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Properties

Label 702.2.g.d
Level $702$
Weight $2$
Character orbit 702.g
Analytic conductor $5.605$
Analytic rank $0$
Dimension $12$
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] = [702,2,Mod(451,702)] 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("702.451"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 702.g (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.60549822189\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: 12.0.157365759791601.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + x^{10} + 11x^{8} - 6x^{7} - 17x^{6} - 12x^{5} + 44x^{4} + 16x^{2} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{5} \)
Twist minimal: no (minimal twist has level 234)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_{2} - 1) q^{4} + (\beta_{10} - \beta_{5}) q^{5} + (\beta_{11} + \beta_{10} + \beta_{8} + \cdots + 1) q^{7} - q^{8} + (\beta_{10} - \beta_{5} - \beta_1) q^{10} + (\beta_{10} - \beta_{2}) q^{11}+ \cdots + (3 \beta_{11} + 2 \beta_{10} + \cdots + 4 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{2} - 6 q^{4} - q^{5} + 10 q^{7} - 12 q^{8} + q^{10} - 8 q^{11} + 8 q^{13} + 5 q^{14} - 6 q^{16} - 3 q^{17} - 7 q^{19} + 2 q^{20} + 8 q^{22} + 2 q^{23} - 13 q^{25} + q^{26} - 5 q^{28} - q^{29}+ \cdots + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{11} + x^{10} + 11x^{8} - 6x^{7} - 17x^{6} - 12x^{5} + 44x^{4} + 16x^{2} - 96x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 3 \nu^{11} - 7 \nu^{10} - 3 \nu^{9} + 2 \nu^{8} + 33 \nu^{7} + 4 \nu^{6} - 63 \nu^{5} - 70 \nu^{4} + \cdots - 224 ) / 32 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{11} + 3 \nu^{10} + 7 \nu^{9} + 20 \nu^{8} - 23 \nu^{7} - 46 \nu^{6} - 11 \nu^{5} + 96 \nu^{4} + \cdots + 128 ) / 32 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 7 \nu^{10} - 5 \nu^{9} - 20 \nu^{8} - 35 \nu^{7} + 50 \nu^{6} + 113 \nu^{5} + 24 \nu^{4} + \cdots + 352 ) / 32 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 11 \nu^{11} + 11 \nu^{10} + 19 \nu^{9} + 30 \nu^{8} - 69 \nu^{7} - 88 \nu^{6} + 35 \nu^{5} + \cdots + 256 ) / 32 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{11} - 15 \nu^{10} - 7 \nu^{9} - 14 \nu^{8} + 73 \nu^{7} + 48 \nu^{6} - 79 \nu^{5} - 202 \nu^{4} + \cdots - 448 ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9 \nu^{11} + 15 \nu^{10} + 15 \nu^{9} + 16 \nu^{8} - 103 \nu^{7} - 70 \nu^{6} + 109 \nu^{5} + \cdots + 480 ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 17 \nu^{11} + 21 \nu^{10} + 25 \nu^{9} + 42 \nu^{8} - 115 \nu^{7} - 140 \nu^{6} + 69 \nu^{5} + \cdots + 512 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 7 \nu^{11} - 13 \nu^{10} - 7 \nu^{9} - 10 \nu^{8} + 67 \nu^{7} + 42 \nu^{6} - 73 \nu^{5} - 180 \nu^{4} + \cdots - 352 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 17 \nu^{11} - 35 \nu^{10} - 11 \nu^{9} - 12 \nu^{8} + 159 \nu^{7} + 58 \nu^{6} - 213 \nu^{5} + \cdots - 992 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27 \nu^{11} - 39 \nu^{10} - 35 \nu^{9} - 54 \nu^{8} + 201 \nu^{7} + 172 \nu^{6} - 167 \nu^{5} + \cdots - 1024 ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43 \nu^{11} + 81 \nu^{10} + 53 \nu^{9} + 56 \nu^{8} - 417 \nu^{7} - 230 \nu^{6} + 475 \nu^{5} + \cdots + 2208 ) / 32 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - 2\beta_{10} - \beta_{8} - 2\beta_{7} + \beta_{6} - \beta_{3} - \beta_{2} - 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{8} - \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} - 2\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{11} - \beta_{10} - 2\beta_{8} - 2\beta_{7} - \beta_{5} + 3\beta_{4} - 3\beta_{2} + \beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 2 \beta_{11} - 3 \beta_{10} - 4 \beta_{9} + 2 \beta_{8} - 4 \beta_{7} + 5 \beta_{6} + 2 \beta_{5} + \cdots - 6 \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{10} - \beta_{9} + 2\beta_{7} - 2\beta_{6} - 4\beta_{5} + 7\beta_{4} + 2\beta_{3} - \beta_{2} - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( \beta_{11} - \beta_{9} + \beta_{8} - 5\beta_{7} - \beta_{6} + 5\beta_{4} + \beta_{2} + 4\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{11} + 6 \beta_{10} - 12 \beta_{9} + 16 \beta_{8} + 7 \beta_{7} + 4 \beta_{6} - 5 \beta_{5} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4 \beta_{11} + 9 \beta_{10} + 5 \beta_{9} - 3 \beta_{7} - 10 \beta_{6} - 14 \beta_{5} + 11 \beta_{4} + \cdots - 37 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 20 \beta_{11} + 27 \beta_{10} + 6 \beta_{9} + 26 \beta_{8} + 28 \beta_{7} - 18 \beta_{6} - 2 \beta_{5} + \cdots + 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - \beta_{11} - 9 \beta_{10} - 2 \beta_{9} + 9 \beta_{8} - 6 \beta_{7} - 8 \beta_{6} - 22 \beta_{5} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 30 \beta_{11} + 37 \beta_{10} + 17 \beta_{9} + 23 \beta_{8} + 17 \beta_{7} - 14 \beta_{6} - 9 \beta_{5} + \cdots - 60 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/702\mathbb{Z}\right)^\times\).

\(n\) \(379\) \(677\)
\(\chi(n)\) \(-\beta_{2}\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
1.40369 − 0.172227i
1.34059 + 0.450356i
−1.29709 − 0.563529i
1.25202 − 0.657613i
−0.681625 − 1.23911i
−0.517581 + 1.31610i
1.40369 + 0.172227i
1.34059 − 0.450356i
−1.29709 + 0.563529i
1.25202 + 0.657613i
−0.681625 + 1.23911i
−0.517581 − 1.31610i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.44068 − 2.49532i 0 4.78015 −1.00000 0 1.44068 − 2.49532i
451.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.09436 − 1.89549i 0 −1.58087 −1.00000 0 1.09436 − 1.89549i
451.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.864869 − 1.49800i 0 −0.326144 −1.00000 0 0.864869 − 1.49800i
451.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.635090 − 1.10001i 0 −1.83842 −1.00000 0 0.635090 − 1.10001i
451.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.57078 + 2.72066i 0 4.02018 −1.00000 0 −1.57078 + 2.72066i
451.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.96422 + 3.40213i 0 −0.0548989 −1.00000 0 −1.96422 + 3.40213i
523.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.44068 + 2.49532i 0 4.78015 −1.00000 0 1.44068 + 2.49532i
523.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.09436 + 1.89549i 0 −1.58087 −1.00000 0 1.09436 + 1.89549i
523.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.864869 + 1.49800i 0 −0.326144 −1.00000 0 0.864869 + 1.49800i
523.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.635090 + 1.10001i 0 −1.83842 −1.00000 0 0.635090 + 1.10001i
523.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.57078 − 2.72066i 0 4.02018 −1.00000 0 −1.57078 − 2.72066i
523.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.96422 − 3.40213i 0 −0.0548989 −1.00000 0 −1.96422 − 3.40213i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.f even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 702.2.g.d 12
3.b odd 2 1 234.2.g.c yes 12
9.c even 3 1 702.2.f.c 12
9.d odd 6 1 234.2.f.d ✓ 12
13.c even 3 1 702.2.f.c 12
39.i odd 6 1 234.2.f.d ✓ 12
117.f even 3 1 inner 702.2.g.d 12
117.u odd 6 1 234.2.g.c yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.f.d ✓ 12 9.d odd 6 1
234.2.f.d ✓ 12 39.i odd 6 1
234.2.g.c yes 12 3.b odd 2 1
234.2.g.c yes 12 117.u odd 6 1
702.2.f.c 12 9.c even 3 1
702.2.f.c 12 13.c even 3 1
702.2.g.d 12 1.a even 1 1 trivial
702.2.g.d 12 117.f even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + T_{5}^{11} + 22 T_{5}^{10} + 55 T_{5}^{9} + 385 T_{5}^{8} + 883 T_{5}^{7} + 3487 T_{5}^{6} + \cdots + 29241 \) acting on \(S_{2}^{\mathrm{new}}(702, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + T^{11} + \cdots + 29241 \) Copy content Toggle raw display
$7$ \( (T^{6} - 5 T^{5} - 10 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 8 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{12} - 8 T^{11} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + 3 T^{11} + \cdots + 69605649 \) Copy content Toggle raw display
$19$ \( T^{12} + 7 T^{11} + \cdots + 77841 \) Copy content Toggle raw display
$23$ \( (T^{6} - T^{5} - 75 T^{4} + \cdots - 81)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + T^{11} + \cdots + 16916769 \) Copy content Toggle raw display
$31$ \( T^{12} + 14 T^{11} + \cdots + 657721 \) Copy content Toggle raw display
$37$ \( T^{12} + 7 T^{11} + \cdots + 81 \) Copy content Toggle raw display
$41$ \( (T^{6} - 9 T^{5} + \cdots - 2349)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 8 T^{5} + \cdots + 18019)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 189692478369 \) Copy content Toggle raw display
$53$ \( (T^{6} - 6 T^{5} + \cdots + 8667)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 4 T^{11} + \cdots + 859329 \) Copy content Toggle raw display
$61$ \( (T^{6} + 28 T^{5} + \cdots - 9893)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 4 T^{5} + \cdots + 71719)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} - 13 T^{11} + \cdots + 431649 \) Copy content Toggle raw display
$73$ \( (T^{6} - 27 T^{5} + \cdots - 1403)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 6 T^{11} + \cdots + 28270489 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 3799612881 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 89771545161 \) Copy content Toggle raw display
$97$ \( (T^{6} - 24 T^{5} + \cdots + 57703)^{2} \) Copy content Toggle raw display
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