# Properties

 Label 1600.2.n.g Level $1600$ Weight $2$ Character orbit 1600.n Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -3 i q^{9} +O(q^{10})$$ $$q -3 i q^{9} + ( -5 + 5 i ) q^{13} + ( 5 + 5 i ) q^{17} + 4 i q^{29} + ( 5 + 5 i ) q^{37} + 8 q^{41} + 7 i q^{49} + ( 5 - 5 i ) q^{53} + 12 q^{61} + ( -5 + 5 i ) q^{73} -9 q^{81} + 16 i q^{89} + ( -5 - 5 i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 10q^{13} + 10q^{17} + 10q^{37} + 16q^{41} + 10q^{53} + 24q^{61} - 10q^{73} - 18q^{81} - 10q^{97} + O(q^{100})$$

## 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 0 0 0 0 0 0 3.00000i 0
1407.1 0 0 0 0 0 0 0 3.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
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.g 2
4.b odd 2 1 CM 1600.2.n.g 2
5.b even 2 1 320.2.n.d 2
5.c odd 4 1 320.2.n.d 2
5.c odd 4 1 inner 1600.2.n.g 2
8.b even 2 1 400.2.n.a 2
8.d odd 2 1 400.2.n.a 2
20.d odd 2 1 320.2.n.d 2
20.e even 4 1 320.2.n.d 2
20.e even 4 1 inner 1600.2.n.g 2
24.f even 2 1 3600.2.x.c 2
24.h odd 2 1 3600.2.x.c 2
40.e odd 2 1 80.2.n.a 2
40.f even 2 1 80.2.n.a 2
40.i odd 4 1 80.2.n.a 2
40.i odd 4 1 400.2.n.a 2
40.k even 4 1 80.2.n.a 2
40.k even 4 1 400.2.n.a 2
80.i odd 4 1 1280.2.o.h 2
80.j even 4 1 1280.2.o.i 2
80.k odd 4 1 1280.2.o.h 2
80.k odd 4 1 1280.2.o.i 2
80.q even 4 1 1280.2.o.h 2
80.q even 4 1 1280.2.o.i 2
80.s even 4 1 1280.2.o.h 2
80.t odd 4 1 1280.2.o.i 2
120.i odd 2 1 720.2.x.a 2
120.m even 2 1 720.2.x.a 2
120.q odd 4 1 720.2.x.a 2
120.q odd 4 1 3600.2.x.c 2
120.w even 4 1 720.2.x.a 2
120.w even 4 1 3600.2.x.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.a 2 40.e odd 2 1
80.2.n.a 2 40.f even 2 1
80.2.n.a 2 40.i odd 4 1
80.2.n.a 2 40.k even 4 1
320.2.n.d 2 5.b even 2 1
320.2.n.d 2 5.c odd 4 1
320.2.n.d 2 20.d odd 2 1
320.2.n.d 2 20.e even 4 1
400.2.n.a 2 8.b even 2 1
400.2.n.a 2 8.d odd 2 1
400.2.n.a 2 40.i odd 4 1
400.2.n.a 2 40.k even 4 1
720.2.x.a 2 120.i odd 2 1
720.2.x.a 2 120.m even 2 1
720.2.x.a 2 120.q odd 4 1
720.2.x.a 2 120.w even 4 1
1280.2.o.h 2 80.i odd 4 1
1280.2.o.h 2 80.k odd 4 1
1280.2.o.h 2 80.q even 4 1
1280.2.o.h 2 80.s even 4 1
1280.2.o.i 2 80.j even 4 1
1280.2.o.i 2 80.k odd 4 1
1280.2.o.i 2 80.q even 4 1
1280.2.o.i 2 80.t odd 4 1
1600.2.n.g 2 1.a even 1 1 trivial
1600.2.n.g 2 4.b odd 2 1 CM
1600.2.n.g 2 5.c odd 4 1 inner
1600.2.n.g 2 20.e even 4 1 inner
3600.2.x.c 2 24.f even 2 1
3600.2.x.c 2 24.h odd 2 1
3600.2.x.c 2 120.q odd 4 1
3600.2.x.c 2 120.w even 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{7}$$ $$T_{11}$$ $$T_{13}^{2} + 10 T_{13} + 50$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$50 + 10 T + T^{2}$$
$17$ $$50 - 10 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$16 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$50 - 10 T + T^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$50 - 10 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -12 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$50 + 10 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$256 + T^{2}$$
$97$ $$50 + 10 T + T^{2}$$