Newform orbit 215.2.i.a

• Label: 215.2.i.a
• Level: $215$
• Weight: $2$
• Character orbit: 215.i
• Analytic conductor: $1.717$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 215 = 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	215.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.71678364346\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 44 q^{4} + q^{5} + 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
49.1		−	2.71163i	−1.88829	−	1.09021i	−5.35294			−0.724925	−	2.11530i	−2.95624	+	5.12036i	2.45772	−	1.41896i			9.09192i	0.877105	+	1.51919i	−5.73590	+	1.96573i
49.2		−	2.34656i	2.87647	+	1.66073i	−3.50635			0.270428	−	2.21966i	3.89700	−	6.74981i	−0.208478	+	0.120365i			3.53476i	4.01604	+	6.95599i	−5.20856	−	0.634576i
49.3		−	2.28980i	0.620444	+	0.358213i	−3.24317			1.86429	+	1.23468i	0.820236	−	1.42069i	2.52509	−	1.45786i			2.84661i	−1.24337	−	2.15357i	2.82716	−	4.26884i
49.4		−	2.12960i	−2.33760	−	1.34961i	−2.53518			0.894650	+	2.04929i	−2.87413	+	4.97814i	−3.45146	+	1.99270i			1.13971i	2.14291	+	3.71162i	4.36416	−	1.90524i
49.5		−	1.81467i	−0.618650	−	0.357178i	−1.29304			−2.19938	−	0.403373i	−0.648162	+	1.12265i	−0.919625	+	0.530946i		−	1.28289i	−1.24485	−	2.15614i	−0.731990	+	3.99117i
49.6		−	1.60368i	0.146007	+	0.0842970i	−0.571779			0.990616	−	2.00466i	0.135185	−	0.234147i	−3.49164	+	2.01590i		−	2.29041i	−1.48579	−	2.57346i	−3.21483	−	1.58863i
49.7		−	1.25369i	2.33661	+	1.34904i	0.428263			−0.991811	+	2.00407i	1.69128	−	2.92938i	−0.568140	+	0.328016i		−	3.04429i	2.13983	+	3.70630i	2.51249	+	1.24342i
49.8		−	0.886389i	−0.968447	−	0.559133i	1.21432			−0.908261	+	2.04330i	−0.495609	+	0.858421i	2.82800	−	1.63275i		−	2.84913i	−0.874740	−	1.51509i	1.81116	+	0.805072i
49.9		−	0.276625i	1.61077	+	0.929976i	1.92348			−1.85606	−	1.24701i	0.257254	−	0.445578i	1.17247	−	0.676925i		−	1.08533i	0.229710	+	0.397870i	−0.344955	+	0.513432i
49.10		−	0.252186i	1.18942	+	0.686713i	1.93640			2.22980	−	0.167330i	0.173179	−	0.299955i	−1.28266	+	0.740544i		−	0.992705i	−0.556849	−	0.964491i	−0.0421983	−	0.562323i
49.11			0.252186i	−1.18942	−	0.686713i	1.93640			−0.969987	−	2.01473i	0.173179	−	0.299955i	1.28266	−	0.740544i			0.992705i	−0.556849	−	0.964491i	0.508086	−	0.244617i
49.12			0.276625i	−1.61077	−	0.929976i	1.92348			2.00797	+	0.983889i	0.257254	−	0.445578i	−1.17247	+	0.676925i			1.08533i	0.229710	+	0.397870i	−0.272168	+	0.555456i
49.13			0.886389i	0.968447	+	0.559133i	1.21432			−1.31542	+	1.80823i	−0.495609	+	0.858421i	−2.82800	+	1.63275i			2.84913i	−0.874740	−	1.51509i	−1.60279	−	1.16597i
49.14			1.25369i	−2.33661	−	1.34904i	0.428263			−1.23967	+	1.86097i	1.69128	−	2.92938i	0.568140	−	0.328016i			3.04429i	2.13983	+	3.70630i	−2.33308	−	1.55416i
49.15			1.60368i	−0.146007	−	0.0842970i	−0.571779			1.24078	−	1.86023i	0.135185	−	0.234147i	3.49164	−	2.01590i			2.29041i	−1.48579	−	2.57346i	2.98321	+	1.98981i
49.16			1.81467i	0.618650	+	0.357178i	−1.29304			1.44902	+	1.70304i	−0.648162	+	1.12265i	0.919625	−	0.530946i			1.28289i	−1.24485	−	2.15614i	−3.09046	+	2.62951i
49.17			2.12960i	2.33760	+	1.34961i	−2.53518			−2.22206	+	0.249856i	−2.87413	+	4.97814i	3.45146	−	1.99270i		−	1.13971i	2.14291	+	3.71162i	−0.532093	−	4.73210i
49.18			2.28980i	−0.620444	−	0.358213i	−3.24317			−2.00141	−	0.997182i	0.820236	−	1.42069i	−2.52509	+	1.45786i		−	2.84661i	−1.24337	−	2.15357i	2.28334	−	4.58281i
49.19			2.34656i	−2.87647	−	1.66073i	−3.50635			1.78706	−	1.34403i	3.89700	−	6.74981i	0.208478	−	0.120365i		−	3.53476i	4.01604	+	6.95599i	3.15384	+	4.19346i
49.20			2.71163i	1.88829	+	1.09021i	−5.35294			2.19436	−	0.429846i	−2.95624	+	5.12036i	−2.45772	+	1.41896i		−	9.09192i	0.877105	+	1.51919i	1.16558	+	5.95030i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
43.c	even	3	1	inner
215.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	215.2.i.a	✓	40
5.b	even	2	1	inner	215.2.i.a	✓	40
43.c	even	3	1	inner	215.2.i.a	✓	40
215.i	even	6	1	inner	215.2.i.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
215.2.i.a	✓	40	1.a	even	1	1	trivial
215.2.i.a	✓	40	5.b	even	2	1	inner
215.2.i.a	✓	40	43.c	even	3	1	inner
215.2.i.a	✓	40	215.i	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(215, [\chi])\).