# Properties

 Label 1600.2.l.e Level $1600$ Weight $2$ Character orbit 1600.l Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 400) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{2} ) q^{3} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( -1 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{9} + ( -2 + \beta_{1} + \beta_{3} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( 2 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{17} + ( -2 - \beta_{1} - \beta_{2} ) q^{19} + ( 5 - 5 \beta_{1} ) q^{21} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + ( -6 + 5 \beta_{1} + \beta_{3} ) q^{27} + ( 2 + 2 \beta_{1} ) q^{29} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{31} + ( -7 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{33} + ( -3 + 5 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -2 + 12 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{39} -5 \beta_{1} q^{41} + ( -3 - \beta_{1} + 4 \beta_{3} ) q^{43} -8 q^{47} + ( -5 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{49} + ( -5 - 4 \beta_{1} - \beta_{2} ) q^{51} + ( 1 + \beta_{1} - 2 \beta_{3} ) q^{53} + ( 1 + 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{57} + ( -1 + 3 \beta_{1} - 2 \beta_{3} ) q^{59} + ( 1 + 5 \beta_{1} - 4 \beta_{2} ) q^{61} + ( 8 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{63} + ( 7 + 8 \beta_{1} - \beta_{2} ) q^{67} + ( 3 - 5 \beta_{1} + 2 \beta_{3} ) q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{71} + ( 1 + 10 \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( 3 + 5 \beta_{1} - 2 \beta_{2} ) q^{77} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{79} + ( -2 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{81} + ( -5 - 6 \beta_{1} + \beta_{2} ) q^{83} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{87} + ( 3 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{89} + ( 10 - 10 \beta_{1} ) q^{91} + ( 5 + 5 \beta_{1} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{97} + ( -7 - 13 \beta_{1} + 6 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + O(q^{10})$$ $$4q + 2q^{3} - 6q^{11} + 4q^{13} + 8q^{17} - 6q^{19} + 20q^{21} - 22q^{27} + 8q^{29} - 4q^{31} - 28q^{33} - 16q^{37} - 4q^{43} - 32q^{47} - 20q^{49} - 18q^{51} - 8q^{59} + 12q^{61} + 32q^{63} + 30q^{67} + 16q^{69} + 16q^{77} - 4q^{79} - 8q^{81} - 22q^{83} + 40q^{91} + 20q^{93} - 40q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 2 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{2} + \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} + 6$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 1.65831 + 0.500000i −1.65831 + 0.500000i 1.65831 − 0.500000i −1.65831 − 0.500000i
0 −1.15831 1.15831i 0 0 0 4.31662i 0 0.316625i 0
401.2 0 2.15831 + 2.15831i 0 0 0 2.31662i 0 6.31662i 0
1201.1 0 −1.15831 + 1.15831i 0 0 0 4.31662i 0 0.316625i 0
1201.2 0 2.15831 2.15831i 0 0 0 2.31662i 0 6.31662i 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
16.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.l.e 4
4.b odd 2 1 400.2.l.d 4
5.b even 2 1 1600.2.l.d 4
5.c odd 4 1 1600.2.q.c 4
5.c odd 4 1 1600.2.q.d 4
16.e even 4 1 inner 1600.2.l.e 4
16.f odd 4 1 400.2.l.d 4
20.d odd 2 1 400.2.l.e yes 4
20.e even 4 1 400.2.q.c 4
20.e even 4 1 400.2.q.d 4
80.i odd 4 1 1600.2.q.c 4
80.j even 4 1 400.2.q.d 4
80.k odd 4 1 400.2.l.e yes 4
80.q even 4 1 1600.2.l.d 4
80.s even 4 1 400.2.q.c 4
80.t odd 4 1 1600.2.q.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.l.d 4 4.b odd 2 1
400.2.l.d 4 16.f odd 4 1
400.2.l.e yes 4 20.d odd 2 1
400.2.l.e yes 4 80.k odd 4 1
400.2.q.c 4 20.e even 4 1
400.2.q.c 4 80.s even 4 1
400.2.q.d 4 20.e even 4 1
400.2.q.d 4 80.j even 4 1
1600.2.l.d 4 5.b even 2 1
1600.2.l.d 4 80.q even 4 1
1600.2.l.e 4 1.a even 1 1 trivial
1600.2.l.e 4 16.e even 4 1 inner
1600.2.q.c 4 5.c odd 4 1
1600.2.q.c 4 80.i odd 4 1
1600.2.q.d 4 5.c odd 4 1
1600.2.q.d 4 80.t odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{4} - 2 T_{3}^{3} + 2 T_{3}^{2} + 10 T_{3} + 25$$ $$T_{7}^{4} + 24 T_{7}^{2} + 100$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$25 + 10 T + 2 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$100 + 24 T^{2} + T^{4}$$
$11$ $$1 - 6 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$13$ $$400 + 80 T + 8 T^{2} - 4 T^{3} + T^{4}$$
$17$ $$( -7 - 4 T + T^{2} )^{2}$$
$19$ $$1 - 6 T + 18 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$4 + 40 T^{2} + T^{4}$$
$29$ $$( 8 - 4 T + T^{2} )^{2}$$
$31$ $$( -10 + 2 T + T^{2} )^{2}$$
$37$ $$100 + 160 T + 128 T^{2} + 16 T^{3} + T^{4}$$
$41$ $$( 25 + T^{2} )^{2}$$
$43$ $$7396 - 344 T + 8 T^{2} + 4 T^{3} + T^{4}$$
$47$ $$( 8 + T )^{4}$$
$53$ $$484 + T^{4}$$
$59$ $$196 - 112 T + 32 T^{2} + 8 T^{3} + T^{4}$$
$61$ $$4900 + 840 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$11449 - 3210 T + 450 T^{2} - 30 T^{3} + T^{4}$$
$71$ $$1600 + 96 T^{2} + T^{4}$$
$73$ $$7921 + 222 T^{2} + T^{4}$$
$79$ $$( -10 + 2 T + T^{2} )^{2}$$
$83$ $$3025 + 1210 T + 242 T^{2} + 22 T^{3} + T^{4}$$
$89$ $$3969 + 270 T^{2} + T^{4}$$
$97$ $$( -44 + T^{2} )^{2}$$