Newform orbit 825.2.df.a

• Label: 825.2.df.a
• Level: $825$
• Weight: $2$
• Character orbit: 825.df
• Analytic conductor: $6.588$
• Analytic rank: $0$
• Dimension: $928$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.df (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.58765816676\)
Analytic rank:	\(0\)
Dimension:	\(928\)
Relative dimension:	\(116\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 928 q - 8 q^{3} - 20 q^{4} - 12 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
47.1	−2.48687	−	1.26712i	−1.65361	−	0.515345i	3.40336	+	4.68432i	1.50735	+	1.65163i	3.45931	+	3.37692i	3.77531	−	1.92362i	−1.65485	−	10.4483i	2.46884	+	1.70436i	−1.65577	−	6.01740i
47.2	−2.44354	−	1.24505i	1.72011	+	0.203064i	3.24518	+	4.46661i	−2.01179	−	0.976071i	−3.95033	−	2.63781i	−0.0338425	+	0.0172436i	−1.51057	−	9.53738i	2.91753	+	0.698582i	3.70063	+	4.88984i
47.3	−2.38420	−	1.21481i	−0.820531	−	1.52536i	3.03308	+	4.17468i	−1.77392	+	1.36133i	0.103285	+	4.63356i	−2.17060	+	1.10598i	−1.32284	−	8.35208i	−1.65346	+	2.50321i	5.88314	−	1.09071i
47.4	−2.36745	−	1.20628i	1.40764	−	1.00923i	2.97414	+	4.09356i	0.867634	+	2.06088i	−4.54993	+	0.691288i	−3.72982	+	1.90044i	−1.27187	−	8.03025i	0.962920	−	2.84126i	0.431906	−	5.92562i
47.5	−2.36159	−	1.20329i	1.26930	+	1.17850i	2.95361	+	4.06530i	2.22097	−	0.259427i	−1.57949	−	4.31047i	−0.841968	+	0.429004i	−1.25423	−	7.91893i	0.222255	+	2.99176i	−5.55717	−	2.05981i
47.6	−2.33049	−	1.18744i	0.968595	−	1.43590i	2.84558	+	3.91661i	1.21379	−	1.87795i	−3.96236	+	2.19621i	4.45325	−	2.26904i	−1.16252	−	7.33985i	−1.12365	−	2.78162i	−5.05869	+	2.93525i
47.7	−2.31589	−	1.18000i	−1.72256	−	0.181118i	2.79536	+	3.84748i	0.0509419	−	2.23549i	3.77553	+	2.45207i	−1.80027	+	0.917286i	−1.12049	−	7.07451i	2.93439	+	0.623972i	−2.75586	+	5.11703i
47.8	−2.25078	−	1.14683i	−1.44175	+	0.959875i	2.57522	+	3.54448i	−2.09173	+	0.790353i	4.34587	−	0.507027i	−1.94030	+	0.988634i	−0.940987	−	5.94116i	1.15728	−	2.76780i	5.61442	+	0.619949i
47.9	−2.19654	−	1.11919i	−0.145071	+	1.72596i	2.39662	+	3.29867i	0.283131	+	2.21807i	2.25034	−	3.62879i	1.86232	−	0.948899i	−0.801134	−	5.05816i	−2.95791	−	0.500775i	1.86054	−	5.18896i
47.10	−2.19278	−	1.11728i	−0.778909	+	1.54703i	2.38440	+	3.28185i	2.17755	−	0.508209i	3.43644	−	2.52204i	−0.805757	+	0.410554i	−0.791758	−	4.99897i	−1.78660	−	2.40999i	−5.34269	−	1.31853i
47.11	−2.19148	−	1.11662i	−0.520963	−	1.65185i	2.38019	+	3.27605i	1.90080	−	1.17770i	−0.702796	+	4.20171i	−2.38823	+	1.21686i	−0.788537	−	4.97862i	−2.45719	+	1.72110i	−5.48060	+	0.458445i
47.12	−2.18427	−	1.11294i	0.764500	−	1.55420i	2.35684	+	3.24391i	−1.59663	−	1.56550i	−3.39962	+	2.54396i	−0.928825	+	0.473260i	−0.770707	−	4.86605i	−1.83108	−	2.37637i	1.74516	+	5.19643i
47.13	−2.11608	−	1.07819i	1.54389	+	0.785110i	2.13970	+	2.94505i	−1.26947	+	1.84077i	−2.42049	−	3.32597i	2.84903	−	1.45165i	−0.609396	−	3.84757i	1.76721	+	2.42425i	4.67101	−	2.52647i
47.14	−1.98361	−	1.01070i	0.772418	+	1.55028i	1.73764	+	2.39165i	0.322563	−	2.21268i	0.0346933	−	3.85584i	3.59764	−	1.83309i	−0.333027	−	2.10265i	−1.80674	+	2.39493i	−2.87620	+	4.06309i
47.15	−1.94987	−	0.993510i	−1.11005	−	1.32958i	1.63937	+	2.25641i	−2.18297	−	0.484412i	0.843506	+	3.69536i	3.74929	−	1.91036i	−0.270131	−	1.70554i	−0.535574	+	2.95181i	3.77524	+	3.11334i
47.16	−1.93694	−	0.986923i	1.58319	−	0.702508i	1.60217	+	2.20520i	0.440812	+	2.19219i	−3.75987	−	0.201764i	1.67916	−	0.855574i	−0.246813	−	1.55832i	2.01296	−	2.22440i	1.30969	−	4.68119i
47.17	−1.88094	−	0.958385i	0.779304	+	1.54683i	1.44385	+	1.98729i	−2.21840	+	0.280559i	0.0166388	−	3.65636i	−1.68901	+	0.860592i	−0.150727	−	0.951656i	−1.78537	+	2.41090i	4.44155	+	1.59836i
47.18	−1.87668	−	0.956217i	−0.0509892	+	1.73130i	1.43201	+	1.97100i	−0.759305	−	2.10320i	1.75119	−	3.20034i	−4.37869	+	2.23105i	−0.143753	−	0.907618i	−2.99480	−	0.176555i	−0.586142	+	4.67310i
47.19	−1.85549	−	0.945422i	−1.72605	+	0.144053i	1.37347	+	1.89042i	−1.57420	−	1.58804i	3.33887	+	1.36456i	1.39961	−	0.713139i	−0.109681	−	0.692500i	2.95850	−	0.497284i	1.41955	+	4.43489i
47.20	−1.84932	−	0.942277i	0.160937	−	1.72456i	1.35654	+	1.86711i	−1.33342	+	1.79499i	−1.92264	+	3.03762i	1.73663	−	0.884856i	−0.0999645	−	0.631151i	−2.94820	−	0.555090i	4.15730	−	2.06307i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
275.bp	odd	20	1	inner
825.df	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	825.2.df.a	yes	928
3.b	odd	2	1	inner	825.2.df.a	yes	928
11.c	even	5	1		825.2.cv.a	✓	928
25.f	odd	20	1		825.2.cv.a	✓	928
33.h	odd	10	1		825.2.cv.a	✓	928
75.l	even	20	1		825.2.cv.a	✓	928
275.bp	odd	20	1	inner	825.2.df.a	yes	928
825.df	even	20	1	inner	825.2.df.a	yes	928

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
825.2.cv.a	✓	928	11.c	even	5	1
825.2.cv.a	✓	928	25.f	odd	20	1
825.2.cv.a	✓	928	33.h	odd	10	1
825.2.cv.a	✓	928	75.l	even	20	1
825.2.df.a	yes	928	1.a	even	1	1	trivial
825.2.df.a	yes	928	3.b	odd	2	1	inner
825.2.df.a	yes	928	275.bp	odd	20	1	inner
825.2.df.a	yes	928	825.df	even	20	1	inner

Hecke kernels

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