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: \(1\)
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 \(1.87939\) 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)
 
\(f(q)\) \(=\) \(q-2.87939 q^{5} -1.00000 q^{7} +1.18479 q^{11} -4.75877 q^{13} +5.41147 q^{17} -1.10607 q^{19} -5.90167 q^{23} +3.29086 q^{25} +4.98545 q^{29} +5.57398 q^{31} +2.87939 q^{35} -2.42602 q^{37} +7.63816 q^{41} +7.57398 q^{43} -0.283119 q^{47} +1.00000 q^{49} -4.22668 q^{53} -3.41147 q^{55} -11.7246 q^{59} +1.98545 q^{61} +13.7023 q^{65} +13.5398 q^{67} +5.11381 q^{71} -0.327696 q^{73} -1.18479 q^{77} +10.4953 q^{79} -7.25402 q^{83} -15.5817 q^{85} -14.7023 q^{89} +4.75877 q^{91} +3.18479 q^{95} +18.0942 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{5} - 3 q^{7} - 3 q^{13} + 6 q^{17} + 9 q^{19} - 6 q^{23} - 6 q^{25} - 3 q^{29} + 9 q^{31} + 3 q^{35} - 15 q^{37} + 6 q^{41} + 15 q^{43} - 9 q^{47} + 3 q^{49} - 6 q^{53} - 3 q^{59} - 12 q^{61}+ \cdots + 15 q^{97}+O(q^{100}) \) 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
\(5\) −2.87939 −1.28770 −0.643850 − 0.765152i \(-0.722664\pi\)
−0.643850 + 0.765152i \(0.722664\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.18479 0.357228 0.178614 − 0.983919i \(-0.442839\pi\)
0.178614 + 0.983919i \(0.442839\pi\)
\(12\) 0 0
\(13\) −4.75877 −1.31985 −0.659923 − 0.751333i \(-0.729411\pi\)
−0.659923 + 0.751333i \(0.729411\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.41147 1.31248 0.656238 − 0.754554i \(-0.272147\pi\)
0.656238 + 0.754554i \(0.272147\pi\)
\(18\) 0 0
\(19\) −1.10607 −0.253749 −0.126875 − 0.991919i \(-0.540495\pi\)
−0.126875 + 0.991919i \(0.540495\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.90167 −1.23058 −0.615292 − 0.788299i \(-0.710962\pi\)
−0.615292 + 0.788299i \(0.710962\pi\)
\(24\) 0 0
\(25\) 3.29086 0.658172
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.98545 0.925775 0.462888 − 0.886417i \(-0.346813\pi\)
0.462888 + 0.886417i \(0.346813\pi\)
\(30\) 0 0
\(31\) 5.57398 1.00112 0.500558 − 0.865703i \(-0.333128\pi\)
0.500558 + 0.865703i \(0.333128\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.87939 0.486705
\(36\) 0 0
\(37\) −2.42602 −0.398836 −0.199418 − 0.979915i \(-0.563905\pi\)
−0.199418 + 0.979915i \(0.563905\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.63816 1.19288 0.596440 − 0.802658i \(-0.296581\pi\)
0.596440 + 0.802658i \(0.296581\pi\)
\(42\) 0 0
\(43\) 7.57398 1.15502 0.577510 − 0.816383i \(-0.304024\pi\)
0.577510 + 0.816383i \(0.304024\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.283119 −0.0412971 −0.0206485 − 0.999787i \(-0.506573\pi\)
−0.0206485 + 0.999787i \(0.506573\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.22668 −0.580579 −0.290290 − 0.956939i \(-0.593752\pi\)
−0.290290 + 0.956939i \(0.593752\pi\)
\(54\) 0 0
\(55\) −3.41147 −0.460003
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.7246 −1.52642 −0.763208 − 0.646153i \(-0.776377\pi\)
−0.763208 + 0.646153i \(0.776377\pi\)
\(60\) 0 0
\(61\) 1.98545 0.254211 0.127106 − 0.991889i \(-0.459431\pi\)
0.127106 + 0.991889i \(0.459431\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 13.7023 1.69957
\(66\) 0 0
\(67\) 13.5398 1.65415 0.827077 − 0.562089i \(-0.190002\pi\)
0.827077 + 0.562089i \(0.190002\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.11381 0.606897 0.303449 − 0.952848i \(-0.401862\pi\)
0.303449 + 0.952848i \(0.401862\pi\)
\(72\) 0 0
\(73\) −0.327696 −0.0383539 −0.0191770 − 0.999816i \(-0.506105\pi\)
−0.0191770 + 0.999816i \(0.506105\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.18479 −0.135020
\(78\) 0 0
\(79\) 10.4953 1.18081 0.590404 − 0.807108i \(-0.298968\pi\)
0.590404 + 0.807108i \(0.298968\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.25402 −0.796232 −0.398116 − 0.917335i \(-0.630336\pi\)
−0.398116 + 0.917335i \(0.630336\pi\)
\(84\) 0 0
\(85\) −15.5817 −1.69007
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −14.7023 −1.55844 −0.779222 − 0.626748i \(-0.784386\pi\)
−0.779222 + 0.626748i \(0.784386\pi\)
\(90\) 0 0
\(91\) 4.75877 0.498855
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.18479 0.326753
\(96\) 0 0
\(97\) 18.0942 1.83719 0.918594 − 0.395202i \(-0.129325\pi\)
0.918594 + 0.395202i \(0.129325\pi\)
\(98\) 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.br.1.1 3
3.2 odd 2 9072.2.a.cc.1.3 3
4.3 odd 2 4536.2.a.s.1.1 3
9.2 odd 6 3024.2.r.h.1009.1 6
9.4 even 3 1008.2.r.i.673.1 6
9.5 odd 6 3024.2.r.h.2017.1 6
9.7 even 3 1008.2.r.i.337.1 6
12.11 even 2 4536.2.a.v.1.3 3
36.7 odd 6 504.2.r.c.337.3 yes 6
36.11 even 6 1512.2.r.c.1009.1 6
36.23 even 6 1512.2.r.c.505.1 6
36.31 odd 6 504.2.r.c.169.3 ✓ 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.c.169.3 ✓ 6 36.31 odd 6
504.2.r.c.337.3 yes 6 36.7 odd 6
1008.2.r.i.337.1 6 9.7 even 3
1008.2.r.i.673.1 6 9.4 even 3
1512.2.r.c.505.1 6 36.23 even 6
1512.2.r.c.1009.1 6 36.11 even 6
3024.2.r.h.1009.1 6 9.2 odd 6
3024.2.r.h.2017.1 6 9.5 odd 6
4536.2.a.s.1.1 3 4.3 odd 2
4536.2.a.v.1.3 3 12.11 even 2
9072.2.a.br.1.1 3 1.1 even 1 trivial
9072.2.a.cc.1.3 3 3.2 odd 2