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Properties

Label 9072.2.a.bq.1.3
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,-5,0,3,0,0,0,2,0,3,0,0,0,-12] 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: 3.3.321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 1 \) 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 \(0.239123\) 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)
 
\(f(q)\) \(=\) \(q+1.18194 q^{5} +1.00000 q^{7} +3.70370 q^{11} +1.00000 q^{13} -6.94282 q^{17} -1.94282 q^{19} +5.60301 q^{23} -3.60301 q^{25} +0.239123 q^{29} -1.66019 q^{31} +1.18194 q^{35} -9.54583 q^{37} -10.1819 q^{41} -2.22545 q^{43} -5.82846 q^{47} +1.00000 q^{49} -11.6030 q^{53} +4.37756 q^{55} -2.60301 q^{59} -7.60301 q^{61} +1.18194 q^{65} -3.50808 q^{67} -8.60301 q^{71} +15.1488 q^{73} +3.70370 q^{77} -7.37756 q^{79} +6.94282 q^{83} -8.20602 q^{85} +2.74720 q^{89} +1.00000 q^{91} -2.29630 q^{95} +7.16827 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{5} + 3 q^{7} + 2 q^{11} + 3 q^{13} - 12 q^{17} + 3 q^{19} + 6 q^{25} + q^{29} + 3 q^{31} - 5 q^{35} - 3 q^{37} - 22 q^{41} + 3 q^{43} + 9 q^{47} + 3 q^{49} - 18 q^{53} + 6 q^{55} + 9 q^{59}+ \cdots + 3 q^{97}+O(q^{100}) \) 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
\(5\) 1.18194 0.528581 0.264291 − 0.964443i \(-0.414862\pi\)
0.264291 + 0.964443i \(0.414862\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.70370 1.11671 0.558353 − 0.829603i \(-0.311433\pi\)
0.558353 + 0.829603i \(0.311433\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.94282 −1.68388 −0.841941 − 0.539570i \(-0.818587\pi\)
−0.841941 + 0.539570i \(0.818587\pi\)
\(18\) 0 0
\(19\) −1.94282 −0.445713 −0.222857 − 0.974851i \(-0.571538\pi\)
−0.222857 + 0.974851i \(0.571538\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.60301 1.16831 0.584154 − 0.811643i \(-0.301426\pi\)
0.584154 + 0.811643i \(0.301426\pi\)
\(24\) 0 0
\(25\) −3.60301 −0.720602
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.239123 0.0444041 0.0222020 − 0.999754i \(-0.492932\pi\)
0.0222020 + 0.999754i \(0.492932\pi\)
\(30\) 0 0
\(31\) −1.66019 −0.298179 −0.149089 − 0.988824i \(-0.547634\pi\)
−0.149089 + 0.988824i \(0.547634\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.18194 0.199785
\(36\) 0 0
\(37\) −9.54583 −1.56932 −0.784662 − 0.619923i \(-0.787164\pi\)
−0.784662 + 0.619923i \(0.787164\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.1819 −1.59015 −0.795076 − 0.606510i \(-0.792569\pi\)
−0.795076 + 0.606510i \(0.792569\pi\)
\(42\) 0 0
\(43\) −2.22545 −0.339378 −0.169689 − 0.985498i \(-0.554276\pi\)
−0.169689 + 0.985498i \(0.554276\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5.82846 −0.850168 −0.425084 − 0.905154i \(-0.639755\pi\)
−0.425084 + 0.905154i \(0.639755\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −11.6030 −1.59380 −0.796898 − 0.604114i \(-0.793527\pi\)
−0.796898 + 0.604114i \(0.793527\pi\)
\(54\) 0 0
\(55\) 4.37756 0.590270
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.60301 −0.338883 −0.169442 − 0.985540i \(-0.554196\pi\)
−0.169442 + 0.985540i \(0.554196\pi\)
\(60\) 0 0
\(61\) −7.60301 −0.973466 −0.486733 − 0.873551i \(-0.661811\pi\)
−0.486733 + 0.873551i \(0.661811\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.18194 0.146602
\(66\) 0 0
\(67\) −3.50808 −0.428580 −0.214290 − 0.976770i \(-0.568744\pi\)
−0.214290 + 0.976770i \(0.568744\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −8.60301 −1.02099 −0.510495 − 0.859881i \(-0.670538\pi\)
−0.510495 + 0.859881i \(0.670538\pi\)
\(72\) 0 0
\(73\) 15.1488 1.77304 0.886519 − 0.462693i \(-0.153117\pi\)
0.886519 + 0.462693i \(0.153117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.70370 0.422075
\(78\) 0 0
\(79\) −7.37756 −0.830040 −0.415020 − 0.909812i \(-0.636225\pi\)
−0.415020 + 0.909812i \(0.636225\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.94282 0.762074 0.381037 − 0.924560i \(-0.375567\pi\)
0.381037 + 0.924560i \(0.375567\pi\)
\(84\) 0 0
\(85\) −8.20602 −0.890068
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.74720 0.291203 0.145602 − 0.989343i \(-0.453488\pi\)
0.145602 + 0.989343i \(0.453488\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.29630 −0.235596
\(96\) 0 0
\(97\) 7.16827 0.727828 0.363914 − 0.931433i \(-0.381440\pi\)
0.363914 + 0.931433i \(0.381440\pi\)
\(98\) 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.bq.1.3 3
3.2 odd 2 9072.2.a.cd.1.1 3
4.3 odd 2 567.2.a.d.1.2 3
9.2 odd 6 3024.2.r.g.1009.3 6
9.4 even 3 1008.2.r.k.673.2 6
9.5 odd 6 3024.2.r.g.2017.3 6
9.7 even 3 1008.2.r.k.337.2 6
12.11 even 2 567.2.a.g.1.2 3
28.27 even 2 3969.2.a.m.1.2 3
36.7 odd 6 63.2.f.b.22.2 ✓ 6
36.11 even 6 189.2.f.a.64.2 6
36.23 even 6 189.2.f.a.127.2 6
36.31 odd 6 63.2.f.b.43.2 yes 6
84.83 odd 2 3969.2.a.p.1.2 3
252.11 even 6 1323.2.h.d.226.2 6
252.23 even 6 1323.2.h.d.802.2 6
252.31 even 6 441.2.g.d.79.2 6
252.47 odd 6 1323.2.g.b.361.2 6
252.59 odd 6 1323.2.g.b.667.2 6
252.67 odd 6 441.2.g.e.79.2 6
252.79 odd 6 441.2.g.e.67.2 6
252.83 odd 6 1323.2.f.c.442.2 6
252.95 even 6 1323.2.g.c.667.2 6
252.103 even 6 441.2.h.b.214.2 6
252.115 even 6 441.2.h.b.373.2 6
252.131 odd 6 1323.2.h.e.802.2 6
252.139 even 6 441.2.f.d.295.2 6
252.151 odd 6 441.2.h.c.373.2 6
252.167 odd 6 1323.2.f.c.883.2 6
252.187 even 6 441.2.g.d.67.2 6
252.191 even 6 1323.2.g.c.361.2 6
252.223 even 6 441.2.f.d.148.2 6
252.227 odd 6 1323.2.h.e.226.2 6
252.247 odd 6 441.2.h.c.214.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.2 ✓ 6 36.7 odd 6
63.2.f.b.43.2 yes 6 36.31 odd 6
189.2.f.a.64.2 6 36.11 even 6
189.2.f.a.127.2 6 36.23 even 6
441.2.f.d.148.2 6 252.223 even 6
441.2.f.d.295.2 6 252.139 even 6
441.2.g.d.67.2 6 252.187 even 6
441.2.g.d.79.2 6 252.31 even 6
441.2.g.e.67.2 6 252.79 odd 6
441.2.g.e.79.2 6 252.67 odd 6
441.2.h.b.214.2 6 252.103 even 6
441.2.h.b.373.2 6 252.115 even 6
441.2.h.c.214.2 6 252.247 odd 6
441.2.h.c.373.2 6 252.151 odd 6
567.2.a.d.1.2 3 4.3 odd 2
567.2.a.g.1.2 3 12.11 even 2
1008.2.r.k.337.2 6 9.7 even 3
1008.2.r.k.673.2 6 9.4 even 3
1323.2.f.c.442.2 6 252.83 odd 6
1323.2.f.c.883.2 6 252.167 odd 6
1323.2.g.b.361.2 6 252.47 odd 6
1323.2.g.b.667.2 6 252.59 odd 6
1323.2.g.c.361.2 6 252.191 even 6
1323.2.g.c.667.2 6 252.95 even 6
1323.2.h.d.226.2 6 252.11 even 6
1323.2.h.d.802.2 6 252.23 even 6
1323.2.h.e.226.2 6 252.227 odd 6
1323.2.h.e.802.2 6 252.131 odd 6
3024.2.r.g.1009.3 6 9.2 odd 6
3024.2.r.g.2017.3 6 9.5 odd 6
3969.2.a.m.1.2 3 28.27 even 2
3969.2.a.p.1.2 3 84.83 odd 2
9072.2.a.bq.1.3 3 1.1 even 1 trivial
9072.2.a.cd.1.1 3 3.2 odd 2