Newform orbit 63.6.f.b

• Label: 63.6.f.b
• Level: $63$
• Weight: $6$
• Character orbit: 63.f
• Analytic conductor: $10.104$
• Analytic rank: $0$
• Dimension: $30$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	63.f (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.1041806482\)
Analytic rank:	\(0\)
Dimension:	\(30\)
Relative dimension:	\(15\) over \(\Q(\zeta_{3})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 30 q + 4 q^{2} + 20 q^{3} - 240 q^{4} + 71 q^{5} - 266 q^{6} + 735 q^{7} + 42 q^{8} - 442 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
22.1	−5.32234	−	9.21856i	−5.57188	−	14.5586i	−40.6546	+	70.4158i	9.68441	−	16.7739i	−104.554	+	128.851i	24.5000	+	42.4352i	524.880			−180.908	+	162.238i	−206.175
22.2	−4.38794	−	7.60013i	−6.48542	+	14.1753i	−22.5080	+	38.9850i	−26.4033	+	45.7318i	136.192	−	12.9103i	24.5000	+	42.4352i	114.226			−158.879	−	183.866i	463.424
22.3	−4.21162	−	7.29473i	13.4591	−	7.86457i	−19.4754	+	33.7324i	25.8420	−	44.7596i	−114.055	−	65.0583i	24.5000	+	42.4352i	58.5489			119.297	−	211.701i	−435.346
22.4	−3.81386	−	6.60581i	12.4507	+	9.37976i	−13.0911	+	22.6745i	−6.17916	+	10.7026i	14.4756	−	118.020i	24.5000	+	42.4352i	−44.3762			67.0401	+	233.569i	94.2659
22.5	−2.21406	−	3.83487i	−11.6210	−	10.3900i	6.19585	−	10.7315i	−30.8296	+	53.3984i	−14.1145	+	67.5692i	24.5000	+	42.4352i	−196.572			27.0967	+	241.485i	273.035
22.6	−0.876872	−	1.51879i	−13.7798	+	7.28808i	14.4622	−	25.0492i	4.30330	−	7.45354i	23.1522	+	14.5379i	24.5000	+	42.4352i	−106.846			136.768	−	200.857i	−15.0938
22.7	−0.772339	−	1.33773i	2.43538	+	15.3970i	14.8070	−	25.6464i	33.3433	−	57.7523i	18.7162	−	15.1496i	24.5000	+	42.4352i	−95.1738			−231.138	+	74.9954i	−103.009
22.8	−0.195179	−	0.338060i	5.35520	−	14.6397i	15.9238	−	27.5808i	48.2760	−	83.6165i	−5.99433	+	1.04699i	24.5000	+	42.4352i	−24.9234			−185.644	−	156.797i	−37.6898
22.9	1.29488	+	2.24280i	14.4289	+	5.89965i	12.6466	−	21.9045i	−20.9318	+	36.2550i	5.45202	+	40.0006i	24.5000	+	42.4352i	148.376			173.388	+	170.251i	−108.417
22.10	2.01180	+	3.48454i	−0.357568	−	15.5844i	7.90534	−	13.6924i	−7.29857	+	12.6415i	53.5849	−	32.5985i	24.5000	+	42.4352i	192.371			−242.744	+	11.1449i	−58.7330
22.11	3.08078	+	5.33606i	−14.8938	−	4.60155i	−2.98236	+	5.16561i	9.09655	−	15.7557i	−21.3303	−	93.6507i	24.5000	+	42.4352i	160.418			200.651	+	137.069i	112.098
22.12	3.31910	+	5.74885i	−7.74251	+	13.5297i	−6.03282	+	10.4491i	−41.6346	+	72.1133i	−103.478	+	0.395999i	24.5000	+	42.4352i	132.328			−123.107	−	209.508i	−552.758
22.13	3.78148	+	6.54972i	14.7418	+	5.06742i	−12.5992	+	21.8225i	51.7579	−	89.6473i	22.5558	+	115.717i	24.5000	+	42.4352i	51.4396			191.643	+	149.406i	782.887
22.14	4.84484	+	8.39151i	13.9648	−	6.92717i	−30.9450	+	53.5982i	−35.2811	+	61.1087i	125.786	+	83.6243i	24.5000	+	42.4352i	−289.624			147.029	−	193.473i	−683.726
22.15	5.46133	+	9.45930i	−6.38387	+	14.2213i	−43.6522	+	75.6079i	21.7547	−	37.6803i	−169.388	+	17.2804i	24.5000	+	42.4352i	−604.072			−161.492	−	181.574i	475.239
43.1	−5.32234	+	9.21856i	−5.57188	+	14.5586i	−40.6546	−	70.4158i	9.68441	+	16.7739i	−104.554	−	128.851i	24.5000	−	42.4352i	524.880			−180.908	−	162.238i	−206.175
43.2	−4.38794	+	7.60013i	−6.48542	−	14.1753i	−22.5080	−	38.9850i	−26.4033	−	45.7318i	136.192	+	12.9103i	24.5000	−	42.4352i	114.226			−158.879	+	183.866i	463.424
43.3	−4.21162	+	7.29473i	13.4591	+	7.86457i	−19.4754	−	33.7324i	25.8420	+	44.7596i	−114.055	+	65.0583i	24.5000	−	42.4352i	58.5489			119.297	+	211.701i	−435.346
43.4	−3.81386	+	6.60581i	12.4507	−	9.37976i	−13.0911	−	22.6745i	−6.17916	−	10.7026i	14.4756	+	118.020i	24.5000	−	42.4352i	−44.3762			67.0401	−	233.569i	94.2659
43.5	−2.21406	+	3.83487i	−11.6210	+	10.3900i	6.19585	+	10.7315i	−30.8296	−	53.3984i	−14.1145	−	67.5692i	24.5000	−	42.4352i	−196.572			27.0967	−	241.485i	273.035

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	63.6.f.b	✓	30
3.b	odd	2	1		189.6.f.a		30
9.c	even	3	1	inner	63.6.f.b	✓	30
9.c	even	3	1		567.6.a.g		15
9.d	odd	6	1		189.6.f.a		30
9.d	odd	6	1		567.6.a.h		15

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
63.6.f.b	✓	30	1.a	even	1	1	trivial
63.6.f.b	✓	30	9.c	even	3	1	inner
189.6.f.a		30	3.b	odd	2	1
189.6.f.a		30	9.d	odd	6	1
567.6.a.g		15	9.c	even	3	1
567.6.a.h		15	9.d	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^30 - 4*T2^29 + 368*T2^28 - 1294*T2^27 + 81464*T2^26 - 268975*T2^25 + 11629347*T2^24 - 35663924*T2^23 + 1218761365*T2^22 - 3522885614*T2^21 + 93734400625*T2^20 - 250270815677*T2^19 + 5491870179797*T2^18 - 13292410014996*T2^17 + 238109596586004*T2^16 - 486766005956880*T2^15 + 7654666116520848*T2^14 - 12237394895628864*T2^13 + 169413412865665664*T2^12 - 151008336006679424*T2^11 + 2532701100821863936*T2^10 - 917539696114855424*T2^9 + 24124271066623485952*T2^8 + 14774428763524470784*T2^7 + 125285634277563752448*T2^6 + 131810108281116311552*T2^5 + 514855249085829201920*T2^4 + 571870238284873662464*T2^3 + 821515617812585578496*T2^2 + 293101335939940352000*T2 + 91930357633024000000