# Properties

 Label 156.2.b.a Level $156$ Weight $2$ Character orbit 156.b Analytic conductor $1.246$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 156.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.24566627153$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} - \beta q^{5} + q^{9} +O(q^{10})$$ q - q^3 - b * q^5 + q^9 $$q - q^{3} - \beta q^{5} + q^{9} - \beta q^{11} + ( - \beta - 1) q^{13} + \beta q^{15} + 6 q^{17} + 2 \beta q^{19} - 7 q^{25} - q^{27} - 6 q^{29} + 2 \beta q^{31} + \beta q^{33} + (\beta + 1) q^{39} + \beta q^{41} + 8 q^{43} - \beta q^{45} - \beta q^{47} + 7 q^{49} - 6 q^{51} + 6 q^{53} - 12 q^{55} - 2 \beta q^{57} + \beta q^{59} + 10 q^{61} + (\beta - 12) q^{65} - 4 \beta q^{67} - 3 \beta q^{71} + 2 \beta q^{73} + 7 q^{75} - 8 q^{79} + q^{81} + \beta q^{83} - 6 \beta q^{85} + 6 q^{87} + 5 \beta q^{89} - 2 \beta q^{93} + 24 q^{95} - 2 \beta q^{97} - \beta q^{99} +O(q^{100})$$ q - q^3 - b * q^5 + q^9 - b * q^11 + (-b - 1) * q^13 + b * q^15 + 6 * q^17 + 2*b * q^19 - 7 * q^25 - q^27 - 6 * q^29 + 2*b * q^31 + b * q^33 + (b + 1) * q^39 + b * q^41 + 8 * q^43 - b * q^45 - b * q^47 + 7 * q^49 - 6 * q^51 + 6 * q^53 - 12 * q^55 - 2*b * q^57 + b * q^59 + 10 * q^61 + (b - 12) * q^65 - 4*b * q^67 - 3*b * q^71 + 2*b * q^73 + 7 * q^75 - 8 * q^79 + q^81 + b * q^83 - 6*b * q^85 + 6 * q^87 + 5*b * q^89 - 2*b * q^93 + 24 * q^95 - 2*b * q^97 - b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + 2 * q^9 $$2 q - 2 q^{3} + 2 q^{9} - 2 q^{13} + 12 q^{17} - 14 q^{25} - 2 q^{27} - 12 q^{29} + 2 q^{39} + 16 q^{43} + 14 q^{49} - 12 q^{51} + 12 q^{53} - 24 q^{55} + 20 q^{61} - 24 q^{65} + 14 q^{75} - 16 q^{79} + 2 q^{81} + 12 q^{87} + 48 q^{95}+O(q^{100})$$ 2 * q - 2 * q^3 + 2 * q^9 - 2 * q^13 + 12 * q^17 - 14 * q^25 - 2 * q^27 - 12 * q^29 + 2 * q^39 + 16 * q^43 + 14 * q^49 - 12 * q^51 + 12 * q^53 - 24 * q^55 + 20 * q^61 - 24 * q^65 + 14 * q^75 - 16 * q^79 + 2 * q^81 + 12 * q^87 + 48 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/156\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.00000 0 3.46410i 0 0 0 1.00000 0
25.2 0 −1.00000 0 3.46410i 0 0 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
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 156.2.b.a 2
3.b odd 2 1 468.2.b.a 2
4.b odd 2 1 624.2.c.f 2
5.b even 2 1 3900.2.c.c 2
5.c odd 4 2 3900.2.j.h 4
7.b odd 2 1 7644.2.e.g 2
8.b even 2 1 2496.2.c.l 2
8.d odd 2 1 2496.2.c.e 2
12.b even 2 1 1872.2.c.c 2
13.b even 2 1 inner 156.2.b.a 2
13.c even 3 1 2028.2.q.b 2
13.c even 3 1 2028.2.q.c 2
13.d odd 4 2 2028.2.a.g 2
13.e even 6 1 2028.2.q.b 2
13.e even 6 1 2028.2.q.c 2
13.f odd 12 4 2028.2.i.i 4
39.d odd 2 1 468.2.b.a 2
39.f even 4 2 6084.2.a.v 2
52.b odd 2 1 624.2.c.f 2
52.f even 4 2 8112.2.a.bs 2
65.d even 2 1 3900.2.c.c 2
65.h odd 4 2 3900.2.j.h 4
91.b odd 2 1 7644.2.e.g 2
104.e even 2 1 2496.2.c.l 2
104.h odd 2 1 2496.2.c.e 2
156.h even 2 1 1872.2.c.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.b.a 2 1.a even 1 1 trivial
156.2.b.a 2 13.b even 2 1 inner
468.2.b.a 2 3.b odd 2 1
468.2.b.a 2 39.d odd 2 1
624.2.c.f 2 4.b odd 2 1
624.2.c.f 2 52.b odd 2 1
1872.2.c.c 2 12.b even 2 1
1872.2.c.c 2 156.h even 2 1
2028.2.a.g 2 13.d odd 4 2
2028.2.i.i 4 13.f odd 12 4
2028.2.q.b 2 13.c even 3 1
2028.2.q.b 2 13.e even 6 1
2028.2.q.c 2 13.c even 3 1
2028.2.q.c 2 13.e even 6 1
2496.2.c.e 2 8.d odd 2 1
2496.2.c.e 2 104.h odd 2 1
2496.2.c.l 2 8.b even 2 1
2496.2.c.l 2 104.e even 2 1
3900.2.c.c 2 5.b even 2 1
3900.2.c.c 2 65.d even 2 1
3900.2.j.h 4 5.c odd 4 2
3900.2.j.h 4 65.h odd 4 2
6084.2.a.v 2 39.f even 4 2
7644.2.e.g 2 7.b odd 2 1
7644.2.e.g 2 91.b odd 2 1
8112.2.a.bs 2 52.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 12$$ acting on $$S_{2}^{\mathrm{new}}(156, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 12$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 12$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$(T - 6)^{2}$$
$19$ $$T^{2} + 48$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 48$$
$37$ $$T^{2}$$
$41$ $$T^{2} + 12$$
$43$ $$(T - 8)^{2}$$
$47$ $$T^{2} + 12$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 12$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2} + 192$$
$71$ $$T^{2} + 108$$
$73$ $$T^{2} + 48$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 300$$
$97$ $$T^{2} + 48$$