Newform orbit 675.2.ba.a

• Label: 675.2.ba.a
• Level: $675$
• Weight: $2$
• Character orbit: 675.ba
• Analytic conductor: $5.390$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.ba (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.38990213644\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(12\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1	−0.233997	+	2.67459i	−1.33933	−	1.09827i	−5.12908	−	0.904395i		0		3.25082	−	3.32517i	1.72808	−	2.46795i	2.22931	−	8.31991i	0.587614	+	2.94189i		0
32.2	−0.197639	+	2.25902i	0.504675	+	1.65690i	−3.09452	−	0.545647i		0		−3.84271	+	0.812605i	−1.46698	+	2.09507i	0.670403	−	2.50198i	−2.49061	+	1.67239i		0
32.3	−0.137404	+	1.57054i	−1.44935	+	0.948353i	−0.478086	−	0.0842994i		0		−1.29028	−	2.40657i	1.21036	−	1.72857i	−0.617988	+	2.30636i	1.20125	−	2.74900i		0
32.4	−0.124648	+	1.42473i	1.69462	+	0.358158i	−0.0447061	−	0.00788289i		0		−0.721510	+	2.36973i	2.32237	−	3.31669i	−0.723509	+	2.70017i	2.74345	+	1.21388i		0
32.5	−0.0919314	+	1.05078i	−1.07151	−	1.36083i	0.873927	+	0.154097i		0		1.52844	−	1.00082i	−1.27896	+	1.82655i	−0.788265	+	2.94185i	−0.703726	+	2.91629i		0
32.6	−0.00680491	+	0.0777805i	0.788185	−	1.54232i	1.96361	+	0.346238i		0		0.114599	+	0.0718008i	0.804356	−	1.14874i	−0.0807087	+	0.301209i	−1.75753	−	2.43127i		0
32.7	0.00680491	−	0.0777805i	−0.788185	+	1.54232i	1.96361	+	0.346238i		0		0.114599	+	0.0718008i	−0.804356	+	1.14874i	0.0807087	−	0.301209i	−1.75753	−	2.43127i		0
32.8	0.0919314	−	1.05078i	1.07151	+	1.36083i	0.873927	+	0.154097i		0		1.52844	−	1.00082i	1.27896	−	1.82655i	0.788265	−	2.94185i	−0.703726	+	2.91629i		0
32.9	0.124648	−	1.42473i	−1.69462	−	0.358158i	−0.0447061	−	0.00788289i		0		−0.721510	+	2.36973i	−2.32237	+	3.31669i	0.723509	−	2.70017i	2.74345	+	1.21388i		0
32.10	0.137404	−	1.57054i	1.44935	−	0.948353i	−0.478086	−	0.0842994i		0		−1.29028	−	2.40657i	−1.21036	+	1.72857i	0.617988	−	2.30636i	1.20125	−	2.74900i		0
32.11	0.197639	−	2.25902i	−0.504675	−	1.65690i	−3.09452	−	0.545647i		0		−3.84271	+	0.812605i	1.46698	−	2.09507i	−0.670403	+	2.50198i	−2.49061	+	1.67239i		0
32.12	0.233997	−	2.67459i	1.33933	+	1.09827i	−5.12908	−	0.904395i		0		3.25082	−	3.32517i	−1.72808	+	2.46795i	−2.22931	+	8.31991i	0.587614	+	2.94189i		0
68.1	−1.54236	+	2.20272i	−0.803366	−	1.53447i	−1.78906	−	4.91539i		0		4.61909	+	0.597120i	−2.24187	+	1.04540i	8.39181	+	2.24858i	−1.70921	+	2.46548i		0
68.2	−1.17664	+	1.68042i	−1.09608	+	1.34112i	−0.755276	−	2.07510i		0		−0.963952	−	3.41989i	−1.82242	+	0.849809i	0.412706	+	0.110584i	−0.597219	−	2.93995i		0
68.3	−1.15330	+	1.64708i	1.68675	−	0.393547i	−0.698744	−	1.91978i		0		−1.29712	+	3.23210i	−0.920373	+	0.429177i	0.0834957	+	0.0223726i	2.69024	−	1.32763i		0
68.4	−0.780943	+	1.11530i	0.249363	−	1.71401i	0.0500132	+	0.137410i		0		1.71690	+	1.61666i	4.57752	−	2.13453i	−2.82259	−	0.756311i	−2.87564	−	0.854819i		0
68.5	−0.347323	+	0.496029i	−1.68158	−	0.415063i	0.558629	+	1.53482i		0		0.789936	−	0.689953i	−0.689128	+	0.321346i	−2.12515	−	0.569433i	2.65544	+	1.39593i		0
68.6	−0.312812	+	0.446741i	0.0191691	+	1.73194i	0.582314	+	1.59989i		0		−0.779727	−	0.533209i	2.81926	−	1.31464i	−1.95047	−	0.522626i	−2.99927	+	0.0663996i		0
68.7	0.312812	−	0.446741i	−0.0191691	−	1.73194i	0.582314	+	1.59989i		0		−0.779727	−	0.533209i	−2.81926	+	1.31464i	1.95047	+	0.522626i	−2.99927	+	0.0663996i		0
68.8	0.347323	−	0.496029i	1.68158	+	0.415063i	0.558629	+	1.53482i		0		0.789936	−	0.689953i	0.689128	−	0.321346i	2.12515	+	0.569433i	2.65544	+	1.39593i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
5.c	odd	4	2	inner
27.f	odd	18	1	inner
135.n	odd	18	1	inner
135.q	even	36	2	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	675.2.ba.a	✓	144
5.b	even	2	1	inner	675.2.ba.a	✓	144
5.c	odd	4	2	inner	675.2.ba.a	✓	144
27.f	odd	18	1	inner	675.2.ba.a	✓	144
135.n	odd	18	1	inner	675.2.ba.a	✓	144
135.q	even	36	2	inner	675.2.ba.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
675.2.ba.a	✓	144	1.a	even	1	1	trivial
675.2.ba.a	✓	144	5.b	even	2	1	inner
675.2.ba.a	✓	144	5.c	odd	4	2	inner
675.2.ba.a	✓	144	27.f	odd	18	1	inner
675.2.ba.a	✓	144	135.n	odd	18	1	inner
675.2.ba.a	✓	144	135.q	even	36	2	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^144 - 414*T2^136 - 141858*T2^132 + 1257471*T2^128 - 4454244*T2^124 + 17631615951*T2^120 + 36153227988*T2^116 + 3453197548257*T2^112 - 304196331133170*T2^108 - 3796619510478213*T2^104 + 17368655741603775*T2^100 + 4291453039366072962*T2^96 + 51068160541145136222*T2^92 + 491356051049171040408*T2^88 - 2805030769771988476020*T2^84 - 12820074281473636045575*T2^80 - 634804202705451441916275*T2^76 + 6451936812251815990661694*T2^72 - 13722253294263629505711282*T2^68 + 36219225824428806040967262*T2^64 - 299714286800649463341115572*T2^60 + 471849241599629614032349212*T2^56 - 863657631360344719873347336*T2^52 + 13103953634794259907313042788*T2^48 - 23000115149094730918008182832*T2^44 + 7803710199628909701147203688*T2^40 + 4100635140751747643729696172*T2^36 + 1739692146173272812711235053*T2^32 + 398124399034700328572393616*T2^28 + 55579110028105278787679106*T2^24 + 4034300677238908102699467*T2^20 + 140895694403982043294977*T2^16 - 7284518845047642303*T2^12 + 400276444839309*T2^8 - 23326539813*T2^4 + 531441