# Properties

 Label 381.2.a.d Level $381$ Weight $2$ Character orbit 381.a Self dual yes Analytic conductor $3.042$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$381 = 3 \cdot 127$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 381.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.04230031701$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.246832.1 Defining polynomial: $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} - q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} + \beta_{3} q^{5} -\beta_{2} q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} - q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{4} + \beta_{3} q^{5} -\beta_{2} q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} + ( 2 \beta_{1} + \beta_{2} ) q^{8} + q^{9} + ( -\beta_{3} - \beta_{4} ) q^{10} + ( 3 - \beta_{1} + \beta_{4} ) q^{11} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{12} + ( -1 - \beta_{1} + \beta_{2} ) q^{13} + ( 3 - \beta_{1} + \beta_{4} ) q^{14} -\beta_{3} q^{15} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} ) q^{16} + ( 2 - 2 \beta_{1} - 2 \beta_{3} ) q^{17} + \beta_{2} q^{18} + ( 2 - 2 \beta_{2} - 2 \beta_{4} ) q^{19} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} ) q^{20} + ( \beta_{1} - \beta_{2} ) q^{21} + ( -3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{22} + ( 3 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} ) q^{23} + ( -2 \beta_{1} - \beta_{2} ) q^{24} + ( -2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{25} + ( 3 - \beta_{1} - \beta_{2} + \beta_{4} ) q^{26} - q^{27} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{28} + ( 2 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{29} + ( \beta_{3} + \beta_{4} ) q^{30} + ( -1 + 3 \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{31} + ( 4 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{32} + ( -3 + \beta_{1} - \beta_{4} ) q^{33} + ( 2 - 4 \beta_{1} + 2 \beta_{4} ) q^{34} + ( 2 - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{35} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{36} + ( -1 - \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{37} + ( -2 - 2 \beta_{2} - 2 \beta_{4} ) q^{38} + ( 1 + \beta_{1} - \beta_{2} ) q^{39} + ( -4 + 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{40} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{41} + ( -3 + \beta_{1} - \beta_{4} ) q^{42} + ( 4 - 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{43} + ( 3 - \beta_{1} + 2 \beta_{3} + 3 \beta_{4} ) q^{44} + \beta_{3} q^{45} + ( -2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{4} ) q^{46} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} ) q^{47} + ( -3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{48} + ( -1 - 2 \beta_{1} - \beta_{3} + \beta_{4} ) q^{49} + ( -3 - \beta_{1} - 3 \beta_{2} - \beta_{4} ) q^{50} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{51} + ( -2 \beta_{1} - 3 \beta_{3} - \beta_{4} ) q^{52} + ( \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{53} -\beta_{2} q^{54} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{55} + ( -3 + \beta_{1} + 2 \beta_{3} + \beta_{4} ) q^{56} + ( -2 + 2 \beta_{2} + 2 \beta_{4} ) q^{57} + ( -2 + 5 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{58} + ( 7 - 3 \beta_{1} - \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{59} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{60} + ( -5 + 6 \beta_{1} - \beta_{3} - \beta_{4} ) q^{61} + ( -6 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} ) q^{62} + ( -\beta_{1} + \beta_{2} ) q^{63} + ( -2 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{64} + ( 2 - \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{65} + ( 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{66} + ( 2 \beta_{1} + 4 \beta_{2} + \beta_{3} + 5 \beta_{4} ) q^{67} + ( -2 - 6 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{69} + ( \beta_{1} - \beta_{2} + 3 \beta_{3} + \beta_{4} ) q^{70} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{71} + ( 2 \beta_{1} + \beta_{2} ) q^{72} + ( -7 - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{73} + ( -3 - 5 \beta_{1} - \beta_{2} - 6 \beta_{3} - 3 \beta_{4} ) q^{74} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} ) q^{75} + ( -6 - 2 \beta_{2} + 2 \beta_{4} ) q^{76} + ( 1 - 5 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{77} + ( -3 + \beta_{1} + \beta_{2} - \beta_{4} ) q^{78} + ( 2 + 2 \beta_{1} + 2 \beta_{3} ) q^{79} + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} ) q^{80} + q^{81} + ( -4 - 8 \beta_{1} - 6 \beta_{3} - 2 \beta_{4} ) q^{82} + ( 1 + 5 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{83} + ( \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{84} + ( -6 + 2 \beta_{2} + 2 \beta_{3} ) q^{85} + ( -3 - 7 \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{86} + ( -2 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{87} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{88} + ( 2 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{4} ) q^{89} + ( -\beta_{3} - \beta_{4} ) q^{90} + ( 6 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} ) q^{91} + ( -2 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{92} + ( 1 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} ) q^{93} + ( -2 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{94} + ( 4 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{95} + ( -4 \beta_{1} - \beta_{2} - 4 \beta_{3} ) q^{96} + ( -6 + 2 \beta_{1} + 4 \beta_{3} + 2 \beta_{4} ) q^{97} + ( 1 - 5 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{98} + ( 3 - \beta_{1} + \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5q + 2q^{2} - 5q^{3} + 6q^{4} + q^{5} - 2q^{6} + 6q^{8} + 5q^{9} + O(q^{10})$$ $$5q + 2q^{2} - 5q^{3} + 6q^{4} + q^{5} - 2q^{6} + 6q^{8} + 5q^{9} - 2q^{10} + 14q^{11} - 6q^{12} - 5q^{13} + 14q^{14} - q^{15} + 8q^{16} + 4q^{17} + 2q^{18} + 4q^{19} + 6q^{20} - 2q^{22} + 15q^{23} - 6q^{24} - 6q^{25} + 12q^{26} - 5q^{27} - 2q^{28} + 9q^{29} + 2q^{30} + 3q^{31} + 14q^{32} - 14q^{33} + 4q^{34} + 4q^{35} + 6q^{36} - 5q^{37} - 16q^{38} + 5q^{39} - 14q^{40} + 4q^{41} - 14q^{42} + 10q^{43} + 18q^{44} + q^{45} - 4q^{46} - 4q^{47} - 8q^{48} - 9q^{49} - 24q^{50} - 4q^{51} - 8q^{52} + 3q^{53} - 2q^{54} + 4q^{55} - 10q^{56} - 4q^{57} + 6q^{58} + 23q^{59} - 6q^{60} - 15q^{61} - 24q^{62} + 3q^{65} + 2q^{66} + 18q^{67} - 24q^{68} - 15q^{69} + 4q^{70} + 12q^{71} + 6q^{72} - 43q^{73} - 36q^{74} + 6q^{75} - 32q^{76} + 4q^{77} - 12q^{78} + 16q^{79} + 4q^{80} + 5q^{81} - 44q^{82} + 11q^{83} + 2q^{84} - 24q^{85} - 28q^{86} - 9q^{87} - 14q^{88} + 9q^{89} - 2q^{90} + 26q^{91} + 14q^{92} - 3q^{93} - 14q^{94} + 16q^{95} - 14q^{96} - 20q^{97} - 10q^{98} + 14q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 5 x^{3} + 6 x^{2} + 7 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 3 \nu^{3} - 2 \nu^{2} + 7 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 2 \beta_{2} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 3 \beta_{3} + 8 \beta_{2} + 10 \beta_{1} + 6$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.245526 1.71250 −1.15351 −1.51908 2.71457
−2.18524 −1.00000 2.77529 2.15766 2.18524 −2.43077 −1.69419 1.00000 −4.71500
1.2 −0.779856 −1.00000 −1.39182 −2.98063 0.779856 −2.49235 2.64513 1.00000 2.32446
1.3 0.484093 −1.00000 −1.76565 2.26452 −0.484093 1.63760 −1.82293 1.00000 1.09624
1.4 1.82669 −1.00000 1.33679 −0.563416 −1.82669 3.34577 −1.21147 1.00000 −1.02918
1.5 2.65432 −1.00000 5.04540 0.121872 −2.65432 −0.0602522 8.08346 1.00000 0.323487
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$127$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 381.2.a.d 5
3.b odd 2 1 1143.2.a.g 5
4.b odd 2 1 6096.2.a.bf 5
5.b even 2 1 9525.2.a.j 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
381.2.a.d 5 1.a even 1 1 trivial
1143.2.a.g 5 3.b odd 2 1
6096.2.a.bf 5 4.b odd 2 1
9525.2.a.j 5 5.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{5} - 2 T_{2}^{4} - 6 T_{2}^{3} + 10 T_{2}^{2} + 5 T_{2} - 4$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(381))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-4 + 5 T + 10 T^{2} - 6 T^{3} - 2 T^{4} + T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$-1 + 7 T + 11 T^{2} - 9 T^{3} - T^{4} + T^{5}$$
$7$ $$2 + 33 T - 4 T^{2} - 13 T^{3} + T^{5}$$
$11$ $$8 + 33 T - 96 T^{2} + 63 T^{3} - 14 T^{4} + T^{5}$$
$13$ $$19 - 9 T - 33 T^{2} - 3 T^{3} + 5 T^{4} + T^{5}$$
$17$ $$64 + 304 T + 64 T^{2} - 32 T^{3} - 4 T^{4} + T^{5}$$
$19$ $$-256 + 192 T + 64 T^{2} - 40 T^{3} - 4 T^{4} + T^{5}$$
$23$ $$592 - 448 T - 4 T^{2} + 64 T^{3} - 15 T^{4} + T^{5}$$
$29$ $$-2279 + 29 T + 347 T^{2} - 31 T^{3} - 9 T^{4} + T^{5}$$
$31$ $$1840 + 1424 T - 4 T^{2} - 100 T^{3} - 3 T^{4} + T^{5}$$
$37$ $$907 - 669 T - 653 T^{2} - 91 T^{3} + 5 T^{4} + T^{5}$$
$41$ $$-2368 + 1424 T + 320 T^{2} - 116 T^{3} - 4 T^{4} + T^{5}$$
$43$ $$-11272 + 1661 T + 746 T^{2} - 89 T^{3} - 10 T^{4} + T^{5}$$
$47$ $$152 + 581 T - 114 T^{2} - 49 T^{3} + 4 T^{4} + T^{5}$$
$53$ $$211 - 475 T + 305 T^{2} - 55 T^{3} - 3 T^{4} + T^{5}$$
$59$ $$-1520 - 2496 T + 284 T^{2} + 128 T^{3} - 23 T^{4} + T^{5}$$
$61$ $$119213 + 7395 T - 2891 T^{2} - 191 T^{3} + 15 T^{4} + T^{5}$$
$67$ $$9836 - 18595 T + 3982 T^{2} - 137 T^{3} - 18 T^{4} + T^{5}$$
$71$ $$-106 - 135 T + 106 T^{2} + 19 T^{3} - 12 T^{4} + T^{5}$$
$73$ $$1369 + 9735 T + 4317 T^{2} + 665 T^{3} + 43 T^{4} + T^{5}$$
$79$ $$-256 - 464 T + 64 T^{2} + 64 T^{3} - 16 T^{4} + T^{5}$$
$83$ $$13120 - 10976 T + 2744 T^{2} - 176 T^{3} - 11 T^{4} + T^{5}$$
$89$ $$13159 + 7699 T + 671 T^{2} - 153 T^{3} - 9 T^{4} + T^{5}$$
$97$ $$-5120 - 6512 T - 1280 T^{2} + 28 T^{3} + 20 T^{4} + T^{5}$$
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