Newform orbit 61.8.i.a

• Label: 61.8.i.a
• Level: $61$
• Weight: $8$
• Character orbit: 61.i
• Analytic conductor: $19.055$
• Analytic rank: $0$
• Dimension: $288$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	61.i (of order \(15\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(19.0554865545\)
Analytic rank:	\(0\)
Dimension:	\(288\)
Relative dimension:	\(36\) over \(\Q(\zeta_{15})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{15}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 288 q - q^{2} - 60 q^{3} + 2307 q^{4} + 1406 q^{5} + 745 q^{6} - 4759 q^{7} - 3334 q^{8} - 56214 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−20.4028	−	9.08390i	33.3642	−	24.2405i	248.107	+	275.551i	−465.800	+	99.0088i	−900.922	+	191.497i	58.0118	−	551.945i	−1675.61	−	5157.01i	−150.252	+	462.427i	10403.0	+	2211.23i
12.2	−18.6898	−	8.32123i	−55.8583	+	40.5835i	194.416	+	215.921i	29.1944	−	6.20547i	1381.68	−	293.686i	−7.63128	+	72.6068i	−1027.65	−	3162.78i	797.317	−	2453.89i	−597.274	−	126.955i
12.3	−18.5166	−	8.24412i	8.00339	−	5.81480i	189.250	+	210.183i	293.691	−	62.4259i	−196.133	+	41.6894i	170.896	−	1625.96i	−969.766	−	2984.63i	−645.578	+	1986.88i	−5952.80	−	1265.31i
12.4	−17.5889	−	7.83109i	73.5506	−	53.4376i	162.395	+	180.358i	358.604	−	76.2237i	−1712.15	+	363.929i	−63.2071	+	601.375i	−682.402	−	2100.22i	1878.29	−	5780.77i	−6904.38	−	1467.57i
12.5	−17.2385	−	7.67509i	10.9898	−	7.98454i	152.611	+	169.492i	102.532	−	21.7938i	−250.730	+	53.2942i	−135.965	+	1293.63i	−583.542	−	1795.96i	−618.798	+	1904.46i	−1934.77	−	411.247i
12.6	−15.3079	−	6.81553i	−24.8382	+	18.0460i	102.232	+	113.540i	−285.899	+	60.7696i	503.214	−	106.961i	−61.2294	+	582.559i	−128.332	−	394.966i	−384.543	+	1183.50i	4790.69	+	1018.29i
12.7	−13.2744	−	5.91015i	−41.0109	+	29.7962i	55.6315	+	61.7851i	545.507	−	115.951i	720.496	−	153.146i	−2.12585	+	20.2261i	201.431	+	619.942i	118.263	−	363.976i	−7926.58	−	1684.85i
12.8	−13.1850	−	5.87036i	−52.5744	+	38.1975i	53.7356	+	59.6795i	−405.418	+	86.1743i	917.429	−	195.006i	176.917	−	1683.26i	212.712	+	654.660i	629.195	−	1936.46i	5851.33	+	1243.74i
12.9	−12.3950	−	5.51862i	55.3611	−	40.2222i	37.5326	+	41.6842i	−358.996	+	76.3069i	−908.173	+	193.038i	−32.9198	+	313.211i	301.494	+	927.904i	771.207	−	2373.53i	4870.86	+	1035.33i
12.10	−12.2681	−	5.46210i	35.4872	−	25.7830i	35.0225	+	38.8964i	37.8832	−	8.05232i	−576.189	+	122.473i	79.3165	−	754.646i	313.974	+	966.312i	−81.2396	+	250.030i	−508.737	−	108.135i
12.11	−9.51516	−	4.23642i	−16.7733	+	12.1865i	−13.0578	−	14.5021i	−5.85308	+	1.24411i	211.228	−	44.8978i	49.8466	−	474.258i	474.792	+	1461.26i	−542.988	+	1671.15i	60.9636	+	12.9582i
12.12	−8.42852	−	3.75262i	−72.3678	+	52.5783i	−28.6909	−	31.8645i	17.6085	−	3.74279i	807.259	−	171.588i	−118.507	+	1127.52i	487.180	+	1499.39i	1796.80	−	5529.98i	−162.458	−	34.5316i
12.13	−5.40585	−	2.40684i	28.2611	−	20.5329i	−62.2184	−	69.1005i	370.053	−	78.6572i	−202.195	+	42.9778i	−135.669	+	1290.81i	404.089	+	1243.66i	−298.731	+	919.398i	−2189.77	−	465.449i
12.14	−4.51820	−	2.01163i	−27.5041	+	19.9829i	−69.2812	−	76.9446i	−465.459	+	98.9364i	164.468	−	34.9587i	−130.992	+	1246.31i	353.869	+	1089.10i	−318.660	+	980.734i	2302.06	+	489.319i
12.15	−3.98687	−	1.77507i	−33.7456	+	24.5176i	−72.9044	−	80.9686i	242.780	−	51.6044i	178.060	−	37.8478i	43.4547	−	413.444i	319.557	+	983.496i	−138.167	+	425.233i	−1059.53	−	225.211i
12.16	−3.98597	−	1.77467i	59.8278	−	43.4675i	−72.9102	−	80.9750i	313.967	−	66.7358i	−315.613	+	67.0855i	112.641	−	1071.71i	319.497	+	983.310i	1014.13	−	3121.17i	−1369.90	−	291.181i
12.17	−3.90975	−	1.74073i	11.1286	−	8.08540i	−73.3927	−	81.5109i	−209.209	+	44.4687i	−57.5845	+	12.2400i	−75.1053	+	714.579i	314.341	+	967.441i	−617.348	+	1900.00i	895.361	+	190.315i
12.18	−0.0163992	−	0.00730138i	18.6576	−	13.5555i	−85.6485	−	95.1223i	−429.162	+	91.2211i	−0.404943	+	0.0860734i	164.743	−	1567.42i	1.42008	+	4.37056i	−511.467	+	1574.13i	7.70394	+	1.63752i
12.19	0.384773	+	0.171312i	−67.6483	+	49.1493i	−85.5300	−	94.9907i	−53.7923	+	11.4339i	−34.4491	+	7.32238i	93.3383	−	888.055i	−33.2963	−	102.475i	1484.81	−	4569.77i	−22.6566	−	4.81581i
12.20	0.998225	+	0.444438i	62.6354	−	45.5073i	−84.8498	−	94.2352i	−198.437	+	42.1791i	82.7494	−	17.5889i	−45.0751	+	428.860i	−86.0380	−	264.798i	1176.46	−	3620.78i	−216.831	−	46.0888i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.i	even	15	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.8.i.a	✓	288
61.i	even	15	1	inner	61.8.i.a	✓	288

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.8.i.a	✓	288	1.a	even	1	1	trivial
61.8.i.a	✓	288	61.i	even	15	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(61, [\chi])\).