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Properties

Label 4608.2.d.p
Level $4608$
Weight $2$
Character orbit 4608.d
Analytic conductor $36.795$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4608,2,Mod(2305,4608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.18939904.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 1536)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} - \beta_{2} q^{7} + (\beta_{6} - \beta_{4}) q^{11} + ( - \beta_{7} + \beta_{5}) q^{13} + (\beta_{3} - 2) q^{17} - \beta_{6} q^{19} + \beta_1 q^{23} + ( - 2 \beta_{3} - 5) q^{25} + ( - 2 \beta_{7} + \beta_{5}) q^{29} - \beta_{2} q^{31} + (3 \beta_{6} + \beta_{4}) q^{35} + (\beta_{7} + \beta_{5}) q^{37} + ( - \beta_{3} + 6) q^{41} + ( - 3 \beta_{6} - 2 \beta_{4}) q^{43} - \beta_1 q^{47} + ( - 4 \beta_{3} + 7) q^{49} + ( - 2 \beta_{7} - \beta_{5}) q^{53} + (2 \beta_{2} + 2 \beta_1) q^{55} + (2 \beta_{6} - 2 \beta_{4}) q^{59} + ( - \beta_{7} - \beta_{5}) q^{61} + ( - 5 \beta_{3} - 8) q^{65} + ( - 2 \beta_{6} + 4 \beta_{4}) q^{67} - 2 \beta_{2} q^{71} - 4 q^{73} + 2 \beta_{5} q^{77} + \beta_{2} q^{79} + ( - \beta_{6} + 3 \beta_{4}) q^{83} - 2 \beta_{7} q^{85} + 10 q^{89} + ( - \beta_{6} - 6 \beta_{4}) q^{91} + ( - 2 \beta_{2} - \beta_1) q^{95} + (2 \beta_{3} + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{17} - 40 q^{25} + 48 q^{41} + 56 q^{49} - 64 q^{65} - 32 q^{73} + 80 q^{89} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( -2\nu^{6} + 6\nu^{5} - 20\nu^{4} + 30\nu^{3} - 34\nu^{2} + 20\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{6} + 3\nu^{5} - 12\nu^{4} + 19\nu^{3} - 31\nu^{2} + 22\nu - 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\nu^{6} + 6\nu^{5} - 20\nu^{4} + 30\nu^{3} - 38\nu^{2} + 24\nu - 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 16\nu^{7} - 56\nu^{6} + 196\nu^{5} - 350\nu^{4} + 512\nu^{3} - 446\nu^{2} + 252\nu - 62 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 18\nu^{7} - 63\nu^{6} + 219\nu^{5} - 390\nu^{4} + 565\nu^{3} - 489\nu^{2} + 272\nu - 66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 20\nu^{7} - 70\nu^{6} + 246\nu^{5} - 440\nu^{4} + 650\nu^{3} - 570\nu^{2} + 332\nu - 84 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 38\nu^{7} - 133\nu^{6} + 465\nu^{5} - 830\nu^{4} + 1215\nu^{3} - 1059\nu^{2} + 608\nu - 152 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} + \beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{7} + 7\beta_{6} + 6\beta_{5} - 6\beta_{4} - 3\beta_{3} + 3\beta _1 - 20 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{7} + 8\beta_{6} + 7\beta_{5} - 6\beta_{4} + 4\beta_{3} - 2\beta_{2} - 3\beta _1 + 14 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 6\beta_{7} - 13\beta_{6} - 16\beta_{5} + 20\beta_{4} + 25\beta_{3} - 10\beta_{2} - 20\beta _1 + 104 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 44\beta_{7} - 80\beta_{6} - 84\beta_{5} + 90\beta_{4} - 16\beta_{3} + 10\beta_{2} + 7\beta _1 - 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 14\beta_{7} - 19\beta_{6} - 8\beta_{5} - 147\beta_{3} + 70\beta_{2} + 98\beta _1 - 472 ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

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

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2305.1
0.500000 + 0.691860i
0.500000 + 2.10607i
0.500000 − 1.44392i
0.500000 − 0.0297061i
0.500000 + 1.44392i
0.500000 + 0.0297061i
0.500000 − 0.691860i
0.500000 − 2.10607i
0 0 0 − 3.95687i 0 −1.63899 0 0 0
2305.2 0 0 0 − 3.95687i 0 1.63899 0 0 0
2305.3 0 0 0 − 2.08402i 0 −5.03127 0 0 0
2305.4 0 0 0 − 2.08402i 0 5.03127 0 0 0
2305.5 0 0 0 2.08402i 0 −5.03127 0 0 0
2305.6 0 0 0 2.08402i 0 5.03127 0 0 0
2305.7 0 0 0 3.95687i 0 −1.63899 0 0 0
2305.8 0 0 0 3.95687i 0 1.63899 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2305.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.d.p 8
3.b odd 2 1 1536.2.d.g 8
4.b odd 2 1 inner 4608.2.d.p 8
8.b even 2 1 inner 4608.2.d.p 8
8.d odd 2 1 inner 4608.2.d.p 8
12.b even 2 1 1536.2.d.g 8
16.e even 4 1 4608.2.a.t 4
16.e even 4 1 4608.2.a.ba 4
16.f odd 4 1 4608.2.a.t 4
16.f odd 4 1 4608.2.a.ba 4
24.f even 2 1 1536.2.d.g 8
24.h odd 2 1 1536.2.d.g 8
48.i odd 4 1 1536.2.a.m ✓ 4
48.i odd 4 1 1536.2.a.n yes 4
48.k even 4 1 1536.2.a.m ✓ 4
48.k even 4 1 1536.2.a.n yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1536.2.a.m ✓ 4 48.i odd 4 1
1536.2.a.m ✓ 4 48.k even 4 1
1536.2.a.n yes 4 48.i odd 4 1
1536.2.a.n yes 4 48.k even 4 1
1536.2.d.g 8 3.b odd 2 1
1536.2.d.g 8 12.b even 2 1
1536.2.d.g 8 24.f even 2 1
1536.2.d.g 8 24.h odd 2 1
4608.2.a.t 4 16.e even 4 1
4608.2.a.t 4 16.f odd 4 1
4608.2.a.ba 4 16.e even 4 1
4608.2.a.ba 4 16.f odd 4 1
4608.2.d.p 8 1.a even 1 1 trivial
4608.2.d.p 8 4.b odd 2 1 inner
4608.2.d.p 8 8.b even 2 1 inner
4608.2.d.p 8 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4608, [\chi])\):

\( T_{5}^{4} + 20T_{5}^{2} + 68 \) Copy content Toggle raw display
\( T_{7}^{4} - 28T_{7}^{2} + 68 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 4 \) Copy content Toggle raw display
\( T_{23}^{4} - 80T_{23}^{2} + 1088 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 20 T^{2} + 68)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 28 T^{2} + 68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 24 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 40 T^{2} + 272)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 116 T^{2} + 3332)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 28 T^{2} + 68)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 56 T^{2} + 272)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12 T + 28)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 176 T^{2} + 3136)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 80 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 148 T^{2} + 68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 96 T^{2} + 256)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 56 T^{2} + 272)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 192 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 112 T^{2} + 1088)^{2} \) Copy content Toggle raw display
$73$ \( (T + 4)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 28 T^{2} + 68)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 88 T^{2} + 784)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{8} \) Copy content Toggle raw display
$97$ \( (T^{2} - 12 T + 4)^{4} \) Copy content Toggle raw display
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