# Properties

 Label 336.2.s.a Level 336 Weight 2 Character orbit 336.s Analytic conductor 2.683 Analytic rank 1 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$1$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{8}^{2} ) q^{2} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{4} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{5} + ( 1 - 2 \zeta_{8} - \zeta_{8}^{2} ) q^{6} - q^{7} + ( 2 + 2 \zeta_{8}^{2} ) q^{8} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{8}^{2} ) q^{2} + ( -1 + \zeta_{8} + \zeta_{8}^{3} ) q^{3} -2 \zeta_{8}^{2} q^{4} + ( -1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{5} + ( 1 - 2 \zeta_{8} - \zeta_{8}^{2} ) q^{6} - q^{7} + ( 2 + 2 \zeta_{8}^{2} ) q^{8} + ( -1 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{9} + ( 2 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{10} + ( -1 + \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{11} + ( 2 \zeta_{8} + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{12} + ( -3 - 2 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{13} + ( 1 - \zeta_{8}^{2} ) q^{14} + ( -1 - 2 \zeta_{8} + 3 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{15} -4 q^{16} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{17} + ( 1 + 4 \zeta_{8} - \zeta_{8}^{2} ) q^{18} + ( -1 + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{19} + ( -2 + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{20} + ( 1 - \zeta_{8} - \zeta_{8}^{3} ) q^{21} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{22} + ( -4 \zeta_{8} - 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{23} + ( -2 - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{24} + ( -4 \zeta_{8} + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{25} + ( 6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{26} + ( 5 + \zeta_{8} + \zeta_{8}^{3} ) q^{27} + 2 \zeta_{8}^{2} q^{28} + ( -3 + 3 \zeta_{8}^{2} ) q^{29} + ( -2 + 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{30} + ( 6 \zeta_{8} - 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{31} + ( 4 - 4 \zeta_{8}^{2} ) q^{32} + ( -3 - 2 \zeta_{8} - 5 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{33} + ( -4 + 4 \zeta_{8} - 4 \zeta_{8}^{2} ) q^{34} + ( 1 - 2 \zeta_{8} + \zeta_{8}^{2} ) q^{35} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{36} + ( -1 + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{38} + ( 5 + 2 \zeta_{8} + \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{39} + ( 4 \zeta_{8} - 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{40} + ( -6 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{41} + ( -1 + 2 \zeta_{8} + \zeta_{8}^{2} ) q^{42} + ( -1 - 4 \zeta_{8} - \zeta_{8}^{2} ) q^{43} + ( 2 + 8 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{44} + ( 5 - 2 \zeta_{8} - 3 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{45} + ( 2 + 8 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{46} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{47} + ( 4 - 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{48} + q^{49} + ( -1 + 8 \zeta_{8} - \zeta_{8}^{2} ) q^{50} + ( 4 - 2 \zeta_{8} - 4 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{51} + ( -6 + 6 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{52} + ( -3 - 4 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{53} + ( -5 - 2 \zeta_{8} + 5 \zeta_{8}^{2} ) q^{54} + ( -6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{55} + ( -2 - 2 \zeta_{8}^{2} ) q^{56} + ( 3 - 2 \zeta_{8} + \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{57} -6 \zeta_{8}^{2} q^{58} + ( -3 + 3 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{59} + ( 6 - 4 \zeta_{8} + 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{60} + ( 1 + 2 \zeta_{8} + \zeta_{8}^{2} ) q^{61} + ( 2 - 12 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{62} + ( 1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{63} + 8 \zeta_{8}^{2} q^{64} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{65} + ( 8 + 6 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{66} + ( -7 + 7 \zeta_{8}^{2} ) q^{67} + ( 8 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{68} + ( 8 + 6 \zeta_{8} + 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{69} + ( -2 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{70} -6 \zeta_{8}^{2} q^{71} + ( -2 - 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{72} + ( -2 \zeta_{8} + 8 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{73} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{74} + ( 8 + 3 \zeta_{8} - \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{75} + ( 2 - 4 \zeta_{8} + 2 \zeta_{8}^{2} ) q^{76} + ( 1 - \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{77} + ( -6 + 4 \zeta_{8} + 4 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{78} + 2 \zeta_{8}^{2} q^{79} + ( 4 - 8 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{80} + ( -7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + ( 6 - 6 \zeta_{8}^{2} + 8 \zeta_{8}^{3} ) q^{82} + ( -1 + 14 \zeta_{8} - \zeta_{8}^{2} ) q^{83} + ( -2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{84} + ( 8 - 8 \zeta_{8}^{2} + 12 \zeta_{8}^{3} ) q^{85} + ( 2 + 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{86} + ( 3 - 6 \zeta_{8} - 3 \zeta_{8}^{2} ) q^{87} + ( -4 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{88} + ( -10 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{89} + ( -2 - 2 \zeta_{8} + 8 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{90} + ( 3 + 2 \zeta_{8} + 3 \zeta_{8}^{2} ) q^{91} + ( -4 - 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{92} + ( -12 - 4 \zeta_{8} + 2 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{93} -8 \zeta_{8}^{3} q^{94} + ( 6 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{95} + ( -4 + 8 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{96} -2 q^{97} + ( -1 + \zeta_{8}^{2} ) q^{98} + ( 9 + 4 \zeta_{8} + 7 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - 4q^{3} - 4q^{5} + 4q^{6} - 4q^{7} + 8q^{8} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} - 4q^{3} - 4q^{5} + 4q^{6} - 4q^{7} + 8q^{8} - 4q^{9} + 8q^{10} - 4q^{11} - 12q^{13} + 4q^{14} - 4q^{15} - 16q^{16} + 4q^{18} - 4q^{19} - 8q^{20} + 4q^{21} - 8q^{24} + 24q^{26} + 20q^{27} - 12q^{29} - 8q^{30} + 16q^{32} - 12q^{33} - 16q^{34} + 4q^{35} - 4q^{37} + 20q^{39} - 24q^{41} - 4q^{42} - 4q^{43} + 8q^{44} + 20q^{45} + 8q^{46} + 16q^{48} + 4q^{49} - 4q^{50} + 16q^{51} - 24q^{52} - 12q^{53} - 20q^{54} - 24q^{55} - 8q^{56} + 12q^{57} - 12q^{59} + 24q^{60} + 4q^{61} + 8q^{62} + 4q^{63} + 32q^{66} - 28q^{67} + 32q^{68} + 32q^{69} - 8q^{70} - 8q^{72} + 32q^{75} + 8q^{76} + 4q^{77} - 24q^{78} + 16q^{80} - 28q^{81} + 24q^{82} - 4q^{83} + 32q^{85} + 8q^{86} + 12q^{87} - 16q^{88} - 40q^{89} - 8q^{90} + 12q^{91} - 16q^{92} - 48q^{93} + 24q^{95} - 16q^{96} - 8q^{97} - 4q^{98} + 36q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$-\zeta_{8}$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
155.1
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−1.00000 1.00000i −1.00000 1.41421i 2.00000i 0.414214 0.414214i −0.414214 + 2.41421i −1.00000 2.00000 2.00000i −1.00000 + 2.82843i −0.828427
155.2 −1.00000 1.00000i −1.00000 + 1.41421i 2.00000i −2.41421 + 2.41421i 2.41421 0.414214i −1.00000 2.00000 2.00000i −1.00000 2.82843i 4.82843
323.1 −1.00000 + 1.00000i −1.00000 1.41421i 2.00000i −2.41421 2.41421i 2.41421 + 0.414214i −1.00000 2.00000 + 2.00000i −1.00000 + 2.82843i 4.82843
323.2 −1.00000 + 1.00000i −1.00000 + 1.41421i 2.00000i 0.414214 + 0.414214i −0.414214 2.41421i −1.00000 2.00000 + 2.00000i −1.00000 2.82843i −0.828427
 $$n$$: 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
48.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.s.a 4
3.b odd 2 1 336.2.s.b yes 4
4.b odd 2 1 1344.2.s.b 4
12.b even 2 1 1344.2.s.a 4
16.e even 4 1 1344.2.s.a 4
16.f odd 4 1 336.2.s.b yes 4
48.i odd 4 1 1344.2.s.b 4
48.k even 4 1 inner 336.2.s.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.s.a 4 1.a even 1 1 trivial
336.2.s.a 4 48.k even 4 1 inner
336.2.s.b yes 4 3.b odd 2 1
336.2.s.b yes 4 16.f odd 4 1
1344.2.s.a 4 12.b even 2 1
1344.2.s.a 4 16.e even 4 1
1344.2.s.b 4 4.b odd 2 1
1344.2.s.b 4 48.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 4 T_{5}^{3} + 8 T_{5}^{2} - 8 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T + 2 T^{2} )^{2}$$
$3$ $$( 1 + 2 T + 3 T^{2} )^{2}$$
$5$ $$1 + 4 T + 8 T^{2} + 12 T^{3} + 14 T^{4} + 60 T^{5} + 200 T^{6} + 500 T^{7} + 625 T^{8}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$1 + 4 T + 8 T^{2} - 12 T^{3} - 178 T^{4} - 132 T^{5} + 968 T^{6} + 5324 T^{7} + 14641 T^{8}$$
$13$ $$1 + 12 T + 72 T^{2} + 324 T^{3} + 1262 T^{4} + 4212 T^{5} + 12168 T^{6} + 26364 T^{7} + 28561 T^{8}$$
$17$ $$1 - 20 T^{2} + 166 T^{4} - 5780 T^{6} + 83521 T^{8}$$
$19$ $$1 + 4 T + 8 T^{2} + 68 T^{3} + 574 T^{4} + 1292 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8}$$
$23$ $$1 - 20 T^{2} + 646 T^{4} - 10580 T^{6} + 279841 T^{8}$$
$29$ $$( 1 - 4 T + 29 T^{2} )^{2}( 1 + 10 T + 29 T^{2} )^{2}$$
$31$ $$1 + 28 T^{2} + 966 T^{4} + 26908 T^{6} + 923521 T^{8}$$
$37$ $$1 + 4 T + 8 T^{2} + 92 T^{3} + 862 T^{4} + 3404 T^{5} + 10952 T^{6} + 202612 T^{7} + 1874161 T^{8}$$
$41$ $$( 1 + 12 T + 86 T^{2} + 492 T^{3} + 1681 T^{4} )^{2}$$
$43$ $$1 + 4 T + 8 T^{2} + 116 T^{3} + 1486 T^{4} + 4988 T^{5} + 14792 T^{6} + 318028 T^{7} + 3418801 T^{8}$$
$47$ $$( 1 + 62 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$1 + 12 T + 72 T^{2} + 660 T^{3} + 6046 T^{4} + 34980 T^{5} + 202248 T^{6} + 1786524 T^{7} + 7890481 T^{8}$$
$59$ $$( 1 + 6 T + 59 T^{2} )^{2}( 1 - 82 T^{2} + 3481 T^{4} )$$
$61$ $$1 - 4 T + 8 T^{2} - 236 T^{3} + 6958 T^{4} - 14396 T^{5} + 29768 T^{6} - 907924 T^{7} + 13845841 T^{8}$$
$67$ $$( 1 + 14 T + 98 T^{2} + 938 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 106 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$1 - 148 T^{2} + 14086 T^{4} - 788692 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 154 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 + 4 T + 8 T^{2} - 444 T^{3} - 12994 T^{4} - 36852 T^{5} + 55112 T^{6} + 2287148 T^{7} + 47458321 T^{8}$$
$89$ $$( 1 + 20 T + 246 T^{2} + 1780 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{4}$$