Newform orbit 570.2.bb.a

• Label: 570.2.bb.a
• 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.

\(\operatorname{Tr}(f)(q) = \) \( 84q - 6q^{3} - 42q^{8} + 6q^{9} + O(q^{10}) \)

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	\( a_{2} \)			\( a_{3} \)			\( a_{4} \)			\( a_{5} \)			\( a_{6} \)			\( a_{7} \)			\( a_{8} \)			\( a_{9} \)			\( a_{10} \)
41.1	0.766044	+	0.642788i	−1.71569	+	0.237506i	0.173648	+	0.984808i	−0.984808	−	0.173648i	−1.46696	−	0.920884i	−2.15391	−	3.73068i	−0.500000	+	0.866025i	2.88718	−	0.814973i	−0.642788	−	0.766044i
41.2	0.766044	+	0.642788i	−1.65947	+	0.496132i	0.173648	+	0.984808i	0.984808	+	0.173648i	−1.59014	−	0.686630i	1.07093	+	1.85491i	−0.500000	+	0.866025i	2.50771	−	1.64664i	0.642788	+	0.766044i
41.3	0.766044	+	0.642788i	−1.64147	−	0.552780i	0.173648	+	0.984808i	0.984808	+	0.173648i	−0.902121	−	1.47857i	−1.68197	−	2.91326i	−0.500000	+	0.866025i	2.38887	+	1.81475i	0.642788	+	0.766044i
41.4	0.766044	+	0.642788i	−1.63627	+	0.567997i	0.173648	+	0.984808i	−0.984808	−	0.173648i	−1.61856	−	0.616663i	1.59792	+	2.76767i	−0.500000	+	0.866025i	2.35476	−	1.85879i	−0.642788	−	0.766044i
41.5	0.766044	+	0.642788i	−0.978339	−	1.42928i	0.173648	+	0.984808i	0.984808	+	0.173648i	0.169275	−	1.72376i	1.36690	+	2.36755i	−0.500000	+	0.866025i	−1.08571	+	2.79665i	0.642788	+	0.766044i
41.6	0.766044	+	0.642788i	−0.565043	+	1.63729i	0.173648	+	0.984808i	−0.984808	−	0.173648i	−1.48528	+	0.891036i	−2.04901	−	3.54899i	−0.500000	+	0.866025i	−2.36145	−	1.85028i	−0.642788	−	0.766044i
41.7	0.766044	+	0.642788i	0.0562681	−	1.73114i	0.173648	+	0.984808i	−0.984808	−	0.173648i	1.15586	−	1.28996i	−0.561471	−	0.972497i	−0.500000	+	0.866025i	−2.99367	−	0.194816i	−0.642788	−	0.766044i
41.8	0.766044	+	0.642788i	0.185551	+	1.72208i	0.173648	+	0.984808i	0.984808	+	0.173648i	−0.964794	+	1.43846i	−1.20034	−	2.07905i	−0.500000	+	0.866025i	−2.93114	+	0.639068i	0.642788	+	0.766044i
41.9	0.766044	+	0.642788i	0.605893	−	1.62262i	0.173648	+	0.984808i	−0.984808	−	0.173648i	1.50714	−	0.853538i	1.94656	+	3.37154i	−0.500000	+	0.866025i	−2.26579	−	1.96627i	−0.642788	−	0.766044i
41.10	0.766044	+	0.642788i	0.733933	+	1.56887i	0.173648	+	0.984808i	0.984808	+	0.173648i	−0.446223	+	1.67358i	2.13041	+	3.68998i	−0.500000	+	0.866025i	−1.92269	+	2.30289i	0.642788	+	0.766044i
41.11	0.766044	+	0.642788i	1.07402	+	1.35885i	0.173648	+	0.984808i	−0.984808	−	0.173648i	−0.0507095	+	1.73131i	0.847833	+	1.46849i	−0.500000	+	0.866025i	−0.692971	+	2.91887i	−0.642788	−	0.766044i
41.12	0.766044	+	0.642788i	1.16246	−	1.28401i	0.173648	+	0.984808i	0.984808	+	0.173648i	1.71584	−	0.236392i	−1.76528	−	3.05755i	−0.500000	+	0.866025i	−0.297363	−	2.98523i	0.642788	+	0.766044i
41.13	0.766044	+	0.642788i	1.68082	+	0.418131i	0.173648	+	0.984808i	−0.984808	−	0.173648i	1.01882	+	1.40072i	0.0247885	+	0.0429350i	−0.500000	+	0.866025i	2.65033	+	1.40561i	−0.642788	−	0.766044i
41.14	0.766044	+	0.642788i	1.69734	+	0.345017i	0.173648	+	0.984808i	0.984808	+	0.173648i	1.07847	+	1.35533i	−0.267952	−	0.464106i	−0.500000	+	0.866025i	2.76193	+	1.17122i	0.642788	+	0.766044i
71.1	−0.939693	+	0.342020i	−1.72889	−	0.104526i	0.766044	−	0.642788i	−0.642788	+	0.766044i	1.66038	−	0.493094i	0.222334	+	0.385093i	−0.500000	+	0.866025i	2.97815	+	0.361430i	0.342020	−	0.939693i
71.2	−0.939693	+	0.342020i	−1.70290	+	0.316445i	0.766044	−	0.642788i	0.642788	−	0.766044i	1.49197	−	0.879786i	−1.40312	−	2.43028i	−0.500000	+	0.866025i	2.79973	−	1.07775i	−0.342020	+	0.939693i
71.3	−0.939693	+	0.342020i	−1.50842	+	0.851282i	0.766044	−	0.642788i	0.642788	−	0.766044i	1.12629	−	1.31585i	2.21271	+	3.83252i	−0.500000	+	0.866025i	1.55064	−	2.56818i	−0.342020	+	0.939693i
71.4	−0.939693	+	0.342020i	−1.33410	−	1.10462i	0.766044	−	0.642788i	0.642788	−	0.766044i	1.63144	+	0.581714i	−0.223800	−	0.387633i	−0.500000	+	0.866025i	0.559632	+	2.94734i	−0.342020	+	0.939693i
71.5	−0.939693	+	0.342020i	−0.939713	+	1.45497i	0.766044	−	0.642788i	−0.642788	+	0.766044i	0.385412	−	1.68863i	−1.27478	−	2.20798i	−0.500000	+	0.866025i	−1.23388	−	2.73451i	0.342020	−	0.939693i
71.6	−0.939693	+	0.342020i	−0.398282	−	1.68564i	0.766044	−	0.642788i	−0.642788	+	0.766044i	0.950784	+	1.44776i	−2.26841	−	3.92899i	−0.500000	+	0.866025i	−2.68274	+	1.34272i	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.a	✓	84
3.b	odd	2	1		570.2.bb.b	yes	84
19.f	odd	18	1		570.2.bb.b	yes	84
57.j	even	18	1	inner	570.2.bb.a	✓	84

    

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

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 \(S_{2}^{\mathrm{new}}(570, [\chi])\).