Properties

Label 784.2.bg.d
Level $784$
Weight $2$
Character orbit 784.bg
Analytic conductor $6.260$
Analytic rank $0$
Dimension $60$
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] = [784,2,Mod(65,784)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(784, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 20]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("784.65");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.bg (of order \(21\), degree \(12\), 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: \(6.26027151847\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(5\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - q^{3} + 3 q^{5} - 4 q^{7} - 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 60 q - q^{3} + 3 q^{5} - 4 q^{7} - 10 q^{9} - 6 q^{11} - 4 q^{13} + 16 q^{15} + 10 q^{17} + 15 q^{19} + q^{21} - 22 q^{23} + 6 q^{25} - 31 q^{27} - 10 q^{29} + 7 q^{31} - 3 q^{33} + 29 q^{35} - 12 q^{37} + 25 q^{39} - 40 q^{41} - 12 q^{43} - 97 q^{45} + 58 q^{47} - 58 q^{49} + 55 q^{51} + 8 q^{53} + 87 q^{55} + 31 q^{57} + 33 q^{59} - 8 q^{61} - 3 q^{63} + 2 q^{65} + 7 q^{67} - 34 q^{69} - 21 q^{71} + 20 q^{73} + 80 q^{75} + 18 q^{77} + 9 q^{79} + 47 q^{81} - 102 q^{83} - 62 q^{85} - 92 q^{87} + 127 q^{89} + 6 q^{91} + 210 q^{93} - 111 q^{95} - 8 q^{97} - 108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −0.216081 2.88341i 0 2.57232 1.75378i 0 −2.26222 1.37200i 0 −5.30084 + 0.798973i 0
65.2 0 −0.158105 2.10976i 0 −2.26833 + 1.54652i 0 2.64238 0.133544i 0 −1.45959 + 0.219998i 0
65.3 0 0.000218519 0.00291593i 0 −0.944972 + 0.644271i 0 −1.63103 2.08320i 0 2.96648 0.447126i 0
65.4 0 0.0826197 + 1.10248i 0 0.0857004 0.0584295i 0 −0.348054 + 2.62276i 0 1.75785 0.264953i 0
65.5 0 0.216618 + 2.89056i 0 3.41911 2.33111i 0 1.03083 2.43668i 0 −5.34193 + 0.805166i 0
81.1 0 −3.19937 0.986875i 0 −2.19185 + 2.03374i 0 0.765446 + 2.53261i 0 6.78332 + 4.62479i 0
81.2 0 −1.60758 0.495873i 0 1.38504 1.28513i 0 −0.118620 2.64309i 0 −0.140286 0.0956455i 0
81.3 0 0.446040 + 0.137585i 0 −3.09708 + 2.87367i 0 −2.15190 1.53927i 0 −2.29869 1.56722i 0
81.4 0 1.14095 + 0.351936i 0 0.218886 0.203097i 0 −1.38333 + 2.25531i 0 −1.30081 0.886879i 0
81.5 0 2.26439 + 0.698472i 0 0.769176 0.713691i 0 2.64575 + 0.00387612i 0 2.16088 + 1.47326i 0
193.1 0 −0.216081 + 2.88341i 0 2.57232 + 1.75378i 0 −2.26222 + 1.37200i 0 −5.30084 0.798973i 0
193.2 0 −0.158105 + 2.10976i 0 −2.26833 1.54652i 0 2.64238 + 0.133544i 0 −1.45959 0.219998i 0
193.3 0 0.000218519 0.00291593i 0 −0.944972 0.644271i 0 −1.63103 + 2.08320i 0 2.96648 + 0.447126i 0
193.4 0 0.0826197 1.10248i 0 0.0857004 + 0.0584295i 0 −0.348054 2.62276i 0 1.75785 + 0.264953i 0
193.5 0 0.216618 2.89056i 0 3.41911 + 2.33111i 0 1.03083 + 2.43668i 0 −5.34193 0.805166i 0
289.1 0 −2.40678 1.64092i 0 −0.0461477 + 0.615798i 0 2.25198 1.38874i 0 2.00397 + 5.10604i 0
289.2 0 −1.11615 0.760977i 0 −0.133970 + 1.78771i 0 −2.40807 + 1.09600i 0 −0.429322 1.09389i 0
289.3 0 −0.230121 0.156894i 0 0.288343 3.84767i 0 2.26023 + 1.37526i 0 −1.06768 2.72041i 0
289.4 0 1.10186 + 0.751237i 0 0.0987472 1.31769i 0 −2.24658 1.39745i 0 −0.446280 1.13710i 0
289.5 0 1.82495 + 1.24423i 0 −0.200332 + 2.67325i 0 1.68003 2.04389i 0 0.686316 + 1.74870i 0
See all 60 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.2.bg.d 60
4.b odd 2 1 196.2.m.a 60
49.g even 21 1 inner 784.2.bg.d 60
196.o odd 42 1 196.2.m.a 60
196.o odd 42 1 9604.2.a.g 30
196.p even 42 1 9604.2.a.f 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.m.a 60 4.b odd 2 1
196.2.m.a 60 196.o odd 42 1
784.2.bg.d 60 1.a even 1 1 trivial
784.2.bg.d 60 49.g even 21 1 inner
9604.2.a.f 30 196.p even 42 1
9604.2.a.g 30 196.o odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{60} + T_{3}^{59} - 2 T_{3}^{58} + 4 T_{3}^{57} - 35 T_{3}^{56} - 207 T_{3}^{55} + 124 T_{3}^{54} - 538 T_{3}^{53} + 1273 T_{3}^{52} + 728 T_{3}^{51} - 17455 T_{3}^{50} - 149413 T_{3}^{49} + 371884 T_{3}^{48} + 1504928 T_{3}^{47} + \cdots + 6889 \) acting on \(S_{2}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display