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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [440,2,Mod(81,440)] 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("440.81"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(440, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 440 = 2^{3} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 440.y (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,1,0,-2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.51341768894\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.13140625.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 5x^{6} - 3x^{5} + 4x^{4} + 3x^{3} + 5x^{2} + 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 81.2
Root \(-0.386111 - 0.280526i\) of defining polynomial
Character \(\chi\) \(=\) 440.81
Dual form 440.2.y.a.201.2

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.0911485 - 0.280526i) q^{3} +(-0.809017 + 0.587785i) q^{5} +(-1.35666 - 4.17538i) q^{7} +(2.35666 + 1.71222i) q^{9} +(3.31118 + 0.189896i) q^{11} +(-4.82402 - 3.50485i) q^{13} +(0.0911485 + 0.280526i) q^{15} +(1.34932 - 0.980336i) q^{17} +(2.37743 - 7.31696i) q^{19} -1.29496 q^{21} -0.904706 q^{23} +(0.309017 - 0.951057i) q^{25} +(1.41102 - 1.02516i) q^{27} +(-1.46443 - 4.50705i) q^{29} +(4.14709 + 3.01303i) q^{31} +(0.355081 - 0.911566i) q^{33} +(3.55179 + 2.58053i) q^{35} +(-0.0571606 - 0.175922i) q^{37} +(-1.42291 + 1.03380i) q^{39} +(-0.810356 + 2.49402i) q^{41} +3.59822 q^{43} -2.91300 q^{45} +(-0.239853 + 0.738191i) q^{47} +(-9.93016 + 7.21469i) q^{49} +(-0.152022 - 0.467875i) q^{51} +(7.76295 + 5.64012i) q^{53} +(-2.79042 + 1.79264i) q^{55} +(-1.83590 - 1.33386i) q^{57} +(-3.47762 - 10.7030i) q^{59} +(-10.6708 + 7.75277i) q^{61} +(3.95196 - 12.1629i) q^{63} +5.96281 q^{65} -7.79954 q^{67} +(-0.0824626 + 0.253794i) q^{69} +(5.63943 - 4.09729i) q^{71} +(3.94122 + 12.1298i) q^{73} +(-0.238630 - 0.173375i) q^{75} +(-3.69927 - 14.0831i) q^{77} +(8.11547 + 5.89624i) q^{79} +(2.54152 + 7.82200i) q^{81} +(-3.31316 + 2.40715i) q^{83} +(-0.515393 + 1.58622i) q^{85} -1.39783 q^{87} -0.466291 q^{89} +(-8.08953 + 24.8970i) q^{91} +(1.22324 - 0.888733i) q^{93} +(2.37743 + 7.31696i) q^{95} +(-5.74372 - 4.17306i) q^{97} +(7.47820 + 6.11699i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 2 q^{5} + q^{7} + 7 q^{9} + 3 q^{11} - 4 q^{13} + q^{15} - 3 q^{17} + 9 q^{19} - 4 q^{21} - 22 q^{23} - 2 q^{25} - 8 q^{27} - 17 q^{29} - 4 q^{31} + 21 q^{33} + 6 q^{35} + 24 q^{37} - 13 q^{39}+ \cdots + 22 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/440\mathbb{Z}\right)^\times\).

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{5}\right)\)

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
\(3\) 0.0911485 − 0.280526i 0.0526246 − 0.161962i −0.921290 − 0.388876i \(-0.872864\pi\)
0.973915 + 0.226914i \(0.0728635\pi\)
\(4\) 0 0
\(5\) −0.809017 + 0.587785i −0.361803 + 0.262866i
\(6\) 0 0
\(7\) −1.35666 − 4.17538i −0.512771 − 1.57815i −0.787302 − 0.616568i \(-0.788523\pi\)
0.274531 − 0.961578i \(-0.411477\pi\)
\(8\) 0 0
\(9\) 2.35666 + 1.71222i 0.785555 + 0.570739i
\(10\) 0 0
\(11\) 3.31118 + 0.189896i 0.998360 + 0.0572559i
\(12\) 0 0
\(13\) −4.82402 − 3.50485i −1.33794 − 0.972071i −0.999517 − 0.0310775i \(-0.990106\pi\)
−0.338424 − 0.940994i \(-0.609894\pi\)
\(14\) 0 0
\(15\) 0.0911485 + 0.280526i 0.0235345 + 0.0724316i
\(16\) 0 0
\(17\) 1.34932 − 0.980336i 0.327258 − 0.237767i −0.412009 − 0.911180i \(-0.635173\pi\)
0.739266 + 0.673413i \(0.235173\pi\)
\(18\) 0 0
\(19\) 2.37743 − 7.31696i 0.545419 − 1.67863i −0.174573 − 0.984644i \(-0.555855\pi\)
0.719992 − 0.693982i \(-0.244145\pi\)
\(20\) 0 0
\(21\) −1.29496 −0.282584
\(22\) 0 0
\(23\) −0.904706 −0.188644 −0.0943221 − 0.995542i \(-0.530068\pi\)
−0.0943221 + 0.995542i \(0.530068\pi\)
\(24\) 0 0
\(25\) 0.309017 − 0.951057i 0.0618034 − 0.190211i
\(26\) 0 0
\(27\) 1.41102 − 1.02516i 0.271551 − 0.197293i
\(28\) 0 0
\(29\) −1.46443 − 4.50705i −0.271938 − 0.836938i −0.990013 − 0.140973i \(-0.954977\pi\)
0.718076 − 0.695965i \(-0.245023\pi\)
\(30\) 0 0
\(31\) 4.14709 + 3.01303i 0.744839 + 0.541157i 0.894223 − 0.447622i \(-0.147729\pi\)
−0.149384 + 0.988779i \(0.547729\pi\)
\(32\) 0 0
\(33\) 0.355081 − 0.911566i 0.0618116 − 0.158683i
\(34\) 0 0
\(35\) 3.55179 + 2.58053i 0.600362 + 0.436189i
\(36\) 0 0
\(37\) −0.0571606 − 0.175922i −0.00939715 − 0.0289214i 0.946248 − 0.323442i \(-0.104840\pi\)
−0.955645 + 0.294521i \(0.904840\pi\)
\(38\) 0 0
\(39\) −1.42291 + 1.03380i −0.227847 + 0.165541i
\(40\) 0 0
\(41\) −0.810356 + 2.49402i −0.126556 + 0.389501i −0.994181 − 0.107719i \(-0.965645\pi\)
0.867625 + 0.497219i \(0.165645\pi\)
\(42\) 0 0
\(43\) 3.59822 0.548723 0.274361 − 0.961627i \(-0.411534\pi\)
0.274361 + 0.961627i \(0.411534\pi\)
\(44\) 0 0
\(45\) −2.91300 −0.434244
\(46\) 0 0
\(47\) −0.239853 + 0.738191i −0.0349861 + 0.107676i −0.967024 − 0.254683i \(-0.918029\pi\)
0.932038 + 0.362360i \(0.118029\pi\)
\(48\) 0 0
\(49\) −9.93016 + 7.21469i −1.41859 + 1.03067i
\(50\) 0 0
\(51\) −0.152022 − 0.467875i −0.0212873 − 0.0655157i
\(52\) 0 0
\(53\) 7.76295 + 5.64012i 1.06632 + 0.774729i 0.975248 − 0.221114i \(-0.0709694\pi\)
0.0910758 + 0.995844i \(0.470969\pi\)
\(54\) 0 0
\(55\) −2.79042 + 1.79264i −0.376260 + 0.241719i
\(56\) 0 0
\(57\) −1.83590 − 1.33386i −0.243171 − 0.176674i
\(58\) 0 0
\(59\) −3.47762 − 10.7030i −0.452748 − 1.39341i −0.873759 − 0.486359i \(-0.838325\pi\)
0.421012 − 0.907055i \(-0.361675\pi\)
\(60\) 0 0
\(61\) −10.6708 + 7.75277i −1.36625 + 0.992640i −0.368233 + 0.929734i \(0.620037\pi\)
−0.998019 + 0.0629067i \(0.979963\pi\)
\(62\) 0 0
\(63\) 3.95196 − 12.1629i 0.497900 − 1.53238i
\(64\) 0 0
\(65\) 5.96281 0.739596
\(66\) 0 0
\(67\) −7.79954 −0.952866 −0.476433 − 0.879211i \(-0.658070\pi\)
−0.476433 + 0.879211i \(0.658070\pi\)
\(68\) 0 0
\(69\) −0.0824626 + 0.253794i −0.00992734 + 0.0305532i
\(70\) 0 0
\(71\) 5.63943 − 4.09729i 0.669278 − 0.486259i −0.200506 − 0.979693i \(-0.564258\pi\)
0.869783 + 0.493434i \(0.164258\pi\)
\(72\) 0 0
\(73\) 3.94122 + 12.1298i 0.461285 + 1.41969i 0.863595 + 0.504186i \(0.168207\pi\)
−0.402310 + 0.915504i \(0.631793\pi\)
\(74\) 0 0
\(75\) −0.238630 − 0.173375i −0.0275546 − 0.0200196i
\(76\) 0 0
\(77\) −3.69927 − 14.0831i −0.421571 − 1.60492i
\(78\) 0 0
\(79\) 8.11547 + 5.89624i 0.913062 + 0.663378i 0.941787 − 0.336209i \(-0.109145\pi\)
−0.0287255 + 0.999587i \(0.509145\pi\)
\(80\) 0 0
\(81\) 2.54152 + 7.82200i 0.282391 + 0.869112i
\(82\) 0 0
\(83\) −3.31316 + 2.40715i −0.363667 + 0.264219i −0.754580 − 0.656208i \(-0.772159\pi\)
0.390913 + 0.920428i \(0.372159\pi\)
\(84\) 0 0
\(85\) −0.515393 + 1.58622i −0.0559023 + 0.172049i
\(86\) 0 0
\(87\) −1.39783 −0.149863
\(88\) 0 0
\(89\) −0.466291 −0.0494267 −0.0247134 − 0.999695i \(-0.507867\pi\)
−0.0247134 + 0.999695i \(0.507867\pi\)
\(90\) 0 0
\(91\) −8.08953 + 24.8970i −0.848013 + 2.60992i
\(92\) 0 0
\(93\) 1.22324 − 0.888733i 0.126844 − 0.0921574i
\(94\) 0 0
\(95\) 2.37743 + 7.31696i 0.243919 + 0.750705i
\(96\) 0 0
\(97\) −5.74372 − 4.17306i −0.583186 − 0.423710i 0.256685 − 0.966495i \(-0.417370\pi\)
−0.839871 + 0.542785i \(0.817370\pi\)
\(98\) 0 0
\(99\) 7.47820 + 6.11699i 0.751588 + 0.614780i
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 440.2.y.a.81.2 ✓ 8
4.3 odd 2 880.2.bo.d.81.1 8
11.3 even 5 inner 440.2.y.a.201.2 yes 8
11.5 even 5 4840.2.a.y.1.3 4
11.6 odd 10 4840.2.a.z.1.3 4
44.3 odd 10 880.2.bo.d.641.1 8
44.27 odd 10 9680.2.a.cu.1.2 4
44.39 even 10 9680.2.a.ct.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.a.81.2 ✓ 8 1.1 even 1 trivial
440.2.y.a.201.2 yes 8 11.3 even 5 inner
880.2.bo.d.81.1 8 4.3 odd 2
880.2.bo.d.641.1 8 44.3 odd 10
4840.2.a.y.1.3 4 11.5 even 5
4840.2.a.z.1.3 4 11.6 odd 10
9680.2.a.ct.1.2 4 44.39 even 10
9680.2.a.cu.1.2 4 44.27 odd 10