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Properties

Label 55.2.l.a
Level $55$
Weight $2$
Character orbit 55.l
Analytic conductor $0.439$
Analytic rank $0$
Dimension $32$
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] = [55,2,Mod(2,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("55.2");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 55.l (of order \(20\), degree \(8\), 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: \(0.439177211117\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(4\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{2} - 4 q^{3} - 2 q^{5} - 20 q^{6} - 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q - 10 q^{2} - 4 q^{3} - 2 q^{5} - 20 q^{6} - 10 q^{8} - 24 q^{11} + 12 q^{12} - 10 q^{13} + 14 q^{15} - 8 q^{16} - 10 q^{18} + 16 q^{20} + 10 q^{22} - 24 q^{23} + 16 q^{25} + 20 q^{26} - 16 q^{27} + 50 q^{28} + 30 q^{30} - 28 q^{31} + 66 q^{33} - 10 q^{35} + 24 q^{36} - 8 q^{37} + 10 q^{38} - 50 q^{40} + 40 q^{41} - 10 q^{42} - 28 q^{45} + 60 q^{46} - 28 q^{47} - 54 q^{48} - 50 q^{50} + 20 q^{51} - 50 q^{52} - 24 q^{53} - 64 q^{55} - 80 q^{56} + 30 q^{57} - 50 q^{58} + 34 q^{60} - 60 q^{61} + 100 q^{62} - 30 q^{63} - 100 q^{66} - 8 q^{67} - 30 q^{68} + 30 q^{70} + 24 q^{71} + 80 q^{72} + 50 q^{73} + 34 q^{75} + 70 q^{77} + 60 q^{78} + 98 q^{80} - 12 q^{81} - 10 q^{82} + 90 q^{83} + 30 q^{85} + 100 q^{86} + 170 q^{88} - 20 q^{90} + 20 q^{91} - 68 q^{92} - 8 q^{93} - 40 q^{95} - 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −1.15100 − 2.25897i 0.313634 − 1.98021i −2.60258 + 3.58214i 1.91314 + 1.15755i −4.83422 + 1.57073i −1.78576 + 0.282837i 6.07934 + 0.962874i −0.969677 − 0.315067i 0.412838 − 5.65406i
2.2 −0.474334 − 0.930933i −0.440550 + 2.78152i 0.533928 − 0.734888i 2.23541 + 0.0540419i 2.79838 − 0.909249i −0.543058 + 0.0860119i −3.00128 − 0.475357i −4.68963 − 1.52375i −1.01002 − 2.10665i
2.3 −0.261423 − 0.513072i 0.120415 − 0.760272i 0.980670 − 1.34978i −1.76986 + 1.36660i −0.421554 + 0.136971i 1.17850 − 0.186656i −2.08639 − 0.330452i 2.28966 + 0.743954i 1.16385 + 0.550801i
2.4 0.665529 + 1.30617i −0.130227 + 0.822224i −0.0875924 + 0.120561i −1.11862 − 1.93615i −1.16064 + 0.377114i −4.16343 + 0.659422i 2.68004 + 0.424477i 2.19408 + 0.712899i 1.78448 − 2.74968i
7.1 −2.40264 + 0.380541i −0.271495 − 0.532840i 3.72576 − 1.21057i 1.85044 + 1.25533i 0.855073 + 1.17691i 2.30033 + 1.17208i −4.15610 + 2.11764i 1.55315 − 2.13772i −4.92366 − 2.31195i
7.2 −1.22307 + 0.193716i 1.15501 + 2.26684i −0.443732 + 0.144177i −2.23029 − 0.160637i −1.85178 − 2.54876i 3.09022 + 1.57455i 2.72149 − 1.38667i −2.04114 + 2.80938i 2.75893 − 0.235572i
7.3 0.482327 − 0.0763931i 0.517260 + 1.01518i −1.67531 + 0.544341i 2.05331 − 0.885381i 0.327041 + 0.450133i −2.59854 − 1.32402i −1.63669 + 0.833936i 1.00032 − 1.37683i 0.922733 − 0.583903i
7.4 1.50135 − 0.237790i −0.361933 − 0.710333i 0.295389 − 0.0959778i −1.89470 + 1.18749i −0.712297 − 0.980393i 0.170602 + 0.0869260i −2.28811 + 1.16585i 1.38978 − 1.91287i −2.56223 + 2.23337i
8.1 −2.40264 − 0.380541i −0.271495 + 0.532840i 3.72576 + 1.21057i 1.85044 − 1.25533i 0.855073 − 1.17691i 2.30033 − 1.17208i −4.15610 − 2.11764i 1.55315 + 2.13772i −4.92366 + 2.31195i
8.2 −1.22307 − 0.193716i 1.15501 − 2.26684i −0.443732 − 0.144177i −2.23029 + 0.160637i −1.85178 + 2.54876i 3.09022 − 1.57455i 2.72149 + 1.38667i −2.04114 − 2.80938i 2.75893 + 0.235572i
8.3 0.482327 + 0.0763931i 0.517260 − 1.01518i −1.67531 − 0.544341i 2.05331 + 0.885381i 0.327041 − 0.450133i −2.59854 + 1.32402i −1.63669 − 0.833936i 1.00032 + 1.37683i 0.922733 + 0.583903i
8.4 1.50135 + 0.237790i −0.361933 + 0.710333i 0.295389 + 0.0959778i −1.89470 − 1.18749i −0.712297 + 0.980393i 0.170602 − 0.0869260i −2.28811 − 1.16585i 1.38978 + 1.91287i −2.56223 − 2.23337i
13.1 −2.25897 + 1.15100i 1.98021 + 0.313634i 2.60258 − 3.58214i −0.867371 + 2.06099i −4.83422 + 1.57073i 0.282837 + 1.78576i −0.962874 + 6.07934i 0.969677 + 0.315067i −0.412838 − 5.65406i
13.2 −0.930933 + 0.474334i −2.78152 − 0.440550i −0.533928 + 0.734888i −1.77672 + 1.35766i 2.79838 − 0.909249i 0.0860119 + 0.543058i 0.475357 − 3.00128i 4.68963 + 1.52375i 1.01002 − 2.10665i
13.3 −0.513072 + 0.261423i 0.760272 + 0.120415i −0.980670 + 1.34978i 2.23511 + 0.0653109i −0.421554 + 0.136971i −0.186656 − 1.17850i 0.330452 − 2.08639i −2.28966 − 0.743954i −1.16385 + 0.550801i
13.4 1.30617 − 0.665529i −0.822224 − 0.130227i 0.0875924 − 0.120561i −0.233059 − 2.22389i −1.16064 + 0.377114i 0.659422 + 4.16343i −0.424477 + 2.68004i −2.19408 − 0.712899i −1.78448 − 2.74968i
17.1 −2.25897 − 1.15100i 1.98021 − 0.313634i 2.60258 + 3.58214i −0.867371 − 2.06099i −4.83422 − 1.57073i 0.282837 − 1.78576i −0.962874 − 6.07934i 0.969677 − 0.315067i −0.412838 + 5.65406i
17.2 −0.930933 − 0.474334i −2.78152 + 0.440550i −0.533928 − 0.734888i −1.77672 − 1.35766i 2.79838 + 0.909249i 0.0860119 − 0.543058i 0.475357 + 3.00128i 4.68963 − 1.52375i 1.01002 + 2.10665i
17.3 −0.513072 − 0.261423i 0.760272 − 0.120415i −0.980670 − 1.34978i 2.23511 − 0.0653109i −0.421554 − 0.136971i −0.186656 + 1.17850i 0.330452 + 2.08639i −2.28966 + 0.743954i −1.16385 − 0.550801i
17.4 1.30617 + 0.665529i −0.822224 + 0.130227i 0.0875924 + 0.120561i −0.233059 + 2.22389i −1.16064 − 0.377114i 0.659422 − 4.16343i −0.424477 − 2.68004i −2.19408 + 0.712899i −1.78448 + 2.74968i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
11.d odd 10 1 inner
55.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 55.2.l.a ✓ 32
3.b odd 2 1 495.2.bj.a 32
4.b odd 2 1 880.2.cm.a 32
5.b even 2 1 275.2.bm.b 32
5.c odd 4 1 inner 55.2.l.a ✓ 32
5.c odd 4 1 275.2.bm.b 32
11.b odd 2 1 605.2.m.e 32
11.c even 5 1 605.2.e.b 32
11.c even 5 1 605.2.m.c 32
11.c even 5 1 605.2.m.d 32
11.c even 5 1 605.2.m.e 32
11.d odd 10 1 inner 55.2.l.a ✓ 32
11.d odd 10 1 605.2.e.b 32
11.d odd 10 1 605.2.m.c 32
11.d odd 10 1 605.2.m.d 32
15.e even 4 1 495.2.bj.a 32
20.e even 4 1 880.2.cm.a 32
33.f even 10 1 495.2.bj.a 32
44.g even 10 1 880.2.cm.a 32
55.e even 4 1 605.2.m.e 32
55.h odd 10 1 275.2.bm.b 32
55.k odd 20 1 605.2.e.b 32
55.k odd 20 1 605.2.m.c 32
55.k odd 20 1 605.2.m.d 32
55.k odd 20 1 605.2.m.e 32
55.l even 20 1 inner 55.2.l.a ✓ 32
55.l even 20 1 275.2.bm.b 32
55.l even 20 1 605.2.e.b 32
55.l even 20 1 605.2.m.c 32
55.l even 20 1 605.2.m.d 32
165.u odd 20 1 495.2.bj.a 32
220.w odd 20 1 880.2.cm.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.l.a ✓ 32 1.a even 1 1 trivial
55.2.l.a ✓ 32 5.c odd 4 1 inner
55.2.l.a ✓ 32 11.d odd 10 1 inner
55.2.l.a ✓ 32 55.l even 20 1 inner
275.2.bm.b 32 5.b even 2 1
275.2.bm.b 32 5.c odd 4 1
275.2.bm.b 32 55.h odd 10 1
275.2.bm.b 32 55.l even 20 1
495.2.bj.a 32 3.b odd 2 1
495.2.bj.a 32 15.e even 4 1
495.2.bj.a 32 33.f even 10 1
495.2.bj.a 32 165.u odd 20 1
605.2.e.b 32 11.c even 5 1
605.2.e.b 32 11.d odd 10 1
605.2.e.b 32 55.k odd 20 1
605.2.e.b 32 55.l even 20 1
605.2.m.c 32 11.c even 5 1
605.2.m.c 32 11.d odd 10 1
605.2.m.c 32 55.k odd 20 1
605.2.m.c 32 55.l even 20 1
605.2.m.d 32 11.c even 5 1
605.2.m.d 32 11.d odd 10 1
605.2.m.d 32 55.k odd 20 1
605.2.m.d 32 55.l even 20 1
605.2.m.e 32 11.b odd 2 1
605.2.m.e 32 11.c even 5 1
605.2.m.e 32 55.e even 4 1
605.2.m.e 32 55.k odd 20 1
880.2.cm.a 32 4.b odd 2 1
880.2.cm.a 32 20.e even 4 1
880.2.cm.a 32 44.g even 10 1
880.2.cm.a 32 220.w odd 20 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(55, [\chi])\).