Newform orbit 570.2.n.a

• Label: 570.2.n.a
• Level: $570$
• Weight: $2$
• Character orbit: 570.n
• Analytic conductor: $4.551$
• Analytic rank: $0$
• Dimension: $80$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.n (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.55147291521\)
Analytic rank:	\(0\)
Dimension:	\(80\)
Relative dimension:	\(40\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 40 q^{4}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
179.1	−0.866025	−	0.500000i	−1.69438	−	0.359253i	0.500000	+	0.866025i	−0.161263	−	2.23025i	1.28775	+	1.15831i			1.18214i		−	1.00000i	2.74187	+	1.21743i	−0.975465	+	2.01208i
179.2	−0.866025	−	0.500000i	−1.68343	+	0.407506i	0.500000	+	0.866025i	2.16632	+	0.554116i	1.66165	+	0.488805i			1.83012i		−	1.00000i	2.66788	−	1.37202i	−1.59903	−	1.56304i
179.3	−0.866025	−	0.500000i	−1.66274	−	0.485072i	0.500000	+	0.866025i	−1.14569	+	1.92026i	1.19744	+	1.25145i		−	4.59876i		−	1.00000i	2.52941	+	1.61310i	1.95233	−	1.09015i
179.4	−0.866025	−	0.500000i	−1.53736	+	0.797834i	0.500000	+	0.866025i	−2.22475	+	0.224660i	1.73031	+	0.0777329i		−	0.0202833i		−	1.00000i	1.72692	−	2.45311i	2.03902	+	0.917815i
179.5	−0.866025	−	0.500000i	−1.25145	−	1.19744i	0.500000	+	0.866025i	−1.09015	+	1.95233i	0.485072	+	1.66274i			4.59876i		−	1.00000i	0.132279	+	2.99708i	1.92026	−	1.14569i
179.6	−0.866025	−	0.500000i	−1.15831	−	1.28775i	0.500000	+	0.866025i	2.01208	−	0.975465i	0.359253	+	1.69438i		−	1.18214i		−	1.00000i	−0.316615	+	2.98325i	−2.23025	−	0.161263i
179.7	−0.866025	−	0.500000i	−0.802721	+	1.53481i	0.500000	+	0.866025i	0.971414	+	2.01404i	1.46258	−	0.927823i		−	3.47395i		−	1.00000i	−1.71128	−	2.46405i	0.165751	−	2.22992i
179.8	−0.866025	−	0.500000i	−0.540671	+	1.64550i	0.500000	+	0.866025i	−0.636160	−	2.14367i	1.29099	−	1.15471i			4.14066i		−	1.00000i	−2.41535	−	1.77935i	−0.520902	+	2.17455i
179.9	−0.866025	−	0.500000i	−0.488805	−	1.66165i	0.500000	+	0.866025i	−1.56304	−	1.59903i	−0.407506	+	1.68343i		−	1.83012i		−	1.00000i	−2.52214	+	1.62444i	0.554116	+	2.16632i
179.10	−0.866025	−	0.500000i	−0.105626	+	1.72883i	0.500000	+	0.866025i	−0.264142	+	2.22041i	0.955888	−	1.44440i			2.53373i		−	1.00000i	−2.97769	−	0.365218i	1.33896	−	1.79086i
179.11	−0.866025	−	0.500000i	−0.0777329	−	1.73031i	0.500000	+	0.866025i	0.917815	+	2.03902i	−0.797834	+	1.53736i			0.0202833i		−	1.00000i	−2.98792	+	0.269003i	0.224660	−	2.22475i
179.12	−0.866025	−	0.500000i	0.457417	+	1.67056i	0.500000	+	0.866025i	−1.92017	−	1.14583i	0.439145	−	1.67546i		−	4.60944i		−	1.00000i	−2.58154	+	1.52829i	1.09000	+	1.95241i
179.13	−0.866025	−	0.500000i	0.813860	+	1.52893i	0.500000	+	0.866025i	0.951780	−	2.02339i	0.0596429	−	1.73102i		−	1.16784i		−	1.00000i	−1.67527	+	2.48867i	−1.83596	+	1.27642i
179.14	−0.866025	−	0.500000i	0.927823	−	1.46258i	0.500000	+	0.866025i	−2.22992	+	0.165751i	−1.53481	+	0.802721i			3.47395i		−	1.00000i	−1.27829	−	2.71403i	2.01404	+	0.971414i
179.15	−0.866025	−	0.500000i	1.08728	+	1.34827i	0.500000	+	0.866025i	2.16634	+	0.554035i	−0.267475	−	1.71127i			1.36390i		−	1.00000i	−0.635658	+	2.93188i	−1.59909	−	1.56298i
179.16	−0.866025	−	0.500000i	1.15471	−	1.29099i	0.500000	+	0.866025i	2.17455	−	0.520902i	−1.64550	+	0.540671i		−	4.14066i		−	1.00000i	−0.333286	−	2.98143i	−2.14367	−	0.636160i
179.17	−0.866025	−	0.500000i	1.44440	−	0.955888i	0.500000	+	0.866025i	−1.79086	+	1.33896i	−1.72883	+	0.105626i		−	2.53373i		−	1.00000i	1.17255	−	2.76136i	2.22041	−	0.264142i
179.18	−0.866025	−	0.500000i	1.67546	−	0.439145i	0.500000	+	0.866025i	1.95241	+	1.09000i	−1.67056	−	0.457417i			4.60944i		−	1.00000i	2.61430	−	1.47153i	−1.14583	−	1.92017i
179.19	−0.866025	−	0.500000i	1.71127	+	0.267475i	0.500000	+	0.866025i	−1.56298	−	1.59909i	−1.34827	−	1.08728i		−	1.36390i		−	1.00000i	2.85691	+	0.915446i	0.554035	+	2.16634i
179.20	−0.866025	−	0.500000i	1.73102	−	0.0596429i	0.500000	+	0.866025i	1.27642	−	1.83596i	−1.52893	−	0.813860i			1.16784i		−	1.00000i	2.99289	−	0.206487i	−2.02339	+	0.951780i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.b	even	2	1	inner
15.d	odd	2	1	inner
19.d	odd	6	1	inner
57.f	even	6	1	inner
95.h	odd	6	1	inner
285.q	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	570.2.n.a	✓	80
3.b	odd	2	1	inner	570.2.n.a	✓	80
5.b	even	2	1	inner	570.2.n.a	✓	80
15.d	odd	2	1	inner	570.2.n.a	✓	80
19.d	odd	6	1	inner	570.2.n.a	✓	80
57.f	even	6	1	inner	570.2.n.a	✓	80
95.h	odd	6	1	inner	570.2.n.a	✓	80
285.q	even	6	1	inner	570.2.n.a	✓	80

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
570.2.n.a	✓	80	1.a	even	1	1	trivial
570.2.n.a	✓	80	3.b	odd	2	1	inner
570.2.n.a	✓	80	5.b	even	2	1	inner
570.2.n.a	✓	80	15.d	odd	2	1	inner
570.2.n.a	✓	80	19.d	odd	6	1	inner
570.2.n.a	✓	80	57.f	even	6	1	inner
570.2.n.a	✓	80	95.h	odd	6	1	inner
570.2.n.a	✓	80	285.q	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(570, [\chi])\).