Properties

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

Related objects

Downloads

Learn more about

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 94.4422 + 163.579i 0 0 0
109.2 0 0 0 28.8141 49.9074i 0 −78.4422 135.866i 0 0 0
217.1 0 0 0 −28.8141 49.9074i 0 94.4422 163.579i 0 0 0
217.2 0 0 0 28.8141 + 49.9074i 0 −78.4422 + 135.866i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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.g 4
3.b odd 2 1 324.6.e.f 4
9.c even 3 1 108.6.a.b 2
9.c even 3 1 inner 324.6.e.g 4
9.d odd 6 1 108.6.a.c yes 2
9.d odd 6 1 324.6.e.f 4
36.f odd 6 1 432.6.a.r 2
36.h even 6 1 432.6.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
108.6.a.b 2 9.c even 3 1
108.6.a.c yes 2 9.d odd 6 1
324.6.e.f 4 3.b odd 2 1
324.6.e.f 4 9.d odd 6 1
324.6.e.g 4 1.a even 1 1 trivial
324.6.e.g 4 9.c even 3 1 inner
432.6.a.q 2 36.h even 6 1
432.6.a.r 2 36.f odd 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
$5$ \( 1 - 2929 T^{2} - 1186584 T^{4} - 28603515625 T^{6} + 95367431640625 T^{8} \)
$7$ \( 1 - 32 T - 2957 T^{2} + 948256 T^{3} - 283837256 T^{4} + 15937338592 T^{5} - 835279311293 T^{6} - 151921968318176 T^{7} + 79792266297612001 T^{8} \)
$11$ \( 1 - 486 T + 67589 T^{2} + 74598570 T^{3} - 35548706148 T^{4} + 12014174297070 T^{5} + 1753084591356989 T^{6} - 2030142610336006386 T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$13$ \( 1 + 208 T - 231914 T^{2} - 97220864 T^{3} - 78199180517 T^{4} - 36097426257152 T^{5} - 31971314278668986 T^{6} + 10646665746930877456 T^{7} + \)\(19\!\cdots\!01\)\( T^{8} \)
$17$ \( ( 1 + 1944 T + 2456098 T^{2} + 2760202008 T^{3} + 2015993900449 T^{4} )^{2} \)
$19$ \( ( 1 + 632 T + 4932498 T^{2} + 1564894568 T^{3} + 6131066257801 T^{4} )^{2} \)
$23$ \( 1 - 6804 T + 21901262 T^{2} - 78385264272 T^{3} + 255632710493379 T^{4} - 504514447000237296 T^{5} + \)\(90\!\cdots\!38\)\( T^{6} - \)\(18\!\cdots\!28\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \)
$29$ \( 1 - 11664 T + 65809898 T^{2} - 340783588800 T^{3} + 1722290429602299 T^{4} - 6989862966631531200 T^{5} + \)\(27\!\cdots\!98\)\( T^{6} - \)\(10\!\cdots\!36\)\( T^{7} + \)\(17\!\cdots\!01\)\( T^{8} \)
$31$ \( 1 + 3328 T - 27162533 T^{2} - 63299175680 T^{3} + 325440737975704 T^{4} - 1812201658718247680 T^{5} - \)\(22\!\cdots\!33\)\( T^{6} + \)\(78\!\cdots\!28\)\( T^{7} + \)\(67\!\cdots\!01\)\( T^{8} \)
$37$ \( ( 1 + 9956 T + 151512798 T^{2} + 690388435892 T^{3} + 4808584372417849 T^{4} )^{2} \)
$41$ \( 1 - 13608 T - 81657310 T^{2} - 477947959776 T^{3} + 36324263378333811 T^{4} - 55373234895348170976 T^{5} - \)\(10\!\cdots\!10\)\( T^{6} - \)\(21\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} \)
$43$ \( 1 + 4960 T - 188409362 T^{2} - 401789383040 T^{3} + 20145544707572395 T^{4} - 59066431614641006720 T^{5} - \)\(40\!\cdots\!38\)\( T^{6} + \)\(15\!\cdots\!20\)\( T^{7} + \)\(46\!\cdots\!01\)\( T^{8} \)
$47$ \( 1 - 18468 T + 28570670 T^{2} + 2699904512880 T^{3} - 33167179979285901 T^{4} + \)\(61\!\cdots\!60\)\( T^{5} + \)\(15\!\cdots\!30\)\( T^{6} - \)\(22\!\cdots\!24\)\( T^{7} + \)\(27\!\cdots\!01\)\( T^{8} \)
$53$ \( ( 1 - 11664 T + 484234009 T^{2} - 4877832230352 T^{3} + 174887470365513049 T^{4} )^{2} \)
$59$ \( 1 - 1944 T - 957504550 T^{2} + 910890188928 T^{3} + 410247775513479531 T^{4} + \)\(65\!\cdots\!72\)\( T^{5} - \)\(48\!\cdots\!50\)\( T^{6} - \)\(71\!\cdots\!56\)\( T^{7} + \)\(26\!\cdots\!01\)\( T^{8} \)
$61$ \( 1 + 8176 T + 82549030 T^{2} - 13939218707456 T^{3} - 769555065825699461 T^{4} - \)\(11\!\cdots\!56\)\( T^{5} + \)\(58\!\cdots\!30\)\( T^{6} + \)\(49\!\cdots\!76\)\( T^{7} + \)\(50\!\cdots\!01\)\( T^{8} \)
$67$ \( 1 + 90064 T + 4055895358 T^{2} + 122070811385536 T^{3} + 3673714317894632923 T^{4} + \)\(16\!\cdots\!52\)\( T^{5} + \)\(73\!\cdots\!42\)\( T^{6} + \)\(22\!\cdots\!52\)\( T^{7} + \)\(33\!\cdots\!01\)\( T^{8} \)
$71$ \( ( 1 - 44712 T + 2046094414 T^{2} - 80670702741912 T^{3} + 3255243551009881201 T^{4} )^{2} \)
$73$ \( ( 1 + 121214 T + 6975764499 T^{2} + 251285300073902 T^{3} + 4297625829703557649 T^{4} )^{2} \)
$79$ \( 1 + 28768 T - 3190116830 T^{2} - 61459901806592 T^{3} + 4127189778201816739 T^{4} - \)\(18\!\cdots\!08\)\( T^{5} - \)\(30\!\cdots\!30\)\( T^{6} + \)\(83\!\cdots\!32\)\( T^{7} + \)\(89\!\cdots\!01\)\( T^{8} \)
$83$ \( 1 + 15066 T + 2485979765 T^{2} - 152725197486870 T^{3} - 11307081532931957076 T^{4} - \)\(60\!\cdots\!10\)\( T^{5} + \)\(38\!\cdots\!85\)\( T^{6} + \)\(92\!\cdots\!62\)\( T^{7} + \)\(24\!\cdots\!01\)\( T^{8} \)
$89$ \( ( 1 - 178848 T + 18739138030 T^{2} - 998697864334752 T^{3} + 31181719929966183601 T^{4} )^{2} \)
$97$ \( 1 + 88942 T - 5868346127 T^{2} - 302016349055666 T^{3} + 48187435429438608484 T^{4} - \)\(25\!\cdots\!62\)\( T^{5} - \)\(43\!\cdots\!23\)\( T^{6} + \)\(56\!\cdots\!06\)\( T^{7} + \)\(54\!\cdots\!01\)\( T^{8} \)
show more
show less