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Properties

Label 510.2.bb.a
Level $510$
Weight $2$
Character orbit 510.bb
Analytic conductor $4.072$
Analytic rank $0$
Dimension $32$
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 algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 8 q^{5} - 8 q^{10} + 16 q^{11} + 8 q^{15} - 32 q^{16} + 16 q^{22} + 16 q^{23} - 24 q^{25} + 24 q^{26} + 24 q^{29} - 8 q^{30} - 8 q^{31} - 8 q^{34} + 16 q^{35} - 16 q^{37} + 8 q^{39} + 8 q^{40} - 8 q^{41} - 16 q^{43} + 8 q^{45} - 16 q^{46} + 80 q^{47} + 32 q^{49} - 16 q^{50} + 24 q^{53} - 16 q^{58} - 16 q^{59} - 8 q^{60} - 16 q^{61} - 16 q^{62} + 40 q^{65} + 8 q^{66} + 24 q^{68} + 16 q^{69} + 16 q^{70} - 16 q^{71} + 32 q^{72} - 72 q^{73} - 56 q^{74} - 32 q^{75} - 64 q^{79} - 8 q^{80} + 8 q^{82} - 16 q^{84} - 80 q^{85} - 16 q^{86} + 8 q^{87} + 64 q^{91} - 8 q^{92} - 32 q^{94} - 96 q^{95} + 32 q^{97} - 16 q^{99}+O(q^{100}) \) 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 −1.16499 + 1.90861i −0.923880 − 0.382683i −2.69425 − 1.11600i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.525815 + 2.17337i
19.2 0.707107 − 0.707107i −0.382683 − 0.923880i − 1.00000i −0.241314 + 2.22301i −0.923880 − 0.382683i 3.75230 + 1.55425i −0.707107 − 0.707107i −0.707107 + 0.707107i 1.40127 + 1.74254i
19.3 0.707107 − 0.707107i −0.382683 − 0.923880i − 1.00000i 0.467897 − 2.18657i −0.923880 − 0.382683i −1.92601 − 0.797780i −0.707107 − 0.707107i −0.707107 + 0.707107i −1.21528 − 1.87699i
19.4 0.707107 − 0.707107i −0.382683 − 0.923880i − 1.00000i 1.98927 − 1.02117i −0.923880 − 0.382683i 2.71572 + 1.12489i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.684551 − 2.12871i
19.5 0.707107 − 0.707107i 0.382683 + 0.923880i − 1.00000i −1.70410 − 1.44777i 0.923880 + 0.382683i −3.83963 − 1.59043i −0.707107 − 0.707107i −0.707107 + 0.707107i −2.22871 + 0.181255i
19.6 0.707107 − 0.707107i 0.382683 + 0.923880i − 1.00000i −1.66566 + 1.49184i 0.923880 + 0.382683i 0.742013 + 0.307352i −0.707107 − 0.707107i −0.707107 + 0.707107i −0.122908 + 2.23269i
19.7 0.707107 − 0.707107i 0.382683 + 0.923880i − 1.00000i −0.372345 − 2.20485i 0.923880 + 0.382683i 2.74129 + 1.13548i −0.707107 − 0.707107i −0.707107 + 0.707107i −1.82235 − 1.29578i
19.8 0.707107 − 0.707107i 0.382683 + 0.923880i − 1.00000i 1.86281 + 1.23690i 0.923880 + 0.382683i −1.49144 − 0.617775i −0.707107 − 0.707107i −0.707107 + 0.707107i 2.19183 − 0.442589i
49.1 −0.707107 + 0.707107i −0.923880 + 0.382683i − 1.00000i −2.20910 − 0.346205i 0.382683 − 0.923880i −0.589677 + 1.42361i 0.707107 + 0.707107i 0.707107 − 0.707107i 1.80688 − 1.31727i
49.2 −0.707107 + 0.707107i −0.923880 + 0.382683i − 1.00000i 0.0199672 + 2.23598i 0.382683 − 0.923880i −0.648047 + 1.56452i 0.707107 + 0.707107i 0.707107 − 0.707107i −1.59519 − 1.56696i
49.3 −0.707107 + 0.707107i −0.923880 + 0.382683i − 1.00000i 0.682622 − 2.12933i 0.382683 − 0.923880i 0.892722 − 2.15522i 0.707107 + 0.707107i 0.707107 − 0.707107i 1.02297 + 1.98835i
49.4 −0.707107 + 0.707107i −0.923880 + 0.382683i − 1.00000i 2.23148 − 0.143132i 0.382683 − 0.923880i −0.420365 + 1.01485i 0.707107 + 0.707107i 0.707107 − 0.707107i −1.47669 + 1.67911i
49.5 −0.707107 + 0.707107i 0.923880 − 0.382683i − 1.00000i −0.261456 + 2.22073i −0.382683 + 0.923880i 0.929745 − 2.24460i 0.707107 + 0.707107i 0.707107 − 0.707107i −1.38542 − 1.75517i
49.6 −0.707107 + 0.707107i 0.923880 − 0.382683i − 1.00000i 0.199236 − 2.22717i −0.382683 + 0.923880i 0.220483 − 0.532294i 0.707107 + 0.707107i 0.707107 − 0.707107i 1.43397 + 1.71573i
49.7 −0.707107 + 0.707107i 0.923880 − 0.382683i − 1.00000i 2.00937 + 0.981028i −0.382683 + 0.923880i 1.18882 − 2.87007i 0.707107 + 0.707107i 0.707107 − 0.707107i −2.11453 + 0.727151i
49.8 −0.707107 + 0.707107i 0.923880 − 0.382683i − 1.00000i 2.15631 − 0.591900i −0.382683 + 0.923880i −1.57368 + 3.79921i 0.707107 + 0.707107i 0.707107 − 0.707107i −1.10620 + 1.94327i
229.1 −0.707107 − 0.707107i −0.923880 − 0.382683i 1.00000i −2.20910 + 0.346205i 0.382683 + 0.923880i −0.589677 − 1.42361i 0.707107 − 0.707107i 0.707107 + 0.707107i 1.80688 + 1.31727i
229.2 −0.707107 − 0.707107i −0.923880 − 0.382683i 1.00000i 0.0199672 − 2.23598i 0.382683 + 0.923880i −0.648047 − 1.56452i 0.707107 − 0.707107i 0.707107 + 0.707107i −1.59519 + 1.56696i
229.3 −0.707107 − 0.707107i −0.923880 − 0.382683i 1.00000i 0.682622 + 2.12933i 0.382683 + 0.923880i 0.892722 + 2.15522i 0.707107 − 0.707107i 0.707107 + 0.707107i 1.02297 − 1.98835i
229.4 −0.707107 − 0.707107i −0.923880 − 0.382683i 1.00000i 2.23148 + 0.143132i 0.382683 + 0.923880i −0.420365 − 1.01485i 0.707107 − 0.707107i 0.707107 + 0.707107i −1.47669 − 1.67911i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.8
Significant digits:
Format:

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

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