Newform orbit 240.2.bk.b

• Label: 240.2.bk.b
• Level: $240$
• Weight: $2$
• Character orbit: 240.bk
• Analytic conductor: $1.916$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	240.bk (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.91640964851\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 4 q^{3} + 8 q^{4} + 8 q^{6} + 16 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.39972	+	0.201959i	1.67583	−	0.437717i	1.91842	−	0.565372i	0.707107	−	0.707107i	−2.25729	+	0.951130i	2.03158			−2.57107	+	1.17881i	2.61681	−	1.46708i	−0.846944	+	1.13256i
11.2	−1.38304	−	0.295312i	−1.72570	+	0.148181i	1.82558	+	0.816854i	−0.707107	+	0.707107i	2.43047	+	0.304680i	−1.12315			−2.28362	−	1.66885i	2.95608	−	0.511431i	1.18677	−	0.769138i
11.3	−1.31930	+	0.509349i	0.956016	+	1.44431i	1.48113	−	1.34397i	−0.707107	+	0.707107i	−1.99693	−	1.41854i	−2.89523			−1.26951	+	2.52752i	−1.17207	+	2.76157i	0.572725	−	1.29305i
11.4	−1.24180	−	0.676700i	−0.861442	+	1.50264i	1.08415	+	1.68066i	0.707107	−	0.707107i	2.08658	−	1.28304i	0.563322			−0.209007	−	2.82069i	−1.51583	−	2.58887i	−1.35659	+	0.399589i
11.5	−1.23685	−	0.685720i	1.73074	+	0.0673020i	1.05958	+	1.69626i	−0.707107	+	0.707107i	−2.09451	−	1.27005i	0.271924			−0.147371	−	2.82459i	2.99094	+	0.232965i	1.35946	−	0.389705i
11.6	−1.20816	+	0.735082i	−1.18902	+	1.25946i	0.919310	−	1.77620i	0.707107	−	0.707107i	0.510724	−	2.39565i	−2.92811			0.194974	+	2.82170i	−0.172464	−	2.99504i	−0.334518	+	1.37408i
11.7	−1.11727	+	0.867014i	0.773899	−	1.54954i	0.496574	−	1.93737i	−0.707107	+	0.707107i	0.478822	+	2.40223i	1.63342			1.12492	+	2.59510i	−1.80216	−	2.39838i	0.176956	−	1.40310i
11.8	−1.10098	−	0.887605i	0.807747	−	1.53217i	0.424316	+	1.95447i	0.707107	−	0.707107i	−2.24928	+	0.969928i	−5.10839			1.26763	−	2.52846i	−1.69509	−	2.47521i	−1.40614	+	0.150879i
11.9	−0.983055	−	1.01666i	−1.40626	−	1.01116i	−0.0672039	+	1.99887i	0.707107	−	0.707107i	0.354421	+	2.42371i	4.53819			2.09824	−	1.89668i	0.955115	+	2.84390i	−1.41401	−	0.0237635i
11.10	−0.932052	+	1.06362i	−1.67834	−	0.427993i	−0.262558	−	1.98269i	−0.707107	+	0.707107i	2.01952	−	1.38620i	2.37757			2.35354	+	1.56871i	2.63364	+	1.43663i	−0.0930299	−	1.41115i
11.11	−0.774214	+	1.18347i	0.764773	+	1.55407i	−0.801186	−	1.83251i	0.707107	−	0.707107i	−2.43128	−	0.298096i	3.70081			2.78901	+	0.470579i	−1.83024	+	2.37702i	0.289385	+	1.38429i
11.12	−0.772787	−	1.18440i	−0.112911	+	1.72837i	−0.805601	+	1.83058i	−0.707107	+	0.707107i	2.13433	−	1.20193i	−2.33502			2.79069	−	0.460493i	−2.97450	−	0.390304i	1.38394	+	0.291053i
11.13	−0.468379	−	1.33440i	1.36262	+	1.06924i	−1.56124	+	1.25001i	0.707107	−	0.707107i	0.788576	−	2.31908i	1.05330			2.39927	+	1.49784i	0.713439	+	2.91393i	−1.27476	−	0.612368i
11.14	−0.371549	+	1.36453i	−1.65322	−	0.516584i	−1.72390	−	1.01398i	0.707107	−	0.707107i	1.31915	−	2.06394i	−1.29120			2.02413	−	1.97558i	2.46628	+	1.70806i	0.702146	+	1.22760i
11.15	−0.0794625	−	1.41198i	−1.53859	+	0.795457i	−1.98737	+	0.224399i	−0.707107	+	0.707107i	1.24543	+	2.10924i	3.51098			0.474768	+	2.78830i	1.73450	−	2.44776i	1.05461	+	0.942232i
11.16	0.0794625	+	1.41198i	0.795457	−	1.53859i	−1.98737	+	0.224399i	0.707107	−	0.707107i	2.23566	+	1.00091i	3.51098			−0.474768	−	2.78830i	−1.73450	−	2.44776i	1.05461	+	0.942232i
11.17	0.371549	−	1.36453i	−0.516584	−	1.65322i	−1.72390	−	1.01398i	−0.707107	+	0.707107i	−2.44781	+	0.0906432i	−1.29120			−2.02413	+	1.97558i	−2.46628	+	1.70806i	0.702146	+	1.22760i
11.18	0.468379	+	1.33440i	1.06924	+	1.36262i	−1.56124	+	1.25001i	−0.707107	+	0.707107i	−1.31746	+	2.06502i	1.05330			−2.39927	−	1.49784i	−0.713439	+	2.91393i	−1.27476	−	0.612368i
11.19	0.772787	+	1.18440i	1.72837	−	0.112911i	−0.805601	+	1.83058i	0.707107	−	0.707107i	1.46939	+	1.95982i	−2.33502			−2.79069	+	0.460493i	2.97450	−	0.390304i	1.38394	+	0.291053i
11.20	0.774214	−	1.18347i	1.55407	+	0.764773i	−0.801186	−	1.83251i	−0.707107	+	0.707107i	2.10826	−	1.24709i	3.70081			−2.78901	−	0.470579i	1.83024	+	2.37702i	0.289385	+	1.38429i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
16.f	odd	4	1	inner
48.k	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	240.2.bk.b	✓	60
3.b	odd	2	1	inner	240.2.bk.b	✓	60
4.b	odd	2	1		960.2.bk.b		60
12.b	even	2	1		960.2.bk.b		60
16.e	even	4	1		960.2.bk.b		60
16.f	odd	4	1	inner	240.2.bk.b	✓	60
48.i	odd	4	1		960.2.bk.b		60
48.k	even	4	1	inner	240.2.bk.b	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
240.2.bk.b	✓	60	1.a	even	1	1	trivial
240.2.bk.b	✓	60	3.b	odd	2	1	inner
240.2.bk.b	✓	60	16.f	odd	4	1	inner
240.2.bk.b	✓	60	48.k	even	4	1	inner
960.2.bk.b		60	4.b	odd	2	1
960.2.bk.b		60	12.b	even	2	1
960.2.bk.b		60	16.e	even	4	1
960.2.bk.b		60	48.i	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T7^15 - 4*T7^14 - 48*T7^13 + 208*T7^12 + 732*T7^11 - 3608*T7^10 - 4272*T7^9 + 27920*T7^8 + 4580*T7^7 - 100288*T7^6 + 36512*T7^5 + 149952*T7^4 - 96176*T7^3 - 68096*T7^2 + 63040*T7 - 11008