Newform orbit 783.2.bf.a

• Label: 783.2.bf.a
• Level: $783$
• Weight: $2$
• Character orbit: 783.bf
• Analytic conductor: $6.252$
• Analytic rank: $0$
• Dimension: $3168$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 783 = 3^{3} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	783.bf (of order \(126\), degree \(36\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.25228647827\)
Analytic rank:	\(0\)
Dimension:	\(3168\)
Relative dimension:	\(88\) over \(\Q(\zeta_{126})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{126}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 3168 * q - 42 * q^2 - 42 * q^3 - 30 * q^4 - 30 * q^5 - 60 * q^6 - 30 * q^7 - 21 * q^8 - 42 * q^9 - 21 * q^10 - 42 * q^11 - 30 * q^13 - 42 * q^14 - 42 * q^15 - 18 * q^16 - 189 * q^18 - 21 * q^19 + 24 * q^20 - 42 * q^21 - 30 * q^22 + 60 * q^23 + 60 * q^24 - 30 * q^25 - 84 * q^26 - 168 * q^27 - 144 * q^28 - 72 * q^29 - 54 * q^30 - 42 * q^31 - 42 * q^32 + 60 * q^33 - 60 * q^34 - 45 * q^35 + 60 * q^36 - 21 * q^37 - 42 * q^39 - 42 * q^40 - 30 * q^42 - 42 * q^43 - 21 * q^44 + 12 * q^45 - 42 * q^47 - 105 * q^48 - 120 * q^49 - 42 * q^50 - 57 * q^51 - 132 * q^53 - 12 * q^54 - 84 * q^55 - 126 * q^56 - 36 * q^57 - 63 * q^58 - 66 * q^59 - 42 * q^60 - 42 * q^61 + 3 * q^62 + 150 * q^63 - 237 * q^64 + 66 * q^65 - 42 * q^66 - 30 * q^67 - 126 * q^68 - 42 * q^69 - 111 * q^71 - 42 * q^72 - 21 * q^73 + 270 * q^74 - 42 * q^76 - 42 * q^77 - 66 * q^78 - 42 * q^79 - 246 * q^80 - 42 * q^81 - 60 * q^82 - 228 * q^83 - 168 * q^84 - 147 * q^85 + 36 * q^86 + 39 * q^87 - 12 * q^88 - 21 * q^89 - 42 * q^90 - 15 * q^91 + 120 * q^92 - 24 * q^93 + 96 * q^94 - 210 * q^95 - 258 * q^96 - 42 * q^97 - 21 * q^98

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
4.1	−2.31815	−	1.49722i	−1.61840	−	0.617070i	2.30957	+	5.11854i	−1.12745	−	0.475261i	2.82781	+	3.85356i	1.11127	−	2.46284i	1.48704	−	9.86590i	2.23845	+	1.99734i	1.90203	+	2.78977i
4.2	−2.29980	−	1.48537i	0.191103	+	1.72148i	2.26020	+	5.00912i	2.36394	+	0.996485i	2.11753	−	4.24292i	−0.465087	+	1.03074i	1.42629	−	9.46282i	−2.92696	+	0.657957i	−3.95645	−	5.80305i
4.3	−2.21274	−	1.42914i	1.69038	−	0.377660i	2.03122	+	4.50164i	1.52196	+	0.641561i	−4.28010	−	1.58012i	−0.818764	+	1.81457i	1.15372	−	7.65445i	2.71475	−	1.27677i	−2.45083	−	3.59471i
4.4	−2.17690	−	1.40599i	−1.59884	+	0.666118i	1.93951	+	4.29840i	−1.03633	−	0.436851i	4.41707	+	0.797882i	−2.00776	+	4.44965i	1.04891	−	6.95909i	2.11257	−	2.13003i	1.64179	+	2.40806i
4.5	−2.16762	−	1.40000i	1.07791	+	1.35577i	1.91601	+	4.24632i	−3.42091	−	1.44203i	−0.438428	−	4.44787i	−1.35513	+	3.00328i	1.02247	−	6.78362i	−0.676217	+	2.92279i	5.39639	+	7.91505i
4.6	−2.16409	−	1.39772i	−0.553725	−	1.64115i	1.90709	+	4.22654i	−1.65832	−	0.699040i	−1.09556	+	4.32555i	−0.678395	+	1.50348i	1.01247	−	6.71729i	−2.38678	+	1.81750i	2.61169	+	3.83064i
4.7	−2.14537	−	1.38563i	1.66689	+	0.470612i	1.86008	+	4.12237i	−2.51815	−	1.06149i	−2.92401	−	3.31933i	1.83118	−	4.05831i	0.960210	−	6.37057i	2.55705	+	1.56892i	3.93154	+	5.76650i
4.8	−2.14186	−	1.38336i	0.885768	−	1.48843i	1.85129	+	4.10289i	2.11365	+	0.890980i	−3.95621	+	1.96266i	1.44917	−	3.21169i	0.950516	−	6.30626i	−1.43083	−	2.63680i	−3.29460	−	4.83229i
4.9	−2.04840	−	1.32300i	−1.48439	+	0.892509i	1.62306	+	3.59706i	1.66007	+	0.699777i	4.22143	+	0.135634i	1.38685	−	3.07357i	0.707358	−	4.69301i	1.40685	−	2.64967i	−2.47468	−	3.62969i
4.10	−2.00959	−	1.29793i	0.816487	−	1.52753i	1.53126	+	3.39361i	−3.11245	−	1.31201i	−3.62344	+	2.00997i	−0.613105	+	1.35878i	0.614372	−	4.07609i	−1.66670	−	2.49442i	4.55185	+	6.67634i
4.11	−1.91602	−	1.23750i	0.932239	+	1.45977i	1.31716	+	2.91914i	1.16159	+	0.489651i	0.0202736	−	3.95060i	1.07427	−	2.38083i	0.408806	−	2.71225i	−1.26186	+	2.72171i	−1.61969	−	2.37565i
4.12	−1.87098	−	1.20841i	−1.55739	−	0.757984i	1.21774	+	2.69879i	3.92962	+	1.65647i	1.99789	+	3.30013i	0.0840813	−	0.186343i	0.318948	−	2.11608i	1.85092	+	2.36095i	−5.35054	−	7.84780i
4.13	−1.86293	−	1.20321i	−1.30729	+	1.13622i	1.20023	+	2.65998i	−3.38341	−	1.42623i	3.80250	−	0.543755i	0.375103	−	0.831313i	0.303504	−	2.01361i	0.418015	−	2.97073i	4.58701	+	6.72791i
4.14	−1.73247	−	1.11895i	−1.02598	+	1.39548i	0.926827	+	2.05406i	2.39352	+	1.00895i	3.33894	−	1.26961i	−0.679465	+	1.50585i	0.0779155	−	0.516936i	−0.894742	−	2.86347i	−3.01773	−	4.42620i
4.15	−1.72710	−	1.11548i	1.65466	−	0.511946i	0.916003	+	2.03007i	−0.987871	−	0.416423i	−3.42883	−	0.961560i	−0.431716	+	0.956780i	0.0696099	−	0.461831i	2.47582	−	1.69420i	1.24164	+	1.82115i
4.16	−1.59639	−	1.03106i	−0.0687713	+	1.73068i	0.662799	+	1.46891i	−1.71473	−	0.722821i	1.89422	−	2.69193i	0.554417	−	1.22871i	−0.110032	+	0.730015i	−2.99054	−	0.238043i	1.99211	+	2.92189i
4.17	−1.59066	−	1.02736i	−1.56230	−	0.747815i	0.652159	+	1.44533i	−3.66732	−	1.54591i	1.71681	+	2.79455i	0.987458	−	2.18843i	−0.116938	+	0.775835i	1.88155	+	2.33662i	4.24526	+	6.22665i
4.18	−1.58421	−	1.02319i	1.57107	+	0.729195i	0.640218	+	1.41887i	2.56954	+	1.08315i	−1.74280	−	2.76270i	−1.47186	+	3.26199i	−0.124625	+	0.826830i	1.93655	+	2.29124i	−2.96241	−	4.34506i
4.19	−1.57501	−	1.01725i	−0.626339	−	1.61484i	0.623282	+	1.38133i	−0.440790	−	0.185809i	−0.656201	+	3.18052i	−0.351653	+	0.779343i	−0.135408	+	0.898374i	−2.21540	+	2.02287i	0.505234	+	0.741042i
4.20	−1.56918	−	1.01348i	−1.65922	−	0.496973i	0.612595	+	1.35765i	0.678126	+	0.285854i	2.09994	+	2.46143i	−1.58084	+	3.50351i	−0.142141	+	0.943043i	2.50604	+	1.64918i	−0.774392	−	1.13582i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
27.e	even	9	1	inner
29.e	even	14	1	inner
783.bf	even	126	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	783.2.bf.a	✓	3168
27.e	even	9	1	inner	783.2.bf.a	✓	3168
29.e	even	14	1	inner	783.2.bf.a	✓	3168
783.bf	even	126	1	inner	783.2.bf.a	✓	3168

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
783.2.bf.a	✓	3168	1.a	even	1	1	trivial
783.2.bf.a	✓	3168	27.e	even	9	1	inner
783.2.bf.a	✓	3168	29.e	even	14	1	inner
783.2.bf.a	✓	3168	783.bf	even	126	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(783, [\chi])\).