# Properties

 Label 546.8.a.r Level $546$ Weight $8$ Character orbit 546.a Self dual yes Analytic conductor $170.562$ Analytic rank $0$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 546.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$170.562223914$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ Defining polynomial: $$x^{6} - x^{5} - 264981 x^{4} + 17519669 x^{3} + 15113237808 x^{2} - 1787613752904 x - 21984668630064$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{3}\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( 34 - \beta_{1} ) q^{5} -216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} +O(q^{10})$$ $$q + 8 q^{2} -27 q^{3} + 64 q^{4} + ( 34 - \beta_{1} ) q^{5} -216 q^{6} -343 q^{7} + 512 q^{8} + 729 q^{9} + ( 272 - 8 \beta_{1} ) q^{10} + ( -448 - 4 \beta_{1} - \beta_{2} + \beta_{4} ) q^{11} -1728 q^{12} + 2197 q^{13} -2744 q^{14} + ( -918 + 27 \beta_{1} ) q^{15} + 4096 q^{16} + ( 487 - 10 \beta_{1} + 3 \beta_{4} - \beta_{5} ) q^{17} + 5832 q^{18} + ( -2173 - 23 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{19} + ( 2176 - 64 \beta_{1} ) q^{20} + 9261 q^{21} + ( -3584 - 32 \beta_{1} - 8 \beta_{2} + 8 \beta_{4} ) q^{22} + ( 1923 + 15 \beta_{1} - 15 \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{23} -13824 q^{24} + ( 11374 - 165 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} - 16 \beta_{4} + 12 \beta_{5} ) q^{25} + 17576 q^{26} -19683 q^{27} -21952 q^{28} + ( -15342 - 139 \beta_{1} + 34 \beta_{2} - 7 \beta_{3} + 16 \beta_{4} - 45 \beta_{5} ) q^{29} + ( -7344 + 216 \beta_{1} ) q^{30} + ( -13849 - 75 \beta_{1} + 6 \beta_{2} - 27 \beta_{3} + 5 \beta_{4} + 23 \beta_{5} ) q^{31} + 32768 q^{32} + ( 12096 + 108 \beta_{1} + 27 \beta_{2} - 27 \beta_{4} ) q^{33} + ( 3896 - 80 \beta_{1} + 24 \beta_{4} - 8 \beta_{5} ) q^{34} + ( -11662 + 343 \beta_{1} ) q^{35} + 46656 q^{36} + ( -44099 - 636 \beta_{1} - 12 \beta_{2} - 12 \beta_{3} - 39 \beta_{4} + 34 \beta_{5} ) q^{37} + ( -17384 - 184 \beta_{1} - 40 \beta_{2} + 32 \beta_{3} + 16 \beta_{4} + 16 \beta_{5} ) q^{38} -59319 q^{39} + ( 17408 - 512 \beta_{1} ) q^{40} + ( -61079 - 976 \beta_{1} - 33 \beta_{2} - 48 \beta_{3} - 22 \beta_{4} - 90 \beta_{5} ) q^{41} + 74088 q^{42} + ( 75998 - 861 \beta_{1} + 36 \beta_{2} + 91 \beta_{3} - 44 \beta_{4} + 41 \beta_{5} ) q^{43} + ( -28672 - 256 \beta_{1} - 64 \beta_{2} + 64 \beta_{4} ) q^{44} + ( 24786 - 729 \beta_{1} ) q^{45} + ( 15384 + 120 \beta_{1} - 120 \beta_{2} + 32 \beta_{4} - 8 \beta_{5} ) q^{46} + ( 122466 - 787 \beta_{1} + 63 \beta_{2} + 23 \beta_{3} + 57 \beta_{4} + 68 \beta_{5} ) q^{47} -110592 q^{48} + 117649 q^{49} + ( 90992 - 1320 \beta_{1} + 72 \beta_{2} + 16 \beta_{3} - 128 \beta_{4} + 96 \beta_{5} ) q^{50} + ( -13149 + 270 \beta_{1} - 81 \beta_{4} + 27 \beta_{5} ) q^{51} + 140608 q^{52} + ( -262839 - 697 \beta_{1} - 100 \beta_{2} - 65 \beta_{3} + 151 \beta_{4} + 11 \beta_{5} ) q^{53} -157464 q^{54} + ( 371449 + 2640 \beta_{1} - 152 \beta_{2} + 178 \beta_{3} - 89 \beta_{4} - 82 \beta_{5} ) q^{55} -175616 q^{56} + ( 58671 + 621 \beta_{1} + 135 \beta_{2} - 108 \beta_{3} - 54 \beta_{4} - 54 \beta_{5} ) q^{57} + ( -122736 - 1112 \beta_{1} + 272 \beta_{2} - 56 \beta_{3} + 128 \beta_{4} - 360 \beta_{5} ) q^{58} + ( 343303 + 2622 \beta_{1} - 37 \beta_{2} + 98 \beta_{3} - 166 \beta_{4} + 195 \beta_{5} ) q^{59} + ( -58752 + 1728 \beta_{1} ) q^{60} + ( -44844 - 1416 \beta_{1} + 219 \beta_{2} - 67 \beta_{3} + 37 \beta_{4} - 243 \beta_{5} ) q^{61} + ( -110792 - 600 \beta_{1} + 48 \beta_{2} - 216 \beta_{3} + 40 \beta_{4} + 184 \beta_{5} ) q^{62} -250047 q^{63} + 262144 q^{64} + ( 74698 - 2197 \beta_{1} ) q^{65} + ( 96768 + 864 \beta_{1} + 216 \beta_{2} - 216 \beta_{4} ) q^{66} + ( -126730 + 938 \beta_{1} + 306 \beta_{2} - 32 \beta_{3} - 44 \beta_{4} + 658 \beta_{5} ) q^{67} + ( 31168 - 640 \beta_{1} + 192 \beta_{4} - 64 \beta_{5} ) q^{68} + ( -51921 - 405 \beta_{1} + 405 \beta_{2} - 108 \beta_{4} + 27 \beta_{5} ) q^{69} + ( -93296 + 2744 \beta_{1} ) q^{70} + ( -1022999 + 1292 \beta_{1} - 85 \beta_{2} + 180 \beta_{3} + 676 \beta_{4} - 662 \beta_{5} ) q^{71} + 373248 q^{72} + ( 1060930 - 2765 \beta_{1} - 178 \beta_{2} + 273 \beta_{3} - 736 \beta_{4} - 323 \beta_{5} ) q^{73} + ( -352792 - 5088 \beta_{1} - 96 \beta_{2} - 96 \beta_{3} - 312 \beta_{4} + 272 \beta_{5} ) q^{74} + ( -307098 + 4455 \beta_{1} - 243 \beta_{2} - 54 \beta_{3} + 432 \beta_{4} - 324 \beta_{5} ) q^{75} + ( -139072 - 1472 \beta_{1} - 320 \beta_{2} + 256 \beta_{3} + 128 \beta_{4} + 128 \beta_{5} ) q^{76} + ( 153664 + 1372 \beta_{1} + 343 \beta_{2} - 343 \beta_{4} ) q^{77} -474552 q^{78} + ( -148863 - 9683 \beta_{1} + 62 \beta_{2} - 751 \beta_{3} - 85 \beta_{4} + 851 \beta_{5} ) q^{79} + ( 139264 - 4096 \beta_{1} ) q^{80} + 531441 q^{81} + ( -488632 - 7808 \beta_{1} - 264 \beta_{2} - 384 \beta_{3} - 176 \beta_{4} - 720 \beta_{5} ) q^{82} + ( 551344 + 3313 \beta_{1} - 1657 \beta_{2} - 325 \beta_{3} + 423 \beta_{4} - 890 \beta_{5} ) q^{83} + 592704 q^{84} + ( 883885 + 5366 \beta_{1} + 188 \beta_{2} + 228 \beta_{3} - 209 \beta_{4} + 48 \beta_{5} ) q^{85} + ( 607984 - 6888 \beta_{1} + 288 \beta_{2} + 728 \beta_{3} - 352 \beta_{4} + 328 \beta_{5} ) q^{86} + ( 414234 + 3753 \beta_{1} - 918 \beta_{2} + 189 \beta_{3} - 432 \beta_{4} + 1215 \beta_{5} ) q^{87} + ( -229376 - 2048 \beta_{1} - 512 \beta_{2} + 512 \beta_{4} ) q^{88} + ( 1867538 + 5257 \beta_{1} + 817 \beta_{2} + 189 \beta_{3} + 793 \beta_{4} + 1115 \beta_{5} ) q^{89} + ( 198288 - 5832 \beta_{1} ) q^{90} -753571 q^{91} + ( 123072 + 960 \beta_{1} - 960 \beta_{2} + 256 \beta_{4} - 64 \beta_{5} ) q^{92} + ( 373923 + 2025 \beta_{1} - 162 \beta_{2} + 729 \beta_{3} - 135 \beta_{4} - 621 \beta_{5} ) q^{93} + ( 979728 - 6296 \beta_{1} + 504 \beta_{2} + 184 \beta_{3} + 456 \beta_{4} + 544 \beta_{5} ) q^{94} + ( 2151469 + 2167 \beta_{1} - 1907 \beta_{2} + 230 \beta_{3} + 430 \beta_{4} - 970 \beta_{5} ) q^{95} -884736 q^{96} + ( 4778503 + 12413 \beta_{1} + 840 \beta_{2} - 335 \beta_{3} + 701 \beta_{4} + 111 \beta_{5} ) q^{97} + 941192 q^{98} + ( -326592 - 2916 \beta_{1} - 729 \beta_{2} + 729 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + O(q^{10})$$ $$6 q + 48 q^{2} - 162 q^{3} + 384 q^{4} + 203 q^{5} - 1296 q^{6} - 2058 q^{7} + 3072 q^{8} + 4374 q^{9} + 1624 q^{10} - 2690 q^{11} - 10368 q^{12} + 13182 q^{13} - 16464 q^{14} - 5481 q^{15} + 24576 q^{16} + 2910 q^{17} + 34992 q^{18} - 13055 q^{19} + 12992 q^{20} + 55566 q^{21} - 21520 q^{22} + 11581 q^{23} - 82944 q^{24} + 68081 q^{25} + 105456 q^{26} - 118098 q^{27} - 131712 q^{28} - 92335 q^{29} - 43848 q^{30} - 83081 q^{31} + 196608 q^{32} + 72630 q^{33} + 23280 q^{34} - 69629 q^{35} + 279936 q^{36} - 265114 q^{37} - 104440 q^{38} - 355914 q^{39} + 103936 q^{40} - 367468 q^{41} + 444528 q^{42} + 454955 q^{43} - 172160 q^{44} + 147987 q^{45} + 92648 q^{46} + 733973 q^{47} - 663552 q^{48} + 705894 q^{49} + 544648 q^{50} - 78570 q^{51} + 843648 q^{52} - 1577379 q^{53} - 944784 q^{54} + 2231118 q^{55} - 1053696 q^{56} + 352485 q^{57} - 738680 q^{58} + 2062708 q^{59} - 350784 q^{60} - 271270 q^{61} - 664648 q^{62} - 1500282 q^{63} + 1572864 q^{64} + 445991 q^{65} + 581040 q^{66} - 758674 q^{67} + 186240 q^{68} - 312687 q^{69} - 557032 q^{70} - 6138216 q^{71} + 2239488 q^{72} + 6361979 q^{73} - 2120912 q^{74} - 1838187 q^{75} - 835520 q^{76} + 922670 q^{77} - 2847312 q^{78} - 899781 q^{79} + 831488 q^{80} + 3188646 q^{81} - 2939744 q^{82} + 3313561 q^{83} + 3556224 q^{84} + 5307940 q^{85} + 3639640 q^{86} + 2493045 q^{87} - 1377280 q^{88} + 11210703 q^{89} + 1183896 q^{90} - 4521426 q^{91} + 741184 q^{92} + 2243187 q^{93} + 5871784 q^{94} + 12912395 q^{95} - 5308416 q^{96} + 28682643 q^{97} + 5647152 q^{98} - 1961010 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 264981 x^{4} + 17519669 x^{3} + 15113237808 x^{2} - 1787613752904 x - 21984668630064$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$664523 \nu^{5} + 1394854154 \nu^{4} - 75579217837 \nu^{3} - 255406399982096 \nu^{2} + 5894885053583340 \nu + 4879576437327548348$$$$)/ 1355267466296380$$ $$\beta_{3}$$ $$=$$ $$($$$$-46974981 \nu^{5} + 15607782922 \nu^{4} + 22419068741319 \nu^{3} - 2839343780808048 \nu^{2} - 1985763386887708920 \nu + 127070152402416763344$$$$)/ 27105349325927600$$ $$\beta_{4}$$ $$=$$ $$($$$$-1679580891 \nu^{5} - 237883755898 \nu^{4} + 392168012091729 \nu^{3} + 22794539601833512 \nu^{2} - 18222872340416428280 \nu + 622583957722034939344$$$$)/ 108421397303710400$$ $$\beta_{5}$$ $$=$$ $$($$$$-1123997957 \nu^{5} - 205637389526 \nu^{4} + 256239695015823 \nu^{3} + 28322557881942504 \nu^{2} - 11225303835486719880 \nu - 172782684689345739792$$$$)/ 54210698651855200$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$12 \beta_{5} - 16 \beta_{4} + 2 \beta_{3} + 9 \beta_{2} - 97 \beta_{1} + 88343$$ $$\nu^{3}$$ $$=$$ $$-880 \beta_{5} + 800 \beta_{4} + 2540 \beta_{3} - 2959 \beta_{2} + 151193 \beta_{1} - 8652389$$ $$\nu^{4}$$ $$=$$ $$2326902 \beta_{5} - 3046716 \beta_{4} + 223442 \beta_{3} + 2927757 \beta_{2} - 26613237 \beta_{1} + 13322692095$$ $$\nu^{5}$$ $$=$$ $$-372179492 \beta_{5} + 336610536 \beta_{4} + 588565048 \beta_{3} - 983423979 \beta_{2} + 26905505169 \beta_{1} - 2337559494441$$

## 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
 390.179 212.575 170.698 −11.2455 −298.282 −462.924
8.00000 −27.0000 64.0000 −356.179 −216.000 −343.000 512.000 729.000 −2849.43
1.2 8.00000 −27.0000 64.0000 −178.575 −216.000 −343.000 512.000 729.000 −1428.60
1.3 8.00000 −27.0000 64.0000 −136.698 −216.000 −343.000 512.000 729.000 −1093.58
1.4 8.00000 −27.0000 64.0000 45.2455 −216.000 −343.000 512.000 729.000 361.964
1.5 8.00000 −27.0000 64.0000 332.282 −216.000 −343.000 512.000 729.000 2658.26
1.6 8.00000 −27.0000 64.0000 496.924 −216.000 −343.000 512.000 729.000 3975.39
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.8.a.r 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.8.a.r 6 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} - 203 T_{5}^{5} - 247811 T_{5}^{4} + 17743227 T_{5}^{3} + 15081987830 T_{5}^{2} + 740548691900 T_{5} -$$$$64\!\cdots\!00$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(546))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -8 + T )^{6}$$
$3$ $$( 27 + T )^{6}$$
$5$ $$-64956644533000 + 740548691900 T + 15081987830 T^{2} + 17743227 T^{3} - 247811 T^{4} - 203 T^{5} + T^{6}$$
$7$ $$( 343 + T )^{6}$$
$11$ $$-$$$$69\!\cdots\!52$$$$+ 467293647780094464 T + 1186072365703064 T^{2} - 71234800866 T^{3} - 63407793 T^{4} + 2690 T^{5} + T^{6}$$
$13$ $$( -2197 + T )^{6}$$
$17$ $$13\!\cdots\!12$$$$+ 12923679858498932232 T + 361664744079324 T^{2} - 2135545970490 T^{3} - 436403085 T^{4} - 2910 T^{5} + T^{6}$$
$19$ $$-$$$$14\!\cdots\!52$$$$+$$$$38\!\cdots\!48$$$$T + 633698604047882516 T^{2} - 20521895328119 T^{3} - 2322953901 T^{4} + 13055 T^{5} + T^{6}$$
$23$ $$-$$$$94\!\cdots\!84$$$$+$$$$99\!\cdots\!64$$$$T + 10001828879816546540 T^{2} - 1469159533595 T^{3} - 7191320581 T^{4} - 11581 T^{5} + T^{6}$$
$29$ $$-$$$$20\!\cdots\!12$$$$+$$$$21\!\cdots\!64$$$$T +$$$$31\!\cdots\!66$$$$T^{2} - 8804722738455739 T^{3} - 106285129247 T^{4} + 92335 T^{5} + T^{6}$$
$31$ $$14\!\cdots\!16$$$$+$$$$22\!\cdots\!72$$$$T +$$$$17\!\cdots\!00$$$$T^{2} - 11188769503925632 T^{3} - 96684518460 T^{4} + 83081 T^{5} + T^{6}$$
$37$ $$15\!\cdots\!24$$$$+$$$$13\!\cdots\!84$$$$T +$$$$66\!\cdots\!52$$$$T^{2} - 42735663298636686 T^{3} - 204953095689 T^{4} + 265114 T^{5} + T^{6}$$
$41$ $$-$$$$10\!\cdots\!08$$$$+$$$$25\!\cdots\!72$$$$T +$$$$18\!\cdots\!72$$$$T^{2} - 220953788741551216 T^{3} - 843667423664 T^{4} + 367468 T^{5} + T^{6}$$
$43$ $$36\!\cdots\!88$$$$-$$$$21\!\cdots\!32$$$$T +$$$$21\!\cdots\!44$$$$T^{2} + 638083458820433571 T^{3} - 1051212986349 T^{4} - 454955 T^{5} + T^{6}$$
$47$ $$-$$$$10\!\cdots\!92$$$$-$$$$11\!\cdots\!16$$$$T +$$$$52\!\cdots\!92$$$$T^{2} + 449722493141260560 T^{3} - 682852960872 T^{4} - 733973 T^{5} + T^{6}$$
$53$ $$21\!\cdots\!96$$$$+$$$$35\!\cdots\!72$$$$T -$$$$84\!\cdots\!52$$$$T^{2} - 882361712134396648 T^{3} - 733635933242 T^{4} + 1577379 T^{5} + T^{6}$$
$59$ $$-$$$$10\!\cdots\!56$$$$+$$$$48\!\cdots\!76$$$$T +$$$$13\!\cdots\!64$$$$T^{2} + 1065987146736888080 T^{3} - 2680962695552 T^{4} - 2062708 T^{5} + T^{6}$$
$61$ $$-$$$$16\!\cdots\!52$$$$+$$$$42\!\cdots\!52$$$$T +$$$$44\!\cdots\!48$$$$T^{2} - 724031070820117428 T^{3} - 3841162776799 T^{4} + 271270 T^{5} + T^{6}$$
$67$ $$-$$$$92\!\cdots\!28$$$$-$$$$86\!\cdots\!12$$$$T +$$$$97\!\cdots\!52$$$$T^{2} + 2182664235106180920 T^{3} - 19144262181932 T^{4} + 758674 T^{5} + T^{6}$$
$71$ $$10\!\cdots\!52$$$$-$$$$13\!\cdots\!24$$$$T -$$$$32\!\cdots\!04$$$$T^{2} -$$$$17\!\cdots\!56$$$$T^{3} - 16932205358424 T^{4} + 6138216 T^{5} + T^{6}$$
$73$ $$19\!\cdots\!92$$$$-$$$$17\!\cdots\!52$$$$T +$$$$65\!\cdots\!62$$$$T^{2} +$$$$22\!\cdots\!03$$$$T^{3} - 31086878586911 T^{4} - 6361979 T^{5} + T^{6}$$
$79$ $$76\!\cdots\!36$$$$+$$$$93\!\cdots\!08$$$$T +$$$$28\!\cdots\!16$$$$T^{2} -$$$$21\!\cdots\!00$$$$T^{3} - 110761595426696 T^{4} + 899781 T^{5} + T^{6}$$
$83$ $$-$$$$43\!\cdots\!60$$$$-$$$$54\!\cdots\!96$$$$T +$$$$47\!\cdots\!52$$$$T^{2} +$$$$32\!\cdots\!64$$$$T^{3} - 131801520667384 T^{4} - 3313561 T^{5} + T^{6}$$
$89$ $$67\!\cdots\!52$$$$-$$$$33\!\cdots\!44$$$$T +$$$$12\!\cdots\!40$$$$T^{2} +$$$$12\!\cdots\!64$$$$T^{3} - 99734063672310 T^{4} - 11210703 T^{5} + T^{6}$$
$97$ $$-$$$$58\!\cdots\!28$$$$+$$$$51\!\cdots\!96$$$$T -$$$$12\!\cdots\!12$$$$T^{2} +$$$$34\!\cdots\!68$$$$T^{3} + 231379482787078 T^{4} - 28682643 T^{5} + T^{6}$$