Newform orbit 41.4.g.a

• Label: 41.4.g.a
• Level: $41$
• Weight: $4$
• Character orbit: 41.g
• Analytic conductor: $2.419$
• Analytic rank: $0$
• Dimension: $72$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 41 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	41.g (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.41907831024\)
Analytic rank:	\(0\)
Dimension:	\(72\)
Relative dimension:	\(9\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 72 q - 10 q^{2} - 6 q^{3} + 50 q^{4} - 10 q^{5} + 34 q^{6} - 8 q^{7} - 10 q^{8}+O(q^{10}) \)

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	−4.76602	+	1.54857i	0.979111	−	0.979111i	13.8448	−	10.0588i	−1.91087	−	2.63009i	−3.15024	+	6.18269i	11.4456	+	22.4632i	−26.8431	+	36.9464i			25.0827i	13.1801	+	9.57594i
2.2	−3.63877	+	1.18231i	−4.50934	+	4.50934i	5.37067	−	3.90202i	2.26520	+	3.11778i	11.0770	−	21.7399i	−12.4881	−	24.5094i	3.06183	−	4.21425i		−	13.6684i	−11.9287	−	8.66671i
2.3	−2.54890	+	0.828189i	5.83812	−	5.83812i	−0.661124	+	0.480334i	−5.76100	−	7.92934i	−10.0457	+	19.7159i	−5.18321	−	10.1726i	13.8898	−	19.1177i		−	41.1672i	21.2512	+	15.4399i
2.4	−1.06188	+	0.345024i	0.816922	−	0.816922i	−5.46360	+	3.96954i	8.31290	+	11.4417i	−0.585612	+	1.14933i	5.62200	+	11.0338i	9.68228	−	13.3265i			25.6653i	−12.7749	−	9.28154i
2.5	−0.543508	+	0.176597i	−1.89239	+	1.89239i	−6.20792	+	4.51032i	−11.8509	−	16.3113i	0.694340	−	1.36272i	−2.02497	−	3.97423i	5.26480	−	7.24638i			19.8377i	9.32156	+	6.77251i
2.6	1.78953	−	0.581455i	−5.65431	+	5.65431i	−3.60780	+	2.62122i	2.43065	+	3.34550i	−6.83085	+	13.4063i	2.81928	+	5.53315i	−13.7801	+	18.9667i		−	36.9423i	6.29498	+	4.57357i
2.7	2.38542	−	0.775071i	4.89142	−	4.89142i	−1.38262	+	1.00454i	2.59830	+	3.57626i	7.87692	−	15.4593i	−4.25789	−	8.35658i	−14.3137	+	19.7012i		−	20.8521i	8.96991	+	6.51702i
2.8	3.85948	−	1.25402i	1.31887	−	1.31887i	6.85090	−	4.97747i	−9.20186	−	12.6653i	3.43627	−	6.74405i	14.0788	+	27.6311i	1.11674	−	1.53706i			23.5212i	−51.3969	−	37.3421i
2.9	4.61774	−	1.50039i	−2.14636	+	2.14636i	12.6002	−	9.15458i	4.00581	+	5.51352i	−6.69095	+	13.1317i	−10.4236	−	20.4575i	21.6176	−	29.7540i			17.7862i	26.7702	+	19.4497i
5.1	−3.03980	−	4.18392i	0.554855	+	0.554855i	−5.79271	+	17.8281i	−14.8078	−	4.81135i	0.634824	−	4.00812i	4.06905	+	25.6909i	52.8522	−	17.1727i		−	26.3843i	24.8824	+	76.5803i
5.2	−2.41737	−	3.32723i	3.85967	+	3.85967i	−2.75463	+	8.47787i	18.1187	+	5.88712i	3.51174	−	22.1723i	−1.58515	−	10.0082i	3.57563	−	1.16179i			2.79408i	−24.2119	−	74.5165i
5.3	−2.17428	−	2.99263i	−6.21630	−	6.21630i	−1.75625	+	5.40517i	6.08367	+	1.97670i	−5.08716	+	32.1191i	0.120097	+	0.758265i	−8.15016	+	2.64815i			50.2848i	−7.31202	−	22.5041i
5.4	−1.14493	−	1.57586i	−0.218114	−	0.218114i	1.29967	−	3.99998i	−6.88944	−	2.23852i	−0.0939918	+	0.593441i	−2.09736	−	13.2422i	−22.6116	+	7.34697i		−	26.9049i	4.36032	+	13.4197i
5.5	0.133879	+	0.184268i	4.66849	+	4.66849i	2.45610	−	7.55911i	0.683943	+	0.222227i	−0.235243	+	1.48527i	3.23594	+	20.4309i	3.45469	−	1.12250i			16.5896i	0.0506161	+	0.155780i
5.6	0.934571	+	1.28633i	−3.05175	−	3.05175i	1.69092	−	5.20412i	17.6993	+	5.75086i	1.07347	−	6.77763i	−0.840598	−	5.30733i	20.3718	−	6.61921i		−	8.37363i	9.14379	+	28.1417i
5.7	1.06468	+	1.46541i	−5.24109	−	5.24109i	1.45826	−	4.48807i	−15.0586	−	4.89284i	2.10024	−	13.2604i	0.653216	+	4.12424i	21.9109	−	7.11930i			27.9381i	−8.86261	−	27.2763i
5.8	2.06663	+	2.84448i	5.04567	+	5.04567i	−1.34794	+	4.14854i	−7.49217	−	2.43435i	−3.92474	+	24.7798i	−5.56453	−	35.1330i	12.1649	−	3.95263i			23.9175i	−8.55911	−	26.3422i
5.9	2.71006	+	3.73007i	−0.180194	−	0.180194i	−4.09690	+	12.6090i	1.11473	+	0.362199i	0.183801	−	1.16047i	1.96039	+	12.3774i	−23.0555	+	7.49117i		−	26.9351i	1.66996	+	5.13962i
8.1	−2.80698	+	3.86348i	−3.77990	−	3.77990i	−4.57520	−	14.0810i	10.1326	−	3.29229i	25.2137	−	3.99345i	−3.68666	−	0.583909i	30.9099	+	10.0432i			1.57521i	−15.7224	+	48.3887i
8.2	−2.05594	+	2.82975i	−0.288405	−	0.288405i	−1.30849	−	4.02712i	−14.1211	+	4.58822i	1.40906	−	0.223173i	−0.0908005	−	0.0143814i	−12.5267	−	4.07016i		−	26.8336i	16.0485	−	49.3923i

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	41.4.g.a	✓	72
41.g	even	20	1	inner	41.4.g.a	✓	72
41.h	odd	40	2		1681.4.a.o		72

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
41.4.g.a	✓	72	1.a	even	1	1	trivial
41.4.g.a	✓	72	41.g	even	20	1	inner
1681.4.a.o		72	41.h	odd	40	2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(41, [\chi])\).