Properties

 Label 930.2.i.k 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: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
211.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
−1.00000 −0.500000 0.866025i 1.00000 0.500000 0.866025i 0.500000 + 0.866025i −1.18614 2.05446i −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 1.68614 + 2.92048i −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 −1.18614 + 2.05446i −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 1.68614 2.92048i −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.k 4
31.c even 3 1 inner 930.2.i.k 4

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

 $$T_{7}^{4} - T_{7}^{3} + 9T_{7}^{2} + 8T_{7} + 64$$ T7^4 - T7^3 + 9*T7^2 + 8*T7 + 64 $$T_{11}^{2} + 5T_{11} + 25$$ T11^2 + 5*T11 + 25 $$T_{13}^{2} + 2T_{13} + 4$$ T13^2 + 2*T13 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{4}$$
$3$ $$(T^{2} + T + 1)^{2}$$
$5$ $$(T^{2} - T + 1)^{2}$$
$7$ $$T^{4} - T^{3} + 9 T^{2} + 8 T + 64$$
$11$ $$(T^{2} + 5 T + 25)^{2}$$
$13$ $$(T^{2} + 2 T + 4)^{2}$$
$17$ $$T^{4} - 5 T^{3} + 27 T^{2} + 10 T + 4$$
$19$ $$T^{4} + 2 T^{3} + 36 T^{2} + \cdots + 1024$$
$23$ $$(T^{2} + T - 74)^{2}$$
$29$ $$(T^{2} + 13 T + 34)^{2}$$
$31$ $$T^{4} - 5 T^{3} - 6 T^{2} - 155 T + 961$$
$37$ $$T^{4} + 3 T^{3} + 81 T^{2} + \cdots + 5184$$
$41$ $$T^{4} + 2 T^{3} + 36 T^{2} + \cdots + 1024$$
$43$ $$T^{4} + T^{3} + 9 T^{2} - 8 T + 64$$
$47$ $$(T^{2} - 7 T - 62)^{2}$$
$53$ $$T^{4} - 9 T^{3} + 69 T^{2} - 108 T + 144$$
$59$ $$T^{4} - 5 T^{3} + 93 T^{2} + \cdots + 4624$$
$61$ $$(T + 2)^{4}$$
$67$ $$T^{4} - 3 T^{3} + 81 T^{2} + \cdots + 5184$$
$71$ $$T^{4} + 6 T^{3} + 60 T^{2} - 144 T + 576$$
$73$ $$(T^{2} + 14 T + 196)^{2}$$
$79$ $$T^{4} - T^{3} + 9 T^{2} + 8 T + 64$$
$83$ $$T^{4} - T^{3} + 9 T^{2} + 8 T + 64$$
$89$ $$T^{4}$$
$97$ $$(T^{2} - 27 T + 174)^{2}$$