# Properties

 Label 1911.2.a.w Level $1911$ Weight $2$ Character orbit 1911.a Self dual yes Analytic conductor $15.259$ Analytic rank $0$ Dimension $5$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1911 = 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1911.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$15.2594118263$$ Analytic rank: $$0$$ Dimension: $$5$$ Coefficient field: 5.5.2196544.1 Defining polynomial: $$x^{5} - 2 x^{4} - 6 x^{3} + 10 x^{2} + 7 x - 8$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 2 x^{4} - 6 x^{3} + 10 x^{2} + 7 x - 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 2$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 6 \nu^{2} + 3 \nu + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 7 \beta_{2} + \beta_{1} + 13$$

## 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
 −2.04659 −1.10811 0.755150 1.79032 2.60923
−2.04659 1.00000 2.18852 0.155007 −2.04659 0 −0.385824 1.00000 −0.317236
1.2 −1.10811 1.00000 −0.772089 2.82337 −1.10811 0 3.07178 1.00000 −3.12861
1.3 0.755150 1.00000 −1.42975 −3.73850 0.755150 0 −2.58997 1.00000 −2.82313
1.4 1.79032 1.00000 1.20526 4.32535 1.79032 0 −1.42284 1.00000 7.74378
1.5 2.60923 1.00000 4.80806 −0.565224 2.60923 0 7.32686 1.00000 −1.47480
 $$n$$: 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
$$3$$ $$-1$$
$$7$$ $$-1$$
$$13$$ $$-1$$

## 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 1911.2.a.w yes 5
3.b odd 2 1 5733.2.a.bn 5
7.b odd 2 1 1911.2.a.v 5
21.c even 2 1 5733.2.a.bo 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.2.a.v 5 7.b odd 2 1
1911.2.a.w yes 5 1.a even 1 1 trivial
5733.2.a.bn 5 3.b odd 2 1
5733.2.a.bo 5 21.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1911))$$:

 $$T_{2}^{5} - 2 T_{2}^{4} - 6 T_{2}^{3} + 10 T_{2}^{2} + 7 T_{2} - 8$$ $$T_{5}^{5} - 3 T_{5}^{4} - 16 T_{5}^{3} + 40 T_{5}^{2} + 20 T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-8 + 7 T + 10 T^{2} - 6 T^{3} - 2 T^{4} + T^{5}$$
$3$ $$( -1 + T )^{5}$$
$5$ $$-4 + 20 T + 40 T^{2} - 16 T^{3} - 3 T^{4} + T^{5}$$
$7$ $$T^{5}$$
$11$ $$-8 - 180 T + 136 T^{2} - 16 T^{3} - 6 T^{4} + T^{5}$$
$13$ $$( -1 + T )^{5}$$
$17$ $$3872 + 16 T - 448 T^{2} - 32 T^{3} + 10 T^{4} + T^{5}$$
$19$ $$64 - 704 T + 416 T^{2} - 56 T^{3} - 5 T^{4} + T^{5}$$
$23$ $$-416 + 544 T - 24 T^{2} - 60 T^{3} - T^{4} + T^{5}$$
$29$ $$6064 - 4176 T + 744 T^{2} + 32 T^{3} - 17 T^{4} + T^{5}$$
$31$ $$-64 + 256 T + 32 T^{2} - 48 T^{3} - T^{4} + T^{5}$$
$37$ $$64 - 432 T + 448 T^{2} - 88 T^{3} - 4 T^{4} + T^{5}$$
$41$ $$-2888 + 988 T + 648 T^{2} - 32 T^{3} - 14 T^{4} + T^{5}$$
$43$ $$1792 + 896 T - 96 T^{2} - 72 T^{3} + T^{4} + T^{5}$$
$47$ $$37556 + 7324 T - 728 T^{2} - 172 T^{3} + 3 T^{4} + T^{5}$$
$53$ $$-4112 - 1168 T + 616 T^{2} + 16 T^{3} - 17 T^{4} + T^{5}$$
$59$ $$34832 + 8652 T - 1152 T^{2} - 184 T^{3} + 8 T^{4} + T^{5}$$
$61$ $$8608 - 1584 T - 944 T^{2} + 8 T^{3} + 18 T^{4} + T^{5}$$
$67$ $$-11264 - 384 T + 832 T^{2} - 16 T^{3} - 16 T^{4} + T^{5}$$
$71$ $$-1408 - 5156 T - 2136 T^{2} - 96 T^{3} + 16 T^{4} + T^{5}$$
$73$ $$-176 - 592 T - 248 T^{2} + 160 T^{3} - 23 T^{4} + T^{5}$$
$79$ $$25936 + 7056 T - 632 T^{2} - 232 T^{3} - 3 T^{4} + T^{5}$$
$83$ $$52 + 92 T - 40 T^{2} - 84 T^{3} + 11 T^{4} + T^{5}$$
$89$ $$-4796 - 3804 T + 424 T^{2} + 344 T^{3} + 35 T^{4} + T^{5}$$
$97$ $$54992 - 35280 T + 5272 T^{2} - 32 T^{3} - 27 T^{4} + T^{5}$$