Newform orbit 693.2.cb.a

• Label: 693.2.cb.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.cb
• 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.cb (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.

\(\operatorname{Tr}(f)(q) = \) \( 736 q - 3 q^{3} - 182 q^{4} - 9 q^{5} - 20 q^{6} - 5 q^{7} - 9 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
74.1	−2.21838	+	1.61175i	1.57809	−	0.713895i	1.70545	−	5.24883i	1.23801	+	2.78061i	−2.35018	+	4.12717i	2.49196	−	0.888894i	2.98176	+	9.17692i	1.98071	−	2.25317i	−7.22800	−	4.17309i
74.2	−2.20747	+	1.60382i	0.281510	+	1.70902i	1.68266	−	5.17868i	0.0189263	+	0.0425091i	−3.36239	−	3.32113i	2.63758	−	0.207791i	2.90491	+	8.94040i	−2.84150	+	0.962212i	−0.109956	−	0.0634833i
74.3	−2.15907	+	1.56866i	1.53629	+	0.799890i	1.58287	−	4.87158i	−0.839653	−	1.88589i	−4.57171	+	0.682889i	−2.52578	−	0.787679i	2.57492	+	7.92480i	1.72035	+	2.45772i	4.77119	+	2.75465i
74.4	−2.12067	+	1.54076i	−1.66356	+	0.482242i	1.50528	−	4.63277i	0.0990696	+	0.222514i	2.78485	−	3.58583i	−1.70900	−	2.01973i	2.32573	+	7.15786i	2.53488	−	1.60448i	−0.552934	−	0.319237i
74.5	−2.11964	+	1.54001i	−1.24147	−	1.20779i	1.50321	−	4.62641i	−1.55025	−	3.48191i	4.49147	+	0.648217i	2.32009	+	1.27169i	2.31918	+	7.13770i	0.0824749	+	2.99887i	8.64813	+	4.99300i
74.6	−2.04610	+	1.48658i	1.03304	−	1.39026i	1.35858	−	4.18129i	−0.283003	−	0.635634i	−0.0469830	+	4.38032i	−0.631268	+	2.56934i	1.87294	+	5.76432i	−0.865639	−	2.87240i	1.52397	+	0.879867i
74.7	−1.96867	+	1.43032i	−0.298371	+	1.70616i	1.21180	−	3.72952i	1.71274	+	3.84687i	−1.85296	−	3.78562i	−2.61693	−	0.389429i	1.44486	+	4.44683i	−2.82195	−	1.01814i	−8.87405	−	5.12344i
74.8	−1.94499	+	1.41312i	−0.694218	−	1.58684i	1.16805	−	3.59488i	1.06743	+	2.39749i	3.59264	+	2.10537i	−0.358150	+	2.62140i	1.32230	+	4.06963i	−2.03612	+	2.20323i	−5.46408	−	3.15469i
74.9	−1.88049	+	1.36625i	−1.33953	+	1.09803i	1.05155	−	3.23634i	−1.66986	−	3.75057i	1.01877	−	3.89497i	−2.00709	+	1.72383i	1.00767	+	3.10129i	0.588658	−	2.94168i	8.26440	+	4.77145i
74.10	−1.86864	+	1.35765i	0.708732	+	1.58041i	1.03058	−	3.17179i	−0.0257979	−	0.0579431i	−3.47000	−	1.99101i	−1.35595	+	2.27187i	0.952875	+	2.93265i	−1.99540	+	2.24017i	0.126873	+	0.0732503i
74.11	−1.85890	+	1.35057i	−1.73202	−	0.00968558i	1.01343	−	3.11903i	0.0462072	+	0.103783i	3.23274	−	2.32121i	2.55792	−	0.676034i	0.908519	+	2.79613i	2.99981	+	0.0335513i	−0.226061	−	0.130516i
74.12	−1.81675	+	1.31994i	0.544965	−	1.64408i	0.940284	−	2.89390i	−0.895460	−	2.01124i	1.18004	+	3.70621i	0.494031	−	2.59922i	0.723653	+	2.22718i	−2.40603	−	1.79194i	4.28154	+	2.47195i
74.13	−1.75721	+	1.27669i	−1.13078	−	1.31200i	0.839823	−	2.58471i	−0.589096	−	1.32313i	3.66203	+	0.861805i	−2.63072	−	0.281590i	0.481733	+	1.48262i	−0.442678	+	2.96716i	2.72439	+	1.57293i
74.14	−1.75269	+	1.27341i	1.68151	−	0.415362i	0.832334	−	2.56166i	−1.30387	−	2.92855i	−2.41824	+	2.86924i	0.845537	−	2.50700i	0.464269	+	1.42887i	2.65495	−	1.39687i	6.01452	+	3.47248i
74.15	−1.71230	+	1.24406i	1.61105	−	0.636012i	0.766248	−	2.35827i	1.00417	+	2.25540i	−1.96736	+	3.09328i	−2.53112	−	0.770354i	0.313698	+	0.965462i	2.19098	−	2.04930i	−4.52527	−	2.61266i
74.16	−1.59244	+	1.15697i	−0.715340	+	1.57743i	0.579235	−	1.78270i	−0.0492475	−	0.110612i	−0.685911	−	3.33959i	0.678318	−	2.55732i	−0.0763695	−	0.235041i	−1.97658	−	2.25680i	0.206399	+	0.119164i
74.17	−1.56524	+	1.13721i	1.56735	+	0.737161i	0.538684	−	1.65790i	1.04638	+	2.35021i	−3.29159	+	0.628578i	1.24939	+	2.33217i	−0.153521	−	0.472489i	1.91319	+	2.31078i	−4.31052	−	2.48868i
74.18	−1.53684	+	1.11658i	1.42385	+	0.986238i	0.497093	−	1.52989i	0.677685	+	1.52210i	−3.28943	+	0.0741470i	1.35033	−	2.27522i	−0.229745	−	0.707082i	1.05467	+	2.80850i	−2.74104	−	1.58254i
74.19	−1.50703	+	1.09492i	−1.12625	+	1.31589i	0.454256	−	1.39806i	−0.00909224	−	0.0204215i	0.256503	−	3.21624i	1.38805	+	2.25240i	−0.305084	−	0.938953i	−0.463114	−	2.96404i	0.0360623	+	0.0208206i
74.20	−1.48767	+	1.08086i	0.696903	+	1.58566i	0.426879	−	1.31380i	−1.63957	−	3.68254i	−2.75063	−	1.60569i	2.59032	+	0.538760i	−0.351507	−	1.08183i	−2.02865	+	2.21011i	6.41944	+	3.70626i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner
63.j	odd	6	1	inner
693.cb	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.cb.a	✓	736
7.c	even	3	1		693.2.cx.a	yes	736
9.d	odd	6	1		693.2.cx.a	yes	736
11.d	odd	10	1	inner	693.2.cb.a	✓	736
63.j	odd	6	1	inner	693.2.cb.a	✓	736
77.o	odd	30	1		693.2.cx.a	yes	736
99.p	even	30	1		693.2.cx.a	yes	736
693.cb	even	30	1	inner	693.2.cb.a	✓	736

    

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

Hecke kernels

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