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Properties

Label 536.6.a.d
Level $536$
Weight $6$
Character orbit 536.a
Self dual yes
Analytic conductor $85.966$
Analytic rank $0$
Dimension $23$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [536,6,Mod(1,536)]
 
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, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("536.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 536 = 2^{3} \cdot 67 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 536.a (trivial)

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: yes
Analytic conductor: \(85.9657274198\)
Analytic rank: \(0\)
Dimension: \(23\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 23 q + 99 q^{5} - 13 q^{7} + 2553 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q + 99 q^{5} - 13 q^{7} + 2553 q^{9} - 87 q^{11} + 396 q^{13} + 963 q^{15} + 5128 q^{17} + 716 q^{19} + 1432 q^{21} - 3126 q^{23} + 21054 q^{25} - 1200 q^{27} + 1953 q^{29} + 8722 q^{31} + 23149 q^{33} - 20185 q^{35} - 19926 q^{37} - 12662 q^{39} + 17997 q^{41} - 16014 q^{43} + 6392 q^{45} - 30051 q^{47} + 123858 q^{49} + 19524 q^{51} - 11715 q^{53} + 49060 q^{55} + 17669 q^{57} + 18885 q^{59} + 111670 q^{61} + 16588 q^{63} + 181391 q^{65} - 103247 q^{67} + 29900 q^{69} + 153663 q^{71} + 107752 q^{73} + 560424 q^{75} + 154567 q^{77} + 212439 q^{79} + 481395 q^{81} + 36309 q^{83} - 37771 q^{85} + 136881 q^{87} + 373300 q^{89} + 456385 q^{91} + 464603 q^{93} + 655753 q^{95} + 318366 q^{97} + 273465 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 0 −30.5398 0 17.5116 0 47.6432 0 689.681 0
1.2 0 −26.8448 0 −53.8369 0 17.2389 0 477.645 0
1.3 0 −26.7641 0 1.52489 0 −116.665 0 473.319 0
1.4 0 −23.4709 0 61.8498 0 −224.935 0 307.881 0
1.5 0 −22.2635 0 −85.9257 0 206.373 0 252.661 0
1.6 0 −16.7043 0 107.419 0 −26.0569 0 36.0329 0
1.7 0 −11.4048 0 18.8719 0 −8.81252 0 −112.931 0
1.8 0 −10.1173 0 −40.4758 0 −106.894 0 −140.641 0
1.9 0 −9.33192 0 −3.56352 0 90.7794 0 −155.915 0
1.10 0 −6.14307 0 −0.825089 0 194.041 0 −205.263 0
1.11 0 −3.58803 0 82.9495 0 168.220 0 −230.126 0
1.12 0 −0.379561 0 −54.9681 0 48.5169 0 −242.856 0
1.13 0 1.48195 0 −45.5233 0 −135.443 0 −240.804 0
1.14 0 5.26919 0 52.3241 0 −219.512 0 −215.236 0
1.15 0 10.7135 0 −59.6329 0 71.2555 0 −128.221 0
1.16 0 11.5970 0 −4.42847 0 −147.896 0 −108.510 0
1.17 0 14.2772 0 85.4264 0 163.693 0 −39.1602 0
1.18 0 16.7950 0 24.0801 0 74.3545 0 39.0711 0
1.19 0 18.8965 0 −104.849 0 −248.908 0 114.076 0
1.20 0 25.1124 0 70.6003 0 162.881 0 387.631 0
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(67\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 536.6.a.d ✓ 23
4.b odd 2 1 1072.6.a.l 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.6.a.d ✓ 23 1.a even 1 1 trivial
1072.6.a.l 23 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{23} - 4071 T_{3}^{21} + 400 T_{3}^{20} + 7021824 T_{3}^{19} - 1035069 T_{3}^{18} + \cdots - 40\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(536))\). Copy content Toggle raw display