Newform orbit 702.2.bs.a

• Label: 702.2.bs.a
• Level: $702$
• Weight: $2$
• Character orbit: 702.bs
• Analytic conductor: $5.605$
• Analytic rank: $0$
• Dimension: $504$
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 702 = 2 \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	702.bs (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(5.60549822189\)
Analytic rank:	\(0\)
Dimension:	\(504\)
Relative dimension:	\(42\) over \(\Q(\zeta_{36})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) 504 * q - 24 * q^11 + 24 * q^15 + 24 * q^21 - 144 * q^27 + 24 * q^39 + 48 * q^41 + 12 * q^45 + 132 * q^47 + 24 * q^50 + 72 * q^54 - 72 * q^57 + 12 * q^63 - 156 * q^65 + 96 * q^66 - 48 * q^72 + 36 * q^73 - 96 * q^74 - 24 * q^78 + 288 * q^79 + 24 * q^83 + 24 * q^84 - 144 * q^85 - 96 * q^86 - 192 * q^87 + 96 * q^92 - 60 * q^93 - 72 * q^98 - 144 * q^99

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	−0.996195	−	0.0871557i	−1.72829	+	0.114016i	0.984808	+	0.173648i	3.39240	+	1.58190i	1.73165	+	0.0370487i	0.735095	−	1.04983i	−0.965926	−	0.258819i	2.97400	−	0.394106i	−3.24162	−	1.87155i
5.2	−0.996195	−	0.0871557i	−1.71493	+	0.242953i	0.984808	+	0.173648i	−3.44912	−	1.60835i	1.72958	−	0.0925624i	−1.15046	+	1.64303i	−0.965926	−	0.258819i	2.88195	−	0.833292i	3.29582	+	1.90284i
5.3	−0.996195	−	0.0871557i	−1.68860	+	0.385530i	0.984808	+	0.173648i	−0.145511	−	0.0678528i	1.71577	−	0.236891i	2.79229	−	3.98781i	−0.965926	−	0.258819i	2.70273	−	1.30201i	0.139043	+	0.0802766i
5.4	−0.996195	−	0.0871557i	−1.44995	−	0.947444i	0.984808	+	0.173648i	1.82372	+	0.850415i	1.36186	+	1.07021i	−0.615439	+	0.878939i	−0.965926	−	0.258819i	1.20470	+	2.74749i	−1.74266	−	1.00613i
5.5	−0.996195	−	0.0871557i	−1.42868	−	0.979218i	0.984808	+	0.173648i	−2.40537	−	1.12164i	1.33790	+	1.10001i	2.42794	−	3.46745i	−0.965926	−	0.258819i	1.08226	+	2.79798i	2.29846	+	1.32702i
5.6	−0.996195	−	0.0871557i	−1.24829	+	1.20074i	0.984808	+	0.173648i	0.699190	+	0.326038i	1.34819	−	1.08737i	−0.588520	+	0.840494i	−0.965926	−	0.258819i	0.116468	−	2.99774i	−0.668114	−	0.385736i
5.7	−0.996195	−	0.0871557i	−1.17070	−	1.27650i	0.984808	+	0.173648i	−0.486588	−	0.226900i	1.05499	+	1.37368i	−1.73769	+	2.48168i	−0.965926	−	0.258819i	−0.258919	+	2.98881i	0.464961	+	0.268445i
5.8	−0.996195	−	0.0871557i	−1.03486	+	1.38890i	0.984808	+	0.173648i	0.229130	+	0.106845i	1.15198	−	1.29343i	0.372424	−	0.531877i	−0.965926	−	0.258819i	−0.858114	−	2.87465i	−0.218946	−	0.126408i
5.9	−0.996195	−	0.0871557i	−0.624455	+	1.61557i	0.984808	+	0.173648i	−3.34201	−	1.55841i	0.762885	−	1.55499i	−0.371998	+	0.531269i	−0.965926	−	0.258819i	−2.22011	−	2.01770i	3.19347	+	1.84375i
5.10	−0.996195	−	0.0871557i	0.0949325	−	1.72945i	0.984808	+	0.173648i	2.31860	+	1.08118i	−0.245302	+	1.71459i	0.605375	−	0.864565i	−0.965926	−	0.258819i	−2.98198	−	0.328361i	−2.21555	−	1.27915i
5.11	−0.996195	−	0.0871557i	0.326010	−	1.70109i	0.984808	+	0.173648i	1.69330	+	0.789598i	−0.473029	+	1.66621i	−2.87225	+	4.10200i	−0.965926	−	0.258819i	−2.78744	−	1.10915i	−1.61804	−	0.934174i
5.12	−0.996195	−	0.0871557i	0.369695	+	1.69214i	0.984808	+	0.173648i	−0.624213	−	0.291075i	−0.220808	−	1.71792i	2.16691	−	3.09467i	−0.965926	−	0.258819i	−2.72665	+	1.25115i	0.596469	+	0.344372i
5.13	−0.996195	−	0.0871557i	0.416480	−	1.68123i	0.984808	+	0.173648i	−2.51109	−	1.17094i	−0.561424	+	1.63854i	1.30083	−	1.85778i	−0.965926	−	0.258819i	−2.65309	−	1.40040i	2.39948	+	1.38534i
5.14	−0.996195	−	0.0871557i	0.849091	+	1.50965i	0.984808	+	0.173648i	3.34415	+	1.55940i	−0.714285	−	1.57791i	−0.970968	+	1.38669i	−0.965926	−	0.258819i	−1.55809	+	2.56366i	−3.19552	−	1.84493i
5.15	−0.996195	−	0.0871557i	0.958159	+	1.44289i	0.984808	+	0.173648i	−0.313418	−	0.146149i	−0.828757	−	1.52091i	−2.45988	+	3.51307i	−0.965926	−	0.258819i	−1.16386	+	2.76504i	0.299487	+	0.172909i
5.16	−0.996195	−	0.0871557i	1.28976	−	1.15608i	0.984808	+	0.173648i	−1.90388	−	0.887795i	−1.38561	+	1.03927i	−0.418222	+	0.597283i	−0.965926	−	0.258819i	0.326947	−	2.98213i	1.81926	+	1.05035i
5.17	−0.996195	−	0.0871557i	1.44083	−	0.961248i	0.984808	+	0.173648i	−0.508880	−	0.237295i	−1.51913	+	0.832013i	−1.36461	+	1.94887i	−0.965926	−	0.258819i	1.15200	−	2.77000i	0.486262	+	0.280743i
5.18	−0.996195	−	0.0871557i	1.50253	+	0.861624i	0.984808	+	0.173648i	−3.41514	−	1.59251i	−1.42172	−	0.989299i	−0.396544	+	0.566323i	−0.965926	−	0.258819i	1.51521	+	2.58924i	3.26335	+	1.88410i
5.19	−0.996195	−	0.0871557i	1.52526	−	0.820720i	0.984808	+	0.173648i	3.44121	+	1.60466i	−1.59099	+	0.684662i	−0.481228	+	0.687265i	−0.965926	−	0.258819i	1.65284	−	2.50362i	−3.28825	−	1.89847i
5.20	−0.996195	−	0.0871557i	1.58718	+	0.693439i	0.984808	+	0.173648i	1.27501	+	0.594545i	−1.52070	−	0.829132i	0.673612	−	0.962018i	−0.965926	−	0.258819i	2.03829	+	2.20123i	−1.21834	−	0.703407i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.d	odd	4	1	inner
27.f	odd	18	1	inner
351.bt	even	36	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	702.2.bs.a	✓	504
13.d	odd	4	1	inner	702.2.bs.a	✓	504
27.f	odd	18	1	inner	702.2.bs.a	✓	504
351.bt	even	36	1	inner	702.2.bs.a	✓	504

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
702.2.bs.a	✓	504	1.a	even	1	1	trivial
702.2.bs.a	✓	504	13.d	odd	4	1	inner
702.2.bs.a	✓	504	27.f	odd	18	1	inner
702.2.bs.a	✓	504	351.bt	even	36	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(702, [\chi])\).