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Properties

Label 4800.2.k.s
Level $4800$
Weight $2$
Character orbit 4800.k
Analytic conductor $38.328$
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] = [4800,2,Mod(2401,4800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4800.2401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.k (of order \(2\), degree \(1\), 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: \(38.3281929702\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} + \beta_{7} q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} + \beta_{7} q^{7} - q^{9} + ( - \beta_{6} - \beta_{2}) q^{11} + ( - 2 \beta_{3} + \beta_1) q^{13} + (\beta_{4} + 3) q^{17} + \beta_{6} q^{19} - \beta_{3} q^{21} + ( - \beta_{7} - 3 \beta_{5}) q^{23} + \beta_{2} q^{27} + ( - \beta_{3} + \beta_1) q^{29} + ( - \beta_{7} - 4 \beta_{5}) q^{31} + (\beta_{4} - 1) q^{33} - 4 \beta_1 q^{37} + ( - 2 \beta_{7} + \beta_{5}) q^{39} + ( - \beta_{4} - 5) q^{41} + \beta_{6} q^{43} - 2 \beta_{7} q^{47} + (\beta_{6} - 3 \beta_{2}) q^{51} + (3 \beta_{3} - \beta_1) q^{53} - \beta_{4} q^{57} - 2 \beta_{6} q^{59} + (2 \beta_{3} + 3 \beta_1) q^{61} - \beta_{7} q^{63} + \beta_{6} q^{67} + (\beta_{3} + 3 \beta_1) q^{69} + (3 \beta_{7} - 5 \beta_{5}) q^{71} - 2 q^{73} + ( - \beta_{3} - 7 \beta_1) q^{77} + 2 \beta_{7} q^{79} + q^{81} - 10 \beta_{2} q^{83} + ( - \beta_{7} + \beta_{5}) q^{87} + (2 \beta_{4} - 6) q^{89} + (\beta_{6} - 14 \beta_{2}) q^{91} + (\beta_{3} + 4 \beta_1) q^{93} + ( - 2 \beta_{4} + 9) q^{97} + (\beta_{6} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{9} + 24 q^{17} - 8 q^{33} - 40 q^{41} - 16 q^{73} + 8 q^{81} - 48 q^{89} + 72 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 26 ) / 10 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{6} - 9 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 3\nu^{4} + \nu^{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - \nu^{5} + \nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7\nu^{7} + 25\nu^{5} + 55\nu^{3} + 184\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} - 11\nu^{3} - 16\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{4} + \beta_{3} + 3\beta _1 - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 5\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{4} + 3\beta_{3} - \beta _1 - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} - 11\beta_{5} - 11\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{3} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} - 7\beta_{6} + 13\beta_{5} - 13\beta_{2} ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1601\) \(4351\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2401.1
−1.09445 + 0.895644i
−0.228425 − 1.39564i
1.09445 + 0.895644i
0.228425 − 1.39564i
−0.228425 + 1.39564i
−1.09445 − 0.895644i
0.228425 + 1.39564i
1.09445 − 0.895644i
0 − 1.00000i 0 0 0 −2.64575 0 −1.00000 0
2401.2 0 − 1.00000i 0 0 0 −2.64575 0 −1.00000 0
2401.3 0 − 1.00000i 0 0 0 2.64575 0 −1.00000 0
2401.4 0 − 1.00000i 0 0 0 2.64575 0 −1.00000 0
2401.5 0 1.00000i 0 0 0 −2.64575 0 −1.00000 0
2401.6 0 1.00000i 0 0 0 −2.64575 0 −1.00000 0
2401.7 0 1.00000i 0 0 0 2.64575 0 −1.00000 0
2401.8 0 1.00000i 0 0 0 2.64575 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2401.8
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 4800.2.k.s yes 8
4.b odd 2 1 inner 4800.2.k.s yes 8
5.b even 2 1 4800.2.k.p ✓ 8
5.c odd 4 1 4800.2.d.s 8
5.c odd 4 1 4800.2.d.t 8
8.b even 2 1 inner 4800.2.k.s yes 8
8.d odd 2 1 inner 4800.2.k.s yes 8
20.d odd 2 1 4800.2.k.p ✓ 8
20.e even 4 1 4800.2.d.s 8
20.e even 4 1 4800.2.d.t 8
40.e odd 2 1 4800.2.k.p ✓ 8
40.f even 2 1 4800.2.k.p ✓ 8
40.i odd 4 1 4800.2.d.s 8
40.i odd 4 1 4800.2.d.t 8
40.k even 4 1 4800.2.d.s 8
40.k even 4 1 4800.2.d.t 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4800.2.d.s 8 5.c odd 4 1
4800.2.d.s 8 20.e even 4 1
4800.2.d.s 8 40.i odd 4 1
4800.2.d.s 8 40.k even 4 1
4800.2.d.t 8 5.c odd 4 1
4800.2.d.t 8 20.e even 4 1
4800.2.d.t 8 40.i odd 4 1
4800.2.d.t 8 40.k even 4 1
4800.2.k.p ✓ 8 5.b even 2 1
4800.2.k.p ✓ 8 20.d odd 2 1
4800.2.k.p ✓ 8 40.e odd 2 1
4800.2.k.p ✓ 8 40.f even 2 1
4800.2.k.s yes 8 1.a even 1 1 trivial
4800.2.k.s yes 8 4.b odd 2 1 inner
4800.2.k.s yes 8 8.b even 2 1 inner
4800.2.k.s yes 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 \(S_{2}^{\mathrm{new}}(4800, [\chi])\):

\( T_{7}^{2} - 7 \) Copy content Toggle raw display
\( T_{11}^{4} + 44T_{11}^{2} + 400 \) Copy content Toggle raw display
\( T_{17}^{2} - 6T_{17} - 12 \) Copy content Toggle raw display
\( T_{23}^{4} - 68T_{23}^{2} + 400 \) Copy content Toggle raw display
\( T_{73} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 44 T^{2} + 400)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 62 T^{2} + 625)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6 T - 12)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 68 T^{2} + 400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 20 T^{2} + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 110 T^{2} + 1681)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 10 T + 4)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 132 T^{2} + 3600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 84)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 110 T^{2} + 1)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 276 T^{2} + 144)^{2} \) Copy content Toggle raw display
$73$ \( (T + 2)^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 100)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 48)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 18 T - 3)^{4} \) Copy content Toggle raw display
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