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Properties

Label 378.3.j.a
Level $378$
Weight $3$
Character orbit 378.j
Analytic conductor $10.300$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,3,Mod(19,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.j (of order \(6\), degree \(2\), not 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: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 126)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 32 q^{4} - 2 q^{7} - 24 q^{11} - 30 q^{13} + 12 q^{14} - 64 q^{16} - 54 q^{17} - 84 q^{23} - 160 q^{25} + 72 q^{26} - 4 q^{28} + 84 q^{29} - 24 q^{31} + 66 q^{35} - 22 q^{37} - 396 q^{41} - 16 q^{43} + 24 q^{44} + 12 q^{46} - 108 q^{47} - 22 q^{49} + 96 q^{50} + 252 q^{53} - 48 q^{56} + 48 q^{58} + 90 q^{59} - 102 q^{61} + 256 q^{64} + 6 q^{65} + 70 q^{67} - 108 q^{70} - 300 q^{71} - 144 q^{74} + 114 q^{77} + 106 q^{79} + 756 q^{83} - 60 q^{85} - 240 q^{86} - 414 q^{89} - 186 q^{91} + 84 q^{92} + 552 q^{95} + 114 q^{97} + 96 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i − 7.27565i 0 −1.35244 + 6.86811i 2.82843 0 −8.91081 + 5.14466i
19.2 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i − 5.63396i 0 −6.65509 − 2.17021i 2.82843 0 −6.90016 + 3.98381i
19.3 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i − 5.57223i 0 1.87450 − 6.74435i 2.82843 0 −6.82456 + 3.94016i
19.4 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i − 1.84415i 0 5.75478 + 3.98528i 2.82843 0 −2.25861 + 1.30401i
19.5 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i − 0.247532i 0 −5.49108 + 4.34143i 2.82843 0 −0.303164 + 0.175032i
19.6 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 3.99138i 0 1.76986 − 6.77256i 2.82843 0 4.88842 − 2.82233i
19.7 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 7.47316i 0 5.96847 + 3.65751i 2.82843 0 9.15272 − 5.28432i
19.8 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 9.10897i 0 −6.61164 − 2.29918i 2.82843 0 11.1562 − 6.44102i
19.9 0.707107 + 1.22474i 0 −1.00000 + 1.73205i − 7.22162i 0 0.767934 + 6.95775i −2.82843 0 8.84464 − 5.10646i
19.10 0.707107 + 1.22474i 0 −1.00000 + 1.73205i − 4.33537i 0 1.70451 − 6.78930i −2.82843 0 5.30973 − 3.06557i
19.11 0.707107 + 1.22474i 0 −1.00000 + 1.73205i − 3.07057i 0 −3.72067 + 5.92930i −2.82843 0 3.76067 − 2.17122i
19.12 0.707107 + 1.22474i 0 −1.00000 + 1.73205i − 1.81116i 0 −5.07376 − 4.82254i −2.82843 0 2.21821 − 1.28068i
19.13 0.707107 + 1.22474i 0 −1.00000 + 1.73205i − 1.80253i 0 6.29258 − 3.06650i −2.82843 0 2.20764 − 1.27458i
19.14 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 3.76202i 0 −6.90261 − 1.16362i −2.82843 0 −4.60752 + 2.66015i
19.15 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 6.94216i 0 3.95343 + 5.77671i −2.82843 0 −8.50237 + 4.90885i
19.16 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 7.53707i 0 6.72123 − 1.95578i −2.82843 0 −9.23099 + 5.32952i
199.1 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i − 9.10897i 0 −6.61164 + 2.29918i 2.82843 0 11.1562 + 6.44102i
199.2 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i − 7.47316i 0 5.96847 − 3.65751i 2.82843 0 9.15272 + 5.28432i
199.3 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i − 3.99138i 0 1.76986 + 6.77256i 2.82843 0 4.88842 + 2.82233i
199.4 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i 0.247532i 0 −5.49108 − 4.34143i 2.82843 0 −0.303164 − 0.175032i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.3.j.a 32
3.b odd 2 1 126.3.j.a ✓ 32
7.d odd 6 1 378.3.p.a 32
9.c even 3 1 378.3.p.a 32
9.d odd 6 1 126.3.p.a yes 32
21.g even 6 1 126.3.p.a yes 32
63.k odd 6 1 inner 378.3.j.a 32
63.s even 6 1 126.3.j.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.j.a ✓ 32 3.b odd 2 1
126.3.j.a ✓ 32 63.s even 6 1
126.3.p.a yes 32 9.d odd 6 1
126.3.p.a yes 32 21.g even 6 1
378.3.j.a 32 1.a even 1 1 trivial
378.3.j.a 32 63.k odd 6 1 inner
378.3.p.a 32 7.d odd 6 1
378.3.p.a 32 9.c even 3 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(378, [\chi])\).