# Properties

 Label 430.2.a.g Level 430 Weight 2 Character orbit 430.a Self dual yes Analytic conductor 3.434 Analytic rank 0 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
1.00000 −1.41421 1.00000 1.00000 −1.41421 1.00000 1.00000 −1.00000 1.00000
1.2 1.00000 1.41421 1.00000 1.00000 1.41421 1.00000 1.00000 −1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.a.g 2
3.b odd 2 1 3870.2.a.bc 2
4.b odd 2 1 3440.2.a.j 2
5.b even 2 1 2150.2.a.v 2
5.c odd 4 2 2150.2.b.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.a.g 2 1.a even 1 1 trivial
2150.2.a.v 2 5.b even 2 1
2150.2.b.o 4 5.c odd 4 2
3440.2.a.j 2 4.b odd 2 1
3870.2.a.bc 2 3.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$43$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(430))$$:

 $$T_{3}^{2} - 2$$ $$T_{7} - 1$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 + 4 T^{2} + 9 T^{4}$$
$5$ $$( 1 - T )^{2}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$1 - 4 T + 24 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$1 - 2 T + 19 T^{2} - 26 T^{3} + 169 T^{4}$$
$17$ $$1 + 32 T^{2} + 289 T^{4}$$
$19$ $$( 1 + T + 19 T^{2} )^{2}$$
$23$ $$1 - 4 T - 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 6 T + 49 T^{2} + 174 T^{3} + 841 T^{4}$$
$31$ $$1 - 2 T + 61 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$1 + 4 T + 60 T^{2} + 148 T^{3} + 1369 T^{4}$$
$41$ $$1 - 2 T + 75 T^{2} - 82 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - T )^{2}$$
$47$ $$1 + 44 T^{2} + 2209 T^{4}$$
$53$ $$1 + 74 T^{2} + 2809 T^{4}$$
$59$ $$1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4}$$
$61$ $$1 + 14 T + 121 T^{2} + 854 T^{3} + 3721 T^{4}$$
$67$ $$1 - 6 T + 125 T^{2} - 402 T^{3} + 4489 T^{4}$$
$71$ $$1 - 4 T + 48 T^{2} - 284 T^{3} + 5041 T^{4}$$
$73$ $$1 + 10 T + 153 T^{2} + 730 T^{3} + 5329 T^{4}$$
$79$ $$1 + 14 T + 205 T^{2} + 1106 T^{3} + 6241 T^{4}$$
$83$ $$1 + 4 T + 138 T^{2} + 332 T^{3} + 6889 T^{4}$$
$89$ $$1 + 4 T + 132 T^{2} + 356 T^{3} + 7921 T^{4}$$
$97$ $$1 + 4 T - 44 T^{2} + 388 T^{3} + 9409 T^{4}$$