Newform orbit 644.2.p.a

• Label: 644.2.p.a
• Level: $644$
• Weight: $2$
• Character orbit: 644.p
• Analytic conductor: $5.142$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	644.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.14236589017\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 16 q^{6} - 12 q^{8} + 80 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
275.1	−1.41369	+	0.0385524i	1.87896	−	1.08482i	1.99703	−	0.109002i	−3.17885	−	1.83531i	−2.61444	+	1.60603i	−2.43317	+	1.03908i	−2.81897	+	0.231085i	0.853653	−	1.47857i	4.56466	+	2.47200i
275.2	−1.41369	+	0.0385524i	1.87896	−	1.08482i	1.99703	−	0.109002i	3.17885	+	1.83531i	−2.61444	+	1.60603i	2.43317	−	1.03908i	−2.81897	+	0.231085i	0.853653	−	1.47857i	−4.56466	−	2.47200i
275.3	−1.41155	+	0.0868146i	−1.08906	+	0.628769i	1.98493	−	0.245086i	−2.08761	−	1.20528i	1.48267	−	0.982083i	1.14696	−	2.38421i	−2.78054	+	0.518271i	−0.709298	+	1.22854i	3.05139	+	1.52008i
275.4	−1.41155	+	0.0868146i	−1.08906	+	0.628769i	1.98493	−	0.245086i	2.08761	+	1.20528i	1.48267	−	0.982083i	−1.14696	+	2.38421i	−2.78054	+	0.518271i	−0.709298	+	1.22854i	−3.05139	−	1.52008i
275.5	−1.40925	+	0.118348i	0.734393	−	0.424002i	1.97199	−	0.333564i	−0.573053	−	0.330852i	−0.984766	+	0.684440i	2.12740	+	1.57295i	−2.73955	+	0.703456i	−1.14044	+	1.97531i	0.846732	+	0.398435i
275.6	−1.40925	+	0.118348i	0.734393	−	0.424002i	1.97199	−	0.333564i	0.573053	+	0.330852i	−0.984766	+	0.684440i	−2.12740	−	1.57295i	−2.73955	+	0.703456i	−1.14044	+	1.97531i	−0.846732	−	0.398435i
275.7	−1.38314	+	0.294816i	−2.76853	+	1.59841i	1.82617	−	0.815544i	−2.55782	−	1.47676i	3.35803	−	3.02704i	1.05294	+	2.42720i	−2.28542	+	1.66640i	3.60983	−	6.25241i	3.97320	+	1.28848i
275.8	−1.38314	+	0.294816i	−2.76853	+	1.59841i	1.82617	−	0.815544i	2.55782	+	1.47676i	3.35803	−	3.02704i	−1.05294	−	2.42720i	−2.28542	+	1.66640i	3.60983	−	6.25241i	−3.97320	−	1.28848i
275.9	−1.35347	−	0.410021i	−1.74211	+	1.00581i	1.66377	+	1.10990i	−2.31311	−	1.33548i	2.77030	−	0.647029i	−0.652711	−	2.56398i	−1.79677	−	2.18440i	0.523296	−	0.906375i	2.58316	+	2.75596i
275.10	−1.35347	−	0.410021i	−1.74211	+	1.00581i	1.66377	+	1.10990i	2.31311	+	1.33548i	2.77030	−	0.647029i	0.652711	+	2.56398i	−1.79677	−	2.18440i	0.523296	−	0.906375i	−2.58316	−	2.75596i
275.11	−1.29604	−	0.565941i	2.54119	−	1.46716i	1.35942	+	1.46696i	−0.116036	−	0.0669933i	−4.12380	+	0.463324i	0.703603	+	2.55048i	−0.931646	−	2.67059i	2.80509	−	4.85856i	0.112472	+	0.152495i
275.12	−1.29604	−	0.565941i	2.54119	−	1.46716i	1.35942	+	1.46696i	0.116036	+	0.0669933i	−4.12380	+	0.463324i	−0.703603	−	2.55048i	−0.931646	−	2.67059i	2.80509	−	4.85856i	−0.112472	−	0.152495i
275.13	−1.26519	−	0.631906i	0.0795555	−	0.0459314i	1.20139	+	1.59896i	−1.85389	−	1.07034i	−0.129677	+	0.00784013i	2.51708	+	0.815058i	−0.509590	−	2.78214i	−1.49578	+	2.59077i	1.66916	+	2.52567i
275.14	−1.26519	−	0.631906i	0.0795555	−	0.0459314i	1.20139	+	1.59896i	1.85389	+	1.07034i	−0.129677	+	0.00784013i	−2.51708	−	0.815058i	−0.509590	−	2.78214i	−1.49578	+	2.59077i	−1.66916	−	2.52567i
275.15	−1.21773	+	0.719127i	−1.46045	+	0.843193i	0.965713	−	1.75140i	−0.795863	−	0.459491i	1.17207	−	2.07703i	−2.45343	+	0.990303i	0.0835040	+	2.82719i	−0.0780526	+	0.135191i	1.29958	−	0.0127914i
275.16	−1.21773	+	0.719127i	−1.46045	+	0.843193i	0.965713	−	1.75140i	0.795863	+	0.459491i	1.17207	−	2.07703i	2.45343	−	0.990303i	0.0835040	+	2.82719i	−0.0780526	+	0.135191i	−1.29958	+	0.0127914i
275.17	−1.19173	+	0.761437i	2.59986	−	1.50103i	0.840427	−	1.81485i	−1.74640	−	1.00828i	−1.95538	+	3.76845i	1.78278	−	1.95492i	0.380335	+	2.80274i	3.00619	−	5.20687i	2.84898	−	0.128174i
275.18	−1.19173	+	0.761437i	2.59986	−	1.50103i	0.840427	−	1.81485i	1.74640	+	1.00828i	−1.95538	+	3.76845i	−1.78278	+	1.95492i	0.380335	+	2.80274i	3.00619	−	5.20687i	−2.84898	+	0.128174i
275.19	−1.16587	+	0.800469i	0.743286	−	0.429136i	0.718499	−	1.86648i	−1.59184	−	0.919050i	−0.523063	+	1.09529i	0.881870	−	2.49445i	0.656387	+	2.75121i	−1.13168	+	1.96013i	2.59155	−	0.202728i
275.20	−1.16587	+	0.800469i	0.743286	−	0.429136i	0.718499	−	1.86648i	1.59184	+	0.919050i	−0.523063	+	1.09529i	−0.881870	+	2.49445i	0.656387	+	2.75121i	−1.13168	+	1.96013i	−2.59155	+	0.202728i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
7.c	even	3	1	inner
23.b	odd	2	1	inner
28.g	odd	6	1	inner
92.b	even	2	1	inner
161.f	odd	6	1	inner
644.p	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	644.2.p.a	✓	184
4.b	odd	2	1	inner	644.2.p.a	✓	184
7.c	even	3	1	inner	644.2.p.a	✓	184
23.b	odd	2	1	inner	644.2.p.a	✓	184
28.g	odd	6	1	inner	644.2.p.a	✓	184
92.b	even	2	1	inner	644.2.p.a	✓	184
161.f	odd	6	1	inner	644.2.p.a	✓	184
644.p	even	6	1	inner	644.2.p.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
644.2.p.a	✓	184	1.a	even	1	1	trivial
644.2.p.a	✓	184	4.b	odd	2	1	inner
644.2.p.a	✓	184	7.c	even	3	1	inner
644.2.p.a	✓	184	23.b	odd	2	1	inner
644.2.p.a	✓	184	28.g	odd	6	1	inner
644.2.p.a	✓	184	92.b	even	2	1	inner
644.2.p.a	✓	184	161.f	odd	6	1	inner
644.2.p.a	✓	184	644.p	even	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).