Newform orbit 704.2.bm.a

• Label: 704.2.bm.a
• Level: $704$
• Weight: $2$
• Character orbit: 704.bm
• Analytic conductor: $5.621$
• Analytic rank: $0$
• Dimension: $3008$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.bm (of order \(80\), degree \(32\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.62146830230\)
Analytic rank:	\(0\)
Dimension:	\(3008\)
Relative dimension:	\(94\) over \(\Q(\zeta_{80})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{80}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 3008 q - 24 q^{2} - 24 q^{3} - 24 q^{4} - 24 q^{5} - 24 q^{6} - 24 q^{7} - 24 q^{8} - 24 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
5.1	−1.41412	+	0.0158961i	−0.590603	−	1.60090i	1.99949	−	0.0449580i	1.17000	−	2.08918i	0.860635	+	2.25448i	−0.660133	−	2.74965i	−2.82682	+	0.0953603i	0.0671469	−	0.0573489i	−1.62132	+	2.97297i
5.2	−1.41134	−	0.0901067i	−0.260874	−	0.707131i	1.98376	+	0.254342i	−2.01493	+	3.59792i	0.304465	+	1.02151i	0.635172	+	2.64568i	−2.77684	−	0.537714i	1.84924	−	1.57940i	3.16795	−	4.89633i
5.3	−1.40854	+	0.126586i	0.606433	+	1.64381i	1.96795	−	0.356603i	1.83131	−	3.27004i	−1.06227	−	2.23860i	−0.154961	−	0.645458i	−2.72679	+	0.751404i	−0.0531242	+	0.0453723i	−2.16552	+	4.83778i
5.4	−1.40233	+	0.182920i	0.105469	+	0.285885i	1.93308	−	0.513028i	0.314231	−	0.561100i	−0.200196	−	0.381614i	0.591790	+	2.46498i	−2.61698	+	1.07304i	2.21061	−	1.88804i	−0.338021	+	0.844328i
5.5	−1.39875	−	0.208551i	−0.971313	−	2.63286i	1.91301	+	0.583423i	−0.726585	+	1.29741i	0.809540	+	3.88529i	0.774721	+	3.22695i	−2.55416	−	1.21503i	−3.70729	+	3.16632i	1.28689	−	1.66322i
5.6	−1.39538	−	0.230017i	1.04052	+	2.82047i	1.89418	+	0.641924i	−1.59006	+	2.83926i	−0.803174	−	4.17497i	−0.247623	−	1.03142i	−2.49546	−	1.33142i	−4.59112	+	3.92118i	2.87183	−	3.59612i
5.7	−1.38664	+	0.277926i	0.480092	+	1.30135i	1.84551	−	0.770763i	−0.207772	+	0.371004i	−1.02739	−	1.67106i	−0.542343	−	2.25902i	−2.34484	+	1.58168i	0.818203	−	0.698812i	0.184993	−	0.572193i
5.8	−1.38436	−	0.289023i	−0.284372	−	0.770823i	1.83293	+	0.800228i	1.28225	−	2.28961i	0.170888	+	1.14929i	0.734875	+	3.06098i	−2.30616	−	1.63757i	1.76792	−	1.50994i	−2.43685	+	2.79906i
5.9	−1.35409	+	0.407958i	−0.639191	−	1.73260i	1.66714	−	1.10483i	−0.432141	+	0.771643i	1.57235	+	2.08534i	−0.200153	−	0.833698i	−1.80674	+	2.17616i	−0.312127	+	0.266582i	0.270362	−	1.22117i
5.10	−1.34341	+	0.441863i	0.940028	+	2.54806i	1.60951	−	1.18721i	−0.585051	+	1.04468i	−2.38874	−	3.00773i	1.07254	+	4.46745i	−1.63766	+	2.30610i	−3.32773	+	2.84215i	0.324357	−	1.66195i
5.11	−1.33579	−	0.464386i	−0.378597	−	1.02623i	1.56869	+	1.24065i	−0.771309	+	1.37727i	0.0291601	+	1.54665i	−0.952975	−	3.96943i	−1.51931	−	2.38573i	1.37140	−	1.17129i	1.66990	−	1.48157i
5.12	−1.33263	−	0.473375i	0.885175	+	2.39937i	1.55183	+	1.26167i	0.600407	−	1.07210i	−0.0438112	−	3.61651i	−0.612749	−	2.55228i	−1.47078	−	2.41595i	−2.69223	+	2.29938i	−1.30763	+	1.14450i
5.13	−1.28363	+	0.593538i	−1.06829	−	2.89572i	1.29543	−	1.52377i	1.97964	−	3.53490i	3.09001	+	3.08297i	0.280309	+	1.16757i	−0.758436	+	2.72484i	−4.96273	+	4.23857i	−0.443032	+	5.71250i
5.14	−1.26092	−	0.640374i	0.327451	+	0.887596i	1.17984	+	1.61492i	1.07073	−	1.91193i	0.155503	−	1.32888i	−0.163297	−	0.680179i	−0.453533	−	2.79183i	1.60062	−	1.36706i	−2.57446	+	1.72512i
5.15	−1.25920	+	0.643754i	−0.976331	−	2.64646i	1.17116	−	1.62123i	−0.489784	+	0.874572i	2.93306	+	2.70390i	−0.306505	−	1.27669i	−0.431055	+	2.79539i	−3.76931	+	3.21930i	0.0537263	−	1.41656i
5.16	−1.23366	+	0.691437i	0.445605	+	1.20786i	1.04383	−	1.70600i	−1.31074	+	2.34050i	−1.38489	−	1.18198i	−0.752586	−	3.13475i	−0.108140	+	2.82636i	1.02085	−	0.871884i	−0.00129908	−	3.79368i
5.17	−1.22922	−	0.699304i	−0.104544	−	0.283379i	1.02195	+	1.71919i	−1.00401	+	1.79278i	−0.0696608	+	0.421443i	−0.664726	−	2.76878i	−0.0539582	−	2.82791i	2.21184	−	1.88909i	2.48784	−	1.50161i
5.18	−1.13341	−	0.845803i	0.247391	+	0.670583i	0.569233	+	1.91728i	−0.372350	+	0.664879i	0.286786	−	0.969290i	0.668131	+	2.78297i	0.976470	−	2.65453i	1.89274	−	1.61655i	0.984381	−	0.438645i
5.19	−1.11537	+	0.869453i	−0.0594789	−	0.161225i	0.488102	−	1.93952i	1.07367	−	1.91717i	0.206518	+	0.128111i	0.712855	+	2.96925i	1.14191	+	2.58767i	2.25876	−	1.92917i	0.469354	+	3.07185i
5.20	−1.11165	+	0.874207i	0.112430	+	0.304754i	0.471525	−	1.94362i	−1.86928	+	3.33784i	−0.391400	−	0.240492i	0.739796	+	3.08147i	1.17496	+	2.57283i	2.20098	−	1.87982i	−0.839980	−	5.34465i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.c	even	5	1	inner
64.i	even	16	1	inner
704.bm	even	80	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	704.2.bm.a	✓	3008
11.c	even	5	1	inner	704.2.bm.a	✓	3008
64.i	even	16	1	inner	704.2.bm.a	✓	3008
704.bm	even	80	1	inner	704.2.bm.a	✓	3008

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
704.2.bm.a	✓	3008	1.a	even	1	1	trivial
704.2.bm.a	✓	3008	11.c	even	5	1	inner
704.2.bm.a	✓	3008	64.i	even	16	1	inner
704.2.bm.a	✓	3008	704.bm	even	80	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(704, [\chi])\).