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Properties

Label 760.2.e.a
Level $760$
Weight $2$
Character orbit 760.e
Analytic conductor $6.069$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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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] = [760,2,Mod(531,760)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(760, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 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("760.531"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.e (of order \(2\), degree \(1\), 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: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(80\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q - 4 q^{4} + 4 q^{6} - 80 q^{9} + 4 q^{16} + 8 q^{19} - 12 q^{24} - 80 q^{25} + 28 q^{26} - 20 q^{28} - 8 q^{30} + 48 q^{36} + 40 q^{38} + 4 q^{42} + 72 q^{44} - 112 q^{49} - 52 q^{54} - 8 q^{57} - 4 q^{58}+ \cdots - 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
531.1 −1.40987 − 0.110740i 1.77027i 1.97547 + 0.312257i − 1.00000i 0.196039 − 2.49585i − 0.723208i −2.75058 − 0.659005i −0.133844 −0.110740 + 1.40987i
531.2 −1.40987 + 0.110740i − 1.77027i 1.97547 − 0.312257i 1.00000i 0.196039 + 2.49585i 0.723208i −2.75058 + 0.659005i −0.133844 −0.110740 − 1.40987i
531.3 −1.39730 − 0.218062i 0.523219i 1.90490 + 0.609397i 1.00000i 0.114094 − 0.731094i − 3.38603i −2.52883 − 1.26690i 2.72624 0.218062 − 1.39730i
531.4 −1.39730 + 0.218062i − 0.523219i 1.90490 − 0.609397i − 1.00000i 0.114094 + 0.731094i 3.38603i −2.52883 + 1.26690i 2.72624 0.218062 + 1.39730i
531.5 −1.37636 − 0.325005i − 1.49424i 1.78874 + 0.894649i − 1.00000i −0.485636 + 2.05662i − 4.01304i −2.17119 − 1.81271i 0.767241 −0.325005 + 1.37636i
531.6 −1.37636 + 0.325005i 1.49424i 1.78874 − 0.894649i 1.00000i −0.485636 − 2.05662i 4.01304i −2.17119 + 1.81271i 0.767241 −0.325005 − 1.37636i
531.7 −1.35069 − 0.419085i − 3.23880i 1.64874 + 1.13211i 1.00000i −1.35733 + 4.37462i − 0.457390i −1.75248 − 2.22009i −7.48983 0.419085 − 1.35069i
531.8 −1.35069 + 0.419085i 3.23880i 1.64874 − 1.13211i − 1.00000i −1.35733 − 4.37462i 0.457390i −1.75248 + 2.22009i −7.48983 0.419085 + 1.35069i
531.9 −1.34496 − 0.437133i − 1.08908i 1.61783 + 1.17585i − 1.00000i −0.476071 + 1.46477i 1.26860i −1.66191 − 2.28868i 1.81391 −0.437133 + 1.34496i
531.10 −1.34496 + 0.437133i 1.08908i 1.61783 − 1.17585i 1.00000i −0.476071 − 1.46477i − 1.26860i −1.66191 + 2.28868i 1.81391 −0.437133 − 1.34496i
531.11 −1.29307 − 0.572689i 2.70495i 1.34405 + 1.48105i 1.00000i 1.54909 − 3.49768i 3.65554i −0.889770 − 2.68483i −4.31673 0.572689 − 1.29307i
531.12 −1.29307 + 0.572689i − 2.70495i 1.34405 − 1.48105i − 1.00000i 1.54909 + 3.49768i − 3.65554i −0.889770 + 2.68483i −4.31673 0.572689 + 1.29307i
531.13 −1.28866 − 0.582537i − 0.0430597i 1.32130 + 1.50139i 1.00000i −0.0250839 + 0.0554895i 3.49579i −0.828098 − 2.70449i 2.99815 0.582537 − 1.28866i
531.14 −1.28866 + 0.582537i 0.0430597i 1.32130 − 1.50139i − 1.00000i −0.0250839 − 0.0554895i − 3.49579i −0.828098 + 2.70449i 2.99815 0.582537 + 1.28866i
531.15 −1.08362 − 0.908720i 2.06995i 0.348457 + 1.96941i − 1.00000i 1.88100 − 2.24303i − 4.61911i 1.41205 − 2.45074i −1.28468 −0.908720 + 1.08362i
531.16 −1.08362 + 0.908720i − 2.06995i 0.348457 − 1.96941i 1.00000i 1.88100 + 2.24303i 4.61911i 1.41205 + 2.45074i −1.28468 −0.908720 − 1.08362i
531.17 −1.04999 − 0.947380i − 0.242176i 0.204944 + 1.98947i 1.00000i −0.229432 + 0.254281i − 0.0837738i 1.66960 − 2.28308i 2.94135 0.947380 − 1.04999i
531.18 −1.04999 + 0.947380i 0.242176i 0.204944 − 1.98947i − 1.00000i −0.229432 − 0.254281i 0.0837738i 1.66960 + 2.28308i 2.94135 0.947380 + 1.04999i
531.19 −0.984479 − 1.01528i − 2.38464i −0.0616041 + 1.99905i − 1.00000i −2.42109 + 2.34763i 4.57467i 2.09025 − 1.90548i −2.68651 −1.01528 + 0.984479i
531.20 −0.984479 + 1.01528i 2.38464i −0.0616041 − 1.99905i 1.00000i −2.42109 − 2.34763i − 4.57467i 2.09025 + 1.90548i −2.68651 −1.01528 − 0.984479i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 531.80
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
19.b odd 2 1 inner
152.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.e.a ✓ 80
4.b odd 2 1 3040.2.e.a 80
8.b even 2 1 3040.2.e.a 80
8.d odd 2 1 inner 760.2.e.a ✓ 80
19.b odd 2 1 inner 760.2.e.a ✓ 80
76.d even 2 1 3040.2.e.a 80
152.b even 2 1 inner 760.2.e.a ✓ 80
152.g odd 2 1 3040.2.e.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.e.a ✓ 80 1.a even 1 1 trivial
760.2.e.a ✓ 80 8.d odd 2 1 inner
760.2.e.a ✓ 80 19.b odd 2 1 inner
760.2.e.a ✓ 80 152.b even 2 1 inner
3040.2.e.a 80 4.b odd 2 1
3040.2.e.a 80 8.b even 2 1
3040.2.e.a 80 76.d even 2 1
3040.2.e.a 80 152.g odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(760, [\chi])\).