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Properties

Label 680.2.l.b
Level $680$
Weight $2$
Character orbit 680.l
Analytic conductor $5.430$
Analytic rank $0$
Dimension $36$
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(101,680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(680, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) 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("680.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 680.l (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [36,0,4] 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: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{3} + 2 q^{4} - 36 q^{5} + 36 q^{9} + 8 q^{11} + 2 q^{12} - 2 q^{14} - 4 q^{15} + 6 q^{16} - 10 q^{18} - 2 q^{20} - 26 q^{24} + 36 q^{25} + 6 q^{26} + 16 q^{27} - 14 q^{28} - 10 q^{32} - 8 q^{33}+ \cdots + 40 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1 −1.40871 − 0.124621i −0.904376 1.96894 + 0.351110i −1.00000 1.27400 + 0.112704i − 1.75754i −2.72991 − 0.739983i −2.18210 1.40871 + 0.124621i
101.2 −1.40871 + 0.124621i −0.904376 1.96894 − 0.351110i −1.00000 1.27400 − 0.112704i 1.75754i −2.72991 + 0.739983i −2.18210 1.40871 − 0.124621i
101.3 −1.38575 − 0.282316i 2.78895 1.84059 + 0.782439i −1.00000 −3.86478 − 0.787366i − 3.69706i −2.32971 − 1.60389i 4.77824 1.38575 + 0.282316i
101.4 −1.38575 + 0.282316i 2.78895 1.84059 − 0.782439i −1.00000 −3.86478 + 0.787366i 3.69706i −2.32971 + 1.60389i 4.77824 1.38575 − 0.282316i
101.5 −1.27109 − 0.619947i −3.03317 1.23133 + 1.57602i −1.00000 3.85542 + 1.88040i 1.14609i −0.588085 − 2.76661i 6.20011 1.27109 + 0.619947i
101.6 −1.27109 + 0.619947i −3.03317 1.23133 − 1.57602i −1.00000 3.85542 − 1.88040i − 1.14609i −0.588085 + 2.76661i 6.20011 1.27109 − 0.619947i
101.7 −1.26408 − 0.634112i 1.91553 1.19580 + 1.60314i −1.00000 −2.42138 − 1.21466i 3.32813i −0.495023 − 2.78477i 0.669252 1.26408 + 0.634112i
101.8 −1.26408 + 0.634112i 1.91553 1.19580 − 1.60314i −1.00000 −2.42138 + 1.21466i − 3.32813i −0.495023 + 2.78477i 0.669252 1.26408 − 0.634112i
101.9 −0.985753 − 1.01405i 1.46409 −0.0565822 + 1.99920i −1.00000 −1.44323 − 1.48466i − 1.02762i 2.08306 − 1.91334i −0.856436 0.985753 + 1.01405i
101.10 −0.985753 + 1.01405i 1.46409 −0.0565822 − 1.99920i −1.00000 −1.44323 + 1.48466i 1.02762i 2.08306 + 1.91334i −0.856436 0.985753 − 1.01405i
101.11 −0.775550 − 1.18259i 0.0722782 −0.797044 + 1.83432i −1.00000 −0.0560554 − 0.0854756i − 0.600402i 2.78740 − 0.480027i −2.99478 0.775550 + 1.18259i
101.12 −0.775550 + 1.18259i 0.0722782 −0.797044 − 1.83432i −1.00000 −0.0560554 + 0.0854756i 0.600402i 2.78740 + 0.480027i −2.99478 0.775550 − 1.18259i
101.13 −0.615521 − 1.27324i −2.68470 −1.24227 + 1.56741i −1.00000 1.65249 + 3.41825i − 4.42201i 2.76033 + 0.616927i 4.20759 0.615521 + 1.27324i
101.14 −0.615521 + 1.27324i −2.68470 −1.24227 − 1.56741i −1.00000 1.65249 − 3.41825i 4.42201i 2.76033 − 0.616927i 4.20759 0.615521 − 1.27324i
101.15 −0.353091 − 1.36943i −1.30568 −1.75065 + 0.967065i −1.00000 0.461025 + 1.78803i 5.06709i 1.94246 + 2.05593i −1.29519 0.353091 + 1.36943i
101.16 −0.353091 + 1.36943i −1.30568 −1.75065 − 0.967065i −1.00000 0.461025 − 1.78803i − 5.06709i 1.94246 − 2.05593i −1.29519 0.353091 − 1.36943i
101.17 −0.0976378 − 1.41084i 0.700054 −1.98093 + 0.275503i −1.00000 −0.0683518 − 0.987663i − 0.472369i 0.582104 + 2.76788i −2.50992 0.0976378 + 1.41084i
101.18 −0.0976378 + 1.41084i 0.700054 −1.98093 − 0.275503i −1.00000 −0.0683518 + 0.987663i 0.472369i 0.582104 − 2.76788i −2.50992 0.0976378 − 1.41084i
101.19 −0.0399866 − 1.41365i 2.61668 −1.99680 + 0.113054i −1.00000 −0.104632 − 3.69906i 3.54885i 0.239664 + 2.81825i 3.84701 0.0399866 + 1.41365i
101.20 −0.0399866 + 1.41365i 2.61668 −1.99680 − 0.113054i −1.00000 −0.104632 + 3.69906i − 3.54885i 0.239664 − 2.81825i 3.84701 0.0399866 − 1.41365i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.36
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
136.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 680.2.l.b yes 36
4.b odd 2 1 2720.2.l.a 36
8.b even 2 1 680.2.l.a ✓ 36
8.d odd 2 1 2720.2.l.b 36
17.b even 2 1 680.2.l.a ✓ 36
68.d odd 2 1 2720.2.l.b 36
136.e odd 2 1 2720.2.l.a 36
136.h even 2 1 inner 680.2.l.b yes 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
680.2.l.a ✓ 36 8.b even 2 1
680.2.l.a ✓ 36 17.b even 2 1
680.2.l.b yes 36 1.a even 1 1 trivial
680.2.l.b yes 36 136.h even 2 1 inner
2720.2.l.a 36 4.b odd 2 1
2720.2.l.a 36 136.e odd 2 1
2720.2.l.b 36 8.d odd 2 1
2720.2.l.b 36 68.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{18} - 2 T_{3}^{17} - 34 T_{3}^{16} + 64 T_{3}^{15} + 463 T_{3}^{14} - 822 T_{3}^{13} - 3226 T_{3}^{12} + \cdots + 64 \) acting on \(S_{2}^{\mathrm{new}}(680, [\chi])\). Copy content Toggle raw display