# Properties

 Label 156.2.k.d Level $156$ Weight $2$ Character orbit 156.k 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.k (of order $$4$$, degree $$2$$, 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - i + 1) q^{2} - i q^{3} - 2 i q^{4} + (2 i + 2) q^{5} + ( - i - 1) q^{6} + ( - 2 i - 2) q^{8} - q^{9} +O(q^{10})$$ q + (-i + 1) * q^2 - i * q^3 - 2*i * q^4 + (2*i + 2) * q^5 + (-i - 1) * q^6 + (-2*i - 2) * q^8 - q^9 $$q + ( - i + 1) q^{2} - i q^{3} - 2 i q^{4} + (2 i + 2) q^{5} + ( - i - 1) q^{6} + ( - 2 i - 2) q^{8} - q^{9} + 4 q^{10} + ( - i - 1) q^{11} - 2 q^{12} + (2 i - 3) q^{13} + ( - 2 i + 2) q^{15} - 4 q^{16} + 2 i q^{17} + (i - 1) q^{18} + (2 i - 2) q^{19} + ( - 4 i + 4) q^{20} - 2 q^{22} + 6 q^{23} + (2 i - 2) q^{24} + 3 i q^{25} + (5 i - 1) q^{26} + i q^{27} + 10 q^{29} - 4 i q^{30} + (6 i - 6) q^{31} + (4 i - 4) q^{32} + (i - 1) q^{33} + (2 i + 2) q^{34} + 2 i q^{36} + ( - 3 i + 3) q^{37} + 4 i q^{38} + (3 i + 2) q^{39} - 8 i q^{40} + ( - 4 i - 4) q^{41} - 4 q^{43} + (2 i - 2) q^{44} + ( - 2 i - 2) q^{45} + ( - 6 i + 6) q^{46} + ( - 5 i - 5) q^{47} + 4 i q^{48} - 7 i q^{49} + (3 i + 3) q^{50} + 2 q^{51} + (6 i + 4) q^{52} - 6 q^{53} + (i + 1) q^{54} - 4 i q^{55} + (2 i + 2) q^{57} + ( - 10 i + 10) q^{58} + (7 i + 7) q^{59} + ( - 4 i - 4) q^{60} - 8 q^{61} + 12 i q^{62} + 8 i q^{64} + ( - 2 i - 10) q^{65} + 2 i q^{66} + (8 i - 8) q^{67} + 4 q^{68} - 6 i q^{69} + ( - 9 i + 9) q^{71} + (2 i + 2) q^{72} + ( - 5 i + 5) q^{73} - 6 i q^{74} + 3 q^{75} + (4 i + 4) q^{76} + (i + 5) q^{78} - 16 i q^{79} + ( - 8 i - 8) q^{80} + q^{81} - 8 q^{82} + (5 i - 5) q^{83} + (4 i - 4) q^{85} + (4 i - 4) q^{86} - 10 i q^{87} + 4 i q^{88} + ( - 2 i + 2) q^{89} - 4 q^{90} - 12 i q^{92} + (6 i + 6) q^{93} - 10 q^{94} - 8 q^{95} + (4 i + 4) q^{96} + (5 i + 5) q^{97} + ( - 7 i - 7) q^{98} + (i + 1) q^{99} +O(q^{100})$$ q + (-i + 1) * q^2 - i * q^3 - 2*i * q^4 + (2*i + 2) * q^5 + (-i - 1) * q^6 + (-2*i - 2) * q^8 - q^9 + 4 * q^10 + (-i - 1) * q^11 - 2 * q^12 + (2*i - 3) * q^13 + (-2*i + 2) * q^15 - 4 * q^16 + 2*i * q^17 + (i - 1) * q^18 + (2*i - 2) * q^19 + (-4*i + 4) * q^20 - 2 * q^22 + 6 * q^23 + (2*i - 2) * q^24 + 3*i * q^25 + (5*i - 1) * q^26 + i * q^27 + 10 * q^29 - 4*i * q^30 + (6*i - 6) * q^31 + (4*i - 4) * q^32 + (i - 1) * q^33 + (2*i + 2) * q^34 + 2*i * q^36 + (-3*i + 3) * q^37 + 4*i * q^38 + (3*i + 2) * q^39 - 8*i * q^40 + (-4*i - 4) * q^41 - 4 * q^43 + (2*i - 2) * q^44 + (-2*i - 2) * q^45 + (-6*i + 6) * q^46 + (-5*i - 5) * q^47 + 4*i * q^48 - 7*i * q^49 + (3*i + 3) * q^50 + 2 * q^51 + (6*i + 4) * q^52 - 6 * q^53 + (i + 1) * q^54 - 4*i * q^55 + (2*i + 2) * q^57 + (-10*i + 10) * q^58 + (7*i + 7) * q^59 + (-4*i - 4) * q^60 - 8 * q^61 + 12*i * q^62 + 8*i * q^64 + (-2*i - 10) * q^65 + 2*i * q^66 + (8*i - 8) * q^67 + 4 * q^68 - 6*i * q^69 + (-9*i + 9) * q^71 + (2*i + 2) * q^72 + (-5*i + 5) * q^73 - 6*i * q^74 + 3 * q^75 + (4*i + 4) * q^76 + (i + 5) * q^78 - 16*i * q^79 + (-8*i - 8) * q^80 + q^81 - 8 * q^82 + (5*i - 5) * q^83 + (4*i - 4) * q^85 + (4*i - 4) * q^86 - 10*i * q^87 + 4*i * q^88 + (-2*i + 2) * q^89 - 4 * q^90 - 12*i * q^92 + (6*i + 6) * q^93 - 10 * q^94 - 8 * q^95 + (4*i + 4) * q^96 + (5*i + 5) * q^97 + (-7*i - 7) * q^98 + (i + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 4 q^{5} - 2 q^{6} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 4 * q^5 - 2 * q^6 - 4 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 4 q^{5} - 2 q^{6} - 4 q^{8} - 2 q^{9} + 8 q^{10} - 2 q^{11} - 4 q^{12} - 6 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{18} - 4 q^{19} + 8 q^{20} - 4 q^{22} + 12 q^{23} - 4 q^{24} - 2 q^{26} + 20 q^{29} - 12 q^{31} - 8 q^{32} - 2 q^{33} + 4 q^{34} + 6 q^{37} + 4 q^{39} - 8 q^{41} - 8 q^{43} - 4 q^{44} - 4 q^{45} + 12 q^{46} - 10 q^{47} + 6 q^{50} + 4 q^{51} + 8 q^{52} - 12 q^{53} + 2 q^{54} + 4 q^{57} + 20 q^{58} + 14 q^{59} - 8 q^{60} - 16 q^{61} - 20 q^{65} - 16 q^{67} + 8 q^{68} + 18 q^{71} + 4 q^{72} + 10 q^{73} + 6 q^{75} + 8 q^{76} + 10 q^{78} - 16 q^{80} + 2 q^{81} - 16 q^{82} - 10 q^{83} - 8 q^{85} - 8 q^{86} + 4 q^{89} - 8 q^{90} + 12 q^{93} - 20 q^{94} - 16 q^{95} + 8 q^{96} + 10 q^{97} - 14 q^{98} + 2 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 4 * q^5 - 2 * q^6 - 4 * q^8 - 2 * q^9 + 8 * q^10 - 2 * q^11 - 4 * q^12 - 6 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^18 - 4 * q^19 + 8 * q^20 - 4 * q^22 + 12 * q^23 - 4 * q^24 - 2 * q^26 + 20 * q^29 - 12 * q^31 - 8 * q^32 - 2 * q^33 + 4 * q^34 + 6 * q^37 + 4 * q^39 - 8 * q^41 - 8 * q^43 - 4 * q^44 - 4 * q^45 + 12 * q^46 - 10 * q^47 + 6 * q^50 + 4 * q^51 + 8 * q^52 - 12 * q^53 + 2 * q^54 + 4 * q^57 + 20 * q^58 + 14 * q^59 - 8 * q^60 - 16 * q^61 - 20 * q^65 - 16 * q^67 + 8 * q^68 + 18 * q^71 + 4 * q^72 + 10 * q^73 + 6 * q^75 + 8 * q^76 + 10 * q^78 - 16 * q^80 + 2 * q^81 - 16 * q^82 - 10 * q^83 - 8 * q^85 - 8 * q^86 + 4 * q^89 - 8 * q^90 + 12 * q^93 - 20 * q^94 - 16 * q^95 + 8 * q^96 + 10 * q^97 - 14 * q^98 + 2 * q^99

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 − 1.00000i 1.00000i
1.00000 + 1.00000i 1.00000i 2.00000i 2.00000 2.00000i −1.00000 + 1.00000i 0 −2.00000 + 2.00000i −1.00000 4.00000
151.1 1.00000 1.00000i 1.00000i 2.00000i 2.00000 + 2.00000i −1.00000 1.00000i 0 −2.00000 2.00000i −1.00000 4.00000
 $$n$$: 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
52.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.2.k.d yes 2
3.b odd 2 1 468.2.n.b 2
4.b odd 2 1 156.2.k.c 2
12.b even 2 1 468.2.n.d 2
13.d odd 4 1 156.2.k.c 2
39.f even 4 1 468.2.n.d 2
52.f even 4 1 inner 156.2.k.d yes 2
156.l odd 4 1 468.2.n.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.2.k.c 2 4.b odd 2 1
156.2.k.c 2 13.d odd 4 1
156.2.k.d yes 2 1.a even 1 1 trivial
156.2.k.d yes 2 52.f even 4 1 inner
468.2.n.b 2 3.b odd 2 1
468.2.n.b 2 156.l odd 4 1
468.2.n.d 2 12.b even 2 1
468.2.n.d 2 39.f even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(156, [\chi])$$:

 $$T_{5}^{2} - 4T_{5} + 8$$ T5^2 - 4*T5 + 8 $$T_{7}$$ T7 $$T_{11}^{2} + 2T_{11} + 2$$ T11^2 + 2*T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 2$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2} - 4T + 8$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 2T + 2$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$T^{2} + 4$$
$19$ $$T^{2} + 4T + 8$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T - 10)^{2}$$
$31$ $$T^{2} + 12T + 72$$
$37$ $$T^{2} - 6T + 18$$
$41$ $$T^{2} + 8T + 32$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 10T + 50$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 14T + 98$$
$61$ $$(T + 8)^{2}$$
$67$ $$T^{2} + 16T + 128$$
$71$ $$T^{2} - 18T + 162$$
$73$ $$T^{2} - 10T + 50$$
$79$ $$T^{2} + 256$$
$83$ $$T^{2} + 10T + 50$$
$89$ $$T^{2} - 4T + 8$$
$97$ $$T^{2} - 10T + 50$$