Newform orbit 37.6.e.a

• Label: 37.6.e.a
• Level: $37$
• Weight: $6$
• Character orbit: 37.e
• Analytic conductor: $5.934$
• Analytic rank: $0$
• Dimension: $32$
• CM: no
• Inner twists: $2$

Newspace parameters

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

Newform invariants

Self dual:	no
Analytic conductor:	\(5.93420133308\)
Analytic rank:	\(0\)
Dimension:	\(32\)
Relative dimension:	\(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 6 q^{2} + 18 q^{3} + 298 q^{4} + 144 q^{5} - 52 q^{7} - 1490 q^{9}+O(q^{10}) \)

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	−9.33962	+	5.39223i	14.0993	−	24.4208i	42.1523	−	73.0099i	−41.0060	−	23.6748i			304.108i	−18.9093	+	32.7518i			564.077i	−276.083	−	478.189i	510.641
11.2	−9.18598	+	5.30353i	−6.26292	+	10.8477i	40.2548	−	69.7233i	63.9646	+	36.9300i		−	132.862i	113.255	−	196.163i			514.543i	43.0517	+	74.5678i	−783.436
11.3	−7.60377	+	4.39004i	−8.72617	+	15.1142i	22.5449	−	39.0489i	−72.9853	−	42.1381i		−	153.233i	−77.9335	+	134.985i			114.930i	−30.7921	−	53.3336i	739.952
11.4	−6.08219	+	3.51156i	3.15654	−	5.46728i	8.66206	−	15.0031i	30.9238	+	17.8538i			44.3374i	−47.9780	+	83.1003i		−	103.070i	101.573	+	175.929i	−250.779
11.5	−5.33497	+	3.08014i	3.71739	−	6.43870i	2.97457	−	5.15211i	−14.5389	−	8.39402i			45.8003i	41.9646	−	72.6849i		−	160.481i	93.8621	+	162.574i	103.419
11.6	−3.47160	+	2.00433i	−13.3195	+	23.0700i	−7.96531	+	13.7963i	85.3532	+	49.2787i		−	106.786i	−91.8980	+	159.172i		−	192.138i	−233.316	−	404.115i	−395.083
11.7	−2.20874	+	1.27522i	14.9045	−	25.8153i	−12.7476	+	22.0796i	67.9636	+	39.2388i			76.0257i	34.5008	−	59.7572i		−	146.638i	−322.785	−	559.081i	−200.152
11.8	−1.48416	+	0.856882i	−9.17590	+	15.8931i	−14.5315	+	25.1693i	−29.0287	−	16.7598i		−	31.4507i	80.8273	−	139.997i		−	104.648i	−46.8942	−	81.2231i	57.4445
11.9	0.312322	−	0.180319i	9.26592	−	16.0490i	−15.9350	+	27.6002i	−90.8164	−	52.4329i		−	6.68329i	−48.1376	+	83.3769i			23.0340i	−50.2145	−	86.9741i	−37.8186
11.10	0.987887	−	0.570357i	2.15443	−	3.73158i	−15.3494	+	26.5859i	19.1329	+	11.0464i		−	4.91517i	−51.7707	+	89.6695i			71.5213i	112.217	+	194.365i	25.2015
11.11	4.33309	−	2.50171i	−3.26958	+	5.66309i	−3.48290	+	6.03256i	54.4047	+	31.4106i			32.7182i	22.9091	−	39.6798i			194.962i	100.120	+	173.412i	314.320
11.12	5.18743	−	2.99497i	9.51908	−	16.4875i	1.93964	−	3.35955i	−4.99548	−	2.88414i		−	114.037i	118.181	−	204.696i			168.441i	−59.7259	−	103.448i	−34.5516
11.13	5.44256	−	3.14226i	−10.4803	+	18.1524i	3.74762	−	6.49106i	−46.0737	−	26.6006i			131.727i	−56.9410	+	98.6246i			154.001i	−98.1728	−	170.040i	−334.345
11.14	7.81751	−	4.51344i	10.9075	−	18.8923i	24.7424	−	42.8550i	44.4487	+	25.6624i		−	196.921i	−121.740	+	210.859i		−	157.833i	−116.446	−	201.690i	463.304
11.15	8.27772	−	4.77914i	2.34354	−	4.05913i	29.6804	−	51.4080i	−63.0654	−	36.4108i		−	44.8004i	27.0685	−	46.8840i		−	261.522i	110.516	+	191.419i	−696.050
11.16	9.35252	−	5.39968i	−9.83385	+	17.0327i	42.3131	−	73.2884i	68.3185	+	39.4437i			212.399i	50.6016	−	87.6446i		−	568.329i	−71.9093	−	124.551i	851.933
27.1	−9.33962	−	5.39223i	14.0993	+	24.4208i	42.1523	+	73.0099i	−41.0060	+	23.6748i		−	304.108i	−18.9093	−	32.7518i		−	564.077i	−276.083	+	478.189i	510.641
27.2	−9.18598	−	5.30353i	−6.26292	−	10.8477i	40.2548	+	69.7233i	63.9646	−	36.9300i			132.862i	113.255	+	196.163i		−	514.543i	43.0517	−	74.5678i	−783.436
27.3	−7.60377	−	4.39004i	−8.72617	−	15.1142i	22.5449	+	39.0489i	−72.9853	+	42.1381i			153.233i	−77.9335	−	134.985i		−	114.930i	−30.7921	+	53.3336i	739.952
27.4	−6.08219	−	3.51156i	3.15654	+	5.46728i	8.66206	+	15.0031i	30.9238	−	17.8538i		−	44.3374i	−47.9780	−	83.1003i			103.070i	101.573	−	175.929i	−250.779

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	37.6.e.a	✓	32
37.e	even	6	1	inner	37.6.e.a	✓	32

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.6.e.a	✓	32	1.a	even	1	1	trivial
37.6.e.a	✓	32	37.e	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(37, [\chi])\).