LMFDB - Embedded newform 84.12.f.b.41.3

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			41.3
Character	$\chi$	$=$	84.41
Dual form			84.12.f.b.41.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-402.361 - 123.503i) q^{3} -10055.0 q^{5} +(27333.5 + 35074.3i) q^{7} +(146641. + 99385.4i) q^{9} +O(q^{10})$	\(q+(-402.361 - 123.503i) q^{3} -10055.0 q^{5} +(27333.5 + 35074.3i) q^{7} +(146641. + 99385.4i) q^{9} +246408. i q^{11} -1.47185e6i q^{13} +(4.04574e6 + 1.24182e6i) q^{15} -2.04511e6 q^{17} -2.01281e7i q^{19} +(-6.66613e6 - 1.74883e7i) q^{21} -3.93427e7i q^{23} +5.22749e7 q^{25} +(-4.67282e7 - 5.80993e7i) q^{27} +1.09725e8i q^{29} +1.28488e8i q^{31} +(3.04321e7 - 9.91448e7i) q^{33} +(-2.74838e8 - 3.52672e8i) q^{35} -1.17329e8 q^{37} +(-1.81778e8 + 5.92216e8i) q^{39} +1.13985e9 q^{41} +1.02636e9 q^{43} +(-1.47448e9 - 9.99320e8i) q^{45} -1.19320e9 q^{47} +(-4.83089e8 + 1.91741e9i) q^{49} +(8.22873e8 + 2.52577e8i) q^{51} -2.82869e9i q^{53} -2.47763e9i q^{55} +(-2.48588e9 + 8.09876e9i) q^{57} +4.66887e9 q^{59} +5.89500e9i q^{61} +(5.22337e8 + 7.85988e9i) q^{63} +1.47995e10i q^{65} -1.10320e10 q^{67} +(-4.85894e9 + 1.58300e10i) q^{69} -1.11935e9i q^{71} -6.64599e9i q^{73} +(-2.10334e10 - 6.45610e9i) q^{75} +(-8.64259e9 + 6.73518e9i) q^{77} +3.28051e10 q^{79} +(1.16262e10 + 2.91480e10i) q^{81} -5.28639e10 q^{83} +2.05636e10 q^{85} +(1.35514e10 - 4.41491e10i) q^{87} -6.39035e10 q^{89} +(5.16243e10 - 4.02309e10i) q^{91} +(1.58687e10 - 5.16987e10i) q^{93} +2.02388e11i q^{95} -3.80382e10i q^{97} +(-2.44893e10 + 3.61335e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−402.361	−	123.503i	−0.955979	−	0.293434i
$3$	−402.361	−	123.503i	−0.955979	−	0.293434i
$4$		0			0
$5$	−10055.0			−1.43895			−0.719477	−	0.694516i	$-0.755619\pi$
$5$	−10055.0			−1.43895			−0.719477	+	0.694516i	$0.755619\pi$
$6$		0			0
$7$	27333.5	+	35074.3i	0.614689	+	0.788769i
$7$	27333.5	+	35074.3i	0.614689	+	0.788769i
$8$		0			0
$9$	146641.	+	99385.4i	0.827793	+	0.561033i
$10$		0			0
$11$			246408.i			0.461312i	0.973035	+	0.230656i	$0.0740872\pi$
$11$			246408.i			0.461312i	−0.973035	+	0.230656i	$0.925913\pi$
$12$		0			0
$13$		−	1.47185e6i		−	1.09945i	−0.835345	−	0.549726i	$-0.814732\pi$
$13$		−	1.47185e6i		−	1.09945i	0.835345	−	0.549726i	$-0.185268\pi$
$14$		0			0
$15$	4.04574e6	+	1.24182e6i	1.37561	+	0.422238i
$16$		0			0
$17$	−2.04511e6			−0.349340			−0.174670	−	0.984627i	$-0.555886\pi$
$17$	−2.04511e6			−0.349340			−0.174670	+	0.984627i	$0.555886\pi$
$18$		0			0
$19$		−	2.01281e7i		−	1.86491i	−0.361284	−	0.932456i	$-0.617662\pi$
$19$		−	2.01281e7i		−	1.86491i	0.361284	−	0.932456i	$-0.382338\pi$
$20$		0			0
$21$	−6.66613e6	−	1.74883e7i	−0.356179	−	0.934418i
$22$		0			0
$23$		−	3.93427e7i		−	1.27456i	−0.770631	−	0.637282i	$-0.780059\pi$
$23$		−	3.93427e7i		−	1.27456i	0.770631	−	0.637282i	$-0.219941\pi$
$24$		0			0
$25$	5.22749e7			1.07059
$26$		0			0
$27$	−4.67282e7	−	5.80993e7i	−0.626727	−	0.779239i
$28$		0			0
$29$			1.09725e8i			0.993385i	0.867927	+	0.496692i	$0.165452\pi$
$29$			1.09725e8i			0.993385i	−0.867927	+	0.496692i	$0.834548\pi$
$30$		0			0
$31$			1.28488e8i			0.806073i	0.915184	+	0.403037i	$0.132045\pi$
$31$			1.28488e8i			0.806073i	−0.915184	+	0.403037i	$0.867955\pi$
$32$		0			0
$33$	3.04321e7	−	9.91448e7i	0.135365	−	0.441005i
$34$		0			0
$35$	−2.74838e8	−	3.52672e8i	−0.884510	−	1.13500i
$36$		0			0
$37$	−1.17329e8			−0.278161			−0.139081	−	0.990281i	$-0.544415\pi$
$37$	−1.17329e8			−0.278161			−0.139081	+	0.990281i	$0.544415\pi$
$38$		0			0
$39$	−1.81778e8	+	5.92216e8i	−0.322617	+	1.05105i
$40$		0			0
$41$	1.13985e9			1.53652			0.768260	−	0.640138i	$-0.221123\pi$
$41$	1.13985e9			1.53652			0.768260	+	0.640138i	$0.221123\pi$
$42$		0			0
$43$	1.02636e9			1.06469			0.532344	−	0.846528i	$-0.321311\pi$
$43$	1.02636e9			1.06469			0.532344	+	0.846528i	$0.321311\pi$
$44$		0			0
$45$	−1.47448e9	−	9.99320e8i	−1.19116	−	0.807301i
$46$		0			0
$47$	−1.19320e9			−0.758885			−0.379443	−	0.925215i	$-0.623884\pi$
$47$	−1.19320e9			−0.758885			−0.379443	+	0.925215i	$0.623884\pi$
$48$		0			0
$49$	−4.83089e8	+	1.91741e9i	−0.244314	+	0.969696i
$50$		0			0
$51$	8.22873e8	+	2.52577e8i	0.333962	+	0.102508i
$52$		0			0
$53$		−	2.82869e9i		−	0.929112i	−0.885544	−	0.464556i	$-0.846214\pi$
$53$		−	2.82869e9i		−	0.929112i	0.885544	−	0.464556i	$-0.153786\pi$
$54$		0			0
$55$		−	2.47763e9i		−	0.663807i
$56$		0			0
$57$	−2.48588e9	+	8.09876e9i	−0.547228	+	1.78282i
$58$		0			0
$59$	4.66887e9			0.850209			0.425104	−	0.905144i	$-0.360237\pi$
$59$	4.66887e9			0.850209			0.425104	+	0.905144i	$0.360237\pi$
$60$		0			0
$61$			5.89500e9i			0.893655i	0.894620	+	0.446828i	$0.147446\pi$
$61$			5.89500e9i			0.893655i	−0.894620	+	0.446828i	$0.852554\pi$
$62$		0			0
$63$	5.22337e8	+	7.85988e9i	0.0663098	+	0.997799i
$64$		0			0
$65$			1.47995e10i			1.58206i
$66$		0			0
$67$	−1.10320e10			−0.998260			−0.499130	−	0.866527i	$-0.666347\pi$
$67$	−1.10320e10			−0.998260			−0.499130	+	0.866527i	$0.666347\pi$
$68$		0			0
$69$	−4.85894e9	+	1.58300e10i	−0.374000	+	1.21846i
$70$		0			0
$71$		−	1.11935e9i		−	0.0736283i	−0.999322	−	0.0368141i	$-0.988279\pi$
$71$		−	1.11935e9i		−	0.0736283i	0.999322	−	0.0368141i	$-0.0117210\pi$
$72$		0			0
$73$		−	6.64599e9i		−	0.375218i	−0.982244	−	0.187609i	$-0.939926\pi$
$73$		−	6.64599e9i		−	0.375218i	0.982244	−	0.187609i	$-0.0600738\pi$
$74$		0			0
$75$	−2.10334e10	−	6.45610e9i	−1.02346	−	0.314147i
$76$		0			0
$77$	−8.64259e9	+	6.73518e9i	−0.363869	+	0.283564i
$78$		0			0
$79$	3.28051e10			1.19948			0.599740	−	0.800195i	$-0.295271\pi$
$79$	3.28051e10			1.19948			0.599740	+	0.800195i	$0.295271\pi$
$80$		0			0
$81$	1.16262e10	+	2.91480e10i	0.370483	+	0.928839i
$82$		0			0
$83$	−5.28639e10			−1.47309			−0.736546	−	0.676387i	$-0.763545\pi$
$83$	−5.28639e10			−1.47309			−0.736546	+	0.676387i	$0.763545\pi$
$84$		0			0
$85$	2.05636e10			0.502684
$86$		0			0
$87$	1.35514e10	−	4.41491e10i	0.291493	−	0.949655i
$88$		0			0
$89$	−6.39035e10			−1.21305			−0.606526	−	0.795064i	$-0.707437\pi$
$89$	−6.39035e10			−1.21305			−0.606526	+	0.795064i	$0.707437\pi$
$90$		0			0
$91$	5.16243e10	−	4.02309e10i	0.867214	−	0.675822i
$92$		0			0
$93$	1.58687e10	−	5.16987e10i	0.236529	−	0.770589i
$94$		0			0
$95$			2.02388e11i			2.68352i
$96$		0			0
$97$		−	3.80382e10i		−	0.449754i	−0.974387	−	0.224877i	$-0.927802\pi$
$97$		−	3.80382e10i		−	0.449754i	0.974387	−	0.224877i	$-0.0721980\pi$
$98$		0			0
$99$	−2.44893e10	+	3.61335e10i	−0.258811	+	0.381871i
$100$		0			0
$101$	−1.20050e11			−1.13656			−0.568282	−	0.822834i	$-0.692392\pi$
$101$	−1.20050e11			−1.13656			−0.568282	+	0.822834i	$0.692392\pi$
$102$		0			0
$103$			2.00798e11i			1.70669i	0.521345	+	0.853346i	$0.325430\pi$
$103$			2.00798e11i			1.70669i	−0.521345	+	0.853346i	$0.674570\pi$
$104$		0			0
$105$	6.70280e10	+	1.75845e11i	0.512525	+	1.34458i
$106$		0			0
$107$			1.96516e11i			1.35453i	0.735740	+	0.677264i	$0.236835\pi$
$107$			1.96516e11i			1.35453i	−0.735740	+	0.677264i	$0.763165\pi$
$108$		0			0
$109$	−2.63879e11			−1.64270			−0.821351	−	0.570423i	$-0.806779\pi$
$109$	−2.63879e11			−1.64270			−0.821351	+	0.570423i	$0.806779\pi$
$110$		0			0
$111$	4.72086e10	+	1.44905e10i	0.265916	+	0.0816219i
$112$		0			0
$113$			1.39560e11i			0.712573i	0.934377	+	0.356287i	$0.115957\pi$
$113$			1.39560e11i			0.712573i	−0.934377	+	0.356287i	$0.884043\pi$
$114$		0			0
$115$			3.95591e11i			1.83404i
$116$		0			0
$117$	1.46281e11	−	2.15834e11i	0.616829	−	0.910119i
$118$		0			0
$119$	−5.59000e10	−	7.17309e10i	−0.214735	−	0.275549i
$120$		0			0
$121$	2.24595e11			0.787191
$122$		0			0
$123$	−4.58633e11	−	1.40775e11i	−1.46888	−	0.450867i
$124$		0			0
$125$	−3.46572e10			−0.101575
$126$		0			0
$127$	−1.51287e11			−0.406333			−0.203167	−	0.979144i	$-0.565123\pi$
$127$	−1.51287e11			−0.406333			−0.203167	+	0.979144i	$0.565123\pi$
$128$		0			0
$129$	−4.12966e11	−	1.26758e11i	−1.01782	−	0.312415i
$130$		0			0
$131$	6.96213e10			0.157670			0.0788351	−	0.996888i	$-0.474880\pi$
$131$	6.96213e10			0.157670			0.0788351	+	0.996888i	$0.474880\pi$
$132$		0			0
$133$	7.05980e11	−	5.50171e11i	1.47098	−	1.14634i
$134$		0			0
$135$	4.69852e11	+	5.84189e11i	0.901832	+	1.12129i
$136$		0			0
$137$			5.25778e9i			0.00930764i	0.999989	+	0.00465382i	$0.00148136\pi$
$137$			5.25778e9i			0.00930764i	−0.999989	+	0.00465382i	$0.998519\pi$
$138$		0			0
$139$			1.51675e11i			0.247933i	0.992286	+	0.123966i	$0.0395615\pi$
$139$			1.51675e11i			0.247933i	−0.992286	+	0.123966i	$0.960439\pi$
$140$		0			0
$141$	4.80098e11	+	1.47364e11i	0.725479	+	0.222683i
$142$		0			0
$143$	3.62677e11			0.507191
$144$		0			0
$145$		−	1.10329e12i		−	1.42944i
$146$		0			0
$147$	4.31181e11	−	7.11826e11i	0.518101	−	0.855319i
$148$		0			0
$149$		−	1.64740e12i		−	1.83770i	−0.394604	−	0.918851i	$-0.629118\pi$
$149$		−	1.64740e12i		−	1.83770i	0.394604	−	0.918851i	$-0.370882\pi$
$150$		0			0
$151$	−1.73945e12			−1.80318			−0.901589	−	0.432593i	$-0.857599\pi$
$151$	−1.73945e12			−1.80318			−0.901589	+	0.432593i	$0.857599\pi$
$152$		0			0
$153$	−2.99897e11	−	2.03254e11i	−0.289181	−	0.195991i
$154$		0			0
$155$		−	1.29195e12i		−	1.15990i
$156$		0			0
$157$			5.64956e11i			0.472679i	0.971671	+	0.236340i	$0.0759478\pi$
$157$			5.64956e11i			0.472679i	−0.971671	+	0.236340i	$0.924052\pi$
$158$		0			0
$159$	−3.49351e11	+	1.13815e12i	−0.272633	+	0.888212i
$160$		0			0
$161$	1.37992e12	−	1.07537e12i	1.00534	−	0.783460i
$162$		0			0
$163$	−1.22445e12			−0.833509			−0.416754	−	0.909019i	$-0.636832\pi$
$163$	−1.22445e12			−0.833509			−0.416754	+	0.909019i	$0.636832\pi$
$164$		0			0
$165$	−3.05994e11	+	9.96901e11i	−0.194783	+	0.634586i
$166$		0			0
$167$	1.00010e12			0.595802			0.297901	−	0.954597i	$-0.403713\pi$
$167$	1.00010e12			0.595802			0.297901	+	0.954597i	$0.403713\pi$
$168$		0			0
$169$	−3.74196e11			−0.208796
$170$		0			0
$171$	2.00044e12	−	2.95161e12i	1.04628	−	1.54376i
$172$		0			0
$173$	1.47370e12			0.723027			0.361513	−	0.932367i	$-0.382260\pi$
$173$	1.47370e12			0.723027			0.361513	+	0.932367i	$0.382260\pi$
$174$		0			0
$175$	1.42885e12	+	1.83351e12i	0.658080	+	0.844448i
$176$		0			0
$177$	−1.87857e12	−	5.76618e11i	−0.812782	−	0.249480i
$178$		0			0
$179$		−	4.60706e11i		−	0.187384i	−0.995601	−	0.0936919i	$-0.970133\pi$
$179$		−	4.60706e11i		−	0.187384i	0.995601	−	0.0936919i	$-0.0298669\pi$
$180$		0			0
$181$			2.77772e12i			1.06281i	0.847117	+	0.531407i	$0.178336\pi$
$181$			2.77772e12i			1.06281i	−0.847117	+	0.531407i	$0.821664\pi$
$182$		0			0
$183$	7.28050e11	−	2.37192e12i	0.262229	−	0.854316i
$184$		0			0
$185$	1.17974e12			0.400261
$186$		0			0
$187$		−	5.03932e11i		−	0.161155i
$188$		0			0
$189$	7.60550e11	−	3.22702e12i	0.229397	−	0.973333i
$190$		0			0
$191$			4.59166e12i			1.30703i	0.756912	+	0.653516i	$0.226707\pi$
$191$			4.59166e12i			1.30703i	−0.756912	+	0.653516i	$0.773293\pi$
$192$		0			0
$193$	−1.71463e12			−0.460898			−0.230449	−	0.973084i	$-0.574019\pi$
$193$	−1.71463e12			−0.460898			−0.230449	+	0.973084i	$0.574019\pi$
$194$		0			0
$195$	1.82778e12	−	5.95473e12i	0.464230	−	1.51242i
$196$		0			0
$197$		−	6.82742e12i		−	1.63943i	−0.572774	−	0.819714i	$-0.694133\pi$
$197$		−	6.82742e12i		−	1.63943i	0.572774	−	0.819714i	$-0.305867\pi$
$198$		0			0
$199$			1.82348e12i			0.414199i	0.978320	+	0.207099i	$0.0664023\pi$
$199$			1.82348e12i			0.414199i	−0.978320	+	0.207099i	$0.933598\pi$
$200$		0			0
$201$	4.43885e12	+	1.36249e12i	0.954316	+	0.292923i
$202$		0			0
$203$	−3.84854e12	+	2.99917e12i	−0.783551	+	0.610623i
$204$		0			0
$205$	−1.14612e13			−2.21098
$206$		0			0
$207$	3.91009e12	−	5.76926e12i	0.715072	−	1.05507i
$208$		0			0
$209$	4.95972e12			0.860306
$210$		0			0
$211$	7.82585e12			1.28818			0.644092	−	0.764948i	$-0.277235\pi$
$211$	7.82585e12			1.28818			0.644092	+	0.764948i	$0.277235\pi$
$212$		0			0
$213$	−1.38243e11	+	4.50382e11i	−0.0216050	+	0.0703871i
$214$		0			0
$215$	−1.03200e13			−1.53204
$216$		0			0
$217$	−4.50664e12	+	3.51203e12i	−0.635806	+	0.495485i
$218$		0			0
$219$	−8.20799e11	+	2.67409e12i	−0.110102	+	0.358701i
$220$		0			0
$221$			3.01011e12i			0.384083i
$222$		0			0
$223$			1.88423e11i			0.0228801i	0.999935	+	0.0114400i	$0.00364156\pi$
$223$			1.88423e11i			0.0228801i	−0.999935	+	0.0114400i	$0.996358\pi$
$224$		0			0
$225$	7.66565e12	+	5.19536e12i	0.886227	+	0.600636i
$226$		0			0
$227$	−7.32729e12			−0.806865			−0.403433	−	0.915009i	$-0.632183\pi$
$227$	−7.32729e12			−0.806865			−0.403433	+	0.915009i	$0.632183\pi$
$228$		0			0
$229$			7.64045e12i			0.801722i	0.916139	+	0.400861i	$0.131289\pi$
$229$			7.64045e12i			0.801722i	−0.916139	+	0.400861i	$0.868711\pi$
$230$		0			0
$231$	4.30925e12	−	1.64259e12i	0.431058	−	0.164310i
$232$		0			0
$233$			1.77658e13i			1.69483i	0.530927	+	0.847417i	$0.321844\pi$
$233$			1.77658e13i			1.69483i	−0.530927	+	0.847417i	$0.678156\pi$
$234$		0			0
$235$	1.19977e13			1.09200
$236$		0			0
$237$	−1.31995e13	−	4.05153e12i	−1.14668	−	0.351968i
$238$		0			0
$239$			8.78358e12i			0.728590i	0.931284	+	0.364295i	$0.118690\pi$
$239$			8.78358e12i			0.728590i	−0.931284	+	0.364295i	$0.881310\pi$
$240$		0			0
$241$		−	1.28818e12i		−	0.102066i	−0.998697	−	0.0510332i	$-0.983749\pi$
$241$		−	1.28818e12i		−	0.102066i	0.998697	−	0.0510332i	$-0.0162514\pi$
$242$		0			0
$243$	−1.07805e12	−	1.31639e13i	−0.0816216	−	0.996663i
$244$		0			0
$245$	4.85746e12	−	1.92795e13i	0.351557	−	1.39535i
$246$		0			0
$247$	−2.96257e13			−2.05038
$248$		0			0
$249$	2.12704e13	+	6.52885e12i	1.40825	+	0.432255i
$250$		0			0
$251$	−9.98790e12			−0.632803			−0.316402	−	0.948625i	$-0.602475\pi$
$251$	−9.98790e12			−0.632803			−0.316402	+	0.948625i	$0.602475\pi$
$252$		0			0
$253$	9.69436e12			0.587971
$254$		0			0
$255$	−8.27398e12	−	2.53966e12i	−0.480556	−	0.147504i
$256$		0			0
$257$	3.02454e12			0.168278			0.0841389	−	0.996454i	$-0.473186\pi$
$257$	3.02454e12			0.168278			0.0841389	+	0.996454i	$0.473186\pi$
$258$		0			0
$259$	−3.20701e12	−	4.11524e12i	−0.170983	−	0.219405i
$260$		0			0
$261$	−1.09051e13	+	1.60902e13i	−0.557322	+	0.822317i
$262$		0			0
$263$			1.48077e13i			0.725654i	0.931857	+	0.362827i	$0.118188\pi$
$263$			1.48077e13i			0.725654i	−0.931857	+	0.362827i	$0.881812\pi$
$264$		0			0
$265$			2.84425e13i			1.33695i
$266$		0			0
$267$	2.57122e13	+	7.89226e12i	1.15965	+	0.355951i
$268$		0			0
$269$	−7.75197e12			−0.335563			−0.167782	−	0.985824i	$-0.553660\pi$
$269$	−7.75197e12			−0.335563			−0.167782	+	0.985824i	$0.553660\pi$
$270$		0			0
$271$			4.26097e13i			1.77083i	0.464801	+	0.885415i	$0.346126\pi$
$271$			4.26097e13i			1.77083i	−0.464801	+	0.885415i	$0.653874\pi$
$272$		0			0
$273$	−2.57402e13	+	9.81158e12i	−1.02735	+	0.391602i
$274$		0			0
$275$			1.28809e13i			0.493876i
$276$		0			0
$277$	−1.36579e13			−0.503205			−0.251602	−	0.967831i	$-0.580958\pi$
$277$	−1.36579e13			−0.503205			−0.251602	+	0.967831i	$0.580958\pi$
$278$		0			0
$279$	−1.27699e13	+	1.88417e13i	−0.452234	+	0.667262i
$280$		0			0
$281$		−	2.18850e13i		−	0.745181i	−0.927996	−	0.372591i	$-0.878470\pi$
$281$		−	2.18850e13i		−	0.745181i	0.927996	−	0.372591i	$-0.121530\pi$
$282$		0			0
$283$			5.57453e13i			1.82550i	0.408515	+	0.912752i	$0.366047\pi$
$283$			5.57453e13i			1.82550i	−0.408515	+	0.912752i	$0.633953\pi$
$284$		0			0
$285$	2.49955e13	−	8.14330e13i	0.787436	−	2.56539i
$286$		0			0
$287$	3.11562e13	+	3.99796e13i	0.944482	+	1.21196i
$288$		0			0
$289$	−3.00894e13			−0.877962
$290$		0			0
$291$	−4.69782e12	+	1.53051e13i	−0.131973	+	0.429956i
$292$		0			0
$293$	4.92480e13			1.33235			0.666173	−	0.745797i	$-0.267931\pi$
$293$	4.92480e13			1.33235			0.666173	+	0.745797i	$0.267931\pi$
$294$		0			0
$295$	−4.69455e13			−1.22341
$296$		0			0
$297$	1.43161e13	−	1.15142e13i	0.359472	−	0.289117i
$298$		0			0
$299$	−5.79068e13			−1.40132
$300$		0			0
$301$	2.80539e13	+	3.59988e13i	0.654452	+	0.839793i
$302$		0			0
$303$	4.83033e13	+	1.48265e13i	1.08653	+	0.333506i
$304$		0			0
$305$		−	5.92743e13i		−	1.28593i
$306$		0			0
$307$		−	6.65789e13i		−	1.39340i	−0.717364	−	0.696699i	$-0.754651\pi$
$307$		−	6.65789e13i		−	1.39340i	0.717364	−	0.696699i	$-0.245349\pi$
$308$		0			0
$309$	2.47991e13	−	8.07933e13i	0.500801	−	1.63156i
$310$		0			0
$311$	−5.65257e13			−1.10170			−0.550851	−	0.834604i	$-0.685697\pi$
$311$	−5.65257e13			−1.10170			−0.550851	+	0.834604i	$0.685697\pi$
$312$		0			0
$313$			2.24278e13i			0.421980i	0.977488	+	0.210990i	$0.0676688\pi$
$313$			2.24278e13i			0.421980i	−0.977488	+	0.210990i	$0.932331\pi$
$314$		0			0
$315$	−5.25209e12	−	7.90311e13i	−0.0954167	−	1.43579i
$316$		0			0
$317$		−	5.94458e13i		−	1.04303i	−0.853243	−	0.521513i	$-0.825368\pi$
$317$		−	5.94458e13i		−	1.04303i	0.853243	−	0.521513i	$-0.174632\pi$
$318$		0			0
$319$	−2.70372e13			−0.458260
$320$		0			0
$321$	2.42703e13	−	7.90705e13i	0.397464	−	1.29490i
$322$		0			0
$323$			4.11642e13i			0.651488i
$324$		0			0
$325$		−	7.69410e13i		−	1.17706i
$326$		0			0
$327$	1.06174e14	+	3.25898e13i	1.57039	+	0.482024i
$328$		0			0
$329$	−3.26144e13	−	4.18508e13i	−0.466479	−	0.598585i
$330$		0			0
$331$	8.95007e13			1.23815			0.619074	−	0.785333i	$-0.287508\pi$
$331$	8.95007e13			1.23815			0.619074	+	0.785333i	$0.287508\pi$
$332$		0			0
$333$	−1.72053e13	−	1.16608e13i	−0.230260	−	0.156058i
$334$		0			0
$335$	1.10927e14			1.43645
$336$		0			0
$337$	−4.90357e13			−0.614537			−0.307269	−	0.951623i	$-0.599415\pi$
$337$	−4.90357e13			−0.614537			−0.307269	+	0.951623i	$0.599415\pi$
$338$		0			0
$339$	1.72361e13	−	5.61534e13i	0.209093	−	0.681205i
$340$		0			0
$341$	−3.16605e13			−0.371851
$342$		0			0
$343$	−8.04562e13	+	3.54654e13i	−0.915044	+	0.403354i
$344$		0			0
$345$	4.88566e13	−	1.59170e14i	0.538169	−	1.75330i
$346$		0			0
$347$			7.11685e13i			0.759409i	0.925108	+	0.379705i	$0.123974\pi$
$347$			7.11685e13i			0.759409i	−0.925108	+	0.379705i	$0.876026\pi$
$348$		0			0
$349$			1.14856e14i			1.18744i	0.804670	+	0.593722i	$0.202342\pi$
$349$			1.14856e14i			1.18744i	−0.804670	+	0.593722i	$0.797658\pi$
$350$		0			0
$351$	−8.55138e13	+	6.87771e13i	−0.856736	+	0.689057i
$352$		0			0
$353$	1.60638e14			1.55986			0.779931	−	0.625866i	$-0.215254\pi$
$353$	1.60638e14			1.55986			0.779931	+	0.625866i	$0.215254\pi$
$354$		0			0
$355$			1.12551e13i			0.105948i
$356$		0			0
$357$	1.36330e13	+	3.57655e13i	0.124427	+	0.326429i
$358$		0			0
$359$		−	1.03156e14i		−	0.913005i	−0.889722	−	0.456503i	$-0.849102\pi$
$359$		−	1.03156e14i		−	0.913005i	0.889722	−	0.456503i	$-0.150898\pi$
$360$		0			0
$361$	−2.88651e14			−2.47789
$362$		0			0
$363$	−9.03681e13	−	2.77381e13i	−0.752539	−	0.230988i
$364$		0			0
$365$			6.68254e13i			0.539922i
$366$		0			0
$367$			9.86203e13i			0.773220i	0.922243	+	0.386610i	$0.126354\pi$
$367$			9.86203e13i			0.773220i	−0.922243	+	0.386610i	$0.873646\pi$
$368$		0			0
$369$	1.67150e14	+	1.13285e14i	1.27192	+	0.862039i
$370$		0			0
$371$	9.92144e13	−	7.73180e13i	0.732855	−	0.571115i
$372$		0			0
$373$	7.43988e13			0.533540			0.266770	−	0.963760i	$-0.414044\pi$
$373$	7.43988e13			0.533540			0.266770	+	0.963760i	$0.414044\pi$
$374$		0			0
$375$	1.39447e13	+	4.28026e12i	0.0971040	+	0.0298057i
$376$		0			0
$377$	1.61500e14			1.09218
$378$		0			0
$379$	−1.09905e14			−0.721943			−0.360971	−	0.932577i	$-0.617555\pi$
$379$	−1.09905e14			−0.721943			−0.360971	+	0.932577i	$0.617555\pi$
$380$		0			0
$381$	6.08721e13	+	1.86844e13i	0.388446	+	0.119232i
$382$		0			0
$383$	−1.71854e14			−1.06553			−0.532767	−	0.846262i	$-0.678848\pi$
$383$	−1.71854e14			−1.06553			−0.532767	+	0.846262i	$0.678848\pi$
$384$		0			0
$385$	8.69012e13	−	6.77223e13i	0.523591	−	0.408035i
$386$		0			0
$387$	1.50506e14	+	1.02005e14i	0.881341	+	0.597325i
$388$		0			0
$389$		−	2.20643e14i		−	1.25593i	−0.778240	−	0.627967i	$-0.783887\pi$
$389$		−	2.20643e14i		−	1.25593i	0.778240	−	0.627967i	$-0.216113\pi$
$390$		0			0
$391$			8.04603e13i			0.445256i
$392$		0			0
$393$	−2.80128e13	−	8.59842e12i	−0.150729	−	0.0462658i
$394$		0			0
$395$	−3.29856e14			−1.72600
$396$		0			0
$397$		−	2.72052e14i		−	1.38454i	−0.721640	−	0.692268i	$-0.756611\pi$
$397$		−	2.72052e14i		−	1.38454i	0.721640	−	0.692268i	$-0.243389\pi$
$398$		0			0
$399$	−3.52006e14	+	1.34177e14i	−1.74261	+	0.664242i
$400$		0			0
$401$		−	2.88412e13i		−	0.138905i	−0.997585	−	0.0694526i	$-0.977875\pi$
$401$		−	2.88412e13i		−	0.138905i	0.997585	−	0.0694526i	$-0.0221253\pi$
$402$		0			0
$403$	1.89116e14			0.886239
$404$		0			0
$405$	−1.16901e14	−	2.93083e14i	−0.533109	−	1.33656i
$406$		0			0
$407$		−	2.89108e13i		−	0.128319i
$408$		0			0
$409$			1.39015e14i			0.600599i	0.953845	+	0.300299i	$0.0970866\pi$
$409$			1.39015e14i			0.600599i	−0.953845	+	0.300299i	$0.902913\pi$
$410$		0			0
$411$	6.49351e11	−	2.11552e12i	0.00273118	−	0.00889791i
$412$		0			0
$413$	1.27616e14	+	1.63757e14i	0.522614	+	0.670618i
$414$		0			0
$415$	5.31547e14			2.11971
$416$		0			0
$417$	1.87324e13	−	6.10282e13i	0.0727518	−	0.237019i
$418$		0			0
$419$	4.03168e14			1.52514			0.762569	−	0.646907i	$-0.223938\pi$
$419$	4.03168e14			1.52514			0.762569	+	0.646907i	$0.223938\pi$
$420$		0			0
$421$	3.16585e13			0.116664			0.0583322	−	0.998297i	$-0.481422\pi$
$421$	3.16585e13			0.116664			0.0583322	+	0.998297i	$0.481422\pi$
$422$		0			0
$423$	−1.74973e14	−	1.18587e14i	−0.628200	−	0.425760i
$424$		0			0
$425$	−1.06908e14			−0.374000
$426$		0			0
$427$	−2.06763e14	+	1.61131e14i	−0.704888	+	0.549320i
$428$		0			0
$429$	−1.45927e14	−	4.47916e13i	−0.484864	−	0.148827i
$430$		0			0
$431$			4.60562e14i			1.49164i	0.666149	+	0.745819i	$0.267942\pi$
$431$			4.60562e14i			1.49164i	−0.666149	+	0.745819i	$0.732058\pi$
$432$		0			0
$433$		−	1.37469e14i		−	0.434030i	−0.976168	−	0.217015i	$-0.930368\pi$
$433$		−	1.37469e14i		−	0.434030i	0.976168	−	0.217015i	$-0.0696321\pi$
$434$		0			0
$435$	−1.36259e14	+	4.43919e14i	−0.419445	+	1.36651i
$436$		0			0
$437$	−7.91895e14			−2.37695
$438$		0			0
$439$		−	1.97674e14i		−	0.578621i	−0.957235	−	0.289311i	$-0.906574\pi$
$439$		−	1.97674e14i		−	0.578621i	0.957235	−	0.289311i	$-0.0934261\pi$
$440$		0			0
$441$	−2.61403e14	+	2.33159e14i	−0.746273	+	0.665640i
$442$		0			0
$443$			4.92634e13i			0.137184i	0.997645	+	0.0685921i	$0.0218507\pi$
$443$			4.92634e13i			0.137184i	−0.997645	+	0.0685921i	$0.978149\pi$
$444$		0			0
$445$	6.42550e14			1.74553
$446$		0			0
$447$	−2.03459e14	+	6.62850e14i	−0.539244	+	1.75681i
$448$		0			0
$449$		−	6.38373e13i		−	0.165090i	−0.996587	−	0.0825448i	$-0.973695\pi$
$449$		−	6.38373e13i		−	0.165090i	0.996587	−	0.0825448i	$-0.0263048\pi$
$450$		0			0
$451$			2.80869e14i			0.708815i
$452$		0			0
$453$	6.99886e14	+	2.14827e14i	1.72380	+	0.529113i
$454$		0			0
$455$	−5.19082e14	+	4.04522e14i	−1.24788	+	0.972477i
$456$		0			0
$457$	−6.38250e14			−1.49779			−0.748896	−	0.662688i	$-0.769416\pi$
$457$	−6.38250e14			−1.49779			−0.748896	+	0.662688i	$0.769416\pi$
$458$		0			0
$459$	9.55644e13	+	1.18820e14i	0.218941	+	0.272219i
$460$		0			0
$461$	7.32576e14			1.63869			0.819346	−	0.573299i	$-0.194337\pi$
$461$	7.32576e14			1.63869			0.819346	+	0.573299i	$0.194337\pi$
$462$		0			0
$463$	−2.02878e14			−0.443139			−0.221569	−	0.975145i	$-0.571118\pi$
$463$	−2.02878e14			−0.443139			−0.221569	+	0.975145i	$0.571118\pi$
$464$		0			0
$465$	−1.59560e14	+	5.19830e14i	−0.340355	+	1.10884i
$466$		0			0
$467$	−4.43652e14			−0.924271			−0.462136	−	0.886809i	$-0.652917\pi$
$467$	−4.43652e14			−0.924271			−0.462136	+	0.886809i	$0.652917\pi$
$468$		0			0
$469$	−3.01544e14	−	3.86941e14i	−0.613620	−	0.787397i
$470$		0			0
$471$	6.97737e13	−	2.27316e14i	0.138700	−	0.451872i
$472$		0			0
$473$			2.52903e14i			0.491153i
$474$		0			0
$475$		−	1.05219e15i		−	1.99656i
$476$		0			0
$477$	2.81131e14	−	4.14802e14i	0.521263	−	0.769113i
$478$		0			0
$479$	−1.20802e13			−0.0218891			−0.0109446	−	0.999940i	$-0.503484\pi$
$479$	−1.20802e13			−0.0218891			−0.0109446	+	0.999940i	$0.503484\pi$
$480$		0			0
$481$			1.72691e14i			0.305825i
$482$		0			0
$483$	−6.88037e14	+	2.62264e14i	−1.19097	+	0.453972i
$484$		0			0
$485$			3.82474e14i			0.647176i
$486$		0			0
$487$	1.01418e15			1.67767			0.838837	−	0.544383i	$-0.183236\pi$
$487$	1.01418e15			1.67767			0.838837	+	0.544383i	$0.183236\pi$
$488$		0			0
$489$	4.92672e14	+	1.51223e14i	0.796817	+	0.244580i
$490$		0			0
$491$			1.00175e15i			1.58421i	0.610384	+	0.792106i	$0.291015\pi$
$491$			1.00175e15i			1.58421i	−0.610384	+	0.792106i	$0.708985\pi$
$492$		0			0
$493$		−	2.24400e14i		−	0.347029i
$494$		0			0
$495$	2.46240e14	−	3.63322e14i	0.372418	−	0.549495i
$496$		0			0
$497$	3.92604e13	−	3.05957e13i	0.0580757	−	0.0452585i
$498$		0			0
$499$	2.15484e14			0.311790			0.155895	−	0.987774i	$-0.450174\pi$
$499$	2.15484e14			0.311790			0.155895	+	0.987774i	$0.450174\pi$
$500$		0			0
$501$	−4.02400e14	−	1.23515e14i	−0.569575	−	0.174828i
$502$		0			0
$503$	1.72804e14			0.239293			0.119646	−	0.992817i	$-0.461824\pi$
$503$	1.72804e14			0.239293			0.119646	+	0.992817i	$0.461824\pi$
$504$		0			0
$505$	1.20710e15			1.63546
$506$		0			0
$507$	1.50562e14	+	4.62142e13i	0.199604	+	0.0612677i
$508$		0			0
$509$	−5.17653e14			−0.671569			−0.335785	−	0.941939i	$-0.609001\pi$
$509$	−5.17653e14			−0.671569			−0.335785	+	0.941939i	$0.609001\pi$
$510$		0			0
$511$	2.33104e14	−	1.81658e14i	0.295961	−	0.230643i
$512$		0			0
$513$	−1.16943e15	+	9.40551e14i	−1.45321	+	1.16879i
$514$		0			0
$515$		−	2.01903e15i		−	2.45585i
$516$		0			0
$517$		−	2.94015e14i		−	0.350083i
$518$		0			0
$519$	−5.92957e14	−	1.82006e14i	−0.691199	−	0.212160i
$520$		0			0
$521$	6.22134e13			0.0710029			0.0355015	−	0.999370i	$-0.488697\pi$
$521$	6.22134e13			0.0710029			0.0355015	+	0.999370i	$0.488697\pi$
$522$		0			0
$523$			9.05238e14i			1.01159i	0.862654	+	0.505794i	$0.168800\pi$
$523$			9.05238e14i			1.01159i	−0.862654	+	0.505794i	$0.831200\pi$
$524$		0			0
$525$	−3.48471e14	−	9.14198e14i	−0.381321	−	1.00038i
$526$		0			0
$527$		−	2.62773e14i		−	0.281593i
$528$		0			0
$529$	−5.95041e14			−0.624512
$530$		0			0
$531$	6.84648e14	+	4.64017e14i	0.703797	+	0.476995i
$532$		0			0
$533$		−	1.67770e15i		−	1.68933i
$534$		0			0
$535$		−	1.97597e15i		−	1.94910i
$536$		0			0
$537$	−5.68985e13	+	1.85370e14i	−0.0549848	+	0.179135i
$538$		0			0
$539$	−4.72464e14	−	1.19037e14i	−0.447332	−	0.112705i
$540$		0			0
$541$	−7.67189e14			−0.711733			−0.355867	−	0.934537i	$-0.615814\pi$
$541$	−7.67189e14			−0.711733			−0.355867	+	0.934537i	$0.615814\pi$
$542$		0			0
$543$	3.43057e14	−	1.11765e15i	0.311865	−	1.01603i
$544$		0			0
$545$	2.65330e15			2.36377
$546$		0			0
$547$	−1.15020e13			−0.0100425			−0.00502126	−	0.999987i	$-0.501598\pi$
$547$	−1.15020e13			−0.0100425			−0.00502126	+	0.999987i	$0.501598\pi$
$548$		0			0
$549$	−5.85877e14	+	8.64450e14i	−0.501370	+	0.739762i
$550$		0			0
$551$	2.20856e15			1.85257
$552$		0			0
$553$	8.96678e14	+	1.15062e15i	0.737307	+	0.946113i
$554$		0			0
$555$	−4.74683e14	−	1.45702e14i	−0.382642	−	0.117450i
$556$		0			0
$557$			9.47035e14i			0.748449i	0.927338	+	0.374225i	$0.122091\pi$
$557$			9.47035e14i			0.748449i	−0.927338	+	0.374225i	$0.877909\pi$
$558$		0			0
$559$		−	1.51065e15i		−	1.17057i
$560$		0			0
$561$	−6.22370e13	+	2.02762e14i	−0.0472882	+	0.154061i
$562$		0			0
$563$	−1.44084e15			−1.07355			−0.536773	−	0.843727i	$-0.680357\pi$
$563$	−1.44084e15			−1.07355			−0.536773	+	0.843727i	$0.680357\pi$
$564$		0			0
$565$		−	1.40328e15i		−	1.02536i
$566$		0			0
$567$	−7.04561e14	+	1.20449e15i	−0.504908	+	0.863173i
$568$		0			0
$569$		−	1.18836e15i		−	0.835275i	−0.908614	−	0.417638i	$-0.862858\pi$
$569$		−	1.18836e15i		−	0.835275i	0.908614	−	0.417638i	$-0.137142\pi$
$570$		0			0
$571$	−1.18051e15			−0.813902			−0.406951	−	0.913450i	$-0.633408\pi$
$571$	−1.18051e15			−0.813902			−0.406951	+	0.913450i	$0.633408\pi$
$572$		0			0
$573$	5.67083e14	−	1.84750e15i	0.383528	−	1.24950i
$574$		0			0
$575$		−	2.05664e15i		−	1.36453i
$576$		0			0
$577$		−	1.80548e14i		−	0.117524i	−0.998272	−	0.0587619i	$-0.981285\pi$
$577$		−	1.80548e14i		−	0.117524i	0.998272	−	0.0587619i	$-0.0187153\pi$
$578$		0			0
$579$	6.89898e14	+	2.11761e14i	0.440609	+	0.135243i
$580$		0			0
$581$	−1.44496e15	−	1.85417e15i	−0.905494	−	1.16193i
$582$		0			0
$583$	6.97012e14			0.428611
$584$		0			0
$585$	−1.47085e15	+	2.17021e15i	−0.887589	+	1.30962i
$586$		0			0
$587$	3.35133e15			1.98476			0.992380	−	0.123216i	$-0.0393209\pi$
$587$	3.35133e15			1.98476			0.992380	+	0.123216i	$0.0393209\pi$
$588$		0			0
$589$	2.58623e15			1.50326
$590$		0			0
$591$	−8.43205e14	+	2.74708e15i	−0.481063	+	1.56726i
$592$		0			0
$593$	1.18515e15			0.663702			0.331851	−	0.943332i	$-0.392327\pi$
$593$	1.18515e15			0.663702			0.331851	+	0.943332i	$0.392327\pi$
$594$		0			0
$595$	5.62075e14	+	7.21254e14i	0.308994	+	0.396502i
$596$		0			0
$597$	2.25205e14	−	7.33696e14i	0.121540	−	0.395965i
$598$		0			0
$599$			2.78060e15i			1.47330i	0.676273	+	0.736651i	$0.263594\pi$
$599$			2.78060e15i			1.47330i	−0.676273	+	0.736651i	$0.736406\pi$
$600$		0			0
$601$			2.56856e14i			0.133623i	0.997766	+	0.0668114i	$0.0212826\pi$
$601$			2.56856e14i			0.133623i	−0.997766	+	0.0668114i	$0.978717\pi$
$602$		0			0
$603$	−1.61775e15	−	1.09642e15i	−0.826353	−	0.560057i
$604$		0			0
$605$	−2.25830e15			−1.13273
$606$		0			0
$607$		−	3.67770e15i		−	1.81150i	−0.423811	−	0.905751i	$-0.639308\pi$
$607$		−	3.67770e15i		−	1.81150i	0.423811	−	0.905751i	$-0.360692\pi$
$608$		0			0
$609$	1.91891e15	−	7.31443e14i	0.928236	−	0.353822i
$610$		0			0
$611$			1.75622e15i			0.834358i
$612$		0			0
$613$	−1.58899e15			−0.741462			−0.370731	−	0.928740i	$-0.620893\pi$
$613$	−1.58899e15			−0.741462			−0.370731	+	0.928740i	$0.620893\pi$
$614$		0			0
$615$	4.61155e15	+	1.41550e15i	2.11365	+	0.648777i
$616$		0			0
$617$			3.36438e14i			0.151474i	0.997128	+	0.0757368i	$0.0241309\pi$
$617$			3.36438e14i			0.151474i	−0.997128	+	0.0757368i	$0.975869\pi$
$618$		0			0
$619$			4.29473e15i			1.89949i	0.313023	+	0.949746i	$0.398658\pi$
$619$			4.29473e15i			1.89949i	−0.313023	+	0.949746i	$0.601342\pi$
$620$		0			0
$621$	−2.28579e15	+	1.83842e15i	−0.993189	+	0.798803i
$622$		0			0
$623$	−1.74670e15	−	2.24137e15i	−0.745650	−	0.956819i
$624$		0			0
$625$	−2.20401e15			−0.924427
$626$		0			0
$627$	−1.99560e15	−	6.12540e14i	−0.822435	−	0.252443i
$628$		0			0
$629$	2.39951e14			0.0971728
$630$		0			0
$631$	−4.14367e15			−1.64901			−0.824505	−	0.565855i	$-0.808546\pi$
$631$	−4.14367e15			−1.64901			−0.824505	+	0.565855i	$0.808546\pi$
$632$		0			0
$633$	−3.14881e15	−	9.66515e14i	−1.23148	−	0.377997i
$634$		0			0
$635$	1.52119e15			0.584695
$636$		0			0
$637$	2.82214e15	+	7.11037e14i	1.06613	+	0.268612i
$638$		0			0
$639$	1.11247e14	−	1.64143e14i	0.0413079	−	0.0609490i
$640$		0			0
$641$		−	6.48851e13i		−	0.0236824i	−0.999930	−	0.0118412i	$-0.996231\pi$
$641$		−	6.48851e13i		−	0.0236824i	0.999930	−	0.0118412i	$-0.00376926\pi$
$642$		0			0
$643$		−	3.39426e13i		−	0.0121783i	−0.999981	−	0.00608913i	$-0.998062\pi$
$643$		−	3.39426e13i		−	0.0121783i	0.999981	−	0.00608913i	$-0.00193824\pi$
$644$		0			0
$645$	4.15237e15	+	1.27455e15i	1.46460	+	0.449551i
$646$		0			0
$647$	−3.13791e15			−1.08810			−0.544048	−	0.839054i	$-0.683109\pi$
$647$	−3.13791e15			−1.08810			−0.544048	+	0.839054i	$0.683109\pi$
$648$		0			0
$649$			1.15045e15i			0.392211i
$650$		0			0
$651$	2.24704e15	−	8.56521e14i	0.753209	−	0.287106i
$652$		0			0
$653$			5.44375e15i			1.79422i	0.441807	+	0.897110i	$0.354338\pi$
$653$			5.44375e15i			1.79422i	−0.441807	+	0.897110i	$0.645662\pi$
$654$		0			0
$655$	−7.00042e14			−0.226880
$656$		0			0
$657$	6.60514e14	−	9.74575e14i	0.210510	−	0.310603i
$658$		0			0
$659$			6.17614e15i			1.93574i	0.251448	+	0.967871i	$0.419093\pi$
$659$			6.17614e15i			1.93574i	−0.251448	+	0.967871i	$0.580907\pi$
$660$		0			0
$661$		−	5.06197e14i		−	0.156031i	−0.996952	−	0.0780157i	$-0.975142\pi$
$661$		−	5.06197e14i		−	0.156031i	0.996952	−	0.0780157i	$-0.0248584\pi$
$662$		0			0
$663$	3.71757e14	−	1.21115e15i	0.112703	−	0.367175i
$664$		0			0
$665$	−7.09863e15	+	5.53197e15i	−2.11668	+	1.64953i
$666$		0			0
$667$	4.31689e15			1.26613
$668$		0			0
$669$	2.32708e13	−	7.58140e13i	0.00671379	−	0.0218729i
$670$		0			0
$671$	−1.45258e15			−0.412254
$672$		0			0
$673$	4.28322e15			1.19588			0.597939	−	0.801541i	$-0.295986\pi$
$673$	4.28322e15			1.19588			0.597939	+	0.801541i	$0.295986\pi$
$674$		0			0
$675$	−2.44271e15	−	3.03714e15i	−0.670968	−	0.834245i
$676$		0			0
$677$	1.36634e15			0.369251			0.184626	−	0.982809i	$-0.440893\pi$
$677$	1.36634e15			0.369251			0.184626	+	0.982809i	$0.440893\pi$
$678$		0			0
$679$	1.33416e15	−	1.03972e15i	0.354752	−	0.276459i
$680$		0			0
$681$	2.94821e15	+	9.04941e14i	0.771346	+	0.236762i
$682$		0			0
$683$		−	3.93349e15i		−	1.01266i	−0.862339	−	0.506331i	$-0.831001\pi$
$683$		−	3.93349e15i		−	1.01266i	0.862339	−	0.506331i	$-0.168999\pi$
$684$		0			0
$685$		−	5.28670e13i		−	0.0133933i
$686$		0			0
$687$	9.43618e14	−	3.07422e15i	0.235252	−	0.766430i
$688$		0			0
$689$	−4.16342e15			−1.02151
$690$		0			0
$691$			2.90223e15i			0.700813i	0.936598	+	0.350406i	$0.113957\pi$
$691$			2.90223e15i			0.700813i	−0.936598	+	0.350406i	$0.886043\pi$
$692$		0			0
$693$	−1.93674e15	+	1.28708e14i	−0.460297	+	0.0305895i
$694$		0			0
$695$		−	1.52510e15i		−	0.356764i
$696$		0			0
$697$	−2.33113e15			−0.536768
$698$		0			0
$699$	2.19413e15	−	7.14826e15i	0.497322	−	1.62023i
$700$		0			0
$701$		−	9.58521e14i		−	0.213871i	−0.994266	−	0.106936i	$-0.965896\pi$
$701$		−	9.58521e14i		−	0.213871i	0.994266	−	0.106936i	$-0.0341039\pi$
$702$		0			0
$703$			2.36161e15i			0.518746i
$704$		0			0
$705$	−4.82738e15	−	1.48174e15i	−1.04393	−	0.320430i
$706$		0			0
$707$	−3.28138e15	−	4.21066e15i	−0.698634	−	0.896487i
$708$		0			0
$709$	−3.64050e15			−0.763144			−0.381572	−	0.924339i	$-0.624617\pi$
$709$	−3.64050e15			−0.763144			−0.381572	+	0.924339i	$0.624617\pi$
$710$		0			0
$711$	4.81058e15	+	3.26035e15i	0.992921	+	0.672948i
$712$		0			0
$713$	5.05508e15			1.02739
$714$		0			0
$715$	−3.64671e15			−0.729824
$716$		0			0
$717$	1.08480e15	−	3.53417e15i	0.213793	−	0.696517i
$718$		0			0
$719$	5.29749e15			1.02816			0.514081	−	0.857742i	$-0.328133\pi$
$719$	5.29749e15			1.02816			0.514081	+	0.857742i	$0.328133\pi$
$720$		0			0
$721$	−7.04286e15	+	5.48851e15i	−1.34619	+	1.04908i
$722$		0			0
$723$	−1.59094e14	+	5.18313e14i	−0.0299497	+	0.0975734i
$724$		0			0
$725$			5.73587e15i			1.06351i
$726$		0			0
$727$		−	1.43487e15i		−	0.262043i	−0.991380	−	0.131022i	$-0.958174\pi$
$727$		−	1.43487e15i		−	0.262043i	0.991380	−	0.131022i	$-0.0418257\pi$
$728$		0			0
$729$	−1.19201e15	+	5.42976e15i	−0.214426	+	0.976740i
$730$		0			0
$731$	−2.09902e15			−0.371938
$732$		0			0
$733$			5.49896e15i			0.959862i	0.877306	+	0.479931i	$0.159338\pi$
$733$			5.49896e15i			0.959862i	−0.877306	+	0.479931i	$0.840662\pi$
$734$		0			0
$735$	−4.33553e15	+	7.15741e15i	−0.745524	+	1.23077i
$736$		0			0
$737$		−	2.71838e15i		−	0.460509i
$738$		0			0
$739$	1.06288e16			1.77394			0.886970	−	0.461828i	$-0.152806\pi$
$739$	1.06288e16			1.77394			0.886970	+	0.461828i	$0.152806\pi$
$740$		0			0
$741$	1.19202e16	+	3.65885e15i	1.96012	+	0.601651i
$742$		0			0
$743$			5.45302e14i			0.0883484i	0.999024	+	0.0441742i	$0.0140657\pi$
$743$			5.45302e14i			0.0883484i	−0.999024	+	0.0441742i	$0.985934\pi$
$744$		0			0
$745$			1.65646e16i			2.64437i
$746$		0			0
$747$	−7.75202e15	−	5.25390e15i	−1.21942	−	0.826454i
$748$		0			0
$749$	−6.89268e15	+	5.37148e15i	−1.06841	+	0.832614i
$750$		0			0
$751$	−8.15256e15			−1.24530			−0.622650	−	0.782500i	$-0.713944\pi$
$751$	−8.15256e15			−1.24530			−0.622650	+	0.782500i	$0.713944\pi$
$752$		0			0
$753$	4.01874e15	+	1.23353e15i	0.604947	+	0.185686i
$754$		0			0
$755$	1.74902e16			2.59469
$756$		0			0
$757$	−8.95718e15			−1.30962			−0.654808	−	0.755795i	$-0.727250\pi$
$757$	−8.95718e15			−1.30962			−0.654808	+	0.755795i	$0.727250\pi$
$758$		0			0
$759$	−3.90063e15	−	1.19728e15i	−0.562089	−	0.172531i
$760$		0			0
$761$	1.32376e16			1.88016			0.940078	−	0.340959i	$-0.110752\pi$
$761$	1.32376e16			1.88016			0.940078	+	0.340959i	$0.110752\pi$
$762$		0			0
$763$	−7.21273e15	−	9.25537e15i	−1.00975	−	1.29571i
$764$		0			0
$765$	3.01547e15	+	2.04372e15i	0.416118	+	0.282022i
$766$		0			0
$767$		−	6.87189e15i		−	0.934764i
$768$		0			0
$769$		−	3.21951e15i		−	0.431713i	−0.976425	−	0.215857i	$-0.930746\pi$
$769$		−	3.21951e15i		−	0.431713i	0.976425	−	0.215857i	$-0.0692543\pi$
$770$		0			0
$771$	−1.21695e15	−	3.73539e14i	−0.160870	−	0.0493784i
$772$		0			0
$773$	−3.46371e15			−0.451392			−0.225696	−	0.974198i	$-0.572466\pi$
$773$	−3.46371e15			−0.451392			−0.225696	+	0.974198i	$0.572466\pi$
$774$		0			0
$775$			6.71672e15i			0.862974i
$776$		0			0
$777$	7.82132e14	+	2.05189e15i	0.0990751	+	0.259919i
$778$		0			0
$779$		−	2.29431e16i		−	2.86547i
$780$		0			0
$781$	2.75816e14			0.0339656
$782$		0			0
$783$	6.37496e15	−	5.12726e15i	0.774084	−	0.622581i
$784$		0			0
$785$		−	5.68063e15i		−	0.680164i
$786$		0			0
$787$		−	1.31190e16i		−	1.54896i	−0.632599	−	0.774480i	$-0.718012\pi$
$787$		−	1.31190e16i		−	1.54896i	0.632599	−	0.774480i	$-0.281988\pi$
$788$		0			0
$789$	1.82879e15	−	5.95802e15i	0.212931	−	0.693710i
$790$		0			0
$791$	−4.89497e15	+	3.81466e15i	−0.562056	+	0.438011i
$792$		0			0
$793$	8.67659e15			0.982532
$794$		0			0
$795$	3.51273e15	−	1.14441e16i	0.392306	−	1.27810i
$796$		0			0
$797$	−5.21813e15			−0.574770			−0.287385	−	0.957815i	$-0.592786\pi$
$797$	−5.21813e15			−0.574770			−0.287385	+	0.957815i	$0.592786\pi$
$798$		0			0
$799$	2.44023e15			0.265109
$800$		0			0
$801$	−9.37088e15	−	6.35107e15i	−1.00416	−	0.680563i
$802$		0			0
$803$	1.63762e15			0.173093
$804$		0			0
$805$	−1.38751e16	+	1.08129e16i	−1.44663	+	1.12736i
$806$		0			0
$807$	3.11909e15	+	9.57390e14i	0.320792	+	0.0984656i
$808$		0			0
$809$		−	1.19015e16i		−	1.20750i	−0.797174	−	0.603749i	$-0.793673\pi$
$809$		−	1.19015e16i		−	1.20750i	0.797174	−	0.603749i	$-0.206327\pi$
$810$		0			0
$811$		−	3.08125e15i		−	0.308399i	−0.988040	−	0.154199i	$-0.950720\pi$
$811$		−	3.08125e15i		−	0.308399i	0.988040	−	0.154199i	$-0.0492798\pi$
$812$		0			0
$813$	5.26242e15	−	1.71445e16i	0.519622	−	1.69288i
$814$		0			0
$815$	1.23119e16			1.19938
$816$		0			0
$817$		−	2.06586e16i		−	1.98555i
$818$		0			0
$819$	1.15686e16	−	7.68804e14i	1.09703	−	0.0729044i
$820$		0			0
$821$			1.30229e16i			1.21848i	0.792985	+	0.609241i	$0.208526\pi$
$821$			1.30229e16i			1.21848i	−0.792985	+	0.609241i	$0.791474\pi$
$822$		0			0
$823$	6.45744e15			0.596158			0.298079	−	0.954541i	$-0.403654\pi$
$823$	6.45744e15			0.596158			0.298079	+	0.954541i	$0.403654\pi$
$824$		0			0
$825$	1.59083e15	−	5.18278e15i	0.144920	−	0.472135i
$826$		0			0
$827$		−	4.38592e15i		−	0.394258i	−0.980378	−	0.197129i	$-0.936838\pi$
$827$		−	4.38592e15i		−	0.394258i	0.980378	−	0.197129i	$-0.0631617\pi$
$828$		0			0
$829$			2.47954e15i			0.219948i	0.993934	+	0.109974i	$0.0350768\pi$
$829$			2.47954e15i			0.219948i	−0.993934	+	0.109974i	$0.964923\pi$
$830$		0			0
$831$	5.49539e15	+	1.68679e15i	0.481053	+	0.147657i
$832$		0			0
$833$	9.87971e14	−	3.92131e15i	0.0853487	−	0.338753i
$834$		0			0
$835$	−1.00560e16			−0.857332
$836$		0			0
$837$	7.46509e15	−	6.00403e15i	0.628124	−	0.505188i
$838$		0			0
$839$	−4.67888e15			−0.388554			−0.194277	−	0.980947i	$-0.562236\pi$
$839$	−4.67888e15			−0.388554			−0.194277	+	0.980947i	$0.562236\pi$
$840$		0			0
$841$	1.60888e14			0.0131870
$842$		0			0
$843$	−2.70286e15	+	8.80567e15i	−0.218661	+	0.712378i
$844$		0			0
$845$	3.76254e15			0.300448
$846$		0			0
$847$	6.13896e15	+	7.87751e15i	0.483878	+	0.620912i
$848$		0			0
$849$	6.88470e15	−	2.24297e16i	0.535664	−	1.74514i
$850$		0			0
$851$			4.61605e15i			0.354534i
$852$		0			0
$853$			1.42911e16i			1.08355i	0.840525	+	0.541773i	$0.182247\pi$
$853$			1.42911e16i			1.08355i	−0.840525	+	0.541773i	$0.817753\pi$
$854$		0			0
$855$	−2.01144e16	+	2.96784e16i	−1.50555	+	2.22140i
$856$		0			0
$857$	−1.79811e16			−1.32868			−0.664341	−	0.747429i	$-0.731288\pi$
$857$	−1.79811e16			−1.32868			−0.664341	+	0.747429i	$0.731288\pi$
$858$		0			0
$859$			1.31077e16i			0.956231i	0.878297	+	0.478116i	$0.158680\pi$
$859$			1.31077e16i			0.956231i	−0.878297	+	0.478116i	$0.841320\pi$
$860$		0			0
$861$	−7.59842e15	−	1.99341e16i	−0.547276	−	1.43575i
$862$		0			0
$863$			1.52135e16i			1.08186i	0.841068	+	0.540930i	$0.181927\pi$
$863$			1.52135e16i			1.08186i	−0.841068	+	0.540930i	$0.818073\pi$
$864$		0			0
$865$	−1.48180e16			−1.04040
$866$		0			0
$867$	1.21068e16	+	3.71613e15i	0.839313	+	0.257624i
$868$		0			0
$869$			8.08344e15i			0.553334i
$870$		0			0
$871$			1.62375e16i			1.09754i
$872$		0			0
$873$	3.78044e15	−	5.57796e15i	0.252327	−	0.372303i
$874$		0			0
$875$	−9.47302e14	−	1.21558e15i	−0.0624373	−	0.0801196i
$876$		0			0
$877$	2.21784e16			1.44355			0.721777	−	0.692125i	$-0.243325\pi$
$877$	2.21784e16			1.44355			0.721777	+	0.692125i	$0.243325\pi$
$878$		0			0
$879$	−1.98155e16	−	6.08227e15i	−1.27369	−	0.390955i
$880$		0			0
$881$	1.15893e16			0.735683			0.367842	−	0.929888i	$-0.380097\pi$
$881$	1.15893e16			0.735683			0.367842	+	0.929888i	$0.380097\pi$
$882$		0			0
$883$	−1.08982e16			−0.683238			−0.341619	−	0.939839i	$-0.610975\pi$
$883$	−1.08982e16			−0.683238			−0.341619	+	0.939839i	$0.610975\pi$
$884$		0			0
$885$	1.88890e16	+	5.79790e15i	1.16956	+	0.358990i
$886$		0			0
$887$	−8.47795e14			−0.0518455			−0.0259227	−	0.999664i	$-0.508252\pi$
$887$	−8.47795e14			−0.0518455			−0.0259227	+	0.999664i	$0.508252\pi$
$888$		0			0
$889$	−4.13521e15	−	5.30630e15i	−0.249769	−	0.320503i
$890$		0			0
$891$	−7.18229e15	+	2.86478e15i	−0.428485	+	0.170908i
$892$		0			0
$893$			2.40169e16i			1.41525i
$894$		0			0
$895$			4.63240e15i			0.269637i
$896$		0			0
$897$	2.32994e16	+	7.15165e15i	1.33963	+	0.411195i
$898$		0			0
$899$	−1.40984e16			−0.800741
$900$		0			0
$901$			5.78499e15i			0.324576i
$902$		0			0
$903$	−6.84184e15	−	1.79492e16i	−0.379219	−	0.994863i
$904$		0			0
$905$		−	2.79300e16i		−	1.52934i
$906$		0			0
$907$	5.49592e15			0.297304			0.148652	−	0.988890i	$-0.452507\pi$
$907$	5.49592e15			0.297304			0.148652	+	0.988890i	$0.452507\pi$
$908$		0			0
$909$	−1.76042e16	−	1.19312e16i	−0.940840	−	0.637650i
$910$		0			0
$911$			1.21470e16i			0.641384i	0.947184	+	0.320692i	$0.103915\pi$
$911$			1.21470e16i			0.641384i	−0.947184	+	0.320692i	$0.896085\pi$
$912$		0			0
$913$		−	1.30261e16i		−	0.679555i
$914$		0			0
$915$	−7.32054e15	+	2.38496e16i	−0.377335	+	1.22932i
$916$		0			0
$917$	1.90299e15	+	2.44192e15i	0.0969182	+	0.124365i
$918$		0			0
$919$	−1.68697e15			−0.0848931			−0.0424466	−	0.999099i	$-0.513515\pi$
$919$	−1.68697e15			−0.0848931			−0.0424466	+	0.999099i	$0.513515\pi$
$920$		0			0
$921$	−8.22268e15	+	2.67887e16i	−0.408870	+	1.33206i
$922$		0			0
$923$	−1.64752e15			−0.0809508
$924$		0			0
$925$	−6.13337e15			−0.297797
$926$		0			0
$927$	−1.99564e16	+	2.94453e16i	−0.957511	+	1.41279i
$928$		0			0
$929$	−2.24618e16			−1.06502			−0.532511	−	0.846423i	$-0.678751\pi$
$929$	−2.24618e16			−1.06502			−0.532511	+	0.846423i	$0.678751\pi$
$930$		0			0
$931$	3.85938e16	+	9.72367e15i	1.80840	+	0.455624i
$932$		0			0
$933$	2.27437e16	+	6.98109e15i	1.05320	+	0.323277i
$934$		0			0
$935$			5.06703e15i			0.231894i
$936$		0			0
$937$		−	2.70591e16i		−	1.22390i	−0.790896	−	0.611950i	$-0.790385\pi$
$937$		−	2.70591e16i		−	1.22390i	0.790896	−	0.611950i	$-0.209615\pi$
$938$		0			0
$939$	2.76989e15	−	9.02405e15i	0.123823	−	0.403405i
$940$		0			0
$941$	4.91171e15			0.217015			0.108507	−	0.994096i	$-0.465393\pi$
$941$	4.91171e15			0.217015			0.108507	+	0.994096i	$0.465393\pi$
$942$		0			0
$943$		−	4.48450e16i		−	1.95839i
$944$		0			0
$945$	−7.64733e15	+	3.24477e16i	−0.330092	+	1.40058i
$946$		0			0
$947$			4.47287e16i			1.90836i	0.299228	+	0.954182i	$0.403271\pi$
$947$			4.47287e16i			1.90836i	−0.299228	+	0.954182i	$0.596729\pi$
$948$		0			0
$949$	−9.78193e15			−0.412535
$950$		0			0
$951$	−7.34173e15	+	2.39187e16i	−0.306059	+	0.997112i
$952$		0			0
$953$		−	5.48421e15i		−	0.225997i	−0.993595	−	0.112999i	$-0.963954\pi$
$953$		−	5.48421e15i		−	0.225997i	0.993595	−	0.112999i	$-0.0360455\pi$
$954$		0			0
$955$		−	4.61692e16i		−	1.88076i
$956$		0			0
$957$	1.08787e16	+	3.33917e15i	0.438087	+	0.134469i
$958$		0			0
$959$	−1.84413e14	+	1.43713e14i	−0.00734158	+	0.00572131i
$960$		0			0
$961$	8.89922e15			0.350246
$962$		0			0
$963$	−1.95309e16	+	2.88174e16i	−0.759936	+	1.12127i
$964$		0			0
$965$	1.72406e16			0.663210
$966$		0			0
$967$	2.61938e16			0.996216			0.498108	−	0.867115i	$-0.334028\pi$
$967$	2.61938e16			0.996216			0.498108	+	0.867115i	$0.334028\pi$
$968$		0			0
$969$	5.08390e15	−	1.65629e16i	0.191169	−	0.622809i
$970$		0			0
$971$	−1.07919e16			−0.401229			−0.200614	−	0.979670i	$-0.564294\pi$
$971$	−1.07919e16			−0.401229			−0.200614	+	0.979670i	$0.564294\pi$
$972$		0			0
$973$	−5.31991e15	+	4.14582e15i	−0.195562	+	0.152402i
$974$		0			0
$975$	−9.50244e15	+	3.09580e16i	−0.345390	+	1.12525i
$976$		0			0
$977$		−	1.46794e16i		−	0.527580i	−0.964580	−	0.263790i	$-0.915027\pi$
$977$		−	1.46794e16i		−	0.527580i	0.964580	−	0.263790i	$-0.0849726\pi$
$978$		0			0
$979$		−	1.57463e16i		−	0.559596i
$980$		0			0
$981$	−3.86955e16	−	2.62257e16i	−1.35982	−	0.921611i
$982$		0			0
$983$	−3.73871e15			−0.129920			−0.0649602	−	0.997888i	$-0.520692\pi$
$983$	−3.73871e15			−0.129920			−0.0649602	+	0.997888i	$0.520692\pi$
$984$		0			0
$985$			6.86497e16i			2.35906i
$986$		0			0
$987$	7.95405e15	+	2.08671e16i	0.270299	+	0.709116i
$988$		0			0
$989$		−	4.03797e16i		−	1.35701i
$990$		0			0
$991$	1.08583e16			0.360874			0.180437	−	0.983587i	$-0.442249\pi$
$991$	1.08583e16			0.360874			0.180437	+	0.983587i	$0.442249\pi$
$992$		0			0
$993$	−3.60116e16	−	1.10536e16i	−1.18364	−	0.363314i
$994$		0			0
$995$		−	1.83351e16i		−	0.596013i
$996$		0			0
$997$			6.75593e15i			0.217201i	0.994085	+	0.108600i	$0.0346369\pi$
$997$			6.75593e15i			0.217201i	−0.994085	+	0.108600i	$0.965363\pi$
$998$		0			0
$999$	5.48258e15	+	6.81675e15i	0.174331	+	0.216754i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.f.b.41.3	✓	28
3.2	odd	2	inner	84.12.f.b.41.25	yes	28
7.6	odd	2	inner	84.12.f.b.41.26	yes	28
21.20	even	2	inner	84.12.f.b.41.4	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.f.b.41.3	✓	28	1.1	even	1	trivial
84.12.f.b.41.4	yes	28	21.20	even	2	inner
84.12.f.b.41.25	yes	28	3.2	odd	2	inner
84.12.f.b.41.26	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)

\(f(q)\)	\(=\)	\(q+(-402.361 - 123.503i) q^{3} -10055.0 q^{5} +(27333.5 + 35074.3i) q^{7} +(146641. + 99385.4i) q^{9} +O(q^{10})\)	\(q+(-402.361 - 123.503i) q^{3} -10055.0 q^{5} +(27333.5 + 35074.3i) q^{7} +(146641. + 99385.4i) q^{9} +246408. i q^{11} -1.47185e6i q^{13} +(4.04574e6 + 1.24182e6i) q^{15} -2.04511e6 q^{17} -2.01281e7i q^{19} +(-6.66613e6 - 1.74883e7i) q^{21} -3.93427e7i q^{23} +5.22749e7 q^{25} +(-4.67282e7 - 5.80993e7i) q^{27} +1.09725e8i q^{29} +1.28488e8i q^{31} +(3.04321e7 - 9.91448e7i) q^{33} +(-2.74838e8 - 3.52672e8i) q^{35} -1.17329e8 q^{37} +(-1.81778e8 + 5.92216e8i) q^{39} +1.13985e9 q^{41} +1.02636e9 q^{43} +(-1.47448e9 - 9.99320e8i) q^{45} -1.19320e9 q^{47} +(-4.83089e8 + 1.91741e9i) q^{49} +(8.22873e8 + 2.52577e8i) q^{51} -2.82869e9i q^{53} -2.47763e9i q^{55} +(-2.48588e9 + 8.09876e9i) q^{57} +4.66887e9 q^{59} +5.89500e9i q^{61} +(5.22337e8 + 7.85988e9i) q^{63} +1.47995e10i q^{65} -1.10320e10 q^{67} +(-4.85894e9 + 1.58300e10i) q^{69} -1.11935e9i q^{71} -6.64599e9i q^{73} +(-2.10334e10 - 6.45610e9i) q^{75} +(-8.64259e9 + 6.73518e9i) q^{77} +3.28051e10 q^{79} +(1.16262e10 + 2.91480e10i) q^{81} -5.28639e10 q^{83} +2.05636e10 q^{85} +(1.35514e10 - 4.41491e10i) q^{87} -6.39035e10 q^{89} +(5.16243e10 - 4.02309e10i) q^{91} +(1.58687e10 - 5.16987e10i) q^{93} +2.02388e11i q^{95} -3.80382e10i q^{97} +(-2.44893e10 + 3.61335e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more