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(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 = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,24] 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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.3
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2303
Dual form 4608.2.f.n.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)
 
\(f(q)\) \(=\) \(q-1.53073 q^{5} -0.585786i q^{7} +5.22625i q^{11} -5.22625i q^{13} +1.41421i q^{17} +3.06147 q^{19} +0.828427 q^{23} -2.65685 q^{25} -5.86030 q^{29} -6.24264i q^{31} +0.896683i q^{35} +3.06147i q^{37} -3.75736i q^{41} -4.32957 q^{43} +8.82843 q^{47} +6.65685 q^{49} +5.86030 q^{53} -8.00000i q^{55} -4.32957i q^{59} +7.39104i q^{61} +8.00000i q^{65} -13.5140 q^{67} +10.4853 q^{71} -7.65685 q^{73} +3.06147 q^{77} -2.92893i q^{79} +9.55582i q^{83} -2.16478i q^{85} -7.07107i q^{89} -3.06147 q^{91} -4.68629 q^{95} +11.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{23} + 24 q^{25} + 48 q^{47} + 8 q^{49} + 16 q^{71} - 16 q^{73} - 128 q^{95}+O(q^{100}) \) 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
\(5\) −1.53073 −0.684565 −0.342282 − 0.939597i \(-0.611200\pi\)
−0.342282 + 0.939597i \(0.611200\pi\)
\(6\) 0 0
\(7\) − 0.585786i − 0.221406i −0.993854 − 0.110703i \(-0.964690\pi\)
0.993854 − 0.110703i \(-0.0353103\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.22625i 1.57577i 0.615820 + 0.787887i \(0.288825\pi\)
−0.615820 + 0.787887i \(0.711175\pi\)
\(12\) 0 0
\(13\) − 5.22625i − 1.44950i −0.689011 − 0.724751i \(-0.741955\pi\)
0.689011 − 0.724751i \(-0.258045\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) 3.06147 0.702349 0.351174 − 0.936310i \(-0.385782\pi\)
0.351174 + 0.936310i \(0.385782\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.828427 0.172739 0.0863695 − 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 0 0
\(25\) −2.65685 −0.531371
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.86030 −1.08823 −0.544115 − 0.839010i \(-0.683135\pi\)
−0.544115 + 0.839010i \(0.683135\pi\)
\(30\) 0 0
\(31\) − 6.24264i − 1.12121i −0.828083 − 0.560606i \(-0.810568\pi\)
0.828083 − 0.560606i \(-0.189432\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.896683i 0.151567i
\(36\) 0 0
\(37\) 3.06147i 0.503302i 0.967818 + 0.251651i \(0.0809735\pi\)
−0.967818 + 0.251651i \(0.919026\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.75736i − 0.586801i −0.955990 − 0.293400i \(-0.905213\pi\)
0.955990 − 0.293400i \(-0.0947869\pi\)
\(42\) 0 0
\(43\) −4.32957 −0.660253 −0.330127 − 0.943937i \(-0.607091\pi\)
−0.330127 + 0.943937i \(0.607091\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.82843 1.28776 0.643879 − 0.765127i \(-0.277324\pi\)
0.643879 + 0.765127i \(0.277324\pi\)
\(48\) 0 0
\(49\) 6.65685 0.950979
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.86030 0.804974 0.402487 − 0.915426i \(-0.368146\pi\)
0.402487 + 0.915426i \(0.368146\pi\)
\(54\) 0 0
\(55\) − 8.00000i − 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 4.32957i − 0.563662i −0.959464 − 0.281831i \(-0.909058\pi\)
0.959464 − 0.281831i \(-0.0909417\pi\)
\(60\) 0 0
\(61\) 7.39104i 0.946325i 0.880975 + 0.473163i \(0.156888\pi\)
−0.880975 + 0.473163i \(0.843112\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.00000i 0.992278i
\(66\) 0 0
\(67\) −13.5140 −1.65099 −0.825497 − 0.564406i \(-0.809105\pi\)
−0.825497 + 0.564406i \(0.809105\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.4853 1.24437 0.622187 − 0.782869i \(-0.286244\pi\)
0.622187 + 0.782869i \(0.286244\pi\)
\(72\) 0 0
\(73\) −7.65685 −0.896167 −0.448084 − 0.893992i \(-0.647893\pi\)
−0.448084 + 0.893992i \(0.647893\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.06147 0.348887
\(78\) 0 0
\(79\) − 2.92893i − 0.329531i −0.986333 − 0.164765i \(-0.947313\pi\)
0.986333 − 0.164765i \(-0.0526867\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.55582i 1.04889i 0.851445 + 0.524444i \(0.175727\pi\)
−0.851445 + 0.524444i \(0.824273\pi\)
\(84\) 0 0
\(85\) − 2.16478i − 0.234804i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 7.07107i − 0.749532i −0.927119 − 0.374766i \(-0.877723\pi\)
0.927119 − 0.374766i \(-0.122277\pi\)
\(90\) 0 0
\(91\) −3.06147 −0.320929
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.68629 −0.480803
\(96\) 0 0
\(97\) 11.3137 1.14873 0.574367 − 0.818598i \(-0.305248\pi\)
0.574367 + 0.818598i \(0.305248\pi\)
\(98\) 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.n.2303.3 8
3.2 odd 2 4608.2.f.o.2303.5 8
4.3 odd 2 4608.2.f.o.2303.4 8
8.3 odd 2 4608.2.f.o.2303.6 8
8.5 even 2 inner 4608.2.f.n.2303.5 8
12.11 even 2 inner 4608.2.f.n.2303.6 8
16.3 odd 4 4608.2.c.q.4607.5 yes 8
16.5 even 4 4608.2.c.r.4607.4 yes 8
16.11 odd 4 4608.2.c.q.4607.3 ✓ 8
16.13 even 4 4608.2.c.r.4607.6 yes 8
24.5 odd 2 4608.2.f.o.2303.3 8
24.11 even 2 inner 4608.2.f.n.2303.4 8
48.5 odd 4 4608.2.c.q.4607.6 yes 8
48.11 even 4 4608.2.c.r.4607.5 yes 8
48.29 odd 4 4608.2.c.q.4607.4 yes 8
48.35 even 4 4608.2.c.r.4607.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.q.4607.3 ✓ 8 16.11 odd 4
4608.2.c.q.4607.4 yes 8 48.29 odd 4
4608.2.c.q.4607.5 yes 8 16.3 odd 4
4608.2.c.q.4607.6 yes 8 48.5 odd 4
4608.2.c.r.4607.3 yes 8 48.35 even 4
4608.2.c.r.4607.4 yes 8 16.5 even 4
4608.2.c.r.4607.5 yes 8 48.11 even 4
4608.2.c.r.4607.6 yes 8 16.13 even 4
4608.2.f.n.2303.3 8 1.1 even 1 trivial
4608.2.f.n.2303.4 8 24.11 even 2 inner
4608.2.f.n.2303.5 8 8.5 even 2 inner
4608.2.f.n.2303.6 8 12.11 even 2 inner
4608.2.f.o.2303.3 8 24.5 odd 2
4608.2.f.o.2303.4 8 4.3 odd 2
4608.2.f.o.2303.5 8 3.2 odd 2
4608.2.f.o.2303.6 8 8.3 odd 2