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(2305,4608)] 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("4608.2305"); 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, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,0,0,0,0,0,0,-4] 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: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2305.4
Root \(-0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2305
Dual form 4608.2.d.e.2305.2

$q$-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)
 
\(f(q)\) \(=\) \(q+2.44949i q^{5} +1.41421 q^{7} +3.46410i q^{11} +4.89898i q^{13} -4.00000 q^{17} +6.92820i q^{19} -5.65685 q^{23} -1.00000 q^{25} -2.44949i q^{29} +1.41421 q^{31} +3.46410i q^{35} -4.89898i q^{37} +4.00000 q^{41} +6.92820i q^{43} -5.65685 q^{47} -5.00000 q^{49} -7.34847i q^{53} -8.48528 q^{55} -13.8564i q^{59} +4.89898i q^{61} -12.0000 q^{65} -11.3137 q^{71} +4.00000 q^{73} +4.89898i q^{77} -7.07107 q^{79} -10.3923i q^{83} -9.79796i q^{85} +16.0000 q^{89} +6.92820i q^{91} -16.9706 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 16 q^{17} - 4 q^{25} + 16 q^{41} - 20 q^{49} - 48 q^{65} + 16 q^{73} + 64 q^{89} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.44949i 1.09545i 0.836660 + 0.547723i \(0.184505\pi\)
−0.836660 + 0.547723i \(0.815495\pi\)
\(6\) 0 0
\(7\) 1.41421 0.534522 0.267261 − 0.963624i \(-0.413881\pi\)
0.267261 + 0.963624i \(0.413881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 4.89898i 1.35873i 0.733799 + 0.679366i \(0.237745\pi\)
−0.733799 + 0.679366i \(0.762255\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i 0.606977 + 0.794719i \(0.292382\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.65685 −1.17954 −0.589768 − 0.807573i \(-0.700781\pi\)
−0.589768 + 0.807573i \(0.700781\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 2.44949i − 0.454859i −0.973795 − 0.227429i \(-0.926968\pi\)
0.973795 − 0.227429i \(-0.0730321\pi\)
\(30\) 0 0
\(31\) 1.41421 0.254000 0.127000 − 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.46410i 0.585540i
\(36\) 0 0
\(37\) − 4.89898i − 0.805387i −0.915335 − 0.402694i \(-0.868074\pi\)
0.915335 − 0.402694i \(-0.131926\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.00000 0.624695 0.312348 − 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 6.92820i 1.05654i 0.849076 + 0.528271i \(0.177159\pi\)
−0.849076 + 0.528271i \(0.822841\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.65685 −0.825137 −0.412568 − 0.910927i \(-0.635368\pi\)
−0.412568 + 0.910927i \(0.635368\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 7.34847i − 1.00939i −0.863298 − 0.504695i \(-0.831605\pi\)
0.863298 − 0.504695i \(-0.168395\pi\)
\(54\) 0 0
\(55\) −8.48528 −1.14416
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 13.8564i − 1.80395i −0.431788 − 0.901975i \(-0.642117\pi\)
0.431788 − 0.901975i \(-0.357883\pi\)
\(60\) 0 0
\(61\) 4.89898i 0.627250i 0.949547 + 0.313625i \(0.101543\pi\)
−0.949547 + 0.313625i \(0.898457\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.0000 −1.48842
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 − 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.89898i 0.558291i
\(78\) 0 0
\(79\) −7.07107 −0.795557 −0.397779 − 0.917481i \(-0.630219\pi\)
−0.397779 + 0.917481i \(0.630219\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 10.3923i − 1.14070i −0.821401 − 0.570352i \(-0.806807\pi\)
0.821401 − 0.570352i \(-0.193193\pi\)
\(84\) 0 0
\(85\) − 9.79796i − 1.06274i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.0000 1.69600 0.847998 − 0.529999i \(-0.177808\pi\)
0.847998 + 0.529999i \(0.177808\pi\)
\(90\) 0 0
\(91\) 6.92820i 0.726273i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −16.9706 −1.74114
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 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.d.e.2305.4 4
3.2 odd 2 4608.2.d.m.2305.2 4
4.3 odd 2 inner 4608.2.d.e.2305.3 4
8.3 odd 2 inner 4608.2.d.e.2305.1 4
8.5 even 2 inner 4608.2.d.e.2305.2 4
12.11 even 2 4608.2.d.m.2305.1 4
16.3 odd 4 4608.2.a.v.1.4 yes 4
16.5 even 4 4608.2.a.v.1.1 ✓ 4
16.11 odd 4 4608.2.a.v.1.2 yes 4
16.13 even 4 4608.2.a.v.1.3 yes 4
24.5 odd 2 4608.2.d.m.2305.4 4
24.11 even 2 4608.2.d.m.2305.3 4
48.5 odd 4 4608.2.a.z.1.3 yes 4
48.11 even 4 4608.2.a.z.1.4 yes 4
48.29 odd 4 4608.2.a.z.1.1 yes 4
48.35 even 4 4608.2.a.z.1.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.v.1.1 ✓ 4 16.5 even 4
4608.2.a.v.1.2 yes 4 16.11 odd 4
4608.2.a.v.1.3 yes 4 16.13 even 4
4608.2.a.v.1.4 yes 4 16.3 odd 4
4608.2.a.z.1.1 yes 4 48.29 odd 4
4608.2.a.z.1.2 yes 4 48.35 even 4
4608.2.a.z.1.3 yes 4 48.5 odd 4
4608.2.a.z.1.4 yes 4 48.11 even 4
4608.2.d.e.2305.1 4 8.3 odd 2 inner
4608.2.d.e.2305.2 4 8.5 even 2 inner
4608.2.d.e.2305.3 4 4.3 odd 2 inner
4608.2.d.e.2305.4 4 1.1 even 1 trivial
4608.2.d.m.2305.1 4 12.11 even 2
4608.2.d.m.2305.2 4 3.2 odd 2
4608.2.d.m.2305.3 4 24.11 even 2
4608.2.d.m.2305.4 4 24.5 odd 2