# Properties

 Label 336.4.q.h 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: $$\Q(\sqrt{-3}, \sqrt{-19})$$ Defining polynomial: $$x^{4} - x^{3} - 4 x^{2} - 5 x + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$3^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{2} q^{3} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( 3 - 3 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{7} + ( -9 - 9 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + 3 \beta_{2} q^{3} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{5} + ( 3 - 3 \beta_{1} - 6 \beta_{2} + \beta_{3} ) q^{7} + ( -9 - 9 \beta_{2} ) q^{9} + ( -1 + \beta_{1} + 25 \beta_{2} - 2 \beta_{3} ) q^{11} + ( 33 + 5 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} ) q^{13} + ( -6 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{15} + ( -8 + 8 \beta_{1} + 8 \beta_{2} - 16 \beta_{3} ) q^{17} + ( 84 + 2 \beta_{1} + 85 \beta_{2} - \beta_{3} ) q^{19} + ( 18 + 3 \beta_{1} + 27 \beta_{2} + 6 \beta_{3} ) q^{21} + ( -96 - 96 \beta_{2} ) q^{23} + ( -3 + 3 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} ) q^{25} + 27 q^{27} + ( -28 - 17 \beta_{1} - 17 \beta_{2} - 17 \beta_{3} ) q^{29} + ( -10 + 10 \beta_{1} + 41 \beta_{2} - 20 \beta_{3} ) q^{31} + ( -75 - 6 \beta_{1} - 78 \beta_{2} + 3 \beta_{3} ) q^{33} + ( -28 + 7 \beta_{1} - 203 \beta_{2} + 7 \beta_{3} ) q^{35} + ( 96 - 38 \beta_{1} + 77 \beta_{2} + 19 \beta_{3} ) q^{37} + ( -15 + 15 \beta_{1} + 84 \beta_{2} - 30 \beta_{3} ) q^{39} + ( 70 - 34 \beta_{1} - 34 \beta_{2} - 34 \beta_{3} ) q^{41} + ( 239 - 19 \beta_{1} - 19 \beta_{2} - 19 \beta_{3} ) q^{43} + ( 9 - 9 \beta_{1} - 9 \beta_{2} + 18 \beta_{3} ) q^{45} + ( 82 + 32 \beta_{1} + 98 \beta_{2} - 16 \beta_{3} ) q^{47} + ( 102 - 25 \beta_{1} + 272 \beta_{2} - 15 \beta_{3} ) q^{49} + ( -24 - 48 \beta_{1} - 48 \beta_{2} + 24 \beta_{3} ) q^{51} + ( -27 + 27 \beta_{1} + 129 \beta_{2} - 54 \beta_{3} ) q^{53} + ( -180 - 27 \beta_{1} - 27 \beta_{2} - 27 \beta_{3} ) q^{55} + ( -255 - 3 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{57} + ( -3 + 3 \beta_{1} + 633 \beta_{2} - 6 \beta_{3} ) q^{59} + ( -182 + 72 \beta_{1} - 146 \beta_{2} - 36 \beta_{3} ) q^{61} + ( -81 + 18 \beta_{1} - 27 \beta_{2} - 27 \beta_{3} ) q^{63} + ( 668 + 76 \beta_{1} + 706 \beta_{2} - 38 \beta_{3} ) q^{65} + ( 9 - 9 \beta_{1} + 442 \beta_{2} + 18 \beta_{3} ) q^{67} + 288 q^{69} + ( 686 - 32 \beta_{1} - 32 \beta_{2} - 32 \beta_{3} ) q^{71} + ( 21 - 21 \beta_{1} - 670 \beta_{2} + 42 \beta_{3} ) q^{73} + ( -12 - 18 \beta_{1} - 21 \beta_{2} + 9 \beta_{3} ) q^{75} + ( 319 + 38 \beta_{1} + 188 \beta_{2} + 41 \beta_{3} ) q^{77} + ( -89 - 8 \beta_{1} - 93 \beta_{2} + 4 \beta_{3} ) q^{79} + 81 \beta_{2} q^{81} + ( 178 - 43 \beta_{1} - 43 \beta_{2} - 43 \beta_{3} ) q^{83} + ( -1056 - 24 \beta_{1} - 24 \beta_{2} - 24 \beta_{3} ) q^{85} + ( 51 - 51 \beta_{1} - 33 \beta_{2} + 102 \beta_{3} ) q^{87} + ( -422 + 44 \beta_{1} - 400 \beta_{2} - 22 \beta_{3} ) q^{89} + ( -946 - 104 \beta_{1} - 1013 \beta_{2} + 93 \beta_{3} ) q^{91} + ( -123 - 60 \beta_{1} - 153 \beta_{2} + 30 \beta_{3} ) q^{93} + ( -86 + 86 \beta_{1} + 212 \beta_{2} - 172 \beta_{3} ) q^{95} + ( 410 - 21 \beta_{1} - 21 \beta_{2} - 21 \beta_{3} ) q^{97} + ( 234 + 9 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{3} + 3q^{5} + 20q^{7} - 18q^{9} + O(q^{10})$$ $$4q - 6q^{3} + 3q^{5} + 20q^{7} - 18q^{9} - 51q^{11} + 122q^{13} - 18q^{15} - 24q^{17} + 169q^{19} + 15q^{21} - 192q^{23} - 11q^{25} + 108q^{27} - 78q^{29} - 92q^{31} - 153q^{33} + 294q^{35} + 173q^{37} - 183q^{39} + 348q^{41} + 994q^{43} + 27q^{45} + 180q^{47} - 146q^{49} - 72q^{51} - 285q^{53} - 666q^{55} - 1014q^{57} - 1269q^{59} - 328q^{61} - 225q^{63} + 1374q^{65} - 875q^{67} + 1152q^{69} + 2808q^{71} + 1361q^{73} - 33q^{75} + 897q^{77} - 182q^{79} - 162q^{81} + 798q^{83} - 4176q^{85} + 117q^{87} - 822q^{89} - 1955q^{91} - 276q^{93} - 510q^{95} + 1682q^{97} + 918q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} + 56 \nu - 25$$$$)/20$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu^{2} - 4 \nu - 25$$$$)/20$$ $$\beta_{3}$$ $$=$$ $$($$$$-7 \nu^{3} + 2 \nu^{2} + 28 \nu + 35$$$$)/10$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 14 \beta_{2} + 14$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{3} + 4 \beta_{1} + 19$$$$)/3$$

## Character values

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

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

## 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
 −1.63746 − 1.52274i 2.13746 + 0.656712i −1.63746 + 1.52274i 2.13746 − 0.656712i
0 −1.50000 + 2.59808i 0 −4.91238 8.50848i 0 16.3248 + 8.74657i 0 −4.50000 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 6.41238 + 11.1066i 0 −6.32475 17.4068i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 −4.91238 + 8.50848i 0 16.3248 8.74657i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 2.59808i 0 6.41238 11.1066i 0 −6.32475 + 17.4068i 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.h 4
4.b odd 2 1 84.4.i.b 4
7.c even 3 1 inner 336.4.q.h 4
7.c even 3 1 2352.4.a.cb 2
7.d odd 6 1 2352.4.a.bp 2
12.b even 2 1 252.4.k.d 4
28.d even 2 1 588.4.i.i 4
28.f even 6 1 588.4.a.h 2
28.f even 6 1 588.4.i.i 4
28.g odd 6 1 84.4.i.b 4
28.g odd 6 1 588.4.a.g 2
84.h odd 2 1 1764.4.k.z 4
84.j odd 6 1 1764.4.a.p 2
84.j odd 6 1 1764.4.k.z 4
84.n even 6 1 252.4.k.d 4
84.n even 6 1 1764.4.a.x 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 3 T_{5}^{3} + 135 T_{5}^{2} + 378 T_{5} + 15876$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 + 3 T + T^{2} )^{2}$$
$5$ $$15876 + 378 T + 135 T^{2} - 3 T^{3} + T^{4}$$
$7$ $$117649 - 6860 T + 273 T^{2} - 20 T^{3} + T^{4}$$
$11$ $$272484 + 26622 T + 2079 T^{2} + 51 T^{3} + T^{4}$$
$13$ $$( -2276 - 61 T + T^{2} )^{2}$$
$17$ $$65028096 - 193536 T + 8640 T^{2} + 24 T^{3} + T^{4}$$
$19$ $$49168144 - 1185028 T + 21549 T^{2} - 169 T^{3} + T^{4}$$
$23$ $$( 9216 + 96 T + T^{2} )^{2}$$
$29$ $$( -36684 + 39 T + T^{2} )^{2}$$
$31$ $$114682681 - 985228 T + 19173 T^{2} + 92 T^{3} + T^{4}$$
$37$ $$1506681856 + 6715168 T + 68745 T^{2} - 173 T^{3} + T^{4}$$
$41$ $$( -140688 - 174 T + T^{2} )^{2}$$
$43$ $$( 15454 - 497 T + T^{2} )^{2}$$
$47$ $$611671824 + 4451760 T + 57132 T^{2} - 180 T^{3} + T^{4}$$
$53$ $$5356483344 - 20858580 T + 154413 T^{2} + 285 T^{3} + T^{4}$$
$59$ $$161150862096 + 509422284 T + 1208925 T^{2} + 1269 T^{3} + T^{4}$$
$61$ $$19408947856 - 45695648 T + 246900 T^{2} + 328 T^{3} + T^{4}$$
$67$ $$32767516324 + 158390750 T + 584607 T^{2} + 875 T^{3} + T^{4}$$
$71$ $$( 361476 - 1404 T + T^{2} )^{2}$$
$73$ $$165260136484 - 553276442 T + 1445799 T^{2} - 1361 T^{3} + T^{4}$$
$79$ $$38800441 + 1133678 T + 26895 T^{2} + 182 T^{3} + T^{4}$$
$83$ $$( -197334 - 399 T + T^{2} )^{2}$$
$89$ $$11416495104 + 87829056 T + 568836 T^{2} + 822 T^{3} + T^{4}$$
$97$ $$( 120262 - 841 T + T^{2} )^{2}$$