# 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 $4$

# Related objects

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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 10 q^{2} - 4 q^{3} - 2 q^{5} - 20 q^{6} - 10 q^{8}+O(q^{10})$$ 32 * q - 10 * q^2 - 4 * q^3 - 2 * q^5 - 20 * q^6 - 10 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 10 q^{2} - 4 q^{3} - 2 q^{5} - 20 q^{6} - 10 q^{8} - 24 q^{11} + 12 q^{12} - 10 q^{13} + 14 q^{15} - 8 q^{16} - 10 q^{18} + 16 q^{20} + 10 q^{22} - 24 q^{23} + 16 q^{25} + 20 q^{26} - 16 q^{27} + 50 q^{28} + 30 q^{30} - 28 q^{31} + 66 q^{33} - 10 q^{35} + 24 q^{36} - 8 q^{37} + 10 q^{38} - 50 q^{40} + 40 q^{41} - 10 q^{42} - 28 q^{45} + 60 q^{46} - 28 q^{47} - 54 q^{48} - 50 q^{50} + 20 q^{51} - 50 q^{52} - 24 q^{53} - 64 q^{55} - 80 q^{56} + 30 q^{57} - 50 q^{58} + 34 q^{60} - 60 q^{61} + 100 q^{62} - 30 q^{63} - 100 q^{66} - 8 q^{67} - 30 q^{68} + 30 q^{70} + 24 q^{71} + 80 q^{72} + 50 q^{73} + 34 q^{75} + 70 q^{77} + 60 q^{78} + 98 q^{80} - 12 q^{81} - 10 q^{82} + 90 q^{83} + 30 q^{85} + 100 q^{86} + 170 q^{88} - 20 q^{90} + 20 q^{91} - 68 q^{92} - 8 q^{93} - 40 q^{95} - 8 q^{97}+O(q^{100})$$ 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(55, [\chi])$$.