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Properties

Label 9072.2.a.bq.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,-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 1.2
Root \(2.46050\) 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)
 
\(f(q)\) \(=\) \(q-2.59358 q^{5} +1.00000 q^{7} -4.51459 q^{11} +1.00000 q^{13} -0.945916 q^{17} +4.05408 q^{19} +0.273346 q^{23} +1.72665 q^{25} +2.46050 q^{29} -2.32743 q^{31} -2.59358 q^{35} +1.78074 q^{37} -6.40642 q^{41} +10.4356 q^{43} +12.1623 q^{47} +1.00000 q^{49} -6.27335 q^{53} +11.7089 q^{55} +2.72665 q^{59} -2.27335 q^{61} -2.59358 q^{65} +15.8171 q^{67} -3.27335 q^{71} -1.50739 q^{73} -4.51459 q^{77} -14.7089 q^{79} +0.945916 q^{83} +2.45331 q^{85} -14.3566 q^{89} +1.00000 q^{91} -10.5146 q^{95} -11.4897 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 5 q^{5} + 3 q^{7} + 2 q^{11} + 3 q^{13} - 12 q^{17} + 3 q^{19} + 6 q^{25} + q^{29} + 3 q^{31} - 5 q^{35} - 3 q^{37} - 22 q^{41} + 3 q^{43} + 9 q^{47} + 3 q^{49} - 18 q^{53} + 6 q^{55} + 9 q^{59}+ \cdots + 3 q^{97}+O(q^{100}) \) 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
\(5\) −2.59358 −1.15988 −0.579942 − 0.814658i \(-0.696925\pi\)
−0.579942 + 0.814658i \(0.696925\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4.51459 −1.36120 −0.680600 − 0.732655i \(-0.738281\pi\)
−0.680600 + 0.732655i \(0.738281\pi\)
\(12\) 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
\(17\) −0.945916 −0.229418 −0.114709 − 0.993399i \(-0.536594\pi\)
−0.114709 + 0.993399i \(0.536594\pi\)
\(18\) 0 0
\(19\) 4.05408 0.930071 0.465035 − 0.885292i \(-0.346042\pi\)
0.465035 + 0.885292i \(0.346042\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.273346 0.0569966 0.0284983 − 0.999594i \(-0.490927\pi\)
0.0284983 + 0.999594i \(0.490927\pi\)
\(24\) 0 0
\(25\) 1.72665 0.345331
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.46050 0.456904 0.228452 − 0.973555i \(-0.426634\pi\)
0.228452 + 0.973555i \(0.426634\pi\)
\(30\) 0 0
\(31\) −2.32743 −0.418019 −0.209009 − 0.977914i \(-0.567024\pi\)
−0.209009 + 0.977914i \(0.567024\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.59358 −0.438395
\(36\) 0 0
\(37\) 1.78074 0.292752 0.146376 − 0.989229i \(-0.453239\pi\)
0.146376 + 0.989229i \(0.453239\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.40642 −1.00051 −0.500257 − 0.865877i \(-0.666761\pi\)
−0.500257 + 0.865877i \(0.666761\pi\)
\(42\) 0 0
\(43\) 10.4356 1.59141 0.795707 − 0.605682i \(-0.207100\pi\)
0.795707 + 0.605682i \(0.207100\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.1623 1.77405 0.887023 − 0.461724i \(-0.152769\pi\)
0.887023 + 0.461724i \(0.152769\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.27335 −0.861710 −0.430855 − 0.902421i \(-0.641788\pi\)
−0.430855 + 0.902421i \(0.641788\pi\)
\(54\) 0 0
\(55\) 11.7089 1.57883
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.72665 0.354980 0.177490 − 0.984123i \(-0.443202\pi\)
0.177490 + 0.984123i \(0.443202\pi\)
\(60\) 0 0
\(61\) −2.27335 −0.291072 −0.145536 − 0.989353i \(-0.546491\pi\)
−0.145536 + 0.989353i \(0.546491\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.59358 −0.321694
\(66\) 0 0
\(67\) 15.8171 1.93237 0.966184 − 0.257854i \(-0.0830152\pi\)
0.966184 + 0.257854i \(0.0830152\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.27335 −0.388475 −0.194237 − 0.980955i \(-0.562223\pi\)
−0.194237 + 0.980955i \(0.562223\pi\)
\(72\) 0 0
\(73\) −1.50739 −0.176427 −0.0882134 − 0.996102i \(-0.528116\pi\)
−0.0882134 + 0.996102i \(0.528116\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.51459 −0.514485
\(78\) 0 0
\(79\) −14.7089 −1.65489 −0.827443 − 0.561550i \(-0.810205\pi\)
−0.827443 + 0.561550i \(0.810205\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0.945916 0.103828 0.0519139 − 0.998652i \(-0.483468\pi\)
0.0519139 + 0.998652i \(0.483468\pi\)
\(84\) 0 0
\(85\) 2.45331 0.266099
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.3566 −1.52180 −0.760899 − 0.648871i \(-0.775242\pi\)
−0.760899 + 0.648871i \(0.775242\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −10.5146 −1.07877
\(96\) 0 0
\(97\) −11.4897 −1.16660 −0.583300 − 0.812257i \(-0.698239\pi\)
−0.583300 + 0.812257i \(0.698239\pi\)
\(98\) 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.1.2 3
3.2 odd 2 9072.2.a.cd.1.2 3
4.3 odd 2 567.2.a.d.1.1 3
9.2 odd 6 3024.2.r.g.1009.2 6
9.4 even 3 1008.2.r.k.673.3 6
9.5 odd 6 3024.2.r.g.2017.2 6
9.7 even 3 1008.2.r.k.337.3 6
12.11 even 2 567.2.a.g.1.3 3
28.27 even 2 3969.2.a.m.1.1 3
36.7 odd 6 63.2.f.b.22.3 ✓ 6
36.11 even 6 189.2.f.a.64.1 6
36.23 even 6 189.2.f.a.127.1 6
36.31 odd 6 63.2.f.b.43.3 yes 6
84.83 odd 2 3969.2.a.p.1.3 3
252.11 even 6 1323.2.h.d.226.3 6
252.23 even 6 1323.2.h.d.802.3 6
252.31 even 6 441.2.g.d.79.3 6
252.47 odd 6 1323.2.g.b.361.1 6
252.59 odd 6 1323.2.g.b.667.1 6
252.67 odd 6 441.2.g.e.79.3 6
252.79 odd 6 441.2.g.e.67.3 6
252.83 odd 6 1323.2.f.c.442.1 6
252.95 even 6 1323.2.g.c.667.1 6
252.103 even 6 441.2.h.b.214.1 6
252.115 even 6 441.2.h.b.373.1 6
252.131 odd 6 1323.2.h.e.802.3 6
252.139 even 6 441.2.f.d.295.3 6
252.151 odd 6 441.2.h.c.373.1 6
252.167 odd 6 1323.2.f.c.883.1 6
252.187 even 6 441.2.g.d.67.3 6
252.191 even 6 1323.2.g.c.361.1 6
252.223 even 6 441.2.f.d.148.3 6
252.227 odd 6 1323.2.h.e.226.3 6
252.247 odd 6 441.2.h.c.214.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.f.b.22.3 ✓ 6 36.7 odd 6
63.2.f.b.43.3 yes 6 36.31 odd 6
189.2.f.a.64.1 6 36.11 even 6
189.2.f.a.127.1 6 36.23 even 6
441.2.f.d.148.3 6 252.223 even 6
441.2.f.d.295.3 6 252.139 even 6
441.2.g.d.67.3 6 252.187 even 6
441.2.g.d.79.3 6 252.31 even 6
441.2.g.e.67.3 6 252.79 odd 6
441.2.g.e.79.3 6 252.67 odd 6
441.2.h.b.214.1 6 252.103 even 6
441.2.h.b.373.1 6 252.115 even 6
441.2.h.c.214.1 6 252.247 odd 6
441.2.h.c.373.1 6 252.151 odd 6
567.2.a.d.1.1 3 4.3 odd 2
567.2.a.g.1.3 3 12.11 even 2
1008.2.r.k.337.3 6 9.7 even 3
1008.2.r.k.673.3 6 9.4 even 3
1323.2.f.c.442.1 6 252.83 odd 6
1323.2.f.c.883.1 6 252.167 odd 6
1323.2.g.b.361.1 6 252.47 odd 6
1323.2.g.b.667.1 6 252.59 odd 6
1323.2.g.c.361.1 6 252.191 even 6
1323.2.g.c.667.1 6 252.95 even 6
1323.2.h.d.226.3 6 252.11 even 6
1323.2.h.d.802.3 6 252.23 even 6
1323.2.h.e.226.3 6 252.227 odd 6
1323.2.h.e.802.3 6 252.131 odd 6
3024.2.r.g.1009.2 6 9.2 odd 6
3024.2.r.g.2017.2 6 9.5 odd 6
3969.2.a.m.1.1 3 28.27 even 2
3969.2.a.p.1.3 3 84.83 odd 2
9072.2.a.bq.1.2 3 1.1 even 1 trivial
9072.2.a.cd.1.2 3 3.2 odd 2