Newform orbit 671.2.bf.a

• Label: 671.2.bf.a
• Level: $671$
• Weight: $2$
• Character orbit: 671.bf
• Analytic conductor: $5.358$
• Analytic rank: $0$
• Dimension: $240$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 671 = 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	671.bf (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.35796197563\)
Analytic rank:	\(0\)
Dimension:	\(240\)
Relative dimension:	\(60\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 5 q^{2} - 6 q^{3} + 57 q^{4} - 5 q^{5} - 5 q^{6} - 15 q^{8} + 218 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−2.62422	+	0.852661i	−0.405012			4.54146	−	3.29957i	−0.810673	−	2.49499i	1.06284	−	0.345338i		−	3.50066i	−5.86067	+	8.06652i	−2.83597			4.25477	+	5.85618i
3.2	−2.48545	+	0.807571i	−2.48310			3.90724	−	2.83878i	−0.0125394	−	0.0385923i	6.17161	−	2.00528i			2.54923i	−4.34655	+	5.98252i	3.16577			0.0623321	+	0.0857928i
3.3	−2.46722	+	0.801650i	1.56678			3.82652	−	2.78013i	0.825061	+	2.53928i	−3.86561	+	1.25601i		−	1.65448i	−4.16254	+	5.72924i	−0.545186			−4.07122	−	5.60356i
3.4	−2.42547	+	0.788084i	−2.48087			3.64380	−	2.64738i	1.11477	+	3.43092i	6.01727	−	1.95513i		−	2.62680i	−3.75354	+	5.16630i	3.15470			−5.40771	−	7.44307i
3.5	−2.35266	+	0.764425i	−0.388378			3.33262	−	2.42129i	0.633288	+	1.94906i	0.913719	−	0.296885i			1.33868i	−3.08157	+	4.24141i	−2.84916			−2.97982	−	4.10137i
3.6	−2.34934	+	0.763347i	2.58727			3.31867	−	2.41116i	−0.373145	−	1.14842i	−6.07839	+	1.97499i		−	3.22657i	−3.05220	+	4.20099i	3.69399			1.75329	+	2.41320i
3.7	−2.30175	+	0.747884i	3.06134			3.12069	−	2.26731i	−0.581156	−	1.78861i	−7.04644	+	2.28953i			4.63656i	−2.64223	+	3.63672i	6.37182			2.67535	+	3.68230i
3.8	−2.28041	+	0.740949i	−0.542633			3.03321	−	2.20376i	−1.32197	−	4.06860i	1.23742	−	0.402063i			3.07449i	−2.46534	+	3.39326i	−2.70555			6.02924	+	8.29854i
3.9	−2.03198	+	0.660230i	−2.07835			2.07500	−	1.50758i	−0.418628	−	1.28841i	4.22316	−	1.37219i			0.959472i	−0.709351	+	0.976337i	1.31953			1.70129	+	2.34162i
3.10	−1.98413	+	0.644683i	1.35252			1.90312	−	1.38270i	0.299038	+	0.920345i	−2.68357	+	0.871943i			0.913873i	−0.432117	+	0.594758i	−1.17070			−1.18666	−	1.63330i
3.11	−1.92017	+	0.623902i	−0.705145			1.67977	−	1.22043i	−0.286636	−	0.882176i	1.35400	−	0.439941i		−	1.77796i	−0.0905617	+	0.124648i	−2.50277			1.10078	+	1.51510i
3.12	−1.91660	+	0.622741i	2.97696			1.66752	−	1.21152i	1.34172	+	4.12939i	−5.70565	+	1.85388i			1.11864i	−0.0724523	+	0.0997220i	5.86230			−5.14309	−	7.07886i
3.13	−1.79444	+	0.583050i	−2.79814			1.26205	−	0.916930i	−0.244291	−	0.751849i	5.02111	−	1.63146i		−	4.22739i	0.488001	−	0.671676i	4.82959			0.876731	+	1.20672i
3.14	−1.64915	+	0.535840i	−0.411027			0.814524	−	0.591787i	0.644687	+	1.98414i	0.677844	−	0.220245i			3.95750i	1.01229	−	1.39330i	−2.83106			−2.12637	−	2.92669i
3.15	−1.51734	+	0.493013i	1.72213			0.441218	−	0.320563i	−0.439917	−	1.35393i	−2.61306	+	0.849034i		−	0.281865i	1.36410	−	1.87752i	−0.0342583			1.33501	+	1.83748i
3.16	−1.49061	+	0.484327i	1.16968			0.369299	−	0.268312i	−1.05025	−	3.23233i	−1.74353	+	0.566507i		−	1.74141i	1.42196	−	1.95716i	−1.63185			3.13101	+	4.30946i
3.17	−1.32968	+	0.432039i	−3.10032			−0.0366444	+	0.0266237i	0.596603	+	1.83615i	4.12243	−	1.33946i			3.46480i	1.68080	−	2.31342i	6.61195			−1.58658	−	2.18374i
3.18	−1.31280	+	0.426555i	0.844742			−0.0765358	+	0.0556065i	0.717660	+	2.20873i	−1.10898	+	0.360329i		−	4.26500i	1.69947	−	2.33912i	−2.28641			−1.88429	−	2.59350i
3.19	−1.10092	+	0.357712i	1.47015			−0.533956	+	0.387942i	−0.409645	−	1.26076i	−1.61853	+	0.525892i			5.03616i	1.80989	−	2.49110i	−0.838647			0.901977	+	1.24146i
3.20	−1.04528	+	0.339631i	−1.08963			−0.640777	+	0.465552i	0.946142	+	2.91192i	1.13897	−	0.370073i		−	0.954402i	1.80371	−	2.48259i	−1.81270			−1.97796	−	2.72243i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
671.bf	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	671.2.bf.a	yes	240
11.c	even	5	1		671.2.bd.a	✓	240
61.g	even	10	1		671.2.bd.a	✓	240
671.bf	even	10	1	inner	671.2.bf.a	yes	240

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
671.2.bd.a	✓	240	11.c	even	5	1
671.2.bd.a	✓	240	61.g	even	10	1
671.2.bf.a	yes	240	1.a	even	1	1	trivial
671.2.bf.a	yes	240	671.bf	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(671, [\chi])\).