# Properties

 Label 8464.2.a.bd Level $8464$ Weight $2$ Character orbit 8464.a Self dual yes Analytic conductor $67.585$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8464 = 2^{4} \cdot 23^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8464.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$67.5853802708$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 184) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} -2 q^{5} + ( 1 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} -2 q^{5} + ( 1 + \beta ) q^{9} + 2 \beta q^{11} + ( 2 + \beta ) q^{13} -2 \beta q^{15} + ( -2 + 2 \beta ) q^{17} + 2 \beta q^{19} - q^{25} + ( 4 - \beta ) q^{27} + ( 2 - \beta ) q^{29} + ( 4 + \beta ) q^{31} + ( 8 + 2 \beta ) q^{33} + ( -2 + 4 \beta ) q^{37} + ( 4 + 3 \beta ) q^{39} + ( -2 + 5 \beta ) q^{41} -8 q^{43} + ( -2 - 2 \beta ) q^{45} + ( -4 - 3 \beta ) q^{47} -7 q^{49} + 8 q^{51} -2 q^{53} -4 \beta q^{55} + ( 8 + 2 \beta ) q^{57} + ( -4 + 4 \beta ) q^{59} + ( -2 - 4 \beta ) q^{61} + ( -4 - 2 \beta ) q^{65} -2 \beta q^{67} + ( -12 + \beta ) q^{71} + ( -10 + 3 \beta ) q^{73} -\beta q^{75} -2 \beta q^{79} -7 q^{81} + ( 8 - 4 \beta ) q^{83} + ( 4 - 4 \beta ) q^{85} + ( -4 + \beta ) q^{87} + ( -2 + 6 \beta ) q^{89} + ( 4 + 5 \beta ) q^{93} -4 \beta q^{95} + ( -2 + 6 \beta ) q^{97} + ( 8 + 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 4 q^{5} + 3 q^{9} + O(q^{10})$$ $$2 q + q^{3} - 4 q^{5} + 3 q^{9} + 2 q^{11} + 5 q^{13} - 2 q^{15} - 2 q^{17} + 2 q^{19} - 2 q^{25} + 7 q^{27} + 3 q^{29} + 9 q^{31} + 18 q^{33} + 11 q^{39} + q^{41} - 16 q^{43} - 6 q^{45} - 11 q^{47} - 14 q^{49} + 16 q^{51} - 4 q^{53} - 4 q^{55} + 18 q^{57} - 4 q^{59} - 8 q^{61} - 10 q^{65} - 2 q^{67} - 23 q^{71} - 17 q^{73} - q^{75} - 2 q^{79} - 14 q^{81} + 12 q^{83} + 4 q^{85} - 7 q^{87} + 2 q^{89} + 13 q^{93} - 4 q^{95} + 2 q^{97} + 20 q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
0 −1.56155 0 −2.00000 0 0 0 −0.561553 0
1.2 0 2.56155 0 −2.00000 0 0 0 3.56155 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
$$2$$ $$1$$
$$23$$ $$-1$$

## 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 8464.2.a.bd 2
4.b odd 2 1 4232.2.a.o 2
23.b odd 2 1 368.2.a.i 2
69.c even 2 1 3312.2.a.t 2
92.b even 2 1 184.2.a.e 2
115.c odd 2 1 9200.2.a.br 2
184.e odd 2 1 1472.2.a.p 2
184.h even 2 1 1472.2.a.u 2
276.h odd 2 1 1656.2.a.j 2
460.g even 2 1 4600.2.a.s 2
460.k odd 4 2 4600.2.e.o 4
644.h odd 2 1 9016.2.a.w 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
184.2.a.e 2 92.b even 2 1
368.2.a.i 2 23.b odd 2 1
1472.2.a.p 2 184.e odd 2 1
1472.2.a.u 2 184.h even 2 1
1656.2.a.j 2 276.h odd 2 1
3312.2.a.t 2 69.c even 2 1
4232.2.a.o 2 4.b odd 2 1
4600.2.a.s 2 460.g even 2 1
4600.2.e.o 4 460.k odd 4 2
8464.2.a.bd 2 1.a even 1 1 trivial
9016.2.a.w 2 644.h odd 2 1
9200.2.a.br 2 115.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8464))$$:

 $$T_{3}^{2} - T_{3} - 4$$ $$T_{5} + 2$$ $$T_{7}$$ $$T_{13}^{2} - 5 T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - T + T^{2}$$
$5$ $$( 2 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$-16 - 2 T + T^{2}$$
$13$ $$2 - 5 T + T^{2}$$
$17$ $$-16 + 2 T + T^{2}$$
$19$ $$-16 - 2 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$-2 - 3 T + T^{2}$$
$31$ $$16 - 9 T + T^{2}$$
$37$ $$-68 + T^{2}$$
$41$ $$-106 - T + T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$-8 + 11 T + T^{2}$$
$53$ $$( 2 + T )^{2}$$
$59$ $$-64 + 4 T + T^{2}$$
$61$ $$-52 + 8 T + T^{2}$$
$67$ $$-16 + 2 T + T^{2}$$
$71$ $$128 + 23 T + T^{2}$$
$73$ $$34 + 17 T + T^{2}$$
$79$ $$-16 + 2 T + T^{2}$$
$83$ $$-32 - 12 T + T^{2}$$
$89$ $$-152 - 2 T + T^{2}$$
$97$ $$-152 - 2 T + T^{2}$$