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Properties

Label 90.3.j.a
Level $90$
Weight $3$
Character orbit 90.j
Analytic conductor $2.452$
Analytic rank $0$
Dimension $8$
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] = [90,3,Mod(29,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 90.j (of order \(6\), 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: \(2.45232237924\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + 3 \beta_1 q^{3} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \cdots - 1) q^{5}+ \cdots + 9 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + 3 \beta_1 q^{3} + (2 \beta_{2} - 2) q^{4} + ( - \beta_{7} - 2 \beta_{6} + \beta_{4} + \cdots - 1) q^{5}+ \cdots + (63 \beta_{6} - 18 \beta_{2} - 18) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{4} - 12 q^{5} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{4} - 12 q^{5} + 36 q^{9} + 16 q^{10} - 24 q^{11} - 48 q^{14} + 72 q^{15} - 16 q^{16} - 144 q^{19} + 24 q^{20} + 168 q^{21} - 12 q^{25} - 36 q^{29} - 48 q^{30} - 88 q^{31} + 136 q^{34} - 144 q^{36} - 72 q^{39} - 16 q^{40} + 36 q^{41} + 32 q^{46} + 96 q^{49} + 144 q^{50} + 72 q^{51} - 176 q^{55} + 96 q^{56} + 192 q^{59} - 20 q^{61} + 64 q^{64} - 312 q^{65} - 168 q^{66} + 252 q^{69} + 160 q^{70} + 72 q^{74} - 96 q^{75} + 144 q^{76} - 288 q^{79} - 324 q^{81} - 168 q^{84} + 184 q^{85} - 384 q^{86} + 72 q^{90} + 336 q^{91} + 80 q^{94} + 264 q^{95} - 216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(1 - \beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−0.258819 − 0.965926i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 + 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.965926 − 0.258819i
−0.707107 + 1.22474i −2.59808 + 1.50000i −1.00000 − 1.73205i −1.48967 − 4.77293i − 4.24264i −1.81954 − 1.05051i 2.82843 4.50000 − 7.79423i 6.89898 + 1.55051i
29.2 −0.707107 + 1.22474i 2.59808 − 1.50000i −1.00000 − 1.73205i −2.92455 + 4.05549i 4.24264i 10.3048 + 5.94949i 2.82843 4.50000 − 7.79423i −2.89898 − 6.44949i
29.3 0.707107 − 1.22474i −2.59808 + 1.50000i −1.00000 − 1.73205i −4.97443 + 0.504984i 4.24264i −10.3048 − 5.94949i −2.82843 4.50000 − 7.79423i −2.89898 + 6.44949i
29.4 0.707107 − 1.22474i 2.59808 − 1.50000i −1.00000 − 1.73205i 3.38865 + 3.67656i − 4.24264i 1.81954 + 1.05051i −2.82843 4.50000 − 7.79423i 6.89898 − 1.55051i
59.1 −0.707107 − 1.22474i −2.59808 − 1.50000i −1.00000 + 1.73205i −1.48967 + 4.77293i 4.24264i −1.81954 + 1.05051i 2.82843 4.50000 + 7.79423i 6.89898 − 1.55051i
59.2 −0.707107 − 1.22474i 2.59808 + 1.50000i −1.00000 + 1.73205i −2.92455 − 4.05549i − 4.24264i 10.3048 − 5.94949i 2.82843 4.50000 + 7.79423i −2.89898 + 6.44949i
59.3 0.707107 + 1.22474i −2.59808 − 1.50000i −1.00000 + 1.73205i −4.97443 − 0.504984i − 4.24264i −10.3048 + 5.94949i −2.82843 4.50000 + 7.79423i −2.89898 − 6.44949i
59.4 0.707107 + 1.22474i 2.59808 + 1.50000i −1.00000 + 1.73205i 3.38865 − 3.67656i 4.24264i 1.81954 − 1.05051i −2.82843 4.50000 + 7.79423i 6.89898 + 1.55051i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.d odd 6 1 inner
45.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.3.j.a ✓ 8
3.b odd 2 1 270.3.j.a 8
5.b even 2 1 inner 90.3.j.a ✓ 8
5.c odd 4 1 450.3.i.a 4
5.c odd 4 1 450.3.i.c 4
9.c even 3 1 270.3.j.a 8
9.c even 3 1 810.3.b.a 8
9.d odd 6 1 inner 90.3.j.a ✓ 8
9.d odd 6 1 810.3.b.a 8
15.d odd 2 1 270.3.j.a 8
15.e even 4 1 1350.3.i.a 4
15.e even 4 1 1350.3.i.c 4
45.h odd 6 1 inner 90.3.j.a ✓ 8
45.h odd 6 1 810.3.b.a 8
45.j even 6 1 270.3.j.a 8
45.j even 6 1 810.3.b.a 8
45.k odd 12 1 1350.3.i.a 4
45.k odd 12 1 1350.3.i.c 4
45.l even 12 1 450.3.i.a 4
45.l even 12 1 450.3.i.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.3.j.a ✓ 8 1.a even 1 1 trivial
90.3.j.a ✓ 8 5.b even 2 1 inner
90.3.j.a ✓ 8 9.d odd 6 1 inner
90.3.j.a ✓ 8 45.h odd 6 1 inner
270.3.j.a 8 3.b odd 2 1
270.3.j.a 8 9.c even 3 1
270.3.j.a 8 15.d odd 2 1
270.3.j.a 8 45.j even 6 1
450.3.i.a 4 5.c odd 4 1
450.3.i.a 4 45.l even 12 1
450.3.i.c 4 5.c odd 4 1
450.3.i.c 4 45.l even 12 1
810.3.b.a 8 9.c even 3 1
810.3.b.a 8 9.d odd 6 1
810.3.b.a 8 45.h odd 6 1
810.3.b.a 8 45.j even 6 1
1350.3.i.a 4 15.e even 4 1
1350.3.i.a 4 45.k odd 12 1
1350.3.i.c 4 15.e even 4 1
1350.3.i.c 4 45.k odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 146T_{7}^{6} + 20691T_{7}^{4} - 91250T_{7}^{2} + 390625 \) acting on \(S_{3}^{\mathrm{new}}(90, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} - 9 T^{2} + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} + 12 T^{7} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{8} - 146 T^{6} + \cdots + 390625 \) Copy content Toggle raw display
$11$ \( (T^{4} + 12 T^{3} + \cdots + 7396)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 4430766096 \) Copy content Toggle raw display
$17$ \( (T^{4} - 1180 T^{2} + 320356)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 36 T + 318)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 310 T^{6} + \cdots + 373301041 \) Copy content Toggle raw display
$29$ \( (T^{4} + 18 T^{3} + \cdots + 2025)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 44 T^{3} + \cdots + 280900)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1460 T^{2} + 386884)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 18 T^{3} + \cdots + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 5337948160000 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 29120366676241 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2832 T^{2} + 14400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 96 T^{3} + \cdots + 21529600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} + 99 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 30549950894401 \) Copy content Toggle raw display
$71$ \( (T^{4} + 21084 T^{2} + 92968164)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 560 T^{2} + 53824)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 144 T^{3} + \cdots + 46656)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 1982119441 \) Copy content Toggle raw display
$89$ \( (T^{4} + 28774 T^{2} + 125731369)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 87219461226496 \) Copy content Toggle raw display
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