# Properties

 Label 1600.2.n.u Level $1600$ Weight $2$ Character orbit 1600.n Analytic conductor $12.776$ Analytic rank $0$ Dimension $8$ CM no 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: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 400) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} - 4 \beta_1 q^{7} + 2 \beta_{3} q^{9}+O(q^{10})$$ q + b5 * q^3 - 4*b1 * q^7 + 2*b3 * q^9 $$q + \beta_{5} q^{3} - 4 \beta_1 q^{7} + 2 \beta_{3} q^{9} + 3 \beta_{4} q^{11} + 2 \beta_{2} q^{13} - 3 \beta_{7} q^{17} - \beta_{6} q^{19} - 4 q^{21} - 6 \beta_{5} q^{23} + 5 \beta_1 q^{27} + 6 \beta_{3} q^{29} - 2 \beta_{4} q^{31} + 3 \beta_{2} q^{33} + 2 \beta_{6} q^{39} - 3 q^{41} - 4 \beta_{5} q^{43} + 6 \beta_1 q^{47} + 9 \beta_{3} q^{49} - 3 \beta_{4} q^{51} + 6 \beta_{2} q^{53} + \beta_{7} q^{57} + 6 \beta_{6} q^{59} + 4 q^{61} + 8 \beta_{5} q^{63} - 7 \beta_1 q^{67} + 6 \beta_{3} q^{69} + 6 \beta_{4} q^{71} + 9 \beta_{2} q^{73} - 12 \beta_{7} q^{77} - q^{81} + 3 \beta_{5} q^{83} + 6 \beta_1 q^{87} + 3 \beta_{3} q^{89} - 8 \beta_{4} q^{91} - 2 \beta_{2} q^{93} - 4 \beta_{7} q^{97} - 6 \beta_{6} q^{99}+O(q^{100})$$ q + b5 * q^3 - 4*b1 * q^7 + 2*b3 * q^9 + 3*b4 * q^11 + 2*b2 * q^13 - 3*b7 * q^17 - b6 * q^19 - 4 * q^21 - 6*b5 * q^23 + 5*b1 * q^27 + 6*b3 * q^29 - 2*b4 * q^31 + 3*b2 * q^33 + 2*b6 * q^39 - 3 * q^41 - 4*b5 * q^43 + 6*b1 * q^47 + 9*b3 * q^49 - 3*b4 * q^51 + 6*b2 * q^53 + b7 * q^57 + 6*b6 * q^59 + 4 * q^61 + 8*b5 * q^63 - 7*b1 * q^67 + 6*b3 * q^69 + 6*b4 * q^71 + 9*b2 * q^73 - 12*b7 * q^77 - q^81 + 3*b5 * q^83 + 6*b1 * q^87 + 3*b3 * q^89 - 8*b4 * q^91 - 2*b2 * q^93 - 4*b7 * q^97 - 6*b6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 32 q^{21} - 24 q^{41} + 32 q^{61} - 8 q^{81}+O(q^{100})$$ 8 * q - 32 * q^21 - 24 * q^41 + 32 * q^61 - 8 * q^81

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$\zeta_{24}^{5} + \zeta_{24}$$ v^5 + v $$\beta_{3}$$ $$=$$ $$\zeta_{24}^{6}$$ v^6 $$\beta_{4}$$ $$=$$ $$2\zeta_{24}^{4} - 1$$ 2*v^4 - 1 $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}$$ -v^5 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{2}$$ -v^6 + 2*v^2 $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} - \zeta_{24}^{3}$$ 2*v^7 - v^3
 $$\zeta_{24}$$ $$=$$ $$( \beta_{5} + \beta_{2} ) / 2$$ (b5 + b2) / 2 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{6} + \beta_{3} ) / 2$$ (b6 + b3) / 2 $$\zeta_{24}^{3}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{4} + 1 ) / 2$$ (b4 + 1) / 2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{5} + \beta_{2} ) / 2$$ (-b5 + b2) / 2 $$\zeta_{24}^{6}$$ $$=$$ $$\beta_{3}$$ b3 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_1 ) / 2$$ (b7 + b1) / 2

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$-\beta_{3}$$ $$1$$ $$-1$$

## 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
 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 − 0.965926i 0.965926 + 0.258819i −0.965926 + 0.258819i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.258819 + 0.965926i
0 −0.707107 + 0.707107i 0 0 0 2.82843 + 2.82843i 0 2.00000i 0
1343.2 0 −0.707107 + 0.707107i 0 0 0 2.82843 + 2.82843i 0 2.00000i 0
1343.3 0 0.707107 0.707107i 0 0 0 −2.82843 2.82843i 0 2.00000i 0
1343.4 0 0.707107 0.707107i 0 0 0 −2.82843 2.82843i 0 2.00000i 0
1407.1 0 −0.707107 0.707107i 0 0 0 2.82843 2.82843i 0 2.00000i 0
1407.2 0 −0.707107 0.707107i 0 0 0 2.82843 2.82843i 0 2.00000i 0
1407.3 0 0.707107 + 0.707107i 0 0 0 −2.82843 + 2.82843i 0 2.00000i 0
1407.4 0 0.707107 + 0.707107i 0 0 0 −2.82843 + 2.82843i 0 2.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1407.4 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.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 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.u 8
4.b odd 2 1 inner 1600.2.n.u 8
5.b even 2 1 inner 1600.2.n.u 8
5.c odd 4 2 inner 1600.2.n.u 8
8.b even 2 1 400.2.n.d 8
8.d odd 2 1 400.2.n.d 8
20.d odd 2 1 inner 1600.2.n.u 8
20.e even 4 2 inner 1600.2.n.u 8
24.f even 2 1 3600.2.x.m 8
24.h odd 2 1 3600.2.x.m 8
40.e odd 2 1 400.2.n.d 8
40.f even 2 1 400.2.n.d 8
40.i odd 4 2 400.2.n.d 8
40.k even 4 2 400.2.n.d 8
120.i odd 2 1 3600.2.x.m 8
120.m even 2 1 3600.2.x.m 8
120.q odd 4 2 3600.2.x.m 8
120.w even 4 2 3600.2.x.m 8

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

 $$T_{3}^{4} + 1$$ T3^4 + 1 $$T_{7}^{4} + 256$$ T7^4 + 256 $$T_{11}^{2} + 27$$ T11^2 + 27 $$T_{13}^{4} + 144$$ T13^4 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} + 1)^{2}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 256)^{2}$$
$11$ $$(T^{2} + 27)^{4}$$
$13$ $$(T^{4} + 144)^{2}$$
$17$ $$(T^{4} + 729)^{2}$$
$19$ $$(T^{2} - 3)^{4}$$
$23$ $$(T^{4} + 1296)^{2}$$
$29$ $$(T^{2} + 36)^{4}$$
$31$ $$(T^{2} + 12)^{4}$$
$37$ $$T^{8}$$
$41$ $$(T + 3)^{8}$$
$43$ $$(T^{4} + 256)^{2}$$
$47$ $$(T^{4} + 1296)^{2}$$
$53$ $$(T^{4} + 11664)^{2}$$
$59$ $$(T^{2} - 108)^{4}$$
$61$ $$(T - 4)^{8}$$
$67$ $$(T^{4} + 2401)^{2}$$
$71$ $$(T^{2} + 108)^{4}$$
$73$ $$(T^{4} + 59049)^{2}$$
$79$ $$T^{8}$$
$83$ $$(T^{4} + 81)^{2}$$
$89$ $$(T^{2} + 9)^{4}$$
$97$ $$(T^{4} + 2304)^{2}$$