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Properties

Label 637.2.u.j
Level $637$
Weight $2$
Character orbit 637.u
Analytic conductor $5.086$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [637,2,Mod(30,637)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(637, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("637.30");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.u (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: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 20 q^{4} + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 20 q^{4} + 56 q^{9} - 12 q^{15} - 28 q^{16} - 24 q^{18} + 8 q^{22} + 24 q^{23} + 20 q^{25} - 24 q^{29} - 48 q^{30} + 60 q^{32} + 92 q^{36} + 52 q^{39} + 12 q^{43} + 24 q^{46} + 12 q^{50} - 36 q^{53} - 276 q^{60} + 32 q^{64} + 12 q^{65} - 48 q^{71} + 24 q^{74} - 156 q^{78} - 48 q^{79} + 128 q^{81} + 12 q^{85} + 48 q^{86} - 112 q^{88} + 168 q^{92} + 48 q^{93} + 84 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1 −2.30510 + 1.33085i −3.18962 2.54233 − 4.40345i 0.285259 + 0.164694i 7.35241 − 4.24492i 0 8.21047i 7.17371 −0.876736
30.2 −2.30510 + 1.33085i 3.18962 2.54233 − 4.40345i −0.285259 − 0.164694i −7.35241 + 4.24492i 0 8.21047i 7.17371 0.876736
30.3 −1.81104 + 1.04560i −1.32799 1.18657 − 2.05519i 3.02157 + 1.74450i 2.40503 − 1.38855i 0 0.780297i −1.23645 −7.29622
30.4 −1.81104 + 1.04560i 1.32799 1.18657 − 2.05519i −3.02157 − 1.74450i −2.40503 + 1.38855i 0 0.780297i −1.23645 7.29622
30.5 −0.900699 + 0.520019i −0.769363 −0.459161 + 0.795291i 1.45206 + 0.838345i 0.692964 − 0.400083i 0 − 3.03516i −2.40808 −1.74382
30.6 −0.900699 + 0.520019i 0.769363 −0.459161 + 0.795291i −1.45206 − 0.838345i −0.692964 + 0.400083i 0 − 3.03516i −2.40808 1.74382
30.7 −0.489742 + 0.282753i −3.09111 −0.840102 + 1.45510i −1.56330 − 0.902570i 1.51385 − 0.874021i 0 − 2.08118i 6.55499 1.02082
30.8 −0.489742 + 0.282753i 3.09111 −0.840102 + 1.45510i 1.56330 + 0.902570i −1.51385 + 0.874021i 0 − 2.08118i 6.55499 −1.02082
30.9 0.250157 − 0.144428i −1.93984 −0.958281 + 1.65979i −3.71818 − 2.14669i −0.485264 + 0.280167i 0 1.13132i 0.762985 −1.24017
30.10 0.250157 − 0.144428i 1.93984 −0.958281 + 1.65979i 3.71818 + 2.14669i 0.485264 − 0.280167i 0 1.13132i 0.762985 1.24017
30.11 1.12902 − 0.651838i −0.269938 −0.150215 + 0.260179i −1.35808 − 0.784090i −0.304765 + 0.175956i 0 2.99901i −2.92713 −2.04440
30.12 1.12902 − 0.651838i 0.269938 −0.150215 + 0.260179i 1.35808 + 0.784090i 0.304765 − 0.175956i 0 2.99901i −2.92713 2.04440
30.13 2.04719 − 1.18194i −2.98693 1.79399 − 3.10727i 2.77356 + 1.60131i −6.11481 + 3.53039i 0 − 3.75379i 5.92175 7.57066
30.14 2.04719 − 1.18194i 2.98693 1.79399 − 3.10727i −2.77356 − 1.60131i 6.11481 − 3.53039i 0 − 3.75379i 5.92175 −7.57066
30.15 2.08022 − 1.20101i −1.77714 1.88487 − 3.26470i 0.611970 + 0.353321i −3.69684 + 2.13437i 0 − 4.25098i 0.158233 1.69737
30.16 2.08022 − 1.20101i 1.77714 1.88487 − 3.26470i −0.611970 − 0.353321i 3.69684 − 2.13437i 0 − 4.25098i 0.158233 −1.69737
361.1 −2.30510 − 1.33085i −3.18962 2.54233 + 4.40345i 0.285259 − 0.164694i 7.35241 + 4.24492i 0 − 8.21047i 7.17371 −0.876736
361.2 −2.30510 − 1.33085i 3.18962 2.54233 + 4.40345i −0.285259 + 0.164694i −7.35241 − 4.24492i 0 − 8.21047i 7.17371 0.876736
361.3 −1.81104 − 1.04560i −1.32799 1.18657 + 2.05519i 3.02157 − 1.74450i 2.40503 + 1.38855i 0 − 0.780297i −1.23645 −7.29622
361.4 −1.81104 − 1.04560i 1.32799 1.18657 + 2.05519i −3.02157 + 1.74450i −2.40503 − 1.38855i 0 − 0.780297i −1.23645 7.29622
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
91.p odd 6 1 inner
91.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.u.j 32
7.b odd 2 1 inner 637.2.u.j 32
7.c even 3 1 637.2.k.j 32
7.c even 3 1 637.2.q.j ✓ 32
7.d odd 6 1 637.2.k.j 32
7.d odd 6 1 637.2.q.j ✓ 32
13.e even 6 1 637.2.k.j 32
91.k even 6 1 637.2.q.j ✓ 32
91.l odd 6 1 637.2.q.j ✓ 32
91.p odd 6 1 inner 637.2.u.j 32
91.t odd 6 1 637.2.k.j 32
91.u even 6 1 inner 637.2.u.j 32
91.w even 12 2 8281.2.a.cx 32
91.bd odd 12 2 8281.2.a.cx 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.j 32 7.c even 3 1
637.2.k.j 32 7.d odd 6 1
637.2.k.j 32 13.e even 6 1
637.2.k.j 32 91.t odd 6 1
637.2.q.j ✓ 32 7.c even 3 1
637.2.q.j ✓ 32 7.d odd 6 1
637.2.q.j ✓ 32 91.k even 6 1
637.2.q.j ✓ 32 91.l odd 6 1
637.2.u.j 32 1.a even 1 1 trivial
637.2.u.j 32 7.b odd 2 1 inner
637.2.u.j 32 91.p odd 6 1 inner
637.2.u.j 32 91.u even 6 1 inner
8281.2.a.cx 32 91.w even 12 2
8281.2.a.cx 32 91.bd odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\( T_{2}^{16} - 13 T_{2}^{14} + 119 T_{2}^{12} - 6 T_{2}^{11} - 544 T_{2}^{10} + 18 T_{2}^{9} + 1804 T_{2}^{8} + \cdots + 49 \) Copy content Toggle raw display
\( T_{3}^{16} - 38T_{3}^{14} + 571T_{3}^{12} - 4316T_{3}^{10} + 17365T_{3}^{8} - 36572T_{3}^{6} + 36508T_{3}^{4} - 13232T_{3}^{2} + 784 \) Copy content Toggle raw display