Newform orbit 196.2.j.a

• Label: 196.2.j.a
• Level: $196$
• Weight: $2$
• Character orbit: 196.j
• Analytic conductor: $1.565$
• Analytic rank: $0$
• Dimension: $156$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 196 = 2^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	196.j (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(1.56506787962\)
Analytic rank:	\(0\)
Dimension:	\(156\)
Relative dimension:	\(26\) over \(\Q(\zeta_{14})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 156 q - 5 q^{2} - 5 q^{4} - 14 q^{5} - 7 q^{6} - 11 q^{8} - 32 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
27.1	−1.41412	+	0.0163478i	−0.0687858	+	0.301370i	1.99947	−	0.0462354i	−0.644334	−	0.147065i	0.0923445	−	0.427298i	1.15895	−	2.37841i	−2.82673	+	0.0980691i	2.61681	+	1.26019i	0.913570	+	0.197434i
27.2	−1.41194	+	0.0802101i	0.320649	−	1.40485i	1.98713	−	0.226503i	3.61677	+	0.825505i	−0.340052	+	2.00928i	−1.15779	+	2.37897i	−2.78754	+	0.479196i	0.832108	+	0.400722i	−5.17287	−	0.875459i
27.3	−1.34277	+	0.443800i	−0.570354	+	2.49888i	1.60608	−	1.19185i	−1.90324	−	0.434402i	−0.343147	−	3.60856i	−2.60517	−	0.461609i	−1.62767	+	2.31316i	−3.21621	−	1.54884i	2.74841	−	0.261353i
27.4	−1.26947	−	0.623262i	−0.743013	+	3.25535i	1.22309	+	1.58242i	2.48779	+	0.567822i	2.97216	−	3.66947i	2.27356	+	1.35311i	−0.566410	−	2.77113i	−7.34233	−	3.53588i	−2.80426	−	2.27137i
27.5	−1.23433	−	0.690243i	−0.0211920	+	0.0928483i	1.04713	+	1.70397i	−3.74506	−	0.854785i	0.0902457	−	0.0999775i	1.51239	+	2.17087i	−0.116345	−	2.82603i	2.69473	+	1.29772i	4.03262	+	3.64008i
27.6	−1.18418	+	0.773118i	0.570354	−	2.49888i	0.804576	−	1.83103i	−1.90324	−	0.434402i	1.25653	+	3.40009i	2.60517	+	0.461609i	0.462835	+	2.79030i	−3.21621	−	1.54884i	2.58963	−	0.957018i
27.7	−1.04449	−	0.953440i	0.451180	−	1.97675i	0.181906	+	1.99171i	1.77951	+	0.406161i	−2.35596	+	1.63452i	1.22122	−	2.34705i	1.70898	−	2.25375i	−1.00107	−	0.482089i	−1.47142	−	2.12088i
27.8	−0.943039	+	1.05389i	−0.320649	+	1.40485i	−0.221354	−	1.98771i	3.61677	+	0.825505i	−1.17817	−	1.66276i	1.15779	−	2.37897i	2.30357	+	1.64121i	0.832108	+	0.400722i	−4.28075	+	3.03318i
27.9	−0.894470	+	1.09541i	0.0687858	−	0.301370i	−0.399847	−	1.95962i	−0.644334	−	0.147065i	0.268597	+	0.344915i	−1.15895	+	2.37841i	2.50424	+	1.31483i	2.61681	+	1.26019i	0.737435	−	0.574265i
27.10	−0.530578	−	1.31091i	−0.325950	+	1.42808i	−1.43697	+	1.39108i	0.906872	+	0.206988i	2.04503	−	0.330416i	−2.40631	−	1.09985i	2.58601	+	1.14567i	0.769737	+	0.370686i	−0.209824	−	1.29865i
27.11	−0.521710	−	1.31447i	0.555098	−	2.43205i	−1.45564	+	1.37154i	−2.13916	−	0.488249i	−3.48644	+	0.539164i	−2.09149	+	1.62039i	2.56226	+	1.19784i	−2.90380	−	1.39840i	0.474234	+	3.06657i
27.12	−0.304214	+	1.38111i	0.743013	−	3.25535i	−1.81491	−	0.840303i	2.48779	+	0.567822i	4.26995	+	2.01650i	−2.27356	−	1.35311i	1.71267	−	2.25095i	−7.34233	−	3.53588i	−1.54104	+	3.26316i
27.13	−0.229937	+	1.39540i	0.0211920	−	0.0928483i	−1.89426	−	0.641706i	−3.74506	−	0.854785i	0.124687	+	0.0509205i	−1.51239	−	2.17087i	1.33099	−	2.49569i	2.69473	+	1.29772i	2.05389	−	5.02929i
27.14	−0.119871	−	1.40912i	−0.148687	+	0.651440i	−1.97126	+	0.337826i	2.00983	+	0.458730i	0.935782	+	0.131430i	2.05174	+	1.67044i	0.712336	+	2.73726i	2.30064	+	1.10793i	0.405488	−	2.88709i
27.15	0.0942021	+	1.41107i	−0.451180	+	1.97675i	−1.98225	+	0.265852i	1.77951	+	0.406161i	−2.83184	−	0.450434i	−1.22122	+	2.34705i	−0.561869	−	2.77206i	−1.00107	−	0.482089i	−0.405490	+	2.54928i
27.16	0.454626	−	1.33915i	−0.636184	+	2.78731i	−1.58663	−	1.21762i	−3.28016	−	0.748676i	3.44339	+	2.11913i	−1.09871	+	2.40683i	−2.35190	+	1.57117i	−4.66144	−	2.24483i	−2.49384	+	4.05225i
27.17	0.584470	−	1.28779i	0.166766	−	0.730650i	−1.31679	−	1.50534i	−1.97974	−	0.451863i	−0.843452	−	0.641802i	−1.05109	−	2.42800i	−2.70819	+	0.815917i	2.19687	+	1.05796i	−1.73900	+	2.28538i
27.18	0.658802	−	1.25139i	0.637509	−	2.79311i	−1.13196	−	1.64884i	1.25203	+	0.285768i	−3.07528	−	2.63788i	1.52974	+	2.15868i	−2.80908	+	0.330264i	−4.69213	−	2.25961i	1.18245	−	1.37852i
27.19	0.694101	+	1.23216i	0.325950	−	1.42808i	−1.03645	+	1.71049i	0.906872	+	0.206988i	1.98587	−	0.589609i	2.40631	+	1.09985i	−2.82700	−	0.0898155i	0.769737	+	0.370686i	0.374419	+	1.26108i
27.20	0.702410	+	1.22744i	−0.555098	+	2.43205i	−1.01324	+	1.72434i	−2.13916	−	0.488249i	−3.37511	+	1.02694i	2.09149	−	1.62039i	−2.82824	−	0.0325071i	−2.90380	−	1.39840i	−0.903267	−	2.96865i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
49.f	odd	14	1	inner
196.j	even	14	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	196.2.j.a	✓	156
4.b	odd	2	1	inner	196.2.j.a	✓	156
49.f	odd	14	1	inner	196.2.j.a	✓	156
196.j	even	14	1	inner	196.2.j.a	✓	156

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
196.2.j.a	✓	156	1.a	even	1	1	trivial
196.2.j.a	✓	156	4.b	odd	2	1	inner
196.2.j.a	✓	156	49.f	odd	14	1	inner
196.2.j.a	✓	156	196.j	even	14	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(196, [\chi])\).