Properties

 Label 336.6.q.j Level $336$ Weight $6$ Character orbit 336.q Analytic conductor $53.889$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 336.q (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$53.8889634572$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Defining polynomial: $$x^{8} - x^{7} + 98 x^{6} + 83 x^{5} + 9122 x^{4} - 91 x^{3} + 28567 x^{2} + 2058 x + 86436$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{8}\cdot 3\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 21) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

 $$f(q)$$ $$=$$ $$q + ( 9 - 9 \beta_{1} ) q^{3} + ( \beta_{4} - \beta_{6} ) q^{5} + ( -9 - 45 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{7} -81 \beta_{1} q^{9} +O(q^{10})$$ $$q + ( 9 - 9 \beta_{1} ) q^{3} + ( \beta_{4} - \beta_{6} ) q^{5} + ( -9 - 45 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{7} ) q^{7} -81 \beta_{1} q^{9} + ( 102 - 102 \beta_{1} - \beta_{2} + 3 \beta_{3} + 6 \beta_{4} ) q^{11} + ( 118 + 13 \beta_{2} - 13 \beta_{5} + 4 \beta_{6} + 5 \beta_{7} ) q^{13} -9 \beta_{6} q^{15} + ( -66 + 66 \beta_{1} + 11 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} ) q^{17} + ( -7 + 131 \beta_{1} - 7 \beta_{3} + 8 \beta_{4} - 22 \beta_{5} - 8 \beta_{6} - 7 \beta_{7} ) q^{19} + ( -486 + 90 \beta_{1} - 9 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} + 9 \beta_{7} ) q^{21} + ( 6 + 1722 \beta_{1} + 6 \beta_{3} - 14 \beta_{4} + 31 \beta_{5} + 14 \beta_{6} + 6 \beta_{7} ) q^{23} + ( -691 + 691 \beta_{1} - 5 \beta_{2} + 25 \beta_{3} + 70 \beta_{4} ) q^{25} -729 q^{27} + ( 162 - 17 \beta_{2} + 17 \beta_{5} - 49 \beta_{6} + 54 \beta_{7} ) q^{29} + ( -1585 + 1585 \beta_{1} - 19 \beta_{2} + 35 \beta_{3} + 59 \beta_{4} ) q^{31} + ( 27 - 918 \beta_{1} + 27 \beta_{3} + 54 \beta_{4} - 9 \beta_{5} - 54 \beta_{6} + 27 \beta_{7} ) q^{33} + ( 5001 - 1728 \beta_{1} + 34 \beta_{2} - 48 \beta_{3} - 56 \beta_{4} - 27 \beta_{5} - 50 \beta_{6} + 21 \beta_{7} ) q^{35} + ( -43 - 3791 \beta_{1} - 43 \beta_{3} - 52 \beta_{4} + 137 \beta_{5} + 52 \beta_{6} - 43 \beta_{7} ) q^{37} + ( 1017 - 1017 \beta_{1} + 117 \beta_{2} - 45 \beta_{3} + 36 \beta_{4} ) q^{39} + ( 1128 + 20 \beta_{2} - 20 \beta_{5} + 40 \beta_{6} + 102 \beta_{7} ) q^{41} + ( -7268 - 78 \beta_{2} + 78 \beta_{5} - 30 \beta_{6} + 63 \beta_{7} ) q^{43} -81 \beta_{4} q^{45} + ( -42 - 3744 \beta_{1} - 42 \beta_{3} - 14 \beta_{4} - 23 \beta_{5} + 14 \beta_{6} - 42 \beta_{7} ) q^{47} + ( -10702 + 5241 \beta_{1} + 183 \beta_{2} - 22 \beta_{3} - 72 \beta_{4} - 171 \beta_{5} + 150 \beta_{6} - 89 \beta_{7} ) q^{49} + ( 54 + 594 \beta_{1} + 54 \beta_{3} + 18 \beta_{4} + 99 \beta_{5} - 18 \beta_{6} + 54 \beta_{7} ) q^{51} + ( -3486 + 3486 \beta_{1} - 76 \beta_{2} - 126 \beta_{3} + 113 \beta_{4} ) q^{53} + ( -18286 + 13 \beta_{2} - 13 \beta_{5} + 325 \beta_{6} - 10 \beta_{7} ) q^{55} + ( 1116 + 198 \beta_{2} - 198 \beta_{5} - 72 \beta_{6} - 63 \beta_{7} ) q^{57} + ( 8766 - 8766 \beta_{1} - 158 \beta_{2} + 117 \beta_{3} + 46 \beta_{4} ) q^{59} + ( -146 + 1414 \beta_{1} - 146 \beta_{3} + 94 \beta_{4} + 196 \beta_{5} - 94 \beta_{6} - 146 \beta_{7} ) q^{61} + ( -3645 + 4455 \beta_{1} + 81 \beta_{2} - 81 \beta_{3} - 81 \beta_{5} + 81 \beta_{6} ) q^{63} + ( -288 - 16572 \beta_{1} - 288 \beta_{3} - 42 \beta_{4} + 341 \beta_{5} + 42 \beta_{6} - 288 \beta_{7} ) q^{65} + ( -1663 + 1663 \beta_{1} + 544 \beta_{2} - 329 \beta_{3} + 136 \beta_{4} ) q^{67} + ( 15552 - 279 \beta_{2} + 279 \beta_{5} + 126 \beta_{6} + 54 \beta_{7} ) q^{69} + ( -22278 - 393 \beta_{2} + 393 \beta_{5} + 102 \beta_{6} + 78 \beta_{7} ) q^{71} + ( -14897 + 14897 \beta_{1} - 365 \beta_{2} + 25 \beta_{3} + 118 \beta_{4} ) q^{73} + ( 225 + 6219 \beta_{1} + 225 \beta_{3} + 630 \beta_{4} - 45 \beta_{5} - 630 \beta_{6} + 225 \beta_{7} ) q^{75} + ( -17850 + 17136 \beta_{1} + 8 \beta_{2} + 6 \beta_{3} + 177 \beta_{4} - 433 \beta_{5} - 715 \beta_{6} + 348 \beta_{7} ) q^{77} + ( -345 - 10843 \beta_{1} - 345 \beta_{3} + 441 \beta_{4} - 3 \beta_{5} - 441 \beta_{6} - 345 \beta_{7} ) q^{79} + ( -6561 + 6561 \beta_{1} ) q^{81} + ( 52395 + 978 \beta_{2} - 978 \beta_{5} - 438 \beta_{6} + 567 \beta_{7} ) q^{83} + ( 9222 - 210 \beta_{2} + 210 \beta_{5} + 90 \beta_{6} + 282 \beta_{7} ) q^{85} + ( 972 - 972 \beta_{1} - 153 \beta_{2} - 486 \beta_{3} - 441 \beta_{4} ) q^{87} + ( 480 + 19140 \beta_{1} + 480 \beta_{3} + 1102 \beta_{4} + 1224 \beta_{5} - 1102 \beta_{6} + 480 \beta_{7} ) q^{89} + ( -48585 + 71403 \beta_{1} - 270 \beta_{2} + 193 \beta_{3} - 240 \beta_{4} + 1332 \beta_{5} - 114 \beta_{6} + 317 \beta_{7} ) q^{91} + ( 315 + 14265 \beta_{1} + 315 \beta_{3} + 531 \beta_{4} - 171 \beta_{5} - 531 \beta_{6} + 315 \beta_{7} ) q^{93} + ( -55032 + 55032 \beta_{1} + 395 \beta_{2} + 624 \beta_{3} + 918 \beta_{4} ) q^{95} + ( -47109 - 85 \beta_{2} + 85 \beta_{5} - 766 \beta_{6} + 97 \beta_{7} ) q^{97} + ( -8019 + 81 \beta_{2} - 81 \beta_{5} - 486 \beta_{6} + 243 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 36q^{3} - 258q^{7} - 324q^{9} + O(q^{10})$$ $$8q + 36q^{3} - 258q^{7} - 324q^{9} + 402q^{11} + 924q^{13} - 276q^{17} + 510q^{19} - 3564q^{21} + 6900q^{23} - 2814q^{25} - 5832q^{27} + 1080q^{29} - 6410q^{31} - 3618q^{33} + 33108q^{35} - 15250q^{37} + 4158q^{39} + 8616q^{41} - 58396q^{43} - 15060q^{47} - 64252q^{49} + 2484q^{51} - 13692q^{53} - 146248q^{55} + 9180q^{57} + 34830q^{59} + 5364q^{61} - 11178q^{63} - 66864q^{65} - 5994q^{67} + 124200q^{69} - 178536q^{71} - 59638q^{73} + 25326q^{75} - 75660q^{77} - 44062q^{79} - 26244q^{81} + 416892q^{83} + 72648q^{85} + 4860q^{87} + 77520q^{89} - 104722q^{91} + 57690q^{93} - 221376q^{95} - 377260q^{97} - 65124q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 98 x^{6} + 83 x^{5} + 9122 x^{4} - 91 x^{3} + 28567 x^{2} + 2058 x + 86436$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 141627885 \nu - 307508418$$$$)/ 9888988410$$ $$\beta_{2}$$ $$=$$ $$($$$$8905 \nu^{7} - 366059 \nu^{6} + 1191820 \nu^{5} - 33153695 \nu^{4} + 47979268 \nu^{3} - 3262309295 \nu^{2} + 39697581525 \nu - 10196496828$$$$)/ 4944494205$$ $$\beta_{3}$$ $$=$$ $$($$$$65603 \nu^{7} - 25662916 \nu^{6} + 83553680 \nu^{5} - 2627557456 \nu^{4} + 3363632432 \nu^{3} - 228707310580 \nu^{2} + 464946681171 \nu - 714835153872$$$$)/ 9888988410$$ $$\beta_{4}$$ $$=$$ $$($$$$731039 \nu^{7} - 7994302 \nu^{6} + 26027960 \nu^{5} - 628584958 \nu^{4} + 1047811304 \nu^{3} - 71245033510 \nu^{2} - 451136527737 \nu - 222679608984$$$$)/ 9888988410$$ $$\beta_{5}$$ $$=$$ $$($$$$1437521 \nu^{7} - 1642579 \nu^{6} + 136763060 \nu^{5} + 99854873 \nu^{4} + 13093975028 \nu^{3} - 2811264295 \nu^{2} + 41000921157 \nu + 2771341902$$$$)/ 4944494205$$ $$\beta_{6}$$ $$=$$ $$($$$$484709 \nu^{7} - 433105 \nu^{6} + 45997385 \nu^{5} + 45127907 \nu^{4} + 4285048727 \nu^{3} + 153033125 \nu^{2} + 442204518 \nu - 5171325642$$$$)/ 282542526$$ $$\beta_{7}$$ $$=$$ $$($$$$2434409 \nu^{7} - 2482177 \nu^{6} + 231017885 \nu^{5} + 226651007 \nu^{4} + 21466747139 \nu^{3} + 768595625 \nu^{2} + 2220933918 \nu + 102356221716$$$$)/ 1412712630$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1} + 2$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - 98 \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-16 \beta_{6} + 95 \beta_{5} - 95 \beta_{2} - 542$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-101 \beta_{4} - 97 \beta_{3} - 22 \beta_{2} + 9050 \beta_{1} - 9050$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-360 \beta_{7} + 1928 \beta_{6} - 9009 \beta_{5} - 1928 \beta_{4} - 360 \beta_{3} + 85442 \beta_{1} - 360$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-9205 \beta_{7} + 9957 \beta_{6} - 4220 \beta_{5} + 4220 \beta_{2} + 860245$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$219312 \beta_{4} + 69024 \beta_{3} + 862207 \beta_{2} - 11352926 \beta_{1} + 11352926$$$$)/8$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$$.

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 −0.874091 + 1.51397i −4.61193 + 7.98809i 5.09061 − 8.81720i 0.895402 − 1.55088i −0.874091 − 1.51397i −4.61193 − 7.98809i 5.09061 + 8.81720i 0.895402 + 1.55088i
0 4.50000 7.79423i 0 −29.1836 50.5475i 0 21.4366 + 127.857i 0 −40.5000 70.1481i 0
193.2 0 4.50000 7.79423i 0 −11.8764 20.5705i 0 −30.6840 125.958i 0 −40.5000 70.1481i 0
193.3 0 4.50000 7.79423i 0 −11.0358 19.1146i 0 −126.882 26.6059i 0 −40.5000 70.1481i 0
193.4 0 4.50000 7.79423i 0 52.0958 + 90.2327i 0 7.12980 129.446i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −29.1836 + 50.5475i 0 21.4366 127.857i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 −11.8764 + 20.5705i 0 −30.6840 + 125.958i 0 −40.5000 + 70.1481i 0
289.3 0 4.50000 + 7.79423i 0 −11.0358 + 19.1146i 0 −126.882 + 26.6059i 0 −40.5000 + 70.1481i 0
289.4 0 4.50000 + 7.79423i 0 52.0958 90.2327i 0 7.12980 + 129.446i 0 −40.5000 + 70.1481i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.6.q.j 8
4.b odd 2 1 21.6.e.c 8
7.c even 3 1 inner 336.6.q.j 8
12.b even 2 1 63.6.e.e 8
28.d even 2 1 147.6.e.o 8
28.f even 6 1 147.6.a.l 4
28.f even 6 1 147.6.e.o 8
28.g odd 6 1 21.6.e.c 8
28.g odd 6 1 147.6.a.m 4
84.j odd 6 1 441.6.a.v 4
84.n even 6 1 63.6.e.e 8
84.n even 6 1 441.6.a.w 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.e.c 8 4.b odd 2 1
21.6.e.c 8 28.g odd 6 1
63.6.e.e 8 12.b even 2 1
63.6.e.e 8 84.n even 6 1
147.6.a.l 4 28.f even 6 1
147.6.a.m 4 28.g odd 6 1
147.6.e.o 8 28.d even 2 1
147.6.e.o 8 28.f even 6 1
336.6.q.j 8 1.a even 1 1 trivial
336.6.q.j 8 7.c even 3 1 inner
441.6.a.v 4 84.j odd 6 1
441.6.a.w 4 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} + 7657 T_{5}^{6} + 605400 T_{5}^{5} + 61817893 T_{5}^{4} + 2317773900 T_{5}^{3} + 67214905692 T_{5}^{2} + 965081458800 T_{5} +$$$$10\!\cdots\!36$$ acting on $$S_{6}^{\mathrm{new}}(336, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 81 - 9 T + T^{2} )^{4}$$
$5$ $$10164899803536 + 965081458800 T + 67214905692 T^{2} + 2317773900 T^{3} + 61817893 T^{4} + 605400 T^{5} + 7657 T^{6} + T^{8}$$
$7$ $$79792266297612001 + 1224870869565294 T + 18476141086592 T^{2} + 206920120008 T^{3} + 1643194665 T^{4} + 12311544 T^{5} + 65408 T^{6} + 258 T^{7} + T^{8}$$
$11$ $$2829568482592751376 - 169576701600014928 T + 9761813367494172 T^{2} - 25381944601332 T^{3} + 99024901837 T^{4} - 105799218 T^{5} + 399967 T^{6} - 402 T^{7} + T^{8}$$
$13$ $$( 149501563456 + 515112852 T - 1148423 T^{2} - 462 T^{3} + T^{4} )^{2}$$
$17$ $$25\!\cdots\!24$$$$+ 44837110146970681344 T + 720903316973027328 T^{2} + 1454766301642752 T^{3} + 2840324474880 T^{4} + 1349604864 T^{5} + 1670928 T^{6} + 276 T^{7} + T^{8}$$
$19$ $$54\!\cdots\!76$$$$+$$$$58\!\cdots\!80$$$$T + 44221112569626422096 T^{2} + 2834297273515140 T^{3} + 27788839301865 T^{4} + 1411756650 T^{5} + 6156971 T^{6} - 510 T^{7} + T^{8}$$
$23$ $$90\!\cdots\!76$$$$+$$$$51\!\cdots\!20$$$$T +$$$$27\!\cdots\!44$$$$T^{2} + 163456653366558720 T^{3} + 165861153884160 T^{4} - 83386679040 T^{5} + 40488144 T^{6} - 6900 T^{7} + T^{8}$$
$29$ $$( 408027025117872 + 62527747272 T - 52650397 T^{2} - 540 T^{3} + T^{4} )^{2}$$
$31$ $$75\!\cdots\!09$$$$-$$$$50\!\cdots\!98$$$$T +$$$$49\!\cdots\!52$$$$T^{2} - 73339378003949028 T^{3} + 602809801419817 T^{4} + 3691164188 T^{5} + 58801968 T^{6} + 6410 T^{7} + T^{8}$$
$37$ $$25\!\cdots\!36$$$$+$$$$75\!\cdots\!80$$$$T +$$$$20\!\cdots\!08$$$$T^{2} +$$$$22\!\cdots\!60$$$$T^{3} + 30068709801011305 T^{4} + 2279577843610 T^{5} + 279418707 T^{6} + 15250 T^{7} + T^{8}$$
$41$ $$( -1856858915261952 - 1101575496480 T - 192741244 T^{2} - 4308 T^{3} + T^{4} )^{2}$$
$43$ $$( -991662745581932 + 199921376588 T + 199961493 T^{2} + 29198 T^{3} + T^{4} )^{2}$$
$47$ $$73\!\cdots\!36$$$$+$$$$12\!\cdots\!48$$$$T +$$$$15\!\cdots\!28$$$$T^{2} + 5876954381324343168 T^{3} + 3512848312989072 T^{4} + 852661119456 T^{5} + 176153856 T^{6} + 15060 T^{7} + T^{8}$$
$53$ $$72\!\cdots\!84$$$$+$$$$68\!\cdots\!40$$$$T +$$$$60\!\cdots\!12$$$$T^{2} +$$$$42\!\cdots\!28$$$$T^{3} + 366485627229994953 T^{4} + 9260410268772 T^{5} + 685378293 T^{6} + 13692 T^{7} + T^{8}$$
$59$ $$66\!\cdots\!96$$$$-$$$$11\!\cdots\!60$$$$T +$$$$69\!\cdots\!76$$$$T^{2} - 92739830872274379000 T^{3} + 54097578476747217 T^{4} - 7480212648750 T^{5} + 1024872459 T^{6} - 34830 T^{7} + T^{8}$$
$61$ $$32\!\cdots\!76$$$$+$$$$60\!\cdots\!60$$$$T +$$$$20\!\cdots\!24$$$$T^{2} -$$$$16\!\cdots\!68$$$$T^{3} + 283845325040842512 T^{4} - 3902598174816 T^{5} + 561368672 T^{6} - 5364 T^{7} + T^{8}$$
$67$ $$30\!\cdots\!56$$$$+$$$$27\!\cdots\!00$$$$T +$$$$15\!\cdots\!44$$$$T^{2} -$$$$20\!\cdots\!92$$$$T^{3} + 7519978747196869197 T^{4} - 26942404854654 T^{5} + 2881922927 T^{6} + 5994 T^{7} + T^{8}$$
$71$ $$( 21932335650275568 - 16377596837712 T + 1521744768 T^{2} + 89268 T^{3} + T^{4} )^{2}$$
$73$ $$14\!\cdots\!00$$$$+$$$$24\!\cdots\!00$$$$T +$$$$45\!\cdots\!00$$$$T^{2} +$$$$71\!\cdots\!80$$$$T^{3} + 1466139740779982221 T^{4} + 62299772588518 T^{5} + 3193750863 T^{6} + 59638 T^{7} + T^{8}$$
$79$ $$27\!\cdots\!81$$$$+$$$$24\!\cdots\!06$$$$T +$$$$84\!\cdots\!28$$$$T^{2} -$$$$70\!\cdots\!88$$$$T^{3} + 13482185316532207897 T^{4} - 196348017119564 T^{5} + 5723264752 T^{6} + 44062 T^{7} + T^{8}$$
$83$ $$( -41533908097096407132 + 738604511000820 T + 7607249829 T^{2} - 208446 T^{3} + T^{4} )^{2}$$
$89$ $$22\!\cdots\!24$$$$-$$$$88\!\cdots\!72$$$$T +$$$$26\!\cdots\!72$$$$T^{2} -$$$$38\!\cdots\!52$$$$T^{3} +$$$$47\!\cdots\!12$$$$T^{4} - 2429594520301248 T^{5} + 22815563908 T^{6} - 77520 T^{7} + T^{8}$$
$97$ $$( -11638556269792123644 - 99054118022220 T + 9271508101 T^{2} + 188630 T^{3} + T^{4} )^{2}$$