Newform orbit 54.6.e.a

• Label: 54.6.e.a
• Level: $54$
• Weight: $6$
• Character orbit: 54.e
• Analytic conductor: $8.661$
• Analytic rank: $0$
• Dimension: $42$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	54.e (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.66072626990\)
Analytic rank:	\(0\)
Dimension:	\(42\)
Relative dimension:	\(7\) over \(\Q(\zeta_{9})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 42 q + 87 q^{5} + 36 q^{6} - 15 q^{7} + 1344 q^{8} - 756 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	3.75877	−	1.36808i	−13.2518	−	8.20910i	12.2567	−	10.2846i	15.3072	+	86.8113i	−61.0412	−	12.7266i	66.3245	+	55.6528i	32.0000	−	55.4256i	108.221	+	217.571i	176.301	+	305.362i
7.2	3.75877	−	1.36808i	−13.1664	+	8.34538i	12.2567	−	10.2846i	4.21373	+	23.8972i	−38.0724	+	49.3811i	−177.680	−	149.091i	32.0000	−	55.4256i	103.709	−	219.758i	48.5318	+	84.0595i
7.3	3.75877	−	1.36808i	−7.91690	+	13.4284i	12.2567	−	10.2846i	−16.3275	−	92.5977i	−11.3866	+	61.3053i	101.146	+	84.8713i	32.0000	−	55.4256i	−117.645	−	212.623i	−188.052	−	325.716i
7.4	3.75877	−	1.36808i	−1.40563	−	15.5250i	12.2567	−	10.2846i	−5.44857	−	30.9004i	−26.5228	−	56.4317i	−17.3216	−	14.5345i	32.0000	−	55.4256i	−239.048	+	43.6445i	−62.7541	−	108.693i
7.5	3.75877	−	1.36808i	4.04068	+	15.0557i	12.2567	−	10.2846i	7.13247	+	40.4503i	35.7853	+	51.0628i	22.9271	+	19.2381i	32.0000	−	55.4256i	−210.346	+	121.670i	82.1485	+	142.285i
7.6	3.75877	−	1.36808i	15.0623	−	4.01585i	12.2567	−	10.2846i	5.41912	+	30.7334i	51.1217	−	35.7011i	106.533	+	89.3920i	32.0000	−	55.4256i	210.746	−	120.976i	62.4149	+	108.106i
7.7	3.75877	−	1.36808i	15.5244	+	1.41130i	12.2567	−	10.2846i	−13.6705	−	77.5294i	60.2836	−	15.9339i	−175.671	−	147.405i	32.0000	−	55.4256i	239.016	+	43.8193i	−157.451	−	272.713i
13.1	−0.694593	−	3.93923i	−14.3789	−	6.02064i	−15.0351	+	5.47232i	−49.2928	−	41.3615i	−13.7292	+	60.8236i	−5.96191	−	2.16996i	32.0000	+	55.4256i	170.504	+	173.140i	−128.694	+	222.905i
13.2	−0.694593	−	3.93923i	−10.0412	+	11.9237i	−15.0351	+	5.47232i	1.32277	+	1.10993i	53.9448	+	31.2724i	17.8430	+	6.49433i	32.0000	+	55.4256i	−41.3498	−	239.456i	3.45350	−	5.98163i
13.3	−0.694593	−	3.93923i	−9.03421	−	12.7037i	−15.0351	+	5.47232i	83.9727	+	70.4614i	−43.7676	+	44.4117i	91.0411	+	33.1362i	32.0000	+	55.4256i	−79.7663	+	229.535i	219.237	−	379.730i
13.4	−0.694593	−	3.93923i	5.70292	+	14.5078i	−15.0351	+	5.47232i	−52.5534	−	44.0975i	53.1884	−	32.5421i	163.977	+	59.6829i	32.0000	+	55.4256i	−177.953	+	165.474i	−137.207	+	237.650i
13.5	−0.694593	−	3.93923i	5.82228	+	14.4603i	−15.0351	+	5.47232i	63.6116	+	53.3765i	52.9185	−	32.9793i	−201.426	−	73.3131i	32.0000	+	55.4256i	−175.202	+	168.384i	166.078	−	287.656i
13.6	−0.694593	−	3.93923i	5.83321	−	14.4559i	−15.0351	+	5.47232i	−11.4963	−	9.64656i	−60.9969	−	12.9374i	−96.8610	−	35.2545i	32.0000	+	55.4256i	−174.947	−	168.649i	−30.0148	+	51.9871i
13.7	−0.694593	−	3.93923i	15.5034	−	1.62578i	−15.0351	+	5.47232i	25.3376	+	21.2608i	−17.1729	−	59.9424i	116.279	+	42.3220i	32.0000	+	55.4256i	237.714	−	50.4103i	66.1517	−	114.578i
25.1	−0.694593	+	3.93923i	−14.3789	+	6.02064i	−15.0351	−	5.47232i	−49.2928	+	41.3615i	−13.7292	−	60.8236i	−5.96191	+	2.16996i	32.0000	−	55.4256i	170.504	−	173.140i	−128.694	−	222.905i
25.2	−0.694593	+	3.93923i	−10.0412	−	11.9237i	−15.0351	−	5.47232i	1.32277	−	1.10993i	53.9448	−	31.2724i	17.8430	−	6.49433i	32.0000	−	55.4256i	−41.3498	+	239.456i	3.45350	+	5.98163i
25.3	−0.694593	+	3.93923i	−9.03421	+	12.7037i	−15.0351	−	5.47232i	83.9727	−	70.4614i	−43.7676	−	44.4117i	91.0411	−	33.1362i	32.0000	−	55.4256i	−79.7663	−	229.535i	219.237	+	379.730i
25.4	−0.694593	+	3.93923i	5.70292	−	14.5078i	−15.0351	−	5.47232i	−52.5534	+	44.0975i	53.1884	+	32.5421i	163.977	−	59.6829i	32.0000	−	55.4256i	−177.953	−	165.474i	−137.207	−	237.650i
25.5	−0.694593	+	3.93923i	5.82228	−	14.4603i	−15.0351	−	5.47232i	63.6116	−	53.3765i	52.9185	+	32.9793i	−201.426	+	73.3131i	32.0000	−	55.4256i	−175.202	−	168.384i	166.078	+	287.656i
25.6	−0.694593	+	3.93923i	5.83321	+	14.4559i	−15.0351	−	5.47232i	−11.4963	+	9.64656i	−60.9969	+	12.9374i	−96.8610	+	35.2545i	32.0000	−	55.4256i	−174.947	+	168.649i	−30.0148	−	51.9871i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	54.6.e.a	✓	42
3.b	odd	2	1		162.6.e.a		42
27.e	even	9	1	inner	54.6.e.a	✓	42
27.f	odd	18	1		162.6.e.a		42

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
54.6.e.a	✓	42	1.a	even	1	1	trivial
54.6.e.a	✓	42	27.e	even	9	1	inner
162.6.e.a		42	3.b	odd	2	1
162.6.e.a		42	27.f	odd	18	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^42 - 87*T5^41 + 11124*T5^40 - 423687*T5^39 + 63335799*T5^38 + 5312415969*T5^37 + 335922145227*T5^36 - 651624118929*T5^35 + 2506759100660268*T5^34 - 94639102552713195*T5^33 + 10235048037516923958*T5^32 + 1436665267895038286301*T5^31 + 36341361355801959565497*T5^30 - 5161298688496546514987949*T5^29 - 155332965820167996653964213*T5^28 - 31000550486408610393014413119*T5^27 + 1698666266087955556962567932169*T5^26 + 241635043276610053644416365399422*T5^25 + 340277537812339811846561881463829*T5^24 - 717620462621053357679471943978379335*T5^23 + 11787046051554693257085458248845717513*T5^22 + 207666531508992479604714973250056514796*T5^21 - 37475902315993267627020020688690016695033*T5^20 + 1842131684148589107755125220261005175408490*T5^19 + 161768755064815574734616010016638891500548956*T5^18 - 9415297346552385276351243218175422465052283506*T5^17 + 306801276642594395669278596576034793190368302659*T5^16 - 4283955425862104585490003676685538252078645690582*T5^15 - 122215295730506851716400544996484456595542560212470*T5^14 + 10675051162049812833858295629006647901568623705704112*T5^13 - 133467238221057744504058551190071553562912440605172515*T5^12 - 3741987416841820129454796808620269321915248185488543136*T5^11 + 393780529414564579055332836536519389540028578904618305740*T5^10 - 14693291919406514870955946960300258893517638203482564787508*T5^9 + 364267870809986462914884169100568573920409864762445339146894*T5^8 - 2633945306421630873218813732255415876085937454397822749071409*T5^7 + 86934593532852134118522981999069484072759059474789956135565552*T5^6 + 860487567867110927706974121762002621991074345186598674566568037*T5^5 + 26801279405072126054761956014319437654526445967994400880522806815*T5^4 - 282083800750714041678002273893391456692806995182033031895634977786*T5^3 + 1008492748484534630949337250213594274509443594244731932735997450272*T5^2 - 1602462630519953431067291379304895472281882344640836675425828566721*T5 + 1108828815114652520381868879941701394517549080132966611483601916849