# Properties

 Label 1600.3.b.s Level $1600$ Weight $3$ Character orbit 1600.b Analytic conductor $43.597$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1600.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$43.5968422976$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{7}$$ Twist minimal: no (minimal twist has level 20) 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 $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( -2 + 4 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{7} + ( 1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( -2 + 4 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{7} + ( 1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{9} + ( -4 + 8 \zeta_{10} + 4 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{11} + ( -6 - 4 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{13} + ( -2 - 16 \zeta_{10}^{2} + 16 \zeta_{10}^{3} ) q^{17} + ( -8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{19} + ( 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{21} + ( -6 + 12 \zeta_{10} - 10 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{23} + ( -12 + 24 \zeta_{10} - 4 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{27} + ( 6 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{29} + ( -20 + 40 \zeta_{10} - 28 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{31} + ( -32 - 24 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{33} + ( -6 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{37} + ( 4 - 8 \zeta_{10} - 4 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{39} + ( -22 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{41} + ( 6 - 12 \zeta_{10} - 2 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{43} + ( -26 + 52 \zeta_{10} - 14 \zeta_{10}^{2} + 38 \zeta_{10}^{3} ) q^{47} + ( 9 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{49} + ( -28 + 56 \zeta_{10} - 60 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{51} + ( -54 - 20 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{53} + ( 16 + 32 \zeta_{10}^{2} - 32 \zeta_{10}^{3} ) q^{57} + ( 24 - 48 \zeta_{10} + 48 \zeta_{10}^{2} ) q^{59} + ( -58 - 52 \zeta_{10}^{2} + 52 \zeta_{10}^{3} ) q^{61} + ( 22 - 44 \zeta_{10} + 26 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{63} + ( -22 + 44 \zeta_{10} - 62 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{67} + ( -16 + 28 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{69} + ( -4 + 8 \zeta_{10} + 36 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{71} + ( -34 + 64 \zeta_{10}^{2} - 64 \zeta_{10}^{3} ) q^{73} + ( 80 + 40 \zeta_{10}^{2} - 40 \zeta_{10}^{3} ) q^{77} + ( 40 - 80 \zeta_{10} + 72 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{79} + ( -55 + 28 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{81} + ( -26 + 52 \zeta_{10} - 2 \zeta_{10}^{2} + 50 \zeta_{10}^{3} ) q^{83} + ( 4 - 8 \zeta_{10} + 20 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{87} + ( -62 - 80 \zeta_{10}^{2} + 80 \zeta_{10}^{3} ) q^{89} + ( -12 + 24 \zeta_{10} + 4 \zeta_{10}^{2} + 28 \zeta_{10}^{3} ) q^{91} + ( -64 + 72 \zeta_{10}^{2} - 72 \zeta_{10}^{3} ) q^{93} + ( 78 + 24 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{97} + ( -20 + 40 \zeta_{10} + 4 \zeta_{10}^{2} + 44 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 16q^{13} + 24q^{17} - 40q^{21} + 8q^{29} - 80q^{33} + 16q^{37} - 112q^{41} - 4q^{49} - 176q^{53} - 128q^{61} - 120q^{69} - 264q^{73} + 240q^{77} - 276q^{81} - 88q^{89} - 400q^{93} + 264q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 −0.309017 − 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i
0 3.80423i 0 0 0 8.50651i 0 −5.47214 0
1151.2 0 2.35114i 0 0 0 5.25731i 0 3.47214 0
1151.3 0 2.35114i 0 0 0 5.25731i 0 3.47214 0
1151.4 0 3.80423i 0 0 0 8.50651i 0 −5.47214 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.b.s 4
4.b odd 2 1 inner 1600.3.b.s 4
5.b even 2 1 320.3.b.c 4
5.c odd 4 2 1600.3.h.n 8
8.b even 2 1 100.3.b.f 4
8.d odd 2 1 100.3.b.f 4
15.d odd 2 1 2880.3.e.e 4
20.d odd 2 1 320.3.b.c 4
20.e even 4 2 1600.3.h.n 8
24.f even 2 1 900.3.c.k 4
24.h odd 2 1 900.3.c.k 4
40.e odd 2 1 20.3.b.a 4
40.f even 2 1 20.3.b.a 4
40.i odd 4 2 100.3.d.b 8
40.k even 4 2 100.3.d.b 8
60.h even 2 1 2880.3.e.e 4
80.k odd 4 2 1280.3.g.e 8
80.q even 4 2 1280.3.g.e 8
120.i odd 2 1 180.3.c.a 4
120.m even 2 1 180.3.c.a 4
120.q odd 4 2 900.3.f.e 8
120.w even 4 2 900.3.f.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 40.e odd 2 1
20.3.b.a 4 40.f even 2 1
100.3.b.f 4 8.b even 2 1
100.3.b.f 4 8.d odd 2 1
100.3.d.b 8 40.i odd 4 2
100.3.d.b 8 40.k even 4 2
180.3.c.a 4 120.i odd 2 1
180.3.c.a 4 120.m even 2 1
320.3.b.c 4 5.b even 2 1
320.3.b.c 4 20.d odd 2 1
900.3.c.k 4 24.f even 2 1
900.3.c.k 4 24.h odd 2 1
900.3.f.e 8 120.q odd 4 2
900.3.f.e 8 120.w even 4 2
1280.3.g.e 8 80.k odd 4 2
1280.3.g.e 8 80.q even 4 2
1600.3.b.s 4 1.a even 1 1 trivial
1600.3.b.s 4 4.b odd 2 1 inner
1600.3.h.n 8 5.c odd 4 2
1600.3.h.n 8 20.e even 4 2
2880.3.e.e 4 15.d odd 2 1
2880.3.e.e 4 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1600, [\chi])$$:

 $$T_{3}^{4} + 20 T_{3}^{2} + 80$$ $$T_{7}^{4} + 100 T_{7}^{2} + 2000$$ $$T_{13}^{2} + 8 T_{13} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$80 + 20 T^{2} + T^{4}$$
$5$ $$T^{4}$$
$7$ $$2000 + 100 T^{2} + T^{4}$$
$11$ $$1280 + 400 T^{2} + T^{4}$$
$13$ $$( -4 + 8 T + T^{2} )^{2}$$
$17$ $$( -284 - 12 T + T^{2} )^{2}$$
$19$ $$20480 + 320 T^{2} + T^{4}$$
$23$ $$80 + 260 T^{2} + T^{4}$$
$29$ $$( -76 - 4 T + T^{2} )^{2}$$
$31$ $$154880 + 2320 T^{2} + T^{4}$$
$37$ $$( -484 - 8 T + T^{2} )^{2}$$
$41$ $$( 604 + 56 T + T^{2} )^{2}$$
$43$ $$2000 + 500 T^{2} + T^{4}$$
$47$ $$3561680 + 4100 T^{2} + T^{4}$$
$53$ $$( 1436 + 88 T + T^{2} )^{2}$$
$59$ $$1658880 + 5760 T^{2} + T^{4}$$
$61$ $$( -2356 + 64 T + T^{2} )^{2}$$
$67$ $$19920080 + 10420 T^{2} + T^{4}$$
$71$ $$10138880 + 8080 T^{2} + T^{4}$$
$73$ $$( -764 + 132 T + T^{2} )^{2}$$
$79$ $$2478080 + 13120 T^{2} + T^{4}$$
$83$ $$2620880 + 6260 T^{2} + T^{4}$$
$89$ $$( -7516 + 44 T + T^{2} )^{2}$$
$97$ $$( 3636 - 132 T + T^{2} )^{2}$$