↑

Properties

Label 2304.3.g.z
Level $2304$
Weight $3$
Character orbit 2304.g
Analytic conductor $62.779$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(1279,2304)]
 
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, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1279");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.g (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: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} - \beta_{7} q^{7} + \beta_{6} q^{11} + ( - \beta_{4} + \beta_1) q^{13} + (\beta_{5} + 2) q^{17} + ( - \beta_{6} - \beta_{3}) q^{19} - 2 \beta_{7} q^{23} + ( - 2 \beta_{5} + 11) q^{25} + ( - 2 \beta_{4} - \beta_1) q^{29} - \beta_{2} q^{31} + (3 \beta_{6} - \beta_{3}) q^{35} + (\beta_{4} - 3 \beta_1) q^{37} + (3 \beta_{5} + 10) q^{41} + (\beta_{6} + 3 \beta_{3}) q^{43} + ( - \beta_{7} - \beta_{2}) q^{47} + ( - 4 \beta_{5} - 11) q^{49} + ( - 2 \beta_{4} + 3 \beta_1) q^{53} + ( - 3 \beta_{7} - \beta_{2}) q^{55} + (4 \beta_{6} + 3 \beta_{3}) q^{59} + (\beta_{4} - 7 \beta_1) q^{61} + (5 \beta_{5} - 24) q^{65} + (8 \beta_{6} + 3 \beta_{3}) q^{67} + (7 \beta_{7} - \beta_{2}) q^{71} + ( - 4 \beta_{5} - 50) q^{73} + (4 \beta_{4} + 4 \beta_1) q^{77} + ( - 6 \beta_{7} + \beta_{2}) q^{79} + (5 \beta_{6} + 4 \beta_{3}) q^{83} + ( - 4 \beta_{4} + 10 \beta_1) q^{85} + (6 \beta_{5} - 50) q^{89} + (9 \beta_{6} - 14 \beta_{3}) q^{91} + (2 \beta_{7} + 2 \beta_{2}) q^{95} + ( - 6 \beta_{5} + 14) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{17} + 88 q^{25} + 80 q^{41} - 88 q^{49} - 192 q^{65} - 400 q^{73} - 400 q^{89} + 112 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{7} - 2\nu^{6} - \nu^{5} + 4\nu^{4} - \nu^{3} - 6\nu^{2} + 10\nu - 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -10\nu^{7} + 18\nu^{6} + 2\nu^{5} - 38\nu^{4} + 22\nu^{3} + 70\nu^{2} - 84\nu + 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -6\nu^{7} + 14\nu^{6} - 6\nu^{5} - 22\nu^{4} + 30\nu^{3} + 22\nu^{2} - 80\nu + 60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -5\nu^{7} + 15\nu^{6} - 9\nu^{5} - 19\nu^{4} + 33\nu^{3} + 15\nu^{2} - 76\nu + 66 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -6\nu^{7} + 18\nu^{6} - 10\nu^{5} - 26\nu^{4} + 42\nu^{3} + 26\nu^{2} - 108\nu + 80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 8\nu^{7} - 22\nu^{6} + 12\nu^{5} + 34\nu^{4} - 48\nu^{3} - 30\nu^{2} + 124\nu - 96 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 10\nu^{7} - 26\nu^{6} + 14\nu^{5} + 38\nu^{4} - 54\nu^{3} - 38\nu^{2} + 148\nu - 114 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} - 2\beta_{6} - 2\beta_{5} + 4\beta_{4} + \beta_{2} + 4\beta _1 + 32 ) / 64 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 2\beta_{5} - 2\beta_{3} + \beta_{2} + 4\beta _1 + 24 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 11\beta_{7} + 2\beta_{6} + 6\beta_{5} + 4\beta_{4} + 12\beta_{3} + \beta_{2} + 12\beta _1 - 16 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{7} + 6\beta_{6} + 2\beta_{5} + 2\beta_{4} + \beta_{3} - 4 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 9\beta_{7} + 4\beta_{6} + 6\beta_{5} + 4\beta_{4} + 10\beta_{3} - \beta_{2} - 16\beta _1 + 56 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{6} + 4\beta_{4} - 3\beta_{3} - 6\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 19\beta_{7} - 26\beta_{6} + 30\beta_{5} + 4\beta_{4} - 56\beta_{3} + \beta_{2} - 44\beta _1 + 256 ) / 64 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1279.1
1.20036 + 0.747754i
1.20036 − 0.747754i
1.40994 + 0.109843i
1.40994 − 0.109843i
−1.27597 + 0.609843i
−1.27597 − 0.609843i
0.665665 + 1.24775i
0.665665 − 1.24775i
0 0 0 −7.98203 0 − 2.13878i 0 0 0
1279.2 0 0 0 −7.98203 0 2.13878i 0 0 0
1279.3 0 0 0 −2.87875 0 − 10.7436i 0 0 0
1279.4 0 0 0 −2.87875 0 10.7436i 0 0 0
1279.5 0 0 0 2.87875 0 − 10.7436i 0 0 0
1279.6 0 0 0 2.87875 0 10.7436i 0 0 0
1279.7 0 0 0 7.98203 0 − 2.13878i 0 0 0
1279.8 0 0 0 7.98203 0 2.13878i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1279.8
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 2304.3.g.z 8
3.b odd 2 1 768.3.g.h 8
4.b odd 2 1 inner 2304.3.g.z 8
8.b even 2 1 inner 2304.3.g.z 8
8.d odd 2 1 inner 2304.3.g.z 8
12.b even 2 1 768.3.g.h 8
16.e even 4 1 72.3.b.b 4
16.e even 4 1 288.3.b.b 4
16.f odd 4 1 72.3.b.b 4
16.f odd 4 1 288.3.b.b 4
24.f even 2 1 768.3.g.h 8
24.h odd 2 1 768.3.g.h 8
48.i odd 4 1 24.3.b.a ✓ 4
48.i odd 4 1 96.3.b.a 4
48.k even 4 1 24.3.b.a ✓ 4
48.k even 4 1 96.3.b.a 4
240.t even 4 1 600.3.g.a 4
240.t even 4 1 2400.3.g.a 4
240.z odd 4 1 600.3.p.a 8
240.z odd 4 1 2400.3.p.a 8
240.bb even 4 1 600.3.p.a 8
240.bb even 4 1 2400.3.p.a 8
240.bd odd 4 1 600.3.p.a 8
240.bd odd 4 1 2400.3.p.a 8
240.bf even 4 1 600.3.p.a 8
240.bf even 4 1 2400.3.p.a 8
240.bm odd 4 1 600.3.g.a 4
240.bm odd 4 1 2400.3.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a ✓ 4 48.i odd 4 1
24.3.b.a ✓ 4 48.k even 4 1
72.3.b.b 4 16.e even 4 1
72.3.b.b 4 16.f odd 4 1
96.3.b.a 4 48.i odd 4 1
96.3.b.a 4 48.k even 4 1
288.3.b.b 4 16.e even 4 1
288.3.b.b 4 16.f odd 4 1
600.3.g.a 4 240.t even 4 1
600.3.g.a 4 240.bm odd 4 1
600.3.p.a 8 240.z odd 4 1
600.3.p.a 8 240.bb even 4 1
600.3.p.a 8 240.bd odd 4 1
600.3.p.a 8 240.bf even 4 1
768.3.g.h 8 3.b odd 2 1
768.3.g.h 8 12.b even 2 1
768.3.g.h 8 24.f even 2 1
768.3.g.h 8 24.h odd 2 1
2304.3.g.z 8 1.a even 1 1 trivial
2304.3.g.z 8 4.b odd 2 1 inner
2304.3.g.z 8 8.b even 2 1 inner
2304.3.g.z 8 8.d odd 2 1 inner
2400.3.g.a 4 240.t even 4 1
2400.3.g.a 4 240.bm odd 4 1
2400.3.p.a 8 240.z odd 4 1
2400.3.p.a 8 240.bb even 4 1
2400.3.p.a 8 240.bd odd 4 1
2400.3.p.a 8 240.bf even 4 1

Hecke kernels

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

\( T_{5}^{4} - 72T_{5}^{2} + 528 \) Copy content Toggle raw display
\( T_{7}^{4} + 120T_{7}^{2} + 528 \) Copy content Toggle raw display
\( T_{11}^{2} + 64 \) Copy content Toggle raw display
\( T_{13}^{4} - 384T_{13}^{2} + 33792 \) 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
$5$ \( (T^{4} - 72 T^{2} + 528)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 120 T^{2} + 528)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 384 T^{2} + 33792)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 188)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 224 T^{2} + 256)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 480 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 1608 T^{2} + 528)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3384 T^{2} + 279312)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 864 T^{2} + 76032)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 20 T - 1628)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 992 T^{2} + 135424)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 3552 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 1800 T^{2} + 803088)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2912 T^{2} + 350464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 3552 T^{2} + 8448)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 9056 T^{2} + 13424896)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8928 T^{2} + 12849408)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 100 T - 572)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 7416 T^{2} + 10797072)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 4736 T^{2} + 692224)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 100 T - 4412)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28 T - 6716)^{4} \) Copy content Toggle raw display
show more
show less