# Properties

 Label 1911.2.a.a Level $1911$ Weight $2$ Character orbit 1911.a Self dual yes Analytic conductor $15.259$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 273) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
−2.00000 1.00000 2.00000 1.00000 −2.00000 0 0 1.00000 −2.00000
 $$n$$: 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.a 1
3.b odd 2 1 5733.2.a.m 1
7.b odd 2 1 273.2.a.a 1
21.c even 2 1 819.2.a.e 1
28.d even 2 1 4368.2.a.q 1
35.c odd 2 1 6825.2.a.l 1
91.b odd 2 1 3549.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.a.a 1 7.b odd 2 1
819.2.a.e 1 21.c even 2 1
1911.2.a.a 1 1.a even 1 1 trivial
3549.2.a.d 1 91.b odd 2 1
4368.2.a.q 1 28.d even 2 1
5733.2.a.m 1 3.b odd 2 1
6825.2.a.l 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1911))$$:

 $$T_{2} + 2$$ $$T_{5} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$-1 + T$$
$5$ $$-1 + T$$
$7$ $$T$$
$11$ $$2 + T$$
$13$ $$1 + T$$
$17$ $$-4 + T$$
$19$ $$3 + T$$
$23$ $$9 + T$$
$29$ $$1 + T$$
$31$ $$-5 + T$$
$37$ $$8 + T$$
$41$ $$6 + T$$
$43$ $$9 + T$$
$47$ $$-3 + T$$
$53$ $$-3 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$2 + T$$
$71$ $$-12 + T$$
$73$ $$5 + T$$
$79$ $$13 + T$$
$83$ $$-11 + T$$
$89$ $$1 + T$$
$97$ $$1 + T$$