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Properties

Label 680.2.z.d
Level $680$
Weight $2$
Character orbit 680.z
Analytic conductor $5.430$
Analytic rank $0$
Dimension $26$
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(89,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 2, 3])) 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("680.89"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.z (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [26,0,2] 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: \(5.42982733745\)
Analytic rank: \(0\)
Dimension: \(26\)
Relative dimension: \(13\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 26 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{15} + 2 q^{17} - 12 q^{21} - 2 q^{23} + 10 q^{25} + 20 q^{27} + 10 q^{29} - 10 q^{31} + 22 q^{35} - 6 q^{37} + 8 q^{39} - 10 q^{41} + 32 q^{43} + 28 q^{45}+ \cdots - 22 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
89.1 0 −2.42900 − 2.42900i 0 1.32395 − 1.80198i 0 −1.41698 + 1.41698i 0 8.80012i 0
89.2 0 −1.79869 − 1.79869i 0 −1.32711 + 1.79966i 0 2.72695 − 2.72695i 0 3.47057i 0
89.3 0 −1.69850 − 1.69850i 0 −2.23426 + 0.0897794i 0 −2.68873 + 2.68873i 0 2.76980i 0
89.4 0 −1.15809 − 1.15809i 0 −0.636179 − 2.14366i 0 2.87596 − 2.87596i 0 − 0.317643i 0
89.5 0 −0.803587 − 0.803587i 0 2.02944 − 0.938821i 0 0.286674 − 0.286674i 0 − 1.70850i 0
89.6 0 0.253240 + 0.253240i 0 1.48946 − 1.66779i 0 −3.26262 + 3.26262i 0 − 2.87174i 0
89.7 0 0.258828 + 0.258828i 0 1.90450 + 1.17169i 0 2.27901 − 2.27901i 0 − 2.86602i 0
89.8 0 0.513320 + 0.513320i 0 −2.10308 + 0.759643i 0 −0.542760 + 0.542760i 0 − 2.47300i 0
89.9 0 0.731849 + 0.731849i 0 −2.20723 + 0.357977i 0 1.16819 − 1.16819i 0 − 1.92879i 0
89.10 0 1.39569 + 1.39569i 0 −0.412948 − 2.19761i 0 2.19248 − 2.19248i 0 0.895892i 0
89.11 0 1.68588 + 1.68588i 0 2.22887 + 0.179214i 0 −0.196476 + 0.196476i 0 2.68435i 0
89.12 0 1.83090 + 1.83090i 0 0.225574 + 2.22466i 0 0.619132 − 0.619132i 0 3.70437i 0
89.13 0 2.21817 + 2.21817i 0 −1.28099 − 1.83277i 0 −3.04081 + 3.04081i 0 6.84059i 0
489.1 0 −2.42900 + 2.42900i 0 1.32395 + 1.80198i 0 −1.41698 − 1.41698i 0 − 8.80012i 0
489.2 0 −1.79869 + 1.79869i 0 −1.32711 − 1.79966i 0 2.72695 + 2.72695i 0 − 3.47057i 0
489.3 0 −1.69850 + 1.69850i 0 −2.23426 − 0.0897794i 0 −2.68873 − 2.68873i 0 − 2.76980i 0
489.4 0 −1.15809 + 1.15809i 0 −0.636179 + 2.14366i 0 2.87596 + 2.87596i 0 0.317643i 0
489.5 0 −0.803587 + 0.803587i 0 2.02944 + 0.938821i 0 0.286674 + 0.286674i 0 1.70850i 0
489.6 0 0.253240 − 0.253240i 0 1.48946 + 1.66779i 0 −3.26262 − 3.26262i 0 2.87174i 0
489.7 0 0.258828 − 0.258828i 0 1.90450 − 1.17169i 0 2.27901 + 2.27901i 0 2.86602i 0
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 89.13
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.z.d yes 26
5.b even 2 1 680.2.z.c ✓ 26
17.c even 4 1 680.2.z.c ✓ 26
85.j even 4 1 inner 680.2.z.d yes 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.z.c ✓ 26 5.b even 2 1
680.2.z.c ✓ 26 17.c even 4 1
680.2.z.d yes 26 1.a even 1 1 trivial
680.2.z.d yes 26 85.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{26} - 2 T_{3}^{25} + 2 T_{3}^{24} - 4 T_{3}^{23} + 213 T_{3}^{22} - 462 T_{3}^{21} + 506 T_{3}^{20} + \cdots + 21632 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\). Copy content Toggle raw display