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Properties

Label 2304.3.h.i
Level $2304$
Weight $3$
Character orbit 2304.h
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(2177,2304)]
 
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, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (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: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{41}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1152)
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{5} + (\beta_{7} - 2) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{5} + \beta_{3}) q^{5} + (\beta_{7} - 2) q^{7} + (2 \beta_{5} + 2 \beta_{3}) q^{11} + (3 \beta_{6} - 4 \beta_1) q^{13} + (3 \beta_{4} + \beta_{2}) q^{17} + (2 \beta_{6} - 4 \beta_1) q^{19} - 10 \beta_{2} q^{23} + ( - \beta_{7} - 11) q^{25} + (2 \beta_{5} + 7 \beta_{3}) q^{29} + (\beta_{7} - 30) q^{31} + ( - 16 \beta_{5} - 6 \beta_{3}) q^{35} + (4 \beta_{6} + 11 \beta_1) q^{37} + (5 \beta_{4} + 21 \beta_{2}) q^{41} + (8 \beta_{6} + 18 \beta_1) q^{43} + (4 \beta_{4} + 10 \beta_{2}) q^{47} + ( - 4 \beta_{7} + 51) q^{49} + (16 \beta_{5} + 5 \beta_{3}) q^{53} + ( - 2 \beta_{7} + 28) q^{55} + ( - 22 \beta_{5} - 24 \beta_{3}) q^{59} + ( - 8 \beta_{6} - 25 \beta_1) q^{61} + ( - 7 \beta_{4} - 44 \beta_{2}) q^{65} + ( - 2 \beta_{6} - 14 \beta_1) q^{67} + (6 \beta_{4} - 54 \beta_{2}) q^{71} + (2 \beta_{7} - 68) q^{73} + ( - 32 \beta_{5} - 12 \beta_{3}) q^{77} + (3 \beta_{7} - 122) q^{79} + ( - 32 \beta_{5} + 14 \beta_{3}) q^{83} + ( - 7 \beta_{6} + 37 \beta_1) q^{85} + ( - 14 \beta_{4} + 17 \beta_{2}) q^{89} + ( - 22 \beta_{6} + 80 \beta_1) q^{91} + ( - 6 \beta_{4} - 32 \beta_{2}) q^{95} + (13 \beta_{7} + 40) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{7} - 88 q^{25} - 240 q^{31} + 408 q^{49} + 224 q^{55} - 544 q^{73} - 976 q^{79} + 320 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{24}^{6} - \zeta_{24}^{5} + \zeta_{24}^{3} + 4\zeta_{24}^{2} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 8\zeta_{24}^{4} - 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{24}^{5} - 2\zeta_{24}^{3} - 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\zeta_{24}^{7} + 4\zeta_{24}^{5} + 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - 2\beta_{5} + 4\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{5} + 2\beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{5} - 2\beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{4} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + 2\beta_{6} + 2\beta_{5} - 4\beta_{2} ) / 16 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} - 2\beta_{5} - 4\beta_{2} ) / 16 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2177.1
−0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 − 0.965926i
0.258819 + 0.965926i
0.965926 − 0.258819i
0.965926 + 0.258819i
−0.965926 − 0.258819i
−0.965926 + 0.258819i
0 0 0 −4.87832 0 −11.7980 0 0 0
2177.2 0 0 0 −4.87832 0 −11.7980 0 0 0
2177.3 0 0 0 −2.04989 0 7.79796 0 0 0
2177.4 0 0 0 −2.04989 0 7.79796 0 0 0
2177.5 0 0 0 2.04989 0 7.79796 0 0 0
2177.6 0 0 0 2.04989 0 7.79796 0 0 0
2177.7 0 0 0 4.87832 0 −11.7980 0 0 0
2177.8 0 0 0 4.87832 0 −11.7980 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2177.8
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.h.i 8
3.b odd 2 1 inner 2304.3.h.i 8
4.b odd 2 1 2304.3.h.k 8
8.b even 2 1 inner 2304.3.h.i 8
8.d odd 2 1 2304.3.h.k 8
12.b even 2 1 2304.3.h.k 8
16.e even 4 1 1152.3.e.f yes 4
16.e even 4 1 1152.3.e.h yes 4
16.f odd 4 1 1152.3.e.b ✓ 4
16.f odd 4 1 1152.3.e.d yes 4
24.f even 2 1 2304.3.h.k 8
24.h odd 2 1 inner 2304.3.h.i 8
48.i odd 4 1 1152.3.e.f yes 4
48.i odd 4 1 1152.3.e.h yes 4
48.k even 4 1 1152.3.e.b ✓ 4
48.k even 4 1 1152.3.e.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.e.b ✓ 4 16.f odd 4 1
1152.3.e.b ✓ 4 48.k even 4 1
1152.3.e.d yes 4 16.f odd 4 1
1152.3.e.d yes 4 48.k even 4 1
1152.3.e.f yes 4 16.e even 4 1
1152.3.e.f yes 4 48.i odd 4 1
1152.3.e.h yes 4 16.e even 4 1
1152.3.e.h yes 4 48.i odd 4 1
2304.3.h.i 8 1.a even 1 1 trivial
2304.3.h.i 8 3.b odd 2 1 inner
2304.3.h.i 8 8.b even 2 1 inner
2304.3.h.i 8 24.h odd 2 1 inner
2304.3.h.k 8 4.b odd 2 1
2304.3.h.k 8 8.d odd 2 1
2304.3.h.k 8 12.b even 2 1
2304.3.h.k 8 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(2304, [\chi])\):

\( T_{5}^{4} - 28T_{5}^{2} + 100 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} - 92 \) 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
$5$ \( (T^{4} - 28 T^{2} + 100)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T - 92)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 112 T^{2} + 1600)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 560 T^{2} + 23104)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 868 T^{2} + 184900)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 320 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 200)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1212 T^{2} + 324900)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 60 T + 804)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1736 T^{2} + 10000)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 4164 T^{2} + 101124)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 5664 T^{2} + 57600)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 1936 T^{2} + 322624)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 3516 T^{2} + 1340964)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 15424 T^{2} + 37356544)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8072 T^{2} + 929296)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1760 T^{2} + 473344)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 15120 T^{2} + 16842816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 136 T + 4240)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 244 T + 14020)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 29040 T^{2} + 96353856)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 19972 T^{2} + 77968900)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 80 T - 14624)^{4} \) Copy content Toggle raw display
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