# Properties

 Label 380.2.i.b Level $380$ Weight $2$ Character orbit 380.i Analytic conductor $3.034$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

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## 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: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1783323.2 Defining polynomial: $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{5} ) q^{3} + ( 1 + \beta_{4} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{5} ) q^{3} + ( 1 + \beta_{4} ) q^{5} + ( 1 - \beta_{3} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} + ( 2 - \beta_{3} ) q^{11} -\beta_{4} q^{13} -\beta_{5} q^{15} + ( -2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} ) q^{19} + ( -6 - 2 \beta_{1} - \beta_{2} - 6 \beta_{4} + \beta_{5} ) q^{21} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{23} + \beta_{4} q^{25} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{27} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{29} + ( 4 + \beta_{1} - \beta_{2} ) q^{31} + ( -6 - 2 \beta_{1} - \beta_{2} + \beta_{3} - 6 \beta_{4} ) q^{33} + ( 1 - \beta_{3} + \beta_{4} + \beta_{5} ) q^{35} + ( 1 - \beta_{3} ) q^{37} + \beta_{3} q^{39} + ( -3 + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{41} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{43} + ( -3 - \beta_{1} + \beta_{2} ) q^{45} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{47} + ( \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{49} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 9 \beta_{4} ) q^{51} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{53} + ( 2 - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{55} + ( 9 + 3 \beta_{1} + \beta_{2} + \beta_{3} + 9 \beta_{4} - 2 \beta_{5} ) q^{57} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{59} + ( 3 \beta_{4} + 2 \beta_{5} ) q^{61} + 5 \beta_{5} q^{63} + q^{65} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{5} ) q^{67} + ( 3 + \beta_{1} - \beta_{2} + 7 \beta_{3} ) q^{69} + ( 3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{71} + ( 7 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 7 \beta_{4} ) q^{73} -\beta_{3} q^{75} + ( 8 + \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{77} + ( -10 - 2 \beta_{3} - 10 \beta_{4} + 2 \beta_{5} ) q^{79} + ( -6 - 5 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} ) q^{81} + ( -10 + \beta_{1} - \beta_{2} ) q^{83} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{85} + ( -\beta_{1} + \beta_{2} - 7 \beta_{3} ) q^{87} + ( 2 \beta_{4} + 4 \beta_{5} ) q^{89} + ( -\beta_{4} - \beta_{5} ) q^{91} + ( 3 + 2 \beta_{1} + \beta_{2} + 7 \beta_{3} + 3 \beta_{4} - 8 \beta_{5} ) q^{93} + ( -1 - \beta_{2} + \beta_{3} - \beta_{5} ) q^{95} + ( -4 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 4 \beta_{4} + 5 \beta_{5} ) q^{97} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{3} + 3q^{5} + 4q^{7} - 8q^{9} + O(q^{10})$$ $$6q + q^{3} + 3q^{5} + 4q^{7} - 8q^{9} + 10q^{11} + 3q^{13} - q^{15} + 3q^{17} - 16q^{21} + 14q^{23} - 3q^{25} - 20q^{27} - 6q^{29} + 22q^{31} - 15q^{33} + 2q^{35} + 4q^{37} + 2q^{39} - 6q^{41} + 5q^{43} - 16q^{45} + 6q^{47} - 6q^{49} - 24q^{51} + 13q^{53} + 5q^{55} + 25q^{57} + 6q^{59} - 7q^{61} + 5q^{63} + 6q^{65} - 4q^{67} + 30q^{69} + 9q^{71} + 18q^{73} - 2q^{75} + 40q^{77} - 32q^{79} - 23q^{81} - 62q^{83} - 3q^{85} - 12q^{87} - 2q^{89} + 2q^{91} + 14q^{93} - 6q^{95} - 11q^{97} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{5} + 15 \nu^{4} + 8 \nu^{3} + 57 \nu^{2} + 47 \nu + 180$$$$)/83$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 131 \nu - 240$$$$)/83$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{5} + 25 \nu^{4} - 42 \nu^{3} + 95 \nu^{2} - 60 \nu + 300$$$$)/83$$ $$\beta_{4}$$ $$=$$ $$($$$$-20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu - 45$$$$)/249$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{5} + 3 \nu^{4} + 68 \nu^{3} + 28 \nu^{2} + 358 \nu + 36$$$$)/83$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 9$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{3} - 5 \beta_{2} + 5 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$18 \beta_{5} + 9 \beta_{4} + 5 \beta_{3} - 23 \beta_{2} - 46 \beta_{1} + 9$$$$)/3$$

## Character values

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

 $$n$$ $$21$$ $$77$$ $$191$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$

## 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
 −0.956115 + 1.65604i 1.09935 − 1.90412i 0.356769 − 0.617942i −0.956115 − 1.65604i 1.09935 + 1.90412i 0.356769 + 0.617942i
0 −1.28442 2.22469i 0 0.500000 + 0.866025i 0 3.56885 0 −1.79949 + 3.11682i 0
121.2 0 0.182224 + 0.315621i 0 0.500000 + 0.866025i 0 0.635552 0 1.43359 2.48305i 0
121.3 0 1.60220 + 2.77509i 0 0.500000 + 0.866025i 0 −2.20440 0 −3.63409 + 6.29444i 0
201.1 0 −1.28442 + 2.22469i 0 0.500000 0.866025i 0 3.56885 0 −1.79949 3.11682i 0
201.2 0 0.182224 0.315621i 0 0.500000 0.866025i 0 0.635552 0 1.43359 + 2.48305i 0
201.3 0 1.60220 2.77509i 0 0.500000 0.866025i 0 −2.20440 0 −3.63409 6.29444i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 201.3 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.b 6
3.b odd 2 1 3420.2.t.v 6
4.b odd 2 1 1520.2.q.i 6
5.b even 2 1 1900.2.i.c 6
5.c odd 4 2 1900.2.s.c 12
19.c even 3 1 inner 380.2.i.b 6
19.c even 3 1 7220.2.a.n 3
19.d odd 6 1 7220.2.a.o 3
57.h odd 6 1 3420.2.t.v 6
76.g odd 6 1 1520.2.q.i 6
95.i even 6 1 1900.2.i.c 6
95.m odd 12 2 1900.2.s.c 12

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} - T_{3}^{5} + 9 T_{3}^{4} + 2 T_{3}^{3} + 67 T_{3}^{2} - 24 T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(380, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$9 - 24 T + 67 T^{2} + 2 T^{3} + 9 T^{4} - T^{5} + T^{6}$$
$5$ $$( 1 - T + T^{2} )^{3}$$
$7$ $$( 5 - 7 T - 2 T^{2} + T^{3} )^{2}$$
$11$ $$( 9 - 5 T^{2} + T^{3} )^{2}$$
$13$ $$( 1 - T + T^{2} )^{3}$$
$17$ $$6561 - 2916 T + 1539 T^{2} - 54 T^{3} + 45 T^{4} - 3 T^{5} + T^{6}$$
$19$ $$6859 - 342 T^{2} + 7 T^{3} - 18 T^{4} + T^{6}$$
$23$ $$2025 + 1755 T + 2151 T^{2} - 636 T^{3} + 157 T^{4} - 14 T^{5} + T^{6}$$
$29$ $$263169 + 41553 T + 9639 T^{2} + 540 T^{3} + 117 T^{4} + 6 T^{5} + T^{6}$$
$31$ $$( 71 + 14 T - 11 T^{2} + T^{3} )^{2}$$
$37$ $$( 5 - 7 T - 2 T^{2} + T^{3} )^{2}$$
$41$ $$18225 + 8505 T + 4779 T^{2} - 108 T^{3} + 99 T^{4} + 6 T^{5} + T^{6}$$
$43$ $$11025 + 6510 T + 3319 T^{2} + 520 T^{3} + 87 T^{4} - 5 T^{5} + T^{6}$$
$47$ $$263169 - 41553 T + 9639 T^{2} - 540 T^{3} + 117 T^{4} - 6 T^{5} + T^{6}$$
$53$ $$81 + 270 T + 1017 T^{2} - 408 T^{3} + 139 T^{4} - 13 T^{5} + T^{6}$$
$59$ $$263169 - 41553 T + 9639 T^{2} - 540 T^{3} + 117 T^{4} - 6 T^{5} + T^{6}$$
$61$ $$3969 + 1071 T + 730 T^{2} + 7 T^{3} + 66 T^{4} + 7 T^{5} + T^{6}$$
$67$ $$28224 - 15456 T + 7792 T^{2} - 704 T^{3} + 108 T^{4} + 4 T^{5} + T^{6}$$
$71$ $$729 - 486 T + 567 T^{2} + 108 T^{3} + 99 T^{4} - 9 T^{5} + T^{6}$$
$73$ $$625 - 1725 T + 4311 T^{2} - 1192 T^{3} + 255 T^{4} - 18 T^{5} + T^{6}$$
$79$ $$732736 + 263648 T + 67472 T^{2} + 8144 T^{3} + 716 T^{4} + 32 T^{5} + T^{6}$$
$83$ $$( 855 + 294 T + 31 T^{2} + T^{3} )^{2}$$
$89$ $$5184 + 9504 T + 17568 T^{2} - 120 T^{3} + 136 T^{4} + 2 T^{5} + T^{6}$$
$97$ $$93025 + 28670 T + 12191 T^{2} - 424 T^{3} + 215 T^{4} + 11 T^{5} + T^{6}$$
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