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Properties

Label 9072.2.a.ca.1.3
Level $9072$
Weight $2$
Character 9072.1
Self dual yes
Analytic conductor $72.440$
Analytic rank $0$
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: \(0\)
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 1.3
Root \(-1.53209\) 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)
 
\(f(q)\) \(=\) \(q+2.53209 q^{5} -1.00000 q^{7} -0.467911 q^{11} +5.82295 q^{13} +3.87939 q^{17} +2.18479 q^{19} +0.106067 q^{23} +1.41147 q^{25} +8.78106 q^{29} +7.68004 q^{31} -2.53209 q^{35} -7.68004 q^{37} -2.22668 q^{41} -1.22668 q^{43} +5.33275 q^{47} +1.00000 q^{49} -0.716881 q^{53} -1.18479 q^{55} -0.736482 q^{59} +0.958111 q^{61} +14.7442 q^{65} +9.63816 q^{67} -13.2344 q^{71} -10.2686 q^{73} +0.467911 q^{77} +12.6382 q^{79} +2.73143 q^{83} +9.82295 q^{85} -8.11381 q^{89} -5.82295 q^{91} +5.53209 q^{95} -13.6040 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7} - 6 q^{11} - 3 q^{13} + 6 q^{17} + 3 q^{19} - 12 q^{23} - 6 q^{25} + 9 q^{29} + 3 q^{31} - 3 q^{35} - 3 q^{37} + 3 q^{43} - 3 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} + 6 q^{61} + 15 q^{65}+ \cdots - 3 q^{97}+O(q^{100}) \) 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
\(5\) 2.53209 1.13238 0.566192 − 0.824273i \(-0.308416\pi\)
0.566192 + 0.824273i \(0.308416\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.467911 −0.141081 −0.0705403 − 0.997509i \(-0.522472\pi\)
−0.0705403 + 0.997509i \(0.522472\pi\)
\(12\) 0 0
\(13\) 5.82295 1.61500 0.807498 − 0.589871i \(-0.200821\pi\)
0.807498 + 0.589871i \(0.200821\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.87939 0.940889 0.470445 − 0.882430i \(-0.344094\pi\)
0.470445 + 0.882430i \(0.344094\pi\)
\(18\) 0 0
\(19\) 2.18479 0.501226 0.250613 − 0.968087i \(-0.419368\pi\)
0.250613 + 0.968087i \(0.419368\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.106067 0.0221165 0.0110582 − 0.999939i \(-0.496480\pi\)
0.0110582 + 0.999939i \(0.496480\pi\)
\(24\) 0 0
\(25\) 1.41147 0.282295
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.78106 1.63060 0.815301 − 0.579038i \(-0.196572\pi\)
0.815301 + 0.579038i \(0.196572\pi\)
\(30\) 0 0
\(31\) 7.68004 1.37938 0.689688 − 0.724106i \(-0.257748\pi\)
0.689688 + 0.724106i \(0.257748\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.53209 −0.428001
\(36\) 0 0
\(37\) −7.68004 −1.26259 −0.631296 − 0.775542i \(-0.717477\pi\)
−0.631296 + 0.775542i \(0.717477\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.22668 −0.347749 −0.173875 − 0.984768i \(-0.555629\pi\)
−0.173875 + 0.984768i \(0.555629\pi\)
\(42\) 0 0
\(43\) −1.22668 −0.187067 −0.0935336 − 0.995616i \(-0.529816\pi\)
−0.0935336 + 0.995616i \(0.529816\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.33275 0.777861 0.388931 − 0.921267i \(-0.372845\pi\)
0.388931 + 0.921267i \(0.372845\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.716881 −0.0984712 −0.0492356 − 0.998787i \(-0.515679\pi\)
−0.0492356 + 0.998787i \(0.515679\pi\)
\(54\) 0 0
\(55\) −1.18479 −0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.736482 −0.0958818 −0.0479409 − 0.998850i \(-0.515266\pi\)
−0.0479409 + 0.998850i \(0.515266\pi\)
\(60\) 0 0
\(61\) 0.958111 0.122674 0.0613368 − 0.998117i \(-0.480464\pi\)
0.0613368 + 0.998117i \(0.480464\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.7442 1.82880
\(66\) 0 0
\(67\) 9.63816 1.17749 0.588744 − 0.808320i \(-0.299623\pi\)
0.588744 + 0.808320i \(0.299623\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.2344 −1.57064 −0.785318 − 0.619092i \(-0.787501\pi\)
−0.785318 + 0.619092i \(0.787501\pi\)
\(72\) 0 0
\(73\) −10.2686 −1.20185 −0.600923 − 0.799307i \(-0.705200\pi\)
−0.600923 + 0.799307i \(0.705200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.467911 0.0533234
\(78\) 0 0
\(79\) 12.6382 1.42190 0.710952 − 0.703241i \(-0.248264\pi\)
0.710952 + 0.703241i \(0.248264\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.73143 0.299813 0.149907 − 0.988700i \(-0.452103\pi\)
0.149907 + 0.988700i \(0.452103\pi\)
\(84\) 0 0
\(85\) 9.82295 1.06545
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.11381 −0.860062 −0.430031 − 0.902814i \(-0.641497\pi\)
−0.430031 + 0.902814i \(0.641497\pi\)
\(90\) 0 0
\(91\) −5.82295 −0.610411
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.53209 0.567580
\(96\) 0 0
\(97\) −13.6040 −1.38128 −0.690639 − 0.723200i \(-0.742671\pi\)
−0.690639 + 0.723200i \(0.742671\pi\)
\(98\) 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.1.3 3
3.2 odd 2 9072.2.a.bs.1.1 3
4.3 odd 2 567.2.a.h.1.2 3
9.2 odd 6 3024.2.r.k.1009.3 6
9.4 even 3 1008.2.r.h.673.1 6
9.5 odd 6 3024.2.r.k.2017.3 6
9.7 even 3 1008.2.r.h.337.1 6
12.11 even 2 567.2.a.c.1.2 3
28.27 even 2 3969.2.a.q.1.2 3
36.7 odd 6 63.2.f.a.22.2 ✓ 6
36.11 even 6 189.2.f.b.64.2 6
36.23 even 6 189.2.f.b.127.2 6
36.31 odd 6 63.2.f.a.43.2 yes 6
84.83 odd 2 3969.2.a.l.1.2 3
252.11 even 6 1323.2.h.c.226.2 6
252.23 even 6 1323.2.h.c.802.2 6
252.31 even 6 441.2.g.b.79.2 6
252.47 odd 6 1323.2.g.e.361.2 6
252.59 odd 6 1323.2.g.e.667.2 6
252.67 odd 6 441.2.g.c.79.2 6
252.79 odd 6 441.2.g.c.67.2 6
252.83 odd 6 1323.2.f.d.442.2 6
252.95 even 6 1323.2.g.d.667.2 6
252.103 even 6 441.2.h.e.214.2 6
252.115 even 6 441.2.h.e.373.2 6
252.131 odd 6 1323.2.h.b.802.2 6
252.139 even 6 441.2.f.c.295.2 6
252.151 odd 6 441.2.h.d.373.2 6
252.167 odd 6 1323.2.f.d.883.2 6
252.187 even 6 441.2.g.b.67.2 6
252.191 even 6 1323.2.g.d.361.2 6
252.223 even 6 441.2.f.c.148.2 6
252.227 odd 6 1323.2.h.b.226.2 6
252.247 odd 6 441.2.h.d.214.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.a.22.2 ✓ 6 36.7 odd 6
63.2.f.a.43.2 yes 6 36.31 odd 6
189.2.f.b.64.2 6 36.11 even 6
189.2.f.b.127.2 6 36.23 even 6
441.2.f.c.148.2 6 252.223 even 6
441.2.f.c.295.2 6 252.139 even 6
441.2.g.b.67.2 6 252.187 even 6
441.2.g.b.79.2 6 252.31 even 6
441.2.g.c.67.2 6 252.79 odd 6
441.2.g.c.79.2 6 252.67 odd 6
441.2.h.d.214.2 6 252.247 odd 6
441.2.h.d.373.2 6 252.151 odd 6
441.2.h.e.214.2 6 252.103 even 6
441.2.h.e.373.2 6 252.115 even 6
567.2.a.c.1.2 3 12.11 even 2
567.2.a.h.1.2 3 4.3 odd 2
1008.2.r.h.337.1 6 9.7 even 3
1008.2.r.h.673.1 6 9.4 even 3
1323.2.f.d.442.2 6 252.83 odd 6
1323.2.f.d.883.2 6 252.167 odd 6
1323.2.g.d.361.2 6 252.191 even 6
1323.2.g.d.667.2 6 252.95 even 6
1323.2.g.e.361.2 6 252.47 odd 6
1323.2.g.e.667.2 6 252.59 odd 6
1323.2.h.b.226.2 6 252.227 odd 6
1323.2.h.b.802.2 6 252.131 odd 6
1323.2.h.c.226.2 6 252.11 even 6
1323.2.h.c.802.2 6 252.23 even 6
3024.2.r.k.1009.3 6 9.2 odd 6
3024.2.r.k.2017.3 6 9.5 odd 6
3969.2.a.l.1.2 3 84.83 odd 2
3969.2.a.q.1.2 3 28.27 even 2
9072.2.a.bs.1.1 3 3.2 odd 2
9072.2.a.ca.1.3 3 1.1 even 1 trivial