Newform orbit 525.2.bm.a

• Label: 525.2.bm.a
• Level: $525$
• Weight: $2$
• Character orbit: 525.bm
• Analytic conductor: $4.192$
• Analytic rank: $0$
• Dimension: $608$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	525.bm (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.19214610612\)
Analytic rank:	\(0\)
Dimension:	\(608\)
Relative dimension:	\(76\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 608 q - 9 q^{3} - 78 q^{4} - 36 q^{7} - 3 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
131.1	−0.578671	+	2.72243i	1.50155	+	0.863338i	−5.24969	−	2.33731i	2.12367	+	0.700010i	−3.21928	+	3.58828i	2.47291	+	0.940586i	6.12911	−	8.43599i	1.50929	+	2.59269i	−3.13464	+	5.37648i
131.2	−0.572271	+	2.69232i	−1.59928	+	0.665065i	−5.09402	−	2.26800i	−1.14863	+	1.91850i	−0.875350	−	4.68637i	−2.63822	−	0.199474i	5.78563	−	7.96324i	2.11538	−	2.12725i	−4.50790	−	4.19039i
131.3	−0.548566	+	2.58080i	1.24592	−	1.20320i	−4.53252	−	2.01801i	−1.83369	+	1.27969i	2.42176	+	3.87550i	1.29192	−	2.30888i	4.59278	−	6.32141i	0.104613	−	2.99818i	−2.29672	−	5.43437i
131.4	−0.540793	+	2.54423i	−1.70841	−	0.285183i	−4.35356	−	1.93833i	0.488675	−	2.18202i	1.64947	−	4.19237i	1.22413	+	2.34553i	4.22820	−	5.81962i	2.83734	+	0.974421i	5.28728	+	2.42332i
131.5	−0.530626	+	2.49640i	−0.257292	+	1.71283i	−4.12336	−	1.83584i	1.58898	−	1.57326i	−4.13939	−	1.55118i	−1.47493	−	2.19649i	3.77069	−	5.18991i	−2.86760	−	0.881397i	3.08432	+	4.80154i
131.6	−0.528347	+	2.48568i	1.52678	−	0.817885i	−4.07236	−	1.81313i	−0.273491	−	2.21928i	1.22633	+	4.22722i	−2.41288	+	1.08537i	3.67110	−	5.05284i	1.66213	−	2.49746i	5.66091	+	0.492741i
131.7	−0.505077	+	2.37620i	0.0606037	−	1.73099i	−3.56413	−	1.58685i	2.07975	+	0.821374i	4.08257	+	1.01829i	−2.37564	+	1.16461i	2.71505	−	3.73695i	−2.99265	−	0.209809i	−3.00218	+	4.52704i
131.8	−0.489122	+	2.30114i	−1.23199	−	1.21745i	−3.22892	−	1.43761i	1.02927	+	1.98509i	3.40412	−	2.23951i	1.46353	−	2.20410i	2.12188	−	2.92052i	0.0356207	+	2.99979i	−5.07142	+	1.39755i
131.9	−0.478695	+	2.25208i	−0.600734	−	1.62454i	−3.01563	−	1.34265i	−2.22631	+	0.208681i	3.94616	−	0.575244i	0.695446	+	2.55272i	1.76069	−	2.42338i	−2.27824	+	1.95183i	0.595755	−	5.11372i
131.10	−0.462774	+	2.17718i	1.15009	+	1.29510i	−2.69886	−	1.20161i	−2.18803	−	0.460993i	−3.35190	+	1.90461i	−0.462395	−	2.60503i	1.24847	−	1.71838i	−0.354583	+	2.97897i	2.01623	−	4.55040i
131.11	−0.457158	+	2.15076i	−0.177450	+	1.72294i	−2.58969	−	1.15300i	−0.910547	+	2.04228i	−3.62450	−	1.16931i	2.32202	+	1.26816i	1.07888	−	1.48495i	−2.93702	−	0.611470i	−3.97619	−	2.89201i
131.12	−0.427547	+	2.01145i	−1.51386	+	0.841555i	−2.03605	−	0.906508i	2.19435	+	0.429892i	−1.04550	−	3.40487i	1.64883	−	2.06914i	0.276476	−	0.380537i	1.58357	−	2.54800i	−1.80290	+	4.23004i
131.13	−0.421130	+	1.98126i	0.963172	+	1.43955i	−1.92096	−	0.855264i	0.372730	+	2.20478i	−3.25774	+	1.30206i	−2.59804	−	0.500191i	0.122328	−	0.168370i	−1.14460	+	2.77306i	−4.52522	−	0.190026i
131.14	−0.414854	+	1.95173i	−1.61597	−	0.623422i	−1.81007	−	0.805895i	−1.81656	−	1.30389i	1.88714	−	2.89531i	−0.995488	−	2.45133i	−0.0218510	+	0.0300753i	2.22269	+	2.01486i	3.29845	−	3.00451i
131.15	−0.406505	+	1.91245i	1.66519	+	0.476610i	−1.66514	−	0.741369i	−0.898652	−	2.04754i	−1.58840	+	2.99085i	2.59781	+	0.501390i	−0.203728	+	0.280407i	2.54569	+	1.58729i	4.28113	−	0.886296i
131.16	−0.392327	+	1.84575i	0.000111749		1.73205i	−1.42579	−	0.634803i	0.827237	−	2.07742i	−3.19698	−	0.679324i	−0.468357	+	2.60397i	−0.487221	+	0.670602i	−3.00000		0.000387111i	3.50986	+	2.34190i
131.17	−0.352641	+	1.65904i	−1.23560	+	1.21379i	−0.800979	−	0.356619i	−2.12372	−	0.699871i	−1.57801	−	2.47795i	−1.86600	+	1.87565i	−1.11979	+	1.54126i	0.0534131	−	2.99952i	1.91003	−	3.27654i
131.18	−0.350482	+	1.64889i	1.30612	−	1.13757i	−0.768903	−	0.342338i	0.782565	+	2.09466i	1.41795	+	2.55234i	1.46624	+	2.20230i	−1.14773	+	1.57971i	0.411886	−	2.97159i	−3.72813	+	0.556222i
131.19	−0.343956	+	1.61819i	1.58851	−	0.690379i	−0.673133	−	0.299698i	2.12496	−	0.696100i	0.570782	+	2.80797i	0.539750	−	2.59011i	−1.22830	+	1.69060i	2.04676	−	2.19335i	0.395527	+	3.67801i
131.20	−0.318971	+	1.50064i	0.109335	−	1.72860i	−0.323084	−	0.143846i	0.267996	−	2.21995i	2.55912	+	0.715443i	−1.46253	−	2.20477i	−1.48460	+	2.04338i	−2.97609	−	0.377991i	3.24586	+	1.11026i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
21.g	even	6	1	inner
25.d	even	5	1	inner
75.j	odd	10	1	inner
175.v	odd	30	1	inner
525.bm	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	525.2.bm.a	✓	608
3.b	odd	2	1	inner	525.2.bm.a	✓	608
7.d	odd	6	1	inner	525.2.bm.a	✓	608
21.g	even	6	1	inner	525.2.bm.a	✓	608
25.d	even	5	1	inner	525.2.bm.a	✓	608
75.j	odd	10	1	inner	525.2.bm.a	✓	608
175.v	odd	30	1	inner	525.2.bm.a	✓	608
525.bm	even	30	1	inner	525.2.bm.a	✓	608

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
525.2.bm.a	✓	608	1.a	even	1	1	trivial
525.2.bm.a	✓	608	3.b	odd	2	1	inner
525.2.bm.a	✓	608	7.d	odd	6	1	inner
525.2.bm.a	✓	608	21.g	even	6	1	inner
525.2.bm.a	✓	608	25.d	even	5	1	inner
525.2.bm.a	✓	608	75.j	odd	10	1	inner
525.2.bm.a	✓	608	175.v	odd	30	1	inner
525.2.bm.a	✓	608	525.bm	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(525, [\chi])\).