Newform orbit 828.2.bc.a

• Label: 828.2.bc.a
• Level: $828$
• Weight: $2$
• Character orbit: 828.bc
• Analytic conductor: $6.612$
• Analytic rank: $0$
• Dimension: $2800$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	828.bc (of order \(66\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.61161328736\)
Analytic rank:	\(0\)
Dimension:	\(2800\)
Relative dimension:	\(140\) over \(\Q(\zeta_{66})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{66}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2800 q - 9 q^{2} - 9 q^{4} - 22 q^{5} - 11 q^{6} - 30 q^{8} - 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−1.41378	−	0.0349503i	0.879752	+	1.49199i	1.99756	+	0.0988242i	3.58986	+	2.55633i	−1.19163	−	2.14010i	1.10213	−	4.54304i	−2.82066	−	0.209531i	−1.45207	+	2.62516i	−4.98593	−	3.73955i
7.2	−1.41333	+	0.0498581i	−1.63313	−	0.576962i	1.99503	−	0.140932i	2.13191	+	1.51813i	2.33693	+	0.734016i	−0.181224	+	0.747014i	−2.81262	+	0.298653i	2.33423	+	1.88451i	−3.08879	−	2.03933i
7.3	−1.41307	+	0.0567420i	0.952273	+	1.44678i	1.99356	−	0.160361i	−2.29028	−	1.63090i	−1.42773	−	1.99038i	0.425048	−	1.75207i	−2.80795	+	0.339721i	−1.18635	+	2.75546i	3.32887	+	2.17463i
7.4	−1.41271	−	0.0651497i	−1.45418	+	0.940936i	1.99151	+	0.184076i	1.59544	+	1.13611i	2.11564	−	1.23453i	−0.222022	+	0.915188i	−2.80144	−	0.389792i	1.22928	−	2.73658i	−2.17988	−	1.70894i
7.5	−1.41107	−	0.0942838i	0.897019	−	1.48167i	1.98222	+	0.266082i	2.23100	+	1.58869i	−1.40545	+	2.00617i	−0.735480	+	3.03169i	−2.77196	−	0.562350i	−1.39071	−	2.65818i	−2.99831	−	2.45210i
7.6	−1.40973	−	0.112564i	−0.254837	+	1.71320i	1.97466	+	0.317369i	−1.92549	−	1.37113i	0.552095	−	2.38646i	−1.16890	+	4.81827i	−2.74800	−	0.669679i	−2.87012	−	0.873173i	2.56007	+	2.14966i
7.7	−1.40499	−	0.161213i	1.15882	−	1.28730i	1.94802	+	0.453007i	0.518240	+	0.369037i	−1.83567	+	1.62183i	0.972868	−	4.01022i	−2.66393	−	0.950519i	−0.314267	−	2.98349i	−0.668631	−	0.602042i
7.8	−1.40293	+	0.178259i	−0.305651	−	1.70487i	1.93645	−	0.500171i	−2.46845	−	1.75778i	0.732716	+	2.33733i	−0.312606	+	1.28858i	−2.62755	+	1.04690i	−2.81316	+	1.04219i	3.77642	+	2.02602i
7.9	−1.40161	−	0.188397i	1.72568	−	0.148373i	1.92901	+	0.528117i	−0.503255	−	0.358366i	−2.44669	−	0.117153i	−0.0226963	+	0.0935555i	−2.60423	−	1.10363i	2.95597	−	0.512088i	0.637852	+	0.597101i
7.10	−1.38227	+	0.298859i	−1.28953	+	1.15634i	1.82137	−	0.826210i	0.199061	+	0.141751i	1.43690	−	1.98376i	0.636225	−	2.62255i	−2.27071	+	1.68638i	0.325777	−	2.98226i	−0.317521	−	0.136448i
7.11	−1.37475	+	0.331740i	1.69049	+	0.377145i	1.77990	−	0.912121i	1.84506	+	1.31386i	−2.44912	+	0.0423209i	−0.644505	+	2.65669i	−2.14434	+	1.84440i	2.71552	+	1.27512i	−2.97237	−	1.19416i
7.12	−1.36251	+	0.378916i	−1.71392	−	0.249964i	1.71285	−	1.03255i	−2.37058	−	1.68808i	2.42994	−	0.308853i	−0.415469	+	1.71258i	−1.94251	+	2.05588i	2.87504	+	0.856836i	3.86957	+	1.40177i
7.13	−1.34732	−	0.429788i	−1.34777	+	1.08789i	1.63057	+	1.15813i	−3.17290	−	2.25941i	2.28345	−	0.886492i	0.934274	−	3.85113i	−1.69915	−	2.26117i	0.632971	−	2.93246i	3.30386	+	4.40784i
7.14	−1.34580	+	0.434530i	1.59703	−	0.670443i	1.62237	−	1.16958i	−2.36401	−	1.68340i	−1.85796	+	1.59624i	−0.562009	+	2.31663i	−1.67517	+	2.27899i	2.10101	−	2.14144i	3.91297	+	1.23829i
7.15	−1.34119	+	0.448564i	0.287118	−	1.70809i	1.59758	−	1.20322i	0.338060	+	0.240732i	0.381106	+	2.41966i	0.592868	−	2.44384i	−1.60294	+	2.33036i	−2.83513	−	0.980846i	−0.561386	−	0.171225i
7.16	−1.31972	−	0.508266i	−1.07230	−	1.36021i	1.48333	+	1.34154i	−1.29077	−	0.919151i	0.723792	+	2.34011i	0.955331	−	3.93793i	−1.27573	−	2.52439i	−0.700337	+	2.91711i	1.23628	+	1.86908i
7.17	−1.31526	−	0.519713i	1.72096	+	0.195659i	1.45980	+	1.36711i	−0.287396	−	0.204654i	−2.16182	−	1.15175i	0.481166	−	1.98339i	−1.20950	−	2.55678i	2.92343	+	0.673446i	0.271638	+	0.418536i
7.18	−1.31208	+	0.527684i	0.0176969	+	1.73196i	1.44310	−	1.38473i	−0.621476	−	0.442551i	−0.937148	−	2.26313i	0.0952313	−	0.392549i	−1.16276	+	2.57837i	−2.99937	+	0.0613006i	1.04895	+	0.252718i
7.19	−1.30675	+	0.540750i	−0.928423	−	1.46220i	1.41518	−	1.41325i	2.61476	+	1.86196i	2.00390	+	1.40868i	0.489950	−	2.01960i	−1.08507	+	2.61202i	−1.27606	+	2.71508i	−4.42369	−	1.01918i
7.20	−1.28682	−	0.586596i	−0.683111	+	1.59165i	1.31181	+	1.50969i	1.55644	+	1.10834i	1.81270	−	1.64746i	−0.0487718	+	0.201040i	−0.802485	−	2.71220i	−2.06672	−	2.17455i	−1.35271	−	2.33923i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.c	even	3	1	inner
23.d	odd	22	1	inner
36.f	odd	6	1	inner
92.h	even	22	1	inner
207.p	odd	66	1	inner
828.bc	even	66	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	828.2.bc.a	✓	2800
4.b	odd	2	1	inner	828.2.bc.a	✓	2800
9.c	even	3	1	inner	828.2.bc.a	✓	2800
23.d	odd	22	1	inner	828.2.bc.a	✓	2800
36.f	odd	6	1	inner	828.2.bc.a	✓	2800
92.h	even	22	1	inner	828.2.bc.a	✓	2800
207.p	odd	66	1	inner	828.2.bc.a	✓	2800
828.bc	even	66	1	inner	828.2.bc.a	✓	2800

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
828.2.bc.a	✓	2800	1.a	even	1	1	trivial
828.2.bc.a	✓	2800	4.b	odd	2	1	inner
828.2.bc.a	✓	2800	9.c	even	3	1	inner
828.2.bc.a	✓	2800	23.d	odd	22	1	inner
828.2.bc.a	✓	2800	36.f	odd	6	1	inner
828.2.bc.a	✓	2800	92.h	even	22	1	inner
828.2.bc.a	✓	2800	207.p	odd	66	1	inner
828.2.bc.a	✓	2800	828.bc	even	66	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(828, [\chi])\).