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Properties

Label 741.2.d.a
Level $741$
Weight $2$
Character orbit 741.d
Analytic conductor $5.917$
Analytic rank $0$
Dimension $88$
Inner twists $8$

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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] = [741,2,Mod(740,741)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("741.740"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(741, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 741 = 3 \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 741.d (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: \(5.91691478978\)
Analytic rank: \(0\)
Dimension: \(88\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q - 88 q^{4} - 4 q^{9} + 72 q^{16} + 64 q^{25} - 60 q^{30} - 48 q^{36} + 20 q^{39} + 48 q^{42} - 64 q^{43} - 112 q^{49} - 24 q^{55} - 64 q^{61} - 104 q^{64} + 12 q^{66} + 60 q^{81} + 56 q^{82} - 32 q^{87}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
740.1 − 2.65210i −1.73061 − 0.0705726i −5.03365 2.98448 −0.187166 + 4.58976i − 2.41210i 8.04555i 2.99004 + 0.244268i − 7.91516i
740.2 − 2.65210i −1.73061 + 0.0705726i −5.03365 −2.98448 0.187166 + 4.58976i − 2.41210i 8.04555i 2.99004 − 0.244268i 7.91516i
740.3 − 2.65210i 1.73061 − 0.0705726i −5.03365 2.98448 −0.187166 − 4.58976i 2.41210i 8.04555i 2.99004 − 0.244268i − 7.91516i
740.4 − 2.65210i 1.73061 + 0.0705726i −5.03365 −2.98448 0.187166 − 4.58976i 2.41210i 8.04555i 2.99004 + 0.244268i 7.91516i
740.5 − 2.62189i −0.572756 − 1.63461i −4.87431 1.64471 −4.28577 + 1.50170i 4.37710i 7.53612i −2.34390 + 1.87247i − 4.31226i
740.6 − 2.62189i −0.572756 + 1.63461i −4.87431 −1.64471 4.28577 + 1.50170i 4.37710i 7.53612i −2.34390 − 1.87247i 4.31226i
740.7 − 2.62189i 0.572756 − 1.63461i −4.87431 1.64471 −4.28577 − 1.50170i − 4.37710i 7.53612i −2.34390 − 1.87247i − 4.31226i
740.8 − 2.62189i 0.572756 + 1.63461i −4.87431 −1.64471 4.28577 − 1.50170i − 4.37710i 7.53612i −2.34390 + 1.87247i 4.31226i
740.9 − 2.19415i −1.62296 − 0.604979i −2.81428 −1.77589 −1.32741 + 3.56101i 2.44560i 1.78666i 2.26800 + 1.96372i 3.89657i
740.10 − 2.19415i −1.62296 + 0.604979i −2.81428 1.77589 1.32741 + 3.56101i 2.44560i 1.78666i 2.26800 − 1.96372i − 3.89657i
740.11 − 2.19415i 1.62296 − 0.604979i −2.81428 −1.77589 −1.32741 − 3.56101i − 2.44560i 1.78666i 2.26800 − 1.96372i 3.89657i
740.12 − 2.19415i 1.62296 + 0.604979i −2.81428 1.77589 1.32741 − 3.56101i − 2.44560i 1.78666i 2.26800 + 1.96372i − 3.89657i
740.13 − 2.15821i −0.966847 − 1.43708i −2.65789 0.857839 −3.10153 + 2.08666i − 1.81675i 1.41987i −1.13041 + 2.77888i − 1.85140i
740.14 − 2.15821i −0.966847 + 1.43708i −2.65789 −0.857839 3.10153 + 2.08666i − 1.81675i 1.41987i −1.13041 − 2.77888i 1.85140i
740.15 − 2.15821i 0.966847 − 1.43708i −2.65789 0.857839 −3.10153 − 2.08666i 1.81675i 1.41987i −1.13041 − 2.77888i − 1.85140i
740.16 − 2.15821i 0.966847 + 1.43708i −2.65789 −0.857839 3.10153 − 2.08666i 1.81675i 1.41987i −1.13041 + 2.77888i 1.85140i
740.17 − 1.74803i −0.589028 − 1.62882i −1.05561 −3.20148 −2.84722 + 1.02964i − 2.86061i − 1.65082i −2.30609 + 1.91884i 5.59629i
740.18 − 1.74803i −0.589028 + 1.62882i −1.05561 3.20148 2.84722 + 1.02964i − 2.86061i − 1.65082i −2.30609 − 1.91884i − 5.59629i
740.19 − 1.74803i 0.589028 − 1.62882i −1.05561 −3.20148 −2.84722 − 1.02964i 2.86061i − 1.65082i −2.30609 − 1.91884i 5.59629i
740.20 − 1.74803i 0.589028 + 1.62882i −1.05561 3.20148 2.84722 − 1.02964i 2.86061i − 1.65082i −2.30609 + 1.91884i − 5.59629i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 740.88
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.b even 2 1 inner
19.b odd 2 1 inner
39.d odd 2 1 inner
57.d even 2 1 inner
247.d odd 2 1 inner
741.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 741.2.d.a ✓ 88
3.b odd 2 1 inner 741.2.d.a ✓ 88
13.b even 2 1 inner 741.2.d.a ✓ 88
19.b odd 2 1 inner 741.2.d.a ✓ 88
39.d odd 2 1 inner 741.2.d.a ✓ 88
57.d even 2 1 inner 741.2.d.a ✓ 88
247.d odd 2 1 inner 741.2.d.a ✓ 88
741.d even 2 1 inner 741.2.d.a ✓ 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
741.2.d.a ✓ 88 1.a even 1 1 trivial
741.2.d.a ✓ 88 3.b odd 2 1 inner
741.2.d.a ✓ 88 13.b even 2 1 inner
741.2.d.a ✓ 88 19.b odd 2 1 inner
741.2.d.a ✓ 88 39.d odd 2 1 inner
741.2.d.a ✓ 88 57.d even 2 1 inner
741.2.d.a ✓ 88 247.d odd 2 1 inner
741.2.d.a ✓ 88 741.d even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(741, [\chi])\).