# Properties

 Label 39.4.b.a 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: 4.0.5054412.1 Defining polynomial: $$x^{4} + 29x^{2} + 48$$ x^4 + 29*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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{3} - 7) q^{4} + \beta_{2} q^{5} - 3 \beta_1 q^{6} + 6 \beta_1 q^{7} + (2 \beta_{2} - 9 \beta_1) q^{8} + 9 q^{9}+O(q^{10})$$ q + b1 * q^2 - 3 * q^3 + (b3 - 7) * q^4 + b2 * q^5 - 3*b1 * q^6 + 6*b1 * q^7 + (2*b2 - 9*b1) * q^8 + 9 * q^9 $$q + \beta_1 q^{2} - 3 q^{3} + (\beta_{3} - 7) q^{4} + \beta_{2} q^{5} - 3 \beta_1 q^{6} + 6 \beta_1 q^{7} + (2 \beta_{2} - 9 \beta_1) q^{8} + 9 q^{9} + ( - 2 \beta_{3} + 6) q^{10} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - 3 \beta_{3} + 21) q^{12} + (2 \beta_{3} - 3 \beta_{2} - 19) q^{13} + (6 \beta_{3} - 90) q^{14} - 3 \beta_{2} q^{15} + ( - 5 \beta_{3} + 91) q^{16} + 54 q^{17} + 9 \beta_1 q^{18} + ( - 6 \beta_{2} + 6 \beta_1) q^{19} + (4 \beta_{2} + 26 \beta_1) q^{20} - 18 \beta_1 q^{21} + (4 \beta_{3} - 36) q^{22} + ( - 12 \beta_{3} + 36) q^{23} + ( - 6 \beta_{2} + 27 \beta_1) q^{24} + ( - 8 \beta_{3} - 7) q^{25} + (6 \beta_{3} + 4 \beta_{2} - 39 \beta_1 - 18) q^{26} - 27 q^{27} + (12 \beta_{2} - 102 \beta_1) q^{28} + (12 \beta_{3} - 18) q^{29} + (6 \beta_{3} - 18) q^{30} + ( - 6 \beta_{2} - 18 \beta_1) q^{31} + (6 \beta_{2} + 69 \beta_1) q^{32} + (3 \beta_{2} - 6 \beta_1) q^{33} + 54 \beta_1 q^{34} + ( - 12 \beta_{3} + 36) q^{35} + (9 \beta_{3} - 63) q^{36} + (24 \beta_{2} + 48 \beta_1) q^{37} + (18 \beta_{3} - 126) q^{38} + ( - 6 \beta_{3} + 9 \beta_{2} + 57) q^{39} + (2 \beta_{3} - 318) q^{40} + ( - 13 \beta_{2} - 4 \beta_1) q^{41} + ( - 18 \beta_{3} + 270) q^{42} + (12 \beta_{3} + 260) q^{43} - 60 \beta_1 q^{44} + 9 \beta_{2} q^{45} + ( - 24 \beta_{2} + 156 \beta_1) q^{46} + ( - 21 \beta_{2} - 122 \beta_1) q^{47} + (15 \beta_{3} - 273) q^{48} + (36 \beta_{3} - 197) q^{49} + ( - 16 \beta_{2} + 73 \beta_1) q^{50} - 162 q^{51} + ( - 31 \beta_{3} - 12 \beta_{2} - 78 \beta_1 + 457) q^{52} + ( - 36 \beta_{3} - 198) q^{53} - 27 \beta_1 q^{54} + (4 \beta_{3} + 144) q^{55} + ( - 78 \beta_{3} + 882) q^{56} + (18 \beta_{2} - 18 \beta_1) q^{57} + (24 \beta_{2} - 138 \beta_1) q^{58} + ( - 15 \beta_{2} + 34 \beta_1) q^{59} + ( - 12 \beta_{2} - 78 \beta_1) q^{60} + ( - 8 \beta_{3} - 566) q^{61} + ( - 6 \beta_{3} + 234) q^{62} + 54 \beta_1 q^{63} + (17 \beta_{3} - 271) q^{64} + (24 \beta_{3} + 3 \beta_{2} + 52 \beta_1 + 396) q^{65} + ( - 12 \beta_{3} + 108) q^{66} + ( - 12 \beta_{2} + 150 \beta_1) q^{67} + (54 \beta_{3} - 378) q^{68} + (36 \beta_{3} - 108) q^{69} + ( - 24 \beta_{2} + 156 \beta_1) q^{70} + ( - 23 \beta_{2} + 6 \beta_1) q^{71} + (18 \beta_{2} - 81 \beta_1) q^{72} + 54 \beta_{2} q^{73} - 576 q^{74} + (24 \beta_{3} + 21) q^{75} + ( - 12 \beta_{2} - 258 \beta_1) q^{76} + (24 \beta_{3} - 216) q^{77} + ( - 18 \beta_{3} - 12 \beta_{2} + 117 \beta_1 + 54) q^{78} + ( - 32 \beta_{3} + 88) q^{79} + (36 \beta_{2} - 130 \beta_1) q^{80} + 81 q^{81} + (22 \beta_{3} - 18) q^{82} + (25 \beta_{2} + 138 \beta_1) q^{83} + ( - 36 \beta_{2} + 306 \beta_1) q^{84} + 54 \beta_{2} q^{85} + (24 \beta_{2} + 140 \beta_1) q^{86} + ( - 36 \beta_{3} + 54) q^{87} + ( - 28 \beta_{3} + 612) q^{88} + (15 \beta_{2} + 4 \beta_1) q^{89} + ( - 18 \beta_{3} + 54) q^{90} + (36 \beta_{3} + 24 \beta_{2} - 234 \beta_1 - 108) q^{91} + (108 \beta_{3} - 2196) q^{92} + (18 \beta_{2} + 54 \beta_1) q^{93} + ( - 80 \beta_{3} + 1704) q^{94} + (36 \beta_{3} + 828) q^{95} + ( - 18 \beta_{2} - 207 \beta_1) q^{96} + ( - 54 \beta_{2} - 336 \beta_1) q^{97} + (72 \beta_{2} - 557 \beta_1) q^{98} + ( - 9 \beta_{2} + 18 \beta_1) q^{99}+O(q^{100})$$ q + b1 * q^2 - 3 * q^3 + (b3 - 7) * q^4 + b2 * q^5 - 3*b1 * q^6 + 6*b1 * q^7 + (2*b2 - 9*b1) * q^8 + 9 * q^9 + (-2*b3 + 6) * q^10 + (-b2 + 2*b1) * q^11 + (-3*b3 + 21) * q^12 + (2*b3 - 3*b2 - 19) * q^13 + (6*b3 - 90) * q^14 - 3*b2 * q^15 + (-5*b3 + 91) * q^16 + 54 * q^17 + 9*b1 * q^18 + (-6*b2 + 6*b1) * q^19 + (4*b2 + 26*b1) * q^20 - 18*b1 * q^21 + (4*b3 - 36) * q^22 + (-12*b3 + 36) * q^23 + (-6*b2 + 27*b1) * q^24 + (-8*b3 - 7) * q^25 + (6*b3 + 4*b2 - 39*b1 - 18) * q^26 - 27 * q^27 + (12*b2 - 102*b1) * q^28 + (12*b3 - 18) * q^29 + (6*b3 - 18) * q^30 + (-6*b2 - 18*b1) * q^31 + (6*b2 + 69*b1) * q^32 + (3*b2 - 6*b1) * q^33 + 54*b1 * q^34 + (-12*b3 + 36) * q^35 + (9*b3 - 63) * q^36 + (24*b2 + 48*b1) * q^37 + (18*b3 - 126) * q^38 + (-6*b3 + 9*b2 + 57) * q^39 + (2*b3 - 318) * q^40 + (-13*b2 - 4*b1) * q^41 + (-18*b3 + 270) * q^42 + (12*b3 + 260) * q^43 - 60*b1 * q^44 + 9*b2 * q^45 + (-24*b2 + 156*b1) * q^46 + (-21*b2 - 122*b1) * q^47 + (15*b3 - 273) * q^48 + (36*b3 - 197) * q^49 + (-16*b2 + 73*b1) * q^50 - 162 * q^51 + (-31*b3 - 12*b2 - 78*b1 + 457) * q^52 + (-36*b3 - 198) * q^53 - 27*b1 * q^54 + (4*b3 + 144) * q^55 + (-78*b3 + 882) * q^56 + (18*b2 - 18*b1) * q^57 + (24*b2 - 138*b1) * q^58 + (-15*b2 + 34*b1) * q^59 + (-12*b2 - 78*b1) * q^60 + (-8*b3 - 566) * q^61 + (-6*b3 + 234) * q^62 + 54*b1 * q^63 + (17*b3 - 271) * q^64 + (24*b3 + 3*b2 + 52*b1 + 396) * q^65 + (-12*b3 + 108) * q^66 + (-12*b2 + 150*b1) * q^67 + (54*b3 - 378) * q^68 + (36*b3 - 108) * q^69 + (-24*b2 + 156*b1) * q^70 + (-23*b2 + 6*b1) * q^71 + (18*b2 - 81*b1) * q^72 + 54*b2 * q^73 - 576 * q^74 + (24*b3 + 21) * q^75 + (-12*b2 - 258*b1) * q^76 + (24*b3 - 216) * q^77 + (-18*b3 - 12*b2 + 117*b1 + 54) * q^78 + (-32*b3 + 88) * q^79 + (36*b2 - 130*b1) * q^80 + 81 * q^81 + (22*b3 - 18) * q^82 + (25*b2 + 138*b1) * q^83 + (-36*b2 + 306*b1) * q^84 + 54*b2 * q^85 + (24*b2 + 140*b1) * q^86 + (-36*b3 + 54) * q^87 + (-28*b3 + 612) * q^88 + (15*b2 + 4*b1) * q^89 + (-18*b3 + 54) * q^90 + (36*b3 + 24*b2 - 234*b1 - 108) * q^91 + (108*b3 - 2196) * q^92 + (18*b2 + 54*b1) * q^93 + (-80*b3 + 1704) * q^94 + (36*b3 + 828) * q^95 + (-18*b2 - 207*b1) * q^96 + (-54*b2 - 336*b1) * q^97 + (72*b2 - 557*b1) * q^98 + (-9*b2 + 18*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 12 q^{3} - 26 q^{4} + 36 q^{9}+O(q^{10})$$ 4 * q - 12 * q^3 - 26 * q^4 + 36 * q^9 $$4 q - 12 q^{3} - 26 q^{4} + 36 q^{9} + 20 q^{10} + 78 q^{12} - 72 q^{13} - 348 q^{14} + 354 q^{16} + 216 q^{17} - 136 q^{22} + 120 q^{23} - 44 q^{25} - 60 q^{26} - 108 q^{27} - 48 q^{29} - 60 q^{30} + 120 q^{35} - 234 q^{36} - 468 q^{38} + 216 q^{39} - 1268 q^{40} + 1044 q^{42} + 1064 q^{43} - 1062 q^{48} - 716 q^{49} - 648 q^{51} + 1766 q^{52} - 864 q^{53} + 584 q^{55} + 3372 q^{56} - 2280 q^{61} + 924 q^{62} - 1050 q^{64} + 1632 q^{65} + 408 q^{66} - 1404 q^{68} - 360 q^{69} - 2304 q^{74} + 132 q^{75} - 816 q^{77} + 180 q^{78} + 288 q^{79} + 324 q^{81} - 28 q^{82} + 144 q^{87} + 2392 q^{88} + 180 q^{90} - 360 q^{91} - 8568 q^{92} + 6656 q^{94} + 3384 q^{95}+O(q^{100})$$ 4 * q - 12 * q^3 - 26 * q^4 + 36 * q^9 + 20 * q^10 + 78 * q^12 - 72 * q^13 - 348 * q^14 + 354 * q^16 + 216 * q^17 - 136 * q^22 + 120 * q^23 - 44 * q^25 - 60 * q^26 - 108 * q^27 - 48 * q^29 - 60 * q^30 + 120 * q^35 - 234 * q^36 - 468 * q^38 + 216 * q^39 - 1268 * q^40 + 1044 * q^42 + 1064 * q^43 - 1062 * q^48 - 716 * q^49 - 648 * q^51 + 1766 * q^52 - 864 * q^53 + 584 * q^55 + 3372 * q^56 - 2280 * q^61 + 924 * q^62 - 1050 * q^64 + 1632 * q^65 + 408 * q^66 - 1404 * q^68 - 360 * q^69 - 2304 * q^74 + 132 * q^75 - 816 * q^77 + 180 * q^78 + 288 * q^79 + 324 * q^81 - 28 * q^82 + 144 * q^87 + 2392 * q^88 + 180 * q^90 - 360 * q^91 - 8568 * q^92 + 6656 * q^94 + 3384 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 29x^{2} + 48$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 25\nu ) / 2$$ (v^3 + 25*v) / 2 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 15$$ v^2 + 15
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 15$$ b3 - 15 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 25\beta_1$$ 2*b2 - 25*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
 − 5.21898i − 1.32750i 1.32750i 5.21898i
5.21898i −3.00000 −19.2377 5.83936i 15.6569i 31.3139i 58.6495i 9.00000 30.4755
25.2 1.32750i −3.00000 6.23774 15.4241i 3.98251i 7.96501i 18.9006i 9.00000 −20.4755
25.3 1.32750i −3.00000 6.23774 15.4241i 3.98251i 7.96501i 18.9006i 9.00000 −20.4755
25.4 5.21898i −3.00000 −19.2377 5.83936i 15.6569i 31.3139i 58.6495i 9.00000 30.4755
 $$n$$: 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.a 4
3.b odd 2 1 117.4.b.d 4
4.b odd 2 1 624.4.c.e 4
13.b even 2 1 inner 39.4.b.a 4
13.d odd 4 2 507.4.a.j 4
39.d odd 2 1 117.4.b.d 4
39.f even 4 2 1521.4.a.x 4
52.b odd 2 1 624.4.c.e 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 29T_{2}^{2} + 48$$ acting on $$S_{4}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 29T^{2} + 48$$
$3$ $$(T + 3)^{4}$$
$5$ $$T^{4} + 272T^{2} + 8112$$
$7$ $$T^{4} + 1044 T^{2} + 62208$$
$11$ $$T^{4} + 428 T^{2} + 43200$$
$13$ $$T^{4} + 72 T^{3} + 3094 T^{2} + \cdots + 4826809$$
$17$ $$(T - 54)^{4}$$
$19$ $$T^{4} + 11556 T^{2} + \cdots + 31492800$$
$23$ $$(T^{2} - 60 T - 22464)^{2}$$
$29$ $$(T^{2} + 24 T - 23220)^{2}$$
$31$ $$T^{4} + 17028 T^{2} + \cdots + 47044800$$
$37$ $$T^{4} + 200448 T^{2} + \cdots + 2293235712$$
$41$ $$T^{4} + 45392 T^{2} + \cdots + 128314800$$
$43$ $$(T^{2} - 532 T + 47392)^{2}$$
$47$ $$T^{4} + 500348 T^{2} + \cdots + 62387841792$$
$53$ $$(T^{2} + 432 T - 163620)^{2}$$
$59$ $$T^{4} + 104924 T^{2} + \cdots + 2436066048$$
$61$ $$(T^{2} + 1140 T + 314516)^{2}$$
$67$ $$T^{4} + 727668 T^{2} + \cdots + 143327232$$
$71$ $$T^{4} + 147692 T^{2} + \cdots + 3298756800$$
$73$ $$T^{4} + 793152 T^{2} + \cdots + 68976790272$$
$79$ $$(T^{2} - 144 T - 160960)^{2}$$
$83$ $$T^{4} + 653276 T^{2} + \cdots + 106682723328$$
$89$ $$T^{4} + 60464 T^{2} + \cdots + 249304368$$
$97$ $$T^{4} + 3704256 T^{2} + \cdots + 3383532000000$$