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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8100,2,Mod(649,8100)] 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("8100.649"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8100, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,0,0,-12,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(29)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(64.6788256372\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 324)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 8100.649
Dual form 8100.2.d.a.649.1

$q$-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)
 
\(f(q)\) \(=\) \(q+2.00000i q^{7} -6.00000 q^{11} -5.00000i q^{13} -3.00000i q^{17} -2.00000 q^{19} -6.00000i q^{23} -3.00000 q^{29} -4.00000 q^{31} +5.00000i q^{37} -6.00000 q^{41} +10.0000i q^{43} +3.00000 q^{49} +6.00000i q^{53} +12.0000 q^{59} +5.00000 q^{61} +2.00000i q^{67} +6.00000 q^{71} +1.00000i q^{73} -12.0000i q^{77} +10.0000 q^{79} +3.00000 q^{89} +10.0000 q^{91} -10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{11} - 4 q^{19} - 6 q^{29} - 8 q^{31} - 12 q^{41} + 6 q^{49} + 24 q^{59} + 10 q^{61} + 12 q^{71} + 20 q^{79} + 6 q^{89} + 20 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −6.00000 −1.80907 −0.904534 − 0.426401i \(-0.859781\pi\)
−0.904534 + 0.426401i \(0.859781\pi\)
\(12\) 0 0
\(13\) − 5.00000i − 1.38675i −0.720577 − 0.693375i \(-0.756123\pi\)
0.720577 − 0.693375i \(-0.243877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.00000i − 0.727607i −0.931476 − 0.363803i \(-0.881478\pi\)
0.931476 − 0.363803i \(-0.118522\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 − 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 6.00000i − 1.25109i −0.780189 − 0.625543i \(-0.784877\pi\)
0.780189 − 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 − 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.00000i 0.821995i 0.911636 + 0.410997i \(0.134819\pi\)
−0.911636 + 0.410997i \(0.865181\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 10.0000i 1.52499i 0.646997 + 0.762493i \(0.276025\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 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
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 − 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 − 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000i 0.244339i 0.992509 + 0.122169i \(0.0389851\pi\)
−0.992509 + 0.122169i \(0.961015\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 − 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 0 0
\(73\) 1.00000i 0.117041i 0.998286 + 0.0585206i \(0.0186383\pi\)
−0.998286 + 0.0585206i \(0.981362\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) 10.0000 1.12509 0.562544 − 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.00000 0.317999 0.159000 − 0.987279i \(-0.449173\pi\)
0.159000 + 0.987279i \(0.449173\pi\)
\(90\) 0 0
\(91\) 10.0000 1.04828
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 10.0000i − 1.01535i −0.861550 − 0.507673i \(-0.830506\pi\)
0.861550 − 0.507673i \(-0.169494\pi\)
\(98\) 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 8100.2.d.a.649.2 2
3.2 odd 2 8100.2.d.j.649.2 2
5.2 odd 4 8100.2.a.a.1.1 1
5.3 odd 4 324.2.a.d.1.1 yes 1
5.4 even 2 inner 8100.2.d.a.649.1 2
15.2 even 4 8100.2.a.f.1.1 1
15.8 even 4 324.2.a.b.1.1 ✓ 1
15.14 odd 2 8100.2.d.j.649.1 2
20.3 even 4 1296.2.a.j.1.1 1
40.3 even 4 5184.2.a.d.1.1 1
40.13 odd 4 5184.2.a.g.1.1 1
45.13 odd 12 324.2.e.a.217.1 2
45.23 even 12 324.2.e.d.217.1 2
45.38 even 12 324.2.e.d.109.1 2
45.43 odd 12 324.2.e.a.109.1 2
60.23 odd 4 1296.2.a.a.1.1 1
120.53 even 4 5184.2.a.bc.1.1 1
120.83 odd 4 5184.2.a.z.1.1 1
180.23 odd 12 1296.2.i.p.865.1 2
180.43 even 12 1296.2.i.d.433.1 2
180.83 odd 12 1296.2.i.p.433.1 2
180.103 even 12 1296.2.i.d.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
324.2.a.b.1.1 ✓ 1 15.8 even 4
324.2.a.d.1.1 yes 1 5.3 odd 4
324.2.e.a.109.1 2 45.43 odd 12
324.2.e.a.217.1 2 45.13 odd 12
324.2.e.d.109.1 2 45.38 even 12
324.2.e.d.217.1 2 45.23 even 12
1296.2.a.a.1.1 1 60.23 odd 4
1296.2.a.j.1.1 1 20.3 even 4
1296.2.i.d.433.1 2 180.43 even 12
1296.2.i.d.865.1 2 180.103 even 12
1296.2.i.p.433.1 2 180.83 odd 12
1296.2.i.p.865.1 2 180.23 odd 12
5184.2.a.d.1.1 1 40.3 even 4
5184.2.a.g.1.1 1 40.13 odd 4
5184.2.a.z.1.1 1 120.83 odd 4
5184.2.a.bc.1.1 1 120.53 even 4
8100.2.a.a.1.1 1 5.2 odd 4
8100.2.a.f.1.1 1 15.2 even 4
8100.2.d.a.649.1 2 5.4 even 2 inner
8100.2.d.a.649.2 2 1.1 even 1 trivial
8100.2.d.j.649.1 2 15.14 odd 2
8100.2.d.j.649.2 2 3.2 odd 2