# Properties

 Label 76.3.b.a Level $76$ Weight $3$ Character orbit 76.b Analytic conductor $2.071$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$76 = 2^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 76.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.07085000914$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ x^4 - x^3 - 4*x^2 - 5*x + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (2 \beta_1 - 2) q^{4} + (2 \beta_{2} - 2) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{6} + (\beta_{3} + \beta_1) q^{7} + 8 q^{8} + (\beta_{2} + 3) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (-b3 - b1) * q^3 + (2*b1 - 2) * q^4 + (2*b2 - 2) * q^5 + (b3 + b2 + b1 - 2) * q^6 + (b3 + b1) * q^7 + 8 * q^8 + (b2 + 3) * q^9 $$q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} - \beta_1) q^{3} + (2 \beta_1 - 2) q^{4} + (2 \beta_{2} - 2) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{6} + (\beta_{3} + \beta_1) q^{7} + 8 q^{8} + (\beta_{2} + 3) q^{9} + (6 \beta_{3} - 2 \beta_{2} + 4 \beta_1 + 2) q^{10} + ( - 2 \beta_{3} - 6 \beta_1) q^{11} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{12} + (3 \beta_{2} - 12) q^{13} + ( - \beta_{3} - \beta_{2} - \beta_1 + 2) q^{14} + (4 \beta_{3} + 12 \beta_1) q^{15} + ( - 8 \beta_1 - 8) q^{16} + (\beta_{2} - 4) q^{17} + (3 \beta_{3} - \beta_{2} - 2 \beta_1 - 3) q^{18} + ( - 2 \beta_{3} - \beta_1) q^{19} + ( - 12 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 4) q^{20} + ( - \beta_{2} + 6) q^{21} + (2 \beta_{3} + 2 \beta_{2} + 6 \beta_1 - 16) q^{22} + ( - 15 \beta_{3} - 7 \beta_1) q^{23} + ( - 8 \beta_{3} - 8 \beta_1) q^{24} + ( - 4 \beta_{2} + 35) q^{25} + (9 \beta_{3} - 3 \beta_{2} + 15 \beta_1 + 12) q^{26} + ( - 11 \beta_{3} - 7 \beta_1) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{28} + ( - 5 \beta_{2} - 12) q^{29} + ( - 4 \beta_{3} - 4 \beta_{2} - 12 \beta_1 + 32) q^{30} + 16 \beta_{3} q^{31} + (16 \beta_1 - 16) q^{32} + (6 \beta_{2} - 20) q^{33} + (3 \beta_{3} - \beta_{2} + 5 \beta_1 + 4) q^{34} + ( - 4 \beta_{3} - 12 \beta_1) q^{35} + ( - 6 \beta_{3} - 2 \beta_{2} + 4 \beta_1 - 6) q^{36} + ( - 8 \beta_{2} - 2) q^{37} + (2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{38} + (15 \beta_{3} + 27 \beta_1) q^{39} + (16 \beta_{2} - 16) q^{40} + ( - 10 \beta_{2} - 26) q^{41} + ( - 3 \beta_{3} + \beta_{2} - 7 \beta_1 - 6) q^{42} + (16 \beta_{3} - 4 \beta_1) q^{43} + (4 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 32) q^{44} + (6 \beta_{2} + 22) q^{45} + (15 \beta_{3} + 15 \beta_{2} + 7 \beta_1 - 6) q^{46} + ( - 8 \beta_{3} - 28 \beta_1) q^{47} + (8 \beta_{3} + 8 \beta_{2} + 8 \beta_1 - 16) q^{48} + (\beta_{2} + 43) q^{49} + ( - 12 \beta_{3} + 4 \beta_{2} - 39 \beta_1 - 35) q^{50} + (5 \beta_{3} + 9 \beta_1) q^{51} + ( - 18 \beta_{3} - 6 \beta_{2} - 30 \beta_1 + 24) q^{52} + (7 \beta_{2} - 4) q^{53} + (11 \beta_{3} + 11 \beta_{2} + 7 \beta_1 - 10) q^{54} + (32 \beta_{3} + 40 \beta_1) q^{55} + (8 \beta_{3} + 8 \beta_1) q^{56} + (\beta_{2} - 10) q^{57} + ( - 15 \beta_{3} + 5 \beta_{2} + 7 \beta_1 + 12) q^{58} + (7 \beta_{3} + 39 \beta_1) q^{59} + ( - 8 \beta_{3} + 8 \beta_{2} - 24 \beta_1 - 64) q^{60} + ( - 4 \beta_{2} - 30) q^{61} + ( - 16 \beta_{3} - 16 \beta_{2} - 16) q^{62} + (2 \beta_{3} - 2 \beta_1) q^{63} + 64 q^{64} + ( - 24 \beta_{2} + 108) q^{65} + (18 \beta_{3} - 6 \beta_{2} + 26 \beta_1 + 20) q^{66} + ( - 33 \beta_{3} + 7 \beta_1) q^{67} + ( - 6 \beta_{3} - 2 \beta_{2} - 10 \beta_1 + 8) q^{68} + (7 \beta_{2} - 74) q^{69} + (4 \beta_{3} + 4 \beta_{2} + 12 \beta_1 - 32) q^{70} + (22 \beta_{3} - 2 \beta_1) q^{71} + (8 \beta_{2} + 24) q^{72} + (7 \beta_{2} - 88) q^{73} + ( - 24 \beta_{3} + 8 \beta_{2} - 6 \beta_1 + 2) q^{74} + ( - 39 \beta_{3} - 55 \beta_1) q^{75} + (4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 + 2) q^{76} + ( - 6 \beta_{2} + 20) q^{77} + ( - 15 \beta_{3} - 15 \beta_{2} - 27 \beta_1 + 66) q^{78} + (2 \beta_{3} - 62 \beta_1) q^{79} + (48 \beta_{3} - 16 \beta_{2} + 32 \beta_1 + 16) q^{80} + (16 \beta_{2} - 31) q^{81} + ( - 30 \beta_{3} + 10 \beta_{2} + 16 \beta_1 + 26) q^{82} + ( - 34 \beta_{3} + 26 \beta_1) q^{83} + (6 \beta_{3} + 2 \beta_{2} + 14 \beta_1 - 12) q^{84} + ( - 8 \beta_{2} + 36) q^{85} + ( - 16 \beta_{3} - 16 \beta_{2} + 4 \beta_1 - 28) q^{86} + (7 \beta_{3} - 13 \beta_1) q^{87} + ( - 16 \beta_{3} - 48 \beta_1) q^{88} + (14 \beta_{2} - 2) q^{89} + (18 \beta_{3} - 6 \beta_{2} - 16 \beta_1 - 22) q^{90} + ( - 15 \beta_{3} - 27 \beta_1) q^{91} + (30 \beta_{3} - 30 \beta_{2} + 14 \beta_1 + 12) q^{92} + 64 q^{93} + (8 \beta_{3} + 8 \beta_{2} + 28 \beta_1 - 76) q^{94} + (2 \beta_{3} + 20 \beta_1) q^{95} + (16 \beta_{3} - 16 \beta_{2} + 16 \beta_1 + 32) q^{96} + ( - 28 \beta_{2} + 2) q^{97} + (3 \beta_{3} - \beta_{2} - 42 \beta_1 - 43) q^{98} + (8 \beta_{3} - 4 \beta_1) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (-b3 - b1) * q^3 + (2*b1 - 2) * q^4 + (2*b2 - 2) * q^5 + (b3 + b2 + b1 - 2) * q^6 + (b3 + b1) * q^7 + 8 * q^8 + (b2 + 3) * q^9 + (6*b3 - 2*b2 + 4*b1 + 2) * q^10 + (-2*b3 - 6*b1) * q^11 + (2*b3 - 2*b2 + 2*b1 + 4) * q^12 + (3*b2 - 12) * q^13 + (-b3 - b2 - b1 + 2) * q^14 + (4*b3 + 12*b1) * q^15 + (-8*b1 - 8) * q^16 + (b2 - 4) * q^17 + (3*b3 - b2 - 2*b1 - 3) * q^18 + (-2*b3 - b1) * q^19 + (-12*b3 - 4*b2 - 8*b1 + 4) * q^20 + (-b2 + 6) * q^21 + (2*b3 + 2*b2 + 6*b1 - 16) * q^22 + (-15*b3 - 7*b1) * q^23 + (-8*b3 - 8*b1) * q^24 + (-4*b2 + 35) * q^25 + (9*b3 - 3*b2 + 15*b1 + 12) * q^26 + (-11*b3 - 7*b1) * q^27 + (-2*b3 + 2*b2 - 2*b1 - 4) * q^28 + (-5*b2 - 12) * q^29 + (-4*b3 - 4*b2 - 12*b1 + 32) * q^30 + 16*b3 * q^31 + (16*b1 - 16) * q^32 + (6*b2 - 20) * q^33 + (3*b3 - b2 + 5*b1 + 4) * q^34 + (-4*b3 - 12*b1) * q^35 + (-6*b3 - 2*b2 + 4*b1 - 6) * q^36 + (-8*b2 - 2) * q^37 + (2*b3 + 2*b2 + b1 - 1) * q^38 + (15*b3 + 27*b1) * q^39 + (16*b2 - 16) * q^40 + (-10*b2 - 26) * q^41 + (-3*b3 + b2 - 7*b1 - 6) * q^42 + (16*b3 - 4*b1) * q^43 + (4*b3 - 4*b2 + 12*b1 + 32) * q^44 + (6*b2 + 22) * q^45 + (15*b3 + 15*b2 + 7*b1 - 6) * q^46 + (-8*b3 - 28*b1) * q^47 + (8*b3 + 8*b2 + 8*b1 - 16) * q^48 + (b2 + 43) * q^49 + (-12*b3 + 4*b2 - 39*b1 - 35) * q^50 + (5*b3 + 9*b1) * q^51 + (-18*b3 - 6*b2 - 30*b1 + 24) * q^52 + (7*b2 - 4) * q^53 + (11*b3 + 11*b2 + 7*b1 - 10) * q^54 + (32*b3 + 40*b1) * q^55 + (8*b3 + 8*b1) * q^56 + (b2 - 10) * q^57 + (-15*b3 + 5*b2 + 7*b1 + 12) * q^58 + (7*b3 + 39*b1) * q^59 + (-8*b3 + 8*b2 - 24*b1 - 64) * q^60 + (-4*b2 - 30) * q^61 + (-16*b3 - 16*b2 - 16) * q^62 + (2*b3 - 2*b1) * q^63 + 64 * q^64 + (-24*b2 + 108) * q^65 + (18*b3 - 6*b2 + 26*b1 + 20) * q^66 + (-33*b3 + 7*b1) * q^67 + (-6*b3 - 2*b2 - 10*b1 + 8) * q^68 + (7*b2 - 74) * q^69 + (4*b3 + 4*b2 + 12*b1 - 32) * q^70 + (22*b3 - 2*b1) * q^71 + (8*b2 + 24) * q^72 + (7*b2 - 88) * q^73 + (-24*b3 + 8*b2 - 6*b1 + 2) * q^74 + (-39*b3 - 55*b1) * q^75 + (4*b3 - 4*b2 + 2*b1 + 2) * q^76 + (-6*b2 + 20) * q^77 + (-15*b3 - 15*b2 - 27*b1 + 66) * q^78 + (2*b3 - 62*b1) * q^79 + (48*b3 - 16*b2 + 32*b1 + 16) * q^80 + (16*b2 - 31) * q^81 + (-30*b3 + 10*b2 + 16*b1 + 26) * q^82 + (-34*b3 + 26*b1) * q^83 + (6*b3 + 2*b2 + 14*b1 - 12) * q^84 + (-8*b2 + 36) * q^85 + (-16*b3 - 16*b2 + 4*b1 - 28) * q^86 + (7*b3 - 13*b1) * q^87 + (-16*b3 - 48*b1) * q^88 + (14*b2 - 2) * q^89 + (18*b3 - 6*b2 - 16*b1 - 22) * q^90 + (-15*b3 - 27*b1) * q^91 + (30*b3 - 30*b2 + 14*b1 + 12) * q^92 + 64 * q^93 + (8*b3 + 8*b2 + 28*b1 - 76) * q^94 + (2*b3 + 20*b1) * q^95 + (16*b3 - 16*b2 + 16*b1 + 32) * q^96 + (-28*b2 + 2) * q^97 + (3*b3 - b2 - 42*b1 - 43) * q^98 + (8*b3 - 4*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 8 q^{4} - 4 q^{5} - 6 q^{6} + 32 q^{8} + 14 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 - 8 * q^4 - 4 * q^5 - 6 * q^6 + 32 * q^8 + 14 * q^9 $$4 q - 4 q^{2} - 8 q^{4} - 4 q^{5} - 6 q^{6} + 32 q^{8} + 14 q^{9} + 4 q^{10} + 12 q^{12} - 42 q^{13} + 6 q^{14} - 32 q^{16} - 14 q^{17} - 14 q^{18} + 8 q^{20} + 22 q^{21} - 60 q^{22} + 132 q^{25} + 42 q^{26} - 12 q^{28} - 58 q^{29} + 120 q^{30} - 64 q^{32} - 68 q^{33} + 14 q^{34} - 28 q^{36} - 24 q^{37} - 32 q^{40} - 124 q^{41} - 22 q^{42} + 120 q^{44} + 100 q^{45} + 6 q^{46} - 48 q^{48} + 174 q^{49} - 132 q^{50} + 84 q^{52} - 2 q^{53} - 18 q^{54} - 38 q^{57} + 58 q^{58} - 240 q^{60} - 128 q^{61} - 96 q^{62} + 256 q^{64} + 384 q^{65} + 68 q^{66} + 28 q^{68} - 282 q^{69} - 120 q^{70} + 112 q^{72} - 338 q^{73} + 24 q^{74} + 68 q^{77} + 234 q^{78} + 32 q^{80} - 92 q^{81} + 124 q^{82} - 44 q^{84} + 128 q^{85} - 144 q^{86} + 20 q^{89} - 100 q^{90} - 12 q^{92} + 256 q^{93} - 288 q^{94} + 96 q^{96} - 48 q^{97} - 174 q^{98}+O(q^{100})$$ 4 * q - 4 * q^2 - 8 * q^4 - 4 * q^5 - 6 * q^6 + 32 * q^8 + 14 * q^9 + 4 * q^10 + 12 * q^12 - 42 * q^13 + 6 * q^14 - 32 * q^16 - 14 * q^17 - 14 * q^18 + 8 * q^20 + 22 * q^21 - 60 * q^22 + 132 * q^25 + 42 * q^26 - 12 * q^28 - 58 * q^29 + 120 * q^30 - 64 * q^32 - 68 * q^33 + 14 * q^34 - 28 * q^36 - 24 * q^37 - 32 * q^40 - 124 * q^41 - 22 * q^42 + 120 * q^44 + 100 * q^45 + 6 * q^46 - 48 * q^48 + 174 * q^49 - 132 * q^50 + 84 * q^52 - 2 * q^53 - 18 * q^54 - 38 * q^57 + 58 * q^58 - 240 * q^60 - 128 * q^61 - 96 * q^62 + 256 * q^64 + 384 * q^65 + 68 * q^66 + 28 * q^68 - 282 * q^69 - 120 * q^70 + 112 * q^72 - 338 * q^73 + 24 * q^74 + 68 * q^77 + 234 * q^78 + 32 * q^80 - 92 * q^81 + 124 * q^82 - 44 * q^84 + 128 * q^85 - 144 * q^86 + 20 * q^89 - 100 * q^90 - 12 * q^92 + 256 * q^93 - 288 * q^94 + 96 * q^96 - 48 * q^97 - 174 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 4x^{2} - 5x + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 4\nu^{2} - 4\nu - 15 ) / 10$$ (v^3 + 4*v^2 - 4*v - 15) / 10 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5$$ (-v^3 + v^2 + 9*v + 5) / 5 $$\beta_{3}$$ $$=$$ $$( -3\nu^{3} - 2\nu^{2} + 2\nu + 25 ) / 10$$ (-3*v^3 - 2*v^2 + 2*v + 25) / 10
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} - \beta_1 ) / 2$$ (-b3 + b2 - b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 5\beta _1 + 4 ) / 2$$ (b3 + b2 + 5*b1 + 4) / 2 $$\nu^{3}$$ $$=$$ $$-4\beta_{3} - 2\beta _1 + 7$$ -4*b3 - 2*b1 + 7

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$39$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
39.1
 −1.63746 − 1.52274i 2.13746 + 0.656712i 2.13746 − 0.656712i −1.63746 + 1.52274i
−1.00000 1.73205i 3.04547i −2.00000 + 3.46410i −8.54983 −5.27492 + 3.04547i 3.04547i 8.00000 −0.274917 8.54983 + 14.8087i
39.2 −1.00000 1.73205i 1.31342i −2.00000 + 3.46410i 6.54983 2.27492 1.31342i 1.31342i 8.00000 7.27492 −6.54983 11.3446i
39.3 −1.00000 + 1.73205i 1.31342i −2.00000 3.46410i 6.54983 2.27492 + 1.31342i 1.31342i 8.00000 7.27492 −6.54983 + 11.3446i
39.4 −1.00000 + 1.73205i 3.04547i −2.00000 3.46410i −8.54983 −5.27492 3.04547i 3.04547i 8.00000 −0.274917 8.54983 14.8087i
 $$n$$: 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.b.a 4
3.b odd 2 1 684.3.g.a 4
4.b odd 2 1 inner 76.3.b.a 4
8.b even 2 1 1216.3.d.a 4
8.d odd 2 1 1216.3.d.a 4
12.b even 2 1 684.3.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.b.a 4 1.a even 1 1 trivial
76.3.b.a 4 4.b odd 2 1 inner
684.3.g.a 4 3.b odd 2 1
684.3.g.a 4 12.b even 2 1
1216.3.d.a 4 8.b even 2 1
1216.3.d.a 4 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 11T_{3}^{2} + 16$$ acting on $$S_{3}^{\mathrm{new}}(76, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2 T + 4)^{2}$$
$3$ $$T^{4} + 11T^{2} + 16$$
$5$ $$(T^{2} + 2 T - 56)^{2}$$
$7$ $$T^{4} + 11T^{2} + 16$$
$11$ $$T^{4} + 188T^{2} + 3136$$
$13$ $$(T^{2} + 21 T - 18)^{2}$$
$17$ $$(T^{2} + 7 T - 2)^{2}$$
$19$ $$(T^{2} + 19)^{2}$$
$23$ $$T^{4} + 2139 T^{2} + \cdots + 1140624$$
$29$ $$(T^{2} + 29 T - 146)^{2}$$
$31$ $$T^{4} + 2816 T^{2} + \cdots + 1048576$$
$37$ $$(T^{2} + 12 T - 876)^{2}$$
$41$ $$(T^{2} + 62 T - 464)^{2}$$
$43$ $$T^{4} + 3296 T^{2} + 614656$$
$47$ $$T^{4} + 4064 T^{2} + \cdots + 2027776$$
$53$ $$(T^{2} + T - 698)^{2}$$
$59$ $$T^{4} + 8027 T^{2} + \cdots + 12588304$$
$61$ $$(T^{2} + 64 T + 796)^{2}$$
$67$ $$T^{4} + 13659 T^{2} + \cdots + 12362256$$
$71$ $$T^{4} + 5612 T^{2} + \cdots + 3211264$$
$73$ $$(T^{2} + 169 T + 6442)^{2}$$
$79$ $$T^{4} + 23852 T^{2} + \cdots + 141324544$$
$83$ $$T^{4} + 22076 T^{2} + 3136$$
$89$ $$(T^{2} - 10 T - 2768)^{2}$$
$97$ $$(T^{2} + 24 T - 11028)^{2}$$