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Properties

Label 81.8.a.a.1.1
Level $81$
Weight $8$
Character 81.1
Self dual yes
Analytic conductor $25.303$
Analytic rank $0$
Dimension $4$
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 = [4,-15] 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: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 342x^{2} - 352x + 2512 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-17.2001\) 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)
 
\(f(q)\) \(=\) \(q-21.2001 q^{2} +321.445 q^{4} -352.223 q^{5} +762.913 q^{7} -4101.05 q^{8} +7467.17 q^{10} +7149.89 q^{11} -7136.69 q^{13} -16173.8 q^{14} +45797.9 q^{16} -10229.9 q^{17} -31729.0 q^{19} -113220. q^{20} -151578. q^{22} -49336.6 q^{23} +45936.1 q^{25} +151299. q^{26} +245234. q^{28} -60580.2 q^{29} +156352. q^{31} -445985. q^{32} +216875. q^{34} -268716. q^{35} -112163. q^{37} +672659. q^{38} +1.44449e6 q^{40} +586257. q^{41} +161964. q^{43} +2.29829e6 q^{44} +1.04594e6 q^{46} +130863. q^{47} -241507. q^{49} -973851. q^{50} -2.29405e6 q^{52} +1.63679e6 q^{53} -2.51836e6 q^{55} -3.12875e6 q^{56} +1.28431e6 q^{58} +80425.4 q^{59} +1.72502e6 q^{61} -3.31469e6 q^{62} +3.59281e6 q^{64} +2.51371e6 q^{65} -2.34085e6 q^{67} -3.28834e6 q^{68} +5.69680e6 q^{70} -469125. q^{71} +3.76695e6 q^{73} +2.37787e6 q^{74} -1.01991e7 q^{76} +5.45474e6 q^{77} +8.31434e6 q^{79} -1.61311e7 q^{80} -1.24287e7 q^{82} -988959. q^{83} +3.60320e6 q^{85} -3.43365e6 q^{86} -2.93221e7 q^{88} +9.33980e6 q^{89} -5.44467e6 q^{91} -1.58590e7 q^{92} -2.77430e6 q^{94} +1.11757e7 q^{95} +8.88487e6 q^{97} +5.11998e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 15 q^{2} + 229 q^{4} - 192 q^{5} + 800 q^{7} - 2505 q^{8} + 5469 q^{10} + 5016 q^{11} - 2200 q^{13} - 19452 q^{14} + 40849 q^{16} + 19620 q^{17} + 11240 q^{19} - 96951 q^{20} - 41280 q^{22} - 154560 q^{23}+ \cdots - 15771015 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.2001 −1.87384 −0.936922 − 0.349540i \(-0.886338\pi\)
−0.936922 + 0.349540i \(0.886338\pi\)
\(3\) 0 0
\(4\) 321.445 2.51129
\(5\) −352.223 −1.26015 −0.630076 − 0.776534i \(-0.716976\pi\)
−0.630076 + 0.776534i \(0.716976\pi\)
\(6\) 0 0
\(7\) 762.913 0.840682 0.420341 − 0.907366i \(-0.361910\pi\)
0.420341 + 0.907366i \(0.361910\pi\)
\(8\) −4101.05 −2.83192
\(9\) 0 0
\(10\) 7467.17 2.36133
\(11\) 7149.89 1.61966 0.809832 − 0.586662i \(-0.199558\pi\)
0.809832 + 0.586662i \(0.199558\pi\)
\(12\) 0 0
\(13\) −7136.69 −0.900938 −0.450469 − 0.892792i \(-0.648743\pi\)
−0.450469 + 0.892792i \(0.648743\pi\)
\(14\) −16173.8 −1.57531
\(15\) 0 0
\(16\) 45797.9 2.79528
\(17\) −10229.9 −0.505009 −0.252505 − 0.967596i \(-0.581254\pi\)
−0.252505 + 0.967596i \(0.581254\pi\)
\(18\) 0 0
\(19\) −31729.0 −1.06125 −0.530627 − 0.847606i \(-0.678043\pi\)
−0.530627 + 0.847606i \(0.678043\pi\)
\(20\) −113220. −3.16460
\(21\) 0 0
\(22\) −151578. −3.03499
\(23\) −49336.6 −0.845517 −0.422758 − 0.906242i \(-0.638938\pi\)
−0.422758 + 0.906242i \(0.638938\pi\)
\(24\) 0 0
\(25\) 45936.1 0.587982
\(26\) 151299. 1.68822
\(27\) 0 0
\(28\) 245234. 2.11119
\(29\) −60580.2 −0.461252 −0.230626 − 0.973043i \(-0.574077\pi\)
−0.230626 + 0.973043i \(0.574077\pi\)
\(30\) 0 0
\(31\) 156352. 0.942625 0.471312 − 0.881966i \(-0.343780\pi\)
0.471312 + 0.881966i \(0.343780\pi\)
\(32\) −445985. −2.40600
\(33\) 0 0
\(34\) 216875. 0.946308
\(35\) −268716. −1.05939
\(36\) 0 0
\(37\) −112163. −0.364036 −0.182018 − 0.983295i \(-0.558263\pi\)
−0.182018 + 0.983295i \(0.558263\pi\)
\(38\) 672659. 1.98862
\(39\) 0 0
\(40\) 1.44449e6 3.56864
\(41\) 586257. 1.32845 0.664224 − 0.747534i \(-0.268762\pi\)
0.664224 + 0.747534i \(0.268762\pi\)
\(42\) 0 0
\(43\) 161964. 0.310655 0.155328 − 0.987863i \(-0.450357\pi\)
0.155328 + 0.987863i \(0.450357\pi\)
\(44\) 2.29829e6 4.06744
\(45\) 0 0
\(46\) 1.04594e6 1.58437
\(47\) 130863. 0.183854 0.0919271 − 0.995766i \(-0.470697\pi\)
0.0919271 + 0.995766i \(0.470697\pi\)
\(48\) 0 0
\(49\) −241507. −0.293254
\(50\) −973851. −1.10179
\(51\) 0 0
\(52\) −2.29405e6 −2.26252
\(53\) 1.63679e6 1.51018 0.755090 − 0.655621i \(-0.227593\pi\)
0.755090 + 0.655621i \(0.227593\pi\)
\(54\) 0 0
\(55\) −2.51836e6 −2.04102
\(56\) −3.12875e6 −2.38074
\(57\) 0 0
\(58\) 1.28431e6 0.864313
\(59\) 80425.4 0.0509813 0.0254907 − 0.999675i \(-0.491885\pi\)
0.0254907 + 0.999675i \(0.491885\pi\)
\(60\) 0 0
\(61\) 1.72502e6 0.973058 0.486529 − 0.873664i \(-0.338263\pi\)
0.486529 + 0.873664i \(0.338263\pi\)
\(62\) −3.31469e6 −1.76633
\(63\) 0 0
\(64\) 3.59281e6 1.71318
\(65\) 2.51371e6 1.13532
\(66\) 0 0
\(67\) −2.34085e6 −0.950851 −0.475426 − 0.879756i \(-0.657706\pi\)
−0.475426 + 0.879756i \(0.657706\pi\)
\(68\) −3.28834e6 −1.26822
\(69\) 0 0
\(70\) 5.69680e6 1.98512
\(71\) −469125. −0.155555 −0.0777775 − 0.996971i \(-0.524782\pi\)
−0.0777775 + 0.996971i \(0.524782\pi\)
\(72\) 0 0
\(73\) 3.76695e6 1.13334 0.566669 − 0.823945i \(-0.308232\pi\)
0.566669 + 0.823945i \(0.308232\pi\)
\(74\) 2.37787e6 0.682145
\(75\) 0 0
\(76\) −1.01991e7 −2.66511
\(77\) 5.45474e6 1.36162
\(78\) 0 0
\(79\) 8.31434e6 1.89729 0.948643 − 0.316349i \(-0.102457\pi\)
0.948643 + 0.316349i \(0.102457\pi\)
\(80\) −1.61311e7 −3.52248
\(81\) 0 0
\(82\) −1.24287e7 −2.48930
\(83\) −988959. −0.189847 −0.0949237 − 0.995485i \(-0.530261\pi\)
−0.0949237 + 0.995485i \(0.530261\pi\)
\(84\) 0 0
\(85\) 3.60320e6 0.636388
\(86\) −3.43365e6 −0.582119
\(87\) 0 0
\(88\) −2.93221e7 −4.58675
\(89\) 9.33980e6 1.40434 0.702170 − 0.712009i \(-0.252214\pi\)
0.702170 + 0.712009i \(0.252214\pi\)
\(90\) 0 0
\(91\) −5.44467e6 −0.757403
\(92\) −1.58590e7 −2.12334
\(93\) 0 0
\(94\) −2.77430e6 −0.344514
\(95\) 1.11757e7 1.33734
\(96\) 0 0
\(97\) 8.88487e6 0.988439 0.494220 − 0.869337i \(-0.335454\pi\)
0.494220 + 0.869337i \(0.335454\pi\)
\(98\) 5.11998e6 0.549511
\(99\) 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.a.1.1 ✓ 4
3.2 odd 2 81.8.a.b.1.4 yes 4
9.2 odd 6 81.8.c.i.28.1 8
9.4 even 3 81.8.c.j.55.4 8
9.5 odd 6 81.8.c.i.55.1 8
9.7 even 3 81.8.c.j.28.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
81.8.a.a.1.1 ✓ 4 1.1 even 1 trivial
81.8.a.b.1.4 yes 4 3.2 odd 2
81.8.c.i.28.1 8 9.2 odd 6
81.8.c.i.55.1 8 9.5 odd 6
81.8.c.j.28.4 8 9.7 even 3
81.8.c.j.55.4 8 9.4 even 3