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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 = [16,0,-3,0,-4,0,8] 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: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 14 x^{14} - 32 x^{13} + 141 x^{12} - 220 x^{11} + 1105 x^{10} - 1935 x^{9} + \cdots + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

Embedding invariants

Embedding label 201.1
Root \(0.856564 + 2.63623i\) of defining polynomial
Character \(\chi\) \(=\) 440.201
Dual form 440.2.y.d.81.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-0.856564 - 2.63623i) q^{3} +(-0.809017 - 0.587785i) q^{5} +(-1.40759 + 4.33212i) q^{7} +(-3.78896 + 2.75284i) q^{9} +(-3.01954 + 1.37200i) q^{11} +(-2.34636 + 1.70473i) q^{13} +(-0.856564 + 2.63623i) q^{15} +(5.73433 + 4.16623i) q^{17} +(-2.20745 - 6.79384i) q^{19} +12.6262 q^{21} -1.86699 q^{23} +(0.309017 + 0.951057i) q^{25} +(3.77509 + 2.74276i) q^{27} +(-0.312324 + 0.961233i) q^{29} +(-3.56804 + 2.59234i) q^{31} +(6.20333 + 6.78500i) q^{33} +(3.68512 - 2.67740i) q^{35} +(-1.66589 + 5.12709i) q^{37} +(6.50386 + 4.72533i) q^{39} +(-1.38684 - 4.26825i) q^{41} -6.73002 q^{43} +4.68342 q^{45} +(2.42584 + 7.46596i) q^{47} +(-11.1228 - 8.08120i) q^{49} +(6.07134 - 18.6857i) q^{51} +(-0.578008 + 0.419948i) q^{53} +(3.24930 + 0.664870i) q^{55} +(-16.0193 + 11.6387i) q^{57} +(-1.17528 + 3.61715i) q^{59} +(-1.71184 - 1.24373i) q^{61} +(-6.59234 - 20.2891i) q^{63} +2.90026 q^{65} -12.8384 q^{67} +(1.59920 + 4.92182i) q^{69} +(-3.40673 - 2.47513i) q^{71} +(0.422042 - 1.29891i) q^{73} +(2.24251 - 1.62928i) q^{75} +(-1.69339 - 15.0122i) q^{77} +(9.90032 - 7.19300i) q^{79} +(-0.344817 + 1.06124i) q^{81} +(13.2153 + 9.60151i) q^{83} +(-2.19032 - 6.74110i) q^{85} +2.80156 q^{87} -11.8165 q^{89} +(-4.08237 - 12.5643i) q^{91} +(9.89025 + 7.18569i) q^{93} +(-2.20745 + 6.79384i) q^{95} +(8.78139 - 6.38005i) q^{97} +(7.66403 - 13.5108i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{3} - 4 q^{5} + 8 q^{7} - 7 q^{9} - 7 q^{11} - 11 q^{13} - 3 q^{15} + 9 q^{17} - 2 q^{19} + 12 q^{21} + 20 q^{23} - 4 q^{25} - 9 q^{27} + q^{29} - 2 q^{31} - 32 q^{33} - 2 q^{35} - 16 q^{37}+ \cdots - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(111\) \(177\) \(221\) \(321\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{4}{5}\right)\)

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
\(3\) −0.856564 − 2.63623i −0.494537 − 1.52203i −0.817677 − 0.575678i \(-0.804738\pi\)
0.323140 − 0.946351i \(-0.395262\pi\)
\(4\) 0 0
\(5\) −0.809017 − 0.587785i −0.361803 − 0.262866i
\(6\) 0 0
\(7\) −1.40759 + 4.33212i −0.532019 + 1.63739i 0.217984 + 0.975952i \(0.430052\pi\)
−0.750003 + 0.661434i \(0.769948\pi\)
\(8\) 0 0
\(9\) −3.78896 + 2.75284i −1.26299 + 0.917615i
\(10\) 0 0
\(11\) −3.01954 + 1.37200i −0.910425 + 0.413673i
\(12\) 0 0
\(13\) −2.34636 + 1.70473i −0.650762 + 0.472807i −0.863531 − 0.504296i \(-0.831752\pi\)
0.212768 + 0.977103i \(0.431752\pi\)
\(14\) 0 0
\(15\) −0.856564 + 2.63623i −0.221164 + 0.680672i
\(16\) 0 0
\(17\) 5.73433 + 4.16623i 1.39078 + 1.01046i 0.995779 + 0.0917781i \(0.0292550\pi\)
0.394999 + 0.918682i \(0.370745\pi\)
\(18\) 0 0
\(19\) −2.20745 − 6.79384i −0.506424 − 1.55861i −0.798363 − 0.602177i \(-0.794300\pi\)
0.291938 − 0.956437i \(-0.405700\pi\)
\(20\) 0 0
\(21\) 12.6262 2.75525
\(22\) 0 0
\(23\) −1.86699 −0.389295 −0.194647 − 0.980873i \(-0.562356\pi\)
−0.194647 + 0.980873i \(0.562356\pi\)
\(24\) 0 0
\(25\) 0.309017 + 0.951057i 0.0618034 + 0.190211i
\(26\) 0 0
\(27\) 3.77509 + 2.74276i 0.726516 + 0.527844i
\(28\) 0 0
\(29\) −0.312324 + 0.961233i −0.0579970 + 0.178497i −0.975858 − 0.218405i \(-0.929915\pi\)
0.917861 + 0.396902i \(0.129915\pi\)
\(30\) 0 0
\(31\) −3.56804 + 2.59234i −0.640840 + 0.465597i −0.860139 − 0.510061i \(-0.829623\pi\)
0.219299 + 0.975658i \(0.429623\pi\)
\(32\) 0 0
\(33\) 6.20333 + 6.78500i 1.07986 + 1.18112i
\(34\) 0 0
\(35\) 3.68512 − 2.67740i 0.622899 − 0.452563i
\(36\) 0 0
\(37\) −1.66589 + 5.12709i −0.273871 + 0.842888i 0.715645 + 0.698464i \(0.246133\pi\)
−0.989516 + 0.144424i \(0.953867\pi\)
\(38\) 0 0
\(39\) 6.50386 + 4.72533i 1.04145 + 0.756659i
\(40\) 0 0
\(41\) −1.38684 − 4.26825i −0.216588 − 0.666588i −0.999037 − 0.0438739i \(-0.986030\pi\)
0.782450 − 0.622714i \(-0.213970\pi\)
\(42\) 0 0
\(43\) −6.73002 −1.02632 −0.513159 − 0.858293i \(-0.671525\pi\)
−0.513159 + 0.858293i \(0.671525\pi\)
\(44\) 0 0
\(45\) 4.68342 0.698163
\(46\) 0 0
\(47\) 2.42584 + 7.46596i 0.353845 + 1.08902i 0.956676 + 0.291154i \(0.0940390\pi\)
−0.602832 + 0.797868i \(0.705961\pi\)
\(48\) 0 0
\(49\) −11.1228 − 8.08120i −1.58897 − 1.15446i
\(50\) 0 0
\(51\) 6.07134 − 18.6857i 0.850157 − 2.61651i
\(52\) 0 0
\(53\) −0.578008 + 0.419948i −0.0793956 + 0.0576843i −0.626775 − 0.779200i \(-0.715625\pi\)
0.547379 + 0.836885i \(0.315625\pi\)
\(54\) 0 0
\(55\) 3.24930 + 0.664870i 0.438135 + 0.0896511i
\(56\) 0 0
\(57\) −16.0193 + 11.6387i −2.12181 + 1.54159i
\(58\) 0 0
\(59\) −1.17528 + 3.61715i −0.153009 + 0.470913i −0.997954 − 0.0639406i \(-0.979633\pi\)
0.844945 + 0.534853i \(0.179633\pi\)
\(60\) 0 0
\(61\) −1.71184 − 1.24373i −0.219179 − 0.159243i 0.472778 − 0.881181i \(-0.343251\pi\)
−0.691957 + 0.721939i \(0.743251\pi\)
\(62\) 0 0
\(63\) −6.59234 − 20.2891i −0.830556 − 2.55619i
\(64\) 0 0
\(65\) 2.90026 0.359733
\(66\) 0 0
\(67\) −12.8384 −1.56847 −0.784233 − 0.620467i \(-0.786943\pi\)
−0.784233 + 0.620467i \(0.786943\pi\)
\(68\) 0 0
\(69\) 1.59920 + 4.92182i 0.192521 + 0.592518i
\(70\) 0 0
\(71\) −3.40673 − 2.47513i −0.404304 − 0.293744i 0.366988 − 0.930226i \(-0.380389\pi\)
−0.771292 + 0.636482i \(0.780389\pi\)
\(72\) 0 0
\(73\) 0.422042 − 1.29891i 0.0493963 − 0.152026i −0.923316 − 0.384042i \(-0.874532\pi\)
0.972712 + 0.232015i \(0.0745319\pi\)
\(74\) 0 0
\(75\) 2.24251 − 1.62928i 0.258943 − 0.188133i
\(76\) 0 0
\(77\) −1.69339 − 15.0122i −0.192979 − 1.71080i
\(78\) 0 0
\(79\) 9.90032 − 7.19300i 1.11387 − 0.809276i 0.130604 − 0.991435i \(-0.458308\pi\)
0.983269 + 0.182159i \(0.0583084\pi\)
\(80\) 0 0
\(81\) −0.344817 + 1.06124i −0.0383130 + 0.117915i
\(82\) 0 0
\(83\) 13.2153 + 9.60151i 1.45057 + 1.05390i 0.985697 + 0.168527i \(0.0539011\pi\)
0.464876 + 0.885376i \(0.346099\pi\)
\(84\) 0 0
\(85\) −2.19032 − 6.74110i −0.237573 − 0.731175i
\(86\) 0 0
\(87\) 2.80156 0.300359
\(88\) 0 0
\(89\) −11.8165 −1.25255 −0.626273 − 0.779604i \(-0.715420\pi\)
−0.626273 + 0.779604i \(0.715420\pi\)
\(90\) 0 0
\(91\) −4.08237 − 12.5643i −0.427949 − 1.31709i
\(92\) 0 0
\(93\) 9.89025 + 7.18569i 1.02557 + 0.745121i
\(94\) 0 0
\(95\) −2.20745 + 6.79384i −0.226480 + 0.697034i
\(96\) 0 0
\(97\) 8.78139 − 6.38005i 0.891615 − 0.647796i −0.0446836 − 0.999001i \(-0.514228\pi\)
0.936299 + 0.351205i \(0.114228\pi\)
\(98\) 0 0
\(99\) 7.66403 − 13.5108i 0.770264 − 1.35788i
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.d.201.1 yes 16
4.3 odd 2 880.2.bo.k.641.4 16
11.2 odd 10 4840.2.a.bh.1.2 8
11.4 even 5 inner 440.2.y.d.81.1 ✓ 16
11.9 even 5 4840.2.a.bg.1.2 8
44.15 odd 10 880.2.bo.k.81.4 16
44.31 odd 10 9680.2.a.df.1.7 8
44.35 even 10 9680.2.a.de.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.2.y.d.81.1 ✓ 16 11.4 even 5 inner
440.2.y.d.201.1 yes 16 1.1 even 1 trivial
880.2.bo.k.81.4 16 44.15 odd 10
880.2.bo.k.641.4 16 4.3 odd 2
4840.2.a.bg.1.2 8 11.9 even 5
4840.2.a.bh.1.2 8 11.2 odd 10
9680.2.a.de.1.7 8 44.35 even 10
9680.2.a.df.1.7 8 44.31 odd 10