Newform orbit 41.6.g.a

• Label: 41.6.g.a
• Level: $41$
• Weight: $6$
• Character orbit: 41.g
• Analytic conductor: $6.576$
• Analytic rank: $0$
• Dimension: $136$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	41.g (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.57573661233\)
Analytic rank:	\(0\)
Dimension:	\(136\)
Relative dimension:	\(17\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 136 q - 10 q^{2} - 30 q^{3} + 538 q^{4} - 10 q^{5} + 154 q^{6} - 8 q^{7} - 10 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−10.4787	+	3.40473i	13.9573	−	13.9573i	72.3223	−	52.5452i	7.70981	+	10.6116i	−98.7333	+	193.775i	−50.4682	−	99.0495i	−371.702	+	511.605i		−	146.612i	−116.919	−	84.9464i
2.2	−10.1949	+	3.31251i	−20.1133	+	20.1133i	67.0740	−	48.7321i	−41.0968	−	56.5649i	138.427	−	271.678i	6.21373	+	12.1951i	−320.760	+	441.488i		−	566.090i	606.348	+	440.538i
2.3	−7.82759	+	2.54334i	−6.73936	+	6.73936i	28.9141	−	21.0073i	50.5405	+	69.5630i	35.6125	−	69.8934i	49.5410	+	97.2297i	−18.0918	+	24.9013i			152.162i	−572.532	−	415.969i
2.4	−7.12764	+	2.31591i	−1.15288	+	1.15288i	19.5512	−	14.2048i	−13.1241	−	18.0637i	5.54734	−	10.8873i	−56.4438	−	110.777i	34.5069	−	47.4946i			240.342i	135.377	+	98.3574i
2.5	−6.65360	+	2.16189i	12.0415	−	12.0415i	13.7081	−	9.95954i	−42.5494	−	58.5643i	−54.0869	+	106.151i	69.7388	+	136.870i	61.9118	−	85.2143i		−	46.9941i	409.716	+	297.676i
2.6	−4.22789	+	1.37373i	18.2619	−	18.2619i	−9.90059	+	7.19320i	40.6448	+	55.9427i	−52.1225	+	102.296i	16.9491	+	33.2645i	115.593	−	159.099i		−	423.994i	−248.692	−	180.685i
2.7	−3.16128	+	1.02716i	−10.7247	+	10.7247i	−16.9499	+	12.3148i	−37.3899	−	51.4628i	22.8877	−	44.9196i	30.8447	+	60.5362i	103.455	−	142.394i			12.9637i	171.061	+	124.283i
2.8	−1.52085	+	0.494154i	−18.2423	+	18.2423i	−23.8198	+	17.3061i	27.9753	+	38.5047i	18.7293	−	36.7584i	−21.7857	−	42.7568i	57.7523	−	79.4893i		−	422.567i	−61.5734	−	44.7357i
2.9	−0.901566	+	0.292936i	6.14732	−	6.14732i	−25.1615	+	18.2809i	8.14737	+	11.2139i	−3.74144	+	7.34299i	−113.599	−	222.950i	35.1600	−	48.3935i			167.421i	−10.6303	−	7.72340i
2.10	0.877377	−	0.285077i	5.23564	−	5.23564i	−25.2000	+	18.3089i	10.1607	+	13.9850i	3.10107	−	6.08619i	63.0303	+	123.704i	−34.2424	+	47.1307i			188.176i	12.9016	+	9.37355i
2.11	2.89825	−	0.941700i	15.7552	−	15.7552i	−18.3755	+	13.3506i	−57.3489	−	78.9339i	30.8260	−	60.4994i	−26.2342	−	51.4874i	−98.0036	+	134.890i		−	253.455i	−240.544	−	174.765i
2.12	4.63908	−	1.50733i	−7.30274	+	7.30274i	−6.63955	+	4.82391i	−7.71647	−	10.6208i	−22.8703	+	44.8856i	64.1837	+	125.968i	−115.278	+	158.666i			136.340i	−51.8063	−	37.6395i
2.13	5.74268	−	1.86591i	−4.58268	+	4.58268i	3.60825	−	2.62155i	64.4480	+	88.7050i	−17.7660	+	34.8677i	−58.4030	−	114.622i	−97.7441	+	134.533i			200.998i	535.620	+	389.151i
2.14	6.05529	−	1.96748i	−14.3792	+	14.3792i	6.90704	−	5.01826i	−43.2988	−	59.5957i	−58.7792	+	115.361i	−78.0065	−	153.096i	−87.8053	+	120.854i		−	170.520i	−379.441	−	275.680i
2.15	6.95824	−	2.26087i	18.2456	−	18.2456i	17.4170	−	12.6542i	30.9205	+	42.5584i	85.7064	−	168.208i	22.3126	+	43.7910i	−45.0315	+	61.9805i		−	422.806i	311.371	+	226.224i
2.16	9.23628	−	3.00105i	5.23103	−	5.23103i	50.4140	−	36.6279i	−14.1479	−	19.4729i	32.6167	−	64.0138i	−11.6199	−	22.8053i	173.049	−	238.182i			188.273i	−189.113	−	137.398i
2.17	10.0735	−	3.27309i	−19.7008	+	19.7008i	64.8744	−	47.1340i	27.0233	+	37.1944i	−133.974	+	262.940i	93.3337	+	183.178i	300.015	−	412.936i		−	533.245i	393.961	+	286.229i
5.1	−6.27783	−	8.64070i	12.9081	+	12.9081i	−25.3619	+	78.0559i	−7.01604	−	2.27965i	30.5001	−	192.570i	−20.6765	−	130.546i	508.627	−	165.263i			90.2377i	24.3478	+	74.9347i
5.2	−5.97478	−	8.22358i	−10.5857	−	10.5857i	−22.0407	+	67.8343i	89.6616	+	29.1328i	−23.8051	+	150.299i	15.9786	+	100.885i	380.173	−	123.526i		−	18.8871i	−296.132	−	911.401i
5.3	−4.81859	−	6.63222i	−11.7986	−	11.7986i	−10.8790	+	33.4821i	−57.1787	−	18.5785i	−21.3984	+	135.104i	−1.37821	−	8.70170i	24.9895	−	8.11959i			35.4156i	152.304	+	468.744i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
41.g	even	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	41.6.g.a	✓	136
41.g	even	20	1	inner	41.6.g.a	✓	136

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
41.6.g.a	✓	136	1.a	even	1	1	trivial
41.6.g.a	✓	136	41.g	even	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(41, [\chi])\).