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Properties

Label 680.2.cg.b
Level $680$
Weight $2$
Character orbit 680.cg
Analytic conductor $5.430$
Analytic rank $0$
Dimension $112$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [680,2,Mod(57,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(16)) chi = DirichletCharacter(H, H._module([0, 0, 4, 15])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("680.57"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.cg (of order \(16\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [112] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(112\)
Relative dimension: \(14\) over \(\Q(\zeta_{16})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 112 q + 16 q^{15} - 24 q^{27} - 8 q^{29} + 16 q^{31} - 48 q^{33} - 32 q^{35} - 16 q^{37} - 16 q^{41} + 48 q^{43} - 24 q^{45} - 16 q^{47} - 80 q^{49} + 32 q^{51} - 8 q^{53} + 24 q^{55} - 80 q^{59} - 24 q^{61}+ \cdots + 48 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
57.1 0 −1.67884 + 2.51256i 0 0.731459 − 2.11305i 0 0.654989 − 3.29285i 0 −2.34641 − 5.66473i 0
57.2 0 −1.57782 + 2.36138i 0 1.71355 + 1.43658i 0 −0.875769 + 4.40279i 0 −1.93854 − 4.68004i 0
57.3 0 −1.25748 + 1.88195i 0 1.85120 + 1.25422i 0 0.697919 − 3.50868i 0 −0.812435 − 1.96139i 0
57.4 0 −1.00723 + 1.50743i 0 −1.73942 + 1.40514i 0 −0.216959 + 1.09073i 0 −0.109773 − 0.265016i 0
57.5 0 −0.716619 + 1.07250i 0 −2.20954 − 0.343430i 0 0.511686 − 2.57242i 0 0.511345 + 1.23450i 0
57.6 0 −0.561368 + 0.840147i 0 −0.645428 − 2.14089i 0 −0.790553 + 3.97438i 0 0.757338 + 1.82838i 0
57.7 0 −0.187109 + 0.280028i 0 0.753771 − 2.10519i 0 −0.0433030 + 0.217699i 0 1.10464 + 2.66685i 0
57.8 0 0.203615 − 0.304732i 0 −0.154785 + 2.23070i 0 −0.0416810 + 0.209544i 0 1.09665 + 2.64754i 0
57.9 0 0.453931 − 0.679356i 0 1.87174 + 1.22336i 0 0.141249 − 0.710106i 0 0.892579 + 2.15488i 0
57.10 0 0.565223 − 0.845916i 0 1.96453 − 1.06800i 0 0.574952 − 2.89048i 0 0.751953 + 1.81538i 0
57.11 0 1.19803 − 1.79298i 0 −2.17385 + 0.523803i 0 −0.696254 + 3.50030i 0 −0.631450 − 1.52445i 0
57.12 0 1.24956 − 1.87011i 0 −1.06474 − 1.96630i 0 0.175316 − 0.881373i 0 −0.787834 − 1.90200i 0
57.13 0 1.46734 − 2.19603i 0 2.21098 + 0.334048i 0 −0.900654 + 4.52789i 0 −1.52140 − 3.67298i 0
57.14 0 1.84877 − 2.76687i 0 −0.388676 + 2.20203i 0 0.809062 − 4.06743i 0 −3.08961 − 7.45897i 0
73.1 0 −0.596630 + 2.99946i 0 2.04191 + 0.911372i 0 1.06574 + 0.712103i 0 −5.86916 − 2.43109i 0
73.2 0 −0.539541 + 2.71245i 0 −1.51271 + 1.64673i 0 −1.85121 − 1.23694i 0 −4.29466 − 1.77891i 0
73.3 0 −0.446513 + 2.24477i 0 0.682045 − 2.12951i 0 −1.62658 − 1.08684i 0 −2.06799 − 0.856589i 0
73.4 0 −0.431185 + 2.16772i 0 −1.72137 − 1.42720i 0 3.13047 + 2.09171i 0 −1.74143 − 0.721324i 0
73.5 0 −0.148366 + 0.745884i 0 −1.88881 − 1.19683i 0 −2.96249 − 1.97947i 0 2.23731 + 0.926723i 0
73.6 0 −0.126113 + 0.634012i 0 2.23108 + 0.149225i 0 −3.28454 − 2.19466i 0 2.38557 + 0.988136i 0
See next 80 embeddings (of 112 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 57.14
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.o even 16 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.cg.b ✓ 112
5.c odd 4 1 680.2.cq.b yes 112
17.e odd 16 1 680.2.cq.b yes 112
85.o even 16 1 inner 680.2.cg.b ✓ 112
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.cg.b ✓ 112 1.a even 1 1 trivial
680.2.cg.b ✓ 112 85.o even 16 1 inner
680.2.cq.b yes 112 5.c odd 4 1
680.2.cq.b yes 112 17.e odd 16 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{112} + 8 T_{3}^{109} - 96 T_{3}^{107} + 216 T_{3}^{106} + 184 T_{3}^{105} - 768 T_{3}^{104} + \cdots + 592973922304 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\). Copy content Toggle raw display