# Properties

 Label 195.2.a.a Level $195$ Weight $2$ Character orbit 195.a Self dual yes Analytic conductor $1.557$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,2,Mod(1,195)]

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([0, 0, 0]))

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

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

chi := DirichletCharacter("195.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.a (trivial)

## 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: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 + q^5 - q^6 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} + q^{13} + q^{15} - q^{16} + 2 q^{17} - q^{18} - 4 q^{19} - q^{20} - 4 q^{22} + 8 q^{23} + 3 q^{24} + q^{25} - q^{26} + q^{27} - 2 q^{29} - q^{30} - 8 q^{31} - 5 q^{32} + 4 q^{33} - 2 q^{34} - q^{36} + 6 q^{37} + 4 q^{38} + q^{39} + 3 q^{40} - 6 q^{41} - 4 q^{43} - 4 q^{44} + q^{45} - 8 q^{46} - 8 q^{47} - q^{48} - 7 q^{49} - q^{50} + 2 q^{51} - q^{52} + 6 q^{53} - q^{54} + 4 q^{55} - 4 q^{57} + 2 q^{58} - 12 q^{59} - q^{60} - 2 q^{61} + 8 q^{62} + 7 q^{64} + q^{65} - 4 q^{66} - 4 q^{67} - 2 q^{68} + 8 q^{69} + 3 q^{72} - 6 q^{73} - 6 q^{74} + q^{75} + 4 q^{76} - q^{78} + 16 q^{79} - q^{80} + q^{81} + 6 q^{82} - 4 q^{83} + 2 q^{85} + 4 q^{86} - 2 q^{87} + 12 q^{88} + 10 q^{89} - q^{90} - 8 q^{92} - 8 q^{93} + 8 q^{94} - 4 q^{95} - 5 q^{96} + 18 q^{97} + 7 q^{98} + 4 q^{99}+O(q^{100})$$ q - q^2 + q^3 - q^4 + q^5 - q^6 + 3 * q^8 + q^9 - q^10 + 4 * q^11 - q^12 + q^13 + q^15 - q^16 + 2 * q^17 - q^18 - 4 * q^19 - q^20 - 4 * q^22 + 8 * q^23 + 3 * q^24 + q^25 - q^26 + q^27 - 2 * q^29 - q^30 - 8 * q^31 - 5 * q^32 + 4 * q^33 - 2 * q^34 - q^36 + 6 * q^37 + 4 * q^38 + q^39 + 3 * q^40 - 6 * q^41 - 4 * q^43 - 4 * q^44 + q^45 - 8 * q^46 - 8 * q^47 - q^48 - 7 * q^49 - q^50 + 2 * q^51 - q^52 + 6 * q^53 - q^54 + 4 * q^55 - 4 * q^57 + 2 * q^58 - 12 * q^59 - q^60 - 2 * q^61 + 8 * q^62 + 7 * q^64 + q^65 - 4 * q^66 - 4 * q^67 - 2 * q^68 + 8 * q^69 + 3 * q^72 - 6 * q^73 - 6 * q^74 + q^75 + 4 * q^76 - q^78 + 16 * q^79 - q^80 + q^81 + 6 * q^82 - 4 * q^83 + 2 * q^85 + 4 * q^86 - 2 * q^87 + 12 * q^88 + 10 * q^89 - q^90 - 8 * q^92 - 8 * q^93 + 8 * q^94 - 4 * q^95 - 5 * q^96 + 18 * q^97 + 7 * q^98 + 4 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 −1.00000 1.00000 −1.00000 0 3.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.a.a 1
3.b odd 2 1 585.2.a.g 1
4.b odd 2 1 3120.2.a.k 1
5.b even 2 1 975.2.a.i 1
5.c odd 4 2 975.2.c.e 2
7.b odd 2 1 9555.2.a.b 1
12.b even 2 1 9360.2.a.o 1
13.b even 2 1 2535.2.a.k 1
15.d odd 2 1 2925.2.a.d 1
15.e even 4 2 2925.2.c.f 2
39.d odd 2 1 7605.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.a.a 1 1.a even 1 1 trivial
585.2.a.g 1 3.b odd 2 1
975.2.a.i 1 5.b even 2 1
975.2.c.e 2 5.c odd 4 2
2535.2.a.k 1 13.b even 2 1
2925.2.a.d 1 15.d odd 2 1
2925.2.c.f 2 15.e even 4 2
3120.2.a.k 1 4.b odd 2 1
7605.2.a.h 1 39.d odd 2 1
9360.2.a.o 1 12.b even 2 1
9555.2.a.b 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(195))$$:

 $$T_{2} + 1$$ T2 + 1 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T - 4$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T + 4$$
$23$ $$T - 8$$
$29$ $$T + 2$$
$31$ $$T + 8$$
$37$ $$T - 6$$
$41$ $$T + 6$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T - 6$$
$59$ $$T + 12$$
$61$ $$T + 2$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T - 16$$
$83$ $$T + 4$$
$89$ $$T - 10$$
$97$ $$T - 18$$