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Properties

Label 352.2.u.a
Level $352$
Weight $2$
Character orbit 352.u
Analytic conductor $2.811$
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] = [352,2,Mod(63,352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(352, 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("352.63");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 352 = 2^{5} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 352.u (of order \(10\), degree \(4\), 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: \(2.81073415115\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
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
95.1 0 −2.59733 − 0.843923i 0 −3.24636 + 2.35862i 0 −0.712525 − 2.19293i 0 3.60686 + 2.62054i 0
95.2 0 −2.56359 − 0.832959i 0 1.18682 − 0.862276i 0 0.845653 + 2.60265i 0 3.45110 + 2.50737i 0
95.3 0 −2.26076 − 0.734564i 0 0.0210968 − 0.0153277i 0 −0.140366 − 0.432001i 0 2.14438 + 1.55799i 0
95.4 0 −1.20227 − 0.390641i 0 3.22527 − 2.34329i 0 −0.861490 − 2.65139i 0 −1.13420 − 0.824045i 0
95.5 0 −0.821048 − 0.266775i 0 0.946699 − 0.687817i 0 1.39243 + 4.28545i 0 −1.82410 − 1.32529i 0
95.6 0 −0.203695 − 0.0661845i 0 −2.13352 + 1.55009i 0 −0.527338 − 1.62298i 0 −2.38994 − 1.73639i 0
95.7 0 0.203695 + 0.0661845i 0 −2.13352 + 1.55009i 0 0.527338 + 1.62298i 0 −2.38994 − 1.73639i 0
95.8 0 0.821048 + 0.266775i 0 0.946699 − 0.687817i 0 −1.39243 − 4.28545i 0 −1.82410 − 1.32529i 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 352.2.u.a ✓ 48
4.b odd 2 1 inner 352.2.u.a ✓ 48
8.b even 2 1 704.2.u.d 48
8.d odd 2 1 704.2.u.d 48
11.d odd 10 1 inner 352.2.u.a ✓ 48
44.g even 10 1 inner 352.2.u.a ✓ 48
88.k even 10 1 704.2.u.d 48
88.p odd 10 1 704.2.u.d 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.u.a ✓ 48 1.a even 1 1 trivial
352.2.u.a ✓ 48 4.b odd 2 1 inner
352.2.u.a ✓ 48 11.d odd 10 1 inner
352.2.u.a ✓ 48 44.g even 10 1 inner
704.2.u.d 48 8.b even 2 1
704.2.u.d 48 8.d odd 2 1
704.2.u.d 48 88.k even 10 1
704.2.u.d 48 88.p odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(352, [\chi])\).