Newform orbit 784.2.bu.a

• Label: 784.2.bu.a
• Level: $784$
• Weight: $2$
• Character orbit: 784.bu
• Analytic conductor: $6.260$
• Analytic rank: $0$
• Dimension: $2640$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.bu (of order \(84\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.26027151847\)
Analytic rank:	\(0\)
Dimension:	\(2640\)
Relative dimension:	\(110\) over \(\Q(\zeta_{84})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{84}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 2640 q - 26 q^{2} - 22 q^{3} - 26 q^{4} - 22 q^{5} - 28 q^{6} - 48 q^{7} - 32 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−1.41295	−	0.0598329i	−0.477750	+	0.411137i	1.99284	+	0.169081i	0.341215	+	1.80337i	0.699635	−	0.552330i	−0.278928	+	2.63101i	−2.80566	−	0.358140i	−0.387915	+	2.57365i	−0.374219	−	2.56848i
3.2	−1.41289	+	0.0611892i	−0.423528	+	0.364476i	1.99251	−	0.172907i	0.286550	+	1.51445i	0.576097	−	0.540879i	−2.11921	−	1.58397i	−2.80462	+	0.366219i	−0.400593	+	2.65776i	−0.497531	−	2.12222i
3.3	−1.41058	−	0.101327i	−2.35531	+	2.02691i	1.97947	+	0.285861i	−0.761045	−	4.02222i	3.52773	−	2.62046i	−2.07791	+	1.63777i	−2.76323	−	0.603803i	0.992002	−	6.58151i	0.665953	+	5.75077i
3.4	−1.40991	−	0.110296i	−2.38429	+	2.05185i	1.97567	+	0.311014i	0.173214	+	0.915456i	3.58794	−	2.62994i	−0.104451	−	2.64369i	−2.75120	−	0.656409i	1.02764	−	6.81793i	−0.143244	−	1.30981i
3.5	−1.40498	−	0.161383i	−0.738363	+	0.635413i	1.94791	+	0.453478i	−0.482842	−	2.55188i	1.13993	−	0.773580i	2.64044	−	0.167511i	−2.66358	−	0.951485i	−0.305696	+	2.02816i	0.266551	+	3.66325i
3.6	−1.39796	+	0.213812i	0.740104	−	0.636911i	1.90857	−	0.597801i	−0.464197	−	2.45334i	−0.898454	+	1.04862i	0.934344	−	2.47528i	−2.54028	+	1.24378i	−0.305029	+	2.02373i	1.17348	+	3.33042i
3.7	−1.39794	+	0.213917i	2.52866	−	2.17608i	1.90848	−	0.598088i	−0.429699	−	2.27101i	−3.06941	+	3.58296i	−0.0376352	−	2.64548i	−2.54000	+	1.24435i	1.21163	−	8.03865i	1.08650	+	3.08282i
3.8	−1.38880	+	0.266912i	−0.580980	+	0.499973i	1.85752	−	0.741374i	0.731707	+	3.86717i	0.673414	−	0.849432i	2.43222	−	1.04129i	−2.38183	+	1.52541i	−0.359563	+	2.38554i	−2.04839	−	5.17541i
3.9	−1.37814	+	0.317370i	1.46191	−	1.25808i	1.79855	−	0.874761i	−0.230178	−	1.21652i	−1.61545	+	2.19778i	0.853569	+	2.50428i	−2.20104	+	1.77635i	0.107303	−	0.711907i	0.703305	+	1.60349i
3.10	−1.37426	−	0.333791i	1.16710	−	1.00437i	1.77717	+	0.917431i	−0.00272970	−	0.0144268i	−1.93914	+	0.990692i	−1.96323	+	1.77362i	−2.13605	−	1.85399i	−0.0937677	+	0.622108i	−0.00106424	+	0.0207373i
3.11	−1.36569	−	0.367288i	−0.363930	+	0.313187i	1.73020	+	1.00320i	−0.537071	−	2.83849i	0.612044	−	0.294048i	−2.43222	−	1.04130i	−1.99445	−	2.00554i	−0.412768	+	2.73854i	−0.309071	+	4.07374i
3.12	−1.34869	+	0.425495i	2.29474	−	1.97478i	1.63791	−	1.14772i	0.603650	+	3.19037i	−2.25462	+	3.63977i	2.06426	+	1.65494i	−1.72067	+	2.24483i	0.918937	−	6.09675i	−2.17162	−	4.04595i
3.13	−1.32578	−	0.492241i	1.49281	−	1.28467i	1.51540	+	1.30521i	0.0922089	+	0.487335i	−2.61151	+	0.968369i	2.62654	−	0.318237i	−1.36661	−	2.47636i	0.130987	−	0.869045i	0.117637	−	0.691490i
3.14	−1.32410	+	0.496746i	1.31464	−	1.13134i	1.50649	−	1.31548i	0.487008	+	2.57390i	−1.17873	+	2.15105i	−2.64005	−	0.173595i	−1.34128	+	2.49018i	0.00122341	−	0.00811677i	−1.92342	−	3.16618i
3.15	−1.31896	−	0.510245i	−1.79927	+	1.54840i	1.47930	+	1.34598i	0.388792	+	2.05482i	3.16323	−	1.12420i	2.63106	+	0.278398i	−1.26436	−	2.53010i	0.392715	−	2.60549i	0.535658	−	2.90859i
3.16	−1.30393	+	0.547519i	−1.10641	+	0.952145i	1.40044	−	1.42785i	−0.181655	−	0.960069i	0.921362	−	1.84731i	−2.09792	+	1.61206i	−1.04430	+	2.62858i	−0.129558	+	0.859562i	0.762521	+	1.15240i
3.17	−1.20352	+	0.742655i	−1.60912	+	1.38476i	0.896926	−	1.78760i	−0.641898	−	3.39251i	0.908211	−	2.86161i	1.74901	−	1.98518i	0.248103	+	2.81752i	0.224582	−	1.49000i	3.29201	+	3.60625i
3.18	−1.19429	−	0.757417i	−2.14471	+	1.84567i	0.852639	+	1.80915i	0.102880	+	0.543734i	3.95934	−	0.579821i	0.0443640	+	2.64538i	0.351981	−	2.80644i	0.746145	−	4.95035i	0.288965	−	0.727298i
3.19	−1.18904	+	0.765625i	−1.67174	+	1.43865i	0.827638	−	1.82072i	0.112121	+	0.592576i	0.886302	−	2.99054i	−1.54747	−	2.14601i	0.409892	+	2.79857i	0.277877	−	1.84359i	−0.587007	−	0.618754i
3.20	−1.18337	−	0.774362i	2.42314	−	2.08528i	0.800726	+	1.83271i	0.376724	+	1.99104i	−4.48224	+	0.591270i	−2.64488	−	0.0679851i	0.471629	−	2.78883i	1.07609	−	7.13938i	1.09598	−	2.64785i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.f	odd	4	1	inner
49.h	odd	42	1	inner
784.bu	even	84	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	784.2.bu.a	✓	2640
16.f	odd	4	1	inner	784.2.bu.a	✓	2640
49.h	odd	42	1	inner	784.2.bu.a	✓	2640
784.bu	even	84	1	inner	784.2.bu.a	✓	2640

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
784.2.bu.a	✓	2640	1.a	even	1	1	trivial
784.2.bu.a	✓	2640	16.f	odd	4	1	inner
784.2.bu.a	✓	2640	49.h	odd	42	1	inner
784.2.bu.a	✓	2640	784.bu	even	84	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(784, [\chi])\).