# Properties

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

# Related objects

## 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( -1 + \beta ) q^{7} + ( -2 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} + ( -1 + \beta ) q^{7} + ( -2 + \beta ) q^{9} + q^{11} + ( -1 + 2 \beta ) q^{13} + 3 \beta q^{17} + ( -1 + 4 \beta ) q^{19} + q^{21} + ( 5 - 7 \beta ) q^{23} + ( 1 - 4 \beta ) q^{27} + ( -6 + \beta ) q^{29} + ( 1 + 2 \beta ) q^{31} + \beta q^{33} + ( 5 - 2 \beta ) q^{37} + ( 2 + \beta ) q^{39} + ( -5 + 8 \beta ) q^{41} + ( 2 + 4 \beta ) q^{43} + ( -5 + 6 \beta ) q^{47} + ( -5 - \beta ) q^{49} + ( 3 + 3 \beta ) q^{51} + ( 1 - 3 \beta ) q^{53} + ( 4 + 3 \beta ) q^{57} + ( 7 - 2 \beta ) q^{59} + ( -1 + 5 \beta ) q^{61} + ( 3 - 2 \beta ) q^{63} + ( -7 - 2 \beta ) q^{69} + ( 10 - 6 \beta ) q^{71} + ( 1 - 5 \beta ) q^{73} + ( -1 + \beta ) q^{77} + ( 2 - 3 \beta ) q^{79} + ( 2 - 6 \beta ) q^{81} + ( 6 - 3 \beta ) q^{83} + ( 1 - 5 \beta ) q^{87} + ( 2 - 9 \beta ) q^{89} + ( 3 - \beta ) q^{91} + ( 2 + 3 \beta ) q^{93} + ( 13 - 5 \beta ) q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{7} - 3 q^{9} + O(q^{10})$$ $$2 q + q^{3} - q^{7} - 3 q^{9} + 2 q^{11} + 3 q^{17} + 2 q^{19} + 2 q^{21} + 3 q^{23} - 2 q^{27} - 11 q^{29} + 4 q^{31} + q^{33} + 8 q^{37} + 5 q^{39} - 2 q^{41} + 8 q^{43} - 4 q^{47} - 11 q^{49} + 9 q^{51} - q^{53} + 11 q^{57} + 12 q^{59} + 3 q^{61} + 4 q^{63} - 16 q^{69} + 14 q^{71} - 3 q^{73} - q^{77} + q^{79} - 2 q^{81} + 9 q^{83} - 3 q^{87} - 5 q^{89} + 5 q^{91} + 7 q^{93} + 21 q^{97} - 3 q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 −0.618034 0 0 0 −1.61803 0 −2.61803 0
1.2 0 1.61803 0 0 0 0.618034 0 −0.381966 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.bq 2
4.b odd 2 1 2200.2.a.n 2
5.b even 2 1 4400.2.a.bk 2
5.c odd 4 2 4400.2.b.ba 4
20.d odd 2 1 2200.2.a.r yes 2
20.e even 4 2 2200.2.b.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.n 2 4.b odd 2 1
2200.2.a.r yes 2 20.d odd 2 1
2200.2.b.j 4 20.e even 4 2
4400.2.a.bk 2 5.b even 2 1
4400.2.a.bq 2 1.a even 1 1 trivial
4400.2.b.ba 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3}^{2} - T_{3} - 1$$ $$T_{7}^{2} + T_{7} - 1$$ $$T_{13}^{2} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-1 - T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-1 + T + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-5 + T^{2}$$
$17$ $$-9 - 3 T + T^{2}$$
$19$ $$-19 - 2 T + T^{2}$$
$23$ $$-59 - 3 T + T^{2}$$
$29$ $$29 + 11 T + T^{2}$$
$31$ $$-1 - 4 T + T^{2}$$
$37$ $$11 - 8 T + T^{2}$$
$41$ $$-79 + 2 T + T^{2}$$
$43$ $$-4 - 8 T + T^{2}$$
$47$ $$-41 + 4 T + T^{2}$$
$53$ $$-11 + T + T^{2}$$
$59$ $$31 - 12 T + T^{2}$$
$61$ $$-29 - 3 T + T^{2}$$
$67$ $$T^{2}$$
$71$ $$4 - 14 T + T^{2}$$
$73$ $$-29 + 3 T + T^{2}$$
$79$ $$-11 - T + T^{2}$$
$83$ $$9 - 9 T + T^{2}$$
$89$ $$-95 + 5 T + T^{2}$$
$97$ $$79 - 21 T + T^{2}$$