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Properties

Label 510.2.z.d
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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 68 q^{4} + 8 q^{5} - 4 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{3} - 68 q^{4} + 8 q^{5} - 4 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{12} - 4 q^{13} + 8 q^{14} + 8 q^{15} + 68 q^{16} - 8 q^{17} - 4 q^{19} - 8 q^{20} + 32 q^{21} + 8 q^{22} - 4 q^{23} - 8 q^{25} - 4 q^{26} - 28 q^{27} + 4 q^{28} + 16 q^{29} + 8 q^{31} - 24 q^{33} - 12 q^{34} - 36 q^{35} - 4 q^{36} + 4 q^{37} - 4 q^{38} - 12 q^{39} + 44 q^{41} + 12 q^{42} + 24 q^{43} + 4 q^{44} - 16 q^{45} + 8 q^{46} - 12 q^{47} - 4 q^{48} + 44 q^{49} - 20 q^{50} + 56 q^{51} + 4 q^{52} + 16 q^{53} + 32 q^{55} - 8 q^{56} - 36 q^{57} - 4 q^{58} + 20 q^{59} - 8 q^{60} - 24 q^{61} + 4 q^{62} + 12 q^{63} - 68 q^{64} + 16 q^{65} - 12 q^{66} + 4 q^{67} + 8 q^{68} - 104 q^{69} - 8 q^{70} + 24 q^{71} - 64 q^{73} - 24 q^{74} - 16 q^{75} + 4 q^{76} - 24 q^{77} + 20 q^{78} + 8 q^{79} + 8 q^{80} + 92 q^{81} - 32 q^{82} - 24 q^{83} - 32 q^{84} - 16 q^{85} - 36 q^{87} - 8 q^{88} + 16 q^{90} - 100 q^{91} + 4 q^{92} - 60 q^{93} - 12 q^{94} - 8 q^{97} + 44 q^{98} + 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 1.00000i −1.72022 + 0.202084i −1.00000 −0.218231 − 2.22539i −0.202084 − 1.72022i 0.593305 + 1.43237i − 1.00000i 2.91832 − 0.695259i 2.22539 − 0.218231i
53.2 1.00000i −1.69010 − 0.378898i −1.00000 −1.45144 + 1.70098i 0.378898 − 1.69010i 1.80715 + 4.36284i − 1.00000i 2.71287 + 1.28075i −1.70098 − 1.45144i
53.3 1.00000i −1.65597 + 0.507701i −1.00000 1.21731 + 1.87567i −0.507701 − 1.65597i −1.50864 − 3.64218i − 1.00000i 2.48448 − 1.68147i −1.87567 + 1.21731i
53.4 1.00000i −1.64000 − 0.557142i −1.00000 −2.18223 + 0.487720i 0.557142 − 1.64000i −1.00160 − 2.41808i − 1.00000i 2.37919 + 1.82742i −0.487720 − 2.18223i
53.5 1.00000i −1.19997 + 1.24903i −1.00000 0.656081 − 2.13765i −1.24903 − 1.19997i 0.470731 + 1.13645i − 1.00000i −0.120155 − 2.99759i 2.13765 + 0.656081i
53.6 1.00000i −0.953848 − 1.44574i −1.00000 2.23448 + 0.0843538i 1.44574 − 0.953848i 0.829784 + 2.00328i − 1.00000i −1.18035 + 2.75804i −0.0843538 + 2.23448i
53.7 1.00000i −0.467905 − 1.66765i −1.00000 −1.47611 − 1.67961i 1.66765 − 0.467905i −0.200520 − 0.484099i − 1.00000i −2.56213 + 1.56060i 1.67961 − 1.47611i
53.8 1.00000i −0.426449 + 1.67873i −1.00000 2.23369 − 0.103039i −1.67873 − 0.426449i −0.382491 − 0.923415i − 1.00000i −2.63628 − 1.43179i 0.103039 + 2.23369i
53.9 1.00000i −0.120611 + 1.72785i −1.00000 −1.55549 + 1.60638i −1.72785 − 0.120611i −0.701219 − 1.69289i − 1.00000i −2.97091 − 0.416795i −1.60638 − 1.55549i
53.10 1.00000i 0.226185 − 1.71722i −1.00000 −1.51363 + 1.64589i 1.71722 + 0.226185i 0.0649895 + 0.156899i − 1.00000i −2.89768 − 0.776818i −1.64589 − 1.51363i
53.11 1.00000i 0.613732 + 1.61967i −1.00000 −0.396495 − 2.20063i −1.61967 + 0.613732i −1.80461 − 4.35672i − 1.00000i −2.24667 + 1.98809i 2.20063 − 0.396495i
53.12 1.00000i 0.691970 − 1.58782i −1.00000 1.30152 − 1.81826i 1.58782 + 0.691970i −1.23846 − 2.98990i − 1.00000i −2.04235 − 2.19745i 1.81826 + 1.30152i
53.13 1.00000i 1.02188 − 1.39848i −1.00000 1.40096 + 1.74279i 1.39848 + 1.02188i 1.52584 + 3.68370i − 1.00000i −0.911505 − 2.85817i −1.74279 + 1.40096i
53.14 1.00000i 1.41296 + 1.00177i −1.00000 −1.81961 − 1.29962i −1.00177 + 1.41296i 0.737866 + 1.78137i − 1.00000i 0.992907 + 2.83092i 1.29962 − 1.81961i
53.15 1.00000i 1.55810 − 0.756525i −1.00000 −1.83136 − 1.28301i 0.756525 + 1.55810i 0.0711949 + 0.171880i − 1.00000i 1.85534 − 2.35748i 1.28301 − 1.83136i
53.16 1.00000i 1.66098 + 0.491059i −1.00000 −0.266360 + 2.22015i −0.491059 + 1.66098i 0.113597 + 0.274249i − 1.00000i 2.51772 + 1.63128i −2.22015 − 0.266360i
53.17 1.00000i 1.68926 − 0.382625i −1.00000 2.13138 + 0.676190i 0.382625 + 1.68926i −1.08402 − 2.61706i − 1.00000i 2.70720 − 1.29271i −0.676190 + 2.13138i
77.1 − 1.00000i −1.72022 − 0.202084i −1.00000 −0.218231 + 2.22539i −0.202084 + 1.72022i 0.593305 − 1.43237i 1.00000i 2.91832 + 0.695259i 2.22539 + 0.218231i
77.2 − 1.00000i −1.69010 + 0.378898i −1.00000 −1.45144 − 1.70098i 0.378898 + 1.69010i 1.80715 − 4.36284i 1.00000i 2.71287 − 1.28075i −1.70098 + 1.45144i
77.3 − 1.00000i −1.65597 − 0.507701i −1.00000 1.21731 − 1.87567i −0.507701 + 1.65597i −1.50864 + 3.64218i 1.00000i 2.48448 + 1.68147i −1.87567 − 1.21731i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.17
Significant digits:
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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.d yes 68
3.b odd 2 1 510.2.z.c yes 68
5.c odd 4 1 510.2.w.c ✓ 68
15.e even 4 1 510.2.w.d yes 68
17.d even 8 1 510.2.w.d yes 68
51.g odd 8 1 510.2.w.c ✓ 68
85.n odd 8 1 510.2.z.c yes 68
255.v even 8 1 inner 510.2.z.d yes 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
510.2.w.c ✓ 68 5.c odd 4 1
510.2.w.c ✓ 68 51.g odd 8 1
510.2.w.d yes 68 15.e even 4 1
510.2.w.d yes 68 17.d even 8 1
510.2.z.c yes 68 3.b odd 2 1
510.2.z.c yes 68 85.n odd 8 1
510.2.z.d yes 68 1.a even 1 1 trivial
510.2.z.d yes 68 255.v even 8 1 inner

Hecke kernels

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

\( T_{7}^{68} + 4 T_{7}^{67} - 14 T_{7}^{66} - 92 T_{7}^{65} + 18 T_{7}^{64} + 704 T_{7}^{63} + \cdots + 90\!\cdots\!72 \) 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