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Properties

Label 990.2.m.g
Level $990$
Weight $2$
Character orbit 990.m
Analytic conductor $7.905$
Analytic rank $0$
Dimension $8$
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(307,990)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(990, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 2])) 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("990.307"); 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.m (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,8,0,8,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.822083584.4
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{6} + 8x^{4} + 28x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 330)
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.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{4} q^{4} + (\beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{4} - \beta_{3} - \beta_1 + 1) q^{7} - \beta_{2} q^{8} + (\beta_{7} - \beta_{6} + 1) q^{10} + ( - \beta_{7} - \beta_{6} + \cdots - \beta_1) q^{11}+ \cdots + (2 \beta_{7} + 2 \beta_{5} - \beta_{4} + \cdots - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{5} + 8 q^{7} + 8 q^{10} + 8 q^{13} - 8 q^{16} - 8 q^{17} + 16 q^{19} - 8 q^{22} - 8 q^{23} + 16 q^{25} - 8 q^{28} + 16 q^{29} + 16 q^{35} + 16 q^{37} + 16 q^{38} + 16 q^{43} + 8 q^{44} + 8 q^{47}+ \cdots - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{6} + 8x^{4} + 28x^{2} + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{6} + 15\nu^{4} - 107\nu^{2} + 49 ) / 231 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 15\nu^{5} - 107\nu^{3} + 49\nu ) / 231 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 15\nu^{4} - 30\nu^{2} - 28 ) / 77 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{7} + 15\nu^{5} - 30\nu^{3} - 28\nu ) / 77 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -9\nu^{6} + 29\nu^{4} - 58\nu^{2} - 203 ) / 231 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -9\nu^{7} + 29\nu^{5} - 58\nu^{3} - 203\nu ) / 231 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 3\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 3\beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} + 11\beta_{4} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{7} + 11\beta_{5} - 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -45\beta_{6} + 29\beta_{4} - 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -45\beta_{7} + 29\beta_{5} - 29\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(\beta_{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
1.94107 + 0.804019i
−1.94107 − 0.804019i
0.481906 − 1.16342i
−0.481906 + 1.16342i
1.94107 − 0.804019i
−1.94107 + 0.804019i
0.481906 + 1.16342i
−0.481906 − 1.16342i
−0.707107 + 0.707107i 0 − 1.00000i −1.64818 − 1.51113i 0 −0.137055 + 0.137055i 0.707107 + 0.707107i 0 2.23397 − 0.0969123i
307.2 −0.707107 + 0.707107i 0 − 1.00000i 2.23397 + 0.0969123i 0 2.13705 − 2.13705i 0.707107 + 0.707107i 0 −1.64818 + 1.51113i
307.3 0.707107 − 0.707107i 0 − 1.00000i 1.22520 + 1.87053i 0 −0.645329 + 0.645329i −0.707107 − 0.707107i 0 2.18901 + 0.456316i
307.4 0.707107 − 0.707107i 0 − 1.00000i 2.18901 − 0.456316i 0 2.64533 − 2.64533i −0.707107 − 0.707107i 0 1.22520 − 1.87053i
703.1 −0.707107 − 0.707107i 0 1.00000i −1.64818 + 1.51113i 0 −0.137055 − 0.137055i 0.707107 − 0.707107i 0 2.23397 + 0.0969123i
703.2 −0.707107 − 0.707107i 0 1.00000i 2.23397 − 0.0969123i 0 2.13705 + 2.13705i 0.707107 − 0.707107i 0 −1.64818 − 1.51113i
703.3 0.707107 + 0.707107i 0 1.00000i 1.22520 − 1.87053i 0 −0.645329 − 0.645329i −0.707107 + 0.707107i 0 2.18901 − 0.456316i
703.4 0.707107 + 0.707107i 0 1.00000i 2.18901 + 0.456316i 0 2.64533 + 2.64533i −0.707107 + 0.707107i 0 1.22520 + 1.87053i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 990.2.m.g 8
3.b odd 2 1 330.2.l.d yes 8
5.c odd 4 1 990.2.m.f 8
11.b odd 2 1 990.2.m.f 8
15.d odd 2 1 1650.2.l.c 8
15.e even 4 1 330.2.l.c ✓ 8
15.e even 4 1 1650.2.l.f 8
33.d even 2 1 330.2.l.c ✓ 8
55.e even 4 1 inner 990.2.m.g 8
165.d even 2 1 1650.2.l.f 8
165.l odd 4 1 330.2.l.d yes 8
165.l odd 4 1 1650.2.l.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.l.c ✓ 8 15.e even 4 1
330.2.l.c ✓ 8 33.d even 2 1
330.2.l.d yes 8 3.b odd 2 1
330.2.l.d yes 8 165.l odd 4 1
990.2.m.f 8 5.c odd 4 1
990.2.m.f 8 11.b odd 2 1
990.2.m.g 8 1.a even 1 1 trivial
990.2.m.g 8 55.e even 4 1 inner
1650.2.l.c 8 15.d odd 2 1
1650.2.l.c 8 165.l odd 4 1
1650.2.l.f 8 15.e even 4 1
1650.2.l.f 8 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(990, [\chi])\):

\( T_{7}^{8} - 8T_{7}^{7} + 32T_{7}^{6} - 48T_{7}^{5} + 12T_{7}^{4} + 80T_{7}^{3} + 128T_{7}^{2} + 32T_{7} + 4 \) Copy content Toggle raw display
\( T_{13}^{8} - 8T_{13}^{7} + 32T_{13}^{6} + 48T_{13}^{5} + 492T_{13}^{4} - 2608T_{13}^{3} + 6272T_{13}^{2} - 5152T_{13} + 2116 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 8 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{8} - 20 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots + 2116 \) Copy content Toggle raw display
$17$ \( T^{8} + 8 T^{7} + \cdots + 26896 \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 4)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + \cdots + 56644 \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 40 T^{2} + \cdots - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 32 T^{2} + 224)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 136 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$43$ \( T^{8} - 16 T^{7} + \cdots + 614656 \) Copy content Toggle raw display
$47$ \( T^{8} - 8 T^{7} + \cdots + 315844 \) Copy content Toggle raw display
$53$ \( T^{8} - 24 T^{7} + \cdots + 122500 \) Copy content Toggle raw display
$59$ \( T^{8} + 256 T^{6} + \cdots + 2458624 \) Copy content Toggle raw display
$61$ \( T^{8} + 392 T^{6} + \cdots + 19909444 \) Copy content Toggle raw display
$67$ \( T^{8} + 24 T^{7} + \cdots + 8464 \) Copy content Toggle raw display
$71$ \( (T^{4} + 16 T^{3} + \cdots + 4850)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 32 T^{7} + \cdots + 50176 \) Copy content Toggle raw display
$79$ \( (T^{4} + 16 T^{3} + \cdots - 158)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 16 T^{7} + \cdots + 4129024 \) Copy content Toggle raw display
$89$ \( T^{8} + 272 T^{6} + \cdots + 160000 \) Copy content Toggle raw display
$97$ \( T^{8} + 512 T^{5} + \cdots + 3211264 \) Copy content Toggle raw display
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