Newform orbit 172.2.f.b

• Label: 172.2.f.b
• Level: $172$
• Weight: $2$
• Character orbit: 172.f
• Analytic conductor: $1.373$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 172 = 2^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	172.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.37342691477\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 14 q^{4} - 18 q^{5} - 3 q^{6} - 6 q^{9}+O(q^{10}) \)

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	−1.41270	−	0.0653657i	0.761217	+	1.31847i	1.99145	+	0.184684i	−1.78958	+	1.03322i	−0.989190	−	1.91236i	−2.30129	+	3.98595i	−2.80126	−	0.391077i	0.341098	−	0.590800i	2.59569	−	1.34265i
7.2	−1.36494	−	0.370058i	1.34577	+	2.33095i	1.72611	+	1.01021i	1.44097	−	0.831945i	−0.974311	−	3.67961i	2.17107	−	3.76040i	−1.98220	−	2.01764i	−2.12221	+	3.67577i	−2.27470	+	0.602310i
7.3	−1.31270	+	0.526126i	−0.227649	−	0.394300i	1.44638	−	1.38129i	−1.33534	+	0.770959i	0.506288	+	0.397828i	1.34450	−	2.32874i	−1.17194	+	2.57421i	1.39635	−	2.41855i	1.34729	−	1.71460i
7.4	−1.21687	−	0.720574i	−1.46995	−	2.54603i	0.961545	+	1.75369i	−2.46507	+	1.42321i	−0.0458646	+	4.15740i	−0.455338	+	0.788668i	0.0935892	−	2.82688i	−2.82152	+	4.88701i	4.02520	+	0.0444062i
7.5	−0.958976	+	1.03941i	0.555042	+	0.961360i	−0.160730	−	1.99353i	2.85214	−	1.64669i	−1.53152	−	0.345008i	0.143888	−	0.249222i	2.22622	+	1.74468i	0.883857	−	1.53089i	−1.02356	+	4.54367i
7.6	−0.903622	−	1.08787i	−0.184925	−	0.320300i	−0.366936	+	1.96605i	1.02957	−	0.594425i	−0.181343	+	0.490606i	0.287788	−	0.498463i	2.47039	−	1.37739i	1.43161	−	2.47961i	−1.57700	−	0.582911i
7.7	−0.840598	+	1.13728i	−1.03031	−	1.78455i	−0.586789	−	1.91198i	−1.00759	+	0.581731i	2.89560	+	0.328344i	−2.14970	+	3.72338i	2.66770	+	0.939869i	−0.623084	+	1.07921i	0.185388	−	1.63491i
7.8	−0.494448	−	1.32496i	0.862004	+	1.49303i	−1.51104	+	1.31025i	−3.22510	+	1.86201i	1.55200	−	1.88035i	−0.0286931	+	0.0496979i	2.48316	+	1.35422i	0.0138980	−	0.0240721i	4.06174	+	3.35246i
7.9	0.494448	−	1.32496i	−0.862004	−	1.49303i	−1.51104	−	1.31025i	−3.22510	+	1.86201i	−2.40443	+	0.403893i	0.0286931	−	0.0496979i	−2.48316	+	1.35422i	0.0138980	−	0.0240721i	0.872448	+	5.19380i
7.10	0.840598	+	1.13728i	1.03031	+	1.78455i	−0.586789	+	1.91198i	−1.00759	+	0.581731i	−1.16345	+	2.67184i	2.14970	−	3.72338i	−2.66770	+	0.939869i	−0.623084	+	1.07921i	−1.50857	−	0.656903i
7.11	0.903622	−	1.08787i	0.184925	+	0.320300i	−0.366936	−	1.96605i	1.02957	−	0.594425i	0.515549	+	0.0882548i	−0.287788	+	0.498463i	−2.47039	−	1.37739i	1.43161	−	2.47961i	0.283686	−	1.65718i
7.12	0.958976	+	1.03941i	−0.555042	−	0.961360i	−0.160730	+	1.99353i	2.85214	−	1.64669i	0.466972	−	1.49884i	−0.143888	+	0.249222i	−2.22622	+	1.74468i	0.883857	−	1.53089i	4.44671	+	1.38540i
7.13	1.21687	−	0.720574i	1.46995	+	2.54603i	0.961545	−	1.75369i	−2.46507	+	1.42321i	3.62335	+	2.03898i	0.455338	−	0.788668i	−0.0935892	−	2.82688i	−2.82152	+	4.88701i	−1.97414	+	3.50813i
7.14	1.31270	+	0.526126i	0.227649	+	0.394300i	1.44638	+	1.38129i	−1.33534	+	0.770959i	0.0913848	+	0.637372i	−1.34450	+	2.32874i	1.17194	+	2.57421i	1.39635	−	2.41855i	−2.15853	+	0.309485i
7.15	1.36494	−	0.370058i	−1.34577	−	2.33095i	1.72611	−	1.01021i	1.44097	−	0.831945i	−2.69948	−	2.68358i	−2.17107	+	3.76040i	1.98220	−	2.01764i	−2.12221	+	3.67577i	1.65897	−	1.66880i
7.16	1.41270	−	0.0653657i	−0.761217	−	1.31847i	1.99145	−	0.184684i	−1.78958	+	1.03322i	−1.16155	−	1.81284i	2.30129	−	3.98595i	2.80126	−	0.391077i	0.341098	−	0.590800i	−2.46061	+	1.57660i
123.1	−1.41270	+	0.0653657i	0.761217	−	1.31847i	1.99145	−	0.184684i	−1.78958	−	1.03322i	−0.989190	+	1.91236i	−2.30129	−	3.98595i	−2.80126	+	0.391077i	0.341098	+	0.590800i	2.59569	+	1.34265i
123.2	−1.36494	+	0.370058i	1.34577	−	2.33095i	1.72611	−	1.01021i	1.44097	+	0.831945i	−0.974311	+	3.67961i	2.17107	+	3.76040i	−1.98220	+	2.01764i	−2.12221	−	3.67577i	−2.27470	−	0.602310i
123.3	−1.31270	−	0.526126i	−0.227649	+	0.394300i	1.44638	+	1.38129i	−1.33534	−	0.770959i	0.506288	−	0.397828i	1.34450	+	2.32874i	−1.17194	−	2.57421i	1.39635	+	2.41855i	1.34729	+	1.71460i
123.4	−1.21687	+	0.720574i	−1.46995	+	2.54603i	0.961545	−	1.75369i	−2.46507	−	1.42321i	−0.0458646	−	4.15740i	−0.455338	−	0.788668i	0.0935892	+	2.82688i	−2.82152	−	4.88701i	4.02520	−	0.0444062i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
43.d	odd	6	1	inner
172.f	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	172.2.f.b	✓	32
4.b	odd	2	1	inner	172.2.f.b	✓	32
43.d	odd	6	1	inner	172.2.f.b	✓	32
172.f	even	6	1	inner	172.2.f.b	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
172.2.f.b	✓	32	1.a	even	1	1	trivial
172.2.f.b	✓	32	4.b	odd	2	1	inner
172.2.f.b	✓	32	43.d	odd	6	1	inner
172.2.f.b	✓	32	172.f	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3^32 + 27*T3^30 + 445*T3^28 + 4692*T3^26 + 36344*T3^24 + 205212*T3^22 + 886426*T3^20 + 2874834*T3^18 + 7154458*T3^16 + 13121196*T3^14 + 17845544*T3^12 + 16235172*T3^10 + 10051109*T3^8 + 2842611*T3^6 + 572393*T3^4 + 58176*T3^2 + 4096