LMFDB - Embedded newform 1008.6.a.bj.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,-28,0,98,0,0,0,-596,0,532] 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{114}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 114 $ x^2 - 114
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.6771$ 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)

$f(q)$	$=$	$q+50.0625 q^{5} +49.0000 q^{7} +150.437 q^{11} +522.250 q^{13} -345.313 q^{17} +2140.75 q^{19} -484.688 q^{23} -618.749 q^{25} +3947.50 q^{29} +3905.75 q^{31} +2453.06 q^{35} -5574.75 q^{37} +7703.56 q^{41} -2565.00 q^{43} +20901.9 q^{47} +2401.00 q^{49} -29684.9 q^{53} +7531.26 q^{55} +14887.6 q^{59} +4981.74 q^{61} +26145.1 q^{65} -35931.5 q^{67} +13176.9 q^{71} +49108.5 q^{73} +7371.43 q^{77} +17296.5 q^{79} -17900.5 q^{83} -17287.2 q^{85} -71460.7 q^{89} +25590.2 q^{91} +107171. q^{95} +74744.5 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 28 q^{5} + 98 q^{7} - 596 q^{11} + 532 q^{13} - 2100 q^{17} + 2744 q^{19} - 3660 q^{23} + 2350 q^{25} + 720 q^{29} - 3976 q^{31} - 1372 q^{35} + 6788 q^{37} + 4004 q^{41} - 19480 q^{43} + 21560 q^{47}+ \cdots - 36036 q^{97}+O(q^{100}) $ 2 * q - 28 * q^5 + 98 * q^7 - 596 * q^11 + 532 * q^13 - 2100 * q^17 + 2744 * q^19 - 3660 * q^23 + 2350 * q^25 + 720 * q^29 - 3976 * q^31 - 1372 * q^35 + 6788 * q^37 + 4004 * q^41 - 19480 * q^43 + 21560 * q^47 + 4802 * q^49 - 57576 * q^53 + 65800 * q^55 + 22344 * q^59 - 38724 * q^61 + 25384 * q^65 - 7288 * q^67 + 3932 * q^71 + 19292 * q^73 - 29204 * q^77 - 15632 * q^79 + 22624 * q^83 + 119688 * q^85 - 112812 * q^89 + 26068 * q^91 + 60080 * q^95 - 36036 * 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
$5$	50.0625	0.895545	0.447772	−	0.894148i	$-0.352218\pi$
$5$	50.0625	0.895545	0.447772	+	0.894148i	$0.352218\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	150.437	0.374864	0.187432	−	0.982278i	$-0.439984\pi$
$11$	150.437	0.374864	0.187432	+	0.982278i	$0.439984\pi$
$12$	0	0
$13$	522.250	0.857077	0.428539	−	0.903523i	$-0.359029\pi$
$13$	522.250	0.857077	0.428539	+	0.903523i	$0.359029\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−345.313	−0.289795	−0.144897	−	0.989447i	$-0.546285\pi$
$17$	−345.313	−0.289795	−0.144897	+	0.989447i	$0.546285\pi$
$18$	0	0
$19$	2140.75	1.36045	0.680224	−	0.733004i	$-0.261883\pi$
$19$	2140.75	1.36045	0.680224	+	0.733004i	$0.261883\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−484.688	−0.191048	−0.0955241	−	0.995427i	$-0.530453\pi$
$23$	−484.688	−0.191048	−0.0955241	+	0.995427i	$0.530453\pi$
$24$	0	0
$25$	−618.749	−0.198000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3947.50	0.871620	0.435810	−	0.900039i	$-0.356462\pi$
$29$	3947.50	0.871620	0.435810	+	0.900039i	$0.356462\pi$
$30$	0	0
$31$	3905.75	0.729961	0.364981	−	0.931015i	$-0.381076\pi$
$31$	3905.75	0.729961	0.364981	+	0.931015i	$0.381076\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2453.06	0.338484
$36$	0	0
$37$	−5574.75	−0.669454	−0.334727	−	0.942315i	$-0.608644\pi$
$37$	−5574.75	−0.669454	−0.334727	+	0.942315i	$0.608644\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7703.56	0.715701	0.357851	−	0.933779i	$-0.383510\pi$
$41$	7703.56	0.715701	0.357851	+	0.933779i	$0.383510\pi$
$42$	0	0
$43$	−2565.00	−0.211552	−0.105776	−	0.994390i	$-0.533733\pi$
$43$	−2565.00	−0.211552	−0.105776	+	0.994390i	$0.533733\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	20901.9	1.38020	0.690098	−	0.723716i	$-0.257568\pi$
$47$	20901.9	1.38020	0.690098	+	0.723716i	$0.257568\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−29684.9	−1.45160	−0.725798	−	0.687908i	$-0.758529\pi$
$53$	−29684.9	−1.45160	−0.725798	+	0.687908i	$0.758529\pi$
$54$	0	0
$55$	7531.26	0.335707
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14887.6	0.556795	0.278398	−	0.960466i	$-0.410197\pi$
$59$	14887.6	0.556795	0.278398	+	0.960466i	$0.410197\pi$
$60$	0	0
$61$	4981.74	0.171418	0.0857090	−	0.996320i	$-0.472684\pi$
$61$	4981.74	0.171418	0.0857090	+	0.996320i	$0.472684\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	26145.1	0.767551
$66$	0	0
$67$	−35931.5	−0.977886	−0.488943	−	0.872316i	$-0.662617\pi$
$67$	−35931.5	−0.977886	−0.488943	+	0.872316i	$0.662617\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13176.9	0.310219	0.155110	−	0.987897i	$-0.450427\pi$
$71$	13176.9	0.310219	0.155110	+	0.987897i	$0.450427\pi$
$72$	0	0
$73$	49108.5	1.07857	0.539286	−	0.842123i	$-0.318694\pi$
$73$	49108.5	1.07857	0.539286	+	0.842123i	$0.318694\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7371.43	0.141685
$78$	0	0
$79$	17296.5	0.311810	0.155905	−	0.987772i	$-0.450171\pi$
$79$	17296.5	0.311810	0.155905	+	0.987772i	$0.450171\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−17900.5	−0.285213	−0.142607	−	0.989779i	$-0.545548\pi$
$83$	−17900.5	−0.285213	−0.142607	+	0.989779i	$0.545548\pi$
$84$	0	0
$85$	−17287.2	−0.259524
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−71460.7	−0.956296	−0.478148	−	0.878279i	$-0.658692\pi$
$89$	−71460.7	−0.956296	−0.478148	+	0.878279i	$0.658692\pi$
$90$	0	0
$91$	25590.2	0.323945
$92$	0	0
$93$	0	0
$94$	0	0
$95$	107171.	1.21834
$96$	0	0
$97$	74744.5	0.806584	0.403292	−	0.915071i	$-0.367866\pi$
$97$	74744.5	0.806584	0.403292	+	0.915071i	$0.367866\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−24726.6	−0.241191	−0.120595	−	0.992702i	$-0.538480\pi$
$101$	−24726.6	−0.241191	−0.120595	+	0.992702i	$0.538480\pi$
$102$	0	0
$103$	−1889.77	−0.0175516	−0.00877580	−	0.999961i	$-0.502793\pi$
$103$	−1889.77	−0.0175516	−0.00877580	+	0.999961i	$0.502793\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	176838.	1.49319	0.746596	−	0.665278i	$-0.231687\pi$
$107$	176838.	1.49319	0.746596	+	0.665278i	$0.231687\pi$
$108$	0	0
$109$	−76023.5	−0.612889	−0.306444	−	0.951889i	$-0.599139\pi$
$109$	−76023.5	−0.612889	−0.306444	+	0.951889i	$0.599139\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−70053.5	−0.516100	−0.258050	−	0.966132i	$-0.583080\pi$
$113$	−70053.5	−0.516100	−0.258050	+	0.966132i	$0.583080\pi$
$114$	0	0
$115$	−24264.7	−0.171092
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−16920.3	−0.109532
$120$	0	0
$121$	−138420.	−0.859477
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−187421.	−1.07286
$126$	0	0
$127$	90666.0	0.498810	0.249405	−	0.968399i	$-0.419765\pi$
$127$	90666.0	0.498810	0.249405	+	0.968399i	$0.419765\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−26597.2	−0.135412	−0.0677060	−	0.997705i	$-0.521568\pi$
$131$	−26597.2	−0.135412	−0.0677060	+	0.997705i	$0.521568\pi$
$132$	0	0
$133$	104897.	0.514201
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−130412.	−0.593632	−0.296816	−	0.954935i	$-0.595925\pi$
$137$	−130412.	−0.593632	−0.296816	+	0.954935i	$0.595925\pi$
$138$	0	0
$139$	295933.	1.29914	0.649572	−	0.760300i	$-0.274948\pi$
$139$	295933.	1.29914	0.649572	+	0.760300i	$0.274948\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	78565.9	0.321287
$144$	0	0
$145$	197622.	0.780574
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−441885.	−1.63059	−0.815293	−	0.579049i	$-0.803424\pi$
$149$	−441885.	−1.63059	−0.815293	+	0.579049i	$0.803424\pi$
$150$	0	0
$151$	−247605.	−0.883726	−0.441863	−	0.897083i	$-0.645682\pi$
$151$	−247605.	−0.883726	−0.441863	+	0.897083i	$0.645682\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	195531.	0.653713
$156$	0	0
$157$	67662.2	0.219077	0.109539	−	0.993983i	$-0.465063\pi$
$157$	67662.2	0.219077	0.109539	+	0.993983i	$0.465063\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−23749.7	−0.0722094
$162$	0	0
$163$	595355.	1.75512	0.877561	−	0.479466i	$-0.159169\pi$
$163$	595355.	1.75512	0.877561	+	0.479466i	$0.159169\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	541587.	1.50272	0.751358	−	0.659895i	$-0.229399\pi$
$167$	541587.	1.50272	0.751358	+	0.659895i	$0.229399\pi$
$168$	0	0
$169$	−98548.1	−0.265419
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−407207.	−1.03443	−0.517213	−	0.855857i	$-0.673031\pi$
$173$	−407207.	−1.03443	−0.517213	+	0.855857i	$0.673031\pi$
$174$	0	0
$175$	−30318.7	−0.0748369
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−309285.	−0.721483	−0.360742	−	0.932666i	$-0.617476\pi$
$179$	−309285.	−0.721483	−0.360742	+	0.932666i	$0.617476\pi$
$180$	0	0
$181$	300949.	0.682804	0.341402	−	0.939917i	$-0.389098\pi$
$181$	300949.	0.682804	0.341402	+	0.939917i	$0.389098\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−279086.	−0.599526
$186$	0	0
$187$	−51947.9	−0.108634
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−41016.3	−0.0813529	−0.0406764	−	0.999172i	$-0.512951\pi$
$191$	−41016.3	−0.0813529	−0.0406764	+	0.999172i	$0.512951\pi$
$192$	0	0
$193$	454328.	0.877963	0.438982	−	0.898496i	$-0.355339\pi$
$193$	454328.	0.877963	0.438982	+	0.898496i	$0.355339\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−477754.	−0.877078	−0.438539	−	0.898712i	$-0.644504\pi$
$197$	−477754.	−0.877078	−0.438539	+	0.898712i	$0.644504\pi$
$198$	0	0
$199$	−260092.	−0.465580	−0.232790	−	0.972527i	$-0.574785\pi$
$199$	−260092.	−0.465580	−0.232790	+	0.972527i	$0.574785\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	193427.	0.329441
$204$	0	0
$205$	385659.	0.640943
$206$	0	0
$207$	0	0
$208$	0	0
$209$	322049.	0.509983
$210$	0	0
$211$	−74213.3	−0.114756	−0.0573780	−	0.998353i	$-0.518274\pi$
$211$	−74213.3	−0.114756	−0.0573780	+	0.998353i	$0.518274\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−128410.	−0.189454
$216$	0	0
$217$	191382.	0.275899
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−180340.	−0.248376
$222$	0	0
$223$	792373.	1.06701	0.533504	−	0.845798i	$-0.320875\pi$
$223$	792373.	1.06701	0.533504	+	0.845798i	$0.320875\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.32051e6	1.70090	0.850449	−	0.526058i	$-0.176330\pi$
$227$	1.32051e6	1.70090	0.850449	+	0.526058i	$0.176330\pi$
$228$	0	0
$229$	1.03037e6	1.29839	0.649197	−	0.760620i	$-0.275105\pi$
$229$	1.03037e6	1.29839	0.649197	+	0.760620i	$0.275105\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	809463.	0.976804	0.488402	−	0.872619i	$-0.337580\pi$
$233$	809463.	0.976804	0.488402	+	0.872619i	$0.337580\pi$
$234$	0	0
$235$	1.04640e6	1.23603
$236$	0	0
$237$	0	0
$238$	0	0
$239$	46281.9	0.0524103	0.0262052	−	0.999657i	$-0.491658\pi$
$239$	46281.9	0.0524103	0.0262052	+	0.999657i	$0.491658\pi$
$240$	0	0
$241$	1.16053e6	1.28711	0.643554	−	0.765401i	$-0.277459\pi$
$241$	1.16053e6	1.28711	0.643554	+	0.765401i	$0.277459\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	120200.	0.127935
$246$	0	0
$247$	1.11801e6	1.16601
$248$	0	0
$249$	0	0
$250$	0	0
$251$	52898.3	0.0529978	0.0264989	−	0.999649i	$-0.491564\pi$
$251$	52898.3	0.0529978	0.0264989	+	0.999649i	$0.491564\pi$
$252$	0	0
$253$	−72915.2	−0.0716171
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−315247.	−0.297727	−0.148863	−	0.988858i	$-0.547561\pi$
$257$	−315247.	−0.297727	−0.148863	+	0.988858i	$0.547561\pi$
$258$	0	0
$259$	−273163.	−0.253030
$260$	0	0
$261$	0	0
$262$	0	0
$263$	222707.	0.198538	0.0992692	−	0.995061i	$-0.468349\pi$
$263$	222707.	0.198538	0.0992692	+	0.995061i	$0.468349\pi$
$264$	0	0
$265$	−1.48610e6	−1.29997
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10498.1	0.00884563	0.00442281	−	0.999990i	$-0.498592\pi$
$269$	10498.1	0.00884563	0.00442281	+	0.999990i	$0.498592\pi$
$270$	0	0
$271$	933583.	0.772200	0.386100	−	0.922457i	$-0.373822\pi$
$271$	933583.	0.772200	0.386100	+	0.922457i	$0.373822\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−93082.9	−0.0742230
$276$	0	0
$277$	1.36264e6	1.06704	0.533522	−	0.845786i	$-0.320868\pi$
$277$	1.36264e6	1.06704	0.533522	+	0.845786i	$0.320868\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.56211e6	1.18018	0.590088	−	0.807339i	$-0.299093\pi$
$281$	1.56211e6	1.18018	0.590088	+	0.807339i	$0.299093\pi$
$282$	0	0
$283$	191256.	0.141954	0.0709772	−	0.997478i	$-0.477388\pi$
$283$	191256.	0.141954	0.0709772	+	0.997478i	$0.477388\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	377474.	0.270510
$288$	0	0
$289$	−1.30062e6	−0.916019
$290$	0	0
$291$	0	0
$292$	0	0
$293$	462576.	0.314785	0.157393	−	0.987536i	$-0.449691\pi$
$293$	462576.	0.314785	0.157393	+	0.987536i	$0.449691\pi$
$294$	0	0
$295$	745311.	0.498635
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−253128.	−0.163743
$300$	0	0
$301$	−125685.	−0.0799591
$302$	0	0
$303$	0	0
$304$	0	0
$305$	249398.	0.153512
$306$	0	0
$307$	−221395.	−0.134067	−0.0670335	−	0.997751i	$-0.521353\pi$
$307$	−221395.	−0.134067	−0.0670335	+	0.997751i	$0.521353\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.03372e6	−0.606039	−0.303019	−	0.952984i	$-0.597995\pi$
$311$	−1.03372e6	−0.606039	−0.303019	+	0.952984i	$0.597995\pi$
$312$	0	0
$313$	1.71162e6	0.987519	0.493759	−	0.869599i	$-0.335622\pi$
$313$	1.71162e6	0.987519	0.493759	+	0.869599i	$0.335622\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−872214.	−0.487500	−0.243750	−	0.969838i	$-0.578378\pi$
$317$	−872214.	−0.487500	−0.243750	+	0.969838i	$0.578378\pi$
$318$	0	0
$319$	593851.	0.326739
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−739228.	−0.394251
$324$	0	0
$325$	−323142.	−0.169701
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.02419e6	0.521665
$330$	0	0
$331$	999733.	0.501550	0.250775	−	0.968045i	$-0.419315\pi$
$331$	999733.	0.501550	0.250775	+	0.968045i	$0.419315\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.79882e6	−0.875740
$336$	0	0
$337$	1.96881e6	0.944340	0.472170	−	0.881507i	$-0.343471\pi$
$337$	1.96881e6	0.944340	0.472170	+	0.881507i	$0.343471\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	587570.	0.273636
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.48114e6	−1.55202	−0.776011	−	0.630719i	$-0.782760\pi$
$347$	−3.48114e6	−1.55202	−0.776011	+	0.630719i	$0.782760\pi$
$348$	0	0
$349$	2.38332e6	1.04741	0.523707	−	0.851899i	$-0.324549\pi$
$349$	2.38332e6	1.04741	0.523707	+	0.851899i	$0.324549\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.12172e6	1.33339	0.666696	−	0.745330i	$-0.267708\pi$
$353$	3.12172e6	1.33339	0.666696	+	0.745330i	$0.267708\pi$
$354$	0	0
$355$	659670.	0.277815
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−607472.	−0.248765	−0.124383	−	0.992234i	$-0.539695\pi$
$359$	−607472.	−0.248765	−0.124383	+	0.992234i	$0.539695\pi$
$360$	0	0
$361$	2.10671e6	0.850818
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.45849e6	0.965910
$366$	0	0
$367$	1.65712e6	0.642229	0.321115	−	0.947040i	$-0.395943\pi$
$367$	1.65712e6	0.642229	0.321115	+	0.947040i	$0.395943\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.45456e6	−0.548651
$372$	0	0
$373$	3.92072e6	1.45913	0.729564	−	0.683913i	$-0.239723\pi$
$373$	3.92072e6	1.45913	0.729564	+	0.683913i	$0.239723\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.06158e6	0.747045
$378$	0	0
$379$	2.50020e6	0.894082	0.447041	−	0.894513i	$-0.352478\pi$
$379$	2.50020e6	0.894082	0.447041	+	0.894513i	$0.352478\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.61937e6	−0.564091	−0.282045	−	0.959401i	$-0.591013\pi$
$383$	−1.61937e6	−0.564091	−0.282045	+	0.959401i	$0.591013\pi$
$384$	0	0
$385$	369032.	0.126886
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3.18930e6	−1.06861	−0.534307	−	0.845290i	$-0.679427\pi$
$389$	−3.18930e6	−1.06861	−0.534307	+	0.845290i	$0.679427\pi$
$390$	0	0
$391$	167369.	0.0553647
$392$	0	0
$393$	0	0
$394$	0	0
$395$	865905.	0.279240
$396$	0	0
$397$	−1.98115e6	−0.630870	−0.315435	−	0.948947i	$-0.602150\pi$
$397$	−1.98115e6	−0.630870	−0.315435	+	0.948947i	$0.602150\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.95405e6	−0.917396	−0.458698	−	0.888592i	$-0.651684\pi$
$401$	−2.95405e6	−0.917396	−0.458698	+	0.888592i	$0.651684\pi$
$402$	0	0
$403$	2.03978e6	0.625633
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−838650.	−0.250954
$408$	0	0
$409$	2.63901e6	0.780067	0.390034	−	0.920801i	$-0.372463\pi$
$409$	2.63901e6	0.780067	0.390034	+	0.920801i	$0.372463\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	729494.	0.210449
$414$	0	0
$415$	−896143.	−0.255421
$416$	0	0
$417$	0	0
$418$	0	0
$419$	625639.	0.174096	0.0870480	−	0.996204i	$-0.472257\pi$
$419$	625639.	0.174096	0.0870480	+	0.996204i	$0.472257\pi$
$420$	0	0
$421$	3.30853e6	0.909767	0.454883	−	0.890551i	$-0.349681\pi$
$421$	3.30853e6	0.909767	0.454883	+	0.890551i	$0.349681\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	213662.	0.0573793
$426$	0	0
$427$	244105.	0.0647899
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.27141e6	−1.88549	−0.942747	−	0.333508i	$-0.891768\pi$
$431$	−7.27141e6	−1.88549	−0.942747	+	0.333508i	$0.891768\pi$
$432$	0	0
$433$	922867.	0.236548	0.118274	−	0.992981i	$-0.462264\pi$
$433$	922867.	0.236548	0.118274	+	0.992981i	$0.462264\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.03760e6	−0.259911
$438$	0	0
$439$	−1.17979e6	−0.292176	−0.146088	−	0.989272i	$-0.546668\pi$
$439$	−1.17979e6	−0.292176	−0.146088	+	0.989272i	$0.546668\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−1.45194e6	−0.351512	−0.175756	−	0.984434i	$-0.556237\pi$
$443$	−1.45194e6	−0.351512	−0.175756	+	0.984434i	$0.556237\pi$
$444$	0	0
$445$	−3.57750e6	−0.856405
$446$	0	0
$447$	0	0
$448$	0	0
$449$	1.52131e6	0.356126	0.178063	−	0.984019i	$-0.443017\pi$
$449$	1.52131e6	0.356126	0.178063	+	0.984019i	$0.443017\pi$
$450$	0	0
$451$	1.15890e6	0.268291
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.28111e6	0.290107
$456$	0	0
$457$	−2.65993e6	−0.595771	−0.297886	−	0.954602i	$-0.596281\pi$
$457$	−2.65993e6	−0.595771	−0.297886	+	0.954602i	$0.596281\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.26417e6	1.81112	0.905559	−	0.424220i	$-0.139452\pi$
$461$	8.26417e6	1.81112	0.905559	+	0.424220i	$0.139452\pi$
$462$	0	0
$463$	5.76340e6	1.24947	0.624736	−	0.780836i	$-0.285207\pi$
$463$	5.76340e6	1.24947	0.624736	+	0.780836i	$0.285207\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.44068e6	−0.730050	−0.365025	−	0.930998i	$-0.618939\pi$
$467$	−3.44068e6	−0.730050	−0.365025	+	0.930998i	$0.618939\pi$
$468$	0	0
$469$	−1.76064e6	−0.369606
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−385872.	−0.0793032
$474$	0	0
$475$	−1.32459e6	−0.269368
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.39817e6	1.27414	0.637070	−	0.770806i	$-0.280146\pi$
$479$	6.39817e6	1.27414	0.637070	+	0.770806i	$0.280146\pi$
$480$	0	0
$481$	−2.91141e6	−0.573774
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.74189e6	0.722332
$486$	0	0
$487$	−5.01077e6	−0.957375	−0.478687	−	0.877985i	$-0.658887\pi$
$487$	−5.01077e6	−0.957375	−0.478687	+	0.877985i	$0.658887\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2.92530e6	−0.547603	−0.273802	−	0.961786i	$-0.588281\pi$
$491$	−2.92530e6	−0.547603	−0.273802	+	0.961786i	$0.588281\pi$
$492$	0	0
$493$	−1.36312e6	−0.252591
$494$	0	0
$495$	0	0
$496$	0	0
$497$	645670.	0.117252
$498$	0	0
$499$	−3.75480e6	−0.675050	−0.337525	−	0.941317i	$-0.609590\pi$
$499$	−3.75480e6	−0.675050	−0.337525	+	0.941317i	$0.609590\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4.49794e6	0.792673	0.396337	−	0.918105i	$-0.370281\pi$
$503$	4.49794e6	0.792673	0.396337	+	0.918105i	$0.370281\pi$
$504$	0	0
$505$	−1.23787e6	−0.215997
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−4.89885e6	−0.838107	−0.419054	−	0.907962i	$-0.637638\pi$
$509$	−4.89885e6	−0.838107	−0.419054	+	0.907962i	$0.637638\pi$
$510$	0	0
$511$	2.40632e6	0.407662
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−94606.8	−0.0157182
$516$	0	0
$517$	3.14442e6	0.517386
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−8.46792e6	−1.36673	−0.683364	−	0.730077i	$-0.739484\pi$
$521$	−8.46792e6	−1.36673	−0.683364	+	0.730077i	$0.739484\pi$
$522$	0	0
$523$	4.23571e6	0.677130	0.338565	−	0.940943i	$-0.390059\pi$
$523$	4.23571e6	0.677130	0.338565	+	0.940943i	$0.390059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.34870e6	−0.211539
$528$	0	0
$529$	−6.20142e6	−0.963501
$530$	0	0
$531$	0	0
$532$	0	0
$533$	4.02318e6	0.613411
$534$	0	0
$535$	8.85293e6	1.33722
$536$	0	0
$537$	0	0
$538$	0	0
$539$	361200.	0.0535520
$540$	0	0
$541$	1.26890e7	1.86395	0.931977	−	0.362517i	$-0.118082\pi$
$541$	1.26890e7	1.86395	0.931977	+	0.362517i	$0.118082\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−3.80593e6	−0.548869
$546$	0	0
$547$	−8.45389e6	−1.20806	−0.604030	−	0.796962i	$-0.706439\pi$
$547$	−8.45389e6	−1.20806	−0.604030	+	0.796962i	$0.706439\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.45061e6	1.18579
$552$	0	0
$553$	847528.	0.117853
$554$	0	0
$555$	0	0
$556$	0	0
$557$	8.19598e6	1.11934	0.559671	−	0.828715i	$-0.310928\pi$
$557$	8.19598e6	1.11934	0.559671	+	0.828715i	$0.310928\pi$
$558$	0	0
$559$	−1.33957e6	−0.181316
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.06213e6	1.20492	0.602462	−	0.798148i	$-0.294187\pi$
$563$	9.06213e6	1.20492	0.602462	+	0.798148i	$0.294187\pi$
$564$	0	0
$565$	−3.50705e6	−0.462191
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.51890e6	−0.455645	−0.227822	−	0.973703i	$-0.573161\pi$
$569$	−3.51890e6	−0.455645	−0.227822	+	0.973703i	$0.573161\pi$
$570$	0	0
$571$	912516.	0.117125	0.0585626	−	0.998284i	$-0.481348\pi$
$571$	912516.	0.117125	0.0585626	+	0.998284i	$0.481348\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	299900.	0.0378275
$576$	0	0
$577$	−5.44889e6	−0.681347	−0.340674	−	0.940182i	$-0.610655\pi$
$577$	−5.44889e6	−0.681347	−0.340674	+	0.940182i	$0.610655\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−877124.	−0.107800
$582$	0	0
$583$	−4.46571e6	−0.544151
$584$	0	0
$585$	0	0
$586$	0	0
$587$	482435.	0.0577888	0.0288944	−	0.999582i	$-0.490801\pi$
$587$	482435.	0.0577888	0.0288944	+	0.999582i	$0.490801\pi$
$588$	0	0
$589$	8.36123e6	0.993074
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−1.60547e6	−0.187484	−0.0937422	−	0.995597i	$-0.529883\pi$
$593$	−1.60547e6	−0.187484	−0.0937422	+	0.995597i	$0.529883\pi$
$594$	0	0
$595$	−847073.	−0.0980909
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−5.66288e6	−0.644867	−0.322434	−	0.946592i	$-0.604501\pi$
$599$	−5.66288e6	−0.644867	−0.322434	+	0.946592i	$0.604501\pi$
$600$	0	0
$601$	−6.72653e6	−0.759635	−0.379817	−	0.925062i	$-0.624013\pi$
$601$	−6.72653e6	−0.759635	−0.379817	+	0.925062i	$0.624013\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.92963e6	−0.769700
$606$	0	0
$607$	−5.25145e6	−0.578505	−0.289253	−	0.957253i	$-0.593407\pi$
$607$	−5.25145e6	−0.578505	−0.289253	+	0.957253i	$0.593407\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.09160e7	1.18293
$612$	0	0
$613$	325286.	0.0349634	0.0174817	−	0.999847i	$-0.494435\pi$
$613$	325286.	0.0349634	0.0174817	+	0.999847i	$0.494435\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3.46400e6	0.366324	0.183162	−	0.983083i	$-0.441367\pi$
$617$	3.46400e6	0.366324	0.183162	+	0.983083i	$0.441367\pi$
$618$	0	0
$619$	−8.92401e6	−0.936124	−0.468062	−	0.883696i	$-0.655048\pi$
$619$	−8.92401e6	−0.936124	−0.468062	+	0.883696i	$0.655048\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.50157e6	−0.361446
$624$	0	0
$625$	−7.44918e6	−0.762796
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.92503e6	0.194004
$630$	0	0
$631$	−1.00882e7	−1.00865	−0.504325	−	0.863514i	$-0.668259\pi$
$631$	−1.00882e7	−1.00865	−0.504325	+	0.863514i	$0.668259\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	4.53896e6	0.446706
$636$	0	0
$637$	1.25392e6	0.122440
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.18337e7	−1.13756	−0.568781	−	0.822489i	$-0.692585\pi$
$641$	−1.18337e7	−1.13756	−0.568781	+	0.822489i	$0.692585\pi$
$642$	0	0
$643$	−1.13185e7	−1.07959	−0.539797	−	0.841795i	$-0.681499\pi$
$643$	−1.13185e7	−1.07959	−0.539797	+	0.841795i	$0.681499\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.59034e7	1.49359	0.746793	−	0.665056i	$-0.231592\pi$
$647$	1.59034e7	1.49359	0.746793	+	0.665056i	$0.231592\pi$
$648$	0	0
$649$	2.23965e6	0.208722
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−2.03780e7	−1.87016	−0.935082	−	0.354432i	$-0.884674\pi$
$653$	−2.03780e7	−1.87016	−0.935082	+	0.354432i	$0.884674\pi$
$654$	0	0
$655$	−1.33152e6	−0.121267
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.51593e7	1.35977	0.679884	−	0.733319i	$-0.262030\pi$
$659$	1.51593e7	1.35977	0.679884	+	0.733319i	$0.262030\pi$
$660$	0	0
$661$	−4.02611e6	−0.358412	−0.179206	−	0.983812i	$-0.557353\pi$
$661$	−4.02611e6	−0.358412	−0.179206	+	0.983812i	$0.557353\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.25139e6	0.460490
$666$	0	0
$667$	−1.91331e6	−0.166521
$668$	0	0
$669$	0	0
$670$	0	0
$671$	749439.	0.0642584
$672$	0	0
$673$	−2.19098e7	−1.86466	−0.932330	−	0.361608i	$-0.882228\pi$
$673$	−2.19098e7	−1.86466	−0.932330	+	0.361608i	$0.882228\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−406122.	−0.0340553	−0.0170277	−	0.999855i	$-0.505420\pi$
$677$	−406122.	−0.0340553	−0.0170277	+	0.999855i	$0.505420\pi$
$678$	0	0
$679$	3.66248e6	0.304860
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8.99699e6	−0.737981	−0.368991	−	0.929433i	$-0.620297\pi$
$683$	−8.99699e6	−0.737981	−0.368991	+	0.929433i	$0.620297\pi$
$684$	0	0
$685$	−6.52877e6	−0.531624
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−1.55029e7	−1.24413
$690$	0	0
$691$	1.13549e7	0.904669	0.452335	−	0.891848i	$-0.350591\pi$
$691$	1.13549e7	0.904669	0.452335	+	0.891848i	$0.350591\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.48152e7	1.16344
$696$	0	0
$697$	−2.66014e6	−0.207406
$698$	0	0
$699$	0	0
$700$	0	0
$701$	4.70090e6	0.361315	0.180657	−	0.983546i	$-0.442177\pi$
$701$	4.70090e6	0.361315	0.180657	+	0.983546i	$0.442177\pi$
$702$	0	0
$703$	−1.19341e7	−0.910757
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.21160e6	−0.0911615
$708$	0	0
$709$	1.51946e6	0.113520	0.0567602	−	0.998388i	$-0.481923\pi$
$709$	1.51946e6	0.113520	0.0567602	+	0.998388i	$0.481923\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.89307e6	−0.139458
$714$	0	0
$715$	3.93320e6	0.287727
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−5.03946e6	−0.363548	−0.181774	−	0.983340i	$-0.558184\pi$
$719$	−5.03946e6	−0.363548	−0.181774	+	0.983340i	$0.558184\pi$
$720$	0	0
$721$	−92599.0	−0.00663388
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.44251e6	−0.172580
$726$	0	0
$727$	1.81583e6	0.127420	0.0637101	−	0.997968i	$-0.479707\pi$
$727$	1.81583e6	0.127420	0.0637101	+	0.997968i	$0.479707\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	885729.	0.0613066
$732$	0	0
$733$	9.26839e6	0.637154	0.318577	−	0.947897i	$-0.396795\pi$
$733$	9.26839e6	0.637154	0.318577	+	0.947897i	$0.396795\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.40544e6	−0.366574
$738$	0	0
$739$	9.41597e6	0.634241	0.317120	−	0.948385i	$-0.397284\pi$
$739$	9.41597e6	0.634241	0.317120	+	0.948385i	$0.397284\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−2.68590e7	−1.78491	−0.892457	−	0.451132i	$-0.851020\pi$
$743$	−2.68590e7	−1.78491	−0.892457	+	0.451132i	$0.851020\pi$
$744$	0	0
$745$	−2.21219e7	−1.46026
$746$	0	0
$747$	0	0
$748$	0	0
$749$	8.66505e6	0.564373
$750$	0	0
$751$	−4.29030e6	−0.277580	−0.138790	−	0.990322i	$-0.544321\pi$
$751$	−4.29030e6	−0.277580	−0.138790	+	0.990322i	$0.544321\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.23957e7	−0.791416
$756$	0	0
$757$	−2.68729e6	−0.170442	−0.0852208	−	0.996362i	$-0.527160\pi$
$757$	−2.68729e6	−0.170442	−0.0852208	+	0.996362i	$0.527160\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−2.65935e7	−1.66461	−0.832307	−	0.554315i	$-0.812980\pi$
$761$	−2.65935e7	−1.66461	−0.832307	+	0.554315i	$0.812980\pi$
$762$	0	0
$763$	−3.72515e6	−0.231650
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.77506e6	0.477216
$768$	0	0
$769$	−2.18614e7	−1.33310	−0.666549	−	0.745461i	$-0.732229\pi$
$769$	−2.18614e7	−1.33310	−0.666549	+	0.745461i	$0.732229\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−7.17747e6	−0.432039	−0.216019	−	0.976389i	$-0.569307\pi$
$773$	−7.17747e6	−0.432039	−0.216019	+	0.976389i	$0.569307\pi$
$774$	0	0
$775$	−2.41668e6	−0.144532
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.64914e7	0.973674
$780$	0	0
$781$	1.98230e6	0.116290
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.38734e6	0.196193
$786$	0	0
$787$	−1.34004e7	−0.771225	−0.385612	−	0.922661i	$-0.626010\pi$
$787$	−1.34004e7	−0.771225	−0.385612	+	0.922661i	$0.626010\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.43262e6	−0.195067
$792$	0	0
$793$	2.60171e6	0.146918
$794$	0	0
$795$	0	0
$796$	0	0
$797$	503745.	0.0280909	0.0140454	−	0.999901i	$-0.495529\pi$
$797$	503745.	0.0280909	0.0140454	+	0.999901i	$0.495529\pi$
$798$	0	0
$799$	−7.21768e6	−0.399973
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7.38775e6	0.404318
$804$	0	0
$805$	−1.18897e6	−0.0646668
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.36891e7	1.27256	0.636279	−	0.771459i	$-0.280473\pi$
$809$	2.36891e7	1.27256	0.636279	+	0.771459i	$0.280473\pi$
$810$	0	0
$811$	−1.47421e7	−0.787057	−0.393528	−	0.919312i	$-0.628746\pi$
$811$	−1.47421e7	−0.787057	−0.393528	+	0.919312i	$0.628746\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2.98050e7	1.57179
$816$	0	0
$817$	−5.49103e6	−0.287805
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.54488e6	−0.338878	−0.169439	−	0.985541i	$-0.554196\pi$
$821$	−6.54488e6	−0.338878	−0.169439	+	0.985541i	$0.554196\pi$
$822$	0	0
$823$	−3.14409e7	−1.61806	−0.809031	−	0.587766i	$-0.800008\pi$
$823$	−3.14409e7	−1.61806	−0.809031	+	0.587766i	$0.800008\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.94944e7	−1.49960	−0.749801	−	0.661663i	$-0.769851\pi$
$827$	−2.94944e7	−1.49960	−0.749801	+	0.661663i	$0.769851\pi$
$828$	0	0
$829$	8.27844e6	0.418371	0.209186	−	0.977876i	$-0.432919\pi$
$829$	8.27844e6	0.418371	0.209186	+	0.977876i	$0.432919\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−829096.	−0.0413992
$834$	0	0
$835$	2.71132e7	1.34575
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.53550e7	−1.24353	−0.621767	−	0.783202i	$-0.713585\pi$
$839$	−2.53550e7	−1.24353	−0.621767	+	0.783202i	$0.713585\pi$
$840$	0	0
$841$	−4.92841e6	−0.240279
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−4.93356e6	−0.237694
$846$	0	0
$847$	−6.78256e6	−0.324852
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.70201e6	0.127898
$852$	0	0
$853$	6.68641e6	0.314645	0.157322	−	0.987547i	$-0.449714\pi$
$853$	6.68641e6	0.314645	0.157322	+	0.987547i	$0.449714\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.50753e7	−1.63136	−0.815678	−	0.578506i	$-0.803636\pi$
$857$	−3.50753e7	−1.63136	−0.815678	+	0.578506i	$0.803636\pi$
$858$	0	0
$859$	−2.38202e7	−1.10144	−0.550722	−	0.834689i	$-0.685647\pi$
$859$	−2.38202e7	−1.10144	−0.550722	+	0.834689i	$0.685647\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.18907e7	−1.00054	−0.500269	−	0.865870i	$-0.666765\pi$
$863$	−2.18907e7	−1.00054	−0.500269	+	0.865870i	$0.666765\pi$
$864$	0	0
$865$	−2.03858e7	−0.926375
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.60204e6	0.116886
$870$	0	0
$871$	−1.87652e7	−0.838124
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−9.18365e6	−0.405504
$876$	0	0
$877$	−2.55178e7	−1.12032	−0.560162	−	0.828383i	$-0.689261\pi$
$877$	−2.55178e7	−1.12032	−0.560162	+	0.828383i	$0.689261\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.08125e7	0.903410	0.451705	−	0.892167i	$-0.350816\pi$
$881$	2.08125e7	0.903410	0.451705	+	0.892167i	$0.350816\pi$
$882$	0	0
$883$	−4.62199e6	−0.199493	−0.0997465	−	0.995013i	$-0.531803\pi$
$883$	−4.62199e6	−0.199493	−0.0997465	+	0.995013i	$0.531803\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.37519e7	1.44042	0.720210	−	0.693756i	$-0.244045\pi$
$887$	3.37519e7	1.44042	0.720210	+	0.693756i	$0.244045\pi$
$888$	0	0
$889$	4.44263e6	0.188532
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.47457e7	1.87768
$894$	0	0
$895$	−1.54836e7	−0.646121
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.54179e7	0.636249
$900$	0	0
$901$	1.02506e7	0.420665
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.50662e7	0.611482
$906$	0	0
$907$	−1.55386e7	−0.627183	−0.313591	−	0.949558i	$-0.601532\pi$
$907$	−1.55386e7	−0.627183	−0.313591	+	0.949558i	$0.601532\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−2.69477e7	−1.07578	−0.537892	−	0.843014i	$-0.680779\pi$
$911$	−2.69477e7	−1.07578	−0.537892	+	0.843014i	$0.680779\pi$
$912$	0	0
$913$	−2.69290e6	−0.106916
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.30326e6	−0.0511809
$918$	0	0
$919$	2.77206e7	1.08272	0.541358	−	0.840792i	$-0.317910\pi$
$919$	2.77206e7	1.08272	0.541358	+	0.840792i	$0.317910\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6.88165e6	0.265882
$924$	0	0
$925$	3.44937e6	0.132552
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.06274e7	1.54447	0.772236	−	0.635336i	$-0.219138\pi$
$929$	4.06274e7	1.54447	0.772236	+	0.635336i	$0.219138\pi$
$930$	0	0
$931$	5.13994e6	0.194350
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.60064e6	−0.0972862
$936$	0	0
$937$	−2.23464e7	−0.831493	−0.415747	−	0.909480i	$-0.636480\pi$
$937$	−2.23464e7	−0.831493	−0.415747	+	0.909480i	$0.636480\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.05299e7	−1.12396	−0.561981	−	0.827150i	$-0.689961\pi$
$941$	−3.05299e7	−1.12396	−0.561981	+	0.827150i	$0.689961\pi$
$942$	0	0
$943$	−3.73382e6	−0.136733
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.02758e6	0.327112	0.163556	−	0.986534i	$-0.447704\pi$
$947$	9.02758e6	0.327112	0.163556	+	0.986534i	$0.447704\pi$
$948$	0	0
$949$	2.56469e7	0.924420
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.20115e7	1.85510	0.927550	−	0.373699i	$-0.121911\pi$
$953$	5.20115e7	1.85510	0.927550	+	0.373699i	$0.121911\pi$
$954$	0	0
$955$	−2.05338e6	−0.0728551
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.39021e6	−0.224372
$960$	0	0
$961$	−1.33743e7	−0.467156
$962$	0	0
$963$	0	0
$964$	0	0
$965$	2.27448e7	0.786255
$966$	0	0
$967$	3.40751e7	1.17185	0.585924	−	0.810366i	$-0.300732\pi$
$967$	3.40751e7	1.17185	0.585924	+	0.810366i	$0.300732\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.90682e7	−0.989396	−0.494698	−	0.869065i	$-0.664721\pi$
$971$	−2.90682e7	−0.989396	−0.494698	+	0.869065i	$0.664721\pi$
$972$	0	0
$973$	1.45007e7	0.491030
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.11776e7	1.38015	0.690073	−	0.723740i	$-0.257578\pi$
$977$	4.11776e7	1.38015	0.690073	+	0.723740i	$0.257578\pi$
$978$	0	0
$979$	−1.07504e7	−0.358481
$980$	0	0
$981$	0	0
$982$	0	0
$983$	3.65465e7	1.20632	0.603160	−	0.797621i	$-0.293908\pi$
$983$	3.65465e7	1.20632	0.603160	+	0.797621i	$0.293908\pi$
$984$	0	0
$985$	−2.39175e7	−0.785463
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.24323e6	0.0404166
$990$	0	0
$991$	−1.79164e7	−0.579518	−0.289759	−	0.957100i	$-0.593575\pi$
$991$	−1.79164e7	−0.579518	−0.289759	+	0.957100i	$0.593575\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.30208e7	−0.416947
$996$	0	0
$997$	4.44808e7	1.41721	0.708606	−	0.705604i	$-0.249324\pi$
$997$	4.44808e7	1.41721	0.708606	+	0.705604i	$0.249324\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.6.a.bj.1.2		2
3.2	odd	2		1008.6.a.br.1.1		2
4.3	odd	2		504.6.a.n.1.2	✓	2
12.11	even	2		504.6.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.6.a.n.1.2	✓	2	4.3	odd	2
504.6.a.q.1.1	yes	2	12.11	even	2
1008.6.a.bj.1.2		2	1.1	even	1	trivial
1008.6.a.br.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+50.0625 q^{5} +49.0000 q^{7} +150.437 q^{11} +522.250 q^{13} -345.313 q^{17} +2140.75 q^{19} -484.688 q^{23} -618.749 q^{25} +3947.50 q^{29} +3905.75 q^{31} +2453.06 q^{35} -5574.75 q^{37} +7703.56 q^{41} -2565.00 q^{43} +20901.9 q^{47} +2401.00 q^{49} -29684.9 q^{53} +7531.26 q^{55} +14887.6 q^{59} +4981.74 q^{61} +26145.1 q^{65} -35931.5 q^{67} +13176.9 q^{71} +49108.5 q^{73} +7371.43 q^{77} +17296.5 q^{79} -17900.5 q^{83} -17287.2 q^{85} -71460.7 q^{89} +25590.2 q^{91} +107171. q^{95} +74744.5 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 28 q^{5} + 98 q^{7} - 596 q^{11} + 532 q^{13} - 2100 q^{17} + 2744 q^{19} - 3660 q^{23} + 2350 q^{25} + 720 q^{29} - 3976 q^{31} - 1372 q^{35} + 6788 q^{37} + 4004 q^{41} - 19480 q^{43} + 21560 q^{47}+ \cdots - 36036 q^{97}+O(q^{100}) \) 2 * q - 28 * q^5 + 98 * q^7 - 596 * q^11 + 532 * q^13 - 2100 * q^17 + 2744 * q^19 - 3660 * q^23 + 2350 * q^25 + 720 * q^29 - 3976 * q^31 - 1372 * q^35 + 6788 * q^37 + 4004 * q^41 - 19480 * q^43 + 21560 * q^47 + 4802 * q^49 - 57576 * q^53 + 65800 * q^55 + 22344 * q^59 - 38724 * q^61 + 25384 * q^65 - 7288 * q^67 + 3932 * q^71 + 19292 * q^73 - 29204 * q^77 - 15632 * q^79 + 22624 * q^83 + 119688 * q^85 - 112812 * q^89 + 26068 * q^91 + 60080 * q^95 - 36036 * q^97

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