# Properties

 Label 430.2.b.a Level 430 Weight 2 Character orbit 430.b Analytic conductor 3.434 Analytic rank 0 Dimension 6 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.350464.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 2 x^{3} + 4 x^{2} - 4 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{5} + 8 \nu^{4} - 4 \nu^{3} - \nu^{2} + 2 \nu + 38$$$$)/23$$ $$\beta_{2}$$ $$=$$ $$($$$$-5 \nu^{5} + 17 \nu^{4} - 20 \nu^{3} - 5 \nu^{2} + 10 \nu + 29$$$$)/23$$ $$\beta_{3}$$ $$=$$ $$($$$$7 \nu^{5} - 10 \nu^{4} + 5 \nu^{3} + 30 \nu^{2} + 32 \nu - 13$$$$)/23$$ $$\beta_{4}$$ $$=$$ $$($$$$-11 \nu^{5} + 19 \nu^{4} - 21 \nu^{3} - 11 \nu^{2} - 70 \nu + 27$$$$)/23$$ $$\beta_{5}$$ $$=$$ $$($$$$-14 \nu^{5} + 20 \nu^{4} - 10 \nu^{3} - 37 \nu^{2} - 64 \nu + 26$$$$)/23$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{5} - \beta_{4} + 2 \beta_{3} - \beta_{2} + 2 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$-\beta_{2} + 5 \beta_{1} - 7$$ $$\nu^{5}$$ $$=$$ $$-8 \beta_{5} + 3 \beta_{4} - 9 \beta_{3} - 3 \beta_{2} + 8 \beta_{1} - 9$$

## Character values

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

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
259.1
 0.403032 + 0.403032i 1.45161 + 1.45161i −0.854638 − 0.854638i −0.854638 + 0.854638i 1.45161 − 1.45161i 0.403032 − 0.403032i
1.00000i 2.48119i −1.00000 −1.48119 1.67513i −2.48119 0.193937i 1.00000i −3.15633 −1.67513 + 1.48119i
259.2 1.00000i 0.688892i −1.00000 0.311108 + 2.21432i −0.688892 1.90321i 1.00000i 2.52543 2.21432 0.311108i
259.3 1.00000i 1.17009i −1.00000 2.17009 0.539189i 1.17009 2.70928i 1.00000i 1.63090 −0.539189 2.17009i
259.4 1.00000i 1.17009i −1.00000 2.17009 + 0.539189i 1.17009 2.70928i 1.00000i 1.63090 −0.539189 + 2.17009i
259.5 1.00000i 0.688892i −1.00000 0.311108 2.21432i −0.688892 1.90321i 1.00000i 2.52543 2.21432 + 0.311108i
259.6 1.00000i 2.48119i −1.00000 −1.48119 + 1.67513i −2.48119 0.193937i 1.00000i −3.15633 −1.67513 1.48119i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 259.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.b.a 6
5.b even 2 1 inner 430.2.b.a 6
5.c odd 4 1 2150.2.a.bc 3
5.c odd 4 1 2150.2.a.bd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.b.a 6 1.a even 1 1 trivial
430.2.b.a 6 5.b even 2 1 inner
2150.2.a.bc 3 5.c odd 4 1
2150.2.a.bd 3 5.c odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{6} + 8 T_{3}^{4} + 12 T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T^{2} )^{3}$$
$3$ $$1 - 10 T^{2} + 51 T^{4} - 176 T^{6} + 459 T^{8} - 810 T^{10} + 729 T^{12}$$
$5$ $$1 - 2 T + 3 T^{2} - 12 T^{3} + 15 T^{4} - 50 T^{5} + 125 T^{6}$$
$7$ $$1 - 31 T^{2} + 454 T^{4} - 4003 T^{6} + 22246 T^{8} - 74431 T^{10} + 117649 T^{12}$$
$11$ $$( 1 + 6 T + 41 T^{2} + 130 T^{3} + 451 T^{4} + 726 T^{5} + 1331 T^{6} )^{2}$$
$13$ $$1 - 43 T^{2} + 878 T^{4} - 12687 T^{6} + 148382 T^{8} - 1228123 T^{10} + 4826809 T^{12}$$
$17$ $$1 - 42 T^{2} + 1155 T^{4} - 24816 T^{6} + 333795 T^{8} - 3507882 T^{10} + 24137569 T^{12}$$
$19$ $$( 1 - 17 T + 148 T^{2} - 797 T^{3} + 2812 T^{4} - 6137 T^{5} + 6859 T^{6} )^{2}$$
$23$ $$1 - 118 T^{2} + 6135 T^{4} - 181696 T^{6} + 3245415 T^{8} - 33021238 T^{10} + 148035889 T^{12}$$
$29$ $$( 1 - 5 T + 82 T^{2} - 291 T^{3} + 2378 T^{4} - 4205 T^{5} + 24389 T^{6} )^{2}$$
$31$ $$( 1 + 9 T + 84 T^{2} + 423 T^{3} + 2604 T^{4} + 8649 T^{5} + 29791 T^{6} )^{2}$$
$37$ $$1 - 82 T^{2} + 4015 T^{4} - 159016 T^{6} + 5496535 T^{8} - 153681202 T^{10} + 2565726409 T^{12}$$
$41$ $$( 1 + 5 T + 46 T^{2} + 225 T^{3} + 1886 T^{4} + 8405 T^{5} + 68921 T^{6} )^{2}$$
$43$ $$( 1 + T^{2} )^{3}$$
$47$ $$1 - 170 T^{2} + 15279 T^{4} - 888968 T^{6} + 33751311 T^{8} - 829545770 T^{10} + 10779215329 T^{12}$$
$53$ $$1 - 138 T^{2} + 14295 T^{4} - 854556 T^{6} + 40154655 T^{8} - 1088886378 T^{10} + 22164361129 T^{12}$$
$59$ $$( 1 + 20 T + 249 T^{2} + 2088 T^{3} + 14691 T^{4} + 69620 T^{5} + 205379 T^{6} )^{2}$$
$61$ $$( 1 + 21 T + 272 T^{2} + 2329 T^{3} + 16592 T^{4} + 78141 T^{5} + 226981 T^{6} )^{2}$$
$67$ $$1 - 87 T^{2} + 978 T^{4} + 259189 T^{6} + 4390242 T^{8} - 1753147527 T^{10} + 90458382169 T^{12}$$
$71$ $$( 1 + 8 T + 159 T^{2} + 930 T^{3} + 11289 T^{4} + 40328 T^{5} + 357911 T^{6} )^{2}$$
$73$ $$1 - 51 T^{2} + 5858 T^{4} - 385823 T^{6} + 31217282 T^{8} - 1448310291 T^{10} + 151334226289 T^{12}$$
$79$ $$( 1 - 9 T + 234 T^{2} - 1361 T^{3} + 18486 T^{4} - 56169 T^{5} + 493039 T^{6} )^{2}$$
$83$ $$1 + 34 T^{2} + 7719 T^{4} + 448204 T^{6} + 53176191 T^{8} + 1613582914 T^{10} + 326940373369 T^{12}$$
$89$ $$( 1 - 14 T - T^{2} + 1286 T^{3} - 89 T^{4} - 110894 T^{5} + 704969 T^{6} )^{2}$$
$97$ $$1 - 510 T^{2} + 114171 T^{4} - 14374424 T^{6} + 1074234939 T^{8} - 45149933310 T^{10} + 832972004929 T^{12}$$