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Properties

Label 9.96.a.c.1.4
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 1.4
Root \(4.79736e12\) 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)
 
\(f(q)\) \(=\) \(q-1.14407e14 q^{2} -2.65251e28 q^{4} +1.07741e33 q^{5} +1.86215e40 q^{7} +7.56680e42 q^{8} -1.23264e47 q^{10} -1.86898e49 q^{11} -5.82728e52 q^{13} -2.13044e54 q^{14} +1.85069e56 q^{16} +1.12709e58 q^{17} +8.30862e60 q^{19} -2.85785e61 q^{20} +2.13824e63 q^{22} -1.09549e64 q^{23} -1.36353e66 q^{25} +6.66683e66 q^{26} -4.93937e68 q^{28} -2.28760e69 q^{29} +3.05026e70 q^{31} -3.20925e71 q^{32} -1.28947e72 q^{34} +2.00631e73 q^{35} +3.99750e74 q^{37} -9.50566e74 q^{38} +8.15257e75 q^{40} +3.81589e76 q^{41} +2.50487e77 q^{43} +4.95747e77 q^{44} +1.25332e78 q^{46} -2.28370e79 q^{47} +1.54313e80 q^{49} +1.55998e80 q^{50} +1.54569e81 q^{52} +5.87919e80 q^{53} -2.01366e82 q^{55} +1.40905e83 q^{56} +2.61719e83 q^{58} -1.48881e84 q^{59} -5.46013e84 q^{61} -3.48972e84 q^{62} +2.93848e85 q^{64} -6.27839e85 q^{65} -6.62877e86 q^{67} -2.98960e86 q^{68} -2.29536e87 q^{70} -5.25075e87 q^{71} -2.09882e88 q^{73} -4.57343e88 q^{74} -2.20387e89 q^{76} -3.48032e89 q^{77} +6.42811e89 q^{79} +1.99396e89 q^{80} -4.36566e90 q^{82} -1.28783e91 q^{83} +1.21434e91 q^{85} -2.86575e91 q^{86} -1.41422e92 q^{88} +5.79584e92 q^{89} -1.08513e93 q^{91} +2.90580e92 q^{92} +2.61272e93 q^{94} +8.95182e93 q^{95} -1.99071e94 q^{97} -1.76545e94 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 5835659138280 q^{2} + 20\!\cdots\!84 q^{4} - 19\!\cdots\!60 q^{5} + 31\!\cdots\!00 q^{7} + 14\!\cdots\!60 q^{8} - 35\!\cdots\!40 q^{10} - 53\!\cdots\!16 q^{11} + 11\!\cdots\!40 q^{13} - 88\!\cdots\!08 q^{14}+ \cdots - 14\!\cdots\!60 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.14407e14 −0.574816 −0.287408 − 0.957808i \(-0.592794\pi\)
−0.287408 + 0.957808i \(0.592794\pi\)
\(3\) 0 0
\(4\) −2.65251e28 −0.669587
\(5\) 1.07741e33 0.678121 0.339061 − 0.940765i \(-0.389891\pi\)
0.339061 + 0.940765i \(0.389891\pi\)
\(6\) 0 0
\(7\) 1.86215e40 1.34233 0.671163 − 0.741309i \(-0.265795\pi\)
0.671163 + 0.741309i \(0.265795\pi\)
\(8\) 7.56680e42 0.959705
\(9\) 0 0
\(10\) −1.23264e47 −0.389795
\(11\) −1.86898e49 −0.638927 −0.319464 − 0.947599i \(-0.603503\pi\)
−0.319464 + 0.947599i \(0.603503\pi\)
\(12\) 0 0
\(13\) −5.82728e52 −0.713110 −0.356555 − 0.934274i \(-0.616049\pi\)
−0.356555 + 0.934274i \(0.616049\pi\)
\(14\) −2.13044e54 −0.771591
\(15\) 0 0
\(16\) 1.85069e56 0.117933
\(17\) 1.12709e58 0.403305 0.201652 − 0.979457i \(-0.435369\pi\)
0.201652 + 0.979457i \(0.435369\pi\)
\(18\) 0 0
\(19\) 8.30862e60 1.50915 0.754576 − 0.656213i \(-0.227843\pi\)
0.754576 + 0.656213i \(0.227843\pi\)
\(20\) −2.85785e61 −0.454061
\(21\) 0 0
\(22\) 2.13824e63 0.367265
\(23\) −1.09549e64 −0.227791 −0.113896 − 0.993493i \(-0.536333\pi\)
−0.113896 + 0.993493i \(0.536333\pi\)
\(24\) 0 0
\(25\) −1.36353e66 −0.540152
\(26\) 6.66683e66 0.409907
\(27\) 0 0
\(28\) −4.93937e68 −0.898804
\(29\) −2.28760e69 −0.786097 −0.393048 − 0.919518i \(-0.628579\pi\)
−0.393048 + 0.919518i \(0.628579\pi\)
\(30\) 0 0
\(31\) 3.05026e70 0.441221 0.220610 − 0.975362i \(-0.429195\pi\)
0.220610 + 0.975362i \(0.429195\pi\)
\(32\) −3.20925e71 −1.02749
\(33\) 0 0
\(34\) −1.28947e72 −0.231826
\(35\) 2.00631e73 0.910260
\(36\) 0 0
\(37\) 3.99750e74 1.29481 0.647406 − 0.762145i \(-0.275854\pi\)
0.647406 + 0.762145i \(0.275854\pi\)
\(38\) −9.50566e74 −0.867484
\(39\) 0 0
\(40\) 8.15257e75 0.650796
\(41\) 3.81589e76 0.942677 0.471338 − 0.881952i \(-0.343771\pi\)
0.471338 + 0.881952i \(0.343771\pi\)
\(42\) 0 0
\(43\) 2.50487e77 0.644219 0.322110 − 0.946702i \(-0.395608\pi\)
0.322110 + 0.946702i \(0.395608\pi\)
\(44\) 4.95747e77 0.427817
\(45\) 0 0
\(46\) 1.25332e78 0.130938
\(47\) −2.28370e79 −0.858994 −0.429497 − 0.903068i \(-0.641309\pi\)
−0.429497 + 0.903068i \(0.641309\pi\)
\(48\) 0 0
\(49\) 1.54313e80 0.801841
\(50\) 1.55998e80 0.310488
\(51\) 0 0
\(52\) 1.54569e81 0.477489
\(53\) 5.87919e80 0.0734875 0.0367438 − 0.999325i \(-0.488301\pi\)
0.0367438 + 0.999325i \(0.488301\pi\)
\(54\) 0 0
\(55\) −2.01366e82 −0.433270
\(56\) 1.40905e83 1.28824
\(57\) 0 0
\(58\) 2.61719e83 0.451861
\(59\) −1.48881e84 −1.14122 −0.570610 − 0.821221i \(-0.693293\pi\)
−0.570610 + 0.821221i \(0.693293\pi\)
\(60\) 0 0
\(61\) −5.46013e84 −0.859084 −0.429542 − 0.903047i \(-0.641325\pi\)
−0.429542 + 0.903047i \(0.641325\pi\)
\(62\) −3.48972e84 −0.253621
\(63\) 0 0
\(64\) 2.93848e85 0.472688
\(65\) −6.27839e85 −0.483575
\(66\) 0 0
\(67\) −6.62877e86 −1.21026 −0.605132 − 0.796125i \(-0.706880\pi\)
−0.605132 + 0.796125i \(0.706880\pi\)
\(68\) −2.98960e86 −0.270047
\(69\) 0 0
\(70\) −2.29536e87 −0.523232
\(71\) −5.25075e87 −0.610168 −0.305084 − 0.952325i \(-0.598685\pi\)
−0.305084 + 0.952325i \(0.598685\pi\)
\(72\) 0 0
\(73\) −2.09882e88 −0.651838 −0.325919 − 0.945398i \(-0.605674\pi\)
−0.325919 + 0.945398i \(0.605674\pi\)
\(74\) −4.57343e88 −0.744279
\(75\) 0 0
\(76\) −2.20387e89 −1.01051
\(77\) −3.48032e89 −0.857649
\(78\) 0 0
\(79\) 6.42811e89 0.468594 0.234297 − 0.972165i \(-0.424721\pi\)
0.234297 + 0.972165i \(0.424721\pi\)
\(80\) 1.99396e89 0.0799727
\(81\) 0 0
\(82\) −4.36566e90 −0.541866
\(83\) −1.28783e91 −0.898770 −0.449385 − 0.893338i \(-0.648357\pi\)
−0.449385 + 0.893338i \(0.648357\pi\)
\(84\) 0 0
\(85\) 1.21434e91 0.273489
\(86\) −2.86575e91 −0.370307
\(87\) 0 0
\(88\) −1.41422e92 −0.613182
\(89\) 5.79584e92 1.46923 0.734617 − 0.678482i \(-0.237362\pi\)
0.734617 + 0.678482i \(0.237362\pi\)
\(90\) 0 0
\(91\) −1.08513e93 −0.957227
\(92\) 2.90580e92 0.152526
\(93\) 0 0
\(94\) 2.61272e93 0.493764
\(95\) 8.95182e93 1.02339
\(96\) 0 0
\(97\) −1.99071e94 −0.845960 −0.422980 − 0.906139i \(-0.639016\pi\)
−0.422980 + 0.906139i \(0.639016\pi\)
\(98\) −1.76545e94 −0.460911
\(99\) 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.1.4 8
3.2 odd 2 1.96.a.a.1.5 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.96.a.a.1.5 ✓ 8 3.2 odd 2
9.96.a.c.1.4 8 1.1 even 1 trivial