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Properties

Label 1000.2.bd.a.29.20
Level $1000$
Weight $2$
Character 1000.29
Analytic conductor $7.985$
Analytic rank $0$
Dimension $2960$
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1000,2,Mod(29,1000)] 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("1000.29"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1000, base_ring=CyclotomicField(50)) chi = DirichletCharacter(H, H._module([0, 25, 31])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.bd (of order \(50\), degree \(20\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(2960\)
Relative dimension: \(148\) over \(\Q(\zeta_{50})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{50}]$

Embedding invariants

Embedding label 29.20
Character \(\chi\) \(=\) 1000.29
Dual form 1000.2.bd.a.69.20

$q$-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+(-1.29946 - 0.558045i) q^{2} +(2.89362 - 1.59078i) q^{3} +(1.37717 + 1.45031i) q^{4} +(-1.75231 + 1.38903i) q^{5} +(-4.64785 + 0.452378i) q^{6} +(-0.949757 + 1.30723i) q^{7} +(-0.980234 - 2.65314i) q^{8} +(4.23495 - 6.67322i) q^{9} +(3.05219 - 0.827108i) q^{10} +(-1.49265 + 3.77000i) q^{11} +(6.29213 + 2.00587i) q^{12} +(-2.85028 + 4.49132i) q^{13} +(1.96366 - 1.16868i) q^{14} +(-2.86089 + 6.80685i) q^{15} +(-0.206800 + 3.99465i) q^{16} +(3.44965 + 3.67351i) q^{17} +(-9.22710 + 6.30826i) q^{18} +(-2.33177 + 4.24148i) q^{19} +(-4.42776 - 0.628472i) q^{20} +(-0.668721 + 5.29347i) q^{21} +(4.04346 - 4.06599i) q^{22} +(2.57449 + 2.12981i) q^{23} +(-7.05698 - 6.11783i) q^{24} +(1.14121 - 4.86802i) q^{25} +(6.21017 - 4.24568i) q^{26} +(1.01670 - 16.1600i) q^{27} +(-3.20386 + 0.422834i) q^{28} +(6.67500 - 1.27332i) q^{29} +(7.51613 - 7.24870i) q^{30} +(-1.09148 + 1.02497i) q^{31} +(2.49792 - 5.07547i) q^{32} +(1.67809 + 13.2834i) q^{33} +(-2.43269 - 6.69862i) q^{34} +(-0.151502 - 3.60991i) q^{35} +(15.5105 - 3.04817i) q^{36} +(0.160405 + 2.54956i) q^{37} +(5.39697 - 4.21038i) q^{38} +(-1.10291 + 17.5303i) q^{39} +(5.40296 + 3.28756i) q^{40} +(-4.55914 - 5.51106i) q^{41} +(3.82297 - 6.50545i) q^{42} +(0.995791 + 3.06473i) q^{43} +(-7.52331 + 3.02713i) q^{44} +(1.84831 + 17.5760i) q^{45} +(-2.15691 - 4.20427i) q^{46} +(3.70866 + 1.74516i) q^{47} +(5.75620 + 11.8880i) q^{48} +(1.35631 + 4.17430i) q^{49} +(-4.19953 + 5.68893i) q^{50} +(15.8257 + 5.14208i) q^{51} +(-10.4391 + 2.05153i) q^{52} +(-6.25865 - 0.790651i) q^{53} +(-10.3392 + 20.4318i) q^{54} +(-2.62104 - 8.67956i) q^{55} +(4.39924 + 1.23845i) q^{56} +15.9825i q^{57} +(-9.38443 - 2.07032i) q^{58} +(3.05848 - 11.9120i) q^{59} +(-13.8120 + 5.22502i) q^{60} +(-6.62113 - 5.47748i) q^{61} +(1.99030 - 0.722803i) q^{62} +(4.70124 + 11.8740i) q^{63} +(-6.07828 + 5.20139i) q^{64} +(-1.24398 - 11.8293i) q^{65} +(5.23215 - 18.1977i) q^{66} +(-0.505597 + 2.65043i) q^{67} +(-0.576964 + 10.0621i) q^{68} +(10.8376 + 2.06739i) q^{69} +(-1.81762 + 4.77547i) q^{70} +(4.49962 - 9.56217i) q^{71} +(-21.8562 - 4.69460i) q^{72} +(3.62867 + 14.1327i) q^{73} +(1.21433 - 3.40255i) q^{74} +(-4.44172 - 15.9016i) q^{75} +(-9.36270 + 2.45945i) q^{76} +(-3.51060 - 5.53182i) q^{77} +(11.2159 - 22.1644i) q^{78} +(-0.572719 + 0.314855i) q^{79} +(-5.18630 - 7.28713i) q^{80} +(-12.6695 - 26.9240i) q^{81} +(2.84899 + 9.70559i) q^{82} +(8.29511 + 4.56028i) q^{83} +(-8.59812 + 6.32016i) q^{84} +(-11.1475 - 1.64548i) q^{85} +(0.416271 - 4.53818i) q^{86} +(17.2893 - 14.3029i) q^{87} +(11.4655 + 0.264720i) q^{88} +(1.96801 - 0.505298i) q^{89} +(7.40643 - 23.8707i) q^{90} +(-3.16411 - 7.99162i) q^{91} +(0.456638 + 6.66692i) q^{92} +(-1.52782 + 4.70215i) q^{93} +(-3.84536 - 4.33736i) q^{94} +(-1.80553 - 10.6713i) q^{95} +(-0.845916 - 18.6601i) q^{96} +(-12.2736 + 2.34132i) q^{97} +(0.566980 - 6.18120i) q^{98} +(18.8368 + 25.9266i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2960 q - 20 q^{2} - 20 q^{4} - 20 q^{6} - 50 q^{7} - 35 q^{8} - 40 q^{9} - 20 q^{10} - 20 q^{12} - 30 q^{14} - 40 q^{15} - 40 q^{17} - 25 q^{18} - 20 q^{20} + 5 q^{22} - 40 q^{23} - 15 q^{24} - 40 q^{25}+ \cdots - 115 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(e\left(\frac{31}{50}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.29946 − 0.558045i −0.918854 − 0.394598i
\(3\) 2.89362 − 1.59078i 1.67063 − 0.918437i 0.689695 − 0.724100i \(-0.257745\pi\)
0.980935 − 0.194336i \(-0.0622554\pi\)
\(4\) 1.37717 + 1.45031i 0.688585 + 0.725155i
\(5\) −1.75231 + 1.38903i −0.783659 + 0.621192i
\(6\) −4.64785 + 0.452378i −1.89748 + 0.184683i
\(7\) −0.949757 + 1.30723i −0.358974 + 0.494086i −0.949863 − 0.312668i \(-0.898777\pi\)
0.590888 + 0.806753i \(0.298777\pi\)
\(8\) −0.980234 − 2.65314i −0.346565 − 0.938026i
\(9\) 4.23495 − 6.67322i 1.41165 − 2.22441i
\(10\) 3.05219 − 0.827108i 0.965189 − 0.261555i
\(11\) −1.49265 + 3.77000i −0.450051 + 1.13670i 0.510817 + 0.859690i \(0.329343\pi\)
−0.960867 + 0.277009i \(0.910657\pi\)
\(12\) 6.29213 + 2.00587i 1.81638 + 0.579044i
\(13\) −2.85028 + 4.49132i −0.790524 + 1.24567i 0.174798 + 0.984604i \(0.444073\pi\)
−0.965322 + 0.261063i \(0.915927\pi\)
\(14\) 1.96366 − 1.16868i 0.524810 − 0.312342i
\(15\) −2.86089 + 6.80685i −0.738679 + 1.75752i
\(16\) −0.206800 + 3.99465i −0.0517001 + 0.998663i
\(17\) 3.44965 + 3.67351i 0.836664 + 0.890956i 0.995345 − 0.0963735i \(-0.0307243\pi\)
−0.158682 + 0.987330i \(0.550724\pi\)
\(18\) −9.22710 + 6.30826i −2.17485 + 1.48687i
\(19\) −2.33177 + 4.24148i −0.534945 + 0.973061i 0.461363 + 0.887211i \(0.347361\pi\)
−0.996308 + 0.0858498i \(0.972639\pi\)
\(20\) −4.42776 − 0.628472i −0.990076 − 0.140531i
\(21\) −0.668721 + 5.29347i −0.145927 + 1.15513i
\(22\) 4.04346 − 4.06599i 0.862069 − 0.866871i
\(23\) 2.57449 + 2.12981i 0.536819 + 0.444095i 0.865908 − 0.500204i \(-0.166742\pi\)
−0.329089 + 0.944299i \(0.606742\pi\)
\(24\) −7.05698 − 6.11783i −1.44050 − 1.24880i
\(25\) 1.14121 − 4.86802i 0.228242 − 0.973604i
\(26\) 6.21017 − 4.24568i 1.21791 − 0.832647i
\(27\) 1.01670 − 16.1600i 0.195664 − 3.10999i
\(28\) −3.20386 + 0.422834i −0.605473 + 0.0799082i
\(29\) 6.67500 − 1.27332i 1.23952 − 0.236450i 0.474380 − 0.880320i \(-0.342672\pi\)
0.765135 + 0.643870i \(0.222672\pi\)
\(30\) 7.51613 − 7.24870i 1.37225 − 1.32343i
\(31\) −1.09148 + 1.02497i −0.196035 + 0.184089i −0.776348 − 0.630304i \(-0.782930\pi\)
0.580313 + 0.814393i \(0.302930\pi\)
\(32\) 2.49792 − 5.07547i 0.441575 − 0.897225i
\(33\) 1.67809 + 13.2834i 0.292117 + 2.31235i
\(34\) −2.43269 − 6.69862i −0.417203 − 1.14880i
\(35\) −0.151502 − 3.60991i −0.0256086 − 0.610186i
\(36\) 15.5105 − 3.04817i 2.58508 − 0.508028i
\(37\) 0.160405 + 2.54956i 0.0263704 + 0.419145i 0.988960 + 0.148182i \(0.0473422\pi\)
−0.962590 + 0.270963i \(0.912658\pi\)
\(38\) 5.39697 − 4.21038i 0.875504 − 0.683013i
\(39\) −1.10291 + 17.5303i −0.176607 + 2.80709i
\(40\) 5.40296 + 3.28756i 0.854283 + 0.519809i
\(41\) −4.55914 − 5.51106i −0.712019 − 0.860683i 0.283128 − 0.959082i \(-0.408628\pi\)
−0.995147 + 0.0983993i \(0.968628\pi\)
\(42\) 3.82297 − 6.50545i 0.589897 − 1.00381i
\(43\) 0.995791 + 3.06473i 0.151857 + 0.467367i 0.997829 − 0.0658602i \(-0.0209791\pi\)
−0.845972 + 0.533227i \(0.820979\pi\)
\(44\) −7.52331 + 3.02713i −1.13418 + 0.456357i
\(45\) 1.84831 + 17.5760i 0.275530 + 2.62008i
\(46\) −2.15691 − 4.20427i −0.318019 − 0.619886i
\(47\) 3.70866 + 1.74516i 0.540964 + 0.254558i 0.676792 − 0.736174i \(-0.263370\pi\)
−0.135829 + 0.990732i \(0.543370\pi\)
\(48\) 5.75620 + 11.8880i 0.830837 + 1.71588i
\(49\) 1.35631 + 4.17430i 0.193759 + 0.596328i
\(50\) −4.19953 + 5.68893i −0.593903 + 0.804537i
\(51\) 15.8257 + 5.14208i 2.21604 + 0.720036i
\(52\) −10.4391 + 2.05153i −1.44765 + 0.284495i
\(53\) −6.25865 − 0.790651i −0.859691 − 0.108604i −0.316887 − 0.948463i \(-0.602638\pi\)
−0.542805 + 0.839859i \(0.682638\pi\)
\(54\) −10.3392 + 20.4318i −1.40698 + 2.78042i
\(55\) −2.62104 − 8.67956i −0.353421 − 1.17035i
\(56\) 4.39924 + 1.23845i 0.587873 + 0.165494i
\(57\) 15.9825i 2.11694i
\(58\) −9.38443 − 2.07032i −1.23224 − 0.271847i
\(59\) 3.05848 − 11.9120i 0.398180 − 1.55081i −0.381343 − 0.924434i \(-0.624538\pi\)
0.779523 − 0.626374i \(-0.215462\pi\)
\(60\) −13.8120 + 5.22502i −1.78312 + 0.674548i
\(61\) −6.62113 − 5.47748i −0.847749 − 0.701319i 0.108429 − 0.994104i \(-0.465418\pi\)
−0.956178 + 0.292785i \(0.905418\pi\)
\(62\) 1.99030 − 0.722803i 0.252769 − 0.0917961i
\(63\) 4.70124 + 11.8740i 0.592301 + 1.49598i
\(64\) −6.07828 + 5.20139i −0.759785 + 0.650174i
\(65\) −1.24398 − 11.8293i −0.154297 − 1.46724i
\(66\) 5.23215 − 18.1977i 0.644033 − 2.23998i
\(67\) −0.505597 + 2.65043i −0.0617685 + 0.323802i −0.999737 − 0.0229176i \(-0.992704\pi\)
0.937969 + 0.346720i \(0.112704\pi\)
\(68\) −0.576964 + 10.0621i −0.0699671 + 1.22021i
\(69\) 10.8376 + 2.06739i 1.30470 + 0.248885i
\(70\) −1.81762 + 4.77547i −0.217248 + 0.570777i
\(71\) 4.49962 − 9.56217i 0.534006 − 1.13482i −0.437807 − 0.899069i \(-0.644245\pi\)
0.971813 − 0.235752i \(-0.0757554\pi\)
\(72\) −21.8562 − 4.69460i −2.57578 − 0.553264i
\(73\) 3.62867 + 14.1327i 0.424704 + 1.65411i 0.716467 + 0.697621i \(0.245758\pi\)
−0.291763 + 0.956491i \(0.594242\pi\)
\(74\) 1.21433 − 3.40255i 0.141163 − 0.395539i
\(75\) −4.44172 − 15.9016i −0.512886 − 1.83616i
\(76\) −9.36270 + 2.45945i −1.07398 + 0.282118i
\(77\) −3.51060 − 5.53182i −0.400070 − 0.630409i
\(78\) 11.2159 − 22.1644i 1.26995 − 2.50962i
\(79\) −0.572719 + 0.314855i −0.0644360 + 0.0354240i −0.513646 − 0.858002i \(-0.671706\pi\)
0.449210 + 0.893426i \(0.351706\pi\)
\(80\) −5.18630 − 7.28713i −0.579846 − 0.814726i
\(81\) −12.6695 − 26.9240i −1.40772 − 2.99155i
\(82\) 2.84899 + 9.70559i 0.314618 + 1.07180i
\(83\) 8.29511 + 4.56028i 0.910507 + 0.500555i 0.866894 − 0.498493i \(-0.166113\pi\)
0.0436131 + 0.999048i \(0.486113\pi\)
\(84\) −8.59812 + 6.32016i −0.938131 + 0.689586i
\(85\) −11.1475 − 1.64548i −1.20911 − 0.178477i
\(86\) 0.416271 − 4.53818i 0.0448877 − 0.489364i
\(87\) 17.2893 − 14.3029i 1.85361 − 1.53344i
\(88\) 11.4655 + 0.264720i 1.22222 + 0.0282193i
\(89\) 1.96801 − 0.505298i 0.208608 − 0.0535615i −0.142940 − 0.989731i \(-0.545655\pi\)
0.351548 + 0.936170i \(0.385655\pi\)
\(90\) 7.40643 − 23.8707i 0.780706 − 2.51620i
\(91\) −3.16411 − 7.99162i −0.331688 − 0.837749i
\(92\) 0.456638 + 6.66692i 0.0476078 + 0.695075i
\(93\) −1.52782 + 4.70215i −0.158428 + 0.487591i
\(94\) −3.84536 − 4.33736i −0.396619 − 0.447365i
\(95\) −1.80553 − 10.6713i −0.185243 − 1.09485i
\(96\) −0.845916 − 18.6601i −0.0863359 − 1.90449i
\(97\) −12.2736 + 2.34132i −1.24620 + 0.237725i −0.767955 − 0.640504i \(-0.778725\pi\)
−0.478241 + 0.878229i \(0.658725\pi\)
\(98\) 0.566980 − 6.18120i 0.0572736 − 0.624395i
\(99\) 18.8368 + 25.9266i 1.89317 + 2.60572i
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 1000.2.bd.a.29.20 ✓ 2960
8.5 even 2 inner 1000.2.bd.a.29.132 yes 2960
125.69 even 50 inner 1000.2.bd.a.69.132 yes 2960
1000.69 even 50 inner 1000.2.bd.a.69.20 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.29.20 ✓ 2960 1.1 even 1 trivial
1000.2.bd.a.29.132 yes 2960 8.5 even 2 inner
1000.2.bd.a.69.20 yes 2960 1000.69 even 50 inner
1000.2.bd.a.69.132 yes 2960 125.69 even 50 inner