Properties

Label 4608.2.a.bc.1.4
Level $4608$
Weight $2$
Character 4608.1
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM discriminant -24
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] = [4608,2,Mod(1,4608)] 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("4608.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4608, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.4
Root \(1.93185\) of defining polynomial
Character \(\chi\) \(=\) 4608.1

$q$-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)
 
\(f(q)\) \(=\) \(q+4.44949 q^{5} +4.87832 q^{7} +3.46410 q^{11} +14.7980 q^{25} -5.34847 q^{29} -10.5352 q^{31} +21.7060 q^{35} +16.7980 q^{49} +12.4495 q^{53} +15.4135 q^{55} -11.3137 q^{59} -9.79796 q^{73} +16.8990 q^{77} -3.32124 q^{79} -17.3205 q^{83} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5} + 20 q^{25} + 8 q^{29} + 28 q^{49} + 40 q^{53} + 48 q^{77} - 8 q^{97}+O(q^{100}) \) 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
\(5\) 4.44949 1.98987 0.994936 − 0.100509i \(-0.0320471\pi\)
0.994936 + 0.100509i \(0.0320471\pi\)
\(6\) 0 0
\(7\) 4.87832 1.84383 0.921915 − 0.387392i \(-0.126624\pi\)
0.921915 + 0.387392i \(0.126624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 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
\(25\) 14.7980 2.95959
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.34847 −0.993186 −0.496593 − 0.867984i \(-0.665416\pi\)
−0.496593 + 0.867984i \(0.665416\pi\)
\(30\) 0 0
\(31\) −10.5352 −1.89217 −0.946086 − 0.323915i \(-0.895001\pi\)
−0.946086 + 0.323915i \(0.895001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 21.7060 3.66899
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 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
\(49\) 16.7980 2.39971
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.4495 1.71007 0.855034 − 0.518571i \(-0.173536\pi\)
0.855034 + 0.518571i \(0.173536\pi\)
\(54\) 0 0
\(55\) 15.4135 2.07835
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.3137 −1.47292 −0.736460 − 0.676481i \(-0.763504\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 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
\(73\) −9.79796 −1.14676 −0.573382 − 0.819288i \(-0.694369\pi\)
−0.573382 + 0.819288i \(0.694369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.8990 1.92582
\(78\) 0 0
\(79\) −3.32124 −0.373668 −0.186834 − 0.982391i \(-0.559823\pi\)
−0.186834 + 0.982391i \(0.559823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −17.3205 −1.90117 −0.950586 − 0.310460i \(-0.899517\pi\)
−0.950586 + 0.310460i \(0.899517\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 − 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 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 4608.2.a.bc.1.4 yes 4
3.2 odd 2 4608.2.a.s.1.2 yes 4
4.3 odd 2 inner 4608.2.a.bc.1.3 yes 4
8.3 odd 2 4608.2.a.s.1.1 ✓ 4
8.5 even 2 4608.2.a.s.1.2 yes 4
12.11 even 2 4608.2.a.s.1.1 ✓ 4
16.3 odd 4 4608.2.d.q.2305.2 8
16.5 even 4 4608.2.d.q.2305.7 8
16.11 odd 4 4608.2.d.q.2305.8 8
16.13 even 4 4608.2.d.q.2305.1 8
24.5 odd 2 CM 4608.2.a.bc.1.4 yes 4
24.11 even 2 inner 4608.2.a.bc.1.3 yes 4
48.5 odd 4 4608.2.d.q.2305.1 8
48.11 even 4 4608.2.d.q.2305.2 8
48.29 odd 4 4608.2.d.q.2305.7 8
48.35 even 4 4608.2.d.q.2305.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.s.1.1 ✓ 4 8.3 odd 2
4608.2.a.s.1.1 ✓ 4 12.11 even 2
4608.2.a.s.1.2 yes 4 3.2 odd 2
4608.2.a.s.1.2 yes 4 8.5 even 2
4608.2.a.bc.1.3 yes 4 4.3 odd 2 inner
4608.2.a.bc.1.3 yes 4 24.11 even 2 inner
4608.2.a.bc.1.4 yes 4 1.1 even 1 trivial
4608.2.a.bc.1.4 yes 4 24.5 odd 2 CM
4608.2.d.q.2305.1 8 16.13 even 4
4608.2.d.q.2305.1 8 48.5 odd 4
4608.2.d.q.2305.2 8 16.3 odd 4
4608.2.d.q.2305.2 8 48.11 even 4
4608.2.d.q.2305.7 8 16.5 even 4
4608.2.d.q.2305.7 8 48.29 odd 4
4608.2.d.q.2305.8 8 16.11 odd 4
4608.2.d.q.2305.8 8 48.35 even 4