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,-1,0,3,0,0,0,2,0,3,0,0,0,-2] 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: 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 252)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.69963\) 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)
 
\(f(q)\) \(=\) \(q+1.69963 q^{5} +1.00000 q^{7} +2.47710 q^{11} +0.777472 q^{13} -2.81089 q^{17} +4.98762 q^{19} +0.712008 q^{23} -2.11126 q^{25} -4.51052 q^{29} -5.09888 q^{31} +1.69963 q^{35} -6.87636 q^{37} +5.87636 q^{41} +4.65383 q^{43} +12.9876 q^{47} +1.00000 q^{49} -1.88874 q^{53} +4.21015 q^{55} +14.2880 q^{59} +14.3090 q^{61} +1.32141 q^{65} -7.98762 q^{67} -10.2632 q^{71} +4.98762 q^{73} +2.47710 q^{77} +9.21015 q^{79} +8.81089 q^{83} -4.77747 q^{85} -9.65383 q^{89} +0.777472 q^{91} +8.47710 q^{95} +8.64145 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{5} + 3 q^{7} + 2 q^{11} + 3 q^{13} - 2 q^{17} - 3 q^{19} + 14 q^{23} - 6 q^{25} - q^{29} + 3 q^{31} - q^{35} - 3 q^{37} - 3 q^{43} + 21 q^{47} + 3 q^{49} - 6 q^{53} - 6 q^{55} + 31 q^{59} + 6 q^{61}+ \cdots - 9 q^{97}+O(q^{100}) \) 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
\(5\) 1.69963 0.760097 0.380048 − 0.924967i \(-0.375907\pi\)
0.380048 + 0.924967i \(0.375907\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.47710 0.746874 0.373437 − 0.927656i \(-0.378179\pi\)
0.373437 + 0.927656i \(0.378179\pi\)
\(12\) 0 0
\(13\) 0.777472 0.215632 0.107816 − 0.994171i \(-0.465614\pi\)
0.107816 + 0.994171i \(0.465614\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.81089 −0.681742 −0.340871 − 0.940110i \(-0.610722\pi\)
−0.340871 + 0.940110i \(0.610722\pi\)
\(18\) 0 0
\(19\) 4.98762 1.14424 0.572119 − 0.820170i \(-0.306121\pi\)
0.572119 + 0.820170i \(0.306121\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.712008 0.148464 0.0742320 − 0.997241i \(-0.476349\pi\)
0.0742320 + 0.997241i \(0.476349\pi\)
\(24\) 0 0
\(25\) −2.11126 −0.422253
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.51052 −0.837583 −0.418791 − 0.908083i \(-0.637546\pi\)
−0.418791 + 0.908083i \(0.637546\pi\)
\(30\) 0 0
\(31\) −5.09888 −0.915787 −0.457893 − 0.889007i \(-0.651396\pi\)
−0.457893 + 0.889007i \(0.651396\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.69963 0.287290
\(36\) 0 0
\(37\) −6.87636 −1.13047 −0.565233 − 0.824931i \(-0.691214\pi\)
−0.565233 + 0.824931i \(0.691214\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.87636 0.917733 0.458866 − 0.888505i \(-0.348256\pi\)
0.458866 + 0.888505i \(0.348256\pi\)
\(42\) 0 0
\(43\) 4.65383 0.709702 0.354851 − 0.934923i \(-0.384532\pi\)
0.354851 + 0.934923i \(0.384532\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.9876 1.89444 0.947220 − 0.320586i \(-0.103880\pi\)
0.947220 + 0.320586i \(0.103880\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.88874 −0.259438 −0.129719 − 0.991551i \(-0.541407\pi\)
−0.129719 + 0.991551i \(0.541407\pi\)
\(54\) 0 0
\(55\) 4.21015 0.567696
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.2880 1.86014 0.930069 − 0.367385i \(-0.119747\pi\)
0.930069 + 0.367385i \(0.119747\pi\)
\(60\) 0 0
\(61\) 14.3090 1.83208 0.916042 − 0.401082i \(-0.131366\pi\)
0.916042 + 0.401082i \(0.131366\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.32141 0.163901
\(66\) 0 0
\(67\) −7.98762 −0.975843 −0.487922 − 0.872887i \(-0.662245\pi\)
−0.487922 + 0.872887i \(0.662245\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.2632 −1.21802 −0.609011 − 0.793162i \(-0.708433\pi\)
−0.609011 + 0.793162i \(0.708433\pi\)
\(72\) 0 0
\(73\) 4.98762 0.583757 0.291878 − 0.956455i \(-0.405720\pi\)
0.291878 + 0.956455i \(0.405720\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.47710 0.282292
\(78\) 0 0
\(79\) 9.21015 1.03622 0.518111 − 0.855313i \(-0.326635\pi\)
0.518111 + 0.855313i \(0.326635\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.81089 0.967121 0.483561 − 0.875311i \(-0.339343\pi\)
0.483561 + 0.875311i \(0.339343\pi\)
\(84\) 0 0
\(85\) −4.77747 −0.518190
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.65383 −1.02330 −0.511652 − 0.859193i \(-0.670966\pi\)
−0.511652 + 0.859193i \(0.670966\pi\)
\(90\) 0 0
\(91\) 0.777472 0.0815012
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.47710 0.869732
\(96\) 0 0
\(97\) 8.64145 0.877406 0.438703 − 0.898632i \(-0.355438\pi\)
0.438703 + 0.898632i \(0.355438\pi\)
\(98\) 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.bv.1.3 3
3.2 odd 2 9072.2.a.by.1.1 3
4.3 odd 2 2268.2.a.h.1.3 3
9.2 odd 6 1008.2.r.j.337.1 6
9.4 even 3 3024.2.r.j.2017.1 6
9.5 odd 6 1008.2.r.j.673.1 6
9.7 even 3 3024.2.r.j.1009.1 6
12.11 even 2 2268.2.a.i.1.1 3
36.7 odd 6 756.2.j.b.253.1 6
36.11 even 6 252.2.j.a.85.3 ✓ 6
36.23 even 6 252.2.j.a.169.3 yes 6
36.31 odd 6 756.2.j.b.505.1 6
252.11 even 6 1764.2.i.g.373.1 6
252.23 even 6 1764.2.i.g.1537.1 6
252.31 even 6 5292.2.l.f.3313.1 6
252.47 odd 6 1764.2.l.f.949.3 6
252.59 odd 6 1764.2.l.f.961.3 6
252.67 odd 6 5292.2.l.e.3313.3 6
252.79 odd 6 5292.2.l.e.361.3 6
252.83 odd 6 1764.2.j.e.589.1 6
252.95 even 6 1764.2.l.e.961.1 6
252.103 even 6 5292.2.i.e.2125.3 6
252.115 even 6 5292.2.i.e.1549.3 6
252.131 odd 6 1764.2.i.d.1537.3 6
252.139 even 6 5292.2.j.d.3529.3 6
252.151 odd 6 5292.2.i.f.1549.1 6
252.167 odd 6 1764.2.j.e.1177.1 6
252.187 even 6 5292.2.l.f.361.1 6
252.191 even 6 1764.2.l.e.949.1 6
252.223 even 6 5292.2.j.d.1765.3 6
252.227 odd 6 1764.2.i.d.373.3 6
252.247 odd 6 5292.2.i.f.2125.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.2.j.a.85.3 ✓ 6 36.11 even 6
252.2.j.a.169.3 yes 6 36.23 even 6
756.2.j.b.253.1 6 36.7 odd 6
756.2.j.b.505.1 6 36.31 odd 6
1008.2.r.j.337.1 6 9.2 odd 6
1008.2.r.j.673.1 6 9.5 odd 6
1764.2.i.d.373.3 6 252.227 odd 6
1764.2.i.d.1537.3 6 252.131 odd 6
1764.2.i.g.373.1 6 252.11 even 6
1764.2.i.g.1537.1 6 252.23 even 6
1764.2.j.e.589.1 6 252.83 odd 6
1764.2.j.e.1177.1 6 252.167 odd 6
1764.2.l.e.949.1 6 252.191 even 6
1764.2.l.e.961.1 6 252.95 even 6
1764.2.l.f.949.3 6 252.47 odd 6
1764.2.l.f.961.3 6 252.59 odd 6
2268.2.a.h.1.3 3 4.3 odd 2
2268.2.a.i.1.1 3 12.11 even 2
3024.2.r.j.1009.1 6 9.7 even 3
3024.2.r.j.2017.1 6 9.4 even 3
5292.2.i.e.1549.3 6 252.115 even 6
5292.2.i.e.2125.3 6 252.103 even 6
5292.2.i.f.1549.1 6 252.151 odd 6
5292.2.i.f.2125.1 6 252.247 odd 6
5292.2.j.d.1765.3 6 252.223 even 6
5292.2.j.d.3529.3 6 252.139 even 6
5292.2.l.e.361.3 6 252.79 odd 6
5292.2.l.e.3313.3 6 252.67 odd 6
5292.2.l.f.361.1 6 252.187 even 6
5292.2.l.f.3313.1 6 252.31 even 6
9072.2.a.bv.1.3 3 1.1 even 1 trivial
9072.2.a.by.1.1 3 3.2 odd 2