# Properties

 Label 4400.2.b.b Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4400 = 2^{4} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4400.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$35.1341768894$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 88) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} -2 i q^{7} -6 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} -2 i q^{7} -6 q^{9} + q^{11} + 6 i q^{17} + 4 q^{19} + 6 q^{21} -i q^{23} -9 i q^{27} + 8 q^{29} + 7 q^{31} + 3 i q^{33} + i q^{37} + 4 q^{41} -6 i q^{43} -8 i q^{47} + 3 q^{49} -18 q^{51} + 2 i q^{53} + 12 i q^{57} - q^{59} + 4 q^{61} + 12 i q^{63} -5 i q^{67} + 3 q^{69} -3 q^{71} + 16 i q^{73} -2 i q^{77} + 2 q^{79} + 9 q^{81} + 2 i q^{83} + 24 i q^{87} -15 q^{89} + 21 i q^{93} + 7 i q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 12 q^{9} + O(q^{10})$$ $$2 q - 12 q^{9} + 2 q^{11} + 8 q^{19} + 12 q^{21} + 16 q^{29} + 14 q^{31} + 8 q^{41} + 6 q^{49} - 36 q^{51} - 2 q^{59} + 8 q^{61} + 6 q^{69} - 6 q^{71} + 4 q^{79} + 18 q^{81} - 30 q^{89} - 12 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$$.

 $$n$$ $$177$$ $$1201$$ $$2751$$ $$3301$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 2.00000i 0 −6.00000 0
4049.2 0 3.00000i 0 0 0 2.00000i 0 −6.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4400.2.b.b 2
4.b odd 2 1 2200.2.b.a 2
5.b even 2 1 inner 4400.2.b.b 2
5.c odd 4 1 176.2.a.c 1
5.c odd 4 1 4400.2.a.a 1
15.e even 4 1 1584.2.a.q 1
20.d odd 2 1 2200.2.b.a 2
20.e even 4 1 88.2.a.a 1
20.e even 4 1 2200.2.a.k 1
35.f even 4 1 8624.2.a.c 1
40.i odd 4 1 704.2.a.b 1
40.k even 4 1 704.2.a.l 1
55.e even 4 1 1936.2.a.l 1
60.l odd 4 1 792.2.a.g 1
80.i odd 4 1 2816.2.c.d 2
80.j even 4 1 2816.2.c.i 2
80.s even 4 1 2816.2.c.i 2
80.t odd 4 1 2816.2.c.d 2
120.q odd 4 1 6336.2.a.h 1
120.w even 4 1 6336.2.a.k 1
140.j odd 4 1 4312.2.a.l 1
220.i odd 4 1 968.2.a.a 1
220.v even 20 4 968.2.i.j 4
220.w odd 20 4 968.2.i.i 4
440.t even 4 1 7744.2.a.b 1
440.w odd 4 1 7744.2.a.bk 1
660.q even 4 1 8712.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
88.2.a.a 1 20.e even 4 1
176.2.a.c 1 5.c odd 4 1
704.2.a.b 1 40.i odd 4 1
704.2.a.l 1 40.k even 4 1
792.2.a.g 1 60.l odd 4 1
968.2.a.a 1 220.i odd 4 1
968.2.i.i 4 220.w odd 20 4
968.2.i.j 4 220.v even 20 4
1584.2.a.q 1 15.e even 4 1
1936.2.a.l 1 55.e even 4 1
2200.2.a.k 1 20.e even 4 1
2200.2.b.a 2 4.b odd 2 1
2200.2.b.a 2 20.d odd 2 1
2816.2.c.d 2 80.i odd 4 1
2816.2.c.d 2 80.t odd 4 1
2816.2.c.i 2 80.j even 4 1
2816.2.c.i 2 80.s even 4 1
4312.2.a.l 1 140.j odd 4 1
4400.2.a.a 1 5.c odd 4 1
4400.2.b.b 2 1.a even 1 1 trivial
4400.2.b.b 2 5.b even 2 1 inner
6336.2.a.h 1 120.q odd 4 1
6336.2.a.k 1 120.w even 4 1
7744.2.a.b 1 440.t even 4 1
7744.2.a.bk 1 440.w odd 4 1
8624.2.a.c 1 35.f even 4 1
8712.2.a.x 1 660.q even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 9$$ $$T_{7}^{2} + 4$$ $$T_{13}$$ $$T_{17}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$T^{2}$$
$17$ $$36 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$1 + T^{2}$$
$29$ $$( -8 + T )^{2}$$
$31$ $$( -7 + T )^{2}$$
$37$ $$1 + T^{2}$$
$41$ $$( -4 + T )^{2}$$
$43$ $$36 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$4 + T^{2}$$
$59$ $$( 1 + T )^{2}$$
$61$ $$( -4 + T )^{2}$$
$67$ $$25 + T^{2}$$
$71$ $$( 3 + T )^{2}$$
$73$ $$256 + T^{2}$$
$79$ $$( -2 + T )^{2}$$
$83$ $$4 + T^{2}$$
$89$ $$( 15 + T )^{2}$$
$97$ $$49 + T^{2}$$