# Properties

 Label 931.2.a.g Level $931$ Weight $2$ Character orbit 931.a Self dual yes Analytic conductor $7.434$ Analytic rank $0$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} + ( 2 - \beta ) q^{3} + ( 1 + \beta ) q^{4} + 3 q^{5} + ( 3 - \beta ) q^{6} -3 q^{8} + ( 4 - 3 \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{2} + ( 2 - \beta ) q^{3} + ( 1 + \beta ) q^{4} + 3 q^{5} + ( 3 - \beta ) q^{6} -3 q^{8} + ( 4 - 3 \beta ) q^{9} -3 \beta q^{10} + ( -3 + \beta ) q^{11} - q^{12} + ( 1 + 2 \beta ) q^{13} + ( 6 - 3 \beta ) q^{15} + ( -2 + \beta ) q^{16} + ( 3 + \beta ) q^{17} + ( 9 - \beta ) q^{18} - q^{19} + ( 3 + 3 \beta ) q^{20} + ( -3 + 2 \beta ) q^{22} -3 q^{23} + ( -6 + 3 \beta ) q^{24} + 4 q^{25} + ( -6 - 3 \beta ) q^{26} + ( 11 - 4 \beta ) q^{27} + ( 3 + 3 \beta ) q^{29} + ( 9 - 3 \beta ) q^{30} + ( 1 - \beta ) q^{31} + ( 3 + \beta ) q^{32} + ( -9 + 4 \beta ) q^{33} + ( -3 - 4 \beta ) q^{34} + ( -5 - 2 \beta ) q^{36} + ( -1 + 2 \beta ) q^{37} + \beta q^{38} + ( -4 + \beta ) q^{39} -9 q^{40} + ( -3 + \beta ) q^{41} -10 q^{43} -\beta q^{44} + ( 12 - 9 \beta ) q^{45} + 3 \beta q^{46} + ( 3 - 4 \beta ) q^{47} + ( -7 + 3 \beta ) q^{48} -4 \beta q^{50} + ( 3 - 2 \beta ) q^{51} + ( 7 + 5 \beta ) q^{52} -3 \beta q^{53} + ( 12 - 7 \beta ) q^{54} + ( -9 + 3 \beta ) q^{55} + ( -2 + \beta ) q^{57} + ( -9 - 6 \beta ) q^{58} + ( -3 + 4 \beta ) q^{59} -3 q^{60} + ( -5 + 4 \beta ) q^{61} + 3 q^{62} + ( 1 - 6 \beta ) q^{64} + ( 3 + 6 \beta ) q^{65} + ( -12 + 5 \beta ) q^{66} + ( 5 - 3 \beta ) q^{67} + ( 6 + 5 \beta ) q^{68} + ( -6 + 3 \beta ) q^{69} + ( 3 + 4 \beta ) q^{71} + ( -12 + 9 \beta ) q^{72} + ( 10 - 5 \beta ) q^{73} + ( -6 - \beta ) q^{74} + ( 8 - 4 \beta ) q^{75} + ( -1 - \beta ) q^{76} + ( -3 + 3 \beta ) q^{78} + ( 2 + 4 \beta ) q^{79} + ( -6 + 3 \beta ) q^{80} + ( 22 - 6 \beta ) q^{81} + ( -3 + 2 \beta ) q^{82} + ( 6 + 3 \beta ) q^{83} + ( 9 + 3 \beta ) q^{85} + 10 \beta q^{86} -3 q^{87} + ( 9 - 3 \beta ) q^{88} + ( 6 + 2 \beta ) q^{89} + ( 27 - 3 \beta ) q^{90} + ( -3 - 3 \beta ) q^{92} + ( 5 - 2 \beta ) q^{93} + ( 12 + \beta ) q^{94} -3 q^{95} + ( 3 - 2 \beta ) q^{96} + ( -5 - 2 \beta ) q^{97} + ( -21 + 10 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{3} + 3 q^{4} + 6 q^{5} + 5 q^{6} - 6 q^{8} + 5 q^{9} + O(q^{10})$$ $$2 q - q^{2} + 3 q^{3} + 3 q^{4} + 6 q^{5} + 5 q^{6} - 6 q^{8} + 5 q^{9} - 3 q^{10} - 5 q^{11} - 2 q^{12} + 4 q^{13} + 9 q^{15} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{19} + 9 q^{20} - 4 q^{22} - 6 q^{23} - 9 q^{24} + 8 q^{25} - 15 q^{26} + 18 q^{27} + 9 q^{29} + 15 q^{30} + q^{31} + 7 q^{32} - 14 q^{33} - 10 q^{34} - 12 q^{36} + q^{38} - 7 q^{39} - 18 q^{40} - 5 q^{41} - 20 q^{43} - q^{44} + 15 q^{45} + 3 q^{46} + 2 q^{47} - 11 q^{48} - 4 q^{50} + 4 q^{51} + 19 q^{52} - 3 q^{53} + 17 q^{54} - 15 q^{55} - 3 q^{57} - 24 q^{58} - 2 q^{59} - 6 q^{60} - 6 q^{61} + 6 q^{62} - 4 q^{64} + 12 q^{65} - 19 q^{66} + 7 q^{67} + 17 q^{68} - 9 q^{69} + 10 q^{71} - 15 q^{72} + 15 q^{73} - 13 q^{74} + 12 q^{75} - 3 q^{76} - 3 q^{78} + 8 q^{79} - 9 q^{80} + 38 q^{81} - 4 q^{82} + 15 q^{83} + 21 q^{85} + 10 q^{86} - 6 q^{87} + 15 q^{88} + 14 q^{89} + 51 q^{90} - 9 q^{92} + 8 q^{93} + 25 q^{94} - 6 q^{95} + 4 q^{96} - 12 q^{97} - 32 q^{99} + O(q^{100})$$

## 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
 2.30278 −1.30278
−2.30278 −0.302776 3.30278 3.00000 0.697224 0 −3.00000 −2.90833 −6.90833
1.2 1.30278 3.30278 −0.302776 3.00000 4.30278 0 −3.00000 7.90833 3.90833
 $$n$$: 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.g 2
3.b odd 2 1 8379.2.a.bf 2
7.b odd 2 1 133.2.a.b 2
7.c even 3 2 931.2.f.g 4
7.d odd 6 2 931.2.f.h 4
21.c even 2 1 1197.2.a.h 2
28.d even 2 1 2128.2.a.l 2
35.c odd 2 1 3325.2.a.n 2
56.e even 2 1 8512.2.a.l 2
56.h odd 2 1 8512.2.a.bh 2
133.c even 2 1 2527.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.a.b 2 7.b odd 2 1
931.2.a.g 2 1.a even 1 1 trivial
931.2.f.g 4 7.c even 3 2
931.2.f.h 4 7.d odd 6 2
1197.2.a.h 2 21.c even 2 1
2128.2.a.l 2 28.d even 2 1
2527.2.a.d 2 133.c even 2 1
3325.2.a.n 2 35.c odd 2 1
8379.2.a.bf 2 3.b odd 2 1
8512.2.a.l 2 56.e even 2 1
8512.2.a.bh 2 56.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(931))$$:

 $$T_{2}^{2} + T_{2} - 3$$ $$T_{3}^{2} - 3 T_{3} - 1$$ $$T_{5} - 3$$ $$T_{13}^{2} - 4 T_{13} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-3 + T + T^{2}$$
$3$ $$-1 - 3 T + T^{2}$$
$5$ $$( -3 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$3 + 5 T + T^{2}$$
$13$ $$-9 - 4 T + T^{2}$$
$17$ $$9 - 7 T + T^{2}$$
$19$ $$( 1 + T )^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$-9 - 9 T + T^{2}$$
$31$ $$-3 - T + T^{2}$$
$37$ $$-13 + T^{2}$$
$41$ $$3 + 5 T + T^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$-51 - 2 T + T^{2}$$
$53$ $$-27 + 3 T + T^{2}$$
$59$ $$-51 + 2 T + T^{2}$$
$61$ $$-43 + 6 T + T^{2}$$
$67$ $$-17 - 7 T + T^{2}$$
$71$ $$-27 - 10 T + T^{2}$$
$73$ $$-25 - 15 T + T^{2}$$
$79$ $$-36 - 8 T + T^{2}$$
$83$ $$27 - 15 T + T^{2}$$
$89$ $$36 - 14 T + T^{2}$$
$97$ $$23 + 12 T + T^{2}$$