# Properties

 Label 336.4.q.g 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: $$\Q(\sqrt{-3}, \sqrt{505})$$ Defining polynomial: $$x^{4} - x^{3} + 127 x^{2} + 126 x + 15876$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \beta_{2} ) q^{3} + ( -\beta_{1} - 4 \beta_{2} ) q^{5} + ( -9 + 17 \beta_{2} + \beta_{3} ) q^{7} -9 \beta_{2} q^{9} +O(q^{10})$$ $$q + ( -3 + 3 \beta_{2} ) q^{3} + ( -\beta_{1} - 4 \beta_{2} ) q^{5} + ( -9 + 17 \beta_{2} + \beta_{3} ) q^{7} -9 \beta_{2} q^{9} + ( 5 - 5 \beta_{1} - 5 \beta_{3} ) q^{11} + ( -10 + \beta_{3} ) q^{13} + ( 15 - 3 \beta_{3} ) q^{15} + ( 4 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{17} + ( 7 \beta_{1} + 55 \beta_{2} ) q^{19} + ( -24 - 3 \beta_{1} - 24 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -4 \beta_{1} + 76 \beta_{2} ) q^{23} + ( -26 + 9 \beta_{1} + 17 \beta_{2} + 9 \beta_{3} ) q^{25} + 27 q^{27} + ( -45 - 5 \beta_{3} ) q^{29} + ( -181 + 2 \beta_{1} + 179 \beta_{2} + 2 \beta_{3} ) q^{31} + 15 \beta_{1} q^{33} + ( 85 - 4 \beta_{1} + 90 \beta_{2} - 17 \beta_{3} ) q^{35} + ( -\beta_{1} + 27 \beta_{2} ) q^{37} + ( 30 - 3 \beta_{1} - 27 \beta_{2} - 3 \beta_{3} ) q^{39} + ( -76 - 18 \beta_{3} ) q^{41} + ( -188 - 27 \beta_{3} ) q^{43} + ( -45 + 9 \beta_{1} + 36 \beta_{2} + 9 \beta_{3} ) q^{45} + ( 36 \beta_{1} + 166 \beta_{2} ) q^{47} + ( -82 - 34 \beta_{1} + 17 \beta_{2} - 17 \beta_{3} ) q^{49} + ( -12 \beta_{1} + 12 \beta_{2} ) q^{51} + ( -351 + 5 \beta_{1} + 346 \beta_{2} + 5 \beta_{3} ) q^{53} + ( -655 + 25 \beta_{3} ) q^{55} + ( -186 + 21 \beta_{3} ) q^{57} + ( -279 - 27 \beta_{1} + 306 \beta_{2} - 27 \beta_{3} ) q^{59} + ( 28 \beta_{1} + 566 \beta_{2} ) q^{61} + ( 153 + 9 \beta_{1} - 81 \beta_{2} ) q^{63} + ( 14 \beta_{1} + 162 \beta_{2} ) q^{65} + ( 80 + 73 \beta_{1} - 153 \beta_{2} + 73 \beta_{3} ) q^{67} + ( -216 - 12 \beta_{3} ) q^{69} + ( -346 + 76 \beta_{3} ) q^{71} + ( 440 - 63 \beta_{1} - 377 \beta_{2} - 63 \beta_{3} ) q^{73} + ( -27 \beta_{1} - 51 \beta_{2} ) q^{75} + ( -590 + 45 \beta_{1} + 630 \beta_{2} - 40 \beta_{3} ) q^{77} + ( 48 \beta_{1} - 425 \beta_{2} ) q^{79} + ( -81 + 81 \beta_{2} ) q^{81} + ( 175 - 67 \beta_{3} ) q^{83} + ( 504 - 16 \beta_{3} ) q^{85} + ( 135 + 15 \beta_{1} - 150 \beta_{2} + 15 \beta_{3} ) q^{87} + ( -22 \beta_{1} + 940 \beta_{2} ) q^{89} + ( 216 - 17 \beta_{1} - 153 \beta_{2} - 18 \beta_{3} ) q^{91} + ( -6 \beta_{1} - 537 \beta_{2} ) q^{93} + ( 1192 - 90 \beta_{1} - 1102 \beta_{2} - 90 \beta_{3} ) q^{95} + ( 837 + 55 \beta_{3} ) q^{97} + ( -45 + 45 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{3} - 9q^{5} - 18q^{9} + O(q^{10})$$ $$4q - 6q^{3} - 9q^{5} - 18q^{9} + 5q^{11} - 38q^{13} + 54q^{15} + 4q^{17} + 117q^{19} - 153q^{21} + 148q^{23} - 43q^{25} + 108q^{27} - 190q^{29} - 360q^{31} + 15q^{33} + 482q^{35} + 53q^{37} + 57q^{39} - 340q^{41} - 806q^{43} - 81q^{45} + 368q^{47} - 362q^{49} + 12q^{51} - 697q^{53} - 2570q^{55} - 702q^{57} - 585q^{59} + 1160q^{61} + 459q^{63} + 338q^{65} + 233q^{67} - 888q^{69} - 1232q^{71} + 817q^{73} - 129q^{75} - 1135q^{77} - 802q^{79} - 162q^{81} + 566q^{83} + 1984q^{85} + 285q^{87} + 1858q^{89} + 505q^{91} - 1080q^{93} + 2294q^{95} + 3458q^{97} - 90q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 127 \nu^{2} - 127 \nu + 15876$$$$)/16002$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 253$$$$)/127$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 126 \beta_{2} + \beta_{1} - 127$$ $$\nu^{3}$$ $$=$$ $$127 \beta_{3} - 253$$

## 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
 5.86805 + 10.1638i −5.36805 − 9.29774i 5.86805 − 10.1638i −5.36805 + 9.29774i
0 −1.50000 + 2.59808i 0 −7.86805 13.6279i 0 −11.2361 + 14.7224i 0 −4.50000 7.79423i 0
193.2 0 −1.50000 + 2.59808i 0 3.36805 + 5.83364i 0 11.2361 + 14.7224i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 −7.86805 + 13.6279i 0 −11.2361 14.7224i 0 −4.50000 + 7.79423i 0
289.2 0 −1.50000 2.59808i 0 3.36805 5.83364i 0 11.2361 14.7224i 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.g 4
4.b odd 2 1 168.4.q.d 4
7.c even 3 1 inner 336.4.q.g 4
7.c even 3 1 2352.4.a.cc 2
7.d odd 6 1 2352.4.a.bo 2
12.b even 2 1 504.4.s.f 4
28.f even 6 1 1176.4.a.u 2
28.g odd 6 1 168.4.q.d 4
28.g odd 6 1 1176.4.a.r 2
84.n even 6 1 504.4.s.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d 4 4.b odd 2 1
168.4.q.d 4 28.g odd 6 1
336.4.q.g 4 1.a even 1 1 trivial
336.4.q.g 4 7.c even 3 1 inner
504.4.s.f 4 12.b even 2 1
504.4.s.f 4 84.n even 6 1
1176.4.a.r 2 28.g odd 6 1
1176.4.a.u 2 28.f even 6 1
2352.4.a.bo 2 7.d odd 6 1
2352.4.a.cc 2 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 9 T_{5}^{3} + 187 T_{5}^{2} - 954 T_{5} + 11236$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 + 3 T + T^{2} )^{2}$$
$5$ $$11236 - 954 T + 187 T^{2} + 9 T^{3} + T^{4}$$
$7$ $$117649 + 181 T^{2} + T^{4}$$
$11$ $$9922500 + 15750 T + 3175 T^{2} - 5 T^{3} + T^{4}$$
$13$ $$( -36 + 19 T + T^{2} )^{2}$$
$17$ $$4064256 + 8064 T + 2032 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$7639696 + 323388 T + 16453 T^{2} - 117 T^{3} + T^{4}$$
$23$ $$11943936 - 511488 T + 18448 T^{2} - 148 T^{3} + T^{4}$$
$29$ $$( -900 + 95 T + T^{2} )^{2}$$
$31$ $$1017291025 + 11482200 T + 97705 T^{2} + 360 T^{3} + T^{4}$$
$37$ $$331776 - 30528 T + 2233 T^{2} - 53 T^{3} + T^{4}$$
$41$ $$( -33680 + 170 T + T^{2} )^{2}$$
$43$ $$( -51434 + 403 T + T^{2} )^{2}$$
$47$ $$16838695696 + 47753152 T + 265188 T^{2} - 368 T^{3} + T^{4}$$
$53$ $$13993943616 + 82452312 T + 367513 T^{2} + 697 T^{3} + T^{4}$$
$59$ $$41990400 - 3790800 T + 348705 T^{2} + 585 T^{3} + T^{4}$$
$61$ $$56368256400 - 275407200 T + 1108180 T^{2} - 1160 T^{3} + T^{4}$$
$67$ $$434563097796 + 153596862 T + 713503 T^{2} - 233 T^{3} + T^{4}$$
$71$ $$( -634356 + 616 T + T^{2} )^{2}$$
$73$ $$111698997796 + 273052838 T + 1001703 T^{2} - 817 T^{3} + T^{4}$$
$79$ $$16920546241 - 104323358 T + 773283 T^{2} + 802 T^{3} + T^{4}$$
$83$ $$( -546714 - 283 T + T^{2} )^{2}$$
$89$ $$643101348096 - 1489997088 T + 2650228 T^{2} - 1858 T^{3} + T^{4}$$
$97$ $$( 365454 - 1729 T + T^{2} )^{2}$$