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(1,4400)] 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("4400.1"); 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, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-1,0,0,0,-5,0,5,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(35.1341768894\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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 1100)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.79129\) of defining polynomial
Character \(\chi\) \(=\) 4400.1

$q$-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)
 
\(f(q)\) \(=\) \(q-2.79129 q^{3} -0.208712 q^{7} +4.79129 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.791288 q^{17} -6.58258 q^{19} +0.582576 q^{21} -3.79129 q^{23} -5.00000 q^{27} +6.79129 q^{29} +8.58258 q^{31} -2.79129 q^{33} +2.58258 q^{37} -2.79129 q^{39} -1.41742 q^{41} -10.0000 q^{43} -1.41742 q^{47} -6.95644 q^{49} +2.20871 q^{51} -11.3739 q^{53} +18.3739 q^{57} +10.5826 q^{59} +4.20871 q^{61} -1.00000 q^{63} -4.00000 q^{67} +10.5826 q^{69} +10.7477 q^{71} +7.79129 q^{73} -0.208712 q^{77} +15.5390 q^{79} -0.417424 q^{81} -9.95644 q^{83} -18.9564 q^{87} -0.791288 q^{89} -0.208712 q^{91} -23.9564 q^{93} +6.20871 q^{97} +4.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 2 q^{13} + 3 q^{17} - 4 q^{19} - 8 q^{21} - 3 q^{23} - 10 q^{27} + 9 q^{29} + 8 q^{31} - q^{33} - 4 q^{37} - q^{39} - 12 q^{41} - 20 q^{43} - 12 q^{47} + 9 q^{49}+ \cdots + 5 q^{99}+O(q^{100}) \) 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
\(3\) −2.79129 −1.61155 −0.805775 − 0.592221i \(-0.798251\pi\)
−0.805775 + 0.592221i \(0.798251\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −0.208712 −0.0788858 −0.0394429 − 0.999222i \(-0.512558\pi\)
−0.0394429 + 0.999222i \(0.512558\pi\)
\(8\) 0 0
\(9\) 4.79129 1.59710
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 − 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.791288 −0.191915 −0.0959577 − 0.995385i \(-0.530591\pi\)
−0.0959577 + 0.995385i \(0.530591\pi\)
\(18\) 0 0
\(19\) −6.58258 −1.51015 −0.755073 − 0.655640i \(-0.772399\pi\)
−0.755073 + 0.655640i \(0.772399\pi\)
\(20\) 0 0
\(21\) 0.582576 0.127128
\(22\) 0 0
\(23\) −3.79129 −0.790538 −0.395269 − 0.918565i \(-0.629349\pi\)
−0.395269 + 0.918565i \(0.629349\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.00000 −0.962250
\(28\) 0 0
\(29\) 6.79129 1.26111 0.630555 − 0.776144i \(-0.282827\pi\)
0.630555 + 0.776144i \(0.282827\pi\)
\(30\) 0 0
\(31\) 8.58258 1.54148 0.770738 − 0.637152i \(-0.219888\pi\)
0.770738 + 0.637152i \(0.219888\pi\)
\(32\) 0 0
\(33\) −2.79129 −0.485901
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.58258 0.424573 0.212286 − 0.977207i \(-0.431909\pi\)
0.212286 + 0.977207i \(0.431909\pi\)
\(38\) 0 0
\(39\) −2.79129 −0.446964
\(40\) 0 0
\(41\) −1.41742 −0.221364 −0.110682 − 0.993856i \(-0.535304\pi\)
−0.110682 + 0.993856i \(0.535304\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 − 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.41742 −0.206753 −0.103376 − 0.994642i \(-0.532965\pi\)
−0.103376 + 0.994642i \(0.532965\pi\)
\(48\) 0 0
\(49\) −6.95644 −0.993777
\(50\) 0 0
\(51\) 2.20871 0.309282
\(52\) 0 0
\(53\) −11.3739 −1.56232 −0.781160 − 0.624331i \(-0.785372\pi\)
−0.781160 + 0.624331i \(0.785372\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 18.3739 2.43368
\(58\) 0 0
\(59\) 10.5826 1.37773 0.688867 − 0.724888i \(-0.258108\pi\)
0.688867 + 0.724888i \(0.258108\pi\)
\(60\) 0 0
\(61\) 4.20871 0.538870 0.269435 − 0.963019i \(-0.413163\pi\)
0.269435 + 0.963019i \(0.413163\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) 0 0
\(69\) 10.5826 1.27399
\(70\) 0 0
\(71\) 10.7477 1.27552 0.637760 − 0.770235i \(-0.279861\pi\)
0.637760 + 0.770235i \(0.279861\pi\)
\(72\) 0 0
\(73\) 7.79129 0.911901 0.455951 − 0.890005i \(-0.349299\pi\)
0.455951 + 0.890005i \(0.349299\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −0.208712 −0.0237850
\(78\) 0 0
\(79\) 15.5390 1.74828 0.874138 − 0.485678i \(-0.161427\pi\)
0.874138 + 0.485678i \(0.161427\pi\)
\(80\) 0 0
\(81\) −0.417424 −0.0463805
\(82\) 0 0
\(83\) −9.95644 −1.09286 −0.546431 − 0.837504i \(-0.684014\pi\)
−0.546431 + 0.837504i \(0.684014\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −18.9564 −2.03234
\(88\) 0 0
\(89\) −0.791288 −0.0838763 −0.0419382 − 0.999120i \(-0.513353\pi\)
−0.0419382 + 0.999120i \(0.513353\pi\)
\(90\) 0 0
\(91\) −0.208712 −0.0218790
\(92\) 0 0
\(93\) −23.9564 −2.48417
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.20871 0.630399 0.315200 − 0.949025i \(-0.397929\pi\)
0.315200 + 0.949025i \(0.397929\pi\)
\(98\) 0 0
\(99\) 4.79129 0.481543
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.a.bi.1.1 2
4.3 odd 2 1100.2.a.h.1.2 yes 2
5.2 odd 4 4400.2.b.s.4049.4 4
5.3 odd 4 4400.2.b.s.4049.1 4
5.4 even 2 4400.2.a.bu.1.2 2
12.11 even 2 9900.2.a.bz.1.1 2
20.3 even 4 1100.2.b.d.749.4 4
20.7 even 4 1100.2.b.d.749.1 4
20.19 odd 2 1100.2.a.g.1.1 ✓ 2
60.23 odd 4 9900.2.c.x.5149.2 4
60.47 odd 4 9900.2.c.x.5149.3 4
60.59 even 2 9900.2.a.bh.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1100.2.a.g.1.1 ✓ 2 20.19 odd 2
1100.2.a.h.1.2 yes 2 4.3 odd 2
1100.2.b.d.749.1 4 20.7 even 4
1100.2.b.d.749.4 4 20.3 even 4
4400.2.a.bi.1.1 2 1.1 even 1 trivial
4400.2.a.bu.1.2 2 5.4 even 2
4400.2.b.s.4049.1 4 5.3 odd 4
4400.2.b.s.4049.4 4 5.2 odd 4
9900.2.a.bh.1.2 2 60.59 even 2
9900.2.a.bz.1.1 2 12.11 even 2
9900.2.c.x.5149.2 4 60.23 odd 4
9900.2.c.x.5149.3 4 60.47 odd 4