# Properties

 Label 1600.4.a.t Level $1600$ Weight $4$ Character orbit 1600.a Self dual yes Analytic conductor $94.403$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$94.4030560092$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 10) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{3} + 26q^{7} - 23q^{9} + O(q^{10})$$ $$q - 2q^{3} + 26q^{7} - 23q^{9} - 28q^{11} + 12q^{13} + 64q^{17} - 60q^{19} - 52q^{21} - 58q^{23} + 100q^{27} - 90q^{29} + 128q^{31} + 56q^{33} + 236q^{37} - 24q^{39} + 242q^{41} - 362q^{43} + 226q^{47} + 333q^{49} - 128q^{51} - 108q^{53} + 120q^{57} - 20q^{59} - 542q^{61} - 598q^{63} + 434q^{67} + 116q^{69} + 1128q^{71} - 632q^{73} - 728q^{77} + 720q^{79} + 421q^{81} + 478q^{83} + 180q^{87} - 490q^{89} + 312q^{91} - 256q^{93} - 1456q^{97} + 644q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −2.00000 0 0 0 26.0000 0 −23.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.4.a.t 1
4.b odd 2 1 1600.4.a.bh 1
5.b even 2 1 1600.4.a.bg 1
5.c odd 4 2 320.4.c.c 2
8.b even 2 1 400.4.a.n 1
8.d odd 2 1 50.4.a.b 1
20.d odd 2 1 1600.4.a.u 1
20.e even 4 2 320.4.c.d 2
24.f even 2 1 450.4.a.k 1
40.e odd 2 1 50.4.a.d 1
40.f even 2 1 400.4.a.h 1
40.i odd 4 2 80.4.c.a 2
40.k even 4 2 10.4.b.a 2
56.e even 2 1 2450.4.a.o 1
120.m even 2 1 450.4.a.j 1
120.q odd 4 2 90.4.c.b 2
120.w even 4 2 720.4.f.f 2
280.n even 2 1 2450.4.a.bb 1
280.y odd 4 2 490.4.c.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.4.b.a 2 40.k even 4 2
50.4.a.b 1 8.d odd 2 1
50.4.a.d 1 40.e odd 2 1
80.4.c.a 2 40.i odd 4 2
90.4.c.b 2 120.q odd 4 2
320.4.c.c 2 5.c odd 4 2
320.4.c.d 2 20.e even 4 2
400.4.a.h 1 40.f even 2 1
400.4.a.n 1 8.b even 2 1
450.4.a.j 1 120.m even 2 1
450.4.a.k 1 24.f even 2 1
490.4.c.b 2 280.y odd 4 2
720.4.f.f 2 120.w even 4 2
1600.4.a.t 1 1.a even 1 1 trivial
1600.4.a.u 1 20.d odd 2 1
1600.4.a.bg 1 5.b even 2 1
1600.4.a.bh 1 4.b odd 2 1
2450.4.a.o 1 56.e even 2 1
2450.4.a.bb 1 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3} + 2$$ $$T_{7} - 26$$ $$T_{11} + 28$$ $$T_{13} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$T$$
$7$ $$-26 + T$$
$11$ $$28 + T$$
$13$ $$-12 + T$$
$17$ $$-64 + T$$
$19$ $$60 + T$$
$23$ $$58 + T$$
$29$ $$90 + T$$
$31$ $$-128 + T$$
$37$ $$-236 + T$$
$41$ $$-242 + T$$
$43$ $$362 + T$$
$47$ $$-226 + T$$
$53$ $$108 + T$$
$59$ $$20 + T$$
$61$ $$542 + T$$
$67$ $$-434 + T$$
$71$ $$-1128 + T$$
$73$ $$632 + T$$
$79$ $$-720 + T$$
$83$ $$-478 + T$$
$89$ $$490 + T$$
$97$ $$1456 + T$$