Newform orbit 333.3.ba.a

• Label: 333.3.ba.a
• Level: $333$
• Weight: $3$
• Character orbit: 333.ba
• Analytic conductor: $9.074$
• Analytic rank: $0$
• Dimension: $296$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 333 = 3^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	333.ba (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(9.07359280320\)
Analytic rank:	\(0\)
Dimension:	\(296\)
Relative dimension:	\(74\) over \(\Q(\zeta_{12})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 296 q - 2 q^{2} - 6 q^{3} - 14 q^{5} - 12 q^{6} + 2 q^{7} + 18 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
214.1	−2.70050	−	2.70050i	−2.34406	−	1.87227i			10.5854i	−2.04638	+	2.04638i	1.27408	+	11.3862i	−5.37722	−	9.31361i	17.7840	−	17.7840i	1.98923	+	8.77741i	11.0525
214.2	−2.69376	−	2.69376i	2.87247	+	0.865412i			10.5127i	1.88389	−	1.88389i	−5.40652	−	10.0689i	0.300945	+	0.521252i	17.5436	−	17.5436i	7.50212	+	4.97174i	−10.1495
214.3	−2.62114	−	2.62114i	−2.32332	+	1.89795i			9.74079i	−4.24476	+	4.24476i	11.0645	+	1.11495i	4.68754	+	8.11905i	15.0474	−	15.0474i	1.79559	−	8.81906i	22.2523
214.4	−2.60826	−	2.60826i	−2.66699	−	1.37375i			9.60601i	4.80047	−	4.80047i	3.37309	+	10.5393i	5.60471	+	9.70763i	14.6219	−	14.6219i	5.22562	+	7.32754i	−25.0417
214.5	−2.50365	−	2.50365i	2.46372	−	1.71175i			8.53657i	−6.85454	+	6.85454i	−10.4539	−	1.88269i	1.34112	+	2.32289i	11.3580	−	11.3580i	3.13986	−	8.43453i	34.3228
214.6	−2.49044	−	2.49044i	−0.541405	+	2.95074i			8.40463i	5.29460	−	5.29460i	8.69700	−	6.00032i	−0.539044	−	0.933651i	10.9695	−	10.9695i	−8.41376	−	3.19510i	−26.3718
214.7	−2.49041	−	2.49041i	0.215422	−	2.99226i			8.40424i	−0.287471	+	0.287471i	−7.98842	+	6.91544i	−0.611083	−	1.05843i	10.9684	−	10.9684i	−8.90719	−	1.28920i	1.43184
214.8	−2.31900	−	2.31900i	−2.78215	+	1.12235i			6.75554i	−0.0840405	+	0.0840405i	9.05453	+	3.84908i	−3.70959	−	6.42520i	6.39010	−	6.39010i	6.48068	−	6.24506i	0.389780
214.9	−2.27728	−	2.27728i	1.18948	+	2.75411i			6.37196i	−1.27652	+	1.27652i	3.56308	−	8.98065i	−1.51858	−	2.63025i	5.40161	−	5.40161i	−6.17025	+	6.55194i	5.81398
214.10	−2.19221	−	2.19221i	1.61795	+	2.52631i			5.61155i	−2.92656	+	2.92656i	1.99132	−	9.08507i	5.80991	+	10.0631i	3.53285	−	3.53285i	−3.76450	+	8.17487i	12.8313
214.11	−2.14877	−	2.14877i	0.982752	−	2.83447i			5.23441i	6.75136	−	6.75136i	−8.20232	+	3.97890i	−3.65477	−	6.33025i	2.65245	−	2.65245i	−7.06840	−	5.57115i	−29.0142
214.12	−2.13022	−	2.13022i	2.82540	−	1.00853i			5.07572i	0.449233	−	0.449233i	−8.16712	−	3.87035i	−5.19996	−	9.00660i	2.29152	−	2.29152i	6.96575	−	5.69898i	−1.91393
214.13	−1.91468	−	1.91468i	−0.907910	−	2.85932i			3.33201i	−0.937348	+	0.937348i	−3.73633	+	7.21304i	4.24213	+	7.34758i	−1.27898	+	1.27898i	−7.35140	+	5.19201i	3.58944
214.14	−1.90774	−	1.90774i	2.38199	−	1.82376i			3.27897i	0.575299	−	0.575299i	−8.02350	−	1.06495i	5.12163	+	8.87092i	−1.37555	+	1.37555i	2.34777	−	8.68838i	−2.19504
214.15	−1.88389	−	1.88389i	−1.04623	+	2.81165i			3.09811i	−3.64917	+	3.64917i	7.26785	−	3.32586i	−3.71184	−	6.42909i	−1.69906	+	1.69906i	−6.81079	−	5.88330i	13.7493
214.16	−1.83830	−	1.83830i	−2.58906	−	1.51551i			2.75867i	−6.28198	+	6.28198i	1.97350	+	7.54542i	0.293790	+	0.508859i	−2.28193	+	2.28193i	4.40645	+	7.84750i	23.0963
214.17	−1.78257	−	1.78257i	2.96827	−	0.435136i			2.35512i	5.55127	−	5.55127i	−6.06682	−	4.51550i	4.54627	+	7.87438i	−2.93212	+	2.93212i	8.62131	−	2.58321i	−19.7911
214.18	−1.76197	−	1.76197i	−2.91390	+	0.713576i			2.20909i	4.39968	−	4.39968i	6.39151	+	3.87691i	−0.682136	−	1.18149i	−3.15554	+	3.15554i	7.98162	−	4.15858i	−15.5042
214.19	−1.54728	−	1.54728i	2.74586	+	1.20840i			0.788150i	−2.81404	+	2.81404i	−2.37888	−	6.11836i	−2.09142	−	3.62245i	−4.96963	+	4.96963i	6.07953	+	6.63621i	8.70821
214.20	−1.53149	−	1.53149i	−2.60590	−	1.48636i			0.690939i	1.95972	−	1.95972i	1.71456	+	6.26727i	−1.38457	−	2.39815i	−5.06780	+	5.06780i	4.58145	+	7.74664i	−6.00259

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
333.ba	odd	12	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	333.3.ba.a	✓	296
9.c	even	3	1		333.3.bg.a	yes	296
37.g	odd	12	1		333.3.bg.a	yes	296
333.ba	odd	12	1	inner	333.3.ba.a	✓	296

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
333.3.ba.a	✓	296	1.a	even	1	1	trivial
333.3.ba.a	✓	296	333.ba	odd	12	1	inner
333.3.bg.a	yes	296	9.c	even	3	1
333.3.bg.a	yes	296	37.g	odd	12	1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(333, [\chi])\).