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Properties

Label 252.3.p.a
Level $252$
Weight $3$
Character orbit 252.p
Analytic conductor $6.867$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [252,3,Mod(61,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 5])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.61"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 252.p (of order \(6\), degree \(2\), 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: \(6.86650266188\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + q^{7} - 6 q^{9} - 12 q^{11} + 15 q^{13} + 9 q^{15} - 27 q^{17} - 36 q^{21} + 30 q^{23} - 160 q^{25} - 9 q^{27} + 24 q^{29} - 24 q^{31} + 81 q^{33} + 141 q^{35} + 11 q^{37} - 21 q^{39} - 90 q^{41}+ \cdots - 45 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
61.1 0 −2.99667 − 0.141306i 0 8.86566i 0 4.97462 − 4.92475i 0 8.96007 + 0.846895i 0
61.2 0 −2.91706 − 0.700537i 0 − 2.30326i 0 −1.15122 + 6.90469i 0 8.01849 + 4.08702i 0
61.3 0 −2.48876 + 1.67514i 0 − 0.196075i 0 −5.97368 − 3.64898i 0 3.38784 − 8.33802i 0
61.4 0 −2.37299 − 1.83546i 0 − 5.22643i 0 −2.36655 − 6.58783i 0 2.26216 + 8.71106i 0
61.5 0 −1.90375 + 2.31856i 0 4.03768i 0 2.36524 + 6.58829i 0 −1.75144 − 8.82794i 0
61.6 0 −1.17251 + 2.76138i 0 − 3.16374i 0 4.69832 − 5.18900i 0 −6.25045 − 6.47548i 0
61.7 0 −1.14119 − 2.77447i 0 3.08422i 0 6.74289 + 1.87973i 0 −6.39539 + 6.33238i 0
61.8 0 0.108849 − 2.99802i 0 2.13702i 0 −6.60308 + 2.32365i 0 −8.97630 − 0.652664i 0
61.9 0 0.200856 + 2.99327i 0 − 9.02073i 0 −4.77619 + 5.11742i 0 −8.91931 + 1.20243i 0
61.10 0 0.980823 − 2.83513i 0 − 8.39011i 0 5.61921 + 4.17426i 0 −7.07597 − 5.56153i 0
61.11 0 1.17091 + 2.76206i 0 6.76413i 0 −6.54281 − 2.48830i 0 −6.25793 + 6.46825i 0
61.12 0 1.94529 + 2.28383i 0 − 0.963344i 0 6.86265 − 1.37988i 0 −1.43172 + 8.88539i 0
61.13 0 1.96955 − 2.26293i 0 7.47619i 0 −1.51829 − 6.83336i 0 −1.24173 − 8.91393i 0
61.14 0 2.63155 − 1.44047i 0 − 4.01620i 0 1.61222 − 6.81181i 0 4.85011 − 7.58132i 0
61.15 0 2.98647 + 0.284637i 0 − 4.90249i 0 −6.05364 + 3.51475i 0 8.83796 + 1.70012i 0
61.16 0 2.99863 − 0.0905301i 0 5.81748i 0 2.61032 + 6.49509i 0 8.98361 − 0.542933i 0
157.1 0 −2.99667 + 0.141306i 0 − 8.86566i 0 4.97462 + 4.92475i 0 8.96007 − 0.846895i 0
157.2 0 −2.91706 + 0.700537i 0 2.30326i 0 −1.15122 − 6.90469i 0 8.01849 − 4.08702i 0
157.3 0 −2.48876 − 1.67514i 0 0.196075i 0 −5.97368 + 3.64898i 0 3.38784 + 8.33802i 0
157.4 0 −2.37299 + 1.83546i 0 5.22643i 0 −2.36655 + 6.58783i 0 2.26216 − 8.71106i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 61.16
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.3.p.a ✓ 32
3.b odd 2 1 756.3.p.a 32
7.d odd 6 1 252.3.bd.a yes 32
9.c even 3 1 252.3.bd.a yes 32
9.d odd 6 1 756.3.bd.a 32
21.g even 6 1 756.3.bd.a 32
63.k odd 6 1 inner 252.3.p.a ✓ 32
63.s even 6 1 756.3.p.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.p.a ✓ 32 1.a even 1 1 trivial
252.3.p.a ✓ 32 63.k odd 6 1 inner
252.3.bd.a yes 32 7.d odd 6 1
252.3.bd.a yes 32 9.c even 3 1
756.3.p.a 32 3.b odd 2 1
756.3.p.a 32 63.s even 6 1
756.3.bd.a 32 9.d odd 6 1
756.3.bd.a 32 21.g even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(252, [\chi])\).