# Properties

 Label 39.2.k.a Level $39$ Weight $2$ Character orbit 39.k Analytic conductor $0.311$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$39 = 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 39.k (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.311416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{12}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9}+O(q^{10})$$ q + (-z^3 - z) * q^3 + 2*z * q^4 + (3*z^3 - z^2 - 2*z - 2) * q^7 + (3*z^2 - 3) * q^9 $$q + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} + 2 \zeta_{12} q^{4} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{7} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 4 \zeta_{12}^{2} + 2) q^{12} + ( - 4 \zeta_{12}^{3} + 3 \zeta_{12}) q^{13} + 4 \zeta_{12}^{2} q^{16} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{19} + (4 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 4) q^{21} - 5 \zeta_{12}^{3} q^{25} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 4 \zeta_{12} - 6) q^{28} + (\zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} - 1) q^{31} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{36} + (7 \zeta_{12}^{3} + 7 \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{37} + ( - 2 \zeta_{12}^{2} - 5) q^{39} + ( - \zeta_{12}^{2} - 1) q^{43} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + ( - 7 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 7 \zeta_{12} + 6) q^{49} + ( - 2 \zeta_{12}^{2} + 8) q^{52} + (\zeta_{12}^{3} + 8 \zeta_{12}^{2} - 8 \zeta_{12} - 1) q^{57} + (10 \zeta_{12}^{3} - 5 \zeta_{12}) q^{61} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} - 3 \zeta_{12} + 9) q^{63} + 8 \zeta_{12}^{3} q^{64} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 9 \zeta_{12} + 7) q^{67} + ( - 9 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 9) q^{73} + (5 \zeta_{12}^{2} - 10) q^{75} + ( - 6 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 10 \zeta_{12} - 4) q^{76} + (7 \zeta_{12}^{3} - 14 \zeta_{12}) q^{79} - 9 \zeta_{12}^{2} q^{81} + (2 \zeta_{12}^{3} + 10 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{84} + (9 \zeta_{12}^{3} + 11 \zeta_{12}^{2} - 10 \zeta_{12} - 5) q^{91} + (11 \zeta_{12}^{3} - 11 \zeta_{12}^{2} - 4 \zeta_{12} + 7) q^{93} + ( - 11 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 8 \zeta_{12} - 8) q^{97} +O(q^{100})$$ q + (-z^3 - z) * q^3 + 2*z * q^4 + (3*z^3 - z^2 - 2*z - 2) * q^7 + (3*z^2 - 3) * q^9 + (-4*z^2 + 2) * q^12 + (-4*z^3 + 3*z) * q^13 + 4*z^2 * q^16 + (2*z^3 - 3*z^2 - 5*z + 5) * q^19 + (4*z^3 + z^2 + z + 4) * q^21 - 5*z^3 * q^25 + (-3*z^3 + 6*z) * q^27 + (-2*z^3 + 2*z^2 - 4*z - 6) * q^28 + (z^3 - 5*z^2 + 5*z - 1) * q^31 + (6*z^3 - 6*z) * q^36 + (7*z^3 + 7*z^2 - 3*z - 4) * q^37 + (-2*z^2 - 5) * q^39 + (-z^2 - 1) * q^43 + (-8*z^3 + 4*z) * q^48 + (-7*z^3 - 3*z^2 + 7*z + 6) * q^49 + (-2*z^2 + 8) * q^52 + (z^3 + 8*z^2 - 8*z - 1) * q^57 + (10*z^3 - 5*z) * q^61 + (-6*z^3 - 6*z^2 - 3*z + 9) * q^63 + 8*z^3 * q^64 + (-2*z^3 + 2*z^2 + 9*z + 7) * q^67 + (-9*z^3 + z^2 + z - 9) * q^73 + (5*z^2 - 10) * q^75 + (-6*z^3 - 6*z^2 + 10*z - 4) * q^76 + (7*z^3 - 14*z) * q^79 - 9*z^2 * q^81 + (2*z^3 + 10*z^2 + 8*z - 8) * q^84 + (9*z^3 + 11*z^2 - 10*z - 5) * q^91 + (11*z^3 - 11*z^2 - 4*z + 7) * q^93 + (-11*z^3 - 3*z^2 + 8*z - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{7} - 6 q^{9}+O(q^{10})$$ 4 * q - 10 * q^7 - 6 * q^9 $$4 q - 10 q^{7} - 6 q^{9} + 8 q^{16} + 14 q^{19} + 18 q^{21} - 20 q^{28} - 14 q^{31} - 2 q^{37} - 24 q^{39} - 6 q^{43} + 18 q^{49} + 28 q^{52} + 12 q^{57} + 24 q^{63} + 32 q^{67} - 34 q^{73} - 30 q^{75} - 28 q^{76} - 18 q^{81} - 12 q^{84} + 2 q^{91} + 6 q^{93} - 38 q^{97}+O(q^{100})$$ 4 * q - 10 * q^7 - 6 * q^9 + 8 * q^16 + 14 * q^19 + 18 * q^21 - 20 * q^28 - 14 * q^31 - 2 * q^37 - 24 * q^39 - 6 * q^43 + 18 * q^49 + 28 * q^52 + 12 * q^57 + 24 * q^63 + 32 * q^67 - 34 * q^73 - 30 * q^75 - 28 * q^76 - 18 * q^81 - 12 * q^84 + 2 * q^91 + 6 * q^93 - 38 * q^97

## Character values

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

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 1.50000i 1.73205 + 1.00000i 0 0 −4.23205 + 1.13397i 0 −1.50000 + 2.59808i 0
11.1 0 0.866025 + 1.50000i −1.73205 1.00000i 0 0 −0.767949 2.86603i 0 −1.50000 + 2.59808i 0
20.1 0 −0.866025 + 1.50000i 1.73205 1.00000i 0 0 −4.23205 1.13397i 0 −1.50000 2.59808i 0
32.1 0 0.866025 1.50000i −1.73205 + 1.00000i 0 0 −0.767949 + 2.86603i 0 −1.50000 2.59808i 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
13.f odd 12 1 inner
39.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.k.a 4
3.b odd 2 1 CM 39.2.k.a 4
4.b odd 2 1 624.2.cn.b 4
5.b even 2 1 975.2.bo.c 4
5.c odd 4 1 975.2.bp.a 4
5.c odd 4 1 975.2.bp.d 4
12.b even 2 1 624.2.cn.b 4
13.b even 2 1 507.2.k.c 4
13.c even 3 1 507.2.f.c 4
13.c even 3 1 507.2.k.b 4
13.d odd 4 1 507.2.k.a 4
13.d odd 4 1 507.2.k.b 4
13.e even 6 1 507.2.f.b 4
13.e even 6 1 507.2.k.a 4
13.f odd 12 1 inner 39.2.k.a 4
13.f odd 12 1 507.2.f.b 4
13.f odd 12 1 507.2.f.c 4
13.f odd 12 1 507.2.k.c 4
15.d odd 2 1 975.2.bo.c 4
15.e even 4 1 975.2.bp.a 4
15.e even 4 1 975.2.bp.d 4
39.d odd 2 1 507.2.k.c 4
39.f even 4 1 507.2.k.a 4
39.f even 4 1 507.2.k.b 4
39.h odd 6 1 507.2.f.b 4
39.h odd 6 1 507.2.k.a 4
39.i odd 6 1 507.2.f.c 4
39.i odd 6 1 507.2.k.b 4
39.k even 12 1 inner 39.2.k.a 4
39.k even 12 1 507.2.f.b 4
39.k even 12 1 507.2.f.c 4
39.k even 12 1 507.2.k.c 4
52.l even 12 1 624.2.cn.b 4
65.o even 12 1 975.2.bp.a 4
65.s odd 12 1 975.2.bo.c 4
65.t even 12 1 975.2.bp.d 4
156.v odd 12 1 624.2.cn.b 4
195.bc odd 12 1 975.2.bp.d 4
195.bh even 12 1 975.2.bo.c 4
195.bn odd 12 1 975.2.bp.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.k.a 4 1.a even 1 1 trivial
39.2.k.a 4 3.b odd 2 1 CM
39.2.k.a 4 13.f odd 12 1 inner
39.2.k.a 4 39.k even 12 1 inner
507.2.f.b 4 13.e even 6 1
507.2.f.b 4 13.f odd 12 1
507.2.f.b 4 39.h odd 6 1
507.2.f.b 4 39.k even 12 1
507.2.f.c 4 13.c even 3 1
507.2.f.c 4 13.f odd 12 1
507.2.f.c 4 39.i odd 6 1
507.2.f.c 4 39.k even 12 1
507.2.k.a 4 13.d odd 4 1
507.2.k.a 4 13.e even 6 1
507.2.k.a 4 39.f even 4 1
507.2.k.a 4 39.h odd 6 1
507.2.k.b 4 13.c even 3 1
507.2.k.b 4 13.d odd 4 1
507.2.k.b 4 39.f even 4 1
507.2.k.b 4 39.i odd 6 1
507.2.k.c 4 13.b even 2 1
507.2.k.c 4 13.f odd 12 1
507.2.k.c 4 39.d odd 2 1
507.2.k.c 4 39.k even 12 1
624.2.cn.b 4 4.b odd 2 1
624.2.cn.b 4 12.b even 2 1
624.2.cn.b 4 52.l even 12 1
624.2.cn.b 4 156.v odd 12 1
975.2.bo.c 4 5.b even 2 1
975.2.bo.c 4 15.d odd 2 1
975.2.bo.c 4 65.s odd 12 1
975.2.bo.c 4 195.bh even 12 1
975.2.bp.a 4 5.c odd 4 1
975.2.bp.a 4 15.e even 4 1
975.2.bp.a 4 65.o even 12 1
975.2.bp.a 4 195.bn odd 12 1
975.2.bp.d 4 5.c odd 4 1
975.2.bp.d 4 15.e even 4 1
975.2.bp.d 4 65.t even 12 1
975.2.bp.d 4 195.bc odd 12 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 3T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 10 T^{3} + 41 T^{2} + \cdots + 169$$
$11$ $$T^{4}$$
$13$ $$T^{4} - T^{2} + 169$$
$17$ $$T^{4}$$
$19$ $$T^{4} - 14 T^{3} + 50 T^{2} + \cdots + 676$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} + 14 T^{3} + 98 T^{2} + \cdots + 169$$
$37$ $$T^{4} + 2 T^{3} + 122 T^{2} + \cdots + 676$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 3 T + 3)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$T^{4} + 75T^{2} + 5625$$
$67$ $$T^{4} - 32 T^{3} + 281 T^{2} + \cdots + 169$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 34 T^{3} + 578 T^{2} + \cdots + 20449$$
$79$ $$(T^{2} - 147)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 38 T^{3} + 557 T^{2} + \cdots + 28561$$