Newform orbit 693.2.cz.a

• Label: 693.2.cz.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.cz
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $736$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.cz (of order \(30\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(736\)
Relative dimension:	\(92\) over \(\Q(\zeta_{30})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{30}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 10 q^{2} - 9 q^{3} + 170 q^{4} - 9 q^{5} - 5 q^{7} - 40 q^{8} - 9 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
40.1	−1.60788	+	2.21305i	1.63320	+	0.576759i	−1.69430	−	5.21451i	−1.07589	+	2.41649i	−3.90238	+	2.68700i	0.929452	+	2.47712i	9.06099	+	2.94410i	2.33470	+	1.88393i	−3.61792	−	6.26642i
40.2	−1.60044	+	2.20281i	0.745058	−	1.56361i	−1.67295	−	5.14882i	0.0602198	−	0.135256i	2.25193	+	4.14369i	−2.53757	+	0.748808i	8.84021	+	2.87236i	−1.88978	−	2.32997i	0.201565	+	0.349122i
40.3	−1.53693	+	2.11540i	−0.604591	−	1.62311i	−1.49473	−	4.60032i	1.41569	−	3.17970i	4.36273	+	1.21564i	2.64515	−	0.0566120i	7.05521	+	2.29238i	−2.26894	+	1.96263i	4.55051	+	7.88172i
40.4	−1.52706	+	2.10182i	−0.984117	−	1.42531i	−1.46770	−	4.51710i	−1.24932	+	2.80603i	4.49856	+	0.108102i	0.276660	−	2.63125i	6.79373	+	2.20742i	−1.06303	+	2.80535i	−3.98997	−	6.91083i
40.5	−1.52509	+	2.09910i	0.891773	+	1.48484i	−1.46231	−	4.50052i	0.0775806	−	0.174249i	−4.47686	−	0.392582i	−1.08633	−	2.41244i	6.74190	+	2.19058i	−1.40948	+	2.64828i	0.247449	+	0.428594i
40.6	−1.49367	+	2.05586i	1.73120	+	0.0543348i	−1.37747	−	4.23943i	1.43767	−	3.22907i	−2.69754	+	3.47794i	1.60450	−	2.10371i	5.93953	+	1.92987i	2.99410	+	0.188129i	4.49110	+	7.77881i
40.7	−1.48351	+	2.04188i	−1.39545	+	1.02603i	−1.35043	−	4.15620i	−1.75881	+	3.95035i	−0.0248582	−	4.37146i	−2.56075	+	0.665266i	5.68909	+	1.84850i	0.894543	−	2.86353i	−5.45693	−	9.45168i
40.8	−1.47456	+	2.02956i	−0.0141910	+	1.73199i	−1.32674	−	4.08328i	0.594614	−	1.33552i	−3.49425	−	2.58273i	1.58908	+	2.11537i	5.47183	+	1.77791i	−2.99960	−	0.0491573i	1.83373	+	3.17611i
40.9	−1.41242	+	1.94403i	−0.973949	+	1.43228i	−1.16629	−	3.58948i	−0.0894302	+	0.200863i	−1.40877	−	3.91637i	2.44899	−	1.00122i	4.05466	+	1.31744i	−1.10284	−	2.78993i	−0.264172	−	0.457559i
40.10	−1.37794	+	1.89657i	−1.21428	−	1.23512i	−1.08023	−	3.32462i	0.233825	−	0.525180i	4.01570	−	0.601054i	−2.33623	+	1.24178i	3.33477	+	1.08353i	−0.0510367	+	2.99957i	0.673845	+	1.16713i
40.11	−1.30689	+	1.79877i	1.34505	−	1.09125i	−0.909602	−	2.79947i	0.431262	−	0.968629i	0.205084	+	3.84559i	0.756682	+	2.53524i	1.99519	+	0.648275i	0.618339	−	2.93558i	1.17874	+	2.04163i
40.12	−1.30569	+	1.79713i	0.909017	−	1.47434i	−0.906812	−	2.79088i	−1.20224	+	2.70028i	1.46269	+	3.55866i	2.49472	−	0.881133i	1.97428	+	0.641483i	−1.34738	−	2.68041i	−3.28299	−	5.68631i
40.13	−1.30074	+	1.79032i	−1.70416	−	0.309559i	−0.895271	−	2.75536i	1.10016	−	2.47100i	2.77088	−	2.64833i	−1.86010	−	1.88149i	1.88820	+	0.613513i	2.80835	+	1.05508i	2.99284	+	5.18376i
40.14	−1.29723	+	1.78548i	−1.66661	−	0.471591i	−0.887106	−	2.73023i	−0.696165	+	1.56361i	3.00399	−	2.36394i	1.92568	+	1.81432i	1.82763	+	0.593834i	2.55520	+	1.57192i	−1.88871	−	3.27135i
40.15	−1.26389	+	1.73959i	−1.38353	+	1.04203i	−0.810732	−	2.49518i	0.146921	−	0.329991i	−0.0640788	−	3.72380i	−0.886744	−	2.49273i	1.27524	+	0.414350i	0.828337	−	2.88338i	0.388357	+	0.672654i
40.16	−1.19619	+	1.64642i	1.18728	+	1.26110i	−0.661782	−	2.03676i	1.45042	−	3.25770i	−3.49651	+	0.446243i	−1.86399	+	1.87764i	0.274009	+	0.0890309i	−0.180735	+	2.99455i	3.62855	+	6.28484i
40.17	−1.17708	+	1.62011i	1.69890	−	0.337276i	−0.621203	−	1.91187i	−1.24108	+	2.78751i	−1.45331	+	3.14939i	−2.59793	−	0.500762i	0.0195302	+	0.00634573i	2.77249	−	1.14599i	−3.05522	−	5.29180i
40.18	−1.12390	+	1.54691i	1.73057	−	0.0715365i	−0.511753	−	1.57501i	0.418107	−	0.939084i	−1.83432	+	2.75744i	−2.60456	−	0.465033i	−0.625442	−	0.203219i	2.98977	−	0.247598i	0.982769	+	1.70221i
40.19	−1.05013	+	1.44538i	1.12136	+	1.32006i	−0.368316	−	1.13356i	−1.16615	+	2.61922i	−3.08556	+	0.234553i	0.0178801	−	2.64569i	−1.37309	−	0.446143i	−0.485115	+	2.96052i	−2.56115	−	4.43605i
40.20	−1.04862	+	1.44330i	−0.202171	+	1.72021i	−0.365484	−	1.12484i	−0.402060	+	0.903042i	−2.27079	−	2.09564i	−1.93582	+	1.80350i	−1.38667	−	0.450555i	−2.91825	−	0.695554i	−0.881755	−	1.52724i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner
63.t	odd	6	1	inner
693.cz	even	30	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.cz.a	yes	736
7.d	odd	6	1		693.2.co.a	✓	736
9.c	even	3	1		693.2.co.a	✓	736
11.d	odd	10	1	inner	693.2.cz.a	yes	736
63.t	odd	6	1	inner	693.2.cz.a	yes	736
77.n	even	30	1		693.2.co.a	✓	736
99.o	odd	30	1		693.2.co.a	✓	736
693.cz	even	30	1	inner	693.2.cz.a	yes	736

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.co.a	✓	736	7.d	odd	6	1
693.2.co.a	✓	736	9.c	even	3	1
693.2.co.a	✓	736	77.n	even	30	1
693.2.co.a	✓	736	99.o	odd	30	1
693.2.cz.a	yes	736	1.a	even	1	1	trivial
693.2.cz.a	yes	736	11.d	odd	10	1	inner
693.2.cz.a	yes	736	63.t	odd	6	1	inner
693.2.cz.a	yes	736	693.cz	even	30	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(693, [\chi])\).