Newform orbit 369.2.bc.a

• Label: 369.2.bc.a
• Level: $369$
• Weight: $2$
• Character orbit: 369.bc
• Analytic conductor: $2.946$
• Analytic rank: $0$
• Dimension: $640$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 369 = 3^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	369.bc (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.94647983459\)
Analytic rank:	\(0\)
Dimension:	\(640\)
Relative dimension:	\(40\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 640 q - 10 q^{2} - 18 q^{3} - 82 q^{4} - 10 q^{5} - 34 q^{6} - 8 q^{7} - 40 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−0.544434	+	2.56136i	−1.68625	+	0.395678i	−4.43707	−	1.97551i	0.360626	−	0.809978i	−0.0954210	−	4.53452i	3.25181	+	0.170420i	4.39737	−	6.05246i	2.68688	−	1.33442i	1.87831	+	1.36467i
43.2	−0.531703	+	2.50147i	−0.155710	+	1.72504i	−4.14753	−	1.84660i	−0.528848	+	1.18781i	−4.23233	−	1.30671i	−0.105112	−	0.00550867i	3.81812	−	5.25519i	−2.95151	−	0.537210i	−2.69008	−	1.95446i
43.3	−0.522732	+	2.45926i	1.66102	+	0.490934i	−3.94763	−	1.75760i	1.15686	−	2.59834i	−2.07560	+	3.82825i	2.06387	+	0.108163i	3.43032	−	4.72143i	2.51797	+	1.63090i	5.78527	+	4.20325i
43.4	−0.502963	+	2.36625i	0.709920	−	1.57988i	−3.51909	−	1.56680i	0.634542	−	1.42520i	3.38133	+	2.47447i	−2.89174	−	0.151550i	2.63358	−	3.62481i	−1.99203	−	2.24317i	3.05325	+	2.21831i
43.5	−0.494263	+	2.32532i	0.0362599	−	1.73167i	−3.33575	−	1.48517i	−1.17661	+	2.64270i	4.00878	+	0.940217i	3.83645	+	0.201060i	2.30759	−	3.17612i	−2.99737	−	0.125580i	−5.56358	−	4.04218i
43.6	−0.447214	+	2.10398i	−1.34761	−	1.08809i	−2.39963	−	1.06838i	0.982397	−	2.20650i	2.89199	−	2.34874i	−1.22441	−	0.0641684i	0.792372	−	1.09061i	0.632126	+	2.93265i	4.20309	+	3.05372i
43.7	−0.446295	+	2.09965i	0.951466	+	1.44731i	−2.38228	−	1.06066i	−0.370613	+	0.832411i	−3.46349	+	1.35182i	−3.96846	−	0.207978i	0.766781	−	1.05538i	−1.18943	+	2.75414i	−1.58237	−	1.14966i
43.8	−0.391804	+	1.84329i	−1.61137	+	0.635210i	−1.41713	−	0.630948i	−1.57381	+	3.53483i	−0.539537	−	3.21910i	−1.42264	−	0.0745575i	−0.497071	+	0.684160i	2.19302	−	2.04711i	−5.89911	−	4.28595i
43.9	−0.366477	+	1.72414i	−1.39026	+	1.03304i	−1.01125	−	0.450239i	1.48061	−	3.32551i	−1.27161	−	2.77558i	−2.93951	−	0.154053i	−0.925252	+	1.27350i	0.865638	−	2.87240i	5.19102	+	3.77150i
43.10	−0.339857	+	1.59890i	1.63587	−	0.569138i	−0.613896	−	0.273324i	−0.991988	+	2.22804i	0.354033	+	2.80903i	−0.783154	−	0.0410434i	−1.27596	+	1.75621i	2.35216	−	1.86208i	−3.22529	−	2.34331i
43.11	−0.303734	+	1.42895i	−0.260060	+	1.71242i	−0.122567	−	0.0545704i	0.886065	−	1.99013i	−2.36798	−	0.891733i	3.14082	+	0.164604i	−1.60216	+	2.20518i	−2.86474	−	0.890663i	2.57468	+	1.87062i
43.12	−0.285873	+	1.34493i	1.39759	+	1.02311i	0.0999818	+	0.0445148i	−0.567491	+	1.27461i	−1.77554	+	1.58718i	2.75902	+	0.144594i	−1.70483	+	2.34650i	0.906504	+	2.85976i	−1.55202	−	1.12761i
43.13	−0.263634	+	1.24030i	−0.941987	−	1.45350i	0.358253	+	0.159505i	0.351098	−	0.788578i	2.05111	−	0.785154i	1.75268	+	0.0918539i	−1.78291	+	2.45397i	−1.22532	+	2.73836i	0.885511	+	0.643361i
43.14	−0.219565	+	1.03297i	−0.318490	−	1.70252i	0.808270	+	0.359865i	−1.15394	+	2.59180i	1.82858	+	0.0448216i	−4.74506	−	0.248678i	−1.79066	+	2.46463i	−2.79713	+	1.08447i	−2.42389	−	1.76106i
43.15	−0.169202	+	0.796033i	1.72252	−	0.181495i	1.22205	+	0.544093i	1.33764	−	3.00438i	−0.146977	+	1.40189i	−2.04189	−	0.107011i	−1.59659	+	2.19752i	2.93412	−	0.625255i	2.16526	+	1.57315i
43.16	−0.168560	+	0.793011i	0.307522	−	1.70453i	1.22664	+	0.546134i	1.31668	−	2.95731i	1.29988	+	0.531184i	4.12745	+	0.216310i	−1.59292	+	2.19247i	−2.81086	−	1.04836i	2.12324	+	1.54263i
43.17	−0.103043	+	0.484779i	−1.71221	−	0.261385i	1.60270	+	0.713567i	0.193805	−	0.435294i	0.303145	−	0.803111i	−3.51466	−	0.184195i	−1.09369	+	1.50534i	2.86336	+	0.895093i	0.191051	+	0.138807i
43.18	−0.0948924	+	0.446434i	0.640329	+	1.60934i	1.63679	+	0.728747i	0.145519	−	0.326841i	−0.779226	+	0.133150i	−3.99497	−	0.209368i	−1.01720	+	1.40005i	−2.17996	+	2.06102i	0.132104	+	0.0959794i
43.19	−0.0904523	+	0.425545i	−1.63543	+	0.570406i	1.65418	+	0.736490i	−0.165542	+	0.371814i	−0.0948049	−	0.747544i	3.34102	+	0.175096i	−0.974468	+	1.34124i	2.34927	−	1.86572i	−0.143250	−	0.104077i
43.20	−0.0412419	+	0.194028i	1.03554	−	1.38840i	1.79114	+	0.797469i	−0.638838	+	1.43485i	0.226681	+	0.258184i	1.70315	+	0.0892585i	−0.461791	+	0.635600i	−0.855315	−	2.87549i	−0.252055	−	0.183128i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
41.g	even	20	1	inner
369.bc	even	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	369.2.bc.a	✓	640
9.c	even	3	1	inner	369.2.bc.a	✓	640
41.g	even	20	1	inner	369.2.bc.a	✓	640
369.bc	even	60	1	inner	369.2.bc.a	✓	640

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
369.2.bc.a	✓	640	1.a	even	1	1	trivial
369.2.bc.a	✓	640	9.c	even	3	1	inner
369.2.bc.a	✓	640	41.g	even	20	1	inner
369.2.bc.a	✓	640	369.bc	even	60	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(369, [\chi])\).