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Properties

Label 1000.2.bd.a.109.41
Level $1000$
Weight $2$
Character 1000.109
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 109.41
Character \(\chi\) \(=\) 1000.109
Dual form 1000.2.bd.a.789.41

$q$-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+(-0.894681 + 1.09524i) q^{2} +(0.859509 + 1.03897i) q^{3} +(-0.399093 - 1.95978i) q^{4} +(0.254836 - 2.22150i) q^{5} +(-1.90690 + 0.0118226i) q^{6} +(4.65603 - 1.51284i) q^{7} +(2.50348 + 1.31627i) q^{8} +(0.221445 - 1.16086i) q^{9} +(2.20507 + 2.26664i) q^{10} +(-0.350697 + 2.77605i) q^{11} +(1.69312 - 2.09909i) q^{12} +(-0.189832 + 0.995136i) q^{13} +(-2.50875 + 6.45297i) q^{14} +(2.52710 - 1.64463i) q^{15} +(-3.68145 + 1.56427i) q^{16} +(-1.66441 - 6.48244i) q^{17} +(1.07329 + 1.28113i) q^{18} +(-1.29260 - 1.06933i) q^{19} +(-4.45535 + 0.387163i) q^{20} +(5.57369 + 3.53717i) q^{21} +(-2.72668 - 2.86778i) q^{22} +(0.102920 - 0.109598i) q^{23} +(0.784202 + 3.73239i) q^{24} +(-4.87012 - 1.13224i) q^{25} +(-0.920072 - 1.09824i) q^{26} +(4.94129 - 2.71650i) q^{27} +(-4.82301 - 8.52102i) q^{28} +(-2.65084 + 0.166777i) q^{29} +(-0.459685 + 4.23920i) q^{30} +(4.75898 - 1.22190i) q^{31} +(1.58048 - 5.43158i) q^{32} +(-3.18566 + 2.02168i) q^{33} +(8.58892 + 3.97679i) q^{34} +(-2.17424 - 10.7289i) q^{35} +(-2.36340 + 0.0293068i) q^{36} +(-4.28495 - 2.35567i) q^{37} +(2.32764 - 0.458995i) q^{38} +(-1.19708 + 0.658099i) q^{39} +(3.56208 - 5.22605i) q^{40} +(2.05023 - 1.92529i) q^{41} +(-8.86072 + 2.93988i) q^{42} +(0.729437 - 0.529967i) q^{43} +(5.58041 - 0.420616i) q^{44} +(-2.52241 - 0.787769i) q^{45} +(0.0279560 + 0.210777i) q^{46} +(2.32115 + 5.86254i) q^{47} +(-4.78946 - 2.48041i) q^{48} +(13.7268 - 9.97314i) q^{49} +(5.59727 - 4.32095i) q^{50} +(5.30447 - 7.30098i) q^{51} +(2.02601 - 0.0251231i) q^{52} +(-4.68128 + 7.37652i) q^{53} +(-1.44567 + 7.84229i) q^{54} +(6.07763 + 1.48651i) q^{55} +(13.6476 + 2.34124i) q^{56} -2.26207i q^{57} +(2.18900 - 3.05252i) q^{58} +(-3.28620 + 1.54637i) q^{59} +(-4.23166 - 4.29619i) q^{60} +(8.01345 - 8.53346i) q^{61} +(-2.91950 + 6.30542i) q^{62} +(-0.725130 - 5.74000i) q^{63} +(4.53486 + 6.59053i) q^{64} +(2.16232 + 0.675310i) q^{65} +(0.635925 - 5.29781i) q^{66} +(-0.269023 + 4.27600i) q^{67} +(-12.0399 + 5.84896i) q^{68} +(0.202329 + 0.0127295i) q^{69} +(13.6959 + 7.21763i) q^{70} +(-8.35992 + 3.30993i) q^{71} +(2.08239 - 2.61470i) q^{72} +(-0.209163 - 0.0984244i) q^{73} +(6.41368 - 2.58547i) q^{74} +(-3.00955 - 6.03307i) q^{75} +(-1.57978 + 2.95997i) q^{76} +(2.56686 + 13.4559i) q^{77} +(0.350227 - 1.89987i) q^{78} +(4.41669 + 5.33886i) q^{79} +(2.53685 + 8.57697i) q^{80} +(3.77304 + 1.49385i) q^{81} +(0.274354 + 3.96801i) q^{82} +(-0.0671268 + 0.0811423i) q^{83} +(4.70765 - 12.3349i) q^{84} +(-14.8249 + 2.04552i) q^{85} +(-0.0721730 + 1.27306i) q^{86} +(-2.45170 - 2.61080i) q^{87} +(-4.53201 + 6.48819i) q^{88} +(-5.81576 + 12.3591i) q^{89} +(3.11954 - 2.05784i) q^{90} +(0.621613 + 4.92057i) q^{91} +(-0.255863 - 0.157960i) q^{92} +(5.35990 + 3.89419i) q^{93} +(-8.49756 - 2.70290i) q^{94} +(-2.70492 + 2.59901i) q^{95} +(7.00168 - 3.02643i) q^{96} +(11.8143 - 0.743293i) q^{97} +(-1.35818 + 23.9569i) q^{98} +(3.14494 + 1.02185i) 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 \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(e\left(\frac{27}{50}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.894681 + 1.09524i −0.632635 + 0.774450i
\(3\) 0.859509 + 1.03897i 0.496238 + 0.599849i 0.957863 − 0.287227i \(-0.0927333\pi\)
−0.461625 + 0.887075i \(0.652733\pi\)
\(4\) −0.399093 − 1.95978i −0.199547 − 0.979888i
\(5\) 0.254836 − 2.22150i 0.113966 − 0.993485i
\(6\) −1.90690 + 0.0118226i −0.778490 + 0.00482657i
\(7\) 4.65603 − 1.51284i 1.75981 − 0.571799i 0.762634 − 0.646830i \(-0.223906\pi\)
0.997181 + 0.0750320i \(0.0239059\pi\)
\(8\) 2.50348 + 1.31627i 0.885115 + 0.465372i
\(9\) 0.221445 − 1.16086i 0.0738151 − 0.386952i
\(10\) 2.20507 + 2.26664i 0.697305 + 0.716774i
\(11\) −0.350697 + 2.77605i −0.105739 + 0.837012i 0.846655 + 0.532142i \(0.178613\pi\)
−0.952394 + 0.304869i \(0.901387\pi\)
\(12\) 1.69312 − 2.09909i 0.488762 − 0.605955i
\(13\) −0.189832 + 0.995136i −0.0526500 + 0.276001i −0.998870 − 0.0475223i \(-0.984867\pi\)
0.946220 + 0.323524i \(0.104867\pi\)
\(14\) −2.50875 + 6.45297i −0.670490 + 1.72463i
\(15\) 2.52710 − 1.64463i 0.652495 − 0.424642i
\(16\) −3.68145 + 1.56427i −0.920362 + 0.391067i
\(17\) −1.66441 − 6.48244i −0.403678 − 1.57222i −0.767686 − 0.640826i \(-0.778592\pi\)
0.364008 − 0.931396i \(-0.381408\pi\)
\(18\) 1.07329 + 1.28113i 0.252977 + 0.301965i
\(19\) −1.29260 − 1.06933i −0.296543 − 0.245322i 0.477239 − 0.878773i \(-0.341637\pi\)
−0.773782 + 0.633452i \(0.781637\pi\)
\(20\) −4.45535 + 0.387163i −0.996246 + 0.0865722i
\(21\) 5.57369 + 3.53717i 1.21628 + 0.771874i
\(22\) −2.72668 − 2.86778i −0.581330 − 0.611412i
\(23\) 0.102920 − 0.109598i 0.0214602 − 0.0228528i −0.718149 − 0.695889i \(-0.755010\pi\)
0.739609 + 0.673037i \(0.235010\pi\)
\(24\) 0.784202 + 3.73239i 0.160075 + 0.761870i
\(25\) −4.87012 − 1.13224i −0.974023 − 0.226448i
\(26\) −0.920072 − 1.09824i −0.180441 − 0.215383i
\(27\) 4.94129 − 2.71650i 0.950952 − 0.522790i
\(28\) −4.82301 − 8.52102i −0.911464 − 1.61032i
\(29\) −2.65084 + 0.166777i −0.492249 + 0.0309697i −0.306976 − 0.951717i \(-0.599317\pi\)
−0.185273 + 0.982687i \(0.559317\pi\)
\(30\) −0.459685 + 4.23920i −0.0839265 + 0.773968i
\(31\) 4.75898 − 1.22190i 0.854738 − 0.219459i 0.204196 − 0.978930i \(-0.434542\pi\)
0.650542 + 0.759471i \(0.274542\pi\)
\(32\) 1.58048 − 5.43158i 0.279391 − 0.960177i
\(33\) −3.18566 + 2.02168i −0.554552 + 0.351929i
\(34\) 8.58892 + 3.97679i 1.47299 + 0.682014i
\(35\) −2.17424 − 10.7289i −0.367513 − 1.81351i
\(36\) −2.36340 + 0.0293068i −0.393899 + 0.00488447i
\(37\) −4.28495 − 2.35567i −0.704442 − 0.387270i 0.0889052 − 0.996040i \(-0.471663\pi\)
−0.793347 + 0.608770i \(0.791663\pi\)
\(38\) 2.32764 − 0.458995i 0.377593 − 0.0744587i
\(39\) −1.19708 + 0.658099i −0.191686 + 0.105380i
\(40\) 3.56208 − 5.22605i 0.563214 − 0.826311i
\(41\) 2.05023 − 1.92529i 0.320192 − 0.300680i −0.507738 − 0.861511i \(-0.669518\pi\)
0.827930 + 0.560831i \(0.189518\pi\)
\(42\) −8.86072 + 2.93988i −1.36724 + 0.453633i
\(43\) 0.729437 − 0.529967i 0.111238 − 0.0808192i −0.530776 − 0.847512i \(-0.678099\pi\)
0.642014 + 0.766693i \(0.278099\pi\)
\(44\) 5.58041 − 0.420616i 0.841278 − 0.0634103i
\(45\) −2.52241 − 0.787769i −0.376019 − 0.117434i
\(46\) 0.0279560 + 0.210777i 0.00412189 + 0.0310774i
\(47\) 2.32115 + 5.86254i 0.338574 + 0.855140i 0.994919 + 0.100679i \(0.0321015\pi\)
−0.656345 + 0.754461i \(0.727898\pi\)
\(48\) −4.78946 − 2.48041i −0.691300 − 0.358016i
\(49\) 13.7268 − 9.97314i 1.96098 − 1.42473i
\(50\) 5.59727 − 4.32095i 0.791573 − 0.611074i
\(51\) 5.30447 − 7.30098i 0.742775 − 1.02234i
\(52\) 2.02601 − 0.0251231i 0.280956 − 0.00348395i
\(53\) −4.68128 + 7.37652i −0.643024 + 1.01324i 0.353932 + 0.935271i \(0.384845\pi\)
−0.996956 + 0.0779723i \(0.975155\pi\)
\(54\) −1.44567 + 7.84229i −0.196730 + 1.06720i
\(55\) 6.07763 + 1.48651i 0.819507 + 0.200441i
\(56\) 13.6476 + 2.34124i 1.82374 + 0.312862i
\(57\) − 2.26207i − 0.299619i
\(58\) 2.18900 − 3.05252i 0.287430 − 0.400815i
\(59\) −3.28620 + 1.54637i −0.427827 + 0.201320i −0.627623 − 0.778518i \(-0.715972\pi\)
0.199796 + 0.979837i \(0.435972\pi\)
\(60\) −4.23166 − 4.29619i −0.546305 − 0.554636i
\(61\) 8.01345 − 8.53346i 1.02602 − 1.09260i 0.0303569 − 0.999539i \(-0.490336\pi\)
0.995661 − 0.0930586i \(-0.0296644\pi\)
\(62\) −2.91950 + 6.30542i −0.370776 + 0.800789i
\(63\) −0.725130 − 5.74000i −0.0913578 − 0.723171i
\(64\) 4.53486 + 6.59053i 0.566857 + 0.823816i
\(65\) 2.16232 + 0.675310i 0.268203 + 0.0837619i
\(66\) 0.635925 − 5.29781i 0.0782770 − 0.652116i
\(67\) −0.269023 + 4.27600i −0.0328664 + 0.522396i 0.946808 + 0.321800i \(0.104288\pi\)
−0.979674 + 0.200596i \(0.935712\pi\)
\(68\) −12.0399 + 5.84896i −1.46005 + 0.709291i
\(69\) 0.202329 + 0.0127295i 0.0243576 + 0.00153245i
\(70\) 13.6959 + 7.21763i 1.63698 + 0.862672i
\(71\) −8.35992 + 3.30993i −0.992140 + 0.392816i −0.807421 − 0.589976i \(-0.799137\pi\)
−0.184719 + 0.982791i \(0.559137\pi\)
\(72\) 2.08239 − 2.61470i 0.245412 − 0.308146i
\(73\) −0.209163 − 0.0984244i −0.0244806 − 0.0115197i 0.413500 − 0.910504i \(-0.364306\pi\)
−0.437981 + 0.898984i \(0.644306\pi\)
\(74\) 6.41368 − 2.58547i 0.745576 − 0.300555i
\(75\) −3.00955 − 6.03307i −0.347513 − 0.696638i
\(76\) −1.57978 + 2.95997i −0.181214 + 0.339532i
\(77\) 2.56686 + 13.4559i 0.292521 + 1.53345i
\(78\) 0.350227 − 1.89987i 0.0396554 − 0.215118i
\(79\) 4.41669 + 5.33886i 0.496916 + 0.600668i 0.958028 − 0.286674i \(-0.0925495\pi\)
−0.461112 + 0.887342i \(0.652549\pi\)
\(80\) 2.53685 + 8.57697i 0.283629 + 0.958934i
\(81\) 3.77304 + 1.49385i 0.419227 + 0.165984i
\(82\) 0.274354 + 3.96801i 0.0302974 + 0.438193i
\(83\) −0.0671268 + 0.0811423i −0.00736812 + 0.00890653i −0.774184 − 0.632960i \(-0.781840\pi\)
0.766816 + 0.641867i \(0.221840\pi\)
\(84\) 4.70765 − 12.3349i 0.513646 − 1.34584i
\(85\) −14.8249 + 2.04552i −1.60798 + 0.221868i
\(86\) −0.0721730 + 1.27306i −0.00778262 + 0.137277i
\(87\) −2.45170 − 2.61080i −0.262850 − 0.279907i
\(88\) −4.53201 + 6.48819i −0.483113 + 0.691643i
\(89\) −5.81576 + 12.3591i −0.616469 + 1.31006i 0.315981 + 0.948765i \(0.397666\pi\)
−0.932450 + 0.361298i \(0.882334\pi\)
\(90\) 3.11954 − 2.05784i 0.328829 − 0.216915i
\(91\) 0.621613 + 4.92057i 0.0651627 + 0.515816i
\(92\) −0.255863 − 0.157960i −0.0266755 − 0.0164684i
\(93\) 5.35990 + 3.89419i 0.555796 + 0.403809i
\(94\) −8.49756 − 2.70290i −0.876457 − 0.278783i
\(95\) −2.70492 + 2.59901i −0.277519 + 0.266652i
\(96\) 7.00168 − 3.02643i 0.714606 − 0.308884i
\(97\) 11.8143 − 0.743293i 1.19956 − 0.0754700i 0.549594 − 0.835432i \(-0.314782\pi\)
0.649968 + 0.759962i \(0.274782\pi\)
\(98\) −1.35818 + 23.9569i −0.137197 + 2.42002i
\(99\) 3.14494 + 1.02185i 0.316078 + 0.102700i
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.109.41 ✓ 2960
8.5 even 2 inner 1000.2.bd.a.109.95 yes 2960
125.39 even 50 inner 1000.2.bd.a.789.95 yes 2960
1000.789 even 50 inner 1000.2.bd.a.789.41 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.109.41 ✓ 2960 1.1 even 1 trivial
1000.2.bd.a.109.95 yes 2960 8.5 even 2 inner
1000.2.bd.a.789.41 yes 2960 1000.789 even 50 inner
1000.2.bd.a.789.95 yes 2960 125.39 even 50 inner