↑

Properties

Label 2312.2.b.g
Level $2312$
Weight $2$
Character orbit 2312.b
Analytic conductor $18.461$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2312,2,Mod(577,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2312.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2312.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: \(18.4614129473\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 136)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + \beta_1 q^{7} + (\beta_{3} - 3) q^{9} + \beta_1 q^{11} + \beta_{3} q^{13} + ( - \beta_{3} + 2) q^{15} + (\beta_{3} + 2) q^{19} + (\beta_{3} - 6) q^{21} + \beta_1 q^{23}+ \cdots + (4 \beta_{2} - 5 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 8 q^{15} + 8 q^{19} - 24 q^{21} + 4 q^{25} - 24 q^{33} + 8 q^{35} + 24 q^{43} + 16 q^{47} + 4 q^{49} + 8 q^{53} + 8 q^{55} - 40 q^{59} - 48 q^{67} - 24 q^{69} - 24 q^{77} + 44 q^{81} - 24 q^{83}+ \cdots - 24 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{3} + 4\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2312\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
− 1.61803i
− 0.618034i
0.618034i
1.61803i
0 − 3.23607i 0 2.00000i 0 − 3.23607i 0 −7.47214 0
577.2 0 − 1.23607i 0 − 2.00000i 0 − 1.23607i 0 1.47214 0
577.3 0 1.23607i 0 2.00000i 0 1.23607i 0 1.47214 0
577.4 0 3.23607i 0 − 2.00000i 0 3.23607i 0 −7.47214 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
17.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2312.2.b.g 4
17.b even 2 1 inner 2312.2.b.g 4
17.c even 4 1 136.2.a.c ✓ 2
17.c even 4 1 2312.2.a.m 2
51.f odd 4 1 1224.2.a.i 2
68.f odd 4 1 272.2.a.f 2
68.f odd 4 1 4624.2.a.h 2
85.f odd 4 1 3400.2.e.f 4
85.i odd 4 1 3400.2.e.f 4
85.j even 4 1 3400.2.a.i 2
119.f odd 4 1 6664.2.a.i 2
136.i even 4 1 1088.2.a.s 2
136.j odd 4 1 1088.2.a.o 2
204.l even 4 1 2448.2.a.u 2
340.n odd 4 1 6800.2.a.bd 2
408.q even 4 1 9792.2.a.da 2
408.t odd 4 1 9792.2.a.db 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.a.c ✓ 2 17.c even 4 1
272.2.a.f 2 68.f odd 4 1
1088.2.a.o 2 136.j odd 4 1
1088.2.a.s 2 136.i even 4 1
1224.2.a.i 2 51.f odd 4 1
2312.2.a.m 2 17.c even 4 1
2312.2.b.g 4 1.a even 1 1 trivial
2312.2.b.g 4 17.b even 2 1 inner
2448.2.a.u 2 204.l even 4 1
3400.2.a.i 2 85.j even 4 1
3400.2.e.f 4 85.f odd 4 1
3400.2.e.f 4 85.i odd 4 1
4624.2.a.h 2 68.f odd 4 1
6664.2.a.i 2 119.f odd 4 1
6800.2.a.bd 2 340.n odd 4 1
9792.2.a.da 2 408.q even 4 1
9792.2.a.db 2 408.t odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2312, [\chi])\):

\( T_{3}^{4} + 12T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$5$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 12T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T - 64)^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 20 T + 80)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 168T^{2} + 5776 \) Copy content Toggle raw display
$67$ \( (T + 12)^{4} \) Copy content Toggle raw display
$71$ \( T^{4} + 108T^{2} + 1936 \) Copy content Toggle raw display
$73$ \( T^{4} + 232T^{2} + 1936 \) Copy content Toggle raw display
$79$ \( T^{4} + 140T^{2} + 400 \) Copy content Toggle raw display
$83$ \( (T^{2} + 12 T + 16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 24 T + 124)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
show more
show less