Newform orbit 61.7.l.a

• Label: 61.7.l.a
• Level: $61$
• Weight: $7$
• Character orbit: 61.l
• Analytic conductor: $14.033$
• Analytic rank: $0$
• Dimension: $480$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	61.l (of order \(60\), degree \(16\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0332991008\)
Analytic rank:	\(0\)
Dimension:	\(480\)
Relative dimension:	\(30\) over \(\Q(\zeta_{60})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{60}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 480 q - 8 q^{2} - 20 q^{3} - 14 q^{4} + 74 q^{5} - 1120 q^{6} - 1664 q^{7} + 1648 q^{8} + 27204 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
2.1	−15.0162	−	0.786968i	−41.5680	−	13.5063i	161.219	+	16.9448i	−98.6424	−	221.555i	613.567	+	235.526i	−120.428	+	78.2069i	−1457.05	−	230.775i	955.710	+	694.364i	1306.88	+	3404.54i
2.2	−15.0079	−	0.786530i	32.3944	+	10.5256i	160.969	+	16.9185i	−32.3938	−	72.7578i	−477.892	−	183.446i	−209.322	+	135.936i	−1452.51	−	230.055i	348.834	+	253.443i	428.937	+	1117.42i
2.3	−13.8985	−	0.728387i	−18.8981	−	6.14035i	128.987	+	13.5571i	48.8157	+	109.642i	258.181	+	99.1065i	69.2345	−	44.9614i	−903.092	−	143.036i	−270.341	−	196.414i	−598.602	−	1559.41i
2.4	−12.3967	−	0.649684i	14.1187	+	4.58743i	89.6069	+	9.41806i	−8.98442	−	20.1793i	−172.045	−	66.0417i	265.975	−	172.726i	−320.015	−	50.6853i	−411.481	−	298.959i	98.2671	+	255.995i
2.5	−10.5121	−	0.550918i	28.5577	+	9.27897i	46.5521	+	4.89282i	86.3931	+	194.042i	−295.091	−	113.275i	−535.961	+	348.057i	178.739	+	28.3095i	139.672	+	101.478i	−801.275	−	2087.39i
2.6	−9.14572	−	0.479307i	−12.0546	−	3.91679i	19.7651	+	2.07739i	−35.5048	−	79.7451i	108.371	+	41.5997i	−528.707	+	343.347i	399.143	+	63.2181i	−459.800	−	334.064i	286.495	+	746.344i
2.7	−9.08349	−	0.476046i	44.6559	+	14.5096i	18.6338	+	1.95849i	22.0884	+	49.6113i	−398.724	−	153.056i	326.954	−	212.326i	406.647	+	64.4065i	1193.85	+	867.381i	−177.022	−	461.159i
2.8	−8.72893	−	0.457464i	3.13000	+	1.01700i	12.3355	+	1.29652i	−83.1315	−	186.717i	−26.8563	−	10.3092i	147.847	−	96.0132i	445.448	+	70.5520i	−581.011	−	422.129i	640.233	+	1667.86i
2.9	−8.38298	−	0.439333i	−35.8150	−	11.6370i	6.43191	+	0.676021i	20.8756	+	46.8874i	295.124	+	113.288i	−132.443	+	86.0095i	477.011	+	75.5511i	557.523	+	405.064i	−154.401	−	402.227i
2.10	−7.24037	−	0.379452i	−44.3412	−	14.4073i	−11.3704	−	1.19508i	−15.1162	−	33.9514i	315.580	+	121.140i	514.529	−	334.139i	540.179	+	85.5560i	1168.80	+	849.180i	96.5636	+	251.557i
2.11	−4.60656	−	0.241420i	0.546462	+	0.177556i	−42.4873	−	4.46559i	94.9128	+	213.178i	−2.47445	−	0.949851i	323.324	−	209.969i	486.232	+	77.0116i	−589.506	−	428.301i	−385.756	−	1004.93i
2.12	−4.25258	−	0.222868i	44.2937	+	14.3919i	−45.6147	−	4.79430i	−81.1217	−	182.202i	−185.155	−	71.0743i	−363.687	+	236.181i	462.094	+	73.1885i	1165.03	+	846.445i	304.369	+	792.909i
2.13	−3.47812	−	0.182281i	15.1851	+	4.93393i	−51.5853	−	5.42183i	12.3483	+	27.7346i	−51.9162	−	19.9287i	−59.5254	+	38.6563i	398.592	+	63.1308i	−383.530	−	278.651i	−37.8932	−	98.7152i
2.14	−0.768748	−	0.0402884i	29.9760	+	9.73980i	−63.0601	−	6.62788i	−7.46379	−	16.7640i	−22.6516	−	8.69513i	−37.5744	+	24.4011i	96.8710	+	15.3429i	213.924	+	155.425i	5.06238	+	13.1880i
2.15	−0.326722	−	0.0171228i	−13.8545	−	4.50161i	−63.5429	−	6.67863i	−2.89792	−	6.50884i	4.44950	+	1.70800i	−158.511	+	102.938i	41.3276	+	6.54565i	−418.090	−	303.760i	0.835366	+	2.17620i
2.16	0.167874	+	0.00879789i	−15.3310	−	4.98134i	−63.6213	−	6.68687i	−51.3373	−	115.306i	−2.52985	−	0.971117i	426.182	−	276.766i	−21.2477	−	3.36531i	−379.548	−	275.758i	−7.60374	−	19.8084i
2.17	1.31726	+	0.0690345i	−39.3103	−	12.7727i	−61.9190	−	6.50795i	78.8648	+	177.133i	−50.9000	−	19.5387i	−249.117	+	161.778i	−164.495	−	26.0534i	792.383	+	575.700i	91.6569	+	238.775i
2.18	3.01970	+	0.158256i	−38.6289	−	12.5513i	−54.5559	−	5.73405i	−76.0038	−	170.707i	−114.661	−	44.0143i	−204.675	+	132.918i	−354.978	−	56.2230i	744.883	+	541.189i	−202.493	−	527.512i
2.19	5.09446	+	0.266989i	40.9417	+	13.3028i	−37.7672	−	3.96949i	49.2048	+	110.516i	205.024	+	78.7013i	211.813	−	137.553i	−513.816	−	81.3805i	909.485	+	660.779i	221.165	+	576.154i
2.20	5.66488	+	0.296884i	17.4206	+	5.66030i	−31.6467	−	3.32620i	47.3184	+	106.279i	97.0053	+	37.2368i	−407.805	+	264.832i	−536.867	−	85.0314i	−318.334	−	231.283i	236.500	+	616.104i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.l	odd	60	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.7.l.a	✓	480
61.l	odd	60	1	inner	61.7.l.a	✓	480

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.7.l.a	✓	480	1.a	even	1	1	trivial
61.7.l.a	✓	480	61.l	odd	60	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(61, [\chi])\).