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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4608,2,Mod(2303,4608)] 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("4608.2303"); 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([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,24,0,-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: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.1
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2303
Dual form 4608.2.f.k.2303.4

$q$-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)
 
\(f(q)\) \(=\) \(q-4.24264i q^{7} -4.00000i q^{11} -6.00000i q^{13} -4.24264i q^{17} +2.82843 q^{19} +6.00000 q^{23} -5.00000 q^{25} +8.48528 q^{29} -4.24264i q^{31} +6.00000i q^{37} -1.41421i q^{41} -2.82843 q^{43} +6.00000 q^{47} -11.0000 q^{49} -8.48528 q^{53} +4.00000i q^{59} +6.00000i q^{61} +11.3137 q^{67} -6.00000 q^{71} -6.00000 q^{73} -16.9706 q^{77} -4.24264i q^{79} +16.0000i q^{83} -12.7279i q^{89} -25.4558 q^{91} -12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 24 q^{23} - 20 q^{25} + 24 q^{47} - 44 q^{49} - 24 q^{71} - 24 q^{73} - 48 q^{97}+O(q^{100}) \) 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
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) − 4.24264i − 1.60357i −0.597614 − 0.801784i \(-0.703885\pi\)
0.597614 − 0.801784i \(-0.296115\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 − 0.603023i \(-0.793963\pi\)
0.797724 − 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) − 6.00000i − 1.66410i −0.554700 − 0.832050i \(-0.687167\pi\)
0.554700 − 0.832050i \(-0.312833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.24264i − 1.02899i −0.857493 − 0.514496i \(-0.827979\pi\)
0.857493 − 0.514496i \(-0.172021\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 − 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.48528 1.57568 0.787839 − 0.615882i \(-0.211200\pi\)
0.787839 + 0.615882i \(0.211200\pi\)
\(30\) 0 0
\(31\) − 4.24264i − 0.762001i −0.924575 − 0.381000i \(-0.875580\pi\)
0.924575 − 0.381000i \(-0.124420\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 6.00000i 0.986394i 0.869918 + 0.493197i \(0.164172\pi\)
−0.869918 + 0.493197i \(0.835828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 1.41421i − 0.220863i −0.993884 − 0.110432i \(-0.964777\pi\)
0.993884 − 0.110432i \(-0.0352233\pi\)
\(42\) 0 0
\(43\) −2.82843 −0.431331 −0.215666 − 0.976467i \(-0.569192\pi\)
−0.215666 + 0.976467i \(0.569192\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −11.0000 −1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.48528 −1.16554 −0.582772 − 0.812636i \(-0.698032\pi\)
−0.582772 + 0.812636i \(0.698032\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.00000i 0.520756i 0.965507 + 0.260378i \(0.0838471\pi\)
−0.965507 + 0.260378i \(0.916153\pi\)
\(60\) 0 0
\(61\) 6.00000i 0.768221i 0.923287 + 0.384111i \(0.125492\pi\)
−0.923287 + 0.384111i \(0.874508\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 11.3137 1.38219 0.691095 − 0.722764i \(-0.257129\pi\)
0.691095 + 0.722764i \(0.257129\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 − 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −16.9706 −1.93398
\(78\) 0 0
\(79\) − 4.24264i − 0.477334i −0.971101 − 0.238667i \(-0.923290\pi\)
0.971101 − 0.238667i \(-0.0767105\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 16.0000i 1.75623i 0.478451 + 0.878114i \(0.341198\pi\)
−0.478451 + 0.878114i \(0.658802\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 12.7279i − 1.34916i −0.738203 − 0.674579i \(-0.764325\pi\)
0.738203 − 0.674579i \(-0.235675\pi\)
\(90\) 0 0
\(91\) −25.4558 −2.66850
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.0000 −1.21842 −0.609208 − 0.793011i \(-0.708512\pi\)
−0.609208 + 0.793011i \(0.708512\pi\)
\(98\) 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.f.k.2303.1 4
3.2 odd 2 4608.2.f.i.2303.2 4
4.3 odd 2 4608.2.f.i.2303.4 4
8.3 odd 2 4608.2.f.i.2303.3 4
8.5 even 2 inner 4608.2.f.k.2303.2 4
12.11 even 2 inner 4608.2.f.k.2303.3 4
16.3 odd 4 4608.2.c.h.4607.1 yes 2
16.5 even 4 4608.2.c.g.4607.2 yes 2
16.11 odd 4 4608.2.c.a.4607.1 ✓ 2
16.13 even 4 4608.2.c.b.4607.2 yes 2
24.5 odd 2 4608.2.f.i.2303.1 4
24.11 even 2 inner 4608.2.f.k.2303.4 4
48.5 odd 4 4608.2.c.a.4607.2 yes 2
48.11 even 4 4608.2.c.g.4607.1 yes 2
48.29 odd 4 4608.2.c.h.4607.2 yes 2
48.35 even 4 4608.2.c.b.4607.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.a.4607.1 ✓ 2 16.11 odd 4
4608.2.c.a.4607.2 yes 2 48.5 odd 4
4608.2.c.b.4607.1 yes 2 48.35 even 4
4608.2.c.b.4607.2 yes 2 16.13 even 4
4608.2.c.g.4607.1 yes 2 48.11 even 4
4608.2.c.g.4607.2 yes 2 16.5 even 4
4608.2.c.h.4607.1 yes 2 16.3 odd 4
4608.2.c.h.4607.2 yes 2 48.29 odd 4
4608.2.f.i.2303.1 4 24.5 odd 2
4608.2.f.i.2303.2 4 3.2 odd 2
4608.2.f.i.2303.3 4 8.3 odd 2
4608.2.f.i.2303.4 4 4.3 odd 2
4608.2.f.k.2303.1 4 1.1 even 1 trivial
4608.2.f.k.2303.2 4 8.5 even 2 inner
4608.2.f.k.2303.3 4 12.11 even 2 inner
4608.2.f.k.2303.4 4 24.11 even 2 inner