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Properties

Label 510.2.w.c
Level $510$
Weight $2$
Character orbit 510.w
Analytic conductor $4.072$
Analytic rank $0$
Dimension $68$
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(257,510)]
 
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, 2, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("510.257");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 510.w (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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 68 q^{2} + 68 q^{4} - 8 q^{5} + 8 q^{7} - 68 q^{8} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 68 q^{2} + 68 q^{4} - 8 q^{5} + 8 q^{7} - 68 q^{8} - 4 q^{9} + 8 q^{10} - 4 q^{11} + 4 q^{13} - 8 q^{14} + 16 q^{15} + 68 q^{16} - 12 q^{17} + 4 q^{18} + 4 q^{19} - 8 q^{20} + 32 q^{21} + 4 q^{22} - 8 q^{23} - 20 q^{25} - 4 q^{26} + 8 q^{28} - 16 q^{29} - 16 q^{30} + 8 q^{31} - 68 q^{32} + 12 q^{33} + 12 q^{34} + 36 q^{35} - 4 q^{36} - 24 q^{37} - 4 q^{38} + 12 q^{39} + 8 q^{40} + 44 q^{41} - 32 q^{42} - 4 q^{44} + 40 q^{45} + 8 q^{46} - 12 q^{47} - 44 q^{49} + 20 q^{50} + 56 q^{51} + 4 q^{52} - 32 q^{55} - 8 q^{56} - 12 q^{57} + 16 q^{58} - 20 q^{59} + 16 q^{60} - 24 q^{61} - 8 q^{62} + 16 q^{63} + 68 q^{64} + 36 q^{65} - 12 q^{66} + 4 q^{67} - 12 q^{68} + 104 q^{69} - 36 q^{70} + 24 q^{71} + 4 q^{72} - 60 q^{73} + 24 q^{74} + 64 q^{75} + 4 q^{76} - 12 q^{78} - 8 q^{79} - 8 q^{80} + 92 q^{81} - 44 q^{82} + 32 q^{84} - 112 q^{85} + 40 q^{87} + 4 q^{88} - 40 q^{90} - 100 q^{91} - 8 q^{92} + 24 q^{93} + 12 q^{94} - 84 q^{95} - 52 q^{97} + 44 q^{98} - 12 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1 −1.00000 −1.71722 − 0.226185i 1.00000 −2.23411 − 0.0935218i 1.71722 + 0.226185i −0.156899 + 0.0649895i −1.00000 2.89768 + 0.776818i 2.23411 + 0.0935218i
257.2 −1.00000 −1.66765 + 0.467905i 1.00000 0.143899 + 2.23143i 1.66765 − 0.467905i 0.484099 − 0.200520i −1.00000 2.56213 − 1.56060i −0.143899 − 2.23143i
257.3 −1.00000 −1.58782 − 0.691970i 1.00000 2.20601 + 0.365391i 1.58782 + 0.691970i 2.98990 − 1.23846i −1.00000 2.04235 + 2.19745i −2.20601 − 0.365391i
257.4 −1.00000 −1.44574 + 0.953848i 1.00000 1.52037 − 1.63966i 1.44574 − 0.953848i −2.00328 + 0.829784i −1.00000 1.18035 − 2.75804i −1.52037 + 1.63966i
257.5 −1.00000 −1.39848 − 1.02188i 1.00000 −0.241714 − 2.22297i 1.39848 + 1.02188i −3.68370 + 1.52584i −1.00000 0.911505 + 2.85817i 0.241714 + 2.22297i
257.6 −1.00000 −0.756525 − 1.55810i 1.00000 −0.387738 + 2.20219i 0.756525 + 1.55810i −0.171880 + 0.0711949i −1.00000 −1.85534 + 2.35748i 0.387738 − 2.20219i
257.7 −1.00000 −0.557142 + 1.64000i 1.00000 −1.88794 + 1.19820i 0.557142 − 1.64000i 2.41808 − 1.00160i −1.00000 −2.37919 − 1.82742i 1.88794 − 1.19820i
257.8 −1.00000 −0.382625 − 1.68926i 1.00000 1.02897 − 1.98525i 0.382625 + 1.68926i 2.61706 − 1.08402i −1.00000 −2.70720 + 1.29271i −1.02897 + 1.98525i
257.9 −1.00000 −0.378898 + 1.69010i 1.00000 −2.22909 − 0.176455i 0.378898 − 1.69010i −4.36284 + 1.80715i −1.00000 −2.71287 − 1.28075i 2.22909 + 0.176455i
257.10 −1.00000 0.202084 + 1.72022i 1.00000 1.41928 + 1.72790i −0.202084 − 1.72022i −1.43237 + 0.593305i −1.00000 −2.91832 + 0.695259i −1.41928 − 1.72790i
257.11 −1.00000 0.491059 − 1.66098i 1.00000 −1.75823 − 1.38154i −0.491059 + 1.66098i −0.274249 + 0.113597i −1.00000 −2.51772 − 1.63128i 1.75823 + 1.38154i
257.12 −1.00000 0.507701 + 1.65597i 1.00000 −0.465530 − 2.18707i −0.507701 − 1.65597i 3.64218 − 1.50864i −1.00000 −2.48448 + 1.68147i 0.465530 + 2.18707i
257.13 −1.00000 1.00177 − 1.41296i 1.00000 −0.367687 + 2.20563i −1.00177 + 1.41296i −1.78137 + 0.737866i −1.00000 −0.992907 − 2.83092i 0.367687 − 2.20563i
257.14 −1.00000 1.24903 + 1.19997i 1.00000 1.97547 + 1.04763i −1.24903 − 1.19997i −1.13645 + 0.470731i −1.00000 0.120155 + 2.99759i −1.97547 − 1.04763i
257.15 −1.00000 1.61967 − 0.613732i 1.00000 1.27572 + 1.83645i −1.61967 + 0.613732i 4.35672 − 1.80461i −1.00000 2.24667 − 1.98809i −1.27572 − 1.83645i
257.16 −1.00000 1.67873 + 0.426449i 1.00000 1.65232 − 1.50660i −1.67873 − 0.426449i 0.923415 − 0.382491i −1.00000 2.63628 + 1.43179i −1.65232 + 1.50660i
257.17 −1.00000 1.72785 + 0.120611i 1.00000 −2.23578 − 0.0359829i −1.72785 − 0.120611i 1.69289 − 0.701219i −1.00000 2.97091 + 0.416795i 2.23578 + 0.0359829i
263.1 −1.00000 −1.69307 − 0.365381i 1.00000 0.359023 + 2.20706i 1.69307 + 0.365381i 0.502145 − 1.21229i −1.00000 2.73299 + 1.23724i −0.359023 − 2.20706i
263.2 −1.00000 −1.64247 − 0.549808i 1.00000 −0.750236 − 2.10645i 1.64247 + 0.549808i −0.934419 + 2.25589i −1.00000 2.39542 + 1.80609i 0.750236 + 2.10645i
263.3 −1.00000 −1.53139 + 0.809221i 1.00000 −1.94500 + 1.10317i 1.53139 − 0.809221i −1.98298 + 4.78735i −1.00000 1.69032 − 2.47847i 1.94500 − 1.10317i
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 257.17
Significant digits:
Format:

Inner twists

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

Hecke kernels

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

\( T_{7}^{68} - 8 T_{7}^{67} + 54 T_{7}^{66} - 324 T_{7}^{65} + 1618 T_{7}^{64} - 6336 T_{7}^{63} + \cdots + 90\!\cdots\!72 \) 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