Newform orbit 684.2.o.a

• Label: 684.2.o.a
• Level: $684$
• Weight: $2$
• Character orbit: 684.o
• 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.o (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.

\(\operatorname{Tr}(f)(q) = \) \( 232 q - 3 q^{2} + q^{4} + 3 q^{6}+O(q^{10}) \)

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	−1.41421	+	0.00455230i	1.10804	−	1.33126i	1.99996	−	0.0128758i		−	4.21996i	−1.56094	+	1.88772i	1.74097	+	1.00515i	−2.82830	+	0.0273134i	−0.544497	−	2.95017i	0.0192105	+	5.96789i
11.2	−1.41394	−	0.0277479i	−0.908967	−	1.47437i	1.99846	+	0.0784677i			0.306310i	1.24431	+	2.10990i	0.747988	+	0.431851i	−2.82353	−	0.166402i	−1.34756	+	2.68031i	0.00849945	−	0.433105i
11.3	−1.41338	−	0.0484033i	−1.15926	+	1.28691i	1.99531	+	0.136825i		−	0.365397i	1.70076	−	1.76278i	−3.58119	−	2.06760i	−2.81352	−	0.289966i	−0.312254	−	2.98371i	−0.0176864	+	0.516447i
11.4	−1.41247	−	0.0701983i	1.42043	−	0.991151i	1.99014	+	0.198306i			4.19860i	−2.07589	+	1.30026i	3.34657	+	1.93215i	−2.79710	−	0.419806i	1.03524	−	2.81572i	0.294734	−	5.93039i
11.5	−1.41241	−	0.0714431i	1.22511	+	1.22438i	1.98979	+	0.201814i			1.44578i	−1.64289	−	1.81684i	2.47038	+	1.42627i	−2.79598	−	0.427200i	0.00181117	+	3.00000i	0.103291	−	2.04203i
11.6	−1.41239	−	0.0718796i	−1.68620	−	0.395888i	1.98967	+	0.203043i			2.95180i	2.35311	+	0.680350i	−0.414473	−	0.239296i	−2.79558	−	0.429792i	2.68655	+	1.33509i	0.212174	−	4.16908i
11.7	−1.40988	+	0.110618i	−0.0126385	+	1.73200i	1.97553	−	0.311917i		−	2.54159i	−0.173773	−	2.44332i	−1.35871	−	0.784451i	−2.75075	+	0.658296i	−2.99968	−	0.0437799i	0.281146	+	3.58333i
11.8	−1.38778	+	0.272176i	1.69723	+	0.345580i	1.85184	−	0.755437i		−	0.372058i	−2.44943	−	0.0176435i	0.164660	+	0.0950666i	−2.36433	+	1.55240i	2.76115	+	1.17305i	0.101265	+	0.516332i
11.9	−1.36579	+	0.366886i	1.11809	+	1.32283i	1.73079	−	1.00218i			3.10398i	−2.01241	−	1.39650i	−4.42597	−	2.55533i	−1.99622	+	2.00378i	−0.499760	+	2.95808i	−1.13881	−	4.23939i
11.10	−1.35542	+	0.403535i	−1.43572	+	0.968869i	1.67432	−	1.09392i		−	2.62337i	1.55503	−	1.89259i	4.39328	+	2.53646i	−1.82797	+	2.15836i	1.12259	−	2.78205i	1.05862	+	3.55576i
11.11	−1.33998	+	0.452168i	−0.0387989	−	1.73162i	1.59109	−	1.21179i		−	1.06785i	0.834971	+	2.30279i	0.640536	+	0.369814i	−1.58409	+	2.34321i	−2.99699	+	0.134370i	0.482846	+	1.43089i
11.12	−1.33660	−	0.462057i	0.610129	+	1.62103i	1.57301	+	1.23517i		−	0.399409i	−0.0664904	−	2.44859i	2.94561	+	1.70065i	−1.53176	−	2.37775i	−2.25549	+	1.97808i	−0.184550	+	0.533851i
11.13	−1.33496	−	0.466773i	−1.66738	−	0.468866i	1.56425	+	1.24625i		−	1.96275i	2.00704	+	1.40421i	2.10937	+	1.21784i	−1.50649	−	2.39384i	2.56033	+	1.56356i	−0.916161	+	2.62020i
11.14	−1.32793	+	0.486408i	−1.56525	−	0.741608i	1.52681	−	1.29184i		−	2.79966i	2.43928	+	0.223454i	−3.69046	−	2.13069i	−1.39915	+	2.45813i	1.90004	+	2.32161i	1.36178	+	3.71777i
11.15	−1.30529	−	0.544263i	1.25014	−	1.19881i	1.40756	+	1.42084i			1.79274i	−2.28426	+	0.884390i	−2.43772	−	1.40742i	−1.06395	−	2.62069i	0.125699	−	2.99737i	0.975721	−	2.34004i
11.16	−1.29647	−	0.564950i	1.46938	+	0.917012i	1.36166	+	1.46488i		−	4.02699i	−1.38695	−	2.01901i	−2.34202	−	1.35217i	−0.937772	−	2.66844i	1.31818	+	2.69488i	−2.27505	+	5.22087i
11.17	−1.29024	−	0.579026i	−1.20544	+	1.24375i	1.32946	+	1.49417i			1.18126i	2.27547	−	0.906761i	−0.0841210	−	0.0485673i	−0.850165	−	2.69763i	−0.0938310	−	2.99853i	0.683980	−	1.52411i
11.18	−1.28188	+	0.597308i	0.151881	−	1.72538i	1.28645	−	1.53136i			2.85550i	0.835889	+	2.30245i	−1.95665	−	1.12967i	−0.734381	+	2.73143i	−2.95386	−	0.524105i	−1.70562	−	3.66042i
11.19	−1.23750	−	0.684543i	−0.706340	−	1.58148i	1.06280	+	1.69424i		−	3.02562i	−0.208498	+	2.44060i	−3.37115	−	1.94633i	−0.155435	−	2.82415i	−2.00217	+	2.23413i	−2.07117	+	3.74420i
11.20	−1.22733	+	0.702609i	−1.62382	+	0.602662i	1.01268	−	1.72467i			0.824073i	1.56953	−	1.88058i	−0.328516	−	0.189669i	−0.0311255	+	2.82826i	2.27360	−	1.95723i	−0.579001	−	1.01141i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
171.n	odd	6	1	inner
684.o	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.o.a	✓	232
4.b	odd	2	1	inner	684.2.o.a	✓	232
9.d	odd	6	1		684.2.bi.a	yes	232
19.c	even	3	1		684.2.bi.a	yes	232
36.h	even	6	1		684.2.bi.a	yes	232
76.g	odd	6	1		684.2.bi.a	yes	232
171.n	odd	6	1	inner	684.2.o.a	✓	232
684.o	even	6	1	inner	684.2.o.a	✓	232

    

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

Hecke kernels

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