Newform orbit 504.2.bt.a

• Label: 504.2.bt.a
• Level: $504$
• Weight: $2$
• Character orbit: 504.bt
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $184$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.bt (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
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.

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

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
11.1	−1.41347	+	0.0459458i	1.18561	−	1.26267i	1.99578	−	0.129886i	−0.120967	+	0.209521i	−1.61780	+	1.83922i	2.21818	+	1.44212i	−2.81500	+	0.275287i	−0.188680	−	2.99406i	0.161356	−	0.301709i
11.2	−1.41299	+	0.0589246i	−0.617940	−	1.61807i	1.99306	−	0.166519i	−1.47387	+	2.55282i	0.968485	+	2.24990i	−2.64405	+	0.0949942i	−2.80635	+	0.352729i	−2.23630	+	1.99974i	1.93213	−	3.69394i
11.3	−1.40622	+	0.150130i	−1.62573	+	0.597489i	1.95492	−	0.422232i	1.01528	−	1.75852i	2.19644	−	1.08427i	−2.61403	−	0.408451i	−2.68567	+	0.887244i	2.28601	−	1.94271i	−1.16371	+	2.62530i
11.4	−1.40502	+	0.160980i	0.138508	+	1.72650i	1.94817	−	0.452362i	0.152023	−	0.263311i	−0.472540	−	2.40348i	2.13879	−	1.55742i	−2.66440	+	0.949196i	−2.96163	+	0.478270i	−0.171207	+	0.394430i
11.5	−1.40362	+	0.172746i	−1.32668	+	1.11351i	1.94032	−	0.484940i	−1.78605	+	3.09352i	1.66981	−	1.79213i	1.43753	−	2.22115i	−2.63970	+	1.01585i	0.520180	−	2.95456i	1.97254	−	4.65068i
11.6	−1.39667	−	0.222043i	0.996010	+	1.41703i	1.90139	+	0.620242i	2.02288	−	3.50373i	−1.07646	−	2.20028i	−1.02043	+	2.44105i	−2.51791	−	1.28847i	−1.01593	+	2.82274i	−3.60328	+	4.44440i
11.7	−1.38769	−	0.272632i	1.73194	+	0.0199803i	1.85134	+	0.756656i	0.662921	−	1.14821i	−2.39793	−	0.499908i	0.655004	−	2.56339i	−2.36279	−	1.55474i	2.99920	+	0.0692093i	−1.23297	+	1.41263i
11.8	−1.33582	+	0.464308i	−0.893431	−	1.48384i	1.56884	−	1.24047i	0.431911	−	0.748092i	1.88242	+	1.56732i	−0.623850	+	2.57115i	−1.51972	+	2.38546i	−1.40356	+	2.65142i	−0.229611	+	1.19986i
11.9	−1.33451	−	0.468067i	−1.72686	−	0.133969i	1.56183	+	1.24928i	−0.658616	+	1.14076i	2.24181	+	0.987070i	1.31090	+	2.29816i	−1.49952	−	2.39821i	2.96410	+	0.462692i	1.41288	−	1.21407i
11.10	−1.33426	−	0.468774i	−1.54596	−	0.781037i	1.56050	+	1.25093i	1.27843	−	2.21431i	1.69658	+	1.76681i	0.102649	−	2.64376i	−1.49571	−	2.40059i	1.77996	+	2.41490i	−2.74377	+	2.35517i
11.11	−1.32352	+	0.498286i	1.29060	+	1.15514i	1.50342	−	1.31898i	−1.45478	+	2.51976i	−2.28373	−	0.885761i	0.649643	+	2.56475i	−1.33258	+	2.49484i	0.331314	+	2.98165i	0.669877	−	4.05985i
11.12	−1.31497	−	0.520448i	0.732856	−	1.56937i	1.45827	+	1.36874i	1.20934	−	2.09463i	−1.78046	+	1.68225i	−2.58361	+	0.570073i	−1.20521	−	2.55880i	−1.92584	−	2.30025i	−2.68038	+	2.12497i
11.13	−1.28291	−	0.595101i	0.671530	+	1.59657i	1.29171	+	1.52692i	−1.33380	+	2.31020i	0.0886094	−	2.44789i	−2.19987	−	1.46989i	−0.748476	−	2.72760i	−2.09809	+	2.14430i	3.08594	−	2.17004i
11.14	−1.25253	+	0.656631i	0.406083	−	1.68377i	1.13767	−	1.64490i	1.72880	−	2.99437i	0.596986	+	2.37563i	−0.995123	−	2.45148i	−0.344876	+	2.80732i	−2.67019	−	1.36751i	−0.199181	+	4.88573i
11.15	−1.22535	−	0.706063i	−0.326067	−	1.70108i	1.00295	+	1.73034i	−1.27390	+	2.20646i	−0.801525	+	2.31464i	2.38039	−	1.15488i	−0.00723210	−	2.82842i	−2.78736	+	1.10933i	3.11887	−	1.80422i
11.16	−1.21905	+	0.716886i	−0.0776217	+	1.73031i	0.972150	−	1.74783i	0.664172	−	1.15038i	−1.14581	−	2.16498i	−2.29630	−	1.31415i	0.0679007	+	2.82761i	−2.98795	−	0.268619i	0.0150339	+	1.87850i
11.17	−1.21438	+	0.724765i	1.22201	−	1.22748i	0.949431	−	1.76028i	−1.73881	+	3.01171i	−0.594345	+	2.37629i	−0.680594	−	2.55672i	0.122820	+	2.82576i	−0.0134011	−	2.99997i	−0.0712072	−	4.91759i
11.18	−1.20074	+	0.747147i	−1.53340	−	0.805409i	0.883544	−	1.79425i	−0.00857237	+	0.0148478i	2.44297	−	0.178589i	2.28903	−	1.32678i	0.279666	+	2.81457i	1.70263	+	2.47003i	−0.000800297	−	0.0242331i
11.19	−1.13389	−	0.845156i	−0.862894	+	1.50180i	0.571423	+	1.91663i	1.48267	−	2.56806i	2.24769	−	0.973603i	2.51854	−	0.810517i	0.971921	−	2.65619i	−1.51083	−	2.59179i	−3.85161	+	1.65882i
11.20	−1.08666	+	0.905084i	1.73140	−	0.0474789i	0.361645	−	1.96703i	1.91888	−	3.32360i	−1.83846	+	1.61866i	2.42950	+	1.04762i	1.38734	+	2.46481i	2.99549	−	0.164410i	0.922971	+	5.34836i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	inner
63.j	odd	6	1	inner
504.bt	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.2.bt.a	✓	184
7.c	even	3	1		504.2.cy.a	yes	184
8.d	odd	2	1	inner	504.2.bt.a	✓	184
9.d	odd	6	1		504.2.cy.a	yes	184
56.k	odd	6	1		504.2.cy.a	yes	184
63.j	odd	6	1	inner	504.2.bt.a	✓	184
72.l	even	6	1		504.2.cy.a	yes	184
504.bt	even	6	1	inner	504.2.bt.a	✓	184

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.bt.a	✓	184	1.a	even	1	1	trivial
504.2.bt.a	✓	184	8.d	odd	2	1	inner
504.2.bt.a	✓	184	63.j	odd	6	1	inner
504.2.bt.a	✓	184	504.bt	even	6	1	inner
504.2.cy.a	yes	184	7.c	even	3	1
504.2.cy.a	yes	184	9.d	odd	6	1
504.2.cy.a	yes	184	56.k	odd	6	1
504.2.cy.a	yes	184	72.l	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(504, [\chi])\).