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Properties

Label 1000.2.bd.a.69.94
Level $1000$
Weight $2$
Character 1000.69
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 69.94
Character \(\chi\) \(=\) 1000.69
Dual form 1000.2.bd.a.29.94

$q$-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+(0.573908 - 1.29253i) q^{2} +(0.623770 + 0.342921i) q^{3} +(-1.34126 - 1.48358i) q^{4} +(-2.15211 + 0.606986i) q^{5} +(0.801221 - 0.609436i) q^{6} +(-0.524496 - 0.721907i) q^{7} +(-2.68733 + 0.882176i) q^{8} +(-1.33599 - 2.10518i) q^{9} +(-0.450565 + 3.13001i) q^{10} +(1.45707 + 3.68014i) q^{11} +(-0.327886 - 1.38536i) q^{12} +(-0.417942 - 0.658571i) q^{13} +(-1.23410 + 0.263618i) q^{14} +(-1.55057 - 0.359382i) q^{15} +(-0.402045 + 3.97974i) q^{16} +(-2.44981 + 2.60878i) q^{17} +(-3.48773 + 0.518622i) q^{18} +(2.33204 + 4.24197i) q^{19} +(3.78705 + 2.37871i) q^{20} +(-0.0796083 - 0.630165i) q^{21} +(5.59292 + 0.228756i) q^{22} +(-6.66995 + 5.51787i) q^{23} +(-1.97880 - 0.371267i) q^{24} +(4.26314 - 2.61260i) q^{25} +(-1.09108 + 0.162243i) q^{26} +(-0.245526 - 3.90252i) q^{27} +(-0.367525 + 1.74640i) q^{28} +(-0.139718 - 0.0266527i) q^{29} +(-1.35440 + 1.79790i) q^{30} +(-0.592587 - 0.556476i) q^{31} +(4.91320 + 2.80366i) q^{32} +(-0.353119 + 2.79522i) q^{33} +(1.96596 + 4.66364i) q^{34} +(1.56696 + 1.23526i) q^{35} +(-1.33130 + 4.80564i) q^{36} +(0.0453327 - 0.720543i) q^{37} +(6.82124 - 0.579732i) q^{38} +(-0.0348621 - 0.554118i) q^{39} +(5.24797 - 3.52971i) q^{40} +(-5.73434 + 6.93163i) q^{41} +(-0.860194 - 0.258761i) q^{42} +(-1.31210 + 4.03822i) q^{43} +(3.50549 - 7.09772i) q^{44} +(4.15300 + 3.71964i) q^{45} +(3.30406 + 11.7879i) q^{46} +(-2.01298 + 0.947238i) q^{47} +(-1.61552 + 2.34458i) q^{48} +(1.91707 - 5.90012i) q^{49} +(-0.930211 - 7.00962i) q^{50} +(-2.42272 + 0.787190i) q^{51} +(-0.416477 + 1.50337i) q^{52} +(-1.56414 + 0.197597i) q^{53} +(-5.18503 - 1.92234i) q^{54} +(-5.36957 - 7.03564i) q^{55} +(2.04635 + 1.47731i) q^{56} +3.44572i q^{57} +(-0.114635 + 0.165294i) q^{58} +(1.63158 + 6.35460i) q^{59} +(1.54654 + 2.78242i) q^{60} +(-3.69926 + 3.06029i) q^{61} +(-1.05935 + 0.446569i) q^{62} +(-0.819023 + 2.06862i) q^{63} +(6.44353 - 4.74140i) q^{64} +(1.29920 + 1.16363i) q^{65} +(3.41025 + 2.06062i) q^{66} +(-1.03873 - 5.44519i) q^{67} +(7.15617 + 0.135445i) q^{68} +(-6.05271 + 1.15462i) q^{69} +(2.49590 - 1.31642i) q^{70} +(-2.92239 - 6.21039i) q^{71} +(5.44738 + 4.47874i) q^{72} +(-3.05460 + 11.8969i) q^{73} +(-0.905306 - 0.472119i) q^{74} +(3.55513 - 0.167744i) q^{75} +(3.16545 - 9.14936i) q^{76} +(1.89249 - 2.98209i) q^{77} +(-0.736221 - 0.272952i) q^{78} +(-1.34553 - 0.739710i) q^{79} +(-1.55040 - 8.80887i) q^{80} +(-1.99970 + 4.24959i) q^{81} +(5.66834 + 11.3899i) q^{82} +(4.97340 - 2.73415i) q^{83} +(-0.828127 + 0.963321i) q^{84} +(3.68876 - 7.10137i) q^{85} +(4.46649 + 4.01349i) q^{86} +(-0.0780124 - 0.0645375i) q^{87} +(-7.16217 - 8.60438i) q^{88} +(-7.48548 - 1.92194i) q^{89} +(7.19118 - 3.23314i) q^{90} +(-0.256218 + 0.647133i) q^{91} +(17.1324 + 2.49455i) q^{92} +(-0.178811 - 0.550324i) q^{93} +(0.0690653 + 3.14547i) q^{94} +(-7.59362 - 7.71366i) q^{95} +(2.10327 + 3.43368i) q^{96} +(-4.10180 - 0.782460i) q^{97} +(-6.52585 - 5.86399i) q^{98} +(5.80073 - 7.98401i) 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 \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(e\left(\frac{19}{50}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.573908 − 1.29253i 0.405814 − 0.913956i
\(3\) 0.623770 + 0.342921i 0.360134 + 0.197985i 0.651540 − 0.758614i \(-0.274123\pi\)
−0.291406 + 0.956599i \(0.594123\pi\)
\(4\) −1.34126 − 1.48358i −0.670630 − 0.741792i
\(5\) −2.15211 + 0.606986i −0.962452 + 0.271452i
\(6\) 0.801221 − 0.609436i 0.327097 − 0.248801i
\(7\) −0.524496 − 0.721907i −0.198241 − 0.272855i 0.698310 − 0.715795i \(-0.253936\pi\)
−0.896551 + 0.442940i \(0.853936\pi\)
\(8\) −2.68733 + 0.882176i −0.950116 + 0.311896i
\(9\) −1.33599 − 2.10518i −0.445329 − 0.701725i
\(10\) −0.450565 + 3.13001i −0.142481 + 0.989798i
\(11\) 1.45707 + 3.68014i 0.439324 + 1.10961i 0.965771 + 0.259397i \(0.0835239\pi\)
−0.526447 + 0.850208i \(0.676476\pi\)
\(12\) −0.327886 − 1.38536i −0.0946526 − 0.399919i
\(13\) −0.417942 − 0.658571i −0.115916 − 0.182655i 0.781416 − 0.624010i \(-0.214498\pi\)
−0.897332 + 0.441356i \(0.854498\pi\)
\(14\) −1.23410 + 0.263618i −0.329827 + 0.0704549i
\(15\) −1.55057 − 0.359382i −0.400355 − 0.0927921i
\(16\) −0.402045 + 3.97974i −0.100511 + 0.994936i
\(17\) −2.44981 + 2.60878i −0.594165 + 0.632722i −0.953825 − 0.300362i \(-0.902893\pi\)
0.359660 + 0.933083i \(0.382893\pi\)
\(18\) −3.48773 + 0.518622i −0.822067 + 0.122240i
\(19\) 2.33204 + 4.24197i 0.535007 + 0.973175i 0.996301 + 0.0859273i \(0.0273853\pi\)
−0.461294 + 0.887247i \(0.652615\pi\)
\(20\) 3.78705 + 2.37871i 0.846810 + 0.531895i
\(21\) −0.0796083 − 0.630165i −0.0173720 − 0.137513i
\(22\) 5.59292 + 0.228756i 1.19241 + 0.0487710i
\(23\) −6.66995 + 5.51787i −1.39078 + 1.15055i −0.420863 + 0.907124i \(0.638273\pi\)
−0.969918 + 0.243431i \(0.921727\pi\)
\(24\) −1.97880 − 0.371267i −0.403920 − 0.0757846i
\(25\) 4.26314 − 2.61260i 0.852627 − 0.522520i
\(26\) −1.09108 + 0.162243i −0.213979 + 0.0318184i
\(27\) −0.245526 − 3.90252i −0.0472515 − 0.751041i
\(28\) −0.367525 + 1.74640i −0.0694556 + 0.330039i
\(29\) −0.139718 − 0.0266527i −0.0259450 − 0.00494928i 0.174392 − 0.984676i \(-0.444204\pi\)
−0.200337 + 0.979727i \(0.564204\pi\)
\(30\) −1.35440 + 1.79790i −0.247278 + 0.328250i
\(31\) −0.592587 − 0.556476i −0.106432 − 0.0999461i 0.629505 − 0.776997i \(-0.283258\pi\)
−0.735936 + 0.677051i \(0.763258\pi\)
\(32\) 4.91320 + 2.80366i 0.868538 + 0.495622i
\(33\) −0.353119 + 2.79522i −0.0614701 + 0.486586i
\(34\) 1.96596 + 4.66364i 0.337159 + 0.799808i
\(35\) 1.56696 + 1.23526i 0.264865 + 0.208797i
\(36\) −1.33130 + 4.80564i −0.221884 + 0.800939i
\(37\) 0.0453327 − 0.720543i 0.00745266 − 0.118457i −0.992541 − 0.121908i \(-0.961099\pi\)
0.999994 + 0.00345157i \(0.00109867\pi\)
\(38\) 6.82124 − 0.579732i 1.10655 − 0.0940449i
\(39\) −0.0348621 − 0.554118i −0.00558241 − 0.0887298i
\(40\) 5.24797 − 3.52971i 0.829776 − 0.558096i
\(41\) −5.73434 + 6.93163i −0.895554 + 1.08254i 0.100601 + 0.994927i \(0.467923\pi\)
−0.996155 + 0.0876115i \(0.972077\pi\)
\(42\) −0.860194 − 0.258761i −0.132731 − 0.0399276i
\(43\) −1.31210 + 4.03822i −0.200093 + 0.615823i 0.799786 + 0.600285i \(0.204946\pi\)
−0.999879 + 0.0155381i \(0.995054\pi\)
\(44\) 3.50549 − 7.09772i 0.528473 − 1.07002i
\(45\) 4.15300 + 3.71964i 0.619092 + 0.554492i
\(46\) 3.30406 + 11.7879i 0.487157 + 1.73802i
\(47\) −2.01298 + 0.947238i −0.293624 + 0.138169i −0.566949 − 0.823753i \(-0.691876\pi\)
0.273325 + 0.961922i \(0.411876\pi\)
\(48\) −1.61552 + 2.34458i −0.233180 + 0.338410i
\(49\) 1.91707 − 5.90012i 0.273866 − 0.842874i
\(50\) −0.930211 − 7.00962i −0.131552 − 0.991309i
\(51\) −2.42272 + 0.787190i −0.339249 + 0.110229i
\(52\) −0.416477 + 1.50337i −0.0577550 + 0.208479i
\(53\) −1.56414 + 0.197597i −0.214851 + 0.0271420i −0.232022 − 0.972711i \(-0.574534\pi\)
0.0171709 + 0.999853i \(0.494534\pi\)
\(54\) −5.18503 − 1.92234i −0.705593 − 0.261597i
\(55\) −5.36957 − 7.03564i −0.724033 − 0.948686i
\(56\) 2.04635 + 1.47731i 0.273455 + 0.197414i
\(57\) 3.44572i 0.456397i
\(58\) −0.114635 + 0.165294i −0.0150523 + 0.0217041i
\(59\) 1.63158 + 6.35460i 0.212414 + 0.827298i 0.982020 + 0.188779i \(0.0604529\pi\)
−0.769606 + 0.638520i \(0.779547\pi\)
\(60\) 1.54654 + 2.78242i 0.199658 + 0.359209i
\(61\) −3.69926 + 3.06029i −0.473641 + 0.391830i −0.843390 − 0.537302i \(-0.819444\pi\)
0.369749 + 0.929132i \(0.379444\pi\)
\(62\) −1.05935 + 0.446569i −0.134538 + 0.0567144i
\(63\) −0.819023 + 2.06862i −0.103187 + 0.260621i
\(64\) 6.44353 − 4.74140i 0.805442 − 0.592675i
\(65\) 1.29920 + 1.16363i 0.161146 + 0.144331i
\(66\) 3.41025 + 2.06062i 0.419773 + 0.253644i
\(67\) −1.03873 − 5.44519i −0.126901 − 0.665236i −0.987709 − 0.156306i \(-0.950041\pi\)
0.860808 − 0.508930i \(-0.169959\pi\)
\(68\) 7.15617 + 0.135445i 0.867813 + 0.0164251i
\(69\) −6.05271 + 1.15462i −0.728661 + 0.138999i
\(70\) 2.49590 − 1.31642i 0.298317 − 0.157342i
\(71\) −2.92239 − 6.21039i −0.346823 − 0.737037i 0.652970 − 0.757384i \(-0.273523\pi\)
−0.999793 + 0.0203469i \(0.993523\pi\)
\(72\) 5.44738 + 4.47874i 0.641979 + 0.527824i
\(73\) −3.05460 + 11.8969i −0.357514 + 1.39243i 0.494623 + 0.869107i \(0.335306\pi\)
−0.852137 + 0.523318i \(0.824694\pi\)
\(74\) −0.905306 − 0.472119i −0.105240 − 0.0548827i
\(75\) 3.55513 − 0.167744i 0.410511 − 0.0193694i
\(76\) 3.16545 − 9.14936i 0.363101 − 1.04950i
\(77\) 1.89249 − 2.98209i 0.215670 − 0.339841i
\(78\) −0.736221 − 0.272952i −0.0833606 − 0.0309057i
\(79\) −1.34553 − 0.739710i −0.151384 − 0.0832238i 0.404261 − 0.914643i \(-0.367529\pi\)
−0.555645 + 0.831420i \(0.687529\pi\)
\(80\) −1.55040 − 8.80887i −0.173340 − 0.984862i
\(81\) −1.99970 + 4.24959i −0.222189 + 0.472176i
\(82\) 5.66834 + 11.3899i 0.625964 + 1.25781i
\(83\) 4.97340 − 2.73415i 0.545902 − 0.300112i −0.184837 − 0.982769i \(-0.559176\pi\)
0.730739 + 0.682657i \(0.239176\pi\)
\(84\) −0.828127 + 0.963321i −0.0903561 + 0.105107i
\(85\) 3.68876 − 7.10137i 0.400102 − 0.770252i
\(86\) 4.46649 + 4.01349i 0.481634 + 0.432786i
\(87\) −0.0780124 − 0.0645375i −0.00836380 − 0.00691914i
\(88\) −7.16217 − 8.60438i −0.763490 − 0.917230i
\(89\) −7.48548 − 1.92194i −0.793459 − 0.203726i −0.169858 − 0.985468i \(-0.554331\pi\)
−0.623601 + 0.781743i \(0.714331\pi\)
\(90\) 7.19118 − 3.23314i 0.758017 − 0.340802i
\(91\) −0.256218 + 0.647133i −0.0268590 + 0.0678380i
\(92\) 17.1324 + 2.49455i 1.78617 + 0.260075i
\(93\) −0.178811 − 0.550324i −0.0185418 − 0.0570659i
\(94\) 0.0690653 + 3.14547i 0.00712354 + 0.324430i
\(95\) −7.59362 − 7.71366i −0.779089 − 0.791405i
\(96\) 2.10327 + 3.43368i 0.214664 + 0.350448i
\(97\) −4.10180 − 0.782460i −0.416475 − 0.0794468i −0.0251149 − 0.999685i \(-0.507995\pi\)
−0.391360 + 0.920238i \(0.627995\pi\)
\(98\) −6.52585 − 5.86399i −0.659211 − 0.592352i
\(99\) 5.80073 − 7.98401i 0.582995 − 0.802424i
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.69.94 yes 2960
8.5 even 2 inner 1000.2.bd.a.69.19 yes 2960
125.29 even 50 inner 1000.2.bd.a.29.19 ✓ 2960
1000.29 even 50 inner 1000.2.bd.a.29.94 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.29.19 ✓ 2960 125.29 even 50 inner
1000.2.bd.a.29.94 yes 2960 1000.29 even 50 inner
1000.2.bd.a.69.19 yes 2960 8.5 even 2 inner
1000.2.bd.a.69.94 yes 2960 1.1 even 1 trivial