# Properties

 Label 195.3.e.a Level $195$ Weight $3$ Character orbit 195.e Self dual yes Analytic conductor $5.313$ Analytic rank $0$ Dimension $1$ CM discriminant -195 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,3,Mod(194,195)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(195, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("195.194");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 195.e (of order $$2$$, degree $$1$$, 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: yes Analytic conductor: $$5.31336515503$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + 4 q^{4} - 5 q^{5} + q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + 4 * q^4 - 5 * q^5 + q^7 + 9 * q^9 $$q - 3 q^{3} + 4 q^{4} - 5 q^{5} + q^{7} + 9 q^{9} + 17 q^{11} - 12 q^{12} - 13 q^{13} + 15 q^{15} + 16 q^{16} + 31 q^{17} - 20 q^{20} - 3 q^{21} + 19 q^{23} + 25 q^{25} - 27 q^{27} + 4 q^{28} - 51 q^{33} - 5 q^{35} + 36 q^{36} + 61 q^{37} + 39 q^{39} - 43 q^{41} + 68 q^{44} - 45 q^{45} - 48 q^{48} - 48 q^{49} - 93 q^{51} - 52 q^{52} - 41 q^{53} - 85 q^{55} + 38 q^{59} + 60 q^{60} - 73 q^{61} + 9 q^{63} + 64 q^{64} + 65 q^{65} - 74 q^{67} + 124 q^{68} - 57 q^{69} - 103 q^{71} + 94 q^{73} - 75 q^{75} + 17 q^{77} - 37 q^{79} - 80 q^{80} + 81 q^{81} - 12 q^{84} - 155 q^{85} + 173 q^{89} - 13 q^{91} + 76 q^{92} + 181 q^{97} + 153 q^{99}+O(q^{100})$$ q - 3 * q^3 + 4 * q^4 - 5 * q^5 + q^7 + 9 * q^9 + 17 * q^11 - 12 * q^12 - 13 * q^13 + 15 * q^15 + 16 * q^16 + 31 * q^17 - 20 * q^20 - 3 * q^21 + 19 * q^23 + 25 * q^25 - 27 * q^27 + 4 * q^28 - 51 * q^33 - 5 * q^35 + 36 * q^36 + 61 * q^37 + 39 * q^39 - 43 * q^41 + 68 * q^44 - 45 * q^45 - 48 * q^48 - 48 * q^49 - 93 * q^51 - 52 * q^52 - 41 * q^53 - 85 * q^55 + 38 * q^59 + 60 * q^60 - 73 * q^61 + 9 * q^63 + 64 * q^64 + 65 * q^65 - 74 * q^67 + 124 * q^68 - 57 * q^69 - 103 * q^71 + 94 * q^73 - 75 * q^75 + 17 * q^77 - 37 * q^79 - 80 * q^80 + 81 * q^81 - 12 * q^84 - 155 * q^85 + 173 * q^89 - 13 * q^91 + 76 * q^92 + 181 * q^97 + 153 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/195\mathbb{Z}\right)^\times$$.

 $$n$$ $$106$$ $$131$$ $$157$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
194.1
 0
0 −3.00000 4.00000 −5.00000 0 1.00000 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
195.e odd 2 1 CM by $$\Q(\sqrt{-195})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.3.e.a 1
3.b odd 2 1 195.3.e.d yes 1
5.b even 2 1 195.3.e.c yes 1
13.b even 2 1 195.3.e.b yes 1
15.d odd 2 1 195.3.e.b yes 1
39.d odd 2 1 195.3.e.c yes 1
65.d even 2 1 195.3.e.d yes 1
195.e odd 2 1 CM 195.3.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.3.e.a 1 1.a even 1 1 trivial
195.3.e.a 1 195.e odd 2 1 CM
195.3.e.b yes 1 13.b even 2 1
195.3.e.b yes 1 15.d odd 2 1
195.3.e.c yes 1 5.b even 2 1
195.3.e.c yes 1 39.d odd 2 1
195.3.e.d yes 1 3.b odd 2 1
195.3.e.d yes 1 65.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(195, [\chi])$$:

 $$T_{2}$$ T2 $$T_{7} - 1$$ T7 - 1 $$T_{11} - 17$$ T11 - 17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3$$
$5$ $$T + 5$$
$7$ $$T - 1$$
$11$ $$T - 17$$
$13$ $$T + 13$$
$17$ $$T - 31$$
$19$ $$T$$
$23$ $$T - 19$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T - 61$$
$41$ $$T + 43$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T + 41$$
$59$ $$T - 38$$
$61$ $$T + 73$$
$67$ $$T + 74$$
$71$ $$T + 103$$
$73$ $$T - 94$$
$79$ $$T + 37$$
$83$ $$T$$
$89$ $$T - 173$$
$97$ $$T - 181$$