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Properties

Label 120.3.q.a
Level $120$
Weight $3$
Character orbit 120.q
Analytic conductor $3.270$
Analytic rank $0$
Dimension $88$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [120,3,Mod(83,120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(120, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("120.83");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 120.q (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: \(3.26976317232\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 4 q^{3} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q - 4 q^{3} - 4 q^{6} + 4 q^{10} + 16 q^{12} - 36 q^{16} + 4 q^{18} - 44 q^{22} - 8 q^{25} - 4 q^{27} - 52 q^{28} - 32 q^{30} + 32 q^{33} - 52 q^{36} - 72 q^{40} - 96 q^{42} - 136 q^{43} + 136 q^{46} - 148 q^{48} - 8 q^{51} - 104 q^{52} + 32 q^{57} + 164 q^{58} - 224 q^{60} + 128 q^{66} - 8 q^{67} + 268 q^{70} - 80 q^{72} - 8 q^{73} - 228 q^{75} - 8 q^{76} + 344 q^{78} - 40 q^{81} + 224 q^{82} + 460 q^{88} + 452 q^{90} - 16 q^{91} + 156 q^{96} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
83.1 −1.99559 + 0.132757i 1.11565 − 2.78484i 3.96475 − 0.529856i 2.73501 − 4.18565i −1.85666 + 5.70551i 6.73333 + 6.73333i −7.84167 + 1.58372i −6.51067 − 6.21379i −4.90229 + 8.71594i
83.2 −1.99008 − 0.198906i −2.99481 + 0.176465i 3.92087 + 0.791681i −2.66353 − 4.23150i 5.99502 + 0.244505i 0.218910 + 0.218910i −7.64540 − 2.35540i 8.93772 − 1.05696i 4.45897 + 8.95084i
83.3 −1.96924 + 0.349404i −2.48215 − 1.68491i 3.75583 − 1.37612i 3.46139 + 3.60816i 5.47667 + 2.45073i −4.09876 − 4.09876i −6.91533 + 4.02222i 3.32214 + 8.36441i −8.07701 − 5.89592i
83.4 −1.94563 + 0.463178i 2.93269 + 0.631914i 3.57093 − 1.80234i −1.30258 + 4.82735i −5.99862 + 0.128889i 2.01797 + 2.01797i −6.11290 + 5.16067i 8.20137 + 3.70642i 0.298411 − 9.99555i
83.5 −1.86798 − 0.714606i 1.62817 + 2.51974i 2.97868 + 2.66974i 4.92106 − 0.884950i −1.24076 − 5.87031i −0.658723 − 0.658723i −3.65629 − 7.11558i −3.69814 + 8.20510i −9.82482 − 1.86356i
83.6 −1.75977 − 0.950368i 2.79687 − 1.08514i 2.19360 + 3.34486i −4.31782 − 2.52119i −5.95314 − 0.748452i −5.13958 − 5.13958i −0.681393 − 7.97093i 6.64494 − 6.06999i 5.20233 + 8.54025i
83.7 −1.75886 − 0.952057i −1.79955 + 2.40034i 2.18718 + 3.34907i −0.927549 + 4.91321i 5.45042 − 2.50858i −3.37784 − 3.37784i −0.658431 − 7.97286i −2.52322 − 8.63906i 6.30909 − 7.75857i
83.8 −1.74189 + 0.982755i 0.779461 + 2.89697i 2.06839 − 3.42371i −1.92063 − 4.61640i −4.20475 − 4.28020i −8.48501 − 8.48501i −0.238241 + 7.99645i −7.78488 + 4.51615i 7.88232 + 6.15378i
83.9 −1.65520 + 1.12264i 0.246691 − 2.98984i 1.47938 − 3.71637i −4.82897 + 1.29657i 2.94818 + 5.22573i −2.54317 − 2.54317i 1.72346 + 7.81215i −8.87829 − 1.47513i 6.53734 − 7.56725i
83.10 −1.65420 + 1.12411i −1.73338 + 2.44855i 1.47275 − 3.71901i 4.98577 + 0.376903i 0.114908 − 5.99890i 6.42678 + 6.42678i 1.74436 + 7.80751i −2.99080 − 8.48853i −8.67114 + 4.98109i
83.11 −1.55469 − 1.25815i −1.32962 − 2.68926i 0.834117 + 3.91206i −3.62780 + 3.44079i −1.31635 + 5.85382i 7.91939 + 7.91939i 3.62517 − 7.13149i −5.46424 + 7.15137i 9.96914 − 0.785043i
83.12 −1.25815 − 1.55469i −1.32962 − 2.68926i −0.834117 + 3.91206i 3.62780 − 3.44079i −2.50811 + 5.45063i −7.91939 − 7.91939i 7.13149 − 3.62517i −5.46424 + 7.15137i −9.91368 − 1.31107i
83.13 −1.12411 + 1.65420i 2.44855 − 1.73338i −1.47275 − 3.71901i 4.98577 + 0.376903i 0.114908 + 5.99890i −6.42678 − 6.42678i 7.80751 + 1.74436i 2.99080 − 8.48853i −6.22804 + 7.82378i
83.14 −1.12264 + 1.65520i −2.98984 + 0.246691i −1.47938 − 3.71637i −4.82897 + 1.29657i 2.94818 − 5.22573i 2.54317 + 2.54317i 7.81215 + 1.72346i 8.87829 − 1.47513i 3.27509 − 9.44848i
83.15 −0.982755 + 1.74189i 2.89697 + 0.779461i −2.06839 − 3.42371i −1.92063 − 4.61640i −4.20475 + 4.28020i 8.48501 + 8.48501i 7.99645 − 0.238241i 7.78488 + 4.51615i 9.92879 + 1.19127i
83.16 −0.952057 − 1.75886i −1.79955 + 2.40034i −2.18718 + 3.34907i 0.927549 − 4.91321i 5.93513 + 0.879906i 3.37784 + 3.37784i 7.97286 + 0.658431i −2.52322 − 8.63906i −9.52473 + 3.04623i
83.17 −0.950368 − 1.75977i 2.79687 − 1.08514i −2.19360 + 3.34486i 4.31782 + 2.52119i −4.56765 − 3.89057i 5.13958 + 5.13958i 7.97093 + 0.681393i 6.64494 − 6.06999i 0.333207 − 9.99445i
83.18 −0.714606 − 1.86798i 1.62817 + 2.51974i −2.97868 + 2.66974i −4.92106 + 0.884950i 3.54331 − 4.84200i 0.658723 + 0.658723i 7.11558 + 3.65629i −3.69814 + 8.20510i 5.16969 + 8.56004i
83.19 −0.463178 + 1.94563i 0.631914 + 2.93269i −3.57093 − 1.80234i −1.30258 + 4.82735i −5.99862 − 0.128889i −2.01797 − 2.01797i 5.16067 − 6.11290i −8.20137 + 3.70642i −8.78889 − 4.77026i
83.20 −0.349404 + 1.96924i −1.68491 − 2.48215i −3.75583 − 1.37612i 3.46139 + 3.60816i 5.47667 − 2.45073i 4.09876 + 4.09876i 4.02222 − 6.91533i −3.32214 + 8.36441i −8.31476 + 5.55560i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 83.44
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
8.d odd 2 1 inner
15.e even 4 1 inner
24.f even 2 1 inner
40.k even 4 1 inner
120.q odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 120.3.q.a ✓ 88
3.b odd 2 1 inner 120.3.q.a ✓ 88
4.b odd 2 1 480.3.u.a 88
5.c odd 4 1 inner 120.3.q.a ✓ 88
8.b even 2 1 480.3.u.a 88
8.d odd 2 1 inner 120.3.q.a ✓ 88
12.b even 2 1 480.3.u.a 88
15.e even 4 1 inner 120.3.q.a ✓ 88
20.e even 4 1 480.3.u.a 88
24.f even 2 1 inner 120.3.q.a ✓ 88
24.h odd 2 1 480.3.u.a 88
40.i odd 4 1 480.3.u.a 88
40.k even 4 1 inner 120.3.q.a ✓ 88
60.l odd 4 1 480.3.u.a 88
120.q odd 4 1 inner 120.3.q.a ✓ 88
120.w even 4 1 480.3.u.a 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.q.a ✓ 88 1.a even 1 1 trivial
120.3.q.a ✓ 88 3.b odd 2 1 inner
120.3.q.a ✓ 88 5.c odd 4 1 inner
120.3.q.a ✓ 88 8.d odd 2 1 inner
120.3.q.a ✓ 88 15.e even 4 1 inner
120.3.q.a ✓ 88 24.f even 2 1 inner
120.3.q.a ✓ 88 40.k even 4 1 inner
120.3.q.a ✓ 88 120.q odd 4 1 inner
480.3.u.a 88 4.b odd 2 1
480.3.u.a 88 8.b even 2 1
480.3.u.a 88 12.b even 2 1
480.3.u.a 88 20.e even 4 1
480.3.u.a 88 24.h odd 2 1
480.3.u.a 88 40.i odd 4 1
480.3.u.a 88 60.l odd 4 1
480.3.u.a 88 120.w even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(120, [\chi])\).