Properties

 Label 931.2.a.m Level $931$ Weight $2$ Character orbit 931.a Self dual yes Analytic conductor $7.434$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.5744.1 Defining polynomial: $$x^{4} - 5 x^{2} - 2 x + 1$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 2 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + \beta_{2} q^{3} + ( \beta_{1} + \beta_{2} + \beta_{3} ) q^{4} + ( 2 + \beta_{1} - \beta_{3} ) q^{5} + ( -1 + \beta_{1} ) q^{6} + ( 1 + \beta_{1} + \beta_{2} ) q^{8} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{9} + ( 2 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{10} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + ( 2 - \beta_{2} + \beta_{3} ) q^{12} + ( 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{15} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{16} + ( 3 - 3 \beta_{1} - 2 \beta_{2} ) q^{17} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{18} - q^{19} + ( -1 + 5 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{20} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{22} + ( -3 + 2 \beta_{2} + 2 \beta_{3} ) q^{23} + ( 3 - \beta_{3} ) q^{24} + ( 4 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{25} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{26} + ( -2 - \beta_{2} - \beta_{3} ) q^{27} + ( 1 - \beta_{2} + 2 \beta_{3} ) q^{29} + ( \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{30} + ( -2 \beta_{1} - \beta_{3} ) q^{31} + ( 1 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{32} + ( 6 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -4 - 2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{34} + ( -5 - 2 \beta_{1} ) q^{36} + ( 3 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{37} -\beta_{1} q^{38} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{39} + ( 4 + 4 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{40} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{41} + ( 2 - 2 \beta_{2} + 2 \beta_{3} ) q^{43} + ( -1 - 5 \beta_{1} - 5 \beta_{2} - \beta_{3} ) q^{44} + ( 3 - 4 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{45} + ( -2 + \beta_{1} - 2 \beta_{3} ) q^{46} + ( 4 + \beta_{1} + 5 \beta_{3} ) q^{47} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{48} + ( 2 + 6 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{50} + ( -5 - \beta_{1} + 5 \beta_{2} + 2 \beta_{3} ) q^{51} + ( 6 + 6 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{52} + ( 2 + 3 \beta_{1} + 2 \beta_{2} ) q^{53} + ( 1 - 4 \beta_{1} + \beta_{3} ) q^{54} + ( -3 - 5 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{55} -\beta_{2} q^{57} + ( 1 + 2 \beta_{1} - 2 \beta_{3} ) q^{58} + ( 4 - \beta_{2} - 3 \beta_{3} ) q^{59} + ( 1 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{60} + ( 6 + \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{61} + ( -4 - 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + ( -5 - 2 \beta_{1} - 2 \beta_{3} ) q^{64} + ( 1 + 7 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{65} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{66} + ( 3 - 4 \beta_{1} - 5 \beta_{2} ) q^{67} + ( -7 - 6 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{68} + ( 6 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{69} + ( 1 - 3 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{71} + ( -2 - 3 \beta_{1} - 2 \beta_{3} ) q^{72} + ( -3 - 2 \beta_{2} + \beta_{3} ) q^{73} + ( -5 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{74} + ( 8 + 2 \beta_{2} - 6 \beta_{3} ) q^{75} + ( -\beta_{1} - \beta_{2} - \beta_{3} ) q^{76} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{78} + ( -2 + 2 \beta_{1} - 4 \beta_{3} ) q^{79} + ( 7 + \beta_{3} ) q^{80} + ( -6 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{81} + ( -2 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{82} + ( 3 + 2 \beta_{1} + 4 \beta_{2} + 5 \beta_{3} ) q^{83} + ( -5 \beta_{1} - 7 \beta_{2} - 5 \beta_{3} ) q^{85} + ( 2 + 2 \beta_{1} - 2 \beta_{3} ) q^{86} + ( -6 + \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{87} + ( 1 - 6 \beta_{1} - 3 \beta_{2} - 4 \beta_{3} ) q^{88} + ( 2 + 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -6 - 6 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{90} + ( 8 - 3 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{92} + ( 3 - 2 \beta_{1} - 2 \beta_{3} ) q^{93} + ( 2 + 10 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{94} + ( -2 - \beta_{1} + \beta_{3} ) q^{95} + ( -4 - \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{96} + ( -2 - 2 \beta_{1} + 5 \beta_{2} + 3 \beta_{3} ) q^{97} + ( -1 + 4 \beta_{1} + 6 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 2q^{4} + 8q^{5} - 4q^{6} + 6q^{8} + 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 2q^{4} + 8q^{5} - 4q^{6} + 6q^{8} + 2q^{9} + 10q^{10} - 6q^{11} + 6q^{12} + 2q^{13} + 4q^{15} + 2q^{16} + 8q^{17} - 6q^{18} - 4q^{19} - 14q^{22} - 8q^{23} + 12q^{24} + 20q^{25} + 16q^{26} - 10q^{27} + 2q^{29} + 2q^{30} + 2q^{32} + 18q^{33} - 22q^{34} - 20q^{36} + 10q^{37} + 2q^{39} + 22q^{40} + 12q^{41} + 4q^{43} - 14q^{44} + 8q^{45} - 8q^{46} + 16q^{47} - 12q^{48} + 12q^{50} - 10q^{51} + 28q^{52} + 12q^{53} + 4q^{54} - 8q^{55} - 2q^{57} + 4q^{58} + 14q^{59} - 2q^{60} + 20q^{61} - 20q^{62} - 20q^{64} + 10q^{65} - 8q^{66} + 2q^{67} - 24q^{68} + 14q^{69} - 2q^{71} - 8q^{72} - 16q^{73} - 26q^{74} + 36q^{75} - 2q^{76} + 14q^{78} - 8q^{79} + 28q^{80} - 20q^{81} - 12q^{82} + 20q^{83} - 14q^{85} + 8q^{86} - 20q^{87} - 2q^{88} + 16q^{89} - 32q^{90} + 26q^{92} + 12q^{93} + 10q^{94} - 8q^{95} - 12q^{96} + 2q^{97} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 5 \nu - 1$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + \nu^{2} + 4 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 5 \beta_{1} + 1$$

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
 −1.92022 −0.751024 0.291367 2.37988
−1.92022 1.52077 1.68725 −2.00692 −2.92022 0 0.600553 −0.687248 3.85372
1.2 −0.751024 2.33152 −1.43596 4.26543 −1.75102 0 2.58049 2.43596 −3.20344
1.3 0.291367 −2.43210 −1.91511 2.06574 −0.708633 0 −1.14073 2.91511 0.601888
1.4 2.37988 0.579810 3.66382 3.67575 1.37988 0 3.95969 −2.66382 8.74783
 $$n$$: 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.m yes 4
3.b odd 2 1 8379.2.a.bu 4
7.b odd 2 1 931.2.a.l 4
7.c even 3 2 931.2.f.n 8
7.d odd 6 2 931.2.f.o 8
21.c even 2 1 8379.2.a.bv 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
931.2.a.l 4 7.b odd 2 1
931.2.a.m yes 4 1.a even 1 1 trivial
931.2.f.n 8 7.c even 3 2
931.2.f.o 8 7.d odd 6 2
8379.2.a.bu 4 3.b odd 2 1
8379.2.a.bv 4 21.c even 2 1

Hecke kernels

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

 $$T_{2}^{4} - 5 T_{2}^{2} - 2 T_{2} + 1$$ $$T_{3}^{4} - 2 T_{3}^{3} - 5 T_{3}^{2} + 12 T_{3} - 5$$ $$T_{5}^{4} - 8 T_{5}^{3} + 12 T_{5}^{2} + 32 T_{5} - 65$$ $$T_{13}^{4} - 2 T_{13}^{3} - 20 T_{13}^{2} - 26 T_{13} - 5$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T - 5 T^{2} + T^{4}$$
$3$ $$-5 + 12 T - 5 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$-65 + 32 T + 12 T^{2} - 8 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$-59 - 102 T - 13 T^{2} + 6 T^{3} + T^{4}$$
$13$ $$-5 - 26 T - 20 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$-125 + 228 T - 23 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$( 1 + T )^{4}$$
$23$ $$-59 - 144 T - 14 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$-191 + 180 T - 41 T^{2} - 2 T^{3} + T^{4}$$
$31$ $$35 + 22 T - 27 T^{2} + T^{4}$$
$37$ $$-691 + 358 T - 22 T^{2} - 10 T^{3} + T^{4}$$
$41$ $$155 + 158 T + 9 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$-448 + 352 T - 64 T^{2} - 4 T^{3} + T^{4}$$
$47$ $$-7945 + 2232 T - 84 T^{2} - 16 T^{3} + T^{4}$$
$53$ $$65 + 102 T + 7 T^{2} - 12 T^{3} + T^{4}$$
$59$ $$-305 + 234 T + 16 T^{2} - 14 T^{3} + T^{4}$$
$61$ $$-685 + 84 T + 96 T^{2} - 20 T^{3} + T^{4}$$
$67$ $$2293 - 260 T - 161 T^{2} - 2 T^{3} + T^{4}$$
$71$ $$3569 - 62 T - 130 T^{2} + 2 T^{3} + T^{4}$$
$73$ $$-305 - 48 T + 55 T^{2} + 16 T^{3} + T^{4}$$
$79$ $$-16 - 288 T - 108 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$4555 + 1352 T - 37 T^{2} - 20 T^{3} + T^{4}$$
$89$ $$-2240 + 736 T + 8 T^{2} - 16 T^{3} + T^{4}$$
$97$ $$815 + 86 T - 224 T^{2} - 2 T^{3} + T^{4}$$