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,-3,0,0,0,0,0,-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 504)
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)
 
\(f(q)\) \(=\) \(q+0.652704 q^{5} -1.00000 q^{7} +3.41147 q^{11} -0.305407 q^{13} +0.226682 q^{17} +2.16250 q^{19} -6.70233 q^{23} -4.57398 q^{25} +0.509800 q^{29} +5.71688 q^{31} -0.652704 q^{35} -2.28312 q^{37} -0.958111 q^{41} +7.71688 q^{43} +8.29086 q^{47} +1.00000 q^{49} +3.18479 q^{53} +2.22668 q^{55} -12.1557 q^{59} -3.50980 q^{61} -0.199340 q^{65} -5.74422 q^{67} +14.0273 q^{71} +12.4192 q^{73} -3.41147 q^{77} +11.9659 q^{79} +4.27126 q^{83} +0.147956 q^{85} +1.19934 q^{89} +0.305407 q^{91} +1.41147 q^{95} -14.9786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{13} - 6 q^{17} + 9 q^{19} + 6 q^{23} - 6 q^{25} + 3 q^{29} + 9 q^{31} - 3 q^{35} - 15 q^{37} - 6 q^{41} + 15 q^{43} + 9 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} - 12 q^{61}+ \cdots + 15 q^{97}+O(q^{100}) \) 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
\(5\) 0.652704 0.291898 0.145949 − 0.989292i \(-0.453376\pi\)
0.145949 + 0.989292i \(0.453376\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.41147 1.02860 0.514299 − 0.857611i \(-0.328052\pi\)
0.514299 + 0.857611i \(0.328052\pi\)
\(12\) 0 0
\(13\) −0.305407 −0.0847047 −0.0423524 − 0.999103i \(-0.513485\pi\)
−0.0423524 + 0.999103i \(0.513485\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.226682 0.0549784 0.0274892 − 0.999622i \(-0.491249\pi\)
0.0274892 + 0.999622i \(0.491249\pi\)
\(18\) 0 0
\(19\) 2.16250 0.496112 0.248056 − 0.968746i \(-0.420208\pi\)
0.248056 + 0.968746i \(0.420208\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −6.70233 −1.39753 −0.698767 − 0.715350i \(-0.746267\pi\)
−0.698767 + 0.715350i \(0.746267\pi\)
\(24\) 0 0
\(25\) −4.57398 −0.914796
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.509800 0.0946675 0.0473338 − 0.998879i \(-0.484928\pi\)
0.0473338 + 0.998879i \(0.484928\pi\)
\(30\) 0 0
\(31\) 5.71688 1.02678 0.513391 − 0.858155i \(-0.328389\pi\)
0.513391 + 0.858155i \(0.328389\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.652704 −0.110327
\(36\) 0 0
\(37\) −2.28312 −0.375342 −0.187671 − 0.982232i \(-0.560094\pi\)
−0.187671 + 0.982232i \(0.560094\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.958111 −0.149632 −0.0748159 − 0.997197i \(-0.523837\pi\)
−0.0748159 + 0.997197i \(0.523837\pi\)
\(42\) 0 0
\(43\) 7.71688 1.17681 0.588407 − 0.808565i \(-0.299755\pi\)
0.588407 + 0.808565i \(0.299755\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.29086 1.20935 0.604673 − 0.796474i \(-0.293304\pi\)
0.604673 + 0.796474i \(0.293304\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.18479 0.437465 0.218732 − 0.975785i \(-0.429808\pi\)
0.218732 + 0.975785i \(0.429808\pi\)
\(54\) 0 0
\(55\) 2.22668 0.300246
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −12.1557 −1.58254 −0.791268 − 0.611469i \(-0.790579\pi\)
−0.791268 + 0.611469i \(0.790579\pi\)
\(60\) 0 0
\(61\) −3.50980 −0.449384 −0.224692 − 0.974430i \(-0.572138\pi\)
−0.224692 + 0.974430i \(0.572138\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.199340 −0.0247251
\(66\) 0 0
\(67\) −5.74422 −0.701768 −0.350884 − 0.936419i \(-0.614119\pi\)
−0.350884 + 0.936419i \(0.614119\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 14.0273 1.66474 0.832370 − 0.554221i \(-0.186984\pi\)
0.832370 + 0.554221i \(0.186984\pi\)
\(72\) 0 0
\(73\) 12.4192 1.45356 0.726780 − 0.686871i \(-0.241016\pi\)
0.726780 + 0.686871i \(0.241016\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.41147 −0.388774
\(78\) 0 0
\(79\) 11.9659 1.34626 0.673132 − 0.739523i \(-0.264949\pi\)
0.673132 + 0.739523i \(0.264949\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.27126 0.468832 0.234416 − 0.972136i \(-0.424682\pi\)
0.234416 + 0.972136i \(0.424682\pi\)
\(84\) 0 0
\(85\) 0.147956 0.0160481
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.19934 0.127130 0.0635649 − 0.997978i \(-0.479753\pi\)
0.0635649 + 0.997978i \(0.479753\pi\)
\(90\) 0 0
\(91\) 0.305407 0.0320154
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.41147 0.144814
\(96\) 0 0
\(97\) −14.9786 −1.52085 −0.760425 − 0.649425i \(-0.775010\pi\)
−0.760425 + 0.649425i \(0.775010\pi\)
\(98\) 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.cc.1.2 3
3.2 odd 2 9072.2.a.br.1.2 3
4.3 odd 2 4536.2.a.v.1.2 3
9.2 odd 6 1008.2.r.i.337.2 6
9.4 even 3 3024.2.r.h.2017.2 6
9.5 odd 6 1008.2.r.i.673.2 6
9.7 even 3 3024.2.r.h.1009.2 6
12.11 even 2 4536.2.a.s.1.2 3
36.7 odd 6 1512.2.r.c.1009.2 6
36.11 even 6 504.2.r.c.337.2 yes 6
36.23 even 6 504.2.r.c.169.2 ✓ 6
36.31 odd 6 1512.2.r.c.505.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.2 ✓ 6 36.23 even 6
504.2.r.c.337.2 yes 6 36.11 even 6
1008.2.r.i.337.2 6 9.2 odd 6
1008.2.r.i.673.2 6 9.5 odd 6
1512.2.r.c.505.2 6 36.31 odd 6
1512.2.r.c.1009.2 6 36.7 odd 6
3024.2.r.h.1009.2 6 9.7 even 3
3024.2.r.h.2017.2 6 9.4 even 3
4536.2.a.s.1.2 3 12.11 even 2
4536.2.a.v.1.2 3 4.3 odd 2
9072.2.a.br.1.2 3 3.2 odd 2
9072.2.a.cc.1.2 3 1.1 even 1 trivial