Newform orbit 675.2.bg.a

• Label: 675.2.bg.a
• Level: $675$
• Weight: $2$
• Character orbit: 675.bg
• Analytic conductor: $5.390$
• Analytic rank: $0$
• Dimension: $2112$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	675.bg (of order \(90\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.38990213644\)
Analytic rank:	\(0\)
Dimension:	\(2112\)
Relative dimension:	\(88\) over \(\Q(\zeta_{90})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{90}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2112 q - 30 q^{2} - 30 q^{3} - 18 q^{4} - 36 q^{5} - 18 q^{6} - 15 q^{8} - 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.17025	−	1.69559i	1.71464	+	0.244954i	1.35114	+	5.41912i	−1.41922	+	1.72795i	−3.30587	−	3.43894i	2.73355	+	3.25772i	4.01590	−	9.01986i	2.87999	+	0.840019i	6.00996	−	1.34367i
4.2	−2.16863	−	1.69432i	−1.46864	+	0.918203i	1.34839	+	5.40810i	1.51238	−	1.64703i	4.74066	+	0.497103i	1.82880	+	2.17948i	4.00018	−	8.98454i	1.31381	−	2.69702i	−6.07038	+	1.00934i
4.3	−2.12187	−	1.65779i	1.17080	+	1.27642i	1.27023	+	5.09462i	−1.45805	−	1.69531i	−0.368250	−	4.64932i	−2.63234	−	3.13711i	3.56010	−	7.99611i	−0.258476	+	2.98884i	0.283328	+	6.01437i
4.4	−2.10740	−	1.64648i	1.26036	−	1.18806i	1.24639	+	4.99899i	1.72516	−	1.42261i	−4.61220	+	0.428561i	−0.219270	−	0.261315i	3.42860	−	7.70076i	0.177018	−	2.99477i	−5.97790	+	0.157552i
4.5	−2.04163	−	1.59510i	−0.643449	−	1.60810i	1.14008	+	4.57262i	−1.48118	−	1.67514i	−1.25138	+	4.30951i	1.41623	+	1.68779i	2.85854	−	6.42039i	−2.17195	+	2.06946i	0.352010	+	5.78265i
4.6	−2.04136	−	1.59489i	−1.36645	−	1.06434i	1.13965	+	4.57089i	2.06615	+	0.855006i	1.09192	+	4.35203i	−1.82922	−	2.17997i	2.85629	−	6.41533i	0.734370	+	2.90873i	−2.85412	−	5.04065i
4.7	−2.03253	−	1.58798i	−0.166259	+	1.72405i	1.12563	+	4.51464i	0.666923	+	2.13429i	3.07569	−	3.24016i	−1.52009	−	1.81157i	2.78311	−	6.25097i	−2.94472	−	0.573280i	2.03369	−	5.39707i
4.8	−1.92693	−	1.50548i	−1.73093	−	0.0621942i	0.962736	+	3.86132i	−2.18967	+	0.453171i	3.24175	+	2.72573i	−2.03107	−	2.42053i	1.96883	−	4.42207i	2.99226	+	0.215308i	4.90157	+	2.42327i
4.9	−1.88777	−	1.47488i	−0.937001	+	1.45672i	0.904533	+	3.62788i	−2.15186	−	0.607870i	3.91733	−	1.36798i	1.25346	+	1.49381i	1.69440	−	3.80568i	−1.24406	−	2.72989i	3.16567	+	4.32126i
4.10	−1.84364	−	1.44041i	1.21310	−	1.23628i	0.840388	+	3.37061i	−0.540402	+	2.16978i	−4.01727	+	0.531890i	−2.01370	−	2.39983i	1.40248	−	3.15002i	−0.0567692	−	2.99946i	4.12169	−	3.22190i
4.11	−1.78103	−	1.39150i	0.577752	+	1.63285i	0.751976	+	3.01601i	2.00306	+	0.993864i	1.24311	−	3.71210i	2.49177	+	2.96958i	1.01889	−	2.28846i	−2.33241	+	1.88677i	−2.18455	−	4.55735i
4.12	−1.77884	−	1.38978i	−1.44655	−	0.952625i	0.748930	+	3.00379i	−1.12469	+	1.93263i	1.24924	+	3.70495i	2.10613	+	2.50999i	1.00607	−	2.25967i	1.18501	+	2.75604i	4.68658	−	1.87476i
4.13	−1.72301	−	1.34616i	1.70578	+	0.300501i	0.672765	+	2.69831i	2.23368	−	0.103368i	−2.53456	−	2.81403i	−1.18003	−	1.40631i	0.694505	−	1.55988i	2.81940	+	1.02518i	−3.98780	−	2.82879i
4.14	−1.69319	−	1.32286i	1.00661	−	1.40951i	0.633072	+	2.53911i	−1.96360	−	1.06972i	−3.56898	+	1.05496i	0.411466	+	0.490366i	0.539092	−	1.21082i	−0.973463	−	2.83767i	1.90965	+	4.40880i
4.15	−1.67398	−	1.30785i	1.14939	+	1.29572i	0.607872	+	2.43804i	0.0919794	−	2.23418i	−0.229437	−	3.67225i	1.70229	+	2.02871i	0.442971	−	0.994928i	−0.357799	+	2.97859i	−3.07595	+	3.61966i
4.16	−1.52673	−	1.19281i	−0.449063	−	1.67282i	0.424259	+	1.70161i	0.556289	−	2.16577i	−1.30977	+	3.08960i	−3.03041	−	3.61150i	−0.194090	+	0.435933i	−2.59669	+	1.50241i	−3.43265	+	2.64299i
4.17	−1.47422	−	1.15178i	−1.62677	+	0.594666i	0.362865	+	1.45537i	2.22797	−	0.190111i	3.08314	+	0.997019i	−0.789409	−	0.940781i	−0.380520	+	0.854662i	2.29274	−	1.93477i	−3.50348	−	2.28588i
4.18	−1.42877	−	1.11627i	−1.46563	+	0.923005i	0.311460	+	1.24920i	0.446673	+	2.19100i	3.12437	+	0.317283i	0.721846	+	0.860263i	−0.525488	+	1.18026i	1.29612	−	2.70556i	1.80757	−	3.62904i
4.19	−1.39821	−	1.09240i	−1.51674	−	0.836356i	0.277808	+	1.11423i	1.48083	−	1.67545i	1.20709	+	2.82630i	2.62887	+	3.13296i	−0.614640	+	1.38051i	1.60102	+	2.53707i	−3.90078	+	0.724969i
4.20	−1.39327	−	1.08854i	1.12495	+	1.31700i	0.272433	+	1.09267i	−2.11682	+	0.720470i	−0.133742	−	3.05949i	−0.859623	−	1.02446i	−0.628447	+	1.41152i	−0.468991	+	2.96311i	3.73356	+	1.30044i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.e	even	10	1	inner
27.e	even	9	1	inner
675.bg	even	90	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	675.2.bg.a	✓	2112
25.e	even	10	1	inner	675.2.bg.a	✓	2112
27.e	even	9	1	inner	675.2.bg.a	✓	2112
675.bg	even	90	1	inner	675.2.bg.a	✓	2112

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
675.2.bg.a	✓	2112	1.a	even	1	1	trivial
675.2.bg.a	✓	2112	25.e	even	10	1	inner
675.2.bg.a	✓	2112	27.e	even	9	1	inner
675.2.bg.a	✓	2112	675.bg	even	90	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(675, [\chi])\).