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Properties

Label 510.2.l.g
Level $510$
Weight $2$
Character orbit 510.l
Analytic conductor $4.072$
Analytic rank $0$
Dimension $28$
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] = [510,2,Mod(137,510)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(510, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.l (of order \(4\), degree \(2\), 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: \(28\)
Relative dimension: \(14\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 4 q^{7} - 12 q^{10} - 8 q^{13} + 8 q^{15} - 28 q^{16} - 16 q^{18} + 48 q^{21} + 20 q^{22} + 40 q^{25} - 36 q^{27} + 4 q^{28} - 12 q^{30} + 16 q^{31} + 8 q^{33} + 8 q^{37} - 8 q^{40} - 48 q^{42} + 48 q^{43} + 36 q^{45} - 64 q^{46} + 8 q^{52} - 36 q^{55} - 32 q^{57} + 20 q^{58} + 12 q^{60} - 16 q^{61} - 56 q^{63} + 32 q^{66} + 112 q^{67} - 28 q^{70} + 16 q^{72} + 12 q^{73} + 16 q^{75} - 8 q^{76} - 32 q^{78} - 48 q^{81} + 24 q^{82} - 12 q^{85} - 44 q^{87} + 20 q^{88} - 20 q^{90} - 80 q^{91} + 28 q^{93} + 60 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −0.707107 + 0.707107i −1.70258 − 0.318145i − 1.00000i 2.18694 − 0.466155i 1.42887 − 0.978944i −2.68155 − 2.68155i 0.707107 + 0.707107i 2.79757 + 1.08334i −1.21678 + 1.87602i
137.2 −0.707107 + 0.707107i −1.58179 + 0.705653i − 1.00000i −2.23394 + 0.0974809i 0.619522 − 1.61746i −3.25954 − 3.25954i 0.707107 + 0.707107i 2.00411 − 2.23239i 1.51071 − 1.64857i
137.3 −0.707107 + 0.707107i −1.26117 − 1.18720i − 1.00000i −2.17027 + 0.538469i 1.73126 − 0.0523066i 0.908386 + 0.908386i 0.707107 + 0.707107i 0.181113 + 2.99453i 1.15385 − 1.91536i
137.4 −0.707107 + 0.707107i 0.209343 + 1.71935i − 1.00000i −1.22996 + 1.86740i −1.36379 − 1.06774i 1.96777 + 1.96777i 0.707107 + 0.707107i −2.91235 + 0.719870i −0.450736 − 2.19017i
137.5 −0.707107 + 0.707107i 1.33031 − 1.10918i − 1.00000i 1.95694 + 1.08185i −0.156362 + 1.72498i 2.14663 + 2.14663i 0.707107 + 0.707107i 0.539441 − 2.95110i −2.14875 + 0.618778i
137.6 −0.707107 + 0.707107i 1.44910 + 0.948737i − 1.00000i 1.45929 − 1.69425i −1.69553 + 0.353812i 3.19962 + 3.19962i 0.707107 + 0.707107i 1.19980 + 2.74963i 0.166138 + 2.22989i
137.7 −0.707107 + 0.707107i 1.55679 − 0.759217i − 1.00000i 0.738113 + 2.11073i −0.563968 + 1.63766i −1.28132 − 1.28132i 0.707107 + 0.707107i 1.84718 − 2.36388i −2.01444 − 0.970588i
137.8 0.707107 − 0.707107i −1.71935 − 0.209343i − 1.00000i 1.22996 − 1.86740i −1.36379 + 1.06774i 1.96777 + 1.96777i −0.707107 − 0.707107i 2.91235 + 0.719870i −0.450736 − 2.19017i
137.9 0.707107 − 0.707107i −0.948737 − 1.44910i − 1.00000i −1.45929 + 1.69425i −1.69553 − 0.353812i 3.19962 + 3.19962i −0.707107 − 0.707107i −1.19980 + 2.74963i 0.166138 + 2.22989i
137.10 0.707107 − 0.707107i −0.705653 + 1.58179i − 1.00000i 2.23394 − 0.0974809i 0.619522 + 1.61746i −3.25954 − 3.25954i −0.707107 − 0.707107i −2.00411 − 2.23239i 1.51071 − 1.64857i
137.11 0.707107 − 0.707107i 0.318145 + 1.70258i − 1.00000i −2.18694 + 0.466155i 1.42887 + 0.978944i −2.68155 − 2.68155i −0.707107 − 0.707107i −2.79757 + 1.08334i −1.21678 + 1.87602i
137.12 0.707107 − 0.707107i 0.759217 − 1.55679i − 1.00000i −0.738113 − 2.11073i −0.563968 − 1.63766i −1.28132 − 1.28132i −0.707107 − 0.707107i −1.84718 − 2.36388i −2.01444 − 0.970588i
137.13 0.707107 − 0.707107i 1.10918 − 1.33031i − 1.00000i −1.95694 − 1.08185i −0.156362 − 1.72498i 2.14663 + 2.14663i −0.707107 − 0.707107i −0.539441 − 2.95110i −2.14875 + 0.618778i
137.14 0.707107 − 0.707107i 1.18720 + 1.26117i − 1.00000i 2.17027 − 0.538469i 1.73126 + 0.0523066i 0.908386 + 0.908386i −0.707107 − 0.707107i −0.181113 + 2.99453i 1.15385 − 1.91536i
443.1 −0.707107 − 0.707107i −1.70258 + 0.318145i 1.00000i 2.18694 + 0.466155i 1.42887 + 0.978944i −2.68155 + 2.68155i 0.707107 − 0.707107i 2.79757 − 1.08334i −1.21678 − 1.87602i
443.2 −0.707107 − 0.707107i −1.58179 − 0.705653i 1.00000i −2.23394 − 0.0974809i 0.619522 + 1.61746i −3.25954 + 3.25954i 0.707107 − 0.707107i 2.00411 + 2.23239i 1.51071 + 1.64857i
443.3 −0.707107 − 0.707107i −1.26117 + 1.18720i 1.00000i −2.17027 − 0.538469i 1.73126 + 0.0523066i 0.908386 − 0.908386i 0.707107 − 0.707107i 0.181113 − 2.99453i 1.15385 + 1.91536i
443.4 −0.707107 − 0.707107i 0.209343 − 1.71935i 1.00000i −1.22996 − 1.86740i −1.36379 + 1.06774i 1.96777 − 1.96777i 0.707107 − 0.707107i −2.91235 − 0.719870i −0.450736 + 2.19017i
443.5 −0.707107 − 0.707107i 1.33031 + 1.10918i 1.00000i 1.95694 − 1.08185i −0.156362 − 1.72498i 2.14663 − 2.14663i 0.707107 − 0.707107i 0.539441 + 2.95110i −2.14875 − 0.618778i
443.6 −0.707107 − 0.707107i 1.44910 − 0.948737i 1.00000i 1.45929 + 1.69425i −1.69553 − 0.353812i 3.19962 − 3.19962i 0.707107 − 0.707107i 1.19980 − 2.74963i 0.166138 − 2.22989i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 510.2.l.g ✓ 28
3.b odd 2 1 inner 510.2.l.g ✓ 28
5.c odd 4 1 inner 510.2.l.g ✓ 28
15.e even 4 1 inner 510.2.l.g ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.l.g ✓ 28 1.a even 1 1 trivial
510.2.l.g ✓ 28 3.b odd 2 1 inner
510.2.l.g ✓ 28 5.c odd 4 1 inner
510.2.l.g ✓ 28 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\):

\( T_{7}^{14} - 2 T_{7}^{13} + 2 T_{7}^{12} - 8 T_{7}^{11} + 632 T_{7}^{10} - 1408 T_{7}^{9} + \cdots + 2420000 \) Copy content Toggle raw display
\( T_{23}^{28} + 5392 T_{23}^{24} + 7875112 T_{23}^{20} + 2956073312 T_{23}^{16} + 309746053520 T_{23}^{12} + \cdots + 959512576 \) Copy content Toggle raw display