Properties

Label 688.2.bg.b
Level $688$
Weight $2$
Character orbit 688.bg
Analytic conductor $5.494$
Analytic rank $0$
Dimension $24$
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] = [688,2,Mod(17,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([0, 0, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.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: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(2\) over \(\Q(\zeta_{21})\)
Twist minimal: no (minimal twist has level 86)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} + 3 q^{5} - 3 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 6 q^{3} + 3 q^{5} - 3 q^{7} - 4 q^{9} - 2 q^{11} - 12 q^{13} + 42 q^{15} + 24 q^{17} + 18 q^{19} + 14 q^{21} + 4 q^{23} - 11 q^{25} - 23 q^{29} + 22 q^{31} + 50 q^{33} - 2 q^{35} + 2 q^{37} + 5 q^{39} - 14 q^{41} - 24 q^{43} + 71 q^{45} - 26 q^{47} + 13 q^{49} + 58 q^{51} + 31 q^{53} - 60 q^{55} - 61 q^{57} + 67 q^{59} + 15 q^{61} + 67 q^{63} - 17 q^{65} + 39 q^{67} - 15 q^{69} - 56 q^{71} - 81 q^{73} - 6 q^{75} + 53 q^{77} + 4 q^{79} + 70 q^{81} - 19 q^{83} - 38 q^{85} - 130 q^{87} - 21 q^{89} - 40 q^{91} + 50 q^{93} + 17 q^{95} + 39 q^{97} + 91 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 0.200549 0.0618612i 0 0.961158 2.44899i 0 −1.57428 + 2.72674i 0 −2.44232 + 1.66515i 0
17.2 0 2.92230 0.901411i 0 −0.758409 + 1.93239i 0 1.24220 2.15155i 0 5.24859 3.57843i 0
81.1 0 0.200549 + 0.0618612i 0 0.961158 + 2.44899i 0 −1.57428 2.72674i 0 −2.44232 1.66515i 0
81.2 0 2.92230 + 0.901411i 0 −0.758409 1.93239i 0 1.24220 + 2.15155i 0 5.24859 + 3.57843i 0
225.1 0 −1.67079 1.55027i 0 0.0705384 0.0106319i 0 −0.174372 0.302021i 0 0.164022 + 2.18872i 0
225.2 0 1.75084 + 1.62455i 0 −0.619298 + 0.0933442i 0 1.53091 + 2.65162i 0 0.202115 + 2.69704i 0
273.1 0 −0.145090 + 0.369683i 0 −0.0922707 1.23127i 0 0.695839 1.20523i 0 2.08354 + 1.93324i 0
273.2 0 1.03020 2.62491i 0 0.260188 + 3.47196i 0 −1.96215 + 3.39854i 0 −3.62970 3.36787i 0
289.1 0 −1.83026 + 1.24785i 0 −0.518590 0.481181i 0 0.925176 + 1.60245i 0 0.696697 1.77516i 0
289.2 0 0.676467 0.461207i 0 1.10645 + 1.02664i 0 −1.97394 3.41896i 0 −0.851128 + 2.16864i 0
353.1 0 −1.37320 0.206976i 0 −1.49011 1.01594i 0 −0.0705203 + 0.122145i 0 −1.02388 0.315826i 0
353.2 0 1.65565 + 0.249549i 0 3.34665 + 2.28171i 0 0.158382 0.274326i 0 −0.187822 0.0579354i 0
369.1 0 −1.83026 1.24785i 0 −0.518590 + 0.481181i 0 0.925176 1.60245i 0 0.696697 + 1.77516i 0
369.2 0 0.676467 + 0.461207i 0 1.10645 1.02664i 0 −1.97394 + 3.41896i 0 −0.851128 2.16864i 0
401.1 0 −0.215103 2.87035i 0 −3.08306 + 0.950998i 0 −1.19592 + 2.07139i 0 −5.22617 + 0.787718i 0
401.2 0 −0.00157347 0.0209965i 0 2.31675 0.714623i 0 0.898666 1.55654i 0 2.96605 0.447061i 0
497.1 0 −1.37320 + 0.206976i 0 −1.49011 + 1.01594i 0 −0.0705203 0.122145i 0 −1.02388 + 0.315826i 0
497.2 0 1.65565 0.249549i 0 3.34665 2.28171i 0 0.158382 + 0.274326i 0 −0.187822 + 0.0579354i 0
513.1 0 −0.215103 + 2.87035i 0 −3.08306 0.950998i 0 −1.19592 2.07139i 0 −5.22617 0.787718i 0
513.2 0 −0.00157347 + 0.0209965i 0 2.31675 + 0.714623i 0 0.898666 + 1.55654i 0 2.96605 + 0.447061i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.2
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.bg.b 24
4.b odd 2 1 86.2.g.b 24
12.b even 2 1 774.2.z.f 24
43.g even 21 1 inner 688.2.bg.b 24
172.o odd 42 1 86.2.g.b 24
172.o odd 42 1 3698.2.a.t 12
172.p even 42 1 3698.2.a.s 12
516.bb even 42 1 774.2.z.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
86.2.g.b 24 4.b odd 2 1
86.2.g.b 24 172.o odd 42 1
688.2.bg.b 24 1.a even 1 1 trivial
688.2.bg.b 24 43.g even 21 1 inner
774.2.z.f 24 12.b even 2 1
774.2.z.f 24 516.bb even 42 1
3698.2.a.s 12 172.p even 42 1
3698.2.a.t 12 172.o odd 42 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} - 6 T_{3}^{23} + 17 T_{3}^{22} - 34 T_{3}^{21} + 43 T_{3}^{20} - 60 T_{3}^{19} + 422 T_{3}^{18} - 1532 T_{3}^{17} + 4016 T_{3}^{16} - 2618 T_{3}^{15} - 11961 T_{3}^{14} + 15786 T_{3}^{13} + 94040 T_{3}^{12} - 179484 T_{3}^{11} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\). Copy content Toggle raw display