Properties

 Label 430.2.e.a Level 430 Weight 2 Character orbit 430.e Analytic conductor 3.434 Analytic rank 1 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

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 −1.50000 + 2.59808i 1.00000 −0.500000 + 0.866025i 1.50000 2.59808i 0 −1.00000 −3.00000 5.19615i 0.500000 0.866025i
251.1 −1.00000 −1.50000 2.59808i 1.00000 −0.500000 0.866025i 1.50000 + 2.59808i 0 −1.00000 −3.00000 + 5.19615i 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.a 2
43.c even 3 1 inner 430.2.e.a 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + 3 T^{2} )( 1 + 3 T + 3 T^{2} )$$
$5$ $$1 + T + T^{2}$$
$7$ $$1 - 7 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 6 T + 11 T^{2} )^{2}$$
$13$ $$1 - 13 T^{2} + 169 T^{4}$$
$17$ $$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 23 T^{2} + 529 T^{4}$$
$29$ $$1 + 9 T + 52 T^{2} + 261 T^{3} + 841 T^{4}$$
$31$ $$( 1 - 7 T + 31 T^{2} )( 1 + 11 T + 31 T^{2} )$$
$37$ $$1 - 8 T + 27 T^{2} - 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 5 T + 41 T^{2} )^{2}$$
$43$ $$1 + 13 T + 43 T^{2}$$
$47$ $$( 1 + 3 T + 47 T^{2} )^{2}$$
$53$ $$1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - 2 T + 59 T^{2} )^{2}$$
$61$ $$1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4}$$
$67$ $$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$1 + 8 T - 7 T^{2} + 568 T^{3} + 5041 T^{4}$$
$73$ $$1 + 8 T - 9 T^{2} + 584 T^{3} + 5329 T^{4}$$
$79$ $$1 - 79 T^{2} + 6241 T^{4}$$
$83$ $$1 + 11 T + 38 T^{2} + 913 T^{3} + 6889 T^{4}$$
$89$ $$1 + 13 T + 80 T^{2} + 1157 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 8 T + 97 T^{2} )^{2}$$