Newform orbit 712.2.z.a

• Label: 712.2.z.a
• Level: $712$
• Weight: $2$
• Character orbit: 712.z
• Analytic conductor: $5.685$
• Analytic rank: $0$
• Dimension: $1760$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	712.z (of order \(44\), degree \(20\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.68534862392\)
Analytic rank:	\(0\)
Dimension:	\(1760\)
Relative dimension:	\(88\) over \(\Q(\zeta_{44})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{44}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 1760 q - 18 q^{2} - 18 q^{4} - 24 q^{6} - 40 q^{7} - 18 q^{8} - 44 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.41269	+	0.0657255i	0.185650	+	0.247999i	1.99136	−	0.185699i	−3.57439	−	2.29712i	−0.278565	−	0.338143i	−0.707473	−	3.25220i	−2.80096	+	0.393217i	0.818160	−	2.78640i	5.20047	+	3.01018i
5.2	−1.41068	−	0.0998698i	−1.89369	−	2.52967i	1.98005	+	0.281769i	3.11427	+	2.00142i	2.41876	+	3.75769i	−0.542982	−	2.49605i	−2.76509	−	0.595234i	−1.96799	+	6.70235i	−4.19337	−	3.13439i
5.3	−1.40734	−	0.139245i	−1.70529	−	2.27800i	1.96122	+	0.391929i	−0.154355	−	0.0991976i	2.08273	+	3.44338i	0.915148	+	4.20687i	−2.70554	−	0.824668i	−1.43607	+	4.89082i	0.203417	+	0.161098i
5.4	−1.40277	−	0.179535i	1.63730	+	2.18718i	1.93553	+	0.503693i	−3.30181	−	2.12195i	−1.90408	−	3.36206i	0.968417	+	4.45174i	−2.62468	−	1.05406i	−1.25779	+	4.28365i	4.25072	+	3.56939i
5.5	−1.40193	+	0.186027i	−0.728833	−	0.973607i	1.93079	−	0.521591i	1.76410	+	1.13372i	1.20289	+	1.22934i	0.251840	+	1.15769i	−2.60979	+	1.09041i	0.428485	−	1.45929i	−2.68404	−	1.26122i
5.6	−1.39492	+	0.232776i	0.670447	+	0.895612i	1.89163	−	0.649409i	0.314819	+	0.202322i	−1.14370	−	1.09325i	0.423193	+	1.94539i	−2.48752	+	1.34620i	0.492576	−	1.67756i	−0.486244	−	0.208942i
5.7	−1.39061	+	0.257303i	1.20971	+	1.61598i	1.86759	−	0.715615i	0.0714897	+	0.0459436i	−2.09803	−	1.93594i	−0.988216	−	4.54275i	−2.41296	+	1.47568i	−0.302806	+	1.03126i	−0.111236	−	0.0454952i
5.8	−1.38396	−	0.290948i	0.442165	+	0.590663i	1.83070	+	0.805321i	−0.258658	−	0.166229i	−0.440087	−	0.946101i	−0.0445955	−	0.205002i	−2.29931	−	1.64717i	0.691825	−	2.35614i	0.309608	+	0.305311i
5.9	−1.38368	−	0.292279i	1.56751	+	2.09395i	1.82915	+	0.808841i	3.28038	+	2.10817i	−1.55692	−	3.35552i	0.105832	+	0.486500i	−2.29455	−	1.65380i	−1.08235	+	3.68613i	−3.92283	−	3.87583i
5.10	−1.36806	−	0.358343i	−0.960323	−	1.28284i	1.74318	+	0.980470i	−1.65741	−	1.06515i	0.854083	+	2.09913i	0.00614263	+	0.0282372i	−2.03343	−	1.96600i	0.121735	−	0.414593i	1.88574	+	2.05111i
5.11	−1.33040	+	0.479618i	−0.516178	−	0.689534i	1.53993	−	1.27617i	2.44507	+	1.57135i	1.01744	+	0.669788i	0.00524368	+	0.0241048i	−1.43665	+	2.43640i	0.636181	−	2.16663i	−4.00658	−	0.917828i
5.12	−1.29272	+	0.573470i	−1.14765	−	1.53308i	1.34227	−	1.48267i	−2.62139	−	1.68467i	2.36277	+	1.32371i	0.505235	+	2.32253i	−0.884908	+	2.68644i	−0.188043	+	0.640416i	4.35484	+	0.674517i
5.13	−1.25716	+	0.647723i	−0.954713	−	1.27535i	1.16091	−	1.62858i	−0.543700	−	0.349415i	2.02630	+	0.984928i	−0.755927	−	3.47494i	−0.404582	+	2.79934i	0.130164	−	0.443299i	0.909843	+	0.0871041i
5.14	−1.25708	−	0.647879i	−0.313795	−	0.419182i	1.16051	+	1.62887i	2.04508	+	1.31429i	0.122887	+	0.730247i	−0.809786	−	3.72252i	−0.403539	−	2.79949i	0.767952	−	2.61540i	−1.71933	−	2.97713i
5.15	−1.25579	+	0.650376i	1.73223	+	2.31399i	1.15402	−	1.63347i	0.623593	+	0.400759i	−3.68028	−	1.77928i	0.374270	+	1.72049i	−0.386839	+	2.80185i	−1.50872	+	5.13822i	−1.04375	−	0.0976995i
5.16	−1.17059	−	0.793547i	1.70658	+	2.27973i	0.740568	+	1.85784i	−0.719538	−	0.462419i	−0.188641	−	4.02288i	−0.573882	−	2.63809i	0.607378	−	2.76244i	−1.43954	+	4.90261i	0.475334	+	1.11229i
5.17	−1.16966	−	0.794916i	−0.464236	−	0.620146i	0.736216	+	1.85957i	2.27997	+	1.46525i	0.0500343	+	1.09439i	1.02206	+	4.69834i	0.617075	−	2.76029i	0.676131	−	2.30269i	−1.50204	−	3.52622i
5.18	−1.15621	−	0.814351i	−1.82803	−	2.44196i	0.673665	+	1.88313i	−2.65192	−	1.70428i	0.124981	+	4.31209i	−0.844517	−	3.88218i	0.754627	−	2.72590i	−1.77629	+	6.04949i	1.67830	+	4.13011i
5.19	−1.15244	+	0.819685i	−2.00041	−	2.67223i	0.656232	−	1.88928i	−1.18847	−	0.763785i	4.49573	+	1.43988i	−0.461078	−	2.11954i	0.792344	+	2.71518i	−2.29399	+	7.81261i	1.99571	−	0.0939580i
5.20	−1.08572	−	0.906209i	0.903177	+	1.20650i	0.357572	+	1.96778i	−1.55814	−	1.00135i	0.112747	−	2.12839i	0.175460	+	0.806576i	1.39499	−	2.46049i	0.205276	−	0.699106i	0.784263	+	2.49918i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.b	even	2	1	inner
89.g	even	44	1	inner
712.z	even	44	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	712.2.z.a	✓	1760
8.b	even	2	1	inner	712.2.z.a	✓	1760
89.g	even	44	1	inner	712.2.z.a	✓	1760
712.z	even	44	1	inner	712.2.z.a	✓	1760

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
712.2.z.a	✓	1760	1.a	even	1	1	trivial
712.2.z.a	✓	1760	8.b	even	2	1	inner
712.2.z.a	✓	1760	89.g	even	44	1	inner
712.2.z.a	✓	1760	712.z	even	44	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(712, [\chi])\).