# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.311416567883$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$14$$ $$28$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{8}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
−0.707107 0.707107i −1.00000 1.41421i 1.00000i 1.41421 + 1.41421i −0.292893 + 1.70711i 1.00000 + 1.00000i −2.12132 + 2.12132i −1.00000 + 2.82843i 2.00000i
5.2 0.707107 + 0.707107i −1.00000 + 1.41421i 1.00000i −1.41421 1.41421i −1.70711 + 0.292893i 1.00000 + 1.00000i 2.12132 2.12132i −1.00000 2.82843i 2.00000i
8.1 −0.707107 + 0.707107i −1.00000 + 1.41421i 1.00000i 1.41421 1.41421i −0.292893 1.70711i 1.00000 1.00000i −2.12132 2.12132i −1.00000 2.82843i 2.00000i
8.2 0.707107 0.707107i −1.00000 1.41421i 1.00000i −1.41421 + 1.41421i −1.70711 0.292893i 1.00000 1.00000i 2.12132 + 2.12132i −1.00000 + 2.82843i 2.00000i
 $$n$$: 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 inner
13.d odd 4 1 inner
39.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 39.2.f.a 4
3.b odd 2 1 inner 39.2.f.a 4
4.b odd 2 1 624.2.bf.d 4
5.b even 2 1 975.2.o.j 4
5.c odd 4 1 975.2.n.c 4
5.c odd 4 1 975.2.n.d 4
12.b even 2 1 624.2.bf.d 4
13.b even 2 1 507.2.f.a 4
13.c even 3 2 507.2.k.j 8
13.d odd 4 1 inner 39.2.f.a 4
13.d odd 4 1 507.2.f.a 4
13.e even 6 2 507.2.k.i 8
13.f odd 12 2 507.2.k.i 8
13.f odd 12 2 507.2.k.j 8
15.d odd 2 1 975.2.o.j 4
15.e even 4 1 975.2.n.c 4
15.e even 4 1 975.2.n.d 4
39.d odd 2 1 507.2.f.a 4
39.f even 4 1 inner 39.2.f.a 4
39.f even 4 1 507.2.f.a 4
39.h odd 6 2 507.2.k.i 8
39.i odd 6 2 507.2.k.j 8
39.k even 12 2 507.2.k.i 8
39.k even 12 2 507.2.k.j 8
52.f even 4 1 624.2.bf.d 4
65.f even 4 1 975.2.n.d 4
65.g odd 4 1 975.2.o.j 4
65.k even 4 1 975.2.n.c 4
156.l odd 4 1 624.2.bf.d 4
195.j odd 4 1 975.2.n.c 4
195.n even 4 1 975.2.o.j 4
195.u odd 4 1 975.2.n.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.f.a 4 1.a even 1 1 trivial
39.2.f.a 4 3.b odd 2 1 inner
39.2.f.a 4 13.d odd 4 1 inner
39.2.f.a 4 39.f even 4 1 inner
507.2.f.a 4 13.b even 2 1
507.2.f.a 4 13.d odd 4 1
507.2.f.a 4 39.d odd 2 1
507.2.f.a 4 39.f even 4 1
507.2.k.i 8 13.e even 6 2
507.2.k.i 8 13.f odd 12 2
507.2.k.i 8 39.h odd 6 2
507.2.k.i 8 39.k even 12 2
507.2.k.j 8 13.c even 3 2
507.2.k.j 8 13.f odd 12 2
507.2.k.j 8 39.i odd 6 2
507.2.k.j 8 39.k even 12 2
624.2.bf.d 4 4.b odd 2 1
624.2.bf.d 4 12.b even 2 1
624.2.bf.d 4 52.f even 4 1
624.2.bf.d 4 156.l odd 4 1
975.2.n.c 4 5.c odd 4 1
975.2.n.c 4 15.e even 4 1
975.2.n.c 4 65.k even 4 1
975.2.n.c 4 195.j odd 4 1
975.2.n.d 4 5.c odd 4 1
975.2.n.d 4 15.e even 4 1
975.2.n.d 4 65.f even 4 1
975.2.n.d 4 195.u odd 4 1
975.2.o.j 4 5.b even 2 1
975.2.o.j 4 15.d odd 2 1
975.2.o.j 4 65.g odd 4 1
975.2.o.j 4 195.n even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(39, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$(T^{2} + 2 T + 3)^{2}$$
$5$ $$T^{4} + 16$$
$7$ $$(T^{2} - 2 T + 2)^{2}$$
$11$ $$T^{4} + 256$$
$13$ $$(T^{2} + 4 T + 13)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} - 2 T + 2)^{2}$$
$23$ $$(T^{2} - 72)^{2}$$
$29$ $$(T^{2} + 8)^{2}$$
$31$ $$(T^{2} + 10 T + 50)^{2}$$
$37$ $$(T^{2} - 2 T + 2)^{2}$$
$41$ $$T^{4} + 16$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$T^{4} + 256$$
$53$ $$(T^{2} + 32)^{2}$$
$59$ $$T^{4} + 256$$
$61$ $$(T - 8)^{4}$$
$67$ $$(T^{2} + 10 T + 50)^{2}$$
$71$ $$T^{4} + 256$$
$73$ $$(T^{2} - 2 T + 2)^{2}$$
$79$ $$(T + 10)^{4}$$
$83$ $$T^{4} + 4096$$
$89$ $$T^{4} + 38416$$
$97$ $$(T^{2} - 14 T + 98)^{2}$$