Newform orbit 272.2.r.a

• Label: 272.2.r.a
• Level: $272$
• Weight: $2$
• Character orbit: 272.r
• Analytic conductor: $2.172$
• Analytic rank: $0$
• Dimension: $68$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	272.r (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.

\(\operatorname{Tr}(f)(q) = \) \( 68 q - 4 q^{2} - 4 q^{4} - 4 q^{8}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
101.1	−1.38152	−	0.302342i	−0.319805	−	0.319805i	1.81718	+	0.835382i	−2.51704	+	2.51704i	0.345125	+	0.538506i	−1.75502			−2.25789	−	1.70350i		−	2.79545i	4.23834	−	2.71633i
101.2	−1.38152	−	0.302342i	0.319805	+	0.319805i	1.81718	+	0.835382i	2.51704	−	2.51704i	−0.345125	−	0.538506i	1.75502			−2.25789	−	1.70350i		−	2.79545i	−4.23834	+	2.71633i
101.3	−1.38120	+	0.303797i	−1.97734	−	1.97734i	1.81541	−	0.839207i	0.636168	−	0.636168i	3.33180	+	2.13038i	4.57537			−2.25250	+	1.71063i			4.81971i	−0.685408	+	1.07194i
101.4	−1.38120	+	0.303797i	1.97734	+	1.97734i	1.81541	−	0.839207i	−0.636168	+	0.636168i	−3.33180	−	2.13038i	−4.57537			−2.25250	+	1.71063i			4.81971i	0.685408	−	1.07194i
101.5	−1.27646	−	0.608803i	−1.64190	−	1.64190i	1.25872	+	1.55423i	0.138160	−	0.138160i	1.09623	+	3.09542i	−2.35523			−0.660488	−	2.75023i			2.39166i	−0.260468	+	0.0922440i
101.6	−1.27646	−	0.608803i	1.64190	+	1.64190i	1.25872	+	1.55423i	−0.138160	+	0.138160i	−1.09623	−	3.09542i	2.35523			−0.660488	−	2.75023i			2.39166i	0.260468	−	0.0922440i
101.7	−1.26925	+	0.623693i	−0.726910	−	0.726910i	1.22201	−	1.58325i	−1.23083	+	1.23083i	1.37600	+	0.469265i	−0.254563			−0.563585	+	2.77171i		−	1.94320i	0.794579	−	2.32990i
101.8	−1.26925	+	0.623693i	0.726910	+	0.726910i	1.22201	−	1.58325i	1.23083	−	1.23083i	−1.37600	−	0.469265i	0.254563			−0.563585	+	2.77171i		−	1.94320i	−0.794579	+	2.32990i
101.9	−0.910413	+	1.08220i	−1.67302	−	1.67302i	−0.342297	−	1.97049i	2.87951	−	2.87951i	3.33368	−	0.287398i	−4.68425			2.44409	+	1.42353i			2.59801i	0.494652	+	5.73773i
101.10	−0.910413	+	1.08220i	1.67302	+	1.67302i	−0.342297	−	1.97049i	−2.87951	+	2.87951i	−3.33368	+	0.287398i	4.68425			2.44409	+	1.42353i			2.59801i	−0.494652	−	5.73773i
101.11	−0.721531	−	1.21630i	−0.659351	−	0.659351i	−0.958786	+	1.75520i	−1.75731	+	1.75731i	−0.326229	+	1.27771i	4.34538			2.82665	−	0.100257i		−	2.13051i	3.40538	+	0.869468i
101.12	−0.721531	−	1.21630i	0.659351	+	0.659351i	−0.958786	+	1.75520i	1.75731	−	1.75731i	0.326229	−	1.27771i	−4.34538			2.82665	−	0.100257i		−	2.13051i	−3.40538	−	0.869468i
101.13	−0.579930	+	1.28984i	−2.06462	−	2.06462i	−1.32736	−	1.49603i	−2.55057	+	2.55057i	3.86037	−	1.46569i	−0.123756			2.69941	−	0.844490i			5.52535i	−1.81067	−	4.76898i
101.14	−0.579930	+	1.28984i	2.06462	+	2.06462i	−1.32736	−	1.49603i	2.55057	−	2.55057i	−3.86037	+	1.46569i	0.123756			2.69941	−	0.844490i			5.52535i	1.81067	+	4.76898i
101.15	−0.290246	−	1.38411i	−2.06867	−	2.06867i	−1.83151	+	0.803464i	−0.674380	+	0.674380i	−2.26284	+	3.46369i	−1.72950			1.64367	+	2.30181i			5.55878i	1.12915	+	0.737679i
101.16	−0.290246	−	1.38411i	2.06867	+	2.06867i	−1.83151	+	0.803464i	0.674380	−	0.674380i	2.26284	−	3.46369i	1.72950			1.64367	+	2.30181i			5.55878i	−1.12915	−	0.737679i
101.17	−0.150304	+	1.40620i	−0.0355432	−	0.0355432i	−1.95482	−	0.422717i	1.15813	−	1.15813i	0.0553232	−	0.0446386i	3.49333			0.888244	−	2.68533i		−	2.99747i	1.45449	+	1.80263i
101.18	−0.150304	+	1.40620i	0.0355432	+	0.0355432i	−1.95482	−	0.422717i	−1.15813	+	1.15813i	−0.0553232	+	0.0446386i	−3.49333			0.888244	−	2.68533i		−	2.99747i	−1.45449	−	1.80263i
101.19	0.0160889	−	1.41412i	−0.790194	−	0.790194i	−1.99948	−	0.0455033i	2.80315	−	2.80315i	−1.13014	+	1.10472i	2.35139			−0.0965166	+	2.82678i		−	1.75119i	−3.91889	−	4.00909i
101.20	0.0160889	−	1.41412i	0.790194	+	0.790194i	−1.99948	−	0.0455033i	−2.80315	+	2.80315i	1.13014	−	1.10472i	−2.35139			−0.0965166	+	2.82678i		−	1.75119i	3.91889	+	4.00909i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
16.e	even	4	1	inner
17.b	even	2	1	inner
272.r	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.r.a	✓	68
4.b	odd	2	1		1088.2.r.a		68
16.e	even	4	1	inner	272.2.r.a	✓	68
16.f	odd	4	1		1088.2.r.a		68
17.b	even	2	1	inner	272.2.r.a	✓	68
68.d	odd	2	1		1088.2.r.a		68
272.k	odd	4	1		1088.2.r.a		68
272.r	even	4	1	inner	272.2.r.a	✓	68

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
272.2.r.a	✓	68	1.a	even	1	1	trivial
272.2.r.a	✓	68	16.e	even	4	1	inner
272.2.r.a	✓	68	17.b	even	2	1	inner
272.2.r.a	✓	68	272.r	even	4	1	inner
1088.2.r.a		68	4.b	odd	2	1
1088.2.r.a		68	16.f	odd	4	1
1088.2.r.a		68	68.d	odd	2	1
1088.2.r.a		68	272.k	odd	4	1

Hecke kernels

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