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Properties

Label 536.3.k.a
Level $536$
Weight $3$
Character orbit 536.k
Analytic conductor $14.605$
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] = [536,3,Mod(97,536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(536, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.97");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 536.k (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: \(14.6049421697\)
Analytic rank: \(0\)
Dimension: \(68\)
Relative dimension: \(34\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 176 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 68 q - 176 q^{9} - 12 q^{13} + 48 q^{15} - 10 q^{17} + 38 q^{19} - 44 q^{21} - 340 q^{25} - 52 q^{29} - 96 q^{31} - 24 q^{33} + 8 q^{35} - 40 q^{37} - 48 q^{39} - 72 q^{41} - 28 q^{47} + 230 q^{49} + 114 q^{51} - 32 q^{55} + 282 q^{57} - 164 q^{59} - 24 q^{61} + 180 q^{63} + 24 q^{65} - 272 q^{67} - 240 q^{69} - 24 q^{71} - 26 q^{73} - 140 q^{77} - 288 q^{79} + 468 q^{81} + 72 q^{83} - 192 q^{85} + 72 q^{87} - 316 q^{89} + 528 q^{91} - 144 q^{93} - 336 q^{95} - 354 q^{97} - 348 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
97.1 0 − 5.38937i 0 5.61927i 0 −10.7293 − 6.19454i 0 −20.0453 0
97.2 0 − 5.29933i 0 4.10344i 0 7.12838 + 4.11557i 0 −19.0829 0
97.3 0 − 5.14125i 0 − 5.99207i 0 −5.37388 − 3.10261i 0 −17.4324 0
97.4 0 − 4.84105i 0 − 5.40848i 0 0.276120 + 0.159418i 0 −14.4358 0
97.5 0 − 4.59288i 0 0.799670i 0 7.21899 + 4.16788i 0 −12.0946 0
97.6 0 − 3.64953i 0 − 0.110021i 0 5.11808 + 2.95493i 0 −4.31905 0
97.7 0 − 3.21872i 0 9.07838i 0 −5.94552 − 3.43265i 0 −1.36013 0
97.8 0 − 2.92885i 0 4.21205i 0 2.65611 + 1.53351i 0 0.421839 0
97.9 0 − 2.61194i 0 2.36207i 0 −1.81467 − 1.04770i 0 2.17775 0
97.10 0 − 2.40643i 0 − 2.53110i 0 −9.58856 − 5.53596i 0 3.20910 0
97.11 0 − 2.15984i 0 − 0.331648i 0 −5.86085 − 3.38376i 0 4.33510 0
97.12 0 − 1.89953i 0 − 9.20938i 0 −5.39673 − 3.11580i 0 5.39177 0
97.13 0 − 1.80379i 0 − 5.13639i 0 7.64398 + 4.41325i 0 5.74634 0
97.14 0 − 1.73052i 0 − 6.66040i 0 5.19379 + 2.99864i 0 6.00530 0
97.15 0 − 1.07175i 0 8.07297i 0 3.26444 + 1.88473i 0 7.85136 0
97.16 0 − 0.0242371i 0 4.58291i 0 −6.25232 − 3.60978i 0 8.99941 0
97.17 0 0.321546i 0 8.82247i 0 10.9552 + 6.32500i 0 8.89661 0
97.18 0 0.675451i 0 0.421763i 0 6.13473 + 3.54189i 0 8.54377 0
97.19 0 0.900496i 0 2.87930i 0 −7.48911 − 4.32384i 0 8.18911 0
97.20 0 0.944539i 0 − 4.49047i 0 3.06843 + 1.77156i 0 8.10785 0
See all 68 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 97.34
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.3.k.a ✓ 68
67.d odd 6 1 inner 536.3.k.a ✓ 68
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.3.k.a ✓ 68 1.a even 1 1 trivial
536.3.k.a ✓ 68 67.d odd 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(536, [\chi])\).