Newform orbit 731.2.z.a

• Label: 731.2.z.a
• Level: $731$
• Weight: $2$
• Character orbit: 731.z
• Analytic conductor: $5.837$
• Analytic rank: $0$
• Dimension: $768$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 731 = 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	731.z (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.83706438776\)
Analytic rank:	\(0\)
Dimension:	\(768\)
Relative dimension:	\(64\) over \(\Q(\zeta_{42})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{42}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 768 q - 24 q^{2} - 144 q^{4} - 16 q^{8} - 98 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
67.1	−1.69621	+	2.12698i	−0.339828	−	2.25461i	−1.20187	−	5.26573i	0.0789863	+	0.115852i	5.37192	+	3.10148i	−0.309020	+	0.178413i	8.33653	+	4.01466i	−2.10106	+	0.648091i	−0.380391	−	0.0285064i
67.2	−1.69621	+	2.12698i	0.339828	+	2.25461i	−1.20187	−	5.26573i	−0.0789863	−	0.115852i	−5.37192	−	3.10148i	0.309020	−	0.178413i	8.33653	+	4.01466i	−2.10106	+	0.648091i	0.380391	+	0.0285064i
67.3	−1.67449	+	2.09974i	−0.149191	−	0.989818i	−1.15996	−	5.08212i	−2.44528	−	3.58657i	2.32818	+	1.34418i	0.298243	−	0.172191i	7.77406	+	3.74379i	1.90924	−	0.588921i	11.6255	+	0.871209i
67.4	−1.67449	+	2.09974i	0.149191	+	0.989818i	−1.15996	−	5.08212i	2.44528	+	3.58657i	−2.32818	−	1.34418i	−0.298243	+	0.172191i	7.77406	+	3.74379i	1.90924	−	0.588921i	−11.6255	−	0.871209i
67.5	−1.53756	+	1.92804i	−0.193458	−	1.28351i	−0.908197	−	3.97907i	0.441633	+	0.647756i	2.77211	+	1.60048i	4.06857	−	2.34899i	4.62452	+	2.22705i	1.25675	−	0.387655i	−1.92793	−	0.144478i
67.6	−1.53756	+	1.92804i	0.193458	+	1.28351i	−0.908197	−	3.97907i	−0.441633	−	0.647756i	−2.77211	−	1.60048i	−4.06857	+	2.34899i	4.62452	+	2.22705i	1.25675	−	0.387655i	1.92793	+	0.144478i
67.7	−1.43695	+	1.80187i	−0.348188	−	2.31008i	−0.736893	−	3.22854i	0.523790	+	0.768259i	4.66280	+	2.69207i	−2.81082	+	1.62283i	2.72340	+	1.31152i	−2.34851	+	0.724419i	−2.13696	−	0.160143i
67.8	−1.43695	+	1.80187i	0.348188	+	2.31008i	−0.736893	−	3.22854i	−0.523790	−	0.768259i	−4.66280	−	2.69207i	2.81082	−	1.62283i	2.72340	+	1.31152i	−2.34851	+	0.724419i	2.13696	+	0.160143i
67.9	−1.29990	+	1.63003i	−0.470647	−	3.12254i	−0.522200	−	2.28791i	−1.19265	−	1.74929i	5.70163	+	3.29184i	0.150209	−	0.0867232i	0.651331	+	0.313664i	−6.66204	+	2.05497i	4.40173	+	0.329864i
67.10	−1.29990	+	1.63003i	0.470647	+	3.12254i	−0.522200	−	2.28791i	1.19265	+	1.74929i	−5.70163	−	3.29184i	−0.150209	+	0.0867232i	0.651331	+	0.313664i	−6.66204	+	2.05497i	−4.40173	−	0.329864i
67.11	−1.29543	+	1.62442i	−0.00546009	−	0.0362253i	−0.515560	−	2.25882i	−2.06352	−	3.02662i	0.0659185	+	0.0380581i	0.0270939	−	0.0156427i	0.593233	+	0.285686i	2.86544	−	0.883870i	7.58967	+	0.568767i
67.12	−1.29543	+	1.62442i	0.00546009	+	0.0362253i	−0.515560	−	2.25882i	2.06352	+	3.02662i	−0.0659185	−	0.0380581i	−0.0270939	+	0.0156427i	0.593233	+	0.285686i	2.86544	−	0.883870i	−7.58967	−	0.568767i
67.13	−1.16355	+	1.45905i	−0.00644316	−	0.0427476i	−0.329925	−	1.44549i	0.740612	+	1.08628i	0.0698677	+	0.0403381i	2.40880	−	1.39072i	−0.869832	−	0.418889i	2.86493	−	0.883715i	−2.44667	−	0.183353i
67.14	−1.16355	+	1.45905i	0.00644316	+	0.0427476i	−0.329925	−	1.44549i	−0.740612	−	1.08628i	−0.0698677	−	0.0403381i	−2.40880	+	1.39072i	−0.869832	−	0.418889i	2.86493	−	0.883715i	2.44667	+	0.183353i
67.15	−1.11327	+	1.39600i	−0.311016	−	2.06346i	−0.264394	−	1.15839i	1.52042	+	2.23004i	3.22682	+	1.86301i	−2.35848	+	1.36167i	−1.30600	−	0.628935i	−1.29440	+	0.399269i	−4.80577	−	0.360142i
67.16	−1.11327	+	1.39600i	0.311016	+	2.06346i	−0.264394	−	1.15839i	−1.52042	−	2.23004i	−3.22682	−	1.86301i	2.35848	−	1.36167i	−1.30600	−	0.628935i	−1.29440	+	0.399269i	4.80577	+	0.360142i
67.17	−0.792393	+	0.993630i	−0.413423	−	2.74288i	0.0856288	+	0.375164i	1.48606	+	2.17965i	3.05300	+	1.76265i	0.708079	−	0.408809i	−2.73071	−	1.31504i	−4.48577	+	1.38368i	−3.34332	−	0.250547i
67.18	−0.792393	+	0.993630i	0.413423	+	2.74288i	0.0856288	+	0.375164i	−1.48606	−	2.17965i	−3.05300	−	1.76265i	−0.708079	+	0.408809i	−2.73071	−	1.31504i	−4.48577	+	1.38368i	3.34332	+	0.250547i
67.19	−0.755649	+	0.947554i	−0.215818	−	1.43186i	0.118189	+	0.517820i	−1.58922	−	2.33095i	1.51984	+	0.877483i	1.01277	−	0.584720i	−2.76386	−	1.33100i	0.863080	−	0.266225i	3.40959	+	0.255514i
67.20	−0.755649	+	0.947554i	0.215818	+	1.43186i	0.118189	+	0.517820i	1.58922	+	2.33095i	−1.51984	−	0.877483i	−1.01277	+	0.584720i	−2.76386	−	1.33100i	0.863080	−	0.266225i	−3.40959	−	0.255514i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
17.b	even	2	1	inner
43.g	even	21	1	inner
731.z	even	42	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	731.2.z.a	✓	768
17.b	even	2	1	inner	731.2.z.a	✓	768
43.g	even	21	1	inner	731.2.z.a	✓	768
731.z	even	42	1	inner	731.2.z.a	✓	768

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
731.2.z.a	✓	768	1.a	even	1	1	trivial
731.2.z.a	✓	768	17.b	even	2	1	inner
731.2.z.a	✓	768	43.g	even	21	1	inner
731.2.z.a	✓	768	731.z	even	42	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(731, [\chi])\).