# Properties

 Label 1911.2.a.b 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

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

## 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
−1.00000 −1.00000 −1.00000 −4.00000 1.00000 0 3.00000 1.00000 4.00000
 $$n$$: 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.b 1
3.b odd 2 1 5733.2.a.k 1
7.b odd 2 1 1911.2.a.c 1
7.d odd 6 2 273.2.i.a 2
21.c even 2 1 5733.2.a.i 1
21.g even 6 2 819.2.j.a 2

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

 $$T_{2} + 1$$ $$T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$1 + T$$
$5$ $$4 + T$$
$7$ $$T$$
$11$ $$5 + T$$
$13$ $$-1 + T$$
$17$ $$3 + T$$
$19$ $$-5 + T$$
$23$ $$-6 + T$$
$29$ $$-7 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$8 + T$$
$43$ $$-2 + T$$
$47$ $$9 + T$$
$53$ $$-9 + T$$
$59$ $$-9 + T$$
$61$ $$1 + T$$
$67$ $$-7 + T$$
$71$ $$3 + T$$
$73$ $$-6 + T$$
$79$ $$10 + T$$
$83$ $$T$$
$89$ $$-8 + T$$
$97$ $$18 + T$$