# Properties

 Label 230.4.a.f Level $230$ Weight $4$ Character orbit 230.a Self dual yes Analytic conductor $13.570$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{73})$$ Defining polynomial: $$x^{2} - x - 18$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{73})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -2 q^{2} + ( -1 - \beta ) q^{3} + 4 q^{4} + 5 q^{5} + ( 2 + 2 \beta ) q^{6} + ( -8 - \beta ) q^{7} -8 q^{8} + ( -8 + 3 \beta ) q^{9} +O(q^{10})$$ $$q -2 q^{2} + ( -1 - \beta ) q^{3} + 4 q^{4} + 5 q^{5} + ( 2 + 2 \beta ) q^{6} + ( -8 - \beta ) q^{7} -8 q^{8} + ( -8 + 3 \beta ) q^{9} -10 q^{10} + ( 4 + 10 \beta ) q^{11} + ( -4 - 4 \beta ) q^{12} + ( -9 + 9 \beta ) q^{13} + ( 16 + 2 \beta ) q^{14} + ( -5 - 5 \beta ) q^{15} + 16 q^{16} + ( -34 - 11 \beta ) q^{17} + ( 16 - 6 \beta ) q^{18} + ( 12 + 10 \beta ) q^{19} + 20 q^{20} + ( 26 + 10 \beta ) q^{21} + ( -8 - 20 \beta ) q^{22} -23 q^{23} + ( 8 + 8 \beta ) q^{24} + 25 q^{25} + ( 18 - 18 \beta ) q^{26} + ( -19 + 29 \beta ) q^{27} + ( -32 - 4 \beta ) q^{28} + ( -55 - 2 \beta ) q^{29} + ( 10 + 10 \beta ) q^{30} + ( -33 - 26 \beta ) q^{31} -32 q^{32} + ( -184 - 24 \beta ) q^{33} + ( 68 + 22 \beta ) q^{34} + ( -40 - 5 \beta ) q^{35} + ( -32 + 12 \beta ) q^{36} + ( -242 - 7 \beta ) q^{37} + ( -24 - 20 \beta ) q^{38} + ( -153 - 9 \beta ) q^{39} -40 q^{40} + ( -129 - 74 \beta ) q^{41} + ( -52 - 20 \beta ) q^{42} + ( -166 - 22 \beta ) q^{43} + ( 16 + 40 \beta ) q^{44} + ( -40 + 15 \beta ) q^{45} + 46 q^{46} + ( -297 - 5 \beta ) q^{47} + ( -16 - 16 \beta ) q^{48} + ( -261 + 17 \beta ) q^{49} -50 q^{50} + ( 232 + 56 \beta ) q^{51} + ( -36 + 36 \beta ) q^{52} + ( -156 + 7 \beta ) q^{53} + ( 38 - 58 \beta ) q^{54} + ( 20 + 50 \beta ) q^{55} + ( 64 + 8 \beta ) q^{56} + ( -192 - 32 \beta ) q^{57} + ( 110 + 4 \beta ) q^{58} + ( 126 + 105 \beta ) q^{59} + ( -20 - 20 \beta ) q^{60} + ( -92 + 12 \beta ) q^{61} + ( 66 + 52 \beta ) q^{62} + ( 10 - 19 \beta ) q^{63} + 64 q^{64} + ( -45 + 45 \beta ) q^{65} + ( 368 + 48 \beta ) q^{66} + ( -286 + 41 \beta ) q^{67} + ( -136 - 44 \beta ) q^{68} + ( 23 + 23 \beta ) q^{69} + ( 80 + 10 \beta ) q^{70} + ( 679 - 104 \beta ) q^{71} + ( 64 - 24 \beta ) q^{72} + ( -183 + 23 \beta ) q^{73} + ( 484 + 14 \beta ) q^{74} + ( -25 - 25 \beta ) q^{75} + ( 48 + 40 \beta ) q^{76} + ( -212 - 94 \beta ) q^{77} + ( 306 + 18 \beta ) q^{78} + ( -16 - 56 \beta ) q^{79} + 80 q^{80} + ( -287 - 120 \beta ) q^{81} + ( 258 + 148 \beta ) q^{82} + ( 578 + 117 \beta ) q^{83} + ( 104 + 40 \beta ) q^{84} + ( -170 - 55 \beta ) q^{85} + ( 332 + 44 \beta ) q^{86} + ( 91 + 59 \beta ) q^{87} + ( -32 - 80 \beta ) q^{88} + ( 562 - 18 \beta ) q^{89} + ( 80 - 30 \beta ) q^{90} + ( -90 - 72 \beta ) q^{91} -92 q^{92} + ( 501 + 85 \beta ) q^{93} + ( 594 + 10 \beta ) q^{94} + ( 60 + 50 \beta ) q^{95} + ( 32 + 32 \beta ) q^{96} + ( -1038 - 164 \beta ) q^{97} + ( 522 - 34 \beta ) q^{98} + ( 508 - 38 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} - 17 q^{7} - 16 q^{8} - 13 q^{9} + O(q^{10})$$ $$2 q - 4 q^{2} - 3 q^{3} + 8 q^{4} + 10 q^{5} + 6 q^{6} - 17 q^{7} - 16 q^{8} - 13 q^{9} - 20 q^{10} + 18 q^{11} - 12 q^{12} - 9 q^{13} + 34 q^{14} - 15 q^{15} + 32 q^{16} - 79 q^{17} + 26 q^{18} + 34 q^{19} + 40 q^{20} + 62 q^{21} - 36 q^{22} - 46 q^{23} + 24 q^{24} + 50 q^{25} + 18 q^{26} - 9 q^{27} - 68 q^{28} - 112 q^{29} + 30 q^{30} - 92 q^{31} - 64 q^{32} - 392 q^{33} + 158 q^{34} - 85 q^{35} - 52 q^{36} - 491 q^{37} - 68 q^{38} - 315 q^{39} - 80 q^{40} - 332 q^{41} - 124 q^{42} - 354 q^{43} + 72 q^{44} - 65 q^{45} + 92 q^{46} - 599 q^{47} - 48 q^{48} - 505 q^{49} - 100 q^{50} + 520 q^{51} - 36 q^{52} - 305 q^{53} + 18 q^{54} + 90 q^{55} + 136 q^{56} - 416 q^{57} + 224 q^{58} + 357 q^{59} - 60 q^{60} - 172 q^{61} + 184 q^{62} + q^{63} + 128 q^{64} - 45 q^{65} + 784 q^{66} - 531 q^{67} - 316 q^{68} + 69 q^{69} + 170 q^{70} + 1254 q^{71} + 104 q^{72} - 343 q^{73} + 982 q^{74} - 75 q^{75} + 136 q^{76} - 518 q^{77} + 630 q^{78} - 88 q^{79} + 160 q^{80} - 694 q^{81} + 664 q^{82} + 1273 q^{83} + 248 q^{84} - 395 q^{85} + 708 q^{86} + 241 q^{87} - 144 q^{88} + 1106 q^{89} + 130 q^{90} - 252 q^{91} - 184 q^{92} + 1087 q^{93} + 1198 q^{94} + 170 q^{95} + 96 q^{96} - 2240 q^{97} + 1010 q^{98} + 978 q^{99} + O(q^{100})$$

## 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
 4.77200 −3.77200
−2.00000 −5.77200 4.00000 5.00000 11.5440 −12.7720 −8.00000 6.31601 −10.0000
1.2 −2.00000 2.77200 4.00000 5.00000 −5.54400 −4.22800 −8.00000 −19.3160 −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.f 2
3.b odd 2 1 2070.4.a.s 2
4.b odd 2 1 1840.4.a.i 2
5.b even 2 1 1150.4.a.l 2
5.c odd 4 2 1150.4.b.k 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 3 T_{3} - 16$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T )^{2}$$
$3$ $$-16 + 3 T + T^{2}$$
$5$ $$( -5 + T )^{2}$$
$7$ $$54 + 17 T + T^{2}$$
$11$ $$-1744 - 18 T + T^{2}$$
$13$ $$-1458 + 9 T + T^{2}$$
$17$ $$-648 + 79 T + T^{2}$$
$19$ $$-1536 - 34 T + T^{2}$$
$23$ $$( 23 + T )^{2}$$
$29$ $$3063 + 112 T + T^{2}$$
$31$ $$-10221 + 92 T + T^{2}$$
$37$ $$59376 + 491 T + T^{2}$$
$41$ $$-72381 + 332 T + T^{2}$$
$43$ $$22496 + 354 T + T^{2}$$
$47$ $$89244 + 599 T + T^{2}$$
$53$ $$22362 + 305 T + T^{2}$$
$59$ $$-169344 - 357 T + T^{2}$$
$61$ $$4768 + 172 T + T^{2}$$
$67$ $$39812 + 531 T + T^{2}$$
$71$ $$195737 - 1254 T + T^{2}$$
$73$ $$19758 + 343 T + T^{2}$$
$79$ $$-55296 + 88 T + T^{2}$$
$83$ $$155308 - 1273 T + T^{2}$$
$89$ $$299896 - 1106 T + T^{2}$$
$97$ $$763548 + 2240 T + T^{2}$$