LMFDB - Embedded newform 252.12.k.e.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, minimal)

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:	no
Analytic conductor:	$193.622481501$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.2
Character	$\chi$	$=$	252.37
Dual form			252.12.k.e.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5383.85 + 9325.11i) q^{5} +(38734.1 - 21840.2i) q^{7} +O(q^{10})$	$q+(-5383.85 + 9325.11i) q^{5} +(38734.1 - 21840.2i) q^{7} +(179830. + 311475. i) q^{11} +489543. q^{13} +(-414037. - 717133. i) q^{17} +(-6.07057e6 + 1.05145e7i) q^{19} +(-1.43855e7 + 2.49163e7i) q^{23} +(-3.35577e7 - 5.81237e7i) q^{25} +2.49350e7 q^{29} +(-5.40289e7 - 9.35808e7i) q^{31} +(-4.87680e6 + 4.78784e8i) q^{35} +(-8.13360e7 + 1.40878e8i) q^{37} +7.65737e8 q^{41} -9.06743e8 q^{43} +(-4.37414e7 + 7.57624e7i) q^{47} +(1.02334e9 - 1.69192e9i) q^{49} +(2.32347e9 + 4.02436e9i) q^{53} -3.87271e9 q^{55} +(5.06339e9 + 8.77004e9i) q^{59} +(1.72799e9 - 2.99296e9i) q^{61} +(-2.63563e9 + 4.56504e9i) q^{65} +(9.39406e9 + 1.62710e10i) q^{67} -4.92529e9 q^{71} +(4.37328e9 + 7.57474e9i) q^{73} +(1.37682e10 + 8.13718e9i) q^{77} +(1.00709e10 - 1.74434e10i) q^{79} -1.15756e10 q^{83} +8.91646e9 q^{85} +(1.83426e10 - 3.17703e10i) q^{89} +(1.89620e10 - 1.06917e10i) q^{91} +(-6.53661e10 - 1.13217e11i) q^{95} -1.53637e11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 85666 q^{7}+O(q^{10}) $ 28 * q + 85666 * q^7	$ 28 q + 85666 q^{7} + 2794700 q^{13} + 37414826 q^{19} - 135469250 q^{25} + 71052098 q^{31} - 395898874 q^{37} + 980170268 q^{43} - 1253333270 q^{49} - 8844657360 q^{55} + 16692685100 q^{61} + 10934009870 q^{67} + 25578690650 q^{73} - 113974174090 q^{79} + 111789733440 q^{85} - 267181531078 q^{91} + 279658007384 q^{97}+O(q^{100}) $ 28 * q + 85666 * q^7 + 2794700 * q^13 + 37414826 * q^19 - 135469250 * q^25 + 71052098 * q^31 - 395898874 * q^37 + 980170268 * q^43 - 1253333270 * q^49 - 8844657360 * q^55 + 16692685100 * q^61 + 10934009870 * q^67 + 25578690650 * q^73 - 113974174090 * q^79 + 111789733440 * q^85 - 267181531078 * q^91 + 279658007384 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$5$	−5383.85	+	9325.11i	−0.770475	+	1.33450i	0.166829	+	0.985986i	$0.446647\pi$
$5$	−5383.85	+	9325.11i	−0.770475	+	1.33450i	−0.937303	+	0.348515i	$0.886686\pi$
$6$		0			0
$7$	38734.1	−	21840.2i	0.871073	−	0.491153i
$7$	38734.1	−	21840.2i	0.871073	−	0.491153i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	179830.	+	311475.i	0.336668	+	0.583127i	0.983804	−	0.179248i	$-0.0573665\pi$
$11$	179830.	+	311475.i	0.336668	+	0.583127i	−0.647135	+	0.762375i	$0.724033\pi$
$12$		0			0
$13$	489543.			0.365681			0.182840	−	0.983143i	$-0.441471\pi$
$13$	489543.			0.365681			0.182840	+	0.983143i	$0.441471\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−414037.	−	717133.i	−0.0707245	−	0.122498i	0.828495	−	0.559997i	$-0.189198\pi$
$17$	−414037.	−	717133.i	−0.0707245	−	0.122498i	−0.899219	+	0.437499i	$0.855864\pi$
$18$		0			0
$19$	−6.07057e6	+	1.05145e7i	−0.562451	+	0.974194i	0.434831	+	0.900512i	$0.356808\pi$
$19$	−6.07057e6	+	1.05145e7i	−0.562451	+	0.974194i	−0.997282	+	0.0736814i	$0.976525\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.43855e7	+	2.49163e7i	−0.466037	+	0.807200i	−0.999248	−	0.0387825i	$-0.987652\pi$
$23$	−1.43855e7	+	2.49163e7i	−0.466037	+	0.807200i	0.533210	+	0.845983i	$0.320985\pi$
$24$		0			0
$25$	−3.35577e7	−	5.81237e7i	−0.687262	−	1.19037i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.49350e7			0.225746			0.112873	−	0.993609i	$-0.463995\pi$
$29$	2.49350e7			0.225746			0.112873	+	0.993609i	$0.463995\pi$
$30$		0			0
$31$	−5.40289e7	−	9.35808e7i	−0.338951	−	0.587080i	0.645285	−	0.763942i	$-0.276739\pi$
$31$	−5.40289e7	−	9.35808e7i	−0.338951	−	0.587080i	−0.984236	+	0.176862i	$0.943405\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−4.87680e6	+	4.78784e8i	−0.0156950	+	1.54087i
$36$		0			0
$37$	−8.13360e7	+	1.40878e8i	−0.192829	+	0.333990i	−0.946187	−	0.323621i	$-0.895100\pi$
$37$	−8.13360e7	+	1.40878e8i	−0.192829	+	0.333990i	0.753357	+	0.657611i	$0.228433\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	7.65737e8			1.03221			0.516105	−	0.856525i	$-0.327381\pi$
$41$	7.65737e8			1.03221			0.516105	+	0.856525i	$0.327381\pi$
$42$		0			0
$43$	−9.06743e8			−0.940606			−0.470303	−	0.882505i	$-0.655855\pi$
$43$	−9.06743e8			−0.940606			−0.470303	+	0.882505i	$0.655855\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.37414e7	+	7.57624e7i	−0.0278198	+	0.0481854i	−0.879600	−	0.475714i	$-0.842190\pi$
$47$	−4.37414e7	+	7.57624e7i	−0.0278198	+	0.0481854i	0.851780	+	0.523899i	$0.175523\pi$
$48$		0			0
$49$	1.02334e9	−	1.69192e9i	0.517537	−	0.855661i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.32347e9	+	4.02436e9i	0.763167	+	1.32184i	0.941210	+	0.337821i	$0.109690\pi$
$53$	2.32347e9	+	4.02436e9i	0.763167	+	1.32184i	−0.178044	+	0.984023i	$0.556977\pi$
$54$		0			0
$55$	−3.87271e9			−1.03758
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.06339e9	+	8.77004e9i	0.922051	+	1.59704i	0.796236	+	0.604986i	$0.206821\pi$
$59$	5.06339e9	+	8.77004e9i	0.922051	+	1.59704i	0.125815	+	0.992054i	$0.459846\pi$
$60$		0			0
$61$	1.72799e9	−	2.99296e9i	0.261955	−	0.453720i	−0.704806	−	0.709400i	$-0.748966\pi$
$61$	1.72799e9	−	2.99296e9i	0.261955	−	0.453720i	0.966761	+	0.255680i	$0.0822994\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.63563e9	+	4.56504e9i	−0.281748	+	0.488002i
$66$		0			0
$67$	9.39406e9	+	1.62710e10i	0.850045	+	1.47232i	0.881167	+	0.472805i	$0.156759\pi$
$67$	9.39406e9	+	1.62710e10i	0.850045	+	1.47232i	−0.0311222	+	0.999516i	$0.509908\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.92529e9			−0.323975			−0.161987	−	0.986793i	$-0.551790\pi$
$71$	−4.92529e9			−0.323975			−0.161987	+	0.986793i	$0.551790\pi$
$72$		0			0
$73$	4.37328e9	+	7.57474e9i	0.246906	+	0.427653i	0.962666	−	0.270693i	$-0.0872528\pi$
$73$	4.37328e9	+	7.57474e9i	0.246906	+	0.427653i	−0.715760	+	0.698346i	$0.753919\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.37682e10	+	8.13718e9i	0.579667	+	0.342590i
$78$		0			0
$79$	1.00709e10	−	1.74434e10i	0.368232	−	0.637796i	−0.621057	−	0.783765i	$-0.713297\pi$
$79$	1.00709e10	−	1.74434e10i	0.368232	−	0.637796i	0.989289	+	0.145969i	$0.0466300\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.15756e10			−0.322564			−0.161282	−	0.986908i	$-0.551563\pi$
$83$	−1.15756e10			−0.322564			−0.161282	+	0.986908i	$0.551563\pi$
$84$		0			0
$85$	8.91646e9			0.217966
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.83426e10	−	3.17703e10i	0.348189	−	0.603081i	−0.637739	−	0.770253i	$-0.720130\pi$
$89$	1.83426e10	−	3.17703e10i	0.348189	−	0.603081i	0.985928	+	0.167172i	$0.0534634\pi$
$90$		0			0
$91$	1.89620e10	−	1.06917e10i	0.318535	−	0.179605i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−6.53661e10	−	1.13217e11i	−0.866708	−	1.50118i
$96$		0			0
$97$	−1.53637e11			−1.81656			−0.908282	−	0.418358i	$-0.862606\pi$
$97$	−1.53637e11			−1.81656			−0.908282	+	0.418358i	$0.862606\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.20614e10	−	5.55320e10i	−0.303539	−	0.525746i	0.673396	−	0.739282i	$-0.264835\pi$
$101$	−3.20614e10	−	5.55320e10i	−0.303539	−	0.525746i	−0.976935	+	0.213537i	$0.931502\pi$
$102$		0			0
$103$	−2.50380e9	+	4.33671e9i	−0.0212812	+	0.0368601i	−0.876470	−	0.481457i	$-0.840108\pi$
$103$	−2.50380e9	+	4.33671e9i	−0.0212812	+	0.0368601i	0.855189	+	0.518317i	$0.173441\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	4.88029e10	−	8.45290e10i	0.336383	−	0.582633i	−0.647366	−	0.762179i	$-0.724130\pi$
$107$	4.88029e10	−	8.45290e10i	0.336383	−	0.582633i	0.983750	+	0.179546i	$0.0574629\pi$
$108$		0			0
$109$	5.93686e10	+	1.02829e11i	0.369582	+	0.640135i	0.989500	−	0.144532i	$-0.0461675\pi$
$109$	5.93686e10	+	1.02829e11i	0.369582	+	0.640135i	−0.619918	+	0.784667i	$0.712834\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.11036e11			−1.07752			−0.538761	−	0.842459i	$-0.681107\pi$
$113$	−2.11036e11			−1.07752			−0.538761	+	0.842459i	$0.681107\pi$
$114$		0			0
$115$	−1.54898e11	−	2.68292e11i	−0.718140	−	1.24385i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.16997e10	−	1.87349e10i	−0.121772	−	0.0719685i
$120$		0			0
$121$	7.79782e10	−	1.35062e11i	0.273309	−	0.473385i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1.96913e11			0.577123
$126$		0			0
$127$	−3.16813e11			−0.850909			−0.425454	−	0.904980i	$-0.639886\pi$
$127$	−3.16813e11			−0.850909			−0.425454	+	0.904980i	$0.639886\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.15737e11	+	2.00462e11i	−0.262107	+	0.453983i	−0.966802	−	0.255528i	$-0.917751\pi$
$131$	−1.15737e11	+	2.00462e11i	−0.262107	+	0.453983i	0.704695	+	0.709511i	$0.251084\pi$
$132$		0			0
$133$	−5.49884e9	+	5.39854e11i	−0.0114574	+	1.12484i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.94160e11	+	5.09500e11i	0.520740	+	0.901948i	0.999709	+	0.0241160i	$0.00767712\pi$
$137$	2.94160e11	+	5.09500e11i	0.520740	+	0.901948i	−0.478969	+	0.877832i	$0.658990\pi$
$138$		0			0
$139$	6.03283e11			0.986143			0.493071	−	0.869989i	$-0.335874\pi$
$139$	6.03283e11			0.986143			0.493071	+	0.869989i	$0.335874\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.80345e10	+	1.52480e11i	0.123113	+	0.213238i
$144$		0			0
$145$	−1.34247e11	+	2.32522e11i	−0.173932	+	0.301259i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.98235e10	+	3.43352e10i	−0.0221134	+	0.0383015i	−0.876870	−	0.480727i	$-0.840373\pi$
$149$	−1.98235e10	+	3.43352e10i	−0.0221134	+	0.0383015i	0.854757	+	0.519028i	$0.173706\pi$
$150$		0			0
$151$	−3.01318e11	−	5.21899e11i	−0.312358	−	0.541020i	0.666514	−	0.745492i	$-0.267785\pi$
$151$	−3.01318e11	−	5.21899e11i	−0.312358	−	0.541020i	−0.978872	+	0.204472i	$0.934452\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.16354e12			1.04461
$156$		0			0
$157$	3.06382e11	+	5.30670e11i	0.256339	+	0.443993i	0.965258	−	0.261297i	$-0.0841502\pi$
$157$	3.06382e11	+	5.30670e11i	0.256339	+	0.443993i	−0.708919	+	0.705290i	$0.750817\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.30306e10	+	1.27929e12i	−0.00949343	+	0.932026i
$162$		0			0
$163$	−1.65200e11	+	2.86135e11i	−0.112455	+	0.194778i	−0.916760	−	0.399440i	$-0.869205\pi$
$163$	−1.65200e11	+	2.86135e11i	−0.112455	+	0.194778i	0.804305	+	0.594217i	$0.202538\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.41627e12			0.843732			0.421866	−	0.906658i	$-0.361375\pi$
$167$	1.41627e12			0.843732			0.421866	+	0.906658i	$0.361375\pi$
$168$		0			0
$169$	−1.55251e12			−0.866277
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.72032e12	+	2.97969e12i	−0.844027	+	1.46190i	0.0424367	+	0.999099i	$0.486488\pi$
$173$	−1.72032e12	+	2.97969e12i	−0.844027	+	1.46190i	−0.886464	+	0.462798i	$0.846845\pi$
$174$		0			0
$175$	−2.56926e12	−	1.51846e12i	−1.18331	−	0.699351i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.30006e12	−	2.25177e12i	−0.528775	−	0.915865i	−0.999437	−	0.0335517i	$-0.989318\pi$
$179$	−1.30006e12	−	2.25177e12i	−0.528775	−	0.915865i	0.470662	−	0.882314i	$-0.344015\pi$
$180$		0			0
$181$	−2.31752e12			−0.886729			−0.443364	−	0.896341i	$-0.646215\pi$
$181$	−2.31752e12			−0.886729			−0.443364	+	0.896341i	$0.646215\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−8.75803e11	−	1.51693e12i	−0.297140	−	0.514662i
$186$		0			0
$187$	1.48912e11	−	2.57924e11i	0.0476214	−	0.0824827i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.38360e12	−	4.12851e12i	0.678500	−	1.17520i	−0.296933	−	0.954898i	$-0.595964\pi$
$191$	2.38360e12	−	4.12851e12i	0.678500	−	1.17520i	0.975433	−	0.220297i	$-0.0707028\pi$
$192$		0			0
$193$	−9.72325e11	−	1.68412e12i	−0.261364	−	0.452696i	0.705240	−	0.708968i	$-0.250839\pi$
$193$	−9.72325e11	−	1.68412e12i	−0.261364	−	0.452696i	−0.966605	+	0.256272i	$0.917506\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.91400e12			−1.17997			−0.589984	−	0.807415i	$-0.700866\pi$
$197$	−4.91400e12			−1.17997			−0.589984	+	0.807415i	$0.700866\pi$
$198$		0			0
$199$	−2.01328e12	−	3.48711e12i	−0.457312	−	0.792088i	0.541506	−	0.840697i	$-0.317854\pi$
$199$	−2.01328e12	−	3.48711e12i	−0.457312	−	0.792088i	−0.998818	+	0.0486093i	$0.984521\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	9.65837e11	−	5.44586e11i	0.196642	−	0.110876i
$204$		0			0
$205$	−4.12262e12	+	7.14058e12i	−0.795292	+	1.37749i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.36668e12			−0.757438
$210$		0			0
$211$	−5.78478e12			−0.952212			−0.476106	−	0.879388i	$-0.657952\pi$
$211$	−5.78478e12			−0.952212			−0.476106	+	0.879388i	$0.657952\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.88177e12	−	8.45548e12i	0.724713	−	1.25524i
$216$		0			0
$217$	−4.13659e12	−	2.44477e12i	−0.583598	−	0.344913i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−2.02689e11	−	3.51067e11i	−0.0258626	−	0.0447953i
$222$		0			0
$223$	−9.45853e12			−1.14854			−0.574271	−	0.818665i	$-0.694714\pi$
$223$	−9.45853e12			−1.14854			−0.574271	+	0.818665i	$0.694714\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−3.18643e12	−	5.51906e12i	−0.350883	−	0.607747i	0.635522	−	0.772083i	$-0.280785\pi$
$227$	−3.18643e12	−	5.51906e12i	−0.350883	−	0.607747i	−0.986404	+	0.164336i	$0.947452\pi$
$228$		0			0
$229$	−4.44586e11	+	7.70046e11i	−0.0466510	+	0.0808019i	−0.888408	−	0.459055i	$-0.848188\pi$
$229$	−4.44586e11	+	7.70046e11i	−0.0466510	+	0.0808019i	0.841757	+	0.539857i	$0.181522\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.97860e11	−	1.20873e12i	0.0665749	−	0.115311i	−0.830817	−	0.556546i	$-0.812126\pi$
$233$	6.97860e11	−	1.20873e12i	0.0665749	−	0.115311i	0.897392	+	0.441235i	$0.145460\pi$
$234$		0			0
$235$	−4.70995e11	−	8.15787e11i	−0.0428690	−	0.0742512i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−4.05762e11			−0.0336576			−0.0168288	−	0.999858i	$-0.505357\pi$
$239$	−4.05762e11			−0.0336576			−0.0168288	+	0.999858i	$0.505357\pi$
$240$		0			0
$241$	−1.93769e12	−	3.35619e12i	−0.153529	−	0.265921i	0.778993	−	0.627032i	$-0.215731\pi$
$241$	−1.93769e12	−	3.35619e12i	−0.153529	−	0.265921i	−0.932523	+	0.361112i	$0.882397\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.02678e13	+	1.86518e13i	0.743132	+	1.34992i
$246$		0			0
$247$	−2.97180e12	+	5.14732e12i	−0.205678	+	0.356244i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.11528e12			−0.134018			−0.0670090	−	0.997752i	$-0.521346\pi$
$251$	−2.11528e12			−0.134018			−0.0670090	+	0.997752i	$0.521346\pi$
$252$		0			0
$253$	−1.03477e13			−0.627600
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.27368e13	+	2.20608e13i	−0.708644	+	1.22741i	0.256716	+	0.966487i	$0.417360\pi$
$257$	−1.27368e13	+	2.20608e13i	−0.708644	+	1.22741i	−0.965360	+	0.260921i	$0.915974\pi$
$258$		0			0
$259$	−7.36757e10	+	7.23318e12i	−0.00392804	+	0.385639i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.47350e13	+	2.55218e13i	0.722094	+	1.25070i	0.960159	+	0.279454i	$0.0901534\pi$
$263$	1.47350e13	+	2.55218e13i	0.722094	+	1.25070i	−0.238066	+	0.971249i	$0.576513\pi$
$264$		0			0
$265$	−5.00369e13			−2.35200
$266$		0			0
$267$		0			0
$268$		0			0
$269$	9.02720e12	+	1.56356e13i	0.390765	+	0.676824i	0.992551	−	0.121833i	$-0.0388773\pi$
$269$	9.02720e12	+	1.56356e13i	0.390765	+	0.676824i	−0.601786	+	0.798657i	$0.705544\pi$
$270$		0			0
$271$	−2.08410e13	+	3.60976e13i	−0.866137	+	1.50019i	−0.000223718		1.00000i	$0.500071\pi$
$271$	−2.08410e13	+	3.60976e13i	−0.866137	+	1.50019i	−0.865914	+	0.500194i	$0.833262\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.20694e13	−	2.09048e13i	0.462759	−	0.801522i
$276$		0			0
$277$	−2.64442e13	−	4.58027e13i	−0.974298	−	1.68753i	−0.682231	−	0.731137i	$-0.738990\pi$
$277$	−2.64442e13	−	4.58027e13i	−0.974298	−	1.68753i	−0.292068	−	0.956398i	$-0.594343\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	4.65083e11			0.0158360			0.00791800	−	0.999969i	$-0.497480\pi$
$281$	4.65083e11			0.0158360			0.00791800	+	0.999969i	$0.497480\pi$
$282$		0			0
$283$	−1.91255e13	−	3.31263e13i	−0.626307	−	1.08479i	−0.988287	−	0.152609i	$-0.951232\pi$
$283$	−1.91255e13	−	3.31263e13i	−0.626307	−	1.08479i	0.361980	−	0.932186i	$-0.382101\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.96602e13	−	1.67238e13i	0.899131	−	0.506974i
$288$		0			0
$289$	1.67931e13	−	2.90865e13i	0.489996	−	0.848698i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−4.88602e13			−1.32185			−0.660927	−	0.750450i	$-0.729837\pi$
$293$	−4.88602e13			−1.32185			−0.660927	+	0.750450i	$0.729837\pi$
$294$		0			0
$295$	−1.09042e14			−2.84167
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−7.04230e12	+	1.21976e13i	−0.170421	+	0.295178i
$300$		0			0
$301$	−3.51219e13	+	1.98034e13i	−0.819336	+	0.461982i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.86065e13	+	3.22274e13i	0.403659	+	0.699159i
$306$		0			0
$307$	−7.09086e12			−0.148401			−0.0742007	−	0.997243i	$-0.523641\pi$
$307$	−7.09086e12			−0.148401			−0.0742007	+	0.997243i	$0.523641\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.79649e13	−	6.57571e13i	−0.739946	−	1.28162i	−0.952519	−	0.304479i	$-0.901518\pi$
$311$	−3.79649e13	−	6.57571e13i	−0.739946	−	1.28162i	0.212573	−	0.977145i	$-0.431816\pi$
$312$		0			0
$313$	−2.21472e13	+	3.83600e13i	−0.416701	+	0.721747i	−0.995605	−	0.0936484i	$-0.970147\pi$
$313$	−2.21472e13	+	3.83600e13i	−0.416701	+	0.721747i	0.578905	+	0.815395i	$0.303480\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.17018e13	−	8.95501e13i	0.907151	−	1.57123i	0.0891478	−	0.996018i	$-0.471586\pi$
$317$	5.17018e13	−	8.95501e13i	0.907151	−	1.57123i	0.818003	−	0.575213i	$-0.195081\pi$
$318$		0			0
$319$	4.48407e12	+	7.76663e12i	0.0760017	+	0.131639i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.00538e13			0.159116
$324$		0			0
$325$	−1.64279e13	−	2.84540e13i	−0.251319	−	0.435297i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3.96218e10	+	3.88991e12i	−0.000566705	+	0.0556368i
$330$		0			0
$331$	−5.29818e13	+	9.17672e13i	−0.732948	+	1.26950i	0.222670	+	0.974894i	$0.428523\pi$
$331$	−5.29818e13	+	9.17672e13i	−0.732948	+	1.26950i	−0.955618	+	0.294609i	$0.904811\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.02305e14			−2.61975
$336$		0			0
$337$	5.01044e13			0.627931			0.313965	−	0.949434i	$-0.398342\pi$
$337$	5.01044e13			0.627931			0.313965	+	0.949434i	$0.398342\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.94320e13	−	3.36573e13i	0.228228	−	0.395303i
$342$		0			0
$343$	2.68628e12	−	8.78850e13i	0.0305516	−	0.999533i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.38207e13	+	1.10541e14i	0.681004	+	1.17953i	0.974675	+	0.223627i	$0.0717897\pi$
$347$	6.38207e13	+	1.10541e14i	0.681004	+	1.17953i	−0.293671	+	0.955907i	$0.594877\pi$
$348$		0			0
$349$	−1.23146e14			−1.27315			−0.636576	−	0.771214i	$-0.719650\pi$
$349$	−1.23146e14			−1.27315			−0.636576	+	0.771214i	$0.719650\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−6.77227e13	−	1.17299e14i	−0.657618	−	1.13903i	−0.981231	−	0.192838i	$-0.938231\pi$
$353$	−6.77227e13	−	1.17299e14i	−0.657618	−	1.13903i	0.323613	−	0.946190i	$-0.395102\pi$
$354$		0			0
$355$	2.65171e13	−	4.59289e13i	0.249614	−	0.432345i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	7.09961e13	−	1.22969e14i	0.628369	−	1.08837i	−0.359510	−	0.933141i	$-0.617056\pi$
$359$	7.09961e13	−	1.22969e14i	0.628369	−	1.08837i	0.987879	−	0.155226i	$-0.0496106\pi$
$360$		0			0
$361$	−1.54585e13	−	2.67749e13i	−0.132702	−	0.229847i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−9.41804e13			−0.760939
$366$		0			0
$367$	2.83576e13	+	4.91168e13i	0.222334	+	0.385094i	0.955516	−	0.294938i	$-0.0952991\pi$
$367$	2.83576e13	+	4.91168e13i	0.222334	+	0.385094i	−0.733182	+	0.680032i	$0.761966\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.77890e14	+	1.05135e14i	1.31400	+	0.776590i
$372$		0			0
$373$	−5.62946e13	+	9.75052e13i	−0.403709	+	0.699244i	−0.994170	−	0.107821i	$-0.965613\pi$
$373$	−5.62946e13	+	9.75052e13i	−0.403709	+	0.699244i	0.590461	+	0.807066i	$0.298946\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.22068e13			0.0825512
$378$		0			0
$379$	−6.79186e13			−0.446142			−0.223071	−	0.974802i	$-0.571608\pi$
$379$	−6.79186e13			−0.446142			−0.223071	+	0.974802i	$0.571608\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.59199e13	+	1.66138e14i	−0.594724	+	1.03009i	0.398862	+	0.917011i	$0.369405\pi$
$383$	−9.59199e13	+	1.66138e14i	−0.594724	+	1.03009i	−0.993586	+	0.113081i	$0.963928\pi$
$384$		0			0
$385$	−1.50006e14	+	8.45808e13i	−0.903806	+	0.509610i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7.53577e13	+	1.30523e14i	0.428948	+	0.742960i	0.996780	−	0.0801843i	$-0.0255509\pi$
$389$	7.53577e13	+	1.30523e14i	0.428948	+	0.742960i	−0.567832	+	0.823145i	$0.692218\pi$
$390$		0			0
$391$	2.38244e13			0.131841
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.08441e14	+	1.87825e14i	0.567426	+	0.982811i
$396$		0			0
$397$	9.08349e13	−	1.57331e14i	0.462280	−	0.800692i	−0.536794	−	0.843713i	$-0.680365\pi$
$397$	9.08349e13	−	1.57331e14i	0.462280	−	0.800692i	0.999074	+	0.0430209i	$0.0136982\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.35174e14	−	2.34128e14i	0.651026	−	1.12761i	−0.331848	−	0.943333i	$-0.607672\pi$
$401$	1.35174e14	−	2.34128e14i	0.651026	−	1.12761i	0.982874	−	0.184278i	$-0.0589947\pi$
$402$		0			0
$403$	−2.64495e13	−	4.58118e13i	−0.123948	−	0.214684i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−5.85066e13			−0.259678
$408$		0			0
$409$	−3.62499e13	−	6.27866e13i	−0.156613	−	0.271262i	0.777032	−	0.629461i	$-0.216724\pi$
$409$	−3.62499e13	−	6.27866e13i	−0.156613	−	0.271262i	−0.933645	+	0.358199i	$0.883391\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	3.87665e14	+	2.29115e14i	1.58757	+	0.938270i
$414$		0			0
$415$	6.23216e13	−	1.07944e14i	0.248527	−	0.430462i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−4.94126e14			−1.86922			−0.934610	−	0.355673i	$-0.884252\pi$
$419$	−4.94126e14			−1.86922			−0.934610	+	0.355673i	$0.884252\pi$
$420$		0			0
$421$	4.21323e13			0.155261			0.0776307	−	0.996982i	$-0.475264\pi$
$421$	4.21323e13			0.155261			0.0776307	+	0.996982i	$0.475264\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.77883e13	+	4.81307e13i	−0.0972126	+	0.168377i
$426$		0			0
$427$	1.56525e12	−	1.53669e14i	0.00533617	−	0.523883i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	1.84105e14	+	3.18880e14i	0.596267	+	1.03277i	0.993367	+	0.114990i	$0.0366835\pi$
$431$	1.84105e14	+	3.18880e14i	0.596267	+	1.03277i	−0.397099	+	0.917776i	$0.629983\pi$
$432$		0			0
$433$	3.26612e14			1.03122			0.515608	−	0.856825i	$-0.327566\pi$
$433$	3.26612e14			1.03122			0.515608	+	0.856825i	$0.327566\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.74656e14	−	3.02513e14i	−0.524246	−	0.908021i
$438$		0			0
$439$	−7.15319e13	+	1.23897e14i	−0.209385	+	0.362665i	−0.951521	−	0.307584i	$-0.900479\pi$
$439$	−7.15319e13	+	1.23897e14i	−0.209385	+	0.362665i	0.742136	+	0.670249i	$0.233813\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.91755e13	−	1.19815e14i	0.192633	−	0.333651i	−0.753489	−	0.657461i	$-0.771631\pi$
$443$	6.91755e13	−	1.19815e14i	0.192633	−	0.333651i	0.946122	+	0.323810i	$0.104964\pi$
$444$		0			0
$445$	1.97507e14	+	3.42093e14i	0.536542	+	0.929317i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.12469e14			0.290857			0.145429	−	0.989369i	$-0.453544\pi$
$449$	1.12469e14			0.290857			0.145429	+	0.989369i	$0.453544\pi$
$450$		0			0
$451$	1.37702e14	+	2.38508e14i	0.347513	+	0.601910i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.38740e12	+	2.34386e14i	−0.00573936	+	0.563467i
$456$		0			0
$457$	1.42643e14	−	2.47064e14i	0.334742	−	0.579790i	−0.648693	−	0.761050i	$-0.724684\pi$
$457$	1.42643e14	−	2.47064e14i	0.334742	−	0.579790i	0.983435	+	0.181260i	$0.0580175\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.54583e14			−0.569475			−0.284737	−	0.958606i	$-0.591906\pi$
$461$	−2.54583e14			−0.569475			−0.284737	+	0.958606i	$0.591906\pi$
$462$		0			0
$463$	3.60258e14			0.786897			0.393449	−	0.919347i	$-0.371282\pi$
$463$	3.60258e14			0.786897			0.393449	+	0.919347i	$0.371282\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−6.46196e13	+	1.11924e14i	−0.134624	+	0.233175i	−0.925454	−	0.378861i	$-0.876316\pi$
$467$	−6.46196e13	+	1.11924e14i	−0.134624	+	0.233175i	0.790830	+	0.612036i	$0.209649\pi$
$468$		0			0
$469$	7.19232e14	+	4.25075e14i	1.46359	+	0.864997i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1.63060e14	−	2.82427e14i	−0.316672	−	0.548492i
$474$		0			0
$475$	8.14858e14			1.54621
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5.40574e14	−	9.36302e14i	−0.979512	−	1.69656i	−0.664161	−	0.747590i	$-0.731211\pi$
$479$	−5.40574e14	−	9.36302e14i	−0.979512	−	1.69656i	−0.315351	−	0.948975i	$-0.602122\pi$
$480$		0			0
$481$	−3.98175e13	+	6.89659e13i	−0.0705141	+	0.122134i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.27159e14	−	1.43268e15i	1.39962	−	2.42421i
$486$		0			0
$487$	1.02018e14	+	1.76701e14i	0.168760	+	0.292301i	0.937984	−	0.346678i	$-0.112690\pi$
$487$	1.02018e14	+	1.76701e14i	0.168760	+	0.292301i	−0.769224	+	0.638979i	$0.779357\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.05067e14			−1.27316			−0.636582	−	0.771209i	$-0.719652\pi$
$491$	−8.05067e14			−1.27316			−0.636582	+	0.771209i	$0.719652\pi$
$492$		0			0
$493$	−1.03240e13	−	1.78817e13i	−0.0159658	−	0.0276536i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.90777e14	+	1.07569e14i	−0.282206	+	0.159121i
$498$		0			0
$499$	5.44983e14	−	9.43939e14i	0.788552	−	1.36581i	−0.138303	−	0.990390i	$-0.544165\pi$
$499$	5.44983e14	−	9.43939e14i	0.788552	−	1.36581i	0.926854	−	0.375421i	$-0.122502\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	7.08875e14			0.981625			0.490813	−	0.871265i	$-0.336700\pi$
$503$	7.08875e14			0.981625			0.490813	+	0.871265i	$0.336700\pi$
$504$		0			0
$505$	6.90456e14			0.935477
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−3.85428e14	+	6.67581e14i	−0.500029	+	0.866076i	0.499971	+	0.866042i	$0.333344\pi$
$509$	−3.85428e14	+	6.67581e14i	−0.500029	+	0.866076i	−1.00000		3.38110e-5i	$0.999989\pi$
$510$		0			0
$511$	3.34829e14	+	1.97888e14i	0.425117	+	0.251249i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.69602e13	−	4.66965e13i	−0.0327932	−	0.0567995i
$516$		0			0
$517$	−3.14641e13			−0.0374642
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.86461e14	+	6.69371e14i	0.441061	+	0.763940i	0.997768	−	0.0667686i	$-0.0212689\pi$
$521$	3.86461e14	+	6.69371e14i	0.441061	+	0.763940i	−0.556708	+	0.830709i	$0.687936\pi$
$522$		0			0
$523$	1.50306e14	−	2.60337e14i	0.167964	−	0.290922i	−0.769740	−	0.638358i	$-0.779614\pi$
$523$	1.50306e14	−	2.60337e14i	0.167964	−	0.290922i	0.937704	+	0.347436i	$0.112947\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4.47399e13	+	7.74918e13i	−0.0479443	+	0.0830419i
$528$		0			0
$529$	6.25220e13	+	1.08291e14i	0.0656186	+	0.113655i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.74861e14			0.377460
$534$		0			0
$535$	5.25495e14	+	9.10184e14i	0.518350	+	0.897808i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	7.11018e14	+	1.44861e13i	0.673197	+	0.0137155i
$540$		0			0
$541$	−1.27191e14	+	2.20301e14i	−0.117997	+	0.204377i	−0.918974	−	0.394319i	$-0.870981\pi$
$541$	−1.27191e14	+	2.20301e14i	−0.117997	+	0.204377i	0.800977	+	0.598695i	$0.204314\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.27853e15			−1.13901
$546$		0			0
$547$	1.33532e15			1.16589			0.582944	−	0.812513i	$-0.301901\pi$
$547$	1.33532e15			1.16589			0.582944	+	0.812513i	$0.301901\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1.51370e14	+	2.62180e14i	−0.126971	+	0.219921i
$552$		0			0
$553$	9.12246e12	−	8.95606e14i	0.00750108	−	0.736425i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	8.19711e14	+	1.41978e15i	0.647824	+	1.12206i	0.983641	+	0.180138i	$0.0576543\pi$
$557$	8.19711e14	+	1.41978e15i	0.647824	+	1.12206i	−0.335817	+	0.941927i	$0.609012\pi$
$558$		0			0
$559$	−4.43889e14			−0.343962
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.76461e14	+	3.05639e14i	0.131478	+	0.227726i	0.924246	−	0.381797i	$-0.124695\pi$
$563$	1.76461e14	+	3.05639e14i	0.131478	+	0.227726i	−0.792769	+	0.609523i	$0.791361\pi$
$564$		0			0
$565$	1.13619e15	−	1.96794e15i	0.830203	−	1.43795i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.26033e15	+	2.18296e15i	−0.885866	+	1.53436i	−0.0411474	+	0.999153i	$0.513101\pi$
$569$	−1.26033e15	+	2.18296e15i	−0.885866	+	1.53436i	−0.844718	+	0.535211i	$0.820232\pi$
$570$		0			0
$571$	6.28637e14	+	1.08883e15i	0.433412	+	0.750692i	0.997165	−	0.0752520i	$-0.0239761\pi$
$571$	6.28637e14	+	1.08883e15i	0.433412	+	0.750692i	−0.563752	+	0.825944i	$0.690643\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.93097e15			1.28116
$576$		0			0
$577$	−1.61312e14	−	2.79401e14i	−0.105003	−	0.181870i	0.808737	−	0.588171i	$-0.200152\pi$
$577$	−1.61312e14	−	2.79401e14i	−0.105003	−	0.181870i	−0.913739	+	0.406301i	$0.866818\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.48373e14	+	2.52814e14i	−0.280977	+	0.158428i
$582$		0			0
$583$	−8.35658e14	+	1.44740e15i	−0.513868	+	0.890046i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.85412e15			1.09807			0.549033	−	0.835800i	$-0.314996\pi$
$587$	1.85412e15			1.09807			0.549033	+	0.835800i	$0.314996\pi$
$588$		0			0
$589$	1.31194e15			0.762573
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5.53596e14	−	9.58856e14i	0.310022	−	0.536973i	−0.668345	−	0.743851i	$-0.732997\pi$
$593$	5.53596e14	−	9.58856e14i	0.310022	−	0.536973i	0.978367	+	0.206878i	$0.0663303\pi$
$594$		0			0
$595$	3.45371e14	−	1.94737e14i	0.189864	−	0.107055i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.52112e14	−	1.64911e15i	−0.504476	−	0.873778i	−0.999987	−	0.00517614i	$-0.998352\pi$
$599$	−9.52112e14	−	1.64911e15i	−0.504476	−	0.873778i	0.495511	−	0.868602i	$-0.334981\pi$
$600$		0			0
$601$	2.04900e15			1.06594			0.532969	−	0.846135i	$-0.321076\pi$
$601$	2.04900e15			1.06594			0.532969	+	0.846135i	$0.321076\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.39647e14	+	1.45431e15i	0.421155	+	0.729462i
$606$		0			0
$607$	−6.85369e14	+	1.18709e15i	−0.337588	+	0.584719i	−0.983978	−	0.178288i	$-0.942944\pi$
$607$	−6.85369e14	+	1.18709e15i	−0.337588	+	0.584719i	0.646391	+	0.763006i	$0.276278\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2.14133e13	+	3.70889e13i	−0.0101732	+	0.0176205i
$612$		0			0
$613$	−5.46947e14	−	9.47339e14i	−0.255219	−	0.442052i	0.709736	−	0.704468i	$-0.248814\pi$
$613$	−5.46947e14	−	9.47339e14i	−0.255219	−	0.442052i	−0.964955	+	0.262416i	$0.915481\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.75679e15			−1.24118			−0.620590	−	0.784135i	$-0.713107\pi$
$617$	−2.75679e15			−1.24118			−0.620590	+	0.784135i	$0.713107\pi$
$618$		0			0
$619$	1.56328e15	+	2.70768e15i	0.691414	+	1.19756i	0.971375	+	0.237553i	$0.0763453\pi$
$619$	1.56328e15	+	2.70768e15i	0.691414	+	1.19756i	−0.279960	+	0.960012i	$0.590321\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.66151e13	−	1.63120e15i	0.00709280	−	0.696342i
$624$		0			0
$625$	5.78412e14	−	1.00184e15i	0.242603	−	0.420202i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	1.34704e14			0.0545511
$630$		0			0
$631$	−1.20556e14			−0.0479764			−0.0239882	−	0.999712i	$-0.507636\pi$
$631$	−1.20556e14			−0.0479764			−0.0239882	+	0.999712i	$0.507636\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1.70568e15	−	2.95432e15i	0.655604	−	1.13554i
$636$		0			0
$637$	5.00968e14	−	8.28268e14i	0.189253	−	0.312899i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8.55694e14	−	1.48211e15i	−0.312320	−	0.540953i	0.666544	−	0.745465i	$-0.267773\pi$
$641$	−8.55694e14	−	1.48211e15i	−0.312320	−	0.540953i	−0.978864	+	0.204512i	$0.934439\pi$
$642$		0			0
$643$	2.81163e15			1.00878			0.504391	−	0.863475i	$-0.331717\pi$
$643$	2.81163e15			1.00878			0.504391	+	0.863475i	$0.331717\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−4.77969e14	−	8.27867e14i	−0.165740	−	0.287070i	0.771178	−	0.636620i	$-0.219668\pi$
$647$	−4.77969e14	−	8.27867e14i	−0.165740	−	0.287070i	−0.936918	+	0.349550i	$0.886334\pi$
$648$		0			0
$649$	−1.82110e15	+	3.15423e15i	−0.620851	+	1.07535i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−1.27312e15	+	2.20511e15i	−0.419612	+	0.726789i	−0.995900	−	0.0904571i	$-0.971167\pi$
$653$	−1.27312e15	+	2.20511e15i	−0.419612	+	0.726789i	0.576288	+	0.817246i	$0.304501\pi$
$654$		0			0
$655$	−1.24622e15	−	2.15851e15i	−0.403894	−	0.699564i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−1.35021e15			−0.423186			−0.211593	−	0.977358i	$-0.567865\pi$
$659$	−1.35021e15			−0.423186			−0.211593	+	0.977358i	$0.567865\pi$
$660$		0			0
$661$	−2.06955e15	−	3.58457e15i	−0.637924	−	1.10492i	−0.985888	−	0.167408i	$-0.946460\pi$
$661$	−2.06955e15	−	3.58457e15i	−0.637924	−	1.10492i	0.347964	−	0.937508i	$-0.386873\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.00459e15	−	2.95777e15i	−1.49228	−	0.881953i
$666$		0			0
$667$	−3.58702e14	+	6.21290e14i	−0.105206	+	0.182223i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.24298e15			0.352768
$672$		0			0
$673$	−6.85518e15			−1.91397			−0.956987	−	0.290132i	$-0.906301\pi$
$673$	−6.85518e15			−1.91397			−0.956987	+	0.290132i	$0.906301\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.44020e15	−	2.49449e15i	0.389210	−	0.674132i	−0.603133	−	0.797640i	$-0.706081\pi$
$677$	1.44020e15	−	2.49449e15i	0.389210	−	0.674132i	0.992344	+	0.123508i	$0.0394146\pi$
$678$		0			0
$679$	−5.95099e15	+	3.35546e15i	−1.58236	+	0.892212i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.57125e15	−	2.72148e15i	−0.404511	−	0.700634i	0.589753	−	0.807584i	$-0.299225\pi$
$683$	−1.57125e15	−	2.72148e15i	−0.404511	−	0.700634i	−0.994264	+	0.106949i	$0.965892\pi$
$684$		0			0
$685$	−6.33486e15			−1.60487
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.13744e15	+	1.97010e15i	0.279075	+	0.483373i
$690$		0			0
$691$	2.70946e15	−	4.69293e15i	0.654265	−	1.13322i	−0.327812	−	0.944743i	$-0.606311\pi$
$691$	2.70946e15	−	4.69293e15i	0.654265	−	1.13322i	0.982077	−	0.188478i	$-0.0603554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.24799e15	+	5.62568e15i	−0.759798	+	1.31601i
$696$		0			0
$697$	−3.17043e14	−	5.49135e14i	−0.0730026	−	0.126444i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.18493e15			1.82627			0.913137	−	0.407652i	$-0.133652\pi$
$701$	8.18493e15			1.82627			0.913137	+	0.407652i	$0.133652\pi$
$702$		0			0
$703$	−9.87512e14	−	1.71042e15i	−0.216914	−	0.375706i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.45470e15	−	1.45076e15i	−0.522627	−	0.308878i
$708$		0			0
$709$	1.23813e15	−	2.14451e15i	0.259546	−	0.449546i	−0.706575	−	0.707638i	$-0.749761\pi$
$709$	1.23813e15	−	2.14451e15i	0.259546	−	0.449546i	0.966120	+	0.258092i	$0.0830939\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	3.10892e15			0.631855
$714$		0			0
$715$	−1.89586e15			−0.379422
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.37037e14	+	1.62300e15i	−0.181864	+	0.314998i	−0.942515	−	0.334163i	$-0.891547\pi$
$719$	−9.37037e14	+	1.62300e15i	−0.181864	+	0.314998i	0.760651	+	0.649161i	$0.224880\pi$
$720$		0			0
$721$	−2.26799e12	+	2.22662e14i	−0.000433509	+	0.0425601i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.36763e14	−	1.44932e15i	−0.155147	−	0.268722i
$726$		0			0
$727$	−7.59096e15			−1.38630			−0.693150	−	0.720793i	$-0.743778\pi$
$727$	−7.59096e15			−1.38630			−0.693150	+	0.720793i	$0.743778\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.75425e14	+	6.50255e14i	0.0665239	+	0.115223i
$732$		0			0
$733$	−2.81259e15	+	4.87156e15i	−0.490948	+	0.850347i	−0.999946	−	0.0104212i	$-0.996683\pi$
$733$	−2.81259e15	+	4.87156e15i	−0.490948	+	0.850347i	0.508998	+	0.860768i	$0.330016\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−3.37867e15	+	5.85202e15i	−0.572366	+	0.991368i
$738$		0			0
$739$	−1.00738e14	−	1.74483e14i	−0.0168131	−	0.0291211i	0.857496	−	0.514490i	$-0.172019\pi$
$739$	−1.00738e14	−	1.74483e14i	−0.0168131	−	0.0291211i	−0.874310	+	0.485369i	$0.838685\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.97146e15			0.967480			0.483740	−	0.875212i	$-0.339278\pi$
$743$	5.97146e15			0.967480			0.483740	+	0.875212i	$0.339278\pi$
$744$		0			0
$745$	−2.13453e14	−	3.69712e14i	−0.0340756	−	0.0590206i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	4.42066e13	−	4.34002e15i	0.00685231	−	0.672732i
$750$		0			0
$751$	−1.77227e15	+	3.06966e15i	−0.270714	+	0.468890i	−0.969045	−	0.246885i	$-0.920593\pi$
$751$	−1.77227e15	+	3.06966e15i	−0.270714	+	0.468890i	0.698331	+	0.715775i	$0.253926\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.48902e15			0.962655
$756$		0			0
$757$	−4.36540e15			−0.638259			−0.319129	−	0.947711i	$-0.603390\pi$
$757$	−4.36540e15			−0.638259			−0.319129	+	0.947711i	$0.603390\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.14591e15	+	1.98477e15i	−0.162755	+	0.281899i	−0.935856	−	0.352384i	$-0.885371\pi$
$761$	−1.14591e15	+	1.98477e15i	−0.162755	+	0.281899i	0.773101	+	0.634283i	$0.218705\pi$
$762$		0			0
$763$	4.54540e15	+	2.68639e15i	0.636337	+	0.376083i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.47874e15	+	4.29331e15i	0.337177	+	0.584007i
$768$		0			0
$769$	−6.77356e15			−0.908285			−0.454142	−	0.890929i	$-0.650054\pi$
$769$	−6.77356e15			−0.908285			−0.454142	+	0.890929i	$0.650054\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.31500e15	+	1.09379e16i	0.822975	+	1.42543i	0.903457	+	0.428678i	$0.141020\pi$
$773$	6.31500e15	+	1.09379e16i	0.822975	+	1.42543i	−0.0804827	+	0.996756i	$0.525646\pi$
$774$		0			0
$775$	−3.62617e15	+	6.28072e15i	−0.465896	+	0.806956i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4.64846e15	+	8.05137e15i	−0.580568	+	1.00557i
$780$		0			0
$781$	−8.85715e14	−	1.53410e15i	−0.109072	−	0.188918i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.59807e15			−0.790012
$786$		0			0
$787$	−5.50497e15	−	9.53489e15i	−0.649971	−	1.12578i	−0.983129	−	0.182912i	$-0.941448\pi$
$787$	−5.50497e15	−	9.53489e15i	−0.649971	−	1.12578i	0.333158	−	0.942871i	$-0.391886\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−8.17431e15	+	4.60907e15i	−0.938600	+	0.529228i
$792$		0			0
$793$	8.45925e14	−	1.46518e15i	0.0957920	−	0.165917i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1.65215e16			1.81982			0.909909	−	0.414808i	$-0.136151\pi$
$797$	1.65215e16			1.81982			0.909909	+	0.414808i	$0.136151\pi$
$798$		0			0
$799$	7.24422e13			0.00787018
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1.57289e15	+	2.72433e15i	−0.166251	+	0.287955i
$804$		0			0
$805$	−1.18594e16	−	7.00905e15i	−1.23648	−	0.730771i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−9.65540e15	−	1.67236e16i	−0.979610	−	1.69673i	−0.663797	−	0.747913i	$-0.731056\pi$
$809$	−9.65540e15	−	1.67236e16i	−0.979610	−	1.69673i	−0.315813	−	0.948821i	$-0.602277\pi$
$810$		0			0
$811$	−7.65147e15			−0.765826			−0.382913	−	0.923784i	$-0.625079\pi$
$811$	−7.65147e15			−0.765826			−0.382913	+	0.923784i	$0.625079\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.77883e15	−	3.08102e15i	−0.173287	−	0.300142i
$816$		0			0
$817$	5.50444e15	−	9.53398e15i	0.529044	−	0.916332i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	6.56134e15	−	1.13646e16i	0.613911	−	1.06332i	−0.376664	−	0.926350i	$-0.622929\pi$
$821$	6.56134e15	−	1.13646e16i	0.613911	−	1.06332i	0.990575	−	0.136975i	$-0.0437379\pi$
$822$		0			0
$823$	6.74492e15	+	1.16825e16i	0.622698	+	1.07854i	0.988981	+	0.148041i	$0.0472968\pi$
$823$	6.74492e15	+	1.16825e16i	0.622698	+	1.07854i	−0.366283	+	0.930503i	$0.619370\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.39787e16			−1.25657			−0.628285	−	0.777983i	$-0.716243\pi$
$827$	−1.39787e16			−1.25657			−0.628285	+	0.777983i	$0.716243\pi$
$828$		0			0
$829$	5.38754e15	+	9.33149e15i	0.477904	+	0.827754i	0.999679	−	0.0253293i	$-0.00806343\pi$
$829$	5.38754e15	+	9.33149e15i	0.477904	+	0.827754i	−0.521775	+	0.853083i	$0.674730\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.63703e15	−	3.33524e13i	−0.141420	−	0.00288124i
$834$		0			0
$835$	−7.62498e15	+	1.32068e16i	−0.650074	+	1.12596i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	2.46498e15			0.204702			0.102351	−	0.994748i	$-0.467363\pi$
$839$	2.46498e15			0.204702			0.102351	+	0.994748i	$0.467363\pi$
$840$		0			0
$841$	−1.15788e16			−0.949039
$842$		0			0
$843$		0			0
$844$		0			0
$845$	8.35848e15	−	1.44773e16i	0.667445	−	1.15605i
$846$		0			0
$847$	7.06342e13	−	6.93457e15i	0.00556745	−	0.546589i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2.34011e15	−	4.05319e15i	−0.179731	−	0.311304i
$852$		0			0
$853$	−1.09410e16			−0.829540			−0.414770	−	0.909926i	$-0.636138\pi$
$853$	−1.09410e16			−0.829540			−0.414770	+	0.909926i	$0.636138\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7.30685e15	−	1.26558e16i	−0.539928	−	0.935182i	−0.998907	−	0.0467355i	$-0.985118\pi$
$857$	−7.30685e15	−	1.26558e16i	−0.539928	−	0.935182i	0.458980	−	0.888447i	$-0.348215\pi$
$858$		0			0
$859$	−3.67285e15	+	6.36156e15i	−0.267942	+	0.464089i	−0.968330	−	0.249673i	$-0.919677\pi$
$859$	−3.67285e15	+	6.36156e15i	−0.267942	+	0.464089i	0.700388	+	0.713762i	$0.253010\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−8.26553e15	+	1.43163e16i	−0.587775	+	1.01806i	0.406748	+	0.913541i	$0.366663\pi$
$863$	−8.26553e15	+	1.43163e16i	−0.587775	+	1.01806i	−0.994523	+	0.104516i	$0.966671\pi$
$864$		0			0
$865$	−1.85239e16	−	3.20844e16i	−1.30060	−	2.25271i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.24423e15			0.495888
$870$		0			0
$871$	4.59879e15	+	7.96535e15i	0.310845	+	0.538400i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	7.62724e15	−	4.30061e15i	0.502716	−	0.283456i
$876$		0			0
$877$	8.18007e15	−	1.41683e16i	0.532426	−	0.922189i	−0.466857	−	0.884333i	$-0.654614\pi$
$877$	8.18007e15	−	1.41683e16i	0.532426	−	0.922189i	0.999283	−	0.0378561i	$-0.0120529\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.23968e16			−1.42174			−0.710868	−	0.703325i	$-0.751698\pi$
$881$	−2.23968e16			−1.42174			−0.710868	+	0.703325i	$0.751698\pi$
$882$		0			0
$883$	9.29442e15			0.582691			0.291346	−	0.956618i	$-0.405897\pi$
$883$	9.29442e15			0.582691			0.291346	+	0.956618i	$0.405897\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.51454e15	−	9.55146e15i	0.337232	−	0.584104i	−0.646679	−	0.762763i	$-0.723843\pi$
$887$	5.51454e15	−	9.55146e15i	0.337232	−	0.584104i	0.983911	+	0.178659i	$0.0571758\pi$
$888$		0			0
$889$	−1.22715e16	+	6.91926e15i	−0.741204	+	0.417927i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−5.31071e14	−	9.19841e14i	−0.0312946	−	0.0542038i
$894$		0			0
$895$	2.79973e16			1.62963
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.34721e15	−	2.33344e15i	−0.0765170	−	0.132531i
$900$		0			0
$901$	1.92400e15	−	3.33247e15i	0.107949	−	0.186973i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.24772e16	−	2.16111e16i	0.683202	−	1.18334i
$906$		0			0
$907$	−1.47756e15	−	2.55921e15i	−0.0799291	−	0.138441i	0.823290	−	0.567621i	$-0.192136\pi$
$907$	−1.47756e15	−	2.55921e15i	−0.0799291	−	0.138441i	−0.903219	+	0.429180i	$0.858803\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	9.86795e15			0.521046			0.260523	−	0.965468i	$-0.416105\pi$
$911$	9.86795e15			0.521046			0.260523	+	0.965468i	$0.416105\pi$
$912$		0			0
$913$	−2.08165e15	−	3.60552e15i	−0.108597	−	0.188096i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1.04837e14	+	1.02924e16i	−0.00533926	+	0.524187i
$918$		0			0
$919$	−5.59118e14	+	9.68421e14i	−0.0281364	+	0.0487337i	−0.879751	−	0.475435i	$-0.842291\pi$
$919$	−5.59118e14	+	9.68421e14i	−0.0281364	+	0.0487337i	0.851614	+	0.524169i	$0.175624\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.41114e15			−0.118471
$924$		0			0
$925$	1.09178e16			0.530098
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.92369e16	+	3.33193e16i	−0.912113	+	1.57983i	−0.101039	+	0.994882i	$0.532217\pi$
$929$	−1.92369e16	+	3.33193e16i	−0.912113	+	1.57983i	−0.811074	+	0.584943i	$0.801117\pi$
$930$		0			0
$931$	1.15775e16	+	2.10309e16i	0.542490	+	0.985448i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.60345e15	+	2.77725e15i	0.0733822	+	0.127102i
$936$		0			0
$937$	1.99216e16			0.901067			0.450534	−	0.892759i	$-0.351234\pi$
$937$	1.99216e16			0.901067			0.450534	+	0.892759i	$0.351234\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.74940e16	+	3.03004e16i	0.772939	+	1.33877i	0.935946	+	0.352144i	$0.114547\pi$
$941$	1.74940e16	+	3.03004e16i	0.772939	+	1.33877i	−0.163007	+	0.986625i	$0.552119\pi$
$942$		0			0
$943$	−1.10155e16	+	1.90794e16i	−0.481049	+	0.833201i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	3.99337e14	−	6.91672e14i	0.0170379	−	0.0295104i	−0.857381	−	0.514683i	$-0.827910\pi$
$947$	3.99337e14	−	6.91672e14i	0.0170379	−	0.0295104i	0.874419	+	0.485172i	$0.161243\pi$
$948$		0			0
$949$	2.14091e15	+	3.70816e15i	0.0902888	+	0.156385i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.15730e16			−1.30108			−0.650542	−	0.759470i	$-0.725458\pi$
$953$	−3.15730e16			−1.30108			−0.650542	+	0.759470i	$0.725458\pi$
$954$		0			0
$955$	2.56659e16	+	4.44546e16i	1.04553	+	1.81092i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2.25216e16	+	1.33105e16i	0.896597	+	0.529899i
$960$		0			0
$961$	6.86599e15	−	1.18922e16i	0.270225	−	0.468043i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	2.09394e16			0.805498
$966$		0			0
$967$	3.61237e16			1.37387			0.686937	−	0.726717i	$-0.258955\pi$
$967$	3.61237e16			1.37387			0.686937	+	0.726717i	$0.258955\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−9.81353e15	+	1.69975e16i	−0.364854	+	0.631946i	−0.988753	−	0.149559i	$-0.952215\pi$
$971$	−9.81353e15	+	1.69975e16i	−0.364854	+	0.631946i	0.623899	+	0.781505i	$0.285548\pi$
$972$		0			0
$973$	2.33677e16	−	1.31758e16i	0.859003	−	0.484347i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	2.68031e16	+	4.64243e16i	0.963307	+	1.66850i	0.714097	+	0.700047i	$0.246838\pi$
$977$	2.68031e16	+	4.64243e16i	0.963307	+	1.66850i	0.249210	+	0.968450i	$0.419829\pi$
$978$		0			0
$979$	1.31942e16			0.468897
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.16040e16	−	2.00987e16i	−0.403240	−	0.698431i	0.590875	−	0.806763i	$-0.298783\pi$
$983$	−1.16040e16	−	2.00987e16i	−0.403240	−	0.698431i	−0.994115	+	0.108332i	$0.965449\pi$
$984$		0			0
$985$	2.64562e16	−	4.58236e16i	0.909136	−	1.57467i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.30439e16	−	2.25927e16i	0.438357	−	0.759257i
$990$		0			0
$991$	1.65685e16	+	2.86974e16i	0.550652	+	0.953757i	0.998228	+	0.0595110i	$0.0189541\pi$
$991$	1.65685e16	+	2.86974e16i	0.550652	+	0.953757i	−0.447576	+	0.894246i	$0.647713\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.33569e16			1.40939
$996$		0			0
$997$	−2.29623e16	−	3.97719e16i	−0.738232	−	1.27865i	−0.953291	−	0.302054i	$-0.902328\pi$
$997$	−2.29623e16	−	3.97719e16i	−0.738232	−	1.27865i	0.215059	−	0.976601i	$-0.431006\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.k.e.37.2	✓	28
3.2	odd	2	inner	252.12.k.e.37.13	yes	28
7.4	even	3	inner	252.12.k.e.109.2	yes	28
21.11	odd	6	inner	252.12.k.e.109.13	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.12.k.e.37.2	✓	28	1.1	even	1	trivial
252.12.k.e.37.13	yes	28	3.2	odd	2	inner
252.12.k.e.109.2	yes	28	7.4	even	3	inner
252.12.k.e.109.13	yes	28	21.11	odd	6	inner

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:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-5383.85 + 9325.11i) q^{5} +(38734.1 - 21840.2i) q^{7} +O(q^{10})\)	\(q+(-5383.85 + 9325.11i) q^{5} +(38734.1 - 21840.2i) q^{7} +(179830. + 311475. i) q^{11} +489543. q^{13} +(-414037. - 717133. i) q^{17} +(-6.07057e6 + 1.05145e7i) q^{19} +(-1.43855e7 + 2.49163e7i) q^{23} +(-3.35577e7 - 5.81237e7i) q^{25} +2.49350e7 q^{29} +(-5.40289e7 - 9.35808e7i) q^{31} +(-4.87680e6 + 4.78784e8i) q^{35} +(-8.13360e7 + 1.40878e8i) q^{37} +7.65737e8 q^{41} -9.06743e8 q^{43} +(-4.37414e7 + 7.57624e7i) q^{47} +(1.02334e9 - 1.69192e9i) q^{49} +(2.32347e9 + 4.02436e9i) q^{53} -3.87271e9 q^{55} +(5.06339e9 + 8.77004e9i) q^{59} +(1.72799e9 - 2.99296e9i) q^{61} +(-2.63563e9 + 4.56504e9i) q^{65} +(9.39406e9 + 1.62710e10i) q^{67} -4.92529e9 q^{71} +(4.37328e9 + 7.57474e9i) q^{73} +(1.37682e10 + 8.13718e9i) q^{77} +(1.00709e10 - 1.74434e10i) q^{79} -1.15756e10 q^{83} +8.91646e9 q^{85} +(1.83426e10 - 3.17703e10i) q^{89} +(1.89620e10 - 1.06917e10i) q^{91} +(-6.53661e10 - 1.13217e11i) q^{95} -1.53637e11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 85666 q^{7}+O(q^{10}) \) 28 * q + 85666 * q^7	\( 28 q + 85666 q^{7} + 2794700 q^{13} + 37414826 q^{19} - 135469250 q^{25} + 71052098 q^{31} - 395898874 q^{37} + 980170268 q^{43} - 1253333270 q^{49} - 8844657360 q^{55} + 16692685100 q^{61} + 10934009870 q^{67} + 25578690650 q^{73} - 113974174090 q^{79} + 111789733440 q^{85} - 267181531078 q^{91} + 279658007384 q^{97}+O(q^{100}) \) 28 * q + 85666 * q^7 + 2794700 * q^13 + 37414826 * q^19 - 135469250 * q^25 + 71052098 * q^31 - 395898874 * q^37 + 980170268 * q^43 - 1253333270 * q^49 - 8844657360 * q^55 + 16692685100 * q^61 + 10934009870 * q^67 + 25578690650 * q^73 - 113974174090 * q^79 + 111789733440 * q^85 - 267181531078 * q^91 + 279658007384 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

Properties

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