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Properties

Label 90.6.f.c
Level $90$
Weight $6$
Character orbit 90.f
Analytic conductor $14.435$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,6,Mod(17,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.17");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 90.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.4345437832\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 3457x^{8} + 2937456x^{4} + 12960000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q + (2 \beta_{3} + 2 \beta_{2}) q^{2} + 16 \beta_1 q^{4} + (\beta_{8} + 3 \beta_{3} - 5 \beta_{2}) q^{5} + (\beta_{6} + \beta_{5} - 12 \beta_1 + 12) q^{7} + ( - 32 \beta_{3} + 32 \beta_{2}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{3} + 2 \beta_{2}) q^{2} + 16 \beta_1 q^{4} + (\beta_{8} + 3 \beta_{3} - 5 \beta_{2}) q^{5} + (\beta_{6} + \beta_{5} - 12 \beta_1 + 12) q^{7} + ( - 32 \beta_{3} + 32 \beta_{2}) q^{8} + ( - 2 \beta_{6} + 2 \beta_{4} + \cdots + 36) q^{10}+ \cdots + ( - 376 \beta_{11} + \cdots - 9386 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 144 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 144 q^{7} + 432 q^{10} - 276 q^{13} - 3072 q^{16} + 384 q^{22} + 10704 q^{25} + 2304 q^{28} - 58512 q^{31} + 25764 q^{37} + 1536 q^{40} + 16080 q^{43} - 60672 q^{46} + 4416 q^{52} + 206544 q^{55} + 23952 q^{58} - 145200 q^{61} + 33552 q^{67} + 155712 q^{70} - 158988 q^{73} - 86016 q^{76} - 75024 q^{82} + 759432 q^{85} + 6144 q^{88} - 465024 q^{91} - 631116 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 3457x^{8} + 2937456x^{4} + 12960000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 17\nu^{10} + 59489\nu^{6} + 53782272\nu^{2} ) / 112708800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 523\nu^{11} - 750\nu^{9} + 1885411\nu^{7} + 690450\nu^{5} + 1667910888\nu^{3} + 3498055200\nu ) / 5071896000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -523\nu^{11} - 750\nu^{9} - 1885411\nu^{7} + 690450\nu^{5} - 1667910888\nu^{3} + 3498055200\nu ) / 5071896000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 122\nu^{10} + 6615\nu^{8} + 620294\nu^{6} + 14197815\nu^{4} + 686164572\nu^{2} + 2963174400 ) / 42265800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -122\nu^{10} + 1185\nu^{8} - 620294\nu^{6} + 3980985\nu^{4} - 686164572\nu^{2} + 2187426600 ) / 21132900 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -823\nu^{10} - 5660\nu^{8} - 2548471\nu^{6} - 8314460\nu^{4} - 1847551248\nu^{2} + 1882238400 ) / 56354400 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2957\nu^{10} + 33960\nu^{8} - 10126589\nu^{6} + 49886760\nu^{4} - 8287312032\nu^{2} - 11293430400 ) / 169063200 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5683 \nu^{11} + 56025 \nu^{9} - 19113331 \nu^{7} + 151299225 \nu^{5} + \cdots + 90380545200 \nu ) / 2535948000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31069 \nu^{11} + 54150 \nu^{9} + 105295933 \nu^{7} + 153025350 \nu^{5} + 87322518864 \nu^{3} + 61086463200 \nu ) / 5071896000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 19703 \nu^{11} - 112050 \nu^{9} + 67069271 \nu^{7} - 302598450 \nu^{5} + \cdots - 180761090400 \nu ) / 2535948000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5091 \nu^{11} + 9650 \nu^{9} + 17235087 \nu^{7} + 24928850 \nu^{5} + 14275767996 \nu^{3} + 19945771200 \nu ) / 422658000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - 2\beta_{9} + 4\beta_{3} + 6\beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{7} + 4\beta_{6} + 2\beta_{5} - 2\beta_{4} + 850\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{11} + 40\beta_{10} - 6\beta_{9} + 86\beta_{8} + 219\beta_{3} - 219\beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -104\beta_{7} + 104\beta_{6} + 129\beta_{5} + 154\beta_{4} - 34570 ) / 30 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1696\beta_{11} - 75\beta_{10} + 3542\beta_{9} + 150\beta_{8} + 21041\beta_{3} + 17649\beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5313\beta_{7} - 5554\beta_{6} - 5072\beta_{5} + 5072\beta_{4} - 1445050\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 723\beta_{11} - 72400\beta_{10} + 1446\beta_{9} - 146246\beta_{8} - 1053219\beta_{3} + 1053219\beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 242384\beta_{7} - 242384\beta_{6} - 219369\beta_{5} - 196354\beta_{4} + 60759370 ) / 30 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3102736 \beta_{11} - 69045 \beta_{10} - 6067382 \beta_{9} + 138090 \beta_{8} + \cdots - 57205809 \beta_{2} ) / 30 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 9101073\beta_{7} + 6780754\beta_{6} + 11421392\beta_{5} - 11421392\beta_{4} + 2566524250\beta_1 ) / 30 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 6960957 \beta_{11} + 133436560 \beta_{10} + 13921914 \beta_{9} + 252951206 \beta_{8} + \cdots - 2952963219 \beta_{2} ) / 30 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−1.02615 + 1.02615i
4.70903 − 4.70903i
−4.38999 + 4.38999i
4.38999 − 4.38999i
−4.70903 + 4.70903i
1.02615 − 1.02615i
−1.02615 − 1.02615i
4.70903 + 4.70903i
−4.38999 − 4.38999i
4.38999 + 4.38999i
−4.70903 − 4.70903i
1.02615 + 1.02615i
−2.82843 + 2.82843i 0 − 16.0000i −53.4724 + 16.3005i 0 148.726 + 148.726i 45.2548 + 45.2548i 0 105.138 − 197.347i
17.2 −2.82843 + 2.82843i 0 − 16.0000i −15.4759 − 53.7168i 0 −103.048 − 103.048i 45.2548 + 45.2548i 0 195.707 + 108.162i
17.3 −2.82843 + 2.82843i 0 − 16.0000i 54.0990 + 14.0818i 0 −9.67825 − 9.67825i 45.2548 + 45.2548i 0 −192.844 + 113.186i
17.4 2.82843 − 2.82843i 0 − 16.0000i −54.0990 − 14.0818i 0 −9.67825 − 9.67825i −45.2548 − 45.2548i 0 −192.844 + 113.186i
17.5 2.82843 − 2.82843i 0 − 16.0000i 15.4759 + 53.7168i 0 −103.048 − 103.048i −45.2548 − 45.2548i 0 195.707 + 108.162i
17.6 2.82843 − 2.82843i 0 − 16.0000i 53.4724 − 16.3005i 0 148.726 + 148.726i −45.2548 − 45.2548i 0 105.138 − 197.347i
53.1 −2.82843 − 2.82843i 0 16.0000i −53.4724 − 16.3005i 0 148.726 − 148.726i 45.2548 − 45.2548i 0 105.138 + 197.347i
53.2 −2.82843 − 2.82843i 0 16.0000i −15.4759 + 53.7168i 0 −103.048 + 103.048i 45.2548 − 45.2548i 0 195.707 − 108.162i
53.3 −2.82843 − 2.82843i 0 16.0000i 54.0990 − 14.0818i 0 −9.67825 + 9.67825i 45.2548 − 45.2548i 0 −192.844 − 113.186i
53.4 2.82843 + 2.82843i 0 16.0000i −54.0990 + 14.0818i 0 −9.67825 + 9.67825i −45.2548 + 45.2548i 0 −192.844 − 113.186i
53.5 2.82843 + 2.82843i 0 16.0000i 15.4759 − 53.7168i 0 −103.048 + 103.048i −45.2548 + 45.2548i 0 195.707 − 108.162i
53.6 2.82843 + 2.82843i 0 16.0000i 53.4724 + 16.3005i 0 148.726 − 148.726i −45.2548 + 45.2548i 0 105.138 + 197.347i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
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 90.6.f.c ✓ 12
3.b odd 2 1 inner 90.6.f.c ✓ 12
5.b even 2 1 450.6.f.e 12
5.c odd 4 1 inner 90.6.f.c ✓ 12
5.c odd 4 1 450.6.f.e 12
15.d odd 2 1 450.6.f.e 12
15.e even 4 1 inner 90.6.f.c ✓ 12
15.e even 4 1 450.6.f.e 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.f.c ✓ 12 1.a even 1 1 trivial
90.6.f.c ✓ 12 3.b odd 2 1 inner
90.6.f.c ✓ 12 5.c odd 4 1 inner
90.6.f.c ✓ 12 15.e even 4 1 inner
450.6.f.e 12 5.b even 2 1
450.6.f.e 12 5.c odd 4 1
450.6.f.e 12 15.d odd 2 1
450.6.f.e 12 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 72T_{7}^{5} + 2592T_{7}^{4} + 2863904T_{7}^{3} + 994519296T_{7}^{2} + 18710687232T_{7} + 176009564672 \) acting on \(S_{6}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 256)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{6} - 72 T^{5} + \cdots + 176009564672)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 11\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 858377073723912)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 35\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 33\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 14\!\cdots\!08)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + 14628 T^{2} + \cdots - 24637853824)^{4} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 26\!\cdots\!32)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 42\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 61\!\cdots\!12)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 27672667200000)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 44\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 14\!\cdots\!28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 24\!\cdots\!08)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 40\!\cdots\!44)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 34\!\cdots\!52)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 20\!\cdots\!92)^{2} \) Copy content Toggle raw display
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