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Properties

Label 2304.4.a.j.1.1
Level $2304$
Weight $4$
Character 2304.1
Self dual yes
Analytic conductor $135.940$
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] = [2304,4,Mod(1,2304)] 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("2304.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2304, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2304.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,4,0,0,0,0,0,0,0,92,0,0,0,-94,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(135.940400653\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 128)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2304.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.00000 q^{5} +92.0000 q^{13} -94.0000 q^{17} -109.000 q^{25} -284.000 q^{29} +396.000 q^{37} -230.000 q^{41} -343.000 q^{49} -572.000 q^{53} -468.000 q^{61} +368.000 q^{65} +1098.00 q^{73} -376.000 q^{85} +1670.00 q^{89} -594.000 q^{97} +O(q^{100})\)

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
\(5\) 4.00000 0.357771 0.178885 − 0.983870i \(-0.442751\pi\)
0.178885 + 0.983870i \(0.442751\pi\)
\(6\) 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
\(13\) 92.0000 1.96279 0.981393 − 0.192012i \(-0.0615011\pi\)
0.981393 + 0.192012i \(0.0615011\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −94.0000 −1.34108 −0.670540 − 0.741874i \(-0.733937\pi\)
−0.670540 + 0.741874i \(0.733937\pi\)
\(18\) 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
\(25\) −109.000 −0.872000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −284.000 −1.81853 −0.909267 − 0.416214i \(-0.863357\pi\)
−0.909267 + 0.416214i \(0.863357\pi\)
\(30\) 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
\(37\) 396.000 1.75951 0.879757 − 0.475424i \(-0.157705\pi\)
0.879757 + 0.475424i \(0.157705\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −230.000 −0.876097 −0.438048 − 0.898951i \(-0.644330\pi\)
−0.438048 + 0.898951i \(0.644330\pi\)
\(42\) 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
\(49\) −343.000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −572.000 −1.48246 −0.741229 − 0.671253i \(-0.765757\pi\)
−0.741229 + 0.671253i \(0.765757\pi\)
\(54\) 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
\(61\) −468.000 −0.982316 −0.491158 − 0.871071i \(-0.663426\pi\)
−0.491158 + 0.871071i \(0.663426\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 368.000 0.702227
\(66\) 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
\(73\) 1098.00 1.76043 0.880214 − 0.474578i \(-0.157399\pi\)
0.880214 + 0.474578i \(0.157399\pi\)
\(74\) 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
\(85\) −376.000 −0.479799
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1670.00 1.98898 0.994492 − 0.104809i \(-0.0334231\pi\)
0.994492 + 0.104809i \(0.0334231\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −594.000 −0.621769 −0.310884 − 0.950448i \(-0.600625\pi\)
−0.310884 + 0.950448i \(0.600625\pi\)
\(98\) 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 2304.4.a.j.1.1 1
3.2 odd 2 256.4.a.d.1.1 1
4.3 odd 2 CM 2304.4.a.j.1.1 1
8.3 odd 2 2304.4.a.g.1.1 1
8.5 even 2 2304.4.a.g.1.1 1
12.11 even 2 256.4.a.d.1.1 1
16.3 odd 4 1152.4.d.d.577.1 2
16.5 even 4 1152.4.d.d.577.2 2
16.11 odd 4 1152.4.d.d.577.2 2
16.13 even 4 1152.4.d.d.577.1 2
24.5 odd 2 256.4.a.e.1.1 1
24.11 even 2 256.4.a.e.1.1 1
48.5 odd 4 128.4.b.c.65.1 ✓ 2
48.11 even 4 128.4.b.c.65.1 ✓ 2
48.29 odd 4 128.4.b.c.65.2 yes 2
48.35 even 4 128.4.b.c.65.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
128.4.b.c.65.1 ✓ 2 48.5 odd 4
128.4.b.c.65.1 ✓ 2 48.11 even 4
128.4.b.c.65.2 yes 2 48.29 odd 4
128.4.b.c.65.2 yes 2 48.35 even 4
256.4.a.d.1.1 1 3.2 odd 2
256.4.a.d.1.1 1 12.11 even 2
256.4.a.e.1.1 1 24.5 odd 2
256.4.a.e.1.1 1 24.11 even 2
1152.4.d.d.577.1 2 16.3 odd 4
1152.4.d.d.577.1 2 16.13 even 4
1152.4.d.d.577.2 2 16.5 even 4
1152.4.d.d.577.2 2 16.11 odd 4
2304.4.a.g.1.1 1 8.3 odd 2
2304.4.a.g.1.1 1 8.5 even 2
2304.4.a.j.1.1 1 1.1 even 1 trivial
2304.4.a.j.1.1 1 4.3 odd 2 CM