Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4400,2,Mod(4049,4400)] 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("4400.4049"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,0,0,-2,0,4,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 7x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 275)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.3
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.y.4049.2

$q$-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)
 
\(f(q)\) \(=\) \(q+1.30278i q^{3} +4.30278i q^{7} +1.30278 q^{9} +1.00000 q^{11} +5.00000i q^{13} +3.90833i q^{17} -1.00000 q^{19} -5.60555 q^{21} +3.69722i q^{23} +5.60555i q^{27} +9.90833 q^{29} +4.21110 q^{31} +1.30278i q^{33} -9.60555i q^{37} -6.51388 q^{39} +1.60555 q^{41} +7.21110i q^{43} -3.00000i q^{47} -11.5139 q^{49} -5.09167 q^{51} +2.30278i q^{53} -1.30278i q^{57} +0.211103 q^{59} +2.90833 q^{61} +5.60555i q^{63} -4.00000i q^{67} -4.81665 q^{69} -4.60555 q^{71} +2.90833i q^{73} +4.30278i q^{77} -0.0916731 q^{79} -3.39445 q^{81} -14.5139i q^{83} +12.9083i q^{87} -5.30278 q^{89} -21.5139 q^{91} +5.48612i q^{93} -11.6972i q^{97} +1.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{9} + 4 q^{11} - 4 q^{19} - 8 q^{21} + 18 q^{29} - 12 q^{31} + 10 q^{39} - 8 q^{41} - 10 q^{49} - 42 q^{51} - 28 q^{59} - 10 q^{61} + 24 q^{69} - 4 q^{71} - 22 q^{79} - 28 q^{81} - 14 q^{89} - 50 q^{91}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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
\(3\) 1.30278i 0.752158i 0.926588 + 0.376079i \(0.122728\pi\)
−0.926588 + 0.376079i \(0.877272\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.30278i 1.62630i 0.582057 + 0.813148i \(0.302248\pi\)
−0.582057 + 0.813148i \(0.697752\pi\)
\(8\) 0 0
\(9\) 1.30278 0.434259
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 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
\(17\) 3.90833i 0.947909i 0.880549 + 0.473954i \(0.157174\pi\)
−0.880549 + 0.473954i \(0.842826\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −5.60555 −1.22323
\(22\) 0 0
\(23\) 3.69722i 0.770925i 0.922724 + 0.385462i \(0.125958\pi\)
−0.922724 + 0.385462i \(0.874042\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.60555i 1.07879i
\(28\) 0 0
\(29\) 9.90833 1.83993 0.919965 − 0.392000i \(-0.128217\pi\)
0.919965 + 0.392000i \(0.128217\pi\)
\(30\) 0 0
\(31\) 4.21110 0.756336 0.378168 − 0.925737i \(-0.376554\pi\)
0.378168 + 0.925737i \(0.376554\pi\)
\(32\) 0 0
\(33\) 1.30278i 0.226784i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.60555i − 1.57914i −0.613659 − 0.789571i \(-0.710303\pi\)
0.613659 − 0.789571i \(-0.289697\pi\)
\(38\) 0 0
\(39\) −6.51388 −1.04306
\(40\) 0 0
\(41\) 1.60555 0.250745 0.125372 − 0.992110i \(-0.459987\pi\)
0.125372 + 0.992110i \(0.459987\pi\)
\(42\) 0 0
\(43\) 7.21110i 1.09968i 0.835269 + 0.549841i \(0.185312\pi\)
−0.835269 + 0.549841i \(0.814688\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 3.00000i − 0.437595i −0.975770 − 0.218797i \(-0.929787\pi\)
0.975770 − 0.218797i \(-0.0702134\pi\)
\(48\) 0 0
\(49\) −11.5139 −1.64484
\(50\) 0 0
\(51\) −5.09167 −0.712977
\(52\) 0 0
\(53\) 2.30278i 0.316311i 0.987414 + 0.158155i \(0.0505547\pi\)
−0.987414 + 0.158155i \(0.949445\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.30278i − 0.172557i
\(58\) 0 0
\(59\) 0.211103 0.0274832 0.0137416 − 0.999906i \(-0.495626\pi\)
0.0137416 + 0.999906i \(0.495626\pi\)
\(60\) 0 0
\(61\) 2.90833 0.372373 0.186187 − 0.982514i \(-0.440387\pi\)
0.186187 + 0.982514i \(0.440387\pi\)
\(62\) 0 0
\(63\) 5.60555i 0.706233i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 − 0.244339i \(-0.921429\pi\)
0.969690 − 0.244339i \(-0.0785709\pi\)
\(68\) 0 0
\(69\) −4.81665 −0.579857
\(70\) 0 0
\(71\) −4.60555 −0.546578 −0.273289 − 0.961932i \(-0.588112\pi\)
−0.273289 + 0.961932i \(0.588112\pi\)
\(72\) 0 0
\(73\) 2.90833i 0.340394i 0.985410 + 0.170197i \(0.0544404\pi\)
−0.985410 + 0.170197i \(0.945560\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.30278i 0.490347i
\(78\) 0 0
\(79\) −0.0916731 −0.0103140 −0.00515701 − 0.999987i \(-0.501642\pi\)
−0.00515701 + 0.999987i \(0.501642\pi\)
\(80\) 0 0
\(81\) −3.39445 −0.377161
\(82\) 0 0
\(83\) − 14.5139i − 1.59311i −0.604569 − 0.796553i \(-0.706655\pi\)
0.604569 − 0.796553i \(-0.293345\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 12.9083i 1.38392i
\(88\) 0 0
\(89\) −5.30278 −0.562093 −0.281047 − 0.959694i \(-0.590682\pi\)
−0.281047 + 0.959694i \(0.590682\pi\)
\(90\) 0 0
\(91\) −21.5139 −2.25527
\(92\) 0 0
\(93\) 5.48612i 0.568884i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 11.6972i − 1.18767i −0.804586 − 0.593837i \(-0.797613\pi\)
0.804586 − 0.593837i \(-0.202387\pi\)
\(98\) 0 0
\(99\) 1.30278 0.130934
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 4400.2.b.y.4049.3 4
4.3 odd 2 275.2.b.c.199.1 4
5.2 odd 4 4400.2.a.bh.1.2 2
5.3 odd 4 4400.2.a.bs.1.1 2
5.4 even 2 inner 4400.2.b.y.4049.2 4
12.11 even 2 2475.2.c.k.199.4 4
20.3 even 4 275.2.a.e.1.1 ✓ 2
20.7 even 4 275.2.a.f.1.2 yes 2
20.19 odd 2 275.2.b.c.199.4 4
60.23 odd 4 2475.2.a.t.1.2 2
60.47 odd 4 2475.2.a.o.1.1 2
60.59 even 2 2475.2.c.k.199.1 4
220.43 odd 4 3025.2.a.n.1.2 2
220.87 odd 4 3025.2.a.h.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 ✓ 2 20.3 even 4
275.2.a.f.1.2 yes 2 20.7 even 4
275.2.b.c.199.1 4 4.3 odd 2
275.2.b.c.199.4 4 20.19 odd 2
2475.2.a.o.1.1 2 60.47 odd 4
2475.2.a.t.1.2 2 60.23 odd 4
2475.2.c.k.199.1 4 60.59 even 2
2475.2.c.k.199.4 4 12.11 even 2
3025.2.a.h.1.1 2 220.87 odd 4
3025.2.a.n.1.2 2 220.43 odd 4
4400.2.a.bh.1.2 2 5.2 odd 4
4400.2.a.bs.1.1 2 5.3 odd 4
4400.2.b.y.4049.2 4 5.4 even 2 inner
4400.2.b.y.4049.3 4 1.1 even 1 trivial