# Properties

 Label 1600.2.n.v Level $1600$ Weight $2$ Character orbit 1600.n 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.n (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.3317760000.5 Defining polynomial: $$x^{8} - 7 x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 100) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + 2 \beta_{3} q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + 2 \beta_{3} q^{9} + \beta_{7} q^{11} + 2 \beta_{4} q^{13} -\beta_{1} q^{17} -\beta_{5} q^{19} + 2 \beta_{2} q^{23} + \beta_{6} q^{27} + 6 \beta_{3} q^{29} + 2 \beta_{7} q^{31} + 5 \beta_{4} q^{33} -4 \beta_{1} q^{37} -2 \beta_{5} q^{39} -3 q^{41} + 2 \beta_{6} q^{47} + 7 \beta_{3} q^{49} -\beta_{7} q^{51} + 2 \beta_{4} q^{53} -5 \beta_{1} q^{57} + 2 \beta_{5} q^{59} + 8 q^{61} -3 \beta_{6} q^{67} + 10 \beta_{3} q^{69} -2 \beta_{7} q^{71} + 3 \beta_{4} q^{73} + 4 \beta_{5} q^{79} + 11 q^{81} -\beta_{2} q^{83} -6 \beta_{6} q^{87} + 9 \beta_{3} q^{89} + 10 \beta_{4} q^{93} + 4 \beta_{1} q^{97} -2 \beta_{5} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{41} + 64q^{61} + 88q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 7 x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{7} + \nu^{3}$$$$)/8$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{5} - \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} - 3 \nu^{2}$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{5} - 5 \nu$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} + 11 \nu^{2}$$$$)/4$$ $$\beta_{6}$$ $$=$$ $$($$$$-3 \nu^{7} + 13 \nu^{3}$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$2 \nu^{4} - 7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{6} + 3 \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$\beta_{7} + 7$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{4} + 5 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$($$$$3 \beta_{5} + 11 \beta_{3}$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{6} + 13 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1343.1
 −1.40294 + 0.178197i −0.178197 + 1.40294i 1.40294 − 0.178197i 0.178197 − 1.40294i −0.178197 − 1.40294i −1.40294 − 0.178197i 0.178197 + 1.40294i 1.40294 + 0.178197i
0 −1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
1343.2 0 −1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
1343.3 0 1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1343.4 0 1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1407.1 0 −1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1407.2 0 −1.58114 1.58114i 0 0 0 0 0 2.00000i 0
1407.3 0 1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
1407.4 0 1.58114 + 1.58114i 0 0 0 0 0 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1407.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e 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.n.v 8
4.b odd 2 1 inner 1600.2.n.v 8
5.b even 2 1 inner 1600.2.n.v 8
5.c odd 4 2 inner 1600.2.n.v 8
8.b even 2 1 100.2.e.d 8
8.d odd 2 1 100.2.e.d 8
20.d odd 2 1 inner 1600.2.n.v 8
20.e even 4 2 inner 1600.2.n.v 8
24.f even 2 1 900.2.k.j 8
24.h odd 2 1 900.2.k.j 8
40.e odd 2 1 100.2.e.d 8
40.f even 2 1 100.2.e.d 8
40.i odd 4 2 100.2.e.d 8
40.k even 4 2 100.2.e.d 8
120.i odd 2 1 900.2.k.j 8
120.m even 2 1 900.2.k.j 8
120.q odd 4 2 900.2.k.j 8
120.w even 4 2 900.2.k.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.e.d 8 8.b even 2 1
100.2.e.d 8 8.d odd 2 1
100.2.e.d 8 40.e odd 2 1
100.2.e.d 8 40.f even 2 1
100.2.e.d 8 40.i odd 4 2
100.2.e.d 8 40.k even 4 2
900.2.k.j 8 24.f even 2 1
900.2.k.j 8 24.h odd 2 1
900.2.k.j 8 120.i odd 2 1
900.2.k.j 8 120.m even 2 1
900.2.k.j 8 120.q odd 4 2
900.2.k.j 8 120.w even 4 2
1600.2.n.v 8 1.a even 1 1 trivial
1600.2.n.v 8 4.b odd 2 1 inner
1600.2.n.v 8 5.b even 2 1 inner
1600.2.n.v 8 5.c odd 4 2 inner
1600.2.n.v 8 20.d odd 2 1 inner
1600.2.n.v 8 20.e 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 $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{4} + 25$$ $$T_{7}$$ $$T_{11}^{2} + 15$$ $$T_{13}^{4} + 144$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 25 + T^{4} )^{2}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 15 + T^{2} )^{4}$$
$13$ $$( 144 + T^{4} )^{2}$$
$17$ $$( 9 + T^{4} )^{2}$$
$19$ $$( -15 + T^{2} )^{4}$$
$23$ $$( 400 + T^{4} )^{2}$$
$29$ $$( 36 + T^{2} )^{4}$$
$31$ $$( 60 + T^{2} )^{4}$$
$37$ $$( 2304 + T^{4} )^{2}$$
$41$ $$( 3 + T )^{8}$$
$43$ $$T^{8}$$
$47$ $$( 400 + T^{4} )^{2}$$
$53$ $$( 144 + T^{4} )^{2}$$
$59$ $$( -60 + T^{2} )^{4}$$
$61$ $$( -8 + T )^{8}$$
$67$ $$( 2025 + T^{4} )^{2}$$
$71$ $$( 60 + T^{2} )^{4}$$
$73$ $$( 729 + T^{4} )^{2}$$
$79$ $$( -240 + T^{2} )^{4}$$
$83$ $$( 25 + T^{4} )^{2}$$
$89$ $$( 81 + T^{2} )^{4}$$
$97$ $$( 2304 + T^{4} )^{2}$$