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Properties

Label 2312.2.b.i
Level $2312$
Weight $2$
Character orbit 2312.b
Analytic conductor $18.461$
Analytic rank $0$
Dimension $4$
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 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);
 
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")
 
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 = [4,0,0,0,0,0,0,0,-4,0,0,0,-16,0,8] 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: \(4\)
Coefficient field: \(\Q(\sqrt{-2 + \sqrt{2}})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 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.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{3} + ( - 2 \beta_{3} + \beta_1) q^{5} + ( - \beta_{3} + \beta_1) q^{7} + ( - 2 \beta_{2} - 1) q^{9} + (\beta_{3} + 3 \beta_1) q^{11} + (\beta_{2} - 4) q^{13} + (4 \beta_{2} + 2) q^{15}+ \cdots + ( - 9 \beta_{3} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} - 16 q^{13} + 8 q^{15} + 8 q^{19} - 20 q^{25} - 32 q^{33} - 24 q^{35} + 24 q^{43} + 16 q^{47} + 12 q^{49} + 16 q^{53} - 8 q^{55} + 40 q^{59} + 32 q^{69} - 16 q^{77} - 12 q^{81} + 40 q^{83}+ \cdots - 24 q^{87}+O(q^{100}) \) Copy content Toggle raw display

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

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

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

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

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
− 0.765367i
− 1.84776i
1.84776i
0.765367i
0 − 2.61313i 0 2.93015i 0 1.08239i 0 −3.82843 0
577.2 0 − 1.08239i 0 − 3.37849i 0 − 2.61313i 0 1.82843 0
577.3 0 1.08239i 0 3.37849i 0 2.61313i 0 1.82843 0
577.4 0 2.61313i 0 − 2.93015i 0 − 1.08239i 0 −3.82843 0
\(n\): e.g. 2-40 or 80-90
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.i 4
17.b even 2 1 inner 2312.2.b.i 4
17.c even 4 2 2312.2.a.t 4
17.e odd 16 2 136.2.n.b ✓ 4
51.i even 16 2 1224.2.bq.b 4
68.f odd 4 2 4624.2.a.bo 4
68.i even 16 2 272.2.v.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
136.2.n.b ✓ 4 17.e odd 16 2
272.2.v.a 4 68.i even 16 2
1224.2.bq.b 4 51.i even 16 2
2312.2.a.t 4 17.c even 4 2
2312.2.b.i 4 1.a even 1 1 trivial
2312.2.b.i 4 17.b even 2 1 inner
4624.2.a.bo 4 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 \(S_{2}^{\mathrm{new}}(2312, [\chi])\):

\( T_{3}^{4} + 8T_{3}^{2} + 8 \) Copy content Toggle raw display
\( T_{5}^{4} + 20T_{5}^{2} + 98 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 8T^{2} + 8 \) Copy content Toggle raw display
$5$ \( T^{4} + 20T^{2} + 98 \) Copy content Toggle raw display
$7$ \( T^{4} + 8T^{2} + 8 \) Copy content Toggle raw display
$11$ \( T^{4} + 40T^{2} + 392 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8 T + 14)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 28)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 40T^{2} + 8 \) Copy content Toggle raw display
$29$ \( T^{4} + 36T^{2} + 162 \) Copy content Toggle raw display
$31$ \( T^{4} + 8T^{2} + 8 \) Copy content Toggle raw display
$37$ \( T^{4} + 116T^{2} + 2 \) Copy content Toggle raw display
$41$ \( T^{4} + 100T^{2} + 1250 \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 8 T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20 T + 92)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 20T^{2} + 2 \) Copy content Toggle raw display
$67$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 296 T^{2} + 17672 \) Copy content Toggle raw display
$73$ \( T^{4} + 260 T^{2} + 12482 \) Copy content Toggle raw display
$79$ \( T^{4} + 40T^{2} + 392 \) Copy content Toggle raw display
$83$ \( (T^{2} - 20 T + 92)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 20T^{2} + 2 \) Copy content Toggle raw display
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