# Properties

 Label 230.4.a.j Level $230$ Weight $4$ Character orbit 230.a Self dual yes Analytic conductor $13.570$ Analytic rank $0$ Dimension $4$ 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ Defining polynomial: $$x^{4} - 2 x^{3} - 60 x^{2} - 45 x + 108$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + ( 4 - \beta_{1} ) q^{3} + 4 q^{4} + 5 q^{5} + ( 8 - 2 \beta_{1} ) q^{6} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{7} + 8 q^{8} + ( 18 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 2 q^{2} + ( 4 - \beta_{1} ) q^{3} + 4 q^{4} + 5 q^{5} + ( 8 - 2 \beta_{1} ) q^{6} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{7} + 8 q^{8} + ( 18 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{9} + 10 q^{10} + ( 6 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{11} + ( 16 - 4 \beta_{1} ) q^{12} + ( 15 + 4 \beta_{1} + \beta_{3} ) q^{13} + ( 2 + 2 \beta_{1} + 4 \beta_{2} ) q^{14} + ( 20 - 5 \beta_{1} ) q^{15} + 16 q^{16} + ( 18 - 2 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} ) q^{17} + ( 36 - 8 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{18} + ( 39 + 8 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{19} + 20 q^{20} + ( -37 + 7 \beta_{1} + 16 \beta_{2} - \beta_{3} ) q^{21} + ( 12 + 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{22} -23 q^{23} + ( 32 - 8 \beta_{1} ) q^{24} + 25 q^{25} + ( 30 + 8 \beta_{1} + 2 \beta_{3} ) q^{26} + ( 105 - 11 \beta_{1} - 23 \beta_{2} + 7 \beta_{3} ) q^{27} + ( 4 + 4 \beta_{1} + 8 \beta_{2} ) q^{28} + ( -34 + 9 \beta_{1} + 10 \beta_{2} - 5 \beta_{3} ) q^{29} + ( 40 - 10 \beta_{1} ) q^{30} + ( -17 + 27 \beta_{1} + 15 \beta_{2} + 8 \beta_{3} ) q^{31} + 32 q^{32} + ( -25 + 14 \beta_{1} - 3 \beta_{2} - 7 \beta_{3} ) q^{33} + ( 36 - 4 \beta_{1} - 12 \beta_{2} - 6 \beta_{3} ) q^{34} + ( 5 + 5 \beta_{1} + 10 \beta_{2} ) q^{35} + ( 72 - 16 \beta_{1} - 8 \beta_{2} + 4 \beta_{3} ) q^{36} + ( -96 + 3 \beta_{1} + 33 \beta_{2} + \beta_{3} ) q^{37} + ( 78 + 16 \beta_{1} - 10 \beta_{2} + 6 \beta_{3} ) q^{38} + ( -43 - 27 \beta_{1} + 7 \beta_{2} - \beta_{3} ) q^{39} + 40 q^{40} + ( 28 + 32 \beta_{1} - 15 \beta_{2} - 11 \beta_{3} ) q^{41} + ( -74 + 14 \beta_{1} + 32 \beta_{2} - 2 \beta_{3} ) q^{42} + ( -32 + 52 \beta_{1} - 18 \beta_{2} - 4 \beta_{3} ) q^{43} + ( 24 + 4 \beta_{1} - 4 \beta_{2} - 8 \beta_{3} ) q^{44} + ( 90 - 20 \beta_{1} - 10 \beta_{2} + 5 \beta_{3} ) q^{45} -46 q^{46} + ( 106 - 16 \beta_{1} - 33 \beta_{2} + 4 \beta_{3} ) q^{47} + ( 64 - 16 \beta_{1} ) q^{48} + ( -17 - 42 \beta_{1} + 10 \beta_{2} + 9 \beta_{3} ) q^{49} + 50 q^{50} + ( 127 - 6 \beta_{1} - 43 \beta_{2} - 7 \beta_{3} ) q^{51} + ( 60 + 16 \beta_{1} + 4 \beta_{3} ) q^{52} + ( -114 + 37 \beta_{1} + 7 \beta_{2} + 11 \beta_{3} ) q^{53} + ( 210 - 22 \beta_{1} - 46 \beta_{2} + 14 \beta_{3} ) q^{54} + ( 30 + 5 \beta_{1} - 5 \beta_{2} - 10 \beta_{3} ) q^{55} + ( 8 + 8 \beta_{1} + 16 \beta_{2} ) q^{56} + ( -7 - 95 \beta_{1} - 22 \beta_{2} + \beta_{3} ) q^{57} + ( -68 + 18 \beta_{1} + 20 \beta_{2} - 10 \beta_{3} ) q^{58} + ( -76 - 79 \beta_{1} + 7 \beta_{2} + \beta_{3} ) q^{59} + ( 80 - 20 \beta_{1} ) q^{60} + ( -66 - 47 \beta_{1} + 31 \beta_{2} ) q^{61} + ( -34 + 54 \beta_{1} + 30 \beta_{2} + 16 \beta_{3} ) q^{62} + ( -487 + 86 \beta_{1} + 73 \beta_{2} - 10 \beta_{3} ) q^{63} + 64 q^{64} + ( 75 + 20 \beta_{1} + 5 \beta_{3} ) q^{65} + ( -50 + 28 \beta_{1} - 6 \beta_{2} - 14 \beta_{3} ) q^{66} + ( -140 + 43 \beta_{1} + 33 \beta_{2} - 7 \beta_{3} ) q^{67} + ( 72 - 8 \beta_{1} - 24 \beta_{2} - 12 \beta_{3} ) q^{68} + ( -92 + 23 \beta_{1} ) q^{69} + ( 10 + 10 \beta_{1} + 20 \beta_{2} ) q^{70} + ( 68 - 72 \beta_{1} + 21 \beta_{2} + 23 \beta_{3} ) q^{71} + ( 144 - 32 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} ) q^{72} + ( -140 - 44 \beta_{1} - 17 \beta_{2} + 16 \beta_{3} ) q^{73} + ( -192 + 6 \beta_{1} + 66 \beta_{2} + 2 \beta_{3} ) q^{74} + ( 100 - 25 \beta_{1} ) q^{75} + ( 156 + 32 \beta_{1} - 20 \beta_{2} + 12 \beta_{3} ) q^{76} + ( 19 - 9 \beta_{1} - 46 \beta_{2} - 7 \beta_{3} ) q^{77} + ( -86 - 54 \beta_{1} + 14 \beta_{2} - 2 \beta_{3} ) q^{78} + ( -220 - 92 \beta_{1} - 20 \beta_{2} - 28 \beta_{3} ) q^{79} + 80 q^{80} + ( 482 - 173 \beta_{1} - 136 \beta_{2} + 5 \beta_{3} ) q^{81} + ( 56 + 64 \beta_{1} - 30 \beta_{2} - 22 \beta_{3} ) q^{82} + ( -290 + 13 \beta_{1} - 47 \beta_{2} - 9 \beta_{3} ) q^{83} + ( -148 + 28 \beta_{1} + 64 \beta_{2} - 4 \beta_{3} ) q^{84} + ( 90 - 10 \beta_{1} - 30 \beta_{2} - 15 \beta_{3} ) q^{85} + ( -64 + 104 \beta_{1} - 36 \beta_{2} - 8 \beta_{3} ) q^{86} + ( -522 + 134 \beta_{1} + 93 \beta_{2} - 24 \beta_{3} ) q^{87} + ( 48 + 8 \beta_{1} - 8 \beta_{2} - 16 \beta_{3} ) q^{88} + ( -512 - 92 \beta_{1} + 30 \beta_{2} + 16 \beta_{3} ) q^{89} + ( 180 - 40 \beta_{1} - 20 \beta_{2} + 10 \beta_{3} ) q^{90} + ( 122 + 19 \beta_{1} + 19 \beta_{2} + 6 \beta_{3} ) q^{91} -92 q^{92} + ( -837 - 19 \beta_{1} + 151 \beta_{2} - 3 \beta_{3} ) q^{93} + ( 212 - 32 \beta_{1} - 66 \beta_{2} + 8 \beta_{3} ) q^{94} + ( 195 + 40 \beta_{1} - 25 \beta_{2} + 15 \beta_{3} ) q^{95} + ( 128 - 32 \beta_{1} ) q^{96} + ( -198 - 13 \beta_{1} + 55 \beta_{2} + 30 \beta_{3} ) q^{97} + ( -34 - 84 \beta_{1} + 20 \beta_{2} + 18 \beta_{3} ) q^{98} + ( -741 + 70 \beta_{1} + 41 \beta_{2} + 19 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{2} + 14 q^{3} + 16 q^{4} + 20 q^{5} + 28 q^{6} + 8 q^{7} + 32 q^{8} + 64 q^{9} + O(q^{10})$$ $$4 q + 8 q^{2} + 14 q^{3} + 16 q^{4} + 20 q^{5} + 28 q^{6} + 8 q^{7} + 32 q^{8} + 64 q^{9} + 40 q^{10} + 21 q^{11} + 56 q^{12} + 70 q^{13} + 16 q^{14} + 70 q^{15} + 64 q^{16} + 56 q^{17} + 128 q^{18} + 173 q^{19} + 80 q^{20} - 120 q^{21} + 42 q^{22} - 92 q^{23} + 112 q^{24} + 100 q^{25} + 140 q^{26} + 389 q^{27} + 32 q^{28} - 118 q^{29} + 140 q^{30} + 17 q^{31} + 128 q^{32} - 89 q^{33} + 112 q^{34} + 40 q^{35} + 256 q^{36} - 343 q^{37} + 346 q^{38} - 221 q^{39} + 160 q^{40} + 139 q^{41} - 240 q^{42} - 50 q^{43} + 84 q^{44} + 320 q^{45} - 184 q^{46} + 367 q^{47} + 224 q^{48} - 124 q^{49} + 200 q^{50} + 439 q^{51} + 280 q^{52} - 353 q^{53} + 778 q^{54} + 105 q^{55} + 64 q^{56} - 238 q^{57} - 236 q^{58} - 453 q^{59} + 280 q^{60} - 327 q^{61} + 34 q^{62} - 1723 q^{63} + 256 q^{64} + 350 q^{65} - 178 q^{66} - 455 q^{67} + 224 q^{68} - 322 q^{69} + 80 q^{70} + 195 q^{71} + 512 q^{72} - 633 q^{73} - 686 q^{74} + 350 q^{75} + 692 q^{76} - 2 q^{77} - 442 q^{78} - 1140 q^{79} + 320 q^{80} + 1456 q^{81} + 278 q^{82} - 1199 q^{83} - 480 q^{84} + 280 q^{85} - 100 q^{86} - 1775 q^{87} + 168 q^{88} - 2170 q^{89} + 640 q^{90} + 557 q^{91} - 368 q^{92} - 3241 q^{93} + 734 q^{94} + 865 q^{95} + 448 q^{96} - 703 q^{97} - 248 q^{98} - 2745 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 60 x^{2} - 45 x + 108$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu^{2} - 45 \nu + 54$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$2 \nu^{3} - \nu^{2} - 126 \nu - 153$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 2 \beta_{2} + 4 \beta_{1} + 29$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3} - \beta_{2} + 65 \beta_{1} + 91$$

## 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
 9.04090 1.01192 −1.92711 −6.12571
2.00000 −5.04090 4.00000 5.00000 −10.0818 5.03071 8.00000 −1.58932 10.0000
1.2 2.00000 2.98808 4.00000 5.00000 5.97615 2.98517 8.00000 −18.0714 10.0000
1.3 2.00000 5.92711 4.00000 5.00000 11.8542 24.6272 8.00000 8.13066 10.0000
1.4 2.00000 10.1257 4.00000 5.00000 20.2514 −24.6431 8.00000 75.5301 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.j 4
3.b odd 2 1 2070.4.a.bg 4
4.b odd 2 1 1840.4.a.k 4
5.b even 2 1 1150.4.a.n 4
5.c odd 4 2 1150.4.b.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.a.j 4 1.a even 1 1 trivial
1150.4.a.n 4 5.b even 2 1
1150.4.b.o 8 5.c odd 4 2
1840.4.a.k 4 4.b odd 2 1
2070.4.a.bg 4 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 14 T_{3}^{3} + 12 T_{3}^{2} + 365 T_{3} - 904$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T )^{4}$$
$3$ $$-904 + 365 T + 12 T^{2} - 14 T^{3} + T^{4}$$
$5$ $$( -5 + T )^{4}$$
$7$ $$-9114 + 4865 T - 592 T^{2} - 8 T^{3} + T^{4}$$
$11$ $$-164232 - 43134 T - 2605 T^{2} - 21 T^{3} + T^{4}$$
$13$ $$-46062 + 11409 T + 286 T^{2} - 70 T^{3} + T^{4}$$
$17$ $$7970400 + 264771 T - 8532 T^{2} - 56 T^{3} + T^{4}$$
$19$ $$9983784 + 626822 T - 2059 T^{2} - 173 T^{3} + T^{4}$$
$23$ $$( 23 + T )^{4}$$
$29$ $$44611452 - 118638 T - 35805 T^{2} + 118 T^{3} + T^{4}$$
$31$ $$2678191911 - 120799 T - 107992 T^{2} - 17 T^{3} + T^{4}$$
$37$ $$-2095186944 - 32938816 T - 99964 T^{2} + 343 T^{3} + T^{4}$$
$41$ $$4789051317 + 10257183 T - 157626 T^{2} - 139 T^{3} + T^{4}$$
$43$ $$5363873792 - 14655968 T - 207336 T^{2} + 50 T^{3} + T^{4}$$
$47$ $$668142000 + 7331796 T - 138234 T^{2} - 367 T^{3} + T^{4}$$
$53$ $$-289523592 - 36583380 T - 112122 T^{2} + 353 T^{3} + T^{4}$$
$59$ $$8963853984 - 16917012 T - 308646 T^{2} + 453 T^{3} + T^{4}$$
$61$ $$-1898667392 - 41660040 T - 204755 T^{2} + 327 T^{3} + T^{4}$$
$67$ $$-1366509568 + 33923056 T - 255912 T^{2} + 455 T^{3} + T^{4}$$
$71$ $$106065123651 + 103060557 T - 706330 T^{2} - 195 T^{3} + T^{4}$$
$73$ $$-15845099784 - 152518736 T - 226884 T^{2} + 633 T^{3} + T^{4}$$
$79$ $$60635801088 - 326815808 T - 526896 T^{2} + 1140 T^{3} + T^{4}$$
$83$ $$103460976 + 9039492 T + 218008 T^{2} + 1199 T^{3} + T^{4}$$
$89$ $$-5919819264 - 213089088 T + 986248 T^{2} + 2170 T^{3} + T^{4}$$
$97$ $$-22808262684 - 296475040 T - 708127 T^{2} + 703 T^{3} + T^{4}$$