Newform orbit 440.2.bd.a

• Label: 440.2.bd.a
• Level: $440$
• Weight: $2$
• Character orbit: 440.bd
• Analytic conductor: $3.513$
• Analytic rank: $0$
• Dimension: $272$
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	440.bd (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.51341768894\)
Analytic rank:	\(0\)
Dimension:	\(272\)
Relative dimension:	\(68\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 272 * q - 6 * q^4 + 2 * q^6 - 72 * q^9 - 12 * q^10 - 18 * q^14 + 6 * q^15 - 34 * q^16 + 4 * q^20 - 10 * q^24 - 6 * q^25 + 44 * q^26 - 38 * q^30 - 12 * q^31 - 40 * q^34 + 16 * q^36 - 20 * q^39 - 12 * q^40 - 12 * q^41 - 42 * q^44 + 42 * q^46 + 32 * q^49 - 18 * q^50 - 64 * q^54 - 34 * q^55 - 32 * q^56 - 46 * q^60 - 18 * q^64 + 4 * q^65 - 34 * q^66 + 22 * q^70 + 4 * q^71 - 20 * q^74 - 36 * q^76 + 28 * q^79 + 74 * q^80 - 24 * q^81 + 68 * q^84 - 18 * q^86 - 32 * q^89 + 32 * q^90 - 96 * q^94 - 86 * q^95 + 110 * 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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
69.1	−1.41421	−	0.00201151i	0.363134	+	1.11761i	1.99999	+	0.00568940i	−1.66040	+	1.49768i	−0.511301	−	1.58127i	−0.495674	−	0.161054i	−2.82840	−	0.0120690i	1.30986	−	0.951669i	2.35118	−	2.11470i
69.2	−1.41416	−	0.0120545i	−0.00450228	−	0.0138566i	1.99971	+	0.0340941i	−0.514302	−	2.17612i	0.00619992	+	0.0196498i	1.45391	+	0.472403i	−2.82750	−	0.0723201i	2.42688	−	1.76323i	0.701074	+	3.08358i
69.3	−1.39377	+	0.239585i	0.897907	+	2.76347i	1.88520	−	0.667852i	1.18714	+	1.89491i	−1.91356	−	3.63653i	1.03680	+	0.336878i	−2.46753	+	1.38250i	−4.40350	+	3.19933i	−2.10860	−	2.35666i
69.4	−1.39363	−	0.240425i	−0.703945	−	2.16652i	1.88439	+	0.670126i	−1.76148	+	1.37738i	0.460151	+	3.18857i	2.75216	+	0.894232i	−2.46502	−	1.38696i	−1.77122	+	1.28686i	2.78601	−	1.49605i
69.5	−1.38721	+	0.275042i	−1.05118	−	3.23521i	1.84870	−	0.763081i	2.15276	−	0.604677i	2.34803	+	4.19879i	−2.50710	−	0.814606i	−2.35466	+	1.56703i	−6.93453	+	5.03823i	−2.82002	+	1.43091i
69.6	−1.37727	−	0.321136i	0.139352	+	0.428881i	1.79374	+	0.884581i	1.76999	+	1.36643i	−0.0541962	−	0.635436i	−2.33222	−	0.757784i	−2.18640	−	1.79434i	2.26253	−	1.64382i	−1.99895	−	2.45035i
69.7	−1.32994	+	0.480907i	−0.313562	−	0.965045i	1.53746	−	1.27915i	2.23460	+	0.0809787i	0.881114	+	1.13265i	3.37030	+	1.09508i	−1.42956	+	2.44056i	1.59406	−	1.15815i	−3.01082	+	0.966940i
69.8	−1.26368	+	0.634921i	0.786928	+	2.42192i	1.19375	−	1.60467i	−2.11481	−	0.726346i	−2.53215	−	2.56088i	−3.88154	−	1.26119i	−0.489673	+	2.78572i	−2.81937	+	2.04839i	3.13361	−	0.424873i
69.9	−1.24780	+	0.665584i	−0.592589	−	1.82380i	1.11399	−	1.66103i	−2.23406	+	0.0946529i	1.95333	+	1.88132i	−1.32376	−	0.430115i	−0.284485	+	2.81408i	−0.548040	+	0.398175i	2.72466	−	1.60507i
69.10	−1.23898	−	0.681851i	−0.578005	−	1.77892i	1.07016	+	1.68960i	1.54673	−	1.61481i	−0.496817	+	2.59816i	1.21265	+	0.394015i	−0.173853	−	2.82308i	−0.403404	+	0.293090i	−3.01744	+	0.946085i
69.11	−1.20900	−	0.733710i	−0.662317	−	2.03840i	0.923339	+	1.77410i	0.896103	+	2.04866i	−0.694858	+	2.95037i	−0.776340	−	0.252248i	0.185365	−	2.82235i	−1.28937	+	0.936782i	0.419736	−	3.13430i
69.12	−1.20853	−	0.734470i	0.494804	+	1.52285i	0.921108	+	1.77526i	−1.76108	−	1.37789i	0.520501	−	2.20384i	1.87765	+	0.610085i	0.190687	−	2.82199i	0.352807	−	0.256329i	1.11631	+	2.95869i
69.13	−1.20813	−	0.735138i	0.882355	+	2.71561i	0.919144	+	1.77628i	0.988697	−	2.00561i	0.930351	−	3.92946i	−4.43032	−	1.43950i	0.195369	−	2.82167i	−4.16893	+	3.02891i	−2.66887	+	1.69621i
69.14	−1.15789	+	0.811963i	0.250044	+	0.769556i	0.681432	−	1.88033i	2.05403	−	0.883720i	−0.914375	−	0.688037i	−3.11966	−	1.01364i	0.737734	+	2.73052i	1.89736	−	1.37851i	−1.66080	+	2.69105i
69.15	−1.13003	+	0.850311i	0.250044	+	0.769556i	0.553941	−	1.92176i	−0.205737	+	2.22658i	−0.936920	−	0.657007i	3.11966	+	1.01364i	1.00812	+	2.64267i	1.89736	−	1.37851i	−1.66080	−	2.69105i
69.16	−1.07881	−	0.914421i	0.402650	+	1.23923i	0.327670	+	1.97298i	2.22341	−	0.237575i	0.698794	−	1.70509i	4.67210	+	1.51806i	1.45064	−	2.42810i	1.05349	−	0.765403i	−2.61589	−	1.77683i
69.17	−1.01860	+	0.981049i	−0.592589	−	1.82380i	0.0750873	−	1.99859i	−0.600343	−	2.15397i	2.39285	+	1.27636i	1.32376	+	0.430115i	1.88423	+	2.10943i	−0.548040	+	0.398175i	2.72466	+	1.60507i
69.18	−0.994343	+	1.00562i	0.786928	+	2.42192i	−0.0225629	−	1.99987i	−1.34431	−	1.78685i	−3.21802	−	1.61686i	3.88154	+	1.26119i	2.03356	+	1.96587i	−2.81937	+	2.04839i	3.13361	+	0.424873i
69.19	−0.976466	−	1.02299i	0.140964	+	0.433841i	−0.0930269	+	1.99784i	−1.73106	+	1.41543i	0.306170	−	0.567836i	−2.83764	−	0.922004i	2.13461	−	1.85565i	2.25870	−	1.64104i	3.13829	+	0.388745i
69.20	−0.868343	+	1.11624i	−0.313562	−	0.965045i	−0.491962	−	1.93855i	0.767545	+	2.10021i	1.34950	+	0.487980i	−3.37030	−	1.09508i	2.59107	+	1.13418i	1.59406	−	1.15815i	−3.01082	−	0.966940i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
8.b	even	2	1	inner
11.c	even	5	1	inner
40.f	even	2	1	inner
55.j	even	10	1	inner
88.o	even	10	1	inner
440.bd	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	440.2.bd.a	✓	272
5.b	even	2	1	inner	440.2.bd.a	✓	272
8.b	even	2	1	inner	440.2.bd.a	✓	272
11.c	even	5	1	inner	440.2.bd.a	✓	272
40.f	even	2	1	inner	440.2.bd.a	✓	272
55.j	even	10	1	inner	440.2.bd.a	✓	272
88.o	even	10	1	inner	440.2.bd.a	✓	272
440.bd	even	10	1	inner	440.2.bd.a	✓	272

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
440.2.bd.a	✓	272	1.a	even	1	1	trivial
440.2.bd.a	✓	272	5.b	even	2	1	inner
440.2.bd.a	✓	272	8.b	even	2	1	inner
440.2.bd.a	✓	272	11.c	even	5	1	inner
440.2.bd.a	✓	272	40.f	even	2	1	inner
440.2.bd.a	✓	272	55.j	even	10	1	inner
440.2.bd.a	✓	272	88.o	even	10	1	inner
440.2.bd.a	✓	272	440.bd	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(440, [\chi])\).