Newform orbit 416.2.bd.a

• Label: 416.2.bd.a
• Level: $416$
• Weight: $2$
• Character orbit: 416.bd
• Analytic conductor: $3.322$
• Analytic rank: $0$
• Dimension: $216$
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	416.bd (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(3.32177672409\)
Analytic rank:	\(0\)
Dimension:	\(216\)
Relative dimension:	\(54\) over \(\Q(\zeta_{8})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 216 q - 4 q^{2} - 8 q^{3} - 4 q^{5} + 8 q^{6} - 4 q^{8} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
83.1	−1.41400	−	0.0247176i	1.03934	+	0.430508i	1.99878	+	0.0699014i	−0.387075	+	0.934483i	−1.45898	−	0.634427i		−	4.00321i	−2.82454	−	0.148245i	−1.22643	−	1.22643i	0.570422	−	1.31179i
83.2	−1.41215	−	0.0764043i	2.23541	+	0.925935i	1.98832	+	0.215788i	0.344928	−	0.832729i	−3.08598	−	1.47835i			1.15405i	−2.79132	−	0.456642i	2.01836	+	2.01836i	−0.550713	+	1.14958i
83.3	−1.41176	−	0.0832778i	−2.32196	−	0.961788i	1.98613	+	0.235136i	0.932501	−	2.25126i	3.19796	+	1.55118i			4.08128i	−2.78436	−	0.497357i	2.34515	+	2.34515i	−1.50395	+	3.10058i
83.4	−1.37663	−	0.323857i	−2.65712	−	1.10062i	1.79023	+	0.891665i	−1.41696	+	3.42085i	3.30144	+	2.37567i			0.353777i	−2.17572	−	1.80728i	3.72762	+	3.72762i	3.05851	−	4.25036i
83.5	−1.35152	+	0.416401i	−2.25523	−	0.934148i	1.65322	−	1.12555i	0.261748	−	0.631915i	3.43698	+	0.323439i		−	4.76655i	−1.76568	+	2.20961i	2.09213	+	2.09213i	−0.0906275	+	0.963039i
83.6	−1.32811	+	0.485919i	−1.25935	−	0.521639i	1.52777	−	1.29071i	0.194227	−	0.468906i	1.92603	+	0.0808542i			2.08466i	−1.40186	+	2.45658i	−0.807469	−	0.807469i	−0.0301053	+	0.717139i
83.7	−1.29289	−	0.573103i	−0.377788	−	0.156485i	1.34311	+	1.48191i	−0.605904	+	1.46278i	0.398755	+	0.418829i		−	0.514771i	−0.887196	−	2.68568i	−2.00308	−	2.00308i	1.62169	−	1.54396i
83.8	−1.28670	+	0.586855i	−0.191326	−	0.0792497i	1.31120	−	1.51021i	−1.64053	+	3.96058i	0.292687	−	0.0103097i			2.32652i	−0.800850	+	2.71268i	−2.09100	−	2.09100i	−0.213419	−	6.05884i
83.9	−1.26014	−	0.641900i	−0.400717	−	0.165982i	1.17593	+	1.61777i	1.56341	−	3.77440i	0.398417	+	0.466382i		−	2.76672i	−0.443390	−	2.79346i	−1.98830	−	1.98830i	−4.39291	+	3.75274i
83.10	−1.24384	+	0.672951i	1.34047	+	0.555241i	1.09427	−	1.67409i	1.54705	−	3.73491i	−2.04098	+	0.211441i		−	0.167444i	−0.234521	+	2.81869i	−0.632754	−	0.632754i	0.589131	+	5.68672i
83.11	−1.22104	+	0.713478i	2.00588	+	0.830863i	0.981898	−	1.74238i	−1.19939	+	2.89559i	−3.04207	+	0.416631i		−	2.39832i	0.0442070	+	2.82808i	1.21190	+	1.21190i	−0.601429	−	4.39139i
83.12	−1.13073	−	0.849387i	0.761899	+	0.315589i	0.557082	+	1.92085i	−0.560536	+	1.35325i	−0.593442	−	1.00399i			4.42814i	1.00164	−	2.64513i	−1.64043	−	1.64043i	1.78325	−	1.05405i
83.13	−1.01464	−	0.985142i	2.77489	+	1.14940i	0.0589886	+	1.99913i	0.336989	−	0.813563i	−1.68319	−	3.89989i		−	2.04545i	1.90958	−	2.08651i	4.25758	+	4.25758i	−1.14340	+	0.493492i
83.14	−0.978420	+	1.02112i	2.81450	+	1.16580i	−0.0853883	−	1.99818i	−0.131889	+	0.318408i	−3.94419	+	1.73331i			4.33289i	2.12393	+	1.86786i	4.44098	+	4.44098i	−0.196091	−	0.446212i
83.15	−0.877210	−	1.10928i	−1.94716	−	0.806541i	−0.461004	+	1.94614i	−0.00996722	+	0.0240630i	0.813391	+	2.86746i		−	2.96755i	2.56322	−	1.19579i	1.01961	+	1.01961i	0.0354360	−	0.0100519i
83.16	−0.873613	+	1.11212i	−0.260922	−	0.108078i	−0.473600	−	1.94312i	0.260195	−	0.628165i	0.348140	−	0.195758i		−	0.282606i	2.57471	+	1.17084i	−2.06492	−	2.06492i	0.471283	+	0.838140i
83.17	−0.825979	−	1.14794i	0.518989	+	0.214972i	−0.635517	+	1.89634i	1.14210	−	2.75727i	−0.181899	−	0.773329i			2.29289i	2.70181	−	0.836807i	−1.89818	−	1.89818i	−4.10852	+	0.966391i
83.18	−0.798622	+	1.16713i	−1.28933	−	0.534060i	−0.724407	−	1.86420i	−0.341082	+	0.823446i	1.65301	−	1.07832i			0.829877i	2.75430	+	0.643308i	−0.744158	−	0.744158i	−0.688676	−	1.05571i
83.19	−0.757760	−	1.19407i	−2.82515	−	1.17022i	−0.851600	+	1.80963i	0.899503	−	2.17159i	0.743468	+	4.26017i			2.64760i	2.80614	−	0.354399i	4.49077	+	4.49077i	−3.27464	+	0.571477i
83.20	−0.649485	+	1.25625i	−2.66851	−	1.10533i	−1.15634	−	1.63183i	1.60908	−	3.88465i	3.12173	−	2.63443i		−	1.17091i	2.80102	−	0.392804i	3.77787	+	3.77787i	3.83503	+	4.54443i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
416.bd	even	8	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	416.2.bd.a	✓	216
13.d	odd	4	1		416.2.bi.a	yes	216
32.h	odd	8	1		416.2.bi.a	yes	216
416.bd	even	8	1	inner	416.2.bd.a	✓	216

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
416.2.bd.a	✓	216	1.a	even	1	1	trivial
416.2.bd.a	✓	216	416.bd	even	8	1	inner
416.2.bi.a	yes	216	13.d	odd	4	1
416.2.bi.a	yes	216	32.h	odd	8	1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(416, [\chi])\).