LMFDB - Embedded newform 1160.4.a.j.1.10

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:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 6 x^{12} - 231 x^{11} + 1278 x^{10} + 18973 x^{9} - 95838 x^{8} - 662972 x^{7} + 3060212 x^{6} + \cdots - 8760320 $ x^13 - 6x^12 - 231x^11 + 1278x^10 + 18973x^9 - 95838x^8 - 662972x^7 + 3060212x^6 + 9493200x^5 - 41097120x^4 - 46163584x^3 + 190570944x^2 + 4234240x - 8760320
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{15} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-3.96683$ 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)

$f(q)$	$=$	$q+3.96683 q^{3} +5.00000 q^{5} +23.5989 q^{7} -11.2642 q^{9} +19.1675 q^{11} +70.7812 q^{13} +19.8342 q^{15} +2.14027 q^{17} +53.0282 q^{19} +93.6131 q^{21} +58.6885 q^{23} +25.0000 q^{25} -151.788 q^{27} +29.0000 q^{29} -46.5724 q^{31} +76.0341 q^{33} +117.995 q^{35} -322.832 q^{37} +280.777 q^{39} -92.9979 q^{41} +212.070 q^{43} -56.3212 q^{45} +392.089 q^{47} +213.910 q^{49} +8.49011 q^{51} -72.6402 q^{53} +95.8373 q^{55} +210.354 q^{57} +443.474 q^{59} -584.193 q^{61} -265.824 q^{63} +353.906 q^{65} +980.440 q^{67} +232.808 q^{69} -808.253 q^{71} +963.185 q^{73} +99.1708 q^{75} +452.332 q^{77} -1254.62 q^{79} -297.983 q^{81} -86.1666 q^{83} +10.7014 q^{85} +115.038 q^{87} -1526.64 q^{89} +1670.36 q^{91} -184.745 q^{93} +265.141 q^{95} +587.479 q^{97} -215.907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 6 q^{3} + 65 q^{5} + 2 q^{7} + 147 q^{9} - 42 q^{11} + 100 q^{13} - 30 q^{15} + 136 q^{17} + 38 q^{19} + 376 q^{21} + 86 q^{23} + 325 q^{25} - 216 q^{27} + 377 q^{29} + 230 q^{31} + 512 q^{33} + 10 q^{35}+ \cdots - 846 q^{99}+O(q^{100}) $ 13 * q - 6 * q^3 + 65 * q^5 + 2 * q^7 + 147 * q^9 - 42 * q^11 + 100 * q^13 - 30 * q^15 + 136 * q^17 + 38 * q^19 + 376 * q^21 + 86 * q^23 + 325 * q^25 - 216 * q^27 + 377 * q^29 + 230 * q^31 + 512 * q^33 + 10 * q^35 + 614 * q^37 - 188 * q^39 + 506 * q^41 + 198 * q^43 + 735 * q^45 - 582 * q^47 + 1383 * q^49 + 336 * q^51 + 772 * q^53 - 210 * q^55 + 1104 * q^57 - 72 * q^59 + 1040 * q^61 - 502 * q^63 + 500 * q^65 - 374 * q^67 - 258 * q^69 + 820 * q^71 + 648 * q^73 - 150 * q^75 + 952 * q^77 + 378 * q^79 + 3857 * q^81 + 478 * q^83 + 680 * q^85 - 174 * q^87 + 2378 * q^89 + 888 * q^91 + 3656 * q^93 + 190 * q^95 + 1760 * q^97 - 846 * 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
$3$	3.96683	0.763417	0.381709	−	0.924283i	$-0.375336\pi$
$3$	3.96683	0.763417	0.381709	+	0.924283i	$0.375336\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	23.5989	1.27422	0.637111	−	0.770772i	$-0.280129\pi$
$7$	23.5989	1.27422	0.637111	+	0.770772i	$0.280129\pi$
$8$	0	0
$9$	−11.2642	−0.417194
$10$	0	0
$11$	19.1675	0.525382	0.262691	−	0.964880i	$-0.415390\pi$
$11$	19.1675	0.525382	0.262691	+	0.964880i	$0.415390\pi$
$12$	0	0
$13$	70.7812	1.51009	0.755045	−	0.655673i	$-0.227615\pi$
$13$	70.7812	1.51009	0.755045	+	0.655673i	$0.227615\pi$
$14$	0	0
$15$	19.8342	0.341411
$16$	0	0
$17$	2.14027	0.0305349	0.0152674	−	0.999883i	$-0.495140\pi$
$17$	2.14027	0.0305349	0.0152674	+	0.999883i	$0.495140\pi$
$18$	0	0
$19$	53.0282	0.640289	0.320145	−	0.947369i	$-0.396269\pi$
$19$	53.0282	0.640289	0.320145	+	0.947369i	$0.396269\pi$
$20$	0	0
$21$	93.6131	0.972764
$22$	0	0
$23$	58.6885	0.532061	0.266030	−	0.963965i	$-0.414288\pi$
$23$	58.6885	0.532061	0.266030	+	0.963965i	$0.414288\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−151.788	−1.08191
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−46.5724	−0.269827	−0.134914	−	0.990857i	$-0.543076\pi$
$31$	−46.5724	−0.269827	−0.134914	+	0.990857i	$0.543076\pi$
$32$	0	0
$33$	76.0341	0.401086
$34$	0	0
$35$	117.995	0.569850
$36$	0	0
$37$	−322.832	−1.43441	−0.717205	−	0.696862i	$-0.754579\pi$
$37$	−322.832	−1.43441	−0.717205	+	0.696862i	$0.754579\pi$
$38$	0	0
$39$	280.777	1.15283
$40$	0	0
$41$	−92.9979	−0.354240	−0.177120	−	0.984189i	$-0.556678\pi$
$41$	−92.9979	−0.354240	−0.177120	+	0.984189i	$0.556678\pi$
$42$	0	0
$43$	212.070	0.752100	0.376050	−	0.926599i	$-0.377282\pi$
$43$	212.070	0.752100	0.376050	+	0.926599i	$0.377282\pi$
$44$	0	0
$45$	−56.3212	−0.186575
$46$	0	0
$47$	392.089	1.21685	0.608426	−	0.793611i	$-0.291801\pi$
$47$	392.089	1.21685	0.608426	+	0.793611i	$0.291801\pi$
$48$	0	0
$49$	213.910	0.623644
$50$	0	0
$51$	8.49011	0.0233108
$52$	0	0
$53$	−72.6402	−0.188262	−0.0941311	−	0.995560i	$-0.530007\pi$
$53$	−72.6402	−0.188262	−0.0941311	+	0.995560i	$0.530007\pi$
$54$	0	0
$55$	95.8373	0.234958
$56$	0	0
$57$	210.354	0.488808
$58$	0	0
$59$	443.474	0.978567	0.489283	−	0.872125i	$-0.337258\pi$
$59$	443.474	0.978567	0.489283	+	0.872125i	$0.337258\pi$
$60$	0	0
$61$	−584.193	−1.22620	−0.613100	−	0.790005i	$-0.710078\pi$
$61$	−584.193	−1.22620	−0.613100	+	0.790005i	$0.710078\pi$
$62$	0	0
$63$	−265.824	−0.531598
$64$	0	0
$65$	353.906	0.675333
$66$	0	0
$67$	980.440	1.78776	0.893879	−	0.448308i	$-0.147973\pi$
$67$	980.440	1.78776	0.893879	+	0.448308i	$0.147973\pi$
$68$	0	0
$69$	232.808	0.406185
$70$	0	0
$71$	−808.253	−1.35101	−0.675507	−	0.737353i	$-0.736075\pi$
$71$	−808.253	−1.35101	−0.675507	+	0.737353i	$0.736075\pi$
$72$	0	0
$73$	963.185	1.54428	0.772139	−	0.635454i	$-0.219187\pi$
$73$	963.185	1.54428	0.772139	+	0.635454i	$0.219187\pi$
$74$	0	0
$75$	99.1708	0.152683
$76$	0	0
$77$	452.332	0.669454
$78$	0	0
$79$	−1254.62	−1.78679	−0.893393	−	0.449276i	$-0.851682\pi$
$79$	−1254.62	−1.78679	−0.893393	+	0.449276i	$0.851682\pi$
$80$	0	0
$81$	−297.983	−0.408755
$82$	0	0
$83$	−86.1666	−0.113952	−0.0569760	−	0.998376i	$-0.518146\pi$
$83$	−86.1666	−0.113952	−0.0569760	+	0.998376i	$0.518146\pi$
$84$	0	0
$85$	10.7014	0.0136556
$86$	0	0
$87$	115.038	0.141763
$88$	0	0
$89$	−1526.64	−1.81824	−0.909118	−	0.416538i	$-0.863243\pi$
$89$	−1526.64	−1.81824	−0.909118	+	0.416538i	$0.863243\pi$
$90$	0	0
$91$	1670.36	1.92419
$92$	0	0
$93$	−184.745	−0.205991
$94$	0	0
$95$	265.141	0.286346
$96$	0	0
$97$	587.479	0.614943	0.307471	−	0.951557i	$-0.400517\pi$
$97$	587.479	0.614943	0.307471	+	0.951557i	$0.400517\pi$
$98$	0	0
$99$	−215.907	−0.219186
$100$	0	0
$101$	1236.16	1.21785	0.608923	−	0.793230i	$-0.291602\pi$
$101$	1236.16	1.21785	0.608923	+	0.793230i	$0.291602\pi$
$102$	0	0
$103$	522.160	0.499514	0.249757	−	0.968309i	$-0.419649\pi$
$103$	522.160	0.499514	0.249757	+	0.968309i	$0.419649\pi$
$104$	0	0
$105$	468.065	0.435033
$106$	0	0
$107$	388.001	0.350556	0.175278	−	0.984519i	$-0.443918\pi$
$107$	388.001	0.350556	0.175278	+	0.984519i	$0.443918\pi$
$108$	0	0
$109$	1279.79	1.12460	0.562299	−	0.826934i	$-0.309917\pi$
$109$	1279.79	1.12460	0.562299	+	0.826934i	$0.309917\pi$
$110$	0	0
$111$	−1280.62	−1.09505
$112$	0	0
$113$	−956.635	−0.796395	−0.398198	−	0.917300i	$-0.630364\pi$
$113$	−956.635	−0.796395	−0.398198	+	0.917300i	$0.630364\pi$
$114$	0	0
$115$	293.443	0.237945
$116$	0	0
$117$	−797.296	−0.630000
$118$	0	0
$119$	50.5082	0.0389082
$120$	0	0
$121$	−963.609	−0.723974
$122$	0	0
$123$	−368.907	−0.270433
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1198.59	0.837459	0.418729	−	0.908111i	$-0.362476\pi$
$127$	1198.59	0.837459	0.418729	+	0.908111i	$0.362476\pi$
$128$	0	0
$129$	841.245	0.574166
$130$	0	0
$131$	−1961.82	−1.30844	−0.654218	−	0.756306i	$-0.727002\pi$
$131$	−1961.82	−1.30844	−0.654218	+	0.756306i	$0.727002\pi$
$132$	0	0
$133$	1251.41	0.815871
$134$	0	0
$135$	−758.939	−0.483845
$136$	0	0
$137$	1435.99	0.895508	0.447754	−	0.894157i	$-0.352224\pi$
$137$	1435.99	0.895508	0.447754	+	0.894157i	$0.352224\pi$
$138$	0	0
$139$	2284.28	1.39389	0.696944	−	0.717126i	$-0.254543\pi$
$139$	2284.28	1.39389	0.696944	+	0.717126i	$0.254543\pi$
$140$	0	0
$141$	1555.35	0.928966
$142$	0	0
$143$	1356.69	0.793374
$144$	0	0
$145$	145.000	0.0830455
$146$	0	0
$147$	848.545	0.476101
$148$	0	0
$149$	−2060.94	−1.13315	−0.566574	−	0.824011i	$-0.691731\pi$
$149$	−2060.94	−1.13315	−0.566574	+	0.824011i	$0.691731\pi$
$150$	0	0
$151$	−1559.66	−0.840555	−0.420277	−	0.907396i	$-0.638067\pi$
$151$	−1559.66	−0.840555	−0.420277	+	0.907396i	$0.638067\pi$
$152$	0	0
$153$	−24.1086	−0.0127390
$154$	0	0
$155$	−232.862	−0.120670
$156$	0	0
$157$	−1090.99	−0.554591	−0.277295	−	0.960785i	$-0.589438\pi$
$157$	−1090.99	−0.554591	−0.277295	+	0.960785i	$0.589438\pi$
$158$	0	0
$159$	−288.152	−0.143723
$160$	0	0
$161$	1384.99	0.677964
$162$	0	0
$163$	−627.921	−0.301734	−0.150867	−	0.988554i	$-0.548206\pi$
$163$	−627.921	−0.301734	−0.150867	+	0.988554i	$0.548206\pi$
$164$	0	0
$165$	380.170	0.179371
$166$	0	0
$167$	−265.319	−0.122940	−0.0614701	−	0.998109i	$-0.519579\pi$
$167$	−265.319	−0.122940	−0.0614701	+	0.998109i	$0.519579\pi$
$168$	0	0
$169$	2812.97	1.28037
$170$	0	0
$171$	−597.322	−0.267125
$172$	0	0
$173$	3659.84	1.60840	0.804198	−	0.594362i	$-0.202595\pi$
$173$	3659.84	1.60840	0.804198	+	0.594362i	$0.202595\pi$
$174$	0	0
$175$	589.973	0.254845
$176$	0	0
$177$	1759.19	0.747055
$178$	0	0
$179$	−2059.86	−0.860118	−0.430059	−	0.902801i	$-0.641507\pi$
$179$	−2059.86	−0.860118	−0.430059	+	0.902801i	$0.641507\pi$
$180$	0	0
$181$	1514.03	0.621753	0.310877	−	0.950450i	$-0.399377\pi$
$181$	1514.03	0.621753	0.310877	+	0.950450i	$0.399377\pi$
$182$	0	0
$183$	−2317.39	−0.936102
$184$	0	0
$185$	−1614.16	−0.641488
$186$	0	0
$187$	41.0236	0.0160425
$188$	0	0
$189$	−3582.03	−1.37860
$190$	0	0
$191$	842.762	0.319268	0.159634	−	0.987176i	$-0.448969\pi$
$191$	842.762	0.319268	0.159634	+	0.987176i	$0.448969\pi$
$192$	0	0
$193$	4440.20	1.65602	0.828012	−	0.560711i	$-0.189472\pi$
$193$	4440.20	1.65602	0.828012	+	0.560711i	$0.189472\pi$
$194$	0	0
$195$	1403.89	0.515561
$196$	0	0
$197$	4057.24	1.46734	0.733671	−	0.679505i	$-0.237805\pi$
$197$	4057.24	1.46734	0.733671	+	0.679505i	$0.237805\pi$
$198$	0	0
$199$	1777.04	0.633020	0.316510	−	0.948589i	$-0.397489\pi$
$199$	1777.04	0.633020	0.316510	+	0.948589i	$0.397489\pi$
$200$	0	0
$201$	3889.24	1.36481
$202$	0	0
$203$	684.369	0.236617
$204$	0	0
$205$	−464.990	−0.158421
$206$	0	0
$207$	−661.081	−0.221973
$208$	0	0
$209$	1016.41	0.336396
$210$	0	0
$211$	2180.83	0.711537	0.355769	−	0.934574i	$-0.384219\pi$
$211$	2180.83	0.711537	0.355769	+	0.934574i	$0.384219\pi$
$212$	0	0
$213$	−3206.21	−1.03139
$214$	0	0
$215$	1060.35	0.336349
$216$	0	0
$217$	−1099.06	−0.343820
$218$	0	0
$219$	3820.79	1.17893
$220$	0	0
$221$	151.491	0.0461104
$222$	0	0
$223$	−2778.56	−0.834379	−0.417189	−	0.908820i	$-0.636985\pi$
$223$	−2778.56	−0.834379	−0.417189	+	0.908820i	$0.636985\pi$
$224$	0	0
$225$	−281.606	−0.0834388
$226$	0	0
$227$	−1366.77	−0.399629	−0.199814	−	0.979834i	$-0.564034\pi$
$227$	−1366.77	−0.399629	−0.199814	+	0.979834i	$0.564034\pi$
$228$	0	0
$229$	829.004	0.239223	0.119612	−	0.992821i	$-0.461835\pi$
$229$	829.004	0.239223	0.119612	+	0.992821i	$0.461835\pi$
$230$	0	0
$231$	1794.32	0.511073
$232$	0	0
$233$	−6605.62	−1.85729	−0.928645	−	0.370968i	$-0.879026\pi$
$233$	−6605.62	−1.85729	−0.928645	+	0.370968i	$0.879026\pi$
$234$	0	0
$235$	1960.44	0.544193
$236$	0	0
$237$	−4976.88	−1.36406
$238$	0	0
$239$	6536.17	1.76900	0.884498	−	0.466545i	$-0.154501\pi$
$239$	6536.17	1.76900	0.884498	+	0.466545i	$0.154501\pi$
$240$	0	0
$241$	−5623.06	−1.50296	−0.751480	−	0.659756i	$-0.770660\pi$
$241$	−5623.06	−1.50296	−0.751480	+	0.659756i	$0.770660\pi$
$242$	0	0
$243$	2916.22	0.769859
$244$	0	0
$245$	1069.55	0.278902
$246$	0	0
$247$	3753.39	0.966894
$248$	0	0
$249$	−341.808	−0.0869929
$250$	0	0
$251$	−6063.17	−1.52472	−0.762359	−	0.647155i	$-0.775959\pi$
$251$	−6063.17	−1.52472	−0.762359	+	0.647155i	$0.775959\pi$
$252$	0	0
$253$	1124.91	0.279535
$254$	0	0
$255$	42.4506	0.0104249
$256$	0	0
$257$	4690.99	1.13858	0.569292	−	0.822136i	$-0.307218\pi$
$257$	4690.99	1.13858	0.569292	+	0.822136i	$0.307218\pi$
$258$	0	0
$259$	−7618.48	−1.82776
$260$	0	0
$261$	−326.663	−0.0774710
$262$	0	0
$263$	−7251.18	−1.70010	−0.850051	−	0.526701i	$-0.823429\pi$
$263$	−7251.18	−1.70010	−0.850051	+	0.526701i	$0.823429\pi$
$264$	0	0
$265$	−363.201	−0.0841934
$266$	0	0
$267$	−6055.91	−1.38807
$268$	0	0
$269$	1971.03	0.446751	0.223375	−	0.974733i	$-0.428292\pi$
$269$	1971.03	0.446751	0.223375	+	0.974733i	$0.428292\pi$
$270$	0	0
$271$	−4998.08	−1.12034	−0.560170	−	0.828378i	$-0.689264\pi$
$271$	−4998.08	−1.12034	−0.560170	+	0.828378i	$0.689264\pi$
$272$	0	0
$273$	6626.04	1.46896
$274$	0	0
$275$	479.186	0.105076
$276$	0	0
$277$	2912.82	0.631822	0.315911	−	0.948789i	$-0.397690\pi$
$277$	2912.82	0.631822	0.315911	+	0.948789i	$0.397690\pi$
$278$	0	0
$279$	524.602	0.112570
$280$	0	0
$281$	−1094.80	−0.232421	−0.116210	−	0.993225i	$-0.537075\pi$
$281$	−1094.80	−0.232421	−0.116210	+	0.993225i	$0.537075\pi$
$282$	0	0
$283$	1672.22	0.351247	0.175624	−	0.984457i	$-0.443806\pi$
$283$	1672.22	0.351247	0.175624	+	0.984457i	$0.443806\pi$
$284$	0	0
$285$	1051.77	0.218602
$286$	0	0
$287$	−2194.65	−0.451380
$288$	0	0
$289$	−4908.42	−0.999068
$290$	0	0
$291$	2330.43	0.469458
$292$	0	0
$293$	1679.59	0.334889	0.167445	−	0.985881i	$-0.446448\pi$
$293$	1679.59	0.334889	0.167445	+	0.985881i	$0.446448\pi$
$294$	0	0
$295$	2217.37	0.437628
$296$	0	0
$297$	−2909.39	−0.568416
$298$	0	0
$299$	4154.04	0.803460
$300$	0	0
$301$	5004.62	0.958343
$302$	0	0
$303$	4903.64	0.929724
$304$	0	0
$305$	−2920.96	−0.548373
$306$	0	0
$307$	−9564.56	−1.77810	−0.889052	−	0.457806i	$-0.848636\pi$
$307$	−9564.56	−1.77810	−0.889052	+	0.457806i	$0.848636\pi$
$308$	0	0
$309$	2071.32	0.381338
$310$	0	0
$311$	3504.65	0.639005	0.319502	−	0.947585i	$-0.396484\pi$
$311$	3504.65	0.639005	0.319502	+	0.947585i	$0.396484\pi$
$312$	0	0
$313$	7739.98	1.39773	0.698865	−	0.715254i	$-0.253689\pi$
$313$	7739.98	1.39773	0.698865	+	0.715254i	$0.253689\pi$
$314$	0	0
$315$	−1329.12	−0.237738
$316$	0	0
$317$	−5176.08	−0.917090	−0.458545	−	0.888671i	$-0.651629\pi$
$317$	−5176.08	−0.917090	−0.458545	+	0.888671i	$0.651629\pi$
$318$	0	0
$319$	555.856	0.0975610
$320$	0	0
$321$	1539.13	0.267620
$322$	0	0
$323$	113.495	0.0195511
$324$	0	0
$325$	1769.53	0.302018
$326$	0	0
$327$	5076.69	0.858538
$328$	0	0
$329$	9252.88	1.55054
$330$	0	0
$331$	8390.95	1.39338	0.696690	−	0.717373i	$-0.254656\pi$
$331$	8390.95	1.39338	0.696690	+	0.717373i	$0.254656\pi$
$332$	0	0
$333$	3636.45	0.598427
$334$	0	0
$335$	4902.20	0.799510
$336$	0	0
$337$	5357.63	0.866020	0.433010	−	0.901389i	$-0.357451\pi$
$337$	5357.63	0.866020	0.433010	+	0.901389i	$0.357451\pi$
$338$	0	0
$339$	−3794.81	−0.607982
$340$	0	0
$341$	−892.674	−0.141762
$342$	0	0
$343$	−3046.39	−0.479561
$344$	0	0
$345$	1164.04	0.181651
$346$	0	0
$347$	−3084.25	−0.477150	−0.238575	−	0.971124i	$-0.576680\pi$
$347$	−3084.25	−0.477150	−0.238575	+	0.971124i	$0.576680\pi$
$348$	0	0
$349$	4693.21	0.719833	0.359916	−	0.932985i	$-0.382805\pi$
$349$	4693.21	0.719833	0.359916	+	0.932985i	$0.382805\pi$
$350$	0	0
$351$	−10743.7	−1.63378
$352$	0	0
$353$	9479.76	1.42934	0.714670	−	0.699462i	$-0.246577\pi$
$353$	9479.76	1.42934	0.714670	+	0.699462i	$0.246577\pi$
$354$	0	0
$355$	−4041.27	−0.604192
$356$	0	0
$357$	200.358	0.0297032
$358$	0	0
$359$	284.925	0.0418879	0.0209440	−	0.999781i	$-0.493333\pi$
$359$	284.925	0.0418879	0.0209440	+	0.999781i	$0.493333\pi$
$360$	0	0
$361$	−4047.01	−0.590030
$362$	0	0
$363$	−3822.48	−0.552694
$364$	0	0
$365$	4815.92	0.690622
$366$	0	0
$367$	−10974.1	−1.56088	−0.780440	−	0.625230i	$-0.785005\pi$
$367$	−10974.1	−1.56088	−0.780440	+	0.625230i	$0.785005\pi$
$368$	0	0
$369$	1047.55	0.147787
$370$	0	0
$371$	−1714.23	−0.239888
$372$	0	0
$373$	2571.23	0.356925	0.178463	−	0.983947i	$-0.442888\pi$
$373$	2571.23	0.356925	0.178463	+	0.983947i	$0.442888\pi$
$374$	0	0
$375$	495.854	0.0682821
$376$	0	0
$377$	2052.65	0.280417
$378$	0	0
$379$	7053.33	0.955950	0.477975	−	0.878373i	$-0.341371\pi$
$379$	7053.33	0.955950	0.477975	+	0.878373i	$0.341371\pi$
$380$	0	0
$381$	4754.59	0.639331
$382$	0	0
$383$	706.482	0.0942547	0.0471273	−	0.998889i	$-0.484993\pi$
$383$	706.482	0.0942547	0.0471273	+	0.998889i	$0.484993\pi$
$384$	0	0
$385$	2261.66	0.299389
$386$	0	0
$387$	−2388.80	−0.313772
$388$	0	0
$389$	3573.96	0.465828	0.232914	−	0.972497i	$-0.425174\pi$
$389$	3573.96	0.465828	0.232914	+	0.972497i	$0.425174\pi$
$390$	0	0
$391$	125.610	0.0162464
$392$	0	0
$393$	−7782.21	−0.998882
$394$	0	0
$395$	−6273.11	−0.799075
$396$	0	0
$397$	−3291.08	−0.416057	−0.208028	−	0.978123i	$-0.566705\pi$
$397$	−3291.08	−0.416057	−0.208028	+	0.978123i	$0.566705\pi$
$398$	0	0
$399$	4964.13	0.622850
$400$	0	0
$401$	−12674.7	−1.57841	−0.789206	−	0.614128i	$-0.789508\pi$
$401$	−12674.7	−1.57841	−0.789206	+	0.614128i	$0.789508\pi$
$402$	0	0
$403$	−3296.45	−0.407463
$404$	0	0
$405$	−1489.91	−0.182801
$406$	0	0
$407$	−6187.86	−0.753614
$408$	0	0
$409$	8132.39	0.983180	0.491590	−	0.870827i	$-0.336416\pi$
$409$	8132.39	0.983180	0.491590	+	0.870827i	$0.336416\pi$
$410$	0	0
$411$	5696.32	0.683646
$412$	0	0
$413$	10465.5	1.24691
$414$	0	0
$415$	−430.833	−0.0509608
$416$	0	0
$417$	9061.37	1.06412
$418$	0	0
$419$	−5662.84	−0.660258	−0.330129	−	0.943936i	$-0.607092\pi$
$419$	−5662.84	−0.660258	−0.330129	+	0.943936i	$0.607092\pi$
$420$	0	0
$421$	−10559.3	−1.22239	−0.611196	−	0.791480i	$-0.709311\pi$
$421$	−10559.3	−1.22239	−0.611196	+	0.791480i	$0.709311\pi$
$422$	0	0
$423$	−4416.58	−0.507663
$424$	0	0
$425$	53.5069	0.00610697
$426$	0	0
$427$	−13786.3	−1.56245
$428$	0	0
$429$	5381.78	0.605676
$430$	0	0
$431$	−5997.30	−0.670255	−0.335127	−	0.942173i	$-0.608779\pi$
$431$	−5997.30	−0.670255	−0.335127	+	0.942173i	$0.608779\pi$
$432$	0	0
$433$	4309.43	0.478286	0.239143	−	0.970984i	$-0.423134\pi$
$433$	4309.43	0.478286	0.239143	+	0.970984i	$0.423134\pi$
$434$	0	0
$435$	575.191	0.0633984
$436$	0	0
$437$	3112.14	0.340673
$438$	0	0
$439$	−5441.67	−0.591610	−0.295805	−	0.955248i	$-0.595588\pi$
$439$	−5441.67	−0.591610	−0.295805	+	0.955248i	$0.595588\pi$
$440$	0	0
$441$	−2409.53	−0.260181
$442$	0	0
$443$	−11293.4	−1.21121	−0.605605	−	0.795766i	$-0.707069\pi$
$443$	−11293.4	−1.21121	−0.605605	+	0.795766i	$0.707069\pi$
$444$	0	0
$445$	−7633.18	−0.813140
$446$	0	0
$447$	−8175.42	−0.865065
$448$	0	0
$449$	−2462.85	−0.258862	−0.129431	−	0.991588i	$-0.541315\pi$
$449$	−2462.85	−0.258862	−0.129431	+	0.991588i	$0.541315\pi$
$450$	0	0
$451$	−1782.53	−0.186111
$452$	0	0
$453$	−6186.93	−0.641694
$454$	0	0
$455$	8351.80	0.860524
$456$	0	0
$457$	3887.80	0.397951	0.198975	−	0.980005i	$-0.436239\pi$
$457$	3887.80	0.397951	0.198975	+	0.980005i	$0.436239\pi$
$458$	0	0
$459$	−324.868	−0.0330360
$460$	0	0
$461$	−14181.7	−1.43277	−0.716386	−	0.697704i	$-0.754205\pi$
$461$	−14181.7	−1.43277	−0.716386	+	0.697704i	$0.754205\pi$
$462$	0	0
$463$	−448.093	−0.0449777	−0.0224888	−	0.999747i	$-0.507159\pi$
$463$	−448.093	−0.0449777	−0.0224888	+	0.999747i	$0.507159\pi$
$464$	0	0
$465$	−923.724	−0.0921219
$466$	0	0
$467$	452.628	0.0448504	0.0224252	−	0.999749i	$-0.492861\pi$
$467$	452.628	0.0448504	0.0224252	+	0.999749i	$0.492861\pi$
$468$	0	0
$469$	23137.3	2.27800
$470$	0	0
$471$	−4327.79	−0.423384
$472$	0	0
$473$	4064.83	0.395140
$474$	0	0
$475$	1325.70	0.128058
$476$	0	0
$477$	818.236	0.0785419
$478$	0	0
$479$	−18089.3	−1.72552	−0.862759	−	0.505616i	$-0.831265\pi$
$479$	−18089.3	−1.72552	−0.862759	+	0.505616i	$0.831265\pi$
$480$	0	0
$481$	−22850.4	−2.16609
$482$	0	0
$483$	5494.01	0.517570
$484$	0	0
$485$	2937.40	0.275011
$486$	0	0
$487$	−9117.33	−0.848348	−0.424174	−	0.905581i	$-0.639436\pi$
$487$	−9117.33	−0.848348	−0.424174	+	0.905581i	$0.639436\pi$
$488$	0	0
$489$	−2490.86	−0.230349
$490$	0	0
$491$	−7019.77	−0.645209	−0.322605	−	0.946534i	$-0.604558\pi$
$491$	−7019.77	−0.645209	−0.322605	+	0.946534i	$0.604558\pi$
$492$	0	0
$493$	62.0680	0.00567018
$494$	0	0
$495$	−1079.53	−0.0980231
$496$	0	0
$497$	−19073.9	−1.72149
$498$	0	0
$499$	−18234.2	−1.63582	−0.817912	−	0.575344i	$-0.804868\pi$
$499$	−18234.2	−1.63582	−0.817912	+	0.575344i	$0.804868\pi$
$500$	0	0
$501$	−1052.48	−0.0938547
$502$	0	0
$503$	−12432.7	−1.10208	−0.551039	−	0.834479i	$-0.685769\pi$
$503$	−12432.7	−1.10208	−0.551039	+	0.834479i	$0.685769\pi$
$504$	0	0
$505$	6180.79	0.544637
$506$	0	0
$507$	11158.6	0.977457
$508$	0	0
$509$	8000.10	0.696656	0.348328	−	0.937373i	$-0.386750\pi$
$509$	8000.10	0.696656	0.348328	+	0.937373i	$0.386750\pi$
$510$	0	0
$511$	22730.1	1.96775
$512$	0	0
$513$	−8049.03	−0.692735
$514$	0	0
$515$	2610.80	0.223389
$516$	0	0
$517$	7515.34	0.639312
$518$	0	0
$519$	14518.0	1.22788
$520$	0	0
$521$	−18899.7	−1.58927	−0.794635	−	0.607087i	$-0.792338\pi$
$521$	−18899.7	−1.58927	−0.794635	+	0.607087i	$0.792338\pi$
$522$	0	0
$523$	−18571.8	−1.55275	−0.776374	−	0.630273i	$-0.782943\pi$
$523$	−18571.8	−1.55275	−0.776374	+	0.630273i	$0.782943\pi$
$524$	0	0
$525$	2340.33	0.194553
$526$	0	0
$527$	−99.6777	−0.00823914
$528$	0	0
$529$	−8722.66	−0.716911
$530$	0	0
$531$	−4995.40	−0.408252
$532$	0	0
$533$	−6582.50	−0.534934
$534$	0	0
$535$	1940.00	0.156773
$536$	0	0
$537$	−8171.12	−0.656629
$538$	0	0
$539$	4100.11	0.327652
$540$	0	0
$541$	13942.1	1.10798	0.553990	−	0.832523i	$-0.313105\pi$
$541$	13942.1	1.10798	0.553990	+	0.832523i	$0.313105\pi$
$542$	0	0
$543$	6005.92	0.474657
$544$	0	0
$545$	6398.93	0.502936
$546$	0	0
$547$	−8368.97	−0.654171	−0.327085	−	0.944995i	$-0.606066\pi$
$547$	−8368.97	−0.654171	−0.327085	+	0.944995i	$0.606066\pi$
$548$	0	0
$549$	6580.48	0.511563
$550$	0	0
$551$	1537.82	0.118899
$552$	0	0
$553$	−29607.8	−2.27676
$554$	0	0
$555$	−6403.10	−0.489723
$556$	0	0
$557$	−5976.94	−0.454670	−0.227335	−	0.973817i	$-0.573001\pi$
$557$	−5976.94	−0.454670	−0.227335	+	0.973817i	$0.573001\pi$
$558$	0	0
$559$	15010.5	1.13574
$560$	0	0
$561$	162.734	0.0122471
$562$	0	0
$563$	−12376.7	−0.926496	−0.463248	−	0.886229i	$-0.653316\pi$
$563$	−12376.7	−0.926496	−0.463248	+	0.886229i	$0.653316\pi$
$564$	0	0
$565$	−4783.18	−0.356159
$566$	0	0
$567$	−7032.07	−0.520845
$568$	0	0
$569$	−12847.3	−0.946549	−0.473274	−	0.880915i	$-0.656928\pi$
$569$	−12847.3	−0.946549	−0.473274	+	0.880915i	$0.656928\pi$
$570$	0	0
$571$	−21342.6	−1.56420	−0.782102	−	0.623151i	$-0.785852\pi$
$571$	−21342.6	−1.56420	−0.782102	+	0.623151i	$0.785852\pi$
$572$	0	0
$573$	3343.10	0.243734
$574$	0	0
$575$	1467.21	0.106412
$576$	0	0
$577$	13579.1	0.979731	0.489866	−	0.871798i	$-0.337046\pi$
$577$	13579.1	0.979731	0.489866	+	0.871798i	$0.337046\pi$
$578$	0	0
$579$	17613.5	1.26424
$580$	0	0
$581$	−2033.44	−0.145200
$582$	0	0
$583$	−1392.33	−0.0989096
$584$	0	0
$585$	−3986.48	−0.281745
$586$	0	0
$587$	10153.3	0.713924	0.356962	−	0.934119i	$-0.383813\pi$
$587$	10153.3	0.713924	0.356962	+	0.934119i	$0.383813\pi$
$588$	0	0
$589$	−2469.65	−0.172767
$590$	0	0
$591$	16094.4	1.12019
$592$	0	0
$593$	12148.1	0.841254	0.420627	−	0.907234i	$-0.361810\pi$
$593$	12148.1	0.841254	0.420627	+	0.907234i	$0.361810\pi$
$594$	0	0
$595$	252.541	0.0174003
$596$	0	0
$597$	7049.22	0.483259
$598$	0	0
$599$	10234.8	0.698135	0.349067	−	0.937098i	$-0.386498\pi$
$599$	10234.8	0.698135	0.349067	+	0.937098i	$0.386498\pi$
$600$	0	0
$601$	−2380.59	−0.161574	−0.0807871	−	0.996731i	$-0.525743\pi$
$601$	−2380.59	−0.161574	−0.0807871	+	0.996731i	$0.525743\pi$
$602$	0	0
$603$	−11043.9	−0.745842
$604$	0	0
$605$	−4818.04	−0.323771
$606$	0	0
$607$	−9305.61	−0.622246	−0.311123	−	0.950370i	$-0.600705\pi$
$607$	−9305.61	−0.622246	−0.311123	+	0.950370i	$0.600705\pi$
$608$	0	0
$609$	2714.78	0.180638
$610$	0	0
$611$	27752.5	1.83756
$612$	0	0
$613$	−18136.0	−1.19495	−0.597475	−	0.801887i	$-0.703829\pi$
$613$	−18136.0	−1.19495	−0.597475	+	0.801887i	$0.703829\pi$
$614$	0	0
$615$	−1844.54	−0.120941
$616$	0	0
$617$	−10399.0	−0.678520	−0.339260	−	0.940693i	$-0.610177\pi$
$617$	−10399.0	−0.678520	−0.339260	+	0.940693i	$0.610177\pi$
$618$	0	0
$619$	4286.80	0.278354	0.139177	−	0.990268i	$-0.455554\pi$
$619$	4286.80	0.278354	0.139177	+	0.990268i	$0.455554\pi$
$620$	0	0
$621$	−8908.20	−0.575642
$622$	0	0
$623$	−36027.0	−2.31684
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	4031.95	0.256811
$628$	0	0
$629$	−690.948	−0.0437995
$630$	0	0
$631$	3077.17	0.194137	0.0970683	−	0.995278i	$-0.469053\pi$
$631$	3077.17	0.194137	0.0970683	+	0.995278i	$0.469053\pi$
$632$	0	0
$633$	8650.98	0.543200
$634$	0	0
$635$	5992.93	0.374523
$636$	0	0
$637$	15140.8	0.941759
$638$	0	0
$639$	9104.35	0.563635
$640$	0	0
$641$	−20569.8	−1.26749	−0.633744	−	0.773543i	$-0.718483\pi$
$641$	−20569.8	−1.26749	−0.633744	+	0.773543i	$0.718483\pi$
$642$	0	0
$643$	−16825.3	−1.03192	−0.515962	−	0.856612i	$-0.672565\pi$
$643$	−16825.3	−1.03192	−0.515962	+	0.856612i	$0.672565\pi$
$644$	0	0
$645$	4206.22	0.256775
$646$	0	0
$647$	15753.1	0.957213	0.478607	−	0.878029i	$-0.341142\pi$
$647$	15753.1	0.957213	0.478607	+	0.878029i	$0.341142\pi$
$648$	0	0
$649$	8500.27	0.514122
$650$	0	0
$651$	−4359.78	−0.262478
$652$	0	0
$653$	−4653.09	−0.278850	−0.139425	−	0.990233i	$-0.544525\pi$
$653$	−4653.09	−0.278850	−0.139425	+	0.990233i	$0.544525\pi$
$654$	0	0
$655$	−9809.10	−0.585150
$656$	0	0
$657$	−10849.5	−0.644263
$658$	0	0
$659$	−17024.0	−1.00631	−0.503157	−	0.864195i	$-0.667828\pi$
$659$	−17024.0	−1.00631	−0.503157	+	0.864195i	$0.667828\pi$
$660$	0	0
$661$	31266.5	1.83983	0.919914	−	0.392121i	$-0.128259\pi$
$661$	31266.5	1.83983	0.919914	+	0.392121i	$0.128259\pi$
$662$	0	0
$663$	600.940	0.0352015
$664$	0	0
$665$	6257.04	0.364869
$666$	0	0
$667$	1701.97	0.0988012
$668$	0	0
$669$	−11022.1	−0.636979
$670$	0	0
$671$	−11197.5	−0.644224
$672$	0	0
$673$	−21149.7	−1.21138	−0.605691	−	0.795700i	$-0.707103\pi$
$673$	−21149.7	−1.21138	−0.605691	+	0.795700i	$0.707103\pi$
$674$	0	0
$675$	−3794.70	−0.216382
$676$	0	0
$677$	−23156.0	−1.31456	−0.657280	−	0.753646i	$-0.728293\pi$
$677$	−23156.0	−1.31456	−0.657280	+	0.753646i	$0.728293\pi$
$678$	0	0
$679$	13863.9	0.783574
$680$	0	0
$681$	−5421.75	−0.305084
$682$	0	0
$683$	22444.1	1.25739	0.628696	−	0.777651i	$-0.283589\pi$
$683$	22444.1	1.25739	0.628696	+	0.777651i	$0.283589\pi$
$684$	0	0
$685$	7179.93	0.400483
$686$	0	0
$687$	3288.52	0.182627
$688$	0	0
$689$	−5141.56	−0.284293
$690$	0	0
$691$	2399.87	0.132121	0.0660604	−	0.997816i	$-0.478957\pi$
$691$	2399.87	0.132121	0.0660604	+	0.997816i	$0.478957\pi$
$692$	0	0
$693$	−5095.17	−0.279292
$694$	0	0
$695$	11421.4	0.623366
$696$	0	0
$697$	−199.041	−0.0108167
$698$	0	0
$699$	−26203.4	−1.41789
$700$	0	0
$701$	−17664.6	−0.951759	−0.475880	−	0.879510i	$-0.657870\pi$
$701$	−17664.6	−0.951759	−0.475880	+	0.879510i	$0.657870\pi$
$702$	0	0
$703$	−17119.2	−0.918438
$704$	0	0
$705$	7776.76	0.415446
$706$	0	0
$707$	29172.0	1.55181
$708$	0	0
$709$	−5982.89	−0.316914	−0.158457	−	0.987366i	$-0.550652\pi$
$709$	−5982.89	−0.316914	−0.158457	+	0.987366i	$0.550652\pi$
$710$	0	0
$711$	14132.4	0.745436
$712$	0	0
$713$	−2733.26	−0.143565
$714$	0	0
$715$	6783.47	0.354808
$716$	0	0
$717$	25927.9	1.35048
$718$	0	0
$719$	7405.01	0.384089	0.192045	−	0.981386i	$-0.438488\pi$
$719$	7405.01	0.384089	0.192045	+	0.981386i	$0.438488\pi$
$720$	0	0
$721$	12322.4	0.636492
$722$	0	0
$723$	−22305.7	−1.14739
$724$	0	0
$725$	725.000	0.0371391
$726$	0	0
$727$	24400.6	1.24480	0.622399	−	0.782700i	$-0.286158\pi$
$727$	24400.6	1.24480	0.622399	+	0.782700i	$0.286158\pi$
$728$	0	0
$729$	19613.7	0.996480
$730$	0	0
$731$	453.887	0.0229653
$732$	0	0
$733$	−1339.06	−0.0674750	−0.0337375	−	0.999431i	$-0.510741\pi$
$733$	−1339.06	−0.0674750	−0.0337375	+	0.999431i	$0.510741\pi$
$734$	0	0
$735$	4242.73	0.212919
$736$	0	0
$737$	18792.5	0.939256
$738$	0	0
$739$	−12325.3	−0.613523	−0.306762	−	0.951786i	$-0.599245\pi$
$739$	−12325.3	−0.613523	−0.306762	+	0.951786i	$0.599245\pi$
$740$	0	0
$741$	14889.1	0.738144
$742$	0	0
$743$	−2735.59	−0.135073	−0.0675364	−	0.997717i	$-0.521514\pi$
$743$	−2735.59	−0.135073	−0.0675364	+	0.997717i	$0.521514\pi$
$744$	0	0
$745$	−10304.7	−0.506759
$746$	0	0
$747$	970.600	0.0475400
$748$	0	0
$749$	9156.40	0.446686
$750$	0	0
$751$	−9673.48	−0.470027	−0.235014	−	0.971992i	$-0.575513\pi$
$751$	−9673.48	−0.470027	−0.235014	+	0.971992i	$0.575513\pi$
$752$	0	0
$753$	−24051.6	−1.16400
$754$	0	0
$755$	−7798.32	−0.375907
$756$	0	0
$757$	−26460.3	−1.27043	−0.635215	−	0.772336i	$-0.719088\pi$
$757$	−26460.3	−1.27043	−0.635215	+	0.772336i	$0.719088\pi$
$758$	0	0
$759$	4462.33	0.213402
$760$	0	0
$761$	−21093.9	−1.00480	−0.502399	−	0.864636i	$-0.667549\pi$
$761$	−21093.9	−1.00480	−0.502399	+	0.864636i	$0.667549\pi$
$762$	0	0
$763$	30201.6	1.43299
$764$	0	0
$765$	−120.543	−0.00569704
$766$	0	0
$767$	31389.6	1.47772
$768$	0	0
$769$	31006.9	1.45402	0.727008	−	0.686629i	$-0.240910\pi$
$769$	31006.9	1.45402	0.727008	+	0.686629i	$0.240910\pi$
$770$	0	0
$771$	18608.4	0.869214
$772$	0	0
$773$	−21013.1	−0.977735	−0.488867	−	0.872358i	$-0.662590\pi$
$773$	−21013.1	−0.977735	−0.488867	+	0.872358i	$0.662590\pi$
$774$	0	0
$775$	−1164.31	−0.0539655
$776$	0	0
$777$	−30221.3	−1.39534
$778$	0	0
$779$	−4931.51	−0.226816
$780$	0	0
$781$	−15492.2	−0.709799
$782$	0	0
$783$	−4401.85	−0.200906
$784$	0	0
$785$	−5454.96	−0.248020
$786$	0	0
$787$	22061.2	0.999236	0.499618	−	0.866246i	$-0.333474\pi$
$787$	22061.2	0.999236	0.499618	+	0.866246i	$0.333474\pi$
$788$	0	0
$789$	−28764.2	−1.29789
$790$	0	0
$791$	−22575.6	−1.01479
$792$	0	0
$793$	−41349.8	−1.85167
$794$	0	0
$795$	−1440.76	−0.0642747
$796$	0	0
$797$	15748.3	0.699916	0.349958	−	0.936765i	$-0.386196\pi$
$797$	15748.3	0.699916	0.349958	+	0.936765i	$0.386196\pi$
$798$	0	0
$799$	839.178	0.0371564
$800$	0	0
$801$	17196.4	0.758557
$802$	0	0
$803$	18461.8	0.811336
$804$	0	0
$805$	6924.93	0.303195
$806$	0	0
$807$	7818.76	0.341057
$808$	0	0
$809$	−10312.6	−0.448174	−0.224087	−	0.974569i	$-0.571940\pi$
$809$	−10312.6	−0.448174	−0.224087	+	0.974569i	$0.571940\pi$
$810$	0	0
$811$	11608.5	0.502627	0.251314	−	0.967906i	$-0.419137\pi$
$811$	11608.5	0.502627	0.251314	+	0.967906i	$0.419137\pi$
$812$	0	0
$813$	−19826.6	−0.855286
$814$	0	0
$815$	−3139.61	−0.134939
$816$	0	0
$817$	11245.7	0.481562
$818$	0	0
$819$	−18815.3	−0.802761
$820$	0	0
$821$	32578.0	1.38487	0.692436	−	0.721479i	$-0.256537\pi$
$821$	32578.0	1.38487	0.692436	+	0.721479i	$0.256537\pi$
$822$	0	0
$823$	17164.6	0.726999	0.363500	−	0.931594i	$-0.381582\pi$
$823$	17164.6	0.726999	0.363500	+	0.931594i	$0.381582\pi$
$824$	0	0
$825$	1900.85	0.0802172
$826$	0	0
$827$	−17509.0	−0.736214	−0.368107	−	0.929783i	$-0.619994\pi$
$827$	−17509.0	−0.736214	−0.368107	+	0.929783i	$0.619994\pi$
$828$	0	0
$829$	18328.0	0.767862	0.383931	−	0.923362i	$-0.374570\pi$
$829$	18328.0	0.767862	0.383931	+	0.923362i	$0.374570\pi$
$830$	0	0
$831$	11554.7	0.482344
$832$	0	0
$833$	457.826	0.0190429
$834$	0	0
$835$	−1326.60	−0.0549806
$836$	0	0
$837$	7069.12	0.291929
$838$	0	0
$839$	28719.4	1.18177	0.590884	−	0.806757i	$-0.298779\pi$
$839$	28719.4	1.18177	0.590884	+	0.806757i	$0.298779\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−4342.88	−0.177434
$844$	0	0
$845$	14064.9	0.572599
$846$	0	0
$847$	−22740.1	−0.922504
$848$	0	0
$849$	6633.40	0.268148
$850$	0	0
$851$	−18946.5	−0.763194
$852$	0	0
$853$	8729.75	0.350411	0.175206	−	0.984532i	$-0.443941\pi$
$853$	8729.75	0.350411	0.175206	+	0.984532i	$0.443941\pi$
$854$	0	0
$855$	−2986.61	−0.119462
$856$	0	0
$857$	22846.3	0.910634	0.455317	−	0.890329i	$-0.349526\pi$
$857$	22846.3	0.910634	0.455317	+	0.890329i	$0.349526\pi$
$858$	0	0
$859$	34370.5	1.36520	0.682599	−	0.730793i	$-0.260849\pi$
$859$	34370.5	1.36520	0.682599	+	0.730793i	$0.260849\pi$
$860$	0	0
$861$	−8705.82	−0.344592
$862$	0	0
$863$	43981.1	1.73480	0.867402	−	0.497608i	$-0.165788\pi$
$863$	43981.1	1.73480	0.867402	+	0.497608i	$0.165788\pi$
$864$	0	0
$865$	18299.2	0.719297
$866$	0	0
$867$	−19470.9	−0.762706
$868$	0	0
$869$	−24047.9	−0.938746
$870$	0	0
$871$	69396.7	2.69967
$872$	0	0
$873$	−6617.50	−0.256550
$874$	0	0
$875$	2949.87	0.113970
$876$	0	0
$877$	−19543.7	−0.752501	−0.376250	−	0.926518i	$-0.622787\pi$
$877$	−19543.7	−0.752501	−0.376250	+	0.926518i	$0.622787\pi$
$878$	0	0
$879$	6662.64	0.255660
$880$	0	0
$881$	−22684.2	−0.867481	−0.433740	−	0.901038i	$-0.642806\pi$
$881$	−22684.2	−0.867481	−0.433740	+	0.901038i	$0.642806\pi$
$882$	0	0
$883$	5394.00	0.205575	0.102787	−	0.994703i	$-0.467224\pi$
$883$	5394.00	0.205575	0.102787	+	0.994703i	$0.467224\pi$
$884$	0	0
$885$	8795.94	0.334093
$886$	0	0
$887$	−17358.4	−0.657091	−0.328545	−	0.944488i	$-0.606558\pi$
$887$	−17358.4	−0.657091	−0.328545	+	0.944488i	$0.606558\pi$
$888$	0	0
$889$	28285.3	1.06711
$890$	0	0
$891$	−5711.57	−0.214753
$892$	0	0
$893$	20791.7	0.779137
$894$	0	0
$895$	−10299.3	−0.384656
$896$	0	0
$897$	16478.4	0.613375
$898$	0	0
$899$	−1350.60	−0.0501057
$900$	0	0
$901$	−155.470	−0.00574856
$902$	0	0
$903$	19852.5	0.731616
$904$	0	0
$905$	7570.17	0.278056
$906$	0	0
$907$	13260.5	0.485455	0.242728	−	0.970094i	$-0.421958\pi$
$907$	13260.5	0.485455	0.242728	+	0.970094i	$0.421958\pi$
$908$	0	0
$909$	−13924.4	−0.508078
$910$	0	0
$911$	8277.28	0.301030	0.150515	−	0.988608i	$-0.451907\pi$
$911$	8277.28	0.301030	0.150515	+	0.988608i	$0.451907\pi$
$912$	0	0
$913$	−1651.59	−0.0598683
$914$	0	0
$915$	−11587.0	−0.418638
$916$	0	0
$917$	−46296.9	−1.66724
$918$	0	0
$919$	18282.0	0.656223	0.328111	−	0.944639i	$-0.393588\pi$
$919$	18282.0	0.656223	0.328111	+	0.944639i	$0.393588\pi$
$920$	0	0
$921$	−37941.0	−1.35744
$922$	0	0
$923$	−57209.1	−2.04015
$924$	0	0
$925$	−8070.79	−0.286882
$926$	0	0
$927$	−5881.73	−0.208394
$928$	0	0
$929$	30690.7	1.08389	0.541943	−	0.840415i	$-0.317689\pi$
$929$	30690.7	1.08389	0.541943	+	0.840415i	$0.317689\pi$
$930$	0	0
$931$	11343.2	0.399313
$932$	0	0
$933$	13902.4	0.487827
$934$	0	0
$935$	205.118	0.00717441
$936$	0	0
$937$	25526.2	0.889972	0.444986	−	0.895537i	$-0.353209\pi$
$937$	25526.2	0.889972	0.444986	+	0.895537i	$0.353209\pi$
$938$	0	0
$939$	30703.2	1.06705
$940$	0	0
$941$	−9533.59	−0.330272	−0.165136	−	0.986271i	$-0.552806\pi$
$941$	−9533.59	−0.330272	−0.165136	+	0.986271i	$0.552806\pi$
$942$	0	0
$943$	−5457.91	−0.188477
$944$	0	0
$945$	−17910.2	−0.616526
$946$	0	0
$947$	−11431.0	−0.392247	−0.196123	−	0.980579i	$-0.562835\pi$
$947$	−11431.0	−0.392247	−0.196123	+	0.980579i	$0.562835\pi$
$948$	0	0
$949$	68175.3	2.33200
$950$	0	0
$951$	−20532.6	−0.700123
$952$	0	0
$953$	−22865.7	−0.777222	−0.388611	−	0.921402i	$-0.627045\pi$
$953$	−22865.7	−0.777222	−0.388611	+	0.921402i	$0.627045\pi$
$954$	0	0
$955$	4213.81	0.142781
$956$	0	0
$957$	2204.99	0.0744798
$958$	0	0
$959$	33887.8	1.14108
$960$	0	0
$961$	−27622.0	−0.927193
$962$	0	0
$963$	−4370.53	−0.146250
$964$	0	0
$965$	22201.0	0.740596
$966$	0	0
$967$	34059.5	1.13266	0.566329	−	0.824179i	$-0.308363\pi$
$967$	34059.5	1.13266	0.566329	+	0.824179i	$0.308363\pi$
$968$	0	0
$969$	450.215	0.0149257
$970$	0	0
$971$	22831.8	0.754589	0.377294	−	0.926093i	$-0.376854\pi$
$971$	22831.8	0.754589	0.377294	+	0.926093i	$0.376854\pi$
$972$	0	0
$973$	53906.7	1.77612
$974$	0	0
$975$	7019.43	0.230566
$976$	0	0
$977$	21776.8	0.713103	0.356551	−	0.934276i	$-0.383952\pi$
$977$	21776.8	0.713103	0.356551	+	0.934276i	$0.383952\pi$
$978$	0	0
$979$	−29261.7	−0.955269
$980$	0	0
$981$	−14415.8	−0.469175
$982$	0	0
$983$	−21311.1	−0.691472	−0.345736	−	0.938332i	$-0.612371\pi$
$983$	−21311.1	−0.691472	−0.345736	+	0.938332i	$0.612371\pi$
$984$	0	0
$985$	20286.2	0.656215
$986$	0	0
$987$	36704.6	1.18371
$988$	0	0
$989$	12446.0	0.400163
$990$	0	0
$991$	30304.7	0.971404	0.485702	−	0.874125i	$-0.338564\pi$
$991$	30304.7	0.971404	0.485702	+	0.874125i	$0.338564\pi$
$992$	0	0
$993$	33285.5	1.06373
$994$	0	0
$995$	8885.20	0.283095
$996$	0	0
$997$	−17844.3	−0.566835	−0.283418	−	0.958997i	$-0.591468\pi$
$997$	−17844.3	−0.566835	−0.283418	+	0.958997i	$0.591468\pi$
$998$	0	0
$999$	49001.9	1.55190

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1160.4.a.j.1.10	✓	13
4.3	odd	2		2320.4.a.bc.1.4		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.4.a.j.1.10	✓	13	1.1	even	1	trivial
2320.4.a.bc.1.4		13	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)

\(f(q)\)	\(=\)	\(q+3.96683 q^{3} +5.00000 q^{5} +23.5989 q^{7} -11.2642 q^{9} +19.1675 q^{11} +70.7812 q^{13} +19.8342 q^{15} +2.14027 q^{17} +53.0282 q^{19} +93.6131 q^{21} +58.6885 q^{23} +25.0000 q^{25} -151.788 q^{27} +29.0000 q^{29} -46.5724 q^{31} +76.0341 q^{33} +117.995 q^{35} -322.832 q^{37} +280.777 q^{39} -92.9979 q^{41} +212.070 q^{43} -56.3212 q^{45} +392.089 q^{47} +213.910 q^{49} +8.49011 q^{51} -72.6402 q^{53} +95.8373 q^{55} +210.354 q^{57} +443.474 q^{59} -584.193 q^{61} -265.824 q^{63} +353.906 q^{65} +980.440 q^{67} +232.808 q^{69} -808.253 q^{71} +963.185 q^{73} +99.1708 q^{75} +452.332 q^{77} -1254.62 q^{79} -297.983 q^{81} -86.1666 q^{83} +10.7014 q^{85} +115.038 q^{87} -1526.64 q^{89} +1670.36 q^{91} -184.745 q^{93} +265.141 q^{95} +587.479 q^{97} -215.907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 6 q^{3} + 65 q^{5} + 2 q^{7} + 147 q^{9} - 42 q^{11} + 100 q^{13} - 30 q^{15} + 136 q^{17} + 38 q^{19} + 376 q^{21} + 86 q^{23} + 325 q^{25} - 216 q^{27} + 377 q^{29} + 230 q^{31} + 512 q^{33} + 10 q^{35}+ \cdots - 846 q^{99}+O(q^{100}) \) 13 * q - 6 * q^3 + 65 * q^5 + 2 * q^7 + 147 * q^9 - 42 * q^11 + 100 * q^13 - 30 * q^15 + 136 * q^17 + 38 * q^19 + 376 * q^21 + 86 * q^23 + 325 * q^25 - 216 * q^27 + 377 * q^29 + 230 * q^31 + 512 * q^33 + 10 * q^35 + 614 * q^37 - 188 * q^39 + 506 * q^41 + 198 * q^43 + 735 * q^45 - 582 * q^47 + 1383 * q^49 + 336 * q^51 + 772 * q^53 - 210 * q^55 + 1104 * q^57 - 72 * q^59 + 1040 * q^61 - 502 * q^63 + 500 * q^65 - 374 * q^67 - 258 * q^69 + 820 * q^71 + 648 * q^73 - 150 * q^75 + 952 * q^77 + 378 * q^79 + 3857 * q^81 + 478 * q^83 + 680 * q^85 - 174 * q^87 + 2378 * q^89 + 888 * q^91 + 3656 * q^93 + 190 * q^95 + 1760 * q^97 - 846 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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