Newform orbit 357.2.bf.a

• Label: 357.2.bf.a
• Level: $357$
• Weight: $2$
• Character orbit: 357.bf
• Analytic conductor: $2.851$
• Analytic rank: $0$
• Dimension: $288$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 357 = 3 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	357.bf (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.85065935216\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 288 * q - 48 * q^12 - 48 * q^24 - 64 * q^25 - 64 * q^31 - 224 * q^34 + 16 * q^39 + 64 * q^40 - 64 * q^43 + 64 * q^45 + 32 * q^51 + 128 * q^52 + 112 * q^54 - 32 * q^57 + 32 * q^58 - 144 * q^60 + 32 * q^61 + 64 * q^64 - 192 * q^66 - 32 * q^70 - 208 * q^72 - 128 * q^76 - 208 * q^78 - 64 * q^79 - 112 * q^81 - 256 * q^82 + 32 * q^85 + 32 * q^88 + 240 * q^90 - 96 * q^91 + 64 * q^93 + 304 * q^96

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
29.1	−2.48994	+	1.03137i	1.58249	+	0.704079i	3.72186	−	3.72186i	0.718104	−	3.61015i	−4.66646	−	0.120987i	0.980785	−	0.195090i	−3.36586	+	8.12591i	2.00855	+	2.22840i	1.93535	+	9.72968i
29.2	−2.38783	+	0.989070i	−0.945455	−	1.45125i	3.30924	−	3.30924i	0.133245	−	0.669869i	3.69296	+	2.53020i	−0.980785	+	0.195090i	−2.65068	+	6.39930i	−1.21223	+	2.74418i	0.344381	+	1.73132i
29.3	−2.36925	+	0.981377i	−0.983386	+	1.42582i	3.23605	−	3.23605i	−0.206402	+	1.03765i	0.930628	−	4.34319i	0.980785	−	0.195090i	−2.52849	+	6.10430i	−1.06590	−	2.80426i	−0.529310	−	2.66102i
29.4	−2.15985	+	0.894639i	0.732732	+	1.56943i	2.45036	−	2.45036i	−0.423607	+	2.12961i	−2.98666	−	2.73420i	−0.980785	+	0.195090i	−1.31095	+	3.16490i	−1.92621	+	2.29994i	−0.990310	−	4.97862i
29.5	−2.13685	+	0.885113i	−1.63908	+	0.559851i	2.36850	−	2.36850i	0.750620	−	3.77362i	3.00693	−	2.64709i	−0.980785	+	0.195090i	−1.19452	+	2.88382i	2.37313	−	1.83528i	1.73612	+	8.72806i
29.6	−2.09079	+	0.866034i	−1.45186	−	0.944517i	2.20718	−	2.20718i	−0.342644	+	1.72259i	3.85351	+	0.717430i	0.980785	−	0.195090i	−0.971185	+	2.34465i	1.21578	+	2.74261i	−0.775421	−	3.89831i
29.7	−1.98193	+	0.820944i	1.72218	−	0.184679i	1.83990	−	1.83990i	−0.0197905	+	0.0994937i	−3.26163	+	1.77983i	−0.980785	+	0.195090i	−0.494214	+	1.19314i	2.93179	−	0.636101i	−0.0424552	−	0.213437i
29.8	−1.65580	+	0.685853i	0.581390	−	1.63156i	0.857051	−	0.857051i	0.523196	−	2.63028i	0.156347	+	3.10028i	0.980785	−	0.195090i	0.540415	−	1.30468i	−2.32397	−	1.89714i	0.937683	+	4.71405i
29.9	−1.48790	+	0.616310i	1.67664	−	0.434588i	0.419805	−	0.419805i	−0.347854	+	1.74878i	−2.22684	+	1.67996i	0.980785	−	0.195090i	0.866720	−	2.09245i	2.62227	−	1.45730i	−0.560218	−	2.81641i
29.10	−1.47181	+	0.609645i	0.286314	−	1.70822i	0.380353	−	0.380353i	−0.751337	+	3.77723i	0.620008	+	2.68873i	−0.980785	+	0.195090i	0.891362	−	2.15194i	−2.83605	−	0.978178i	−1.19694	−	6.01742i
29.11	−1.34674	+	0.557836i	−0.301109	+	1.70568i	0.0883016	−	0.0883016i	0.256673	−	1.29038i	−0.545974	−	2.46506i	0.980785	−	0.195090i	1.04601	−	2.52529i	−2.81867	−	1.02719i	0.374152	+	1.88099i
29.12	−1.26033	+	0.522046i	0.768461	+	1.55225i	−0.0983113	+	0.0983113i	0.660437	−	3.32024i	−1.77886	−	1.55517i	−0.980785	+	0.195090i	1.11667	−	2.69589i	−1.81893	+	2.38568i	0.900949	+	4.52938i
29.13	−0.927040	+	0.383993i	−1.08976	−	1.34627i	−0.702260	+	0.702260i	0.206206	−	1.03667i	1.52720	+	0.829583i	−0.980785	+	0.195090i	1.14935	−	2.77477i	−0.624860	+	2.93420i	0.206911	+	1.04021i
29.14	−0.685253	+	0.283841i	0.562722	+	1.63809i	−1.02521	+	1.02521i	−0.790762	+	3.97543i	−0.850565	−	0.962784i	0.980785	−	0.195090i	0.979213	−	2.36403i	−2.36669	+	1.84358i	−0.586518	−	2.94863i
29.15	−0.452544	+	0.187450i	1.19594	−	1.25289i	−1.24456	+	1.24456i	0.529229	−	2.66061i	−0.306359	+	0.791166i	−0.980785	+	0.195090i	0.704823	−	1.70159i	−0.139473	−	2.99676i	0.259232	+	1.30325i
29.16	−0.401587	+	0.166343i	−0.420869	−	1.68014i	−1.28061	+	1.28061i	−0.144922	+	0.728574i	0.448495	+	0.604714i	0.980785	−	0.195090i	0.633942	−	1.53047i	−2.64574	+	1.41424i	−0.0629941	−	0.316693i
29.17	−0.338546	+	0.140230i	−1.31719	+	1.12473i	−1.31926	+	1.31926i	0.566934	−	2.85017i	0.288207	−	0.565482i	0.980785	−	0.195090i	0.542091	−	1.30872i	0.469963	−	2.96296i	0.207747	+	1.04441i
29.18	−0.139618	+	0.0578315i	−1.72945	−	0.0948153i	−1.39806	+	1.39806i	0.140402	−	0.705848i	0.246946	−	0.0867791i	−0.980785	+	0.195090i	0.230005	−	0.555282i	2.98202	+	0.327957i	0.0212177	+	0.106668i
29.19	0.139618	−	0.0578315i	−0.574235	+	1.63409i	−1.39806	+	1.39806i	−0.140402	+	0.705848i	0.0143286	+	0.261357i	−0.980785	+	0.195090i	−0.230005	+	0.555282i	−2.34051	−	1.87671i	0.0212177	+	0.106668i
29.20	0.338546	−	0.140230i	−1.54318	+	0.786506i	−1.31926	+	1.31926i	−0.566934	+	2.85017i	−0.412145	+	0.482669i	0.980785	−	0.195090i	−0.542091	+	1.30872i	1.76282	−	2.42744i	0.207747	+	1.04441i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
17.e	odd	16	1	inner
51.i	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	357.2.bf.a	✓	288
3.b	odd	2	1	inner	357.2.bf.a	✓	288
17.e	odd	16	1	inner	357.2.bf.a	✓	288
51.i	even	16	1	inner	357.2.bf.a	✓	288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
357.2.bf.a	✓	288	1.a	even	1	1	trivial
357.2.bf.a	✓	288	3.b	odd	2	1	inner
357.2.bf.a	✓	288	17.e	odd	16	1	inner
357.2.bf.a	✓	288	51.i	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(357, [\chi])\).