# Properties

 Label 1911.2.a.j Level $1911$ Weight $2$ Character orbit 1911.a Self dual yes Analytic conductor $15.259$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

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## 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−1.41421 1.00000 0 −1.00000 −1.41421 0 2.82843 1.00000 1.41421
1.2 1.41421 1.00000 0 −1.00000 1.41421 0 −2.82843 1.00000 −1.41421
 $$n$$: 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.j yes 2
3.b odd 2 1 5733.2.a.r 2
7.b odd 2 1 1911.2.a.i 2
21.c even 2 1 5733.2.a.q 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$23 + 10 T + T^{2}$$
$23$ $$-1 + 2 T + T^{2}$$
$29$ $$-9 + 6 T + T^{2}$$
$31$ $$-1 - 2 T + T^{2}$$
$37$ $$( 4 + T )^{2}$$
$41$ $$56 - 16 T + T^{2}$$
$43$ $$-7 - 2 T + T^{2}$$
$47$ $$-31 - 2 T + T^{2}$$
$53$ $$-1 - 2 T + T^{2}$$
$59$ $$-28 - 4 T + T^{2}$$
$61$ $$4 + 12 T + T^{2}$$
$67$ $$34 + 12 T + T^{2}$$
$71$ $$8 + 8 T + T^{2}$$
$73$ $$47 + 14 T + T^{2}$$
$79$ $$-23 + 14 T + T^{2}$$
$83$ $$-63 - 6 T + T^{2}$$
$89$ $$-23 + 6 T + T^{2}$$
$97$ $$-49 + 14 T + T^{2}$$
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