Newform orbit 231.3.k.a

• Label: 231.3.k.a
• Level: $231$
• Weight: $3$
• Character orbit: 231.k
• Analytic conductor: $6.294$
• Analytic rank: $0$
• Dimension: $120$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 231 = 3 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	231.k (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.29429410672\)
Analytic rank:	\(0\)
Dimension:	\(120\)
Relative dimension:	\(60\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 6 q^{3} - 116 q^{4} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
131.1	−1.94037	+	3.36082i	−2.97029	−	0.421185i	−5.53008	−	9.57839i	3.08113	−	5.33667i	7.17899	−	9.16535i	−4.45296	+	5.40103i	27.3987			8.64521	+	2.50208i	11.9571	+	20.7103i
131.2	−1.94037	+	3.36082i	−1.12039	+	2.78294i	−5.53008	−	9.57839i	−3.08113	+	5.33667i	−7.17899	−	9.16535i	4.45296	−	5.40103i	27.3987			−6.48947	−	6.23593i	−11.9571	−	20.7103i
131.3	−1.85086	+	3.20578i	−0.183968	−	2.99435i	−4.85133	−	8.40275i	−0.683314	+	1.18353i	9.93972	+	4.95236i	−4.06233	−	5.70066i	21.1096			−8.93231	+	1.10173i	−2.52943	−	4.38110i
131.4	−1.85086	+	3.20578i	2.50120	+	1.65650i	−4.85133	−	8.40275i	0.683314	−	1.18353i	−9.93972	+	4.95236i	4.06233	+	5.70066i	21.1096			3.51203	+	8.28647i	2.52943	+	4.38110i
131.5	−1.69747	+	2.94010i	2.22789	−	2.00911i	−3.76281	−	6.51738i	−1.32321	+	2.29186i	2.12523	+	9.96062i	6.22846	+	3.19473i	11.9693			0.926949	−	8.95214i	−4.49221	−	7.78073i
131.6	−1.69747	+	2.94010i	2.85388	−	0.924850i	−3.76281	−	6.51738i	1.32321	−	2.29186i	−2.12523	+	9.96062i	−6.22846	−	3.19473i	11.9693			7.28930	−	5.27883i	4.49221	+	7.78073i
131.7	−1.60960	+	2.78790i	−2.06235	−	2.17870i	−3.18160	−	5.51069i	−4.25642	+	7.37233i	9.39354	−	2.24279i	3.57877	+	6.01601i	7.60758			−0.493463	+	8.98646i	−13.7022	−	23.7329i
131.8	−1.60960	+	2.78790i	0.855636	+	2.87539i	−3.18160	−	5.51069i	4.25642	−	7.37233i	−9.39354	−	2.24279i	−3.57877	−	6.01601i	7.60758			−7.53577	+	4.92058i	13.7022	+	23.7329i
131.9	−1.44639	+	2.50522i	−2.60021	+	1.49629i	−2.18407	−	3.78292i	2.06497	−	3.57663i	0.0123854	−	8.67831i	6.99487	+	0.267880i	1.06495			4.52223	−	7.78135i	5.97349	+	10.3464i
131.10	−1.44639	+	2.50522i	−2.59593	+	1.50371i	−2.18407	−	3.78292i	−2.06497	+	3.57663i	−0.0123854	−	8.67831i	−6.99487	−	0.267880i	1.06495			4.47773	−	7.80704i	−5.97349	−	10.3464i
131.11	−1.42363	+	2.46580i	0.0335667	−	2.99981i	−2.05344	−	3.55666i	3.79600	−	6.57486i	7.34914	+	4.35339i	6.50498	−	2.58558i	0.304317			−8.99775	−	0.201388i	10.8082	+	18.7203i
131.12	−1.42363	+	2.46580i	2.61470	+	1.47084i	−2.05344	−	3.55666i	−3.79600	+	6.57486i	−7.34914	+	4.35339i	−6.50498	+	2.58558i	0.304317			4.67328	+	7.69158i	−10.8082	−	18.7203i
131.13	−1.23819	+	2.14461i	−2.90725	−	0.740200i	−1.06625	−	1.84679i	−1.54873	+	2.68248i	5.18718	−	5.31842i	−0.785376	−	6.95580i	−4.62467			7.90421	+	4.30390i	−3.83525	−	6.64285i
131.14	−1.23819	+	2.14461i	−0.812593	+	2.88785i	−1.06625	−	1.84679i	1.54873	−	2.68248i	−5.18718	−	5.31842i	0.785376	+	6.95580i	−4.62467			−7.67939	−	4.69330i	3.83525	+	6.64285i
131.15	−1.18344	+	2.04977i	−1.07169	−	2.80205i	−0.801051	−	1.38746i	1.75890	−	3.04651i	7.01185	+	1.11933i	−3.32197	+	6.16153i	−5.67553			−6.70297	+	6.00585i	4.16310	+	7.21070i
131.16	−1.18344	+	2.04977i	1.89080	+	2.32913i	−0.801051	−	1.38746i	−1.75890	+	3.04651i	−7.01185	+	1.11933i	3.32197	−	6.16153i	−5.67553			−1.84973	+	8.80787i	−4.16310	−	7.21070i
131.17	−0.956646	+	1.65696i	1.90591	−	2.31679i	0.169656	+	0.293852i	−1.21819	+	2.10997i	2.01554	+	5.37436i	−4.59407	+	5.28152i	−8.30237			−1.73501	−	8.83118i	−2.33075	−	4.03698i
131.18	−0.956646	+	1.65696i	2.95935	−	0.492173i	0.169656	+	0.293852i	1.21819	−	2.10997i	−2.01554	+	5.37436i	4.59407	−	5.28152i	−8.30237			8.51553	−	2.91303i	2.33075	+	4.03698i
131.19	−0.757476	+	1.31199i	0.427840	−	2.96934i	0.852461	+	1.47651i	−4.04732	+	7.01016i	3.57165	+	2.81052i	−1.47739	−	6.84232i	−8.64268			−8.63391	−	2.54080i	−6.13149	−	10.6201i
131.20	−0.757476	+	1.31199i	2.78544	+	1.11415i	0.852461	+	1.47651i	4.04732	−	7.01016i	−3.57165	+	2.81052i	1.47739	+	6.84232i	−8.64268			6.51735	+	6.20678i	6.13149	+	10.6201i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
7.d	odd	6	1	inner
11.b	odd	2	1	inner
21.g	even	6	1	inner
33.d	even	2	1	inner
77.i	even	6	1	inner
231.k	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	231.3.k.a	✓	120
3.b	odd	2	1	inner	231.3.k.a	✓	120
7.d	odd	6	1	inner	231.3.k.a	✓	120
11.b	odd	2	1	inner	231.3.k.a	✓	120
21.g	even	6	1	inner	231.3.k.a	✓	120
33.d	even	2	1	inner	231.3.k.a	✓	120
77.i	even	6	1	inner	231.3.k.a	✓	120
231.k	odd	6	1	inner	231.3.k.a	✓	120

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
231.3.k.a	✓	120	1.a	even	1	1	trivial
231.3.k.a	✓	120	3.b	odd	2	1	inner
231.3.k.a	✓	120	7.d	odd	6	1	inner
231.3.k.a	✓	120	11.b	odd	2	1	inner
231.3.k.a	✓	120	21.g	even	6	1	inner
231.3.k.a	✓	120	33.d	even	2	1	inner
231.3.k.a	✓	120	77.i	even	6	1	inner
231.3.k.a	✓	120	231.k	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(231, [\chi])\).