# Properties

 Label 195.2.bh.a Level $195$ Weight $2$ Character orbit 195.bh Analytic conductor $1.557$ Analytic rank $0$ Dimension $96$ CM no Inner twists $8$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,2,Mod(59,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([6, 6, 11]))

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

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

chi := DirichletCharacter("195.59");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.bh (of order $$12$$, degree $$4$$, 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: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$24$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$96 q - 24 q^{4} - 4 q^{9}+O(q^{10})$$ 96 * q - 24 * q^4 - 4 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$96 q - 24 q^{4} - 4 q^{9} - 36 q^{10} + 10 q^{15} - 16 q^{21} - 28 q^{24} + 18 q^{30} - 40 q^{31} - 32 q^{34} - 60 q^{36} - 4 q^{39} + 40 q^{40} + 4 q^{45} - 128 q^{46} + 60 q^{54} + 28 q^{55} + 20 q^{60} - 48 q^{61} - 8 q^{66} + 72 q^{69} + 168 q^{70} + 66 q^{75} - 80 q^{76} - 80 q^{79} + 48 q^{81} - 132 q^{84} - 20 q^{85} - 56 q^{91} + 72 q^{94} + 204 q^{96} - 48 q^{99}+O(q^{100})$$ 96 * q - 24 * q^4 - 4 * q^9 - 36 * q^10 + 10 * q^15 - 16 * q^21 - 28 * q^24 + 18 * q^30 - 40 * q^31 - 32 * q^34 - 60 * q^36 - 4 * q^39 + 40 * q^40 + 4 * q^45 - 128 * q^46 + 60 * q^54 + 28 * q^55 + 20 * q^60 - 48 * q^61 - 8 * q^66 + 72 * q^69 + 168 * q^70 + 66 * q^75 - 80 * q^76 - 80 * q^79 + 48 * q^81 - 132 * q^84 - 20 * q^85 - 56 * q^91 + 72 * q^94 + 204 * q^96 - 48 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
59.1 −2.37850 + 0.637316i 0.0570156 1.73111i 3.51903 2.03171i −1.24285 + 1.85885i 0.967655 + 4.15378i −0.462194 + 1.72493i −3.59279 + 3.59279i −2.99350 0.197401i 1.77144 5.21336i
59.2 −2.37850 + 0.637316i 1.47068 0.914933i 3.51903 2.03171i 1.85885 1.24285i −2.91490 + 3.11345i 0.462194 1.72493i −3.59279 + 3.59279i 1.32580 2.69115i −3.62918 + 4.14079i
59.3 −1.91529 + 0.513200i −1.72511 0.154893i 1.67290 0.965849i 1.31191 + 1.81077i 3.38357 0.588662i 0.434913 1.62312i 0.0957651 0.0957651i 2.95202 + 0.534415i −3.44197 2.79487i
59.4 −1.91529 + 0.513200i 0.996697 + 1.41654i 1.67290 0.965849i 1.81077 + 1.31191i −2.63593 2.20158i −0.434913 + 1.62312i 0.0957651 0.0957651i −1.01319 + 2.82373i −4.14142 1.58340i
59.5 −1.66849 + 0.447071i −1.28242 + 1.16422i 0.851938 0.491866i −2.11813 + 0.716618i 1.61922 2.51582i −0.830623 + 3.09993i 1.24129 1.24129i 0.289197 2.98603i 3.21369 2.14262i
59.6 −1.66849 + 0.447071i −0.367032 + 1.69272i 0.851938 0.491866i 0.716618 2.11813i −0.144374 2.98837i 0.830623 3.09993i 1.24129 1.24129i −2.73057 1.24256i −0.248718 + 3.85445i
59.7 −1.25247 + 0.335598i −0.377636 1.69038i −0.276000 + 0.159349i −0.638573 2.14295i 1.04027 + 1.99042i −0.589843 + 2.20132i 2.12594 2.12594i −2.71478 + 1.27670i 1.51896 + 2.46967i
59.8 −1.25247 + 0.335598i 1.65273 0.518148i −0.276000 + 0.159349i −2.14295 0.638573i −1.89610 + 1.20362i 0.589843 2.20132i 2.12594 2.12594i 2.46304 1.71272i 2.89828 + 0.0806242i
59.9 −0.578879 + 0.155110i −0.110304 1.72853i −1.42101 + 0.820420i 1.54681 + 1.61473i 0.331966 + 0.983504i 1.15087 4.29509i 1.54287 1.54287i −2.97567 + 0.381329i −1.14588 0.694809i
59.10 −0.578879 + 0.155110i 1.55211 0.768741i −1.42101 + 0.820420i 1.61473 + 1.54681i −0.779243 + 0.685756i −1.15087 + 4.29509i 1.54287 1.54287i 1.81807 2.38634i −1.17466 0.644957i
59.11 −0.481640 + 0.129055i −1.71178 0.264216i −1.51673 + 0.875684i 1.67675 1.47936i 0.858561 0.0936571i −0.339617 + 1.26747i 1.32268 1.32268i 2.86038 + 0.904558i −0.616670 + 0.928914i
59.12 −0.481640 + 0.129055i 1.08471 + 1.35034i −1.51673 + 0.875684i −1.47936 + 1.67675i −0.696707 0.510390i 0.339617 1.26747i 1.32268 1.32268i −0.646820 + 2.92944i 0.496128 0.998509i
59.13 0.481640 0.129055i −1.08471 1.35034i −1.51673 + 0.875684i −1.67675 + 1.47936i −0.696707 0.510390i −0.339617 + 1.26747i −1.32268 + 1.32268i −0.646820 + 2.92944i −0.616670 + 0.928914i
59.14 0.481640 0.129055i 1.71178 + 0.264216i −1.51673 + 0.875684i 1.47936 1.67675i 0.858561 0.0936571i 0.339617 1.26747i −1.32268 + 1.32268i 2.86038 + 0.904558i 0.496128 0.998509i
59.15 0.578879 0.155110i −1.55211 + 0.768741i −1.42101 + 0.820420i −1.54681 1.61473i −0.779243 + 0.685756i 1.15087 4.29509i −1.54287 + 1.54287i 1.81807 2.38634i −1.14588 0.694809i
59.16 0.578879 0.155110i 0.110304 + 1.72853i −1.42101 + 0.820420i −1.61473 1.54681i 0.331966 + 0.983504i −1.15087 + 4.29509i −1.54287 + 1.54287i −2.97567 + 0.381329i −1.17466 0.644957i
59.17 1.25247 0.335598i −1.65273 + 0.518148i −0.276000 + 0.159349i 0.638573 + 2.14295i −1.89610 + 1.20362i −0.589843 + 2.20132i −2.12594 + 2.12594i 2.46304 1.71272i 1.51896 + 2.46967i
59.18 1.25247 0.335598i 0.377636 + 1.69038i −0.276000 + 0.159349i 2.14295 + 0.638573i 1.04027 + 1.99042i 0.589843 2.20132i −2.12594 + 2.12594i −2.71478 + 1.27670i 2.89828 + 0.0806242i
59.19 1.66849 0.447071i 0.367032 1.69272i 0.851938 0.491866i 2.11813 0.716618i −0.144374 2.98837i −0.830623 + 3.09993i −1.24129 + 1.24129i −2.73057 1.24256i 3.21369 2.14262i
59.20 1.66849 0.447071i 1.28242 1.16422i 0.851938 0.491866i −0.716618 + 2.11813i 1.61922 2.51582i 0.830623 3.09993i −1.24129 + 1.24129i 0.289197 2.98603i −0.248718 + 3.85445i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
13.f odd 12 1 inner
15.d odd 2 1 inner
39.k even 12 1 inner
65.s odd 12 1 inner
195.bh even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.bh.a 96
3.b odd 2 1 inner 195.2.bh.a 96
5.b even 2 1 inner 195.2.bh.a 96
5.c odd 4 2 975.2.bo.h 96
13.f odd 12 1 inner 195.2.bh.a 96
15.d odd 2 1 inner 195.2.bh.a 96
15.e even 4 2 975.2.bo.h 96
39.k even 12 1 inner 195.2.bh.a 96
65.o even 12 1 975.2.bo.h 96
65.s odd 12 1 inner 195.2.bh.a 96
65.t even 12 1 975.2.bo.h 96
195.bc odd 12 1 975.2.bo.h 96
195.bh even 12 1 inner 195.2.bh.a 96
195.bn odd 12 1 975.2.bo.h 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.bh.a 96 1.a even 1 1 trivial
195.2.bh.a 96 3.b odd 2 1 inner
195.2.bh.a 96 5.b even 2 1 inner
195.2.bh.a 96 13.f odd 12 1 inner
195.2.bh.a 96 15.d odd 2 1 inner
195.2.bh.a 96 39.k even 12 1 inner
195.2.bh.a 96 65.s odd 12 1 inner
195.2.bh.a 96 195.bh even 12 1 inner
975.2.bo.h 96 5.c odd 4 2
975.2.bo.h 96 15.e even 4 2
975.2.bo.h 96 65.o even 12 1
975.2.bo.h 96 65.t even 12 1
975.2.bo.h 96 195.bc odd 12 1
975.2.bo.h 96 195.bn odd 12 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(195, [\chi])$$.