LMFDB - Embedded newform 1008.6.a.bi.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,-42,0,98,0,0,0,-716,0,-714] 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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 56)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.44622$ 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)

$f(q)$	$=$	$q+48.4622 q^{5} +49.0000 q^{7} -163.506 q^{11} -120.828 q^{13} -78.1920 q^{17} +2265.00 q^{19} -2451.95 q^{23} -776.413 q^{25} -6985.27 q^{29} -2794.03 q^{31} +2374.65 q^{35} +9459.44 q^{37} -10088.5 q^{41} +6934.29 q^{43} +1165.76 q^{47} +2401.00 q^{49} -8562.42 q^{53} -7923.85 q^{55} +6220.26 q^{59} -41928.9 q^{61} -5855.62 q^{65} -1812.09 q^{67} -56823.3 q^{71} -44299.5 q^{73} -8011.78 q^{77} -34912.4 q^{79} -39652.9 q^{83} -3789.36 q^{85} +126299. q^{89} -5920.59 q^{91} +109767. q^{95} -145513. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 42 q^{5} + 98 q^{7} - 716 q^{11} - 714 q^{13} + 1344 q^{17} + 1946 q^{19} - 1792 q^{23} + 4282 q^{25} + 1200 q^{29} + 6804 q^{31} - 2058 q^{35} + 14640 q^{37} - 7896 q^{41} - 524 q^{43} - 18396 q^{47}+ \cdots - 44240 q^{97}+O(q^{100}) $ 2 * q - 42 * q^5 + 98 * q^7 - 716 * q^11 - 714 * q^13 + 1344 * q^17 + 1946 * q^19 - 1792 * q^23 + 4282 * q^25 + 1200 * q^29 + 6804 * q^31 - 2058 * q^35 + 14640 * q^37 - 7896 * q^41 - 524 * q^43 - 18396 * q^47 + 4802 * q^49 - 45132 * q^53 + 42056 * q^55 + 22582 * q^59 - 52822 * q^61 + 47804 * q^65 - 9848 * q^67 - 840 * q^71 - 122052 * q^73 - 35084 * q^77 - 31704 * q^79 + 36974 * q^83 - 132444 * q^85 + 210588 * q^89 - 34986 * q^91 + 138624 * q^95 - 44240 * 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
$5$	48.4622	0.866919	0.433459	−	0.901173i	$-0.357293\pi$
$5$	48.4622	0.866919	0.433459	+	0.901173i	$0.357293\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−163.506	−0.407429	−0.203714	−	0.979030i	$-0.565301\pi$
$11$	−163.506	−0.407429	−0.203714	+	0.979030i	$0.565301\pi$
$12$	0	0
$13$	−120.828	−0.198295	−0.0991473	−	0.995073i	$-0.531611\pi$
$13$	−120.828	−0.198295	−0.0991473	+	0.995073i	$0.531611\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−78.1920	−0.0656206	−0.0328103	−	0.999462i	$-0.510446\pi$
$17$	−78.1920	−0.0656206	−0.0328103	+	0.999462i	$0.510446\pi$
$18$	0	0
$19$	2265.00	1.43941	0.719704	−	0.694281i	$-0.244278\pi$
$19$	2265.00	1.43941	0.719704	+	0.694281i	$0.244278\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2451.95	−0.966480	−0.483240	−	0.875488i	$-0.660540\pi$
$23$	−2451.95	−0.966480	−0.483240	+	0.875488i	$0.660540\pi$
$24$	0	0
$25$	−776.413	−0.248452
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−6985.27	−1.54237	−0.771185	−	0.636611i	$-0.780336\pi$
$29$	−6985.27	−1.54237	−0.771185	+	0.636611i	$0.780336\pi$
$30$	0	0
$31$	−2794.03	−0.522188	−0.261094	−	0.965313i	$-0.584083\pi$
$31$	−2794.03	−0.522188	−0.261094	+	0.965313i	$0.584083\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2374.65	0.327664
$36$	0	0
$37$	9459.44	1.13595	0.567977	−	0.823044i	$-0.307726\pi$
$37$	9459.44	1.13595	0.567977	+	0.823044i	$0.307726\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10088.5	−0.937271	−0.468636	−	0.883392i	$-0.655254\pi$
$41$	−10088.5	−0.937271	−0.468636	+	0.883392i	$0.655254\pi$
$42$	0	0
$43$	6934.29	0.571914	0.285957	−	0.958242i	$-0.407689\pi$
$43$	6934.29	0.571914	0.285957	+	0.958242i	$0.407689\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1165.76	0.0769778	0.0384889	−	0.999259i	$-0.487746\pi$
$47$	1165.76	0.0769778	0.0384889	+	0.999259i	$0.487746\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8562.42	−0.418704	−0.209352	−	0.977840i	$-0.567135\pi$
$53$	−8562.42	−0.418704	−0.209352	+	0.977840i	$0.567135\pi$
$54$	0	0
$55$	−7923.85	−0.353207
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6220.26	0.232637	0.116318	−	0.993212i	$-0.462891\pi$
$59$	6220.26	0.232637	0.116318	+	0.993212i	$0.462891\pi$
$60$	0	0
$61$	−41928.9	−1.44274	−0.721371	−	0.692549i	$-0.756488\pi$
$61$	−41928.9	−1.44274	−0.721371	+	0.692549i	$0.756488\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5855.62	−0.171905
$66$	0	0
$67$	−1812.09	−0.0493166	−0.0246583	−	0.999696i	$-0.507850\pi$
$67$	−1812.09	−0.0493166	−0.0246583	+	0.999696i	$0.507850\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−56823.3	−1.33777	−0.668884	−	0.743367i	$-0.733228\pi$
$71$	−56823.3	−1.33777	−0.668884	+	0.743367i	$0.733228\pi$
$72$	0	0
$73$	−44299.5	−0.972953	−0.486476	−	0.873694i	$-0.661718\pi$
$73$	−44299.5	−0.972953	−0.486476	+	0.873694i	$0.661718\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8011.78	−0.153993
$78$	0	0
$79$	−34912.4	−0.629379	−0.314690	−	0.949195i	$-0.601900\pi$
$79$	−34912.4	−0.629379	−0.314690	+	0.949195i	$0.601900\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−39652.9	−0.631800	−0.315900	−	0.948793i	$-0.602306\pi$
$83$	−39652.9	−0.631800	−0.315900	+	0.948793i	$0.602306\pi$
$84$	0	0
$85$	−3789.36	−0.0568877
$86$	0	0
$87$	0	0
$88$	0	0
$89$	126299.	1.69015	0.845077	−	0.534645i	$-0.179555\pi$
$89$	126299.	1.69015	0.845077	+	0.534645i	$0.179555\pi$
$90$	0	0
$91$	−5920.59	−0.0749483
$92$	0	0
$93$	0	0
$94$	0	0
$95$	109767.	1.24785
$96$	0	0
$97$	−145513.	−1.57026	−0.785130	−	0.619331i	$-0.787404\pi$
$97$	−145513.	−1.57026	−0.785130	+	0.619331i	$0.787404\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−99545.5	−0.970998	−0.485499	−	0.874237i	$-0.661362\pi$
$101$	−99545.5	−0.970998	−0.485499	+	0.874237i	$0.661362\pi$
$102$	0	0
$103$	129791.	1.20546	0.602729	−	0.797946i	$-0.294080\pi$
$103$	129791.	1.20546	0.602729	+	0.797946i	$0.294080\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	218811.	1.84761	0.923804	−	0.382865i	$-0.125062\pi$
$107$	218811.	1.84761	0.923804	+	0.382865i	$0.125062\pi$
$108$	0	0
$109$	201627.	1.62549	0.812743	−	0.582623i	$-0.197974\pi$
$109$	201627.	1.62549	0.812743	+	0.582623i	$0.197974\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−31803.9	−0.234306	−0.117153	−	0.993114i	$-0.537377\pi$
$113$	−31803.9	−0.234306	−0.117153	+	0.993114i	$0.537377\pi$
$114$	0	0
$115$	−118827.	−0.837859
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3831.41	−0.0248022
$120$	0	0
$121$	−134317.	−0.834002
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−189071.	−1.08231
$126$	0	0
$127$	318115.	1.75015	0.875075	−	0.483988i	$-0.160812\pi$
$127$	318115.	1.75015	0.875075	+	0.483988i	$0.160812\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−305194.	−1.55381	−0.776905	−	0.629618i	$-0.783212\pi$
$131$	−305194.	−1.55381	−0.776905	+	0.629618i	$0.783212\pi$
$132$	0	0
$133$	110985.	0.544045
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−180045.	−0.819556	−0.409778	−	0.912185i	$-0.634394\pi$
$137$	−180045.	−0.819556	−0.409778	+	0.912185i	$0.634394\pi$
$138$	0	0
$139$	323204.	1.41886	0.709430	−	0.704776i	$-0.248953\pi$
$139$	323204.	1.41886	0.709430	+	0.704776i	$0.248953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	19756.2	0.0807909
$144$	0	0
$145$	−338522.	−1.33711
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−294159.	−1.08547	−0.542733	−	0.839905i	$-0.682610\pi$
$149$	−294159.	−1.08547	−0.542733	+	0.839905i	$0.682610\pi$
$150$	0	0
$151$	334962.	1.19551	0.597754	−	0.801679i	$-0.296060\pi$
$151$	334962.	1.19551	0.597754	+	0.801679i	$0.296060\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−135405.	−0.452694
$156$	0	0
$157$	−145150.	−0.469967	−0.234984	−	0.971999i	$-0.575504\pi$
$157$	−145150.	−0.469967	−0.234984	+	0.971999i	$0.575504\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−120146.	−0.365295
$162$	0	0
$163$	176221.	0.519504	0.259752	−	0.965675i	$-0.416359\pi$
$163$	176221.	0.519504	0.259752	+	0.965675i	$0.416359\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−45460.7	−0.126138	−0.0630689	−	0.998009i	$-0.520089\pi$
$167$	−45460.7	−0.126138	−0.0630689	+	0.998009i	$0.520089\pi$
$168$	0	0
$169$	−356693.	−0.960679
$170$	0	0
$171$	0	0
$172$	0	0
$173$	205842.	0.522900	0.261450	−	0.965217i	$-0.415799\pi$
$173$	205842.	0.522900	0.261450	+	0.965217i	$0.415799\pi$
$174$	0	0
$175$	−38044.2	−0.0939061
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−199205.	−0.464695	−0.232348	−	0.972633i	$-0.574641\pi$
$179$	−199205.	−0.464695	−0.232348	+	0.972633i	$0.574641\pi$
$180$	0	0
$181$	198411.	0.450162	0.225081	−	0.974340i	$-0.427735\pi$
$181$	198411.	0.450162	0.225081	+	0.974340i	$0.427735\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	458425.	0.984780
$186$	0	0
$187$	12784.8	0.0267357
$188$	0	0
$189$	0	0
$190$	0	0
$191$	372947.	0.739714	0.369857	−	0.929089i	$-0.379407\pi$
$191$	372947.	0.739714	0.369857	+	0.929089i	$0.379407\pi$
$192$	0	0
$193$	175020.	0.338215	0.169108	−	0.985598i	$-0.445911\pi$
$193$	175020.	0.338215	0.169108	+	0.985598i	$0.445911\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−883770.	−1.62246	−0.811229	−	0.584728i	$-0.801201\pi$
$197$	−883770.	−1.62246	−0.811229	+	0.584728i	$0.801201\pi$
$198$	0	0
$199$	−785843.	−1.40671	−0.703353	−	0.710841i	$-0.748314\pi$
$199$	−785843.	−1.40671	−0.703353	+	0.710841i	$0.748314\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−342278.	−0.582961
$204$	0	0
$205$	−488909.	−0.812538
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−370340.	−0.586456
$210$	0	0
$211$	−763861.	−1.18116	−0.590579	−	0.806980i	$-0.701101\pi$
$211$	−763861.	−1.18116	−0.590579	+	0.806980i	$0.701101\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	336051.	0.495803
$216$	0	0
$217$	−136907.	−0.197368
$218$	0	0
$219$	0	0
$220$	0	0
$221$	9447.82	0.0130122
$222$	0	0
$223$	204400.	0.275245	0.137623	−	0.990485i	$-0.456054\pi$
$223$	204400.	0.275245	0.137623	+	0.990485i	$0.456054\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.00806e6	1.29844	0.649218	−	0.760603i	$-0.275096\pi$
$227$	1.00806e6	1.29844	0.649218	+	0.760603i	$0.275096\pi$
$228$	0	0
$229$	−244324.	−0.307877	−0.153938	−	0.988080i	$-0.549196\pi$
$229$	−244324.	−0.307877	−0.153938	+	0.988080i	$0.549196\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20822.9	−0.0251276	−0.0125638	−	0.999921i	$-0.503999\pi$
$233$	−20822.9	−0.0251276	−0.0125638	+	0.999921i	$0.503999\pi$
$234$	0	0
$235$	56495.5	0.0667335
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−474033.	−0.536801	−0.268401	−	0.963307i	$-0.586495\pi$
$239$	−474033.	−0.536801	−0.268401	+	0.963307i	$0.586495\pi$
$240$	0	0
$241$	−1.70390e6	−1.88974	−0.944869	−	0.327450i	$-0.893811\pi$
$241$	−1.70390e6	−1.88974	−0.944869	+	0.327450i	$0.893811\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	116358.	0.123846
$246$	0	0
$247$	−273676.	−0.285427
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−810918.	−0.812442	−0.406221	−	0.913775i	$-0.633154\pi$
$251$	−810918.	−0.812442	−0.406221	+	0.913775i	$0.633154\pi$
$252$	0	0
$253$	400909.	0.393771
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.26657e6	−1.19618	−0.598088	−	0.801431i	$-0.704073\pi$
$257$	−1.26657e6	−1.19618	−0.598088	+	0.801431i	$0.704073\pi$
$258$	0	0
$259$	463512.	0.429350
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−1.96048e6	−1.74773	−0.873863	−	0.486172i	$-0.838393\pi$
$263$	−1.96048e6	−1.74773	−0.873863	+	0.486172i	$0.838393\pi$
$264$	0	0
$265$	−414954.	−0.362982
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.49358e6	1.25848	0.629242	−	0.777209i	$-0.283365\pi$
$269$	1.49358e6	1.25848	0.629242	+	0.777209i	$0.283365\pi$
$270$	0	0
$271$	−1.23626e6	−1.02255	−0.511275	−	0.859417i	$-0.670827\pi$
$271$	−1.23626e6	−1.02255	−0.511275	+	0.859417i	$0.670827\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	126948.	0.101227
$276$	0	0
$277$	−170150.	−0.133239	−0.0666197	−	0.997778i	$-0.521221\pi$
$277$	−170150.	−0.133239	−0.0666197	+	0.997778i	$0.521221\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1.29957e6	−0.981824	−0.490912	−	0.871209i	$-0.663336\pi$
$281$	−1.29957e6	−0.981824	−0.490912	+	0.871209i	$0.663336\pi$
$282$	0	0
$283$	374032.	0.277615	0.138807	−	0.990319i	$-0.455673\pi$
$283$	374032.	0.277615	0.138807	+	0.990319i	$0.455673\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−494335.	−0.354255
$288$	0	0
$289$	−1.41374e6	−0.995694
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.15022e6	−0.782728	−0.391364	−	0.920236i	$-0.627997\pi$
$293$	−1.15022e6	−0.782728	−0.391364	+	0.920236i	$0.627997\pi$
$294$	0	0
$295$	301448.	0.201677
$296$	0	0
$297$	0	0
$298$	0	0
$299$	296266.	0.191648
$300$	0	0
$301$	339780.	0.216163
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.03197e6	−1.25074
$306$	0	0
$307$	−2.61214e6	−1.58179	−0.790897	−	0.611949i	$-0.790386\pi$
$307$	−2.61214e6	−1.58179	−0.790897	+	0.611949i	$0.790386\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.40646e6	−0.824566	−0.412283	−	0.911056i	$-0.635269\pi$
$311$	−1.40646e6	−0.824566	−0.412283	+	0.911056i	$0.635269\pi$
$312$	0	0
$313$	2.37930e6	1.37274	0.686369	−	0.727253i	$-0.259203\pi$
$313$	2.37930e6	1.37274	0.686369	+	0.727253i	$0.259203\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	563199.	0.314785	0.157393	−	0.987536i	$-0.449691\pi$
$317$	563199.	0.314785	0.157393	+	0.987536i	$0.449691\pi$
$318$	0	0
$319$	1.14213e6	0.628405
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−177105.	−0.0944547
$324$	0	0
$325$	93812.8	0.0492667
$326$	0	0
$327$	0	0
$328$	0	0
$329$	57122.4	0.0290949
$330$	0	0
$331$	850649.	0.426757	0.213378	−	0.976970i	$-0.431553\pi$
$331$	850649.	0.426757	0.213378	+	0.976970i	$0.431553\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−87818.0	−0.0427535
$336$	0	0
$337$	4.01506e6	1.92582	0.962912	−	0.269814i	$-0.0869623\pi$
$337$	4.01506e6	1.92582	0.962912	+	0.269814i	$0.0869623\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	456840.	0.212754
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.39961e6	−1.06984	−0.534918	−	0.844904i	$-0.679657\pi$
$347$	−2.39961e6	−1.06984	−0.534918	+	0.844904i	$0.679657\pi$
$348$	0	0
$349$	−919171.	−0.403955	−0.201977	−	0.979390i	$-0.564737\pi$
$349$	−919171.	−0.403955	−0.201977	+	0.979390i	$0.564737\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.07684e6	−0.459953	−0.229976	−	0.973196i	$-0.573865\pi$
$353$	−1.07684e6	−0.459953	−0.229976	+	0.973196i	$0.573865\pi$
$354$	0	0
$355$	−2.75378e6	−1.15974
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.24219e6	−0.508688	−0.254344	−	0.967114i	$-0.581860\pi$
$359$	−1.24219e6	−0.508688	−0.254344	+	0.967114i	$0.581860\pi$
$360$	0	0
$361$	2.65411e6	1.07189
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.14685e6	−0.843471
$366$	0	0
$367$	−4.17293e6	−1.61725	−0.808624	−	0.588326i	$-0.799787\pi$
$367$	−4.17293e6	−1.61725	−0.808624	+	0.588326i	$0.799787\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−419558.	−0.158255
$372$	0	0
$373$	−1.72238e6	−0.640999	−0.320500	−	0.947249i	$-0.603851\pi$
$373$	−1.72238e6	−0.640999	−0.320500	+	0.947249i	$0.603851\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	844020.	0.305844
$378$	0	0
$379$	−1.72998e6	−0.618647	−0.309324	−	0.950957i	$-0.600103\pi$
$379$	−1.72998e6	−0.618647	−0.309324	+	0.950957i	$0.600103\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.34881e6	−1.16652	−0.583262	−	0.812284i	$-0.698224\pi$
$383$	−3.34881e6	−1.16652	−0.583262	+	0.812284i	$0.698224\pi$
$384$	0	0
$385$	−388269.	−0.133500
$386$	0	0
$387$	0	0
$388$	0	0
$389$	5.18691e6	1.73794	0.868970	−	0.494864i	$-0.164782\pi$
$389$	5.18691e6	1.73794	0.868970	+	0.494864i	$0.164782\pi$
$390$	0	0
$391$	191723.	0.0634209
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.69193e6	−0.545621
$396$	0	0
$397$	−1.30271e6	−0.414830	−0.207415	−	0.978253i	$-0.566505\pi$
$397$	−1.30271e6	−0.414830	−0.207415	+	0.978253i	$0.566505\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1.53036e6	0.475261	0.237631	−	0.971356i	$-0.423629\pi$
$401$	1.53036e6	0.475261	0.237631	+	0.971356i	$0.423629\pi$
$402$	0	0
$403$	337598.	0.103547
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.54667e6	−0.462820
$408$	0	0
$409$	−2.93690e6	−0.868123	−0.434062	−	0.900883i	$-0.642920\pi$
$409$	−2.93690e6	−0.868123	−0.434062	+	0.900883i	$0.642920\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	304793.	0.0879284
$414$	0	0
$415$	−1.92167e6	−0.547719
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.16465e6	−0.324085	−0.162043	−	0.986784i	$-0.551808\pi$
$419$	−1.16465e6	−0.324085	−0.162043	+	0.986784i	$0.551808\pi$
$420$	0	0
$421$	−1.58204e6	−0.435022	−0.217511	−	0.976058i	$-0.569794\pi$
$421$	−1.58204e6	−0.435022	−0.217511	+	0.976058i	$0.569794\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	60709.3	0.0163036
$426$	0	0
$427$	−2.05451e6	−0.545305
$428$	0	0
$429$	0	0
$430$	0	0
$431$	6.10233e6	1.58235	0.791174	−	0.611591i	$-0.209470\pi$
$431$	6.10233e6	1.58235	0.791174	+	0.611591i	$0.209470\pi$
$432$	0	0
$433$	1.08115e6	0.277118	0.138559	−	0.990354i	$-0.455753\pi$
$433$	1.08115e6	0.277118	0.138559	+	0.990354i	$0.455753\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.55367e6	−1.39116
$438$	0	0
$439$	7.04807e6	1.74546	0.872728	−	0.488207i	$-0.162349\pi$
$439$	7.04807e6	1.74546	0.872728	+	0.488207i	$0.162349\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−2.32031e6	−0.561742	−0.280871	−	0.959746i	$-0.590623\pi$
$443$	−2.32031e6	−0.561742	−0.280871	+	0.959746i	$0.590623\pi$
$444$	0	0
$445$	6.12075e6	1.46523
$446$	0	0
$447$	0	0
$448$	0	0
$449$	5.45039e6	1.27589	0.637943	−	0.770084i	$-0.279786\pi$
$449$	5.45039e6	1.27589	0.637943	+	0.770084i	$0.279786\pi$
$450$	0	0
$451$	1.64952e6	0.381871
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−286925.	−0.0649741
$456$	0	0
$457$	6.62676e6	1.48426	0.742132	−	0.670254i	$-0.233815\pi$
$457$	6.62676e6	1.48426	0.742132	+	0.670254i	$0.233815\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−606397.	−0.132894	−0.0664469	−	0.997790i	$-0.521166\pi$
$461$	−606397.	−0.132894	−0.0664469	+	0.997790i	$0.521166\pi$
$462$	0	0
$463$	−3.65855e6	−0.793153	−0.396576	−	0.918002i	$-0.629802\pi$
$463$	−3.65855e6	−0.793153	−0.396576	+	0.918002i	$0.629802\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.62712e6	1.19397	0.596986	−	0.802252i	$-0.296365\pi$
$467$	5.62712e6	1.19397	0.596986	+	0.802252i	$0.296365\pi$
$468$	0	0
$469$	−88792.5	−0.0186399
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.13380e6	−0.233014
$474$	0	0
$475$	−1.75857e6	−0.357624
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.25107e6	0.846563	0.423281	−	0.905998i	$-0.360878\pi$
$479$	4.25107e6	0.846563	0.423281	+	0.905998i	$0.360878\pi$
$480$	0	0
$481$	−1.14297e6	−0.225254
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.05187e6	−1.36129
$486$	0	0
$487$	−5.28756e6	−1.01026	−0.505130	−	0.863043i	$-0.668555\pi$
$487$	−5.28756e6	−1.01026	−0.505130	+	0.863043i	$0.668555\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−3.33842e6	−0.624939	−0.312469	−	0.949928i	$-0.601156\pi$
$491$	−3.33842e6	−0.624939	−0.312469	+	0.949928i	$0.601156\pi$
$492$	0	0
$493$	546192.	0.101211
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.78434e6	−0.505629
$498$	0	0
$499$	−2.52847e6	−0.454576	−0.227288	−	0.973828i	$-0.572986\pi$
$499$	−2.52847e6	−0.454576	−0.227288	+	0.973828i	$0.572986\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.94498e6	−1.04768	−0.523842	−	0.851815i	$-0.675502\pi$
$503$	−5.94498e6	−1.04768	−0.523842	+	0.851815i	$0.675502\pi$
$504$	0	0
$505$	−4.82420e6	−0.841776
$506$	0	0
$507$	0	0
$508$	0	0
$509$	603525.	0.103253	0.0516263	−	0.998666i	$-0.483560\pi$
$509$	603525.	0.103253	0.0516263	+	0.998666i	$0.483560\pi$
$510$	0	0
$511$	−2.17068e6	−0.367741
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.28997e6	1.04503
$516$	0	0
$517$	−190609.	−0.0313630
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−1.03389e7	−1.66870	−0.834350	−	0.551235i	$-0.814157\pi$
$521$	−1.03389e7	−1.66870	−0.834350	+	0.551235i	$0.814157\pi$
$522$	0	0
$523$	8.82310e6	1.41048	0.705240	−	0.708968i	$-0.250839\pi$
$523$	8.82310e6	1.41048	0.705240	+	0.708968i	$0.250839\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	218471.	0.0342663
$528$	0	0
$529$	−424266.	−0.0659172
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.21897e6	0.185856
$534$	0	0
$535$	1.06041e7	1.60173
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−392577.	−0.0582041
$540$	0	0
$541$	1.16811e6	0.171590	0.0857949	−	0.996313i	$-0.472657\pi$
$541$	1.16811e6	0.171590	0.0857949	+	0.996313i	$0.472657\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.77130e6	1.40916
$546$	0	0
$547$	−9.17075e6	−1.31050	−0.655249	−	0.755413i	$-0.727436\pi$
$547$	−9.17075e6	−1.31050	−0.655249	+	0.755413i	$0.727436\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.58216e7	−2.22010
$552$	0	0
$553$	−1.71071e6	−0.237883
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−6.22421e6	−0.850054	−0.425027	−	0.905181i	$-0.639735\pi$
$557$	−6.22421e6	−0.850054	−0.425027	+	0.905181i	$0.639735\pi$
$558$	0	0
$559$	−837859.	−0.113407
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.64042e6	−0.749964	−0.374982	−	0.927032i	$-0.622351\pi$
$563$	−5.64042e6	−0.749964	−0.374982	+	0.927032i	$0.622351\pi$
$564$	0	0
$565$	−1.54129e6	−0.203125
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.05286e7	1.36330	0.681650	−	0.731678i	$-0.261263\pi$
$569$	1.05286e7	1.36330	0.681650	+	0.731678i	$0.261263\pi$
$570$	0	0
$571$	7.31395e6	0.938776	0.469388	−	0.882992i	$-0.344475\pi$
$571$	7.31395e6	0.938776	0.469388	+	0.882992i	$0.344475\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.90373e6	0.240124
$576$	0	0
$577$	2.66534e6	0.333282	0.166641	−	0.986018i	$-0.446708\pi$
$577$	2.66534e6	0.333282	0.166641	+	0.986018i	$0.446708\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.94299e6	−0.238798
$582$	0	0
$583$	1.40000e6	0.170592
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.92431e6	0.589861	0.294931	−	0.955519i	$-0.404703\pi$
$587$	4.92431e6	0.589861	0.294931	+	0.955519i	$0.404703\pi$
$588$	0	0
$589$	−6.32847e6	−0.751641
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5.08735e6	0.594094	0.297047	−	0.954863i	$-0.403998\pi$
$593$	5.08735e6	0.594094	0.297047	+	0.954863i	$0.403998\pi$
$594$	0	0
$595$	−185678.	−0.0215015
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.64678e6	0.529157	0.264579	−	0.964364i	$-0.414767\pi$
$599$	4.64678e6	0.529157	0.264579	+	0.964364i	$0.414767\pi$
$600$	0	0
$601$	−4.35424e6	−0.491730	−0.245865	−	0.969304i	$-0.579072\pi$
$601$	−4.35424e6	−0.491730	−0.245865	+	0.969304i	$0.579072\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.50929e6	−0.723012
$606$	0	0
$607$	1.25461e7	1.38209	0.691043	−	0.722813i	$-0.257151\pi$
$607$	1.25461e7	1.38209	0.691043	+	0.722813i	$0.257151\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−140857.	−0.0152643
$612$	0	0
$613$	−4.07637e6	−0.438149	−0.219075	−	0.975708i	$-0.570304\pi$
$613$	−4.07637e6	−0.438149	−0.219075	+	0.975708i	$0.570304\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.88533e6	0.939639	0.469819	−	0.882763i	$-0.344319\pi$
$617$	8.88533e6	0.939639	0.469819	+	0.882763i	$0.344319\pi$
$618$	0	0
$619$	1.28530e7	1.34827	0.674136	−	0.738607i	$-0.264516\pi$
$619$	1.28530e7	1.34827	0.674136	+	0.738607i	$0.264516\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.18867e6	0.638818
$624$	0	0
$625$	−6.73652e6	−0.689819
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−739652.	−0.0745420
$630$	0	0
$631$	288616.	0.0288567	0.0144283	−	0.999896i	$-0.495407\pi$
$631$	288616.	0.0288567	0.0144283	+	0.999896i	$0.495407\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.54166e7	1.51724
$636$	0	0
$637$	−290109.	−0.0283278
$638$	0	0
$639$	0	0
$640$	0	0
$641$	1.94591e7	1.87058	0.935291	−	0.353880i	$-0.115138\pi$
$641$	1.94591e7	1.87058	0.935291	+	0.353880i	$0.115138\pi$
$642$	0	0
$643$	−1.30171e7	−1.24161	−0.620805	−	0.783965i	$-0.713194\pi$
$643$	−1.30171e7	−1.24161	−0.620805	+	0.783965i	$0.713194\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6.42864e6	0.603751	0.301876	−	0.953347i	$-0.402387\pi$
$647$	6.42864e6	0.603751	0.301876	+	0.953347i	$0.402387\pi$
$648$	0	0
$649$	−1.01705e6	−0.0947829
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−1.00202e7	−0.919592	−0.459796	−	0.888025i	$-0.652077\pi$
$653$	−1.00202e7	−0.919592	−0.459796	+	0.888025i	$0.652077\pi$
$654$	0	0
$655$	−1.47904e7	−1.34703
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.49356e7	1.33971	0.669854	−	0.742493i	$-0.266357\pi$
$659$	1.49356e7	1.33971	0.669854	+	0.742493i	$0.266357\pi$
$660$	0	0
$661$	−1.35055e6	−0.120229	−0.0601143	−	0.998191i	$-0.519147\pi$
$661$	−1.35055e6	−0.120229	−0.0601143	+	0.998191i	$0.519147\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.37857e6	0.471643
$666$	0	0
$667$	1.71276e7	1.49067
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.85561e6	0.587814
$672$	0	0
$673$	6.59401e6	0.561193	0.280596	−	0.959826i	$-0.409468\pi$
$673$	6.59401e6	0.561193	0.280596	+	0.959826i	$0.409468\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.42517e7	−1.19508	−0.597538	−	0.801840i	$-0.703854\pi$
$677$	−1.42517e7	−1.19508	−0.597538	+	0.801840i	$0.703854\pi$
$678$	0	0
$679$	−7.13012e6	−0.593502
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−7.65006e6	−0.627499	−0.313749	−	0.949506i	$-0.601585\pi$
$683$	−7.65006e6	−0.627499	−0.313749	+	0.949506i	$0.601585\pi$
$684$	0	0
$685$	−8.72536e6	−0.710489
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1.03458e6	0.0830266
$690$	0	0
$691$	−1.38655e7	−1.10469	−0.552344	−	0.833616i	$-0.686267\pi$
$691$	−1.38655e7	−1.10469	−0.552344	+	0.833616i	$0.686267\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.56632e7	1.23004
$696$	0	0
$697$	788837.	0.0615042
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.38825e6	0.183563	0.0917814	−	0.995779i	$-0.470744\pi$
$701$	2.38825e6	0.183563	0.0917814	+	0.995779i	$0.470744\pi$
$702$	0	0
$703$	2.14256e7	1.63510
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.87773e6	−0.367003
$708$	0	0
$709$	−1.55641e7	−1.16281	−0.581404	−	0.813615i	$-0.697496\pi$
$709$	−1.55641e7	−1.16281	−0.581404	+	0.813615i	$0.697496\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6.85083e6	0.504684
$714$	0	0
$715$	957427.	0.0700391
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.82555e7	1.31696	0.658478	−	0.752600i	$-0.271200\pi$
$719$	1.82555e7	1.31696	0.658478	+	0.752600i	$0.271200\pi$
$720$	0	0
$721$	6.35977e6	0.455620
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.42346e6	0.383205
$726$	0	0
$727$	2.11427e6	0.148362	0.0741812	−	0.997245i	$-0.476366\pi$
$727$	2.11427e6	0.148362	0.0741812	+	0.997245i	$0.476366\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−542206.	−0.0375293
$732$	0	0
$733$	2.12932e6	0.146380	0.0731898	−	0.997318i	$-0.476682\pi$
$733$	2.12932e6	0.146380	0.0731898	+	0.997318i	$0.476682\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	296288.	0.0200930
$738$	0	0
$739$	8.08671e6	0.544704	0.272352	−	0.962198i	$-0.412198\pi$
$739$	8.08671e6	0.544704	0.272352	+	0.962198i	$0.412198\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.18944e7	−0.790442	−0.395221	−	0.918586i	$-0.629332\pi$
$743$	−1.18944e7	−0.790442	−0.395221	+	0.918586i	$0.629332\pi$
$744$	0	0
$745$	−1.42556e7	−0.941010
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.07217e7	0.698331
$750$	0	0
$751$	1.85538e7	1.20042	0.600211	−	0.799842i	$-0.295083\pi$
$751$	1.85538e7	1.20042	0.600211	+	0.799842i	$0.295083\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.62330e7	1.03641
$756$	0	0
$757$	9.32534e6	0.591459	0.295730	−	0.955272i	$-0.404437\pi$
$757$	9.32534e6	0.591459	0.295730	+	0.955272i	$0.404437\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−8.00756e6	−0.501232	−0.250616	−	0.968087i	$-0.580633\pi$
$761$	−8.00756e6	−0.501232	−0.250616	+	0.968087i	$0.580633\pi$
$762$	0	0
$763$	9.87974e6	0.614376
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−751584.	−0.0461306
$768$	0	0
$769$	2.02720e7	1.23618	0.618089	−	0.786108i	$-0.287907\pi$
$769$	2.02720e7	1.23618	0.618089	+	0.786108i	$0.287907\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−2.87881e6	−0.173286	−0.0866431	−	0.996239i	$-0.527614\pi$
$773$	−2.87881e6	−0.173286	−0.0866431	+	0.996239i	$0.527614\pi$
$774$	0	0
$775$	2.16932e6	0.129739
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2.28503e7	−1.34911
$780$	0	0
$781$	9.29094e6	0.545045
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.03428e6	−0.407423
$786$	0	0
$787$	2.03508e7	1.17124	0.585618	−	0.810587i	$-0.300852\pi$
$787$	2.03508e7	1.17124	0.585618	+	0.810587i	$0.300852\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.55839e6	−0.0885595
$792$	0	0
$793$	5.06620e6	0.286088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.24765e7	1.25338	0.626689	−	0.779269i	$-0.284410\pi$
$797$	2.24765e7	1.25338	0.626689	+	0.779269i	$0.284410\pi$
$798$	0	0
$799$	−91153.3	−0.00505133
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.24322e6	0.396409
$804$	0	0
$805$	−5.82253e6	−0.316681
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.15427e6	0.0620062	0.0310031	−	0.999519i	$-0.490130\pi$
$809$	1.15427e6	0.0620062	0.0310031	+	0.999519i	$0.490130\pi$
$810$	0	0
$811$	2.26698e7	1.21031	0.605154	−	0.796108i	$-0.293112\pi$
$811$	2.26698e7	1.21031	0.605154	+	0.796108i	$0.293112\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.54007e6	0.450368
$816$	0	0
$817$	1.57061e7	0.823217
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.41657e6	0.125124	0.0625622	−	0.998041i	$-0.480073\pi$
$821$	2.41657e6	0.125124	0.0625622	+	0.998041i	$0.480073\pi$
$822$	0	0
$823$	2.10169e6	0.108161	0.0540804	−	0.998537i	$-0.482777\pi$
$823$	2.10169e6	0.108161	0.0540804	+	0.998537i	$0.482777\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.53997e7	0.782977	0.391489	−	0.920183i	$-0.371960\pi$
$827$	1.53997e7	0.782977	0.391489	+	0.920183i	$0.371960\pi$
$828$	0	0
$829$	−2.33839e7	−1.18177	−0.590883	−	0.806757i	$-0.701221\pi$
$829$	−2.33839e7	−1.18177	−0.590883	+	0.806757i	$0.701221\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−187739.	−0.00937436
$834$	0	0
$835$	−2.20313e6	−0.109351
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.03016e7	0.995695	0.497848	−	0.867265i	$-0.334124\pi$
$839$	2.03016e7	0.995695	0.497848	+	0.867265i	$0.334124\pi$
$840$	0	0
$841$	2.82829e7	1.37890
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.72862e7	−0.832831
$846$	0	0
$847$	−6.58153e6	−0.315223
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.31941e7	−1.09788
$852$	0	0
$853$	3.35163e7	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$853$	3.35163e7	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.30661e7	1.07281	0.536405	−	0.843961i	$-0.319782\pi$
$857$	2.30661e7	1.07281	0.536405	+	0.843961i	$0.319782\pi$
$858$	0	0
$859$	−3.85609e7	−1.78305	−0.891527	−	0.452968i	$-0.850365\pi$
$859$	−3.85609e7	−1.78305	−0.891527	+	0.452968i	$0.850365\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	3.86892e7	1.76833	0.884163	−	0.467179i	$-0.154729\pi$
$863$	3.86892e7	1.76833	0.884163	+	0.467179i	$0.154729\pi$
$864$	0	0
$865$	9.97555e6	0.453311
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.70838e6	0.256427
$870$	0	0
$871$	218952.	0.00977922
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.26449e6	−0.409073
$876$	0	0
$877$	−2.17369e7	−0.954332	−0.477166	−	0.878813i	$-0.658336\pi$
$877$	−2.17369e7	−0.954332	−0.477166	+	0.878813i	$0.658336\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−8.42086e6	−0.365525	−0.182762	−	0.983157i	$-0.558504\pi$
$881$	−8.42086e6	−0.365525	−0.182762	+	0.983157i	$0.558504\pi$
$882$	0	0
$883$	−3.22954e7	−1.39392	−0.696962	−	0.717108i	$-0.745465\pi$
$883$	−3.22954e7	−1.39392	−0.696962	+	0.717108i	$0.745465\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−4.21800e7	−1.80010	−0.900052	−	0.435782i	$-0.856472\pi$
$887$	−4.21800e7	−1.80010	−0.900052	+	0.435782i	$0.856472\pi$
$888$	0	0
$889$	1.55876e7	0.661494
$890$	0	0
$891$	0	0
$892$	0	0
$893$	2.64045e6	0.110802
$894$	0	0
$895$	−9.65393e6	−0.402853
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.95171e7	0.805407
$900$	0	0
$901$	669512.	0.0274756
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.61542e6	0.390254
$906$	0	0
$907$	7.64517e6	0.308581	0.154291	−	0.988026i	$-0.450691\pi$
$907$	7.64517e6	0.308581	0.154291	+	0.988026i	$0.450691\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−3.93310e7	−1.57014	−0.785072	−	0.619405i	$-0.787374\pi$
$911$	−3.93310e7	−1.57014	−0.785072	+	0.619405i	$0.787374\pi$
$912$	0	0
$913$	6.48347e6	0.257413
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.49545e7	−0.587285
$918$	0	0
$919$	−3.52204e7	−1.37564	−0.687822	−	0.725880i	$-0.741433\pi$
$919$	−3.52204e7	−1.37564	−0.687822	+	0.725880i	$0.741433\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.86587e6	0.265272
$924$	0	0
$925$	−7.34443e6	−0.282230
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.66300e7	0.632199	0.316099	−	0.948726i	$-0.397627\pi$
$929$	1.66300e7	0.632199	0.316099	+	0.948726i	$0.397627\pi$
$930$	0	0
$931$	5.43826e6	0.205630
$932$	0	0
$933$	0	0
$934$	0	0
$935$	619582.	0.0231777
$936$	0	0
$937$	2.07121e7	0.770681	0.385341	−	0.922774i	$-0.374084\pi$
$937$	2.07121e7	0.770681	0.385341	+	0.922774i	$0.374084\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.09797e7	0.404219	0.202110	−	0.979363i	$-0.435220\pi$
$941$	1.09797e7	0.404219	0.202110	+	0.979363i	$0.435220\pi$
$942$	0	0
$943$	2.47364e7	0.905853
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.71723e6	0.0984581	0.0492290	−	0.998788i	$-0.484324\pi$
$947$	2.71723e6	0.0984581	0.0492290	+	0.998788i	$0.484324\pi$
$948$	0	0
$949$	5.35264e6	0.192931
$950$	0	0
$951$	0	0
$952$	0	0
$953$	9.24083e6	0.329594	0.164797	−	0.986328i	$-0.447303\pi$
$953$	9.24083e6	0.329594	0.164797	+	0.986328i	$0.447303\pi$
$954$	0	0
$955$	1.80739e7	0.641272
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.82219e6	−0.309763
$960$	0	0
$961$	−2.08225e7	−0.727320
$962$	0	0
$963$	0	0
$964$	0	0
$965$	8.48183e6	0.293205
$966$	0	0
$967$	1.50444e7	0.517379	0.258690	−	0.965960i	$-0.416709\pi$
$967$	1.50444e7	0.517379	0.258690	+	0.965960i	$0.416709\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.27894e7	0.775685	0.387842	−	0.921726i	$-0.373220\pi$
$971$	2.27894e7	0.775685	0.387842	+	0.921726i	$0.373220\pi$
$972$	0	0
$973$	1.58370e7	0.536279
$974$	0	0
$975$	0	0
$976$	0	0
$977$	3.96682e7	1.32955	0.664777	−	0.747042i	$-0.268526\pi$
$977$	3.96682e7	1.32955	0.664777	+	0.747042i	$0.268526\pi$
$978$	0	0
$979$	−2.06507e7	−0.688617
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.77093e7	0.914621	0.457310	−	0.889307i	$-0.348813\pi$
$983$	2.77093e7	0.914621	0.457310	+	0.889307i	$0.348813\pi$
$984$	0	0
$985$	−4.28294e7	−1.40654
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.70025e7	−0.552743
$990$	0	0
$991$	−4.78535e7	−1.54785	−0.773926	−	0.633276i	$-0.781710\pi$
$991$	−4.78535e7	−1.54785	−0.773926	+	0.633276i	$0.781710\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.80837e7	−1.21950
$996$	0	0
$997$	−5.48289e7	−1.74691	−0.873457	−	0.486902i	$-0.838127\pi$
$997$	−5.48289e7	−1.74691	−0.873457	+	0.486902i	$0.838127\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.a.bi.1.2		2
3.2	odd	2		112.6.a.j.1.1		2
4.3	odd	2		504.6.a.m.1.2		2
12.11	even	2		56.6.a.d.1.2	✓	2
21.20	even	2		784.6.a.q.1.2		2
24.5	odd	2		448.6.a.r.1.2		2
24.11	even	2		448.6.a.x.1.1		2
84.11	even	6		392.6.i.k.177.1		4
84.23	even	6		392.6.i.k.361.1		4
84.47	odd	6		392.6.i.h.361.2		4
84.59	odd	6		392.6.i.h.177.2		4
84.83	odd	2		392.6.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
56.6.a.d.1.2	✓	2	12.11	even	2
112.6.a.j.1.1		2	3.2	odd	2
392.6.a.e.1.1		2	84.83	odd	2
392.6.i.h.177.2		4	84.59	odd	6
392.6.i.h.361.2		4	84.47	odd	6
392.6.i.k.177.1		4	84.11	even	6
392.6.i.k.361.1		4	84.23	even	6
448.6.a.r.1.2		2	24.5	odd	2
448.6.a.x.1.1		2	24.11	even	2
504.6.a.m.1.2		2	4.3	odd	2
784.6.a.q.1.2		2	21.20	even	2
1008.6.a.bi.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)

\(f(q)\)	\(=\)	\(q+48.4622 q^{5} +49.0000 q^{7} -163.506 q^{11} -120.828 q^{13} -78.1920 q^{17} +2265.00 q^{19} -2451.95 q^{23} -776.413 q^{25} -6985.27 q^{29} -2794.03 q^{31} +2374.65 q^{35} +9459.44 q^{37} -10088.5 q^{41} +6934.29 q^{43} +1165.76 q^{47} +2401.00 q^{49} -8562.42 q^{53} -7923.85 q^{55} +6220.26 q^{59} -41928.9 q^{61} -5855.62 q^{65} -1812.09 q^{67} -56823.3 q^{71} -44299.5 q^{73} -8011.78 q^{77} -34912.4 q^{79} -39652.9 q^{83} -3789.36 q^{85} +126299. q^{89} -5920.59 q^{91} +109767. q^{95} -145513. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 42 q^{5} + 98 q^{7} - 716 q^{11} - 714 q^{13} + 1344 q^{17} + 1946 q^{19} - 1792 q^{23} + 4282 q^{25} + 1200 q^{29} + 6804 q^{31} - 2058 q^{35} + 14640 q^{37} - 7896 q^{41} - 524 q^{43} - 18396 q^{47}+ \cdots - 44240 q^{97}+O(q^{100}) \) 2 * q - 42 * q^5 + 98 * q^7 - 716 * q^11 - 714 * q^13 + 1344 * q^17 + 1946 * q^19 - 1792 * q^23 + 4282 * q^25 + 1200 * q^29 + 6804 * q^31 - 2058 * q^35 + 14640 * q^37 - 7896 * q^41 - 524 * q^43 - 18396 * q^47 + 4802 * q^49 - 45132 * q^53 + 42056 * q^55 + 22582 * q^59 - 52822 * q^61 + 47804 * q^65 - 9848 * q^67 - 840 * q^71 - 122052 * q^73 - 35084 * q^77 - 31704 * q^79 + 36974 * q^83 - 132444 * q^85 + 210588 * q^89 - 34986 * q^91 + 138624 * q^95 - 44240 * q^97

Introduction

L-functions

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