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Properties

Label 90.4.l.a
Level $90$
Weight $4$
Character orbit 90.l
Analytic conductor $5.310$
Analytic rank $0$
Dimension $72$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [90,4,Mod(23,90)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(90, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("90.23");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 90.l (of order \(12\), degree \(4\), 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: \(5.31017190052\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(18\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 8 q^{3} + 16 q^{6} + 48 q^{11} + 32 q^{12} + 172 q^{15} + 576 q^{16} + 16 q^{18} - 192 q^{20} - 464 q^{21} - 312 q^{23} + 288 q^{25} + 244 q^{27} + 336 q^{30} + 268 q^{33} - 32 q^{36} - 288 q^{37}+ \cdots - 1512 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
23.1 −0.517638 + 1.93185i −4.93707 + 1.62028i −3.46410 − 2.00000i 5.77745 − 9.57189i −0.574521 − 10.3764i −1.66858 − 0.447095i 5.65685 − 5.65685i 21.7494 − 15.9989i 15.5008 + 16.1159i
23.2 −0.517638 + 1.93185i −4.22825 − 3.02025i −3.46410 − 2.00000i −11.1170 + 1.18817i 8.02337 − 6.60496i 23.8855 + 6.40009i 5.65685 − 5.65685i 8.75622 + 25.5407i 3.45924 − 22.0915i
23.3 −0.517638 + 1.93185i −3.91985 − 3.41098i −3.46410 − 2.00000i 8.87803 + 6.79563i 8.61857 − 5.80691i −1.19100 − 0.319128i 5.65685 − 5.65685i 3.73043 + 26.7411i −17.7238 + 13.6334i
23.4 −0.517638 + 1.93185i −3.78220 + 3.56300i −3.46410 − 2.00000i −2.13754 + 10.9741i −4.92537 − 9.15100i −11.0581 − 2.96302i 5.65685 − 5.65685i 1.61012 − 26.9519i −20.0939 − 9.81002i
23.5 −0.517638 + 1.93185i 1.36699 + 5.01312i −3.46410 − 2.00000i −6.13463 − 9.34700i −10.3922 + 0.0458498i −25.4835 − 6.82827i 5.65685 − 5.65685i −23.2627 + 13.7058i 21.2325 − 7.01283i
23.6 −0.517638 + 1.93185i 1.63624 + 4.93181i −3.46410 − 2.00000i 11.1207 + 1.15355i −10.3745 + 0.608091i 29.8452 + 7.99699i 5.65685 − 5.65685i −21.6454 + 16.1393i −7.98497 + 20.8864i
23.7 −0.517638 + 1.93185i 2.43231 − 4.59172i −3.46410 − 2.00000i −11.0786 + 1.50487i 7.61146 + 7.07571i −10.0583 − 2.69511i 5.65685 − 5.65685i −15.1677 − 22.3370i 2.82751 − 22.1812i
23.8 −0.517638 + 1.93185i 3.25274 − 4.05212i −3.46410 − 2.00000i 10.9773 + 2.12101i 6.14435 + 8.38134i −5.89626 − 1.57990i 5.65685 − 5.65685i −5.83936 − 26.3610i −9.77975 + 20.1086i
23.9 −0.517638 + 1.93185i 5.18822 − 0.287005i −3.46410 − 2.00000i −2.82156 − 10.8184i −2.13117 + 10.1714i 20.0285 + 5.36661i 5.65685 − 5.65685i 26.8353 − 2.97809i 22.3602 + 0.149197i
23.10 0.517638 − 1.93185i −5.19613 − 0.0142387i −3.46410 − 2.00000i 1.85147 + 11.0260i −2.71722 + 10.0308i 17.5853 + 4.71196i −5.65685 + 5.65685i 26.9996 + 0.147972i 22.2589 + 2.13070i
23.11 0.517638 − 1.93185i −4.13988 − 3.14028i −3.46410 − 2.00000i −6.18941 − 9.31081i −8.20952 + 6.37210i 11.2370 + 3.01094i −5.65685 + 5.65685i 7.27723 + 26.0008i −21.1910 + 7.13740i
23.12 0.517638 − 1.93185i −4.11577 + 3.17182i −3.46410 − 2.00000i 11.0644 − 1.60590i 3.99702 + 9.59291i −31.7299 − 8.50201i −5.65685 + 5.65685i 6.87905 − 26.1090i 2.62500 − 22.2061i
23.13 0.517638 − 1.93185i −1.42405 + 4.99721i −3.46410 − 2.00000i 1.51899 − 11.0767i 8.91673 + 5.33779i 23.9804 + 6.42552i −5.65685 + 5.65685i −22.9442 − 14.2325i −20.6122 − 8.66817i
23.14 0.517638 − 1.93185i −0.495164 − 5.17251i −3.46410 − 2.00000i −7.31344 + 8.45657i −10.2488 − 1.72090i −17.1411 − 4.59296i −5.65685 + 5.65685i −26.5096 + 5.12247i 12.5511 + 18.5059i
23.15 0.517638 − 1.93185i 0.759735 + 5.14031i −3.46410 − 2.00000i −9.83011 + 5.32626i 10.3236 + 1.19313i −8.51682 − 2.28208i −5.65685 + 5.65685i −25.8456 + 7.81055i 5.20110 + 21.7474i
23.16 0.517638 − 1.93185i 2.43185 − 4.59196i −3.46410 − 2.00000i 9.28847 − 6.22288i −7.61218 − 7.07494i 14.2017 + 3.80534i −5.65685 + 5.65685i −15.1723 − 22.3339i −7.21361 − 21.1652i
23.17 0.517638 − 1.93185i 4.62106 + 2.37609i −3.46410 − 2.00000i 8.62775 + 7.11070i 6.98229 − 7.69725i 3.96829 + 1.06330i −5.65685 + 5.65685i 15.7084 + 21.9601i 18.2029 − 12.9868i
23.18 0.517638 − 1.93185i 5.08511 − 1.06846i −3.46410 − 2.00000i −5.55402 − 9.70324i 0.568135 − 10.3768i −31.9881 − 8.57118i −5.65685 + 5.65685i 24.7168 − 10.8665i −21.6202 + 5.70678i
47.1 −0.517638 − 1.93185i −4.93707 − 1.62028i −3.46410 + 2.00000i 5.77745 + 9.57189i −0.574521 + 10.3764i −1.66858 + 0.447095i 5.65685 + 5.65685i 21.7494 + 15.9989i 15.5008 − 16.1159i
47.2 −0.517638 − 1.93185i −4.22825 + 3.02025i −3.46410 + 2.00000i −11.1170 − 1.18817i 8.02337 + 6.60496i 23.8855 − 6.40009i 5.65685 + 5.65685i 8.75622 − 25.5407i 3.45924 + 22.0915i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 23.18
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
9.d odd 6 1 inner
45.l even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 90.4.l.a ✓ 72
3.b odd 2 1 270.4.m.a 72
5.c odd 4 1 inner 90.4.l.a ✓ 72
9.c even 3 1 270.4.m.a 72
9.d odd 6 1 inner 90.4.l.a ✓ 72
15.e even 4 1 270.4.m.a 72
45.k odd 12 1 270.4.m.a 72
45.l even 12 1 inner 90.4.l.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.l.a ✓ 72 1.a even 1 1 trivial
90.4.l.a ✓ 72 5.c odd 4 1 inner
90.4.l.a ✓ 72 9.d odd 6 1 inner
90.4.l.a ✓ 72 45.l even 12 1 inner
270.4.m.a 72 3.b odd 2 1
270.4.m.a 72 9.c even 3 1
270.4.m.a 72 15.e even 4 1
270.4.m.a 72 45.k odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(90, [\chi])\).