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Properties

Label 90.11.g.g
Level $90$
Weight $11$
Character orbit 90.g
Analytic conductor $57.182$
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] = [90,11,Mod(37,90)] 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("90.37"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-192,0,0,5880] 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: \(57.1821527406\)
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: \( x^{12} - 1321585038 x^{10} - 4746832718600 x^{9} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{17}\cdot 3^{28}\cdot 5^{8} \)
Twist minimal: no (minimal twist has level 30)
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.

\(f(q)\) \(=\) \( q + ( - 16 \beta_1 - 16) q^{2} + 512 \beta_1 q^{4} + (\beta_{6} - 3 \beta_1 + 490) q^{5} + ( - \beta_{7} + 2 \beta_{6} + \cdots + 946) q^{7} + ( - 8192 \beta_1 + 8192) q^{8} + ( - 16 \beta_{6} + 16 \beta_{2} + \cdots - 7888) q^{10}+ \cdots + (84800 \beta_{11} + 212544 \beta_{10} + \cdots + 2557486896) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 192 q^{2} + 5880 q^{5} + 11352 q^{7} + 98304 q^{8} - 94656 q^{10} - 406032 q^{11} - 112404 q^{13} - 3145728 q^{16} + 4207428 q^{17} + 18432 q^{20} + 6496512 q^{22} - 22928832 q^{23} - 28197828 q^{25}+ \cdots + 30689842752 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 1321585038 x^{10} - 4746832718600 x^{9} + \cdots + 30\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 17\!\cdots\!33 \nu^{11} + \cdots - 62\!\cdots\!00 ) / 82\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27\!\cdots\!53 \nu^{11} + \cdots + 13\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 37\!\cdots\!03 \nu^{11} + \cdots + 19\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 39\!\cdots\!97 \nu^{11} + \cdots - 20\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46\!\cdots\!93 \nu^{11} + \cdots + 15\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!17 \nu^{11} + \cdots - 54\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 29\!\cdots\!41 \nu^{11} + \cdots + 10\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!39 \nu^{11} + \cdots + 13\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 55\!\cdots\!21 \nu^{11} + \cdots + 21\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 75\!\cdots\!67 \nu^{11} + \cdots - 26\!\cdots\!00 ) / 32\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 81\!\cdots\!69 \nu^{11} + \cdots - 28\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} + \beta_{7} - 3\beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} - 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2038 \beta_{11} + 6905 \beta_{10} - 4339 \beta_{9} + 1775 \beta_{8} + 10163 \beta_{7} + \cdots + 440528346 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7229430 \beta_{11} + 379973653 \beta_{10} + 17700657 \beta_{9} + 39462681 \beta_{8} + \cdots + 2373416359300 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 714949313314 \beta_{11} + 2889119062463 \beta_{10} - 1281631322129 \beta_{9} + 862193442613 \beta_{8} + \cdots + 15\!\cdots\!66 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 47\!\cdots\!50 \beta_{11} + \cdots + 11\!\cdots\!00 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 28\!\cdots\!26 \beta_{11} + \cdots + 60\!\cdots\!86 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 22\!\cdots\!70 \beta_{11} + \cdots + 50\!\cdots\!00 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11\!\cdots\!58 \beta_{11} + \cdots + 23\!\cdots\!06 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10\!\cdots\!90 \beta_{11} + \cdots + 21\!\cdots\!00 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 47\!\cdots\!10 \beta_{11} + \cdots + 95\!\cdots\!26 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 48\!\cdots\!10 \beta_{11} + \cdots + 93\!\cdots\!00 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
18817.8 − 1.00000i
−8899.16 − 1.00000i
−3112.91 − 1.00000i
20839.9 − 1.00000i
−8248.68 − 1.00000i
−19397.0 − 1.00000i
18817.8 + 1.00000i
−8899.16 + 1.00000i
−3112.91 + 1.00000i
20839.9 + 1.00000i
−8248.68 + 1.00000i
−19397.0 + 1.00000i
−16.0000 − 16.0000i 0 512.000i −2015.56 + 2388.12i 0 −17871.8 − 17871.8i 8192.00 − 8192.00i 0 70459.0 − 5960.93i
37.2 −16.0000 − 16.0000i 0 512.000i −1477.10 + 2753.87i 0 9845.16 + 9845.16i 8192.00 − 8192.00i 0 67695.6 − 20428.2i
37.3 −16.0000 − 16.0000i 0 512.000i −124.618 − 3122.51i 0 4058.91 + 4058.91i 8192.00 − 8192.00i 0 −47966.3 + 51954.1i
37.4 −16.0000 − 16.0000i 0 512.000i 1172.83 − 2896.57i 0 −19893.9 − 19893.9i 8192.00 − 8192.00i 0 −65110.3 + 27579.8i
37.5 −16.0000 − 16.0000i 0 512.000i 2451.57 + 1937.89i 0 9194.68 + 9194.68i 8192.00 − 8192.00i 0 −8218.93 − 70231.4i
37.6 −16.0000 − 16.0000i 0 512.000i 2932.89 − 1078.80i 0 20343.0 + 20343.0i 8192.00 − 8192.00i 0 −64186.9 − 29665.4i
73.1 −16.0000 + 16.0000i 0 − 512.000i −2015.56 − 2388.12i 0 −17871.8 + 17871.8i 8192.00 + 8192.00i 0 70459.0 + 5960.93i
73.2 −16.0000 + 16.0000i 0 − 512.000i −1477.10 − 2753.87i 0 9845.16 − 9845.16i 8192.00 + 8192.00i 0 67695.6 + 20428.2i
73.3 −16.0000 + 16.0000i 0 − 512.000i −124.618 + 3122.51i 0 4058.91 − 4058.91i 8192.00 + 8192.00i 0 −47966.3 − 51954.1i
73.4 −16.0000 + 16.0000i 0 − 512.000i 1172.83 + 2896.57i 0 −19893.9 + 19893.9i 8192.00 + 8192.00i 0 −65110.3 − 27579.8i
73.5 −16.0000 + 16.0000i 0 − 512.000i 2451.57 − 1937.89i 0 9194.68 − 9194.68i 8192.00 + 8192.00i 0 −8218.93 + 70231.4i
73.6 −16.0000 + 16.0000i 0 − 512.000i 2932.89 + 1078.80i 0 20343.0 − 20343.0i 8192.00 + 8192.00i 0 −64186.9 + 29665.4i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.11.g.g 12
3.b odd 2 1 30.11.f.b ✓ 12
5.c odd 4 1 inner 90.11.g.g 12
15.d odd 2 1 150.11.f.h 12
15.e even 4 1 30.11.f.b ✓ 12
15.e even 4 1 150.11.f.h 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.11.f.b ✓ 12 3.b odd 2 1
30.11.f.b ✓ 12 15.e even 4 1
90.11.g.g 12 1.a even 1 1 trivial
90.11.g.g 12 5.c odd 4 1 inner
150.11.f.h 12 15.d odd 2 1
150.11.f.h 12 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{11}^{\mathrm{new}}(90, [\chi])\):

\( T_{7}^{12} - 11352 T_{7}^{11} + 64433952 T_{7}^{10} - 4729832980784 T_{7}^{9} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
\( T_{11}^{6} + 203016 T_{11}^{5} - 115446592188 T_{11}^{4} + \cdots - 42\!\cdots\!00 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 32 T + 512)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 14\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 40\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 83\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 13\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 19\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 53\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 22\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
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