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Properties

Label 484.2.g.a
Level $484$
Weight $2$
Character orbit 484.g
Analytic conductor $3.865$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [484,2,Mod(215,484)] 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("484.215"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(484, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([5, 9])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 484 = 2^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 484.g (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-10,4,-2,0,0,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.86475945783\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} + 2 \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} - \beta_{2} q^{5} + ( - \beta_{5} + 2 \beta_{3} - 2 \beta_1) q^{6} + (2 \beta_{7} + 2 \beta_{3} - \beta_1) q^{7} + 2 \beta_{3} q^{8}+ \cdots + ( - 8 \beta_{5} + \beta_{3} - 8 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{3} + 4 q^{4} - 2 q^{5} - 16 q^{9} - 20 q^{12} - 20 q^{14} + 10 q^{15} - 8 q^{16} + 4 q^{20} + 8 q^{25} - 24 q^{26} + 50 q^{27} - 10 q^{31} + 20 q^{34} + 32 q^{36} + 6 q^{37} + 20 q^{42} - 16 q^{45}+ \cdots + 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/484\mathbb{Z}\right)^\times\).

\(n\) \(243\) \(365\)
\(\chi(n)\) \(-1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
−0.831254 − 1.14412i
0.831254 + 1.14412i
−1.34500 − 0.437016i
1.34500 + 0.437016i
−1.34500 + 0.437016i
1.34500 − 0.437016i
−0.831254 + 1.14412i
0.831254 − 1.14412i
−0.831254 − 1.14412i −1.80902 + 2.48990i −0.618034 + 1.90211i 0.309017 − 0.951057i 4.35250 0.831254 − 0.603941i 2.68999 − 0.874032i −2.00000 − 6.15537i −1.34500 + 0.437016i
215.2 0.831254 + 1.14412i −1.80902 + 2.48990i −0.618034 + 1.90211i 0.309017 − 0.951057i −4.35250 −0.831254 + 0.603941i −2.68999 + 0.874032i −2.00000 − 6.15537i 1.34500 − 0.437016i
239.1 −1.34500 − 0.437016i −0.690983 + 0.224514i 1.61803 + 1.17557i −0.809017 − 0.587785i 1.02749 1.34500 − 4.13948i −1.66251 − 2.28825i −2.00000 + 1.45309i 0.831254 + 1.14412i
239.2 1.34500 + 0.437016i −0.690983 + 0.224514i 1.61803 + 1.17557i −0.809017 − 0.587785i −1.02749 −1.34500 + 4.13948i 1.66251 + 2.28825i −2.00000 + 1.45309i −0.831254 − 1.14412i
403.1 −1.34500 + 0.437016i −0.690983 − 0.224514i 1.61803 − 1.17557i −0.809017 + 0.587785i 1.02749 1.34500 + 4.13948i −1.66251 + 2.28825i −2.00000 − 1.45309i 0.831254 − 1.14412i
403.2 1.34500 − 0.437016i −0.690983 − 0.224514i 1.61803 − 1.17557i −0.809017 + 0.587785i −1.02749 −1.34500 − 4.13948i 1.66251 − 2.28825i −2.00000 − 1.45309i −0.831254 + 1.14412i
475.1 −0.831254 + 1.14412i −1.80902 − 2.48990i −0.618034 − 1.90211i 0.309017 + 0.951057i 4.35250 0.831254 + 0.603941i 2.68999 + 0.874032i −2.00000 + 6.15537i −1.34500 − 0.437016i
475.2 0.831254 − 1.14412i −1.80902 − 2.48990i −0.618034 − 1.90211i 0.309017 + 0.951057i −4.35250 −0.831254 − 0.603941i −2.68999 − 0.874032i −2.00000 + 6.15537i 1.34500 + 0.437016i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
44.g even 10 1 inner
44.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 484.2.g.a 8
4.b odd 2 1 484.2.g.d 8
11.b odd 2 1 inner 484.2.g.a 8
11.c even 5 1 484.2.c.c ✓ 8
11.c even 5 1 484.2.g.b 8
11.c even 5 1 484.2.g.d 8
11.c even 5 1 484.2.g.e 8
11.d odd 10 1 484.2.c.c ✓ 8
11.d odd 10 1 484.2.g.b 8
11.d odd 10 1 484.2.g.d 8
11.d odd 10 1 484.2.g.e 8
44.c even 2 1 484.2.g.d 8
44.g even 10 1 484.2.c.c ✓ 8
44.g even 10 1 inner 484.2.g.a 8
44.g even 10 1 484.2.g.b 8
44.g even 10 1 484.2.g.e 8
44.h odd 10 1 484.2.c.c ✓ 8
44.h odd 10 1 inner 484.2.g.a 8
44.h odd 10 1 484.2.g.b 8
44.h odd 10 1 484.2.g.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
484.2.c.c ✓ 8 11.c even 5 1
484.2.c.c ✓ 8 11.d odd 10 1
484.2.c.c ✓ 8 44.g even 10 1
484.2.c.c ✓ 8 44.h odd 10 1
484.2.g.a 8 1.a even 1 1 trivial
484.2.g.a 8 11.b odd 2 1 inner
484.2.g.a 8 44.g even 10 1 inner
484.2.g.a 8 44.h odd 10 1 inner
484.2.g.b 8 11.c even 5 1
484.2.g.b 8 11.d odd 10 1
484.2.g.b 8 44.g even 10 1
484.2.g.b 8 44.h odd 10 1
484.2.g.d 8 4.b odd 2 1
484.2.g.d 8 11.c even 5 1
484.2.g.d 8 11.d odd 10 1
484.2.g.d 8 44.c even 2 1
484.2.g.e 8 11.c even 5 1
484.2.g.e 8 11.d odd 10 1
484.2.g.e 8 44.g even 10 1
484.2.g.e 8 44.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(484, [\chi])\):

\( T_{3}^{4} + 5T_{3}^{3} + 15T_{3}^{2} + 15T_{3} + 5 \) Copy content Toggle raw display
\( T_{5}^{4} + T_{5}^{3} + T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{13}^{8} - 32T_{13}^{6} + 384T_{13}^{4} + 512T_{13}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + 5 T^{3} + 15 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 30 T^{6} + \cdots + 400 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 32 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$17$ \( T^{8} - 10 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$19$ \( T^{8} + 40 T^{6} + \cdots + 102400 \) Copy content Toggle raw display
$23$ \( (T^{4} + 50 T^{2} + 605)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 128 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$31$ \( (T^{4} + 5 T^{3} + \cdots + 125)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( (T^{4} - 100 T^{2} + 2420)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 320 T + 1280)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 25 T^{3} + \cdots + 4205)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} - 32 T^{6} + \cdots + 1048576 \) Copy content Toggle raw display
$67$ \( (T^{4} + 50 T^{2} + 605)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 5 T^{3} + 5 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 40 T^{6} + \cdots + 2560000 \) Copy content Toggle raw display
$79$ \( T^{8} + 270 T^{6} + \cdots + 2624400 \) Copy content Toggle raw display
$83$ \( T^{8} + 110 T^{6} + \cdots + 282912400 \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T + 5)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} - 17 T^{3} + \cdots + 5041)^{2} \) Copy content Toggle raw display
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