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Properties

Label 510.2.bb.b
Level $510$
Weight $2$
Character orbit 510.bb
Analytic conductor $4.072$
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] = [510,2,Mod(19,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 4, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.bb (of order \(8\), 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: \(4.07237050309\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(8\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5} + 16 q^{11} - 8 q^{15} - 32 q^{16} - 8 q^{20} - 16 q^{22} - 16 q^{23} + 16 q^{25} + 24 q^{26} + 24 q^{29} + 8 q^{30} - 8 q^{31} - 8 q^{34} + 16 q^{35} + 16 q^{37} + 8 q^{39} - 8 q^{40} - 8 q^{41} + 16 q^{43} - 16 q^{46} - 80 q^{47} + 32 q^{49} - 16 q^{50} - 24 q^{53} + 16 q^{58} - 16 q^{59} + 8 q^{60} - 16 q^{61} + 16 q^{62} + 32 q^{65} + 8 q^{66} - 24 q^{68} + 16 q^{69} - 16 q^{71} - 32 q^{72} + 72 q^{73} - 56 q^{74} - 64 q^{79} - 8 q^{80} - 8 q^{82} - 16 q^{84} + 8 q^{85} - 16 q^{86} - 8 q^{87} - 8 q^{90} + 64 q^{91} + 8 q^{92} - 32 q^{94} - 16 q^{95} - 32 q^{97} - 16 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −0.707107 + 0.707107i −0.382683 − 0.923880i − 1.00000i −1.29578 − 1.82235i 0.923880 + 0.382683i −2.74129 − 1.13548i 0.707107 + 0.707107i −0.707107 + 0.707107i 2.20485 + 0.372345i
19.2 −0.707107 + 0.707107i −0.382683 − 0.923880i − 1.00000i −0.442589 + 2.19183i 0.923880 + 0.382683i 1.49144 + 0.617775i 0.707107 + 0.707107i −0.707107 + 0.707107i −1.23690 − 1.86281i
19.3 −0.707107 + 0.707107i −0.382683 − 0.923880i − 1.00000i 0.181255 − 2.22871i 0.923880 + 0.382683i 3.83963 + 1.59043i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.44777 + 1.70410i
19.4 −0.707107 + 0.707107i −0.382683 − 0.923880i − 1.00000i 2.23269 − 0.122908i 0.923880 + 0.382683i −0.742013 − 0.307352i 0.707107 + 0.707107i −0.707107 + 0.707107i −1.49184 + 1.66566i
19.5 −0.707107 + 0.707107i 0.382683 + 0.923880i − 1.00000i −2.12871 + 0.684551i −0.923880 − 0.382683i −2.71572 − 1.12489i 0.707107 + 0.707107i −0.707107 + 0.707107i 1.02117 − 1.98927i
19.6 −0.707107 + 0.707107i 0.382683 + 0.923880i − 1.00000i −1.87699 − 1.21528i −0.923880 − 0.382683i 1.92601 + 0.797780i 0.707107 + 0.707107i −0.707107 + 0.707107i 2.18657 − 0.467897i
19.7 −0.707107 + 0.707107i 0.382683 + 0.923880i − 1.00000i 1.74254 + 1.40127i −0.923880 − 0.382683i −3.75230 − 1.55425i 0.707107 + 0.707107i −0.707107 + 0.707107i −2.22301 + 0.241314i
19.8 −0.707107 + 0.707107i 0.382683 + 0.923880i − 1.00000i 2.17337 + 0.525815i −0.923880 − 0.382683i 2.69425 + 1.11600i 0.707107 + 0.707107i −0.707107 + 0.707107i −1.90861 + 1.16499i
49.1 0.707107 − 0.707107i −0.923880 + 0.382683i − 1.00000i −1.75517 − 1.38542i −0.382683 + 0.923880i −0.929745 + 2.24460i −0.707107 − 0.707107i 0.707107 − 0.707107i −2.22073 + 0.261456i
49.2 0.707107 − 0.707107i −0.923880 + 0.382683i − 1.00000i 0.727151 − 2.11453i −0.382683 + 0.923880i −1.18882 + 2.87007i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.981028 − 2.00937i
49.3 0.707107 − 0.707107i −0.923880 + 0.382683i − 1.00000i 1.71573 + 1.43397i −0.382683 + 0.923880i −0.220483 + 0.532294i −0.707107 − 0.707107i 0.707107 − 0.707107i 2.22717 − 0.199236i
49.4 0.707107 − 0.707107i −0.923880 + 0.382683i − 1.00000i 1.94327 − 1.10620i −0.382683 + 0.923880i 1.57368 − 3.79921i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.591900 − 2.15631i
49.5 0.707107 − 0.707107i 0.923880 − 0.382683i − 1.00000i −1.56696 − 1.59519i 0.382683 − 0.923880i 0.648047 − 1.56452i −0.707107 − 0.707107i 0.707107 − 0.707107i −2.23598 − 0.0199672i
49.6 0.707107 − 0.707107i 0.923880 − 0.382683i − 1.00000i −1.31727 + 1.80688i 0.382683 − 0.923880i 0.589677 − 1.42361i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.346205 + 2.20910i
49.7 0.707107 − 0.707107i 0.923880 − 0.382683i − 1.00000i 1.67911 − 1.47669i 0.382683 − 0.923880i 0.420365 − 1.01485i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.143132 − 2.23148i
49.8 0.707107 − 0.707107i 0.923880 − 0.382683i − 1.00000i 1.98835 + 1.02297i 0.382683 − 0.923880i −0.892722 + 2.15522i −0.707107 − 0.707107i 0.707107 − 0.707107i 2.12933 − 0.682622i
229.1 0.707107 + 0.707107i −0.923880 − 0.382683i 1.00000i −1.75517 + 1.38542i −0.382683 − 0.923880i −0.929745 − 2.24460i −0.707107 + 0.707107i 0.707107 + 0.707107i −2.22073 − 0.261456i
229.2 0.707107 + 0.707107i −0.923880 − 0.382683i 1.00000i 0.727151 + 2.11453i −0.382683 − 0.923880i −1.18882 − 2.87007i −0.707107 + 0.707107i 0.707107 + 0.707107i −0.981028 + 2.00937i
229.3 0.707107 + 0.707107i −0.923880 − 0.382683i 1.00000i 1.71573 − 1.43397i −0.382683 − 0.923880i −0.220483 − 0.532294i −0.707107 + 0.707107i 0.707107 + 0.707107i 2.22717 + 0.199236i
229.4 0.707107 + 0.707107i −0.923880 − 0.382683i 1.00000i 1.94327 + 1.10620i −0.382683 − 0.923880i 1.57368 + 3.79921i −0.707107 + 0.707107i 0.707107 + 0.707107i 0.591900 + 2.15631i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
85.m even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 510.2.bb.b yes 32
5.b even 2 1 510.2.bb.a ✓ 32
17.d even 8 1 510.2.bb.a ✓ 32
85.m even 8 1 inner 510.2.bb.b yes 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.bb.a ✓ 32 5.b even 2 1
510.2.bb.a ✓ 32 17.d even 8 1
510.2.bb.b yes 32 1.a even 1 1 trivial
510.2.bb.b yes 32 85.m even 8 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{32} - 16 T_{7}^{30} + 16 T_{7}^{29} + 128 T_{7}^{28} - 544 T_{7}^{27} + 752 T_{7}^{26} + \cdots + 19252117504 \) acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\). Copy content Toggle raw display