# Properties

 Label 380.2.i.a Level $380$ Weight $2$ Character orbit 380.i Analytic conductor $3.034$ Analytic rank $1$ Dimension $2$ 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.03431527681$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{5} - 4 q^{7} - \zeta_{6} q^{9} +O(q^{10})$$ q + (2*z - 2) * q^3 + (-z + 1) * q^5 - 4 * q^7 - z * q^9 $$q + (2 \zeta_{6} - 2) q^{3} + ( - \zeta_{6} + 1) q^{5} - 4 q^{7} - \zeta_{6} q^{9} - 3 q^{11} - 6 \zeta_{6} q^{13} + 2 \zeta_{6} q^{15} + (2 \zeta_{6} - 2) q^{17} + (3 \zeta_{6} + 2) q^{19} + ( - 8 \zeta_{6} + 8) q^{21} - 4 \zeta_{6} q^{23} - \zeta_{6} q^{25} - 4 q^{27} - \zeta_{6} q^{29} - 5 q^{31} + ( - 6 \zeta_{6} + 6) q^{33} + (4 \zeta_{6} - 4) q^{35} - 4 q^{37} + 12 q^{39} + (2 \zeta_{6} - 2) q^{41} - q^{45} - 6 \zeta_{6} q^{47} + 9 q^{49} - 4 \zeta_{6} q^{51} + 6 \zeta_{6} q^{53} + (3 \zeta_{6} - 3) q^{55} + (4 \zeta_{6} - 10) q^{57} + ( - \zeta_{6} + 1) q^{59} + 7 \zeta_{6} q^{61} + 4 \zeta_{6} q^{63} - 6 q^{65} + 14 \zeta_{6} q^{67} + 8 q^{69} + (15 \zeta_{6} - 15) q^{71} + (12 \zeta_{6} - 12) q^{73} + 2 q^{75} + 12 q^{77} + ( - \zeta_{6} + 1) q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 16 q^{83} + 2 \zeta_{6} q^{85} + 2 q^{87} - 17 \zeta_{6} q^{89} + 24 \zeta_{6} q^{91} + ( - 10 \zeta_{6} + 10) q^{93} + ( - 2 \zeta_{6} + 5) q^{95} + (12 \zeta_{6} - 12) q^{97} + 3 \zeta_{6} q^{99} +O(q^{100})$$ q + (2*z - 2) * q^3 + (-z + 1) * q^5 - 4 * q^7 - z * q^9 - 3 * q^11 - 6*z * q^13 + 2*z * q^15 + (2*z - 2) * q^17 + (3*z + 2) * q^19 + (-8*z + 8) * q^21 - 4*z * q^23 - z * q^25 - 4 * q^27 - z * q^29 - 5 * q^31 + (-6*z + 6) * q^33 + (4*z - 4) * q^35 - 4 * q^37 + 12 * q^39 + (2*z - 2) * q^41 - q^45 - 6*z * q^47 + 9 * q^49 - 4*z * q^51 + 6*z * q^53 + (3*z - 3) * q^55 + (4*z - 10) * q^57 + (-z + 1) * q^59 + 7*z * q^61 + 4*z * q^63 - 6 * q^65 + 14*z * q^67 + 8 * q^69 + (15*z - 15) * q^71 + (12*z - 12) * q^73 + 2 * q^75 + 12 * q^77 + (-z + 1) * q^79 + (-11*z + 11) * q^81 + 16 * q^83 + 2*z * q^85 + 2 * q^87 - 17*z * q^89 + 24*z * q^91 + (-10*z + 10) * q^93 + (-2*z + 5) * q^95 + (12*z - 12) * q^97 + 3*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + q^{5} - 8 q^{7} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 + q^5 - 8 * q^7 - q^9 $$2 q - 2 q^{3} + q^{5} - 8 q^{7} - q^{9} - 6 q^{11} - 6 q^{13} + 2 q^{15} - 2 q^{17} + 7 q^{19} + 8 q^{21} - 4 q^{23} - q^{25} - 8 q^{27} - q^{29} - 10 q^{31} + 6 q^{33} - 4 q^{35} - 8 q^{37} + 24 q^{39} - 2 q^{41} - 2 q^{45} - 6 q^{47} + 18 q^{49} - 4 q^{51} + 6 q^{53} - 3 q^{55} - 16 q^{57} + q^{59} + 7 q^{61} + 4 q^{63} - 12 q^{65} + 14 q^{67} + 16 q^{69} - 15 q^{71} - 12 q^{73} + 4 q^{75} + 24 q^{77} + q^{79} + 11 q^{81} + 32 q^{83} + 2 q^{85} + 4 q^{87} - 17 q^{89} + 24 q^{91} + 10 q^{93} + 8 q^{95} - 12 q^{97} + 3 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 + q^5 - 8 * q^7 - q^9 - 6 * q^11 - 6 * q^13 + 2 * q^15 - 2 * q^17 + 7 * q^19 + 8 * q^21 - 4 * q^23 - q^25 - 8 * q^27 - q^29 - 10 * q^31 + 6 * q^33 - 4 * q^35 - 8 * q^37 + 24 * q^39 - 2 * q^41 - 2 * q^45 - 6 * q^47 + 18 * q^49 - 4 * q^51 + 6 * q^53 - 3 * q^55 - 16 * q^57 + q^59 + 7 * q^61 + 4 * q^63 - 12 * q^65 + 14 * q^67 + 16 * q^69 - 15 * q^71 - 12 * q^73 + 4 * q^75 + 24 * q^77 + q^79 + 11 * q^81 + 32 * q^83 + 2 * q^85 + 4 * q^87 - 17 * q^89 + 24 * q^91 + 10 * q^93 + 8 * q^95 - 12 * q^97 + 3 * q^99

## Character values

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

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
121.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.00000 1.73205i 0 0.500000 + 0.866025i 0 −4.00000 0 −0.500000 + 0.866025i 0
201.1 0 −1.00000 + 1.73205i 0 0.500000 0.866025i 0 −4.00000 0 −0.500000 0.866025i 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
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 380.2.i.a 2
3.b odd 2 1 3420.2.t.b 2
4.b odd 2 1 1520.2.q.g 2
5.b even 2 1 1900.2.i.b 2
5.c odd 4 2 1900.2.s.b 4
19.c even 3 1 inner 380.2.i.a 2
19.c even 3 1 7220.2.a.e 1
19.d odd 6 1 7220.2.a.a 1
57.h odd 6 1 3420.2.t.b 2
76.g odd 6 1 1520.2.q.g 2
95.i even 6 1 1900.2.i.b 2
95.m odd 12 2 1900.2.s.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.a 2 1.a even 1 1 trivial
380.2.i.a 2 19.c even 3 1 inner
1520.2.q.g 2 4.b odd 2 1
1520.2.q.g 2 76.g odd 6 1
1900.2.i.b 2 5.b even 2 1
1900.2.i.b 2 95.i even 6 1
1900.2.s.b 4 5.c odd 4 2
1900.2.s.b 4 95.m odd 12 2
3420.2.t.b 2 3.b odd 2 1
3420.2.t.b 2 57.h odd 6 1
7220.2.a.a 1 19.d odd 6 1
7220.2.a.e 1 19.c even 3 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 4$$
$5$ $$T^{2} - T + 1$$
$7$ $$(T + 4)^{2}$$
$11$ $$(T + 3)^{2}$$
$13$ $$T^{2} + 6T + 36$$
$17$ $$T^{2} + 2T + 4$$
$19$ $$T^{2} - 7T + 19$$
$23$ $$T^{2} + 4T + 16$$
$29$ $$T^{2} + T + 1$$
$31$ $$(T + 5)^{2}$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 2T + 4$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$T^{2} - 6T + 36$$
$59$ $$T^{2} - T + 1$$
$61$ $$T^{2} - 7T + 49$$
$67$ $$T^{2} - 14T + 196$$
$71$ $$T^{2} + 15T + 225$$
$73$ $$T^{2} + 12T + 144$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T - 16)^{2}$$
$89$ $$T^{2} + 17T + 289$$
$97$ $$T^{2} + 12T + 144$$