# Properties

 Label 1600.2.c.l Level $1600$ Weight $2$ Character orbit 1600.c Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 32) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{9} +O(q^{10})$$ $$q + 3 q^{9} -6 i q^{13} -2 i q^{17} -10 q^{29} -2 i q^{37} + 10 q^{41} + 7 q^{49} -14 i q^{53} + 10 q^{61} -6 i q^{73} + 9 q^{81} -10 q^{89} -18 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{9} + O(q^{10})$$ $$2q + 6q^{9} - 20q^{29} + 20q^{41} + 14q^{49} + 20q^{61} + 18q^{81} - 20q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 1.00000i − 1.00000i
0 0 0 0 0 0 0 3.00000 0
449.2 0 0 0 0 0 0 0 3.00000 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 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
20.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.c.l 2
4.b odd 2 1 CM 1600.2.c.l 2
5.b even 2 1 inner 1600.2.c.l 2
5.c odd 4 1 64.2.a.a 1
5.c odd 4 1 1600.2.a.n 1
8.b even 2 1 800.2.c.e 2
8.d odd 2 1 800.2.c.e 2
15.e even 4 1 576.2.a.c 1
20.d odd 2 1 inner 1600.2.c.l 2
20.e even 4 1 64.2.a.a 1
20.e even 4 1 1600.2.a.n 1
24.f even 2 1 7200.2.f.m 2
24.h odd 2 1 7200.2.f.m 2
35.f even 4 1 3136.2.a.m 1
40.e odd 2 1 800.2.c.e 2
40.f even 2 1 800.2.c.e 2
40.i odd 4 1 32.2.a.a 1
40.i odd 4 1 800.2.a.d 1
40.k even 4 1 32.2.a.a 1
40.k even 4 1 800.2.a.d 1
55.e even 4 1 7744.2.a.v 1
60.l odd 4 1 576.2.a.c 1
80.i odd 4 1 256.2.b.b 2
80.j even 4 1 256.2.b.b 2
80.s even 4 1 256.2.b.b 2
80.t odd 4 1 256.2.b.b 2
120.i odd 2 1 7200.2.f.m 2
120.m even 2 1 7200.2.f.m 2
120.q odd 4 1 288.2.a.d 1
120.q odd 4 1 7200.2.a.v 1
120.w even 4 1 288.2.a.d 1
120.w even 4 1 7200.2.a.v 1
140.j odd 4 1 3136.2.a.m 1
160.u even 8 2 1024.2.e.j 4
160.v odd 8 2 1024.2.e.j 4
160.ba even 8 2 1024.2.e.j 4
160.bb odd 8 2 1024.2.e.j 4
220.i odd 4 1 7744.2.a.v 1
240.z odd 4 1 2304.2.d.j 2
240.bb even 4 1 2304.2.d.j 2
240.bd odd 4 1 2304.2.d.j 2
240.bf even 4 1 2304.2.d.j 2
280.s even 4 1 1568.2.a.e 1
280.y odd 4 1 1568.2.a.e 1
280.bp odd 12 2 1568.2.i.f 2
280.br even 12 2 1568.2.i.g 2
280.bt odd 12 2 1568.2.i.g 2
280.bv even 12 2 1568.2.i.f 2
360.bo even 12 2 2592.2.i.t 2
360.br even 12 2 2592.2.i.e 2
360.bt odd 12 2 2592.2.i.e 2
360.bu odd 12 2 2592.2.i.t 2
440.t even 4 1 3872.2.a.f 1
440.w odd 4 1 3872.2.a.f 1
520.bc even 4 1 5408.2.a.g 1
520.bg odd 4 1 5408.2.a.g 1
680.u even 4 1 9248.2.a.f 1
680.bi odd 4 1 9248.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 40.i odd 4 1
32.2.a.a 1 40.k even 4 1
64.2.a.a 1 5.c odd 4 1
64.2.a.a 1 20.e even 4 1
256.2.b.b 2 80.i odd 4 1
256.2.b.b 2 80.j even 4 1
256.2.b.b 2 80.s even 4 1
256.2.b.b 2 80.t odd 4 1
288.2.a.d 1 120.q odd 4 1
288.2.a.d 1 120.w even 4 1
576.2.a.c 1 15.e even 4 1
576.2.a.c 1 60.l odd 4 1
800.2.a.d 1 40.i odd 4 1
800.2.a.d 1 40.k even 4 1
800.2.c.e 2 8.b even 2 1
800.2.c.e 2 8.d odd 2 1
800.2.c.e 2 40.e odd 2 1
800.2.c.e 2 40.f even 2 1
1024.2.e.j 4 160.u even 8 2
1024.2.e.j 4 160.v odd 8 2
1024.2.e.j 4 160.ba even 8 2
1024.2.e.j 4 160.bb odd 8 2
1568.2.a.e 1 280.s even 4 1
1568.2.a.e 1 280.y odd 4 1
1568.2.i.f 2 280.bp odd 12 2
1568.2.i.f 2 280.bv even 12 2
1568.2.i.g 2 280.br even 12 2
1568.2.i.g 2 280.bt odd 12 2
1600.2.a.n 1 5.c odd 4 1
1600.2.a.n 1 20.e even 4 1
1600.2.c.l 2 1.a even 1 1 trivial
1600.2.c.l 2 4.b odd 2 1 CM
1600.2.c.l 2 5.b even 2 1 inner
1600.2.c.l 2 20.d odd 2 1 inner
2304.2.d.j 2 240.z odd 4 1
2304.2.d.j 2 240.bb even 4 1
2304.2.d.j 2 240.bd odd 4 1
2304.2.d.j 2 240.bf even 4 1
2592.2.i.e 2 360.br even 12 2
2592.2.i.e 2 360.bt odd 12 2
2592.2.i.t 2 360.bo even 12 2
2592.2.i.t 2 360.bu odd 12 2
3136.2.a.m 1 35.f even 4 1
3136.2.a.m 1 140.j odd 4 1
3872.2.a.f 1 440.t even 4 1
3872.2.a.f 1 440.w odd 4 1
5408.2.a.g 1 520.bc even 4 1
5408.2.a.g 1 520.bg odd 4 1
7200.2.a.v 1 120.q odd 4 1
7200.2.a.v 1 120.w even 4 1
7200.2.f.m 2 24.f even 2 1
7200.2.f.m 2 24.h odd 2 1
7200.2.f.m 2 120.i odd 2 1
7200.2.f.m 2 120.m even 2 1
7744.2.a.v 1 55.e even 4 1
7744.2.a.v 1 220.i odd 4 1
9248.2.a.f 1 680.u even 4 1
9248.2.a.f 1 680.bi odd 4 1

## Hecke kernels

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

 $$T_{3}$$ $$T_{7}$$ $$T_{11}$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$36 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$4 + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$196 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( 10 + T )^{2}$$
$97$ $$324 + T^{2}$$