↑

Properties

Label 81.8.a.c.1.5
Level $81$
Weight $8$
Character 81.1
Self dual yes
Analytic conductor $25.303$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.5
Root \(-16.1993\) 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)
 
\(f(q)\) \(=\) \(q+14.2319 q^{2} +74.5469 q^{4} -290.607 q^{5} +1111.88 q^{7} -760.739 q^{8} -4135.89 q^{10} -4490.71 q^{11} +2436.58 q^{13} +15824.2 q^{14} -20368.8 q^{16} -15905.4 q^{17} -49949.6 q^{19} -21663.9 q^{20} -63911.4 q^{22} -69385.1 q^{23} +6327.45 q^{25} +34677.2 q^{26} +82887.2 q^{28} +94071.6 q^{29} -19927.2 q^{31} -192512. q^{32} -226364. q^{34} -323120. q^{35} +331750. q^{37} -710878. q^{38} +221076. q^{40} -242266. q^{41} +831426. q^{43} -334769. q^{44} -987481. q^{46} +160011. q^{47} +412732. q^{49} +90051.7 q^{50} +181639. q^{52} -311589. q^{53} +1.30503e6 q^{55} -845850. q^{56} +1.33882e6 q^{58} +312353. q^{59} -57447.8 q^{61} -283601. q^{62} -132603. q^{64} -708087. q^{65} -4.10199e6 q^{67} -1.18570e6 q^{68} -4.59861e6 q^{70} +403110. q^{71} -823496. q^{73} +4.72144e6 q^{74} -3.72359e6 q^{76} -4.99313e6 q^{77} +978827. q^{79} +5.91931e6 q^{80} -3.44791e6 q^{82} +3.70408e6 q^{83} +4.62222e6 q^{85} +1.18328e7 q^{86} +3.41626e6 q^{88} -2.09023e6 q^{89} +2.70918e6 q^{91} -5.17244e6 q^{92} +2.27726e6 q^{94} +1.45157e7 q^{95} +3.50249e6 q^{97} +5.87396e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 9 q^{2} + 321 q^{4} - 180 q^{5} + 84 q^{7} - 2961 q^{8} + 126 q^{10} - 8460 q^{11} + 1848 q^{13} - 16272 q^{14} + 12417 q^{16} - 15282 q^{17} + 12216 q^{19} - 40788 q^{20} + 35001 q^{22} - 51588 q^{23}+ \cdots + 47916657 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.2319 1.25793 0.628967 − 0.777432i \(-0.283478\pi\)
0.628967 + 0.777432i \(0.283478\pi\)
\(3\) 0 0
\(4\) 74.5469 0.582398
\(5\) −290.607 −1.03971 −0.519854 − 0.854255i \(-0.674014\pi\)
−0.519854 + 0.854255i \(0.674014\pi\)
\(6\) 0 0
\(7\) 1111.88 1.22522 0.612611 − 0.790385i \(-0.290119\pi\)
0.612611 + 0.790385i \(0.290119\pi\)
\(8\) −760.739 −0.525316
\(9\) 0 0
\(10\) −4135.89 −1.30788
\(11\) −4490.71 −1.01728 −0.508640 − 0.860979i \(-0.669852\pi\)
−0.508640 + 0.860979i \(0.669852\pi\)
\(12\) 0 0
\(13\) 2436.58 0.307595 0.153797 − 0.988102i \(-0.450850\pi\)
0.153797 + 0.988102i \(0.450850\pi\)
\(14\) 15824.2 1.54125
\(15\) 0 0
\(16\) −20368.8 −1.24321
\(17\) −15905.4 −0.785187 −0.392593 − 0.919712i \(-0.628422\pi\)
−0.392593 + 0.919712i \(0.628422\pi\)
\(18\) 0 0
\(19\) −49949.6 −1.67069 −0.835343 − 0.549730i \(-0.814731\pi\)
−0.835343 + 0.549730i \(0.814731\pi\)
\(20\) −21663.9 −0.605523
\(21\) 0 0
\(22\) −63911.4 −1.27967
\(23\) −69385.1 −1.18910 −0.594550 − 0.804058i \(-0.702670\pi\)
−0.594550 + 0.804058i \(0.702670\pi\)
\(24\) 0 0
\(25\) 6327.45 0.0809914
\(26\) 34677.2 0.386934
\(27\) 0 0
\(28\) 82887.2 0.713566
\(29\) 94071.6 0.716251 0.358126 − 0.933673i \(-0.383416\pi\)
0.358126 + 0.933673i \(0.383416\pi\)
\(30\) 0 0
\(31\) −19927.2 −0.120138 −0.0600689 − 0.998194i \(-0.519132\pi\)
−0.0600689 + 0.998194i \(0.519132\pi\)
\(32\) −192512. −1.03856
\(33\) 0 0
\(34\) −226364. −0.987713
\(35\) −323120. −1.27387
\(36\) 0 0
\(37\) 331750. 1.07673 0.538363 − 0.842713i \(-0.319043\pi\)
0.538363 + 0.842713i \(0.319043\pi\)
\(38\) −710878. −2.10161
\(39\) 0 0
\(40\) 221076. 0.546175
\(41\) −242266. −0.548971 −0.274486 − 0.961591i \(-0.588508\pi\)
−0.274486 + 0.961591i \(0.588508\pi\)
\(42\) 0 0
\(43\) 831426. 1.59472 0.797359 − 0.603505i \(-0.206230\pi\)
0.797359 + 0.603505i \(0.206230\pi\)
\(44\) −334769. −0.592462
\(45\) 0 0
\(46\) −987481. −1.49581
\(47\) 160011. 0.224805 0.112403 − 0.993663i \(-0.464145\pi\)
0.112403 + 0.993663i \(0.464145\pi\)
\(48\) 0 0
\(49\) 412732. 0.501167
\(50\) 90051.7 0.101882
\(51\) 0 0
\(52\) 181639. 0.179142
\(53\) −311589. −0.287486 −0.143743 − 0.989615i \(-0.545914\pi\)
−0.143743 + 0.989615i \(0.545914\pi\)
\(54\) 0 0
\(55\) 1.30503e6 1.05767
\(56\) −845850. −0.643628
\(57\) 0 0
\(58\) 1.33882e6 0.900997
\(59\) 312353. 0.197999 0.0989997 − 0.995087i \(-0.468436\pi\)
0.0989997 + 0.995087i \(0.468436\pi\)
\(60\) 0 0
\(61\) −57447.8 −0.0324055 −0.0162028 − 0.999869i \(-0.505158\pi\)
−0.0162028 + 0.999869i \(0.505158\pi\)
\(62\) −283601. −0.151125
\(63\) 0 0
\(64\) −132603. −0.0632302
\(65\) −708087. −0.319808
\(66\) 0 0
\(67\) −4.10199e6 −1.66622 −0.833111 − 0.553105i \(-0.813443\pi\)
−0.833111 + 0.553105i \(0.813443\pi\)
\(68\) −1.18570e6 −0.457291
\(69\) 0 0
\(70\) −4.59861e6 −1.60245
\(71\) 403110. 0.133666 0.0668328 − 0.997764i \(-0.478711\pi\)
0.0668328 + 0.997764i \(0.478711\pi\)
\(72\) 0 0
\(73\) −823496. −0.247760 −0.123880 − 0.992297i \(-0.539534\pi\)
−0.123880 + 0.992297i \(0.539534\pi\)
\(74\) 4.72144e6 1.35445
\(75\) 0 0
\(76\) −3.72359e6 −0.973003
\(77\) −4.99313e6 −1.24639
\(78\) 0 0
\(79\) 978827. 0.223363 0.111682 − 0.993744i \(-0.464376\pi\)
0.111682 + 0.993744i \(0.464376\pi\)
\(80\) 5.91931e6 1.29258
\(81\) 0 0
\(82\) −3.44791e6 −0.690569
\(83\) 3.70408e6 0.711060 0.355530 − 0.934665i \(-0.384300\pi\)
0.355530 + 0.934665i \(0.384300\pi\)
\(84\) 0 0
\(85\) 4.62222e6 0.816365
\(86\) 1.18328e7 2.00605
\(87\) 0 0
\(88\) 3.41626e6 0.534394
\(89\) −2.09023e6 −0.314289 −0.157145 − 0.987576i \(-0.550229\pi\)
−0.157145 + 0.987576i \(0.550229\pi\)
\(90\) 0 0
\(91\) 2.70918e6 0.376872
\(92\) −5.17244e6 −0.692530
\(93\) 0 0
\(94\) 2.27726e6 0.282790
\(95\) 1.45157e7 1.73702
\(96\) 0 0
\(97\) 3.50249e6 0.389651 0.194826 − 0.980838i \(-0.437586\pi\)
0.194826 + 0.980838i \(0.437586\pi\)
\(98\) 5.87396e6 0.630435
\(99\) 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.1.5 6
3.2 odd 2 81.8.a.e.1.2 6
9.2 odd 6 9.8.c.a.4.5 ✓ 12
9.4 even 3 27.8.c.a.19.2 12
9.5 odd 6 9.8.c.a.7.5 yes 12
9.7 even 3 27.8.c.a.10.2 12
36.7 odd 6 432.8.i.c.145.5 12
36.11 even 6 144.8.i.c.49.1 12
36.23 even 6 144.8.i.c.97.1 12
36.31 odd 6 432.8.i.c.289.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.5 ✓ 12 9.2 odd 6
9.8.c.a.7.5 yes 12 9.5 odd 6
27.8.c.a.10.2 12 9.7 even 3
27.8.c.a.19.2 12 9.4 even 3
81.8.a.c.1.5 6 1.1 even 1 trivial
81.8.a.e.1.2 6 3.2 odd 2
144.8.i.c.49.1 12 36.11 even 6
144.8.i.c.97.1 12 36.23 even 6
432.8.i.c.145.5 12 36.7 odd 6
432.8.i.c.289.5 12 36.31 odd 6