# Properties

 Label 3234.2.a.bg Level $3234$ Weight $2$ Character orbit 3234.a Self dual yes Analytic conductor $25.824$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.621.1 Defining polynomial: $$x^{3} - 6x - 3$$ x^3 - 6*x - 3 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 462) 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6x - 3$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$ v^2 - v - 4
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( 2\beta_{2} + \beta _1 + 8 ) / 2$$ (2*b2 + b1 + 8) / 2

## 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
 −2.14510 2.66908 −0.523976
1.00000 −1.00000 1.00000 −2.74657 −1.00000 0 1.00000 1.00000 −2.74657
1.2 1.00000 −1.00000 1.00000 −0.454904 −1.00000 0 1.00000 1.00000 −0.454904
1.3 1.00000 −1.00000 1.00000 3.20147 −1.00000 0 1.00000 1.00000 3.20147
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 3234.2.a.bg 3
3.b odd 2 1 9702.2.a.du 3
7.b odd 2 1 3234.2.a.bi 3
7.d odd 6 2 462.2.i.f 6
21.c even 2 1 9702.2.a.dt 3
21.g even 6 2 1386.2.k.w 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.f 6 7.d odd 6 2
1386.2.k.w 6 21.g even 6 2
3234.2.a.bg 3 1.a even 1 1 trivial
3234.2.a.bi 3 7.b odd 2 1
9702.2.a.dt 3 21.c even 2 1
9702.2.a.du 3 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5}^{3} - 9T_{5} - 4$$ T5^3 - 9*T5 - 4 $$T_{13}^{3} - 36T_{13} - 32$$ T13^3 - 36*T13 - 32 $$T_{17} + 1$$ T17 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{3}$$
$3$ $$(T + 1)^{3}$$
$5$ $$T^{3} - 9T - 4$$
$7$ $$T^{3}$$
$11$ $$(T - 1)^{3}$$
$13$ $$T^{3} - 36T - 32$$
$17$ $$(T + 1)^{3}$$
$19$ $$T^{3} - 3 T^{2} - 36 T - 36$$
$23$ $$T^{3} - 9 T^{2} + 3 T + 21$$
$29$ $$T^{3} - 9T^{2} + 92$$
$31$ $$T^{3} + 6 T^{2} - 96 T - 512$$
$37$ $$T^{3} - 9 T^{2} - 12 T + 164$$
$41$ $$T^{3} + 12 T^{2} + 21 T - 82$$
$43$ $$T^{3} - 9 T^{2} - 12 T + 164$$
$47$ $$T^{3} - 15 T^{2} + 51 T + 19$$
$53$ $$T^{3} - 36T + 32$$
$59$ $$T^{3} + 9 T^{2} - 96 T - 768$$
$61$ $$T^{3} - 6 T^{2} - 129 T + 226$$
$67$ $$T^{3} - 6 T^{2} - 15 T + 84$$
$71$ $$T^{3} - 3 T^{2} - 24 T + 64$$
$73$ $$T^{3} - 96T + 192$$
$79$ $$T^{3} - 12 T^{2} - 33 T + 368$$
$83$ $$T^{3} - 6 T^{2} - 111 T - 188$$
$89$ $$T^{3} - 36 T^{2} + 396 T - 1328$$
$97$ $$T^{3} + 9 T^{2} - 81 T + 7$$