Newform orbit 520.2.cg.a

• Label: 520.2.cg.a
• Level: $520$
• Weight: $2$
• Character orbit: 520.cg
• Analytic conductor: $4.152$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.cg (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.15222090511\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(56\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 224 q + 12 q^{6} + 12 q^{8} - 112 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41409	+	0.0188603i	0.880879	+	1.52573i	1.99929	−	0.0533402i	−0.707107	+	0.707107i	−1.27442	−	2.14090i	4.22804	−	1.13290i	−2.82616	+	0.113135i	−0.0518970	+	0.0898883i	0.986575	−	1.01325i
11.2	−1.41089	−	0.0969163i	−0.786544	−	1.36233i	1.98121	+	0.273476i	0.707107	−	0.707107i	0.977694	+	1.99833i	1.51746	−	0.406601i	−2.76877	−	0.577857i	0.262697	−	0.455004i	−1.06618	+	0.929119i
11.3	−1.41088	+	0.0970360i	1.08846	+	1.88527i	1.98117	−	0.273812i	0.707107	−	0.707107i	−1.71863	−	2.55428i	0.957204	−	0.256482i	−2.76862	+	0.578561i	−0.869503	+	1.50602i	−0.929028	+	1.06626i
11.4	−1.40280	+	0.179305i	−0.0859264	−	0.148829i	1.93570	−	0.503057i	−0.707107	+	0.707107i	0.147223	+	0.193370i	−3.67994	+	0.986036i	−2.62520	+	1.05277i	1.48523	−	2.57250i	0.865142	−	1.11872i
11.5	−1.35423	−	0.407517i	0.559775	+	0.969559i	1.66786	+	1.10374i	0.707107	−	0.707107i	−0.362951	−	1.54112i	−3.19335	+	0.855655i	−1.80887	−	2.17440i	0.873304	−	1.51261i	−1.24574	+	0.669425i
11.6	−1.35320	−	0.410921i	−0.818276	−	1.41729i	1.66229	+	1.11211i	−0.707107	+	0.707107i	0.524893	+	2.25413i	−0.00954340	+	0.00255715i	−1.79241	−	2.18798i	0.160850	−	0.278600i	1.24742	−	0.666290i
11.7	−1.27648	+	0.608772i	1.29662	+	2.24580i	1.25879	−	1.55417i	−0.707107	+	0.707107i	−3.02228	−	2.07738i	−1.86140	+	0.498760i	−0.660691	+	2.75018i	−1.86243	+	3.22582i	0.472140	−	1.33307i
11.8	−1.24117	+	0.677854i	−0.887908	−	1.53790i	1.08103	−	1.68267i	−0.707107	+	0.707107i	2.14452	+	1.30693i	4.43967	−	1.18961i	−0.201142	+	2.82127i	−0.0767617	+	0.132955i	0.398328	−	1.35696i
11.9	−1.21426	−	0.724959i	−1.44049	−	2.49500i	0.948870	+	1.76058i	0.707107	−	0.707107i	−0.0596391	+	4.07388i	−5.01827	+	1.34464i	0.124171	−	2.82570i	−2.65001	+	4.58996i	−1.37124	+	0.345991i
11.10	−1.21181	−	0.729050i	1.61112	+	2.79054i	0.936972	+	1.76694i	−0.707107	+	0.707107i	0.0820713	−	4.55619i	−0.337782	+	0.0905083i	0.152756	−	2.82430i	−3.69141	+	6.39371i	1.37240	−	0.341364i
11.11	−1.18499	+	0.771884i	0.172051	+	0.298001i	0.808390	−	1.82935i	0.707107	−	0.707107i	−0.433900	−	0.220324i	3.43079	−	0.919278i	0.454111	+	2.79173i	1.44080	−	2.49553i	−0.292108	+	1.38372i
11.12	−1.07395	+	0.920128i	−0.928695	−	1.60855i	0.306730	−	1.97634i	0.707107	−	0.707107i	2.47744	+	0.872979i	−2.93094	+	0.785344i	1.48907	+	2.40472i	−0.224950	+	0.389625i	−0.108768	+	1.41002i
11.13	−1.01590	−	0.983848i	0.472512	+	0.818414i	0.0640860	+	1.99897i	−0.707107	+	0.707107i	0.325173	−	1.29630i	2.69320	−	0.721641i	1.90158	−	2.09380i	1.05347	−	1.82466i	1.41403	−	0.0226607i
11.14	−0.954491	−	1.04353i	1.24351	+	2.15382i	−0.177892	+	1.99207i	0.707107	−	0.707107i	1.06065	−	3.35344i	0.192908	−	0.0516897i	2.24858	−	1.71578i	−1.59264	+	2.75853i	−1.41281	−	0.0629569i
11.15	−0.927526	−	1.06756i	−1.47141	−	2.54855i	−0.279390	+	1.98039i	0.707107	−	0.707107i	−1.35598	+	3.93467i	3.79646	−	1.01726i	2.37334	−	1.53860i	−2.83007	+	4.90183i	−1.41074	−	0.0990223i
11.16	−0.905772	−	1.08608i	−0.0968629	−	0.167771i	−0.359155	+	1.96749i	0.707107	−	0.707107i	−0.0944781	+	0.257164i	−0.305231	+	0.0817864i	2.46217	−	1.39202i	1.48124	−	2.56557i	−1.40845	−	0.127500i
11.17	−0.869825	−	1.11508i	−0.797142	−	1.38069i	−0.486808	+	1.93985i	−0.707107	+	0.707107i	−0.846206	+	2.08984i	−0.0941914	+	0.0252385i	2.58653	−	1.14450i	0.229130	−	0.396865i	1.40354	+	0.173422i
11.18	−0.809916	+	1.15933i	−1.49420	−	2.58802i	−0.688072	−	1.87791i	0.707107	−	0.707107i	4.21054	+	0.363822i	1.81405	−	0.486074i	2.73439	+	0.723252i	−2.96525	+	5.13596i	0.247070	+	1.39246i
11.19	−0.742802	+	1.20343i	0.132427	+	0.229371i	−0.896491	−	1.78782i	−0.707107	+	0.707107i	−0.374399	−	0.0110100i	−1.86776	+	0.500464i	2.81743	+	0.249132i	1.46493	−	2.53733i	−0.325714	−	1.37619i
11.20	−0.704311	+	1.22635i	1.57294	+	2.72441i	−1.00789	−	1.72747i	0.707107	−	0.707107i	−4.44893	+	0.0101495i	3.30556	−	0.885723i	2.82836	−	0.0193575i	−3.44827	+	5.97257i	0.369141	+	1.36519i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
13.f	odd	12	1	inner
104.u	even	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	520.2.cg.a	✓	224
8.d	odd	2	1	inner	520.2.cg.a	✓	224
13.f	odd	12	1	inner	520.2.cg.a	✓	224
104.u	even	12	1	inner	520.2.cg.a	✓	224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.cg.a	✓	224	1.a	even	1	1	trivial
520.2.cg.a	✓	224	8.d	odd	2	1	inner
520.2.cg.a	✓	224	13.f	odd	12	1	inner
520.2.cg.a	✓	224	104.u	even	12	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(520, [\chi])\).