# Properties

 Label 1600.2.n.t Level $1600$ Weight $2$ Character orbit 1600.n Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{7} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{9}+O(q^{10})$$ q + (b2 + b1 + 1) * q^3 + (-2*b2 + 2) * q^7 + (2*b3 + 2*b2 + 2*b1) * q^9 $$q + (\beta_{2} + \beta_1 + 1) q^{3} + ( - 2 \beta_{2} + 2) q^{7} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{9} + ( - \beta_{3} - \beta_{2} - \beta_1) q^{11} + ( - 2 \beta_{3} - 2 \beta_{2} + 2) q^{13} + (2 \beta_{2} - \beta_1 + 2) q^{17} + (\beta_{3} - \beta_1 + 5) q^{19} + ( - 2 \beta_{3} + 2 \beta_1 + 4) q^{21} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{23} + (3 \beta_{3} + 5 \beta_{2} - 5) q^{27} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{29} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{31} + ( - 3 \beta_{3} - 4 \beta_{2} + 4) q^{33} + ( - 6 \beta_{2} - 6) q^{37} + ( - 4 \beta_{3} + 4 \beta_1 + 10) q^{39} + 5 q^{41} + ( - 2 \beta_{2} + 4 \beta_1 - 2) q^{43} + 6 \beta_{3} q^{47} - \beta_{2} q^{49} + (\beta_{3} + \beta_{2} + \beta_1) q^{51} + (2 \beta_{3} + 4 \beta_{2} - 4) q^{53} + (2 \beta_{2} + 3 \beta_1 + 2) q^{57} + (4 \beta_{3} - 4 \beta_1 + 2) q^{59} + ( - 2 \beta_{3} + 2 \beta_1 - 4) q^{61} + (4 \beta_{2} + 8 \beta_1 + 4) q^{63} + ( - \beta_{3} - 3 \beta_{2} + 3) q^{67} + 2 \beta_{2} q^{69} + 2 \beta_{2} q^{71} + (3 \beta_{3} + 2 \beta_{2} - 2) q^{73} + ( - 2 \beta_{2} - 4 \beta_1 - 2) q^{77} + (2 \beta_{3} - 2 \beta_1 + 4) q^{79} + (2 \beta_{3} - 2 \beta_1 - 13) q^{81} + (\beta_{2} + 3 \beta_1 + 1) q^{83} + ( - 2 \beta_{3} - 4 \beta_{2} + 4) q^{87} + ( - 2 \beta_{3} - 7 \beta_{2} - 2 \beta_1) q^{89} + ( - 4 \beta_{3} - 8 \beta_{2} - 4 \beta_1) q^{91} + (2 \beta_{3} + 4 \beta_{2} - 4) q^{93} - 4 \beta_1 q^{97} + ( - 4 \beta_{3} + 4 \beta_1 + 14) q^{99}+O(q^{100})$$ q + (b2 + b1 + 1) * q^3 + (-2*b2 + 2) * q^7 + (2*b3 + 2*b2 + 2*b1) * q^9 + (-b3 - b2 - b1) * q^11 + (-2*b3 - 2*b2 + 2) * q^13 + (2*b2 - b1 + 2) * q^17 + (b3 - b1 + 5) * q^19 + (-2*b3 + 2*b1 + 4) * q^21 + (-2*b2 + 2*b1 - 2) * q^23 + (3*b3 + 5*b2 - 5) * q^27 + (-2*b3 + 2*b2 - 2*b1) * q^29 + (2*b3 - 2*b2 + 2*b1) * q^31 + (-3*b3 - 4*b2 + 4) * q^33 + (-6*b2 - 6) * q^37 + (-4*b3 + 4*b1 + 10) * q^39 + 5 * q^41 + (-2*b2 + 4*b1 - 2) * q^43 + 6*b3 * q^47 - b2 * q^49 + (b3 + b2 + b1) * q^51 + (2*b3 + 4*b2 - 4) * q^53 + (2*b2 + 3*b1 + 2) * q^57 + (4*b3 - 4*b1 + 2) * q^59 + (-2*b3 + 2*b1 - 4) * q^61 + (4*b2 + 8*b1 + 4) * q^63 + (-b3 - 3*b2 + 3) * q^67 + 2*b2 * q^69 + 2*b2 * q^71 + (3*b3 + 2*b2 - 2) * q^73 + (-2*b2 - 4*b1 - 2) * q^77 + (2*b3 - 2*b1 + 4) * q^79 + (2*b3 - 2*b1 - 13) * q^81 + (b2 + 3*b1 + 1) * q^83 + (-2*b3 - 4*b2 + 4) * q^87 + (-2*b3 - 7*b2 - 2*b1) * q^89 + (-4*b3 - 8*b2 - 4*b1) * q^91 + (2*b3 + 4*b2 - 4) * q^93 - 4*b1 * q^97 + (-4*b3 + 4*b1 + 14) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} + 8 q^{7}+O(q^{10})$$ 4 * q + 4 * q^3 + 8 * q^7 $$4 q + 4 q^{3} + 8 q^{7} + 8 q^{13} + 8 q^{17} + 20 q^{19} + 16 q^{21} - 8 q^{23} - 20 q^{27} + 16 q^{33} - 24 q^{37} + 40 q^{39} + 20 q^{41} - 8 q^{43} - 16 q^{53} + 8 q^{57} + 8 q^{59} - 16 q^{61} + 16 q^{63} + 12 q^{67} - 8 q^{73} - 8 q^{77} + 16 q^{79} - 52 q^{81} + 4 q^{83} + 16 q^{87} - 16 q^{93} + 56 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 + 8 * q^7 + 8 * q^13 + 8 * q^17 + 20 * q^19 + 16 * q^21 - 8 * q^23 - 20 * q^27 + 16 * q^33 - 24 * q^37 + 40 * q^39 + 20 * q^41 - 8 * q^43 - 16 * q^53 + 8 * q^57 + 8 * q^59 - 16 * q^61 + 16 * q^63 + 12 * q^67 - 8 * q^73 - 8 * q^77 + 16 * q^79 - 52 * q^81 + 4 * q^83 + 16 * q^87 - 16 * q^93 + 56 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

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

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$\beta_{2}$$ $$1$$ $$-1$$

## 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
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
0 −0.224745 + 0.224745i 0 0 0 2.00000 + 2.00000i 0 2.89898i 0
1343.2 0 2.22474 2.22474i 0 0 0 2.00000 + 2.00000i 0 6.89898i 0
1407.1 0 −0.224745 0.224745i 0 0 0 2.00000 2.00000i 0 2.89898i 0
1407.2 0 2.22474 + 2.22474i 0 0 0 2.00000 2.00000i 0 6.89898i 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.t 4
4.b odd 2 1 1600.2.n.p 4
5.b even 2 1 1600.2.n.o 4
5.c odd 4 1 1600.2.n.p 4
5.c odd 4 1 1600.2.n.s 4
8.b even 2 1 800.2.n.k 4
8.d odd 2 1 800.2.n.m yes 4
20.d odd 2 1 1600.2.n.s 4
20.e even 4 1 1600.2.n.o 4
20.e even 4 1 inner 1600.2.n.t 4
40.e odd 2 1 800.2.n.l yes 4
40.f even 2 1 800.2.n.n yes 4
40.i odd 4 1 800.2.n.l yes 4
40.i odd 4 1 800.2.n.m yes 4
40.k even 4 1 800.2.n.k 4
40.k even 4 1 800.2.n.n yes 4

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

## Hecke kernels

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

 $$T_{3}^{4} - 4T_{3}^{3} + 8T_{3}^{2} + 4T_{3} + 1$$ T3^4 - 4*T3^3 + 8*T3^2 + 4*T3 + 1 $$T_{7}^{2} - 4T_{7} + 8$$ T7^2 - 4*T7 + 8 $$T_{11}^{4} + 14T_{11}^{2} + 25$$ T11^4 + 14*T11^2 + 25 $$T_{13}^{4} - 8T_{13}^{3} + 32T_{13}^{2} + 32T_{13} + 16$$ T13^4 - 8*T13^3 + 32*T13^2 + 32*T13 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4 T^{3} + 8 T^{2} + 4 T + 1$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 4 T + 8)^{2}$$
$11$ $$T^{4} + 14T^{2} + 25$$
$13$ $$T^{4} - 8 T^{3} + 32 T^{2} + 32 T + 16$$
$17$ $$T^{4} - 8 T^{3} + 32 T^{2} - 40 T + 25$$
$19$ $$(T^{2} - 10 T + 19)^{2}$$
$23$ $$T^{4} + 8 T^{3} + 32 T^{2} - 32 T + 16$$
$29$ $$T^{4} + 56T^{2} + 400$$
$31$ $$T^{4} + 56T^{2} + 400$$
$37$ $$(T^{2} + 12 T + 72)^{2}$$
$41$ $$(T - 5)^{4}$$
$43$ $$T^{4} + 8 T^{3} + 32 T^{2} + \cdots + 1600$$
$47$ $$T^{4} + 11664$$
$53$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 400$$
$59$ $$(T^{2} - 4 T - 92)^{2}$$
$61$ $$(T^{2} + 8 T - 8)^{2}$$
$67$ $$T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 225$$
$71$ $$(T^{2} + 4)^{2}$$
$73$ $$T^{4} + 8 T^{3} + 32 T^{2} - 152 T + 361$$
$79$ $$(T^{2} - 8 T - 8)^{2}$$
$83$ $$T^{4} - 4 T^{3} + 8 T^{2} + 100 T + 625$$
$89$ $$T^{4} + 146T^{2} + 625$$
$97$ $$T^{4} + 2304$$