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Properties

Label 1218.2.i.b
Level $1218$
Weight $2$
Character orbit 1218.i
Analytic conductor $9.726$
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] = [1218,2,Mod(697,1218)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1218, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 0])) 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("1218.697"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1218 = 2 \cdot 3 \cdot 7 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1218.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,-3,3,-3,-1] 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: \(9.72577896619\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.1783323.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{4} + 1) q^{3} + (\beta_{4} - 1) q^{4} + ( - \beta_{5} + \beta_{3} + \cdots + \beta_1) q^{5} - q^{6} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{7} + q^{8} - \beta_{4} q^{9}+ \cdots + ( - \beta_{2} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{2} + 3 q^{3} - 3 q^{4} - q^{5} - 6 q^{6} + 2 q^{7} + 6 q^{8} - 3 q^{9} - q^{10} + 11 q^{11} + 3 q^{12} + 10 q^{13} - 4 q^{14} - 2 q^{15} - 3 q^{16} - 6 q^{17} - 3 q^{18} + 4 q^{19} + 2 q^{20}+ \cdots - 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 5x^{4} - 2x^{3} + 19x^{2} - 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} + 25\nu^{3} - 19\nu^{2} + 12\nu - 60 ) / 83 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{5} - 20\nu^{4} + 17\nu^{3} - 76\nu^{2} + 48\nu - 240 ) / 83 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -20\nu^{5} + 17\nu^{4} - 85\nu^{3} - 35\nu^{2} - 323\nu + 204 ) / 249 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -16\nu^{5} - 3\nu^{4} - 68\nu^{3} - 28\nu^{2} - 275\nu - 36 ) / 83 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 3\beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{5} + 12\beta_{4} - \beta _1 - 12 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{5} + 3\beta_{4} + 6\beta_{3} - 17\beta_{2} - 17\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1218\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(407\) \(871\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
697.1
0.356769 − 0.617942i
1.09935 − 1.90412i
−0.956115 + 1.65604i
0.356769 + 0.617942i
1.09935 + 1.90412i
−0.956115 − 1.65604i
−0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i −1.60220 − 2.77509i −1.00000 2.63409 − 0.248083i 1.00000 −0.500000 − 0.866025i −1.60220 + 2.77509i
697.2 −0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i −0.182224 − 0.315621i −1.00000 −2.43359 + 1.03810i 1.00000 −0.500000 − 0.866025i −0.182224 + 0.315621i
697.3 −0.500000 − 0.866025i 0.500000 − 0.866025i −0.500000 + 0.866025i 1.28442 + 2.22469i −1.00000 0.799494 − 2.52206i 1.00000 −0.500000 − 0.866025i 1.28442 − 2.22469i
1045.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i −1.60220 + 2.77509i −1.00000 2.63409 + 0.248083i 1.00000 −0.500000 + 0.866025i −1.60220 − 2.77509i
1045.2 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i −0.182224 + 0.315621i −1.00000 −2.43359 − 1.03810i 1.00000 −0.500000 + 0.866025i −0.182224 − 0.315621i
1045.3 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 − 0.866025i 1.28442 − 2.22469i −1.00000 0.799494 + 2.52206i 1.00000 −0.500000 + 0.866025i 1.28442 + 2.22469i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 697.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1218.2.i.b ✓ 6
7.c even 3 1 inner 1218.2.i.b ✓ 6
7.c even 3 1 8526.2.a.by 3
7.d odd 6 1 8526.2.a.ca 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1218.2.i.b ✓ 6 1.a even 1 1 trivial
1218.2.i.b ✓ 6 7.c even 3 1 inner
8526.2.a.by 3 7.c even 3 1
8526.2.a.ca 3 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + T_{5}^{5} + 9T_{5}^{4} - 2T_{5}^{3} + 67T_{5}^{2} + 24T_{5} + 9 \) acting on \(S_{2}^{\mathrm{new}}(1218, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + T^{5} + 9 T^{4} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots + 343 \) Copy content Toggle raw display
$11$ \( T^{6} - 11 T^{5} + \cdots + 1225 \) Copy content Toggle raw display
$13$ \( (T^{3} - 5 T^{2} - 2 T + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 6 T^{5} + \cdots + 26569 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$23$ \( T^{6} - 14 T^{5} + \cdots + 3969 \) Copy content Toggle raw display
$29$ \( (T - 1)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} + \cdots + 27889 \) Copy content Toggle raw display
$37$ \( T^{6} - 3 T^{5} + \cdots + 19881 \) Copy content Toggle raw display
$41$ \( (T^{3} + 9 T^{2} + 22 T + 15)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 13 T^{2} + \cdots - 525)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 13 T^{5} + \cdots + 194481 \) Copy content Toggle raw display
$53$ \( T^{6} - 14 T^{5} + \cdots + 419904 \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 419904 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots + 974169 \) Copy content Toggle raw display
$67$ \( T^{6} - 13 T^{5} + \cdots + 17161 \) Copy content Toggle raw display
$71$ \( (T^{3} + 3 T^{2} + \cdots - 749)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 28 T^{5} + \cdots + 540225 \) Copy content Toggle raw display
$79$ \( T^{6} - 19 T^{5} + \cdots + 24649 \) Copy content Toggle raw display
$83$ \( (T^{3} + 17 T^{2} + \cdots + 159)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - T^{5} + \cdots + 196249 \) Copy content Toggle raw display
$97$ \( (T^{3} - 19 T^{2} + \cdots + 1637)^{2} \) Copy content Toggle raw display
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