# Properties

 Label 39.4.a.c Level $39$ Weight $4$ Character orbit 39.a Self dual yes Analytic conductor $2.301$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.3144.1 Defining polynomial: $$x^{3} - x^{2} - 16x - 8$$ x^3 - x^2 - 16*x - 8 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} + 3) q^{4} + ( - 2 \beta_{2} + 2) q^{5} + ( - 3 \beta_1 + 3) q^{6} + (6 \beta_1 + 8) q^{7} + (2 \beta_{2} + \beta_1 - 3) q^{8} + 9 q^{9}+O(q^{10})$$ q + (-b1 + 1) * q^2 + 3 * q^3 + (b2 + 3) * q^4 + (-2*b2 + 2) * q^5 + (-3*b1 + 3) * q^6 + (6*b1 + 8) * q^7 + (2*b2 + b1 - 3) * q^8 + 9 * q^9 $$q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} + 3) q^{4} + ( - 2 \beta_{2} + 2) q^{5} + ( - 3 \beta_1 + 3) q^{6} + (6 \beta_1 + 8) q^{7} + (2 \beta_{2} + \beta_1 - 3) q^{8} + 9 q^{9} + ( - 4 \beta_{2} + 6 \beta_1 - 2) q^{10} + (6 \beta_{2} + 2 \beta_1 - 8) q^{11} + (3 \beta_{2} + 9) q^{12} + 13 q^{13} + ( - 6 \beta_{2} - 14 \beta_1 - 52) q^{14} + ( - 6 \beta_{2} + 6) q^{15} + ( - 5 \beta_{2} - 6 \beta_1 - 33) q^{16} + ( - 8 \beta_1 - 46) q^{17} + ( - 9 \beta_1 + 9) q^{18} + (16 \beta_{2} - 6 \beta_1 + 28) q^{19} + (2 \beta_{2} + 12 \beta_1 - 86) q^{20} + (18 \beta_1 + 24) q^{21} + (10 \beta_{2} - 18 \beta_1 - 16) q^{22} + ( - 8 \beta_{2} + 32 \beta_1 - 24) q^{23} + (6 \beta_{2} + 3 \beta_1 - 9) q^{24} + ( - 20 \beta_{2} - 24 \beta_1 + 63) q^{25} + ( - 13 \beta_1 + 13) q^{26} + 27 q^{27} + (2 \beta_{2} + 42 \beta_1 + 12) q^{28} + (8 \beta_{2} + 20 \beta_1 - 10) q^{29} + ( - 12 \beta_{2} + 18 \beta_1 - 6) q^{30} + ( - 4 \beta_{2} - 54 \beta_1 + 120) q^{31} + ( - 20 \beta_{2} + 51 \beta_1 + 41) q^{32} + (18 \beta_{2} + 6 \beta_1 - 24) q^{33} + (8 \beta_{2} + 54 \beta_1 + 34) q^{34} + ( - 4 \beta_{2} - 36 \beta_1 + 40) q^{35} + (9 \beta_{2} + 27) q^{36} + ( - 28 \beta_{2} - 48 \beta_1 + 150) q^{37} + (38 \beta_{2} - 86 \beta_1 + 120) q^{38} + 39 q^{39} + (24 \beta_{2} + 18 \beta_1 - 186) q^{40} + (34 \beta_{2} - 4 \beta_1 + 150) q^{41} + ( - 18 \beta_{2} - 42 \beta_1 - 156) q^{42} + (4 \beta_{2} - 60 \beta_1 - 68) q^{43} + ( - 10 \beta_{2} - 22 \beta_1 + 248) q^{44} + ( - 18 \beta_{2} + 18) q^{45} + ( - 48 \beta_{2} + 24 \beta_1 - 360) q^{46} + ( - 42 \beta_{2} + 54 \beta_1 - 12) q^{47} + ( - 15 \beta_{2} - 18 \beta_1 - 99) q^{48} + (36 \beta_{2} + 168 \beta_1 + 81) q^{49} + ( - 16 \beta_{2} + 41 \beta_1 + 263) q^{50} + ( - 24 \beta_1 - 138) q^{51} + (13 \beta_{2} + 39) q^{52} + ( - 12 \beta_{2} - 108 \beta_1 - 186) q^{53} + ( - 27 \beta_1 + 27) q^{54} + (68 \beta_{2} + 60 \beta_1 - 560) q^{55} + (10 \beta_{2} + 50 \beta_1 + 12) q^{56} + (48 \beta_{2} - 18 \beta_1 + 84) q^{57} + ( - 4 \beta_{2} - 42 \beta_1 - 194) q^{58} + (2 \beta_{2} + 82 \beta_1 - 624) q^{59} + (6 \beta_{2} + 36 \beta_1 - 258) q^{60} + ( - 28 \beta_{2} + 24 \beta_1 + 78) q^{61} + (46 \beta_{2} - 50 \beta_1 + 652) q^{62} + (54 \beta_1 + 72) q^{63} + ( - 51 \beta_{2} + 36 \beta_1 - 245) q^{64} + ( - 26 \beta_{2} + 26) q^{65} + (30 \beta_{2} - 54 \beta_1 - 48) q^{66} + ( - 76 \beta_{2} + 42 \beta_1 + 36) q^{67} + ( - 38 \beta_{2} - 56 \beta_1 - 122) q^{68} + ( - 24 \beta_{2} + 96 \beta_1 - 72) q^{69} + (28 \beta_{2} + 12 \beta_1 + 392) q^{70} + (14 \beta_{2} - 134 \beta_1 - 276) q^{71} + (18 \beta_{2} + 9 \beta_1 - 27) q^{72} + (12 \beta_{2} - 240 \beta_1 + 2) q^{73} + ( - 8 \beta_{2} + 10 \beta_1 + 574) q^{74} + ( - 60 \beta_{2} - 72 \beta_1 + 189) q^{75} + (34 \beta_{2} - 138 \beta_1 + 832) q^{76} + (24 \beta_{2} + 136 \beta_1 - 16) q^{77} + ( - 39 \beta_1 + 39) q^{78} + ( - 24 \beta_{2} + 48 \beta_1 - 16) q^{79} + (14 \beta_{2} - 24 \beta_1 + 370) q^{80} + 81 q^{81} + (72 \beta_{2} - 282 \beta_1 + 258) q^{82} + ( - 10 \beta_{2} + 30 \beta_1 - 272) q^{83} + (6 \beta_{2} + 126 \beta_1 + 36) q^{84} + (76 \beta_{2} + 48 \beta_1 - 124) q^{85} + (68 \beta_{2} + 112 \beta_1 + 540) q^{86} + (24 \beta_{2} + 60 \beta_1 - 30) q^{87} + ( - 78 \beta_{2} - 42 \beta_1 + 576) q^{88} + (30 \beta_{2} + 116 \beta_1 + 430) q^{89} + ( - 36 \beta_{2} + 54 \beta_1 - 18) q^{90} + (78 \beta_1 + 104) q^{91} + ( - 56 \beta_{2} + 272 \beta_1 - 504) q^{92} + ( - 12 \beta_{2} - 162 \beta_1 + 360) q^{93} + ( - 138 \beta_{2} + 126 \beta_1 - 636) q^{94} + (60 \beta_{2} + 228 \beta_1 - 1440) q^{95} + ( - 60 \beta_{2} + 153 \beta_1 + 123) q^{96} + ( - 4 \beta_{2} - 48 \beta_1 + 1098) q^{97} + ( - 96 \beta_{2} - 393 \beta_1 - 1527) q^{98} + (54 \beta_{2} + 18 \beta_1 - 72) q^{99}+O(q^{100})$$ q + (-b1 + 1) * q^2 + 3 * q^3 + (b2 + 3) * q^4 + (-2*b2 + 2) * q^5 + (-3*b1 + 3) * q^6 + (6*b1 + 8) * q^7 + (2*b2 + b1 - 3) * q^8 + 9 * q^9 + (-4*b2 + 6*b1 - 2) * q^10 + (6*b2 + 2*b1 - 8) * q^11 + (3*b2 + 9) * q^12 + 13 * q^13 + (-6*b2 - 14*b1 - 52) * q^14 + (-6*b2 + 6) * q^15 + (-5*b2 - 6*b1 - 33) * q^16 + (-8*b1 - 46) * q^17 + (-9*b1 + 9) * q^18 + (16*b2 - 6*b1 + 28) * q^19 + (2*b2 + 12*b1 - 86) * q^20 + (18*b1 + 24) * q^21 + (10*b2 - 18*b1 - 16) * q^22 + (-8*b2 + 32*b1 - 24) * q^23 + (6*b2 + 3*b1 - 9) * q^24 + (-20*b2 - 24*b1 + 63) * q^25 + (-13*b1 + 13) * q^26 + 27 * q^27 + (2*b2 + 42*b1 + 12) * q^28 + (8*b2 + 20*b1 - 10) * q^29 + (-12*b2 + 18*b1 - 6) * q^30 + (-4*b2 - 54*b1 + 120) * q^31 + (-20*b2 + 51*b1 + 41) * q^32 + (18*b2 + 6*b1 - 24) * q^33 + (8*b2 + 54*b1 + 34) * q^34 + (-4*b2 - 36*b1 + 40) * q^35 + (9*b2 + 27) * q^36 + (-28*b2 - 48*b1 + 150) * q^37 + (38*b2 - 86*b1 + 120) * q^38 + 39 * q^39 + (24*b2 + 18*b1 - 186) * q^40 + (34*b2 - 4*b1 + 150) * q^41 + (-18*b2 - 42*b1 - 156) * q^42 + (4*b2 - 60*b1 - 68) * q^43 + (-10*b2 - 22*b1 + 248) * q^44 + (-18*b2 + 18) * q^45 + (-48*b2 + 24*b1 - 360) * q^46 + (-42*b2 + 54*b1 - 12) * q^47 + (-15*b2 - 18*b1 - 99) * q^48 + (36*b2 + 168*b1 + 81) * q^49 + (-16*b2 + 41*b1 + 263) * q^50 + (-24*b1 - 138) * q^51 + (13*b2 + 39) * q^52 + (-12*b2 - 108*b1 - 186) * q^53 + (-27*b1 + 27) * q^54 + (68*b2 + 60*b1 - 560) * q^55 + (10*b2 + 50*b1 + 12) * q^56 + (48*b2 - 18*b1 + 84) * q^57 + (-4*b2 - 42*b1 - 194) * q^58 + (2*b2 + 82*b1 - 624) * q^59 + (6*b2 + 36*b1 - 258) * q^60 + (-28*b2 + 24*b1 + 78) * q^61 + (46*b2 - 50*b1 + 652) * q^62 + (54*b1 + 72) * q^63 + (-51*b2 + 36*b1 - 245) * q^64 + (-26*b2 + 26) * q^65 + (30*b2 - 54*b1 - 48) * q^66 + (-76*b2 + 42*b1 + 36) * q^67 + (-38*b2 - 56*b1 - 122) * q^68 + (-24*b2 + 96*b1 - 72) * q^69 + (28*b2 + 12*b1 + 392) * q^70 + (14*b2 - 134*b1 - 276) * q^71 + (18*b2 + 9*b1 - 27) * q^72 + (12*b2 - 240*b1 + 2) * q^73 + (-8*b2 + 10*b1 + 574) * q^74 + (-60*b2 - 72*b1 + 189) * q^75 + (34*b2 - 138*b1 + 832) * q^76 + (24*b2 + 136*b1 - 16) * q^77 + (-39*b1 + 39) * q^78 + (-24*b2 + 48*b1 - 16) * q^79 + (14*b2 - 24*b1 + 370) * q^80 + 81 * q^81 + (72*b2 - 282*b1 + 258) * q^82 + (-10*b2 + 30*b1 - 272) * q^83 + (6*b2 + 126*b1 + 36) * q^84 + (76*b2 + 48*b1 - 124) * q^85 + (68*b2 + 112*b1 + 540) * q^86 + (24*b2 + 60*b1 - 30) * q^87 + (-78*b2 - 42*b1 + 576) * q^88 + (30*b2 + 116*b1 + 430) * q^89 + (-36*b2 + 54*b1 - 18) * q^90 + (78*b1 + 104) * q^91 + (-56*b2 + 272*b1 - 504) * q^92 + (-12*b2 - 162*b1 + 360) * q^93 + (-138*b2 + 126*b1 - 636) * q^94 + (60*b2 + 228*b1 - 1440) * q^95 + (-60*b2 + 153*b1 + 123) * q^96 + (-4*b2 - 48*b1 + 1098) * q^97 + (-96*b2 - 393*b1 - 1527) * q^98 + (54*b2 + 18*b1 - 72) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9}+O(q^{10})$$ 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 $$3 q + 2 q^{2} + 9 q^{3} + 10 q^{4} + 4 q^{5} + 6 q^{6} + 30 q^{7} - 6 q^{8} + 27 q^{9} - 4 q^{10} - 16 q^{11} + 30 q^{12} + 39 q^{13} - 176 q^{14} + 12 q^{15} - 110 q^{16} - 146 q^{17} + 18 q^{18} + 94 q^{19} - 244 q^{20} + 90 q^{21} - 56 q^{22} - 48 q^{23} - 18 q^{24} + 145 q^{25} + 26 q^{26} + 81 q^{27} + 80 q^{28} - 2 q^{29} - 12 q^{30} + 302 q^{31} + 154 q^{32} - 48 q^{33} + 164 q^{34} + 80 q^{35} + 90 q^{36} + 374 q^{37} + 312 q^{38} + 117 q^{39} - 516 q^{40} + 480 q^{41} - 528 q^{42} - 260 q^{43} + 712 q^{44} + 36 q^{45} - 1104 q^{46} - 24 q^{47} - 330 q^{48} + 447 q^{49} + 814 q^{50} - 438 q^{51} + 130 q^{52} - 678 q^{53} + 54 q^{54} - 1552 q^{55} + 96 q^{56} + 282 q^{57} - 628 q^{58} - 1788 q^{59} - 732 q^{60} + 230 q^{61} + 1952 q^{62} + 270 q^{63} - 750 q^{64} + 52 q^{65} - 168 q^{66} + 74 q^{67} - 460 q^{68} - 144 q^{69} + 1216 q^{70} - 948 q^{71} - 54 q^{72} - 222 q^{73} + 1724 q^{74} + 435 q^{75} + 2392 q^{76} + 112 q^{77} + 78 q^{78} - 24 q^{79} + 1100 q^{80} + 243 q^{81} + 564 q^{82} - 796 q^{83} + 240 q^{84} - 248 q^{85} + 1800 q^{86} - 6 q^{87} + 1608 q^{88} + 1436 q^{89} - 36 q^{90} + 390 q^{91} - 1296 q^{92} + 906 q^{93} - 1920 q^{94} - 4032 q^{95} + 462 q^{96} + 3242 q^{97} - 5070 q^{98} - 144 q^{99}+O(q^{100})$$ 3 * q + 2 * q^2 + 9 * q^3 + 10 * q^4 + 4 * q^5 + 6 * q^6 + 30 * q^7 - 6 * q^8 + 27 * q^9 - 4 * q^10 - 16 * q^11 + 30 * q^12 + 39 * q^13 - 176 * q^14 + 12 * q^15 - 110 * q^16 - 146 * q^17 + 18 * q^18 + 94 * q^19 - 244 * q^20 + 90 * q^21 - 56 * q^22 - 48 * q^23 - 18 * q^24 + 145 * q^25 + 26 * q^26 + 81 * q^27 + 80 * q^28 - 2 * q^29 - 12 * q^30 + 302 * q^31 + 154 * q^32 - 48 * q^33 + 164 * q^34 + 80 * q^35 + 90 * q^36 + 374 * q^37 + 312 * q^38 + 117 * q^39 - 516 * q^40 + 480 * q^41 - 528 * q^42 - 260 * q^43 + 712 * q^44 + 36 * q^45 - 1104 * q^46 - 24 * q^47 - 330 * q^48 + 447 * q^49 + 814 * q^50 - 438 * q^51 + 130 * q^52 - 678 * q^53 + 54 * q^54 - 1552 * q^55 + 96 * q^56 + 282 * q^57 - 628 * q^58 - 1788 * q^59 - 732 * q^60 + 230 * q^61 + 1952 * q^62 + 270 * q^63 - 750 * q^64 + 52 * q^65 - 168 * q^66 + 74 * q^67 - 460 * q^68 - 144 * q^69 + 1216 * q^70 - 948 * q^71 - 54 * q^72 - 222 * q^73 + 1724 * q^74 + 435 * q^75 + 2392 * q^76 + 112 * q^77 + 78 * q^78 - 24 * q^79 + 1100 * q^80 + 243 * q^81 + 564 * q^82 - 796 * q^83 + 240 * q^84 - 248 * q^85 + 1800 * q^86 - 6 * q^87 + 1608 * q^88 + 1436 * q^89 - 36 * q^90 + 390 * q^91 - 1296 * q^92 + 906 * q^93 - 1920 * q^94 - 4032 * q^95 + 462 * q^96 + 3242 * q^97 - 5070 * q^98 - 144 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 16x - 8$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ 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
 4.73549 −0.526440 −3.20905
−3.73549 3.00000 5.95388 −3.90776 −11.2065 36.4129 7.64325 9.00000 14.5974
1.2 1.52644 3.00000 −5.66998 19.3400 4.57932 4.84136 −20.8664 9.00000 29.5213
1.3 4.20905 3.00000 9.71610 −11.4322 12.6271 −11.2543 7.22315 9.00000 −48.1187
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$13$$ $$-1$$

## 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 39.4.a.c 3
3.b odd 2 1 117.4.a.f 3
4.b odd 2 1 624.4.a.t 3
5.b even 2 1 975.4.a.l 3
7.b odd 2 1 1911.4.a.k 3
8.b even 2 1 2496.4.a.bl 3
8.d odd 2 1 2496.4.a.bp 3
12.b even 2 1 1872.4.a.bk 3
13.b even 2 1 507.4.a.h 3
13.d odd 4 2 507.4.b.g 6
39.d odd 2 1 1521.4.a.u 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.a.c 3 1.a even 1 1 trivial
117.4.a.f 3 3.b odd 2 1
507.4.a.h 3 13.b even 2 1
507.4.b.g 6 13.d odd 4 2
624.4.a.t 3 4.b odd 2 1
975.4.a.l 3 5.b even 2 1
1521.4.a.u 3 39.d odd 2 1
1872.4.a.bk 3 12.b even 2 1
1911.4.a.k 3 7.b odd 2 1
2496.4.a.bl 3 8.b even 2 1
2496.4.a.bp 3 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2T_{2}^{2} - 15T_{2} + 24$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(39))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 2 T^{2} - 15 T + 24$$
$3$ $$(T - 3)^{3}$$
$5$ $$T^{3} - 4 T^{2} - 252 T - 864$$
$7$ $$T^{3} - 30 T^{2} - 288 T + 1984$$
$11$ $$T^{3} + 16 T^{2} - 2256 T + 30336$$
$13$ $$(T - 13)^{3}$$
$17$ $$T^{3} + 146 T^{2} + 6060 T + 71256$$
$19$ $$T^{3} - 94 T^{2} - 14432 T + 779616$$
$23$ $$T^{3} + 48 T^{2} - 20928 T + 534528$$
$29$ $$T^{3} + 2 T^{2} - 10116 T - 199176$$
$31$ $$T^{3} - 302 T^{2} - 17536 T + 7197248$$
$37$ $$T^{3} - 374 T^{2} - 36964 T + 7758104$$
$41$ $$T^{3} - 480 T^{2} + \cdots + 12919824$$
$43$ $$T^{3} + 260 T^{2} - 38096 T - 3663168$$
$47$ $$T^{3} + 24 T^{2} - 168480 T + 18102528$$
$53$ $$T^{3} + 678 T^{2} - 42228 T - 1471608$$
$59$ $$T^{3} + 1788 T^{2} + \cdots + 137423808$$
$61$ $$T^{3} - 230 T^{2} - 44452 T + 6279512$$
$67$ $$T^{3} - 74 T^{2} - 409216 T + 4260896$$
$71$ $$T^{3} + 948 T^{2} + \cdots - 70464384$$
$73$ $$T^{3} + 222 T^{2} + \cdots + 22780552$$
$79$ $$T^{3} + 24 T^{2} - 78336 T + 7757824$$
$83$ $$T^{3} + 796 T^{2} + \cdots + 13963968$$
$89$ $$T^{3} - 1436 T^{2} + \cdots - 30129888$$
$97$ $$T^{3} - 3242 T^{2} + \cdots - 1218481048$$