# Properties

 Label 336.4.q.d Level $336$ Weight $4$ Character orbit 336.q Analytic conductor $19.825$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} + 15 \zeta_{6} q^{5} + ( -21 + 7 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -3 + 3 \zeta_{6} ) q^{3} + 15 \zeta_{6} q^{5} + ( -21 + 7 \zeta_{6} ) q^{7} -9 \zeta_{6} q^{9} + ( -9 + 9 \zeta_{6} ) q^{11} -88 q^{13} -45 q^{15} + ( 84 - 84 \zeta_{6} ) q^{17} + 104 \zeta_{6} q^{19} + ( 42 - 63 \zeta_{6} ) q^{21} -84 \zeta_{6} q^{23} + ( -100 + 100 \zeta_{6} ) q^{25} + 27 q^{27} + 51 q^{29} + ( 185 - 185 \zeta_{6} ) q^{31} -27 \zeta_{6} q^{33} + ( -105 - 210 \zeta_{6} ) q^{35} -44 \zeta_{6} q^{37} + ( 264 - 264 \zeta_{6} ) q^{39} -168 q^{41} -326 q^{43} + ( 135 - 135 \zeta_{6} ) q^{45} -138 \zeta_{6} q^{47} + ( 392 - 245 \zeta_{6} ) q^{49} + 252 \zeta_{6} q^{51} + ( -639 + 639 \zeta_{6} ) q^{53} -135 q^{55} -312 q^{57} + ( 159 - 159 \zeta_{6} ) q^{59} -722 \zeta_{6} q^{61} + ( 63 + 126 \zeta_{6} ) q^{63} -1320 \zeta_{6} q^{65} + ( -166 + 166 \zeta_{6} ) q^{67} + 252 q^{69} -1086 q^{71} + ( -218 + 218 \zeta_{6} ) q^{73} -300 \zeta_{6} q^{75} + ( 126 - 189 \zeta_{6} ) q^{77} -583 \zeta_{6} q^{79} + ( -81 + 81 \zeta_{6} ) q^{81} + 597 q^{83} + 1260 q^{85} + ( -153 + 153 \zeta_{6} ) q^{87} + 1038 \zeta_{6} q^{89} + ( 1848 - 616 \zeta_{6} ) q^{91} + 555 \zeta_{6} q^{93} + ( -1560 + 1560 \zeta_{6} ) q^{95} -169 q^{97} + 81 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} + 15q^{5} - 35q^{7} - 9q^{9} + O(q^{10})$$ $$2q - 3q^{3} + 15q^{5} - 35q^{7} - 9q^{9} - 9q^{11} - 176q^{13} - 90q^{15} + 84q^{17} + 104q^{19} + 21q^{21} - 84q^{23} - 100q^{25} + 54q^{27} + 102q^{29} + 185q^{31} - 27q^{33} - 420q^{35} - 44q^{37} + 264q^{39} - 336q^{41} - 652q^{43} + 135q^{45} - 138q^{47} + 539q^{49} + 252q^{51} - 639q^{53} - 270q^{55} - 624q^{57} + 159q^{59} - 722q^{61} + 252q^{63} - 1320q^{65} - 166q^{67} + 504q^{69} - 2172q^{71} - 218q^{73} - 300q^{75} + 63q^{77} - 583q^{79} - 81q^{81} + 1194q^{83} + 2520q^{85} - 153q^{87} + 1038q^{89} + 3080q^{91} + 555q^{93} - 1560q^{95} - 338q^{97} + 162q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 + 2.59808i 0 7.50000 + 12.9904i 0 −17.5000 + 6.06218i 0 −4.50000 7.79423i 0
289.1 0 −1.50000 2.59808i 0 7.50000 12.9904i 0 −17.5000 6.06218i 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.d 2
4.b odd 2 1 42.4.e.b 2
7.c even 3 1 inner 336.4.q.d 2
7.c even 3 1 2352.4.a.u 1
7.d odd 6 1 2352.4.a.q 1
12.b even 2 1 126.4.g.a 2
28.d even 2 1 294.4.e.e 2
28.f even 6 1 294.4.a.g 1
28.f even 6 1 294.4.e.e 2
28.g odd 6 1 42.4.e.b 2
28.g odd 6 1 294.4.a.a 1
84.h odd 2 1 882.4.g.l 2
84.j odd 6 1 882.4.a.h 1
84.j odd 6 1 882.4.g.l 2
84.n even 6 1 126.4.g.a 2
84.n even 6 1 882.4.a.r 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 15 T_{5} + 225$$ acting on $$S_{4}^{\mathrm{new}}(336, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$225 - 15 T + T^{2}$$
$7$ $$343 + 35 T + T^{2}$$
$11$ $$81 + 9 T + T^{2}$$
$13$ $$( 88 + T )^{2}$$
$17$ $$7056 - 84 T + T^{2}$$
$19$ $$10816 - 104 T + T^{2}$$
$23$ $$7056 + 84 T + T^{2}$$
$29$ $$( -51 + T )^{2}$$
$31$ $$34225 - 185 T + T^{2}$$
$37$ $$1936 + 44 T + T^{2}$$
$41$ $$( 168 + T )^{2}$$
$43$ $$( 326 + T )^{2}$$
$47$ $$19044 + 138 T + T^{2}$$
$53$ $$408321 + 639 T + T^{2}$$
$59$ $$25281 - 159 T + T^{2}$$
$61$ $$521284 + 722 T + T^{2}$$
$67$ $$27556 + 166 T + T^{2}$$
$71$ $$( 1086 + T )^{2}$$
$73$ $$47524 + 218 T + T^{2}$$
$79$ $$339889 + 583 T + T^{2}$$
$83$ $$( -597 + T )^{2}$$
$89$ $$1077444 - 1038 T + T^{2}$$
$97$ $$( 169 + T )^{2}$$