# Properties

 Label 4400.2.b.k 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 44) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} - 2 i q^{7} + 2 q^{9} +O(q^{10})$$ q + i * q^3 - 2*i * q^7 + 2 * q^9 $$q + i q^{3} - 2 i q^{7} + 2 q^{9} + q^{11} + 4 i q^{13} + 6 i q^{17} + 8 q^{19} + 2 q^{21} - 3 i q^{23} + 5 i q^{27} - 5 q^{31} + i q^{33} - i q^{37} - 4 q^{39} - 10 i q^{43} + 3 q^{49} - 6 q^{51} + 6 i q^{53} + 8 i q^{57} + 3 q^{59} - 4 q^{61} - 4 i q^{63} + i q^{67} + 3 q^{69} - 15 q^{71} + 4 i q^{73} - 2 i q^{77} + 2 q^{79} + q^{81} + 6 i q^{83} + 9 q^{89} + 8 q^{91} - 5 i q^{93} - 7 i q^{97} + 2 q^{99} +O(q^{100})$$ q + i * q^3 - 2*i * q^7 + 2 * q^9 + q^11 + 4*i * q^13 + 6*i * q^17 + 8 * q^19 + 2 * q^21 - 3*i * q^23 + 5*i * q^27 - 5 * q^31 + i * q^33 - i * q^37 - 4 * q^39 - 10*i * q^43 + 3 * q^49 - 6 * q^51 + 6*i * q^53 + 8*i * q^57 + 3 * q^59 - 4 * q^61 - 4*i * q^63 + i * q^67 + 3 * q^69 - 15 * q^71 + 4*i * q^73 - 2*i * q^77 + 2 * q^79 + q^81 + 6*i * q^83 + 9 * q^89 + 8 * q^91 - 5*i * q^93 - 7*i * q^97 + 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} + 2 q^{11} + 16 q^{19} + 4 q^{21} - 10 q^{31} - 8 q^{39} + 6 q^{49} - 12 q^{51} + 6 q^{59} - 8 q^{61} + 6 q^{69} - 30 q^{71} + 4 q^{79} + 2 q^{81} + 18 q^{89} + 16 q^{91} + 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 + 2 * q^11 + 16 * q^19 + 4 * q^21 - 10 * q^31 - 8 * q^39 + 6 * q^49 - 12 * q^51 + 6 * q^59 - 8 * q^61 + 6 * q^69 - 30 * q^71 + 4 * q^79 + 2 * q^81 + 18 * q^89 + 16 * q^91 + 4 * q^99

## 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 1.00000i 0 0 0 2.00000i 0 2.00000 0
4049.2 0 1.00000i 0 0 0 2.00000i 0 2.00000 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.k 2
4.b odd 2 1 1100.2.b.c 2
5.b even 2 1 inner 4400.2.b.k 2
5.c odd 4 1 176.2.a.a 1
5.c odd 4 1 4400.2.a.v 1
12.b even 2 1 9900.2.c.g 2
15.e even 4 1 1584.2.a.p 1
20.d odd 2 1 1100.2.b.c 2
20.e even 4 1 44.2.a.a 1
20.e even 4 1 1100.2.a.b 1
35.f even 4 1 8624.2.a.w 1
40.i odd 4 1 704.2.a.i 1
40.k even 4 1 704.2.a.f 1
55.e even 4 1 1936.2.a.c 1
60.h even 2 1 9900.2.c.g 2
60.l odd 4 1 396.2.a.c 1
60.l odd 4 1 9900.2.a.h 1
80.i odd 4 1 2816.2.c.k 2
80.j even 4 1 2816.2.c.e 2
80.s even 4 1 2816.2.c.e 2
80.t odd 4 1 2816.2.c.k 2
120.q odd 4 1 6336.2.a.j 1
120.w even 4 1 6336.2.a.i 1
140.j odd 4 1 2156.2.a.a 1
140.w even 12 2 2156.2.i.b 2
140.x odd 12 2 2156.2.i.c 2
180.v odd 12 2 3564.2.i.a 2
180.x even 12 2 3564.2.i.j 2
220.i odd 4 1 484.2.a.a 1
220.v even 20 4 484.2.e.a 4
220.w odd 20 4 484.2.e.b 4
260.p even 4 1 7436.2.a.d 1
440.t even 4 1 7744.2.a.bc 1
440.w odd 4 1 7744.2.a.m 1
660.q even 4 1 4356.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.2.a.a 1 20.e even 4 1
176.2.a.a 1 5.c odd 4 1
396.2.a.c 1 60.l odd 4 1
484.2.a.a 1 220.i odd 4 1
484.2.e.a 4 220.v even 20 4
484.2.e.b 4 220.w odd 20 4
704.2.a.f 1 40.k even 4 1
704.2.a.i 1 40.i odd 4 1
1100.2.a.b 1 20.e even 4 1
1100.2.b.c 2 4.b odd 2 1
1100.2.b.c 2 20.d odd 2 1
1584.2.a.p 1 15.e even 4 1
1936.2.a.c 1 55.e even 4 1
2156.2.a.a 1 140.j odd 4 1
2156.2.i.b 2 140.w even 12 2
2156.2.i.c 2 140.x odd 12 2
2816.2.c.e 2 80.j even 4 1
2816.2.c.e 2 80.s even 4 1
2816.2.c.k 2 80.i odd 4 1
2816.2.c.k 2 80.t odd 4 1
3564.2.i.a 2 180.v odd 12 2
3564.2.i.j 2 180.x even 12 2
4356.2.a.j 1 660.q even 4 1
4400.2.a.v 1 5.c odd 4 1
4400.2.b.k 2 1.a even 1 1 trivial
4400.2.b.k 2 5.b even 2 1 inner
6336.2.a.i 1 120.w even 4 1
6336.2.a.j 1 120.q odd 4 1
7436.2.a.d 1 260.p even 4 1
7744.2.a.m 1 440.w odd 4 1
7744.2.a.bc 1 440.t even 4 1
8624.2.a.w 1 35.f even 4 1
9900.2.a.h 1 60.l odd 4 1
9900.2.c.g 2 12.b even 2 1
9900.2.c.g 2 60.h even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{13}^{2} + 16$$ T13^2 + 16 $$T_{17}^{2} + 36$$ T17^2 + 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 36$$
$19$ $$(T - 8)^{2}$$
$23$ $$T^{2} + 9$$
$29$ $$T^{2}$$
$31$ $$(T + 5)^{2}$$
$37$ $$T^{2} + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 100$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 3)^{2}$$
$61$ $$(T + 4)^{2}$$
$67$ $$T^{2} + 1$$
$71$ $$(T + 15)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$(T - 2)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 9)^{2}$$
$97$ $$T^{2} + 49$$