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Properties

Label 688.2.bb.a
Level $688$
Weight $2$
Character orbit 688.bb
Analytic conductor $5.494$
Analytic rank $0$
Dimension $36$
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] = [688,2,Mod(223,688)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(688, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("688.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.bb (of order \(14\), degree \(6\), 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: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 2 q^{9} + 26 q^{13} - 2 q^{17} - 16 q^{21} - 10 q^{25} - 28 q^{29} - 42 q^{33} - 14 q^{45} + 12 q^{49} - 14 q^{53} - 60 q^{57} - 14 q^{73} - 14 q^{77} - 44 q^{81} + 42 q^{89} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1 0 −0.544206 − 2.38432i 0 2.21977 + 1.77021i 0 1.77535 0 −2.68593 + 1.29347i 0
223.2 0 −0.304826 − 1.33553i 0 0.930525 + 0.742069i 0 −4.73289 0 1.01218 − 0.487442i 0
223.3 0 −0.210551 − 0.922483i 0 −1.62583 − 1.29656i 0 1.09033 0 1.89626 − 0.913192i 0
223.4 0 0.210551 + 0.922483i 0 −1.62583 − 1.29656i 0 −1.09033 0 1.89626 − 0.913192i 0
223.5 0 0.304826 + 1.33553i 0 0.930525 + 0.742069i 0 4.73289 0 1.01218 − 0.487442i 0
223.6 0 0.544206 + 2.38432i 0 2.21977 + 1.77021i 0 −1.77535 0 −2.68593 + 1.29347i 0
303.1 0 −2.44774 − 1.17877i 0 1.46426 + 0.334207i 0 −1.50985 0 2.73147 + 3.42515i 0
303.2 0 −1.50976 − 0.727063i 0 −2.79950 − 0.638967i 0 2.26473 0 −0.119709 − 0.150110i 0
303.3 0 −0.455953 − 0.219575i 0 0.489230 + 0.111664i 0 3.77574 0 −1.71079 − 2.14526i 0
303.4 0 0.455953 + 0.219575i 0 0.489230 + 0.111664i 0 −3.77574 0 −1.71079 − 2.14526i 0
303.5 0 1.50976 + 0.727063i 0 −2.79950 − 0.638967i 0 −2.26473 0 −0.119709 − 0.150110i 0
303.6 0 2.44774 + 1.17877i 0 1.46426 + 0.334207i 0 1.50985 0 2.73147 + 3.42515i 0
383.1 0 −1.84328 − 2.31139i 0 −0.113093 + 0.234840i 0 1.02557 0 −1.27732 + 5.59629i 0
383.2 0 −1.04959 − 1.31615i 0 0.759013 − 1.57611i 0 −0.281989 0 0.0369605 − 0.161935i 0
383.3 0 −0.297602 − 0.373181i 0 −1.32437 + 2.75008i 0 4.05745 0 0.616866 − 2.70267i 0
383.4 0 0.297602 + 0.373181i 0 −1.32437 + 2.75008i 0 −4.05745 0 0.616866 − 2.70267i 0
383.5 0 1.04959 + 1.31615i 0 0.759013 − 1.57611i 0 0.281989 0 0.0369605 − 0.161935i 0
383.6 0 1.84328 + 2.31139i 0 −0.113093 + 0.234840i 0 −1.02557 0 −1.27732 + 5.59629i 0
495.1 0 −2.44774 + 1.17877i 0 1.46426 − 0.334207i 0 −1.50985 0 2.73147 − 3.42515i 0
495.2 0 −1.50976 + 0.727063i 0 −2.79950 + 0.638967i 0 2.26473 0 −0.119709 + 0.150110i 0
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
43.f odd 14 1 inner
172.j even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.bb.a ✓ 36
4.b odd 2 1 inner 688.2.bb.a ✓ 36
43.f odd 14 1 inner 688.2.bb.a ✓ 36
172.j even 14 1 inner 688.2.bb.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
688.2.bb.a ✓ 36 1.a even 1 1 trivial
688.2.bb.a ✓ 36 4.b odd 2 1 inner
688.2.bb.a ✓ 36 43.f odd 14 1 inner
688.2.bb.a ✓ 36 172.j even 14 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{36} + 8 T_{3}^{34} + 97 T_{3}^{32} + 606 T_{3}^{30} + 4038 T_{3}^{28} + 38638 T_{3}^{26} + \cdots + 90601 \) acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\). Copy content Toggle raw display