Newform orbit 192.4.s.a

• Label: 192.4.s.a
• Level: $192$
• Weight: $4$
• Character orbit: 192.s
• Analytic conductor: $11.328$
• Analytic rank: $0$
• Dimension: $752$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 192 = 2^{6} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	192.s (of order \(16\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(11.3283667211\)
Analytic rank:	\(0\)
Dimension:	\(752\)
Relative dimension:	\(94\) over \(\Q(\zeta_{16})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{16}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 752 q - 8 q^{3} - 16 q^{4} - 8 q^{6} - 16 q^{7} - 8 q^{9}+O(q^{10}) \)

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	−2.82644	+	0.106092i	4.98683	−	1.45996i	7.97749	−	0.599723i	8.11148	−	5.41992i	−13.9401	+	4.65555i	4.72329	−	11.4030i	−22.4842	+	2.54142i	22.7370	−	14.5612i	−22.3516	+	16.1796i
11.2	−2.82562	−	0.125892i	−3.99484	−	3.32283i	7.96830	+	0.711447i	3.03116	−	2.02536i	10.8696	+	9.89200i	1.85677	−	4.48263i	−22.4259	−	3.01343i	4.91755	+	26.5484i	−8.81990	+	5.34130i
11.3	−2.81354	−	0.289782i	2.11359	+	4.74687i	7.83205	+	1.63063i	4.22313	−	2.82180i	−4.57113	−	13.9680i	−5.87541	+	14.1845i	−21.5633	−	6.85744i	−18.0655	+	20.0659i	−12.6997	+	6.71548i
11.4	−2.81349	−	0.290342i	3.76814	−	3.57787i	7.83140	+	1.63374i	−10.5748	+	7.06585i	−11.6404	+	8.97223i	−12.9916	+	31.3646i	−21.5592	−	6.87030i	1.39775	−	26.9638i	31.8035	−	16.8094i
11.5	−2.80607	−	0.354949i	−2.23014	−	4.69324i	7.74802	+	1.99202i	−16.0970	+	10.7557i	4.59206	+	13.9611i	4.44468	−	10.7304i	−21.0344	−	8.33989i	−17.0530	+	20.9331i	48.9870	−	24.4675i
11.6	−2.79109	+	0.458054i	−5.17801	+	0.433851i	7.58037	−	2.55694i	−2.16623	+	1.44743i	14.2536	−	3.58272i	−2.45272	+	5.92138i	−19.9863	+	10.6089i	26.6235	−	4.49297i	5.38315	−	5.03216i
11.7	−2.72225	−	0.767695i	1.92425	−	4.82673i	6.82129	+	4.17972i	8.42410	−	5.62880i	−8.94373	+	11.6623i	10.2106	−	24.6506i	−15.3605	−	16.6149i	−19.5946	−	18.5756i	−27.2537	+	8.85587i
11.8	−2.71727	+	0.785136i	−0.0376904	−	5.19602i	6.76712	−	4.26685i	12.1903	−	8.14528i	4.18199	+	14.0894i	−12.9250	+	31.2038i	−15.0380	+	16.9073i	−26.9972	+	0.391680i	−26.7291	+	31.7039i
11.9	−2.70717	−	0.819277i	4.94338	+	1.60094i	6.65757	+	4.43585i	−13.3167	+	8.89796i	−12.0710	−	8.38401i	6.37772	−	15.3972i	−14.3890	−	17.4630i	21.8740	+	15.8281i	43.3406	−	13.1782i
11.10	−2.69889	+	0.846155i	−3.41820	+	3.91356i	6.56804	−	4.56736i	4.68115	−	3.12784i	5.91388	−	13.4546i	11.0247	−	26.6159i	−13.8618	+	17.8844i	−3.63183	−	26.7546i	−9.98728	+	12.4027i
11.11	−2.59849	−	1.11707i	−4.71591	+	2.18179i	5.50432	+	5.80538i	16.8773	−	11.2770i	14.6915	−	0.401369i	−3.47579	+	8.39129i	−7.81794	−	21.2339i	17.4796	−	20.5782i	−56.4526	+	10.4502i
11.12	−2.59781	+	1.11866i	−1.55589	+	4.95774i	5.49718	−	5.81214i	−15.8264	+	10.5749i	−1.50414	−	14.6198i	−8.00150	+	19.3173i	−7.77877	+	21.2483i	−22.1584	−	15.4274i	29.2842	−	45.1758i
11.13	−2.59648	−	1.12174i	−2.00614	+	4.79327i	5.48340	+	5.82515i	−6.13514	+	4.09937i	10.5857	−	10.1952i	4.90774	−	11.8483i	−7.70321	−	21.2758i	−18.9508	−	19.2319i	20.5282	−	3.76189i
11.14	−2.56488	+	1.19221i	4.98095	+	1.47990i	5.15726	−	6.11577i	12.7649	−	8.52926i	−14.5399	+	2.14258i	−1.73730	+	4.19422i	−5.93649	+	21.8348i	22.6198	+	14.7426i	−22.5719	+	37.0951i
11.15	−2.53126	+	1.26203i	2.98426	+	4.25373i	4.81459	−	6.38904i	−6.77522	+	4.52706i	−12.9223	−	7.00108i	10.2826	−	24.8243i	−4.12386	+	22.2485i	−9.18836	+	25.3885i	11.4366	−	20.0097i
11.16	−2.50247	+	1.31819i	4.70340	−	2.20863i	4.52472	−	6.59749i	−8.22088	+	5.49302i	−8.85871	+	11.7270i	1.15969	−	2.79974i	−2.62622	+	22.4745i	17.2439	−	20.7761i	13.3316	−	24.5828i
11.17	−2.39079	+	1.51133i	0.794247	−	5.13509i	3.43175	−	7.22656i	−7.83205	+	5.23321i	5.86196	+	13.4773i	4.69205	−	11.3276i	2.71714	+	22.4637i	−25.7383	−	8.15706i	10.8157	−	24.3483i
11.18	−2.29649	−	1.65109i	−2.66306	−	4.46185i	2.54777	+	7.58346i	8.14700	−	5.44365i	−1.25123	+	14.6436i	−3.56707	+	8.61168i	6.67005	−	21.6220i	−12.8162	+	23.7644i	−27.6975	−	0.950150i
11.19	−2.29206	−	1.65725i	−5.14407	+	0.733846i	2.50704	+	7.59702i	−12.1134	+	8.09395i	13.0067	+	6.84300i	−12.0256	+	29.0323i	6.84387	−	21.5676i	25.9229	−	7.54991i	41.1784	+	1.52323i
11.20	−2.26848	+	1.68938i	−2.32414	+	4.64741i	2.29202	−	7.66464i	12.0667	−	8.06274i	−2.57895	−	14.4689i	−9.72015	+	23.4665i	7.74906	+	21.2592i	−16.1968	−	21.6024i	−13.7522	+	38.6755i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
64.j	odd	16	1	inner
192.s	even	16	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	192.4.s.a	✓	752
3.b	odd	2	1	inner	192.4.s.a	✓	752
64.j	odd	16	1	inner	192.4.s.a	✓	752
192.s	even	16	1	inner	192.4.s.a	✓	752

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
192.4.s.a	✓	752	1.a	even	1	1	trivial
192.4.s.a	✓	752	3.b	odd	2	1	inner
192.4.s.a	✓	752	64.j	odd	16	1	inner
192.4.s.a	✓	752	192.s	even	16	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(192, [\chi])\).