# Properties

 Label 336.2.bj.g Level 336 Weight 2 Character orbit 336.bj Analytic conductor 2.683 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 336.bj (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.8275904784.2 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{4} - \beta_{7} ) q^{5} + ( 3 - \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( -\beta_{1} - \beta_{4} - \beta_{7} ) q^{5} + ( 3 - \beta_{1} + \beta_{3} - \beta_{5} + 2 \beta_{6} ) q^{7} + ( -1 + \beta_{1} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{9} + ( 1 - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} ) q^{11} + 2 q^{13} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{15} + ( \beta_{1} - 2 \beta_{2} - \beta_{7} ) q^{17} + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{19} + ( -\beta_{2} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{21} + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} ) q^{23} + ( -4 + 2 \beta_{1} - 4 \beta_{3} + 4 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{25} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{27} + ( -2 + 3 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{29} + ( 2 + \beta_{6} ) q^{31} + ( -4 + 2 \beta_{3} - \beta_{4} - \beta_{5} - 4 \beta_{6} - 5 \beta_{7} ) q^{33} + ( -3 + 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{35} + ( 3 - \beta_{1} - \beta_{3} + \beta_{5} + 3 \beta_{6} + 2 \beta_{7} ) q^{37} + 2 \beta_{1} q^{39} + ( -2 \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{41} + ( -2 + 2 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} - 2 \beta_{7} ) q^{43} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} - \beta_{7} ) q^{45} + ( 3 - 3 \beta_{1} - 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{47} + ( 3 - 3 \beta_{1} - \beta_{6} + 3 \beta_{7} ) q^{49} + ( -2 + \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{51} + ( 5 - 2 \beta_{1} - \beta_{2} - 5 \beta_{3} - 5 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -5 - 4 \beta_{3} + 4 \beta_{5} - 6 \beta_{6} + 4 \beta_{7} ) q^{55} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{57} + ( 1 - 4 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{59} + ( 7 + \beta_{1} + \beta_{3} - \beta_{5} + 7 \beta_{6} - 2 \beta_{7} ) q^{61} + ( -3 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 4 \beta_{7} ) q^{63} + ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{7} ) q^{65} + ( -8 - 3 \beta_{1} - 4 \beta_{6} + 3 \beta_{7} ) q^{67} + ( -5 + \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{69} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} ) q^{71} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} + \beta_{7} ) q^{73} + ( 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 4 \beta_{5} - 8 \beta_{6} + 2 \beta_{7} ) q^{75} + ( -1 + 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 4 \beta_{7} ) q^{77} + ( 3 - 3 \beta_{6} ) q^{79} + ( 3 - \beta_{1} - \beta_{2} - 5 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{81} + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{83} + ( -1 - 6 \beta_{1} + 3 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{85} + ( -12 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 7 \beta_{6} + \beta_{7} ) q^{87} + ( -3 + \beta_{1} + 3 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} - 2 \beta_{7} ) q^{89} + ( 6 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} + 4 \beta_{6} ) q^{91} + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} ) q^{93} + ( -2 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{95} + ( -2 + 2 \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{97} + ( 2 + 3 \beta_{3} + 4 \beta_{5} + 8 \beta_{6} - 3 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 3q^{3} + 6q^{7} - q^{9} + O(q^{10})$$ $$8q + 3q^{3} + 6q^{7} - q^{9} + 16q^{13} - 6q^{19} - 19q^{21} + 12q^{25} + 12q^{31} - 11q^{33} + 10q^{37} + 6q^{39} - 17q^{45} + 10q^{49} - 9q^{51} - 22q^{57} + 30q^{61} - 27q^{63} - 66q^{67} - 26q^{69} + 14q^{73} + 66q^{75} + 36q^{79} + 7q^{81} - 68q^{85} - 54q^{87} + 12q^{91} + 3q^{93} + 4q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - 18 x^{3} + 45 x^{2} - 81 x + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 3 \nu^{6} - 14 \nu^{5} + 6 \nu^{4} - 22 \nu^{3} + 18 \nu^{2} - 27 \nu + 189$$$$)/54$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - 3 \nu^{6} + 5 \nu^{5} - 6 \nu^{4} + 4 \nu^{3} - 18 \nu^{2} + 45 \nu - 81$$$$)/27$$ $$\beta_{4}$$ $$=$$ $$($$$$-5 \nu^{7} + 6 \nu^{6} - 16 \nu^{5} + 12 \nu^{4} - 2 \nu^{3} + 108 \nu^{2} - 135 \nu + 270$$$$)/54$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{7} + 9 \nu^{6} - 16 \nu^{5} - 2 \nu^{3} + 66 \nu^{2} - 153 \nu + 297$$$$)/54$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} - 9 \nu^{6} + 17 \nu^{5} - 9 \nu^{4} + 19 \nu^{3} - 105 \nu^{2} + 144 \nu - 324$$$$)/54$$ $$\beta_{7}$$ $$=$$ $$($$$$-10 \nu^{7} + 9 \nu^{6} - 23 \nu^{5} + 9 \nu^{4} - 13 \nu^{3} + 123 \nu^{2} - 135 \nu + 378$$$$)/54$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} + \beta_{1} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} + 3 \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} - 5 \beta_{3} - \beta_{2} - \beta_{1} + 3$$ $$\nu^{5}$$ $$=$$ $$-2 \beta_{7} - 6 \beta_{6} - 2 \beta_{5} - 2 \beta_{4} - \beta_{3} - 4 \beta_{2} + 10$$ $$\nu^{6}$$ $$=$$ $$-6 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} - 4 \beta_{4} - 6 \beta_{3} - 4 \beta_{2} + 16 \beta_{1} - 11$$ $$\nu^{7}$$ $$=$$ $$-18 \beta_{7} - 6 \beta_{6} + 6 \beta_{5} + 12 \beta_{4} + 6 \beta_{3} + 6 \beta_{2} + 11 \beta_{1} - 6$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
95.1
 −1.37009 + 1.05965i 0.232633 − 1.71636i 0.906034 + 1.47618i 1.73142 + 0.0465589i −1.37009 − 1.05965i 0.232633 + 1.71636i 0.906034 − 1.47618i 1.73142 − 0.0465589i
0 −1.37009 + 1.05965i 0 0.581054 0.335472i 0 2.63746 + 0.209313i 0 0.754305 2.90362i 0
95.2 0 0.232633 1.71636i 0 −0.581054 + 0.335472i 0 2.63746 + 0.209313i 0 −2.89176 0.798564i 0
95.3 0 0.906034 + 1.47618i 0 3.41502 1.97166i 0 −1.13746 + 2.38876i 0 −1.35821 + 2.67493i 0
95.4 0 1.73142 + 0.0465589i 0 −3.41502 + 1.97166i 0 −1.13746 + 2.38876i 0 2.99566 + 0.161227i 0
191.1 0 −1.37009 1.05965i 0 0.581054 + 0.335472i 0 2.63746 0.209313i 0 0.754305 + 2.90362i 0
191.2 0 0.232633 + 1.71636i 0 −0.581054 0.335472i 0 2.63746 0.209313i 0 −2.89176 + 0.798564i 0
191.3 0 0.906034 1.47618i 0 3.41502 + 1.97166i 0 −1.13746 2.38876i 0 −1.35821 2.67493i 0
191.4 0 1.73142 0.0465589i 0 −3.41502 1.97166i 0 −1.13746 2.38876i 0 2.99566 0.161227i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 191.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
28.g odd 6 1 inner
84.n even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.bj.g yes 8
3.b odd 2 1 inner 336.2.bj.g yes 8
4.b odd 2 1 336.2.bj.e 8
7.c even 3 1 336.2.bj.e 8
7.c even 3 1 2352.2.h.n 8
7.d odd 6 1 2352.2.h.m 8
12.b even 2 1 336.2.bj.e 8
21.g even 6 1 2352.2.h.m 8
21.h odd 6 1 336.2.bj.e 8
21.h odd 6 1 2352.2.h.n 8
28.f even 6 1 2352.2.h.m 8
28.g odd 6 1 inner 336.2.bj.g yes 8
28.g odd 6 1 2352.2.h.n 8
84.j odd 6 1 2352.2.h.m 8
84.n even 6 1 inner 336.2.bj.g yes 8
84.n even 6 1 2352.2.h.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
336.2.bj.e 8 4.b odd 2 1
336.2.bj.e 8 7.c even 3 1
336.2.bj.e 8 12.b even 2 1
336.2.bj.e 8 21.h odd 6 1
336.2.bj.g yes 8 1.a even 1 1 trivial
336.2.bj.g yes 8 3.b odd 2 1 inner
336.2.bj.g yes 8 28.g odd 6 1 inner
336.2.bj.g yes 8 84.n even 6 1 inner
2352.2.h.m 8 7.d odd 6 1
2352.2.h.m 8 21.g even 6 1
2352.2.h.m 8 28.f even 6 1
2352.2.h.m 8 84.j odd 6 1
2352.2.h.n 8 7.c even 3 1
2352.2.h.n 8 21.h odd 6 1
2352.2.h.n 8 28.g odd 6 1
2352.2.h.n 8 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}^{8} - 16 T_{5}^{6} + 249 T_{5}^{4} - 112 T_{5}^{2} + 49$$ $$T_{13} - 2$$ $$T_{19}^{4} + 3 T_{19}^{3} - T_{19}^{2} - 12 T_{19} + 16$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 3 T + 5 T^{2} - 6 T^{3} + 4 T^{4} - 18 T^{5} + 45 T^{6} - 81 T^{7} + 81 T^{8}$$
$5$ $$1 + 4 T^{2} + 19 T^{4} - 212 T^{6} - 1016 T^{8} - 5300 T^{10} + 11875 T^{12} + 62500 T^{14} + 390625 T^{16}$$
$7$ $$( 1 - 3 T + 2 T^{2} - 21 T^{3} + 49 T^{4} )^{2}$$
$11$ $$1 - 4 T^{2} - 173 T^{4} + 212 T^{6} + 19144 T^{8} + 25652 T^{10} - 2532893 T^{12} - 7086244 T^{14} + 214358881 T^{16}$$
$13$ $$( 1 - 2 T + 13 T^{2} )^{8}$$
$17$ $$1 + 25 T^{2} - 95 T^{4} + 3550 T^{6} + 254254 T^{8} + 1025950 T^{10} - 7934495 T^{12} + 603439225 T^{14} + 6975757441 T^{16}$$
$19$ $$( 1 + 3 T + 37 T^{2} + 102 T^{3} + 852 T^{4} + 1938 T^{5} + 13357 T^{6} + 20577 T^{7} + 130321 T^{8} )^{2}$$
$23$ $$1 - 79 T^{2} + 3637 T^{4} - 122134 T^{6} + 3199486 T^{8} - 64608886 T^{10} + 1017781717 T^{12} - 11694835231 T^{14} + 78310985281 T^{16}$$
$29$ $$( 1 - 31 T^{2} + 1908 T^{4} - 26071 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 7 T + 31 T^{2} )^{4}( 1 + 4 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 5 T - 41 T^{2} + 40 T^{3} + 2062 T^{4} + 1480 T^{5} - 56129 T^{6} - 253265 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 64 T^{2} + 2334 T^{4} - 107584 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 80 T^{2} + 3246 T^{4} - 147920 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$1 - 71 T^{2} + 517 T^{4} - 7526 T^{6} + 4451326 T^{8} - 16624934 T^{10} + 2522795077 T^{12} - 765324288359 T^{14} + 23811286661761 T^{16}$$
$53$ $$1 - 20 T^{2} - 4805 T^{4} + 8260 T^{6} + 18077944 T^{8} + 23202340 T^{10} - 37913761205 T^{12} - 443287222580 T^{14} + 62259690411361 T^{16}$$
$59$ $$1 - 76 T^{2} - 2117 T^{4} - 70756 T^{6} + 30075832 T^{8} - 246301636 T^{10} - 25652453237 T^{12} - 3205720556716 T^{14} + 146830437604321 T^{16}$$
$61$ $$( 1 - 15 T + 61 T^{2} )^{4}( 1 + 15 T + 164 T^{2} + 915 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 33 T + 545 T^{2} + 6006 T^{3} + 52956 T^{4} + 402402 T^{5} + 2446505 T^{6} + 9925179 T^{7} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 + 160 T^{2} + 14430 T^{4} + 806560 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 7 T - 95 T^{2} + 14 T^{3} + 11830 T^{4} + 1022 T^{5} - 506255 T^{6} - 2723119 T^{7} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 - 13 T + 79 T^{2} )^{4}( 1 + 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 + 301 T^{2} + 36300 T^{4} + 2073589 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$1 + 109 T^{2} - 1787 T^{4} - 236966 T^{6} + 29962582 T^{8} - 1877007686 T^{10} - 112120384667 T^{12} + 54170960714749 T^{14} + 3936588805702081 T^{16}$$
$97$ $$( 1 - T + 180 T^{2} - 97 T^{3} + 9409 T^{4} )^{4}$$