# Properties

 Label 966.4.a.f Level $966$ Weight $4$ Character orbit 966.a Self dual yes Analytic conductor $56.996$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 966.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.9958450655$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.12197.1 Defining polynomial: $$x^{3} - x^{2} - 15x - 4$$ x^3 - x^2 - 15*x - 4 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - 3 \beta_1 + 4) q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9}+O(q^{10})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 + (-3*b1 + 4) * q^5 - 6 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 $$q + 2 q^{2} - 3 q^{3} + 4 q^{4} + ( - 3 \beta_1 + 4) q^{5} - 6 q^{6} + 7 q^{7} + 8 q^{8} + 9 q^{9} + ( - 6 \beta_1 + 8) q^{10} + ( - 2 \beta_{2} - 12 \beta_1 + 21) q^{11} - 12 q^{12} + ( - 3 \beta_{2} - 2 \beta_1 + 20) q^{13} + 14 q^{14} + (9 \beta_1 - 12) q^{15} + 16 q^{16} + (3 \beta_{2} - 5 \beta_1 + 2) q^{17} + 18 q^{18} + ( - 4 \beta_{2} - 10 \beta_1 + 23) q^{19} + ( - 12 \beta_1 + 16) q^{20} - 21 q^{21} + ( - 4 \beta_{2} - 24 \beta_1 + 42) q^{22} - 23 q^{23} - 24 q^{24} + (9 \beta_{2} - 6 \beta_1 - 19) q^{25} + ( - 6 \beta_{2} - 4 \beta_1 + 40) q^{26} - 27 q^{27} + 28 q^{28} + (9 \beta_{2} + 45 \beta_1 + 92) q^{29} + (18 \beta_1 - 24) q^{30} + (15 \beta_{2} + 43 \beta_1 + 64) q^{31} + 32 q^{32} + (6 \beta_{2} + 36 \beta_1 - 63) q^{33} + (6 \beta_{2} - 10 \beta_1 + 4) q^{34} + ( - 21 \beta_1 + 28) q^{35} + 36 q^{36} + (21 \beta_{2} - 59 \beta_1 + 12) q^{37} + ( - 8 \beta_{2} - 20 \beta_1 + 46) q^{38} + (9 \beta_{2} + 6 \beta_1 - 60) q^{39} + ( - 24 \beta_1 + 32) q^{40} + ( - 48 \beta_{2} + 42 \beta_1 + 27) q^{41} - 42 q^{42} + ( - 49 \beta_{2} - 34 \beta_1 - 214) q^{43} + ( - 8 \beta_{2} - 48 \beta_1 + 84) q^{44} + ( - 27 \beta_1 + 36) q^{45} - 46 q^{46} + (69 \beta_{2} + 11 \beta_1 + 137) q^{47} - 48 q^{48} + 49 q^{49} + (18 \beta_{2} - 12 \beta_1 - 38) q^{50} + ( - 9 \beta_{2} + 15 \beta_1 - 6) q^{51} + ( - 12 \beta_{2} - 8 \beta_1 + 80) q^{52} + ( - 65 \beta_{2} - 14 \beta_1 - 185) q^{53} - 54 q^{54} + (22 \beta_{2} - 21 \beta_1 + 408) q^{55} + 56 q^{56} + (12 \beta_{2} + 30 \beta_1 - 69) q^{57} + (18 \beta_{2} + 90 \beta_1 + 184) q^{58} + ( - 7 \beta_{2} + 138 \beta_1 + 131) q^{59} + (36 \beta_1 - 48) q^{60} + (20 \beta_{2} - 111 \beta_1 + 129) q^{61} + (30 \beta_{2} + 86 \beta_1 + 128) q^{62} + 63 q^{63} + 64 q^{64} + ( - 15 \beta_{2} - 29 \beta_1 + 86) q^{65} + (12 \beta_{2} + 72 \beta_1 - 126) q^{66} + ( - 50 \beta_{2} + 43 \beta_1 + 340) q^{67} + (12 \beta_{2} - 20 \beta_1 + 8) q^{68} + 69 q^{69} + ( - 42 \beta_1 + 56) q^{70} + (75 \beta_{2} + 194 \beta_1 - 22) q^{71} + 72 q^{72} + ( - 53 \beta_{2} + 121 \beta_1 + 62) q^{73} + (42 \beta_{2} - 118 \beta_1 + 24) q^{74} + ( - 27 \beta_{2} + 18 \beta_1 + 57) q^{75} + ( - 16 \beta_{2} - 40 \beta_1 + 92) q^{76} + ( - 14 \beta_{2} - 84 \beta_1 + 147) q^{77} + (18 \beta_{2} + 12 \beta_1 - 120) q^{78} + (27 \beta_{2} - 151 \beta_1 + 104) q^{79} + ( - 48 \beta_1 + 64) q^{80} + 81 q^{81} + ( - 96 \beta_{2} + 84 \beta_1 + 54) q^{82} + (13 \beta_{2} - 41 \beta_1 + 520) q^{83} - 84 q^{84} + (36 \beta_{2} - 23 \beta_1 + 212) q^{85} + ( - 98 \beta_{2} - 68 \beta_1 - 428) q^{86} + ( - 27 \beta_{2} - 135 \beta_1 - 276) q^{87} + ( - 16 \beta_{2} - 96 \beta_1 + 168) q^{88} + ( - 23 \beta_{2} + 186 \beta_1 + 724) q^{89} + ( - 54 \beta_1 + 72) q^{90} + ( - 21 \beta_{2} - 14 \beta_1 + 140) q^{91} - 92 q^{92} + ( - 45 \beta_{2} - 129 \beta_1 - 192) q^{93} + (138 \beta_{2} + 22 \beta_1 + 274) q^{94} + (2 \beta_{2} - 13 \beta_1 + 320) q^{95} - 96 q^{96} + (29 \beta_{2} - 59 \beta_1) q^{97} + 98 q^{98} + ( - 18 \beta_{2} - 108 \beta_1 + 189) q^{99}+O(q^{100})$$ q + 2 * q^2 - 3 * q^3 + 4 * q^4 + (-3*b1 + 4) * q^5 - 6 * q^6 + 7 * q^7 + 8 * q^8 + 9 * q^9 + (-6*b1 + 8) * q^10 + (-2*b2 - 12*b1 + 21) * q^11 - 12 * q^12 + (-3*b2 - 2*b1 + 20) * q^13 + 14 * q^14 + (9*b1 - 12) * q^15 + 16 * q^16 + (3*b2 - 5*b1 + 2) * q^17 + 18 * q^18 + (-4*b2 - 10*b1 + 23) * q^19 + (-12*b1 + 16) * q^20 - 21 * q^21 + (-4*b2 - 24*b1 + 42) * q^22 - 23 * q^23 - 24 * q^24 + (9*b2 - 6*b1 - 19) * q^25 + (-6*b2 - 4*b1 + 40) * q^26 - 27 * q^27 + 28 * q^28 + (9*b2 + 45*b1 + 92) * q^29 + (18*b1 - 24) * q^30 + (15*b2 + 43*b1 + 64) * q^31 + 32 * q^32 + (6*b2 + 36*b1 - 63) * q^33 + (6*b2 - 10*b1 + 4) * q^34 + (-21*b1 + 28) * q^35 + 36 * q^36 + (21*b2 - 59*b1 + 12) * q^37 + (-8*b2 - 20*b1 + 46) * q^38 + (9*b2 + 6*b1 - 60) * q^39 + (-24*b1 + 32) * q^40 + (-48*b2 + 42*b1 + 27) * q^41 - 42 * q^42 + (-49*b2 - 34*b1 - 214) * q^43 + (-8*b2 - 48*b1 + 84) * q^44 + (-27*b1 + 36) * q^45 - 46 * q^46 + (69*b2 + 11*b1 + 137) * q^47 - 48 * q^48 + 49 * q^49 + (18*b2 - 12*b1 - 38) * q^50 + (-9*b2 + 15*b1 - 6) * q^51 + (-12*b2 - 8*b1 + 80) * q^52 + (-65*b2 - 14*b1 - 185) * q^53 - 54 * q^54 + (22*b2 - 21*b1 + 408) * q^55 + 56 * q^56 + (12*b2 + 30*b1 - 69) * q^57 + (18*b2 + 90*b1 + 184) * q^58 + (-7*b2 + 138*b1 + 131) * q^59 + (36*b1 - 48) * q^60 + (20*b2 - 111*b1 + 129) * q^61 + (30*b2 + 86*b1 + 128) * q^62 + 63 * q^63 + 64 * q^64 + (-15*b2 - 29*b1 + 86) * q^65 + (12*b2 + 72*b1 - 126) * q^66 + (-50*b2 + 43*b1 + 340) * q^67 + (12*b2 - 20*b1 + 8) * q^68 + 69 * q^69 + (-42*b1 + 56) * q^70 + (75*b2 + 194*b1 - 22) * q^71 + 72 * q^72 + (-53*b2 + 121*b1 + 62) * q^73 + (42*b2 - 118*b1 + 24) * q^74 + (-27*b2 + 18*b1 + 57) * q^75 + (-16*b2 - 40*b1 + 92) * q^76 + (-14*b2 - 84*b1 + 147) * q^77 + (18*b2 + 12*b1 - 120) * q^78 + (27*b2 - 151*b1 + 104) * q^79 + (-48*b1 + 64) * q^80 + 81 * q^81 + (-96*b2 + 84*b1 + 54) * q^82 + (13*b2 - 41*b1 + 520) * q^83 - 84 * q^84 + (36*b2 - 23*b1 + 212) * q^85 + (-98*b2 - 68*b1 - 428) * q^86 + (-27*b2 - 135*b1 - 276) * q^87 + (-16*b2 - 96*b1 + 168) * q^88 + (-23*b2 + 186*b1 + 724) * q^89 + (-54*b1 + 72) * q^90 + (-21*b2 - 14*b1 + 140) * q^91 - 92 * q^92 + (-45*b2 - 129*b1 - 192) * q^93 + (138*b2 + 22*b1 + 274) * q^94 + (2*b2 - 13*b1 + 320) * q^95 - 96 * q^96 + (29*b2 - 59*b1) * q^97 + 98 * q^98 + (-18*b2 - 108*b1 + 189) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 9 * q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9 $$3 q + 6 q^{2} - 9 q^{3} + 12 q^{4} + 9 q^{5} - 18 q^{6} + 21 q^{7} + 24 q^{8} + 27 q^{9} + 18 q^{10} + 53 q^{11} - 36 q^{12} + 61 q^{13} + 42 q^{14} - 27 q^{15} + 48 q^{16} - 2 q^{17} + 54 q^{18} + 63 q^{19} + 36 q^{20} - 63 q^{21} + 106 q^{22} - 69 q^{23} - 72 q^{24} - 72 q^{25} + 122 q^{26} - 81 q^{27} + 84 q^{28} + 312 q^{29} - 54 q^{30} + 220 q^{31} + 96 q^{32} - 159 q^{33} - 4 q^{34} + 63 q^{35} + 108 q^{36} - 44 q^{37} + 126 q^{38} - 183 q^{39} + 72 q^{40} + 171 q^{41} - 126 q^{42} - 627 q^{43} + 212 q^{44} + 81 q^{45} - 138 q^{46} + 353 q^{47} - 144 q^{48} + 147 q^{49} - 144 q^{50} + 6 q^{51} + 244 q^{52} - 504 q^{53} - 162 q^{54} + 1181 q^{55} + 168 q^{56} - 189 q^{57} + 624 q^{58} + 538 q^{59} - 108 q^{60} + 256 q^{61} + 440 q^{62} + 189 q^{63} + 192 q^{64} + 244 q^{65} - 318 q^{66} + 1113 q^{67} - 8 q^{68} + 207 q^{69} + 126 q^{70} + 53 q^{71} + 216 q^{72} + 360 q^{73} - 88 q^{74} + 216 q^{75} + 252 q^{76} + 371 q^{77} - 366 q^{78} + 134 q^{79} + 144 q^{80} + 243 q^{81} + 342 q^{82} + 1506 q^{83} - 252 q^{84} + 577 q^{85} - 1254 q^{86} - 936 q^{87} + 424 q^{88} + 2381 q^{89} + 162 q^{90} + 427 q^{91} - 276 q^{92} - 660 q^{93} + 706 q^{94} + 945 q^{95} - 288 q^{96} - 88 q^{97} + 294 q^{98} + 477 q^{99}+O(q^{100})$$ 3 * q + 6 * q^2 - 9 * q^3 + 12 * q^4 + 9 * q^5 - 18 * q^6 + 21 * q^7 + 24 * q^8 + 27 * q^9 + 18 * q^10 + 53 * q^11 - 36 * q^12 + 61 * q^13 + 42 * q^14 - 27 * q^15 + 48 * q^16 - 2 * q^17 + 54 * q^18 + 63 * q^19 + 36 * q^20 - 63 * q^21 + 106 * q^22 - 69 * q^23 - 72 * q^24 - 72 * q^25 + 122 * q^26 - 81 * q^27 + 84 * q^28 + 312 * q^29 - 54 * q^30 + 220 * q^31 + 96 * q^32 - 159 * q^33 - 4 * q^34 + 63 * q^35 + 108 * q^36 - 44 * q^37 + 126 * q^38 - 183 * q^39 + 72 * q^40 + 171 * q^41 - 126 * q^42 - 627 * q^43 + 212 * q^44 + 81 * q^45 - 138 * q^46 + 353 * q^47 - 144 * q^48 + 147 * q^49 - 144 * q^50 + 6 * q^51 + 244 * q^52 - 504 * q^53 - 162 * q^54 + 1181 * q^55 + 168 * q^56 - 189 * q^57 + 624 * q^58 + 538 * q^59 - 108 * q^60 + 256 * q^61 + 440 * q^62 + 189 * q^63 + 192 * q^64 + 244 * q^65 - 318 * q^66 + 1113 * q^67 - 8 * q^68 + 207 * q^69 + 126 * q^70 + 53 * q^71 + 216 * q^72 + 360 * q^73 - 88 * q^74 + 216 * q^75 + 252 * q^76 + 371 * q^77 - 366 * q^78 + 134 * q^79 + 144 * q^80 + 243 * q^81 + 342 * q^82 + 1506 * q^83 - 252 * q^84 + 577 * q^85 - 1254 * q^86 - 936 * q^87 + 424 * q^88 + 2381 * q^89 + 162 * q^90 + 427 * q^91 - 276 * q^92 - 660 * q^93 + 706 * q^94 + 945 * q^95 - 288 * q^96 - 88 * q^97 + 294 * q^98 + 477 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 15x - 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2\nu - 10$$ v^2 - 2*v - 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2\beta _1 + 10$$ b2 + 2*b1 + 10

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 4.51691 −0.272991 −3.24392
2.00000 −3.00000 4.00000 −9.55073 −6.00000 7.00000 8.00000 9.00000 −19.1015
1.2 2.00000 −3.00000 4.00000 4.81897 −6.00000 7.00000 8.00000 9.00000 9.63795
1.3 2.00000 −3.00000 4.00000 13.7318 −6.00000 7.00000 8.00000 9.00000 27.4635
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.4.a.f 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.4.a.f 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - 9T_{5}^{2} - 111T_{5} + 632$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(966))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 2)^{3}$$
$3$ $$(T + 3)^{3}$$
$5$ $$T^{3} - 9 T^{2} - 111 T + 632$$
$7$ $$(T - 7)^{3}$$
$11$ $$T^{3} - 53 T^{2} - 1221 T + 71001$$
$13$ $$T^{3} - 61 T^{2} + 637 T - 1822$$
$17$ $$T^{3} + 2 T^{2} - 1211 T - 16024$$
$19$ $$T^{3} - 63 T^{2} - 773 T + 47899$$
$23$ $$(T + 23)^{3}$$
$29$ $$T^{3} - 312 T^{2} + 1317 T + 13186$$
$31$ $$T^{3} - 220 T^{2} - 19003 T + 731490$$
$37$ $$T^{3} + 44 T^{2} - 100239 T - 13366222$$
$41$ $$T^{3} - 171 T^{2} + \cdots + 31352967$$
$43$ $$T^{3} + 627 T^{2} + \cdots - 49544756$$
$47$ $$T^{3} - 353 T^{2} + \cdots + 84406692$$
$53$ $$T^{3} + 504 T^{2} + \cdots - 86011099$$
$59$ $$T^{3} - 538 T^{2} + \cdots + 43304511$$
$61$ $$T^{3} - 256 T^{2} - 225150 T - 6141519$$
$67$ $$T^{3} - 1113 T^{2} + \cdots + 55713572$$
$71$ $$T^{3} - 53 T^{2} - 767299 T - 93489564$$
$73$ $$T^{3} - 360 T^{2} + \cdots + 197977404$$
$79$ $$T^{3} - 134 T^{2} + \cdots - 45775266$$
$83$ $$T^{3} - 1506 T^{2} + \cdots - 107382958$$
$89$ $$T^{3} - 2381 T^{2} + \cdots + 55340850$$
$97$ $$T^{3} + 88 T^{2} - 132487 T - 22908546$$