↑

Properties

Label 2312.1.p.e
Level $2312$
Weight $1$
Character orbit 2312.p
Analytic conductor $1.154$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -8
Inner twists $16$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2312,1,Mod(155,2312)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2312, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2312.155");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2312 = 2^{3} \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2312.p (of order \(8\), degree \(4\), 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: \(1.15383830921\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 136)
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.314432.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{16}^{6} q^{2} + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{3} - \zeta_{16}^{4} q^{4} + (\zeta_{16}^{5} + \zeta_{16}) q^{6} + \zeta_{16}^{2} q^{8} - \zeta_{16}^{2} q^{9} + (\zeta_{16}^{5} + \zeta_{16}) q^{11} + \cdots + ( - \zeta_{16}^{7} - \zeta_{16}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{16} + 8 q^{18} + 16 q^{33} - 8 q^{50}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1157\) \(1735\) \(1737\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{16}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
−0.382683 − 0.923880i
0.382683 + 0.923880i
−0.382683 + 0.923880i
0.382683 − 0.923880i
−0.923880 − 0.382683i
0.923880 + 0.382683i
−0.923880 + 0.382683i
0.923880 − 0.382683i
0.707107 + 0.707107i −1.30656 + 0.541196i 1.00000i 0 −1.30656 − 0.541196i 0 −0.707107 + 0.707107i 0.707107 − 0.707107i 0
155.2 0.707107 + 0.707107i 1.30656 − 0.541196i 1.00000i 0 1.30656 + 0.541196i 0 −0.707107 + 0.707107i 0.707107 − 0.707107i 0
179.1 0.707107 − 0.707107i −1.30656 − 0.541196i − 1.00000i 0 −1.30656 + 0.541196i 0 −0.707107 − 0.707107i 0.707107 + 0.707107i 0
179.2 0.707107 − 0.707107i 1.30656 + 0.541196i − 1.00000i 0 1.30656 − 0.541196i 0 −0.707107 − 0.707107i 0.707107 + 0.707107i 0
1555.1 −0.707107 + 0.707107i −0.541196 + 1.30656i − 1.00000i 0 −0.541196 − 1.30656i 0 0.707107 + 0.707107i −0.707107 − 0.707107i 0
1555.2 −0.707107 + 0.707107i 0.541196 − 1.30656i − 1.00000i 0 0.541196 + 1.30656i 0 0.707107 + 0.707107i −0.707107 − 0.707107i 0
1579.1 −0.707107 − 0.707107i −0.541196 − 1.30656i 1.00000i 0 −0.541196 + 1.30656i 0 0.707107 − 0.707107i −0.707107 + 0.707107i 0
1579.2 −0.707107 − 0.707107i 0.541196 + 1.30656i 1.00000i 0 0.541196 − 1.30656i 0 0.707107 − 0.707107i −0.707107 + 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
17.b even 2 1 inner
17.c even 4 2 inner
17.d even 8 4 inner
136.e odd 2 1 inner
136.j odd 4 2 inner
136.p odd 8 4 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2312.1.p.e 8
8.d odd 2 1 CM 2312.1.p.e 8
17.b even 2 1 inner 2312.1.p.e 8
17.c even 4 2 inner 2312.1.p.e 8
17.d even 8 4 inner 2312.1.p.e 8
17.e odd 16 2 136.1.j.a ✓ 2
17.e odd 16 2 2312.1.e.a 2
17.e odd 16 2 2312.1.f.b 2
17.e odd 16 2 2312.1.j.b 2
51.i even 16 2 1224.1.s.a 2
68.i even 16 2 544.1.n.a 2
85.o even 16 1 3400.1.bc.a 2
85.o even 16 1 3400.1.bc.b 2
85.p odd 16 2 3400.1.y.a 2
85.r even 16 1 3400.1.bc.a 2
85.r even 16 1 3400.1.bc.b 2
136.e odd 2 1 inner 2312.1.p.e 8
136.j odd 4 2 inner 2312.1.p.e 8
136.p odd 8 4 inner 2312.1.p.e 8
136.q odd 16 2 544.1.n.a 2
136.s even 16 2 136.1.j.a ✓ 2
136.s even 16 2 2312.1.e.a 2
136.s even 16 2 2312.1.f.b 2
136.s even 16 2 2312.1.j.b 2
408.bg odd 16 2 1224.1.s.a 2
680.ch odd 16 1 3400.1.bc.a 2
680.ch odd 16 1 3400.1.bc.b 2
680.co even 16 2 3400.1.y.a 2
680.cr odd 16 1 3400.1.bc.a 2
680.cr odd 16 1 3400.1.bc.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.1.j.a ✓ 2 17.e odd 16 2
136.1.j.a ✓ 2 136.s even 16 2
544.1.n.a 2 68.i even 16 2
544.1.n.a 2 136.q odd 16 2
1224.1.s.a 2 51.i even 16 2
1224.1.s.a 2 408.bg odd 16 2
2312.1.e.a 2 17.e odd 16 2
2312.1.e.a 2 136.s even 16 2
2312.1.f.b 2 17.e odd 16 2
2312.1.f.b 2 136.s even 16 2
2312.1.j.b 2 17.e odd 16 2
2312.1.j.b 2 136.s even 16 2
2312.1.p.e 8 1.a even 1 1 trivial
2312.1.p.e 8 8.d odd 2 1 CM
2312.1.p.e 8 17.b even 2 1 inner
2312.1.p.e 8 17.c even 4 2 inner
2312.1.p.e 8 17.d even 8 4 inner
2312.1.p.e 8 136.e odd 2 1 inner
2312.1.p.e 8 136.j odd 4 2 inner
2312.1.p.e 8 136.p odd 8 4 inner
3400.1.y.a 2 85.p odd 16 2
3400.1.y.a 2 680.co even 16 2
3400.1.bc.a 2 85.o even 16 1
3400.1.bc.a 2 85.r even 16 1
3400.1.bc.a 2 680.ch odd 16 1
3400.1.bc.a 2 680.cr odd 16 1
3400.1.bc.b 2 85.o even 16 1
3400.1.bc.b 2 85.r even 16 1
3400.1.bc.b 2 680.ch odd 16 1
3400.1.bc.b 2 680.cr odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 16 \) acting on \(S_{1}^{\mathrm{new}}(2312, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 16 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 16 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 16 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} + 16 \) Copy content Toggle raw display
show more
show less