# Properties

 Label 1911.2.a.t Level $1911$ Weight $2$ Character orbit 1911.a Self dual yes Analytic conductor $15.259$ Analytic rank $1$ 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: $$1$$ 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})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 5 q^{3} + 6 q^{4} - 3 q^{5} + 3 q^{8} + 5 q^{9} + O(q^{10})$$ $$5 q - 5 q^{3} + 6 q^{4} - 3 q^{5} + 3 q^{8} + 5 q^{9} - 2 q^{10} - q^{11} - 6 q^{12} - 5 q^{13} + 3 q^{15} - 13 q^{17} - 7 q^{19} - 13 q^{20} - 19 q^{22} - 4 q^{23} - 3 q^{24} + 16 q^{25} - 5 q^{27} - 12 q^{29} + 2 q^{30} - 6 q^{31} + 21 q^{32} + q^{33} - 7 q^{34} + 6 q^{36} + 11 q^{37} - 14 q^{38} + 5 q^{39} - 11 q^{40} - 10 q^{41} + 10 q^{43} - 29 q^{44} - 3 q^{45} + q^{46} + 4 q^{47} - 29 q^{50} + 13 q^{51} - 6 q^{52} - 9 q^{53} + 12 q^{55} + 7 q^{57} + 34 q^{58} - 7 q^{59} + 13 q^{60} - 23 q^{61} + 24 q^{62} - 13 q^{64} + 3 q^{65} + 19 q^{66} + 25 q^{67} - 20 q^{68} + 4 q^{69} - 27 q^{71} + 3 q^{72} - 18 q^{73} + 15 q^{74} - 16 q^{75} - 2 q^{76} + 8 q^{79} - 41 q^{80} + 5 q^{81} - 26 q^{82} - 12 q^{83} + 10 q^{85} - 19 q^{86} + 12 q^{87} - 36 q^{88} - 29 q^{89} - 2 q^{90} - 50 q^{92} + 6 q^{93} + 2 q^{94} - 33 q^{95} - 21 q^{96} - 13 q^{97} - q^{99} + O(q^{100})$$

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.t 5
3.b odd 2 1 5733.2.a.bq 5
7.b odd 2 1 1911.2.a.u 5
7.c even 3 2 273.2.i.e 10
21.c even 2 1 5733.2.a.bp 5
21.h odd 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.c even 3 2
819.2.j.g 10 21.h odd 6 2
1911.2.a.t 5 1.a even 1 1 trivial
1911.2.a.u 5 7.b odd 2 1
5733.2.a.bp 5 21.c even 2 1
5733.2.a.bq 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1911))$$:

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

## Hecke characteristic polynomials

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