# Properties

 Label 4400.2.b.ba 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2200) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{3} ) q^{7} + ( 2 + \beta_{2} ) q^{9} + q^{11} + ( 2 \beta_{1} + \beta_{3} ) q^{13} -3 \beta_{1} q^{17} + ( 1 + 4 \beta_{2} ) q^{19} + q^{21} + ( -7 \beta_{1} - 5 \beta_{3} ) q^{23} + ( 4 \beta_{1} + \beta_{3} ) q^{27} + ( 6 + \beta_{2} ) q^{29} + ( 1 - 2 \beta_{2} ) q^{31} + \beta_{1} q^{33} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{37} + ( -2 + \beta_{2} ) q^{39} + ( -5 - 8 \beta_{2} ) q^{41} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{43} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{47} + ( 5 - \beta_{2} ) q^{49} + ( 3 - 3 \beta_{2} ) q^{51} + ( -3 \beta_{1} - \beta_{3} ) q^{53} + ( -3 \beta_{1} + 4 \beta_{3} ) q^{57} + ( -7 - 2 \beta_{2} ) q^{59} + ( -1 - 5 \beta_{2} ) q^{61} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{63} + ( 7 - 2 \beta_{2} ) q^{69} + ( 10 + 6 \beta_{2} ) q^{71} + ( -5 \beta_{1} - \beta_{3} ) q^{73} + ( -\beta_{1} - \beta_{3} ) q^{77} + ( -2 - 3 \beta_{2} ) q^{79} + ( 2 + 6 \beta_{2} ) q^{81} + ( -3 \beta_{1} - 6 \beta_{3} ) q^{83} + ( 5 \beta_{1} + \beta_{3} ) q^{87} + ( -2 - 9 \beta_{2} ) q^{89} + ( 3 + \beta_{2} ) q^{91} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{93} + ( 5 \beta_{1} + 13 \beta_{3} ) q^{97} + ( 2 + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{9} + O(q^{10})$$ $$4 q + 6 q^{9} + 4 q^{11} - 4 q^{19} + 4 q^{21} + 22 q^{29} + 8 q^{31} - 10 q^{39} - 4 q^{41} + 22 q^{49} + 18 q^{51} - 24 q^{59} + 6 q^{61} + 32 q^{69} + 28 q^{71} - 2 q^{79} - 4 q^{81} + 10 q^{89} + 10 q^{91} + 6 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2 \beta_{1}$$

## 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
 − 1.61803i − 0.618034i 0.618034i 1.61803i
0 1.61803i 0 0 0 0.618034i 0 0.381966 0
4049.2 0 0.618034i 0 0 0 1.61803i 0 2.61803 0
4049.3 0 0.618034i 0 0 0 1.61803i 0 2.61803 0
4049.4 0 1.61803i 0 0 0 0.618034i 0 0.381966 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.ba 4
4.b odd 2 1 2200.2.b.j 4
5.b even 2 1 inner 4400.2.b.ba 4
5.c odd 4 1 4400.2.a.bk 2
5.c odd 4 1 4400.2.a.bq 2
20.d odd 2 1 2200.2.b.j 4
20.e even 4 1 2200.2.a.n 2
20.e even 4 1 2200.2.a.r yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.2.a.n 2 20.e even 4 1
2200.2.a.r yes 2 20.e even 4 1
2200.2.b.j 4 4.b odd 2 1
2200.2.b.j 4 20.d odd 2 1
4400.2.a.bk 2 5.c odd 4 1
4400.2.a.bq 2 5.c odd 4 1
4400.2.b.ba 4 1.a even 1 1 trivial
4400.2.b.ba 4 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{4} + 3 T_{3}^{2} + 1$$ $$T_{7}^{4} + 3 T_{7}^{2} + 1$$ $$T_{13}^{2} + 5$$ $$T_{17}^{4} + 27 T_{17}^{2} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1 + 3 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$1 + 3 T^{2} + T^{4}$$
$11$ $$( -1 + T )^{4}$$
$13$ $$( 5 + T^{2} )^{2}$$
$17$ $$81 + 27 T^{2} + T^{4}$$
$19$ $$( -19 + 2 T + T^{2} )^{2}$$
$23$ $$3481 + 127 T^{2} + T^{4}$$
$29$ $$( 29 - 11 T + T^{2} )^{2}$$
$31$ $$( -1 - 4 T + T^{2} )^{2}$$
$37$ $$121 + 42 T^{2} + T^{4}$$
$41$ $$( -79 + 2 T + T^{2} )^{2}$$
$43$ $$16 + 72 T^{2} + T^{4}$$
$47$ $$1681 + 98 T^{2} + T^{4}$$
$53$ $$121 + 23 T^{2} + T^{4}$$
$59$ $$( 31 + 12 T + T^{2} )^{2}$$
$61$ $$( -29 - 3 T + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( 4 - 14 T + T^{2} )^{2}$$
$73$ $$841 + 67 T^{2} + T^{4}$$
$79$ $$( -11 + T + T^{2} )^{2}$$
$83$ $$81 + 63 T^{2} + T^{4}$$
$89$ $$( -95 - 5 T + T^{2} )^{2}$$
$97$ $$6241 + 283 T^{2} + T^{4}$$