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Properties

Label 252.3.bh.a
Level $252$
Weight $3$
Character orbit 252.bh
Analytic conductor $6.867$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,3,Mod(137,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.137");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 252.bh (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - q^{7} - 14 q^{9} + 18 q^{11} - 5 q^{13} - 29 q^{15} + 27 q^{17} - 14 q^{19} - 8 q^{21} - 45 q^{23} + 80 q^{25} + 45 q^{27} + 36 q^{29} + 16 q^{31} + 116 q^{33} + 45 q^{35} - 11 q^{37} + 55 q^{39} + 72 q^{41} + 16 q^{43} - 154 q^{45} - 37 q^{49} - 133 q^{51} - 180 q^{53} - 24 q^{55} + 57 q^{57} + 82 q^{61} - 99 q^{63} + 70 q^{67} - 97 q^{69} - 98 q^{73} - 149 q^{75} - 135 q^{77} + 142 q^{79} - 98 q^{81} - 30 q^{85} - 86 q^{87} + 189 q^{89} + 109 q^{91} + 109 q^{93} + 19 q^{97} - 101 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
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 −2.91627 + 0.703822i 0 5.42354 + 3.13128i 0 −2.59062 − 6.50298i 0 8.00927 − 4.10507i 0
137.2 0 −2.83139 − 0.991573i 0 −1.22253 − 0.705830i 0 6.87631 − 1.31010i 0 7.03356 + 5.61507i 0
137.3 0 −2.56340 − 1.55851i 0 −4.61560 − 2.66482i 0 −5.90728 + 3.75553i 0 4.14209 + 7.99019i 0
137.4 0 −2.25353 + 1.98031i 0 −0.786362 − 0.454006i 0 −1.89219 + 6.73941i 0 1.15678 − 8.92535i 0
137.5 0 −1.68231 + 2.48391i 0 −5.63091 − 3.25101i 0 6.72412 − 1.94582i 0 −3.33964 − 8.35744i 0
137.6 0 −1.40238 − 2.65204i 0 6.97157 + 4.02504i 0 1.23048 + 6.89100i 0 −5.06667 + 7.43834i 0
137.7 0 −0.457963 + 2.96484i 0 −0.178612 − 0.103122i 0 −6.49543 − 2.60948i 0 −8.58054 − 2.71557i 0
137.8 0 −0.369633 − 2.97714i 0 2.99244 + 1.72768i 0 −5.40412 − 4.44921i 0 −8.72674 + 2.20090i 0
137.9 0 0.0431547 − 2.99969i 0 −4.30914 − 2.48788i 0 0.646685 − 6.97006i 0 −8.99628 − 0.258901i 0
137.10 0 0.156453 + 2.99592i 0 8.53889 + 4.92993i 0 6.92226 + 1.04035i 0 −8.95104 + 0.937442i 0
137.11 0 1.65008 − 2.50544i 0 1.01490 + 0.585951i 0 4.44266 + 5.40951i 0 −3.55445 − 8.26837i 0
137.12 0 1.96463 + 2.26721i 0 −2.30580 − 1.33126i 0 3.52367 + 6.04845i 0 −1.28045 + 8.90845i 0
137.13 0 2.44265 + 1.74168i 0 −6.63283 − 3.82946i 0 −1.42259 − 6.85392i 0 2.93311 + 8.50864i 0
137.14 0 2.50034 − 1.65779i 0 −6.77026 − 3.90881i 0 −4.95821 + 4.94127i 0 3.50345 − 8.29011i 0
137.15 0 2.82760 + 1.00234i 0 4.75727 + 2.74661i 0 −6.69466 + 2.04489i 0 6.99063 + 5.66843i 0
137.16 0 2.89197 − 0.797831i 0 2.75344 + 1.58970i 0 4.49892 − 5.36281i 0 7.72693 − 4.61460i 0
149.1 0 −2.91627 − 0.703822i 0 5.42354 − 3.13128i 0 −2.59062 + 6.50298i 0 8.00927 + 4.10507i 0
149.2 0 −2.83139 + 0.991573i 0 −1.22253 + 0.705830i 0 6.87631 + 1.31010i 0 7.03356 − 5.61507i 0
149.3 0 −2.56340 + 1.55851i 0 −4.61560 + 2.66482i 0 −5.90728 − 3.75553i 0 4.14209 − 7.99019i 0
149.4 0 −2.25353 − 1.98031i 0 −0.786362 + 0.454006i 0 −1.89219 − 6.73941i 0 1.15678 + 8.92535i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.j 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.bh.a yes 32
3.b odd 2 1 756.3.bh.a 32
7.c even 3 1 252.3.m.a ✓ 32
9.c even 3 1 756.3.m.a 32
9.d odd 6 1 252.3.m.a ✓ 32
21.h odd 6 1 756.3.m.a 32
63.h even 3 1 756.3.bh.a 32
63.j odd 6 1 inner 252.3.bh.a yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.3.m.a ✓ 32 7.c even 3 1
252.3.m.a ✓ 32 9.d odd 6 1
252.3.bh.a yes 32 1.a even 1 1 trivial
252.3.bh.a yes 32 63.j odd 6 1 inner
756.3.m.a 32 9.c even 3 1
756.3.m.a 32 21.h odd 6 1
756.3.bh.a 32 3.b odd 2 1
756.3.bh.a 32 63.h even 3 1

Hecke kernels

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