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Properties

Label 728.2.c.a
Level $728$
Weight $2$
Character orbit 728.c
Analytic conductor $5.813$
Analytic rank $0$
Dimension $34$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [728,2,Mod(365,728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 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("728.365"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 728.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.81310926715\)
Analytic rank: \(0\)
Dimension: \(34\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 2 q^{2} - 2 q^{4} - 6 q^{6} - 34 q^{7} + 8 q^{8} - 26 q^{9} - 4 q^{12} - 2 q^{14} - 8 q^{15} - 6 q^{16} - 20 q^{17} + 14 q^{18} - 4 q^{20} - 10 q^{22} - 20 q^{23} + 10 q^{24} - 22 q^{25} + 2 q^{28}+ \cdots + 2 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
365.1 −1.41239 − 0.0717634i 1.60179i 1.98970 + 0.202716i − 2.04118i 0.114950 − 2.26236i −1.00000 −2.79569 − 0.429102i 0.434268 −0.146482 + 2.88295i
365.2 −1.41239 + 0.0717634i − 1.60179i 1.98970 − 0.202716i 2.04118i 0.114950 + 2.26236i −1.00000 −2.79569 + 0.429102i 0.434268 −0.146482 − 2.88295i
365.3 −1.23674 − 0.685911i 0.302261i 1.05905 + 1.69659i 1.42871i 0.207324 − 0.373818i −1.00000 −0.146067 − 2.82465i 2.90864 0.979968 − 1.76694i
365.4 −1.23674 + 0.685911i − 0.302261i 1.05905 − 1.69659i − 1.42871i 0.207324 + 0.373818i −1.00000 −0.146067 + 2.82465i 2.90864 0.979968 + 1.76694i
365.5 −1.14018 − 0.836660i 2.38465i 0.600001 + 1.90788i 1.59665i 1.99514 − 2.71892i −1.00000 0.912139 − 2.67731i −2.68654 1.33585 − 1.82046i
365.6 −1.14018 + 0.836660i − 2.38465i 0.600001 − 1.90788i − 1.59665i 1.99514 + 2.71892i −1.00000 0.912139 + 2.67731i −2.68654 1.33585 + 1.82046i
365.7 −1.11456 − 0.870498i − 2.73244i 0.484466 + 1.94044i 4.29306i −2.37859 + 3.04546i −1.00000 1.14918 − 2.58445i −4.46625 3.73710 − 4.78486i
365.8 −1.11456 + 0.870498i 2.73244i 0.484466 − 1.94044i − 4.29306i −2.37859 − 3.04546i −1.00000 1.14918 + 2.58445i −4.46625 3.73710 + 4.78486i
365.9 −0.720007 − 1.21721i − 2.66332i −0.963179 + 1.75279i − 2.77422i −3.24180 + 1.91761i −1.00000 2.82701 − 0.0896372i −4.09325 −3.37680 + 1.99746i
365.10 −0.720007 + 1.21721i 2.66332i −0.963179 − 1.75279i 2.77422i −3.24180 − 1.91761i −1.00000 2.82701 + 0.0896372i −4.09325 −3.37680 − 1.99746i
365.11 −0.687700 − 1.23575i 3.16783i −1.05414 + 1.69965i − 1.11795i 3.91463 − 2.17851i −1.00000 2.82526 + 0.133800i −7.03512 −1.38150 + 0.768812i
365.12 −0.687700 + 1.23575i − 3.16783i −1.05414 − 1.69965i 1.11795i 3.91463 + 2.17851i −1.00000 2.82526 − 0.133800i −7.03512 −1.38150 − 0.768812i
365.13 −0.624850 − 1.26869i − 1.12898i −1.21912 + 1.58548i 0.544172i −1.43232 + 0.705444i −1.00000 2.77324 + 0.556000i 1.72540 0.690383 − 0.340026i
365.14 −0.624850 + 1.26869i 1.12898i −1.21912 − 1.58548i − 0.544172i −1.43232 − 0.705444i −1.00000 2.77324 − 0.556000i 1.72540 0.690383 + 0.340026i
365.15 −0.106298 − 1.41021i 1.65437i −1.97740 + 0.299806i − 2.15163i 2.33302 − 0.175857i −1.00000 0.632984 + 2.75669i 0.263045 −3.03425 + 0.228714i
365.16 −0.106298 + 1.41021i − 1.65437i −1.97740 − 0.299806i 2.15163i 2.33302 + 0.175857i −1.00000 0.632984 − 2.75669i 0.263045 −3.03425 − 0.228714i
365.17 0.182514 − 1.40239i − 0.275879i −1.93338 − 0.511911i − 3.61620i −0.386890 − 0.0503520i −1.00000 −1.07077 + 2.61791i 2.92389 −5.07131 − 0.660008i
365.18 0.182514 + 1.40239i 0.275879i −1.93338 + 0.511911i 3.61620i −0.386890 + 0.0503520i −1.00000 −1.07077 − 2.61791i 2.92389 −5.07131 + 0.660008i
365.19 0.220384 − 1.39694i − 1.54889i −1.90286 − 0.615724i 3.27466i −2.16370 − 0.341350i −1.00000 −1.27949 + 2.52248i 0.600939 4.57450 + 0.721683i
365.20 0.220384 + 1.39694i 1.54889i −1.90286 + 0.615724i − 3.27466i −2.16370 + 0.341350i −1.00000 −1.27949 − 2.52248i 0.600939 4.57450 − 0.721683i
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 365.34
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 728.2.c.a ✓ 34
4.b odd 2 1 2912.2.c.a 34
8.b even 2 1 inner 728.2.c.a ✓ 34
8.d odd 2 1 2912.2.c.a 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.c.a ✓ 34 1.a even 1 1 trivial
728.2.c.a ✓ 34 8.b even 2 1 inner
2912.2.c.a 34 4.b odd 2 1
2912.2.c.a 34 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{34} + 64 T_{3}^{32} + 1838 T_{3}^{30} + 31312 T_{3}^{28} + 352481 T_{3}^{26} + 2764784 T_{3}^{24} + \cdots + 1024 \) acting on \(S_{2}^{\mathrm{new}}(728, [\chi])\). Copy content Toggle raw display