# Properties

 Label 4400.2.a.bg 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 275) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta ) q^{3} + ( -2 + 3 \beta ) q^{7} + ( -1 + 3 \beta ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta ) q^{3} + ( -2 + 3 \beta ) q^{7} + ( -1 + 3 \beta ) q^{9} - q^{11} + ( 3 + 2 \beta ) q^{13} + ( 1 - \beta ) q^{17} + ( -3 + 6 \beta ) q^{19} + ( -1 - 4 \beta ) q^{21} + ( -4 + 5 \beta ) q^{23} + ( 1 - 2 \beta ) q^{27} + ( -3 + \beta ) q^{29} + 3 q^{31} + ( 1 + \beta ) q^{33} + ( 7 + 2 \beta ) q^{37} + ( -5 - 7 \beta ) q^{39} -3 q^{41} + 6 q^{43} + ( 1 - 8 \beta ) q^{47} + ( 6 - 3 \beta ) q^{49} + \beta q^{51} + ( -2 + 7 \beta ) q^{53} + ( -3 - 9 \beta ) q^{57} + ( -7 + 4 \beta ) q^{59} + ( -8 + 5 \beta ) q^{61} + 11 q^{63} -8 q^{67} + ( -1 - 6 \beta ) q^{69} + ( 8 - 10 \beta ) q^{71} + ( 12 - \beta ) q^{73} + ( 2 - 3 \beta ) q^{77} + ( -1 - 3 \beta ) q^{79} + ( 4 - 6 \beta ) q^{81} + ( 15 - 3 \beta ) q^{83} + ( 2 + \beta ) q^{87} + ( -15 + 5 \beta ) q^{89} + 11 \beta q^{91} + ( -3 - 3 \beta ) q^{93} + \beta q^{97} + ( 1 - 3 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - q^{7} + q^{9} + O(q^{10})$$ $$2 q - 3 q^{3} - q^{7} + q^{9} - 2 q^{11} + 8 q^{13} + q^{17} - 6 q^{21} - 3 q^{23} - 5 q^{29} + 6 q^{31} + 3 q^{33} + 16 q^{37} - 17 q^{39} - 6 q^{41} + 12 q^{43} - 6 q^{47} + 9 q^{49} + q^{51} + 3 q^{53} - 15 q^{57} - 10 q^{59} - 11 q^{61} + 22 q^{63} - 16 q^{67} - 8 q^{69} + 6 q^{71} + 23 q^{73} + q^{77} - 5 q^{79} + 2 q^{81} + 27 q^{83} + 5 q^{87} - 25 q^{89} + 11 q^{91} - 9 q^{93} + q^{97} - q^{99} + O(q^{100})$$

## 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 −2.61803 0 0 0 2.85410 0 3.85410 0
1.2 0 −0.381966 0 0 0 −3.85410 0 −2.85410 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.bg 2
4.b odd 2 1 275.2.a.g yes 2
5.b even 2 1 4400.2.a.bv 2
5.c odd 4 2 4400.2.b.x 4
12.b even 2 1 2475.2.a.n 2
20.d odd 2 1 275.2.a.d 2
20.e even 4 2 275.2.b.e 4
44.c even 2 1 3025.2.a.i 2
60.h even 2 1 2475.2.a.s 2
60.l odd 4 2 2475.2.c.p 4
220.g even 2 1 3025.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.d 2 20.d odd 2 1
275.2.a.g yes 2 4.b odd 2 1
275.2.b.e 4 20.e even 4 2
2475.2.a.n 2 12.b even 2 1
2475.2.a.s 2 60.h even 2 1
2475.2.c.p 4 60.l odd 4 2
3025.2.a.i 2 44.c even 2 1
3025.2.a.m 2 220.g even 2 1
4400.2.a.bg 2 1.a even 1 1 trivial
4400.2.a.bv 2 5.b even 2 1
4400.2.b.x 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4400))$$:

 $$T_{3}^{2} + 3 T_{3} + 1$$ $$T_{7}^{2} + T_{7} - 11$$ $$T_{13}^{2} - 8 T_{13} + 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 + 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-11 + T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$11 - 8 T + T^{2}$$
$17$ $$-1 - T + T^{2}$$
$19$ $$-45 + T^{2}$$
$23$ $$-29 + 3 T + T^{2}$$
$29$ $$5 + 5 T + T^{2}$$
$31$ $$( -3 + T )^{2}$$
$37$ $$59 - 16 T + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$-71 + 6 T + T^{2}$$
$53$ $$-59 - 3 T + T^{2}$$
$59$ $$5 + 10 T + T^{2}$$
$61$ $$-1 + 11 T + T^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$-116 - 6 T + T^{2}$$
$73$ $$131 - 23 T + T^{2}$$
$79$ $$-5 + 5 T + T^{2}$$
$83$ $$171 - 27 T + T^{2}$$
$89$ $$125 + 25 T + T^{2}$$
$97$ $$-1 - T + T^{2}$$