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Properties

Label 990.2.k.c
Level $990$
Weight $2$
Character orbit 990.k
Analytic conductor $7.905$
Analytic rank $0$
Dimension $12$
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(287,990)] 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([2, 1, 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("990.287"); 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.k (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,-12,0,0,-12,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.90518980011\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 32x^{9} + 9x^{8} - 16x^{7} + 464x^{6} - 80x^{5} + 225x^{4} - 4000x^{3} + 15625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} - \beta_{2} q^{4} + \beta_{11} q^{5} + (\beta_{11} - \beta_{3}) q^{7} + \beta_{5} q^{8} - \beta_{4} q^{10} - \beta_{2} q^{11} + ( - \beta_{11} - 2 \beta_{6} + \cdots - 1) q^{13} + (\beta_{7} - \beta_{4}) q^{14}+ \cdots + ( - \beta_{10} - \beta_{9} + 2 \beta_{8} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{13} - 12 q^{16} - 8 q^{17} + 8 q^{23} - 16 q^{31} - 4 q^{37} + 8 q^{38} + 4 q^{40} + 8 q^{43} - 12 q^{44} + 16 q^{46} + 24 q^{47} + 8 q^{50} + 12 q^{52} - 16 q^{53} + 4 q^{58} - 32 q^{59} + 16 q^{61}+ \cdots + 24 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 32x^{9} + 9x^{8} - 16x^{7} + 464x^{6} - 80x^{5} + 225x^{4} - 4000x^{3} + 15625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 12 \nu^{11} + 3985 \nu^{10} - 9900 \nu^{9} + 8009 \nu^{8} - 95128 \nu^{7} + 237232 \nu^{6} + \cdots + 11890625 ) / 256250 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{11} + 32\nu^{8} - 9\nu^{7} + 16\nu^{6} - 464\nu^{5} + 80\nu^{4} - 225\nu^{3} + 4000\nu^{2} ) / 3125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 257 \nu^{11} + 1045 \nu^{10} - 4250 \nu^{9} - 1974 \nu^{8} - 26752 \nu^{7} + 95668 \nu^{6} + \cdots + 5318750 ) / 51250 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1702 \nu^{11} - 1285 \nu^{10} - 5225 \nu^{9} - 33214 \nu^{8} + 25188 \nu^{7} + 106528 \nu^{6} + \cdots + 10293750 ) / 256250 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2029 \nu^{11} - 3225 \nu^{10} - 1950 \nu^{9} - 45178 \nu^{8} + 75836 \nu^{7} + 22786 \nu^{6} + \cdots + 8156250 ) / 256250 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 522 \nu^{11} + 2029 \nu^{10} - 3225 \nu^{9} + 14754 \nu^{8} - 49876 \nu^{7} + 84188 \nu^{6} + \cdots + 2448750 ) / 51250 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 129 \nu^{11} + 78 \nu^{10} - 790 \nu^{9} - 2303 \nu^{8} - 2210 \nu^{7} + 17668 \nu^{6} + \cdots + 1268125 ) / 10250 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3294 \nu^{11} + 8510 \nu^{10} - 6425 \nu^{9} + 79283 \nu^{8} - 195716 \nu^{7} + 178644 \nu^{6} + \cdots - 1865625 ) / 256250 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 761 \nu^{11} + 12 \nu^{10} - 3985 \nu^{9} - 14452 \nu^{8} - 1160 \nu^{7} + 82952 \nu^{6} + \cdots + 6522500 ) / 51250 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 797 \nu^{11} + 1980 \nu^{10} - 1525 \nu^{9} + 19004 \nu^{8} - 47408 \nu^{7} + 46872 \nu^{6} + \cdots - 37500 ) / 51250 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - 2\beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{11} + 2\beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{4} - 4\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3 \beta_{11} - 2 \beta_{10} + 4 \beta_{9} + \beta_{8} - 3 \beta_{7} + 4 \beta_{6} + 8 \beta_{5} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3 \beta_{11} + 12 \beta_{10} + \beta_{9} - 7 \beta_{8} - 2 \beta_{7} - 24 \beta_{6} + 32 \beta_{5} + \cdots + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 22 \beta_{11} + \beta_{10} + 21 \beta_{9} + 21 \beta_{8} + 28 \beta_{7} + 3 \beta_{6} + 6 \beta_{5} + \cdots + 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 5 \beta_{11} - 33 \beta_{10} + 75 \beta_{9} + 19 \beta_{8} - 15 \beta_{7} + 96 \beta_{6} + 72 \beta_{5} + \cdots - 140 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 3 \beta_{11} + 13 \beta_{10} + 53 \beta_{9} - 101 \beta_{8} - 21 \beta_{7} - 76 \beta_{6} + 388 \beta_{5} + \cdots + 167 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 219 \beta_{11} - 153 \beta_{10} + 49 \beta_{9} + 73 \beta_{8} + 277 \beta_{7} + 216 \beta_{6} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 441 \beta_{11} - 170 \beta_{10} + 531 \beta_{9} + 3 \beta_{8} + 83 \beta_{7} + 630 \beta_{6} + \cdots - 2136 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1228 \beta_{11} - 999 \beta_{10} + 763 \beta_{9} - 189 \beta_{8} + 374 \beta_{7} + 208 \beta_{6} + \cdots + 3012 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(397\) \(541\) \(551\)
\(\chi(n)\) \(\beta_{2}\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−1.35386 − 1.77963i
2.22132 + 0.256421i
−0.160352 + 2.23031i
−1.46707 − 1.68751i
2.07793 − 0.825965i
−1.31796 + 1.80637i
−1.35386 + 1.77963i
2.22132 − 0.256421i
−0.160352 − 2.23031i
−1.46707 + 1.68751i
2.07793 + 0.825965i
−1.31796 − 1.80637i
−0.707107 + 0.707107i 0 − 1.00000i −1.77963 + 1.35386i 0 −0.425767 − 0.425767i 0.707107 + 0.707107i 0 0.301063 − 2.21571i
287.2 −0.707107 + 0.707107i 0 − 1.00000i 0.256421 − 2.22132i 0 −1.96490 − 1.96490i 0.707107 + 0.707107i 0 1.38939 + 1.75203i
287.3 −0.707107 + 0.707107i 0 − 1.00000i 2.23031 + 0.160352i 0 2.39066 + 2.39066i 0.707107 + 0.707107i 0 −1.69045 + 1.46368i
287.4 0.707107 − 0.707107i 0 − 1.00000i −1.68751 + 1.46707i 0 −0.220441 − 0.220441i −0.707107 − 0.707107i 0 −0.155875 + 2.23063i
287.5 0.707107 − 0.707107i 0 − 1.00000i −0.825965 − 2.07793i 0 −2.90389 − 2.90389i −0.707107 − 0.707107i 0 −2.05336 − 0.885271i
287.6 0.707107 − 0.707107i 0 − 1.00000i 1.80637 + 1.31796i 0 3.12433 + 3.12433i −0.707107 − 0.707107i 0 2.20924 − 0.345357i
683.1 −0.707107 − 0.707107i 0 1.00000i −1.77963 − 1.35386i 0 −0.425767 + 0.425767i 0.707107 − 0.707107i 0 0.301063 + 2.21571i
683.2 −0.707107 − 0.707107i 0 1.00000i 0.256421 + 2.22132i 0 −1.96490 + 1.96490i 0.707107 − 0.707107i 0 1.38939 − 1.75203i
683.3 −0.707107 − 0.707107i 0 1.00000i 2.23031 − 0.160352i 0 2.39066 − 2.39066i 0.707107 − 0.707107i 0 −1.69045 − 1.46368i
683.4 0.707107 + 0.707107i 0 1.00000i −1.68751 − 1.46707i 0 −0.220441 + 0.220441i −0.707107 + 0.707107i 0 −0.155875 − 2.23063i
683.5 0.707107 + 0.707107i 0 1.00000i −0.825965 + 2.07793i 0 −2.90389 + 2.90389i −0.707107 + 0.707107i 0 −2.05336 + 0.885271i
683.6 0.707107 + 0.707107i 0 1.00000i 1.80637 − 1.31796i 0 3.12433 − 3.12433i −0.707107 + 0.707107i 0 2.20924 + 0.345357i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.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.k.c ✓ 12
3.b odd 2 1 990.2.k.d yes 12
5.c odd 4 1 990.2.k.d yes 12
15.e even 4 1 inner 990.2.k.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
990.2.k.c ✓ 12 1.a even 1 1 trivial
990.2.k.c ✓ 12 15.e even 4 1 inner
990.2.k.d yes 12 3.b odd 2 1
990.2.k.d yes 12 5.c odd 4 1

Hecke kernels

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

\( T_{7}^{12} + 16 T_{7}^{9} + 428 T_{7}^{8} + 224 T_{7}^{7} + 128 T_{7}^{6} + 3424 T_{7}^{5} + 33476 T_{7}^{4} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{17}^{12} + 8 T_{17}^{11} + 32 T_{17}^{10} + 80 T_{17}^{9} + 1176 T_{17}^{8} + 9120 T_{17}^{7} + \cdots + 802816 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 16 T^{9} + \cdots + 15625 \) Copy content Toggle raw display
$7$ \( T^{12} + 16 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} + 12 T^{11} + \cdots + 784 \) Copy content Toggle raw display
$17$ \( T^{12} + 8 T^{11} + \cdots + 802816 \) Copy content Toggle raw display
$19$ \( T^{12} + 104 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} + \cdots + 6390784 \) Copy content Toggle raw display
$29$ \( (T^{6} - 42 T^{4} + \cdots - 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 8 T^{5} + \cdots - 20224)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 2651014144 \) Copy content Toggle raw display
$41$ \( T^{12} + 220 T^{10} + \cdots + 153664 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 6795034624 \) Copy content Toggle raw display
$47$ \( T^{12} - 24 T^{11} + \cdots + 16516096 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 19743622144 \) Copy content Toggle raw display
$59$ \( (T^{6} + 16 T^{5} + \cdots - 1792)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 8 T^{5} + \cdots - 37408)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 8 T^{11} + \cdots + 802816 \) Copy content Toggle raw display
$71$ \( T^{12} + 336 T^{10} + \cdots + 8667136 \) Copy content Toggle raw display
$73$ \( T^{12} - 20 T^{11} + \cdots + 802816 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 6466733056 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 59395538944 \) Copy content Toggle raw display
$89$ \( (T^{6} - 278 T^{4} + \cdots + 33208)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2994062672896 \) Copy content Toggle raw display
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