Newform orbit 381.3.t.a

• Label: 381.3.t.a
• Level: $381$
• Weight: $3$
• Character orbit: 381.t
• Analytic conductor: $10.381$
• Analytic rank: $0$
• Dimension: $1008$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 381 = 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	381.t (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(10.3814980721\)
Analytic rank:	\(0\)
Dimension:	\(1008\)
Relative dimension:	\(84\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1008 q - 13 q^{3} + 316 q^{4} + 15 q^{6} - 52 q^{7} + 27 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
38.1	−1.70300	+	3.53631i	0.111433	−	2.99793i	−7.11132	−	8.91732i	−3.35254	−	2.67356i	10.4118	+	5.49952i	3.11463	+	7.93596i	28.3386	−	6.46809i	−8.97517	−	0.668134i	15.1639	−	7.30255i
38.2	−1.68394	+	3.49674i	2.83521	+	0.980612i	−6.89758	−	8.64929i	3.31527	+	2.64384i	−8.20327	+	8.26269i	−2.98495	−	7.60553i	26.7243	−	6.09965i	7.07680	+	5.56047i	−14.8276	+	7.14057i
38.3	−1.66977	+	3.46731i	−0.397106	+	2.97360i	−6.74013	−	8.45186i	−7.32054	−	5.83794i	−9.64731	−	6.34211i	−0.316400	−	0.806174i	25.5519	−	5.83206i	−8.68461	−	2.36167i	32.4655	−	15.6346i
38.4	−1.64088	+	3.40732i	−2.39663	+	1.80448i	−6.42338	−	8.05467i	5.11334	+	4.07775i	−2.21586	−	11.1270i	4.57155	+	11.6481i	23.2367	−	5.30363i	2.48769	−	8.64936i	−22.2846	+	10.7317i
38.5	−1.63616	+	3.39753i	−1.24783	−	2.72817i	−6.37222	−	7.99051i	4.98742	+	3.97734i	11.3107	+	0.224190i	−1.47769	−	3.76508i	22.8683	−	5.21954i	−5.88583	+	6.80860i	−21.6734	+	10.4373i
38.6	−1.54365	+	3.20541i	−2.98778	−	0.270482i	−5.39788	−	6.76872i	−1.98551	−	1.58339i	5.47908	−	9.15955i	−2.13545	−	5.44103i	16.1548	−	3.68723i	8.85368	+	1.61628i	8.14035	−	3.92019i
38.7	−1.51263	+	3.14100i	2.69867	−	1.31041i	−5.08389	−	6.37499i	−1.02764	−	0.819515i	0.0339175	+	10.4587i	0.408947	+	1.04198i	14.1185	−	3.22246i	5.56565	−	7.07273i	4.12853	−	1.98820i
38.8	−1.45588	+	3.02316i	−0.647060	+	2.92939i	−4.52597	−	5.67539i	2.51543	+	2.00599i	−7.91398	−	6.22100i	−2.13092	−	5.42948i	10.6616	−	2.43344i	−8.16263	−	3.79098i	−9.72661	+	4.68409i
38.9	−1.39985	+	2.90681i	1.84268	+	2.36739i	−3.99601	−	5.01084i	−0.446468	−	0.356046i	−9.46102	+	2.04232i	1.84354	+	4.69727i	7.57767	−	1.72955i	−2.20909	+	8.72467i	1.65995	−	0.799388i
38.10	−1.36250	+	2.82927i	2.89380	+	0.791136i	−3.65438	−	4.58245i	−5.68788	−	4.53593i	−6.18116	+	7.10942i	0.821364	+	2.09280i	5.69803	−	1.30054i	7.74821	+	4.57878i	20.5831	−	9.91231i
38.11	−1.33072	+	2.76328i	1.59229	−	2.54256i	−3.37092	−	4.22700i	−1.19680	−	0.954417i	4.90690	+	7.78339i	0.492640	+	1.25523i	4.20570	−	0.959924i	−3.92922	−	8.09699i	4.22993	−	2.03703i
38.12	−1.32446	+	2.75027i	−2.47567	−	1.69442i	−3.31583	−	4.15792i	3.75667	+	2.99584i	7.93905	−	4.56456i	2.16664	+	5.52050i	3.92296	−	0.895391i	3.25786	+	8.38966i	−13.2149	+	6.36398i
38.13	−1.32156	+	2.74425i	2.87808	−	0.846565i	−3.29044	−	4.12608i	6.37432	+	5.08335i	−1.48037	+	9.01696i	4.39496	+	11.1982i	3.79342	−	0.865823i	7.56665	−	4.87296i	−22.3741	+	10.7748i
38.14	−1.27763	+	2.65302i	0.977524	−	2.83627i	−2.91222	−	3.65181i	4.30171	+	3.43050i	6.27578	+	6.21709i	−3.55816	−	9.06603i	1.92586	−	0.439565i	−7.08889	−	5.54505i	−14.5972	+	7.02963i
38.15	−1.25535	+	2.60677i	−0.984202	−	2.83396i	−2.72538	−	3.41752i	−5.69976	−	4.54541i	8.62301	+	0.992041i	−3.73117	−	9.50687i	1.04698	−	0.238966i	−7.06269	+	5.57838i	19.0041	−	9.15187i
38.16	−1.25254	+	2.60092i	−2.46370	−	1.71177i	−2.70197	−	3.38816i	−4.80701	−	3.83346i	7.53805	−	4.26384i	4.01938	+	10.2412i	0.938937	−	0.214306i	3.13968	+	8.43460i	15.9915	−	7.70109i
38.17	−1.16504	+	2.41924i	−2.40703	+	1.79059i	−2.00142	−	2.50970i	−5.61308	−	4.47628i	−1.52758	−	7.90928i	3.98375	+	10.1504i	−2.06800	+	0.472008i	2.58754	−	8.62001i	17.3687	−	8.36430i
38.18	−1.08811	+	2.25949i	0.586735	+	2.94206i	−1.42734	−	1.78983i	−0.290398	−	0.231585i	−7.28599	−	1.87558i	2.85158	+	7.26570i	−4.18266	+	0.954665i	−8.31148	+	3.45242i	0.839249	−	0.404161i
38.19	−1.08716	+	2.25752i	−2.47469	+	1.69586i	−1.42051	−	1.78127i	1.97762	+	1.57710i	−1.13804	−	7.43033i	−0.234434	−	0.597327i	−4.20577	+	0.959939i	3.24814	−	8.39342i	−5.71034	+	2.74995i
38.20	−1.06863	+	2.21903i	1.97390	+	2.25914i	−1.28817	−	1.61531i	7.68489	+	6.12850i	−7.12247	+	1.96596i	−0.854449	−	2.17710i	−4.64376	+	1.05991i	−1.20745	+	8.91864i	−21.8116	+	10.5039i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
127.i	even	21	1	inner
381.t	odd	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	381.3.t.a	✓	1008
3.b	odd	2	1	inner	381.3.t.a	✓	1008
127.i	even	21	1	inner	381.3.t.a	✓	1008
381.t	odd	42	1	inner	381.3.t.a	✓	1008

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
381.3.t.a	✓	1008	1.a	even	1	1	trivial
381.3.t.a	✓	1008	3.b	odd	2	1	inner
381.3.t.a	✓	1008	127.i	even	21	1	inner
381.3.t.a	✓	1008	381.t	odd	42	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(381, [\chi])\).