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Properties

Label 81.9.b.a.80.9
Level $81$
Weight $9$
Character 81.80
Analytic conductor $32.998$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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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] = [81,9,Mod(80,81)] 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("81.80"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 427 x^{12} - 1362 x^{11} + 135762 x^{10} - 371244 x^{9} + 18261508 x^{8} + \cdots + 872385888256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{63} \)
Twist minimal: no (minimal twist has level 9)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 80.9
Root \(-2.00397 + 3.47098i\) of defining polynomial
Character \(\chi\) \(=\) 81.80
Dual form 81.9.b.a.80.6

$q$-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)
 
\(f(q)\) \(=\) \(q+6.94197i q^{2} +207.809 q^{4} -382.663i q^{5} -935.032 q^{7} +3219.75i q^{8} +2656.43 q^{10} +5646.61i q^{11} -35326.1 q^{13} -6490.97i q^{14} +30847.7 q^{16} +152914. i q^{17} +191248. q^{19} -79520.8i q^{20} -39198.6 q^{22} -152781. i q^{23} +244194. q^{25} -245233. i q^{26} -194308. q^{28} +463741. i q^{29} +787911. q^{31} +1.03840e6i q^{32} -1.06153e6 q^{34} +357802. i q^{35} -1.10561e6 q^{37} +1.32764e6i q^{38} +1.23208e6 q^{40} +3.63633e6i q^{41} +3.02637e6 q^{43} +1.17342e6i q^{44} +1.06060e6 q^{46} +5.48838e6i q^{47} -4.89052e6 q^{49} +1.69519e6i q^{50} -7.34109e6 q^{52} +1.41381e7i q^{53} +2.16075e6 q^{55} -3.01057e6i q^{56} -3.21928e6 q^{58} -8.75358e6i q^{59} +6.94204e6 q^{61} +5.46965e6i q^{62} +688484. q^{64} +1.35180e7i q^{65} -1.41780e7 q^{67} +3.17770e7i q^{68} -2.48385e6 q^{70} -7.97888e6i q^{71} -4.61414e6 q^{73} -7.67508e6i q^{74} +3.97430e7 q^{76} -5.27976e6i q^{77} +2.75242e6 q^{79} -1.18043e7i q^{80} -2.52433e7 q^{82} +4.06790e7i q^{83} +5.85146e7 q^{85} +2.10089e7i q^{86} -1.81807e7 q^{88} -2.90621e7i q^{89} +3.30311e7 q^{91} -3.17494e7i q^{92} -3.81002e7 q^{94} -7.31834e7i q^{95} +9.16123e7 q^{97} -3.39498e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 1534 q^{4} - 1844 q^{7} - 516 q^{10} - 3368 q^{13} + 130562 q^{16} - 269630 q^{19} - 122622 q^{22} - 130354 q^{25} + 1075708 q^{28} + 328264 q^{31} - 1309986 q^{34} - 1671668 q^{37} - 1226652 q^{40}+ \cdots - 254098322 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

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

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.94197i 0.433873i 0.976186 + 0.216937i \(0.0696065\pi\)
−0.976186 + 0.216937i \(0.930394\pi\)
\(3\) 0 0
\(4\) 207.809 0.811754
\(5\) − 382.663i − 0.612261i −0.951990 − 0.306130i \(-0.900966\pi\)
0.951990 − 0.306130i \(-0.0990343\pi\)
\(6\) 0 0
\(7\) −935.032 −0.389435 −0.194717 − 0.980859i \(-0.562379\pi\)
−0.194717 + 0.980859i \(0.562379\pi\)
\(8\) 3219.75i 0.786071i
\(9\) 0 0
\(10\) 2656.43 0.265643
\(11\) 5646.61i 0.385671i 0.981231 + 0.192835i \(0.0617683\pi\)
−0.981231 + 0.192835i \(0.938232\pi\)
\(12\) 0 0
\(13\) −35326.1 −1.23687 −0.618433 − 0.785838i \(-0.712232\pi\)
−0.618433 + 0.785838i \(0.712232\pi\)
\(14\) − 6490.97i − 0.168965i
\(15\) 0 0
\(16\) 30847.7 0.470699
\(17\) 152914.i 1.83085i 0.402491 + 0.915424i \(0.368144\pi\)
−0.402491 + 0.915424i \(0.631856\pi\)
\(18\) 0 0
\(19\) 191248. 1.46751 0.733756 − 0.679413i \(-0.237766\pi\)
0.733756 + 0.679413i \(0.237766\pi\)
\(20\) − 79520.8i − 0.497005i
\(21\) 0 0
\(22\) −39198.6 −0.167332
\(23\) − 152781.i − 0.545958i −0.962020 − 0.272979i \(-0.911991\pi\)
0.962020 − 0.272979i \(-0.0880090\pi\)
\(24\) 0 0
\(25\) 244194. 0.625137
\(26\) − 245233.i − 0.536643i
\(27\) 0 0
\(28\) −194308. −0.316125
\(29\) 463741.i 0.655667i 0.944735 + 0.327834i \(0.106319\pi\)
−0.944735 + 0.327834i \(0.893681\pi\)
\(30\) 0 0
\(31\) 787911. 0.853160 0.426580 − 0.904450i \(-0.359718\pi\)
0.426580 + 0.904450i \(0.359718\pi\)
\(32\) 1.03840e6i 0.990295i
\(33\) 0 0
\(34\) −1.06153e6 −0.794355
\(35\) 357802.i 0.238435i
\(36\) 0 0
\(37\) −1.10561e6 −0.589921 −0.294960 − 0.955509i \(-0.595306\pi\)
−0.294960 + 0.955509i \(0.595306\pi\)
\(38\) 1.32764e6i 0.636714i
\(39\) 0 0
\(40\) 1.23208e6 0.481280
\(41\) 3.63633e6i 1.28685i 0.765509 + 0.643425i \(0.222487\pi\)
−0.765509 + 0.643425i \(0.777513\pi\)
\(42\) 0 0
\(43\) 3.02637e6 0.885213 0.442607 − 0.896716i \(-0.354054\pi\)
0.442607 + 0.896716i \(0.354054\pi\)
\(44\) 1.17342e6i 0.313070i
\(45\) 0 0
\(46\) 1.06060e6 0.236876
\(47\) 5.48838e6i 1.12474i 0.826885 + 0.562371i \(0.190111\pi\)
−0.826885 + 0.562371i \(0.809889\pi\)
\(48\) 0 0
\(49\) −4.89052e6 −0.848341
\(50\) 1.69519e6i 0.271230i
\(51\) 0 0
\(52\) −7.34109e6 −1.00403
\(53\) 1.41381e7i 1.79179i 0.444269 + 0.895894i \(0.353464\pi\)
−0.444269 + 0.895894i \(0.646536\pi\)
\(54\) 0 0
\(55\) 2.16075e6 0.236131
\(56\) − 3.01057e6i − 0.306123i
\(57\) 0 0
\(58\) −3.21928e6 −0.284476
\(59\) − 8.75358e6i − 0.722400i −0.932488 − 0.361200i \(-0.882367\pi\)
0.932488 − 0.361200i \(-0.117633\pi\)
\(60\) 0 0
\(61\) 6.94204e6 0.501381 0.250690 − 0.968067i \(-0.419342\pi\)
0.250690 + 0.968067i \(0.419342\pi\)
\(62\) 5.46965e6i 0.370163i
\(63\) 0 0
\(64\) 688484. 0.0410368
\(65\) 1.35180e7i 0.757284i
\(66\) 0 0
\(67\) −1.41780e7 −0.703583 −0.351791 − 0.936078i \(-0.614427\pi\)
−0.351791 + 0.936078i \(0.614427\pi\)
\(68\) 3.17770e7i 1.48620i
\(69\) 0 0
\(70\) −2.48385e6 −0.103451
\(71\) − 7.97888e6i − 0.313985i −0.987600 − 0.156992i \(-0.949820\pi\)
0.987600 − 0.156992i \(-0.0501798\pi\)
\(72\) 0 0
\(73\) −4.61414e6 −0.162480 −0.0812399 − 0.996695i \(-0.525888\pi\)
−0.0812399 + 0.996695i \(0.525888\pi\)
\(74\) − 7.67508e6i − 0.255951i
\(75\) 0 0
\(76\) 3.97430e7 1.19126
\(77\) − 5.27976e6i − 0.150194i
\(78\) 0 0
\(79\) 2.75242e6 0.0706652 0.0353326 − 0.999376i \(-0.488751\pi\)
0.0353326 + 0.999376i \(0.488751\pi\)
\(80\) − 1.18043e7i − 0.288191i
\(81\) 0 0
\(82\) −2.52433e7 −0.558329
\(83\) 4.06790e7i 0.857153i 0.903505 + 0.428577i \(0.140985\pi\)
−0.903505 + 0.428577i \(0.859015\pi\)
\(84\) 0 0
\(85\) 5.85146e7 1.12096
\(86\) 2.10089e7i 0.384070i
\(87\) 0 0
\(88\) −1.81807e7 −0.303165
\(89\) − 2.90621e7i − 0.463199i −0.972811 − 0.231599i \(-0.925604\pi\)
0.972811 − 0.231599i \(-0.0743958\pi\)
\(90\) 0 0
\(91\) 3.30311e7 0.481678
\(92\) − 3.17494e7i − 0.443184i
\(93\) 0 0
\(94\) −3.81002e7 −0.487995
\(95\) − 7.31834e7i − 0.898500i
\(96\) 0 0
\(97\) 9.16123e7 1.03482 0.517412 − 0.855736i \(-0.326895\pi\)
0.517412 + 0.855736i \(0.326895\pi\)
\(98\) − 3.39498e7i − 0.368072i
\(99\) 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 81.9.b.a.80.9 14
3.2 odd 2 inner 81.9.b.a.80.6 14
9.2 odd 6 27.9.d.a.17.3 14
9.4 even 3 27.9.d.a.8.3 14
9.5 odd 6 9.9.d.a.2.5 ✓ 14
9.7 even 3 9.9.d.a.5.5 yes 14
36.7 odd 6 144.9.q.a.113.4 14
36.11 even 6 432.9.q.a.17.3 14
36.23 even 6 144.9.q.a.65.4 14
36.31 odd 6 432.9.q.a.305.3 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.9.d.a.2.5 ✓ 14 9.5 odd 6
9.9.d.a.5.5 yes 14 9.7 even 3
27.9.d.a.8.3 14 9.4 even 3
27.9.d.a.17.3 14 9.2 odd 6
81.9.b.a.80.6 14 3.2 odd 2 inner
81.9.b.a.80.9 14 1.1 even 1 trivial
144.9.q.a.65.4 14 36.23 even 6
144.9.q.a.113.4 14 36.7 odd 6
432.9.q.a.17.3 14 36.11 even 6
432.9.q.a.305.3 14 36.31 odd 6