Newform orbit 600.2.bk.a

• Label: 600.2.bk.a
• Level: $600$
• Weight: $2$
• Character orbit: 600.bk
• Analytic conductor: $4.791$
• Analytic rank: $0$
• Dimension: $464$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	600.bk (of order \(10\), degree \(4\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.79102412128\)
Analytic rank:	\(0\)
Dimension:	\(464\)
Relative dimension:	\(116\) over \(\Q(\zeta_{10})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{10}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 464 q - 10 q^{3} - 6 q^{4} + q^{6} - 6 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
59.1	−1.41378	+	0.0350741i	1.68189	−	0.413828i	1.99754	−	0.0991741i	−0.0280397	+	2.23589i	−2.36330	+	0.644051i	−3.19488			−2.82060	+	0.210272i	2.65749	−	1.39202i	−0.0387801	−	3.16204i
59.2	−1.41337	+	0.0488228i	0.550722	−	1.64216i	1.99523	−	0.138009i	1.98623	+	1.02708i	−0.698199	+	2.34788i	0.938089			−2.81327	+	0.292471i	−2.39341	−	1.80875i	−2.85742	−	1.35467i
59.3	−1.41185	−	0.0816491i	−0.420915	−	1.68013i	1.98667	+	0.230553i	−2.13682	+	0.658799i	0.457089	+	2.40646i	−1.43736			−2.78606	−	0.487717i	−2.64566	+	1.41438i	3.07066	−	0.755659i
59.4	−1.40968	+	0.113177i	−1.52666	+	0.818114i	1.97438	−	0.319086i	2.23565	+	0.0434281i	2.05951	−	1.32606i	−1.18229			−2.74713	+	0.673262i	1.66138	−	2.49796i	−3.15646	+	0.191804i
59.5	−1.40516	−	0.159738i	0.223577	+	1.71756i	1.94897	+	0.448917i	0.305686	−	2.21507i	−0.0398025	−	2.44917i	1.47604			−2.66691	−	0.942126i	−2.90003	+	0.768015i	−0.783371	+	3.06371i
59.6	−1.38548	+	0.283608i	0.802404	+	1.53497i	1.83913	−	0.785869i	−2.18529	+	0.473809i	−1.54705	−	1.89911i	−4.19014			−2.32521	+	1.61040i	−1.71229	+	2.46334i	2.89331	−	1.27622i
59.7	−1.38429	−	0.289391i	1.47130	+	0.913939i	1.83251	+	0.801201i	0.0473381	+	2.23557i	−1.77221	−	1.69093i	3.46171			−2.30486	−	1.63940i	1.32943	+	2.68935i	0.581423	−	3.10837i
59.8	−1.38243	−	0.298125i	−1.70754	+	0.290346i	1.82224	+	0.824276i	−0.232683	−	2.22393i	2.44712	+	0.107677i	1.95460			−2.27339	−	1.68276i	2.83140	−	0.991555i	−0.341340	+	3.14380i
59.9	−1.38045	+	0.307170i	−0.792051	+	1.54034i	1.81129	−	0.848068i	−2.19234	+	0.440049i	0.620240	−	2.36966i	3.60889			−2.23990	+	1.72709i	−1.74531	−	2.44006i	2.89125	−	1.28089i
59.10	−1.37271	−	0.340095i	−1.70095	−	0.326753i	1.76867	+	0.933703i	−1.35194	+	1.78109i	2.22379	+	1.02702i	0.716957			−2.11033	−	1.88322i	2.78647	+	1.11158i	2.46156	−	1.98513i
59.11	−1.34762	+	0.428873i	1.29310	−	1.15234i	1.63214	−	1.15591i	−0.557333	−	2.16550i	−1.24839	+	2.10749i	−1.67503			−1.70375	+	2.25770i	0.344214	−	2.98019i	1.67979	+	2.67923i
59.12	−1.34256	−	0.444448i	0.365765	+	1.69299i	1.60493	+	1.19339i	2.23065	+	0.155573i	0.261384	−	2.43550i	−3.73244			−1.62432	−	2.31551i	−2.73243	+	1.23847i	−2.92564	−	1.20027i
59.13	−1.34238	+	0.445000i	1.62927	+	0.587762i	1.60395	−	1.19471i	1.82378	−	1.29377i	−2.44865	−	0.0639716i	2.29660			−1.62146	+	2.31751i	2.30907	+	1.91525i	−1.87247	+	2.54831i
59.14	−1.33942	+	0.453823i	−1.47999	−	0.899792i	1.58809	−	1.21572i	1.50979	−	1.64940i	2.39068	+	0.533545i	−3.47761			−1.57540	+	2.34907i	1.38075	+	2.66337i	−1.27371	+	2.89442i
59.15	−1.27389	+	0.614170i	−1.56146	−	0.749570i	1.24559	−	1.56477i	1.26771	+	1.84199i	2.44949	−	0.00412900i	4.42667			−0.625712	+	2.75835i	1.87629	+	2.34084i	−2.74621	−	1.56790i
59.16	−1.26954	+	0.623117i	−1.00007	−	1.41417i	1.22345	−	1.58214i	−1.73406	−	1.41175i	2.15081	+	1.17218i	1.35405			−0.567359	+	2.77094i	−0.999729	+	2.82852i	3.08114	+	0.711751i
59.17	−1.26837	−	0.625480i	1.49613	+	0.872691i	1.21755	+	1.58669i	−1.78593	−	1.34554i	−1.35180	−	2.04270i	1.06666			−0.551868	−	2.77407i	1.47682	+	2.61132i	1.42362	+	2.82371i
59.18	−1.24723	−	0.666648i	0.0947089	−	1.72946i	1.11116	+	1.66293i	0.266616	−	2.22012i	−1.27106	+	2.09389i	−4.09535			−0.277285	−	2.81480i	−2.98206	−	0.327591i	−1.81257	+	2.59125i
59.19	−1.24651	−	0.667996i	−1.21174	−	1.23761i	1.10756	+	1.66532i	1.60435	+	1.55758i	0.683727	+	2.35213i	−2.85872			−0.268158	−	2.81569i	−0.0633642	+	2.99933i	−0.959373	−	3.01324i
59.20	−1.20643	−	0.737927i	1.51281	−	0.843447i	0.910928	+	1.78051i	1.90067	−	1.17790i	−2.44750	−	0.0987875i	1.56189			0.214918	−	2.82025i	1.57719	−	2.55195i	−3.16222	+	0.0184939i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner
8.d	odd	2	1	inner
24.f	even	2	1	inner
25.e	even	10	1	inner
75.h	odd	10	1	inner
200.s	odd	10	1	inner
600.bk	even	10	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.2.bk.a	✓	464
3.b	odd	2	1	inner	600.2.bk.a	✓	464
8.d	odd	2	1	inner	600.2.bk.a	✓	464
24.f	even	2	1	inner	600.2.bk.a	✓	464
25.e	even	10	1	inner	600.2.bk.a	✓	464
75.h	odd	10	1	inner	600.2.bk.a	✓	464
200.s	odd	10	1	inner	600.2.bk.a	✓	464
600.bk	even	10	1	inner	600.2.bk.a	✓	464

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.2.bk.a	✓	464	1.a	even	1	1	trivial
600.2.bk.a	✓	464	3.b	odd	2	1	inner
600.2.bk.a	✓	464	8.d	odd	2	1	inner
600.2.bk.a	✓	464	24.f	even	2	1	inner
600.2.bk.a	✓	464	25.e	even	10	1	inner
600.2.bk.a	✓	464	75.h	odd	10	1	inner
600.2.bk.a	✓	464	200.s	odd	10	1	inner
600.2.bk.a	✓	464	600.bk	even	10	1	inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(600, [\chi])\).