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Properties

Label 9072.2.a.ci.1.3
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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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] = [9072,2,Mod(1,9072)] 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("9072.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,4,0,0,0,0,0,16,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 567)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.18890\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-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)
 
\(f(q)\) \(=\) \(q+0.913701 q^{5} +1.00000 q^{7} -2.64575 q^{11} +4.00000 q^{13} +3.46410 q^{17} -5.58258 q^{19} +3.46410 q^{23} -4.16515 q^{25} -8.75560 q^{29} +9.16515 q^{31} +0.913701 q^{35} +3.00000 q^{37} -0.913701 q^{41} -0.582576 q^{43} +13.1334 q^{47} +1.00000 q^{49} -8.66025 q^{53} -2.41742 q^{55} -3.46410 q^{59} +11.5826 q^{61} +3.65480 q^{65} -8.58258 q^{67} +4.47315 q^{71} +15.1652 q^{73} -2.64575 q^{77} -0.582576 q^{79} +9.66930 q^{83} +3.16515 q^{85} -1.82740 q^{89} +4.00000 q^{91} -5.10080 q^{95} -1.58258 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 16 q^{13} - 4 q^{19} + 20 q^{25} + 12 q^{37} + 16 q^{43} + 4 q^{49} - 28 q^{55} + 28 q^{61} - 16 q^{67} + 24 q^{73} + 16 q^{79} - 24 q^{85} + 16 q^{91} + 12 q^{97}+O(q^{100}) \) 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
\(5\) 0.913701 0.408619 0.204310 − 0.978906i \(-0.434505\pi\)
0.204310 + 0.978906i \(0.434505\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.64575 −0.797724 −0.398862 − 0.917011i \(-0.630595\pi\)
−0.398862 + 0.917011i \(0.630595\pi\)
\(12\) 0 0
\(13\) 4.00000 1.10940 0.554700 − 0.832050i \(-0.312833\pi\)
0.554700 + 0.832050i \(0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.46410 0.840168 0.420084 − 0.907485i \(-0.362001\pi\)
0.420084 + 0.907485i \(0.362001\pi\)
\(18\) 0 0
\(19\) −5.58258 −1.28073 −0.640365 − 0.768070i \(-0.721217\pi\)
−0.640365 + 0.768070i \(0.721217\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.46410 0.722315 0.361158 − 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 0 0
\(25\) −4.16515 −0.833030
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −8.75560 −1.62587 −0.812937 − 0.582351i \(-0.802133\pi\)
−0.812937 + 0.582351i \(0.802133\pi\)
\(30\) 0 0
\(31\) 9.16515 1.64611 0.823055 − 0.567962i \(-0.192268\pi\)
0.823055 + 0.567962i \(0.192268\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.913701 0.154444
\(36\) 0 0
\(37\) 3.00000 0.493197 0.246598 − 0.969118i \(-0.420687\pi\)
0.246598 + 0.969118i \(0.420687\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.913701 −0.142696 −0.0713480 − 0.997451i \(-0.522730\pi\)
−0.0713480 + 0.997451i \(0.522730\pi\)
\(42\) 0 0
\(43\) −0.582576 −0.0888420 −0.0444210 − 0.999013i \(-0.514144\pi\)
−0.0444210 + 0.999013i \(0.514144\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.1334 1.91570 0.957852 − 0.287262i \(-0.0927450\pi\)
0.957852 + 0.287262i \(0.0927450\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.66025 −1.18958 −0.594789 − 0.803882i \(-0.702764\pi\)
−0.594789 + 0.803882i \(0.702764\pi\)
\(54\) 0 0
\(55\) −2.41742 −0.325965
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.46410 −0.450988 −0.225494 − 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 11.5826 1.48300 0.741498 − 0.670955i \(-0.234115\pi\)
0.741498 + 0.670955i \(0.234115\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.65480 0.453322
\(66\) 0 0
\(67\) −8.58258 −1.04853 −0.524264 − 0.851556i \(-0.675660\pi\)
−0.524264 + 0.851556i \(0.675660\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.47315 0.530866 0.265433 − 0.964129i \(-0.414485\pi\)
0.265433 + 0.964129i \(0.414485\pi\)
\(72\) 0 0
\(73\) 15.1652 1.77495 0.887473 − 0.460859i \(-0.152459\pi\)
0.887473 + 0.460859i \(0.152459\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.64575 −0.301511
\(78\) 0 0
\(79\) −0.582576 −0.0655449 −0.0327724 − 0.999463i \(-0.510434\pi\)
−0.0327724 + 0.999463i \(0.510434\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.66930 1.06134 0.530672 − 0.847577i \(-0.321940\pi\)
0.530672 + 0.847577i \(0.321940\pi\)
\(84\) 0 0
\(85\) 3.16515 0.343309
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.82740 −0.193704 −0.0968521 − 0.995299i \(-0.530877\pi\)
−0.0968521 + 0.995299i \(0.530877\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.10080 −0.523331
\(96\) 0 0
\(97\) −1.58258 −0.160686 −0.0803431 − 0.996767i \(-0.525602\pi\)
−0.0803431 + 0.996767i \(0.525602\pi\)
\(98\) 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 9072.2.a.ci.1.3 4
3.2 odd 2 inner 9072.2.a.ci.1.2 4
4.3 odd 2 567.2.a.i.1.4 yes 4
12.11 even 2 567.2.a.i.1.1 ✓ 4
28.27 even 2 3969.2.a.u.1.4 4
36.7 odd 6 567.2.f.n.190.1 8
36.11 even 6 567.2.f.n.190.4 8
36.23 even 6 567.2.f.n.379.4 8
36.31 odd 6 567.2.f.n.379.1 8
84.83 odd 2 3969.2.a.u.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.1 ✓ 4 12.11 even 2
567.2.a.i.1.4 yes 4 4.3 odd 2
567.2.f.n.190.1 8 36.7 odd 6
567.2.f.n.190.4 8 36.11 even 6
567.2.f.n.379.1 8 36.31 odd 6
567.2.f.n.379.4 8 36.23 even 6
3969.2.a.u.1.1 4 84.83 odd 2
3969.2.a.u.1.4 4 28.27 even 2
9072.2.a.ci.1.2 4 3.2 odd 2 inner
9072.2.a.ci.1.3 4 1.1 even 1 trivial