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Properties

Label 990.2.i.j
Level $990$
Weight $2$
Character orbit 990.i
Analytic conductor $7.905$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [990,2,Mod(331,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 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("990.331"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 990 = 2 \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 990.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-7,-1] 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: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - x^{13} + 3 x^{12} - 4 x^{11} + 4 x^{10} - 27 x^{8} - 9 x^{7} - 81 x^{6} + 108 x^{4} + \cdots + 2187 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - x^{13} + 3 x^{12} - 4 x^{11} + 4 x^{10} - 27 x^{8} - 9 x^{7} - 81 x^{6} + 108 x^{4} + \cdots + 2187 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - \nu^{13} - 2 \nu^{12} - 5 \nu^{10} + 8 \nu^{9} - 12 \nu^{8} + 27 \nu^{7} + 90 \nu^{6} + 108 \nu^{5} + \cdots - 1458 ) / 729 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{13} - \nu^{12} + 3 \nu^{11} - 4 \nu^{10} + 4 \nu^{9} - 27 \nu^{7} - 9 \nu^{6} - 81 \nu^{5} + \cdots - 729 ) / 729 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{13} + 2 \nu^{12} + 18 \nu^{11} + 50 \nu^{10} + 37 \nu^{9} + 48 \nu^{8} - 18 \nu^{7} + \cdots + 2187 ) / 1458 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - \nu^{13} + 7 \nu^{12} + 18 \nu^{11} + 40 \nu^{10} + 35 \nu^{9} + 51 \nu^{8} + 9 \nu^{7} + \cdots + 1458 ) / 1458 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} + 6 \nu^{9} + 8 \nu^{8} + 10 \nu^{7} + 6 \nu^{6} - 12 \nu^{5} - 36 \nu^{4} + \cdots - 486 ) / 54 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2 \nu^{13} + 4 \nu^{11} - 2 \nu^{10} + 9 \nu^{9} + 26 \nu^{8} - 90 \nu^{5} - 108 \nu^{4} + 108 \nu^{3} + \cdots - 2187 ) / 486 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2 \nu^{13} + 2 \nu^{12} + 3 \nu^{11} + 17 \nu^{10} + 28 \nu^{9} + 18 \nu^{8} + 72 \nu^{7} + \cdots + 486 ) / 486 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{13} + \nu^{12} + 3 \nu^{11} + 2 \nu^{9} + 9 \nu^{8} - \nu^{7} - 9 \nu^{6} - 81 \nu^{5} + \cdots - 1053 ) / 162 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2 \nu^{13} - 2 \nu^{12} + 6 \nu^{11} + \nu^{10} + 26 \nu^{9} + 54 \nu^{8} + 18 \nu^{7} + 72 \nu^{6} + \cdots - 4374 ) / 486 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2 \nu^{13} - 2 \nu^{12} + 6 \nu^{11} + \nu^{10} + 26 \nu^{9} + 54 \nu^{8} + 18 \nu^{7} + 72 \nu^{6} + \cdots - 4374 ) / 486 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7 \nu^{13} - 29 \nu^{12} - 75 \nu^{11} - 143 \nu^{10} - 190 \nu^{9} - 162 \nu^{8} - 18 \nu^{7} + \cdots + 1458 ) / 1458 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3 \nu^{13} - \nu^{12} - 11 \nu^{11} - 17 \nu^{9} - 37 \nu^{8} + 9 \nu^{7} - 15 \nu^{6} + \cdots + 3888 ) / 486 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 5 \nu^{13} + 6 \nu^{12} + 22 \nu^{11} + 13 \nu^{10} + 12 \nu^{9} + 26 \nu^{8} - 54 \nu^{7} + \cdots - 2916 ) / 486 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - \beta_{9} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + 2\beta_{9} + 3\beta_{7} - 3\beta_{5} - 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{13} + 3\beta_{12} + \beta_{10} + 2\beta_{9} - 3\beta_{8} - 3\beta_{3} - 3\beta_{2} + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3 \beta_{13} + 6 \beta_{12} - 2 \beta_{10} + 2 \beta_{9} + 12 \beta_{8} - 3 \beta_{7} - 9 \beta_{6} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -5\beta_{10} - 4\beta_{9} - 9\beta_{8} - 3\beta_{7} + 27\beta_{6} + 9\beta_{4} - 9\beta_{2} + 9\beta _1 - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 9 \beta_{12} - 6 \beta_{11} + 10 \beta_{10} + 11 \beta_{9} - 6 \beta_{8} - 3 \beta_{7} - 18 \beta_{6} + \cdots + 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 12 \beta_{13} + 12 \beta_{12} + 12 \beta_{11} + 4 \beta_{10} - 22 \beta_{9} + 18 \beta_{8} + 12 \beta_{7} + \cdots + 72 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 12 \beta_{13} + 6 \beta_{12} + 12 \beta_{11} + 34 \beta_{10} + 20 \beta_{9} + 24 \beta_{8} + \cdots + 30 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6 \beta_{13} - 12 \beta_{12} - 48 \beta_{11} + 64 \beta_{10} + 2 \beta_{9} - 126 \beta_{8} + 48 \beta_{7} + \cdots - 51 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 36 \beta_{13} + 162 \beta_{12} - 12 \beta_{11} + 55 \beta_{10} + 125 \beta_{9} + 114 \beta_{8} + \cdots - 144 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 138 \beta_{13} + 228 \beta_{12} + 132 \beta_{11} - 197 \beta_{10} + 224 \beta_{9} + 138 \beta_{8} + \cdots + 489 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 141 \beta_{13} - 87 \beta_{12} - 234 \beta_{11} + 37 \beta_{10} - 340 \beta_{9} - 75 \beta_{8} + \cdots + 381 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 435 \beta_{13} - 156 \beta_{12} - 216 \beta_{11} - 128 \beta_{10} - 178 \beta_{9} + 498 \beta_{8} + \cdots + 513 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/990\mathbb{Z}\right)^\times\).

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
331.1
1.73201 + 0.0120874i
1.06504 − 1.36590i
1.01796 + 1.40134i
−0.131312 + 1.72707i
−0.513778 − 1.65410i
−0.992802 − 1.41928i
−1.67712 + 0.432753i
1.73201 − 0.0120874i
1.06504 + 1.36590i
1.01796 − 1.40134i
−0.131312 − 1.72707i
−0.513778 + 1.65410i
−0.992802 + 1.41928i
−1.67712 − 0.432753i
−0.500000 + 0.866025i −1.73201 + 0.0120874i −0.500000 − 0.866025i −0.500000 − 0.866025i 0.855536 − 1.50601i −1.08662 + 1.88208i 1.00000 2.99971 − 0.0418709i 1.00000
331.2 −0.500000 + 0.866025i −1.06504 − 1.36590i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.71543 − 0.239402i 0.420124 − 0.727676i 1.00000 −0.731375 + 2.90948i 1.00000
331.3 −0.500000 + 0.866025i −1.01796 + 1.40134i −0.500000 − 0.866025i −0.500000 − 0.866025i −0.704617 − 1.58225i 0.170965 − 0.296120i 1.00000 −0.927514 − 2.85302i 1.00000
331.4 −0.500000 + 0.866025i 0.131312 + 1.72707i −0.500000 − 0.866025i −0.500000 − 0.866025i −1.56134 − 0.749813i −2.37312 + 4.11036i 1.00000 −2.96551 + 0.453570i 1.00000
331.5 −0.500000 + 0.866025i 0.513778 − 1.65410i −0.500000 − 0.866025i −0.500000 − 0.866025i 1.17560 + 1.27199i 2.32261 − 4.02288i 1.00000 −2.47206 − 1.69968i 1.00000
331.6 −0.500000 + 0.866025i 0.992802 − 1.41928i −0.500000 − 0.866025i −0.500000 − 0.866025i 0.732728 + 1.56943i −1.65827 + 2.87221i 1.00000 −1.02869 − 2.81812i 1.00000
331.7 −0.500000 + 0.866025i 1.67712 + 0.432753i −0.500000 − 0.866025i −0.500000 − 0.866025i −1.21333 + 1.23605i −0.295691 + 0.512152i 1.00000 2.62545 + 1.45156i 1.00000
661.1 −0.500000 − 0.866025i −1.73201 − 0.0120874i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.855536 + 1.50601i −1.08662 − 1.88208i 1.00000 2.99971 + 0.0418709i 1.00000
661.2 −0.500000 − 0.866025i −1.06504 + 1.36590i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.71543 + 0.239402i 0.420124 + 0.727676i 1.00000 −0.731375 − 2.90948i 1.00000
661.3 −0.500000 − 0.866025i −1.01796 − 1.40134i −0.500000 + 0.866025i −0.500000 + 0.866025i −0.704617 + 1.58225i 0.170965 + 0.296120i 1.00000 −0.927514 + 2.85302i 1.00000
661.4 −0.500000 − 0.866025i 0.131312 − 1.72707i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.56134 + 0.749813i −2.37312 − 4.11036i 1.00000 −2.96551 − 0.453570i 1.00000
661.5 −0.500000 − 0.866025i 0.513778 + 1.65410i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.17560 − 1.27199i 2.32261 + 4.02288i 1.00000 −2.47206 + 1.69968i 1.00000
661.6 −0.500000 − 0.866025i 0.992802 + 1.41928i −0.500000 + 0.866025i −0.500000 + 0.866025i 0.732728 − 1.56943i −1.65827 − 2.87221i 1.00000 −1.02869 + 2.81812i 1.00000
661.7 −0.500000 − 0.866025i 1.67712 − 0.432753i −0.500000 + 0.866025i −0.500000 + 0.866025i −1.21333 − 1.23605i −0.295691 − 0.512152i 1.00000 2.62545 − 1.45156i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 331.7
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.i.j ✓ 14
3.b odd 2 1 2970.2.i.k 14
9.c even 3 1 inner 990.2.i.j ✓ 14
9.c even 3 1 8910.2.a.cd 7
9.d odd 6 1 2970.2.i.k 14
9.d odd 6 1 8910.2.a.ca 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.i.j ✓ 14 1.a even 1 1 trivial
990.2.i.j ✓ 14 9.c even 3 1 inner
2970.2.i.k 14 3.b odd 2 1
2970.2.i.k 14 9.d odd 6 1
8910.2.a.ca 7 9.d odd 6 1
8910.2.a.cd 7 9.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} + 5 T_{7}^{13} + 43 T_{7}^{12} + 138 T_{7}^{11} + 975 T_{7}^{10} + 3003 T_{7}^{9} + \cdots + 729 \) acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + T^{13} + \cdots + 2187 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{14} + 5 T^{13} + \cdots + 729 \) Copy content Toggle raw display
$11$ \( (T^{2} + T + 1)^{7} \) Copy content Toggle raw display
$13$ \( T^{14} + 5 T^{13} + \cdots + 6718464 \) Copy content Toggle raw display
$17$ \( (T^{7} - 3 T^{6} - 69 T^{5} + \cdots + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{7} - 11 T^{6} + \cdots - 40178)^{2} \) Copy content Toggle raw display
$23$ \( T^{14} + 3 T^{13} + \cdots + 3779136 \) Copy content Toggle raw display
$29$ \( T^{14} + 12 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$31$ \( T^{14} + 5 T^{13} + \cdots + 23097636 \) Copy content Toggle raw display
$37$ \( (T^{7} - 2 T^{6} + \cdots + 4644)^{2} \) Copy content Toggle raw display
$41$ \( T^{14} + 6 T^{13} + \cdots + 96353856 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 1026217624576 \) Copy content Toggle raw display
$47$ \( T^{14} - 3 T^{13} + \cdots + 50466816 \) Copy content Toggle raw display
$53$ \( (T^{7} - 3 T^{6} + \cdots + 21744)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 20983142301696 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 317089249 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 17\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{7} - 12 T^{6} + \cdots - 164064)^{2} \) Copy content Toggle raw display
$73$ \( (T^{7} - 2 T^{6} + \cdots + 22888)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 4481499136 \) Copy content Toggle raw display
$83$ \( T^{14} + 219 T^{12} + \cdots + 70963776 \) Copy content Toggle raw display
$89$ \( (T^{7} + 3 T^{6} + \cdots - 227448)^{2} \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 685107154944 \) Copy content Toggle raw display
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