# Properties

 Label 195.2.ba.a Level $195$ Weight $2$ Character orbit 195.ba Analytic conductor $1.557$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,2,Mod(94,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("195.94");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.ba (of order $$6$$, 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: $$24$$ Relative dimension: $$12$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{4} + 4 q^{5} + 12 q^{9}+O(q^{10})$$ 24 * q + 8 * q^4 + 4 * q^5 + 12 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 8 q^{4} + 4 q^{5} + 12 q^{9} - 4 q^{10} + 4 q^{11} + 24 q^{14} - 2 q^{15} + 16 q^{16} - 16 q^{19} - 16 q^{20} - 8 q^{21} - 16 q^{25} - 48 q^{26} - 12 q^{29} - 4 q^{30} + 8 q^{31} - 32 q^{34} + 10 q^{35} - 8 q^{36} + 8 q^{39} - 48 q^{40} - 40 q^{41} + 40 q^{44} + 2 q^{45} - 24 q^{46} - 16 q^{49} + 20 q^{50} - 24 q^{51} + 20 q^{55} - 24 q^{56} + 12 q^{59} + 48 q^{60} + 20 q^{61} + 48 q^{64} + 14 q^{65} - 56 q^{66} - 8 q^{69} - 56 q^{70} + 4 q^{71} - 12 q^{74} + 16 q^{75} + 8 q^{76} + 136 q^{79} - 4 q^{80} - 12 q^{81} - 16 q^{84} - 4 q^{85} + 48 q^{86} - 64 q^{89} - 8 q^{90} + 60 q^{91} - 48 q^{94} - 28 q^{95} + 40 q^{96} + 8 q^{99}+O(q^{100})$$ 24 * q + 8 * q^4 + 4 * q^5 + 12 * q^9 - 4 * q^10 + 4 * q^11 + 24 * q^14 - 2 * q^15 + 16 * q^16 - 16 * q^19 - 16 * q^20 - 8 * q^21 - 16 * q^25 - 48 * q^26 - 12 * q^29 - 4 * q^30 + 8 * q^31 - 32 * q^34 + 10 * q^35 - 8 * q^36 + 8 * q^39 - 48 * q^40 - 40 * q^41 + 40 * q^44 + 2 * q^45 - 24 * q^46 - 16 * q^49 + 20 * q^50 - 24 * q^51 + 20 * q^55 - 24 * q^56 + 12 * q^59 + 48 * q^60 + 20 * q^61 + 48 * q^64 + 14 * q^65 - 56 * q^66 - 8 * q^69 - 56 * q^70 + 4 * q^71 - 12 * q^74 + 16 * q^75 + 8 * q^76 + 136 * q^79 - 4 * q^80 - 12 * q^81 - 16 * q^84 - 4 * q^85 + 48 * q^86 - 64 * q^89 - 8 * q^90 + 60 * q^91 - 48 * q^94 - 28 * q^95 + 40 * q^96 + 8 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
94.1 −2.01317 + 1.16230i 0.866025 0.500000i 1.70191 2.94779i −0.174568 + 2.22924i −1.16230 + 2.01317i −0.473110 0.273150i 3.26331i 0.500000 0.866025i −2.23963 4.69075i
94.2 −1.85914 + 1.07337i −0.866025 + 0.500000i 1.30426 2.25904i −2.16557 + 0.557052i 1.07337 1.85914i −0.635729 0.367038i 1.30633i 0.500000 0.866025i 3.42817 3.36010i
94.3 −1.52669 + 0.881436i −0.866025 + 0.500000i 0.553860 0.959313i 1.52636 1.63408i 0.881436 1.52669i 1.92736 + 1.11276i 1.57298i 0.500000 0.866025i −0.889946 + 3.84013i
94.4 −1.16430 + 0.672211i 0.866025 0.500000i −0.0962645 + 0.166735i −0.868136 2.06066i −0.672211 + 1.16430i −3.39681 1.96115i 2.94768i 0.500000 0.866025i 2.39598 + 1.81567i
94.5 −0.729738 + 0.421315i 0.866025 0.500000i −0.644988 + 1.11715i 2.23540 0.0545741i −0.421315 + 0.729738i 0.347589 + 0.200681i 2.77223i 0.500000 0.866025i −1.60827 + 0.981632i
94.6 −0.521384 + 0.301021i −0.866025 + 0.500000i −0.818772 + 1.41816i 0.446511 + 2.19103i 0.301021 0.521384i −3.08191 1.77934i 2.18996i 0.500000 0.866025i −0.892352 1.00796i
94.7 0.521384 0.301021i 0.866025 0.500000i −0.818772 + 1.41816i 0.446511 2.19103i 0.301021 0.521384i 3.08191 + 1.77934i 2.18996i 0.500000 0.866025i −0.426744 1.27678i
94.8 0.729738 0.421315i −0.866025 + 0.500000i −0.644988 + 1.11715i 2.23540 + 0.0545741i −0.421315 + 0.729738i −0.347589 0.200681i 2.77223i 0.500000 0.866025i 1.65425 0.901983i
94.9 1.16430 0.672211i −0.866025 + 0.500000i −0.0962645 + 0.166735i −0.868136 + 2.06066i −0.672211 + 1.16430i 3.39681 + 1.96115i 2.94768i 0.500000 0.866025i 0.374428 + 2.98281i
94.10 1.52669 0.881436i 0.866025 0.500000i 0.553860 0.959313i 1.52636 + 1.63408i 0.881436 1.52669i −1.92736 1.11276i 1.57298i 0.500000 0.866025i 3.77062 + 1.14935i
94.11 1.85914 1.07337i 0.866025 0.500000i 1.30426 2.25904i −2.16557 0.557052i 1.07337 1.85914i 0.635729 + 0.367038i 1.30633i 0.500000 0.866025i −4.62401 + 1.28883i
94.12 2.01317 1.16230i −0.866025 + 0.500000i 1.70191 2.94779i −0.174568 2.22924i −1.16230 + 2.01317i 0.473110 + 0.273150i 3.26331i 0.500000 0.866025i −2.94250 4.28495i
139.1 −2.01317 1.16230i 0.866025 + 0.500000i 1.70191 + 2.94779i −0.174568 2.22924i −1.16230 2.01317i −0.473110 + 0.273150i 3.26331i 0.500000 + 0.866025i −2.23963 + 4.69075i
139.2 −1.85914 1.07337i −0.866025 0.500000i 1.30426 + 2.25904i −2.16557 0.557052i 1.07337 + 1.85914i −0.635729 + 0.367038i 1.30633i 0.500000 + 0.866025i 3.42817 + 3.36010i
139.3 −1.52669 0.881436i −0.866025 0.500000i 0.553860 + 0.959313i 1.52636 + 1.63408i 0.881436 + 1.52669i 1.92736 1.11276i 1.57298i 0.500000 + 0.866025i −0.889946 3.84013i
139.4 −1.16430 0.672211i 0.866025 + 0.500000i −0.0962645 0.166735i −0.868136 + 2.06066i −0.672211 1.16430i −3.39681 + 1.96115i 2.94768i 0.500000 + 0.866025i 2.39598 1.81567i
139.5 −0.729738 0.421315i 0.866025 + 0.500000i −0.644988 1.11715i 2.23540 + 0.0545741i −0.421315 0.729738i 0.347589 0.200681i 2.77223i 0.500000 + 0.866025i −1.60827 0.981632i
139.6 −0.521384 0.301021i −0.866025 0.500000i −0.818772 1.41816i 0.446511 2.19103i 0.301021 + 0.521384i −3.08191 + 1.77934i 2.18996i 0.500000 + 0.866025i −0.892352 + 1.00796i
139.7 0.521384 + 0.301021i 0.866025 + 0.500000i −0.818772 1.41816i 0.446511 + 2.19103i 0.301021 + 0.521384i 3.08191 1.77934i 2.18996i 0.500000 + 0.866025i −0.426744 + 1.27678i
139.8 0.729738 + 0.421315i −0.866025 0.500000i −0.644988 1.11715i 2.23540 0.0545741i −0.421315 0.729738i −0.347589 + 0.200681i 2.77223i 0.500000 + 0.866025i 1.65425 + 0.901983i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 94.12 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.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.ba.a 24
3.b odd 2 1 585.2.bs.b 24
5.b even 2 1 inner 195.2.ba.a 24
5.c odd 4 1 975.2.i.o 12
5.c odd 4 1 975.2.i.q 12
13.c even 3 1 inner 195.2.ba.a 24
15.d odd 2 1 585.2.bs.b 24
39.i odd 6 1 585.2.bs.b 24
65.n even 6 1 inner 195.2.ba.a 24
65.q odd 12 1 975.2.i.o 12
65.q odd 12 1 975.2.i.q 12
195.x odd 6 1 585.2.bs.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.ba.a 24 1.a even 1 1 trivial
195.2.ba.a 24 5.b even 2 1 inner
195.2.ba.a 24 13.c even 3 1 inner
195.2.ba.a 24 65.n even 6 1 inner
585.2.bs.b 24 3.b odd 2 1
585.2.bs.b 24 15.d odd 2 1
585.2.bs.b 24 39.i odd 6 1
585.2.bs.b 24 195.x odd 6 1
975.2.i.o 12 5.c odd 4 1
975.2.i.o 12 65.q odd 12 1
975.2.i.q 12 5.c odd 4 1
975.2.i.q 12 65.q odd 12 1

## Hecke kernels

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