Newform orbit 684.2.bn.a

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

\(\operatorname{Tr}(f)(q) = \) \( 232 q - 3 q^{2} + q^{4} - 4 q^{5} + 3 q^{6}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
31.1	−1.41348	+	0.0455575i	−1.68140	−	0.415818i	1.99585	−	0.128789i	−1.94560			2.39556	+	0.511150i	2.21940	−	1.28137i	−2.81522	+	0.272967i	2.65419	+	1.39831i	2.75006	−	0.0886365i
31.2	−1.41339	+	0.0481210i	1.67449	+	0.442815i	1.99537	−	0.136028i	1.89516			−2.38802	−	0.545295i	−0.449885	+	0.259741i	−2.81370	+	0.288280i	2.60783	+	1.48298i	−2.67860	+	0.0911967i
31.3	−1.41163	−	0.0854790i	−0.701022	−	1.58385i	1.98539	+	0.241329i	−0.877371			0.854196	+	2.29572i	0.0464500	−	0.0268179i	−2.78200	−	0.510376i	−2.01714	+	2.22062i	1.23852	+	0.0749968i
31.4	−1.40066	−	0.195298i	−1.07325	+	1.35946i	1.92372	+	0.547093i	−0.0693155			1.76876	−	1.69455i	0.877367	−	0.506548i	−2.58764	−	1.14199i	−0.696279	−	2.91808i	0.0970877	+	0.0135372i
31.5	−1.39102	+	0.255055i	0.711266	−	1.57927i	1.86989	−	0.709576i	4.24079			−0.586586	+	2.37822i	−3.37313	+	1.94748i	−2.42009	+	1.46396i	−1.98820	−	2.24656i	−5.89904	+	1.08164i
31.6	−1.38824	+	0.269790i	1.37096	+	1.05852i	1.85443	−	0.749067i	−1.14543			−2.18880	−	1.09962i	−1.74456	+	1.00722i	−2.37230	+	1.54019i	0.759054	+	2.90238i	1.59013	−	0.309025i
31.7	−1.38783	+	0.271913i	1.53662	−	0.799258i	1.85213	−	0.754736i	−3.33643			−1.91523	+	1.52706i	1.88032	−	1.08560i	−2.36521	+	1.55106i	1.72237	−	2.45630i	4.63038	−	0.907218i
31.8	−1.38404	+	0.290577i	−1.72476	+	0.158770i	1.83113	−	0.804341i	0.325215			2.34100	−	0.720920i	−4.29993	+	2.48257i	−2.30063	+	1.64532i	2.94958	−	0.547681i	−0.450111	+	0.0945002i
31.9	−1.37871	−	0.314887i	1.48521	−	0.891156i	1.80169	+	0.868278i	1.83719			−2.32829	+	0.760974i	3.30785	−	1.90979i	−2.21060	−	1.76443i	1.41168	−	2.64710i	−2.53296	−	0.578509i
31.10	−1.37304	+	0.338744i	−1.69246	−	0.368225i	1.77050	−	0.930222i	2.96209			2.44855	−	0.0677202i	2.97232	−	1.71607i	−2.11588	+	1.87698i	2.72882	+	1.24641i	−4.06708	+	1.00339i
31.11	−1.36998	−	0.350937i	0.219066	−	1.71814i	1.75369	+	0.961553i	−1.42258			−0.903075	+	2.27694i	−1.99986	+	1.15462i	−2.06507	−	1.93274i	−2.90402	−	0.752771i	1.94891	+	0.499236i
31.12	−1.35040	−	0.420012i	0.119934	+	1.72789i	1.64718	+	1.13437i	−3.57263			0.563777	−	2.38373i	−4.14718	+	2.39438i	−1.74791	−	2.22370i	−2.97123	+	0.414467i	4.82449	+	1.50055i
31.13	−1.31766	+	0.513596i	−0.620087	+	1.61725i	1.47244	−	1.35349i	1.82149			−0.0135508	−	2.44945i	−2.01024	+	1.16061i	−1.24502	+	2.53967i	−2.23098	−	2.00567i	−2.40010	+	0.935510i
31.14	−1.29721	−	0.563247i	−1.38284	−	1.04295i	1.36551	+	1.46130i	1.60178			1.20640	+	2.13181i	−1.76120	+	1.01683i	−0.948277	−	2.66473i	0.824508	+	2.88447i	−2.07785	−	0.902198i
31.15	−1.28986	+	0.579871i	−0.971557	+	1.43390i	1.32750	−	1.49591i	−3.77286			0.421698	−	2.41292i	1.44946	−	0.836844i	−0.844857	+	2.69930i	−1.11215	−	2.78624i	4.86648	−	2.18777i
31.16	−1.28693	+	0.586348i	0.470510	+	1.66692i	1.31239	−	1.50918i	2.84777			−1.58291	−	1.86933i	4.57571	−	2.64179i	−0.804054	+	2.71173i	−2.55724	+	1.56860i	−3.66488	+	1.66978i
31.17	−1.28658	−	0.587131i	0.549313	+	1.64264i	1.31055	+	1.51078i	3.61928			0.257711	−	2.43590i	−2.29219	+	1.32340i	−0.799103	−	2.71320i	−2.39651	+	1.80464i	−4.65648	−	2.12499i
31.18	−1.26892	+	0.624379i	0.326697	−	1.70096i	1.22030	−	1.58457i	−0.733758			0.647493	+	2.36236i	3.02297	−	1.74532i	−0.559090	+	2.77262i	−2.78654	−	1.11140i	0.931079	−	0.458143i
31.19	−1.24335	−	0.673851i	1.73200	−	0.0128754i	1.09185	+	1.67567i	−3.43126			−2.16217	−	1.15110i	−0.273850	+	0.158108i	−0.228402	−	2.81919i	2.99967	−	0.0446004i	4.26626	+	2.31216i
31.20	−1.24194	−	0.676458i	−1.33192	+	1.10724i	1.08481	+	1.68023i	0.393450			2.40316	−	0.474131i	2.41357	−	1.39347i	−0.210655	−	2.82057i	0.548039	−	2.94952i	−0.488639	−	0.266152i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
4.b	odd	2	1	inner
171.s	odd	6	1	inner
684.bn	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.bn.a	yes	232
4.b	odd	2	1	inner	684.2.bn.a	yes	232
9.c	even	3	1		684.2.u.a	✓	232
19.d	odd	6	1		684.2.u.a	✓	232
36.f	odd	6	1		684.2.u.a	✓	232
76.f	even	6	1		684.2.u.a	✓	232
171.s	odd	6	1	inner	684.2.bn.a	yes	232
684.bn	even	6	1	inner	684.2.bn.a	yes	232

    

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

Hecke kernels

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