# Properties

 Label 4400.2.a.bj Level $4400$ Weight $2$ Character orbit 4400.a Self dual yes Analytic conductor $35.134$ Analytic rank $1$ 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: $$1$$ 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 440) 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.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} -\beta q^{7} + ( 1 + \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{3} -\beta q^{7} + ( 1 + \beta ) q^{9} - q^{11} -2 q^{13} + ( -2 + \beta ) q^{17} -\beta q^{19} + ( 4 + \beta ) q^{21} + 2 \beta q^{23} + ( -4 + \beta ) q^{27} + ( 2 + 3 \beta ) q^{29} + ( -4 - \beta ) q^{31} + \beta q^{33} + ( -2 + 3 \beta ) q^{37} + 2 \beta q^{39} + 2 q^{41} + 4 \beta q^{43} + ( -8 - 2 \beta ) q^{47} + ( -3 + \beta ) q^{49} + ( -4 + \beta ) q^{51} + ( -2 - \beta ) q^{53} + ( 4 + \beta ) q^{57} + ( 4 - 2 \beta ) q^{59} + ( 10 - 3 \beta ) q^{61} + ( -4 - 2 \beta ) q^{63} + ( -4 + 4 \beta ) q^{67} + ( -8 - 2 \beta ) q^{69} + ( 4 - 3 \beta ) q^{71} + 2 q^{73} + \beta q^{77} + 6 \beta q^{79} -7 q^{81} + 2 \beta q^{83} + ( -12 - 5 \beta ) q^{87} + ( -10 - \beta ) q^{89} + 2 \beta q^{91} + ( 4 + 5 \beta ) q^{93} + ( -2 - 2 \beta ) q^{97} + ( -1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} - q^{7} + 3 q^{9} + O(q^{10})$$ $$2 q - q^{3} - q^{7} + 3 q^{9} - 2 q^{11} - 4 q^{13} - 3 q^{17} - q^{19} + 9 q^{21} + 2 q^{23} - 7 q^{27} + 7 q^{29} - 9 q^{31} + q^{33} - q^{37} + 2 q^{39} + 4 q^{41} + 4 q^{43} - 18 q^{47} - 5 q^{49} - 7 q^{51} - 5 q^{53} + 9 q^{57} + 6 q^{59} + 17 q^{61} - 10 q^{63} - 4 q^{67} - 18 q^{69} + 5 q^{71} + 4 q^{73} + q^{77} + 6 q^{79} - 14 q^{81} + 2 q^{83} - 29 q^{87} - 21 q^{89} + 2 q^{91} + 13 q^{93} - 6 q^{97} - 3 q^{99} + O(q^{100})$$

## 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
 2.56155 −1.56155
0 −2.56155 0 0 0 −2.56155 0 3.56155 0
1.2 0 1.56155 0 0 0 1.56155 0 −0.561553 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.bj 2
4.b odd 2 1 2200.2.a.s 2
5.b even 2 1 880.2.a.o 2
5.c odd 4 2 4400.2.b.t 4
15.d odd 2 1 7920.2.a.bu 2
20.d odd 2 1 440.2.a.e 2
20.e even 4 2 2200.2.b.i 4
40.e odd 2 1 3520.2.a.bp 2
40.f even 2 1 3520.2.a.bk 2
55.d odd 2 1 9680.2.a.bs 2
60.h even 2 1 3960.2.a.w 2
220.g even 2 1 4840.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
440.2.a.e 2 20.d odd 2 1
880.2.a.o 2 5.b even 2 1
2200.2.a.s 2 4.b odd 2 1
2200.2.b.i 4 20.e even 4 2
3520.2.a.bk 2 40.f even 2 1
3520.2.a.bp 2 40.e odd 2 1
3960.2.a.w 2 60.h even 2 1
4400.2.a.bj 2 1.a even 1 1 trivial
4400.2.b.t 4 5.c odd 4 2
4840.2.a.j 2 220.g even 2 1
7920.2.a.bu 2 15.d odd 2 1
9680.2.a.bs 2 55.d odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 + T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-4 + T + T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-2 + 3 T + T^{2}$$
$19$ $$-4 + T + T^{2}$$
$23$ $$-16 - 2 T + T^{2}$$
$29$ $$-26 - 7 T + T^{2}$$
$31$ $$16 + 9 T + T^{2}$$
$37$ $$-38 + T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-64 - 4 T + T^{2}$$
$47$ $$64 + 18 T + T^{2}$$
$53$ $$2 + 5 T + T^{2}$$
$59$ $$-8 - 6 T + T^{2}$$
$61$ $$34 - 17 T + T^{2}$$
$67$ $$-64 + 4 T + T^{2}$$
$71$ $$-32 - 5 T + T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$-144 - 6 T + T^{2}$$
$83$ $$-16 - 2 T + T^{2}$$
$89$ $$106 + 21 T + T^{2}$$
$97$ $$-8 + 6 T + T^{2}$$