Newform orbit 845.2.bf.a

• Label: 845.2.bf.a
• Level: $845$
• Weight: $2$
• Character orbit: 845.bf
• Analytic conductor: $6.747$
• Analytic rank: $0$
• Dimension: $2160$
• CM: no
• Inner twists: $4$

Newspace parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.bf (of order \(78\), degree \(24\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(6.74735897080\)
Analytic rank:	\(0\)
Dimension:	\(2160\)
Relative dimension:	\(90\) over \(\Q(\zeta_{78})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{78}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 2160 q - 140 q^{4} - 20 q^{5} - 42 q^{6} - 142 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
9.1	−2.61184	−	0.756529i	−1.02839	+	2.41372i	4.55900	+	2.88294i	2.22632	+	0.208577i	4.51205	−	5.52625i	0.0768631	−	0.230233i	−6.12005	−	6.90810i	−2.69029	−	2.80089i	−5.65700	−	2.22905i
9.2	−2.59468	−	0.751559i	0.915798	−	2.14946i	4.47715	+	2.83118i	−2.22930	+	0.173838i	−3.99165	+	4.88888i	0.337823	−	1.01190i	−5.90636	−	6.66690i	−1.70330	−	1.77333i	5.91498	+	1.22440i
9.3	−2.51372	−	0.728109i	0.321292	−	0.754101i	4.09828	+	2.59160i	1.15164	−	1.91670i	−1.35671	+	1.66167i	−0.867762	+	2.59926i	−4.94413	−	5.58077i	1.61273	+	1.67903i	−4.29047	+	3.97953i
9.4	−2.50351	−	0.725150i	−0.752714	+	1.76669i	4.05133	+	2.56190i	−0.676790	−	2.13119i	3.16554	−	3.87708i	−0.349159	+	1.04586i	−4.82802	−	5.44970i	−0.476427	−	0.496013i	0.148919	+	5.82621i
9.5	−2.47833	−	0.717856i	0.592567	−	1.39081i	3.93641	+	2.48923i	2.21016	−	0.339401i	−2.46697	+	3.02150i	0.402523	−	1.20570i	−4.54681	−	5.13229i	0.494966	+	0.515314i	−5.72114	−	0.745430i
9.6	−2.45929	−	0.712341i	−0.204762	+	0.480594i	3.85028	+	2.43477i	−2.14655	−	0.626357i	0.845915	−	1.03606i	1.36297	−	4.08259i	−4.33886	−	4.89756i	1.88913	+	1.96679i	4.83280	+	3.06947i
9.7	−2.44238	−	0.707443i	1.24176	−	2.91451i	3.77435	+	2.38675i	0.939794	+	2.02899i	−5.09468	+	6.23985i	−1.08765	+	3.25791i	−4.15754	−	4.69290i	−4.87423	−	5.07461i	−0.859939	−	5.62040i
9.8	−2.38880	−	0.691923i	−0.0647049	+	0.151868i	3.53721	+	2.23680i	−2.05590	+	0.879358i	0.259648	−	0.318011i	−1.09297	+	3.27383i	−3.60363	−	4.06765i	2.05930	+	2.14395i	5.51958	−	0.678082i
9.9	−2.26524	−	0.656135i	−1.30627	+	3.06592i	3.01043	+	1.90368i	−1.86213	+	1.23793i	4.97067	−	6.08797i	1.36632	−	4.09263i	−2.44252	−	2.75703i	−5.61538	−	5.84623i	5.03043	−	1.58239i
9.10	−2.26053	−	0.654772i	1.18142	−	2.77290i	2.99091	+	1.89134i	0.401844	−	2.19966i	−4.48626	+	5.49468i	1.07247	−	3.21245i	−2.40141	−	2.71063i	−4.21505	−	4.38833i	−2.34866	+	4.70930i
9.11	−2.24471	−	0.650189i	0.551308	−	1.29397i	2.92562	+	1.85005i	−0.0234030	+	2.23595i	−2.07885	+	2.54613i	0.230153	−	0.689391i	−2.26488	−	2.55652i	0.707760	+	0.736856i	1.50632	−	5.00384i
9.12	−2.16941	−	0.628377i	−0.430675	+	1.01083i	2.62110	+	1.65749i	1.97247	+	1.05326i	1.56949	−	1.92228i	0.450185	−	1.34847i	−1.64929	−	1.86166i	1.24188	+	1.29293i	−3.61725	−	3.52441i
9.13	−1.99231	−	0.577080i	−1.11523	+	2.61754i	1.94590	+	1.23052i	0.519951	+	2.17478i	3.73242	−	4.57138i	−1.04664	+	3.13507i	−0.415836	−	0.469382i	−3.52961	−	3.67472i	0.219117	−	4.63288i
9.14	−1.96180	−	0.568242i	−0.973499	+	2.28489i	1.83538	+	1.16062i	−0.619896	−	2.14842i	3.20818	−	3.92931i	−0.651201	+	1.95059i	−0.232359	−	0.262279i	−2.19483	−	2.28506i	−0.00471445	+	4.56703i
9.15	−1.94341	−	0.562915i	−0.427271	+	1.00284i	1.76958	+	1.11901i	1.78386	−	1.34827i	1.39488	−	1.70842i	1.43753	−	4.30593i	−0.125726	−	0.141915i	1.25504	+	1.30664i	−4.22573	+	1.61608i
9.16	−1.92840	−	0.558567i	−0.0965483	+	0.226607i	1.71633	+	1.08534i	1.70755	+	1.44371i	0.312759	−	0.383060i	−1.42698	+	4.27432i	−0.0408783	−	0.0461421i	2.03614	+	2.11985i	−2.48642	−	3.73781i
9.17	−1.91452	−	0.554548i	0.614684	−	1.44272i	1.66749	+	1.05446i	−1.24005	−	1.86072i	−1.97688	+	2.42124i	0.749055	−	2.24369i	0.0357914	+	0.0404001i	0.374580	+	0.389980i	1.34225	+	4.25006i
9.18	−1.85375	−	0.536945i	1.06086	−	2.48992i	1.45770	+	0.921792i	−1.47421	−	1.68128i	−3.30352	+	4.04607i	−1.56846	+	4.69810i	0.352329	+	0.397697i	−2.99613	−	3.11930i	1.83007	+	3.90824i
9.19	−1.75736	−	0.509026i	0.290775	−	0.682475i	1.13883	+	0.720154i	−0.300640	+	2.21577i	−0.858395	+	1.05134i	1.04583	−	3.13265i	0.791733	+	0.893681i	1.69695	+	1.76671i	1.65622	−	3.74087i
9.20	−1.72612	−	0.499976i	−0.738332	+	1.73293i	1.03913	+	0.657103i	−2.23357	+	0.105669i	2.14087	−	2.62209i	−0.164276	+	0.492068i	0.918238	+	1.03648i	−0.379734	−	0.395345i	3.90824	+	0.934335i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
169.i	even	39	1	inner
845.bf	even	78	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	845.2.bf.a	✓	2160
5.b	even	2	1	inner	845.2.bf.a	✓	2160
169.i	even	39	1	inner	845.2.bf.a	✓	2160
845.bf	even	78	1	inner	845.2.bf.a	✓	2160

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
845.2.bf.a	✓	2160	1.a	even	1	1	trivial
845.2.bf.a	✓	2160	5.b	even	2	1	inner
845.2.bf.a	✓	2160	169.i	even	39	1	inner
845.2.bf.a	✓	2160	845.bf	even	78	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(845, [\chi])\).