# Properties

 Label 230.4.a.d 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

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - q^{3} + 4 q^{4} + 5 q^{5} - 2 q^{6} - 32 q^{7} + 8 q^{8} - 26 q^{9} + O(q^{10})$$ $$q + 2 q^{2} - q^{3} + 4 q^{4} + 5 q^{5} - 2 q^{6} - 32 q^{7} + 8 q^{8} - 26 q^{9} + 10 q^{10} - 30 q^{11} - 4 q^{12} + 19 q^{13} - 64 q^{14} - 5 q^{15} + 16 q^{16} - 60 q^{17} - 52 q^{18} - 58 q^{19} + 20 q^{20} + 32 q^{21} - 60 q^{22} + 23 q^{23} - 8 q^{24} + 25 q^{25} + 38 q^{26} + 53 q^{27} - 128 q^{28} + 85 q^{29} - 10 q^{30} - 65 q^{31} + 32 q^{32} + 30 q^{33} - 120 q^{34} - 160 q^{35} - 104 q^{36} - 34 q^{37} - 116 q^{38} - 19 q^{39} + 40 q^{40} + 143 q^{41} + 64 q^{42} - 332 q^{43} - 120 q^{44} - 130 q^{45} + 46 q^{46} - 561 q^{47} - 16 q^{48} + 681 q^{49} + 50 q^{50} + 60 q^{51} + 76 q^{52} - 422 q^{53} + 106 q^{54} - 150 q^{55} - 256 q^{56} + 58 q^{57} + 170 q^{58} + 392 q^{59} - 20 q^{60} - 246 q^{61} - 130 q^{62} + 832 q^{63} + 64 q^{64} + 95 q^{65} + 60 q^{66} + 894 q^{67} - 240 q^{68} - 23 q^{69} - 320 q^{70} - 737 q^{71} - 208 q^{72} + 1041 q^{73} - 68 q^{74} - 25 q^{75} - 232 q^{76} + 960 q^{77} - 38 q^{78} + 1114 q^{79} + 80 q^{80} + 649 q^{81} + 286 q^{82} - 936 q^{83} + 128 q^{84} - 300 q^{85} - 664 q^{86} - 85 q^{87} - 240 q^{88} + 824 q^{89} - 260 q^{90} - 608 q^{91} + 92 q^{92} + 65 q^{93} - 1122 q^{94} - 290 q^{95} - 32 q^{96} - 868 q^{97} + 1362 q^{98} + 780 q^{99} + O(q^{100})$$

## 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 −1.00000 4.00000 5.00000 −2.00000 −32.0000 8.00000 −26.0000 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.d 1
3.b odd 2 1 2070.4.a.a 1
4.b odd 2 1 1840.4.a.e 1
5.b even 2 1 1150.4.a.c 1
5.c odd 4 2 1150.4.b.d 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-2 + T$$
$3$ $$1 + T$$
$5$ $$-5 + T$$
$7$ $$32 + T$$
$11$ $$30 + T$$
$13$ $$-19 + T$$
$17$ $$60 + T$$
$19$ $$58 + T$$
$23$ $$-23 + T$$
$29$ $$-85 + T$$
$31$ $$65 + T$$
$37$ $$34 + T$$
$41$ $$-143 + T$$
$43$ $$332 + T$$
$47$ $$561 + T$$
$53$ $$422 + T$$
$59$ $$-392 + T$$
$61$ $$246 + T$$
$67$ $$-894 + T$$
$71$ $$737 + T$$
$73$ $$-1041 + T$$
$79$ $$-1114 + T$$
$83$ $$936 + T$$
$89$ $$-824 + T$$
$97$ $$868 + T$$