# Properties

 Label 336.4.q.j 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: $$\Q(\sqrt{-3}, \sqrt{1345})$$ Defining polynomial: $$x^{4} - x^{3} + 337 x^{2} + 336 x + 112896$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} + ( -1 + 3 \beta_{2} - \beta_{3} ) q^{7} -9 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( 3 - 3 \beta_{2} ) q^{3} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} + ( -1 + 3 \beta_{2} - \beta_{3} ) q^{7} -9 \beta_{2} q^{9} + ( 33 + \beta_{1} - 34 \beta_{2} + \beta_{3} ) q^{11} + ( 20 + \beta_{3} ) q^{13} + ( -9 + 3 \beta_{3} ) q^{15} + ( 48 - 4 \beta_{1} - 44 \beta_{2} - 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} + 23 \beta_{2} ) q^{19} + ( 6 - 3 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -4 \beta_{1} + 76 \beta_{2} ) q^{23} + ( -220 + 5 \beta_{1} + 215 \beta_{2} + 5 \beta_{3} ) q^{25} -27 q^{27} + ( 33 + 11 \beta_{3} ) q^{29} + ( -259 - 2 \beta_{1} + 261 \beta_{2} - 2 \beta_{3} ) q^{31} + ( 3 \beta_{1} - 102 \beta_{2} ) q^{33} + ( 9 - 4 \beta_{1} - 338 \beta_{2} - 3 \beta_{3} ) q^{35} + ( -9 \beta_{1} + \beta_{2} ) q^{37} + ( 60 + 3 \beta_{1} - 63 \beta_{2} + 3 \beta_{3} ) q^{39} + ( -216 + 6 \beta_{3} ) q^{41} + ( 46 + 15 \beta_{3} ) q^{43} + ( -27 + 9 \beta_{1} + 18 \beta_{2} + 9 \beta_{3} ) q^{45} + ( 12 \beta_{1} + 282 \beta_{2} ) q^{47} + ( 328 + 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{49} + ( -12 \beta_{1} - 132 \beta_{2} ) q^{51} + ( 123 - 3 \beta_{1} - 120 \beta_{2} - 3 \beta_{3} ) q^{53} + ( 237 + 31 \beta_{3} ) q^{55} + ( 60 + 9 \beta_{3} ) q^{57} + ( 9 - 25 \beta_{1} + 16 \beta_{2} - 25 \beta_{3} ) q^{59} + ( 4 \beta_{1} - 114 \beta_{2} ) q^{61} + ( 27 - 9 \beta_{1} - 9 \beta_{2} ) q^{63} + ( -18 \beta_{1} + 294 \beta_{2} ) q^{65} + ( 338 + 11 \beta_{1} - 349 \beta_{2} + 11 \beta_{3} ) q^{67} + ( 216 + 12 \beta_{3} ) q^{69} + ( -246 + 20 \beta_{3} ) q^{71} + ( 478 - 35 \beta_{1} - 443 \beta_{2} - 35 \beta_{3} ) q^{73} + ( 15 \beta_{1} + 645 \beta_{2} ) q^{75} + ( -270 - 35 \beta_{1} + 404 \beta_{2} - 32 \beta_{3} ) q^{77} + ( 8 \beta_{1} - 267 \beta_{2} ) q^{79} + ( -81 + 81 \beta_{2} ) q^{81} + ( 123 - 25 \beta_{3} ) q^{83} + ( -1488 + 56 \beta_{3} ) q^{85} + ( 99 + 33 \beta_{1} - 132 \beta_{2} + 33 \beta_{3} ) q^{87} + ( 42 \beta_{1} - 408 \beta_{2} ) q^{89} + ( -356 - 3 \beta_{1} + 63 \beta_{2} - 22 \beta_{3} ) q^{91} + ( -6 \beta_{1} + 783 \beta_{2} ) q^{93} + ( -948 - 14 \beta_{1} + 962 \beta_{2} - 14 \beta_{3} ) q^{95} + ( 959 + 35 \beta_{3} ) q^{97} + ( -297 - 9 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{3} - 5q^{5} - 18q^{9} + O(q^{10})$$ $$4q + 6q^{3} - 5q^{5} - 18q^{9} + 67q^{11} + 82q^{13} - 30q^{15} + 92q^{17} + 43q^{19} + 27q^{21} + 148q^{23} - 435q^{25} - 108q^{27} + 154q^{29} - 520q^{31} - 201q^{33} - 650q^{35} - 7q^{37} + 123q^{39} - 852q^{41} + 214q^{43} - 45q^{45} + 576q^{47} + 1318q^{49} - 276q^{51} + 243q^{53} + 1010q^{55} + 258q^{57} - 7q^{59} - 224q^{61} + 81q^{63} + 570q^{65} + 687q^{67} + 888q^{69} - 944q^{71} + 921q^{73} + 1305q^{75} - 371q^{77} - 526q^{79} - 162q^{81} + 442q^{83} - 5840q^{85} + 231q^{87} - 774q^{89} - 1345q^{91} + 1560q^{93} - 1910q^{95} + 3906q^{97} - 1206q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 337 x^{2} + 336 x + 112896$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 337 \nu^{2} - 337 \nu + 112896$$$$)/113232$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 673$$$$)/337$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 336 \beta_{2} + \beta_{1} - 337$$ $$\nu^{3}$$ $$=$$ $$337 \beta_{3} - 673$$

## 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
 9.41856 + 16.3134i −8.91856 − 15.4474i 9.41856 − 16.3134i −8.91856 + 15.4474i
0 1.50000 2.59808i 0 −10.4186 18.0455i 0 18.3371 + 2.59808i 0 −4.50000 7.79423i 0
193.2 0 1.50000 2.59808i 0 7.91856 + 13.7153i 0 −18.3371 + 2.59808i 0 −4.50000 7.79423i 0
289.1 0 1.50000 + 2.59808i 0 −10.4186 + 18.0455i 0 18.3371 2.59808i 0 −4.50000 + 7.79423i 0
289.2 0 1.50000 + 2.59808i 0 7.91856 13.7153i 0 −18.3371 2.59808i 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.j 4
4.b odd 2 1 42.4.e.c 4
7.c even 3 1 inner 336.4.q.j 4
7.c even 3 1 2352.4.a.bq 2
7.d odd 6 1 2352.4.a.ca 2
12.b even 2 1 126.4.g.g 4
28.d even 2 1 294.4.e.l 4
28.f even 6 1 294.4.a.m 2
28.f even 6 1 294.4.e.l 4
28.g odd 6 1 42.4.e.c 4
28.g odd 6 1 294.4.a.n 2
84.h odd 2 1 882.4.g.bf 4
84.j odd 6 1 882.4.a.z 2
84.j odd 6 1 882.4.g.bf 4
84.n even 6 1 126.4.g.g 4
84.n even 6 1 882.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.e.c 4 4.b odd 2 1
42.4.e.c 4 28.g odd 6 1
126.4.g.g 4 12.b even 2 1
126.4.g.g 4 84.n even 6 1
294.4.a.m 2 28.f even 6 1
294.4.a.n 2 28.g odd 6 1
294.4.e.l 4 28.d even 2 1
294.4.e.l 4 28.f even 6 1
336.4.q.j 4 1.a even 1 1 trivial
336.4.q.j 4 7.c even 3 1 inner
882.4.a.v 2 84.n even 6 1
882.4.a.z 2 84.j odd 6 1
882.4.g.bf 4 84.h odd 2 1
882.4.g.bf 4 84.j odd 6 1
2352.4.a.bq 2 7.c even 3 1
2352.4.a.ca 2 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 5 T_{5}^{3} + 355 T_{5}^{2} - 1650 T_{5} + 108900$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 - 3 T + T^{2} )^{2}$$
$5$ $$108900 - 1650 T + 355 T^{2} + 5 T^{3} + T^{4}$$
$7$ $$117649 - 659 T^{2} + T^{4}$$
$11$ $$617796 - 52662 T + 3703 T^{2} - 67 T^{3} + T^{4}$$
$13$ $$( 84 - 41 T + T^{2} )^{2}$$
$17$ $$10653696 + 300288 T + 11728 T^{2} - 92 T^{3} + T^{4}$$
$19$ $$6574096 + 110252 T + 4413 T^{2} - 43 T^{3} + T^{4}$$
$23$ $$9216 - 14208 T + 21808 T^{2} - 148 T^{3} + T^{4}$$
$29$ $$( -39204 - 77 T + T^{2} )^{2}$$
$31$ $$4389725025 + 34452600 T + 204145 T^{2} + 520 T^{3} + T^{4}$$
$37$ $$741146176 - 190568 T + 27273 T^{2} + 7 T^{3} + T^{4}$$
$41$ $$( 33264 + 426 T + T^{2} )^{2}$$
$43$ $$( -72794 - 107 T + T^{2} )^{2}$$
$47$ $$1191906576 - 19885824 T + 297252 T^{2} - 576 T^{3} + T^{4}$$
$53$ $$137733696 - 2851848 T + 47313 T^{2} - 243 T^{3} + T^{4}$$
$59$ $$44160500736 - 1471008 T + 210193 T^{2} + 7 T^{3} + T^{4}$$
$61$ $$51322896 + 1604736 T + 43012 T^{2} + 224 T^{3} + T^{4}$$
$67$ $$5976217636 - 53109222 T + 394663 T^{2} - 687 T^{3} + T^{4}$$
$71$ $$( -78804 + 472 T + T^{2} )^{2}$$
$73$ $$39938423716 + 184058166 T + 1048087 T^{2} - 921 T^{3} + T^{4}$$
$79$ $$2270427201 + 25063374 T + 229027 T^{2} + 526 T^{3} + T^{4}$$
$83$ $$( -197946 - 221 T + T^{2} )^{2}$$
$89$ $$196582277376 - 343173024 T + 1042452 T^{2} + 774 T^{3} + T^{4}$$
$97$ $$( 541646 - 1953 T + T^{2} )^{2}$$