LMFDB - Embedded newform 27.9.b.d.26.5

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:	$6$
Coefficient field:	6.0.6171673600.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 40x^{3} + 225x^{2} + 150x + 50 $ x^6 - 2x^5 + 2x^4 + 40x^3 + 225x^2 + 150*x + 50
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3}\cdot 3^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.5
Root			$-0.356289 + 0.356289i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.9.b.d.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.9629i q^{2} +1.18662 q^{4} -230.735i q^{5} +3842.93 q^{7} +4105.44i q^{8} +O(q^{10})$	\(q+15.9629i q^{2} +1.18662 q^{4} -230.735i q^{5} +3842.93 q^{7} +4105.44i q^{8} +3683.19 q^{10} +1958.08i q^{11} +9938.39 q^{13} +61344.1i q^{14} -65230.8 q^{16} +112083. i q^{17} -75459.4 q^{19} -273.795i q^{20} -31256.6 q^{22} +345203. i q^{23} +337386. q^{25} +158645. i q^{26} +4560.10 q^{28} -1.34995e6i q^{29} +1.51691e6 q^{31} +9720.74i q^{32} -1.78917e6 q^{34} -886697. i q^{35} -1.77132e6 q^{37} -1.20455e6i q^{38} +947267. q^{40} -3.33829e6i q^{41} +71375.0 q^{43} +2323.50i q^{44} -5.51043e6 q^{46} +5.72845e6i q^{47} +9.00328e6 q^{49} +5.38566e6i q^{50} +11793.1 q^{52} -4.25497e6i q^{53} +451797. q^{55} +1.57769e7i q^{56} +2.15490e7 q^{58} -7.30106e6i q^{59} -1.95088e7 q^{61} +2.42142e7i q^{62} -1.68543e7 q^{64} -2.29313e6i q^{65} -8.03403e6 q^{67} +133001. i q^{68} +1.41542e7 q^{70} -3.86030e7i q^{71} -2.24543e7 q^{73} -2.82754e7i q^{74} -89541.9 q^{76} +7.52475e6i q^{77} +5.63558e6 q^{79} +1.50510e7i q^{80} +5.32887e7 q^{82} -7.31076e7i q^{83} +2.58615e7 q^{85} +1.13935e6i q^{86} -8.03877e6 q^{88} -4.03551e7i q^{89} +3.81925e7 q^{91} +409625. i q^{92} -9.14426e7 q^{94} +1.74111e7i q^{95} +2.17067e7 q^{97} +1.43718e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 786 q^{4} - 1698 q^{7}+O(q^{10}) $ 6 * q - 786 * q^4 - 1698 * q^7	$ 6 q - 786 q^{4} - 1698 q^{7} - 46278 q^{10} + 41844 q^{13} + 170466 q^{16} - 36384 q^{19} + 647298 q^{22} - 1646688 q^{25} - 838986 q^{28} + 2058474 q^{31} - 1320300 q^{34} - 9395880 q^{37} + 29560086 q^{40} - 2737284 q^{43} - 31514616 q^{46} + 28900656 q^{49} - 34081692 q^{52} - 26674542 q^{55} + 75244572 q^{58} - 40180776 q^{61} - 48541458 q^{64} + 111355284 q^{67} + 17727282 q^{70} + 12821718 q^{73} - 38623776 q^{76} + 21820404 q^{79} - 138785832 q^{82} - 83844396 q^{85} + 57552174 q^{88} + 156632964 q^{91} - 502013916 q^{94} + 53735106 q^{97}+O(q^{100}) $ 6 * q - 786 * q^4 - 1698 * q^7 - 46278 * q^10 + 41844 * q^13 + 170466 * q^16 - 36384 * q^19 + 647298 * q^22 - 1646688 * q^25 - 838986 * q^28 + 2058474 * q^31 - 1320300 * q^34 - 9395880 * q^37 + 29560086 * q^40 - 2737284 * q^43 - 31514616 * q^46 + 28900656 * q^49 - 34081692 * q^52 - 26674542 * q^55 + 75244572 * q^58 - 40180776 * q^61 - 48541458 * q^64 + 111355284 * q^67 + 17727282 * q^70 + 12821718 * q^73 - 38623776 * q^76 + 21820404 * q^79 - 138785832 * q^82 - 83844396 * q^85 + 57552174 * q^88 + 156632964 * q^91 - 502013916 * q^94 + 53735106 * 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 $

$2$	15.9629i	0.997680i	0.866694	+	0.498840i	$0.166240\pi$
$2$	15.9629i	0.997680i	−0.866694	+	0.498840i	$0.833760\pi$
$3$	0	0
$3$	0	0
$4$	1.18662	0.00463525
$5$	− 230.735i	− 0.369176i	−0.982816	−	0.184588i	$-0.940905\pi$
$5$	− 230.735i	− 0.369176i	0.982816	−	0.184588i	$-0.0590950\pi$
$6$	0	0
$7$	3842.93	1.60055	0.800276	−	0.599632i	$-0.204686\pi$
$7$	3842.93	1.60055	0.800276	+	0.599632i	$0.204686\pi$
$8$	4105.44i	1.00230i
$9$	0	0
$10$	3683.19	0.368319
$11$	1958.08i	0.133739i	0.997762	+	0.0668697i	$0.0213012\pi$
$11$	1958.08i	0.133739i	−0.997762	+	0.0668697i	$0.978699\pi$
$12$	0	0
$13$	9938.39	0.347971	0.173985	−	0.984748i	$-0.444335\pi$
$13$	9938.39	0.347971	0.173985	+	0.984748i	$0.444335\pi$
$14$	61344.1i	1.59684i
$15$	0	0
$16$	−65230.8	−0.995343
$17$	112083.i	1.34198i	0.741467	+	0.670989i	$0.234130\pi$
$17$	112083.i	1.34198i	−0.741467	+	0.670989i	$0.765870\pi$
$18$	0	0
$19$	−75459.4	−0.579027	−0.289514	−	0.957174i	$-0.593494\pi$
$19$	−75459.4	−0.579027	−0.289514	+	0.957174i	$0.593494\pi$
$20$	− 273.795i	− 0.00171122i
$21$	0	0
$22$	−31256.6	−0.133429
$23$	345203.i	1.23357i	0.787133	+	0.616783i	$0.211565\pi$
$23$	345203.i	1.23357i	−0.787133	+	0.616783i	$0.788435\pi$
$24$	0	0
$25$	337386.	0.863709
$26$	158645.i	0.347163i
$27$	0	0
$28$	4560.10	0.00741895
$29$	− 1.34995e6i	− 1.90864i	−0.298780	−	0.954322i	$-0.596580\pi$
$29$	− 1.34995e6i	− 1.90864i	0.298780	−	0.954322i	$-0.403420\pi$
$30$	0	0
$31$	1.51691e6	1.64253	0.821263	−	0.570549i	$-0.193270\pi$
$31$	1.51691e6	1.64253	0.821263	+	0.570549i	$0.193270\pi$
$32$	9720.74i	0.00927042i
$33$	0	0
$34$	−1.78917e6	−1.33886
$35$	− 886697.i	− 0.590885i
$36$	0	0
$37$	−1.77132e6	−0.945127	−0.472564	−	0.881297i	$-0.656671\pi$
$37$	−1.77132e6	−0.945127	−0.472564	+	0.881297i	$0.656671\pi$
$38$	− 1.20455e6i	− 0.577684i
$39$	0	0
$40$	947267.	0.370026
$41$	− 3.33829e6i	− 1.18138i	−0.806899	−	0.590689i	$-0.798856\pi$
$41$	− 3.33829e6i	− 1.18138i	0.806899	−	0.590689i	$-0.201144\pi$
$42$	0	0
$43$	71375.0	0.0208772	0.0104386	−	0.999946i	$-0.496677\pi$
$43$	71375.0	0.0208772	0.0104386	+	0.999946i	$0.496677\pi$
$44$	2323.50i	0 0.000619915i
$45$	0	0
$46$	−5.51043e6	−1.23070
$47$	5.72845e6i	1.17394i	0.809609	+	0.586970i	$0.199679\pi$
$47$	5.72845e6i	1.17394i	−0.809609	+	0.586970i	$0.800321\pi$
$48$	0	0
$49$	9.00328e6	1.56177
$50$	5.38566e6i	0.861705i
$51$	0	0
$52$	11793.1	0.00161293
$53$	− 4.25497e6i	− 0.539254i	−0.962965	−	0.269627i	$-0.913100\pi$
$53$	− 4.25497e6i	− 0.539254i	0.962965	−	0.269627i	$-0.0869003\pi$
$54$	0	0
$55$	451797.	0.0493733
$56$	1.57769e7i	1.60424i
$57$	0	0
$58$	2.15490e7	1.90422
$59$	− 7.30106e6i	− 0.602529i	−0.953541	−	0.301264i	$-0.902591\pi$
$59$	− 7.30106e6i	− 0.602529i	0.953541	−	0.301264i	$-0.0974087\pi$
$60$	0	0
$61$	−1.95088e7	−1.40900	−0.704500	−	0.709704i	$-0.748829\pi$
$61$	−1.95088e7	−1.40900	−0.704500	+	0.709704i	$0.748829\pi$
$62$	2.42142e7i	1.63872i
$63$	0	0
$64$	−1.68543e7	−1.00459
$65$	− 2.29313e6i	− 0.128462i
$66$	0	0
$67$	−8.03403e6	−0.398689	−0.199344	−	0.979929i	$-0.563881\pi$
$67$	−8.03403e6	−0.398689	−0.199344	+	0.979929i	$0.563881\pi$
$68$	133001.i	0.00622040i
$69$	0	0
$70$	1.41542e7	0.589514
$71$	− 3.86030e7i	− 1.51911i	−0.650445	−	0.759553i	$-0.725418\pi$
$71$	− 3.86030e7i	− 1.51911i	0.650445	−	0.759553i	$-0.274582\pi$
$72$	0	0
$73$	−2.24543e7	−0.790695	−0.395347	−	0.918532i	$-0.629376\pi$
$73$	−2.24543e7	−0.790695	−0.395347	+	0.918532i	$0.629376\pi$
$74$	− 2.82754e7i	− 0.942934i
$75$	0	0
$76$	−89541.9	−0.00268393
$77$	7.52475e6i	0.214057i
$78$	0	0
$79$	5.63558e6	0.144687	0.0723436	−	0.997380i	$-0.476952\pi$
$79$	5.63558e6	0.144687	0.0723436	+	0.997380i	$0.476952\pi$
$80$	1.50510e7i	0.367456i
$81$	0	0
$82$	5.32887e7	1.17864
$83$	− 7.31076e7i	− 1.54046i	−0.637767	−	0.770229i	$-0.720142\pi$
$83$	− 7.31076e7i	− 1.54046i	0.637767	−	0.770229i	$-0.279858\pi$
$84$	0	0
$85$	2.58615e7	0.495426
$86$	1.13935e6i	0.0208288i
$87$	0	0
$88$	−8.03877e6	−0.134048
$89$	− 4.03551e7i	− 0.643189i	−0.946877	−	0.321595i	$-0.895781\pi$
$89$	− 4.03551e7i	− 0.643189i	0.946877	−	0.321595i	$-0.104219\pi$
$90$	0	0
$91$	3.81925e7	0.556945
$92$	409625.i	0.00571789i
$93$	0	0
$94$	−9.14426e7	−1.17122
$95$	1.74111e7i	0.213763i
$96$	0	0
$97$	2.17067e7	0.245193	0.122596	−	0.992457i	$-0.460878\pi$
$97$	2.17067e7	0.245193	0.122596	+	0.992457i	$0.460878\pi$
$98$	1.43718e8i	1.55814i
$99$	0	0
$100$	400351.	0.00400351
$101$	9.10228e7i	0.874712i	0.899289	+	0.437356i	$0.144085\pi$
$101$	9.10228e7i	0.874712i	−0.899289	+	0.437356i	$0.855915\pi$
$102$	0	0
$103$	−1.51537e8	−1.34639	−0.673193	−	0.739467i	$-0.735078\pi$
$103$	−1.51537e8	−1.34639	−0.673193	+	0.739467i	$0.735078\pi$
$104$	4.08014e7i	0.348772i
$105$	0	0
$106$	6.79216e7	0.538002
$107$	− 4.22642e7i	− 0.322432i	−0.986919	−	0.161216i	$-0.948458\pi$
$107$	− 4.22642e7i	− 0.322432i	0.986919	−	0.161216i	$-0.0515415\pi$
$108$	0	0
$109$	5.28053e7	0.374086	0.187043	−	0.982352i	$-0.440110\pi$
$109$	5.28053e7	0.374086	0.187043	+	0.982352i	$0.440110\pi$
$110$	7.21198e6i	0.0492588i
$111$	0	0
$112$	−2.50677e8	−1.59310
$113$	8.85124e7i	0.542863i	0.962458	+	0.271432i	$0.0874971\pi$
$113$	8.85124e7i	0.542863i	−0.962458	+	0.271432i	$0.912503\pi$
$114$	0	0
$115$	7.96502e7	0.455403
$116$	− 1.60188e6i	− 0.00884703i
$117$	0	0
$118$	1.16546e8	0.601131
$119$	4.30728e8i	2.14791i
$120$	0	0
$121$	2.10525e8	0.982114
$122$	− 3.11416e8i	− 1.40573i
$123$	0	0
$124$	1.80000e6	0.00761352
$125$	− 1.67978e8i	− 0.688036i
$126$	0	0
$127$	3.72133e8	1.43048	0.715242	−	0.698876i	$-0.246316\pi$
$127$	3.72133e8	1.43048	0.715242	+	0.698876i	$0.246316\pi$
$128$	− 2.66554e8i	− 0.992991i
$129$	0	0
$130$	3.66050e7	0.128164
$131$	− 2.00203e8i	− 0.679806i	−0.940461	−	0.339903i	$-0.889606\pi$
$131$	− 2.00203e8i	− 0.679806i	0.940461	−	0.339903i	$-0.110394\pi$
$132$	0	0
$133$	−2.89985e8	−0.926763
$134$	− 1.28246e8i	− 0.397764i
$135$	0	0
$136$	−4.60151e8	−1.34507
$137$	− 1.15345e8i	− 0.327429i	−0.986508	−	0.163715i	$-0.947652\pi$
$137$	− 1.15345e8i	− 0.327429i	0.986508	−	0.163715i	$-0.0523476\pi$
$138$	0	0
$139$	−2.42122e8	−0.648597	−0.324299	−	0.945955i	$-0.605128\pi$
$139$	−2.42122e8	−0.648597	−0.324299	+	0.945955i	$0.605128\pi$
$140$	− 1.05217e6i	− 0.00273890i
$141$	0	0
$142$	6.16216e8	1.51558
$143$	1.94601e7i	0.0465374i
$144$	0	0
$145$	−3.11480e8	−0.704625
$146$	− 3.58436e8i	− 0.788860i
$147$	0	0
$148$	−2.10189e6	−0.00438090
$149$	3.76837e8i	0.764554i	0.924048	+	0.382277i	$0.124860\pi$
$149$	3.76837e8i	0.764554i	−0.924048	+	0.382277i	$0.875140\pi$
$150$	0	0
$151$	−6.94381e8	−1.33564	−0.667821	−	0.744322i	$-0.732773\pi$
$151$	−6.94381e8	−1.33564	−0.667821	+	0.744322i	$0.732773\pi$
$152$	− 3.09794e8i	− 0.580361i
$153$	0	0
$154$	−1.20117e8	−0.213560
$155$	− 3.50003e8i	− 0.606381i
$156$	0	0
$157$	7.97666e8	1.31287	0.656436	−	0.754382i	$-0.272063\pi$
$157$	7.97666e8	1.31287	0.656436	+	0.754382i	$0.272063\pi$
$158$	8.99601e7i	0.144352i
$159$	0	0
$160$	2.24291e6	0.00342241
$161$	1.32659e9i	1.97439i
$162$	0	0
$163$	−1.24362e8	−0.176173	−0.0880864	−	0.996113i	$-0.528075\pi$
$163$	−1.24362e8	−0.176173	−0.0880864	+	0.996113i	$0.528075\pi$
$164$	− 3.96129e6i	− 0.00547598i
$165$	0	0
$166$	1.16701e9	1.53688
$167$	8.56253e8i	1.10087i	0.834878	+	0.550435i	$0.185538\pi$
$167$	8.56253e8i	1.10087i	−0.834878	+	0.550435i	$0.814462\pi$
$168$	0	0
$169$	−7.16959e8	−0.878917
$170$	4.12824e8i	0.494276i
$171$	0	0
$172$	84695.2	9.67709e−5 0
$173$	6.11101e8i	0.682227i	0.940022	+	0.341113i	$0.110804\pi$
$173$	6.11101e8i	0.682227i	−0.940022	+	0.341113i	$0.889196\pi$
$174$	0	0
$175$	1.29655e9	1.38241
$176$	− 1.27727e8i	− 0.133117i
$177$	0	0
$178$	6.44184e8	0.641697
$179$	− 1.08189e9i	− 1.05383i	−0.849917	−	0.526916i	$-0.823348\pi$
$179$	− 1.08189e9i	− 1.05383i	0.849917	−	0.526916i	$-0.176652\pi$
$180$	0	0
$181$	−9.65349e8	−0.899436	−0.449718	−	0.893171i	$-0.648476\pi$
$181$	−9.65349e8	−0.899436	−0.449718	+	0.893171i	$0.648476\pi$
$182$	6.09662e8i	0.555653i
$183$	0	0
$184$	−1.41721e9	−1.23641
$185$	4.08705e8i	0.348918i
$186$	0	0
$187$	−2.19468e8	−0.179475
$188$	6.79751e6i	0.00544150i
$189$	0	0
$190$	−2.77931e8	−0.213267
$191$	− 1.17971e9i	− 0.886421i	−0.896417	−	0.443211i	$-0.853839\pi$
$191$	− 1.17971e9i	− 0.886421i	0.896417	−	0.443211i	$-0.146161\pi$
$192$	0	0
$193$	4.49079e8	0.323664	0.161832	−	0.986818i	$-0.448260\pi$
$193$	4.49079e8	0.323664	0.161832	+	0.986818i	$0.448260\pi$
$194$	3.46502e8i	0.244624i
$195$	0	0
$196$	1.06835e7	0.00723918
$197$	4.57782e8i	0.303944i	0.988385	+	0.151972i	$0.0485623\pi$
$197$	4.57782e8i	0.303944i	−0.988385	+	0.151972i	$0.951438\pi$
$198$	0	0
$199$	−4.56110e8	−0.290842	−0.145421	−	0.989370i	$-0.546454\pi$
$199$	−4.56110e8	−0.290842	−0.145421	+	0.989370i	$0.546454\pi$
$200$	1.38512e9i	0.865700i
$201$	0	0
$202$	−1.45299e9	−0.872682
$203$	− 5.18775e9i	− 3.05488i
$204$	0	0
$205$	−7.70260e8	−0.436136
$206$	− 2.41896e9i	− 1.34326i
$207$	0	0
$208$	−6.48289e8	−0.346350
$209$	− 1.47755e8i	− 0.0774387i
$210$	0	0
$211$	1.61104e9	0.812789	0.406394	−	0.913698i	$-0.366786\pi$
$211$	1.61104e9	0.812789	0.406394	+	0.913698i	$0.366786\pi$
$212$	− 5.04905e6i	− 0.00249957i
$213$	0	0
$214$	6.74658e8	0.321684
$215$	− 1.64687e7i	− 0.00770735i
$216$	0	0
$217$	5.82936e9	2.62895
$218$	8.42924e8i	0.373218i
$219$	0	0
$220$	536113.	0.000228858 0
$221$	1.11393e9i	0.466969i
$222$	0	0
$223$	−9.14517e8	−0.369805	−0.184902	−	0.982757i	$-0.559197\pi$
$223$	−9.14517e8	−0.369805	−0.184902	+	0.982757i	$0.559197\pi$
$224$	3.73561e7i	0.0148378i
$225$	0	0
$226$	−1.41291e9	−0.541604
$227$	− 5.56882e8i	− 0.209730i	−0.994486	−	0.104865i	$-0.966559\pi$
$227$	− 5.56882e8i	− 0.209730i	0.994486	−	0.104865i	$-0.0334410\pi$
$228$	0	0
$229$	−2.99605e9	−1.08945	−0.544725	−	0.838615i	$-0.683366\pi$
$229$	−2.99605e9	−1.08945	−0.544725	+	0.838615i	$0.683366\pi$
$230$	1.27145e9i	0.454346i
$231$	0	0
$232$	5.54213e9	1.91304
$233$	− 3.57490e8i	− 0.121294i	−0.998159	−	0.0606470i	$-0.980684\pi$
$233$	− 3.57490e8i	− 0.121294i	0.998159	−	0.0606470i	$-0.0193164\pi$
$234$	0	0
$235$	1.32175e9	0.433390
$236$	− 8.66361e6i	− 0.00279287i
$237$	0	0
$238$	−6.87566e9	−2.14292
$239$	9.02108e8i	0.276482i	0.990399	+	0.138241i	$0.0441449\pi$
$239$	9.02108e8i	0.276482i	−0.990399	+	0.138241i	$0.955855\pi$
$240$	0	0
$241$	−1.37470e9	−0.407511	−0.203756	−	0.979022i	$-0.565315\pi$
$241$	−1.37470e9	−0.407511	−0.203756	+	0.979022i	$0.565315\pi$
$242$	3.36058e9i	0.979835i
$243$	0	0
$244$	−2.31496e7	−0.00653106
$245$	− 2.07737e9i	− 0.576566i
$246$	0	0
$247$	−7.49945e8	−0.201484
$248$	6.22757e9i	1.64631i
$249$	0	0
$250$	2.68140e9	0.686440
$251$	1.89972e9i	0.478623i	0.970943	+	0.239312i	$0.0769218\pi$
$251$	1.89972e9i	0.478623i	−0.970943	+	0.239312i	$0.923078\pi$
$252$	0	0
$253$	−6.75934e8	−0.164976
$254$	5.94031e9i	1.42717i
$255$	0	0
$256$	−5.97234e7	−0.0139054
$257$	8.24063e8i	0.188898i	0.995530	+	0.0944491i	$0.0301090\pi$
$257$	8.24063e8i	0.188898i	−0.995530	+	0.0944491i	$0.969891\pi$
$258$	0	0
$259$	−6.80706e9	−1.51273
$260$	− 2.72108e6i	0 0.000595454i
$261$	0	0
$262$	3.19581e9	0.678229
$263$	− 8.25047e9i	− 1.72447i	−0.506508	−	0.862235i	$-0.669064\pi$
$263$	− 8.25047e9i	− 1.72447i	0.506508	−	0.862235i	$-0.330936\pi$
$264$	0	0
$265$	−9.81769e8	−0.199079
$266$	− 4.62899e9i	− 0.924613i
$267$	0	0
$268$	−9.53336e6	−0.00184802
$269$	6.97137e9i	1.33140i	0.746218	+	0.665701i	$0.231867\pi$
$269$	6.97137e9i	1.33140i	−0.746218	+	0.665701i	$0.768133\pi$
$270$	0	0
$271$	1.17379e9	0.217627	0.108814	−	0.994062i	$-0.465295\pi$
$271$	1.17379e9	0.217627	0.108814	+	0.994062i	$0.465295\pi$
$272$	− 7.31129e9i	− 1.33573i
$273$	0	0
$274$	1.84124e9	0.326669
$275$	6.60629e8i	0.115512i
$276$	0	0
$277$	5.94761e9	1.01024	0.505119	−	0.863050i	$-0.331449\pi$
$277$	5.94761e9	1.01024	0.505119	+	0.863050i	$0.331449\pi$
$278$	− 3.86496e9i	− 0.647092i
$279$	0	0
$280$	3.64028e9	0.592246
$281$	9.08369e9i	1.45693i	0.685086	+	0.728463i	$0.259765\pi$
$281$	9.08369e9i	1.45693i	−0.685086	+	0.728463i	$0.740235\pi$
$282$	0	0
$283$	6.73752e9	1.05040	0.525199	−	0.850979i	$-0.323991\pi$
$283$	6.73752e9	1.05040	0.525199	+	0.850979i	$0.323991\pi$
$284$	− 4.58073e7i	− 0.00704143i
$285$	0	0
$286$	−3.10640e8	−0.0464294
$287$	− 1.28288e10i	− 1.89086i
$288$	0	0
$289$	−5.58692e9	−0.800905
$290$	− 4.97211e9i	− 0.702990i
$291$	0	0
$292$	−2.66448e7	−0.00366507
$293$	5.43554e8i	0.0737518i	0.999320	+	0.0368759i	$0.0117406\pi$
$293$	5.43554e8i	0.0737518i	−0.999320	+	0.0368759i	$0.988259\pi$
$294$	0	0
$295$	−1.68461e9	−0.222439
$296$	− 7.27205e9i	− 0.947305i
$297$	0	0
$298$	−6.01540e9	−0.762780
$299$	3.43076e9i	0.429245i
$300$	0	0
$301$	2.74289e8	0.0334150
$302$	− 1.10843e10i	− 1.33254i
$303$	0	0
$304$	4.92228e9	0.576331
$305$	4.50135e9i	0.520168i
$306$	0	0
$307$	−1.04267e10	−1.17380	−0.586899	−	0.809660i	$-0.699651\pi$
$307$	−1.04267e10	−1.17380	−0.586899	+	0.809660i	$0.699651\pi$
$308$	8.92904e6i	0 0.000992206i
$309$	0	0
$310$	5.58706e9	0.604974
$311$	− 1.24157e10i	− 1.32718i	−0.748095	−	0.663591i	$-0.769031\pi$
$311$	− 1.24157e10i	− 1.32718i	0.748095	−	0.663591i	$-0.230969\pi$
$312$	0	0
$313$	−9.91227e9	−1.03275	−0.516376	−	0.856362i	$-0.672719\pi$
$313$	−9.91227e9	−1.03275	−0.516376	+	0.856362i	$0.672719\pi$
$314$	1.27330e10i	1.30983i
$315$	0	0
$316$	6.68731e6	0.000670661 0
$317$	1.92732e10i	1.90861i	0.298833	+	0.954305i	$0.403403\pi$
$317$	1.92732e10i	1.90861i	−0.298833	+	0.954305i	$0.596597\pi$
$318$	0	0
$319$	2.64330e9	0.255261
$320$	3.88886e9i	0.370871i
$321$	0	0
$322$	−2.11762e10	−1.96981
$323$	− 8.45774e9i	− 0.777042i
$324$	0	0
$325$	3.35308e9	0.300545
$326$	− 1.98518e9i	− 0.175764i
$327$	0	0
$328$	1.37052e10	1.18410
$329$	2.20140e10i	1.87895i
$330$	0	0
$331$	1.04247e10	0.868467	0.434233	−	0.900800i	$-0.357019\pi$
$331$	1.04247e10	0.868467	0.434233	+	0.900800i	$0.357019\pi$
$332$	− 8.67511e7i	− 0.00714040i
$333$	0	0
$334$	−1.36683e10	−1.09832
$335$	1.85373e9i	0.147186i
$336$	0	0
$337$	2.24880e10	1.74354	0.871769	−	0.489918i	$-0.162973\pi$
$337$	2.24880e10	1.74354	0.871769	+	0.489918i	$0.162973\pi$
$338$	− 1.14447e10i	− 0.876877i
$339$	0	0
$340$	3.06879e7	0.00229642
$341$	2.97022e9i	0.219670i
$342$	0	0
$343$	1.24452e10	0.899138
$344$	2.93026e8i	0.0209253i
$345$	0	0
$346$	−9.75493e9	−0.680644
$347$	1.79591e10i	1.23870i	0.785114	+	0.619351i	$0.212604\pi$
$347$	1.79591e10i	1.23870i	−0.785114	+	0.619351i	$0.787396\pi$
$348$	0	0
$349$	6.97654e9	0.470260	0.235130	−	0.971964i	$-0.424448\pi$
$349$	6.97654e9	0.470260	0.235130	+	0.971964i	$0.424448\pi$
$350$	2.06967e10i	1.37920i
$351$	0	0
$352$	−1.90340e7	−0.00123982
$353$	− 2.68255e9i	− 0.172762i	−0.996262	−	0.0863811i	$-0.972470\pi$
$353$	− 2.68255e9i	− 0.172762i	0.996262	−	0.0863811i	$-0.0275303\pi$
$354$	0	0
$355$	−8.90706e9	−0.560817
$356$	− 4.78864e7i	− 0.00298134i
$357$	0	0
$358$	1.72701e10	1.05139
$359$	7.90093e9i	0.475664i	0.971306	+	0.237832i	$0.0764368\pi$
$359$	7.90093e9i	0.475664i	−0.971306	+	0.237832i	$0.923563\pi$
$360$	0	0
$361$	−1.12894e10	−0.664728
$362$	− 1.54098e10i	− 0.897349i
$363$	0	0
$364$	4.53201e7	0.00258158
$365$	5.18100e9i	0.291905i
$366$	0	0
$367$	−9.29479e8	−0.0512360	−0.0256180	−	0.999672i	$-0.508155\pi$
$367$	−9.29479e8	−0.0512360	−0.0256180	+	0.999672i	$0.508155\pi$
$368$	− 2.25178e10i	− 1.22782i
$369$	0	0
$370$	−6.52411e9	−0.348108
$371$	− 1.63515e10i	− 0.863103i
$372$	0	0
$373$	−2.50705e10	−1.29518	−0.647588	−	0.761991i	$-0.724222\pi$
$373$	−2.50705e10	−1.29518	−0.647588	+	0.761991i	$0.724222\pi$
$374$	− 3.50334e9i	− 0.179059i
$375$	0	0
$376$	−2.35178e10	−1.17664
$377$	− 1.34163e10i	− 0.664152i
$378$	0	0
$379$	3.13100e10	1.51749	0.758745	−	0.651387i	$-0.225813\pi$
$379$	3.13100e10	1.51749	0.758745	+	0.651387i	$0.225813\pi$
$380$	2.06604e7i	0 0.000990843i
$381$	0	0
$382$	1.88315e10	0.884364
$383$	− 2.83514e10i	− 1.31759i	−0.752324	−	0.658793i	$-0.771067\pi$
$383$	− 2.83514e10i	− 1.31759i	0.752324	−	0.658793i	$-0.228933\pi$
$384$	0	0
$385$	1.73622e9	0.0790246
$386$	7.16860e9i	0.322913i
$387$	0	0
$388$	2.57577e7	0.00113653
$389$	− 1.70542e10i	− 0.744788i	−0.928075	−	0.372394i	$-0.878537\pi$
$389$	− 1.70542e10i	− 0.744788i	0.928075	−	0.372394i	$-0.121463\pi$
$390$	0	0
$391$	−3.86915e10	−1.65542
$392$	3.69624e10i	1.56537i
$393$	0	0
$394$	−7.30751e9	−0.303239
$395$	− 1.30032e9i	− 0.0534150i
$396$	0	0
$397$	−3.13771e10	−1.26314	−0.631569	−	0.775320i	$-0.717589\pi$
$397$	−3.13771e10	−1.26314	−0.631569	+	0.775320i	$0.717589\pi$
$398$	− 7.28082e9i	− 0.290167i
$399$	0	0
$400$	−2.20080e10	−0.859687
$401$	− 2.88603e10i	− 1.11615i	−0.829790	−	0.558075i	$-0.811540\pi$
$401$	− 2.88603e10i	− 1.11615i	0.829790	−	0.558075i	$-0.188460\pi$
$402$	0	0
$403$	1.50756e10	0.571551
$404$	1.08010e8i	0.00405450i
$405$	0	0
$406$	8.28114e10	3.04780
$407$	− 3.46839e9i	− 0.126401i
$408$	0	0
$409$	2.36107e10	0.843754	0.421877	−	0.906653i	$-0.361371\pi$
$409$	2.36107e10	0.843754	0.421877	+	0.906653i	$0.361371\pi$
$410$	− 1.22956e10i	− 0.435124i
$411$	0	0
$412$	−1.79817e8	−0.00624083
$413$	− 2.80574e10i	− 0.964379i
$414$	0	0
$415$	−1.68685e10	−0.568700
$416$	9.66085e7i	0.00322583i
$417$	0	0
$418$	2.35860e9	0.0772590
$419$	2.44970e10i	0.794797i	0.917646	+	0.397398i	$0.130087\pi$
$419$	2.44970e10i	0.794797i	−0.917646	+	0.397398i	$0.869913\pi$
$420$	0	0
$421$	6.02726e10	1.91863	0.959316	−	0.282335i	$-0.0911089\pi$
$421$	6.02726e10	1.91863	0.959316	+	0.282335i	$0.0911089\pi$
$422$	2.57169e10i	0.810903i
$423$	0	0
$424$	1.74685e10	0.540496
$425$	3.78154e10i	1.15908i
$426$	0	0
$427$	−7.49708e10	−2.25518
$428$	− 5.01517e7i	− 0.00149455i
$429$	0	0
$430$	2.62888e8	0.00768947
$431$	7.00871e9i	0.203109i	0.994830	+	0.101554i	$0.0323816\pi$
$431$	7.00871e9i	0.203109i	−0.994830	+	0.101554i	$0.967618\pi$
$432$	0	0
$433$	5.91840e7	0.00168365	0.000841827	−	1.00000i	$-0.499732\pi$
$433$	5.91840e7	0.00168365	0.000841827		1.00000i	$0.499732\pi$
$434$	9.30534e10i	2.62285i
$435$	0	0
$436$	6.26599e7	0.00173398
$437$	− 2.60488e10i	− 0.714269i
$438$	0	0
$439$	3.48999e10	0.939649	0.469824	−	0.882760i	$-0.344317\pi$
$439$	3.48999e10	0.939649	0.469824	+	0.882760i	$0.344317\pi$
$440$	1.85482e9i	0.0494871i
$441$	0	0
$442$	−1.77815e10	−0.465885
$443$	− 9.06460e9i	− 0.235361i	−0.993052	−	0.117680i	$-0.962454\pi$
$443$	− 9.06460e9i	− 0.235361i	0.993052	−	0.117680i	$-0.0375458\pi$
$444$	0	0
$445$	−9.31134e9	−0.237450
$446$	− 1.45983e10i	− 0.368947i
$447$	0	0
$448$	−6.47697e10	−1.60790
$449$	5.91282e10i	1.45482i	0.686204	+	0.727410i	$0.259276\pi$
$449$	5.91282e10i	1.45482i	−0.686204	+	0.727410i	$0.740724\pi$
$450$	0	0
$451$	6.53664e9	0.157997
$452$	1.05031e8i	0.00251631i
$453$	0	0
$454$	8.88944e9	0.209243
$455$	− 8.81233e9i	− 0.205611i
$456$	0	0
$457$	−5.92814e10	−1.35911	−0.679553	−	0.733626i	$-0.737826\pi$
$457$	−5.92814e10	−1.35911	−0.679553	+	0.733626i	$0.737826\pi$
$458$	− 4.78256e10i	− 1.08692i
$459$	0	0
$460$	9.45148e7	0.00211090
$461$	− 1.59270e10i	− 0.352638i	−0.984333	−	0.176319i	$-0.943581\pi$
$461$	− 1.59270e10i	− 0.352638i	0.984333	−	0.176319i	$-0.0564190\pi$
$462$	0	0
$463$	4.91696e10	1.06997	0.534987	−	0.844860i	$-0.320317\pi$
$463$	4.91696e10	1.06997	0.534987	+	0.844860i	$0.320317\pi$
$464$	8.80582e10i	1.89976i
$465$	0	0
$466$	5.70656e9	0.121013
$467$	− 3.86411e10i	− 0.812423i	−0.913779	−	0.406211i	$-0.866850\pi$
$467$	− 3.86411e10i	− 0.812423i	0.913779	−	0.406211i	$-0.133150\pi$
$468$	0	0
$469$	−3.08742e10	−0.638122
$470$	2.10990e10i	0.432384i
$471$	0	0
$472$	2.99740e10	0.603917
$473$	1.39758e8i	0.00279210i
$474$	0	0
$475$	−2.54590e10	−0.500111
$476$	5.11112e8i	0.00995607i
$477$	0	0
$478$	−1.44002e10	−0.275841
$479$	1.14362e10i	0.217239i	0.994083	+	0.108620i	$0.0346431\pi$
$479$	1.14362e10i	0.217239i	−0.994083	+	0.108620i	$0.965357\pi$
$480$	0	0
$481$	−1.76041e10	−0.328877
$482$	− 2.19442e10i	− 0.406566i
$483$	0	0
$484$	2.49814e8	0.00455234
$485$	− 5.00850e9i	− 0.0905192i
$486$	0	0
$487$	4.05025e10	0.720055	0.360027	−	0.932942i	$-0.382767\pi$
$487$	4.05025e10	0.720055	0.360027	+	0.932942i	$0.382767\pi$
$488$	− 8.00921e10i	− 1.41225i
$489$	0	0
$490$	3.31608e10	0.575229
$491$	− 6.85589e10i	− 1.17961i	−0.807546	−	0.589805i	$-0.799205\pi$
$491$	− 6.85589e10i	− 1.17961i	0.807546	−	0.589805i	$-0.200795\pi$
$492$	0	0
$493$	1.51307e11	2.56136
$494$	− 1.19713e10i	− 0.201017i
$495$	0	0
$496$	−9.89491e10	−1.63488
$497$	− 1.48349e11i	− 2.43141i
$498$	0	0
$499$	8.48821e10	1.36903	0.684516	−	0.728998i	$-0.260013\pi$
$499$	8.48821e10	1.36903	0.684516	+	0.728998i	$0.260013\pi$
$500$	− 1.99326e8i	− 0.00318922i
$501$	0	0
$502$	−3.03249e10	−0.477513
$503$	1.00206e11i	1.56538i	0.622411	+	0.782691i	$0.286153\pi$
$503$	1.00206e11i	1.56538i	−0.622411	+	0.782691i	$0.713847\pi$
$504$	0	0
$505$	2.10021e10	0.322922
$506$	− 1.07898e10i	− 0.164594i
$507$	0	0
$508$	4.41582e8	0.00663065
$509$	5.73518e10i	0.854428i	0.904151	+	0.427214i	$0.140505\pi$
$509$	5.73518e10i	0.854428i	−0.904151	+	0.427214i	$0.859495\pi$
$510$	0	0
$511$	−8.62904e10	−1.26555
$512$	− 6.91912e10i	− 1.00686i
$513$	0	0
$514$	−1.31544e10	−0.188460
$515$	3.49648e10i	0.497053i
$516$	0	0
$517$	−1.12168e10	−0.157002
$518$	− 1.08660e11i	− 1.50922i
$519$	0	0
$520$	9.41431e9	0.128758
$521$	9.71593e9i	0.131866i	0.997824	+	0.0659331i	$0.0210024\pi$
$521$	9.71593e9i	0.131866i	−0.997824	+	0.0659331i	$0.978998\pi$
$522$	0	0
$523$	−3.77383e10	−0.504400	−0.252200	−	0.967675i	$-0.581154\pi$
$523$	−3.77383e10	−0.504400	−0.252200	+	0.967675i	$0.581154\pi$
$524$	− 2.37565e8i	− 0.00315107i
$525$	0	0
$526$	1.31701e11	1.72047
$527$	1.70020e11i	2.20423i
$528$	0	0
$529$	−4.08538e10	−0.521687
$530$	− 1.56719e10i	− 0.198617i
$531$	0	0
$532$	−3.44103e8	−0.00429578
$533$	− 3.31772e10i	− 0.411085i
$534$	0	0
$535$	−9.75182e9	−0.119034
$536$	− 3.29832e10i	− 0.399608i
$537$	0	0
$538$	−1.11283e11	−1.32831
$539$	1.76291e10i	0.208870i
$540$	0	0
$541$	−8.54675e10	−0.997728	−0.498864	−	0.866680i	$-0.666249\pi$
$541$	−8.54675e10	−0.997728	−0.498864	+	0.866680i	$0.666249\pi$
$542$	1.87371e10i	0.217122i
$543$	0	0
$544$	−1.08953e9	−0.0124407
$545$	− 1.21840e10i	− 0.138103i
$546$	0	0
$547$	−3.29466e9	−0.0368011	−0.0184006	−	0.999831i	$-0.505857\pi$
$547$	−3.29466e9	−0.0368011	−0.0184006	+	0.999831i	$0.505857\pi$
$548$	− 1.36871e8i	− 0.00151771i
$549$	0	0
$550$	−1.05455e10	−0.115244
$551$	1.01866e11i	1.10516i
$552$	0	0
$553$	2.16571e10	0.231580
$554$	9.49410e10i	1.00789i
$555$	0	0
$556$	−2.87308e8	−0.00300641
$557$	− 3.99074e10i	− 0.414603i	−0.978277	−	0.207302i	$-0.933532\pi$
$557$	− 3.99074e10i	− 0.414603i	0.978277	−	0.207302i	$-0.0664682\pi$
$558$	0	0
$559$	7.09352e8	0.00726465
$560$	5.78399e10i	0.588133i
$561$	0	0
$562$	−1.45002e11	−1.45354
$563$	3.53016e10i	0.351367i	0.984447	+	0.175683i	$0.0562135\pi$
$563$	3.53016e10i	0.351367i	−0.984447	+	0.175683i	$0.943787\pi$
$564$	0	0
$565$	2.04229e10	0.200412
$566$	1.07550e11i	1.04796i
$567$	0	0
$568$	1.58482e11	1.52261
$569$	1.49795e11i	1.42905i	0.699607	+	0.714527i	$0.253358\pi$
$569$	1.49795e11i	1.42905i	−0.699607	+	0.714527i	$0.746642\pi$
$570$	0	0
$571$	6.82354e10	0.641897	0.320949	−	0.947097i	$-0.395998\pi$
$571$	6.82354e10	0.641897	0.320949	+	0.947097i	$0.395998\pi$
$572$	2.30919e7i	0 0.000215712i
$573$	0	0
$574$	2.04785e11	1.88647
$575$	1.16467e11i	1.06544i
$576$	0	0
$577$	−7.94055e9	−0.0716386	−0.0358193	−	0.999358i	$-0.511404\pi$
$577$	−7.94055e9	−0.0716386	−0.0358193	+	0.999358i	$0.511404\pi$
$578$	− 8.91833e10i	− 0.799047i
$579$	0	0
$580$	−3.69609e8	−0.00326611
$581$	− 2.80947e11i	− 2.46558i
$582$	0	0
$583$	8.33157e9	0.0721194
$584$	− 9.21849e10i	− 0.792517i
$585$	0	0
$586$	−8.67669e9	−0.0735807
$587$	1.83541e11i	1.54589i	0.634470	+	0.772947i	$0.281218\pi$
$587$	1.83541e11i	1.54589i	−0.634470	+	0.772947i	$0.718782\pi$
$588$	0	0
$589$	−1.14465e11	−0.951067
$590$	− 2.68912e10i	− 0.221923i
$591$	0	0
$592$	1.15545e11	0.940726
$593$	− 1.59741e11i	− 1.29180i	−0.763420	−	0.645902i	$-0.776481\pi$
$593$	− 1.59741e11i	− 1.29180i	0.763420	−	0.645902i	$-0.223519\pi$
$594$	0	0
$595$	9.93839e10	0.792954
$596$	4.47163e8i	0.00354390i
$597$	0	0
$598$	−5.47647e10	−0.428249
$599$	4.21319e10i	0.327268i	0.986521	+	0.163634i	$0.0523216\pi$
$599$	4.21319e10i	0.327268i	−0.986521	+	0.163634i	$0.947678\pi$
$600$	0	0
$601$	−1.64694e10	−0.126235	−0.0631174	−	0.998006i	$-0.520104\pi$
$601$	−1.64694e10	−0.126235	−0.0631174	+	0.998006i	$0.520104\pi$
$602$	4.37844e9i	0.0333375i
$603$	0	0
$604$	−8.23968e8	−0.00619103
$605$	− 4.85754e10i	− 0.362572i
$606$	0	0
$607$	−1.06560e11	−0.784946	−0.392473	−	0.919764i	$-0.628380\pi$
$607$	−1.06560e11	−0.784946	−0.392473	+	0.919764i	$0.628380\pi$
$608$	− 7.33521e8i	− 0.00536782i
$609$	0	0
$610$	−7.18546e10	−0.518961
$611$	5.69316e10i	0.408496i
$612$	0	0
$613$	−5.59032e10	−0.395908	−0.197954	−	0.980211i	$-0.563430\pi$
$613$	−5.59032e10	−0.395908	−0.197954	+	0.980211i	$0.563430\pi$
$614$	− 1.66440e11i	− 1.17107i
$615$	0	0
$616$	−3.08924e10	−0.214550
$617$	1.19750e11i	0.826296i	0.910664	+	0.413148i	$0.135571\pi$
$617$	1.19750e11i	0.826296i	−0.910664	+	0.413148i	$0.864429\pi$
$618$	0	0
$619$	−2.29562e11	−1.56364	−0.781820	−	0.623504i	$-0.785709\pi$
$619$	−2.29562e11	−1.56364	−0.781820	+	0.623504i	$0.785709\pi$
$620$	− 4.15322e8i	− 0.00281072i
$621$	0	0
$622$	1.98191e11	1.32410
$623$	− 1.55082e11i	− 1.02946i
$624$	0	0
$625$	9.30333e10	0.609703
$626$	− 1.58228e11i	− 1.03036i
$627$	0	0
$628$	9.46528e8	0.00608548
$629$	− 1.98536e11i	− 1.26834i
$630$	0	0
$631$	1.10553e11	0.697353	0.348677	−	0.937243i	$-0.386631\pi$
$631$	1.10553e11	0.697353	0.348677	+	0.937243i	$0.386631\pi$
$632$	2.31365e10i	0.145021i
$633$	0	0
$634$	−3.07656e11	−1.90418
$635$	− 8.58640e10i	− 0.528100i
$636$	0	0
$637$	8.94781e10	0.543449
$638$	4.21947e10i	0.254669i
$639$	0	0
$640$	−6.15033e10	−0.366588
$641$	− 8.74222e10i	− 0.517833i	−0.965900	−	0.258917i	$-0.916635\pi$
$641$	− 8.74222e10i	− 0.517833i	0.965900	−	0.258917i	$-0.0833655\pi$
$642$	0	0
$643$	−1.61836e11	−0.946742	−0.473371	−	0.880863i	$-0.656963\pi$
$643$	−1.61836e11	−0.946742	−0.473371	+	0.880863i	$0.656963\pi$
$644$	1.57416e9i	0.00915178i
$645$	0	0
$646$	1.35010e11	0.775239
$647$	− 1.79280e11i	− 1.02309i	−0.859256	−	0.511546i	$-0.829073\pi$
$647$	− 1.79280e11i	− 1.02309i	0.859256	−	0.511546i	$-0.170927\pi$
$648$	0	0
$649$	1.42960e10	0.0805818
$650$	5.35248e10i	0.299848i
$651$	0	0
$652$	−1.47571e8	−0.000816605 0
$653$	1.95545e11i	1.07546i	0.843117	+	0.537730i	$0.180718\pi$
$653$	1.95545e11i	1.07546i	−0.843117	+	0.537730i	$0.819282\pi$
$654$	0	0
$655$	−4.61937e10	−0.250968
$656$	2.17760e11i	1.17588i
$657$	0	0
$658$	−3.51407e11	−1.87459
$659$	2.88981e8i	0.00153224i	1.00000		0.000766120i	$0.000243864\pi$
$659$	2.88981e8i	0.00153224i	−1.00000		0.000766120i	$0.999756\pi$
$660$	0	0
$661$	1.23207e11	0.645399	0.322700	−	0.946501i	$-0.395410\pi$
$661$	1.23207e11	0.645399	0.322700	+	0.946501i	$0.395410\pi$
$662$	1.66409e11i	0.866451i
$663$	0	0
$664$	3.00139e11	1.54401
$665$	6.69096e10i	0.342138i
$666$	0	0
$667$	4.66005e11	2.35444
$668$	1.01605e9i	0.00510281i
$669$	0	0
$670$	−2.95909e10	−0.146845
$671$	− 3.81997e10i	− 0.188439i
$672$	0	0
$673$	8.13801e10	0.396696	0.198348	−	0.980132i	$-0.436442\pi$
$673$	8.13801e10	0.396696	0.198348	+	0.980132i	$0.436442\pi$
$674$	3.58973e11i	1.73949i
$675$	0	0
$676$	−8.50760e8	−0.00407399
$677$	− 3.40869e11i	− 1.62268i	−0.584575	−	0.811340i	$-0.698739\pi$
$677$	− 3.40869e11i	− 1.62268i	0.584575	−	0.811340i	$-0.301261\pi$
$678$	0	0
$679$	8.34174e10	0.392444
$680$	1.06173e11i	0.496567i
$681$	0	0
$682$	−4.74133e10	−0.219161
$683$	− 7.34802e10i	− 0.337666i	−0.985645	−	0.168833i	$-0.946000\pi$
$683$	− 7.34802e10i	− 0.337666i	0.985645	−	0.168833i	$-0.0539999\pi$
$684$	0	0
$685$	−2.66141e10	−0.120879
$686$	1.98662e11i	0.897052i
$687$	0	0
$688$	−4.65585e9	−0.0207800
$689$	− 4.22875e10i	− 0.187644i
$690$	0	0
$691$	−2.39210e11	−1.04922	−0.524610	−	0.851343i	$-0.675789\pi$
$691$	−2.39210e11	−1.04922	−0.524610	+	0.851343i	$0.675789\pi$
$692$	7.25147e8i	0.00316229i
$693$	0	0
$694$	−2.86679e11	−1.23583
$695$	5.58660e10i	0.239446i
$696$	0	0
$697$	3.74167e11	1.58538
$698$	1.11366e11i	0.469169i
$699$	0	0
$700$	1.53852e9	0.00640782
$701$	1.91418e11i	0.792703i	0.918099	+	0.396351i	$0.129724\pi$
$701$	1.91418e11i	0.792703i	−0.918099	+	0.396351i	$0.870276\pi$
$702$	0	0
$703$	1.33663e11	0.547254
$704$	− 3.30020e10i	− 0.134354i
$705$	0	0
$706$	4.28212e10	0.172361
$707$	3.49794e11i	1.40002i
$708$	0	0
$709$	−2.46861e11	−0.976937	−0.488469	−	0.872581i	$-0.662444\pi$
$709$	−2.46861e11	−0.976937	−0.488469	+	0.872581i	$0.662444\pi$
$710$	− 1.42182e11i	− 0.559516i
$711$	0	0
$712$	1.65676e11	0.644671
$713$	5.23640e11i	2.02617i
$714$	0	0
$715$	4.49013e9	0.0171805
$716$	− 1.28380e9i	− 0.00488477i
$717$	0	0
$718$	−1.26122e11	−0.474561
$719$	− 2.56345e11i	− 0.959200i	−0.877487	−	0.479600i	$-0.840782\pi$
$719$	− 2.56345e11i	− 0.959200i	0.877487	−	0.479600i	$-0.159218\pi$
$720$	0	0
$721$	−5.82345e11	−2.15496
$722$	− 1.80212e11i	− 0.663185i
$723$	0	0
$724$	−1.14551e9	−0.00416911
$725$	− 4.55454e11i	− 1.64851i
$726$	0	0
$727$	−1.98382e11	−0.710175	−0.355087	−	0.934833i	$-0.615549\pi$
$727$	−1.98382e11	−0.710175	−0.355087	+	0.934833i	$0.615549\pi$
$728$	1.56797e11i	0.558228i
$729$	0	0
$730$	−8.27036e10	−0.291228
$731$	7.99995e9i	0.0280167i
$732$	0	0
$733$	2.15501e11	0.746506	0.373253	−	0.927730i	$-0.378242\pi$
$733$	2.15501e11	0.746506	0.373253	+	0.927730i	$0.378242\pi$
$734$	− 1.48372e10i	− 0.0511172i
$735$	0	0
$736$	−3.35562e9	−0.0114357
$737$	− 1.57313e10i	− 0.0533204i
$738$	0	0
$739$	−1.49718e11	−0.501990	−0.250995	−	0.967988i	$-0.580758\pi$
$739$	−1.49718e11	−0.501990	−0.250995	+	0.967988i	$0.580758\pi$
$740$	4.84979e8i	0.00161732i
$741$	0	0
$742$	2.61017e11	0.861101
$743$	− 4.03331e11i	− 1.32345i	−0.749748	−	0.661723i	$-0.769825\pi$
$743$	− 4.03331e11i	− 1.32345i	0.749748	−	0.661723i	$-0.230175\pi$
$744$	0	0
$745$	8.69494e10	0.282255
$746$	− 4.00198e11i	− 1.29217i
$747$	0	0
$748$	−2.60426e8	−0.000831912 0
$749$	− 1.62418e11i	− 0.516069i
$750$	0	0
$751$	−5.28951e10	−0.166286	−0.0831430	−	0.996538i	$-0.526496\pi$
$751$	−5.28951e10	−0.166286	−0.0831430	+	0.996538i	$0.526496\pi$
$752$	− 3.73672e11i	− 1.16847i
$753$	0	0
$754$	2.14163e11	0.662611
$755$	1.60218e11i	0.493086i
$756$	0	0
$757$	−4.85199e11	−1.47753	−0.738765	−	0.673963i	$-0.764591\pi$
$757$	−4.85199e11	−1.47753	−0.738765	+	0.673963i	$0.764591\pi$
$758$	4.99797e11i	1.51397i
$759$	0	0
$760$	−7.14802e10	−0.214255
$761$	− 8.33991e10i	− 0.248670i	−0.992240	−	0.124335i	$-0.960320\pi$
$761$	− 8.33991e10i	− 0.248670i	0.992240	−	0.124335i	$-0.0396797\pi$
$762$	0	0
$763$	2.02927e11	0.598744
$764$	− 1.39987e9i	− 0.00410878i
$765$	0	0
$766$	4.52569e11	1.31453
$767$	− 7.25607e10i	− 0.209662i
$768$	0	0
$769$	4.71183e11	1.34736	0.673681	−	0.739022i	$-0.264712\pi$
$769$	4.71183e11	1.34736	0.673681	+	0.739022i	$0.264712\pi$
$770$	2.77151e10i	0.0788412i
$771$	0	0
$772$	5.32888e8	0.00150026
$773$	− 3.90308e11i	− 1.09317i	−0.837403	−	0.546587i	$-0.815927\pi$
$773$	− 3.90308e11i	− 1.09317i	0.837403	−	0.546587i	$-0.184073\pi$
$774$	0	0
$775$	5.11784e11	1.41867
$776$	8.91157e10i	0.245758i
$777$	0	0
$778$	2.72234e11	0.743060
$779$	2.51905e11i	0.684050i
$780$	0	0
$781$	7.55878e10	0.203164
$782$	− 6.17627e11i	− 1.65158i
$783$	0	0
$784$	−5.87291e11	−1.55449
$785$	− 1.84049e11i	− 0.484680i
$786$	0	0
$787$	−3.51300e11	−0.915755	−0.457877	−	0.889015i	$-0.651390\pi$
$787$	−3.51300e11	−0.915755	−0.457877	+	0.889015i	$0.651390\pi$
$788$	5.43214e8i	0.00140886i
$789$	0	0
$790$	2.07569e10	0.0532911
$791$	3.40147e11i	0.868881i
$792$	0	0
$793$	−1.93886e11	−0.490290
$794$	− 5.00868e11i	− 1.26021i
$795$	0	0
$796$	−5.41230e8	−0.00134812
$797$	1.39220e11i	0.345039i	0.985006	+	0.172520i	$0.0551908\pi$
$797$	1.39220e11i	0.345039i	−0.985006	+	0.172520i	$0.944809\pi$
$798$	0	0
$799$	−6.42064e11	−1.57540
$800$	3.27965e9i	0.00800695i
$801$	0	0
$802$	4.60693e11	1.11356
$803$	− 4.39674e10i	− 0.105747i
$804$	0	0
$805$	3.06090e11	0.728896
$806$	2.40650e11i	0.570225i
$807$	0	0
$808$	−3.73689e11	−0.876727
$809$	6.81001e11i	1.58984i	0.606713	+	0.794921i	$0.292488\pi$
$809$	6.81001e11i	1.58984i	−0.606713	+	0.794921i	$0.707512\pi$
$810$	0	0
$811$	1.95186e11	0.451196	0.225598	−	0.974220i	$-0.427566\pi$
$811$	1.95186e11	0.451196	0.225598	+	0.974220i	$0.427566\pi$
$812$	− 6.15590e9i	− 0.0141601i
$813$	0	0
$814$	5.53654e10	0.126107
$815$	2.86947e10i	0.0650387i
$816$	0	0
$817$	−5.38591e9	−0.0120885
$818$	3.76895e11i	0.841796i
$819$	0	0
$820$	−9.14008e8	−0.00202160
$821$	6.51505e11i	1.43399i	0.697080	+	0.716994i	$0.254482\pi$
$821$	6.51505e11i	1.43399i	−0.697080	+	0.716994i	$0.745518\pi$
$822$	0	0
$823$	−3.56993e11	−0.778145	−0.389073	−	0.921207i	$-0.627204\pi$
$823$	−3.56993e11	−0.778145	−0.389073	+	0.921207i	$0.627204\pi$
$824$	− 6.22125e11i	− 1.34949i
$825$	0	0
$826$	4.47877e11	0.962141
$827$	5.64712e11i	1.20727i	0.797260	+	0.603636i	$0.206282\pi$
$827$	5.64712e11i	1.20727i	−0.797260	+	0.603636i	$0.793718\pi$
$828$	0	0
$829$	−3.69401e11	−0.782131	−0.391066	−	0.920363i	$-0.627893\pi$
$829$	−3.69401e11	−0.782131	−0.391066	+	0.920363i	$0.627893\pi$
$830$	− 2.69269e11i	− 0.567380i
$831$	0	0
$832$	−1.67504e11	−0.349568
$833$	1.00912e12i	2.09586i
$834$	0	0
$835$	1.97567e11	0.406415
$836$	− 1.75330e8i	0 0.000358948i
$837$	0	0
$838$	−3.91042e11	−0.792953
$839$	− 7.68099e11i	− 1.55013i	−0.631879	−	0.775067i	$-0.717716\pi$
$839$	− 7.68099e11i	− 1.55013i	0.631879	−	0.775067i	$-0.282284\pi$
$840$	0	0
$841$	−1.32211e12	−2.64292
$842$	9.62124e11i	1.91418i
$843$	0	0
$844$	1.91170e9	0.00376748
$845$	1.65427e11i	0.324475i
$846$	0	0
$847$	8.09031e11	1.57192
$848$	2.77555e11i	0.536742i
$849$	0	0
$850$	−6.03643e11	−1.15639
$851$	− 6.11465e11i	− 1.16588i
$852$	0	0
$853$	8.12050e11	1.53386	0.766931	−	0.641729i	$-0.221783\pi$
$853$	8.12050e11	1.53386	0.766931	+	0.641729i	$0.221783\pi$
$854$	− 1.19675e12i	− 2.24994i
$855$	0	0
$856$	1.73513e11	0.323175
$857$	5.23692e11i	0.970851i	0.874278	+	0.485426i	$0.161335\pi$
$857$	5.23692e11i	0.970851i	−0.874278	+	0.485426i	$0.838665\pi$
$858$	0	0
$859$	4.04879e11	0.743623	0.371811	−	0.928308i	$-0.378737\pi$
$859$	4.04879e11	0.743623	0.371811	+	0.928308i	$0.378737\pi$
$860$	− 1.95421e7i	0 3.57255e-5i
$861$	0	0
$862$	−1.11879e11	−0.202638
$863$	2.89294e11i	0.521551i	0.965399	+	0.260775i	$0.0839782\pi$
$863$	2.89294e11i	0.521551i	−0.965399	+	0.260775i	$0.916022\pi$
$864$	0	0
$865$	1.41002e11	0.251861
$866$	9.44747e8i	0.00167975i
$867$	0	0
$868$	6.91726e9	0.0121858
$869$	1.10349e10i	0.0193504i
$870$	0	0
$871$	−7.98453e10	−0.138732
$872$	2.16789e11i	0.374948i
$873$	0	0
$874$	4.15813e11	0.712611
$875$	− 6.45525e11i	− 1.10124i
$876$	0	0
$877$	−7.70757e10	−0.130292	−0.0651462	−	0.997876i	$-0.520751\pi$
$877$	−7.70757e10	−0.130292	−0.0651462	+	0.997876i	$0.520751\pi$
$878$	5.57102e11i	0.937469i
$879$	0	0
$880$	−2.94711e10	−0.0491434
$881$	1.03933e12i	1.72525i	0.505846	+	0.862624i	$0.331180\pi$
$881$	1.03933e12i	1.72525i	−0.505846	+	0.862624i	$0.668820\pi$
$882$	0	0
$883$	4.24171e11	0.697748	0.348874	−	0.937170i	$-0.386564\pi$
$883$	4.24171e11	0.697748	0.348874	+	0.937170i	$0.386564\pi$
$884$	1.32181e9i	0.00216452i
$885$	0	0
$886$	1.44697e11	0.234815
$887$	− 1.35911e11i	− 0.219564i	−0.993956	−	0.109782i	$-0.964985\pi$
$887$	− 1.35911e11i	− 0.219564i	0.993956	−	0.109782i	$-0.0350152\pi$
$888$	0	0
$889$	1.43008e12	2.28957
$890$	− 1.48636e11i	− 0.236899i
$891$	0	0
$892$	−1.08519e9	−0.00171414
$893$	− 4.32265e11i	− 0.679743i
$894$	0	0
$895$	−2.49630e11	−0.389049
$896$	− 1.02435e12i	− 1.58933i
$897$	0	0
$898$	−9.43855e11	−1.45144
$899$	− 2.04775e12i	− 3.13500i
$900$	0	0
$901$	4.76911e11	0.723666
$902$	1.04344e11i	0.157630i
$903$	0	0
$904$	−3.63382e11	−0.544114
$905$	2.22740e11i	0.332050i
$906$	0	0
$907$	−8.51226e11	−1.25781	−0.628906	−	0.777481i	$-0.716497\pi$
$907$	−8.51226e11	−1.25781	−0.628906	+	0.777481i	$0.716497\pi$
$908$	− 6.60810e8i	0 0.000972149i
$909$	0	0
$910$	1.40670e11	0.205133
$911$	3.35090e11i	0.486505i	0.969963	+	0.243253i	$0.0782144\pi$
$911$	3.35090e11i	0.486505i	−0.969963	+	0.243253i	$0.921786\pi$
$912$	0	0
$913$	1.43150e11	0.206020
$914$	− 9.46301e11i	− 1.35595i
$915$	0	0
$916$	−3.55519e9	−0.00504987
$917$	− 7.69364e11i	− 1.08806i
$918$	0	0
$919$	−3.87997e11	−0.543959	−0.271980	−	0.962303i	$-0.587678\pi$
$919$	−3.87997e11	−0.543959	−0.271980	+	0.962303i	$0.587678\pi$
$920$	3.26999e11i	0.456452i
$921$	0	0
$922$	2.54240e11	0.351820
$923$	− 3.83652e11i	− 0.528604i
$924$	0	0
$925$	−5.97620e11	−0.816315
$926$	7.84889e11i	1.06749i
$927$	0	0
$928$	1.31225e10	0.0176939
$929$	4.61876e11i	0.620101i	0.950720	+	0.310050i	$0.100346\pi$
$929$	4.61876e11i	0.620101i	−0.950720	+	0.310050i	$0.899654\pi$
$930$	0	0
$931$	−6.79382e11	−0.904305
$932$	− 4.24205e8i	0 0.000562228i
$933$	0	0
$934$	6.16823e11	0.810538
$935$	5.06389e10i	0.0662579i
$936$	0	0
$937$	−9.03116e11	−1.17162	−0.585808	−	0.810450i	$-0.699223\pi$
$937$	−9.03116e11	−1.17162	−0.585808	+	0.810450i	$0.699223\pi$
$938$	− 4.92841e11i	− 0.636642i
$939$	0	0
$940$	1.56842e9	0.00200887
$941$	− 5.24206e11i	− 0.668565i	−0.942473	−	0.334283i	$-0.891506\pi$
$941$	− 5.24206e11i	− 0.668565i	0.942473	−	0.334283i	$-0.108494\pi$
$942$	0	0
$943$	1.15239e12	1.45731
$944$	4.76254e11i	0.599723i
$945$	0	0
$946$	−2.23094e9	−0.00278562
$947$	7.76577e11i	0.965572i	0.875738	+	0.482786i	$0.160375\pi$
$947$	7.76577e11i	0.965572i	−0.875738	+	0.482786i	$0.839625\pi$
$948$	0	0
$949$	−2.23160e11	−0.275138
$950$	− 4.06398e11i	− 0.498951i
$951$	0	0
$952$	−1.76833e12	−2.15286
$953$	7.74054e11i	0.938426i	0.883085	+	0.469213i	$0.155462\pi$
$953$	7.74054e11i	0.938426i	−0.883085	+	0.469213i	$0.844538\pi$
$954$	0	0
$955$	−2.72199e11	−0.327245
$956$	1.07046e9i	0.00128156i
$957$	0	0
$958$	−1.82554e11	−0.216735
$959$	− 4.43263e11i	− 0.524067i
$960$	0	0
$961$	1.44812e12	1.69789
$962$	− 2.81012e11i	− 0.328113i
$963$	0	0
$964$	−1.63125e9	−0.00188892
$965$	− 1.03618e11i	− 0.119489i
$966$	0	0
$967$	1.34934e11	0.154318	0.0771590	−	0.997019i	$-0.475415\pi$
$967$	1.34934e11	0.154318	0.0771590	+	0.997019i	$0.475415\pi$
$968$	8.64297e11i	0.984377i
$969$	0	0
$970$	7.99500e10	0.0903091
$971$	− 1.44117e12i	− 1.62121i	−0.585592	−	0.810606i	$-0.699138\pi$
$971$	− 1.44117e12i	− 1.62121i	0.585592	−	0.810606i	$-0.300862\pi$
$972$	0	0
$973$	−9.30457e11	−1.03811
$974$	6.46536e11i	0.718384i
$975$	0	0
$976$	1.27257e12	1.40244
$977$	5.44207e11i	0.597291i	0.954364	+	0.298646i	$0.0965349\pi$
$977$	5.44207e11i	0.597291i	−0.954364	+	0.298646i	$0.903465\pi$
$978$	0	0
$979$	7.90185e10	0.0860198
$980$	− 2.46505e9i	− 0.00267253i
$981$	0	0
$982$	1.09440e12	1.17687
$983$	2.03148e11i	0.217569i	0.994065	+	0.108785i	$0.0346959\pi$
$983$	2.03148e11i	0.217569i	−0.994065	+	0.108785i	$0.965304\pi$
$984$	0	0
$985$	1.05626e11	0.112209
$986$	2.41529e12i	2.55541i
$987$	0	0
$988$	−8.89902e8	−0.000933930 0
$989$	2.46388e10i	0.0257534i
$990$	0	0
$991$	−4.09543e11	−0.424624	−0.212312	−	0.977202i	$-0.568099\pi$
$991$	−4.09543e11	−0.424624	−0.212312	+	0.977202i	$0.568099\pi$
$992$	1.47455e10i	0.0152269i
$993$	0	0
$994$	2.36807e12	2.42577
$995$	1.05240e11i	0.107372i
$996$	0	0
$997$	−7.91890e11	−0.801465	−0.400732	−	0.916195i	$-0.631244\pi$
$997$	−7.91890e11	−0.801465	−0.400732	+	0.916195i	$0.631244\pi$
$998$	1.35496e12i	1.36586i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	27.9.b.d.26.5	yes	6
3.2	odd	2	inner	27.9.b.d.26.2	✓	6
4.3	odd	2		432.9.e.k.161.3		6
9.2	odd	6		81.9.d.f.53.2		12
9.4	even	3		81.9.d.f.26.2		12
9.5	odd	6		81.9.d.f.26.5		12
9.7	even	3		81.9.d.f.53.5		12
12.11	even	2		432.9.e.k.161.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.b.d.26.2	✓	6	3.2	odd	2	inner
27.9.b.d.26.5	yes	6	1.1	even	1	trivial
81.9.d.f.26.2		12	9.4	even	3
81.9.d.f.26.5		12	9.5	odd	6
81.9.d.f.53.2		12	9.2	odd	6
81.9.d.f.53.5		12	9.7	even	3
432.9.e.k.161.3		6	4.3	odd	2
432.9.e.k.161.4		6	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)

\(f(q)\)	\(=\)	\(q+15.9629i q^{2} +1.18662 q^{4} -230.735i q^{5} +3842.93 q^{7} +4105.44i q^{8} +O(q^{10})\)	\(q+15.9629i q^{2} +1.18662 q^{4} -230.735i q^{5} +3842.93 q^{7} +4105.44i q^{8} +3683.19 q^{10} +1958.08i q^{11} +9938.39 q^{13} +61344.1i q^{14} -65230.8 q^{16} +112083. i q^{17} -75459.4 q^{19} -273.795i q^{20} -31256.6 q^{22} +345203. i q^{23} +337386. q^{25} +158645. i q^{26} +4560.10 q^{28} -1.34995e6i q^{29} +1.51691e6 q^{31} +9720.74i q^{32} -1.78917e6 q^{34} -886697. i q^{35} -1.77132e6 q^{37} -1.20455e6i q^{38} +947267. q^{40} -3.33829e6i q^{41} +71375.0 q^{43} +2323.50i q^{44} -5.51043e6 q^{46} +5.72845e6i q^{47} +9.00328e6 q^{49} +5.38566e6i q^{50} +11793.1 q^{52} -4.25497e6i q^{53} +451797. q^{55} +1.57769e7i q^{56} +2.15490e7 q^{58} -7.30106e6i q^{59} -1.95088e7 q^{61} +2.42142e7i q^{62} -1.68543e7 q^{64} -2.29313e6i q^{65} -8.03403e6 q^{67} +133001. i q^{68} +1.41542e7 q^{70} -3.86030e7i q^{71} -2.24543e7 q^{73} -2.82754e7i q^{74} -89541.9 q^{76} +7.52475e6i q^{77} +5.63558e6 q^{79} +1.50510e7i q^{80} +5.32887e7 q^{82} -7.31076e7i q^{83} +2.58615e7 q^{85} +1.13935e6i q^{86} -8.03877e6 q^{88} -4.03551e7i q^{89} +3.81925e7 q^{91} +409625. i q^{92} -9.14426e7 q^{94} +1.74111e7i q^{95} +2.17067e7 q^{97} +1.43718e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 786 q^{4} - 1698 q^{7}+O(q^{10}) \) 6 * q - 786 * q^4 - 1698 * q^7	\( 6 q - 786 q^{4} - 1698 q^{7} - 46278 q^{10} + 41844 q^{13} + 170466 q^{16} - 36384 q^{19} + 647298 q^{22} - 1646688 q^{25} - 838986 q^{28} + 2058474 q^{31} - 1320300 q^{34} - 9395880 q^{37} + 29560086 q^{40} - 2737284 q^{43} - 31514616 q^{46} + 28900656 q^{49} - 34081692 q^{52} - 26674542 q^{55} + 75244572 q^{58} - 40180776 q^{61} - 48541458 q^{64} + 111355284 q^{67} + 17727282 q^{70} + 12821718 q^{73} - 38623776 q^{76} + 21820404 q^{79} - 138785832 q^{82} - 83844396 q^{85} + 57552174 q^{88} + 156632964 q^{91} - 502013916 q^{94} + 53735106 q^{97}+O(q^{100}) \) 6 * q - 786 * q^4 - 1698 * q^7 - 46278 * q^10 + 41844 * q^13 + 170466 * q^16 - 36384 * q^19 + 647298 * q^22 - 1646688 * q^25 - 838986 * q^28 + 2058474 * q^31 - 1320300 * q^34 - 9395880 * q^37 + 29560086 * q^40 - 2737284 * q^43 - 31514616 * q^46 + 28900656 * q^49 - 34081692 * q^52 - 26674542 * q^55 + 75244572 * q^58 - 40180776 * q^61 - 48541458 * q^64 + 111355284 * q^67 + 17727282 * q^70 + 12821718 * q^73 - 38623776 * q^76 + 21820404 * q^79 - 138785832 * q^82 - 83844396 * q^85 + 57552174 * q^88 + 156632964 * q^91 - 502013916 * q^94 + 53735106 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more