# Properties

 Label 380.3.g.a Level $380$ Weight $3$ Character orbit 380.g Analytic conductor $10.354$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$380 = 2^{2} \cdot 5 \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 380.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.3542500457$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-6}, \sqrt{14})$$ Defining polynomial: $$x^{4} - 4x^{2} + 25$$ x^4 - 4*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{5}$$ 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{3} - 5 q^{5} + \beta_{2} q^{7} + 5 q^{9}+O(q^{10})$$ q - b1 * q^3 - 5 * q^5 + b2 * q^7 + 5 * q^9 $$q - \beta_1 q^{3} - 5 q^{5} + \beta_{2} q^{7} + 5 q^{9} + 4 q^{11} - 3 \beta_1 q^{13} + 5 \beta_1 q^{15} + 2 \beta_{2} q^{17} + ( - \beta_{3} - 5) q^{19} - 2 \beta_{3} q^{21} - \beta_{2} q^{23} + 25 q^{25} + 4 \beta_1 q^{27} - 2 \beta_{3} q^{29} - 2 \beta_{3} q^{31} - 4 \beta_1 q^{33} - 5 \beta_{2} q^{35} - 9 \beta_1 q^{37} + 42 q^{39} + 2 \beta_{3} q^{41} - 7 \beta_{2} q^{43} - 25 q^{45} - \beta_{2} q^{47} - 47 q^{49} - 4 \beta_{3} q^{51} + 15 \beta_1 q^{53} - 20 q^{55} + (7 \beta_{2} + 5 \beta_1) q^{57} + 4 \beta_{3} q^{59} + 100 q^{61} + 5 \beta_{2} q^{63} + 15 \beta_1 q^{65} - 3 \beta_1 q^{67} + 2 \beta_{3} q^{69} - 2 \beta_{3} q^{71} - 2 \beta_{2} q^{73} - 25 \beta_1 q^{75} + 4 \beta_{2} q^{77} - 6 \beta_{3} q^{79} - 101 q^{81} - 3 \beta_{2} q^{83} - 10 \beta_{2} q^{85} + 14 \beta_{2} q^{87} + 8 \beta_{3} q^{89} - 6 \beta_{3} q^{91} + 14 \beta_{2} q^{93} + (5 \beta_{3} + 25) q^{95} - 33 \beta_1 q^{97} + 20 q^{99}+O(q^{100})$$ q - b1 * q^3 - 5 * q^5 + b2 * q^7 + 5 * q^9 + 4 * q^11 - 3*b1 * q^13 + 5*b1 * q^15 + 2*b2 * q^17 + (-b3 - 5) * q^19 - 2*b3 * q^21 - b2 * q^23 + 25 * q^25 + 4*b1 * q^27 - 2*b3 * q^29 - 2*b3 * q^31 - 4*b1 * q^33 - 5*b2 * q^35 - 9*b1 * q^37 + 42 * q^39 + 2*b3 * q^41 - 7*b2 * q^43 - 25 * q^45 - b2 * q^47 - 47 * q^49 - 4*b3 * q^51 + 15*b1 * q^53 - 20 * q^55 + (7*b2 + 5*b1) * q^57 + 4*b3 * q^59 + 100 * q^61 + 5*b2 * q^63 + 15*b1 * q^65 - 3*b1 * q^67 + 2*b3 * q^69 - 2*b3 * q^71 - 2*b2 * q^73 - 25*b1 * q^75 + 4*b2 * q^77 - 6*b3 * q^79 - 101 * q^81 - 3*b2 * q^83 - 10*b2 * q^85 + 14*b2 * q^87 + 8*b3 * q^89 - 6*b3 * q^91 + 14*b2 * q^93 + (5*b3 + 25) * q^95 - 33*b1 * q^97 + 20 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{5} + 20 q^{9}+O(q^{10})$$ 4 * q - 20 * q^5 + 20 * q^9 $$4 q - 20 q^{5} + 20 q^{9} + 16 q^{11} - 20 q^{19} + 100 q^{25} + 168 q^{39} - 100 q^{45} - 188 q^{49} - 80 q^{55} + 400 q^{61} - 404 q^{81} + 100 q^{95} + 80 q^{99}+O(q^{100})$$ 4 * q - 20 * q^5 + 20 * q^9 + 16 * q^11 - 20 * q^19 + 100 * q^25 + 168 * q^39 - 100 * q^45 - 188 * q^49 - 80 * q^55 + 400 * q^61 - 404 * q^81 + 100 * q^95 + 80 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 9\nu ) / 5$$ (-v^3 + 9*v) / 5 $$\beta_{2}$$ $$=$$ $$( 4\nu^{3} + 4\nu ) / 5$$ (4*v^3 + 4*v) / 5 $$\beta_{3}$$ $$=$$ $$4\nu^{2} - 8$$ 4*v^2 - 8
 $$\nu$$ $$=$$ $$( \beta_{2} + 4\beta_1 ) / 8$$ (b2 + 4*b1) / 8 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 8 ) / 4$$ (b3 + 8) / 4 $$\nu^{3}$$ $$=$$ $$( 9\beta_{2} - 4\beta_1 ) / 8$$ (9*b2 - 4*b1) / 8

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
189.1
 1.87083 − 1.22474i 1.87083 + 1.22474i −1.87083 − 1.22474i −1.87083 + 1.22474i
0 −3.74166 0 −5.00000 0 9.79796i 0 5.00000 0
189.2 0 −3.74166 0 −5.00000 0 9.79796i 0 5.00000 0
189.3 0 3.74166 0 −5.00000 0 9.79796i 0 5.00000 0
189.4 0 3.74166 0 −5.00000 0 9.79796i 0 5.00000 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
19.b odd 2 1 inner
95.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.3.g.a 4
3.b odd 2 1 3420.3.h.c 4
5.b even 2 1 inner 380.3.g.a 4
5.c odd 4 2 1900.3.e.c 4
15.d odd 2 1 3420.3.h.c 4
19.b odd 2 1 inner 380.3.g.a 4
57.d even 2 1 3420.3.h.c 4
95.d odd 2 1 inner 380.3.g.a 4
95.g even 4 2 1900.3.e.c 4
285.b even 2 1 3420.3.h.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.3.g.a 4 1.a even 1 1 trivial
380.3.g.a 4 5.b even 2 1 inner
380.3.g.a 4 19.b odd 2 1 inner
380.3.g.a 4 95.d odd 2 1 inner
1900.3.e.c 4 5.c odd 4 2
1900.3.e.c 4 95.g even 4 2
3420.3.h.c 4 3.b odd 2 1
3420.3.h.c 4 15.d odd 2 1
3420.3.h.c 4 57.d even 2 1
3420.3.h.c 4 285.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 14$$ acting on $$S_{3}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 14)^{2}$$
$5$ $$(T + 5)^{4}$$
$7$ $$(T^{2} + 96)^{2}$$
$11$ $$(T - 4)^{4}$$
$13$ $$(T^{2} - 126)^{2}$$
$17$ $$(T^{2} + 384)^{2}$$
$19$ $$(T^{2} + 10 T + 361)^{2}$$
$23$ $$(T^{2} + 96)^{2}$$
$29$ $$(T^{2} + 1344)^{2}$$
$31$ $$(T^{2} + 1344)^{2}$$
$37$ $$(T^{2} - 1134)^{2}$$
$41$ $$(T^{2} + 1344)^{2}$$
$43$ $$(T^{2} + 4704)^{2}$$
$47$ $$(T^{2} + 96)^{2}$$
$53$ $$(T^{2} - 3150)^{2}$$
$59$ $$(T^{2} + 5376)^{2}$$
$61$ $$(T - 100)^{4}$$
$67$ $$(T^{2} - 126)^{2}$$
$71$ $$(T^{2} + 1344)^{2}$$
$73$ $$(T^{2} + 384)^{2}$$
$79$ $$(T^{2} + 12096)^{2}$$
$83$ $$(T^{2} + 864)^{2}$$
$89$ $$(T^{2} + 21504)^{2}$$
$97$ $$(T^{2} - 15246)^{2}$$