Properties

Label 702.2.bb.a
Level 702702
Weight 22
Character orbit 702.bb
Analytic conductor 5.6055.605
Analytic rank 00
Dimension 5656
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [702,2,Mod(71,702)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(702, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([10, 5])) 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("702.71"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 702=23313 702 = 2 \cdot 3^{3} \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 702.bb (of order 1212, degree 44, not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 5.605498221895.60549822189
Analytic rank: 00
Dimension: 5656
Relative dimension: 1414 over Q(ζ12)\Q(\zeta_{12})
Twist minimal: no (minimal twist has level 234)
Sato-Tate group: SU(2)[C12]\mathrm{SU}(2)[C_{12}]

qq-expansion

The algebraic qq-expansion of this newform has not been computed, but we have computed the trace expansion.

Tr(f)(q)=\operatorname{Tr}(f)(q) = 56q+4q756q168q194q28+8q3124q354q37+36q38+48q41+12q4360q47+24q504q52+120q6556q6724q71+28q73++48q98+O(q100) 56 q + 4 q^{7} - 56 q^{16} - 8 q^{19} - 4 q^{28} + 8 q^{31} - 24 q^{35} - 4 q^{37} + 36 q^{38} + 48 q^{41} + 12 q^{43} - 60 q^{47} + 24 q^{50} - 4 q^{52} + 120 q^{65} - 56 q^{67} - 24 q^{71} + 28 q^{73}+ \cdots + 48 q^{98}+O(q^{100}) Copy content Toggle raw display

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\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   a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
71.1 −0.707107 0.707107i 0 1.00000i −3.04921 + 0.817032i 0 2.84235 0.761606i 0.707107 0.707107i 0 2.73384 + 1.57838i
71.2 −0.707107 0.707107i 0 1.00000i −1.62665 + 0.435860i 0 0.290365 0.0778030i 0.707107 0.707107i 0 1.45842 + 0.842018i
71.3 −0.707107 0.707107i 0 1.00000i −1.35053 + 0.361873i 0 0.977097 0.261812i 0.707107 0.707107i 0 1.21085 + 0.699086i
71.4 −0.707107 0.707107i 0 1.00000i −0.650628 + 0.174335i 0 −1.73068 + 0.463733i 0.707107 0.707107i 0 0.583337 + 0.336790i
71.5 −0.707107 0.707107i 0 1.00000i 0.670386 0.179629i 0 −4.43374 + 1.18802i 0.707107 0.707107i 0 −0.601052 0.347017i
71.6 −0.707107 0.707107i 0 1.00000i 2.64098 0.707650i 0 3.36644 0.902034i 0.707107 0.707107i 0 −2.36784 1.36707i
71.7 −0.707107 0.707107i 0 1.00000i 3.36565 0.901822i 0 0.0541850 0.0145188i 0.707107 0.707107i 0 −3.01756 1.74219i
71.8 0.707107 + 0.707107i 0 1.00000i −3.83221 + 1.02684i 0 −1.55176 + 0.415793i −0.707107 + 0.707107i 0 −3.43586 1.98370i
71.9 0.707107 + 0.707107i 0 1.00000i −1.51706 + 0.406496i 0 4.52587 1.21270i −0.707107 + 0.707107i 0 −1.36016 0.785290i
71.10 0.707107 + 0.707107i 0 1.00000i −0.994452 + 0.266463i 0 0.339163 0.0908784i −0.707107 + 0.707107i 0 −0.891601 0.514766i
71.11 0.707107 + 0.707107i 0 1.00000i −0.653109 + 0.175000i 0 −3.90017 + 1.04505i −0.707107 + 0.707107i 0 −0.585562 0.338074i
71.12 0.707107 + 0.707107i 0 1.00000i −0.339151 + 0.0908752i 0 2.97685 0.797644i −0.707107 + 0.707107i 0 −0.304074 0.175557i
71.13 0.707107 + 0.707107i 0 1.00000i 3.62177 0.970449i 0 −3.13748 + 0.840685i −0.707107 + 0.707107i 0 3.24719 + 1.87476i
71.14 0.707107 + 0.707107i 0 1.00000i 3.71422 0.995221i 0 2.11355 0.566325i −0.707107 + 0.707107i 0 3.33007 + 1.92262i
89.1 −0.707107 + 0.707107i 0 1.00000i −3.04921 0.817032i 0 2.84235 + 0.761606i 0.707107 + 0.707107i 0 2.73384 1.57838i
89.2 −0.707107 + 0.707107i 0 1.00000i −1.62665 0.435860i 0 0.290365 + 0.0778030i 0.707107 + 0.707107i 0 1.45842 0.842018i
89.3 −0.707107 + 0.707107i 0 1.00000i −1.35053 0.361873i 0 0.977097 + 0.261812i 0.707107 + 0.707107i 0 1.21085 0.699086i
89.4 −0.707107 + 0.707107i 0 1.00000i −0.650628 0.174335i 0 −1.73068 0.463733i 0.707107 + 0.707107i 0 0.583337 0.336790i
89.5 −0.707107 + 0.707107i 0 1.00000i 0.670386 + 0.179629i 0 −4.43374 1.18802i 0.707107 + 0.707107i 0 −0.601052 + 0.347017i
89.6 −0.707107 + 0.707107i 0 1.00000i 2.64098 + 0.707650i 0 3.36644 + 0.902034i 0.707107 + 0.707107i 0 −2.36784 + 1.36707i
See all 56 embeddings
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
117.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 702.2.bb.a 56
3.b odd 2 1 234.2.y.a 56
9.c even 3 1 234.2.z.a yes 56
9.d odd 6 1 702.2.bc.a 56
13.f odd 12 1 702.2.bc.a 56
39.k even 12 1 234.2.z.a yes 56
117.w odd 12 1 234.2.y.a 56
117.x even 12 1 inner 702.2.bb.a 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
234.2.y.a 56 3.b odd 2 1
234.2.y.a 56 117.w odd 12 1
234.2.z.a yes 56 9.c even 3 1
234.2.z.a yes 56 39.k even 12 1
702.2.bb.a 56 1.a even 1 1 trivial
702.2.bb.a 56 117.x even 12 1 inner
702.2.bc.a 56 9.d odd 6 1
702.2.bc.a 56 13.f odd 12 1

Hecke kernels

This newform subspace is the entire newspace S2new(702,[χ])S_{2}^{\mathrm{new}}(702, [\chi]).