Newform orbit 507.2.s.a

• Label: 507.2.s.a
• Level: $507$
• Weight: $2$
• Character orbit: 507.s
• Analytic conductor: $4.048$
• Analytic rank: $0$
• Dimension: $1392$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	507.s (of order \(52\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.04841538248\)
Analytic rank:	\(0\)
Dimension:	\(1392\)
Relative dimension:	\(58\) over \(\Q(\zeta_{52})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{52}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 1392 * q - 22 * q^3 - 52 * q^4 - 22 * q^6 - 56 * q^7 - 22 * q^9 - 52 * q^10 - 26 * q^12 - 44 * q^13 - 34 * q^15 + 48 * q^16 - 34 * q^18 - 56 * q^19 - 22 * q^21 - 96 * q^22 - 194 * q^24 - 52 * q^25 - 34 * q^27 - 56 * q^28 + 78 * q^30 - 32 * q^31 - 10 * q^33 - 52 * q^34 - 26 * q^36 - 56 * q^37 + 70 * q^39 - 36 * q^40 - 10 * q^42 - 52 * q^43 - 140 * q^45 - 180 * q^46 - 2 * q^48 - 52 * q^49 - 26 * q^51 - 64 * q^52 - 22 * q^54 + 132 * q^55 - 22 * q^57 - 60 * q^58 - 18 * q^60 - 92 * q^61 + 108 * q^63 - 52 * q^64 - 124 * q^66 - 240 * q^67 - 104 * q^69 - 268 * q^70 - 2 * q^72 - 56 * q^73 + 104 * q^75 - 48 * q^76 - 22 * q^78 - 12 * q^79 + 2 * q^81 + 208 * q^82 - 22 * q^84 - 52 * q^85 - 26 * q^87 - 260 * q^88 - 26 * q^90 - 32 * q^91 - 176 * q^93 - 240 * q^94 + 292 * q^96 - 80 * q^97 - 58 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−0.504808	+	2.75465i	1.69376	−	0.362196i	−5.46324	−	2.07193i	2.58865	+	0.156585i	0.142700	+	4.84855i	0.0295242	+	0.0178480i	5.56770	−	9.21011i	2.73763	−	1.22694i	−1.73811	+	7.05179i
5.2	−0.493960	+	2.69545i	−1.66195	−	0.487770i	−5.15143	−	1.95368i	−2.69150	−	0.162806i	2.13570	−	4.23877i	−3.53808	−	2.13884i	4.97528	−	8.23013i	2.52416	+	1.62130i	1.76833	−	7.17439i
5.3	−0.465956	+	2.54264i	−0.218600	+	1.71820i	−4.37786	−	1.66030i	1.00192	+	0.0606047i	−4.26690	−	1.35643i	−1.49633	−	0.904565i	3.58682	−	5.93332i	−2.90443	−	0.751197i	−0.620944	+	2.51927i
5.4	−0.450509	+	2.45835i	0.339251	−	1.69850i	−3.97049	−	1.50581i	−1.53833	−	0.0930519i	4.02267	+	1.59919i	0.917465	+	0.554627i	2.90459	−	4.80477i	−2.76982	−	1.15244i	0.921786	−	3.73983i
5.5	−0.436480	+	2.38179i	1.15819	+	1.28787i	−3.61239	−	1.37000i	−0.777816	−	0.0470492i	−3.57296	+	2.19643i	3.90248	+	2.35913i	2.33436	−	3.86150i	−0.317208	+	2.98318i	0.451562	−	1.83206i
5.6	−0.422557	+	2.30582i	1.72797	+	0.118863i	−3.26821	−	1.23947i	−4.34895	−	0.263063i	−1.00424	+	3.93415i	−1.68324	−	1.01756i	1.81349	−	2.99989i	2.97174	+	0.410783i	2.44426	−	9.91674i
5.7	−0.407325	+	2.22270i	−1.46444	+	0.924888i	−2.90444	−	1.10151i	3.11001	+	0.188121i	−1.45925	−	3.63174i	1.33442	+	0.806684i	1.29330	−	2.13939i	1.28916	−	2.70888i	−1.68492	+	6.83599i
5.8	−0.400794	+	2.18706i	0.580417	−	1.63191i	−2.75258	−	1.04392i	0.356016	+	0.0215350i	3.33645	+	1.92347i	−1.89154	−	1.14347i	1.08575	−	1.79605i	−2.32623	−	1.89437i	−0.189788	+	0.769999i
5.9	−0.392585	+	2.14227i	−1.02029	−	1.39965i	−2.56516	−	0.972837i	−1.56541	−	0.0946898i	3.39897	−	1.63625i	2.97174	+	1.79648i	0.837658	−	1.38566i	−0.918027	+	2.85609i	0.817408	−	3.31635i
5.10	−0.378713	+	2.06657i	−1.55175	−	0.769460i	−2.25726	−	0.856068i	2.99779	+	0.181333i	2.17781	−	2.91540i	−2.77544	−	1.67781i	0.450143	−	0.744627i	1.81586	+	2.38802i	−1.51004	+	6.12649i
5.11	−0.367556	+	2.00569i	1.27896	+	1.16801i	−2.01765	−	0.765194i	1.15023	+	0.0695764i	−2.81276	+	2.13589i	−2.33236	−	1.40996i	0.166545	−	0.275500i	0.271498	+	2.98769i	−0.562324	+	2.28144i
5.12	−0.320681	+	1.74990i	−1.48788	+	0.886686i	−1.08928	−	0.413111i	−3.06762	−	0.185557i	−1.07448	−	2.88799i	1.60183	+	0.968339i	−0.768515	+	1.27128i	1.42758	−	2.63857i	1.30844	−	5.30853i
5.13	−0.315367	+	1.72090i	1.06756	−	1.36393i	−0.992015	−	0.376222i	3.73990	+	0.226222i	2.01052	+	2.26731i	2.39720	+	1.44916i	−0.849938	+	1.40597i	−0.720612	−	2.91217i	−1.56875	+	6.36465i
5.14	−0.308001	+	1.68071i	−1.68938	−	0.382115i	−0.859883	−	0.326111i	−0.599116	−	0.0362399i	1.16255	−	2.72165i	0.705424	+	0.426444i	−0.955005	+	1.57977i	2.70798	+	1.29107i	0.245437	−	0.995778i
5.15	−0.289269	+	1.57849i	−1.18129	+	1.26671i	−0.537915	−	0.204004i	−1.19167	−	0.0720827i	−1.65777	−	2.23107i	−3.74112	−	2.26159i	−1.18280	+	1.95659i	−0.209095	−	2.99270i	0.458494	−	1.86018i
5.16	−0.287822	+	1.57060i	1.61828	−	0.617401i	−0.513895	−	0.194895i	−0.563686	−	0.0340967i	0.503912	+	2.71936i	2.05136	+	1.24009i	−1.19811	+	1.98191i	2.23763	−	1.99825i	0.215793	−	0.875508i
5.17	−0.259021	+	1.41343i	0.863683	+	1.50135i	−0.0606581	−	0.0230046i	−2.17786	−	0.131736i	−2.34576	+	0.831874i	−1.46195	−	0.883779i	−1.43857	+	2.37968i	−1.50810	+	2.59338i	0.750309	−	3.04412i
5.18	−0.250240	+	1.36552i	0.0535762	+	1.73122i	0.0680148	+	0.0257946i	1.85720	+	0.112340i	−2.37742	−	0.360062i	3.00163	+	1.81455i	−1.48864	+	2.46251i	−2.99426	+	0.185505i	−0.618149	+	2.50793i
5.19	−0.230989	+	1.26047i	−0.545148	−	1.64402i	0.334608	+	0.126900i	2.48769	+	0.150478i	2.19816	−	0.307389i	0.0931907	+	0.0563358i	−1.56314	+	2.58575i	−2.40563	+	1.79247i	−0.764303	+	3.10090i
5.20	−0.205529	+	1.12154i	0.653740	−	1.60394i	0.654428	+	0.248192i	−3.58890	−	0.217089i	1.66452	+	1.06285i	−1.80817	−	1.09308i	−1.59261	+	2.63450i	−2.14525	−	2.09712i	0.981098	−	3.98047i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
169.j	odd	52	1	inner
507.s	even	52	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	507.2.s.a	✓	1392
3.b	odd	2	1	inner	507.2.s.a	✓	1392
169.j	odd	52	1	inner	507.2.s.a	✓	1392
507.s	even	52	1	inner	507.2.s.a	✓	1392

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
507.2.s.a	✓	1392	1.a	even	1	1	trivial
507.2.s.a	✓	1392	3.b	odd	2	1	inner
507.2.s.a	✓	1392	169.j	odd	52	1	inner
507.2.s.a	✓	1392	507.s	even	52	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(507, [\chi])\).