0.745636\pi$
$402$	0	0
$403$	−3242.28	−0.400768
$404$	0	0
$405$	−7421.36	−0.910545
$406$	0	0
$407$	130.069	0.0158410
$408$	0	0
$409$	11582.6	1.40030	0.700152	−	0.713993i	$-0.253115\pi$
$409$	11582.6	1.40030	0.700152	+	0.713993i	$0.253115\pi$
$410$	0	0
$411$	13033.9	1.56427
$412$	0	0
$413$	2291.00	0.272960
$414$	0	0
$415$	−5258.62	−0.622013
$416$	0	0
$417$	−8869.72	−1.04161
$418$	0	0
$419$	−929.899	−0.108421	−0.0542106	−	0.998530i	$-0.517264\pi$
$419$	−929.899	−0.108421	−0.0542106	+	0.998530i	$0.517264\pi$
$420$	0	0
$421$	−2527.80	−0.292630	−0.146315	−	0.989238i	$-0.546741\pi$
$421$	−2527.80	−0.292630	−0.146315	+	0.989238i	$0.546741\pi$
$422$	0	0
$423$	−5404.07	−0.621169
$424$	0	0
$425$	1590.00	0.181474
$426$	0	0
$427$	1674.68	0.189797
$428$	0	0
$429$	10718.8	1.20632
$430$	0	0
$431$	7852.67	0.877610	0.438805	−	0.898582i	$-0.355402\pi$
$431$	7852.67	0.877610	0.438805	+	0.898582i	$0.355402\pi$
$432$	0	0
$433$	−2762.74	−0.306625	−0.153313	−	0.988178i	$-0.548994\pi$
$433$	−2762.74	−0.306625	−0.153313	+	0.988178i	$0.548994\pi$
$434$	0	0
$435$	8623.49	0.950494
$436$	0	0
$437$	−4359.47	−0.477212
$438$	0	0
$439$	−3479.59	−0.378295	−0.189148	−	0.981949i	$-0.560572\pi$
$439$	−3479.59	−0.378295	−0.189148	+	0.981949i	$0.560572\pi$
$440$	0	0
$441$	1601.64	0.172945
$442$	0	0
$443$	−7548.54	−0.809576	−0.404788	−	0.914411i	$-0.632655\pi$
$443$	−7548.54	−0.809576	−0.404788	+	0.914411i	$0.632655\pi$
$444$	0	0
$445$	−804.652	−0.0857172
$446$	0	0
$447$	5483.67	0.580244
$448$	0	0
$449$	13223.2	1.38985	0.694924	−	0.719084i	$-0.255438\pi$
$449$	13223.2	1.38985	0.694924	+	0.719084i	$0.255438\pi$
$450$	0	0
$451$	980.189	0.102340
$452$	0	0
$453$	7918.44	0.821282
$454$	0	0
$455$	−5550.92	−0.571936
$456$	0	0
$457$	−7102.91	−0.727046	−0.363523	−	0.931585i	$-0.618426\pi$
$457$	−7102.91	−0.727046	−0.363523	+	0.931585i	$0.618426\pi$
$458$	0	0
$459$	−1131.96	−0.115110
$460$	0	0
$461$	−13585.4	−1.37252	−0.686261	−	0.727355i	$-0.740749\pi$
$461$	−13585.4	−1.37252	−0.686261	+	0.727355i	$0.740749\pi$
$462$	0	0
$463$	252.389	0.0253337	0.0126669	−	0.999920i	$-0.495968\pi$
$463$	252.389	0.0253337	0.0126669	+	0.999920i	$0.495968\pi$
$464$	0	0
$465$	5897.80	0.588181
$466$	0	0
$467$	4520.35	0.447916	0.223958	−	0.974599i	$-0.428102\pi$
$467$	4520.35	0.447916	0.223958	+	0.974599i	$0.428102\pi$
$468$	0	0
$469$	−1021.13	−0.100536
$470$	0	0
$471$	20410.7	1.99676
$472$	0	0
$473$	−8990.66	−0.873977
$474$	0	0
$475$	−8671.27	−0.837611
$476$	0	0
$477$	12170.9	1.16827
$478$	0	0
$479$	−13158.2	−1.25515	−0.627573	−	0.778558i	$-0.715952\pi$
$479$	−13158.2	−1.25515	−0.627573	+	0.778558i	$0.715952\pi$
$480$	0	0
$481$	−315.741	−0.0299305
$482$	0	0
$483$	1677.81	0.158060
$484$	0	0
$485$	914.431	0.0856128
$486$	0	0
$487$	−8469.26	−0.788046	−0.394023	−	0.919100i	$-0.628917\pi$
$487$	−8469.26	−0.788046	−0.394023	+	0.919100i	$0.628917\pi$
$488$	0	0
$489$	16718.2	1.54606
$490$	0	0
$491$	−5302.38	−0.487358	−0.243679	−	0.969856i	$-0.578354\pi$
$491$	−5302.38	−0.487358	−0.243679	+	0.969856i	$0.578354\pi$
$492$	0	0
$493$	−2104.76	−0.192279
$494$	0	0
$495$	−10677.7	−0.969553
$496$	0	0
$497$	−6740.38	−0.608345
$498$	0	0
$499$	−12991.4	−1.16548	−0.582741	−	0.812658i	$-0.698020\pi$
$499$	−12991.4	−1.16548	−0.582741	+	0.812658i	$0.698020\pi$
$500$	0	0
$501$	−13066.1	−1.16517
$502$	0	0
$503$	1800.70	0.159621	0.0798105	−	0.996810i	$-0.474568\pi$
$503$	1800.70	0.159621	0.0798105	+	0.996810i	$0.474568\pi$
$504$	0	0
$505$	−1756.44	−0.154774
$506$	0	0
$507$	−9046.41	−0.792437
$508$	0	0
$509$	8596.48	0.748590	0.374295	−	0.927310i	$-0.377885\pi$
$509$	8596.48	0.748590	0.374295	+	0.927310i	$0.377885\pi$
$510$	0	0
$511$	−5329.59	−0.461384
$512$	0	0
$513$	6173.28	0.531300
$514$	0	0
$515$	−8760.67	−0.749594
$516$	0	0
$517$	3952.55	0.336234
$518$	0	0
$519$	18953.9	1.60306
$520$	0	0
$521$	−13246.5	−1.11390	−0.556948	−	0.830547i	$-0.688028\pi$
$521$	−13246.5	−1.11390	−0.556948	+	0.830547i	$0.688028\pi$
$522$	0	0
$523$	−8189.42	−0.684700	−0.342350	−	0.939572i	$-0.611223\pi$
$523$	−8189.42	−0.684700	−0.342350	+	0.939572i	$0.611223\pi$
$524$	0	0
$525$	3337.28	0.277430
$526$	0	0
$527$	−1439.49	−0.118985
$528$	0	0
$529$	−11204.5	−0.920890
$530$	0	0
$531$	−10697.8	−0.874287
$532$	0	0
$533$	−2379.40	−0.193364
$534$	0	0
$535$	−7361.56	−0.594894
$536$	0	0
$537$	26372.9	2.11932
$538$	0	0
$539$	−1171.45	−0.0936136
$540$	0	0
$541$	−22254.8	−1.76859	−0.884297	−	0.466926i	$-0.845362\pi$
$541$	−22254.8	−1.76859	−0.884297	+	0.466926i	$0.845362\pi$
$542$	0	0
$543$	−10913.9	−0.862542
$544$	0	0
$545$	−8538.74	−0.671118
$546$	0	0
$547$	−21688.3	−1.69529	−0.847645	−	0.530565i	$-0.821980\pi$
$547$	−21688.3	−1.69529	−0.847645	+	0.530565i	$0.821980\pi$
$548$	0	0
$549$	−7819.93	−0.607917
$550$	0	0
$551$	11478.6	0.887484
$552$	0	0
$553$	4961.26	0.381509
$554$	0	0
$555$	574.342	0.0439269
$556$	0	0
$557$	−5678.29	−0.431952	−0.215976	−	0.976399i	$-0.569293\pi$
$557$	−5678.29	−0.431952	−0.215976	+	0.976399i	$0.569293\pi$
$558$	0	0
$559$	21824.7	1.65132
$560$	0	0
$561$	4758.89	0.358147
$562$	0	0
$563$	−19214.9	−1.43839	−0.719195	−	0.694808i	$-0.755489\pi$
$563$	−19214.9	−1.43839	−0.719195	+	0.694808i	$0.755489\pi$
$564$	0	0
$565$	−13797.5	−1.02737
$566$	0	0
$567$	3801.88	0.281594
$568$	0	0
$569$	−19667.1	−1.44901	−0.724507	−	0.689267i	$-0.757933\pi$
$569$	−19667.1	−1.44901	−0.724507	+	0.689267i	$0.757933\pi$
$570$	0	0
$571$	14502.9	1.06292	0.531461	−	0.847083i	$-0.321643\pi$
$571$	14502.9	1.06292	0.531461	+	0.847083i	$0.321643\pi$
$572$	0	0
$573$	10170.8	0.741523
$574$	0	0
$575$	1914.53	0.138855
$576$	0	0
$577$	6120.42	0.441588	0.220794	−	0.975320i	$-0.429135\pi$
$577$	6120.42	0.441588	0.220794	+	0.975320i	$0.429135\pi$
$578$	0	0
$579$	35314.6	2.53476
$580$	0	0
$581$	2693.93	0.192363
$582$	0	0
$583$	−8901.83	−0.632378
$584$	0	0
$585$	25920.1	1.83190
$586$	0	0
$587$	−25067.5	−1.76260	−0.881301	−	0.472556i	$-0.843331\pi$
$587$	−25067.5	−1.76260	−0.881301	+	0.472556i	$0.843331\pi$
$588$	0	0
$589$	7850.45	0.549189
$590$	0	0
$591$	24038.6	1.67312
$592$	0	0
$593$	12143.4	0.840929	0.420464	−	0.907309i	$-0.361867\pi$
$593$	12143.4	0.840929	0.420464	+	0.907309i	$0.361867\pi$
$594$	0	0
$595$	−2464.47	−0.169804
$596$	0	0
$597$	31793.0	2.17956
$598$	0	0
$599$	15972.5	1.08952	0.544758	−	0.838593i	$-0.316621\pi$
$599$	15972.5	1.08952	0.544758	+	0.838593i	$0.316621\pi$
$600$	0	0
$601$	−20775.5	−1.41007	−0.705034	−	0.709174i	$-0.749068\pi$
$601$	−20775.5	−1.41007	−0.705034	+	0.709174i	$0.749068\pi$
$602$	0	0
$603$	4768.18	0.322015
$604$	0	0
$605$	−10377.3	−0.697351
$606$	0	0
$607$	338.381	0.0226268	0.0113134	−	0.999936i	$-0.496399\pi$
$607$	338.381	0.0226268	0.0113134	+	0.999936i	$0.496399\pi$
$608$	0	0
$609$	−4417.71	−0.293949
$610$	0	0
$611$	−9594.76	−0.635290
$612$	0	0
$613$	−3481.71	−0.229404	−0.114702	−	0.993400i	$-0.536591\pi$
$613$	−3481.71	−0.229404	−0.114702	+	0.993400i	$0.536591\pi$
$614$	0	0
$615$	4328.19	0.283788
$616$	0	0
$617$	8704.67	0.567969	0.283984	−	0.958829i	$-0.408344\pi$
$617$	8704.67	0.567969	0.283984	+	0.958829i	$0.408344\pi$
$618$	0	0
$619$	3033.62	0.196981	0.0984907	−	0.995138i	$-0.468599\pi$
$619$	3033.62	0.196981	0.0984907	+	0.995138i	$0.468599\pi$
$620$	0	0
$621$	−1363.00	−0.0880763
$622$	0	0
$623$	412.213	0.0265088
$624$	0	0
$625$	−19530.6	−1.24996
$626$	0	0
$627$	−25953.2	−1.65307
$628$	0	0
$629$	−140.181	−0.00888614
$630$	0	0
$631$	−10589.9	−0.668108	−0.334054	−	0.942554i	$-0.608417\pi$
$631$	−10589.9	−0.668108	−0.334054	+	0.942554i	$0.608417\pi$
$632$	0	0
$633$	9606.88	0.603222
$634$	0	0
$635$	12246.0	0.765304
$636$	0	0
$637$	2843.67	0.176876
$638$	0	0
$639$	31474.3	1.94852
$640$	0	0
$641$	23425.1	1.44343	0.721714	−	0.692192i	$-0.243355\pi$
$641$	23425.1	1.44343	0.721714	+	0.692192i	$0.243355\pi$
$642$	0	0
$643$	27560.0	1.69030	0.845149	−	0.534531i	$-0.179512\pi$
$643$	27560.0	1.69030	0.845149	+	0.534531i	$0.179512\pi$
$644$	0	0
$645$	−39699.7	−2.42353
$646$	0	0
$647$	27693.1	1.68273	0.841366	−	0.540466i	$-0.181752\pi$
$647$	27693.1	1.68273	0.841366	+	0.540466i	$0.181752\pi$
$648$	0	0
$649$	7824.43	0.473245
$650$	0	0
$651$	−3021.37	−0.181900
$652$	0	0
$653$	15238.6	0.913221	0.456610	−	0.889667i	$-0.349063\pi$
$653$	15238.6	0.913221	0.456610	+	0.889667i	$0.349063\pi$
$654$	0	0
$655$	12860.4	0.767171
$656$	0	0
$657$	24886.6	1.47781
$658$	0	0
$659$	6006.76	0.355069	0.177534	−	0.984115i	$-0.443188\pi$
$659$	6006.76	0.355069	0.177534	+	0.984115i	$0.443188\pi$
$660$	0	0
$661$	4840.17	0.284812	0.142406	−	0.989808i	$-0.454516\pi$
$661$	4840.17	0.284812	0.142406	+	0.989808i	$0.454516\pi$
$662$	0	0
$663$	−11552.1	−0.676693
$664$	0	0
$665$	13440.3	0.783748
$666$	0	0
$667$	−2534.36	−0.147122
$668$	0	0
$669$	−18348.3	−1.06037
$670$	0	0
$671$	5719.52	0.329061
$672$	0	0
$673$	−17412.4	−0.997321	−0.498660	−	0.866797i	$-0.666174\pi$
$673$	−17412.4	−0.997321	−0.498660	+	0.866797i	$0.666174\pi$
$674$	0	0
$675$	−2711.10	−0.154593
$676$	0	0
$677$	11695.9	0.663974	0.331987	−	0.943284i	$-0.392281\pi$
$677$	11695.9	0.663974	0.331987	+	0.943284i	$0.392281\pi$
$678$	0	0
$679$	−468.452	−0.0264765
$680$	0	0
$681$	44151.5	2.48442
$682$	0	0
$683$	−6964.57	−0.390178	−0.195089	−	0.980786i	$-0.562500\pi$
$683$	−6964.57	−0.390178	−0.195089	+	0.980786i	$0.562500\pi$
$684$	0	0
$685$	−23052.6	−1.28583
$686$	0	0
$687$	14269.4	0.792450
$688$	0	0
$689$	21609.1	1.19483
$690$	0	0
$691$	−25243.1	−1.38972	−0.694858	−	0.719147i	$-0.744533\pi$
$691$	−25243.1	−1.38972	−0.694858	+	0.719147i	$0.744533\pi$
$692$	0	0
$693$	5470.08	0.299843
$694$	0	0
$695$	15687.6	0.856206
$696$	0	0
$697$	−1056.39	−0.0574084
$698$	0	0
$699$	23978.7	1.29751
$700$	0	0
$701$	16562.1	0.892359	0.446179	−	0.894944i	$-0.352784\pi$
$701$	16562.1	0.892359	0.446179	+	0.894944i	$0.352784\pi$
$702$	0	0
$703$	764.496	0.0410149
$704$	0	0
$705$	17453.1	0.932373
$706$	0	0
$707$	899.804	0.0478651
$708$	0	0
$709$	37212.8	1.97116	0.985581	−	0.169203i	$-0.0541192\pi$
$709$	37212.8	1.97116	0.985581	+	0.169203i	$0.0541192\pi$
$710$	0	0
$711$	−23166.7	−1.22197
$712$	0	0
$713$	−1733.30	−0.0910417
$714$	0	0
$715$	−18958.0	−0.991594
$716$	0	0
$717$	−10234.2	−0.533058
$718$	0	0
$719$	28054.5	1.45515	0.727576	−	0.686027i	$-0.240647\pi$
$719$	28054.5	1.45515	0.727576	+	0.686027i	$0.240647\pi$
$720$	0	0
$721$	4487.99	0.231819
$722$	0	0
$723$	−6085.86	−0.313050
$724$	0	0
$725$	−5041.00	−0.258232
$726$	0	0
$727$	33508.4	1.70943	0.854717	−	0.519094i	$-0.173730\pi$
$727$	33508.4	1.70943	0.854717	+	0.519094i	$0.173730\pi$
$728$	0	0
$729$	−26917.0	−1.36753
$730$	0	0
$731$	9689.61	0.490264
$732$	0	0
$733$	−11343.9	−0.571616	−0.285808	−	0.958287i	$-0.592262\pi$
$733$	−11343.9	−0.571616	−0.285808	+	0.958287i	$0.592262\pi$
$734$	0	0
$735$	−5172.71	−0.259590
$736$	0	0
$737$	−3487.46	−0.174304
$738$	0	0
$739$	−34535.4	−1.71909	−0.859545	−	0.511061i	$-0.829253\pi$
$739$	−34535.4	−1.71909	−0.859545	+	0.511061i	$0.829253\pi$
$740$	0	0
$741$	63001.1	3.12335
$742$	0	0
$743$	−17157.4	−0.847166	−0.423583	−	0.905857i	$-0.639228\pi$
$743$	−17157.4	−0.847166	−0.423583	+	0.905857i	$0.639228\pi$
$744$	0	0
$745$	−9698.78	−0.476961
$746$	0	0
$747$	−12579.3	−0.616136
$748$	0	0
$749$	3771.24	0.183976
$750$	0	0
$751$	−16894.0	−0.820868	−0.410434	−	0.911890i	$-0.634623\pi$
$751$	−16894.0	−0.820868	−0.410434	+	0.911890i	$0.634623\pi$
$752$	0	0
$753$	−38665.1	−1.87123
$754$	0	0
$755$	−14005.1	−0.675094
$756$	0	0
$757$	−34399.6	−1.65162	−0.825808	−	0.563951i	$-0.809281\pi$
$757$	−34399.6	−1.65162	−0.825808	+	0.563951i	$0.809281\pi$
$758$	0	0
$759$	5730.22	0.274037
$760$	0	0
$761$	30747.8	1.46466	0.732330	−	0.680950i	$-0.238433\pi$
$761$	30747.8	1.46466	0.732330	+	0.680950i	$0.238433\pi$
$762$	0	0
$763$	4374.30	0.207549
$764$	0	0
$765$	11507.9	0.543879
$766$	0	0
$767$	−18993.7	−0.894162
$768$	0	0
$769$	−28641.0	−1.34307	−0.671534	−	0.740973i	$-0.734364\pi$
$769$	−28641.0	−1.34307	−0.671534	+	0.740973i	$0.734364\pi$
$770$	0	0
$771$	15727.3	0.734635
$772$	0	0
$773$	14885.6	0.692624	0.346312	−	0.938119i	$-0.387434\pi$
$773$	14885.6	0.692624	0.346312	+	0.938119i	$0.387434\pi$
$774$	0	0
$775$	−3447.65	−0.159798
$776$	0	0
$777$	−294.228	−0.0135848
$778$	0	0
$779$	5761.17	0.264975
$780$	0	0
$781$	−23020.4	−1.05472
$782$	0	0
$783$	3588.81	0.163798
$784$	0	0
$785$	−36099.7	−1.64134
$786$	0	0
$787$	−6389.19	−0.289390	−0.144695	−	0.989476i	$-0.546220\pi$
$787$	−6389.19	−0.289390	−0.144695	+	0.989476i	$0.546220\pi$
$788$	0	0
$789$	40901.0	1.84552
$790$	0	0
$791$	7068.27	0.317723
$792$	0	0
$793$	−13884.0	−0.621737
$794$	0	0
$795$	−39307.5	−1.75358
$796$	0	0
$797$	5274.02	0.234398	0.117199	−	0.993108i	$-0.462608\pi$
$797$	5274.02	0.234398	0.117199	+	0.993108i	$0.462608\pi$
$798$	0	0
$799$	−4259.83	−0.188613
$800$	0	0
$801$	−1924.83	−0.0849072
$802$	0	0
$803$	−18202.1	−0.799924
$804$	0	0
$805$	−2967.49	−0.129926
$806$	0	0
$807$	−28122.4	−1.22671
$808$	0	0
$809$	18316.0	0.795989	0.397995	−	0.917388i	$-0.369706\pi$
$809$	18316.0	0.795989	0.397995	+	0.917388i	$0.369706\pi$
$810$	0	0
$811$	19804.8	0.857511	0.428756	−	0.903420i	$-0.358952\pi$
$811$	19804.8	0.857511	0.428756	+	0.903420i	$0.358952\pi$
$812$	0	0
$813$	−24674.1	−1.06440
$814$	0	0
$815$	−29568.9	−1.27086
$816$	0	0
$817$	−52843.6	−2.26287
$818$	0	0
$819$	−13278.5	−0.566532
$820$	0	0
$821$	27674.4	1.17642	0.588212	−	0.808707i	$-0.299832\pi$
$821$	27674.4	1.17642	0.588212	+	0.808707i	$0.299832\pi$
$822$	0	0
$823$	32978.3	1.39678	0.698392	−	0.715716i	$-0.253899\pi$
$823$	32978.3	1.39678	0.698392	+	0.715716i	$0.253899\pi$
$824$	0	0
$825$	11397.8	0.480994
$826$	0	0
$827$	−22662.6	−0.952911	−0.476456	−	0.879199i	$-0.658079\pi$
$827$	−22662.6	−0.952911	−0.476456	+	0.879199i	$0.658079\pi$
$828$	0	0
$829$	−12229.4	−0.512357	−0.256179	−	0.966629i	$-0.582464\pi$
$829$	−12229.4	−0.512357	−0.256179	+	0.966629i	$0.582464\pi$
$830$	0	0
$831$	−35797.6	−1.49435
$832$	0	0
$833$	1262.52	0.0525133
$834$	0	0
$835$	23109.6	0.957774
$836$	0	0
$837$	2454.47	0.101361
$838$	0	0
$839$	20487.8	0.843047	0.421524	−	0.906817i	$-0.361495\pi$
$839$	20487.8	0.843047	0.421524	+	0.906817i	$0.361495\pi$
$840$	0	0
$841$	−17716.0	−0.726392
$842$	0	0
$843$	13582.3	0.554921
$844$	0	0
$845$	16000.1	0.651384
$846$	0	0
$847$	5316.17	0.215662
$848$	0	0
$849$	−22423.4	−0.906443
$850$	0	0
$851$	−168.793	−0.00679924
$852$	0	0
$853$	6634.87	0.266323	0.133161	−	0.991094i	$-0.457487\pi$
$853$	6634.87	0.266323	0.133161	+	0.991094i	$0.457487\pi$
$854$	0	0
$855$	−62759.6	−2.51033
$856$	0	0
$857$	−11309.7	−0.450797	−0.225398	−	0.974267i	$-0.572368\pi$
$857$	−11309.7	−0.450797	−0.225398	+	0.974267i	$0.572368\pi$
$858$	0	0
$859$	−1547.75	−0.0614767	−0.0307383	−	0.999527i	$-0.509786\pi$
$859$	−1547.75	−0.0614767	−0.0307383	+	0.999527i	$0.509786\pi$
$860$	0	0
$861$	−2217.28	−0.0877638
$862$	0	0
$863$	26123.5	1.03042	0.515210	−	0.857064i	$-0.327714\pi$
$863$	26123.5	1.03042	0.515210	+	0.857064i	$0.327714\pi$
$864$	0	0
$865$	−33523.2	−1.31771
$866$	0	0
$867$	32827.6	1.28591
$868$	0	0
$869$	16944.2	0.661440
$870$	0	0
$871$	8465.76	0.329335
$872$	0	0
$873$	2187.44	0.0848038
$874$	0	0
$875$	6053.64	0.233886
$876$	0	0
$877$	540.715	0.0208195	0.0104097	−	0.999946i	$-0.496686\pi$
$877$	540.715	0.0208195	0.0104097	+	0.999946i	$0.496686\pi$
$878$	0	0
$879$	20698.1	0.794233
$880$	0	0
$881$	−26511.0	−1.01382	−0.506911	−	0.861998i	$-0.669213\pi$
$881$	−26511.0	−1.01382	−0.506911	+	0.861998i	$0.669213\pi$
$882$	0	0
$883$	−30223.7	−1.15188	−0.575938	−	0.817493i	$-0.695363\pi$
$883$	−30223.7	−1.15188	−0.575938	+	0.817493i	$0.695363\pi$
$884$	0	0
$885$	34550.0	1.31230
$886$	0	0
$887$	41338.7	1.56485	0.782423	−	0.622748i	$-0.213984\pi$
$887$	41338.7	1.56485	0.782423	+	0.622748i	$0.213984\pi$
$888$	0	0
$889$	−6273.48	−0.236677
$890$	0	0
$891$	12984.5	0.488213
$892$	0	0
$893$	23231.5	0.870564
$894$	0	0
$895$	−46644.7	−1.74208
$896$	0	0
$897$	−13910.0	−0.517773
$898$	0	0
$899$	4563.82	0.169313
$900$	0	0
$901$	9593.87	0.354737
$902$	0	0
$903$	20337.7	0.749498
$904$	0	0
$905$	19303.0	0.709010
$906$	0	0
$907$	−50462.0	−1.84737	−0.923684	−	0.383154i	$-0.874838\pi$
$907$	−50462.0	−1.84737	−0.923684	+	0.383154i	$0.874838\pi$
$908$	0	0
$909$	−4201.64	−0.153311
$910$	0	0
$911$	23936.3	0.870521	0.435261	−	0.900304i	$-0.356656\pi$
$911$	23936.3	0.870521	0.435261	+	0.900304i	$0.356656\pi$
$912$	0	0
$913$	9200.55	0.333509
$914$	0	0
$915$	25255.5	0.912481
$916$	0	0
$917$	−6588.22	−0.237254
$918$	0	0
$919$	−2125.68	−0.0763002	−0.0381501	−	0.999272i	$-0.512146\pi$
$919$	−2125.68	−0.0763002	−0.0381501	+	0.999272i	$0.512146\pi$
$920$	0	0
$921$	22130.7	0.791781
$922$	0	0
$923$	55881.6	1.99281
$924$	0	0
$925$	−335.741	−0.0119341
$926$	0	0
$927$	−20956.7	−0.742512
$928$	0	0
$929$	26939.3	0.951399	0.475699	−	0.879608i	$-0.342195\pi$
$929$	26939.3	0.951399	0.475699	+	0.879608i	$0.342195\pi$
$930$	0	0
$931$	−6885.30	−0.242381
$932$	0	0
$933$	53717.1	1.88491
$934$	0	0
$935$	−8416.88	−0.294397
$936$	0	0
$937$	−10211.6	−0.356028	−0.178014	−	0.984028i	$-0.556967\pi$
$937$	−10211.6	−0.356028	−0.178014	+	0.984028i	$0.556967\pi$
$938$	0	0
$939$	−47540.0	−1.65219
$940$	0	0
$941$	39092.3	1.35428	0.677138	−	0.735856i	$-0.263220\pi$
$941$	39092.3	1.35428	0.677138	+	0.735856i	$0.263220\pi$
$942$	0	0
$943$	−1272.01	−0.0439261
$944$	0	0
$945$	4202.15	0.144652
$946$	0	0
$947$	−45806.4	−1.57182	−0.785908	−	0.618344i	$-0.787804\pi$
$947$	−45806.4	−1.57182	−0.785908	+	0.618344i	$0.787804\pi$
$948$	0	0
$949$	44185.4	1.51140
$950$	0	0
$951$	30967.5	1.05593
$952$	0	0
$953$	5375.13	0.182705	0.0913524	−	0.995819i	$-0.470881\pi$
$953$	5375.13	0.182705	0.0913524	+	0.995819i	$0.470881\pi$
$954$	0	0
$955$	−17988.8	−0.609533
$956$	0	0
$957$	−15087.8	−0.509633
$958$	0	0
$959$	11809.6	0.397656
$960$	0	0
$961$	−26669.7	−0.895227
$962$	0	0
$963$	−17609.9	−0.589273
$964$	0	0
$965$	−62459.8	−2.08358
$966$	0	0
$967$	8810.36	0.292991	0.146495	−	0.989211i	$-0.453201\pi$
$967$	8810.36	0.292991	0.146495	+	0.989211i	$0.453201\pi$
$968$	0	0
$969$	27970.9	0.927300
$970$	0	0
$971$	4049.47	0.133835	0.0669174	−	0.997759i	$-0.478684\pi$
$971$	4049.47	0.133835	0.0669174	+	0.997759i	$0.478684\pi$
$972$	0	0
$973$	−8036.55	−0.264789
$974$	0	0
$975$	−27667.9	−0.908804
$976$	0	0
$977$	58636.1	1.92010	0.960048	−	0.279835i	$-0.0902796\pi$
$977$	58636.1	1.92010	0.960048	+	0.279835i	$0.0902796\pi$
$978$	0	0
$979$	1407.83	0.0459596
$980$	0	0
$981$	−20425.8	−0.664777
$982$	0	0
$983$	21237.9	0.689099	0.344550	−	0.938768i	$-0.388032\pi$
$983$	21237.9	0.689099	0.344550	+	0.938768i	$0.388032\pi$
$984$	0	0
$985$	−42516.2	−1.37531
$986$	0	0
$987$	−8941.03	−0.288345
$988$	0	0
$989$	11667.4	0.375127
$990$	0	0
$991$	18896.5	0.605718	0.302859	−	0.953035i	$-0.402059\pi$
$991$	18896.5	0.605718	0.302859	+	0.953035i	$0.402059\pi$
$992$	0	0
$993$	−2844.43	−0.0909015
$994$	0	0
$995$	−56231.1	−1.79160
$996$	0	0
$997$	−18344.4	−0.582720	−0.291360	−	0.956614i	$-0.594108\pi$
$997$	−18344.4	−0.582720	−0.291360	+	0.956614i	$0.594108\pi$
$998$	0	0
$999$	239.022	0.00756989

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.4.a.b.1.2	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.4.a.b.1.2	✓	15	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:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q-7.72571 q^{3} +13.6642 q^{5} -7.00000 q^{7} +32.6866 q^{9} +O(q^{10})\)	\(q-7.72571 q^{3} +13.6642 q^{5} -7.00000 q^{7} +32.6866 q^{9} -23.9071 q^{11} +58.0340 q^{13} -105.566 q^{15} +25.7656 q^{17} -140.516 q^{19} +54.0800 q^{21} +31.0246 q^{23} +61.7100 q^{25} -43.9329 q^{27} -81.6885 q^{29} -55.8686 q^{31} +184.699 q^{33} -95.6493 q^{35} -5.44062 q^{37} -448.354 q^{39} -41.0000 q^{41} +376.067 q^{43} +446.636 q^{45} -165.330 q^{47} +49.0000 q^{49} -199.058 q^{51} +372.351 q^{53} -326.671 q^{55} +1085.59 q^{57} -327.285 q^{59} -239.240 q^{61} -228.806 q^{63} +792.988 q^{65} +145.876 q^{67} -239.687 q^{69} +962.911 q^{71} +761.370 q^{73} -476.754 q^{75} +167.349 q^{77} -708.752 q^{79} -543.125 q^{81} -384.847 q^{83} +352.067 q^{85} +631.102 q^{87} -58.8876 q^{89} -406.238 q^{91} +431.625 q^{93} -1920.04 q^{95} +66.9217 q^{97} -781.440 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9}+O(q^{10}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9	\( 15 q + 6 q^{5} - 105 q^{7} + 89 q^{9} - 20 q^{11} - 70 q^{13} - 20 q^{15} + 160 q^{17} - 6 q^{19} - 118 q^{23} + 569 q^{25} + 18 q^{27} - 162 q^{29} - 164 q^{31} - 292 q^{33} - 42 q^{35} - 410 q^{37} - 206 q^{39} - 615 q^{41} - 1022 q^{43} + 196 q^{45} - 628 q^{47} + 735 q^{49} - 1994 q^{51} - 512 q^{53} - 1128 q^{55} - 266 q^{57} - 144 q^{59} - 256 q^{61} - 623 q^{63} - 1000 q^{65} - 2670 q^{67} + 108 q^{69} - 1048 q^{71} - 606 q^{73} - 3796 q^{75} + 140 q^{77} - 1386 q^{79} - 2541 q^{81} - 2022 q^{83} - 2848 q^{85} - 3700 q^{87} - 500 q^{89} + 490 q^{91} - 2194 q^{93} - 5230 q^{95} + 1326 q^{97} - 2732 q^{99}+O(q^{100}) \) 15 * q + 6 * q^5 - 105 * q^7 + 89 * q^9 - 20 * q^11 - 70 * q^13 - 20 * q^15 + 160 * q^17 - 6 * q^19 - 118 * q^23 + 569 * q^25 + 18 * q^27 - 162 * q^29 - 164 * q^31 - 292 * q^33 - 42 * q^35 - 410 * q^37 - 206 * q^39 - 615 * q^41 - 1022 * q^43 + 196 * q^45 - 628 * q^47 + 735 * q^49 - 1994 * q^51 - 512 * q^53 - 1128 * q^55 - 266 * q^57 - 144 * q^59 - 256 * q^61 - 623 * q^63 - 1000 * q^65 - 2670 * q^67 + 108 * q^69 - 1048 * q^71 - 606 * q^73 - 3796 * q^75 + 140 * q^77 - 1386 * q^79 - 2541 * q^81 - 2022 * q^83 - 2848 * q^85 - 3700 * q^87 - 500 * q^89 + 490 * q^91 - 2194 * q^93 - 5230 * q^95 + 1326 * q^97 - 2732 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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