Newform orbit 510.2.t.a

• Label: 510.2.t.a
• Level: $510$
• Weight: $2$
• Character orbit: 510.t
• Analytic conductor: $4.072$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.t (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.07237050309\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(36\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 16 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−0.707107	−	0.707107i	−1.73205		0.000693695i			1.00000i	−1.64316	+	1.51658i	1.22524	+	1.22425i	0.207712			0.707107	−	0.707107i	3.00000	−	0.00240303i	2.23428	+	0.0895028i
47.2	−0.707107	−	0.707107i	−1.56719	−	0.737514i			1.00000i	−1.14662	−	1.91970i	0.586667	+	1.62967i	4.16512			0.707107	−	0.707107i	1.91215	+	2.31164i	−0.546648	+	2.16822i
47.3	−0.707107	−	0.707107i	−1.50604	+	0.855483i			1.00000i	−1.20450	−	1.88393i	1.66985	+	0.460012i	−2.33679			0.707107	−	0.707107i	1.53630	−	2.57678i	−0.480428	+	2.18385i
47.4	−0.707107	−	0.707107i	−1.38505	−	1.04002i			1.00000i	2.20871	−	0.348733i	0.243968	+	1.71478i	0.835259			0.707107	−	0.707107i	0.836706	+	2.88096i	−1.80838	−	1.31520i
47.5	−0.707107	−	0.707107i	−0.853413	−	1.50721i			1.00000i	0.531042	+	2.17209i	−0.462306	+	1.66921i	−4.54722			0.707107	−	0.707107i	−1.54337	+	2.57255i	1.16040	−	1.91141i
47.6	−0.707107	−	0.707107i	−0.751757	+	1.56040i			1.00000i	−2.18634	+	0.468963i	1.63494	−	0.571800i	−0.425589			0.707107	−	0.707107i	−1.86972	−	2.34609i	1.87758	+	1.21437i
47.7	−0.707107	−	0.707107i	−0.666742	+	1.59858i			1.00000i	1.95907	−	1.07797i	1.60182	−	0.658908i	4.69884			0.707107	−	0.707107i	−2.11091	−	2.13168i	−2.14752	−	0.623031i
47.8	−0.707107	−	0.707107i	−0.535979	+	1.64704i			1.00000i	0.292938	+	2.21680i	1.54362	−	0.785636i	−2.55527			0.707107	−	0.707107i	−2.42545	−	1.76555i	1.36037	−	1.77465i
47.9	−0.707107	−	0.707107i	−0.352453	−	1.69581i			1.00000i	−0.660502	+	2.13629i	−0.949898	+	1.44834i	2.44675			0.707107	−	0.707107i	−2.75155	+	1.19539i	1.97763	−	1.04354i
47.10	−0.707107	−	0.707107i	−0.0883062	−	1.72980i			1.00000i	1.21459	−	1.87744i	−1.16071	+	1.28559i	−0.280829			0.707107	−	0.707107i	−2.98440	+	0.305504i	−2.18639	+	0.468707i
47.11	−0.707107	−	0.707107i	0.196898	+	1.72082i			1.00000i	1.74798	−	1.39448i	1.07758	−	1.35603i	−4.23041			0.707107	−	0.707107i	−2.92246	+	0.677654i	−2.22205	−	0.249959i
47.12	−0.707107	−	0.707107i	0.609385	−	1.62131i			1.00000i	−1.84595	−	1.26193i	−1.57734	+	0.715540i	−3.28181			0.707107	−	0.707107i	−2.25730	−	1.97601i	0.412964	+	2.19760i
47.13	−0.707107	−	0.707107i	1.04605	+	1.38050i			1.00000i	−2.22214	+	0.249214i	0.236491	−	1.71583i	3.45755			0.707107	−	0.707107i	−0.811555	+	2.88814i	1.74751	+	1.39507i
47.14	−0.707107	−	0.707107i	1.19239	+	1.25627i			1.00000i	1.33852	+	1.79119i	0.0451679	−	1.73146i	1.13144			0.707107	−	0.707107i	−0.156413	+	2.99592i	0.320086	−	2.21304i
47.15	−0.707107	−	0.707107i	1.40742	−	1.00954i			1.00000i	−1.68334	+	1.47186i	−1.70905	−	0.281345i	1.15184			0.707107	−	0.707107i	0.961664	−	2.84169i	2.23106	+	0.149539i
47.16	−0.707107	−	0.707107i	1.61310	−	0.630790i			1.00000i	2.07976	+	0.821352i	−1.58667	−	0.694600i	0.773307			0.707107	−	0.707107i	2.20421	−	2.03506i	−0.889826	−	2.05139i
47.17	−0.707107	−	0.707107i	1.68130	−	0.416190i			1.00000i	0.443485	−	2.19165i	−1.48315	−	0.894571i	2.58566			0.707107	−	0.707107i	2.65357	−	1.39949i	−1.86332	+	1.23614i
47.18	−0.707107	−	0.707107i	1.69242	+	0.368405i			1.00000i	−2.05196	−	0.888509i	−0.936219	−	1.45722i	−3.79557			0.707107	−	0.707107i	2.72856	+	1.24699i	0.822686	+	2.07923i
47.19	0.707107	+	0.707107i	−1.73205		0.000693695i			1.00000i	1.64316	−	1.51658i	−1.22425	−	1.22524i	0.207712			−0.707107	+	0.707107i	3.00000	+	0.00240303i	2.23428	+	0.0895028i
47.20	0.707107	+	0.707107i	−1.56719	+	0.737514i			1.00000i	1.14662	+	1.91970i	−1.62967	−	0.586667i	4.16512			−0.707107	+	0.707107i	1.91215	−	2.31164i	−0.546648	+	2.16822i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
85.i	odd	4	1	inner
255.r	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	510.2.t.a	yes	72
3.b	odd	2	1	inner	510.2.t.a	yes	72
5.c	odd	4	1		510.2.i.a	✓	72
15.e	even	4	1		510.2.i.a	✓	72
17.c	even	4	1		510.2.i.a	✓	72
51.f	odd	4	1		510.2.i.a	✓	72
85.i	odd	4	1	inner	510.2.t.a	yes	72
255.r	even	4	1	inner	510.2.t.a	yes	72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
510.2.i.a	✓	72	5.c	odd	4	1
510.2.i.a	✓	72	15.e	even	4	1
510.2.i.a	✓	72	17.c	even	4	1
510.2.i.a	✓	72	51.f	odd	4	1
510.2.t.a	yes	72	1.a	even	1	1	trivial
510.2.t.a	yes	72	3.b	odd	2	1	inner
510.2.t.a	yes	72	85.i	odd	4	1	inner
510.2.t.a	yes	72	255.r	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(510, [\chi])\).