Newform orbit 77.5.h.a

• Label: 77.5.h.a
• Level: $77$
• Weight: $5$
• Character orbit: 77.h
• Analytic conductor: $7.959$
• Analytic rank: $0$
• Dimension: $60$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	77.h (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.95948715746\)
Analytic rank:	\(0\)
Dimension:	\(60\)
Relative dimension:	\(30\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 60 q - 2 q^{3} + 216 q^{4} + 16 q^{5} - 636 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1	−6.82377	+	3.93970i	−3.61857	+	6.26755i	23.0425	−	39.9109i	15.6547	+	27.1148i		−	57.0244i	−39.8610	+	28.4974i			237.053i	14.3119	+	24.7889i	−213.649	−	123.350i
32.2	−6.25873	+	3.61348i	5.81528	−	10.0724i	18.1145	−	31.3752i	8.76034	+	15.1734i			84.0536i	34.8449	−	34.4504i			146.194i	−27.1349	−	46.9991i	−109.657	−	63.3107i
32.3	−6.00433	+	3.46660i	−1.66525	+	2.88430i	16.0346	−	27.7728i	−17.4727	−	30.2636i		−	23.0911i	34.8225	+	34.4731i			111.412i	34.9539	+	60.5419i	209.824	+	121.142i
32.4	−5.25416	+	3.03349i	4.33580	−	7.50983i	10.4041	−	18.0205i	−15.6488	−	27.1045i			52.6104i	−39.6695	−	28.7633i			29.1714i	2.90167	+	5.02585i	164.442	+	94.9409i
32.5	−5.08729	+	2.93715i	−8.80540	+	15.2514i	9.25367	−	16.0278i	−3.17506	−	5.49937i		−	103.451i	45.3563	−	18.5421i			14.7289i	−114.570	−	198.441i	32.3049	+	18.6513i
32.6	−4.65580	+	2.68802i	5.71680	−	9.90179i	6.45095	−	11.1734i	4.29211	+	7.43414i			61.4676i	−13.7997	+	47.0167i		−	16.6555i	−24.8636	−	43.0650i	−39.9663	−	23.0746i
32.7	−4.59488	+	2.65286i	−4.29326	+	7.43615i	6.07531	−	10.5227i	−0.158664	−	0.274813i		−	45.5576i	−38.1787	−	30.7146i		−	20.4237i	3.63582	+	6.29743i	1.45808	+	0.841824i
32.8	−4.20660	+	2.42868i	−0.868621	+	1.50450i	3.79697	−	6.57655i	20.3429	+	35.2350i		−	8.43841i	41.9802	−	25.2718i		−	40.8313i	38.9910	+	67.5344i	−171.149	−	98.8129i
32.9	−2.92831	+	1.69066i	−1.64165	+	2.84343i	−2.28332	+	3.95482i	6.38015	+	11.0507i		−	11.1019i	−6.88963	+	48.5132i		−	69.5425i	35.1099	+	60.8122i	−37.3662	−	21.5734i
32.10	−2.10254	+	1.21390i	8.03759	−	13.9215i	−5.05289	+	8.75186i	−13.1288	−	22.7397i			39.0273i	46.0771	+	16.6703i		−	63.3797i	−88.7056	−	153.643i	55.2075	+	31.8740i
32.11	−1.98845	+	1.14803i	−6.01968	+	10.4264i	−5.36404	+	9.29079i	−18.8017	−	32.5656i		−	27.6432i	−34.7802	+	34.5157i		−	61.3694i	−31.9730	−	55.3789i	74.7727	+	43.1700i
32.12	−1.98190	+	1.14425i	0.293019	−	0.507525i	−5.38138	+	9.32082i	−13.9275	−	24.1232i			1.34115i	43.2215	−	23.0846i		−	61.2467i	40.3283	+	69.8506i	55.2060	+	31.8732i
32.13	−1.70335	+	0.983432i	5.87781	−	10.1807i	−6.06572	+	10.5061i	19.4212	+	33.6385i			23.1217i	−48.7468	−	4.97513i		−	55.3307i	−28.5972	−	49.5319i	−66.1624	−	38.1989i
32.14	−1.15704	+	0.668017i	2.66708	−	4.61951i	−7.10751	+	12.3106i	−1.27698	−	2.21179i			7.12661i	−11.3484	−	47.6677i		−	40.3683i	26.2734	+	45.5069i	2.95503	+	1.70609i
32.15	−0.351032	+	0.202669i	−6.33093	+	10.9655i	−7.91785	+	13.7141i	12.7388	+	22.0642i		−	5.13233i	39.3410	+	29.2111i		−	12.9042i	−39.6614	−	68.6956i	−8.94344	−	5.16350i
32.16	0.351032	−	0.202669i	−6.33093	+	10.9655i	−7.91785	+	13.7141i	12.7388	+	22.0642i			5.13233i	−39.3410	−	29.2111i			12.9042i	−39.6614	−	68.6956i	8.94344	+	5.16350i
32.17	1.15704	−	0.668017i	2.66708	−	4.61951i	−7.10751	+	12.3106i	−1.27698	−	2.21179i		−	7.12661i	11.3484	+	47.6677i			40.3683i	26.2734	+	45.5069i	−2.95503	−	1.70609i
32.18	1.70335	−	0.983432i	5.87781	−	10.1807i	−6.06572	+	10.5061i	19.4212	+	33.6385i		−	23.1217i	48.7468	+	4.97513i			55.3307i	−28.5972	−	49.5319i	66.1624	+	38.1989i
32.19	1.98190	−	1.14425i	0.293019	−	0.507525i	−5.38138	+	9.32082i	−13.9275	−	24.1232i		−	1.34115i	−43.2215	+	23.0846i			61.2467i	40.3283	+	69.8506i	−55.2060	−	31.8732i
32.20	1.98845	−	1.14803i	−6.01968	+	10.4264i	−5.36404	+	9.29079i	−18.8017	−	32.5656i			27.6432i	34.7802	−	34.5157i			61.3694i	−31.9730	−	55.3789i	−74.7727	−	43.1700i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.c	even	3	1	inner
11.b	odd	2	1	inner
77.h	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	77.5.h.a	✓	60
7.c	even	3	1	inner	77.5.h.a	✓	60
11.b	odd	2	1	inner	77.5.h.a	✓	60
77.h	odd	6	1	inner	77.5.h.a	✓	60

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.5.h.a	✓	60	1.a	even	1	1	trivial
77.5.h.a	✓	60	7.c	even	3	1	inner
77.5.h.a	✓	60	11.b	odd	2	1	inner
77.5.h.a	✓	60	77.h	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(77, [\chi])\).