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Properties

Label 536.2.q.a
Level $536$
Weight $2$
Character orbit 536.q
Analytic conductor $4.280$
Analytic rank $0$
Dimension $80$
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(9,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 536.q (of order \(11\), degree \(10\), 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: \(80\)
Relative dimension: \(8\) over \(\Q(\zeta_{11})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q - q^{3} + q^{5} - 6 q^{7} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q - q^{3} + q^{5} - 6 q^{7} - 7 q^{9} - q^{11} - q^{13} - 6 q^{15} + 8 q^{17} + 7 q^{19} + 71 q^{21} - 18 q^{23} - 5 q^{25} - 4 q^{27} - 14 q^{29} - 34 q^{31} - 24 q^{33} + 20 q^{35} - 40 q^{37} + 21 q^{39} - 70 q^{41} + 10 q^{43} + 34 q^{45} - 8 q^{47} + 10 q^{49} + 10 q^{51} - 39 q^{53} - 50 q^{55} + 35 q^{57} - 66 q^{59} + 4 q^{61} + 21 q^{63} + 20 q^{65} - 24 q^{67} - 8 q^{69} - 13 q^{71} + 25 q^{73} + 44 q^{75} + 15 q^{77} + 59 q^{79} + 16 q^{81} - 51 q^{83} + 59 q^{85} - 26 q^{87} + 58 q^{89} + 16 q^{91} + 87 q^{93} - 56 q^{95} - 30 q^{97} + 93 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1 0 −3.01935 − 0.886560i 0 0.286208 − 0.330301i 0 −0.276040 − 0.177400i 0 5.80671 + 3.73174i 0
9.2 0 −1.66245 − 0.488140i 0 2.03510 − 2.34863i 0 −4.22795 − 2.71714i 0 0.00170444 + 0.00109538i 0
9.3 0 −1.43103 − 0.420187i 0 −2.46104 + 2.84020i 0 −1.25412 − 0.805977i 0 −0.652484 − 0.419326i 0
9.4 0 −1.40348 − 0.412098i 0 −0.153426 + 0.177063i 0 2.63565 + 1.69383i 0 −0.723838 − 0.465182i 0
9.5 0 −0.141353 − 0.0415048i 0 0.945327 − 1.09097i 0 1.49726 + 0.962231i 0 −2.50550 − 1.61019i 0
9.6 0 0.913280 + 0.268163i 0 −1.62985 + 1.88095i 0 −2.49107 − 1.60092i 0 −1.76159 − 1.13211i 0
9.7 0 2.11968 + 0.622395i 0 2.67746 − 3.08996i 0 2.16673 + 1.39247i 0 1.58192 + 1.01664i 0
9.8 0 2.59492 + 0.761938i 0 −1.04492 + 1.20590i 0 0.794694 + 0.510719i 0 3.62931 + 2.33242i 0
25.1 0 −1.73579 + 2.00321i 0 −2.32352 + 1.49324i 0 −0.509277 − 3.54210i 0 −0.572931 − 3.98482i 0
25.2 0 −1.46179 + 1.68700i 0 0.390046 − 0.250668i 0 0.516248 + 3.59058i 0 −0.282181 − 1.96261i 0
25.3 0 −0.0837208 + 0.0966190i 0 −1.91873 + 1.23309i 0 −0.0678491 − 0.471901i 0 0.424618 + 2.95329i 0
25.4 0 −0.0602838 + 0.0695712i 0 1.86622 − 1.19934i 0 0.0840190 + 0.584365i 0 0.425738 + 2.96108i 0
25.5 0 0.158529 − 0.182952i 0 1.60876 − 1.03389i 0 −0.394603 − 2.74452i 0 0.418604 + 2.91146i 0
25.6 0 0.838461 − 0.967635i 0 −2.02552 + 1.30172i 0 0.0308627 + 0.214655i 0 0.193643 + 1.34682i 0
25.7 0 1.62153 − 1.87135i 0 −1.39168 + 0.894379i 0 0.711954 + 4.95175i 0 −0.445633 − 3.09945i 0
25.8 0 1.86895 − 2.15688i 0 2.95318 − 1.89789i 0 −0.0301005 − 0.209354i 0 −0.732224 − 5.09273i 0
81.1 0 −2.32677 − 1.49533i 0 −0.508686 − 3.53799i 0 −0.703004 − 1.53936i 0 1.93163 + 4.22968i 0
81.2 0 −1.78476 − 1.14700i 0 0.350357 + 2.43679i 0 −0.479052 − 1.04898i 0 0.623526 + 1.36533i 0
81.3 0 −1.43219 − 0.920413i 0 −0.0560810 − 0.390052i 0 1.45919 + 3.19517i 0 −0.0422358 − 0.0924835i 0
81.4 0 0.684139 + 0.439669i 0 0.545934 + 3.79705i 0 0.683141 + 1.49587i 0 −0.971508 − 2.12731i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.e even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.2.q.a ✓ 80
67.e even 11 1 inner 536.2.q.a ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.2.q.a ✓ 80 1.a even 1 1 trivial
536.2.q.a ✓ 80 67.e even 11 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{80} + T_{3}^{79} + 16 T_{3}^{78} + 19 T_{3}^{77} + 179 T_{3}^{76} + 355 T_{3}^{75} + \cdots + 27426169 \) acting on \(S_{2}^{\mathrm{new}}(536, [\chi])\). Copy content Toggle raw display