# Properties

 Label 195.2.b.b Level $195$ Weight $2$ Character orbit 195.b Analytic conductor $1.557$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [195,2,Mod(181,195)]

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, 1]))

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

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

chi := DirichletCharacter("195.181");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.b (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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{3} + q^{4} + i q^{5} + i q^{6} - 2 i q^{7} + 3 i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 + q^3 + q^4 + i * q^5 + i * q^6 - 2*i * q^7 + 3*i * q^8 + q^9 $$q + i q^{2} + q^{3} + q^{4} + i q^{5} + i q^{6} - 2 i q^{7} + 3 i q^{8} + q^{9} - q^{10} + q^{12} + ( - 2 i - 3) q^{13} + 2 q^{14} + i q^{15} - q^{16} + 2 q^{17} + i q^{18} + 2 i q^{19} + i q^{20} - 2 i q^{21} - 8 q^{23} + 3 i q^{24} - q^{25} + ( - 3 i + 2) q^{26} + q^{27} - 2 i q^{28} + 2 q^{29} - q^{30} - 2 i q^{31} + 5 i q^{32} + 2 i q^{34} + 2 q^{35} + q^{36} - 8 i q^{37} - 2 q^{38} + ( - 2 i - 3) q^{39} - 3 q^{40} - 2 i q^{41} + 2 q^{42} - 4 q^{43} + i q^{45} - 8 i q^{46} - 4 i q^{47} - q^{48} + 3 q^{49} - i q^{50} + 2 q^{51} + ( - 2 i - 3) q^{52} - 6 q^{53} + i q^{54} + 6 q^{56} + 2 i q^{57} + 2 i q^{58} - 12 i q^{59} + i q^{60} + 10 q^{61} + 2 q^{62} - 2 i q^{63} - 7 q^{64} + ( - 3 i + 2) q^{65} - 6 i q^{67} + 2 q^{68} - 8 q^{69} + 2 i q^{70} + 8 i q^{71} + 3 i q^{72} + 16 i q^{73} + 8 q^{74} - q^{75} + 2 i q^{76} + ( - 3 i + 2) q^{78} - 8 q^{79} - i q^{80} + q^{81} + 2 q^{82} + 12 i q^{83} - 2 i q^{84} + 2 i q^{85} - 4 i q^{86} + 2 q^{87} + 6 i q^{89} - q^{90} + (6 i - 4) q^{91} - 8 q^{92} - 2 i q^{93} + 4 q^{94} - 2 q^{95} + 5 i q^{96} + 16 i q^{97} + 3 i q^{98} +O(q^{100})$$ q + i * q^2 + q^3 + q^4 + i * q^5 + i * q^6 - 2*i * q^7 + 3*i * q^8 + q^9 - q^10 + q^12 + (-2*i - 3) * q^13 + 2 * q^14 + i * q^15 - q^16 + 2 * q^17 + i * q^18 + 2*i * q^19 + i * q^20 - 2*i * q^21 - 8 * q^23 + 3*i * q^24 - q^25 + (-3*i + 2) * q^26 + q^27 - 2*i * q^28 + 2 * q^29 - q^30 - 2*i * q^31 + 5*i * q^32 + 2*i * q^34 + 2 * q^35 + q^36 - 8*i * q^37 - 2 * q^38 + (-2*i - 3) * q^39 - 3 * q^40 - 2*i * q^41 + 2 * q^42 - 4 * q^43 + i * q^45 - 8*i * q^46 - 4*i * q^47 - q^48 + 3 * q^49 - i * q^50 + 2 * q^51 + (-2*i - 3) * q^52 - 6 * q^53 + i * q^54 + 6 * q^56 + 2*i * q^57 + 2*i * q^58 - 12*i * q^59 + i * q^60 + 10 * q^61 + 2 * q^62 - 2*i * q^63 - 7 * q^64 + (-3*i + 2) * q^65 - 6*i * q^67 + 2 * q^68 - 8 * q^69 + 2*i * q^70 + 8*i * q^71 + 3*i * q^72 + 16*i * q^73 + 8 * q^74 - q^75 + 2*i * q^76 + (-3*i + 2) * q^78 - 8 * q^79 - i * q^80 + q^81 + 2 * q^82 + 12*i * q^83 - 2*i * q^84 + 2*i * q^85 - 4*i * q^86 + 2 * q^87 + 6*i * q^89 - q^90 + (6*i - 4) * q^91 - 8 * q^92 - 2*i * q^93 + 4 * q^94 - 2 * q^95 + 5*i * q^96 + 16*i * q^97 + 3*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^9 $$2 q + 2 q^{3} + 2 q^{4} + 2 q^{9} - 2 q^{10} + 2 q^{12} - 6 q^{13} + 4 q^{14} - 2 q^{16} + 4 q^{17} - 16 q^{23} - 2 q^{25} + 4 q^{26} + 2 q^{27} + 4 q^{29} - 2 q^{30} + 4 q^{35} + 2 q^{36} - 4 q^{38} - 6 q^{39} - 6 q^{40} + 4 q^{42} - 8 q^{43} - 2 q^{48} + 6 q^{49} + 4 q^{51} - 6 q^{52} - 12 q^{53} + 12 q^{56} + 20 q^{61} + 4 q^{62} - 14 q^{64} + 4 q^{65} + 4 q^{68} - 16 q^{69} + 16 q^{74} - 2 q^{75} + 4 q^{78} - 16 q^{79} + 2 q^{81} + 4 q^{82} + 4 q^{87} - 2 q^{90} - 8 q^{91} - 16 q^{92} + 8 q^{94} - 4 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^4 + 2 * q^9 - 2 * q^10 + 2 * q^12 - 6 * q^13 + 4 * q^14 - 2 * q^16 + 4 * q^17 - 16 * q^23 - 2 * q^25 + 4 * q^26 + 2 * q^27 + 4 * q^29 - 2 * q^30 + 4 * q^35 + 2 * q^36 - 4 * q^38 - 6 * q^39 - 6 * q^40 + 4 * q^42 - 8 * q^43 - 2 * q^48 + 6 * q^49 + 4 * q^51 - 6 * q^52 - 12 * q^53 + 12 * q^56 + 20 * q^61 + 4 * q^62 - 14 * q^64 + 4 * q^65 + 4 * q^68 - 16 * q^69 + 16 * q^74 - 2 * q^75 + 4 * q^78 - 16 * q^79 + 2 * q^81 + 4 * q^82 + 4 * q^87 - 2 * q^90 - 8 * q^91 - 16 * q^92 + 8 * q^94 - 4 * q^95

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
181.1
 − 1.00000i 1.00000i
1.00000i 1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 −1.00000
181.2 1.00000i 1.00000 1.00000 1.00000i 1.00000i 2.00000i 3.00000i 1.00000 −1.00000
 $$n$$: 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
13.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.b.b 2
3.b odd 2 1 585.2.b.a 2
4.b odd 2 1 3120.2.g.a 2
5.b even 2 1 975.2.b.b 2
5.c odd 4 1 975.2.h.a 2
5.c odd 4 1 975.2.h.d 2
13.b even 2 1 inner 195.2.b.b 2
13.d odd 4 1 2535.2.a.e 1
13.d odd 4 1 2535.2.a.l 1
39.d odd 2 1 585.2.b.a 2
39.f even 4 1 7605.2.a.d 1
39.f even 4 1 7605.2.a.p 1
52.b odd 2 1 3120.2.g.a 2
65.d even 2 1 975.2.b.b 2
65.h odd 4 1 975.2.h.a 2
65.h odd 4 1 975.2.h.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.b.b 2 1.a even 1 1 trivial
195.2.b.b 2 13.b even 2 1 inner
585.2.b.a 2 3.b odd 2 1
585.2.b.a 2 39.d odd 2 1
975.2.b.b 2 5.b even 2 1
975.2.b.b 2 65.d even 2 1
975.2.h.a 2 5.c odd 4 1
975.2.h.a 2 65.h odd 4 1
975.2.h.d 2 5.c odd 4 1
975.2.h.d 2 65.h odd 4 1
2535.2.a.e 1 13.d odd 4 1
2535.2.a.l 1 13.d odd 4 1
3120.2.g.a 2 4.b odd 2 1
3120.2.g.a 2 52.b odd 2 1
7605.2.a.d 1 39.f even 4 1
7605.2.a.p 1 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(195, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2} + 4$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$(T + 8)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 4$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 4$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 16$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} + 144$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 36$$
$71$ $$T^{2} + 64$$
$73$ $$T^{2} + 256$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 256$$