Newform orbit 684.2.u.a

• Label: 684.2.u.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.u
• Analytic conductor: $5.462$
• Analytic rank: $0$
• Dimension: $232$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	684.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.46176749826\)
Analytic rank:	\(0\)
Dimension:	\(232\)
Relative dimension:	\(116\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 232 q - 2 q^{4} + 2 q^{5} - 9 q^{6} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
103.1	−1.41404	−	0.0219130i	0.376309	−	1.69068i	1.99904	+	0.0619719i	0.966473	+	1.67398i	−0.569165	+	2.38245i	3.77843	−	2.18148i	−2.82537	−	0.131436i	−2.71678	−	1.27243i	−1.32995	−	2.38826i
103.2	−1.41281	+	0.0628903i	1.73022	−	0.0795794i	1.99209	−	0.177705i	−1.21439	−	2.10339i	−2.43948	+	0.221245i	3.30611	−	1.90878i	−2.80328	+	0.376347i	2.98733	−	0.275380i	1.84799	+	2.89532i
103.3	−1.41227	−	0.0740289i	−1.31516	+	1.12710i	1.98904	+	0.209098i	0.476002	+	0.824459i	1.94081	−	1.49441i	−0.323567	+	0.186812i	−2.79359	−	0.442551i	0.459304	−	2.96463i	−0.611212	−	1.19960i
103.4	−1.41118	−	0.0926061i	1.36857	+	1.06161i	1.98285	+	0.261368i	0.878161	+	1.52102i	−1.83299	−	1.62486i	−3.54617	+	2.04738i	−2.77395	−	0.552460i	0.745972	+	2.90577i	−1.09839	−	2.22775i
103.5	−1.40607	−	0.151593i	0.404662	+	1.68412i	1.95404	+	0.426301i	1.70160	+	2.94726i	−0.313680	−	2.42932i	2.52422	−	1.45736i	−2.68288	−	0.895626i	−2.67250	+	1.36300i	−1.94578	−	4.40199i
103.6	−1.39568	+	0.228187i	−1.73114	−	0.0562840i	1.89586	−	0.636954i	−2.03169	−	3.51898i	2.42896	−	0.316469i	−1.88122	+	1.08612i	−2.50068	+	1.32160i	2.99366	+	0.194870i	3.63858	+	4.44778i
103.7	−1.38847	+	0.268618i	−0.878655	−	1.49264i	1.85569	−	0.745936i	0.530861	+	0.919477i	1.62093	+	1.83646i	−1.57523	+	0.909461i	−2.37619	+	1.53418i	−1.45593	+	2.62303i	−0.984072	−	1.13407i
103.8	−1.38415	+	0.290052i	1.46465	−	0.924555i	1.83174	−	0.802949i	−0.261320	−	0.452620i	−1.75913	+	1.70455i	−1.74257	+	1.00607i	−2.30251	+	1.64270i	1.29040	−	2.70830i	0.492990	+	0.550698i
103.9	−1.37938	−	0.311943i	0.284931	−	1.70845i	1.80538	+	0.860576i	−1.62976	−	2.82282i	−0.925969	+	2.26773i	−1.02096	+	0.589452i	−2.22186	−	1.75024i	−2.83763	−	0.973584i	1.36750	+	4.40214i
103.10	−1.37532	+	0.329405i	−0.579583	+	1.63220i	1.78298	−	0.906073i	−1.74171	−	3.01673i	0.259453	−	2.43571i	1.28895	−	0.744176i	−2.15370	+	1.83346i	−2.32817	−	1.89199i	3.38913	+	3.57523i
103.11	−1.35239	−	0.413564i	−1.50097	−	0.864346i	1.65793	+	1.11860i	−0.774298	−	1.34112i	1.67244	+	1.78968i	0.970572	−	0.560360i	−1.77956	−	2.19845i	1.50581	+	2.59471i	0.492514	+	2.13395i
103.12	−1.34805	−	0.427508i	1.00032	+	1.41399i	1.63447	+	1.15260i	−1.41358	−	2.44839i	−0.743990	−	2.33377i	0.231171	−	0.133467i	−1.71061	−	2.25252i	−0.998721	+	2.82888i	0.858869	+	3.90487i
103.13	−1.30620	+	0.542072i	−1.68963	+	0.380980i	1.41232	−	1.41611i	0.981035	+	1.69920i	2.00048	−	1.41354i	3.84020	−	2.21714i	−1.07713	+	2.61530i	2.70971	−	1.28743i	−2.20252	−	1.68771i
103.14	−1.29728	−	0.563091i	1.73163	−	0.0381409i	1.36586	+	1.46097i	0.253330	+	0.438781i	−2.26788	−	0.925586i	−1.26177	+	0.728486i	−0.949239	−	2.66438i	2.99709	−	0.132092i	−0.0815663	−	0.711869i
103.15	−1.29059	+	0.578245i	0.433570	+	1.67691i	1.33126	−	1.49256i	−0.377212	−	0.653350i	−1.52923	−	1.91350i	−3.40486	+	1.96580i	−0.855057	+	2.69609i	−2.62403	+	1.45411i	0.864624	+	0.625089i
103.16	−1.28922	+	0.581296i	1.50856	−	0.851021i	1.32419	−	1.49884i	1.83931	+	3.18577i	−1.45018	+	1.97408i	−1.10914	+	0.640364i	−0.835904	+	2.70209i	1.55153	−	2.56764i	−4.22315	−	3.03799i
103.17	−1.28682	−	0.586588i	−0.661437	+	1.60078i	1.31183	+	1.50967i	0.605760	+	1.04921i	1.79015	−	1.67193i	−0.602656	+	0.347944i	−0.802535	−	2.71218i	−2.12500	−	2.11763i	−0.164053	−	1.70548i
103.18	−1.26022	+	0.641743i	−1.42038	−	0.991227i	1.17633	−	1.61748i	−0.831446	−	1.44011i	2.42611	+	0.337653i	2.43962	−	1.40851i	−0.444438	+	2.79329i	1.03494	+	2.81583i	1.97199	+	1.28128i
103.19	−1.25331	−	0.655146i	0.547227	−	1.64333i	1.14157	+	1.64220i	2.02197	+	3.50216i	−1.76247	+	1.70109i	−3.31443	+	1.91359i	−0.354853	−	2.80608i	−2.40109	−	1.79855i	−0.239729	−	5.71397i
103.20	−1.21415	+	0.725153i	0.729938	+	1.57073i	0.948305	−	1.76089i	0.383835	+	0.664823i	−2.02527	−	1.37778i	2.61934	−	1.51228i	0.125530	+	2.82564i	−1.93438	+	2.29307i	−0.948131	−	0.528853i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
171.i	odd	6	1	inner
684.u	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	684.2.u.a	✓	232
4.b	odd	2	1	inner	684.2.u.a	✓	232
9.c	even	3	1		684.2.bn.a	yes	232
19.d	odd	6	1		684.2.bn.a	yes	232
36.f	odd	6	1		684.2.bn.a	yes	232
76.f	even	6	1		684.2.bn.a	yes	232
171.i	odd	6	1	inner	684.2.u.a	✓	232
684.u	even	6	1	inner	684.2.u.a	✓	232

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
684.2.u.a	✓	232	1.a	even	1	1	trivial
684.2.u.a	✓	232	4.b	odd	2	1	inner
684.2.u.a	✓	232	171.i	odd	6	1	inner
684.2.u.a	✓	232	684.u	even	6	1	inner
684.2.bn.a	yes	232	9.c	even	3	1
684.2.bn.a	yes	232	19.d	odd	6	1
684.2.bn.a	yes	232	36.f	odd	6	1
684.2.bn.a	yes	232	76.f	even	6	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(684, [\chi])\).