Newform orbit 570.2.bb.b

• Label: 570.2.bb.b
• Level: $570$
• Weight: $2$
• Character orbit: 570.bb
• Analytic conductor: $4.551$
• Analytic rank: $0$
• Dimension: $84$
• CM: no
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	570.bb (of order \(18\), degree \(6\), minimal)

Newform invariants

Self dual:	no
Analytic conductor:	\(4.55147291521\)
Analytic rank:	\(0\)
Dimension:	\(84\)
Relative dimension:	\(14\) over \(\Q(\zeta_{18})\)
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 84q - 6q^{6} + 42q^{8} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
41.1	−0.766044	−	0.642788i	−1.71584	−	0.236392i	0.173648	+	0.984808i	−0.984808	−	0.173648i	1.16246	+	1.28401i	−1.76528	−	3.05755i	0.500000	−	0.866025i	2.88824	+	0.811224i	0.642788	+	0.766044i
41.2	−0.766044	−	0.642788i	−1.50714	−	0.853538i	0.173648	+	0.984808i	0.984808	+	0.173648i	0.605893	+	1.62262i	1.94656	+	3.37154i	0.500000	−	0.866025i	1.54295	+	2.57280i	−0.642788	−	0.766044i
41.3	−0.766044	−	0.642788i	−1.15586	−	1.28996i	0.173648	+	0.984808i	0.984808	+	0.173648i	0.0562681	+	1.73114i	−0.561471	−	0.972497i	0.500000	−	0.866025i	−0.327989	+	2.98202i	−0.642788	−	0.766044i
41.4	−0.766044	−	0.642788i	−1.07847	+	1.35533i	0.173648	+	0.984808i	−0.984808	−	0.173648i	1.69734	−	0.345017i	−0.267952	−	0.464106i	0.500000	−	0.866025i	−0.673825	−	2.92335i	0.642788	+	0.766044i
41.5	−0.766044	−	0.642788i	−1.01882	+	1.40072i	0.173648	+	0.984808i	0.984808	+	0.173648i	1.68082	−	0.418131i	0.0247885	+	0.0429350i	0.500000	−	0.866025i	−0.924029	−	2.85415i	−0.642788	−	0.766044i
41.6	−0.766044	−	0.642788i	−0.169275	−	1.72376i	0.173648	+	0.984808i	−0.984808	−	0.173648i	−0.978339	+	1.42928i	1.36690	+	2.36755i	0.500000	−	0.866025i	−2.94269	+	0.583578i	0.642788	+	0.766044i
41.7	−0.766044	−	0.642788i	0.0507095	+	1.73131i	0.173648	+	0.984808i	0.984808	+	0.173648i	1.07402	−	1.35885i	0.847833	+	1.46849i	0.500000	−	0.866025i	−2.99486	+	0.175587i	−0.642788	−	0.766044i
41.8	−0.766044	−	0.642788i	0.446223	+	1.67358i	0.173648	+	0.984808i	−0.984808	−	0.173648i	0.733933	−	1.56887i	2.13041	+	3.68998i	0.500000	−	0.866025i	−2.60177	+	1.49358i	0.642788	+	0.766044i
41.9	−0.766044	−	0.642788i	0.902121	−	1.47857i	0.173648	+	0.984808i	−0.984808	−	0.173648i	−1.64147	+	0.552780i	−1.68197	−	2.91326i	0.500000	−	0.866025i	−1.37235	−	2.66770i	0.642788	+	0.766044i
41.10	−0.766044	−	0.642788i	0.964794	+	1.43846i	0.173648	+	0.984808i	−0.984808	−	0.173648i	0.185551	−	1.72208i	−1.20034	−	2.07905i	0.500000	−	0.866025i	−1.13835	+	2.77564i	0.642788	+	0.766044i
41.11	−0.766044	−	0.642788i	1.46696	−	0.920884i	0.173648	+	0.984808i	0.984808	+	0.173648i	−1.71569	−	0.237506i	−2.15391	−	3.73068i	0.500000	−	0.866025i	1.30395	−	2.70180i	−0.642788	−	0.766044i
41.12	−0.766044	−	0.642788i	1.48528	+	0.891036i	0.173648	+	0.984808i	0.984808	+	0.173648i	−0.565043	−	1.63729i	−2.04901	−	3.54899i	0.500000	−	0.866025i	1.41211	+	2.64687i	−0.642788	−	0.766044i
41.13	−0.766044	−	0.642788i	1.59014	−	0.686630i	0.173648	+	0.984808i	−0.984808	−	0.173648i	−1.65947	−	0.496132i	1.07093	+	1.85491i	0.500000	−	0.866025i	2.05708	−	2.18367i	0.642788	+	0.766044i
41.14	−0.766044	−	0.642788i	1.61856	−	0.616663i	0.173648	+	0.984808i	0.984808	+	0.173648i	−1.63627	−	0.567997i	1.59792	+	2.76767i	0.500000	−	0.866025i	2.23945	−	1.99621i	−0.642788	−	0.766044i
71.1	0.939693	−	0.342020i	−1.66038	−	0.493094i	0.766044	−	0.642788i	0.642788	−	0.766044i	−1.72889	+	0.104526i	0.222334	+	0.385093i	0.500000	−	0.866025i	2.51372	+	1.63745i	0.342020	−	0.939693i
71.2	0.939693	−	0.342020i	−1.63144	+	0.581714i	0.766044	−	0.642788i	−0.642788	+	0.766044i	−1.33410	+	1.10462i	−0.223800	−	0.387633i	0.500000	−	0.866025i	2.32322	−	1.89807i	−0.342020	+	0.939693i
71.3	0.939693	−	0.342020i	−1.49197	−	0.879786i	0.766044	−	0.642788i	−0.642788	+	0.766044i	−1.70290	−	0.316445i	−1.40312	−	2.43028i	0.500000	−	0.866025i	1.45195	+	2.62523i	−0.342020	+	0.939693i
71.4	0.939693	−	0.342020i	−1.12629	−	1.31585i	0.766044	−	0.642788i	−0.642788	+	0.766044i	−1.50842	−	0.851282i	2.21271	+	3.83252i	0.500000	−	0.866025i	−0.462935	+	2.96407i	−0.342020	+	0.939693i
71.5	0.939693	−	0.342020i	−0.950784	+	1.44776i	0.766044	−	0.642788i	0.642788	−	0.766044i	−0.398282	+	1.68564i	−2.26841	−	3.92899i	0.500000	−	0.866025i	−1.19202	−	2.75301i	0.342020	−	0.939693i
71.6	0.939693	−	0.342020i	−0.936498	+	1.45704i	0.766044	−	0.642788i	0.642788	−	0.766044i	−0.381683	+	1.68947i	0.199417	+	0.345401i	0.500000	−	0.866025i	−1.24594	−	2.72903i	0.342020	−	0.939693i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
57.j	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	570.2.bb.b	yes	84
3.b	odd	2	1		570.2.bb.a	✓	84
19.f	odd	18	1		570.2.bb.a	✓	84
57.j	even	18	1	inner	570.2.bb.b	yes	84

    

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
570.2.bb.a	✓	84	3.b	odd	2	1
570.2.bb.a	✓	84	19.f	odd	18	1
570.2.bb.b	yes	84	1.a	even	1	1	trivial
570.2.bb.b	yes	84	57.j	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(37\!\cdots\!79\)\( T_{11}^{70} - \)\(32\!\cdots\!96\)\( T_{11}^{69} + \)\(13\!\cdots\!38\)\( T_{11}^{68} + \)\(14\!\cdots\!78\)\( T_{11}^{67} - \)\(41\!\cdots\!89\)\( T_{11}^{66} - \)\(54\!\cdots\!78\)\( T_{11}^{65} + \)\(11\!\cdots\!14\)\( T_{11}^{64} + \)\(17\!\cdots\!92\)\( T_{11}^{63} - \)\(25\!\cdots\!80\)\( T_{11}^{62} - \)\(46\!\cdots\!52\)\( T_{11}^{61} + \)\(52\!\cdots\!57\)\( T_{11}^{60} + \)\(10\!\cdots\!18\)\( T_{11}^{59} - \)\(92\!\cdots\!84\)\( T_{11}^{58} - \)\(22\!\cdots\!02\)\( T_{11}^{57} + \)\(14\!\cdots\!27\)\( T_{11}^{56} + \)\(39\!\cdots\!82\)\( T_{11}^{55} - \)\(18\!\cdots\!41\)\( T_{11}^{54} - \)\(60\!\cdots\!86\)\( T_{11}^{53} + \)\(21\!\cdots\!77\)\( T_{11}^{52} + \)\(81\!\cdots\!28\)\( T_{11}^{51} - \)\(21\!\cdots\!66\)\( T_{11}^{50} - \)\(95\!\cdots\!20\)\( T_{11}^{49} + \)\(16\!\cdots\!53\)\( T_{11}^{48} + \)\(98\!\cdots\!80\)\( T_{11}^{47} - \)\(10\!\cdots\!02\)\( T_{11}^{46} - \)\(86\!\cdots\!96\)\( T_{11}^{45} + \)\(43\!\cdots\!02\)\( T_{11}^{44} + \)\(66\!\cdots\!82\)\( T_{11}^{43} + \)\(10\!\cdots\!16\)\( T_{11}^{42} - \)\(43\!\cdots\!86\)\( T_{11}^{41} - \)\(21\!\cdots\!99\)\( T_{11}^{40} + \)\(24\!\cdots\!60\)\( T_{11}^{39} + \)\(23\!\cdots\!81\)\( T_{11}^{38} - \)\(11\!\cdots\!08\)\( T_{11}^{37} - \)\(16\!\cdots\!90\)\( T_{11}^{36} + \)\(40\!\cdots\!56\)\( T_{11}^{35} + \)\(91\!\cdots\!51\)\( T_{11}^{34} - \)\(11\!\cdots\!00\)\( T_{11}^{33} - \)\(39\!\cdots\!14\)\( T_{11}^{32} + \)\(17\!\cdots\!36\)\( T_{11}^{31} + \)\(13\!\cdots\!77\)\( T_{11}^{30} + \)\(20\!\cdots\!80\)\( T_{11}^{29} - \)\(35\!\cdots\!72\)\( T_{11}^{28} - \)\(24\!\cdots\!80\)\( T_{11}^{27} + \)\(69\!\cdots\!15\)\( T_{11}^{26} + \)\(88\!\cdots\!48\)\( T_{11}^{25} - \)\(88\!\cdots\!15\)\( T_{11}^{24} - \)\(20\!\cdots\!76\)\( T_{11}^{23} + \)\(37\!\cdots\!09\)\( T_{11}^{22} + \)\(33\!\cdots\!72\)\( T_{11}^{21} + \)\(12\!\cdots\!28\)\( T_{11}^{20} - \)\(33\!\cdots\!10\)\( T_{11}^{19} - \)\(29\!\cdots\!71\)\( T_{11}^{18} + \)\(20\!\cdots\!86\)\( T_{11}^{17} + \)\(35\!\cdots\!45\)\( T_{11}^{16} + \)\(90\!\cdots\!32\)\( T_{11}^{15} - \)\(23\!\cdots\!38\)\( T_{11}^{14} - \)\(86\!\cdots\!52\)\( T_{11}^{13} + \)\(97\!\cdots\!87\)\( T_{11}^{12} + \)\(77\!\cdots\!68\)\( T_{11}^{11} - \)\(11\!\cdots\!30\)\( T_{11}^{10} - \)\(28\!\cdots\!44\)\( T_{11}^{9} - \)\(34\!\cdots\!27\)\( T_{11}^{8} + \)\(72\!\cdots\!98\)\( T_{11}^{7} + \)\(30\!\cdots\!04\)\( T_{11}^{6} - \)\(34\!\cdots\!70\)\( T_{11}^{5} - \)\(41\!\cdots\!21\)\( T_{11}^{4} - \)\(11\!\cdots\!02\)\( T_{11}^{3} + \)\(47\!\cdots\!83\)\( T_{11}^{2} + \)\(13\!\cdots\!02\)\( T_{11} + \)\(12\!\cdots\!49\)\( \)">\(T_{11}^{84} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\).