# Properties

 Label 430.2.g.a Level 430 Weight 2 Character orbit 430.g Analytic conductor 3.434 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$430 = 2 \cdot 5 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 430.g (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.43356728692$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$87$$ $$261$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
257.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i −1.41421 1.41421i 1.00000i −0.707107 2.12132i 2.00000 −2.82843 + 2.82843i 0.707107 + 0.707107i 1.00000i 2.00000 + 1.00000i
257.2 0.707107 0.707107i 1.41421 + 1.41421i 1.00000i 0.707107 + 2.12132i 2.00000 2.82843 2.82843i −0.707107 0.707107i 1.00000i 2.00000 + 1.00000i
343.1 −0.707107 0.707107i −1.41421 + 1.41421i 1.00000i −0.707107 + 2.12132i 2.00000 −2.82843 2.82843i 0.707107 0.707107i 1.00000i 2.00000 1.00000i
343.2 0.707107 + 0.707107i 1.41421 1.41421i 1.00000i 0.707107 2.12132i 2.00000 2.82843 + 2.82843i −0.707107 + 0.707107i 1.00000i 2.00000 1.00000i
 $$n$$: 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
5.c odd 4 1 inner
43.b odd 2 1 inner
215.g even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 430.2.g.a 4
5.c odd 4 1 inner 430.2.g.a 4
43.b odd 2 1 inner 430.2.g.a 4
215.g even 4 1 inner 430.2.g.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
430.2.g.a 4 1.a even 1 1 trivial
430.2.g.a 4 5.c odd 4 1 inner
430.2.g.a 4 43.b odd 2 1 inner
430.2.g.a 4 215.g even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(430, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$( 1 - 4 T + 8 T^{2} - 12 T^{3} + 9 T^{4} )( 1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4} )$$
$5$ $$1 + 8 T^{2} + 25 T^{4}$$
$7$ $$1 - 94 T^{4} + 2401 T^{8}$$
$11$ $$( 1 + 4 T + 11 T^{2} )^{4}$$
$13$ $$( 1 - 4 T + 8 T^{2} - 52 T^{3} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 - 2 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 20 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 10 T + 50 T^{2} + 230 T^{3} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 40 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 8 T + 31 T^{2} )^{4}$$
$37$ $$1 - 1294 T^{4} + 1874161 T^{8}$$
$41$ $$( 1 + 12 T + 41 T^{2} )^{4}$$
$43$ $$1 + 12 T + 72 T^{2} + 516 T^{3} + 1849 T^{4}$$
$47$ $$( 1 + 10 T + 50 T^{2} + 470 T^{3} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 2809 T^{4} )^{2}$$
$59$ $$( 1 - 59 T^{2} )^{4}$$
$61$ $$( 1 - 104 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 4 T + 8 T^{2} + 268 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 24 T + 288 T^{2} - 1752 T^{3} + 5329 T^{4} )( 1 + 24 T + 288 T^{2} + 1752 T^{3} + 5329 T^{4} )$$
$79$ $$( 1 - 58 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 12 T + 72 T^{2} - 996 T^{3} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 22 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 18 T + 162 T^{2} - 1746 T^{3} + 9409 T^{4} )^{2}$$