# Properties

 Label 4400.2.b.i 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 110) 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.

 $$f(q)$$ $$=$$ $$q + i q^{3} + 3 i q^{7} + 2 q^{9}+O(q^{10})$$ q + i * q^3 + 3*i * q^7 + 2 * q^9 $$q + i q^{3} + 3 i q^{7} + 2 q^{9} - q^{11} - 6 i q^{13} + 7 i q^{17} + 5 q^{19} - 3 q^{21} + 6 i q^{23} + 5 i q^{27} - 5 q^{29} + 3 q^{31} - i q^{33} - 3 i q^{37} + 6 q^{39} + 2 q^{41} - 4 i q^{43} - 2 i q^{47} - 2 q^{49} - 7 q^{51} - i q^{53} + 5 i q^{57} - 10 q^{59} + 7 q^{61} + 6 i q^{63} + 8 i q^{67} - 6 q^{69} - 7 q^{71} + 14 i q^{73} - 3 i q^{77} + 10 q^{79} + q^{81} + 6 i q^{83} - 5 i q^{87} + 15 q^{89} + 18 q^{91} + 3 i q^{93} + 12 i q^{97} - 2 q^{99} +O(q^{100})$$ q + i * q^3 + 3*i * q^7 + 2 * q^9 - q^11 - 6*i * q^13 + 7*i * q^17 + 5 * q^19 - 3 * q^21 + 6*i * q^23 + 5*i * q^27 - 5 * q^29 + 3 * q^31 - i * q^33 - 3*i * q^37 + 6 * q^39 + 2 * q^41 - 4*i * q^43 - 2*i * q^47 - 2 * q^49 - 7 * q^51 - i * q^53 + 5*i * q^57 - 10 * q^59 + 7 * q^61 + 6*i * q^63 + 8*i * q^67 - 6 * q^69 - 7 * q^71 + 14*i * q^73 - 3*i * q^77 + 10 * q^79 + q^81 + 6*i * q^83 - 5*i * q^87 + 15 * q^89 + 18 * q^91 + 3*i * q^93 + 12*i * q^97 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{9}+O(q^{10})$$ 2 * q + 4 * q^9 $$2 q + 4 q^{9} - 2 q^{11} + 10 q^{19} - 6 q^{21} - 10 q^{29} + 6 q^{31} + 12 q^{39} + 4 q^{41} - 4 q^{49} - 14 q^{51} - 20 q^{59} + 14 q^{61} - 12 q^{69} - 14 q^{71} + 20 q^{79} + 2 q^{81} + 30 q^{89} + 36 q^{91} - 4 q^{99}+O(q^{100})$$ 2 * q + 4 * q^9 - 2 * q^11 + 10 * q^19 - 6 * q^21 - 10 * q^29 + 6 * q^31 + 12 * q^39 + 4 * q^41 - 4 * q^49 - 14 * q^51 - 20 * q^59 + 14 * q^61 - 12 * q^69 - 14 * q^71 + 20 * q^79 + 2 * q^81 + 30 * q^89 + 36 * 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 3.00000i 0 2.00000 0
4049.2 0 1.00000i 0 0 0 3.00000i 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.i 2
4.b odd 2 1 550.2.b.a 2
5.b even 2 1 inner 4400.2.b.i 2
5.c odd 4 1 880.2.a.i 1
5.c odd 4 1 4400.2.a.l 1
12.b even 2 1 4950.2.c.m 2
15.e even 4 1 7920.2.a.d 1
20.d odd 2 1 550.2.b.a 2
20.e even 4 1 110.2.a.b 1
20.e even 4 1 550.2.a.f 1
40.i odd 4 1 3520.2.a.h 1
40.k even 4 1 3520.2.a.y 1
55.e even 4 1 9680.2.a.x 1
60.h even 2 1 4950.2.c.m 2
60.l odd 4 1 990.2.a.d 1
60.l odd 4 1 4950.2.a.bc 1
140.j odd 4 1 5390.2.a.bf 1
220.i odd 4 1 1210.2.a.b 1
220.i odd 4 1 6050.2.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 20.e even 4 1
550.2.a.f 1 20.e even 4 1
550.2.b.a 2 4.b odd 2 1
550.2.b.a 2 20.d odd 2 1
880.2.a.i 1 5.c odd 4 1
990.2.a.d 1 60.l odd 4 1
1210.2.a.b 1 220.i odd 4 1
3520.2.a.h 1 40.i odd 4 1
3520.2.a.y 1 40.k even 4 1
4400.2.a.l 1 5.c odd 4 1
4400.2.b.i 2 1.a even 1 1 trivial
4400.2.b.i 2 5.b even 2 1 inner
4950.2.a.bc 1 60.l odd 4 1
4950.2.c.m 2 12.b even 2 1
4950.2.c.m 2 60.h even 2 1
5390.2.a.bf 1 140.j odd 4 1
6050.2.a.bj 1 220.i odd 4 1
7920.2.a.d 1 15.e even 4 1
9680.2.a.x 1 55.e even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 1$$ T3^2 + 1 $$T_{7}^{2} + 9$$ T7^2 + 9 $$T_{13}^{2} + 36$$ T13^2 + 36 $$T_{17}^{2} + 49$$ T17^2 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 9$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 36$$
$17$ $$T^{2} + 49$$
$19$ $$(T - 5)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T + 5)^{2}$$
$31$ $$(T - 3)^{2}$$
$37$ $$T^{2} + 9$$
$41$ $$(T - 2)^{2}$$
$43$ $$T^{2} + 16$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 1$$
$59$ $$(T + 10)^{2}$$
$61$ $$(T - 7)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T + 7)^{2}$$
$73$ $$T^{2} + 196$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 15)^{2}$$
$97$ $$T^{2} + 144$$