# Properties

 Label 4096.2.a.f Level $4096$ Weight $2$ Character orbit 4096.a Self dual yes Analytic conductor $32.707$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4096 = 2^{12}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4096.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.7067246679$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{16})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 32) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( \beta_{1} + 2 \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( \beta_{1} + 2 \beta_{3} ) q^{5} + \beta_{2} q^{7} + ( -1 + \beta_{2} ) q^{9} + ( -\beta_{1} + 2 \beta_{3} ) q^{11} -\beta_{1} q^{13} + ( 2 + 3 \beta_{2} ) q^{15} + 2 \beta_{2} q^{17} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{19} + ( \beta_{1} + \beta_{3} ) q^{21} + ( 4 - 3 \beta_{2} ) q^{23} + ( 5 + \beta_{2} ) q^{25} + ( -3 \beta_{1} + \beta_{3} ) q^{27} + ( 2 \beta_{1} - \beta_{3} ) q^{29} -4 q^{31} + ( -2 + \beta_{2} ) q^{33} + ( 3 \beta_{1} - \beta_{3} ) q^{35} + \beta_{1} q^{37} + ( -2 - \beta_{2} ) q^{39} + ( -4 - 3 \beta_{2} ) q^{41} + ( 3 \beta_{1} + 4 \beta_{3} ) q^{43} + ( 2 \beta_{1} - 3 \beta_{3} ) q^{45} + ( 6 + 4 \beta_{2} ) q^{47} -5 q^{49} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{51} + ( 4 \beta_{1} + \beta_{3} ) q^{53} + ( 6 - 5 \beta_{2} ) q^{55} + ( 4 + 5 \beta_{2} ) q^{57} + ( -4 \beta_{1} + \beta_{3} ) q^{59} -\beta_{3} q^{61} + ( 2 - \beta_{2} ) q^{63} + ( -2 - 3 \beta_{2} ) q^{65} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{67} + ( \beta_{1} - 3 \beta_{3} ) q^{69} + ( 4 - 3 \beta_{2} ) q^{71} -7 \beta_{2} q^{73} + ( 6 \beta_{1} + \beta_{3} ) q^{75} + ( \beta_{1} - 3 \beta_{3} ) q^{77} + 6 q^{79} + ( -3 - 5 \beta_{2} ) q^{81} + ( -4 \beta_{1} + \beta_{3} ) q^{83} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{85} + ( 4 + \beta_{2} ) q^{87} + ( -8 + 3 \beta_{2} ) q^{89} + ( -\beta_{1} - \beta_{3} ) q^{91} -4 \beta_{1} q^{93} + ( 16 + 3 \beta_{2} ) q^{95} + ( 10 - 6 \beta_{2} ) q^{97} + ( 2 \beta_{1} - 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{9} + O(q^{10})$$ $$4 q - 4 q^{9} + 8 q^{15} + 16 q^{23} + 20 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} - 16 q^{41} + 24 q^{47} - 20 q^{49} + 24 q^{55} + 16 q^{57} + 8 q^{63} - 8 q^{65} + 16 q^{71} + 24 q^{79} - 12 q^{81} + 16 q^{87} - 32 q^{89} + 64 q^{95} + 40 q^{97} + O(q^{100})$$

## 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
 −1.84776 −0.765367 0.765367 1.84776
0 −1.84776 0 −3.37849 0 1.41421 0 0.414214 0
1.2 0 −0.765367 0 2.93015 0 −1.41421 0 −2.41421 0
1.3 0 0.765367 0 −2.93015 0 −1.41421 0 −2.41421 0
1.4 0 1.84776 0 3.37849 0 1.41421 0 0.414214 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
$$2$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4096.2.a.f 4
4.b odd 2 1 4096.2.a.e 4
8.b even 2 1 inner 4096.2.a.f 4
8.d odd 2 1 4096.2.a.e 4
64.i even 16 2 128.2.g.a 4
64.i even 16 2 256.2.g.a 4
64.i even 16 2 512.2.g.b 4
64.i even 16 2 512.2.g.c 4
64.j odd 16 2 32.2.g.a 4
64.j odd 16 2 256.2.g.b 4
64.j odd 16 2 512.2.g.a 4
64.j odd 16 2 512.2.g.d 4
192.q odd 16 2 1152.2.v.a 4
192.s even 16 2 288.2.v.a 4
320.bd even 16 2 800.2.ba.a 4
320.bh odd 16 2 800.2.y.a 4
320.bj even 16 2 800.2.ba.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.a 4 64.j odd 16 2
128.2.g.a 4 64.i even 16 2
256.2.g.a 4 64.i even 16 2
256.2.g.b 4 64.j odd 16 2
288.2.v.a 4 192.s even 16 2
512.2.g.a 4 64.j odd 16 2
512.2.g.b 4 64.i even 16 2
512.2.g.c 4 64.i even 16 2
512.2.g.d 4 64.j odd 16 2
800.2.y.a 4 320.bh odd 16 2
800.2.ba.a 4 320.bd even 16 2
800.2.ba.b 4 320.bj even 16 2
1152.2.v.a 4 192.q odd 16 2
4096.2.a.e 4 4.b odd 2 1
4096.2.a.e 4 8.d odd 2 1
4096.2.a.f 4 1.a even 1 1 trivial
4096.2.a.f 4 8.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4096))$$:

 $$T_{3}^{4} - 4 T_{3}^{2} + 2$$ $$T_{5}^{4} - 20 T_{5}^{2} + 98$$ $$T_{7}^{2} - 2$$ $$T_{23}^{2} - 8 T_{23} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$2 - 4 T^{2} + T^{4}$$
$5$ $$98 - 20 T^{2} + T^{4}$$
$7$ $$( -2 + T^{2} )^{2}$$
$11$ $$2 - 20 T^{2} + T^{4}$$
$13$ $$2 - 4 T^{2} + T^{4}$$
$17$ $$( -8 + T^{2} )^{2}$$
$19$ $$578 - 52 T^{2} + T^{4}$$
$23$ $$( -2 - 8 T + T^{2} )^{2}$$
$29$ $$98 - 20 T^{2} + T^{4}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$2 - 4 T^{2} + T^{4}$$
$41$ $$( -2 + 8 T + T^{2} )^{2}$$
$43$ $$1922 - 100 T^{2} + T^{4}$$
$47$ $$( 4 - 12 T + T^{2} )^{2}$$
$53$ $$98 - 68 T^{2} + T^{4}$$
$59$ $$1058 - 68 T^{2} + T^{4}$$
$61$ $$2 - 4 T^{2} + T^{4}$$
$67$ $$578 - 52 T^{2} + T^{4}$$
$71$ $$( -2 - 8 T + T^{2} )^{2}$$
$73$ $$( -98 + T^{2} )^{2}$$
$79$ $$( -6 + T )^{4}$$
$83$ $$1058 - 68 T^{2} + T^{4}$$
$89$ $$( 46 + 16 T + T^{2} )^{2}$$
$97$ $$( 28 - 20 T + T^{2} )^{2}$$