# Properties

 Label 336.4.bc.b Level $336$ Weight $4$ Character orbit 336.bc Analytic conductor $19.825$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$336 = 2^{4} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 336.bc (of order $$6$$, 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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (3 \zeta_{6} + 3) q^{3} + ( - 19 \zeta_{6} + 18) q^{7} + 27 \zeta_{6} q^{9}+O(q^{10})$$ q + (3*z + 3) * q^3 + (-19*z + 18) * q^7 + 27*z * q^9 $$q + (3 \zeta_{6} + 3) q^{3} + ( - 19 \zeta_{6} + 18) q^{7} + 27 \zeta_{6} q^{9} + (106 \zeta_{6} - 53) q^{13} + ( - 73 \zeta_{6} + 146) q^{19} + ( - 60 \zeta_{6} + 111) q^{21} + ( - 125 \zeta_{6} + 125) q^{25} + (162 \zeta_{6} - 81) q^{27} + (109 \zeta_{6} + 109) q^{31} + 323 \zeta_{6} q^{37} + (477 \zeta_{6} - 477) q^{39} + 71 q^{43} + ( - 323 \zeta_{6} - 37) q^{49} + 657 q^{57} + (540 \zeta_{6} - 1080) q^{61} + ( - 27 \zeta_{6} + 513) q^{63} + (127 \zeta_{6} - 127) q^{67} + (703 \zeta_{6} + 703) q^{73} + ( - 375 \zeta_{6} + 750) q^{75} - 1387 \zeta_{6} q^{79} + (729 \zeta_{6} - 729) q^{81} + (901 \zeta_{6} + 1060) q^{91} + 981 \zeta_{6} q^{93} + ( - 1584 \zeta_{6} + 792) q^{97} +O(q^{100})$$ q + (3*z + 3) * q^3 + (-19*z + 18) * q^7 + 27*z * q^9 + (106*z - 53) * q^13 + (-73*z + 146) * q^19 + (-60*z + 111) * q^21 + (-125*z + 125) * q^25 + (162*z - 81) * q^27 + (109*z + 109) * q^31 + 323*z * q^37 + (477*z - 477) * q^39 + 71 * q^43 + (-323*z - 37) * q^49 + 657 * q^57 + (540*z - 1080) * q^61 + (-27*z + 513) * q^63 + (127*z - 127) * q^67 + (703*z + 703) * q^73 + (-375*z + 750) * q^75 - 1387*z * q^79 + (729*z - 729) * q^81 + (901*z + 1060) * q^91 + 981*z * q^93 + (-1584*z + 792) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 9 q^{3} + 17 q^{7} + 27 q^{9}+O(q^{10})$$ 2 * q + 9 * q^3 + 17 * q^7 + 27 * q^9 $$2 q + 9 q^{3} + 17 q^{7} + 27 q^{9} + 219 q^{19} + 162 q^{21} + 125 q^{25} + 327 q^{31} + 323 q^{37} - 477 q^{39} + 142 q^{43} - 397 q^{49} + 1314 q^{57} - 1620 q^{61} + 999 q^{63} - 127 q^{67} + 2109 q^{73} + 1125 q^{75} - 1387 q^{79} - 729 q^{81} + 3021 q^{91} + 981 q^{93}+O(q^{100})$$ 2 * q + 9 * q^3 + 17 * q^7 + 27 * q^9 + 219 * q^19 + 162 * q^21 + 125 * q^25 + 327 * q^31 + 323 * q^37 - 477 * q^39 + 142 * q^43 - 397 * q^49 + 1314 * q^57 - 1620 * q^61 + 999 * q^63 - 127 * q^67 + 2109 * q^73 + 1125 * q^75 - 1387 * q^79 - 729 * q^81 + 3021 * q^91 + 981 * q^93

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 4.50000 + 2.59808i 0 0 0 8.50000 16.4545i 0 13.5000 + 23.3827i 0
257.1 0 4.50000 2.59808i 0 0 0 8.50000 + 16.4545i 0 13.5000 23.3827i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 336.4.bc.b 2
3.b odd 2 1 CM 336.4.bc.b 2
4.b odd 2 1 84.4.k.a 2
7.d odd 6 1 inner 336.4.bc.b 2
12.b even 2 1 84.4.k.a 2
21.g even 6 1 inner 336.4.bc.b 2
28.d even 2 1 588.4.k.b 2
28.f even 6 1 84.4.k.a 2
28.f even 6 1 588.4.f.a 2
28.g odd 6 1 588.4.f.a 2
28.g odd 6 1 588.4.k.b 2
84.h odd 2 1 588.4.k.b 2
84.j odd 6 1 84.4.k.a 2
84.j odd 6 1 588.4.f.a 2
84.n even 6 1 588.4.f.a 2
84.n even 6 1 588.4.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.k.a 2 4.b odd 2 1
84.4.k.a 2 12.b even 2 1
84.4.k.a 2 28.f even 6 1
84.4.k.a 2 84.j odd 6 1
336.4.bc.b 2 1.a even 1 1 trivial
336.4.bc.b 2 3.b odd 2 1 CM
336.4.bc.b 2 7.d odd 6 1 inner
336.4.bc.b 2 21.g even 6 1 inner
588.4.f.a 2 28.f even 6 1
588.4.f.a 2 28.g odd 6 1
588.4.f.a 2 84.j odd 6 1
588.4.f.a 2 84.n even 6 1
588.4.k.b 2 28.d even 2 1
588.4.k.b 2 28.g odd 6 1
588.4.k.b 2 84.h odd 2 1
588.4.k.b 2 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 $$S_{4}^{\mathrm{new}}(336, [\chi])$$:

 $$T_{5}$$ T5 $$T_{13}^{2} + 8427$$ T13^2 + 8427

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 9T + 27$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 17T + 343$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 8427$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 219T + 15987$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 327T + 35643$$
$37$ $$T^{2} - 323T + 104329$$
$41$ $$T^{2}$$
$43$ $$(T - 71)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 1620 T + 874800$$
$67$ $$T^{2} + 127T + 16129$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 2109 T + 1482627$$
$79$ $$T^{2} + 1387 T + 1923769$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 1881792$$