# Properties

 Label 1600.2.n.q Level $1600$ Weight $2$ Character orbit 1600.n Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM discriminant -20 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 400) Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{3} - \beta_{2} q^{7} - 7 \beta_1 q^{9}+O(q^{10})$$ q - b3 * q^3 - b2 * q^7 - 7*b1 * q^9 $$q - \beta_{3} q^{3} - \beta_{2} q^{7} - 7 \beta_1 q^{9} - 10 q^{21} - 3 \beta_{3} q^{23} - 4 \beta_{2} q^{27} + 6 \beta_1 q^{29} - 12 q^{41} + \beta_{3} q^{43} - 3 \beta_{2} q^{47} + 3 \beta_1 q^{49} - 8 q^{61} + 7 \beta_{3} q^{63} + 5 \beta_{2} q^{67} - 30 \beta_1 q^{69} - 19 q^{81} - 3 \beta_{3} q^{83} + 6 \beta_{2} q^{87} - 6 \beta_1 q^{89}+O(q^{100})$$ q - b3 * q^3 - b2 * q^7 - 7*b1 * q^9 - 10 * q^21 - 3*b3 * q^23 - 4*b2 * q^27 + 6*b1 * q^29 - 12 * q^41 + b3 * q^43 - 3*b2 * q^47 + 3*b1 * q^49 - 8 * q^61 + 7*b3 * q^63 + 5*b2 * q^67 - 30*b1 * q^69 - 19 * q^81 - 3*b3 * q^83 + 6*b2 * q^87 - 6*b1 * q^89 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q - 40 q^{21} - 48 q^{41} - 32 q^{61} - 76 q^{81}+O(q^{100})$$ 4 * q - 40 * q^21 - 48 * q^41 - 32 * q^61 - 76 * q^81

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v $$\beta_{2}$$ $$=$$ $$\nu^{3} + 2\nu^{2} + 4\nu + 3$$ v^3 + 2*v^2 + 4*v + 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2\nu^{2} + 4\nu - 3$$ v^3 - 2*v^2 + 4*v - 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4$$ (b3 + b2 - 2*b1) / 4 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} - 6 ) / 4$$ (-b3 + b2 - 6) / 4 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + 4\beta_1 ) / 2$$ (-b3 - b2 + 4*b1) / 2

## Character values

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

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

## 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
 − 1.61803i 0.618034i 1.61803i − 0.618034i
0 −2.23607 + 2.23607i 0 0 0 2.23607 + 2.23607i 0 7.00000i 0
1343.2 0 2.23607 2.23607i 0 0 0 −2.23607 2.23607i 0 7.00000i 0
1407.1 0 −2.23607 2.23607i 0 0 0 2.23607 2.23607i 0 7.00000i 0
1407.2 0 2.23607 + 2.23607i 0 0 0 −2.23607 + 2.23607i 0 7.00000i 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 $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 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.q 4
4.b odd 2 1 inner 1600.2.n.q 4
5.b even 2 1 inner 1600.2.n.q 4
5.c odd 4 2 inner 1600.2.n.q 4
8.b even 2 1 400.2.n.c 4
8.d odd 2 1 400.2.n.c 4
20.d odd 2 1 CM 1600.2.n.q 4
20.e even 4 2 inner 1600.2.n.q 4
24.f even 2 1 3600.2.x.f 4
24.h odd 2 1 3600.2.x.f 4
40.e odd 2 1 400.2.n.c 4
40.f even 2 1 400.2.n.c 4
40.i odd 4 2 400.2.n.c 4
40.k even 4 2 400.2.n.c 4
120.i odd 2 1 3600.2.x.f 4
120.m even 2 1 3600.2.x.f 4
120.q odd 4 2 3600.2.x.f 4
120.w even 4 2 3600.2.x.f 4

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

## Hecke kernels

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

 $$T_{3}^{4} + 100$$ T3^4 + 100 $$T_{7}^{4} + 100$$ T7^4 + 100 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 100$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 100$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4} + 8100$$
$29$ $$(T^{2} + 36)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T + 12)^{4}$$
$43$ $$T^{4} + 100$$
$47$ $$T^{4} + 8100$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T + 8)^{4}$$
$67$ $$T^{4} + 62500$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 8100$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$T^{4}$$