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Properties

Label 510.2.z.c
Level $510$
Weight $2$
Character orbit 510.z
Analytic conductor $4.072$
Analytic rank $0$
Dimension $68$
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(53,510)]
 
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([4, 6, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.z (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: \(68\)
Relative dimension: \(17\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 68 q^{4} - 8 q^{5} + 4 q^{6} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 68 q^{4} - 8 q^{5} + 4 q^{6} - 4 q^{7} + 4 q^{11} + 4 q^{12} - 4 q^{13} - 8 q^{14} + 20 q^{15} + 68 q^{16} + 8 q^{17} + 4 q^{18} - 4 q^{19} + 8 q^{20} - 32 q^{21} + 8 q^{22} + 4 q^{23} - 4 q^{24} - 8 q^{25} + 4 q^{26} - 4 q^{27} + 4 q^{28} - 16 q^{29} + 16 q^{30} + 8 q^{31} + 12 q^{33} - 12 q^{34} + 36 q^{35} + 4 q^{37} + 4 q^{38} - 44 q^{41} - 12 q^{42} + 24 q^{43} - 4 q^{44} + 56 q^{45} + 8 q^{46} + 12 q^{47} - 4 q^{48} + 44 q^{49} + 20 q^{50} - 60 q^{51} + 4 q^{52} - 16 q^{53} - 4 q^{54} + 32 q^{55} + 8 q^{56} + 16 q^{57} - 4 q^{58} - 20 q^{59} - 20 q^{60} - 24 q^{61} - 4 q^{62} - 32 q^{63} - 68 q^{64} - 16 q^{65} - 24 q^{66} + 4 q^{67} - 8 q^{68} + 104 q^{69} - 8 q^{70} - 24 q^{71} - 4 q^{72} - 64 q^{73} + 24 q^{74} - 4 q^{75} + 4 q^{76} + 24 q^{77} + 8 q^{79} - 8 q^{80} - 92 q^{81} - 32 q^{82} + 24 q^{83} + 32 q^{84} - 16 q^{85} - 36 q^{87} - 8 q^{88} + 24 q^{90} - 100 q^{91} - 4 q^{92} - 60 q^{93} - 12 q^{94} + 4 q^{96} - 8 q^{97} - 44 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 − 1.00000i −1.69677 + 0.347822i −1.00000 −2.23448 − 0.0843538i 0.347822 + 1.69677i 0.829784 + 2.00328i 1.00000i 2.75804 − 1.18035i −0.0843538 + 2.23448i
53.2 − 1.00000i −1.55361 − 0.765694i −1.00000 2.18223 − 0.487720i −0.765694 + 1.55361i −1.00160 − 2.41808i 1.00000i 1.82742 + 2.37919i −0.487720 − 2.18223i
53.3 − 1.00000i −1.51007 + 0.848350i −1.00000 1.47611 + 1.67961i 0.848350 + 1.51007i −0.200520 − 0.484099i 1.00000i 1.56060 − 2.56213i 1.67961 − 1.47611i
53.4 − 1.00000i −1.46300 − 0.927160i −1.00000 1.45144 − 1.70098i −0.927160 + 1.46300i 1.80715 + 4.36284i 1.00000i 1.28075 + 2.71287i −1.70098 − 1.45144i
53.5 − 1.00000i −1.07349 − 1.35928i −1.00000 0.218231 + 2.22539i −1.35928 + 1.07349i 0.593305 + 1.43237i 1.00000i −0.695259 + 2.91832i 2.22539 − 0.218231i
53.6 − 1.00000i −1.05432 + 1.37419i −1.00000 1.51363 − 1.64589i 1.37419 + 1.05432i 0.0649895 + 0.156899i 1.00000i −0.776818 − 2.89768i −1.64589 − 1.51363i
53.7 − 1.00000i −0.811950 − 1.52995i −1.00000 −1.21731 − 1.87567i −1.52995 + 0.811950i −1.50864 − 3.64218i 1.00000i −1.68147 + 2.48448i −1.87567 + 1.21731i
53.8 − 1.00000i −0.633463 + 1.61206i −1.00000 −1.30152 + 1.81826i 1.61206 + 0.633463i −1.23846 − 2.98990i 1.00000i −2.19745 − 2.04235i 1.81826 + 1.30152i
53.9 − 1.00000i −0.266295 + 1.71146i −1.00000 −1.40096 − 1.74279i 1.71146 + 0.266295i 1.52584 + 3.68370i 1.00000i −2.85817 − 0.911505i −1.74279 + 1.40096i
53.10 − 1.00000i 0.0346928 − 1.73170i −1.00000 −0.656081 + 2.13765i −1.73170 − 0.0346928i 0.470731 + 1.13645i 1.00000i −2.99759 − 0.120155i 2.13765 + 0.656081i
53.11 − 1.00000i 0.566798 + 1.63669i −1.00000 1.83136 + 1.28301i 1.63669 − 0.566798i 0.0711949 + 0.171880i 1.00000i −2.35748 + 1.85534i 1.28301 − 1.83136i
53.12 − 1.00000i 0.885498 − 1.48859i −1.00000 −2.23369 + 0.103039i −1.48859 − 0.885498i −0.382491 − 0.923415i 1.00000i −1.43179 − 2.63628i 0.103039 + 2.23369i
53.13 − 1.00000i 0.923930 + 1.46504i −1.00000 −2.13138 − 0.676190i 1.46504 − 0.923930i −1.08402 − 2.61706i 1.00000i −1.29271 + 2.70720i −0.676190 + 2.13138i
53.14 − 1.00000i 1.13649 − 1.30706i −1.00000 1.55549 − 1.60638i −1.30706 − 1.13649i −0.701219 − 1.69289i 1.00000i −0.416795 − 2.97091i −1.60638 − 1.55549i
53.15 − 1.00000i 1.52172 + 0.827260i −1.00000 0.266360 − 2.22015i 0.827260 − 1.52172i 0.113597 + 0.274249i 1.00000i 1.63128 + 2.51772i −2.22015 − 0.266360i
53.16 − 1.00000i 1.57925 − 0.711306i −1.00000 0.396495 + 2.20063i −0.711306 − 1.57925i −1.80461 − 4.35672i 1.00000i 1.98809 − 2.24667i 2.20063 − 0.396495i
53.17 − 1.00000i 1.70747 + 0.290754i −1.00000 1.81961 + 1.29962i 0.290754 − 1.70747i 0.737866 + 1.78137i 1.00000i 2.83092 + 0.992907i 1.29962 − 1.81961i
77.1 1.00000i −1.69677 − 0.347822i −1.00000 −2.23448 + 0.0843538i 0.347822 − 1.69677i 0.829784 − 2.00328i − 1.00000i 2.75804 + 1.18035i −0.0843538 − 2.23448i
77.2 1.00000i −1.55361 + 0.765694i −1.00000 2.18223 + 0.487720i −0.765694 − 1.55361i −1.00160 + 2.41808i − 1.00000i 1.82742 − 2.37919i −0.487720 + 2.18223i
77.3 1.00000i −1.51007 − 0.848350i −1.00000 1.47611 − 1.67961i 0.848350 − 1.51007i −0.200520 + 0.484099i − 1.00000i 1.56060 + 2.56213i 1.67961 + 1.47611i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.17
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
255.v 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.z.c yes 68
3.b odd 2 1 510.2.z.d yes 68
5.c odd 4 1 510.2.w.d yes 68
15.e even 4 1 510.2.w.c ✓ 68
17.d even 8 1 510.2.w.c ✓ 68
51.g odd 8 1 510.2.w.d yes 68
85.n odd 8 1 510.2.z.d yes 68
255.v even 8 1 inner 510.2.z.c yes 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.w.c ✓ 68 15.e even 4 1
510.2.w.c ✓ 68 17.d even 8 1
510.2.w.d yes 68 5.c odd 4 1
510.2.w.d yes 68 51.g odd 8 1
510.2.z.c yes 68 1.a even 1 1 trivial
510.2.z.c yes 68 255.v even 8 1 inner
510.2.z.d yes 68 3.b odd 2 1
510.2.z.d yes 68 85.n odd 8 1

Hecke kernels

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

\( T_{7}^{68} + 4 T_{7}^{67} - 14 T_{7}^{66} - 92 T_{7}^{65} + 18 T_{7}^{64} + 704 T_{7}^{63} + \cdots + 90\!\cdots\!72 \) Copy content Toggle raw display
\( T_{11}^{68} - 4 T_{11}^{67} + 50 T_{11}^{66} - 284 T_{11}^{65} + 1650 T_{11}^{64} + \cdots + 16\!\cdots\!52 \) Copy content Toggle raw display