Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5100,2,Mod(2449,5100)] 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("5100.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 5100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5100.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-2,0,10,0,0,0,0,0,0,0,-2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(40.7237050309\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 204)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2449.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 5100.2449
Dual form 5100.2.g.l.2449.2

$q$-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)
 
\(f(q)\) \(=\) \(q-1.00000i q^{3} -1.00000 q^{9} +5.00000 q^{11} +5.00000i q^{13} +1.00000i q^{17} -1.00000 q^{19} +3.00000i q^{23} +1.00000i q^{27} -2.00000 q^{29} +2.00000 q^{31} -5.00000i q^{33} -8.00000i q^{37} +5.00000 q^{39} -5.00000 q^{41} +9.00000i q^{43} +6.00000i q^{47} +7.00000 q^{49} +1.00000 q^{51} +6.00000i q^{53} +1.00000i q^{57} -6.00000 q^{59} -4.00000 q^{61} +12.0000i q^{67} +3.00000 q^{69} -12.0000 q^{71} +2.00000i q^{73} -10.0000 q^{79} +1.00000 q^{81} +2.00000i q^{83} +2.00000i q^{87} -12.0000 q^{89} -2.00000i q^{93} +16.0000i q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{9} + 10 q^{11} - 2 q^{19} - 4 q^{29} + 4 q^{31} + 10 q^{39} - 10 q^{41} + 14 q^{49} + 2 q^{51} - 12 q^{59} - 8 q^{61} + 6 q^{69} - 24 q^{71} - 20 q^{79} + 2 q^{81} - 24 q^{89} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5100\mathbb{Z}\right)^\times\).

\(n\) \(2551\) \(3301\) \(3401\) \(3877\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.00000i − 0.577350i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 − 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.00000i 0.242536i
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.00000i 0.625543i 0.949828 + 0.312772i \(0.101257\pi\)
−0.949828 + 0.312772i \(0.898743\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 − 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 − 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) 0 0
\(33\) − 5.00000i − 0.870388i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 5.00000 0.800641
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 − 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 9.00000i 1.37249i 0.727372 + 0.686244i \(0.240742\pi\)
−0.727372 + 0.686244i \(0.759258\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.00000i 0.132453i
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 − 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 − 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 0 0
\(69\) 3.00000 0.361158
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 − 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 − 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.00000i 0.219529i 0.993958 + 0.109764i \(0.0350096\pi\)
−0.993958 + 0.109764i \(0.964990\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 2.00000i − 0.207390i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.0000i 1.62455i 0.583272 + 0.812277i \(0.301772\pi\)
−0.583272 + 0.812277i \(0.698228\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
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 5100.2.g.l.2449.1 2
5.2 odd 4 5100.2.a.f.1.1 1
5.3 odd 4 204.2.a.b.1.1 ✓ 1
5.4 even 2 inner 5100.2.g.l.2449.2 2
15.8 even 4 612.2.a.b.1.1 1
20.3 even 4 816.2.a.e.1.1 1
35.13 even 4 9996.2.a.b.1.1 1
40.3 even 4 3264.2.a.u.1.1 1
40.13 odd 4 3264.2.a.f.1.1 1
60.23 odd 4 2448.2.a.e.1.1 1
85.8 odd 8 3468.2.j.f.829.1 4
85.13 odd 4 3468.2.b.a.577.2 2
85.33 odd 4 3468.2.a.a.1.1 1
85.38 odd 4 3468.2.b.a.577.1 2
85.43 odd 8 3468.2.j.f.829.2 4
85.53 odd 8 3468.2.j.f.3217.2 4
85.83 odd 8 3468.2.j.f.3217.1 4
120.53 even 4 9792.2.a.bo.1.1 1
120.83 odd 4 9792.2.a.bn.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
204.2.a.b.1.1 ✓ 1 5.3 odd 4
612.2.a.b.1.1 1 15.8 even 4
816.2.a.e.1.1 1 20.3 even 4
2448.2.a.e.1.1 1 60.23 odd 4
3264.2.a.f.1.1 1 40.13 odd 4
3264.2.a.u.1.1 1 40.3 even 4
3468.2.a.a.1.1 1 85.33 odd 4
3468.2.b.a.577.1 2 85.38 odd 4
3468.2.b.a.577.2 2 85.13 odd 4
3468.2.j.f.829.1 4 85.8 odd 8
3468.2.j.f.829.2 4 85.43 odd 8
3468.2.j.f.3217.1 4 85.83 odd 8
3468.2.j.f.3217.2 4 85.53 odd 8
5100.2.a.f.1.1 1 5.2 odd 4
5100.2.g.l.2449.1 2 1.1 even 1 trivial
5100.2.g.l.2449.2 2 5.4 even 2 inner
9792.2.a.bn.1.1 1 120.83 odd 4
9792.2.a.bo.1.1 1 120.53 even 4
9996.2.a.b.1.1 1 35.13 even 4