# Properties

 Label 195.2.n.a Level $195$ Weight $2$ Character orbit 195.n Analytic conductor $1.557$ Analytic rank $0$ Dimension $48$ 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(44,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 2, 3]))

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

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

chi := DirichletCharacter("195.44");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.n (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: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 12 q^{6} - 8 q^{9}+O(q^{10})$$ 48 * q - 12 * q^6 - 8 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$48 q - 12 q^{6} - 8 q^{9} - 16 q^{15} - 24 q^{16} - 24 q^{19} + 4 q^{21} - 8 q^{24} - 8 q^{31} + 32 q^{34} + 16 q^{39} + 8 q^{40} + 20 q^{45} - 40 q^{46} - 24 q^{54} + 8 q^{55} + 76 q^{60} - 120 q^{61} + 32 q^{66} - 48 q^{70} + 104 q^{76} + 8 q^{79} - 48 q^{81} + 108 q^{84} + 56 q^{85} + 32 q^{91} - 192 q^{94} - 12 q^{96} - 60 q^{99}+O(q^{100})$$ 48 * q - 12 * q^6 - 8 * q^9 - 16 * q^15 - 24 * q^16 - 24 * q^19 + 4 * q^21 - 8 * q^24 - 8 * q^31 + 32 * q^34 + 16 * q^39 + 8 * q^40 + 20 * q^45 - 40 * q^46 - 24 * q^54 + 8 * q^55 + 76 * q^60 - 120 * q^61 + 32 * q^66 - 48 * q^70 + 104 * q^76 + 8 * q^79 - 48 * q^81 + 108 * q^84 + 56 * q^85 + 32 * q^91 - 192 * q^94 - 12 * q^96 - 60 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
44.1 −1.72658 + 1.72658i 1.24653 1.20257i 3.96217i −2.19956 0.402404i −0.0759004 + 4.22856i −0.495582 + 0.495582i 3.38784 + 3.38784i 0.107662 2.99807i 4.49251 3.10294i
44.2 −1.72658 + 1.72658i 1.24653 + 1.20257i 3.96217i 0.402404 + 2.19956i −4.22856 + 0.0759004i 0.495582 0.495582i 3.38784 + 3.38784i 0.107662 + 2.99807i −4.49251 3.10294i
44.3 −1.69452 + 1.69452i −1.14037 1.30367i 3.74279i 0.990671 2.00464i 4.14147 + 0.276713i −2.57529 + 2.57529i 2.95318 + 2.95318i −0.399109 + 2.97333i 1.71818 + 5.07561i
44.4 −1.69452 + 1.69452i −1.14037 + 1.30367i 3.74279i 2.00464 0.990671i −0.276713 4.14147i 2.57529 2.57529i 2.95318 + 2.95318i −0.399109 2.97333i −1.71818 + 5.07561i
44.5 −1.26780 + 1.26780i 0.428519 1.67820i 1.21464i 1.71852 + 1.43063i 1.58435 + 2.67091i 1.40105 1.40105i −0.995685 0.995685i −2.63274 1.43829i −3.99249 + 0.364987i
44.6 −1.26780 + 1.26780i 0.428519 + 1.67820i 1.21464i −1.43063 1.71852i −2.67091 1.58435i −1.40105 + 1.40105i −0.995685 0.995685i −2.63274 + 1.43829i 3.99249 + 0.364987i
44.7 −0.800574 + 0.800574i 1.71512 0.241578i 0.718163i 0.0148595 2.23602i −1.17968 + 1.56648i 3.41858 3.41858i −2.17609 2.17609i 2.88328 0.828672i 1.77820 + 1.80199i
44.8 −0.800574 + 0.800574i 1.71512 + 0.241578i 0.718163i 2.23602 0.0148595i −1.56648 + 1.17968i −3.41858 + 3.41858i −2.17609 2.17609i 2.88328 + 0.828672i −1.77820 + 1.80199i
44.9 −0.590281 + 0.590281i −0.502466 1.65757i 1.30314i −2.08713 + 0.802426i 1.27503 + 0.681834i −0.315643 + 0.315643i −1.94978 1.94978i −2.49506 + 1.66574i 0.758337 1.70565i
44.10 −0.590281 + 0.590281i −0.502466 + 1.65757i 1.30314i −0.802426 + 2.08713i −0.681834 1.27503i 0.315643 0.315643i −1.94978 1.94978i −2.49506 1.66574i −0.758337 1.70565i
44.11 −0.225510 + 0.225510i −1.50598 0.855580i 1.89829i 0.293103 2.21677i 0.532556 0.146672i 0.934339 0.934339i −0.879103 0.879103i 1.53596 + 2.57698i 0.433807 + 0.566002i
44.12 −0.225510 + 0.225510i −1.50598 + 0.855580i 1.89829i 2.21677 0.293103i 0.146672 0.532556i −0.934339 + 0.934339i −0.879103 0.879103i 1.53596 2.57698i −0.433807 + 0.566002i
44.13 0.225510 0.225510i 1.50598 0.855580i 1.89829i −0.293103 + 2.21677i 0.146672 0.532556i 0.934339 0.934339i 0.879103 + 0.879103i 1.53596 2.57698i 0.433807 + 0.566002i
44.14 0.225510 0.225510i 1.50598 + 0.855580i 1.89829i −2.21677 + 0.293103i 0.532556 0.146672i −0.934339 + 0.934339i 0.879103 + 0.879103i 1.53596 + 2.57698i −0.433807 + 0.566002i
44.15 0.590281 0.590281i 0.502466 1.65757i 1.30314i 2.08713 0.802426i −0.681834 1.27503i −0.315643 + 0.315643i 1.94978 + 1.94978i −2.49506 1.66574i 0.758337 1.70565i
44.16 0.590281 0.590281i 0.502466 + 1.65757i 1.30314i 0.802426 2.08713i 1.27503 + 0.681834i 0.315643 0.315643i 1.94978 + 1.94978i −2.49506 + 1.66574i −0.758337 1.70565i
44.17 0.800574 0.800574i −1.71512 0.241578i 0.718163i −0.0148595 + 2.23602i −1.56648 + 1.17968i 3.41858 3.41858i 2.17609 + 2.17609i 2.88328 + 0.828672i 1.77820 + 1.80199i
44.18 0.800574 0.800574i −1.71512 + 0.241578i 0.718163i −2.23602 + 0.0148595i −1.17968 + 1.56648i −3.41858 + 3.41858i 2.17609 + 2.17609i 2.88328 0.828672i −1.77820 + 1.80199i
44.19 1.26780 1.26780i −0.428519 1.67820i 1.21464i −1.71852 1.43063i −2.67091 1.58435i 1.40105 1.40105i 0.995685 + 0.995685i −2.63274 + 1.43829i −3.99249 + 0.364987i
44.20 1.26780 1.26780i −0.428519 + 1.67820i 1.21464i 1.43063 + 1.71852i 1.58435 + 2.67091i −1.40105 + 1.40105i 0.995685 + 0.995685i −2.63274 1.43829i 3.99249 + 0.364987i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 44.24 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.d odd 4 1 inner
15.d odd 2 1 inner
39.f even 4 1 inner
65.g odd 4 1 inner
195.n even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.n.a 48
3.b odd 2 1 inner 195.2.n.a 48
5.b even 2 1 inner 195.2.n.a 48
5.c odd 4 2 975.2.o.q 48
13.d odd 4 1 inner 195.2.n.a 48
15.d odd 2 1 inner 195.2.n.a 48
15.e even 4 2 975.2.o.q 48
39.f even 4 1 inner 195.2.n.a 48
65.f even 4 1 975.2.o.q 48
65.g odd 4 1 inner 195.2.n.a 48
65.k even 4 1 975.2.o.q 48
195.j odd 4 1 975.2.o.q 48
195.n even 4 1 inner 195.2.n.a 48
195.u odd 4 1 975.2.o.q 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.n.a 48 1.a even 1 1 trivial
195.2.n.a 48 3.b odd 2 1 inner
195.2.n.a 48 5.b even 2 1 inner
195.2.n.a 48 13.d odd 4 1 inner
195.2.n.a 48 15.d odd 2 1 inner
195.2.n.a 48 39.f even 4 1 inner
195.2.n.a 48 65.g odd 4 1 inner
195.2.n.a 48 195.n even 4 1 inner
975.2.o.q 48 5.c odd 4 2
975.2.o.q 48 15.e even 4 2
975.2.o.q 48 65.f even 4 1
975.2.o.q 48 65.k even 4 1
975.2.o.q 48 195.j odd 4 1
975.2.o.q 48 195.u odd 4 1

## Hecke kernels

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