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Properties

Label 4600.2.e.s
Level $4600$
Weight $2$
Character orbit 4600.e
Analytic conductor $36.731$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4600,2,Mod(4049,4600)] 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("4600.4049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + (\beta_{5} + \beta_{3}) q^{3} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{7} + ( - \beta_{4} - 2 \beta_{2} + 2) q^{9} + (4 \beta_{4} + 2 \beta_{2} - 2) q^{11} + (\beta_{5} + 2 \beta_{3} - \beta_1) q^{13} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{17}+ \cdots + (6 \beta_{4} - 2 \beta_{2} - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{9} - 8 q^{19} + 24 q^{21} + 8 q^{29} + 20 q^{31} - 14 q^{39} - 8 q^{41} - 6 q^{49} + 24 q^{51} + 22 q^{59} - 8 q^{61} + 4 q^{69} + 8 q^{71} + 8 q^{79} - 2 q^{81} + 40 q^{89} + 28 q^{91} - 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : 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
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) 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
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4049.1
− 0.445042i
− 1.24698i
− 1.80194i
1.80194i
1.24698i
0.445042i
0 − 2.24698i 0 0 0 4.49396i 0 −2.04892 0
4049.2 0 − 0.801938i 0 0 0 1.60388i 0 2.35690 0
4049.3 0 − 0.554958i 0 0 0 1.10992i 0 2.69202 0
4049.4 0 0.554958i 0 0 0 − 1.10992i 0 2.69202 0
4049.5 0 0.801938i 0 0 0 − 1.60388i 0 2.35690 0
4049.6 0 2.24698i 0 0 0 − 4.49396i 0 −2.04892 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 4049.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4600.2.e.s 6
5.b even 2 1 inner 4600.2.e.s 6
5.c odd 4 1 4600.2.a.w ✓ 3
5.c odd 4 1 4600.2.a.z yes 3
20.e even 4 1 9200.2.a.cb 3
20.e even 4 1 9200.2.a.ch 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4600.2.a.w ✓ 3 5.c odd 4 1
4600.2.a.z yes 3 5.c odd 4 1
4600.2.e.s 6 1.a even 1 1 trivial
4600.2.e.s 6 5.b even 2 1 inner
9200.2.a.cb 3 20.e even 4 1
9200.2.a.ch 3 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4600, [\chi])\):

\( T_{3}^{6} + 6T_{3}^{4} + 5T_{3}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{6} + 24T_{7}^{4} + 80T_{7}^{2} + 64 \) Copy content Toggle raw display
\( T_{11}^{3} - 28T_{11} + 56 \) Copy content Toggle raw display
\( T_{13}^{6} + 14T_{13}^{4} + 49T_{13}^{2} + 49 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 6 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 24 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( (T^{3} - 28 T + 56)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 14 T^{4} + \cdots + 49 \) Copy content Toggle raw display
$17$ \( T^{6} + 24 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$19$ \( (T^{3} + 4 T^{2} - 32 T - 64)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$29$ \( (T^{3} - 4 T^{2} + \cdots + 169)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 10 T^{2} + \cdots + 41)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 164 T^{4} + \cdots + 107584 \) Copy content Toggle raw display
$41$ \( (T^{3} + 4 T^{2} - 39 T + 41)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 248 T^{4} + \cdots + 53824 \) Copy content Toggle raw display
$47$ \( T^{6} + 118 T^{4} + \cdots + 6889 \) Copy content Toggle raw display
$53$ \( T^{6} + 108 T^{4} + \cdots + 10816 \) Copy content Toggle raw display
$59$ \( (T^{3} - 11 T^{2} + \cdots + 379)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 4 T^{2} + \cdots + 104)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 272 T^{4} + \cdots + 692224 \) Copy content Toggle raw display
$71$ \( (T^{3} - 4 T^{2} + \cdots + 533)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 318 T^{4} + \cdots + 121801 \) Copy content Toggle raw display
$79$ \( (T^{3} - 4 T^{2} + \cdots + 568)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 544 T^{4} + \cdots + 3444736 \) Copy content Toggle raw display
$89$ \( (T^{3} - 20 T^{2} + 68 T - 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 500 T^{4} + \cdots + 3655744 \) Copy content Toggle raw display
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