# Properties

 Label 320.6.d.b Level 320 Weight 6 Character orbit 320.d Analytic conductor 51.323 Analytic rank 0 Dimension 8 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - 4 x^{7} - 384 x^{6} + 506 x^{5} + 49869 x^{4} + 29654 x^{3} - 2235516 x^{2} - 1528906 x + 34180205$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{3} -25 \beta_{1} q^{5} + ( 40 + 3 \beta_{2} + \beta_{6} ) q^{7} + ( -149 - 5 \beta_{2} + \beta_{3} - \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{3} -25 \beta_{1} q^{5} + ( 40 + 3 \beta_{2} + \beta_{6} ) q^{7} + ( -149 - 5 \beta_{2} + \beta_{3} - \beta_{6} ) q^{9} + ( -92 \beta_{1} + 2 \beta_{5} - 3 \beta_{7} ) q^{11} + ( -220 \beta_{1} - 3 \beta_{4} + 5 \beta_{5} - \beta_{7} ) q^{13} + 25 \beta_{2} q^{15} + ( -300 - 3 \beta_{2} - 5 \beta_{3} - 5 \beta_{6} ) q^{17} + ( 204 \beta_{1} - 30 \beta_{4} - 14 \beta_{5} + \beta_{7} ) q^{19} + ( 1090 \beta_{1} - 120 \beta_{4} - 20 \beta_{5} + 10 \beta_{7} ) q^{21} + ( -220 + 117 \beta_{2} - 10 \beta_{3} - 11 \beta_{6} ) q^{23} -625 q^{25} + ( -1980 \beta_{1} + 58 \beta_{4} + 30 \beta_{5} ) q^{27} + ( 1480 \beta_{1} - 210 \beta_{4} + 10 \beta_{5} - 20 \beta_{7} ) q^{29} + ( -3880 + 150 \beta_{2} + 40 \beta_{3} ) q^{31} + ( 470 + 21 \beta_{2} + 35 \beta_{3} + 55 \beta_{6} ) q^{33} + ( -1000 \beta_{1} + 75 \beta_{4} + 25 \beta_{7} ) q^{35} + ( 1230 \beta_{1} - 312 \beta_{4} + 40 \beta_{5} - 4 \beta_{7} ) q^{37} + ( -560 + 450 \beta_{2} - 20 \beta_{3} + 50 \beta_{6} ) q^{39} + ( 2698 + 165 \beta_{2} - 33 \beta_{3} - 47 \beta_{6} ) q^{41} + ( 7020 \beta_{1} - 135 \beta_{4} - 70 \beta_{5} + 50 \beta_{7} ) q^{43} + ( 3725 \beta_{1} - 125 \beta_{4} - 25 \beta_{5} - 25 \beta_{7} ) q^{45} + ( -5960 + 321 \beta_{2} - 80 \beta_{3} - 43 \beta_{6} ) q^{47} + ( 10353 + 195 \beta_{2} - 95 \beta_{3} + 35 \beta_{6} ) q^{49} + ( -216 \beta_{1} + 330 \beta_{4} + 48 \beta_{5} - 102 \beta_{7} ) q^{51} + ( 10020 \beta_{1} - 141 \beta_{4} - 45 \beta_{5} + 153 \beta_{7} ) q^{53} + ( -2300 + 50 \beta_{3} + 75 \beta_{6} ) q^{55} + ( -13330 - 1157 \beta_{2} + 125 \beta_{3} - 155 \beta_{6} ) q^{57} + ( 13524 \beta_{1} - 570 \beta_{4} + 30 \beta_{5} - 135 \beta_{7} ) q^{59} + ( 17350 \beta_{1} + 1290 \beta_{4} + 50 \beta_{5} + 30 \beta_{7} ) q^{61} + ( -40300 - 1551 \beta_{2} + 110 \beta_{3} - 167 \beta_{6} ) q^{63} + ( -5500 + 75 \beta_{2} + 125 \beta_{3} + 25 \beta_{6} ) q^{65} + ( -18300 \beta_{1} - 723 \beta_{4} - 10 \beta_{5} + 110 \beta_{7} ) q^{67} + ( 47870 \beta_{1} - 270 \beta_{4} - 10 \beta_{5} - 340 \beta_{7} ) q^{69} + ( -30840 - 1290 \beta_{2} - 180 \beta_{3} - 180 \beta_{6} ) q^{71} + ( 5800 + 141 \beta_{2} - 205 \beta_{3} - 5 \beta_{6} ) q^{73} + 625 \beta_{4} q^{75} + ( 49990 \beta_{1} + 2709 \beta_{4} - 155 \beta_{5} + 103 \beta_{7} ) q^{77} + ( -40720 - 1020 \beta_{2} + 60 \beta_{3} + 410 \beta_{6} ) q^{79} + ( -10291 + 2915 \beta_{2} - 55 \beta_{3} + 55 \beta_{6} ) q^{81} + ( -14300 \beta_{1} - 69 \beta_{4} + 110 \beta_{5} - 60 \beta_{7} ) q^{83} + ( 7500 \beta_{1} - 75 \beta_{4} + 125 \beta_{5} - 125 \beta_{7} ) q^{85} + ( -79540 - 3210 \beta_{2} + 470 \beta_{3} + 130 \beta_{6} ) q^{87} + ( -9774 - 1470 \beta_{2} + 254 \beta_{3} + 206 \beta_{6} ) q^{89} + ( -16240 \beta_{1} + 4530 \beta_{4} - 220 \beta_{5} + 440 \beta_{7} ) q^{91} + ( 54560 \beta_{1} + 5610 \beta_{4} + 170 \beta_{5} + 170 \beta_{7} ) q^{93} + ( 5100 + 750 \beta_{2} - 350 \beta_{3} - 25 \beta_{6} ) q^{95} + ( 3120 - 3015 \beta_{2} - 145 \beta_{3} + 475 \beta_{6} ) q^{97} + ( -22564 \beta_{1} - 1980 \beta_{4} - 190 \beta_{5} + 245 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 320q^{7} - 1192q^{9} + O(q^{10})$$ $$8q + 320q^{7} - 1192q^{9} - 2400q^{17} - 1760q^{23} - 5000q^{25} - 31040q^{31} + 3760q^{33} - 4480q^{39} + 21584q^{41} - 47680q^{47} + 82824q^{49} - 18400q^{55} - 106640q^{57} - 322400q^{63} - 44000q^{65} - 246720q^{71} + 46400q^{73} - 325760q^{79} - 82328q^{81} - 636320q^{87} - 78192q^{89} + 40800q^{95} + 24960q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{7} - 384 x^{6} + 506 x^{5} + 49869 x^{4} + 29654 x^{3} - 2235516 x^{2} - 1528906 x + 34180205$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$1351312 \nu^{7} - 13653287 \nu^{6} - 394224929 \nu^{5} + 2862879190 \nu^{4} + 38223976863 \nu^{3} - 163085600918 \nu^{2} - 987814371481 \nu + 3699160158415$$$$)/ 281446960630$$ $$\beta_{2}$$ $$=$$ $$($$$$-1351312 \nu^{7} + 13653287 \nu^{6} + 394224929 \nu^{5} - 2862879190 \nu^{4} - 38223976863 \nu^{3} + 163085600918 \nu^{2} + 1550708292741 \nu - 3980607119045$$$$)/ 281446960630$$ $$\beta_{3}$$ $$=$$ $$($$$$46680604 \nu^{7} - 595364489 \nu^{6} - 11237259467 \nu^{5} + 129626329330 \nu^{4} + 680074862701 \nu^{3} - 8470823112146 \nu^{2} + 6684424237957 \nu + 233594355624595$$$$)/ 844340881890$$ $$\beta_{4}$$ $$=$$ $$($$$$324498 \nu^{7} - 4782019 \nu^{6} - 86408283 \nu^{5} + 1112583304 \nu^{4} + 8082523469 \nu^{3} - 76894837308 \nu^{2} - 227603504321 \nu + 1741708282831$$$$)/ 5117217466$$ $$\beta_{5}$$ $$=$$ $$($$$$-8770602 \nu^{7} + 204376019 \nu^{6} + 1922738643 \nu^{5} - 61548799060 \nu^{4} - 143197714893 \nu^{3} + 5939121203576 \nu^{2} + 7063580525697 \nu - 148559382301075$$$$)/ 76758261990$$ $$\beta_{6}$$ $$=$$ $$($$$$87017312 \nu^{7} - 1189115032 \nu^{6} - 22832683396 \nu^{5} + 269861003750 \nu^{4} + 1960333630208 \nu^{3} - 16457519954998 \nu^{2} - 47531360053324 \nu + 269387125903700$$$$)/ 422170440945$$ $$\beta_{7}$$ $$=$$ $$($$$$109345770 \nu^{7} - 502754801 \nu^{6} - 43714650957 \nu^{5} + 138978838510 \nu^{4} + 5044385010495 \nu^{3} - 8601384396614 \nu^{2} - 140104059489243 \nu + 305925739736305$$$$)/ 422170440945$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{6} - 2 \beta_{4} - \beta_{3} + 7 \beta_{2} + 2 \beta_{1} + 392$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{7} + 3 \beta_{6} - 3 \beta_{5} - 21 \beta_{4} - 33 \beta_{3} + 559 \beta_{2} + 1178 \beta_{1} + 3154$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-3 \beta_{7} + 196 \beta_{6} - 33 \beta_{5} - 561 \beta_{4} - 226 \beta_{3} + 2182 \beta_{2} + 3156 \beta_{1} + 56086$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$-985 \beta_{7} + 1635 \beta_{6} - 1135 \beta_{5} - 10945 \beta_{4} - 7515 \beta_{3} + 95369 \beta_{2} + 282394 \beta_{1} + 915374$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-2460 \beta_{7} + 33721 \beta_{6} - 11355 \beta_{5} - 144457 \beta_{4} - 47416 \beta_{3} + 512212 \beta_{2} + 1380952 \beta_{1} + 9668402$$$$)/4$$ $$\nu^{7}$$ $$=$$ $$($$$$-119749 \beta_{7} + 242214 \beta_{6} - 167944 \beta_{5} - 1811908 \beta_{4} - 750429 \beta_{3} + 8864437 \beta_{2} + 34334284 \beta_{1} + 105600322$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1
 14.2856 − 0.500000i −10.3099 + 0.500000i −7.33903 + 0.500000i 5.36332 − 0.500000i 5.36332 + 0.500000i −7.33903 − 0.500000i −10.3099 − 0.500000i 14.2856 + 0.500000i
0 27.5713i 0 25.0000i 0 220.359 0 −517.174 0
161.2 0 21.6198i 0 25.0000i 0 169.466 0 −224.417 0
161.3 0 15.6781i 0 25.0000i 0 −164.681 0 −2.80186 0
161.4 0 9.72664i 0 25.0000i 0 −65.1432 0 148.393 0
161.5 0 9.72664i 0 25.0000i 0 −65.1432 0 148.393 0
161.6 0 15.6781i 0 25.0000i 0 −164.681 0 −2.80186 0
161.7 0 21.6198i 0 25.0000i 0 169.466 0 −224.417 0
161.8 0 27.5713i 0 25.0000i 0 220.359 0 −517.174 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 161.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.d.b yes 8
4.b odd 2 1 320.6.d.a 8
8.b even 2 1 inner 320.6.d.b yes 8
8.d odd 2 1 320.6.d.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.d.a 8 4.b odd 2 1
320.6.d.a 8 8.d odd 2 1
320.6.d.b yes 8 1.a even 1 1 trivial
320.6.d.b yes 8 8.b 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 $$S_{6}^{\mathrm{new}}(320, [\chi])$$:

 $$T_{3}^{8} + 1568 T_{3}^{6} + 796456 T_{3}^{4} + 149500800 T_{3}^{2} + 8262810000$$ $$T_{7}^{4} - 160 T_{7}^{3} - 41520 T_{7}^{2} + 4400400 T_{7} + 400612500$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 376 T^{2} + 163684 T^{4} - 39360744 T^{6} + 11878187814 T^{8} - 2324212572456 T^{10} + 570730817893284 T^{12} - 77415065667588024 T^{14} + 12157665459056928801 T^{16}$$
$5$ $$( 1 + 625 T^{2} )^{4}$$
$7$ $$( 1 - 160 T + 25708 T^{2} - 3666960 T^{3} + 699810714 T^{4} - 61630596720 T^{5} + 7261873701292 T^{6} - 759609841590880 T^{7} + 79792266297612001 T^{8} )^{2}$$
$11$ $$1 - 461928 T^{2} + 80897449276 T^{4} - 9751521734727448 T^{6} +$$$$14\!\cdots\!94$$$$T^{8} -$$$$25\!\cdots\!48$$$$T^{10} +$$$$54\!\cdots\!76$$$$T^{12} -$$$$80\!\cdots\!28$$$$T^{14} +$$$$45\!\cdots\!01$$$$T^{16}$$
$13$ $$1 - 1173304 T^{2} + 912998161852 T^{4} - 514796045425358792 T^{6} +$$$$21\!\cdots\!70$$$$T^{8} -$$$$70\!\cdots\!08$$$$T^{10} +$$$$17\!\cdots\!52$$$$T^{12} -$$$$30\!\cdots\!96$$$$T^{14} +$$$$36\!\cdots\!01$$$$T^{16}$$
$17$ $$( 1 + 1200 T + 4842092 T^{2} + 4006103760 T^{3} + 9533675124390 T^{4} + 5688094466362320 T^{5} + 9761627937412899308 T^{6} +$$$$34\!\cdots\!00$$$$T^{7} +$$$$40\!\cdots\!01$$$$T^{8} )^{2}$$
$19$ $$1 - 4628552 T^{2} + 15129915669436 T^{4} - 43933554813794877112 T^{6} +$$$$13\!\cdots\!54$$$$T^{8} -$$$$26\!\cdots\!12$$$$T^{10} +$$$$56\!\cdots\!36$$$$T^{12} -$$$$10\!\cdots\!52$$$$T^{14} +$$$$14\!\cdots\!01$$$$T^{16}$$
$23$ $$( 1 + 880 T + 8141852 T^{2} + 31216146720 T^{3} + 19333174339674 T^{4} + 200917827428244960 T^{5} +$$$$33\!\cdots\!48$$$$T^{6} +$$$$23\!\cdots\!60$$$$T^{7} +$$$$17\!\cdots\!01$$$$T^{8} )^{2}$$
$29$ $$1 - 57762792 T^{2} + 1773990452604028 T^{4} -$$$$43\!\cdots\!44$$$$T^{6} +$$$$95\!\cdots\!70$$$$T^{8} -$$$$18\!\cdots\!44$$$$T^{10} +$$$$31\!\cdots\!28$$$$T^{12} -$$$$43\!\cdots\!92$$$$T^{14} +$$$$31\!\cdots\!01$$$$T^{16}$$
$31$ $$( 1 + 15520 T + 133260604 T^{2} + 763025830560 T^{3} + 3983065379572806 T^{4} + 21844781720002654560 T^{5} +$$$$10\!\cdots\!04$$$$T^{6} +$$$$36\!\cdots\!20$$$$T^{7} +$$$$67\!\cdots\!01$$$$T^{8} )^{2}$$
$37$ $$1 - 301131416 T^{2} + 47822074973756092 T^{4} -$$$$50\!\cdots\!08$$$$T^{6} +$$$$40\!\cdots\!70$$$$T^{8} -$$$$24\!\cdots\!92$$$$T^{10} +$$$$11\!\cdots\!92$$$$T^{12} -$$$$33\!\cdots\!84$$$$T^{14} +$$$$53\!\cdots\!01$$$$T^{16}$$
$41$ $$( 1 - 10792 T + 381689636 T^{2} - 2595640517112 T^{3} + 58822716652823334 T^{4} -$$$$30\!\cdots\!12$$$$T^{5} +$$$$51\!\cdots\!36$$$$T^{6} -$$$$16\!\cdots\!92$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{2}$$
$43$ $$1 - 542148344 T^{2} + 153567024284830372 T^{4} -$$$$32\!\cdots\!92$$$$T^{6} +$$$$53\!\cdots\!70$$$$T^{8} -$$$$69\!\cdots\!08$$$$T^{10} +$$$$71\!\cdots\!72$$$$T^{12} -$$$$54\!\cdots\!56$$$$T^{14} +$$$$21\!\cdots\!01$$$$T^{16}$$
$47$ $$( 1 + 23840 T + 811201148 T^{2} + 13600000745040 T^{3} + 275458842496788474 T^{4} +$$$$31\!\cdots\!80$$$$T^{5} +$$$$42\!\cdots\!52$$$$T^{6} +$$$$28\!\cdots\!20$$$$T^{7} +$$$$27\!\cdots\!01$$$$T^{8} )^{2}$$
$53$ $$1 - 984292984 T^{2} + 650040420165524092 T^{4} -$$$$31\!\cdots\!92$$$$T^{6} +$$$$12\!\cdots\!70$$$$T^{8} -$$$$55\!\cdots\!08$$$$T^{10} +$$$$19\!\cdots\!92$$$$T^{12} -$$$$52\!\cdots\!16$$$$T^{14} +$$$$93\!\cdots\!01$$$$T^{16}$$
$59$ $$1 - 3042039688 T^{2} + 4949915655184301308 T^{4} -$$$$54\!\cdots\!56$$$$T^{6} +$$$$44\!\cdots\!70$$$$T^{8} -$$$$27\!\cdots\!56$$$$T^{10} +$$$$12\!\cdots\!08$$$$T^{12} -$$$$40\!\cdots\!88$$$$T^{14} +$$$$68\!\cdots\!01$$$$T^{16}$$
$61$ $$1 - 2582224408 T^{2} + 4867570283316136828 T^{4} -$$$$60\!\cdots\!56$$$$T^{6} +$$$$59\!\cdots\!70$$$$T^{8} -$$$$42\!\cdots\!56$$$$T^{10} +$$$$24\!\cdots\!28$$$$T^{12} -$$$$93\!\cdots\!08$$$$T^{14} +$$$$25\!\cdots\!01$$$$T^{16}$$
$67$ $$1 - 7537851704 T^{2} + 25204334234273950564 T^{4} -$$$$51\!\cdots\!16$$$$T^{6} +$$$$78\!\cdots\!94$$$$T^{8} -$$$$94\!\cdots\!84$$$$T^{10} +$$$$83\!\cdots\!64$$$$T^{12} -$$$$45\!\cdots\!96$$$$T^{14} +$$$$11\!\cdots\!01$$$$T^{16}$$
$71$ $$( 1 + 123360 T + 10006413404 T^{2} + 529309926298080 T^{3} + 24936343678058479206 T^{4} +$$$$95\!\cdots\!80$$$$T^{5} +$$$$32\!\cdots\!04$$$$T^{6} +$$$$72\!\cdots\!60$$$$T^{7} +$$$$10\!\cdots\!01$$$$T^{8} )^{2}$$
$73$ $$( 1 - 23200 T + 7007019628 T^{2} - 115342151623520 T^{3} + 20469715216228479110 T^{4} -$$$$23\!\cdots\!60$$$$T^{5} +$$$$30\!\cdots\!72$$$$T^{6} -$$$$20\!\cdots\!00$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8} )^{2}$$
$79$ $$( 1 + 162880 T + 14181869596 T^{2} + 858599705367360 T^{3} + 48361267115636995206 T^{4} +$$$$26\!\cdots\!40$$$$T^{5} +$$$$13\!\cdots\!96$$$$T^{6} +$$$$47\!\cdots\!20$$$$T^{7} +$$$$89\!\cdots\!01$$$$T^{8} )^{2}$$
$83$ $$1 - 29848826136 T^{2} +$$$$39\!\cdots\!24$$$$T^{4} -$$$$30\!\cdots\!64$$$$T^{6} +$$$$14\!\cdots\!54$$$$T^{8} -$$$$47\!\cdots\!36$$$$T^{10} +$$$$95\!\cdots\!24$$$$T^{12} -$$$$11\!\cdots\!64$$$$T^{14} +$$$$57\!\cdots\!01$$$$T^{16}$$
$89$ $$( 1 + 39096 T + 18076226204 T^{2} + 489497603288904 T^{3} +$$$$13\!\cdots\!94$$$$T^{4} +$$$$27\!\cdots\!96$$$$T^{5} +$$$$56\!\cdots\!04$$$$T^{6} +$$$$68\!\cdots\!04$$$$T^{7} +$$$$97\!\cdots\!01$$$$T^{8} )^{2}$$
$97$ $$( 1 - 12480 T + 14325999628 T^{2} - 824172284150080 T^{3} + 95392462085617046694 T^{4} -$$$$70\!\cdots\!60$$$$T^{5} +$$$$10\!\cdots\!72$$$$T^{6} -$$$$79\!\cdots\!40$$$$T^{7} +$$$$54\!\cdots\!01$$$$T^{8} )^{2}$$