# Properties

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

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,2,Mod(79,195)]

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, 1, 0]))

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

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

chi := DirichletCharacter("195.79");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.c (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: no Analytic conductor: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 2 q^{4} + (2 i - 1) q^{5} + i q^{7} - q^{9}+O(q^{10})$$ q + i * q^3 + 2 * q^4 + (2*i - 1) * q^5 + i * q^7 - q^9 $$q + i q^{3} + 2 q^{4} + (2 i - 1) q^{5} + i q^{7} - q^{9} - q^{11} + 2 i q^{12} - i q^{13} + ( - i - 2) q^{15} + 4 q^{16} + i q^{17} + 4 q^{19} + (4 i - 2) q^{20} - q^{21} - 3 i q^{23} + ( - 4 i - 3) q^{25} - i q^{27} + 2 i q^{28} + 8 q^{29} - 4 q^{31} - i q^{33} + ( - i - 2) q^{35} - 2 q^{36} - 3 i q^{37} + q^{39} - 9 q^{41} - 8 i q^{43} - 2 q^{44} + ( - 2 i + 1) q^{45} - 10 i q^{47} + 4 i q^{48} + 6 q^{49} - q^{51} - 2 i q^{52} - i q^{53} + ( - 2 i + 1) q^{55} + 4 i q^{57} - 4 q^{59} + ( - 2 i - 4) q^{60} - 11 q^{61} - i q^{63} + 8 q^{64} + (i + 2) q^{65} + 4 i q^{67} + 2 i q^{68} + 3 q^{69} - q^{71} + 14 i q^{73} + ( - 3 i + 4) q^{75} + 8 q^{76} - i q^{77} - q^{79} + (8 i - 4) q^{80} + q^{81} + 6 i q^{83} - 2 q^{84} + ( - i - 2) q^{85} + 8 i q^{87} + 15 q^{89} + q^{91} - 6 i q^{92} - 4 i q^{93} + (8 i - 4) q^{95} + 15 i q^{97} + q^{99} +O(q^{100})$$ q + i * q^3 + 2 * q^4 + (2*i - 1) * q^5 + i * q^7 - q^9 - q^11 + 2*i * q^12 - i * q^13 + (-i - 2) * q^15 + 4 * q^16 + i * q^17 + 4 * q^19 + (4*i - 2) * q^20 - q^21 - 3*i * q^23 + (-4*i - 3) * q^25 - i * q^27 + 2*i * q^28 + 8 * q^29 - 4 * q^31 - i * q^33 + (-i - 2) * q^35 - 2 * q^36 - 3*i * q^37 + q^39 - 9 * q^41 - 8*i * q^43 - 2 * q^44 + (-2*i + 1) * q^45 - 10*i * q^47 + 4*i * q^48 + 6 * q^49 - q^51 - 2*i * q^52 - i * q^53 + (-2*i + 1) * q^55 + 4*i * q^57 - 4 * q^59 + (-2*i - 4) * q^60 - 11 * q^61 - i * q^63 + 8 * q^64 + (i + 2) * q^65 + 4*i * q^67 + 2*i * q^68 + 3 * q^69 - q^71 + 14*i * q^73 + (-3*i + 4) * q^75 + 8 * q^76 - i * q^77 - q^79 + (8*i - 4) * q^80 + q^81 + 6*i * q^83 - 2 * q^84 + (-i - 2) * q^85 + 8*i * q^87 + 15 * q^89 + q^91 - 6*i * q^92 - 4*i * q^93 + (8*i - 4) * q^95 + 15*i * q^97 + q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 2 q^{5} - 2 q^{9}+O(q^{10})$$ 2 * q + 4 * q^4 - 2 * q^5 - 2 * q^9 $$2 q + 4 q^{4} - 2 q^{5} - 2 q^{9} - 2 q^{11} - 4 q^{15} + 8 q^{16} + 8 q^{19} - 4 q^{20} - 2 q^{21} - 6 q^{25} + 16 q^{29} - 8 q^{31} - 4 q^{35} - 4 q^{36} + 2 q^{39} - 18 q^{41} - 4 q^{44} + 2 q^{45} + 12 q^{49} - 2 q^{51} + 2 q^{55} - 8 q^{59} - 8 q^{60} - 22 q^{61} + 16 q^{64} + 4 q^{65} + 6 q^{69} - 2 q^{71} + 8 q^{75} + 16 q^{76} - 2 q^{79} - 8 q^{80} + 2 q^{81} - 4 q^{84} - 4 q^{85} + 30 q^{89} + 2 q^{91} - 8 q^{95} + 2 q^{99}+O(q^{100})$$ 2 * q + 4 * q^4 - 2 * q^5 - 2 * q^9 - 2 * q^11 - 4 * q^15 + 8 * q^16 + 8 * q^19 - 4 * q^20 - 2 * q^21 - 6 * q^25 + 16 * q^29 - 8 * q^31 - 4 * q^35 - 4 * q^36 + 2 * q^39 - 18 * q^41 - 4 * q^44 + 2 * q^45 + 12 * q^49 - 2 * q^51 + 2 * q^55 - 8 * q^59 - 8 * q^60 - 22 * q^61 + 16 * q^64 + 4 * q^65 + 6 * q^69 - 2 * q^71 + 8 * q^75 + 16 * q^76 - 2 * q^79 - 8 * q^80 + 2 * q^81 - 4 * q^84 - 4 * q^85 + 30 * q^89 + 2 * q^91 - 8 * q^95 + 2 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 − 1.00000i 1.00000i
0 1.00000i 2.00000 −1.00000 2.00000i 0 1.00000i 0 −1.00000 0
79.2 0 1.00000i 2.00000 −1.00000 + 2.00000i 0 1.00000i 0 −1.00000 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.c.a 2
3.b odd 2 1 585.2.c.a 2
4.b odd 2 1 3120.2.l.c 2
5.b even 2 1 inner 195.2.c.a 2
5.c odd 4 1 975.2.a.g 1
5.c odd 4 1 975.2.a.h 1
15.d odd 2 1 585.2.c.a 2
15.e even 4 1 2925.2.a.h 1
15.e even 4 1 2925.2.a.j 1
20.d odd 2 1 3120.2.l.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.c.a 2 1.a even 1 1 trivial
195.2.c.a 2 5.b even 2 1 inner
585.2.c.a 2 3.b odd 2 1
585.2.c.a 2 15.d odd 2 1
975.2.a.g 1 5.c odd 4 1
975.2.a.h 1 5.c odd 4 1
2925.2.a.h 1 15.e even 4 1
2925.2.a.j 1 15.e even 4 1
3120.2.l.c 2 4.b odd 2 1
3120.2.l.c 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{2}^{\mathrm{new}}(195, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} + 2T + 5$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 1$$
$17$ $$T^{2} + 1$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$(T - 8)^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$(T + 9)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 100$$
$53$ $$T^{2} + 1$$
$59$ $$(T + 4)^{2}$$
$61$ $$(T + 11)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$(T + 1)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T + 1)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 225$$