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Properties

Label 738.2.v.a
Level $738$
Weight $2$
Character orbit 738.v
Analytic conductor $5.893$
Analytic rank $0$
Dimension $168$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [738,2,Mod(137,738)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(738, base_ring=CyclotomicField(24)) chi = DirichletCharacter(H, H._module([4, 15])) 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("738.137"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 738 = 2 \cdot 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 738.v (of order \(24\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [168,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.89295966917\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(21\) over \(\Q(\zeta_{24})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 4 q^{3} - 12 q^{7} - 4 q^{9} + 8 q^{15} + 84 q^{16} - 12 q^{17} - 8 q^{21} - 12 q^{22} + 48 q^{25} - 40 q^{27} + 16 q^{30} - 64 q^{33} - 36 q^{34} + 24 q^{35} - 4 q^{36} + 32 q^{39} - 32 q^{42}+ \cdots + 68 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 0.258819 + 0.965926i −1.71071 − 0.271030i −0.866025 + 0.500000i −0.965567 + 3.60354i −0.180971 − 1.72257i 5.09454 + 0.670709i −0.707107 − 0.707107i 2.85309 + 0.927308i −3.73066
137.2 0.258819 + 0.965926i −1.70144 − 0.324200i −0.866025 + 0.500000i −0.744537 + 2.77865i −0.127212 − 1.72737i −2.44350 − 0.321693i −0.707107 − 0.707107i 2.78979 + 1.10321i −2.87667
137.3 0.258819 + 0.965926i −1.68221 − 0.412525i −0.866025 + 0.500000i 0.872167 − 3.25497i −0.0369189 − 1.73166i 2.62871 + 0.346077i −0.707107 − 0.707107i 2.65965 + 1.38791i 3.36980
137.4 0.258819 + 0.965926i −1.65104 + 0.523507i −0.866025 + 0.500000i 0.0867317 − 0.323687i −0.932990 − 1.45929i −1.99857 − 0.263117i −0.707107 − 0.707107i 2.45188 − 1.72866i 0.335106
137.5 0.258819 + 0.965926i −1.56158 + 0.749313i −0.866025 + 0.500000i 0.334633 − 1.24887i −1.12795 − 1.31443i 1.11308 + 0.146539i −0.707107 − 0.707107i 1.87706 − 2.34022i 1.29292
137.6 0.258819 + 0.965926i −1.22407 − 1.22542i −0.866025 + 0.500000i 0.747125 − 2.78831i 0.866855 − 1.49952i −4.39686 − 0.578858i −0.707107 − 0.707107i −0.00331887 + 3.00000i 2.88667
137.7 0.258819 + 0.965926i −0.910506 + 1.47342i −0.866025 + 0.500000i −0.482354 + 1.80017i −1.65887 − 0.498132i −1.92542 − 0.253487i −0.707107 − 0.707107i −1.34196 − 2.68312i −1.86367
137.8 0.258819 + 0.965926i −0.548900 − 1.64277i −0.866025 + 0.500000i −0.322689 + 1.20429i 1.44473 − 0.955378i 1.10027 + 0.144854i −0.707107 − 0.707107i −2.39742 + 1.80344i −1.24678
137.9 0.258819 + 0.965926i −0.533681 − 1.64778i −0.866025 + 0.500000i −0.503592 + 1.87943i 1.45351 − 0.941973i −3.38676 − 0.445875i −0.707107 − 0.707107i −2.43037 + 1.75878i −1.94573
137.10 0.258819 + 0.965926i −0.445870 − 1.67368i −0.866025 + 0.500000i −0.396469 + 1.47964i 1.50125 − 0.863857i 2.63748 + 0.347231i −0.707107 − 0.707107i −2.60240 + 1.49249i −1.53184
137.11 0.258819 + 0.965926i −0.442169 + 1.67466i −0.866025 + 0.500000i 0.752912 − 2.80991i −1.73204 + 0.00633171i −1.62918 − 0.214486i −0.707107 − 0.707107i −2.60897 − 1.48096i 2.90903
137.12 0.258819 + 0.965926i 0.195081 + 1.72103i −0.866025 + 0.500000i 0.330687 − 1.23414i −1.61190 + 0.633869i 3.38424 + 0.445544i −0.707107 − 0.707107i −2.92389 + 0.671481i 1.27767
137.13 0.258819 + 0.965926i 0.407149 + 1.68352i −0.866025 + 0.500000i −0.822534 + 3.06974i −1.52077 + 0.829002i 0.231562 + 0.0304857i −0.707107 − 0.707107i −2.66846 + 1.37088i −3.17803
137.14 0.258819 + 0.965926i 0.512711 − 1.65443i −0.866025 + 0.500000i 0.888433 − 3.31568i 1.73075 + 0.0670433i 1.11582 + 0.146901i −0.707107 − 0.707107i −2.47426 − 1.69648i 3.43264
137.15 0.258819 + 0.965926i 0.954604 − 1.44524i −0.866025 + 0.500000i 0.492787 − 1.83911i 1.64307 + 0.548020i −2.37039 − 0.312067i −0.707107 − 0.707107i −1.17746 − 2.75927i 1.90398
137.16 0.258819 + 0.965926i 1.05437 + 1.37416i −0.866025 + 0.500000i 0.130284 − 0.486227i −1.05444 + 1.37410i −4.08950 − 0.538393i −0.707107 − 0.707107i −0.776605 + 2.89774i 0.503379
137.17 0.258819 + 0.965926i 1.17165 − 1.27563i −0.866025 + 0.500000i −0.235162 + 0.877635i 1.53541 + 0.801572i 1.72923 + 0.227658i −0.707107 − 0.707107i −0.254463 − 2.98919i −0.908595
137.18 0.258819 + 0.965926i 1.45241 − 0.943659i −0.866025 + 0.500000i −1.08080 + 4.03360i 1.28742 + 1.15869i −4.10812 − 0.540844i −0.707107 − 0.707107i 1.21902 − 2.74117i −4.17589
137.19 0.258819 + 0.965926i 1.53005 + 0.811761i −0.866025 + 0.500000i 0.647338 − 2.41590i −0.388095 + 1.68801i 4.54768 + 0.598714i −0.707107 − 0.707107i 1.68209 + 2.48406i 2.50112
137.20 0.258819 + 0.965926i 1.67744 + 0.431512i −0.866025 + 0.500000i −0.198370 + 0.740327i 0.0173440 + 1.73196i 1.24252 + 0.163580i −0.707107 − 0.707107i 2.62759 + 1.44767i −0.766443
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.21
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
369.v even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 738.2.v.a ✓ 168
9.d odd 6 1 738.2.v.b yes 168
41.e odd 8 1 738.2.v.b yes 168
369.v even 24 1 inner 738.2.v.a ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.v.a ✓ 168 1.a even 1 1 trivial
738.2.v.a ✓ 168 369.v even 24 1 inner
738.2.v.b yes 168 9.d odd 6 1
738.2.v.b yes 168 41.e odd 8 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{168} - 24 T_{5}^{166} - 16 T_{5}^{165} - 1110 T_{5}^{164} + 1152 T_{5}^{163} + \cdots + 14\!\cdots\!96 \) acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\). Copy content Toggle raw display