# Properties

 Label 430.2.e.d Level 430 Weight 2 Character orbit 430.e Analytic conductor 3.434 Analytic rank 0 Dimension 4 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{10})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 10 x^{2} + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/10$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$10 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{3}$$

## Character values

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

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$1$$ $$-1 - \beta_{2}$$

## 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
 −1.58114 − 2.73861i 1.58114 + 2.73861i −1.58114 + 2.73861i 1.58114 − 2.73861i
−1.00000 1.00000 1.73205i 1.00000 −0.500000 + 0.866025i −1.00000 + 1.73205i −1.58114 2.73861i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
221.2 −1.00000 1.00000 1.73205i 1.00000 −0.500000 + 0.866025i −1.00000 + 1.73205i 1.58114 + 2.73861i −1.00000 −0.500000 0.866025i 0.500000 0.866025i
251.1 −1.00000 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i −1.00000 1.73205i −1.58114 + 2.73861i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i
251.2 −1.00000 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i −1.00000 1.73205i 1.58114 2.73861i −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
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.d 4
43.c even 3 1 inner 430.2.e.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.e.d 4 1.a even 1 1 trivial
430.2.e.d 4 43.c even 3 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 2 T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$1 - 4 T^{2} - 33 T^{4} - 196 T^{6} + 2401 T^{8}$$
$11$ $$( 1 + 2 T + 13 T^{2} + 22 T^{3} + 121 T^{4} )^{2}$$
$13$ $$1 - 16 T^{2} + 87 T^{4} - 2704 T^{6} + 28561 T^{8}$$
$17$ $$1 + 2 T - 21 T^{2} - 18 T^{3} + 268 T^{4} - 306 T^{5} - 6069 T^{6} + 9826 T^{7} + 83521 T^{8}$$
$19$ $$1 + 2 T - 25 T^{2} - 18 T^{3} + 404 T^{4} - 342 T^{5} - 9025 T^{6} + 13718 T^{7} + 130321 T^{8}$$
$23$ $$( 1 - 23 T^{2} + 529 T^{4} )^{2}$$
$29$ $$1 + 8 T + 30 T^{2} - 192 T^{3} - 1541 T^{4} - 5568 T^{5} + 25230 T^{6} + 195112 T^{7} + 707281 T^{8}$$
$31$ $$1 - 12 T + 56 T^{2} - 312 T^{3} + 2319 T^{4} - 9672 T^{5} + 53816 T^{6} - 357492 T^{7} + 923521 T^{8}$$
$37$ $$( 1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 + 10 T + 67 T^{2} + 410 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 6 T + 55 T^{2} + 258 T^{3} + 1849 T^{4}$$
$47$ $$( 1 - 4 T + 58 T^{2} - 188 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$1 - 8 T - 18 T^{2} + 192 T^{3} + 523 T^{4} + 10176 T^{5} - 50562 T^{6} - 1191016 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 - 14 T + 157 T^{2} - 826 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$1 - 4 T - 100 T^{2} + 24 T^{3} + 8759 T^{4} + 1464 T^{5} - 372100 T^{6} - 907924 T^{7} + 13845841 T^{8}$$
$67$ $$1 + 2 T + 29 T^{2} - 318 T^{3} - 4132 T^{4} - 21306 T^{5} + 130181 T^{6} + 601526 T^{7} + 20151121 T^{8}$$
$71$ $$1 + 16 T + 60 T^{2} + 864 T^{3} + 15199 T^{4} + 61344 T^{5} + 302460 T^{6} + 5726576 T^{7} + 25411681 T^{8}$$
$73$ $$1 + 6 T - 109 T^{2} - 6 T^{3} + 13068 T^{4} - 438 T^{5} - 580861 T^{6} + 2334102 T^{7} + 28398241 T^{8}$$
$79$ $$1 - 148 T^{2} + 15663 T^{4} - 923668 T^{6} + 38950081 T^{8}$$
$83$ $$( 1 + 6 T - 47 T^{2} + 498 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 14 T + 233 T^{2} - 1358 T^{3} + 9409 T^{4} )^{2}$$