Newform orbit 211.3.h.a

• Label: 211.3.h.a
• Level: $211$
• Weight: $3$
• Character orbit: 211.h
• Analytic conductor: $5.749$
• Analytic rank: $0$
• Dimension: $210$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 211 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	211.h (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.74933357800\)
Analytic rank:	\(0\)
Dimension:	\(210\)
Relative dimension:	\(35\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 210 q - 14 q^{2} - 7 q^{3} + 86 q^{4} - 5 q^{5} - 29 q^{6} - 7 q^{7} + 28 q^{8} + 84 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−3.12152	−	2.48933i	3.72817	+	2.97311i	2.65704	+	11.6413i	4.90715	+	6.15337i	−4.23648	−	18.5613i	−2.97145	+	2.36965i	13.7556	−	28.5639i	3.05713	+	13.3942i		−	31.4234i
12.2	−2.96821	−	2.36707i	−1.29374	−	1.03172i	2.31717	+	10.1522i	−4.74098	−	5.94501i	1.39793	+	6.12474i	−10.2269	+	8.15569i	10.5641	−	21.9366i	−1.39338	−	6.10478i			28.8682i
12.3	−2.91060	−	2.32113i	−3.42232	−	2.72921i	2.19388	+	9.61203i	−0.335702	−	0.420958i	3.62617	+	15.8873i	10.2742	−	8.19339i	9.46416	−	19.6525i	2.26101	+	9.90612i			2.00445i
12.4	−2.67985	−	2.13711i	3.05747	+	2.43825i	1.72428	+	7.55455i	−4.70665	−	5.90195i	−2.98275	−	13.0683i	6.00293	−	4.78718i	5.57528	−	11.5772i	1.40037	+	6.13542i			25.8750i
12.5	−2.61508	−	2.08545i	−0.273653	−	0.218231i	1.59942	+	7.00753i	2.22469	+	2.78967i	0.260513	+	1.14138i	3.11887	−	2.48722i	4.62624	−	9.60648i	−1.97543	−	8.65491i		−	11.9347i
12.6	−2.36565	−	1.88654i	−1.66421	−	1.32716i	1.14717	+	5.02610i	3.38176	+	4.24059i	1.43319	+	6.27920i	−3.57148	+	2.84816i	1.51679	−	3.14966i	−0.994458	−	4.35701i		−	16.4116i
12.7	−2.22173	−	1.77177i	1.93212	+	1.54082i	0.906833	+	3.97309i	−0.575519	−	0.721678i	−1.56268	−	6.84657i	−0.298098	+	0.237725i	0.0928055	−	0.192713i	−0.643704	−	2.82025i			2.62306i
12.8	−2.01169	−	1.60427i	−3.98391	−	3.17706i	0.583135	+	2.55488i	−1.88955	−	2.36942i	2.91753	+	12.7825i	−2.70632	+	2.15822i	−1.53998	+	3.19781i	3.77512	+	16.5399i			7.79791i
12.9	−1.77964	−	1.41922i	3.86307	+	3.08069i	0.262864	+	1.15168i	−1.51575	−	1.90069i	−2.50270	−	10.9651i	−7.98720	+	6.36958i	−2.78382	+	5.78067i	3.42993	+	15.0275i			5.53373i
12.10	−1.54448	−	1.23168i	2.84980	+	2.27264i	−0.0217028	−	0.0950863i	4.21706	+	5.28803i	−1.60229	−	7.02011i	7.97995	−	6.36380i	−3.51209	+	7.29293i	0.953787	+	4.17881i		−	13.3614i
12.11	−1.49658	−	1.19348i	−0.942502	−	0.751620i	−0.0747329	−	0.327426i	−4.97722	−	6.24123i	0.513484	+	2.24972i	9.31868	−	7.43140i	−3.60109	+	7.47774i	−1.67931	−	7.35754i			15.2807i
12.12	−1.38668	−	1.10584i	−1.06303	−	0.847742i	−0.190082	−	0.832804i	−3.20669	−	4.02107i	0.536623	+	2.35110i	−3.21364	+	2.56279i	−3.73557	+	7.75699i	−1.59131	−	6.97199i			9.12204i
12.13	−1.20508	−	0.961017i	−3.73520	−	2.97872i	−0.361426	−	1.58351i	3.52229	+	4.41681i	1.63860	+	7.17917i	−0.130461	+	0.104039i	−3.76131	+	7.81043i	3.07623	+	13.4778i		−	8.70757i
12.14	−0.767347	−	0.611939i	1.11383	+	0.888246i	−0.675731	−	2.96057i	1.45951	+	1.83017i	−0.311138	−	1.36319i	0.354098	−	0.282384i	−2.99655	+	6.22240i	−1.55106	−	6.79565i		−	2.29750i
12.15	−0.753224	−	0.600676i	0.456916	+	0.364379i	−0.683549	−	2.99483i	5.42411	+	6.80162i	−0.125287	−	0.548917i	−10.2335	+	8.16092i	−2.95608	+	6.13837i	−1.92669	−	8.44137i		−	8.38128i
12.16	−0.282677	−	0.225427i	4.38408	+	3.49618i	−0.860995	−	3.77227i	−3.10488	−	3.89339i	−0.451142	−	1.97658i	7.22476	−	5.76155i	−1.23448	+	2.56343i	4.99413	+	21.8807i			1.80050i
12.17	−0.219987	−	0.175433i	−2.28732	−	1.82408i	−0.872467	−	3.82253i	2.08434	+	2.61368i	0.183175	+	0.802544i	9.38675	−	7.48568i	−0.967001	+	2.00800i	−0.0981140	−	0.429865i		−	0.940639i
12.18	−0.194930	−	0.155452i	1.47328	+	1.17490i	−0.876251	−	3.83911i	−3.43433	−	4.30652i	−0.104546	−	0.458047i	−1.81792	+	1.44974i	−0.858701	+	1.78311i	−1.21253	−	5.31245i			1.37334i
12.19	0.160901	+	0.128314i	−2.83794	−	2.26318i	−0.880659	−	3.85842i	−0.621369	−	0.779173i	−0.166228	−	0.728294i	−2.38865	+	1.90488i	0.710564	−	1.47550i	0.929215	+	4.07116i		−	0.205100i
12.20	0.442013	+	0.352494i	−4.17377	−	3.32847i	−0.818960	−	3.58810i	−5.65264	−	7.08818i	−0.671596	−	2.94245i	1.93871	−	1.54607i	1.88399	−	3.91214i	4.33895	+	19.0102i		−	5.12559i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
211.h	odd	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	211.3.h.a	✓	210
211.h	odd	14	1	inner	211.3.h.a	✓	210

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
211.3.h.a	✓	210	1.a	even	1	1	trivial
211.3.h.a	✓	210	211.h	odd	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(211, [\chi])\).