# Properties

 Label 39.2.e.b Level $39$ Weight $2$ Character orbit 39.e Analytic conductor $0.311$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.311416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{17})$$ Defining polynomial: $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 5 x^{2} + 4 x + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu^{2} - 5 \nu + 16$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 4 \beta_{2} + \beta_{1} - 4$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - 4$$

## Character values

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

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1
 1.28078 + 2.21837i −0.780776 − 1.35234i 1.28078 − 2.21837i −0.780776 + 1.35234i
−1.28078 2.21837i −0.500000 0.866025i −2.28078 + 3.95042i 0.561553 −1.28078 + 2.21837i 1.78078 3.08440i 6.56155 −0.500000 + 0.866025i −0.719224 1.24573i
16.2 0.780776 + 1.35234i −0.500000 0.866025i −0.219224 + 0.379706i −3.56155 0.780776 1.35234i −0.280776 + 0.486319i 2.43845 −0.500000 + 0.866025i −2.78078 4.81645i
22.1 −1.28078 + 2.21837i −0.500000 + 0.866025i −2.28078 3.95042i 0.561553 −1.28078 2.21837i 1.78078 + 3.08440i 6.56155 −0.500000 0.866025i −0.719224 + 1.24573i
22.2 0.780776 1.35234i −0.500000 + 0.866025i −0.219224 0.379706i −3.56155 0.780776 + 1.35234i −0.280776 0.486319i 2.43845 −0.500000 0.866025i −2.78078 + 4.81645i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.e.b 4
3.b odd 2 1 117.2.g.c 4
4.b odd 2 1 624.2.q.h 4
5.b even 2 1 975.2.i.k 4
5.c odd 4 2 975.2.bb.i 8
12.b even 2 1 1872.2.t.r 4
13.b even 2 1 507.2.e.g 4
13.c even 3 1 inner 39.2.e.b 4
13.c even 3 1 507.2.a.g 2
13.d odd 4 2 507.2.j.g 8
13.e even 6 1 507.2.a.d 2
13.e even 6 1 507.2.e.g 4
13.f odd 12 2 507.2.b.d 4
13.f odd 12 2 507.2.j.g 8
39.h odd 6 1 1521.2.a.m 2
39.i odd 6 1 117.2.g.c 4
39.i odd 6 1 1521.2.a.g 2
39.k even 12 2 1521.2.b.h 4
52.i odd 6 1 8112.2.a.bo 2
52.j odd 6 1 624.2.q.h 4
52.j odd 6 1 8112.2.a.bk 2
65.n even 6 1 975.2.i.k 4
65.q odd 12 2 975.2.bb.i 8
156.p even 6 1 1872.2.t.r 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.e.b 4 1.a even 1 1 trivial
39.2.e.b 4 13.c even 3 1 inner
117.2.g.c 4 3.b odd 2 1
117.2.g.c 4 39.i odd 6 1
507.2.a.d 2 13.e even 6 1
507.2.a.g 2 13.c even 3 1
507.2.b.d 4 13.f odd 12 2
507.2.e.g 4 13.b even 2 1
507.2.e.g 4 13.e even 6 1
507.2.j.g 8 13.d odd 4 2
507.2.j.g 8 13.f odd 12 2
624.2.q.h 4 4.b odd 2 1
624.2.q.h 4 52.j odd 6 1
975.2.i.k 4 5.b even 2 1
975.2.i.k 4 65.n even 6 1
975.2.bb.i 8 5.c odd 4 2
975.2.bb.i 8 65.q odd 12 2
1521.2.a.g 2 39.i odd 6 1
1521.2.a.m 2 39.h odd 6 1
1521.2.b.h 4 39.k even 12 2
1872.2.t.r 4 12.b even 2 1
1872.2.t.r 4 156.p even 6 1
8112.2.a.bk 2 52.j odd 6 1
8112.2.a.bo 2 52.i odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + T_{2}^{3} + 5 T_{2}^{2} - 4 T_{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( -2 + 3 T + T^{2} )^{2}$$
$7$ $$4 + 6 T + 11 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$( 4 - 2 T + T^{2} )^{2}$$
$13$ $$( 13 - T + T^{2} )^{2}$$
$17$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$19$ $$64 - 48 T + 44 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$( 4 + 2 T + T^{2} )^{2}$$
$29$ $$1444 - 38 T + 39 T^{2} + T^{3} + T^{4}$$
$31$ $$( -4 - T + T^{2} )^{2}$$
$37$ $$676 + 286 T + 95 T^{2} + 11 T^{3} + T^{4}$$
$41$ $$16 - 4 T + 5 T^{2} + T^{3} + T^{4}$$
$43$ $$4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4}$$
$47$ $$( -68 + T^{2} )^{2}$$
$53$ $$( -8 - 11 T + T^{2} )^{2}$$
$59$ $$1024 - 448 T + 164 T^{2} - 14 T^{3} + T^{4}$$
$61$ $$2209 + 752 T + 209 T^{2} + 16 T^{3} + T^{4}$$
$67$ $$4 + 10 T + 23 T^{2} + 5 T^{3} + T^{4}$$
$71$ $$( 196 + 14 T + T^{2} )^{2}$$
$73$ $$( 19 + 12 T + T^{2} )^{2}$$
$79$ $$( 52 - 15 T + T^{2} )^{2}$$
$83$ $$( 8 + 10 T + T^{2} )^{2}$$
$89$ $$4096 + 1152 T + 260 T^{2} + 18 T^{3} + T^{4}$$
$97$ $$1444 - 494 T + 131 T^{2} - 13 T^{3} + T^{4}$$