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Properties

Label 9072.2.a.bs.1.2
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(1\)
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 1.2
Root \(-0.347296\) 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)
 
\(f(q)\) \(=\) \(q-1.34730 q^{5} -1.00000 q^{7} +1.65270 q^{11} -3.36959 q^{13} -0.467911 q^{17} +3.22668 q^{19} +8.94356 q^{23} -3.18479 q^{25} -6.26857 q^{29} -9.23442 q^{31} +1.34730 q^{35} +9.23442 q^{37} -3.41147 q^{41} +4.41147 q^{43} +9.35504 q^{47} +1.00000 q^{49} +0.573978 q^{53} -2.22668 q^{55} -10.3969 q^{59} +7.63816 q^{61} +4.53983 q^{65} -0.596267 q^{67} -0.554378 q^{71} +2.04963 q^{73} -1.65270 q^{77} +2.40373 q^{79} -15.0496 q^{83} +0.630415 q^{85} -9.08647 q^{89} +3.36959 q^{91} -4.34730 q^{95} -1.89899 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 3 q^{7} + 6 q^{11} - 3 q^{13} - 6 q^{17} + 3 q^{19} + 12 q^{23} - 6 q^{25} - 9 q^{29} + 3 q^{31} + 3 q^{35} - 3 q^{37} + 3 q^{43} + 3 q^{47} + 3 q^{49} - 6 q^{53} - 3 q^{59} + 6 q^{61} - 15 q^{65}+ \cdots - 3 q^{97}+O(q^{100}) \) 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
\(5\) −1.34730 −0.602529 −0.301265 − 0.953541i \(-0.597409\pi\)
−0.301265 + 0.953541i \(0.597409\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.65270 0.498309 0.249154 − 0.968464i \(-0.419847\pi\)
0.249154 + 0.968464i \(0.419847\pi\)
\(12\) 0 0
\(13\) −3.36959 −0.934555 −0.467277 − 0.884111i \(-0.654765\pi\)
−0.467277 + 0.884111i \(0.654765\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.467911 −0.113485 −0.0567426 − 0.998389i \(-0.518071\pi\)
−0.0567426 + 0.998389i \(0.518071\pi\)
\(18\) 0 0
\(19\) 3.22668 0.740252 0.370126 − 0.928982i \(-0.379315\pi\)
0.370126 + 0.928982i \(0.379315\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.94356 1.86486 0.932431 − 0.361348i \(-0.117683\pi\)
0.932431 + 0.361348i \(0.117683\pi\)
\(24\) 0 0
\(25\) −3.18479 −0.636959
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.26857 −1.16404 −0.582022 − 0.813173i \(-0.697738\pi\)
−0.582022 + 0.813173i \(0.697738\pi\)
\(30\) 0 0
\(31\) −9.23442 −1.65855 −0.829276 − 0.558840i \(-0.811247\pi\)
−0.829276 + 0.558840i \(0.811247\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.34730 0.227735
\(36\) 0 0
\(37\) 9.23442 1.51813 0.759065 − 0.651015i \(-0.225657\pi\)
0.759065 + 0.651015i \(0.225657\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.41147 −0.532783 −0.266391 − 0.963865i \(-0.585831\pi\)
−0.266391 + 0.963865i \(0.585831\pi\)
\(42\) 0 0
\(43\) 4.41147 0.672743 0.336372 − 0.941729i \(-0.390800\pi\)
0.336372 + 0.941729i \(0.390800\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.35504 1.36457 0.682286 − 0.731085i \(-0.260986\pi\)
0.682286 + 0.731085i \(0.260986\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.573978 0.0788419 0.0394210 − 0.999223i \(-0.487449\pi\)
0.0394210 + 0.999223i \(0.487449\pi\)
\(54\) 0 0
\(55\) −2.22668 −0.300246
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.3969 −1.35356 −0.676782 − 0.736183i \(-0.736626\pi\)
−0.676782 + 0.736183i \(0.736626\pi\)
\(60\) 0 0
\(61\) 7.63816 0.977966 0.488983 − 0.872293i \(-0.337368\pi\)
0.488983 + 0.872293i \(0.337368\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.53983 0.563097
\(66\) 0 0
\(67\) −0.596267 −0.0728456 −0.0364228 − 0.999336i \(-0.511596\pi\)
−0.0364228 + 0.999336i \(0.511596\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.554378 −0.0657925 −0.0328963 − 0.999459i \(-0.510473\pi\)
−0.0328963 + 0.999459i \(0.510473\pi\)
\(72\) 0 0
\(73\) 2.04963 0.239891 0.119946 − 0.992780i \(-0.461728\pi\)
0.119946 + 0.992780i \(0.461728\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.65270 −0.188343
\(78\) 0 0
\(79\) 2.40373 0.270441 0.135221 − 0.990816i \(-0.456826\pi\)
0.135221 + 0.990816i \(0.456826\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.0496 −1.65191 −0.825956 − 0.563735i \(-0.809364\pi\)
−0.825956 + 0.563735i \(0.809364\pi\)
\(84\) 0 0
\(85\) 0.630415 0.0683781
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.08647 −0.963164 −0.481582 − 0.876401i \(-0.659938\pi\)
−0.481582 + 0.876401i \(0.659938\pi\)
\(90\) 0 0
\(91\) 3.36959 0.353228
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.34730 −0.446023
\(96\) 0 0
\(97\) −1.89899 −0.192813 −0.0964064 − 0.995342i \(-0.530735\pi\)
−0.0964064 + 0.995342i \(0.530735\pi\)
\(98\) 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.bs.1.2 3
3.2 odd 2 9072.2.a.ca.1.2 3
4.3 odd 2 567.2.a.c.1.3 3
9.2 odd 6 1008.2.r.h.337.2 6
9.4 even 3 3024.2.r.k.2017.2 6
9.5 odd 6 1008.2.r.h.673.2 6
9.7 even 3 3024.2.r.k.1009.2 6
12.11 even 2 567.2.a.h.1.1 3
28.27 even 2 3969.2.a.l.1.3 3
36.7 odd 6 189.2.f.b.64.1 6
36.11 even 6 63.2.f.a.22.3 ✓ 6
36.23 even 6 63.2.f.a.43.3 yes 6
36.31 odd 6 189.2.f.b.127.1 6
84.83 odd 2 3969.2.a.q.1.1 3
252.11 even 6 441.2.h.d.373.1 6
252.23 even 6 441.2.h.d.214.1 6
252.31 even 6 1323.2.g.e.667.1 6
252.47 odd 6 441.2.g.b.67.3 6
252.59 odd 6 441.2.g.b.79.3 6
252.67 odd 6 1323.2.g.d.667.1 6
252.79 odd 6 1323.2.g.d.361.1 6
252.83 odd 6 441.2.f.c.148.3 6
252.95 even 6 441.2.g.c.79.3 6
252.103 even 6 1323.2.h.b.802.3 6
252.115 even 6 1323.2.h.b.226.3 6
252.131 odd 6 441.2.h.e.214.1 6
252.139 even 6 1323.2.f.d.883.1 6
252.151 odd 6 1323.2.h.c.226.3 6
252.167 odd 6 441.2.f.c.295.3 6
252.187 even 6 1323.2.g.e.361.1 6
252.191 even 6 441.2.g.c.67.3 6
252.223 even 6 1323.2.f.d.442.1 6
252.227 odd 6 441.2.h.e.373.1 6
252.247 odd 6 1323.2.h.c.802.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.3 ✓ 6 36.11 even 6
63.2.f.a.43.3 yes 6 36.23 even 6
189.2.f.b.64.1 6 36.7 odd 6
189.2.f.b.127.1 6 36.31 odd 6
441.2.f.c.148.3 6 252.83 odd 6
441.2.f.c.295.3 6 252.167 odd 6
441.2.g.b.67.3 6 252.47 odd 6
441.2.g.b.79.3 6 252.59 odd 6
441.2.g.c.67.3 6 252.191 even 6
441.2.g.c.79.3 6 252.95 even 6
441.2.h.d.214.1 6 252.23 even 6
441.2.h.d.373.1 6 252.11 even 6
441.2.h.e.214.1 6 252.131 odd 6
441.2.h.e.373.1 6 252.227 odd 6
567.2.a.c.1.3 3 4.3 odd 2
567.2.a.h.1.1 3 12.11 even 2
1008.2.r.h.337.2 6 9.2 odd 6
1008.2.r.h.673.2 6 9.5 odd 6
1323.2.f.d.442.1 6 252.223 even 6
1323.2.f.d.883.1 6 252.139 even 6
1323.2.g.d.361.1 6 252.79 odd 6
1323.2.g.d.667.1 6 252.67 odd 6
1323.2.g.e.361.1 6 252.187 even 6
1323.2.g.e.667.1 6 252.31 even 6
1323.2.h.b.226.3 6 252.115 even 6
1323.2.h.b.802.3 6 252.103 even 6
1323.2.h.c.226.3 6 252.151 odd 6
1323.2.h.c.802.3 6 252.247 odd 6
3024.2.r.k.1009.2 6 9.7 even 3
3024.2.r.k.2017.2 6 9.4 even 3
3969.2.a.l.1.3 3 28.27 even 2
3969.2.a.q.1.1 3 84.83 odd 2
9072.2.a.bs.1.2 3 1.1 even 1 trivial
9072.2.a.ca.1.2 3 3.2 odd 2