# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$195 = 3 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 195.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.55708283941$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (z - 1) * q^3 + 2*z * q^4 - q^5 + z * q^7 - z * q^9 $$q + (\zeta_{6} - 1) q^{3} + 2 \zeta_{6} q^{4} - q^{5} + \zeta_{6} q^{7} - \zeta_{6} q^{9} + (6 \zeta_{6} - 6) q^{11} - 2 q^{12} + ( - 3 \zeta_{6} + 4) q^{13} + ( - \zeta_{6} + 1) q^{15} + (4 \zeta_{6} - 4) q^{16} + 4 \zeta_{6} q^{19} - 2 \zeta_{6} q^{20} - q^{21} + ( - 6 \zeta_{6} + 6) q^{23} + q^{25} + q^{27} + (2 \zeta_{6} - 2) q^{28} + ( - 6 \zeta_{6} + 6) q^{29} + 5 q^{31} - 6 \zeta_{6} q^{33} - \zeta_{6} q^{35} + ( - 2 \zeta_{6} + 2) q^{36} + (2 \zeta_{6} - 2) q^{37} + (4 \zeta_{6} - 1) q^{39} - 11 \zeta_{6} q^{43} - 12 q^{44} + \zeta_{6} q^{45} + 6 q^{47} - 4 \zeta_{6} q^{48} + ( - 6 \zeta_{6} + 6) q^{49} + (2 \zeta_{6} + 6) q^{52} + ( - 6 \zeta_{6} + 6) q^{55} - 4 q^{57} - 6 \zeta_{6} q^{59} + 2 q^{60} + \zeta_{6} q^{61} + ( - \zeta_{6} + 1) q^{63} - 8 q^{64} + (3 \zeta_{6} - 4) q^{65} + (11 \zeta_{6} - 11) q^{67} + 6 \zeta_{6} q^{69} + 6 \zeta_{6} q^{71} + 5 q^{73} + (\zeta_{6} - 1) q^{75} + (8 \zeta_{6} - 8) q^{76} - 6 q^{77} + 11 q^{79} + ( - 4 \zeta_{6} + 4) q^{80} + (\zeta_{6} - 1) q^{81} + 12 q^{83} - 2 \zeta_{6} q^{84} + 6 \zeta_{6} q^{87} + (12 \zeta_{6} - 12) q^{89} + (\zeta_{6} + 3) q^{91} + 12 q^{92} + (5 \zeta_{6} - 5) q^{93} - 4 \zeta_{6} q^{95} - 17 \zeta_{6} q^{97} + 6 q^{99} +O(q^{100})$$ q + (z - 1) * q^3 + 2*z * q^4 - q^5 + z * q^7 - z * q^9 + (6*z - 6) * q^11 - 2 * q^12 + (-3*z + 4) * q^13 + (-z + 1) * q^15 + (4*z - 4) * q^16 + 4*z * q^19 - 2*z * q^20 - q^21 + (-6*z + 6) * q^23 + q^25 + q^27 + (2*z - 2) * q^28 + (-6*z + 6) * q^29 + 5 * q^31 - 6*z * q^33 - z * q^35 + (-2*z + 2) * q^36 + (2*z - 2) * q^37 + (4*z - 1) * q^39 - 11*z * q^43 - 12 * q^44 + z * q^45 + 6 * q^47 - 4*z * q^48 + (-6*z + 6) * q^49 + (2*z + 6) * q^52 + (-6*z + 6) * q^55 - 4 * q^57 - 6*z * q^59 + 2 * q^60 + z * q^61 + (-z + 1) * q^63 - 8 * q^64 + (3*z - 4) * q^65 + (11*z - 11) * q^67 + 6*z * q^69 + 6*z * q^71 + 5 * q^73 + (z - 1) * q^75 + (8*z - 8) * q^76 - 6 * q^77 + 11 * q^79 + (-4*z + 4) * q^80 + (z - 1) * q^81 + 12 * q^83 - 2*z * q^84 + 6*z * q^87 + (12*z - 12) * q^89 + (z + 3) * q^91 + 12 * q^92 + (5*z - 5) * q^93 - 4*z * q^95 - 17*z * q^97 + 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 2 q^{4} - 2 q^{5} + q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 + 2 * q^4 - 2 * q^5 + q^7 - q^9 $$2 q - q^{3} + 2 q^{4} - 2 q^{5} + q^{7} - q^{9} - 6 q^{11} - 4 q^{12} + 5 q^{13} + q^{15} - 4 q^{16} + 4 q^{19} - 2 q^{20} - 2 q^{21} + 6 q^{23} + 2 q^{25} + 2 q^{27} - 2 q^{28} + 6 q^{29} + 10 q^{31} - 6 q^{33} - q^{35} + 2 q^{36} - 2 q^{37} + 2 q^{39} - 11 q^{43} - 24 q^{44} + q^{45} + 12 q^{47} - 4 q^{48} + 6 q^{49} + 14 q^{52} + 6 q^{55} - 8 q^{57} - 6 q^{59} + 4 q^{60} + q^{61} + q^{63} - 16 q^{64} - 5 q^{65} - 11 q^{67} + 6 q^{69} + 6 q^{71} + 10 q^{73} - q^{75} - 8 q^{76} - 12 q^{77} + 22 q^{79} + 4 q^{80} - q^{81} + 24 q^{83} - 2 q^{84} + 6 q^{87} - 12 q^{89} + 7 q^{91} + 24 q^{92} - 5 q^{93} - 4 q^{95} - 17 q^{97} + 12 q^{99}+O(q^{100})$$ 2 * q - q^3 + 2 * q^4 - 2 * q^5 + q^7 - q^9 - 6 * q^11 - 4 * q^12 + 5 * q^13 + q^15 - 4 * q^16 + 4 * q^19 - 2 * q^20 - 2 * q^21 + 6 * q^23 + 2 * q^25 + 2 * q^27 - 2 * q^28 + 6 * q^29 + 10 * q^31 - 6 * q^33 - q^35 + 2 * q^36 - 2 * q^37 + 2 * q^39 - 11 * q^43 - 24 * q^44 + q^45 + 12 * q^47 - 4 * q^48 + 6 * q^49 + 14 * q^52 + 6 * q^55 - 8 * q^57 - 6 * q^59 + 4 * q^60 + q^61 + q^63 - 16 * q^64 - 5 * q^65 - 11 * q^67 + 6 * q^69 + 6 * q^71 + 10 * q^73 - q^75 - 8 * q^76 - 12 * q^77 + 22 * q^79 + 4 * q^80 - q^81 + 24 * q^83 - 2 * q^84 + 6 * q^87 - 12 * q^89 + 7 * q^91 + 24 * q^92 - 5 * q^93 - 4 * q^95 - 17 * q^97 + 12 * q^99

## Character values

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

 $$n$$ $$106$$ $$131$$ $$157$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −0.500000 0.866025i 1.00000 1.73205i −1.00000 0 0.500000 0.866025i 0 −0.500000 + 0.866025i 0
61.1 0 −0.500000 + 0.866025i 1.00000 + 1.73205i −1.00000 0 0.500000 + 0.866025i 0 −0.500000 0.866025i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 195.2.i.b 2
3.b odd 2 1 585.2.j.a 2
5.b even 2 1 975.2.i.d 2
5.c odd 4 2 975.2.bb.b 4
13.c even 3 1 inner 195.2.i.b 2
13.c even 3 1 2535.2.a.h 1
13.e even 6 1 2535.2.a.i 1
39.h odd 6 1 7605.2.a.k 1
39.i odd 6 1 585.2.j.a 2
39.i odd 6 1 7605.2.a.l 1
65.n even 6 1 975.2.i.d 2
65.q odd 12 2 975.2.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.2.i.b 2 1.a even 1 1 trivial
195.2.i.b 2 13.c even 3 1 inner
585.2.j.a 2 3.b odd 2 1
585.2.j.a 2 39.i odd 6 1
975.2.i.d 2 5.b even 2 1
975.2.i.d 2 65.n even 6 1
975.2.bb.b 4 5.c odd 4 2
975.2.bb.b 4 65.q odd 12 2
2535.2.a.h 1 13.c even 3 1
2535.2.a.i 1 13.e even 6 1
7605.2.a.k 1 39.h odd 6 1
7605.2.a.l 1 39.i odd 6 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 6T + 36$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - 6T + 36$$
$31$ $$(T - 5)^{2}$$
$37$ $$T^{2} + 2T + 4$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 11T + 121$$
$47$ $$(T - 6)^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 6T + 36$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2} - 6T + 36$$
$73$ $$(T - 5)^{2}$$
$79$ $$(T - 11)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 12T + 144$$
$97$ $$T^{2} + 17T + 289$$