LMFDB - Embedded newform 252.8.a.h.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,8,Mod(1,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("252.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	252.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$78.7210264220$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4648x^{2} + 2987775 $ x^4 - 4648*x^2 + 2987775
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$62.2692$ of defining polynomial
Character	$\chi$	$=$	252.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+373.615 q^{5} +343.000 q^{7} +4603.99 q^{11} -5287.16 q^{13} +16065.4 q^{17} +28391.5 q^{19} +39011.9 q^{23} +61463.1 q^{25} +97392.9 q^{29} -241746. q^{31} +128150. q^{35} +249203. q^{37} +108198. q^{41} -228813. q^{43} -352722. q^{47} +117649. q^{49} -778685. q^{53} +1.72012e6 q^{55} -1.76721e6 q^{59} +260500. q^{61} -1.97536e6 q^{65} -1.24096e6 q^{67} +1.36800e6 q^{71} +1.75423e6 q^{73} +1.57917e6 q^{77} +6.81583e6 q^{79} +7.92440e6 q^{83} +6.00229e6 q^{85} +2.49883e6 q^{89} -1.81350e6 q^{91} +1.06075e7 q^{95} +693361. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 1372 q^{7} + 10808 q^{13} + 17696 q^{19} + 22156 q^{25} + 87584 q^{31} - 121672 q^{37} - 20464 q^{43} + 470596 q^{49} + 840672 q^{55} + 4844840 q^{61} + 6220976 q^{67} + 10659992 q^{73} + 4446272 q^{79}+ \cdots + 25590488 q^{97}+O(q^{100}) $ 4 * q + 1372 * q^7 + 10808 * q^13 + 17696 * q^19 + 22156 * q^25 + 87584 * q^31 - 121672 * q^37 - 20464 * q^43 + 470596 * q^49 + 840672 * q^55 + 4844840 * q^61 + 6220976 * q^67 + 10659992 * q^73 + 4446272 * q^79 + 14390208 * q^85 + 3707144 * q^91 + 25590488 * 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
$5$	373.615	1.33669	0.668343	−	0.743853i	$-0.267004\pi$
$5$	373.615	1.33669	0.668343	+	0.743853i	$0.267004\pi$
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4603.99	1.04294	0.521471	−	0.853269i	$-0.325384\pi$
$11$	4603.99	1.04294	0.521471	+	0.853269i	$0.325384\pi$
$12$	0	0
$13$	−5287.16	−0.667453	−0.333727	−	0.942670i	$-0.608306\pi$
$13$	−5287.16	−0.667453	−0.333727	+	0.942670i	$0.608306\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	16065.4	0.793088	0.396544	−	0.918016i	$-0.370209\pi$
$17$	16065.4	0.793088	0.396544	+	0.918016i	$0.370209\pi$
$18$	0	0
$19$	28391.5	0.949621	0.474811	−	0.880088i	$-0.342517\pi$
$19$	28391.5	0.949621	0.474811	+	0.880088i	$0.342517\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	39011.9	0.668575	0.334287	−	0.942471i	$-0.391504\pi$
$23$	39011.9	0.668575	0.334287	+	0.942471i	$0.391504\pi$
$24$	0	0
$25$	61463.1	0.786728
$26$	0	0
$27$	0	0
$28$	0	0
$29$	97392.9	0.741539	0.370770	−	0.928725i	$-0.379094\pi$
$29$	97392.9	0.741539	0.370770	+	0.928725i	$0.379094\pi$
$30$	0	0
$31$	−241746.	−1.45745	−0.728726	−	0.684806i	$-0.759887\pi$
$31$	−241746.	−1.45745	−0.728726	+	0.684806i	$0.759887\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	128150.	0.505220
$36$	0	0
$37$	249203.	0.808810	0.404405	−	0.914580i	$-0.367479\pi$
$37$	249203.	0.808810	0.404405	+	0.914580i	$0.367479\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	108198.	0.245175	0.122587	−	0.992458i	$-0.460881\pi$
$41$	108198.	0.245175	0.122587	+	0.992458i	$0.460881\pi$
$42$	0	0
$43$	−228813.	−0.438874	−0.219437	−	0.975627i	$-0.570422\pi$
$43$	−228813.	−0.438874	−0.219437	+	0.975627i	$0.570422\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−352722.	−0.495552	−0.247776	−	0.968817i	$-0.579700\pi$
$47$	−352722.	−0.495552	−0.247776	+	0.968817i	$0.579700\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−778685.	−0.718449	−0.359225	−	0.933251i	$-0.616959\pi$
$53$	−778685.	−0.718449	−0.359225	+	0.933251i	$0.616959\pi$
$54$	0	0
$55$	1.72012e6	1.39408
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.76721e6	−1.12023	−0.560113	−	0.828416i	$-0.689242\pi$
$59$	−1.76721e6	−1.12023	−0.560113	+	0.828416i	$0.689242\pi$
$60$	0	0
$61$	260500.	0.146944	0.0734722	−	0.997297i	$-0.476592\pi$
$61$	260500.	0.146944	0.0734722	+	0.997297i	$0.476592\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.97536e6	−0.892175
$66$	0	0
$67$	−1.24096e6	−0.504077	−0.252038	−	0.967717i	$-0.581101\pi$
$67$	−1.24096e6	−0.504077	−0.252038	+	0.967717i	$0.581101\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.36800e6	0.453610	0.226805	−	0.973940i	$-0.427172\pi$
$71$	1.36800e6	0.453610	0.226805	+	0.973940i	$0.427172\pi$
$72$	0	0
$73$	1.75423e6	0.527785	0.263893	−	0.964552i	$-0.414994\pi$
$73$	1.75423e6	0.527785	0.263893	+	0.964552i	$0.414994\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.57917e6	0.394195
$78$	0	0
$79$	6.81583e6	1.55533	0.777667	−	0.628676i	$-0.216403\pi$
$79$	6.81583e6	1.55533	0.777667	+	0.628676i	$0.216403\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.92440e6	1.52122	0.760612	−	0.649207i	$-0.224899\pi$
$83$	7.92440e6	1.52122	0.760612	+	0.649207i	$0.224899\pi$
$84$	0	0
$85$	6.00229e6	1.06011
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.49883e6	0.375726	0.187863	−	0.982195i	$-0.439844\pi$
$89$	2.49883e6	0.375726	0.187863	+	0.982195i	$0.439844\pi$
$90$	0	0
$91$	−1.81350e6	−0.252274
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.06075e7	1.26935
$96$	0	0
$97$	693361.	0.0771362	0.0385681	−	0.999256i	$-0.487720\pi$
$97$	693361.	0.0771362	0.0385681	+	0.999256i	$0.487720\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.52730e7	−1.47503	−0.737515	−	0.675331i	$-0.764001\pi$
$101$	−1.52730e7	−1.47503	−0.737515	+	0.675331i	$0.764001\pi$
$102$	0	0
$103$	6.03134e6	0.543855	0.271928	−	0.962318i	$-0.412339\pi$
$103$	6.03134e6	0.543855	0.271928	+	0.962318i	$0.412339\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.57901e6	0.519179	0.259589	−	0.965719i	$-0.416413\pi$
$107$	6.57901e6	0.519179	0.259589	+	0.965719i	$0.416413\pi$
$108$	0	0
$109$	2.56522e7	1.89728	0.948639	−	0.316360i	$-0.102461\pi$
$109$	2.56522e7	1.89728	0.948639	+	0.316360i	$0.102461\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−2.22526e7	−1.45079	−0.725397	−	0.688330i	$-0.758344\pi$
$113$	−2.22526e7	−1.45079	−0.725397	+	0.688330i	$0.758344\pi$
$114$	0	0
$115$	1.45754e7	0.893674
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.51045e6	0.299759
$120$	0	0
$121$	1.70955e6	0.0877272
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.22512e6	−0.285077
$126$	0	0
$127$	2.93911e7	1.27322	0.636609	−	0.771187i	$-0.280336\pi$
$127$	2.93911e7	1.27322	0.636609	+	0.771187i	$0.280336\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	2.27085e7	0.882550	0.441275	−	0.897372i	$-0.354526\pi$
$131$	2.27085e7	0.882550	0.441275	+	0.897372i	$0.354526\pi$
$132$	0	0
$133$	9.73828e6	0.358923
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.89164e7	−0.960775	−0.480388	−	0.877056i	$-0.659504\pi$
$137$	−2.89164e7	−0.960775	−0.480388	+	0.877056i	$0.659504\pi$
$138$	0	0
$139$	4.55584e7	1.43886	0.719428	−	0.694567i	$-0.244404\pi$
$139$	4.55584e7	1.43886	0.719428	+	0.694567i	$0.244404\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.43420e7	−0.696115
$144$	0	0
$145$	3.63874e7	0.991205
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2.84743e7	0.705183	0.352591	−	0.935777i	$-0.385301\pi$
$149$	2.84743e7	0.705183	0.352591	+	0.935777i	$0.385301\pi$
$150$	0	0
$151$	−5.87770e6	−0.138928	−0.0694638	−	0.997584i	$-0.522129\pi$
$151$	−5.87770e6	−0.138928	−0.0694638	+	0.997584i	$0.522129\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.03200e7	−1.94815
$156$	0	0
$157$	4.55696e7	0.939780	0.469890	−	0.882725i	$-0.344294\pi$
$157$	4.55696e7	0.939780	0.469890	+	0.882725i	$0.344294\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.33811e7	0.252697
$162$	0	0
$163$	2.71069e7	0.490257	0.245129	−	0.969491i	$-0.421170\pi$
$163$	2.71069e7	0.490257	0.245129	+	0.969491i	$0.421170\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.20283e7	1.36287	0.681437	−	0.731877i	$-0.261355\pi$
$167$	8.20283e7	1.36287	0.681437	+	0.731877i	$0.261355\pi$
$168$	0	0
$169$	−3.47944e7	−0.554506
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.32062e8	1.93917	0.969583	−	0.244763i	$-0.0787102\pi$
$173$	1.32062e8	1.93917	0.969583	+	0.244763i	$0.0787102\pi$
$174$	0	0
$175$	2.10819e7	0.297355
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.33808e7	−1.08663	−0.543314	−	0.839530i	$-0.682831\pi$
$179$	−8.33808e7	−1.08663	−0.543314	+	0.839530i	$0.682831\pi$
$180$	0	0
$181$	−3.96597e7	−0.497135	−0.248567	−	0.968615i	$-0.579960\pi$
$181$	−3.96597e7	−0.497135	−0.248567	+	0.968615i	$0.579960\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.31058e7	1.08112
$186$	0	0
$187$	7.39651e7	0.827145
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.21078e8	1.25733	0.628663	−	0.777678i	$-0.283603\pi$
$191$	1.21078e8	1.25733	0.628663	+	0.777678i	$0.283603\pi$
$192$	0	0
$193$	8.14383e7	0.815414	0.407707	−	0.913113i	$-0.366328\pi$
$193$	8.14383e7	0.815414	0.407707	+	0.913113i	$0.366328\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.10941e8	−1.96575	−0.982877	−	0.184260i	$-0.941011\pi$
$197$	−2.10941e8	−1.96575	−0.982877	+	0.184260i	$0.941011\pi$
$198$	0	0
$199$	−604097.	−0.00543402	−0.00271701	−	0.999996i	$-0.500865\pi$
$199$	−604097.	−0.00543402	−0.00271701	+	0.999996i	$0.500865\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.34058e7	0.280276
$204$	0	0
$205$	4.04244e7	0.327721
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.30714e8	0.990400
$210$	0	0
$211$	−2.09618e7	−0.153618	−0.0768088	−	0.997046i	$-0.524473\pi$
$211$	−2.09618e7	−0.153618	−0.0768088	+	0.997046i	$0.524473\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.54878e7	−0.586637
$216$	0	0
$217$	−8.29190e7	−0.550865
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−8.49406e7	−0.529349
$222$	0	0
$223$	3.81109e7	0.230135	0.115067	−	0.993358i	$-0.463292\pi$
$223$	3.81109e7	0.230135	0.115067	+	0.993358i	$0.463292\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2.07890e8	−1.17963	−0.589813	−	0.807540i	$-0.700798\pi$
$227$	−2.07890e8	−1.17963	−0.589813	+	0.807540i	$0.700798\pi$
$228$	0	0
$229$	−3.03688e8	−1.67111	−0.835553	−	0.549410i	$-0.814853\pi$
$229$	−3.03688e8	−1.67111	−0.835553	+	0.549410i	$0.814853\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.03389e8	−0.535459	−0.267730	−	0.963494i	$-0.586273\pi$
$233$	−1.03389e8	−0.535459	−0.267730	+	0.963494i	$0.586273\pi$
$234$	0	0
$235$	−1.31782e8	−0.662398
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.53581e8	−1.20150	−0.600749	−	0.799437i	$-0.705131\pi$
$239$	−2.53581e8	−1.20150	−0.600749	+	0.799437i	$0.705131\pi$
$240$	0	0
$241$	−2.28295e8	−1.05060	−0.525299	−	0.850918i	$-0.676046\pi$
$241$	−2.28295e8	−1.05060	−0.525299	+	0.850918i	$0.676046\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.39554e7	0.190955
$246$	0	0
$247$	−1.50110e8	−0.633828
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−1.39619e8	−0.557296	−0.278648	−	0.960393i	$-0.589886\pi$
$251$	−1.39619e8	−0.557296	−0.278648	+	0.960393i	$0.589886\pi$
$252$	0	0
$253$	1.79610e8	0.697284
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.94011e8	1.44791	0.723956	−	0.689847i	$-0.242322\pi$
$257$	3.94011e8	1.44791	0.723956	+	0.689847i	$0.242322\pi$
$258$	0	0
$259$	8.54765e7	0.305702
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2.86267e8	0.970347	0.485173	−	0.874418i	$-0.338757\pi$
$263$	2.86267e8	0.970347	0.485173	+	0.874418i	$0.338757\pi$
$264$	0	0
$265$	−2.90928e8	−0.960341
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.60724e8	−1.12991	−0.564953	−	0.825123i	$-0.691106\pi$
$269$	−3.60724e8	−1.12991	−0.564953	+	0.825123i	$0.691106\pi$
$270$	0	0
$271$	−3.27558e8	−0.999761	−0.499881	−	0.866094i	$-0.666623\pi$
$271$	−3.27558e8	−0.999761	−0.499881	+	0.866094i	$0.666623\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.82976e8	0.820511
$276$	0	0
$277$	−4.07506e8	−1.15201	−0.576003	−	0.817448i	$-0.695388\pi$
$277$	−4.07506e8	−1.15201	−0.576003	+	0.817448i	$0.695388\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.41823e8	1.45675	0.728376	−	0.685178i	$-0.240275\pi$
$281$	5.41823e8	1.45675	0.728376	+	0.685178i	$0.240275\pi$
$282$	0	0
$283$	−3.43393e7	−0.0900615	−0.0450307	−	0.998986i	$-0.514339\pi$
$283$	−3.43393e7	−0.0900615	−0.0450307	+	0.998986i	$0.514339\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.71119e7	0.0926673
$288$	0	0
$289$	−1.52240e8	−0.371011
$290$	0	0
$291$	0	0
$292$	0	0
$293$	2.69394e8	0.625679	0.312839	−	0.949806i	$-0.398720\pi$
$293$	2.69394e8	0.625679	0.312839	+	0.949806i	$0.398720\pi$
$294$	0	0
$295$	−6.60256e8	−1.49739
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.06262e8	−0.446242
$300$	0	0
$301$	−7.84827e7	−0.165879
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.73266e7	0.196418
$306$	0	0
$307$	−1.50431e8	−0.296723	−0.148362	−	0.988933i	$-0.547400\pi$
$307$	−1.50431e8	−0.296723	−0.148362	+	0.988933i	$0.547400\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.08437e8	0.204417	0.102208	−	0.994763i	$-0.467409\pi$
$311$	1.08437e8	0.204417	0.102208	+	0.994763i	$0.467409\pi$
$312$	0	0
$313$	−8.51922e7	−0.157034	−0.0785172	−	0.996913i	$-0.525019\pi$
$313$	−8.51922e7	−0.157034	−0.0785172	+	0.996913i	$0.525019\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−7.33084e8	−1.29255	−0.646274	−	0.763106i	$-0.723674\pi$
$317$	−7.33084e8	−1.29255	−0.646274	+	0.763106i	$0.723674\pi$
$318$	0	0
$319$	4.48396e8	0.773382
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.56122e8	0.753134
$324$	0	0
$325$	−3.24966e8	−0.525104
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.20984e8	−0.187301
$330$	0	0
$331$	3.83682e8	0.581531	0.290766	−	0.956794i	$-0.406090\pi$
$331$	3.83682e8	0.581531	0.290766	+	0.956794i	$0.406090\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−4.63642e8	−0.673792
$336$	0	0
$337$	−1.00460e9	−1.42984	−0.714922	−	0.699205i	$-0.753538\pi$
$337$	−1.00460e9	−1.42984	−0.714922	+	0.699205i	$0.753538\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.11300e9	−1.52004
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.27919e8	0.678288	0.339144	−	0.940734i	$-0.389863\pi$
$347$	5.27919e8	0.678288	0.339144	+	0.940734i	$0.389863\pi$
$348$	0	0
$349$	−4.01377e8	−0.505433	−0.252717	−	0.967540i	$-0.581324\pi$
$349$	−4.01377e8	−0.505433	−0.252717	+	0.967540i	$0.581324\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.58894e8	0.434264	0.217132	−	0.976142i	$-0.430330\pi$
$353$	3.58894e8	0.434264	0.217132	+	0.976142i	$0.430330\pi$
$354$	0	0
$355$	5.11106e8	0.606333
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.49771e8	−0.284913	−0.142456	−	0.989801i	$-0.545500\pi$
$359$	−2.49771e8	−0.284913	−0.142456	+	0.989801i	$0.545500\pi$
$360$	0	0
$361$	−8.77953e7	−0.0982191
$362$	0	0
$363$	0	0
$364$	0	0
$365$	6.55408e8	0.705483
$366$	0	0
$367$	7.47532e8	0.789403	0.394701	−	0.918809i	$-0.370848\pi$
$367$	7.47532e8	0.789403	0.394701	+	0.918809i	$0.370848\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2.67089e8	−0.271548
$372$	0	0
$373$	−1.40371e9	−1.40055	−0.700274	−	0.713874i	$-0.746939\pi$
$373$	−1.40371e9	−1.40055	−0.700274	+	0.713874i	$0.746939\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.14932e8	−0.494943
$378$	0	0
$379$	8.25191e8	0.778605	0.389303	−	0.921110i	$-0.372716\pi$
$379$	8.25191e8	0.778605	0.389303	+	0.921110i	$0.372716\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−1.01369e9	−0.921955	−0.460978	−	0.887412i	$-0.652501\pi$
$383$	−1.01369e9	−0.921955	−0.460978	+	0.887412i	$0.652501\pi$
$384$	0	0
$385$	5.90001e8	0.526915
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.89783e9	−1.63468	−0.817342	−	0.576153i	$-0.804553\pi$
$389$	−1.89783e9	−1.63468	−0.817342	+	0.576153i	$0.804553\pi$
$390$	0	0
$391$	6.26744e8	0.530239
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.54650e9	2.07899
$396$	0	0
$397$	−1.67665e9	−1.34486	−0.672429	−	0.740162i	$-0.734749\pi$
$397$	−1.67665e9	−1.34486	−0.672429	+	0.740162i	$0.734749\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.80128e8	−0.216946	−0.108473	−	0.994099i	$-0.534596\pi$
$401$	−2.80128e8	−0.216946	−0.108473	+	0.994099i	$0.534596\pi$
$402$	0	0
$403$	1.27815e9	0.972780
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.14733e9	0.843542
$408$	0	0
$409$	1.16783e9	0.844008	0.422004	−	0.906594i	$-0.361327\pi$
$409$	1.16783e9	0.844008	0.422004	+	0.906594i	$0.361327\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.06153e8	−0.423406
$414$	0	0
$415$	2.96068e9	2.03340
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.17872e9	−0.782819	−0.391410	−	0.920217i	$-0.628012\pi$
$419$	−1.17872e9	−0.782819	−0.391410	+	0.920217i	$0.628012\pi$
$420$	0	0
$421$	1.86093e9	1.21547	0.607733	−	0.794141i	$-0.292079\pi$
$421$	1.86093e9	1.21547	0.607733	+	0.794141i	$0.292079\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.87432e8	0.623945
$426$	0	0
$427$	8.93514e7	0.0555397
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.33812e9	1.40668	0.703341	−	0.710853i	$-0.251691\pi$
$431$	2.33812e9	1.40668	0.703341	+	0.710853i	$0.251691\pi$
$432$	0	0
$433$	7.11857e7	0.0421391	0.0210695	−	0.999778i	$-0.493293\pi$
$433$	7.11857e7	0.0421391	0.0210695	+	0.999778i	$0.493293\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.10761e9	0.634893
$438$	0	0
$439$	6.25224e7	0.0352704	0.0176352	−	0.999844i	$-0.494386\pi$
$439$	6.25224e7	0.0352704	0.0176352	+	0.999844i	$0.494386\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	8.74681e8	0.478009	0.239005	−	0.971018i	$-0.423179\pi$
$443$	8.74681e8	0.478009	0.239005	+	0.971018i	$0.423179\pi$
$444$	0	0
$445$	9.33600e8	0.502228
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.33350e9	1.73795	0.868977	−	0.494852i	$-0.164778\pi$
$449$	3.33350e9	1.73795	0.868977	+	0.494852i	$0.164778\pi$
$450$	0	0
$451$	4.98142e8	0.255703
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.77549e8	−0.337210
$456$	0	0
$457$	1.26840e9	0.621653	0.310827	−	0.950467i	$-0.399394\pi$
$457$	1.26840e9	0.621653	0.310827	+	0.950467i	$0.399394\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−1.33971e9	−0.636879	−0.318440	−	0.947943i	$-0.603159\pi$
$461$	−1.33971e9	−0.636879	−0.318440	+	0.947943i	$0.603159\pi$
$462$	0	0
$463$	6.04332e8	0.282971	0.141485	−	0.989940i	$-0.454812\pi$
$463$	6.04332e8	0.282971	0.141485	+	0.989940i	$0.454812\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.09562e7	0.0367825	0.0183912	−	0.999831i	$-0.494146\pi$
$467$	8.09562e7	0.0367825	0.0183912	+	0.999831i	$0.494146\pi$
$468$	0	0
$469$	−4.25650e8	−0.190523
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.05345e9	−0.457720
$474$	0	0
$475$	1.74503e9	0.747094
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−2.21153e9	−0.919430	−0.459715	−	0.888067i	$-0.652048\pi$
$479$	−2.21153e9	−0.919430	−0.459715	+	0.888067i	$0.652048\pi$
$480$	0	0
$481$	−1.31757e9	−0.539843
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.59050e8	0.103107
$486$	0	0
$487$	4.76077e9	1.86778	0.933890	−	0.357559i	$-0.116391\pi$
$487$	4.76077e9	1.86778	0.933890	+	0.357559i	$0.116391\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−4.60736e9	−1.75657	−0.878287	−	0.478134i	$-0.841313\pi$
$491$	−4.60736e9	−1.75657	−0.878287	+	0.478134i	$0.841313\pi$
$492$	0	0
$493$	1.56466e9	0.588106
$494$	0	0
$495$	0	0
$496$	0	0
$497$	4.69224e8	0.171448
$498$	0	0
$499$	2.73796e9	0.986449	0.493224	−	0.869902i	$-0.335818\pi$
$499$	2.73796e9	0.986449	0.493224	+	0.869902i	$0.335818\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.77627e9	−0.972688	−0.486344	−	0.873767i	$-0.661670\pi$
$503$	−2.77627e9	−0.972688	−0.486344	+	0.873767i	$0.661670\pi$
$504$	0	0
$505$	−5.70624e9	−1.97165
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.05026e9	−1.02524	−0.512618	−	0.858616i	$-0.671324\pi$
$509$	−3.05026e9	−1.02524	−0.512618	+	0.858616i	$0.671324\pi$
$510$	0	0
$511$	6.01702e8	0.199484
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.25340e9	0.726963
$516$	0	0
$517$	−1.62393e9	−0.516832
$518$	0	0
$519$	0	0
$520$	0	0
$521$	8.53878e8	0.264523	0.132262	−	0.991215i	$-0.457776\pi$
$521$	8.53878e8	0.264523	0.132262	+	0.991215i	$0.457776\pi$
$522$	0	0
$523$	3.41762e8	0.104464	0.0522321	−	0.998635i	$-0.483366\pi$
$523$	3.41762e8	0.104464	0.0522321	+	0.998635i	$0.483366\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.88376e9	−1.15589
$528$	0	0
$529$	−1.88290e9	−0.553008
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−5.72060e8	−0.163643
$534$	0	0
$535$	2.45801e9	0.693979
$536$	0	0
$537$	0	0
$538$	0	0
$539$	5.41655e8	0.148992
$540$	0	0
$541$	5.46752e9	1.48457	0.742283	−	0.670086i	$-0.233743\pi$
$541$	5.46752e9	1.48457	0.742283	+	0.670086i	$0.233743\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	9.58403e9	2.53606
$546$	0	0
$547$	2.88623e9	0.754007	0.377004	−	0.926212i	$-0.376954\pi$
$547$	2.88623e9	0.754007	0.377004	+	0.926212i	$0.376954\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.76513e9	0.704182
$552$	0	0
$553$	2.33783e9	0.587861
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.51107e9	−0.370502	−0.185251	−	0.982691i	$-0.559310\pi$
$557$	−1.51107e9	−0.370502	−0.185251	+	0.982691i	$0.559310\pi$
$558$	0	0
$559$	1.20977e9	0.292928
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7.35341e9	−1.73664	−0.868319	−	0.496006i	$-0.834799\pi$
$563$	−7.35341e9	−1.73664	−0.868319	+	0.496006i	$0.834799\pi$
$564$	0	0
$565$	−8.31390e9	−1.93926
$566$	0	0
$567$	0	0
$568$	0	0
$569$	3.81175e9	0.867424	0.433712	−	0.901052i	$-0.357204\pi$
$569$	3.81175e9	0.867424	0.433712	+	0.901052i	$0.357204\pi$
$570$	0	0
$571$	2.07708e8	0.0466902	0.0233451	−	0.999727i	$-0.492568\pi$
$571$	2.07708e8	0.0466902	0.0233451	+	0.999727i	$0.492568\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.39779e9	0.525986
$576$	0	0
$577$	−5.18328e9	−1.12328	−0.561642	−	0.827380i	$-0.689830\pi$
$577$	−5.18328e9	−1.12328	−0.561642	+	0.827380i	$0.689830\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	2.71807e9	0.574969
$582$	0	0
$583$	−3.58506e9	−0.749301
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.43761e9	0.293365	0.146683	−	0.989184i	$-0.453140\pi$
$587$	1.43761e9	0.293365	0.146683	+	0.989184i	$0.453140\pi$
$588$	0	0
$589$	−6.86354e9	−1.38403
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.73473e9	0.341618	0.170809	−	0.985304i	$-0.445362\pi$
$593$	1.73473e9	0.341618	0.170809	+	0.985304i	$0.445362\pi$
$594$	0	0
$595$	2.05879e9	0.400684
$596$	0	0
$597$	0	0
$598$	0	0
$599$	2.71944e9	0.516995	0.258497	−	0.966012i	$-0.416773\pi$
$599$	2.71944e9	0.516995	0.258497	+	0.966012i	$0.416773\pi$
$600$	0	0
$601$	−3.37029e9	−0.633295	−0.316647	−	0.948543i	$-0.602557\pi$
$601$	−3.37029e9	−0.633295	−0.316647	+	0.948543i	$0.602557\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.38715e8	0.117264
$606$	0	0
$607$	−7.98423e9	−1.44901	−0.724507	−	0.689267i	$-0.757933\pi$
$607$	−7.98423e9	−1.44901	−0.724507	+	0.689267i	$0.757933\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.86490e9	0.330758
$612$	0	0
$613$	4.42191e9	0.775351	0.387676	−	0.921796i	$-0.373278\pi$
$613$	4.42191e9	0.775351	0.387676	+	0.921796i	$0.373278\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.72972e9	−1.15345	−0.576725	−	0.816938i	$-0.695670\pi$
$617$	−6.72972e9	−1.15345	−0.576725	+	0.816938i	$0.695670\pi$
$618$	0	0
$619$	1.01216e10	1.71527	0.857633	−	0.514263i	$-0.171934\pi$
$619$	1.01216e10	1.71527	0.857633	+	0.514263i	$0.171934\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.57098e8	0.142011
$624$	0	0
$625$	−7.12761e9	−1.16779
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.00355e9	0.641458
$630$	0	0
$631$	−3.36807e9	−0.533677	−0.266838	−	0.963741i	$-0.585979\pi$
$631$	−3.36807e9	−0.533677	−0.266838	+	0.963741i	$0.585979\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.09810e10	1.70189
$636$	0	0
$637$	−6.22029e8	−0.0953505
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.27624e10	−1.91394	−0.956971	−	0.290183i	$-0.906284\pi$
$641$	−1.27624e10	−1.91394	−0.956971	+	0.290183i	$0.906284\pi$
$642$	0	0
$643$	−1.15776e10	−1.71743	−0.858715	−	0.512453i	$-0.828737\pi$
$643$	−1.15776e10	−1.71743	−0.858715	+	0.512453i	$0.828737\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.34167e9	−0.920532	−0.460266	−	0.887781i	$-0.652246\pi$
$647$	−6.34167e9	−0.920532	−0.460266	+	0.887781i	$0.652246\pi$
$648$	0	0
$649$	−8.13621e9	−1.16833
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.08753e9	−0.574466	−0.287233	−	0.957861i	$-0.592735\pi$
$653$	−4.08753e9	−0.574466	−0.287233	+	0.957861i	$0.592735\pi$
$654$	0	0
$655$	8.48424e9	1.17969
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8.38043e9	−1.14069	−0.570345	−	0.821405i	$-0.693190\pi$
$659$	−8.38043e9	−1.14069	−0.570345	+	0.821405i	$0.693190\pi$
$660$	0	0
$661$	−8.35309e9	−1.12497	−0.562487	−	0.826806i	$-0.690155\pi$
$661$	−8.35309e9	−1.12497	−0.562487	+	0.826806i	$0.690155\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.63837e9	0.479767
$666$	0	0
$667$	3.79948e9	0.495774
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.19934e9	0.153254
$672$	0	0
$673$	−7.99625e9	−1.01119	−0.505596	−	0.862770i	$-0.668727\pi$
$673$	−7.99625e9	−1.01119	−0.505596	+	0.862770i	$0.668727\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−7.26296e9	−0.899608	−0.449804	−	0.893127i	$-0.648506\pi$
$677$	−7.26296e9	−0.899608	−0.449804	+	0.893127i	$0.648506\pi$
$678$	0	0
$679$	2.37823e8	0.0291547
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.84973e8	0.0222145	0.0111073	−	0.999938i	$-0.496464\pi$
$683$	1.84973e8	0.0222145	0.0111073	+	0.999938i	$0.496464\pi$
$684$	0	0
$685$	−1.08036e10	−1.28425
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.11703e9	0.479531
$690$	0	0
$691$	−1.69205e10	−1.95092	−0.975462	−	0.220170i	$-0.929339\pi$
$691$	−1.69205e10	−1.95092	−0.975462	+	0.220170i	$0.929339\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.70213e10	1.92330
$696$	0	0
$697$	1.73825e9	0.194445
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.39267e10	−1.52699	−0.763494	−	0.645815i	$-0.776518\pi$
$701$	−1.39267e10	−1.52699	−0.763494	+	0.645815i	$0.776518\pi$
$702$	0	0
$703$	7.07523e9	0.768064
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.23865e9	−0.557509
$708$	0	0
$709$	1.04796e10	1.10429	0.552144	−	0.833749i	$-0.313810\pi$
$709$	1.04796e10	1.10429	0.552144	+	0.833749i	$0.313810\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9.43099e9	−0.974415
$714$	0	0
$715$	−9.09455e9	−0.930486
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.18930e10	1.19327	0.596637	−	0.802511i	$-0.296503\pi$
$719$	1.18930e10	1.19327	0.596637	+	0.802511i	$0.296503\pi$
$720$	0	0
$721$	2.06875e9	0.205558
$722$	0	0
$723$	0	0
$724$	0	0
$725$	5.98607e9	0.583390
$726$	0	0
$727$	7.07722e9	0.683113	0.341556	−	0.939861i	$-0.389046\pi$
$727$	7.07722e9	0.683113	0.341556	+	0.939861i	$0.389046\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−3.67597e9	−0.348066
$732$	0	0
$733$	−1.73914e10	−1.63106	−0.815532	−	0.578712i	$-0.803555\pi$
$733$	−1.73914e10	−1.63106	−0.815532	+	0.578712i	$0.803555\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.71338e9	−0.525723
$738$	0	0
$739$	−8.22127e8	−0.0749347	−0.0374674	−	0.999298i	$-0.511929\pi$
$739$	−8.22127e8	−0.0749347	−0.0374674	+	0.999298i	$0.511929\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.44720e9	−0.576647	−0.288324	−	0.957533i	$-0.593098\pi$
$743$	−6.44720e9	−0.576647	−0.288324	+	0.957533i	$0.593098\pi$
$744$	0	0
$745$	1.06384e10	0.942608
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.25660e9	0.196231
$750$	0	0
$751$	1.83855e10	1.58393	0.791966	−	0.610565i	$-0.209058\pi$
$751$	1.83855e10	1.58393	0.791966	+	0.610565i	$0.209058\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−2.19600e9	−0.185702
$756$	0	0
$757$	−7.10090e9	−0.594946	−0.297473	−	0.954730i	$-0.596144\pi$
$757$	−7.10090e9	−0.594946	−0.297473	+	0.954730i	$0.596144\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5.39249e9	−0.443551	−0.221775	−	0.975098i	$-0.571185\pi$
$761$	−5.39249e9	−0.443551	−0.221775	+	0.975098i	$0.571185\pi$
$762$	0	0
$763$	8.79869e9	0.717104
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.34352e9	0.747699
$768$	0	0
$769$	−7.24359e9	−0.574396	−0.287198	−	0.957871i	$-0.592724\pi$
$769$	−7.24359e9	−0.574396	−0.287198	+	0.957871i	$0.592724\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.67950e10	1.30783	0.653916	−	0.756568i	$-0.273125\pi$
$773$	1.67950e10	1.30783	0.653916	+	0.756568i	$0.273125\pi$
$774$	0	0
$775$	−1.48585e10	−1.14662
$776$	0	0
$777$	0	0
$778$	0	0
$779$	3.07190e9	0.232823
$780$	0	0
$781$	6.29826e9	0.473088
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.70255e10	1.25619
$786$	0	0
$787$	−8.09876e9	−0.592253	−0.296126	−	0.955149i	$-0.595695\pi$
$787$	−8.09876e9	−0.592253	−0.296126	+	0.955149i	$0.595695\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−7.63264e9	−0.548349
$792$	0	0
$793$	−1.37730e9	−0.0980785
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.34199e10	1.63863	0.819315	−	0.573344i	$-0.194354\pi$
$797$	2.34199e10	1.63863	0.819315	+	0.573344i	$0.194354\pi$
$798$	0	0
$799$	−5.66663e9	−0.393017
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.07647e9	0.550449
$804$	0	0
$805$	4.99937e9	0.337777
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.76599e10	−1.17265	−0.586326	−	0.810075i	$-0.699426\pi$
$809$	−1.76599e10	−1.17265	−0.586326	+	0.810075i	$0.699426\pi$
$810$	0	0
$811$	1.02969e10	0.677847	0.338924	−	0.940814i	$-0.389937\pi$
$811$	1.02969e10	0.677847	0.338924	+	0.940814i	$0.389937\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.01276e10	0.655320
$816$	0	0
$817$	−6.49633e9	−0.416764
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.55020e9	0.223899	0.111949	−	0.993714i	$-0.464291\pi$
$821$	3.55020e9	0.223899	0.111949	+	0.993714i	$0.464291\pi$
$822$	0	0
$823$	1.85370e10	1.15915	0.579574	−	0.814920i	$-0.303219\pi$
$823$	1.85370e10	1.15915	0.579574	+	0.814920i	$0.303219\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.04188e9	0.0640543	0.0320271	−	0.999487i	$-0.489804\pi$
$827$	1.04188e9	0.0640543	0.0320271	+	0.999487i	$0.489804\pi$
$828$	0	0
$829$	1.57283e10	0.958830	0.479415	−	0.877588i	$-0.340849\pi$
$829$	1.57283e10	0.958830	0.479415	+	0.877588i	$0.340849\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.89008e9	0.113298
$834$	0	0
$835$	3.06470e10	1.82173
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−2.09602e10	−1.22526	−0.612632	−	0.790368i	$-0.709889\pi$
$839$	−2.09602e10	−1.22526	−0.612632	+	0.790368i	$0.709889\pi$
$840$	0	0
$841$	−7.76450e9	−0.450119
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.29997e10	−0.741200
$846$	0	0
$847$	5.86377e8	0.0331578
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.72187e9	0.540750
$852$	0	0
$853$	2.15633e10	1.18958	0.594789	−	0.803882i	$-0.297236\pi$
$853$	2.15633e10	1.18958	0.594789	+	0.803882i	$0.297236\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.49287e9	−0.515187	−0.257593	−	0.966253i	$-0.582929\pi$
$857$	−9.49287e9	−0.515187	−0.257593	+	0.966253i	$0.582929\pi$
$858$	0	0
$859$	−3.57195e10	−1.92278	−0.961389	−	0.275193i	$-0.911258\pi$
$859$	−3.57195e10	−1.92278	−0.961389	+	0.275193i	$0.911258\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−3.36759e10	−1.78353	−0.891766	−	0.452496i	$-0.850534\pi$
$863$	−3.36759e10	−1.78353	−0.891766	+	0.452496i	$0.850534\pi$
$864$	0	0
$865$	4.93402e10	2.59205
$866$	0	0
$867$	0	0
$868$	0	0
$869$	3.13800e10	1.62212
$870$	0	0
$871$	6.56117e9	0.336448
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.13522e9	−0.107749
$876$	0	0
$877$	3.20092e10	1.60242	0.801210	−	0.598384i	$-0.204190\pi$
$877$	3.20092e10	1.60242	0.801210	+	0.598384i	$0.204190\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−2.52093e10	−1.24207	−0.621035	−	0.783783i	$-0.713287\pi$
$881$	−2.52093e10	−1.24207	−0.621035	+	0.783783i	$0.713287\pi$
$882$	0	0
$883$	−1.03766e9	−0.0507215	−0.0253608	−	0.999678i	$-0.508073\pi$
$883$	−1.03766e9	−0.0507215	−0.0253608	+	0.999678i	$0.508073\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.95294e10	−1.42076	−0.710381	−	0.703818i	$-0.751477\pi$
$887$	−2.95294e10	−1.42076	−0.710381	+	0.703818i	$0.751477\pi$
$888$	0	0
$889$	1.00811e10	0.481231
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.00143e10	−0.470587
$894$	0	0
$895$	−3.11523e10	−1.45248
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.35444e10	−1.08076
$900$	0	0
$901$	−1.25099e10	−0.569794
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.48174e10	−0.664513
$906$	0	0
$907$	3.81643e10	1.69837	0.849184	−	0.528097i	$-0.177094\pi$
$907$	3.81643e10	1.69837	0.849184	+	0.528097i	$0.177094\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.65595e10	1.60209	0.801044	−	0.598606i	$-0.204278\pi$
$911$	3.65595e10	1.60209	0.801044	+	0.598606i	$0.204278\pi$
$912$	0	0
$913$	3.64839e10	1.58655
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.78902e9	0.333572
$918$	0	0
$919$	−1.48793e9	−0.0632379	−0.0316189	−	0.999500i	$-0.510066\pi$
$919$	−1.48793e9	−0.0632379	−0.0316189	+	0.999500i	$0.510066\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7.23284e9	−0.302763
$924$	0	0
$925$	1.53168e10	0.636314
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.22706e10	−1.32054	−0.660271	−	0.751027i	$-0.729559\pi$
$929$	−3.22706e10	−1.32054	−0.660271	+	0.751027i	$0.729559\pi$
$930$	0	0
$931$	3.34023e9	0.135660
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.76345e10	1.10563
$936$	0	0
$937$	−3.58587e10	−1.42399	−0.711994	−	0.702185i	$-0.752208\pi$
$937$	−3.58587e10	−1.42399	−0.711994	+	0.702185i	$0.752208\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	3.56425e10	1.39446	0.697228	−	0.716849i	$-0.254416\pi$
$941$	3.56425e10	1.39446	0.697228	+	0.716849i	$0.254416\pi$
$942$	0	0
$943$	4.22101e9	0.163918
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.98005e10	0.757621	0.378810	−	0.925474i	$-0.376333\pi$
$947$	1.98005e10	0.757621	0.378810	+	0.925474i	$0.376333\pi$
$948$	0	0
$949$	−9.27492e9	−0.352272
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3.17008e10	1.18644	0.593219	−	0.805041i	$-0.297857\pi$
$953$	3.17008e10	1.18644	0.593219	+	0.805041i	$0.297857\pi$
$954$	0	0
$955$	4.52365e10	1.68065
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−9.91832e9	−0.363139
$960$	0	0
$961$	3.09287e10	1.12416
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.04266e10	1.08995
$966$	0	0
$967$	−3.87099e10	−1.37667	−0.688334	−	0.725394i	$-0.741657\pi$
$967$	−3.87099e10	−1.37667	−0.688334	+	0.725394i	$0.741657\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.92284e9	0.242671	0.121335	−	0.992612i	$-0.461282\pi$
$971$	6.92284e9	0.242671	0.121335	+	0.992612i	$0.461282\pi$
$972$	0	0
$973$	1.56265e10	0.543836
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.23668e10	0.424254	0.212127	−	0.977242i	$-0.431961\pi$
$977$	1.23668e10	0.424254	0.212127	+	0.977242i	$0.431961\pi$
$978$	0	0
$979$	1.15046e10	0.391860
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−5.38110e9	−0.180690	−0.0903449	−	0.995911i	$-0.528797\pi$
$983$	−5.38110e9	−0.180690	−0.0903449	+	0.995911i	$0.528797\pi$
$984$	0	0
$985$	−7.88107e10	−2.62760
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.92641e9	−0.293420
$990$	0	0
$991$	2.07275e10	0.676533	0.338267	−	0.941050i	$-0.390159\pi$
$991$	2.07275e10	0.676533	0.338267	+	0.941050i	$0.390159\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−2.25700e8	−0.00726357
$996$	0	0
$997$	3.85694e10	1.23257	0.616283	−	0.787525i	$-0.288638\pi$
$997$	3.85694e10	1.23257	0.616283	+	0.787525i	$0.288638\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.8.a.h.1.4	yes	4
3.2	odd	2	inner	252.8.a.h.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.8.a.h.1.1	✓	4	3.2	odd	2	inner
252.8.a.h.1.4	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	252.a (trivial)

\(f(q)\)	\(=\)	\(q+373.615 q^{5} +343.000 q^{7} +4603.99 q^{11} -5287.16 q^{13} +16065.4 q^{17} +28391.5 q^{19} +39011.9 q^{23} +61463.1 q^{25} +97392.9 q^{29} -241746. q^{31} +128150. q^{35} +249203. q^{37} +108198. q^{41} -228813. q^{43} -352722. q^{47} +117649. q^{49} -778685. q^{53} +1.72012e6 q^{55} -1.76721e6 q^{59} +260500. q^{61} -1.97536e6 q^{65} -1.24096e6 q^{67} +1.36800e6 q^{71} +1.75423e6 q^{73} +1.57917e6 q^{77} +6.81583e6 q^{79} +7.92440e6 q^{83} +6.00229e6 q^{85} +2.49883e6 q^{89} -1.81350e6 q^{91} +1.06075e7 q^{95} +693361. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 1372 q^{7} + 10808 q^{13} + 17696 q^{19} + 22156 q^{25} + 87584 q^{31} - 121672 q^{37} - 20464 q^{43} + 470596 q^{49} + 840672 q^{55} + 4844840 q^{61} + 6220976 q^{67} + 10659992 q^{73} + 4446272 q^{79}+ \cdots + 25590488 q^{97}+O(q^{100}) \) 4 * q + 1372 * q^7 + 10808 * q^13 + 17696 * q^19 + 22156 * q^25 + 87584 * q^31 - 121672 * q^37 - 20464 * q^43 + 470596 * q^49 + 840672 * q^55 + 4844840 * q^61 + 6220976 * q^67 + 10659992 * q^73 + 4446272 * q^79 + 14390208 * q^85 + 3707144 * q^91 + 25590488 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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