Newform orbit 28.7.g.a

• Label: 28.7.g.a
• Level: $28$
• Weight: $7$
• Character orbit: 28.g
• Analytic conductor: $6.442$
• Analytic rank: $0$
• Dimension: $44$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 28 = 2^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	28.g (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.44151434136\)
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 dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 6 q^{2} + 44 q^{4} - 2 q^{5} - 132 q^{6} + 792 q^{8} + 4372 q^{9}+O(q^{10}) \)

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	−7.89877	−	1.26864i	−2.43660	−	1.40677i	60.7811	+	20.0414i	−19.2345	−	33.3152i	17.4614	+	14.2029i	274.756	+	205.325i	−454.671	−	235.412i	−360.542	−	624.477i	109.664	+	287.551i
11.2	−7.69548	+	2.18621i	−42.3811	−	24.4687i	54.4409	−	33.6479i	−76.3886	−	132.309i	379.637	+	95.6446i	−289.700	−	183.639i	−345.388	+	377.957i	832.936	+	1442.69i	877.103	+	851.180i
11.3	−7.53145	−	2.69764i	40.4534	+	23.3558i	49.4455	+	40.6343i	20.5775	+	35.6413i	−241.668	−	285.032i	−318.771	+	126.625i	−262.780	−	439.421i	726.487	+	1258.31i	−58.8312	−	323.942i
11.4	−7.21097	+	3.46438i	18.9511	+	10.9414i	39.9961	−	49.9631i	−31.3249	−	54.2563i	−174.561	−	13.2444i	226.343	−	257.717i	−115.320	+	498.844i	−125.071	−	216.630i	413.847	+	282.719i
11.5	−6.69360	−	4.38130i	−20.9088	−	12.0717i	25.6084	+	58.6533i	97.9064	+	169.579i	87.0653	+	172.411i	−66.1665	−	336.558i	85.5650	−	504.800i	−73.0485	−	126.524i	87.6298	−	1564.05i
11.6	−6.35138	+	4.86415i	−1.29774	−	0.749250i	16.6800	−	61.7882i	85.8124	+	148.631i	11.8869	−	1.55363i	−307.530	+	151.901i	194.606	+	473.574i	−363.377	−	629.388i	−1267.99	−	526.610i
11.7	−4.45621	−	6.64396i	−20.9712	−	12.1077i	−24.2844	+	59.2137i	−61.8703	−	107.162i	13.0088	+	193.286i	−79.3188	+	333.703i	501.630	−	102.524i	−71.3061	−	123.506i	−436.276	+	888.602i
11.8	−3.52573	−	7.18117i	20.9712	+	12.1077i	−39.1384	+	50.6378i	−61.8703	−	107.162i	13.0088	−	193.286i	79.3188	−	333.703i	501.630	+	102.524i	−71.3061	−	123.506i	−551.414	+	822.127i
11.9	−2.75359	+	7.51117i	−28.0528	−	16.1963i	−48.8354	−	41.3655i	14.2313	+	24.6493i	198.899	−	166.112i	311.758	+	143.024i	445.176	−	252.908i	160.141	+	277.372i	−224.332	+	39.0194i
11.10	−2.37213	+	7.64022i	19.1737	+	11.0699i	−52.7460	−	36.2472i	−117.928	−	204.258i	−130.059	+	120.232i	−278.420	+	200.329i	402.057	−	317.008i	−119.414	−	206.831i	1840.32	−	416.471i
11.11	−1.03641	+	7.93258i	39.2900	+	22.6841i	−61.8517	−	16.4428i	86.1951	+	149.294i	−220.664	+	288.161i	197.176	−	280.661i	194.538	−	473.602i	664.636	+	1151.18i	−1273.62	+	529.019i
11.12	−0.447518	−	7.98747i	20.9088	+	12.0717i	−63.5995	+	7.14908i	97.9064	+	169.579i	87.0653	−	172.411i	66.1665	+	336.558i	85.5650	+	504.800i	−73.0485	−	126.524i	1310.69	−	857.914i
11.13	1.42950	−	7.87125i	−40.4534	−	23.3558i	−59.9131	−	22.5039i	20.5775	+	35.6413i	−241.668	+	285.032i	318.771	−	126.625i	−262.780	+	439.421i	726.487	+	1258.31i	309.957	−	111.022i
11.14	2.77592	+	7.50295i	−16.7059	−	9.64517i	−48.5886	+	41.6551i	1.52383	+	2.63935i	25.9930	−	152.118i	−178.748	−	292.742i	−447.414	−	248.927i	−178.442	−	309.070i	−15.5729	+	18.7598i
11.15	2.85071	−	7.47485i	2.43660	+	1.40677i	−47.7469	−	42.6173i	−19.2345	−	33.3152i	17.4614	−	14.2029i	−274.756	−	205.325i	−454.671	+	235.412i	−360.542	−	624.477i	−303.859	+	48.8034i
11.16	5.10979	+	6.15549i	16.7059	+	9.64517i	−11.7801	+	62.9065i	1.52383	+	2.63935i	25.9930	+	152.118i	178.748	+	292.742i	−447.414	+	248.927i	−178.442	−	309.070i	−8.46005	+	22.8664i
11.17	5.74106	−	5.57138i	42.3811	+	24.4687i	1.91950	−	63.9712i	−76.3886	−	132.309i	379.637	−	95.6446i	289.700	+	183.639i	−345.388	−	377.957i	832.936	+	1442.69i	−1175.69	−	334.004i
11.18	6.60573	−	4.51269i	−18.9511	−	10.9414i	23.2712	−	59.6192i	−31.3249	−	54.2563i	−174.561	+	13.2444i	−226.343	+	257.717i	−115.320	−	498.844i	−125.071	−	216.630i	−451.766	−	217.043i
11.19	7.38802	+	3.06873i	−39.2900	−	22.6841i	45.1658	+	45.3437i	86.1951	+	149.294i	−220.664	−	288.161i	−197.176	+	280.661i	194.538	+	473.602i	664.636	+	1151.18i	178.667	+	1367.50i
11.20	7.38817	−	3.06838i	1.29774	+	0.749250i	45.1701	−	45.3394i	85.8124	+	148.631i	11.8869	+	1.55363i	307.530	−	151.901i	194.606	−	473.574i	−363.377	−	629.388i	1090.05	+	834.809i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	28.7.g.a	✓	44
4.b	odd	2	1	inner	28.7.g.a	✓	44
7.c	even	3	1	inner	28.7.g.a	✓	44
28.g	odd	6	1	inner	28.7.g.a	✓	44

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
28.7.g.a	✓	44	1.a	even	1	1	trivial
28.7.g.a	✓	44	4.b	odd	2	1	inner
28.7.g.a	✓	44	7.c	even	3	1	inner
28.7.g.a	✓	44	28.g	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(28, [\chi])\).