Newform orbit 50.6.e.a

• Label: 50.6.e.a
• Level: $50$
• Weight: $6$
• Character orbit: 50.e
• Analytic conductor: $8.019$
• Analytic rank: $0$
• Dimension: $48$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	50.e (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(8.01919099065\)
Analytic rank:	\(0\)
Dimension:	\(48\)
Relative dimension:	\(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 192 q^{4} + 180 q^{5} + 72 q^{6} + 686 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1	−2.35114	+	3.23607i	−19.7385	+	6.41342i	−4.94427	−	15.2169i	−52.1632	−	20.0999i	25.6537	−	78.9540i			5.87904i	60.8676	+	19.7771i	151.885	−	110.351i	187.687	−	121.546i
9.2	−2.35114	+	3.23607i	−11.7770	+	3.82658i	−4.94427	−	15.2169i	55.5298	−	6.43773i	15.3063	−	47.1080i		−	133.881i	60.8676	+	19.7771i	−72.5363	+	52.7007i	−109.725	+	194.834i
9.3	−2.35114	+	3.23607i	−2.02410	+	0.657668i	−4.94427	−	15.2169i	24.5492	−	50.2229i	2.63067	−	8.09638i			244.793i	60.8676	+	19.7771i	−192.927	+	140.169i	104.806	+	197.524i
9.4	−2.35114	+	3.23607i	−0.401275	+	0.130382i	−4.94427	−	15.2169i	−3.09691	+	55.8159i	0.521529	−	1.60510i		−	69.1609i	60.8676	+	19.7771i	−196.447	+	142.727i	−173.343	−	141.253i
9.5	−2.35114	+	3.23607i	17.7513	−	5.76774i	−4.94427	−	15.2169i	−50.3601	+	24.2664i	−23.0710	+	71.0051i			135.417i	60.8676	+	19.7771i	85.2500	−	61.9377i	39.8760	−	220.023i
9.6	−2.35114	+	3.23607i	25.8671	−	8.40474i	−4.94427	−	15.2169i	52.0281	+	20.4469i	−33.6189	+	103.468i		−	51.8294i	60.8676	+	19.7771i	401.877	−	291.981i	−188.493	+	120.293i
9.7	2.35114	−	3.23607i	−27.7680	+	9.02236i	−4.94427	−	15.2169i	12.4445	−	54.4989i	−36.0894	+	111.072i			185.806i	−60.8676	−	19.7771i	493.066	−	358.234i	−147.103	−	168.406i
9.8	2.35114	−	3.23607i	−12.2615	+	3.98399i	−4.94427	−	15.2169i	55.8772	+	1.65580i	−15.9360	+	49.0459i		−	145.423i	−60.8676	−	19.7771i	−62.1197	+	45.1326i	136.733	−	176.929i
9.9	2.35114	−	3.23607i	−6.79071	+	2.20643i	−4.94427	−	15.2169i	6.92330	+	55.4713i	−8.82574	+	27.1628i			47.7219i	−60.8676	−	19.7771i	−155.346	+	112.865i	195.787	+	108.017i
9.10	2.35114	−	3.23607i	−1.61240	+	0.523901i	−4.94427	−	15.2169i	−47.4044	−	29.6281i	−2.09560	+	6.44960i			48.2855i	−60.8676	−	19.7771i	−194.266	+	141.142i	−207.333	+	83.7438i
9.11	2.35114	−	3.23607i	18.4329	−	5.98921i	−4.94427	−	15.2169i	49.2396	−	26.4662i	23.9569	−	73.7316i			28.3664i	−60.8676	−	19.7771i	107.310	−	77.9654i	30.1231	−	221.569i
9.12	2.35114	−	3.23607i	22.5582	−	7.32959i	−4.94427	−	15.2169i	−54.0951	+	14.0970i	29.3184	−	90.2327i		−	229.539i	−60.8676	−	19.7771i	258.557	−	187.853i	−81.5663	+	208.199i
19.1	−3.80423	+	1.23607i	−14.7520	−	20.3044i	12.9443	−	9.40456i	−53.1272	−	17.3924i	81.2178	+	59.0082i		−	248.570i	−37.6183	+	51.7771i	−119.557	+	367.957i	223.606	+	0.495932i
19.2	−3.80423	+	1.23607i	−9.38088	−	12.9117i	12.9443	−	9.40456i	48.5017	−	27.7954i	51.6467	+	37.5235i			40.6861i	−37.6183	+	51.7771i	−3.61930	+	11.1391i	−150.154	+	165.691i
19.3	−3.80423	+	1.23607i	−6.76547	−	9.31187i	12.9443	−	9.40456i	−7.78168	+	55.3574i	37.2475	+	27.0619i			51.2744i	−37.6183	+	51.7771i	34.1518	−	105.108i	−38.8223	−	220.211i
19.4	−3.80423	+	1.23607i	4.87367	+	6.70803i	12.9443	−	9.40456i	44.0032	+	34.4778i	−26.8321	−	19.4947i		−	63.1969i	−37.6183	+	51.7771i	53.8461	−	165.721i	−210.015	−	76.7702i
19.5	−3.80423	+	1.23607i	9.57478	+	13.1786i	12.9443	−	9.40456i	−0.421737	−	55.9001i	−52.7142	−	38.2991i		−	120.466i	−37.6183	+	51.7771i	−6.90673	+	21.2567i	70.7007	+	212.135i
19.6	−3.80423	+	1.23607i	10.0418	+	13.8214i	12.9443	−	9.40456i	−52.7366	+	18.5432i	−55.2856	−	40.1674i			107.711i	−37.6183	+	51.7771i	−15.1017	+	46.4781i	177.701	−	135.729i
19.7	3.80423	−	1.23607i	−16.9585	−	23.3414i	12.9443	−	9.40456i	−8.25144	−	55.2894i	−93.3657	−	67.8341i			138.453i	37.6183	−	51.7771i	−182.139	+	560.566i	−99.7318	−	200.134i
19.8	3.80423	−	1.23607i	−8.70645	−	11.9834i	12.9443	−	9.40456i	45.9076	+	31.8981i	−47.9336	−	34.8258i		−	137.000i	37.6183	−	51.7771i	7.29147	−	22.4408i	214.071	+	64.6026i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
25.e	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	50.6.e.a	✓	48
25.e	even	10	1	inner	50.6.e.a	✓	48

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
50.6.e.a	✓	48	1.a	even	1	1	trivial
50.6.e.a	✓	48	25.e	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(50, [\chi])\).