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Properties

Label 1000.2.bd.a.29.19
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 29.19
Character \(\chi\) \(=\) 1000.29
Dual form 1000.2.bd.a.69.19

$q$-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+(-1.30316 + 0.549346i) q^{2} +(-0.623770 + 0.342921i) q^{3} +(1.39644 - 1.43177i) q^{4} +(2.15211 + 0.606986i) q^{5} +(0.624489 - 0.789545i) q^{6} +(-0.524496 + 0.721907i) q^{7} +(-1.03324 + 2.63295i) q^{8} +(-1.33599 + 2.10518i) q^{9} +(-3.13798 + 0.391253i) q^{10} +(-1.45707 + 3.68014i) q^{11} +(-0.380074 + 1.37196i) q^{12} +(0.417942 - 0.658571i) q^{13} +(0.286925 - 1.22889i) q^{14} +(-1.55057 + 0.359382i) q^{15} +(-0.0999192 - 3.99875i) q^{16} +(-2.44981 - 2.60878i) q^{17} +(0.584530 - 3.47729i) q^{18} +(-2.33204 + 4.24197i) q^{19} +(3.87435 - 2.23370i) q^{20} +(0.0796083 - 0.630165i) q^{21} +(-0.122877 - 5.59624i) q^{22} +(-6.66995 - 5.51787i) q^{23} +(-0.258385 - 1.99667i) q^{24} +(4.26314 + 2.61260i) q^{25} +(-0.182861 + 1.08782i) q^{26} +(0.245526 - 3.90252i) q^{27} +(0.301177 + 1.75906i) q^{28} +(0.139718 - 0.0266527i) q^{29} +(1.82321 - 1.32013i) q^{30} +(-0.592587 + 0.556476i) q^{31} +(2.32691 + 5.15611i) q^{32} +(-0.353119 - 2.79522i) q^{33} +(4.62561 + 2.05386i) q^{34} +(-1.56696 + 1.23526i) q^{35} +(1.14850 + 4.85257i) q^{36} +(-0.0453327 - 0.720543i) q^{37} +(0.708710 - 6.80905i) q^{38} +(-0.0348621 + 0.554118i) q^{39} +(-3.82181 + 5.03922i) q^{40} +(-5.73434 - 6.93163i) q^{41} +(0.242436 + 0.864937i) q^{42} +(1.31210 + 4.03822i) q^{43} +(3.23440 + 7.22528i) q^{44} +(-4.15300 + 3.71964i) q^{45} +(11.7232 + 3.52654i) q^{46} +(-2.01298 - 0.947238i) q^{47} +(1.43358 + 2.46004i) q^{48} +(1.91707 + 5.90012i) q^{49} +(-6.99076 - 1.06269i) q^{50} +(2.42272 + 0.787190i) q^{51} +(-0.359291 - 1.51805i) q^{52} +(1.56414 + 0.197597i) q^{53} +(1.82387 + 5.22048i) q^{54} +(-5.36957 + 7.03564i) q^{55} +(-1.35881 - 2.12688i) q^{56} -3.44572i q^{57} +(-0.167433 + 0.111486i) q^{58} +(-1.63158 + 6.35460i) q^{59} +(-1.65072 + 2.72191i) q^{60} +(3.69926 + 3.06029i) q^{61} +(0.466536 - 1.05071i) q^{62} +(-0.819023 - 2.06862i) q^{63} +(-5.86482 - 5.44095i) q^{64} +(1.29920 - 1.16363i) q^{65} +(1.99571 + 3.44863i) q^{66} +(1.03873 - 5.44519i) q^{67} +(-7.15617 - 0.135445i) q^{68} +(6.05271 + 1.15462i) q^{69} +(1.36341 - 2.47054i) q^{70} +(-2.92239 + 6.21039i) q^{71} +(-4.16242 - 5.69274i) q^{72} +(-3.05460 - 11.8969i) q^{73} +(0.454903 + 0.914078i) q^{74} +(-3.55513 - 0.167744i) q^{75} +(2.81696 + 9.26259i) q^{76} +(-1.89249 - 2.98209i) q^{77} +(-0.258971 - 0.741254i) q^{78} +(-1.34553 + 0.739710i) q^{79} +(2.21215 - 8.66639i) q^{80} +(-1.99970 - 4.24959i) q^{81} +(11.2806 + 5.88287i) q^{82} +(-4.97340 - 2.73415i) q^{83} +(-0.791082 - 0.993967i) q^{84} +(-3.68876 - 7.10137i) q^{85} +(-3.92825 - 4.54164i) q^{86} +(-0.0780124 + 0.0645375i) q^{87} +(-8.18411 - 7.63888i) q^{88} +(-7.48548 + 1.92194i) q^{89} +(3.36864 - 7.12871i) q^{90} +(0.256218 + 0.647133i) q^{91} +(-17.2145 + 1.84447i) q^{92} +(0.178811 - 0.550324i) q^{93} +(3.14360 + 0.128576i) q^{94} +(-7.59362 + 7.71366i) q^{95} +(-3.21959 - 2.41829i) q^{96} +(-4.10180 + 0.782460i) q^{97} +(-5.73944 - 6.63565i) 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 \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.30316 + 0.549346i −0.921471 + 0.388446i
\(3\) −0.623770 + 0.342921i −0.360134 + 0.197985i −0.651540 − 0.758614i \(-0.725877\pi\)
0.291406 + 0.956599i \(0.405877\pi\)
\(4\) 1.39644 − 1.43177i 0.698219 − 0.715884i
\(5\) 2.15211 + 0.606986i 0.962452 + 0.271452i
\(6\) 0.624489 − 0.789545i 0.254946 − 0.322330i
\(7\) −0.524496 + 0.721907i −0.198241 + 0.272855i −0.896551 − 0.442940i \(-0.853936\pi\)
0.698310 + 0.715795i \(0.253936\pi\)
\(8\) −1.03324 + 2.63295i −0.365307 + 0.930887i
\(9\) −1.33599 + 2.10518i −0.445329 + 0.701725i
\(10\) −3.13798 + 0.391253i −0.992317 + 0.123725i
\(11\) −1.45707 + 3.68014i −0.439324 + 1.10961i 0.526447 + 0.850208i \(0.323524\pi\)
−0.965771 + 0.259397i \(0.916476\pi\)
\(12\) −0.380074 + 1.37196i −0.109718 + 0.396051i
\(13\) 0.417942 − 0.658571i 0.115916 − 0.182655i −0.781416 − 0.624010i \(-0.785502\pi\)
0.897332 + 0.441356i \(0.145502\pi\)
\(14\) 0.286925 − 1.22889i 0.0766838 − 0.328434i
\(15\) −1.55057 + 0.359382i −0.400355 + 0.0927921i
\(16\) −0.0999192 − 3.99875i −0.0249798 − 0.999688i
\(17\) −2.44981 − 2.60878i −0.594165 − 0.632722i 0.359660 − 0.933083i \(-0.382893\pi\)
−0.953825 + 0.300362i \(0.902893\pi\)
\(18\) 0.584530 − 3.47729i 0.137775 − 0.819606i
\(19\) −2.33204 + 4.24197i −0.535007 + 0.973175i 0.461294 + 0.887247i \(0.347385\pi\)
−0.996301 + 0.0859273i \(0.972615\pi\)
\(20\) 3.87435 − 2.23370i 0.866331 − 0.499471i
\(21\) 0.0796083 − 0.630165i 0.0173720 − 0.137513i
\(22\) −0.122877 − 5.59624i −0.0261975 − 1.19312i
\(23\) −6.66995 − 5.51787i −1.39078 − 1.15055i −0.969918 − 0.243431i \(-0.921727\pi\)
−0.420863 − 0.907124i \(-0.638273\pi\)
\(24\) −0.258385 − 1.99667i −0.0527426 − 0.407569i
\(25\) 4.26314 + 2.61260i 0.852627 + 0.522520i
\(26\) −0.182861 + 1.08782i −0.0358620 + 0.213338i
\(27\) 0.245526 − 3.90252i 0.0472515 − 0.751041i
\(28\) 0.301177 + 1.75906i 0.0569171 + 0.332430i
\(29\) 0.139718 − 0.0266527i 0.0259450 − 0.00494928i −0.174392 − 0.984676i \(-0.555796\pi\)
0.200337 + 0.979727i \(0.435796\pi\)
\(30\) 1.82321 − 1.32013i 0.332871 − 0.241022i
\(31\) −0.592587 + 0.556476i −0.106432 + 0.0999461i −0.735936 − 0.677051i \(-0.763258\pi\)
0.629505 + 0.776997i \(0.283258\pi\)
\(32\) 2.32691 + 5.15611i 0.411343 + 0.911481i
\(33\) −0.353119 − 2.79522i −0.0614701 − 0.486586i
\(34\) 4.62561 + 2.05386i 0.793285 + 0.352234i
\(35\) −1.56696 + 1.23526i −0.264865 + 0.208797i
\(36\) 1.14850 + 4.85257i 0.191417 + 0.808762i
\(37\) −0.0453327 − 0.720543i −0.00745266 − 0.118457i 0.992541 − 0.121908i \(-0.0389013\pi\)
−0.999994 + 0.00345157i \(0.998901\pi\)
\(38\) 0.708710 − 6.80905i 0.114968 − 1.10457i
\(39\) −0.0348621 + 0.554118i −0.00558241 + 0.0887298i
\(40\) −3.82181 + 5.03922i −0.604282 + 0.796771i
\(41\) −5.73434 − 6.93163i −0.895554 − 1.08254i −0.996155 − 0.0876115i \(-0.972077\pi\)
0.100601 − 0.994927i \(-0.467923\pi\)
\(42\) 0.242436 + 0.864937i 0.0374087 + 0.133463i
\(43\) 1.31210 + 4.03822i 0.200093 + 0.615823i 0.999879 + 0.0155381i \(0.00494612\pi\)
−0.799786 + 0.600285i \(0.795054\pi\)
\(44\) 3.23440 + 7.22528i 0.487604 + 1.08925i
\(45\) −4.15300 + 3.71964i −0.619092 + 0.554492i
\(46\) 11.7232 + 3.52654i 1.72849 + 0.519960i
\(47\) −2.01298 − 0.947238i −0.293624 − 0.138169i 0.273325 − 0.961922i \(-0.411876\pi\)
−0.566949 + 0.823753i \(0.691876\pi\)
\(48\) 1.43358 + 2.46004i 0.206920 + 0.355076i
\(49\) 1.91707 + 5.90012i 0.273866 + 0.842874i
\(50\) −6.99076 − 1.06269i −0.988642 − 0.150287i
\(51\) 2.42272 + 0.787190i 0.339249 + 0.110229i
\(52\) −0.359291 − 1.51805i −0.0498247 − 0.210516i
\(53\) 1.56414 + 0.197597i 0.214851 + 0.0271420i 0.232022 − 0.972711i \(-0.425466\pi\)
−0.0171709 + 0.999853i \(0.505466\pi\)
\(54\) 1.82387 + 5.22048i 0.248198 + 0.710417i
\(55\) −5.36957 + 7.03564i −0.724033 + 0.948686i
\(56\) −1.35881 − 2.12688i −0.181579 − 0.284216i
\(57\) − 3.44572i − 0.456397i
\(58\) −0.167433 + 0.111486i −0.0219851 + 0.0146389i
\(59\) −1.63158 + 6.35460i −0.212414 + 0.827298i 0.769606 + 0.638520i \(0.220453\pi\)
−0.982020 + 0.188779i \(0.939547\pi\)
\(60\) −1.65072 + 2.72191i −0.213107 + 0.351397i
\(61\) 3.69926 + 3.06029i 0.473641 + 0.391830i 0.843390 − 0.537302i \(-0.180556\pi\)
−0.369749 + 0.929132i \(0.620556\pi\)
\(62\) 0.466536 − 1.05071i 0.0592502 − 0.133440i
\(63\) −0.819023 − 2.06862i −0.103187 − 0.260621i
\(64\) −5.86482 − 5.44095i −0.733102 − 0.680119i
\(65\) 1.29920 − 1.16363i 0.161146 − 0.144331i
\(66\) 1.99571 + 3.44863i 0.245655 + 0.424497i
\(67\) 1.03873 − 5.44519i 0.126901 − 0.665236i −0.860808 − 0.508930i \(-0.830041\pi\)
0.987709 − 0.156306i \(-0.0499588\pi\)
\(68\) −7.15617 − 0.135445i −0.867813 − 0.0164251i
\(69\) 6.05271 + 1.15462i 0.728661 + 0.138999i
\(70\) 1.36341 − 2.47054i 0.162959 − 0.295286i
\(71\) −2.92239 + 6.21039i −0.346823 + 0.737037i −0.999793 − 0.0203469i \(-0.993523\pi\)
0.652970 + 0.757384i \(0.273523\pi\)
\(72\) −4.16242 − 5.69274i −0.490546 − 0.670896i
\(73\) −3.05460 − 11.8969i −0.357514 − 1.39243i −0.852137 − 0.523318i \(-0.824694\pi\)
0.494623 − 0.869107i \(-0.335306\pi\)
\(74\) 0.454903 + 0.914078i 0.0528814 + 0.106259i
\(75\) −3.55513 − 0.167744i −0.410511 − 0.0193694i
\(76\) 2.81696 + 9.26259i 0.323128 + 1.06249i
\(77\) −1.89249 − 2.98209i −0.215670 − 0.339841i
\(78\) −0.258971 − 0.741254i −0.0293227 − 0.0839305i
\(79\) −1.34553 + 0.739710i −0.151384 + 0.0832238i −0.555645 − 0.831420i \(-0.687529\pi\)
0.404261 + 0.914643i \(0.367529\pi\)
\(80\) 2.21215 − 8.66639i 0.247326 − 0.968932i
\(81\) −1.99970 − 4.24959i −0.222189 − 0.472176i
\(82\) 11.2806 + 5.88287i 1.24573 + 0.649654i
\(83\) −4.97340 − 2.73415i −0.545902 − 0.300112i 0.184837 − 0.982769i \(-0.440824\pi\)
−0.730739 + 0.682657i \(0.760824\pi\)
\(84\) −0.791082 − 0.993967i −0.0863141 − 0.108451i
\(85\) −3.68876 − 7.10137i −0.400102 − 0.770252i
\(86\) −3.92825 − 4.54164i −0.423594 − 0.489738i
\(87\) −0.0780124 + 0.0645375i −0.00836380 + 0.00691914i
\(88\) −8.18411 − 7.63888i −0.872429 − 0.814307i
\(89\) −7.48548 + 1.92194i −0.793459 + 0.203726i −0.623601 − 0.781743i \(-0.714331\pi\)
−0.169858 + 0.985468i \(0.554331\pi\)
\(90\) 3.36864 − 7.12871i 0.355086 − 0.751432i
\(91\) 0.256218 + 0.647133i 0.0268590 + 0.0678380i
\(92\) −17.2145 + 1.84447i −1.79473 + 0.192299i
\(93\) 0.178811 − 0.550324i 0.0185418 − 0.0570659i
\(94\) 3.14360 + 0.128576i 0.324237 + 0.0132616i
\(95\) −7.59362 + 7.71366i −0.779089 + 0.791405i
\(96\) −3.21959 − 2.41829i −0.328598 − 0.246815i
\(97\) −4.10180 + 0.782460i −0.416475 + 0.0794468i −0.391360 − 0.920238i \(-0.627995\pi\)
−0.0251149 + 0.999685i \(0.507995\pi\)
\(98\) −5.73944 − 6.63565i −0.579771 − 0.670302i
\(99\) −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.29.19 ✓ 2960
8.5 even 2 inner 1000.2.bd.a.29.94 yes 2960
125.69 even 50 inner 1000.2.bd.a.69.94 yes 2960
1000.69 even 50 inner 1000.2.bd.a.69.19 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.29.19 ✓ 2960 1.1 even 1 trivial
1000.2.bd.a.29.94 yes 2960 8.5 even 2 inner
1000.2.bd.a.69.19 yes 2960 1000.69 even 50 inner
1000.2.bd.a.69.94 yes 2960 125.69 even 50 inner