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Properties

Label 90.11.b.a
Level $90$
Weight $11$
Character orbit 90.b
Analytic conductor $57.182$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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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(89,90)] 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("90.89"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 4 x^{19} - 369578 x^{18} - 662660 x^{17} + 57311021974 x^{16} + 660003027888 x^{15} + \cdots + 27\!\cdots\!02 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{64}\cdot 3^{52}\cdot 5^{20} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 512 q^{4} + (\beta_{4} + 8 \beta_1) q^{5} + \beta_{5} q^{7} + 512 \beta_1 q^{8} + ( - \beta_{9} + \beta_{6} + \cdots + 4112) q^{10} + (\beta_{11} - 3 \beta_{8} + \cdots - 90 \beta_{3}) q^{11}+ \cdots + (3312 \beta_{19} + \cdots - 2063245 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10240 q^{4} + 82240 q^{10} + 5242880 q^{16} - 6418080 q^{19} - 27678000 q^{25} + 103749680 q^{31} - 23308160 q^{34} + 42106880 q^{40} - 659422720 q^{46} - 41560140 q^{49} + 3032188480 q^{55} - 3937749800 q^{61}+ \cdots - 23966195200 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} - 369578 x^{18} - 662660 x^{17} + 57311021974 x^{16} + 660003027888 x^{15} + \cdots + 27\!\cdots\!02 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 25\!\cdots\!85 \nu^{19} + \cdots - 99\!\cdots\!28 ) / 21\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 56\!\cdots\!40 \nu^{19} + \cdots - 48\!\cdots\!00 ) / 16\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 61\!\cdots\!21 \nu^{19} + \cdots + 23\!\cdots\!24 ) / 33\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 47\!\cdots\!27 \nu^{19} + \cdots - 28\!\cdots\!24 ) / 68\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11\!\cdots\!56 \nu^{19} + \cdots - 32\!\cdots\!48 ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 18\!\cdots\!60 \nu^{19} + \cdots - 62\!\cdots\!94 ) / 17\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 29\!\cdots\!52 \nu^{19} + \cdots - 97\!\cdots\!88 ) / 26\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 52\!\cdots\!46 \nu^{19} + \cdots - 39\!\cdots\!78 ) / 34\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 60\!\cdots\!64 \nu^{19} + \cdots - 21\!\cdots\!92 ) / 13\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 59\!\cdots\!40 \nu^{19} + \cdots - 22\!\cdots\!88 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 12\!\cdots\!41 \nu^{19} + \cdots + 14\!\cdots\!00 ) / 12\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 84\!\cdots\!56 \nu^{19} + \cdots - 30\!\cdots\!52 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 93\!\cdots\!44 \nu^{19} + \cdots - 33\!\cdots\!44 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 17\!\cdots\!20 \nu^{19} + \cdots + 64\!\cdots\!64 ) / 76\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25\!\cdots\!49 \nu^{19} + \cdots + 50\!\cdots\!26 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 34\!\cdots\!14 \nu^{19} + \cdots + 13\!\cdots\!00 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 14\!\cdots\!42 \nu^{19} + \cdots - 95\!\cdots\!78 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 21\!\cdots\!14 \nu^{19} + \cdots - 87\!\cdots\!94 ) / 68\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 41\!\cdots\!42 \nu^{19} + \cdots + 16\!\cdots\!94 ) / 68\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 162\beta_{9} - 810\beta_{6} + 162\beta_{5} + 400\beta_{3} + 162\beta_{2} + 6075\beta _1 + 38880 ) / 194400 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 24 \beta_{19} + 24 \beta_{18} - 90 \beta_{16} + 3024 \beta_{14} + 12960 \beta_{13} + \cdots + 7184751840 ) / 194400 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 864 \beta_{19} - 5616 \beta_{18} - 6480 \beta_{17} - 29160 \beta_{16} + 12960 \beta_{15} + \cdots + 62431365600 ) / 194400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 138236 \beta_{19} + 390476 \beta_{18} - 122160 \beta_{17} - 512625 \beta_{16} + \cdots + 35612430904800 ) / 16200 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 70518240 \beta_{19} - 707534640 \beta_{18} - 872888400 \beta_{17} - 3374233200 \beta_{16} + \cdots - 47\!\cdots\!20 ) / 194400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 121080170040 \beta_{19} + 534068634840 \beta_{18} - 209947240800 \beta_{17} - 471726209250 \beta_{16} + \cdots + 28\!\cdots\!40 ) / 194400 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 16848179836992 \beta_{19} - 73419612254352 \beta_{18} - 91289526446160 \beta_{17} + \cdots - 83\!\cdots\!00 ) / 194400 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 401511290427492 \beta_{19} + \cdots + 82\!\cdots\!00 ) / 8100 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 19\!\cdots\!32 \beta_{19} + \cdots - 86\!\cdots\!80 ) / 194400 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 80\!\cdots\!36 \beta_{19} + \cdots + 13\!\cdots\!40 ) / 194400 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 19\!\cdots\!48 \beta_{19} + \cdots - 77\!\cdots\!00 ) / 194400 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 56\!\cdots\!00 \beta_{19} + \cdots + 82\!\cdots\!00 ) / 16200 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 17\!\cdots\!32 \beta_{19} + \cdots - 65\!\cdots\!00 ) / 194400 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 57\!\cdots\!00 \beta_{19} + \cdots + 70\!\cdots\!80 ) / 194400 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 14\!\cdots\!64 \beta_{19} + \cdots - 53\!\cdots\!00 ) / 194400 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 98\!\cdots\!84 \beta_{19} + \cdots + 10\!\cdots\!00 ) / 4050 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 12\!\cdots\!48 \beta_{19} + \cdots - 43\!\cdots\!00 ) / 194400 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 38\!\cdots\!00 \beta_{19} + \cdots + 35\!\cdots\!20 ) / 194400 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 10\!\cdots\!76 \beta_{19} + \cdots - 34\!\cdots\!00 ) / 194400 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1
−271.650 + 0.707107i
−271.650 − 0.707107i
−103.306 − 0.707107i
−103.306 + 0.707107i
−58.8011 + 0.707107i
−58.8011 − 0.707107i
171.484 − 0.707107i
171.484 + 0.707107i
259.738 + 0.707107i
259.738 − 0.707107i
261.152 + 0.707107i
261.152 − 0.707107i
172.898 − 0.707107i
172.898 + 0.707107i
−57.3869 + 0.707107i
−57.3869 − 0.707107i
−101.892 − 0.707107i
−101.892 + 0.707107i
−270.236 + 0.707107i
−270.236 − 0.707107i
−22.6274 0 512.000 −3057.63 − 645.365i 0 − 3570.42i −11585.2 0 69186.4 + 14602.9i
89.2 −22.6274 0 512.000 −3057.63 + 645.365i 0 3570.42i −11585.2 0 69186.4 − 14602.9i
89.3 −22.6274 0 512.000 −1272.07 − 2854.37i 0 − 32441.9i −11585.2 0 28783.7 + 64587.1i
89.4 −22.6274 0 512.000 −1272.07 + 2854.37i 0 32441.9i −11585.2 0 28783.7 − 64587.1i
89.5 −22.6274 0 512.000 −800.027 − 3020.86i 0 3679.45i −11585.2 0 18102.6 + 68354.2i
89.6 −22.6274 0 512.000 −800.027 + 3020.86i 0 − 3679.45i −11585.2 0 18102.6 − 68354.2i
89.7 −22.6274 0 512.000 1642.51 − 2658.53i 0 13965.4i −11585.2 0 −37165.9 + 60155.6i
89.8 −22.6274 0 512.000 1642.51 + 2658.53i 0 − 13965.4i −11585.2 0 −37165.9 − 60155.6i
89.9 −22.6274 0 512.000 2578.59 − 1765.36i 0 − 12205.5i −11585.2 0 −58346.8 + 39945.6i
89.10 −22.6274 0 512.000 2578.59 + 1765.36i 0 12205.5i −11585.2 0 −58346.8 − 39945.6i
89.11 22.6274 0 512.000 −2578.59 − 1765.36i 0 12205.5i 11585.2 0 −58346.8 − 39945.6i
89.12 22.6274 0 512.000 −2578.59 + 1765.36i 0 − 12205.5i 11585.2 0 −58346.8 + 39945.6i
89.13 22.6274 0 512.000 −1642.51 − 2658.53i 0 − 13965.4i 11585.2 0 −37165.9 − 60155.6i
89.14 22.6274 0 512.000 −1642.51 + 2658.53i 0 13965.4i 11585.2 0 −37165.9 + 60155.6i
89.15 22.6274 0 512.000 800.027 − 3020.86i 0 − 3679.45i 11585.2 0 18102.6 − 68354.2i
89.16 22.6274 0 512.000 800.027 + 3020.86i 0 3679.45i 11585.2 0 18102.6 + 68354.2i
89.17 22.6274 0 512.000 1272.07 − 2854.37i 0 32441.9i 11585.2 0 28783.7 − 64587.1i
89.18 22.6274 0 512.000 1272.07 + 2854.37i 0 − 32441.9i 11585.2 0 28783.7 + 64587.1i
89.19 22.6274 0 512.000 3057.63 − 645.365i 0 3570.42i 11585.2 0 69186.4 − 14602.9i
89.20 22.6274 0 512.000 3057.63 + 645.365i 0 − 3570.42i 11585.2 0 69186.4 + 14602.9i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 89.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.11.b.a ✓ 20
3.b odd 2 1 inner 90.11.b.a ✓ 20
5.b even 2 1 inner 90.11.b.a ✓ 20
15.d odd 2 1 inner 90.11.b.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.11.b.a ✓ 20 1.a even 1 1 trivial
90.11.b.a ✓ 20 3.b odd 2 1 inner
90.11.b.a ✓ 20 5.b even 2 1 inner
90.11.b.a ✓ 20 15.d odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(90, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 512)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 78\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 52\!\cdots\!76)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 22\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 59\!\cdots\!24)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 96\!\cdots\!32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 42\!\cdots\!76)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 20\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 31\!\cdots\!24)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 69\!\cdots\!76)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 51\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 74\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 13\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 34\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots + 11\!\cdots\!00)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 39\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 72\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 32\!\cdots\!76)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 18\!\cdots\!76)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 30\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 28\!\cdots\!76)^{2} \) Copy content Toggle raw display
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