Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [9072,2,Mod(1,9072)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9072.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9072, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9072.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,3,0,-3,0,0,0,-6,0,-3,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(72.4402847137\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.87939\) of defining polynomial
Character \(\chi\) \(=\) 9072.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.879385 q^{5} -1.00000 q^{7} -3.87939 q^{11} -5.45336 q^{13} +1.65270 q^{17} -2.41147 q^{19} -3.16250 q^{23} -4.22668 q^{25} -6.04963 q^{29} +4.55438 q^{31} +0.879385 q^{35} -4.55438 q^{37} -1.18479 q^{41} -0.184793 q^{43} +1.02229 q^{47} +1.00000 q^{49} +7.29086 q^{53} +3.41147 q^{55} -6.66044 q^{59} -2.59627 q^{61} +4.79561 q^{65} +2.95811 q^{67} +3.68004 q^{71} -12.7811 q^{73} +3.87939 q^{77} +5.95811 q^{79} +0.218941 q^{83} -1.45336 q^{85} +11.0273 q^{89} +5.45336 q^{91} +2.12061 q^{95} +12.5030 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7} - 6 q^{11} - 3 q^{13} + 6 q^{17} + 3 q^{19} - 12 q^{23} - 6 q^{25} + 9 q^{29} + 3 q^{31} - 3 q^{35} - 3 q^{37} + 3 q^{43} - 3 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} + 6 q^{61} + 15 q^{65}+ \cdots - 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.879385 −0.393273 −0.196637 − 0.980476i \(-0.563002\pi\)
−0.196637 + 0.980476i \(0.563002\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.87939 −1.16968 −0.584839 − 0.811149i \(-0.698842\pi\)
−0.584839 + 0.811149i \(0.698842\pi\)
\(12\) 0 0
\(13\) −5.45336 −1.51249 −0.756245 − 0.654288i \(-0.772968\pi\)
−0.756245 + 0.654288i \(0.772968\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.65270 0.400840 0.200420 − 0.979710i \(-0.435769\pi\)
0.200420 + 0.979710i \(0.435769\pi\)
\(18\) 0 0
\(19\) −2.41147 −0.553230 −0.276615 − 0.960981i \(-0.589213\pi\)
−0.276615 + 0.960981i \(0.589213\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.16250 −0.659428 −0.329714 − 0.944081i \(-0.606952\pi\)
−0.329714 + 0.944081i \(0.606952\pi\)
\(24\) 0 0
\(25\) −4.22668 −0.845336
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.04963 −1.12339 −0.561694 − 0.827345i \(-0.689850\pi\)
−0.561694 + 0.827345i \(0.689850\pi\)
\(30\) 0 0
\(31\) 4.55438 0.817990 0.408995 − 0.912537i \(-0.365879\pi\)
0.408995 + 0.912537i \(0.365879\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.879385 0.148643
\(36\) 0 0
\(37\) −4.55438 −0.748735 −0.374368 − 0.927280i \(-0.622140\pi\)
−0.374368 + 0.927280i \(0.622140\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.18479 −0.185034 −0.0925168 − 0.995711i \(-0.529491\pi\)
−0.0925168 + 0.995711i \(0.529491\pi\)
\(42\) 0 0
\(43\) −0.184793 −0.0281806 −0.0140903 − 0.999901i \(-0.504485\pi\)
−0.0140903 + 0.999901i \(0.504485\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.02229 0.149116 0.0745581 − 0.997217i \(-0.476245\pi\)
0.0745581 + 0.997217i \(0.476245\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.29086 1.00148 0.500738 − 0.865599i \(-0.333062\pi\)
0.500738 + 0.865599i \(0.333062\pi\)
\(54\) 0 0
\(55\) 3.41147 0.460003
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.66044 −0.867116 −0.433558 − 0.901126i \(-0.642742\pi\)
−0.433558 + 0.901126i \(0.642742\pi\)
\(60\) 0 0
\(61\) −2.59627 −0.332418 −0.166209 − 0.986091i \(-0.553153\pi\)
−0.166209 + 0.986091i \(0.553153\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.79561 0.594822
\(66\) 0 0
\(67\) 2.95811 0.361391 0.180695 − 0.983539i \(-0.442165\pi\)
0.180695 + 0.983539i \(0.442165\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.68004 0.436741 0.218370 − 0.975866i \(-0.429926\pi\)
0.218370 + 0.975866i \(0.429926\pi\)
\(72\) 0 0
\(73\) −12.7811 −1.49591 −0.747955 − 0.663750i \(-0.768964\pi\)
−0.747955 + 0.663750i \(0.768964\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.87939 0.442097
\(78\) 0 0
\(79\) 5.95811 0.670340 0.335170 − 0.942158i \(-0.391206\pi\)
0.335170 + 0.942158i \(0.391206\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.218941 0.0240319 0.0120159 − 0.999928i \(-0.496175\pi\)
0.0120159 + 0.999928i \(0.496175\pi\)
\(84\) 0 0
\(85\) −1.45336 −0.157639
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.0273 1.16890 0.584448 − 0.811431i \(-0.301311\pi\)
0.584448 + 0.811431i \(0.301311\pi\)
\(90\) 0 0
\(91\) 5.45336 0.571668
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.12061 0.217570
\(96\) 0 0
\(97\) 12.5030 1.26949 0.634743 − 0.772723i \(-0.281106\pi\)
0.634743 + 0.772723i \(0.281106\pi\)
\(98\) 0 0
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 9072.2.a.ca.1.1 3
3.2 odd 2 9072.2.a.bs.1.3 3
4.3 odd 2 567.2.a.h.1.3 3
9.2 odd 6 3024.2.r.k.1009.1 6
9.4 even 3 1008.2.r.h.673.3 6
9.5 odd 6 3024.2.r.k.2017.1 6
9.7 even 3 1008.2.r.h.337.3 6
12.11 even 2 567.2.a.c.1.1 3
28.27 even 2 3969.2.a.q.1.3 3
36.7 odd 6 63.2.f.a.22.1 ✓ 6
36.11 even 6 189.2.f.b.64.3 6
36.23 even 6 189.2.f.b.127.3 6
36.31 odd 6 63.2.f.a.43.1 yes 6
84.83 odd 2 3969.2.a.l.1.1 3
252.11 even 6 1323.2.h.c.226.1 6
252.23 even 6 1323.2.h.c.802.1 6
252.31 even 6 441.2.g.b.79.1 6
252.47 odd 6 1323.2.g.e.361.3 6
252.59 odd 6 1323.2.g.e.667.3 6
252.67 odd 6 441.2.g.c.79.1 6
252.79 odd 6 441.2.g.c.67.1 6
252.83 odd 6 1323.2.f.d.442.3 6
252.95 even 6 1323.2.g.d.667.3 6
252.103 even 6 441.2.h.e.214.3 6
252.115 even 6 441.2.h.e.373.3 6
252.131 odd 6 1323.2.h.b.802.1 6
252.139 even 6 441.2.f.c.295.1 6
252.151 odd 6 441.2.h.d.373.3 6
252.167 odd 6 1323.2.f.d.883.3 6
252.187 even 6 441.2.g.b.67.1 6
252.191 even 6 1323.2.g.d.361.3 6
252.223 even 6 441.2.f.c.148.1 6
252.227 odd 6 1323.2.h.b.226.1 6
252.247 odd 6 441.2.h.d.214.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.1 ✓ 6 36.7 odd 6
63.2.f.a.43.1 yes 6 36.31 odd 6
189.2.f.b.64.3 6 36.11 even 6
189.2.f.b.127.3 6 36.23 even 6
441.2.f.c.148.1 6 252.223 even 6
441.2.f.c.295.1 6 252.139 even 6
441.2.g.b.67.1 6 252.187 even 6
441.2.g.b.79.1 6 252.31 even 6
441.2.g.c.67.1 6 252.79 odd 6
441.2.g.c.79.1 6 252.67 odd 6
441.2.h.d.214.3 6 252.247 odd 6
441.2.h.d.373.3 6 252.151 odd 6
441.2.h.e.214.3 6 252.103 even 6
441.2.h.e.373.3 6 252.115 even 6
567.2.a.c.1.1 3 12.11 even 2
567.2.a.h.1.3 3 4.3 odd 2
1008.2.r.h.337.3 6 9.7 even 3
1008.2.r.h.673.3 6 9.4 even 3
1323.2.f.d.442.3 6 252.83 odd 6
1323.2.f.d.883.3 6 252.167 odd 6
1323.2.g.d.361.3 6 252.191 even 6
1323.2.g.d.667.3 6 252.95 even 6
1323.2.g.e.361.3 6 252.47 odd 6
1323.2.g.e.667.3 6 252.59 odd 6
1323.2.h.b.226.1 6 252.227 odd 6
1323.2.h.b.802.1 6 252.131 odd 6
1323.2.h.c.226.1 6 252.11 even 6
1323.2.h.c.802.1 6 252.23 even 6
3024.2.r.k.1009.1 6 9.2 odd 6
3024.2.r.k.2017.1 6 9.5 odd 6
3969.2.a.l.1.1 3 84.83 odd 2
3969.2.a.q.1.3 3 28.27 even 2
9072.2.a.bs.1.3 3 3.2 odd 2
9072.2.a.ca.1.1 3 1.1 even 1 trivial