Properties

 Label 930.2.a.r Level $930$ Weight $2$ Character orbit 930.a Self dual yes Analytic conductor $7.426$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$930 = 2 \cdot 3 \cdot 5 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 930.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$7.42608738798$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + \beta q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 + q^3 + q^4 + q^5 + q^6 + b * q^7 + q^8 + q^9 $$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + \beta q^{7} + q^{8} + q^{9} + q^{10} + ( - \beta + 4) q^{11} + q^{12} - 2 q^{13} + \beta q^{14} + q^{15} + q^{16} + ( - 2 \beta + 2) q^{17} + q^{18} + (\beta - 4) q^{19} + q^{20} + \beta q^{21} + ( - \beta + 4) q^{22} + \beta q^{23} + q^{24} + q^{25} - 2 q^{26} + q^{27} + \beta q^{28} + ( - 2 \beta - 2) q^{29} + q^{30} - q^{31} + q^{32} + ( - \beta + 4) q^{33} + ( - 2 \beta + 2) q^{34} + \beta q^{35} + q^{36} + ( - 2 \beta + 6) q^{37} + (\beta - 4) q^{38} - 2 q^{39} + q^{40} + (2 \beta - 6) q^{41} + \beta q^{42} + ( - \beta + 4) q^{43} + ( - \beta + 4) q^{44} + q^{45} + \beta q^{46} - 2 \beta q^{47} + q^{48} + (\beta + 1) q^{49} + q^{50} + ( - 2 \beta + 2) q^{51} - 2 q^{52} + (\beta - 2) q^{53} + q^{54} + ( - \beta + 4) q^{55} + \beta q^{56} + (\beta - 4) q^{57} + ( - 2 \beta - 2) q^{58} + (2 \beta - 4) q^{59} + q^{60} + (4 \beta - 2) q^{61} - q^{62} + \beta q^{63} + q^{64} - 2 q^{65} + ( - \beta + 4) q^{66} + (2 \beta + 4) q^{67} + ( - 2 \beta + 2) q^{68} + \beta q^{69} + \beta q^{70} + \beta q^{71} + q^{72} + ( - 3 \beta + 2) q^{73} + ( - 2 \beta + 6) q^{74} + q^{75} + (\beta - 4) q^{76} + (3 \beta - 8) q^{77} - 2 q^{78} + (\beta - 8) q^{79} + q^{80} + q^{81} + (2 \beta - 6) q^{82} - 12 q^{83} + \beta q^{84} + ( - 2 \beta + 2) q^{85} + ( - \beta + 4) q^{86} + ( - 2 \beta - 2) q^{87} + ( - \beta + 4) q^{88} + ( - \beta + 2) q^{89} + q^{90} - 2 \beta q^{91} + \beta q^{92} - q^{93} - 2 \beta q^{94} + (\beta - 4) q^{95} + q^{96} + 2 q^{97} + (\beta + 1) q^{98} + ( - \beta + 4) q^{99} +O(q^{100})$$ q + q^2 + q^3 + q^4 + q^5 + q^6 + b * q^7 + q^8 + q^9 + q^10 + (-b + 4) * q^11 + q^12 - 2 * q^13 + b * q^14 + q^15 + q^16 + (-2*b + 2) * q^17 + q^18 + (b - 4) * q^19 + q^20 + b * q^21 + (-b + 4) * q^22 + b * q^23 + q^24 + q^25 - 2 * q^26 + q^27 + b * q^28 + (-2*b - 2) * q^29 + q^30 - q^31 + q^32 + (-b + 4) * q^33 + (-2*b + 2) * q^34 + b * q^35 + q^36 + (-2*b + 6) * q^37 + (b - 4) * q^38 - 2 * q^39 + q^40 + (2*b - 6) * q^41 + b * q^42 + (-b + 4) * q^43 + (-b + 4) * q^44 + q^45 + b * q^46 - 2*b * q^47 + q^48 + (b + 1) * q^49 + q^50 + (-2*b + 2) * q^51 - 2 * q^52 + (b - 2) * q^53 + q^54 + (-b + 4) * q^55 + b * q^56 + (b - 4) * q^57 + (-2*b - 2) * q^58 + (2*b - 4) * q^59 + q^60 + (4*b - 2) * q^61 - q^62 + b * q^63 + q^64 - 2 * q^65 + (-b + 4) * q^66 + (2*b + 4) * q^67 + (-2*b + 2) * q^68 + b * q^69 + b * q^70 + b * q^71 + q^72 + (-3*b + 2) * q^73 + (-2*b + 6) * q^74 + q^75 + (b - 4) * q^76 + (3*b - 8) * q^77 - 2 * q^78 + (b - 8) * q^79 + q^80 + q^81 + (2*b - 6) * q^82 - 12 * q^83 + b * q^84 + (-2*b + 2) * q^85 + (-b + 4) * q^86 + (-2*b - 2) * q^87 + (-b + 4) * q^88 + (-b + 2) * q^89 + q^90 - 2*b * q^91 + b * q^92 - q^93 - 2*b * q^94 + (b - 4) * q^95 + q^96 + 2 * q^97 + (b + 1) * q^98 + (-b + 4) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 2 * q^8 + 2 * q^9 $$2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 7 q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 7 q^{19} + 2 q^{20} + q^{21} + 7 q^{22} + q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} + 2 q^{27} + q^{28} - 6 q^{29} + 2 q^{30} - 2 q^{31} + 2 q^{32} + 7 q^{33} + 2 q^{34} + q^{35} + 2 q^{36} + 10 q^{37} - 7 q^{38} - 4 q^{39} + 2 q^{40} - 10 q^{41} + q^{42} + 7 q^{43} + 7 q^{44} + 2 q^{45} + q^{46} - 2 q^{47} + 2 q^{48} + 3 q^{49} + 2 q^{50} + 2 q^{51} - 4 q^{52} - 3 q^{53} + 2 q^{54} + 7 q^{55} + q^{56} - 7 q^{57} - 6 q^{58} - 6 q^{59} + 2 q^{60} - 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + 7 q^{66} + 10 q^{67} + 2 q^{68} + q^{69} + q^{70} + q^{71} + 2 q^{72} + q^{73} + 10 q^{74} + 2 q^{75} - 7 q^{76} - 13 q^{77} - 4 q^{78} - 15 q^{79} + 2 q^{80} + 2 q^{81} - 10 q^{82} - 24 q^{83} + q^{84} + 2 q^{85} + 7 q^{86} - 6 q^{87} + 7 q^{88} + 3 q^{89} + 2 q^{90} - 2 q^{91} + q^{92} - 2 q^{93} - 2 q^{94} - 7 q^{95} + 2 q^{96} + 4 q^{97} + 3 q^{98} + 7 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^5 + 2 * q^6 + q^7 + 2 * q^8 + 2 * q^9 + 2 * q^10 + 7 * q^11 + 2 * q^12 - 4 * q^13 + q^14 + 2 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 7 * q^19 + 2 * q^20 + q^21 + 7 * q^22 + q^23 + 2 * q^24 + 2 * q^25 - 4 * q^26 + 2 * q^27 + q^28 - 6 * q^29 + 2 * q^30 - 2 * q^31 + 2 * q^32 + 7 * q^33 + 2 * q^34 + q^35 + 2 * q^36 + 10 * q^37 - 7 * q^38 - 4 * q^39 + 2 * q^40 - 10 * q^41 + q^42 + 7 * q^43 + 7 * q^44 + 2 * q^45 + q^46 - 2 * q^47 + 2 * q^48 + 3 * q^49 + 2 * q^50 + 2 * q^51 - 4 * q^52 - 3 * q^53 + 2 * q^54 + 7 * q^55 + q^56 - 7 * q^57 - 6 * q^58 - 6 * q^59 + 2 * q^60 - 2 * q^62 + q^63 + 2 * q^64 - 4 * q^65 + 7 * q^66 + 10 * q^67 + 2 * q^68 + q^69 + q^70 + q^71 + 2 * q^72 + q^73 + 10 * q^74 + 2 * q^75 - 7 * q^76 - 13 * q^77 - 4 * q^78 - 15 * q^79 + 2 * q^80 + 2 * q^81 - 10 * q^82 - 24 * q^83 + q^84 + 2 * q^85 + 7 * q^86 - 6 * q^87 + 7 * q^88 + 3 * q^89 + 2 * q^90 - 2 * q^91 + q^92 - 2 * q^93 - 2 * q^94 - 7 * q^95 + 2 * q^96 + 4 * q^97 + 3 * q^98 + 7 * q^99

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −2.37228 3.37228
1.00000 1.00000 1.00000 1.00000 1.00000 −2.37228 1.00000 1.00000 1.00000
1.2 1.00000 1.00000 1.00000 1.00000 1.00000 3.37228 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$31$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 930.2.a.r 2
3.b odd 2 1 2790.2.a.bd 2
4.b odd 2 1 7440.2.a.bg 2
5.b even 2 1 4650.2.a.by 2
5.c odd 4 2 4650.2.d.bh 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
930.2.a.r 2 1.a even 1 1 trivial
2790.2.a.bd 2 3.b odd 2 1
4650.2.a.by 2 5.b even 2 1
4650.2.d.bh 4 5.c odd 4 2
7440.2.a.bg 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(930))$$:

 $$T_{7}^{2} - T_{7} - 8$$ T7^2 - T7 - 8 $$T_{11}^{2} - 7T_{11} + 4$$ T11^2 - 7*T11 + 4 $$T_{19}^{2} + 7T_{19} + 4$$ T19^2 + 7*T19 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$(T - 1)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - T - 8$$
$11$ $$T^{2} - 7T + 4$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 2T - 32$$
$19$ $$T^{2} + 7T + 4$$
$23$ $$T^{2} - T - 8$$
$29$ $$T^{2} + 6T - 24$$
$31$ $$(T + 1)^{2}$$
$37$ $$T^{2} - 10T - 8$$
$41$ $$T^{2} + 10T - 8$$
$43$ $$T^{2} - 7T + 4$$
$47$ $$T^{2} + 2T - 32$$
$53$ $$T^{2} + 3T - 6$$
$59$ $$T^{2} + 6T - 24$$
$61$ $$T^{2} - 132$$
$67$ $$T^{2} - 10T - 8$$
$71$ $$T^{2} - T - 8$$
$73$ $$T^{2} - T - 74$$
$79$ $$T^{2} + 15T + 48$$
$83$ $$(T + 12)^{2}$$
$89$ $$T^{2} - 3T - 6$$
$97$ $$(T - 2)^{2}$$