# Properties

 Label 1600.2.l.h Level $1600$ Weight $2$ Character orbit 1600.l Analytic conductor $12.776$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.l (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: 16.0.534694406811304329216.1 Defining polynomial: $$x^{16} - 2 x^{14} - 2 x^{12} + 4 x^{10} + 4 x^{8} + 16 x^{6} - 32 x^{4} - 128 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{14} q^{3} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{14} ) q^{7} + ( 1 - \beta_{4} - \beta_{10} - \beta_{15} ) q^{9} +O(q^{10})$$ $$q -\beta_{14} q^{3} + ( -\beta_{2} + \beta_{3} + \beta_{5} - \beta_{8} - \beta_{14} ) q^{7} + ( 1 - \beta_{4} - \beta_{10} - \beta_{15} ) q^{9} + ( -1 + \beta_{4} + \beta_{9} + \beta_{10} ) q^{11} + ( -\beta_{5} - \beta_{14} ) q^{13} + ( -\beta_{5} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{17} + ( -\beta_{9} - \beta_{13} ) q^{19} + ( -1 - \beta_{7} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{21} + ( -\beta_{2} + \beta_{3} - \beta_{14} ) q^{23} + ( \beta_{1} + \beta_{3} + \beta_{11} + \beta_{14} ) q^{27} + ( 1 + \beta_{9} + \beta_{10} - \beta_{13} + \beta_{15} ) q^{29} + ( -1 - \beta_{4} + 2 \beta_{12} + \beta_{15} ) q^{31} + ( -\beta_{3} + \beta_{6} - \beta_{11} - \beta_{14} ) q^{33} + ( -2 \beta_{3} + 2 \beta_{8} ) q^{37} + ( 1 - \beta_{4} + 4 \beta_{9} - \beta_{15} ) q^{39} + ( 1 - \beta_{4} + \beta_{7} - \beta_{9} - 3 \beta_{10} - \beta_{12} - \beta_{13} ) q^{41} + ( \beta_{1} - 2 \beta_{8} + \beta_{11} + \beta_{14} ) q^{43} + ( -2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{8} + 2 \beta_{11} + \beta_{14} ) q^{47} + ( -1 + \beta_{4} - \beta_{7} - \beta_{13} ) q^{49} + ( 2 - 3 \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{15} ) q^{51} + ( \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{11} + \beta_{14} ) q^{53} + ( -\beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{8} + \beta_{14} ) q^{57} + ( -1 - \beta_{4} - \beta_{7} + \beta_{9} - \beta_{12} ) q^{59} + ( 2 \beta_{10} + 2 \beta_{12} ) q^{61} + ( -2 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} + 2 \beta_{8} + 2 \beta_{11} + 3 \beta_{14} ) q^{63} + ( \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - 2 \beta_{6} - \beta_{11} - 2 \beta_{14} ) q^{67} + ( -2 - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{12} ) q^{69} + ( 2 - 2 \beta_{4} - 2 \beta_{15} ) q^{71} + ( -\beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} - 2 \beta_{8} + 2 \beta_{14} ) q^{73} + ( -3 \beta_{5} + \beta_{14} ) q^{77} + ( 1 + \beta_{4} - \beta_{7} + 3 \beta_{12} - \beta_{13} ) q^{79} + ( -1 + \beta_{4} + 3 \beta_{12} - \beta_{15} ) q^{81} + ( 3 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{11} ) q^{83} + ( -2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{8} ) q^{87} + ( \beta_{7} + 4 \beta_{9} - \beta_{12} - \beta_{13} + \beta_{15} ) q^{89} + ( 2 - 2 \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} - 3 \beta_{12} ) q^{91} + ( -\beta_{1} - 3 \beta_{2} + 2 \beta_{3} - 3 \beta_{6} + \beta_{11} + \beta_{14} ) q^{93} + ( \beta_{3} + 2 \beta_{5} - 3 \beta_{6} + 2 \beta_{8} - \beta_{11} - 3 \beta_{14} ) q^{97} + ( 6 + 5 \beta_{9} - 2 \beta_{10} - 2 \beta_{12} - \beta_{13} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 8q^{11} - 8q^{19} - 16q^{21} + 16q^{29} - 16q^{31} - 16q^{49} + 16q^{51} - 24q^{59} - 32q^{69} + 16q^{79} - 16q^{81} + 16q^{91} + 88q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{14} - 2 x^{12} + 4 x^{10} + 4 x^{8} + 16 x^{6} - 32 x^{4} - 128 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{15} + 2 \nu^{13} + 6 \nu^{11} - 20 \nu^{9} - 76 \nu^{7} + 192 \nu^{5} - 224 \nu^{3} - 448 \nu$$$$)/576$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{13} + \nu^{11} - 2 \nu^{7} + 8 \nu^{5} - 36 \nu^{3} + 48 \nu$$$$)/48$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{15} - \nu^{13} + 6 \nu^{11} - 2 \nu^{9} + 20 \nu^{7} + 12 \nu^{5} - 8 \nu^{3} + 224 \nu$$$$)/288$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{14} + 2 \nu^{12} + 6 \nu^{10} + 4 \nu^{8} - 28 \nu^{6} + 48 \nu^{4} - 32 \nu^{2} + 80$$$$)/144$$ $$\beta_{5}$$ $$=$$ $$($$$$-2 \nu^{15} - \nu^{13} - 2 \nu^{9} + 32 \nu^{7} + 36 \nu^{5} - 104 \nu^{3} + 32 \nu$$$$)/288$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} + 2 \nu^{13} + 2 \nu^{11} + 12 \nu^{9} - 4 \nu^{7} + 16 \nu^{5} - 32 \nu^{3} + 320 \nu$$$$)/192$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{14} + 6 \nu^{12} + 6 \nu^{10} - 28 \nu^{8} + 20 \nu^{6} + 144 \nu^{4} + 192 \nu^{2} - 128$$$$)/192$$ $$\beta_{8}$$ $$=$$ $$($$$$-\nu^{15} + \nu^{13} + 3 \nu^{11} + 8 \nu^{9} + 10 \nu^{7} - 48 \nu^{5} - 52 \nu^{3} + 160 \nu$$$$)/144$$ $$\beta_{9}$$ $$=$$ $$($$$$-5 \nu^{14} + 2 \nu^{12} + 18 \nu^{10} - 20 \nu^{8} - 4 \nu^{6} - 144 \nu^{4} - 128 \nu^{2} + 704$$$$)/576$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{14} - 2 \nu^{10} + 4 \nu^{6} + 8 \nu^{4} - 32 \nu^{2} + 32$$$$)/96$$ $$\beta_{11}$$ $$=$$ $$($$$$\nu^{15} - 2 \nu^{13} + 2 \nu^{11} + 4 \nu^{9} - 4 \nu^{7} + 16 \nu^{5} + 16 \nu^{3} - 96 \nu$$$$)/96$$ $$\beta_{12}$$ $$=$$ $$($$$$-3 \nu^{14} + 2 \nu^{12} + 6 \nu^{10} + 12 \nu^{8} + 20 \nu^{6} - 96 \nu^{4} + 320$$$$)/192$$ $$\beta_{13}$$ $$=$$ $$($$$$-2 \nu^{14} + 5 \nu^{12} + 6 \nu^{10} - 2 \nu^{8} - 76 \nu^{6} - 60 \nu^{4} + 256 \nu^{2} + 368$$$$)/144$$ $$\beta_{14}$$ $$=$$ $$($$$$-5 \nu^{15} + 2 \nu^{13} + 12 \nu^{11} - 8 \nu^{9} + 8 \nu^{7} - 120 \nu^{5} + 40 \nu^{3} + 512 \nu$$$$)/288$$ $$\beta_{15}$$ $$=$$ $$($$$$5 \nu^{14} + 10 \nu^{12} - 18 \nu^{10} - 4 \nu^{8} + 4 \nu^{6} + 224 \nu^{2} - 320$$$$)/288$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{11} - \beta_{8} + \beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{15} + \beta_{13} + \beta_{12} - 2 \beta_{10} - 4 \beta_{9} + \beta_{7} - 2 \beta_{4} + 2$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{14} + \beta_{11} - 2 \beta_{8} + \beta_{6} - \beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{10} - 3 \beta_{9} + \beta_{7} + \beta_{4} + 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{11} - 3 \beta_{8} + 3 \beta_{6} + 3 \beta_{5} + \beta_{3} - \beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-\beta_{13} + 2 \beta_{12} - \beta_{10} - \beta_{9} + \beta_{7} - \beta_{4} + 2$$ $$\nu^{7}$$ $$=$$ $$-\beta_{14} + \beta_{8} - \beta_{6} + 4 \beta_{5} + 4 \beta_{3} - 2 \beta_{2} - \beta_{1}$$ $$\nu^{8}$$ $$=$$ $$-\beta_{15} + 6 \beta_{12} - 11 \beta_{9} - \beta_{7} + 3 \beta_{4}$$ $$\nu^{9}$$ $$=$$ $$-2 \beta_{14} + 6 \beta_{11} + 4 \beta_{8} + 6 \beta_{6} + 2 \beta_{5} - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_{1}$$ $$\nu^{10}$$ $$=$$ $$-5 \beta_{15} - \beta_{13} + 5 \beta_{12} - 12 \beta_{10} + 2 \beta_{9} + 3 \beta_{7} + 4 \beta_{4} - 10$$ $$\nu^{11}$$ $$=$$ $$8 \beta_{14} + 12 \beta_{11} + 8 \beta_{8} - 2 \beta_{6} + 2 \beta_{5} + 14 \beta_{3} - 8 \beta_{2} + 14 \beta_{1}$$ $$\nu^{12}$$ $$=$$ $$18 \beta_{15} - 2 \beta_{13} + 10 \beta_{12} + 16 \beta_{10} + 16 \beta_{9} + 2 \beta_{7} + 16 \beta_{4} - 24$$ $$\nu^{13}$$ $$=$$ $$-8 \beta_{14} - 14 \beta_{11} + 18 \beta_{8} + 6 \beta_{6} - 6 \beta_{5} + 22 \beta_{3} - 26 \beta_{2} + 26 \beta_{1}$$ $$\nu^{14}$$ $$=$$ $$14 \beta_{15} - 10 \beta_{13} - 10 \beta_{12} - 52 \beta_{10} + 12 \beta_{9} - 6 \beta_{7} + 8 \beta_{4} + 52$$ $$\nu^{15}$$ $$=$$ $$-36 \beta_{14} - 20 \beta_{11} + 24 \beta_{8} - 20 \beta_{6} - 56 \beta_{5} + 72 \beta_{3} + 4 \beta_{2} + 4 \beta_{1}$$

## Character values

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

 $$n$$ $$577$$ $$901$$ $$1151$$ $$\chi(n)$$ $$1$$ $$-\beta_{9}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 −0.238945 − 1.39388i 1.40501 − 0.161069i 0.841995 − 1.13624i 1.32661 − 0.490008i −1.32661 + 0.490008i −0.841995 + 1.13624i −1.40501 + 0.161069i 0.238945 + 1.39388i −0.238945 + 1.39388i 1.40501 + 0.161069i 0.841995 + 1.13624i 1.32661 + 0.490008i −1.32661 − 0.490008i −0.841995 − 1.13624i −1.40501 − 0.161069i 0.238945 − 1.39388i
0 −1.99154 1.99154i 0 0 0 1.09033i 0 4.93244i 0
401.2 0 −1.86033 1.86033i 0 0 0 3.61392i 0 3.92163i 0
401.3 0 −0.734294 0.734294i 0 0 0 1.71452i 0 1.92163i 0
401.4 0 −0.183790 0.183790i 0 0 0 3.84853i 0 2.93244i 0
401.5 0 0.183790 + 0.183790i 0 0 0 3.84853i 0 2.93244i 0
401.6 0 0.734294 + 0.734294i 0 0 0 1.71452i 0 1.92163i 0
401.7 0 1.86033 + 1.86033i 0 0 0 3.61392i 0 3.92163i 0
401.8 0 1.99154 + 1.99154i 0 0 0 1.09033i 0 4.93244i 0
1201.1 0 −1.99154 + 1.99154i 0 0 0 1.09033i 0 4.93244i 0
1201.2 0 −1.86033 + 1.86033i 0 0 0 3.61392i 0 3.92163i 0
1201.3 0 −0.734294 + 0.734294i 0 0 0 1.71452i 0 1.92163i 0
1201.4 0 −0.183790 + 0.183790i 0 0 0 3.84853i 0 2.93244i 0
1201.5 0 0.183790 0.183790i 0 0 0 3.84853i 0 2.93244i 0
1201.6 0 0.734294 0.734294i 0 0 0 1.71452i 0 1.92163i 0
1201.7 0 1.86033 1.86033i 0 0 0 3.61392i 0 3.92163i 0
1201.8 0 1.99154 1.99154i 0 0 0 1.09033i 0 4.93244i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1201.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.l.h 16
4.b odd 2 1 400.2.l.i 16
5.b even 2 1 inner 1600.2.l.h 16
5.c odd 4 2 320.2.q.c 16
16.e even 4 1 inner 1600.2.l.h 16
16.f odd 4 1 400.2.l.i 16
20.d odd 2 1 400.2.l.i 16
20.e even 4 2 80.2.q.c 16
40.i odd 4 2 640.2.q.f 16
40.k even 4 2 640.2.q.e 16
60.l odd 4 2 720.2.bm.f 16
80.i odd 4 1 320.2.q.c 16
80.i odd 4 1 640.2.q.f 16
80.j even 4 1 80.2.q.c 16
80.j even 4 1 640.2.q.e 16
80.k odd 4 1 400.2.l.i 16
80.q even 4 1 inner 1600.2.l.h 16
80.s even 4 1 80.2.q.c 16
80.s even 4 1 640.2.q.e 16
80.t odd 4 1 320.2.q.c 16
80.t odd 4 1 640.2.q.f 16
240.z odd 4 1 720.2.bm.f 16
240.bd odd 4 1 720.2.bm.f 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.q.c 16 20.e even 4 2
80.2.q.c 16 80.j even 4 1
80.2.q.c 16 80.s even 4 1
320.2.q.c 16 5.c odd 4 2
320.2.q.c 16 80.i odd 4 1
320.2.q.c 16 80.t odd 4 1
400.2.l.i 16 4.b odd 2 1
400.2.l.i 16 16.f odd 4 1
400.2.l.i 16 20.d odd 2 1
400.2.l.i 16 80.k odd 4 1
640.2.q.e 16 40.k even 4 2
640.2.q.e 16 80.j even 4 1
640.2.q.e 16 80.s even 4 1
640.2.q.f 16 40.i odd 4 2
640.2.q.f 16 80.i odd 4 1
640.2.q.f 16 80.t odd 4 1
720.2.bm.f 16 60.l odd 4 2
720.2.bm.f 16 240.z odd 4 1
720.2.bm.f 16 240.bd odd 4 1
1600.2.l.h 16 1.a even 1 1 trivial
1600.2.l.h 16 5.b even 2 1 inner
1600.2.l.h 16 16.e even 4 1 inner
1600.2.l.h 16 80.q even 4 1 inner

## Hecke kernels

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

 $$T_{3}^{16} + 112 T_{3}^{12} + 3144 T_{3}^{8} + 3520 T_{3}^{4} + 16$$ $$T_{7}^{8} + 32 T_{7}^{6} + 312 T_{7}^{4} + 896 T_{7}^{2} + 676$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$16 + 3520 T^{4} + 3144 T^{8} + 112 T^{12} + T^{16}$$
$5$ $$T^{16}$$
$7$ $$( 676 + 896 T^{2} + 312 T^{4} + 32 T^{6} + T^{8} )^{2}$$
$11$ $$( 16 - 160 T + 800 T^{2} + 464 T^{3} + 136 T^{4} - 8 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$13$ $$( 7744 + 224 T^{4} + T^{8} )^{2}$$
$17$ $$( 88 - 28 T^{2} + T^{4} )^{4}$$
$19$ $$( 59536 - 9760 T + 800 T^{2} + 464 T^{3} + 808 T^{4} - 104 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$23$ $$( 4 + 224 T^{2} + 168 T^{4} + 32 T^{6} + T^{8} )^{2}$$
$29$ $$( 2704 - 832 T + 128 T^{2} + 992 T^{3} + 1192 T^{4} + 304 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$31$ $$( 208 - 80 T - 72 T^{2} + 4 T^{3} + T^{4} )^{4}$$
$37$ $$( 147456 + 1536 T^{4} + T^{8} )^{2}$$
$41$ $$( 219024 + 110592 T^{2} + 10152 T^{4} + 192 T^{6} + T^{8} )^{2}$$
$43$ $$9721171216 + 502726720 T^{4} + 5803656 T^{8} + 17296 T^{12} + T^{16}$$
$47$ $$( 27556 - 12512 T^{2} + 1752 T^{4} - 80 T^{6} + T^{8} )^{2}$$
$53$ $$4096 + 495616 T^{4} + 2212992 T^{8} + 4672 T^{12} + T^{16}$$
$59$ $$( 144 + 1440 T + 7200 T^{2} + 4464 T^{3} + 1320 T^{4} - 312 T^{5} + 72 T^{6} + 12 T^{7} + T^{8} )^{2}$$
$61$ $$( 576 + T^{4} )^{4}$$
$67$ $$36804120336 + 1578000960 T^{4} + 17125128 T^{8} + 8976 T^{12} + T^{16}$$
$71$ $$( 65536 + 458752 T^{2} + 19968 T^{4} + 256 T^{6} + T^{8} )^{2}$$
$73$ $$( 48776256 + 3160512 T^{2} + 62208 T^{4} + 456 T^{6} + T^{8} )^{2}$$
$79$ $$( -368 + 464 T - 120 T^{2} - 4 T^{3} + T^{4} )^{4}$$
$83$ $$1475727898361616 + 2351652312384 T^{4} + 660646152 T^{8} + 47568 T^{12} + T^{16}$$
$89$ $$( 135424 + 120064 T^{2} + 10080 T^{4} + 208 T^{6} + T^{8} )^{2}$$
$97$ $$( 2383936 - 310208 T^{2} + 12672 T^{4} - 200 T^{6} + T^{8} )^{2}$$