LMFDB - Embedded newform 84.12.f.b.41.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(41,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 1]))

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

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

chi := DirichletCharacter("84.41");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.f (of order $2$, degree $1$, 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:	$64.5408271670$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			41.8
Character	$\chi$	$=$	84.41
Dual form			84.12.f.b.41.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-300.501 + 294.696i) q^{3} -5023.98 q^{5} +(-28355.3 + 34253.5i) q^{7} +(3454.97 - 177113. i) q^{9} +O(q^{10})$	\(q+(-300.501 + 294.696i) q^{3} -5023.98 q^{5} +(-28355.3 + 34253.5i) q^{7} +(3454.97 - 177113. i) q^{9} -376150. i q^{11} -135332. i q^{13} +(1.50971e6 - 1.48055e6i) q^{15} +7.23110e6 q^{17} -5.82572e6i q^{19} +(-1.57358e6 - 1.86494e7i) q^{21} +1.30105e6i q^{23} -2.35877e7 q^{25} +(5.11564e7 + 5.42409e7i) q^{27} -8.52912e7i q^{29} -8.26815e7i q^{31} +(1.10850e8 + 1.13033e8i) q^{33} +(1.42457e8 - 1.72089e8i) q^{35} +5.72087e8 q^{37} +(3.98818e7 + 4.06674e7i) q^{39} +2.33736e8 q^{41} -1.16444e9 q^{43} +(-1.73577e7 + 8.89814e8i) q^{45} -2.56622e8 q^{47} +(-3.69278e8 - 1.94254e9i) q^{49} +(-2.17295e9 + 2.13098e9i) q^{51} +1.13843e9i q^{53} +1.88977e9i q^{55} +(1.71682e9 + 1.75063e9i) q^{57} -5.04699e9 q^{59} +4.16882e9i q^{61} +(5.96878e9 + 5.14045e9i) q^{63} +6.79904e8i q^{65} -5.89248e9 q^{67} +(-3.83415e8 - 3.90967e8i) q^{69} +1.31962e10i q^{71} +3.07327e10i q^{73} +(7.08814e9 - 6.95122e9i) q^{75} +(1.28844e10 + 1.06658e10i) q^{77} -2.61505e10 q^{79} +(-3.13572e10 - 1.22384e9i) q^{81} +4.28312e10 q^{83} -3.63289e10 q^{85} +(2.51350e10 + 2.56301e10i) q^{87} +7.80249e10 q^{89} +(4.63559e9 + 3.83738e9i) q^{91} +(2.43660e10 + 2.48459e10i) q^{93} +2.92683e10i q^{95} +4.16792e10i q^{97} +(-6.66211e10 - 1.29958e9i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) $ 28 * q + 9632 * q^7 + 267660 * q^9	$ 28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100}) $ 28 * q + 9632 * q^7 + 267660 * q^9 - 3434160 * q^15 - 18804156 * q^21 + 397876900 * q^25 - 2059460504 * q^37 + 2276313936 * q^39 + 607100560 * q^43 + 1145242588 * q^49 + 1424787216 * q^51 - 32512522344 * q^57 + 16390616256 * q^63 - 48876957136 * q^67 - 1293110368 * q^79 + 82706814108 * q^81 + 197440859760 * q^85 - 329206232880 * q^91 - 243855044280 * q^93 - 81383696064 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−300.501	+	294.696i	−0.713969	+	0.700177i
$3$	−300.501	+	294.696i	−0.713969	+	0.700177i
$4$		0			0
$5$	−5023.98			−0.718974			−0.359487	−	0.933150i	$-0.617048\pi$
$5$	−5023.98			−0.718974			−0.359487	+	0.933150i	$0.617048\pi$
$6$		0			0
$7$	−28355.3	+	34253.5i	−0.637669	+	0.770310i
$7$	−28355.3	+	34253.5i	−0.637669	+	0.770310i
$8$		0			0
$9$	3454.97	−	177113.i	0.0195034	−	0.999810i
$10$		0			0
$11$		−	376150.i		−	0.704208i	−0.935961	−	0.352104i	$-0.885466\pi$
$11$		−	376150.i		−	0.704208i	0.935961	−	0.352104i	$-0.114534\pi$
$12$		0			0
$13$		−	135332.i		−	0.101091i	−0.998722	−	0.0505454i	$-0.983904\pi$
$13$		−	135332.i		−	0.101091i	0.998722	−	0.0505454i	$-0.0160959\pi$
$14$		0			0
$15$	1.50971e6	−	1.48055e6i	0.513325	−	0.503409i
$16$		0			0
$17$	7.23110e6			1.23519			0.617597	−	0.786494i	$-0.288106\pi$
$17$	7.23110e6			1.23519			0.617597	+	0.786494i	$0.288106\pi$
$18$		0			0
$19$		−	5.82572e6i		−	0.539765i	−0.962893	−	0.269882i	$-0.913015\pi$
$19$		−	5.82572e6i		−	0.539765i	0.962893	−	0.269882i	$-0.0869848\pi$
$20$		0			0
$21$	−1.57358e6	−	1.86494e7i	−0.0840778	−	0.996459i
$22$		0			0
$23$			1.30105e6i			0.0421493i	0.999778	+	0.0210747i	$0.00670877\pi$
$23$			1.30105e6i			0.0421493i	−0.999778	+	0.0210747i	$0.993291\pi$
$24$		0			0
$25$	−2.35877e7			−0.483077
$26$		0			0
$27$	5.11564e7	+	5.42409e7i	0.686119	+	0.727489i
$28$		0			0
$29$		−	8.52912e7i		−	0.772174i	−0.922462	−	0.386087i	$-0.873826\pi$
$29$		−	8.52912e7i		−	0.772174i	0.922462	−	0.386087i	$-0.126174\pi$
$30$		0			0
$31$		−	8.26815e7i		−	0.518704i	−0.965783	−	0.259352i	$-0.916491\pi$
$31$		−	8.26815e7i		−	0.518704i	0.965783	−	0.259352i	$-0.0835089\pi$
$32$		0			0
$33$	1.10850e8	+	1.13033e8i	0.493070	+	0.502783i
$34$		0			0
$35$	1.42457e8	−	1.72089e8i	0.458467	−	0.553833i
$36$		0			0
$37$	5.72087e8			1.35629			0.678145	−	0.734928i	$-0.262784\pi$
$37$	5.72087e8			1.35629			0.678145	+	0.734928i	$0.262784\pi$
$38$		0			0
$39$	3.98818e7	+	4.06674e7i	0.0707814	+	0.0721756i
$40$		0			0
$41$	2.33736e8			0.315075			0.157538	−	0.987513i	$-0.449644\pi$
$41$	2.33736e8			0.315075			0.157538	+	0.987513i	$0.449644\pi$
$42$		0			0
$43$	−1.16444e9			−1.20793			−0.603964	−	0.797012i	$-0.706413\pi$
$43$	−1.16444e9			−1.20793			−0.603964	+	0.797012i	$0.706413\pi$
$44$		0			0
$45$	−1.73577e7	+	8.89814e8i	−0.0140224	+	0.718837i
$46$		0			0
$47$	−2.56622e8			−0.163214			−0.0816068	−	0.996665i	$-0.526005\pi$
$47$	−2.56622e8			−0.163214			−0.0816068	+	0.996665i	$0.526005\pi$
$48$		0			0
$49$	−3.69278e8	−	1.94254e9i	−0.186756	−	0.982406i
$50$		0			0
$51$	−2.17295e9	+	2.13098e9i	−0.881891	+	0.864855i
$52$		0			0
$53$			1.13843e9i			0.373929i	0.982367	+	0.186964i	$0.0598649\pi$
$53$			1.13843e9i			0.373929i	−0.982367	+	0.186964i	$0.940135\pi$
$54$		0			0
$55$			1.88977e9i			0.506307i
$56$		0			0
$57$	1.71682e9	+	1.75063e9i	0.377931	+	0.385375i
$58$		0			0
$59$	−5.04699e9			−0.919065			−0.459533	−	0.888161i	$-0.651983\pi$
$59$	−5.04699e9			−0.919065			−0.459533	+	0.888161i	$0.651983\pi$
$60$		0			0
$61$			4.16882e9i			0.631973i	0.948764	+	0.315987i	$0.102335\pi$
$61$			4.16882e9i			0.631973i	−0.948764	+	0.315987i	$0.897665\pi$
$62$		0			0
$63$	5.96878e9	+	5.14045e9i	0.757727	+	0.652572i
$64$		0			0
$65$			6.79904e8i			0.0726816i
$66$		0			0
$67$	−5.89248e9			−0.533196			−0.266598	−	0.963808i	$-0.585900\pi$
$67$	−5.89248e9			−0.533196			−0.266598	+	0.963808i	$0.585900\pi$
$68$		0			0
$69$	−3.83415e8	−	3.90967e8i	−0.0295120	−	0.0300933i
$70$		0			0
$71$			1.31962e10i			0.868013i	0.900910	+	0.434007i	$0.142901\pi$
$71$			1.31962e10i			0.868013i	−0.900910	+	0.434007i	$0.857099\pi$
$72$		0			0
$73$			3.07327e10i			1.73510i	0.497349	+	0.867550i	$0.334307\pi$
$73$			3.07327e10i			1.73510i	−0.497349	+	0.867550i	$0.665693\pi$
$74$		0			0
$75$	7.08814e9	−	6.95122e9i	0.344902	−	0.338239i
$76$		0			0
$77$	1.28844e10	+	1.06658e10i	0.542459	+	0.449052i
$78$		0			0
$79$	−2.61505e10			−0.956162			−0.478081	−	0.878316i	$-0.658668\pi$
$79$	−2.61505e10			−0.956162			−0.478081	+	0.878316i	$0.658668\pi$
$80$		0			0
$81$	−3.13572e10	−	1.22384e9i	−0.999239	−	0.0389994i
$82$		0			0
$83$	4.28312e10			1.19352			0.596762	−	0.802418i	$-0.296454\pi$
$83$	4.28312e10			1.19352			0.596762	+	0.802418i	$0.296454\pi$
$84$		0			0
$85$	−3.63289e10			−0.888073
$86$		0			0
$87$	2.51350e10	+	2.56301e10i	0.540659	+	0.551308i
$88$		0			0
$89$	7.80249e10			1.48111			0.740557	−	0.671994i	$-0.234562\pi$
$89$	7.80249e10			1.48111			0.740557	+	0.671994i	$0.234562\pi$
$90$		0			0
$91$	4.63559e9	+	3.83738e9i	0.0778712	+	0.0644624i
$92$		0			0
$93$	2.43660e10	+	2.48459e10i	0.363185	+	0.370338i
$94$		0			0
$95$			2.92683e10i			0.388077i
$96$		0			0
$97$			4.16792e10i			0.492805i	0.969168	+	0.246402i	$0.0792485\pi$
$97$			4.16792e10i			0.492805i	−0.969168	+	0.246402i	$0.920752\pi$
$98$		0			0
$99$	−6.66211e10	−	1.29958e9i	−0.704074	−	0.0137344i
$100$		0			0
$101$	1.52373e10			0.144258			0.0721290	−	0.997395i	$-0.477021\pi$
$101$	1.52373e10			0.144258			0.0721290	+	0.997395i	$0.477021\pi$
$102$		0			0
$103$			6.11780e10i			0.519985i	0.965611	+	0.259993i	$0.0837201\pi$
$103$			6.11780e10i			0.519985i	−0.965611	+	0.259993i	$0.916280\pi$
$104$		0			0
$105$	7.90562e9	+	9.36944e10i	0.0604498	+	0.716428i
$106$		0			0
$107$			1.91301e11i			1.31858i	0.751888	+	0.659291i	$0.229143\pi$
$107$			1.91301e11i			1.31858i	−0.751888	+	0.659291i	$0.770857\pi$
$108$		0			0
$109$	−2.13955e11			−1.33192			−0.665959	−	0.745989i	$-0.731977\pi$
$109$	−2.13955e11			−1.33192			−0.665959	+	0.745989i	$0.731977\pi$
$110$		0			0
$111$	−1.71913e11	+	1.68592e11i	−0.968349	+	0.949644i
$112$		0			0
$113$			1.92300e11i			0.981854i	0.871201	+	0.490927i	$0.163342\pi$
$113$			1.92300e11i			0.981854i	−0.871201	+	0.490927i	$0.836658\pi$
$114$		0			0
$115$		−	6.53645e9i		−	0.0303043i
$116$		0			0
$117$	−2.39691e10	−	4.67567e8i	−0.101071	−	0.00197161i
$118$		0			0
$119$	−2.05040e11	+	2.47691e11i	−0.787646	+	0.951483i
$120$		0			0
$121$	1.43823e11			0.504091
$122$		0			0
$123$	−7.02380e10	+	6.88812e10i	−0.224954	+	0.220609i
$124$		0			0
$125$	3.63816e11			1.06629
$126$		0			0
$127$	−6.74839e10			−0.181251			−0.0906253	−	0.995885i	$-0.528887\pi$
$127$	−6.74839e10			−0.181251			−0.0906253	+	0.995885i	$0.528887\pi$
$128$		0			0
$129$	3.49916e11	−	3.43157e11i	0.862423	−	0.845763i
$130$		0			0
$131$	−1.36773e11			−0.309748			−0.154874	−	0.987934i	$-0.549497\pi$
$131$	−1.36773e11			−0.309748			−0.154874	+	0.987934i	$0.549497\pi$
$132$		0			0
$133$	1.99551e11	+	1.65190e11i	0.415786	+	0.344191i
$134$		0			0
$135$	−2.57009e11	−	2.72505e11i	−0.493302	−	0.523045i
$136$		0			0
$137$		−	1.10723e12i		−	1.96009i	−0.198777	−	0.980045i	$-0.563697\pi$
$137$		−	1.10723e12i		−	1.96009i	0.198777	−	0.980045i	$-0.436303\pi$
$138$		0			0
$139$		−	9.75169e10i		−	0.159404i	−0.996819	−	0.0797019i	$-0.974603\pi$
$139$		−	9.75169e10i		−	0.159404i	0.996819	−	0.0797019i	$-0.0253968\pi$
$140$		0			0
$141$	7.71153e10	−	7.56257e10i	0.116529	−	0.114278i
$142$		0			0
$143$	−5.09050e10			−0.0711889
$144$		0			0
$145$			4.28501e11i			0.555173i
$146$		0			0
$147$	6.83428e11	+	4.74910e11i	0.821197	+	0.570645i
$148$		0			0
$149$		−	1.30548e12i		−	1.45629i	−0.685424	−	0.728144i	$-0.740383\pi$
$149$		−	1.30548e12i		−	1.45629i	0.685424	−	0.728144i	$-0.259617\pi$
$150$		0			0
$151$	8.46580e10			0.0877596			0.0438798	−	0.999037i	$-0.486028\pi$
$151$	8.46580e10			0.0877596			0.0438798	+	0.999037i	$0.486028\pi$
$152$		0			0
$153$	2.49832e10	−	1.28072e12i	0.0240905	−	1.23496i
$154$		0			0
$155$			4.15391e11i			0.372934i
$156$		0			0
$157$		−	1.58428e11i		−	0.132551i	−0.997801	−	0.0662757i	$-0.978888\pi$
$157$		−	1.58428e11i		−	0.132551i	0.997801	−	0.0662757i	$-0.0211117\pi$
$158$		0			0
$159$	−3.35491e11	−	3.42099e11i	−0.261816	−	0.266974i
$160$		0			0
$161$	−4.45655e10	−	3.68917e10i	−0.0324681	−	0.0268773i
$162$		0			0
$163$	1.69152e12			1.15145			0.575726	−	0.817643i	$-0.304720\pi$
$163$	1.69152e12			1.15145			0.575726	+	0.817643i	$0.304720\pi$
$164$		0			0
$165$	−5.56908e11	−	5.67878e11i	−0.354505	−	0.361488i
$166$		0			0
$167$	−1.15100e12			−0.685702			−0.342851	−	0.939390i	$-0.611393\pi$
$167$	−1.15100e12			−0.685702			−0.342851	+	0.939390i	$0.611393\pi$
$168$		0			0
$169$	1.77385e12			0.989781
$170$		0			0
$171$	−1.03181e12	−	2.01277e10i	−0.539662	−	0.0105272i
$172$		0			0
$173$	−1.77556e12			−0.871127			−0.435563	−	0.900158i	$-0.643451\pi$
$173$	−1.77556e12			−0.871127			−0.435563	+	0.900158i	$0.643451\pi$
$174$		0			0
$175$	6.68838e11	−	8.07962e11i	0.308043	−	0.372119i
$176$		0			0
$177$	1.51663e12	−	1.48733e12i	0.656184	−	0.643508i
$178$		0			0
$179$		−	1.40743e12i		−	0.572447i	−0.958163	−	0.286223i	$-0.907600\pi$
$179$		−	1.40743e12i		−	0.572447i	0.958163	−	0.286223i	$-0.0923999\pi$
$180$		0			0
$181$			1.23910e11i			0.0474103i	0.999719	+	0.0237052i	$0.00754630\pi$
$181$			1.23910e11i			0.0474103i	−0.999719	+	0.0237052i	$0.992454\pi$
$182$		0			0
$183$	−1.22854e12	−	1.25273e12i	−0.442493	−	0.451209i
$184$		0			0
$185$	−2.87416e12			−0.975137
$186$		0			0
$187$		−	2.71998e12i		−	0.869834i
$188$		0			0
$189$	−3.30850e12	+	2.14268e11i	−0.997909	+	0.0646275i
$190$		0			0
$191$		−	4.52478e12i		−	1.28799i	−0.765028	−	0.643997i	$-0.777275\pi$
$191$		−	4.52478e12i		−	1.28799i	0.765028	−	0.643997i	$-0.222725\pi$
$192$		0			0
$193$	3.91365e12			1.05200			0.526001	−	0.850484i	$-0.323691\pi$
$193$	3.91365e12			1.05200			0.526001	+	0.850484i	$0.323691\pi$
$194$		0			0
$195$	−2.00365e11	−	2.04312e11i	−0.0508900	−	0.0518924i
$196$		0			0
$197$			5.97128e12i			1.43385i	0.697151	+	0.716924i	$0.254451\pi$
$197$			5.97128e12i			1.43385i	−0.697151	+	0.716924i	$0.745549\pi$
$198$		0			0
$199$			2.27127e12i			0.515914i	0.966156	+	0.257957i	$0.0830493\pi$
$199$			2.27127e12i			0.515914i	−0.966156	+	0.257957i	$0.916951\pi$
$200$		0			0
$201$	1.77070e12	−	1.73649e12i	0.380685	−	0.373332i
$202$		0			0
$203$	2.92152e12	+	2.41846e12i	0.594813	+	0.492391i
$204$		0			0
$205$	−1.17429e12			−0.226531
$206$		0			0
$207$	2.30433e11	+	4.49508e9i	0.0421413	+	0.000822055i
$208$		0			0
$209$	−2.19134e12			−0.380107
$210$		0			0
$211$	1.58935e12			0.261616			0.130808	−	0.991408i	$-0.458243\pi$
$211$	1.58935e12			0.261616			0.130808	+	0.991408i	$0.458243\pi$
$212$		0			0
$213$	−3.88886e12	−	3.96546e12i	−0.607763	−	0.619735i
$214$		0			0
$215$	5.85013e12			0.868468
$216$		0			0
$217$	2.83213e12	+	2.34446e12i	0.399563	+	0.330761i
$218$		0			0
$219$	−9.05681e12	−	9.23521e12i	−1.21488	−	1.23881i
$220$		0			0
$221$		−	9.78598e11i		−	0.124867i
$222$		0			0
$223$			1.34356e13i			1.63148i	0.578421	+	0.815738i	$0.303669\pi$
$223$			1.34356e13i			1.63148i	−0.578421	+	0.815738i	$0.696331\pi$
$224$		0			0
$225$	−8.14948e10	+	4.17770e12i	−0.00942163	+	0.482985i
$226$		0			0
$227$	1.24221e13			1.36790			0.683949	−	0.729530i	$-0.260261\pi$
$227$	1.24221e13			1.36790			0.683949	+	0.729530i	$0.260261\pi$
$228$		0			0
$229$			9.24530e12i			0.970121i	0.874481	+	0.485060i	$0.161202\pi$
$229$			9.24530e12i			0.970121i	−0.874481	+	0.485060i	$0.838798\pi$
$230$		0			0
$231$	−7.01498e12	+	5.91900e11i	−0.701714	+	0.0592083i
$232$		0			0
$233$		−	1.77859e13i		−	1.69675i	−0.529392	−	0.848377i	$-0.677580\pi$
$233$		−	1.77859e13i		−	1.69675i	0.529392	−	0.848377i	$-0.322420\pi$
$234$		0			0
$235$	1.28927e12			0.117346
$236$		0			0
$237$	7.85827e12	−	7.70647e12i	0.682670	−	0.669483i
$238$		0			0
$239$			2.07226e13i			1.71892i	0.511200	+	0.859462i	$0.329201\pi$
$239$			2.07226e13i			1.71892i	−0.511200	+	0.859462i	$0.670799\pi$
$240$		0			0
$241$			1.24855e13i			0.989268i	0.869101	+	0.494634i	$0.164698\pi$
$241$			1.24855e13i			0.989268i	−0.869101	+	0.494634i	$0.835302\pi$
$242$		0			0
$243$	9.78353e12	−	8.87309e12i	0.740732	−	0.671800i
$244$		0			0
$245$	1.85524e12	+	9.75928e12i	0.134273	+	0.706324i
$246$		0			0
$247$	−7.88404e11			−0.0545652
$248$		0			0
$249$	−1.28708e13	+	1.26222e13i	−0.852139	+	0.835678i
$250$		0			0
$251$	9.04658e12			0.573164			0.286582	−	0.958056i	$-0.407481\pi$
$251$	9.04658e12			0.573164			0.286582	+	0.958056i	$0.407481\pi$
$252$		0			0
$253$	4.89390e11			0.0296819
$254$		0			0
$255$	1.09169e13	−	1.07060e13i	0.634056	−	0.621808i
$256$		0			0
$257$	−1.92010e13			−1.06830			−0.534148	−	0.845391i	$-0.679368\pi$
$257$	−1.92010e13			−1.06830			−0.534148	+	0.845391i	$0.679368\pi$
$258$		0			0
$259$	−1.62217e13	+	1.95960e13i	−0.864865	+	1.04476i
$260$		0			0
$261$	−1.51062e13	−	2.94678e11i	−0.772027	−	0.0150600i
$262$		0			0
$263$			1.57360e13i			0.771147i	0.922677	+	0.385574i	$0.125996\pi$
$263$			1.57360e13i			0.771147i	−0.922677	+	0.385574i	$0.874004\pi$
$264$		0			0
$265$		−	5.71945e12i		−	0.268845i
$266$		0			0
$267$	−2.34466e13	+	2.29937e13i	−1.05747	+	1.03704i
$268$		0			0
$269$	−1.39267e13			−0.602852			−0.301426	−	0.953490i	$-0.597463\pi$
$269$	−1.39267e13			−0.602852			−0.301426	+	0.953490i	$0.597463\pi$
$270$		0			0
$271$			4.35616e13i			1.81039i	0.424994	+	0.905196i	$0.360276\pi$
$271$			4.35616e13i			1.81039i	−0.424994	+	0.905196i	$0.639724\pi$
$272$		0			0
$273$	−2.52386e12	+	2.12955e11i	−0.100733	+	0.00849949i
$274$		0			0
$275$			8.87252e12i			0.340186i
$276$		0			0
$277$	3.09477e13			1.14022			0.570111	−	0.821567i	$-0.306900\pi$
$277$	3.09477e13			1.14022			0.570111	+	0.821567i	$0.306900\pi$
$278$		0			0
$279$	−1.46440e13	−	2.85662e11i	−0.518605	−	0.0101165i
$280$		0			0
$281$		−	8.85374e12i		−	0.301469i	−0.988574	−	0.150734i	$-0.951836\pi$
$281$		−	8.85374e12i		−	0.301469i	0.988574	−	0.150734i	$-0.0481638\pi$
$282$		0			0
$283$			1.10662e12i			0.0362387i	0.999836	+	0.0181194i	$0.00576789\pi$
$283$			1.10662e12i			0.0362387i	−0.999836	+	0.0181194i	$0.994232\pi$
$284$		0			0
$285$	−8.62526e12	−	8.79516e12i	−0.271722	−	0.277075i
$286$		0			0
$287$	−6.62766e12	+	8.00628e12i	−0.200914	+	0.242706i
$288$		0			0
$289$	1.80169e13			0.525706
$290$		0			0
$291$	−1.22827e13	−	1.25247e13i	−0.345051	−	0.351847i
$292$		0			0
$293$	4.92340e13			1.33197			0.665984	−	0.745966i	$-0.268012\pi$
$293$	4.92340e13			1.33197			0.665984	+	0.745966i	$0.268012\pi$
$294$		0			0
$295$	2.53560e13			0.660784
$296$		0			0
$297$	2.04027e13	−	1.92425e13i	0.512304	−	0.483171i
$298$		0			0
$299$	1.76073e11			0.00426091
$300$		0			0
$301$	3.30181e13	−	3.98862e13i	0.770258	−	0.930479i
$302$		0			0
$303$	−4.57882e12	+	4.49037e12i	−0.102996	+	0.101006i
$304$		0			0
$305$		−	2.09441e13i		−	0.454372i
$306$		0			0
$307$			6.64923e13i			1.39159i	0.718242	+	0.695794i	$0.244947\pi$
$307$			6.64923e13i			1.39159i	−0.718242	+	0.695794i	$0.755053\pi$
$308$		0			0
$309$	−1.80290e13	−	1.83841e13i	−0.364082	−	0.371253i
$310$		0			0
$311$	−1.54351e13			−0.300835			−0.150417	−	0.988623i	$-0.548062\pi$
$311$	−1.54351e13			−0.300835			−0.150417	+	0.988623i	$0.548062\pi$
$312$		0			0
$313$			3.34280e11i			0.00628951i	0.999995	+	0.00314475i	$0.00100101\pi$
$313$			3.34280e11i			0.00628951i	−0.999995	+	0.00314475i	$0.998999\pi$
$314$		0			0
$315$	−2.99871e13	−	2.58255e13i	−0.544786	−	0.469182i
$316$		0			0
$317$		−	5.86507e13i		−	1.02907i	−0.857468	−	0.514537i	$-0.827964\pi$
$317$		−	5.86507e13i		−	1.02907i	0.857468	−	0.514537i	$-0.172036\pi$
$318$		0			0
$319$	−3.20822e13			−0.543771
$320$		0			0
$321$	−5.63758e13	−	5.74863e13i	−0.923241	−	0.941426i
$322$		0			0
$323$		−	4.21263e13i		−	0.666714i
$324$		0			0
$325$			3.19217e12i			0.0488346i
$326$		0			0
$327$	6.42938e13	−	6.30519e13i	0.950948	−	0.932578i
$328$		0			0
$329$	7.27661e12	−	8.79021e12i	0.104076	−	0.125725i
$330$		0			0
$331$	4.37226e13			0.604855			0.302428	−	0.953172i	$-0.402203\pi$
$331$	4.37226e13			0.604855			0.302428	+	0.953172i	$0.402203\pi$
$332$		0			0
$333$	1.97654e12	−	1.01324e14i	0.0264523	−	1.35603i
$334$		0			0
$335$	2.96037e13			0.383354
$336$		0			0
$337$	1.01610e14			1.27342			0.636711	−	0.771102i	$-0.280294\pi$
$337$	1.01610e14			1.27342			0.636711	+	0.771102i	$0.280294\pi$
$338$		0			0
$339$	−5.66700e13	−	5.77863e13i	−0.687472	−	0.701013i
$340$		0			0
$341$	−3.11006e13			−0.365275
$342$		0			0
$343$	7.70097e13	+	4.24323e13i	0.875846	+	0.482590i
$344$		0			0
$345$	1.92627e12	+	1.96421e12i	0.0212184	+	0.0216363i
$346$		0			0
$347$			2.33774e13i			0.249450i	0.992191	+	0.124725i	$0.0398049\pi$
$347$			2.33774e13i			0.249450i	−0.992191	+	0.124725i	$0.960195\pi$
$348$		0			0
$349$			3.49134e13i			0.360955i	0.983579	+	0.180477i	$0.0577642\pi$
$349$			3.49134e13i			0.360955i	−0.983579	+	0.180477i	$0.942236\pi$
$350$		0			0
$351$	7.34052e12	−	6.92309e12i	0.0735424	−	0.0693603i
$352$		0			0
$353$	−1.80827e14			−1.75591			−0.877956	−	0.478741i	$-0.841093\pi$
$353$	−1.80827e14			−1.75591			−0.877956	+	0.478741i	$0.841093\pi$
$354$		0			0
$355$		−	6.62972e13i		−	0.624079i
$356$		0			0
$357$	−1.13787e13	−	1.34856e14i	−0.103853	−	1.23082i
$358$		0			0
$359$			1.99756e14i			1.76799i	0.467496	+	0.883995i	$0.345156\pi$
$359$			1.99756e14i			1.76799i	−0.467496	+	0.883995i	$0.654844\pi$
$360$		0			0
$361$	8.25513e13			0.708654
$362$		0			0
$363$	−4.32190e13	+	4.23842e13i	−0.359905	+	0.352953i
$364$		0			0
$365$		−	1.54400e14i		−	1.24749i
$366$		0			0
$367$			7.75941e13i			0.608366i	0.952614	+	0.304183i	$0.0983835\pi$
$367$			7.75941e13i			0.608366i	−0.952614	+	0.304183i	$0.901617\pi$
$368$		0			0
$369$	8.07550e11	−	4.13978e13i	0.00614504	−	0.315015i
$370$		0			0
$371$	−3.89952e13	−	3.22805e13i	−0.288041	−	0.238443i
$372$		0			0
$373$	1.66824e14			1.19635			0.598177	−	0.801364i	$-0.295892\pi$
$373$	1.66824e14			1.19635			0.598177	+	0.801364i	$0.295892\pi$
$374$		0			0
$375$	−1.09327e14	+	1.07215e14i	−0.761300	+	0.746594i
$376$		0			0
$377$	−1.15426e13			−0.0780596
$378$		0			0
$379$	−2.21954e14			−1.45797			−0.728985	−	0.684530i	$-0.760007\pi$
$379$	−2.21954e14			−1.45797			−0.728985	+	0.684530i	$0.760007\pi$
$380$		0			0
$381$	2.02790e13	−	1.98873e13i	0.129407	−	0.126908i
$382$		0			0
$383$	−8.37328e13			−0.519161			−0.259581	−	0.965721i	$-0.583584\pi$
$383$	−8.37328e13			−0.519161			−0.259581	+	0.965721i	$0.583584\pi$
$384$		0			0
$385$	−6.47312e13	−	5.35850e13i	−0.390014	−	0.322856i
$386$		0			0
$387$	−4.02310e12	+	2.06238e14i	−0.0235587	+	1.20770i
$388$		0			0
$389$		−	1.21366e14i		−	0.690833i	−0.938449	−	0.345417i	$-0.887738\pi$
$389$		−	1.21366e14i		−	0.690833i	0.938449	−	0.345417i	$-0.112262\pi$
$390$		0			0
$391$			9.40802e12i			0.0520627i
$392$		0			0
$393$	4.11005e13	−	4.03066e13i	0.221151	−	0.216879i
$394$		0			0
$395$	1.31380e14			0.687456
$396$		0			0
$397$			4.47843e13i			0.227918i	0.993485	+	0.113959i	$0.0363532\pi$
$397$			4.47843e13i			0.227918i	−0.993485	+	0.113959i	$0.963647\pi$
$398$		0			0
$399$	−1.08646e14	+	9.16721e12i	−0.537854	+	0.0453823i
$400$		0			0
$401$			1.96586e14i			0.946800i	0.880848	+	0.473400i	$0.156973\pi$
$401$			1.96586e14i			0.946800i	−0.880848	+	0.473400i	$0.843027\pi$
$402$		0			0
$403$	−1.11894e13			−0.0524361
$404$		0			0
$405$	1.57538e14	+	6.14855e12i	0.718427	+	0.0280395i
$406$		0			0
$407$		−	2.15190e14i		−	0.955111i
$408$		0			0
$409$		−	3.40505e14i		−	1.47111i	−0.677465	−	0.735555i	$-0.736921\pi$
$409$		−	3.40505e14i		−	1.47111i	0.677465	−	0.735555i	$-0.263079\pi$
$410$		0			0
$411$	3.26298e14	+	3.32725e14i	1.37241	+	1.39944i
$412$		0			0
$413$	1.43109e14	−	1.72877e14i	0.586059	−	0.707965i
$414$		0			0
$415$	−2.15183e14			−0.858112
$416$		0			0
$417$	2.87379e13	+	2.93040e13i	0.111611	+	0.113809i
$418$		0			0
$419$	4.76804e14			1.80370			0.901848	−	0.432054i	$-0.142211\pi$
$419$	4.76804e14			1.80370			0.901848	+	0.432054i	$0.142211\pi$
$420$		0			0
$421$	1.96446e14			0.723922			0.361961	−	0.932193i	$-0.382107\pi$
$421$	1.96446e14			0.723922			0.361961	+	0.932193i	$0.382107\pi$
$422$		0			0
$423$	−8.86622e11	+	4.54512e13i	−0.00318322	+	0.163183i
$424$		0			0
$425$	−1.70565e14			−0.596694
$426$		0			0
$427$	−1.42797e14	−	1.18208e14i	−0.486816	−	0.402990i
$428$		0			0
$429$	1.52970e13	−	1.50015e13i	0.0508266	−	0.0498448i
$430$		0			0
$431$			2.72849e14i			0.883686i	0.897092	+	0.441843i	$0.145675\pi$
$431$			2.72849e14i			0.883686i	−0.897092	+	0.441843i	$0.854325\pi$
$432$		0			0
$433$			3.79305e14i			1.19758i	0.800906	+	0.598790i	$0.204352\pi$
$433$			3.79305e14i			1.19758i	−0.800906	+	0.598790i	$0.795648\pi$
$434$		0			0
$435$	−1.26278e14	−	1.28765e14i	−0.388719	−	0.396376i
$436$		0			0
$437$	7.57955e12			0.0227507
$438$		0			0
$439$			3.17456e14i			0.929242i	0.885510	+	0.464621i	$0.153809\pi$
$439$			3.17456e14i			0.929242i	−0.885510	+	0.464621i	$0.846191\pi$
$440$		0			0
$441$	−3.45325e14	+	5.86926e13i	−0.985862	+	0.167560i
$442$		0			0
$443$			2.81427e14i			0.783691i	0.920031	+	0.391846i	$0.128163\pi$
$443$			2.81427e14i			0.783691i	−0.920031	+	0.391846i	$0.871837\pi$
$444$		0			0
$445$	−3.91996e14			−1.06488
$446$		0			0
$447$	3.84722e14	+	3.92300e14i	1.01966	+	1.03974i
$448$		0			0
$449$			5.23816e14i			1.35464i	0.735689	+	0.677319i	$0.236858\pi$
$449$			5.23816e14i			1.35464i	−0.735689	+	0.677319i	$0.763142\pi$
$450$		0			0
$451$		−	8.79197e13i		−	0.221879i
$452$		0			0
$453$	−2.54398e13	+	2.49484e13i	−0.0626576	+	0.0614473i
$454$		0			0
$455$	−2.32891e13	−	1.92789e13i	−0.0559874	−	0.0463468i
$456$		0			0
$457$	−6.59236e14			−1.54704			−0.773520	−	0.633771i	$-0.781506\pi$
$457$	−6.59236e14			−1.54704			−0.773520	+	0.633771i	$0.781506\pi$
$458$		0			0
$459$	3.69917e14	+	3.92222e14i	0.847491	+	0.898591i
$460$		0			0
$461$	1.46474e14			0.327647			0.163823	−	0.986490i	$-0.447617\pi$
$461$	1.46474e14			0.327647			0.163823	+	0.986490i	$0.447617\pi$
$462$		0			0
$463$	−3.84421e14			−0.839677			−0.419838	−	0.907599i	$-0.637913\pi$
$463$	−3.84421e14			−0.839677			−0.419838	+	0.907599i	$0.637913\pi$
$464$		0			0
$465$	−1.22414e14	−	1.24825e14i	−0.261120	−	0.266264i
$466$		0			0
$467$	−7.03181e13			−0.146496			−0.0732478	−	0.997314i	$-0.523336\pi$
$467$	−7.03181e13			−0.146496			−0.0732478	+	0.997314i	$0.523336\pi$
$468$		0			0
$469$	1.67083e14	−	2.01838e14i	0.340003	−	0.410726i
$470$		0			0
$471$	4.66882e13	+	4.76079e13i	0.0928095	+	0.0946376i
$472$		0			0
$473$			4.38004e14i			0.850632i
$474$		0			0
$475$			1.37415e14i			0.260748i
$476$		0			0
$477$	2.01631e14	+	3.93324e12i	0.373858	+	0.00729288i
$478$		0			0
$479$	−1.91186e14			−0.346426			−0.173213	−	0.984884i	$-0.555415\pi$
$479$	−1.91186e14			−0.346426			−0.173213	+	0.984884i	$0.555415\pi$
$480$		0			0
$481$		−	7.74216e13i		−	0.137108i
$482$		0			0
$483$	2.42638e13	−	2.04730e12i	0.0420001	−	0.00354383i
$484$		0			0
$485$		−	2.09396e14i		−	0.354314i
$486$		0			0
$487$	−3.91764e13			−0.0648059			−0.0324030	−	0.999475i	$-0.510316\pi$
$487$	−3.91764e13			−0.0648059			−0.0324030	+	0.999475i	$0.510316\pi$
$488$		0			0
$489$	−5.08304e14	+	4.98486e14i	−0.822101	+	0.806220i
$490$		0			0
$491$			1.30769e14i			0.206804i	0.994640	+	0.103402i	$0.0329727\pi$
$491$			1.30769e14i			0.206804i	−0.994640	+	0.103402i	$0.967027\pi$
$492$		0			0
$493$		−	6.16749e14i		−	0.953785i
$494$		0			0
$495$	3.34703e14	+	6.52909e12i	0.506211	+	0.00987470i
$496$		0			0
$497$	−4.52014e14	−	3.74181e14i	−0.668640	−	0.553505i
$498$		0			0
$499$	3.05572e14			0.442141			0.221071	−	0.975258i	$-0.429045\pi$
$499$	3.05572e14			0.442141			0.221071	+	0.975258i	$0.429045\pi$
$500$		0			0
$501$	3.45877e14	−	3.39196e14i	0.489570	−	0.480113i
$502$		0			0
$503$	1.08426e15			1.50144			0.750722	−	0.660618i	$-0.229706\pi$
$503$	1.08426e15			1.50144			0.750722	+	0.660618i	$0.229706\pi$
$504$		0			0
$505$	−7.65518e13			−0.103718
$506$		0			0
$507$	−5.33043e14	+	5.22746e14i	−0.706673	+	0.693022i
$508$		0			0
$509$	8.33801e14			1.08172			0.540860	−	0.841113i	$-0.318099\pi$
$509$	8.33801e14			1.08172			0.540860	+	0.841113i	$0.318099\pi$
$510$		0			0
$511$	−1.05270e15	−	8.71435e14i	−1.33657	−	1.10642i
$512$		0			0
$513$	3.15992e14	−	2.98023e14i	0.392673	−	0.370343i
$514$		0			0
$515$		−	3.07357e14i		−	0.373856i
$516$		0			0
$517$			9.65284e13i			0.114936i
$518$		0			0
$519$	5.33557e14	−	5.23251e14i	0.621957	−	0.609943i
$520$		0			0
$521$	−1.65663e15			−1.89068			−0.945341	−	0.326083i	$-0.894271\pi$
$521$	−1.65663e15			−1.89068			−0.945341	+	0.326083i	$0.894271\pi$
$522$		0			0
$523$		−	1.04332e15i		−	1.16590i	−0.812509	−	0.582948i	$-0.801899\pi$
$523$		−	1.04332e15i		−	1.16590i	0.812509	−	0.582948i	$-0.198101\pi$
$524$		0			0
$525$	3.71171e13	+	4.39898e14i	0.0406160	+	0.481366i
$526$		0			0
$527$		−	5.97879e14i		−	0.640700i
$528$		0			0
$529$	9.51117e14			0.998223
$530$		0			0
$531$	−1.74372e13	+	8.93889e14i	−0.0179249	+	0.918890i
$532$		0			0
$533$		−	3.16319e13i		−	0.0318512i
$534$		0			0
$535$		−	9.61094e14i		−	0.948025i
$536$		0			0
$537$	4.14765e14	+	4.22934e14i	0.400814	+	0.408709i
$538$		0			0
$539$	−7.30685e14	+	1.38904e14i	−0.691818	+	0.131515i
$540$		0			0
$541$	−2.32635e14			−0.215820			−0.107910	−	0.994161i	$-0.534416\pi$
$541$	−2.32635e14			−0.215820			−0.107910	+	0.994161i	$0.534416\pi$
$542$		0			0
$543$	−3.65157e13	−	3.72350e13i	−0.0331956	−	0.0338495i
$544$		0			0
$545$	1.07491e15			0.957614
$546$		0			0
$547$	−1.57902e13			−0.0137866			−0.00689332	−	0.999976i	$-0.502194\pi$
$547$	−1.57902e13			−0.0137866			−0.00689332	+	0.999976i	$0.502194\pi$
$548$		0			0
$549$	7.38353e14	+	1.44031e13i	0.631853	+	0.0123256i
$550$		0			0
$551$	−4.96882e14			−0.416792
$552$		0			0
$553$	7.41507e14	−	8.95747e14i	0.609715	−	0.736542i
$554$		0			0
$555$	8.63687e14	−	8.47004e14i	0.696218	−	0.682769i
$556$		0			0
$557$		−	1.79723e15i		−	1.42037i	−0.704017	−	0.710183i	$-0.748612\pi$
$557$		−	1.79723e15i		−	1.42037i	0.704017	−	0.710183i	$-0.251388\pi$
$558$		0			0
$559$			1.57586e14i			0.122110i
$560$		0			0
$561$	8.01568e14	+	8.17356e14i	0.609038	+	0.621034i
$562$		0			0
$563$	−1.65239e15			−1.23116			−0.615581	−	0.788074i	$-0.711079\pi$
$563$	−1.65239e15			−1.23116			−0.615581	+	0.788074i	$0.711079\pi$
$564$		0			0
$565$		−	9.66110e14i		−	0.705927i
$566$		0			0
$567$	9.31064e14	−	1.03939e15i	0.667226	−	0.744856i
$568$		0			0
$569$		−	2.54146e15i		−	1.78634i	−0.449715	−	0.893172i	$-0.648474\pi$
$569$		−	2.54146e15i		−	1.78634i	0.449715	−	0.893172i	$-0.351526\pi$
$570$		0			0
$571$	−5.68905e14			−0.392230			−0.196115	−	0.980581i	$-0.562833\pi$
$571$	−5.68905e14			−0.392230			−0.196115	+	0.980581i	$0.562833\pi$
$572$		0			0
$573$	1.33344e15	+	1.35970e15i	0.901824	+	0.919588i
$574$		0			0
$575$		−	3.06888e13i		−	0.0203614i
$576$		0			0
$577$		−	1.93488e15i		−	1.25947i	−0.776811	−	0.629734i	$-0.783164\pi$
$577$		−	1.93488e15i		−	1.25947i	0.776811	−	0.629734i	$-0.216836\pi$
$578$		0			0
$579$	−1.17606e15	+	1.15334e15i	−0.751096	+	0.736587i
$580$		0			0
$581$	−1.21449e15	+	1.46712e15i	−0.761073	+	0.919384i
$582$		0			0
$583$	4.28220e14			0.263324
$584$		0			0
$585$	1.20420e14	+	2.34905e12i	0.0726677	+	0.00141754i
$586$		0			0
$587$	7.72722e14			0.457629			0.228814	−	0.973470i	$-0.426515\pi$
$587$	7.72722e14			0.457629			0.228814	+	0.973470i	$0.426515\pi$
$588$		0			0
$589$	−4.81679e14			−0.279978
$590$		0			0
$591$	−1.75972e15	−	1.79438e15i	−1.00395	−	1.02372i
$592$		0			0
$593$	2.06901e14			0.115868			0.0579338	−	0.998320i	$-0.481549\pi$
$593$	2.06901e14			0.115868			0.0579338	+	0.998320i	$0.481549\pi$
$594$		0			0
$595$	1.03012e15	−	1.24439e15i	0.566296	−	0.684091i
$596$		0			0
$597$	−6.69336e14	−	6.82520e14i	−0.361231	−	0.368347i
$598$		0			0
$599$		−	5.95647e14i		−	0.315603i	−0.987471	−	0.157802i	$-0.949559\pi$
$599$		−	5.95647e14i		−	0.315603i	0.987471	−	0.157802i	$-0.0504406\pi$
$600$		0			0
$601$			3.05150e15i			1.58746i	0.608268	+	0.793732i	$0.291865\pi$
$601$			3.05150e15i			1.58746i	−0.608268	+	0.793732i	$0.708135\pi$
$602$		0			0
$603$	−2.03583e13	+	1.04364e15i	−0.0103991	+	0.533095i
$604$		0			0
$605$	−7.22565e14			−0.362428
$606$		0			0
$607$			3.30985e15i			1.63031i	0.579241	+	0.815156i	$0.303349\pi$
$607$			3.30985e15i			1.63031i	−0.579241	+	0.815156i	$0.696651\pi$
$608$		0			0
$609$	−1.59063e15	+	1.34212e14i	−0.769440	+	0.0649227i
$610$		0			0
$611$			3.47292e13i			0.0164994i
$612$		0			0
$613$	−2.77517e15			−1.29496			−0.647481	−	0.762082i	$-0.724177\pi$
$613$	−2.77517e15			−1.29496			−0.647481	+	0.762082i	$0.724177\pi$
$614$		0			0
$615$	3.52874e14	−	3.46058e14i	0.161736	−	0.158612i
$616$		0			0
$617$			2.89628e15i			1.30398i	0.758227	+	0.651991i	$0.226066\pi$
$617$			2.89628e15i			1.30398i	−0.758227	+	0.651991i	$0.773934\pi$
$618$		0			0
$619$		−	3.27975e15i		−	1.45058i	−0.688441	−	0.725292i	$-0.741705\pi$
$619$		−	3.27975e15i		−	1.45058i	0.688441	−	0.725292i	$-0.258295\pi$
$620$		0			0
$621$	−7.05702e13	+	6.65571e13i	−0.0306632	+	0.0289195i
$622$		0			0
$623$	−2.21242e15	+	2.67263e15i	−0.944461	+	1.14092i
$624$		0			0
$625$	−6.76060e14			−0.283560
$626$		0			0
$627$	6.58501e14	−	6.45780e14i	0.271384	−	0.266142i
$628$		0			0
$629$	4.13682e15			1.67528
$630$		0			0
$631$	9.65016e14			0.384037			0.192018	−	0.981391i	$-0.438497\pi$
$631$	9.65016e14			0.384037			0.192018	+	0.981391i	$0.438497\pi$
$632$		0			0
$633$	−4.77600e14	+	4.68374e14i	−0.186786	+	0.183178i
$634$		0			0
$635$	3.39038e14			0.130314
$636$		0			0
$637$	−2.62887e14	+	4.99750e13i	−0.0993121	+	0.0188793i
$638$		0			0
$639$	2.33721e15	+	4.55923e13i	0.867848	+	0.0169292i
$640$		0			0
$641$			1.54672e15i			0.564538i	0.959335	+	0.282269i	$0.0910871\pi$
$641$			1.54672e15i			0.564538i	−0.959335	+	0.282269i	$0.908913\pi$
$642$		0			0
$643$			3.27491e15i			1.17500i	0.809223	+	0.587501i	$0.199888\pi$
$643$			3.27491e15i			1.17500i	−0.809223	+	0.587501i	$0.800112\pi$
$644$		0			0
$645$	−1.75797e15	+	1.72401e15i	−0.620059	+	0.608082i
$646$		0			0
$647$	1.43171e15			0.496459			0.248229	−	0.968701i	$-0.420151\pi$
$647$	1.43171e15			0.496459			0.248229	+	0.968701i	$0.420151\pi$
$648$		0			0
$649$			1.89842e15i			0.647213i
$650$		0			0
$651$	−1.54196e15	+	1.30106e14i	−0.516867	+	0.0436115i
$652$		0			0
$653$			1.27538e15i			0.420357i	0.977663	+	0.210179i	$0.0674046\pi$
$653$			1.27538e15i			0.420357i	−0.977663	+	0.210179i	$0.932595\pi$
$654$		0			0
$655$	6.87146e14			0.222701
$656$		0			0
$657$	5.44317e15	+	1.06180e14i	1.73477	+	0.0338403i
$658$		0			0
$659$		−	4.35156e15i		−	1.36388i	−0.731409	−	0.681939i	$-0.761137\pi$
$659$		−	4.35156e15i		−	1.36388i	0.731409	−	0.681939i	$-0.238863\pi$
$660$		0			0
$661$			2.25760e15i			0.695886i	0.937516	+	0.347943i	$0.113120\pi$
$661$			2.25760e15i			0.695886i	−0.937516	+	0.347943i	$0.886880\pi$
$662$		0			0
$663$	2.88389e14	+	2.94070e14i	0.0874288	+	0.0891510i
$664$		0			0
$665$	−1.00254e15	−	8.29912e14i	−0.298939	−	0.247465i
$666$		0			0
$667$	1.10968e14			0.0325466
$668$		0			0
$669$	−3.95943e15	−	4.03742e15i	−1.14232	−	1.16482i
$670$		0			0
$671$	1.56810e15			0.445041
$672$		0			0
$673$	−2.01032e15			−0.561283			−0.280642	−	0.959813i	$-0.590547\pi$
$673$	−2.01032e15			−0.561283			−0.280642	+	0.959813i	$0.590547\pi$
$674$		0			0
$675$	−1.20666e15	−	1.27942e15i	−0.331448	−	0.351433i
$676$		0			0
$677$	7.16554e15			1.93647			0.968235	−	0.250040i	$-0.0804439\pi$
$677$	7.16554e15			1.93647			0.968235	+	0.250040i	$0.0804439\pi$
$678$		0			0
$679$	−1.42766e15	−	1.18183e15i	−0.379613	−	0.314246i
$680$		0			0
$681$	−3.73287e15	+	3.66076e15i	−0.976637	+	0.957772i
$682$		0			0
$683$			2.06591e15i			0.531860i	0.963992	+	0.265930i	$0.0856790\pi$
$683$			2.06591e15i			0.531860i	−0.963992	+	0.265930i	$0.914321\pi$
$684$		0			0
$685$			5.56272e15i			1.40925i
$686$		0			0
$687$	−2.72456e15	−	2.77822e15i	−0.679257	−	0.692636i
$688$		0			0
$689$	1.54066e14			0.0378007
$690$		0			0
$691$		−	3.16105e15i		−	0.763313i	−0.924304	−	0.381656i	$-0.875354\pi$
$691$		−	3.16105e15i		−	0.763313i	0.924304	−	0.381656i	$-0.124646\pi$
$692$		0			0
$693$	1.93358e15	−	2.24516e15i	0.459546	−	0.533597i
$694$		0			0
$695$			4.89923e14i			0.114607i
$696$		0			0
$697$	1.69017e15			0.389179
$698$		0			0
$699$	5.24145e15	+	5.34469e15i	1.18803	+	1.21143i
$700$		0			0
$701$			7.11588e15i			1.58774i	0.608086	+	0.793871i	$0.291937\pi$
$701$			7.11588e15i			1.58774i	−0.608086	+	0.793871i	$0.708063\pi$
$702$		0			0
$703$		−	3.33282e15i		−	0.732078i
$704$		0			0
$705$	−3.87426e14	+	3.79942e14i	−0.0837816	+	0.0821632i
$706$		0			0
$707$	−4.32058e14	+	5.21930e14i	−0.0919888	+	0.111123i
$708$		0			0
$709$	1.72354e15			0.361300			0.180650	−	0.983547i	$-0.442180\pi$
$709$	1.72354e15			0.361300			0.180650	+	0.983547i	$0.442180\pi$
$710$		0			0
$711$	−9.03492e13	+	4.63161e15i	−0.0186484	+	0.955980i
$712$		0			0
$713$	1.07573e14			0.0218630
$714$		0			0
$715$	2.55746e14			0.0511829
$716$		0			0
$717$	−6.10689e15	−	6.22718e15i	−1.20355	−	1.22726i
$718$		0			0
$719$	3.84227e15			0.745725			0.372862	−	0.927887i	$-0.378376\pi$
$719$	3.84227e15			0.745725			0.372862	+	0.927887i	$0.378376\pi$
$720$		0			0
$721$	−2.09556e15	−	1.73472e15i	−0.400550	−	0.331578i
$722$		0			0
$723$	−3.67945e15	−	3.75192e15i	−0.692663	−	0.706306i
$724$		0			0
$725$			2.01183e15i			0.373019i
$726$		0			0
$727$			6.68478e15i			1.22081i	0.792090	+	0.610404i	$0.208993\pi$
$727$			6.68478e15i			1.22081i	−0.792090	+	0.610404i	$0.791007\pi$
$728$		0			0
$729$	−3.25097e14	+	5.54955e15i	−0.0584805	+	0.998289i
$730$		0			0
$731$	−8.42019e15			−1.49203
$732$		0			0
$733$			8.58757e15i			1.49899i	0.662010	+	0.749495i	$0.269704\pi$
$733$			8.58757e15i			1.49899i	−0.662010	+	0.749495i	$0.730296\pi$
$734$		0			0
$735$	−3.43353e15	−	2.38594e15i	−0.590419	−	0.410279i
$736$		0			0
$737$			2.21646e15i			0.375481i
$738$		0			0
$739$	4.87279e15			0.813267			0.406634	−	0.913591i	$-0.366703\pi$
$739$	4.87279e15			0.813267			0.406634	+	0.913591i	$0.366703\pi$
$740$		0			0
$741$	2.36916e14	−	2.32340e14i	0.0389579	−	0.0382053i
$742$		0			0
$743$		−	1.08733e16i		−	1.76166i	−0.473435	−	0.880829i	$-0.656986\pi$
$743$		−	1.08733e16i		−	1.76166i	0.473435	−	0.880829i	$-0.343014\pi$
$744$		0			0
$745$			6.55873e15i			1.04703i
$746$		0			0
$747$	1.47980e14	−	7.58598e15i	0.0232778	−	1.19330i
$748$		0			0
$749$	−6.55274e15	−	5.42441e15i	−1.01572	−	0.840819i
$750$		0			0
$751$	2.76843e13			0.00422876			0.00211438	−	0.999998i	$-0.499327\pi$
$751$	2.76843e13			0.00422876			0.00211438	+	0.999998i	$0.499327\pi$
$752$		0			0
$753$	−2.71851e15	+	2.66600e15i	−0.409222	+	0.401317i
$754$		0			0
$755$	−4.25320e14			−0.0630969
$756$		0			0
$757$	1.04108e16			1.52214			0.761071	−	0.648668i	$-0.224674\pi$
$757$	1.04108e16			1.52214			0.761071	+	0.648668i	$0.224674\pi$
$758$		0			0
$759$	−1.47062e14	+	1.44221e14i	−0.0211920	+	0.0207826i
$760$		0			0
$761$	−6.02072e15			−0.855131			−0.427566	−	0.903984i	$-0.640629\pi$
$761$	−6.02072e15			−0.855131			−0.427566	+	0.903984i	$0.640629\pi$
$762$		0			0
$763$	6.06677e15	−	7.32872e15i	0.849323	−	1.02599i
$764$		0			0
$765$	−1.25515e14	+	6.43434e15i	−0.0173204	+	0.887904i
$766$		0			0
$767$			6.83018e14i			0.0929089i
$768$		0			0
$769$			7.89401e15i			1.05853i	0.848457	+	0.529265i	$0.177532\pi$
$769$			7.89401e15i			1.05853i	−0.848457	+	0.529265i	$0.822468\pi$
$770$		0			0
$771$	5.76992e15	−	5.65847e15i	0.762730	−	0.747997i
$772$		0			0
$773$	−6.36819e15			−0.829906			−0.414953	−	0.909843i	$-0.636202\pi$
$773$	−6.36819e15			−0.829906			−0.414953	+	0.909843i	$0.636202\pi$
$774$		0			0
$775$			1.95027e15i			0.250574i
$776$		0			0
$777$	−9.00223e14	−	1.06691e16i	−0.114034	−	1.35149i
$778$		0			0
$779$		−	1.36168e15i		−	0.170067i
$780$		0			0
$781$	4.96373e15			0.611262
$782$		0			0
$783$	4.62627e15	−	4.36319e15i	0.561748	−	0.529803i
$784$		0			0
$785$			7.95941e14i			0.0953010i
$786$		0			0
$787$		−	8.64263e15i		−	1.02043i	−0.860046	−	0.510217i	$-0.829565\pi$
$787$		−	8.64263e15i		−	1.02043i	0.860046	−	0.510217i	$-0.170435\pi$
$788$		0			0
$789$	−4.63734e15	−	4.72868e15i	−0.539940	−	0.550575i
$790$		0			0
$791$	−6.58694e15	−	5.45272e15i	−0.756332	−	0.626098i
$792$		0			0
$793$	5.64173e14			0.0638866
$794$		0			0
$795$	1.68550e15	+	1.71870e15i	0.188239	+	0.191947i
$796$		0			0
$797$	1.50473e16			1.65744			0.828720	−	0.559664i	$-0.189070\pi$
$797$	1.50473e16			1.65744			0.828720	+	0.559664i	$0.189070\pi$
$798$		0			0
$799$	−1.85566e15			−0.201601
$800$		0			0
$801$	2.69574e14	−	1.38193e16i	0.0288867	−	1.48083i
$802$		0			0
$803$	1.15601e16			1.22187
$804$		0			0
$805$	2.23896e14	+	1.85343e14i	0.0233437	+	0.0193241i
$806$		0			0
$807$	4.18499e15	−	4.10415e15i	0.430417	−	0.422103i
$808$		0			0
$809$		−	6.05841e15i		−	0.614670i	−0.951601	−	0.307335i	$-0.900563\pi$
$809$		−	6.05841e15i		−	0.614670i	0.951601	−	0.307335i	$-0.0994372\pi$
$810$		0			0
$811$			2.08023e14i			0.0208208i	0.999946	+	0.0104104i	$0.00331379\pi$
$811$			2.08023e14i			0.0208208i	−0.999946	+	0.0104104i	$0.996686\pi$
$812$		0			0
$813$	−1.28374e16	−	1.30903e16i	−1.26760	−	1.29256i
$814$		0			0
$815$	−8.49817e15			−0.827864
$816$		0			0
$817$			6.78370e15i			0.651997i
$818$		0			0
$819$	6.95666e14	−	8.07766e14i	0.0659689	−	0.0765992i
$820$		0			0
$821$			1.24685e16i			1.16661i	0.812252	+	0.583307i	$0.198241\pi$
$821$			1.24685e16i			1.16661i	−0.812252	+	0.583307i	$0.801759\pi$
$822$		0			0
$823$	9.75073e15			0.900198			0.450099	−	0.892979i	$-0.351389\pi$
$823$	9.75073e15			0.900198			0.450099	+	0.892979i	$0.351389\pi$
$824$		0			0
$825$	−2.61470e15	−	2.66620e15i	−0.238191	−	0.242883i
$826$		0			0
$827$		−	1.53859e16i		−	1.38306i	−0.722347	−	0.691531i	$-0.756937\pi$
$827$		−	1.53859e16i		−	1.38306i	0.722347	−	0.691531i	$-0.243063\pi$
$828$		0			0
$829$		−	7.18285e15i		−	0.637158i	−0.947896	−	0.318579i	$-0.896794\pi$
$829$		−	7.18285e15i		−	0.637158i	0.947896	−	0.318579i	$-0.103206\pi$
$830$		0			0
$831$	−9.29982e15	+	9.12018e15i	−0.814084	+	0.798358i
$832$		0			0
$833$	−2.67028e15	−	1.40467e16i	−0.230680	−	1.21346i
$834$		0			0
$835$	5.78261e15			0.493001
$836$		0			0
$837$	4.48472e15	−	4.22969e15i	0.377351	−	0.355893i
$838$		0			0
$839$	2.55206e15			0.211934			0.105967	−	0.994370i	$-0.466206\pi$
$839$	2.55206e15			0.211934			0.105967	+	0.994370i	$0.466206\pi$
$840$		0			0
$841$	4.92593e15			0.403748
$842$		0			0
$843$	2.60917e15	+	2.66056e15i	0.211082	+	0.215239i
$844$		0			0
$845$	−8.91177e15			−0.711626
$846$		0			0
$847$	−4.07815e15	+	4.92644e15i	−0.321443	+	0.388307i
$848$		0			0
$849$	−3.26117e14	−	3.32541e14i	−0.0253735	−	0.0258733i
$850$		0			0
$851$			7.44314e14i			0.0571668i
$852$		0			0
$853$		−	2.86934e14i		−	0.0217552i	−0.999941	−	0.0108776i	$-0.996537\pi$
$853$		−	2.86934e14i		−	0.0217552i	0.999941	−	0.0108776i	$-0.00346251\pi$
$854$		0			0
$855$	5.18380e15	+	1.01121e14i	0.388003	+	0.00756881i
$856$		0			0
$857$	1.43199e16			1.05814			0.529072	−	0.848577i	$-0.322540\pi$
$857$	1.43199e16			1.05814			0.529072	+	0.848577i	$0.322540\pi$
$858$		0			0
$859$		−	1.18340e16i		−	0.863313i	−0.902038	−	0.431656i	$-0.857929\pi$
$859$		−	1.18340e16i		−	0.863313i	0.902038	−	0.431656i	$-0.142071\pi$
$860$		0			0
$861$	−3.67801e14	−	4.35904e15i	−0.0264909	−	0.313960i
$862$		0			0
$863$			8.33358e15i			0.592614i	0.955093	+	0.296307i	$0.0957552\pi$
$863$			8.33358e15i			0.592614i	−0.955093	+	0.296307i	$0.904245\pi$
$864$		0			0
$865$	8.92037e15			0.626317
$866$		0			0
$867$	−5.41411e15	+	5.30953e15i	−0.375338	+	0.368087i
$868$		0			0
$869$			9.83652e15i			0.673337i
$870$		0			0
$871$			7.97440e14i			0.0539012i
$872$		0			0
$873$	7.38194e15	+	1.44000e14i	0.492711	+	0.00961136i
$874$		0			0
$875$	−1.03161e16	+	1.24620e16i	−0.679942	+	0.821377i
$876$		0			0
$877$	2.41586e15			0.157244			0.0786220	−	0.996904i	$-0.474948\pi$
$877$	2.41586e15			0.157244			0.0786220	+	0.996904i	$0.474948\pi$
$878$		0			0
$879$	−1.47949e16	+	1.45091e16i	−0.950983	+	0.932613i
$880$		0			0
$881$	4.51210e15			0.286425			0.143213	−	0.989692i	$-0.454257\pi$
$881$	4.51210e15			0.286425			0.143213	+	0.989692i	$0.454257\pi$
$882$		0			0
$883$	9.01421e14			0.0565124			0.0282562	−	0.999601i	$-0.491005\pi$
$883$	9.01421e14			0.0565124			0.0282562	+	0.999601i	$0.491005\pi$
$884$		0			0
$885$	−7.61950e15	+	7.47232e15i	−0.471779	+	0.462666i
$886$		0			0
$887$	−3.92478e15			−0.240013			−0.120007	−	0.992773i	$-0.538292\pi$
$887$	−3.92478e15			−0.240013			−0.120007	+	0.992773i	$0.538292\pi$
$888$		0			0
$889$	1.91353e15	−	2.31156e15i	0.115578	−	0.139619i
$890$		0			0
$891$	−4.60347e14	+	1.17950e16i	−0.0274637	+	0.703672i
$892$		0			0
$893$			1.49501e15i			0.0880969i
$894$		0			0
$895$			7.07090e15i			0.411574i
$896$		0			0
$897$	−5.29103e13	+	5.18882e13i	−0.00304216	+	0.00298339i
$898$		0			0
$899$	−7.05201e15			−0.400529
$900$		0			0
$901$			8.23210e15i			0.461875i
$902$		0			0
$903$	1.83234e15	+	2.17162e16i	0.101560	+	1.20365i
$904$		0			0
$905$		−	6.22520e14i		−	0.0340868i
$906$		0			0
$907$	−3.50371e16			−1.89534			−0.947671	−	0.319248i	$-0.896570\pi$
$907$	−3.50371e16			−1.89534			−0.947671	+	0.319248i	$0.896570\pi$
$908$		0			0
$909$	5.26443e13	−	2.69872e15i	0.00281352	−	0.144230i
$910$		0			0
$911$			2.71016e16i			1.43102i	0.698604	+	0.715508i	$0.253805\pi$
$911$			2.71016e16i			1.43102i	−0.698604	+	0.715508i	$0.746195\pi$
$912$		0			0
$913$		−	1.61110e16i		−	0.840489i
$914$		0			0
$915$	6.17214e15	+	6.29372e15i	0.318141	+	0.324408i
$916$		0			0
$917$	3.87825e15	−	4.68496e15i	0.197517	−	0.238602i
$918$		0			0
$919$	−2.69025e16			−1.35381			−0.676903	−	0.736072i	$-0.736678\pi$
$919$	−2.69025e16			−1.35381			−0.676903	+	0.736072i	$0.736678\pi$
$920$		0			0
$921$	−1.95951e16	−	1.99810e16i	−0.974358	−	0.993550i
$922$		0			0
$923$	1.78586e15			0.0877481
$924$		0			0
$925$	−1.34942e16			−0.655192
$926$		0			0
$927$	1.08354e16	+	2.11368e14i	0.519886	+	0.0101415i
$928$		0			0
$929$	2.51374e16			1.19188			0.595942	−	0.803028i	$-0.296779\pi$
$929$	2.51374e16			1.19188			0.595942	+	0.803028i	$0.296779\pi$
$930$		0			0
$931$	−1.13167e16	+	2.15131e15i	−0.530268	+	0.100804i
$932$		0			0
$933$	4.63827e15	−	4.54868e15i	0.214787	−	0.210638i
$934$		0			0
$935$			1.36651e16i			0.625388i
$936$		0			0
$937$			2.00619e16i			0.907412i	0.891152	+	0.453706i	$0.149898\pi$
$937$			2.00619e16i			0.907412i	−0.891152	+	0.453706i	$0.850102\pi$
$938$		0			0
$939$	−9.85112e13	−	1.00452e14i	−0.00440377	−	0.00449051i
$940$		0			0
$941$	−7.53418e15			−0.332884			−0.166442	−	0.986051i	$-0.553228\pi$
$941$	−7.53418e15			−0.332884			−0.166442	+	0.986051i	$0.553228\pi$
$942$		0			0
$943$			3.04102e14i			0.0132802i
$944$		0			0
$945$	1.66218e16	−	1.07648e15i	0.717471	−	0.0464655i
$946$		0			0
$947$		−	1.43387e16i		−	0.611764i	−0.952069	−	0.305882i	$-0.901049\pi$
$947$		−	1.43387e16i		−	0.611764i	0.952069	−	0.305882i	$-0.0989513\pi$
$948$		0			0
$949$	4.15911e15			0.175403
$950$		0			0
$951$	1.72841e16	+	1.76246e16i	0.720535	+	0.734727i
$952$		0			0
$953$		−	9.15601e15i		−	0.377307i	−0.982044	−	0.188654i	$-0.939588\pi$
$953$		−	9.15601e15i		−	0.377307i	0.982044	−	0.188654i	$-0.0604123\pi$
$954$		0			0
$955$			2.27324e16i			0.926034i
$956$		0			0
$957$	9.64075e15	−	9.45452e15i	0.388236	−	0.380736i
$958$		0			0
$959$	3.79266e16	+	3.13959e16i	1.50988	+	1.24989i
$960$		0			0
$961$	1.85722e16			0.730947
$962$		0			0
$963$	3.38820e16	+	6.60939e14i	1.31833	+	0.0257168i
$964$		0			0
$965$	−1.96621e16			−0.756361
$966$		0			0
$967$	−1.95632e16			−0.744037			−0.372019	−	0.928225i	$-0.621334\pi$
$967$	−1.95632e16			−0.744037			−0.372019	+	0.928225i	$0.621334\pi$
$968$		0			0
$969$	1.24145e16	+	1.26590e16i	0.466818	+	0.476013i
$970$		0			0
$971$	−7.30616e15			−0.271634			−0.135817	−	0.990734i	$-0.543366\pi$
$971$	−7.30616e15			−0.271634			−0.135817	+	0.990734i	$0.543366\pi$
$972$		0			0
$973$	3.34030e15	+	2.76512e15i	0.122790	+	0.101647i
$974$		0			0
$975$	−9.40721e14	−	9.59251e14i	−0.0341929	−	0.0348664i
$976$		0			0
$977$		−	3.70876e16i		−	1.33294i	−0.745534	−	0.666468i	$-0.767805\pi$
$977$		−	3.70876e16i		−	1.33294i	0.745534	−	0.666468i	$-0.232195\pi$
$978$		0			0
$979$		−	2.93491e16i		−	1.04301i
$980$		0			0
$981$	−7.39208e14	+	3.78943e16i	−0.0259769	+	1.33166i
$982$		0			0
$983$	3.42046e16			1.18861			0.594305	−	0.804239i	$-0.297427\pi$
$983$	3.42046e16			1.18861			0.594305	+	0.804239i	$0.297427\pi$
$984$		0			0
$985$		−	2.99996e16i		−	1.03090i
$986$		0			0
$987$	4.03815e14	+	4.78586e15i	0.0137226	+	0.162636i
$988$		0			0
$989$		−	1.51500e15i		−	0.0509134i
$990$		0			0
$991$	−2.82313e16			−0.938265			−0.469133	−	0.883128i	$-0.655433\pi$
$991$	−2.82313e16			−0.938265			−0.469133	+	0.883128i	$0.655433\pi$
$992$		0			0
$993$	−1.31387e16	+	1.28849e16i	−0.431848	+	0.423506i
$994$		0			0
$995$		−	1.14108e16i		−	0.370929i
$996$		0			0
$997$			3.61089e16i			1.16089i	0.814300	+	0.580444i	$0.197121\pi$
$997$			3.61089e16i			1.16089i	−0.814300	+	0.580444i	$0.802879\pi$
$998$		0			0
$999$	2.92659e16	+	3.10305e16i	0.930577	+	0.986687i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.f.b.41.8	yes	28
3.2	odd	2	inner	84.12.f.b.41.22	yes	28
7.6	odd	2	inner	84.12.f.b.41.21	yes	28
21.20	even	2	inner	84.12.f.b.41.7	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.f.b.41.7	✓	28	21.20	even	2	inner
84.12.f.b.41.8	yes	28	1.1	even	1	trivial
84.12.f.b.41.21	yes	28	7.6	odd	2	inner
84.12.f.b.41.22	yes	28	3.2	odd	2	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:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-300.501 + 294.696i) q^{3} -5023.98 q^{5} +(-28355.3 + 34253.5i) q^{7} +(3454.97 - 177113. i) q^{9} +O(q^{10})\)	\(q+(-300.501 + 294.696i) q^{3} -5023.98 q^{5} +(-28355.3 + 34253.5i) q^{7} +(3454.97 - 177113. i) q^{9} -376150. i q^{11} -135332. i q^{13} +(1.50971e6 - 1.48055e6i) q^{15} +7.23110e6 q^{17} -5.82572e6i q^{19} +(-1.57358e6 - 1.86494e7i) q^{21} +1.30105e6i q^{23} -2.35877e7 q^{25} +(5.11564e7 + 5.42409e7i) q^{27} -8.52912e7i q^{29} -8.26815e7i q^{31} +(1.10850e8 + 1.13033e8i) q^{33} +(1.42457e8 - 1.72089e8i) q^{35} +5.72087e8 q^{37} +(3.98818e7 + 4.06674e7i) q^{39} +2.33736e8 q^{41} -1.16444e9 q^{43} +(-1.73577e7 + 8.89814e8i) q^{45} -2.56622e8 q^{47} +(-3.69278e8 - 1.94254e9i) q^{49} +(-2.17295e9 + 2.13098e9i) q^{51} +1.13843e9i q^{53} +1.88977e9i q^{55} +(1.71682e9 + 1.75063e9i) q^{57} -5.04699e9 q^{59} +4.16882e9i q^{61} +(5.96878e9 + 5.14045e9i) q^{63} +6.79904e8i q^{65} -5.89248e9 q^{67} +(-3.83415e8 - 3.90967e8i) q^{69} +1.31962e10i q^{71} +3.07327e10i q^{73} +(7.08814e9 - 6.95122e9i) q^{75} +(1.28844e10 + 1.06658e10i) q^{77} -2.61505e10 q^{79} +(-3.13572e10 - 1.22384e9i) q^{81} +4.28312e10 q^{83} -3.63289e10 q^{85} +(2.51350e10 + 2.56301e10i) q^{87} +7.80249e10 q^{89} +(4.63559e9 + 3.83738e9i) q^{91} +(2.43660e10 + 2.48459e10i) q^{93} +2.92683e10i q^{95} +4.16792e10i q^{97} +(-6.66211e10 - 1.29958e9i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) 28 * q + 9632 * q^7 + 267660 * q^9	\( 28 q + 9632 q^{7} + 267660 q^{9} - 3434160 q^{15} - 18804156 q^{21} + 397876900 q^{25} - 2059460504 q^{37} + 2276313936 q^{39} + 607100560 q^{43} + 1145242588 q^{49} + 1424787216 q^{51} - 32512522344 q^{57} + 16390616256 q^{63} - 48876957136 q^{67} - 1293110368 q^{79} + 82706814108 q^{81} + 197440859760 q^{85} - 329206232880 q^{91} - 243855044280 q^{93} - 81383696064 q^{99}+O(q^{100}) \) 28 * q + 9632 * q^7 + 267660 * q^9 - 3434160 * q^15 - 18804156 * q^21 + 397876900 * q^25 - 2059460504 * q^37 + 2276313936 * q^39 + 607100560 * q^43 + 1145242588 * q^49 + 1424787216 * q^51 - 32512522344 * q^57 + 16390616256 * q^63 - 48876957136 * q^67 - 1293110368 * q^79 + 82706814108 * q^81 + 197440859760 * q^85 - 329206232880 * q^91 - 243855044280 * q^93 - 81383696064 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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