# Properties

 Label 39.4.b.b Level $39$ Weight $4$ Character orbit 39.b Analytic conductor $2.301$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 39.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.1362828.1 Defining polynomial: $$x^{4} + 23x^{2} + 48$$ x^4 + 23*x^2 + 48 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + 3 q^{3} + (\beta_{3} - 4) q^{4} + ( - \beta_{2} + 2 \beta_1) q^{5} + 3 \beta_1 q^{6} + (2 \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{2} - 3 \beta_1) q^{8} + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 + 3 * q^3 + (b3 - 4) * q^4 + (-b2 + 2*b1) * q^5 + 3*b1 * q^6 + (2*b2 - 2*b1) * q^7 + (2*b2 - 3*b1) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} + 3 q^{3} + (\beta_{3} - 4) q^{4} + ( - \beta_{2} + 2 \beta_1) q^{5} + 3 \beta_1 q^{6} + (2 \beta_{2} - 2 \beta_1) q^{7} + (2 \beta_{2} - 3 \beta_1) q^{8} + 9 q^{9} + (4 \beta_{3} - 24) q^{10} + ( - 3 \beta_{2} - 8 \beta_1) q^{11} + (3 \beta_{3} - 12) q^{12} + ( - 4 \beta_{3} + \beta_{2} + 8 \beta_1 - 1) q^{13} + ( - 6 \beta_{3} + 24) q^{14} + ( - 3 \beta_{2} + 6 \beta_1) q^{15} + (\beta_{3} + 4) q^{16} + ( - 12 \beta_{3} + 30) q^{17} + 9 \beta_1 q^{18} + ( - 8 \beta_{2} - 10 \beta_1) q^{19} - 36 \beta_1 q^{20} + (6 \beta_{2} - 6 \beta_1) q^{21} + ( - 2 \beta_{3} + 96) q^{22} - 72 q^{23} + (6 \beta_{2} - 9 \beta_1) q^{24} + (4 \beta_{3} - 7) q^{25} + (6 \beta_{3} - 8 \beta_{2} + 27 \beta_1 - 96) q^{26} + 27 q^{27} + (4 \beta_{2} + 50 \beta_1) q^{28} + (12 \beta_{3} - 102) q^{29} + (12 \beta_{3} - 72) q^{30} + (20 \beta_{2} - 38 \beta_1) q^{31} + (18 \beta_{2} - 27 \beta_1) q^{32} + ( - 9 \beta_{2} - 24 \beta_1) q^{33} + ( - 24 \beta_{2} + 114 \beta_1) q^{34} + 216 q^{35} + (9 \beta_{3} - 36) q^{36} + ( - 4 \beta_{2} - 68 \beta_1) q^{37} + (6 \beta_{3} + 120) q^{38} + ( - 12 \beta_{3} + 3 \beta_{2} + 24 \beta_1 - 3) q^{39} + ( - 4 \beta_{3} + 240) q^{40} + (13 \beta_{2} + 66 \beta_1) q^{41} + ( - 18 \beta_{3} + 72) q^{42} + (12 \beta_{3} - 328) q^{43} + ( - 28 \beta_{2} + 46 \beta_1) q^{44} + ( - 9 \beta_{2} + 18 \beta_1) q^{45} - 72 \beta_1 q^{46} + (\beta_{2} + 36 \beta_1) q^{47} + (3 \beta_{3} + 12) q^{48} + ( - 12 \beta_{3} - 41) q^{49} + (8 \beta_{2} - 35 \beta_1) q^{50} + ( - 36 \beta_{3} + 90) q^{51} + (11 \beta_{3} + 20 \beta_{2} - 74 \beta_1 - 332) q^{52} + (48 \beta_{3} + 222) q^{53} + 27 \beta_1 q^{54} + ( - 44 \beta_{3} - 60) q^{55} + ( - 6 \beta_{3} - 408) q^{56} + ( - 24 \beta_{2} - 30 \beta_1) q^{57} + (24 \beta_{2} - 186 \beta_1) q^{58} + ( - \beta_{2} - 52 \beta_1) q^{59} - 108 \beta_1 q^{60} + (28 \beta_{3} + 58) q^{61} + ( - 78 \beta_{3} + 456) q^{62} + (18 \beta_{2} - 18 \beta_1) q^{63} + ( - 55 \beta_{3} + 356) q^{64} + (36 \beta_{3} + 17 \beta_{2} + 110 \beta_1 - 108) q^{65} + ( - 6 \beta_{3} + 288) q^{66} + ( - 54 \beta_{2} - 54 \beta_1) q^{67} + (66 \beta_{3} - 1128) q^{68} - 216 q^{69} + 216 \beta_1 q^{70} + ( - 41 \beta_{2} + 168 \beta_1) q^{71} + (18 \beta_{2} - 27 \beta_1) q^{72} + (42 \beta_{2} + 156 \beta_1) q^{73} + ( - 60 \beta_{3} + 816) q^{74} + (12 \beta_{3} - 21) q^{75} + ( - 52 \beta_{2} - 2 \beta_1) q^{76} + (84 \beta_{3} + 312) q^{77} + (18 \beta_{3} - 24 \beta_{2} + 81 \beta_1 - 288) q^{78} + (16 \beta_{3} + 1072) q^{79} + ( - 8 \beta_{2} - 20 \beta_1) q^{80} + 81 q^{81} + (40 \beta_{3} - 792) q^{82} + (63 \beta_{2} - 188 \beta_1) q^{83} + (12 \beta_{2} + 150 \beta_1) q^{84} + (18 \beta_{2} + 396 \beta_1) q^{85} + (24 \beta_{2} - 412 \beta_1) q^{86} + (36 \beta_{3} - 306) q^{87} + (86 \beta_{3} + 216) q^{88} + (25 \beta_{2} + 138 \beta_1) q^{89} + (36 \beta_{3} - 216) q^{90} + ( - 60 \beta_{3} - 50 \beta_{2} - 166 \beta_1 + 24) q^{91} + ( - 72 \beta_{3} + 288) q^{92} + (60 \beta_{2} - 114 \beta_1) q^{93} + (34 \beta_{3} - 432) q^{94} + ( - 72 \beta_{3} - 432) q^{95} + (54 \beta_{2} - 81 \beta_1) q^{96} + (14 \beta_{2} - 68 \beta_1) q^{97} + ( - 24 \beta_{2} + 43 \beta_1) q^{98} + ( - 27 \beta_{2} - 72 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 + 3 * q^3 + (b3 - 4) * q^4 + (-b2 + 2*b1) * q^5 + 3*b1 * q^6 + (2*b2 - 2*b1) * q^7 + (2*b2 - 3*b1) * q^8 + 9 * q^9 + (4*b3 - 24) * q^10 + (-3*b2 - 8*b1) * q^11 + (3*b3 - 12) * q^12 + (-4*b3 + b2 + 8*b1 - 1) * q^13 + (-6*b3 + 24) * q^14 + (-3*b2 + 6*b1) * q^15 + (b3 + 4) * q^16 + (-12*b3 + 30) * q^17 + 9*b1 * q^18 + (-8*b2 - 10*b1) * q^19 - 36*b1 * q^20 + (6*b2 - 6*b1) * q^21 + (-2*b3 + 96) * q^22 - 72 * q^23 + (6*b2 - 9*b1) * q^24 + (4*b3 - 7) * q^25 + (6*b3 - 8*b2 + 27*b1 - 96) * q^26 + 27 * q^27 + (4*b2 + 50*b1) * q^28 + (12*b3 - 102) * q^29 + (12*b3 - 72) * q^30 + (20*b2 - 38*b1) * q^31 + (18*b2 - 27*b1) * q^32 + (-9*b2 - 24*b1) * q^33 + (-24*b2 + 114*b1) * q^34 + 216 * q^35 + (9*b3 - 36) * q^36 + (-4*b2 - 68*b1) * q^37 + (6*b3 + 120) * q^38 + (-12*b3 + 3*b2 + 24*b1 - 3) * q^39 + (-4*b3 + 240) * q^40 + (13*b2 + 66*b1) * q^41 + (-18*b3 + 72) * q^42 + (12*b3 - 328) * q^43 + (-28*b2 + 46*b1) * q^44 + (-9*b2 + 18*b1) * q^45 - 72*b1 * q^46 + (b2 + 36*b1) * q^47 + (3*b3 + 12) * q^48 + (-12*b3 - 41) * q^49 + (8*b2 - 35*b1) * q^50 + (-36*b3 + 90) * q^51 + (11*b3 + 20*b2 - 74*b1 - 332) * q^52 + (48*b3 + 222) * q^53 + 27*b1 * q^54 + (-44*b3 - 60) * q^55 + (-6*b3 - 408) * q^56 + (-24*b2 - 30*b1) * q^57 + (24*b2 - 186*b1) * q^58 + (-b2 - 52*b1) * q^59 - 108*b1 * q^60 + (28*b3 + 58) * q^61 + (-78*b3 + 456) * q^62 + (18*b2 - 18*b1) * q^63 + (-55*b3 + 356) * q^64 + (36*b3 + 17*b2 + 110*b1 - 108) * q^65 + (-6*b3 + 288) * q^66 + (-54*b2 - 54*b1) * q^67 + (66*b3 - 1128) * q^68 - 216 * q^69 + 216*b1 * q^70 + (-41*b2 + 168*b1) * q^71 + (18*b2 - 27*b1) * q^72 + (42*b2 + 156*b1) * q^73 + (-60*b3 + 816) * q^74 + (12*b3 - 21) * q^75 + (-52*b2 - 2*b1) * q^76 + (84*b3 + 312) * q^77 + (18*b3 - 24*b2 + 81*b1 - 288) * q^78 + (16*b3 + 1072) * q^79 + (-8*b2 - 20*b1) * q^80 + 81 * q^81 + (40*b3 - 792) * q^82 + (63*b2 - 188*b1) * q^83 + (12*b2 + 150*b1) * q^84 + (18*b2 + 396*b1) * q^85 + (24*b2 - 412*b1) * q^86 + (36*b3 - 306) * q^87 + (86*b3 + 216) * q^88 + (25*b2 + 138*b1) * q^89 + (36*b3 - 216) * q^90 + (-60*b3 - 50*b2 - 166*b1 + 24) * q^91 + (-72*b3 + 288) * q^92 + (60*b2 - 114*b1) * q^93 + (34*b3 - 432) * q^94 + (-72*b3 - 432) * q^95 + (54*b2 - 81*b1) * q^96 + (14*b2 - 68*b1) * q^97 + (-24*b2 + 43*b1) * q^98 + (-27*b2 - 72*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{3} - 14 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q + 12 * q^3 - 14 * q^4 + 36 * q^9 $$4 q + 12 q^{3} - 14 q^{4} + 36 q^{9} - 88 q^{10} - 42 q^{12} - 12 q^{13} + 84 q^{14} + 18 q^{16} + 96 q^{17} + 380 q^{22} - 288 q^{23} - 20 q^{25} - 372 q^{26} + 108 q^{27} - 384 q^{29} - 264 q^{30} + 864 q^{35} - 126 q^{36} + 492 q^{38} - 36 q^{39} + 952 q^{40} + 252 q^{42} - 1288 q^{43} + 54 q^{48} - 188 q^{49} + 288 q^{51} - 1306 q^{52} + 984 q^{53} - 328 q^{55} - 1644 q^{56} + 288 q^{61} + 1668 q^{62} + 1314 q^{64} - 360 q^{65} + 1140 q^{66} - 4380 q^{68} - 864 q^{69} + 3144 q^{74} - 60 q^{75} + 1416 q^{77} - 1116 q^{78} + 4320 q^{79} + 324 q^{81} - 3088 q^{82} - 1152 q^{87} + 1036 q^{88} - 792 q^{90} - 24 q^{91} + 1008 q^{92} - 1660 q^{94} - 1872 q^{95}+O(q^{100})$$ 4 * q + 12 * q^3 - 14 * q^4 + 36 * q^9 - 88 * q^10 - 42 * q^12 - 12 * q^13 + 84 * q^14 + 18 * q^16 + 96 * q^17 + 380 * q^22 - 288 * q^23 - 20 * q^25 - 372 * q^26 + 108 * q^27 - 384 * q^29 - 264 * q^30 + 864 * q^35 - 126 * q^36 + 492 * q^38 - 36 * q^39 + 952 * q^40 + 252 * q^42 - 1288 * q^43 + 54 * q^48 - 188 * q^49 + 288 * q^51 - 1306 * q^52 + 984 * q^53 - 328 * q^55 - 1644 * q^56 + 288 * q^61 + 1668 * q^62 + 1314 * q^64 - 360 * q^65 + 1140 * q^66 - 4380 * q^68 - 864 * q^69 + 3144 * q^74 - 60 * q^75 + 1416 * q^77 - 1116 * q^78 + 4320 * q^79 + 324 * q^81 - 3088 * q^82 - 1152 * q^87 + 1036 * q^88 - 792 * q^90 - 24 * q^91 + 1008 * q^92 - 1660 * q^94 - 1872 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 23x^{2} + 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 19\nu ) / 2$$ (v^3 + 19*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 12$$ v^2 + 12
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 12$$ b3 - 12 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 19\beta_1$$ 2*b2 - 19*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$$.

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 − 4.54739i − 1.52356i 1.52356i 4.54739i
4.54739i 3.00000 −12.6788 12.9118i 13.6422i 16.7289i 21.2762i 9.00000 −58.7151
25.2 1.52356i 3.00000 5.67878 9.65841i 4.57067i 22.3639i 20.8404i 9.00000 14.7151
25.3 1.52356i 3.00000 5.67878 9.65841i 4.57067i 22.3639i 20.8404i 9.00000 14.7151
25.4 4.54739i 3.00000 −12.6788 12.9118i 13.6422i 16.7289i 21.2762i 9.00000 −58.7151
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.4.b.b 4
3.b odd 2 1 117.4.b.e 4
4.b odd 2 1 624.4.c.c 4
13.b even 2 1 inner 39.4.b.b 4
13.d odd 4 2 507.4.a.l 4
39.d odd 2 1 117.4.b.e 4
39.f even 4 2 1521.4.a.w 4
52.b odd 2 1 624.4.c.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.b.b 4 1.a even 1 1 trivial
39.4.b.b 4 13.b even 2 1 inner
117.4.b.e 4 3.b odd 2 1
117.4.b.e 4 39.d odd 2 1
507.4.a.l 4 13.d odd 4 2
624.4.c.c 4 4.b odd 2 1
624.4.c.c 4 52.b odd 2 1
1521.4.a.w 4 39.f even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 23T_{2}^{2} + 48$$ acting on $$S_{4}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 23T^{2} + 48$$
$3$ $$(T - 3)^{4}$$
$5$ $$T^{4} + 260 T^{2} + 15552$$
$7$ $$T^{4} + 780 T^{2} + 139968$$
$11$ $$T^{4} + 3152 T^{2} + \cdots + 1572528$$
$13$ $$T^{4} + 12 T^{3} - 962 T^{2} + \cdots + 4826809$$
$17$ $$(T^{2} - 48 T - 11556)^{2}$$
$19$ $$T^{4} + 13884 T^{2} + \cdots + 3048192$$
$23$ $$(T + 72)^{4}$$
$29$ $$(T^{2} + 192 T - 2916)^{2}$$
$31$ $$T^{4} + 100572 T^{2} + \cdots + 2389782528$$
$37$ $$T^{4} + 110256 T^{2} + \cdots + 2060577792$$
$41$ $$T^{4} + 133364 T^{2} + \cdots + 4431055872$$
$43$ $$(T^{2} + 644 T + 91552)^{2}$$
$47$ $$T^{4} + 30128 T^{2} + \cdots + 116663088$$
$53$ $$(T^{2} - 492 T - 133596)^{2}$$
$59$ $$T^{4} + 62576 T^{2} + \cdots + 457419312$$
$61$ $$(T^{2} - 144 T - 60868)^{2}$$
$67$ $$T^{4} + 591948 T^{2} + \cdots + 918330048$$
$71$ $$T^{4} + 917456 T^{2} + \cdots + 59484058032$$
$73$ $$T^{4} + 896400 T^{2} + \cdots + 179358354432$$
$79$ $$(T^{2} - 2160 T + 1144832)^{2}$$
$83$ $$T^{4} + 1464080 T^{2} + \cdots + 317023217328$$
$89$ $$T^{4} + 561812 T^{2} + \cdots + 78903164928$$
$97$ $$T^{4} + 137040 T^{2} + \cdots + 725594112$$