# Properties

 Label 156.3.n.d Level $156$ Weight $3$ Character orbit 156.n Analytic conductor $4.251$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 156.n (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.25069212402$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{142})$$ Defining polynomial: $$x^{4} + 142x^{2} + 20164$$ x^4 + 142*x^2 + 20164 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \beta_{2} - 2) q^{2} + ( - \beta_{2} - 2) q^{3} + 4 \beta_{2} q^{4} + ( - 2 \beta_{2} - 1) q^{5} + (4 \beta_{2} + 2) q^{6} + (2 \beta_{2} + \beta_1 + 2) q^{7} + 8 q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10})$$ q + (-2*b2 - 2) * q^2 + (-b2 - 2) * q^3 + 4*b2 * q^4 + (-2*b2 - 1) * q^5 + (4*b2 + 2) * q^6 + (2*b2 + b1 + 2) * q^7 + 8 * q^8 + (3*b2 + 3) * q^9 $$q + ( - 2 \beta_{2} - 2) q^{2} + ( - \beta_{2} - 2) q^{3} + 4 \beta_{2} q^{4} + ( - 2 \beta_{2} - 1) q^{5} + (4 \beta_{2} + 2) q^{6} + (2 \beta_{2} + \beta_1 + 2) q^{7} + 8 q^{8} + (3 \beta_{2} + 3) q^{9} + (2 \beta_{2} - 2) q^{10} + ( - \beta_{3} + 6 \beta_{2} - \beta_1) q^{11} + ( - 4 \beta_{2} + 4) q^{12} + ( - \beta_{3} - 3 \beta_{2} - \beta_1 - 6) q^{13} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1) q^{14} + 3 \beta_{2} q^{15} + ( - 16 \beta_{2} - 16) q^{16} + (13 \beta_{2} - \beta_1 + 13) q^{17} - 6 \beta_{2} q^{18} + (8 \beta_{2} + \beta_1 + 8) q^{19} + (4 \beta_{2} + 8) q^{20} + ( - \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 2) q^{21} + (2 \beta_{3} + 12) q^{22} + ( - \beta_{3} + 4 \beta_{2} + \beta_1 + 8) q^{23} + ( - 8 \beta_{2} - 16) q^{24} + 22 q^{25} + (2 \beta_{3} + 12 \beta_{2} + 6) q^{26} + ( - 6 \beta_{2} - 3) q^{27} + (4 \beta_{3} - 8) q^{28} + ( - 2 \beta_{3} - 7 \beta_{2} - 2 \beta_1) q^{29} + 6 q^{30} + ( - 2 \beta_{3} - 32) q^{31} + 32 \beta_{2} q^{32} + (2 \beta_{3} - 6 \beta_{2} + \beta_1 + 6) q^{33} + (2 \beta_{3} - 26 \beta_{2} + 2 \beta_1) q^{34} + ( - 2 \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{35} - 12 q^{36} + ( - \beta_{3} - 7 \beta_{2} + \beta_1 - 14) q^{37} + ( - 2 \beta_{3} - 16 \beta_{2} - 2 \beta_1) q^{38} + (2 \beta_{3} + 9 \beta_{2} + \beta_1 + 9) q^{39} + ( - 16 \beta_{2} - 8) q^{40} + ( - 3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 6) q^{41} + (4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{42} + (2 \beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{43} + ( - 24 \beta_{2} + 4 \beta_1 - 24) q^{44} + ( - 3 \beta_{2} + 3) q^{45} + ( - 2 \beta_{3} - 16 \beta_{2} - 4 \beta_1 - 8) q^{46} + (\beta_{3} + 12) q^{47} + (32 \beta_{2} + 16) q^{48} + (4 \beta_{3} + 97 \beta_{2} + 4 \beta_1) q^{49} + ( - 44 \beta_{2} - 44) q^{50} + (\beta_{3} - 26 \beta_{2} + 2 \beta_1 - 13) q^{51} + ( - 12 \beta_{2} + 4 \beta_1 + 12) q^{52} + (2 \beta_{3} - 25) q^{53} + (6 \beta_{2} - 6) q^{54} + (\beta_{3} + 6 \beta_{2} - \beta_1 + 12) q^{55} + (16 \beta_{2} + 8 \beta_1 + 16) q^{56} + ( - \beta_{3} - 16 \beta_{2} - 2 \beta_1 - 8) q^{57} + (4 \beta_{3} - 14) q^{58} + (56 \beta_{2} + 56) q^{59} + ( - 12 \beta_{2} - 12) q^{60} + ( - 27 \beta_{2} + 5 \beta_1 - 27) q^{61} + (64 \beta_{2} - 4 \beta_1 + 64) q^{62} + (3 \beta_{3} + 6 \beta_{2} + 3 \beta_1) q^{63} + 64 q^{64} + (\beta_{3} + 9 \beta_{2} - \beta_1) q^{65} + ( - 2 \beta_{3} - 12 \beta_{2} + 2 \beta_1 - 24) q^{66} + ( - 7 \beta_{3} - 20 \beta_{2} - 7 \beta_1) q^{67} + ( - 4 \beta_{3} - 52) q^{68} + ( - 12 \beta_{2} - 3 \beta_1 - 12) q^{69} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 8) q^{70} + ( - 88 \beta_{2} - 3 \beta_1 - 88) q^{71} + (24 \beta_{2} + 24) q^{72} + (2 \beta_{3} - 98 \beta_{2} + 4 \beta_1 - 49) q^{73} + ( - 2 \beta_{3} + 28 \beta_{2} - 4 \beta_1 + 14) q^{74} + ( - 22 \beta_{2} - 44) q^{75} + (4 \beta_{3} - 32) q^{76} + (4 \beta_{3} + 130) q^{77} + ( - 2 \beta_{3} - 18 \beta_{2} + 2 \beta_1) q^{78} + (2 \beta_{3} + 8 \beta_{2} + 4 \beta_1 + 4) q^{79} + (16 \beta_{2} - 16) q^{80} + 9 \beta_{2} q^{81} + ( - 6 \beta_{3} + 12 \beta_{2} - 12 \beta_1 + 6) q^{82} + (9 \beta_{3} - 26) q^{83} + ( - 4 \beta_{3} + 8 \beta_{2} + 4 \beta_1 + 16) q^{84} + (2 \beta_{3} - 13 \beta_{2} + \beta_1 + 13) q^{85} + ( - 2 \beta_{3} + 8 \beta_{2} + 2 \beta_1 + 16) q^{86} + (4 \beta_{3} + 7 \beta_{2} + 2 \beta_1 - 7) q^{87} + ( - 8 \beta_{3} + 48 \beta_{2} - 8 \beta_1) q^{88} + (2 \beta_{3} - 48 \beta_{2} - 2 \beta_1 - 96) q^{89} + ( - 6 \beta_{2} - 12) q^{90} + ( - 5 \beta_{3} - 12 \beta_{2} - 6 \beta_1 + 136) q^{91} + (8 \beta_{3} + 16 \beta_{2} + 4 \beta_1 - 16) q^{92} + (2 \beta_{3} + 32 \beta_{2} - 2 \beta_1 + 64) q^{93} + ( - 24 \beta_{2} + 2 \beta_1 - 24) q^{94} + ( - 2 \beta_{3} - 8 \beta_{2} - \beta_1 + 8) q^{95} + ( - 32 \beta_{2} + 32) q^{96} + ( - 8 \beta_{3} + 2 \beta_{2} - 4 \beta_1 - 2) q^{97} + ( - 8 \beta_{3} + 194) q^{98} + ( - 3 \beta_{3} - 18) q^{99}+O(q^{100})$$ q + (-2*b2 - 2) * q^2 + (-b2 - 2) * q^3 + 4*b2 * q^4 + (-2*b2 - 1) * q^5 + (4*b2 + 2) * q^6 + (2*b2 + b1 + 2) * q^7 + 8 * q^8 + (3*b2 + 3) * q^9 + (2*b2 - 2) * q^10 + (-b3 + 6*b2 - b1) * q^11 + (-4*b2 + 4) * q^12 + (-b3 - 3*b2 - b1 - 6) * q^13 + (-2*b3 - 4*b2 - 2*b1) * q^14 + 3*b2 * q^15 + (-16*b2 - 16) * q^16 + (13*b2 - b1 + 13) * q^17 - 6*b2 * q^18 + (8*b2 + b1 + 8) * q^19 + (4*b2 + 8) * q^20 + (-b3 - 4*b2 - 2*b1 - 2) * q^21 + (2*b3 + 12) * q^22 + (-b3 + 4*b2 + b1 + 8) * q^23 + (-8*b2 - 16) * q^24 + 22 * q^25 + (2*b3 + 12*b2 + 6) * q^26 + (-6*b2 - 3) * q^27 + (4*b3 - 8) * q^28 + (-2*b3 - 7*b2 - 2*b1) * q^29 + 6 * q^30 + (-2*b3 - 32) * q^31 + 32*b2 * q^32 + (2*b3 - 6*b2 + b1 + 6) * q^33 + (2*b3 - 26*b2 + 2*b1) * q^34 + (-2*b3 - 2*b2 - b1 + 2) * q^35 - 12 * q^36 + (-b3 - 7*b2 + b1 - 14) * q^37 + (-2*b3 - 16*b2 - 2*b1) * q^38 + (2*b3 + 9*b2 + b1 + 9) * q^39 + (-16*b2 - 8) * q^40 + (-3*b3 - 3*b2 + 3*b1 - 6) * q^41 + (4*b3 + 4*b2 + 2*b1 - 4) * q^42 + (2*b3 + 4*b2 + b1 - 4) * q^43 + (-24*b2 + 4*b1 - 24) * q^44 + (-3*b2 + 3) * q^45 + (-2*b3 - 16*b2 - 4*b1 - 8) * q^46 + (b3 + 12) * q^47 + (32*b2 + 16) * q^48 + (4*b3 + 97*b2 + 4*b1) * q^49 + (-44*b2 - 44) * q^50 + (b3 - 26*b2 + 2*b1 - 13) * q^51 + (-12*b2 + 4*b1 + 12) * q^52 + (2*b3 - 25) * q^53 + (6*b2 - 6) * q^54 + (b3 + 6*b2 - b1 + 12) * q^55 + (16*b2 + 8*b1 + 16) * q^56 + (-b3 - 16*b2 - 2*b1 - 8) * q^57 + (4*b3 - 14) * q^58 + (56*b2 + 56) * q^59 + (-12*b2 - 12) * q^60 + (-27*b2 + 5*b1 - 27) * q^61 + (64*b2 - 4*b1 + 64) * q^62 + (3*b3 + 6*b2 + 3*b1) * q^63 + 64 * q^64 + (b3 + 9*b2 - b1) * q^65 + (-2*b3 - 12*b2 + 2*b1 - 24) * q^66 + (-7*b3 - 20*b2 - 7*b1) * q^67 + (-4*b3 - 52) * q^68 + (-12*b2 - 3*b1 - 12) * q^69 + (2*b3 - 4*b2 - 2*b1 - 8) * q^70 + (-88*b2 - 3*b1 - 88) * q^71 + (24*b2 + 24) * q^72 + (2*b3 - 98*b2 + 4*b1 - 49) * q^73 + (-2*b3 + 28*b2 - 4*b1 + 14) * q^74 + (-22*b2 - 44) * q^75 + (4*b3 - 32) * q^76 + (4*b3 + 130) * q^77 + (-2*b3 - 18*b2 + 2*b1) * q^78 + (2*b3 + 8*b2 + 4*b1 + 4) * q^79 + (16*b2 - 16) * q^80 + 9*b2 * q^81 + (-6*b3 + 12*b2 - 12*b1 + 6) * q^82 + (9*b3 - 26) * q^83 + (-4*b3 + 8*b2 + 4*b1 + 16) * q^84 + (2*b3 - 13*b2 + b1 + 13) * q^85 + (-2*b3 + 8*b2 + 2*b1 + 16) * q^86 + (4*b3 + 7*b2 + 2*b1 - 7) * q^87 + (-8*b3 + 48*b2 - 8*b1) * q^88 + (2*b3 - 48*b2 - 2*b1 - 96) * q^89 + (-6*b2 - 12) * q^90 + (-5*b3 - 12*b2 - 6*b1 + 136) * q^91 + (8*b3 + 16*b2 + 4*b1 - 16) * q^92 + (2*b3 + 32*b2 - 2*b1 + 64) * q^93 + (-24*b2 + 2*b1 - 24) * q^94 + (-2*b3 - 8*b2 - b1 + 8) * q^95 + (-32*b2 + 32) * q^96 + (-8*b3 + 2*b2 - 4*b1 - 2) * q^97 + (-8*b3 + 194) * q^98 + (-3*b3 - 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} + 4 q^{7} + 32 q^{8} + 6 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 - 6 * q^3 - 8 * q^4 + 4 * q^7 + 32 * q^8 + 6 * q^9 $$4 q - 4 q^{2} - 6 q^{3} - 8 q^{4} + 4 q^{7} + 32 q^{8} + 6 q^{9} - 12 q^{10} - 12 q^{11} + 24 q^{12} - 18 q^{13} + 8 q^{14} - 6 q^{15} - 32 q^{16} + 26 q^{17} + 12 q^{18} + 16 q^{19} + 24 q^{20} + 48 q^{22} + 24 q^{23} - 48 q^{24} + 88 q^{25} - 32 q^{28} + 14 q^{29} + 24 q^{30} - 128 q^{31} - 64 q^{32} + 36 q^{33} + 52 q^{34} + 12 q^{35} - 48 q^{36} - 42 q^{37} + 32 q^{38} + 18 q^{39} - 18 q^{41} - 24 q^{42} - 24 q^{43} - 48 q^{44} + 18 q^{45} + 48 q^{47} - 194 q^{49} - 88 q^{50} + 72 q^{52} - 100 q^{53} - 36 q^{54} + 36 q^{55} + 32 q^{56} - 56 q^{58} + 112 q^{59} - 24 q^{60} - 54 q^{61} + 128 q^{62} - 12 q^{63} + 256 q^{64} - 18 q^{65} - 72 q^{66} + 40 q^{67} - 208 q^{68} - 24 q^{69} - 24 q^{70} - 176 q^{71} + 48 q^{72} - 132 q^{75} - 128 q^{76} + 520 q^{77} + 36 q^{78} - 96 q^{80} - 18 q^{81} - 104 q^{83} + 48 q^{84} + 78 q^{85} + 48 q^{86} - 42 q^{87} - 96 q^{88} - 288 q^{89} - 36 q^{90} + 568 q^{91} - 96 q^{92} + 192 q^{93} - 48 q^{94} + 48 q^{95} + 192 q^{96} - 12 q^{97} + 776 q^{98} - 72 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 - 6 * q^3 - 8 * q^4 + 4 * q^7 + 32 * q^8 + 6 * q^9 - 12 * q^10 - 12 * q^11 + 24 * q^12 - 18 * q^13 + 8 * q^14 - 6 * q^15 - 32 * q^16 + 26 * q^17 + 12 * q^18 + 16 * q^19 + 24 * q^20 + 48 * q^22 + 24 * q^23 - 48 * q^24 + 88 * q^25 - 32 * q^28 + 14 * q^29 + 24 * q^30 - 128 * q^31 - 64 * q^32 + 36 * q^33 + 52 * q^34 + 12 * q^35 - 48 * q^36 - 42 * q^37 + 32 * q^38 + 18 * q^39 - 18 * q^41 - 24 * q^42 - 24 * q^43 - 48 * q^44 + 18 * q^45 + 48 * q^47 - 194 * q^49 - 88 * q^50 + 72 * q^52 - 100 * q^53 - 36 * q^54 + 36 * q^55 + 32 * q^56 - 56 * q^58 + 112 * q^59 - 24 * q^60 - 54 * q^61 + 128 * q^62 - 12 * q^63 + 256 * q^64 - 18 * q^65 - 72 * q^66 + 40 * q^67 - 208 * q^68 - 24 * q^69 - 24 * q^70 - 176 * q^71 + 48 * q^72 - 132 * q^75 - 128 * q^76 + 520 * q^77 + 36 * q^78 - 96 * q^80 - 18 * q^81 - 104 * q^83 + 48 * q^84 + 78 * q^85 + 48 * q^86 - 42 * q^87 - 96 * q^88 - 288 * q^89 - 36 * q^90 + 568 * q^91 - 96 * q^92 + 192 * q^93 - 48 * q^94 + 48 * q^95 + 192 * q^96 - 12 * q^97 + 776 * q^98 - 72 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 142x^{2} + 20164$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 142$$ (v^2) / 142 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 142$$ (v^3) / 142
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$142\beta_{2}$$ 142*b2 $$\nu^{3}$$ $$=$$ $$142\beta_{3}$$ 142*b3

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 −5.95819 − 10.3199i 5.95819 + 10.3199i −5.95819 + 10.3199i 5.95819 − 10.3199i
−1.00000 1.73205i −1.50000 0.866025i −2.00000 + 3.46410i 1.73205i 3.46410i −4.95819 8.58783i 8.00000 1.50000 + 2.59808i −3.00000 + 1.73205i
43.2 −1.00000 1.73205i −1.50000 0.866025i −2.00000 + 3.46410i 1.73205i 3.46410i 6.95819 + 12.0519i 8.00000 1.50000 + 2.59808i −3.00000 + 1.73205i
127.1 −1.00000 + 1.73205i −1.50000 + 0.866025i −2.00000 3.46410i 1.73205i 3.46410i −4.95819 + 8.58783i 8.00000 1.50000 2.59808i −3.00000 1.73205i
127.2 −1.00000 + 1.73205i −1.50000 + 0.866025i −2.00000 3.46410i 1.73205i 3.46410i 6.95819 12.0519i 8.00000 1.50000 2.59808i −3.00000 1.73205i
 $$n$$: 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.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.n.d yes 4
4.b odd 2 1 156.3.n.c 4
13.e even 6 1 156.3.n.c 4
52.i odd 6 1 inner 156.3.n.d yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.n.c 4 4.b odd 2 1
156.3.n.c 4 13.e even 6 1
156.3.n.d yes 4 1.a even 1 1 trivial
156.3.n.d yes 4 52.i odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(156, [\chi])$$:

 $$T_{5}^{2} + 3$$ T5^2 + 3 $$T_{7}^{4} - 4T_{7}^{3} + 154T_{7}^{2} + 552T_{7} + 19044$$ T7^4 - 4*T7^3 + 154*T7^2 + 552*T7 + 19044 $$T_{11}^{4} + 12T_{11}^{3} + 250T_{11}^{2} - 1272T_{11} + 11236$$ T11^4 + 12*T11^3 + 250*T11^2 - 1272*T11 + 11236

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{2}$$
$3$ $$(T^{2} + 3 T + 3)^{2}$$
$5$ $$(T^{2} + 3)^{2}$$
$7$ $$T^{4} - 4 T^{3} + 154 T^{2} + \cdots + 19044$$
$11$ $$T^{4} + 12 T^{3} + 250 T^{2} + \cdots + 11236$$
$13$ $$T^{4} + 18 T^{3} + 277 T^{2} + \cdots + 28561$$
$17$ $$T^{4} - 26 T^{3} + 649 T^{2} + \cdots + 729$$
$19$ $$T^{4} - 16 T^{3} + 334 T^{2} + \cdots + 6084$$
$23$ $$T^{4} - 24 T^{3} - 186 T^{2} + \cdots + 142884$$
$29$ $$T^{4} - 14 T^{3} + 715 T^{2} + \cdots + 269361$$
$31$ $$(T^{2} + 64 T + 456)^{2}$$
$37$ $$T^{4} + 42 T^{3} + 309 T^{2} + \cdots + 77841$$
$41$ $$T^{4} + 18 T^{3} - 3699 T^{2} + \cdots + 14493249$$
$43$ $$T^{4} + 24 T^{3} - 186 T^{2} + \cdots + 142884$$
$47$ $$(T^{2} - 24 T + 2)^{2}$$
$53$ $$(T^{2} + 50 T + 57)^{2}$$
$59$ $$(T^{2} - 56 T + 3136)^{2}$$
$61$ $$T^{4} + 54 T^{3} + 5737 T^{2} + \cdots + 7958041$$
$67$ $$T^{4} - 40 T^{3} + 8158 T^{2} + \cdots + 43007364$$
$71$ $$T^{4} + 176 T^{3} + \cdots + 41809156$$
$73$ $$T^{4} + 17814 T^{2} + \cdots + 30239001$$
$79$ $$T^{4} + 3504 T^{2} + \cdots + 2742336$$
$83$ $$(T^{2} + 52 T - 10826)^{2}$$
$89$ $$T^{4} + 288 T^{3} + \cdots + 27123264$$
$97$ $$T^{4} + 12 T^{3} - 6756 T^{2} + \cdots + 46294416$$