Newform orbit 76.5.g.a

• Label: 76.5.g.a
• Level: $76$
• Weight: $5$
• Character orbit: 76.g
• Analytic conductor: $7.856$
• Analytic rank: $0$
• Dimension: $76$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	76.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(7.85611719437\)
Analytic rank:	\(0\)
Dimension:	\(76\)
Relative dimension:	\(38\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q - q^{2} + 13 q^{4} - 2 q^{5} - 15 q^{6} - 94 q^{8} + 972 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
7.1	−3.99928	+	0.0760910i	−4.58614	+	2.64781i	15.9884	−	0.608618i	−5.59416	−	9.68937i	18.1398	−	10.9383i			39.7837i	−63.8958	+	3.65061i	−26.4782	+	45.8616i	23.1099	+	38.3248i
7.2	−3.98849	−	0.303244i	0.697952	−	0.402963i	15.8161	+	2.41897i	22.3482	+	38.7083i	−2.90597	+	1.39556i		−	79.0280i	−62.3487	−	14.4442i	−40.1752	+	69.5856i	−77.3976	−	161.164i
7.3	−3.88155	+	0.966218i	8.18792	−	4.72730i	14.1328	−	7.50085i	−10.0033	−	17.3262i	−27.2142	+	26.2605i		−	65.6661i	−47.6099	+	42.7703i	4.19464	−	7.26533i	55.5691	+	57.5871i
7.4	−3.77438	+	1.32442i	13.7242	−	7.92369i	12.4918	−	9.99773i	14.8198	+	25.6686i	−41.3061	+	48.0836i			78.9843i	−33.9076	+	54.2796i	85.0696	−	147.345i	−89.9315	−	77.2553i
7.5	−3.71892	−	1.47296i	4.05898	−	2.34345i	11.6608	+	10.9556i	−1.00987	−	1.74915i	−18.5469	+	2.73642i			54.3250i	−27.2284	−	57.9190i	−29.5164	+	51.1240i	1.17921	+	7.99246i
7.6	−3.65546	+	1.62407i	−12.2858	+	7.09323i	10.7248	−	11.8735i	−5.51885	−	9.55894i	33.3905	−	45.8821i		−	41.7305i	−19.9207	+	60.8208i	60.1279	−	104.145i	35.6983	+	25.9793i
7.7	−3.54548	−	1.85191i	−14.2858	+	8.24790i	9.14084	+	13.1318i	15.0446	+	26.0580i	65.9243	−	2.78673i			53.2660i	−8.08964	−	63.4867i	95.5557	−	165.507i	−5.08315	−	120.250i
7.8	−3.25774	−	2.32102i	12.2124	−	7.05085i	5.22570	+	15.1226i	−5.66259	−	9.80789i	−56.1501	−	5.37552i		−	30.6646i	18.0759	−	61.3943i	58.9291	−	102.068i	−4.31712	+	45.0945i
7.9	−3.19572	−	2.40570i	−7.19063	+	4.15151i	4.42524	+	15.3759i	−24.3962	−	42.2554i	32.9665	+	4.03141i		−	41.7489i	22.8479	−	59.7827i	−6.02993	+	10.4441i	−23.6904	+	193.726i
7.10	−2.88768	+	2.76791i	−6.22437	+	3.59364i	0.677370	−	15.9857i	17.9743	+	31.1323i	8.02710	−	27.6057i			37.3202i	42.2908	+	48.0363i	−14.6715	+	25.4118i	−138.075	−	40.1490i
7.11	−2.87075	+	2.78547i	2.38822	−	1.37884i	0.482361	−	15.9927i	−16.6192	−	28.7853i	−3.01526	+	10.6106i			28.7543i	43.1625	+	47.2547i	−36.6976	+	63.5621i	127.890	+	36.3431i
7.12	−2.44419	+	3.16638i	7.49168	−	4.32533i	−4.05190	−	15.4784i	5.46741	+	9.46983i	−4.61545	+	34.2934i		−	37.0844i	58.9142	+	25.0023i	−3.08311	+	5.34010i	−43.3484	−	5.83414i
7.13	−2.38146	−	3.21382i	−5.15508	+	2.97629i	−4.65733	+	15.3072i	8.40388	+	14.5559i	21.8419	+	9.47962i		−	36.4093i	60.2858	−	21.4855i	−22.7834	+	39.4621i	26.7668	−	61.6729i
7.14	−1.59252	−	3.66931i	5.15508	−	2.97629i	−10.9277	+	11.6869i	8.40388	+	14.5559i	−19.1305	−	14.1758i			36.4093i	60.2858	+	21.4855i	−22.7834	+	39.4621i	40.0269	−	54.0172i
7.15	−1.17048	+	3.82491i	−8.74558	+	5.04926i	−13.2599	−	8.95400i	1.11866	+	1.93758i	−9.07644	−	39.3612i		−	77.4402i	49.7688	−	40.2376i	10.4901	−	18.1694i	−8.72047	+	2.01089i
7.16	−0.707709	+	3.93690i	−10.7755	+	6.22123i	−14.9983	−	5.57236i	−13.6139	−	23.5800i	−16.8664	−	46.8248i			71.9337i	32.5522	−	55.1031i	36.9073	−	63.9254i	102.467	−	36.9088i
7.17	−0.485536	−	3.97042i	7.19063	−	4.15151i	−15.5285	+	3.85556i	−24.3962	−	42.2554i	−19.9746	−	26.5341i			41.7489i	22.8479	+	59.7827i	−6.02993	+	10.4441i	−155.927	+	117.380i
7.18	−0.381196	−	3.98179i	−12.2124	+	7.05085i	−15.7094	+	3.03569i	−5.66259	−	9.80789i	32.7304	+	45.9397i			30.6646i	18.0759	+	61.3943i	58.9291	−	102.068i	−36.8945	+	26.2860i
7.19	−0.359633	+	3.98380i	4.35470	−	2.51418i	−15.7413	−	2.86542i	9.73967	+	16.8696i	8.44992	+	18.2524i			18.1632i	17.0764	−	61.6798i	−27.8578	+	48.2510i	−70.7079	+	32.7340i
7.20	−0.244844	+	3.99250i	14.5224	−	8.38453i	−15.8801	−	1.95508i	−17.7086	−	30.6723i	29.9195	+	60.0337i			23.2025i	11.6938	−	62.9226i	100.101	−	173.380i	126.795	−	63.1918i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
19.c	even	3	1	inner
76.g	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	76.5.g.a	✓	76
4.b	odd	2	1	inner	76.5.g.a	✓	76
19.c	even	3	1	inner	76.5.g.a	✓	76
76.g	odd	6	1	inner	76.5.g.a	✓	76

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
76.5.g.a	✓	76	1.a	even	1	1	trivial
76.5.g.a	✓	76	4.b	odd	2	1	inner
76.5.g.a	✓	76	19.c	even	3	1	inner
76.5.g.a	✓	76	76.g	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(76, [\chi])\).