Newform orbit 225.2.u.a

• Label: 225.2.u.a
• Level: $225$
• Weight: $2$
• Character orbit: 225.u
• Analytic conductor: $1.797$
• Analytic rank: $0$
• Dimension: $224$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 225 = 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	225.u (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.79663404548\)
Analytic rank:	\(0\)
Dimension:	\(224\)
Relative dimension:	\(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 224 q - 5 q^{2} - 10 q^{3} - 29 q^{4} - 2 q^{6} - 20 q^{8} - 8 q^{9}+O(q^{10}) \)

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	−0.550606	−	2.59040i	1.37211	+	1.05703i	−4.57991	+	2.03911i	0.814817	+	2.08232i	1.98265	−	4.13632i	3.52968	−	2.03786i	4.69059	+	6.45605i	0.765363	+	2.90073i	4.94541	−	3.25724i
4.2	−0.539236	−	2.53690i	−0.336813	+	1.69899i	−4.31802	+	1.92251i	0.556768	−	2.16564i	4.49179	−	0.0616929i	−3.64986	+	2.10725i	4.15670	+	5.72121i	−2.77311	−	1.14448i	−5.79426	−	0.244676i
4.3	−0.517346	−	2.43392i	1.02336	−	1.39740i	−3.82924	+	1.70489i	−1.52579	−	1.63461i	−3.93060	−	1.76785i	2.64101	−	1.52479i	3.20543	+	4.41190i	−0.905452	−	2.86010i	−3.18915	+	4.55932i
4.4	−0.475209	−	2.23568i	−1.65679	+	0.505009i	−2.94537	+	1.31136i	−0.0794165	+	2.23466i	1.91636	+	3.46408i	−1.14229	+	0.659503i	1.64454	+	2.26351i	2.48993	−	1.67339i	5.03373	−	0.884380i
4.5	−0.420095	−	1.97639i	−0.517307	−	1.65300i	−1.90255	+	0.847070i	−1.23487	+	1.86416i	−3.04965	+	1.71682i	−1.21233	+	0.699936i	0.0980997	+	0.135023i	−2.46479	+	1.71021i	4.20307	+	1.65746i
4.6	−0.414255	−	1.94892i	1.36173	−	1.07037i	−1.79958	+	0.801226i	2.23349	−	0.107302i	−2.65016	−	2.21050i	−2.73241	+	1.57756i	−0.0352641	−	0.0485369i	0.708633	−	2.91511i	−1.13436	−	4.30844i
4.7	−0.381989	−	1.79712i	−1.72999	+	0.0845135i	−1.25662	+	0.559483i	1.89156	−	1.19248i	0.812716	+	3.07671i	3.44865	−	1.99108i	−0.674363	−	0.928181i	2.98571	−	0.292414i	−2.86558	−	2.94384i
4.8	−0.344064	−	1.61870i	−0.429541	+	1.67794i	−0.674706	+	0.300398i	−2.10677	−	0.749336i	2.86387	+	0.117975i	2.64448	−	1.52679i	−1.22701	−	1.68883i	−2.63099	−	1.44149i	−0.488082	+	3.66805i
4.9	−0.255527	−	1.20216i	−0.862898	−	1.50180i	0.447195	−	0.199104i	0.173727	−	2.22931i	−1.58491	+	1.42109i	−1.60284	+	0.925401i	−1.79842	−	2.47532i	−1.51081	+	2.59180i	−2.72438	+	0.360801i
4.10	−0.233920	−	1.10051i	0.855105	+	1.50625i	0.670698	−	0.298614i	2.15310	+	0.603467i	1.45761	−	1.29339i	−1.09191	+	0.630415i	−1.80814	−	2.48869i	−1.53759	+	2.57601i	0.160466	−	2.51066i
4.11	−0.192241	−	0.904421i	1.63521	−	0.571043i	1.04607	−	0.465740i	−1.07309	+	1.96175i	−0.830817	−	1.36914i	0.805624	−	0.465127i	−1.70929	−	2.35263i	2.34782	−	1.86755i	1.98054	+	0.593396i
4.12	−0.182883	−	0.860395i	1.53743	+	0.797698i	1.12026	−	0.498771i	−0.712729	−	2.11944i	0.405166	−	1.46868i	1.09544	−	0.632452i	−1.66807	−	2.29590i	1.72736	+	2.45280i	−1.69321	+	1.00084i
4.13	−0.119971	−	0.564417i	−1.64559	+	0.540392i	1.52292	−	0.678046i	−2.21766	−	0.286312i	0.502429	+	0.863970i	−3.44219	+	1.98735i	−1.24374	−	1.71186i	2.41595	−	1.77853i	0.104455	+	1.28604i
4.14	−0.0134680	−	0.0633621i	−1.52430	−	0.822509i	1.82326	−	0.811767i	1.33347	+	1.79495i	−0.0315866	+	0.107660i	0.0617846	−	0.0356713i	−0.152142	−	0.209405i	1.64696	+	2.50749i	0.0957728	−	0.108666i
4.15	0.0524738	+	0.246870i	0.431699	−	1.67739i	1.76890	−	0.787565i	2.18454	−	0.477257i	0.436749	+	0.0185543i	−0.263893	+	0.152358i	0.583943	+	0.803728i	−2.62727	−	1.44825i	0.232451	+	0.514254i
4.16	0.0838034	+	0.394264i	−0.757696	+	1.55753i	1.67867	−	0.747392i	−0.252792	+	2.22173i	−0.677575	−	0.168206i	4.03987	−	2.33242i	0.909187	+	1.25139i	−1.85179	−	2.36027i	−0.897134	+	0.0865220i
4.17	0.0963190	+	0.453145i	0.136257	−	1.72668i	1.63103	−	0.726180i	−2.22769	−	0.193424i	0.795562	−	0.104568i	1.95191	−	1.12694i	1.03077	+	1.41873i	−2.96287	−	0.470545i	−0.126919	−	1.02810i
4.18	0.0965635	+	0.454296i	0.723972	+	1.57349i	1.63003	−	0.725736i	−1.56019	+	1.60181i	−0.644920	+	0.480839i	−3.50662	+	2.02455i	1.03309	+	1.42192i	−1.95173	+	2.27832i	−0.878353	−	0.554113i
4.19	0.188119	+	0.885029i	1.67113	−	0.455340i	1.07920	−	0.480492i	−0.419997	−	2.19627i	0.717359	+	1.39334i	−3.56731	+	2.05959i	1.69193	+	2.32874i	2.58533	−	1.52186i	1.86475	−	0.784869i
4.20	0.250369	+	1.17789i	−1.73132	+	0.0503723i	0.502342	−	0.223657i	−1.04324	−	1.97779i	−0.492802	−	2.02670i	2.32094	−	1.33999i	1.80485	+	2.48416i	2.99493	−	0.174421i	2.06843	−	1.72401i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
9.c	even	3	1	inner
25.e	even	10	1	inner
225.u	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	225.2.u.a	✓	224
3.b	odd	2	1		675.2.y.a		224
9.c	even	3	1	inner	225.2.u.a	✓	224
9.d	odd	6	1		675.2.y.a		224
25.e	even	10	1	inner	225.2.u.a	✓	224
75.h	odd	10	1		675.2.y.a		224
225.u	even	30	1	inner	225.2.u.a	✓	224
225.v	odd	30	1		675.2.y.a		224

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
225.2.u.a	✓	224	1.a	even	1	1	trivial
225.2.u.a	✓	224	9.c	even	3	1	inner
225.2.u.a	✓	224	25.e	even	10	1	inner
225.2.u.a	✓	224	225.u	even	30	1	inner
675.2.y.a		224	3.b	odd	2	1
675.2.y.a		224	9.d	odd	6	1
675.2.y.a		224	75.h	odd	10	1
675.2.y.a		224	225.v	odd	30	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(225, [\chi])\).