# Properties

 Label 1600.2.o.k Level $1600$ Weight $2$ Character orbit 1600.o Analytic conductor $12.776$ Analytic rank $0$ Dimension $8$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.o (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.12745506816.1 Defining polynomial: $$x^{8} + 23 x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} -2 \beta_{6} q^{7} -4 \beta_{3} q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{3} -2 \beta_{6} q^{7} -4 \beta_{3} q^{9} -\beta_{4} q^{11} -2 \beta_{2} q^{13} + \beta_{1} q^{17} + \beta_{5} q^{19} + 2 \beta_{5} q^{21} + 4 \beta_{1} q^{23} + \beta_{2} q^{27} + 2 \beta_{4} q^{29} -4 \beta_{3} q^{31} + 7 \beta_{6} q^{33} + 2 \beta_{7} q^{37} + 14 q^{39} -9 q^{41} + 2 \beta_{7} q^{43} + 2 \beta_{6} q^{47} -5 \beta_{3} q^{49} -\beta_{4} q^{51} + 7 \beta_{1} q^{57} -2 \beta_{5} q^{61} + 8 \beta_{1} q^{63} -\beta_{2} q^{67} -4 \beta_{4} q^{69} + 6 \beta_{3} q^{71} + 5 \beta_{6} q^{73} -6 \beta_{7} q^{77} + 10 q^{79} + 5 q^{81} -3 \beta_{7} q^{83} -14 \beta_{6} q^{87} -15 \beta_{3} q^{89} + 4 \beta_{4} q^{91} + 4 \beta_{2} q^{93} + 4 \beta_{1} q^{97} -4 \beta_{5} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q + 112q^{39} - 72q^{41} + 80q^{79} + 40q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 23 x^{4} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{5} + 19 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} + 29 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 24 \nu^{2}$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$2 \nu^{4} + 23$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$-\nu^{6} - 22 \nu^{2}$$ $$\beta_{6}$$ $$=$$ $$($$$$-4 \nu^{7} - 91 \nu^{3}$$$$)/5$$ $$\beta_{7}$$ $$=$$ $$($$$$6 \nu^{7} + 139 \nu^{3}$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + 5 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{7} + 3 \beta_{6}$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{4} - 23$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-19 \beta_{2} + 29 \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-12 \beta_{5} - 55 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$($$$$-91 \beta_{7} - 139 \beta_{6}$$$$)/2$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$\beta_{3}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
543.1
 0.323042 − 0.323042i 1.54779 − 1.54779i −1.54779 + 1.54779i −0.323042 + 0.323042i 0.323042 + 0.323042i 1.54779 + 1.54779i −1.54779 − 1.54779i −0.323042 − 0.323042i
0 −1.87083 1.87083i 0 0 0 −2.44949 2.44949i 0 4.00000i 0
543.2 0 −1.87083 1.87083i 0 0 0 2.44949 + 2.44949i 0 4.00000i 0
543.3 0 1.87083 + 1.87083i 0 0 0 −2.44949 2.44949i 0 4.00000i 0
543.4 0 1.87083 + 1.87083i 0 0 0 2.44949 + 2.44949i 0 4.00000i 0
607.1 0 −1.87083 + 1.87083i 0 0 0 −2.44949 + 2.44949i 0 4.00000i 0
607.2 0 −1.87083 + 1.87083i 0 0 0 2.44949 2.44949i 0 4.00000i 0
607.3 0 1.87083 1.87083i 0 0 0 −2.44949 + 2.44949i 0 4.00000i 0
607.4 0 1.87083 1.87083i 0 0 0 2.44949 2.44949i 0 4.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 607.4 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
8.b even 2 1 inner
20.e even 4 2 inner
40.f even 2 1 inner
40.k even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.o.k yes 8
4.b odd 2 1 1600.2.o.f 8
5.b even 2 1 inner 1600.2.o.k yes 8
5.c odd 4 2 1600.2.o.f 8
8.b even 2 1 inner 1600.2.o.k yes 8
8.d odd 2 1 1600.2.o.f 8
20.d odd 2 1 1600.2.o.f 8
20.e even 4 2 inner 1600.2.o.k yes 8
40.e odd 2 1 1600.2.o.f 8
40.f even 2 1 inner 1600.2.o.k yes 8
40.i odd 4 2 1600.2.o.f 8
40.k even 4 2 inner 1600.2.o.k yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.o.f 8 4.b odd 2 1
1600.2.o.f 8 5.c odd 4 2
1600.2.o.f 8 8.d odd 2 1
1600.2.o.f 8 20.d odd 2 1
1600.2.o.f 8 40.e odd 2 1
1600.2.o.f 8 40.i odd 4 2
1600.2.o.k yes 8 1.a even 1 1 trivial
1600.2.o.k yes 8 5.b even 2 1 inner
1600.2.o.k yes 8 8.b even 2 1 inner
1600.2.o.k yes 8 20.e even 4 2 inner
1600.2.o.k yes 8 40.f even 2 1 inner
1600.2.o.k yes 8 40.k even 4 2 inner

## Hecke kernels

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

 $$T_{3}^{4} + 49$$ $$T_{7}^{4} + 144$$ $$T_{79} - 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 49 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$( 144 + T^{4} )^{2}$$
$11$ $$( -21 + T^{2} )^{4}$$
$13$ $$( 784 + T^{4} )^{2}$$
$17$ $$( 9 + T^{4} )^{2}$$
$19$ $$( 21 + T^{2} )^{4}$$
$23$ $$( 2304 + T^{4} )^{2}$$
$29$ $$( -84 + T^{2} )^{4}$$
$31$ $$( 16 + T^{2} )^{4}$$
$37$ $$( 784 + T^{4} )^{2}$$
$41$ $$( 9 + T )^{8}$$
$43$ $$( 784 + T^{4} )^{2}$$
$47$ $$( 144 + T^{4} )^{2}$$
$53$ $$T^{8}$$
$59$ $$T^{8}$$
$61$ $$( 84 + T^{2} )^{4}$$
$67$ $$( 49 + T^{4} )^{2}$$
$71$ $$( 36 + T^{2} )^{4}$$
$73$ $$( 5625 + T^{4} )^{2}$$
$79$ $$( -10 + T )^{8}$$
$83$ $$( 3969 + T^{4} )^{2}$$
$89$ $$( 225 + T^{2} )^{4}$$
$97$ $$( 2304 + T^{4} )^{2}$$