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Properties

Label 81.8.a.c.1.6
Level $81$
Weight $8$
Character 81.1
Self dual yes
Analytic conductor $25.303$
Analytic rank $1$
Dimension $6$
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] = [81,8,Mod(1,81)] 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("81.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 81.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-9] 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: \(25.3031870642\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 401x^{4} - 1212x^{3} + 17752x^{2} + 15108x - 22632 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{9} \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(5.85583\) of defining polynomial
Character \(\chi\) \(=\) 81.1

$q$-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)
 
\(f(q)\) \(=\) \(q+15.4889 q^{2} +111.906 q^{4} +105.528 q^{5} -1522.84 q^{7} -249.280 q^{8} +1634.51 q^{10} -2090.17 q^{11} -9335.82 q^{13} -23587.1 q^{14} -18185.0 q^{16} +20641.6 q^{17} -9369.08 q^{19} +11809.2 q^{20} -32374.4 q^{22} +77390.4 q^{23} -66988.8 q^{25} -144602. q^{26} -170415. q^{28} -162956. q^{29} +19659.9 q^{31} -249758. q^{32} +319715. q^{34} -160702. q^{35} +105494. q^{37} -145117. q^{38} -26306.1 q^{40} -137182. q^{41} +12261.5 q^{43} -233902. q^{44} +1.19869e6 q^{46} -540106. q^{47} +1.49549e6 q^{49} -1.03758e6 q^{50} -1.04473e6 q^{52} -17622.4 q^{53} -220572. q^{55} +379613. q^{56} -2.52401e6 q^{58} -388690. q^{59} +2.64758e6 q^{61} +304510. q^{62} -1.54079e6 q^{64} -985192. q^{65} +600683. q^{67} +2.30991e6 q^{68} -2.48910e6 q^{70} -1.63676e6 q^{71} -3.75812e6 q^{73} +1.63398e6 q^{74} -1.04845e6 q^{76} +3.18299e6 q^{77} -2.60479e6 q^{79} -1.91903e6 q^{80} -2.12479e6 q^{82} -385879. q^{83} +2.17826e6 q^{85} +189917. q^{86} +521037. q^{88} +9.20167e6 q^{89} +1.42170e7 q^{91} +8.66045e6 q^{92} -8.36564e6 q^{94} -988701. q^{95} +3.81171e6 q^{97} +2.31636e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{2} + 321 q^{4} - 180 q^{5} + 84 q^{7} - 2961 q^{8} + 126 q^{10} - 8460 q^{11} + 1848 q^{13} - 16272 q^{14} + 12417 q^{16} - 15282 q^{17} + 12216 q^{19} - 40788 q^{20} + 35001 q^{22} - 51588 q^{23}+ \cdots + 47916657 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 15.4889 1.36904 0.684519 − 0.728995i \(-0.260012\pi\)
0.684519 + 0.728995i \(0.260012\pi\)
\(3\) 0 0
\(4\) 111.906 0.874265
\(5\) 105.528 0.377549 0.188774 − 0.982020i \(-0.439549\pi\)
0.188774 + 0.982020i \(0.439549\pi\)
\(6\) 0 0
\(7\) −1522.84 −1.67807 −0.839036 − 0.544075i \(-0.816881\pi\)
−0.839036 + 0.544075i \(0.816881\pi\)
\(8\) −249.280 −0.172136
\(9\) 0 0
\(10\) 1634.51 0.516879
\(11\) −2090.17 −0.473486 −0.236743 − 0.971572i \(-0.576080\pi\)
−0.236743 + 0.971572i \(0.576080\pi\)
\(12\) 0 0
\(13\) −9335.82 −1.17856 −0.589279 − 0.807930i \(-0.700588\pi\)
−0.589279 + 0.807930i \(0.700588\pi\)
\(14\) −23587.1 −2.29735
\(15\) 0 0
\(16\) −18185.0 −1.10993
\(17\) 20641.6 1.01899 0.509497 − 0.860473i \(-0.329832\pi\)
0.509497 + 0.860473i \(0.329832\pi\)
\(18\) 0 0
\(19\) −9369.08 −0.313371 −0.156686 − 0.987649i \(-0.550081\pi\)
−0.156686 + 0.987649i \(0.550081\pi\)
\(20\) 11809.2 0.330078
\(21\) 0 0
\(22\) −32374.4 −0.648220
\(23\) 77390.4 1.32629 0.663147 − 0.748489i \(-0.269220\pi\)
0.663147 + 0.748489i \(0.269220\pi\)
\(24\) 0 0
\(25\) −66988.8 −0.857457
\(26\) −144602. −1.61349
\(27\) 0 0
\(28\) −170415. −1.46708
\(29\) −162956. −1.24073 −0.620366 − 0.784313i \(-0.713016\pi\)
−0.620366 + 0.784313i \(0.713016\pi\)
\(30\) 0 0
\(31\) 19659.9 0.118526 0.0592632 − 0.998242i \(-0.481125\pi\)
0.0592632 + 0.998242i \(0.481125\pi\)
\(32\) −249758. −1.34739
\(33\) 0 0
\(34\) 319715. 1.39504
\(35\) −160702. −0.633554
\(36\) 0 0
\(37\) 105494. 0.342390 0.171195 − 0.985237i \(-0.445237\pi\)
0.171195 + 0.985237i \(0.445237\pi\)
\(38\) −145117. −0.429017
\(39\) 0 0
\(40\) −26306.1 −0.0649899
\(41\) −137182. −0.310851 −0.155426 − 0.987848i \(-0.549675\pi\)
−0.155426 + 0.987848i \(0.549675\pi\)
\(42\) 0 0
\(43\) 12261.5 0.0235182 0.0117591 − 0.999931i \(-0.496257\pi\)
0.0117591 + 0.999931i \(0.496257\pi\)
\(44\) −233902. −0.413952
\(45\) 0 0
\(46\) 1.19869e6 1.81575
\(47\) −540106. −0.758816 −0.379408 − 0.925230i \(-0.623872\pi\)
−0.379408 + 0.925230i \(0.623872\pi\)
\(48\) 0 0
\(49\) 1.49549e6 1.81593
\(50\) −1.03758e6 −1.17389
\(51\) 0 0
\(52\) −1.04473e6 −1.03037
\(53\) −17622.4 −0.0162592 −0.00812962 − 0.999967i \(-0.502588\pi\)
−0.00812962 + 0.999967i \(0.502588\pi\)
\(54\) 0 0
\(55\) −220572. −0.178764
\(56\) 379613. 0.288857
\(57\) 0 0
\(58\) −2.52401e6 −1.69861
\(59\) −388690. −0.246389 −0.123194 − 0.992383i \(-0.539314\pi\)
−0.123194 + 0.992383i \(0.539314\pi\)
\(60\) 0 0
\(61\) 2.64758e6 1.49346 0.746731 − 0.665126i \(-0.231622\pi\)
0.746731 + 0.665126i \(0.231622\pi\)
\(62\) 304510. 0.162267
\(63\) 0 0
\(64\) −1.54079e6 −0.734708
\(65\) −985192. −0.444963
\(66\) 0 0
\(67\) 600683. 0.243996 0.121998 − 0.992530i \(-0.461070\pi\)
0.121998 + 0.992530i \(0.461070\pi\)
\(68\) 2.30991e6 0.890870
\(69\) 0 0
\(70\) −2.48910e6 −0.867360
\(71\) −1.63676e6 −0.542727 −0.271364 − 0.962477i \(-0.587475\pi\)
−0.271364 + 0.962477i \(0.587475\pi\)
\(72\) 0 0
\(73\) −3.75812e6 −1.13068 −0.565341 − 0.824857i \(-0.691255\pi\)
−0.565341 + 0.824857i \(0.691255\pi\)
\(74\) 1.63398e6 0.468744
\(75\) 0 0
\(76\) −1.04845e6 −0.273969
\(77\) 3.18299e6 0.794543
\(78\) 0 0
\(79\) −2.60479e6 −0.594400 −0.297200 − 0.954815i \(-0.596053\pi\)
−0.297200 + 0.954815i \(0.596053\pi\)
\(80\) −1.91903e6 −0.419051
\(81\) 0 0
\(82\) −2.12479e6 −0.425567
\(83\) −385879. −0.0740760 −0.0370380 − 0.999314i \(-0.511792\pi\)
−0.0370380 + 0.999314i \(0.511792\pi\)
\(84\) 0 0
\(85\) 2.17826e6 0.384720
\(86\) 189917. 0.0321972
\(87\) 0 0
\(88\) 521037. 0.0815041
\(89\) 9.20167e6 1.38357 0.691786 − 0.722102i \(-0.256824\pi\)
0.691786 + 0.722102i \(0.256824\pi\)
\(90\) 0 0
\(91\) 1.42170e7 1.97771
\(92\) 8.66045e6 1.15953
\(93\) 0 0
\(94\) −8.36564e6 −1.03885
\(95\) −988701. −0.118313
\(96\) 0 0
\(97\) 3.81171e6 0.424052 0.212026 − 0.977264i \(-0.431994\pi\)
0.212026 + 0.977264i \(0.431994\pi\)
\(98\) 2.31636e7 2.48607
\(99\) 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 81.8.a.c.1.6 6
3.2 odd 2 81.8.a.e.1.1 6
9.2 odd 6 9.8.c.a.4.6 ✓ 12
9.4 even 3 27.8.c.a.19.1 12
9.5 odd 6 9.8.c.a.7.6 yes 12
9.7 even 3 27.8.c.a.10.1 12
36.7 odd 6 432.8.i.c.145.2 12
36.11 even 6 144.8.i.c.49.6 12
36.23 even 6 144.8.i.c.97.6 12
36.31 odd 6 432.8.i.c.289.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.6 ✓ 12 9.2 odd 6
9.8.c.a.7.6 yes 12 9.5 odd 6
27.8.c.a.10.1 12 9.7 even 3
27.8.c.a.19.1 12 9.4 even 3
81.8.a.c.1.6 6 1.1 even 1 trivial
81.8.a.e.1.1 6 3.2 odd 2
144.8.i.c.49.6 12 36.11 even 6
144.8.i.c.97.6 12 36.23 even 6
432.8.i.c.145.2 12 36.7 odd 6
432.8.i.c.289.2 12 36.31 odd 6