# Properties

 Label 1911.2.a.n Level $1911$ Weight $2$ Character orbit 1911.a Self dual yes Analytic conductor $15.259$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 −1.81361 0.470683 2.34292
−2.81361 1.00000 5.91638 1.28917 −2.81361 0 −11.0192 1.00000 −3.62721
1.2 −0.529317 1.00000 −1.71982 −1.77846 −0.529317 0 1.96896 1.00000 0.941367
1.3 1.34292 1.00000 −0.196558 3.48929 1.34292 0 −2.94981 1.00000 4.68585
 $$n$$: e.g. 2-40 or 990-1000 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.n 3
3.b odd 2 1 5733.2.a.bc 3
7.b odd 2 1 273.2.a.d 3
21.c even 2 1 819.2.a.j 3
28.d even 2 1 4368.2.a.bq 3
35.c odd 2 1 6825.2.a.bd 3
91.b odd 2 1 3549.2.a.t 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.d 3 7.b odd 2 1
819.2.a.j 3 21.c even 2 1
1911.2.a.n 3 1.a even 1 1 trivial
3549.2.a.t 3 91.b odd 2 1
4368.2.a.bq 3 28.d even 2 1
5733.2.a.bc 3 3.b odd 2 1
6825.2.a.bd 3 35.c odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} + 2 T_{2}^{2} - 3 T_{2} - 2$$ $$T_{5}^{3} - 3 T_{5}^{2} - 4 T_{5} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 - 3 T + 2 T^{2} + T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$8 - 4 T - 3 T^{2} + T^{3}$$
$7$ $$T^{3}$$
$11$ $$8 - 28 T + 2 T^{2} + T^{3}$$
$13$ $$( -1 + T )^{3}$$
$17$ $$32 + 4 T - 8 T^{2} + T^{3}$$
$19$ $$128 - 16 T - 7 T^{2} + T^{3}$$
$23$ $$8 + 20 T + 9 T^{2} + T^{3}$$
$29$ $$-76 - 32 T + T^{2} + T^{3}$$
$31$ $$272 - 40 T - 7 T^{2} + T^{3}$$
$37$ $$32 + 20 T - 12 T^{2} + T^{3}$$
$41$ $$16 + 16 T - 10 T^{2} + T^{3}$$
$43$ $$16 - 16 T + T^{2} + T^{3}$$
$47$ $$-68 + 80 T - 17 T^{2} + T^{3}$$
$53$ $$148 - 96 T + 5 T^{2} + T^{3}$$
$59$ $$64 + 20 T - 12 T^{2} + T^{3}$$
$61$ $$-232 - 36 T + 10 T^{2} + T^{3}$$
$67$ $$608 - 128 T - 2 T^{2} + T^{3}$$
$71$ $$-496 - 92 T + 4 T^{2} + T^{3}$$
$73$ $$-436 - 144 T - 5 T^{2} + T^{3}$$
$79$ $$32 + 40 T + 13 T^{2} + T^{3}$$
$83$ $$76 - 32 T - T^{2} + T^{3}$$
$89$ $$344 - 4 T - 13 T^{2} + T^{3}$$
$97$ $$524 - 184 T - 9 T^{2} + T^{3}$$