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Properties

Label 660.2.bl.b
Level $660$
Weight $2$
Character orbit 660.bl
Analytic conductor $5.270$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [660,2,Mod(41,660)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(660, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 5, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("660.41");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 660 = 2^{2} \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 660.bl (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: \(5.27012653340\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 4 q^{3} - 10 q^{7} + 18 q^{9} + 6 q^{15} + 60 q^{19} + 12 q^{25} - 2 q^{27} + 26 q^{31} + 26 q^{33} - 6 q^{37} + 20 q^{39} - 32 q^{45} + 22 q^{49} - 10 q^{51} - 22 q^{55} + 10 q^{57} - 10 q^{61} - 120 q^{63} + 76 q^{67} - 110 q^{69} + 10 q^{73} + 6 q^{75} - 20 q^{79} - 42 q^{81} + 20 q^{85} - 54 q^{91} - 56 q^{93} + 138 q^{97} + 84 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1 0 −1.73048 + 0.0737875i 0 −0.587785 + 0.809017i 0 1.00509 + 0.326574i 0 2.98911 − 0.255375i 0
41.2 0 −1.70423 + 0.309181i 0 0.587785 − 0.809017i 0 −4.89894 − 1.59176i 0 2.80881 − 1.05383i 0
41.3 0 −1.38861 − 1.03526i 0 −0.587785 + 0.809017i 0 −0.0362461 − 0.0117771i 0 0.856463 + 2.87515i 0
41.4 0 −1.12179 + 1.31969i 0 0.587785 − 0.809017i 0 0.145172 + 0.0471692i 0 −0.483161 − 2.96084i 0
41.5 0 −0.932745 + 1.45945i 0 −0.587785 + 0.809017i 0 −1.01433 − 0.329575i 0 −1.25997 − 2.72258i 0
41.6 0 0.343109 + 1.69773i 0 0.587785 − 0.809017i 0 4.10827 + 1.33486i 0 −2.76455 + 1.16501i 0
41.7 0 0.514895 − 1.65375i 0 0.587785 − 0.809017i 0 −0.0362461 − 0.0117771i 0 −2.46977 − 1.70301i 0
41.8 0 0.720317 + 1.57516i 0 −0.587785 + 0.809017i 0 4.10827 + 1.33486i 0 −1.96229 + 2.26924i 0
41.9 0 1.44336 − 0.957454i 0 0.587785 − 0.809017i 0 1.00509 + 0.326574i 0 1.16656 − 2.76390i 0
41.10 0 1.56048 − 0.751590i 0 −0.587785 + 0.809017i 0 −4.89894 − 1.59176i 0 1.87023 − 2.34569i 0
41.11 0 1.61245 + 0.632464i 0 0.587785 − 0.809017i 0 −1.01433 − 0.329575i 0 2.19998 + 2.03963i 0
41.12 0 1.68324 + 0.408278i 0 −0.587785 + 0.809017i 0 0.145172 + 0.0471692i 0 2.66662 + 1.37446i 0
101.1 0 −1.69419 + 0.360153i 0 −0.951057 + 0.309017i 0 1.50090 − 2.06580i 0 2.74058 − 1.22034i 0
101.2 0 −1.40794 + 1.00881i 0 0.951057 − 0.309017i 0 0.0804042 − 0.110667i 0 0.964588 − 2.84070i 0
101.3 0 −1.39452 + 1.02729i 0 −0.951057 + 0.309017i 0 0.0804042 − 0.110667i 0 0.889353 − 2.86514i 0
101.4 0 −1.21822 − 1.23123i 0 −0.951057 + 0.309017i 0 −0.445787 + 0.613573i 0 −0.0318721 + 2.99983i 0
101.5 0 −0.866060 + 1.49998i 0 0.951057 − 0.309017i 0 1.50090 − 2.06580i 0 −1.49988 − 2.59815i 0
101.6 0 −0.0609128 − 1.73098i 0 0.951057 − 0.309017i 0 −1.62361 + 2.23471i 0 −2.99258 + 0.210877i 0
101.7 0 0.716063 − 1.57710i 0 −0.951057 + 0.309017i 0 −2.57679 + 3.54665i 0 −1.97451 − 2.25861i 0
101.8 0 0.794521 + 1.53907i 0 0.951057 − 0.309017i 0 −0.445787 + 0.613573i 0 −1.73747 + 2.44565i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
11.d odd 10 1 inner
33.f even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 660.2.bl.b ✓ 48
3.b odd 2 1 inner 660.2.bl.b ✓ 48
11.d odd 10 1 inner 660.2.bl.b ✓ 48
33.f even 10 1 inner 660.2.bl.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.bl.b ✓ 48 1.a even 1 1 trivial
660.2.bl.b ✓ 48 3.b odd 2 1 inner
660.2.bl.b ✓ 48 11.d odd 10 1 inner
660.2.bl.b ✓ 48 33.f even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 5 T_{7}^{23} - 14 T_{7}^{22} - 140 T_{7}^{21} - 60 T_{7}^{20} + 1805 T_{7}^{19} + 4143 T_{7}^{18} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(660, [\chi])\). Copy content Toggle raw display