# Properties

 Label 3696.2.a.bo Level $3696$ Weight $2$ Character orbit 3696.a Self dual yes Analytic conductor $29.513$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3696.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.5127085871$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 231) 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.

 $$f(q)$$ $$=$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + ( 1 - \beta_{1} ) q^{5} + q^{7} + q^{9} + q^{11} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 2 + \beta_{1} - 3 \beta_{2} ) q^{17} + ( 3 + \beta_{2} ) q^{19} - q^{21} + ( -4 - \beta_{1} - \beta_{2} ) q^{23} + ( 4 - \beta_{1} - 2 \beta_{2} ) q^{25} - q^{27} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{29} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{31} - q^{33} + ( 1 - \beta_{1} ) q^{35} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{37} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{39} + ( 4 - 2 \beta_{2} ) q^{41} + ( 4 - 3 \beta_{1} + \beta_{2} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( -1 - 2 \beta_{1} - \beta_{2} ) q^{47} + q^{49} + ( -2 - \beta_{1} + 3 \beta_{2} ) q^{51} + ( -\beta_{1} + \beta_{2} ) q^{53} + ( 1 - \beta_{1} ) q^{55} + ( -3 - \beta_{2} ) q^{57} + ( -1 - 3 \beta_{2} ) q^{59} -2 q^{61} + q^{63} + ( 7 + 3 \beta_{1} + 4 \beta_{2} ) q^{65} + ( 1 + 2 \beta_{1} - 3 \beta_{2} ) q^{67} + ( 4 + \beta_{1} + \beta_{2} ) q^{69} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -7 + \beta_{1} - 2 \beta_{2} ) q^{73} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{75} + q^{77} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{79} + q^{81} + ( -2 + 3 \beta_{1} - 3 \beta_{2} ) q^{83} + ( -6 - 5 \beta_{1} - 7 \beta_{2} ) q^{85} + ( 1 + \beta_{1} - 2 \beta_{2} ) q^{87} + ( 10 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} + ( -1 - \beta_{1} + 2 \beta_{2} ) q^{91} + ( -2 - \beta_{1} - 3 \beta_{2} ) q^{93} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{95} + ( 3 \beta_{1} + \beta_{2} ) q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 3 q^{3} + 4 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 4 q^{13} - 4 q^{15} + 8 q^{17} + 8 q^{19} - 3 q^{21} - 10 q^{23} + 15 q^{25} - 3 q^{27} - 4 q^{29} + 2 q^{31} - 3 q^{33} + 4 q^{35} + 4 q^{39} + 14 q^{41} + 14 q^{43} + 4 q^{45} + 3 q^{49} - 8 q^{51} + 4 q^{55} - 8 q^{57} - 6 q^{61} + 3 q^{63} + 14 q^{65} + 4 q^{67} + 10 q^{69} + 12 q^{71} - 20 q^{73} - 15 q^{75} + 3 q^{77} - 12 q^{79} + 3 q^{81} - 6 q^{83} - 6 q^{85} + 4 q^{87} + 26 q^{89} - 4 q^{91} - 2 q^{93} + 8 q^{95} - 4 q^{97} + 3 q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 6$$$$)/2$$

## 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
 2.11491 −1.86081 −0.254102
0 −1.00000 0 −2.58774 0 1.00000 0 1.00000 0
1.2 0 −1.00000 0 2.39821 0 1.00000 0 1.00000 0
1.3 0 −1.00000 0 4.18953 0 1.00000 0 1.00000 0
 $$n$$: 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 3696.2.a.bo 3
4.b odd 2 1 231.2.a.e 3
12.b even 2 1 693.2.a.l 3
20.d odd 2 1 5775.2.a.bp 3
28.d even 2 1 1617.2.a.t 3
44.c even 2 1 2541.2.a.bg 3
84.h odd 2 1 4851.2.a.bi 3
132.d odd 2 1 7623.2.a.cd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.e 3 4.b odd 2 1
693.2.a.l 3 12.b even 2 1
1617.2.a.t 3 28.d even 2 1
2541.2.a.bg 3 44.c even 2 1
3696.2.a.bo 3 1.a even 1 1 trivial
4851.2.a.bi 3 84.h odd 2 1
5775.2.a.bp 3 20.d odd 2 1
7623.2.a.cd 3 132.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3696))$$:

 $$T_{5}^{3} - 4 T_{5}^{2} - 7 T_{5} + 26$$ $$T_{13}^{3} + 4 T_{13}^{2} - 27 T_{13} - 94$$ $$T_{17}^{3} - 8 T_{17}^{2} - 40 T_{17} + 328$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$26 - 7 T - 4 T^{2} + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-94 - 27 T + 4 T^{2} + T^{3}$$
$17$ $$328 - 40 T - 8 T^{2} + T^{3}$$
$19$ $$-4 + 15 T - 8 T^{2} + T^{3}$$
$23$ $$-64 + 12 T + 10 T^{2} + T^{3}$$
$29$ $$-94 - 27 T + 4 T^{2} + T^{3}$$
$31$ $$256 - 76 T - 2 T^{2} + T^{3}$$
$37$ $$106 - 43 T + T^{3}$$
$41$ $$32 + 40 T - 14 T^{2} + T^{3}$$
$43$ $$848 - 44 T - 14 T^{2} + T^{3}$$
$47$ $$-32 - 61 T + T^{3}$$
$53$ $$8 - 16 T + T^{3}$$
$59$ $$52 - 57 T + T^{3}$$
$61$ $$( 2 + T )^{3}$$
$67$ $$236 - 85 T - 4 T^{2} + T^{3}$$
$71$ $$256 - 16 T - 12 T^{2} + T^{3}$$
$73$ $$134 + 101 T + 20 T^{2} + T^{3}$$
$79$ $$-256 - 16 T + 12 T^{2} + T^{3}$$
$83$ $$-496 - 132 T + 6 T^{2} + T^{3}$$
$89$ $$328 + 140 T - 26 T^{2} + T^{3}$$
$97$ $$-232 - 120 T + 4 T^{2} + T^{3}$$