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Properties

Label 330.2.p.a
Level $330$
Weight $2$
Character orbit 330.p
Analytic conductor $2.635$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [330,2,Mod(29,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("330.29");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.p (of order \(10\), degree \(4\), 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: \(2.63506326670\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 24 q^{4} + 8 q^{9} + 4 q^{15} - 24 q^{16} - 32 q^{25} - 30 q^{30} - 4 q^{31} + 8 q^{34} + 12 q^{36} + 20 q^{39} - 10 q^{40} + 48 q^{45} - 20 q^{46} - 144 q^{49} - 20 q^{51} - 24 q^{55} - 4 q^{60} - 40 q^{61} + 24 q^{64} + 44 q^{66} - 48 q^{69} + 22 q^{70} - 48 q^{75} - 20 q^{79} + 104 q^{81} - 20 q^{84} - 60 q^{85} - 90 q^{90} + 40 q^{91} - 80 q^{94} - 124 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −0.587785 + 0.809017i −1.72218 − 0.184625i −0.309017 − 0.951057i −0.518761 − 2.17506i 1.16164 − 1.28476i 1.44461 + 4.44604i 0.951057 + 0.309017i 2.93183 + 0.635917i 2.06458 + 0.858782i
29.2 −0.587785 + 0.809017i −1.70077 − 0.327702i −0.309017 − 0.951057i −2.23504 + 0.0677374i 1.26480 − 1.18333i −0.856709 − 2.63668i 0.951057 + 0.309017i 2.78522 + 1.11469i 1.25892 − 1.84800i
29.3 −0.587785 + 0.809017i −1.69224 + 0.369215i −0.309017 − 0.951057i 2.07779 + 0.826320i 0.695973 − 1.58607i −0.465600 − 1.43297i 0.951057 + 0.309017i 2.72736 − 1.24960i −1.88980 + 1.19527i
29.4 −0.587785 + 0.809017i −0.910001 + 1.47374i −0.309017 − 0.951057i −0.0123239 + 2.23603i −0.657393 − 1.60245i 0.657342 + 2.02309i 0.951057 + 0.309017i −1.34380 − 2.68220i −1.80175 − 1.32428i
29.5 −0.587785 + 0.809017i −0.475008 − 1.66564i −0.309017 − 0.951057i 1.53488 − 1.62608i 1.62674 + 0.594751i −0.545881 − 1.68005i 0.951057 + 0.309017i −2.54873 + 1.58239i 0.413350 + 2.19753i
29.6 −0.587785 + 0.809017i −0.130035 + 1.72716i −0.309017 − 0.951057i −2.13040 − 0.679252i −1.32087 − 1.12040i −0.657342 − 2.02309i 0.951057 + 0.309017i −2.96618 − 0.449182i 1.80175 − 1.32428i
29.7 −0.587785 + 0.809017i 0.161693 − 1.72449i −0.309017 − 0.951057i −2.19325 + 0.435499i 1.30010 + 1.14444i 1.49298 + 4.59493i 0.951057 + 0.309017i −2.94771 − 0.557675i 0.936834 − 2.03036i
29.8 −0.587785 + 0.809017i 0.882816 − 1.49018i −0.309017 − 0.951057i −1.09193 + 1.95133i 0.686675 + 1.59012i −1.49298 − 4.59493i 0.951057 + 0.309017i −1.44127 − 2.63111i −0.936834 − 2.03036i
29.9 −0.587785 + 0.809017i 1.15203 + 1.29338i −0.309017 − 0.951057i −0.143806 − 2.23144i −1.72351 + 0.171787i 0.465600 + 1.43297i 0.951057 + 0.309017i −0.345642 + 2.98002i 1.88980 + 1.19527i
29.10 −0.587785 + 0.809017i 1.36333 − 1.06833i −0.309017 − 0.951057i 2.02080 − 0.957269i 0.0629525 + 1.73091i 0.545881 + 1.68005i 0.951057 + 0.309017i 0.717338 − 2.91298i −0.413350 + 2.19753i
29.11 −0.587785 + 0.809017i 1.50180 + 0.862909i −0.309017 − 0.951057i 1.90830 + 1.16550i −1.58084 + 0.707773i −1.44461 − 4.44604i 0.951057 + 0.309017i 1.51078 + 2.59182i −2.06458 + 0.858782i
29.12 −0.587785 + 0.809017i 1.56857 + 0.734570i −0.309017 − 0.951057i −0.755088 + 2.10472i −1.51626 + 0.837229i 0.856709 + 2.63668i 0.951057 + 0.309017i 1.92081 + 2.30445i −1.25892 − 1.84800i
29.13 0.587785 − 0.809017i −1.56857 − 0.734570i −0.309017 − 0.951057i 2.23504 − 0.0677374i −1.51626 + 0.837229i −0.856709 − 2.63668i −0.951057 − 0.309017i 1.92081 + 2.30445i 1.25892 − 1.84800i
29.14 0.587785 − 0.809017i −1.50180 − 0.862909i −0.309017 − 0.951057i 0.518761 + 2.17506i −1.58084 + 0.707773i 1.44461 + 4.44604i −0.951057 − 0.309017i 1.51078 + 2.59182i 2.06458 + 0.858782i
29.15 0.587785 − 0.809017i −1.36333 + 1.06833i −0.309017 − 0.951057i −1.53488 + 1.62608i 0.0629525 + 1.73091i −0.545881 − 1.68005i −0.951057 − 0.309017i 0.717338 − 2.91298i 0.413350 + 2.19753i
29.16 0.587785 − 0.809017i −1.15203 − 1.29338i −0.309017 − 0.951057i −2.07779 − 0.826320i −1.72351 + 0.171787i −0.465600 − 1.43297i −0.951057 − 0.309017i −0.345642 + 2.98002i −1.88980 + 1.19527i
29.17 0.587785 − 0.809017i −0.882816 + 1.49018i −0.309017 − 0.951057i 2.19325 − 0.435499i 0.686675 + 1.59012i 1.49298 + 4.59493i −0.951057 − 0.309017i −1.44127 − 2.63111i 0.936834 − 2.03036i
29.18 0.587785 − 0.809017i −0.161693 + 1.72449i −0.309017 − 0.951057i 1.09193 − 1.95133i 1.30010 + 1.14444i −1.49298 − 4.59493i −0.951057 − 0.309017i −2.94771 − 0.557675i −0.936834 − 2.03036i
29.19 0.587785 − 0.809017i 0.130035 − 1.72716i −0.309017 − 0.951057i 0.0123239 − 2.23603i −1.32087 − 1.12040i 0.657342 + 2.02309i −0.951057 − 0.309017i −2.96618 − 0.449182i −1.80175 − 1.32428i
29.20 0.587785 − 0.809017i 0.475008 + 1.66564i −0.309017 − 0.951057i −2.02080 + 0.957269i 1.62674 + 0.594751i 0.545881 + 1.68005i −0.951057 − 0.309017i −2.54873 + 1.58239i −0.413350 + 2.19753i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
11.d odd 10 1 inner
15.d odd 2 1 inner
33.f even 10 1 inner
55.h odd 10 1 inner
165.r even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.p.a ✓ 96
3.b odd 2 1 inner 330.2.p.a ✓ 96
5.b even 2 1 inner 330.2.p.a ✓ 96
11.d odd 10 1 inner 330.2.p.a ✓ 96
15.d odd 2 1 inner 330.2.p.a ✓ 96
33.f even 10 1 inner 330.2.p.a ✓ 96
55.h odd 10 1 inner 330.2.p.a ✓ 96
165.r even 10 1 inner 330.2.p.a ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.p.a ✓ 96 1.a even 1 1 trivial
330.2.p.a ✓ 96 3.b odd 2 1 inner
330.2.p.a ✓ 96 5.b even 2 1 inner
330.2.p.a ✓ 96 11.d odd 10 1 inner
330.2.p.a ✓ 96 15.d odd 2 1 inner
330.2.p.a ✓ 96 33.f even 10 1 inner
330.2.p.a ✓ 96 55.h odd 10 1 inner
330.2.p.a ✓ 96 165.r even 10 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(330, [\chi])\).