# Properties

 Label 3696.2.a.bp 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: 3.3.837.1 Defining polynomial: $$x^{3} - 6 x - 1$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{3} -\beta_{2} q^{5} + q^{7} + q^{9} +O(q^{10})$$ $$q + q^{3} -\beta_{2} q^{5} + q^{7} + q^{9} - q^{11} -\beta_{2} q^{13} -\beta_{2} q^{15} + ( -\beta_{1} + \beta_{2} ) q^{17} + ( -4 + \beta_{1} ) q^{19} + q^{21} + ( 2 + \beta_{1} - \beta_{2} ) q^{23} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{25} + q^{27} + ( 4 + \beta_{2} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} ) q^{31} - q^{33} -\beta_{2} q^{35} + ( 2 \beta_{1} - \beta_{2} ) q^{37} -\beta_{2} q^{39} + ( 2 + 2 \beta_{1} ) q^{41} + ( -2 + \beta_{1} - \beta_{2} ) q^{43} -\beta_{2} q^{45} + ( 8 - \beta_{1} ) q^{47} + q^{49} + ( -\beta_{1} + \beta_{2} ) q^{51} + ( -\beta_{1} - \beta_{2} ) q^{53} + \beta_{2} q^{55} + ( -4 + \beta_{1} ) q^{57} + ( 8 + \beta_{1} + 2 \beta_{2} ) q^{59} + 6 q^{61} + q^{63} + ( 10 - 2 \beta_{1} + \beta_{2} ) q^{65} + ( -4 - \beta_{1} ) q^{67} + ( 2 + \beta_{1} - \beta_{2} ) q^{69} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{71} + ( 8 + \beta_{2} ) q^{73} + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{75} - q^{77} + ( -4 - 2 \beta_{1} + 2 \beta_{2} ) q^{79} + q^{81} + ( 6 - \beta_{1} - \beta_{2} ) q^{83} + ( -6 + \beta_{1} - 3 \beta_{2} ) q^{85} + ( 4 + \beta_{2} ) q^{87} + ( 6 + 2 \beta_{1} + 2 \beta_{2} ) q^{89} -\beta_{2} q^{91} + ( 2 - \beta_{1} - \beta_{2} ) q^{93} + ( -4 + \beta_{1} + 6 \beta_{2} ) q^{95} + ( 8 + \beta_{1} + \beta_{2} ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{3} + 3q^{7} + 3q^{9} + O(q^{10})$$ $$3q + 3q^{3} + 3q^{7} + 3q^{9} - 3q^{11} - 12q^{19} + 3q^{21} + 6q^{23} + 15q^{25} + 3q^{27} + 12q^{29} + 6q^{31} - 3q^{33} + 6q^{41} - 6q^{43} + 24q^{47} + 3q^{49} - 12q^{57} + 24q^{59} + 18q^{61} + 3q^{63} + 30q^{65} - 12q^{67} + 6q^{69} - 12q^{71} + 24q^{73} + 15q^{75} - 3q^{77} - 12q^{79} + 3q^{81} + 18q^{83} - 18q^{85} + 12q^{87} + 18q^{89} + 6q^{93} - 12q^{95} + 24q^{97} - 3q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 6 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 4$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 8$$$$)/2$$

## 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
 −2.36147 2.52892 −0.167449
0 1.00000 0 −3.93800 0 1.00000 0 1.00000 0
1.2 0 1.00000 0 0.133492 0 1.00000 0 1.00000 0
1.3 0 1.00000 0 3.80451 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.bp 3
4.b odd 2 1 231.2.a.d 3
12.b even 2 1 693.2.a.m 3
20.d odd 2 1 5775.2.a.bw 3
28.d even 2 1 1617.2.a.s 3
44.c even 2 1 2541.2.a.bi 3
84.h odd 2 1 4851.2.a.bp 3
132.d odd 2 1 7623.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.d 3 4.b odd 2 1
693.2.a.m 3 12.b even 2 1
1617.2.a.s 3 28.d even 2 1
2541.2.a.bi 3 44.c even 2 1
3696.2.a.bp 3 1.a even 1 1 trivial
4851.2.a.bp 3 84.h odd 2 1
5775.2.a.bw 3 20.d odd 2 1
7623.2.a.cb 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3696))$$:

 $$T_{5}^{3} - 15 T_{5} + 2$$ $$T_{13}^{3} - 15 T_{13} + 2$$ $$T_{17}^{3} - 24 T_{17} + 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$2 - 15 T + T^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$2 - 15 T + T^{3}$$
$17$ $$8 - 24 T + T^{3}$$
$19$ $$-36 + 27 T + 12 T^{2} + T^{3}$$
$23$ $$32 - 12 T - 6 T^{2} + T^{3}$$
$29$ $$-6 + 33 T - 12 T^{2} + T^{3}$$
$31$ $$-32 - 36 T - 6 T^{2} + T^{3}$$
$37$ $$-246 - 75 T + T^{3}$$
$41$ $$32 - 72 T - 6 T^{2} + T^{3}$$
$43$ $$-48 - 12 T + 6 T^{2} + T^{3}$$
$47$ $$-328 + 171 T - 24 T^{2} + T^{3}$$
$53$ $$-120 - 48 T + T^{3}$$
$59$ $$716 + 87 T - 24 T^{2} + T^{3}$$
$61$ $$( -6 + T )^{3}$$
$67$ $$-4 + 27 T + 12 T^{2} + T^{3}$$
$71$ $$-384 - 48 T + 12 T^{2} + T^{3}$$
$73$ $$-394 + 177 T - 24 T^{2} + T^{3}$$
$79$ $$-256 - 48 T + 12 T^{2} + T^{3}$$
$83$ $$-48 + 60 T - 18 T^{2} + T^{3}$$
$89$ $$1896 - 84 T - 18 T^{2} + T^{3}$$
$97$ $$-8 + 144 T - 24 T^{2} + T^{3}$$