# Properties

 Label 1600.3.g.k Level $1600$ Weight $3$ Character orbit 1600.g Analytic conductor $43.597$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1600.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$43.5968422976$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ Defining polynomial: $$x^{16} + 19 x^{14} + 301 x^{12} + 1102 x^{10} + 3238 x^{8} + 1102 x^{6} + 301 x^{4} + 19 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{37}$$ Twist minimal: no (minimal twist has level 320) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( 6 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} + ( 6 + \beta_{3} ) q^{9} + ( -2 \beta_{11} + \beta_{14} ) q^{11} + ( -\beta_{7} - \beta_{10} ) q^{13} + \beta_{15} q^{17} -3 \beta_{11} q^{19} + ( -\beta_{5} - 3 \beta_{6} ) q^{21} + ( \beta_{2} - 3 \beta_{9} ) q^{23} + ( 6 \beta_{1} + \beta_{4} ) q^{27} + ( -3 \beta_{5} + 2 \beta_{6} ) q^{29} -\beta_{13} q^{31} + ( -5 \beta_{12} - 2 \beta_{15} ) q^{33} + ( -4 \beta_{7} + \beta_{10} ) q^{37} + ( 5 \beta_{8} + 2 \beta_{13} ) q^{39} + ( -3 - 3 \beta_{3} ) q^{41} + ( 7 \beta_{1} - \beta_{4} ) q^{43} + ( 7 \beta_{2} - \beta_{9} ) q^{47} + ( 22 - \beta_{3} ) q^{49} + ( -\beta_{11} - 3 \beta_{14} ) q^{51} + ( -7 \beta_{7} - \beta_{10} ) q^{53} -3 \beta_{12} q^{57} + ( -15 \beta_{11} - 4 \beta_{14} ) q^{59} + ( -2 \beta_{5} + 9 \beta_{6} ) q^{61} + ( 13 \beta_{2} - 5 \beta_{9} ) q^{63} + ( -9 \beta_{1} - 3 \beta_{4} ) q^{67} + ( -13 \beta_{5} - 3 \beta_{6} ) q^{69} + ( \beta_{8} + \beta_{13} ) q^{71} + ( -\beta_{12} + 4 \beta_{15} ) q^{73} + ( 11 \beta_{7} + 2 \beta_{10} ) q^{77} + ( 2 \beta_{8} - 5 \beta_{13} ) q^{79} + ( 30 + 3 \beta_{3} ) q^{81} + ( 13 \beta_{1} + 3 \beta_{4} ) q^{83} -4 \beta_{9} q^{87} + ( 72 - 2 \beta_{3} ) q^{89} + ( -17 \beta_{11} + 3 \beta_{14} ) q^{91} + ( 6 \beta_{7} + 8 \beta_{10} ) q^{93} + ( -2 \beta_{12} - 7 \beta_{15} ) q^{97} + ( -55 \beta_{11} + 12 \beta_{14} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 96q^{9} + O(q^{10})$$ $$16q + 96q^{9} - 48q^{41} + 352q^{49} + 480q^{81} + 1152q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 19 x^{14} + 301 x^{12} + 1102 x^{10} + 3238 x^{8} + 1102 x^{6} + 301 x^{4} + 19 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$484 \nu^{14} + 9043 \nu^{12} + 142780 \nu^{10} + 487388 \nu^{8} + 1399796 \nu^{6} + 45980 \nu^{4} + 2904 \nu^{2} - 7647$$$$)/3900$$ $$\beta_{2}$$ $$=$$ $$($$$$80 \nu^{14} + 1513 \nu^{12} + 23950 \nu^{10} + 86110 \nu^{8} + 252226 \nu^{6} + 68750 \nu^{4} + 25230 \nu^{2} + 913$$$$)/300$$ $$\beta_{3}$$ $$=$$ $$($$$$60 \nu^{14} + 1121 \nu^{12} + 17700 \nu^{10} + 60420 \nu^{8} + 173642 \nu^{6} + 5700 \nu^{4} + 360 \nu^{2} - 1729$$$$)/150$$ $$\beta_{4}$$ $$=$$ $$($$$$-306 \nu^{14} - 5717 \nu^{12} - 90270 \nu^{10} - 308142 \nu^{8} - 885934 \nu^{6} - 29070 \nu^{4} - 1836 \nu^{2} + 25383$$$$)/650$$ $$\beta_{5}$$ $$=$$ $$($$$$-1007 \nu^{14} - 19074 \nu^{12} - 302005 \nu^{10} - 1092309 \nu^{8} - 3201253 \nu^{6} - 937805 \nu^{4} - 297502 \nu^{2} - 10979$$$$)/1950$$ $$\beta_{6}$$ $$=$$ $$($$$$3071 \nu^{14} + 58417 \nu^{12} + 925575 \nu^{10} + 3403067 \nu^{8} + 9992809 \nu^{6} + 3514575 \nu^{4} + 724196 \nu^{2} + 28792$$$$)/5850$$ $$\beta_{7}$$ $$=$$ $$($$$$-89 \nu^{15} - 1682 \nu^{13} - 26620 \nu^{11} - 95408 \nu^{9} - 278884 \nu^{7} - 71420 \nu^{5} - 24119 \nu^{3} - 2802 \nu$$$$)/150$$ $$\beta_{8}$$ $$=$$ $$($$$$-2182 \nu^{15} - 42080 \nu^{13} - 668520 \nu^{11} - 2590264 \nu^{9} - 7726640 \nu^{7} - 4329720 \nu^{5} - 1079482 \nu^{3} - 121760 \nu$$$$)/2925$$ $$\beta_{9}$$ $$=$$ $$($$$$100 \nu^{14} + 1899 \nu^{12} + 30080 \nu^{10} + 109880 \nu^{8} + 322398 \nu^{6} + 105880 \nu^{4} + 25980 \nu^{2} + 999$$$$)/75$$ $$\beta_{10}$$ $$=$$ $$($$$$261 \nu^{15} + 5018 \nu^{13} + 79675 \nu^{11} + 305247 \nu^{9} + 908011 \nu^{7} + 470675 \nu^{5} + 120186 \nu^{3} + 13593 \nu$$$$)/300$$ $$\beta_{11}$$ $$=$$ $$($$$$379 \nu^{15} + 7182 \nu^{13} + 113715 \nu^{11} + 411883 \nu^{9} + 1205379 \nu^{7} + 353115 \nu^{5} + 84304 \nu^{3} + 1197 \nu$$$$)/300$$ $$\beta_{12}$$ $$=$$ $$($$$$-979 \nu^{15} - 18540 \nu^{13} - 293550 \nu^{11} - 1061058 \nu^{9} - 3111630 \nu^{7} - 911550 \nu^{5} - 312479 \nu^{3} - 3090 \nu$$$$)/650$$ $$\beta_{13}$$ $$=$$ $$($$$$1796 \nu^{15} + 34204 \nu^{13} + 542118 \nu^{11} + 2003312 \nu^{9} + 5904250 \nu^{7} + 2241078 \nu^{5} + 638156 \nu^{3} + 73006 \nu$$$$)/585$$ $$\beta_{14}$$ $$=$$ $$($$$$381 \nu^{15} + 7218 \nu^{13} + 114285 \nu^{11} + 413597 \nu^{9} + 1211421 \nu^{7} + 354885 \nu^{5} + 100416 \nu^{3} + 1203 \nu$$$$)/100$$ $$\beta_{15}$$ $$=$$ $$($$$$2967 \nu^{15} + 56220 \nu^{13} + 890150 \nu^{11} + 3223394 \nu^{9} + 9435590 \nu^{7} + 2764150 \nu^{5} + 695947 \nu^{3} + 9370 \nu$$$$)/390$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} - 2 \beta_{14} + 2 \beta_{13} - \beta_{12} - 2 \beta_{11} - 4 \beta_{10} - \beta_{8} + 4 \beta_{7}$$$$)/32$$ $$\nu^{2}$$ $$=$$ $$($$$$-10 \beta_{9} + 12 \beta_{6} - 22 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} - 16 \beta_{2} - 2 \beta_{1} - 76$$$$)/32$$ $$\nu^{3}$$ $$=$$ $$($$$$-9 \beta_{15} + 10 \beta_{14} + 5 \beta_{12} + 30 \beta_{11}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$170 \beta_{9} - 228 \beta_{6} + 298 \beta_{5} + 43 \beta_{4} + 76 \beta_{3} + 176 \beta_{2} - 82 \beta_{1} - 964$$$$)/32$$ $$\nu^{5}$$ $$=$$ $$($$$$281 \beta_{15} - 278 \beta_{14} - 278 \beta_{13} - 121 \beta_{12} - 998 \beta_{11} + 1124 \beta_{10} + 679 \beta_{8} - 644 \beta_{7}$$$$)/32$$ $$\nu^{6}$$ $$=$$ $$-40 \beta_{4} - 75 \beta_{3} + 90 \beta_{1} + 874$$ $$\nu^{7}$$ $$=$$ $$($$$$4279 \beta_{15} - 4118 \beta_{14} + 4118 \beta_{13} - 1719 \beta_{12} - 15398 \beta_{11} - 17116 \beta_{10} - 10519 \beta_{8} + 9436 \beta_{7}$$$$)/32$$ $$\nu^{8}$$ $$=$$ $$($$$$-40150 \beta_{9} + 54948 \beta_{6} - 66058 \beta_{5} + 9637 \beta_{4} + 18316 \beta_{3} - 35344 \beta_{2} - 22478 \beta_{1} - 209284$$$$)/32$$ $$\nu^{9}$$ $$=$$ $$($$$$-32391 \beta_{15} + 30970 \beta_{14} + 12795 \beta_{12} + 116910 \beta_{11}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$607670 \beta_{9} - 832332 \beta_{6} + 996982 \beta_{5} + 145517 \beta_{4} + 277444 \beta_{3} + 530864 \beta_{2} - 342238 \beta_{1} - 3155596$$$$)/32$$ $$\nu^{11}$$ $$=$$ $$($$$$979439 \beta_{15} - 935042 \beta_{14} - 935042 \beta_{13} - 385359 \beta_{12} - 3537602 \beta_{11} + 3917756 \beta_{10} + 2419441 \beta_{8} - 2135516 \beta_{7}$$$$)/32$$ $$\nu^{12}$$ $$=$$ $$-137420 \beta_{4} - 262200 \beta_{3} + 323820 \beta_{1} + 2979001$$ $$\nu^{13}$$ $$=$$ $$($$$$14803441 \beta_{15} - 14127362 \beta_{14} + 14127362 \beta_{13} - 5818961 \beta_{12} - 53476802 \beta_{11} - 59213764 \beta_{10} - 36575761 \beta_{8} + 32260324 \beta_{7}$$$$)/32$$ $$\nu^{14}$$ $$=$$ $$($$$$-138847690 \beta_{9} + 190226892 \beta_{6} - 227617462 \beta_{5} + 33227123 \beta_{4} + 63408964 \beta_{3} - 121030096 \beta_{2} - 78332642 \beta_{1} - 720242956$$$$)/32$$ $$\nu^{15}$$ $$=$$ $$($$$$-111862809 \beta_{15} + 106745050 \beta_{14} + 43961605 \beta_{12} + 404115150 \beta_{11}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
351.1
 0.940543 + 1.62907i −0.940543 − 1.62907i 0.940543 − 1.62907i −0.940543 + 1.62907i 0.128617 − 0.222771i −0.128617 + 0.222771i 0.128617 + 0.222771i −0.128617 − 0.222771i 1.94376 + 3.36668i −1.94376 − 3.36668i 1.94376 − 3.36668i −1.94376 + 3.36668i 0.265804 − 0.460386i −0.265804 + 0.460386i 0.265804 + 0.460386i −0.265804 − 0.460386i
0 −5.13399 0 0 0 6.19337i 0 17.3578 0
351.2 0 −5.13399 0 0 0 6.19337i 0 17.3578 0
351.3 0 −5.13399 0 0 0 6.19337i 0 17.3578 0
351.4 0 −5.13399 0 0 0 6.19337i 0 17.3578 0
351.5 0 −1.90845 0 0 0 3.95502i 0 −5.35782 0
351.6 0 −1.90845 0 0 0 3.95502i 0 −5.35782 0
351.7 0 −1.90845 0 0 0 3.95502i 0 −5.35782 0
351.8 0 −1.90845 0 0 0 3.95502i 0 −5.35782 0
351.9 0 1.90845 0 0 0 3.95502i 0 −5.35782 0
351.10 0 1.90845 0 0 0 3.95502i 0 −5.35782 0
351.11 0 1.90845 0 0 0 3.95502i 0 −5.35782 0
351.12 0 1.90845 0 0 0 3.95502i 0 −5.35782 0
351.13 0 5.13399 0 0 0 6.19337i 0 17.3578 0
351.14 0 5.13399 0 0 0 6.19337i 0 17.3578 0
351.15 0 5.13399 0 0 0 6.19337i 0 17.3578 0
351.16 0 5.13399 0 0 0 6.19337i 0 17.3578 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 351.16 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
5.b even 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.3.g.k 16
4.b odd 2 1 inner 1600.3.g.k 16
5.b even 2 1 inner 1600.3.g.k 16
5.c odd 4 2 320.3.e.b 16
8.b even 2 1 inner 1600.3.g.k 16
8.d odd 2 1 inner 1600.3.g.k 16
20.d odd 2 1 inner 1600.3.g.k 16
20.e even 4 2 320.3.e.b 16
40.e odd 2 1 inner 1600.3.g.k 16
40.f even 2 1 inner 1600.3.g.k 16
40.i odd 4 2 320.3.e.b 16
40.k even 4 2 320.3.e.b 16
80.i odd 4 2 1280.3.h.n 16
80.j even 4 2 1280.3.h.n 16
80.s even 4 2 1280.3.h.n 16
80.t odd 4 2 1280.3.h.n 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.3.e.b 16 5.c odd 4 2
320.3.e.b 16 20.e even 4 2
320.3.e.b 16 40.i odd 4 2
320.3.e.b 16 40.k even 4 2
1280.3.h.n 16 80.i odd 4 2
1280.3.h.n 16 80.j even 4 2
1280.3.h.n 16 80.s even 4 2
1280.3.h.n 16 80.t odd 4 2
1600.3.g.k 16 1.a even 1 1 trivial
1600.3.g.k 16 4.b odd 2 1 inner
1600.3.g.k 16 5.b even 2 1 inner
1600.3.g.k 16 8.b even 2 1 inner
1600.3.g.k 16 8.d odd 2 1 inner
1600.3.g.k 16 20.d odd 2 1 inner
1600.3.g.k 16 40.e odd 2 1 inner
1600.3.g.k 16 40.f even 2 1 inner

## Hecke kernels

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

 $$T_{3}^{4} - 30 T_{3}^{2} + 96$$ $$T_{17}^{4} - 552 T_{17}^{2} + 24576$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 96 - 30 T^{2} + T^{4} )^{4}$$
$5$ $$T^{16}$$
$7$ $$( 600 + 54 T^{2} + T^{4} )^{4}$$
$11$ $$( 15376 - 440 T^{2} + T^{4} )^{4}$$
$13$ $$( 6144 + 276 T^{2} + T^{4} )^{4}$$
$17$ $$( 24576 - 552 T^{2} + T^{4} )^{4}$$
$19$ $$( -108 + T^{2} )^{8}$$
$23$ $$( 786264 + 2070 T^{2} + T^{4} )^{4}$$
$29$ $$( 16384 + 944 T^{2} + T^{4} )^{4}$$
$31$ $$( 262144 + 1040 T^{2} + T^{4} )^{4}$$
$37$ $$( 726624 + 1836 T^{2} + T^{4} )^{4}$$
$41$ $$( -1152 + 6 T + T^{2} )^{8}$$
$43$ $$( 1548384 - 3294 T^{2} + T^{4} )^{4}$$
$47$ $$( 411864 + 2550 T^{2} + T^{4} )^{4}$$
$53$ $$( 1935744 + 6324 T^{2} + T^{4} )^{4}$$
$59$ $$( 2704 - 10904 T^{2} + T^{4} )^{4}$$
$61$ $$( 12110400 + 6972 T^{2} + T^{4} )^{4}$$
$67$ $$( 67254624 - 16686 T^{2} + T^{4} )^{4}$$
$71$ $$( 230400 + 1104 T^{2} + T^{4} )^{4}$$
$73$ $$( 960000 - 9384 T^{2} + T^{4} )^{4}$$
$79$ $$( 149426176 + 27152 T^{2} + T^{4} )^{4}$$
$83$ $$( 89397600 - 19038 T^{2} + T^{4} )^{4}$$
$89$ $$( 4668 - 144 T + T^{2} )^{8}$$
$97$ $$( 147609600 - 27816 T^{2} + T^{4} )^{4}$$