# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{193})$$ Defining polynomial: $$x^{4} - x^{3} + 49 x^{2} + 48 x + 2304$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \beta_{2} ) q^{3} + ( \beta_{1} - 6 \beta_{2} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{7} -9 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \beta_{2} ) q^{3} + ( \beta_{1} - 6 \beta_{2} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{7} -9 \beta_{2} q^{9} + ( 1 - 7 \beta_{1} + 6 \beta_{2} - 7 \beta_{3} ) q^{11} + 5 \beta_{3} q^{13} + ( -15 - 3 \beta_{3} ) q^{15} + ( -52 + 4 \beta_{1} + 48 \beta_{2} + 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 35 \beta_{2} ) q^{19} + ( -6 - 9 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -20 \beta_{1} + 48 \beta_{2} ) q^{23} + ( 52 - 11 \beta_{1} - 41 \beta_{2} - 11 \beta_{3} ) q^{25} -27 q^{27} + ( 143 - 11 \beta_{3} ) q^{29} + ( 171 + 20 \beta_{1} - 191 \beta_{2} + 20 \beta_{3} ) q^{31} + ( -21 \beta_{1} + 18 \beta_{2} ) q^{33} + ( 81 - 8 \beta_{1} + 66 \beta_{2} + 9 \beta_{3} ) q^{35} + ( -45 \beta_{1} + 25 \beta_{2} ) q^{37} + ( 15 \beta_{1} - 15 \beta_{2} + 15 \beta_{3} ) q^{39} + ( -72 - 18 \beta_{3} ) q^{41} + ( -362 + 3 \beta_{3} ) q^{43} + ( -45 - 9 \beta_{1} + 54 \beta_{2} - 9 \beta_{3} ) q^{45} + ( -36 \beta_{1} - 90 \beta_{2} ) q^{47} + ( 232 - 2 \beta_{1} - 379 \beta_{2} + 11 \beta_{3} ) q^{49} + ( 12 \beta_{1} + 144 \beta_{2} ) q^{51} + ( -243 - 9 \beta_{1} + 252 \beta_{2} - 9 \beta_{3} ) q^{53} + ( 331 + 41 \beta_{3} ) q^{55} + ( 96 + 9 \beta_{3} ) q^{57} + ( 113 - 53 \beta_{1} - 60 \beta_{2} - 53 \beta_{3} ) q^{59} + ( -40 \beta_{1} + 286 \beta_{2} ) q^{61} + ( -27 - 9 \beta_{1} + 27 \beta_{2} + 18 \beta_{3} ) q^{63} + ( 30 \beta_{1} - 270 \beta_{2} ) q^{65} + ( 94 - 77 \beta_{1} - 17 \beta_{2} - 77 \beta_{3} ) q^{67} + ( 84 + 60 \beta_{3} ) q^{69} + ( -778 - 44 \beta_{3} ) q^{71} + ( -634 + 53 \beta_{1} + 581 \beta_{2} + 53 \beta_{3} ) q^{73} + ( -33 \beta_{1} - 123 \beta_{2} ) q^{75} + ( 334 + 11 \beta_{1} - 1020 \beta_{2} + 20 \beta_{3} ) q^{77} + ( -62 \beta_{1} + 761 \beta_{2} ) q^{79} + ( -81 + 81 \beta_{2} ) q^{81} + ( 755 - 101 \beta_{3} ) q^{83} + ( 68 + 28 \beta_{3} ) q^{85} + ( 429 - 33 \beta_{1} - 396 \beta_{2} - 33 \beta_{3} ) q^{87} + ( -42 \beta_{1} + 1008 \beta_{2} ) q^{89} + ( -720 + 5 \beta_{1} + 475 \beta_{2} - 10 \beta_{3} ) q^{91} + ( 60 \beta_{1} - 573 \beta_{2} ) q^{93} + ( 304 + 50 \beta_{1} - 354 \beta_{2} + 50 \beta_{3} ) q^{95} + ( 295 - 29 \beta_{3} ) q^{97} + ( -9 + 63 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{3} - 11q^{5} - 6q^{7} - 18q^{9} + O(q^{10})$$ $$4q + 6q^{3} - 11q^{5} - 6q^{7} - 18q^{9} - 5q^{11} + 10q^{13} - 66q^{15} - 100q^{17} + 67q^{19} - 27q^{21} + 76q^{23} + 93q^{25} - 108q^{27} + 550q^{29} + 362q^{31} + 15q^{33} + 466q^{35} + 5q^{37} + 15q^{39} - 324q^{41} - 1442q^{43} - 99q^{45} - 216q^{47} + 190q^{49} + 300q^{51} - 495q^{53} + 1406q^{55} + 402q^{57} + 173q^{59} + 532q^{61} - 27q^{63} - 510q^{65} + 111q^{67} + 456q^{69} - 3200q^{71} - 1215q^{73} - 279q^{75} - 653q^{77} + 1460q^{79} - 162q^{81} + 2818q^{83} + 328q^{85} + 825q^{87} + 1974q^{89} - 1945q^{91} - 1086q^{93} + 658q^{95} + 1122q^{97} + 90q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 49 x^{2} + 48 x + 2304$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 49 \nu^{2} - 49 \nu + 2304$$$$)/2352$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 97$$$$)/49$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 48 \beta_{2} + \beta_{1} - 49$$ $$\nu^{3}$$ $$=$$ $$49 \beta_{3} - 97$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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
 −3.22311 − 5.58259i 3.72311 + 6.44862i −3.22311 + 5.58259i 3.72311 − 6.44862i
0 1.50000 2.59808i 0 −6.22311 10.7787i 0 −15.3924 + 10.2992i 0 −4.50000 7.79423i 0
193.2 0 1.50000 2.59808i 0 0.723111 + 1.25246i 0 12.3924 13.7633i 0 −4.50000 7.79423i 0
289.1 0 1.50000 + 2.59808i 0 −6.22311 + 10.7787i 0 −15.3924 10.2992i 0 −4.50000 + 7.79423i 0
289.2 0 1.50000 + 2.59808i 0 0.723111 1.25246i 0 12.3924 + 13.7633i 0 −4.50000 + 7.79423i 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 336.4.q.i 4
4.b odd 2 1 84.4.i.a 4
7.c even 3 1 inner 336.4.q.i 4
7.c even 3 1 2352.4.a.bt 2
7.d odd 6 1 2352.4.a.bx 2
12.b even 2 1 252.4.k.f 4
28.d even 2 1 588.4.i.j 4
28.f even 6 1 588.4.a.f 2
28.f even 6 1 588.4.i.j 4
28.g odd 6 1 84.4.i.a 4
28.g odd 6 1 588.4.a.i 2
84.h odd 2 1 1764.4.k.q 4
84.j odd 6 1 1764.4.a.y 2
84.j odd 6 1 1764.4.k.q 4
84.n even 6 1 252.4.k.f 4
84.n even 6 1 1764.4.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.a 4 4.b odd 2 1
84.4.i.a 4 28.g odd 6 1
252.4.k.f 4 12.b even 2 1
252.4.k.f 4 84.n even 6 1
336.4.q.i 4 1.a even 1 1 trivial
336.4.q.i 4 7.c even 3 1 inner
588.4.a.f 2 28.f even 6 1
588.4.a.i 2 28.g odd 6 1
588.4.i.j 4 28.d even 2 1
588.4.i.j 4 28.f even 6 1
1764.4.a.o 2 84.n even 6 1
1764.4.a.y 2 84.j odd 6 1
1764.4.k.q 4 84.h odd 2 1
1764.4.k.q 4 84.j odd 6 1
2352.4.a.bt 2 7.c even 3 1
2352.4.a.bx 2 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 11 T_{5}^{3} + 139 T_{5}^{2} - 198 T_{5} + 324$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$324 - 198 T + 139 T^{2} + 11 T^{3} + T^{4}$$
$7$ $$117649 + 2058 T - 77 T^{2} + 6 T^{3} + T^{4}$$
$11$ $$5560164 - 11790 T + 2383 T^{2} + 5 T^{3} + T^{4}$$
$13$ $$( -1200 - 5 T + T^{2} )^{2}$$
$17$ $$2985984 + 172800 T + 8272 T^{2} + 100 T^{3} + T^{4}$$
$19$ $$473344 - 46096 T + 3801 T^{2} - 67 T^{3} + T^{4}$$
$23$ $$318836736 + 1357056 T + 23632 T^{2} - 76 T^{3} + T^{4}$$
$29$ $$( 13068 - 275 T + T^{2} )^{2}$$
$31$ $$181198521 - 4872882 T + 117583 T^{2} - 362 T^{3} + T^{4}$$
$37$ $$9545290000 + 488500 T + 97725 T^{2} - 5 T^{3} + T^{4}$$
$41$ $$( -9072 + 162 T + T^{2} )^{2}$$
$43$ $$( 129526 + 721 T + T^{2} )^{2}$$
$47$ $$2587553424 - 10987488 T + 97524 T^{2} + 216 T^{3} + T^{4}$$
$53$ $$3288793104 + 28387260 T + 187677 T^{2} + 495 T^{3} + T^{4}$$
$59$ $$16397314704 + 22152996 T + 157981 T^{2} - 173 T^{3} + T^{4}$$
$61$ $$41525136 + 3428208 T + 289468 T^{2} - 532 T^{3} + T^{4}$$
$67$ $$80085604036 + 31412334 T + 295315 T^{2} - 111 T^{3} + T^{4}$$
$71$ $$( 546588 + 1600 T + T^{2} )^{2}$$
$73$ $$54532524484 + 283729230 T + 1242703 T^{2} + 1215 T^{3} + T^{4}$$
$79$ $$120705520329 - 507243420 T + 1784173 T^{2} - 1460 T^{3} + T^{4}$$
$83$ $$( 4122 - 1409 T + T^{2} )^{2}$$
$89$ $$790420571136 - 1754996544 T + 3007620 T^{2} - 1974 T^{3} + T^{4}$$
$97$ $$( 38102 - 561 T + T^{2} )^{2}$$