LMFDB - Embedded newform 1008.4.a.bh.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(1,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.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:	$59.4739252858$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{22}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 22 $ x^2 - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-4.69042$ of defining polynomial
Character	$\chi$	$=$	1008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.38083 q^{5} +7.00000 q^{7} +O(q^{10})$	$q-3.38083 q^{5} +7.00000 q^{7} -11.3808 q^{11} +7.52333 q^{13} +30.4275 q^{17} -108.570 q^{19} +185.474 q^{23} -113.570 q^{25} -120.000 q^{29} +79.6166 q^{31} -23.6658 q^{35} +37.6166 q^{37} -202.811 q^{41} -45.9067 q^{43} +158.192 q^{47} +49.0000 q^{49} -405.995 q^{53} +38.4767 q^{55} +625.808 q^{59} -567.897 q^{61} -25.4351 q^{65} -415.047 q^{67} +148.049 q^{71} -86.9533 q^{73} -79.6658 q^{77} -807.606 q^{79} +401.140 q^{83} -102.870 q^{85} -1185.85 q^{89} +52.6633 q^{91} +367.057 q^{95} -246.953 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 12 q^{5} + 14 q^{7}+O(q^{10}) $ 2 * q + 12 * q^5 + 14 * q^7	$ 2 q + 12 q^{5} + 14 q^{7} - 4 q^{11} - 60 q^{13} - 108 q^{17} + 8 q^{19} + 52 q^{23} - 2 q^{25} - 240 q^{29} - 216 q^{31} + 84 q^{35} - 300 q^{37} - 612 q^{41} - 392 q^{43} + 504 q^{47} + 98 q^{49} - 24 q^{53} + 152 q^{55} + 1064 q^{59} + 140 q^{61} - 1064 q^{65} - 680 q^{67} + 540 q^{71} - 324 q^{73} - 28 q^{77} + 336 q^{79} + 352 q^{83} - 2232 q^{85} - 852 q^{89} - 420 q^{91} + 2160 q^{95} - 644 q^{97}+O(q^{100}) $ 2 * q + 12 * q^5 + 14 * q^7 - 4 * q^11 - 60 * q^13 - 108 * q^17 + 8 * q^19 + 52 * q^23 - 2 * q^25 - 240 * q^29 - 216 * q^31 + 84 * q^35 - 300 * q^37 - 612 * q^41 - 392 * q^43 + 504 * q^47 + 98 * q^49 - 24 * q^53 + 152 * q^55 + 1064 * q^59 + 140 * q^61 - 1064 * q^65 - 680 * q^67 + 540 * q^71 - 324 * q^73 - 28 * q^77 + 336 * q^79 + 352 * q^83 - 2232 * q^85 - 852 * q^89 - 420 * q^91 + 2160 * q^95 - 644 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−3.38083	−0.302391	−0.151195	−	0.988504i	$-0.548312\pi$
$5$	−3.38083	−0.302391	−0.151195	+	0.988504i	$0.548312\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−11.3808	−0.311950	−0.155975	−	0.987761i	$-0.549852\pi$
$11$	−11.3808	−0.311950	−0.155975	+	0.987761i	$0.549852\pi$
$12$	0	0
$13$	7.52333	0.160507	0.0802537	−	0.996774i	$-0.474427\pi$
$13$	7.52333	0.160507	0.0802537	+	0.996774i	$0.474427\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.4275	0.434103	0.217051	−	0.976160i	$-0.430356\pi$
$17$	30.4275	0.434103	0.217051	+	0.976160i	$0.430356\pi$
$18$	0	0
$19$	−108.570	−1.31093	−0.655465	−	0.755226i	$-0.727527\pi$
$19$	−108.570	−1.31093	−0.655465	+	0.755226i	$0.727527\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	185.474	1.68148	0.840740	−	0.541439i	$-0.182120\pi$
$23$	185.474	1.68148	0.840740	+	0.541439i	$0.182120\pi$
$24$	0	0
$25$	−113.570	−0.908560
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−120.000	−0.768395	−0.384197	−	0.923251i	$-0.625522\pi$
$29$	−120.000	−0.768395	−0.384197	+	0.923251i	$0.625522\pi$
$30$	0	0
$31$	79.6166	0.461276	0.230638	−	0.973040i	$-0.425919\pi$
$31$	79.6166	0.461276	0.230638	+	0.973040i	$0.425919\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−23.6658	−0.114293
$36$	0	0
$37$	37.6166	0.167139	0.0835694	−	0.996502i	$-0.473368\pi$
$37$	37.6166	0.167139	0.0835694	+	0.996502i	$0.473368\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−202.811	−0.772530	−0.386265	−	0.922388i	$-0.626235\pi$
$41$	−202.811	−0.772530	−0.386265	+	0.922388i	$0.626235\pi$
$42$	0	0
$43$	−45.9067	−0.162807	−0.0814036	−	0.996681i	$-0.525940\pi$
$43$	−45.9067	−0.162807	−0.0814036	+	0.996681i	$0.525940\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	158.192	0.490950	0.245475	−	0.969403i	$-0.421056\pi$
$47$	158.192	0.490950	0.245475	+	0.969403i	$0.421056\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−405.995	−1.05222	−0.526110	−	0.850416i	$-0.676350\pi$
$53$	−405.995	−1.05222	−0.526110	+	0.850416i	$0.676350\pi$
$54$	0	0
$55$	38.4767	0.0943308
$56$	0	0
$57$	0	0
$58$	0	0
$59$	625.808	1.38090	0.690452	−	0.723378i	$-0.257412\pi$
$59$	625.808	1.38090	0.690452	+	0.723378i	$0.257412\pi$
$60$	0	0
$61$	−567.897	−1.19200	−0.595998	−	0.802986i	$-0.703243\pi$
$61$	−567.897	−1.19200	−0.595998	+	0.802986i	$0.703243\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−25.4351	−0.0485359
$66$	0	0
$67$	−415.047	−0.756806	−0.378403	−	0.925641i	$-0.623527\pi$
$67$	−415.047	−0.756806	−0.378403	+	0.925641i	$0.623527\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	148.049	0.247468	0.123734	−	0.992315i	$-0.460513\pi$
$71$	148.049	0.247468	0.123734	+	0.992315i	$0.460513\pi$
$72$	0	0
$73$	−86.9533	−0.139413	−0.0697063	−	0.997568i	$-0.522206\pi$
$73$	−86.9533	−0.139413	−0.0697063	+	0.997568i	$0.522206\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−79.6658	−0.117906
$78$	0	0
$79$	−807.606	−1.15016	−0.575081	−	0.818096i	$-0.695030\pi$
$79$	−807.606	−1.15016	−0.575081	+	0.818096i	$0.695030\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	401.140	0.530492	0.265246	−	0.964181i	$-0.414547\pi$
$83$	401.140	0.530492	0.265246	+	0.964181i	$0.414547\pi$
$84$	0	0
$85$	−102.870	−0.131269
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1185.85	−1.41235	−0.706177	−	0.708035i	$-0.749582\pi$
$89$	−1185.85	−1.41235	−0.706177	+	0.708035i	$0.749582\pi$
$90$	0	0
$91$	52.6633	0.0606661
$92$	0	0
$93$	0	0
$94$	0	0
$95$	367.057	0.396413
$96$	0	0
$97$	−246.953	−0.258498	−0.129249	−	0.991612i	$-0.541257\pi$
$97$	−246.953	−0.258498	−0.129249	+	0.991612i	$0.541257\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1171.85	−1.15449	−0.577246	−	0.816570i	$-0.695873\pi$
$101$	−1171.85	−1.15449	−0.577246	+	0.816570i	$0.695873\pi$
$102$	0	0
$103$	−275.057	−0.263128	−0.131564	−	0.991308i	$-0.542000\pi$
$103$	−275.057	−0.263128	−0.131564	+	0.991308i	$0.542000\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	421.956	0.381234	0.190617	−	0.981665i	$-0.438951\pi$
$107$	421.956	0.381234	0.190617	+	0.981665i	$0.438951\pi$
$108$	0	0
$109$	−919.140	−0.807685	−0.403842	−	0.914829i	$-0.632326\pi$
$109$	−919.140	−0.807685	−0.403842	+	0.914829i	$0.632326\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−46.8600	−0.0390108	−0.0195054	−	0.999810i	$-0.506209\pi$
$113$	−46.8600	−0.0390108	−0.0195054	+	0.999810i	$0.506209\pi$
$114$	0	0
$115$	−627.057	−0.508464
$116$	0	0
$117$	0	0
$118$	0	0
$119$	212.992	0.164075
$120$	0	0
$121$	−1201.48	−0.902687
$122$	0	0
$123$	0	0
$124$	0	0
$125$	806.565	0.577131
$126$	0	0
$127$	−1140.19	−0.796655	−0.398328	−	0.917243i	$-0.630409\pi$
$127$	−1140.19	−0.796655	−0.398328	+	0.917243i	$0.630409\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−436.363	−0.291032	−0.145516	−	0.989356i	$-0.546484\pi$
$131$	−436.363	−0.291032	−0.145516	+	0.989356i	$0.546484\pi$
$132$	0	0
$133$	−759.990	−0.495485
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1441.61	0.899016	0.449508	−	0.893276i	$-0.351599\pi$
$137$	1441.61	0.899016	0.449508	+	0.893276i	$0.351599\pi$
$138$	0	0
$139$	−1905.89	−1.16299	−0.581493	−	0.813551i	$-0.697531\pi$
$139$	−1905.89	−1.16299	−0.581493	+	0.813551i	$0.697531\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−85.6217	−0.0500703
$144$	0	0
$145$	405.700	0.232355
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2103.68	−1.15665	−0.578324	−	0.815807i	$-0.696293\pi$
$149$	−2103.68	−1.15665	−0.578324	+	0.815807i	$0.696293\pi$
$150$	0	0
$151$	326.073	0.175731	0.0878657	−	0.996132i	$-0.471995\pi$
$151$	326.073	0.175731	0.0878657	+	0.996132i	$0.471995\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−269.170	−0.139486
$156$	0	0
$157$	−2701.60	−1.37332	−0.686659	−	0.726980i	$-0.740923\pi$
$157$	−2701.60	−1.37332	−0.686659	+	0.726980i	$0.740923\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1298.32	0.635540
$162$	0	0
$163$	−816.953	−0.392569	−0.196284	−	0.980547i	$-0.562888\pi$
$163$	−816.953	−0.392569	−0.196284	+	0.980547i	$0.562888\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−152.689	−0.0707509	−0.0353755	−	0.999374i	$-0.511263\pi$
$167$	−152.689	−0.0707509	−0.0353755	+	0.999374i	$0.511263\pi$
$168$	0	0
$169$	−2140.40	−0.974237
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1857.65	0.816385	0.408192	−	0.912896i	$-0.366159\pi$
$173$	1857.65	0.816385	0.408192	+	0.912896i	$0.366159\pi$
$174$	0	0
$175$	−794.990	−0.343403
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2586.52	−1.08003	−0.540015	−	0.841656i	$-0.681581\pi$
$179$	−2586.52	−1.08003	−0.540015	+	0.841656i	$0.681581\pi$
$180$	0	0
$181$	−677.057	−0.278040	−0.139020	−	0.990290i	$-0.544395\pi$
$181$	−677.057	−0.278040	−0.139020	+	0.990290i	$0.544395\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−127.175	−0.0505412
$186$	0	0
$187$	−346.290	−0.135418
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−865.454	−0.327864	−0.163932	−	0.986472i	$-0.552418\pi$
$191$	−865.454	−0.327864	−0.163932	+	0.986472i	$0.552418\pi$
$192$	0	0
$193$	2026.36	0.755755	0.377878	−	0.925856i	$-0.376654\pi$
$193$	2026.36	0.755755	0.377878	+	0.925856i	$0.376654\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	4253.03	1.53815	0.769076	−	0.639157i	$-0.220717\pi$
$197$	4253.03	1.53815	0.769076	+	0.639157i	$0.220717\pi$
$198$	0	0
$199$	−1705.12	−0.607401	−0.303700	−	0.952768i	$-0.598222\pi$
$199$	−1705.12	−0.607401	−0.303700	+	0.952768i	$0.598222\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−840.000	−0.290426
$204$	0	0
$205$	685.669	0.233606
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1235.62	0.408944
$210$	0	0
$211$	−1082.86	−0.353304	−0.176652	−	0.984273i	$-0.556527\pi$
$211$	−1082.86	−0.353304	−0.176652	+	0.984273i	$0.556527\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	155.203	0.0492314
$216$	0	0
$217$	557.316	0.174346
$218$	0	0
$219$	0	0
$220$	0	0
$221$	228.916	0.0696767
$222$	0	0
$223$	1699.79	0.510433	0.255217	−	0.966884i	$-0.417853\pi$
$223$	1699.79	0.510433	0.255217	+	0.966884i	$0.417853\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4144.63	1.21184	0.605922	−	0.795524i	$-0.292804\pi$
$227$	4144.63	1.21184	0.605922	+	0.795524i	$0.292804\pi$
$228$	0	0
$229$	−4551.32	−1.31336	−0.656680	−	0.754169i	$-0.728040\pi$
$229$	−4551.32	−1.31336	−0.656680	+	0.754169i	$0.728040\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3012.09	0.846903	0.423452	−	0.905919i	$-0.360818\pi$
$233$	3012.09	0.846903	0.423452	+	0.905919i	$0.360818\pi$
$234$	0	0
$235$	−534.819	−0.148459
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5285.89	−1.43061	−0.715306	−	0.698812i	$-0.753713\pi$
$239$	−5285.89	−1.43061	−0.715306	+	0.698812i	$0.753713\pi$
$240$	0	0
$241$	3577.03	0.956085	0.478043	−	0.878337i	$-0.341346\pi$
$241$	3577.03	0.956085	0.478043	+	0.878337i	$0.341346\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−165.661	−0.0431987
$246$	0	0
$247$	−816.807	−0.210414
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6277.78	−1.57868	−0.789342	−	0.613953i	$-0.789578\pi$
$251$	−6277.78	−1.57868	−0.789342	+	0.613953i	$0.789578\pi$
$252$	0	0
$253$	−2110.85	−0.524538
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1844.48	−0.447687	−0.223843	−	0.974625i	$-0.571860\pi$
$257$	−1844.48	−0.447687	−0.223843	+	0.974625i	$0.571860\pi$
$258$	0	0
$259$	263.316	0.0631725
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−6319.45	−1.48165	−0.740825	−	0.671698i	$-0.765565\pi$
$263$	−6319.45	−1.48165	−0.740825	+	0.671698i	$0.765565\pi$
$264$	0	0
$265$	1372.60	0.318182
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1350.64	−0.306135	−0.153067	−	0.988216i	$-0.548915\pi$
$269$	−1350.64	−0.306135	−0.153067	+	0.988216i	$0.548915\pi$
$270$	0	0
$271$	−3607.60	−0.808656	−0.404328	−	0.914614i	$-0.632495\pi$
$271$	−3607.60	−0.808656	−0.404328	+	0.914614i	$0.632495\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1292.52	0.283425
$276$	0	0
$277$	−2990.36	−0.648641	−0.324320	−	0.945947i	$-0.605136\pi$
$277$	−2990.36	−0.648641	−0.324320	+	0.945947i	$0.605136\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2483.74	0.527286	0.263643	−	0.964620i	$-0.415076\pi$
$281$	2483.74	0.527286	0.263643	+	0.964620i	$0.415076\pi$
$282$	0	0
$283$	2281.92	0.479314	0.239657	−	0.970858i	$-0.422965\pi$
$283$	2281.92	0.479314	0.239657	+	0.970858i	$0.422965\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1419.68	−0.291989
$288$	0	0
$289$	−3987.17	−0.811555
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5600.47	1.11667	0.558333	−	0.829617i	$-0.311441\pi$
$293$	5600.47	1.11667	0.558333	+	0.829617i	$0.311441\pi$
$294$	0	0
$295$	−2115.75	−0.417572
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1395.38	0.269890
$300$	0	0
$301$	−321.347	−0.0615353
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1919.96	0.360448
$306$	0	0
$307$	6500.86	1.20855	0.604273	−	0.796777i	$-0.293464\pi$
$307$	6500.86	1.20855	0.604273	+	0.796777i	$0.293464\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9201.61	1.67773	0.838867	−	0.544336i	$-0.183218\pi$
$311$	9201.61	1.67773	0.838867	+	0.544336i	$0.183218\pi$
$312$	0	0
$313$	5800.05	1.04741	0.523703	−	0.851901i	$-0.324550\pi$
$313$	5800.05	1.04741	0.523703	+	0.851901i	$0.324550\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−4582.98	−0.812005	−0.406003	−	0.913872i	$-0.633078\pi$
$317$	−4582.98	−0.812005	−0.406003	+	0.913872i	$0.633078\pi$
$318$	0	0
$319$	1365.70	0.239701
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3303.51	−0.569078
$324$	0	0
$325$	−854.424	−0.145831
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1107.34	0.185561
$330$	0	0
$331$	945.327	0.156978	0.0784892	−	0.996915i	$-0.474990\pi$
$331$	945.327	0.156978	0.0784892	+	0.996915i	$0.474990\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1403.20	0.228851
$336$	0	0
$337$	6444.43	1.04169	0.520846	−	0.853651i	$-0.325617\pi$
$337$	6444.43	1.04169	0.520846	+	0.853651i	$0.325617\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−906.103	−0.143895
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1851.97	−0.286510	−0.143255	−	0.989686i	$-0.545757\pi$
$347$	−1851.97	−0.286510	−0.143255	+	0.989686i	$0.545757\pi$
$348$	0	0
$349$	5787.50	0.887673	0.443837	−	0.896108i	$-0.353617\pi$
$349$	5787.50	0.887673	0.443837	+	0.896108i	$0.353617\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−11976.4	−1.80577	−0.902887	−	0.429879i	$-0.858556\pi$
$353$	−11976.4	−1.80577	−0.902887	+	0.429879i	$0.858556\pi$
$354$	0	0
$355$	−500.529	−0.0748319
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6478.11	0.952372	0.476186	−	0.879344i	$-0.342019\pi$
$359$	6478.11	0.952372	0.476186	+	0.879344i	$0.342019\pi$
$360$	0	0
$361$	4928.44	0.718536
$362$	0	0
$363$	0	0
$364$	0	0
$365$	293.975	0.0421571
$366$	0	0
$367$	4440.37	0.631568	0.315784	−	0.948831i	$-0.397732\pi$
$367$	4440.37	0.631568	0.315784	+	0.948831i	$0.397732\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2841.96	−0.397702
$372$	0	0
$373$	−4959.68	−0.688478	−0.344239	−	0.938882i	$-0.611863\pi$
$373$	−4959.68	−0.688478	−0.344239	+	0.938882i	$0.611863\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−902.799	−0.123333
$378$	0	0
$379$	2741.06	0.371501	0.185750	−	0.982597i	$-0.440528\pi$
$379$	2741.06	0.371501	0.185750	+	0.982597i	$0.440528\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5040.20	0.672433	0.336217	−	0.941785i	$-0.390853\pi$
$383$	5040.20	0.672433	0.336217	+	0.941785i	$0.390853\pi$
$384$	0	0
$385$	269.337	0.0356537
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6783.31	0.884132	0.442066	−	0.896983i	$-0.354246\pi$
$389$	6783.31	0.884132	0.442066	+	0.896983i	$0.354246\pi$
$390$	0	0
$391$	5643.51	0.729935
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2730.38	0.347798
$396$	0	0
$397$	−1912.33	−0.241756	−0.120878	−	0.992667i	$-0.538571\pi$
$397$	−1912.33	−0.241756	−0.120878	+	0.992667i	$0.538571\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−13840.1	−1.72354	−0.861771	−	0.507297i	$-0.830645\pi$
$401$	−13840.1	−1.72354	−0.861771	+	0.507297i	$0.830645\pi$
$402$	0	0
$403$	598.982	0.0740382
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−428.109	−0.0521389
$408$	0	0
$409$	12066.5	1.45880	0.729402	−	0.684085i	$-0.239798\pi$
$409$	12066.5	1.45880	0.729402	+	0.684085i	$0.239798\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4380.66	0.521932
$414$	0	0
$415$	−1356.19	−0.160416
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10962.0	1.27812	0.639058	−	0.769158i	$-0.279324\pi$
$419$	10962.0	1.27812	0.639058	+	0.769158i	$0.279324\pi$
$420$	0	0
$421$	−8637.54	−0.999923	−0.499962	−	0.866048i	$-0.666653\pi$
$421$	−8637.54	−0.999923	−0.499962	+	0.866048i	$0.666653\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3455.65	−0.394408
$426$	0	0
$427$	−3975.28	−0.450532
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−939.835	−0.105035	−0.0525177	−	0.998620i	$-0.516725\pi$
$431$	−939.835	−0.105035	−0.0525177	+	0.998620i	$0.516725\pi$
$432$	0	0
$433$	15429.4	1.71245	0.856226	−	0.516602i	$-0.172803\pi$
$433$	15429.4	1.71245	0.856226	+	0.516602i	$0.172803\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20136.9	−2.20430
$438$	0	0
$439$	10085.7	1.09650	0.548249	−	0.836315i	$-0.315295\pi$
$439$	10085.7	1.09650	0.548249	+	0.836315i	$0.315295\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	11172.8	1.19828	0.599138	−	0.800646i	$-0.295510\pi$
$443$	11172.8	1.19828	0.599138	+	0.800646i	$0.295510\pi$
$444$	0	0
$445$	4009.15	0.427083
$446$	0	0
$447$	0	0
$448$	0	0
$449$	14909.4	1.56708	0.783541	−	0.621340i	$-0.213411\pi$
$449$	14909.4	1.56708	0.783541	+	0.621340i	$0.213411\pi$
$450$	0	0
$451$	2308.16	0.240991
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−178.046	−0.0183449
$456$	0	0
$457$	12117.9	1.24037	0.620186	−	0.784454i	$-0.287057\pi$
$457$	12117.9	1.24037	0.620186	+	0.784454i	$0.287057\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1089.85	0.110107	0.0550534	−	0.998483i	$-0.482467\pi$
$461$	1089.85	0.110107	0.0550534	+	0.998483i	$0.482467\pi$
$462$	0	0
$463$	8843.28	0.887650	0.443825	−	0.896113i	$-0.353621\pi$
$463$	8843.28	0.887650	0.443825	+	0.896113i	$0.353621\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−154.534	−0.0153126	−0.00765632	−	0.999971i	$-0.502437\pi$
$467$	−154.534	−0.0153126	−0.00765632	+	0.999971i	$0.502437\pi$
$468$	0	0
$469$	−2905.33	−0.286046
$470$	0	0
$471$	0	0
$472$	0	0
$473$	522.456	0.0507877
$474$	0	0
$475$	12330.3	1.19106
$476$	0	0
$477$	0	0
$478$	0	0
$479$	17392.3	1.65903	0.829513	−	0.558488i	$-0.188618\pi$
$479$	17392.3	1.65903	0.829513	+	0.558488i	$0.188618\pi$
$480$	0	0
$481$	283.002	0.0268270
$482$	0	0
$483$	0	0
$484$	0	0
$485$	834.908	0.0781674
$486$	0	0
$487$	11408.6	1.06155	0.530775	−	0.847513i	$-0.321901\pi$
$487$	11408.6	1.06155	0.530775	+	0.847513i	$0.321901\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1031.06	0.0947678	0.0473839	−	0.998877i	$-0.484912\pi$
$491$	1031.06	0.0947678	0.0473839	+	0.998877i	$0.484912\pi$
$492$	0	0
$493$	−3651.30	−0.333562
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1036.34	0.0935340
$498$	0	0
$499$	−3877.49	−0.347857	−0.173928	−	0.984758i	$-0.555646\pi$
$499$	−3877.49	−0.347857	−0.173928	+	0.984758i	$0.555646\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21048.5	−1.86582	−0.932910	−	0.360110i	$-0.882739\pi$
$503$	−21048.5	−1.86582	−0.932910	+	0.360110i	$0.882739\pi$
$504$	0	0
$505$	3961.84	0.349108
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−22848.9	−1.98971	−0.994854	−	0.101319i	$-0.967694\pi$
$509$	−22848.9	−1.98971	−0.994854	+	0.101319i	$0.967694\pi$
$510$	0	0
$511$	−608.673	−0.0526930
$512$	0	0
$513$	0	0
$514$	0	0
$515$	929.921	0.0795674
$516$	0	0
$517$	−1800.35	−0.153152
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−17616.4	−1.48136	−0.740678	−	0.671860i	$-0.765496\pi$
$521$	−17616.4	−1.48136	−0.740678	+	0.671860i	$0.765496\pi$
$522$	0	0
$523$	−4497.49	−0.376026	−0.188013	−	0.982167i	$-0.560205\pi$
$523$	−4497.49	−0.376026	−0.188013	+	0.982167i	$0.560205\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2422.53	0.200241
$528$	0	0
$529$	22233.7	1.82737
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1525.81	−0.123997
$534$	0	0
$535$	−1426.56	−0.115282
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−557.661	−0.0445643
$540$	0	0
$541$	−16766.7	−1.33246	−0.666228	−	0.745748i	$-0.732092\pi$
$541$	−16766.7	−1.33246	−0.666228	+	0.745748i	$0.732092\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3107.46	0.244236
$546$	0	0
$547$	−18004.6	−1.40735	−0.703675	−	0.710522i	$-0.748459\pi$
$547$	−18004.6	−1.40735	−0.703675	+	0.710522i	$0.748459\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	13028.4	1.00731
$552$	0	0
$553$	−5653.25	−0.434721
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18121.2	1.37849	0.689245	−	0.724528i	$-0.257942\pi$
$557$	18121.2	1.37849	0.689245	+	0.724528i	$0.257942\pi$
$558$	0	0
$559$	−345.371	−0.0261317
$560$	0	0
$561$	0	0
$562$	0	0
$563$	20209.7	1.51285	0.756426	−	0.654080i	$-0.226944\pi$
$563$	20209.7	1.51285	0.756426	+	0.654080i	$0.226944\pi$
$564$	0	0
$565$	158.426	0.0117965
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−6014.15	−0.443104	−0.221552	−	0.975149i	$-0.571112\pi$
$569$	−6014.15	−0.443104	−0.221552	+	0.975149i	$0.571112\pi$
$570$	0	0
$571$	22780.8	1.66961	0.834806	−	0.550544i	$-0.185580\pi$
$571$	22780.8	1.66961	0.834806	+	0.550544i	$0.185580\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−21064.3	−1.52772
$576$	0	0
$577$	−3034.06	−0.218908	−0.109454	−	0.993992i	$-0.534910\pi$
$577$	−3034.06	−0.218908	−0.109454	+	0.993992i	$0.534910\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2807.98	0.200507
$582$	0	0
$583$	4620.56	0.328240
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16836.7	−1.18386	−0.591928	−	0.805991i	$-0.701633\pi$
$587$	−16836.7	−1.18386	−0.591928	+	0.805991i	$0.701633\pi$
$588$	0	0
$589$	−8643.98	−0.604701
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22152.2	1.53403	0.767016	−	0.641628i	$-0.221741\pi$
$593$	22152.2	1.53403	0.767016	+	0.641628i	$0.221741\pi$
$594$	0	0
$595$	−720.091	−0.0496149
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4335.10	−0.295705	−0.147853	−	0.989009i	$-0.547236\pi$
$599$	−4335.10	−0.295705	−0.147853	+	0.989009i	$0.547236\pi$
$600$	0	0
$601$	24956.3	1.69382	0.846912	−	0.531734i	$-0.178459\pi$
$601$	24956.3	1.69382	0.846912	+	0.531734i	$0.178459\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4061.99	0.272964
$606$	0	0
$607$	−12577.6	−0.841033	−0.420517	−	0.907285i	$-0.638151\pi$
$607$	−12577.6	−0.841033	−0.420517	+	0.907285i	$0.638151\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1190.13	0.0788010
$612$	0	0
$613$	−1421.17	−0.0936386	−0.0468193	−	0.998903i	$-0.514908\pi$
$613$	−1421.17	−0.0936386	−0.0468193	+	0.998903i	$0.514908\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2058.22	0.134296	0.0671482	−	0.997743i	$-0.478610\pi$
$617$	2058.22	0.134296	0.0671482	+	0.997743i	$0.478610\pi$
$618$	0	0
$619$	19129.8	1.24215	0.621074	−	0.783752i	$-0.286696\pi$
$619$	19129.8	1.24215	0.621074	+	0.783752i	$0.286696\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8300.93	−0.533820
$624$	0	0
$625$	11469.4	0.734041
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1144.58	0.0725554
$630$	0	0
$631$	−16806.3	−1.06030	−0.530148	−	0.847905i	$-0.677864\pi$
$631$	−16806.3	−1.06030	−0.530148	+	0.847905i	$0.677864\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3854.78	0.240901
$636$	0	0
$637$	368.643	0.0229296
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2834.99	0.174688	0.0873442	−	0.996178i	$-0.472162\pi$
$641$	2834.99	0.174688	0.0873442	+	0.996178i	$0.472162\pi$
$642$	0	0
$643$	12016.0	0.736959	0.368479	−	0.929636i	$-0.379879\pi$
$643$	12016.0	0.736959	0.368479	+	0.929636i	$0.379879\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10271.7	0.624144	0.312072	−	0.950058i	$-0.398977\pi$
$647$	10271.7	0.624144	0.312072	+	0.950058i	$0.398977\pi$
$648$	0	0
$649$	−7122.22	−0.430773
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2519.17	−0.150969	−0.0754846	−	0.997147i	$-0.524050\pi$
$653$	−2519.17	−0.150969	−0.0754846	+	0.997147i	$0.524050\pi$
$654$	0	0
$655$	1475.27	0.0880054
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30237.6	−1.78739	−0.893694	−	0.448677i	$-0.851895\pi$
$659$	−30237.6	−1.78739	−0.893694	+	0.448677i	$0.851895\pi$
$660$	0	0
$661$	11145.7	0.655854	0.327927	−	0.944703i	$-0.393650\pi$
$661$	11145.7	0.655854	0.327927	+	0.944703i	$0.393650\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2569.40	0.149830
$666$	0	0
$667$	−22256.9	−1.29204
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6463.13	0.371843
$672$	0	0
$673$	1228.04	0.0703379	0.0351690	−	0.999381i	$-0.488803\pi$
$673$	1228.04	0.0703379	0.0351690	+	0.999381i	$0.488803\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	308.833	0.0175324	0.00876619	−	0.999962i	$-0.497210\pi$
$677$	308.833	0.0175324	0.00876619	+	0.999962i	$0.497210\pi$
$678$	0	0
$679$	−1728.67	−0.0977031
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−32567.6	−1.82454	−0.912272	−	0.409585i	$-0.865673\pi$
$683$	−32567.6	−1.82454	−0.912272	+	0.409585i	$0.865673\pi$
$684$	0	0
$685$	−4873.85	−0.271854
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−3054.43	−0.168889
$690$	0	0
$691$	2168.67	0.119392	0.0596960	−	0.998217i	$-0.480987\pi$
$691$	2168.67	0.119392	0.0596960	+	0.998217i	$0.480987\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	6443.48	0.351676
$696$	0	0
$697$	−6171.02	−0.335358
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17485.0	0.942079	0.471040	−	0.882112i	$-0.343879\pi$
$701$	17485.0	0.942079	0.471040	+	0.882112i	$0.343879\pi$
$702$	0	0
$703$	−4084.04	−0.219107
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−8202.97	−0.436357
$708$	0	0
$709$	−26492.7	−1.40332	−0.701661	−	0.712511i	$-0.747558\pi$
$709$	−26492.7	−1.40332	−0.701661	+	0.712511i	$0.747558\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	14766.8	0.775627
$714$	0	0
$715$	289.473	0.0151408
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21342.3	−1.10700	−0.553501	−	0.832848i	$-0.686709\pi$
$719$	−21342.3	−1.10700	−0.553501	+	0.832848i	$0.686709\pi$
$720$	0	0
$721$	−1925.40	−0.0994529
$722$	0	0
$723$	0	0
$724$	0	0
$725$	13628.4	0.698132
$726$	0	0
$727$	38232.1	1.95041	0.975206	−	0.221297i	$-0.0710291\pi$
$727$	38232.1	1.95041	0.975206	+	0.221297i	$0.0710291\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1396.83	−0.0706750
$732$	0	0
$733$	−29067.0	−1.46469	−0.732343	−	0.680936i	$-0.761573\pi$
$733$	−29067.0	−1.46469	−0.732343	+	0.680936i	$0.761573\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	4723.58	0.236086
$738$	0	0
$739$	−21606.2	−1.07550	−0.537751	−	0.843104i	$-0.680726\pi$
$739$	−21606.2	−1.07550	−0.537751	+	0.843104i	$0.680726\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	283.079	0.0139774	0.00698868	−	0.999976i	$-0.497775\pi$
$743$	283.079	0.0139774	0.00698868	+	0.999976i	$0.497775\pi$
$744$	0	0
$745$	7112.20	0.349760
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2953.69	0.144093
$750$	0	0
$751$	−37646.0	−1.82919	−0.914595	−	0.404372i	$-0.867490\pi$
$751$	−37646.0	−1.82919	−0.914595	+	0.404372i	$0.867490\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1102.40	−0.0531396
$756$	0	0
$757$	−33648.5	−1.61555	−0.807777	−	0.589488i	$-0.799329\pi$
$757$	−33648.5	−1.61555	−0.807777	+	0.589488i	$0.799329\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−121.480	−0.00578668	−0.00289334	−	0.999996i	$-0.500921\pi$
$761$	−121.480	−0.00578668	−0.00289334	+	0.999996i	$0.500921\pi$
$762$	0	0
$763$	−6433.98	−0.305276
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4708.16	0.221645
$768$	0	0
$769$	35407.7	1.66038	0.830191	−	0.557479i	$-0.188231\pi$
$769$	35407.7	1.66038	0.830191	+	0.557479i	$0.188231\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5272.72	−0.245338	−0.122669	−	0.992448i	$-0.539145\pi$
$773$	−5272.72	−0.245338	−0.122669	+	0.992448i	$0.539145\pi$
$774$	0	0
$775$	−9042.06	−0.419097
$776$	0	0
$777$	0	0
$778$	0	0
$779$	22019.2	1.01273
$780$	0	0
$781$	−1684.92	−0.0771975
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9133.64	0.415279
$786$	0	0
$787$	−4375.82	−0.198197	−0.0990985	−	0.995078i	$-0.531596\pi$
$787$	−4375.82	−0.198197	−0.0990985	+	0.995078i	$0.531596\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−328.020	−0.0147447
$792$	0	0
$793$	−4272.47	−0.191324
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5163.94	0.229506	0.114753	−	0.993394i	$-0.463392\pi$
$797$	5163.94	0.229506	0.114753	+	0.993394i	$0.463392\pi$
$798$	0	0
$799$	4813.37	0.213123
$800$	0	0
$801$	0	0
$802$	0	0
$803$	989.601	0.0434898
$804$	0	0
$805$	−4389.40	−0.192181
$806$	0	0
$807$	0	0
$808$	0	0
$809$	32576.3	1.41573	0.707863	−	0.706349i	$-0.249659\pi$
$809$	32576.3	1.41573	0.707863	+	0.706349i	$0.249659\pi$
$810$	0	0
$811$	3704.58	0.160401	0.0802005	−	0.996779i	$-0.474444\pi$
$811$	3704.58	0.160401	0.0802005	+	0.996779i	$0.474444\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2761.98	0.118709
$816$	0	0
$817$	4984.09	0.213429
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6710.39	−0.285255	−0.142628	−	0.989776i	$-0.545555\pi$
$821$	−6710.39	−0.285255	−0.142628	+	0.989776i	$0.545555\pi$
$822$	0	0
$823$	−2589.91	−0.109695	−0.0548473	−	0.998495i	$-0.517467\pi$
$823$	−2589.91	−0.109695	−0.0548473	+	0.998495i	$0.517467\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3846.46	0.161735	0.0808673	−	0.996725i	$-0.474231\pi$
$827$	3846.46	0.161735	0.0808673	+	0.996725i	$0.474231\pi$
$828$	0	0
$829$	−15858.0	−0.664380	−0.332190	−	0.943213i	$-0.607787\pi$
$829$	−15858.0	−0.664380	−0.332190	+	0.943213i	$0.607787\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1490.95	0.0620147
$834$	0	0
$835$	516.215	0.0213944
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−24440.6	−1.00570	−0.502851	−	0.864373i	$-0.667715\pi$
$839$	−24440.6	−1.00570	−0.502851	+	0.864373i	$0.667715\pi$
$840$	0	0
$841$	−9989.00	−0.409570
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7236.33	0.294600
$846$	0	0
$847$	−8410.34	−0.341184
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6976.91	0.281040
$852$	0	0
$853$	19819.3	0.795543	0.397772	−	0.917484i	$-0.369784\pi$
$853$	19819.3	0.795543	0.397772	+	0.917484i	$0.369784\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−20745.7	−0.826909	−0.413454	−	0.910525i	$-0.635678\pi$
$857$	−20745.7	−0.826909	−0.413454	+	0.910525i	$0.635678\pi$
$858$	0	0
$859$	−19726.0	−0.783518	−0.391759	−	0.920068i	$-0.628133\pi$
$859$	−19726.0	−0.783518	−0.391759	+	0.920068i	$0.628133\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46583.8	−1.83746	−0.918731	−	0.394884i	$-0.870785\pi$
$863$	−46583.8	−1.83746	−0.918731	+	0.394884i	$0.870785\pi$
$864$	0	0
$865$	−6280.40	−0.246867
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9191.23	0.358793
$870$	0	0
$871$	−3122.53	−0.121473
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5645.95	0.218135
$876$	0	0
$877$	−49365.6	−1.90075	−0.950376	−	0.311105i	$-0.899301\pi$
$877$	−49365.6	−1.90075	−0.950376	+	0.311105i	$0.899301\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	11939.8	0.456599	0.228299	−	0.973591i	$-0.426683\pi$
$881$	11939.8	0.456599	0.228299	+	0.973591i	$0.426683\pi$
$882$	0	0
$883$	−43448.5	−1.65590	−0.827949	−	0.560804i	$-0.810492\pi$
$883$	−43448.5	−1.65590	−0.827949	+	0.560804i	$0.810492\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	20705.6	0.783794	0.391897	−	0.920009i	$-0.371819\pi$
$887$	20705.6	0.783794	0.391897	+	0.920009i	$0.371819\pi$
$888$	0	0
$889$	−7981.31	−0.301107
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−17174.9	−0.643600
$894$	0	0
$895$	8744.57	0.326591
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9554.00	−0.354442
$900$	0	0
$901$	−12353.4	−0.456772
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2289.01	0.0840767
$906$	0	0
$907$	8394.92	0.307331	0.153665	−	0.988123i	$-0.450892\pi$
$907$	8394.92	0.307331	0.153665	+	0.988123i	$0.450892\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	22424.5	0.815540	0.407770	−	0.913085i	$-0.366307\pi$
$911$	22424.5	0.815540	0.407770	+	0.913085i	$0.366307\pi$
$912$	0	0
$913$	−4565.31	−0.165487
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3054.54	−0.110000
$918$	0	0
$919$	−21080.1	−0.756656	−0.378328	−	0.925672i	$-0.623501\pi$
$919$	−21080.1	−0.756656	−0.378328	+	0.925672i	$0.623501\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1113.82	0.0397204
$924$	0	0
$925$	−4272.12	−0.151856
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−41876.6	−1.47893	−0.739466	−	0.673194i	$-0.764922\pi$
$929$	−41876.6	−1.47893	−0.739466	+	0.673194i	$0.764922\pi$
$930$	0	0
$931$	−5319.93	−0.187276
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1170.75	0.0409493
$936$	0	0
$937$	7875.22	0.274570	0.137285	−	0.990532i	$-0.456162\pi$
$937$	7875.22	0.274570	0.137285	+	0.990532i	$0.456162\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46272.7	−1.60302	−0.801512	−	0.597979i	$-0.795971\pi$
$941$	−46272.7	−1.60302	−0.801512	+	0.597979i	$0.795971\pi$
$942$	0	0
$943$	−37616.2	−1.29899
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15445.3	0.529995	0.264998	−	0.964249i	$-0.414629\pi$
$947$	15445.3	0.529995	0.264998	+	0.964249i	$0.414629\pi$
$948$	0	0
$949$	−654.178	−0.0223767
$950$	0	0
$951$	0	0
$952$	0	0
$953$	10435.4	0.354707	0.177354	−	0.984147i	$-0.443246\pi$
$953$	10435.4	0.354707	0.177354	+	0.984147i	$0.443246\pi$
$954$	0	0
$955$	2925.95	0.0991431
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10091.3	0.339796
$960$	0	0
$961$	−23452.2	−0.787224
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6850.79	−0.228533
$966$	0	0
$967$	−38974.1	−1.29609	−0.648047	−	0.761600i	$-0.724414\pi$
$967$	−38974.1	−1.29609	−0.648047	+	0.761600i	$0.724414\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9102.24	0.300829	0.150414	−	0.988623i	$-0.451939\pi$
$971$	9102.24	0.300829	0.150414	+	0.988623i	$0.451939\pi$
$972$	0	0
$973$	−13341.2	−0.439568
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−31300.2	−1.02496	−0.512478	−	0.858700i	$-0.671272\pi$
$977$	−31300.2	−1.02496	−0.512478	+	0.858700i	$0.671272\pi$
$978$	0	0
$979$	13495.9	0.440584
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4603.20	−0.149358	−0.0746792	−	0.997208i	$-0.523793\pi$
$983$	−4603.20	−0.149358	−0.0746792	+	0.997208i	$0.523793\pi$
$984$	0	0
$985$	−14378.8	−0.465123
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8514.50	−0.273757
$990$	0	0
$991$	21580.7	0.691760	0.345880	−	0.938279i	$-0.387580\pi$
$991$	21580.7	0.691760	0.345880	+	0.938279i	$0.387580\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5764.72	0.183672
$996$	0	0
$997$	8912.67	0.283116	0.141558	−	0.989930i	$-0.454789\pi$
$997$	8912.67	0.283116	0.141558	+	0.989930i	$0.454789\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.a.bh.1.1		2
3.2	odd	2		1008.4.a.z.1.2		2
4.3	odd	2		504.4.a.o.1.1	yes	2
12.11	even	2		504.4.a.k.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.a.k.1.2	✓	2	12.11	even	2
504.4.a.o.1.1	yes	2	4.3	odd	2
1008.4.a.z.1.2		2	3.2	odd	2
1008.4.a.bh.1.1		2	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:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q-3.38083 q^{5} +7.00000 q^{7} +O(q^{10})\)	\(q-3.38083 q^{5} +7.00000 q^{7} -11.3808 q^{11} +7.52333 q^{13} +30.4275 q^{17} -108.570 q^{19} +185.474 q^{23} -113.570 q^{25} -120.000 q^{29} +79.6166 q^{31} -23.6658 q^{35} +37.6166 q^{37} -202.811 q^{41} -45.9067 q^{43} +158.192 q^{47} +49.0000 q^{49} -405.995 q^{53} +38.4767 q^{55} +625.808 q^{59} -567.897 q^{61} -25.4351 q^{65} -415.047 q^{67} +148.049 q^{71} -86.9533 q^{73} -79.6658 q^{77} -807.606 q^{79} +401.140 q^{83} -102.870 q^{85} -1185.85 q^{89} +52.6633 q^{91} +367.057 q^{95} -246.953 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 12 q^{5} + 14 q^{7}+O(q^{10}) \) 2 * q + 12 * q^5 + 14 * q^7	\( 2 q + 12 q^{5} + 14 q^{7} - 4 q^{11} - 60 q^{13} - 108 q^{17} + 8 q^{19} + 52 q^{23} - 2 q^{25} - 240 q^{29} - 216 q^{31} + 84 q^{35} - 300 q^{37} - 612 q^{41} - 392 q^{43} + 504 q^{47} + 98 q^{49} - 24 q^{53} + 152 q^{55} + 1064 q^{59} + 140 q^{61} - 1064 q^{65} - 680 q^{67} + 540 q^{71} - 324 q^{73} - 28 q^{77} + 336 q^{79} + 352 q^{83} - 2232 q^{85} - 852 q^{89} - 420 q^{91} + 2160 q^{95} - 644 q^{97}+O(q^{100}) \) 2 * q + 12 * q^5 + 14 * q^7 - 4 * q^11 - 60 * q^13 - 108 * q^17 + 8 * q^19 + 52 * q^23 - 2 * q^25 - 240 * q^29 - 216 * q^31 + 84 * q^35 - 300 * q^37 - 612 * q^41 - 392 * q^43 + 504 * q^47 + 98 * q^49 - 24 * q^53 + 152 * q^55 + 1064 * q^59 + 140 * q^61 - 1064 * q^65 - 680 * q^67 + 540 * q^71 - 324 * q^73 - 28 * q^77 + 336 * q^79 + 352 * q^83 - 2232 * q^85 - 852 * q^89 - 420 * q^91 + 2160 * q^95 - 644 * q^97

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