Newform orbit 138.5.h.a

• Label: 138.5.h.a
• Level: $138$
• Weight: $5$
• Character orbit: 138.h
• Analytic conductor: $14.265$
• Analytic rank: $0$
• Dimension: $160$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.h (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(14.2650549056\)
Analytic rank:	\(0\)
Dimension:	\(160\)
Relative dimension:	\(16\) over \(\Q(\zeta_{22})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 160 q - 128 q^{4} - 432 q^{9}+O(q^{10}) \)

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	−1.85223	+	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	−31.8139	+	14.5289i	2.09159	−	14.5473i	−15.1574	−	51.6215i	19.0354	+	12.2333i	11.2162	−	24.5601i	27.8698	−	94.9158i
7.2	−1.85223	+	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	−16.9332	+	7.73311i	2.09159	−	14.5473i	26.3243	+	89.6523i	19.0354	+	12.2333i	11.2162	−	24.5601i	14.8339	−	50.5195i
7.3	−1.85223	+	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	0.0789724	−	0.0360655i	2.09159	−	14.5473i	−7.37685	−	25.1232i	19.0354	+	12.2333i	11.2162	−	24.5601i	−0.0691817	+	0.235611i
7.4	−1.85223	+	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	33.6623	−	15.3731i	2.09159	−	14.5473i	−9.29808	−	31.6664i	19.0354	+	12.2333i	11.2162	−	24.5601i	−29.4890	+	100.430i
7.5	−1.85223	+	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	−19.2776	+	8.80377i	−2.09159	+	14.5473i	14.6430	+	49.8696i	19.0354	+	12.2333i	11.2162	−	24.5601i	16.8876	−	57.5140i
7.6	−1.85223	+	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	2.65171	−	1.21100i	−2.09159	+	14.5473i	−17.2943	−	58.8990i	19.0354	+	12.2333i	11.2162	−	24.5601i	−2.32296	+	7.91129i
7.7	−1.85223	+	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	5.66414	−	2.58673i	−2.09159	+	14.5473i	9.37267	+	31.9204i	19.0354	+	12.2333i	11.2162	−	24.5601i	−4.96193	+	16.8988i
7.8	−1.85223	+	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	43.6718	−	19.9442i	−2.09159	+	14.5473i	−0.290384	−	0.988957i	19.0354	+	12.2333i	11.2162	−	24.5601i	−38.2575	+	130.293i
7.9	1.85223	−	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	−23.8564	+	10.8948i	−2.09159	+	14.5473i	2.81212	+	9.57718i	−19.0354	−	12.2333i	11.2162	−	24.5601i	−20.8988	+	71.1747i
7.10	1.85223	−	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	0.678234	−	0.309739i	−2.09159	+	14.5473i	−6.16270	−	20.9882i	−19.0354	−	12.2333i	11.2162	−	24.5601i	0.594150	−	2.02349i
7.11	1.85223	−	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	1.14894	−	0.524701i	−2.09159	+	14.5473i	1.77751	+	6.05364i	−19.0354	−	12.2333i	11.2162	−	24.5601i	1.00650	−	3.42781i
7.12	1.85223	−	2.13758i	−4.37128	+	2.80925i	−1.13852	−	7.91857i	42.3107	−	19.3226i	−2.09159	+	14.5473i	22.4575	+	76.4833i	−19.0354	−	12.2333i	11.2162	−	24.5601i	37.0652	−	126.232i
7.13	1.85223	−	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	−26.7504	+	12.2165i	2.09159	−	14.5473i	18.8860	+	64.3200i	−19.0354	−	12.2333i	11.2162	−	24.5601i	−23.4340	+	79.8088i
7.14	1.85223	−	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	−22.2415	+	10.1573i	2.09159	−	14.5473i	−7.02266	−	23.9170i	−19.0354	−	12.2333i	11.2162	−	24.5601i	−19.4841	+	66.3567i
7.15	1.85223	−	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	21.6210	−	9.87396i	2.09159	−	14.5473i	14.7757	+	50.3214i	−19.0354	−	12.2333i	11.2162	−	24.5601i	18.9405	−	64.5054i
7.16	1.85223	−	2.13758i	4.37128	−	2.80925i	−1.13852	−	7.91857i	24.7936	−	11.3229i	2.09159	−	14.5473i	−17.6937	−	60.2593i	−19.0354	−	12.2333i	11.2162	−	24.5601i	21.7198	−	73.9710i
19.1	−1.17497	−	2.57283i	−4.98567	+	1.46393i	−5.23889	+	6.04600i	−20.3659	+	31.6899i	9.62445	+	11.1072i	−81.9918	+	11.7886i	21.7108	+	6.37488i	22.7138	−	14.5973i	105.462	+	15.1631i
19.2	−1.17497	−	2.57283i	−4.98567	+	1.46393i	−5.23889	+	6.04600i	−14.7644	+	22.9738i	9.62445	+	11.1072i	56.7330	−	8.15697i	21.7108	+	6.37488i	22.7138	−	14.5973i	76.4554	+	10.9926i
19.3	−1.17497	−	2.57283i	−4.98567	+	1.46393i	−5.23889	+	6.04600i	7.54204	−	11.7356i	9.62445	+	11.1072i	−12.2007	+	1.75419i	21.7108	+	6.37488i	22.7138	−	14.5973i	−39.0554	−	5.61533i
19.4	−1.17497	−	2.57283i	−4.98567	+	1.46393i	−5.23889	+	6.04600i	19.8768	−	30.9289i	9.62445	+	11.1072i	−20.0835	+	2.88758i	21.7108	+	6.37488i	22.7138	−	14.5973i	−102.929	−	14.7990i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
23.d	odd	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	138.5.h.a	✓	160
23.d	odd	22	1	inner	138.5.h.a	✓	160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
138.5.h.a	✓	160	1.a	even	1	1	trivial
138.5.h.a	✓	160	23.d	odd	22	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(138, [\chi])\).