LMFDB - Embedded newform 1160.4.a.g.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1160,4,Mod(1,1160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1160.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:	$68.4422156067$
Analytic rank:	$0$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 179 x^{9} + 370 x^{8} + 10353 x^{7} - 19394 x^{6} - 210392 x^{5} + 267796 x^{4} + \cdots + 567808 $ x^11 - 2x^10 - 179x^9 + 370x^8 + 10353x^7 - 19394x^6 - 210392x^5 + 267796x^4 + 1107520x^3 - 894176x^2 - 757696x + 567808
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{12}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.901365$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.901365 q^{3} -5.00000 q^{5} -4.01917 q^{7} -26.1875 q^{9} +17.6839 q^{11} +23.0707 q^{13} +4.50682 q^{15} -30.3703 q^{17} -31.8815 q^{19} +3.62274 q^{21} -133.053 q^{23} +25.0000 q^{25} +47.9414 q^{27} -29.0000 q^{29} +138.251 q^{31} -15.9396 q^{33} +20.0959 q^{35} -18.2916 q^{37} -20.7951 q^{39} -299.867 q^{41} +323.842 q^{43} +130.938 q^{45} -411.129 q^{47} -326.846 q^{49} +27.3747 q^{51} +129.190 q^{53} -88.4194 q^{55} +28.7369 q^{57} +118.350 q^{59} +142.293 q^{61} +105.252 q^{63} -115.354 q^{65} +506.102 q^{67} +119.929 q^{69} +1137.48 q^{71} +662.100 q^{73} -22.5341 q^{75} -71.0745 q^{77} -415.685 q^{79} +663.851 q^{81} -398.932 q^{83} +151.851 q^{85} +26.1396 q^{87} +342.392 q^{89} -92.7252 q^{91} -124.614 q^{93} +159.408 q^{95} +1783.40 q^{97} -463.097 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) $ 11 * q + 2 * q^3 - 55 * q^5 + 56 * q^7 + 65 * q^9 - 8 * q^11 + 16 * q^13 - 10 * q^15 + 64 * q^17 + 160 * q^19 - 48 * q^21 + 140 * q^23 + 275 * q^25 - 136 * q^27 - 319 * q^29 - 144 * q^31 - 424 * q^33 - 280 * q^35 - 50 * q^37 + 572 * q^39 + 62 * q^41 + 326 * q^43 - 325 * q^45 + 1074 * q^47 + 1513 * q^49 - 144 * q^51 + 576 * q^53 + 40 * q^55 - 896 * q^57 - 468 * q^59 + 660 * q^61 + 2552 * q^63 - 80 * q^65 + 892 * q^67 - 510 * q^69 + 920 * q^71 - 676 * q^73 + 50 * q^75 + 1708 * q^77 + 2308 * q^79 + 1287 * q^81 - 1548 * q^83 - 320 * q^85 - 58 * q^87 + 1854 * q^89 + 3100 * q^91 + 5424 * q^93 - 800 * q^95 + 4724 * q^97 + 6884 * q^99

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
$3$	−0.901365	−0.173468	−0.0867339	−	0.996232i	$-0.527643\pi$
$3$	−0.901365	−0.173468	−0.0867339	+	0.996232i	$0.527643\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−4.01917	−0.217015	−0.108507	−	0.994096i	$-0.534607\pi$
$7$	−4.01917	−0.217015	−0.108507	+	0.994096i	$0.534607\pi$
$8$	0	0
$9$	−26.1875	−0.969909
$10$	0	0
$11$	17.6839	0.484717	0.242359	−	0.970187i	$-0.422079\pi$
$11$	17.6839	0.484717	0.242359	+	0.970187i	$0.422079\pi$
$12$	0	0
$13$	23.0707	0.492205	0.246103	−	0.969244i	$-0.420850\pi$
$13$	23.0707	0.492205	0.246103	+	0.969244i	$0.420850\pi$
$14$	0	0
$15$	4.50682	0.0775771
$16$	0	0
$17$	−30.3703	−0.433287	−0.216643	−	0.976251i	$-0.569511\pi$
$17$	−30.3703	−0.433287	−0.216643	+	0.976251i	$0.569511\pi$
$18$	0	0
$19$	−31.8815	−0.384954	−0.192477	−	0.981302i	$-0.561652\pi$
$19$	−31.8815	−0.384954	−0.192477	+	0.981302i	$0.561652\pi$
$20$	0	0
$21$	3.62274	0.0376451
$22$	0	0
$23$	−133.053	−1.20623	−0.603117	−	0.797653i	$-0.706075\pi$
$23$	−133.053	−1.20623	−0.603117	+	0.797653i	$0.706075\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	47.9414	0.341716
$28$	0	0
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	138.251	0.800985	0.400493	−	0.916300i	$-0.368839\pi$
$31$	138.251	0.800985	0.400493	+	0.916300i	$0.368839\pi$
$32$	0	0
$33$	−15.9396	−0.0840828
$34$	0	0
$35$	20.0959	0.0970520
$36$	0	0
$37$	−18.2916	−0.0812736	−0.0406368	−	0.999174i	$-0.512939\pi$
$37$	−18.2916	−0.0812736	−0.0406368	+	0.999174i	$0.512939\pi$
$38$	0	0
$39$	−20.7951	−0.0853817
$40$	0	0
$41$	−299.867	−1.14223	−0.571114	−	0.820871i	$-0.693489\pi$
$41$	−299.867	−1.14223	−0.571114	+	0.820871i	$0.693489\pi$
$42$	0	0
$43$	323.842	1.14850	0.574249	−	0.818681i	$-0.305294\pi$
$43$	323.842	1.14850	0.574249	+	0.818681i	$0.305294\pi$
$44$	0	0
$45$	130.938	0.433756
$46$	0	0
$47$	−411.129	−1.27594	−0.637972	−	0.770059i	$-0.720227\pi$
$47$	−411.129	−1.27594	−0.637972	+	0.770059i	$0.720227\pi$
$48$	0	0
$49$	−326.846	−0.952905
$50$	0	0
$51$	27.3747	0.0751613
$52$	0	0
$53$	129.190	0.334822	0.167411	−	0.985887i	$-0.446459\pi$
$53$	129.190	0.334822	0.167411	+	0.985887i	$0.446459\pi$
$54$	0	0
$55$	−88.4194	−0.216772
$56$	0	0
$57$	28.7369	0.0667771
$58$	0	0
$59$	118.350	0.261149	0.130575	−	0.991438i	$-0.458318\pi$
$59$	118.350	0.261149	0.130575	+	0.991438i	$0.458318\pi$
$60$	0	0
$61$	142.293	0.298667	0.149334	−	0.988787i	$-0.452287\pi$
$61$	142.293	0.298667	0.149334	+	0.988787i	$0.452287\pi$
$62$	0	0
$63$	105.252	0.210485
$64$	0	0
$65$	−115.354	−0.220121
$66$	0	0
$67$	506.102	0.922839	0.461419	−	0.887182i	$-0.347340\pi$
$67$	506.102	0.922839	0.461419	+	0.887182i	$0.347340\pi$
$68$	0	0
$69$	119.929	0.209243
$70$	0	0
$71$	1137.48	1.90132	0.950661	−	0.310233i	$-0.100407\pi$
$71$	1137.48	1.90132	0.950661	+	0.310233i	$0.100407\pi$
$72$	0	0
$73$	662.100	1.06155	0.530773	−	0.847514i	$-0.321902\pi$
$73$	662.100	1.06155	0.530773	+	0.847514i	$0.321902\pi$
$74$	0	0
$75$	−22.5341	−0.0346936
$76$	0	0
$77$	−71.0745	−0.105191
$78$	0	0
$79$	−415.685	−0.592003	−0.296001	−	0.955187i	$-0.595653\pi$
$79$	−415.685	−0.592003	−0.296001	+	0.955187i	$0.595653\pi$
$80$	0	0
$81$	663.851	0.910632
$82$	0	0
$83$	−398.932	−0.527571	−0.263786	−	0.964581i	$-0.584971\pi$
$83$	−398.932	−0.527571	−0.263786	+	0.964581i	$0.584971\pi$
$84$	0	0
$85$	151.851	0.193772
$86$	0	0
$87$	26.1396	0.0322122
$88$	0	0
$89$	342.392	0.407792	0.203896	−	0.978993i	$-0.434640\pi$
$89$	342.392	0.407792	0.203896	+	0.978993i	$0.434640\pi$
$90$	0	0
$91$	−92.7252	−0.106816
$92$	0	0
$93$	−124.614	−0.138945
$94$	0	0
$95$	159.408	0.172157
$96$	0	0
$97$	1783.40	1.86677	0.933385	−	0.358875i	$-0.116840\pi$
$97$	1783.40	1.86677	0.933385	+	0.358875i	$0.116840\pi$
$98$	0	0
$99$	−463.097	−0.470131
$100$	0	0
$101$	959.193	0.944983	0.472492	−	0.881335i	$-0.343355\pi$
$101$	959.193	0.944983	0.472492	+	0.881335i	$0.343355\pi$
$102$	0	0
$103$	−471.929	−0.451461	−0.225731	−	0.974190i	$-0.572477\pi$
$103$	−471.929	−0.451461	−0.225731	+	0.974190i	$0.572477\pi$
$104$	0	0
$105$	−18.1137	−0.0168354
$106$	0	0
$107$	−537.002	−0.485177	−0.242588	−	0.970129i	$-0.577996\pi$
$107$	−537.002	−0.485177	−0.242588	+	0.970129i	$0.577996\pi$
$108$	0	0
$109$	1041.18	0.914926	0.457463	−	0.889229i	$-0.348758\pi$
$109$	1041.18	0.914926	0.457463	+	0.889229i	$0.348758\pi$
$110$	0	0
$111$	16.4874	0.0140983
$112$	0	0
$113$	−1025.51	−0.853737	−0.426868	−	0.904314i	$-0.640383\pi$
$113$	−1025.51	−0.853737	−0.426868	+	0.904314i	$0.640383\pi$
$114$	0	0
$115$	665.263	0.539444
$116$	0	0
$117$	−604.166	−0.477394
$118$	0	0
$119$	122.063	0.0940297
$120$	0	0
$121$	−1018.28	−0.765049
$122$	0	0
$123$	270.290	0.198140
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−895.813	−0.625910	−0.312955	−	0.949768i	$-0.601319\pi$
$127$	−895.813	−0.625910	−0.312955	+	0.949768i	$0.601319\pi$
$128$	0	0
$129$	−291.900	−0.199227
$130$	0	0
$131$	1841.66	1.22829	0.614147	−	0.789192i	$-0.289500\pi$
$131$	1841.66	1.22829	0.614147	+	0.789192i	$0.289500\pi$
$132$	0	0
$133$	128.137	0.0835407
$134$	0	0
$135$	−239.707	−0.152820
$136$	0	0
$137$	2385.85	1.48786	0.743930	−	0.668257i	$-0.232959\pi$
$137$	2385.85	1.48786	0.743930	+	0.668257i	$0.232959\pi$
$138$	0	0
$139$	−1034.75	−0.631411	−0.315706	−	0.948857i	$-0.602241\pi$
$139$	−1034.75	−0.631411	−0.315706	+	0.948857i	$0.602241\pi$
$140$	0	0
$141$	370.578	0.221335
$142$	0	0
$143$	407.980	0.238580
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	294.608	0.165298
$148$	0	0
$149$	1903.55	1.04661	0.523305	−	0.852146i	$-0.324699\pi$
$149$	1903.55	1.04661	0.523305	+	0.852146i	$0.324699\pi$
$150$	0	0
$151$	447.755	0.241310	0.120655	−	0.992695i	$-0.461501\pi$
$151$	447.755	0.241310	0.120655	+	0.992695i	$0.461501\pi$
$152$	0	0
$153$	795.323	0.420249
$154$	0	0
$155$	−691.253	−0.358211
$156$	0	0
$157$	−303.006	−0.154029	−0.0770145	−	0.997030i	$-0.524539\pi$
$157$	−303.006	−0.154029	−0.0770145	+	0.997030i	$0.524539\pi$
$158$	0	0
$159$	−116.447	−0.0580809
$160$	0	0
$161$	534.761	0.261771
$162$	0	0
$163$	2707.00	1.30079	0.650394	−	0.759597i	$-0.274604\pi$
$163$	2707.00	1.30079	0.650394	+	0.759597i	$0.274604\pi$
$164$	0	0
$165$	79.6981	0.0376030
$166$	0	0
$167$	−1409.93	−0.653314	−0.326657	−	0.945143i	$-0.605922\pi$
$167$	−1409.93	−0.653314	−0.326657	+	0.945143i	$0.605922\pi$
$168$	0	0
$169$	−1664.74	−0.757734
$170$	0	0
$171$	834.898	0.373370
$172$	0	0
$173$	3281.47	1.44211	0.721056	−	0.692876i	$-0.243657\pi$
$173$	3281.47	1.44211	0.721056	+	0.692876i	$0.243657\pi$
$174$	0	0
$175$	−100.479	−0.0434030
$176$	0	0
$177$	−106.676	−0.0453010
$178$	0	0
$179$	2700.79	1.12774	0.563872	−	0.825862i	$-0.309311\pi$
$179$	2700.79	1.12774	0.563872	+	0.825862i	$0.309311\pi$
$180$	0	0
$181$	1628.09	0.668590	0.334295	−	0.942468i	$-0.391502\pi$
$181$	1628.09	0.668590	0.334295	+	0.942468i	$0.391502\pi$
$182$	0	0
$183$	−128.257	−0.0518091
$184$	0	0
$185$	91.4581	0.0363466
$186$	0	0
$187$	−537.064	−0.210021
$188$	0	0
$189$	−192.685	−0.0741574
$190$	0	0
$191$	−4348.90	−1.64751	−0.823757	−	0.566942i	$-0.808126\pi$
$191$	−4348.90	−1.64751	−0.823757	+	0.566942i	$0.808126\pi$
$192$	0	0
$193$	3742.36	1.39576	0.697878	−	0.716216i	$-0.254128\pi$
$193$	3742.36	1.39576	0.697878	+	0.716216i	$0.254128\pi$
$194$	0	0
$195$	103.976	0.0381839
$196$	0	0
$197$	3121.88	1.12906	0.564529	−	0.825413i	$-0.309058\pi$
$197$	3121.88	1.12906	0.564529	+	0.825413i	$0.309058\pi$
$198$	0	0
$199$	1250.31	0.445387	0.222693	−	0.974889i	$-0.428515\pi$
$199$	1250.31	0.445387	0.222693	+	0.974889i	$0.428515\pi$
$200$	0	0
$201$	−456.183	−0.160083
$202$	0	0
$203$	116.556	0.0402987
$204$	0	0
$205$	1499.34	0.510820
$206$	0	0
$207$	3484.32	1.16994
$208$	0	0
$209$	−563.789	−0.186594
$210$	0	0
$211$	277.499	0.0905395	0.0452697	−	0.998975i	$-0.485585\pi$
$211$	277.499	0.0905395	0.0452697	+	0.998975i	$0.485585\pi$
$212$	0	0
$213$	−1025.28	−0.329818
$214$	0	0
$215$	−1619.21	−0.513624
$216$	0	0
$217$	−555.653	−0.173826
$218$	0	0
$219$	−596.793	−0.184144
$220$	0	0
$221$	−700.664	−0.213266
$222$	0	0
$223$	3144.67	0.944317	0.472158	−	0.881514i	$-0.343475\pi$
$223$	3144.67	0.944317	0.472158	+	0.881514i	$0.343475\pi$
$224$	0	0
$225$	−654.689	−0.193982
$226$	0	0
$227$	−4343.00	−1.26985	−0.634924	−	0.772575i	$-0.718968\pi$
$227$	−4343.00	−1.26985	−0.634924	+	0.772575i	$0.718968\pi$
$228$	0	0
$229$	−2050.39	−0.591674	−0.295837	−	0.955238i	$-0.595599\pi$
$229$	−2050.39	−0.591674	−0.295837	+	0.955238i	$0.595599\pi$
$230$	0	0
$231$	64.0641	0.0182472
$232$	0	0
$233$	4920.15	1.38339	0.691695	−	0.722190i	$-0.256864\pi$
$233$	4920.15	1.38339	0.691695	+	0.722190i	$0.256864\pi$
$234$	0	0
$235$	2055.65	0.570620
$236$	0	0
$237$	374.684	0.102693
$238$	0	0
$239$	5301.89	1.43494	0.717470	−	0.696589i	$-0.245300\pi$
$239$	5301.89	1.43494	0.717470	+	0.696589i	$0.245300\pi$
$240$	0	0
$241$	4900.86	1.30993	0.654963	−	0.755661i	$-0.272684\pi$
$241$	4900.86	1.30993	0.654963	+	0.755661i	$0.272684\pi$
$242$	0	0
$243$	−1892.79	−0.499681
$244$	0	0
$245$	1634.23	0.426152
$246$	0	0
$247$	−735.530	−0.189476
$248$	0	0
$249$	359.583	0.0915166
$250$	0	0
$251$	−7923.84	−1.99262	−0.996311	−	0.0858121i	$-0.972652\pi$
$251$	−7923.84	−1.99262	−0.996311	+	0.0858121i	$0.972652\pi$
$252$	0	0
$253$	−2352.89	−0.584682
$254$	0	0
$255$	−136.873	−0.0336131
$256$	0	0
$257$	2559.86	0.621323	0.310661	−	0.950521i	$-0.399449\pi$
$257$	2559.86	0.621323	0.310661	+	0.950521i	$0.399449\pi$
$258$	0	0
$259$	73.5171	0.0176376
$260$	0	0
$261$	759.439	0.180108
$262$	0	0
$263$	429.586	0.100720	0.0503601	−	0.998731i	$-0.483963\pi$
$263$	429.586	0.100720	0.0503601	+	0.998731i	$0.483963\pi$
$264$	0	0
$265$	−645.949	−0.149737
$266$	0	0
$267$	−308.620	−0.0707387
$268$	0	0
$269$	307.430	0.0696815	0.0348407	−	0.999393i	$-0.488908\pi$
$269$	307.430	0.0696815	0.0348407	+	0.999393i	$0.488908\pi$
$270$	0	0
$271$	1636.65	0.366861	0.183431	−	0.983033i	$-0.441280\pi$
$271$	1636.65	0.366861	0.183431	+	0.983033i	$0.441280\pi$
$272$	0	0
$273$	83.5793	0.0185291
$274$	0	0
$275$	442.097	0.0969434
$276$	0	0
$277$	1340.76	0.290825	0.145413	−	0.989371i	$-0.453549\pi$
$277$	1340.76	0.290825	0.145413	+	0.989371i	$0.453549\pi$
$278$	0	0
$279$	−3620.44	−0.776883
$280$	0	0
$281$	−1421.87	−0.301856	−0.150928	−	0.988545i	$-0.548226\pi$
$281$	−1421.87	−0.301856	−0.150928	+	0.988545i	$0.548226\pi$
$282$	0	0
$283$	−5999.19	−1.26012	−0.630061	−	0.776546i	$-0.716970\pi$
$283$	−5999.19	−1.26012	−0.630061	+	0.776546i	$0.716970\pi$
$284$	0	0
$285$	−143.684	−0.0298636
$286$	0	0
$287$	1205.22	0.247881
$288$	0	0
$289$	−3990.65	−0.812263
$290$	0	0
$291$	−1607.49	−0.323825
$292$	0	0
$293$	−2661.05	−0.530581	−0.265291	−	0.964168i	$-0.585468\pi$
$293$	−2661.05	−0.530581	−0.265291	+	0.964168i	$0.585468\pi$
$294$	0	0
$295$	−591.748	−0.116790
$296$	0	0
$297$	847.789	0.165635
$298$	0	0
$299$	−3069.62	−0.593715
$300$	0	0
$301$	−1301.58	−0.249241
$302$	0	0
$303$	−864.583	−0.163924
$304$	0	0
$305$	−711.463	−0.133568
$306$	0	0
$307$	8439.64	1.56898	0.784488	−	0.620143i	$-0.212926\pi$
$307$	8439.64	1.56898	0.784488	+	0.620143i	$0.212926\pi$
$308$	0	0
$309$	425.380	0.0783140
$310$	0	0
$311$	5648.29	1.02986	0.514928	−	0.857234i	$-0.327819\pi$
$311$	5648.29	1.02986	0.514928	+	0.857234i	$0.327819\pi$
$312$	0	0
$313$	2739.26	0.494671	0.247336	−	0.968930i	$-0.420445\pi$
$313$	2739.26	0.494671	0.247336	+	0.968930i	$0.420445\pi$
$314$	0	0
$315$	−526.261	−0.0941316
$316$	0	0
$317$	2249.25	0.398519	0.199260	−	0.979947i	$-0.436146\pi$
$317$	2249.25	0.398519	0.199260	+	0.979947i	$0.436146\pi$
$318$	0	0
$319$	−512.832	−0.0900097
$320$	0	0
$321$	484.034	0.0841625
$322$	0	0
$323$	968.250	0.166795
$324$	0	0
$325$	576.768	0.0984411
$326$	0	0
$327$	−938.483	−0.158710
$328$	0	0
$329$	1652.40	0.276899
$330$	0	0
$331$	−3222.88	−0.535183	−0.267592	−	0.963532i	$-0.586228\pi$
$331$	−3222.88	−0.535183	−0.267592	+	0.963532i	$0.586228\pi$
$332$	0	0
$333$	479.012	0.0788280
$334$	0	0
$335$	−2530.51	−0.412706
$336$	0	0
$337$	1060.25	0.171381	0.0856904	−	0.996322i	$-0.472690\pi$
$337$	1060.25	0.171381	0.0856904	+	0.996322i	$0.472690\pi$
$338$	0	0
$339$	924.362	0.148096
$340$	0	0
$341$	2444.81	0.388251
$342$	0	0
$343$	2692.23	0.423809
$344$	0	0
$345$	−599.645	−0.0935762
$346$	0	0
$347$	−8539.96	−1.32118	−0.660589	−	0.750748i	$-0.729693\pi$
$347$	−8539.96	−1.32118	−0.660589	+	0.750748i	$0.729693\pi$
$348$	0	0
$349$	9244.22	1.41786	0.708928	−	0.705280i	$-0.249179\pi$
$349$	9244.22	1.41786	0.708928	+	0.705280i	$0.249179\pi$
$350$	0	0
$351$	1106.04	0.168194
$352$	0	0
$353$	7893.63	1.19019	0.595093	−	0.803657i	$-0.297115\pi$
$353$	7893.63	1.19019	0.595093	+	0.803657i	$0.297115\pi$
$354$	0	0
$355$	−5687.39	−0.850297
$356$	0	0
$357$	−110.024	−0.0163111
$358$	0	0
$359$	−5936.27	−0.872715	−0.436357	−	0.899773i	$-0.643732\pi$
$359$	−5936.27	−0.872715	−0.436357	+	0.899773i	$0.643732\pi$
$360$	0	0
$361$	−5842.57	−0.851811
$362$	0	0
$363$	917.842	0.132711
$364$	0	0
$365$	−3310.50	−0.474738
$366$	0	0
$367$	745.125	0.105982	0.0529908	−	0.998595i	$-0.483125\pi$
$367$	745.125	0.105982	0.0529908	+	0.998595i	$0.483125\pi$
$368$	0	0
$369$	7852.78	1.10786
$370$	0	0
$371$	−519.236	−0.0726615
$372$	0	0
$373$	2670.49	0.370705	0.185352	−	0.982672i	$-0.440657\pi$
$373$	2670.49	0.370705	0.185352	+	0.982672i	$0.440657\pi$
$374$	0	0
$375$	112.671	0.0155154
$376$	0	0
$377$	−669.051	−0.0914002
$378$	0	0
$379$	−6882.68	−0.932822	−0.466411	−	0.884568i	$-0.654453\pi$
$379$	−6882.68	−0.932822	−0.466411	+	0.884568i	$0.654453\pi$
$380$	0	0
$381$	807.454	0.108575
$382$	0	0
$383$	−1148.23	−0.153190	−0.0765949	−	0.997062i	$-0.524405\pi$
$383$	−1148.23	−0.153190	−0.0765949	+	0.997062i	$0.524405\pi$
$384$	0	0
$385$	355.373	0.0470428
$386$	0	0
$387$	−8480.62	−1.11394
$388$	0	0
$389$	−9082.56	−1.18381	−0.591907	−	0.806006i	$-0.701625\pi$
$389$	−9082.56	−1.18381	−0.591907	+	0.806006i	$0.701625\pi$
$390$	0	0
$391$	4040.84	0.522645
$392$	0	0
$393$	−1660.01	−0.213069
$394$	0	0
$395$	2078.43	0.264752
$396$	0	0
$397$	9812.01	1.24043	0.620215	−	0.784432i	$-0.287045\pi$
$397$	9812.01	1.24043	0.620215	+	0.784432i	$0.287045\pi$
$398$	0	0
$399$	−115.498	−0.0144916
$400$	0	0
$401$	6347.24	0.790439	0.395220	−	0.918587i	$-0.370669\pi$
$401$	6347.24	0.790439	0.395220	+	0.918587i	$0.370669\pi$
$402$	0	0
$403$	3189.54	0.394249
$404$	0	0
$405$	−3319.25	−0.407247
$406$	0	0
$407$	−323.466	−0.0393947
$408$	0	0
$409$	−6411.65	−0.775149	−0.387574	−	0.921838i	$-0.626687\pi$
$409$	−6411.65	−0.775149	−0.387574	+	0.921838i	$0.626687\pi$
$410$	0	0
$411$	−2150.52	−0.258096
$412$	0	0
$413$	−475.668	−0.0566733
$414$	0	0
$415$	1994.66	0.235937
$416$	0	0
$417$	932.686	0.109530
$418$	0	0
$419$	−6703.98	−0.781649	−0.390825	−	0.920465i	$-0.627810\pi$
$419$	−6703.98	−0.781649	−0.390825	+	0.920465i	$0.627810\pi$
$420$	0	0
$421$	8331.97	0.964549	0.482275	−	0.876020i	$-0.339811\pi$
$421$	8331.97	0.964549	0.482275	+	0.876020i	$0.339811\pi$
$422$	0	0
$423$	10766.5	1.23755
$424$	0	0
$425$	−759.257	−0.0866573
$426$	0	0
$427$	−571.898	−0.0648152
$428$	0	0
$429$	−367.739	−0.0413860
$430$	0	0
$431$	−11127.3	−1.24358	−0.621790	−	0.783184i	$-0.713594\pi$
$431$	−11127.3	−1.24358	−0.621790	+	0.783184i	$0.713594\pi$
$432$	0	0
$433$	−9783.53	−1.08583	−0.542917	−	0.839786i	$-0.682680\pi$
$433$	−9783.53	−1.08583	−0.542917	+	0.839786i	$0.682680\pi$
$434$	0	0
$435$	−130.698	−0.0144057
$436$	0	0
$437$	4241.92	0.464344
$438$	0	0
$439$	−12560.2	−1.36552	−0.682762	−	0.730641i	$-0.739221\pi$
$439$	−12560.2	−1.36552	−0.682762	+	0.730641i	$0.739221\pi$
$440$	0	0
$441$	8559.30	0.924231
$442$	0	0
$443$	−3287.85	−0.352620	−0.176310	−	0.984335i	$-0.556416\pi$
$443$	−3287.85	−0.352620	−0.176310	+	0.984335i	$0.556416\pi$
$444$	0	0
$445$	−1711.96	−0.182370
$446$	0	0
$447$	−1715.79	−0.181553
$448$	0	0
$449$	−10580.5	−1.11208	−0.556042	−	0.831154i	$-0.687681\pi$
$449$	−10580.5	−1.11208	−0.556042	+	0.831154i	$0.687681\pi$
$450$	0	0
$451$	−5302.81	−0.553658
$452$	0	0
$453$	−403.591	−0.0418595
$454$	0	0
$455$	463.626	0.0477695
$456$	0	0
$457$	−12300.7	−1.25909	−0.629545	−	0.776964i	$-0.716758\pi$
$457$	−12300.7	−1.25909	−0.629545	+	0.776964i	$0.716758\pi$
$458$	0	0
$459$	−1455.99	−0.148061
$460$	0	0
$461$	−12356.9	−1.24841	−0.624204	−	0.781261i	$-0.714577\pi$
$461$	−12356.9	−1.24841	−0.624204	+	0.781261i	$0.714577\pi$
$462$	0	0
$463$	529.927	0.0531917	0.0265959	−	0.999646i	$-0.491533\pi$
$463$	529.927	0.0531917	0.0265959	+	0.999646i	$0.491533\pi$
$464$	0	0
$465$	623.071	0.0621381
$466$	0	0
$467$	−10521.2	−1.04254	−0.521268	−	0.853393i	$-0.674541\pi$
$467$	−10521.2	−1.04254	−0.521268	+	0.853393i	$0.674541\pi$
$468$	0	0
$469$	−2034.11	−0.200270
$470$	0	0
$471$	273.119	0.0267190
$472$	0	0
$473$	5726.78	0.556697
$474$	0	0
$475$	−797.038	−0.0769907
$476$	0	0
$477$	−3383.16	−0.324747
$478$	0	0
$479$	680.741	0.0649350	0.0324675	−	0.999473i	$-0.489663\pi$
$479$	680.741	0.0649350	0.0324675	+	0.999473i	$0.489663\pi$
$480$	0	0
$481$	−422.001	−0.0400033
$482$	0	0
$483$	−482.015	−0.0454088
$484$	0	0
$485$	−8917.00	−0.834845
$486$	0	0
$487$	6618.28	0.615817	0.307909	−	0.951416i	$-0.400371\pi$
$487$	6618.28	0.615817	0.307909	+	0.951416i	$0.400371\pi$
$488$	0	0
$489$	−2439.99	−0.225645
$490$	0	0
$491$	12459.3	1.14518	0.572588	−	0.819843i	$-0.305939\pi$
$491$	12459.3	1.14518	0.572588	+	0.819843i	$0.305939\pi$
$492$	0	0
$493$	880.738	0.0804593
$494$	0	0
$495$	2315.49	0.210249
$496$	0	0
$497$	−4571.72	−0.412615
$498$	0	0
$499$	−7196.22	−0.645585	−0.322792	−	0.946470i	$-0.604622\pi$
$499$	−7196.22	−0.645585	−0.322792	+	0.946470i	$0.604622\pi$
$500$	0	0
$501$	1270.86	0.113329
$502$	0	0
$503$	−2636.26	−0.233688	−0.116844	−	0.993150i	$-0.537278\pi$
$503$	−2636.26	−0.233688	−0.116844	+	0.993150i	$0.537278\pi$
$504$	0	0
$505$	−4795.97	−0.422609
$506$	0	0
$507$	1500.54	0.131442
$508$	0	0
$509$	−15934.6	−1.38760	−0.693800	−	0.720168i	$-0.744065\pi$
$509$	−15934.6	−1.38760	−0.693800	+	0.720168i	$0.744065\pi$
$510$	0	0
$511$	−2661.09	−0.230371
$512$	0	0
$513$	−1528.44	−0.131545
$514$	0	0
$515$	2359.64	0.201900
$516$	0	0
$517$	−7270.36	−0.618472
$518$	0	0
$519$	−2957.80	−0.250160
$520$	0	0
$521$	12529.3	1.05358	0.526792	−	0.849994i	$-0.323395\pi$
$521$	12529.3	1.05358	0.526792	+	0.849994i	$0.323395\pi$
$522$	0	0
$523$	17960.6	1.50164	0.750822	−	0.660504i	$-0.229658\pi$
$523$	17960.6	1.50164	0.750822	+	0.660504i	$0.229658\pi$
$524$	0	0
$525$	90.5685	0.00752902
$526$	0	0
$527$	−4198.71	−0.347056
$528$	0	0
$529$	5536.00	0.455001
$530$	0	0
$531$	−3099.29	−0.253291
$532$	0	0
$533$	−6918.15	−0.562211
$534$	0	0
$535$	2685.01	0.216978
$536$	0	0
$537$	−2434.39	−0.195627
$538$	0	0
$539$	−5779.91	−0.461889
$540$	0	0
$541$	1015.79	0.0807247	0.0403624	−	0.999185i	$-0.487149\pi$
$541$	1015.79	0.0807247	0.0403624	+	0.999185i	$0.487149\pi$
$542$	0	0
$543$	−1467.50	−0.115979
$544$	0	0
$545$	−5205.90	−0.409167
$546$	0	0
$547$	−9679.60	−0.756618	−0.378309	−	0.925679i	$-0.623494\pi$
$547$	−9679.60	−0.756618	−0.378309	+	0.925679i	$0.623494\pi$
$548$	0	0
$549$	−3726.29	−0.289680
$550$	0	0
$551$	924.564	0.0714841
$552$	0	0
$553$	1670.71	0.128473
$554$	0	0
$555$	−82.4371	−0.00630497
$556$	0	0
$557$	−16777.3	−1.27626	−0.638130	−	0.769929i	$-0.720292\pi$
$557$	−16777.3	−1.27626	−0.638130	+	0.769929i	$0.720292\pi$
$558$	0	0
$559$	7471.27	0.565297
$560$	0	0
$561$	484.091	0.0364319
$562$	0	0
$563$	−1960.91	−0.146789	−0.0733947	−	0.997303i	$-0.523383\pi$
$563$	−1960.91	−0.146789	−0.0733947	+	0.997303i	$0.523383\pi$
$564$	0	0
$565$	5127.57	0.381803
$566$	0	0
$567$	−2668.13	−0.197621
$568$	0	0
$569$	14104.9	1.03920	0.519602	−	0.854409i	$-0.326080\pi$
$569$	14104.9	1.03920	0.519602	+	0.854409i	$0.326080\pi$
$570$	0	0
$571$	−12669.5	−0.928547	−0.464273	−	0.885692i	$-0.653685\pi$
$571$	−12669.5	−0.928547	−0.464273	+	0.885692i	$0.653685\pi$
$572$	0	0
$573$	3919.95	0.285791
$574$	0	0
$575$	−3326.32	−0.241247
$576$	0	0
$577$	26448.3	1.90824	0.954122	−	0.299418i	$-0.0967925\pi$
$577$	26448.3	1.90824	0.954122	+	0.299418i	$0.0967925\pi$
$578$	0	0
$579$	−3373.23	−0.242119
$580$	0	0
$581$	1603.37	0.114491
$582$	0	0
$583$	2284.58	0.162294
$584$	0	0
$585$	3020.83	0.213497
$586$	0	0
$587$	−19462.8	−1.36851	−0.684255	−	0.729243i	$-0.739872\pi$
$587$	−19462.8	−1.36851	−0.684255	+	0.729243i	$0.739872\pi$
$588$	0	0
$589$	−4407.64	−0.308342
$590$	0	0
$591$	−2813.95	−0.195855
$592$	0	0
$593$	5612.56	0.388668	0.194334	−	0.980935i	$-0.437745\pi$
$593$	5612.56	0.388668	0.194334	+	0.980935i	$0.437745\pi$
$594$	0	0
$595$	−610.317	−0.0420513
$596$	0	0
$597$	−1126.98	−0.0772602
$598$	0	0
$599$	−13832.7	−0.943556	−0.471778	−	0.881717i	$-0.656388\pi$
$599$	−13832.7	−0.943556	−0.471778	+	0.881717i	$0.656388\pi$
$600$	0	0
$601$	−22578.3	−1.53243	−0.766213	−	0.642586i	$-0.777861\pi$
$601$	−22578.3	−1.53243	−0.766213	+	0.642586i	$0.777861\pi$
$602$	0	0
$603$	−13253.6	−0.895070
$604$	0	0
$605$	5091.40	0.342140
$606$	0	0
$607$	21512.4	1.43848	0.719242	−	0.694760i	$-0.244489\pi$
$607$	21512.4	1.43848	0.719242	+	0.694760i	$0.244489\pi$
$608$	0	0
$609$	−105.059	−0.00699052
$610$	0	0
$611$	−9485.05	−0.628026
$612$	0	0
$613$	2667.05	0.175728	0.0878641	−	0.996132i	$-0.471996\pi$
$613$	2667.05	0.175728	0.0878641	+	0.996132i	$0.471996\pi$
$614$	0	0
$615$	−1351.45	−0.0886108
$616$	0	0
$617$	−7892.93	−0.515004	−0.257502	−	0.966278i	$-0.582899\pi$
$617$	−7892.93	−0.515004	−0.257502	+	0.966278i	$0.582899\pi$
$618$	0	0
$619$	−11950.9	−0.776002	−0.388001	−	0.921659i	$-0.626834\pi$
$619$	−11950.9	−0.776002	−0.388001	+	0.921659i	$0.626834\pi$
$620$	0	0
$621$	−6378.73	−0.412189
$622$	0	0
$623$	−1376.13	−0.0884969
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	508.179	0.0323680
$628$	0	0
$629$	555.521	0.0352148
$630$	0	0
$631$	18209.6	1.14883	0.574414	−	0.818565i	$-0.305230\pi$
$631$	18209.6	1.14883	0.574414	+	0.818565i	$0.305230\pi$
$632$	0	0
$633$	−250.128	−0.0157057
$634$	0	0
$635$	4479.06	0.279915
$636$	0	0
$637$	−7540.58	−0.469025
$638$	0	0
$639$	−29787.8	−1.84411
$640$	0	0
$641$	21566.0	1.32887	0.664436	−	0.747345i	$-0.268672\pi$
$641$	21566.0	1.32887	0.664436	+	0.747345i	$0.268672\pi$
$642$	0	0
$643$	24611.9	1.50948	0.754742	−	0.656022i	$-0.227762\pi$
$643$	24611.9	1.50948	0.754742	+	0.656022i	$0.227762\pi$
$644$	0	0
$645$	1459.50	0.0890972
$646$	0	0
$647$	25091.9	1.52467	0.762337	−	0.647180i	$-0.224052\pi$
$647$	25091.9	1.52467	0.762337	+	0.647180i	$0.224052\pi$
$648$	0	0
$649$	2092.88	0.126584
$650$	0	0
$651$	500.846	0.0301532
$652$	0	0
$653$	20915.3	1.25341	0.626706	−	0.779256i	$-0.284403\pi$
$653$	20915.3	1.25341	0.626706	+	0.779256i	$0.284403\pi$
$654$	0	0
$655$	−9208.29	−0.549309
$656$	0	0
$657$	−17338.8	−1.02960
$658$	0	0
$659$	−14530.2	−0.858902	−0.429451	−	0.903090i	$-0.641293\pi$
$659$	−14530.2	−0.858902	−0.429451	+	0.903090i	$0.641293\pi$
$660$	0	0
$661$	21454.4	1.26245	0.631225	−	0.775599i	$-0.282552\pi$
$661$	21454.4	1.26245	0.631225	+	0.775599i	$0.282552\pi$
$662$	0	0
$663$	631.554	0.0369948
$664$	0	0
$665$	−640.686	−0.0373605
$666$	0	0
$667$	3858.53	0.223992
$668$	0	0
$669$	−2834.50	−0.163809
$670$	0	0
$671$	2516.28	0.144769
$672$	0	0
$673$	3880.17	0.222243	0.111122	−	0.993807i	$-0.464556\pi$
$673$	3880.17	0.222243	0.111122	+	0.993807i	$0.464556\pi$
$674$	0	0
$675$	1198.53	0.0683431
$676$	0	0
$677$	−913.324	−0.0518492	−0.0259246	−	0.999664i	$-0.508253\pi$
$677$	−913.324	−0.0518492	−0.0259246	+	0.999664i	$0.508253\pi$
$678$	0	0
$679$	−7167.79	−0.405117
$680$	0	0
$681$	3914.63	0.220278
$682$	0	0
$683$	16766.1	0.939291	0.469645	−	0.882855i	$-0.344382\pi$
$683$	16766.1	0.939291	0.469645	+	0.882855i	$0.344382\pi$
$684$	0	0
$685$	−11929.2	−0.665392
$686$	0	0
$687$	1848.15	0.102636
$688$	0	0
$689$	2980.50	0.164801
$690$	0	0
$691$	4872.52	0.268248	0.134124	−	0.990965i	$-0.457178\pi$
$691$	4872.52	0.268248	0.134124	+	0.990965i	$0.457178\pi$
$692$	0	0
$693$	1861.27	0.102026
$694$	0	0
$695$	5173.74	0.282376
$696$	0	0
$697$	9107.04	0.494912
$698$	0	0
$699$	−4434.85	−0.239974
$700$	0	0
$701$	28386.1	1.52942	0.764712	−	0.644372i	$-0.222881\pi$
$701$	28386.1	1.52942	0.764712	+	0.644372i	$0.222881\pi$
$702$	0	0
$703$	583.164	0.0312866
$704$	0	0
$705$	−1852.89	−0.0989841
$706$	0	0
$707$	−3855.16	−0.205075
$708$	0	0
$709$	−25742.0	−1.36356	−0.681778	−	0.731559i	$-0.738793\pi$
$709$	−25742.0	−1.36356	−0.681778	+	0.731559i	$0.738793\pi$
$710$	0	0
$711$	10885.8	0.574189
$712$	0	0
$713$	−18394.6	−0.966176
$714$	0	0
$715$	−2039.90	−0.106696
$716$	0	0
$717$	−4778.94	−0.248916
$718$	0	0
$719$	−4311.44	−0.223629	−0.111815	−	0.993729i	$-0.535666\pi$
$719$	−4311.44	−0.223629	−0.111815	+	0.993729i	$0.535666\pi$
$720$	0	0
$721$	1896.76	0.0979738
$722$	0	0
$723$	−4417.46	−0.227230
$724$	0	0
$725$	−725.000	−0.0371391
$726$	0	0
$727$	−25991.9	−1.32598	−0.662990	−	0.748628i	$-0.730713\pi$
$727$	−25991.9	−1.32598	−0.662990	+	0.748628i	$0.730713\pi$
$728$	0	0
$729$	−16217.9	−0.823954
$730$	0	0
$731$	−9835.17	−0.497629
$732$	0	0
$733$	−1690.71	−0.0851947	−0.0425973	−	0.999092i	$-0.513563\pi$
$733$	−1690.71	−0.0851947	−0.0425973	+	0.999092i	$0.513563\pi$
$734$	0	0
$735$	−1473.04	−0.0739236
$736$	0	0
$737$	8949.84	0.447316
$738$	0	0
$739$	−19766.2	−0.983914	−0.491957	−	0.870620i	$-0.663718\pi$
$739$	−19766.2	−0.983914	−0.491957	+	0.870620i	$0.663718\pi$
$740$	0	0
$741$	662.981	0.0328680
$742$	0	0
$743$	4692.83	0.231714	0.115857	−	0.993266i	$-0.463039\pi$
$743$	4692.83	0.231714	0.115857	+	0.993266i	$0.463039\pi$
$744$	0	0
$745$	−9517.74	−0.468058
$746$	0	0
$747$	10447.0	0.511696
$748$	0	0
$749$	2158.30	0.105291
$750$	0	0
$751$	3964.19	0.192617	0.0963084	−	0.995352i	$-0.469296\pi$
$751$	3964.19	0.192617	0.0963084	+	0.995352i	$0.469296\pi$
$752$	0	0
$753$	7142.27	0.345656
$754$	0	0
$755$	−2238.78	−0.107917
$756$	0	0
$757$	33516.6	1.60922	0.804612	−	0.593801i	$-0.202373\pi$
$757$	33516.6	1.60922	0.804612	+	0.593801i	$0.202373\pi$
$758$	0	0
$759$	2120.81	0.101424
$760$	0	0
$761$	−16250.5	−0.774085	−0.387042	−	0.922062i	$-0.626503\pi$
$761$	−16250.5	−0.774085	−0.387042	+	0.922062i	$0.626503\pi$
$762$	0	0
$763$	−4184.68	−0.198552
$764$	0	0
$765$	−3976.61	−0.187941
$766$	0	0
$767$	2730.41	0.128539
$768$	0	0
$769$	18777.8	0.880552	0.440276	−	0.897862i	$-0.354881\pi$
$769$	18777.8	0.880552	0.440276	+	0.897862i	$0.354881\pi$
$770$	0	0
$771$	−2307.37	−0.107779
$772$	0	0
$773$	−33581.7	−1.56255	−0.781274	−	0.624188i	$-0.785430\pi$
$773$	−33581.7	−1.56255	−0.781274	+	0.624188i	$0.785430\pi$
$774$	0	0
$775$	3456.26	0.160197
$776$	0	0
$777$	−66.2658	−0.00305955
$778$	0	0
$779$	9560.21	0.439705
$780$	0	0
$781$	20115.0	0.921603
$782$	0	0
$783$	−1390.30	−0.0634550
$784$	0	0
$785$	1515.03	0.0688838
$786$	0	0
$787$	5602.35	0.253751	0.126876	−	0.991919i	$-0.459505\pi$
$787$	5602.35	0.253751	0.126876	+	0.991919i	$0.459505\pi$
$788$	0	0
$789$	−387.214	−0.0174717
$790$	0	0
$791$	4121.72	0.185274
$792$	0	0
$793$	3282.79	0.147005
$794$	0	0
$795$	582.236	0.0259746
$796$	0	0
$797$	26516.7	1.17851	0.589254	−	0.807948i	$-0.299422\pi$
$797$	26516.7	1.17851	0.589254	+	0.807948i	$0.299422\pi$
$798$	0	0
$799$	12486.1	0.552850
$800$	0	0
$801$	−8966.40	−0.395521
$802$	0	0
$803$	11708.5	0.514550
$804$	0	0
$805$	−2673.81	−0.117067
$806$	0	0
$807$	−277.106	−0.0120875
$808$	0	0
$809$	−30041.4	−1.30556	−0.652782	−	0.757546i	$-0.726398\pi$
$809$	−30041.4	−1.30556	−0.652782	+	0.757546i	$0.726398\pi$
$810$	0	0
$811$	−17845.8	−0.772690	−0.386345	−	0.922354i	$-0.626263\pi$
$811$	−17845.8	−0.772690	−0.386345	+	0.922354i	$0.626263\pi$
$812$	0	0
$813$	−1475.22	−0.0636386
$814$	0	0
$815$	−13535.0	−0.581730
$816$	0	0
$817$	−10324.6	−0.442119
$818$	0	0
$819$	2428.25	0.103602
$820$	0	0
$821$	−31288.0	−1.33004	−0.665018	−	0.746827i	$-0.731576\pi$
$821$	−31288.0	−1.33004	−0.665018	+	0.746827i	$0.731576\pi$
$822$	0	0
$823$	14715.6	0.623274	0.311637	−	0.950201i	$-0.399123\pi$
$823$	14715.6	0.623274	0.311637	+	0.950201i	$0.399123\pi$
$824$	0	0
$825$	−398.491	−0.0168166
$826$	0	0
$827$	−9760.47	−0.410405	−0.205202	−	0.978720i	$-0.565785\pi$
$827$	−9760.47	−0.410405	−0.205202	+	0.978720i	$0.565785\pi$
$828$	0	0
$829$	−3813.55	−0.159771	−0.0798855	−	0.996804i	$-0.525455\pi$
$829$	−3813.55	−0.159771	−0.0798855	+	0.996804i	$0.525455\pi$
$830$	0	0
$831$	−1208.52	−0.0504488
$832$	0	0
$833$	9926.41	0.412881
$834$	0	0
$835$	7049.64	0.292171
$836$	0	0
$837$	6627.92	0.273709
$838$	0	0
$839$	−899.368	−0.0370079	−0.0185040	−	0.999829i	$-0.505890\pi$
$839$	−899.368	−0.0370079	−0.0185040	+	0.999829i	$0.505890\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	1281.62	0.0523623
$844$	0	0
$845$	8323.71	0.338869
$846$	0	0
$847$	4092.65	0.166027
$848$	0	0
$849$	5407.45	0.218591
$850$	0	0
$851$	2433.75	0.0980350
$852$	0	0
$853$	8871.64	0.356107	0.178053	−	0.984021i	$-0.443020\pi$
$853$	8871.64	0.356107	0.178053	+	0.984021i	$0.443020\pi$
$854$	0	0
$855$	−4174.49	−0.166976
$856$	0	0
$857$	−35747.7	−1.42488	−0.712439	−	0.701735i	$-0.752409\pi$
$857$	−35747.7	−1.42488	−0.712439	+	0.701735i	$0.752409\pi$
$858$	0	0
$859$	11155.1	0.443080	0.221540	−	0.975151i	$-0.428892\pi$
$859$	11155.1	0.443080	0.221540	+	0.975151i	$0.428892\pi$
$860$	0	0
$861$	−1086.34	−0.0429993
$862$	0	0
$863$	8445.47	0.333125	0.166563	−	0.986031i	$-0.446733\pi$
$863$	8445.47	0.333125	0.166563	+	0.986031i	$0.446733\pi$
$864$	0	0
$865$	−16407.3	−0.644932
$866$	0	0
$867$	3597.03	0.140901
$868$	0	0
$869$	−7350.92	−0.286954
$870$	0	0
$871$	11676.1	0.454226
$872$	0	0
$873$	−46702.8	−1.81060
$874$	0	0
$875$	502.397	0.0194104
$876$	0	0
$877$	36481.5	1.40467	0.702333	−	0.711849i	$-0.252142\pi$
$877$	36481.5	1.40467	0.702333	+	0.711849i	$0.252142\pi$
$878$	0	0
$879$	2398.58	0.0920387
$880$	0	0
$881$	11301.9	0.432203	0.216102	−	0.976371i	$-0.430666\pi$
$881$	11301.9	0.432203	0.216102	+	0.976371i	$0.430666\pi$
$882$	0	0
$883$	34410.2	1.31143	0.655717	−	0.755007i	$-0.272366\pi$
$883$	34410.2	1.31143	0.655717	+	0.755007i	$0.272366\pi$
$884$	0	0
$885$	533.381	0.0202592
$886$	0	0
$887$	27906.7	1.05639	0.528194	−	0.849124i	$-0.322869\pi$
$887$	27906.7	1.05639	0.528194	+	0.849124i	$0.322869\pi$
$888$	0	0
$889$	3600.43	0.135832
$890$	0	0
$891$	11739.5	0.441399
$892$	0	0
$893$	13107.4	0.491179
$894$	0	0
$895$	−13503.9	−0.504342
$896$	0	0
$897$	2766.85	0.102990
$898$	0	0
$899$	−4009.27	−0.148739
$900$	0	0
$901$	−3923.53	−0.145074
$902$	0	0
$903$	1173.20	0.0432353
$904$	0	0
$905$	−8140.44	−0.299003
$906$	0	0
$907$	1791.73	0.0655936	0.0327968	−	0.999462i	$-0.489559\pi$
$907$	1791.73	0.0655936	0.0327968	+	0.999462i	$0.489559\pi$
$908$	0	0
$909$	−25118.9	−0.916547
$910$	0	0
$911$	36745.9	1.33638	0.668192	−	0.743989i	$-0.267068\pi$
$911$	36745.9	1.33638	0.668192	+	0.743989i	$0.267068\pi$
$912$	0	0
$913$	−7054.65	−0.255723
$914$	0	0
$915$	641.287	0.0231697
$916$	0	0
$917$	−7401.94	−0.266558
$918$	0	0
$919$	39360.9	1.41284	0.706419	−	0.707794i	$-0.250310\pi$
$919$	39360.9	1.41284	0.706419	+	0.707794i	$0.250310\pi$
$920$	0	0
$921$	−7607.20	−0.272167
$922$	0	0
$923$	26242.4	0.935840
$924$	0	0
$925$	−457.290	−0.0162547
$926$	0	0
$927$	12358.6	0.437876
$928$	0	0
$929$	25334.7	0.894732	0.447366	−	0.894351i	$-0.352362\pi$
$929$	25334.7	0.894732	0.447366	+	0.894351i	$0.352362\pi$
$930$	0	0
$931$	10420.4	0.366824
$932$	0	0
$933$	−5091.17	−0.178647
$934$	0	0
$935$	2685.32	0.0939244
$936$	0	0
$937$	31989.9	1.11533	0.557665	−	0.830066i	$-0.311697\pi$
$937$	31989.9	1.11533	0.557665	+	0.830066i	$0.311697\pi$
$938$	0	0
$939$	−2469.07	−0.0858095
$940$	0	0
$941$	28945.4	1.00276	0.501379	−	0.865228i	$-0.332826\pi$
$941$	28945.4	1.00276	0.501379	+	0.865228i	$0.332826\pi$
$942$	0	0
$943$	39898.1	1.37780
$944$	0	0
$945$	963.423	0.0331642
$946$	0	0
$947$	20241.3	0.694566	0.347283	−	0.937760i	$-0.387104\pi$
$947$	20241.3	0.694566	0.347283	+	0.937760i	$0.387104\pi$
$948$	0	0
$949$	15275.1	0.522499
$950$	0	0
$951$	−2027.40	−0.0691303
$952$	0	0
$953$	31028.1	1.05467	0.527335	−	0.849658i	$-0.323191\pi$
$953$	31028.1	1.05467	0.527335	+	0.849658i	$0.323191\pi$
$954$	0	0
$955$	21744.5	0.736791
$956$	0	0
$957$	462.249	0.0156138
$958$	0	0
$959$	−9589.14	−0.322888
$960$	0	0
$961$	−10677.8	−0.358423
$962$	0	0
$963$	14062.8	0.470577
$964$	0	0
$965$	−18711.8	−0.624201
$966$	0	0
$967$	−33984.7	−1.13017	−0.565086	−	0.825032i	$-0.691157\pi$
$967$	−33984.7	−1.13017	−0.565086	+	0.825032i	$0.691157\pi$
$968$	0	0
$969$	−872.747	−0.0289336
$970$	0	0
$971$	−49106.1	−1.62295	−0.811477	−	0.584384i	$-0.801336\pi$
$971$	−49106.1	−1.62295	−0.811477	+	0.584384i	$0.801336\pi$
$972$	0	0
$973$	4158.83	0.137026
$974$	0	0
$975$	−519.879	−0.0170763
$976$	0	0
$977$	−4067.12	−0.133182	−0.0665909	−	0.997780i	$-0.521212\pi$
$977$	−4067.12	−0.133182	−0.0665909	+	0.997780i	$0.521212\pi$
$978$	0	0
$979$	6054.81	0.197664
$980$	0	0
$981$	−27265.9	−0.887395
$982$	0	0
$983$	20019.6	0.649570	0.324785	−	0.945788i	$-0.394708\pi$
$983$	20019.6	0.649570	0.324785	+	0.945788i	$0.394708\pi$
$984$	0	0
$985$	−15609.4	−0.504930
$986$	0	0
$987$	−1489.42	−0.0480330
$988$	0	0
$989$	−43088.0	−1.38536
$990$	0	0
$991$	36059.4	1.15587	0.577933	−	0.816084i	$-0.303859\pi$
$991$	36059.4	1.15587	0.577933	+	0.816084i	$0.303859\pi$
$992$	0	0
$993$	2904.99	0.0928370
$994$	0	0
$995$	−6251.54	−0.199183
$996$	0	0
$997$	−2876.50	−0.0913739	−0.0456870	−	0.998956i	$-0.514548\pi$
$997$	−2876.50	−0.0913739	−0.0456870	+	0.998956i	$0.514548\pi$
$998$	0	0
$999$	−876.925	−0.0277725

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.g.1.5	✓	11
4.3	odd	2		2320.4.a.bb.1.7		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.g.1.5	✓	11	1.1	even	1	trivial
2320.4.a.bb.1.7		11	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q-0.901365 q^{3} -5.00000 q^{5} -4.01917 q^{7} -26.1875 q^{9} +17.6839 q^{11} +23.0707 q^{13} +4.50682 q^{15} -30.3703 q^{17} -31.8815 q^{19} +3.62274 q^{21} -133.053 q^{23} +25.0000 q^{25} +47.9414 q^{27} -29.0000 q^{29} +138.251 q^{31} -15.9396 q^{33} +20.0959 q^{35} -18.2916 q^{37} -20.7951 q^{39} -299.867 q^{41} +323.842 q^{43} +130.938 q^{45} -411.129 q^{47} -326.846 q^{49} +27.3747 q^{51} +129.190 q^{53} -88.4194 q^{55} +28.7369 q^{57} +118.350 q^{59} +142.293 q^{61} +105.252 q^{63} -115.354 q^{65} +506.102 q^{67} +119.929 q^{69} +1137.48 q^{71} +662.100 q^{73} -22.5341 q^{75} -71.0745 q^{77} -415.685 q^{79} +663.851 q^{81} -398.932 q^{83} +151.851 q^{85} +26.1396 q^{87} +342.392 q^{89} -92.7252 q^{91} -124.614 q^{93} +159.408 q^{95} +1783.40 q^{97} -463.097 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q + 2 q^{3} - 55 q^{5} + 56 q^{7} + 65 q^{9} - 8 q^{11} + 16 q^{13} - 10 q^{15} + 64 q^{17} + 160 q^{19} - 48 q^{21} + 140 q^{23} + 275 q^{25} - 136 q^{27} - 319 q^{29} - 144 q^{31} - 424 q^{33} - 280 q^{35}+ \cdots + 6884 q^{99}+O(q^{100}) \) 11 * q + 2 * q^3 - 55 * q^5 + 56 * q^7 + 65 * q^9 - 8 * q^11 + 16 * q^13 - 10 * q^15 + 64 * q^17 + 160 * q^19 - 48 * q^21 + 140 * q^23 + 275 * q^25 - 136 * q^27 - 319 * q^29 - 144 * q^31 - 424 * q^33 - 280 * q^35 - 50 * q^37 + 572 * q^39 + 62 * q^41 + 326 * q^43 - 325 * q^45 + 1074 * q^47 + 1513 * q^49 - 144 * q^51 + 576 * q^53 + 40 * q^55 - 896 * q^57 - 468 * q^59 + 660 * q^61 + 2552 * q^63 - 80 * q^65 + 892 * q^67 - 510 * q^69 + 920 * q^71 - 676 * q^73 + 50 * q^75 + 1708 * q^77 + 2308 * q^79 + 1287 * q^81 - 1548 * q^83 - 320 * q^85 - 58 * q^87 + 1854 * q^89 + 3100 * q^91 + 5424 * q^93 - 800 * q^95 + 4724 * q^97 + 6884 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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