# Properties

 Label 156.2.q.a Level $156$ Weight $2$ Character orbit 156.q 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.q (of order $$6$$, degree $$2$$, 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 4) q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 + (2*z - 1) * q^5 + (-2*z + 4) * q^7 - z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} + (2 \zeta_{6} - 1) q^{5} + ( - 2 \zeta_{6} + 4) q^{7} - \zeta_{6} q^{9} + (2 \zeta_{6} + 2) q^{11} + ( - 3 \zeta_{6} - 1) q^{13} + (\zeta_{6} + 1) q^{15} - 3 \zeta_{6} q^{17} + (2 \zeta_{6} - 4) q^{19} + ( - 4 \zeta_{6} + 2) q^{21} + (6 \zeta_{6} - 6) q^{23} + 2 q^{25} - q^{27} + (9 \zeta_{6} - 9) q^{29} + ( - 2 \zeta_{6} + 4) q^{33} + 6 \zeta_{6} q^{35} + ( - 3 \zeta_{6} - 3) q^{37} + (\zeta_{6} - 4) q^{39} + ( - 5 \zeta_{6} - 5) q^{41} - 2 \zeta_{6} q^{43} + ( - \zeta_{6} + 2) q^{45} + ( - 4 \zeta_{6} + 2) q^{47} + ( - 5 \zeta_{6} + 5) q^{49} - 3 q^{51} - 9 q^{53} + (6 \zeta_{6} - 6) q^{55} + (4 \zeta_{6} - 2) q^{57} + ( - 8 \zeta_{6} + 16) q^{59} + 11 \zeta_{6} q^{61} + ( - 2 \zeta_{6} - 2) q^{63} + ( - 5 \zeta_{6} + 7) q^{65} + (6 \zeta_{6} + 6) q^{67} + 6 \zeta_{6} q^{69} + ( - 6 \zeta_{6} + 12) q^{71} + ( - 6 \zeta_{6} + 3) q^{73} + ( - 2 \zeta_{6} + 2) q^{75} + 12 q^{77} - 8 q^{79} + (\zeta_{6} - 1) q^{81} + (4 \zeta_{6} - 2) q^{83} + ( - 3 \zeta_{6} + 6) q^{85} + 9 \zeta_{6} q^{87} + ( - 4 \zeta_{6} - 4) q^{89} + ( - 4 \zeta_{6} - 10) q^{91} - 6 \zeta_{6} q^{95} + ( - 4 \zeta_{6} + 8) q^{97} + ( - 4 \zeta_{6} + 2) q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 + (2*z - 1) * q^5 + (-2*z + 4) * q^7 - z * q^9 + (2*z + 2) * q^11 + (-3*z - 1) * q^13 + (z + 1) * q^15 - 3*z * q^17 + (2*z - 4) * q^19 + (-4*z + 2) * q^21 + (6*z - 6) * q^23 + 2 * q^25 - q^27 + (9*z - 9) * q^29 + (-2*z + 4) * q^33 + 6*z * q^35 + (-3*z - 3) * q^37 + (z - 4) * q^39 + (-5*z - 5) * q^41 - 2*z * q^43 + (-z + 2) * q^45 + (-4*z + 2) * q^47 + (-5*z + 5) * q^49 - 3 * q^51 - 9 * q^53 + (6*z - 6) * q^55 + (4*z - 2) * q^57 + (-8*z + 16) * q^59 + 11*z * q^61 + (-2*z - 2) * q^63 + (-5*z + 7) * q^65 + (6*z + 6) * q^67 + 6*z * q^69 + (-6*z + 12) * q^71 + (-6*z + 3) * q^73 + (-2*z + 2) * q^75 + 12 * q^77 - 8 * q^79 + (z - 1) * q^81 + (4*z - 2) * q^83 + (-3*z + 6) * q^85 + 9*z * q^87 + (-4*z - 4) * q^89 + (-4*z - 10) * q^91 - 6*z * q^95 + (-4*z + 8) * q^97 + (-4*z + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} + 6 q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^3 + 6 * q^7 - q^9 $$2 q + q^{3} + 6 q^{7} - q^{9} + 6 q^{11} - 5 q^{13} + 3 q^{15} - 3 q^{17} - 6 q^{19} - 6 q^{23} + 4 q^{25} - 2 q^{27} - 9 q^{29} + 6 q^{33} + 6 q^{35} - 9 q^{37} - 7 q^{39} - 15 q^{41} - 2 q^{43} + 3 q^{45} + 5 q^{49} - 6 q^{51} - 18 q^{53} - 6 q^{55} + 24 q^{59} + 11 q^{61} - 6 q^{63} + 9 q^{65} + 18 q^{67} + 6 q^{69} + 18 q^{71} + 2 q^{75} + 24 q^{77} - 16 q^{79} - q^{81} + 9 q^{85} + 9 q^{87} - 12 q^{89} - 24 q^{91} - 6 q^{95} + 12 q^{97}+O(q^{100})$$ 2 * q + q^3 + 6 * q^7 - q^9 + 6 * q^11 - 5 * q^13 + 3 * q^15 - 3 * q^17 - 6 * q^19 - 6 * q^23 + 4 * q^25 - 2 * q^27 - 9 * q^29 + 6 * q^33 + 6 * q^35 - 9 * q^37 - 7 * q^39 - 15 * q^41 - 2 * q^43 + 3 * q^45 + 5 * q^49 - 6 * q^51 - 18 * q^53 - 6 * q^55 + 24 * q^59 + 11 * q^61 - 6 * q^63 + 9 * q^65 + 18 * q^67 + 6 * q^69 + 18 * q^71 + 2 * q^75 + 24 * q^77 - 16 * q^79 - q^81 + 9 * q^85 + 9 * q^87 - 12 * q^89 - 24 * q^91 - 6 * q^95 + 12 * q^97

## Character values

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

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 1.73205i 0 3.00000 + 1.73205i 0 −0.500000 + 0.866025i 0
121.1 0 0.500000 0.866025i 0 1.73205i 0 3.00000 1.73205i 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.e even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.q.a 2
3.b odd 2 1 468.2.t.c 2
4.b odd 2 1 624.2.bv.a 2
5.b even 2 1 3900.2.cd.a 2
5.c odd 4 2 3900.2.bw.e 4
12.b even 2 1 1872.2.by.b 2
13.b even 2 1 2028.2.q.a 2
13.c even 3 1 2028.2.b.b 2
13.c even 3 1 2028.2.q.a 2
13.d odd 4 2 2028.2.i.h 4
13.e even 6 1 inner 156.2.q.a 2
13.e even 6 1 2028.2.b.b 2
13.f odd 12 2 2028.2.a.h 2
13.f odd 12 2 2028.2.i.h 4
39.h odd 6 1 468.2.t.c 2
39.h odd 6 1 6084.2.b.c 2
39.i odd 6 1 6084.2.b.c 2
39.k even 12 2 6084.2.a.u 2
52.i odd 6 1 624.2.bv.a 2
52.l even 12 2 8112.2.a.bt 2
65.l even 6 1 3900.2.cd.a 2
65.r odd 12 2 3900.2.bw.e 4
156.r even 6 1 1872.2.by.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.q.a 2 1.a even 1 1 trivial
156.2.q.a 2 13.e even 6 1 inner
468.2.t.c 2 3.b odd 2 1
468.2.t.c 2 39.h odd 6 1
624.2.bv.a 2 4.b odd 2 1
624.2.bv.a 2 52.i odd 6 1
1872.2.by.b 2 12.b even 2 1
1872.2.by.b 2 156.r even 6 1
2028.2.a.h 2 13.f odd 12 2
2028.2.b.b 2 13.c even 3 1
2028.2.b.b 2 13.e even 6 1
2028.2.i.h 4 13.d odd 4 2
2028.2.i.h 4 13.f odd 12 2
2028.2.q.a 2 13.b even 2 1
2028.2.q.a 2 13.c even 3 1
3900.2.bw.e 4 5.c odd 4 2
3900.2.bw.e 4 65.r odd 12 2
3900.2.cd.a 2 5.b even 2 1
3900.2.cd.a 2 65.l even 6 1
6084.2.a.u 2 39.k even 12 2
6084.2.b.c 2 39.h odd 6 1
6084.2.b.c 2 39.i odd 6 1
8112.2.a.bt 2 52.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(156, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 3$$
$7$ $$T^{2} - 6T + 12$$
$11$ $$T^{2} - 6T + 12$$
$13$ $$T^{2} + 5T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 6T + 12$$
$23$ $$T^{2} + 6T + 36$$
$29$ $$T^{2} + 9T + 81$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 9T + 27$$
$41$ $$T^{2} + 15T + 75$$
$43$ $$T^{2} + 2T + 4$$
$47$ $$T^{2} + 12$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2} - 24T + 192$$
$61$ $$T^{2} - 11T + 121$$
$67$ $$T^{2} - 18T + 108$$
$71$ $$T^{2} - 18T + 108$$
$73$ $$T^{2} + 27$$
$79$ $$(T + 8)^{2}$$
$83$ $$T^{2} + 12$$
$89$ $$T^{2} + 12T + 48$$
$97$ $$T^{2} - 12T + 48$$