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Properties

Label 2304.3.b.s
Level $2304$
Weight $3$
Character orbit 2304.b
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(127,2304)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.b (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: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{18} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{6} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{6} - \beta_1) q^{7} + \beta_{4} q^{11} + (2 \beta_{6} + 3 \beta_1) q^{13} + (\beta_{4} - \beta_{3}) q^{17} + (\beta_{7} + 12) q^{19} + ( - \beta_{5} - 5 \beta_{2}) q^{23} + ( - 2 \beta_{7} - 11) q^{25} + ( - 2 \beta_{5} - \beta_{2}) q^{29} + (\beta_{6} + 9 \beta_1) q^{31} + (3 \beta_{4} + 8 \beta_{3}) q^{35} + (2 \beta_{6} - \beta_1) q^{37} + (\beta_{4} + 7 \beta_{3}) q^{41} + ( - \beta_{7} + 20) q^{43} + (5 \beta_{5} - 7 \beta_{2}) q^{47} + ( - 2 \beta_{7} - 11) q^{49} + (6 \beta_{5} + \beta_{2}) q^{53} + (4 \beta_{6} + 28 \beta_1) q^{55} + (2 \beta_{4} - 8 \beta_{3}) q^{59} + ( - 2 \beta_{6} - 7 \beta_1) q^{61} + ( - 7 \beta_{4} - 17 \beta_{3}) q^{65} + ( - 6 \beta_{7} + 24) q^{67} + ( - 8 \beta_{5} - 8 \beta_{2}) q^{71} + (6 \beta_{7} + 38) q^{73} + ( - 6 \beta_{5} - 10 \beta_{2}) q^{77} + ( - 3 \beta_{6} + 29 \beta_1) q^{79} + (\beta_{4} + 16 \beta_{3}) q^{83} + 20 \beta_1 q^{85} + ( - 6 \beta_{4} + 14 \beta_{3}) q^{89} + (5 \beta_{7} + 124) q^{91} + (5 \beta_{5} + 25 \beta_{2}) q^{95} + (4 \beta_{7} + 14) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 96 q^{19} - 88 q^{25} + 160 q^{43} - 88 q^{49} + 192 q^{67} + 304 q^{73} + 992 q^{91} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 16\nu^{4} + 7\nu^{3} - 4\nu + 8 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} - 4\nu^{5} + 7\nu^{3} + 4\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{6} + 3\nu^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 12\nu^{5} - 16\nu^{4} + 21\nu^{3} - 12\nu - 8 ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{7} + 4\nu^{5} + 13\nu^{3} + 44\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 4\nu^{5} - 13\nu^{3} + 44\nu ) / 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{7} + 4\beta_{6} - \beta_{5} + 2\beta_{3} - \beta_{2} ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 6\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 4\beta_{6} + 5\beta_{5} + 10\beta_{3} + 5\beta_{2} ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{5} + 9\beta_{2} - 8 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2\beta_{7} + 4\beta_{6} + 11\beta_{5} - 22\beta_{3} + 11\beta_{2} ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{4} + 18\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -14\beta_{7} + 28\beta_{6} - 13\beta_{5} - 26\beta_{3} - 13\beta_{2} ) / 32 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1
0.581861 + 1.28897i
1.28897 − 0.581861i
−1.28897 + 0.581861i
−0.581861 − 1.28897i
−0.581861 + 1.28897i
−1.28897 − 0.581861i
1.28897 + 0.581861i
0.581861 − 1.28897i
0 0 0 − 8.11993i 0 − 9.48331i 0 0 0
127.2 0 0 0 − 8.11993i 0 9.48331i 0 0 0
127.3 0 0 0 − 2.46308i 0 − 5.48331i 0 0 0
127.4 0 0 0 − 2.46308i 0 5.48331i 0 0 0
127.5 0 0 0 2.46308i 0 − 5.48331i 0 0 0
127.6 0 0 0 2.46308i 0 5.48331i 0 0 0
127.7 0 0 0 8.11993i 0 − 9.48331i 0 0 0
127.8 0 0 0 8.11993i 0 9.48331i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2304.3.b.s 8
3.b odd 2 1 inner 2304.3.b.s 8
4.b odd 2 1 2304.3.b.r 8
8.b even 2 1 2304.3.b.r 8
8.d odd 2 1 inner 2304.3.b.s 8
12.b even 2 1 2304.3.b.r 8
16.e even 4 1 1152.3.g.d ✓ 8
16.e even 4 1 1152.3.g.e yes 8
16.f odd 4 1 1152.3.g.d ✓ 8
16.f odd 4 1 1152.3.g.e yes 8
24.f even 2 1 inner 2304.3.b.s 8
24.h odd 2 1 2304.3.b.r 8
48.i odd 4 1 1152.3.g.d ✓ 8
48.i odd 4 1 1152.3.g.e yes 8
48.k even 4 1 1152.3.g.d ✓ 8
48.k even 4 1 1152.3.g.e yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.g.d ✓ 8 16.e even 4 1
1152.3.g.d ✓ 8 16.f odd 4 1
1152.3.g.d ✓ 8 48.i odd 4 1
1152.3.g.d ✓ 8 48.k even 4 1
1152.3.g.e yes 8 16.e even 4 1
1152.3.g.e yes 8 16.f odd 4 1
1152.3.g.e yes 8 48.i odd 4 1
1152.3.g.e yes 8 48.k even 4 1
2304.3.b.r 8 4.b odd 2 1
2304.3.b.r 8 8.b even 2 1
2304.3.b.r 8 12.b even 2 1
2304.3.b.r 8 24.h odd 2 1
2304.3.b.s 8 1.a even 1 1 trivial
2304.3.b.s 8 3.b odd 2 1 inner
2304.3.b.s 8 8.d odd 2 1 inner
2304.3.b.s 8 24.f even 2 1 inner

Hecke kernels

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

\( T_{5}^{4} + 72T_{5}^{2} + 400 \) Copy content Toggle raw display
\( T_{7}^{4} + 120T_{7}^{2} + 2704 \) Copy content Toggle raw display
\( T_{11}^{2} - 112 \) Copy content Toggle raw display
\( T_{17}^{4} - 288T_{17}^{2} + 6400 \) Copy content Toggle raw display
\( T_{19}^{2} - 24T_{19} - 80 \) 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
$5$ \( (T^{4} + 72 T^{2} + 400)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 120 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 112)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 520 T^{2} + 35344)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 288 T^{2} + 6400)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 24 T - 80)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 1920 T^{2} + 4096)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 840 T^{2} + 132496)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 760 T^{2} + 71824)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 456 T^{2} + 48400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 3360 T^{2} + 2119936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T + 176)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 9088 T^{2} + 12390400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 7176 T^{2} + 4787344)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4992 T^{2} + 2560000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 840 T^{2} + 784)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 48 T - 7488)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8192)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 76 T - 6620)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 7736 T^{2} + 8179600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 16608 T^{2} + 65286400)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 20608 T^{2} + 5017600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28 T - 3388)^{4} \) Copy content Toggle raw display
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