# Properties

 Label 1600.2.n.b Level $1600$ Weight $2$ Character orbit 1600.n Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM discriminant -20 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: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 100) 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.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{3} + (3 i - 3) q^{7} - i q^{9} +O(q^{10})$$ q + (-i - 1) * q^3 + (3*i - 3) * q^7 - i * q^9 $$q + ( - i - 1) q^{3} + (3 i - 3) q^{7} - i q^{9} + 6 q^{21} + (i + 1) q^{23} + (4 i - 4) q^{27} - 6 i q^{29} + 12 q^{41} + (9 i + 9) q^{43} + ( - 7 i + 7) q^{47} - 11 i q^{49} + 8 q^{61} + (3 i + 3) q^{63} + ( - 3 i + 3) q^{67} - 2 i q^{69} + 5 q^{81} + ( - 11 i - 11) q^{83} + (6 i - 6) q^{87} + 6 i q^{89} +O(q^{100})$$ q + (-i - 1) * q^3 + (3*i - 3) * q^7 - i * q^9 + 6 * q^21 + (i + 1) * q^23 + (4*i - 4) * q^27 - 6*i * q^29 + 12 * q^41 + (9*i + 9) * q^43 + (-7*i + 7) * q^47 - 11*i * q^49 + 8 * q^61 + (3*i + 3) * q^63 + (-3*i + 3) * q^67 - 2*i * q^69 + 5 * q^81 + (-11*i - 11) * q^83 + (6*i - 6) * q^87 + 6*i * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 6 q^{7}+O(q^{10})$$ 2 * q - 2 * q^3 - 6 * q^7 $$2 q - 2 q^{3} - 6 q^{7} + 12 q^{21} + 2 q^{23} - 8 q^{27} + 24 q^{41} + 18 q^{43} + 14 q^{47} + 16 q^{61} + 6 q^{63} + 6 q^{67} + 10 q^{81} - 22 q^{83} - 12 q^{87}+O(q^{100})$$ 2 * q - 2 * q^3 - 6 * q^7 + 12 * q^21 + 2 * q^23 - 8 * q^27 + 24 * q^41 + 18 * q^43 + 14 * q^47 + 16 * q^61 + 6 * q^63 + 6 * q^67 + 10 * q^81 - 22 * q^83 - 12 * q^87

## 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 −3.00000 3.00000i 0 1.00000i 0
1407.1 0 −1.00000 1.00000i 0 0 0 −3.00000 + 3.00000i 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.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
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.b 2
4.b odd 2 1 1600.2.n.l 2
5.b even 2 1 1600.2.n.l 2
5.c odd 4 1 inner 1600.2.n.b 2
5.c odd 4 1 1600.2.n.l 2
8.b even 2 1 100.2.e.c yes 2
8.d odd 2 1 100.2.e.a 2
20.d odd 2 1 CM 1600.2.n.b 2
20.e even 4 1 inner 1600.2.n.b 2
20.e even 4 1 1600.2.n.l 2
24.f even 2 1 900.2.k.e 2
24.h odd 2 1 900.2.k.a 2
40.e odd 2 1 100.2.e.c yes 2
40.f even 2 1 100.2.e.a 2
40.i odd 4 1 100.2.e.a 2
40.i odd 4 1 100.2.e.c yes 2
40.k even 4 1 100.2.e.a 2
40.k even 4 1 100.2.e.c yes 2
120.i odd 2 1 900.2.k.e 2
120.m even 2 1 900.2.k.a 2
120.q odd 4 1 900.2.k.a 2
120.q odd 4 1 900.2.k.e 2
120.w even 4 1 900.2.k.a 2
120.w even 4 1 900.2.k.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.e.a 2 8.d odd 2 1
100.2.e.a 2 40.f even 2 1
100.2.e.a 2 40.i odd 4 1
100.2.e.a 2 40.k even 4 1
100.2.e.c yes 2 8.b even 2 1
100.2.e.c yes 2 40.e odd 2 1
100.2.e.c yes 2 40.i odd 4 1
100.2.e.c yes 2 40.k even 4 1
900.2.k.a 2 24.h odd 2 1
900.2.k.a 2 120.m even 2 1
900.2.k.a 2 120.q odd 4 1
900.2.k.a 2 120.w even 4 1
900.2.k.e 2 24.f even 2 1
900.2.k.e 2 120.i odd 2 1
900.2.k.e 2 120.q odd 4 1
900.2.k.e 2 120.w even 4 1
1600.2.n.b 2 1.a even 1 1 trivial
1600.2.n.b 2 5.c odd 4 1 inner
1600.2.n.b 2 20.d odd 2 1 CM
1600.2.n.b 2 20.e even 4 1 inner
1600.2.n.l 2 4.b odd 2 1
1600.2.n.l 2 5.b even 2 1
1600.2.n.l 2 5.c odd 4 1
1600.2.n.l 2 20.e even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 2T_{3} + 2$$ T3^2 + 2*T3 + 2 $$T_{7}^{2} + 6T_{7} + 18$$ T7^2 + 6*T7 + 18 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 2T + 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6T + 18$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 2T + 2$$
$29$ $$T^{2} + 36$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$(T - 12)^{2}$$
$43$ $$T^{2} - 18T + 162$$
$47$ $$T^{2} - 14T + 98$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} - 6T + 18$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 22T + 242$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2}$$