# Properties

 Label 1911.2.a.u 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: 5.5.375116.1 Defining polynomial: $$x^{5} - x^{4} - 6 x^{3} + 7 x^{2} + 2 x - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 273) 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{4} - 5 \nu^{2} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{4} + 6 \nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{4} - 6 \nu^{2} + 2 \nu + 3$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + \nu^{3} + 6 \nu^{2} - 6 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} + 3 \beta_{3} + 2 \beta_{2} - 1$$ $$\nu^{4}$$ $$=$$ $$-6 \beta_{3} - \beta_{2} + 6 \beta_{1} + 15$$

## 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
 −2.44025 1.32173 −0.562376 2.17362 0.507274
−2.14957 1.00000 2.62066 3.73093 −2.14957 0 −1.33415 1.00000 −8.01989
1.2 −1.78646 1.00000 1.19144 −3.42992 −1.78646 0 1.44447 1.00000 6.12741
1.3 0.0776754 1.00000 −1.99397 2.20243 0.0776754 0 −0.310233 1.00000 0.171074
1.4 1.32155 1.00000 −0.253495 −2.02568 1.32155 0 −2.97812 1.00000 −2.67705
1.5 2.53680 1.00000 4.43536 2.52225 2.53680 0 6.17804 1.00000 6.39846
 $$n$$: 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.u 5
3.b odd 2 1 5733.2.a.bp 5
7.b odd 2 1 1911.2.a.t 5
7.d odd 6 2 273.2.i.e 10
21.c even 2 1 5733.2.a.bq 5
21.g even 6 2 819.2.j.g 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.i.e 10 7.d odd 6 2
819.2.j.g 10 21.g even 6 2
1911.2.a.t 5 7.b odd 2 1
1911.2.a.u 5 1.a even 1 1 trivial
5733.2.a.bp 5 3.b odd 2 1
5733.2.a.bq 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1911))$$:

 $$T_{2}^{5} - 8 T_{2}^{3} - T_{2}^{2} + 13 T_{2} - 1$$ $$T_{5}^{5} - 3 T_{5}^{4} - 16 T_{5}^{3} + 47 T_{5}^{2} + 48 T_{5} - 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 13 T - T^{2} - 8 T^{3} + T^{5}$$
$3$ $$( -1 + T )^{5}$$
$5$ $$-144 + 48 T + 47 T^{2} - 16 T^{3} - 3 T^{4} + T^{5}$$
$7$ $$T^{5}$$
$11$ $$-1 - 12 T + 38 T^{2} - 23 T^{3} + T^{4} + T^{5}$$
$13$ $$( -1 + T )^{5}$$
$17$ $$1 + 22 T - 82 T^{2} + 55 T^{3} - 13 T^{4} + T^{5}$$
$19$ $$-19 - 139 T + 106 T^{2} - 6 T^{3} - 7 T^{4} + T^{5}$$
$23$ $$1108 + 408 T - 143 T^{2} - 47 T^{3} + 4 T^{4} + T^{5}$$
$29$ $$-251 - 537 T - 231 T^{2} + 2 T^{3} + 12 T^{4} + T^{5}$$
$31$ $$-1376 + 120 T + 251 T^{2} - 39 T^{3} - 6 T^{4} + T^{5}$$
$37$ $$7456 - 5640 T + 1351 T^{2} - 74 T^{3} - 11 T^{4} + T^{5}$$
$41$ $$-224 - 392 T + 279 T^{2} - 13 T^{3} - 10 T^{4} + T^{5}$$
$43$ $$-76 - 688 T + 413 T^{2} - 37 T^{3} - 10 T^{4} + T^{5}$$
$47$ $$-18392 + 7532 T - 178 T^{2} - 171 T^{3} + 4 T^{4} + T^{5}$$
$53$ $$84029 + 8722 T - 1960 T^{2} - 209 T^{3} + 9 T^{4} + T^{5}$$
$59$ $$-71807 + 11494 T + 1400 T^{2} - 225 T^{3} - 7 T^{4} + T^{5}$$
$61$ $$-1051 + 1506 T - 804 T^{2} + 199 T^{3} - 23 T^{4} + T^{5}$$
$67$ $$11303 - 4320 T + 28 T^{2} + 177 T^{3} - 25 T^{4} + T^{5}$$
$71$ $$761 - 1624 T + 206 T^{2} + 205 T^{3} + 27 T^{4} + T^{5}$$
$73$ $$7468 - 4280 T + 571 T^{2} + 61 T^{3} - 18 T^{4} + T^{5}$$
$79$ $$18572 + 11488 T + 1059 T^{2} - 229 T^{3} - 8 T^{4} + T^{5}$$
$83$ $$-17248 - 2856 T + 2103 T^{2} - 145 T^{3} - 12 T^{4} + T^{5}$$
$89$ $$-2144 + 2984 T - 1421 T^{2} + 302 T^{3} - 29 T^{4} + T^{5}$$
$97$ $$-14308 - 6804 T + 3039 T^{2} - 186 T^{3} - 13 T^{4} + T^{5}$$