Newform orbit 61.7.d.a

• Label: 61.7.d.a
• Level: $61$
• Weight: $7$
• Character orbit: 61.d
• Analytic conductor: $14.033$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	61.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.0332991008\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 60 q + 6 q^{2} + 226 q^{6} - 258 q^{7} - 642 q^{8} - 13612 q^{9}+O(q^{10}) \)

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	−11.1698	−	11.1698i		−	38.5849i			185.530i			48.0827i	−430.987	+	430.987i	150.044	+	150.044i	1357.47	−	1357.47i	−759.794			537.076	−	537.076i
11.2	−10.7154	−	10.7154i			30.6859i			165.641i		−	88.1346i	328.813	−	328.813i	−416.274	−	416.274i	1089.12	−	1089.12i	−212.626			−944.399	+	944.399i
11.3	−9.45449	−	9.45449i			30.4105i			114.775i			110.721i	287.516	−	287.516i	298.165	+	298.165i	480.048	−	480.048i	−195.800			1046.81	−	1046.81i
11.4	−8.64212	−	8.64212i		−	22.7033i			85.3725i		−	111.123i	−196.205	+	196.205i	−123.412	−	123.412i	184.704	−	184.704i	213.560			−960.341	+	960.341i
11.5	−8.43545	−	8.43545i			3.72817i			78.3138i		−	201.962i	31.4488	−	31.4488i	278.619	+	278.619i	120.743	−	120.743i	715.101			−1703.64	+	1703.64i
11.6	−8.22656	−	8.22656i		−	12.9741i			71.3524i			172.914i	−106.732	+	106.732i	−152.901	−	152.901i	60.4851	−	60.4851i	560.674			1422.49	−	1422.49i
11.7	−5.89388	−	5.89388i			27.3539i			5.47566i			34.4850i	161.221	−	161.221i	−278.383	−	278.383i	−344.935	+	344.935i	−19.2386			203.250	−	203.250i
11.8	−5.72007	−	5.72007i			53.2773i			1.43842i		−	131.335i	304.750	−	304.750i	117.816	+	117.816i	−357.857	+	357.857i	−2109.47			−751.245	+	751.245i
11.9	−5.69120	−	5.69120i		−	50.9924i			0.779533i		−	42.8606i	−290.208	+	290.208i	42.0043	+	42.0043i	−359.800	+	359.800i	−1871.23			−243.928	+	243.928i
11.10	−4.44483	−	4.44483i		−	17.3210i		−	24.4869i			38.5916i	−76.9889	+	76.9889i	390.345	+	390.345i	−393.310	+	393.310i	428.984			171.533	−	171.533i
11.11	−4.02577	−	4.02577i			21.3155i		−	31.5863i			23.3432i	85.8115	−	85.8115i	−64.3808	−	64.3808i	−384.809	+	384.809i	274.648			93.9743	−	93.9743i
11.12	−2.54280	−	2.54280i		−	18.4786i		−	51.0683i		−	168.638i	−46.9873	+	46.9873i	−375.684	−	375.684i	−292.596	+	292.596i	387.542			−428.812	+	428.812i
11.13	−1.96084	−	1.96084i		−	36.0161i		−	56.3102i			208.285i	−70.6218	+	70.6218i	−229.938	−	229.938i	−235.909	+	235.909i	−568.157			408.414	−	408.414i
11.14	−1.12919	−	1.12919i			15.2878i		−	61.4499i		−	167.558i	17.2629	−	17.2629i	8.28887	+	8.28887i	−141.657	+	141.657i	495.282			−189.205	+	189.205i
11.15	−1.00607	−	1.00607i			39.4851i		−	61.9757i			238.806i	39.7247	−	39.7247i	89.8973	+	89.8973i	−126.740	+	126.740i	−830.075			240.255	−	240.255i
11.16	−0.771999	−	0.771999i		−	11.7930i		−	62.8080i			70.8521i	−9.10415	+	9.10415i	352.163	+	352.163i	−97.8957	+	97.8957i	589.926			54.6977	−	54.6977i
11.17	2.47016	+	2.47016i		−	32.1226i		−	51.7966i			2.54929i	79.3480	−	79.3480i	−139.702	−	139.702i	286.036	−	286.036i	−302.864			−6.29715	+	6.29715i
11.18	2.80751	+	2.80751i			32.2877i		−	48.2357i		−	102.063i	−90.6482	+	90.6482i	293.024	+	293.024i	315.103	−	315.103i	−313.497			286.545	−	286.545i
11.19	3.18765	+	3.18765i			7.55690i		−	43.6778i			79.7023i	−24.0887	+	24.0887i	−225.135	−	225.135i	343.239	−	343.239i	671.893			−254.063	+	254.063i
11.20	3.29634	+	3.29634i		−	40.2717i		−	42.2682i		−	199.234i	132.750	−	132.750i	277.577	+	277.577i	350.297	−	350.297i	−892.813			656.743	−	656.743i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
61.d	odd	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	61.7.d.a	✓	60
61.d	odd	4	1	inner	61.7.d.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.7.d.a	✓	60	1.a	even	1	1	trivial
61.7.d.a	✓	60	61.d	odd	4	1	inner

Hecke kernels

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