# Properties

 Label 324.6.e.f Level 324 Weight 6 Character orbit 324.e Analytic conductor 51.964 Analytic rank 0 Dimension 4 CM no Inner twists 2

# 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: $$\Q(\sqrt{-3}, \sqrt{41})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} + ( 16 - 16 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{7} +O(q^{10})$$ $$q -\beta_{2} q^{5} + ( 16 - 16 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{7} + ( -243 + 243 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{11} + ( -104 \beta_{1} + 12 \beta_{2} ) q^{13} + ( 972 - 20 \beta_{3} ) q^{17} + ( -316 - 6 \beta_{3} ) q^{19} + ( -3402 \beta_{1} + 4 \beta_{2} ) q^{23} + ( -196 + 196 \beta_{1} ) q^{25} + ( -5832 + 5832 \beta_{1} - 38 \beta_{2} - 38 \beta_{3} ) q^{29} + ( -1664 \beta_{1} - 81 \beta_{2} ) q^{31} + ( 9963 + 16 \beta_{3} ) q^{35} + ( -4978 + 60 \beta_{3} ) q^{37} + ( -6804 \beta_{1} - 58 \beta_{2} ) q^{41} + ( -2480 + 2480 \beta_{1} - 162 \beta_{2} - 162 \beta_{3} ) q^{43} + ( -9234 + 9234 \beta_{1} + 264 \beta_{2} + 264 \beta_{3} ) q^{47} + ( -13338 \beta_{1} + 96 \beta_{2} ) q^{49} + ( -5832 + 341 \beta_{3} ) q^{53} + ( -26568 - 243 \beta_{3} ) q^{55} + ( -972 \beta_{1} + 376 \beta_{2} ) q^{59} + ( -4088 + 4088 \beta_{1} + 720 \beta_{2} + 720 \beta_{3} ) q^{61} + ( 39852 - 39852 \beta_{1} + 104 \beta_{2} + 104 \beta_{3} ) q^{65} + ( -45032 \beta_{1} + 450 \beta_{2} ) q^{67} + ( -22356 - 788 \beta_{3} ) q^{71} + ( -60607 + 504 \beta_{3} ) q^{73} + ( 83592 \beta_{1} - 857 \beta_{2} ) q^{77} + ( -14384 + 14384 \beta_{1} - 840 \beta_{2} - 840 \beta_{3} ) q^{79} + ( 7533 - 7533 \beta_{1} - 1752 \beta_{2} - 1752 \beta_{3} ) q^{83} + ( -66420 \beta_{1} - 972 \beta_{2} ) q^{85} + ( -89424 + 358 \beta_{3} ) q^{89} + ( -121220 - 504 \beta_{3} ) q^{91} + ( -19926 \beta_{1} + 316 \beta_{2} ) q^{95} + ( -44471 + 44471 \beta_{1} - 1272 \beta_{2} - 1272 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{7} + O(q^{10})$$ $$4q + 32q^{7} - 486q^{11} - 208q^{13} + 3888q^{17} - 1264q^{19} - 6804q^{23} - 392q^{25} - 11664q^{29} - 3328q^{31} + 39852q^{35} - 19912q^{37} - 13608q^{41} - 4960q^{43} - 18468q^{47} - 26676q^{49} - 23328q^{53} - 106272q^{55} - 1944q^{59} - 8176q^{61} + 79704q^{65} - 90064q^{67} - 89424q^{71} - 242428q^{73} + 167184q^{77} - 28768q^{79} + 15066q^{83} - 132840q^{85} - 357696q^{89} - 484880q^{91} - 39852q^{95} - 88942q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 11 x^{2} + 10 x + 100$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 11 \nu^{2} - 11 \nu + 100$$$$)/110$$ $$\beta_{2}$$ $$=$$ $$($$$$9 \nu^{3} - 99 \nu^{2} + 2079 \nu - 900$$$$)/110$$ $$\beta_{3}$$ $$=$$ $$($$$$18 \nu^{3} + 279$$$$)/11$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 9 \beta_{1}$$$$)/18$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 189 \beta_{1} - 189$$$$)/18$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{3} - 279$$$$)/18$$

## 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
 1.85078 − 3.20565i −1.35078 + 2.33962i 1.85078 + 3.20565i −1.35078 − 2.33962i
0 0 0 −28.8141 + 49.9074i 0 −78.4422 135.866i 0 0 0
109.2 0 0 0 28.8141 49.9074i 0 94.4422 + 163.579i 0 0 0
217.1 0 0 0 −28.8141 49.9074i 0 −78.4422 + 135.866i 0 0 0
217.2 0 0 0 28.8141 + 49.9074i 0 94.4422 163.579i 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
9.c even 3 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.b 2 9.d odd 6 1
108.6.a.c yes 2 9.c even 3 1
324.6.e.f 4 1.a even 1 1 trivial
324.6.e.f 4 9.c even 3 1 inner
324.6.e.g 4 3.b odd 2 1
324.6.e.g 4 9.d odd 6 1
432.6.a.q 2 36.f odd 6 1
432.6.a.r 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 $$S_{6}^{\mathrm{new}}(324, [\chi])$$:

 $$T_{5}^{4} + 3321 T_{5}^{2} + 11029041$$ $$T_{7}^{4} - 32 T_{7}^{3} + 30657 T_{7}^{2} + 948256 T_{7} + 878114689$$ $$T_{11}^{4} + 486 T_{11}^{3} + 389691 T_{11}^{2} - 74598570 T_{11} + 23560715025$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 2929 T^{2} - 1186584 T^{4} - 28603515625 T^{6} + 95367431640625 T^{8}$$
$7$ $$1 - 32 T - 2957 T^{2} + 948256 T^{3} - 283837256 T^{4} + 15937338592 T^{5} - 835279311293 T^{6} - 151921968318176 T^{7} + 79792266297612001 T^{8}$$
$11$ $$1 + 486 T + 67589 T^{2} - 74598570 T^{3} - 35548706148 T^{4} - 12014174297070 T^{5} + 1753084591356989 T^{6} + 2030142610336006386 T^{7} +$$$$67\!\cdots\!01$$$$T^{8}$$
$13$ $$1 + 208 T - 231914 T^{2} - 97220864 T^{3} - 78199180517 T^{4} - 36097426257152 T^{5} - 31971314278668986 T^{6} + 10646665746930877456 T^{7} +$$$$19\!\cdots\!01$$$$T^{8}$$
$17$ $$( 1 - 1944 T + 2456098 T^{2} - 2760202008 T^{3} + 2015993900449 T^{4} )^{2}$$
$19$ $$( 1 + 632 T + 4932498 T^{2} + 1564894568 T^{3} + 6131066257801 T^{4} )^{2}$$
$23$ $$1 + 6804 T + 21901262 T^{2} + 78385264272 T^{3} + 255632710493379 T^{4} + 504514447000237296 T^{5} +$$$$90\!\cdots\!38$$$$T^{6} +$$$$18\!\cdots\!28$$$$T^{7} +$$$$17\!\cdots\!01$$$$T^{8}$$
$29$ $$1 + 11664 T + 65809898 T^{2} + 340783588800 T^{3} + 1722290429602299 T^{4} + 6989862966631531200 T^{5} +$$$$27\!\cdots\!98$$$$T^{6} +$$$$10\!\cdots\!36$$$$T^{7} +$$$$17\!\cdots\!01$$$$T^{8}$$
$31$ $$1 + 3328 T - 27162533 T^{2} - 63299175680 T^{3} + 325440737975704 T^{4} - 1812201658718247680 T^{5} -$$$$22\!\cdots\!33$$$$T^{6} +$$$$78\!\cdots\!28$$$$T^{7} +$$$$67\!\cdots\!01$$$$T^{8}$$
$37$ $$( 1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4} )^{2}$$
$41$ $$1 + 13608 T - 81657310 T^{2} + 477947959776 T^{3} + 36324263378333811 T^{4} + 55373234895348170976 T^{5} -$$$$10\!\cdots\!10$$$$T^{6} +$$$$21\!\cdots\!08$$$$T^{7} +$$$$18\!\cdots\!01$$$$T^{8}$$
$43$ $$1 + 4960 T - 188409362 T^{2} - 401789383040 T^{3} + 20145544707572395 T^{4} - 59066431614641006720 T^{5} -$$$$40\!\cdots\!38$$$$T^{6} +$$$$15\!\cdots\!20$$$$T^{7} +$$$$46\!\cdots\!01$$$$T^{8}$$
$47$ $$1 + 18468 T + 28570670 T^{2} - 2699904512880 T^{3} - 33167179979285901 T^{4} -$$$$61\!\cdots\!60$$$$T^{5} +$$$$15\!\cdots\!30$$$$T^{6} +$$$$22\!\cdots\!24$$$$T^{7} +$$$$27\!\cdots\!01$$$$T^{8}$$
$53$ $$( 1 + 11664 T + 484234009 T^{2} + 4877832230352 T^{3} + 174887470365513049 T^{4} )^{2}$$
$59$ $$1 + 1944 T - 957504550 T^{2} - 910890188928 T^{3} + 410247775513479531 T^{4} -$$$$65\!\cdots\!72$$$$T^{5} -$$$$48\!\cdots\!50$$$$T^{6} +$$$$71\!\cdots\!56$$$$T^{7} +$$$$26\!\cdots\!01$$$$T^{8}$$
$61$ $$1 + 8176 T + 82549030 T^{2} - 13939218707456 T^{3} - 769555065825699461 T^{4} -$$$$11\!\cdots\!56$$$$T^{5} +$$$$58\!\cdots\!30$$$$T^{6} +$$$$49\!\cdots\!76$$$$T^{7} +$$$$50\!\cdots\!01$$$$T^{8}$$
$67$ $$1 + 90064 T + 4055895358 T^{2} + 122070811385536 T^{3} + 3673714317894632923 T^{4} +$$$$16\!\cdots\!52$$$$T^{5} +$$$$73\!\cdots\!42$$$$T^{6} +$$$$22\!\cdots\!52$$$$T^{7} +$$$$33\!\cdots\!01$$$$T^{8}$$
$71$ $$( 1 + 44712 T + 2046094414 T^{2} + 80670702741912 T^{3} + 3255243551009881201 T^{4} )^{2}$$
$73$ $$( 1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4} )^{2}$$
$79$ $$1 + 28768 T - 3190116830 T^{2} - 61459901806592 T^{3} + 4127189778201816739 T^{4} -$$$$18\!\cdots\!08$$$$T^{5} -$$$$30\!\cdots\!30$$$$T^{6} +$$$$83\!\cdots\!32$$$$T^{7} +$$$$89\!\cdots\!01$$$$T^{8}$$
$83$ $$1 - 15066 T + 2485979765 T^{2} + 152725197486870 T^{3} - 11307081532931957076 T^{4} +$$$$60\!\cdots\!10$$$$T^{5} +$$$$38\!\cdots\!85$$$$T^{6} -$$$$92\!\cdots\!62$$$$T^{7} +$$$$24\!\cdots\!01$$$$T^{8}$$
$89$ $$( 1 + 178848 T + 18739138030 T^{2} + 998697864334752 T^{3} + 31181719929966183601 T^{4} )^{2}$$
$97$ $$1 + 88942 T - 5868346127 T^{2} - 302016349055666 T^{3} + 48187435429438608484 T^{4} -$$$$25\!\cdots\!62$$$$T^{5} -$$$$43\!\cdots\!23$$$$T^{6} +$$$$56\!\cdots\!06$$$$T^{7} +$$$$54\!\cdots\!01$$$$T^{8}$$