LMFDB - Embedded newform 84.12.f.b.41.6

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			41.6
Character	$\chi$	$=$	84.41
Dual form			84.12.f.b.41.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-384.586 + 170.999i) q^{3} +2406.66 q^{5} +(-39402.8 - 20609.4i) q^{7} +(118665. - 131528. i) q^{9} +O(q^{10})$	\(q+(-384.586 + 170.999i) q^{3} +2406.66 q^{5} +(-39402.8 - 20609.4i) q^{7} +(118665. - 131528. i) q^{9} +669348. i q^{11} +528001. i q^{13} +(-925566. + 411537. i) q^{15} -6.71333e6 q^{17} -3.29364e6i q^{19} +(1.86779e7 + 1.18824e6i) q^{21} -5.21969e7i q^{23} -4.30361e7 q^{25} +(-2.31458e7 + 7.08755e7i) q^{27} -8.39935e7i q^{29} +2.41368e8i q^{31} +(-1.14458e8 - 2.57422e8i) q^{33} +(-9.48290e7 - 4.95999e7i) q^{35} -7.47283e8 q^{37} +(-9.02879e7 - 2.03062e8i) q^{39} -1.59170e8 q^{41} +6.53005e8 q^{43} +(2.85587e8 - 3.16543e8i) q^{45} +1.84720e9 q^{47} +(1.12783e9 + 1.62414e9i) q^{49} +(2.58185e9 - 1.14798e9i) q^{51} +1.27100e9i q^{53} +1.61089e9i q^{55} +(5.63211e8 + 1.26669e9i) q^{57} +8.68623e9 q^{59} -1.10675e10i q^{61} +(-7.38646e9 + 2.73694e9i) q^{63} +1.27072e9i q^{65} +8.09405e9 q^{67} +(8.92563e9 + 2.00742e10i) q^{69} -1.92348e10i q^{71} +2.94706e10i q^{73} +(1.65511e10 - 7.35915e9i) q^{75} +(1.37949e10 - 2.63742e10i) q^{77} +8.47582e9 q^{79} +(-3.21811e9 - 3.12156e10i) q^{81} +2.91537e10 q^{83} -1.61567e10 q^{85} +(1.43628e10 + 3.23027e10i) q^{87} +3.33043e10 q^{89} +(1.08818e10 - 2.08047e10i) q^{91} +(-4.12737e10 - 9.28265e10i) q^{93} -7.92667e9i q^{95} +2.25760e10i q^{97} +(8.80380e10 + 7.94285e10i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−384.586	+	170.999i	−0.913748	+	0.406282i
$3$	−384.586	+	170.999i	−0.913748	+	0.406282i
$4$		0			0
$5$	2406.66			0.344413			0.172206	−	0.985061i	$-0.444910\pi$
$5$	2406.66			0.344413			0.172206	+	0.985061i	$0.444910\pi$
$6$		0			0
$7$	−39402.8	−	20609.4i	−0.886110	−	0.463476i
$7$	−39402.8	−	20609.4i	−0.886110	−	0.463476i
$8$		0			0
$9$	118665.	−	131528.i	0.669870	−	0.742479i
$10$		0			0
$11$			669348.i			1.25312i	0.779374	+	0.626559i	$0.215537\pi$
$11$			669348.i			1.25312i	−0.779374	+	0.626559i	$0.784463\pi$
$12$		0			0
$13$			528001.i			0.394409i	0.980362	+	0.197204i	$0.0631863\pi$
$13$			528001.i			0.394409i	−0.980362	+	0.197204i	$0.936814\pi$
$14$		0			0
$15$	−925566.	+	411537.i	−0.314706	+	0.139929i
$16$		0			0
$17$	−6.71333e6			−1.14675			−0.573375	−	0.819293i	$-0.694366\pi$
$17$	−6.71333e6			−1.14675			−0.573375	+	0.819293i	$0.694366\pi$
$18$		0			0
$19$		−	3.29364e6i		−	0.305163i	−0.988291	−	0.152581i	$-0.951241\pi$
$19$		−	3.29364e6i		−	0.305163i	0.988291	−	0.152581i	$-0.0487586\pi$
$20$		0			0
$21$	1.86779e7	+	1.18824e6i	0.997983	+	0.0634891i
$22$		0			0
$23$		−	5.21969e7i		−	1.69099i	−0.533983	−	0.845495i	$-0.679305\pi$
$23$		−	5.21969e7i		−	1.69099i	0.533983	−	0.845495i	$-0.320695\pi$
$24$		0			0
$25$	−4.30361e7			−0.881380
$26$		0			0
$27$	−2.31458e7	+	7.08755e7i	−0.310436	+	0.950594i
$28$		0			0
$29$		−	8.39935e7i		−	0.760425i	−0.924899	−	0.380213i	$-0.875851\pi$
$29$		−	8.39935e7i		−	0.760425i	0.924899	−	0.380213i	$-0.124149\pi$
$30$		0			0
$31$			2.41368e8i			1.51422i	0.653286	+	0.757111i	$0.273390\pi$
$31$			2.41368e8i			1.51422i	−0.653286	+	0.757111i	$0.726610\pi$
$32$		0			0
$33$	−1.14458e8	−	2.57422e8i	−0.509120	−	1.14503i
$34$		0			0
$35$	−9.48290e7	−	4.95999e7i	−0.305188	−	0.159627i
$36$		0			0
$37$	−7.47283e8			−1.77164			−0.885820	−	0.464029i	$-0.846403\pi$
$37$	−7.47283e8			−1.77164			−0.885820	+	0.464029i	$0.846403\pi$
$38$		0			0
$39$	−9.02879e7	−	2.03062e8i	−0.160241	−	0.360390i
$40$		0			0
$41$	−1.59170e8			−0.214560			−0.107280	−	0.994229i	$-0.534214\pi$
$41$	−1.59170e8			−0.214560			−0.107280	+	0.994229i	$0.534214\pi$
$42$		0			0
$43$	6.53005e8			0.677392			0.338696	−	0.940896i	$-0.390014\pi$
$43$	6.53005e8			0.677392			0.338696	+	0.940896i	$0.390014\pi$
$44$		0			0
$45$	2.85587e8	−	3.16543e8i	0.230712	−	0.255719i
$46$		0			0
$47$	1.84720e9			1.17483			0.587416	−	0.809285i	$-0.300145\pi$
$47$	1.84720e9			1.17483			0.587416	+	0.809285i	$0.300145\pi$
$48$		0			0
$49$	1.12783e9	+	1.62414e9i	0.570381	+	0.821380i
$50$		0			0
$51$	2.58185e9	−	1.14798e9i	1.04784	−	0.465904i
$52$		0			0
$53$			1.27100e9i			0.417472i	0.977972	+	0.208736i	$0.0669350\pi$
$53$			1.27100e9i			0.417472i	−0.977972	+	0.208736i	$0.933065\pi$
$54$		0			0
$55$			1.61089e9i			0.431590i
$56$		0			0
$57$	5.63211e8	+	1.26669e9i	0.123982	+	0.278842i
$58$		0			0
$59$	8.68623e9			1.58178			0.790888	−	0.611961i	$-0.209619\pi$
$59$	8.68623e9			1.58178			0.790888	+	0.611961i	$0.209619\pi$
$60$		0			0
$61$		−	1.10675e10i		−	1.67777i	−0.544305	−	0.838887i	$-0.683207\pi$
$61$		−	1.10675e10i		−	1.67777i	0.544305	−	0.838887i	$-0.316793\pi$
$62$		0			0
$63$	−7.38646e9	+	2.73694e9i	−0.937699	+	0.347450i
$64$		0			0
$65$			1.27072e9i			0.135839i
$66$		0			0
$67$	8.09405e9			0.732410			0.366205	−	0.930534i	$-0.380657\pi$
$67$	8.09405e9			0.732410			0.366205	+	0.930534i	$0.380657\pi$
$68$		0			0
$69$	8.92563e9	+	2.00742e10i	0.687019	+	1.54514i
$70$		0			0
$71$		−	1.92348e10i		−	1.26522i	−0.774468	−	0.632612i	$-0.781983\pi$
$71$		−	1.92348e10i		−	1.26522i	0.774468	−	0.632612i	$-0.218017\pi$
$72$		0			0
$73$			2.94706e10i			1.66384i	0.554892	+	0.831922i	$0.312759\pi$
$73$			2.94706e10i			1.66384i	−0.554892	+	0.831922i	$0.687241\pi$
$74$		0			0
$75$	1.65511e10	−	7.35915e9i	0.805359	−	0.358089i
$76$		0			0
$77$	1.37949e10	−	2.63742e10i	0.580790	−	1.11040i
$78$		0			0
$79$	8.47582e9			0.309908			0.154954	−	0.987922i	$-0.450477\pi$
$79$	8.47582e9			0.309908			0.154954	+	0.987922i	$0.450477\pi$
$80$		0			0
$81$	−3.21811e9	−	3.12156e10i	−0.102550	−	0.994728i
$82$		0			0
$83$	2.91537e10			0.812390			0.406195	−	0.913786i	$-0.366855\pi$
$83$	2.91537e10			0.812390			0.406195	+	0.913786i	$0.366855\pi$
$84$		0			0
$85$	−1.61567e10			−0.394955
$86$		0			0
$87$	1.43628e10	+	3.23027e10i	0.308947	+	0.694837i
$88$		0			0
$89$	3.33043e10			0.632201			0.316100	−	0.948726i	$-0.397626\pi$
$89$	3.33043e10			0.632201			0.316100	+	0.948726i	$0.397626\pi$
$90$		0			0
$91$	1.08818e10	−	2.08047e10i	0.182799	−	0.349489i
$92$		0			0
$93$	−4.12737e10	−	9.28265e10i	−0.615202	−	1.38362i
$94$		0			0
$95$		−	7.92667e9i		−	0.105102i
$96$		0			0
$97$			2.25760e10i			0.266933i	0.991053	+	0.133467i	$0.0426109\pi$
$97$			2.25760e10i			0.266933i	−0.991053	+	0.133467i	$0.957389\pi$
$98$		0			0
$99$	8.80380e10	+	7.94285e10i	0.930414	+	0.839426i
$100$		0			0
$101$	1.64691e11			1.55920			0.779600	−	0.626277i	$-0.215422\pi$
$101$	1.64691e11			1.55920			0.779600	+	0.626277i	$0.215422\pi$
$102$		0			0
$103$			2.09978e11i			1.78472i	0.451325	+	0.892360i	$0.350952\pi$
$103$			2.09978e11i			1.78472i	−0.451325	+	0.892360i	$0.649048\pi$
$104$		0			0
$105$	4.49514e10	+	2.85969e9i	0.343718	+	0.0218665i
$106$		0			0
$107$		−	4.28922e10i		−	0.295643i	−0.989014	−	0.147822i	$-0.952774\pi$
$107$		−	4.28922e10i		−	0.295643i	0.989014	−	0.147822i	$-0.0472261\pi$
$108$		0			0
$109$	1.23216e11			0.767048			0.383524	−	0.923531i	$-0.374710\pi$
$109$	1.23216e11			0.767048			0.383524	+	0.923531i	$0.374710\pi$
$110$		0			0
$111$	2.87394e11	−	1.27785e11i	1.61883	−	0.719786i
$112$		0			0
$113$		−	1.53088e11i		−	0.781643i	−0.920466	−	0.390822i	$-0.872191\pi$
$113$		−	1.53088e11i		−	0.781643i	0.920466	−	0.390822i	$-0.127809\pi$
$114$		0			0
$115$		−	1.25620e11i		−	0.582399i
$116$		0			0
$117$	6.94469e10	+	6.26555e10i	0.292840	+	0.264202i
$118$		0			0
$119$	2.64524e11	+	1.38358e11i	1.01615	+	0.531491i
$120$		0			0
$121$	−1.62715e11			−0.570307
$122$		0			0
$123$	6.12143e10	−	2.72179e10i	0.196054	−	0.0871719i
$124$		0			0
$125$	−2.21086e11			−0.647971
$126$		0			0
$127$	4.28811e11			1.15172			0.575858	−	0.817550i	$-0.304668\pi$
$127$	4.28811e11			1.15172			0.575858	+	0.817550i	$0.304668\pi$
$128$		0			0
$129$	−2.51137e11	+	1.11664e11i	−0.618966	+	0.275212i
$130$		0			0
$131$	3.29058e10			0.0745213			0.0372606	−	0.999306i	$-0.488137\pi$
$131$	3.29058e10			0.0745213			0.0372606	+	0.999306i	$0.488137\pi$
$132$		0			0
$133$	−6.78801e10	+	1.29779e11i	−0.141435	+	0.270408i
$134$		0			0
$135$	−5.57041e10	+	1.70573e11i	−0.106918	+	0.327397i
$136$		0			0
$137$			1.89098e11i			0.334753i	0.985893	+	0.167376i	$0.0535295\pi$
$137$			1.89098e11i			0.334753i	−0.985893	+	0.167376i	$0.946470\pi$
$138$		0			0
$139$		−	2.16797e11i		−	0.354382i	−0.984176	−	0.177191i	$-0.943299\pi$
$139$		−	2.16797e11i		−	0.354382i	0.984176	−	0.177191i	$-0.0567011\pi$
$140$		0			0
$141$	−7.10406e11	+	3.15870e11i	−1.07350	+	0.477313i
$142$		0			0
$143$	−3.53417e11			−0.494241
$144$		0			0
$145$		−	2.02144e11i		−	0.261900i
$146$		0			0
$147$	−7.11474e11	−	4.31762e11i	−0.854896	−	0.518799i
$148$		0			0
$149$		−	1.40953e12i		−	1.57235i	−0.618003	−	0.786176i	$-0.712058\pi$
$149$		−	1.40953e12i		−	1.57235i	0.618003	−	0.786176i	$-0.287942\pi$
$150$		0			0
$151$	−1.96012e11			−0.203193			−0.101597	−	0.994826i	$-0.532395\pi$
$151$	−1.96012e11			−0.203193			−0.101597	+	0.994826i	$0.532395\pi$
$152$		0			0
$153$	−7.96640e11	+	8.82990e11i	−0.768173	+	0.851438i
$154$		0			0
$155$			5.80889e11i			0.521518i
$156$		0			0
$157$		−	1.53836e12i		−	1.28710i	−0.765406	−	0.643548i	$-0.777462\pi$
$157$		−	1.53836e12i		−	1.28710i	0.765406	−	0.643548i	$-0.222538\pi$
$158$		0			0
$159$	−2.17340e11	−	4.88808e11i	−0.169612	−	0.381464i
$160$		0			0
$161$	−1.07575e12	+	2.05670e12i	−0.783733	+	1.49840i
$162$		0			0
$163$	−1.24940e12			−0.850494			−0.425247	−	0.905077i	$-0.639813\pi$
$163$	−1.24940e12			−0.850494			−0.425247	+	0.905077i	$0.639813\pi$
$164$		0			0
$165$	−2.75462e11	−	6.19526e11i	−0.175347	−	0.394365i
$166$		0			0
$167$	−1.04116e12			−0.620262			−0.310131	−	0.950694i	$-0.600373\pi$
$167$	−1.04116e12			−0.620262			−0.310131	+	0.950694i	$0.600373\pi$
$168$		0			0
$169$	1.51338e12			0.844442
$170$		0			0
$171$	−4.33206e11	−	3.90841e11i	−0.226577	−	0.204419i
$172$		0			0
$173$	−1.44616e12			−0.709516			−0.354758	−	0.934958i	$-0.615437\pi$
$173$	−1.44616e12			−0.709516			−0.354758	+	0.934958i	$0.615437\pi$
$174$		0			0
$175$	1.69574e12	+	8.86950e11i	0.780999	+	0.408498i
$176$		0			0
$177$	−3.34060e12	+	1.48534e12i	−1.44534	+	0.642648i
$178$		0			0
$179$			2.66412e12i			1.08358i	0.840513	+	0.541791i	$0.182254\pi$
$179$			2.66412e12i			1.08358i	−0.840513	+	0.541791i	$0.817746\pi$
$180$		0			0
$181$		−	1.23125e12i		−	0.471101i	−0.971862	−	0.235550i	$-0.924311\pi$
$181$		−	1.23125e12i		−	0.471101i	0.971862	−	0.235550i	$-0.0756892\pi$
$182$		0			0
$183$	1.89253e12	+	4.25638e12i	0.681650	+	1.53306i
$184$		0			0
$185$	−1.79845e12			−0.610176
$186$		0			0
$187$		−	4.49355e12i		−	1.43701i
$188$		0			0
$189$	2.37271e12	−	2.31567e12i	0.715657	−	0.698451i
$190$		0			0
$191$		−	3.61450e12i		−	1.02888i	−0.857527	−	0.514440i	$-0.828000\pi$
$191$		−	3.61450e12i		−	1.02888i	0.857527	−	0.514440i	$-0.172000\pi$
$192$		0			0
$193$	1.51086e12			0.406125			0.203063	−	0.979166i	$-0.434910\pi$
$193$	1.51086e12			0.406125			0.203063	+	0.979166i	$0.434910\pi$
$194$		0			0
$195$	−2.17292e11	−	4.88700e11i	−0.0551892	−	0.124123i
$196$		0			0
$197$			1.83627e12i			0.440933i	0.975395	+	0.220467i	$0.0707580\pi$
$197$			1.83627e12i			0.440933i	−0.975395	+	0.220467i	$0.929242\pi$
$198$		0			0
$199$			4.63584e12i			1.05302i	0.850169	+	0.526509i	$0.176500\pi$
$199$			4.63584e12i			1.05302i	−0.850169	+	0.526509i	$0.823500\pi$
$200$		0			0
$201$	−3.11286e12	+	1.38408e12i	−0.669238	+	0.297565i
$202$		0			0
$203$	−1.73106e12	+	3.30958e12i	−0.352439	+	0.673820i
$204$		0			0
$205$	−3.83067e11			−0.0738972
$206$		0			0
$207$	−6.86534e12	−	6.19396e12i	−1.25552	−	1.13274i
$208$		0			0
$209$	2.20459e12			0.382405
$210$		0			0
$211$	2.73638e12			0.450426			0.225213	−	0.974310i	$-0.427692\pi$
$211$	2.73638e12			0.450426			0.225213	+	0.974310i	$0.427692\pi$
$212$		0			0
$213$	3.28915e12	+	7.39745e12i	0.514038	+	1.15610i
$214$		0			0
$215$	1.57156e12			0.233303
$216$		0			0
$217$	4.97445e12	−	9.51055e12i	0.701805	−	1.34177i
$218$		0			0
$219$	−5.03945e12	−	1.13340e13i	−0.675990	−	1.52033i
$220$		0			0
$221$		−	3.54465e12i		−	0.452288i
$222$		0			0
$223$		−	7.28085e11i		−	0.0884108i	−0.999022	−	0.0442054i	$-0.985924\pi$
$223$		−	7.28085e11i		−	0.0884108i	0.999022	−	0.0442054i	$-0.0140756\pi$
$224$		0			0
$225$	−5.10690e12	+	5.66045e12i	−0.590409	+	0.654406i
$226$		0			0
$227$	6.97404e10			0.00767967			0.00383983	−	0.999993i	$-0.498778\pi$
$227$	6.97404e10			0.00767967			0.00383983	+	0.999993i	$0.498778\pi$
$228$		0			0
$229$			1.17867e13i			1.23679i	0.785867	+	0.618395i	$0.212217\pi$
$229$			1.17867e13i			1.23679i	−0.785867	+	0.618395i	$0.787783\pi$
$230$		0			0
$231$	−7.95348e11	+	1.25020e13i	−0.0795594	+	1.25059i
$232$		0			0
$233$		−	2.71154e12i		−	0.258678i	−0.991600	−	0.129339i	$-0.958715\pi$
$233$		−	2.71154e12i		−	0.258678i	0.991600	−	0.129339i	$-0.0412855\pi$
$234$		0			0
$235$	4.44558e12			0.404627
$236$		0			0
$237$	−3.25968e12	+	1.44936e12i	−0.283178	+	0.125910i
$238$		0			0
$239$		−	1.50856e13i		−	1.25133i	−0.780090	−	0.625667i	$-0.784827\pi$
$239$		−	1.50856e13i		−	1.25133i	0.780090	−	0.625667i	$-0.215173\pi$
$240$		0			0
$241$		−	1.85713e13i		−	1.47146i	−0.677274	−	0.735731i	$-0.736839\pi$
$241$		−	1.85713e13i		−	1.47146i	0.677274	−	0.735731i	$-0.263161\pi$
$242$		0			0
$243$	6.57549e12	+	1.14548e13i	0.497845	+	0.867266i
$244$		0			0
$245$	2.71430e12	+	3.90874e12i	0.196447	+	0.282894i
$246$		0			0
$247$	1.73905e12			0.120359
$248$		0			0
$249$	−1.12121e13	+	4.98527e12i	−0.742320	+	0.330060i
$250$		0			0
$251$	2.65263e13			1.68063			0.840315	−	0.542099i	$-0.182370\pi$
$251$	2.65263e13			1.68063			0.840315	+	0.542099i	$0.182370\pi$
$252$		0			0
$253$	3.49379e13			2.11901
$254$		0			0
$255$	6.21363e12	−	2.76278e12i	0.360890	−	0.160463i
$256$		0			0
$257$	1.80939e13			1.00670			0.503349	−	0.864083i	$-0.332101\pi$
$257$	1.80939e13			1.00670			0.503349	+	0.864083i	$0.332101\pi$
$258$		0			0
$259$	2.94450e13	+	1.54011e13i	1.56987	+	0.821112i
$260$		0			0
$261$	−1.10475e13	−	9.96712e12i	−0.564600	−	0.509386i
$262$		0			0
$263$		−	1.27803e13i		−	0.626301i	−0.949704	−	0.313150i	$-0.898616\pi$
$263$		−	1.27803e13i		−	0.626301i	0.949704	−	0.313150i	$-0.101384\pi$
$264$		0			0
$265$			3.05886e12i			0.143783i
$266$		0			0
$267$	−1.28083e13	+	5.69501e12i	−0.577672	+	0.256852i
$268$		0			0
$269$	1.43292e13			0.620276			0.310138	−	0.950692i	$-0.399625\pi$
$269$	1.43292e13			0.620276			0.310138	+	0.950692i	$0.399625\pi$
$270$		0			0
$271$		−	2.83627e13i		−	1.17874i	−0.807864	−	0.589369i	$-0.799376\pi$
$271$		−	2.83627e13i		−	1.17874i	0.807864	−	0.589369i	$-0.200624\pi$
$272$		0			0
$273$	−6.27394e11	+	9.86198e12i	−0.0250406	+	0.393613i
$274$		0			0
$275$		−	2.88061e13i		−	1.10447i
$276$		0			0
$277$	2.84897e12			0.104966			0.0524830	−	0.998622i	$-0.483286\pi$
$277$	2.84897e12			0.104966			0.0524830	+	0.998622i	$0.483286\pi$
$278$		0			0
$279$	3.17466e13	+	2.86420e13i	1.12428	+	1.01433i
$280$		0			0
$281$			5.49387e13i			1.87065i	0.353785	+	0.935327i	$0.384894\pi$
$281$			5.49387e13i			1.87065i	−0.353785	+	0.935327i	$0.615106\pi$
$282$		0			0
$283$			1.18390e13i			0.387694i	0.981032	+	0.193847i	$0.0620965\pi$
$283$			1.18390e13i			0.387694i	−0.981032	+	0.193847i	$0.937903\pi$
$284$		0			0
$285$	1.35546e12	+	3.04848e12i	0.0427010	+	0.0960367i
$286$		0			0
$287$	6.27172e12	+	3.28039e12i	0.190124	+	0.0994433i
$288$		0			0
$289$	1.07969e13			0.315036
$290$		0			0
$291$	−3.86048e12	−	8.68241e12i	−0.108450	−	0.243909i
$292$		0			0
$293$	−6.39139e13			−1.72911			−0.864556	−	0.502536i	$-0.832401\pi$
$293$	−6.39139e13			−1.72911			−0.864556	+	0.502536i	$0.832401\pi$
$294$		0			0
$295$	2.09048e13			0.544784
$296$		0			0
$297$	−4.74404e13	−	1.54926e13i	−1.19121	−	0.389013i
$298$		0			0
$299$	2.75600e13			0.666942
$300$		0			0
$301$	−2.57302e13	−	1.34581e13i	−0.600244	−	0.313955i
$302$		0			0
$303$	−6.33378e13	+	2.81621e13i	−1.42472	+	0.633476i
$304$		0			0
$305$		−	2.66356e13i		−	0.577847i
$306$		0			0
$307$		−	2.17541e13i		−	0.455282i	−0.973745	−	0.227641i	$-0.926899\pi$
$307$		−	2.17541e13i		−	0.455282i	0.973745	−	0.227641i	$-0.0731012\pi$
$308$		0			0
$309$	−3.59062e13	−	8.07547e13i	−0.725100	−	1.63078i
$310$		0			0
$311$	−6.54471e12			−0.127558			−0.0637790	−	0.997964i	$-0.520315\pi$
$311$	−6.54471e12			−0.127558			−0.0637790	+	0.997964i	$0.520315\pi$
$312$		0			0
$313$		−	8.01256e12i		−	0.150757i	−0.997155	−	0.0753785i	$-0.975984\pi$
$313$		−	8.01256e12i		−	0.150757i	0.997155	−	0.0753785i	$-0.0240165\pi$
$314$		0			0
$315$	−1.77767e13	+	6.58687e12i	−0.322956	+	0.119666i
$316$		0			0
$317$			1.57050e13i			0.275557i	0.990463	+	0.137779i	$0.0439963\pi$
$317$			1.57050e13i			0.275557i	−0.990463	+	0.137779i	$0.956004\pi$
$318$		0			0
$319$	5.62209e13			0.952903
$320$		0			0
$321$	7.33455e12	+	1.64957e13i	0.120115	+	0.270143i
$322$		0			0
$323$			2.21113e13i			0.349945i
$324$		0			0
$325$		−	2.27231e13i		−	0.347624i
$326$		0			0
$327$	−4.73873e13	+	2.10699e13i	−0.700889	+	0.311638i
$328$		0			0
$329$	−7.27847e13	−	3.80697e13i	−1.04103	−	0.544505i
$330$		0			0
$331$	9.69720e13			1.34150			0.670752	−	0.741681i	$-0.265971\pi$
$331$	9.69720e13			1.34150			0.670752	+	0.741681i	$0.265971\pi$
$332$		0			0
$333$	−8.86766e13	+	9.82885e13i	−1.18677	+	1.31540i
$334$		0			0
$335$	1.94796e13			0.252252
$336$		0			0
$337$	−4.44374e13			−0.556909			−0.278455	−	0.960449i	$-0.589822\pi$
$337$	−4.44374e13			−0.556909			−0.278455	+	0.960449i	$0.589822\pi$
$338$		0			0
$339$	2.61779e13	+	5.88753e13i	0.317568	+	0.714225i
$340$		0			0
$341$	−1.61559e14			−1.89750
$342$		0			0
$343$	−1.09670e13	−	8.72394e13i	−0.124730	−	0.992191i
$344$		0			0
$345$	2.14809e13	+	4.83117e13i	0.236618	+	0.532166i
$346$		0			0
$347$			8.12041e13i			0.866495i	0.901275	+	0.433247i	$0.142632\pi$
$347$			8.12041e13i			0.866495i	−0.901275	+	0.433247i	$0.857368\pi$
$348$		0			0
$349$		−	9.08426e13i		−	0.939181i	−0.882884	−	0.469591i	$-0.844401\pi$
$349$		−	9.08426e13i		−	0.939181i	0.882884	−	0.469591i	$-0.155599\pi$
$350$		0			0
$351$	−3.74223e13	−	1.22210e13i	−0.374923	−	0.122439i
$352$		0			0
$353$	−1.22420e14			−1.18875			−0.594374	−	0.804188i	$-0.702600\pi$
$353$	−1.22420e14			−1.18875			−0.594374	+	0.804188i	$0.702600\pi$
$354$		0			0
$355$		−	4.62917e13i		−	0.435760i
$356$		0			0
$357$	−1.25391e14	−	7.97706e12i	−1.14444	−	0.0728061i
$358$		0			0
$359$			9.85524e13i			0.872264i	0.899883	+	0.436132i	$0.143652\pi$
$359$			9.85524e13i			0.872264i	−0.899883	+	0.436132i	$0.856348\pi$
$360$		0			0
$361$	1.05642e14			0.906876
$362$		0			0
$363$	6.25780e13	−	2.78242e13i	0.521117	−	0.231706i
$364$		0			0
$365$			7.09256e13i			0.573049i
$366$		0			0
$367$		−	6.10737e13i		−	0.478841i	−0.970916	−	0.239420i	$-0.923043\pi$
$367$		−	6.10737e13i		−	0.478841i	0.970916	−	0.239420i	$-0.0769574\pi$
$368$		0			0
$369$	−1.88879e13	+	2.09352e13i	−0.143727	+	0.159306i
$370$		0			0
$371$	2.61945e13	−	5.00808e13i	0.193488	−	0.369926i
$372$		0			0
$373$	−3.99859e13			−0.286753			−0.143377	−	0.989668i	$-0.545796\pi$
$373$	−3.99859e13			−0.286753			−0.143377	+	0.989668i	$0.545796\pi$
$374$		0			0
$375$	8.50265e13	−	3.78056e13i	0.592082	−	0.263259i
$376$		0			0
$377$	4.43487e13			0.299918
$378$		0			0
$379$	7.33197e13			0.481621			0.240810	−	0.970572i	$-0.422587\pi$
$379$	7.33197e13			0.481621			0.240810	+	0.970572i	$0.422587\pi$
$380$		0			0
$381$	−1.64914e14	+	7.33264e13i	−1.05238	+	0.467921i
$382$		0			0
$383$	2.47538e14			1.53479			0.767394	−	0.641176i	$-0.221553\pi$
$383$	2.47538e14			1.53479			0.767394	+	0.641176i	$0.221553\pi$
$384$		0			0
$385$	3.31996e13	−	6.34736e13i	0.200032	−	0.382436i
$386$		0			0
$387$	7.74891e13	−	8.58884e13i	0.453764	−	0.502949i
$388$		0			0
$389$		−	2.02276e13i		−	0.115139i	−0.998342	−	0.0575693i	$-0.981665\pi$
$389$		−	2.02276e13i		−	0.115139i	0.998342	−	0.0575693i	$-0.0183350\pi$
$390$		0			0
$391$			3.50415e14i			1.93914i
$392$		0			0
$393$	−1.26551e13	+	5.62687e12i	−0.0680936	+	0.0302767i
$394$		0			0
$395$	2.03984e13			0.106736
$396$		0			0
$397$			4.26883e13i			0.217251i	0.994083	+	0.108625i	$0.0346449\pi$
$397$			4.26883e13i			0.217251i	−0.994083	+	0.108625i	$0.965355\pi$
$398$		0			0
$399$	3.91364e12	−	6.15184e13i	0.0193745	−	0.304547i
$400$		0			0
$401$			3.46334e14i			1.66802i	0.551748	+	0.834011i	$0.313961\pi$
$401$			3.46334e14i			1.66802i	−0.551748	+	0.834011i	$0.686039\pi$
$402$		0			0
$403$	−1.27442e14			−0.597223
$404$		0			0
$405$	−7.74490e12	−	7.51253e13i	−0.0353194	−	0.342597i
$406$		0			0
$407$		−	5.00192e14i		−	2.22008i
$408$		0			0
$409$		−	1.24342e14i		−	0.537203i	−0.963251	−	0.268601i	$-0.913439\pi$
$409$		−	1.24342e14i		−	0.537203i	0.963251	−	0.268601i	$-0.0865615\pi$
$410$		0			0
$411$	−3.23357e13	−	7.27245e13i	−0.136004	−	0.305880i
$412$		0			0
$413$	−3.42261e14	−	1.79018e14i	−1.40163	−	0.733115i
$414$		0			0
$415$	7.01631e13			0.279798
$416$		0			0
$417$	3.70722e13	+	8.33771e13i	0.143979	+	0.323816i
$418$		0			0
$419$	3.60068e14			1.36210			0.681048	−	0.732238i	$-0.261524\pi$
$419$	3.60068e14			1.36210			0.681048	+	0.732238i	$0.261524\pi$
$420$		0			0
$421$	−3.65790e14			−1.34797			−0.673985	−	0.738745i	$-0.735419\pi$
$421$	−3.65790e14			−1.34797			−0.673985	+	0.738745i	$0.735419\pi$
$422$		0			0
$423$	2.19199e14	−	2.42958e14i	0.786984	−	0.872287i
$424$		0			0
$425$	2.88916e14			1.01072
$426$		0			0
$427$	−2.28094e14	+	4.36088e14i	−0.777607	+	1.48669i
$428$		0			0
$429$	1.35919e14	−	6.04341e13i	0.451612	−	0.200801i
$430$		0			0
$431$		−	1.36534e14i		−	0.442196i	−0.975252	−	0.221098i	$-0.929036\pi$
$431$		−	1.36534e14i		−	0.442196i	0.975252	−	0.221098i	$-0.0709640\pi$
$432$		0			0
$433$		−	6.98493e13i		−	0.220536i	−0.993902	−	0.110268i	$-0.964829\pi$
$433$		−	6.98493e13i		−	0.220536i	0.993902	−	0.110268i	$-0.0351709\pi$
$434$		0			0
$435$	3.45664e13	+	7.77416e13i	0.106405	+	0.239311i
$436$		0			0
$437$	−1.71918e14			−0.516027
$438$		0			0
$439$			4.14958e14i			1.21464i	0.794456	+	0.607322i	$0.207756\pi$
$439$			4.14958e14i			1.21464i	−0.794456	+	0.607322i	$0.792244\pi$
$440$		0			0
$441$	3.47454e14	+	4.43879e13i	0.991938	+	0.126722i
$442$		0			0
$443$		−	2.18721e14i		−	0.609073i	−0.952501	−	0.304536i	$-0.901498\pi$
$443$		−	2.18721e14i		−	0.609073i	0.952501	−	0.304536i	$-0.0985016\pi$
$444$		0			0
$445$	8.01520e13			0.217738
$446$		0			0
$447$	2.41029e14	+	5.42085e14i	0.638819	+	1.43673i
$448$		0			0
$449$		−	1.52766e14i		−	0.395068i	−0.980296	−	0.197534i	$-0.936707\pi$
$449$		−	1.52766e14i		−	0.395068i	0.980296	−	0.197534i	$-0.0632932\pi$
$450$		0			0
$451$		−	1.06540e14i		−	0.268869i
$452$		0			0
$453$	7.53835e13	−	3.35180e13i	0.185668	−	0.0825539i
$454$		0			0
$455$	2.61888e13	−	5.00698e13i	0.0629583	−	0.120369i
$456$		0			0
$457$	6.97169e14			1.63606			0.818029	−	0.575177i	$-0.195067\pi$
$457$	6.97169e14			1.63606			0.818029	+	0.575177i	$0.195067\pi$
$458$		0			0
$459$	1.55385e14	−	4.75810e14i	0.355992	−	1.09009i
$460$		0			0
$461$	7.41148e14			1.65787			0.828934	−	0.559347i	$-0.188948\pi$
$461$	7.41148e14			1.65787			0.828934	+	0.559347i	$0.188948\pi$
$462$		0			0
$463$	−1.86395e14			−0.407135			−0.203567	−	0.979061i	$-0.565254\pi$
$463$	−1.86395e14			−0.407135			−0.203567	+	0.979061i	$0.565254\pi$
$464$		0			0
$465$	−9.93317e13	−	2.23402e14i	−0.211883	−	0.476536i
$466$		0			0
$467$	5.36426e14			1.11755			0.558775	−	0.829319i	$-0.311271\pi$
$467$	5.36426e14			1.11755			0.558775	+	0.829319i	$0.311271\pi$
$468$		0			0
$469$	−3.18928e14	−	1.66814e14i	−0.648996	−	0.339454i
$470$		0			0
$471$	2.63059e14	+	5.91633e14i	0.522924	+	1.17608i
$472$		0			0
$473$			4.37088e14i			0.848853i
$474$		0			0
$475$			1.41745e14i			0.268964i
$476$		0			0
$477$	1.67172e14	+	1.50823e14i	0.309964	+	0.279652i
$478$		0			0
$479$	−3.31010e14			−0.599785			−0.299892	−	0.953973i	$-0.596951\pi$
$479$	−3.31010e14			−0.599785			−0.299892	+	0.953973i	$0.596951\pi$
$480$		0			0
$481$		−	3.94566e14i		−	0.698750i
$482$		0			0
$483$	6.20225e13	−	9.74930e14i	0.107359	−	1.68758i
$484$		0			0
$485$			5.43327e13i			0.0919352i
$486$		0			0
$487$	−4.01717e14			−0.664524			−0.332262	−	0.943187i	$-0.607812\pi$
$487$	−4.01717e14			−0.664524			−0.332262	+	0.943187i	$0.607812\pi$
$488$		0			0
$489$	4.80503e14	−	2.13647e14i	0.777137	−	0.345541i
$490$		0			0
$491$			7.77882e13i			0.123017i	0.998107	+	0.0615086i	$0.0195912\pi$
$491$			7.77882e13i			0.123017i	−0.998107	+	0.0615086i	$0.980409\pi$
$492$		0			0
$493$			5.63876e14i			0.872018i
$494$		0			0
$495$	2.11877e14	+	1.91157e14i	0.320447	+	0.289109i
$496$		0			0
$497$	−3.96419e14	+	7.57906e14i	−0.586401	+	1.12113i
$498$		0			0
$499$	−1.18938e15			−1.72095			−0.860476	−	0.509491i	$-0.829834\pi$
$499$	−1.18938e15			−1.72095			−0.860476	+	0.509491i	$0.829834\pi$
$500$		0			0
$501$	4.00414e14	−	1.78037e14i	0.566763	−	0.252001i
$502$		0			0
$503$	3.02960e14			0.419528			0.209764	−	0.977752i	$-0.432730\pi$
$503$	3.02960e14			0.419528			0.209764	+	0.977752i	$0.432730\pi$
$504$		0			0
$505$	3.96355e14			0.537009
$506$		0			0
$507$	−5.82022e14	+	2.58786e14i	−0.771607	+	0.343082i
$508$		0			0
$509$	−8.28759e14			−1.07518			−0.537589	−	0.843207i	$-0.680665\pi$
$509$	−8.28759e14			−1.07518			−0.537589	+	0.843207i	$0.680665\pi$
$510$		0			0
$511$	6.07372e14	−	1.16122e15i	0.771151	−	1.47435i
$512$		0			0
$513$	2.33438e14	+	7.62340e13i	0.290086	+	0.0947334i
$514$		0			0
$515$			5.05346e14i			0.614680i
$516$		0			0
$517$			1.23642e15i			1.47220i
$518$		0			0
$519$	5.56172e14	−	2.47292e14i	0.648319	−	0.288264i
$520$		0			0
$521$	−5.51909e14			−0.629883			−0.314942	−	0.949111i	$-0.601985\pi$
$521$	−5.51909e14			−0.629883			−0.314942	+	0.949111i	$0.601985\pi$
$522$		0			0
$523$			1.27347e15i			1.42309i	0.702643	+	0.711543i	$0.252003\pi$
$523$			1.27347e15i			1.42309i	−0.702643	+	0.711543i	$0.747997\pi$
$524$		0			0
$525$	−8.03826e14	−	5.11374e13i	−0.879602	−	0.0559580i
$526$		0			0
$527$		−	1.62038e15i		−	1.73643i
$528$		0			0
$529$	−1.77170e15			−1.85945
$530$		0			0
$531$	1.03075e15	−	1.14248e15i	1.05958	−	1.17444i
$532$		0			0
$533$		−	8.40417e13i		−	0.0846244i
$534$		0			0
$535$		−	1.03227e14i		−	0.101823i
$536$		0			0
$537$	−4.55563e14	−	1.02458e15i	−0.440240	−	0.990121i
$538$		0			0
$539$	−1.08711e15	+	7.54910e14i	−1.02929	+	0.714755i
$540$		0			0
$541$	−1.81103e15			−1.68012			−0.840062	−	0.542490i	$-0.817482\pi$
$541$	−1.81103e15			−1.68012			−0.840062	+	0.542490i	$0.817482\pi$
$542$		0			0
$543$	2.10543e14	+	4.73521e14i	0.191400	+	0.430467i
$544$		0			0
$545$	2.96540e14			0.264181
$546$		0			0
$547$	−1.38466e15			−1.20897			−0.604483	−	0.796618i	$-0.706620\pi$
$547$	−1.38466e15			−1.20897			−0.604483	+	0.796618i	$0.706620\pi$
$548$		0			0
$549$	−1.45568e15	−	1.31332e15i	−1.24571	−	1.12389i
$550$		0			0
$551$	−2.76644e14			−0.232053
$552$		0			0
$553$	−3.33971e14	−	1.74682e14i	−0.274612	−	0.143635i
$554$		0			0
$555$	6.91660e14	−	3.07535e14i	0.557546	−	0.247903i
$556$		0			0
$557$		−	1.86834e15i		−	1.47656i	−0.674493	−	0.738281i	$-0.735638\pi$
$557$		−	1.86834e15i		−	1.47656i	0.674493	−	0.738281i	$-0.264362\pi$
$558$		0			0
$559$			3.44788e14i			0.267169i
$560$		0			0
$561$	7.68395e14	+	1.72816e15i	0.583833	+	1.31307i
$562$		0			0
$563$	−2.12169e15			−1.58083			−0.790416	−	0.612570i	$-0.790136\pi$
$563$	−2.12169e15			−1.58083			−0.790416	+	0.612570i	$0.790136\pi$
$564$		0			0
$565$		−	3.68429e14i		−	0.269208i
$566$		0			0
$567$	−5.16533e14	+	1.29631e15i	−0.370162	+	0.928967i
$568$		0			0
$569$			9.75299e14i			0.685520i	0.939423	+	0.342760i	$0.111362\pi$
$569$			9.75299e14i			0.685520i	−0.939423	+	0.342760i	$0.888638\pi$
$570$		0			0
$571$	1.32979e15			0.916821			0.458411	−	0.888741i	$-0.348419\pi$
$571$	1.32979e15			0.916821			0.458411	+	0.888741i	$0.348419\pi$
$572$		0			0
$573$	6.18077e14	+	1.39008e15i	0.418015	+	0.940136i
$574$		0			0
$575$			2.24635e15i			1.49040i
$576$		0			0
$577$		−	1.34056e15i		−	0.872606i	−0.899800	−	0.436303i	$-0.856288\pi$
$577$		−	1.34056e15i		−	0.872606i	0.899800	−	0.436303i	$-0.143712\pi$
$578$		0			0
$579$	−5.81057e14	+	2.58357e14i	−0.371096	+	0.165001i
$580$		0			0
$581$	−1.14874e15	−	6.00842e14i	−0.719867	−	0.376523i
$582$		0			0
$583$	−8.50740e14			−0.523142
$584$		0			0
$585$	1.67135e14	+	1.50790e14i	0.100858	+	0.0909947i
$586$		0			0
$587$	1.64814e15			0.976078			0.488039	−	0.872822i	$-0.337712\pi$
$587$	1.64814e15			0.976078			0.488039	+	0.872822i	$0.337712\pi$
$588$		0			0
$589$	7.94978e14			0.462084
$590$		0			0
$591$	−3.14001e14	−	7.06204e14i	−0.179143	−	0.402902i
$592$		0			0
$593$	1.75773e15			0.984352			0.492176	−	0.870496i	$-0.336202\pi$
$593$	1.75773e15			0.984352			0.492176	+	0.870496i	$0.336202\pi$
$594$		0			0
$595$	6.36618e14	+	3.32980e14i	0.349974	+	0.183052i
$596$		0			0
$597$	−7.92725e14	−	1.78288e15i	−0.427823	−	0.962194i
$598$		0			0
$599$			2.69923e15i			1.43019i	0.699029	+	0.715093i	$0.253616\pi$
$599$			2.69923e15i			1.43019i	−0.699029	+	0.715093i	$0.746384\pi$
$600$		0			0
$601$			2.08618e15i			1.08528i	0.839965	+	0.542641i	$0.182576\pi$
$601$			2.08618e15i			1.08528i	−0.839965	+	0.542641i	$0.817424\pi$
$602$		0			0
$603$	9.60484e14	−	1.06459e15i	0.490619	−	0.543799i
$604$		0			0
$605$	−3.91600e14			−0.196421
$606$		0			0
$607$		−	3.17896e15i		−	1.56584i	−0.622122	−	0.782920i	$-0.713729\pi$
$607$		−	3.17896e15i		−	1.56584i	0.622122	−	0.782920i	$-0.286271\pi$
$608$		0			0
$609$	9.98046e13	−	1.56883e15i	0.0482787	−	0.758891i
$610$		0			0
$611$			9.75323e14i			0.463364i
$612$		0			0
$613$	3.79236e14			0.176961			0.0884803	−	0.996078i	$-0.471799\pi$
$613$	3.79236e14			0.176961			0.0884803	+	0.996078i	$0.471799\pi$
$614$		0			0
$615$	1.47322e14	−	6.55042e13i	0.0675234	−	0.0300231i
$616$		0			0
$617$			1.47866e14i			0.0665733i	0.999446	+	0.0332867i	$0.0105974\pi$
$617$			1.47866e14i			0.0665733i	−0.999446	+	0.0332867i	$0.989403\pi$
$618$		0			0
$619$		−	9.89562e14i		−	0.437668i	−0.975762	−	0.218834i	$-0.929775\pi$
$619$		−	9.89562e14i		−	0.437668i	0.975762	−	0.218834i	$-0.0702253\pi$
$620$		0			0
$621$	3.69948e15	+	1.20814e15i	1.60745	+	0.524944i
$622$		0			0
$623$	−1.31228e15	−	6.86382e14i	−0.560199	−	0.293009i
$624$		0			0
$625$	1.56930e15			0.658210
$626$		0			0
$627$	−8.47855e14	+	3.76984e14i	−0.349422	+	0.155364i
$628$		0			0
$629$	5.01675e15			2.03163
$630$		0			0
$631$	7.61944e14			0.303223			0.151611	−	0.988440i	$-0.451554\pi$
$631$	7.61944e14			0.303223			0.151611	+	0.988440i	$0.451554\pi$
$632$		0			0
$633$	−1.05237e15	+	4.67920e14i	−0.411576	+	0.183000i
$634$		0			0
$635$	1.03200e15			0.396666
$636$		0			0
$637$	−8.57547e14	+	5.95495e14i	−0.323960	+	0.224963i
$638$		0			0
$639$	−2.52992e15	−	2.28251e15i	−0.939403	−	0.847536i
$640$		0			0
$641$		−	3.93622e15i		−	1.43668i	−0.695692	−	0.718340i	$-0.744902\pi$
$641$		−	3.93622e15i		−	1.43668i	0.695692	−	0.718340i	$-0.255098\pi$
$642$		0			0
$643$			5.86223e14i			0.210331i	0.994455	+	0.105165i	$0.0335372\pi$
$643$			5.86223e14i			0.210331i	−0.994455	+	0.105165i	$0.966463\pi$
$644$		0			0
$645$	−6.04400e14	+	2.68736e14i	−0.213180	+	0.0947867i
$646$		0			0
$647$	−4.93677e15			−1.71186			−0.855932	−	0.517089i	$-0.827016\pi$
$647$	−4.93677e15			−1.71186			−0.855932	+	0.517089i	$0.827016\pi$
$648$		0			0
$649$			5.81411e15i			1.98215i
$650$		0			0
$651$	−2.86803e14	+	4.50825e15i	−0.0961366	+	1.51117i
$652$		0			0
$653$		−	4.41572e15i		−	1.45539i	−0.685902	−	0.727694i	$-0.740592\pi$
$653$		−	4.41572e15i		−	1.45539i	0.685902	−	0.727694i	$-0.259408\pi$
$654$		0			0
$655$	7.91930e13			0.0256661
$656$		0			0
$657$	3.87620e15	+	3.49714e15i	1.23537	+	1.11456i
$658$		0			0
$659$		−	2.16648e14i		−	0.0679024i	−0.999423	−	0.0339512i	$-0.989191\pi$
$659$		−	2.16648e14i		−	0.0679024i	0.999423	−	0.0339512i	$-0.0108091\pi$
$660$		0			0
$661$		−	2.76661e15i		−	0.852786i	−0.904538	−	0.426393i	$-0.859784\pi$
$661$		−	2.76661e15i		−	0.852786i	0.904538	−	0.426393i	$-0.140216\pi$
$662$		0			0
$663$	6.06132e14	+	1.36322e15i	0.183757	+	0.413277i
$664$		0			0
$665$	−1.63364e14	+	3.12333e14i	−0.0487122	+	0.0931319i
$666$		0			0
$667$	−4.38420e15			−1.28587
$668$		0			0
$669$	1.24502e14	+	2.80011e14i	0.0359197	+	0.0807851i
$670$		0			0
$671$	7.40798e15			2.10245
$672$		0			0
$673$	4.51487e15			1.26056			0.630278	−	0.776369i	$-0.282941\pi$
$673$	4.51487e15			1.26056			0.630278	+	0.776369i	$0.282941\pi$
$674$		0			0
$675$	9.96106e14	−	3.05021e15i	0.273612	−	0.837835i
$676$		0			0
$677$	1.10565e15			0.298801			0.149400	−	0.988777i	$-0.452266\pi$
$677$	1.10565e15			0.298801			0.149400	+	0.988777i	$0.452266\pi$
$678$		0			0
$679$	4.65278e14	−	8.89557e14i	0.123717	−	0.236532i
$680$		0			0
$681$	−2.68212e13	+	1.19256e13i	−0.00701728	+	0.00312011i
$682$		0			0
$683$		−	2.34034e15i		−	0.602511i	−0.953544	−	0.301255i	$-0.902594\pi$
$683$		−	2.34034e15i		−	0.602511i	0.953544	−	0.301255i	$-0.0974056\pi$
$684$		0			0
$685$			4.55095e14i			0.115293i
$686$		0			0
$687$	−2.01551e15	−	4.53299e15i	−0.502486	−	1.13011i
$688$		0			0
$689$	−6.71089e14			−0.164655
$690$		0			0
$691$			2.22948e15i			0.538362i	0.963090	+	0.269181i	$0.0867530\pi$
$691$			2.22948e15i			0.538362i	−0.963090	+	0.269181i	$0.913247\pi$
$692$		0			0
$693$	−1.83196e15	−	4.94411e15i	−0.435396	−	1.17505i
$694$		0			0
$695$		−	5.21756e14i		−	0.122054i
$696$		0			0
$697$	1.06856e15			0.246047
$698$		0			0
$699$	4.63672e14	+	1.04282e15i	0.105096	+	0.236366i
$700$		0			0
$701$			2.70381e15i			0.603291i	0.953420	+	0.301645i	$0.0975359\pi$
$701$			2.70381e15i			0.603291i	−0.953420	+	0.301645i	$0.902464\pi$
$702$		0			0
$703$			2.46128e15i			0.540638i
$704$		0			0
$705$	−1.70971e15	+	7.60191e14i	−0.369727	+	0.164393i
$706$		0			0
$707$	−6.48928e15	−	3.39419e15i	−1.38162	−	0.722651i
$708$		0			0
$709$	5.94065e15			1.24532			0.622658	−	0.782494i	$-0.286053\pi$
$709$	5.94065e15			1.24532			0.622658	+	0.782494i	$0.286053\pi$
$710$		0			0
$711$	1.00579e15	−	1.11481e15i	0.207598	−	0.230100i
$712$		0			0
$713$	1.25986e16			2.56054
$714$		0			0
$715$	−8.50553e14			−0.170223
$716$		0			0
$717$	2.57963e15	+	5.80170e15i	0.508395	+	1.14340i
$718$		0			0
$719$	−4.79255e13			−0.00930160			−0.00465080	−	0.999989i	$-0.501480\pi$
$719$	−4.79255e13			−0.00930160			−0.00465080	+	0.999989i	$0.501480\pi$
$720$		0			0
$721$	4.32754e15	−	8.27373e15i	0.827174	−	1.58146i
$722$		0			0
$723$	3.17568e15	+	7.14226e15i	0.597829	+	1.34454i
$724$		0			0
$725$			3.61475e15i			0.670224i
$726$		0			0
$727$			4.47294e15i			0.816871i	0.912787	+	0.408436i	$0.133926\pi$
$727$			4.47294e15i			0.816871i	−0.912787	+	0.408436i	$0.866074\pi$
$728$		0			0
$729$	−4.48760e15	−	3.28094e15i	−0.807259	−	0.590197i
$730$		0			0
$731$	−4.38384e15			−0.776800
$732$		0			0
$733$		−	2.32289e15i		−	0.405469i	−0.979234	−	0.202734i	$-0.935017\pi$
$733$		−	2.32289e15i		−	0.405469i	0.979234	−	0.202734i	$-0.0649828\pi$
$734$		0			0
$735$	−1.71227e15	−	1.03910e15i	−0.294437	−	0.178681i
$736$		0			0
$737$			5.41774e15i			0.917797i
$738$		0			0
$739$	1.26867e15			0.211741			0.105870	−	0.994380i	$-0.466237\pi$
$739$	1.26867e15			0.211741			0.105870	+	0.994380i	$0.466237\pi$
$740$		0			0
$741$	−6.68812e14	+	2.97376e14i	−0.109978	+	0.0488996i
$742$		0			0
$743$		−	8.45566e14i		−	0.136996i	−0.997651	−	0.0684982i	$-0.978179\pi$
$743$		−	8.45566e14i		−	0.136996i	0.997651	−	0.0684982i	$-0.0218207\pi$
$744$		0			0
$745$		−	3.39226e15i		−	0.541538i
$746$		0			0
$747$	3.45954e15	−	3.83453e15i	0.544196	−	0.603183i
$748$		0			0
$749$	−8.83984e14	+	1.69007e15i	−0.137023	+	0.261972i
$750$		0			0
$751$	9.80062e15			1.49704			0.748521	−	0.663111i	$-0.230764\pi$
$751$	9.80062e15			1.49704			0.748521	+	0.663111i	$0.230764\pi$
$752$		0			0
$753$	−1.02017e16	+	4.53599e15i	−1.53567	+	0.682810i
$754$		0			0
$755$	−4.71734e14			−0.0699824
$756$		0			0
$757$	2.30235e15			0.336622			0.168311	−	0.985734i	$-0.446169\pi$
$757$	2.30235e15			0.336622			0.168311	+	0.985734i	$0.446169\pi$
$758$		0			0
$759$	−1.34366e16	+	5.97436e15i	−1.93624	+	0.860917i
$760$		0			0
$761$	6.52678e15			0.927008			0.463504	−	0.886095i	$-0.346592\pi$
$761$	6.52678e15			0.927008			0.463504	+	0.886095i	$0.346592\pi$
$762$		0			0
$763$	−4.85507e15	−	2.53942e15i	−0.679689	−	0.355508i
$764$		0			0
$765$	−1.91724e15	+	2.12505e15i	−0.264569	+	0.293246i
$766$		0			0
$767$			4.58634e15i			0.623866i
$768$		0			0
$769$		−	1.25023e15i		−	0.167647i	−0.996481	−	0.0838235i	$-0.973287\pi$
$769$		−	1.25023e15i		−	0.167647i	0.996481	−	0.0838235i	$-0.0267132\pi$
$770$		0			0
$771$	−6.95864e15	+	3.09404e15i	−0.919867	+	0.409003i
$772$		0			0
$773$	−7.36811e15			−0.960216			−0.480108	−	0.877209i	$-0.659403\pi$
$773$	−7.36811e15			−0.960216			−0.480108	+	0.877209i	$0.659403\pi$
$774$		0			0
$775$		−	1.03875e16i		−	1.33460i
$776$		0			0
$777$	−1.39577e16	−	8.87953e14i	−1.76807	−	0.112480i
$778$		0			0
$779$			5.24247e14i			0.0654757i
$780$		0			0
$781$	1.28748e16			1.58548
$782$		0			0
$783$	5.95308e15	+	1.94410e15i	0.722856	+	0.236063i
$784$		0			0
$785$		−	3.70231e15i		−	0.443292i
$786$		0			0
$787$		−	1.43155e16i		−	1.69023i	−0.534587	−	0.845114i	$-0.679533\pi$
$787$		−	1.43155e16i		−	1.69023i	0.534587	−	0.845114i	$-0.320467\pi$
$788$		0			0
$789$	2.18542e15	+	4.91510e15i	0.254455	+	0.572281i
$790$		0			0
$791$	−3.15505e15	+	6.03207e15i	−0.362272	+	0.692622i
$792$		0			0
$793$	5.84363e15			0.661729
$794$		0			0
$795$	−5.23063e14	−	1.17639e15i	−0.0584164	−	0.131381i
$796$		0			0
$797$	−6.01404e15			−0.662438			−0.331219	−	0.943554i	$-0.607460\pi$
$797$	−6.01404e15			−0.662438			−0.331219	+	0.943554i	$0.607460\pi$
$798$		0			0
$799$	−1.24009e16			−1.34724
$800$		0			0
$801$	3.95206e15	−	4.38044e15i	0.423492	−	0.469396i
$802$		0			0
$803$	−1.97261e16			−2.08500
$804$		0			0
$805$	−2.58896e15	+	4.94977e15i	−0.269928	+	0.516069i
$806$		0			0
$807$	−5.51081e15	+	2.45029e15i	−0.566775	+	0.252007i
$808$		0			0
$809$		−	1.56379e16i		−	1.58658i	−0.608844	−	0.793290i	$-0.708366\pi$
$809$		−	1.56379e16i		−	1.58658i	0.608844	−	0.793290i	$-0.291634\pi$
$810$		0			0
$811$			6.69487e15i			0.670081i	0.942204	+	0.335041i	$0.108750\pi$
$811$			6.69487e15i			0.670081i	−0.942204	+	0.335041i	$0.891250\pi$
$812$		0			0
$813$	4.85001e15	+	1.09079e16i	0.478900	+	1.07707i
$814$		0			0
$815$	−3.00689e15			−0.292921
$816$		0			0
$817$		−	2.15076e15i		−	0.206715i
$818$		0			0
$819$	−1.44511e15	−	3.90006e15i	−0.137037	−	0.369837i
$820$		0			0
$821$			1.77809e16i			1.66367i	0.555023	+	0.831835i	$0.312710\pi$
$821$			1.77809e16i			1.66367i	−0.555023	+	0.831835i	$0.687290\pi$
$822$		0			0
$823$	−4.29337e15			−0.396368			−0.198184	−	0.980165i	$-0.563504\pi$
$823$	−4.29337e15			−0.396368			−0.198184	+	0.980165i	$0.563504\pi$
$824$		0			0
$825$	4.92584e15	+	1.10784e16i	0.448728	+	1.00921i
$826$		0			0
$827$		−	1.89047e16i		−	1.69938i	−0.527284	−	0.849689i	$-0.676790\pi$
$827$		−	1.89047e16i		−	1.69938i	0.527284	−	0.849689i	$-0.323210\pi$
$828$		0			0
$829$		−	2.53918e15i		−	0.225239i	−0.993638	−	0.112619i	$-0.964076\pi$
$829$		−	2.53918e15i		−	0.225239i	0.993638	−	0.112619i	$-0.0359241\pi$
$830$		0			0
$831$	−1.09567e15	+	4.87172e14i	−0.0959125	+	0.0426458i
$832$		0			0
$833$	−7.57149e15	−	1.09034e16i	−0.654084	−	0.941918i
$834$		0			0
$835$	−2.50571e15			−0.213626
$836$		0			0
$837$	−1.71070e16	−	5.58665e15i	−1.43941	−	0.470069i
$838$		0			0
$839$	7.15920e15			0.594530			0.297265	−	0.954795i	$-0.403926\pi$
$839$	7.15920e15			0.594530			0.297265	+	0.954795i	$0.403926\pi$
$840$		0			0
$841$	5.14560e15			0.421753
$842$		0			0
$843$	−9.39448e15	−	2.11286e16i	−0.760013	−	1.70931i
$844$		0			0
$845$	3.64218e15			0.290837
$846$		0			0
$847$	6.41143e15	+	3.35347e15i	0.505355	+	0.264323i
$848$		0			0
$849$	−2.02446e15	−	4.55311e15i	−0.157513	−	0.354255i
$850$		0			0
$851$			3.90058e16i			2.99583i
$852$		0			0
$853$		−	1.40236e16i		−	1.06326i	−0.846976	−	0.531631i	$-0.821579\pi$
$853$		−	1.40236e16i		−	1.06326i	0.846976	−	0.531631i	$-0.178421\pi$
$854$		0			0
$855$	−1.04258e15	−	9.40621e14i	−0.0780360	−	0.0704046i
$856$		0			0
$857$	−1.79737e16			−1.32814			−0.664069	−	0.747672i	$-0.731172\pi$
$857$	−1.79737e16			−1.32814			−0.664069	+	0.747672i	$0.731172\pi$
$858$		0			0
$859$			1.74127e16i			1.27030i	0.772391	+	0.635148i	$0.219061\pi$
$859$			1.74127e16i			1.27030i	−0.772391	+	0.635148i	$0.780939\pi$
$860$		0			0
$861$	−2.97296e15	−	1.89132e14i	−0.214127	−	0.0136222i
$862$		0			0
$863$			1.02749e15i			0.0730662i	0.999332	+	0.0365331i	$0.0116314\pi$
$863$			1.02749e15i			0.0730662i	−0.999332	+	0.0365331i	$0.988369\pi$
$864$		0			0
$865$	−3.48041e15			−0.244367
$866$		0			0
$867$	−4.15232e15	+	1.84626e15i	−0.287863	+	0.127993i
$868$		0			0
$869$			5.67327e15i			0.388351i
$870$		0			0
$871$			4.27367e15i			0.288869i
$872$		0			0
$873$	2.96937e15	+	2.67899e15i	0.198192	+	0.178810i
$874$		0			0
$875$	8.71139e15	+	4.55645e15i	0.574174	+	0.300319i
$876$		0			0
$877$	5.93759e15			0.386467			0.193233	−	0.981153i	$-0.438103\pi$
$877$	5.93759e15			0.386467			0.193233	+	0.981153i	$0.438103\pi$
$878$		0			0
$879$	2.45804e16	−	1.09292e16i	1.57997	−	0.702508i
$880$		0			0
$881$	3.09866e15			0.196701			0.0983505	−	0.995152i	$-0.468643\pi$
$881$	3.09866e15			0.196701			0.0983505	+	0.995152i	$0.468643\pi$
$882$		0			0
$883$	−3.61795e15			−0.226819			−0.113409	−	0.993548i	$-0.536177\pi$
$883$	−3.61795e15			−0.226819			−0.113409	+	0.993548i	$0.536177\pi$
$884$		0			0
$885$	−8.03968e15	+	3.57471e15i	−0.497795	+	0.221336i
$886$		0			0
$887$	2.75926e16			1.68738			0.843690	−	0.536831i	$-0.180379\pi$
$887$	2.75926e16			1.68738			0.843690	+	0.536831i	$0.180379\pi$
$888$		0			0
$889$	−1.68963e16	−	8.83755e15i	−1.02055	−	0.533792i
$890$		0			0
$891$	2.08941e16	−	2.15404e15i	1.24651	−	0.128507i
$892$		0			0
$893$		−	6.08401e15i		−	0.358515i
$894$		0			0
$895$			6.41162e15i			0.373200i
$896$		0			0
$897$	−1.05992e16	+	4.71275e15i	−0.609416	+	0.270966i
$898$		0			0
$899$	2.02733e16			1.15145
$900$		0			0
$901$		−	8.53263e15i		−	0.478736i
$902$		0			0
$903$	1.21968e16	+	7.75929e14i	0.676026	+	0.0430070i
$904$		0			0
$905$		−	2.96319e15i		−	0.162253i
$906$		0			0
$907$	1.69365e16			0.916185			0.458093	−	0.888904i	$-0.348533\pi$
$907$	1.69365e16			0.916185			0.458093	+	0.888904i	$0.348533\pi$
$908$		0			0
$909$	1.95431e16	−	2.16614e16i	1.04446	−	1.15767i
$910$		0			0
$911$		−	2.67885e16i		−	1.41448i	−0.706972	−	0.707242i	$-0.749939\pi$
$911$		−	2.67885e16i		−	1.41448i	0.706972	−	0.707242i	$-0.250061\pi$
$912$		0			0
$913$			1.95140e16i			1.01802i
$914$		0			0
$915$	4.55467e15	+	1.02437e16i	0.234769	+	0.528007i
$916$		0			0
$917$	−1.29658e15	−	6.78170e14i	−0.0660340	−	0.0345388i
$918$		0			0
$919$	−6.34800e15			−0.319449			−0.159725	−	0.987162i	$-0.551061\pi$
$919$	−6.34800e15			−0.319449			−0.159725	+	0.987162i	$0.551061\pi$
$920$		0			0
$921$	3.71994e15	+	8.36632e15i	0.184973	+	0.416012i
$922$		0			0
$923$	1.01560e16			0.499016
$924$		0			0
$925$	3.21601e16			1.56149
$926$		0			0
$927$	2.76180e16	+	2.49172e16i	1.32512	+	1.19553i
$928$		0			0
$929$	−1.87489e16			−0.888977			−0.444488	−	0.895785i	$-0.646614\pi$
$929$	−1.87489e16			−0.888977			−0.444488	+	0.895785i	$0.646614\pi$
$930$		0			0
$931$	5.34932e15	−	3.71466e15i	0.250655	−	0.174059i
$932$		0			0
$933$	2.51700e15	−	1.11914e15i	0.116556	−	0.0518246i
$934$		0			0
$935$		−	1.08144e16i		−	0.494926i
$936$		0			0
$937$			9.82879e15i			0.444562i	0.974983	+	0.222281i	$0.0713502\pi$
$937$			9.82879e15i			0.444562i	−0.974983	+	0.222281i	$0.928650\pi$
$938$		0			0
$939$	1.37014e15	+	3.08152e15i	0.0612499	+	0.137754i
$940$		0			0
$941$	1.59854e16			0.706287			0.353143	−	0.935569i	$-0.385113\pi$
$941$	1.59854e16			0.706287			0.353143	+	0.935569i	$0.385113\pi$
$942$		0			0
$943$			8.30815e15i			0.362819i
$944$		0			0
$945$	5.71031e15	−	5.57302e15i	0.246482	−	0.240556i
$946$		0			0
$947$		−	2.57678e16i		−	1.09939i	−0.835365	−	0.549696i	$-0.814743\pi$
$947$		−	2.57678e16i		−	1.09939i	0.835365	−	0.549696i	$-0.185257\pi$
$948$		0			0
$949$	−1.55605e16			−0.656235
$950$		0			0
$951$	−2.68555e15	−	6.03992e15i	−0.111954	−	0.251790i
$952$		0			0
$953$			3.34113e16i			1.37684i	0.725313	+	0.688419i	$0.241695\pi$
$953$			3.34113e16i			1.37684i	−0.725313	+	0.688419i	$0.758305\pi$
$954$		0			0
$955$		−	8.69886e15i		−	0.354359i
$956$		0			0
$957$	−2.16218e16	+	9.61374e15i	−0.870713	+	0.387148i
$958$		0			0
$959$	3.89721e15	−	7.45100e15i	0.155150	−	0.296628i
$960$		0			0
$961$	−3.28498e16			−1.29287
$962$		0			0
$963$	−5.64152e15	−	5.08982e15i	−0.219509	−	0.198042i
$964$		0			0
$965$	3.63613e15			0.139875
$966$		0			0
$967$	−3.46783e16			−1.31890			−0.659451	−	0.751748i	$-0.729211\pi$
$967$	−3.46783e16			−1.31890			−0.659451	+	0.751748i	$0.729211\pi$
$968$		0			0
$969$	−3.78102e15	−	8.50368e15i	−0.142177	−	0.319762i
$970$		0			0
$971$	1.47672e16			0.549027			0.274514	−	0.961583i	$-0.411483\pi$
$971$	1.47672e16			0.549027			0.274514	+	0.961583i	$0.411483\pi$
$972$		0			0
$973$	−4.46806e15	+	8.54240e15i	−0.164248	+	0.314022i
$974$		0			0
$975$	3.88564e15	+	8.73899e15i	0.141233	+	0.317641i
$976$		0			0
$977$		−	2.94417e16i		−	1.05814i	−0.848579	−	0.529069i	$-0.822541\pi$
$977$		−	2.94417e16i		−	1.05814i	0.848579	−	0.529069i	$-0.177459\pi$
$978$		0			0
$979$			2.22922e16i			0.792223i
$980$		0			0
$981$	1.46215e16	−	1.62064e16i	0.513822	−	0.569517i
$982$		0			0
$983$	1.02784e16			0.357176			0.178588	−	0.983924i	$-0.442847\pi$
$983$	1.02784e16			0.357176			0.178588	+	0.983924i	$0.442847\pi$
$984$		0			0
$985$			4.41928e15i			0.151863i
$986$		0			0
$987$	3.45019e16	+	2.19492e15i	1.17246	+	0.0745889i
$988$		0			0
$989$		−	3.40848e16i		−	1.14546i
$990$		0			0
$991$	6.60300e15			0.219450			0.109725	−	0.993962i	$-0.465003\pi$
$991$	6.60300e15			0.219450			0.109725	+	0.993962i	$0.465003\pi$
$992$		0			0
$993$	−3.72940e16	+	1.65822e16i	−1.22580	+	0.545030i
$994$		0			0
$995$			1.11569e16i			0.362673i
$996$		0			0
$997$			1.54402e16i			0.496396i	0.968709	+	0.248198i	$0.0798384\pi$
$997$			1.54402e16i			0.496396i	−0.968709	+	0.248198i	$0.920162\pi$
$998$		0			0
$999$	1.72965e16	−	5.29640e16i	0.549980	−	1.68411i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.f.b.41.6	yes	28
3.2	odd	2	inner	84.12.f.b.41.24	yes	28
7.6	odd	2	inner	84.12.f.b.41.23	yes	28
21.20	even	2	inner	84.12.f.b.41.5	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.f.b.41.5	✓	28	21.20	even	2	inner
84.12.f.b.41.6	yes	28	1.1	even	1	trivial
84.12.f.b.41.23	yes	28	7.6	odd	2	inner
84.12.f.b.41.24	yes	28	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-384.586 + 170.999i) q^{3} +2406.66 q^{5} +(-39402.8 - 20609.4i) q^{7} +(118665. - 131528. i) q^{9} +O(q^{10})\)	\(q+(-384.586 + 170.999i) q^{3} +2406.66 q^{5} +(-39402.8 - 20609.4i) q^{7} +(118665. - 131528. i) q^{9} +669348. i q^{11} +528001. i q^{13} +(-925566. + 411537. i) q^{15} -6.71333e6 q^{17} -3.29364e6i q^{19} +(1.86779e7 + 1.18824e6i) q^{21} -5.21969e7i q^{23} -4.30361e7 q^{25} +(-2.31458e7 + 7.08755e7i) q^{27} -8.39935e7i q^{29} +2.41368e8i q^{31} +(-1.14458e8 - 2.57422e8i) q^{33} +(-9.48290e7 - 4.95999e7i) q^{35} -7.47283e8 q^{37} +(-9.02879e7 - 2.03062e8i) q^{39} -1.59170e8 q^{41} +6.53005e8 q^{43} +(2.85587e8 - 3.16543e8i) q^{45} +1.84720e9 q^{47} +(1.12783e9 + 1.62414e9i) q^{49} +(2.58185e9 - 1.14798e9i) q^{51} +1.27100e9i q^{53} +1.61089e9i q^{55} +(5.63211e8 + 1.26669e9i) q^{57} +8.68623e9 q^{59} -1.10675e10i q^{61} +(-7.38646e9 + 2.73694e9i) q^{63} +1.27072e9i q^{65} +8.09405e9 q^{67} +(8.92563e9 + 2.00742e10i) q^{69} -1.92348e10i q^{71} +2.94706e10i q^{73} +(1.65511e10 - 7.35915e9i) q^{75} +(1.37949e10 - 2.63742e10i) q^{77} +8.47582e9 q^{79} +(-3.21811e9 - 3.12156e10i) q^{81} +2.91537e10 q^{83} -1.61567e10 q^{85} +(1.43628e10 + 3.23027e10i) q^{87} +3.33043e10 q^{89} +(1.08818e10 - 2.08047e10i) q^{91} +(-4.12737e10 - 9.28265e10i) q^{93} -7.92667e9i q^{95} +2.25760e10i q^{97} +(8.80380e10 + 7.94285e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 9632 q^{7} + 267660 q^{9}+O(q^{10}) \) 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

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Database

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