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Properties

Label 660.2.be.b
Level $660$
Weight $2$
Character orbit 660.be
Analytic conductor $5.270$
Analytic rank $0$
Dimension $96$
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(151,660)]
 
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([5, 0, 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("660.151");
 
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.be (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: \(96\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{4} + 24 q^{5} + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 8 q^{4} + 24 q^{5} + 24 q^{9} - 20 q^{13} - 4 q^{14} + 8 q^{16} + 10 q^{18} + 2 q^{20} + 30 q^{22} + 30 q^{24} - 24 q^{25} - 60 q^{28} + 20 q^{29} + 8 q^{34} - 8 q^{36} + 8 q^{37} - 58 q^{38} + 20 q^{41} - 42 q^{42} - 54 q^{44} + 96 q^{45} - 10 q^{46} + 16 q^{48} + 20 q^{49} + 56 q^{53} - 8 q^{56} - 68 q^{58} - 10 q^{64} + 54 q^{66} + 8 q^{69} + 4 q^{70} - 10 q^{72} - 40 q^{73} - 50 q^{74} - 32 q^{77} + 8 q^{78} + 2 q^{80} - 24 q^{81} - 24 q^{82} - 60 q^{84} - 84 q^{86} - 72 q^{88} + 72 q^{89} + 2 q^{92} + 80 q^{94} - 50 q^{96} - 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
151.1 −1.39999 + 0.200103i 0.951057 − 0.309017i 1.91992 − 0.560284i 0.809017 + 0.587785i −1.26963 + 0.622929i 0.767370 − 2.36172i −2.57574 + 1.16857i 0.809017 − 0.587785i −1.25023 − 0.661004i
151.2 −1.39898 − 0.207008i 0.951057 − 0.309017i 1.91430 + 0.579201i 0.809017 + 0.587785i −1.39448 + 0.235432i −0.370335 + 1.13977i −2.55816 − 1.20657i 0.809017 − 0.587785i −1.01012 − 0.989774i
151.3 −1.33488 − 0.467017i −0.951057 + 0.309017i 1.56379 + 1.24682i 0.809017 + 0.587785i 1.41386 + 0.0316600i −0.589915 + 1.81557i −1.50518 − 2.39467i 0.809017 − 0.587785i −0.805432 − 1.16245i
151.4 −1.32024 + 0.506913i −0.951057 + 0.309017i 1.48608 − 1.33850i 0.809017 + 0.587785i 1.09898 − 0.890080i −1.07719 + 3.31525i −1.28348 + 2.52045i 0.809017 − 0.587785i −1.36605 − 0.365918i
151.5 −0.965670 + 1.03319i −0.951057 + 0.309017i −0.134965 − 1.99544i 0.809017 + 0.587785i 0.599133 − 1.28103i 0.660835 − 2.03384i 2.19200 + 1.78749i 0.809017 − 0.587785i −1.38854 + 0.268262i
151.6 −0.909042 + 1.08335i 0.951057 − 0.309017i −0.347286 − 1.96962i 0.809017 + 0.587785i −0.529777 + 1.31123i −0.0472177 + 0.145321i 2.44948 + 1.41423i 0.809017 − 0.587785i −1.37221 + 0.342126i
151.7 −0.891899 − 1.09750i −0.951057 + 0.309017i −0.409031 + 1.95773i 0.809017 + 0.587785i 1.18739 + 0.768176i 0.00165851 − 0.00510436i 2.51343 − 1.29718i 0.809017 − 0.587785i −0.0764651 − 1.41214i
151.8 −0.768176 − 1.18739i 0.951057 − 0.309017i −0.819810 + 1.82426i 0.809017 + 0.587785i −1.09750 − 0.891899i −0.00165851 + 0.00510436i 2.79587 − 0.427913i 0.809017 − 0.587785i 0.0764651 − 1.41214i
151.9 −0.364252 + 1.36650i −0.951057 + 0.309017i −1.73464 − 0.995501i 0.809017 + 0.587785i −0.0758471 − 1.41218i −0.494503 + 1.52193i 1.99220 − 2.00777i 0.809017 − 0.587785i −1.09789 + 0.891419i
151.10 −0.198732 + 1.40018i 0.951057 − 0.309017i −1.92101 − 0.556521i 0.809017 + 0.587785i 0.243674 + 1.39306i −1.45429 + 4.47583i 1.16100 − 2.57916i 0.809017 − 0.587785i −0.983783 + 1.01596i
151.11 −0.0316600 − 1.41386i 0.951057 − 0.309017i −1.99800 + 0.0895256i 0.809017 + 0.587785i −0.467017 − 1.33488i 0.589915 − 1.81557i 0.189833 + 2.82205i 0.809017 − 0.587785i 0.805432 − 1.16245i
151.12 0.235432 − 1.39448i −0.951057 + 0.309017i −1.88914 − 0.656611i 0.809017 + 0.587785i 0.207008 + 1.39898i 0.370335 − 1.13977i −1.36040 + 2.47978i 0.809017 − 0.587785i 1.01012 − 0.989774i
151.13 0.269956 + 1.38821i 0.951057 − 0.309017i −1.85425 + 0.749510i 0.809017 + 0.587785i 0.685723 + 1.23684i 1.07349 − 3.30387i −1.54104 − 2.37175i 0.809017 − 0.587785i −0.597570 + 1.28176i
151.14 0.275572 + 1.38710i −0.951057 + 0.309017i −1.84812 + 0.764494i 0.809017 + 0.587785i −0.690723 − 1.23406i 0.550425 − 1.69403i −1.56972 − 2.35286i 0.809017 − 0.587785i −0.592378 + 1.28417i
151.15 0.622929 − 1.26963i −0.951057 + 0.309017i −1.22392 − 1.58178i 0.809017 + 0.587785i −0.200103 + 1.39999i −0.767370 + 2.36172i −2.77069 + 0.568589i 0.809017 − 0.587785i 1.25023 − 0.661004i
151.16 0.737307 + 1.20681i 0.951057 − 0.309017i −0.912758 + 1.77957i 0.809017 + 0.587785i 1.07414 + 0.919900i −0.917712 + 2.82443i −2.82058 + 0.210569i 0.809017 − 0.587785i −0.112849 + 1.40970i
151.17 0.890080 − 1.09898i 0.951057 − 0.309017i −0.415515 − 1.95636i 0.809017 + 0.587785i 0.506913 − 1.32024i 1.07719 − 3.31525i −2.51984 − 1.28467i 0.809017 − 0.587785i 1.36605 − 0.365918i
151.18 0.919900 + 1.07414i −0.951057 + 0.309017i −0.307569 + 1.97621i 0.809017 + 0.587785i −1.20681 − 0.737307i 0.917712 − 2.82443i −2.40566 + 1.48754i 0.809017 − 0.587785i 0.112849 + 1.40970i
151.19 1.23406 + 0.690723i 0.951057 − 0.309017i 1.04580 + 1.70479i 0.809017 + 0.587785i 1.38710 + 0.275572i −0.550425 + 1.69403i 0.113047 + 2.82617i 0.809017 − 0.587785i 0.592378 + 1.28417i
151.20 1.23684 + 0.685723i −0.951057 + 0.309017i 1.05957 + 1.69627i 0.809017 + 0.587785i −1.38821 − 0.269956i −1.07349 + 3.30387i 0.147351 + 2.82459i 0.809017 − 0.587785i 0.597570 + 1.28176i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 151.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.d odd 10 1 inner
44.g 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.be.b ✓ 96
4.b odd 2 1 inner 660.2.be.b ✓ 96
11.d odd 10 1 inner 660.2.be.b ✓ 96
44.g even 10 1 inner 660.2.be.b ✓ 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.2.be.b ✓ 96 1.a even 1 1 trivial
660.2.be.b ✓ 96 4.b odd 2 1 inner
660.2.be.b ✓ 96 11.d odd 10 1 inner
660.2.be.b ✓ 96 44.g even 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{96} + 74 T_{7}^{94} + 3283 T_{7}^{92} + 112572 T_{7}^{90} + 3322475 T_{7}^{88} + \cdots + 60523872256 \) acting on \(S_{2}^{\mathrm{new}}(660, [\chi])\). Copy content Toggle raw display