Properties

Label 2304.3.b.k
Level $2304$
Weight $3$
Character orbit 2304.b
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 96)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{3} - 2 \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{5} + (\beta_{3} - 2 \beta_1) q^{7} + ( - \beta_{2} - 8) q^{11} + ( - 2 \beta_{3} - 5 \beta_1) q^{13} + (2 \beta_{2} + 6) q^{17} + (\beta_{2} + 24) q^{19} + 4 \beta_1 q^{23} + (4 \beta_{2} - 27) q^{25} + ( - \beta_{3} - 5 \beta_1) q^{29} + (5 \beta_{3} + 2 \beta_1) q^{31} + ( - 6 \beta_{2} + 56) q^{35} + ( - 4 \beta_{3} + 9 \beta_1) q^{37} + (2 \beta_{2} - 22) q^{41} + (5 \beta_{2} - 40) q^{43} + ( - 2 \beta_{3} - 28 \beta_1) q^{47} + (8 \beta_{2} - 15) q^{49} + (\beta_{3} - 3 \beta_1) q^{53} + (6 \beta_{3} + 16 \beta_1) q^{55} + ( - 11 \beta_{2} - 32) q^{59} + 7 \beta_1 q^{61} + ( - 6 \beta_{2} - 76) q^{65} + (7 \beta_{2} - 32) q^{67} + ( - 12 \beta_{3} + 20 \beta_1) q^{71} + ( - 8 \beta_{2} + 30) q^{73} + ( - 4 \beta_{3} - 8 \beta_1) q^{77} + ( - 3 \beta_{3} - 38 \beta_1) q^{79} + ( - \beta_{2} - 56) q^{83} + ( - 2 \beta_{3} - 42 \beta_1) q^{85} + (4 \beta_{2} - 78) q^{89} + (2 \beta_{2} + 56) q^{91} - 22 \beta_{3} q^{95} + ( - 12 \beta_{2} - 62) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{11} + 24 q^{17} + 96 q^{19} - 108 q^{25} + 224 q^{35} - 88 q^{41} - 160 q^{43} - 60 q^{49} - 128 q^{59} - 304 q^{65} - 128 q^{67} + 120 q^{73} - 224 q^{83} - 312 q^{89} + 224 q^{91} - 248 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\zeta_{12}^{3} + 8\zeta_{12} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\zeta_{12}^{2} - 4 \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{3} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

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
−0.866025 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 8.92820i 0 10.9282i 0 0 0
127.2 0 0 0 4.92820i 0 2.92820i 0 0 0
127.3 0 0 0 4.92820i 0 2.92820i 0 0 0
127.4 0 0 0 8.92820i 0 10.9282i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.b.k 4
3.b odd 2 1 768.3.b.d 4
4.b odd 2 1 2304.3.b.o 4
8.b even 2 1 2304.3.b.o 4
8.d odd 2 1 inner 2304.3.b.k 4
12.b even 2 1 768.3.b.a 4
16.e even 4 1 288.3.g.d 4
16.e even 4 1 576.3.g.j 4
16.f odd 4 1 288.3.g.d 4
16.f odd 4 1 576.3.g.j 4
24.f even 2 1 768.3.b.d 4
24.h odd 2 1 768.3.b.a 4
48.i odd 4 1 96.3.g.a 4
48.i odd 4 1 192.3.g.c 4
48.k even 4 1 96.3.g.a 4
48.k even 4 1 192.3.g.c 4
240.t even 4 1 2400.3.e.a 4
240.z odd 4 1 2400.3.j.a 4
240.bb even 4 1 2400.3.j.b 4
240.bd odd 4 1 2400.3.j.b 4
240.bf even 4 1 2400.3.j.a 4
240.bm odd 4 1 2400.3.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.3.g.a 4 48.i odd 4 1
96.3.g.a 4 48.k even 4 1
192.3.g.c 4 48.i odd 4 1
192.3.g.c 4 48.k even 4 1
288.3.g.d 4 16.e even 4 1
288.3.g.d 4 16.f odd 4 1
576.3.g.j 4 16.e even 4 1
576.3.g.j 4 16.f odd 4 1
768.3.b.a 4 12.b even 2 1
768.3.b.a 4 24.h odd 2 1
768.3.b.d 4 3.b odd 2 1
768.3.b.d 4 24.f even 2 1
2304.3.b.k 4 1.a even 1 1 trivial
2304.3.b.k 4 8.d odd 2 1 inner
2304.3.b.o 4 4.b odd 2 1
2304.3.b.o 4 8.b even 2 1
2400.3.e.a 4 240.t even 4 1
2400.3.e.a 4 240.bm odd 4 1
2400.3.j.a 4 240.z odd 4 1
2400.3.j.a 4 240.bf even 4 1
2400.3.j.b 4 240.bb even 4 1
2400.3.j.b 4 240.bd odd 4 1

Hecke kernels

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

\( T_{5}^{4} + 104T_{5}^{2} + 1936 \) Copy content Toggle raw display
\( T_{7}^{4} + 128T_{7}^{2} + 1024 \) Copy content Toggle raw display
\( T_{11}^{2} + 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{17}^{2} - 12T_{17} - 156 \) Copy content Toggle raw display
\( T_{19}^{2} - 48T_{19} + 528 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$7$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( (T^{2} + 16 T + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 584T^{2} + 8464 \) Copy content Toggle raw display
$17$ \( (T^{2} - 12 T - 156)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 48 T + 528)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 296T^{2} + 2704 \) Copy content Toggle raw display
$31$ \( T^{4} + 2432 T^{2} + \cdots + 1401856 \) Copy content Toggle raw display
$37$ \( T^{4} + 2184 T^{2} + 197136 \) Copy content Toggle raw display
$41$ \( (T^{2} + 44 T + 292)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 80 T + 400)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 6656 T^{2} + \cdots + 8667136 \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( (T^{2} + 64 T - 4784)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64 T - 1328)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 17024 T^{2} + \cdots + 28217344 \) Copy content Toggle raw display
$73$ \( (T^{2} - 60 T - 2172)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 12416 T^{2} + \cdots + 28558336 \) Copy content Toggle raw display
$83$ \( (T^{2} + 112 T + 3088)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 156 T + 5316)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 124 T - 3068)^{2} \) Copy content Toggle raw display
show more
show less