LMFDB - Embedded newform 27.9.b.b.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{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			26.1
Root			$-2.44949i$ of defining polynomial
Character	$\chi$	$=$	27.26
Dual form			27.9.b.b.26.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-29.3939i q^{2} -608.000 q^{4} +823.029i q^{5} +1967.00 q^{7} +10346.6i q^{8} +O(q^{10})$	\(q-29.3939i q^{2} -608.000 q^{4} +823.029i q^{5} +1967.00 q^{7} +10346.6i q^{8} +24192.0 q^{10} +12580.6i q^{11} -45505.0 q^{13} -57817.8i q^{14} +148480. q^{16} +59610.8i q^{17} +152399. q^{19} -500401. i q^{20} +369792. q^{22} +131332. i q^{23} -286751. q^{25} +1.33757e6i q^{26} -1.19594e6 q^{28} +588583. i q^{29} -164350. q^{31} -1.71566e6i q^{32} +1.75219e6 q^{34} +1.61890e6i q^{35} -663937. q^{37} -4.47960e6i q^{38} -8.51558e6 q^{40} +938017. i q^{41} +575330. q^{43} -7.64899e6i q^{44} +3.86035e6 q^{46} +9.23426e6i q^{47} -1.89571e6 q^{49} +8.42872e6i q^{50} +2.76670e7 q^{52} -1.03765e7i q^{53} -1.03542e7 q^{55} +2.03519e7i q^{56} +1.73007e7 q^{58} +5.03987e6i q^{59} -1.92130e7 q^{61} +4.83088e6i q^{62} -1.24191e7 q^{64} -3.74519e7i q^{65} -598033. q^{67} -3.62434e7i q^{68} +4.75857e7 q^{70} -2.92721e7i q^{71} +1.28502e7 q^{73} +1.95157e7i q^{74} -9.26586e7 q^{76} +2.47460e7i q^{77} -2.35847e7 q^{79} +1.22203e8i q^{80} +2.75720e7 q^{82} -3.34509e7i q^{83} -4.90614e7 q^{85} -1.69112e7i q^{86} -1.30167e8 q^{88} +2.82848e7i q^{89} -8.95083e7 q^{91} -7.98498e7i q^{92} +2.71431e8 q^{94} +1.25429e8i q^{95} +1.36490e8 q^{97} +5.57223e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1216 q^{4} + 3934 q^{7}+O(q^{10}) $ 2 * q - 1216 * q^4 + 3934 * q^7	$ 2 q - 1216 q^{4} + 3934 q^{7} + 48384 q^{10} - 91010 q^{13} + 296960 q^{16} + 304798 q^{19} + 739584 q^{22} - 573502 q^{25} - 2391872 q^{28} - 328700 q^{31} + 3504384 q^{34} - 1327874 q^{37} - 17031168 q^{40} + 1150660 q^{43} + 7720704 q^{46} - 3791424 q^{49} + 55334080 q^{52} - 20708352 q^{55} + 34601472 q^{58} - 38425922 q^{61} - 24838144 q^{64} - 1196066 q^{67} + 95171328 q^{70} + 25700350 q^{73} - 185317184 q^{76} - 47169314 q^{79} + 55143936 q^{82} - 98122752 q^{85} - 260333568 q^{88} - 179016670 q^{91} + 542861568 q^{94} + 272979262 q^{97}+O(q^{100}) $ 2 * q - 1216 * q^4 + 3934 * q^7 + 48384 * q^10 - 91010 * q^13 + 296960 * q^16 + 304798 * q^19 + 739584 * q^22 - 573502 * q^25 - 2391872 * q^28 - 328700 * q^31 + 3504384 * q^34 - 1327874 * q^37 - 17031168 * q^40 + 1150660 * q^43 + 7720704 * q^46 - 3791424 * q^49 + 55334080 * q^52 - 20708352 * q^55 + 34601472 * q^58 - 38425922 * q^61 - 24838144 * q^64 - 1196066 * q^67 + 95171328 * q^70 + 25700350 * q^73 - 185317184 * q^76 - 47169314 * q^79 + 55143936 * q^82 - 98122752 * q^85 - 260333568 * q^88 - 179016670 * q^91 + 542861568 * q^94 + 272979262 * 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 $

$2$	− 29.3939i	− 1.83712i	−0.395285	−	0.918559i	$-0.629354\pi$
$2$	− 29.3939i	− 1.83712i	0.395285	−	0.918559i	$-0.370646\pi$
$3$	0	0
$3$	0	0
$4$	−608.000	−2.37500
$5$	823.029i	1.31685i	0.752648	+	0.658423i	$0.228776\pi$
$5$	823.029i	1.31685i	−0.752648	+	0.658423i	$0.771224\pi$
$6$	0	0
$7$	1967.00	0.819242	0.409621	−	0.912256i	$-0.365661\pi$
$7$	1967.00	0.819242	0.409621	+	0.912256i	$0.365661\pi$
$8$	10346.6i	2.52604i
$9$	0	0
$10$	24192.0	2.41920
$11$	12580.6i	0.859270i	0.903003	+	0.429635i	$0.141358\pi$
$11$	12580.6i	0.859270i	−0.903003	+	0.429635i	$0.858642\pi$
$12$	0	0
$13$	−45505.0	−1.59326	−0.796628	−	0.604470i	$-0.793385\pi$
$13$	−45505.0	−1.59326	−0.796628	+	0.604470i	$0.793385\pi$
$14$	− 57817.8i	− 1.50504i
$15$	0	0
$16$	148480.	2.26562
$17$	59610.8i	0.713722i	0.934157	+	0.356861i	$0.116153\pi$
$17$	59610.8i	0.713722i	−0.934157	+	0.356861i	$0.883847\pi$
$18$	0	0
$19$	152399.	1.16941	0.584706	−	0.811245i	$-0.301210\pi$
$19$	152399.	1.16941	0.584706	+	0.811245i	$0.301210\pi$
$20$	− 500401.i	− 3.12751i
$21$	0	0
$22$	369792.	1.57858
$23$	131332.i	0.469309i	0.972079	+	0.234654i	$0.0753958\pi$
$23$	131332.i	0.469309i	−0.972079	+	0.234654i	$0.924604\pi$
$24$	0	0
$25$	−286751.	−0.734083
$26$	1.33757e6i	2.92700i
$27$	0	0
$28$	−1.19594e6	−1.94570
$29$	588583.i	0.832177i	0.909324	+	0.416089i	$0.136599\pi$
$29$	588583.i	0.832177i	−0.909324	+	0.416089i	$0.863401\pi$
$30$	0	0
$31$	−164350.	−0.177960	−0.0889801	−	0.996033i	$-0.528361\pi$
$31$	−164350.	−0.177960	−0.0889801	+	0.996033i	$0.528361\pi$
$32$	− 1.71566e6i	− 1.63618i
$33$	0	0
$34$	1.75219e6	1.31119
$35$	1.61890e6i	1.07882i
$36$	0	0
$37$	−663937.	−0.354258	−0.177129	−	0.984188i	$-0.556681\pi$
$37$	−663937.	−0.354258	−0.177129	+	0.984188i	$0.556681\pi$
$38$	− 4.47960e6i	− 2.14835i
$39$	0	0
$40$	−8.51558e6	−3.32640
$41$	938017.i	0.331952i	0.986130	+	0.165976i	$0.0530774\pi$
$41$	938017.i	0.331952i	−0.986130	+	0.165976i	$0.946923\pi$
$42$	0	0
$43$	575330.	0.168284	0.0841421	−	0.996454i	$-0.473185\pi$
$43$	575330.	0.168284	0.0841421	+	0.996454i	$0.473185\pi$
$44$	− 7.64899e6i	− 2.04077i
$45$	0	0
$46$	3.86035e6	0.862175
$47$	9.23426e6i	1.89239i	0.323596	+	0.946195i	$0.395108\pi$
$47$	9.23426e6i	1.89239i	−0.323596	+	0.946195i	$0.604892\pi$
$48$	0	0
$49$	−1.89571e6	−0.328843
$50$	8.42872e6i	1.34860i
$51$	0	0
$52$	2.76670e7	3.78398
$53$	− 1.03765e7i	− 1.31507i	−0.753425	−	0.657533i	$-0.771600\pi$
$53$	− 1.03765e7i	− 1.31507i	0.753425	−	0.657533i	$-0.228400\pi$
$54$	0	0
$55$	−1.03542e7	−1.13153
$56$	2.03519e7i	2.06943i
$57$	0	0
$58$	1.73007e7	1.52881
$59$	5.03987e6i	0.415922i	0.978137	+	0.207961i	$0.0666827\pi$
$59$	5.03987e6i	0.415922i	−0.978137	+	0.207961i	$0.933317\pi$
$60$	0	0
$61$	−1.92130e7	−1.38763	−0.693817	−	0.720151i	$-0.744072\pi$
$61$	−1.92130e7	−1.38763	−0.693817	+	0.720151i	$0.744072\pi$
$62$	4.83088e6i	0.326934i
$63$	0	0
$64$	−1.24191e7	−0.740234
$65$	− 3.74519e7i	− 2.09807i
$66$	0	0
$67$	−598033.	−0.0296774	−0.0148387	−	0.999890i	$-0.504723\pi$
$67$	−598033.	−0.0296774	−0.0148387	+	0.999890i	$0.504723\pi$
$68$	− 3.62434e7i	− 1.69509i
$69$	0	0
$70$	4.75857e7	1.98191
$71$	− 2.92721e7i	− 1.15191i	−0.817480	−	0.575957i	$-0.804630\pi$
$71$	− 2.92721e7i	− 1.15191i	0.817480	−	0.575957i	$-0.195370\pi$
$72$	0	0
$73$	1.28502e7	0.452499	0.226249	−	0.974069i	$-0.427354\pi$
$73$	1.28502e7	0.452499	0.226249	+	0.974069i	$0.427354\pi$
$74$	1.95157e7i	0.650814i
$75$	0	0
$76$	−9.26586e7	−2.77735
$77$	2.47460e7i	0.703950i
$78$	0	0
$79$	−2.35847e7	−0.605510	−0.302755	−	0.953068i	$-0.597906\pi$
$79$	−2.35847e7	−0.605510	−0.302755	+	0.953068i	$0.597906\pi$
$80$	1.22203e8i	2.98348i
$81$	0	0
$82$	2.75720e7	0.609835
$83$	− 3.34509e7i	− 0.704849i	−0.935840	−	0.352424i	$-0.885357\pi$
$83$	− 3.34509e7i	− 0.704849i	0.935840	−	0.352424i	$-0.114643\pi$
$84$	0	0
$85$	−4.90614e7	−0.939862
$86$	− 1.69112e7i	− 0.309158i
$87$	0	0
$88$	−1.30167e8	−2.17055
$89$	2.82848e7i	0.450809i	0.974265	+	0.225405i	$0.0723704\pi$
$89$	2.82848e7i	0.450809i	−0.974265	+	0.225405i	$0.927630\pi$
$90$	0	0
$91$	−8.95083e7	−1.30526
$92$	− 7.98498e7i	− 1.11461i
$93$	0	0
$94$	2.71431e8	3.47654
$95$	1.25429e8i	1.53994i
$96$	0	0
$97$	1.36490e8	1.54175	0.770873	−	0.636989i	$-0.219820\pi$
$97$	1.36490e8	1.54175	0.770873	+	0.636989i	$0.219820\pi$
$98$	5.57223e7i	0.604122i
$99$	0	0
$100$	1.74345e8	1.74345
$101$	− 4.02494e7i	− 0.386789i	−0.981121	−	0.193394i	$-0.938050\pi$
$101$	− 4.02494e7i	− 0.386789i	0.981121	−	0.193394i	$-0.0619497\pi$
$102$	0	0
$103$	3.35907e7	0.298449	0.149225	−	0.988803i	$-0.452322\pi$
$103$	3.35907e7	0.298449	0.149225	+	0.988803i	$0.452322\pi$
$104$	− 4.70824e8i	− 4.02462i
$105$	0	0
$106$	−3.05006e8	−2.41593
$107$	− 1.95483e8i	− 1.49133i	−0.666322	−	0.745664i	$-0.732132\pi$
$107$	− 1.95483e8i	− 1.49133i	0.666322	−	0.745664i	$-0.267868\pi$
$108$	0	0
$109$	−1.01182e8	−0.716800	−0.358400	−	0.933568i	$-0.616678\pi$
$109$	−1.01182e8	−0.716800	−0.358400	+	0.933568i	$0.616678\pi$
$110$	3.04349e8i	2.07875i
$111$	0	0
$112$	2.92060e8	1.85610
$113$	3.14504e8i	1.92891i	0.264238	+	0.964457i	$0.414880\pi$
$113$	3.14504e8i	1.92891i	−0.264238	+	0.964457i	$0.585120\pi$
$114$	0	0
$115$	−1.08090e8	−0.618007
$116$	− 3.57858e8i	− 1.97642i
$117$	0	0
$118$	1.48141e8	0.764097
$119$	1.17254e8i	0.584711i
$120$	0	0
$121$	5.60879e7	0.261654
$122$	5.64743e8i	2.54925i
$123$	0	0
$124$	9.99248e7	0.422656
$125$	8.54913e7i	0.350172i
$126$	0	0
$127$	4.30869e8	1.65627	0.828133	−	0.560532i	$-0.189403\pi$
$127$	4.30869e8	1.65627	0.828133	+	0.560532i	$0.189403\pi$
$128$	− 7.41647e7i	− 0.276285i
$129$	0	0
$130$	−1.10086e9	−3.85441
$131$	1.88933e8i	0.641538i	0.947158	+	0.320769i	$0.103941\pi$
$131$	1.88933e8i	0.641538i	−0.947158	+	0.320769i	$0.896059\pi$
$132$	0	0
$133$	2.99769e8	0.958032
$134$	1.75785e7i	0.0545209i
$135$	0	0
$136$	−6.16772e8	−1.80289
$137$	1.22998e8i	0.349152i	0.984644	+	0.174576i	$0.0558555\pi$
$137$	1.22998e8i	0.349152i	−0.984644	+	0.174576i	$0.944144\pi$
$138$	0	0
$139$	1.72072e8	0.460947	0.230474	−	0.973079i	$-0.425972\pi$
$139$	1.72072e8	0.460947	0.230474	+	0.973079i	$0.425972\pi$
$140$	− 9.84289e8i	− 2.56219i
$141$	0	0
$142$	−8.60420e8	−2.11620
$143$	− 5.72479e8i	− 1.36904i
$144$	0	0
$145$	−4.84421e8	−1.09585
$146$	− 3.77716e8i	− 0.831294i
$147$	0	0
$148$	4.03674e8	0.841363
$149$	− 7.31667e8i	− 1.48446i	−0.670145	−	0.742230i	$-0.733768\pi$
$149$	− 7.31667e8i	− 1.48446i	0.670145	−	0.742230i	$-0.266232\pi$
$150$	0	0
$151$	1.85952e8	0.357679	0.178840	−	0.983878i	$-0.442766\pi$
$151$	1.85952e8	0.357679	0.178840	+	0.983878i	$0.442766\pi$
$152$	1.57682e9i	2.95398i
$153$	0	0
$154$	7.27381e8	1.29324
$155$	− 1.35265e8i	− 0.234346i
$156$	0	0
$157$	9.74007e8	1.60311	0.801556	−	0.597920i	$-0.204006\pi$
$157$	9.74007e8	1.60311	0.801556	+	0.597920i	$0.204006\pi$
$158$	6.93245e8i	1.11239i
$159$	0	0
$160$	1.41204e9	2.15460
$161$	2.58330e8i	0.384477i
$162$	0	0
$163$	−1.15499e9	−1.63616	−0.818082	−	0.575102i	$-0.804962\pi$
$163$	−1.15499e9	−1.63616	−0.818082	+	0.575102i	$0.804962\pi$
$164$	− 5.70315e8i	− 0.788386i
$165$	0	0
$166$	−9.83253e8	−1.29489
$167$	− 1.14302e8i	− 0.146956i	−0.997297	−	0.0734779i	$-0.976590\pi$
$167$	− 1.14302e8i	− 0.146956i	0.997297	−	0.0734779i	$-0.0234098\pi$
$168$	0	0
$169$	1.25497e9	1.53847
$170$	1.44210e9i	1.72664i
$171$	0	0
$172$	−3.49801e8	−0.399675
$173$	− 1.65322e9i	− 1.84564i	−0.385235	−	0.922819i	$-0.625879\pi$
$173$	− 1.65322e9i	− 1.84564i	0.385235	−	0.922819i	$-0.374121\pi$
$174$	0	0
$175$	−5.64039e8	−0.601391
$176$	1.86796e9i	1.94678i
$177$	0	0
$178$	8.31400e8	0.828190
$179$	4.17229e8i	0.406408i	0.979136	+	0.203204i	$0.0651355\pi$
$179$	4.17229e8i	0.406408i	−0.979136	+	0.203204i	$0.934865\pi$
$180$	0	0
$181$	1.16424e9	1.08475	0.542374	−	0.840137i	$-0.317526\pi$
$181$	1.16424e9	1.08475	0.542374	+	0.840137i	$0.317526\pi$
$182$	2.63100e9i	2.39792i
$183$	0	0
$184$	−1.35884e9	−1.18549
$185$	− 5.46439e8i	− 0.466503i
$186$	0	0
$187$	−7.49938e8	−0.613280
$188$	− 5.61443e9i	− 4.49443i
$189$	0	0
$190$	3.68684e9	2.82904
$191$	1.61055e9i	1.21016i	0.796166	+	0.605078i	$0.206858\pi$
$191$	1.61055e9i	1.21016i	−0.796166	+	0.605078i	$0.793142\pi$
$192$	0	0
$193$	−7.10279e8	−0.511917	−0.255959	−	0.966688i	$-0.582391\pi$
$193$	−7.10279e8	−0.511917	−0.255959	+	0.966688i	$0.582391\pi$
$194$	− 4.01196e9i	− 2.83237i
$195$	0	0
$196$	1.15259e9	0.781001
$197$	1.41623e9i	0.940303i	0.882586	+	0.470152i	$0.155801\pi$
$197$	1.41623e9i	0.940303i	−0.882586	+	0.470152i	$0.844199\pi$
$198$	0	0
$199$	−2.34324e9	−1.49419	−0.747093	−	0.664720i	$-0.768551\pi$
$199$	−2.34324e9	−1.49419	−0.747093	+	0.664720i	$0.768551\pi$
$200$	− 2.96691e9i	− 1.85432i
$201$	0	0
$202$	−1.18309e9	−0.710576
$203$	1.15774e9i	0.681754i
$204$	0	0
$205$	−7.72015e8	−0.437130
$206$	− 9.87362e8i	− 0.548286i
$207$	0	0
$208$	−6.75658e9	−3.60972
$209$	1.91727e9i	1.00484i
$210$	0	0
$211$	1.06517e9	0.537389	0.268695	−	0.963225i	$-0.413408\pi$
$211$	1.06517e9	0.537389	0.268695	+	0.963225i	$0.413408\pi$
$212$	6.30892e9i	3.12328i
$213$	0	0
$214$	−5.74600e9	−2.73975
$215$	4.73513e8i	0.221604i
$216$	0	0
$217$	−3.23276e8	−0.145792
$218$	2.97414e9i	1.31685i
$219$	0	0
$220$	6.29534e9	2.68738
$221$	− 2.71259e9i	− 1.13714i
$222$	0	0
$223$	1.99586e9	0.807068	0.403534	−	0.914965i	$-0.367782\pi$
$223$	1.99586e9	0.807068	0.403534	+	0.914965i	$0.367782\pi$
$224$	− 3.37471e9i	− 1.34043i
$225$	0	0
$226$	9.24451e9	3.54364
$227$	1.79923e9i	0.677615i	0.940856	+	0.338807i	$0.110023\pi$
$227$	1.79923e9i	0.677615i	−0.940856	+	0.338807i	$0.889977\pi$
$228$	0	0
$229$	1.59787e9	0.581032	0.290516	−	0.956870i	$-0.406173\pi$
$229$	1.59787e9	0.581032	0.290516	+	0.956870i	$0.406173\pi$
$230$	3.17718e9i	1.13535i
$231$	0	0
$232$	−6.08986e9	−2.10211
$233$	4.98411e9i	1.69108i	0.533912	+	0.845540i	$0.320721\pi$
$233$	4.98411e9i	1.69108i	−0.533912	+	0.845540i	$0.679279\pi$
$234$	0	0
$235$	−7.60006e9	−2.49199
$236$	− 3.06424e9i	− 0.987814i
$237$	0	0
$238$	3.44656e9	1.07418
$239$	− 5.89754e9i	− 1.80751i	−0.428055	−	0.903753i	$-0.640801\pi$
$239$	− 5.89754e9i	− 1.80751i	0.428055	−	0.903753i	$-0.359199\pi$
$240$	0	0
$241$	2.26174e9	0.670463	0.335231	−	0.942136i	$-0.391185\pi$
$241$	2.26174e9	0.670463	0.335231	+	0.942136i	$0.391185\pi$
$242$	− 1.64864e9i	− 0.480689i
$243$	0	0
$244$	1.16815e10	3.29563
$245$	− 1.56023e9i	− 0.433035i
$246$	0	0
$247$	−6.93492e9	−1.86317
$248$	− 1.70047e9i	− 0.449534i
$249$	0	0
$250$	2.51292e9	0.643307
$251$	− 3.31815e9i	− 0.835989i	−0.908450	−	0.417994i	$-0.862733\pi$
$251$	− 3.31815e9i	− 0.835989i	0.908450	−	0.417994i	$-0.137267\pi$
$252$	0	0
$253$	−1.65223e9	−0.403263
$254$	− 1.26649e10i	− 3.04276i
$255$	0	0
$256$	−5.35927e9	−1.24780
$257$	3.05018e9i	0.699185i	0.936902	+	0.349593i	$0.113680\pi$
$257$	3.05018e9i	0.699185i	−0.936902	+	0.349593i	$0.886320\pi$
$258$	0	0
$259$	−1.30596e9	−0.290223
$260$	2.27708e10i	4.98292i
$261$	0	0
$262$	5.55347e9	1.17858
$263$	− 1.97422e9i	− 0.412640i	−0.978485	−	0.206320i	$-0.933851\pi$
$263$	− 1.97422e9i	− 0.412640i	0.978485	−	0.206320i	$-0.0661489\pi$
$264$	0	0
$265$	8.54016e9	1.73174
$266$	− 8.81137e9i	− 1.76002i
$267$	0	0
$268$	3.63604e8	0.0704838
$269$	− 3.49334e9i	− 0.667163i	−0.942721	−	0.333581i	$-0.891743\pi$
$269$	− 3.49334e9i	− 0.667163i	0.942721	−	0.333581i	$-0.108257\pi$
$270$	0	0
$271$	1.43226e9	0.265549	0.132774	−	0.991146i	$-0.457611\pi$
$271$	1.43226e9	0.265549	0.132774	+	0.991146i	$0.457611\pi$
$272$	8.85101e9i	1.61703i
$273$	0	0
$274$	3.61538e9	0.641434
$275$	− 3.60749e9i	− 0.630775i
$276$	0	0
$277$	3.35095e9	0.569179	0.284589	−	0.958650i	$-0.408143\pi$
$277$	3.35095e9	0.569179	0.284589	+	0.958650i	$0.408143\pi$
$278$	− 5.05787e9i	− 0.846814i
$279$	0	0
$280$	−1.67502e10	−2.72513
$281$	4.29040e9i	0.688133i	0.938945	+	0.344067i	$0.111805\pi$
$281$	4.29040e9i	0.688133i	−0.938945	+	0.344067i	$0.888195\pi$
$282$	0	0
$283$	−9.91145e7	−0.0154522	−0.00772612	−	0.999970i	$-0.502459\pi$
$283$	−9.91145e7	−0.0154522	−0.00772612	+	0.999970i	$0.502459\pi$
$284$	1.77974e10i	2.73580i
$285$	0	0
$286$	−1.68274e10	−2.51508
$287$	1.84508e9i	0.271949i
$288$	0	0
$289$	3.42231e9	0.490601
$290$	1.42390e10i	2.01320i
$291$	0	0
$292$	−7.81291e9	−1.07469
$293$	− 9.02972e8i	− 0.122519i	−0.998122	−	0.0612596i	$-0.980488\pi$
$293$	− 9.02972e8i	− 0.122519i	0.998122	−	0.0612596i	$-0.0195117\pi$
$294$	0	0
$295$	−4.14796e9	−0.547705
$296$	− 6.86952e9i	− 0.894869i
$297$	0	0
$298$	−2.15065e10	−2.72713
$299$	− 5.97626e9i	− 0.747729i
$300$	0	0
$301$	1.13167e9	0.137865
$302$	− 5.46586e9i	− 0.657099i
$303$	0	0
$304$	2.26282e10	2.64945
$305$	− 1.58128e10i	− 1.82730i
$306$	0	0
$307$	3.36397e9	0.378703	0.189352	−	0.981909i	$-0.439361\pi$
$307$	3.36397e9	0.378703	0.189352	+	0.981909i	$0.439361\pi$
$308$	− 1.50456e10i	− 1.67188i
$309$	0	0
$310$	−3.97596e9	−0.430521
$311$	1.33023e10i	1.42196i	0.703215	+	0.710978i	$0.251747\pi$
$311$	1.33023e10i	1.42196i	−0.703215	+	0.710978i	$0.748253\pi$
$312$	0	0
$313$	1.31249e10	1.36748	0.683738	−	0.729728i	$-0.260353\pi$
$313$	1.31249e10	1.36748	0.683738	+	0.729728i	$0.260353\pi$
$314$	− 2.86299e10i	− 2.94510i
$315$	0	0
$316$	1.43395e10	1.43809
$317$	− 6.98520e9i	− 0.691739i	−0.938283	−	0.345869i	$-0.887584\pi$
$317$	− 6.98520e9i	− 0.691739i	0.938283	−	0.345869i	$-0.112416\pi$
$318$	0	0
$319$	−7.40472e9	−0.715065
$320$	− 1.02213e10i	− 0.974774i
$321$	0	0
$322$	7.59331e9	0.706330
$323$	9.08462e9i	0.834635i
$324$	0	0
$325$	1.30486e10	1.16958
$326$	3.39495e10i	3.00582i
$327$	0	0
$328$	−9.70533e9	−0.838523
$329$	1.81638e10i	1.55033i
$330$	0	0
$331$	−1.18733e10	−0.989140	−0.494570	−	0.869138i	$-0.664674\pi$
$331$	−1.18733e10	−0.989140	−0.494570	+	0.869138i	$0.664674\pi$
$332$	2.03382e10i	1.67402i
$333$	0	0
$334$	−3.35977e9	−0.269975
$335$	− 4.92198e8i	− 0.0390806i
$336$	0	0
$337$	9.15419e9	0.709741	0.354871	−	0.934915i	$-0.384525\pi$
$337$	9.15419e9	0.709741	0.354871	+	0.934915i	$0.384525\pi$
$338$	− 3.68886e10i	− 2.82634i
$339$	0	0
$340$	2.98293e10	2.23217
$341$	− 2.06762e9i	− 0.152916i
$342$	0	0
$343$	−1.50682e10	−1.08864
$344$	5.95274e9i	0.425092i
$345$	0	0
$346$	−4.85946e10	−3.39065
$347$	2.05081e10i	1.41451i	0.706957	+	0.707257i	$0.250067\pi$
$347$	2.05081e10i	1.41451i	−0.706957	+	0.707257i	$0.749933\pi$
$348$	0	0
$349$	1.34799e10	0.908625	0.454312	−	0.890842i	$-0.349885\pi$
$349$	1.34799e10	0.908625	0.454312	+	0.890842i	$0.349885\pi$
$350$	1.65793e10i	1.10483i
$351$	0	0
$352$	2.15840e10	1.40592
$353$	9.78259e9i	0.630021i	0.949088	+	0.315011i	$0.102008\pi$
$353$	9.78259e9i	0.630021i	−0.949088	+	0.315011i	$0.897992\pi$
$354$	0	0
$355$	2.40917e10	1.51689
$356$	− 1.71972e10i	− 1.07067i
$357$	0	0
$358$	1.22640e10	0.746619
$359$	1.67338e10i	1.00743i	0.863868	+	0.503717i	$0.168035\pi$
$359$	1.67338e10i	1.00743i	−0.863868	+	0.503717i	$0.831965\pi$
$360$	0	0
$361$	6.24189e9	0.367525
$362$	− 3.42216e10i	− 1.99281i
$363$	0	0
$364$	5.44211e10	3.10000
$365$	1.05761e10i	0.595871i
$366$	0	0
$367$	−1.49879e10	−0.826184	−0.413092	−	0.910689i	$-0.635551\pi$
$367$	−1.49879e10	−0.826184	−0.413092	+	0.910689i	$0.635551\pi$
$368$	1.95002e10i	1.06328i
$369$	0	0
$370$	−1.60620e10	−0.857022
$371$	− 2.04106e10i	− 1.07736i
$372$	0	0
$373$	−1.97964e10	−1.02271	−0.511353	−	0.859371i	$-0.670855\pi$
$373$	−1.97964e10	−1.02271	−0.511353	+	0.859371i	$0.670855\pi$
$374$	2.20436e10i	1.12667i
$375$	0	0
$376$	−9.55436e10	−4.78025
$377$	− 2.67835e10i	− 1.32587i
$378$	0	0
$379$	−5.34899e9	−0.259248	−0.129624	−	0.991563i	$-0.541377\pi$
$379$	−5.34899e9	−0.259248	−0.129624	+	0.991563i	$0.541377\pi$
$380$	− 7.62607e10i	− 3.65735i
$381$	0	0
$382$	4.73404e10	2.22320
$383$	− 1.44938e10i	− 0.673577i	−0.941580	−	0.336788i	$-0.890659\pi$
$383$	− 1.44938e10i	− 0.673577i	0.941580	−	0.336788i	$-0.109341\pi$
$384$	0	0
$385$	−2.03667e10	−0.926994
$386$	2.08778e10i	0.940452i
$387$	0	0
$388$	−8.29857e10	−3.66165
$389$	1.80507e10i	0.788308i	0.919044	+	0.394154i	$0.128962\pi$
$389$	1.80507e10i	0.788308i	−0.919044	+	0.394154i	$0.871038\pi$
$390$	0	0
$391$	−7.82879e9	−0.334956
$392$	− 1.96143e10i	− 0.830668i
$393$	0	0
$394$	4.16284e10	1.72745
$395$	− 1.94108e10i	− 0.797363i
$396$	0	0
$397$	3.82640e10	1.54038	0.770191	−	0.637814i	$-0.220161\pi$
$397$	3.82640e10	1.54038	0.770191	+	0.637814i	$0.220161\pi$
$398$	6.88769e10i	2.74499i
$399$	0	0
$400$	−4.25768e10	−1.66316
$401$	− 4.26602e10i	− 1.64985i	−0.565241	−	0.824926i	$-0.691217\pi$
$401$	− 4.26602e10i	− 1.64985i	0.565241	−	0.824926i	$-0.308783\pi$
$402$	0	0
$403$	7.47875e9	0.283536
$404$	2.44716e10i	0.918623i
$405$	0	0
$406$	3.40305e10	1.25246
$407$	− 8.35271e9i	− 0.304404i
$408$	0	0
$409$	−3.53953e10	−1.26489	−0.632444	−	0.774606i	$-0.717948\pi$
$409$	−3.53953e10	−1.26489	−0.632444	+	0.774606i	$0.717948\pi$
$410$	2.26925e10i	0.803059i
$411$	0	0
$412$	−2.04232e10	−0.708817
$413$	9.91343e9i	0.340741i
$414$	0	0
$415$	2.75311e10	0.928177
$416$	7.80712e10i	2.60686i
$417$	0	0
$418$	5.63559e10	1.84601
$419$	− 1.45926e10i	− 0.473453i	−0.971576	−	0.236726i	$-0.923926\pi$
$419$	− 1.45926e10i	− 0.473453i	0.971576	−	0.236726i	$-0.0760745\pi$
$420$	0	0
$421$	−2.65099e10	−0.843879	−0.421939	−	0.906624i	$-0.638650\pi$
$421$	−2.65099e10	−0.843879	−0.421939	+	0.906624i	$0.638650\pi$
$422$	− 3.13095e10i	− 0.987247i
$423$	0	0
$424$	1.07362e11	3.32191
$425$	− 1.70935e10i	− 0.523931i
$426$	0	0
$427$	−3.77919e10	−1.13681
$428$	1.18854e11i	3.54191i
$429$	0	0
$430$	1.39184e10	0.407113
$431$	− 4.61676e9i	− 0.133791i	−0.997760	−	0.0668957i	$-0.978691\pi$
$431$	− 4.61676e9i	− 0.133791i	0.997760	−	0.0668957i	$-0.0213095\pi$
$432$	0	0
$433$	−1.64494e10	−0.467948	−0.233974	−	0.972243i	$-0.575173\pi$
$433$	−1.64494e10	−0.467948	−0.233974	+	0.972243i	$0.575173\pi$
$434$	9.50235e9i	0.267838i
$435$	0	0
$436$	6.15188e10	1.70240
$437$	2.00148e10i	0.548816i
$438$	0	0
$439$	−2.61475e10	−0.703999	−0.351999	−	0.936000i	$-0.614498\pi$
$439$	−2.61475e10	−0.703999	−0.351999	+	0.936000i	$0.614498\pi$
$440$	− 1.07131e11i	− 2.85828i
$441$	0	0
$442$	−7.97335e10	−2.08906
$443$	4.31455e10i	1.12027i	0.828403	+	0.560133i	$0.189250\pi$
$443$	4.31455e10i	1.12027i	−0.828403	+	0.560133i	$0.810750\pi$
$444$	0	0
$445$	−2.32792e10	−0.593646
$446$	− 5.86660e10i	− 1.48268i
$447$	0	0
$448$	−2.44283e10	−0.606431
$449$	2.18287e10i	0.537084i	0.963268	+	0.268542i	$0.0865418\pi$
$449$	2.18287e10i	0.537084i	−0.963268	+	0.268542i	$0.913458\pi$
$450$	0	0
$451$	−1.18008e10	−0.285237
$452$	− 1.91219e11i	− 4.58117i
$453$	0	0
$454$	5.28863e10	1.24486
$455$	− 7.36679e10i	− 1.71883i
$456$	0	0
$457$	2.46905e10	0.566064	0.283032	−	0.959110i	$-0.408660\pi$
$457$	2.46905e10	0.566064	0.283032	+	0.959110i	$0.408660\pi$
$458$	− 4.69676e10i	− 1.06742i
$459$	0	0
$460$	6.57186e10	1.46777
$461$	2.61732e10i	0.579500i	0.957102	+	0.289750i	$0.0935722\pi$
$461$	2.61732e10i	0.579500i	−0.957102	+	0.289750i	$0.906428\pi$
$462$	0	0
$463$	7.04102e10	1.53219	0.766093	−	0.642729i	$-0.222198\pi$
$463$	7.04102e10	1.53219	0.766093	+	0.642729i	$0.222198\pi$
$464$	8.73928e10i	1.88540i
$465$	0	0
$466$	1.46502e11	3.10671
$467$	− 1.07220e10i	− 0.225428i	−0.993627	−	0.112714i	$-0.964046\pi$
$467$	− 1.07220e10i	− 0.225428i	0.993627	−	0.112714i	$-0.0359543\pi$
$468$	0	0
$469$	−1.17633e9	−0.0243130
$470$	2.23395e11i	4.57807i
$471$	0	0
$472$	−5.21458e10	−1.05063
$473$	7.23798e9i	0.144602i
$474$	0	0
$475$	−4.37006e10	−0.858445
$476$	− 7.12907e10i	− 1.38869i
$477$	0	0
$478$	−1.73352e11	−3.32060
$479$	1.40034e10i	0.266005i	0.991116	+	0.133003i	$0.0424619\pi$
$479$	1.40034e10i	0.266005i	−0.991116	+	0.133003i	$0.957538\pi$
$480$	0	0
$481$	3.02125e10	0.564424
$482$	− 6.64814e10i	− 1.23172i
$483$	0	0
$484$	−3.41014e10	−0.621429
$485$	1.12335e11i	2.03024i
$486$	0	0
$487$	1.72271e9	0.0306264	0.0153132	−	0.999883i	$-0.495125\pi$
$487$	1.72271e9	0.0306264	0.0153132	+	0.999883i	$0.495125\pi$
$488$	− 1.98790e11i	− 3.50521i
$489$	0	0
$490$	−4.58611e10	−0.795536
$491$	3.70419e10i	0.637334i	0.947867	+	0.318667i	$0.103235\pi$
$491$	3.70419e10i	0.637334i	−0.947867	+	0.318667i	$0.896765\pi$
$492$	0	0
$493$	−3.50859e10	−0.593943
$494$	2.03844e11i	3.42287i
$495$	0	0
$496$	−2.44027e10	−0.403191
$497$	− 5.75782e10i	− 0.943696i
$498$	0	0
$499$	−1.33497e10	−0.215312	−0.107656	−	0.994188i	$-0.534335\pi$
$499$	−1.33497e10	−0.215312	−0.107656	+	0.994188i	$0.534335\pi$
$500$	− 5.19787e10i	− 0.831659i
$501$	0	0
$502$	−9.75332e10	−1.53581
$503$	5.91175e10i	0.923516i	0.887006	+	0.461758i	$0.152781\pi$
$503$	5.91175e10i	0.923516i	−0.887006	+	0.461758i	$0.847219\pi$
$504$	0	0
$505$	3.31264e10	0.509341
$506$	4.85655e10i	0.740842i
$507$	0	0
$508$	−2.61968e11	−3.93363
$509$	− 3.32051e10i	− 0.494691i	−0.968927	−	0.247345i	$-0.920442\pi$
$509$	− 3.32051e10i	− 0.494691i	0.968927	−	0.247345i	$-0.0795583\pi$
$510$	0	0
$511$	2.52763e10	0.370706
$512$	1.38544e11i	2.01607i
$513$	0	0
$514$	8.96565e10	1.28449
$515$	2.76461e10i	0.393012i
$516$	0	0
$517$	−1.16172e11	−1.62608
$518$	3.83873e10i	0.533174i
$519$	0	0
$520$	3.87502e11	5.29981
$521$	2.53746e10i	0.344388i	0.985063	+	0.172194i	$0.0550856\pi$
$521$	2.53746e10i	0.344388i	−0.985063	+	0.172194i	$0.944914\pi$
$522$	0	0
$523$	1.27775e11	1.70781	0.853903	−	0.520432i	$-0.174229\pi$
$523$	1.27775e11	1.70781	0.853903	+	0.520432i	$0.174229\pi$
$524$	− 1.14871e11i	− 1.52365i
$525$	0	0
$526$	−5.80299e10	−0.758069
$527$	− 9.79703e9i	− 0.127014i
$528$	0	0
$529$	6.10629e10	0.779749
$530$	− 2.51029e11i	− 3.18141i
$531$	0	0
$532$	−1.82259e11	−2.27533
$533$	− 4.26845e10i	− 0.528885i
$534$	0	0
$535$	1.60888e11	1.96385
$536$	− 6.18763e9i	− 0.0749662i
$537$	0	0
$538$	−1.02683e11	−1.22566
$539$	− 2.38492e10i	− 0.282565i
$540$	0	0
$541$	6.16835e9	0.0720078	0.0360039	−	0.999352i	$-0.488537\pi$
$541$	6.16835e9	0.0720078	0.0360039	+	0.999352i	$0.488537\pi$
$542$	− 4.20997e10i	− 0.487845i
$543$	0	0
$544$	1.02272e11	1.16778
$545$	− 8.32758e10i	− 0.943915i
$546$	0	0
$547$	1.02309e11	1.14279	0.571394	−	0.820676i	$-0.306403\pi$
$547$	1.02309e11	1.14279	0.571394	+	0.820676i	$0.306403\pi$
$548$	− 7.47826e10i	− 0.829237i
$549$	0	0
$550$	−1.06038e11	−1.15881
$551$	8.96995e10i	0.973158i
$552$	0	0
$553$	−4.63910e10	−0.496059
$554$	− 9.84974e10i	− 1.04565i
$555$	0	0
$556$	−1.04620e11	−1.09475
$557$	− 1.00553e11i	− 1.04466i	−0.852743	−	0.522331i	$-0.825063\pi$
$557$	− 1.00553e11i	− 1.04466i	0.852743	−	0.522331i	$-0.174937\pi$
$558$	0	0
$559$	−2.61804e10	−0.268120
$560$	2.40374e11i	2.44419i
$561$	0	0
$562$	1.26112e11	1.26418
$563$	1.29004e11i	1.28402i	0.766697	+	0.642009i	$0.221899\pi$
$563$	1.29004e11i	1.28402i	−0.766697	+	0.642009i	$0.778101\pi$
$564$	0	0
$565$	−2.58846e11	−2.54008
$566$	2.91336e9i	0.0283876i
$567$	0	0
$568$	3.02868e11	2.90978
$569$	− 1.38131e10i	− 0.131778i	−0.997827	−	0.0658889i	$-0.979012\pi$
$569$	− 1.38131e10i	− 0.131778i	0.997827	−	0.0658889i	$-0.0209883\pi$
$570$	0	0
$571$	7.83161e10	0.736727	0.368363	−	0.929682i	$-0.379918\pi$
$571$	7.83161e10	0.736727	0.368363	+	0.929682i	$0.379918\pi$
$572$	3.48067e11i	3.25147i
$573$	0	0
$574$	5.42341e10	0.499602
$575$	− 3.76595e10i	− 0.344511i
$576$	0	0
$577$	−6.44834e10	−0.581761	−0.290880	−	0.956759i	$-0.593948\pi$
$577$	−6.44834e10	−0.581761	−0.290880	+	0.956759i	$0.593948\pi$
$578$	− 1.00595e11i	− 0.901291i
$579$	0	0
$580$	2.94528e11	2.60264
$581$	− 6.57980e10i	− 0.577442i
$582$	0	0
$583$	1.30542e11	1.13000
$584$	1.32956e11i	1.14303i
$585$	0	0
$586$	−2.65418e10	−0.225082
$587$	− 1.10003e11i	− 0.926512i	−0.886225	−	0.463256i	$-0.846681\pi$
$587$	− 1.10003e11i	− 0.926512i	0.886225	−	0.463256i	$-0.153319\pi$
$588$	0	0
$589$	−2.50468e10	−0.208109
$590$	1.21925e11i	1.00620i
$591$	0	0
$592$	−9.85814e10	−0.802616
$593$	9.33048e9i	0.0754545i	0.999288	+	0.0377273i	$0.0120118\pi$
$593$	9.33048e9i	0.0754545i	−0.999288	+	0.0377273i	$0.987988\pi$
$594$	0	0
$595$	−9.65037e10	−0.769974
$596$	4.44854e11i	3.52559i
$597$	0	0
$598$	−1.75665e11	−1.37367
$599$	− 8.12693e10i	− 0.631276i	−0.948880	−	0.315638i	$-0.897781\pi$
$599$	− 8.12693e10i	− 0.631276i	0.948880	−	0.315638i	$-0.102219\pi$
$600$	0	0
$601$	1.80583e11	1.38414	0.692069	−	0.721831i	$-0.256699\pi$
$601$	1.80583e11	1.38414	0.692069	+	0.721831i	$0.256699\pi$
$602$	− 3.32643e10i	− 0.253275i
$603$	0	0
$604$	−1.13059e11	−0.849488
$605$	4.61619e10i	0.344558i
$606$	0	0
$607$	1.15750e11	0.852644	0.426322	−	0.904571i	$-0.359809\pi$
$607$	1.15750e11	0.852644	0.426322	+	0.904571i	$0.359809\pi$
$608$	− 2.61465e11i	− 1.91337i
$609$	0	0
$610$	−4.64800e11	−3.35696
$611$	− 4.20205e11i	− 3.01506i
$612$	0	0
$613$	7.01257e10	0.496632	0.248316	−	0.968679i	$-0.420123\pi$
$613$	7.01257e10	0.496632	0.248316	+	0.968679i	$0.420123\pi$
$614$	− 9.88802e10i	− 0.695722i
$615$	0	0
$616$	−2.56038e11	−1.77820
$617$	1.79037e10i	0.123538i	0.998090	+	0.0617691i	$0.0196742\pi$
$617$	1.79037e10i	0.123538i	−0.998090	+	0.0617691i	$0.980326\pi$
$618$	0	0
$619$	1.35992e11	0.926300	0.463150	−	0.886280i	$-0.346719\pi$
$619$	1.35992e11	0.926300	0.463150	+	0.886280i	$0.346719\pi$
$620$	8.22410e10i	0.556572i
$621$	0	0
$622$	3.91007e11	2.61230
$623$	5.56362e10i	0.369322i
$624$	0	0
$625$	−1.82374e11	−1.19521
$626$	− 3.85793e11i	− 2.51221i
$627$	0	0
$628$	−5.92197e11	−3.80739
$629$	− 3.95778e10i	− 0.252842i
$630$	0	0
$631$	−2.50988e11	−1.58320	−0.791600	−	0.611040i	$-0.790752\pi$
$631$	−2.50988e11	−1.58320	−0.791600	+	0.611040i	$0.790752\pi$
$632$	− 2.44022e11i	− 1.52954i
$633$	0	0
$634$	−2.05322e11	−1.27080
$635$	3.54617e11i	2.18105i
$636$	0	0
$637$	8.62644e10	0.523931
$638$	2.17653e11i	1.31366i
$639$	0	0
$640$	6.10397e10	0.363825
$641$	− 2.57286e10i	− 0.152400i	−0.997093	−	0.0761999i	$-0.975721\pi$
$641$	− 2.57286e10i	− 0.152400i	0.997093	−	0.0761999i	$-0.0242787\pi$
$642$	0	0
$643$	4.29706e10	0.251378	0.125689	−	0.992070i	$-0.459886\pi$
$643$	4.29706e10	0.251378	0.125689	+	0.992070i	$0.459886\pi$
$644$	− 1.57064e11i	− 0.913134i
$645$	0	0
$646$	2.67032e11	1.53332
$647$	2.20975e11i	1.26103i	0.776177	+	0.630516i	$0.217157\pi$
$647$	2.20975e11i	1.26103i	−0.776177	+	0.630516i	$0.782843\pi$
$648$	0	0
$649$	−6.34045e10	−0.357389
$650$	− 3.83549e11i	− 2.14866i
$651$	0	0
$652$	7.02232e11	3.88589
$653$	2.99011e11i	1.64450i	0.569125	+	0.822251i	$0.307282\pi$
$653$	2.99011e11i	1.64450i	−0.569125	+	0.822251i	$0.692718\pi$
$654$	0	0
$655$	−1.55497e11	−0.844806
$656$	1.39277e11i	0.752079i
$657$	0	0
$658$	5.33904e11	2.84813
$659$	− 2.99806e11i	− 1.58964i	−0.606846	−	0.794820i	$-0.707565\pi$
$659$	− 2.99806e11i	− 1.58964i	0.606846	−	0.794820i	$-0.292435\pi$
$660$	0	0
$661$	−2.11448e11	−1.10764	−0.553820	−	0.832637i	$-0.686830\pi$
$661$	−2.11448e11	−1.10764	−0.553820	+	0.832637i	$0.686830\pi$
$662$	3.49001e11i	1.81717i
$663$	0	0
$664$	3.46105e11	1.78047
$665$	2.46718e11i	1.26158i
$666$	0	0
$667$	−7.72997e10	−0.390548
$668$	6.94954e10i	0.349020i
$669$	0	0
$670$	−1.44676e10	−0.0717956
$671$	− 2.41710e11i	− 1.19235i
$672$	0	0
$673$	−1.90166e11	−0.926985	−0.463493	−	0.886101i	$-0.653404\pi$
$673$	−1.90166e11	−0.926985	−0.463493	+	0.886101i	$0.653404\pi$
$674$	− 2.69077e11i	− 1.30388i
$675$	0	0
$676$	−7.63024e11	−3.65386
$677$	1.14815e11i	0.546570i	0.961933	+	0.273285i	$0.0881102\pi$
$677$	1.14815e11i	0.546570i	−0.961933	+	0.273285i	$0.911890\pi$
$678$	0	0
$679$	2.68475e11	1.26306
$680$	− 5.07621e11i	− 2.37413i
$681$	0	0
$682$	−6.07753e10	−0.280925
$683$	3.07050e11i	1.41100i	0.708710	+	0.705500i	$0.249277\pi$
$683$	3.07050e11i	1.41100i	−0.708710	+	0.705500i	$0.750723\pi$
$684$	0	0
$685$	−1.01231e11	−0.459780
$686$	4.42914e11i	1.99997i
$687$	0	0
$688$	8.54250e10	0.381269
$689$	4.72183e11i	2.09524i
$690$	0	0
$691$	−2.20597e11	−0.967582	−0.483791	−	0.875184i	$-0.660741\pi$
$691$	−2.20597e11	−0.967582	−0.483791	+	0.875184i	$0.660741\pi$
$692$	1.00516e12i	4.38339i
$693$	0	0
$694$	6.02812e11	2.59863
$695$	1.41620e11i	0.606996i
$696$	0	0
$697$	−5.59160e10	−0.236922
$698$	− 3.96226e11i	− 1.66925i
$699$	0	0
$700$	3.42936e11	1.42830
$701$	− 4.11156e11i	− 1.70269i	−0.524609	−	0.851343i	$-0.675788\pi$
$701$	− 4.11156e11i	− 1.70269i	0.524609	−	0.851343i	$-0.324212\pi$
$702$	0	0
$703$	−1.01183e11	−0.414274
$704$	− 1.56239e11i	− 0.636062i
$705$	0	0
$706$	2.87548e11	1.15742
$707$	− 7.91705e10i	− 0.316874i
$708$	0	0
$709$	1.41043e11	0.558169	0.279084	−	0.960267i	$-0.409969\pi$
$709$	1.41043e11	0.558169	0.279084	+	0.960267i	$0.409969\pi$
$710$	− 7.08150e11i	− 2.78671i
$711$	0	0
$712$	−2.92653e11	−1.13876
$713$	− 2.15844e10i	− 0.0835183i
$714$	0	0
$715$	4.71167e11	1.80281
$716$	− 2.53675e11i	− 0.965219i
$717$	0	0
$718$	4.91871e11	1.85078
$719$	− 1.87514e11i	− 0.701645i	−0.936442	−	0.350823i	$-0.885902\pi$
$719$	− 1.87514e11i	− 0.701645i	0.936442	−	0.350823i	$-0.114098\pi$
$720$	0	0
$721$	6.60730e10	0.244502
$722$	− 1.83473e11i	− 0.675187i
$723$	0	0
$724$	−7.07859e11	−2.57628
$725$	− 1.68777e11i	− 0.610887i
$726$	0	0
$727$	6.74621e10	0.241503	0.120751	−	0.992683i	$-0.461470\pi$
$727$	6.74621e10	0.241503	0.120751	+	0.992683i	$0.461470\pi$
$728$	− 9.26111e11i	− 3.29714i
$729$	0	0
$730$	3.10871e11	1.09469
$731$	3.42959e10i	0.120108i
$732$	0	0
$733$	−5.51348e11	−1.90990	−0.954948	−	0.296773i	$-0.904090\pi$
$733$	−5.51348e11	−1.90990	−0.954948	+	0.296773i	$0.904090\pi$
$734$	4.40553e11i	1.51780i
$735$	0	0
$736$	2.25321e11	0.767875
$737$	− 7.52360e9i	− 0.0255009i
$738$	0	0
$739$	−3.53894e11	−1.18658	−0.593288	−	0.804990i	$-0.702171\pi$
$739$	−3.53894e11	−1.18658	−0.593288	+	0.804990i	$0.702171\pi$
$740$	3.32235e11i	1.10795i
$741$	0	0
$742$	−5.99946e11	−1.97923
$743$	− 4.61456e10i	− 0.151417i	−0.997130	−	0.0757086i	$-0.975878\pi$
$743$	− 4.61456e10i	− 0.151417i	0.997130	−	0.0757086i	$-0.0241219\pi$
$744$	0	0
$745$	6.02183e11	1.95480
$746$	5.81893e11i	1.87883i
$747$	0	0
$748$	4.55962e11	1.45654
$749$	− 3.84515e11i	− 1.22176i
$750$	0	0
$751$	−2.78555e11	−0.875690	−0.437845	−	0.899050i	$-0.644258\pi$
$751$	−2.78555e11	−0.875690	−0.437845	+	0.899050i	$0.644258\pi$
$752$	1.37110e12i	4.28745i
$753$	0	0
$754$	−7.87270e11	−2.43578
$755$	1.53044e11i	0.471008i
$756$	0	0
$757$	−2.54100e11	−0.773786	−0.386893	−	0.922125i	$-0.626452\pi$
$757$	−2.54100e11	−0.773786	−0.386893	+	0.922125i	$0.626452\pi$
$758$	1.57228e11i	0.476268i
$759$	0	0
$760$	−1.29777e12	−3.88993
$761$	− 5.04366e11i	− 1.50386i	−0.659243	−	0.751930i	$-0.729123\pi$
$761$	− 5.04366e11i	− 1.50386i	0.659243	−	0.751930i	$-0.270877\pi$
$762$	0	0
$763$	−1.99025e11	−0.587233
$764$	− 9.79216e11i	− 2.87412i
$765$	0	0
$766$	−4.26029e11	−1.23744
$767$	− 2.29339e11i	− 0.662670i
$768$	0	0
$769$	2.16447e11	0.618937	0.309468	−	0.950910i	$-0.399849\pi$
$769$	2.16447e11	0.618937	0.309468	+	0.950910i	$0.399849\pi$
$770$	5.98655e11i	1.70300i
$771$	0	0
$772$	4.31850e11	1.21580
$773$	− 4.05778e11i	− 1.13650i	−0.822855	−	0.568252i	$-0.807620\pi$
$773$	− 4.05778e11i	− 1.13650i	0.822855	−	0.568252i	$-0.192380\pi$
$774$	0	0
$775$	4.71275e10	0.130637
$776$	1.41221e12i	3.89451i
$777$	0	0
$778$	5.30581e11	1.44821
$779$	1.42953e11i	0.388189i
$780$	0	0
$781$	3.68260e11	0.989806
$782$	2.30119e11i	0.615354i
$783$	0	0
$784$	−2.81475e11	−0.745034
$785$	8.01636e11i	2.11105i
$786$	0	0
$787$	6.28441e11	1.63820	0.819098	−	0.573653i	$-0.194474\pi$
$787$	6.28441e11	1.63820	0.819098	+	0.573653i	$0.194474\pi$
$788$	− 8.61066e11i	− 2.23322i
$789$	0	0
$790$	−5.70560e11	−1.46485
$791$	6.18630e11i	1.58025i
$792$	0	0
$793$	8.74286e11	2.21086
$794$	− 1.12473e12i	− 2.82986i
$795$	0	0
$796$	1.42469e12	3.54869
$797$	− 3.23780e11i	− 0.802447i	−0.915980	−	0.401224i	$-0.868585\pi$
$797$	− 3.23780e11i	− 0.802447i	0.915980	−	0.401224i	$-0.131415\pi$
$798$	0	0
$799$	−5.50462e11	−1.35064
$800$	4.91968e11i	1.20109i
$801$	0	0
$802$	−1.25395e12	−3.03097
$803$	1.61663e11i	0.388819i
$804$	0	0
$805$	−2.12613e11	−0.506297
$806$	− 2.19829e11i	− 0.520889i
$807$	0	0
$808$	4.16446e11	0.977042
$809$	− 4.65514e10i	− 0.108677i	−0.998523	−	0.0543386i	$-0.982695\pi$
$809$	− 4.65514e10i	− 0.108677i	0.998523	−	0.0543386i	$-0.0173050\pi$
$810$	0	0
$811$	9.90434e9	0.0228951	0.0114475	−	0.999934i	$-0.496356\pi$
$811$	9.90434e9	0.0228951	0.0114475	+	0.999934i	$0.496356\pi$
$812$	− 7.03908e11i	− 1.61917i
$813$	0	0
$814$	−2.45519e11	−0.559225
$815$	− 9.50587e11i	− 2.15457i
$816$	0	0
$817$	8.76797e10	0.196794
$818$	1.04040e12i	2.32375i
$819$	0	0
$820$	4.69385e11	1.03818
$821$	− 3.00799e11i	− 0.662069i	−0.943619	−	0.331034i	$-0.892602\pi$
$821$	− 3.00799e11i	− 0.662069i	0.943619	−	0.331034i	$-0.107398\pi$
$822$	0	0
$823$	3.64988e10	0.0795571	0.0397785	−	0.999209i	$-0.487335\pi$
$823$	3.64988e10	0.0795571	0.0397785	+	0.999209i	$0.487335\pi$
$824$	3.47551e11i	0.753894i
$825$	0	0
$826$	2.91394e11	0.625980
$827$	− 3.63233e11i	− 0.776540i	−0.921546	−	0.388270i	$-0.873073\pi$
$827$	− 3.63233e11i	− 0.776540i	0.921546	−	0.388270i	$-0.126927\pi$
$828$	0	0
$829$	−1.99283e11	−0.421941	−0.210971	−	0.977492i	$-0.567662\pi$
$829$	−1.99283e11	−0.421941	−0.210971	+	0.977492i	$0.567662\pi$
$830$	− 8.09245e11i	− 1.70517i
$831$	0	0
$832$	5.65130e11	1.17938
$833$	− 1.13005e11i	− 0.234702i
$834$	0	0
$835$	9.40735e10	0.193518
$836$	− 1.16570e12i	− 2.38650i
$837$	0	0
$838$	−4.28933e11	−0.869788
$839$	4.55688e11i	0.919644i	0.888011	+	0.459822i	$0.152087\pi$
$839$	4.55688e11i	0.919644i	−0.888011	+	0.459822i	$0.847913\pi$
$840$	0	0
$841$	1.53816e11	0.307481
$842$	7.79229e11i	1.55030i
$843$	0	0
$844$	−6.47623e11	−1.27630
$845$	1.03288e12i	2.02592i
$846$	0	0
$847$	1.10325e11	0.214358
$848$	− 1.54070e12i	− 2.97945i
$849$	0	0
$850$	−5.02443e11	−0.962523
$851$	− 8.71961e10i	− 0.166257i
$852$	0	0
$853$	−3.80568e11	−0.718846	−0.359423	−	0.933175i	$-0.617027\pi$
$853$	−3.80568e11	−0.718846	−0.359423	+	0.933175i	$0.617027\pi$
$854$	1.11085e12i	2.08845i
$855$	0	0
$856$	2.02259e12	3.76715
$857$	− 2.04675e11i	− 0.379438i	−0.981838	−	0.189719i	$-0.939242\pi$
$857$	− 2.04675e11i	− 0.379438i	0.981838	−	0.189719i	$-0.0607577\pi$
$858$	0	0
$859$	2.40882e11	0.442417	0.221208	−	0.975227i	$-0.429000\pi$
$859$	2.40882e11	0.442417	0.221208	+	0.975227i	$0.429000\pi$
$860$	− 2.87896e11i	− 0.526310i
$861$	0	0
$862$	−1.35704e11	−0.245790
$863$	8.63484e11i	1.55672i	0.627817	+	0.778361i	$0.283949\pi$
$863$	8.63484e11i	1.55672i	−0.627817	+	0.778361i	$0.716051\pi$
$864$	0	0
$865$	1.36065e12	2.43042
$866$	4.83510e11i	0.859675i
$867$	0	0
$868$	1.96552e11	0.346257
$869$	− 2.96709e11i	− 0.520297i
$870$	0	0
$871$	2.72135e10	0.0472837
$872$	− 1.04690e12i	− 1.81066i
$873$	0	0
$874$	5.88314e11	1.00824
$875$	1.68161e11i	0.286876i
$876$	0	0
$877$	−4.79159e11	−0.809993	−0.404997	−	0.914318i	$-0.632727\pi$
$877$	−4.79159e11	−0.809993	−0.404997	+	0.914318i	$0.632727\pi$
$878$	7.68576e11i	1.29333i
$879$	0	0
$880$	−1.53739e12	−2.56362
$881$	− 6.43291e11i	− 1.06783i	−0.845537	−	0.533917i	$-0.820720\pi$
$881$	− 6.43291e11i	− 1.06783i	0.845537	−	0.533917i	$-0.179280\pi$
$882$	0	0
$883$	−1.08619e12	−1.78674	−0.893371	−	0.449319i	$-0.851667\pi$
$883$	−1.08619e12	−1.78674	−0.893371	+	0.449319i	$0.851667\pi$
$884$	1.64925e12i	2.70071i
$885$	0	0
$886$	1.26821e12	2.05806
$887$	1.68383e11i	0.272022i	0.990707	+	0.136011i	$0.0434283\pi$
$887$	1.68383e11i	0.272022i	−0.990707	+	0.136011i	$0.956572\pi$
$888$	0	0
$889$	8.47519e11	1.35688
$890$	6.84266e11i	1.09060i
$891$	0	0
$892$	−1.21348e12	−1.91679
$893$	1.40729e12i	2.21299i
$894$	0	0
$895$	−3.43391e11	−0.535177
$896$	− 1.45882e11i	− 0.226344i
$897$	0	0
$898$	6.41630e11	0.986687
$899$	− 9.67336e10i	− 0.148094i
$900$	0	0
$901$	6.18552e11	0.938592
$902$	3.46871e11i	0.524013i
$903$	0	0
$904$	−3.25407e12	−4.87251
$905$	9.58204e11i	1.42845i
$906$	0	0
$907$	−6.69645e11	−0.989500	−0.494750	−	0.869035i	$-0.664740\pi$
$907$	−6.69645e11	−0.989500	−0.494750	+	0.869035i	$0.664740\pi$
$908$	− 1.09393e12i	− 1.60933i
$909$	0	0
$910$	−2.16539e12	−3.15769
$911$	− 1.19351e12i	− 1.73282i	−0.499337	−	0.866408i	$-0.666423\pi$
$911$	− 1.19351e12i	− 1.73282i	0.499337	−	0.866408i	$-0.333577\pi$
$912$	0	0
$913$	4.20832e11	0.605656
$914$	− 7.25750e11i	− 1.03993i
$915$	0	0
$916$	−9.71506e11	−1.37995
$917$	3.71631e11i	0.525575i
$918$	0	0
$919$	−7.65817e11	−1.07365	−0.536825	−	0.843693i	$-0.680377\pi$
$919$	−7.65817e11	−1.07365	−0.536825	+	0.843693i	$0.680377\pi$
$920$	− 1.11837e12i	− 1.56111i
$921$	0	0
$922$	7.69333e11	1.06461
$923$	1.33203e12i	1.83529i
$924$	0	0
$925$	1.90385e11	0.260055
$926$	− 2.06963e12i	− 2.81481i
$927$	0	0
$928$	1.00981e12	1.36159
$929$	− 4.26646e11i	− 0.572803i	−0.958110	−	0.286401i	$-0.907541\pi$
$929$	− 4.26646e11i	− 0.572803i	0.958110	−	0.286401i	$-0.0924591\pi$
$930$	0	0
$931$	−2.88905e11	−0.384553
$932$	− 3.03034e12i	− 4.01631i
$933$	0	0
$934$	−3.15160e11	−0.414137
$935$	− 6.17221e11i	− 0.807596i
$936$	0	0
$937$	1.35431e12	1.75695	0.878477	−	0.477784i	$-0.158560\pi$
$937$	1.35431e12	1.75695	0.878477	+	0.477784i	$0.158560\pi$
$938$	3.45769e10i	0.0446658i
$939$	0	0
$940$	4.62084e12	5.91847
$941$	− 1.88655e10i	− 0.0240608i	−0.999928	−	0.0120304i	$-0.996171\pi$
$941$	− 1.88655e10i	− 0.0240608i	0.999928	−	0.0120304i	$-0.00382948\pi$
$942$	0	0
$943$	−1.23192e11	−0.155788
$944$	7.48321e11i	0.942323i
$945$	0	0
$946$	2.12752e11	0.265650
$947$	− 6.03707e11i	− 0.750631i	−0.926897	−	0.375316i	$-0.877534\pi$
$947$	− 6.03707e11i	− 0.750631i	0.926897	−	0.375316i	$-0.122466\pi$
$948$	0	0
$949$	−5.84747e11	−0.720947
$950$	1.28453e12i	1.57706i
$951$	0	0
$952$	−1.21319e12	−1.47700
$953$	6.59941e10i	0.0800080i	0.999200	+	0.0400040i	$0.0127371\pi$
$953$	6.59941e10i	0.0800080i	−0.999200	+	0.0400040i	$0.987263\pi$
$954$	0	0
$955$	−1.32553e12	−1.59359
$956$	3.58571e12i	4.29283i
$957$	0	0
$958$	4.11613e11	0.488683
$959$	2.41937e11i	0.286040i
$960$	0	0
$961$	−8.25880e11	−0.968330
$962$	− 8.88061e11i	− 1.03691i
$963$	0	0
$964$	−1.37514e12	−1.59235
$965$	− 5.84580e11i	− 0.674116i
$966$	0	0
$967$	5.39935e11	0.617498	0.308749	−	0.951144i	$-0.400090\pi$
$967$	5.39935e11	0.617498	0.308749	+	0.951144i	$0.400090\pi$
$968$	5.80322e11i	0.660948i
$969$	0	0
$970$	3.30196e12	3.72979
$971$	− 7.23031e11i	− 0.813355i	−0.913572	−	0.406677i	$-0.866687\pi$
$971$	− 7.23031e11i	− 0.813355i	0.913572	−	0.406677i	$-0.133313\pi$
$972$	0	0
$973$	3.38466e11	0.377627
$974$	− 5.06370e10i	− 0.0562642i
$975$	0	0
$976$	−2.85274e12	−3.14386
$977$	− 4.79457e11i	− 0.526225i	−0.964765	−	0.263112i	$-0.915251\pi$
$977$	− 4.79457e11i	− 0.526225i	0.964765	−	0.263112i	$-0.0847490\pi$
$978$	0	0
$979$	−3.55839e11	−0.387367
$980$	9.48617e11i	1.02846i
$981$	0	0
$982$	1.08880e12	1.17086
$983$	9.92587e11i	1.06305i	0.847042	+	0.531526i	$0.178381\pi$
$983$	9.92587e11i	1.06305i	−0.847042	+	0.531526i	$0.821619\pi$
$984$	0	0
$985$	−1.16560e12	−1.23823
$986$	1.03131e12i	1.09114i
$987$	0	0
$988$	4.21643e12	4.42504
$989$	7.55591e10i	0.0789772i
$990$	0	0
$991$	9.53660e11	0.988778	0.494389	−	0.869241i	$-0.335392\pi$
$991$	9.53660e11	0.988778	0.494389	+	0.869241i	$0.335392\pi$
$992$	2.81969e11i	0.291175i
$993$	0	0
$994$	−1.69245e12	−1.73368
$995$	− 1.92855e12i	− 1.96761i
$996$	0	0
$997$	−1.51632e12	−1.53465	−0.767324	−	0.641259i	$-0.778412\pi$
$997$	−1.51632e12	−1.53465	−0.767324	+	0.641259i	$0.778412\pi$
$998$	3.92399e11i	0.395553i
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-29.3939i q^{2} -608.000 q^{4} +823.029i q^{5} +1967.00 q^{7} +10346.6i q^{8} +O(q^{10})\)	\(q-29.3939i q^{2} -608.000 q^{4} +823.029i q^{5} +1967.00 q^{7} +10346.6i q^{8} +24192.0 q^{10} +12580.6i q^{11} -45505.0 q^{13} -57817.8i q^{14} +148480. q^{16} +59610.8i q^{17} +152399. q^{19} -500401. i q^{20} +369792. q^{22} +131332. i q^{23} -286751. q^{25} +1.33757e6i q^{26} -1.19594e6 q^{28} +588583. i q^{29} -164350. q^{31} -1.71566e6i q^{32} +1.75219e6 q^{34} +1.61890e6i q^{35} -663937. q^{37} -4.47960e6i q^{38} -8.51558e6 q^{40} +938017. i q^{41} +575330. q^{43} -7.64899e6i q^{44} +3.86035e6 q^{46} +9.23426e6i q^{47} -1.89571e6 q^{49} +8.42872e6i q^{50} +2.76670e7 q^{52} -1.03765e7i q^{53} -1.03542e7 q^{55} +2.03519e7i q^{56} +1.73007e7 q^{58} +5.03987e6i q^{59} -1.92130e7 q^{61} +4.83088e6i q^{62} -1.24191e7 q^{64} -3.74519e7i q^{65} -598033. q^{67} -3.62434e7i q^{68} +4.75857e7 q^{70} -2.92721e7i q^{71} +1.28502e7 q^{73} +1.95157e7i q^{74} -9.26586e7 q^{76} +2.47460e7i q^{77} -2.35847e7 q^{79} +1.22203e8i q^{80} +2.75720e7 q^{82} -3.34509e7i q^{83} -4.90614e7 q^{85} -1.69112e7i q^{86} -1.30167e8 q^{88} +2.82848e7i q^{89} -8.95083e7 q^{91} -7.98498e7i q^{92} +2.71431e8 q^{94} +1.25429e8i q^{95} +1.36490e8 q^{97} +5.57223e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1216 q^{4} + 3934 q^{7}+O(q^{10}) \) 2 * q - 1216 * q^4 + 3934 * q^7	\( 2 q - 1216 q^{4} + 3934 q^{7} + 48384 q^{10} - 91010 q^{13} + 296960 q^{16} + 304798 q^{19} + 739584 q^{22} - 573502 q^{25} - 2391872 q^{28} - 328700 q^{31} + 3504384 q^{34} - 1327874 q^{37} - 17031168 q^{40} + 1150660 q^{43} + 7720704 q^{46} - 3791424 q^{49} + 55334080 q^{52} - 20708352 q^{55} + 34601472 q^{58} - 38425922 q^{61} - 24838144 q^{64} - 1196066 q^{67} + 95171328 q^{70} + 25700350 q^{73} - 185317184 q^{76} - 47169314 q^{79} + 55143936 q^{82} - 98122752 q^{85} - 260333568 q^{88} - 179016670 q^{91} + 542861568 q^{94} + 272979262 q^{97}+O(q^{100}) \) 2 * q - 1216 * q^4 + 3934 * q^7 + 48384 * q^10 - 91010 * q^13 + 296960 * q^16 + 304798 * q^19 + 739584 * q^22 - 573502 * q^25 - 2391872 * q^28 - 328700 * q^31 + 3504384 * q^34 - 1327874 * q^37 - 17031168 * q^40 + 1150660 * q^43 + 7720704 * q^46 - 3791424 * q^49 + 55334080 * q^52 - 20708352 * q^55 + 34601472 * q^58 - 38425922 * q^61 - 24838144 * q^64 - 1196066 * q^67 + 95171328 * q^70 + 25700350 * q^73 - 185317184 * q^76 - 47169314 * q^79 + 55143936 * q^82 - 98122752 * q^85 - 260333568 * q^88 - 179016670 * q^91 + 542861568 * q^94 + 272979262 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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