Newform orbit 77.4.i.a

• Label: 77.4.i.a
• Level: $77$
• Weight: $4$
• Character orbit: 77.i
• Analytic conductor: $4.543$
• Analytic rank: $0$
• Dimension: $44$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	77.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.54314707044\)
Analytic rank:	\(0\)
Dimension:	\(44\)
Relative dimension:	\(22\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) 44 * q - 6 * q^3 + 88 * q^4 - 30 * q^5 + 192 * q^9 + 13 * q^11 - 54 * q^12 - 156 * q^14 - 204 * q^15 - 500 * q^16 + 112 * q^22 - 26 * q^23 - 88 * q^25 + 642 * q^26 + 822 * q^31 - 801 * q^33 + 2196 * q^36 + 122 * q^37 + 120 * q^38 - 1940 * q^42 + 256 * q^44 - 2640 * q^45 - 462 * q^47 + 244 * q^49 + 1162 * q^53 + 3706 * q^56 - 918 * q^58 + 462 * q^59 - 3000 * q^60 - 10020 * q^64 - 2400 * q^66 + 950 * q^67 + 3578 * q^70 + 4376 * q^71 + 6552 * q^75 + 877 * q^77 + 4280 * q^78 + 7770 * q^80 + 2350 * q^81 - 5214 * q^82 + 3146 * q^86 + 3764 * q^88 - 3966 * q^89 - 7236 * q^91 + 1116 * q^92 - 2126 * q^93 - 5240 * q^99

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
10.1	−4.81148	+	2.77791i	−6.18845	−	3.57290i	11.4335	−	19.8035i	−2.56430	+	1.48050i	39.7008			1.74480	−	18.4379i			82.5987i	12.0313	+	20.8388i	8.22539	−	14.2468i
10.2	−4.51600	+	2.60731i	6.78401	+	3.91675i	9.59618	−	16.6211i	−12.3853	+	7.15065i	−40.8488			16.2310	+	8.91933i			58.3640i	17.1818	+	29.7598i	37.2880	−	64.5847i
10.3	−3.79143	+	2.18898i	1.35419	+	0.781840i	5.58329	−	9.67055i	−0.165658	+	0.0956428i	−6.84574			−17.1815	+	6.91355i			13.8632i	−12.2775	−	21.2652i	0.418721	−	0.725246i
10.4	−3.49297	+	2.01667i	−4.00405	−	2.31174i	4.13388	−	7.16009i	9.63519	−	5.56288i	18.6480			2.71384	+	18.3203i			1.07996i	−2.81171	−	4.87002i	−22.4369	+	38.8619i
10.5	−3.48074	+	2.00961i	3.19973	+	1.84737i	4.07703	−	7.06163i	8.91138	−	5.14499i	−14.8499			5.55950	−	17.6661i			0.619214i	−6.67447	−	11.5605i	−20.6788	+	35.8167i
10.6	−2.61630	+	1.51052i	−4.45189	−	2.57030i	0.563345	−	0.975742i	−14.9138	+	8.61049i	15.5300			14.6921	+	11.2757i		−	20.7646i	−0.287126	−	0.497318i	26.0127	−	45.0552i
10.7	−1.77427	+	1.02437i	−8.30792	−	4.79658i	−1.90131	+	3.29317i	−0.821073	+	0.474047i	19.6540			−14.8519	−	11.0644i		−	24.1806i	32.5144	+	56.3166i	0.971203	−	1.68217i
10.8	−1.75974	+	1.01599i	6.53171	+	3.77109i	−1.93555	+	3.35247i	−10.7319	+	6.19608i	−15.3255			−18.3393	+	2.58270i		−	24.1217i	14.9422	+	25.8806i	12.5903	−	21.8070i
10.9	−1.44451	+	0.833988i	6.44092	+	3.71867i	−2.60893	+	4.51879i	11.5145	−	6.64788i	−12.4053			12.2288	+	13.9088i		−	22.0471i	14.1570	+	24.5206i	−11.0885	+	19.2058i
10.10	−1.13937	+	0.657817i	0.191280	+	0.110436i	−3.13455	+	5.42920i	−7.66070	+	4.42291i	−0.290586			7.31040	−	17.0164i		−	18.7729i	−13.4756	−	23.3404i	5.81893	−	10.0787i
10.11	−0.538156	+	0.310704i	−3.04952	−	1.76064i	−3.80693	+	6.59379i	11.6817	−	6.74445i	2.18816			−18.4738	+	1.31112i		−	9.70259i	−7.30028	−	12.6445i	−4.19106	+	7.25913i
10.12	0.538156	−	0.310704i	−3.04952	−	1.76064i	−3.80693	+	6.59379i	11.6817	−	6.74445i	−2.18816			18.4738	−	1.31112i			9.70259i	−7.30028	−	12.6445i	4.19106	−	7.25913i
10.13	1.13937	−	0.657817i	0.191280	+	0.110436i	−3.13455	+	5.42920i	−7.66070	+	4.42291i	0.290586			−7.31040	+	17.0164i			18.7729i	−13.4756	−	23.3404i	−5.81893	+	10.0787i
10.14	1.44451	−	0.833988i	6.44092	+	3.71867i	−2.60893	+	4.51879i	11.5145	−	6.64788i	12.4053			−12.2288	−	13.9088i			22.0471i	14.1570	+	24.5206i	11.0885	−	19.2058i
10.15	1.75974	−	1.01599i	6.53171	+	3.77109i	−1.93555	+	3.35247i	−10.7319	+	6.19608i	15.3255			18.3393	−	2.58270i			24.1217i	14.9422	+	25.8806i	−12.5903	+	21.8070i
10.16	1.77427	−	1.02437i	−8.30792	−	4.79658i	−1.90131	+	3.29317i	−0.821073	+	0.474047i	−19.6540			14.8519	+	11.0644i			24.1806i	32.5144	+	56.3166i	−0.971203	+	1.68217i
10.17	2.61630	−	1.51052i	−4.45189	−	2.57030i	0.563345	−	0.975742i	−14.9138	+	8.61049i	−15.5300			−14.6921	−	11.2757i			20.7646i	−0.287126	−	0.497318i	−26.0127	+	45.0552i
10.18	3.48074	−	2.00961i	3.19973	+	1.84737i	4.07703	−	7.06163i	8.91138	−	5.14499i	14.8499			−5.55950	+	17.6661i		−	0.619214i	−6.67447	−	11.5605i	20.6788	−	35.8167i
10.19	3.49297	−	2.01667i	−4.00405	−	2.31174i	4.13388	−	7.16009i	9.63519	−	5.56288i	−18.6480			−2.71384	−	18.3203i		−	1.07996i	−2.81171	−	4.87002i	22.4369	−	38.8619i
10.20	3.79143	−	2.18898i	1.35419	+	0.781840i	5.58329	−	9.67055i	−0.165658	+	0.0956428i	6.84574			17.1815	−	6.91355i		−	13.8632i	−12.2775	−	21.2652i	−0.418721	+	0.725246i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.d	odd	6	1	inner
11.b	odd	2	1	inner
77.i	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	77.4.i.a	✓	44
7.d	odd	6	1	inner	77.4.i.a	✓	44
11.b	odd	2	1	inner	77.4.i.a	✓	44
77.i	even	6	1	inner	77.4.i.a	✓	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
77.4.i.a	✓	44	1.a	even	1	1	trivial
77.4.i.a	✓	44	7.d	odd	6	1	inner
77.4.i.a	✓	44	11.b	odd	2	1	inner
77.4.i.a	✓	44	77.i	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(77, [\chi])\).