# Properties

 Label 336.4.q.k Level $336$ Weight $4$ Character orbit 336.q Analytic conductor $19.825$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$19.8246417619$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.9924270768.1 Defining polynomial: $$x^{6} - x^{5} + 25 x^{4} + 12 x^{3} + 582 x^{2} - 144 x + 36$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{3} q^{3} + ( -4 + \beta_{2} - 4 \beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -9 - 9 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 3 \beta_{3} q^{3} + ( -4 + \beta_{2} - 4 \beta_{3} + \beta_{4} ) q^{5} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{7} + ( -9 - 9 \beta_{3} ) q^{9} + ( \beta_{1} - 11 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{11} + ( 20 - \beta_{1} + 2 \beta_{2} ) q^{13} + ( 12 - 3 \beta_{2} ) q^{15} + ( \beta_{1} + 17 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{17} + ( -67 - \beta_{2} - 67 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{19} + ( 6 - 3 \beta_{2} + 9 \beta_{3} - 3 \beta_{5} ) q^{21} + ( 73 - 3 \beta_{2} + 73 \beta_{3} - 3 \beta_{4} - 7 \beta_{5} ) q^{23} + ( -7 \beta_{1} + 46 \beta_{3} - 8 \beta_{4} - 7 \beta_{5} ) q^{25} + 27 q^{27} + ( 21 + 5 \beta_{1} - 10 \beta_{2} ) q^{29} + ( -5 \beta_{1} + 34 \beta_{3} - 7 \beta_{4} - 5 \beta_{5} ) q^{31} + ( 33 + 6 \beta_{2} + 33 \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{33} + ( 61 - 10 \beta_{1} + 14 \beta_{2} + 151 \beta_{3} - 3 \beta_{4} - 7 \beta_{5} ) q^{35} + ( -86 - 4 \beta_{2} - 86 \beta_{3} - 4 \beta_{4} - 5 \beta_{5} ) q^{37} + ( 3 \beta_{1} + 60 \beta_{3} + 6 \beta_{4} + 3 \beta_{5} ) q^{39} + ( 78 + 4 \beta_{1} + 10 \beta_{2} ) q^{41} + ( -125 - 18 \beta_{1} + 15 \beta_{2} ) q^{43} + ( 36 \beta_{3} - 9 \beta_{4} ) q^{45} + ( -71 + 3 \beta_{2} - 71 \beta_{3} + 3 \beta_{4} - 25 \beta_{5} ) q^{47} + ( -111 - 3 \beta_{1} + 16 \beta_{2} - 107 \beta_{3} - 10 \beta_{4} - 12 \beta_{5} ) q^{49} + ( -51 + 9 \beta_{2} - 51 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} ) q^{51} + ( -20 \beta_{1} + 134 \beta_{3} - 9 \beta_{4} - 20 \beta_{5} ) q^{53} + ( 339 + 11 \beta_{1} + 14 \beta_{2} ) q^{55} + ( 201 - 6 \beta_{1} + 3 \beta_{2} ) q^{57} + ( -10 \beta_{1} - 368 \beta_{3} - 39 \beta_{4} - 10 \beta_{5} ) q^{59} + ( 10 + 40 \beta_{2} + 10 \beta_{3} + 40 \beta_{4} + 20 \beta_{5} ) q^{61} + ( -27 - 9 \beta_{1} - 9 \beta_{3} - 9 \beta_{4} ) q^{63} + ( 157 + \beta_{2} + 157 \beta_{3} + \beta_{4} - 17 \beta_{5} ) q^{65} + ( -16 \beta_{1} + 199 \beta_{3} + 31 \beta_{4} - 16 \beta_{5} ) q^{67} + ( -219 - 21 \beta_{1} + 9 \beta_{2} ) q^{69} + ( -99 + 33 \beta_{1} - 21 \beta_{2} ) q^{71} + ( -31 \beta_{1} + 332 \beta_{3} - 8 \beta_{4} - 31 \beta_{5} ) q^{73} + ( -138 + 24 \beta_{2} - 138 \beta_{3} + 24 \beta_{4} + 21 \beta_{5} ) q^{75} + ( -39 + 16 \beta_{1} - 16 \beta_{2} - 433 \beta_{3} - 40 \beta_{4} + 5 \beta_{5} ) q^{77} + ( 302 - 45 \beta_{2} + 302 \beta_{3} - 45 \beta_{4} + 3 \beta_{5} ) q^{79} + 81 \beta_{3} q^{81} + ( -162 + 6 \beta_{1} - 33 \beta_{2} ) q^{83} + ( 606 + 18 \beta_{1} - 18 \beta_{2} ) q^{85} + ( -15 \beta_{1} + 63 \beta_{3} - 30 \beta_{4} - 15 \beta_{5} ) q^{87} + ( -580 - 26 \beta_{2} - 580 \beta_{3} - 26 \beta_{4} - 48 \beta_{5} ) q^{89} + ( 257 + 30 \beta_{1} + 11 \beta_{2} + 305 \beta_{3} - 5 \beta_{4} + 18 \beta_{5} ) q^{91} + ( -102 + 21 \beta_{2} - 102 \beta_{3} + 21 \beta_{4} + 15 \beta_{5} ) q^{93} + ( 13 \beta_{1} + 259 \beta_{3} - 41 \beta_{4} + 13 \beta_{5} ) q^{95} + ( 23 + \beta_{1} - 50 \beta_{2} ) q^{97} + ( -99 - 9 \beta_{1} - 18 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 9q^{3} - 11q^{5} + 13q^{7} - 27q^{9} + O(q^{10})$$ $$6q - 9q^{3} - 11q^{5} + 13q^{7} - 27q^{9} + 35q^{11} + 124q^{13} + 66q^{15} - 48q^{17} - 202q^{19} + 3q^{21} + 216q^{23} - 130q^{25} + 162q^{27} + 106q^{29} - 95q^{31} + 105q^{33} - 56q^{35} - 262q^{37} - 186q^{39} + 488q^{41} - 720q^{43} - 99q^{45} - 210q^{47} - 303q^{49} - 144q^{51} - 393q^{53} + 2062q^{55} + 1212q^{57} + 1143q^{59} + 70q^{61} - 126q^{63} + 472q^{65} - 628q^{67} - 1296q^{69} - 636q^{71} - 988q^{73} - 390q^{75} + 1073q^{77} + 861q^{79} - 243q^{81} - 1038q^{83} + 3600q^{85} - 159q^{87} - 1766q^{89} + 654q^{91} - 285q^{93} - 736q^{95} + 38q^{97} - 630q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 25 x^{4} + 12 x^{3} + 582 x^{2} - 144 x + 36$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-11 \nu^{5} + 275 \nu^{4} + 328 \nu^{3} + 6402 \nu^{2} - 1584 \nu + 118833$$$$)/7203$$ $$\beta_{2}$$ $$=$$ $$($$$$-13 \nu^{5} + 325 \nu^{4} - 922 \nu^{3} + 7566 \nu^{2} - 1872 \nu + 118830$$$$)/7203$$ $$\beta_{3}$$ $$=$$ $$($$$$100 \nu^{5} - 99 \nu^{4} + 2475 \nu^{3} + 1825 \nu^{2} + 57618 \nu - 14256$$$$)/14406$$ $$\beta_{4}$$ $$=$$ $$($$$$801 \nu^{5} - 817 \nu^{4} + 20425 \nu^{3} + 6815 \nu^{2} + 475494 \nu - 117648$$$$)/7203$$ $$\beta_{5}$$ $$=$$ $$($$$$-1676 \nu^{5} + 1083 \nu^{4} - 41481 \nu^{3} - 30587 \nu^{2} - 947238 \nu - 2514$$$$)/14406$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + 33 \beta_{3} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-11 \beta_{2} + 13 \beta_{1} - 33$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$-17 \beta_{5} + 8 \beta_{4} - 411 \beta_{3} + 8 \beta_{2} - 411$$ $$\nu^{5}$$ $$=$$ $$-170 \beta_{5} - 127 \beta_{4} - 708 \beta_{3} - 170 \beta_{1}$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1 - \beta_{3}$$

## 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
 −2.27818 − 3.94593i 2.65415 + 4.59712i 0.124036 + 0.214837i −2.27818 + 3.94593i 2.65415 − 4.59712i 0.124036 − 0.214837i
0 −1.50000 + 2.59808i 0 −8.93660 15.4786i 0 −2.26047 18.3818i 0 −4.50000 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 −2.78070 4.81631i 0 −9.67799 + 15.7904i 0 −4.50000 7.79423i 0
193.3 0 −1.50000 + 2.59808i 0 6.21730 + 10.7687i 0 18.4385 1.73873i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 −8.93660 + 15.4786i 0 −2.26047 + 18.3818i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 2.59808i 0 −2.78070 + 4.81631i 0 −9.67799 15.7904i 0 −4.50000 + 7.79423i 0
289.3 0 −1.50000 2.59808i 0 6.21730 10.7687i 0 18.4385 + 1.73873i 0 −4.50000 + 7.79423i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.3 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 336.4.q.k 6
4.b odd 2 1 21.4.e.b 6
7.c even 3 1 inner 336.4.q.k 6
7.c even 3 1 2352.4.a.ci 3
7.d odd 6 1 2352.4.a.cg 3
12.b even 2 1 63.4.e.c 6
28.d even 2 1 147.4.e.n 6
28.f even 6 1 147.4.a.m 3
28.f even 6 1 147.4.e.n 6
28.g odd 6 1 21.4.e.b 6
28.g odd 6 1 147.4.a.l 3
84.h odd 2 1 441.4.e.w 6
84.j odd 6 1 441.4.a.t 3
84.j odd 6 1 441.4.e.w 6
84.n even 6 1 63.4.e.c 6
84.n even 6 1 441.4.a.s 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.b 6 4.b odd 2 1
21.4.e.b 6 28.g odd 6 1
63.4.e.c 6 12.b even 2 1
63.4.e.c 6 84.n even 6 1
147.4.a.l 3 28.g odd 6 1
147.4.a.m 3 28.f even 6 1
147.4.e.n 6 28.d even 2 1
147.4.e.n 6 28.f even 6 1
336.4.q.k 6 1.a even 1 1 trivial
336.4.q.k 6 7.c even 3 1 inner
441.4.a.s 3 84.n even 6 1
441.4.a.t 3 84.j odd 6 1
441.4.e.w 6 84.h odd 2 1
441.4.e.w 6 84.j odd 6 1
2352.4.a.cg 3 7.d odd 6 1
2352.4.a.ci 3 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 11 T_{5}^{5} + 313 T_{5}^{4} + 360 T_{5}^{3} + 50460 T_{5}^{2} + 237312 T_{5} + 1527696$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 9 + 3 T + T^{2} )^{3}$$
$5$ $$1527696 + 237312 T + 50460 T^{2} + 360 T^{3} + 313 T^{4} + 11 T^{5} + T^{6}$$
$7$ $$40353607 - 1529437 T + 80948 T^{2} - 12145 T^{3} + 236 T^{4} - 13 T^{5} + T^{6}$$
$11$ $$91470096 + 13083552 T + 1536684 T^{2} + 67008 T^{3} + 2593 T^{4} - 35 T^{5} + T^{6}$$
$13$ $$( 18452 + 425 T - 62 T^{2} + T^{3} )^{2}$$
$17$ $$12745506816 + 270950400 T + 11179008 T^{2} + 110592 T^{3} + 4704 T^{4} + 48 T^{5} + T^{6}$$
$19$ $$54664310416 + 2871346924 T + 103594553 T^{2} + 2013154 T^{3} + 28523 T^{4} + 202 T^{5} + T^{6}$$
$23$ $$2498119335936 - 1062125568 T + 341849088 T^{2} - 3015936 T^{3} + 47328 T^{4} - 216 T^{5} + T^{6}$$
$29$ $$( -824976 - 20472 T - 53 T^{2} + T^{3} )^{2}$$
$31$ $$139783329 + 118241823 T + 101143186 T^{2} - 926449 T^{3} + 19026 T^{4} + 95 T^{5} + T^{6}$$
$37$ $$2415919104 + 692502528 T + 185622097 T^{2} + 3593014 T^{3} + 54555 T^{4} + 262 T^{5} + T^{6}$$
$41$ $$( -300384 - 18780 T - 244 T^{2} + T^{3} )^{2}$$
$43$ $$( -18269746 - 72363 T + 360 T^{2} + T^{3} )^{2}$$
$47$ $$26205471480384 - 1261946958048 T + 59695121376 T^{2} - 62006616 T^{3} + 290616 T^{4} + 210 T^{5} + T^{6}$$
$53$ $$1100208565649664 + 2677964032512 T + 19553872752 T^{2} + 34609536 T^{3} + 235185 T^{4} + 393 T^{5} + T^{6}$$
$59$ $$10094008708475136 + 13372818322176 T + 132552677808 T^{2} - 353075760 T^{3} + 1173345 T^{4} - 1143 T^{5} + T^{6}$$
$61$ $$7162406161000000 - 28850707900000 T + 122136980000 T^{2} - 145399000 T^{3} + 345800 T^{4} - 70 T^{5} + T^{6}$$
$67$ $$783608160972004 - 8536829868926 T + 75422826113 T^{2} - 247502768 T^{3} + 699347 T^{4} + 628 T^{5} + T^{6}$$
$71$ $$( 28535976 - 330804 T + 318 T^{2} + T^{3} )^{2}$$
$73$ $$20508278645865924 + 623666998890 T + 141507598609 T^{2} + 282111496 T^{3} + 980499 T^{4} + 988 T^{5} + T^{6}$$
$79$ $$37619060662457569 - 50021533268637 T + 233509331958 T^{2} - 165859913 T^{3} + 999222 T^{4} - 861 T^{5} + T^{6}$$
$83$ $$( -47916036 - 131616 T + 519 T^{2} + T^{3} )^{2}$$
$89$ $$169118164647936 - 3614222868480 T + 100205551104 T^{2} + 516815808 T^{3} + 2840836 T^{4} + 1766 T^{5} + T^{6}$$
$97$ $$( 44776452 - 569600 T - 19 T^{2} + T^{3} )^{2}$$