Newform orbit 555.2.n.b

• Label: 555.2.n.b
• Level: $555$
• Weight: $2$
• Character orbit: 555.n
• Analytic conductor: $4.432$
• Analytic rank: $0$
• Dimension: $112$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 555 = 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	555.n (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.43169731218\)
Analytic rank:	\(0\)
Dimension:	\(112\)
Relative dimension:	\(56\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 112 q - 8 q^{3} - 8 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
332.1	−1.76534	−	1.76534i	−0.0437317	−	1.73150i			4.23282i	0.0481278	−	2.23555i	−2.97948	+	3.13388i	−0.629145	−	0.629145i	3.94168	−	3.94168i	−2.99618	+	0.151443i	−4.03146	+	3.86153i
332.2	−1.76534	−	1.76534i	1.73150	+	0.0437317i			4.23282i	0.0481278	−	2.23555i	−2.97948	−	3.13388i	−0.629145	−	0.629145i	3.94168	−	3.94168i	2.99618	+	0.151443i	−4.03146	+	3.86153i
332.3	−1.76107	−	1.76107i	−1.40789	−	1.00888i			4.20272i	−0.404051	+	2.19926i	0.702678	+	4.25610i	1.18402	+	1.18402i	3.87914	−	3.87914i	0.964309	+	2.84079i	4.58461	−	3.16148i
332.4	−1.76107	−	1.76107i	1.00888	+	1.40789i			4.20272i	−0.404051	+	2.19926i	0.702678	−	4.25610i	1.18402	+	1.18402i	3.87914	−	3.87914i	−0.964309	+	2.84079i	4.58461	−	3.16148i
332.5	−1.71658	−	1.71658i	−1.61958	−	0.613983i			3.89331i	2.15823	−	0.584861i	1.72618	+	3.83409i	−2.39240	−	2.39240i	3.25001	−	3.25001i	2.24605	+	1.98878i	−4.70873	−	2.70081i
332.6	−1.71658	−	1.71658i	0.613983	+	1.61958i			3.89331i	2.15823	−	0.584861i	1.72618	−	3.83409i	−2.39240	−	2.39240i	3.25001	−	3.25001i	−2.24605	+	1.98878i	−4.70873	−	2.70081i
332.7	−1.46007	−	1.46007i	−1.73093	+	0.0623716i			2.26360i	−0.961753	−	2.01867i	2.61834	+	2.43621i	1.26369	+	1.26369i	0.384871	−	0.384871i	2.99222	−	0.215921i	−1.54317	+	4.35162i
332.8	−1.46007	−	1.46007i	−0.0623716	+	1.73093i			2.26360i	−0.961753	−	2.01867i	2.61834	−	2.43621i	1.26369	+	1.26369i	0.384871	−	0.384871i	−2.99222	−	0.215921i	−1.54317	+	4.35162i
332.9	−1.38041	−	1.38041i	−0.647455	−	1.60649i			1.81107i	−2.22755	−	0.194962i	−1.32386	+	3.11137i	−2.95997	−	2.95997i	−0.260798	+	0.260798i	−2.16161	+	2.08026i	2.80581	+	3.34407i
332.10	−1.38041	−	1.38041i	1.60649	+	0.647455i			1.81107i	−2.22755	−	0.194962i	−1.32386	−	3.11137i	−2.95997	−	2.95997i	−0.260798	+	0.260798i	2.16161	+	2.08026i	2.80581	+	3.34407i
332.11	−1.15880	−	1.15880i	−1.65192	+	0.520727i			0.685629i	−1.98788	+	1.02387i	2.51766	+	1.31083i	0.692262	+	0.692262i	−1.52309	+	1.52309i	2.45769	−	1.72040i	3.49002	+	1.11710i
332.12	−1.15880	−	1.15880i	−0.520727	+	1.65192i			0.685629i	−1.98788	+	1.02387i	2.51766	−	1.31083i	0.692262	+	0.692262i	−1.52309	+	1.52309i	−2.45769	−	1.72040i	3.49002	+	1.11710i
332.13	−1.15212	−	1.15212i	−0.335392	−	1.69927i			0.654759i	1.32724	+	1.79957i	−1.57135	+	2.34417i	0.904861	+	0.904861i	−1.54988	+	1.54988i	−2.77502	+	1.13984i	0.544182	−	3.60245i
332.14	−1.15212	−	1.15212i	1.69927	+	0.335392i			0.654759i	1.32724	+	1.79957i	−1.57135	−	2.34417i	0.904861	+	0.904861i	−1.54988	+	1.54988i	2.77502	+	1.13984i	0.544182	−	3.60245i
332.15	−1.02818	−	1.02818i	−1.33760	−	1.10038i			0.114318i	1.70482	−	1.44692i	0.243906	+	2.50668i	1.17953	+	1.17953i	−1.93882	+	1.93882i	0.578337	+	2.94373i	−3.24057	−	0.265173i
332.16	−1.02818	−	1.02818i	1.10038	+	1.33760i			0.114318i	1.70482	−	1.44692i	0.243906	−	2.50668i	1.17953	+	1.17953i	−1.93882	+	1.93882i	−0.578337	+	2.94373i	−3.24057	−	0.265173i
332.17	−0.917523	−	0.917523i	−1.60906	+	0.641036i		−	0.316301i	2.08385	+	0.810915i	2.06452	+	0.888185i	−0.319231	−	0.319231i	−2.12526	+	2.12526i	2.17815	−	2.06293i	−1.16795	−	2.65601i
332.18	−0.917523	−	0.917523i	−0.641036	+	1.60906i		−	0.316301i	2.08385	+	0.810915i	2.06452	−	0.888185i	−0.319231	−	0.319231i	−2.12526	+	2.12526i	−2.17815	−	2.06293i	−1.16795	−	2.65601i
332.19	−0.583934	−	0.583934i	−1.50818	−	0.851692i		−	1.31804i	−0.905328	+	2.04460i	0.383348	+	1.37801i	−2.28069	−	2.28069i	−1.93752	+	1.93752i	1.54924	+	2.56902i	1.72256	−	0.665258i
332.20	−0.583934	−	0.583934i	0.851692	+	1.50818i		−	1.31804i	−0.905328	+	2.04460i	0.383348	−	1.37801i	−2.28069	−	2.28069i	−1.93752	+	1.93752i	−1.54924	+	2.56902i	1.72256	−	0.665258i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner
37.b	even	2	1	inner
111.d	odd	2	1	inner
185.h	odd	4	1	inner
555.n	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	555.2.n.b	✓	112
3.b	odd	2	1	inner	555.2.n.b	✓	112
5.c	odd	4	1	inner	555.2.n.b	✓	112
15.e	even	4	1	inner	555.2.n.b	✓	112
37.b	even	2	1	inner	555.2.n.b	✓	112
111.d	odd	2	1	inner	555.2.n.b	✓	112
185.h	odd	4	1	inner	555.2.n.b	✓	112
555.n	even	4	1	inner	555.2.n.b	✓	112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
555.2.n.b	✓	112	1.a	even	1	1	trivial
555.2.n.b	✓	112	3.b	odd	2	1	inner
555.2.n.b	✓	112	5.c	odd	4	1	inner
555.2.n.b	✓	112	15.e	even	4	1	inner
555.2.n.b	✓	112	37.b	even	2	1	inner
555.2.n.b	✓	112	111.d	odd	2	1	inner
555.2.n.b	✓	112	185.h	odd	4	1	inner
555.2.n.b	✓	112	555.n	even	4	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2^56 + 167*T2^52 + 11511*T2^48 + 425643*T2^44 + 9231982*T2^40 + 121395054*T2^36 + 970246043*T2^32 + 4603184679*T2^28 + 12177753199*T2^24 + 15645219137*T2^20 + 7071840288*T2^16 + 1048845032*T2^12 + 55065312*T2^8 + 886544*T2^4 + 64