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Properties

Label 90.12.c.d
Level $90$
Weight $12$
Character orbit 90.c
Analytic conductor $69.151$
Analytic rank $0$
Dimension $12$
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] = [90,12,Mod(19,90)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 12, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("90.19"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 90.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-12288,0] 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: \(69.1508862504\)
Analytic rank: \(0\)
Dimension: \(12\)
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: \( x^{12} + 2686231540547x^{8} + 128219731460991388255453x^{4} + 14060999354420335522970873124 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{48}\cdot 3^{16}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 1024 q^{4} + (\beta_{2} - 5 \beta_1) q^{5} + \beta_{5} q^{7} - 1024 \beta_1 q^{8} + (\beta_{5} - \beta_{3} + 4704) q^{10} + ( - \beta_{7} + \beta_{6} + \cdots + 8 \beta_1) q^{11} + (\beta_{10} + 2 \beta_{5} + \beta_{3}) q^{13}+ \cdots + (139840 \beta_{6} + \cdots - 516674313 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12288 q^{4} + 56448 q^{10} + 12582912 q^{16} - 489648 q^{19} - 125575548 q^{25} + 47933952 q^{31} + 341743872 q^{34} - 57802752 q^{40} + 3346675200 q^{46} - 6178917036 q^{49} - 11944070688 q^{55}+ \cdots - 10041254400 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2686231540547x^{8} + 128219731460991388255453x^{4} + 14060999354420335522970873124 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 29518192 \nu^{10} + \cdots - 37\!\cdots\!72 \nu^{2} ) / 39\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 41\!\cdots\!93 \nu^{11} + \cdots + 17\!\cdots\!94 \nu ) / 14\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 95\!\cdots\!63 \nu^{11} + \cdots - 79\!\cdots\!60 ) / 40\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!51 \nu^{11} + \cdots - 12\!\cdots\!58 \nu ) / 73\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24\!\cdots\!73 \nu^{11} + \cdots - 10\!\cdots\!66 \nu ) / 36\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 20\!\cdots\!65 \nu^{11} + \cdots - 88\!\cdots\!70 \nu ) / 24\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!81 \nu^{11} + \cdots - 47\!\cdots\!42 \nu ) / 73\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 12\!\cdots\!41 \nu^{11} + \cdots + 21\!\cdots\!20 ) / 36\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 71\!\cdots\!89 \nu^{11} + \cdots - 82\!\cdots\!60 ) / 12\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 96\!\cdots\!60 \nu^{11} + \cdots + 14\!\cdots\!68 ) / 73\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 35\!\cdots\!73 \nu^{11} + \cdots + 14\!\cdots\!06 \nu ) / 24\!\cdots\!50 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 85 \beta_{11} - 69 \beta_{10} - 163 \beta_{8} + 65 \beta_{7} + 758 \beta_{6} + 247 \beta_{5} + \cdots + 6149 \beta_1 ) / 345600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2224\beta_{6} - 4000\beta_{4} + 10720\beta_{2} - 830203283\beta_1 ) / 43200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 53320960 \beta_{11} - 34577061 \beta_{10} - 141821107 \beta_{8} + 59744560 \beta_{7} + \cdots - 4844926976 \beta_1 ) / 172800 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 792837627 \beta_{10} - 2723877200 \beta_{9} - 2621768211 \beta_{8} - 2621768211 \beta_{5} + \cdots - 15\!\cdots\!20 ) / 17280 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 167045997828715 \beta_{11} + 103056992328741 \beta_{10} + 464548311591907 \beta_{8} + \cdots - 15\!\cdots\!11 \beta_1 ) / 345600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 22\!\cdots\!08 \beta_{6} + \cdots + 96\!\cdots\!61 \beta_1 ) / 21600 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26\!\cdots\!55 \beta_{11} + \cdots + 25\!\cdots\!03 \beta_1 ) / 345600 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 20\!\cdots\!61 \beta_{10} + \cdots + 40\!\cdots\!80 ) / 17280 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 21\!\cdots\!80 \beta_{11} + \cdots + 20\!\cdots\!52 \beta_1 ) / 172800 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12\!\cdots\!44 \beta_{6} + \cdots - 50\!\cdots\!23 \beta_1 ) / 43200 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 71\!\cdots\!45 \beta_{11} + \cdots - 67\!\cdots\!17 \beta_1 ) / 345600 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
901.131 + 901.131i
−12.8677 − 12.8677i
332.025 + 332.025i
−332.025 − 332.025i
12.8677 + 12.8677i
−901.131 − 901.131i
901.131 − 901.131i
−12.8677 + 12.8677i
332.025 − 332.025i
−332.025 + 332.025i
12.8677 − 12.8677i
−901.131 + 901.131i
− 32.0000i 0 −1024.00 −6895.58 + 1130.97i 0 − 14383.7i 32768.0i 0 36191.1 + 220659.i
19.2 − 32.0000i 0 −1024.00 −3091.89 + 6266.44i 0 73452.1i 32768.0i 0 200526. + 98940.6i
19.3 − 32.0000i 0 −1024.00 −660.610 − 6956.42i 0 43296.8i 32768.0i 0 −222605. + 21139.5i
19.4 − 32.0000i 0 −1024.00 660.610 − 6956.42i 0 − 43296.8i 32768.0i 0 −222605. − 21139.5i
19.5 − 32.0000i 0 −1024.00 3091.89 + 6266.44i 0 − 73452.1i 32768.0i 0 200526. − 98940.6i
19.6 − 32.0000i 0 −1024.00 6895.58 + 1130.97i 0 14383.7i 32768.0i 0 36191.1 − 220659.i
19.7 32.0000i 0 −1024.00 −6895.58 − 1130.97i 0 14383.7i − 32768.0i 0 36191.1 − 220659.i
19.8 32.0000i 0 −1024.00 −3091.89 − 6266.44i 0 − 73452.1i − 32768.0i 0 200526. − 98940.6i
19.9 32.0000i 0 −1024.00 −660.610 + 6956.42i 0 − 43296.8i − 32768.0i 0 −222605. − 21139.5i
19.10 32.0000i 0 −1024.00 660.610 + 6956.42i 0 43296.8i − 32768.0i 0 −222605. + 21139.5i
19.11 32.0000i 0 −1024.00 3091.89 − 6266.44i 0 73452.1i − 32768.0i 0 200526. + 98940.6i
19.12 32.0000i 0 −1024.00 6895.58 − 1130.97i 0 − 14383.7i − 32768.0i 0 36191.1 + 220659.i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
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 90.12.c.d ✓ 12
3.b odd 2 1 inner 90.12.c.d ✓ 12
5.b even 2 1 inner 90.12.c.d ✓ 12
15.d odd 2 1 inner 90.12.c.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.12.c.d ✓ 12 1.a even 1 1 trivial
90.12.c.d ✓ 12 3.b odd 2 1 inner
90.12.c.d ✓ 12 5.b even 2 1 inner
90.12.c.d ✓ 12 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} + 7476709488T_{7}^{4} + 11617977439224834048T_{7}^{2} + 2092470522763866018758311936 \) acting on \(S_{12}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 1024)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 20\!\cdots\!36)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 49\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 14\!\cdots\!44)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + \cdots - 34\!\cdots\!36)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 40\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 20\!\cdots\!36)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 32\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 64\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 37\!\cdots\!44)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 99\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 63\!\cdots\!08)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 66\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots - 38\!\cdots\!36)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 15\!\cdots\!16)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 47\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 41\!\cdots\!36)^{2} \) Copy content Toggle raw display
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