Properties

 Label 930.2.j.d Level $930$ Weight $2$ Character orbit 930.j Analytic conductor $7.426$ Analytic rank $0$ Dimension $8$ 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.j (of order $$4$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{24}^{3} q^{2} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{3} + \zeta_{24}^{6} q^{4} + ( 2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + ( -1 + 2 \zeta_{24}^{4} ) q^{6} + ( -1 + 2 \zeta_{24}^{3} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{7} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{8} + 3 \zeta_{24}^{6} q^{9} +O(q^{10})$$ $$q + \zeta_{24}^{3} q^{2} + ( \zeta_{24} + \zeta_{24}^{5} ) q^{3} + \zeta_{24}^{6} q^{4} + ( 2 \zeta_{24} + \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{5} + ( -1 + 2 \zeta_{24}^{4} ) q^{6} + ( -1 + 2 \zeta_{24}^{3} + \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{7} + ( -\zeta_{24} + \zeta_{24}^{5} ) q^{8} + 3 \zeta_{24}^{6} q^{9} + ( 2 + \zeta_{24}^{6} ) q^{10} + ( -2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{11} + ( -\zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{12} + ( -2 + 2 \zeta_{24} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{13} + ( -\zeta_{24} + 4 \zeta_{24}^{2} - \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{14} + ( -1 + 4 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{15} - q^{16} + ( 1 - 2 \zeta_{24}^{2} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{17} + ( -3 \zeta_{24} + 3 \zeta_{24}^{5} ) q^{18} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{19} + ( -\zeta_{24} + 2 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{20} + ( 6 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{21} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{22} + ( 1 + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{23} + ( -2 \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{24} + ( 4 - 3 \zeta_{24}^{6} ) q^{25} + ( -2 + 2 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{5} ) q^{26} + ( -3 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{27} + ( -1 + 2 \zeta_{24} + 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{28} + ( 2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{29} + ( 2 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{30} - q^{31} -\zeta_{24}^{3} q^{32} + ( 2 - 6 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{33} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{34} + ( 4 - 3 \zeta_{24} + 4 \zeta_{24}^{2} + \zeta_{24}^{3} - 8 \zeta_{24}^{4} + 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{35} -3 q^{36} + ( -2 - 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{38} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{39} + ( -1 + 2 \zeta_{24}^{6} ) q^{40} + ( 2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} ) q^{41} + ( 1 - 2 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 2 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{42} + ( 2 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{43} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{44} + ( -3 \zeta_{24} + 6 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{45} + ( \zeta_{24} + \zeta_{24}^{3} + \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{46} -8 \zeta_{24}^{3} q^{47} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{48} + ( 4 \zeta_{24} - 4 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 7 \zeta_{24}^{6} + 8 \zeta_{24}^{7} ) q^{49} + ( 3 \zeta_{24} + 4 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{50} + ( 2 + 3 \zeta_{24} - 3 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 3 \zeta_{24}^{5} ) q^{51} + ( 2 - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{52} + ( 6 \zeta_{24} - 6 \zeta_{24}^{5} ) q^{53} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{54} + ( -2 + 4 \zeta_{24} - 2 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{55} + ( -2 + \zeta_{24} - \zeta_{24}^{3} + 4 \zeta_{24}^{4} - \zeta_{24}^{5} ) q^{56} + ( -6 + 6 \zeta_{24}^{6} ) q^{57} + ( 2 + 2 \zeta_{24}^{6} ) q^{58} + ( -5 \zeta_{24} + 4 \zeta_{24}^{2} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{59} + ( -2 - 2 \zeta_{24}^{2} + 4 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{60} + ( -2 - \zeta_{24} - \zeta_{24}^{3} - \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{61} -\zeta_{24}^{3} q^{62} + ( -3 + 6 \zeta_{24} + 6 \zeta_{24}^{5} - 3 \zeta_{24}^{6} ) q^{63} -\zeta_{24}^{6} q^{64} + ( -2 - 2 \zeta_{24} + 8 \zeta_{24}^{2} - 6 \zeta_{24}^{3} + 4 \zeta_{24}^{4} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{65} + ( -6 + 2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 4 \zeta_{24}^{7} ) q^{66} + ( -6 - 2 \zeta_{24}^{3} + 6 \zeta_{24}^{6} + 4 \zeta_{24}^{7} ) q^{67} + ( 1 + 2 \zeta_{24} + 2 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{68} + ( 3 \zeta_{24} + 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} ) q^{69} + ( -3 + 2 \zeta_{24} + 4 \zeta_{24}^{3} + 2 \zeta_{24}^{5} + \zeta_{24}^{6} - 8 \zeta_{24}^{7} ) q^{70} + ( 2 + 4 \zeta_{24} - 4 \zeta_{24}^{3} - 4 \zeta_{24}^{4} - 4 \zeta_{24}^{5} ) q^{71} -3 \zeta_{24}^{3} q^{72} + ( -4 \zeta_{24} - 4 \zeta_{24}^{5} ) q^{73} + ( 4 \zeta_{24} + 3 \zeta_{24}^{3} + 4 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{75} + ( -2 \zeta_{24} - 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} + 4 \zeta_{24}^{7} ) q^{76} + ( 6 - 12 \zeta_{24}^{2} + 16 \zeta_{24}^{3} - 12 \zeta_{24}^{4} + 6 \zeta_{24}^{6} ) q^{77} + ( 2 - 6 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 6 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{78} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 4 \zeta_{24}^{6} - 4 \zeta_{24}^{7} ) q^{79} + ( -2 \zeta_{24} - \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{80} -9 q^{81} + ( 2 - 2 \zeta_{24}^{6} ) q^{82} + ( 2 - 4 \zeta_{24} + 4 \zeta_{24}^{2} - 4 \zeta_{24}^{4} + 4 \zeta_{24}^{5} - 2 \zeta_{24}^{6} ) q^{83} + ( -\zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + 6 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{84} + ( -4 - 3 \zeta_{24} - \zeta_{24}^{3} - 3 \zeta_{24}^{5} - 2 \zeta_{24}^{6} + 2 \zeta_{24}^{7} ) q^{85} + ( -2 + 4 \zeta_{24}^{4} ) q^{86} + ( -2 + 4 \zeta_{24}^{2} + 4 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{87} + ( 2 - 2 \zeta_{24} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{6} ) q^{88} + ( -8 \zeta_{24} - 8 \zeta_{24}^{2} - 8 \zeta_{24}^{3} + 8 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{89} + ( -3 + 6 \zeta_{24}^{6} ) q^{90} + ( 16 - 6 \zeta_{24} - 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 12 \zeta_{24}^{7} ) q^{91} + ( -1 + 2 \zeta_{24}^{2} + 2 \zeta_{24}^{4} - \zeta_{24}^{6} ) q^{92} + ( -\zeta_{24} - \zeta_{24}^{5} ) q^{93} -8 \zeta_{24}^{6} q^{94} + ( -6 + 4 \zeta_{24}^{2} + 12 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{95} + ( 1 - 2 \zeta_{24}^{4} ) q^{96} + ( -3 + 3 \zeta_{24}^{6} ) q^{97} + ( -4 + 7 \zeta_{24} - 8 \zeta_{24}^{2} + 8 \zeta_{24}^{4} - 7 \zeta_{24}^{5} + 4 \zeta_{24}^{6} ) q^{98} + ( 6 \zeta_{24} - 12 \zeta_{24}^{2} + 6 \zeta_{24}^{3} - 6 \zeta_{24}^{5} + 6 \zeta_{24}^{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{7} + O(q^{10})$$ $$8q - 8q^{7} + 16q^{10} - 16q^{13} - 8q^{16} + 48q^{21} + 16q^{22} + 32q^{25} - 8q^{28} - 8q^{31} - 24q^{36} - 8q^{40} + 16q^{52} - 16q^{55} - 48q^{57} + 16q^{58} - 16q^{61} - 24q^{63} - 48q^{66} - 48q^{67} - 24q^{70} - 72q^{81} + 16q^{82} - 32q^{85} + 16q^{88} - 24q^{90} + 128q^{91} - 24q^{97} + O(q^{100})$$

Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$-\zeta_{24}^{3}$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
497.1
 −0.965926 + 0.258819i 0.258819 − 0.965926i −0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 − 0.258819i 0.258819 + 0.965926i −0.258819 − 0.965926i 0.965926 + 0.258819i
−0.707107 + 0.707107i −1.22474 + 1.22474i 1.00000i −2.12132 0.707107i 1.73205i −3.44949 3.44949i 0.707107 + 0.707107i 3.00000i 2.00000 1.00000i
497.2 −0.707107 + 0.707107i 1.22474 1.22474i 1.00000i −2.12132 0.707107i 1.73205i 1.44949 + 1.44949i 0.707107 + 0.707107i 3.00000i 2.00000 1.00000i
497.3 0.707107 0.707107i −1.22474 + 1.22474i 1.00000i 2.12132 + 0.707107i 1.73205i −3.44949 3.44949i −0.707107 0.707107i 3.00000i 2.00000 1.00000i
497.4 0.707107 0.707107i 1.22474 1.22474i 1.00000i 2.12132 + 0.707107i 1.73205i 1.44949 + 1.44949i −0.707107 0.707107i 3.00000i 2.00000 1.00000i
683.1 −0.707107 0.707107i −1.22474 1.22474i 1.00000i −2.12132 + 0.707107i 1.73205i −3.44949 + 3.44949i 0.707107 0.707107i 3.00000i 2.00000 + 1.00000i
683.2 −0.707107 0.707107i 1.22474 + 1.22474i 1.00000i −2.12132 + 0.707107i 1.73205i 1.44949 1.44949i 0.707107 0.707107i 3.00000i 2.00000 + 1.00000i
683.3 0.707107 + 0.707107i −1.22474 1.22474i 1.00000i 2.12132 0.707107i 1.73205i −3.44949 + 3.44949i −0.707107 + 0.707107i 3.00000i 2.00000 + 1.00000i
683.4 0.707107 + 0.707107i 1.22474 + 1.22474i 1.00000i 2.12132 0.707107i 1.73205i 1.44949 1.44949i −0.707107 + 0.707107i 3.00000i 2.00000 + 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 683.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.j.d 8
3.b odd 2 1 inner 930.2.j.d 8
5.c odd 4 1 inner 930.2.j.d 8
15.e even 4 1 inner 930.2.j.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.d 8 1.a even 1 1 trivial
930.2.j.d 8 3.b odd 2 1 inner
930.2.j.d 8 5.c odd 4 1 inner
930.2.j.d 8 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{4} + 4 T_{7}^{3} + 8 T_{7}^{2} - 40 T_{7} + 100$$ $$T_{11}^{4} + 40 T_{11}^{2} + 16$$ $$T_{17}^{8} + 392 T_{17}^{4} + 16$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{4} )^{2}$$
$3$ $$( 9 + T^{4} )^{2}$$
$5$ $$( 25 - 8 T^{2} + T^{4} )^{2}$$
$7$ $$( 100 - 40 T + 8 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$11$ $$( 16 + 40 T^{2} + T^{4} )^{2}$$
$13$ $$( 16 - 32 T + 32 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$17$ $$16 + 392 T^{4} + T^{8}$$
$19$ $$( 24 + T^{2} )^{4}$$
$23$ $$( 36 + T^{4} )^{2}$$
$29$ $$( -8 + T^{2} )^{4}$$
$31$ $$( 1 + T )^{8}$$
$37$ $$T^{8}$$
$41$ $$( 8 + T^{2} )^{4}$$
$43$ $$( 144 + T^{4} )^{2}$$
$47$ $$( 4096 + T^{4} )^{2}$$
$53$ $$( 1296 + T^{4} )^{2}$$
$59$ $$( 1444 - 124 T^{2} + T^{4} )^{2}$$
$61$ $$( -2 + 4 T + T^{2} )^{4}$$
$67$ $$( 3600 + 1440 T + 288 T^{2} + 24 T^{3} + T^{4} )^{2}$$
$71$ $$( 400 + 88 T^{2} + T^{4} )^{2}$$
$73$ $$( 2304 + T^{4} )^{2}$$
$79$ $$( 64 + 80 T^{2} + T^{4} )^{2}$$
$83$ $$4096 + 6272 T^{4} + T^{8}$$
$89$ $$( 6400 - 352 T^{2} + T^{4} )^{2}$$
$97$ $$( 18 + 6 T + T^{2} )^{4}$$