Newform orbit 272.2.j.a

• Label: 272.2.j.a
• Level: $272$
• Weight: $2$
• Character orbit: 272.j
• Analytic conductor: $2.172$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	272.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(2.17193093498\)
Analytic rank:	\(0\)
Dimension:	\(68\)
Relative dimension:	\(34\) over \(\Q(i)\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{4} - 4 q^{5} + 6 q^{6} - 60 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
13.1	−1.39459	+	0.234745i			1.53523i	1.88979	−	0.654748i	−1.14091			−0.360387	−	2.14102i	0.381270	+	0.381270i	−2.48179	+	1.35673i	0.643071			1.59111	−	0.267823i
13.2	−1.38600	−	0.281086i		−	0.583850i	1.84198	+	0.779169i	−2.97401			−0.164112	+	0.809216i	1.20406	+	1.20406i	−2.33397	−	1.59768i	2.65912			4.12198	+	0.835953i
13.3	−1.37201	+	0.342907i		−	3.15633i	1.76483	−	0.940945i	−2.32665			1.08233	+	4.33052i	−1.27746	−	1.27746i	−2.09871	+	1.89616i	−6.96241			3.19219	−	0.797823i
13.4	−1.35590	−	0.401923i		−	0.696614i	1.67692	+	1.08993i	2.41672			−0.279985	+	0.944537i	−3.18472	−	3.18472i	−1.83566	−	2.15183i	2.51473			−3.27683	−	0.971335i
13.5	−1.31970	+	0.508326i		−	2.04663i	1.48321	−	1.34167i	3.76923			1.04036	+	2.70094i	1.84127	+	1.84127i	−1.27538	+	2.52456i	−1.18870			−4.97425	+	1.91600i
13.6	−1.18765	+	0.767782i			0.304704i	0.821022	−	1.82371i	0.222425			−0.233946	−	0.361881i	−0.942993	−	0.942993i	0.425126	+	2.79630i	2.90716			−0.264163	+	0.170774i
13.7	−1.12578	−	0.855937i			2.16143i	0.534743	+	1.92719i	1.75971			1.85005	−	2.43329i	0.943570	+	0.943570i	1.04755	−	2.62729i	−1.67179			−1.98104	−	1.50620i
13.8	−1.10071	−	0.887946i		−	1.75767i	0.423105	+	1.95473i	0.854468			−1.56072	+	1.93468i	3.25373	+	3.25373i	1.26998	−	2.52728i	−0.0894064			−0.940518	−	0.758722i
13.9	−0.897653	+	1.09280i			3.03209i	−0.388439	−	1.96192i	1.47873			−3.31348	−	2.72176i	3.44575	+	3.44575i	2.49267	+	1.33663i	−6.19356			−1.32739	+	1.61597i
13.10	−0.789510	−	1.17332i		−	2.66798i	−0.753348	+	1.85269i	−2.32940			−3.13038	+	2.10639i	−1.22784	−	1.22784i	2.76857	−	0.578801i	−4.11809			1.83908	+	2.73312i
13.11	−0.740658	−	1.20475i			1.71343i	−0.902850	+	1.78462i	−1.98068			2.06426	−	1.26907i	−2.55435	−	2.55435i	2.81872	−	0.234083i	0.0641544			1.46701	+	2.38623i
13.12	−0.684620	+	1.23746i		−	1.24285i	−1.06259	−	1.69437i	−3.96950			1.53797	+	0.850877i	1.97557	+	1.97557i	2.82418	−	0.154908i	1.45534			2.71760	−	4.91208i
13.13	−0.579280	+	1.29013i			2.05711i	−1.32887	−	1.49469i	−2.40124			−2.65394	−	1.19164i	−2.53680	−	2.53680i	2.69814	−	0.848567i	−1.23170			1.39099	−	3.09791i
13.14	−0.447665	−	1.34149i		−	1.69724i	−1.59919	+	1.20108i	3.05115			−2.27683	+	0.759795i	−1.42577	−	1.42577i	2.32714	+	1.60762i	0.119377			−1.36589	−	4.09308i
13.15	−0.374003	+	1.36386i			0.378959i	−1.72024	−	1.02018i	3.16725			−0.516848	−	0.141732i	−0.346423	−	0.346423i	2.03476	−	1.96463i	2.85639			−1.18456	+	4.31970i
13.16	−0.292310	−	1.38367i			0.572337i	−1.82911	+	0.808925i	−1.54731			0.791928	−	0.167300i	2.11387	+	2.11387i	1.65396	+	2.29443i	2.67243			0.452294	+	2.14097i
13.17	−0.187619	+	1.40171i		−	2.98411i	−1.92960	−	0.525976i	0.923892			4.18286	+	0.559875i	2.36413	+	2.36413i	1.09930	−	2.60606i	−5.90489			−0.173340	+	1.29503i
13.18	0.0393821	−	1.41367i			2.55478i	−1.99690	−	0.111346i	3.78051			3.61161	+	0.100613i	−1.48991	−	1.48991i	−0.236048	+	2.81856i	−3.52692			0.148884	−	5.34437i
13.19	0.412083	+	1.35284i		−	1.71768i	−1.66038	+	1.11497i	−2.12772			2.32375	−	0.707826i	−3.08083	−	3.08083i	−2.19259	−	1.78677i	0.0495749			−0.876798	−	2.87848i
13.20	0.435351	+	1.34554i		−	0.730083i	−1.62094	+	1.17156i	0.867081			0.982353	−	0.317842i	0.860326	+	0.860326i	−2.28206	−	1.67099i	2.46698			0.377484	+	1.16669i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
272.j	even	4	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	272.2.j.a	✓	68
4.b	odd	2	1		1088.2.j.a		68
16.e	even	4	1		272.2.s.a	yes	68
16.f	odd	4	1		1088.2.s.a		68
17.c	even	4	1		272.2.s.a	yes	68
68.f	odd	4	1		1088.2.s.a		68
272.j	even	4	1	inner	272.2.j.a	✓	68
272.t	odd	4	1		1088.2.j.a		68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
272.2.j.a	✓	68	1.a	even	1	1	trivial
272.2.j.a	✓	68	272.j	even	4	1	inner
272.2.s.a	yes	68	16.e	even	4	1
272.2.s.a	yes	68	17.c	even	4	1
1088.2.j.a		68	4.b	odd	2	1
1088.2.j.a		68	272.t	odd	4	1
1088.2.s.a		68	16.f	odd	4	1
1088.2.s.a		68	68.f	odd	4	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(272, [\chi])\).