Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4536,2,Mod(1,4536)] 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("4536.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4536, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4536 = 2^{3} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4536.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-3,0,-4,0,0,0,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(36.2201423569\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.45729.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 11x^{2} + 12x - 3 \) 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 \(-3.38095\) of defining polynomial
Character \(\chi\) \(=\) 4536.1

$q$-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)
 
\(f(q)\) \(=\) \(q-4.38095 q^{5} -1.00000 q^{7} +5.38095 q^{11} -2.54348 q^{13} +2.58058 q^{17} -6.72479 q^{19} +0.800370 q^{23} +14.1927 q^{25} -3.74311 q^{29} +3.39370 q^{31} +4.38095 q^{35} +4.38095 q^{37} +6.39370 q^{41} -0.763270 q^{43} +8.26827 q^{47} +1.00000 q^{49} +4.94877 q^{53} -23.5736 q^{55} -5.57502 q^{59} -8.28659 q^{61} +11.1428 q^{65} -1.89288 q^{67} -1.34385 q^{71} -8.65060 q^{73} -5.38095 q^{77} -13.2810 q^{79} -3.72479 q^{83} -11.3054 q^{85} +6.99862 q^{89} +2.54348 q^{91} +29.4610 q^{95} +2.96152 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 3 q^{5} - 4 q^{7} + 7 q^{11} - 3 q^{13} - 3 q^{17} - 4 q^{19} + 2 q^{23} + 5 q^{25} - 9 q^{29} - 3 q^{31} + 3 q^{35} + 3 q^{37} + 9 q^{41} - 8 q^{43} + 3 q^{47} + 4 q^{49} - 6 q^{53} - 28 q^{55} + 10 q^{59}+ \cdots - 16 q^{97}+O(q^{100}) \) 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
\(5\) −4.38095 −1.95922 −0.979610 − 0.200911i \(-0.935610\pi\)
−0.979610 + 0.200911i \(0.935610\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.38095 1.62242 0.811208 − 0.584757i \(-0.198810\pi\)
0.811208 + 0.584757i \(0.198810\pi\)
\(12\) 0 0
\(13\) −2.54348 −0.705434 −0.352717 − 0.935730i \(-0.614742\pi\)
−0.352717 + 0.935730i \(0.614742\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.58058 0.625882 0.312941 − 0.949773i \(-0.398686\pi\)
0.312941 + 0.949773i \(0.398686\pi\)
\(18\) 0 0
\(19\) −6.72479 −1.54277 −0.771387 − 0.636366i \(-0.780437\pi\)
−0.771387 + 0.636366i \(0.780437\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.800370 0.166889 0.0834443 − 0.996512i \(-0.473408\pi\)
0.0834443 + 0.996512i \(0.473408\pi\)
\(24\) 0 0
\(25\) 14.1927 2.83854
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.74311 −0.695078 −0.347539 − 0.937666i \(-0.612983\pi\)
−0.347539 + 0.937666i \(0.612983\pi\)
\(30\) 0 0
\(31\) 3.39370 0.609527 0.304764 − 0.952428i \(-0.401423\pi\)
0.304764 + 0.952428i \(0.401423\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.38095 0.740515
\(36\) 0 0
\(37\) 4.38095 0.720223 0.360112 − 0.932909i \(-0.382739\pi\)
0.360112 + 0.932909i \(0.382739\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.39370 0.998529 0.499264 − 0.866450i \(-0.333604\pi\)
0.499264 + 0.866450i \(0.333604\pi\)
\(42\) 0 0
\(43\) −0.763270 −0.116398 −0.0581988 − 0.998305i \(-0.518536\pi\)
−0.0581988 + 0.998305i \(0.518536\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.26827 1.20605 0.603026 − 0.797722i \(-0.293962\pi\)
0.603026 + 0.797722i \(0.293962\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.94877 0.679766 0.339883 − 0.940468i \(-0.389613\pi\)
0.339883 + 0.940468i \(0.389613\pi\)
\(54\) 0 0
\(55\) −23.5736 −3.17867
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −5.57502 −0.725806 −0.362903 − 0.931827i \(-0.618214\pi\)
−0.362903 + 0.931827i \(0.618214\pi\)
\(60\) 0 0
\(61\) −8.28659 −1.06099 −0.530494 − 0.847689i \(-0.677994\pi\)
−0.530494 + 0.847689i \(0.677994\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.1428 1.38210
\(66\) 0 0
\(67\) −1.89288 −0.231252 −0.115626 − 0.993293i \(-0.536887\pi\)
−0.115626 + 0.993293i \(0.536887\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −1.34385 −0.159485 −0.0797427 − 0.996815i \(-0.525410\pi\)
−0.0797427 + 0.996815i \(0.525410\pi\)
\(72\) 0 0
\(73\) −8.65060 −1.01248 −0.506238 − 0.862394i \(-0.668964\pi\)
−0.506238 + 0.862394i \(0.668964\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.38095 −0.613216
\(78\) 0 0
\(79\) −13.2810 −1.49423 −0.747116 − 0.664693i \(-0.768562\pi\)
−0.747116 + 0.664693i \(0.768562\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.72479 −0.408849 −0.204425 − 0.978882i \(-0.565532\pi\)
−0.204425 + 0.978882i \(0.565532\pi\)
\(84\) 0 0
\(85\) −11.3054 −1.22624
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.99862 0.741853 0.370926 − 0.928662i \(-0.379040\pi\)
0.370926 + 0.928662i \(0.379040\pi\)
\(90\) 0 0
\(91\) 2.54348 0.266629
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 29.4610 3.02263
\(96\) 0 0
\(97\) 2.96152 0.300697 0.150349 − 0.988633i \(-0.451960\pi\)
0.150349 + 0.988633i \(0.451960\pi\)
\(98\) 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 4536.2.a.y.1.1 4
3.2 odd 2 4536.2.a.z.1.4 4
4.3 odd 2 9072.2.a.cg.1.1 4
9.2 odd 6 504.2.r.e.337.2 yes 8
9.4 even 3 1512.2.r.e.505.4 8
9.5 odd 6 504.2.r.e.169.2 ✓ 8
9.7 even 3 1512.2.r.e.1009.4 8
12.11 even 2 9072.2.a.cj.1.4 4
36.7 odd 6 3024.2.r.m.1009.4 8
36.11 even 6 1008.2.r.l.337.3 8
36.23 even 6 1008.2.r.l.673.3 8
36.31 odd 6 3024.2.r.m.2017.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.2.r.e.169.2 ✓ 8 9.5 odd 6
504.2.r.e.337.2 yes 8 9.2 odd 6
1008.2.r.l.337.3 8 36.11 even 6
1008.2.r.l.673.3 8 36.23 even 6
1512.2.r.e.505.4 8 9.4 even 3
1512.2.r.e.1009.4 8 9.7 even 3
3024.2.r.m.1009.4 8 36.7 odd 6
3024.2.r.m.2017.4 8 36.31 odd 6
4536.2.a.y.1.1 4 1.1 even 1 trivial
4536.2.a.z.1.4 4 3.2 odd 2
9072.2.a.cg.1.1 4 4.3 odd 2
9072.2.a.cj.1.4 4 12.11 even 2