# Properties

 Label 3381.2.a.y Level $3381$ Weight $2$ Character orbit 3381.a Self dual yes Analytic conductor $26.997$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3381 = 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3381.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$26.9974209234$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: 5.5.3176240.1 Defining polynomial: $$x^{5} - x^{4} - 9 x^{3} + 7 x^{2} + 19 x - 9$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 9 x^{3} + 7 x^{2} + 19 x - 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 4$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - 6 \nu^{2} - \nu + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 4$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + 6 \beta_{2} + \beta_{1} + 20$$

## 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
 −2.34113 −1.67261 0.443666 2.08269 2.48738
−2.34113 −1.00000 3.48089 1.12582 2.34113 0 −3.46695 1.00000 −2.63569
1.2 −1.67261 −1.00000 0.797614 −3.68373 1.67261 0 2.01112 1.00000 6.16143
1.3 0.443666 −1.00000 −1.80316 2.13100 −0.443666 0 −1.68733 1.00000 0.945453
1.4 2.08269 −1.00000 2.33760 1.37958 −2.08269 0 0.703109 1.00000 2.87324
1.5 2.48738 −1.00000 4.18706 −2.95267 −2.48738 0 5.44006 1.00000 −7.34443
 $$n$$: 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
$$3$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## 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 3381.2.a.y 5
7.b odd 2 1 3381.2.a.z yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3381.2.a.y 5 1.a even 1 1 trivial
3381.2.a.z yes 5 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3381))$$:

 $$T_{2}^{5} - T_{2}^{4} - 9 T_{2}^{3} + 7 T_{2}^{2} + 19 T_{2} - 9$$ $$T_{5}^{5} + 2 T_{5}^{4} - 13 T_{5}^{3} - 8 T_{5}^{2} + 53 T_{5} - 36$$ $$T_{11}^{5} + 2 T_{11}^{4} - 16 T_{11}^{3} - 26 T_{11}^{2} + 7 T_{11} + 4$$ $$T_{13}^{5} + 4 T_{13}^{4} - 17 T_{13}^{3} - 104 T_{13}^{2} - 145 T_{13} - 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-9 + 19 T + 7 T^{2} - 9 T^{3} - T^{4} + T^{5}$$
$3$ $$( 1 + T )^{5}$$
$5$ $$-36 + 53 T - 8 T^{2} - 13 T^{3} + 2 T^{4} + T^{5}$$
$7$ $$T^{5}$$
$11$ $$4 + 7 T - 26 T^{2} - 16 T^{3} + 2 T^{4} + T^{5}$$
$13$ $$-40 - 145 T - 104 T^{2} - 17 T^{3} + 4 T^{4} + T^{5}$$
$17$ $$-4 - 27 T + 78 T^{2} - 38 T^{3} + 2 T^{4} + T^{5}$$
$19$ $$336 - 155 T - 192 T^{2} - 14 T^{3} + 8 T^{4} + T^{5}$$
$23$ $$( -1 + T )^{5}$$
$29$ $$-1162 + 481 T + 202 T^{2} - 58 T^{3} - 4 T^{4} + T^{5}$$
$31$ $$-148 - 157 T + 40 T^{3} + 12 T^{4} + T^{5}$$
$37$ $$-334 + 3219 T - 648 T^{2} - 144 T^{3} + 6 T^{4} + T^{5}$$
$41$ $$-448 - 877 T - 426 T^{2} - 52 T^{3} + 6 T^{4} + T^{5}$$
$43$ $$-1324 + 511 T + 716 T^{2} + 207 T^{3} + 24 T^{4} + T^{5}$$
$47$ $$-5376 - 3776 T - 352 T^{2} + 136 T^{3} + 24 T^{4} + T^{5}$$
$53$ $$-7554 + 1471 T + 560 T^{2} - 123 T^{3} - 2 T^{4} + T^{5}$$
$59$ $$500 - 125 T - 310 T^{2} + 29 T^{3} + 16 T^{4} + T^{5}$$
$61$ $$30564 - 3523 T - 1726 T^{2} + 19 T^{3} + 22 T^{4} + T^{5}$$
$67$ $$-1084 - 2965 T - 1664 T^{2} - 83 T^{3} + 16 T^{4} + T^{5}$$
$71$ $$-960 - 535 T + 154 T^{2} + 55 T^{3} - 16 T^{4} + T^{5}$$
$73$ $$100600 + 25115 T - 576 T^{2} - 328 T^{3} + T^{5}$$
$79$ $$-435816 + 34343 T + 4812 T^{2} - 388 T^{3} - 12 T^{4} + T^{5}$$
$83$ $$-216 + 425 T - 174 T^{2} - 18 T^{3} + 10 T^{4} + T^{5}$$
$89$ $$7388 + 17757 T + 2022 T^{2} - 209 T^{3} - 16 T^{4} + T^{5}$$
$97$ $$36 - 275 T - 344 T^{2} - 102 T^{3} - 4 T^{4} + T^{5}$$