# Properties

 Label 930.2.s.b Level $930$ Weight $2$ Character orbit 930.s 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.s (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{2} - \beta_{4} ) q^{2} -\beta_{2} q^{3} - q^{4} + ( 2 \beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{6} -\beta_{6} q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} -\beta_{3} q^{9} +O(q^{10})$$ $$q + ( -\beta_{2} - \beta_{4} ) q^{2} -\beta_{2} q^{3} - q^{4} + ( 2 \beta_{3} + \beta_{4} ) q^{5} + ( -1 - \beta_{3} ) q^{6} -\beta_{6} q^{7} + ( \beta_{2} + \beta_{4} ) q^{8} -\beta_{3} q^{9} + ( -\beta_{3} + 2 \beta_{4} ) q^{10} + ( 2 \beta_{3} - \beta_{5} ) q^{11} + \beta_{2} q^{12} + 2 \beta_{1} q^{13} + \beta_{7} q^{14} + ( 1 + 2 \beta_{2} + 2 \beta_{4} ) q^{15} + q^{16} + 6 \beta_{2} q^{17} -\beta_{4} q^{18} -2 \beta_{7} q^{19} + ( -2 \beta_{3} - \beta_{4} ) q^{20} + \beta_{5} q^{21} + ( \beta_{1} + 2 \beta_{4} ) q^{22} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{23} + ( 1 + \beta_{3} ) q^{24} + ( -3 + 4 \beta_{2} - 3 \beta_{3} ) q^{25} + 2 \beta_{5} q^{26} + ( -\beta_{2} - \beta_{4} ) q^{27} + \beta_{6} q^{28} + ( \beta_{5} - \beta_{7} ) q^{29} + ( 2 - \beta_{2} - \beta_{4} ) q^{30} + ( -1 + \beta_{3} + 2 \beta_{7} ) q^{31} + ( -\beta_{2} - \beta_{4} ) q^{32} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{33} + ( 6 + 6 \beta_{3} ) q^{34} + ( 2 \beta_{1} + \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{35} + \beta_{3} q^{36} -4 \beta_{2} q^{37} -2 \beta_{6} q^{38} + ( 2 \beta_{5} - 2 \beta_{7} ) q^{39} + ( \beta_{3} - 2 \beta_{4} ) q^{40} + ( -2 \beta_{3} + 2 \beta_{5} ) q^{41} -\beta_{1} q^{42} + ( 6 \beta_{2} - 2 \beta_{6} ) q^{43} + ( -2 \beta_{3} + \beta_{5} ) q^{44} + ( 2 - \beta_{2} + 2 \beta_{3} ) q^{45} + ( -4 + 2 \beta_{5} - 2 \beta_{7} ) q^{46} + ( 2 \beta_{1} - 4 \beta_{2} - 4 \beta_{4} + 2 \beta_{6} ) q^{47} -\beta_{2} q^{48} + ( 4 + 3 \beta_{2} + 4 \beta_{3} ) q^{50} + 6 \beta_{3} q^{51} -2 \beta_{1} q^{52} + ( 2 \beta_{1} + 3 \beta_{4} ) q^{53} - q^{54} + ( -4 + 2 \beta_{2} - 4 \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{55} -\beta_{7} q^{56} + 2 \beta_{1} q^{57} + ( -\beta_{1} - \beta_{6} ) q^{58} + ( 2 + 2 \beta_{3} - \beta_{7} ) q^{59} + ( -1 - 2 \beta_{2} - 2 \beta_{4} ) q^{60} + 10 q^{61} + ( \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{62} + ( -\beta_{1} - \beta_{6} ) q^{63} - q^{64} + ( 4 \beta_{6} - 2 \beta_{7} ) q^{65} + ( 2 + \beta_{5} - \beta_{7} ) q^{66} + ( -2 \beta_{1} + 6 \beta_{4} ) q^{67} -6 \beta_{2} q^{68} + ( -4 - 4 \beta_{3} - 2 \beta_{7} ) q^{69} + ( -\beta_{1} + 2 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{70} + ( -4 \beta_{3} + 4 \beta_{5} ) q^{71} + \beta_{4} q^{72} + ( 4 \beta_{1} + 4 \beta_{4} ) q^{73} + ( -4 - 4 \beta_{3} ) q^{74} + ( 4 \beta_{3} - 3 \beta_{4} ) q^{75} + 2 \beta_{7} q^{76} + ( 2 \beta_{1} + 7 \beta_{2} + 7 \beta_{4} + 2 \beta_{6} ) q^{77} + ( -2 \beta_{1} - 2 \beta_{6} ) q^{78} + ( 4 + 4 \beta_{3} + 4 \beta_{7} ) q^{79} + ( 2 \beta_{3} + \beta_{4} ) q^{80} + ( -1 - \beta_{3} ) q^{81} + ( -2 \beta_{1} - 2 \beta_{4} ) q^{82} + ( -4 \beta_{1} - \beta_{4} ) q^{83} -\beta_{5} q^{84} + ( -6 - 12 \beta_{2} - 12 \beta_{4} ) q^{85} + ( 6 + 6 \beta_{3} + 2 \beta_{7} ) q^{86} -\beta_{6} q^{87} + ( -\beta_{1} - 2 \beta_{4} ) q^{88} + ( -2 - 2 \beta_{5} + 2 \beta_{7} ) q^{89} + ( -1 - 2 \beta_{2} - \beta_{3} ) q^{90} + 14 q^{91} + ( -2 \beta_{1} + 4 \beta_{2} + 4 \beta_{4} - 2 \beta_{6} ) q^{92} + ( -2 \beta_{1} + 2 \beta_{2} + \beta_{4} ) q^{93} + ( -4 + 2 \beta_{5} - 2 \beta_{7} ) q^{94} + ( 2 \beta_{1} - 4 \beta_{5} + 2 \beta_{6} + 4 \beta_{7} ) q^{95} + ( -1 - \beta_{3} ) q^{96} + ( -5 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} - 5 \beta_{6} ) q^{97} + ( 2 + 2 \beta_{3} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 8q^{4} - 8q^{5} - 4q^{6} + 4q^{9} + O(q^{10})$$ $$8q - 8q^{4} - 8q^{5} - 4q^{6} + 4q^{9} + 4q^{10} - 8q^{11} + 8q^{15} + 8q^{16} + 8q^{20} + 4q^{24} - 12q^{25} + 16q^{30} - 12q^{31} + 24q^{34} - 4q^{36} - 4q^{40} + 8q^{41} + 8q^{44} + 8q^{45} - 32q^{46} + 16q^{50} - 24q^{51} - 8q^{54} - 16q^{55} + 8q^{59} - 8q^{60} + 80q^{61} - 8q^{64} + 16q^{66} - 16q^{69} + 16q^{71} - 16q^{74} - 16q^{75} + 16q^{79} - 8q^{80} - 4q^{81} - 48q^{85} + 24q^{86} - 16q^{89} - 4q^{90} + 112q^{91} - 32q^{94} - 4q^{96} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 15 \nu^{4} + 5 \nu^{2} + 12$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{5} + 5 \nu^{3} + 12 \nu$$$$)/40$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{6} - 5 \nu^{4} - 15 \nu^{2} - 36$$$$)/20$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - 7 \nu$$$$)/10$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + 13 \nu$$$$)/10$$ $$\beta_{6}$$ $$=$$ $$($$$$-9 \nu^{6} - 15 \nu^{4} - 5 \nu^{2} - 48$$$$)/20$$ $$\beta_{7}$$ $$=$$ $$($$$$11 \nu^{7} + 25 \nu^{5} + 55 \nu^{3} + 132 \nu$$$$)/40$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 3 \beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{7} - \beta_{5} + 5 \beta_{4} + 5 \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{1}$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} - 11 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{6} - 5 \beta_{1} - 9$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-7 \beta_{5} - 13 \beta_{4}$$$$)/2$$

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\beta_{3}$$

## 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
 1.09445 + 0.895644i −0.228425 − 1.39564i −1.09445 − 0.895644i 0.228425 + 1.39564i −1.09445 + 0.895644i 0.228425 − 1.39564i 1.09445 − 0.895644i −0.228425 + 1.39564i
1.00000i −0.866025 0.500000i −1.00000 −1.86603 1.23205i −0.500000 + 0.866025i −2.29129 1.32288i 1.00000i 0.500000 + 0.866025i −1.23205 + 1.86603i
439.2 1.00000i −0.866025 0.500000i −1.00000 −1.86603 1.23205i −0.500000 + 0.866025i 2.29129 + 1.32288i 1.00000i 0.500000 + 0.866025i −1.23205 + 1.86603i
439.3 1.00000i 0.866025 + 0.500000i −1.00000 −0.133975 2.23205i −0.500000 + 0.866025i −2.29129 1.32288i 1.00000i 0.500000 + 0.866025i 2.23205 0.133975i
439.4 1.00000i 0.866025 + 0.500000i −1.00000 −0.133975 2.23205i −0.500000 + 0.866025i 2.29129 + 1.32288i 1.00000i 0.500000 + 0.866025i 2.23205 0.133975i
769.1 1.00000i 0.866025 0.500000i −1.00000 −0.133975 + 2.23205i −0.500000 0.866025i −2.29129 + 1.32288i 1.00000i 0.500000 0.866025i 2.23205 + 0.133975i
769.2 1.00000i 0.866025 0.500000i −1.00000 −0.133975 + 2.23205i −0.500000 0.866025i 2.29129 1.32288i 1.00000i 0.500000 0.866025i 2.23205 + 0.133975i
769.3 1.00000i −0.866025 + 0.500000i −1.00000 −1.86603 + 1.23205i −0.500000 0.866025i −2.29129 + 1.32288i 1.00000i 0.500000 0.866025i −1.23205 1.86603i
769.4 1.00000i −0.866025 + 0.500000i −1.00000 −1.86603 + 1.23205i −0.500000 0.866025i 2.29129 1.32288i 1.00000i 0.500000 0.866025i −1.23205 1.86603i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 769.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.b 8
5.b even 2 1 inner 930.2.s.b 8
31.c even 3 1 inner 930.2.s.b 8
155.j even 6 1 inner 930.2.s.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.s.b 8 1.a even 1 1 trivial
930.2.s.b 8 5.b even 2 1 inner
930.2.s.b 8 31.c even 3 1 inner
930.2.s.b 8 155.j even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 7 T_{7}^{2} + 49$$ acting on $$S_{2}^{\mathrm{new}}(930, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{4}$$
$3$ $$( 1 - T^{2} + T^{4} )^{2}$$
$5$ $$( 25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$7$ $$( 49 - 7 T^{2} + T^{4} )^{2}$$
$11$ $$( 9 - 12 T + 19 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$13$ $$( 784 - 28 T^{2} + T^{4} )^{2}$$
$17$ $$( 1296 - 36 T^{2} + T^{4} )^{2}$$
$19$ $$( 784 + 28 T^{2} + T^{4} )^{2}$$
$23$ $$( 144 + 88 T^{2} + T^{4} )^{2}$$
$29$ $$( -7 + T^{2} )^{4}$$
$31$ $$( 961 + 186 T + 43 T^{2} + 6 T^{3} + T^{4} )^{2}$$
$37$ $$( 256 - 16 T^{2} + T^{4} )^{2}$$
$41$ $$( 576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$43$ $$4096 - 8192 T^{2} + 16320 T^{4} - 128 T^{6} + T^{8}$$
$47$ $$( 144 + 88 T^{2} + T^{4} )^{2}$$
$53$ $$130321 - 26714 T^{2} + 5115 T^{4} - 74 T^{6} + T^{8}$$
$59$ $$( 9 + 12 T + 19 T^{2} - 4 T^{3} + T^{4} )^{2}$$
$61$ $$( -10 + T )^{8}$$
$67$ $$4096 - 8192 T^{2} + 16320 T^{4} - 128 T^{6} + T^{8}$$
$71$ $$( 9216 + 768 T + 160 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$73$ $$84934656 - 2359296 T^{2} + 56320 T^{4} - 256 T^{6} + T^{8}$$
$79$ $$( 9216 + 768 T + 160 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$83$ $$151807041 - 2784546 T^{2} + 38755 T^{4} - 226 T^{6} + T^{8}$$
$89$ $$( -24 + 4 T + T^{2} )^{4}$$
$97$ $$( 29241 + 358 T^{2} + T^{4} )^{2}$$