# Properties

 Label 320.6.a.z Level 320 Weight 6 Character orbit 320.a Self dual yes Analytic conductor 51.323 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.81998080.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{11}$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -25 q^{5} + ( 6 \beta_{1} - \beta_{2} ) q^{7} + ( 181 + \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -25 q^{5} + ( 6 \beta_{1} - \beta_{2} ) q^{7} + ( 181 + \beta_{3} ) q^{9} + ( -15 \beta_{1} - \beta_{2} ) q^{11} + ( 146 + 3 \beta_{3} ) q^{13} + 25 \beta_{1} q^{15} + ( -190 - 3 \beta_{3} ) q^{17} + ( -57 \beta_{1} - 9 \beta_{2} ) q^{19} + ( -2456 - 14 \beta_{3} ) q^{21} + ( -6 \beta_{1} + \beta_{2} ) q^{23} + 625 q^{25} + ( -170 \beta_{1} + 24 \beta_{2} ) q^{27} + ( 42 + 18 \beta_{3} ) q^{29} + ( -312 \beta_{1} - 10 \beta_{2} ) q^{31} + ( 6448 + 7 \beta_{3} ) q^{33} + ( -150 \beta_{1} + 25 \beta_{2} ) q^{35} + ( -4934 + 6 \beta_{3} ) q^{37} + ( -842 \beta_{1} + 72 \beta_{2} ) q^{39} + ( 11906 - 21 \beta_{3} ) q^{41} + ( -99 \beta_{1} - 54 \beta_{2} ) q^{43} + ( -4525 - 25 \beta_{3} ) q^{45} + ( -408 \beta_{1} - 79 \beta_{2} ) q^{47} + ( 30337 + 51 \beta_{3} ) q^{49} + ( 886 \beta_{1} - 72 \beta_{2} ) q^{51} + ( -4374 - 15 \beta_{3} ) q^{53} + ( 375 \beta_{1} + 25 \beta_{2} ) q^{55} + ( 24960 - 15 \beta_{3} ) q^{57} + ( -411 \beta_{1} + 185 \beta_{2} ) q^{59} + ( 15506 + 84 \beta_{3} ) q^{61} + ( 4246 \beta_{1} - 93 \beta_{2} ) q^{63} + ( -3650 - 75 \beta_{3} ) q^{65} + ( 471 \beta_{1} - 170 \beta_{2} ) q^{67} + ( 2456 + 14 \beta_{3} ) q^{69} + ( 162 \beta_{1} - 112 \beta_{2} ) q^{71} + ( 29850 + 9 \beta_{3} ) q^{73} -625 \beta_{1} q^{75} + ( -4432 - 243 \beta_{3} ) q^{77} + ( 1020 \beta_{1} + 88 \beta_{2} ) q^{79} + ( 25985 + 119 \beta_{3} ) q^{81} + ( -2997 \beta_{1} - 52 \beta_{2} ) q^{83} + ( 4750 + 75 \beta_{3} ) q^{85} + ( -4218 \beta_{1} + 432 \beta_{2} ) q^{87} + ( 52330 - 90 \beta_{3} ) q^{89} + ( 10356 \beta_{1} + 118 \beta_{2} ) q^{91} + ( 133168 + 232 \beta_{3} ) q^{93} + ( 1425 \beta_{1} + 225 \beta_{2} ) q^{95} + ( -14782 - 165 \beta_{3} ) q^{97} + ( -4427 \beta_{1} + 411 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 100q^{5} + 724q^{9} + O(q^{10})$$ $$4q - 100q^{5} + 724q^{9} + 584q^{13} - 760q^{17} - 9824q^{21} + 2500q^{25} + 168q^{29} + 25792q^{33} - 19736q^{37} + 47624q^{41} - 18100q^{45} + 121348q^{49} - 17496q^{53} + 99840q^{57} + 62024q^{61} - 14600q^{65} + 9824q^{69} + 119400q^{73} - 17728q^{77} + 103940q^{81} + 19000q^{85} + 209320q^{89} + 532672q^{93} - 59128q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 100 x^{2} - 376 x - 367$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} - 4 \nu^{2} - 190 \nu - 364$$ $$\beta_{2}$$ $$=$$ $$($$$$26 \nu^{3} - 68 \nu^{2} - 2406 \nu - 3932$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$-32 \nu^{3} + 64 \nu^{2} + 3104 \nu + 5824$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 16 \beta_{1}$$$$)/64$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3 \beta_{2} + 29 \beta_{1} + 800$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$103 \beta_{3} - 24 \beta_{2} + 1784 \beta_{1} + 18048$$$$)/64$$

## 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
 11.6219 −2.18136 −7.51400 −1.92659
0 −27.0971 0 −25.0000 0 250.932 0 491.252 0
1.2 0 −10.6653 0 −25.0000 0 −176.978 0 −129.252 0
1.3 0 10.6653 0 −25.0000 0 176.978 0 −129.252 0
1.4 0 27.0971 0 −25.0000 0 −250.932 0 491.252 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 320.6.a.z 4
4.b odd 2 1 inner 320.6.a.z 4
8.b even 2 1 160.6.a.h 4
8.d odd 2 1 160.6.a.h 4
40.e odd 2 1 800.6.a.r 4
40.f even 2 1 800.6.a.r 4
40.i odd 4 2 800.6.c.l 8
40.k even 4 2 800.6.c.l 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.a.h 4 8.b even 2 1
160.6.a.h 4 8.d odd 2 1
320.6.a.z 4 1.a even 1 1 trivial
320.6.a.z 4 4.b odd 2 1 inner
800.6.a.r 4 40.e odd 2 1
800.6.a.r 4 40.f even 2 1
800.6.c.l 8 40.i odd 4 2
800.6.c.l 8 40.k even 4 2

## Atkin-Lehner signs

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 848 T_{3}^{2} + 83520$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(320))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 124 T^{2} + 25686 T^{4} + 7322076 T^{6} + 3486784401 T^{8}$$
$5$ $$( 1 + 25 T )^{4}$$
$7$ $$1 - 27060 T^{2} + 497649542 T^{4} - 7643780237940 T^{6} + 79792266297612001 T^{8}$$
$11$ $$1 + 382252 T^{2} + 87516901782 T^{4} + 9914632428581452 T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$
$13$ $$( 1 - 292 T - 102402 T^{2} - 108417556 T^{3} + 137858491849 T^{4} )^{2}$$
$17$ $$( 1 + 380 T + 2009510 T^{2} + 539545660 T^{3} + 2015993900449 T^{4} )^{2}$$
$19$ $$1 + 1633036 T^{2} + 154661897526 T^{4} + 10012251917374313836 T^{6} +$$$$37\!\cdots\!01$$$$T^{8}$$
$23$ $$1 + 25651084 T^{2} + 247347299659206 T^{4} +$$$$10\!\cdots\!16$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$( 1 - 84 T + 9837118 T^{2} - 1722936516 T^{3} + 420707233300201 T^{4} )^{2}$$
$31$ $$1 + 24283452 T^{2} + 1637830022018822 T^{4} +$$$$19\!\cdots\!52$$$$T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$
$37$ $$( 1 + 9868 T + 159567054 T^{2} + 684286167676 T^{3} + 4808584372417849 T^{4} )^{2}$$
$41$ $$( 1 - 23812 T + 331016342 T^{2} - 2758767858212 T^{3} + 13422659310152401 T^{4} )^{2}$$
$43$ $$1 + 385757980 T^{2} + 71059469286049782 T^{4} +$$$$83\!\cdots\!20$$$$T^{6} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 353765740 T^{2} + 66158288717340582 T^{4} +$$$$18\!\cdots\!60$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$( 1 + 8748 T + 833865262 T^{2} + 3658374172764 T^{3} + 174887470365513049 T^{4} )^{2}$$
$59$ $$1 + 488747308 T^{2} + 896846029819225302 T^{4} +$$$$24\!\cdots\!08$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$61$ $$( 1 - 31012 T + 1250446302 T^{2} - 26192620486612 T^{3} + 713342911662882601 T^{4} )^{2}$$
$67$ $$1 + 3336863100 T^{2} + 6361754088145172822 T^{4} +$$$$60\!\cdots\!00$$$$T^{6} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$1 + 6374750812 T^{2} + 16622748841310481702 T^{4} +$$$$20\!\cdots\!12$$$$T^{6} +$$$$10\!\cdots\!01$$$$T^{8}$$
$73$ $$( 1 - 59700 T + 5029368950 T^{2} - 123762374102100 T^{3} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$1 + 10884258108 T^{2} + 48452581383610561862 T^{4} +$$$$10\!\cdots\!08$$$$T^{6} +$$$$89\!\cdots\!01$$$$T^{8}$$
$83$ $$1 + 7906443964 T^{2} + 42876570344126662806 T^{4} +$$$$12\!\cdots\!36$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 - 104660 T + 13126874198 T^{2} - 584427661932340 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 29564 T + 14772618438 T^{2} + 253876127357948 T^{3} + 73742412689492826049 T^{4} )^{2}$$