# Properties

 Label 3234.2.a.bh 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: 3.3.2700.1 Defining polynomial: $$x^{3} - 15 x - 20$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 15 x - 20$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 10$$

## 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
 4.41883 −1.61323 −2.80560
1.00000 1.00000 1.00000 −4.41883 1.00000 0 1.00000 1.00000 −4.41883
1.2 1.00000 1.00000 1.00000 1.61323 1.00000 0 1.00000 1.00000 1.61323
1.3 1.00000 1.00000 1.00000 2.80560 1.00000 0 1.00000 1.00000 2.80560
 $$n$$: 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.bh 3
3.b odd 2 1 9702.2.a.dw 3
7.b odd 2 1 3234.2.a.bf 3
7.d odd 6 2 462.2.i.g 6
21.c even 2 1 9702.2.a.dv 3
21.g even 6 2 1386.2.k.v 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.i.g 6 7.d odd 6 2
1386.2.k.v 6 21.g even 6 2
3234.2.a.bf 3 7.b odd 2 1
3234.2.a.bh 3 1.a even 1 1 trivial
9702.2.a.dv 3 21.c even 2 1
9702.2.a.dw 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3234))$$:

 $$T_{5}^{3} - 15 T_{5} + 20$$ $$T_{13}$$ $$T_{17}^{3} + 3 T_{17}^{2} - 57 T_{17} - 139$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$20 - 15 T + T^{3}$$
$7$ $$T^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$T^{3}$$
$17$ $$-139 - 57 T + 3 T^{2} + T^{3}$$
$19$ $$28 + 12 T - 9 T^{2} + T^{3}$$
$23$ $$89 - 27 T - 3 T^{2} + T^{3}$$
$29$ $$348 - 48 T - 9 T^{2} + T^{3}$$
$31$ $$192 - 48 T - 6 T^{2} + T^{3}$$
$37$ $$-228 + 132 T - 21 T^{2} + T^{3}$$
$41$ $$-306 - 27 T + 12 T^{2} + T^{3}$$
$43$ $$404 - 132 T - 3 T^{2} + T^{3}$$
$47$ $$-9 - 27 T + 3 T^{2} + T^{3}$$
$53$ $$( -8 + T )^{3}$$
$59$ $$48 - 48 T - 9 T^{2} + T^{3}$$
$61$ $$-98 - 63 T - 6 T^{2} + T^{3}$$
$67$ $$212 - 63 T - 6 T^{2} + T^{3}$$
$71$ $$108 - 108 T - 9 T^{2} + T^{3}$$
$73$ $$480 - 120 T + T^{3}$$
$79$ $$16 + 33 T - 12 T^{2} + T^{3}$$
$83$ $$-164 + 33 T + 18 T^{2} + T^{3}$$
$89$ $$16 - 12 T - 12 T^{2} + T^{3}$$
$97$ $$( 7 + T )^{3}$$