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Properties

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

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2312,2,Mod(577,2312)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage: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")
 
Copy content magma://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

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,2,0,0,0,-4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.4614129473\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 136)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - \beta q^{5} - \beta q^{7} + q^{9} + \beta q^{11} - 2 q^{13} + 2 q^{15} - 2 q^{19} + 2 q^{21} + 5 \beta q^{23} + 3 q^{25} + 4 \beta q^{27} + 3 \beta q^{29} - 3 \beta q^{31} - 2 q^{33} + \cdots + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{9} - 4 q^{13} + 4 q^{15} - 4 q^{19} + 4 q^{21} + 6 q^{25} - 4 q^{33} - 4 q^{35} + 20 q^{43} + 10 q^{49} + 24 q^{53} + 4 q^{55} + 4 q^{59} + 24 q^{67} - 20 q^{69} + 4 q^{77} - 10 q^{81} + 12 q^{83}+ \cdots + 12 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
− 1.41421i
1.41421i
0 − 1.41421i 0 1.41421i 0 1.41421i 0 1.00000 0
577.2 0 1.41421i 0 − 1.41421i 0 − 1.41421i 0 1.00000 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.e 2
17.b even 2 1 inner 2312.2.b.e 2
17.c even 4 2 2312.2.a.i 2
17.d even 8 2 136.2.k.c ✓ 2
51.g odd 8 2 1224.2.w.b 2
68.f odd 4 2 4624.2.a.p 2
68.g odd 8 2 272.2.o.b 2
136.o even 8 2 1088.2.o.e 2
136.p odd 8 2 1088.2.o.n 2
204.p even 8 2 2448.2.be.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.k.c ✓ 2 17.d even 8 2
272.2.o.b 2 68.g odd 8 2
1088.2.o.e 2 136.o even 8 2
1088.2.o.n 2 136.p odd 8 2
1224.2.w.b 2 51.g odd 8 2
2312.2.a.i 2 17.c even 4 2
2312.2.b.e 2 1.a even 1 1 trivial
2312.2.b.e 2 17.b even 2 1 inner
2448.2.be.e 2 204.p even 8 2
4624.2.a.p 2 68.f odd 4 2

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 2 \) Copy content Toggle raw display
$5$ \( T^{2} + 2 \) Copy content Toggle raw display
$7$ \( T^{2} + 2 \) Copy content Toggle raw display
$11$ \( T^{2} + 2 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 50 \) Copy content Toggle raw display
$29$ \( T^{2} + 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 18 \) Copy content Toggle raw display
$37$ \( T^{2} + 2 \) Copy content Toggle raw display
$41$ \( T^{2} + 98 \) Copy content Toggle raw display
$43$ \( (T - 10)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 12)^{2} \) Copy content Toggle raw display
$59$ \( (T - 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 50 \) Copy content Toggle raw display
$67$ \( (T - 12)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 162 \) Copy content Toggle raw display
$73$ \( T^{2} + 162 \) Copy content Toggle raw display
$79$ \( T^{2} + 50 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( (T - 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 18 \) Copy content Toggle raw display
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