Properties

 Label 324.6.e.e Level 324 Weight 6 Character orbit 324.e Analytic conductor 51.964 Analytic rank 0 Dimension 4 CM no Inner twists 4

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 324.e (of order $$3$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$51.9643576194$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 3^{5}$$ Twist minimal: no (minimal twist has level 108) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( -29 + 29 \beta_{1} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( -29 + 29 \beta_{1} ) q^{7} + ( \beta_{2} - \beta_{3} ) q^{11} -329 \beta_{1} q^{13} -25 \beta_{3} q^{17} + 1799 q^{19} -41 \beta_{2} q^{23} + ( -4651 + 4651 \beta_{1} ) q^{25} + ( 16 \beta_{2} - 16 \beta_{3} ) q^{29} -5228 \beta_{1} q^{31} + 29 \beta_{3} q^{35} + 8783 q^{37} + 176 \beta_{2} q^{41} + ( -19976 + 19976 \beta_{1} ) q^{43} + ( -123 \beta_{2} + 123 \beta_{3} ) q^{47} + 15966 \beta_{1} q^{49} + 334 \beta_{3} q^{53} + 7776 q^{55} -65 \beta_{2} q^{59} + ( 1069 - 1069 \beta_{1} ) q^{61} + ( 329 \beta_{2} - 329 \beta_{3} ) q^{65} + 62077 \beta_{1} q^{67} -526 \beta_{3} q^{71} -48079 q^{73} -29 \beta_{2} q^{77} + ( -49979 + 49979 \beta_{1} ) q^{79} + ( -654 \beta_{2} + 654 \beta_{3} ) q^{83} + 194400 \beta_{1} q^{85} -997 \beta_{3} q^{89} + 9541 q^{91} -1799 \beta_{2} q^{95} + ( -12917 + 12917 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 58q^{7} + O(q^{10})$$ $$4q - 58q^{7} - 658q^{13} + 7196q^{19} - 9302q^{25} - 10456q^{31} + 35132q^{37} - 39952q^{43} + 31932q^{49} + 31104q^{55} + 2138q^{61} + 124154q^{67} - 192316q^{73} - 99958q^{79} + 388800q^{85} + 38164q^{91} - 25834q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$18 \nu^{3} + 36 \nu$$ $$\beta_{3}$$ $$=$$ $$-18 \nu^{3} + 72 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/108$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} + 2 \beta_{2}$$$$)/54$$

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$$.

 $$n$$ $$163$$ $$245$$ $$\chi(n)$$ $$1$$ $$-\beta_{1}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 1.22474 − 0.707107i −1.22474 + 0.707107i 1.22474 + 0.707107i −1.22474 − 0.707107i
0 0 0 −44.0908 + 76.3675i 0 −14.5000 25.1147i 0 0 0
109.2 0 0 0 44.0908 76.3675i 0 −14.5000 25.1147i 0 0 0
217.1 0 0 0 −44.0908 76.3675i 0 −14.5000 + 25.1147i 0 0 0
217.2 0 0 0 44.0908 + 76.3675i 0 −14.5000 + 25.1147i 0 0 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 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.6.e.e 4
3.b odd 2 1 inner 324.6.e.e 4
9.c even 3 1 108.6.a.d 2
9.c even 3 1 inner 324.6.e.e 4
9.d odd 6 1 108.6.a.d 2
9.d odd 6 1 inner 324.6.e.e 4
36.f odd 6 1 432.6.a.p 2
36.h even 6 1 432.6.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.d 2 9.c even 3 1
108.6.a.d 2 9.d odd 6 1
324.6.e.e 4 1.a even 1 1 trivial
324.6.e.e 4 3.b odd 2 1 inner
324.6.e.e 4 9.c even 3 1 inner
324.6.e.e 4 9.d odd 6 1 inner
432.6.a.p 2 36.f odd 6 1
432.6.a.p 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{4} + 7776 T_{5}^{2} + 60466176$$ $$T_{7}^{2} + 29 T_{7} + 841$$ $$T_{11}^{4} + 7776 T_{11}^{2} + 60466176$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + 1526 T^{2} - 7436949 T^{4} + 14902343750 T^{6} + 95367431640625 T^{8}$$
$7$ $$( 1 + 29 T - 15966 T^{2} + 487403 T^{3} + 282475249 T^{4} )^{2}$$
$11$ $$1 - 314326 T^{2} + 72863409675 T^{4} - 8152806925133926 T^{6} +$$$$67\!\cdots\!01$$$$T^{8}$$
$13$ $$( 1 + 329 T - 263052 T^{2} + 122155397 T^{3} + 137858491849 T^{4} )^{2}$$
$17$ $$( 1 - 2020286 T^{2} + 2015993900449 T^{4} )^{2}$$
$19$ $$( 1 - 1799 T + 2476099 T^{2} )^{4}$$
$23$ $$1 + 198770 T^{2} - 41387001700749 T^{4} + 8234347633937011730 T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$1 - 39031642 T^{2} + 1102761843915963 T^{4} -$$$$16\!\cdots\!42$$$$T^{6} +$$$$17\!\cdots\!01$$$$T^{8}$$
$31$ $$( 1 + 5228 T - 1297167 T^{2} + 149673201428 T^{3} + 819628286980801 T^{4} )^{2}$$
$37$ $$( 1 - 8783 T + 69343957 T^{2} )^{4}$$
$41$ $$1 + 9156974 T^{2} - 13338809137315725 T^{4} +$$$$12\!\cdots\!74$$$$T^{6} +$$$$18\!\cdots\!01$$$$T^{8}$$
$43$ $$( 1 + 19976 T + 252032133 T^{2} + 2936640657368 T^{3} + 21611482313284249 T^{4} )^{2}$$
$47$ $$1 - 341046910 T^{2} + 63713862584718051 T^{4} -$$$$17\!\cdots\!90$$$$T^{6} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$( 1 - 31068470 T^{2} + 174887470365513049 T^{4} )^{2}$$
$59$ $$1 - 1396994998 T^{2} + 1440478271136378603 T^{4} -$$$$71\!\cdots\!98$$$$T^{6} +$$$$26\!\cdots\!01$$$$T^{8}$$
$61$ $$( 1 - 1069 T - 843453540 T^{2} - 902873445769 T^{3} + 713342911662882601 T^{4} )^{2}$$
$67$ $$( 1 - 62077 T + 2503428822 T^{2} - 83811716267239 T^{3} + 1822837804551761449 T^{4} )^{2}$$
$71$ $$( 1 + 1457026126 T^{2} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 + 48079 T + 2073071593 T^{2} )^{4}$$
$79$ $$( 1 + 49979 T - 579155958 T^{2} + 153788201765621 T^{3} + 9468276082626847201 T^{4} )^{2}$$
$83$ $$1 - 4552161670 T^{2} + 5206134682611335451 T^{4} -$$$$70\!\cdots\!30$$$$T^{6} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 + 3438704914 T^{2} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$( 1 + 12917 T - 8420491368 T^{2} + 110922674099669 T^{3} + 73742412689492826049 T^{4} )^{2}$$