# Properties

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

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( -1 - \beta_{2} ) q^{3} + q^{4} + \beta_{2} q^{5} + ( 1 + \beta_{2} ) q^{6} + ( -1 - \beta_{2} ) q^{7} - q^{8} + \beta_{2} q^{9} +O(q^{10})$$ $$q - q^{2} + ( -1 - \beta_{2} ) q^{3} + q^{4} + \beta_{2} q^{5} + ( 1 + \beta_{2} ) q^{6} + ( -1 - \beta_{2} ) q^{7} - q^{8} + \beta_{2} q^{9} -\beta_{2} q^{10} -\beta_{2} q^{11} + ( -1 - \beta_{2} ) q^{12} + 2 \beta_{2} q^{13} + ( 1 + \beta_{2} ) q^{14} + q^{15} + q^{16} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{17} -\beta_{2} q^{18} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{19} + \beta_{2} q^{20} + \beta_{2} q^{21} + \beta_{2} q^{22} + ( 2 - \beta_{3} ) q^{23} + ( 1 + \beta_{2} ) q^{24} + ( -1 - \beta_{2} ) q^{25} -2 \beta_{2} q^{26} + q^{27} + ( -1 - \beta_{2} ) q^{28} + ( -1 - \beta_{3} ) q^{29} - q^{30} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} - q^{32} - q^{33} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{34} + q^{35} + \beta_{2} q^{36} + ( 4 + \beta_{1} + 4 \beta_{2} ) q^{37} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{38} + 2 q^{39} -\beta_{2} q^{40} + ( -\beta_{1} - \beta_{3} ) q^{41} -\beta_{2} q^{42} + ( 4 + 4 \beta_{2} ) q^{43} -\beta_{2} q^{44} + ( -1 - \beta_{2} ) q^{45} + ( -2 + \beta_{3} ) q^{46} + ( -6 - \beta_{3} ) q^{47} + ( -1 - \beta_{2} ) q^{48} -6 \beta_{2} q^{49} + ( 1 + \beta_{2} ) q^{50} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{51} + 2 \beta_{2} q^{52} + 3 \beta_{2} q^{53} - q^{54} + ( 1 + \beta_{2} ) q^{55} + ( 1 + \beta_{2} ) q^{56} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{57} + ( 1 + \beta_{3} ) q^{58} + ( 7 + 7 \beta_{2} ) q^{59} + q^{60} + ( 2 - \beta_{3} ) q^{61} + ( 1 + \beta_{1} - \beta_{2} ) q^{62} + q^{63} + q^{64} + ( -2 - 2 \beta_{2} ) q^{65} + q^{66} + ( -\beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{67} + ( 2 + \beta_{1} + 2 \beta_{2} ) q^{68} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{69} - q^{70} + ( -2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{71} -\beta_{2} q^{72} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} ) q^{73} + ( -4 - \beta_{1} - 4 \beta_{2} ) q^{74} + \beta_{2} q^{75} + ( -2 - \beta_{1} - 2 \beta_{2} ) q^{76} - q^{77} -2 q^{78} + ( 8 + 8 \beta_{2} ) q^{79} + \beta_{2} q^{80} + ( -1 - \beta_{2} ) q^{81} + ( \beta_{1} + \beta_{3} ) q^{82} + ( -\beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{83} + \beta_{2} q^{84} + ( -2 + \beta_{3} ) q^{85} + ( -4 - 4 \beta_{2} ) q^{86} + ( 1 - \beta_{1} + \beta_{2} ) q^{87} + \beta_{2} q^{88} + ( 1 + \beta_{2} ) q^{90} + 2 q^{91} + ( 2 - \beta_{3} ) q^{92} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{93} + ( 6 + \beta_{3} ) q^{94} + ( 2 - \beta_{3} ) q^{95} + ( 1 + \beta_{2} ) q^{96} + ( -5 - \beta_{3} ) q^{97} + 6 \beta_{2} q^{98} + ( 1 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 2q^{3} + 4q^{4} - 2q^{5} + 2q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + 2q^{10} + 2q^{11} - 2q^{12} - 4q^{13} + 2q^{14} + 4q^{15} + 4q^{16} + 4q^{17} + 2q^{18} - 4q^{19} - 2q^{20} - 2q^{21} - 2q^{22} + 8q^{23} + 2q^{24} - 2q^{25} + 4q^{26} + 4q^{27} - 2q^{28} - 4q^{29} - 4q^{30} - 6q^{31} - 4q^{32} - 4q^{33} - 4q^{34} + 4q^{35} - 2q^{36} + 8q^{37} + 4q^{38} + 8q^{39} + 2q^{40} + 2q^{42} + 8q^{43} + 2q^{44} - 2q^{45} - 8q^{46} - 24q^{47} - 2q^{48} + 12q^{49} + 2q^{50} + 4q^{51} - 4q^{52} - 6q^{53} - 4q^{54} + 2q^{55} + 2q^{56} - 4q^{57} + 4q^{58} + 14q^{59} + 4q^{60} + 8q^{61} + 6q^{62} + 4q^{63} + 4q^{64} - 4q^{65} + 4q^{66} + 8q^{67} + 4q^{68} - 4q^{69} - 4q^{70} - 8q^{71} + 2q^{72} + 12q^{73} - 8q^{74} - 2q^{75} - 4q^{76} - 4q^{77} - 8q^{78} + 16q^{79} - 2q^{80} - 2q^{81} - 6q^{83} - 2q^{84} - 8q^{85} - 8q^{86} + 2q^{87} - 2q^{88} + 2q^{90} + 8q^{91} + 8q^{92} + 6q^{93} + 24q^{94} + 8q^{95} + 2q^{96} - 20q^{97} - 12q^{98} + 2q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$$$/2$$

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 −1.32288 − 2.29129i 1.32288 + 2.29129i −1.32288 + 2.29129i 1.32288 − 2.29129i
−1.00000 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i
211.2 −1.00000 −0.500000 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −1.00000 −0.500000 + 0.866025i 0.500000 0.866025i
811.1 −1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
811.2 −1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −1.00000 −0.500000 0.866025i 0.500000 + 0.866025i
 $$n$$: 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
31.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.i.j 4
31.c even 3 1 inner 930.2.i.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.i.j 4 1.a even 1 1 trivial
930.2.i.j 4 31.c even 3 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + T_{7} + 1$$ $$T_{11}^{2} - T_{11} + 1$$ $$T_{13}^{2} + 2 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$( 1 + T + T^{2} )^{2}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$( 4 + 2 T + T^{2} )^{2}$$
$17$ $$576 + 96 T + 40 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$576 - 96 T + 40 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$( -24 - 4 T + T^{2} )^{2}$$
$29$ $$( -27 + 2 T + T^{2} )^{2}$$
$31$ $$961 + 186 T + 43 T^{2} + 6 T^{3} + T^{4}$$
$37$ $$144 + 96 T + 76 T^{2} - 8 T^{3} + T^{4}$$
$41$ $$784 + 28 T^{2} + T^{4}$$
$43$ $$( 16 - 4 T + T^{2} )^{2}$$
$47$ $$( 8 + 12 T + T^{2} )^{2}$$
$53$ $$( 9 + 3 T + T^{2} )^{2}$$
$59$ $$( 49 - 7 T + T^{2} )^{2}$$
$61$ $$( -24 - 4 T + T^{2} )^{2}$$
$67$ $$144 + 96 T + 76 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$9216 - 768 T + 160 T^{2} + 8 T^{3} + T^{4}$$
$73$ $$5776 + 912 T + 220 T^{2} - 12 T^{3} + T^{4}$$
$79$ $$( 64 - 8 T + T^{2} )^{2}$$
$83$ $$361 - 114 T + 55 T^{2} + 6 T^{3} + T^{4}$$
$89$ $$T^{4}$$
$97$ $$( -3 + 10 T + T^{2} )^{2}$$