Properties

 Label 1600.2.d.e Level $1600$ Weight $2$ Character orbit 1600.d 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.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{3} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + 2 q^{9} +O(q^{10})$$ $$q + \zeta_{12}^{3} q^{3} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + 2 q^{9} + 3 \zeta_{12}^{3} q^{11} + ( 2 - 4 \zeta_{12}^{2} ) q^{13} -3 q^{17} + \zeta_{12}^{3} q^{19} + ( 2 - 4 \zeta_{12}^{2} ) q^{21} + 5 \zeta_{12}^{3} q^{27} + ( -6 + 12 \zeta_{12}^{2} ) q^{29} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{31} -3 q^{33} + ( 6 - 12 \zeta_{12}^{2} ) q^{37} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{39} -9 q^{41} -4 \zeta_{12}^{3} q^{43} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{47} + 5 q^{49} -3 \zeta_{12}^{3} q^{51} - q^{57} -12 \zeta_{12}^{3} q^{59} + ( 2 - 4 \zeta_{12}^{2} ) q^{61} + ( -8 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{63} + 11 \zeta_{12}^{3} q^{67} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{71} -7 q^{73} + ( 6 - 12 \zeta_{12}^{2} ) q^{77} + ( 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{79} + q^{81} + 15 \zeta_{12}^{3} q^{83} + ( -12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{87} -3 q^{89} + 12 \zeta_{12}^{3} q^{91} + ( 4 - 8 \zeta_{12}^{2} ) q^{93} -14 q^{97} + 6 \zeta_{12}^{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{9} + O(q^{10})$$ $$4q + 8q^{9} - 12q^{17} - 12q^{33} - 36q^{41} + 20q^{49} - 4q^{57} - 28q^{73} + 4q^{81} - 12q^{89} - 56q^{97} + O(q^{100})$$

Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
801.1
 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i
0 1.00000i 0 0 0 −3.46410 0 2.00000 0
801.2 0 1.00000i 0 0 0 3.46410 0 2.00000 0
801.3 0 1.00000i 0 0 0 −3.46410 0 2.00000 0
801.4 0 1.00000i 0 0 0 3.46410 0 2.00000 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
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.d.e 4
4.b odd 2 1 inner 1600.2.d.e 4
5.b even 2 1 1600.2.d.f yes 4
5.c odd 4 1 1600.2.f.c 4
5.c odd 4 1 1600.2.f.g 4
8.b even 2 1 inner 1600.2.d.e 4
8.d odd 2 1 inner 1600.2.d.e 4
16.e even 4 1 6400.2.a.bb 2
16.e even 4 1 6400.2.a.cf 2
16.f odd 4 1 6400.2.a.bb 2
16.f odd 4 1 6400.2.a.cf 2
20.d odd 2 1 1600.2.d.f yes 4
20.e even 4 1 1600.2.f.c 4
20.e even 4 1 1600.2.f.g 4
40.e odd 2 1 1600.2.d.f yes 4
40.f even 2 1 1600.2.d.f yes 4
40.i odd 4 1 1600.2.f.c 4
40.i odd 4 1 1600.2.f.g 4
40.k even 4 1 1600.2.f.c 4
40.k even 4 1 1600.2.f.g 4
80.k odd 4 1 6400.2.a.ba 2
80.k odd 4 1 6400.2.a.cg 2
80.q even 4 1 6400.2.a.ba 2
80.q even 4 1 6400.2.a.cg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1600.2.d.e 4 1.a even 1 1 trivial
1600.2.d.e 4 4.b odd 2 1 inner
1600.2.d.e 4 8.b even 2 1 inner
1600.2.d.e 4 8.d odd 2 1 inner
1600.2.d.f yes 4 5.b even 2 1
1600.2.d.f yes 4 20.d odd 2 1
1600.2.d.f yes 4 40.e odd 2 1
1600.2.d.f yes 4 40.f even 2 1
1600.2.f.c 4 5.c odd 4 1
1600.2.f.c 4 20.e even 4 1
1600.2.f.c 4 40.i odd 4 1
1600.2.f.c 4 40.k even 4 1
1600.2.f.g 4 5.c odd 4 1
1600.2.f.g 4 20.e even 4 1
1600.2.f.g 4 40.i odd 4 1
1600.2.f.g 4 40.k even 4 1
6400.2.a.ba 2 80.k odd 4 1
6400.2.a.ba 2 80.q even 4 1
6400.2.a.bb 2 16.e even 4 1
6400.2.a.bb 2 16.f odd 4 1
6400.2.a.cf 2 16.e even 4 1
6400.2.a.cf 2 16.f odd 4 1
6400.2.a.cg 2 80.k odd 4 1
6400.2.a.cg 2 80.q even 4 1

Hecke kernels

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

 $$T_{3}^{2} + 1$$ $$T_{7}^{2} - 12$$ $$T_{17} + 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -12 + T^{2} )^{2}$$
$11$ $$( 9 + T^{2} )^{2}$$
$13$ $$( 12 + T^{2} )^{2}$$
$17$ $$( 3 + T )^{4}$$
$19$ $$( 1 + T^{2} )^{2}$$
$23$ $$T^{4}$$
$29$ $$( 108 + T^{2} )^{2}$$
$31$ $$( -48 + T^{2} )^{2}$$
$37$ $$( 108 + T^{2} )^{2}$$
$41$ $$( 9 + T )^{4}$$
$43$ $$( 16 + T^{2} )^{2}$$
$47$ $$( -108 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( 144 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( 121 + T^{2} )^{2}$$
$71$ $$( -108 + T^{2} )^{2}$$
$73$ $$( 7 + T )^{4}$$
$79$ $$( -108 + T^{2} )^{2}$$
$83$ $$( 225 + T^{2} )^{2}$$
$89$ $$( 3 + T )^{4}$$
$97$ $$( 14 + T )^{4}$$