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Properties

Label 370.2.x.a
Level $370$
Weight $2$
Character orbit 370.x
Analytic conductor $2.954$
Analytic rank $0$
Dimension $108$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [370,2,Mod(9,370)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(370, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("370.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 370 = 2 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 370.x (of order \(18\), degree \(6\), 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: \(2.95446487479\)
Analytic rank: \(0\)
Dimension: \(108\)
Relative dimension: \(18\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 108 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 108 q - 12 q^{9} + 12 q^{11} - 6 q^{14} + 6 q^{19} - 24 q^{21} + 30 q^{25} + 48 q^{26} + 24 q^{30} - 192 q^{31} + 12 q^{34} + 30 q^{35} + 84 q^{36} + 24 q^{39} + 12 q^{40} - 108 q^{41} + 6 q^{44} - 30 q^{45} - 54 q^{46} + 60 q^{49} - 12 q^{50} + 30 q^{55} - 48 q^{59} + 24 q^{61} + 54 q^{64} - 6 q^{65} - 120 q^{69} - 12 q^{70} + 12 q^{71} - 60 q^{74} - 156 q^{75} - 6 q^{76} + 12 q^{80} + 120 q^{81} - 24 q^{84} - 30 q^{85} + 12 q^{86} + 54 q^{89} + 18 q^{90} - 138 q^{91} + 6 q^{94} + 84 q^{95} - 90 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 −0.984808 + 0.173648i −2.81720 − 0.496749i 0.939693 − 0.342020i 1.07854 − 1.95876i 2.86066 0.735165 − 0.876136i −0.866025 + 0.500000i 4.87081 + 1.77283i −0.722019 + 2.11629i
9.2 −0.984808 + 0.173648i −2.29177 − 0.404101i 0.939693 − 0.342020i −2.21358 + 0.316298i 2.32713 1.49272 − 1.77895i −0.866025 + 0.500000i 2.26984 + 0.826155i 2.12503 − 0.695878i
9.3 −0.984808 + 0.173648i −1.38745 − 0.244645i 0.939693 − 0.342020i 1.61905 + 1.54229i 1.40886 1.64033 − 1.95487i −0.866025 + 0.500000i −0.953902 − 0.347192i −1.86227 − 1.23771i
9.4 −0.984808 + 0.173648i −0.771593 − 0.136053i 0.939693 − 0.342020i −0.425568 + 2.19520i 0.783496 −0.680207 + 0.810639i −0.866025 + 0.500000i −2.24223 − 0.816106i 0.0379105 − 2.23575i
9.5 −0.984808 + 0.173648i −0.397192 − 0.0700357i 0.939693 − 0.342020i 2.17168 − 0.532725i 0.403320 −3.23589 + 3.85639i −0.866025 + 0.500000i −2.66622 − 0.970425i −2.04618 + 0.901741i
9.6 −0.984808 + 0.173648i 0.906979 + 0.159925i 0.939693 − 0.342020i −1.36673 − 1.76976i −0.920970 −1.33907 + 1.59584i −0.866025 + 0.500000i −2.02204 − 0.735964i 1.65328 + 1.50555i
9.7 −0.984808 + 0.173648i 1.76238 + 0.310755i 0.939693 − 0.342020i −1.56893 + 1.59325i −1.78956 −0.326840 + 0.389513i −0.866025 + 0.500000i 0.190327 + 0.0692735i 1.26843 − 1.84149i
9.8 −0.984808 + 0.173648i 2.38342 + 0.420261i 0.939693 − 0.342020i −0.137432 − 2.23184i −2.42018 0.265330 − 0.316208i −0.866025 + 0.500000i 2.68498 + 0.977252i 0.522899 + 2.17407i
9.9 −0.984808 + 0.173648i 2.61244 + 0.460644i 0.939693 − 0.342020i 1.38577 + 1.75489i −2.65274 2.09125 − 2.49226i −0.866025 + 0.500000i 3.79359 + 1.38075i −1.66945 − 1.48759i
9.10 0.984808 − 0.173648i −2.61244 − 0.460644i 0.939693 − 0.342020i 1.96886 + 1.05999i −2.65274 −2.09125 + 2.49226i 0.866025 − 0.500000i 3.79359 + 1.38075i 2.12302 + 0.701996i
9.11 0.984808 − 0.173648i −2.38342 − 0.420261i 0.939693 − 0.342020i −2.22180 + 0.252211i −2.42018 −0.265330 + 0.316208i 0.866025 − 0.500000i 2.68498 + 0.977252i −2.14425 + 0.634191i
9.12 0.984808 − 0.173648i −1.76238 − 0.310755i 0.939693 − 0.342020i 1.29660 − 1.82176i −1.78956 0.326840 − 0.389513i 0.866025 − 0.500000i 0.190327 + 0.0692735i 0.960558 − 2.01924i
9.13 0.984808 − 0.173648i −0.906979 − 0.159925i 0.939693 − 0.342020i −1.98021 − 1.03865i −0.920970 1.33907 − 1.59584i 0.866025 − 0.500000i −2.02204 − 0.735964i −2.13048 − 0.679008i
9.14 0.984808 − 0.173648i 0.397192 + 0.0700357i 0.939693 − 0.342020i −0.147523 + 2.23120i 0.403320 3.23589 − 3.85639i 0.866025 − 0.500000i −2.66622 − 0.970425i 0.242161 + 2.22292i
9.15 0.984808 − 0.173648i 0.771593 + 0.136053i 0.939693 − 0.342020i 2.08795 − 0.800295i 0.783496 0.680207 − 0.810639i 0.866025 − 0.500000i −2.24223 − 0.816106i 1.91726 − 1.15070i
9.16 0.984808 − 0.173648i 1.38745 + 0.244645i 0.939693 − 0.342020i 1.80001 + 1.32664i 1.40886 −1.64033 + 1.95487i 0.866025 − 0.500000i −0.953902 − 0.347192i 2.00303 + 0.993919i
9.17 0.984808 − 0.173648i 2.29177 + 0.404101i 0.939693 − 0.342020i −0.0728921 − 2.23488i 2.32713 −1.49272 + 1.77895i 0.866025 − 0.500000i 2.26984 + 0.826155i −0.459868 − 2.18827i
9.18 0.984808 − 0.173648i 2.81720 + 0.496749i 0.939693 − 0.342020i −1.74172 + 1.40229i 2.86066 −0.735165 + 0.876136i 0.866025 − 0.500000i 4.87081 + 1.77283i −1.47175 + 1.68343i
49.1 −0.642788 − 0.766044i −1.67544 + 1.99671i −0.173648 + 0.984808i −0.552122 − 2.16683i 2.60652 0.903886 + 2.48341i 0.866025 − 0.500000i −0.658811 − 3.73630i −1.30499 + 1.81576i
49.2 −0.642788 − 0.766044i −1.48887 + 1.77437i −0.173648 + 0.984808i 2.17115 − 0.534875i 2.31627 −0.983118 − 2.70110i 0.866025 − 0.500000i −0.410698 − 2.32918i −1.80533 − 1.31939i
See next 80 embeddings (of 108 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.f even 9 1 inner
185.x even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 370.2.x.a ✓ 108
5.b even 2 1 inner 370.2.x.a ✓ 108
37.f even 9 1 inner 370.2.x.a ✓ 108
185.x even 18 1 inner 370.2.x.a ✓ 108
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.x.a ✓ 108 1.a even 1 1 trivial
370.2.x.a ✓ 108 5.b even 2 1 inner
370.2.x.a ✓ 108 37.f even 9 1 inner
370.2.x.a ✓ 108 185.x even 18 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(370, [\chi])\).