LMFDB - Embedded newform 1008.6.a.bp.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1008,6,Mod(1,1008)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1008.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage: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, 6, names="a")

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1008.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,10,0,-98,0,0,0,54,0,240] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$161.666890371$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{4281}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1070 $ x^2 - x - 1070
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-32.2147$ 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)

$f(q)$	$=$	$q+70.4294 q^{5} -49.0000 q^{7} -169.288 q^{11} -10.8587 q^{13} -2327.02 q^{17} +1508.59 q^{19} +3755.03 q^{23} +1835.29 q^{25} -5648.89 q^{29} -4284.02 q^{31} -3451.04 q^{35} +36.5543 q^{37} -90.6792 q^{41} +8293.26 q^{43} +18570.1 q^{47} +2401.00 q^{49} +11862.5 q^{53} -11922.8 q^{55} +48240.0 q^{59} +22226.2 q^{61} -764.771 q^{65} +10068.0 q^{67} +41797.1 q^{71} +67194.5 q^{73} +8295.11 q^{77} -58313.2 q^{79} -99460.2 q^{83} -163890. q^{85} +107733. q^{89} +532.076 q^{91} +106249. q^{95} -96488.9 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{5} - 98 q^{7} + 54 q^{11} + 240 q^{13} - 1906 q^{17} + 400 q^{19} + 2930 q^{23} + 2362 q^{25} - 5540 q^{29} - 4904 q^{31} - 490 q^{35} + 2952 q^{37} - 6070 q^{41} - 3304 q^{43} + 16988 q^{47}+ \cdots - 226216 q^{97}+O(q^{100}) $ 2 * q + 10 * q^5 - 98 * q^7 + 54 * q^11 + 240 * q^13 - 1906 * q^17 + 400 * q^19 + 2930 * q^23 + 2362 * q^25 - 5540 * q^29 - 4904 * q^31 - 490 * q^35 + 2952 * q^37 - 6070 * q^41 - 3304 * q^43 + 16988 * q^47 + 4802 * q^49 - 6896 * q^53 - 25416 * q^55 + 53820 * q^59 + 4148 * q^61 - 15924 * q^65 - 33516 * q^67 + 73518 * q^71 + 128 * q^73 - 2646 * q^77 - 57740 * q^79 + 23016 * q^83 - 189332 * q^85 + 141530 * q^89 - 11760 * q^91 + 173240 * q^95 - 226216 * q^97

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
$5$	70.4294	1.25988	0.629939	−	0.776644i	$-0.283080\pi$
$5$	70.4294	1.25988	0.629939	+	0.776644i	$0.283080\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−169.288	−0.421837	−0.210918	−	0.977504i	$-0.567645\pi$
$11$	−169.288	−0.421837	−0.210918	+	0.977504i	$0.567645\pi$
$12$	0	0
$13$	−10.8587	−0.0178205	−0.00891024	−	0.999960i	$-0.502836\pi$
$13$	−10.8587	−0.0178205	−0.00891024	+	0.999960i	$0.502836\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2327.02	−1.95289	−0.976444	−	0.215773i	$-0.930773\pi$
$17$	−2327.02	−1.95289	−0.976444	+	0.215773i	$0.930773\pi$
$18$	0	0
$19$	1508.59	0.958708	0.479354	−	0.877622i	$-0.340871\pi$
$19$	1508.59	0.958708	0.479354	+	0.877622i	$0.340871\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3755.03	1.48011	0.740054	−	0.672547i	$-0.234800\pi$
$23$	3755.03	1.48011	0.740054	+	0.672547i	$0.234800\pi$
$24$	0	0
$25$	1835.29	0.587294
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−5648.89	−1.24729	−0.623646	−	0.781707i	$-0.714349\pi$
$29$	−5648.89	−1.24729	−0.623646	+	0.781707i	$0.714349\pi$
$30$	0	0
$31$	−4284.02	−0.800659	−0.400329	−	0.916371i	$-0.631104\pi$
$31$	−4284.02	−0.800659	−0.400329	+	0.916371i	$0.631104\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3451.04	−0.476189
$36$	0	0
$37$	36.5543	0.00438969	0.00219484	−	0.999998i	$-0.499301\pi$
$37$	36.5543	0.00438969	0.00219484	+	0.999998i	$0.499301\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−90.6792	−0.00842458	−0.00421229	−	0.999991i	$-0.501341\pi$
$41$	−90.6792	−0.00842458	−0.00421229	+	0.999991i	$0.501341\pi$
$42$	0	0
$43$	8293.26	0.683997	0.341999	−	0.939700i	$-0.388896\pi$
$43$	8293.26	0.683997	0.341999	+	0.939700i	$0.388896\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	18570.1	1.22622	0.613112	−	0.789996i	$-0.289917\pi$
$47$	18570.1	1.22622	0.613112	+	0.789996i	$0.289917\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	11862.5	0.580077	0.290038	−	0.957015i	$-0.406332\pi$
$53$	11862.5	0.580077	0.290038	+	0.957015i	$0.406332\pi$
$54$	0	0
$55$	−11922.8	−0.531463
$56$	0	0
$57$	0	0
$58$	0	0
$59$	48240.0	1.80417	0.902084	−	0.431560i	$-0.142037\pi$
$59$	48240.0	1.80417	0.902084	+	0.431560i	$0.142037\pi$
$60$	0	0
$61$	22226.2	0.764789	0.382394	−	0.923999i	$-0.375100\pi$
$61$	22226.2	0.764789	0.382394	+	0.923999i	$0.375100\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−764.771	−0.0224516
$66$	0	0
$67$	10068.0	0.274004	0.137002	−	0.990571i	$-0.456253\pi$
$67$	10068.0	0.274004	0.137002	+	0.990571i	$0.456253\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	41797.1	0.984011	0.492005	−	0.870592i	$-0.336264\pi$
$71$	41797.1	0.984011	0.492005	+	0.870592i	$0.336264\pi$
$72$	0	0
$73$	67194.5	1.47580	0.737899	−	0.674912i	$-0.235818\pi$
$73$	67194.5	1.47580	0.737899	+	0.674912i	$0.235818\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	8295.11	0.159439
$78$	0	0
$79$	−58313.2	−1.05123	−0.525617	−	0.850721i	$-0.676165\pi$
$79$	−58313.2	−1.05123	−0.525617	+	0.850721i	$0.676165\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−99460.2	−1.58473	−0.792363	−	0.610050i	$-0.791149\pi$
$83$	−99460.2	−1.58473	−0.792363	+	0.610050i	$0.791149\pi$
$84$	0	0
$85$	−163890.	−2.46040
$86$	0	0
$87$	0	0
$88$	0	0
$89$	107733.	1.44169	0.720845	−	0.693096i	$-0.243754\pi$
$89$	107733.	1.44169	0.720845	+	0.693096i	$0.243754\pi$
$90$	0	0
$91$	532.076	0.00673551
$92$	0	0
$93$	0	0
$94$	0	0
$95$	106249.	1.20786
$96$	0	0
$97$	−96488.9	−1.04123	−0.520617	−	0.853790i	$-0.674298\pi$
$97$	−96488.9	−1.04123	−0.520617	+	0.853790i	$0.674298\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−53571.9	−0.522557	−0.261279	−	0.965263i	$-0.584144\pi$
$101$	−53571.9	−0.522557	−0.261279	+	0.965263i	$0.584144\pi$
$102$	0	0
$103$	111547.	1.03601	0.518004	−	0.855378i	$-0.326675\pi$
$103$	111547.	1.03601	0.518004	+	0.855378i	$0.326675\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	59718.8	0.504257	0.252128	−	0.967694i	$-0.418869\pi$
$107$	59718.8	0.504257	0.252128	+	0.967694i	$0.418869\pi$
$108$	0	0
$109$	−82440.7	−0.664623	−0.332312	−	0.943170i	$-0.607829\pi$
$109$	−82440.7	−0.664623	−0.332312	+	0.943170i	$0.607829\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	204480.	1.50645	0.753225	−	0.657763i	$-0.228497\pi$
$113$	204480.	1.50645	0.753225	+	0.657763i	$0.228497\pi$
$114$	0	0
$115$	264464.	1.86476
$116$	0	0
$117$	0	0
$118$	0	0
$119$	114024.	0.738122
$120$	0	0
$121$	−132393.	−0.822054
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−90833.2	−0.519960
$126$	0	0
$127$	−120810.	−0.664654	−0.332327	−	0.943164i	$-0.607834\pi$
$127$	−120810.	−0.664654	−0.332327	+	0.943164i	$0.607834\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	260775.	1.32766	0.663832	−	0.747882i	$-0.268929\pi$
$131$	260775.	1.32766	0.663832	+	0.747882i	$0.268929\pi$
$132$	0	0
$133$	−73920.8	−0.362358
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2185.69	−0.00994918	−0.00497459	−	0.999988i	$-0.501583\pi$
$137$	−2185.69	−0.00994918	−0.00497459	+	0.999988i	$0.501583\pi$
$138$	0	0
$139$	−4670.22	−0.0205022	−0.0102511	−	0.999947i	$-0.503263\pi$
$139$	−4670.22	−0.0205022	−0.0102511	+	0.999947i	$0.503263\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1838.25	0.00751734
$144$	0	0
$145$	−397848.	−1.57144
$146$	0	0
$147$	0	0
$148$	0	0
$149$	214126.	0.790140	0.395070	−	0.918651i	$-0.370720\pi$
$149$	214126.	0.790140	0.395070	+	0.918651i	$0.370720\pi$
$150$	0	0
$151$	373534.	1.33318	0.666588	−	0.745426i	$-0.267754\pi$
$151$	373534.	1.33318	0.666588	+	0.745426i	$0.267754\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−301721.	−1.00873
$156$	0	0
$157$	−320994.	−1.03932	−0.519658	−	0.854374i	$-0.673941\pi$
$157$	−320994.	−1.03932	−0.519658	+	0.854374i	$0.673941\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−183996.	−0.559428
$162$	0	0
$163$	−297770.	−0.877833	−0.438916	−	0.898528i	$-0.644638\pi$
$163$	−297770.	−0.877833	−0.438916	+	0.898528i	$0.644638\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	108151.	0.300082	0.150041	−	0.988680i	$-0.452059\pi$
$167$	108151.	0.300082	0.150041	+	0.988680i	$0.452059\pi$
$168$	0	0
$169$	−371175.	−0.999682
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−453000.	−1.15076	−0.575378	−	0.817888i	$-0.695145\pi$
$173$	−453000.	−1.15076	−0.575378	+	0.817888i	$0.695145\pi$
$174$	0	0
$175$	−89929.4	−0.221976
$176$	0	0
$177$	0	0
$178$	0	0
$179$	307341.	0.716949	0.358474	−	0.933540i	$-0.383297\pi$
$179$	307341.	0.716949	0.358474	+	0.933540i	$0.383297\pi$
$180$	0	0
$181$	258687.	0.586918	0.293459	−	0.955972i	$-0.405194\pi$
$181$	258687.	0.586918	0.293459	+	0.955972i	$0.405194\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2574.49	0.00553048
$186$	0	0
$187$	393936.	0.823800
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−122006.	−0.241991	−0.120995	−	0.992653i	$-0.538609\pi$
$191$	−122006.	−0.241991	−0.120995	+	0.992653i	$0.538609\pi$
$192$	0	0
$193$	873534.	1.68806	0.844028	−	0.536300i	$-0.180178\pi$
$193$	873534.	1.68806	0.844028	+	0.536300i	$0.180178\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	286734.	0.526398	0.263199	−	0.964742i	$-0.415222\pi$
$197$	286734.	0.526398	0.263199	+	0.964742i	$0.415222\pi$
$198$	0	0
$199$	488646.	0.874706	0.437353	−	0.899290i	$-0.355916\pi$
$199$	488646.	0.874706	0.437353	+	0.899290i	$0.355916\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	276796.	0.471432
$204$	0	0
$205$	−6386.48	−0.0106139
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−255386.	−0.404418
$210$	0	0
$211$	203765.	0.315081	0.157541	−	0.987513i	$-0.449643\pi$
$211$	203765.	0.315081	0.157541	+	0.987513i	$0.449643\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	584089.	0.861753
$216$	0	0
$217$	209917.	0.302621
$218$	0	0
$219$	0	0
$220$	0	0
$221$	25268.4	0.0348014
$222$	0	0
$223$	−120144.	−0.161786	−0.0808929	−	0.996723i	$-0.525777\pi$
$223$	−120144.	−0.161786	−0.0808929	+	0.996723i	$0.525777\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−487855.	−0.628386	−0.314193	−	0.949359i	$-0.601734\pi$
$227$	−487855.	−0.628386	−0.314193	+	0.949359i	$0.601734\pi$
$228$	0	0
$229$	1.13748e6	1.43336	0.716678	−	0.697405i	$-0.245662\pi$
$229$	1.13748e6	1.43336	0.716678	+	0.697405i	$0.245662\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.15703e6	1.39622	0.698109	−	0.715992i	$-0.254025\pi$
$233$	1.15703e6	1.39622	0.698109	+	0.715992i	$0.254025\pi$
$234$	0	0
$235$	1.30788e6	1.54489
$236$	0	0
$237$	0	0
$238$	0	0
$239$	835593.	0.946237	0.473119	−	0.880999i	$-0.343128\pi$
$239$	835593.	0.946237	0.473119	+	0.880999i	$0.343128\pi$
$240$	0	0
$241$	353868.	0.392463	0.196232	−	0.980558i	$-0.437130\pi$
$241$	353868.	0.392463	0.196232	+	0.980558i	$0.437130\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	169101.	0.179983
$246$	0	0
$247$	−16381.3	−0.0170846
$248$	0	0
$249$	0	0
$250$	0	0
$251$	983591.	0.985440	0.492720	−	0.870188i	$-0.336003\pi$
$251$	983591.	0.985440	0.492720	+	0.870188i	$0.336003\pi$
$252$	0	0
$253$	−635681.	−0.624364
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.25338e6	−1.18372	−0.591861	−	0.806040i	$-0.701606\pi$
$257$	−1.25338e6	−1.18372	−0.591861	+	0.806040i	$0.701606\pi$
$258$	0	0
$259$	−1791.16	−0.00165915
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.15241e6	1.02735	0.513673	−	0.857986i	$-0.328284\pi$
$263$	1.15241e6	1.02735	0.513673	+	0.857986i	$0.328284\pi$
$264$	0	0
$265$	835466.	0.730826
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−1.60138e6	−1.34932	−0.674659	−	0.738129i	$-0.735709\pi$
$269$	−1.60138e6	−1.34932	−0.674659	+	0.738129i	$0.735709\pi$
$270$	0	0
$271$	−1.82052e6	−1.50582	−0.752910	−	0.658123i	$-0.771351\pi$
$271$	−1.82052e6	−1.50582	−0.752910	+	0.658123i	$0.771351\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−310693.	−0.247742
$276$	0	0
$277$	−1.30239e6	−1.01986	−0.509929	−	0.860216i	$-0.670328\pi$
$277$	−1.30239e6	−1.01986	−0.509929	+	0.860216i	$0.670328\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.56996e6	1.18610	0.593052	−	0.805164i	$-0.297923\pi$
$281$	1.56996e6	1.18610	0.593052	+	0.805164i	$0.297923\pi$
$282$	0	0
$283$	1.19887e6	0.889831	0.444915	−	0.895573i	$-0.353234\pi$
$283$	1.19887e6	0.889831	0.444915	+	0.895573i	$0.353234\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4443.28	0.00318419
$288$	0	0
$289$	3.99515e6	2.81377
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.93610e6	1.31753	0.658763	−	0.752350i	$-0.271080\pi$
$293$	1.93610e6	1.31753	0.658763	+	0.752350i	$0.271080\pi$
$294$	0	0
$295$	3.39751e6	2.27303
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−40774.7	−0.0263763
$300$	0	0
$301$	−406370.	−0.258527
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.56538e6	0.963541
$306$	0	0
$307$	−977005.	−0.591631	−0.295816	−	0.955245i	$-0.595591\pi$
$307$	−977005.	−0.591631	−0.295816	+	0.955245i	$0.595591\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	416145.	0.243974	0.121987	−	0.992532i	$-0.461073\pi$
$311$	416145.	0.243974	0.121987	+	0.992532i	$0.461073\pi$
$312$	0	0
$313$	2.35309e6	1.35762	0.678809	−	0.734315i	$-0.262496\pi$
$313$	2.35309e6	1.35762	0.678809	+	0.734315i	$0.262496\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−420149.	−0.234831	−0.117415	−	0.993083i	$-0.537461\pi$
$317$	−420149.	−0.234831	−0.117415	+	0.993083i	$0.537461\pi$
$318$	0	0
$319$	956290.	0.526154
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.51051e6	−1.87225
$324$	0	0
$325$	−19928.9	−0.0104659
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−909936.	−0.463469
$330$	0	0
$331$	2.26504e6	1.13633	0.568166	−	0.822914i	$-0.307653\pi$
$331$	2.26504e6	1.13633	0.568166	+	0.822914i	$0.307653\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	709085.	0.345212
$336$	0	0
$337$	−312951.	−0.150107	−0.0750537	−	0.997179i	$-0.523913\pi$
$337$	−312951.	−0.150107	−0.0750537	+	0.997179i	$0.523913\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	725234.	0.337747
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.26857e6	−1.01141	−0.505706	−	0.862706i	$-0.668768\pi$
$347$	−2.26857e6	−1.01141	−0.505706	+	0.862706i	$0.668768\pi$
$348$	0	0
$349$	−1.85308e6	−0.814388	−0.407194	−	0.913342i	$-0.633493\pi$
$349$	−1.85308e6	−0.814388	−0.407194	+	0.913342i	$0.633493\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2.81453e6	1.20218	0.601089	−	0.799182i	$-0.294734\pi$
$353$	2.81453e6	1.20218	0.601089	+	0.799182i	$0.294734\pi$
$354$	0	0
$355$	2.94374e6	1.23973
$356$	0	0
$357$	0	0
$358$	0	0
$359$	2.07160e6	0.848339	0.424169	−	0.905583i	$-0.360566\pi$
$359$	2.07160e6	0.848339	0.424169	+	0.905583i	$0.360566\pi$
$360$	0	0
$361$	−200264.	−0.0808789
$362$	0	0
$363$	0	0
$364$	0	0
$365$	4.73247e6	1.85933
$366$	0	0
$367$	−3.90586e6	−1.51374	−0.756870	−	0.653566i	$-0.773272\pi$
$367$	−3.90586e6	−1.51374	−0.756870	+	0.653566i	$0.773272\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−581261.	−0.219248
$372$	0	0
$373$	1.26552e6	0.470974	0.235487	−	0.971877i	$-0.424331\pi$
$373$	1.26552e6	0.470974	0.235487	+	0.971877i	$0.424331\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	61339.6	0.0222274
$378$	0	0
$379$	−2.73373e6	−0.977592	−0.488796	−	0.872398i	$-0.662564\pi$
$379$	−2.73373e6	−0.977592	−0.488796	+	0.872398i	$0.662564\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	3.59033e6	1.25066	0.625328	−	0.780362i	$-0.284965\pi$
$383$	3.59033e6	1.25066	0.625328	+	0.780362i	$0.284965\pi$
$384$	0	0
$385$	584220.	0.200874
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.58625e6	1.20162	0.600809	−	0.799392i	$-0.294845\pi$
$389$	3.58625e6	1.20162	0.600809	+	0.799392i	$0.294845\pi$
$390$	0	0
$391$	−8.73801e6	−2.89048
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.10696e6	−1.32443
$396$	0	0
$397$	5.23550e6	1.66718	0.833590	−	0.552384i	$-0.186282\pi$
$397$	5.23550e6	1.66718	0.833590	+	0.552384i	$0.186282\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.97474e6	1.54493	0.772467	−	0.635055i	$-0.219023\pi$
$401$	4.97474e6	1.54493	0.772467	+	0.635055i	$0.219023\pi$
$402$	0	0
$403$	46518.9	0.0142681
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6188.20	−0.00185173
$408$	0	0
$409$	−3.04069e6	−0.898801	−0.449401	−	0.893330i	$-0.648362\pi$
$409$	−3.04069e6	−0.898801	−0.449401	+	0.893330i	$0.648362\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.36376e6	−0.681911
$414$	0	0
$415$	−7.00492e6	−1.99656
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5.25931e6	1.46350	0.731752	−	0.681571i	$-0.238703\pi$
$419$	5.25931e6	1.46350	0.731752	+	0.681571i	$0.238703\pi$
$420$	0	0
$421$	−6.51872e6	−1.79249	−0.896245	−	0.443559i	$-0.853716\pi$
$421$	−6.51872e6	−1.79249	−0.896245	+	0.443559i	$0.853716\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.27076e6	−1.14692
$426$	0	0
$427$	−1.08909e6	−0.289063
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.19867e6	0.570120	0.285060	−	0.958510i	$-0.407986\pi$
$431$	2.19867e6	0.570120	0.285060	+	0.958510i	$0.407986\pi$
$432$	0	0
$433$	−899455.	−0.230547	−0.115274	−	0.993334i	$-0.536774\pi$
$433$	−899455.	−0.230547	−0.115274	+	0.993334i	$0.536774\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.66479e6	1.41899
$438$	0	0
$439$	−3.76938e6	−0.933489	−0.466744	−	0.884392i	$-0.654573\pi$
$439$	−3.76938e6	−0.933489	−0.466744	+	0.884392i	$0.654573\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	6.23892e6	1.51043	0.755214	−	0.655479i	$-0.227533\pi$
$443$	6.23892e6	1.51043	0.755214	+	0.655479i	$0.227533\pi$
$444$	0	0
$445$	7.58754e6	1.81636
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.09379e6	−0.724228	−0.362114	−	0.932134i	$-0.617945\pi$
$449$	−3.09379e6	−0.724228	−0.362114	+	0.932134i	$0.617945\pi$
$450$	0	0
$451$	15350.9	0.00355380
$452$	0	0
$453$	0	0
$454$	0	0
$455$	37473.8	0.00848593
$456$	0	0
$457$	−5.90697e6	−1.32304	−0.661522	−	0.749926i	$-0.730089\pi$
$457$	−5.90697e6	−1.32304	−0.661522	+	0.749926i	$0.730089\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.80920e6	−0.396491	−0.198245	−	0.980152i	$-0.563524\pi$
$461$	−1.80920e6	−0.396491	−0.198245	+	0.980152i	$0.563524\pi$
$462$	0	0
$463$	1.75786e6	0.381094	0.190547	−	0.981678i	$-0.438974\pi$
$463$	1.75786e6	0.381094	0.190547	+	0.981678i	$0.438974\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−368945.	−0.0782833	−0.0391417	−	0.999234i	$-0.512462\pi$
$467$	−368945.	−0.0782833	−0.0391417	+	0.999234i	$0.512462\pi$
$468$	0	0
$469$	−493334.	−0.103564
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.40395e6	−0.288535
$474$	0	0
$475$	2.76870e6	0.563043
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.53603e6	−1.50074	−0.750368	−	0.661021i	$-0.770123\pi$
$479$	−7.53603e6	−1.50074	−0.750368	+	0.661021i	$0.770123\pi$
$480$	0	0
$481$	−396.932	−7.82264e−5 0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.79565e6	−1.31183
$486$	0	0
$487$	5.60411e6	1.07074	0.535371	−	0.844617i	$-0.320172\pi$
$487$	5.60411e6	1.07074	0.535371	+	0.844617i	$0.320172\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−1.77047e6	−0.331425	−0.165713	−	0.986174i	$-0.552992\pi$
$491$	−1.77047e6	−0.331425	−0.165713	+	0.986174i	$0.552992\pi$
$492$	0	0
$493$	1.31451e7	2.43582
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.04806e6	−0.371921
$498$	0	0
$499$	−5.51873e6	−0.992174	−0.496087	−	0.868273i	$-0.665230\pi$
$499$	−5.51873e6	−0.992174	−0.496087	+	0.868273i	$0.665230\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.25678e6	1.27886	0.639431	−	0.768848i	$-0.279170\pi$
$503$	7.25678e6	1.27886	0.639431	+	0.768848i	$0.279170\pi$
$504$	0	0
$505$	−3.77304e6	−0.658359
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.80322e6	1.67716	0.838580	−	0.544779i	$-0.183386\pi$
$509$	9.80322e6	1.67716	0.838580	+	0.544779i	$0.183386\pi$
$510$	0	0
$511$	−3.29253e6	−0.557799
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7.85615e6	1.30524
$516$	0	0
$517$	−3.14370e6	−0.517267
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−391729.	−0.0632254	−0.0316127	−	0.999500i	$-0.510064\pi$
$521$	−391729.	−0.0632254	−0.0316127	+	0.999500i	$0.510064\pi$
$522$	0	0
$523$	7.78131e6	1.24394	0.621969	−	0.783042i	$-0.286333\pi$
$523$	7.78131e6	1.24394	0.621969	+	0.783042i	$0.286333\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.96899e6	1.56360
$528$	0	0
$529$	7.66389e6	1.19072
$530$	0	0
$531$	0	0
$532$	0	0
$533$	984.658	0.000150130 0
$534$	0	0
$535$	4.20596e6	0.635302
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−406461.	−0.0602624
$540$	0	0
$541$	−7.05542e6	−1.03641	−0.518203	−	0.855258i	$-0.673399\pi$
$541$	−7.05542e6	−1.03641	−0.518203	+	0.855258i	$0.673399\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.80624e6	−0.837344
$546$	0	0
$547$	1.81191e6	0.258922	0.129461	−	0.991585i	$-0.458675\pi$
$547$	1.81191e6	0.258922	0.129461	+	0.991585i	$0.458675\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.52184e6	−1.19579
$552$	0	0
$553$	2.85735e6	0.397329
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4.06385e6	0.555009	0.277504	−	0.960724i	$-0.410493\pi$
$557$	4.06385e6	0.555009	0.277504	+	0.960724i	$0.410493\pi$
$558$	0	0
$559$	−90054.1	−0.0121892
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−2.44664e6	−0.325312	−0.162656	−	0.986683i	$-0.552006\pi$
$563$	−2.44664e6	−0.325312	−0.162656	+	0.986683i	$0.552006\pi$
$564$	0	0
$565$	1.44014e7	1.89794
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.04615e6	0.394431	0.197215	−	0.980360i	$-0.436810\pi$
$569$	3.04615e6	0.394431	0.197215	+	0.980360i	$0.436810\pi$
$570$	0	0
$571$	1.24745e7	1.60115	0.800574	−	0.599234i	$-0.204528\pi$
$571$	1.24745e7	1.60115	0.800574	+	0.599234i	$0.204528\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.89158e6	0.869259
$576$	0	0
$577$	−8.47677e6	−1.05996	−0.529982	−	0.848009i	$-0.677801\pi$
$577$	−8.47677e6	−1.05996	−0.529982	+	0.848009i	$0.677801\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.87355e6	0.598970
$582$	0	0
$583$	−2.00817e6	−0.244698
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.44764e7	−1.73406	−0.867030	−	0.498256i	$-0.833974\pi$
$587$	−1.44764e7	−1.73406	−0.867030	+	0.498256i	$0.833974\pi$
$588$	0	0
$589$	−6.46282e6	−0.767598
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.37231e6	0.627371	0.313686	−	0.949527i	$-0.398436\pi$
$593$	5.37231e6	0.627371	0.313686	+	0.949527i	$0.398436\pi$
$594$	0	0
$595$	8.03062e6	0.929944
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−9.96968e6	−1.13531	−0.567655	−	0.823267i	$-0.692149\pi$
$599$	−9.96968e6	−1.13531	−0.567655	+	0.823267i	$0.692149\pi$
$600$	0	0
$601$	−6.79612e6	−0.767493	−0.383747	−	0.923438i	$-0.625366\pi$
$601$	−6.79612e6	−0.767493	−0.383747	+	0.923438i	$0.625366\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.32432e6	−1.03569
$606$	0	0
$607$	−3.73507e6	−0.411459	−0.205730	−	0.978609i	$-0.565957\pi$
$607$	−3.73507e6	−0.411459	−0.205730	+	0.978609i	$0.565957\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−201647.	−0.0218519
$612$	0	0
$613$	−6.64420e6	−0.714153	−0.357076	−	0.934075i	$-0.616226\pi$
$613$	−6.64420e6	−0.714153	−0.357076	+	0.934075i	$0.616226\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.85062e6	−0.512961	−0.256480	−	0.966549i	$-0.582563\pi$
$617$	−4.85062e6	−0.512961	−0.256480	+	0.966549i	$0.582563\pi$
$618$	0	0
$619$	−1.73039e7	−1.81517	−0.907585	−	0.419868i	$-0.862077\pi$
$619$	−1.73039e7	−1.81517	−0.907585	+	0.419868i	$0.862077\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.27890e6	−0.544908
$624$	0	0
$625$	−1.21326e7	−1.24238
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−85062.4	−0.00857257
$630$	0	0
$631$	1.41085e7	1.41061	0.705306	−	0.708903i	$-0.250810\pi$
$631$	1.41085e7	1.41061	0.705306	+	0.708903i	$0.250810\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.50860e6	−0.837383
$636$	0	0
$637$	−26071.7	−0.00254578
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.48183e7	−1.42447	−0.712234	−	0.701942i	$-0.752316\pi$
$641$	−1.48183e7	−1.42447	−0.712234	+	0.701942i	$0.752316\pi$
$642$	0	0
$643$	1.36121e7	1.29837	0.649186	−	0.760630i	$-0.275110\pi$
$643$	1.36121e7	1.29837	0.649186	+	0.760630i	$0.275110\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.95253e6	0.277289	0.138645	−	0.990342i	$-0.455725\pi$
$647$	2.95253e6	0.277289	0.138645	+	0.990342i	$0.455725\pi$
$648$	0	0
$649$	−8.16645e6	−0.761065
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.25557e6	0.849416	0.424708	−	0.905330i	$-0.360377\pi$
$653$	9.25557e6	0.849416	0.424708	+	0.905330i	$0.360377\pi$
$654$	0	0
$655$	1.83662e7	1.67269
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.38161e6	−0.751820	−0.375910	−	0.926656i	$-0.622670\pi$
$659$	−8.38161e6	−0.751820	−0.375910	+	0.926656i	$0.622670\pi$
$660$	0	0
$661$	1.42353e7	1.26725	0.633624	−	0.773641i	$-0.281567\pi$
$661$	1.42353e7	1.26725	0.633624	+	0.773641i	$0.281567\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.20619e6	−0.456527
$666$	0	0
$667$	−2.12117e7	−1.84613
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−3.76264e6	−0.322616
$672$	0	0
$673$	−1.22312e7	−1.04095	−0.520477	−	0.853876i	$-0.674246\pi$
$673$	−1.22312e7	−1.04095	−0.520477	+	0.853876i	$0.674246\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.73520e7	−1.45505	−0.727524	−	0.686082i	$-0.759329\pi$
$677$	−1.73520e7	−1.45505	−0.727524	+	0.686082i	$0.759329\pi$
$678$	0	0
$679$	4.72796e6	0.393549
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−1.12597e7	−0.923584	−0.461792	−	0.886988i	$-0.652793\pi$
$683$	−1.12597e7	−0.923584	−0.461792	+	0.886988i	$0.652793\pi$
$684$	0	0
$685$	−153937.	−0.0125348
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−128811.	−0.0103372
$690$	0	0
$691$	1.39801e7	1.11382	0.556912	−	0.830572i	$-0.311986\pi$
$691$	1.39801e7	1.11382	0.556912	+	0.830572i	$0.311986\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−328921.	−0.0258303
$696$	0	0
$697$	211012.	0.0164522
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.27004e6	−0.712503	−0.356251	−	0.934390i	$-0.615945\pi$
$701$	−9.27004e6	−0.712503	−0.356251	+	0.934390i	$0.615945\pi$
$702$	0	0
$703$	55145.3	0.00420843
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.62502e6	0.197508
$708$	0	0
$709$	−6.26469e6	−0.468041	−0.234021	−	0.972232i	$-0.575188\pi$
$709$	−6.26469e6	−0.468041	−0.234021	+	0.972232i	$0.575188\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.60866e7	−1.18506
$714$	0	0
$715$	129467.	0.00947093
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.98860e6	0.143458	0.0717289	−	0.997424i	$-0.477148\pi$
$719$	1.98860e6	0.143458	0.0717289	+	0.997424i	$0.477148\pi$
$720$	0	0
$721$	−5.46578e6	−0.391574
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.03674e7	−0.732527
$726$	0	0
$727$	6.79449e6	0.476783	0.238392	−	0.971169i	$-0.423380\pi$
$727$	6.79449e6	0.476783	0.238392	+	0.971169i	$0.423380\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−1.92986e7	−1.33577
$732$	0	0
$733$	−1.08090e7	−0.743064	−0.371532	−	0.928420i	$-0.621167\pi$
$733$	−1.08090e7	−0.743064	−0.371532	+	0.928420i	$0.621167\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.70440e6	−0.115585
$738$	0	0
$739$	5.71735e6	0.385109	0.192554	−	0.981286i	$-0.438323\pi$
$739$	5.71735e6	0.385109	0.192554	+	0.981286i	$0.438323\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.73241e7	1.15127	0.575636	−	0.817706i	$-0.304754\pi$
$743$	1.73241e7	1.15127	0.575636	+	0.817706i	$0.304754\pi$
$744$	0	0
$745$	1.50808e7	0.995480
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.92622e6	−0.190591
$750$	0	0
$751$	−6.62606e6	−0.428702	−0.214351	−	0.976757i	$-0.568764\pi$
$751$	−6.62606e6	−0.428702	−0.214351	+	0.976757i	$0.568764\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.63078e7	1.67964
$756$	0	0
$757$	7.48660e6	0.474837	0.237419	−	0.971407i	$-0.423699\pi$
$757$	7.48660e6	0.474837	0.237419	+	0.971407i	$0.423699\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.15179e7	−1.34691	−0.673453	−	0.739230i	$-0.735190\pi$
$761$	−2.15179e7	−1.34691	−0.673453	+	0.739230i	$0.735190\pi$
$762$	0	0
$763$	4.03959e6	0.251204
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−523823.	−0.0321512
$768$	0	0
$769$	−2.82284e7	−1.72136	−0.860678	−	0.509149i	$-0.829960\pi$
$769$	−2.82284e7	−1.72136	−0.860678	+	0.509149i	$0.829960\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−8.55229e6	−0.514794	−0.257397	−	0.966306i	$-0.582865\pi$
$773$	−8.55229e6	−0.514794	−0.257397	+	0.966306i	$0.582865\pi$
$774$	0	0
$775$	−7.86244e6	−0.470222
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−136797.	−0.00807671
$780$	0	0
$781$	−7.07574e6	−0.415092
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.26074e7	−1.30941
$786$	0	0
$787$	−1.02638e7	−0.590707	−0.295354	−	0.955388i	$-0.595437\pi$
$787$	−1.02638e7	−0.590707	−0.295354	+	0.955388i	$0.595437\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.00195e7	−0.569385
$792$	0	0
$793$	−241348.	−0.0136289
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.90512e7	1.62001	0.810007	−	0.586421i	$-0.199464\pi$
$797$	2.90512e7	1.62001	0.810007	+	0.586421i	$0.199464\pi$
$798$	0	0
$799$	−4.32130e7	−2.39468
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−1.13752e7	−0.622546
$804$	0	0
$805$	−1.29587e7	−0.704812
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.64973e7	−1.96060	−0.980302	−	0.197506i	$-0.936716\pi$
$809$	−3.64973e7	−1.96060	−0.980302	+	0.197506i	$0.936716\pi$
$810$	0	0
$811$	−1.27111e7	−0.678625	−0.339313	−	0.940674i	$-0.610194\pi$
$811$	−1.27111e7	−0.678625	−0.339313	+	0.940674i	$0.610194\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.09717e7	−1.10596
$816$	0	0
$817$	1.25111e7	0.655753
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.48113e7	1.80245	0.901224	−	0.433353i	$-0.142670\pi$
$821$	3.48113e7	1.80245	0.901224	+	0.433353i	$0.142670\pi$
$822$	0	0
$823$	−2.50392e7	−1.28861	−0.644303	−	0.764770i	$-0.722853\pi$
$823$	−2.50392e7	−1.28861	−0.644303	+	0.764770i	$0.722853\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.90538e7	0.968766	0.484383	−	0.874856i	$-0.339044\pi$
$827$	1.90538e7	0.968766	0.484383	+	0.874856i	$0.339044\pi$
$828$	0	0
$829$	2.85861e7	1.44467	0.722336	−	0.691542i	$-0.243068\pi$
$829$	2.85861e7	1.44467	0.722336	+	0.691542i	$0.243068\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−5.58717e6	−0.278984
$834$	0	0
$835$	7.61702e6	0.378067
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.79713e6	0.431456	0.215728	−	0.976454i	$-0.430788\pi$
$839$	8.79713e6	0.431456	0.215728	+	0.976454i	$0.430788\pi$
$840$	0	0
$841$	1.13988e7	0.555738
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2.61416e7	−1.25948
$846$	0	0
$847$	6.48724e6	0.310707
$848$	0	0
$849$	0	0
$850$	0	0
$851$	137262.	0.00649722
$852$	0	0
$853$	3.29937e6	0.155260	0.0776298	−	0.996982i	$-0.475265\pi$
$853$	3.29937e6	0.155260	0.0776298	+	0.996982i	$0.475265\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−7.79857e6	−0.362713	−0.181356	−	0.983417i	$-0.558049\pi$
$857$	−7.79857e6	−0.362713	−0.181356	+	0.983417i	$0.558049\pi$
$858$	0	0
$859$	−1.03989e7	−0.480845	−0.240422	−	0.970668i	$-0.577286\pi$
$859$	−1.03989e7	−0.480845	−0.240422	+	0.970668i	$0.577286\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.47008e7	−1.12898	−0.564488	−	0.825442i	$-0.690926\pi$
$863$	−2.47008e7	−1.12898	−0.564488	+	0.825442i	$0.690926\pi$
$864$	0	0
$865$	−3.19045e7	−1.44981
$866$	0	0
$867$	0	0
$868$	0	0
$869$	9.87173e6	0.443449
$870$	0	0
$871$	−109326.	−0.00488289
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.45083e6	0.196526
$876$	0	0
$877$	2.47844e7	1.08813	0.544064	−	0.839044i	$-0.316885\pi$
$877$	2.47844e7	1.08813	0.544064	+	0.839044i	$0.316885\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	3.77740e7	1.63966	0.819828	−	0.572610i	$-0.194069\pi$
$881$	3.77740e7	1.63966	0.819828	+	0.572610i	$0.194069\pi$
$882$	0	0
$883$	−1.85386e7	−0.800158	−0.400079	−	0.916481i	$-0.631017\pi$
$883$	−1.85386e7	−0.800158	−0.400079	+	0.916481i	$0.631017\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.45793e7	−1.90250	−0.951249	−	0.308425i	$-0.900198\pi$
$887$	−4.45793e7	−1.90250	−0.951249	+	0.308425i	$0.900198\pi$
$888$	0	0
$889$	5.91971e6	0.251215
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.80146e7	1.17559
$894$	0	0
$895$	2.16458e7	0.903269
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.42000e7	0.998655
$900$	0	0
$901$	−2.76042e7	−1.13282
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.82191e7	0.739446
$906$	0	0
$907$	−2.67809e7	−1.08095	−0.540477	−	0.841359i	$-0.681756\pi$
$907$	−2.67809e7	−1.08095	−0.540477	+	0.841359i	$0.681756\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−5.02429e6	−0.200576	−0.100288	−	0.994958i	$-0.531976\pi$
$911$	−5.02429e6	−0.200576	−0.100288	+	0.994958i	$0.531976\pi$
$912$	0	0
$913$	1.68374e7	0.668496
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.27780e7	−0.501810
$918$	0	0
$919$	2.20858e7	0.862628	0.431314	−	0.902202i	$-0.358050\pi$
$919$	2.20858e7	0.862628	0.431314	+	0.902202i	$0.358050\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−453862.	−0.0175356
$924$	0	0
$925$	67087.8	0.00257804
$926$	0	0
$927$	0	0
$928$	0	0
$929$	3.40060e7	1.29276	0.646378	−	0.763017i	$-0.276283\pi$
$929$	3.40060e7	1.29276	0.646378	+	0.763017i	$0.276283\pi$
$930$	0	0
$931$	3.62212e6	0.136958
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.77447e7	1.03789
$936$	0	0
$937$	1.80468e7	0.671509	0.335754	−	0.941950i	$-0.391009\pi$
$937$	1.80468e7	0.671509	0.335754	+	0.941950i	$0.391009\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.72299e7	0.634321	0.317161	−	0.948372i	$-0.397271\pi$
$941$	1.72299e7	0.634321	0.317161	+	0.948372i	$0.397271\pi$
$942$	0	0
$943$	−340503.	−0.0124693
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.27460e7	−0.461847	−0.230923	−	0.972972i	$-0.574175\pi$
$947$	−1.27460e7	−0.461847	−0.230923	+	0.972972i	$0.574175\pi$
$948$	0	0
$949$	−729645.	−0.0262994
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−4.67300e7	−1.66672	−0.833361	−	0.552729i	$-0.813586\pi$
$953$	−4.67300e7	−1.66672	−0.833361	+	0.552729i	$0.813586\pi$
$954$	0	0
$955$	−8.59282e6	−0.304879
$956$	0	0
$957$	0	0
$958$	0	0
$959$	107099.	0.00376043
$960$	0	0
$961$	−1.02763e7	−0.358946
$962$	0	0
$963$	0	0
$964$	0	0
$965$	6.15224e7	2.12674
$966$	0	0
$967$	4.33085e7	1.48939	0.744693	−	0.667408i	$-0.232596\pi$
$967$	4.33085e7	1.48939	0.744693	+	0.667408i	$0.232596\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.38056e7	−0.810272	−0.405136	−	0.914256i	$-0.632776\pi$
$971$	−2.38056e7	−0.810272	−0.405136	+	0.914256i	$0.632776\pi$
$972$	0	0
$973$	228841.	0.00774910
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.55338e7	−0.855815	−0.427907	−	0.903823i	$-0.640749\pi$
$977$	−2.55338e7	−0.855815	−0.427907	+	0.903823i	$0.640749\pi$
$978$	0	0
$979$	−1.82378e7	−0.608158
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.83504e6	−0.291625	−0.145812	−	0.989312i	$-0.546580\pi$
$983$	−8.83504e6	−0.291625	−0.145812	+	0.989312i	$0.546580\pi$
$984$	0	0
$985$	2.01945e7	0.663198
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.11414e7	1.01239
$990$	0	0
$991$	2.34048e7	0.757045	0.378522	−	0.925592i	$-0.376432\pi$
$991$	2.34048e7	0.757045	0.378522	+	0.925592i	$0.376432\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.44151e7	1.10202
$996$	0	0
$997$	4.81022e7	1.53259	0.766297	−	0.642487i	$-0.222097\pi$
$997$	4.81022e7	1.53259	0.766297	+	0.642487i	$0.222097\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.a.bp.1.2		2
3.2	odd	2		336.6.a.s.1.1		2
4.3	odd	2		504.6.a.p.1.2		2
12.11	even	2		168.6.a.i.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.a.i.1.1	✓	2	12.11	even	2
336.6.a.s.1.1		2	3.2	odd	2
504.6.a.p.1.2		2	4.3	odd	2
1008.6.a.bp.1.2		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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1008.a (trivial)

\(f(q)\)	\(=\)	\(q+70.4294 q^{5} -49.0000 q^{7} -169.288 q^{11} -10.8587 q^{13} -2327.02 q^{17} +1508.59 q^{19} +3755.03 q^{23} +1835.29 q^{25} -5648.89 q^{29} -4284.02 q^{31} -3451.04 q^{35} +36.5543 q^{37} -90.6792 q^{41} +8293.26 q^{43} +18570.1 q^{47} +2401.00 q^{49} +11862.5 q^{53} -11922.8 q^{55} +48240.0 q^{59} +22226.2 q^{61} -764.771 q^{65} +10068.0 q^{67} +41797.1 q^{71} +67194.5 q^{73} +8295.11 q^{77} -58313.2 q^{79} -99460.2 q^{83} -163890. q^{85} +107733. q^{89} +532.076 q^{91} +106249. q^{95} -96488.9 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{5} - 98 q^{7} + 54 q^{11} + 240 q^{13} - 1906 q^{17} + 400 q^{19} + 2930 q^{23} + 2362 q^{25} - 5540 q^{29} - 4904 q^{31} - 490 q^{35} + 2952 q^{37} - 6070 q^{41} - 3304 q^{43} + 16988 q^{47}+ \cdots - 226216 q^{97}+O(q^{100}) \) 2 * q + 10 * q^5 - 98 * q^7 + 54 * q^11 + 240 * q^13 - 1906 * q^17 + 400 * q^19 + 2930 * q^23 + 2362 * q^25 - 5540 * q^29 - 4904 * q^31 - 490 * q^35 + 2952 * q^37 - 6070 * q^41 - 3304 * q^43 + 16988 * q^47 + 4802 * q^49 - 6896 * q^53 - 25416 * q^55 + 53820 * q^59 + 4148 * q^61 - 15924 * q^65 - 33516 * q^67 + 73518 * q^71 + 128 * q^73 - 2646 * q^77 - 57740 * q^79 + 23016 * q^83 - 189332 * q^85 + 141530 * q^89 - 11760 * q^91 + 173240 * q^95 - 226216 * q^97

Introduction

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