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Properties

Label 81.8.a.e.1.5
Level $81$
Weight $8$
Character 81.1
Self dual yes
Analytic conductor $25.303$
Analytic rank $0$
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: \(0\)
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 \(0.799228\) 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)
 
\(f(q)\) \(=\) \(q+12.1944 q^{2} +20.7039 q^{4} +492.052 q^{5} +764.622 q^{7} -1308.41 q^{8} +6000.29 q^{10} -72.7023 q^{11} +6021.53 q^{13} +9324.12 q^{14} -18605.4 q^{16} -5989.93 q^{17} +18676.2 q^{19} +10187.4 q^{20} -886.563 q^{22} +24279.0 q^{23} +163990. q^{25} +73429.1 q^{26} +15830.6 q^{28} +86756.2 q^{29} +211779. q^{31} -59405.6 q^{32} -73043.8 q^{34} +376234. q^{35} -327978. q^{37} +227745. q^{38} -643808. q^{40} +392072. q^{41} -687223. q^{43} -1505.22 q^{44} +296069. q^{46} +641509. q^{47} -238896. q^{49} +1.99977e6 q^{50} +124669. q^{52} -814485. q^{53} -35773.3 q^{55} -1.00044e6 q^{56} +1.05794e6 q^{58} -2.51727e6 q^{59} -443242. q^{61} +2.58252e6 q^{62} +1.65708e6 q^{64} +2.96291e6 q^{65} +592096. q^{67} -124015. q^{68} +4.58795e6 q^{70} +1.48821e6 q^{71} -5.41341e6 q^{73} -3.99951e6 q^{74} +386669. q^{76} -55589.8 q^{77} -889471. q^{79} -9.15485e6 q^{80} +4.78109e6 q^{82} -3.38646e6 q^{83} -2.94736e6 q^{85} -8.38028e6 q^{86} +95124.7 q^{88} +1.17388e6 q^{89} +4.60420e6 q^{91} +502670. q^{92} +7.82284e6 q^{94} +9.18965e6 q^{95} -8.60028e6 q^{97} -2.91320e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 9 q^{2} + 321 q^{4} + 180 q^{5} + 84 q^{7} + 2961 q^{8} + 126 q^{10} + 8460 q^{11} + 1848 q^{13} + 16272 q^{14} + 12417 q^{16} + 15282 q^{17} + 12216 q^{19} + 40788 q^{20} + 35001 q^{22} + 51588 q^{23}+ \cdots - 47916657 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.1944 1.07784 0.538922 − 0.842355i \(-0.318832\pi\)
0.538922 + 0.842355i \(0.318832\pi\)
\(3\) 0 0
\(4\) 20.7039 0.161749
\(5\) 492.052 1.76042 0.880210 − 0.474585i \(-0.157402\pi\)
0.880210 + 0.474585i \(0.157402\pi\)
\(6\) 0 0
\(7\) 764.622 0.842565 0.421283 − 0.906929i \(-0.361580\pi\)
0.421283 + 0.906929i \(0.361580\pi\)
\(8\) −1308.41 −0.903504
\(9\) 0 0
\(10\) 6000.29 1.89746
\(11\) −72.7023 −0.0164693 −0.00823463 − 0.999966i \(-0.502621\pi\)
−0.00823463 + 0.999966i \(0.502621\pi\)
\(12\) 0 0
\(13\) 6021.53 0.760160 0.380080 − 0.924954i \(-0.375896\pi\)
0.380080 + 0.924954i \(0.375896\pi\)
\(14\) 9324.12 0.908155
\(15\) 0 0
\(16\) −18605.4 −1.13559
\(17\) −5989.93 −0.295700 −0.147850 − 0.989010i \(-0.547235\pi\)
−0.147850 + 0.989010i \(0.547235\pi\)
\(18\) 0 0
\(19\) 18676.2 0.624670 0.312335 − 0.949972i \(-0.398889\pi\)
0.312335 + 0.949972i \(0.398889\pi\)
\(20\) 10187.4 0.284746
\(21\) 0 0
\(22\) −886.563 −0.0177513
\(23\) 24279.0 0.416087 0.208043 − 0.978120i \(-0.433290\pi\)
0.208043 + 0.978120i \(0.433290\pi\)
\(24\) 0 0
\(25\) 163990. 2.09908
\(26\) 73429.1 0.819335
\(27\) 0 0
\(28\) 15830.6 0.136284
\(29\) 86756.2 0.660553 0.330276 − 0.943884i \(-0.392858\pi\)
0.330276 + 0.943884i \(0.392858\pi\)
\(30\) 0 0
\(31\) 211779. 1.27678 0.638392 − 0.769712i \(-0.279600\pi\)
0.638392 + 0.769712i \(0.279600\pi\)
\(32\) −59405.6 −0.320481
\(33\) 0 0
\(34\) −73043.8 −0.318718
\(35\) 376234. 1.48327
\(36\) 0 0
\(37\) −327978. −1.06448 −0.532242 − 0.846592i \(-0.678650\pi\)
−0.532242 + 0.846592i \(0.678650\pi\)
\(38\) 227745. 0.673297
\(39\) 0 0
\(40\) −643808. −1.59055
\(41\) 392072. 0.888429 0.444214 − 0.895920i \(-0.353483\pi\)
0.444214 + 0.895920i \(0.353483\pi\)
\(42\) 0 0
\(43\) −687223. −1.31813 −0.659064 − 0.752086i \(-0.729048\pi\)
−0.659064 + 0.752086i \(0.729048\pi\)
\(44\) −1505.22 −0.00266388
\(45\) 0 0
\(46\) 296069. 0.448477
\(47\) 641509. 0.901282 0.450641 − 0.892705i \(-0.351196\pi\)
0.450641 + 0.892705i \(0.351196\pi\)
\(48\) 0 0
\(49\) −238896. −0.290084
\(50\) 1.99977e6 2.26248
\(51\) 0 0
\(52\) 124669. 0.122955
\(53\) −814485. −0.751480 −0.375740 − 0.926725i \(-0.622611\pi\)
−0.375740 + 0.926725i \(0.622611\pi\)
\(54\) 0 0
\(55\) −35773.3 −0.0289928
\(56\) −1.00044e6 −0.761262
\(57\) 0 0
\(58\) 1.05794e6 0.711973
\(59\) −2.51727e6 −1.59569 −0.797843 − 0.602866i \(-0.794025\pi\)
−0.797843 + 0.602866i \(0.794025\pi\)
\(60\) 0 0
\(61\) −443242. −0.250027 −0.125013 − 0.992155i \(-0.539897\pi\)
−0.125013 + 0.992155i \(0.539897\pi\)
\(62\) 2.58252e6 1.37617
\(63\) 0 0
\(64\) 1.65708e6 0.790157
\(65\) 2.96291e6 1.33820
\(66\) 0 0
\(67\) 592096. 0.240508 0.120254 − 0.992743i \(-0.461629\pi\)
0.120254 + 0.992743i \(0.461629\pi\)
\(68\) −124015. −0.0478291
\(69\) 0 0
\(70\) 4.58795e6 1.59873
\(71\) 1.48821e6 0.493469 0.246734 − 0.969083i \(-0.420643\pi\)
0.246734 + 0.969083i \(0.420643\pi\)
\(72\) 0 0
\(73\) −5.41341e6 −1.62870 −0.814350 − 0.580374i \(-0.802906\pi\)
−0.814350 + 0.580374i \(0.802906\pi\)
\(74\) −3.99951e6 −1.14735
\(75\) 0 0
\(76\) 386669. 0.101040
\(77\) −55589.8 −0.0138764
\(78\) 0 0
\(79\) −889471. −0.202972 −0.101486 − 0.994837i \(-0.532360\pi\)
−0.101486 + 0.994837i \(0.532360\pi\)
\(80\) −9.15485e6 −1.99911
\(81\) 0 0
\(82\) 4.78109e6 0.957588
\(83\) −3.38646e6 −0.650089 −0.325044 − 0.945699i \(-0.605379\pi\)
−0.325044 + 0.945699i \(0.605379\pi\)
\(84\) 0 0
\(85\) −2.94736e6 −0.520555
\(86\) −8.38028e6 −1.42074
\(87\) 0 0
\(88\) 95124.7 0.0148800
\(89\) 1.17388e6 0.176506 0.0882531 − 0.996098i \(-0.471872\pi\)
0.0882531 + 0.996098i \(0.471872\pi\)
\(90\) 0 0
\(91\) 4.60420e6 0.640485
\(92\) 502670. 0.0673016
\(93\) 0 0
\(94\) 7.82284e6 0.971442
\(95\) 9.18965e6 1.09968
\(96\) 0 0
\(97\) −8.60028e6 −0.956779 −0.478390 − 0.878148i \(-0.658779\pi\)
−0.478390 + 0.878148i \(0.658779\pi\)
\(98\) −2.91320e6 −0.312665
\(99\) 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.e.1.5 6
3.2 odd 2 81.8.a.c.1.2 6
9.2 odd 6 27.8.c.a.10.5 12
9.4 even 3 9.8.c.a.7.2 yes 12
9.5 odd 6 27.8.c.a.19.5 12
9.7 even 3 9.8.c.a.4.2 ✓ 12
36.7 odd 6 144.8.i.c.49.5 12
36.11 even 6 432.8.i.c.145.6 12
36.23 even 6 432.8.i.c.289.6 12
36.31 odd 6 144.8.i.c.97.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.8.c.a.4.2 ✓ 12 9.7 even 3
9.8.c.a.7.2 yes 12 9.4 even 3
27.8.c.a.10.5 12 9.2 odd 6
27.8.c.a.19.5 12 9.5 odd 6
81.8.a.c.1.2 6 3.2 odd 2
81.8.a.e.1.5 6 1.1 even 1 trivial
144.8.i.c.49.5 12 36.7 odd 6
144.8.i.c.97.5 12 36.31 odd 6
432.8.i.c.145.6 12 36.11 even 6
432.8.i.c.289.6 12 36.23 even 6