# Properties

 Label 231.2.a.e Level 231 Weight 2 Character orbit 231.a Self dual yes Analytic conductor 1.845 Analytic rank 0 Dimension 3 CM no Inner twists 1

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## Newspace parameters

 Level: $$N$$ = $$231 = 3 \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 231.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.84454428669$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 4 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$

## 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
 −0.254102 −1.86081 2.11491
−1.93543 1.00000 1.74590 4.18953 −1.93543 −1.00000 0.491797 1.00000 −8.10856
1.2 1.46260 1.00000 0.139194 2.39821 1.46260 −1.00000 −2.72161 1.00000 3.50761
1.3 2.47283 1.00000 4.11491 −2.58774 2.47283 −1.00000 5.22982 1.00000 −6.39905
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 231.2.a.e 3
3.b odd 2 1 693.2.a.l 3
4.b odd 2 1 3696.2.a.bo 3
5.b even 2 1 5775.2.a.bp 3
7.b odd 2 1 1617.2.a.t 3
11.b odd 2 1 2541.2.a.bg 3
21.c even 2 1 4851.2.a.bi 3
33.d even 2 1 7623.2.a.cd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 1.a even 1 1 trivial
693.2.a.l 3 3.b odd 2 1
1617.2.a.t 3 7.b odd 2 1
2541.2.a.bg 3 11.b odd 2 1
3696.2.a.bo 3 4.b odd 2 1
4851.2.a.bi 3 21.c even 2 1
5775.2.a.bp 3 5.b even 2 1
7623.2.a.cd 3 33.d even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 2 T_{2}^{2} - 4 T_{2} + 7$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(231))$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T + 2 T^{2} - T^{3} + 4 T^{4} - 8 T^{5} + 8 T^{6}$$
$3$ $$( 1 - T )^{3}$$
$5$ $$1 - 4 T + 8 T^{2} - 14 T^{3} + 40 T^{4} - 100 T^{5} + 125 T^{6}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$1 + 4 T + 12 T^{2} + 10 T^{3} + 156 T^{4} + 676 T^{5} + 2197 T^{6}$$
$17$ $$1 - 8 T + 11 T^{2} + 56 T^{3} + 187 T^{4} - 2312 T^{5} + 4913 T^{6}$$
$19$ $$1 + 8 T + 72 T^{2} + 308 T^{3} + 1368 T^{4} + 2888 T^{5} + 6859 T^{6}$$
$23$ $$1 - 10 T + 81 T^{2} - 396 T^{3} + 1863 T^{4} - 5290 T^{5} + 12167 T^{6}$$
$29$ $$1 + 4 T + 60 T^{2} + 138 T^{3} + 1740 T^{4} + 3364 T^{5} + 24389 T^{6}$$
$31$ $$1 + 2 T + 17 T^{2} - 132 T^{3} + 527 T^{4} + 1922 T^{5} + 29791 T^{6}$$
$37$ $$1 + 68 T^{2} + 106 T^{3} + 2516 T^{4} + 50653 T^{6}$$
$41$ $$1 - 14 T + 163 T^{2} - 1116 T^{3} + 6683 T^{4} - 23534 T^{5} + 68921 T^{6}$$
$43$ $$1 + 14 T + 85 T^{2} + 356 T^{3} + 3655 T^{4} + 25886 T^{5} + 79507 T^{6}$$
$47$ $$1 + 80 T^{2} + 32 T^{3} + 3760 T^{4} + 103823 T^{6}$$
$53$ $$1 + 143 T^{2} + 8 T^{3} + 7579 T^{4} + 148877 T^{6}$$
$59$ $$1 + 120 T^{2} - 52 T^{3} + 7080 T^{4} + 205379 T^{6}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{3}$$
$67$ $$1 + 4 T + 116 T^{2} + 300 T^{3} + 7772 T^{4} + 17956 T^{5} + 300763 T^{6}$$
$71$ $$1 + 12 T + 197 T^{2} + 1448 T^{3} + 13987 T^{4} + 60492 T^{5} + 357911 T^{6}$$
$73$ $$1 + 20 T + 320 T^{2} + 3054 T^{3} + 23360 T^{4} + 106580 T^{5} + 389017 T^{6}$$
$79$ $$1 - 12 T + 221 T^{2} - 1640 T^{3} + 17459 T^{4} - 74892 T^{5} + 493039 T^{6}$$
$83$ $$1 - 6 T + 117 T^{2} - 500 T^{3} + 9711 T^{4} - 41334 T^{5} + 571787 T^{6}$$
$89$ $$1 - 26 T + 407 T^{2} - 4300 T^{3} + 36223 T^{4} - 205946 T^{5} + 704969 T^{6}$$
$97$ $$1 + 4 T + 171 T^{2} + 544 T^{3} + 16587 T^{4} + 37636 T^{5} + 912673 T^{6}$$
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