LMFDB - Embedded newform 8400.2.a.dg.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8400,2,Mod(1,8400)] 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("8400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,0,0,-3,0,3,0,-6,0,6,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.0743376979$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	8400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +6.42864 q^{13} -4.42864 q^{17} +2.42864 q^{19} +1.00000 q^{21} +1.37778 q^{23} -1.00000 q^{27} +0.755569 q^{29} -5.18421 q^{31} +2.00000 q^{33} -7.61285 q^{37} -6.42864 q^{39} -8.23506 q^{41} +10.1017 q^{43} +2.75557 q^{47} +1.00000 q^{49} +4.42864 q^{51} -9.18421 q^{53} -2.42864 q^{57} -14.1017 q^{59} +6.85728 q^{61} -1.00000 q^{63} +2.75557 q^{67} -1.37778 q^{69} -2.00000 q^{71} +1.57136 q^{73} +2.00000 q^{77} +4.85728 q^{79} +1.00000 q^{81} +11.6128 q^{83} -0.755569 q^{87} +4.62222 q^{89} -6.42864 q^{91} +5.18421 q^{93} +11.9398 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{13} - 6 q^{19} + 3 q^{21} + 4 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} + 4 q^{37} - 6 q^{39} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 6 * q^13 - 6 * q^19 + 3 * q^21 + 4 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 + 6 * q^33 + 4 * q^37 - 6 * q^39 + 2 * q^41 + 4 * q^43 + 8 * q^47 + 3 * q^49 - 14 * q^53 + 6 * q^57 - 16 * q^59 - 6 * q^61 - 3 * q^63 + 8 * q^67 - 4 * q^69 - 6 * q^71 + 18 * q^73 + 6 * q^77 - 12 * q^79 + 3 * q^81 + 8 * q^83 - 2 * q^87 + 14 * q^89 - 6 * q^91 + 2 * q^93 + 22 * q^97 - 6 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	6.42864	1.78298	0.891492	−	0.453037i	$-0.149659\pi$
$13$	6.42864	1.78298	0.891492	+	0.453037i	$0.149659\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.42864	−1.07410	−0.537051	−	0.843550i	$-0.680462\pi$
$17$	−4.42864	−1.07410	−0.537051	+	0.843550i	$0.680462\pi$
$18$	0	0
$19$	2.42864	0.557168	0.278584	−	0.960412i	$-0.410135\pi$
$19$	2.42864	0.557168	0.278584	+	0.960412i	$0.410135\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.37778	0.287288	0.143644	−	0.989629i	$-0.454118\pi$
$23$	1.37778	0.287288	0.143644	+	0.989629i	$0.454118\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	0.755569	0.140306	0.0701528	−	0.997536i	$-0.477651\pi$
$29$	0.755569	0.140306	0.0701528	+	0.997536i	$0.477651\pi$
$30$	0	0
$31$	−5.18421	−0.931111	−0.465556	−	0.885019i	$-0.654145\pi$
$31$	−5.18421	−0.931111	−0.465556	+	0.885019i	$0.654145\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.61285	−1.25154	−0.625772	−	0.780006i	$-0.715216\pi$
$37$	−7.61285	−1.25154	−0.625772	+	0.780006i	$0.715216\pi$
$38$	0	0
$39$	−6.42864	−1.02941
$40$	0	0
$41$	−8.23506	−1.28610	−0.643050	−	0.765824i	$-0.722331\pi$
$41$	−8.23506	−1.28610	−0.643050	+	0.765824i	$0.722331\pi$
$42$	0	0
$43$	10.1017	1.54050	0.770248	−	0.637744i	$-0.220132\pi$
$43$	10.1017	1.54050	0.770248	+	0.637744i	$0.220132\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.75557	0.401941	0.200971	−	0.979597i	$-0.435590\pi$
$47$	2.75557	0.401941	0.200971	+	0.979597i	$0.435590\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.42864	0.620134
$52$	0	0
$53$	−9.18421	−1.26155	−0.630774	−	0.775967i	$-0.717263\pi$
$53$	−9.18421	−1.26155	−0.630774	+	0.775967i	$0.717263\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.42864	−0.321681
$58$	0	0
$59$	−14.1017	−1.83589	−0.917943	−	0.396712i	$-0.870151\pi$
$59$	−14.1017	−1.83589	−0.917943	+	0.396712i	$0.870151\pi$
$60$	0	0
$61$	6.85728	0.877985	0.438992	−	0.898491i	$-0.355336\pi$
$61$	6.85728	0.877985	0.438992	+	0.898491i	$0.355336\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	2.75557	0.336646	0.168323	−	0.985732i	$-0.446165\pi$
$67$	2.75557	0.336646	0.168323	+	0.985732i	$0.446165\pi$
$68$	0	0
$69$	−1.37778	−0.165866
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	1.57136	0.183914	0.0919569	−	0.995763i	$-0.470688\pi$
$73$	1.57136	0.183914	0.0919569	+	0.995763i	$0.470688\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.00000	0.227921
$78$	0	0
$79$	4.85728	0.546487	0.273243	−	0.961945i	$-0.411904\pi$
$79$	4.85728	0.546487	0.273243	+	0.961945i	$0.411904\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.6128	1.27468	0.637338	−	0.770585i	$-0.280036\pi$
$83$	11.6128	1.27468	0.637338	+	0.770585i	$0.280036\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−0.755569	−0.0810055
$88$	0	0
$89$	4.62222	0.489954	0.244977	−	0.969529i	$-0.421220\pi$
$89$	4.62222	0.489954	0.244977	+	0.969529i	$0.421220\pi$
$90$	0	0
$91$	−6.42864	−0.673905
$92$	0	0
$93$	5.18421	0.537577
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.9398	1.21230	0.606150	−	0.795350i	$-0.292713\pi$
$97$	11.9398	1.21230	0.606150	+	0.795350i	$0.292713\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	1.47949	0.147215	0.0736076	−	0.997287i	$-0.476549\pi$
$101$	1.47949	0.147215	0.0736076	+	0.997287i	$0.476549\pi$
$102$	0	0
$103$	−8.85728	−0.872734	−0.436367	−	0.899769i	$-0.643735\pi$
$103$	−8.85728	−0.872734	−0.436367	+	0.899769i	$0.643735\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.76494	0.170623	0.0853114	−	0.996354i	$-0.472811\pi$
$107$	1.76494	0.170623	0.0853114	+	0.996354i	$0.472811\pi$
$108$	0	0
$109$	5.61285	0.537613	0.268807	−	0.963194i	$-0.413371\pi$
$109$	5.61285	0.537613	0.268807	+	0.963194i	$0.413371\pi$
$110$	0	0
$111$	7.61285	0.722580
$112$	0	0
$113$	11.2859	1.06169	0.530845	−	0.847469i	$-0.321875\pi$
$113$	11.2859	1.06169	0.530845	+	0.847469i	$0.321875\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.42864	0.594328
$118$	0	0
$119$	4.42864	0.405973
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	8.23506	0.742531
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.8573	−1.14090	−0.570450	−	0.821333i	$-0.693231\pi$
$127$	−12.8573	−1.14090	−0.570450	+	0.821333i	$0.693231\pi$
$128$	0	0
$129$	−10.1017	−0.889406
$130$	0	0
$131$	2.10171	0.183627	0.0918136	−	0.995776i	$-0.470734\pi$
$131$	2.10171	0.183627	0.0918136	+	0.995776i	$0.470734\pi$
$132$	0	0
$133$	−2.42864	−0.210590
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15.9398	−1.36183	−0.680914	−	0.732364i	$-0.738417\pi$
$137$	−15.9398	−1.36183	−0.680914	+	0.732364i	$0.738417\pi$
$138$	0	0
$139$	−11.6731	−0.990097	−0.495048	−	0.868865i	$-0.664850\pi$
$139$	−11.6731	−0.990097	−0.495048	+	0.868865i	$0.664850\pi$
$140$	0	0
$141$	−2.75557	−0.232061
$142$	0	0
$143$	−12.8573	−1.07518
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−21.2257	−1.73888	−0.869438	−	0.494041i	$-0.835519\pi$
$149$	−21.2257	−1.73888	−0.869438	+	0.494041i	$0.835519\pi$
$150$	0	0
$151$	−16.8573	−1.37183	−0.685913	−	0.727684i	$-0.740597\pi$
$151$	−16.8573	−1.37183	−0.685913	+	0.727684i	$0.740597\pi$
$152$	0	0
$153$	−4.42864	−0.358034
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.4286	0.832296	0.416148	−	0.909297i	$-0.363380\pi$
$157$	10.4286	0.832296	0.416148	+	0.909297i	$0.363380\pi$
$158$	0	0
$159$	9.18421	0.728355
$160$	0	0
$161$	−1.37778	−0.108585
$162$	0	0
$163$	20.8573	1.63367	0.816834	−	0.576873i	$-0.195727\pi$
$163$	20.8573	1.63367	0.816834	+	0.576873i	$0.195727\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.3461	1.18752	0.593760	−	0.804642i	$-0.297643\pi$
$167$	15.3461	1.18752	0.593760	+	0.804642i	$0.297643\pi$
$168$	0	0
$169$	28.3274	2.17903
$170$	0	0
$171$	2.42864	0.185723
$172$	0	0
$173$	−2.06022	−0.156636	−0.0783179	−	0.996928i	$-0.524955\pi$
$173$	−2.06022	−0.156636	−0.0783179	+	0.996928i	$0.524955\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	14.1017	1.05995
$178$	0	0
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	0	0
$181$	−12.1017	−0.899513	−0.449757	−	0.893151i	$-0.648489\pi$
$181$	−12.1017	−0.899513	−0.449757	+	0.893151i	$0.648489\pi$
$182$	0	0
$183$	−6.85728	−0.506905
$184$	0	0
$185$	0	0
$186$	0	0
$187$	8.85728	0.647708
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−0.488863	−0.0353729	−0.0176864	−	0.999844i	$-0.505630\pi$
$191$	−0.488863	−0.0353729	−0.0176864	+	0.999844i	$0.505630\pi$
$192$	0	0
$193$	22.9590	1.65262	0.826312	−	0.563212i	$-0.190435\pi$
$193$	22.9590	1.65262	0.826312	+	0.563212i	$0.190435\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.18421	0.0843713	0.0421857	−	0.999110i	$-0.486568\pi$
$197$	1.18421	0.0843713	0.0421857	+	0.999110i	$0.486568\pi$
$198$	0	0
$199$	−8.79706	−0.623607	−0.311803	−	0.950147i	$-0.600933\pi$
$199$	−8.79706	−0.623607	−0.311803	+	0.950147i	$0.600933\pi$
$200$	0	0
$201$	−2.75557	−0.194363
$202$	0	0
$203$	−0.755569	−0.0530305
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.37778	0.0957626
$208$	0	0
$209$	−4.85728	−0.335985
$210$	0	0
$211$	−23.2257	−1.59892	−0.799461	−	0.600717i	$-0.794882\pi$
$211$	−23.2257	−1.59892	−0.799461	+	0.600717i	$0.794882\pi$
$212$	0	0
$213$	2.00000	0.137038
$214$	0	0
$215$	0	0
$216$	0	0
$217$	5.18421	0.351927
$218$	0	0
$219$	−1.57136	−0.106183
$220$	0	0
$221$	−28.4701	−1.91511
$222$	0	0
$223$	−15.2257	−1.01959	−0.509794	−	0.860297i	$-0.670278\pi$
$223$	−15.2257	−1.01959	−0.509794	+	0.860297i	$0.670278\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−14.3684	−0.953665	−0.476833	−	0.878994i	$-0.658215\pi$
$227$	−14.3684	−0.953665	−0.476833	+	0.878994i	$0.658215\pi$
$228$	0	0
$229$	−5.61285	−0.370907	−0.185454	−	0.982653i	$-0.559375\pi$
$229$	−5.61285	−0.370907	−0.185454	+	0.982653i	$0.559375\pi$
$230$	0	0
$231$	−2.00000	−0.131590
$232$	0	0
$233$	−23.2859	−1.52551	−0.762756	−	0.646687i	$-0.776154\pi$
$233$	−23.2859	−1.52551	−0.762756	+	0.646687i	$0.776154\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−4.85728	−0.315514
$238$	0	0
$239$	8.48886	0.549099	0.274549	−	0.961573i	$-0.411471\pi$
$239$	8.48886	0.549099	0.274549	+	0.961573i	$0.411471\pi$
$240$	0	0
$241$	−7.24443	−0.466655	−0.233327	−	0.972398i	$-0.574961\pi$
$241$	−7.24443	−0.466655	−0.233327	+	0.972398i	$0.574961\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.6128	0.993422
$248$	0	0
$249$	−11.6128	−0.735934
$250$	0	0
$251$	−27.6128	−1.74291	−0.871454	−	0.490478i	$-0.836822\pi$
$251$	−27.6128	−1.74291	−0.871454	+	0.490478i	$0.836822\pi$
$252$	0	0
$253$	−2.75557	−0.173241
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.428639	0.0267378	0.0133689	−	0.999911i	$-0.495744\pi$
$257$	0.428639	0.0267378	0.0133689	+	0.999911i	$0.495744\pi$
$258$	0	0
$259$	7.61285	0.473039
$260$	0	0
$261$	0.755569	0.0467685
$262$	0	0
$263$	−9.37778	−0.578259	−0.289129	−	0.957290i	$-0.593366\pi$
$263$	−9.37778	−0.578259	−0.289129	+	0.957290i	$0.593366\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.62222	−0.282875
$268$	0	0
$269$	1.74620	0.106468	0.0532339	−	0.998582i	$-0.483047\pi$
$269$	1.74620	0.106468	0.0532339	+	0.998582i	$0.483047\pi$
$270$	0	0
$271$	2.69535	0.163731	0.0818653	−	0.996643i	$-0.473912\pi$
$271$	2.69535	0.163731	0.0818653	+	0.996643i	$0.473912\pi$
$272$	0	0
$273$	6.42864	0.389079
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5.12399	0.307870	0.153935	−	0.988081i	$-0.450805\pi$
$277$	5.12399	0.307870	0.153935	+	0.988081i	$0.450805\pi$
$278$	0	0
$279$	−5.18421	−0.310370
$280$	0	0
$281$	23.9813	1.43060	0.715301	−	0.698816i	$-0.246290\pi$
$281$	23.9813	1.43060	0.715301	+	0.698816i	$0.246290\pi$
$282$	0	0
$283$	2.36842	0.140788	0.0703939	−	0.997519i	$-0.477574\pi$
$283$	2.36842	0.140788	0.0703939	+	0.997519i	$0.477574\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.23506	0.486100
$288$	0	0
$289$	2.61285	0.153697
$290$	0	0
$291$	−11.9398	−0.699922
$292$	0	0
$293$	−8.42864	−0.492406	−0.246203	−	0.969218i	$-0.579183\pi$
$293$	−8.42864	−0.492406	−0.246203	+	0.969218i	$0.579183\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	8.85728	0.512230
$300$	0	0
$301$	−10.1017	−0.582253
$302$	0	0
$303$	−1.47949	−0.0849947
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−22.5718	−1.28824	−0.644121	−	0.764923i	$-0.722777\pi$
$307$	−22.5718	−1.28824	−0.644121	+	0.764923i	$0.722777\pi$
$308$	0	0
$309$	8.85728	0.503873
$310$	0	0
$311$	24.0830	1.36562	0.682810	−	0.730596i	$-0.260758\pi$
$311$	24.0830	1.36562	0.682810	+	0.730596i	$0.260758\pi$
$312$	0	0
$313$	−9.65433	−0.545695	−0.272848	−	0.962057i	$-0.587965\pi$
$313$	−9.65433	−0.545695	−0.272848	+	0.962057i	$0.587965\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.04149	0.339324	0.169662	−	0.985502i	$-0.445732\pi$
$317$	6.04149	0.339324	0.169662	+	0.985502i	$0.445732\pi$
$318$	0	0
$319$	−1.51114	−0.0846075
$320$	0	0
$321$	−1.76494	−0.0985092
$322$	0	0
$323$	−10.7556	−0.598456
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.61285	−0.310391
$328$	0	0
$329$	−2.75557	−0.151919
$330$	0	0
$331$	−13.5111	−0.742639	−0.371320	−	0.928505i	$-0.621095\pi$
$331$	−13.5111	−0.742639	−0.371320	+	0.928505i	$0.621095\pi$
$332$	0	0
$333$	−7.61285	−0.417181
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−10.4889	−0.571365	−0.285682	−	0.958324i	$-0.592220\pi$
$337$	−10.4889	−0.571365	−0.285682	+	0.958324i	$0.592220\pi$
$338$	0	0
$339$	−11.2859	−0.612967
$340$	0	0
$341$	10.3684	0.561481
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.7239	−0.897787	−0.448894	−	0.893585i	$-0.648182\pi$
$347$	−16.7239	−0.897787	−0.448894	+	0.893585i	$0.648182\pi$
$348$	0	0
$349$	−16.3684	−0.876181	−0.438091	−	0.898931i	$-0.644345\pi$
$349$	−16.3684	−0.876181	−0.438091	+	0.898931i	$0.644345\pi$
$350$	0	0
$351$	−6.42864	−0.343135
$352$	0	0
$353$	−0.549086	−0.0292249	−0.0146124	−	0.999893i	$-0.504651\pi$
$353$	−0.549086	−0.0292249	−0.0146124	+	0.999893i	$0.504651\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.42864	−0.234388
$358$	0	0
$359$	0.285442	0.0150651	0.00753253	−	0.999972i	$-0.497602\pi$
$359$	0.285442	0.0150651	0.00753253	+	0.999972i	$0.497602\pi$
$360$	0	0
$361$	−13.1017	−0.689564
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.71456	0.0894992	0.0447496	−	0.998998i	$-0.485751\pi$
$367$	1.71456	0.0894992	0.0447496	+	0.998998i	$0.485751\pi$
$368$	0	0
$369$	−8.23506	−0.428700
$370$	0	0
$371$	9.18421	0.476820
$372$	0	0
$373$	−16.0000	−0.828449	−0.414224	−	0.910175i	$-0.635947\pi$
$373$	−16.0000	−0.828449	−0.414224	+	0.910175i	$0.635947\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.85728	0.250163
$378$	0	0
$379$	4.85728	0.249502	0.124751	−	0.992188i	$-0.460187\pi$
$379$	4.85728	0.249502	0.124751	+	0.992188i	$0.460187\pi$
$380$	0	0
$381$	12.8573	0.658698
$382$	0	0
$383$	−8.38715	−0.428563	−0.214282	−	0.976772i	$-0.568741\pi$
$383$	−8.38715	−0.428563	−0.214282	+	0.976772i	$0.568741\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.1017	0.513499
$388$	0	0
$389$	8.95899	0.454239	0.227119	−	0.973867i	$-0.427069\pi$
$389$	8.95899	0.454239	0.227119	+	0.973867i	$0.427069\pi$
$390$	0	0
$391$	−6.10171	−0.308577
$392$	0	0
$393$	−2.10171	−0.106017
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.54909	0.127935	0.0639675	−	0.997952i	$-0.479625\pi$
$397$	2.54909	0.127935	0.0639675	+	0.997952i	$0.479625\pi$
$398$	0	0
$399$	2.42864	0.121584
$400$	0	0
$401$	0.958989	0.0478896	0.0239448	−	0.999713i	$-0.492377\pi$
$401$	0.958989	0.0478896	0.0239448	+	0.999713i	$0.492377\pi$
$402$	0	0
$403$	−33.3274	−1.66016
$404$	0	0
$405$	0	0
$406$	0	0
$407$	15.2257	0.754710
$408$	0	0
$409$	−31.9813	−1.58137	−0.790686	−	0.612222i	$-0.790276\pi$
$409$	−31.9813	−1.58137	−0.790686	+	0.612222i	$0.790276\pi$
$410$	0	0
$411$	15.9398	0.786251
$412$	0	0
$413$	14.1017	0.693900
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.6731	0.571633
$418$	0	0
$419$	0.470127	0.0229672	0.0114836	−	0.999934i	$-0.496345\pi$
$419$	0.470127	0.0229672	0.0114836	+	0.999934i	$0.496345\pi$
$420$	0	0
$421$	−33.6128	−1.63819	−0.819095	−	0.573658i	$-0.805524\pi$
$421$	−33.6128	−1.63819	−0.819095	+	0.573658i	$0.805524\pi$
$422$	0	0
$423$	2.75557	0.133980
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−6.85728	−0.331847
$428$	0	0
$429$	12.8573	0.620755
$430$	0	0
$431$	−11.7146	−0.564270	−0.282135	−	0.959375i	$-0.591043\pi$
$431$	−11.7146	−0.564270	−0.282135	+	0.959375i	$0.591043\pi$
$432$	0	0
$433$	0.0602231	0.00289414	0.00144707	−	0.999999i	$-0.499539\pi$
$433$	0.0602231	0.00289414	0.00144707	+	0.999999i	$0.499539\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.34614	0.160068
$438$	0	0
$439$	22.4286	1.07046	0.535230	−	0.844706i	$-0.320225\pi$
$439$	22.4286	1.07046	0.535230	+	0.844706i	$0.320225\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−23.9496	−1.13788	−0.568940	−	0.822379i	$-0.692647\pi$
$443$	−23.9496	−1.13788	−0.568940	+	0.822379i	$0.692647\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	21.2257	1.00394
$448$	0	0
$449$	29.4291	1.38885	0.694423	−	0.719567i	$-0.255660\pi$
$449$	29.4291	1.38885	0.694423	+	0.719567i	$0.255660\pi$
$450$	0	0
$451$	16.4701	0.775548
$452$	0	0
$453$	16.8573	0.792024
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.14272	0.147010	0.0735051	−	0.997295i	$-0.476581\pi$
$457$	3.14272	0.147010	0.0735051	+	0.997295i	$0.476581\pi$
$458$	0	0
$459$	4.42864	0.206711
$460$	0	0
$461$	−3.37778	−0.157319	−0.0786596	−	0.996902i	$-0.525064\pi$
$461$	−3.37778	−0.157319	−0.0786596	+	0.996902i	$0.525064\pi$
$462$	0	0
$463$	20.8573	0.969320	0.484660	−	0.874703i	$-0.338943\pi$
$463$	20.8573	0.969320	0.484660	+	0.874703i	$0.338943\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.3684	−0.664891	−0.332446	−	0.943122i	$-0.607874\pi$
$467$	−14.3684	−0.664891	−0.332446	+	0.943122i	$0.607874\pi$
$468$	0	0
$469$	−2.75557	−0.127240
$470$	0	0
$471$	−10.4286	−0.480526
$472$	0	0
$473$	−20.2034	−0.928954
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−9.18421	−0.420516
$478$	0	0
$479$	−6.36842	−0.290980	−0.145490	−	0.989360i	$-0.546476\pi$
$479$	−6.36842	−0.290980	−0.145490	+	0.989360i	$0.546476\pi$
$480$	0	0
$481$	−48.9403	−2.23148
$482$	0	0
$483$	1.37778	0.0626914
$484$	0	0
$485$	0	0
$486$	0	0
$487$	17.3274	0.785180	0.392590	−	0.919714i	$-0.371579\pi$
$487$	17.3274	0.785180	0.392590	+	0.919714i	$0.371579\pi$
$488$	0	0
$489$	−20.8573	−0.943199
$490$	0	0
$491$	−2.00000	−0.0902587	−0.0451294	−	0.998981i	$-0.514370\pi$
$491$	−2.00000	−0.0902587	−0.0451294	+	0.998981i	$0.514370\pi$
$492$	0	0
$493$	−3.34614	−0.150703
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.00000	0.0897123
$498$	0	0
$499$	−23.3461	−1.04512	−0.522558	−	0.852603i	$-0.675022\pi$
$499$	−23.3461	−1.04512	−0.522558	+	0.852603i	$0.675022\pi$
$500$	0	0
$501$	−15.3461	−0.685615
$502$	0	0
$503$	0.387152	0.0172623	0.00863113	−	0.999963i	$-0.497253\pi$
$503$	0.387152	0.0172623	0.00863113	+	0.999963i	$0.497253\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−28.3274	−1.25806
$508$	0	0
$509$	−29.9496	−1.32749	−0.663747	−	0.747957i	$-0.731035\pi$
$509$	−29.9496	−1.32749	−0.663747	+	0.747957i	$0.731035\pi$
$510$	0	0
$511$	−1.57136	−0.0695129
$512$	0	0
$513$	−2.42864	−0.107227
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.51114	−0.242380
$518$	0	0
$519$	2.06022	0.0904338
$520$	0	0
$521$	−18.5205	−0.811398	−0.405699	−	0.914007i	$-0.632972\pi$
$521$	−18.5205	−0.811398	−0.405699	+	0.914007i	$0.632972\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.9590	1.00011
$528$	0	0
$529$	−21.1017	−0.917466
$530$	0	0
$531$	−14.1017	−0.611962
$532$	0	0
$533$	−52.9403	−2.29310
$534$	0	0
$535$	0	0
$536$	0	0
$537$	10.0000	0.431532
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	14.5906	0.627298	0.313649	−	0.949539i	$-0.398449\pi$
$541$	14.5906	0.627298	0.313649	+	0.949539i	$0.398449\pi$
$542$	0	0
$543$	12.1017	0.519334
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.7556	−0.801930	−0.400965	−	0.916093i	$-0.631325\pi$
$547$	−18.7556	−0.801930	−0.400965	+	0.916093i	$0.631325\pi$
$548$	0	0
$549$	6.85728	0.292662
$550$	0	0
$551$	1.83500	0.0781738
$552$	0	0
$553$	−4.85728	−0.206553
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.8765	−1.35065	−0.675325	−	0.737520i	$-0.735997\pi$
$557$	−31.8765	−1.35065	−0.675325	+	0.737520i	$0.735997\pi$
$558$	0	0
$559$	64.9403	2.74668
$560$	0	0
$561$	−8.85728	−0.373955
$562$	0	0
$563$	−2.01874	−0.0850796	−0.0425398	−	0.999095i	$-0.513545\pi$
$563$	−2.01874	−0.0850796	−0.0425398	+	0.999095i	$0.513545\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−28.9590	−1.21402	−0.607012	−	0.794693i	$-0.707632\pi$
$569$	−28.9590	−1.21402	−0.607012	+	0.794693i	$0.707632\pi$
$570$	0	0
$571$	−8.97773	−0.375706	−0.187853	−	0.982197i	$-0.560153\pi$
$571$	−8.97773	−0.375706	−0.187853	+	0.982197i	$0.560153\pi$
$572$	0	0
$573$	0.488863	0.0204225
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.6766	−1.19382	−0.596911	−	0.802307i	$-0.703606\pi$
$577$	−28.6766	−1.19382	−0.596911	+	0.802307i	$0.703606\pi$
$578$	0	0
$579$	−22.9590	−0.954143
$580$	0	0
$581$	−11.6128	−0.481782
$582$	0	0
$583$	18.3684	0.760742
$584$	0	0
$585$	0	0
$586$	0	0
$587$	45.2070	1.86589	0.932945	−	0.360018i	$-0.117229\pi$
$587$	45.2070	1.86589	0.932945	+	0.360018i	$0.117229\pi$
$588$	0	0
$589$	−12.5906	−0.518786
$590$	0	0
$591$	−1.18421	−0.0487118
$592$	0	0
$593$	18.2636	0.749998	0.374999	−	0.927025i	$-0.377643\pi$
$593$	18.2636	0.749998	0.374999	+	0.927025i	$0.377643\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	8.79706	0.360040
$598$	0	0
$599$	22.7368	0.929002	0.464501	−	0.885573i	$-0.346234\pi$
$599$	22.7368	0.929002	0.464501	+	0.885573i	$0.346234\pi$
$600$	0	0
$601$	0.488863	0.0199411	0.00997056	−	0.999950i	$-0.496826\pi$
$601$	0.488863	0.0199411	0.00997056	+	0.999950i	$0.496826\pi$
$602$	0	0
$603$	2.75557	0.112215
$604$	0	0
$605$	0	0
$606$	0	0
$607$	20.2034	0.820032	0.410016	−	0.912078i	$-0.365523\pi$
$607$	20.2034	0.820032	0.410016	+	0.912078i	$0.365523\pi$
$608$	0	0
$609$	0.755569	0.0306172
$610$	0	0
$611$	17.7146	0.716654
$612$	0	0
$613$	10.3684	0.418776	0.209388	−	0.977833i	$-0.432853\pi$
$613$	10.3684	0.418776	0.209388	+	0.977833i	$0.432853\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−39.2859	−1.58159	−0.790796	−	0.612080i	$-0.790333\pi$
$617$	−39.2859	−1.58159	−0.790796	+	0.612080i	$0.790333\pi$
$618$	0	0
$619$	−42.8988	−1.72425	−0.862123	−	0.506698i	$-0.830866\pi$
$619$	−42.8988	−1.72425	−0.862123	+	0.506698i	$0.830866\pi$
$620$	0	0
$621$	−1.37778	−0.0552886
$622$	0	0
$623$	−4.62222	−0.185185
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.85728	0.193981
$628$	0	0
$629$	33.7146	1.34429
$630$	0	0
$631$	−15.3461	−0.610920	−0.305460	−	0.952205i	$-0.598810\pi$
$631$	−15.3461	−0.610920	−0.305460	+	0.952205i	$0.598810\pi$
$632$	0	0
$633$	23.2257	0.923139
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.42864	0.254712
$638$	0	0
$639$	−2.00000	−0.0791188
$640$	0	0
$641$	30.6735	1.21153	0.605766	−	0.795643i	$-0.292867\pi$
$641$	30.6735	1.21153	0.605766	+	0.795643i	$0.292867\pi$
$642$	0	0
$643$	49.0607	1.93477	0.967383	−	0.253320i	$-0.0815225\pi$
$643$	49.0607	1.93477	0.967383	+	0.253320i	$0.0815225\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.3461	0.603319	0.301660	−	0.953416i	$-0.402459\pi$
$647$	15.3461	0.603319	0.301660	+	0.953416i	$0.402459\pi$
$648$	0	0
$649$	28.2034	1.10708
$650$	0	0
$651$	−5.18421	−0.203185
$652$	0	0
$653$	−19.4697	−0.761906	−0.380953	−	0.924594i	$-0.624404\pi$
$653$	−19.4697	−0.761906	−0.380953	+	0.924594i	$0.624404\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.57136	0.0613046
$658$	0	0
$659$	−30.9403	−1.20526	−0.602631	−	0.798020i	$-0.705881\pi$
$659$	−30.9403	−1.20526	−0.602631	+	0.798020i	$0.705881\pi$
$660$	0	0
$661$	47.7975	1.85911	0.929554	−	0.368685i	$-0.120192\pi$
$661$	47.7975	1.85911	0.929554	+	0.368685i	$0.120192\pi$
$662$	0	0
$663$	28.4701	1.10569
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.04101	0.0403081
$668$	0	0
$669$	15.2257	0.588659
$670$	0	0
$671$	−13.7146	−0.529445
$672$	0	0
$673$	27.8163	1.07224	0.536119	−	0.844142i	$-0.319890\pi$
$673$	27.8163	1.07224	0.536119	+	0.844142i	$0.319890\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−19.0005	−0.730248	−0.365124	−	0.930959i	$-0.618973\pi$
$677$	−19.0005	−0.730248	−0.365124	+	0.930959i	$0.618973\pi$
$678$	0	0
$679$	−11.9398	−0.458207
$680$	0	0
$681$	14.3684	0.550599
$682$	0	0
$683$	−4.52051	−0.172972	−0.0864862	−	0.996253i	$-0.527564\pi$
$683$	−4.52051	−0.172972	−0.0864862	+	0.996253i	$0.527564\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.61285	0.214143
$688$	0	0
$689$	−59.0420	−2.24932
$690$	0	0
$691$	1.18421	0.0450494	0.0225247	−	0.999746i	$-0.492830\pi$
$691$	1.18421	0.0450494	0.0225247	+	0.999746i	$0.492830\pi$
$692$	0	0
$693$	2.00000	0.0759737
$694$	0	0
$695$	0	0
$696$	0	0
$697$	36.4701	1.38140
$698$	0	0
$699$	23.2859	0.880754
$700$	0	0
$701$	−26.6735	−1.00745	−0.503723	−	0.863865i	$-0.668037\pi$
$701$	−26.6735	−1.00745	−0.503723	+	0.863865i	$0.668037\pi$
$702$	0	0
$703$	−18.4889	−0.697321
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.47949	−0.0556421
$708$	0	0
$709$	−18.2034	−0.683644	−0.341822	−	0.939765i	$-0.611044\pi$
$709$	−18.2034	−0.683644	−0.341822	+	0.939765i	$0.611044\pi$
$710$	0	0
$711$	4.85728	0.182162
$712$	0	0
$713$	−7.14272	−0.267497
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−8.48886	−0.317022
$718$	0	0
$719$	−4.85728	−0.181146	−0.0905730	−	0.995890i	$-0.528870\pi$
$719$	−4.85728	−0.181146	−0.0905730	+	0.995890i	$0.528870\pi$
$720$	0	0
$721$	8.85728	0.329862
$722$	0	0
$723$	7.24443	0.269423
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21.0607	−0.781098	−0.390549	−	0.920582i	$-0.627715\pi$
$727$	−21.0607	−0.781098	−0.390549	+	0.920582i	$0.627715\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−44.7368	−1.65465
$732$	0	0
$733$	9.45091	0.349077	0.174539	−	0.984650i	$-0.444157\pi$
$733$	9.45091	0.349077	0.174539	+	0.984650i	$0.444157\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.51114	−0.203005
$738$	0	0
$739$	8.20342	0.301768	0.150884	−	0.988551i	$-0.451788\pi$
$739$	8.20342	0.301768	0.150884	+	0.988551i	$0.451788\pi$
$740$	0	0
$741$	−15.6128	−0.573552
$742$	0	0
$743$	−8.33677	−0.305847	−0.152923	−	0.988238i	$-0.548869\pi$
$743$	−8.33677	−0.305847	−0.152923	+	0.988238i	$0.548869\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	11.6128	0.424892
$748$	0	0
$749$	−1.76494	−0.0644894
$750$	0	0
$751$	25.9180	0.945760	0.472880	−	0.881127i	$-0.343214\pi$
$751$	25.9180	0.945760	0.472880	+	0.881127i	$0.343214\pi$
$752$	0	0
$753$	27.6128	1.00627
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.94025	0.324939	0.162470	−	0.986714i	$-0.448054\pi$
$757$	8.94025	0.324939	0.162470	+	0.986714i	$0.448054\pi$
$758$	0	0
$759$	2.75557	0.100021
$760$	0	0
$761$	−0.825636	−0.0299293	−0.0149646	−	0.999888i	$-0.504764\pi$
$761$	−0.825636	−0.0299293	−0.0149646	+	0.999888i	$0.504764\pi$
$762$	0	0
$763$	−5.61285	−0.203199
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−90.6548	−3.27336
$768$	0	0
$769$	21.2257	0.765418	0.382709	−	0.923869i	$-0.374991\pi$
$769$	21.2257	0.765418	0.382709	+	0.923869i	$0.374991\pi$
$770$	0	0
$771$	−0.428639	−0.0154371
$772$	0	0
$773$	29.4893	1.06066	0.530329	−	0.847792i	$-0.322068\pi$
$773$	29.4893	1.06066	0.530329	+	0.847792i	$0.322068\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−7.61285	−0.273109
$778$	0	0
$779$	−20.0000	−0.716574
$780$	0	0
$781$	4.00000	0.143131
$782$	0	0
$783$	−0.755569	−0.0270018
$784$	0	0
$785$	0	0
$786$	0	0
$787$	34.4514	1.22806	0.614030	−	0.789283i	$-0.289547\pi$
$787$	34.4514	1.22806	0.614030	+	0.789283i	$0.289547\pi$
$788$	0	0
$789$	9.37778	0.333858
$790$	0	0
$791$	−11.2859	−0.401281
$792$	0	0
$793$	44.0830	1.56543
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.9175	0.670092	0.335046	−	0.942202i	$-0.391248\pi$
$797$	18.9175	0.670092	0.335046	+	0.942202i	$0.391248\pi$
$798$	0	0
$799$	−12.2034	−0.431726
$800$	0	0
$801$	4.62222	0.163318
$802$	0	0
$803$	−3.14272	−0.110904
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.74620	−0.0614692
$808$	0	0
$809$	21.2257	0.746256	0.373128	−	0.927780i	$-0.378285\pi$
$809$	21.2257	0.746256	0.373128	+	0.927780i	$0.378285\pi$
$810$	0	0
$811$	21.5081	0.755251	0.377625	−	0.925958i	$-0.376741\pi$
$811$	21.5081	0.755251	0.377625	+	0.925958i	$0.376741\pi$
$812$	0	0
$813$	−2.69535	−0.0945299
$814$	0	0
$815$	0	0
$816$	0	0
$817$	24.5334	0.858315
$818$	0	0
$819$	−6.42864	−0.224635
$820$	0	0
$821$	−46.2034	−1.61251	−0.806255	−	0.591568i	$-0.798509\pi$
$821$	−46.2034	−1.61251	−0.806255	+	0.591568i	$0.798509\pi$
$822$	0	0
$823$	17.8350	0.621689	0.310845	−	0.950461i	$-0.399388\pi$
$823$	17.8350	0.621689	0.310845	+	0.950461i	$0.399388\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.2128	−1.22447	−0.612234	−	0.790676i	$-0.709729\pi$
$827$	−35.2128	−1.22447	−0.612234	+	0.790676i	$0.709729\pi$
$828$	0	0
$829$	14.3872	0.499686	0.249843	−	0.968286i	$-0.419621\pi$
$829$	14.3872	0.499686	0.249843	+	0.968286i	$0.419621\pi$
$830$	0	0
$831$	−5.12399	−0.177749
$832$	0	0
$833$	−4.42864	−0.153443
$834$	0	0
$835$	0	0
$836$	0	0
$837$	5.18421	0.179192
$838$	0	0
$839$	−1.51114	−0.0521703	−0.0260851	−	0.999660i	$-0.508304\pi$
$839$	−1.51114	−0.0521703	−0.0260851	+	0.999660i	$0.508304\pi$
$840$	0	0
$841$	−28.4291	−0.980314
$842$	0	0
$843$	−23.9813	−0.825959
$844$	0	0
$845$	0	0
$846$	0	0
$847$	7.00000	0.240523
$848$	0	0
$849$	−2.36842	−0.0812838
$850$	0	0
$851$	−10.4889	−0.359554
$852$	0	0
$853$	−15.4064	−0.527504	−0.263752	−	0.964591i	$-0.584960\pi$
$853$	−15.4064	−0.527504	−0.263752	+	0.964591i	$0.584960\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19.8578	0.678328	0.339164	−	0.940727i	$-0.389856\pi$
$857$	19.8578	0.678328	0.339164	+	0.940727i	$0.389856\pi$
$858$	0	0
$859$	−2.42864	−0.0828641	−0.0414321	−	0.999141i	$-0.513192\pi$
$859$	−2.42864	−0.0828641	−0.0414321	+	0.999141i	$0.513192\pi$
$860$	0	0
$861$	−8.23506	−0.280650
$862$	0	0
$863$	39.2958	1.33764	0.668822	−	0.743423i	$-0.266799\pi$
$863$	39.2958	1.33764	0.668822	+	0.743423i	$0.266799\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−2.61285	−0.0887370
$868$	0	0
$869$	−9.71456	−0.329544
$870$	0	0
$871$	17.7146	0.600235
$872$	0	0
$873$	11.9398	0.404100
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−56.2864	−1.90066	−0.950328	−	0.311249i	$-0.899253\pi$
$877$	−56.2864	−1.90066	−0.950328	+	0.311249i	$0.899253\pi$
$878$	0	0
$879$	8.42864	0.284291
$880$	0	0
$881$	−2.33677	−0.0787279	−0.0393640	−	0.999225i	$-0.512533\pi$
$881$	−2.33677	−0.0787279	−0.0393640	+	0.999225i	$0.512533\pi$
$882$	0	0
$883$	−33.7146	−1.13459	−0.567293	−	0.823516i	$-0.692009\pi$
$883$	−33.7146	−1.13459	−0.567293	+	0.823516i	$0.692009\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−47.8992	−1.60830	−0.804150	−	0.594427i	$-0.797379\pi$
$887$	−47.8992	−1.60830	−0.804150	+	0.594427i	$0.797379\pi$
$888$	0	0
$889$	12.8573	0.431219
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	6.69228	0.223949
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−8.85728	−0.295736
$898$	0	0
$899$	−3.91703	−0.130640
$900$	0	0
$901$	40.6735	1.35503
$902$	0	0
$903$	10.1017	0.336164
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−23.7591	−0.788908	−0.394454	−	0.918916i	$-0.629066\pi$
$907$	−23.7591	−0.788908	−0.394454	+	0.918916i	$0.629066\pi$
$908$	0	0
$909$	1.47949	0.0490717
$910$	0	0
$911$	−22.9403	−0.760045	−0.380022	−	0.924977i	$-0.624084\pi$
$911$	−22.9403	−0.760045	−0.380022	+	0.924977i	$0.624084\pi$
$912$	0	0
$913$	−23.2257	−0.768658
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.10171	−0.0694046
$918$	0	0
$919$	−16.9777	−0.560043	−0.280022	−	0.959994i	$-0.590342\pi$
$919$	−16.9777	−0.560043	−0.280022	+	0.959994i	$0.590342\pi$
$920$	0	0
$921$	22.5718	0.743767
$922$	0	0
$923$	−12.8573	−0.423202
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−8.85728	−0.290911
$928$	0	0
$929$	39.3403	1.29071	0.645357	−	0.763881i	$-0.276709\pi$
$929$	39.3403	1.29071	0.645357	+	0.763881i	$0.276709\pi$
$930$	0	0
$931$	2.42864	0.0795954
$932$	0	0
$933$	−24.0830	−0.788441
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−17.7748	−0.580677	−0.290338	−	0.956924i	$-0.593768\pi$
$937$	−17.7748	−0.580677	−0.290338	+	0.956924i	$0.593768\pi$
$938$	0	0
$939$	9.65433	0.315057
$940$	0	0
$941$	35.5812	1.15991	0.579957	−	0.814647i	$-0.303069\pi$
$941$	35.5812	1.15991	0.579957	+	0.814647i	$0.303069\pi$
$942$	0	0
$943$	−11.3461	−0.369481
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.5018	−0.991174	−0.495587	−	0.868558i	$-0.665047\pi$
$947$	−30.5018	−0.991174	−0.495587	+	0.868558i	$0.665047\pi$
$948$	0	0
$949$	10.1017	0.327915
$950$	0	0
$951$	−6.04149	−0.195909
$952$	0	0
$953$	−51.1655	−1.65741	−0.828706	−	0.559684i	$-0.810923\pi$
$953$	−51.1655	−1.65741	−0.828706	+	0.559684i	$0.810923\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.51114	0.0488481
$958$	0	0
$959$	15.9398	0.514722
$960$	0	0
$961$	−4.12399	−0.133032
$962$	0	0
$963$	1.76494	0.0568743
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−47.8992	−1.54034	−0.770168	−	0.637841i	$-0.779828\pi$
$967$	−47.8992	−1.54034	−0.770168	+	0.637841i	$0.779828\pi$
$968$	0	0
$969$	10.7556	0.345519
$970$	0	0
$971$	−40.6735	−1.30528	−0.652638	−	0.757670i	$-0.726338\pi$
$971$	−40.6735	−1.30528	−0.652638	+	0.757670i	$0.726338\pi$
$972$	0	0
$973$	11.6731	0.374221
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.4893	−0.879462	−0.439731	−	0.898130i	$-0.644926\pi$
$977$	−27.4893	−0.879462	−0.439731	+	0.898130i	$0.644926\pi$
$978$	0	0
$979$	−9.24443	−0.295453
$980$	0	0
$981$	5.61285	0.179204
$982$	0	0
$983$	−24.0000	−0.765481	−0.382741	−	0.923856i	$-0.625020\pi$
$983$	−24.0000	−0.765481	−0.382741	+	0.923856i	$0.625020\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.75557	0.0877107
$988$	0	0
$989$	13.9180	0.442566
$990$	0	0
$991$	34.6923	1.10204	0.551018	−	0.834493i	$-0.314239\pi$
$991$	34.6923	1.10204	0.551018	+	0.834493i	$0.314239\pi$
$992$	0	0
$993$	13.5111	0.428763
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−28.6766	−0.908197	−0.454099	−	0.890951i	$-0.650039\pi$
$997$	−28.6766	−0.908197	−0.454099	+	0.890951i	$0.650039\pi$
$998$	0	0
$999$	7.61285	0.240860

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8400.2.a.dg.1.3		3
4.3	odd	2		525.2.a.j.1.3		3
5.2	odd	4		1680.2.t.k.1009.5		6
5.3	odd	4		1680.2.t.k.1009.2		6
5.4	even	2		8400.2.a.dj.1.1		3
12.11	even	2		1575.2.a.x.1.1		3
15.2	even	4		5040.2.t.v.1009.3		6
15.8	even	4		5040.2.t.v.1009.4		6
20.3	even	4		105.2.d.b.64.2	✓	6
20.7	even	4		105.2.d.b.64.5	yes	6
20.19	odd	2		525.2.a.k.1.1		3
28.27	even	2		3675.2.a.bi.1.3		3
60.23	odd	4		315.2.d.e.64.5		6
60.47	odd	4		315.2.d.e.64.2		6
60.59	even	2		1575.2.a.w.1.3		3
140.3	odd	12		735.2.q.f.79.5		12
140.23	even	12		735.2.q.e.214.2		12
140.27	odd	4		735.2.d.b.589.5		6
140.47	odd	12		735.2.q.f.214.5		12
140.67	even	12		735.2.q.e.79.2		12
140.83	odd	4		735.2.d.b.589.2		6
140.87	odd	12		735.2.q.f.79.2		12
140.103	odd	12		735.2.q.f.214.2		12
140.107	even	12		735.2.q.e.214.5		12
140.123	even	12		735.2.q.e.79.5		12
140.139	even	2		3675.2.a.bj.1.1		3
420.83	even	4		2205.2.d.l.1324.5		6
420.167	even	4		2205.2.d.l.1324.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.2.d.b.64.2	✓	6	20.3	even	4
105.2.d.b.64.5	yes	6	20.7	even	4
315.2.d.e.64.2		6	60.47	odd	4
315.2.d.e.64.5		6	60.23	odd	4
525.2.a.j.1.3		3	4.3	odd	2
525.2.a.k.1.1		3	20.19	odd	2
735.2.d.b.589.2		6	140.83	odd	4
735.2.d.b.589.5		6	140.27	odd	4
735.2.q.e.79.2		12	140.67	even	12
735.2.q.e.79.5		12	140.123	even	12
735.2.q.e.214.2		12	140.23	even	12
735.2.q.e.214.5		12	140.107	even	12
735.2.q.f.79.2		12	140.87	odd	12
735.2.q.f.79.5		12	140.3	odd	12
735.2.q.f.214.2		12	140.103	odd	12
735.2.q.f.214.5		12	140.47	odd	12
1575.2.a.w.1.3		3	60.59	even	2
1575.2.a.x.1.1		3	12.11	even	2
1680.2.t.k.1009.2		6	5.3	odd	4
1680.2.t.k.1009.5		6	5.2	odd	4
2205.2.d.l.1324.2		6	420.167	even	4
2205.2.d.l.1324.5		6	420.83	even	4
3675.2.a.bi.1.3		3	28.27	even	2
3675.2.a.bj.1.1		3	140.139	even	2
5040.2.t.v.1009.3		6	15.2	even	4
5040.2.t.v.1009.4		6	15.8	even	4
8400.2.a.dg.1.3		3	1.1	even	1	trivial
8400.2.a.dj.1.1		3	5.4	even	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:	\( N \)	\(=\)	\( 8400 = 2^{4} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8400.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{7} +1.00000 q^{9} -2.00000 q^{11} +6.42864 q^{13} -4.42864 q^{17} +2.42864 q^{19} +1.00000 q^{21} +1.37778 q^{23} -1.00000 q^{27} +0.755569 q^{29} -5.18421 q^{31} +2.00000 q^{33} -7.61285 q^{37} -6.42864 q^{39} -8.23506 q^{41} +10.1017 q^{43} +2.75557 q^{47} +1.00000 q^{49} +4.42864 q^{51} -9.18421 q^{53} -2.42864 q^{57} -14.1017 q^{59} +6.85728 q^{61} -1.00000 q^{63} +2.75557 q^{67} -1.37778 q^{69} -2.00000 q^{71} +1.57136 q^{73} +2.00000 q^{77} +4.85728 q^{79} +1.00000 q^{81} +11.6128 q^{83} -0.755569 q^{87} +4.62222 q^{89} -6.42864 q^{91} +5.18421 q^{93} +11.9398 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} - 6 q^{11} + 6 q^{13} - 6 q^{19} + 3 q^{21} + 4 q^{23} - 3 q^{27} + 2 q^{29} - 2 q^{31} + 6 q^{33} + 4 q^{37} - 6 q^{39} + 2 q^{41} + 4 q^{43} + 8 q^{47} + 3 q^{49} - 14 q^{53}+ \cdots - 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 3 * q^9 - 6 * q^11 + 6 * q^13 - 6 * q^19 + 3 * q^21 + 4 * q^23 - 3 * q^27 + 2 * q^29 - 2 * q^31 + 6 * q^33 + 4 * q^37 - 6 * q^39 + 2 * q^41 + 4 * q^43 + 8 * q^47 + 3 * q^49 - 14 * q^53 + 6 * q^57 - 16 * q^59 - 6 * q^61 - 3 * q^63 + 8 * q^67 - 4 * q^69 - 6 * q^71 + 18 * q^73 + 6 * q^77 - 12 * q^79 + 3 * q^81 + 8 * q^83 - 2 * q^87 + 14 * q^89 - 6 * q^91 + 2 * q^93 + 22 * q^97 - 6 * q^99

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