# Properties

 Label 260.2.o.a Level $260$ Weight $2$ Character orbit 260.o Analytic conductor $2.076$ Analytic rank $0$ Dimension $72$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [260,2,Mod(27,260)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(260, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("260.27");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$260 = 2^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 260.o (of order $$4$$, degree $$2$$, 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: $$2.07611045255$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The algebraic $$q$$-expansion of this newform has not been computed, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 8 q^{6} - 12 q^{8}+O(q^{10})$$ 72 * q - 8 * q^6 - 12 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$72 q - 8 q^{6} - 12 q^{8} - 8 q^{10} - 8 q^{12} - 8 q^{16} + 28 q^{18} - 16 q^{21} - 8 q^{22} - 20 q^{28} - 32 q^{30} - 40 q^{32} + 16 q^{33} + 32 q^{36} - 12 q^{38} - 8 q^{40} - 40 q^{42} - 8 q^{46} + 60 q^{48} + 40 q^{50} + 8 q^{52} - 48 q^{53} + 8 q^{56} - 60 q^{58} + 20 q^{60} - 64 q^{61} + 60 q^{62} + 8 q^{66} - 16 q^{68} - 60 q^{70} + 40 q^{72} - 16 q^{73} - 72 q^{76} + 48 q^{77} - 20 q^{80} + 8 q^{81} - 12 q^{82} + 48 q^{85} + 48 q^{86} + 12 q^{88} + 44 q^{90} - 36 q^{92} + 16 q^{93} + 32 q^{96} - 80 q^{97} - 32 q^{98}+O(q^{100})$$ 72 * q - 8 * q^6 - 12 * q^8 - 8 * q^10 - 8 * q^12 - 8 * q^16 + 28 * q^18 - 16 * q^21 - 8 * q^22 - 20 * q^28 - 32 * q^30 - 40 * q^32 + 16 * q^33 + 32 * q^36 - 12 * q^38 - 8 * q^40 - 40 * q^42 - 8 * q^46 + 60 * q^48 + 40 * q^50 + 8 * q^52 - 48 * q^53 + 8 * q^56 - 60 * q^58 + 20 * q^60 - 64 * q^61 + 60 * q^62 + 8 * q^66 - 16 * q^68 - 60 * q^70 + 40 * q^72 - 16 * q^73 - 72 * q^76 + 48 * q^77 - 20 * q^80 + 8 * q^81 - 12 * q^82 + 48 * q^85 + 48 * q^86 + 12 * q^88 + 44 * q^90 - 36 * q^92 + 16 * q^93 + 32 * q^96 - 80 * q^97 - 32 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
27.1 −1.41404 + 0.0221303i 0.798638 + 0.798638i 1.99902 0.0625863i −1.57330 1.58893i −1.14698 1.11163i 1.30625 1.30625i −2.82531 + 0.132739i 1.72435i 2.25988 + 2.21200i
27.2 −1.41202 0.0788259i −1.63309 1.63309i 1.98757 + 0.222607i −0.963620 + 2.01778i 2.17721 + 2.43467i 3.04756 3.04756i −2.78894 0.470996i 2.33394i 1.51970 2.77318i
27.3 −1.39950 + 0.203483i 0.0775283 + 0.0775283i 1.91719 0.569548i 0.871245 + 2.05935i −0.124276 0.0927250i −2.89635 + 2.89635i −2.56721 + 1.18720i 2.98798i −1.63835 2.70478i
27.4 −1.34983 0.421862i −1.90910 1.90910i 1.64407 + 1.13888i −0.987103 2.00640i 1.77158 + 3.38234i −3.15102 + 3.15102i −1.73875 2.23086i 4.28935i 0.485997 + 3.12471i
27.5 −1.32767 0.487116i 0.396814 + 0.396814i 1.52544 + 1.29346i 2.02359 0.951360i −0.333545 0.720134i 0.477103 0.477103i −1.39521 2.46036i 2.68508i −3.15009 + 0.277373i
27.6 −1.26622 + 0.629837i 2.21225 + 2.21225i 1.20661 1.59502i 1.36305 1.77259i −4.19454 1.40783i −2.47925 + 2.47925i −0.523229 + 2.77961i 6.78807i −0.609477 + 3.10299i
27.7 −1.23527 + 0.688553i −1.59413 1.59413i 1.05179 1.70110i 1.94514 1.10291i 3.06682 + 0.871538i 0.0942377 0.0942377i −0.127946 + 2.82553i 2.08249i −1.64337 + 2.70173i
27.8 −1.18329 0.774482i 2.05925 + 2.05925i 0.800354 + 1.83288i 1.34695 + 1.78486i −0.841839 4.03155i 0.845670 0.845670i 0.472478 2.78869i 5.48103i −0.211491 3.15520i
27.9 −1.08734 + 0.904262i 1.78782 + 1.78782i 0.364619 1.96648i −1.34049 + 1.78972i −3.56063 0.327311i 2.34459 2.34459i 1.38175 + 2.46795i 3.39260i −0.160802 3.15819i
27.10 −0.942648 + 1.05424i 0.158208 + 0.158208i −0.222831 1.98755i −2.23604 + 0.0111104i −0.315922 + 0.0176543i −2.20671 + 2.20671i 2.30540 + 1.63864i 2.94994i 2.09608 2.36779i
27.11 −0.774482 1.18329i −2.05925 2.05925i −0.800354 + 1.83288i 1.34695 + 1.78486i −0.841839 + 4.03155i −0.845670 + 0.845670i 2.78869 0.472478i 5.48103i 1.06882 2.97618i
27.12 −0.769019 + 1.18685i −0.808783 0.808783i −0.817219 1.82542i 0.885949 + 2.05307i 1.58187 0.337933i 0.771151 0.771151i 2.79495 + 0.433867i 1.69174i −3.11799 0.527363i
27.13 −0.516756 + 1.31642i −2.23963 2.23963i −1.46593 1.36054i −2.20168 0.390628i 4.10564 1.79095i 0.918543 0.918543i 2.54856 1.22671i 7.03189i 1.65196 2.69648i
27.14 −0.505980 + 1.32060i 0.658144 + 0.658144i −1.48797 1.33639i 0.820865 2.07995i −1.20215 + 0.536137i 1.89124 1.89124i 2.51772 1.28882i 2.13369i 2.33144 + 2.13645i
27.15 −0.487116 1.32767i −0.396814 0.396814i −1.52544 + 1.29346i 2.02359 0.951360i −0.333545 + 0.720134i −0.477103 + 0.477103i 2.46036 + 1.39521i 2.68508i −2.24882 2.22324i
27.16 −0.421862 1.34983i 1.90910 + 1.90910i −1.64407 + 1.13888i −0.987103 2.00640i 1.77158 3.38234i 3.15102 3.15102i 2.23086 + 1.73875i 4.28935i −2.29187 + 2.17884i
27.17 −0.0788259 1.41202i 1.63309 + 1.63309i −1.98757 + 0.222607i −0.963620 + 2.01778i 2.17721 2.43467i −3.04756 + 3.04756i 0.470996 + 2.78894i 2.33394i 2.92509 + 1.20159i
27.18 −0.0447185 + 1.41351i 1.60820 + 1.60820i −1.99600 0.126420i 1.86263 + 1.23718i −2.34511 + 2.20128i −0.275770 + 0.275770i 0.267954 2.81571i 2.17259i −1.83206 + 2.57751i
27.19 0.0221303 1.41404i −0.798638 0.798638i −1.99902 0.0625863i −1.57330 1.58893i −1.14698 + 1.11163i −1.30625 + 1.30625i −0.132739 + 2.82531i 1.72435i −2.28164 + 2.18955i
27.20 0.203483 1.39950i −0.0775283 0.0775283i −1.91719 0.569548i 0.871245 + 2.05935i −0.124276 + 0.0927250i 2.89635 2.89635i −1.18720 + 2.56721i 2.98798i 3.05934 0.800263i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 27.36 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 260.2.o.a 72
4.b odd 2 1 inner 260.2.o.a 72
5.c odd 4 1 inner 260.2.o.a 72
20.e even 4 1 inner 260.2.o.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.o.a 72 1.a even 1 1 trivial
260.2.o.a 72 4.b odd 2 1 inner
260.2.o.a 72 5.c odd 4 1 inner
260.2.o.a 72 20.e even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(260, [\chi])$$.