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Properties

Label 9216.2.a.br.1.5
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $1$
Dimension $8$
Inner twists $4$

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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] = [9216,2,Mod(1,9216)] 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("9216.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0, 0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,-48,0, 0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(67)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(73.5901305028\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.3288334336.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 4608)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.724535\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-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)
 
\(f(q)\) \(=\) \(q+1.78089 q^{5} -3.29066 q^{7} -1.53073 q^{11} -0.585786 q^{13} -6.08034 q^{17} +1.92762 q^{19} +5.22625 q^{23} -1.82843 q^{25} +6.81801 q^{29} +6.01673 q^{31} -5.86030 q^{35} -3.41421 q^{37} -1.04322 q^{41} +11.2350 q^{43} +0.896683 q^{47} +3.82843 q^{49} -6.81801 q^{53} -2.72607 q^{55} +10.4525 q^{59} -4.58579 q^{61} -1.04322 q^{65} +9.30739 q^{67} -14.7821 q^{71} -6.48528 q^{73} +5.03712 q^{77} +3.29066 q^{79} +13.2513 q^{83} -10.8284 q^{85} -7.12356 q^{89} +1.92762 q^{91} +3.43289 q^{95} +7.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{13} + 8 q^{25} - 16 q^{37} + 8 q^{49} - 48 q^{61} + 16 q^{73} - 64 q^{85} - 32 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.78089 0.796439 0.398219 − 0.917290i \(-0.369628\pi\)
0.398219 + 0.917290i \(0.369628\pi\)
\(6\) 0 0
\(7\) −3.29066 −1.24375 −0.621876 − 0.783116i \(-0.713629\pi\)
−0.621876 + 0.783116i \(0.713629\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.53073 −0.461534 −0.230767 − 0.973009i \(-0.574123\pi\)
−0.230767 + 0.973009i \(0.574123\pi\)
\(12\) 0 0
\(13\) −0.585786 −0.162468 −0.0812340 − 0.996695i \(-0.525886\pi\)
−0.0812340 + 0.996695i \(0.525886\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.08034 −1.47470 −0.737350 − 0.675511i \(-0.763923\pi\)
−0.737350 + 0.675511i \(0.763923\pi\)
\(18\) 0 0
\(19\) 1.92762 0.442227 0.221113 − 0.975248i \(-0.429031\pi\)
0.221113 + 0.975248i \(0.429031\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.22625 1.08975 0.544874 − 0.838518i \(-0.316577\pi\)
0.544874 + 0.838518i \(0.316577\pi\)
\(24\) 0 0
\(25\) −1.82843 −0.365685
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.81801 1.26607 0.633036 − 0.774122i \(-0.281808\pi\)
0.633036 + 0.774122i \(0.281808\pi\)
\(30\) 0 0
\(31\) 6.01673 1.08064 0.540318 − 0.841461i \(-0.318304\pi\)
0.540318 + 0.841461i \(0.318304\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.86030 −0.990572
\(36\) 0 0
\(37\) −3.41421 −0.561293 −0.280647 − 0.959811i \(-0.590549\pi\)
−0.280647 + 0.959811i \(0.590549\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.04322 −0.162924 −0.0814619 − 0.996676i \(-0.525959\pi\)
−0.0814619 + 0.996676i \(0.525959\pi\)
\(42\) 0 0
\(43\) 11.2350 1.71332 0.856661 − 0.515879i \(-0.172535\pi\)
0.856661 + 0.515879i \(0.172535\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.896683 0.130795 0.0653973 − 0.997859i \(-0.479169\pi\)
0.0653973 + 0.997859i \(0.479169\pi\)
\(48\) 0 0
\(49\) 3.82843 0.546918
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.81801 −0.936526 −0.468263 − 0.883589i \(-0.655120\pi\)
−0.468263 + 0.883589i \(0.655120\pi\)
\(54\) 0 0
\(55\) −2.72607 −0.367583
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.4525 1.36080 0.680400 − 0.732841i \(-0.261806\pi\)
0.680400 + 0.732841i \(0.261806\pi\)
\(60\) 0 0
\(61\) −4.58579 −0.587150 −0.293575 − 0.955936i \(-0.594845\pi\)
−0.293575 + 0.955936i \(0.594845\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.04322 −0.129396
\(66\) 0 0
\(67\) 9.30739 1.13708 0.568539 − 0.822656i \(-0.307509\pi\)
0.568539 + 0.822656i \(0.307509\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −14.7821 −1.75431 −0.877155 − 0.480208i \(-0.840561\pi\)
−0.877155 + 0.480208i \(0.840561\pi\)
\(72\) 0 0
\(73\) −6.48528 −0.759045 −0.379522 − 0.925183i \(-0.623912\pi\)
−0.379522 + 0.925183i \(0.623912\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.03712 0.574033
\(78\) 0 0
\(79\) 3.29066 0.370228 0.185114 − 0.982717i \(-0.440735\pi\)
0.185114 + 0.982717i \(0.440735\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.2513 1.45452 0.727262 − 0.686360i \(-0.240793\pi\)
0.727262 + 0.686360i \(0.240793\pi\)
\(84\) 0 0
\(85\) −10.8284 −1.17451
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.12356 −0.755096 −0.377548 − 0.925990i \(-0.623233\pi\)
−0.377548 + 0.925990i \(0.623233\pi\)
\(90\) 0 0
\(91\) 1.92762 0.202070
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.43289 0.352207
\(96\) 0 0
\(97\) 7.31371 0.742595 0.371297 − 0.928514i \(-0.378913\pi\)
0.371297 + 0.928514i \(0.378913\pi\)
\(98\) 0 0
\(99\) 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 9216.2.a.br.1.5 8
3.2 odd 2 inner 9216.2.a.br.1.3 8
4.3 odd 2 inner 9216.2.a.br.1.6 8
8.3 odd 2 9216.2.a.bs.1.4 8
8.5 even 2 9216.2.a.bs.1.3 8
12.11 even 2 inner 9216.2.a.br.1.4 8
24.5 odd 2 9216.2.a.bs.1.5 8
24.11 even 2 9216.2.a.bs.1.6 8
32.3 odd 8 4608.2.k.bk.1153.3 ✓ 16
32.5 even 8 4608.2.k.bl.3457.5 yes 16
32.11 odd 8 4608.2.k.bk.3457.4 yes 16
32.13 even 8 4608.2.k.bl.1153.6 yes 16
32.19 odd 8 4608.2.k.bl.1153.5 yes 16
32.21 even 8 4608.2.k.bk.3457.3 yes 16
32.27 odd 8 4608.2.k.bl.3457.6 yes 16
32.29 even 8 4608.2.k.bk.1153.4 yes 16
96.5 odd 8 4608.2.k.bl.3457.3 yes 16
96.11 even 8 4608.2.k.bk.3457.6 yes 16
96.29 odd 8 4608.2.k.bk.1153.6 yes 16
96.35 even 8 4608.2.k.bk.1153.5 yes 16
96.53 odd 8 4608.2.k.bk.3457.5 yes 16
96.59 even 8 4608.2.k.bl.3457.4 yes 16
96.77 odd 8 4608.2.k.bl.1153.4 yes 16
96.83 even 8 4608.2.k.bl.1153.3 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.k.bk.1153.3 ✓ 16 32.3 odd 8
4608.2.k.bk.1153.4 yes 16 32.29 even 8
4608.2.k.bk.1153.5 yes 16 96.35 even 8
4608.2.k.bk.1153.6 yes 16 96.29 odd 8
4608.2.k.bk.3457.3 yes 16 32.21 even 8
4608.2.k.bk.3457.4 yes 16 32.11 odd 8
4608.2.k.bk.3457.5 yes 16 96.53 odd 8
4608.2.k.bk.3457.6 yes 16 96.11 even 8
4608.2.k.bl.1153.3 yes 16 96.83 even 8
4608.2.k.bl.1153.4 yes 16 96.77 odd 8
4608.2.k.bl.1153.5 yes 16 32.19 odd 8
4608.2.k.bl.1153.6 yes 16 32.13 even 8
4608.2.k.bl.3457.3 yes 16 96.5 odd 8
4608.2.k.bl.3457.4 yes 16 96.59 even 8
4608.2.k.bl.3457.5 yes 16 32.5 even 8
4608.2.k.bl.3457.6 yes 16 32.27 odd 8
9216.2.a.br.1.3 8 3.2 odd 2 inner
9216.2.a.br.1.4 8 12.11 even 2 inner
9216.2.a.br.1.5 8 1.1 even 1 trivial
9216.2.a.br.1.6 8 4.3 odd 2 inner
9216.2.a.bs.1.3 8 8.5 even 2
9216.2.a.bs.1.4 8 8.3 odd 2
9216.2.a.bs.1.5 8 24.5 odd 2
9216.2.a.bs.1.6 8 24.11 even 2