Newform orbit 80.6.s.a

• Label: 80.6.s.a
• Level: $80$
• Weight: $6$
• Character orbit: 80.s
• Analytic conductor: $12.831$
• Analytic rank: $0$
• Dimension: $116$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	80.s (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(12.8307055850\)
Analytic rank:	\(0\)
Dimension:	\(116\)
Relative dimension:	\(58\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 2 q^{2} - 4 q^{3} + 20 q^{4} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 248 q^{8} + 8748 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
3.1	−5.61166	+	0.713657i	5.58602			30.9814	−	8.00959i	24.7855	−	50.1067i	−31.3468	+	3.98650i	145.876	−	145.876i	−168.141	+	67.0572i	−211.796			−103.329	+	298.870i
3.2	−5.60238	+	0.783133i	14.3156			30.7734	−	8.77482i	23.2238	+	50.8494i	−80.2014	+	11.2110i	−103.868	+	103.868i	−165.533	+	73.2596i	−38.0641			−169.930	−	266.690i
3.3	−5.58900	+	0.873525i	−13.7617			30.4739	−	9.76427i	−55.0423	−	9.76467i	76.9143	−	12.0212i	−58.2575	+	58.2575i	−161.789	+	81.1923i	−53.6151			316.161	+	6.49396i
3.4	−5.54990	−	1.09479i	−11.0755			29.6029	+	12.1520i	24.0182	+	50.4790i	61.4682	+	12.1254i	119.074	−	119.074i	−150.989	−	99.8515i	−120.332			−78.0344	−	306.448i
3.5	−5.53415	−	1.17184i	28.9277			29.2536	+	12.9702i	−47.1500	+	30.0314i	−160.090	−	33.8985i	139.505	−	139.505i	−146.695	−	106.060i	593.809			296.127	−	110.946i
3.6	−5.46635	+	1.45570i	−23.1807			27.7619	−	15.9147i	47.1776	−	29.9879i	126.714	−	33.7441i	−125.489	+	125.489i	−128.589	+	127.408i	294.343			−214.236	+	232.601i
3.7	−5.31218	−	1.94441i	16.7325			24.4385	+	20.6581i	−7.11375	−	55.4472i	−88.8859	−	32.5348i	−104.316	+	104.316i	−89.6538	−	157.258i	36.9755			−70.0228	+	308.378i
3.8	−5.11622	−	2.41336i	−5.06764			20.3514	+	24.6946i	−51.4203	+	21.9305i	25.9272	+	12.2301i	−35.3013	+	35.3013i	−44.5250	−	175.458i	−217.319			316.004	+	11.8945i
3.9	−4.99964	+	2.64643i	−25.1055			17.9928	−	26.4624i	−17.6043	+	53.0574i	125.519	−	66.4399i	92.6452	−	92.6452i	−19.9271	+	179.919i	387.286			−52.3971	−	311.857i
3.10	−4.95241	−	2.73379i	−9.68797			17.0528	+	27.0777i	53.9138	−	14.7751i	47.9788	+	26.4849i	−40.5624	+	40.5624i	−10.4273	−	180.719i	−149.143			−307.395	−	74.2169i
3.11	−4.94622	+	2.74497i	14.4255			16.9303	−	27.1545i	−46.0646	−	31.6710i	−71.3518	+	39.5976i	−46.8577	+	46.8577i	−9.20264	+	180.785i	−34.9048			314.782	+	30.2059i
3.12	−4.88220	−	2.85730i	−30.2943			15.6717	+	27.8998i	−21.4296	−	51.6311i	147.903	+	86.5599i	106.513	−	106.513i	3.20578	−	180.991i	674.746			−42.9018	+	313.304i
3.13	−4.25790	+	3.72428i	−2.96317			4.25947	−	31.7152i	54.1456	+	13.9016i	12.6169	−	11.0357i	6.13739	−	6.13739i	99.9801	+	150.904i	−234.220			−282.320	+	142.462i
3.14	−4.25189	+	3.73115i	30.7007			4.15710	−	31.7288i	47.2290	−	29.9068i	−130.536	+	114.549i	−7.67296	+	7.67296i	100.709	+	150.418i	699.530			−89.2258	+	303.379i
3.15	−4.11912	−	3.87722i	21.0242			1.93432	+	31.9415i	54.6422	+	11.7996i	−86.6012	−	81.5154i	19.5216	−	19.5216i	115.876	−	139.071i	199.016			−179.328	−	260.464i
3.16	−4.09704	+	3.90055i	12.0492			1.57141	−	31.9614i	−35.9351	+	42.8214i	−49.3660	+	46.9985i	63.2042	−	63.2042i	118.229	+	137.076i	−97.8169			−19.7998	−	315.607i
3.17	−3.48504	−	4.45584i	−24.9367			−7.70899	+	31.0576i	−3.26981	+	55.8060i	86.9053	+	111.114i	−112.850	+	112.850i	165.254	−	73.8868i	378.838			260.058	−	179.916i
3.18	−3.31543	+	4.58344i	−14.5939			−10.0158	−	30.3922i	−17.9275	−	52.9491i	48.3851	−	66.8903i	47.3215	−	47.3215i	172.507	+	54.8563i	−30.0178			302.126	+	93.3792i
3.19	−3.26583	−	4.61891i	4.38609			−10.6687	+	30.1692i	−47.6895	−	29.1670i	−14.3242	−	20.2590i	121.663	−	121.663i	174.191	−	49.2495i	−223.762			21.0260	+	315.528i
3.20	−2.93140	−	4.83807i	8.72573			−14.8138	+	28.3646i	−5.22519	+	55.6570i	−25.5786	−	42.2157i	−6.29366	+	6.29366i	180.655	−	11.4775i	−166.862			284.589	−	137.873i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
80.s	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	80.6.s.a	yes	116
5.c	odd	4	1		80.6.j.a	✓	116
16.f	odd	4	1		80.6.j.a	✓	116
80.s	even	4	1	inner	80.6.s.a	yes	116

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
80.6.j.a	✓	116	5.c	odd	4	1
80.6.j.a	✓	116	16.f	odd	4	1
80.6.s.a	yes	116	1.a	even	1	1	trivial
80.6.s.a	yes	116	80.s	even	4	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(80, [\chi])\).