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Properties

Label 728.2.c.b
Level $728$
Weight $2$
Character orbit 728.c
Analytic conductor $5.813$
Analytic rank $0$
Dimension $38$
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 = [38] 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: \(38\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q + 8 q^{4} + 14 q^{6} + 38 q^{7} - 6 q^{8} - 46 q^{9} - 4 q^{12} - 8 q^{15} - 4 q^{16} + 20 q^{17} + 4 q^{18} - 24 q^{20} + 10 q^{22} + 12 q^{23} + 10 q^{24} - 50 q^{25} + 8 q^{28} + 4 q^{30} + 16 q^{31}+ \cdots + 82 q^{96}+O(q^{100}) \) 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 −1.40887 − 0.122797i 0.192506i 1.96984 + 0.346010i 2.98429i 0.0236391 − 0.271216i 1.00000 −2.73277 − 0.729373i 2.96294 0.366461 − 4.20449i
365.2 −1.40887 + 0.122797i − 0.192506i 1.96984 − 0.346010i − 2.98429i 0.0236391 + 0.271216i 1.00000 −2.73277 + 0.729373i 2.96294 0.366461 + 4.20449i
365.3 −1.36059 − 0.385756i 3.17321i 1.70238 + 1.04971i 3.62722i 1.22409 − 4.31743i 1.00000 −1.91131 − 2.08492i −7.06928 1.39923 − 4.93515i
365.4 −1.36059 + 0.385756i − 3.17321i 1.70238 − 1.04971i − 3.62722i 1.22409 + 4.31743i 1.00000 −1.91131 + 2.08492i −7.06928 1.39923 + 4.93515i
365.5 −1.34174 − 0.446901i − 2.10063i 1.60056 + 1.19925i 0.418613i −0.938772 + 2.81851i 1.00000 −1.61159 − 2.32438i −1.41263 0.187079 − 0.561672i
365.6 −1.34174 + 0.446901i 2.10063i 1.60056 − 1.19925i − 0.418613i −0.938772 − 2.81851i 1.00000 −1.61159 + 2.32438i −1.41263 0.187079 + 0.561672i
365.7 −1.31212 − 0.527569i 3.05148i 1.44334 + 1.38447i − 3.70659i 1.60986 − 4.00392i 1.00000 −1.16344 − 2.57806i −6.31151 −1.95548 + 4.86350i
365.8 −1.31212 + 0.527569i − 3.05148i 1.44334 − 1.38447i 3.70659i 1.60986 + 4.00392i 1.00000 −1.16344 + 2.57806i −6.31151 −1.95548 − 4.86350i
365.9 −1.08735 − 0.904247i 0.560000i 0.364676 + 1.96647i 0.293198i 0.506378 − 0.608918i 1.00000 1.38164 − 2.46801i 2.68640 0.265123 − 0.318810i
365.10 −1.08735 + 0.904247i − 0.560000i 0.364676 − 1.96647i − 0.293198i 0.506378 + 0.608918i 1.00000 1.38164 + 2.46801i 2.68640 0.265123 + 0.318810i
365.11 −1.06885 − 0.926043i − 2.33171i 0.284890 + 1.97961i 0.889625i −2.15926 + 2.49225i 1.00000 1.52869 − 2.37973i −2.43685 0.823831 − 0.950878i
365.12 −1.06885 + 0.926043i 2.33171i 0.284890 − 1.97961i − 0.889625i −2.15926 − 2.49225i 1.00000 1.52869 + 2.37973i −2.43685 0.823831 + 0.950878i
365.13 −0.716045 − 1.21954i 0.472093i −0.974560 + 1.74649i 4.35100i 0.575736 − 0.338039i 1.00000 2.82775 − 0.0620491i 2.77713 5.30622 − 3.11551i
365.14 −0.716045 + 1.21954i − 0.472093i −0.974560 − 1.74649i − 4.35100i 0.575736 + 0.338039i 1.00000 2.82775 + 0.0620491i 2.77713 5.30622 + 3.11551i
365.15 −0.581532 − 1.28912i 2.40918i −1.32364 + 1.49933i − 0.687773i 3.10571 − 1.40101i 1.00000 2.70254 + 0.834421i −2.80413 −0.886619 + 0.399962i
365.16 −0.581532 + 1.28912i − 2.40918i −1.32364 − 1.49933i 0.687773i 3.10571 + 1.40101i 1.00000 2.70254 − 0.834421i −2.80413 −0.886619 − 0.399962i
365.17 −0.420523 − 1.35024i − 0.273626i −1.64632 + 1.13562i − 2.12501i −0.369462 + 0.115066i 1.00000 2.22568 + 1.74538i 2.92513 −2.86928 + 0.893617i
365.18 −0.420523 + 1.35024i 0.273626i −1.64632 − 1.13562i 2.12501i −0.369462 − 0.115066i 1.00000 2.22568 − 1.74538i 2.92513 −2.86928 − 0.893617i
365.19 0.242210 − 1.39332i 1.67830i −1.88267 − 0.674950i 0.0254406i 2.33840 + 0.406500i 1.00000 −1.39642 + 2.45968i 0.183319 0.0354469 + 0.00616197i
365.20 0.242210 + 1.39332i − 1.67830i −1.88267 + 0.674950i − 0.0254406i 2.33840 − 0.406500i 1.00000 −1.39642 − 2.45968i 0.183319 0.0354469 − 0.00616197i
See all 38 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 365.38
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.b ✓ 38
4.b odd 2 1 2912.2.c.b 38
8.b even 2 1 inner 728.2.c.b ✓ 38
8.d odd 2 1 2912.2.c.b 38
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.c.b ✓ 38 1.a even 1 1 trivial
728.2.c.b ✓ 38 8.b even 2 1 inner
2912.2.c.b 38 4.b odd 2 1
2912.2.c.b 38 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{38} + 80 T_{3}^{36} + 2910 T_{3}^{34} + 63760 T_{3}^{32} + 939841 T_{3}^{30} + 9859520 T_{3}^{28} + \cdots + 262144 \) acting on \(S_{2}^{\mathrm{new}}(728, [\chi])\). Copy content Toggle raw display