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Properties

Label 7200.2.a.y.1.1
Level $7200$
Weight $2$
Character 7200.1
Self dual yes
Analytic conductor $57.492$
Analytic rank $1$
Dimension $1$
CM discriminant -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] = [7200,2,Mod(1,7200)] 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("7200.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,0,0,0,0,0,0,0,0,-4,0,0,0,8,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(57.4922894553\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 160)
Fricke sign: \(+1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7200.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.00000 q^{13} +8.00000 q^{17} -10.0000 q^{29} -12.0000 q^{37} +10.0000 q^{41} -7.00000 q^{49} -4.00000 q^{53} +10.0000 q^{61} -16.0000 q^{73} +10.0000 q^{89} +8.00000 q^{97} +O(q^{100})\)

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 0
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 − 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.00000 1.94029 0.970143 − 0.242536i \(-0.0779791\pi\)
0.970143 + 0.242536i \(0.0779791\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −10.0000 −1.85695 −0.928477 − 0.371391i \(-0.878881\pi\)
−0.928477 + 0.371391i \(0.878881\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −12.0000 −1.97279 −0.986394 − 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 − 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.00000 −0.549442 −0.274721 − 0.961524i \(-0.588586\pi\)
−0.274721 + 0.961524i \(0.588586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 − 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 − 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) 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 7200.2.a.y.1.1 1
3.2 odd 2 800.2.a.e.1.1 1
4.3 odd 2 CM 7200.2.a.y.1.1 1
5.2 odd 4 1440.2.f.c.289.1 2
5.3 odd 4 1440.2.f.c.289.2 2
5.4 even 2 7200.2.a.bb.1.1 1
12.11 even 2 800.2.a.e.1.1 1
15.2 even 4 160.2.c.a.129.2 yes 2
15.8 even 4 160.2.c.a.129.1 ✓ 2
15.14 odd 2 800.2.a.f.1.1 1
20.3 even 4 1440.2.f.c.289.2 2
20.7 even 4 1440.2.f.c.289.1 2
20.19 odd 2 7200.2.a.bb.1.1 1
24.5 odd 2 1600.2.a.m.1.1 1
24.11 even 2 1600.2.a.m.1.1 1
40.3 even 4 2880.2.f.n.1729.1 2
40.13 odd 4 2880.2.f.n.1729.1 2
40.27 even 4 2880.2.f.n.1729.2 2
40.37 odd 4 2880.2.f.n.1729.2 2
60.23 odd 4 160.2.c.a.129.1 ✓ 2
60.47 odd 4 160.2.c.a.129.2 yes 2
60.59 even 2 800.2.a.f.1.1 1
120.29 odd 2 1600.2.a.l.1.1 1
120.53 even 4 320.2.c.a.129.2 2
120.59 even 2 1600.2.a.l.1.1 1
120.77 even 4 320.2.c.a.129.1 2
120.83 odd 4 320.2.c.a.129.2 2
120.107 odd 4 320.2.c.a.129.1 2
240.53 even 4 1280.2.f.d.129.2 2
240.77 even 4 1280.2.f.d.129.1 2
240.83 odd 4 1280.2.f.c.129.1 2
240.107 odd 4 1280.2.f.c.129.2 2
240.173 even 4 1280.2.f.c.129.1 2
240.197 even 4 1280.2.f.c.129.2 2
240.203 odd 4 1280.2.f.d.129.2 2
240.227 odd 4 1280.2.f.d.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
160.2.c.a.129.1 ✓ 2 15.8 even 4
160.2.c.a.129.1 ✓ 2 60.23 odd 4
160.2.c.a.129.2 yes 2 15.2 even 4
160.2.c.a.129.2 yes 2 60.47 odd 4
320.2.c.a.129.1 2 120.77 even 4
320.2.c.a.129.1 2 120.107 odd 4
320.2.c.a.129.2 2 120.53 even 4
320.2.c.a.129.2 2 120.83 odd 4
800.2.a.e.1.1 1 3.2 odd 2
800.2.a.e.1.1 1 12.11 even 2
800.2.a.f.1.1 1 15.14 odd 2
800.2.a.f.1.1 1 60.59 even 2
1280.2.f.c.129.1 2 240.83 odd 4
1280.2.f.c.129.1 2 240.173 even 4
1280.2.f.c.129.2 2 240.107 odd 4
1280.2.f.c.129.2 2 240.197 even 4
1280.2.f.d.129.1 2 240.77 even 4
1280.2.f.d.129.1 2 240.227 odd 4
1280.2.f.d.129.2 2 240.53 even 4
1280.2.f.d.129.2 2 240.203 odd 4
1440.2.f.c.289.1 2 5.2 odd 4
1440.2.f.c.289.1 2 20.7 even 4
1440.2.f.c.289.2 2 5.3 odd 4
1440.2.f.c.289.2 2 20.3 even 4
1600.2.a.l.1.1 1 120.29 odd 2
1600.2.a.l.1.1 1 120.59 even 2
1600.2.a.m.1.1 1 24.5 odd 2
1600.2.a.m.1.1 1 24.11 even 2
2880.2.f.n.1729.1 2 40.3 even 4
2880.2.f.n.1729.1 2 40.13 odd 4
2880.2.f.n.1729.2 2 40.27 even 4
2880.2.f.n.1729.2 2 40.37 odd 4
7200.2.a.y.1.1 1 1.1 even 1 trivial
7200.2.a.y.1.1 1 4.3 odd 2 CM
7200.2.a.bb.1.1 1 5.4 even 2
7200.2.a.bb.1.1 1 20.19 odd 2