Properties

 Label 930.2.s.a Level $930$ Weight $2$ Character orbit 930.s Analytic conductor $7.426$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.s (of order $$6$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{2} + \zeta_{12} q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -1 + \zeta_{12}^{2} ) q^{6} + 2 \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{2} + \zeta_{12} q^{3} - q^{4} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{5} + ( -1 + \zeta_{12}^{2} ) q^{6} + 2 \zeta_{12} q^{7} -\zeta_{12}^{3} q^{8} + \zeta_{12}^{2} q^{9} + ( -\zeta_{12} - 2 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{10} + 5 \zeta_{12}^{2} q^{11} -\zeta_{12} q^{12} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{13} + ( -2 + 2 \zeta_{12}^{2} ) q^{14} + ( -2 + \zeta_{12}^{3} ) q^{15} + q^{16} -3 \zeta_{12} q^{17} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{18} + ( 4 - 4 \zeta_{12}^{2} ) q^{19} + ( 2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{20} + 2 \zeta_{12}^{2} q^{21} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{22} + 3 \zeta_{12}^{3} q^{23} + ( 1 - \zeta_{12}^{2} ) q^{24} + ( 3 - 4 \zeta_{12} - 3 \zeta_{12}^{2} ) q^{25} -2 \zeta_{12}^{2} q^{26} + \zeta_{12}^{3} q^{27} -2 \zeta_{12} q^{28} + 2 q^{29} + ( -1 - 2 \zeta_{12}^{3} ) q^{30} + ( -5 + 6 \zeta_{12}^{2} ) q^{31} + \zeta_{12}^{3} q^{32} + 5 \zeta_{12}^{3} q^{33} + ( 3 - 3 \zeta_{12}^{2} ) q^{34} + ( -4 + 2 \zeta_{12}^{3} ) q^{35} -\zeta_{12}^{2} q^{36} -5 \zeta_{12} q^{37} + 4 \zeta_{12} q^{38} -2 q^{39} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{40} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{42} + \zeta_{12} q^{43} -5 \zeta_{12}^{2} q^{44} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{45} -3 q^{46} + 9 \zeta_{12}^{3} q^{47} + \zeta_{12} q^{48} -3 \zeta_{12}^{2} q^{49} + ( 4 + 3 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{50} -3 \zeta_{12}^{2} q^{51} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{52} + ( -8 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{53} - q^{54} + ( -5 - 10 \zeta_{12} + 5 \zeta_{12}^{2} ) q^{55} + ( 2 - 2 \zeta_{12}^{2} ) q^{56} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{57} + 2 \zeta_{12}^{3} q^{58} + ( 4 - 4 \zeta_{12}^{2} ) q^{59} + ( 2 - \zeta_{12}^{3} ) q^{60} -2 q^{61} + ( -6 \zeta_{12} + \zeta_{12}^{3} ) q^{62} + 2 \zeta_{12}^{3} q^{63} - q^{64} + ( 4 - 2 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{65} -5 q^{66} + ( -7 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{67} + 3 \zeta_{12} q^{68} + ( -3 + 3 \zeta_{12}^{2} ) q^{69} + ( -2 - 4 \zeta_{12}^{3} ) q^{70} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{72} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{73} + ( 5 - 5 \zeta_{12}^{2} ) q^{74} + ( 3 \zeta_{12} - 4 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{75} + ( -4 + 4 \zeta_{12}^{2} ) q^{76} + 10 \zeta_{12}^{3} q^{77} -2 \zeta_{12}^{3} q^{78} + ( 5 - 5 \zeta_{12}^{2} ) q^{79} + ( -2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{80} + ( -1 + \zeta_{12}^{2} ) q^{81} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{83} -2 \zeta_{12}^{2} q^{84} + ( 6 - 3 \zeta_{12}^{3} ) q^{85} + ( -1 + \zeta_{12}^{2} ) q^{86} + 2 \zeta_{12} q^{87} + ( 5 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{88} + ( 2 - \zeta_{12} - 2 \zeta_{12}^{2} ) q^{90} -4 q^{91} -3 \zeta_{12}^{3} q^{92} + ( -5 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{93} -9 q^{94} + ( 4 + 8 \zeta_{12}^{3} ) q^{95} + ( -1 + \zeta_{12}^{2} ) q^{96} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{98} + ( -5 + 5 \zeta_{12}^{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 2q^{5} - 2q^{6} + 2q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 2q^{5} - 2q^{6} + 2q^{9} - 4q^{10} + 10q^{11} - 4q^{14} - 8q^{15} + 4q^{16} + 8q^{19} - 2q^{20} + 4q^{21} + 2q^{24} + 6q^{25} - 4q^{26} + 8q^{29} - 4q^{30} - 8q^{31} + 6q^{34} - 16q^{35} - 2q^{36} - 8q^{39} + 4q^{40} - 10q^{44} - 2q^{45} - 12q^{46} - 6q^{49} + 8q^{50} - 6q^{51} - 4q^{54} - 10q^{55} + 4q^{56} + 8q^{59} + 8q^{60} - 8q^{61} - 4q^{64} + 8q^{65} - 20q^{66} - 6q^{69} - 8q^{70} + 10q^{74} - 8q^{75} - 8q^{76} + 10q^{79} + 2q^{80} - 2q^{81} - 4q^{84} + 24q^{85} - 2q^{86} + 4q^{90} - 16q^{91} - 36q^{94} + 16q^{95} - 2q^{96} - 10q^{99} + O(q^{100})$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/930\mathbb{Z}\right)^\times$$.

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-\zeta_{12}^{2}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
439.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.00000i −0.866025 0.500000i −1.00000 2.23205 0.133975i −0.500000 + 0.866025i −1.73205 1.00000i 1.00000i 0.500000 + 0.866025i −0.133975 2.23205i
439.2 1.00000i 0.866025 + 0.500000i −1.00000 −1.23205 + 1.86603i −0.500000 + 0.866025i 1.73205 + 1.00000i 1.00000i 0.500000 + 0.866025i −1.86603 1.23205i
769.1 1.00000i 0.866025 0.500000i −1.00000 −1.23205 1.86603i −0.500000 0.866025i 1.73205 1.00000i 1.00000i 0.500000 0.866025i −1.86603 + 1.23205i
769.2 1.00000i −0.866025 + 0.500000i −1.00000 2.23205 + 0.133975i −0.500000 0.866025i −1.73205 + 1.00000i 1.00000i 0.500000 0.866025i −0.133975 + 2.23205i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
31.c even 3 1 inner
155.j even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.s.a 4
5.b even 2 1 inner 930.2.s.a 4
31.c even 3 1 inner 930.2.s.a 4
155.j even 6 1 inner 930.2.s.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.a 4 1.a even 1 1 trivial
930.2.s.a 4 5.b even 2 1 inner
930.2.s.a 4 31.c even 3 1 inner
930.2.s.a 4 155.j even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4 T_{7}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$25 - 10 T - T^{2} - 2 T^{3} + T^{4}$$
$7$ $$16 - 4 T^{2} + T^{4}$$
$11$ $$( 25 - 5 T + T^{2} )^{2}$$
$13$ $$16 - 4 T^{2} + T^{4}$$
$17$ $$81 - 9 T^{2} + T^{4}$$
$19$ $$( 16 - 4 T + T^{2} )^{2}$$
$23$ $$( 9 + T^{2} )^{2}$$
$29$ $$( -2 + T )^{4}$$
$31$ $$( 31 + 4 T + T^{2} )^{2}$$
$37$ $$625 - 25 T^{2} + T^{4}$$
$41$ $$T^{4}$$
$43$ $$1 - T^{2} + T^{4}$$
$47$ $$( 81 + T^{2} )^{2}$$
$53$ $$4096 - 64 T^{2} + T^{4}$$
$59$ $$( 16 - 4 T + T^{2} )^{2}$$
$61$ $$( 2 + T )^{4}$$
$67$ $$2401 - 49 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$1296 - 36 T^{2} + T^{4}$$
$79$ $$( 25 - 5 T + T^{2} )^{2}$$
$83$ $$1296 - 36 T^{2} + T^{4}$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$