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Properties

Label 9.96.a.c.1.1
Level $9$
Weight $96$
Character 9.1
Self dual yes
Analytic conductor $514.382$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9,96,Mod(1,9)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9.1"); S:= CuspForms(chi, 96); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 96, names="a")
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 96 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,5835659138280] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(514.382317934\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{104}\cdot 3^{57}\cdot 5^{12}\cdot 7^{7}\cdot 11\cdot 13\cdot 19^{3} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.62871e13\) of defining polynomial
Character \(\chi\) \(=\) 9.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.90162e14 q^{2} +1.12612e29 q^{4} -1.39914e33 q^{5} +1.75361e40 q^{7} -2.84812e43 q^{8} +5.45891e47 q^{10} +5.41264e48 q^{11} +1.04831e52 q^{13} -6.84194e54 q^{14} +6.65124e57 q^{16} +2.56029e58 q^{17} -4.34204e60 q^{19} -1.57560e62 q^{20} -2.11181e63 q^{22} +7.48379e64 q^{23} -5.66766e65 q^{25} -4.09009e66 q^{26} +1.97479e69 q^{28} -2.03053e69 q^{29} +3.72805e70 q^{31} -1.46681e72 q^{32} -9.98929e72 q^{34} -2.45355e73 q^{35} -1.43964e74 q^{37} +1.69410e75 q^{38} +3.98491e76 q^{40} +4.16311e76 q^{41} +4.94148e76 q^{43} +6.09530e77 q^{44} -2.91989e79 q^{46} -4.71948e79 q^{47} +1.15068e80 q^{49} +2.21131e80 q^{50} +1.18052e81 q^{52} -9.21253e80 q^{53} -7.57304e81 q^{55} -4.99450e83 q^{56} +7.92238e83 q^{58} +1.44917e84 q^{59} +2.68780e84 q^{61} -1.45454e85 q^{62} +3.08809e86 q^{64} -1.46673e85 q^{65} +3.24337e86 q^{67} +2.88321e87 q^{68} +9.57282e87 q^{70} -2.78222e87 q^{71} +2.94316e88 q^{73} +5.61691e88 q^{74} -4.88967e89 q^{76} +9.49169e88 q^{77} +9.20074e89 q^{79} -9.30600e90 q^{80} -1.62429e91 q^{82} +2.26821e90 q^{83} -3.58220e91 q^{85} -1.92798e91 q^{86} -1.54158e92 q^{88} -3.95651e92 q^{89} +1.83833e92 q^{91} +8.42767e93 q^{92} +1.84136e94 q^{94} +6.07512e93 q^{95} -3.21613e94 q^{97} -4.48953e94 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5835659138280 q^{2} + 20\!\cdots\!84 q^{4} - 19\!\cdots\!60 q^{5} + 31\!\cdots\!00 q^{7} + 14\!\cdots\!60 q^{8} - 35\!\cdots\!40 q^{10} - 53\!\cdots\!16 q^{11} + 11\!\cdots\!40 q^{13} - 88\!\cdots\!08 q^{14}+ \cdots - 14\!\cdots\!60 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.90162e14 −1.96029 −0.980145 − 0.198283i \(-0.936464\pi\)
−0.980145 + 0.198283i \(0.936464\pi\)
\(3\) 0 0
\(4\) 1.12612e29 2.84274
\(5\) −1.39914e33 −0.880614 −0.440307 − 0.897847i \(-0.645130\pi\)
−0.440307 + 0.897847i \(0.645130\pi\)
\(6\) 0 0
\(7\) 1.75361e40 1.26409 0.632044 − 0.774932i \(-0.282216\pi\)
0.632044 + 0.774932i \(0.282216\pi\)
\(8\) −2.84812e43 −3.61230
\(9\) 0 0
\(10\) 5.45891e47 1.72626
\(11\) 5.41264e48 0.185036 0.0925181 − 0.995711i \(-0.470508\pi\)
0.0925181 + 0.995711i \(0.470508\pi\)
\(12\) 0 0
\(13\) 1.04831e52 0.128286 0.0641430 − 0.997941i \(-0.479569\pi\)
0.0641430 + 0.997941i \(0.479569\pi\)
\(14\) −6.84194e54 −2.47798
\(15\) 0 0
\(16\) 6.65124e57 4.23841
\(17\) 2.56029e58 0.916148 0.458074 − 0.888914i \(-0.348539\pi\)
0.458074 + 0.888914i \(0.348539\pi\)
\(18\) 0 0
\(19\) −4.34204e60 −0.788674 −0.394337 − 0.918966i \(-0.629026\pi\)
−0.394337 + 0.918966i \(0.629026\pi\)
\(20\) −1.57560e62 −2.50335
\(21\) 0 0
\(22\) −2.11181e63 −0.362725
\(23\) 7.48379e64 1.55614 0.778071 − 0.628177i \(-0.216198\pi\)
0.778071 + 0.628177i \(0.216198\pi\)
\(24\) 0 0
\(25\) −5.66766e65 −0.224519
\(26\) −4.09009e66 −0.251478
\(27\) 0 0
\(28\) 1.97479e69 3.59347
\(29\) −2.03053e69 −0.697759 −0.348880 − 0.937168i \(-0.613438\pi\)
−0.348880 + 0.937168i \(0.613438\pi\)
\(30\) 0 0
\(31\) 3.72805e70 0.539264 0.269632 − 0.962963i \(-0.413098\pi\)
0.269632 + 0.962963i \(0.413098\pi\)
\(32\) −1.46681e72 −4.69622
\(33\) 0 0
\(34\) −9.98929e72 −1.79592
\(35\) −2.45355e73 −1.11317
\(36\) 0 0
\(37\) −1.43964e74 −0.466305 −0.233153 − 0.972440i \(-0.574904\pi\)
−0.233153 + 0.972440i \(0.574904\pi\)
\(38\) 1.69410e75 1.54603
\(39\) 0 0
\(40\) 3.98491e76 3.18104
\(41\) 4.16311e76 1.02845 0.514227 − 0.857654i \(-0.328079\pi\)
0.514227 + 0.857654i \(0.328079\pi\)
\(42\) 0 0
\(43\) 4.94148e76 0.127088 0.0635441 − 0.997979i \(-0.479760\pi\)
0.0635441 + 0.997979i \(0.479760\pi\)
\(44\) 6.09530e77 0.526009
\(45\) 0 0
\(46\) −2.91989e79 −3.05049
\(47\) −4.71948e79 −1.77519 −0.887596 − 0.460622i \(-0.847626\pi\)
−0.887596 + 0.460622i \(0.847626\pi\)
\(48\) 0 0
\(49\) 1.15068e80 0.597919
\(50\) 2.21131e80 0.440122
\(51\) 0 0
\(52\) 1.18052e81 0.364683
\(53\) −9.21253e80 −0.115153 −0.0575765 − 0.998341i \(-0.518337\pi\)
−0.0575765 + 0.998341i \(0.518337\pi\)
\(54\) 0 0
\(55\) −7.57304e81 −0.162945
\(56\) −4.99450e83 −4.56626
\(57\) 0 0
\(58\) 7.92238e83 1.36781
\(59\) 1.44917e84 1.11083 0.555417 − 0.831572i \(-0.312558\pi\)
0.555417 + 0.831572i \(0.312558\pi\)
\(60\) 0 0
\(61\) 2.68780e84 0.422892 0.211446 − 0.977390i \(-0.432183\pi\)
0.211446 + 0.977390i \(0.432183\pi\)
\(62\) −1.45454e85 −1.05711
\(63\) 0 0
\(64\) 3.08809e86 4.96754
\(65\) −1.46673e85 −0.112970
\(66\) 0 0
\(67\) 3.24337e86 0.592165 0.296083 − 0.955162i \(-0.404320\pi\)
0.296083 + 0.955162i \(0.404320\pi\)
\(68\) 2.88321e87 2.60437
\(69\) 0 0
\(70\) 9.57282e87 2.18214
\(71\) −2.78222e87 −0.323311 −0.161656 − 0.986847i \(-0.551683\pi\)
−0.161656 + 0.986847i \(0.551683\pi\)
\(72\) 0 0
\(73\) 2.94316e88 0.914067 0.457033 − 0.889450i \(-0.348912\pi\)
0.457033 + 0.889450i \(0.348912\pi\)
\(74\) 5.61691e88 0.914094
\(75\) 0 0
\(76\) −4.88967e89 −2.24199
\(77\) 9.49169e88 0.233902
\(78\) 0 0
\(79\) 9.20074e89 0.670712 0.335356 − 0.942092i \(-0.391143\pi\)
0.335356 + 0.942092i \(0.391143\pi\)
\(80\) −9.30600e90 −3.73241
\(81\) 0 0
\(82\) −1.62429e91 −2.01607
\(83\) 2.26821e90 0.158298 0.0791488 − 0.996863i \(-0.474780\pi\)
0.0791488 + 0.996863i \(0.474780\pi\)
\(84\) 0 0
\(85\) −3.58220e91 −0.806773
\(86\) −1.92798e91 −0.249130
\(87\) 0 0
\(88\) −1.54158e92 −0.668406
\(89\) −3.95651e92 −1.00297 −0.501484 − 0.865167i \(-0.667212\pi\)
−0.501484 + 0.865167i \(0.667212\pi\)
\(90\) 0 0
\(91\) 1.83833e92 0.162165
\(92\) 8.42767e93 4.42370
\(93\) 0 0
\(94\) 1.84136e94 3.47989
\(95\) 6.07512e93 0.694518
\(96\) 0 0
\(97\) −3.21613e94 −1.36671 −0.683354 − 0.730087i \(-0.739479\pi\)
−0.683354 + 0.730087i \(0.739479\pi\)
\(98\) −4.48953e94 −1.17209
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9.96.a.c.1.1 8
3.2 odd 2 1.96.a.a.1.8 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.8 ✓ 8 3.2 odd 2
9.96.a.c.1.1 8 1.1 even 1 trivial