Newform orbit 165.3.x.a

• Label: 165.3.x.a
• Level: $165$
• Weight: $3$
• Character orbit: 165.x
• Analytic conductor: $4.496$
• Analytic rank: $0$
• Dimension: $192$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	165.x (of order \(20\), degree \(8\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.49592436194\)
Analytic rank:	\(0\)
Dimension:	\(192\)
Relative dimension:	\(24\) over \(\Q(\zeta_{20})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 192 q - 8 q^{5} + 52 q^{7}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
37.1	−0.590553	−	3.72860i	−0.786335	−	1.54327i	−9.74951	+	3.16781i	1.43508	−	4.78963i	−5.28987	+	3.84331i	−5.32374	+	10.4484i	10.7137	+	21.0268i	−1.76336	+	2.42705i	−18.7061	−	2.52232i
37.2	−0.590296	−	3.72698i	0.786335	+	1.54327i	−9.73771	+	3.16398i	2.40087	+	4.38587i	5.28756	−	3.84164i	−1.28882	+	2.52945i	10.6878	+	20.9760i	−1.76336	+	2.42705i	14.9288	−	11.5370i
37.3	−0.498911	−	3.15000i	−0.786335	−	1.54327i	−5.86936	+	1.90707i	−1.76653	+	4.67754i	−4.46898	+	3.24691i	−1.83910	+	3.60944i	3.14398	+	6.17040i	−1.76336	+	2.42705i	15.6156	+	3.23089i
37.4	−0.481948	−	3.04290i	0.786335	+	1.54327i	−5.22274	+	1.69697i	−1.51124	−	4.76615i	4.31704	−	3.13651i	2.90318	−	5.69781i	2.08614	+	4.09427i	−1.76336	+	2.42705i	−13.7746	+	6.89559i
37.5	−0.464620	−	2.93350i	−0.786335	−	1.54327i	−4.58531	+	1.48986i	4.49287	−	2.19412i	−4.16183	+	3.02374i	6.27351	−	12.3125i	1.10740	+	2.17340i	−1.76336	+	2.42705i	−8.52392	−	12.1604i
37.6	−0.402474	−	2.54112i	0.786335	+	1.54327i	−2.49107	+	0.809397i	−4.97645	+	0.484721i	3.60515	−	2.61929i	−5.09135	+	9.99233i	−1.61273	−	3.16516i	−1.76336	+	2.42705i	3.23462	+	12.4507i
37.7	−0.242377	−	1.53031i	0.786335	+	1.54327i	1.52112	−	0.494242i	−3.97608	+	3.03163i	2.17109	−	1.57739i	3.96010	−	7.77213i	−3.93866	−	7.73005i	−1.76336	+	2.42705i	5.60305	+	5.34984i
37.8	−0.234692	−	1.48179i	−0.786335	−	1.54327i	1.66362	−	0.540542i	−4.81592	+	1.34422i	−2.10225	+	1.52737i	1.45471	−	2.85503i	−3.91581	−	7.68522i	−1.76336	+	2.42705i	3.12210	+	6.82068i
37.9	−0.211793	−	1.33721i	−0.786335	−	1.54327i	2.06095	−	0.669645i	−2.14588	−	4.51611i	−1.89713	+	1.37835i	−0.0285578	+	0.0560478i	−3.79054	−	7.43936i	−1.76336	+	2.42705i	−5.58449	+	3.82597i
37.10	−0.182845	−	1.15444i	0.786335	+	1.54327i	2.50493	−	0.813902i	4.99785	−	0.146722i	1.63783	−	1.18995i	−0.149476	+	0.293364i	−3.52016	−	6.90870i	−1.76336	+	2.42705i	−1.08321	−	5.74288i
37.11	−0.0424684	−	0.268135i	−0.786335	−	1.54327i	3.73413	−	1.21329i	4.13132	−	2.81641i	−0.380410	+	0.276384i	−0.778008	+	1.52693i	−0.976902	−	1.91728i	−1.76336	+	2.42705i	−0.930629	−	0.988143i
37.12	−0.0395915	−	0.249971i	0.786335	+	1.54327i	3.74331	−	1.21627i	−0.152819	−	4.99766i	0.354640	−	0.257661i	3.22915	−	6.33756i	−0.911833	−	1.78957i	−1.76336	+	2.42705i	−1.24322	+	0.236065i
37.13	−0.0191809	−	0.121103i	0.786335	+	1.54327i	3.78993	−	1.23142i	1.76250	+	4.67906i	0.171812	−	0.124829i	−5.36080	+	10.5212i	−0.444484	−	0.872348i	−1.76336	+	2.42705i	0.532843	−	0.303193i
37.14	0.0689677	+	0.435445i	−0.786335	−	1.54327i	3.61937	−	1.17600i	−3.11319	+	3.91255i	0.617776	−	0.448841i	−5.57479	+	10.9411i	1.56231	+	3.06621i	−1.76336	+	2.42705i	−1.91841	−	1.08578i
37.15	0.0981531	+	0.619714i	−0.786335	−	1.54327i	3.42981	−	1.11441i	2.72030	+	4.19523i	0.879204	−	0.638779i	3.33900	−	6.55316i	2.16667	+	4.25233i	−1.76336	+	2.42705i	−2.33284	+	2.09758i
37.16	0.270612	+	1.70858i	0.786335	+	1.54327i	0.958222	−	0.311345i	1.31108	−	4.82505i	−2.42400	+	1.76114i	−3.33765	+	6.55051i	3.93265	+	7.71826i	−1.76336	+	2.42705i	8.59876	+	0.934369i
37.17	0.284411	+	1.79570i	0.786335	+	1.54327i	0.660571	−	0.214632i	4.56734	+	2.03455i	−2.54761	+	1.85095i	5.39460	−	10.5875i	3.87486	+	7.60485i	−1.76336	+	2.42705i	−2.35445	+	8.78023i
37.18	0.304568	+	1.92297i	0.786335	+	1.54327i	0.199185	−	0.0647190i	−3.29982	+	3.75649i	−2.72816	+	1.98213i	0.815546	−	1.60060i	3.72068	+	7.30225i	−1.76336	+	2.42705i	−8.22863	−	5.20133i
37.19	0.383947	+	2.42414i	−0.786335	−	1.54327i	−1.92483	+	0.625415i	−4.06401	−	2.91270i	3.43919	−	2.49872i	−5.06846	+	9.94741i	2.20190	+	4.32147i	−1.76336	+	2.42705i	5.50044	−	10.9701i
37.20	0.425569	+	2.68694i	−0.786335	−	1.54327i	−3.23429	+	1.05088i	1.35016	−	4.81426i	3.81202	−	2.76960i	2.21886	−	4.35477i	0.740129	+	1.45258i	−1.76336	+	2.42705i	13.5102	+	1.57899i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.c	odd	4	1	inner
11.c	even	5	1	inner
55.k	odd	20	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	165.3.x.a	✓	192
5.c	odd	4	1	inner	165.3.x.a	✓	192
11.c	even	5	1	inner	165.3.x.a	✓	192
55.k	odd	20	1	inner	165.3.x.a	✓	192

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
165.3.x.a	✓	192	1.a	even	1	1	trivial
165.3.x.a	✓	192	5.c	odd	4	1	inner
165.3.x.a	✓	192	11.c	even	5	1	inner
165.3.x.a	✓	192	55.k	odd	20	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(165, [\chi])\).