Newform orbit 417.2.p.a

• Label: 417.2.p.a
• Level: $417$
• Weight: $2$
• Character orbit: 417.p
• Analytic conductor: $3.330$
• Analytic rank: $0$
• Dimension: $44$
• CM discriminant: -3
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 417 = 3 \cdot 139 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	417.p (of order \(138\), degree \(44\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.32976176429\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{138}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 3 q^{3} - 2 q^{4} - q^{7} - 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1		0		−1.03079	−	1.39193i	1.99793	+	0.0910292i		0			0		1.87795	+	4.06749i		0		−0.874938	+	2.86958i		0
17.1		0		−1.36811	+	1.06221i	0.317365	−	1.97466i		0			0		−1.76459	+	1.57436i		0		0.743423	−	2.90643i		0
26.1		0		−0.966370	−	1.43740i	−1.96692	+	0.362232i		0			0		−0.269479	+	1.67671i		0		−1.13226	+	2.77813i		0
32.1		0		−0.0788336	−	1.73026i	1.94840	+	0.451381i		0			0		3.04437	−	2.04674i		0		−2.98757	+	0.272805i		0
50.1		0		−0.542728	−	1.64482i	−0.0455264	+	1.99948i		0			0		−0.621319	+	0.971204i		0		−2.41089	+	1.78538i		0
53.1		0		−1.72084	+	0.196727i	−1.07780	+	1.68474i		0			0		3.10208	+	0.283260i		0		2.92260	−	0.677071i		0
56.1		0		1.21073	−	1.23861i	−1.49237	+	1.33148i		0			0		−2.85631	−	2.11524i		0		−0.0682896	−	2.99922i		0
68.1		0		−0.617029	−	1.61842i	0.495616	−	1.93762i		0			0		−0.120438	−	5.28956i		0		−2.23855	+	1.99722i		0
92.1		0		−0.617029	+	1.61842i	0.495616	+	1.93762i		0			0		−0.120438	+	5.28956i		0		−2.23855	−	1.99722i		0
98.1		0		1.73160	+	0.0394270i	−0.227160	−	1.98706i		0			0		1.21136	−	3.97295i		0		2.99689	+	0.136544i		0
101.1		0		1.67803	−	0.429216i	1.89928	−	0.626688i		0			0		0.209603	+	1.83348i		0		2.63155	−	1.44047i		0
104.1		0		1.65675	−	0.505146i	−1.99171	+	0.181870i		0			0		−3.40654	−	3.99775i		0		2.48966	−	1.67380i		0
110.1		0		0.390907	+	1.68736i	0.838353	+	1.81581i		0			0		−5.03119	−	0.926554i		0		−2.69438	+	1.31920i		0
119.1		0		0.761643	−	1.55560i	1.29716	+	1.52229i		0			0		0.264567	−	0.100867i		0		−1.83980	−	2.36963i		0
128.1		0		1.67803	+	0.429216i	1.89928	+	0.626688i		0			0		0.209603	−	1.83348i		0		2.63155	+	1.44047i		0
134.1		0		−1.71011	−	0.274846i	1.43022	+	1.39802i		0			0		−4.28417	−	2.34509i		0		2.84892	+	0.940032i		0
158.1		0		−1.60395	−	0.653709i	−1.86879	+	0.712483i		0			0		−0.416880	−	0.137554i		0		2.14533	+	2.09704i		0
161.1		0		−1.60395	+	0.653709i	−1.86879	−	0.712483i		0			0		−0.416880	+	0.137554i		0		2.14533	−	2.09704i		0
179.1		0		0.313702	+	1.70341i	−1.22653	−	1.57975i		0			0		1.79029	+	1.74999i		0		−2.80318	+	1.06873i		0
197.1		0		−1.31834	−	1.12338i	−0.754838	−	1.85209i		0			0		−1.00485	−	3.92846i		0		0.476048	+	2.96199i		0

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by \(\Q(\sqrt{-3}) \)
139.h	odd	138	1	inner
417.p	even	138	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	417.2.p.a	✓	44
3.b	odd	2	1	CM	417.2.p.a	✓	44
139.h	odd	138	1	inner	417.2.p.a	✓	44
417.p	even	138	1	inner	417.2.p.a	✓	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
417.2.p.a	✓	44	1.a	even	1	1	trivial
417.2.p.a	✓	44	3.b	odd	2	1	CM
417.2.p.a	✓	44	139.h	odd	138	1	inner
417.2.p.a	✓	44	417.p	even	138	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(417, [\chi])\).