Newform orbit 504.2.cc.b

• Label: 504.2.cc.b
• Level: $504$
• Weight: $2$
• Character orbit: 504.cc
• Analytic conductor: $4.024$
• Analytic rank: $0$
• Dimension: $168$
• CM: no
• Inner twists: $8$

Newspace parameters

Level:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	504.cc (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.02446026187\)
Analytic rank:	\(0\)
Dimension:	\(168\)
Relative dimension:	\(84\) over \(\Q(\zeta_{6})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 168 q - 6 q^{2} - 18 q^{4} - 2 q^{7} - 8 q^{9}+O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
293.1	−1.41305	+	0.0572344i	−1.67485	−	0.441454i	1.99345	−	0.161751i	−2.14271	−	1.23709i	2.39192	+	0.527939i	−0.933167	+	2.47572i	−2.80759	+	0.342656i	2.61024	+	1.47874i	3.09857	+	1.62545i
293.2	−1.41305	+	0.0572344i	1.67485	+	0.441454i	1.99345	−	0.161751i	2.14271	+	1.23709i	−2.39192	−	0.527939i	2.61062	+	0.429715i	−2.80759	+	0.342656i	2.61024	+	1.47874i	−3.09857	−	1.62545i
293.3	−1.39399	−	0.238316i	−1.00341	−	1.41180i	1.88641	+	0.664419i	1.68580	+	0.973297i	1.06229	+	2.20716i	−2.24361	−	1.40222i	−2.47129	−	1.37576i	−0.986337	+	2.83322i	−2.11804	−	1.75852i
293.4	−1.39399	−	0.238316i	1.00341	+	1.41180i	1.88641	+	0.664419i	−1.68580	−	0.973297i	−1.06229	−	2.20716i	−0.0925508	−	2.64413i	−2.47129	−	1.37576i	−0.986337	+	2.83322i	2.11804	+	1.75852i
293.5	−1.38810	−	0.270536i	−1.21338	+	1.23600i	1.85362	+	0.751060i	2.17541	+	1.25598i	2.01867	−	1.38743i	−2.64254	−	0.130417i	−2.36981	−	1.54401i	−0.0554045	−	2.99949i	−2.67990	−	2.33194i
293.6	−1.38810	−	0.270536i	1.21338	−	1.23600i	1.85362	+	0.751060i	−2.17541	−	1.25598i	−2.01867	+	1.38743i	1.20832	−	2.35371i	−2.36981	−	1.54401i	−0.0554045	−	2.99949i	2.67990	+	2.33194i
293.7	−1.37160	+	0.344550i	−1.55842	+	0.755861i	1.76257	−	0.945170i	−0.673579	−	0.388891i	1.87710	−	1.57369i	0.999298	−	2.44978i	−2.09188	+	1.90369i	1.85735	−	2.35590i	1.05787	+	0.301321i
293.8	−1.37160	+	0.344550i	1.55842	−	0.755861i	1.76257	−	0.945170i	0.673579	+	0.388891i	−1.87710	+	1.57369i	−2.62122	−	0.359471i	−2.09188	+	1.90369i	1.85735	−	2.35590i	−1.05787	−	0.301321i
293.9	−1.36240	−	0.379297i	−0.713494	−	1.57827i	1.71227	+	1.03351i	−0.431334	−	0.249031i	0.373433	+	2.42086i	2.62075	+	0.362852i	−1.94079	−	2.05751i	−1.98185	+	2.25217i	0.493193	+	0.502883i
293.10	−1.36240	−	0.379297i	0.713494	+	1.57827i	1.71227	+	1.03351i	0.431334	+	0.249031i	−0.373433	−	2.42086i	−0.996137	+	2.45106i	−1.94079	−	2.05751i	−1.98185	+	2.25217i	−0.493193	−	0.502883i
293.11	−1.30669	+	0.540891i	−0.0157202	+	1.73198i	1.41487	−	1.41355i	−1.10228	−	0.636400i	−0.916271	−	2.27166i	2.59794	−	0.500682i	−1.08422	+	2.61237i	−2.99951	−	0.0544542i	1.78456	+	0.235365i
293.12	−1.30669	+	0.540891i	0.0157202	−	1.73198i	1.41487	−	1.41355i	1.10228	+	0.636400i	0.916271	+	2.27166i	−1.73258	+	1.99955i	−1.08422	+	2.61237i	−2.99951	−	0.0544542i	−1.78456	−	0.235365i
293.13	−1.21983	−	0.715553i	−1.72909	+	0.101211i	0.975968	+	1.74571i	2.82704	+	1.63219i	2.18162	+	1.11380i	2.40792	−	1.09632i	0.0586299	−	2.82782i	2.97951	−	0.350006i	−2.28058	−	4.01389i
293.14	−1.21983	−	0.715553i	1.72909	−	0.101211i	0.975968	+	1.74571i	−2.82704	−	1.63219i	−2.18162	−	1.11380i	−2.15340	+	1.53716i	0.0586299	−	2.82782i	2.97951	−	0.350006i	2.28058	+	4.01389i
293.15	−1.21170	−	0.729232i	−1.13327	+	1.30985i	0.936442	+	1.76722i	−2.78508	−	1.60797i	2.32837	−	0.760729i	2.37927	+	1.15719i	0.154027	−	2.82423i	−0.431402	−	2.96882i	2.20210	+	3.97934i
293.16	−1.21170	−	0.729232i	1.13327	−	1.30985i	0.936442	+	1.76722i	2.78508	+	1.60797i	−2.32837	+	0.760729i	−0.187474	+	2.63910i	0.154027	−	2.82423i	−0.431402	−	2.96882i	−2.20210	−	3.97934i
293.17	−1.12177	+	0.861180i	−0.0157202	+	1.73198i	0.516736	−	1.93209i	−1.10228	−	0.636400i	−1.47391	−	1.95642i	−1.73258	+	1.99955i	1.08422	+	2.61237i	−2.99951	−	0.0544542i	1.78456	−	0.235365i
293.18	−1.12177	+	0.861180i	0.0157202	−	1.73198i	0.516736	−	1.93209i	1.10228	+	0.636400i	1.47391	+	1.95642i	2.59794	−	0.500682i	1.08422	+	2.61237i	−2.99951	−	0.0544542i	−1.78456	+	0.235365i
293.19	−0.995788	−	1.00419i	−0.539289	+	1.64595i	−0.0168127	+	1.99993i	−0.194059	−	0.112040i	2.18988	−	1.09747i	−1.79621	−	1.94258i	2.02506	−	1.97462i	−2.41834	−	1.77529i	0.0807318	+	0.306441i
293.20	−0.995788	−	1.00419i	0.539289	−	1.64595i	−0.0168127	+	1.99993i	0.194059	+	0.112040i	−2.18988	+	1.09747i	−0.784223	−	2.52685i	2.02506	−	1.97462i	−2.41834	−	1.77529i	−0.0807318	−	0.306441i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
7.b	odd	2	1	inner
8.b	even	2	1	inner
9.d	odd	6	1	inner
56.h	odd	2	1	inner
63.o	even	6	1	inner
72.j	odd	6	1	inner
504.cc	even	6	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	504.2.cc.b	✓	168
7.b	odd	2	1	inner	504.2.cc.b	✓	168
8.b	even	2	1	inner	504.2.cc.b	✓	168
9.d	odd	6	1	inner	504.2.cc.b	✓	168
56.h	odd	2	1	inner	504.2.cc.b	✓	168
63.o	even	6	1	inner	504.2.cc.b	✓	168
72.j	odd	6	1	inner	504.2.cc.b	✓	168
504.cc	even	6	1	inner	504.2.cc.b	✓	168

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
504.2.cc.b	✓	168	1.a	even	1	1	trivial
504.2.cc.b	✓	168	7.b	odd	2	1	inner
504.2.cc.b	✓	168	8.b	even	2	1	inner
504.2.cc.b	✓	168	9.d	odd	6	1	inner
504.2.cc.b	✓	168	56.h	odd	2	1	inner
504.2.cc.b	✓	168	63.o	even	6	1	inner
504.2.cc.b	✓	168	72.j	odd	6	1	inner
504.2.cc.b	✓	168	504.cc	even	6	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T5^84 - 113*T5^82 + 6931*T5^80 - 293962*T5^78 + 9535250*T5^76 - 249198964*T5^74 + 5419241364*T5^72 - 100160206416*T5^70 + 1596853507539*T5^68 - 22195387598859*T5^66 + 271068906081177*T5^64 - 2925453103087806*T5^62 + 28015783043706492*T5^60 - 238756728047154258*T5^58 + 1814062653600424296*T5^56 - 12299794011371450814*T5^54 + 74430638988184797744*T5^52 - 401737561402780210902*T5^50 + 1931476427736070352910*T5^48 - 8254601313732273793908*T5^46 + 31272218909764138803237*T5^44 - 104651209842457763448621*T5^42 + 308035938143423928440223*T5^40 - 793475022573975853983438*T5^38 + 1778472117632197249740768*T5^36 - 3446255186429004078450162*T5^34 + 5734737859656622492863930*T5^32 - 8137919106907144296399412*T5^30 + 9786185307515333394639647*T5^28 - 9910430751500620094322943*T5^26 + 8414191269940139961754795*T5^24 - 5961030955050152105602028*T5^22 + 3516508963918312546656709*T5^20 - 1719926724328641210777849*T5^18 + 695110883316695327998929*T5^16 - 230074618128131408543352*T5^14 + 61733222221268653908120*T5^12 - 13133459161030792948272*T5^10 + 2156680211617104436656*T5^8 - 257039005221544666752*T5^6 + 20656746205665757056*T5^4 - 844538344427623680*T5^2 + 22767608618129664