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Properties

Label 1600.2.f.j.1249.4
Level $1600$
Weight $2$
Character 1600.1249
Analytic conductor $12.776$
Analytic rank $0$
Dimension $4$
CM discriminant -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] = [1600,2,Mod(1249,1600)] 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("1600.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1600 = 2^{6} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,4,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,52,0,0, 0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1249.4
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 1600.1249
Dual form 1600.2.f.j.1249.3

$q$-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)
 
\(f(q)\) \(=\) \(q+3.44949 q^{3} +8.89898 q^{9} +5.44949i q^{11} +1.89898i q^{17} -6.34847i q^{19} +20.3485 q^{27} +18.7980i q^{33} -6.79796 q^{41} -10.0000 q^{43} +7.00000 q^{49} +6.55051i q^{51} -21.8990i q^{57} +6.00000i q^{59} -0.348469 q^{67} -15.6969i q^{73} +43.4949 q^{81} +6.55051 q^{83} +4.10102 q^{89} -10.0000i q^{97} +48.4949i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 16 q^{9} + 52 q^{27} + 12 q^{41} - 40 q^{43} + 28 q^{49} + 28 q^{67} + 76 q^{81} + 36 q^{83} + 36 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1151\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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
\(3\) 3.44949 1.99156 0.995782 − 0.0917517i \(-0.0292466\pi\)
0.995782 + 0.0917517i \(0.0292466\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 8.89898 2.96633
\(10\) 0 0
\(11\) 5.44949i 1.64308i 0.570149 + 0.821541i \(0.306886\pi\)
−0.570149 + 0.821541i \(0.693114\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.89898i 0.460570i 0.973123 + 0.230285i \(0.0739659\pi\)
−0.973123 + 0.230285i \(0.926034\pi\)
\(18\) 0 0
\(19\) − 6.34847i − 1.45644i −0.685344 − 0.728219i \(-0.740348\pi\)
0.685344 − 0.728219i \(-0.259652\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 20.3485 3.91606
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 18.7980i 3.27230i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.79796 −1.06166 −0.530831 − 0.847477i \(-0.678120\pi\)
−0.530831 + 0.847477i \(0.678120\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 6.55051i 0.917255i
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 21.8990i − 2.90059i
\(58\) 0 0
\(59\) 6.00000i 0.781133i 0.920575 + 0.390567i \(0.127721\pi\)
−0.920575 + 0.390567i \(0.872279\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −0.348469 −0.0425723 −0.0212861 − 0.999773i \(-0.506776\pi\)
−0.0212861 + 0.999773i \(0.506776\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) − 15.6969i − 1.83719i −0.395203 − 0.918594i \(-0.629326\pi\)
0.395203 − 0.918594i \(-0.370674\pi\)
\(74\) 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
\(81\) 43.4949 4.83277
\(82\) 0 0
\(83\) 6.55051 0.719012 0.359506 − 0.933143i \(-0.382945\pi\)
0.359506 + 0.933143i \(0.382945\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.10102 0.434707 0.217354 − 0.976093i \(-0.430258\pi\)
0.217354 + 0.976093i \(0.430258\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) 0 0
\(99\) 48.4949i 4.87392i
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 1600.2.f.j.1249.4 4
4.3 odd 2 1600.2.f.f.1249.1 4
5.2 odd 4 1600.2.d.d.801.1 yes 4
5.3 odd 4 1600.2.d.c.801.4 yes 4
5.4 even 2 1600.2.f.f.1249.2 4
8.3 odd 2 CM 1600.2.f.j.1249.4 4
8.5 even 2 1600.2.f.f.1249.1 4
20.3 even 4 1600.2.d.c.801.1 ✓ 4
20.7 even 4 1600.2.d.d.801.4 yes 4
20.19 odd 2 inner 1600.2.f.j.1249.3 4
40.3 even 4 1600.2.d.c.801.4 yes 4
40.13 odd 4 1600.2.d.c.801.1 ✓ 4
40.19 odd 2 1600.2.f.f.1249.2 4
40.27 even 4 1600.2.d.d.801.1 yes 4
40.29 even 2 inner 1600.2.f.j.1249.3 4
40.37 odd 4 1600.2.d.d.801.4 yes 4
80.3 even 4 6400.2.a.ch.1.2 2
80.13 odd 4 6400.2.a.bd.1.1 2
80.27 even 4 6400.2.a.ci.1.2 2
80.37 odd 4 6400.2.a.bc.1.1 2
80.43 even 4 6400.2.a.bd.1.1 2
80.53 odd 4 6400.2.a.ch.1.2 2
80.67 even 4 6400.2.a.bc.1.1 2
80.77 odd 4 6400.2.a.ci.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1600.2.d.c.801.1 ✓ 4 20.3 even 4
1600.2.d.c.801.1 ✓ 4 40.13 odd 4
1600.2.d.c.801.4 yes 4 5.3 odd 4
1600.2.d.c.801.4 yes 4 40.3 even 4
1600.2.d.d.801.1 yes 4 5.2 odd 4
1600.2.d.d.801.1 yes 4 40.27 even 4
1600.2.d.d.801.4 yes 4 20.7 even 4
1600.2.d.d.801.4 yes 4 40.37 odd 4
1600.2.f.f.1249.1 4 4.3 odd 2
1600.2.f.f.1249.1 4 8.5 even 2
1600.2.f.f.1249.2 4 5.4 even 2
1600.2.f.f.1249.2 4 40.19 odd 2
1600.2.f.j.1249.3 4 20.19 odd 2 inner
1600.2.f.j.1249.3 4 40.29 even 2 inner
1600.2.f.j.1249.4 4 1.1 even 1 trivial
1600.2.f.j.1249.4 4 8.3 odd 2 CM
6400.2.a.bc.1.1 2 80.37 odd 4
6400.2.a.bc.1.1 2 80.67 even 4
6400.2.a.bd.1.1 2 80.13 odd 4
6400.2.a.bd.1.1 2 80.43 even 4
6400.2.a.ch.1.2 2 80.3 even 4
6400.2.a.ch.1.2 2 80.53 odd 4
6400.2.a.ci.1.2 2 80.27 even 4
6400.2.a.ci.1.2 2 80.77 odd 4