# Properties

 Label 156.2.b.b 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: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{3} + \beta q^{5} + 2 \beta q^{7} + q^{9}+O(q^{10})$$ q + q^3 + b * q^5 + 2*b * q^7 + q^9 $$q + q^{3} + \beta q^{5} + 2 \beta q^{7} + q^{9} - 3 \beta q^{11} + ( - \beta + 3) q^{13} + \beta q^{15} - 2 q^{17} + 2 \beta q^{21} - 8 q^{23} + q^{25} + q^{27} + 2 q^{29} - 4 \beta q^{31} - 3 \beta q^{33} - 8 q^{35} + 4 \beta q^{37} + ( - \beta + 3) q^{39} - \beta q^{41} - 8 q^{43} + \beta q^{45} - 3 \beta q^{47} - 9 q^{49} - 2 q^{51} + 6 q^{53} + 12 q^{55} - \beta q^{59} + 2 q^{61} + 2 \beta q^{63} + (3 \beta + 4) q^{65} + 2 \beta q^{67} - 8 q^{69} + 3 \beta q^{71} - 2 \beta q^{73} + q^{75} + 24 q^{77} + q^{81} + 7 \beta q^{83} - 2 \beta q^{85} + 2 q^{87} + 3 \beta q^{89} + (6 \beta + 8) q^{91} - 4 \beta q^{93} - 6 \beta q^{97} - 3 \beta q^{99} +O(q^{100})$$ q + q^3 + b * q^5 + 2*b * q^7 + q^9 - 3*b * q^11 + (-b + 3) * q^13 + b * q^15 - 2 * q^17 + 2*b * q^21 - 8 * q^23 + q^25 + q^27 + 2 * q^29 - 4*b * q^31 - 3*b * q^33 - 8 * q^35 + 4*b * q^37 + (-b + 3) * q^39 - b * q^41 - 8 * q^43 + b * q^45 - 3*b * q^47 - 9 * q^49 - 2 * q^51 + 6 * q^53 + 12 * q^55 - b * q^59 + 2 * q^61 + 2*b * q^63 + (3*b + 4) * q^65 + 2*b * q^67 - 8 * q^69 + 3*b * q^71 - 2*b * q^73 + q^75 + 24 * q^77 + q^81 + 7*b * q^83 - 2*b * q^85 + 2 * q^87 + 3*b * q^89 + (6*b + 8) * q^91 - 4*b * q^93 - 6*b * q^97 - 3*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 $$2 q + 2 q^{3} + 2 q^{9} + 6 q^{13} - 4 q^{17} - 16 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} - 16 q^{35} + 6 q^{39} - 16 q^{43} - 18 q^{49} - 4 q^{51} + 12 q^{53} + 24 q^{55} + 4 q^{61} + 8 q^{65} - 16 q^{69} + 2 q^{75} + 48 q^{77} + 2 q^{81} + 4 q^{87} + 16 q^{91}+O(q^{100})$$ 2 * q + 2 * q^3 + 2 * q^9 + 6 * q^13 - 4 * q^17 - 16 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 - 16 * q^35 + 6 * q^39 - 16 * q^43 - 18 * q^49 - 4 * q^51 + 12 * q^53 + 24 * q^55 + 4 * q^61 + 8 * q^65 - 16 * q^69 + 2 * q^75 + 48 * q^77 + 2 * q^81 + 4 * q^87 + 16 * q^91

## 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
 − 1.00000i 1.00000i
0 1.00000 0 2.00000i 0 4.00000i 0 1.00000 0
25.2 0 1.00000 0 2.00000i 0 4.00000i 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
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.b 2
3.b odd 2 1 468.2.b.c 2
4.b odd 2 1 624.2.c.d 2
5.b even 2 1 3900.2.c.a 2
5.c odd 4 1 3900.2.j.b 2
5.c odd 4 1 3900.2.j.e 2
7.b odd 2 1 7644.2.e.b 2
8.b even 2 1 2496.2.c.b 2
8.d odd 2 1 2496.2.c.i 2
12.b even 2 1 1872.2.c.h 2
13.b even 2 1 inner 156.2.b.b 2
13.c even 3 2 2028.2.q.e 4
13.d odd 4 1 2028.2.a.d 1
13.d odd 4 1 2028.2.a.f 1
13.e even 6 2 2028.2.q.e 4
13.f odd 12 2 2028.2.i.a 2
13.f odd 12 2 2028.2.i.d 2
39.d odd 2 1 468.2.b.c 2
39.f even 4 1 6084.2.a.d 1
39.f even 4 1 6084.2.a.n 1
52.b odd 2 1 624.2.c.d 2
52.f even 4 1 8112.2.a.d 1
52.f even 4 1 8112.2.a.l 1
65.d even 2 1 3900.2.c.a 2
65.h odd 4 1 3900.2.j.b 2
65.h odd 4 1 3900.2.j.e 2
91.b odd 2 1 7644.2.e.b 2
104.e even 2 1 2496.2.c.b 2
104.h odd 2 1 2496.2.c.i 2
156.h even 2 1 1872.2.c.h 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(156, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 36$$
$13$ $$T^{2} - 6T + 13$$
$17$ $$(T + 2)^{2}$$
$19$ $$T^{2}$$
$23$ $$(T + 8)^{2}$$
$29$ $$(T - 2)^{2}$$
$31$ $$T^{2} + 64$$
$37$ $$T^{2} + 64$$
$41$ $$T^{2} + 4$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} + 36$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 4$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 16$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 196$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 144$$