Newform orbit 37.10.e.a

• Label: 37.10.e.a
• Level: $37$
• Weight: $10$
• Character orbit: 37.e
• Analytic conductor: $19.056$
• Analytic rank: $0$
• Dimension: $56$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 37 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	37.e (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0563259381\)
Analytic rank:	\(0\)
Dimension:	\(56\)
Relative dimension:	\(28\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 56 q - 54 q^{2} - 75 q^{3} + 7082 q^{4} - 186 q^{5} - 4233 q^{7} - 173645 q^{9}+O(q^{10}) \)

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	−39.1578	+	22.6078i	−118.702	+	205.598i	766.223	−	1327.14i	−1675.70	−	967.463i		−	10734.4i	1849.99	−	3204.27i			46140.0i	−18339.0	−	31764.0i	87488.7
11.2	−37.8505	+	21.8530i	76.7161	−	132.876i	699.105	−	1210.89i	838.490	+	484.103i			6705.90i	260.580	−	451.338i			38732.7i	−1929.22	−	3341.51i	−42316.3
11.3	−31.1498	+	17.9843i	57.1110	−	98.9191i	390.874	−	677.013i	−1648.13	−	951.547i			4108.41i	−5484.32	+	9499.11i			9702.46i	3318.18	+	5747.25i	68451.8
11.4	−30.7576	+	17.7579i	−73.3239	+	127.001i	374.688	−	648.979i	2028.62	+	1171.23i		−	5208.32i	−279.638	+	484.348i			8430.63i	−911.280	−	1578.38i	−83194.2
11.5	−29.7974	+	17.2035i	14.1784	−	24.5577i	335.923	−	581.835i	−83.3280	−	48.1094i			975.673i	4286.41	−	7424.29i			5499.82i	9439.45	+	16349.6i	3310.61
11.6	−27.6882	+	15.9858i	−44.3839	+	76.8751i	255.090	−	441.829i	−158.315	−	91.4030i		−	2838.04i	−1394.49	+	2415.33i		−	58.1829i	5901.64	+	10221.9i	5844.59
11.7	−23.8055	+	13.7441i	131.157	−	227.170i	121.802	−	210.967i	−1058.66	−	611.220i			7210.54i	4250.53	−	7362.13i		−	7377.74i	−24562.7	−	42543.9i	33602.7
11.8	−21.0748	+	12.1675i	98.7278	−	171.002i	40.0973	−	69.4506i	2085.31	+	1203.95i			4805.09i	−3856.72	+	6680.04i		−	10508.0i	−9652.86	−	16719.2i	−58596.6
11.9	−17.2642	+	9.96751i	−121.025	+	209.622i	−57.2974	+	99.2421i	−231.945	−	133.913i		−	4825.28i	−4145.43	+	7180.09i		−	12491.2i	−19452.7	−	33693.1i	5339.13
11.10	−16.0906	+	9.28991i	−49.1825	+	85.1865i	−83.3953	+	144.445i	−2260.08	−	1304.86i		−	1827.60i	2409.21	−	4172.87i		−	12611.8i	5003.67	+	8666.61i	48487.9
11.11	−11.7800	+	6.80116i	32.8904	−	56.9678i	−163.488	+	283.170i	1285.87	+	742.395i			894.771i	2908.10	−	5036.98i		−	11412.0i	7677.95	+	13298.6i	−20196.6
11.12	−11.2408	+	6.48986i	62.0101	−	107.405i	−171.763	+	297.503i	−199.442	−	115.148i			1609.75i	−1112.71	+	1927.27i		−	11104.5i	2151.00	+	3725.64i	2989.18
11.13	−9.56308	+	5.52125i	−112.816	+	195.403i	−195.032	+	337.805i	615.768	+	355.514i		−	2491.54i	5957.98	−	10319.5i		−	9961.03i	−15613.4	−	27043.2i	−7851.52
11.14	−3.82302	+	2.20722i	−29.2527	+	50.6671i	−246.256	+	426.528i	111.286	+	64.2508i		−	258.269i	−3702.66	+	6413.20i		−	4434.37i	8130.06	+	14081.7i	−567.263
11.15	2.40508	−	1.38857i	61.5559	−	106.618i	−252.144	+	436.726i	−1906.94	−	1100.97i		−	341.899i	241.891	−	418.967i			2822.38i	2263.23	+	3920.03i	−6115.11
11.16	4.44570	−	2.56673i	−64.5524	+	111.808i	−242.824	+	420.583i	2105.55	+	1215.64i			662.753i	−1802.14	+	3121.40i			5121.38i	1507.48	+	2611.03i	12480.9
11.17	7.51322	−	4.33776i	133.176	−	230.668i	−218.368	+	378.224i	−134.122	−	77.4352i		−	2310.74i	−3416.07	+	5916.81i			8230.77i	−25630.2	−	44392.8i	−1343.58
11.18	10.8373	−	6.25693i	−17.7229	+	30.6970i	−177.702	+	307.788i	−53.3178	−	30.7830i			443.564i	3924.98	−	6798.27i			10854.6i	9213.30	+	15957.9i	−770.429
11.19	11.4704	−	6.62244i	−93.3595	+	161.703i	−168.287	+	291.481i	−1327.40	−	766.378i			2473.07i	−1020.89	+	1768.24i			11239.3i	−7590.47	−	13147.1i	−20301.2
11.20	12.9927	−	7.50134i	92.3544	−	159.963i	−143.460	+	248.480i	1347.77	+	778.135i		−	2771.13i	3647.45	−	6317.57i			11985.9i	−7217.17	−	12500.5i	23348.2

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	37.10.e.a	✓	56
37.e	even	6	1	inner	37.10.e.a	✓	56

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
37.10.e.a	✓	56	1.a	even	1	1	trivial
37.10.e.a	✓	56	37.e	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(37, [\chi])\).