# Properties

 Label 336.6.q.h Level $336$ Weight $6$ Character orbit 336.q Analytic conductor $53.889$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{505})$$ Defining polynomial: $$x^{4} - x^{3} + 127x^{2} + 126x + 15876$$ x^4 - x^3 + 127*x^2 + 126*x + 15876 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$7$$ Twist minimal: no (minimal twist has level 42) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 8) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} - 18 \beta_1 + 110) q^{7} + (81 \beta_1 - 81) q^{9}+O(q^{10})$$ q + 9*b1 * q^3 + (-3*b3 + 2*b2 + 7*b1 - 8) * q^5 + (-2*b3 + 3*b2 - 18*b1 + 110) * q^7 + (81*b1 - 81) * q^9 $$q + 9 \beta_1 q^{3} + ( - 3 \beta_{3} + 2 \beta_{2} + 7 \beta_1 - 8) q^{5} + ( - 2 \beta_{3} + 3 \beta_{2} - 18 \beta_1 + 110) q^{7} + (81 \beta_1 - 81) q^{9} + (7 \beta_{3} - 21 \beta_{2} + 76 \beta_1 + 7) q^{11} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 - 356) q^{13} + ( - 18 \beta_{3} - 9 \beta_{2} - 9 \beta_1 - 63) q^{15} + (20 \beta_{3} - 60 \beta_{2} - 676 \beta_1 + 20) q^{17} + (81 \beta_{3} - 54 \beta_{2} - 500 \beta_1 + 527) q^{19} + ( - 27 \beta_{3} + 9 \beta_{2} + 828 \beta_1 + 162) q^{21} + ( - 12 \beta_{3} + 8 \beta_{2} + 2248 \beta_1 - 2252) q^{23} + (17 \beta_{3} - 51 \beta_{2} - 3125 \beta_1 + 17) q^{25} - 729 q^{27} + ( - 118 \beta_{3} - 59 \beta_{2} - 59 \beta_1 + 4021) q^{29} + ( - 14 \beta_{3} + 42 \beta_{2} + 4401 \beta_1 - 14) q^{31} + (189 \beta_{3} - 126 \beta_{2} + 747 \beta_1 - 684) q^{33} + ( - 306 \beta_{3} + 221 \beta_{2} - 1865 \beta_1 - 5171) q^{35} + ( - 243 \beta_{3} + 162 \beta_{2} - 7408 \beta_1 + 7327) q^{37} + (9 \beta_{3} - 27 \beta_{2} - 3213 \beta_1 + 9) q^{39} + (132 \beta_{3} + 66 \beta_{2} + 66 \beta_1 + 3576) q^{41} + (126 \beta_{3} + 63 \beta_{2} + 63 \beta_1 + 2866) q^{43} + (81 \beta_{3} - 243 \beta_{2} - 648 \beta_1 + 81) q^{45} + ( - 108 \beta_{3} + 72 \beta_{2} - 22458 \beta_1 + 22422) q^{47} + ( - 340 \beta_{3} + 629 \beta_{2} - 1618 \beta_1 + 4833) q^{49} + (540 \beta_{3} - 360 \beta_{2} - 5904 \beta_1 + 6084) q^{51} + (45 \beta_{3} - 135 \beta_{2} - 4686 \beta_1 + 45) q^{53} + ( - 26 \beta_{3} - 13 \beta_{2} - 13 \beta_1 + 42707) q^{55} + (486 \beta_{3} + 243 \beta_{2} + 243 \beta_1 + 4500) q^{57} + (377 \beta_{3} - 1131 \beta_{2} - 2350 \beta_1 + 377) q^{59} + ( - 276 \beta_{3} + 184 \beta_{2} + 21046 \beta_1 - 21138) q^{61} + ( - 81 \beta_{3} - 162 \beta_{2} + 8910 \beta_1 - 7452) q^{63} + (1098 \beta_{3} - 732 \beta_{2} - 8676 \beta_1 + 9042) q^{65} + (317 \beta_{3} - 951 \beta_{2} + 15409 \beta_1 + 317) q^{67} + ( - 72 \beta_{3} - 36 \beta_{2} - 36 \beta_1 - 20232) q^{69} + (440 \beta_{3} + 220 \beta_{2} + 220 \beta_1 - 46202) q^{71} + (505 \beta_{3} - 1515 \beta_{2} + 43085 \beta_1 + 505) q^{73} + (459 \beta_{3} - 306 \beta_{2} - 27972 \beta_1 + 28125) q^{75} + (199 \beta_{3} - 2010 \beta_{2} - 24053 \beta_1 + 51628) q^{77} + ( - 2064 \beta_{3} + 1376 \beta_{2} - 48355 \beta_1 + 47667) q^{79} - 6561 \beta_1 q^{81} + (62 \beta_{3} + 31 \beta_{2} + 31 \beta_1 - 16967) q^{83} + (1712 \beta_{3} + 856 \beta_{2} + 856 \beta_1 + 128272) q^{85} + (531 \beta_{3} - 1593 \beta_{2} + 35658 \beta_1 + 531) q^{87} + ( - 522 \beta_{3} + 348 \beta_{2} + 13518 \beta_1 - 13692) q^{89} + (477 \beta_{3} - 1132 \beta_{2} - 754 \beta_1 - 36525) q^{91} + ( - 378 \beta_{3} + 252 \beta_{2} + 39483 \beta_1 - 39609) q^{93} + ( - 770 \beta_{3} + 2310 \beta_{2} + 171238 \beta_1 - 770) q^{95} + ( - 1726 \beta_{3} - 863 \beta_{2} - 863 \beta_1 - 22041) q^{97} + (1134 \beta_{3} + 567 \beta_{2} + 567 \beta_1 - 6723) q^{99}+O(q^{100})$$ q + 9*b1 * q^3 + (-3*b3 + 2*b2 + 7*b1 - 8) * q^5 + (-2*b3 + 3*b2 - 18*b1 + 110) * q^7 + (81*b1 - 81) * q^9 + (7*b3 - 21*b2 + 76*b1 + 7) * q^11 + (-2*b3 - b2 - b1 - 356) * q^13 + (-18*b3 - 9*b2 - 9*b1 - 63) * q^15 + (20*b3 - 60*b2 - 676*b1 + 20) * q^17 + (81*b3 - 54*b2 - 500*b1 + 527) * q^19 + (-27*b3 + 9*b2 + 828*b1 + 162) * q^21 + (-12*b3 + 8*b2 + 2248*b1 - 2252) * q^23 + (17*b3 - 51*b2 - 3125*b1 + 17) * q^25 - 729 * q^27 + (-118*b3 - 59*b2 - 59*b1 + 4021) * q^29 + (-14*b3 + 42*b2 + 4401*b1 - 14) * q^31 + (189*b3 - 126*b2 + 747*b1 - 684) * q^33 + (-306*b3 + 221*b2 - 1865*b1 - 5171) * q^35 + (-243*b3 + 162*b2 - 7408*b1 + 7327) * q^37 + (9*b3 - 27*b2 - 3213*b1 + 9) * q^39 + (132*b3 + 66*b2 + 66*b1 + 3576) * q^41 + (126*b3 + 63*b2 + 63*b1 + 2866) * q^43 + (81*b3 - 243*b2 - 648*b1 + 81) * q^45 + (-108*b3 + 72*b2 - 22458*b1 + 22422) * q^47 + (-340*b3 + 629*b2 - 1618*b1 + 4833) * q^49 + (540*b3 - 360*b2 - 5904*b1 + 6084) * q^51 + (45*b3 - 135*b2 - 4686*b1 + 45) * q^53 + (-26*b3 - 13*b2 - 13*b1 + 42707) * q^55 + (486*b3 + 243*b2 + 243*b1 + 4500) * q^57 + (377*b3 - 1131*b2 - 2350*b1 + 377) * q^59 + (-276*b3 + 184*b2 + 21046*b1 - 21138) * q^61 + (-81*b3 - 162*b2 + 8910*b1 - 7452) * q^63 + (1098*b3 - 732*b2 - 8676*b1 + 9042) * q^65 + (317*b3 - 951*b2 + 15409*b1 + 317) * q^67 + (-72*b3 - 36*b2 - 36*b1 - 20232) * q^69 + (440*b3 + 220*b2 + 220*b1 - 46202) * q^71 + (505*b3 - 1515*b2 + 43085*b1 + 505) * q^73 + (459*b3 - 306*b2 - 27972*b1 + 28125) * q^75 + (199*b3 - 2010*b2 - 24053*b1 + 51628) * q^77 + (-2064*b3 + 1376*b2 - 48355*b1 + 47667) * q^79 - 6561*b1 * q^81 + (62*b3 + 31*b2 + 31*b1 - 16967) * q^83 + (1712*b3 + 856*b2 + 856*b1 + 128272) * q^85 + (531*b3 - 1593*b2 + 35658*b1 + 531) * q^87 + (-522*b3 + 348*b2 + 13518*b1 - 13692) * q^89 + (477*b3 - 1132*b2 - 754*b1 - 36525) * q^91 + (-378*b3 + 252*b2 + 39483*b1 - 39609) * q^93 + (-770*b3 + 2310*b2 + 171238*b1 - 770) * q^95 + (-1726*b3 - 863*b2 - 863*b1 - 22041) * q^97 + (1134*b3 + 567*b2 + 567*b1 - 6723) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 18 q^{3} - 17 q^{5} + 408 q^{7} - 162 q^{9}+O(q^{10})$$ 4 * q + 18 * q^3 - 17 * q^5 + 408 * q^7 - 162 * q^9 $$4 q + 18 q^{3} - 17 q^{5} + 408 q^{7} - 162 q^{9} + 145 q^{11} - 1430 q^{13} - 306 q^{15} - 1372 q^{17} + 1081 q^{19} + 2295 q^{21} - 4508 q^{23} - 6267 q^{25} - 2916 q^{27} + 15730 q^{29} + 8816 q^{31} - 1305 q^{33} - 24278 q^{35} + 14573 q^{37} - 6435 q^{39} + 14700 q^{41} + 11842 q^{43} - 1377 q^{45} + 44808 q^{47} + 17014 q^{49} + 12348 q^{51} - 9417 q^{53} + 170750 q^{55} + 19458 q^{57} - 5077 q^{59} - 42368 q^{61} - 12393 q^{63} + 18450 q^{65} + 30501 q^{67} - 81144 q^{69} - 183488 q^{71} + 85665 q^{73} + 56403 q^{75} + 154585 q^{77} + 94646 q^{79} - 13122 q^{81} - 67682 q^{83} + 518224 q^{85} + 70785 q^{87} - 27558 q^{89} - 149395 q^{91} - 79344 q^{93} + 343246 q^{95} - 93342 q^{97} - 23490 q^{99}+O(q^{100})$$ 4 * q + 18 * q^3 - 17 * q^5 + 408 * q^7 - 162 * q^9 + 145 * q^11 - 1430 * q^13 - 306 * q^15 - 1372 * q^17 + 1081 * q^19 + 2295 * q^21 - 4508 * q^23 - 6267 * q^25 - 2916 * q^27 + 15730 * q^29 + 8816 * q^31 - 1305 * q^33 - 24278 * q^35 + 14573 * q^37 - 6435 * q^39 + 14700 * q^41 + 11842 * q^43 - 1377 * q^45 + 44808 * q^47 + 17014 * q^49 + 12348 * q^51 - 9417 * q^53 + 170750 * q^55 + 19458 * q^57 - 5077 * q^59 - 42368 * q^61 - 12393 * q^63 + 18450 * q^65 + 30501 * q^67 - 81144 * q^69 - 183488 * q^71 + 85665 * q^73 + 56403 * q^75 + 154585 * q^77 + 94646 * q^79 - 13122 * q^81 - 67682 * q^83 + 518224 * q^85 + 70785 * q^87 - 27558 * q^89 - 149395 * q^91 - 79344 * q^93 + 343246 * q^95 - 93342 * q^97 - 23490 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 127x^{2} + 126x + 15876$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002$$ (-v^3 + 127*v^2 - 127*v + 15876) / 16002 $$\beta_{2}$$ $$=$$ $$( 125\nu^{3} + 127\nu^{2} - 32131\nu + 47754 ) / 16002$$ (125*v^3 + 127*v^2 - 32131*v + 47754) / 16002 $$\beta_{3}$$ $$=$$ $$( 379\nu^{3} - 127\nu^{2} + 16129\nu + 63756 ) / 16002$$ (379*v^3 - 127*v^2 + 16129*v + 63756) / 16002
 $$\nu$$ $$=$$ $$( \beta_{3} - 3\beta_{2} + 4\beta _1 + 1 ) / 7$$ (b3 - 3*b2 + 4*b1 + 1) / 7 $$\nu^{2}$$ $$=$$ $$( 3\beta_{3} - 2\beta_{2} + 887\beta _1 - 886 ) / 7$$ (3*b3 - 2*b2 + 887*b1 - 886) / 7 $$\nu^{3}$$ $$=$$ $$( 254\beta_{3} + 127\beta_{2} + 127\beta _1 - 1517 ) / 7$$ (254*b3 + 127*b2 + 127*b1 - 1517) / 7

## Character values

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

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

## 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
 −5.36805 + 9.29774i 5.86805 − 10.1638i −5.36805 − 9.29774i 5.86805 + 10.1638i
0 4.50000 7.79423i 0 −43.5764 75.4765i 0 113.236 63.1236i 0 −40.5000 70.1481i 0
193.2 0 4.50000 7.79423i 0 35.0764 + 60.7540i 0 90.7639 + 92.5684i 0 −40.5000 70.1481i 0
289.1 0 4.50000 + 7.79423i 0 −43.5764 + 75.4765i 0 113.236 + 63.1236i 0 −40.5000 + 70.1481i 0
289.2 0 4.50000 + 7.79423i 0 35.0764 60.7540i 0 90.7639 92.5684i 0 −40.5000 + 70.1481i 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
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.h 4
4.b odd 2 1 42.6.e.d 4
7.c even 3 1 inner 336.6.q.h 4
12.b even 2 1 126.6.g.g 4
28.d even 2 1 294.6.e.y 4
28.f even 6 1 294.6.a.o 2
28.f even 6 1 294.6.e.y 4
28.g odd 6 1 42.6.e.d 4
28.g odd 6 1 294.6.a.p 2
84.j odd 6 1 882.6.a.bs 2
84.n even 6 1 126.6.g.g 4
84.n even 6 1 882.6.a.bm 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.d 4 4.b odd 2 1
42.6.e.d 4 28.g odd 6 1
126.6.g.g 4 12.b even 2 1
126.6.g.g 4 84.n even 6 1
294.6.a.o 2 28.f even 6 1
294.6.a.p 2 28.g odd 6 1
294.6.e.y 4 28.d even 2 1
294.6.e.y 4 28.f even 6 1
336.6.q.h 4 1.a even 1 1 trivial
336.6.q.h 4 7.c even 3 1 inner
882.6.a.bm 2 84.n even 6 1
882.6.a.bs 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 17T_{5}^{3} + 6403T_{5}^{2} - 103938T_{5} + 37380996$$ acting on $$S_{6}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 9 T + 81)^{2}$$
$5$ $$T^{4} + 17 T^{3} + 6403 T^{2} + \cdots + 37380996$$
$7$ $$T^{4} - 408 T^{3} + \cdots + 282475249$$
$11$ $$T^{4} - 145 T^{3} + \cdots + 88726536900$$
$13$ $$(T^{2} + 715 T + 121620)^{2}$$
$17$ $$T^{4} + 1372 T^{3} + \cdots + 4015631241216$$
$19$ $$T^{4} - 1081 T^{3} + \cdots + 17788453428496$$
$23$ $$T^{4} + 4508 T^{3} + \cdots + 24815700919296$$
$29$ $$(T^{2} - 7865 T - 6069780)^{2}$$
$31$ $$T^{4} + \cdots + 331894030125681$$
$37$ $$T^{4} + \cdots + 156377425969216$$
$41$ $$(T^{2} - 7350 T - 13441680)^{2}$$
$43$ $$(T^{2} - 5921 T - 15788666)^{2}$$
$47$ $$T^{4} - 44808 T^{3} + \cdots + 24\!\cdots\!96$$
$53$ $$T^{4} + 9417 T^{3} + \cdots + 92983900409856$$
$59$ $$T^{4} + 5077 T^{3} + \cdots + 76\!\cdots\!36$$
$61$ $$T^{4} + 42368 T^{3} + \cdots + 15\!\cdots\!96$$
$67$ $$T^{4} - 30501 T^{3} + \cdots + 15\!\cdots\!76$$
$71$ $$(T^{2} + 91744 T + 1804825884)^{2}$$
$73$ $$T^{4} - 85665 T^{3} + \cdots + 66\!\cdots\!00$$
$79$ $$T^{4} - 94646 T^{3} + \cdots + 47\!\cdots\!81$$
$83$ $$(T^{2} + 33841 T + 280358334)^{2}$$
$89$ $$T^{4} + 27558 T^{3} + \cdots + 6584027556096$$
$97$ $$(T^{2} + 46671 T - 4062781666)^{2}$$