# Properties

 Label 39.4.f.a Level $39$ Weight $4$ Character orbit 39.f Analytic conductor $2.301$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.30107449022$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} + 8 \beta_1 q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} + 10 \beta_1 - 10) q^{7} + 27 q^{9}+O(q^{10})$$ q - b3 * q^3 + 8*b1 * q^4 + (-3*b3 + 3*b2 + 10*b1 - 10) * q^7 + 27 * q^9 $$q - \beta_{3} q^{3} + 8 \beta_1 q^{4} + ( - 3 \beta_{3} + 3 \beta_{2} + 10 \beta_1 - 10) q^{7} + 27 q^{9} - 8 \beta_{2} q^{12} + (6 \beta_{3} - 35 \beta_1) q^{13} - 64 q^{16} + (15 \beta_{3} + 15 \beta_{2} - 28 \beta_1 - 28) q^{19} + (10 \beta_{3} - 10 \beta_{2} - 81 \beta_1 + 81) q^{21} + 125 \beta_1 q^{25} - 27 \beta_{3} q^{27} + ( - 24 \beta_{3} - 24 \beta_{2} - 80 \beta_1 - 80) q^{28} + (15 \beta_{3} + 15 \beta_{2} + 154 \beta_1 + 154) q^{31} + 216 \beta_1 q^{36} + ( - 42 \beta_{3} + 42 \beta_{2} - 55 \beta_1 + 55) q^{37} + (35 \beta_{2} - 162) q^{39} - 42 \beta_{2} q^{43} + 64 \beta_{3} q^{48} + ( - 120 \beta_{2} - 343 \beta_1) q^{49} + (48 \beta_{2} + 280) q^{52} + (28 \beta_{3} + 28 \beta_{2} - 405 \beta_1 - 405) q^{57} + 180 \beta_{3} q^{61} + ( - 81 \beta_{3} + 81 \beta_{2} + 270 \beta_1 - 270) q^{63} - 512 \beta_1 q^{64} + ( - 63 \beta_{3} - 63 \beta_{2} + 440 \beta_1 + 440) q^{67} + (36 \beta_{3} - 36 \beta_{2} + 595 \beta_1 - 595) q^{73} - 125 \beta_{2} q^{75} + ( - 120 \beta_{3} + 120 \beta_{2} - 224 \beta_1 + 224) q^{76} - 210 \beta_{3} q^{79} + 729 q^{81} + (80 \beta_{3} + 80 \beta_{2} + 648 \beta_1 + 648) q^{84} + (45 \beta_{3} + 165 \beta_{2} + 836 \beta_1 - 136) q^{91} + ( - 154 \beta_{3} - 154 \beta_{2} - 405 \beta_1 - 405) q^{93} + (132 \beta_{3} + 132 \beta_{2} - 665 \beta_1 - 665) q^{97}+O(q^{100})$$ q - b3 * q^3 + 8*b1 * q^4 + (-3*b3 + 3*b2 + 10*b1 - 10) * q^7 + 27 * q^9 - 8*b2 * q^12 + (6*b3 - 35*b1) * q^13 - 64 * q^16 + (15*b3 + 15*b2 - 28*b1 - 28) * q^19 + (10*b3 - 10*b2 - 81*b1 + 81) * q^21 + 125*b1 * q^25 - 27*b3 * q^27 + (-24*b3 - 24*b2 - 80*b1 - 80) * q^28 + (15*b3 + 15*b2 + 154*b1 + 154) * q^31 + 216*b1 * q^36 + (-42*b3 + 42*b2 - 55*b1 + 55) * q^37 + (35*b2 - 162) * q^39 - 42*b2 * q^43 + 64*b3 * q^48 + (-120*b2 - 343*b1) * q^49 + (48*b2 + 280) * q^52 + (28*b3 + 28*b2 - 405*b1 - 405) * q^57 + 180*b3 * q^61 + (-81*b3 + 81*b2 + 270*b1 - 270) * q^63 - 512*b1 * q^64 + (-63*b3 - 63*b2 + 440*b1 + 440) * q^67 + (36*b3 - 36*b2 + 595*b1 - 595) * q^73 - 125*b2 * q^75 + (-120*b3 + 120*b2 - 224*b1 + 224) * q^76 - 210*b3 * q^79 + 729 * q^81 + (80*b3 + 80*b2 + 648*b1 + 648) * q^84 + (45*b3 + 165*b2 + 836*b1 - 136) * q^91 + (-154*b3 - 154*b2 - 405*b1 - 405) * q^93 + (132*b3 + 132*b2 - 665*b1 - 665) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 40 q^{7} + 108 q^{9}+O(q^{10})$$ 4 * q - 40 * q^7 + 108 * q^9 $$4 q - 40 q^{7} + 108 q^{9} - 256 q^{16} - 112 q^{19} + 324 q^{21} - 320 q^{28} + 616 q^{31} + 220 q^{37} - 648 q^{39} + 1120 q^{52} - 1620 q^{57} - 1080 q^{63} + 1760 q^{67} - 2380 q^{73} + 896 q^{76} + 2916 q^{81} + 2592 q^{84} - 544 q^{91} - 1620 q^{93} - 2660 q^{97}+O(q^{100})$$ 4 * q - 40 * q^7 + 108 * q^9 - 256 * q^16 - 112 * q^19 + 324 * q^21 - 320 * q^28 + 616 * q^31 + 220 * q^37 - 648 * q^39 + 1120 * q^52 - 1620 * q^57 - 1080 * q^63 + 1760 * q^67 - 2380 * q^73 + 896 * q^76 + 2916 * q^81 + 2592 * q^84 - 544 * q^91 - 1620 * q^93 - 2660 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$6\zeta_{12}^{2} - 3$$ 6*v^2 - 3 $$\beta_{3}$$ $$=$$ $$-3\zeta_{12}^{3} + 6\zeta_{12}$$ -3*v^3 + 6*v
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + 3\beta_1 ) / 6$$ (b3 + 3*b1) / 6 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 3 ) / 6$$ (b2 + 3) / 6 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 −5.19615 8.00000i 0 0 −25.5885 25.5885i 0 27.0000 0
5.2 0 5.19615 8.00000i 0 0 5.58846 + 5.58846i 0 27.0000 0
8.1 0 −5.19615 8.00000i 0 0 −25.5885 + 25.5885i 0 27.0000 0
8.2 0 5.19615 8.00000i 0 0 5.58846 5.58846i 0 27.0000 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.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 39.4.f.a 4
3.b odd 2 1 CM 39.4.f.a 4
13.d odd 4 1 inner 39.4.f.a 4
39.f even 4 1 inner 39.4.f.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.f.a 4 1.a even 1 1 trivial
39.4.f.a 4 3.b odd 2 1 CM
39.4.f.a 4 13.d odd 4 1 inner
39.4.f.a 4 39.f even 4 1 inner

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 27)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 40 T^{3} + 800 T^{2} + \cdots + 81796$$
$11$ $$T^{4}$$
$13$ $$T^{4} + 506 T^{2} + 4826809$$
$17$ $$T^{4}$$
$19$ $$T^{4} + 112 T^{3} + \cdots + 111978724$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4} - 616 T^{3} + \cdots + 1244819524$$
$37$ $$T^{4} - 220 T^{3} + \cdots + 7957710436$$
$41$ $$T^{4}$$
$43$ $$(T^{2} + 47628)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T^{2} - 874800)^{2}$$
$67$ $$T^{4} - 1760 T^{3} + \cdots + 29885419876$$
$71$ $$T^{4}$$
$73$ $$T^{4} + 2380 T^{3} + \cdots + 407128220356$$
$79$ $$(T^{2} - 1190700)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 2660 T^{3} + \cdots + 3186150916$$