Newform orbit 510.2.q.a

• Label: 510.2.q.a
• Level: $510$
• Weight: $2$
• Character orbit: 510.q
• Analytic conductor: $4.072$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	510.q (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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
203.1	−0.707107	−	0.707107i	−1.72710	+	0.130905i			1.00000i	−1.54054	−	1.62072i	1.31381	+	1.12868i	1.47852	+	1.47852i	0.707107	−	0.707107i	2.96573	−	0.452171i	−0.0566968	+	2.23535i
203.2	−0.707107	−	0.707107i	−1.68647	−	0.394727i			1.00000i	−0.495513	+	2.18047i	0.913403	+	1.47163i	2.60655	+	2.60655i	0.707107	−	0.707107i	2.68838	+	1.33139i	1.89221	−	1.19145i
203.3	−0.707107	−	0.707107i	−1.65234	−	0.519393i			1.00000i	2.09933	−	0.769938i	0.801115	+	1.53565i	−1.71379	−	1.71379i	0.707107	−	0.707107i	2.46046	+	1.71643i	−2.02888	−	0.940024i
203.4	−0.707107	−	0.707107i	−1.36705	+	1.06357i			1.00000i	1.21201	+	1.87910i	1.71871	+	0.214587i	−0.205943	−	0.205943i	0.707107	−	0.707107i	0.737624	−	2.90790i	0.471707	−	2.18575i
203.5	−0.707107	−	0.707107i	−0.995001	−	1.41773i			1.00000i	−1.93239	−	1.12510i	−0.298918	+	1.70606i	−1.98852	−	1.98852i	0.707107	−	0.707107i	−1.01994	+	2.82130i	0.570840	+	2.16198i
203.6	−0.707107	−	0.707107i	−0.971398	+	1.43401i			1.00000i	1.05899	−	1.96940i	1.70088	−	0.327117i	2.89682	+	2.89682i	0.707107	−	0.707107i	−1.11277	−	2.78599i	−2.14140	+	0.643756i
203.7	−0.707107	−	0.707107i	−0.935196	+	1.45788i			1.00000i	−1.28768	+	1.82808i	1.69216	−	0.369592i	−1.53615	−	1.53615i	0.707107	−	0.707107i	−1.25082	−	2.72680i	2.20318	−	0.382123i
203.8	−0.707107	−	0.707107i	−0.484377	−	1.66294i			1.00000i	1.50219	+	1.65633i	−0.833372	+	1.51838i	0.715684	+	0.715684i	0.707107	−	0.707107i	−2.53076	+	1.61098i	0.108998	−	2.23341i
203.9	−0.707107	−	0.707107i	−0.176206	+	1.72306i			1.00000i	2.05785	−	0.874781i	1.34299	−	1.09379i	−2.61202	−	2.61202i	0.707107	−	0.707107i	−2.93790	−	0.607230i	−2.07369	−	0.836558i
203.10	−0.707107	−	0.707107i	0.176206	−	1.72306i			1.00000i	−2.05785	+	0.874781i	−1.34299	+	1.09379i	2.61202	+	2.61202i	0.707107	−	0.707107i	−2.93790	−	0.607230i	2.07369	+	0.836558i
203.11	−0.707107	−	0.707107i	0.484377	+	1.66294i			1.00000i	−1.50219	−	1.65633i	0.833372	−	1.51838i	−0.715684	−	0.715684i	0.707107	−	0.707107i	−2.53076	+	1.61098i	−0.108998	+	2.23341i
203.12	−0.707107	−	0.707107i	0.935196	−	1.45788i			1.00000i	1.28768	−	1.82808i	−1.69216	+	0.369592i	1.53615	+	1.53615i	0.707107	−	0.707107i	−1.25082	−	2.72680i	−2.20318	+	0.382123i
203.13	−0.707107	−	0.707107i	0.971398	−	1.43401i			1.00000i	−1.05899	+	1.96940i	−1.70088	+	0.327117i	−2.89682	−	2.89682i	0.707107	−	0.707107i	−1.11277	−	2.78599i	2.14140	−	0.643756i
203.14	−0.707107	−	0.707107i	0.995001	+	1.41773i			1.00000i	1.93239	+	1.12510i	0.298918	−	1.70606i	1.98852	+	1.98852i	0.707107	−	0.707107i	−1.01994	+	2.82130i	−0.570840	−	2.16198i
203.15	−0.707107	−	0.707107i	1.36705	−	1.06357i			1.00000i	−1.21201	−	1.87910i	−1.71871	−	0.214587i	0.205943	+	0.205943i	0.707107	−	0.707107i	0.737624	−	2.90790i	−0.471707	+	2.18575i
203.16	−0.707107	−	0.707107i	1.65234	+	0.519393i			1.00000i	−2.09933	+	0.769938i	−0.801115	−	1.53565i	1.71379	+	1.71379i	0.707107	−	0.707107i	2.46046	+	1.71643i	2.02888	+	0.940024i
203.17	−0.707107	−	0.707107i	1.68647	+	0.394727i			1.00000i	0.495513	−	2.18047i	−0.913403	−	1.47163i	−2.60655	−	2.60655i	0.707107	−	0.707107i	2.68838	+	1.33139i	−1.89221	+	1.19145i
203.18	−0.707107	−	0.707107i	1.72710	−	0.130905i			1.00000i	1.54054	+	1.62072i	−1.31381	−	1.12868i	−1.47852	−	1.47852i	0.707107	−	0.707107i	2.96573	−	0.452171i	0.0566968	−	2.23535i
203.19	0.707107	+	0.707107i	−1.72306	+	0.176206i			1.00000i	−2.05785	+	0.874781i	−1.34299	−	1.09379i	−2.61202	−	2.61202i	−0.707107	+	0.707107i	2.93790	−	0.607230i	−2.07369	−	0.836558i
203.20	0.707107	+	0.707107i	−1.66294	−	0.484377i			1.00000i	1.50219	+	1.65633i	−0.833372	−	1.51838i	−0.715684	−	0.715684i	−0.707107	+	0.707107i	2.53076	+	1.61098i	−0.108998	+	2.23341i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
5.c	odd	4	1	inner
15.e	even	4	1	inner
17.b	even	2	1	inner
51.c	odd	2	1	inner
85.g	odd	4	1	inner
255.o	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.q.a	✓	72
3.b	odd	2	1	inner	510.2.q.a	✓	72
5.c	odd	4	1	inner	510.2.q.a	✓	72
15.e	even	4	1	inner	510.2.q.a	✓	72
17.b	even	2	1	inner	510.2.q.a	✓	72
51.c	odd	2	1	inner	510.2.q.a	✓	72
85.g	odd	4	1	inner	510.2.q.a	✓	72
255.o	even	4	1	inner	510.2.q.a	✓	72

    

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

Hecke kernels

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