Newform orbit 900.2.bn.a

• Label: 900.2.bn.a
• Level: $900$
• Weight: $2$
• Character orbit: 900.bn
• Analytic conductor: $7.187$
• Analytic rank: $0$
• Dimension: $1408$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	900.bn (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.18653618192\)
Analytic rank:	\(0\)
Dimension:	\(1408\)
Relative dimension:	\(176\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 1408 q - 15 q^{2} - 3 q^{4} - 24 q^{5} - 2 q^{6} - 12 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41378	−	0.0351605i	−1.71505	−	0.242067i	1.99753	+	0.0994182i	−0.577017	+	2.16034i	2.41619	+	0.402531i	0.724318	+	1.25456i	−2.82056	−	0.210789i	2.88281	+	0.830316i	0.891731	−	3.03394i
59.2	−1.41368	−	0.0389778i	−1.51581	+	0.838050i	1.99696	+	0.110204i	−2.11446	+	0.727352i	2.17553	−	1.12565i	−1.71500	−	2.97047i	−2.81876	−	0.233630i	1.59535	−	2.54064i	3.01752	−	0.945824i
59.3	−1.41272	−	0.0649136i	1.61305	+	0.630926i	1.99157	+	0.183410i	0.622416	+	2.14770i	−2.23784	−	0.996033i	−1.13866	−	1.97221i	−2.80163	−	0.388387i	2.20386	+	2.03543i	−0.739886	−	3.07450i
59.4	−1.41134	−	0.0901438i	−0.106530	+	1.72877i	1.98375	+	0.254447i	0.246616	−	2.22243i	0.306188	−	2.43028i	0.129619	+	0.224506i	−2.77680	−	0.537933i	−2.97730	−	0.368332i	−0.548396	+	3.11436i
59.5	−1.40896	+	0.121751i	1.69974	+	0.332982i	1.97035	−	0.343085i	1.74968	−	1.39235i	−2.43541	−	0.262214i	2.16913	+	3.75705i	−2.73438	+	0.723285i	2.77825	+	1.13197i	−2.29571	+	2.17479i
59.6	−1.40808	+	0.131536i	0.738114	−	1.56690i	1.96540	−	0.370427i	−1.59718	−	1.56493i	−0.833222	+	2.30342i	1.79884	+	3.11569i	−2.71872	+	0.780113i	−1.91038	−	2.31311i	2.45481	+	1.99346i
59.7	−1.40453	+	0.165210i	0.461556	+	1.66942i	1.94541	−	0.464086i	−2.05760	+	0.875369i	−0.924075	−	2.26850i	2.50617	+	4.34081i	−2.65572	+	0.973225i	−2.57393	+	1.54106i	2.74535	−	1.56942i
59.8	−1.40172	+	0.187537i	−0.761278	+	1.55578i	1.92966	−	0.525751i	2.18712	+	0.465301i	0.775335	−	2.32354i	−0.622212	−	1.07770i	−2.60625	+	1.09884i	−1.84091	−	2.36876i	−3.15300	−	0.242057i
59.9	−1.39842	+	0.210756i	0.989235	−	1.42176i	1.91116	−	0.589450i	−0.827455	+	2.07733i	−1.08372	+	2.19671i	−2.44452	−	4.23404i	−2.54838	+	1.22709i	−1.04283	−	2.81292i	0.719321	−	3.07938i
59.10	−1.39837	−	0.211070i	−1.29509	−	1.15011i	1.91090	+	0.590308i	0.126512	−	2.23249i	1.56827	+	1.88163i	−1.59165	−	2.75682i	−2.54756	−	1.22880i	0.354508	+	2.97898i	−0.648120	+	3.09515i
59.11	−1.39128	−	0.253626i	−0.369897	−	1.69209i	1.87135	+	0.705733i	1.73134	+	1.41509i	0.0854732	+	2.44800i	1.86914	+	3.23744i	−2.42458	−	1.45650i	−2.72635	+	1.25180i	−2.04988	−	2.40790i
59.12	−1.38718	−	0.275189i	−0.339162	−	1.69852i	1.84854	+	0.763474i	−2.14320	−	0.637740i	0.00306453	+	2.44949i	−0.908066	−	1.57282i	−2.35416	−	1.56777i	−2.76994	+	1.15215i	2.79750	+	1.47444i
59.13	−1.37689	+	0.322743i	0.785403	−	1.54374i	1.79167	−	0.888766i	1.47137	−	1.68377i	−0.583184	+	2.37905i	−0.757196	−	1.31150i	−2.18010	+	1.80199i	−1.76628	−	2.42492i	−1.48249	+	2.79324i
59.14	−1.37056	−	0.348652i	1.48666	−	0.888725i	1.75688	+	0.955699i	2.22595	+	0.212507i	−2.34742	+	0.699725i	0.0131030	+	0.0226950i	−2.07471	−	1.92239i	1.42033	−	2.64247i	−2.97671	−	1.06734i
59.15	−1.35716	+	0.397651i	0.658108	+	1.60215i	1.68375	−	1.07935i	−1.81600	−	1.30466i	−1.53025	−	1.91268i	−1.49189	−	2.58403i	−1.85590	+	2.13439i	−2.13379	+	2.10878i	2.98340	+	1.04849i
59.16	−1.35547	−	0.403363i	1.48220	+	0.896146i	1.67460	+	1.09349i	1.52567	−	1.63472i	−1.64761	−	1.81256i	−1.89882	−	3.28885i	−1.82879	−	2.15767i	1.39384	+	2.65654i	−2.72739	+	1.60042i
59.17	−1.35185	+	0.415333i	1.72815	−	0.116168i	1.65500	−	1.12294i	−1.67393	−	1.48255i	−2.28795	+	0.874800i	−0.578938	−	1.00275i	−1.77092	+	2.20542i	2.97301	−	0.401512i	2.87865	+	1.30895i
59.18	−1.34914	+	0.424069i	1.15895	+	1.28718i	1.64033	−	1.14425i	1.22896	+	1.86806i	−2.10943	−	1.24511i	1.01557	+	1.75902i	−1.72779	+	2.23937i	−0.313685	+	2.98356i	−2.45022	−	1.99910i
59.19	−1.34793	−	0.427876i	−1.38923	+	1.03442i	1.63384	+	1.15350i	−1.23890	−	1.86148i	2.31520	−	0.799909i	0.897655	+	1.55478i	−1.70876	−	2.25392i	0.859946	−	2.87411i	0.873471	+	3.03925i
59.20	−1.34539	+	0.435803i	−1.32260	−	1.11835i	1.62015	−	1.17265i	1.93014	+	1.12897i	2.26680	+	0.928229i	−0.0551920	−	0.0955953i	−1.66869	+	2.28374i	0.498567	+	2.95828i	−3.08880	−	0.677740i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
25.e	even	10	1	inner
36.h	even	6	1	inner
100.h	odd	10	1	inner
225.v	odd	30	1	inner
900.bn	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	900.2.bn.a	✓	1408
4.b	odd	2	1	inner	900.2.bn.a	✓	1408
9.d	odd	6	1	inner	900.2.bn.a	✓	1408
25.e	even	10	1	inner	900.2.bn.a	✓	1408
36.h	even	6	1	inner	900.2.bn.a	✓	1408
100.h	odd	10	1	inner	900.2.bn.a	✓	1408
225.v	odd	30	1	inner	900.2.bn.a	✓	1408
900.bn	even	30	1	inner	900.2.bn.a	✓	1408

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
900.2.bn.a	✓	1408	1.a	even	1	1	trivial
900.2.bn.a	✓	1408	4.b	odd	2	1	inner
900.2.bn.a	✓	1408	9.d	odd	6	1	inner
900.2.bn.a	✓	1408	25.e	even	10	1	inner
900.2.bn.a	✓	1408	36.h	even	6	1	inner
900.2.bn.a	✓	1408	100.h	odd	10	1	inner
900.2.bn.a	✓	1408	225.v	odd	30	1	inner
900.2.bn.a	✓	1408	900.bn	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(900, [\chi])\).