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Properties

Label 539.2.q.h
Level $539$
Weight $2$
Character orbit 539.q
Analytic conductor $4.304$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [539,2,Mod(214,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([20, 12]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.214");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.q (of order \(15\), degree \(8\), not 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(5\) over \(\Q(\zeta_{15})\)
Twist minimal: no (minimal twist has level 77)
Sato-Tate group: $\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} + 4 q^{3} - 3 q^{4} - 4 q^{5} + 16 q^{6} - 38 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 3 q^{2} + 4 q^{3} - 3 q^{4} - 4 q^{5} + 16 q^{6} - 38 q^{8} + 7 q^{9} - 14 q^{10} - 9 q^{11} + 18 q^{12} - 6 q^{13} - 14 q^{15} - 5 q^{16} + 7 q^{17} + 24 q^{18} + 4 q^{19} + 30 q^{20} + 44 q^{22} - 14 q^{23} + 12 q^{24} + 21 q^{25} + 16 q^{27} + 16 q^{30} + 17 q^{31} - 30 q^{32} + 15 q^{33} - 48 q^{34} + 14 q^{36} + 24 q^{37} - 12 q^{38} + 28 q^{39} - 10 q^{40} - 60 q^{41} - 72 q^{43} + 18 q^{44} + 16 q^{45} + 8 q^{46} - 13 q^{47} - 128 q^{48} + 6 q^{50} - 7 q^{51} - 2 q^{52} + 33 q^{53} - 34 q^{54} + 6 q^{55} + 44 q^{57} - 17 q^{58} - 21 q^{59} - 48 q^{60} + 52 q^{62} + 94 q^{64} - 40 q^{65} + 49 q^{66} - 38 q^{67} + 23 q^{68} + 124 q^{69} + 20 q^{71} - 38 q^{72} - 11 q^{73} - 41 q^{74} + 11 q^{75} + 96 q^{76} - 100 q^{78} + 21 q^{79} - 12 q^{80} - 58 q^{81} - 6 q^{82} + 46 q^{83} - 78 q^{85} + 7 q^{86} - 48 q^{87} + 32 q^{88} + 10 q^{89} + 18 q^{90} - 110 q^{92} + 12 q^{93} - 37 q^{94} + 7 q^{95} + 53 q^{96} + 54 q^{97} - 36 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1 −1.23711 + 1.37395i −1.07981 + 0.480764i −0.148239 − 1.41040i −2.31107 + 0.491232i 0.675302 − 2.07837i 0 −0.870261 − 0.632281i −1.07253 + 1.19116i 2.18411 − 3.78299i
214.2 −0.450970 + 0.500853i −1.96516 + 0.874946i 0.161577 + 1.53730i 1.65218 − 0.351182i 0.448009 − 1.37883i 0 −1.93333 − 1.40464i 1.08893 − 1.20938i −0.569194 + 0.985873i
214.3 −0.0508685 + 0.0564952i 2.15146 − 0.957893i 0.208453 + 1.98330i 2.52173 − 0.536011i −0.0553254 + 0.170274i 0 −0.245656 − 0.178480i 1.70384 − 1.89231i −0.0979948 + 0.169732i
214.4 0.391628 − 0.434946i 1.26312 − 0.562375i 0.173251 + 1.64837i −3.87755 + 0.824199i 0.250068 − 0.769629i 0 1.73180 + 1.25823i −0.728198 + 0.808746i −1.16007 + 2.00931i
214.5 1.76087 − 1.95564i 2.02209 − 0.900292i −0.514820 − 4.89819i −1.15065 + 0.244578i 1.79998 − 5.53977i 0 −6.22764 − 4.52465i 1.27093 − 1.41151i −1.54783 + 2.68093i
312.1 −0.233841 + 2.22485i −0.893246 − 0.992050i −2.93896 − 0.624696i 0.791435 + 0.352369i 2.41604 − 1.75535i 0 0.694498 − 2.13745i 0.127310 − 1.21128i −0.969038 + 1.67842i
312.2 −0.133281 + 1.26809i 0.0932166 + 0.103528i 0.366016 + 0.0777992i −1.08262 − 0.482012i −0.143706 + 0.104408i 0 −0.935477 + 2.87910i 0.311557 − 2.96426i 0.755525 − 1.30861i
312.3 0.0457018 − 0.434823i 1.72872 + 1.91994i 1.76931 + 0.376079i 1.11068 + 0.494505i 0.913838 − 0.663942i 0 0.514604 − 1.58379i −0.384102 + 3.65449i 0.265782 − 0.460348i
312.4 0.234707 − 2.23309i 1.27743 + 1.41873i −2.97532 − 0.632424i −2.25193 − 1.00262i 3.46797 − 2.51963i 0 −0.722861 + 2.22474i −0.0673794 + 0.641072i −2.76749 + 4.79344i
312.5 0.255843 − 2.43419i −1.95053 − 2.16628i −3.90352 − 0.829718i 0.303226 + 0.135005i −5.77217 + 4.19373i 0 −1.50568 + 4.63401i −0.574630 + 5.46724i 0.406206 − 0.703570i
324.1 −2.57407 − 0.547134i −0.231369 + 2.20133i 4.49937 + 2.00325i 0.787136 + 0.874203i 1.79998 − 5.53977i 0 −6.22764 − 4.52465i −1.85787 − 0.394902i −1.54783 − 2.68093i
324.2 −0.572488 − 0.121686i −0.144526 + 1.37508i −1.51416 − 0.674145i 2.65255 + 2.94596i 0.250068 − 0.769629i 0 1.73180 + 1.25823i 1.06449 + 0.226265i −1.16007 − 2.00931i
324.3 0.0743606 + 0.0158058i −0.246172 + 2.34217i −1.82181 − 0.811123i −1.72507 − 1.91588i −0.0553254 + 0.170274i 0 −0.245656 − 0.178480i −2.49071 − 0.529416i −0.0979948 − 0.169732i
324.4 0.659236 + 0.140125i 0.224855 − 2.13935i −1.41213 − 0.628722i −1.13022 − 1.25524i 0.448009 − 1.37883i 0 −1.93333 − 1.40464i −1.59182 − 0.338352i −0.569194 − 0.985873i
324.5 1.80843 + 0.384393i 0.123553 − 1.17553i 1.29556 + 0.576822i 1.58095 + 1.75583i 0.675302 − 2.07837i 0 −0.870261 − 0.632281i 1.56784 + 0.333255i 2.18411 + 3.78299i
361.1 −2.57407 + 0.547134i −0.231369 − 2.20133i 4.49937 − 2.00325i 0.787136 − 0.874203i 1.79998 + 5.53977i 0 −6.22764 + 4.52465i −1.85787 + 0.394902i −1.54783 + 2.68093i
361.2 −0.572488 + 0.121686i −0.144526 − 1.37508i −1.51416 + 0.674145i 2.65255 − 2.94596i 0.250068 + 0.769629i 0 1.73180 − 1.25823i 1.06449 − 0.226265i −1.16007 + 2.00931i
361.3 0.0743606 − 0.0158058i −0.246172 − 2.34217i −1.82181 + 0.811123i −1.72507 + 1.91588i −0.0553254 − 0.170274i 0 −0.245656 + 0.178480i −2.49071 + 0.529416i −0.0979948 + 0.169732i
361.4 0.659236 − 0.140125i 0.224855 + 2.13935i −1.41213 + 0.628722i −1.13022 + 1.25524i 0.448009 + 1.37883i 0 −1.93333 + 1.40464i −1.59182 + 0.338352i −0.569194 + 0.985873i
361.5 1.80843 − 0.384393i 0.123553 + 1.17553i 1.29556 − 0.576822i 1.58095 − 1.75583i 0.675302 + 2.07837i 0 −0.870261 + 0.632281i 1.56784 − 0.333255i 2.18411 − 3.78299i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 214.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
11.c even 5 1 inner
77.m even 15 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.q.h 40
7.b odd 2 1 77.2.m.b ✓ 40
7.c even 3 1 539.2.f.g 20
7.c even 3 1 inner 539.2.q.h 40
7.d odd 6 1 77.2.m.b ✓ 40
7.d odd 6 1 539.2.f.h 20
11.c even 5 1 inner 539.2.q.h 40
21.c even 2 1 693.2.by.b 40
21.g even 6 1 693.2.by.b 40
77.b even 2 1 847.2.n.j 40
77.i even 6 1 847.2.n.j 40
77.j odd 10 1 77.2.m.b ✓ 40
77.j odd 10 1 847.2.e.i 20
77.j odd 10 2 847.2.n.i 40
77.l even 10 1 847.2.e.h 20
77.l even 10 2 847.2.n.h 40
77.l even 10 1 847.2.n.j 40
77.m even 15 1 539.2.f.g 20
77.m even 15 1 inner 539.2.q.h 40
77.m even 15 1 5929.2.a.bx 10
77.n even 30 1 847.2.e.h 20
77.n even 30 2 847.2.n.h 40
77.n even 30 1 847.2.n.j 40
77.n even 30 1 5929.2.a.by 10
77.o odd 30 1 5929.2.a.bz 10
77.p odd 30 1 77.2.m.b ✓ 40
77.p odd 30 1 539.2.f.h 20
77.p odd 30 1 847.2.e.i 20
77.p odd 30 2 847.2.n.i 40
77.p odd 30 1 5929.2.a.bw 10
231.u even 10 1 693.2.by.b 40
231.bc even 30 1 693.2.by.b 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.m.b ✓ 40 7.b odd 2 1
77.2.m.b ✓ 40 7.d odd 6 1
77.2.m.b ✓ 40 77.j odd 10 1
77.2.m.b ✓ 40 77.p odd 30 1
539.2.f.g 20 7.c even 3 1
539.2.f.g 20 77.m even 15 1
539.2.f.h 20 7.d odd 6 1
539.2.f.h 20 77.p odd 30 1
539.2.q.h 40 1.a even 1 1 trivial
539.2.q.h 40 7.c even 3 1 inner
539.2.q.h 40 11.c even 5 1 inner
539.2.q.h 40 77.m even 15 1 inner
693.2.by.b 40 21.c even 2 1
693.2.by.b 40 21.g even 6 1
693.2.by.b 40 231.u even 10 1
693.2.by.b 40 231.bc even 30 1
847.2.e.h 20 77.l even 10 1
847.2.e.h 20 77.n even 30 1
847.2.e.i 20 77.j odd 10 1
847.2.e.i 20 77.p odd 30 1
847.2.n.h 40 77.l even 10 2
847.2.n.h 40 77.n even 30 2
847.2.n.i 40 77.j odd 10 2
847.2.n.i 40 77.p odd 30 2
847.2.n.j 40 77.b even 2 1
847.2.n.j 40 77.i even 6 1
847.2.n.j 40 77.l even 10 1
847.2.n.j 40 77.n even 30 1
5929.2.a.bw 10 77.p odd 30 1
5929.2.a.bx 10 77.m even 15 1
5929.2.a.by 10 77.n even 30 1
5929.2.a.bz 10 77.o odd 30 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\):

\( T_{2}^{40} + 3 T_{2}^{39} + T_{2}^{38} + 12 T_{2}^{37} + 21 T_{2}^{36} - 50 T_{2}^{35} + 238 T_{2}^{34} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{40} - 4 T_{3}^{39} - 3 T_{3}^{38} + 24 T_{3}^{37} + 8 T_{3}^{36} - 52 T_{3}^{35} - 77 T_{3}^{34} + \cdots + 5764801 \) Copy content Toggle raw display