Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [507,2,Mod(1,507)] 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("507.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(507, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-2,-4,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -3.46410 q^{5} -1.73205 q^{7} +1.00000 q^{9} +3.46410 q^{11} +2.00000 q^{12} +3.46410 q^{15} +4.00000 q^{16} -3.46410 q^{19} +6.92820 q^{20} +1.73205 q^{21} +6.00000 q^{23} +7.00000 q^{25} -1.00000 q^{27} +3.46410 q^{28} +6.00000 q^{29} -1.73205 q^{31} -3.46410 q^{33} +6.00000 q^{35} -2.00000 q^{36} -6.92820 q^{41} +1.00000 q^{43} -6.92820 q^{44} -3.46410 q^{45} +3.46410 q^{47} -4.00000 q^{48} -4.00000 q^{49} +12.0000 q^{53} -12.0000 q^{55} +3.46410 q^{57} -3.46410 q^{59} -6.92820 q^{60} +1.00000 q^{61} -1.73205 q^{63} -8.00000 q^{64} +8.66025 q^{67} -6.00000 q^{69} +10.3923 q^{71} +1.73205 q^{73} -7.00000 q^{75} +6.92820 q^{76} -6.00000 q^{77} -11.0000 q^{79} -13.8564 q^{80} +1.00000 q^{81} +13.8564 q^{83} -3.46410 q^{84} -6.00000 q^{87} -6.92820 q^{89} -12.0000 q^{92} +1.73205 q^{93} +12.0000 q^{95} +5.19615 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 4 q^{4} + 2 q^{9} + 4 q^{12} + 8 q^{16} + 12 q^{23} + 14 q^{25} - 2 q^{27} + 12 q^{29} + 12 q^{35} - 4 q^{36} + 2 q^{43} - 8 q^{48} - 8 q^{49} + 24 q^{53} - 24 q^{55} + 2 q^{61} - 16 q^{64}+ \cdots + 24 q^{95}+O(q^{100}) \) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) −3.46410 −1.54919 −0.774597 − 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) −1.73205 −0.654654 −0.327327 − 0.944911i \(-0.606148\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 − 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 2.00000 0.577350
\(13\) 0 0
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 4.00000 1.00000
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.46410 −0.794719 −0.397360 − 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 6.92820 1.54919
\(21\) 1.73205 0.377964
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 − 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 3.46410 0.654654
\(29\) 6.00000 1.11417 0.557086 − 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −1.73205 −0.311086 −0.155543 − 0.987829i \(-0.549713\pi\)
−0.155543 + 0.987829i \(0.549713\pi\)
\(32\) 0 0
\(33\) −3.46410 −0.603023
\(34\) 0 0
\(35\) 6.00000 1.01419
\(36\) −2.00000 −0.333333
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 − 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) 1.00000 0.152499 0.0762493 − 0.997089i \(-0.475706\pi\)
0.0762493 + 0.997089i \(0.475706\pi\)
\(44\) −6.92820 −1.04447
\(45\) −3.46410 −0.516398
\(46\) 0 0
\(47\) 3.46410 0.505291 0.252646 − 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) −4.00000 −0.577350
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.0000 1.64833 0.824163 − 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −12.0000 −1.61808
\(56\) 0 0
\(57\) 3.46410 0.458831
\(58\) 0 0
\(59\) −3.46410 −0.450988 −0.225494 − 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) −6.92820 −0.894427
\(61\) 1.00000 0.128037 0.0640184 − 0.997949i \(-0.479608\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) −1.73205 −0.218218
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.66025 1.05802 0.529009 − 0.848616i \(-0.322564\pi\)
0.529009 + 0.848616i \(0.322564\pi\)
\(68\) 0 0
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) 10.3923 1.23334 0.616670 − 0.787222i \(-0.288481\pi\)
0.616670 + 0.787222i \(0.288481\pi\)
\(72\) 0 0
\(73\) 1.73205 0.202721 0.101361 − 0.994850i \(-0.467680\pi\)
0.101361 + 0.994850i \(0.467680\pi\)
\(74\) 0 0
\(75\) −7.00000 −0.808290
\(76\) 6.92820 0.794719
\(77\) −6.00000 −0.683763
\(78\) 0 0
\(79\) −11.0000 −1.23760 −0.618798 − 0.785550i \(-0.712380\pi\)
−0.618798 + 0.785550i \(0.712380\pi\)
\(80\) −13.8564 −1.54919
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 13.8564 1.52094 0.760469 − 0.649374i \(-0.224969\pi\)
0.760469 + 0.649374i \(0.224969\pi\)
\(84\) −3.46410 −0.377964
\(85\) 0 0
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) −6.92820 −0.734388 −0.367194 − 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) 1.73205 0.179605
\(94\) 0 0
\(95\) 12.0000 1.23117
\(96\) 0 0
\(97\) 5.19615 0.527589 0.263795 − 0.964579i \(-0.415026\pi\)
0.263795 + 0.964579i \(0.415026\pi\)
\(98\) 0 0
\(99\) 3.46410 0.348155
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 507.2.a.e.1.1 2
3.2 odd 2 1521.2.a.h.1.2 2
4.3 odd 2 8112.2.a.bu.1.1 2
13.2 odd 12 39.2.j.a.4.1 ✓ 2
13.3 even 3 507.2.e.f.22.1 4
13.4 even 6 507.2.e.f.484.2 4
13.5 odd 4 507.2.b.c.337.2 2
13.6 odd 12 507.2.j.b.361.1 2
13.7 odd 12 39.2.j.a.10.1 yes 2
13.8 odd 4 507.2.b.c.337.1 2
13.9 even 3 507.2.e.f.484.1 4
13.10 even 6 507.2.e.f.22.2 4
13.11 odd 12 507.2.j.b.316.1 2
13.12 even 2 inner 507.2.a.e.1.2 2
39.2 even 12 117.2.q.a.82.1 2
39.5 even 4 1521.2.b.f.1351.1 2
39.8 even 4 1521.2.b.f.1351.2 2
39.20 even 12 117.2.q.a.10.1 2
39.38 odd 2 1521.2.a.h.1.1 2
52.7 even 12 624.2.bv.b.49.1 2
52.15 even 12 624.2.bv.b.433.1 2
52.51 odd 2 8112.2.a.bu.1.2 2
65.2 even 12 975.2.w.d.199.1 4
65.7 even 12 975.2.w.d.49.2 4
65.28 even 12 975.2.w.d.199.2 4
65.33 even 12 975.2.w.d.49.1 4
65.54 odd 12 975.2.bc.c.901.1 2
65.59 odd 12 975.2.bc.c.751.1 2
156.59 odd 12 1872.2.by.f.1297.1 2
156.119 odd 12 1872.2.by.f.433.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.j.a.4.1 ✓ 2 13.2 odd 12
39.2.j.a.10.1 yes 2 13.7 odd 12
117.2.q.a.10.1 2 39.20 even 12
117.2.q.a.82.1 2 39.2 even 12
507.2.a.e.1.1 2 1.1 even 1 trivial
507.2.a.e.1.2 2 13.12 even 2 inner
507.2.b.c.337.1 2 13.8 odd 4
507.2.b.c.337.2 2 13.5 odd 4
507.2.e.f.22.1 4 13.3 even 3
507.2.e.f.22.2 4 13.10 even 6
507.2.e.f.484.1 4 13.9 even 3
507.2.e.f.484.2 4 13.4 even 6
507.2.j.b.316.1 2 13.11 odd 12
507.2.j.b.361.1 2 13.6 odd 12
624.2.bv.b.49.1 2 52.7 even 12
624.2.bv.b.433.1 2 52.15 even 12
975.2.w.d.49.1 4 65.33 even 12
975.2.w.d.49.2 4 65.7 even 12
975.2.w.d.199.1 4 65.2 even 12
975.2.w.d.199.2 4 65.28 even 12
975.2.bc.c.751.1 2 65.59 odd 12
975.2.bc.c.901.1 2 65.54 odd 12
1521.2.a.h.1.1 2 39.38 odd 2
1521.2.a.h.1.2 2 3.2 odd 2
1521.2.b.f.1351.1 2 39.5 even 4
1521.2.b.f.1351.2 2 39.8 even 4
1872.2.by.f.433.1 2 156.119 odd 12
1872.2.by.f.1297.1 2 156.59 odd 12
8112.2.a.bu.1.1 2 4.3 odd 2
8112.2.a.bu.1.2 2 52.51 odd 2