Newform orbit 693.2.r.a

• Label: 693.2.r.a
• Level: $693$
• Weight: $2$
• Character orbit: 693.r
• Analytic conductor: $5.534$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.r (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.53363286007\)
Analytic rank:	\(0\)
Dimension:	\(184\)
Relative dimension:	\(92\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 184 q - 2 q^{3} - 86 q^{4} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
32.1	−1.38013	+	2.39046i	1.47497	−	0.907994i	−2.80952	−	4.86624i			2.33301i	0.134863	+	4.77901i	−0.854297	+	2.50403i	9.98953			1.35110	−	2.67853i	−5.57696	−	3.21986i
32.2	−1.37952	+	2.38940i	0.310009	+	1.70408i	−2.80615	−	4.86039i		−	1.17845i	−4.49939	−	1.61008i	1.90327	+	1.83781i	9.96648			−2.80779	+	1.05656i	2.81578	+	1.62569i
32.3	−1.31516	+	2.27793i	−1.72198	−	0.186503i	−2.45930	−	4.25964i		−	3.09923i	2.68953	−	3.67727i	2.47997	+	0.921806i	7.67689			2.93043	+	0.642311i	7.05983	+	4.07599i
32.4	−1.28452	+	2.22485i	1.30058	−	1.14389i	−2.29998	−	3.98368i		−	3.14155i	0.874377	+	4.36295i	−0.353471	−	2.62203i	6.67940			0.383013	−	2.97545i	6.98949	+	4.03539i
32.5	−1.28452	+	2.22485i	−0.522595	+	1.65133i	−2.29997	−	3.98366i			1.91712i	−3.00268	−	3.28386i	1.33384	−	2.28493i	6.67931			−2.45379	−	1.72596i	−4.26530	−	2.46257i
32.6	−1.27299	+	2.20489i	1.43960	+	0.963096i	−2.24103	−	3.88157i		−	1.47270i	−3.95612	+	1.94814i	−2.60051	−	0.487186i	6.31928			1.14489	+	2.77294i	3.24714	+	1.87474i
32.7	−1.24873	+	2.16286i	−1.46670	+	0.921304i	−2.11865	−	3.66961i			2.56481i	−0.161146	−	4.32272i	−1.99852	+	1.73375i	5.58756			1.30240	−	2.70255i	−5.54734	−	3.20276i
32.8	−1.23056	+	2.13139i	−0.546512	−	1.64357i	−2.02854	−	3.51353i		−	1.18855i	4.17560	+	0.857679i	−1.88916	+	1.85232i	5.06268			−2.40265	+	1.79646i	2.53326	+	1.46258i
32.9	−1.20107	+	2.08032i	−0.261612	−	1.71218i	−1.88515	−	3.26517i		−	0.488480i	3.87609	+	1.51221i	2.64507	−	0.0598399i	4.25250			−2.86312	+	0.895855i	1.01619	+	0.586700i
32.10	−1.15050	+	1.99273i	1.59728	+	0.669852i	−1.64732	−	2.85324i			2.83395i	−3.17251	+	2.41228i	−0.128725	−	2.64262i	2.97898			2.10260	+	2.13988i	−5.64730	−	3.26047i
32.11	−1.14599	+	1.98492i	−0.858225	+	1.50448i	−1.62660	−	2.81735i		−	4.06236i	−2.00274	−	3.42762i	−2.60212	+	0.478508i	2.87230			−1.52690	−	2.58236i	8.06344	+	4.65543i
32.12	−1.12947	+	1.95630i	−1.53540	+	0.801594i	−1.55142	−	2.68714i			0.256197i	0.166029	−	3.90909i	−0.237628	−	2.63506i	2.49125			1.71489	−	2.46153i	−0.501199	−	0.289367i
32.13	−1.06743	+	1.84885i	1.34388	+	1.09269i	−1.27883	−	2.21501i			2.69562i	−3.45473	+	1.31826i	2.01333	+	1.71654i	1.19055			0.612049	+	2.93690i	−4.98380	−	2.87740i
32.14	−1.06376	+	1.84248i	−1.72517	−	0.154244i	−1.26315	−	2.18784i			1.71991i	2.11935	−	3.01451i	0.874708	+	2.49698i	1.11971			2.95242	+	0.532195i	−3.16889	−	1.82956i
32.15	−1.04326	+	1.80699i	1.32970	−	1.10991i	−1.17680	−	2.03828i			0.781280i	0.618370	+	3.56068i	2.63996	−	0.174906i	0.737804			0.536192	−	2.95169i	−1.41176	−	0.815082i
32.16	−1.00475	+	1.74028i	1.72811	+	0.116771i	−1.01906	−	1.76506i		−	2.82365i	−1.93954	+	2.89007i	1.45176	+	2.21187i	0.0765903			2.97273	+	0.403587i	4.91395	+	2.83707i
32.17	−0.997772	+	1.72819i	0.870661	−	1.49731i	−0.991098	−	1.71663i			3.70831i	1.71892	+	2.99865i	−1.88891	−	1.85257i	−0.0355293			−1.48390	−	2.60731i	−6.40867	−	3.70005i
32.18	−0.989463	+	1.71380i	−1.24040	−	1.20889i	−0.958072	−	1.65943i			2.85900i	3.29912	−	0.929647i	−2.61998	−	0.368369i	−0.165943			0.0771826	+	2.99901i	−4.89974	−	2.82887i
32.19	−0.888640	+	1.53917i	0.650782	+	1.60514i	−0.579364	−	1.00349i		−	3.18894i	−3.04890	−	0.424730i	1.56859	−	2.13062i	−1.49518			−2.15297	+	2.08920i	4.90832	+	2.83382i
32.20	−0.886651	+	1.53572i	−1.43789	−	0.965645i	−0.572300	−	0.991253i			0.473605i	2.75787	−	1.35201i	1.06688	−	2.42111i	−1.51688			1.13506	+	2.77698i	−0.727327	−	0.419923i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner
63.n	odd	6	1	inner
693.r	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	693.2.r.a	✓	184
7.c	even	3	1		693.2.bn.a	yes	184
9.d	odd	6	1		693.2.bn.a	yes	184
11.b	odd	2	1	inner	693.2.r.a	✓	184
63.n	odd	6	1	inner	693.2.r.a	✓	184
77.h	odd	6	1		693.2.bn.a	yes	184
99.g	even	6	1		693.2.bn.a	yes	184
693.r	even	6	1	inner	693.2.r.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
693.2.r.a	✓	184	1.a	even	1	1	trivial
693.2.r.a	✓	184	11.b	odd	2	1	inner
693.2.r.a	✓	184	63.n	odd	6	1	inner
693.2.r.a	✓	184	693.r	even	6	1	inner
693.2.bn.a	yes	184	7.c	even	3	1
693.2.bn.a	yes	184	9.d	odd	6	1
693.2.bn.a	yes	184	77.h	odd	6	1
693.2.bn.a	yes	184	99.g	even	6	1

Hecke kernels

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