# Properties

 Label 156.3.l.a Level $156$ Weight $3$ Character orbit 156.l Analytic conductor $4.251$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 156.l (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.25069212402$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{3} + 4 \beta_{2} q^{4} - 4 \beta_{3} q^{5} + (\beta_{3} + 4 \beta_{2} + 4) q^{6} + ( - 3 \beta_{2} - 3) q^{7} + 4 \beta_{3} q^{8} + (2 \beta_{3} + 2 \beta_1 + 7) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b3 + b2 + b1) * q^3 + 4*b2 * q^4 - 4*b3 * q^5 + (b3 + 4*b2 + 4) * q^6 + (-3*b2 - 3) * q^7 + 4*b3 * q^8 + (2*b3 + 2*b1 + 7) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{3} + 4 \beta_{2} q^{4} - 4 \beta_{3} q^{5} + (\beta_{3} + 4 \beta_{2} + 4) q^{6} + ( - 3 \beta_{2} - 3) q^{7} + 4 \beta_{3} q^{8} + (2 \beta_{3} + 2 \beta_1 + 7) q^{9} + 16 q^{10} + 2 \beta_{3} q^{11} + (4 \beta_{3} + 4 \beta_1 - 4) q^{12} - 13 \beta_{2} q^{13} + ( - 3 \beta_{3} - 3 \beta_1) q^{14} + ( - 16 \beta_{2} + 4 \beta_1 + 16) q^{15} - 16 q^{16} + (10 \beta_{3} - 10 \beta_1) q^{17} + (8 \beta_{2} + 7 \beta_1 - 8) q^{18} + (17 \beta_{2} - 17) q^{19} + 16 \beta_1 q^{20} + ( - 3 \beta_{2} - 6 \beta_1 + 3) q^{21} - 8 q^{22} + (6 \beta_{3} + 6 \beta_1) q^{23} + (16 \beta_{2} - 4 \beta_1 - 16) q^{24} - 39 \beta_{2} q^{25} - 13 \beta_{3} q^{26} + ( - 5 \beta_{3} + 23 \beta_{2} + 5 \beta_1) q^{27} + ( - 12 \beta_{2} + 12) q^{28} + (8 \beta_{3} + 8 \beta_1) q^{29} + ( - 16 \beta_{3} + 16 \beta_{2} + 16 \beta_1) q^{30} + ( - 21 \beta_{2} + 21) q^{31} - 16 \beta_1 q^{32} + (8 \beta_{2} - 2 \beta_1 - 8) q^{33} + ( - 40 \beta_{2} - 40) q^{34} + (12 \beta_{3} - 12 \beta_1) q^{35} + (8 \beta_{3} + 28 \beta_{2} - 8 \beta_1) q^{36} + ( - 21 \beta_{2} + 21) q^{37} + (17 \beta_{3} - 17 \beta_1) q^{38} + ( - 13 \beta_{3} - 13 \beta_1 + 13) q^{39} + 64 \beta_{2} q^{40} - 28 \beta_{3} q^{41} + ( - 3 \beta_{3} - 24 \beta_{2} + 3 \beta_1) q^{42} + 14 q^{43} - 8 \beta_1 q^{44} + ( - 28 \beta_{3} + 32 \beta_{2} + 32) q^{45} + (24 \beta_{2} - 24) q^{46} + 22 \beta_{3} q^{47} + (16 \beta_{3} - 16 \beta_{2} - 16 \beta_1) q^{48} - 31 \beta_{2} q^{49} - 39 \beta_{3} q^{50} + ( - 10 \beta_{3} - 10 \beta_1 - 80) q^{51} + 52 q^{52} + (18 \beta_{3} + 18 \beta_1) q^{53} + (23 \beta_{3} + 20 \beta_{2} + 20) q^{54} + 32 \beta_{2} q^{55} + ( - 12 \beta_{3} + 12 \beta_1) q^{56} + (34 \beta_{3} - 17 \beta_{2} - 17) q^{57} + (32 \beta_{2} - 32) q^{58} - 26 \beta_{3} q^{59} + (16 \beta_{3} + 64 \beta_{2} + 64) q^{60} + 16 q^{61} + ( - 21 \beta_{3} + 21 \beta_1) q^{62} + ( - 12 \beta_{3} - 21 \beta_{2} - 21) q^{63} - 64 \beta_{2} q^{64} - 52 \beta_1 q^{65} + (8 \beta_{3} - 8 \beta_{2} - 8 \beta_1) q^{66} + ( - \beta_{2} + 1) q^{67} + ( - 40 \beta_{3} - 40 \beta_1) q^{68} + (6 \beta_{3} + 48 \beta_{2} - 6 \beta_1) q^{69} + ( - 48 \beta_{2} - 48) q^{70} - 18 \beta_1 q^{71} + (28 \beta_{3} - 32 \beta_{2} - 32) q^{72} + (7 \beta_{2} - 7) q^{73} + ( - 21 \beta_{3} + 21 \beta_1) q^{74} + ( - 39 \beta_{3} - 39 \beta_1 + 39) q^{75} + ( - 68 \beta_{2} - 68) q^{76} + ( - 6 \beta_{3} + 6 \beta_1) q^{77} + ( - 52 \beta_{2} + 13 \beta_1 + 52) q^{78} - 106 \beta_{2} q^{79} + 64 \beta_{3} q^{80} + (28 \beta_{3} + 28 \beta_1 + 17) q^{81} + 112 q^{82} + 70 \beta_1 q^{83} + ( - 24 \beta_{3} + 12 \beta_{2} + 12) q^{84} + (160 \beta_{2} - 160) q^{85} + 14 \beta_1 q^{86} + (8 \beta_{3} + 64 \beta_{2} - 8 \beta_1) q^{87} - 32 \beta_{2} q^{88} + 56 \beta_1 q^{89} + (32 \beta_{3} + 32 \beta_1 + 112) q^{90} + (39 \beta_{2} - 39) q^{91} + (24 \beta_{3} - 24 \beta_1) q^{92} + ( - 42 \beta_{3} + 21 \beta_{2} + 21) q^{93} - 88 q^{94} + (68 \beta_{3} + 68 \beta_1) q^{95} + ( - 16 \beta_{3} - 64 \beta_{2} - 64) q^{96} + ( - \beta_{2} - 1) q^{97} - 31 \beta_{3} q^{98} + (14 \beta_{3} - 16 \beta_{2} - 16) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b3 + b2 + b1) * q^3 + 4*b2 * q^4 - 4*b3 * q^5 + (b3 + 4*b2 + 4) * q^6 + (-3*b2 - 3) * q^7 + 4*b3 * q^8 + (2*b3 + 2*b1 + 7) * q^9 + 16 * q^10 + 2*b3 * q^11 + (4*b3 + 4*b1 - 4) * q^12 - 13*b2 * q^13 + (-3*b3 - 3*b1) * q^14 + (-16*b2 + 4*b1 + 16) * q^15 - 16 * q^16 + (10*b3 - 10*b1) * q^17 + (8*b2 + 7*b1 - 8) * q^18 + (17*b2 - 17) * q^19 + 16*b1 * q^20 + (-3*b2 - 6*b1 + 3) * q^21 - 8 * q^22 + (6*b3 + 6*b1) * q^23 + (16*b2 - 4*b1 - 16) * q^24 - 39*b2 * q^25 - 13*b3 * q^26 + (-5*b3 + 23*b2 + 5*b1) * q^27 + (-12*b2 + 12) * q^28 + (8*b3 + 8*b1) * q^29 + (-16*b3 + 16*b2 + 16*b1) * q^30 + (-21*b2 + 21) * q^31 - 16*b1 * q^32 + (8*b2 - 2*b1 - 8) * q^33 + (-40*b2 - 40) * q^34 + (12*b3 - 12*b1) * q^35 + (8*b3 + 28*b2 - 8*b1) * q^36 + (-21*b2 + 21) * q^37 + (17*b3 - 17*b1) * q^38 + (-13*b3 - 13*b1 + 13) * q^39 + 64*b2 * q^40 - 28*b3 * q^41 + (-3*b3 - 24*b2 + 3*b1) * q^42 + 14 * q^43 - 8*b1 * q^44 + (-28*b3 + 32*b2 + 32) * q^45 + (24*b2 - 24) * q^46 + 22*b3 * q^47 + (16*b3 - 16*b2 - 16*b1) * q^48 - 31*b2 * q^49 - 39*b3 * q^50 + (-10*b3 - 10*b1 - 80) * q^51 + 52 * q^52 + (18*b3 + 18*b1) * q^53 + (23*b3 + 20*b2 + 20) * q^54 + 32*b2 * q^55 + (-12*b3 + 12*b1) * q^56 + (34*b3 - 17*b2 - 17) * q^57 + (32*b2 - 32) * q^58 - 26*b3 * q^59 + (16*b3 + 64*b2 + 64) * q^60 + 16 * q^61 + (-21*b3 + 21*b1) * q^62 + (-12*b3 - 21*b2 - 21) * q^63 - 64*b2 * q^64 - 52*b1 * q^65 + (8*b3 - 8*b2 - 8*b1) * q^66 + (-b2 + 1) * q^67 + (-40*b3 - 40*b1) * q^68 + (6*b3 + 48*b2 - 6*b1) * q^69 + (-48*b2 - 48) * q^70 - 18*b1 * q^71 + (28*b3 - 32*b2 - 32) * q^72 + (7*b2 - 7) * q^73 + (-21*b3 + 21*b1) * q^74 + (-39*b3 - 39*b1 + 39) * q^75 + (-68*b2 - 68) * q^76 + (-6*b3 + 6*b1) * q^77 + (-52*b2 + 13*b1 + 52) * q^78 - 106*b2 * q^79 + 64*b3 * q^80 + (28*b3 + 28*b1 + 17) * q^81 + 112 * q^82 + 70*b1 * q^83 + (-24*b3 + 12*b2 + 12) * q^84 + (160*b2 - 160) * q^85 + 14*b1 * q^86 + (8*b3 + 64*b2 - 8*b1) * q^87 - 32*b2 * q^88 + 56*b1 * q^89 + (32*b3 + 32*b1 + 112) * q^90 + (39*b2 - 39) * q^91 + (24*b3 - 24*b1) * q^92 + (-42*b3 + 21*b2 + 21) * q^93 - 88 * q^94 + (68*b3 + 68*b1) * q^95 + (-16*b3 - 64*b2 - 64) * q^96 + (-b2 - 1) * q^97 - 31*b3 * q^98 + (14*b3 - 16*b2 - 16) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 16 q^{6} - 12 q^{7} + 28 q^{9}+O(q^{10})$$ 4 * q + 16 * q^6 - 12 * q^7 + 28 * q^9 $$4 q + 16 q^{6} - 12 q^{7} + 28 q^{9} + 64 q^{10} - 16 q^{12} + 64 q^{15} - 64 q^{16} - 32 q^{18} - 68 q^{19} + 12 q^{21} - 32 q^{22} - 64 q^{24} + 48 q^{28} + 84 q^{31} - 32 q^{33} - 160 q^{34} + 84 q^{37} + 52 q^{39} + 56 q^{43} + 128 q^{45} - 96 q^{46} - 320 q^{51} + 208 q^{52} + 80 q^{54} - 68 q^{57} - 128 q^{58} + 256 q^{60} + 64 q^{61} - 84 q^{63} + 4 q^{67} - 192 q^{70} - 128 q^{72} - 28 q^{73} + 156 q^{75} - 272 q^{76} + 208 q^{78} + 68 q^{81} + 448 q^{82} + 48 q^{84} - 640 q^{85} + 448 q^{90} - 156 q^{91} + 84 q^{93} - 352 q^{94} - 256 q^{96} - 4 q^{97} - 64 q^{99}+O(q^{100})$$ 4 * q + 16 * q^6 - 12 * q^7 + 28 * q^9 + 64 * q^10 - 16 * q^12 + 64 * q^15 - 64 * q^16 - 32 * q^18 - 68 * q^19 + 12 * q^21 - 32 * q^22 - 64 * q^24 + 48 * q^28 + 84 * q^31 - 32 * q^33 - 160 * q^34 + 84 * q^37 + 52 * q^39 + 56 * q^43 + 128 * q^45 - 96 * q^46 - 320 * q^51 + 208 * q^52 + 80 * q^54 - 68 * q^57 - 128 * q^58 + 256 * q^60 + 64 * q^61 - 84 * q^63 + 4 * q^67 - 192 * q^70 - 128 * q^72 - 28 * q^73 + 156 * q^75 - 272 * q^76 + 208 * q^78 + 68 * q^81 + 448 * q^82 + 48 * q^84 - 640 * q^85 + 448 * q^90 - 156 * q^91 + 84 * q^93 - 352 * q^94 - 256 * q^96 - 4 * q^97 - 64 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{8}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{8}^{3}$$ 2*v^3
 $$\zeta_{8}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{8}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

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

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−1.41421 1.41421i −2.82843 + 1.00000i 4.00000i −5.65685 + 5.65685i 5.41421 + 2.58579i −3.00000 3.00000i 5.65685 5.65685i 7.00000 5.65685i 16.0000
47.2 1.41421 + 1.41421i 2.82843 + 1.00000i 4.00000i 5.65685 5.65685i 2.58579 + 5.41421i −3.00000 3.00000i −5.65685 + 5.65685i 7.00000 + 5.65685i 16.0000
83.1 −1.41421 + 1.41421i −2.82843 1.00000i 4.00000i −5.65685 5.65685i 5.41421 2.58579i −3.00000 + 3.00000i 5.65685 + 5.65685i 7.00000 + 5.65685i 16.0000
83.2 1.41421 1.41421i 2.82843 1.00000i 4.00000i 5.65685 + 5.65685i 2.58579 5.41421i −3.00000 + 3.00000i −5.65685 5.65685i 7.00000 5.65685i 16.0000
 $$n$$: 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
3.b odd 2 1 inner
52.f even 4 1 inner
156.l odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.l.a 4
3.b odd 2 1 inner 156.3.l.a 4
4.b odd 2 1 156.3.l.b yes 4
12.b even 2 1 156.3.l.b yes 4
13.d odd 4 1 156.3.l.b yes 4
39.f even 4 1 156.3.l.b yes 4
52.f even 4 1 inner 156.3.l.a 4
156.l odd 4 1 inner 156.3.l.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.l.a 4 1.a even 1 1 trivial
156.3.l.a 4 3.b odd 2 1 inner
156.3.l.a 4 52.f even 4 1 inner
156.3.l.a 4 156.l odd 4 1 inner
156.3.l.b yes 4 4.b odd 2 1
156.3.l.b yes 4 12.b even 2 1
156.3.l.b yes 4 13.d odd 4 1
156.3.l.b yes 4 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 $$S_{3}^{\mathrm{new}}(156, [\chi])$$:

 $$T_{5}^{4} + 4096$$ T5^4 + 4096 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 16$$
$3$ $$T^{4} - 14T^{2} + 81$$
$5$ $$T^{4} + 4096$$
$7$ $$(T^{2} + 6 T + 18)^{2}$$
$11$ $$T^{4} + 256$$
$13$ $$(T^{2} + 169)^{2}$$
$17$ $$(T^{2} - 800)^{2}$$
$19$ $$(T^{2} + 34 T + 578)^{2}$$
$23$ $$(T^{2} + 288)^{2}$$
$29$ $$(T^{2} + 512)^{2}$$
$31$ $$(T^{2} - 42 T + 882)^{2}$$
$37$ $$(T^{2} - 42 T + 882)^{2}$$
$41$ $$T^{4} + 9834496$$
$43$ $$(T - 14)^{4}$$
$47$ $$T^{4} + 3748096$$
$53$ $$(T^{2} + 2592)^{2}$$
$59$ $$T^{4} + 7311616$$
$61$ $$(T - 16)^{4}$$
$67$ $$(T^{2} - 2 T + 2)^{2}$$
$71$ $$T^{4} + 1679616$$
$73$ $$(T^{2} + 14 T + 98)^{2}$$
$79$ $$(T^{2} + 11236)^{2}$$
$83$ $$T^{4} + 384160000$$
$89$ $$T^{4} + 157351936$$
$97$ $$(T^{2} + 2 T + 2)^{2}$$