Newform orbit 209.3.i.a

• Label: 209.3.i.a
• Level: $209$
• Weight: $3$
• Character orbit: 209.i
• Analytic conductor: $5.695$
• Analytic rank: $0$
• Dimension: $64$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	209.i (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.69483752513\)
Analytic rank:	\(0\)
Dimension:	\(64\)
Relative dimension:	\(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 64 q + 60 q^{4} - 12 q^{6} - 20 q^{7} + 84 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
12.1	−3.36507	+	1.94283i	2.92948	−	1.69133i	5.54915	−	9.61141i	−1.90274	−	3.29564i	−6.57194	+	11.3829i	3.39030					27.5815i	1.22122	−	2.11522i	12.8057	+	7.39338i
12.2	−3.08967	+	1.78382i	−1.17110	+	0.676135i	4.36404	−	7.55874i	−0.0732669	−	0.126902i	2.41221	−	4.17807i	5.84479					16.8681i	−3.58568	+	6.21059i	0.452741	+	0.261390i
12.3	−3.03501	+	1.75226i	−4.35914	+	2.51675i	4.14086	−	7.17217i	−4.02858	−	6.97770i	8.82002	−	15.2767i	−10.2081					15.0054i	8.16806	−	14.1475i	24.4535	+	14.1183i
12.4	−2.81652	+	1.62612i	2.23307	−	1.28926i	3.28852	−	5.69588i	3.51353	+	6.08561i	−4.19299	+	7.26248i	−6.32546					8.38112i	−1.17559	+	2.03619i	−19.7918	−	11.4268i
12.5	−2.52757	+	1.45929i	0.619016	−	0.357389i	2.25907	−	3.91283i	−3.70910	−	6.42435i	−1.04307	+	1.80665i	−5.13220					1.51224i	−4.24455	+	7.35177i	18.7500	+	10.8253i
12.6	−2.35250	+	1.35822i	4.78684	−	2.76368i	1.68951	−	2.92631i	−1.42180	−	2.46262i	−7.50736	+	13.0031i	−9.10238				−	1.68688i	10.7759	−	18.6644i	6.68955	+	3.86221i
12.7	−2.28952	+	1.32185i	3.09408	−	1.78637i	1.49459	−	2.58871i	2.09190	+	3.62327i	−4.72263	+	8.17984i	10.8912				−	2.67230i	1.88222	−	3.26010i	−9.57886	−	5.53036i
12.8	−1.92589	+	1.11192i	−1.28075	+	0.739442i	0.472713	−	0.818764i	−2.53368	−	4.38847i	1.64440	−	2.84818i	8.24939				−	6.79286i	−3.40645	+	5.90014i	9.75921	+	5.63448i
12.9	−1.85443	+	1.07066i	−0.775959	+	0.448000i	0.292619	−	0.506831i	0.230431	+	0.399118i	0.959311	−	1.66158i	−2.74758				−	7.31209i	−4.09859	+	7.09897i	−0.854639	−	0.493426i
12.10	−1.80512	+	1.04219i	−4.87636	+	2.81537i	0.172315	−	0.298459i	−0.539449	−	0.934353i	5.86829	−	10.1642i	10.0548				−	7.61917i	11.3526	−	19.6633i	1.94754	+	1.12441i
12.11	−1.42959	+	0.825373i	−2.83662	+	1.63772i	−0.637518	+	1.10421i	0.790935	+	1.36994i	2.70346	−	4.68254i	−11.5170				−	8.70775i	0.864263	−	1.49695i	−2.26142	−	1.30563i
12.12	−1.03051	+	0.594962i	3.98507	−	2.30078i	−1.29204	+	2.23788i	0.273054	+	0.472944i	−2.73776	+	4.74193i	5.84303				−	7.83456i	6.08718	−	10.5433i	−0.562768	−	0.324914i
12.13	−0.719527	+	0.415419i	−1.92597	+	1.11196i	−1.65485	+	2.86629i	4.50256	+	7.79867i	0.923858	−	1.60017i	10.4824				−	6.07319i	−2.02710	+	3.51103i	−6.47943	−	3.74090i
12.14	−0.705330	+	0.407223i	1.38826	−	0.801511i	−1.66834	+	2.88965i	3.84883	+	6.66637i	−0.652787	+	1.13066i	−5.15696				−	5.97532i	−3.21516	+	5.56882i	−5.42939	−	3.13466i
12.15	−0.605445	+	0.349554i	1.75525	−	1.01339i	−1.75562	+	3.04083i	−4.40833	−	7.63546i	−0.708472	+	1.22711i	7.00790				−	5.25117i	−2.44606	+	4.23670i	5.33800	+	3.08190i
12.16	−0.120689	+	0.0696801i	−3.56290	+	2.05704i	−1.99029	+	3.44728i	−3.52061	−	6.09787i	0.286670	−	0.496527i	−1.50718				−	1.11218i	3.96285	−	6.86385i	0.849801	+	0.490633i
12.17	0.103941	−	0.0600101i	−2.80388	+	1.61882i	−1.99280	+	3.45163i	0.547663	+	0.948581i	−0.194291	+	0.336522i	1.06297					0.958433i	0.741148	−	1.28371i	0.113849	+	0.0657307i
12.18	0.660094	−	0.381105i	0.520020	−	0.300234i	−1.70952	+	2.96097i	−0.644581	−	1.11645i	0.228841	−	0.396365i	8.43058					5.65487i	−4.31972	+	7.48197i	−0.850968	−	0.491307i
12.19	0.784755	−	0.453078i	−4.89694	+	2.82725i	−1.58944	+	2.75299i	2.85427	+	4.94374i	−2.56193	+	4.43740i	−6.73173					6.50519i	11.4867	−	19.8955i	4.47981	+	2.58642i
12.20	0.851438	−	0.491578i	4.77851	−	2.75887i	−1.51670	+	2.62700i	3.17930	+	5.50671i	2.71240	−	4.69802i	0.601906					6.91494i	10.7228	−	18.5724i	5.41396	+	3.12575i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
19.d	odd	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	209.3.i.a	✓	64
19.d	odd	6	1	inner	209.3.i.a	✓	64

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
209.3.i.a	✓	64	1.a	even	1	1	trivial
209.3.i.a	✓	64	19.d	odd	6	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(209, [\chi])\).