# Properties

 Label 1911.2.a.v 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} - 3 q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9} + O(q^{10})$$ $$5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} - 3 q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9} + 6 q^{11} - 6 q^{12} - 5 q^{13} + 3 q^{15} + 10 q^{17} + 2 q^{18} - 5 q^{19} - 6 q^{20} + 12 q^{22} + q^{23} - 6 q^{24} + 16 q^{25} - 2 q^{26} - 5 q^{27} + 17 q^{29} - q^{31} + 14 q^{32} - 6 q^{33} + 6 q^{36} + 4 q^{37} + 24 q^{38} + 5 q^{39} - 8 q^{40} - 14 q^{41} - q^{43} - 8 q^{44} - 3 q^{45} + 20 q^{46} + 3 q^{47} + 26 q^{50} - 10 q^{51} - 6 q^{52} + 17 q^{53} - 2 q^{54} - 2 q^{55} + 5 q^{57} - 28 q^{58} + 8 q^{59} + 6 q^{60} + 18 q^{61} + 8 q^{62} + 8 q^{64} + 3 q^{65} - 12 q^{66} + 16 q^{67} + 8 q^{68} - q^{69} - 16 q^{71} + 6 q^{72} - 23 q^{73} + 28 q^{74} - 16 q^{75} + 26 q^{76} + 2 q^{78} + 3 q^{79} + 36 q^{80} + 5 q^{81} + 11 q^{83} - 2 q^{85} - 24 q^{86} - 17 q^{87} + 28 q^{88} + 35 q^{89} + 34 q^{92} + q^{93} + 39 q^{95} - 14 q^{96} - 27 q^{97} + 6 q^{99} + O(q^{100})$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1911.2.a.v 5 1.a even 1 1 trivial
1911.2.a.w yes 5 7.b odd 2 1
5733.2.a.bn 5 21.c even 2 1
5733.2.a.bo 5 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1911))$$:

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

## Hecke characteristic polynomials

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