# Properties

 Label 1600.2.j.c Level $1600$ Weight $2$ Character orbit 1600.j 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.j (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: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ Defining polynomial: $$x^{16} - 2 x^{14} + 6 x^{12} - 12 x^{10} + 36 x^{8} - 48 x^{6} + 96 x^{4} - 128 x^{2} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{14}$$ Twist minimal: no (minimal twist has level 400) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} + \beta_{7} q^{7} + ( -1 + \beta_{12} ) q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{3} + \beta_{7} q^{7} + ( -1 + \beta_{12} ) q^{9} + ( -1 - \beta_{9} - \beta_{11} ) q^{11} + ( -\beta_{2} + \beta_{4} + \beta_{7} + \beta_{8} ) q^{13} + ( \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{17} + ( 1 - \beta_{12} + \beta_{14} - \beta_{15} ) q^{19} + ( -\beta_{1} + \beta_{10} ) q^{21} + ( \beta_{2} - \beta_{3} ) q^{23} + ( -2 \beta_{3} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{13} ) q^{27} + ( \beta_{1} - 2 \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{29} + ( 1 - \beta_{1} + \beta_{9} - \beta_{15} ) q^{31} + ( -\beta_{2} + \beta_{3} + 2 \beta_{8} - \beta_{13} ) q^{33} + ( 2 \beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{13} ) q^{37} + ( -3 \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{14} ) q^{39} + ( 1 - \beta_{1} - \beta_{10} - \beta_{11} + \beta_{14} - \beta_{15} ) q^{41} + ( \beta_{4} + \beta_{6} - \beta_{13} ) q^{43} + ( \beta_{8} - 2 \beta_{13} ) q^{47} + ( 1 - \beta_{1} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{14} - \beta_{15} ) q^{49} + ( 3 - \beta_{1} - 3 \beta_{9} + \beta_{10} - \beta_{12} - \beta_{14} ) q^{51} + ( -\beta_{3} - 3 \beta_{5} + \beta_{6} + \beta_{8} + \beta_{13} ) q^{53} + ( -2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - \beta_{8} - \beta_{13} ) q^{57} + ( \beta_{1} - \beta_{9} + \beta_{12} + \beta_{14} ) q^{59} + ( -2 - 2 \beta_{9} - 2 \beta_{11} + \beta_{12} - \beta_{14} ) q^{61} + ( \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} - \beta_{8} ) q^{63} + ( -2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{13} ) q^{67} + ( -1 - 2 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{69} + ( 1 - \beta_{1} + \beta_{10} - \beta_{11} + 2 \beta_{12} + \beta_{15} ) q^{71} + ( \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{8} ) q^{73} + ( -\beta_{3} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} + 3 \beta_{8} - \beta_{13} ) q^{77} + ( -6 - \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} ) q^{79} + ( -1 - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{81} + ( -\beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + \beta_{13} ) q^{83} + ( 3 \beta_{2} + 3 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} - 3 \beta_{8} ) q^{87} + ( 1 + \beta_{1} - \beta_{10} + \beta_{11} - \beta_{15} ) q^{89} + ( 4 + 3 \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} - \beta_{15} ) q^{91} + ( -2 \beta_{4} - \beta_{6} - 3 \beta_{7} - 3 \beta_{8} + \beta_{13} ) q^{93} + ( -\beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + \beta_{8} ) q^{97} + ( 1 + \beta_{12} - \beta_{14} - \beta_{15} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q - 16q^{9} + O(q^{10})$$ $$16q - 16q^{9} - 8q^{11} + 8q^{19} + 16q^{29} + 48q^{51} + 8q^{59} - 16q^{61} - 16q^{69} + 32q^{71} - 80q^{79} + 16q^{81} + 64q^{91} + 8q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 2 x^{14} + 6 x^{12} - 12 x^{10} + 36 x^{8} - 48 x^{6} + 96 x^{4} - 128 x^{2} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{12} + 2 \nu^{10} - 6 \nu^{8} + 12 \nu^{6} - 20 \nu^{4} + 48 \nu^{2} - 48$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{15} + 18 \nu^{13} - 42 \nu^{11} + 68 \nu^{9} - 60 \nu^{7} + 224 \nu^{5} - 432 \nu^{3} + 320 \nu$$$$)/448$$ $$\beta_{3}$$ $$=$$ $$($$$$5 \nu^{15} - 6 \nu^{13} - 42 \nu^{11} + 108 \nu^{9} + 132 \nu^{7} + 480 \nu^{3} + 1088 \nu$$$$)/1344$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{15} - 4 \nu^{13} - 12 \nu^{9} + 32 \nu^{7} - 72 \nu^{3} - 96 \nu$$$$)/224$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{15} + 6 \nu^{11} - 24 \nu^{9} + 36 \nu^{7} - 24 \nu^{5} + 144 \nu^{3} - 128 \nu$$$$)/192$$ $$\beta_{6}$$ $$=$$ $$($$$$2 \nu^{15} + 6 \nu^{13} + 21 \nu^{11} + 18 \nu^{9} + 78 \nu^{7} + 84 \nu^{5} + 276 \nu^{3} + 32 \nu$$$$)/336$$ $$\beta_{7}$$ $$=$$ $$($$$$-3 \nu^{15} - 2 \nu^{13} - 14 \nu^{11} - 20 \nu^{9} + 44 \nu^{7} - 224 \nu^{5} + 48 \nu^{3} - 384 \nu$$$$)/448$$ $$\beta_{8}$$ $$=$$ $$($$$$-3 \nu^{15} + 12 \nu^{13} - 42 \nu^{11} + 64 \nu^{9} - 124 \nu^{7} + 280 \nu^{5} - 176 \nu^{3} + 64 \nu$$$$)/448$$ $$\beta_{9}$$ $$=$$ $$($$$$11 \nu^{14} - 30 \nu^{12} + 42 \nu^{10} - 132 \nu^{8} + 156 \nu^{6} - 336 \nu^{4} + 384 \nu^{2} - 832$$$$)/1344$$ $$\beta_{10}$$ $$=$$ $$($$$$-5 \nu^{14} + 6 \nu^{12} - 14 \nu^{10} + 116 \nu^{8} - 244 \nu^{6} + 448 \nu^{4} - 256 \nu^{2} + 1600$$$$)/448$$ $$\beta_{11}$$ $$=$$ $$($$$$4 \nu^{14} - 9 \nu^{12} + 42 \nu^{10} - 6 \nu^{8} + 156 \nu^{6} - 84 \nu^{4} + 48 \nu^{2} + 64$$$$)/336$$ $$\beta_{12}$$ $$=$$ $$($$$$-\nu^{14} - 2 \nu^{12} - 6 \nu^{10} + 4 \nu^{8} - 4 \nu^{6} - 64 \nu^{2}$$$$)/64$$ $$\beta_{13}$$ $$=$$ $$($$$$-11 \nu^{15} + 9 \nu^{13} - 42 \nu^{11} + 90 \nu^{9} - 156 \nu^{7} + 420 \nu^{5} - 216 \nu^{3} + 1504 \nu$$$$)/672$$ $$\beta_{14}$$ $$=$$ $$($$$$23 \nu^{14} + 6 \nu^{12} + 42 \nu^{10} - 108 \nu^{8} + 540 \nu^{6} + 672 \nu^{4} + 1536 \nu^{2} - 640$$$$)/1344$$ $$\beta_{15}$$ $$=$$ $$($$$$31 \nu^{14} + 30 \nu^{12} + 42 \nu^{10} + 132 \nu^{8} + 348 \nu^{6} + 2304 \nu^{2} + 832$$$$)/1344$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{13} - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{15} - \beta_{11} + \beta_{10} + \beta_{1} - 1$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{8} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$-\beta_{15} + 2 \beta_{14} + \beta_{10} - 2$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{13} + 2 \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 3 \beta_{4} - \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$\beta_{14} + \beta_{12} + \beta_{11} - \beta_{10} - 3 \beta_{9} + 2$$ $$\nu^{7}$$ $$=$$ $$\beta_{13} - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 3 \beta_{4} + \beta_{2}$$ $$\nu^{8}$$ $$=$$ $$3 \beta_{15} - 2 \beta_{14} + 2 \beta_{12} + 2 \beta_{11} + \beta_{10} - 2 \beta_{9} - 8$$ $$\nu^{9}$$ $$=$$ $$-2 \beta_{13} + 2 \beta_{8} + 2 \beta_{7} + 4 \beta_{6} - 8 \beta_{5} + 2 \beta_{3} - 2 \beta_{2}$$ $$\nu^{10}$$ $$=$$ $$-3 \beta_{15} - 2 \beta_{14} - 6 \beta_{12} + 5 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} + \beta_{1} - 9$$ $$\nu^{11}$$ $$=$$ $$-4 \beta_{8} - 2 \beta_{7} + 6 \beta_{6} - 8 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} - 2 \beta_{2}$$ $$\nu^{12}$$ $$=$$ $$-2 \beta_{15} - 12 \beta_{12} - 2 \beta_{11} - 10 \beta_{10} - 28 \beta_{9} - 2 \beta_{1} + 14$$ $$\nu^{13}$$ $$=$$ $$-2 \beta_{13} - 12 \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + 14 \beta_{5} - 22 \beta_{4} - 10 \beta_{3} + 22 \beta_{2}$$ $$\nu^{14}$$ $$=$$ $$18 \beta_{15} - 6 \beta_{11} - 6 \beta_{10} + 72 \beta_{9} - 18 \beta_{1} + 2$$ $$\nu^{15}$$ $$=$$ $$-40 \beta_{13} - 4 \beta_{8} - 56 \beta_{7} - 32 \beta_{6} + 20 \beta_{5} - 20 \beta_{4} + 44 \beta_{3} + 20 \beta_{2}$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 1.24570 − 0.669507i −0.601202 − 1.28006i −0.859408 + 1.12313i −1.35949 − 0.389597i 1.35949 + 0.389597i 0.859408 − 1.12313i 0.601202 + 1.28006i −1.24570 + 0.669507i −1.24570 − 0.669507i 0.601202 − 1.28006i 0.859408 + 1.12313i 1.35949 − 0.389597i −1.35949 + 0.389597i −0.859408 − 1.12313i −0.601202 + 1.28006i 1.24570 + 0.669507i
0 3.07455i 0 0 0 −1.47763 + 1.47763i 0 −6.45288 0
143.2 0 2.02856i 0 0 0 3.26957 3.26957i 0 −1.11507 0
143.3 0 1.54564i 0 0 0 −1.17442 + 1.17442i 0 0.611000 0
143.4 0 0.207468i 0 0 0 −1.32185 + 1.32185i 0 2.95696 0
143.5 0 0.207468i 0 0 0 1.32185 1.32185i 0 2.95696 0
143.6 0 1.54564i 0 0 0 1.17442 1.17442i 0 0.611000 0
143.7 0 2.02856i 0 0 0 −3.26957 + 3.26957i 0 −1.11507 0
143.8 0 3.07455i 0 0 0 1.47763 1.47763i 0 −6.45288 0
1007.1 0 3.07455i 0 0 0 1.47763 + 1.47763i 0 −6.45288 0
1007.2 0 2.02856i 0 0 0 −3.26957 3.26957i 0 −1.11507 0
1007.3 0 1.54564i 0 0 0 1.17442 + 1.17442i 0 0.611000 0
1007.4 0 0.207468i 0 0 0 1.32185 + 1.32185i 0 2.95696 0
1007.5 0 0.207468i 0 0 0 −1.32185 1.32185i 0 2.95696 0
1007.6 0 1.54564i 0 0 0 −1.17442 1.17442i 0 0.611000 0
1007.7 0 2.02856i 0 0 0 3.26957 + 3.26957i 0 −1.11507 0
1007.8 0 3.07455i 0 0 0 −1.47763 1.47763i 0 −6.45288 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1007.8 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
80.j even 4 1 inner
80.s 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.j.c 16
4.b odd 2 1 400.2.j.c 16
5.b even 2 1 inner 1600.2.j.c 16
5.c odd 4 2 1600.2.s.c 16
16.e even 4 1 400.2.s.c yes 16
16.f odd 4 1 1600.2.s.c 16
20.d odd 2 1 400.2.j.c 16
20.e even 4 2 400.2.s.c yes 16
80.i odd 4 1 400.2.j.c 16
80.j even 4 1 inner 1600.2.j.c 16
80.k odd 4 1 1600.2.s.c 16
80.q even 4 1 400.2.s.c yes 16
80.s even 4 1 inner 1600.2.j.c 16
80.t odd 4 1 400.2.j.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.j.c 16 4.b odd 2 1
400.2.j.c 16 20.d odd 2 1
400.2.j.c 16 80.i odd 4 1
400.2.j.c 16 80.t odd 4 1
400.2.s.c yes 16 16.e even 4 1
400.2.s.c yes 16 20.e even 4 2
400.2.s.c yes 16 80.q even 4 1
1600.2.j.c 16 1.a even 1 1 trivial
1600.2.j.c 16 5.b even 2 1 inner
1600.2.j.c 16 80.j even 4 1 inner
1600.2.j.c 16 80.s even 4 1 inner
1600.2.s.c 16 5.c odd 4 2
1600.2.s.c 16 16.f odd 4 1
1600.2.s.c 16 80.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 16 T_{3}^{6} + 72 T_{3}^{4} + 96 T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 4 + 96 T^{2} + 72 T^{4} + 16 T^{6} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$810000 + 217024 T^{4} + 18248 T^{8} + 496 T^{12} + T^{16}$$
$11$ $$( 7056 - 3360 T + 800 T^{2} + 464 T^{3} + 232 T^{4} - 40 T^{5} + 8 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$13$ $$( 5184 - 4160 T^{2} + 896 T^{4} - 56 T^{6} + T^{8} )^{2}$$
$17$ $$1600000000 + 70569984 T^{4} + 793728 T^{8} + 1856 T^{12} + T^{16}$$
$19$ $$( 38416 + 32928 T + 14112 T^{2} - 6608 T^{3} + 1544 T^{4} + 8 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$23$ $$10000 + 2498624 T^{4} + 470280 T^{8} + 2448 T^{12} + T^{16}$$
$29$ $$( 242064 - 244032 T + 123008 T^{2} - 29728 T^{3} + 3688 T^{4} - 80 T^{5} + 32 T^{6} - 8 T^{7} + T^{8} )^{2}$$
$31$ $$( 1849600 + 216576 T^{2} + 9056 T^{4} + 160 T^{6} + T^{8} )^{2}$$
$37$ $$( 9216 - 9728 T^{2} + 2816 T^{4} - 144 T^{6} + T^{8} )^{2}$$
$41$ $$( 90000 + 77824 T^{2} + 8616 T^{4} + 192 T^{6} + T^{8} )^{2}$$
$43$ $$( 36 - 608 T^{2} + 632 T^{4} - 96 T^{6} + T^{8} )^{2}$$
$47$ $$38416 + 47051712 T^{4} + 25159112 T^{8} + 20464 T^{12} + T^{16}$$
$53$ $$( 1915456 + 353472 T^{2} + 13760 T^{4} + 200 T^{6} + T^{8} )^{2}$$
$59$ $$( 1192464 + 637728 T + 170528 T^{2} - 68048 T^{3} + 13192 T^{4} - 88 T^{5} + 8 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$61$ $$( 3625216 + 2741760 T + 1036800 T^{2} + 199552 T^{3} + 20192 T^{4} + 416 T^{5} + 32 T^{6} + 8 T^{7} + T^{8} )^{2}$$
$67$ $$( 9771876 - 1428800 T^{2} + 36792 T^{4} - 336 T^{6} + T^{8} )^{2}$$
$71$ $$( -192 + 1088 T - 128 T^{2} - 8 T^{3} + T^{4} )^{4}$$
$73$ $$14281868906496 + 163305631744 T^{4} + 116635776 T^{8} + 22208 T^{12} + T^{16}$$
$79$ $$( -6000 - 1552 T + 8 T^{2} + 20 T^{3} + T^{4} )^{4}$$
$83$ $$( 4235364 + 509600 T^{2} + 19416 T^{4} + 256 T^{6} + T^{8} )^{2}$$
$89$ $$( -240 - 512 T - 136 T^{2} + T^{4} )^{4}$$
$97$ $$77407703861760000 + 35550049398784 T^{4} + 4644204672 T^{8} + 136640 T^{12} + T^{16}$$