Newform orbit 825.2.bu.a

• Label: 825.2.bu.a
• Level: $825$
• Weight: $2$
• Character orbit: 825.bu
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.bu (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(116\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 464 q + 110 q^{4} - 15 q^{6} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
239.1	−2.62116	−	0.851666i	1.51814	+	0.833820i	4.52711	+	3.28913i	−1.46562	+	1.68877i	−3.26915	−	3.47852i	0.804181	−	2.47502i	−5.82509	−	8.01754i	1.60949	+	2.53171i	5.27990	−	3.17831i
239.2	−2.58002	−	0.838300i	0.123521	−	1.72764i	4.33573	+	3.15009i	1.46170	+	1.69217i	−1.76697	+	4.35380i	0.773533	−	2.38069i	−5.35647	−	7.37255i	−2.96949	−	0.426800i	−2.35268	−	5.59117i
239.3	−2.57863	−	0.837848i	1.19061	+	1.25796i	4.32932	+	3.14544i	2.10485	−	0.754712i	−2.01617	−	4.24135i	−0.254580	+	0.783516i	−5.34096	−	7.35120i	−0.164903	+	2.99546i	−6.05998	+	0.182576i
239.4	−2.50886	−	0.815178i	−0.887971	−	1.48711i	4.01183	+	2.91476i	−1.19696	−	1.88873i	1.01553	+	4.45481i	0.272774	−	0.839513i	−4.58794	−	6.31476i	−1.42301	+	2.64103i	1.46334	+	5.71429i
239.5	−2.49221	−	0.809767i	−0.549050	+	1.64272i	3.93733	+	2.86064i	−0.239811	+	2.22317i	2.69857	−	3.64940i	−1.11984	+	3.44651i	−4.41566	−	6.07763i	−2.39709	−	1.80388i	2.39791	−	5.34641i
239.6	−2.44601	−	0.794756i	0.800051	−	1.53620i	3.73328	+	2.71239i	−1.98182	+	1.03556i	−3.17784	+	3.12172i	−1.28164	+	3.94448i	−3.95252	−	5.44018i	−1.71984	−	2.45808i	5.67057	−	0.957920i
239.7	−2.40616	−	0.781808i	−1.72191	−	0.187119i	3.56034	+	2.58674i	2.23301	−	0.116885i	3.99690	+	1.79644i	−1.00180	+	3.08321i	−3.57023	−	4.91400i	2.92997	+	0.644407i	−5.46436	−	1.46454i
239.8	−2.40378	−	0.781035i	−1.51150	+	0.845802i	3.55010	+	2.57930i	1.11675	−	1.93723i	4.29390	−	0.852590i	1.15500	−	3.55472i	−3.54789	−	4.88326i	1.56924	−	2.55685i	−4.19746	+	3.78446i
239.9	−2.37506	−	0.771705i	1.10876	−	1.33066i	3.42736	+	2.49012i	0.827161	−	2.07745i	−3.66025	+	2.30476i	0.124501	−	0.383174i	−3.28281	−	4.51841i	−0.541294	−	2.95076i	−3.56774	+	4.29575i
239.10	−2.37494	−	0.771665i	−1.62563	−	0.597757i	3.42684	+	2.48974i	−0.156651	+	2.23057i	3.39952	+	2.67408i	−0.0954560	+	0.293783i	−3.28170	−	4.51687i	2.28537	+	1.94347i	2.09329	−	5.17660i
239.11	−2.31995	−	0.753797i	1.70952	+	0.278457i	3.19591	+	2.32197i	−1.24537	−	1.85716i	−3.75610	−	1.93464i	−1.34937	+	4.15292i	−2.79645	−	3.84898i	2.84492	+	0.952057i	1.48928	+	5.24727i
239.12	−2.25473	−	0.732605i	0.391293	+	1.68727i	2.92904	+	2.12807i	−1.68453	−	1.47050i	0.353845	−	4.09100i	0.0288392	−	0.0887579i	−2.25816	−	3.10809i	−2.69378	+	1.32044i	2.72085	+	4.54966i
239.13	−2.23636	−	0.726638i	−0.930229	+	1.46105i	2.85528	+	2.07448i	−2.23395	+	0.0972772i	3.14198	−	2.59150i	0.916026	−	2.81924i	−2.11374	−	2.90932i	−1.26935	−	2.71823i	5.06661	+	1.40573i
239.14	−2.19239	−	0.712351i	−0.267830	+	1.71122i	2.68111	+	1.94794i	1.49642	−	1.66154i	1.80618	−	3.56087i	−0.134692	+	0.414539i	−1.78047	−	2.45061i	−2.85653	−	0.916631i	−4.46435	+	2.57677i
239.15	−2.10519	−	0.684019i	1.66003	−	0.494282i	2.34593	+	1.70441i	1.98017	+	1.03870i	−3.83277	−	0.0949289i	0.815308	−	2.50926i	−1.17061	−	1.61121i	2.51137	−	1.64104i	−3.45816	−	3.54115i
239.16	−2.07461	−	0.674083i	1.72969	−	0.0904317i	2.23160	+	1.62136i	1.61048	+	1.55125i	−3.64940	−	0.978343i	−0.690334	+	2.12463i	−0.972427	−	1.33843i	2.98364	−	0.312837i	−2.29545	−	4.30384i
239.17	−1.96590	−	0.638760i	−1.03281	−	1.39043i	1.83872	+	1.33591i	−2.23353	−	0.106470i	1.14226	+	3.39317i	−0.960313	+	2.95554i	−0.331431	−	0.456176i	−0.866592	+	2.87211i	4.32290	+	1.63600i
239.18	−1.93267	−	0.627964i	1.33383	−	1.10495i	1.72286	+	1.25173i	−1.27512	+	1.83686i	−3.27172	+	1.29791i	0.345392	−	1.06301i	−0.154767	−	0.213019i	0.558182	−	2.94761i	3.61788	−	2.74933i
239.19	−1.93207	−	0.627767i	−1.16113	−	1.28521i	1.72076	+	1.25021i	2.16481	−	0.559993i	1.43657	+	3.21204i	0.915860	−	2.81873i	−0.151621	−	0.208688i	−0.303549	+	2.98460i	−4.53411	−	0.277053i
239.20	−1.88794	−	0.613430i	−1.73067	−	0.0690487i	1.57000	+	1.14067i	−0.286359	−	2.21766i	3.22506	+	1.19201i	−0.463302	+	1.42590i	0.0692743	+	0.0953479i	2.99046	+	0.239002i	−0.819747	+	4.36247i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
275.o	odd	10	1	inner
825.bu	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.bu.a	yes	464
3.b	odd	2	1	inner	825.2.bu.a	yes	464
11.d	odd	10	1		825.2.s.a	✓	464
25.e	even	10	1		825.2.s.a	✓	464
33.f	even	10	1		825.2.s.a	✓	464
75.h	odd	10	1		825.2.s.a	✓	464
275.o	odd	10	1	inner	825.2.bu.a	yes	464
825.bu	even	10	1	inner	825.2.bu.a	yes	464

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.s.a	✓	464	11.d	odd	10	1
825.2.s.a	✓	464	25.e	even	10	1
825.2.s.a	✓	464	33.f	even	10	1
825.2.s.a	✓	464	75.h	odd	10	1
825.2.bu.a	yes	464	1.a	even	1	1	trivial
825.2.bu.a	yes	464	3.b	odd	2	1	inner
825.2.bu.a	yes	464	275.o	odd	10	1	inner
825.2.bu.a	yes	464	825.bu	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(825, [\chi])\).