# Properties

 Label 4400.2.b.f 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: $$2$$ Twist minimal: no (minimal twist has level 220) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 2 \beta q^{7} - q^{9} +O(q^{10})$$ q + b * q^3 - 2*b * q^7 - q^9 $$q + \beta q^{3} - 2 \beta q^{7} - q^{9} + q^{11} - 2 \beta q^{13} - 4 q^{19} + 8 q^{21} + 3 \beta q^{23} + 2 \beta q^{27} + 6 q^{29} - 8 q^{31} + \beta q^{33} - \beta q^{37} + 8 q^{39} + 6 q^{41} - 4 \beta q^{43} + 3 \beta q^{47} - 9 q^{49} - 3 \beta q^{53} - 4 \beta q^{57} - 12 q^{59} + 2 q^{61} + 2 \beta q^{63} - 5 \beta q^{67} - 12 q^{69} + 12 q^{71} - 8 \beta q^{73} - 2 \beta q^{77} + 8 q^{79} - 11 q^{81} + 6 \beta q^{87} - 6 q^{89} - 16 q^{91} - 8 \beta q^{93} - 7 \beta q^{97} - q^{99} +O(q^{100})$$ q + b * q^3 - 2*b * q^7 - q^9 + q^11 - 2*b * q^13 - 4 * q^19 + 8 * q^21 + 3*b * q^23 + 2*b * q^27 + 6 * q^29 - 8 * q^31 + b * q^33 - b * q^37 + 8 * q^39 + 6 * q^41 - 4*b * q^43 + 3*b * q^47 - 9 * q^49 - 3*b * q^53 - 4*b * q^57 - 12 * q^59 + 2 * q^61 + 2*b * q^63 - 5*b * q^67 - 12 * q^69 + 12 * q^71 - 8*b * q^73 - 2*b * q^77 + 8 * q^79 - 11 * q^81 + 6*b * q^87 - 6 * q^89 - 16 * q^91 - 8*b * q^93 - 7*b * q^97 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^9 $$2 q - 2 q^{9} + 2 q^{11} - 8 q^{19} + 16 q^{21} + 12 q^{29} - 16 q^{31} + 16 q^{39} + 12 q^{41} - 18 q^{49} - 24 q^{59} + 4 q^{61} - 24 q^{69} + 24 q^{71} + 16 q^{79} - 22 q^{81} - 12 q^{89} - 32 q^{91} - 2 q^{99}+O(q^{100})$$ 2 * q - 2 * q^9 + 2 * q^11 - 8 * q^19 + 16 * q^21 + 12 * q^29 - 16 * q^31 + 16 * q^39 + 12 * q^41 - 18 * q^49 - 24 * q^59 + 4 * q^61 - 24 * q^69 + 24 * q^71 + 16 * q^79 - 22 * q^81 - 12 * q^89 - 32 * q^91 - 2 * 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 2.00000i 0 0 0 4.00000i 0 −1.00000 0
4049.2 0 2.00000i 0 0 0 4.00000i 0 −1.00000 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.f 2
4.b odd 2 1 1100.2.b.a 2
5.b even 2 1 inner 4400.2.b.f 2
5.c odd 4 1 880.2.a.j 1
5.c odd 4 1 4400.2.a.e 1
12.b even 2 1 9900.2.c.m 2
15.e even 4 1 7920.2.a.o 1
20.d odd 2 1 1100.2.b.a 2
20.e even 4 1 220.2.a.a 1
20.e even 4 1 1100.2.a.e 1
40.i odd 4 1 3520.2.a.d 1
40.k even 4 1 3520.2.a.bd 1
55.e even 4 1 9680.2.a.bb 1
60.h even 2 1 9900.2.c.m 2
60.l odd 4 1 1980.2.a.a 1
60.l odd 4 1 9900.2.a.bd 1
220.i odd 4 1 2420.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.a.a 1 20.e even 4 1
880.2.a.j 1 5.c odd 4 1
1100.2.a.e 1 20.e even 4 1
1100.2.b.a 2 4.b odd 2 1
1100.2.b.a 2 20.d odd 2 1
1980.2.a.a 1 60.l odd 4 1
2420.2.a.b 1 220.i odd 4 1
3520.2.a.d 1 40.i odd 4 1
3520.2.a.bd 1 40.k even 4 1
4400.2.a.e 1 5.c odd 4 1
4400.2.b.f 2 1.a even 1 1 trivial
4400.2.b.f 2 5.b even 2 1 inner
7920.2.a.o 1 15.e even 4 1
9680.2.a.bb 1 55.e even 4 1
9900.2.a.bd 1 60.l odd 4 1
9900.2.c.m 2 12.b even 2 1
9900.2.c.m 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 $$S_{2}^{\mathrm{new}}(4400, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 16$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 36$$
$29$ $$(T - 6)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 4$$
$41$ $$(T - 6)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 36$$
$53$ $$T^{2} + 36$$
$59$ $$(T + 12)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 100$$
$71$ $$(T - 12)^{2}$$
$73$ $$T^{2} + 256$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 6)^{2}$$
$97$ $$T^{2} + 196$$