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Properties

Label 536.6.a.c
Level $536$
Weight $6$
Character orbit 536.a
Self dual yes
Analytic conductor $85.966$
Analytic rank $0$
Dimension $22$
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: \(22\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 36 q^{3} + 25 q^{5} + 171 q^{7} + 1740 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q + 36 q^{3} + 25 q^{5} + 171 q^{7} + 1740 q^{9} + 581 q^{11} + 874 q^{13} + 883 q^{15} - 2407 q^{17} + 3187 q^{19} + 2616 q^{21} + 5587 q^{23} + 14095 q^{25} + 15390 q^{27} + 4190 q^{29} + 15482 q^{31} - 11223 q^{33} - 5741 q^{35} - 7673 q^{37} - 9694 q^{39} - 36347 q^{41} + 16710 q^{43} - 5324 q^{45} + 20064 q^{47} + 62665 q^{49} + 31094 q^{51} + 26795 q^{53} + 94740 q^{55} + 46573 q^{57} + 54226 q^{59} + 82744 q^{61} + 146336 q^{63} + 37575 q^{65} + 98758 q^{67} + 192484 q^{69} + 124063 q^{71} + 94583 q^{73} + 114498 q^{75} + 215979 q^{77} + 5851 q^{79} + 397938 q^{81} + 212613 q^{83} + 302469 q^{85} + 330727 q^{87} + 181565 q^{89} + 202361 q^{91} + 246191 q^{93} + 224203 q^{95} + 172896 q^{97} + 532665 q^{99}+O(q^{100}) \) 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 −30.1333 0 −42.5083 0 −15.3370 0 665.018 0
1.2 0 −23.0376 0 −66.2538 0 130.585 0 287.732 0
1.3 0 −21.3539 0 74.0629 0 111.004 0 212.989 0
1.4 0 −21.0663 0 −22.6189 0 −84.1376 0 200.789 0
1.5 0 −16.7496 0 14.2770 0 29.7315 0 37.5475 0
1.6 0 −14.6716 0 48.7843 0 2.24050 0 −27.7454 0
1.7 0 −8.72108 0 42.3388 0 −233.461 0 −166.943 0
1.8 0 −7.60297 0 109.159 0 −94.4043 0 −185.195 0
1.9 0 −3.79642 0 −110.785 0 125.201 0 −228.587 0
1.10 0 −2.85311 0 −30.8326 0 185.358 0 −234.860 0
1.11 0 −1.71247 0 −8.60878 0 −91.8092 0 −240.067 0
1.12 0 5.07862 0 77.2848 0 211.992 0 −217.208 0
1.13 0 5.54820 0 −22.3055 0 −103.378 0 −212.217 0
1.14 0 6.99898 0 −40.9584 0 −249.237 0 −194.014 0
1.15 0 11.6927 0 −76.2866 0 −98.0054 0 −106.281 0
1.16 0 13.6548 0 −52.4985 0 205.489 0 −56.5475 0
1.17 0 13.7187 0 75.3890 0 8.65849 0 −54.7967 0
1.18 0 21.1742 0 48.7455 0 −6.27510 0 205.346 0
1.19 0 25.2408 0 −87.5782 0 45.4839 0 394.097 0
1.20 0 26.9403 0 92.0849 0 129.071 0 482.779 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.22
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.c ✓ 22
4.b odd 2 1 1072.6.a.k 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
536.6.a.c ✓ 22 1.a even 1 1 trivial
1072.6.a.k 22 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} - 36 T_{3}^{21} - 2895 T_{3}^{20} + 108810 T_{3}^{19} + 3379038 T_{3}^{18} + \cdots - 48\!\cdots\!40 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(536))\). Copy content Toggle raw display