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Properties

Label 840.2.j.e
Level $840$
Weight $2$
Character orbit 840.j
Analytic conductor $6.707$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [840,2,Mod(589,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(840, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1, 0])) 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("840.589"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.j (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,-2,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.70743376979\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 2 q^{2} - 32 q^{3} - 2 q^{4} + 2 q^{6} - 2 q^{8} + 32 q^{9} + 2 q^{12} - 32 q^{13} + 6 q^{16} - 2 q^{18} + 12 q^{20} + 2 q^{24} - 16 q^{25} + 12 q^{26} - 32 q^{27} - 8 q^{28} + 32 q^{31} - 22 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
589.1 −1.41390 − 0.0297183i −1.00000 1.99823 + 0.0840375i −1.06062 + 1.96852i 1.41390 + 0.0297183i 1.00000i −2.82281 − 0.178205i 1.00000 1.55812 − 2.75178i
589.2 −1.41390 + 0.0297183i −1.00000 1.99823 − 0.0840375i −1.06062 − 1.96852i 1.41390 − 0.0297183i − 1.00000i −2.82281 + 0.178205i 1.00000 1.55812 + 2.75178i
589.3 −1.39151 − 0.252412i −1.00000 1.87258 + 0.702466i 2.03532 − 0.926004i 1.39151 + 0.252412i − 1.00000i −2.42839 − 1.45015i 1.00000 −3.06589 + 0.774802i
589.4 −1.39151 + 0.252412i −1.00000 1.87258 − 0.702466i 2.03532 + 0.926004i 1.39151 − 0.252412i 1.00000i −2.42839 + 1.45015i 1.00000 −3.06589 − 0.774802i
589.5 −1.24401 − 0.672631i −1.00000 1.09514 + 1.67352i 0.123450 + 2.23266i 1.24401 + 0.672631i − 1.00000i −0.236701 − 2.81851i 1.00000 1.34818 − 2.86049i
589.6 −1.24401 + 0.672631i −1.00000 1.09514 − 1.67352i 0.123450 − 2.23266i 1.24401 − 0.672631i 1.00000i −0.236701 + 2.81851i 1.00000 1.34818 + 2.86049i
589.7 −1.20719 − 0.736672i −1.00000 0.914630 + 1.77861i −1.64475 − 1.51486i 1.20719 + 0.736672i 1.00000i 0.206117 − 2.82091i 1.00000 0.869577 + 3.04037i
589.8 −1.20719 + 0.736672i −1.00000 0.914630 − 1.77861i −1.64475 + 1.51486i 1.20719 − 0.736672i − 1.00000i 0.206117 + 2.82091i 1.00000 0.869577 − 3.04037i
589.9 −0.870129 − 1.11484i −1.00000 −0.485751 + 1.94012i 1.48054 + 1.67571i 0.870129 + 1.11484i 1.00000i 2.58559 − 1.14661i 1.00000 0.579891 − 3.10865i
589.10 −0.870129 + 1.11484i −1.00000 −0.485751 − 1.94012i 1.48054 − 1.67571i 0.870129 − 1.11484i − 1.00000i 2.58559 + 1.14661i 1.00000 0.579891 + 3.10865i
589.11 −0.538709 − 1.30759i −1.00000 −1.41959 + 1.40882i 0.725275 − 2.11518i 0.538709 + 1.30759i 1.00000i 2.60690 + 1.09729i 1.00000 −3.15650 + 0.191102i
589.12 −0.538709 + 1.30759i −1.00000 −1.41959 − 1.40882i 0.725275 + 2.11518i 0.538709 − 1.30759i − 1.00000i 2.60690 − 1.09729i 1.00000 −3.15650 − 0.191102i
589.13 −0.453365 − 1.33957i −1.00000 −1.58892 + 1.21463i −2.12585 − 0.693382i 0.453365 + 1.33957i − 1.00000i 2.34745 + 1.57780i 1.00000 0.0349481 + 3.16208i
589.14 −0.453365 + 1.33957i −1.00000 −1.58892 − 1.21463i −2.12585 + 0.693382i 0.453365 − 1.33957i 1.00000i 2.34745 − 1.57780i 1.00000 0.0349481 − 3.16208i
589.15 −0.406657 − 1.35449i −1.00000 −1.66926 + 1.10162i 0.836338 − 2.07377i 0.406657 + 1.35449i − 1.00000i 2.17095 + 1.81300i 1.00000 −3.14900 − 0.289491i
589.16 −0.406657 + 1.35449i −1.00000 −1.66926 − 1.10162i 0.836338 + 2.07377i 0.406657 − 1.35449i 1.00000i 2.17095 − 1.81300i 1.00000 −3.14900 + 0.289491i
589.17 0.0454850 − 1.41348i −1.00000 −1.99586 − 0.128585i −1.37352 + 1.76449i −0.0454850 + 1.41348i 1.00000i −0.272534 + 2.81527i 1.00000 2.43160 + 2.02171i
589.18 0.0454850 + 1.41348i −1.00000 −1.99586 + 0.128585i −1.37352 − 1.76449i −0.0454850 − 1.41348i − 1.00000i −0.272534 − 2.81527i 1.00000 2.43160 − 2.02171i
589.19 0.150941 − 1.40614i −1.00000 −1.95443 − 0.424487i 2.20890 + 0.347530i −0.150941 + 1.40614i 1.00000i −0.891891 + 2.68413i 1.00000 0.822088 − 3.05355i
589.20 0.150941 + 1.40614i −1.00000 −1.95443 + 0.424487i 2.20890 − 0.347530i −0.150941 − 1.40614i − 1.00000i −0.891891 − 2.68413i 1.00000 0.822088 + 3.05355i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 840.2.j.e ✓ 32
4.b odd 2 1 3360.2.j.f 32
5.b even 2 1 840.2.j.f yes 32
8.b even 2 1 840.2.j.f yes 32
8.d odd 2 1 3360.2.j.e 32
20.d odd 2 1 3360.2.j.e 32
40.e odd 2 1 3360.2.j.f 32
40.f even 2 1 inner 840.2.j.e ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.2.j.e ✓ 32 1.a even 1 1 trivial
840.2.j.e ✓ 32 40.f even 2 1 inner
840.2.j.f yes 32 5.b even 2 1
840.2.j.f yes 32 8.b even 2 1
3360.2.j.e 32 8.d odd 2 1
3360.2.j.e 32 20.d odd 2 1
3360.2.j.f 32 4.b odd 2 1
3360.2.j.f 32 40.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(840, [\chi])\):

\( T_{11}^{32} + 200 T_{11}^{30} + 17624 T_{11}^{28} + 901760 T_{11}^{26} + 29739664 T_{11}^{24} + \cdots + 1677721600 \) Copy content Toggle raw display
\( T_{13}^{16} + 16 T_{13}^{15} + 24 T_{13}^{14} - 888 T_{13}^{13} - 5072 T_{13}^{12} + 6112 T_{13}^{11} + \cdots - 262144 \) Copy content Toggle raw display