Newform orbit 220.3.i.a

• Label: 220.3.i.a
• Level: $220$
• Weight: $3$
• Character orbit: 220.i
• Analytic conductor: $5.995$
• Analytic rank: $0$
• Dimension: $136$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	220.i (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.99456581593\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(68\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 136 * q - 8 * q^5 + 8 * q^12 + 16 * q^16 + 80 * q^20 - 96 * q^22 - 8 * q^25 - 160 * q^26 + 80 * q^33 - 104 * q^36 - 8 * q^37 - 16 * q^38 - 168 * q^42 + 192 * q^45 + 32 * q^48 + 136 * q^53 + 264 * q^56 - 248 * q^58 - 128 * q^60 + 328 * q^66 + 96 * q^70 - 152 * q^77 - 24 * q^78 - 624 * q^80 - 664 * q^81 - 200 * q^82 + 752 * q^86 - 296 * q^88 - 776 * q^92 + 592 * q^93 - 168 * q^97

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
43.1	−1.99997	+	0.0112748i	3.70405	−	3.70405i	3.99975	−	0.0450984i	−3.59444	+	3.47563i	−7.36622	+	7.44975i	−1.62459	+	1.62459i	−7.99886	+	0.135292i		−	18.4400i	7.14958	−	6.99168i
43.2	−1.99978	−	0.0293959i	1.37211	−	1.37211i	3.99827	+	0.117571i	−3.60976	−	3.45972i	−2.78426	+	2.70359i	−5.40554	+	5.40554i	−7.99222	−	0.352649i			5.23463i	7.11703	+	7.02480i
43.3	−1.97731	−	0.300385i	3.24619	−	3.24619i	3.81954	+	1.18791i	1.98058	−	4.59100i	−7.39385	+	5.44363i	7.61337	−	7.61337i	−7.19559	−	3.49620i		−	12.0755i	−5.29530	+	8.48291i
43.4	−1.97634	−	0.306706i	−0.409452	+	0.409452i	3.81186	+	1.21231i	4.83179	+	1.28601i	0.934798	−	0.683636i	5.71256	−	5.71256i	−7.16172	−	3.56507i			8.66470i	−9.15485	−	4.02353i
43.5	−1.96707	+	0.361417i	−2.82785	+	2.82785i	3.73876	−	1.42187i	3.27930	−	3.77442i	4.54056	−	6.58463i	−1.01424	+	1.01424i	−6.84052	+	4.14816i		−	6.99351i	−5.08648	+	8.60974i
43.6	−1.96076	−	0.394239i	−2.39705	+	2.39705i	3.68915	+	1.54602i	1.59525	+	4.73869i	5.64505	−	3.75503i	−9.72511	+	9.72511i	−6.62404	−	4.48577i		−	2.49170i	−1.25973	−	9.92034i
43.7	−1.91212	+	0.586345i	−1.31103	+	1.31103i	3.31240	−	2.24232i	−3.80088	+	3.24859i	1.73813	−	3.27556i	7.34311	−	7.34311i	−5.01893	+	6.22979i			5.56240i	5.36294	−	8.44031i
43.8	−1.87429	−	0.697868i	−0.715296	+	0.715296i	3.02596	+	2.61602i	−4.99120	+	0.296598i	1.83986	−	0.841493i	2.22084	−	2.22084i	−3.84590	−	7.01492i			7.97670i	9.56196	+	2.92729i
43.9	−1.84841	+	0.763796i	1.18814	−	1.18814i	2.83323	−	2.82361i	1.57917	+	4.74407i	−1.28867	+	3.10367i	−1.57737	+	1.57737i	−3.08030	+	7.38321i			6.17665i	−6.54246	−	7.56282i
43.10	−1.80130	−	0.869100i	−3.80725	+	3.80725i	2.48933	+	3.13101i	−3.43663	−	3.63175i	10.1669	−	3.54910i	2.92353	−	2.92353i	−1.76286	−	7.80335i		−	19.9903i	3.03403	+	9.52862i
43.11	−1.73178	+	1.00047i	1.22071	−	1.22071i	1.99810	−	3.46520i	4.48571	−	2.20872i	−0.892710	+	3.33529i	−3.81872	+	3.81872i	0.00657265	+	8.00000i			6.01972i	−5.55847	+	8.31284i
43.12	−1.71305	+	1.03221i	−2.84767	+	2.84767i	1.86907	−	3.53646i	−4.99552	+	0.211543i	1.93880	−	7.81761i	−7.03107	+	7.03107i	0.448578	+	7.98741i		−	7.21849i	8.33922	−	5.51883i
43.13	−1.67372	−	1.09483i	0.0944542	−	0.0944542i	1.60268	+	3.66489i	1.01027	−	4.89687i	−0.261502	+	0.0546783i	−6.56836	+	6.56836i	1.33000	−	7.88867i			8.98216i	−7.05216	+	7.08992i
43.14	−1.64403	+	1.13893i	1.80964	−	1.80964i	1.40568	−	3.74487i	−2.37711	−	4.39879i	−0.914055	+	5.03615i	4.00592	−	4.00592i	1.95415	+	7.75766i			2.45042i	8.91796	+	4.52439i
43.15	−1.61925	−	1.17390i	2.04929	−	2.04929i	1.24394	+	3.80166i	−1.54252	+	4.75612i	−5.72396	+	0.912660i	0.191923	−	0.191923i	2.44850	−	7.61609i			0.600827i	8.08090	−	5.89059i
43.16	−1.49067	+	1.33338i	−3.87627	+	3.87627i	0.444207	−	3.97526i	3.64792	+	3.41945i	0.609714	−	10.9468i	5.44609	−	5.44609i	4.63836	+	6.51810i		−	21.0509i	−9.99728	−	0.233225i
43.17	−1.48754	−	1.33687i	2.95023	−	2.95023i	0.425555	+	3.97730i	4.93806	+	0.784567i	−8.33266	+	0.444512i	−2.85360	+	2.85360i	4.68410	−	6.48530i		−	8.40772i	−6.29670	−	7.76862i
43.18	−1.33687	−	1.48754i	−2.95023	+	2.95023i	−0.425555	+	3.97730i	4.93806	+	0.784567i	8.33266	+	0.444512i	2.85360	−	2.85360i	6.48530	−	4.68410i		−	8.40772i	−5.43447	−	8.39443i
43.19	−1.33338	+	1.49067i	3.87627	−	3.87627i	−0.444207	−	3.97526i	3.64792	+	3.41945i	0.609714	+	10.9468i	5.44609	−	5.44609i	6.51810	+	4.63836i		−	21.0509i	−9.96134	+	0.878432i
43.20	−1.17390	−	1.61925i	−2.04929	+	2.04929i	−1.24394	+	3.80166i	−1.54252	+	4.75612i	5.72396	+	0.912660i	−0.191923	+	0.191923i	7.61609	−	2.44850i			0.600827i	9.51209	−	3.08546i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
5.c	odd	4	1	inner
11.b	odd	2	1	inner
20.e	even	4	1	inner
44.c	even	2	1	inner
55.e	even	4	1	inner
220.i	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	220.3.i.a	✓	136
4.b	odd	2	1	inner	220.3.i.a	✓	136
5.c	odd	4	1	inner	220.3.i.a	✓	136
11.b	odd	2	1	inner	220.3.i.a	✓	136
20.e	even	4	1	inner	220.3.i.a	✓	136
44.c	even	2	1	inner	220.3.i.a	✓	136
55.e	even	4	1	inner	220.3.i.a	✓	136
220.i	odd	4	1	inner	220.3.i.a	✓	136

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
220.3.i.a	✓	136	1.a	even	1	1	trivial
220.3.i.a	✓	136	4.b	odd	2	1	inner
220.3.i.a	✓	136	5.c	odd	4	1	inner
220.3.i.a	✓	136	11.b	odd	2	1	inner
220.3.i.a	✓	136	20.e	even	4	1	inner
220.3.i.a	✓	136	44.c	even	2	1	inner
220.3.i.a	✓	136	55.e	even	4	1	inner
220.3.i.a	✓	136	220.i	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(220, [\chi])\).