# Properties

 Label 4400.2.a.bm Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2200) 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.

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

## 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
 1.61803 −0.618034
0 −1.61803 0 0 0 2.61803 0 −0.381966 0
1.2 0 0.618034 0 0 0 0.381966 0 −2.61803 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
$$2$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$

## 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 4400.2.a.bm 2
4.b odd 2 1 2200.2.a.q yes 2
5.b even 2 1 4400.2.a.bo 2
5.c odd 4 2 4400.2.b.z 4
20.d odd 2 1 2200.2.a.p 2
20.e even 4 2 2200.2.b.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.p 2 20.d odd 2 1
2200.2.a.q yes 2 4.b odd 2 1
2200.2.b.k 4 20.e even 4 2
4400.2.a.bm 2 1.a even 1 1 trivial
4400.2.a.bo 2 5.b even 2 1
4400.2.b.z 4 5.c odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3}^{2} + T_{3} - 1$$ $$T_{7}^{2} - 3 T_{7} + 1$$ $$T_{13}^{2} - 2 T_{13} - 19$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 - 3 T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-19 - 2 T + T^{2}$$
$17$ $$-31 - T + T^{2}$$
$19$ $$31 - 12 T + T^{2}$$
$23$ $$1 + 3 T + T^{2}$$
$29$ $$-61 - T + T^{2}$$
$31$ $$-41 - 4 T + T^{2}$$
$37$ $$-1 + 4 T + T^{2}$$
$41$ $$-29 + 8 T + T^{2}$$
$43$ $$-76 - 4 T + T^{2}$$
$47$ $$-5 + T^{2}$$
$53$ $$5 - 5 T + T^{2}$$
$59$ $$-29 - 8 T + T^{2}$$
$61$ $$25 + 15 T + T^{2}$$
$67$ $$( -12 + T )^{2}$$
$71$ $$-116 - 6 T + T^{2}$$
$73$ $$109 - 21 T + T^{2}$$
$79$ $$71 - 17 T + T^{2}$$
$83$ $$-89 - 7 T + T^{2}$$
$89$ $$-19 + 13 T + T^{2}$$
$97$ $$131 - 23 T + T^{2}$$
show more
show less