# Properties

 Label 380.2.c.a Level $380$ Weight $2$ Character orbit 380.c Analytic conductor $3.034$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 380.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.03431527681$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Defining polynomial: $$x^{4} + 6x^{2} + 4$$ x^4 + 6*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 + b2 * q^5 + (-b3 + b1) * q^7 + b2 * q^9 $$q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{3} + \beta_1) q^{7} + \beta_{2} q^{9} + (\beta_{2} - 3) q^{11} + ( - \beta_{3} - 2 \beta_1) q^{13} + (\beta_{3} - 2 \beta_1) q^{15} + 2 \beta_{3} q^{17} + q^{19} + (2 \beta_{2} - 2) q^{21} + ( - \beta_{3} + 3 \beta_1) q^{23} + 5 q^{25} + (\beta_{3} + \beta_1) q^{27} + 2 \beta_{2} q^{29} - 4 q^{31} + (\beta_{3} - 5 \beta_1) q^{33} + ( - \beta_{3} - 3 \beta_1) q^{35} + 3 \beta_1 q^{37} + ( - \beta_{2} + 7) q^{39} - 6 q^{41} + (3 \beta_{3} - 3 \beta_1) q^{43} + 5 q^{45} + ( - 3 \beta_{3} + 3 \beta_1) q^{47} - q^{49} + ( - 2 \beta_{2} - 2) q^{51} + 3 \beta_1 q^{53} + ( - 3 \beta_{2} + 5) q^{55} + \beta_1 q^{57} + ( - 2 \beta_{2} + 6) q^{59} + ( - 3 \beta_{2} - 5) q^{61} + ( - \beta_{3} - 3 \beta_1) q^{63} + ( - 4 \beta_{3} + 3 \beta_1) q^{65} - 3 \beta_{3} q^{67} + (4 \beta_{2} - 8) q^{69} + ( - 2 \beta_{2} - 6) q^{71} + 6 \beta_1 q^{73} + 5 \beta_1 q^{75} + (2 \beta_{3} - 6 \beta_1) q^{77} + ( - 6 \beta_{2} - 2) q^{79} + (3 \beta_{2} - 4) q^{81} + (3 \beta_{3} + 3 \beta_1) q^{83} + (4 \beta_{3} + 2 \beta_1) q^{85} + (2 \beta_{3} - 4 \beta_1) q^{87} + ( - 4 \beta_{2} + 6) q^{89} + ( - 6 \beta_{2} - 2) q^{91} - 4 \beta_1 q^{93} + \beta_{2} q^{95} + ( - 4 \beta_{3} + \beta_1) q^{97} + ( - 3 \beta_{2} + 5) q^{99}+O(q^{100})$$ q + b1 * q^3 + b2 * q^5 + (-b3 + b1) * q^7 + b2 * q^9 + (b2 - 3) * q^11 + (-b3 - 2*b1) * q^13 + (b3 - 2*b1) * q^15 + 2*b3 * q^17 + q^19 + (2*b2 - 2) * q^21 + (-b3 + 3*b1) * q^23 + 5 * q^25 + (b3 + b1) * q^27 + 2*b2 * q^29 - 4 * q^31 + (b3 - 5*b1) * q^33 + (-b3 - 3*b1) * q^35 + 3*b1 * q^37 + (-b2 + 7) * q^39 - 6 * q^41 + (3*b3 - 3*b1) * q^43 + 5 * q^45 + (-3*b3 + 3*b1) * q^47 - q^49 + (-2*b2 - 2) * q^51 + 3*b1 * q^53 + (-3*b2 + 5) * q^55 + b1 * q^57 + (-2*b2 + 6) * q^59 + (-3*b2 - 5) * q^61 + (-b3 - 3*b1) * q^63 + (-4*b3 + 3*b1) * q^65 - 3*b3 * q^67 + (4*b2 - 8) * q^69 + (-2*b2 - 6) * q^71 + 6*b1 * q^73 + 5*b1 * q^75 + (2*b3 - 6*b1) * q^77 + (-6*b2 - 2) * q^79 + (3*b2 - 4) * q^81 + (3*b3 + 3*b1) * q^83 + (4*b3 + 2*b1) * q^85 + (2*b3 - 4*b1) * q^87 + (-4*b2 + 6) * q^89 + (-6*b2 - 2) * q^91 - 4*b1 * q^93 + b2 * q^95 + (-4*b3 + b1) * q^97 + (-3*b2 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 12 q^{11} + 4 q^{19} - 8 q^{21} + 20 q^{25} - 16 q^{31} + 28 q^{39} - 24 q^{41} + 20 q^{45} - 4 q^{49} - 8 q^{51} + 20 q^{55} + 24 q^{59} - 20 q^{61} - 32 q^{69} - 24 q^{71} - 8 q^{79} - 16 q^{81} + 24 q^{89} - 8 q^{91} + 20 q^{99}+O(q^{100})$$ 4 * q - 12 * q^11 + 4 * q^19 - 8 * q^21 + 20 * q^25 - 16 * q^31 + 28 * q^39 - 24 * q^41 + 20 * q^45 - 4 * q^49 - 8 * q^51 + 20 * q^55 + 24 * q^59 - 20 * q^61 - 32 * q^69 - 24 * q^71 - 8 * q^79 - 16 * q^81 + 24 * q^89 - 8 * q^91 + 20 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 5\nu$$ v^3 + 5*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 5\beta_1$$ b3 - 5*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/380\mathbb{Z}\right)^\times$$.

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
229.1
 − 2.28825i − 0.874032i 0.874032i 2.28825i
0 2.28825i 0 −2.23607 0 2.82843i 0 −2.23607 0
229.2 0 0.874032i 0 2.23607 0 2.82843i 0 2.23607 0
229.3 0 0.874032i 0 2.23607 0 2.82843i 0 2.23607 0
229.4 0 2.28825i 0 −2.23607 0 2.82843i 0 −2.23607 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
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.c.a 4
3.b odd 2 1 3420.2.f.a 4
4.b odd 2 1 1520.2.d.f 4
5.b even 2 1 inner 380.2.c.a 4
5.c odd 4 2 1900.2.a.j 4
15.d odd 2 1 3420.2.f.a 4
20.d odd 2 1 1520.2.d.f 4
20.e even 4 2 7600.2.a.ce 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.c.a 4 1.a even 1 1 trivial
380.2.c.a 4 5.b even 2 1 inner
1520.2.d.f 4 4.b odd 2 1
1520.2.d.f 4 20.d odd 2 1
1900.2.a.j 4 5.c odd 4 2
3420.2.f.a 4 3.b odd 2 1
3420.2.f.a 4 15.d odd 2 1
7600.2.a.ce 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 6T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 6T^{2} + 4$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} + 6 T + 4)^{2}$$
$13$ $$T^{4} + 46T^{2} + 484$$
$17$ $$T^{4} + 56T^{2} + 64$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 56T^{2} + 64$$
$29$ $$(T^{2} - 20)^{2}$$
$31$ $$(T + 4)^{4}$$
$37$ $$T^{4} + 54T^{2} + 324$$
$41$ $$(T + 6)^{4}$$
$43$ $$(T^{2} + 72)^{2}$$
$47$ $$(T^{2} + 72)^{2}$$
$53$ $$T^{4} + 54T^{2} + 324$$
$59$ $$(T^{2} - 12 T + 16)^{2}$$
$61$ $$(T^{2} + 10 T - 20)^{2}$$
$67$ $$T^{4} + 126T^{2} + 324$$
$71$ $$(T^{2} + 12 T + 16)^{2}$$
$73$ $$T^{4} + 216T^{2} + 5184$$
$79$ $$(T^{2} + 4 T - 176)^{2}$$
$83$ $$T^{4} + 216T^{2} + 5184$$
$89$ $$(T^{2} - 12 T - 44)^{2}$$
$97$ $$T^{4} + 214T^{2} + 3844$$