# 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: $$x^{4} + 7x^{2} + 49$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + \beta_{2} q^{5} + (\beta_{2} + 1) q^{6} + ( - \beta_{2} - 1) q^{7} - q^{8} + \beta_{2} q^{9}+O(q^{10})$$ q - q^2 + (-b2 - 1) * q^3 + q^4 + b2 * q^5 + (b2 + 1) * q^6 + (-b2 - 1) * q^7 - q^8 + b2 * q^9 $$q - q^{2} + ( - \beta_{2} - 1) q^{3} + q^{4} + \beta_{2} q^{5} + (\beta_{2} + 1) q^{6} + ( - \beta_{2} - 1) q^{7} - q^{8} + \beta_{2} q^{9} - \beta_{2} q^{10} - \beta_{2} q^{11} + ( - \beta_{2} - 1) q^{12} + 2 \beta_{2} q^{13} + (\beta_{2} + 1) q^{14} + q^{15} + q^{16} + (2 \beta_{2} + \beta_1 + 2) q^{17} - \beta_{2} q^{18} + ( - 2 \beta_{2} - \beta_1 - 2) q^{19} + \beta_{2} q^{20} + \beta_{2} q^{21} + \beta_{2} q^{22} + ( - \beta_{3} + 2) q^{23} + (\beta_{2} + 1) q^{24} + ( - \beta_{2} - 1) q^{25} - 2 \beta_{2} q^{26} + q^{27} + ( - \beta_{2} - 1) q^{28} + ( - \beta_{3} - 1) q^{29} - q^{30} + (\beta_{2} - \beta_1 - 1) q^{31} - q^{32} - q^{33} + ( - 2 \beta_{2} - \beta_1 - 2) q^{34} + q^{35} + \beta_{2} q^{36} + (4 \beta_{2} + \beta_1 + 4) q^{37} + (2 \beta_{2} + \beta_1 + 2) q^{38} + 2 q^{39} - \beta_{2} q^{40} + ( - \beta_{3} - \beta_1) q^{41} - \beta_{2} q^{42} + (4 \beta_{2} + 4) q^{43} - \beta_{2} q^{44} + ( - \beta_{2} - 1) q^{45} + (\beta_{3} - 2) q^{46} + ( - \beta_{3} - 6) q^{47} + ( - \beta_{2} - 1) q^{48} - 6 \beta_{2} q^{49} + (\beta_{2} + 1) q^{50} + ( - \beta_{3} - 2 \beta_{2} - \beta_1) q^{51} + 2 \beta_{2} q^{52} + 3 \beta_{2} q^{53} - q^{54} + (\beta_{2} + 1) q^{55} + (\beta_{2} + 1) q^{56} + (\beta_{3} + 2 \beta_{2} + \beta_1) q^{57} + (\beta_{3} + 1) q^{58} + (7 \beta_{2} + 7) q^{59} + q^{60} + ( - \beta_{3} + 2) q^{61} + ( - \beta_{2} + \beta_1 + 1) q^{62} + q^{63} + q^{64} + ( - 2 \beta_{2} - 2) q^{65} + q^{66} + ( - \beta_{3} - 4 \beta_{2} - \beta_1) q^{67} + (2 \beta_{2} + \beta_1 + 2) q^{68} + ( - 2 \beta_{2} - \beta_1 - 2) q^{69} - q^{70} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{71} - \beta_{2} q^{72} + ( - 2 \beta_{3} - 6 \beta_{2} - 2 \beta_1) q^{73} + ( - 4 \beta_{2} - \beta_1 - 4) q^{74} + \beta_{2} q^{75} + ( - 2 \beta_{2} - \beta_1 - 2) q^{76} - q^{77} - 2 q^{78} + (8 \beta_{2} + 8) q^{79} + \beta_{2} q^{80} + ( - \beta_{2} - 1) q^{81} + (\beta_{3} + \beta_1) q^{82} + ( - \beta_{3} + 3 \beta_{2} - \beta_1) q^{83} + \beta_{2} q^{84} + (\beta_{3} - 2) q^{85} + ( - 4 \beta_{2} - 4) q^{86} + (\beta_{2} - \beta_1 + 1) q^{87} + \beta_{2} q^{88} + (\beta_{2} + 1) q^{90} + 2 q^{91} + ( - \beta_{3} + 2) q^{92} + (\beta_{3} + \beta_{2} + \beta_1 + 2) q^{93} + (\beta_{3} + 6) q^{94} + ( - \beta_{3} + 2) q^{95} + (\beta_{2} + 1) q^{96} + ( - \beta_{3} - 5) q^{97} + 6 \beta_{2} q^{98} + (\beta_{2} + 1) q^{99}+O(q^{100})$$ q - q^2 + (-b2 - 1) * q^3 + q^4 + b2 * q^5 + (b2 + 1) * q^6 + (-b2 - 1) * q^7 - q^8 + b2 * q^9 - b2 * q^10 - b2 * q^11 + (-b2 - 1) * q^12 + 2*b2 * q^13 + (b2 + 1) * q^14 + q^15 + q^16 + (2*b2 + b1 + 2) * q^17 - b2 * q^18 + (-2*b2 - b1 - 2) * q^19 + b2 * q^20 + b2 * q^21 + b2 * q^22 + (-b3 + 2) * q^23 + (b2 + 1) * q^24 + (-b2 - 1) * q^25 - 2*b2 * q^26 + q^27 + (-b2 - 1) * q^28 + (-b3 - 1) * q^29 - q^30 + (b2 - b1 - 1) * q^31 - q^32 - q^33 + (-2*b2 - b1 - 2) * q^34 + q^35 + b2 * q^36 + (4*b2 + b1 + 4) * q^37 + (2*b2 + b1 + 2) * q^38 + 2 * q^39 - b2 * q^40 + (-b3 - b1) * q^41 - b2 * q^42 + (4*b2 + 4) * q^43 - b2 * q^44 + (-b2 - 1) * q^45 + (b3 - 2) * q^46 + (-b3 - 6) * q^47 + (-b2 - 1) * q^48 - 6*b2 * q^49 + (b2 + 1) * q^50 + (-b3 - 2*b2 - b1) * q^51 + 2*b2 * q^52 + 3*b2 * q^53 - q^54 + (b2 + 1) * q^55 + (b2 + 1) * q^56 + (b3 + 2*b2 + b1) * q^57 + (b3 + 1) * q^58 + (7*b2 + 7) * q^59 + q^60 + (-b3 + 2) * q^61 + (-b2 + b1 + 1) * q^62 + q^63 + q^64 + (-2*b2 - 2) * q^65 + q^66 + (-b3 - 4*b2 - b1) * q^67 + (2*b2 + b1 + 2) * q^68 + (-2*b2 - b1 - 2) * q^69 - q^70 + (-2*b3 + 4*b2 - 2*b1) * q^71 - b2 * q^72 + (-2*b3 - 6*b2 - 2*b1) * q^73 + (-4*b2 - b1 - 4) * q^74 + b2 * q^75 + (-2*b2 - b1 - 2) * q^76 - q^77 - 2 * q^78 + (8*b2 + 8) * q^79 + b2 * q^80 + (-b2 - 1) * q^81 + (b3 + b1) * q^82 + (-b3 + 3*b2 - b1) * q^83 + b2 * q^84 + (b3 - 2) * q^85 + (-4*b2 - 4) * q^86 + (b2 - b1 + 1) * q^87 + b2 * q^88 + (b2 + 1) * q^90 + 2 * q^91 + (-b3 + 2) * q^92 + (b3 + b2 + b1 + 2) * q^93 + (b3 + 6) * q^94 + (-b3 + 2) * q^95 + (b2 + 1) * q^96 + (-b3 - 5) * q^97 + 6*b2 * q^98 + (b2 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} - 2 q^{9}+O(q^{10})$$ 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 4 * q^8 - 2 * q^9 $$4 q - 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{7} - 4 q^{8} - 2 q^{9} + 2 q^{10} + 2 q^{11} - 2 q^{12} - 4 q^{13} + 2 q^{14} + 4 q^{15} + 4 q^{16} + 4 q^{17} + 2 q^{18} - 4 q^{19} - 2 q^{20} - 2 q^{21} - 2 q^{22} + 8 q^{23} + 2 q^{24} - 2 q^{25} + 4 q^{26} + 4 q^{27} - 2 q^{28} - 4 q^{29} - 4 q^{30} - 6 q^{31} - 4 q^{32} - 4 q^{33} - 4 q^{34} + 4 q^{35} - 2 q^{36} + 8 q^{37} + 4 q^{38} + 8 q^{39} + 2 q^{40} + 2 q^{42} + 8 q^{43} + 2 q^{44} - 2 q^{45} - 8 q^{46} - 24 q^{47} - 2 q^{48} + 12 q^{49} + 2 q^{50} + 4 q^{51} - 4 q^{52} - 6 q^{53} - 4 q^{54} + 2 q^{55} + 2 q^{56} - 4 q^{57} + 4 q^{58} + 14 q^{59} + 4 q^{60} + 8 q^{61} + 6 q^{62} + 4 q^{63} + 4 q^{64} - 4 q^{65} + 4 q^{66} + 8 q^{67} + 4 q^{68} - 4 q^{69} - 4 q^{70} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 8 q^{74} - 2 q^{75} - 4 q^{76} - 4 q^{77} - 8 q^{78} + 16 q^{79} - 2 q^{80} - 2 q^{81} - 6 q^{83} - 2 q^{84} - 8 q^{85} - 8 q^{86} + 2 q^{87} - 2 q^{88} + 2 q^{90} + 8 q^{91} + 8 q^{92} + 6 q^{93} + 24 q^{94} + 8 q^{95} + 2 q^{96} - 20 q^{97} - 12 q^{98} + 2 q^{99}+O(q^{100})$$ 4 * q - 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^7 - 4 * q^8 - 2 * q^9 + 2 * q^10 + 2 * q^11 - 2 * q^12 - 4 * q^13 + 2 * q^14 + 4 * q^15 + 4 * q^16 + 4 * q^17 + 2 * q^18 - 4 * q^19 - 2 * q^20 - 2 * q^21 - 2 * q^22 + 8 * q^23 + 2 * q^24 - 2 * q^25 + 4 * q^26 + 4 * q^27 - 2 * q^28 - 4 * q^29 - 4 * q^30 - 6 * q^31 - 4 * q^32 - 4 * q^33 - 4 * q^34 + 4 * q^35 - 2 * q^36 + 8 * q^37 + 4 * q^38 + 8 * q^39 + 2 * q^40 + 2 * q^42 + 8 * q^43 + 2 * q^44 - 2 * q^45 - 8 * q^46 - 24 * q^47 - 2 * q^48 + 12 * q^49 + 2 * q^50 + 4 * q^51 - 4 * q^52 - 6 * q^53 - 4 * q^54 + 2 * q^55 + 2 * q^56 - 4 * q^57 + 4 * q^58 + 14 * q^59 + 4 * q^60 + 8 * q^61 + 6 * q^62 + 4 * q^63 + 4 * q^64 - 4 * q^65 + 4 * q^66 + 8 * q^67 + 4 * q^68 - 4 * q^69 - 4 * q^70 - 8 * q^71 + 2 * q^72 + 12 * q^73 - 8 * q^74 - 2 * q^75 - 4 * q^76 - 4 * q^77 - 8 * q^78 + 16 * q^79 - 2 * q^80 - 2 * q^81 - 6 * q^83 - 2 * q^84 - 8 * q^85 - 8 * q^86 + 2 * q^87 - 2 * q^88 + 2 * q^90 + 8 * q^91 + 8 * q^92 + 6 * q^93 + 24 * q^94 + 8 * q^95 + 2 * q^96 - 20 * q^97 - 12 * q^98 + 2 * q^99

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

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 7$$ (v^2) / 7 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} ) / 7$$ (2*v^3) / 7
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$7\beta_{2}$$ 7*b2 $$\nu^{3}$$ $$=$$ $$( 7\beta_{3} ) / 2$$ (7*b3) / 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 $$S_{2}^{\mathrm{new}}(930, [\chi])$$:

 $$T_{7}^{2} + T_{7} + 1$$ T7^2 + T7 + 1 $$T_{11}^{2} - T_{11} + 1$$ T11^2 - T11 + 1 $$T_{13}^{2} + 2T_{13} + 4$$ T13^2 + 2*T13 + 4

## Hecke characteristic polynomials

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