# Properties

 Label 930.2.j.b Level $930$ Weight $2$ Character orbit 930.j 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.j (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ 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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{3} - \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} + 1) q^{6} + \zeta_{8} q^{8} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{9} +O(q^{10})$$ q + z^3 * q^2 + (-z^2 - z + 1) * q^3 - z^2 * q^4 + (-z^3 + 2*z) * q^5 + (z^3 + z + 1) * q^6 + z * q^8 + (2*z^3 - z^2 - 2*z) * q^9 $$q + \zeta_{8}^{3} q^{2} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{3} - \zeta_{8}^{2} q^{4} + ( - \zeta_{8}^{3} + 2 \zeta_{8}) q^{5} + (\zeta_{8}^{3} + \zeta_{8} + 1) q^{6} + \zeta_{8} q^{8} + (2 \zeta_{8}^{3} - \zeta_{8}^{2} - 2 \zeta_{8}) q^{9} + (\zeta_{8}^{2} - 2) q^{10} + (\zeta_{8}^{3} + \zeta_{8}) q^{11} + (\zeta_{8}^{3} - \zeta_{8}^{2} - 1) q^{12} + ( - 3 \zeta_{8}^{2} + 3) q^{13} + ( - 3 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + \zeta_{8} - 1) q^{15} - q^{16} - 2 \zeta_{8}^{3} q^{17} + ( - 2 \zeta_{8}^{2} + \zeta_{8} + 2) q^{18} - 4 \zeta_{8}^{2} q^{19} + ( - 2 \zeta_{8}^{3} - \zeta_{8}) q^{20} + ( - \zeta_{8}^{2} - 1) q^{22} + 4 \zeta_{8} q^{23} + ( - \zeta_{8}^{3} - \zeta_{8}^{2} + \zeta_{8}) q^{24} + (3 \zeta_{8}^{2} + 4) q^{25} + (3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{26} + (5 \zeta_{8}^{3} + \zeta_{8}^{2} + 1) q^{27} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8}) q^{29} + ( - \zeta_{8}^{3} + 3 \zeta_{8}^{2} + 2 \zeta_{8} - 1) q^{30} + q^{31} - \zeta_{8}^{3} q^{32} + ( - \zeta_{8}^{2} + 2 \zeta_{8} + 1) q^{33} + 2 \zeta_{8}^{2} q^{34} + (2 \zeta_{8}^{3} + 2 \zeta_{8} - 1) q^{36} + ( - 3 \zeta_{8}^{2} - 3) q^{37} + 4 \zeta_{8} q^{38} + (3 \zeta_{8}^{3} - 6 \zeta_{8}^{2} - 3 \zeta_{8}) q^{39} + (2 \zeta_{8}^{2} + 1) q^{40} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}) q^{41} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{44} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - \zeta_{8} - 6) q^{45} - 4 q^{46} + 2 \zeta_{8}^{3} q^{47} + (\zeta_{8}^{2} + \zeta_{8} - 1) q^{48} - 7 \zeta_{8}^{2} q^{49} + (4 \zeta_{8}^{3} - 3 \zeta_{8}) q^{50} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 2) q^{51} + ( - 3 \zeta_{8}^{2} - 3) q^{52} - 2 \zeta_{8} q^{53} + (\zeta_{8}^{3} - 5 \zeta_{8}^{2} - \zeta_{8}) q^{54} + (3 \zeta_{8}^{2} - 1) q^{55} + (4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} - 4) q^{57} + (2 \zeta_{8}^{2} - 2) q^{58} + ( - \zeta_{8}^{3} + \zeta_{8}^{2} - 3 \zeta_{8} - 2) q^{60} + 8 q^{61} + \zeta_{8}^{3} q^{62} + \zeta_{8}^{2} q^{64} + ( - 9 \zeta_{8}^{3} + 3 \zeta_{8}) q^{65} + (\zeta_{8}^{3} + \zeta_{8} - 2) q^{66} + (7 \zeta_{8}^{2} + 7) q^{67} - 2 \zeta_{8} q^{68} + ( - 4 \zeta_{8}^{3} - 4 \zeta_{8}^{2} + 4 \zeta_{8}) q^{69} + ( - 9 \zeta_{8}^{3} - 9 \zeta_{8}) q^{71} + ( - \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2) q^{72} + (2 \zeta_{8}^{2} - 2) q^{73} + ( - 3 \zeta_{8}^{3} + 3 \zeta_{8}) q^{74} + ( - 3 \zeta_{8}^{3} - \zeta_{8}^{2} - 4 \zeta_{8} + 7) q^{75} - 4 q^{76} + ( - 3 \zeta_{8}^{2} + 6 \zeta_{8} + 3) q^{78} + 2 \zeta_{8}^{2} q^{79} + (\zeta_{8}^{3} - 2 \zeta_{8}) q^{80} + (4 \zeta_{8}^{3} + 4 \zeta_{8} + 7) q^{81} + (4 \zeta_{8}^{2} + 4) q^{82} - 6 \zeta_{8} q^{83} + ( - 2 \zeta_{8}^{2} + 4) q^{85} + ( - 4 \zeta_{8}^{3} - 2 \zeta_{8}^{2} - 2) q^{87} + (\zeta_{8}^{2} - 1) q^{88} + (5 \zeta_{8}^{3} - 5 \zeta_{8}) q^{89} + ( - 6 \zeta_{8}^{3} + 2 \zeta_{8}^{2} + 2 \zeta_{8} + 1) q^{90} - 4 \zeta_{8}^{3} q^{92} + ( - \zeta_{8}^{2} - \zeta_{8} + 1) q^{93} - 2 \zeta_{8}^{2} q^{94} + ( - 8 \zeta_{8}^{3} - 4 \zeta_{8}) q^{95} + ( - \zeta_{8}^{3} - \zeta_{8} - 1) q^{96} + (11 \zeta_{8}^{2} + 11) q^{97} + 7 \zeta_{8} q^{98} + ( - \zeta_{8}^{3} - 4 \zeta_{8}^{2} + \zeta_{8}) q^{99} +O(q^{100})$$ q + z^3 * q^2 + (-z^2 - z + 1) * q^3 - z^2 * q^4 + (-z^3 + 2*z) * q^5 + (z^3 + z + 1) * q^6 + z * q^8 + (2*z^3 - z^2 - 2*z) * q^9 + (z^2 - 2) * q^10 + (z^3 + z) * q^11 + (z^3 - z^2 - 1) * q^12 + (-3*z^2 + 3) * q^13 + (-3*z^3 - 2*z^2 + z - 1) * q^15 - q^16 - 2*z^3 * q^17 + (-2*z^2 + z + 2) * q^18 - 4*z^2 * q^19 + (-2*z^3 - z) * q^20 + (-z^2 - 1) * q^22 + 4*z * q^23 + (-z^3 - z^2 + z) * q^24 + (3*z^2 + 4) * q^25 + (3*z^3 + 3*z) * q^26 + (5*z^3 + z^2 + 1) * q^27 + (-2*z^3 + 2*z) * q^29 + (-z^3 + 3*z^2 + 2*z - 1) * q^30 + q^31 - z^3 * q^32 + (-z^2 + 2*z + 1) * q^33 + 2*z^2 * q^34 + (2*z^3 + 2*z - 1) * q^36 + (-3*z^2 - 3) * q^37 + 4*z * q^38 + (3*z^3 - 6*z^2 - 3*z) * q^39 + (2*z^2 + 1) * q^40 + (-4*z^3 - 4*z) * q^41 + (-z^3 + z) * q^44 + (-2*z^3 - 2*z^2 - z - 6) * q^45 - 4 * q^46 + 2*z^3 * q^47 + (z^2 + z - 1) * q^48 - 7*z^2 * q^49 + (4*z^3 - 3*z) * q^50 + (-2*z^3 - 2*z - 2) * q^51 + (-3*z^2 - 3) * q^52 - 2*z * q^53 + (z^3 - 5*z^2 - z) * q^54 + (3*z^2 - 1) * q^55 + (4*z^3 - 4*z^2 - 4) * q^57 + (2*z^2 - 2) * q^58 + (-z^3 + z^2 - 3*z - 2) * q^60 + 8 * q^61 + z^3 * q^62 + z^2 * q^64 + (-9*z^3 + 3*z) * q^65 + (z^3 + z - 2) * q^66 + (7*z^2 + 7) * q^67 - 2*z * q^68 + (-4*z^3 - 4*z^2 + 4*z) * q^69 + (-9*z^3 - 9*z) * q^71 + (-z^3 - 2*z^2 - 2) * q^72 + (2*z^2 - 2) * q^73 + (-3*z^3 + 3*z) * q^74 + (-3*z^3 - z^2 - 4*z + 7) * q^75 - 4 * q^76 + (-3*z^2 + 6*z + 3) * q^78 + 2*z^2 * q^79 + (z^3 - 2*z) * q^80 + (4*z^3 + 4*z + 7) * q^81 + (4*z^2 + 4) * q^82 - 6*z * q^83 + (-2*z^2 + 4) * q^85 + (-4*z^3 - 2*z^2 - 2) * q^87 + (z^2 - 1) * q^88 + (5*z^3 - 5*z) * q^89 + (-6*z^3 + 2*z^2 + 2*z + 1) * q^90 - 4*z^3 * q^92 + (-z^2 - z + 1) * q^93 - 2*z^2 * q^94 + (-8*z^3 - 4*z) * q^95 + (-z^3 - z - 1) * q^96 + (11*z^2 + 11) * q^97 + 7*z * q^98 + (-z^3 - 4*z^2 + z) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 4 q^{6}+O(q^{10})$$ 4 * q + 4 * q^3 + 4 * q^6 $$4 q + 4 q^{3} + 4 q^{6} - 8 q^{10} - 4 q^{12} + 12 q^{13} - 4 q^{15} - 4 q^{16} + 8 q^{18} - 4 q^{22} + 16 q^{25} + 4 q^{27} - 4 q^{30} + 4 q^{31} + 4 q^{33} - 4 q^{36} - 12 q^{37} + 4 q^{40} - 24 q^{45} - 16 q^{46} - 4 q^{48} - 8 q^{51} - 12 q^{52} - 4 q^{55} - 16 q^{57} - 8 q^{58} - 8 q^{60} + 32 q^{61} - 8 q^{66} + 28 q^{67} - 8 q^{72} - 8 q^{73} + 28 q^{75} - 16 q^{76} + 12 q^{78} + 28 q^{81} + 16 q^{82} + 16 q^{85} - 8 q^{87} - 4 q^{88} + 4 q^{90} + 4 q^{93} - 4 q^{96} + 44 q^{97}+O(q^{100})$$ 4 * q + 4 * q^3 + 4 * q^6 - 8 * q^10 - 4 * q^12 + 12 * q^13 - 4 * q^15 - 4 * q^16 + 8 * q^18 - 4 * q^22 + 16 * q^25 + 4 * q^27 - 4 * q^30 + 4 * q^31 + 4 * q^33 - 4 * q^36 - 12 * q^37 + 4 * q^40 - 24 * q^45 - 16 * q^46 - 4 * q^48 - 8 * q^51 - 12 * q^52 - 4 * q^55 - 16 * q^57 - 8 * q^58 - 8 * q^60 + 32 * q^61 - 8 * q^66 + 28 * q^67 - 8 * q^72 - 8 * q^73 + 28 * q^75 - 16 * q^76 + 12 * q^78 + 28 * q^81 + 16 * q^82 + 16 * q^85 - 8 * q^87 - 4 * q^88 + 4 * q^90 + 4 * q^93 - 4 * q^96 + 44 * q^97

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$\zeta_{8}^{2}$$ $$-1$$ $$1$$

## 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
 0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i −0.707107 + 0.707107i
−0.707107 + 0.707107i 0.292893 1.70711i 1.00000i 2.12132 + 0.707107i 1.00000 + 1.41421i 0 0.707107 + 0.707107i −2.82843 1.00000i −2.00000 + 1.00000i
497.2 0.707107 0.707107i 1.70711 0.292893i 1.00000i −2.12132 0.707107i 1.00000 1.41421i 0 −0.707107 0.707107i 2.82843 1.00000i −2.00000 + 1.00000i
683.1 −0.707107 0.707107i 0.292893 + 1.70711i 1.00000i 2.12132 0.707107i 1.00000 1.41421i 0 0.707107 0.707107i −2.82843 + 1.00000i −2.00000 1.00000i
683.2 0.707107 + 0.707107i 1.70711 + 0.292893i 1.00000i −2.12132 + 0.707107i 1.00000 + 1.41421i 0 −0.707107 + 0.707107i 2.82843 + 1.00000i −2.00000 1.00000i
 $$n$$: 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
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.b 4
3.b odd 2 1 inner 930.2.j.b 4
5.c odd 4 1 inner 930.2.j.b 4
15.e even 4 1 inner 930.2.j.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.j.b 4 1.a even 1 1 trivial
930.2.j.b 4 3.b odd 2 1 inner
930.2.j.b 4 5.c odd 4 1 inner
930.2.j.b 4 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}^{2} + 2$$ T11^2 + 2 $$T_{17}^{4} + 16$$ T17^4 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 1$$
$3$ $$T^{4} - 4 T^{3} + 8 T^{2} - 12 T + 9$$
$5$ $$T^{4} - 8T^{2} + 25$$
$7$ $$T^{4}$$
$11$ $$(T^{2} + 2)^{2}$$
$13$ $$(T^{2} - 6 T + 18)^{2}$$
$17$ $$T^{4} + 16$$
$19$ $$(T^{2} + 16)^{2}$$
$23$ $$T^{4} + 256$$
$29$ $$(T^{2} - 8)^{2}$$
$31$ $$(T - 1)^{4}$$
$37$ $$(T^{2} + 6 T + 18)^{2}$$
$41$ $$(T^{2} + 32)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4} + 16$$
$53$ $$T^{4} + 16$$
$59$ $$T^{4}$$
$61$ $$(T - 8)^{4}$$
$67$ $$(T^{2} - 14 T + 98)^{2}$$
$71$ $$(T^{2} + 162)^{2}$$
$73$ $$(T^{2} + 4 T + 8)^{2}$$
$79$ $$(T^{2} + 4)^{2}$$
$83$ $$T^{4} + 1296$$
$89$ $$(T^{2} - 50)^{2}$$
$97$ $$(T^{2} - 22 T + 242)^{2}$$