# Properties

 Label 230.2.a.d Level $230$ Weight $2$ Character orbit 230.a Self dual yes Analytic conductor $1.837$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 Defining polynomial: $$x^{3} - x^{2} - 9x + 12$$ x^3 - x^2 - 9*x + 12 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10})$$ q + q^2 + b1 * q^3 + q^4 - q^5 + b1 * q^6 + (-b2 - b1 + 1) * q^7 + q^8 + (b2 - b1 + 4) * q^9 $$q + q^{2} + \beta_1 q^{3} + q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + q^{8} + (\beta_{2} - \beta_1 + 4) q^{9} - q^{10} + (2 \beta_{2} - \beta_1 + 2) q^{11} + \beta_1 q^{12} + ( - \beta_{2} + \beta_1 - 1) q^{13} + ( - \beta_{2} - \beta_1 + 1) q^{14} - \beta_1 q^{15} + q^{16} + ( - \beta_1 - 2) q^{17} + (\beta_{2} - \beta_1 + 4) q^{18} + ( - \beta_{2} - \beta_1 + 1) q^{19} - q^{20} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{21} + (2 \beta_{2} - \beta_1 + 2) q^{22} - q^{23} + \beta_1 q^{24} + q^{25} + ( - \beta_{2} + \beta_1 - 1) q^{26} + (\beta_{2} + 2 \beta_1 - 5) q^{27} + ( - \beta_{2} - \beta_1 + 1) q^{28} + (2 \beta_1 - 2) q^{29} - \beta_1 q^{30} + (\beta_{2} - \beta_1 - 1) q^{31} + q^{32} + (3 \beta_{2} + 3 \beta_1 - 3) q^{33} + ( - \beta_1 - 2) q^{34} + (\beta_{2} + \beta_1 - 1) q^{35} + (\beta_{2} - \beta_1 + 4) q^{36} + 2 \beta_{2} q^{37} + ( - \beta_{2} - \beta_1 + 1) q^{38} + ( - \beta_{2} - 2 \beta_1 + 5) q^{39} - q^{40} + ( - 2 \beta_{2} - \beta_1) q^{41} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{42} + 8 q^{43} + (2 \beta_{2} - \beta_1 + 2) q^{44} + ( - \beta_{2} + \beta_1 - 4) q^{45} - q^{46} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{47} + \beta_1 q^{48} + (2 \beta_{2} - \beta_1 + 11) q^{49} + q^{50} + ( - \beta_{2} - \beta_1 - 7) q^{51} + ( - \beta_{2} + \beta_1 - 1) q^{52} - 6 q^{53} + (\beta_{2} + 2 \beta_1 - 5) q^{54} + ( - 2 \beta_{2} + \beta_1 - 2) q^{55} + ( - \beta_{2} - \beta_1 + 1) q^{56} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{57} + (2 \beta_1 - 2) q^{58} + (2 \beta_1 + 4) q^{59} - \beta_1 q^{60} + (4 \beta_{2} - \beta_1 + 2) q^{61} + (\beta_{2} - \beta_1 - 1) q^{62} + ( - \beta_{2} - 8 \beta_1 + 5) q^{63} + q^{64} + (\beta_{2} - \beta_1 + 1) q^{65} + (3 \beta_{2} + 3 \beta_1 - 3) q^{66} + (4 \beta_{2} + 4) q^{67} + ( - \beta_1 - 2) q^{68} - \beta_1 q^{69} + (\beta_{2} + \beta_1 - 1) q^{70} + ( - \beta_1 + 4) q^{71} + (\beta_{2} - \beta_1 + 4) q^{72} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{73} + 2 \beta_{2} q^{74} + \beta_1 q^{75} + ( - \beta_{2} - \beta_1 + 1) q^{76} + (\beta_{2} - 8 \beta_1 - 5) q^{77} + ( - \beta_{2} - 2 \beta_1 + 5) q^{78} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{79} - q^{80} + (\beta_{2} - 4 \beta_1 + 4) q^{81} + ( - 2 \beta_{2} - \beta_1) q^{82} + ( - 2 \beta_{2} + 2) q^{83} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{84} + (\beta_1 + 2) q^{85} + 8 q^{86} + (2 \beta_{2} - 4 \beta_1 + 14) q^{87} + (2 \beta_{2} - \beta_1 + 2) q^{88} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{89} + ( - \beta_{2} + \beta_1 - 4) q^{90} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{91} - q^{92} + (\beta_{2} - 5) q^{93} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{94} + (\beta_{2} + \beta_1 - 1) q^{95} + \beta_1 q^{96} + ( - 3 \beta_1 - 10) q^{97} + (2 \beta_{2} - \beta_1 + 11) q^{98} + (3 \beta_{2} - 3 \beta_1 + 21) q^{99}+O(q^{100})$$ q + q^2 + b1 * q^3 + q^4 - q^5 + b1 * q^6 + (-b2 - b1 + 1) * q^7 + q^8 + (b2 - b1 + 4) * q^9 - q^10 + (2*b2 - b1 + 2) * q^11 + b1 * q^12 + (-b2 + b1 - 1) * q^13 + (-b2 - b1 + 1) * q^14 - b1 * q^15 + q^16 + (-b1 - 2) * q^17 + (b2 - b1 + 4) * q^18 + (-b2 - b1 + 1) * q^19 - q^20 + (-3*b2 + 2*b1 - 9) * q^21 + (2*b2 - b1 + 2) * q^22 - q^23 + b1 * q^24 + q^25 + (-b2 + b1 - 1) * q^26 + (b2 + 2*b1 - 5) * q^27 + (-b2 - b1 + 1) * q^28 + (2*b1 - 2) * q^29 - b1 * q^30 + (b2 - b1 - 1) * q^31 + q^32 + (3*b2 + 3*b1 - 3) * q^33 + (-b1 - 2) * q^34 + (b2 + b1 - 1) * q^35 + (b2 - b1 + 4) * q^36 + 2*b2 * q^37 + (-b2 - b1 + 1) * q^38 + (-b2 - 2*b1 + 5) * q^39 - q^40 + (-2*b2 - b1) * q^41 + (-3*b2 + 2*b1 - 9) * q^42 + 8 * q^43 + (2*b2 - b1 + 2) * q^44 + (-b2 + b1 - 4) * q^45 - q^46 + (-2*b2 + 2*b1 - 6) * q^47 + b1 * q^48 + (2*b2 - b1 + 11) * q^49 + q^50 + (-b2 - b1 - 7) * q^51 + (-b2 + b1 - 1) * q^52 - 6 * q^53 + (b2 + 2*b1 - 5) * q^54 + (-2*b2 + b1 - 2) * q^55 + (-b2 - b1 + 1) * q^56 + (-3*b2 + 2*b1 - 9) * q^57 + (2*b1 - 2) * q^58 + (2*b1 + 4) * q^59 - b1 * q^60 + (4*b2 - b1 + 2) * q^61 + (b2 - b1 - 1) * q^62 + (-b2 - 8*b1 + 5) * q^63 + q^64 + (b2 - b1 + 1) * q^65 + (3*b2 + 3*b1 - 3) * q^66 + (4*b2 + 4) * q^67 + (-b1 - 2) * q^68 - b1 * q^69 + (b2 + b1 - 1) * q^70 + (-b1 + 4) * q^71 + (b2 - b1 + 4) * q^72 + (-2*b2 + 2*b1 - 4) * q^73 + 2*b2 * q^74 + b1 * q^75 + (-b2 - b1 + 1) * q^76 + (b2 - 8*b1 - 5) * q^77 + (-b2 - 2*b1 + 5) * q^78 + (-4*b2 + 4*b1 - 4) * q^79 - q^80 + (b2 - 4*b1 + 4) * q^81 + (-2*b2 - b1) * q^82 + (-2*b2 + 2) * q^83 + (-3*b2 + 2*b1 - 9) * q^84 + (b1 + 2) * q^85 + 8 * q^86 + (2*b2 - 4*b1 + 14) * q^87 + (2*b2 - b1 + 2) * q^88 + (-2*b2 + 4*b1 + 4) * q^89 + (-b2 + b1 - 4) * q^90 + (-2*b2 + 5*b1 - 2) * q^91 - q^92 + (b2 - 5) * q^93 + (-2*b2 + 2*b1 - 6) * q^94 + (b2 + b1 - 1) * q^95 + b1 * q^96 + (-3*b1 - 10) * q^97 + (2*b2 - b1 + 11) * q^98 + (3*b2 - 3*b1 + 21) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + q^3 + 3 * q^4 - 3 * q^5 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9 $$3 q + 3 q^{2} + q^{3} + 3 q^{4} - 3 q^{5} + q^{6} + 3 q^{7} + 3 q^{8} + 10 q^{9} - 3 q^{10} + 3 q^{11} + q^{12} - q^{13} + 3 q^{14} - q^{15} + 3 q^{16} - 7 q^{17} + 10 q^{18} + 3 q^{19} - 3 q^{20} - 22 q^{21} + 3 q^{22} - 3 q^{23} + q^{24} + 3 q^{25} - q^{26} - 14 q^{27} + 3 q^{28} - 4 q^{29} - q^{30} - 5 q^{31} + 3 q^{32} - 9 q^{33} - 7 q^{34} - 3 q^{35} + 10 q^{36} - 2 q^{37} + 3 q^{38} + 14 q^{39} - 3 q^{40} + q^{41} - 22 q^{42} + 24 q^{43} + 3 q^{44} - 10 q^{45} - 3 q^{46} - 14 q^{47} + q^{48} + 30 q^{49} + 3 q^{50} - 21 q^{51} - q^{52} - 18 q^{53} - 14 q^{54} - 3 q^{55} + 3 q^{56} - 22 q^{57} - 4 q^{58} + 14 q^{59} - q^{60} + q^{61} - 5 q^{62} + 8 q^{63} + 3 q^{64} + q^{65} - 9 q^{66} + 8 q^{67} - 7 q^{68} - q^{69} - 3 q^{70} + 11 q^{71} + 10 q^{72} - 8 q^{73} - 2 q^{74} + q^{75} + 3 q^{76} - 24 q^{77} + 14 q^{78} - 4 q^{79} - 3 q^{80} + 7 q^{81} + q^{82} + 8 q^{83} - 22 q^{84} + 7 q^{85} + 24 q^{86} + 36 q^{87} + 3 q^{88} + 18 q^{89} - 10 q^{90} + q^{91} - 3 q^{92} - 16 q^{93} - 14 q^{94} - 3 q^{95} + q^{96} - 33 q^{97} + 30 q^{98} + 57 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + q^3 + 3 * q^4 - 3 * q^5 + q^6 + 3 * q^7 + 3 * q^8 + 10 * q^9 - 3 * q^10 + 3 * q^11 + q^12 - q^13 + 3 * q^14 - q^15 + 3 * q^16 - 7 * q^17 + 10 * q^18 + 3 * q^19 - 3 * q^20 - 22 * q^21 + 3 * q^22 - 3 * q^23 + q^24 + 3 * q^25 - q^26 - 14 * q^27 + 3 * q^28 - 4 * q^29 - q^30 - 5 * q^31 + 3 * q^32 - 9 * q^33 - 7 * q^34 - 3 * q^35 + 10 * q^36 - 2 * q^37 + 3 * q^38 + 14 * q^39 - 3 * q^40 + q^41 - 22 * q^42 + 24 * q^43 + 3 * q^44 - 10 * q^45 - 3 * q^46 - 14 * q^47 + q^48 + 30 * q^49 + 3 * q^50 - 21 * q^51 - q^52 - 18 * q^53 - 14 * q^54 - 3 * q^55 + 3 * q^56 - 22 * q^57 - 4 * q^58 + 14 * q^59 - q^60 + q^61 - 5 * q^62 + 8 * q^63 + 3 * q^64 + q^65 - 9 * q^66 + 8 * q^67 - 7 * q^68 - q^69 - 3 * q^70 + 11 * q^71 + 10 * q^72 - 8 * q^73 - 2 * q^74 + q^75 + 3 * q^76 - 24 * q^77 + 14 * q^78 - 4 * q^79 - 3 * q^80 + 7 * q^81 + q^82 + 8 * q^83 - 22 * q^84 + 7 * q^85 + 24 * q^86 + 36 * q^87 + 3 * q^88 + 18 * q^89 - 10 * q^90 + q^91 - 3 * q^92 - 16 * q^93 - 14 * q^94 - 3 * q^95 + q^96 - 33 * q^97 + 30 * q^98 + 57 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$ v^2 + v - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 7$$ b2 - b1 + 7

## 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
 −3.11903 1.43163 2.68740
1.00000 −3.11903 1.00000 −1.00000 −3.11903 4.50973 1.00000 6.72833 −1.00000
1.2 1.00000 1.43163 1.00000 −1.00000 1.43163 3.08719 1.00000 −0.950444 −1.00000
1.3 1.00000 2.68740 1.00000 −1.00000 2.68740 −4.59692 1.00000 4.22212 −1.00000
 $$n$$: 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 230.2.a.d 3
3.b odd 2 1 2070.2.a.z 3
4.b odd 2 1 1840.2.a.r 3
5.b even 2 1 1150.2.a.q 3
5.c odd 4 2 1150.2.b.j 6
8.b even 2 1 7360.2.a.bz 3
8.d odd 2 1 7360.2.a.ce 3
20.d odd 2 1 9200.2.a.cf 3
23.b odd 2 1 5290.2.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 1.a even 1 1 trivial
1150.2.a.q 3 5.b even 2 1
1150.2.b.j 6 5.c odd 4 2
1840.2.a.r 3 4.b odd 2 1
2070.2.a.z 3 3.b odd 2 1
5290.2.a.r 3 23.b odd 2 1
7360.2.a.bz 3 8.b even 2 1
7360.2.a.ce 3 8.d odd 2 1
9200.2.a.cf 3 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} - T_{3}^{2} - 9T_{3} + 12$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$T^{3} - T^{2} - 9T + 12$$
$5$ $$(T + 1)^{3}$$
$7$ $$T^{3} - 3 T^{2} - 21 T + 64$$
$11$ $$T^{3} - 3 T^{2} - 39 T + 144$$
$13$ $$T^{3} + T^{2} - 15 T - 18$$
$17$ $$T^{3} + 7 T^{2} + 7 T - 18$$
$19$ $$T^{3} - 3 T^{2} - 21 T + 64$$
$23$ $$(T + 1)^{3}$$
$29$ $$T^{3} + 4 T^{2} - 32 T + 24$$
$31$ $$T^{3} + 5 T^{2} - 7 T - 8$$
$37$ $$T^{3} + 2 T^{2} - 40 T - 32$$
$41$ $$T^{3} - T^{2} - 59 T + 186$$
$43$ $$(T - 8)^{3}$$
$47$ $$T^{3} + 14 T^{2} + 4 T - 288$$
$53$ $$(T + 6)^{3}$$
$59$ $$T^{3} - 14 T^{2} + 28 T + 144$$
$61$ $$T^{3} - T^{2} - 157 T + 526$$
$67$ $$T^{3} - 8 T^{2} - 144 T + 384$$
$71$ $$T^{3} - 11 T^{2} + 31 T - 24$$
$73$ $$T^{3} + 8 T^{2} - 40 T - 248$$
$79$ $$T^{3} + 4 T^{2} - 240 T - 1152$$
$83$ $$T^{3} - 8 T^{2} - 20 T + 96$$
$89$ $$T^{3} - 18 T^{2} - 48 T + 1152$$
$97$ $$T^{3} + 33 T^{2} + 279 T + 166$$