# Properties

 Label 930.2.d.g Level $930$ Weight $2$ Character orbit 930.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{2} q^{2} + \zeta_{8}^{2} q^{3} - q^{4} + ( 2 + \zeta_{8}^{2} ) q^{5} - q^{6} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{2} q^{8} - q^{9} +O(q^{10})$$ $$q + \zeta_{8}^{2} q^{2} + \zeta_{8}^{2} q^{3} - q^{4} + ( 2 + \zeta_{8}^{2} ) q^{5} - q^{6} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{2} q^{8} - q^{9} + ( -1 + 2 \zeta_{8}^{2} ) q^{10} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{11} -\zeta_{8}^{2} q^{12} + ( -2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{13} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{14} + ( -1 + 2 \zeta_{8}^{2} ) q^{15} + q^{16} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{17} -\zeta_{8}^{2} q^{18} + ( -2 - \zeta_{8}^{2} ) q^{20} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{21} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{22} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{23} + q^{24} + ( 3 + 4 \zeta_{8}^{2} ) q^{25} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} -\zeta_{8}^{2} q^{27} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{28} + ( -4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( -2 - \zeta_{8}^{2} ) q^{30} + q^{31} + \zeta_{8}^{2} q^{32} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{33} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{34} + ( -2 + 2 \zeta_{8} + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{35} + q^{36} + ( -6 \zeta_{8} + 2 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{37} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{39} + ( 1 - 2 \zeta_{8}^{2} ) q^{40} + ( 2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{42} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{43} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{44} + ( -2 - \zeta_{8}^{2} ) q^{45} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{46} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{47} + \zeta_{8}^{2} q^{48} + ( -5 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{49} + ( -4 + 3 \zeta_{8}^{2} ) q^{50} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{51} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{52} + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{53} + q^{54} + ( 4 + 6 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{55} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{56} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{58} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + ( 1 - 2 \zeta_{8}^{2} ) q^{60} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{61} + \zeta_{8}^{2} q^{62} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{63} - q^{64} + ( -2 - 2 \zeta_{8} + 4 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{65} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{66} + ( -2 \zeta_{8} - 12 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{67} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{68} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{69} + ( -4 - 6 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{70} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{71} + \zeta_{8}^{2} q^{72} + ( 4 \zeta_{8} - 8 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{73} + ( -2 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{74} + ( -4 + 3 \zeta_{8}^{2} ) q^{75} + ( 8 \zeta_{8} + 12 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{77} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{78} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{79} + ( 2 + \zeta_{8}^{2} ) q^{80} + q^{81} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{82} + ( 4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{83} + ( 2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{84} + ( -2 + 2 \zeta_{8} + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{85} + ( 4 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{86} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{87} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{88} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} + ( 1 - 2 \zeta_{8}^{2} ) q^{90} + 4 q^{91} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{92} + \zeta_{8}^{2} q^{93} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{94} - q^{96} + ( -8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{97} + ( -8 \zeta_{8} - 5 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{98} + ( -2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{4} + 8q^{5} - 4q^{6} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{4} + 8q^{5} - 4q^{6} - 4q^{9} - 4q^{10} + 8q^{11} - 8q^{14} - 4q^{15} + 4q^{16} - 8q^{20} - 8q^{21} + 4q^{24} + 12q^{25} - 8q^{26} - 16q^{29} - 8q^{30} + 4q^{31} - 8q^{34} - 8q^{35} + 4q^{36} - 8q^{39} + 4q^{40} + 8q^{41} - 8q^{44} - 8q^{45} - 20q^{49} - 16q^{50} - 8q^{51} + 4q^{54} + 16q^{55} + 8q^{56} - 24q^{59} + 4q^{60} - 8q^{61} - 4q^{64} - 8q^{65} - 8q^{66} - 16q^{70} - 8q^{74} - 16q^{75} + 8q^{80} + 4q^{81} + 8q^{84} - 8q^{85} + 16q^{86} - 8q^{89} + 4q^{90} + 16q^{91} - 4q^{96} - 8q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 4.82843i 1.00000i −1.00000 −1.00000 2.00000i
559.2 1.00000i 1.00000i −1.00000 2.00000 1.00000i −1.00000 0.828427i 1.00000i −1.00000 −1.00000 2.00000i
559.3 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i −1.00000 0.828427i 1.00000i −1.00000 −1.00000 + 2.00000i
559.4 1.00000i 1.00000i −1.00000 2.00000 + 1.00000i −1.00000 4.82843i 1.00000i −1.00000 −1.00000 + 2.00000i
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.d.g 4
3.b odd 2 1 2790.2.d.i 4
5.b even 2 1 inner 930.2.d.g 4
5.c odd 4 1 4650.2.a.cc 2
5.c odd 4 1 4650.2.a.cf 2
15.d odd 2 1 2790.2.d.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.d.g 4 1.a even 1 1 trivial
930.2.d.g 4 5.b even 2 1 inner
2790.2.d.i 4 3.b odd 2 1
2790.2.d.i 4 15.d odd 2 1
4650.2.a.cc 2 5.c odd 4 1
4650.2.a.cf 2 5.c odd 4 1

## Hecke kernels

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

 $$T_{7}^{4} + 24 T_{7}^{2} + 16$$ $$T_{11}^{2} - 4 T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{2}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( 5 - 4 T + T^{2} )^{2}$$
$7$ $$16 + 24 T^{2} + T^{4}$$
$11$ $$( -4 - 4 T + T^{2} )^{2}$$
$13$ $$16 + 24 T^{2} + T^{4}$$
$17$ $$16 + 24 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 72 + T^{2} )^{2}$$
$29$ $$( -16 + 8 T + T^{2} )^{2}$$
$31$ $$( -1 + T )^{4}$$
$37$ $$4624 + 152 T^{2} + T^{4}$$
$41$ $$( -28 - 4 T + T^{2} )^{2}$$
$43$ $$256 + 96 T^{2} + T^{4}$$
$47$ $$( 32 + T^{2} )^{2}$$
$53$ $$16 + 136 T^{2} + T^{4}$$
$59$ $$( 28 + 12 T + T^{2} )^{2}$$
$61$ $$( -4 + 4 T + T^{2} )^{2}$$
$67$ $$18496 + 304 T^{2} + T^{4}$$
$71$ $$( -8 + T^{2} )^{2}$$
$73$ $$1024 + 192 T^{2} + T^{4}$$
$79$ $$( -128 + T^{2} )^{2}$$
$83$ $$256 + 96 T^{2} + T^{4}$$
$89$ $$( -4 + 4 T + T^{2} )^{2}$$
$97$ $$( 128 + T^{2} )^{2}$$