# Properties

 Label 3696.2.a.be Level 3696 Weight 2 Character orbit 3696.a Self dual yes Analytic conductor 29.513 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} + q^{5} - q^{7} + q^{9} - q^{11} + ( 1 - 4 \beta ) q^{13} - q^{15} + ( 4 - 2 \beta ) q^{17} + ( 3 - 6 \beta ) q^{19} + q^{21} + ( -2 + 6 \beta ) q^{23} -4 q^{25} - q^{27} + 5 q^{29} + ( 4 - 2 \beta ) q^{31} + q^{33} - q^{35} -7 q^{37} + ( -1 + 4 \beta ) q^{39} + 4 \beta q^{41} + ( -2 + 6 \beta ) q^{43} + q^{45} + ( 1 + 2 \beta ) q^{47} + q^{49} + ( -4 + 2 \beta ) q^{51} + ( -6 + 10 \beta ) q^{53} - q^{55} + ( -3 + 6 \beta ) q^{57} + ( 5 - 10 \beta ) q^{59} + 2 q^{61} - q^{63} + ( 1 - 4 \beta ) q^{65} + ( 11 + 2 \beta ) q^{67} + ( 2 - 6 \beta ) q^{69} -4 \beta q^{71} + ( 7 + 4 \beta ) q^{73} + 4 q^{75} + q^{77} + ( 12 - 4 \beta ) q^{79} + q^{81} + ( -8 - 2 \beta ) q^{83} + ( 4 - 2 \beta ) q^{85} -5 q^{87} + ( -2 + 4 \beta ) q^{89} + ( -1 + 4 \beta ) q^{91} + ( -4 + 2 \beta ) q^{93} + ( 3 - 6 \beta ) q^{95} + ( 6 - 6 \beta ) q^{97} - q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{5} - 2q^{7} + 2q^{9} - 2q^{11} - 2q^{13} - 2q^{15} + 6q^{17} + 2q^{21} + 2q^{23} - 8q^{25} - 2q^{27} + 10q^{29} + 6q^{31} + 2q^{33} - 2q^{35} - 14q^{37} + 2q^{39} + 4q^{41} + 2q^{43} + 2q^{45} + 4q^{47} + 2q^{49} - 6q^{51} - 2q^{53} - 2q^{55} + 4q^{61} - 2q^{63} - 2q^{65} + 24q^{67} - 2q^{69} - 4q^{71} + 18q^{73} + 8q^{75} + 2q^{77} + 20q^{79} + 2q^{81} - 18q^{83} + 6q^{85} - 10q^{87} + 2q^{91} - 6q^{93} + 6q^{97} - 2q^{99} + O(q^{100})$$

## 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
 1.61803 −0.618034
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.00000 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

## 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.be 2
4.b odd 2 1 231.2.a.c 2
12.b even 2 1 693.2.a.f 2
20.d odd 2 1 5775.2.a.be 2
28.d even 2 1 1617.2.a.p 2
44.c even 2 1 2541.2.a.t 2
84.h odd 2 1 4851.2.a.w 2
132.d odd 2 1 7623.2.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
231.2.a.c 2 4.b odd 2 1
693.2.a.f 2 12.b even 2 1
1617.2.a.p 2 28.d even 2 1
2541.2.a.t 2 44.c even 2 1
3696.2.a.be 2 1.a even 1 1 trivial
4851.2.a.w 2 84.h odd 2 1
5775.2.a.be 2 20.d odd 2 1
7623.2.a.bm 2 132.d odd 2 1

## Atkin-Lehner signs

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

## Hecke kernels

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

 $$T_{5} - 1$$ $$T_{13}^{2} + 2 T_{13} - 19$$ $$T_{17}^{2} - 6 T_{17} + 4$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 - T + 5 T^{2} )^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$1 - 6 T + 38 T^{2} - 102 T^{3} + 289 T^{4}$$
$19$ $$1 - 7 T^{2} + 361 T^{4}$$
$23$ $$1 - 2 T + 2 T^{2} - 46 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 5 T + 29 T^{2} )^{2}$$
$31$ $$1 - 6 T + 66 T^{2} - 186 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 7 T + 37 T^{2} )^{2}$$
$41$ $$1 - 4 T + 66 T^{2} - 164 T^{3} + 1681 T^{4}$$
$43$ $$1 - 2 T + 42 T^{2} - 86 T^{3} + 1849 T^{4}$$
$47$ $$1 - 4 T + 93 T^{2} - 188 T^{3} + 2209 T^{4}$$
$53$ $$1 + 2 T - 18 T^{2} + 106 T^{3} + 2809 T^{4}$$
$59$ $$1 - 7 T^{2} + 3481 T^{4}$$
$61$ $$( 1 - 2 T + 61 T^{2} )^{2}$$
$67$ $$1 - 24 T + 273 T^{2} - 1608 T^{3} + 4489 T^{4}$$
$71$ $$1 + 4 T + 126 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 - 18 T + 207 T^{2} - 1314 T^{3} + 5329 T^{4}$$
$79$ $$1 - 20 T + 238 T^{2} - 1580 T^{3} + 6241 T^{4}$$
$83$ $$1 + 18 T + 242 T^{2} + 1494 T^{3} + 6889 T^{4}$$
$89$ $$1 + 158 T^{2} + 7921 T^{4}$$
$97$ $$1 - 6 T + 158 T^{2} - 582 T^{3} + 9409 T^{4}$$