# Properties

 Label 130.2.e.b Level $130$ Weight $2$ Character orbit 130.e Analytic conductor $1.038$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 130.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.03805522628$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/130\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$41$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 1.00000 1.73205i −0.500000 0.866025i −1.00000 −1.00000 1.73205i 0.500000 + 0.866025i −1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
81.1 0.500000 + 0.866025i 1.00000 + 1.73205i −0.500000 + 0.866025i −1.00000 −1.00000 + 1.73205i 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.e.b 2
3.b odd 2 1 1170.2.i.f 2
4.b odd 2 1 1040.2.q.c 2
5.b even 2 1 650.2.e.a 2
5.c odd 4 2 650.2.o.b 4
13.b even 2 1 1690.2.e.e 2
13.c even 3 1 inner 130.2.e.b 2
13.c even 3 1 1690.2.a.a 1
13.d odd 4 2 1690.2.l.i 4
13.e even 6 1 1690.2.a.g 1
13.e even 6 1 1690.2.e.e 2
13.f odd 12 2 1690.2.d.a 2
13.f odd 12 2 1690.2.l.i 4
39.i odd 6 1 1170.2.i.f 2
52.j odd 6 1 1040.2.q.c 2
65.l even 6 1 8450.2.a.k 1
65.n even 6 1 650.2.e.a 2
65.n even 6 1 8450.2.a.w 1
65.q odd 12 2 650.2.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.e.b 2 1.a even 1 1 trivial
130.2.e.b 2 13.c even 3 1 inner
650.2.e.a 2 5.b even 2 1
650.2.e.a 2 65.n even 6 1
650.2.o.b 4 5.c odd 4 2
650.2.o.b 4 65.q odd 12 2
1040.2.q.c 2 4.b odd 2 1
1040.2.q.c 2 52.j odd 6 1
1170.2.i.f 2 3.b odd 2 1
1170.2.i.f 2 39.i odd 6 1
1690.2.a.a 1 13.c even 3 1
1690.2.a.g 1 13.e even 6 1
1690.2.d.a 2 13.f odd 12 2
1690.2.e.e 2 13.b even 2 1
1690.2.e.e 2 13.e even 6 1
1690.2.l.i 4 13.d odd 4 2
1690.2.l.i 4 13.f odd 12 2
8450.2.a.k 1 65.l even 6 1
8450.2.a.w 1 65.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(130, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2} - 2T + 4$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} - 5T + 13$$
$17$ $$T^{2} - 6T + 36$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 11T + 121$$
$41$ $$T^{2} + 6T + 36$$
$43$ $$T^{2} + 2T + 4$$
$47$ $$(T + 3)^{2}$$
$53$ $$(T + 9)^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 8T + 64$$
$67$ $$T^{2} - 16T + 256$$
$71$ $$T^{2} + 6T + 36$$
$73$ $$(T - 14)^{2}$$
$79$ $$(T + 16)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} + 9T + 81$$
$97$ $$T^{2} - 10T + 100$$