# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 800) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - i ) q^{3} + ( 1 - i ) q^{7} -i q^{9} +O(q^{10})$$ $$q + ( -1 - i ) q^{3} + ( 1 - i ) q^{7} -i q^{9} -4 i q^{11} + ( 4 - 4 i ) q^{13} + ( 4 + 4 i ) q^{17} -4 q^{19} -2 q^{21} + ( 5 + 5 i ) q^{23} + ( -4 + 4 i ) q^{27} + 2 i q^{29} -8 i q^{31} + ( -4 + 4 i ) q^{33} -8 q^{39} -4 q^{41} + ( -7 - 7 i ) q^{43} + ( 3 - 3 i ) q^{47} + 5 i q^{49} -8 i q^{51} + ( 4 - 4 i ) q^{53} + ( 4 + 4 i ) q^{57} -4 q^{59} + 8 q^{61} + ( -1 - i ) q^{63} + ( 3 - 3 i ) q^{67} -10 i q^{69} -16 i q^{71} + ( -4 + 4 i ) q^{73} + ( -4 - 4 i ) q^{77} -8 q^{79} + 5 q^{81} + ( 5 + 5 i ) q^{83} + ( 2 - 2 i ) q^{87} -10 i q^{89} -8 i q^{91} + ( -8 + 8 i ) q^{93} + ( -12 - 12 i ) q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} + 2q^{7} + O(q^{10})$$ $$2q - 2q^{3} + 2q^{7} + 8q^{13} + 8q^{17} - 8q^{19} - 4q^{21} + 10q^{23} - 8q^{27} - 8q^{33} - 16q^{39} - 8q^{41} - 14q^{43} + 6q^{47} + 8q^{53} + 8q^{57} - 8q^{59} + 16q^{61} - 2q^{63} + 6q^{67} - 8q^{73} - 8q^{77} - 16q^{79} + 10q^{81} + 10q^{83} + 4q^{87} - 16q^{93} - 24q^{97} - 8q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$i$$ $$1$$ $$-1$$

## 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
 − 1.00000i 1.00000i
0 −1.00000 + 1.00000i 0 0 0 1.00000 + 1.00000i 0 1.00000i 0
1407.1 0 −1.00000 1.00000i 0 0 0 1.00000 1.00000i 0 1.00000i 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
20.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.n.f 2
4.b odd 2 1 1600.2.n.k 2
5.b even 2 1 1600.2.n.i 2
5.c odd 4 1 1600.2.n.d 2
5.c odd 4 1 1600.2.n.k 2
8.b even 2 1 800.2.n.g yes 2
8.d odd 2 1 800.2.n.b 2
20.d odd 2 1 1600.2.n.d 2
20.e even 4 1 inner 1600.2.n.f 2
20.e even 4 1 1600.2.n.i 2
40.e odd 2 1 800.2.n.i yes 2
40.f even 2 1 800.2.n.d yes 2
40.i odd 4 1 800.2.n.b 2
40.i odd 4 1 800.2.n.i yes 2
40.k even 4 1 800.2.n.d yes 2
40.k even 4 1 800.2.n.g yes 2

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

## Hecke kernels

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

 $$T_{3}^{2} + 2 T_{3} + 2$$ $$T_{7}^{2} - 2 T_{7} + 2$$ $$T_{11}^{2} + 16$$ $$T_{13}^{2} - 8 T_{13} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$2 + 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$2 - 2 T + T^{2}$$
$11$ $$16 + T^{2}$$
$13$ $$32 - 8 T + T^{2}$$
$17$ $$32 - 8 T + T^{2}$$
$19$ $$( 4 + T )^{2}$$
$23$ $$50 - 10 T + T^{2}$$
$29$ $$4 + T^{2}$$
$31$ $$64 + T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 4 + T )^{2}$$
$43$ $$98 + 14 T + T^{2}$$
$47$ $$18 - 6 T + T^{2}$$
$53$ $$32 - 8 T + T^{2}$$
$59$ $$( 4 + T )^{2}$$
$61$ $$( -8 + T )^{2}$$
$67$ $$18 - 6 T + T^{2}$$
$71$ $$256 + T^{2}$$
$73$ $$32 + 8 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$50 - 10 T + T^{2}$$
$89$ $$100 + T^{2}$$
$97$ $$288 + 24 T + T^{2}$$