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Properties

Label 2952.2.j.c
Level $2952$
Weight $2$
Character orbit 2952.j
Analytic conductor $23.572$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2952,2,Mod(2377,2952)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2952, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 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("2952.2377"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2952 = 2^{3} \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2952.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.5718386767\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.1229312.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 10x^{4} + 24x^{2} + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 328)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{5} - \beta_{3} q^{7} + \beta_1 q^{11} + ( - \beta_{3} + \beta_1) q^{13} + (2 \beta_{4} - 2 \beta_1) q^{17} + \beta_{4} q^{19} + ( - \beta_{5} + \beta_{2} - 1) q^{23} + ( - \beta_{5} + \beta_{2} + 2) q^{25}+ \cdots + ( - 4 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{5} - 8 q^{23} + 10 q^{25} - 8 q^{31} - 4 q^{37} + 2 q^{41} - 16 q^{43} - 14 q^{49} + 36 q^{61} - 36 q^{73} - 8 q^{83} - 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 10x^{4} + 24x^{2} + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} - 6\nu^{3} ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 6\nu^{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} - 10\nu^{3} - 16\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + 10\nu^{3} + 24\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{4} + 10\nu^{2} + 16 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{2} - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{4} - 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{5} + 5\beta_{2} + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{4} + 18\beta_{3} - 10\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2952\mathbb{Z}\right)^\times\).

\(n\) \(1441\) \(1477\) \(2215\) \(2297\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2377.1
1.76350i
− 1.76350i
− 0.629384i
0.629384i
2.54832i
− 2.54832i
0 0 0 −2.49396 0 − 2.39288i 0 0 0
2377.2 0 0 0 −2.49396 0 2.39288i 0 0 0
2377.3 0 0 0 0.890084 0 − 1.91894i 0 0 0
2377.4 0 0 0 0.890084 0 1.91894i 0 0 0
2377.5 0 0 0 3.60388 0 − 4.31182i 0 0 0
2377.6 0 0 0 3.60388 0 4.31182i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2377.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
41.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2952.2.j.c 6
3.b odd 2 1 328.2.d.b ✓ 6
12.b even 2 1 656.2.d.f 6
24.f even 2 1 2624.2.d.n 6
24.h odd 2 1 2624.2.d.o 6
41.b even 2 1 inner 2952.2.j.c 6
123.b odd 2 1 328.2.d.b ✓ 6
492.d even 2 1 656.2.d.f 6
984.m odd 2 1 2624.2.d.o 6
984.p even 2 1 2624.2.d.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
328.2.d.b ✓ 6 3.b odd 2 1
328.2.d.b ✓ 6 123.b odd 2 1
656.2.d.f 6 12.b even 2 1
656.2.d.f 6 492.d even 2 1
2624.2.d.n 6 24.f even 2 1
2624.2.d.n 6 984.p even 2 1
2624.2.d.o 6 24.h odd 2 1
2624.2.d.o 6 984.m odd 2 1
2952.2.j.c 6 1.a even 1 1 trivial
2952.2.j.c 6 41.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 2T_{5}^{2} - 8T_{5} + 8 \) acting on \(S_{2}^{\mathrm{new}}(2952, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( (T^{3} - 2 T^{2} - 8 T + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} + 28 T^{4} + \cdots + 392 \) Copy content Toggle raw display
$11$ \( T^{6} + 20 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$13$ \( T^{6} + 48 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$17$ \( (T^{2} + 32)^{3} \) Copy content Toggle raw display
$19$ \( T^{6} + 12 T^{4} + \cdots + 8 \) Copy content Toggle raw display
$23$ \( (T^{3} + 4 T^{2} - 32 T - 64)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 80 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$31$ \( (T^{3} + 4 T^{2} - 32 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 2 T^{2} - 8 T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 2 T^{5} + \cdots + 68921 \) Copy content Toggle raw display
$43$ \( (T^{3} + 8 T^{2} - 16 T - 64)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 180 T^{4} + \cdots + 129032 \) Copy content Toggle raw display
$53$ \( T^{6} + 208 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$59$ \( (T^{3} - 112 T - 448)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} - 18 T^{2} + \cdots + 904)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 132 T^{4} + \cdots + 1352 \) Copy content Toggle raw display
$71$ \( T^{6} + 244 T^{4} + \cdots + 262088 \) Copy content Toggle raw display
$73$ \( (T^{3} + 18 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + 180 T^{4} + \cdots + 129032 \) Copy content Toggle raw display
$83$ \( (T^{3} + 4 T^{2} - 144 T - 64)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + 544 T^{4} + \cdots + 5537792 \) Copy content Toggle raw display
$97$ \( T^{6} + 448 T^{4} + \cdots + 1605632 \) Copy content Toggle raw display
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