Newform orbit 693.2.cj.a

• Label: 693.2.cj.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.cj
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $736$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.cj (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.

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 18 q^{2} - 94 q^{4} - 3 q^{7} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
20.1	−2.05625	+	1.85145i	−0.798552	+	1.53698i	0.591216	−	5.62505i	1.62799	−	1.80806i	−1.20363	−	4.63890i	−0.000437917	−	2.64575i	5.94608	+	8.18408i	−1.72463	−	2.45472i			6.73196i
20.2	−2.05625	+	1.85145i	0.798552	−	1.53698i	0.591216	−	5.62505i	−1.62799	+	1.80806i	1.20363	+	4.63890i	1.96588	+	1.77068i	5.94608	+	8.18408i	−1.72463	−	2.45472i		−	6.73196i
20.3	−2.05117	+	1.84689i	−1.06980	−	1.36217i	0.587272	−	5.58752i	1.73234	−	1.92396i	4.71013	+	0.818250i	−1.54803	+	2.14560i	5.87019	+	8.07963i	−0.711037	+	2.91452i			7.14579i
20.4	−2.05117	+	1.84689i	1.06980	+	1.36217i	0.587272	−	5.58752i	−1.73234	+	1.92396i	−4.71013	−	0.818250i	−2.63033	−	0.285280i	5.87019	+	8.07963i	−0.711037	+	2.91452i		−	7.14579i
20.5	−1.89577	+	1.70696i	−1.63510	−	0.571356i	0.471177	−	4.48295i	−2.42255	+	2.69051i	4.07505	−	1.70789i	−0.994461	−	2.45174i	3.76008	+	5.17530i	2.34710	+	1.86845i		−	9.23577i
20.6	−1.89577	+	1.70696i	1.63510	+	0.571356i	0.471177	−	4.48295i	2.42255	−	2.69051i	−4.07505	+	1.70789i	1.15658	+	2.37957i	3.76008	+	5.17530i	2.34710	+	1.86845i			9.23577i
20.7	−1.72786	+	1.55578i	−1.21958	+	1.22989i	0.356019	−	3.38730i	−2.44481	+	2.71523i	0.193834	−	4.02247i	−0.489245	+	2.60012i	1.92144	+	2.64463i	−0.0252566	−	2.99989i		−	8.49512i
20.8	−1.72786	+	1.55578i	1.21958	−	1.22989i	0.356019	−	3.38730i	2.44481	−	2.71523i	−0.193834	+	4.02247i	−2.25964	−	1.37624i	1.92144	+	2.64463i	−0.0252566	−	2.99989i			8.49512i
20.9	−1.72293	+	1.55133i	−1.69229	−	0.368979i	0.352794	−	3.35661i	0.684384	−	0.760085i	3.48810	−	1.98958i	2.45038	+	0.997826i	1.87389	+	2.57919i	2.72771	+	1.24884i			2.37127i
20.10	−1.72293	+	1.55133i	1.69229	+	0.368979i	0.352794	−	3.35661i	−0.684384	+	0.760085i	−3.48810	+	1.98958i	0.898093	−	2.48866i	1.87389	+	2.57919i	2.72771	+	1.24884i		−	2.37127i
20.11	−1.72022	+	1.54889i	−0.448644	−	1.67294i	0.351030	−	3.33983i	−0.553101	+	0.614281i	3.36297	+	2.18292i	1.37029	−	2.26325i	1.84799	+	2.54354i	−2.59744	+	1.50111i		−	1.91339i
20.12	−1.72022	+	1.54889i	0.448644	+	1.67294i	0.351030	−	3.33983i	0.553101	−	0.614281i	−3.36297	−	2.18292i	2.59883	+	0.496082i	1.84799	+	2.54354i	−2.59744	+	1.50111i			1.91339i
20.13	−1.69910	+	1.52987i	−1.26945	+	1.17834i	0.337359	−	3.20976i	0.970021	−	1.07732i	0.354211	−	3.94422i	−1.83572	+	1.90529i	1.64954	+	2.27040i	0.223024	−	2.99170i			3.31447i
20.14	−1.69910	+	1.52987i	1.26945	−	1.17834i	0.337359	−	3.20976i	−0.970021	+	1.07732i	−0.354211	+	3.94422i	−2.64424	+	0.0893125i	1.64954	+	2.27040i	0.223024	−	2.99170i		−	3.31447i
20.15	−1.53236	+	1.37974i	−0.313414	+	1.70346i	0.235378	−	2.23947i	−1.81061	+	2.01089i	−1.87007	−	3.04274i	2.10506	−	1.60272i	0.305195	+	0.420065i	−2.80354	−	1.06777i		−	5.57958i
20.16	−1.53236	+	1.37974i	0.313414	−	1.70346i	0.235378	−	2.23947i	1.81061	−	2.01089i	1.87007	+	3.04274i	2.59961	−	0.491938i	0.305195	+	0.420065i	−2.80354	−	1.06777i			5.57958i
20.17	−1.47219	+	1.32557i	−1.16925	−	1.27783i	0.201163	−	1.91393i	−0.793153	+	0.880886i	3.41521	+	0.331291i	−1.37308	+	2.26156i	−0.0879400	−	0.121039i	−0.265703	+	2.98821i		−	2.34821i
20.18	−1.47219	+	1.32557i	1.16925	+	1.27783i	0.201163	−	1.91393i	0.793153	−	0.880886i	−3.41521	−	0.331291i	−2.59944	−	0.492883i	−0.0879400	−	0.121039i	−0.265703	+	2.98821i			2.34821i
20.19	−1.42250	+	1.28082i	−1.70520	−	0.303782i	0.173935	−	1.65488i	2.69773	−	2.99613i	2.81473	−	1.75193i	0.518805	−	2.59439i	−0.378045	−	0.520334i	2.81543	+	1.03602i			7.71729i
20.20	−1.42250	+	1.28082i	1.70520	+	0.303782i	0.173935	−	1.65488i	−2.69773	+	2.99613i	−2.81473	+	1.75193i	2.27515	+	1.35044i	−0.378045	−	0.520334i	2.81543	+	1.03602i		−	7.71729i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
9.d	odd	6	1	inner
11.c	even	5	1	inner
63.o	even	6	1	inner
77.j	odd	10	1	inner
99.n	odd	30	1	inner
693.cj	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.cj.a	✓	736
7.b	odd	2	1	inner	693.2.cj.a	✓	736
9.d	odd	6	1	inner	693.2.cj.a	✓	736
11.c	even	5	1	inner	693.2.cj.a	✓	736
63.o	even	6	1	inner	693.2.cj.a	✓	736
77.j	odd	10	1	inner	693.2.cj.a	✓	736
99.n	odd	30	1	inner	693.2.cj.a	✓	736
693.cj	even	30	1	inner	693.2.cj.a	✓	736

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.cj.a	✓	736	1.a	even	1	1	trivial
693.2.cj.a	✓	736	7.b	odd	2	1	inner
693.2.cj.a	✓	736	9.d	odd	6	1	inner
693.2.cj.a	✓	736	11.c	even	5	1	inner
693.2.cj.a	✓	736	63.o	even	6	1	inner
693.2.cj.a	✓	736	77.j	odd	10	1	inner
693.2.cj.a	✓	736	99.n	odd	30	1	inner
693.2.cj.a	✓	736	693.cj	even	30	1	inner

Hecke kernels

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