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Properties

Label 380.3.x.a
Level $380$
Weight $3$
Character orbit 380.x
Analytic conductor $10.354$
Analytic rank $0$
Dimension $80$
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] = [380,3,Mod(197,380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(380, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("380.197");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 380.x (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: \(10.3542500457\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(20\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 2 q^{5} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 2 q^{5} - 12 q^{7} - 16 q^{11} - 8 q^{13} - 6 q^{15} - 32 q^{17} + 24 q^{21} - 60 q^{23} - 50 q^{25} - 84 q^{27} + 112 q^{31} + 12 q^{33} + 76 q^{35} - 128 q^{37} - 40 q^{41} + 58 q^{43} - 116 q^{45} + 70 q^{47} - 8 q^{51} - 82 q^{53} - 136 q^{55} + 322 q^{57} + 104 q^{61} - 86 q^{63} + 280 q^{65} - 72 q^{67} - 32 q^{71} - 292 q^{73} + 108 q^{75} + 420 q^{77} + 336 q^{81} - 100 q^{83} - 296 q^{85} + 200 q^{87} + 112 q^{91} + 16 q^{93} - 308 q^{95} - 122 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
197.1 0 −5.56071 − 1.48999i 0 −1.80231 − 4.66387i 0 3.53180 + 3.53180i 0 20.9072 + 12.0708i 0
197.2 0 −4.44050 − 1.18983i 0 −4.73990 + 1.59165i 0 −8.09471 − 8.09471i 0 10.5081 + 6.06686i 0
197.3 0 −4.39499 − 1.17763i 0 4.76910 − 1.50189i 0 2.17731 + 2.17731i 0 10.1349 + 5.85137i 0
197.4 0 −4.16676 − 1.11648i 0 −2.44518 + 4.36132i 0 5.86227 + 5.86227i 0 8.32115 + 4.80422i 0
197.5 0 −3.86197 − 1.03481i 0 3.12952 + 3.89950i 0 −5.06938 − 5.06938i 0 6.04977 + 3.49284i 0
197.6 0 −2.09218 − 0.560598i 0 −1.48606 − 4.77406i 0 −8.46084 − 8.46084i 0 −3.73129 − 2.15426i 0
197.7 0 −1.93603 − 0.518757i 0 4.03887 − 2.94745i 0 −0.717661 − 0.717661i 0 −4.31514 − 2.49135i 0
197.8 0 −1.54038 − 0.412744i 0 −4.58249 − 2.00019i 0 4.58241 + 4.58241i 0 −5.59181 − 3.22844i 0
197.9 0 −0.683173 − 0.183056i 0 −3.28548 + 3.76904i 0 −0.698514 − 0.698514i 0 −7.36101 − 4.24988i 0
197.10 0 −0.354279 − 0.0949287i 0 3.26125 + 3.79002i 0 9.65988 + 9.65988i 0 −7.67773 − 4.43274i 0
197.11 0 0.388313 + 0.104048i 0 −0.554701 − 4.96914i 0 2.85694 + 2.85694i 0 −7.65427 − 4.41919i 0
197.12 0 0.575088 + 0.154094i 0 3.56393 + 3.50691i 0 −4.71619 − 4.71619i 0 −7.48725 − 4.32276i 0
197.13 0 1.80356 + 0.483264i 0 4.99577 − 0.205732i 0 −2.22113 − 2.22113i 0 −4.77493 − 2.75681i 0
197.14 0 1.99282 + 0.533974i 0 −0.602538 + 4.96356i 0 −5.19204 − 5.19204i 0 −4.10804 − 2.37178i 0
197.15 0 2.72602 + 0.730434i 0 −4.98107 − 0.434701i 0 −1.98605 − 1.98605i 0 −0.896601 − 0.517653i 0
197.16 0 3.21202 + 0.860659i 0 1.86810 − 4.63791i 0 7.38473 + 7.38473i 0 1.78213 + 1.02891i 0
197.17 0 4.03682 + 1.08166i 0 −0.515984 + 4.97330i 0 3.76343 + 3.76343i 0 7.33166 + 4.23293i 0
197.18 0 4.09116 + 1.09622i 0 0.0453168 − 4.99979i 0 −8.62513 − 8.62513i 0 7.74162 + 4.46963i 0
197.19 0 4.96065 + 1.32920i 0 −4.99668 − 0.182094i 0 2.01834 + 2.01834i 0 15.0471 + 8.68742i 0
197.20 0 5.24452 + 1.40527i 0 4.82054 + 1.32754i 0 0.944538 + 0.944538i 0 17.7360 + 10.2399i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 197.20
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
19.c even 3 1 inner
95.m odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.x.a ✓ 80
5.c odd 4 1 inner 380.3.x.a ✓ 80
19.c even 3 1 inner 380.3.x.a ✓ 80
95.m odd 12 1 inner 380.3.x.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.x.a ✓ 80 1.a even 1 1 trivial
380.3.x.a ✓ 80 5.c odd 4 1 inner
380.3.x.a ✓ 80 19.c even 3 1 inner
380.3.x.a ✓ 80 95.m odd 12 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(380, [\chi])\).