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Properties

Label 2394.2.by.a
Level $2394$
Weight $2$
Character orbit 2394.by
Analytic conductor $19.116$
Analytic rank $0$
Dimension $48$
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] = [2394,2,Mod(647,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.647");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.by (of order \(6\), 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: \(19.1161862439\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 24 q^{4} - 4 q^{7} - 12 q^{10} - 4 q^{14} - 24 q^{16} - 16 q^{17} + 8 q^{22} - 28 q^{25} + 4 q^{28} + 12 q^{31} - 24 q^{35} + 24 q^{38} - 12 q^{40} + 16 q^{41} + 8 q^{43} + 8 q^{46} - 32 q^{49} - 48 q^{53} + 4 q^{56} - 4 q^{58} + 16 q^{59} + 32 q^{62} - 48 q^{64} - 24 q^{67} + 16 q^{68} + 28 q^{70} + 12 q^{73} + 8 q^{77} - 20 q^{79} - 48 q^{82} + 128 q^{83} - 40 q^{85} + 4 q^{88} + 32 q^{89} + 40 q^{91} - 12 q^{95} - 16 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
647.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.68591 + 2.92009i 0 −2.39615 − 1.12181i 1.00000i 0 −2.92009 − 1.68591i
647.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.33805 − 2.31757i 0 −2.59469 + 0.517288i 1.00000i 0 2.31757 + 1.33805i
647.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.551892 + 0.955905i 0 0.398562 + 2.61556i 1.00000i 0 −0.955905 − 0.551892i
647.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.591287 + 1.02414i 0 2.63492 − 0.239189i 1.00000i 0 −1.02414 − 0.591287i
647.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.0632452 + 0.109544i 0 0.676596 − 2.55778i 1.00000i 0 −0.109544 − 0.0632452i
647.6 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.473253 − 0.819698i 0 0.694114 − 2.55308i 1.00000i 0 0.819698 + 0.473253i
647.7 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.580202 − 1.00494i 0 2.42321 + 1.06211i 1.00000i 0 1.00494 + 0.580202i
647.8 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.49224 + 2.58463i 0 −2.55485 + 0.687563i 1.00000i 0 −2.58463 − 1.49224i
647.9 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.45477 + 2.51973i 0 −1.02043 + 2.44105i 1.00000i 0 −2.51973 − 1.45477i
647.10 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.55788 − 2.69832i 0 −1.49413 + 2.18348i 1.00000i 0 2.69832 + 1.55788i
647.11 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.91362 − 3.31449i 0 1.31918 − 2.29342i 1.00000i 0 3.31449 + 1.91362i
647.12 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.75570 + 3.04097i 0 2.64573 − 0.00972055i 1.00000i 0 −3.04097 − 1.75570i
647.13 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.99239 + 3.45092i 0 1.19252 + 2.36176i − 1.00000i 0 3.45092 + 1.99239i
647.14 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.10791 − 1.91896i 0 −0.709401 + 2.54887i − 1.00000i 0 −1.91896 − 1.10791i
647.15 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.10985 − 1.92231i 0 0.475240 − 2.60272i − 1.00000i 0 −1.92231 − 1.10985i
647.16 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.566607 + 0.981393i 0 −0.203647 − 2.63790i − 1.00000i 0 0.981393 + 0.566607i
647.17 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.681982 − 1.18123i 0 −1.02630 − 2.43859i − 1.00000i 0 −1.18123 − 0.681982i
647.18 0.866025 − 0.500000i 0 0.500000 − 0.866025i −0.0320868 − 0.0555760i 0 −2.60358 − 0.470501i − 1.00000i 0 −0.0555760 − 0.0320868i
647.19 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.140890 + 0.244028i 0 −1.30708 + 2.30034i − 1.00000i 0 0.244028 + 0.140890i
647.20 0.866025 − 0.500000i 0 0.500000 − 0.866025i 0.858944 + 1.48773i 0 1.78243 + 1.95524i − 1.00000i 0 1.48773 + 0.858944i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 647.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.by.a ✓ 48
3.b odd 2 1 2394.2.by.b yes 48
7.d odd 6 1 2394.2.by.b yes 48
21.g even 6 1 inner 2394.2.by.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.by.a ✓ 48 1.a even 1 1 trivial
2394.2.by.a ✓ 48 21.g even 6 1 inner
2394.2.by.b yes 48 3.b odd 2 1
2394.2.by.b yes 48 7.d odd 6 1