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Properties

Label 9072.2.a.ci.1.1
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 1.1
Root \(0.456850\) 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)
 
\(f(q)\) \(=\) \(q-4.37780 q^{5} +1.00000 q^{7} +2.64575 q^{11} +4.00000 q^{13} +3.46410 q^{17} +3.58258 q^{19} +3.46410 q^{23} +14.1652 q^{25} +1.82740 q^{29} -9.16515 q^{31} -4.37780 q^{35} +3.00000 q^{37} +4.37780 q^{41} +8.58258 q^{43} -2.74110 q^{47} +1.00000 q^{49} -8.66025 q^{53} -11.5826 q^{55} -3.46410 q^{59} +2.41742 q^{61} -17.5112 q^{65} +0.582576 q^{67} -11.4014 q^{71} -3.16515 q^{73} +2.64575 q^{77} +8.58258 q^{79} -6.20520 q^{83} -15.1652 q^{85} +8.75560 q^{89} +4.00000 q^{91} -15.6838 q^{95} +7.58258 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{7} + 16 q^{13} - 4 q^{19} + 20 q^{25} + 12 q^{37} + 16 q^{43} + 4 q^{49} - 28 q^{55} + 28 q^{61} - 16 q^{67} + 24 q^{73} + 16 q^{79} - 24 q^{85} + 16 q^{91} + 12 q^{97}+O(q^{100}) \) 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
\(5\) −4.37780 −1.95781 −0.978906 − 0.204310i \(-0.934505\pi\)
−0.978906 + 0.204310i \(0.934505\pi\)
\(6\) 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 \(-0.369405\pi\)
0.398862 + 0.917011i \(0.369405\pi\)
\(12\) 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
\(19\) 3.58258 0.821899 0.410950 − 0.911658i \(-0.365197\pi\)
0.410950 + 0.911658i \(0.365197\pi\)
\(20\) 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
\(25\) 14.1652 2.83303
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.82740 0.339340 0.169670 − 0.985501i \(-0.445730\pi\)
0.169670 + 0.985501i \(0.445730\pi\)
\(30\) 0 0
\(31\) −9.16515 −1.64611 −0.823055 − 0.567962i \(-0.807732\pi\)
−0.823055 + 0.567962i \(0.807732\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −4.37780 −0.739984
\(36\) 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
\(41\) 4.37780 0.683698 0.341849 − 0.939755i \(-0.388947\pi\)
0.341849 + 0.939755i \(0.388947\pi\)
\(42\) 0 0
\(43\) 8.58258 1.30883 0.654415 − 0.756135i \(-0.272915\pi\)
0.654415 + 0.756135i \(0.272915\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.74110 −0.399831 −0.199915 − 0.979813i \(-0.564067\pi\)
−0.199915 + 0.979813i \(0.564067\pi\)
\(48\) 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
\(55\) −11.5826 −1.56179
\(56\) 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
\(61\) 2.41742 0.309519 0.154760 − 0.987952i \(-0.450540\pi\)
0.154760 + 0.987952i \(0.450540\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −17.5112 −2.17200
\(66\) 0 0
\(67\) 0.582576 0.0711729 0.0355865 − 0.999367i \(-0.488670\pi\)
0.0355865 + 0.999367i \(0.488670\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.4014 −1.35309 −0.676546 − 0.736400i \(-0.736524\pi\)
−0.676546 + 0.736400i \(0.736524\pi\)
\(72\) 0 0
\(73\) −3.16515 −0.370453 −0.185226 − 0.982696i \(-0.559302\pi\)
−0.185226 + 0.982696i \(0.559302\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.64575 0.301511
\(78\) 0 0
\(79\) 8.58258 0.965615 0.482808 − 0.875726i \(-0.339617\pi\)
0.482808 + 0.875726i \(0.339617\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −6.20520 −0.681110 −0.340555 − 0.940225i \(-0.610615\pi\)
−0.340555 + 0.940225i \(0.610615\pi\)
\(84\) 0 0
\(85\) −15.1652 −1.64489
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.75560 0.928092 0.464046 − 0.885811i \(-0.346397\pi\)
0.464046 + 0.885811i \(0.346397\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −15.6838 −1.60912
\(96\) 0 0
\(97\) 7.58258 0.769894 0.384947 − 0.922939i \(-0.374220\pi\)
0.384947 + 0.922939i \(0.374220\pi\)
\(98\) 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.1.1 4
3.2 odd 2 inner 9072.2.a.ci.1.4 4
4.3 odd 2 567.2.a.i.1.2 ✓ 4
12.11 even 2 567.2.a.i.1.3 yes 4
28.27 even 2 3969.2.a.u.1.2 4
36.7 odd 6 567.2.f.n.190.3 8
36.11 even 6 567.2.f.n.190.2 8
36.23 even 6 567.2.f.n.379.2 8
36.31 odd 6 567.2.f.n.379.3 8
84.83 odd 2 3969.2.a.u.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
567.2.a.i.1.2 ✓ 4 4.3 odd 2
567.2.a.i.1.3 yes 4 12.11 even 2
567.2.f.n.190.2 8 36.11 even 6
567.2.f.n.190.3 8 36.7 odd 6
567.2.f.n.379.2 8 36.23 even 6
567.2.f.n.379.3 8 36.31 odd 6
3969.2.a.u.1.2 4 28.27 even 2
3969.2.a.u.1.3 4 84.83 odd 2
9072.2.a.ci.1.1 4 1.1 even 1 trivial
9072.2.a.ci.1.4 4 3.2 odd 2 inner