LMFDB - Embedded newform 27.9.b.c.26.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [27,9,Mod(26,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("27.26");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 27 = 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	27.b (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:	$10.9992224717$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-30}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 30 $ x^2 + 30
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-5.47723i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.9.b.c.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-16.4317i q^{2} -14.0000 q^{4} -427.224i q^{5} -679.000 q^{7} -3976.47i q^{8} +O(q^{10})$	\(q-16.4317i q^{2} -14.0000 q^{4} -427.224i q^{5} -679.000 q^{7} -3976.47i q^{8} -7020.00 q^{10} -13375.4i q^{11} -30817.0 q^{13} +11157.1i q^{14} -68924.0 q^{16} +128266. i q^{17} -138391. q^{19} +5981.13i q^{20} -219780. q^{22} -303690. i q^{23} +208105. q^{25} +506375. i q^{26} +9506.00 q^{28} -1.32669e6i q^{29} +352214. q^{31} +114562. i q^{32} +2.10762e6 q^{34} +290085. i q^{35} +1.18999e6 q^{37} +2.27400e6i q^{38} -1.69884e6 q^{40} +1.09402e6i q^{41} +6.24609e6 q^{43} +187255. i q^{44} -4.99014e6 q^{46} +2399.02i q^{47} -5.30376e6 q^{49} -3.41951e6i q^{50} +431438. q^{52} -1.25779e7i q^{53} -5.71428e6 q^{55} +2.70002e6i q^{56} -2.17998e7 q^{58} +1.05361e7i q^{59} +1.65804e7 q^{61} -5.78747e6i q^{62} -1.57621e7 q^{64} +1.31657e7i q^{65} +7.66715e6 q^{67} -1.79572e6i q^{68} +4.76658e6 q^{70} +2.32211e7i q^{71} +2.49496e7 q^{73} -1.95535e7i q^{74} +1.93747e6 q^{76} +9.08189e6i q^{77} +4.16851e7 q^{79} +2.94460e7i q^{80} +1.79766e7 q^{82} -4.47280e7i q^{83} +5.47981e7 q^{85} -1.02634e8i q^{86} -5.31868e7 q^{88} +740707. i q^{89} +2.09247e7 q^{91} +4.25166e6i q^{92} +39420.0 q^{94} +5.91239e7i q^{95} -1.05926e8 q^{97} +8.71497e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 28 q^{4} - 1358 q^{7}+O(q^{10}) $ 2 * q - 28 * q^4 - 1358 * q^7	$ 2 q - 28 q^{4} - 1358 q^{7} - 14040 q^{10} - 61634 q^{13} - 137848 q^{16} - 276782 q^{19} - 439560 q^{22} + 416210 q^{25} + 19012 q^{28} + 704428 q^{31} + 4215240 q^{34} + 2379982 q^{37} - 3397680 q^{40} + 12492172 q^{43} - 9980280 q^{46} - 10607520 q^{49} + 862876 q^{52} - 11428560 q^{55} - 43599600 q^{58} + 33160798 q^{61} - 31524208 q^{64} + 15334306 q^{67} + 9533160 q^{70} + 49899262 q^{73} + 3874948 q^{76} + 83370178 q^{79} + 35953200 q^{82} + 109596240 q^{85} - 106373520 q^{88} + 41849486 q^{91} + 78840 q^{94} - 211852178 q^{97}+O(q^{100}) $ 2 * q - 28 * q^4 - 1358 * q^7 - 14040 * q^10 - 61634 * q^13 - 137848 * q^16 - 276782 * q^19 - 439560 * q^22 + 416210 * q^25 + 19012 * q^28 + 704428 * q^31 + 4215240 * q^34 + 2379982 * q^37 - 3397680 * q^40 + 12492172 * q^43 - 9980280 * q^46 - 10607520 * q^49 + 862876 * q^52 - 11428560 * q^55 - 43599600 * q^58 + 33160798 * q^61 - 31524208 * q^64 + 15334306 * q^67 + 9533160 * q^70 + 49899262 * q^73 + 3874948 * q^76 + 83370178 * q^79 + 35953200 * q^82 + 109596240 * q^85 - 106373520 * q^88 + 41849486 * q^91 + 78840 * q^94 - 211852178 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/27\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	− 16.4317i	− 1.02698i	−0.858096	−	0.513490i	$-0.828352\pi$
$2$	− 16.4317i	− 1.02698i	0.858096	−	0.513490i	$-0.171648\pi$
$3$	0	0
$3$	0	0
$4$	−14.0000	−0.0546875
$5$	− 427.224i	− 0.683558i	−0.939780	−	0.341779i	$-0.888971\pi$
$5$	− 427.224i	− 0.683558i	0.939780	−	0.341779i	$-0.111029\pi$
$6$	0	0
$7$	−679.000	−0.282799	−0.141399	−	0.989953i	$-0.545160\pi$
$7$	−679.000	−0.282799	−0.141399	+	0.989953i	$0.545160\pi$
$8$	− 3976.47i	− 0.970817i
$9$	0	0
$10$	−7020.00	−0.702000
$11$	− 13375.4i	− 0.913557i	−0.889581	−	0.456778i	$-0.849003\pi$
$11$	− 13375.4i	− 0.913557i	0.889581	−	0.456778i	$-0.150997\pi$
$12$	0	0
$13$	−30817.0	−1.07899	−0.539494	−	0.841989i	$-0.681385\pi$
$13$	−30817.0	−1.07899	−0.539494	+	0.841989i	$0.681385\pi$
$14$	11157.1i	0.290429i
$15$	0	0
$16$	−68924.0	−1.05170
$17$	128266.i	1.53573i	0.640612	+	0.767865i	$0.278681\pi$
$17$	128266.i	1.53573i	−0.640612	+	0.767865i	$0.721319\pi$
$18$	0	0
$19$	−138391.	−1.06192	−0.530962	−	0.847396i	$-0.678169\pi$
$19$	−138391.	−1.06192	−0.530962	+	0.847396i	$0.678169\pi$
$20$	5981.13i	0.0373821i
$21$	0	0
$22$	−219780.	−0.938204
$23$	− 303690.i	− 1.08522i	−0.839983	−	0.542612i	$-0.817435\pi$
$23$	− 303690.i	− 1.08522i	0.839983	−	0.542612i	$-0.182565\pi$
$24$	0	0
$25$	208105.	0.532749
$26$	506375.i	1.10810i
$27$	0	0
$28$	9506.00	0.0154656
$29$	− 1.32669e6i	− 1.87577i	−0.346952	−	0.937883i	$-0.612783\pi$
$29$	− 1.32669e6i	− 1.87577i	0.346952	−	0.937883i	$-0.387217\pi$
$30$	0	0
$31$	352214.	0.381382	0.190691	−	0.981650i	$-0.438927\pi$
$31$	352214.	0.381382	0.190691	+	0.981650i	$0.438927\pi$
$32$	114562.i	0.109255i
$33$	0	0
$34$	2.10762e6	1.57716
$35$	290085.i	0.193309i
$36$	0	0
$37$	1.18999e6	0.634946	0.317473	−	0.948267i	$-0.397166\pi$
$37$	1.18999e6	0.634946	0.317473	+	0.948267i	$0.397166\pi$
$38$	2.27400e6i	1.09057i
$39$	0	0
$40$	−1.69884e6	−0.663609
$41$	1.09402e6i	0.387160i	0.981085	+	0.193580i	$0.0620099\pi$
$41$	1.09402e6i	0.387160i	−0.981085	+	0.193580i	$0.937990\pi$
$42$	0	0
$43$	6.24609e6	1.82698	0.913491	−	0.406860i	$-0.133376\pi$
$43$	6.24609e6	1.82698	0.913491	+	0.406860i	$0.133376\pi$
$44$	187255.i	0.0499601i
$45$	0	0
$46$	−4.99014e6	−1.11450
$47$	2399.02i	0 0.000491636i	1.00000		0.000245818i	$7.82462e-5\pi$
$47$	2399.02i	0 0.000491636i	−1.00000		0.000245818i	$0.999922\pi$
$48$	0	0
$49$	−5.30376e6	−0.920025
$50$	− 3.41951e6i	− 0.547122i
$51$	0	0
$52$	431438.	0.0590072
$53$	− 1.25779e7i	− 1.59406i	−0.603938	−	0.797031i	$-0.706402\pi$
$53$	− 1.25779e7i	− 1.59406i	0.603938	−	0.797031i	$-0.293598\pi$
$54$	0	0
$55$	−5.71428e6	−0.624469
$56$	2.70002e6i	0.274546i
$57$	0	0
$58$	−2.17998e7	−1.92637
$59$	1.05361e7i	0.869504i	0.900550	+	0.434752i	$0.143164\pi$
$59$	1.05361e7i	0.869504i	−0.900550	+	0.434752i	$0.856836\pi$
$60$	0	0
$61$	1.65804e7	1.19750	0.598750	−	0.800936i	$-0.295664\pi$
$61$	1.65804e7	1.19750	0.598750	+	0.800936i	$0.295664\pi$
$62$	− 5.78747e6i	− 0.391671i
$63$	0	0
$64$	−1.57621e7	−0.939495
$65$	1.31657e7i	0.737551i
$66$	0	0
$67$	7.66715e6	0.380483	0.190241	−	0.981737i	$-0.439073\pi$
$67$	7.66715e6	0.380483	0.190241	+	0.981737i	$0.439073\pi$
$68$	− 1.79572e6i	− 0.0839852i
$69$	0	0
$70$	4.76658e6	0.198525
$71$	2.32211e7i	0.913797i	0.889519	+	0.456898i	$0.151040\pi$
$71$	2.32211e7i	0.913797i	−0.889519	+	0.456898i	$0.848960\pi$
$72$	0	0
$73$	2.49496e7	0.878563	0.439281	−	0.898350i	$-0.355233\pi$
$73$	2.49496e7	0.878563	0.439281	+	0.898350i	$0.355233\pi$
$74$	− 1.95535e7i	− 0.652077i
$75$	0	0
$76$	1.93747e6	0.0580740
$77$	9.08189e6i	0.258353i
$78$	0	0
$79$	4.16851e7	1.07022	0.535109	−	0.844783i	$-0.320270\pi$
$79$	4.16851e7	1.07022	0.535109	+	0.844783i	$0.320270\pi$
$80$	2.94460e7i	0.718895i
$81$	0	0
$82$	1.79766e7	0.397605
$83$	− 4.47280e7i	− 0.942469i	−0.882008	−	0.471235i	$-0.843809\pi$
$83$	− 4.47280e7i	− 0.942469i	0.882008	−	0.471235i	$-0.156191\pi$
$84$	0	0
$85$	5.47981e7	1.04976
$86$	− 1.02634e8i	− 1.87627i
$87$	0	0
$88$	−5.31868e7	−0.886896
$89$	740707.i	0.0118056i	0.999983	+	0.00590278i	$0.00187892\pi$
$89$	740707.i	0.0118056i	−0.999983	+	0.00590278i	$0.998121\pi$
$90$	0	0
$91$	2.09247e7	0.305137
$92$	4.25166e6i	0.0593482i
$93$	0	0
$94$	39420.0	0.000504900 0
$95$	5.91239e7i	0.725886i
$96$	0	0
$97$	−1.05926e8	−1.19651	−0.598255	−	0.801306i	$-0.704139\pi$
$97$	−1.05926e8	−1.19651	−0.598255	+	0.801306i	$0.704139\pi$
$98$	8.71497e7i	0.944847i
$99$	0	0
$100$	−2.91347e6	−0.0291347
$101$	2.09923e7i	0.201732i	0.994900	+	0.100866i	$0.0321614\pi$
$101$	2.09923e7i	0.201732i	−0.994900	+	0.100866i	$0.967839\pi$
$102$	0	0
$103$	2.58536e7	0.229706	0.114853	−	0.993383i	$-0.463360\pi$
$103$	2.58536e7	0.229706	0.114853	+	0.993383i	$0.463360\pi$
$104$	1.22543e8i	1.04750i
$105$	0	0
$106$	−2.06676e8	−1.63707
$107$	− 1.92732e8i	− 1.47035i	−0.677879	−	0.735173i	$-0.737101\pi$
$107$	− 1.92732e8i	− 1.47035i	0.677879	−	0.735173i	$-0.262899\pi$
$108$	0	0
$109$	−1.09904e8	−0.778585	−0.389292	−	0.921114i	$-0.627280\pi$
$109$	−1.09904e8	−0.778585	−0.389292	+	0.921114i	$0.627280\pi$
$110$	9.38952e7i	0.641317i
$111$	0	0
$112$	4.67994e7	0.297419
$113$	2.68827e7i	0.164877i	0.996596	+	0.0824384i	$0.0262708\pi$
$113$	2.68827e7i	0.164877i	−0.996596	+	0.0824384i	$0.973729\pi$
$114$	0	0
$115$	−1.29744e8	−0.741813
$116$	1.85737e7i	0.102581i
$117$	0	0
$118$	1.73126e8	0.892963
$119$	− 8.70924e7i	− 0.434303i
$120$	0	0
$121$	3.54580e7	0.165414
$122$	− 2.72444e8i	− 1.22981i
$123$	0	0
$124$	−4.93100e6	−0.0208568
$125$	− 2.55792e8i	− 1.04772i
$126$	0	0
$127$	−2.16766e7	−0.0833253	−0.0416627	−	0.999132i	$-0.513265\pi$
$127$	−2.16766e7	−0.0833253	−0.0416627	+	0.999132i	$0.513265\pi$
$128$	2.88326e8i	1.07410i
$129$	0	0
$130$	2.16335e8	0.757450
$131$	1.35791e8i	0.461088i	0.973062	+	0.230544i	$0.0740506\pi$
$131$	1.35791e8i	0.461088i	−0.973062	+	0.230544i	$0.925949\pi$
$132$	0	0
$133$	9.39675e7	0.300311
$134$	− 1.25984e8i	− 0.390748i
$135$	0	0
$136$	5.10044e8	1.49091
$137$	3.29294e6i	0.00934763i	0.999989	+	0.00467382i	$0.00148773\pi$
$137$	3.29294e6i	0.00934763i	−0.999989	+	0.00467382i	$0.998512\pi$
$138$	0	0
$139$	8.20959e7	0.219919	0.109959	−	0.993936i	$-0.464928\pi$
$139$	8.20959e7	0.219919	0.109959	+	0.993936i	$0.464928\pi$
$140$	− 4.06119e6i	− 0.0105716i
$141$	0	0
$142$	3.81562e8	0.938451
$143$	4.12189e8i	0.985718i
$144$	0	0
$145$	−5.66795e8	−1.28219
$146$	− 4.09964e8i	− 0.902266i
$147$	0	0
$148$	−1.66599e7	−0.0347236
$149$	7.18143e8i	1.45702i	0.685034	+	0.728511i	$0.259787\pi$
$149$	7.18143e8i	1.45702i	−0.685034	+	0.728511i	$0.740213\pi$
$150$	0	0
$151$	4.99935e8	0.961625	0.480812	−	0.876824i	$-0.340342\pi$
$151$	4.99935e8	0.961625	0.480812	+	0.876824i	$0.340342\pi$
$152$	5.50307e8i	1.03093i
$153$	0	0
$154$	1.49231e8	0.265323
$155$	− 1.50474e8i	− 0.260696i
$156$	0	0
$157$	−8.36039e8	−1.37603	−0.688015	−	0.725696i	$-0.741518\pi$
$157$	−8.36039e8	−1.37603	−0.688015	+	0.725696i	$0.741518\pi$
$158$	− 6.84956e8i	− 1.09909i
$159$	0	0
$160$	4.89434e7	0.0746818
$161$	2.06206e8i	0.306900i
$162$	0	0
$163$	−7.36451e8	−1.04326	−0.521631	−	0.853171i	$-0.674676\pi$
$163$	−7.36451e8	−1.04326	−0.521631	+	0.853171i	$0.674676\pi$
$164$	− 1.53163e7i	− 0.0211728i
$165$	0	0
$166$	−7.34956e8	−0.967897
$167$	− 1.27612e9i	− 1.64069i	−0.571872	−	0.820343i	$-0.693782\pi$
$167$	− 1.27612e9i	− 1.64069i	0.571872	−	0.820343i	$-0.306218\pi$
$168$	0	0
$169$	1.33957e8	0.164217
$170$	− 9.00425e8i	− 1.07808i
$171$	0	0
$172$	−8.74452e7	−0.0999130
$173$	− 4.02601e8i	− 0.449460i	−0.974421	−	0.224730i	$-0.927850\pi$
$173$	− 4.02601e8i	− 0.449460i	0.974421	−	0.224730i	$-0.0721500\pi$
$174$	0	0
$175$	−1.41303e8	−0.150661
$176$	9.21885e8i	0.960785i
$177$	0	0
$178$	1.21711e7	0.0121241
$179$	1.44429e9i	1.40683i	0.710780	+	0.703414i	$0.248342\pi$
$179$	1.44429e9i	1.40683i	−0.710780	+	0.703414i	$0.751658\pi$
$180$	0	0
$181$	−1.32830e9	−1.23760	−0.618800	−	0.785548i	$-0.712381\pi$
$181$	−1.32830e9	−1.23760	−0.618800	+	0.785548i	$0.712381\pi$
$182$	− 3.43829e8i	− 0.313369i
$183$	0	0
$184$	−1.20761e9	−1.05355
$185$	− 5.08392e8i	− 0.434022i
$186$	0	0
$187$	1.71560e9	1.40298
$188$	− 33586.3i	0 2.68863e-5i
$189$	0	0
$190$	9.71505e8	0.745471
$191$	1.90148e9i	1.42876i	0.699759	+	0.714379i	$0.253291\pi$
$191$	1.90148e9i	1.42876i	−0.699759	+	0.714379i	$0.746709\pi$
$192$	0	0
$193$	7.81373e8	0.563156	0.281578	−	0.959538i	$-0.409142\pi$
$193$	7.81373e8	0.563156	0.281578	+	0.959538i	$0.409142\pi$
$194$	1.74054e9i	1.22879i
$195$	0	0
$196$	7.42526e7	0.0503139
$197$	− 2.18762e9i	− 1.45247i	−0.687448	−	0.726234i	$-0.741269\pi$
$197$	− 2.18762e9i	− 1.45247i	0.687448	−	0.726234i	$-0.258731\pi$
$198$	0	0
$199$	2.94513e8	0.187798	0.0938991	−	0.995582i	$-0.470067\pi$
$199$	2.94513e8	0.187798	0.0938991	+	0.995582i	$0.470067\pi$
$200$	− 8.27522e8i	− 0.517202i
$201$	0	0
$202$	3.44939e8	0.207175
$203$	9.00825e8i	0.530464i
$204$	0	0
$205$	4.67392e8	0.264646
$206$	− 4.24817e8i	− 0.235903i
$207$	0	0
$208$	2.12403e9	1.13477
$209$	1.85103e9i	0.970128i
$210$	0	0
$211$	3.34050e9	1.68532	0.842659	−	0.538447i	$-0.180989\pi$
$211$	3.34050e9	1.68532	0.842659	+	0.538447i	$0.180989\pi$
$212$	1.76091e8i	0.0871753i
$213$	0	0
$214$	−3.16692e9	−1.51002
$215$	− 2.66848e9i	− 1.24885i
$216$	0	0
$217$	−2.39153e8	−0.107854
$218$	1.80590e9i	0.799591i
$219$	0	0
$220$	7.99999e7	0.0341506
$221$	− 3.95276e9i	− 1.65704i
$222$	0	0
$223$	2.70744e9	1.09481	0.547405	−	0.836868i	$-0.315616\pi$
$223$	2.70744e9	1.09481	0.547405	+	0.836868i	$0.315616\pi$
$224$	− 7.77874e7i	− 0.0308970i
$225$	0	0
$226$	4.41728e8	0.169325
$227$	− 5.22445e9i	− 1.96760i	−0.179267	−	0.983800i	$-0.557373\pi$
$227$	− 5.22445e9i	− 1.96760i	0.179267	−	0.983800i	$-0.442627\pi$
$228$	0	0
$229$	−2.18619e8	−0.0794960	−0.0397480	−	0.999210i	$-0.512656\pi$
$229$	−2.18619e8	−0.0794960	−0.0397480	+	0.999210i	$0.512656\pi$
$230$	2.13191e9i	0.761827i
$231$	0	0
$232$	−5.27555e9	−1.82103
$233$	4.81293e9i	1.63300i	0.577346	+	0.816499i	$0.304088\pi$
$233$	4.81293e9i	1.63300i	−0.577346	+	0.816499i	$0.695912\pi$
$234$	0	0
$235$	1.02492e6	0.000336061 0
$236$	− 1.47505e8i	− 0.0475510i
$237$	0	0
$238$	−1.43107e9	−0.446020
$239$	− 1.14772e8i	− 0.0351759i	−0.999845	−	0.0175880i	$-0.994401\pi$
$239$	− 1.14772e8i	− 0.0351759i	0.999845	−	0.0175880i	$-0.00559871\pi$
$240$	0	0
$241$	−4.11453e9	−1.21970	−0.609848	−	0.792518i	$-0.708770\pi$
$241$	−4.11453e9	−1.21970	−0.609848	+	0.792518i	$0.708770\pi$
$242$	− 5.82634e8i	− 0.169877i
$243$	0	0
$244$	−2.32126e8	−0.0654883
$245$	2.26589e9i	0.628890i
$246$	0	0
$247$	4.26480e9	1.14580
$248$	− 1.40057e9i	− 0.370252i
$249$	0	0
$250$	−4.20308e9	−1.07599
$251$	5.13492e9i	1.29371i	0.762611	+	0.646857i	$0.223917\pi$
$251$	5.13492e9i	1.29371i	−0.762611	+	0.646857i	$0.776083\pi$
$252$	0	0
$253$	−4.06197e9	−0.991414
$254$	3.56184e8i	0.0855734i
$255$	0	0
$256$	7.02574e8	0.163581
$257$	2.59521e9i	0.594895i	0.954738	+	0.297448i	$0.0961354\pi$
$257$	2.59521e9i	0.594895i	−0.954738	+	0.297448i	$0.903865\pi$
$258$	0	0
$259$	−8.08004e8	−0.179562
$260$	− 1.84320e8i	− 0.0403348i
$261$	0	0
$262$	2.23127e9	0.473529
$263$	4.85698e8i	0.101518i	0.998711	+	0.0507590i	$0.0161641\pi$
$263$	4.85698e8i	0.101518i	−0.998711	+	0.0507590i	$0.983836\pi$
$264$	0	0
$265$	−5.37359e9	−1.08963
$266$	− 1.54404e9i	− 0.308413i
$267$	0	0
$268$	−1.07340e8	−0.0208076
$269$	− 2.94120e8i	− 0.0561714i	−0.999606	−	0.0280857i	$-0.991059\pi$
$269$	− 2.94120e8i	− 0.0561714i	0.999606	−	0.0280857i	$-0.00894114\pi$
$270$	0	0
$271$	5.47200e9	1.01454	0.507270	−	0.861787i	$-0.330655\pi$
$271$	5.47200e9	1.01454	0.507270	+	0.861787i	$0.330655\pi$
$272$	− 8.84058e9i	− 1.61512i
$273$	0	0
$274$	5.41085e7	0.00959983
$275$	− 2.78348e9i	− 0.486696i
$276$	0	0
$277$	3.67450e9	0.624137	0.312068	−	0.950060i	$-0.398978\pi$
$277$	3.67450e9	0.624137	0.312068	+	0.950060i	$0.398978\pi$
$278$	− 1.34897e9i	− 0.225852i
$279$	0	0
$280$	1.15351e9	0.187668
$281$	− 4.81620e9i	− 0.772465i	−0.922401	−	0.386233i	$-0.873776\pi$
$281$	− 4.81620e9i	− 0.772465i	0.922401	−	0.386233i	$-0.126224\pi$
$282$	0	0
$283$	9.24187e9	1.44083	0.720417	−	0.693541i	$-0.243950\pi$
$283$	9.24187e9	1.44083	0.720417	+	0.693541i	$0.243950\pi$
$284$	− 3.25096e8i	− 0.0499733i
$285$	0	0
$286$	6.77296e9	1.01231
$287$	− 7.42840e8i	− 0.109488i
$288$	0	0
$289$	−9.47632e9	−1.35847
$290$	9.31339e9i	1.31679i
$291$	0	0
$292$	−3.49295e8	−0.0480464
$293$	6.34803e9i	0.861328i	0.902512	+	0.430664i	$0.141721\pi$
$293$	6.34803e9i	0.861328i	−0.902512	+	0.430664i	$0.858279\pi$
$294$	0	0
$295$	4.50127e9	0.594356
$296$	− 4.73196e9i	− 0.616416i
$297$	0	0
$298$	1.18003e10	1.49633
$299$	9.35882e9i	1.17094i
$300$	0	0
$301$	−4.24109e9	−0.516668
$302$	− 8.21477e9i	− 0.987569i
$303$	0	0
$304$	9.53846e9	1.11682
$305$	− 7.08354e9i	− 0.818561i
$306$	0	0
$307$	−5.37906e9	−0.605554	−0.302777	−	0.953061i	$-0.597914\pi$
$307$	−5.37906e9	−0.605554	−0.302777	+	0.953061i	$0.597914\pi$
$308$	− 1.27146e8i	− 0.0141287i
$309$	0	0
$310$	−2.47254e9	−0.267730
$311$	6.35655e9i	0.679485i	0.940518	+	0.339743i	$0.110340\pi$
$311$	6.35655e9i	0.679485i	−0.940518	+	0.339743i	$0.889660\pi$
$312$	0	0
$313$	1.11484e9	0.116154	0.0580772	−	0.998312i	$-0.481503\pi$
$313$	1.11484e9	0.116154	0.0580772	+	0.998312i	$0.481503\pi$
$314$	1.37375e10i	1.41316i
$315$	0	0
$316$	−5.83591e8	−0.0585276
$317$	1.38485e9i	0.137140i	0.997646	+	0.0685702i	$0.0218437\pi$
$317$	1.38485e9i	0.137140i	−0.997646	+	0.0685702i	$0.978156\pi$
$318$	0	0
$319$	−1.77450e10	−1.71362
$320$	6.73394e9i	0.642199i
$321$	0	0
$322$	3.38831e9	0.315180
$323$	− 1.77508e10i	− 1.63083i
$324$	0	0
$325$	−6.41317e9	−0.574830
$326$	1.21011e10i	1.07141i
$327$	0	0
$328$	4.35034e9	0.375861
$329$	− 1.62894e6i	0 0.000139034i
$330$	0	0
$331$	9.36529e9	0.780206	0.390103	−	0.920771i	$-0.372439\pi$
$331$	9.36529e9	0.780206	0.390103	+	0.920771i	$0.372439\pi$
$332$	6.26192e8i	0.0515413i
$333$	0	0
$334$	−2.09688e10	−1.68495
$335$	− 3.27559e9i	− 0.260082i
$336$	0	0
$337$	−1.54504e10	−1.19790	−0.598948	−	0.800788i	$-0.704414\pi$
$337$	−1.54504e10	−1.19790	−0.598948	+	0.800788i	$0.704414\pi$
$338$	− 2.20113e9i	− 0.168647i
$339$	0	0
$340$	−7.67174e8	−0.0574087
$341$	− 4.71100e9i	− 0.348414i
$342$	0	0
$343$	7.51555e9	0.542981
$344$	− 2.48373e10i	− 1.77366i
$345$	0	0
$346$	−6.61541e9	−0.461586
$347$	− 1.46874e8i	− 0.0101304i	−0.999987	−	0.00506521i	$-0.998388\pi$
$347$	− 1.46874e8i	− 0.0101304i	0.999987	−	0.00506521i	$-0.00161231\pi$
$348$	0	0
$349$	−2.74904e10	−1.85302	−0.926508	−	0.376276i	$-0.877204\pi$
$349$	−2.74904e10	−1.85302	−0.926508	+	0.376276i	$0.877204\pi$
$350$	2.32185e9i	0.154726i
$351$	0	0
$352$	1.53231e9	0.0998102
$353$	− 1.90766e10i	− 1.22857i	−0.789083	−	0.614287i	$-0.789444\pi$
$353$	− 1.90766e10i	− 1.22857i	0.789083	−	0.614287i	$-0.210556\pi$
$354$	0	0
$355$	9.92061e9	0.624633
$356$	− 1.03699e7i	0 0.000645616i
$357$	0	0
$358$	2.37320e10	1.44478
$359$	6.80030e9i	0.409402i	0.978825	+	0.204701i	$0.0656222\pi$
$359$	6.80030e9i	0.409402i	−0.978825	+	0.204701i	$0.934378\pi$
$360$	0	0
$361$	2.16851e9	0.127683
$362$	2.18261e10i	1.27099i
$363$	0	0
$364$	−2.92946e8	−0.0166872
$365$	− 1.06591e10i	− 0.600548i
$366$	0	0
$367$	−1.55772e9	−0.0858668	−0.0429334	−	0.999078i	$-0.513670\pi$
$367$	−1.55772e9	−0.0858668	−0.0429334	+	0.999078i	$0.513670\pi$
$368$	2.09315e10i	1.14133i
$369$	0	0
$370$	−8.35374e9	−0.445732
$371$	8.54041e9i	0.450799i
$372$	0	0
$373$	−2.05254e9	−0.106037	−0.0530184	−	0.998594i	$-0.516884\pi$
$373$	−2.05254e9	−0.106037	−0.0530184	+	0.998594i	$0.516884\pi$
$374$	− 2.81902e10i	− 1.44083i
$375$	0	0
$376$	9.53964e6	0.000477288 0
$377$	4.08847e10i	2.02393i
$378$	0	0
$379$	2.21941e10	1.07567	0.537837	−	0.843049i	$-0.319241\pi$
$379$	2.21941e10	1.07567	0.537837	+	0.843049i	$0.319241\pi$
$380$	− 8.27735e8i	− 0.0396969i
$381$	0	0
$382$	3.12445e10	1.46731
$383$	1.82640e10i	0.848792i	0.905477	+	0.424396i	$0.139513\pi$
$383$	1.82640e10i	0.848792i	−0.905477	+	0.424396i	$0.860487\pi$
$384$	0	0
$385$	3.88000e9	0.176599
$386$	− 1.28393e10i	− 0.578350i
$387$	0	0
$388$	1.48297e9	0.0654341
$389$	− 7.59438e9i	− 0.331661i	−0.986154	−	0.165830i	$-0.946970\pi$
$389$	− 7.59438e9i	− 0.331661i	0.986154	−	0.165830i	$-0.0530304\pi$
$390$	0	0
$391$	3.89530e10	1.66661
$392$	2.10902e10i	0.893176i
$393$	0	0
$394$	−3.59462e10	−1.49166
$395$	− 1.78089e10i	− 0.731556i
$396$	0	0
$397$	3.83153e10	1.54245	0.771223	−	0.636564i	$-0.219645\pi$
$397$	3.83153e10	1.54245	0.771223	+	0.636564i	$0.219645\pi$
$398$	− 4.83933e9i	− 0.192865i
$399$	0	0
$400$	−1.43434e10	−0.560290
$401$	8.55644e8i	0.0330914i	0.999863	+	0.0165457i	$0.00526690\pi$
$401$	8.55644e8i	0.0330914i	−0.999863	+	0.0165457i	$0.994733\pi$
$402$	0	0
$403$	−1.08542e10	−0.411507
$404$	− 2.93893e8i	− 0.0110322i
$405$	0	0
$406$	1.48021e10	0.544776
$407$	− 1.59166e10i	− 0.580059i
$408$	0	0
$409$	−3.15643e9	−0.112798	−0.0563992	−	0.998408i	$-0.517962\pi$
$409$	−3.15643e9	−0.112798	−0.0563992	+	0.998408i	$0.517962\pi$
$410$	− 7.68003e9i	− 0.271786i
$411$	0	0
$412$	−3.61950e8	−0.0125620
$413$	− 7.15400e9i	− 0.245895i
$414$	0	0
$415$	−1.91089e10	−0.644232
$416$	− 3.53045e9i	− 0.117884i
$417$	0	0
$418$	3.04156e10	0.996302
$419$	− 6.87440e9i	− 0.223038i	−0.993762	−	0.111519i	$-0.964428\pi$
$419$	− 6.87440e9i	− 0.223038i	0.993762	−	0.111519i	$-0.0355716\pi$
$420$	0	0
$421$	−1.53006e10	−0.487057	−0.243528	−	0.969894i	$-0.578305\pi$
$421$	−1.53006e10	−0.487057	−0.243528	+	0.969894i	$0.578305\pi$
$422$	− 5.48901e10i	− 1.73079i
$423$	0	0
$424$	−5.00157e10	−1.54754
$425$	2.66927e10i	0.818158i
$426$	0	0
$427$	−1.12581e10	−0.338652
$428$	2.69825e9i	0.0804096i
$429$	0	0
$430$	−4.38475e10	−1.28254
$431$	− 3.16429e9i	− 0.0916996i	−0.998948	−	0.0458498i	$-0.985400\pi$
$431$	− 3.16429e9i	− 0.0916996i	0.998948	−	0.0458498i	$-0.0145996\pi$
$432$	0	0
$433$	6.22045e10	1.76958	0.884791	−	0.465989i	$-0.154301\pi$
$433$	6.22045e10	1.76958	0.884791	+	0.465989i	$0.154301\pi$
$434$	3.92969e9i	0.110764i
$435$	0	0
$436$	1.53865e9	0.0425789
$437$	4.20280e10i	1.15243i
$438$	0	0
$439$	−2.30286e10	−0.620024	−0.310012	−	0.950733i	$-0.600333\pi$
$439$	−2.30286e10	−0.620024	−0.310012	+	0.950733i	$0.600333\pi$
$440$	2.27226e10i	0.606245i
$441$	0	0
$442$	−6.49505e10	−1.70174
$443$	3.93754e10i	1.02237i	0.859470	+	0.511187i	$0.170794\pi$
$443$	3.93754e10i	1.02237i	−0.859470	+	0.511187i	$0.829206\pi$
$444$	0	0
$445$	3.16448e8	0.00806978
$446$	− 4.44877e10i	− 1.12435i
$447$	0	0
$448$	1.07025e10	0.265688
$449$	− 5.66523e10i	− 1.39390i	−0.717119	−	0.696951i	$-0.754539\pi$
$449$	− 5.66523e10i	− 1.39390i	0.717119	−	0.696951i	$-0.245461\pi$
$450$	0	0
$451$	1.46330e10	0.353692
$452$	− 3.76358e8i	− 0.00901670i
$453$	0	0
$454$	−8.58464e10	−2.02069
$455$	− 8.93954e9i	− 0.208579i
$456$	0	0
$457$	−3.34855e10	−0.767702	−0.383851	−	0.923395i	$-0.625402\pi$
$457$	−3.34855e10	−0.767702	−0.383851	+	0.923395i	$0.625402\pi$
$458$	3.59227e9i	0.0816408i
$459$	0	0
$460$	1.81641e9	0.0405679
$461$	4.89291e10i	1.08334i	0.840593	+	0.541668i	$0.182207\pi$
$461$	4.89291e10i	1.08334i	−0.840593	+	0.541668i	$0.817793\pi$
$462$	0	0
$463$	3.40379e10	0.740695	0.370347	−	0.928893i	$-0.379239\pi$
$463$	3.40379e10	0.740695	0.370347	+	0.928893i	$0.379239\pi$
$464$	9.14410e10i	1.97274i
$465$	0	0
$466$	7.90845e10	1.67706
$467$	7.93796e10i	1.66894i	0.551052	+	0.834471i	$0.314227\pi$
$467$	7.93796e10i	1.66894i	−0.551052	+	0.834471i	$0.685773\pi$
$468$	0	0
$469$	−5.20600e9	−0.107600
$470$	− 1.68412e7i	0 0.000345128i
$471$	0	0
$472$	4.18964e10	0.844129
$473$	− 8.35438e10i	− 1.66905i
$474$	0	0
$475$	−2.87999e10	−0.565739
$476$	1.21929e9i	0.0237509i
$477$	0	0
$478$	−1.88590e9	−0.0361250
$479$	− 9.07702e10i	− 1.72425i	−0.506692	−	0.862127i	$-0.669132\pi$
$479$	− 9.07702e10i	− 1.72425i	0.506692	−	0.862127i	$-0.330868\pi$
$480$	0	0
$481$	−3.66720e10	−0.685100
$482$	6.76086e10i	1.25260i
$483$	0	0
$484$	−4.96411e8	−0.00904608
$485$	4.52541e10i	0.817883i
$486$	0	0
$487$	2.51505e10	0.447127	0.223563	−	0.974689i	$-0.428231\pi$
$487$	2.51505e10	0.447127	0.223563	+	0.974689i	$0.428231\pi$
$488$	− 6.59314e10i	− 1.16255i
$489$	0	0
$490$	3.72324e10	0.645857
$491$	− 9.74682e10i	− 1.67702i	−0.544890	−	0.838508i	$-0.683428\pi$
$491$	− 9.74682e10i	− 1.67702i	0.544890	−	0.838508i	$-0.316572\pi$
$492$	0	0
$493$	1.70169e11	2.88067
$494$	− 7.00777e10i	− 1.17672i
$495$	0	0
$496$	−2.42760e10	−0.401098
$497$	− 1.57671e10i	− 0.258421i
$498$	0	0
$499$	−6.95297e10	−1.12142	−0.560709	−	0.828013i	$-0.689472\pi$
$499$	−6.95297e10	−1.12142	−0.560709	+	0.828013i	$0.689472\pi$
$500$	3.58108e9i	0.0572973i
$501$	0	0
$502$	8.43753e10	1.32862
$503$	− 2.66836e10i	− 0.416842i	−0.978039	−	0.208421i	$-0.933168\pi$
$503$	− 2.66836e10i	− 0.416842i	0.978039	−	0.208421i	$-0.0668325\pi$
$504$	0	0
$505$	8.96842e9	0.137896
$506$	6.67450e10i	1.01816i
$507$	0	0
$508$	3.03473e8	0.00455685
$509$	− 1.54473e10i	− 0.230134i	−0.993358	−	0.115067i	$-0.963292\pi$
$509$	− 1.54473e10i	− 0.230134i	0.993358	−	0.115067i	$-0.0367082\pi$
$510$	0	0
$511$	−1.69408e10	−0.248456
$512$	6.22669e10i	0.906102i
$513$	0	0
$514$	4.26437e10	0.610945
$515$	− 1.10453e10i	− 0.157017i
$516$	0	0
$517$	3.20879e7	0.000449137 0
$518$	1.32769e10i	0.184407i
$519$	0	0
$520$	5.23532e10	0.716027
$521$	7.64491e9i	0.103758i	0.998653	+	0.0518790i	$0.0165210\pi$
$521$	7.64491e9i	0.103758i	−0.998653	+	0.0518790i	$0.983479\pi$
$522$	0	0
$523$	−7.26324e10	−0.970787	−0.485393	−	0.874296i	$-0.661324\pi$
$523$	−7.26324e10	−0.970787	−0.485393	+	0.874296i	$0.661324\pi$
$524$	− 1.90107e9i	− 0.0252158i
$525$	0	0
$526$	7.98083e9	0.104257
$527$	4.51770e10i	0.585699i
$528$	0	0
$529$	−1.39168e10	−0.177712
$530$	8.82970e10i	1.11903i
$531$	0	0
$532$	−1.31554e9	−0.0164233
$533$	− 3.37144e10i	− 0.417741i
$534$	0	0
$535$	−8.23398e10	−1.00507
$536$	− 3.04882e10i	− 0.369379i
$537$	0	0
$538$	−4.83289e9	−0.0576869
$539$	7.09398e10i	0.840495i
$540$	0	0
$541$	3.77715e10	0.440935	0.220468	−	0.975394i	$-0.429242\pi$
$541$	3.77715e10	0.440935	0.220468	+	0.975394i	$0.429242\pi$
$542$	− 8.99141e10i	− 1.04191i
$543$	0	0
$544$	−1.46943e10	−0.167785
$545$	4.69534e10i	0.532208i
$546$	0	0
$547$	−6.24076e10	−0.697089	−0.348544	−	0.937292i	$-0.613324\pi$
$547$	−6.24076e10	−0.697089	−0.348544	+	0.937292i	$0.613324\pi$
$548$	− 4.61012e7i	0 0.000511199i
$549$	0	0
$550$	−4.57373e10	−0.499827
$551$	1.83602e11i	1.99192i
$552$	0	0
$553$	−2.83042e10	−0.302656
$554$	− 6.03783e10i	− 0.640976i
$555$	0	0
$556$	−1.14934e9	−0.0120268
$557$	− 6.15741e10i	− 0.639702i	−0.947468	−	0.319851i	$-0.896367\pi$
$557$	− 6.15741e10i	− 0.639702i	0.947468	−	0.319851i	$-0.103633\pi$
$558$	0	0
$559$	−1.92486e11	−1.97129
$560$	− 1.99938e10i	− 0.203303i
$561$	0	0
$562$	−7.91382e10	−0.793306
$563$	2.94000e10i	0.292626i	0.989238	+	0.146313i	$0.0467407\pi$
$563$	2.94000e10i	0.292626i	−0.989238	+	0.146313i	$0.953259\pi$
$564$	0	0
$565$	1.14849e10	0.112703
$566$	− 1.51859e11i	− 1.47971i
$567$	0	0
$568$	9.23380e10	0.887129
$569$	4.27721e9i	0.0408049i	0.999792	+	0.0204024i	$0.00649475\pi$
$569$	4.27721e9i	0.0408049i	−0.999792	+	0.0204024i	$0.993505\pi$
$570$	0	0
$571$	7.14732e10	0.672356	0.336178	−	0.941799i	$-0.390866\pi$
$571$	7.14732e10	0.672356	0.336178	+	0.941799i	$0.390866\pi$
$572$	− 5.77065e9i	− 0.0539064i
$573$	0	0
$574$	−1.22061e10	−0.112442
$575$	− 6.31995e10i	− 0.578152i
$576$	0	0
$577$	−1.09889e10	−0.0991404	−0.0495702	−	0.998771i	$-0.515785\pi$
$577$	−1.09889e10	−0.0991404	−0.0495702	+	0.998771i	$0.515785\pi$
$578$	1.55712e11i	1.39512i
$579$	0	0
$580$	7.93513e9	0.0701200
$581$	3.03703e10i	0.266529i
$582$	0	0
$583$	−1.68235e11	−1.45627
$584$	− 9.92114e10i	− 0.852923i
$585$	0	0
$586$	1.04309e11	0.884567
$587$	− 1.22636e11i	− 1.03292i	−0.856312	−	0.516459i	$-0.827250\pi$
$587$	− 1.22636e11i	− 1.03292i	0.856312	−	0.516459i	$-0.172750\pi$
$588$	0	0
$589$	−4.87432e10	−0.404998
$590$	− 7.39633e10i	− 0.610392i
$591$	0	0
$592$	−8.20189e10	−0.667771
$593$	− 1.00435e11i	− 0.812206i	−0.913827	−	0.406103i	$-0.866887\pi$
$593$	− 1.00435e11i	− 0.812206i	0.913827	−	0.406103i	$-0.133113\pi$
$594$	0	0
$595$	−3.72079e10	−0.296871
$596$	− 1.00540e10i	− 0.0796809i
$597$	0	0
$598$	1.53781e11	1.20254
$599$	1.14078e11i	0.886125i	0.896491	+	0.443063i	$0.146108\pi$
$599$	1.14078e11i	0.886125i	−0.896491	+	0.443063i	$0.853892\pi$
$600$	0	0
$601$	6.33781e10	0.485782	0.242891	−	0.970054i	$-0.421904\pi$
$601$	6.33781e10	0.485782	0.242891	+	0.970054i	$0.421904\pi$
$602$	6.96883e10i	0.530608i
$603$	0	0
$604$	−6.99909e9	−0.0525889
$605$	− 1.51485e10i	− 0.113070i
$606$	0	0
$607$	2.66309e11	1.96169	0.980846	−	0.194786i	$-0.0624011\pi$
$607$	2.66309e11	1.96169	0.980846	+	0.194786i	$0.0624011\pi$
$608$	− 1.58543e10i	− 0.116020i
$609$	0	0
$610$	−1.16394e11	−0.840645
$611$	− 7.39307e7i	0 0.000530469i
$612$	0	0
$613$	2.39412e11	1.69552	0.847761	−	0.530378i	$-0.177950\pi$
$613$	2.39412e11	1.69552	0.847761	+	0.530378i	$0.177950\pi$
$614$	8.83870e10i	0.621892i
$615$	0	0
$616$	3.61138e10	0.250813
$617$	1.05956e11i	0.731112i	0.930789	+	0.365556i	$0.119121\pi$
$617$	1.05956e11i	0.731112i	−0.930789	+	0.365556i	$0.880879\pi$
$618$	0	0
$619$	−2.13868e10	−0.145674	−0.0728372	−	0.997344i	$-0.523205\pi$
$619$	−2.13868e10	−0.145674	−0.0728372	+	0.997344i	$0.523205\pi$
$620$	2.10664e9i	0.0142568i
$621$	0	0
$622$	1.04449e11	0.697818
$623$	− 5.02940e8i	− 0.00333860i
$624$	0	0
$625$	−2.79892e10	−0.183430
$626$	− 1.83187e10i	− 0.119288i
$627$	0	0
$628$	1.17046e10	0.0752517
$629$	1.52635e11i	0.975105i
$630$	0	0
$631$	−1.12422e11	−0.709145	−0.354573	−	0.935028i	$-0.615374\pi$
$631$	−1.12422e11	−0.709145	−0.354573	+	0.935028i	$0.615374\pi$
$632$	− 1.65759e11i	− 1.03899i
$633$	0	0
$634$	2.27554e10	0.140840
$635$	9.26077e9i	0.0569577i
$636$	0	0
$637$	1.63446e11	0.992697
$638$	2.91581e11i	1.75985i
$639$	0	0
$640$	1.23179e11	0.734207
$641$	1.44527e11i	0.856086i	0.903758	+	0.428043i	$0.140797\pi$
$641$	1.44527e11i	0.856086i	−0.903758	+	0.428043i	$0.859203\pi$
$642$	0	0
$643$	5.57505e10	0.326140	0.163070	−	0.986614i	$-0.447860\pi$
$643$	5.57505e10	0.326140	0.163070	+	0.986614i	$0.447860\pi$
$644$	− 2.88688e9i	− 0.0167836i
$645$	0	0
$646$	−2.91676e11	−1.67483
$647$	− 5.47913e10i	− 0.312676i	−0.987704	−	0.156338i	$-0.950031\pi$
$647$	− 5.47913e10i	− 0.312676i	0.987704	−	0.156338i	$-0.0499689\pi$
$648$	0	0
$649$	1.40924e11	0.794341
$650$	1.05379e11i	0.590339i
$651$	0	0
$652$	1.03103e10	0.0570534
$653$	− 1.37326e11i	− 0.755264i	−0.925956	−	0.377632i	$-0.876739\pi$
$653$	− 1.37326e11i	− 0.755264i	0.925956	−	0.377632i	$-0.123261\pi$
$654$	0	0
$655$	5.80129e10	0.315181
$656$	− 7.54043e10i	− 0.407175i
$657$	0	0
$658$	−2.67662e7	−0.000142785 0
$659$	− 2.32347e11i	− 1.23196i	−0.787763	−	0.615978i	$-0.788761\pi$
$659$	− 2.32347e11i	− 1.23196i	0.787763	−	0.615978i	$-0.211239\pi$
$660$	0	0
$661$	1.27504e11	0.667912	0.333956	−	0.942589i	$-0.391616\pi$
$661$	1.27504e11	0.667912	0.333956	+	0.942589i	$0.391616\pi$
$662$	− 1.53887e11i	− 0.801256i
$663$	0	0
$664$	−1.77859e11	−0.914965
$665$	− 4.01451e10i	− 0.205280i
$666$	0	0
$667$	−4.02904e11	−2.03563
$668$	1.78657e10i	0.0897250i
$669$	0	0
$670$	−5.38234e10	−0.267099
$671$	− 2.21769e11i	− 1.09398i
$672$	0	0
$673$	−6.54382e9	−0.0318986	−0.0159493	−	0.999873i	$-0.505077\pi$
$673$	−6.54382e9	−0.0318986	−0.0159493	+	0.999873i	$0.505077\pi$
$674$	2.53875e11i	1.23021i
$675$	0	0
$676$	−1.87539e9	−0.00898061
$677$	− 2.32186e11i	− 1.10530i	−0.833412	−	0.552652i	$-0.813616\pi$
$677$	− 2.32186e11i	− 1.10530i	0.833412	−	0.552652i	$-0.186384\pi$
$678$	0	0
$679$	7.19238e10	0.338371
$680$	− 2.17903e11i	− 1.01912i
$681$	0	0
$682$	−7.74096e10	−0.357814
$683$	− 4.94502e10i	− 0.227240i	−0.993524	−	0.113620i	$-0.963755\pi$
$683$	− 4.94502e10i	− 0.227240i	0.993524	−	0.113620i	$-0.0362446\pi$
$684$	0	0
$685$	1.40682e9	0.00638965
$686$	− 1.23493e11i	− 0.557630i
$687$	0	0
$688$	−4.30505e11	−1.92143
$689$	3.87614e11i	1.71998i
$690$	0	0
$691$	1.15283e11	0.505652	0.252826	−	0.967512i	$-0.418640\pi$
$691$	1.15283e11	0.505652	0.252826	+	0.967512i	$0.418640\pi$
$692$	5.63642e9i	0.0245798i
$693$	0	0
$694$	−2.41339e9	−0.0104037
$695$	− 3.50733e10i	− 0.150327i
$696$	0	0
$697$	−1.40325e11	−0.594573
$698$	4.51713e11i	1.90301i
$699$	0	0
$700$	1.97825e9	0.00823926
$701$	− 2.53572e10i	− 0.105009i	−0.998621	−	0.0525047i	$-0.983280\pi$
$701$	− 2.53572e10i	− 0.105009i	0.998621	−	0.0525047i	$-0.0167205\pi$
$702$	0	0
$703$	−1.64684e11	−0.674264
$704$	2.10824e11i	0.858282i
$705$	0	0
$706$	−3.13460e11	−1.26172
$707$	− 1.42538e10i	− 0.0570496i
$708$	0	0
$709$	2.47270e11	0.978558	0.489279	−	0.872127i	$-0.337260\pi$
$709$	2.47270e11	0.978558	0.489279	+	0.872127i	$0.337260\pi$
$710$	− 1.63012e11i	− 0.641485i
$711$	0	0
$712$	2.94540e9	0.0114610
$713$	− 1.06964e11i	− 0.413885i
$714$	0	0
$715$	1.76097e11	0.673795
$716$	− 2.02200e10i	− 0.0769359i
$717$	0	0
$718$	1.11740e11	0.420448
$719$	1.96339e10i	0.0734668i	0.999325	+	0.0367334i	$0.0116952\pi$
$719$	1.96339e10i	0.0734668i	−0.999325	+	0.0367334i	$0.988305\pi$
$720$	0	0
$721$	−1.75546e10	−0.0649605
$722$	− 3.56322e10i	− 0.131127i
$723$	0	0
$724$	1.85961e10	0.0676813
$725$	− 2.76092e11i	− 0.999312i
$726$	0	0
$727$	6.17310e10	0.220986	0.110493	−	0.993877i	$-0.464757\pi$
$727$	6.17310e10	0.220986	0.110493	+	0.993877i	$0.464757\pi$
$728$	− 8.32065e10i	− 0.296232i
$729$	0	0
$730$	−1.75146e11	−0.616751
$731$	8.01158e11i	2.80575i
$732$	0	0
$733$	−1.78735e11	−0.619145	−0.309573	−	0.950876i	$-0.600186\pi$
$733$	−1.78735e11	−0.619145	−0.309573	+	0.950876i	$0.600186\pi$
$734$	2.55960e10i	0.0881835i
$735$	0	0
$736$	3.47913e10	0.118566
$737$	− 1.02551e11i	− 0.347593i
$738$	0	0
$739$	2.19706e11	0.736655	0.368328	−	0.929696i	$-0.379931\pi$
$739$	2.19706e11	0.736655	0.368328	+	0.929696i	$0.379931\pi$
$740$	7.11749e9i	0.0237356i
$741$	0	0
$742$	1.40333e11	0.462962
$743$	3.55374e11i	1.16608i	0.812442	+	0.583042i	$0.198138\pi$
$743$	3.55374e11i	1.16608i	−0.812442	+	0.583042i	$0.801862\pi$
$744$	0	0
$745$	3.06808e11	0.995958
$746$	3.37267e10i	0.108898i
$747$	0	0
$748$	−2.40184e10	−0.0767253
$749$	1.30865e11i	0.415812i
$750$	0	0
$751$	8.30521e10	0.261090	0.130545	−	0.991442i	$-0.458327\pi$
$751$	8.30521e10	0.261090	0.130545	+	0.991442i	$0.458327\pi$
$752$	− 1.65350e8i	0 0.000517052i
$753$	0	0
$754$	6.71804e11	2.07854
$755$	− 2.13584e11i	− 0.657326i
$756$	0	0
$757$	−1.47694e11	−0.449759	−0.224879	−	0.974387i	$-0.572199\pi$
$757$	−1.47694e11	−0.449759	−0.224879	+	0.974387i	$0.572199\pi$
$758$	− 3.64686e11i	− 1.10470i
$759$	0	0
$760$	2.35104e11	0.704703
$761$	− 3.95812e11i	− 1.18019i	−0.807336	−	0.590093i	$-0.799091\pi$
$761$	− 3.95812e11i	− 1.18019i	0.807336	−	0.590093i	$-0.200909\pi$
$762$	0	0
$763$	7.46245e10	0.220183
$764$	− 2.66207e10i	− 0.0781352i
$765$	0	0
$766$	3.00108e11	0.871692
$767$	− 3.24691e11i	− 0.938185i
$768$	0	0
$769$	−6.19413e11	−1.77123	−0.885615	−	0.464420i	$-0.846263\pi$
$769$	−6.19413e11	−1.77123	−0.885615	+	0.464420i	$0.846263\pi$
$770$	− 6.37548e10i	− 0.181364i
$771$	0	0
$772$	−1.09392e10	−0.0307976
$773$	8.71283e10i	0.244029i	0.992528	+	0.122014i	$0.0389354\pi$
$773$	8.71283e10i	0.244029i	−0.992528	+	0.122014i	$0.961065\pi$
$774$	0	0
$775$	7.32975e10	0.203181
$776$	4.21211e11i	1.16159i
$777$	0	0
$778$	−1.24788e11	−0.340609
$779$	− 1.51403e11i	− 0.411134i
$780$	0	0
$781$	3.10591e11	0.834805
$782$	− 6.40064e11i	− 1.71158i
$783$	0	0
$784$	3.65556e11	0.967587
$785$	3.57176e11i	0.940597i
$786$	0	0
$787$	4.87095e11	1.26974	0.634870	−	0.772619i	$-0.281054\pi$
$787$	4.87095e11	1.26974	0.634870	+	0.772619i	$0.281054\pi$
$788$	3.06267e10i	0.0794318i
$789$	0	0
$790$	−2.92629e11	−0.751293
$791$	− 1.82534e10i	− 0.0466269i
$792$	0	0
$793$	−5.10958e11	−1.29209
$794$	− 6.29585e11i	− 1.58406i
$795$	0	0
$796$	−4.12318e9	−0.0102702
$797$	3.92490e11i	0.972737i	0.873754	+	0.486368i	$0.161679\pi$
$797$	3.92490e11i	0.972737i	−0.873754	+	0.486368i	$0.838321\pi$
$798$	0	0
$799$	−3.07713e8	−0.000755019 0
$800$	2.38409e10i	0.0582052i
$801$	0	0
$802$	1.40597e10	0.0339842
$803$	− 3.33711e11i	− 0.802617i
$804$	0	0
$805$	8.80959e10	0.209784
$806$	1.78352e11i	0.422609i
$807$	0	0
$808$	8.34752e10	0.195845
$809$	6.95054e11i	1.62265i	0.584597	+	0.811324i	$0.301253\pi$
$809$	6.95054e11i	1.62265i	−0.584597	+	0.811324i	$0.698747\pi$
$810$	0	0
$811$	−5.75064e11	−1.32933	−0.664665	−	0.747142i	$-0.731426\pi$
$811$	−5.75064e11	−1.32933	−0.664665	+	0.747142i	$0.731426\pi$
$812$	− 1.26115e10i	− 0.0290098i
$813$	0	0
$814$	−2.61536e11	−0.595709
$815$	3.14629e11i	0.713129i
$816$	0	0
$817$	−8.64402e11	−1.94012
$818$	5.18655e10i	0.115842i
$819$	0	0
$820$	−6.54348e9	−0.0144728
$821$	9.35693e8i	0.00205949i	0.999999	+	0.00102975i	$0.000327779\pi$
$821$	9.35693e8i	0.00205949i	−0.999999	+	0.00102975i	$0.999672\pi$
$822$	0	0
$823$	−6.59429e9	−0.0143737	−0.00718685	−	0.999974i	$-0.502288\pi$
$823$	−6.59429e9	−0.0143737	−0.00718685	+	0.999974i	$0.502288\pi$
$824$	− 1.02806e11i	− 0.223002i
$825$	0	0
$826$	−1.17552e11	−0.252529
$827$	8.55938e11i	1.82987i	0.403601	+	0.914935i	$0.367758\pi$
$827$	8.55938e11i	1.82987i	−0.403601	+	0.914935i	$0.632242\pi$
$828$	0	0
$829$	8.38807e11	1.77600	0.888002	−	0.459839i	$-0.152093\pi$
$829$	8.38807e11	1.77600	0.888002	+	0.459839i	$0.152093\pi$
$830$	3.13991e11i	0.661613i
$831$	0	0
$832$	4.85741e11	1.01370
$833$	− 6.80290e11i	− 1.41291i
$834$	0	0
$835$	−5.45188e11	−1.12150
$836$	− 2.59145e10i	− 0.0530539i
$837$	0	0
$838$	−1.12958e11	−0.229056
$839$	− 1.07846e11i	− 0.217649i	−0.994061	−	0.108825i	$-0.965291\pi$
$839$	− 1.07846e11i	− 0.217649i	0.994061	−	0.108825i	$-0.0347087\pi$
$840$	0	0
$841$	−1.25987e12	−2.51850
$842$	2.51414e11i	0.500197i
$843$	0	0
$844$	−4.67670e10	−0.0921658
$845$	− 5.72295e10i	− 0.112252i
$846$	0	0
$847$	−2.40760e10	−0.0467789
$848$	8.66921e11i	1.67647i
$849$	0	0
$850$	4.38606e11	0.840232
$851$	− 3.61389e11i	− 0.689059i
$852$	0	0
$853$	−6.34437e11	−1.19837	−0.599187	−	0.800609i	$-0.704509\pi$
$853$	−6.34437e11	−1.19837	−0.599187	+	0.800609i	$0.704509\pi$
$854$	1.84989e11i	0.347788i
$855$	0	0
$856$	−7.66394e11	−1.42744
$857$	− 7.56535e11i	− 1.40251i	−0.712911	−	0.701254i	$-0.752624\pi$
$857$	− 7.56535e11i	− 1.40251i	0.712911	−	0.701254i	$-0.247376\pi$
$858$	0	0
$859$	1.05633e12	1.94011	0.970055	−	0.242883i	$-0.0780932\pi$
$859$	1.05633e12	1.94011	0.970055	+	0.242883i	$0.0780932\pi$
$860$	3.73587e10i	0.0682963i
$861$	0	0
$862$	−5.19946e10	−0.0941736
$863$	− 4.06826e11i	− 0.733441i	−0.930331	−	0.366720i	$-0.880481\pi$
$863$	− 4.06826e11i	− 0.733441i	0.930331	−	0.366720i	$-0.119519\pi$
$864$	0	0
$865$	−1.72001e11	−0.307232
$866$	− 1.02212e12i	− 1.81732i
$867$	0	0
$868$	3.34815e9	0.00589828
$869$	− 5.57554e11i	− 0.977705i
$870$	0	0
$871$	−2.36279e11	−0.410537
$872$	4.37028e11i	0.755863i
$873$	0	0
$874$	6.90590e11	1.18352
$875$	1.73682e11i	0.296295i
$876$	0	0
$877$	−3.11948e11	−0.527332	−0.263666	−	0.964614i	$-0.584932\pi$
$877$	−3.11948e11	−0.527332	−0.263666	+	0.964614i	$0.584932\pi$
$878$	3.78398e11i	0.636753i
$879$	0	0
$880$	3.93851e11	0.656752
$881$	− 1.97724e11i	− 0.328213i	−0.986443	−	0.164107i	$-0.947526\pi$
$881$	− 1.97724e11i	− 0.328213i	0.986443	−	0.164107i	$-0.0524741\pi$
$882$	0	0
$883$	1.84908e11	0.304168	0.152084	−	0.988368i	$-0.451402\pi$
$883$	1.84908e11	0.304168	0.152084	+	0.988368i	$0.451402\pi$
$884$	5.53387e10i	0.0906191i
$885$	0	0
$886$	6.47003e11	1.04996
$887$	− 1.38337e11i	− 0.223482i	−0.993737	−	0.111741i	$-0.964357\pi$
$887$	− 1.38337e11i	− 0.223482i	0.993737	−	0.111741i	$-0.0356427\pi$
$888$	0	0
$889$	1.47184e10	0.0235643
$890$	− 5.19976e9i	− 0.00828750i
$891$	0	0
$892$	−3.79041e10	−0.0598724
$893$	− 3.32003e8i	0 0.000522080i
$894$	0	0
$895$	6.17033e11	0.961648
$896$	− 1.95773e11i	− 0.303753i
$897$	0	0
$898$	−9.30893e11	−1.43151
$899$	− 4.67280e11i	− 0.715383i
$900$	0	0
$901$	1.61332e12	2.44805
$902$	− 2.40444e11i	− 0.363235i
$903$	0	0
$904$	1.06898e11	0.160065
$905$	5.67479e11i	0.845971i
$906$	0	0
$907$	−6.41021e10	−0.0947203	−0.0473601	−	0.998878i	$-0.515081\pi$
$907$	−6.41021e10	−0.0947203	−0.0473601	+	0.998878i	$0.515081\pi$
$908$	7.31423e10i	0.107603i
$909$	0	0
$910$	−1.46892e11	−0.214206
$911$	6.83733e11i	0.992688i	0.868126	+	0.496344i	$0.165325\pi$
$911$	6.83733e11i	0.992688i	−0.868126	+	0.496344i	$0.834675\pi$
$912$	0	0
$913$	−5.98254e11	−0.860999
$914$	5.50224e11i	0.788414i
$915$	0	0
$916$	3.06066e9	0.00434744
$917$	− 9.22018e10i	− 0.130395i
$918$	0	0
$919$	1.30518e11	0.182982	0.0914908	−	0.995806i	$-0.470837\pi$
$919$	1.30518e11	0.182982	0.0914908	+	0.995806i	$0.470837\pi$
$920$	5.15921e11i	0.720165i
$921$	0	0
$922$	8.03987e11	1.11256
$923$	− 7.15605e11i	− 0.985977i
$924$	0	0
$925$	2.47643e11	0.338267
$926$	− 5.59300e11i	− 0.760679i
$927$	0	0
$928$	1.51988e11	0.204936
$929$	5.69894e11i	0.765123i	0.923930	+	0.382562i	$0.124958\pi$
$929$	5.69894e11i	0.765123i	−0.923930	+	0.382562i	$0.875042\pi$
$930$	0	0
$931$	7.33993e11	0.976996
$932$	− 6.73810e10i	− 0.0893046i
$933$	0	0
$934$	1.30434e12	1.71397
$935$	− 7.32946e11i	− 0.959015i
$936$	0	0
$937$	−4.64545e11	−0.602656	−0.301328	−	0.953521i	$-0.597430\pi$
$937$	−4.64545e11	−0.602656	−0.301328	+	0.953521i	$0.597430\pi$
$938$	8.55433e10i	0.110503i
$939$	0	0
$940$	−1.43489e7	−1.83784e−5 0
$941$	1.44687e12i	1.84532i	0.385617	+	0.922659i	$0.373989\pi$
$941$	1.44687e12i	1.84532i	−0.385617	+	0.922659i	$0.626011\pi$
$942$	0	0
$943$	3.32244e11	0.420155
$944$	− 7.26189e11i	− 0.914454i
$945$	0	0
$946$	−1.37276e12	−1.71408
$947$	7.23887e11i	0.900059i	0.893014	+	0.450030i	$0.148587\pi$
$947$	7.23887e11i	0.900059i	−0.893014	+	0.450030i	$0.851413\pi$
$948$	0	0
$949$	−7.68873e11	−0.947959
$950$	4.73230e11i	0.581002i
$951$	0	0
$952$	−3.46320e11	−0.421628
$953$	3.62461e11i	0.439430i	0.975564	+	0.219715i	$0.0705127\pi$
$953$	3.62461e11i	0.439430i	−0.975564	+	0.219715i	$0.929487\pi$
$954$	0	0
$955$	8.12358e11	0.976639
$956$	1.60681e9i	0.00192368i
$957$	0	0
$958$	−1.49151e12	−1.77077
$959$	− 2.23591e9i	− 0.00264350i
$960$	0	0
$961$	−7.28836e11	−0.854548
$962$	6.02582e11i	0.703583i
$963$	0	0
$964$	5.76034e10	0.0667021
$965$	− 3.33821e11i	− 0.384950i
$966$	0	0
$967$	1.37227e12	1.56939	0.784697	−	0.619879i	$-0.212818\pi$
$967$	1.37227e12	1.56939	0.784697	+	0.619879i	$0.212818\pi$
$968$	− 1.40997e11i	− 0.160587i
$969$	0	0
$970$	7.43601e11	0.839949
$971$	4.29812e11i	0.483505i	0.970338	+	0.241753i	$0.0777223\pi$
$971$	4.29812e11i	0.483505i	−0.970338	+	0.241753i	$0.922278\pi$
$972$	0	0
$973$	−5.57431e10	−0.0621928
$974$	− 4.13265e11i	− 0.459190i
$975$	0	0
$976$	−1.14279e12	−1.25941
$977$	4.14853e11i	0.455319i	0.973741	+	0.227659i	$0.0731073\pi$
$977$	4.14853e11i	0.455319i	−0.973741	+	0.227659i	$0.926893\pi$
$978$	0	0
$979$	9.90724e9	0.0107850
$980$	− 3.17225e10i	− 0.0343924i
$981$	0	0
$982$	−1.60157e12	−1.72226
$983$	− 8.03059e11i	− 0.860069i	−0.902812	−	0.430035i	$-0.858501\pi$
$983$	− 8.03059e11i	− 0.860069i	0.902812	−	0.430035i	$-0.141499\pi$
$984$	0	0
$985$	−9.34602e11	−0.992846
$986$	− 2.79617e12i	− 2.95839i
$987$	0	0
$988$	−5.97071e10	−0.0626612
$989$	− 1.89688e12i	− 1.98268i
$990$	0	0
$991$	6.15925e11	0.638606	0.319303	−	0.947653i	$-0.396551\pi$
$991$	6.15925e11	0.638606	0.319303	+	0.947653i	$0.396551\pi$
$992$	4.03502e10i	0.0416677i
$993$	0	0
$994$	−2.59080e11	−0.265393
$995$	− 1.25823e11i	− 0.128371i
$996$	0	0
$997$	8.73230e11	0.883788	0.441894	−	0.897067i	$-0.354307\pi$
$997$	8.73230e11	0.883788	0.441894	+	0.897067i	$0.354307\pi$
$998$	1.14249e12i	1.15167i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.9.b.c.26.1	✓	2
3.2	odd	2	inner	27.9.b.c.26.2	yes	2
4.3	odd	2		432.9.e.f.161.1		2
9.2	odd	6		81.9.d.c.53.2		4
9.4	even	3		81.9.d.c.26.2		4
9.5	odd	6		81.9.d.c.26.1		4
9.7	even	3		81.9.d.c.53.1		4
12.11	even	2		432.9.e.f.161.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.b.c.26.1	✓	2	1.1	even	1	trivial
27.9.b.c.26.2	yes	2	3.2	odd	2	inner
81.9.d.c.26.1		4	9.5	odd	6
81.9.d.c.26.2		4	9.4	even	3
81.9.d.c.53.1		4	9.7	even	3
81.9.d.c.53.2		4	9.2	odd	6
432.9.e.f.161.1		2	4.3	odd	2
432.9.e.f.161.2		2	12.11	even	2

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:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	27.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.4317i q^{2} -14.0000 q^{4} -427.224i q^{5} -679.000 q^{7} -3976.47i q^{8} +O(q^{10})\)	\(q-16.4317i q^{2} -14.0000 q^{4} -427.224i q^{5} -679.000 q^{7} -3976.47i q^{8} -7020.00 q^{10} -13375.4i q^{11} -30817.0 q^{13} +11157.1i q^{14} -68924.0 q^{16} +128266. i q^{17} -138391. q^{19} +5981.13i q^{20} -219780. q^{22} -303690. i q^{23} +208105. q^{25} +506375. i q^{26} +9506.00 q^{28} -1.32669e6i q^{29} +352214. q^{31} +114562. i q^{32} +2.10762e6 q^{34} +290085. i q^{35} +1.18999e6 q^{37} +2.27400e6i q^{38} -1.69884e6 q^{40} +1.09402e6i q^{41} +6.24609e6 q^{43} +187255. i q^{44} -4.99014e6 q^{46} +2399.02i q^{47} -5.30376e6 q^{49} -3.41951e6i q^{50} +431438. q^{52} -1.25779e7i q^{53} -5.71428e6 q^{55} +2.70002e6i q^{56} -2.17998e7 q^{58} +1.05361e7i q^{59} +1.65804e7 q^{61} -5.78747e6i q^{62} -1.57621e7 q^{64} +1.31657e7i q^{65} +7.66715e6 q^{67} -1.79572e6i q^{68} +4.76658e6 q^{70} +2.32211e7i q^{71} +2.49496e7 q^{73} -1.95535e7i q^{74} +1.93747e6 q^{76} +9.08189e6i q^{77} +4.16851e7 q^{79} +2.94460e7i q^{80} +1.79766e7 q^{82} -4.47280e7i q^{83} +5.47981e7 q^{85} -1.02634e8i q^{86} -5.31868e7 q^{88} +740707. i q^{89} +2.09247e7 q^{91} +4.25166e6i q^{92} +39420.0 q^{94} +5.91239e7i q^{95} -1.05926e8 q^{97} +8.71497e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 28 q^{4} - 1358 q^{7}+O(q^{10}) \) 2 * q - 28 * q^4 - 1358 * q^7	\( 2 q - 28 q^{4} - 1358 q^{7} - 14040 q^{10} - 61634 q^{13} - 137848 q^{16} - 276782 q^{19} - 439560 q^{22} + 416210 q^{25} + 19012 q^{28} + 704428 q^{31} + 4215240 q^{34} + 2379982 q^{37} - 3397680 q^{40} + 12492172 q^{43} - 9980280 q^{46} - 10607520 q^{49} + 862876 q^{52} - 11428560 q^{55} - 43599600 q^{58} + 33160798 q^{61} - 31524208 q^{64} + 15334306 q^{67} + 9533160 q^{70} + 49899262 q^{73} + 3874948 q^{76} + 83370178 q^{79} + 35953200 q^{82} + 109596240 q^{85} - 106373520 q^{88} + 41849486 q^{91} + 78840 q^{94} - 211852178 q^{97}+O(q^{100}) \) 2 * q - 28 * q^4 - 1358 * q^7 - 14040 * q^10 - 61634 * q^13 - 137848 * q^16 - 276782 * q^19 - 439560 * q^22 + 416210 * q^25 + 19012 * q^28 + 704428 * q^31 + 4215240 * q^34 + 2379982 * q^37 - 3397680 * q^40 + 12492172 * q^43 - 9980280 * q^46 - 10607520 * q^49 + 862876 * q^52 - 11428560 * q^55 - 43599600 * q^58 + 33160798 * q^61 - 31524208 * q^64 + 15334306 * q^67 + 9533160 * q^70 + 49899262 * q^73 + 3874948 * q^76 + 83370178 * q^79 + 35953200 * q^82 + 109596240 * q^85 - 106373520 * q^88 + 41849486 * q^91 + 78840 * q^94 - 211852178 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more