# Properties

 Label 230.4.a.b Level $230$ Weight $4$ Character orbit 230.a Self dual yes Analytic conductor $13.570$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 230.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.5704393013$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} + 3q^{7} - 8q^{8} - 11q^{9} + O(q^{10})$$ $$q - 2q^{2} + 4q^{3} + 4q^{4} - 5q^{5} - 8q^{6} + 3q^{7} - 8q^{8} - 11q^{9} + 10q^{10} - 2q^{11} + 16q^{12} - 38q^{13} - 6q^{14} - 20q^{15} + 16q^{16} - 45q^{17} + 22q^{18} - 74q^{19} - 20q^{20} + 12q^{21} + 4q^{22} + 23q^{23} - 32q^{24} + 25q^{25} + 76q^{26} - 152q^{27} + 12q^{28} + 283q^{29} + 40q^{30} - 303q^{31} - 32q^{32} - 8q^{33} + 90q^{34} - 15q^{35} - 44q^{36} + 79q^{37} + 148q^{38} - 152q^{39} + 40q^{40} - 407q^{41} - 24q^{42} - 328q^{43} - 8q^{44} + 55q^{45} - 46q^{46} + 360q^{47} + 64q^{48} - 334q^{49} - 50q^{50} - 180q^{51} - 152q^{52} - 561q^{53} + 304q^{54} + 10q^{55} - 24q^{56} - 296q^{57} - 566q^{58} + 101q^{59} - 80q^{60} - 268q^{61} + 606q^{62} - 33q^{63} + 64q^{64} + 190q^{65} + 16q^{66} - 69q^{67} - 180q^{68} + 92q^{69} + 30q^{70} - 641q^{71} + 88q^{72} + 994q^{73} - 158q^{74} + 100q^{75} - 296q^{76} - 6q^{77} + 304q^{78} - 884q^{79} - 80q^{80} - 311q^{81} + 814q^{82} + 503q^{83} + 48q^{84} + 225q^{85} + 656q^{86} + 1132q^{87} + 16q^{88} + 1608q^{89} - 110q^{90} - 114q^{91} + 92q^{92} - 1212q^{93} - 720q^{94} + 370q^{95} - 128q^{96} + 1082q^{97} + 668q^{98} + 22q^{99} + O(q^{100})$$

## 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
 0
−2.00000 4.00000 4.00000 −5.00000 −8.00000 3.00000 −8.00000 −11.0000 10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$23$$ $$-1$$

## 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 230.4.a.b 1
3.b odd 2 1 2070.4.a.n 1
4.b odd 2 1 1840.4.a.b 1
5.b even 2 1 1150.4.a.f 1
5.c odd 4 2 1150.4.b.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.b 1 1.a even 1 1 trivial
1150.4.a.f 1 5.b even 2 1
1150.4.b.c 2 5.c odd 4 2
1840.4.a.b 1 4.b odd 2 1
2070.4.a.n 1 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 4$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T$$
$3$ $$-4 + T$$
$5$ $$5 + T$$
$7$ $$-3 + T$$
$11$ $$2 + T$$
$13$ $$38 + T$$
$17$ $$45 + T$$
$19$ $$74 + T$$
$23$ $$-23 + T$$
$29$ $$-283 + T$$
$31$ $$303 + T$$
$37$ $$-79 + T$$
$41$ $$407 + T$$
$43$ $$328 + T$$
$47$ $$-360 + T$$
$53$ $$561 + T$$
$59$ $$-101 + T$$
$61$ $$268 + T$$
$67$ $$69 + T$$
$71$ $$641 + T$$
$73$ $$-994 + T$$
$79$ $$884 + T$$
$83$ $$-503 + T$$
$89$ $$-1608 + T$$
$97$ $$-1082 + T$$