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Properties

Label 704.2.u.d
Level $704$
Weight $2$
Character orbit 704.u
Analytic conductor $5.621$
Analytic rank $0$
Dimension $48$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [704,2,Mod(63,704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(704, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("704.63");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 704 = 2^{6} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 704.u (of order \(10\), degree \(4\), 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: \(5.62146830230\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: no (minimal twist has level 352)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{9} + 4 q^{25} + 36 q^{33} + 40 q^{41} + 96 q^{45} - 4 q^{49} + 8 q^{53} + 20 q^{57} + 8 q^{69} - 40 q^{73} + 72 q^{77} - 72 q^{81} + 80 q^{85} - 40 q^{89} - 8 q^{93} + 4 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
63.1 0 −2.59733 + 0.843923i 0 3.24636 + 2.35862i 0 0.712525 − 2.19293i 0 3.60686 − 2.62054i 0
63.2 0 −2.56359 + 0.832959i 0 −1.18682 − 0.862276i 0 −0.845653 + 2.60265i 0 3.45110 − 2.50737i 0
63.3 0 −2.26076 + 0.734564i 0 −0.0210968 − 0.0153277i 0 0.140366 − 0.432001i 0 2.14438 − 1.55799i 0
63.4 0 −1.20227 + 0.390641i 0 −3.22527 − 2.34329i 0 0.861490 − 2.65139i 0 −1.13420 + 0.824045i 0
63.5 0 −0.821048 + 0.266775i 0 −0.946699 − 0.687817i 0 −1.39243 + 4.28545i 0 −1.82410 + 1.32529i 0
63.6 0 −0.203695 + 0.0661845i 0 2.13352 + 1.55009i 0 0.527338 − 1.62298i 0 −2.38994 + 1.73639i 0
63.7 0 0.203695 − 0.0661845i 0 2.13352 + 1.55009i 0 −0.527338 + 1.62298i 0 −2.38994 + 1.73639i 0
63.8 0 0.821048 − 0.266775i 0 −0.946699 − 0.687817i 0 1.39243 − 4.28545i 0 −1.82410 + 1.32529i 0
63.9 0 1.20227 − 0.390641i 0 −3.22527 − 2.34329i 0 −0.861490 + 2.65139i 0 −1.13420 + 0.824045i 0
63.10 0 2.26076 − 0.734564i 0 −0.0210968 − 0.0153277i 0 −0.140366 + 0.432001i 0 2.14438 − 1.55799i 0
63.11 0 2.56359 − 0.832959i 0 −1.18682 − 0.862276i 0 0.845653 − 2.60265i 0 3.45110 − 2.50737i 0
63.12 0 2.59733 − 0.843923i 0 3.24636 + 2.35862i 0 −0.712525 + 2.19293i 0 3.60686 − 2.62054i 0
127.1 0 −1.92996 + 2.65636i 0 −0.755086 + 2.32392i 0 −3.01001 + 2.18690i 0 −2.40446 − 7.40017i 0
127.2 0 −1.68102 + 2.31373i 0 0.612659 − 1.88557i 0 1.05680 − 0.767808i 0 −1.60046 − 4.92570i 0
127.3 0 −1.24141 + 1.70866i 0 −0.547096 + 1.68379i 0 3.48626 − 2.53292i 0 −0.451352 − 1.38912i 0
127.4 0 −0.896074 + 1.23334i 0 1.21876 − 3.75097i 0 −1.48073 + 1.07581i 0 0.208872 + 0.642842i 0
127.5 0 −0.708608 + 0.975315i 0 −0.671667 + 2.06718i 0 1.18220 − 0.858919i 0 0.477936 + 1.47094i 0
127.6 0 −0.114336 + 0.157369i 0 0.142428 − 0.438348i 0 1.69623 − 1.23238i 0 0.915358 + 2.81718i 0
127.7 0 0.114336 − 0.157369i 0 0.142428 − 0.438348i 0 −1.69623 + 1.23238i 0 0.915358 + 2.81718i 0
127.8 0 0.708608 − 0.975315i 0 −0.671667 + 2.06718i 0 −1.18220 + 0.858919i 0 0.477936 + 1.47094i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 63.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 704.2.u.d 48
4.b odd 2 1 inner 704.2.u.d 48
8.b even 2 1 352.2.u.a ✓ 48
8.d odd 2 1 352.2.u.a ✓ 48
11.d odd 10 1 inner 704.2.u.d 48
44.g even 10 1 inner 704.2.u.d 48
88.k even 10 1 352.2.u.a ✓ 48
88.p odd 10 1 352.2.u.a ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.u.a ✓ 48 8.b even 2 1
352.2.u.a ✓ 48 8.d odd 2 1
352.2.u.a ✓ 48 88.k even 10 1
352.2.u.a ✓ 48 88.p odd 10 1
704.2.u.d 48 1.a even 1 1 trivial
704.2.u.d 48 4.b odd 2 1 inner
704.2.u.d 48 11.d odd 10 1 inner
704.2.u.d 48 44.g even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{48} - 20 T_{3}^{46} + 326 T_{3}^{44} - 4678 T_{3}^{42} + 56247 T_{3}^{40} - 527972 T_{3}^{38} + \cdots + 707281 \) acting on \(S_{2}^{\mathrm{new}}(704, [\chi])\). Copy content Toggle raw display