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Properties

Label 90.11.g.f
Level $90$
Weight $11$
Character orbit 90.g
Analytic conductor $57.182$
Analytic rank $0$
Dimension $10$
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] = [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 = [10,160,0,0,1206] 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: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 176578 x^{8} + 10882598715 x^{6} + 272310626037682 x^{4} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{13}\cdot 3^{12}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (16 \beta_1 + 16) q^{2} + 512 \beta_1 q^{4} + (\beta_{2} - 105 \beta_1 + 121) q^{5} + ( - \beta_{8} - 1698 \beta_1 - 1698) q^{7} + (8192 \beta_1 - 8192) q^{8} + (16 \beta_{3} + 16 \beta_{2} + \cdots + 3616) q^{10}+ \cdots + (2080 \beta_{9} - 59840 \beta_{7} + \cdots - 2008546864) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 160 q^{2} + 1206 q^{5} - 16984 q^{7} - 81920 q^{8} + 36064 q^{10} + 422072 q^{11} + 530634 q^{13} - 2621440 q^{16} + 2551610 q^{17} + 536576 q^{20} + 6753152 q^{22} - 7948600 q^{23} + 10878866 q^{25}+ \cdots - 20080812640 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 176578 x^{8} + 10882598715 x^{6} + 272310626037682 x^{4} + \cdots + 57\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 1934789351 \nu^{9} - 432764584778918 \nu^{7} + \cdots - 10\!\cdots\!89 \nu ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 15\!\cdots\!62 \nu^{9} + \cdots - 78\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 47\!\cdots\!59 \nu^{9} + \cdots + 13\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47\!\cdots\!59 \nu^{9} + \cdots + 12\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 44\!\cdots\!72 \nu^{9} + \cdots - 18\!\cdots\!50 ) / 32\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19\!\cdots\!21 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22\!\cdots\!26 \nu^{9} + \cdots - 92\!\cdots\!00 ) / 32\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22\!\cdots\!26 \nu^{9} + \cdots - 92\!\cdots\!00 ) / 32\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 57\!\cdots\!47 \nu^{9} + \cdots - 78\!\cdots\!00 ) / 65\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -5\beta_{9} + 30\beta_{8} - 30\beta_{7} - 7\beta_{6} + \beta_{5} - 24\beta_{3} - 13\beta_{2} + 2130\beta _1 - 15 ) / 5400 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 165 \beta_{8} + 165 \beta_{7} + 5798 \beta_{6} + 1526 \beta_{5} + 215 \beta_{4} - 26459 \beta_{3} + \cdots - 190714485 ) / 5400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 142405 \beta_{9} - 826305 \beta_{8} + 826305 \beta_{7} - 46 \beta_{6} + 346018 \beta_{5} + \cdots - 346110 ) / 2700 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 96175875 \beta_{8} - 96175875 \beta_{7} - 410468612 \beta_{6} - 104557184 \beta_{5} + \cdots + 10168557529935 ) / 5400 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 17450351405 \beta_{9} + 99458332680 \beta_{8} - 99458332680 \beta_{7} + 11055721343 \beta_{6} + \cdots + 85652299335 ) / 5400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2700630418935 \beta_{8} + 2700630418935 \beta_{7} + 6823206147962 \beta_{6} + 1580743818194 \beta_{5} + \cdots - 15\!\cdots\!40 ) / 1350 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 11\!\cdots\!35 \beta_{9} + \cdots - 72\!\cdots\!45 ) / 5400 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 94\!\cdots\!25 \beta_{8} + \cdots + 37\!\cdots\!85 ) / 5400 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 36\!\cdots\!15 \beta_{9} + \cdots + 28\!\cdots\!30 ) / 2700 \) 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
− 64.5902i
209.777i
− 95.6623i
− 266.784i
219.260i
64.5902i
− 209.777i
95.6623i
266.784i
− 219.260i
16.0000 + 16.0000i 0 512.000i −2728.25 + 1523.90i 0 12126.1 + 12126.1i −8192.00 + 8192.00i 0 −68034.4 − 19269.7i
37.2 16.0000 + 16.0000i 0 512.000i −2447.61 − 1942.89i 0 −20354.5 − 20354.5i −8192.00 + 8192.00i 0 −8075.58 − 70248.0i
37.3 16.0000 + 16.0000i 0 512.000i 782.788 + 3025.37i 0 −3235.29 − 3235.29i −8192.00 + 8192.00i 0 −35881.3 + 60930.5i
37.4 16.0000 + 16.0000i 0 512.000i 1947.38 − 2444.04i 0 16380.8 + 16380.8i −8192.00 + 8192.00i 0 70262.7 − 7946.61i
37.5 16.0000 + 16.0000i 0 512.000i 3048.70 − 686.336i 0 −13409.1 − 13409.1i −8192.00 + 8192.00i 0 59760.6 + 37797.8i
73.1 16.0000 − 16.0000i 0 − 512.000i −2728.25 − 1523.90i 0 12126.1 − 12126.1i −8192.00 − 8192.00i 0 −68034.4 + 19269.7i
73.2 16.0000 − 16.0000i 0 − 512.000i −2447.61 + 1942.89i 0 −20354.5 + 20354.5i −8192.00 − 8192.00i 0 −8075.58 + 70248.0i
73.3 16.0000 − 16.0000i 0 − 512.000i 782.788 − 3025.37i 0 −3235.29 + 3235.29i −8192.00 − 8192.00i 0 −35881.3 − 60930.5i
73.4 16.0000 − 16.0000i 0 − 512.000i 1947.38 + 2444.04i 0 16380.8 − 16380.8i −8192.00 − 8192.00i 0 70262.7 + 7946.61i
73.5 16.0000 − 16.0000i 0 − 512.000i 3048.70 + 686.336i 0 −13409.1 + 13409.1i −8192.00 − 8192.00i 0 59760.6 − 37797.8i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 37.5
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.f yes 10
3.b odd 2 1 90.11.g.e ✓ 10
5.c odd 4 1 inner 90.11.g.f yes 10
15.e even 4 1 90.11.g.e ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.11.g.e ✓ 10 3.b odd 2 1
90.11.g.e ✓ 10 15.e even 4 1
90.11.g.f yes 10 1.a even 1 1 trivial
90.11.g.f yes 10 5.c odd 4 1 inner

Hecke kernels

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

\( T_{7}^{10} + 16984 T_{7}^{9} + 144228128 T_{7}^{8} - 5449301510000 T_{7}^{7} + \cdots + 98\!\cdots\!68 \) Copy content Toggle raw display
\( T_{11}^{5} - 211036 T_{11}^{4} - 14714347892 T_{11}^{3} + \cdots - 22\!\cdots\!76 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 32 T + 512)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 88\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 98\!\cdots\!68 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 22\!\cdots\!76)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 27\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 39\!\cdots\!32 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 92\!\cdots\!24 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 11\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 12\!\cdots\!24)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 14\!\cdots\!68 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 83\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 11\!\cdots\!32 \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 18\!\cdots\!68 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 27\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 46\!\cdots\!32)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 42\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 21\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 52\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 10\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 32\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 12\!\cdots\!68 \) Copy content Toggle raw display
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