# Properties

 Label 930.2.i.c Level $930$ Weight $2$ Character orbit 930.i Analytic conductor $7.426$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$187$$ $$311$$ $$871$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 0.500000 + 0.866025i 1.00000 0.500000 0.866025i −0.500000 0.866025i 1.50000 + 2.59808i −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 1.50000 2.59808i −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.c 2
31.c even 3 1 inner 930.2.i.c 2

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

 $$T_{7}^{2} - 3T_{7} + 9$$ T7^2 - 3*T7 + 9 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{13}^{2} + 4T_{13} + 16$$ T13^2 + 4*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - 3T + 9$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} + 4T + 16$$
$17$ $$T^{2} - 4T + 16$$
$19$ $$T^{2}$$
$23$ $$(T - 2)^{2}$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} + 7T + 31$$
$37$ $$T^{2} - 6T + 36$$
$41$ $$T^{2} - 2T + 4$$
$43$ $$T^{2} + 4T + 16$$
$47$ $$(T - 4)^{2}$$
$53$ $$T^{2} - 3T + 9$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 2T + 4$$
$71$ $$T^{2} + 4T + 16$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} - 4T + 16$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$(T - 10)^{2}$$
$97$ $$(T - 11)^{2}$$