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Properties

Label 1000.2.bd.a.109.6
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 109.6
Character \(\chi\) \(=\) 1000.109
Dual form 1000.2.bd.a.789.6

$q$-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+(-1.41094 + 0.0961446i) q^{2} +(-0.549982 - 0.664814i) q^{3} +(1.98151 - 0.271309i) q^{4} +(-2.23574 - 0.0382621i) q^{5} +(0.839911 + 0.885136i) q^{6} +(-1.98189 + 0.643956i) q^{7} +(-2.76971 + 0.573312i) q^{8} +(0.422647 - 2.21559i) q^{9} +(3.15818 - 0.160969i) q^{10} +(0.679530 - 5.37904i) q^{11} +(-1.27017 - 1.16812i) q^{12} +(0.357757 - 1.87543i) q^{13} +(2.73442 - 1.09913i) q^{14} +(1.20418 + 1.50739i) q^{15} +(3.85278 - 1.07520i) q^{16} +(0.0742898 + 0.289340i) q^{17} +(-0.383313 + 3.16670i) q^{18} +(4.58684 + 3.79456i) q^{19} +(-4.44053 + 0.530759i) q^{20} +(1.51812 + 0.963426i) q^{21} +(-0.441613 + 7.65484i) q^{22} +(1.91509 - 2.03936i) q^{23} +(1.90444 + 1.52603i) q^{24} +(4.99707 + 0.171088i) q^{25} +(-0.324462 + 2.68052i) q^{26} +(-3.97369 + 2.18455i) q^{27} +(-3.75244 + 1.81371i) q^{28} +(-6.62372 + 0.416729i) q^{29} +(-1.84395 - 2.01107i) q^{30} +(-7.44931 + 1.91266i) q^{31} +(-5.33268 + 1.88747i) q^{32} +(-3.94979 + 2.50661i) q^{33} +(-0.132637 - 0.401099i) q^{34} +(4.45564 - 1.36389i) q^{35} +(0.236370 - 4.50489i) q^{36} +(-0.554468 - 0.304822i) q^{37} +(-6.83659 - 4.91291i) q^{38} +(-1.44357 + 0.793610i) q^{39} +(6.21430 - 1.17580i) q^{40} +(7.70586 - 7.23629i) q^{41} +(-2.23460 - 1.21338i) q^{42} +(-5.95401 + 4.32584i) q^{43} +(-0.112882 - 10.8430i) q^{44} +(-1.02970 + 4.93731i) q^{45} +(-2.50600 + 3.06154i) q^{46} +(-1.05390 - 2.66184i) q^{47} +(-2.83377 - 1.97004i) q^{48} +(-2.14990 + 1.56199i) q^{49} +(-7.06703 + 0.239046i) q^{50} +(0.151499 - 0.208520i) q^{51} +(0.200080 - 3.81325i) q^{52} +(-3.71589 + 5.85531i) q^{53} +(5.39661 - 3.46432i) q^{54} +(-1.72507 + 12.0001i) q^{55} +(5.12009 - 2.91982i) q^{56} -5.13633i q^{57} +(9.30562 - 1.22482i) q^{58} +(-4.96248 + 2.33517i) q^{59} +(2.79507 + 2.66022i) q^{60} +(2.69791 - 2.87298i) q^{61} +(10.3267 - 3.41486i) q^{62} +(0.589103 + 4.66323i) q^{63} +(7.34263 - 3.17582i) q^{64} +(-0.871610 + 4.17928i) q^{65} +(5.33192 - 3.91643i) q^{66} +(-0.663533 + 10.5466i) q^{67} +(0.225707 + 0.553175i) q^{68} +(-2.40906 - 0.151565i) q^{69} +(-6.15552 + 2.35275i) q^{70} +(-10.4554 + 4.13960i) q^{71} +(0.0996157 + 6.37886i) q^{72} +(-13.0067 - 6.12049i) q^{73} +(0.811629 + 0.376776i) q^{74} +(-2.63456 - 3.41622i) q^{75} +(10.1184 + 6.27452i) q^{76} +(2.11711 + 11.0983i) q^{77} +(1.96049 - 1.25853i) q^{78} +(1.87607 + 2.26778i) q^{79} +(-8.65496 + 2.25646i) q^{80} +(-2.65367 - 1.05066i) q^{81} +(-10.1768 + 10.9509i) q^{82} +(6.00246 - 7.25573i) q^{83} +(3.26955 + 1.49716i) q^{84} +(-0.155022 - 0.649731i) q^{85} +(7.98486 - 6.67596i) q^{86} +(3.91997 + 4.17435i) q^{87} +(1.20176 + 15.2880i) q^{88} +(-1.76094 + 3.74220i) q^{89} +(0.978152 - 7.06526i) q^{90} +(0.498658 + 3.94728i) q^{91} +(3.24147 - 4.56059i) q^{92} +(5.36855 + 3.90048i) q^{93} +(1.74291 + 3.65437i) q^{94} +(-10.1098 - 8.65916i) q^{95} +(4.18769 + 2.50716i) q^{96} +(3.94208 - 0.248014i) q^{97} +(2.88320 - 2.41058i) q^{98} +(-11.6305 - 3.77899i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41094 + 0.0961446i −0.997686 + 0.0679845i
\(3\) −0.549982 − 0.664814i −0.317532 − 0.383830i 0.587464 − 0.809250i \(-0.300126\pi\)
−0.904996 + 0.425420i \(0.860126\pi\)
\(4\) 1.98151 − 0.271309i 0.990756 − 0.135654i
\(5\) −2.23574 − 0.0382621i −0.999854 − 0.0171113i
\(6\) 0.839911 + 0.885136i 0.342892 + 0.361355i
\(7\) −1.98189 + 0.643956i −0.749085 + 0.243393i −0.658588 − 0.752504i \(-0.728846\pi\)
−0.0904978 + 0.995897i \(0.528846\pi\)
\(8\) −2.76971 + 0.573312i −0.979242 + 0.202697i
\(9\) 0.422647 − 2.21559i 0.140882 − 0.738530i
\(10\) 3.15818 − 0.160969i 0.998704 − 0.0509028i
\(11\) 0.679530 − 5.37904i 0.204886 − 1.62184i −0.469133 − 0.883127i \(-0.655434\pi\)
0.674019 − 0.738714i \(-0.264566\pi\)
\(12\) −1.27017 − 1.16812i −0.366665 − 0.337208i
\(13\) 0.357757 − 1.87543i 0.0992240 − 0.520150i −0.897620 − 0.440769i \(-0.854706\pi\)
0.996844 − 0.0793810i \(-0.0252944\pi\)
\(14\) 2.73442 − 1.09913i 0.730805 − 0.293756i
\(15\) 1.20418 + 1.50739i 0.310918 + 0.389208i
\(16\) 3.85278 − 1.07520i 0.963196 − 0.268801i
\(17\) 0.0742898 + 0.289340i 0.0180179 + 0.0701752i 0.976956 − 0.213439i \(-0.0684665\pi\)
−0.958938 + 0.283614i \(0.908466\pi\)
\(18\) −0.383313 + 3.16670i −0.0903476 + 0.746399i
\(19\) 4.58684 + 3.79456i 1.05229 + 0.870532i 0.991821 − 0.127638i \(-0.0407395\pi\)
0.0604718 + 0.998170i \(0.480739\pi\)
\(20\) −4.44053 + 0.530759i −0.992932 + 0.118681i
\(21\) 1.51812 + 0.963426i 0.331280 + 0.210237i
\(22\) −0.441613 + 7.65484i −0.0941521 + 1.63202i
\(23\) 1.91509 − 2.03936i 0.399323 − 0.425236i −0.497611 − 0.867400i \(-0.665789\pi\)
0.896934 + 0.442165i \(0.145789\pi\)
\(24\) 1.90444 + 1.52603i 0.388742 + 0.311500i
\(25\) 4.99707 + 0.171088i 0.999414 + 0.0342176i
\(26\) −0.324462 + 2.68052i −0.0636323 + 0.525693i
\(27\) −3.97369 + 2.18455i −0.764736 + 0.420417i
\(28\) −3.75244 + 1.81371i −0.709144 + 0.342759i
\(29\) −6.62372 + 0.416729i −1.22999 + 0.0773847i −0.664450 − 0.747333i \(-0.731334\pi\)
−0.565545 + 0.824718i \(0.691334\pi\)
\(30\) −1.84395 − 2.01107i −0.336659 − 0.367170i
\(31\) −7.44931 + 1.91266i −1.33794 + 0.343524i −0.848814 − 0.528692i \(-0.822683\pi\)
−0.489122 + 0.872215i \(0.662683\pi\)
\(32\) −5.33268 + 1.88747i −0.942693 + 0.333661i
\(33\) −3.94979 + 2.50661i −0.687570 + 0.436345i
\(34\) −0.132637 − 0.401099i −0.0227471 − 0.0687879i
\(35\) 4.45564 − 1.36389i 0.753140 − 0.230539i
\(36\) 0.236370 − 4.50489i 0.0393951 − 0.750815i
\(37\) −0.554468 − 0.304822i −0.0911540 − 0.0501124i 0.435525 − 0.900177i \(-0.356563\pi\)
−0.526679 + 0.850064i \(0.676563\pi\)
\(38\) −6.83659 − 4.91291i −1.10904 − 0.796979i
\(39\) −1.44357 + 0.793610i −0.231156 + 0.127079i
\(40\) 6.21430 − 1.17580i 0.982567 − 0.185911i
\(41\) 7.70586 − 7.23629i 1.20345 − 1.13012i 0.215316 − 0.976544i \(-0.430922\pi\)
0.988137 − 0.153574i \(-0.0490783\pi\)
\(42\) −2.23460 − 1.21338i −0.344807 − 0.187229i
\(43\) −5.95401 + 4.32584i −0.907978 + 0.659685i −0.940503 − 0.339786i \(-0.889645\pi\)
0.0325244 + 0.999471i \(0.489645\pi\)
\(44\) −0.112882 − 10.8430i −0.0170175 − 1.63464i
\(45\) −1.02970 + 4.93731i −0.153499 + 0.736011i
\(46\) −2.50600 + 3.06154i −0.369490 + 0.451400i
\(47\) −1.05390 − 2.66184i −0.153727 − 0.388269i 0.832420 − 0.554145i \(-0.186955\pi\)
−0.986147 + 0.165876i \(0.946955\pi\)
\(48\) −2.83377 − 1.97004i −0.409020 − 0.284351i
\(49\) −2.14990 + 1.56199i −0.307128 + 0.223142i
\(50\) −7.06703 + 0.239046i −0.999428 + 0.0338062i
\(51\) 0.151499 − 0.208520i 0.0212141 − 0.0291987i
\(52\) 0.200080 − 3.81325i 0.0277461 − 0.528802i
\(53\) −3.71589 + 5.85531i −0.510417 + 0.804288i −0.997616 − 0.0690024i \(-0.978018\pi\)
0.487200 + 0.873291i \(0.338018\pi\)
\(54\) 5.39661 − 3.46432i 0.734385 − 0.471435i
\(55\) −1.72507 + 12.0001i −0.232608 + 1.61810i
\(56\) 5.12009 − 2.91982i 0.684201 − 0.390177i
\(57\) − 5.13633i − 0.680324i
\(58\) 9.30562 − 1.22482i 1.22189 − 0.160826i
\(59\) −4.96248 + 2.33517i −0.646060 + 0.304013i −0.720751 − 0.693194i \(-0.756203\pi\)
0.0746912 + 0.997207i \(0.476203\pi\)
\(60\) 2.79507 + 2.66022i 0.360842 + 0.343433i
\(61\) 2.69791 − 2.87298i 0.345432 − 0.367848i −0.532668 − 0.846324i \(-0.678811\pi\)
0.878101 + 0.478476i \(0.158811\pi\)
\(62\) 10.3267 − 3.41486i 1.31149 − 0.433688i
\(63\) 0.589103 + 4.66323i 0.0742200 + 0.587512i
\(64\) 7.34263 − 3.17582i 0.917828 − 0.396978i
\(65\) −0.871610 + 4.17928i −0.108110 + 0.518376i
\(66\) 5.33192 − 3.91643i 0.656314 − 0.482080i
\(67\) −0.663533 + 10.5466i −0.0810635 + 1.28847i 0.721715 + 0.692190i \(0.243354\pi\)
−0.802779 + 0.596277i \(0.796646\pi\)
\(68\) 0.225707 + 0.553175i 0.0273709 + 0.0670823i
\(69\) −2.40906 − 0.151565i −0.290016 − 0.0182463i
\(70\) −6.15552 + 2.35275i −0.735725 + 0.281208i
\(71\) −10.4554 + 4.13960i −1.24083 + 0.491280i −0.894599 − 0.446869i \(-0.852539\pi\)
−0.346232 + 0.938149i \(0.612539\pi\)
\(72\) 0.0996157 + 6.37886i 0.0117398 + 0.751756i
\(73\) −13.0067 − 6.12049i −1.52232 − 0.716349i −0.530944 − 0.847407i \(-0.678163\pi\)
−0.991375 + 0.131058i \(0.958163\pi\)
\(74\) 0.811629 + 0.376776i 0.0943500 + 0.0437994i
\(75\) −2.63456 − 3.41622i −0.304212 − 0.394471i
\(76\) 10.1184 + 6.27452i 1.16066 + 0.719737i
\(77\) 2.11711 + 11.0983i 0.241267 + 1.26477i
\(78\) 1.96049 − 1.25853i 0.221982 − 0.142500i
\(79\) 1.87607 + 2.26778i 0.211075 + 0.255145i 0.865416 − 0.501054i \(-0.167054\pi\)
−0.654342 + 0.756199i \(0.727054\pi\)
\(80\) −8.65496 + 2.25646i −0.967654 + 0.252280i
\(81\) −2.65367 − 1.05066i −0.294853 − 0.116740i
\(82\) −10.1768 + 10.9509i −1.12384 + 1.20932i
\(83\) 6.00246 − 7.25573i 0.658856 − 0.796420i −0.330208 − 0.943908i \(-0.607119\pi\)
0.989064 + 0.147488i \(0.0471189\pi\)
\(84\) 3.26955 + 1.49716i 0.356737 + 0.163354i
\(85\) −0.155022 − 0.649731i −0.0168145 − 0.0704732i
\(86\) 7.98486 − 6.67596i 0.861029 − 0.719887i
\(87\) 3.91997 + 4.17435i 0.420265 + 0.447537i
\(88\) 1.20176 + 15.2880i 0.128108 + 1.62970i
\(89\) −1.76094 + 3.74220i −0.186660 + 0.396672i −0.976152 − 0.217088i \(-0.930344\pi\)
0.789492 + 0.613761i \(0.210344\pi\)
\(90\) 0.978152 − 7.06526i 0.103106 − 0.744744i
\(91\) 0.498658 + 3.94728i 0.0522735 + 0.413787i
\(92\) 3.24147 − 4.56059i 0.337947 − 0.475475i
\(93\) 5.36855 + 3.90048i 0.556692 + 0.404461i
\(94\) 1.74291 + 3.65437i 0.179767 + 0.376920i
\(95\) −10.1098 − 8.65916i −1.03724 − 0.888411i
\(96\) 4.18769 + 2.50716i 0.427405 + 0.255886i
\(97\) 3.94208 − 0.248014i 0.400257 − 0.0251820i 0.138613 − 0.990347i \(-0.455735\pi\)
0.261644 + 0.965165i \(0.415735\pi\)
\(98\) 2.88320 − 2.41058i 0.291247 − 0.243505i
\(99\) −11.6305 − 3.77899i −1.16891 − 0.379803i
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.109.6 ✓ 2960
8.5 even 2 inner 1000.2.bd.a.109.133 yes 2960
125.39 even 50 inner 1000.2.bd.a.789.133 yes 2960
1000.789 even 50 inner 1000.2.bd.a.789.6 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.109.6 ✓ 2960 1.1 even 1 trivial
1000.2.bd.a.109.133 yes 2960 8.5 even 2 inner
1000.2.bd.a.789.6 yes 2960 1000.789 even 50 inner
1000.2.bd.a.789.133 yes 2960 125.39 even 50 inner