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Properties

Label 361.2.c.j
Level $361$
Weight $2$
Character orbit 361.c
Analytic conductor $2.883$
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] = [361,2,Mod(68,361)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(361, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) 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("361.68"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 361.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.88259951297\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.324000000.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 5x^{6} + 20x^{4} + 25x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{3} q^{3} + (\beta_{6} + \beta_{4} + \beta_{2}) q^{4} + (2 \beta_{6} + 2 \beta_{4} + 2) q^{5} + ( - \beta_{6} - 3 \beta_{4} - \beta_{2}) q^{6} + ( - 2 \beta_{2} - 1) q^{7} + (\beta_{7} - \beta_{5} + \beta_{3}) q^{8}+ \cdots + ( - 3 \beta_{6} + \beta_{4} - 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 4 q^{5} + 10 q^{6} - 16 q^{7} + 2 q^{9} - 20 q^{11} + 14 q^{16} + 8 q^{17} - 24 q^{20} + 6 q^{23} - 10 q^{24} - 4 q^{25} + 20 q^{26} - 6 q^{28} + 40 q^{30} + 12 q^{35} - 4 q^{36} - 20 q^{39}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 5x^{6} + 20x^{4} + 25x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 15 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 35\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 4\nu^{4} + 20\nu^{2} + 5 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 4\nu^{5} + 20\nu^{3} + 5\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 20\nu^{2} + 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 6\nu^{5} + 20\nu^{3} + 25\nu ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + 2\beta_{4} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + 3\beta_{5} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{6} - 5\beta_{4} - 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 5\beta_{7} - 10\beta_{5} - 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 20\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 20\beta_{3} + 35\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
68.1
0.587785 − 1.01807i
−0.951057 + 1.64728i
0.951057 − 1.64728i
−0.587785 + 1.01807i
0.587785 + 1.01807i
−0.951057 − 1.64728i
0.951057 + 1.64728i
−0.587785 − 1.01807i
−0.951057 + 1.64728i 0.951057 − 1.64728i −0.809017 − 1.40126i 1.61803 − 2.80252i 1.80902 + 3.13331i 0.236068 −0.726543 −0.309017 − 0.535233i 3.07768 + 5.33070i
68.2 −0.587785 + 1.01807i 0.587785 − 1.01807i 0.309017 + 0.535233i −0.618034 + 1.07047i 0.690983 + 1.19682i −4.23607 −3.07768 0.809017 + 1.40126i −0.726543 − 1.25841i
68.3 0.587785 − 1.01807i −0.587785 + 1.01807i 0.309017 + 0.535233i −0.618034 + 1.07047i 0.690983 + 1.19682i −4.23607 3.07768 0.809017 + 1.40126i 0.726543 + 1.25841i
68.4 0.951057 − 1.64728i −0.951057 + 1.64728i −0.809017 − 1.40126i 1.61803 − 2.80252i 1.80902 + 3.13331i 0.236068 0.726543 −0.309017 − 0.535233i −3.07768 − 5.33070i
292.1 −0.951057 − 1.64728i 0.951057 + 1.64728i −0.809017 + 1.40126i 1.61803 + 2.80252i 1.80902 − 3.13331i 0.236068 −0.726543 −0.309017 + 0.535233i 3.07768 − 5.33070i
292.2 −0.587785 − 1.01807i 0.587785 + 1.01807i 0.309017 − 0.535233i −0.618034 − 1.07047i 0.690983 − 1.19682i −4.23607 −3.07768 0.809017 − 1.40126i −0.726543 + 1.25841i
292.3 0.587785 + 1.01807i −0.587785 − 1.01807i 0.309017 − 0.535233i −0.618034 − 1.07047i 0.690983 − 1.19682i −4.23607 3.07768 0.809017 − 1.40126i 0.726543 − 1.25841i
292.4 0.951057 + 1.64728i −0.951057 − 1.64728i −0.809017 + 1.40126i 1.61803 + 2.80252i 1.80902 − 3.13331i 0.236068 0.726543 −0.309017 + 0.535233i −3.07768 + 5.33070i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 68.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner
19.c even 3 1 inner
19.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.2.c.j 8
19.b odd 2 1 inner 361.2.c.j 8
19.c even 3 1 361.2.a.i ✓ 4
19.c even 3 1 inner 361.2.c.j 8
19.d odd 6 1 361.2.a.i ✓ 4
19.d odd 6 1 inner 361.2.c.j 8
19.e even 9 6 361.2.e.m 24
19.f odd 18 6 361.2.e.m 24
57.f even 6 1 3249.2.a.bc 4
57.h odd 6 1 3249.2.a.bc 4
76.f even 6 1 5776.2.a.bu 4
76.g odd 6 1 5776.2.a.bu 4
95.h odd 6 1 9025.2.a.bj 4
95.i even 6 1 9025.2.a.bj 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.i ✓ 4 19.c even 3 1
361.2.a.i ✓ 4 19.d odd 6 1
361.2.c.j 8 1.a even 1 1 trivial
361.2.c.j 8 19.b odd 2 1 inner
361.2.c.j 8 19.c even 3 1 inner
361.2.c.j 8 19.d odd 6 1 inner
361.2.e.m 24 19.e even 9 6
361.2.e.m 24 19.f odd 18 6
3249.2.a.bc 4 57.f even 6 1
3249.2.a.bc 4 57.h odd 6 1
5776.2.a.bu 4 76.f even 6 1
5776.2.a.bu 4 76.g odd 6 1
9025.2.a.bj 4 95.h odd 6 1
9025.2.a.bj 4 95.i even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):

\( T_{2}^{8} + 5T_{2}^{6} + 20T_{2}^{4} + 25T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{3}^{8} + 5T_{3}^{6} + 20T_{3}^{4} + 25T_{3}^{2} + 25 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$3$ \( T^{8} + 5 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{4} - 2 T^{3} + 8 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T - 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 5 T + 5)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 10 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$17$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{4} - 3 T^{3} + 8 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 25 T^{6} + \cdots + 15625 \) Copy content Toggle raw display
$31$ \( (T^{4} - 5 T^{2} + 5)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 85 T^{2} + 1805)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 90 T^{6} + \cdots + 164025 \) Copy content Toggle raw display
$43$ \( (T^{4} + 3 T^{3} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} + \cdots + 841)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 25 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$59$ \( T^{8} + 85 T^{6} + \cdots + 366025 \) Copy content Toggle raw display
$61$ \( (T^{4} - 10 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 170 T^{6} + \cdots + 25 \) Copy content Toggle raw display
$71$ \( T^{8} + 250 T^{6} + \cdots + 228765625 \) Copy content Toggle raw display
$73$ \( (T^{2} + 9 T + 81)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} + 80 T^{6} + \cdots + 1638400 \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T - 4)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} + 370 T^{6} + \cdots + 302934025 \) Copy content Toggle raw display
$97$ \( T^{8} + 265 T^{6} + \cdots + 23088025 \) Copy content Toggle raw display
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