Newform orbit 37.8.e.a

• Label: 37.8.e.a
• Level: $37$
• Weight: $8$
• Character orbit: 37.e
• Analytic conductor: $11.558$
• Analytic rank: $0$
• Dimension: $40$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.5582459429\)
Analytic rank:	\(0\)
Dimension:	\(40\)
Relative dimension:	\(20\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q + 18 q^{2} - 42 q^{3} + 1114 q^{4} - 672 q^{5} + 156 q^{7} - 9128 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−18.1421	+	10.4744i	−15.3322	+	26.5561i	155.424	−	269.203i	181.047	+	104.528i		−	642.380i	−800.230	+	1386.04i			3830.44i	623.347	+	1079.67i	−4379.44
11.2	−16.8690	+	9.73931i	5.24451	−	9.08376i	125.708	−	217.733i	−269.659	−	155.688i			204.312i	385.556	−	667.802i			2403.99i	1038.49	+	1798.72i	6065.16
11.3	−14.9331	+	8.62160i	30.8914	−	53.5054i	84.6641	−	146.643i	237.219	+	136.958i			1065.33i	106.642	−	184.710i			712.630i	−815.054	−	1411.72i	−4723.20
11.4	−13.4758	+	7.78027i	−38.3502	+	66.4244i	57.0652	−	98.8398i	4.26713	+	2.46363i		−	1193.50i	243.028	−	420.937i		−	215.818i	−1847.97	−	3200.78i	−76.6708
11.5	−10.0545	+	5.80498i	31.9798	−	55.3907i	3.39554	−	5.88124i	−274.161	−	158.287i			742.569i	−610.528	+	1057.47i		−	1407.23i	−951.919	−	1648.77i	3675.40
11.6	−8.60443	+	4.96777i	−5.90461	+	10.2271i	−14.6425	+	25.3615i	369.746	+	213.473i		−	117.331i	428.306	−	741.848i		−	1562.71i	1023.77	+	1773.22i	−4241.93
11.7	−8.00216	+	4.62005i	−13.9119	+	24.0960i	−21.3103	+	36.9106i	−360.513	−	208.143i		−	257.094i	67.3669	−	116.683i		−	1576.55i	706.421	+	1223.56i	3846.51
11.8	−6.09195	+	3.51719i	−7.67088	+	13.2864i	−39.2587	+	67.9981i	66.8903	+	38.6191i		−	107.920i	−602.681	+	1043.87i		−	1452.72i	975.815	+	1690.16i	−543.323
11.9	−2.67688	+	1.54550i	32.7398	−	56.7070i	−59.2229	+	102.577i	−63.8099	−	36.8407i			202.397i	457.071	−	791.671i		−	761.764i	−1050.29	−	1819.15i	227.749
11.10	0.806849	−	0.465835i	−38.5559	+	66.7808i	−63.5660	+	110.100i	−299.174	−	172.728i			71.8427i	−372.891	+	645.866i			237.699i	−1879.62	−	3255.59i	−321.851
11.11	2.05422	−	1.18601i	3.39795	−	5.88542i	−61.1868	+	105.979i	−176.269	−	101.769i		−	16.1199i	721.166	−	1249.10i			593.889i	1070.41	+	1854.00i	−482.794
11.12	3.57062	−	2.06150i	−32.2488	+	55.8566i	−55.5005	+	96.1296i	228.187	+	131.744i			265.923i	124.613	−	215.836i			985.399i	−986.473	−	1708.62i	1086.36
11.13	3.63260	−	2.09728i	24.6553	−	42.7042i	−55.2028	+	95.6141i	292.816	+	169.058i		−	206.836i	−480.087	+	831.535i			1000.01i	−122.265	−	211.770i	1418.24
11.14	9.14535	−	5.28007i	2.08216	−	3.60640i	−8.24170	+	14.2750i	−276.472	−	159.621i		−	43.9758i	−330.781	+	572.930i			1525.77i	1084.83	+	1878.98i	−3371.24
11.15	12.1223	−	6.99882i	43.2891	−	74.9789i	33.9669	−	58.8324i	−221.481	−	127.872i		−	1211.89i	−32.7203	+	56.6733i			840.785i	−2654.39	−	4597.54i	−3579.82
11.16	12.1985	−	7.04279i	−13.3616	+	23.1430i	35.2017	−	60.9712i	158.100	+	91.2791i			376.413i	380.940	−	659.807i			811.281i	736.434	+	1275.54i	2571.44
11.17	14.2692	−	8.23834i	23.0376	−	39.9023i	71.7405	−	124.258i	363.263	+	209.730i		−	759.167i	415.828	−	720.236i		−	255.076i	32.0370	+	55.4897i	6911.31
11.18	15.6973	−	9.06286i	−28.3079	+	49.0307i	100.271	−	173.674i	169.896	+	98.0895i			1026.20i	−758.009	+	1312.91i		−	1314.87i	−509.171	−	881.910i	3555.89
11.19	15.8575	−	9.15532i	−37.9133	+	65.6677i	103.640	−	179.510i	−366.586	−	211.649i			1388.43i	808.308	−	1400.03i		−	1451.67i	−1781.33	−	3085.36i	−7750.85
11.20	18.4955	−	10.6784i	13.2396	−	22.9317i	164.055	−	284.151i	−99.3053	−	57.3339i		−	565.509i	−72.8986	+	126.264i		−	4273.68i	742.926	+	1286.79i	−2448.93

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
37.e	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	37.8.e.a	✓	40
37.e	even	6	1	inner	37.8.e.a	✓	40

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.8.e.a	✓	40	1.a	even	1	1	trivial
37.8.e.a	✓	40	37.e	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(37, [\chi])\).