Newform orbit 61.5.d.a

• Label: 61.5.d.a
• Level: $61$
• Weight: $5$
• Character orbit: 61.d
• Analytic conductor: $6.306$
• Analytic rank: $0$
• Dimension: $38$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	61.d (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.30556774811\)
Analytic rank:	\(0\)
Dimension:	\(38\)
Relative dimension:	\(19\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 38 q - 2 q^{2} + 130 q^{6} - 22 q^{7} + 30 q^{8} - 814 q^{9}+O(q^{10}) \)

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	−5.30047	−	5.30047i			1.20855i			40.1900i		−	7.04968i	6.40588	−	6.40588i	15.3758	+	15.3758i	128.219	−	128.219i	79.5394			−37.3667	+	37.3667i
11.2	−4.38864	−	4.38864i			16.9799i			22.5204i			37.6409i	74.5188	−	74.5188i	−33.6298	−	33.6298i	28.6156	−	28.6156i	−207.318			165.193	−	165.193i
11.3	−4.35920	−	4.35920i		−	11.7825i			22.0052i		−	8.77445i	−51.3624	+	51.3624i	−57.1115	−	57.1115i	26.1779	−	26.1779i	−57.8279			−38.2496	+	38.2496i
11.4	−3.96587	−	3.96587i		−	13.2596i			15.4563i			40.7461i	−52.5857	+	52.5857i	54.8892	+	54.8892i	−2.15644	+	2.15644i	−94.8160			161.594	−	161.594i
11.5	−3.82040	−	3.82040i			11.3890i			13.1909i		−	33.8543i	43.5106	−	43.5106i	10.5533	+	10.5533i	−10.7317	+	10.7317i	−48.7097			−129.337	+	129.337i
11.6	−3.01388	−	3.01388i			2.94620i			2.16690i			12.9250i	8.87947	−	8.87947i	13.2800	+	13.2800i	−41.6913	+	41.6913i	72.3199			38.9542	−	38.9542i
11.7	−1.79600	−	1.79600i		−	8.07253i		−	9.54875i		−	32.6061i	−14.4983	+	14.4983i	16.6576	+	16.6576i	−45.8856	+	45.8856i	15.8343			−58.5607	+	58.5607i
11.8	−1.38139	−	1.38139i			1.03130i		−	12.1835i			26.7208i	1.42463	−	1.42463i	−61.4473	−	61.4473i	−38.9325	+	38.9325i	79.9364			36.9118	−	36.9118i
11.9	−0.119288	−	0.119288i			11.4261i		−	15.9715i		−	26.1038i	1.36299	−	1.36299i	−19.6169	−	19.6169i	−3.81382	+	3.81382i	−49.5551			−3.11387	+	3.11387i
11.10	−0.0201709	−	0.0201709i			14.1025i		−	15.9992i			14.0804i	0.284460	−	0.284460i	39.6550	+	39.6550i	−0.645452	+	0.645452i	−117.881			0.284014	−	0.284014i
11.11	0.386185	+	0.386185i		−	14.9162i		−	15.7017i			5.95285i	5.76041	−	5.76041i	2.59705	+	2.59705i	12.2427	−	12.2427i	−141.493			−2.29890	+	2.29890i
11.12	0.782142	+	0.782142i		−	1.00323i		−	14.7765i			36.8216i	0.784672	−	0.784672i	25.4478	+	25.4478i	24.0716	−	24.0716i	79.9935			−28.7997	+	28.7997i
11.13	2.44821	+	2.44821i			0.303739i		−	4.01257i		−	30.1604i	−0.743617	+	0.743617i	−53.3115	−	53.3115i	48.9949	−	48.9949i	80.9077			73.8388	−	73.8388i
11.14	2.65588	+	2.65588i			0.183545i		−	1.89261i		−	20.8872i	−0.487472	+	0.487472i	64.7949	+	64.7949i	47.5206	−	47.5206i	80.9663			55.4738	−	55.4738i
11.15	2.75988	+	2.75988i		−	9.83583i		−	0.766074i			8.24520i	27.1457	−	27.1457i	−32.3228	−	32.3228i	46.2724	−	46.2724i	−15.7435			−22.7558	+	22.7558i
11.16	3.53451	+	3.53451i			12.5540i			8.98550i			18.5994i	−44.3721	+	44.3721i	−13.3736	−	13.3736i	24.7928	−	24.7928i	−76.6019			−65.7396	+	65.7396i
11.17	4.71899	+	4.71899i		−	6.47171i			28.5377i			32.8182i	30.5399	−	30.5399i	4.12836	+	4.12836i	−59.1654	+	59.1654i	39.1169			−154.869	+	154.869i
11.18	4.71915	+	4.71915i		−	14.8951i			28.5407i		−	38.2800i	70.2924	−	70.2924i	7.20832	+	7.20832i	−59.1814	+	59.1814i	−140.865			180.649	−	180.649i
11.19	5.16037	+	5.16037i			8.11192i			37.2588i		−	18.8344i	−41.8605	+	41.8605i	5.22629	+	5.22629i	−109.703	+	109.703i	15.1967			97.1925	−	97.1925i
50.1	−5.30047	+	5.30047i		−	1.20855i		−	40.1900i			7.04968i	6.40588	+	6.40588i	15.3758	−	15.3758i	128.219	+	128.219i	79.5394			−37.3667	−	37.3667i

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	61.5.d.a	✓	38
61.d	odd	4	1	inner	61.5.d.a	✓	38

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
61.5.d.a	✓	38	1.a	even	1	1	trivial
61.5.d.a	✓	38	61.d	odd	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(61, [\chi])\).