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Properties

Label 536.2.c.a
Level $536$
Weight $2$
Character orbit 536.c
Analytic conductor $4.280$
Analytic rank $0$
Dimension $66$
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] = [536,2,Mod(269,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("536.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 536.c (of order \(2\), degree \(1\), 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: \(4.27998154834\)
Analytic rank: \(0\)
Dimension: \(66\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 66 q - 4 q^{4} + 2 q^{6} - 6 q^{8} - 66 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q - 4 q^{4} + 2 q^{6} - 6 q^{8} - 66 q^{9} - 4 q^{10} + 4 q^{12} - 2 q^{14} + 8 q^{15} - 12 q^{16} - 4 q^{17} - 10 q^{18} + 22 q^{20} + 8 q^{22} - 12 q^{23} - 62 q^{25} - 24 q^{26} + 10 q^{28} + 22 q^{30} - 16 q^{31} + 10 q^{32} - 10 q^{34} + 2 q^{36} + 8 q^{38} + 8 q^{39} - 18 q^{40} + 4 q^{41} + 28 q^{42} + 34 q^{44} + 18 q^{46} + 20 q^{47} - 38 q^{48} + 66 q^{49} - 32 q^{50} - 6 q^{52} + 38 q^{54} + 16 q^{55} - 30 q^{56} + 2 q^{58} + 18 q^{60} + 22 q^{62} - 40 q^{63} + 26 q^{64} + 16 q^{65} - 34 q^{66} + 26 q^{68} - 18 q^{70} + 4 q^{71} - 2 q^{72} - 20 q^{73} - 4 q^{74} - 30 q^{76} - 36 q^{78} + 16 q^{79} - 32 q^{80} + 66 q^{81} + 16 q^{82} - 6 q^{84} + 2 q^{86} + 32 q^{87} + 22 q^{88} - 20 q^{89} - 18 q^{90} + 8 q^{92} - 16 q^{94} - 64 q^{96} + 12 q^{97} - 46 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1 −1.41318 − 0.0539790i 0.637598i 1.99417 + 0.152564i 3.76727i 0.0344169 − 0.901042i 3.61196 −2.80990 − 0.323245i 2.59347 0.203353 − 5.32384i
269.2 −1.41318 + 0.0539790i − 0.637598i 1.99417 − 0.152564i − 3.76727i 0.0344169 + 0.901042i 3.61196 −2.80990 + 0.323245i 2.59347 0.203353 + 5.32384i
269.3 −1.38335 − 0.293851i 0.169712i 1.82730 + 0.812996i − 1.96500i 0.0498701 − 0.234771i −2.19112 −2.28890 − 1.66161i 2.97120 −0.577416 + 2.71828i
269.4 −1.38335 + 0.293851i − 0.169712i 1.82730 − 0.812996i 1.96500i 0.0498701 + 0.234771i −2.19112 −2.28890 + 1.66161i 2.97120 −0.577416 − 2.71828i
269.5 −1.36197 − 0.380844i − 2.66849i 1.70992 + 1.03740i 0.500023i −1.01628 + 3.63440i 2.91987 −1.93376 − 2.06411i −4.12086 0.190431 − 0.681016i
269.6 −1.36197 + 0.380844i 2.66849i 1.70992 − 1.03740i − 0.500023i −1.01628 − 3.63440i 2.91987 −1.93376 + 2.06411i −4.12086 0.190431 + 0.681016i
269.7 −1.33251 − 0.473728i − 2.42962i 1.55116 + 1.26249i 1.44729i −1.15098 + 3.23749i −3.31305 −1.46886 − 2.41711i −2.90304 0.685621 − 1.92853i
269.8 −1.33251 + 0.473728i 2.42962i 1.55116 − 1.26249i − 1.44729i −1.15098 − 3.23749i −3.31305 −1.46886 + 2.41711i −2.90304 0.685621 + 1.92853i
269.9 −1.28456 − 0.591532i 2.92432i 1.30018 + 1.51971i 1.98992i 1.72983 − 3.75645i −1.89891 −0.771199 − 2.72126i −5.55162 1.17710 − 2.55616i
269.10 −1.28456 + 0.591532i − 2.92432i 1.30018 − 1.51971i − 1.98992i 1.72983 + 3.75645i −1.89891 −0.771199 + 2.72126i −5.55162 1.17710 + 2.55616i
269.11 −1.24445 − 0.671828i 0.811064i 1.09729 + 1.67211i 1.48237i 0.544895 − 1.00933i −0.447567 −0.242154 − 2.81804i 2.34218 0.995897 − 1.84473i
269.12 −1.24445 + 0.671828i − 0.811064i 1.09729 − 1.67211i − 1.48237i 0.544895 + 1.00933i −0.447567 −0.242154 + 2.81804i 2.34218 0.995897 + 1.84473i
269.13 −1.07826 − 0.915067i − 0.609143i 0.325306 + 1.97337i − 0.917662i −0.557406 + 0.656816i 2.48712 1.45500 − 2.42549i 2.62895 −0.839722 + 0.989482i
269.14 −1.07826 + 0.915067i 0.609143i 0.325306 − 1.97337i 0.917662i −0.557406 − 0.656816i 2.48712 1.45500 + 2.42549i 2.62895 −0.839722 − 0.989482i
269.15 −1.04337 − 0.954665i 2.76147i 0.177228 + 1.99213i − 2.61039i 2.63628 − 2.88123i 5.13828 1.71690 − 2.24772i −4.62573 −2.49205 + 2.72360i
269.16 −1.04337 + 0.954665i − 2.76147i 0.177228 − 1.99213i 2.61039i 2.63628 + 2.88123i 5.13828 1.71690 + 2.24772i −4.62573 −2.49205 − 2.72360i
269.17 −0.959378 − 1.03904i − 2.60130i −0.159188 + 1.99365i − 4.14613i −2.70285 + 2.49563i −0.431856 2.22420 − 1.74727i −3.76678 −4.30798 + 3.97771i
269.18 −0.959378 + 1.03904i 2.60130i −0.159188 − 1.99365i 4.14613i −2.70285 − 2.49563i −0.431856 2.22420 + 1.74727i −3.76678 −4.30798 − 3.97771i
269.19 −0.839692 − 1.13794i 1.89011i −0.589836 + 1.91105i − 3.64397i 2.15084 − 1.58711i −4.03010 2.66994 − 0.933488i −0.572522 −4.14663 + 3.05981i
269.20 −0.839692 + 1.13794i − 1.89011i −0.589836 − 1.91105i 3.64397i 2.15084 + 1.58711i −4.03010 2.66994 + 0.933488i −0.572522 −4.14663 − 3.05981i
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.66
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.2.c.a ✓ 66
4.b odd 2 1 2144.2.c.a 66
8.b even 2 1 inner 536.2.c.a ✓ 66
8.d odd 2 1 2144.2.c.a 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.2.c.a ✓ 66 1.a even 1 1 trivial
536.2.c.a ✓ 66 8.b even 2 1 inner
2144.2.c.a 66 4.b odd 2 1
2144.2.c.a 66 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(536, [\chi])\).