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 1.1
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)
 
\(f(q)\) \(=\) \(q-0.532089 q^{5} -1.00000 q^{7} -2.22668 q^{11} +2.06418 q^{13} -0.815207 q^{17} +7.94356 q^{19} +6.80066 q^{23} -4.71688 q^{25} +7.47565 q^{29} -2.29086 q^{31} +0.532089 q^{35} -10.2909 q^{37} +2.59627 q^{41} -0.290859 q^{43} +0.426022 q^{47} +1.00000 q^{49} -1.41147 q^{53} +1.18479 q^{55} +3.43107 q^{59} -10.4757 q^{61} -1.09833 q^{65} +4.20439 q^{67} +12.0865 q^{71} -9.09152 q^{73} +2.22668 q^{77} -7.46110 q^{79} -17.5253 q^{83} +0.433763 q^{85} +2.09833 q^{89} -2.06418 q^{91} -4.22668 q^{95} +11.8844 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{5} - 3 q^{7} - 3 q^{13} - 6 q^{17} + 9 q^{19} + 6 q^{23} - 6 q^{25} + 3 q^{29} + 9 q^{31} - 3 q^{35} - 15 q^{37} - 6 q^{41} + 15 q^{43} + 9 q^{47} + 3 q^{49} + 6 q^{53} + 3 q^{59} - 12 q^{61}+ \cdots + 15 q^{97}+O(q^{100}) \) 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
\(5\) −0.532089 −0.237957 −0.118979 − 0.992897i \(-0.537962\pi\)
−0.118979 + 0.992897i \(0.537962\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.22668 −0.671370 −0.335685 − 0.941974i \(-0.608968\pi\)
−0.335685 + 0.941974i \(0.608968\pi\)
\(12\) 0 0
\(13\) 2.06418 0.572500 0.286250 − 0.958155i \(-0.407591\pi\)
0.286250 + 0.958155i \(0.407591\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.815207 −0.197717 −0.0988584 − 0.995102i \(-0.531519\pi\)
−0.0988584 + 0.995102i \(0.531519\pi\)
\(18\) 0 0
\(19\) 7.94356 1.82238 0.911189 − 0.411988i \(-0.135166\pi\)
0.911189 + 0.411988i \(0.135166\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.80066 1.41804 0.709018 − 0.705191i \(-0.249139\pi\)
0.709018 + 0.705191i \(0.249139\pi\)
\(24\) 0 0
\(25\) −4.71688 −0.943376
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.47565 1.38819 0.694097 − 0.719882i \(-0.255804\pi\)
0.694097 + 0.719882i \(0.255804\pi\)
\(30\) 0 0
\(31\) −2.29086 −0.411450 −0.205725 − 0.978610i \(-0.565955\pi\)
−0.205725 + 0.978610i \(0.565955\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.532089 0.0899394
\(36\) 0 0
\(37\) −10.2909 −1.69181 −0.845903 − 0.533336i \(-0.820938\pi\)
−0.845903 + 0.533336i \(0.820938\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.59627 0.405469 0.202734 − 0.979234i \(-0.435017\pi\)
0.202734 + 0.979234i \(0.435017\pi\)
\(42\) 0 0
\(43\) −0.290859 −0.0443556 −0.0221778 − 0.999754i \(-0.507060\pi\)
−0.0221778 + 0.999754i \(0.507060\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.426022 0.0621417 0.0310709 − 0.999517i \(-0.490108\pi\)
0.0310709 + 0.999517i \(0.490108\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.41147 −0.193881 −0.0969404 − 0.995290i \(-0.530906\pi\)
−0.0969404 + 0.995290i \(0.530906\pi\)
\(54\) 0 0
\(55\) 1.18479 0.159757
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 3.43107 0.446688 0.223344 − 0.974740i \(-0.428303\pi\)
0.223344 + 0.974740i \(0.428303\pi\)
\(60\) 0 0
\(61\) −10.4757 −1.34127 −0.670635 − 0.741788i \(-0.733978\pi\)
−0.670635 + 0.741788i \(0.733978\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.09833 −0.136231
\(66\) 0 0
\(67\) 4.20439 0.513648 0.256824 − 0.966458i \(-0.417324\pi\)
0.256824 + 0.966458i \(0.417324\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0865 1.43440 0.717200 − 0.696868i \(-0.245423\pi\)
0.717200 + 0.696868i \(0.245423\pi\)
\(72\) 0 0
\(73\) −9.09152 −1.06408 −0.532041 − 0.846719i \(-0.678575\pi\)
−0.532041 + 0.846719i \(0.678575\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.22668 0.253754
\(78\) 0 0
\(79\) −7.46110 −0.839440 −0.419720 − 0.907654i \(-0.637872\pi\)
−0.419720 + 0.907654i \(0.637872\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −17.5253 −1.92365 −0.961825 − 0.273666i \(-0.911764\pi\)
−0.961825 + 0.273666i \(0.911764\pi\)
\(84\) 0 0
\(85\) 0.433763 0.0470482
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.09833 0.222422 0.111211 − 0.993797i \(-0.464527\pi\)
0.111211 + 0.993797i \(0.464527\pi\)
\(90\) 0 0
\(91\) −2.06418 −0.216385
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.22668 −0.433648
\(96\) 0 0
\(97\) 11.8844 1.20668 0.603341 − 0.797483i \(-0.293836\pi\)
0.603341 + 0.797483i \(0.293836\pi\)
\(98\) 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.1.1 3
3.2 odd 2 9072.2.a.br.1.3 3
4.3 odd 2 4536.2.a.v.1.1 3
9.2 odd 6 1008.2.r.i.337.3 6
9.4 even 3 3024.2.r.h.2017.3 6
9.5 odd 6 1008.2.r.i.673.3 6
9.7 even 3 3024.2.r.h.1009.3 6
12.11 even 2 4536.2.a.s.1.3 3
36.7 odd 6 1512.2.r.c.1009.3 6
36.11 even 6 504.2.r.c.337.1 yes 6
36.23 even 6 504.2.r.c.169.1 ✓ 6
36.31 odd 6 1512.2.r.c.505.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.1 ✓ 6 36.23 even 6
504.2.r.c.337.1 yes 6 36.11 even 6
1008.2.r.i.337.3 6 9.2 odd 6
1008.2.r.i.673.3 6 9.5 odd 6
1512.2.r.c.505.3 6 36.31 odd 6
1512.2.r.c.1009.3 6 36.7 odd 6
3024.2.r.h.1009.3 6 9.7 even 3
3024.2.r.h.2017.3 6 9.4 even 3
4536.2.a.s.1.3 3 12.11 even 2
4536.2.a.v.1.1 3 4.3 odd 2
9072.2.a.br.1.3 3 3.2 odd 2
9072.2.a.cc.1.1 3 1.1 even 1 trivial