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 = [6,0,0,0,0,0,0,0,-12,0,6,0,0,0,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(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: \(6\)
Coefficient field: 6.0.44836416.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 12x^{4} + 36x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2200)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.3
Root \(-0.167449i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.bc.4049.4

$q$-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)
 
\(f(q)\) \(=\) \(q-1.16745i q^{3} +4.97196i q^{7} +1.63706 q^{9} +1.00000 q^{11} -0.665102i q^{13} -6.77647i q^{17} +1.00000 q^{19} +5.80451 q^{21} -2.16745i q^{23} -5.41353i q^{27} -7.97196 q^{29} +8.94392 q^{31} -1.16745i q^{33} +0.139410i q^{37} -0.776472 q^{39} -1.80451 q^{41} -2.80451i q^{43} +0.530387i q^{47} -17.7204 q^{49} -7.91119 q^{51} -6.30216i q^{53} -1.16745i q^{57} +11.4696 q^{59} +10.5810 q^{61} +8.13941i q^{63} +9.60902i q^{67} -2.53039 q^{69} +9.80921 q^{71} -7.02804i q^{73} +4.97196i q^{77} +5.50235 q^{79} -1.40884 q^{81} -13.5810i q^{83} +9.30686i q^{87} +9.58098 q^{89} +3.30686 q^{91} -10.4416i q^{93} +14.7812i q^{97} +1.63706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{9} + 6 q^{11} + 6 q^{19} + 12 q^{21} - 24 q^{29} + 6 q^{31} + 42 q^{39} + 12 q^{41} - 12 q^{49} + 18 q^{51} + 48 q^{59} - 6 q^{61} - 36 q^{69} + 30 q^{71} + 30 q^{79} + 54 q^{81} - 12 q^{89}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
\(3\) − 1.16745i − 0.674027i −0.941500 − 0.337014i \(-0.890583\pi\)
0.941500 − 0.337014i \(-0.109417\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.97196i 1.87922i 0.342241 + 0.939612i \(0.388814\pi\)
−0.342241 + 0.939612i \(0.611186\pi\)
\(8\) 0 0
\(9\) 1.63706 0.545687
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) − 0.665102i − 0.184466i −0.995737 − 0.0922330i \(-0.970600\pi\)
0.995737 − 0.0922330i \(-0.0294005\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.77647i − 1.64354i −0.569822 − 0.821768i \(-0.692988\pi\)
0.569822 − 0.821768i \(-0.307012\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 − 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 5.80451 1.26665
\(22\) 0 0
\(23\) − 2.16745i − 0.451944i −0.974134 − 0.225972i \(-0.927444\pi\)
0.974134 − 0.225972i \(-0.0725559\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.41353i − 1.04184i
\(28\) 0 0
\(29\) −7.97196 −1.48036 −0.740178 − 0.672411i \(-0.765259\pi\)
−0.740178 + 0.672411i \(0.765259\pi\)
\(30\) 0 0
\(31\) 8.94392 1.60638 0.803188 − 0.595726i \(-0.203136\pi\)
0.803188 + 0.595726i \(0.203136\pi\)
\(32\) 0 0
\(33\) − 1.16745i − 0.203227i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.139410i 0.0229189i 0.999934 + 0.0114594i \(0.00364773\pi\)
−0.999934 + 0.0114594i \(0.996352\pi\)
\(38\) 0 0
\(39\) −0.776472 −0.124335
\(40\) 0 0
\(41\) −1.80451 −0.281817 −0.140909 − 0.990023i \(-0.545002\pi\)
−0.140909 + 0.990023i \(0.545002\pi\)
\(42\) 0 0
\(43\) − 2.80451i − 0.427684i −0.976868 − 0.213842i \(-0.931402\pi\)
0.976868 − 0.213842i \(-0.0685978\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0.530387i 0.0773649i 0.999252 + 0.0386824i \(0.0123161\pi\)
−0.999252 + 0.0386824i \(0.987684\pi\)
\(48\) 0 0
\(49\) −17.7204 −2.53148
\(50\) 0 0
\(51\) −7.91119 −1.10779
\(52\) 0 0
\(53\) − 6.30216i − 0.865669i −0.901473 − 0.432834i \(-0.857513\pi\)
0.901473 − 0.432834i \(-0.142487\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.16745i − 0.154632i
\(58\) 0 0
\(59\) 11.4696 1.49322 0.746608 − 0.665264i \(-0.231681\pi\)
0.746608 + 0.665264i \(0.231681\pi\)
\(60\) 0 0
\(61\) 10.5810 1.35476 0.677378 − 0.735635i \(-0.263116\pi\)
0.677378 + 0.735635i \(0.263116\pi\)
\(62\) 0 0
\(63\) 8.13941i 1.02547i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.60902i 1.17393i 0.809613 + 0.586965i \(0.199677\pi\)
−0.809613 + 0.586965i \(0.800323\pi\)
\(68\) 0 0
\(69\) −2.53039 −0.304623
\(70\) 0 0
\(71\) 9.80921 1.16414 0.582069 − 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(72\) 0 0
\(73\) − 7.02804i − 0.822570i −0.911507 − 0.411285i \(-0.865080\pi\)
0.911507 − 0.411285i \(-0.134920\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.97196i 0.566608i
\(78\) 0 0
\(79\) 5.50235 0.619062 0.309531 − 0.950889i \(-0.399828\pi\)
0.309531 + 0.950889i \(0.399828\pi\)
\(80\) 0 0
\(81\) −1.40884 −0.156538
\(82\) 0 0
\(83\) − 13.5810i − 1.49071i −0.666670 − 0.745353i \(-0.732281\pi\)
0.666670 − 0.745353i \(-0.267719\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.30686i 0.997800i
\(88\) 0 0
\(89\) 9.58098 1.01558 0.507791 − 0.861480i \(-0.330462\pi\)
0.507791 + 0.861480i \(0.330462\pi\)
\(90\) 0 0
\(91\) 3.30686 0.346653
\(92\) 0 0
\(93\) − 10.4416i − 1.08274i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 14.7812i 1.50080i 0.660984 + 0.750400i \(0.270139\pi\)
−0.660984 + 0.750400i \(0.729861\pi\)
\(98\) 0 0
\(99\) 1.63706 0.164531
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.bc.4049.3 6
4.3 odd 2 2200.2.b.l.1849.4 6
5.2 odd 4 4400.2.a.bx.1.2 3
5.3 odd 4 4400.2.a.ca.1.2 3
5.4 even 2 inner 4400.2.b.bc.4049.4 6
20.3 even 4 2200.2.a.t.1.2 ✓ 3
20.7 even 4 2200.2.a.w.1.2 yes 3
20.19 odd 2 2200.2.b.l.1849.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2200.2.a.t.1.2 ✓ 3 20.3 even 4
2200.2.a.w.1.2 yes 3 20.7 even 4
2200.2.b.l.1849.3 6 20.19 odd 2
2200.2.b.l.1849.4 6 4.3 odd 2
4400.2.a.bx.1.2 3 5.2 odd 4
4400.2.a.ca.1.2 3 5.3 odd 4
4400.2.b.bc.4049.3 6 1.1 even 1 trivial
4400.2.b.bc.4049.4 6 5.4 even 2 inner