Newform orbit 825.2.bf.a

• Label: 825.2.bf.a
• Level: $825$
• Weight: $2$
• Character orbit: 825.bf
• 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.bf (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.

\(\operatorname{Tr}(f)(q) = \) \( 464 q + 2 q^{3} - 114 q^{4} - 10 q^{7} - 4 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
266.1	−2.25620	+	1.63922i	−1.49630	−	0.872411i	1.78534	−	5.49470i	−1.15839	−	1.91262i	4.80601	−	0.484430i	1.16450	−	1.60279i	3.25538	+	10.0190i	1.47780	+	2.61077i	5.74877	+	2.41640i
266.2	−2.22435	+	1.61609i	1.17750	+	1.27023i	1.71797	−	5.28738i	2.19234	−	0.440033i	−4.67199	−	0.922496i	1.88094	−	2.58889i	3.02423	+	9.30763i	−0.226975	+	2.99140i	−4.16542	+	4.52181i
266.3	−2.18447	+	1.58711i	1.40829	−	1.00833i	1.63496	−	5.03188i	1.17104	+	1.90491i	−1.47604	+	4.43777i	−0.0216204	+	0.0297579i	2.74585	+	8.45086i	0.966559	−	2.84003i	−5.58140	−	2.30264i
266.4	−2.16886	+	1.57577i	−1.14770	+	1.29722i	1.60288	−	4.93316i	2.23175	−	0.138941i	0.445089	−	4.62201i	−2.55569	+	3.51761i	2.64024	+	8.12583i	−0.365558	−	2.97764i	−4.62142	+	3.81807i
266.5	−2.10628	+	1.53030i	0.238050	+	1.71561i	1.47655	−	4.54435i	−2.13943	+	0.650272i	−3.12680	−	3.24927i	−2.03311	+	2.79834i	2.23514	+	6.87906i	−2.88666	+	0.816805i	3.51111	−	4.64361i
266.6	−2.10401	+	1.52865i	0.213424	−	1.71885i	1.47204	−	4.53047i	−2.22953	+	0.170837i	2.17848	+	3.94273i	−1.92982	+	2.65617i	2.22101	+	6.83556i	−2.90890	−	0.733689i	4.42980	−	3.76762i
266.7	−2.06649	+	1.50139i	−1.72381	+	0.168794i	1.39817	−	4.30313i	−1.33050	+	1.79716i	3.30881	−	2.93693i	−0.494051	+	0.680003i	1.99272	+	6.13298i	2.94302	−	0.581936i	0.0512230	−	5.71142i
266.8	−2.05849	+	1.49558i	1.66999	−	0.459495i	1.38259	−	4.25518i	−0.381974	−	2.20320i	−2.75045	+	3.44347i	−1.99125	+	2.74071i	1.94537	+	5.98722i	2.57773	−	1.53470i	4.08136	+	3.96400i
266.9	−2.02207	+	1.46912i	1.58562	+	0.696997i	1.31242	−	4.03922i	−2.00820	−	0.983422i	−4.23021	+	0.920091i	0.785970	−	1.08180i	1.73557	+	5.34153i	2.02839	+	2.21035i	5.50550	−	0.961745i
266.10	−1.99557	+	1.44986i	−1.15292	+	1.29258i	1.26215	−	3.88449i	−0.776305	−	2.09699i	0.426668	−	4.25101i	0.624786	−	0.859945i	1.58880	+	4.88983i	−0.341534	−	2.98050i	4.58951	+	3.05914i
266.11	−1.98282	+	1.44060i	−0.218240	−	1.71825i	1.23821	−	3.81081i	−0.674172	+	2.13202i	2.90804	+	3.09258i	2.13976	−	2.94512i	1.51998	+	4.67803i	−2.90474	+	0.749980i	−1.73463	−	5.19862i
266.12	−1.91048	+	1.38804i	1.58187	+	0.705480i	1.10523	−	3.40154i	−0.185262	+	2.22838i	−4.00135	+	0.847893i	−0.0746061	+	0.102687i	1.15049	+	3.54086i	2.00460	+	2.23195i	−2.73915	−	4.51442i
266.13	−1.87608	+	1.36305i	0.915952	−	1.47005i	1.04372	−	3.21225i	1.84934	−	1.25695i	0.285349	+	4.00640i	2.38827	−	3.28717i	0.987155	+	3.03815i	−1.32207	−	2.69298i	−1.75622	+	4.87888i
266.14	−1.85148	+	1.34518i	−1.65291	−	0.517578i	1.00043	−	3.07902i	2.04696	+	0.899966i	3.75656	−	1.26517i	2.29866	−	3.16383i	0.875140	+	2.69340i	2.46423	+	1.71102i	−5.00052	+	1.08726i
266.15	−1.78305	+	1.29546i	−0.204988	+	1.71988i	0.883006	−	2.71761i	0.816400	−	2.08170i	−1.86253	−	3.33217i	1.54088	−	2.12084i	0.583988	+	1.79733i	−2.91596	−	0.705109i	1.24108	+	4.76939i
266.16	−1.74985	+	1.27134i	−0.994474	−	1.41810i	0.827640	−	2.54721i	0.884754	−	2.05358i	3.54308	+	1.21716i	−0.340967	+	0.469301i	0.453364	+	1.39531i	−1.02204	+	2.82054i	1.06262	+	4.71830i
266.17	−1.72707	+	1.25479i	−1.20589	+	1.24332i	0.790234	−	2.43209i	1.36631	+	1.77008i	0.522549	−	3.66042i	1.20879	−	1.66376i	0.367608	+	1.13138i	−0.0916737	−	2.99860i	−4.58079	−	1.34261i
266.18	−1.71699	+	1.24746i	1.44098	−	0.961034i	0.773845	−	2.38165i	−1.68695	−	1.46772i	−1.27528	+	3.44765i	2.75730	−	3.79509i	0.330681	+	1.01773i	1.15283	−	2.76965i	4.72740	+	0.415643i
266.19	−1.64362	+	1.19416i	−1.73079	−	0.0661690i	0.657441	−	2.02339i	1.64077	−	1.51917i	2.92378	−	1.95808i	−1.27164	+	1.75026i	0.0800599	+	0.246399i	2.99124	+	0.229049i	−0.882668	+	4.45629i
266.20	−1.64023	+	1.19170i	1.44240	−	0.958890i	0.652175	−	2.00719i	1.64469	+	1.51492i	−1.22317	+	3.29170i	−1.94569	+	2.67801i	0.0692190	+	0.213034i	1.16106	−	2.76621i	−4.50299	−	0.524847i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
275.u	odd	10	1	inner
825.bf	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.bf.a	✓	464
3.b	odd	2	1	inner	825.2.bf.a	✓	464
11.d	odd	10	1		825.2.bj.a	yes	464
25.d	even	5	1		825.2.bj.a	yes	464
33.f	even	10	1		825.2.bj.a	yes	464
75.j	odd	10	1		825.2.bj.a	yes	464
275.u	odd	10	1	inner	825.2.bf.a	✓	464
825.bf	even	10	1	inner	825.2.bf.a	✓	464

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.bf.a	✓	464	1.a	even	1	1	trivial
825.2.bf.a	✓	464	3.b	odd	2	1	inner
825.2.bf.a	✓	464	275.u	odd	10	1	inner
825.2.bf.a	✓	464	825.bf	even	10	1	inner
825.2.bj.a	yes	464	11.d	odd	10	1
825.2.bj.a	yes	464	25.d	even	5	1
825.2.bj.a	yes	464	33.f	even	10	1
825.2.bj.a	yes	464	75.j	odd	10	1

Hecke kernels

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