# Properties

 Label 931.2.a.e Level $931$ Weight $2$ Character orbit 931.a Self dual yes Analytic conductor $7.434$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$931 = 7^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 931.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$7.43407242818$$ 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: no (minimal twist has level 133) 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.

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

## 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
 −1.41421 1.41421
−2.41421 −1.41421 3.82843 −1.00000 3.41421 0 −4.41421 −1.00000 2.41421
1.2 0.414214 1.41421 −1.82843 −1.00000 0.585786 0 −1.58579 −1.00000 −0.414214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$19$$ $$-1$$

## 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 931.2.a.e 2
3.b odd 2 1 8379.2.a.bl 2
7.b odd 2 1 931.2.a.f 2
7.c even 3 2 931.2.f.i 4
7.d odd 6 2 133.2.f.c 4
21.c even 2 1 8379.2.a.bi 2
21.g even 6 2 1197.2.j.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.c 4 7.d odd 6 2
931.2.a.e 2 1.a even 1 1 trivial
931.2.a.f 2 7.b odd 2 1
931.2.f.i 4 7.c even 3 2
1197.2.j.e 4 21.g even 6 2
8379.2.a.bi 2 21.c even 2 1
8379.2.a.bl 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(931))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{3}^{2} - 2$$ $$T_{5} + 1$$ $$T_{13}^{2} + 4 T_{13} - 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$-2 + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-1 - 2 T + T^{2}$$
$13$ $$-14 + 4 T + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$( -1 + T )^{2}$$
$23$ $$47 + 14 T + T^{2}$$
$29$ $$62 + 16 T + T^{2}$$
$31$ $$-14 - 4 T + T^{2}$$
$37$ $$-46 - 4 T + T^{2}$$
$41$ $$28 - 12 T + T^{2}$$
$43$ $$-49 - 2 T + T^{2}$$
$47$ $$7 - 6 T + T^{2}$$
$53$ $$-18 + T^{2}$$
$59$ $$8 + 8 T + T^{2}$$
$61$ $$73 + 18 T + T^{2}$$
$67$ $$8 - 8 T + T^{2}$$
$71$ $$82 + 20 T + T^{2}$$
$73$ $$-199 + 2 T + T^{2}$$
$79$ $$-2 + T^{2}$$
$83$ $$7 + 10 T + T^{2}$$
$89$ $$98 - 20 T + T^{2}$$
$97$ $$28 - 12 T + T^{2}$$