# Properties

 Label 4400.2.b.q Level $4400$ Weight $2$ Character orbit 4400.b Analytic conductor $35.134$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 55) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{7} -5 q^{9} +O(q^{10})$$ $$q + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{7} -5 q^{9} - q^{11} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{13} + ( -2 \zeta_{8} - 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{21} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{29} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{33} + ( 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{37} + ( -8 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{39} + 6 q^{41} + 6 \zeta_{8}^{2} q^{43} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{47} + 3 q^{49} + ( 8 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{51} + ( 4 \zeta_{8} + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{53} + ( -4 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{59} + ( 2 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{61} + 10 \zeta_{8}^{2} q^{63} + ( 6 \zeta_{8} + 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{67} -8 q^{69} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{71} + ( 2 \zeta_{8} - 4 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{73} + 2 \zeta_{8}^{2} q^{77} + 4 q^{79} + q^{81} + 6 \zeta_{8}^{2} q^{83} + ( -4 \zeta_{8} + 16 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{87} + ( 2 + 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{89} + ( -8 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{91} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{97} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 20 q^{9} + O(q^{10})$$ $$4 q - 20 q^{9} - 4 q^{11} - 8 q^{29} - 32 q^{39} + 24 q^{41} + 12 q^{49} + 32 q^{51} - 16 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} + 4 q^{81} + 8 q^{89} - 32 q^{91} + 20 q^{99} + O(q^{100})$$

## 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
 −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
0 2.82843i 0 0 0 2.00000i 0 −5.00000 0
4049.2 0 2.82843i 0 0 0 2.00000i 0 −5.00000 0
4049.3 0 2.82843i 0 0 0 2.00000i 0 −5.00000 0
4049.4 0 2.82843i 0 0 0 2.00000i 0 −5.00000 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.q 4
4.b odd 2 1 275.2.b.d 4
5.b even 2 1 inner 4400.2.b.q 4
5.c odd 4 1 880.2.a.m 2
5.c odd 4 1 4400.2.a.bn 2
12.b even 2 1 2475.2.c.l 4
15.e even 4 1 7920.2.a.ch 2
20.d odd 2 1 275.2.b.d 4
20.e even 4 1 55.2.a.b 2
20.e even 4 1 275.2.a.c 2
40.i odd 4 1 3520.2.a.bo 2
40.k even 4 1 3520.2.a.bn 2
55.e even 4 1 9680.2.a.bn 2
60.h even 2 1 2475.2.c.l 4
60.l odd 4 1 495.2.a.b 2
60.l odd 4 1 2475.2.a.x 2
140.j odd 4 1 2695.2.a.f 2
220.i odd 4 1 605.2.a.d 2
220.i odd 4 1 3025.2.a.o 2
220.v even 20 4 605.2.g.f 8
220.w odd 20 4 605.2.g.l 8
260.p even 4 1 9295.2.a.g 2
660.q even 4 1 5445.2.a.y 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 20.e even 4 1
275.2.a.c 2 20.e even 4 1
275.2.b.d 4 4.b odd 2 1
275.2.b.d 4 20.d odd 2 1
495.2.a.b 2 60.l odd 4 1
605.2.a.d 2 220.i odd 4 1
605.2.g.f 8 220.v even 20 4
605.2.g.l 8 220.w odd 20 4
880.2.a.m 2 5.c odd 4 1
2475.2.a.x 2 60.l odd 4 1
2475.2.c.l 4 12.b even 2 1
2475.2.c.l 4 60.h even 2 1
2695.2.a.f 2 140.j odd 4 1
3025.2.a.o 2 220.i odd 4 1
3520.2.a.bn 2 40.k even 4 1
3520.2.a.bo 2 40.i odd 4 1
4400.2.a.bn 2 5.c odd 4 1
4400.2.b.q 4 1.a even 1 1 trivial
4400.2.b.q 4 5.b even 2 1 inner
5445.2.a.y 2 660.q even 4 1
7920.2.a.ch 2 15.e even 4 1
9295.2.a.g 2 260.p even 4 1
9680.2.a.bn 2 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 $$S_{2}^{\mathrm{new}}(4400, [\chi])$$:

 $$T_{3}^{2} + 8$$ $$T_{7}^{2} + 4$$ $$T_{13}^{4} + 48 T_{13}^{2} + 64$$ $$T_{17}^{4} + 48 T_{17}^{2} + 64$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 8 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 4 + T^{2} )^{2}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$64 + 48 T^{2} + T^{4}$$
$17$ $$64 + 48 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$( 8 + T^{2} )^{2}$$
$29$ $$( -28 + 4 T + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$784 + 72 T^{2} + T^{4}$$
$41$ $$( -6 + T )^{4}$$
$43$ $$( 36 + T^{2} )^{2}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$16 + 136 T^{2} + T^{4}$$
$59$ $$( -16 + 8 T + T^{2} )^{2}$$
$61$ $$( -124 - 4 T + T^{2} )^{2}$$
$67$ $$3136 + 176 T^{2} + T^{4}$$
$71$ $$( -128 + T^{2} )^{2}$$
$73$ $$64 + 48 T^{2} + T^{4}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( 36 + T^{2} )^{2}$$
$89$ $$( -124 - 4 T + T^{2} )^{2}$$
$97$ $$784 + 72 T^{2} + T^{4}$$