Newform orbit 168.3.y.a

• Label: 168.3.y.a
• Level: $168$
• Weight: $3$
• Character orbit: 168.y
• Analytic conductor: $4.578$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	168.y (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.57766844125\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q - 2 q^{2} + 6 q^{4} - 32 q^{8} - 96 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.99624	−	0.122505i	0.866025	−	1.50000i	3.96999	+	0.489099i	6.23419	−	3.59931i	−1.91256	+	2.88827i	6.98064	+	0.520227i	−7.86514	−	1.46270i	−1.50000	−	2.59808i	−12.8859	+	6.42139i
67.2	−1.99178	+	0.181149i	−0.866025	+	1.50000i	3.93437	−	0.721616i	7.87806	−	4.54840i	1.45321	−	3.14455i	−5.37265	−	4.48717i	−7.70568	+	2.15001i	−1.50000	−	2.59808i	−14.8674	+	10.4865i
67.3	−1.98339	+	0.257253i	−0.866025	+	1.50000i	3.86764	−	1.02046i	0.378950	−	0.218787i	1.33178	−	3.19787i	2.60440	+	6.49747i	−7.40851	+	3.01894i	−1.50000	−	2.59808i	−0.695320	+	0.531425i
67.4	−1.94295	+	0.474282i	0.866025	−	1.50000i	3.55011	−	1.84301i	−2.76728	+	1.59769i	−0.971221	+	3.32517i	−6.29103	−	3.06969i	−6.02358	+	5.26464i	−1.50000	−	2.59808i	4.61894	−	4.41671i
67.5	−1.85529	−	0.746918i	0.866025	−	1.50000i	2.88423	+	2.77150i	−7.62406	+	4.40175i	−2.72711	+	2.13609i	6.99490	−	0.267227i	−3.28100	−	7.29623i	−1.50000	−	2.59808i	17.4326	−	2.47199i
67.6	−1.75189	−	0.964820i	−0.866025	+	1.50000i	2.13824	+	3.38052i	−1.41690	+	0.818048i	2.96441	−	1.79228i	5.98157	−	3.63605i	−0.484377	−	7.98532i	−1.50000	−	2.59808i	3.27153	−	0.0660766i
67.7	−1.73973	+	0.986579i	−0.866025	+	1.50000i	2.05332	−	3.43276i	−6.59300	+	3.80647i	0.0267816	−	3.46400i	1.06552	−	6.91843i	−0.185533	+	7.99785i	−1.50000	−	2.59808i	7.71465	−	13.1267i
67.8	−1.57038	−	1.23850i	−0.866025	+	1.50000i	0.932215	+	3.88986i	−1.31650	+	0.760084i	3.21775	−	1.28300i	−6.66783	+	2.13075i	3.35367	−	7.26312i	−1.50000	−	2.59808i	3.00878	+	0.436871i
67.9	−1.56642	+	1.24351i	0.866025	−	1.50000i	0.907356	−	3.89573i	3.61947	−	2.08970i	0.508707	+	3.42655i	−2.31505	+	6.60610i	3.42308	+	7.23066i	−1.50000	−	2.59808i	−3.07105	+	7.77420i
67.10	−1.06715	+	1.69151i	0.866025	−	1.50000i	−1.72240	−	3.61017i	−4.80721	+	2.77545i	1.61309	+	3.06561i	6.80756	+	1.63006i	7.94469	+	0.939119i	−1.50000	−	2.59808i	0.435305	−	11.0932i
67.11	−1.03920	−	1.70882i	0.866025	−	1.50000i	−1.84011	+	3.55162i	−2.67250	+	1.54297i	−3.46320	+	0.0789285i	−1.72536	+	6.78403i	7.98131	−	0.546454i	−1.50000	−	2.59808i	5.41393	+	2.96336i
67.12	−0.960276	−	1.75439i	0.866025	−	1.50000i	−2.15574	+	3.36939i	2.67250	−	1.54297i	−3.46320	−	0.0789285i	1.72536	−	6.78403i	7.98131	+	0.546454i	−1.50000	−	2.59808i	−5.27331	−	3.20693i
67.13	−0.646089	+	1.89277i	−0.866025	+	1.50000i	−3.16514	−	2.44579i	−5.18546	+	2.99383i	−2.27962	−	2.60832i	−0.175343	+	6.99780i	6.67428	−	4.41067i	−1.50000	−	2.59808i	−2.31635	−	11.7491i
67.14	−0.506495	+	1.93480i	−0.866025	+	1.50000i	−3.48693	−	1.95994i	6.11791	−	3.53218i	−2.46357	−	2.43533i	6.16163	−	3.32180i	5.55820	−	5.75382i	−1.50000	−	2.59808i	3.73537	+	13.6260i
67.15	−0.365129	+	1.96639i	0.866025	−	1.50000i	−3.73336	−	1.43597i	−0.153269	+	0.0884897i	2.63337	+	2.25064i	−2.72695	−	6.44700i	4.18684	−	6.81692i	−1.50000	−	2.59808i	−0.118042	−	0.333696i
67.16	−0.287384	−	1.97924i	−0.866025	+	1.50000i	−3.83482	+	1.13761i	1.31650	−	0.760084i	3.21775	+	1.28300i	6.66783	−	2.13075i	3.35367	+	7.26312i	−1.50000	−	2.59808i	−1.88273	−	2.38725i
67.17	0.0403867	−	1.99959i	−0.866025	+	1.50000i	−3.99674	−	0.161514i	1.41690	−	0.818048i	2.96441	+	1.79228i	−5.98157	+	3.63605i	−0.484377	+	7.98532i	−1.50000	−	2.59808i	−1.57854	−	2.86626i
67.18	0.280797	−	1.98019i	0.866025	−	1.50000i	−3.84231	−	1.11206i	7.62406	−	4.40175i	−2.72711	−	2.13609i	−6.99490	+	0.267227i	−3.28100	+	7.29623i	−1.50000	−	2.59808i	−6.57550	−	16.3331i
67.19	0.608416	+	1.90521i	−0.866025	+	1.50000i	−3.25966	+	2.31832i	−2.50327	+	1.44526i	−3.38472	−	0.737337i	−3.77733	−	5.89337i	−6.40012	−	4.79984i	−1.50000	−	2.59808i	−4.27656	−	3.88994i
67.20	0.857569	+	1.80681i	0.866025	−	1.50000i	−2.52915	+	3.09893i	−5.65466	+	3.26472i	3.45290	+	0.278393i	−5.03611	+	4.86185i	−7.76812	−	1.91215i	−1.50000	−	2.59808i	−10.7480	−	7.41719i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
8.d	odd	2	1	inner
56.k	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	168.3.y.a	✓	64
4.b	odd	2	1		672.3.bg.a		64
7.c	even	3	1	inner	168.3.y.a	✓	64
8.b	even	2	1		672.3.bg.a		64
8.d	odd	2	1	inner	168.3.y.a	✓	64
28.g	odd	6	1		672.3.bg.a		64
56.k	odd	6	1	inner	168.3.y.a	✓	64
56.p	even	6	1		672.3.bg.a		64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
168.3.y.a	✓	64	1.a	even	1	1	trivial
168.3.y.a	✓	64	7.c	even	3	1	inner
168.3.y.a	✓	64	8.d	odd	2	1	inner
168.3.y.a	✓	64	56.k	odd	6	1	inner
672.3.bg.a		64	4.b	odd	2	1
672.3.bg.a		64	8.b	even	2	1
672.3.bg.a		64	28.g	odd	6	1
672.3.bg.a		64	56.p	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(168, [\chi])\).