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Properties

Label 676.3.g.c
Level $676$
Weight $3$
Character orbit 676.g
Analytic conductor $18.420$
Analytic rank $0$
Dimension $8$
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] = [676,3,Mod(437,676)] 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("676.437"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(676, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 676.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,-6] 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: \(18.4196658708\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.44991500544.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 38x^{6} + 555x^{4} - 3674x^{2} + 9409 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 52)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + ( - \beta_{7} - \beta_{4} + \beta_{2}) q^{5} + (\beta_{6} - \beta_{5} - \beta_{4} + \cdots + 1) q^{7} + ( - \beta_{7} + \beta_{6} - \beta_{3} + \cdots + 2) q^{9} + ( - \beta_{6} - 2 \beta_{5} + \cdots - 3 \beta_1) q^{11}+ \cdots + (8 \beta_{6} - 10 \beta_{5} + \cdots - 32) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5} + 14 q^{7} + 12 q^{9} + 18 q^{11} - 24 q^{15} + 14 q^{19} + 54 q^{21} + 36 q^{27} - 216 q^{29} - 176 q^{31} + 198 q^{33} + 60 q^{35} + 40 q^{37} - 24 q^{41} + 102 q^{45} + 96 q^{47} - 72 q^{53}+ \cdots + 130 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 38x^{6} + 555x^{4} - 3674x^{2} + 9409 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{7} + 97\nu^{6} + 173\nu^{5} - 2716\nu^{4} - 2565\nu^{3} + 25414\nu^{2} + 11325\nu - 80801 ) / 5044 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} - 97\nu^{6} + 173\nu^{5} + 2716\nu^{4} - 2565\nu^{3} - 25414\nu^{2} + 11325\nu + 75757 ) / 5044 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 11\nu^{7} + 97\nu^{6} - 321\nu^{5} - 2716\nu^{4} + 3389\nu^{3} + 25414\nu^{2} - 17522\nu - 78279 ) / 5044 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\nu^{7} - 321\nu^{5} + 3389\nu^{3} - 12478\nu ) / 2522 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{7} + 194\nu^{6} - 494\nu^{5} - 5432\nu^{4} + 5954\nu^{3} + 55872\nu^{2} - 23803\nu - 204476 ) / 5044 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -84\nu^{7} + 97\nu^{6} + 2222\nu^{5} - 3977\nu^{4} - 19460\nu^{3} + 51895\nu^{2} + 51954\nu - 223294 ) / 5044 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 84\nu^{7} + 97\nu^{6} - 2222\nu^{5} - 3977\nu^{4} + 19460\nu^{3} + 51895\nu^{2} - 51954\nu - 223294 ) / 5044 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} - 2\beta_{3} - \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{5} - \beta_{4} + 3\beta_{2} - \beta _1 + 20 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 13\beta_{4} - 10\beta_{3} - 3\beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -4\beta_{7} - 4\beta_{6} + 42\beta_{5} - 21\beta_{4} + 59\beta_{2} - 17\beta _1 + 190 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -34\beta_{7} + 34\beta_{6} + 371\beta_{4} - 182\beta_{3} + 21\beta_{2} + 203\beta _1 + 112 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -56\beta_{7} - 56\beta_{6} + 326\beta_{5} - 163\beta_{4} + 407\beta_{2} - 81\beta _1 + 847 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -376\beta_{7} + 376\beta_{6} + 4409\beta_{4} - 1418\beta_{3} + 1327\beta_{2} + 2745\beta _1 + 2036 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/676\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(509\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
437.1
−2.83160 + 0.500000i
−3.38852 + 0.500000i
2.83160 + 0.500000i
3.38852 + 0.500000i
−2.83160 − 0.500000i
−3.38852 − 0.500000i
2.83160 − 0.500000i
3.38852 − 0.500000i
0 −3.69762 0 2.68502 + 2.68502i 0 2.35342 − 2.35342i 0 4.67242 0
437.2 0 −2.52249 0 −1.55727 − 1.55727i 0 −2.44579 + 2.44579i 0 −2.63703 0
437.3 0 1.96557 0 −5.05105 − 5.05105i 0 0.280550 − 0.280550i 0 −5.13652 0
437.4 0 4.25454 0 0.923296 + 0.923296i 0 6.81181 − 6.81181i 0 9.10114 0
577.1 0 −3.69762 0 2.68502 − 2.68502i 0 2.35342 + 2.35342i 0 4.67242 0
577.2 0 −2.52249 0 −1.55727 + 1.55727i 0 −2.44579 − 2.44579i 0 −2.63703 0
577.3 0 1.96557 0 −5.05105 + 5.05105i 0 0.280550 + 0.280550i 0 −5.13652 0
577.4 0 4.25454 0 0.923296 − 0.923296i 0 6.81181 + 6.81181i 0 9.10114 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 437.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 676.3.g.c 8
13.b even 2 1 676.3.g.d 8
13.d odd 4 1 inner 676.3.g.c 8
13.d odd 4 1 676.3.g.d 8
13.e even 6 1 52.3.k.a ✓ 8
13.f odd 12 1 52.3.k.a ✓ 8
39.h odd 6 1 468.3.cd.b 8
39.k even 12 1 468.3.cd.b 8
52.i odd 6 1 208.3.bd.e 8
52.l even 12 1 208.3.bd.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.3.k.a ✓ 8 13.e even 6 1
52.3.k.a ✓ 8 13.f odd 12 1
208.3.bd.e 8 52.i odd 6 1
208.3.bd.e 8 52.l even 12 1
468.3.cd.b 8 39.h odd 6 1
468.3.cd.b 8 39.k even 12 1
676.3.g.c 8 1.a even 1 1 trivial
676.3.g.c 8 13.d odd 4 1 inner
676.3.g.d 8 13.b even 2 1
676.3.g.d 8 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(676, [\chi])\):

\( T_{3}^{4} - 21T_{3}^{2} - 6T_{3} + 78 \) Copy content Toggle raw display
\( T_{5}^{8} + 6T_{5}^{7} + 18T_{5}^{6} - 114T_{5}^{5} + 573T_{5}^{4} + 828T_{5}^{3} + 1152T_{5}^{2} - 3744T_{5} + 6084 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 21 T^{2} + \cdots + 78)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 6 T^{7} + \cdots + 6084 \) Copy content Toggle raw display
$7$ \( T^{8} - 14 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{8} - 18 T^{7} + \cdots + 16451136 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} + 1344 T^{6} + \cdots + 229613409 \) Copy content Toggle raw display
$19$ \( T^{8} - 14 T^{7} + \cdots + 64609444 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 96160769604 \) Copy content Toggle raw display
$29$ \( (T^{4} + 108 T^{3} + \cdots + 262797)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 206628430096 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 45522062881 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 60225577281 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 67422688721424 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 351853021584 \) Copy content Toggle raw display
$53$ \( (T^{4} + 36 T^{3} + \cdots - 613164)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 48466826323344 \) Copy content Toggle raw display
$61$ \( (T^{4} - 18 T^{3} + \cdots - 2375967)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 178293048380164 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 59610474691524 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 574467612546916 \) Copy content Toggle raw display
$79$ \( (T^{4} - 96 T^{3} + \cdots + 11795424)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 246401334374976 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 26147207272356 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 151521790564 \) Copy content Toggle raw display
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