# Properties

 Label 195.2.v.a Level $195$ Weight $2$ Character orbit 195.v Analytic conductor $1.557$ 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] = [195,2,Mod(4,195)]

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, 1]))

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

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

chi := DirichletCharacter("195.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.v (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: $$32$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 20 q^{4} + 16 q^{9}+O(q^{10})$$ 32 * q - 20 * q^4 + 16 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$32 q - 20 q^{4} + 16 q^{9} + 2 q^{10} - 12 q^{11} + 8 q^{14} - 6 q^{15} - 28 q^{16} - 30 q^{20} - 4 q^{25} + 52 q^{26} - 24 q^{29} + 4 q^{30} - 2 q^{35} + 20 q^{36} + 4 q^{40} - 36 q^{41} + 12 q^{45} - 48 q^{46} - 28 q^{49} + 54 q^{50} - 40 q^{51} + 24 q^{55} - 56 q^{56} + 84 q^{59} - 32 q^{61} + 136 q^{64} + 20 q^{65} + 8 q^{66} - 24 q^{69} + 12 q^{71} + 40 q^{74} - 16 q^{75} + 48 q^{76} - 104 q^{79} + 66 q^{80} - 16 q^{81} - 48 q^{84} - 54 q^{85} - 48 q^{89} + 4 q^{90} + 12 q^{91} - 8 q^{94} + 12 q^{95}+O(q^{100})$$ 32 * q - 20 * q^4 + 16 * q^9 + 2 * q^10 - 12 * q^11 + 8 * q^14 - 6 * q^15 - 28 * q^16 - 30 * q^20 - 4 * q^25 + 52 * q^26 - 24 * q^29 + 4 * q^30 - 2 * q^35 + 20 * q^36 + 4 * q^40 - 36 * q^41 + 12 * q^45 - 48 * q^46 - 28 * q^49 + 54 * q^50 - 40 * q^51 + 24 * q^55 - 56 * q^56 + 84 * q^59 - 32 * q^61 + 136 * q^64 + 20 * q^65 + 8 * q^66 - 24 * q^69 + 12 * q^71 + 40 * q^74 - 16 * q^75 + 48 * q^76 - 104 * q^79 + 66 * q^80 - 16 * q^81 - 48 * q^84 - 54 * q^85 - 48 * q^89 + 4 * q^90 + 12 * q^91 - 8 * q^94 + 12 * q^95

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −1.36408 + 2.36265i 0.866025 + 0.500000i −2.72141 4.71362i −1.91068 1.16160i −2.36265 + 1.36408i −1.86649 3.23285i 9.39252 0.500000 + 0.866025i 5.35076 2.92975i
4.2 −1.30511 + 2.26052i −0.866025 0.500000i −2.40664 4.16842i 2.09035 0.794006i 2.26052 1.30511i 0.372536 + 0.645251i 7.34329 0.500000 + 0.866025i −0.933271 + 5.76154i
4.3 −1.00561 + 1.74177i −0.866025 0.500000i −1.02251 1.77105i −1.28334 + 1.83113i 1.74177 1.00561i 1.11175 + 1.92561i 0.0905636 0.500000 + 0.866025i −1.89887 4.07669i
4.4 −0.946062 + 1.63863i 0.866025 + 0.500000i −0.790065 1.36843i 1.73576 1.40965i −1.63863 + 0.946062i 1.37597 + 2.38324i −0.794445 0.500000 + 0.866025i 0.667754 + 4.17789i
4.5 −0.733363 + 1.27022i −0.866025 0.500000i −0.0756426 0.131017i 0.387771 2.20219i 1.27022 0.733363i −2.16559 3.75091i −2.71156 0.500000 + 0.866025i 2.51289 + 2.10756i
4.6 −0.611600 + 1.05932i 0.866025 + 0.500000i 0.251890 + 0.436286i −2.22227 + 0.247984i −1.05932 + 0.611600i 0.997778 + 1.72820i −3.06263 0.500000 + 0.866025i 1.09645 2.50577i
4.7 −0.296043 + 0.512762i 0.866025 + 0.500000i 0.824717 + 1.42845i 0.903471 + 2.04542i −0.512762 + 0.296043i −0.828705 1.43536i −2.16078 0.500000 + 0.866025i −1.31628 0.142267i
4.8 −0.173693 + 0.300844i −0.866025 0.500000i 0.939662 + 1.62754i −0.956446 2.02119i 0.300844 0.173693i 2.09191 + 3.62329i −1.34762 0.500000 + 0.866025i 0.774192 + 0.0633244i
4.9 0.173693 0.300844i 0.866025 + 0.500000i 0.939662 + 1.62754i 0.956446 2.02119i 0.300844 0.173693i −2.09191 3.62329i 1.34762 0.500000 + 0.866025i −0.441936 0.638807i
4.10 0.296043 0.512762i −0.866025 0.500000i 0.824717 + 1.42845i −0.903471 + 2.04542i −0.512762 + 0.296043i 0.828705 + 1.43536i 2.16078 0.500000 + 0.866025i 0.781346 + 1.06880i
4.11 0.611600 1.05932i −0.866025 0.500000i 0.251890 + 0.436286i 2.22227 + 0.247984i −1.05932 + 0.611600i −0.997778 1.72820i 3.06263 0.500000 + 0.866025i 1.62184 2.20244i
4.12 0.733363 1.27022i 0.866025 + 0.500000i −0.0756426 0.131017i −0.387771 2.20219i 1.27022 0.733363i 2.16559 + 3.75091i 2.71156 0.500000 + 0.866025i −3.08164 1.12245i
4.13 0.946062 1.63863i −0.866025 0.500000i −0.790065 1.36843i −1.73576 1.40965i −1.63863 + 0.946062i −1.37597 2.38324i 0.794445 0.500000 + 0.866025i −3.95204 + 1.51065i
4.14 1.00561 1.74177i 0.866025 + 0.500000i −1.02251 1.77105i 1.28334 + 1.83113i 1.74177 1.00561i −1.11175 1.92561i −0.0905636 0.500000 + 0.866025i 4.47996 0.393873i
4.15 1.30511 2.26052i 0.866025 + 0.500000i −2.40664 4.16842i −2.09035 0.794006i 2.26052 1.30511i −0.372536 0.645251i −7.34329 0.500000 + 0.866025i −4.52301 + 3.68901i
4.16 1.36408 2.36265i −0.866025 0.500000i −2.72141 4.71362i 1.91068 1.16160i −2.36265 + 1.36408i 1.86649 + 3.23285i −9.39252 0.500000 + 0.866025i −0.138139 6.09877i
49.1 −1.36408 2.36265i 0.866025 0.500000i −2.72141 + 4.71362i −1.91068 + 1.16160i −2.36265 1.36408i −1.86649 + 3.23285i 9.39252 0.500000 0.866025i 5.35076 + 2.92975i
49.2 −1.30511 2.26052i −0.866025 + 0.500000i −2.40664 + 4.16842i 2.09035 + 0.794006i 2.26052 + 1.30511i 0.372536 0.645251i 7.34329 0.500000 0.866025i −0.933271 5.76154i
49.3 −1.00561 1.74177i −0.866025 + 0.500000i −1.02251 + 1.77105i −1.28334 1.83113i 1.74177 + 1.00561i 1.11175 1.92561i 0.0905636 0.500000 0.866025i −1.89887 + 4.07669i
49.4 −0.946062 1.63863i 0.866025 0.500000i −0.790065 + 1.36843i 1.73576 + 1.40965i −1.63863 0.946062i 1.37597 2.38324i −0.794445 0.500000 0.866025i 0.667754 4.17789i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 49.16 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.e even 6 1 inner
65.l 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.v.a 32
3.b odd 2 1 585.2.bf.c 32
5.b even 2 1 inner 195.2.v.a 32
5.c odd 4 1 975.2.bc.m 16
5.c odd 4 1 975.2.bc.n 16
13.e even 6 1 inner 195.2.v.a 32
15.d odd 2 1 585.2.bf.c 32
39.h odd 6 1 585.2.bf.c 32
65.l even 6 1 inner 195.2.v.a 32
65.r odd 12 1 975.2.bc.m 16
65.r odd 12 1 975.2.bc.n 16
195.y odd 6 1 585.2.bf.c 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.v.a 32 1.a even 1 1 trivial
195.2.v.a 32 5.b even 2 1 inner
195.2.v.a 32 13.e even 6 1 inner
195.2.v.a 32 65.l even 6 1 inner
585.2.bf.c 32 3.b odd 2 1
585.2.bf.c 32 15.d odd 2 1
585.2.bf.c 32 39.h odd 6 1
585.2.bf.c 32 195.y odd 6 1
975.2.bc.m 16 5.c odd 4 1
975.2.bc.m 16 65.r odd 12 1
975.2.bc.n 16 5.c odd 4 1
975.2.bc.n 16 65.r odd 12 1

## Hecke kernels

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