# Properties

 Label 336.2.k.c Level 336 Weight 2 Character orbit 336.k 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.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.68297350792$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.342102016.5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{5}$$ Twist minimal: no (minimal twist has level 168) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{5} ) q^{7} + ( -\beta_{5} - \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + ( 1 + \beta_{5} ) q^{7} + ( -\beta_{5} - \beta_{7} ) q^{9} + ( -\beta_{3} + \beta_{6} - \beta_{7} ) q^{11} + ( \beta_{1} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{13} + ( -1 - \beta_{3} - \beta_{5} - \beta_{6} ) q^{15} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{17} + ( 3 \beta_{1} - 3 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{6} ) q^{21} + \beta_{3} q^{23} + ( 1 + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{25} + ( 2 \beta_{2} + \beta_{4} - \beta_{6} - \beta_{7} ) q^{27} + ( -2 \beta_{3} - \beta_{6} + \beta_{7} ) q^{29} + ( -2 \beta_{1} + 2 \beta_{4} + \beta_{6} + \beta_{7} ) q^{31} + ( -\beta_{1} + \beta_{2} - \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{33} + ( -\beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} ) q^{35} -2 q^{37} + ( -1 + 2 \beta_{3} - \beta_{5} - \beta_{6} ) q^{39} + ( \beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{41} + 4 q^{43} + ( -2 \beta_{1} - \beta_{2} + 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{45} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} ) q^{47} + ( 3 + 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{49} + ( 2 - \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{51} + ( 2 \beta_{3} + \beta_{6} - \beta_{7} ) q^{53} + ( -4 \beta_{1} + 4 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{55} + ( -5 + 4 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{57} + ( 3 \beta_{1} + 3 \beta_{4} ) q^{59} + ( -3 \beta_{1} + 3 \beta_{4} - 3 \beta_{6} - 3 \beta_{7} ) q^{61} + ( -4 - 2 \beta_{1} + 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{63} + ( -\beta_{6} + \beta_{7} ) q^{65} + ( 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{67} + ( \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} ) q^{69} + ( 5 \beta_{3} - \beta_{6} + \beta_{7} ) q^{71} + ( -4 \beta_{1} + 4 \beta_{4} ) q^{73} + ( \beta_{1} - 2 \beta_{2} - 4 \beta_{4} + \beta_{6} + \beta_{7} ) q^{75} + ( -3 \beta_{1} - \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{77} + ( -6 + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{79} + ( -1 - 4 \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{81} + ( \beta_{1} + \beta_{4} ) q^{83} + 4 q^{85} + ( -2 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} - \beta_{6} - \beta_{7} ) q^{87} + ( -5 \beta_{1} + \beta_{2} - 5 \beta_{4} ) q^{89} + ( 2 + \beta_{1} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{91} + ( 8 + 2 \beta_{3} + 2 \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{93} + ( -2 \beta_{3} - 3 \beta_{6} + 3 \beta_{7} ) q^{95} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{97} + ( -8 - 5 \beta_{3} + \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{7} + 4q^{9} + O(q^{10})$$ $$8q + 4q^{7} + 4q^{9} - 4q^{15} - 8q^{21} - 16q^{37} - 4q^{39} + 32q^{43} + 16q^{49} + 24q^{51} - 28q^{57} - 32q^{63} - 16q^{67} - 56q^{79} + 32q^{85} + 24q^{91} + 56q^{93} - 64q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + \nu^{5} + 3 \nu^{4} - 6 \nu^{3} + 10 \nu^{2} + 8 \nu + 8$$$$)/16$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 3 \nu^{5} + 2 \nu^{3}$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3}$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} - \nu^{6} + \nu^{5} - 3 \nu^{4} - 6 \nu^{3} - 10 \nu^{2} + 8 \nu - 8$$$$)/16$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} + 6 \nu^{2} - 8$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} + 3 \nu^{5} + 2 \nu^{4} + 10 \nu^{3} - 12 \nu^{2} + 24 \nu$$$$)/16$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} - 2 \nu^{6} - 3 \nu^{5} + 2 \nu^{4} - 10 \nu^{3} - 12 \nu^{2} - 24 \nu$$$$)/16$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} - \beta_{4} + \beta_{1}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - 3 \beta_{4} + 3 \beta_{3} - \beta_{2} - 3 \beta_{1}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 3 \beta_{4} + 3 \beta_{1} - 4$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + 3 \beta_{4} + 5 \beta_{3} + 9 \beta_{2} + 3 \beta_{1}$$$$)/4$$ $$\nu^{6}$$ $$=$$ $$($$$$-6 \beta_{7} - 6 \beta_{6} - 7 \beta_{5} + 3 \beta_{4} - 3 \beta_{1} - 4$$$$)/2$$ $$\nu^{7}$$ $$=$$ $$($$$$-\beta_{7} + \beta_{6} - 3 \beta_{4} - 21 \beta_{3} + 7 \beta_{2} - 3 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$85$$ $$113$$ $$127$$ $$241$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 −0.599676 + 1.28078i −0.599676 − 1.28078i −1.17915 + 0.780776i −1.17915 − 0.780776i 1.17915 − 0.780776i 1.17915 + 0.780776i 0.599676 − 1.28078i 0.599676 + 1.28078i
0 −1.66757 0.468213i 0 −0.936426 0 −1.56155 2.13578i 0 2.56155 + 1.56155i 0
209.2 0 −1.66757 + 0.468213i 0 −0.936426 0 −1.56155 + 2.13578i 0 2.56155 1.56155i 0
209.3 0 −0.848071 1.51022i 0 3.02045 0 2.56155 0.662153i 0 −1.56155 + 2.56155i 0
209.4 0 −0.848071 + 1.51022i 0 3.02045 0 2.56155 + 0.662153i 0 −1.56155 2.56155i 0
209.5 0 0.848071 1.51022i 0 −3.02045 0 2.56155 0.662153i 0 −1.56155 2.56155i 0
209.6 0 0.848071 + 1.51022i 0 −3.02045 0 2.56155 + 0.662153i 0 −1.56155 + 2.56155i 0
209.7 0 1.66757 0.468213i 0 0.936426 0 −1.56155 2.13578i 0 2.56155 1.56155i 0
209.8 0 1.66757 + 0.468213i 0 0.936426 0 −1.56155 + 2.13578i 0 2.56155 + 1.56155i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.8 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
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.2.k.c 8
3.b odd 2 1 inner 336.2.k.c 8
4.b odd 2 1 168.2.k.a 8
7.b odd 2 1 inner 336.2.k.c 8
8.b even 2 1 1344.2.k.i 8
8.d odd 2 1 1344.2.k.f 8
12.b even 2 1 168.2.k.a 8
21.c even 2 1 inner 336.2.k.c 8
24.f even 2 1 1344.2.k.f 8
24.h odd 2 1 1344.2.k.i 8
28.d even 2 1 168.2.k.a 8
28.f even 6 2 1176.2.u.a 16
28.g odd 6 2 1176.2.u.a 16
56.e even 2 1 1344.2.k.f 8
56.h odd 2 1 1344.2.k.i 8
84.h odd 2 1 168.2.k.a 8
84.j odd 6 2 1176.2.u.a 16
84.n even 6 2 1176.2.u.a 16
168.e odd 2 1 1344.2.k.f 8
168.i even 2 1 1344.2.k.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.k.a 8 4.b odd 2 1
168.2.k.a 8 12.b even 2 1
168.2.k.a 8 28.d even 2 1
168.2.k.a 8 84.h odd 2 1
336.2.k.c 8 1.a even 1 1 trivial
336.2.k.c 8 3.b odd 2 1 inner
336.2.k.c 8 7.b odd 2 1 inner
336.2.k.c 8 21.c even 2 1 inner
1176.2.u.a 16 28.f even 6 2
1176.2.u.a 16 28.g odd 6 2
1176.2.u.a 16 84.j odd 6 2
1176.2.u.a 16 84.n even 6 2
1344.2.k.f 8 8.d odd 2 1
1344.2.k.f 8 24.f even 2 1
1344.2.k.f 8 56.e even 2 1
1344.2.k.f 8 168.e odd 2 1
1344.2.k.i 8 8.b even 2 1
1344.2.k.i 8 24.h odd 2 1
1344.2.k.i 8 56.h odd 2 1
1344.2.k.i 8 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 10 T_{5}^{2} + 8$$ acting on $$S_{2}^{\mathrm{new}}(336, [\chi])$$.

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 - 2 T^{2} + 2 T^{4} - 18 T^{6} + 81 T^{8}$$
$5$ $$( 1 + 10 T^{2} + 58 T^{4} + 250 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 8 T^{2} + 190 T^{4} - 968 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 38 T^{2} + 682 T^{4} - 6422 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 48 T^{2} + 1086 T^{4} + 13872 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 - 2 T^{2} + 706 T^{4} - 722 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 42 T^{2} + 529 T^{4} )^{4}$$
$29$ $$( 1 - 32 T^{2} + 238 T^{4} - 26912 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 - 32 T^{2} + 2110 T^{4} - 30752 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 2 T + 37 T^{2} )^{8}$$
$41$ $$( 1 + 48 T^{2} + 606 T^{4} + 80688 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 4 T + 43 T^{2} )^{8}$$
$47$ $$( 1 + 4 T^{2} + 4150 T^{4} + 8836 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 128 T^{2} + 8014 T^{4} - 359552 T^{6} + 7890481 T^{8} )^{2}$$
$59$ $$( 1 + 110 T^{2} + 8610 T^{4} + 382910 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 118 T^{2} + 9546 T^{4} - 439078 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 4 T + 70 T^{2} + 268 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 88 T^{2} + 6510 T^{4} - 443608 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 132 T^{2} + 10662 T^{4} - 703428 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 14 T + 190 T^{2} + 1106 T^{3} + 6241 T^{4} )^{4}$$
$83$ $$( 1 + 318 T^{2} + 39042 T^{4} + 2190702 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 24 T^{2} + 12654 T^{4} - 190104 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} )^{2}$$