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Properties

Label 728.2.h.b
Level $728$
Weight $2$
Character orbit 728.h
Analytic conductor $5.813$
Analytic rank $0$
Dimension $48$
Inner twists $2$

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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] = [728,2,Mod(27,728)] 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([1, 1, 1, 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("728.27"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 728.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,1,0,1,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.81310926715\)
Analytic rank: \(0\)
Dimension: \(48\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + q^{2} + q^{4} + 10 q^{6} - 5 q^{8} - 48 q^{9} - 4 q^{11} - 10 q^{12} + 48 q^{13} - 6 q^{14} + 5 q^{16} - 15 q^{18} + 22 q^{20} - 6 q^{22} + 48 q^{25} + q^{26} - 26 q^{28} - 26 q^{30} - 19 q^{32}+ \cdots + 20 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
27.1 −1.40591 − 0.153058i 1.61368i 1.95315 + 0.430370i 3.74465 0.246987 − 2.26868i −0.220320 + 2.63656i −2.68007 − 0.904005i 0.396033 −5.26463 − 0.573149i
27.2 −1.40591 + 0.153058i − 1.61368i 1.95315 − 0.430370i 3.74465 0.246987 + 2.26868i −0.220320 − 2.63656i −2.68007 + 0.904005i 0.396033 −5.26463 + 0.573149i
27.3 −1.39580 − 0.227483i − 1.35635i 1.89650 + 0.635042i −2.01767 −0.308546 + 1.89318i 2.18534 − 1.49140i −2.50267 − 1.31781i 1.16033 2.81626 + 0.458986i
27.4 −1.39580 + 0.227483i 1.35635i 1.89650 − 0.635042i −2.01767 −0.308546 − 1.89318i 2.18534 + 1.49140i −2.50267 + 1.31781i 1.16033 2.81626 − 0.458986i
27.5 −1.32721 − 0.488382i − 0.610846i 1.52297 + 1.29637i −0.780706 −0.298326 + 0.810720i −2.18555 − 1.49110i −1.38817 − 2.46434i 2.62687 1.03616 + 0.381283i
27.6 −1.32721 + 0.488382i 0.610846i 1.52297 − 1.29637i −0.780706 −0.298326 − 0.810720i −2.18555 + 1.49110i −1.38817 + 2.46434i 2.62687 1.03616 − 0.381283i
27.7 −1.29766 − 0.562216i 3.37415i 1.36783 + 1.45913i 0.523773 1.89700 − 4.37848i 2.30913 − 1.29148i −0.954623 − 2.66246i −8.38487 −0.679678 − 0.294474i
27.8 −1.29766 + 0.562216i − 3.37415i 1.36783 − 1.45913i 0.523773 1.89700 + 4.37848i 2.30913 + 1.29148i −0.954623 + 2.66246i −8.38487 −0.679678 + 0.294474i
27.9 −1.25982 − 0.642544i 2.44281i 1.17427 + 1.61897i −3.85137 1.56961 − 3.07749i −2.49284 + 0.886436i −0.439107 − 2.79413i −2.96733 4.85202 + 2.47467i
27.10 −1.25982 + 0.642544i − 2.44281i 1.17427 − 1.61897i −3.85137 1.56961 + 3.07749i −2.49284 − 0.886436i −0.439107 + 2.79413i −2.96733 4.85202 − 2.47467i
27.11 −1.10110 − 0.887461i − 0.140163i 0.424825 + 1.95436i 0.986963 −0.124389 + 0.154333i 1.39484 + 2.24820i 1.26665 − 2.52895i 2.98035 −1.08674 − 0.875891i
27.12 −1.10110 + 0.887461i 0.140163i 0.424825 − 1.95436i 0.986963 −0.124389 − 0.154333i 1.39484 − 2.24820i 1.26665 + 2.52895i 2.98035 −1.08674 + 0.875891i
27.13 −0.879218 − 1.10769i − 1.92788i −0.453951 + 1.94780i 2.16680 −2.13549 + 1.69503i −1.74001 − 1.99308i 2.55668 − 1.20971i −0.716713 −1.90509 − 2.40014i
27.14 −0.879218 + 1.10769i 1.92788i −0.453951 − 1.94780i 2.16680 −2.13549 − 1.69503i −1.74001 + 1.99308i 2.55668 + 1.20971i −0.716713 −1.90509 + 2.40014i
27.15 −0.635480 − 1.26339i 1.40360i −1.19233 + 1.60572i 3.11493 1.77331 − 0.891962i 2.64532 − 0.0479799i 2.78636 + 0.485980i 1.02989 −1.97948 − 3.93539i
27.16 −0.635480 + 1.26339i − 1.40360i −1.19233 − 1.60572i 3.11493 1.77331 + 0.891962i 2.64532 + 0.0479799i 2.78636 − 0.485980i 1.02989 −1.97948 + 3.93539i
27.17 −0.629285 − 1.26649i − 2.38356i −1.20800 + 1.59397i 1.68057 −3.01876 + 1.49994i −1.10198 + 2.40533i 2.77892 + 0.526865i −2.68138 −1.05755 − 2.12842i
27.18 −0.629285 + 1.26649i 2.38356i −1.20800 − 1.59397i 1.68057 −3.01876 − 1.49994i −1.10198 − 2.40533i 2.77892 − 0.526865i −2.68138 −1.05755 + 2.12842i
27.19 −0.270952 − 1.38801i − 0.332595i −1.85317 + 0.752171i −4.11591 −0.461647 + 0.0901174i 1.93546 + 1.80388i 1.54614 + 2.36843i 2.88938 1.11521 + 5.71294i
27.20 −0.270952 + 1.38801i 0.332595i −1.85317 − 0.752171i −4.11591 −0.461647 − 0.0901174i 1.93546 − 1.80388i 1.54614 − 2.36843i 2.88938 1.11521 − 5.71294i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 27.48
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.e 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.h.b yes 48
4.b odd 2 1 2912.2.h.b 48
7.b odd 2 1 728.2.h.a ✓ 48
8.b even 2 1 2912.2.h.a 48
8.d odd 2 1 728.2.h.a ✓ 48
28.d even 2 1 2912.2.h.a 48
56.e even 2 1 inner 728.2.h.b yes 48
56.h odd 2 1 2912.2.h.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
728.2.h.a ✓ 48 7.b odd 2 1
728.2.h.a ✓ 48 8.d odd 2 1
728.2.h.b yes 48 1.a even 1 1 trivial
728.2.h.b yes 48 56.e even 2 1 inner
2912.2.h.a 48 8.b even 2 1
2912.2.h.a 48 28.d even 2 1
2912.2.h.b 48 4.b odd 2 1
2912.2.h.b 48 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 72 T_{5}^{22} + 4 T_{5}^{21} + 2196 T_{5}^{20} - 220 T_{5}^{19} - 37124 T_{5}^{18} + \cdots - 75264 \) acting on \(S_{2}^{\mathrm{new}}(728, [\chi])\). Copy content Toggle raw display