# Properties

 Label 1600.2.n.r Level $1600$ Weight $2$ Character orbit 1600.n Analytic conductor $12.776$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 80) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - \beta_{3} q^{7} + 3 \beta_1 q^{9}+O(q^{10})$$ q - b2 * q^3 - b3 * q^7 + 3*b1 * q^9 $$q - \beta_{2} q^{3} - \beta_{3} q^{7} + 3 \beta_1 q^{9} + (\beta_{3} - \beta_{2}) q^{11} + ( - \beta_1 + 1) q^{13} + ( - \beta_1 - 1) q^{17} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{19} + 6 q^{21} + \beta_{2} q^{23} + 4 \beta_1 q^{29} + (\beta_{3} - \beta_{2}) q^{31} + (6 \beta_1 - 6) q^{33} + (5 \beta_1 + 5) q^{37} + ( - \beta_{3} - \beta_{2}) q^{39} + 2 q^{41} - \beta_{2} q^{43} - \beta_{3} q^{47} + \beta_1 q^{49} + ( - \beta_{3} + \beta_{2}) q^{51} + (7 \beta_1 - 7) q^{53} + (12 \beta_1 + 12) q^{57} + (2 \beta_{3} + 2 \beta_{2}) q^{59} - 6 q^{61} - 3 \beta_{2} q^{63} - 3 \beta_{3} q^{67} - 6 \beta_1 q^{69} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{71} + ( - 7 \beta_1 + 7) q^{73} + (6 \beta_1 + 6) q^{77} + 9 q^{81} + 7 \beta_{2} q^{83} + 4 \beta_{3} q^{87} - 8 \beta_1 q^{89} + ( - \beta_{3} + \beta_{2}) q^{91} + (6 \beta_1 - 6) q^{93} + (7 \beta_1 + 7) q^{97} + (3 \beta_{3} + 3 \beta_{2}) q^{99}+O(q^{100})$$ q - b2 * q^3 - b3 * q^7 + 3*b1 * q^9 + (b3 - b2) * q^11 + (-b1 + 1) * q^13 + (-b1 - 1) * q^17 + (-2*b3 - 2*b2) * q^19 + 6 * q^21 + b2 * q^23 + 4*b1 * q^29 + (b3 - b2) * q^31 + (6*b1 - 6) * q^33 + (5*b1 + 5) * q^37 + (-b3 - b2) * q^39 + 2 * q^41 - b2 * q^43 - b3 * q^47 + b1 * q^49 + (-b3 + b2) * q^51 + (7*b1 - 7) * q^53 + (12*b1 + 12) * q^57 + (2*b3 + 2*b2) * q^59 - 6 * q^61 - 3*b2 * q^63 - 3*b3 * q^67 - 6*b1 * q^69 + (-3*b3 + 3*b2) * q^71 + (-7*b1 + 7) * q^73 + (6*b1 + 6) * q^77 + 9 * q^81 + 7*b2 * q^83 + 4*b3 * q^87 - 8*b1 * q^89 + (-b3 + b2) * q^91 + (6*b1 - 6) * q^93 + (7*b1 + 7) * q^97 + (3*b3 + 3*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q+O(q^{10})$$ 4 * q $$4 q + 4 q^{13} - 4 q^{17} + 24 q^{21} - 24 q^{33} + 20 q^{37} + 8 q^{41} - 28 q^{53} + 48 q^{57} - 24 q^{61} + 28 q^{73} + 24 q^{77} + 36 q^{81} - 24 q^{93} + 28 q^{97}+O(q^{100})$$ 4 * q + 4 * q^13 - 4 * q^17 + 24 * q^21 - 24 * q^33 + 20 * q^37 + 8 * q^41 - 28 * q^53 + 48 * q^57 - 24 * q^61 + 28 * q^73 + 24 * q^77 + 36 * q^81 - 24 * q^93 + 28 * q^97

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}^{2} + 2\zeta_{12} - 1$$ -v^3 + 2*v^2 + 2*v - 1 $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} - 2\zeta_{12}^{2} + 2\zeta_{12} + 1$$ -v^3 - 2*v^2 + 2*v + 1
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 2\beta_1 ) / 4$$ (b3 + b2 + 2*b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 2 ) / 4$$ (-b3 + b2 + 2) / 4 $$\zeta_{12}^{3}$$ $$=$$ $$\beta_1$$ b1

## 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
 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 −1.73205 + 1.73205i 0 0 0 −1.73205 1.73205i 0 3.00000i 0
1343.2 0 1.73205 1.73205i 0 0 0 1.73205 + 1.73205i 0 3.00000i 0
1407.1 0 −1.73205 1.73205i 0 0 0 −1.73205 + 1.73205i 0 3.00000i 0
1407.2 0 1.73205 + 1.73205i 0 0 0 1.73205 1.73205i 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 inner
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.r 4
4.b odd 2 1 inner 1600.2.n.r 4
5.b even 2 1 320.2.n.i 4
5.c odd 4 1 320.2.n.i 4
5.c odd 4 1 inner 1600.2.n.r 4
8.b even 2 1 400.2.n.b 4
8.d odd 2 1 400.2.n.b 4
20.d odd 2 1 320.2.n.i 4
20.e even 4 1 320.2.n.i 4
20.e even 4 1 inner 1600.2.n.r 4
24.f even 2 1 3600.2.x.e 4
24.h odd 2 1 3600.2.x.e 4
40.e odd 2 1 80.2.n.b 4
40.f even 2 1 80.2.n.b 4
40.i odd 4 1 80.2.n.b 4
40.i odd 4 1 400.2.n.b 4
40.k even 4 1 80.2.n.b 4
40.k even 4 1 400.2.n.b 4
80.i odd 4 1 1280.2.o.r 4
80.j even 4 1 1280.2.o.q 4
80.k odd 4 1 1280.2.o.q 4
80.k odd 4 1 1280.2.o.r 4
80.q even 4 1 1280.2.o.q 4
80.q even 4 1 1280.2.o.r 4
80.s even 4 1 1280.2.o.r 4
80.t odd 4 1 1280.2.o.q 4
120.i odd 2 1 720.2.x.d 4
120.m even 2 1 720.2.x.d 4
120.q odd 4 1 720.2.x.d 4
120.q odd 4 1 3600.2.x.e 4
120.w even 4 1 720.2.x.d 4
120.w even 4 1 3600.2.x.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.n.b 4 40.e odd 2 1
80.2.n.b 4 40.f even 2 1
80.2.n.b 4 40.i odd 4 1
80.2.n.b 4 40.k even 4 1
320.2.n.i 4 5.b even 2 1
320.2.n.i 4 5.c odd 4 1
320.2.n.i 4 20.d odd 2 1
320.2.n.i 4 20.e even 4 1
400.2.n.b 4 8.b even 2 1
400.2.n.b 4 8.d odd 2 1
400.2.n.b 4 40.i odd 4 1
400.2.n.b 4 40.k even 4 1
720.2.x.d 4 120.i odd 2 1
720.2.x.d 4 120.m even 2 1
720.2.x.d 4 120.q odd 4 1
720.2.x.d 4 120.w even 4 1
1280.2.o.q 4 80.j even 4 1
1280.2.o.q 4 80.k odd 4 1
1280.2.o.q 4 80.q even 4 1
1280.2.o.q 4 80.t odd 4 1
1280.2.o.r 4 80.i odd 4 1
1280.2.o.r 4 80.k odd 4 1
1280.2.o.r 4 80.q even 4 1
1280.2.o.r 4 80.s even 4 1
1600.2.n.r 4 1.a even 1 1 trivial
1600.2.n.r 4 4.b odd 2 1 inner
1600.2.n.r 4 5.c odd 4 1 inner
1600.2.n.r 4 20.e even 4 1 inner
3600.2.x.e 4 24.f even 2 1
3600.2.x.e 4 24.h odd 2 1
3600.2.x.e 4 120.q odd 4 1
3600.2.x.e 4 120.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 $$S_{2}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{4} + 36$$ T3^4 + 36 $$T_{7}^{4} + 36$$ T7^4 + 36 $$T_{11}^{2} + 12$$ T11^2 + 12 $$T_{13}^{2} - 2T_{13} + 2$$ T13^2 - 2*T13 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 36$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 36$$
$11$ $$(T^{2} + 12)^{2}$$
$13$ $$(T^{2} - 2 T + 2)^{2}$$
$17$ $$(T^{2} + 2 T + 2)^{2}$$
$19$ $$(T^{2} - 48)^{2}$$
$23$ $$T^{4} + 36$$
$29$ $$(T^{2} + 16)^{2}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$(T^{2} - 10 T + 50)^{2}$$
$41$ $$(T - 2)^{4}$$
$43$ $$T^{4} + 36$$
$47$ $$T^{4} + 36$$
$53$ $$(T^{2} + 14 T + 98)^{2}$$
$59$ $$(T^{2} - 48)^{2}$$
$61$ $$(T + 6)^{4}$$
$67$ $$T^{4} + 2916$$
$71$ $$(T^{2} + 108)^{2}$$
$73$ $$(T^{2} - 14 T + 98)^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4} + 86436$$
$89$ $$(T^{2} + 64)^{2}$$
$97$ $$(T^{2} - 14 T + 98)^{2}$$