# Properties

 Label 39.2.e.a Level $39$ Weight $2$ Character orbit 39.e Analytic conductor $0.311$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i 0.500000 0.866025i −1.00000 0.500000 0.866025i −1.00000 + 1.73205i −3.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
22.1 −0.500000 + 0.866025i 0.500000 0.866025i 0.500000 + 0.866025i −1.00000 0.500000 + 0.866025i −1.00000 1.73205i −3.00000 −0.500000 0.866025i 0.500000 0.866025i
 $$n$$: 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.a 2
3.b odd 2 1 117.2.g.b 2
4.b odd 2 1 624.2.q.c 2
5.b even 2 1 975.2.i.f 2
5.c odd 4 2 975.2.bb.d 4
12.b even 2 1 1872.2.t.j 2
13.b even 2 1 507.2.e.c 2
13.c even 3 1 inner 39.2.e.a 2
13.c even 3 1 507.2.a.c 1
13.d odd 4 2 507.2.j.d 4
13.e even 6 1 507.2.a.b 1
13.e even 6 1 507.2.e.c 2
13.f odd 12 2 507.2.b.b 2
13.f odd 12 2 507.2.j.d 4
39.h odd 6 1 1521.2.a.d 1
39.i odd 6 1 117.2.g.b 2
39.i odd 6 1 1521.2.a.a 1
39.k even 12 2 1521.2.b.c 2
52.i odd 6 1 8112.2.a.bc 1
52.j odd 6 1 624.2.q.c 2
52.j odd 6 1 8112.2.a.w 1
65.n even 6 1 975.2.i.f 2
65.q odd 12 2 975.2.bb.d 4
156.p even 6 1 1872.2.t.j 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - T + 1$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} + 2T + 4$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$T^{2} + 7T + 13$$
$17$ $$T^{2} - 7T + 49$$
$19$ $$T^{2} - 6T + 36$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} - T + 1$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} + T + 1$$
$41$ $$T^{2} + 9T + 81$$
$43$ $$T^{2} + 6T + 36$$
$47$ $$(T - 6)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + T + 1$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$(T - 11)^{2}$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T + 14)^{2}$$
$89$ $$T^{2} - 14T + 196$$
$97$ $$T^{2} - 2T + 4$$