# Properties

 Label 430.2.e.b Level 430 Weight 2 Character orbit 430.e Analytic conductor 3.434 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/430\mathbb{Z}\right)^\times$$.

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 −0.500000 + 0.866025i 1.00000 −0.500000 + 0.866025i 0.500000 0.866025i −2.00000 3.46410i −1.00000 1.00000 + 1.73205i 0.500000 0.866025i
251.1 −1.00000 −0.500000 0.866025i 1.00000 −0.500000 0.866025i 0.500000 + 0.866025i −2.00000 + 3.46410i −1.00000 1.00000 1.73205i 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
43.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.e.b 2
43.c even 3 1 inner 430.2.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.b 2 1.a even 1 1 trivial
430.2.e.b 2 43.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4}$$
$5$ $$1 + T + T^{2}$$
$7$ $$( 1 - T + 7 T^{2} )( 1 + 5 T + 7 T^{2} )$$
$11$ $$( 1 - 4 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 7 T + 13 T^{2} )( 1 + 5 T + 13 T^{2} )$$
$17$ $$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$
$19$ $$1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$
$31$ $$1 + 10 T + 69 T^{2} + 310 T^{3} + 961 T^{4}$$
$37$ $$1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 5 T + 41 T^{2} )^{2}$$
$43$ $$1 - 13 T + 43 T^{2}$$
$47$ $$( 1 - T + 47 T^{2} )^{2}$$
$53$ $$1 + 10 T + 47 T^{2} + 530 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - 14 T + 59 T^{2} )^{2}$$
$61$ $$1 - 6 T - 25 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 3 T - 58 T^{2} - 201 T^{3} + 4489 T^{4}$$
$71$ $$1 + 10 T + 29 T^{2} + 710 T^{3} + 5041 T^{4}$$
$73$ $$1 - 14 T + 123 T^{2} - 1022 T^{3} + 5329 T^{4}$$
$79$ $$1 + 16 T + 177 T^{2} + 1264 T^{3} + 6241 T^{4}$$
$83$ $$1 - 15 T + 142 T^{2} - 1245 T^{3} + 6889 T^{4}$$
$89$ $$1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 12 T + 97 T^{2} )^{2}$$