Newform orbit 209.3.l.a

• Label: 209.3.l.a
• Level: $209$
• Weight: $3$
• Character orbit: 209.l
• Analytic conductor: $5.695$
• Analytic rank: $0$
• Dimension: $144$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	209.l (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.69483752513\)
Analytic rank:	\(0\)
Dimension:	\(144\)
Relative dimension:	\(36\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 144 q + 2 q^{3} + 84 q^{4} + 4 q^{5} - 15 q^{7} - 40 q^{8} - 110 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
39.1	−3.63817	+	1.18211i	−2.27527	−	1.65308i	8.60280	−	6.25030i	−1.88012	+	5.78641i	10.2319	+	3.32456i	−6.60288	−	9.08808i	−14.9158	+	20.5299i	−0.336975	−	1.03710i		−	23.2745i
39.2	−3.54882	+	1.15308i	−0.678923	−	0.493267i	8.02848	−	5.83303i	1.48354	−	4.56588i	2.97816	+	0.967662i	7.38799	+	10.1687i	−12.9925	+	17.8827i	−2.56353	−	7.88973i			17.9141i
39.3	−3.37974	+	1.09814i	4.28832	+	3.11565i	6.98066	−	5.07175i	0.902063	−	2.77626i	−17.9148	−	5.82088i	−2.47100	−	3.40104i	−9.66814	+	13.3071i	5.90127	+	18.1623i			10.3736i
39.4	−3.30955	+	1.07534i	2.00527	+	1.45691i	6.56071	−	4.76664i	−2.82707	+	8.70084i	−8.20323	−	2.66539i	5.53152	+	7.61348i	−8.40561	+	11.5693i	−0.882641	−	2.71649i		−	31.8359i
39.5	−3.13274	+	1.01789i	−4.70794	−	3.42052i	5.54188	−	4.02641i	1.08231	−	3.33101i	18.2305	+	5.92344i	2.19237	+	3.01753i	−5.51826	+	7.59524i	7.68362	+	23.6477i			11.5369i
39.6	−2.99476	+	0.973058i	−1.00914	−	0.733185i	4.78570	−	3.47702i	2.44982	−	7.53977i	3.73558	+	1.21376i	−4.78346	−	6.58387i	−3.54525	+	4.87962i	−2.30034	−	7.07973i			24.9636i
39.7	−2.98669	+	0.970433i	1.90607	+	1.38484i	4.74248	−	3.44562i	−0.161438	+	0.496856i	−7.03673	−	2.28637i	−1.41506	−	1.94766i	−3.43708	+	4.73073i	−1.06584	−	3.28030i		−	1.64062i
39.8	−2.58031	+	0.838392i	−2.16625	−	1.57388i	2.71901	−	1.97548i	−1.10730	+	3.40793i	6.90913	+	2.24491i	−1.16925	−	1.60933i	1.01921	−	1.40282i	−0.565580	−	1.74068i		−	9.72186i
39.9	−2.13065	+	0.692291i	3.51561	+	2.55424i	0.824345	−	0.598921i	2.10768	−	6.48678i	−9.25881	−	3.00837i	4.99886	+	6.88034i	3.92550	−	5.40298i	3.05421	+	9.39988i			15.2802i
39.10	−2.07711	+	0.674895i	−3.56502	−	2.59014i	0.622847	−	0.452525i	−1.92437	+	5.92261i	9.15302	+	2.97400i	3.48015	+	4.79002i	4.14659	−	5.70729i	3.21940	+	9.90829i		−	13.6007i
39.11	−2.05801	+	0.668689i	0.374132	+	0.271822i	0.552205	−	0.401200i	0.107615	−	0.331204i	−0.951732	−	0.309237i	1.09516	+	1.50736i	4.21952	−	5.80767i	−2.71507	−	8.35611i			0.753584i
39.12	−1.70220	+	0.553079i	1.61487	+	1.17327i	−0.644475	+	0.468239i	−0.908993	+	2.79759i	−3.39775	−	1.10399i	−6.55700	−	9.02494i	5.04613	−	6.94540i	−1.54991	−	4.77014i		−	5.26481i
39.13	−1.53314	+	0.498146i	−3.08261	−	2.23965i	−1.13371	+	0.823689i	2.12805	−	6.54946i	5.84174	+	1.89810i	−0.597007	−	0.821710i	5.11794	−	7.04424i	1.70532	+	5.24844i			11.1013i
39.14	−1.26148	+	0.409881i	4.59679	+	3.33976i	−1.81273	+	1.31703i	−1.59645	+	4.91338i	−7.16767	−	2.32892i	−2.86934	−	3.94930i	4.86546	−	6.69674i	7.19529	+	22.1448i		−	6.85250i
39.15	−1.06254	+	0.345239i	2.76093	+	2.00593i	−2.22628	+	1.61748i	−1.72324	+	5.30360i	−3.62611	−	1.17820i	6.09142	+	8.38412i	4.43381	−	6.10262i	0.817816	+	2.51698i		−	6.23019i
39.16	−0.535652	+	0.174044i	−1.01540	−	0.737733i	−2.97944	+	2.16469i	−2.68620	+	8.26726i	0.672300	+	0.218443i	0.240454	+	0.330956i	2.54339	−	3.50068i	−2.29436	−	7.06132i		−	4.89589i
39.17	−0.521786	+	0.169539i	2.72258	+	1.97807i	−2.99255	+	2.17422i	2.65785	−	8.18002i	−1.75596	−	0.570548i	−6.50335	−	8.95110i	2.48278	−	3.41726i	0.718532	+	2.21142i			4.71883i
39.18	−0.365510	+	0.118761i	−3.82332	−	2.77781i	−3.11657	+	2.26432i	−0.649291	+	1.99831i	1.72736	+	0.561253i	−7.20495	−	9.91677i	1.77382	−	2.44145i	4.12044	+	12.6814i		−	0.807515i
39.19	0.0750461	−	0.0243840i	0.558090	+	0.405476i	−3.23103	+	2.34748i	1.40914	−	4.33688i	0.0517696	+	0.0168210i	−0.714382	−	0.983263i	−0.370760	+	0.510307i	−2.63410	−	8.10692i		−	0.359827i
39.20	0.533255	−	0.173265i	0.595876	+	0.432929i	−2.98173	+	2.16635i	−0.936824	+	2.88325i	0.392765	+	0.127617i	−3.09188	−	4.25561i	−2.53295	+	3.48630i	−2.61351	−	8.04356i			1.69982i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.d	odd	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	209.3.l.a	✓	144
11.d	odd	10	1	inner	209.3.l.a	✓	144

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
209.3.l.a	✓	144	1.a	even	1	1	trivial
209.3.l.a	✓	144	11.d	odd	10	1	inner

Hecke kernels

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