↑

Properties

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

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,3,Mod(179,378)]
 
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([5, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.179");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.i (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
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{4} + 2 q^{7} + 10 q^{13} - 36 q^{14} - 64 q^{16} - 54 q^{17} + 28 q^{19} - 160 q^{25} - 72 q^{26} - 4 q^{28} - 36 q^{29} - 8 q^{31} - 90 q^{35} + 22 q^{37} - 72 q^{41} + 16 q^{43} + 72 q^{44} - 12 q^{46} + 108 q^{47} + 74 q^{49} + 288 q^{50} + 40 q^{52} - 72 q^{53} - 24 q^{55} + 48 q^{58} + 90 q^{59} - 62 q^{61} - 256 q^{64} - 378 q^{65} + 70 q^{67} - 108 q^{70} + 196 q^{73} - 56 q^{76} - 630 q^{77} - 38 q^{79} + 60 q^{85} - 486 q^{89} - 122 q^{91} - 252 q^{92} + 168 q^{94} + 72 q^{95} - 38 q^{97} + 288 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
179.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i − 9.69788i 0 6.99572 − 0.244854i − 2.82843i 0 −6.85744 + 11.8774i
179.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i − 6.30630i 0 −3.75927 − 5.90490i − 2.82843i 0 −4.45923 + 7.72361i
179.3 −1.22474 − 0.707107i 0 1.00000 + 1.73205i − 2.96160i 0 −2.14099 + 6.66455i − 2.82843i 0 −2.09417 + 3.62721i
179.4 −1.22474 − 0.707107i 0 1.00000 + 1.73205i − 2.12829i 0 −5.95128 + 3.68541i − 2.82843i 0 −1.50493 + 2.60662i
179.5 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 1.75162i 0 6.58841 + 2.36493i − 2.82843i 0 1.23858 − 2.14528i
179.6 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 2.45915i 0 1.16622 − 6.90217i − 2.82843i 0 1.73888 − 3.01183i
179.7 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 8.31909i 0 0.934868 + 6.93729i − 2.82843i 0 5.88249 − 10.1888i
179.8 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 8.56422i 0 4.01479 − 5.73423i − 2.82843i 0 6.05582 − 10.4890i
179.9 1.22474 + 0.707107i 0 1.00000 + 1.73205i − 7.66608i 0 −6.97124 − 0.633929i 2.82843i 0 5.42073 − 9.38899i
179.10 1.22474 + 0.707107i 0 1.00000 + 1.73205i − 3.35218i 0 −6.40395 + 2.82655i 2.82843i 0 2.37035 − 4.10556i
179.11 1.22474 + 0.707107i 0 1.00000 + 1.73205i − 2.72742i 0 −4.37650 − 5.46317i 2.82843i 0 1.92857 − 3.34039i
179.12 1.22474 + 0.707107i 0 1.00000 + 1.73205i − 1.41430i 0 3.88340 + 5.82402i 2.82843i 0 1.00006 − 1.73216i
179.13 1.22474 + 0.707107i 0 1.00000 + 1.73205i − 0.989713i 0 4.95314 − 4.94635i 2.82843i 0 0.699833 − 1.21215i
179.14 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.39465i 0 6.52929 − 2.52357i 2.82843i 0 −1.69327 + 2.93283i
179.15 1.22474 + 0.707107i 0 1.00000 + 1.73205i 6.40162i 0 2.49631 + 6.53976i 2.82843i 0 −4.52663 + 7.84035i
179.16 1.22474 + 0.707107i 0 1.00000 + 1.73205i 7.35342i 0 −6.95892 − 0.757275i 2.82843i 0 −5.19965 + 9.00606i
359.1 −1.22474 + 0.707107i 0 1.00000 − 1.73205i − 8.56422i 0 4.01479 + 5.73423i 2.82843i 0 6.05582 + 10.4890i
359.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i − 8.31909i 0 0.934868 − 6.93729i 2.82843i 0 5.88249 + 10.1888i
359.3 −1.22474 + 0.707107i 0 1.00000 − 1.73205i − 2.45915i 0 1.16622 + 6.90217i 2.82843i 0 1.73888 + 3.01183i
359.4 −1.22474 + 0.707107i 0 1.00000 − 1.73205i − 1.75162i 0 6.58841 − 2.36493i 2.82843i 0 1.23858 + 2.14528i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 179.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.n 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.i.a 32
3.b odd 2 1 126.3.i.a ✓ 32
7.c even 3 1 378.3.r.a 32
9.c even 3 1 126.3.r.a yes 32
9.d odd 6 1 378.3.r.a 32
21.h odd 6 1 126.3.r.a yes 32
63.g even 3 1 126.3.i.a ✓ 32
63.n odd 6 1 inner 378.3.i.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.i.a ✓ 32 3.b odd 2 1
126.3.i.a ✓ 32 63.g even 3 1
126.3.r.a yes 32 9.c even 3 1
126.3.r.a yes 32 21.h odd 6 1
378.3.i.a 32 1.a even 1 1 trivial
378.3.i.a 32 63.n odd 6 1 inner
378.3.r.a 32 7.c even 3 1
378.3.r.a 32 9.d odd 6 1

Hecke kernels

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