LMFDB - Embedded newform 84.12.f.b.41.1

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			41.1
Character	$\chi$	$=$	84.41
Dual form			84.12.f.b.41.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-419.400 - 35.3635i) q^{3} +9428.81 q^{5} +(19241.0 + 40088.8i) q^{7} +(174646. + 29662.9i) q^{9} +O(q^{10})$	\(q+(-419.400 - 35.3635i) q^{3} +9428.81 q^{5} +(19241.0 + 40088.8i) q^{7} +(174646. + 29662.9i) q^{9} +905231. i q^{11} +942810. i q^{13} +(-3.95444e6 - 333436. i) q^{15} +5.13698e6 q^{17} +6.72467e6i q^{19} +(-6.65201e6 - 1.74937e7i) q^{21} +1.46770e7i q^{23} +4.00744e7 q^{25} +(-7.21975e7 - 1.86167e7i) q^{27} -4.78194e7i q^{29} -2.52543e8i q^{31} +(3.20121e7 - 3.79654e8i) q^{33} +(1.81420e8 + 3.77990e8i) q^{35} +6.12574e8 q^{37} +(3.33410e7 - 3.95414e8i) q^{39} -6.04843e8 q^{41} -2.28564e8 q^{43} +(1.64670e9 + 2.79686e8i) q^{45} -2.41283e9 q^{47} +(-1.23689e9 + 1.54270e9i) q^{49} +(-2.15445e9 - 1.81662e8i) q^{51} +5.67867e9i q^{53} +8.53525e9i q^{55} +(2.37808e8 - 2.82033e9i) q^{57} +1.15550e9 q^{59} -9.53454e9i q^{61} +(2.17121e9 + 7.57208e9i) q^{63} +8.88958e9i q^{65} +3.19178e9 q^{67} +(5.19029e8 - 6.15552e9i) q^{69} +2.41537e10i q^{71} -1.89129e10i q^{73} +(-1.68072e10 - 1.41717e9i) q^{75} +(-3.62896e10 + 1.74176e10i) q^{77} -2.36389e10 q^{79} +(2.96213e10 + 1.03610e10i) q^{81} +1.59837e10 q^{83} +4.84356e10 q^{85} +(-1.69106e9 + 2.00555e10i) q^{87} -4.14113e10 q^{89} +(-3.77961e10 + 1.81406e10i) q^{91} +(-8.93079e9 + 1.05916e11i) q^{93} +6.34056e10i q^{95} -4.99510e10i q^{97} +(-2.68518e10 + 1.58095e11i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−419.400	−	35.3635i	−0.996464	−	0.0840211i
$3$	−419.400	−	35.3635i	−0.996464	−	0.0840211i
$4$		0			0
$5$	9428.81			1.34934			0.674671	−	0.738119i	$-0.264286\pi$
$5$	9428.81			1.34934			0.674671	+	0.738119i	$0.264286\pi$
$6$		0			0
$7$	19241.0	+	40088.8i	0.432702	+	0.901537i
$7$	19241.0	+	40088.8i	0.432702	+	0.901537i
$8$		0			0
$9$	174646.	+	29662.9i	0.985881	+	0.167448i
$10$		0			0
$11$			905231.i			1.69473i	0.531014	+	0.847363i	$0.321811\pi$
$11$			905231.i			1.69473i	−0.531014	+	0.847363i	$0.678189\pi$
$12$		0			0
$13$			942810.i			0.704264i	0.935950	+	0.352132i	$0.114543\pi$
$13$			942810.i			0.704264i	−0.935950	+	0.352132i	$0.885457\pi$
$14$		0			0
$15$	−3.95444e6	−	333436.i	−1.34457	−	0.113373i
$16$		0			0
$17$	5.13698e6			0.877483			0.438742	−	0.898613i	$-0.355424\pi$
$17$	5.13698e6			0.877483			0.438742	+	0.898613i	$0.355424\pi$
$18$		0			0
$19$			6.72467e6i			0.623054i	0.950237	+	0.311527i	$0.100840\pi$
$19$			6.72467e6i			0.623054i	−0.950237	+	0.311527i	$0.899160\pi$
$20$		0			0
$21$	−6.65201e6	−	1.74937e7i	−0.355424	−	0.934705i
$22$		0			0
$23$			1.46770e7i			0.475481i	0.971329	+	0.237740i	$0.0764068\pi$
$23$			1.46770e7i			0.475481i	−0.971329	+	0.237740i	$0.923593\pi$
$24$		0			0
$25$	4.00744e7			0.820723
$26$		0			0
$27$	−7.21975e7	−	1.86167e7i	−0.968326	−	0.249691i
$28$		0			0
$29$		−	4.78194e7i		−	0.432927i	−0.976291	−	0.216464i	$-0.930548\pi$
$29$		−	4.78194e7i		−	0.432927i	0.976291	−	0.216464i	$-0.0694523\pi$
$30$		0			0
$31$		−	2.52543e8i		−	1.58433i	−0.610307	−	0.792165i	$-0.708954\pi$
$31$		−	2.52543e8i		−	1.58433i	0.610307	−	0.792165i	$-0.291046\pi$
$32$		0			0
$33$	3.20121e7	−	3.79654e8i	0.142393	−	1.68873i
$34$		0			0
$35$	1.81420e8	+	3.77990e8i	0.583863	+	1.21648i
$36$		0			0
$37$	6.12574e8			1.45227			0.726137	−	0.687550i	$-0.241314\pi$
$37$	6.12574e8			1.45227			0.726137	+	0.687550i	$0.241314\pi$
$38$		0			0
$39$	3.33410e7	−	3.95414e8i	0.0591730	−	0.701774i
$40$		0			0
$41$	−6.04843e8			−0.815326			−0.407663	−	0.913132i	$-0.633656\pi$
$41$	−6.04843e8			−0.815326			−0.407663	+	0.913132i	$0.633656\pi$
$42$		0			0
$43$	−2.28564e8			−0.237100			−0.118550	−	0.992948i	$-0.537825\pi$
$43$	−2.28564e8			−0.237100			−0.118550	+	0.992948i	$0.537825\pi$
$44$		0			0
$45$	1.64670e9	+	2.79686e8i	1.33029	+	0.225945i
$46$		0			0
$47$	−2.41283e9			−1.53458			−0.767288	−	0.641303i	$-0.778394\pi$
$47$	−2.41283e9			−1.53458			−0.767288	+	0.641303i	$0.778394\pi$
$48$		0			0
$49$	−1.23689e9	+	1.54270e9i	−0.625538	+	0.780194i
$50$		0			0
$51$	−2.15445e9	−	1.81662e8i	−0.874380	−	0.0737271i
$52$		0			0
$53$			5.67867e9i			1.86522i	0.360889	+	0.932609i	$0.382473\pi$
$53$			5.67867e9i			1.86522i	−0.360889	+	0.932609i	$0.617527\pi$
$54$		0			0
$55$			8.53525e9i			2.28677i
$56$		0			0
$57$	2.37808e8	−	2.82033e9i	0.0523497	−	0.620851i
$58$		0			0
$59$	1.15550e9			0.210418			0.105209	−	0.994450i	$-0.466449\pi$
$59$	1.15550e9			0.210418			0.105209	+	0.994450i	$0.466449\pi$
$60$		0			0
$61$		−	9.53454e9i		−	1.44539i	−0.691166	−	0.722696i	$-0.742903\pi$
$61$		−	9.53454e9i		−	1.44539i	0.691166	−	0.722696i	$-0.257097\pi$
$62$		0			0
$63$	2.17121e9	+	7.57208e9i	0.275632	+	0.961263i
$64$		0			0
$65$			8.88958e9i			0.950293i
$66$		0			0
$67$	3.19178e9			0.288816			0.144408	−	0.989518i	$-0.453872\pi$
$67$	3.19178e9			0.288816			0.144408	+	0.989518i	$0.453872\pi$
$68$		0			0
$69$	5.19029e8	−	6.15552e9i	0.0399504	−	0.473800i
$70$		0			0
$71$			2.41537e10i			1.58878i	0.607411	+	0.794388i	$0.292208\pi$
$71$			2.41537e10i			1.58878i	−0.607411	+	0.794388i	$0.707792\pi$
$72$		0			0
$73$		−	1.89129e10i		−	1.06778i	−0.845553	−	0.533891i	$-0.820729\pi$
$73$		−	1.89129e10i		−	1.06778i	0.845553	−	0.533891i	$-0.179271\pi$
$74$		0			0
$75$	−1.68072e10	−	1.41717e9i	−0.817821	−	0.0689581i
$76$		0			0
$77$	−3.62896e10	+	1.74176e10i	−1.52786	+	0.733312i
$78$		0			0
$79$	−2.36389e10			−0.864326			−0.432163	−	0.901796i	$-0.642249\pi$
$79$	−2.36389e10			−0.864326			−0.432163	+	0.901796i	$0.642249\pi$
$80$		0			0
$81$	2.96213e10	+	1.03610e10i	0.943922	+	0.330167i
$82$		0			0
$83$	1.59837e10			0.445397			0.222699	−	0.974887i	$-0.428513\pi$
$83$	1.59837e10			0.445397			0.222699	+	0.974887i	$0.428513\pi$
$84$		0			0
$85$	4.84356e10			1.18402
$86$		0			0
$87$	−1.69106e9	+	2.00555e10i	−0.0363750	+	0.431397i
$88$		0			0
$89$	−4.14113e10			−0.786093			−0.393046	−	0.919519i	$-0.628579\pi$
$89$	−4.14113e10			−0.786093			−0.393046	+	0.919519i	$0.628579\pi$
$90$		0			0
$91$	−3.77961e10	+	1.81406e10i	−0.634920	+	0.304736i
$92$		0			0
$93$	−8.93079e9	+	1.05916e11i	−0.133117	+	1.57873i
$94$		0			0
$95$			6.34056e10i			0.840713i
$96$		0			0
$97$		−	4.99510e10i		−	0.590609i	−0.955403	−	0.295304i	$-0.904579\pi$
$97$		−	4.99510e10i		−	0.590609i	0.955403	−	0.295304i	$-0.0954210\pi$
$98$		0			0
$99$	−2.68518e10	+	1.58095e11i	−0.283778	+	1.67080i
$100$		0			0
$101$	−5.52327e10			−0.522912			−0.261456	−	0.965215i	$-0.584203\pi$
$101$	−5.52327e10			−0.522912			−0.261456	+	0.965215i	$0.584203\pi$
$102$		0			0
$103$			1.37426e11i			1.16806i	0.811733	+	0.584028i	$0.198524\pi$
$103$			1.37426e11i			1.16806i	−0.811733	+	0.584028i	$0.801476\pi$
$104$		0			0
$105$	−6.27205e10	−	1.64944e11i	−0.479588	−	1.26124i
$106$		0			0
$107$			8.65239e10i			0.596383i	0.954506	+	0.298192i	$0.0963835\pi$
$107$			8.65239e10i			0.596383i	−0.954506	+	0.298192i	$0.903617\pi$
$108$		0			0
$109$	2.75980e11			1.71804			0.859019	−	0.511944i	$-0.171075\pi$
$109$	2.75980e11			1.71804			0.859019	+	0.511944i	$0.171075\pi$
$110$		0			0
$111$	−2.56913e11	−	2.16627e10i	−1.44714	−	0.122022i
$112$		0			0
$113$		−	5.49719e10i		−	0.280679i	−0.990103	−	0.140339i	$-0.955181\pi$
$113$		−	5.49719e10i		−	0.280679i	0.990103	−	0.140339i	$-0.0448194\pi$
$114$		0			0
$115$			1.38386e11i			0.641586i
$116$		0			0
$117$	−2.79665e10	+	1.64658e11i	−0.117928	+	0.694321i
$118$		0			0
$119$	9.88407e10	+	2.05935e11i	0.379689	+	0.791083i
$120$		0			0
$121$	−5.34132e11			−1.87210
$122$		0			0
$123$	2.53671e11	+	2.13893e10i	0.812443	+	0.0685046i
$124$		0			0
$125$	−8.25374e10			−0.241905
$126$		0			0
$127$	1.70590e11			0.458177			0.229088	−	0.973406i	$-0.426426\pi$
$127$	1.70590e11			0.458177			0.229088	+	0.973406i	$0.426426\pi$
$128$		0			0
$129$	9.58596e10	+	8.08281e9i	0.236261	+	0.0199214i
$130$		0			0
$131$	3.60458e11			0.816324			0.408162	−	0.912910i	$-0.366170\pi$
$131$	3.60458e11			0.816324			0.408162	+	0.912910i	$0.366170\pi$
$132$		0			0
$133$	−2.69584e11	+	1.29389e11i	−0.561707	+	0.269597i
$134$		0			0
$135$	−6.80737e11	−	1.75533e11i	−1.30660	−	0.336918i
$136$		0			0
$137$			3.63783e11i			0.643991i	0.946741	+	0.321995i	$0.104354\pi$
$137$			3.63783e11i			0.643991i	−0.946741	+	0.321995i	$0.895646\pi$
$138$		0			0
$139$			5.35844e10i			0.0875905i	0.999041	+	0.0437953i	$0.0139449\pi$
$139$			5.35844e10i			0.0875905i	−0.999041	+	0.0437953i	$0.986055\pi$
$140$		0			0
$141$	1.01194e12	+	8.53261e10i	1.52915	+	0.128937i
$142$		0			0
$143$	−8.53461e11			−1.19354
$144$		0			0
$145$		−	4.50880e11i		−	0.584167i
$146$		0			0
$147$	5.73308e11	−	6.03267e11i	0.688879	−	0.724877i
$148$		0			0
$149$		−	1.16137e11i		−	0.129552i	−0.997900	−	0.0647762i	$-0.979367\pi$
$149$		−	1.16137e11i		−	0.129552i	0.997900	−	0.0647762i	$-0.0206333\pi$
$150$		0			0
$151$	1.23125e12			1.27636			0.638180	−	0.769887i	$-0.279688\pi$
$151$	1.23125e12			1.27636			0.638180	+	0.769887i	$0.279688\pi$
$152$		0			0
$153$	8.97152e11	+	1.52378e11i	0.865094	+	0.146933i
$154$		0			0
$155$		−	2.38118e12i		−	2.13780i
$156$		0			0
$157$		−	1.22558e12i		−	1.02540i	−0.858568	−	0.512700i	$-0.828645\pi$
$157$		−	1.22558e12i		−	1.02540i	0.858568	−	0.512700i	$-0.171355\pi$
$158$		0			0
$159$	2.00818e11	−	2.38164e12i	0.156718	−	1.85862i
$160$		0			0
$161$	−5.88382e11	+	2.82400e11i	−0.428664	+	0.205742i
$162$		0			0
$163$	−2.92669e12			−1.99225			−0.996127	−	0.0879271i	$-0.971976\pi$
$163$	−2.92669e12			−1.99225			−0.996127	+	0.0879271i	$0.971976\pi$
$164$		0			0
$165$	3.01836e11	−	3.57969e12i	0.192136	−	2.27868i
$166$		0			0
$167$	1.44207e12			0.859105			0.429552	−	0.903042i	$-0.358671\pi$
$167$	1.44207e12			0.859105			0.429552	+	0.903042i	$0.358671\pi$
$168$		0			0
$169$	9.03270e11			0.504012
$170$		0			0
$171$	−1.99473e11	+	1.17444e12i	−0.104329	+	0.614257i
$172$		0			0
$173$	4.86331e11			0.238604			0.119302	−	0.992858i	$-0.461934\pi$
$173$	4.86331e11			0.238604			0.119302	+	0.992858i	$0.461934\pi$
$174$		0			0
$175$	7.71072e11	+	1.60653e12i	0.355129	+	0.739913i
$176$		0			0
$177$	−4.84616e11	−	4.08624e10i	−0.209674	−	0.0176796i
$178$		0			0
$179$			3.55318e10i			0.0144519i	0.999974	+	0.00722596i	$0.00230011\pi$
$179$			3.55318e10i			0.0144519i	−0.999974	+	0.00722596i	$0.997700\pi$
$180$		0			0
$181$			3.40714e12i			1.30364i	0.758374	+	0.651820i	$0.225994\pi$
$181$			3.40714e12i			1.30364i	−0.758374	+	0.651820i	$0.774006\pi$
$182$		0			0
$183$	−3.37175e11	+	3.99879e12i	−0.121443	+	1.44028i
$184$		0			0
$185$	5.77584e12			1.95962
$186$		0			0
$187$			4.65015e12i			1.48709i
$188$		0			0
$189$	−6.42832e11	−	3.25251e12i	−0.193891	−	0.981023i
$190$		0			0
$191$		−	3.72339e12i		−	1.05988i	−0.848037	−	0.529938i	$-0.822215\pi$
$191$		−	3.72339e12i		−	1.05988i	0.848037	−	0.529938i	$-0.177785\pi$
$192$		0			0
$193$	−5.44657e12			−1.46406			−0.732028	−	0.681274i	$-0.761426\pi$
$193$	−5.44657e12			−1.46406			−0.732028	+	0.681274i	$0.761426\pi$
$194$		0			0
$195$	3.14366e11	−	3.72829e12i	0.0798446	−	0.946933i
$196$		0			0
$197$		−	6.53458e12i		−	1.56911i	−0.620059	−	0.784556i	$-0.712891\pi$
$197$		−	6.53458e12i		−	1.56911i	0.620059	−	0.784556i	$-0.287109\pi$
$198$		0			0
$199$			3.50299e12i			0.795696i	0.917451	+	0.397848i	$0.130243\pi$
$199$			3.50299e12i			0.795696i	−0.917451	+	0.397848i	$0.869757\pi$
$200$		0			0
$201$	−1.33863e12	−	1.12872e11i	−0.287795	−	0.0242666i
$202$		0			0
$203$	1.91702e12	−	9.20094e11i	0.390300	−	0.187329i
$204$		0			0
$205$	−5.70295e12			−1.10015
$206$		0			0
$207$	−4.35361e11	+	2.56327e12i	−0.0796183	+	0.468768i
$208$		0			0
$209$	−6.08738e12			−1.05591
$210$		0			0
$211$	7.88195e12			1.29742			0.648709	−	0.761036i	$-0.275309\pi$
$211$	7.88195e12			1.29742			0.648709	+	0.761036i	$0.275309\pi$
$212$		0			0
$213$	8.54159e11	−	1.01301e13i	0.133491	−	1.58316i
$214$		0			0
$215$	−2.15508e12			−0.319928
$216$		0			0
$217$	1.01241e13	−	4.85918e12i	1.42833	−	0.685542i
$218$		0			0
$219$	−6.68827e11	+	7.93208e12i	−0.0897162	+	1.06401i
$220$		0			0
$221$			4.84319e12i			0.617980i
$222$		0			0
$223$		−	1.14396e13i		−	1.38910i	−0.719443	−	0.694552i	$-0.755603\pi$
$223$		−	1.14396e13i		−	1.38910i	0.719443	−	0.694552i	$-0.244397\pi$
$224$		0			0
$225$	6.99882e12	+	1.18872e12i	0.809136	+	0.137428i
$226$		0			0
$227$	−3.09774e12			−0.341117			−0.170558	−	0.985348i	$-0.554557\pi$
$227$	−3.09774e12			−0.341117			−0.170558	+	0.985348i	$0.554557\pi$
$228$		0			0
$229$			9.29820e12i			0.975672i	0.872935	+	0.487836i	$0.162214\pi$
$229$			9.29820e12i			0.975672i	−0.872935	+	0.487836i	$0.837786\pi$
$230$		0			0
$231$	1.58358e13	−	6.02160e12i	1.58407	−	0.602346i
$232$		0			0
$233$			7.25338e12i			0.691963i	0.938241	+	0.345982i	$0.112454\pi$
$233$			7.25338e12i			0.691963i	−0.938241	+	0.345982i	$0.887546\pi$
$234$		0			0
$235$	−2.27501e13			−2.07067
$236$		0			0
$237$	9.91414e12	+	8.35952e11i	0.861269	+	0.0726216i
$238$		0			0
$239$			7.23709e12i			0.600310i	0.953890	+	0.300155i	$0.0970384\pi$
$239$			7.23709e12i			0.600310i	−0.953890	+	0.300155i	$0.902962\pi$
$240$		0			0
$241$			1.58692e13i			1.25736i	0.777664	+	0.628681i	$0.216405\pi$
$241$			1.58692e13i			1.25736i	−0.777664	+	0.628681i	$0.783595\pi$
$242$		0			0
$243$	−1.20568e13	−	5.39292e12i	−0.912844	−	0.408309i
$244$		0			0
$245$	−1.16624e13	+	1.45458e13i	−0.844065	+	1.05275i
$246$		0			0
$247$	−6.34008e12			−0.438795
$248$		0			0
$249$	−6.70356e12	−	5.65239e11i	−0.443822	−	0.0374227i
$250$		0			0
$251$	−7.18785e12			−0.455400			−0.227700	−	0.973731i	$-0.573121\pi$
$251$	−7.18785e12			−0.455400			−0.227700	+	0.973731i	$0.573121\pi$
$252$		0			0
$253$	−1.32860e13			−0.805810
$254$		0			0
$255$	−2.03139e13	−	1.71285e12i	−1.17984	−	0.0994830i
$256$		0			0
$257$	2.70093e13			1.50273			0.751366	−	0.659885i	$-0.229395\pi$
$257$	2.70093e13			1.50273			0.751366	+	0.659885i	$0.229395\pi$
$258$		0			0
$259$	1.17865e13	+	2.45573e13i	0.628402	+	1.30928i
$260$		0			0
$261$	1.41846e12	−	8.35146e12i	0.0724928	−	0.426815i
$262$		0			0
$263$			2.89025e13i			1.41638i	0.706024	+	0.708188i	$0.250487\pi$
$263$			2.89025e13i			1.41638i	−0.706024	+	0.708188i	$0.749513\pi$
$264$		0			0
$265$			5.35431e13i			2.51682i
$266$		0			0
$267$	1.73679e13	+	1.46445e12i	0.783313	+	0.0660483i
$268$		0			0
$269$	−3.26794e13			−1.41461			−0.707305	−	0.706909i	$-0.750089\pi$
$269$	−3.26794e13			−1.41461			−0.707305	+	0.706909i	$0.750089\pi$
$270$		0			0
$271$		−	1.67549e13i		−	0.696324i	−0.937434	−	0.348162i	$-0.886806\pi$
$271$		−	1.67549e13i		−	0.696324i	0.937434	−	0.348162i	$-0.113194\pi$
$272$		0			0
$273$	1.64932e13	−	6.27158e12i	0.658279	−	0.250312i
$274$		0			0
$275$			3.62766e13i			1.39090i
$276$		0			0
$277$	−1.24308e13			−0.457995			−0.228997	−	0.973427i	$-0.573545\pi$
$277$	−1.24308e13			−0.457995			−0.228997	+	0.973427i	$0.573545\pi$
$278$		0			0
$279$	7.49115e12	−	4.41055e13i	0.265293	−	1.56196i
$280$		0			0
$281$			2.63675e13i			0.897810i	0.893580	+	0.448905i	$0.148186\pi$
$281$			2.63675e13i			0.897810i	−0.893580	+	0.448905i	$0.851814\pi$
$282$		0			0
$283$			1.94917e13i			0.638300i	0.947704	+	0.319150i	$0.103397\pi$
$283$			1.94917e13i			0.638300i	−0.947704	+	0.319150i	$0.896603\pi$
$284$		0			0
$285$	2.24224e12	−	2.65923e13i	0.0706376	−	0.837741i
$286$		0			0
$287$	−1.16378e13	−	2.42474e13i	−0.352793	−	0.735046i
$288$		0			0
$289$	−7.88334e12			−0.230024
$290$		0			0
$291$	−1.76644e12	+	2.09495e13i	−0.0496236	+	0.588520i
$292$		0			0
$293$	−5.08759e11			−0.0137639			−0.00688193	−	0.999976i	$-0.502191\pi$
$293$	−5.08759e11			−0.0137639			−0.00688193	+	0.999976i	$0.502191\pi$
$294$		0			0
$295$	1.08950e13			0.283926
$296$		0			0
$297$	1.68524e13	−	6.53554e13i	0.423157	−	1.64105i
$298$		0			0
$299$	−1.38376e13			−0.334864
$300$		0			0
$301$	−4.39780e12	−	9.16284e12i	−0.102593	−	0.213754i
$302$		0			0
$303$	2.31646e13	+	1.95322e12i	0.521063	+	0.0439357i
$304$		0			0
$305$		−	8.98994e13i		−	1.95033i
$306$		0			0
$307$			5.04948e13i			1.05678i	0.849001	+	0.528391i	$0.177205\pi$
$307$			5.04948e13i			1.05678i	−0.849001	+	0.528391i	$0.822795\pi$
$308$		0			0
$309$	4.85986e12	−	5.76365e13i	0.0981414	−	1.16393i
$310$		0			0
$311$	4.82736e13			0.940866			0.470433	−	0.882436i	$-0.344098\pi$
$311$	4.82736e13			0.940866			0.470433	+	0.882436i	$0.344098\pi$
$312$		0			0
$313$			1.49468e13i			0.281225i	0.990065	+	0.140613i	$0.0449072\pi$
$313$			1.49468e13i			0.281225i	−0.990065	+	0.140613i	$0.955093\pi$
$314$		0			0
$315$	2.04720e13	+	7.13957e13i	0.371922	+	1.29707i
$316$		0			0
$317$		−	7.91166e13i		−	1.38817i	−0.719895	−	0.694083i	$-0.755810\pi$
$317$		−	7.91166e13i		−	1.38817i	0.719895	−	0.694083i	$-0.244190\pi$
$318$		0			0
$319$	4.32876e13			0.733694
$320$		0			0
$321$	3.05979e12	−	3.62882e13i	0.0501088	−	0.594275i
$322$		0			0
$323$			3.45445e13i			0.546720i
$324$		0			0
$325$			3.77825e13i			0.578006i
$326$		0			0
$327$	−1.15746e14	−	9.75963e12i	−1.71196	−	0.144351i
$328$		0			0
$329$	−4.64253e13	−	9.67274e13i	−0.664014	−	1.38348i
$330$		0			0
$331$	−1.30136e14			−1.80030			−0.900149	−	0.435581i	$-0.856543\pi$
$331$	−1.30136e14			−1.80030			−0.900149	+	0.435581i	$0.856543\pi$
$332$		0			0
$333$	1.06983e14	+	1.81707e13i	1.43177	+	0.243180i
$334$		0			0
$335$	3.00947e13			0.389712
$336$		0			0
$337$	−3.61454e13			−0.452990			−0.226495	−	0.974012i	$-0.572727\pi$
$337$	−3.61454e13			−0.452990			−0.226495	+	0.974012i	$0.572727\pi$
$338$		0			0
$339$	−1.94400e12	+	2.30552e13i	−0.0235829	+	0.279686i
$340$		0			0
$341$	2.28609e14			2.68501
$342$		0			0
$343$	−8.56440e13	−	1.99024e13i	−0.974045	−	0.226354i
$344$		0			0
$345$	4.89382e12	−	5.80392e13i	0.0539068	−	0.639318i
$346$		0			0
$347$		−	8.26658e13i		−	0.882092i	−0.897485	−	0.441046i	$-0.854608\pi$
$347$		−	8.26658e13i		−	0.882092i	0.897485	−	0.441046i	$-0.145392\pi$
$348$		0			0
$349$		−	1.43806e14i		−	1.48674i	−0.668879	−	0.743372i	$-0.733225\pi$
$349$		−	1.43806e14i		−	1.48674i	0.668879	−	0.743372i	$-0.266775\pi$
$350$		0			0
$351$	1.75520e13	−	6.80685e13i	0.175848	−	0.681957i
$352$		0			0
$353$	5.84688e13			0.567758			0.283879	−	0.958860i	$-0.408379\pi$
$353$	5.84688e13			0.567758			0.283879	+	0.958860i	$0.408379\pi$
$354$		0			0
$355$			2.27741e14i			2.14380i
$356$		0			0
$357$	−3.41712e13	−	8.98646e13i	−0.311878	−	0.820188i
$358$		0			0
$359$			3.10412e13i			0.274738i	0.990520	+	0.137369i	$0.0438647\pi$
$359$			3.10412e13i			0.274738i	−0.990520	+	0.137369i	$0.956135\pi$
$360$		0			0
$361$	7.12691e13			0.611803
$362$		0			0
$363$	2.24015e14	+	1.88888e13i	1.86548	+	0.157296i
$364$		0			0
$365$		−	1.78326e14i		−	1.44080i
$366$		0			0
$367$			2.43232e12i			0.0190703i	0.999955	+	0.00953516i	$0.00303518\pi$
$367$			2.43232e12i			0.0190703i	−0.999955	+	0.00953516i	$0.996965\pi$
$368$		0			0
$369$	−1.05633e14	−	1.79414e13i	−0.803814	−	0.136525i
$370$		0			0
$371$	−2.27651e14	+	1.09263e14i	−1.68156	+	0.807083i
$372$		0			0
$373$	2.64914e14			1.89979			0.949895	−	0.312569i	$-0.101189\pi$
$373$	2.64914e14			1.89979			0.949895	+	0.312569i	$0.101189\pi$
$374$		0			0
$375$	3.46162e13	+	2.91881e12i	0.241050	+	0.0203252i
$376$		0			0
$377$	4.50846e13			0.304895
$378$		0			0
$379$	−9.57107e13			−0.628702			−0.314351	−	0.949307i	$-0.601787\pi$
$379$	−9.57107e13			−0.628702			−0.314351	+	0.949307i	$0.601787\pi$
$380$		0			0
$381$	−7.15454e13	−	6.03266e12i	−0.456557	−	0.0384965i
$382$		0			0
$383$	4.03897e13			0.250425			0.125212	−	0.992130i	$-0.460039\pi$
$383$	4.03897e13			0.250425			0.125212	+	0.992130i	$0.460039\pi$
$384$		0			0
$385$	−3.42168e14	+	1.64227e14i	−2.06160	+	0.989488i
$386$		0			0
$387$	−3.99177e13	−	6.77986e12i	−0.233752	−	0.0397018i
$388$		0			0
$389$		−	7.87616e12i		−	0.0448324i	−0.999749	−	0.0224162i	$-0.992864\pi$
$389$		−	7.87616e12i		−	0.0448324i	0.999749	−	0.0224162i	$-0.00713589\pi$
$390$		0			0
$391$			7.53953e13i			0.417226i
$392$		0			0
$393$	−1.51176e14	−	1.27470e13i	−0.813437	−	0.0685884i
$394$		0			0
$395$	−2.22886e14			−1.16627
$396$		0			0
$397$		−	1.43086e14i		−	0.728196i	−0.931361	−	0.364098i	$-0.881377\pi$
$397$		−	1.43086e14i		−	0.728196i	0.931361	−	0.364098i	$-0.118623\pi$
$398$		0			0
$399$	1.17639e14	−	4.47325e13i	0.582372	−	0.221448i
$400$		0			0
$401$		−	1.07986e14i		−	0.520082i	−0.965598	−	0.260041i	$-0.916264\pi$
$401$		−	1.07986e14i		−	0.520082i	0.965598	−	0.260041i	$-0.0837361\pi$
$402$		0			0
$403$	2.38100e14			1.11579
$404$		0			0
$405$	2.79294e14	+	9.76920e13i	1.27367	+	0.445509i
$406$		0			0
$407$			5.54521e14i			2.46121i
$408$		0			0
$409$			5.22456e13i			0.225721i	0.993611	+	0.112860i	$0.0360013\pi$
$409$			5.22456e13i			0.225721i	−0.993611	+	0.112860i	$0.963999\pi$
$410$		0			0
$411$	1.28646e13	−	1.52571e14i	0.0541088	−	0.641714i
$412$		0			0
$413$	2.22330e13	+	4.63225e13i	0.0910483	+	0.189700i
$414$		0			0
$415$	1.50707e14			0.600993
$416$		0			0
$417$	1.89493e12	−	2.24733e13i	0.00735945	−	0.0872808i
$418$		0			0
$419$	−2.79280e14			−1.05648			−0.528241	−	0.849094i	$-0.677148\pi$
$419$	−2.79280e14			−1.05648			−0.528241	+	0.849094i	$0.677148\pi$
$420$		0			0
$421$	−1.40126e14			−0.516378			−0.258189	−	0.966094i	$-0.583126\pi$
$421$	−1.40126e14			−0.516378			−0.258189	+	0.966094i	$0.583126\pi$
$422$		0			0
$423$	−4.21391e14	−	7.15715e13i	−1.51291	−	0.256962i
$424$		0			0
$425$	2.05861e14			0.720171
$426$		0			0
$427$	3.82228e14	−	1.83454e14i	1.30307	−	0.625424i
$428$		0			0
$429$	3.57941e14	+	3.01813e13i	1.18931	+	0.100282i
$430$		0			0
$431$			4.84787e14i			1.57009i	0.619436	+	0.785047i	$0.287361\pi$
$431$			4.84787e14i			1.57009i	−0.619436	+	0.785047i	$0.712639\pi$
$432$		0			0
$433$		−	4.05023e13i		−	0.127878i	−0.997954	−	0.0639390i	$-0.979634\pi$
$433$		−	4.05023e13i		−	0.127878i	0.997954	−	0.0639390i	$-0.0203663\pi$
$434$		0			0
$435$	−1.59447e13	+	1.89099e14i	−0.0490823	+	0.582101i
$436$		0			0
$437$	−9.86977e13			−0.296250
$438$		0			0
$439$		−	5.17260e14i		−	1.51410i	−0.653358	−	0.757049i	$-0.726640\pi$
$439$		−	5.17260e14i		−	1.51410i	0.653358	−	0.757049i	$-0.273360\pi$
$440$		0			0
$441$	−2.61779e14	+	2.32736e14i	−0.747348	+	0.664433i
$442$		0			0
$443$			4.56833e14i			1.27215i	0.771629	+	0.636073i	$0.219442\pi$
$443$			4.56833e14i			1.27215i	−0.771629	+	0.636073i	$0.780558\pi$
$444$		0			0
$445$	−3.90459e14			−1.06071
$446$		0			0
$447$	−4.10700e12	+	4.87078e13i	−0.0108851	+	0.129094i
$448$		0			0
$449$		−	2.72548e14i		−	0.704835i	−0.935843	−	0.352418i	$-0.885360\pi$
$449$		−	2.72548e14i		−	0.704835i	0.935843	−	0.352418i	$-0.114640\pi$
$450$		0			0
$451$		−	5.47522e14i		−	1.38175i
$452$		0			0
$453$	−5.16387e14	−	4.35413e13i	−1.27185	−	0.107241i
$454$		0			0
$455$	−3.56372e14	+	1.71045e14i	−0.856724	+	0.411194i
$456$		0			0
$457$	4.82915e14			1.13327			0.566633	−	0.823970i	$-0.308246\pi$
$457$	4.82915e14			1.13327			0.566633	+	0.823970i	$0.308246\pi$
$458$		0			0
$459$	−3.70877e14	−	9.56337e13i	−0.849689	−	0.219099i
$460$		0			0
$461$	6.26006e13			0.140031			0.0700154	−	0.997546i	$-0.477695\pi$
$461$	6.26006e13			0.140031			0.0700154	+	0.997546i	$0.477695\pi$
$462$		0			0
$463$	8.34527e13			0.182282			0.0911412	−	0.995838i	$-0.470949\pi$
$463$	8.34527e13			0.182282			0.0911412	+	0.995838i	$0.470949\pi$
$464$		0			0
$465$	−8.42068e13	+	9.98666e14i	−0.179620	+	2.13024i
$466$		0			0
$467$	6.52502e14			1.35937			0.679687	−	0.733502i	$-0.262115\pi$
$467$	6.52502e14			1.35937			0.679687	+	0.733502i	$0.262115\pi$
$468$		0			0
$469$	6.14131e13	+	1.27955e14i	0.124971	+	0.260378i
$470$		0			0
$471$	−4.33408e13	+	5.14008e14i	−0.0861552	+	1.02177i
$472$		0			0
$473$		−	2.06903e14i		−	0.401819i
$474$		0			0
$475$			2.69487e14i			0.511355i
$476$		0			0
$477$	−1.68446e14	+	9.91756e14i	−0.312327	+	1.83888i
$478$		0			0
$479$	−4.65482e14			−0.843447			−0.421723	−	0.906725i	$-0.638575\pi$
$479$	−4.65482e14			−0.843447			−0.421723	+	0.906725i	$0.638575\pi$
$480$		0			0
$481$			5.77540e14i			1.02279i
$482$		0			0
$483$	2.56754e14	−	9.76313e13i	0.444435	−	0.168997i
$484$		0			0
$485$		−	4.70979e14i		−	0.796933i
$486$		0			0
$487$	−5.23974e14			−0.866763			−0.433381	−	0.901211i	$-0.642680\pi$
$487$	−5.23974e14			−0.866763			−0.433381	+	0.901211i	$0.642680\pi$
$488$		0			0
$489$	1.22745e15	+	1.03498e14i	1.98521	+	0.167391i
$490$		0			0
$491$		−	1.07726e15i		−	1.70362i	−0.523854	−	0.851808i	$-0.675506\pi$
$491$		−	1.07726e15i		−	1.70362i	0.523854	−	0.851808i	$-0.324494\pi$
$492$		0			0
$493$		−	2.45647e14i		−	0.379886i
$494$		0			0
$495$	−2.53180e14	+	1.49065e15i	−0.382914	+	2.25448i
$496$		0			0
$497$	−9.68292e14	+	4.64742e14i	−1.43234	+	0.687466i
$498$		0			0
$499$	1.05938e13			0.0153285			0.00766425	−	0.999971i	$-0.497560\pi$
$499$	1.05938e13			0.0153285			0.00766425	+	0.999971i	$0.497560\pi$
$500$		0			0
$501$	−6.04805e14	−	5.09967e13i	−0.856067	−	0.0721829i
$502$		0			0
$503$	1.40489e15			1.94544			0.972721	−	0.231976i	$-0.0745191\pi$
$503$	1.40489e15			1.94544			0.972721	+	0.231976i	$0.0745191\pi$
$504$		0			0
$505$	−5.20779e14			−0.705588
$506$		0			0
$507$	−3.78832e14	−	3.19428e13i	−0.502230	−	0.0423476i
$508$		0			0
$509$	3.83886e14			0.498029			0.249015	−	0.968500i	$-0.419893\pi$
$509$	3.83886e14			0.498029			0.249015	+	0.968500i	$0.419893\pi$
$510$		0			0
$511$	7.58196e14	−	3.63904e14i	0.962645	−	0.462032i
$512$		0			0
$513$	1.25191e14	−	4.85504e14i	0.155571	−	0.603320i
$514$		0			0
$515$			1.29576e15i			1.57611i
$516$		0			0
$517$		−	2.18417e15i		−	2.60069i
$518$		0			0
$519$	−2.03967e14	−	1.71984e13i	−0.237761	−	0.0200478i
$520$		0			0
$521$	−5.59774e14			−0.638860			−0.319430	−	0.947610i	$-0.603491\pi$
$521$	−5.59774e14			−0.638860			−0.319430	+	0.947610i	$0.603491\pi$
$522$		0			0
$523$		−	1.10701e15i		−	1.23707i	−0.785759	−	0.618533i	$-0.787727\pi$
$523$		−	1.10701e15i		−	1.23707i	0.785759	−	0.618533i	$-0.212273\pi$
$524$		0			0
$525$	−2.66575e14	−	7.01048e14i	−0.291705	−	0.767134i
$526$		0			0
$527$		−	1.29731e15i		−	1.39022i
$528$		0			0
$529$	7.37396e14			0.773918
$530$		0			0
$531$	2.01803e14	+	3.42754e13i	0.207447	+	0.0352341i
$532$		0			0
$533$		−	5.70252e14i		−	0.574205i
$534$		0			0
$535$			8.15818e14i			0.804725i
$536$		0			0
$537$	1.25653e12	−	1.49020e13i	0.00121427	−	0.0144008i
$538$		0			0
$539$	−1.39650e15	−	1.11967e15i	−1.32222	−	1.06012i
$540$		0			0
$541$	1.10349e15			1.02372			0.511861	−	0.859068i	$-0.328956\pi$
$541$	1.10349e15			1.02372			0.511861	+	0.859068i	$0.328956\pi$
$542$		0			0
$543$	1.20488e14	−	1.42895e15i	0.109533	−	1.29903i
$544$		0			0
$545$	2.60217e15			2.31822
$546$		0			0
$547$	2.00102e15			1.74711			0.873555	−	0.486726i	$-0.161809\pi$
$547$	2.00102e15			1.74711			0.873555	+	0.486726i	$0.161809\pi$
$548$		0			0
$549$	2.82822e14	−	1.66517e15i	0.242028	−	1.42498i
$550$		0			0
$551$	3.21569e14			0.269737
$552$		0			0
$553$	−4.54836e14	−	9.47653e14i	−0.373995	−	0.779222i
$554$		0			0
$555$	−2.42239e15	−	2.04254e14i	−1.95269	−	0.164649i
$556$		0			0
$557$		−	1.85582e15i		−	1.46667i	−0.679866	−	0.733337i	$-0.737962\pi$
$557$		−	1.85582e15i		−	1.46667i	0.679866	−	0.733337i	$-0.262038\pi$
$558$		0			0
$559$		−	2.15492e14i		−	0.166981i
$560$		0			0
$561$	1.64446e14	−	1.95027e15i	0.124947	−	1.48184i
$562$		0			0
$563$	2.22927e13			0.0166099			0.00830495	−	0.999966i	$-0.497356\pi$
$563$	2.22927e13			0.0166099			0.00830495	+	0.999966i	$0.497356\pi$
$564$		0			0
$565$		−	5.18320e14i		−	0.378732i
$566$		0			0
$567$	1.54584e14	+	1.38684e15i	0.110779	+	0.993845i
$568$		0			0
$569$			2.72246e15i			1.91357i	0.290802	+	0.956783i	$0.406078\pi$
$569$			2.72246e15i			1.91357i	−0.290802	+	0.956783i	$0.593922\pi$
$570$		0			0
$571$	−4.05194e14			−0.279360			−0.139680	−	0.990197i	$-0.544607\pi$
$571$	−4.05194e14			−0.279360			−0.139680	+	0.990197i	$0.544607\pi$
$572$		0			0
$573$	−1.31672e14	+	1.56159e15i	−0.0890518	+	1.05613i
$574$		0			0
$575$			5.88170e14i			0.390238i
$576$		0			0
$577$		−	8.30149e13i		−	0.0540368i	−0.999635	−	0.0270184i	$-0.991399\pi$
$577$		−	8.30149e13i		−	0.0540368i	0.999635	−	0.0270184i	$-0.00860126\pi$
$578$		0			0
$579$	2.28429e15	+	1.92610e14i	1.45888	+	0.123012i
$580$		0			0
$581$	3.07542e14	+	6.40766e14i	0.192724	+	0.401542i
$582$		0			0
$583$	−5.14051e15			−3.16103
$584$		0			0
$585$	−2.63691e14	+	1.55253e15i	−0.159125	+	0.936876i
$586$		0			0
$587$	1.62078e15			0.959877			0.479939	−	0.877302i	$-0.340659\pi$
$587$	1.62078e15			0.959877			0.479939	+	0.877302i	$0.340659\pi$
$588$		0			0
$589$	1.69827e15			0.987123
$590$		0			0
$591$	−2.31086e14	+	2.74060e15i	−0.131838	+	1.56356i
$592$		0			0
$593$	−5.51749e14			−0.308988			−0.154494	−	0.987994i	$-0.549375\pi$
$593$	−5.51749e14			−0.308988			−0.154494	+	0.987994i	$0.549375\pi$
$594$		0			0
$595$	9.31951e14	+	1.94172e15i	0.512330	+	1.06744i
$596$		0			0
$597$	1.23878e14	−	1.46916e15i	0.0668553	−	0.792883i
$598$		0			0
$599$		−	2.07250e15i		−	1.09811i	−0.835786	−	0.549056i	$-0.814988\pi$
$599$		−	2.07250e15i		−	1.09811i	0.835786	−	0.549056i	$-0.185012\pi$
$600$		0			0
$601$			2.94951e15i			1.53441i	0.641403	+	0.767204i	$0.278352\pi$
$601$			2.94951e15i			1.53441i	−0.641403	+	0.767204i	$0.721648\pi$
$602$		0			0
$603$	5.57431e14	+	9.46774e13i	0.284738	+	0.0483617i
$604$		0			0
$605$	−5.03623e15			−2.52610
$606$		0			0
$607$		−	1.49459e15i		−	0.736180i	−0.929790	−	0.368090i	$-0.880012\pi$
$607$		−	1.49459e15i		−	0.736180i	0.929790	−	0.368090i	$-0.119988\pi$
$608$		0			0
$609$	−8.36536e14	+	3.18095e14i	−0.404660	+	0.153873i
$610$		0			0
$611$		−	2.27484e15i		−	1.08075i
$612$		0			0
$613$	1.71316e15			0.799403			0.399702	−	0.916645i	$-0.369114\pi$
$613$	1.71316e15			0.799403			0.399702	+	0.916645i	$0.369114\pi$
$614$		0			0
$615$	2.39182e15	+	2.01676e14i	1.09626	+	0.0924361i
$616$		0			0
$617$			2.33346e15i			1.05059i	0.850921	+	0.525294i	$0.176045\pi$
$617$			2.33346e15i			1.05059i	−0.850921	+	0.525294i	$0.823955\pi$
$618$		0			0
$619$		−	2.30939e15i		−	1.02140i	−0.859758	−	0.510702i	$-0.829385\pi$
$619$		−	2.30939e15i		−	1.02140i	0.859758	−	0.510702i	$-0.170615\pi$
$620$		0			0
$621$	2.73237e14	−	1.05964e15i	0.118723	−	0.460420i
$622$		0			0
$623$	−7.96795e14	−	1.66013e15i	−0.340144	−	0.708692i
$624$		0			0
$625$	−2.73499e15			−1.14714
$626$		0			0
$627$	2.55305e15	+	2.15271e14i	1.05217	+	0.0887184i
$628$		0			0
$629$	3.14678e15			1.27435
$630$		0			0
$631$	4.22396e14			0.168096			0.0840481	−	0.996462i	$-0.473215\pi$
$631$	4.22396e14			0.168096			0.0840481	+	0.996462i	$0.473215\pi$
$632$		0			0
$633$	−3.30569e15	−	2.78733e14i	−1.29283	−	0.109010i
$634$		0			0
$635$	1.60846e15			0.618237
$636$		0			0
$637$	−1.45447e15	−	1.16615e15i	−0.549462	−	0.440544i
$638$		0			0
$639$	−7.16469e14	+	4.21834e15i	−0.266037	+	1.56634i
$640$		0			0
$641$			2.63881e15i			0.963140i	0.876408	+	0.481570i	$0.159933\pi$
$641$			2.63881e15i			0.963140i	−0.876408	+	0.481570i	$0.840067\pi$
$642$		0			0
$643$		−	3.82438e15i		−	1.37215i	−0.727533	−	0.686073i	$-0.759333\pi$
$643$		−	3.82438e15i		−	1.37215i	0.727533	−	0.686073i	$-0.240667\pi$
$644$		0			0
$645$	9.03842e14	+	7.62113e13i	0.318797	+	0.0268807i
$646$		0			0
$647$	−2.94372e15			−1.02076			−0.510379	−	0.859950i	$-0.670495\pi$
$647$	−2.94372e15			−1.02076			−0.510379	+	0.859950i	$0.670495\pi$
$648$		0			0
$649$			1.04599e15i			0.356601i
$650$		0			0
$651$	−4.41790e15	+	1.67992e15i	−1.48088	+	0.563108i
$652$		0			0
$653$			3.09434e15i			1.01987i	0.860212	+	0.509936i	$0.170331\pi$
$653$			3.09434e15i			1.01987i	−0.860212	+	0.509936i	$0.829669\pi$
$654$		0			0
$655$	3.39869e15			1.10150
$656$		0			0
$657$	5.61012e14	−	3.30306e15i	0.178798	−	1.05271i
$658$		0			0
$659$		−	1.56786e14i		−	0.0491402i	−0.999698	−	0.0245701i	$-0.992178\pi$
$659$		−	1.56786e14i		−	0.0491402i	0.999698	−	0.0245701i	$-0.00782170\pi$
$660$		0			0
$661$			2.85873e15i			0.881182i	0.897708	+	0.440591i	$0.145231\pi$
$661$			2.85873e15i			0.881182i	−0.897708	+	0.440591i	$0.854769\pi$
$662$		0			0
$663$	1.71272e14	−	2.03124e15i	0.0519233	−	0.615795i
$664$		0			0
$665$	−2.54185e15	+	1.21999e15i	−0.757934	+	0.363778i
$666$		0			0
$667$	7.01844e14			0.205849
$668$		0			0
$669$	−4.04545e14	+	4.79778e15i	−0.116714	+	1.38419i
$670$		0			0
$671$	8.63096e15			2.44954
$672$		0			0
$673$	7.65854e14			0.213827			0.106914	−	0.994268i	$-0.465903\pi$
$673$	7.65854e14			0.213827			0.106914	+	0.994268i	$0.465903\pi$
$674$		0			0
$675$	−2.89327e15	−	7.46053e14i	−0.794728	−	0.204927i
$676$		0			0
$677$	−1.16871e15			−0.315840			−0.157920	−	0.987452i	$-0.550479\pi$
$677$	−1.16871e15			−0.315840			−0.157920	+	0.987452i	$0.550479\pi$
$678$		0			0
$679$	2.00247e15	−	9.61108e14i	0.532456	−	0.255558i
$680$		0			0
$681$	1.29919e15	+	1.09547e14i	0.339911	+	0.0286610i
$682$		0			0
$683$			2.32426e14i			0.0598371i	0.999552	+	0.0299186i	$0.00952480\pi$
$683$			2.32426e14i			0.0598371i	−0.999552	+	0.0299186i	$0.990475\pi$
$684$		0			0
$685$			3.43004e15i			0.868964i
$686$		0			0
$687$	3.28817e14	−	3.89966e15i	0.0819770	−	0.972222i
$688$		0			0
$689$	−5.35391e15			−1.31361
$690$		0			0
$691$			6.32582e15i			1.52752i	0.645499	+	0.763761i	$0.276650\pi$
$691$			6.32582e15i			1.52752i	−0.645499	+	0.763761i	$0.723350\pi$
$692$		0			0
$693$	−6.85448e15	+	1.96545e15i	−1.62908	+	0.467121i
$694$		0			0
$695$			5.05237e14i			0.118190i
$696$		0			0
$697$	−3.10706e15			−0.715435
$698$		0			0
$699$	2.56505e14	−	3.04207e15i	0.0581395	−	0.689516i
$700$		0			0
$701$			6.26430e15i			1.39773i	0.715253	+	0.698865i	$0.246311\pi$
$701$			6.26430e15i			1.39773i	−0.715253	+	0.698865i	$0.753689\pi$
$702$		0			0
$703$			4.11935e15i			0.904846i
$704$		0			0
$705$	9.54140e15	+	8.04524e14i	2.06335	+	0.173980i
$706$		0			0
$707$	−1.06273e15	−	2.21421e15i	−0.226265	−	0.471425i
$708$		0			0
$709$	4.08087e15			0.855457			0.427728	−	0.903907i	$-0.359314\pi$
$709$	4.08087e15			0.855457			0.427728	+	0.903907i	$0.359314\pi$
$710$		0			0
$711$	−4.12843e15	−	7.01197e14i	−0.852122	−	0.144730i
$712$		0			0
$713$	3.70656e15			0.753319
$714$		0			0
$715$	−8.04712e15			−1.61049
$716$		0			0
$717$	2.55929e14	−	3.03524e15i	0.0504387	−	0.598188i
$718$		0			0
$719$	−9.32093e15			−1.80905			−0.904525	−	0.426421i	$-0.859774\pi$
$719$	−9.32093e15			−1.80905			−0.904525	+	0.426421i	$0.859774\pi$
$720$		0			0
$721$	−5.50924e15	+	2.64422e15i	−1.05305	+	0.505421i
$722$		0			0
$723$	5.61189e14	−	6.65552e15i	0.105645	−	1.25292i
$724$		0			0
$725$		−	1.91633e15i		−	0.355314i
$726$		0			0
$727$		−	3.48367e15i		−	0.636206i	−0.948056	−	0.318103i	$-0.896954\pi$
$727$		−	3.48367e15i		−	0.636206i	0.948056	−	0.318103i	$-0.103046\pi$
$728$		0			0
$729$	4.86590e15	+	2.68816e15i	0.875309	+	0.483564i
$730$		0			0
$731$	−1.17413e15			−0.208051
$732$		0			0
$733$			5.82428e15i			1.01665i	0.861166	+	0.508324i	$0.169735\pi$
$733$			5.82428e15i			1.01665i	−0.861166	+	0.508324i	$0.830265\pi$
$734$		0			0
$735$	5.40562e15	−	5.68809e15i	0.929533	−	0.978106i
$736$		0			0
$737$			2.88930e15i			0.489464i
$738$		0			0
$739$	−6.56216e14			−0.109522			−0.0547612	−	0.998499i	$-0.517440\pi$
$739$	−6.56216e14			−0.109522			−0.0547612	+	0.998499i	$0.517440\pi$
$740$		0			0
$741$	2.65903e15	+	2.24207e14i	0.437243	+	0.0368680i
$742$		0			0
$743$		−	6.61666e12i		−	0.00107201i	−1.00000		0.000536007i	$-0.999829\pi$
$743$		−	6.61666e12i		−	0.00107201i	1.00000		0.000536007i	$-0.000170616\pi$
$744$		0			0
$745$		−	1.09503e15i		−	0.174810i
$746$		0			0
$747$	2.79148e15	+	4.74122e14i	0.439108	+	0.0745808i
$748$		0			0
$749$	−3.46864e15	+	1.66481e15i	−0.537662	+	0.258056i
$750$		0			0
$751$	2.76366e15			0.422148			0.211074	−	0.977470i	$-0.432304\pi$
$751$	2.76366e15			0.422148			0.211074	+	0.977470i	$0.432304\pi$
$752$		0			0
$753$	3.01458e15	+	2.54187e14i	0.453790	+	0.0382632i
$754$		0			0
$755$	1.16092e16			1.72225
$756$		0			0
$757$	−1.68541e14			−0.0246421			−0.0123210	−	0.999924i	$-0.503922\pi$
$757$	−1.68541e14			−0.0246421			−0.0123210	+	0.999924i	$0.503922\pi$
$758$		0			0
$759$	5.57217e15	+	4.69841e14i	0.802961	+	0.0677050i
$760$		0			0
$761$	−6.26293e15			−0.889533			−0.444767	−	0.895647i	$-0.646713\pi$
$761$	−6.26293e15			−0.889533			−0.444767	+	0.895647i	$0.646713\pi$
$762$		0			0
$763$	5.31015e15	+	1.10637e16i	0.743398	+	1.54887i
$764$		0			0
$765$	8.45908e15	+	1.43674e15i	1.16731	+	0.198262i
$766$		0			0
$767$			1.08941e15i			0.148190i
$768$		0			0
$769$			4.09451e15i			0.549044i	0.961581	+	0.274522i	$0.0885197\pi$
$769$			4.09451e15i			0.549044i	−0.961581	+	0.274522i	$0.911480\pi$
$770$		0			0
$771$	−1.13277e16	−	9.55144e14i	−1.49742	−	0.126261i
$772$		0			0
$773$	4.65483e15			0.606620			0.303310	−	0.952892i	$-0.401908\pi$
$773$	4.65483e15			0.606620			0.303310	+	0.952892i	$0.401908\pi$
$774$		0			0
$775$		−	1.01205e16i		−	1.30030i
$776$		0			0
$777$	−4.07484e15	−	1.07162e16i	−0.516173	−	1.35745i
$778$		0			0
$779$		−	4.06737e15i		−	0.507992i
$780$		0			0
$781$	−2.18647e16			−2.69254
$782$		0			0
$783$	−8.90240e14	+	3.45244e15i	−0.108098	+	0.419215i
$784$		0			0
$785$		−	1.15558e16i		−	1.38362i
$786$		0			0
$787$		−	2.93252e15i		−	0.346242i	−0.984901	−	0.173121i	$-0.944615\pi$
$787$		−	2.93252e15i		−	0.346242i	0.984901	−	0.173121i	$-0.0553852\pi$
$788$		0			0
$789$	1.02209e15	−	1.21217e16i	0.119005	−	1.41137i
$790$		0			0
$791$	2.20376e15	−	1.05772e15i	0.253042	−	0.121450i
$792$		0			0
$793$	8.98926e15			1.01794
$794$		0			0
$795$	1.89347e15	−	2.24560e16i	0.211466	−	2.50792i
$796$		0			0
$797$	1.20745e16			1.32999			0.664994	−	0.746848i	$-0.268434\pi$
$797$	1.20745e16			1.32999			0.664994	+	0.746848i	$0.268434\pi$
$798$		0			0
$799$	−1.23947e16			−1.34656
$800$		0			0
$801$	−7.23231e15	−	1.22838e15i	−0.774994	−	0.131630i
$802$		0			0
$803$	1.71206e16			1.80960
$804$		0			0
$805$	−5.54774e15	+	2.66269e15i	−0.578414	+	0.277616i
$806$		0			0
$807$	1.37057e16	+	1.15566e15i	1.40961	+	0.118857i
$808$		0			0
$809$		−	2.83801e15i		−	0.287936i	−0.989582	−	0.143968i	$-0.954014\pi$
$809$		−	2.83801e15i		−	0.287936i	0.989582	−	0.143968i	$-0.0459863\pi$
$810$		0			0
$811$			1.35011e16i			1.35131i	0.737219	+	0.675654i	$0.236138\pi$
$811$			1.35011e16i			1.35131i	−0.737219	+	0.675654i	$0.763862\pi$
$812$		0			0
$813$	−5.92512e14	+	7.02701e15i	−0.0585059	+	0.693862i
$814$		0			0
$815$	−2.75952e16			−2.68823
$816$		0			0
$817$		−	1.53701e15i		−	0.147726i
$818$		0			0
$819$	−7.13903e15	+	2.04704e15i	−0.676983	+	0.194118i
$820$		0			0
$821$		−	1.84959e16i		−	1.73056i	−0.501286	−	0.865282i	$-0.667139\pi$
$821$		−	1.84959e16i		−	1.73056i	0.501286	−	0.865282i	$-0.332861\pi$
$822$		0			0
$823$	1.43048e16			1.32063			0.660316	−	0.750988i	$-0.270422\pi$
$823$	1.43048e16			1.32063			0.660316	+	0.750988i	$0.270422\pi$
$824$		0			0
$825$	1.28287e15	−	1.52144e16i	0.116865	−	1.38598i
$826$		0			0
$827$		−	6.37130e15i		−	0.572727i	−0.958121	−	0.286364i	$-0.907553\pi$
$827$		−	6.37130e15i		−	0.572727i	0.958121	−	0.286364i	$-0.0924465\pi$
$828$		0			0
$829$			1.16574e16i			1.03407i	0.855963	+	0.517037i	$0.172965\pi$
$829$			1.16574e16i			1.03407i	−0.855963	+	0.517037i	$0.827035\pi$
$830$		0			0
$831$	5.21348e15	+	4.39597e14i	0.456375	+	0.0384812i
$832$		0			0
$833$	−6.35389e15	+	7.92481e15i	−0.548899	+	0.684607i
$834$		0			0
$835$	1.35970e16			1.15923
$836$		0			0
$837$	−4.70151e15	+	1.82330e16i	−0.395592	+	1.53415i
$838$		0			0
$839$	1.77788e16			1.47643			0.738213	−	0.674567i	$-0.235670\pi$
$839$	1.77788e16			1.47643			0.738213	+	0.674567i	$0.235670\pi$
$840$		0			0
$841$	9.91382e15			0.812574
$842$		0			0
$843$	9.32447e14	−	1.10585e16i	0.0754349	−	0.894635i
$844$		0			0
$845$	8.51677e15			0.680085
$846$		0			0
$847$	−1.02772e16	−	2.14127e16i	−0.810061	−	1.68777i
$848$		0			0
$849$	6.89295e14	−	8.17483e15i	0.0536307	−	0.636043i
$850$		0			0
$851$			8.99072e15i			0.690529i
$852$		0			0
$853$		−	5.62749e15i		−	0.426673i	−0.976979	−	0.213336i	$-0.931567\pi$
$853$		−	5.62749e15i		−	0.426673i	0.976979	−	0.213336i	$-0.0684330\pi$
$854$		0			0
$855$	−1.88079e15	+	1.10735e16i	−0.140776	+	0.828843i
$856$		0			0
$857$	1.08777e15			0.0803787			0.0401893	−	0.999192i	$-0.487204\pi$
$857$	1.08777e15			0.0803787			0.0401893	+	0.999192i	$0.487204\pi$
$858$		0			0
$859$			1.30624e15i			0.0952930i	0.998864	+	0.0476465i	$0.0151721\pi$
$859$			1.30624e15i			0.0952930i	−0.998864	+	0.0476465i	$0.984828\pi$
$860$		0			0
$861$	4.02342e15	+	1.05809e16i	0.289786	+	0.762089i
$862$		0			0
$863$		−	9.79395e15i		−	0.696464i	−0.937408	−	0.348232i	$-0.886782\pi$
$863$		−	9.79395e15i		−	0.696464i	0.937408	−	0.348232i	$-0.113218\pi$
$864$		0			0
$865$	4.58552e15			0.321959
$866$		0			0
$867$	3.30627e15	+	2.78782e14i	0.229210	+	0.0193268i
$868$		0			0
$869$		−	2.13986e16i		−	1.46480i
$870$		0			0
$871$			3.00924e15i			0.203403i
$872$		0			0
$873$	1.48169e15	−	8.72374e15i	0.0988962	−	0.582270i
$874$		0			0
$875$	−1.58810e15	−	3.30882e15i	−0.104673	−	0.218087i
$876$		0			0
$877$	−1.69760e16			−1.10494			−0.552470	−	0.833533i	$-0.686314\pi$
$877$	−1.69760e16			−1.10494			−0.552470	+	0.833533i	$0.686314\pi$
$878$		0			0
$879$	2.13374e14	+	1.79915e13i	0.0137152	+	0.00115645i
$880$		0			0
$881$	3.43711e15			0.218186			0.109093	−	0.994032i	$-0.465205\pi$
$881$	3.43711e15			0.218186			0.109093	+	0.994032i	$0.465205\pi$
$882$		0			0
$883$	−2.15455e16			−1.35074			−0.675372	−	0.737477i	$-0.736017\pi$
$883$	−2.15455e16			−1.35074			−0.675372	+	0.737477i	$0.736017\pi$
$884$		0			0
$885$	−4.56935e15	−	3.85284e14i	−0.282922	−	0.0238558i
$886$		0			0
$887$	−1.65582e16			−1.01259			−0.506295	−	0.862361i	$-0.668985\pi$
$887$	−1.65582e16			−1.01259			−0.506295	+	0.862361i	$0.668985\pi$
$888$		0			0
$889$	3.28233e15	+	6.83874e15i	0.198254	+	0.413063i
$890$		0			0
$891$	−9.37910e15	+	2.68141e16i	−0.559544	+	1.59969i
$892$		0			0
$893$		−	1.62255e16i		−	0.956124i
$894$		0			0
$895$			3.35023e14i			0.0195006i
$896$		0			0
$897$	5.80348e15	+	4.89345e14i	0.333680	+	0.0281357i
$898$		0			0
$899$	−1.20764e16			−0.685900
$900$		0			0
$901$			2.91712e16i			1.63670i
$902$		0			0
$903$	1.52041e15	+	3.99842e15i	0.0842708	+	0.221618i
$904$		0			0
$905$			3.21253e16i			1.75906i
$906$		0			0
$907$	2.44557e16			1.32294			0.661470	−	0.749971i	$-0.269933\pi$
$907$	2.44557e16			1.32294			0.661470	+	0.749971i	$0.269933\pi$
$908$		0			0
$909$	−9.64617e15	−	1.63836e15i	−0.515529	−	0.0875606i
$910$		0			0
$911$			2.30579e16i			1.21750i	0.793362	+	0.608750i	$0.208329\pi$
$911$			2.30579e16i			1.21750i	−0.793362	+	0.608750i	$0.791671\pi$
$912$		0			0
$913$			1.44689e16i			0.754826i
$914$		0			0
$915$	−3.17916e15	+	3.77038e16i	−0.163869	+	1.94343i
$916$		0			0
$917$	6.93558e15	+	1.44503e16i	0.353225	+	0.735946i
$918$		0			0
$919$	−8.79627e15			−0.442653			−0.221326	−	0.975200i	$-0.571039\pi$
$919$	−8.79627e15			−0.442653			−0.221326	+	0.975200i	$0.571039\pi$
$920$		0			0
$921$	1.78567e15	−	2.11775e16i	0.0887920	−	1.05305i
$922$		0			0
$923$	−2.27723e16			−1.11892
$924$		0			0
$925$	2.45485e16			1.19192
$926$		0			0
$927$	−4.07645e15	+	2.40009e16i	−0.195589	+	1.15157i
$928$		0			0
$929$	4.66655e15			0.221264			0.110632	−	0.993861i	$-0.464713\pi$
$929$	4.66655e15			0.221264			0.110632	+	0.993861i	$0.464713\pi$
$930$		0			0
$931$	−1.03741e16	−	8.31769e15i	−0.486103	−	0.389744i
$932$		0			0
$933$	−2.02460e16	−	1.70712e15i	−0.937539	−	0.0790526i
$934$		0			0
$935$			4.38454e16i			2.00660i
$936$		0			0
$937$		−	2.38050e16i		−	1.07671i	−0.842717	−	0.538357i	$-0.819045\pi$
$937$		−	2.38050e16i		−	1.07671i	0.842717	−	0.538357i	$-0.180955\pi$
$938$		0			0
$939$	5.28571e14	−	6.26869e15i	0.0236288	−	0.280231i
$940$		0			0
$941$	3.23581e16			1.42968			0.714842	−	0.699286i	$-0.246499\pi$
$941$	3.23581e16			1.42968			0.714842	+	0.699286i	$0.246499\pi$
$942$		0			0
$943$		−	8.87725e15i		−	0.387672i
$944$		0			0
$945$	−6.06115e15	−	3.06673e16i	−0.261625	−	1.32374i
$946$		0			0
$947$			9.53074e15i			0.406632i	0.979113	+	0.203316i	$0.0651719\pi$
$947$			9.53074e15i			0.406632i	−0.979113	+	0.203316i	$0.934828\pi$
$948$		0			0
$949$	1.78313e16			0.752001
$950$		0			0
$951$	−2.79784e15	+	3.31815e16i	−0.116635	+	1.38326i
$952$		0			0
$953$			2.09691e16i			0.864111i	0.901847	+	0.432055i	$0.142211\pi$
$953$			2.09691e16i			0.864111i	−0.901847	+	0.432055i	$0.857789\pi$
$954$		0			0
$955$		−	3.51071e16i		−	1.43013i
$956$		0			0
$957$	−1.81548e16	−	1.53080e15i	−0.731099	−	0.0616457i
$958$		0			0
$959$	−1.45836e16	+	6.99956e15i	−0.580582	+	0.278656i
$960$		0			0
$961$	−3.83693e16			−1.51010
$962$		0			0
$963$	−2.56655e15	+	1.51110e16i	−0.0998632	+	0.587963i
$964$		0			0
$965$	−5.13547e16			−1.97551
$966$		0			0
$967$	1.71022e16			0.650437			0.325219	−	0.945639i	$-0.394562\pi$
$967$	1.71022e16			0.650437			0.325219	+	0.945639i	$0.394562\pi$
$968$		0			0
$969$	1.22161e15	−	1.44880e16i	0.0459360	−	0.544786i
$970$		0			0
$971$	7.16745e14			0.0266477			0.0133238	−	0.999911i	$-0.495759\pi$
$971$	7.16745e14			0.0266477			0.0133238	+	0.999911i	$0.495759\pi$
$972$		0			0
$973$	−2.14813e15	+	1.03102e15i	−0.0789661	+	0.0379006i
$974$		0			0
$975$	1.33612e15	−	1.58460e16i	0.0485647	−	0.575962i
$976$		0			0
$977$			1.19534e16i			0.429607i	0.976657	+	0.214804i	$0.0689111\pi$
$977$			1.19534e16i			0.429607i	−0.976657	+	0.214804i	$0.931089\pi$
$978$		0			0
$979$		−	3.74868e16i		−	1.33221i
$980$		0			0
$981$	4.81988e16	+	8.18638e15i	1.69378	+	0.287682i
$982$		0			0
$983$	3.78192e16			1.31422			0.657110	−	0.753795i	$-0.271779\pi$
$983$	3.78192e16			1.31422			0.657110	+	0.753795i	$0.271779\pi$
$984$		0			0
$985$		−	6.16134e16i		−	2.11727i
$986$		0			0
$987$	1.60502e16	+	4.22092e16i	0.545425	+	1.43438i
$988$		0			0
$989$		−	3.35462e15i		−	0.112736i
$990$		0			0
$991$	3.36920e16			1.11975			0.559876	−	0.828577i	$-0.310849\pi$
$991$	3.36920e16			1.11975			0.559876	+	0.828577i	$0.310849\pi$
$992$		0			0
$993$	5.45792e16	+	4.60208e15i	1.79393	+	0.151263i
$994$		0			0
$995$			3.30291e16i			1.07367i
$996$		0			0
$997$		−	2.18487e16i		−	0.702429i	−0.936295	−	0.351215i	$-0.885769\pi$
$997$		−	2.18487e16i		−	0.702429i	0.936295	−	0.351215i	$-0.114231\pi$
$998$		0			0
$999$	−4.42263e16	−	1.14041e16i	−1.40627	−	0.362619i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.f.b.41.1	✓	28
3.2	odd	2	inner	84.12.f.b.41.27	yes	28
7.6	odd	2	inner	84.12.f.b.41.28	yes	28
21.20	even	2	inner	84.12.f.b.41.2	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.f.b.41.1	✓	28	1.1	even	1	trivial
84.12.f.b.41.2	yes	28	21.20	even	2	inner
84.12.f.b.41.27	yes	28	3.2	odd	2	inner
84.12.f.b.41.28	yes	28	7.6	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)

\(f(q)\)	\(=\)	\(q+(-419.400 - 35.3635i) q^{3} +9428.81 q^{5} +(19241.0 + 40088.8i) q^{7} +(174646. + 29662.9i) q^{9} +O(q^{10})\)	\(q+(-419.400 - 35.3635i) q^{3} +9428.81 q^{5} +(19241.0 + 40088.8i) q^{7} +(174646. + 29662.9i) q^{9} +905231. i q^{11} +942810. i q^{13} +(-3.95444e6 - 333436. i) q^{15} +5.13698e6 q^{17} +6.72467e6i q^{19} +(-6.65201e6 - 1.74937e7i) q^{21} +1.46770e7i q^{23} +4.00744e7 q^{25} +(-7.21975e7 - 1.86167e7i) q^{27} -4.78194e7i q^{29} -2.52543e8i q^{31} +(3.20121e7 - 3.79654e8i) q^{33} +(1.81420e8 + 3.77990e8i) q^{35} +6.12574e8 q^{37} +(3.33410e7 - 3.95414e8i) q^{39} -6.04843e8 q^{41} -2.28564e8 q^{43} +(1.64670e9 + 2.79686e8i) q^{45} -2.41283e9 q^{47} +(-1.23689e9 + 1.54270e9i) q^{49} +(-2.15445e9 - 1.81662e8i) q^{51} +5.67867e9i q^{53} +8.53525e9i q^{55} +(2.37808e8 - 2.82033e9i) q^{57} +1.15550e9 q^{59} -9.53454e9i q^{61} +(2.17121e9 + 7.57208e9i) q^{63} +8.88958e9i q^{65} +3.19178e9 q^{67} +(5.19029e8 - 6.15552e9i) q^{69} +2.41537e10i q^{71} -1.89129e10i q^{73} +(-1.68072e10 - 1.41717e9i) q^{75} +(-3.62896e10 + 1.74176e10i) q^{77} -2.36389e10 q^{79} +(2.96213e10 + 1.03610e10i) q^{81} +1.59837e10 q^{83} +4.84356e10 q^{85} +(-1.69106e9 + 2.00555e10i) q^{87} -4.14113e10 q^{89} +(-3.77961e10 + 1.81406e10i) q^{91} +(-8.93079e9 + 1.05916e11i) q^{93} +6.34056e10i q^{95} -4.99510e10i q^{97} +(-2.68518e10 + 1.58095e11i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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