Newform orbit 828.2.ba.a

• Label: 828.2.ba.a
• Level: $828$
• Weight: $2$
• Character orbit: 828.ba
• 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.ba (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.

\(\operatorname{Tr}(f)(q) = \) \( 2800 q - 27 q^{2} - 9 q^{4} - 54 q^{5} - 19 q^{6} - 36 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41419	−	0.00742597i	1.20428	+	1.24487i	1.99989	+	0.0210035i	0.405591	−	0.288820i	−1.69385	−	1.76943i	2.02668	−	0.491667i	−2.82808	−	0.0445542i	−0.0994049	+	2.99835i	−0.575729	+	0.405435i
59.2	−1.41376	+	0.0359176i	0.881599	+	1.49090i	1.99742	−	0.101557i	−1.11223	+	0.792014i	−1.29992	−	2.07611i	−3.99398	+	0.968928i	−2.82022	+	0.215320i	−1.44557	+	2.62875i	1.54397	−	1.15966i
59.3	−1.41252	+	0.0690924i	−1.67880	+	0.426175i	1.99045	−	0.195189i	−1.14817	+	0.817609i	2.34190	−	0.717975i	2.89752	−	0.702930i	−2.79808	+	0.413235i	2.63675	−	1.43093i	1.56533	−	1.23422i
59.4	−1.41186	−	0.0815026i	−0.549751	−	1.64249i	1.98671	+	0.230141i	3.42570	−	2.43943i	0.642306	+	2.36378i	3.30628	−	0.802096i	−2.78621	−	0.486850i	−2.39555	+	1.80592i	−5.03544	+	3.16494i
59.5	−1.40649	+	0.147582i	−1.34305	+	1.09372i	1.95644	−	0.415144i	1.79683	−	1.27951i	1.72757	−	1.73652i	−2.62013	+	0.635637i	−2.69045	+	0.872632i	0.607546	−	2.93784i	−2.33839	+	2.06481i
59.6	−1.40567	+	0.155256i	−0.402723	+	1.68458i	1.95179	−	0.436476i	2.57131	−	1.83102i	0.304552	−	2.43048i	4.35419	−	1.05632i	−2.67580	+	0.916567i	−2.67563	−	1.35684i	−3.33013	+	2.97302i
59.7	−1.39352	−	0.241060i	1.73204	+	0.00514474i	1.88378	+	0.671844i	0.536645	−	0.382143i	−2.41239	−	0.424696i	−4.30696	+	1.04486i	−2.46312	−	1.39033i	2.99995	+	0.0178218i	−0.839944	+	0.403159i
59.8	−1.39058	+	0.257447i	1.70312	−	0.315252i	1.86744	−	0.716003i	−2.01898	+	1.43771i	−2.28717	+	0.876848i	0.372850	−	0.0904525i	−2.41250	+	1.47643i	2.80123	−	1.07383i	2.43743	−	2.51904i
59.9	−1.39016	−	0.259747i	−0.620438	−	1.61711i	1.86506	+	0.722176i	0.929742	−	0.662066i	0.442465	+	2.40920i	−1.05251	+	0.255335i	−2.40514	−	1.48838i	−2.23011	+	2.00664i	−1.46445	+	0.678877i
59.10	−1.37822	+	0.317023i	−1.61030	−	0.637915i	1.79899	−	0.873855i	1.59294	−	1.13433i	2.42158	+	0.368687i	−2.98255	+	0.723560i	−2.20238	+	1.77469i	2.18613	+	2.05447i	−1.83582	+	2.06835i
59.11	−1.37350	+	0.336880i	0.400965	−	1.68500i	1.77302	−	0.925410i	−0.538518	+	0.383477i	0.0169153	+	2.44943i	0.845417	−	0.205096i	−2.12350	+	1.86835i	−2.67845	−	1.35125i	0.610471	−	0.708123i
59.12	−1.35368	−	0.409330i	−1.51686	−	0.836135i	1.66490	+	1.10820i	−2.85696	+	2.03443i	1.71109	+	1.75276i	−3.35039	+	0.812797i	−1.80012	−	2.18165i	1.60176	+	2.53661i	4.70016	−	1.58453i
59.13	−1.34631	−	0.432949i	1.13809	−	1.30566i	1.62511	+	1.16577i	0.713120	−	0.507811i	−2.09751	+	1.26509i	1.28415	−	0.311532i	−1.68319	−	2.27308i	−0.409489	−	2.97192i	−1.17994	+	0.374927i
59.14	−1.33984	−	0.452594i	−0.776969	+	1.54801i	1.59032	+	1.21280i	−2.37178	+	1.68894i	1.74163	−	1.72242i	−0.608887	+	0.147714i	−1.58186	−	2.34472i	−1.79264	−	2.40550i	3.94220	−	1.18945i
59.15	−1.33296	−	0.472468i	0.776969	−	1.54801i	1.55355	+	1.25956i	−2.37178	+	1.68894i	−1.76705	+	1.69633i	0.608887	−	0.147714i	−1.47571	−	2.41294i	−1.79264	−	2.40550i	3.95946	−	1.13069i
59.16	−1.32590	−	0.491914i	−1.13809	+	1.30566i	1.51604	+	1.30446i	0.713120	−	0.507811i	2.15127	−	1.17133i	−1.28415	+	0.311532i	−1.36844	−	2.47535i	−0.409489	−	2.97192i	−1.19533	+	0.322515i
59.17	−1.31710	−	0.515034i	1.51686	+	0.836135i	1.46948	+	1.35670i	−2.85696	+	2.03443i	−1.56722	−	1.88251i	3.35039	−	0.812797i	−1.23670	−	2.54373i	1.60176	+	2.53661i	4.81069	−	1.20811i
59.18	−1.30866	+	0.536111i	1.70061	−	0.328506i	1.42517	−	1.40317i	0.212902	−	0.151607i	−2.04940	+	1.34162i	3.12467	−	0.758037i	−1.11281	+	2.60032i	2.78417	−	1.11732i	−0.197338	+	0.312540i
59.19	−1.30864	+	0.536157i	−0.0276331	+	1.73183i	1.42507	−	1.40327i	−2.69157	+	1.91666i	−0.892371	−	2.28116i	2.82419	−	0.685141i	−1.11253	+	2.60044i	−2.99847	−	0.0957117i	2.49467	−	3.95132i
59.20	−1.30745	+	0.539051i	−0.112656	−	1.72838i	1.41885	−	1.40956i	−2.00581	+	1.42833i	1.07898	+	2.19905i	−4.63754	+	1.12506i	−1.09524	+	2.60777i	−2.97462	+	0.389424i	1.85255	−	2.94870i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
9.d	odd	6	1	inner
23.c	even	11	1	inner
36.h	even	6	1	inner
92.g	odd	22	1	inner
207.n	odd	66	1	inner
828.ba	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.ba.a	✓	2800
4.b	odd	2	1	inner	828.2.ba.a	✓	2800
9.d	odd	6	1	inner	828.2.ba.a	✓	2800
23.c	even	11	1	inner	828.2.ba.a	✓	2800
36.h	even	6	1	inner	828.2.ba.a	✓	2800
92.g	odd	22	1	inner	828.2.ba.a	✓	2800
207.n	odd	66	1	inner	828.2.ba.a	✓	2800
828.ba	even	66	1	inner	828.2.ba.a	✓	2800

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
828.2.ba.a	✓	2800	1.a	even	1	1	trivial
828.2.ba.a	✓	2800	4.b	odd	2	1	inner
828.2.ba.a	✓	2800	9.d	odd	6	1	inner
828.2.ba.a	✓	2800	23.c	even	11	1	inner
828.2.ba.a	✓	2800	36.h	even	6	1	inner
828.2.ba.a	✓	2800	92.g	odd	22	1	inner
828.2.ba.a	✓	2800	207.n	odd	66	1	inner
828.2.ba.a	✓	2800	828.ba	even	66	1	inner

Hecke kernels

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