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Properties

Label 536.2.bd.a
Level $536$
Weight $2$
Character orbit 536.bd
Analytic conductor $4.280$
Analytic rank $0$
Dimension $40$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [536,2,Mod(11,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(66))
 
chi = DirichletCharacter(H, H._module([33, 33, 59]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.11");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 536.bd (of order \(66\), degree \(20\), 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: \(40\)
Relative dimension: \(2\) over \(\Q(\zeta_{66})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} - 4 q^{6} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 4 q^{4} - 4 q^{6} + 8 q^{9} - 32 q^{12} + 8 q^{16} + 72 q^{17} - 24 q^{18} + 2 q^{19} - 16 q^{22} - 16 q^{24} + 20 q^{25} + 8 q^{33} - 24 q^{34} + 8 q^{36} + 36 q^{38} + 24 q^{48} - 14 q^{49} + 42 q^{51} - 244 q^{54} - 30 q^{57} - 54 q^{59} + 32 q^{64} - 28 q^{67} + 24 q^{68} - 2 q^{73} + 110 q^{75} + 8 q^{76} - 52 q^{81} + 288 q^{82} - 162 q^{83} - 12 q^{86} - 16 q^{88} + 36 q^{89} - 16 q^{96} + 30 q^{97} - 48 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −1.33643 + 0.462544i −1.93962 + 0.885796i 1.57211 − 1.23632i 0 2.18246 − 2.08097i 0 −1.52916 + 2.37942i 1.01292 − 1.16897i 0
11.2 1.33643 − 0.462544i −1.69478 + 0.773982i 1.57211 − 1.23632i 0 −1.90696 + 1.81829i 0 1.52916 − 2.37942i 0.308661 − 0.356213i 0
51.1 −0.267642 + 1.38866i −1.45319 + 0.663648i −1.85674 − 0.743325i 0 −0.532646 − 2.19560i 0 1.52916 − 2.37942i −0.293262 + 0.338443i 0
51.2 0.267642 − 1.38866i 3.12046 − 1.42506i −1.85674 − 0.743325i 0 −1.14376 − 4.71465i 0 −1.52916 + 2.37942i 5.74185 − 6.62645i 0
99.1 −1.37435 − 0.333413i 2.37439 − 0.341386i 1.77767 + 0.916453i 0 −3.37707 − 0.322471i 0 −2.13758 − 1.85223i 2.64273 − 0.775975i 0
99.2 1.37435 + 0.333413i −3.12369 + 0.449119i 1.77767 + 0.916453i 0 −4.44279 − 0.424235i 0 2.13758 + 1.85223i 6.67728 − 1.96063i 0
115.1 −0.648030 + 1.25700i −0.530360 − 1.80624i −1.16011 − 1.62915i 0 2.61414 + 0.503834i 0 2.79964 − 0.402527i −0.457467 + 0.293996i 0
115.2 0.648030 − 1.25700i 0.949198 + 3.23267i −1.16011 − 1.62915i 0 4.67859 + 0.901724i 0 −2.79964 + 0.402527i −7.02543 + 4.51497i 0
147.1 −0.874209 − 1.11165i −2.27158 − 1.96834i −0.471518 + 1.94362i 0 −0.202259 + 4.24593i 0 2.57283 − 1.17497i 0.858788 + 5.97300i 0
147.2 0.874209 + 1.11165i 1.98423 + 1.71934i −0.471518 + 1.94362i 0 −0.176673 + 3.70882i 0 −2.57283 + 1.17497i 0.554073 + 3.85366i 0
195.1 −1.33643 − 0.462544i −1.93962 − 0.885796i 1.57211 + 1.23632i 0 2.18246 + 2.08097i 0 −1.52916 − 2.37942i 1.01292 + 1.16897i 0
195.2 1.33643 + 0.462544i −1.69478 − 0.773982i 1.57211 + 1.23632i 0 −1.90696 − 1.81829i 0 1.52916 + 2.37942i 0.308661 + 0.356213i 0
203.1 −1.41261 − 0.0672910i −0.278063 + 0.946996i 1.99094 + 0.190112i 0 0.456520 − 1.31903i 0 −2.79964 − 0.402527i 1.70428 + 1.09527i 0
203.2 1.41261 + 0.0672910i 0.974685 − 3.31947i 1.99094 + 0.190112i 0 1.60022 − 4.62354i 0 2.79964 + 0.402527i −7.54513 − 4.84896i 0
219.1 −1.15198 − 0.820324i −1.85288 + 2.88313i 0.654136 + 1.89000i 0 4.49959 − 1.80136i 0 0.796860 − 2.71386i −3.63306 − 7.95529i 0
219.2 1.15198 + 0.820324i 0.360538 − 0.561008i 0.654136 + 1.89000i 0 0.875542 − 0.350514i 0 −0.796860 + 2.71386i 1.06150 + 2.32437i 0
235.1 −1.41261 + 0.0672910i −0.278063 − 0.946996i 1.99094 − 0.190112i 0 0.456520 + 1.31903i 0 −2.79964 + 0.402527i 1.70428 − 1.09527i 0
235.2 1.41261 − 0.0672910i 0.974685 + 3.31947i 1.99094 − 0.190112i 0 1.60022 + 4.62354i 0 2.79964 − 0.402527i −7.54513 + 4.84896i 0
251.1 −0.134430 − 1.40781i 0.690288 + 1.07411i −1.96386 + 0.378502i 0 1.41935 − 1.11619i 0 0.796860 + 2.71386i 0.569032 − 1.24601i 0
251.2 0.134430 + 1.40781i 1.41132 + 2.19605i −1.96386 + 0.378502i 0 −2.90190 + 2.28208i 0 −0.796860 − 2.71386i −1.58458 + 3.46975i 0
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.2
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
67.h odd 66 1 inner
536.bd even 66 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.2.bd.a ✓ 40
8.d odd 2 1 CM 536.2.bd.a ✓ 40
67.h odd 66 1 inner 536.2.bd.a ✓ 40
536.bd even 66 1 inner 536.2.bd.a ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.2.bd.a ✓ 40 1.a even 1 1 trivial
536.2.bd.a ✓ 40 8.d odd 2 1 CM
536.2.bd.a ✓ 40 67.h odd 66 1 inner
536.2.bd.a ✓ 40 536.bd even 66 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} - 10 T_{3}^{38} + 99 T_{3}^{36} - 396 T_{3}^{35} - 980 T_{3}^{34} + 9900 T_{3}^{33} + \cdots + 480091921 \) acting on \(S_{2}^{\mathrm{new}}(536, [\chi])\). Copy content Toggle raw display