↑

Properties

Label 494.2.m.b
Level $494$
Weight $2$
Character orbit 494.m
Analytic conductor $3.945$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [494,2,Mod(153,494)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(494, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("494.153");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 494 = 2 \cdot 13 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 494.m (of order \(6\), 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: \(3.94460985985\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{3} + 14 q^{4} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 4 q^{3} + 14 q^{4} - 18 q^{9} - 2 q^{10} - 6 q^{11} - 8 q^{12} + 8 q^{13} + 4 q^{14} + 42 q^{15} - 14 q^{16} + 2 q^{17} - 10 q^{22} + 12 q^{23} - 88 q^{25} + 2 q^{26} + 20 q^{27} - 4 q^{29} - 4 q^{30} - 54 q^{33} - 14 q^{35} + 18 q^{36} + 6 q^{37} - 28 q^{38} + 60 q^{39} - 4 q^{40} + 42 q^{41} + 18 q^{42} + 28 q^{43} - 30 q^{45} - 12 q^{46} - 4 q^{48} + 28 q^{49} + 12 q^{50} - 20 q^{51} + 4 q^{52} + 24 q^{53} + 54 q^{54} + 2 q^{55} + 2 q^{56} - 6 q^{58} - 96 q^{59} - 30 q^{61} + 4 q^{62} + 12 q^{63} - 28 q^{64} + 82 q^{65} + 24 q^{66} - 6 q^{67} - 2 q^{68} - 42 q^{69} - 6 q^{71} + 12 q^{72} - 18 q^{74} + 46 q^{75} - 136 q^{77} + 50 q^{78} + 68 q^{79} + 6 q^{81} + 6 q^{82} + 12 q^{84} - 72 q^{85} - 52 q^{87} + 10 q^{88} + 48 q^{89} - 16 q^{90} + 44 q^{91} + 24 q^{92} + 30 q^{93} + 20 q^{94} - 2 q^{95} + 12 q^{97} - 72 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
153.1 −0.866025 + 0.500000i −1.40902 − 2.44050i 0.500000 − 0.866025i 3.87767i 2.44050 + 1.40902i −2.48598 − 1.43528i 1.00000i −2.47070 + 4.27937i −1.93884 − 3.35817i
153.2 −0.866025 + 0.500000i −0.785510 − 1.36054i 0.500000 − 0.866025i − 2.81950i 1.36054 + 0.785510i 4.00004 + 2.30942i 1.00000i 0.265948 − 0.460635i 1.40975 + 2.44175i
153.3 −0.866025 + 0.500000i −0.750797 − 1.30042i 0.500000 − 0.866025i − 2.98870i 1.30042 + 0.750797i −3.23154 − 1.86573i 1.00000i 0.372609 − 0.645378i 1.49435 + 2.58829i
153.4 −0.866025 + 0.500000i −0.654655 − 1.13390i 0.500000 − 0.866025i 4.16024i 1.13390 + 0.654655i 4.23825 + 2.44695i 1.00000i 0.642852 − 1.11345i −2.08012 − 3.60287i
153.5 −0.866025 + 0.500000i 0.171852 + 0.297656i 0.500000 − 0.866025i 2.06376i −0.297656 − 0.171852i −0.942528 − 0.544169i 1.00000i 1.44093 − 2.49577i −1.03188 − 1.78727i
153.6 −0.866025 + 0.500000i 1.03817 + 1.79816i 0.500000 − 0.866025i − 2.52870i −1.79816 − 1.03817i −3.26870 − 1.88719i 1.00000i −0.655585 + 1.13551i 1.26435 + 2.18992i
153.7 −0.866025 + 0.500000i 1.38997 + 2.40749i 0.500000 − 0.866025i − 0.764778i −2.40749 − 1.38997i 0.824441 + 0.475991i 1.00000i −2.36401 + 4.09459i 0.382389 + 0.662317i
153.8 0.866025 − 0.500000i −1.68216 − 2.91359i 0.500000 − 0.866025i 3.01748i −2.91359 − 1.68216i −1.39082 − 0.802988i − 1.00000i −4.15934 + 7.20420i 1.50874 + 2.61321i
153.9 0.866025 − 0.500000i −1.33253 − 2.30801i 0.500000 − 0.866025i − 2.40281i −2.30801 − 1.33253i −0.588885 − 0.339993i − 1.00000i −2.05126 + 3.55289i −1.20141 − 2.08090i
153.10 0.866025 − 0.500000i −0.792694 − 1.37299i 0.500000 − 0.866025i 2.09051i −1.37299 − 0.792694i 2.83583 + 1.63727i − 1.00000i 0.243272 − 0.421359i 1.04526 + 1.81044i
153.11 0.866025 − 0.500000i −0.115664 − 0.200335i 0.500000 − 0.866025i − 2.62395i −0.200335 − 0.115664i −2.39869 − 1.38488i − 1.00000i 1.47324 − 2.55173i −1.31197 − 2.27240i
153.12 0.866025 − 0.500000i 0.589345 + 1.02077i 0.500000 − 0.866025i − 0.970755i 1.02077 + 0.589345i −1.68102 − 0.970537i − 1.00000i 0.805346 − 1.39490i −0.485378 − 0.840699i
153.13 0.866025 − 0.500000i 1.01102 + 1.75115i 0.500000 − 0.866025i 3.58162i 1.75115 + 1.01102i 0.562149 + 0.324557i − 1.00000i −0.544341 + 0.942826i 1.79081 + 3.10177i
153.14 0.866025 − 0.500000i 1.32268 + 2.29095i 0.500000 − 0.866025i − 3.69209i 2.29095 + 1.32268i 3.52745 + 2.03657i − 1.00000i −1.99896 + 3.46231i −1.84605 − 3.19745i
381.1 −0.866025 − 0.500000i −1.40902 + 2.44050i 0.500000 + 0.866025i − 3.87767i 2.44050 − 1.40902i −2.48598 + 1.43528i − 1.00000i −2.47070 − 4.27937i −1.93884 + 3.35817i
381.2 −0.866025 − 0.500000i −0.785510 + 1.36054i 0.500000 + 0.866025i 2.81950i 1.36054 − 0.785510i 4.00004 − 2.30942i − 1.00000i 0.265948 + 0.460635i 1.40975 − 2.44175i
381.3 −0.866025 − 0.500000i −0.750797 + 1.30042i 0.500000 + 0.866025i 2.98870i 1.30042 − 0.750797i −3.23154 + 1.86573i − 1.00000i 0.372609 + 0.645378i 1.49435 − 2.58829i
381.4 −0.866025 − 0.500000i −0.654655 + 1.13390i 0.500000 + 0.866025i − 4.16024i 1.13390 − 0.654655i 4.23825 − 2.44695i − 1.00000i 0.642852 + 1.11345i −2.08012 + 3.60287i
381.5 −0.866025 − 0.500000i 0.171852 − 0.297656i 0.500000 + 0.866025i − 2.06376i −0.297656 + 0.171852i −0.942528 + 0.544169i − 1.00000i 1.44093 + 2.49577i −1.03188 + 1.78727i
381.6 −0.866025 − 0.500000i 1.03817 − 1.79816i 0.500000 + 0.866025i 2.52870i −1.79816 + 1.03817i −3.26870 + 1.88719i − 1.00000i −0.655585 − 1.13551i 1.26435 − 2.18992i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 153.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 494.2.m.b ✓ 28
13.e even 6 1 inner 494.2.m.b ✓ 28
13.f odd 12 1 6422.2.a.bm 14
13.f odd 12 1 6422.2.a.bn 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
494.2.m.b ✓ 28 1.a even 1 1 trivial
494.2.m.b ✓ 28 13.e even 6 1 inner
6422.2.a.bm 14 13.f odd 12 1
6422.2.a.bn 14 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{28} + 4 T_{3}^{27} + 38 T_{3}^{26} + 108 T_{3}^{25} + 712 T_{3}^{24} + 1756 T_{3}^{23} + \cdots + 128164 \) acting on \(S_{2}^{\mathrm{new}}(494, [\chi])\). Copy content Toggle raw display