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Properties

Label 2304.3.e.o
Level $2304$
Weight $3$
Character orbit 2304.e
Analytic conductor $62.779$
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] = [2304,3,Mod(1025,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1025");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.e (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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.540942598144.16
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 71x^{4} + 56x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} - \beta_{3} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} - \beta_{3} q^{7} + ( - \beta_{4} - \beta_{2}) q^{11} - \beta_{5} q^{13} + (2 \beta_{6} + \beta_1) q^{17} + ( - \beta_{7} - \beta_{5}) q^{19} + ( - \beta_{6} - 3 \beta_1) q^{23} + ( - 4 \beta_{3} - 5) q^{25} + (2 \beta_{4} - \beta_{2}) q^{29} + (3 \beta_{3} + 16) q^{31} + (\beta_{4} - 7 \beta_{2}) q^{35} + (\beta_{7} + \beta_{5}) q^{37} + (2 \beta_{6} - \beta_1) q^{41} + ( - 2 \beta_{7} + 2 \beta_{5}) q^{43} + (3 \beta_{6} - 7 \beta_1) q^{47} + 3 q^{49} + ( - 2 \beta_{4} - 5 \beta_{2}) q^{53} + (2 \beta_{3} + 32) q^{55} + (4 \beta_{4} - 4 \beta_{2}) q^{59} + ( - \beta_{7} + 3 \beta_{5}) q^{61} - 2 \beta_{6} q^{65} + ( - \beta_{7} + 3 \beta_{5}) q^{67} + (3 \beta_{6} - 15 \beta_1) q^{71} + (8 \beta_{3} + 20) q^{73} + ( - 8 \beta_{4} + 4 \beta_{2}) q^{77} + ( - 11 \beta_{3} + 48) q^{79} + ( - \beta_{4} + 7 \beta_{2}) q^{83} + ( - 5 \beta_{7} - 2 \beta_{5}) q^{85} + ( - 4 \beta_{6} + 7 \beta_1) q^{89} + (\beta_{7} - 7 \beta_{5}) q^{91} + ( - 14 \beta_{6} - 34 \beta_1) q^{95} + ( - 4 \beta_{3} - 24) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{25} + 128 q^{31} + 24 q^{49} + 256 q^{55} + 160 q^{73} + 384 q^{79} - 192 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 16x^{6} + 71x^{4} + 56x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{5} + 9\nu^{3} + 11\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{7} + 29\nu^{5} + 115\nu^{3} + 73\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{4} + 32\nu^{2} + 14 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} - 35\nu^{5} - 157\nu^{3} - 43\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{6} + 24\nu^{4} + 68\nu^{2} + 16 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{7} + 61\nu^{5} + 245\nu^{3} + 107\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\nu^{6} + 24\nu^{4} + 76\nu^{2} + 48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{4} - \beta_{2} + \beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} - 32 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{6} - 3\beta_{4} + 5\beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} + 4\beta_{5} + \beta_{3} + 114 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 61\beta_{6} + 43\beta_{4} - 79\beta_{2} + 33\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 31\beta_{7} - 29\beta_{5} - 12\beta_{3} - 856 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -461\beta_{6} - 315\beta_{4} + 619\beta_{2} - 285\beta_1 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
− 0.467011i
2.78961i
− 2.69642i
− 0.854013i
2.69642i
0.854013i
0.467011i
− 2.78961i
0 0 0 − 7.67101i 0 −7.21110 0 0 0
1025.2 0 0 0 − 7.67101i 0 −7.21110 0 0 0
1025.3 0 0 0 − 1.07498i 0 7.21110 0 0 0
1025.4 0 0 0 − 1.07498i 0 7.21110 0 0 0
1025.5 0 0 0 1.07498i 0 7.21110 0 0 0
1025.6 0 0 0 1.07498i 0 7.21110 0 0 0
1025.7 0 0 0 7.67101i 0 −7.21110 0 0 0
1025.8 0 0 0 7.67101i 0 −7.21110 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1025.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.e.o 8
3.b odd 2 1 inner 2304.3.e.o 8
4.b odd 2 1 2304.3.e.n 8
8.b even 2 1 inner 2304.3.e.o 8
8.d odd 2 1 2304.3.e.n 8
12.b even 2 1 2304.3.e.n 8
16.e even 4 2 288.3.h.a 8
16.f odd 4 2 72.3.h.a ✓ 8
24.f even 2 1 2304.3.e.n 8
24.h odd 2 1 inner 2304.3.e.o 8
48.i odd 4 2 288.3.h.a 8
48.k even 4 2 72.3.h.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.3.h.a ✓ 8 16.f odd 4 2
72.3.h.a ✓ 8 48.k even 4 2
288.3.h.a 8 16.e even 4 2
288.3.h.a 8 48.i odd 4 2
2304.3.e.n 8 4.b odd 2 1
2304.3.e.n 8 8.d odd 2 1
2304.3.e.n 8 12.b even 2 1
2304.3.e.n 8 24.f even 2 1
2304.3.e.o 8 1.a even 1 1 trivial
2304.3.e.o 8 3.b odd 2 1 inner
2304.3.e.o 8 8.b even 2 1 inner
2304.3.e.o 8 24.h 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 \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} + 60T_{5}^{2} + 68 \) Copy content Toggle raw display
\( T_{7}^{2} - 52 \) Copy content Toggle raw display
\( T_{13}^{4} - 472T_{13}^{2} + 2448 \) Copy content Toggle raw display
\( T_{19}^{4} - 1504T_{19}^{2} + 352512 \) Copy content Toggle raw display
\( T_{31}^{2} - 32T_{31} - 212 \) 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
$5$ \( (T^{4} + 60 T^{2} + 68)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 52)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 304 T^{2} + 9792)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 472 T^{2} + 2448)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 836 T^{2} + 171396)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1504 T^{2} + 352512)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 464 T^{2} + 576)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 988 T^{2} + 103428)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 32 T - 212)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 1504 T^{2} + 352512)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 932 T^{2} + 133956)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 6400 T^{2} + 10027008)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 4176 T^{2} + 46656)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2524 T^{2} + 1591812)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 4608 T^{2} + 4456448)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 5472 T^{2} + 3172608)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 5472 T^{2} + 3172608)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 11088 T^{2} + 13483584)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 40 T - 2928)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 96 T - 3988)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 3120 T^{2} + 183872)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5828 T^{2} + 171396)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 48 T - 256)^{4} \) Copy content Toggle raw display
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