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Properties

Label 1000.2.bd.a.109.79
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 109.79
Character \(\chi\) \(=\) 1000.109
Dual form 1000.2.bd.a.789.79

$q$-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+(0.126392 + 1.40855i) q^{2} +(-0.698121 - 0.843883i) q^{3} +(-1.96805 + 0.356059i) q^{4} +(2.03025 - 0.937060i) q^{5} +(1.10042 - 1.09000i) q^{6} +(1.23613 - 0.401644i) q^{7} +(-0.750273 - 2.72710i) q^{8} +(0.337378 - 1.76860i) q^{9} +(1.57651 + 2.74128i) q^{10} +(-0.219083 + 1.73422i) q^{11} +(1.67441 + 1.41223i) q^{12} +(0.0419820 - 0.220077i) q^{13} +(0.721974 + 1.69040i) q^{14} +(-2.20813 - 1.05911i) q^{15} +(3.74644 - 1.40148i) q^{16} +(-0.945912 - 3.68408i) q^{17} +(2.53381 + 0.251679i) q^{18} +(-4.99827 - 4.13493i) q^{19} +(-3.66199 + 2.56707i) q^{20} +(-1.20191 - 0.762756i) q^{21} +(-2.47044 - 0.0893995i) q^{22} +(-1.85158 + 1.97173i) q^{23} +(-1.77758 + 2.53699i) q^{24} +(3.24384 - 3.80494i) q^{25} +(0.315297 + 0.0313180i) q^{26} +(-4.60727 + 2.53287i) q^{27} +(-2.28976 + 1.23059i) q^{28} +(-6.22807 + 0.391837i) q^{29} +(1.21273 - 3.24413i) q^{30} +(4.02911 - 1.03450i) q^{31} +(2.44758 + 5.09993i) q^{32} +(1.61643 - 1.02582i) q^{33} +(5.06968 - 1.79801i) q^{34} +(2.13329 - 1.97377i) q^{35} +(-0.0342522 + 3.60081i) q^{36} +(-3.21413 - 1.76699i) q^{37} +(5.19253 - 7.56296i) q^{38} +(-0.215028 + 0.118213i) q^{39} +(-4.07870 - 4.83365i) q^{40} +(7.66491 - 7.19783i) q^{41} +(0.922471 - 1.78936i) q^{42} +(3.84943 - 2.79678i) q^{43} +(-0.186318 - 3.49104i) q^{44} +(-0.972320 - 3.90684i) q^{45} +(-3.01131 - 2.35884i) q^{46} +(-1.13384 - 2.86375i) q^{47} +(-3.79816 - 2.18316i) q^{48} +(-4.29641 + 3.12153i) q^{49} +(5.76945 + 4.08821i) q^{50} +(-2.44858 + 3.37017i) q^{51} +(-0.00426221 + 0.448071i) q^{52} +(3.38258 - 5.33010i) q^{53} +(-4.15000 - 6.16946i) q^{54} +(1.18028 + 3.72620i) q^{55} +(-2.02276 - 3.06972i) q^{56} +7.10464i q^{57} +(-1.33910 - 8.72305i) q^{58} +(-0.877787 + 0.413055i) q^{59} +(4.72282 + 1.29816i) q^{60} +(5.80787 - 6.18476i) q^{61} +(1.96640 + 5.54447i) q^{62} +(-0.293302 - 2.32173i) q^{63} +(-6.87418 + 4.09214i) q^{64} +(-0.120992 - 0.486152i) q^{65} +(1.64922 + 2.14717i) q^{66} +(0.0875745 - 1.39196i) q^{67} +(3.17335 + 6.91366i) q^{68} +(2.95653 + 0.186009i) q^{69} +(3.04979 + 2.75539i) q^{70} +(8.36832 - 3.31325i) q^{71} +(-5.07627 + 0.406866i) q^{72} +(3.45272 + 1.62473i) q^{73} +(2.08266 - 4.75062i) q^{74} +(-5.47551 - 0.0811132i) q^{75} +(11.3091 + 6.35807i) q^{76} +(0.425724 + 2.23172i) q^{77} +(-0.193687 - 0.287937i) q^{78} +(8.56130 + 10.3488i) q^{79} +(6.29295 - 6.35601i) q^{80} +(0.331723 + 0.131339i) q^{81} +(11.1073 + 9.88669i) q^{82} +(-3.77354 + 4.56142i) q^{83} +(2.63701 + 1.07319i) q^{84} +(-5.37265 - 6.59324i) q^{85} +(4.42595 + 5.06864i) q^{86} +(4.67861 + 4.98222i) q^{87} +(4.89378 - 0.703678i) q^{88} +(5.89626 - 12.5302i) q^{89} +(5.38010 - 1.86336i) q^{90} +(-0.0364974 - 0.288906i) q^{91} +(2.94194 - 4.53973i) q^{92} +(-3.68581 - 2.67789i) q^{93} +(3.89044 - 1.95903i) q^{94} +(-14.0224 - 3.71126i) q^{95} +(2.59504 - 5.62585i) q^{96} +(-1.94213 + 0.122189i) q^{97} +(-4.93987 - 5.65720i) q^{98} +(2.99323 + 0.972558i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.126392 + 1.40855i 0.0893723 + 0.995998i
\(3\) −0.698121 − 0.843883i −0.403060 − 0.487216i 0.529459 − 0.848335i \(-0.322395\pi\)
−0.932520 + 0.361119i \(0.882395\pi\)
\(4\) −1.96805 + 0.356059i −0.984025 + 0.178029i
\(5\) 2.03025 − 0.937060i 0.907956 − 0.419066i
\(6\) 1.10042 − 1.09000i 0.449244 − 0.444991i
\(7\) 1.23613 − 0.401644i 0.467214 − 0.151807i −0.0659428 − 0.997823i \(-0.521006\pi\)
0.533157 + 0.846016i \(0.321006\pi\)
\(8\) −0.750273 − 2.72710i −0.265262 − 0.964176i
\(9\) 0.337378 − 1.76860i 0.112459 − 0.589532i
\(10\) 1.57651 + 2.74128i 0.498535 + 0.866869i
\(11\) −0.219083 + 1.73422i −0.0660561 + 0.522888i 0.924019 + 0.382347i \(0.124884\pi\)
−0.990075 + 0.140541i \(0.955116\pi\)
\(12\) 1.67441 + 1.41223i 0.483360 + 0.407676i
\(13\) 0.0419820 − 0.220077i 0.0116437 − 0.0610384i −0.975991 − 0.217810i \(-0.930109\pi\)
0.987635 + 0.156772i \(0.0501087\pi\)
\(14\) 0.721974 + 1.69040i 0.192956 + 0.451777i
\(15\) −2.20813 − 1.05911i −0.570137 − 0.273462i
\(16\) 3.74644 − 1.40148i 0.936611 − 0.350371i
\(17\) −0.945912 − 3.68408i −0.229417 − 0.893521i −0.974103 − 0.226106i \(-0.927400\pi\)
0.744685 − 0.667416i \(-0.232600\pi\)
\(18\) 2.53381 + 0.251679i 0.597224 + 0.0593214i
\(19\) −4.99827 − 4.13493i −1.14668 − 0.948618i −0.147500 − 0.989062i \(-0.547123\pi\)
−0.999182 + 0.0404442i \(0.987123\pi\)
\(20\) −3.66199 + 2.56707i −0.818845 + 0.574014i
\(21\) −1.20191 − 0.762756i −0.262278 − 0.166447i
\(22\) −2.47044 − 0.0893995i −0.526699 − 0.0190600i
\(23\) −1.85158 + 1.97173i −0.386080 + 0.411134i −0.892424 − 0.451198i \(-0.850997\pi\)
0.506343 + 0.862332i \(0.330997\pi\)
\(24\) −1.77758 + 2.53699i −0.362846 + 0.517861i
\(25\) 3.24384 − 3.80494i 0.648767 − 0.760987i
\(26\) 0.315297 + 0.0313180i 0.0618348 + 0.00614196i
\(27\) −4.60727 + 2.53287i −0.886670 + 0.487451i
\(28\) −2.28976 + 1.23059i −0.432724 + 0.232560i
\(29\) −6.22807 + 0.391837i −1.15652 + 0.0727623i −0.629439 − 0.777050i \(-0.716715\pi\)
−0.527085 + 0.849812i \(0.676715\pi\)
\(30\) 1.21273 − 3.24413i 0.221413 − 0.592295i
\(31\) 4.02911 − 1.03450i 0.723650 − 0.185802i 0.131139 − 0.991364i \(-0.458137\pi\)
0.592511 + 0.805562i \(0.298137\pi\)
\(32\) 2.44758 + 5.09993i 0.432676 + 0.901550i
\(33\) 1.61643 − 1.02582i 0.281384 − 0.178572i
\(34\) 5.06968 − 1.79801i 0.869442 − 0.308355i
\(35\) 2.13329 − 1.97377i 0.360593 − 0.333628i
\(36\) −0.0342522 + 3.60081i −0.00570870 + 0.600136i
\(37\) −3.21413 − 1.76699i −0.528400 − 0.290491i 0.195137 − 0.980776i \(-0.437485\pi\)
−0.723537 + 0.690285i \(0.757485\pi\)
\(38\) 5.19253 − 7.56296i 0.842340 − 1.22687i
\(39\) −0.215028 + 0.118213i −0.0344320 + 0.0189292i
\(40\) −4.07870 − 4.83365i −0.644900 − 0.764267i
\(41\) 7.66491 − 7.19783i 1.19706 − 1.12411i 0.207763 − 0.978179i \(-0.433382\pi\)
0.989295 − 0.145933i \(-0.0466183\pi\)
\(42\) 0.922471 − 1.78936i 0.142340 − 0.276105i
\(43\) 3.84943 − 2.79678i 0.587033 − 0.426504i −0.254220 − 0.967146i \(-0.581819\pi\)
0.841253 + 0.540642i \(0.181819\pi\)
\(44\) −0.186318 − 3.49104i −0.0280885 − 0.526295i
\(45\) −0.972320 − 3.90684i −0.144945 − 0.582397i
\(46\) −3.01131 − 2.35884i −0.443994 − 0.347791i
\(47\) −1.13384 − 2.86375i −0.165387 − 0.417721i 0.823321 − 0.567577i \(-0.192119\pi\)
−0.988708 + 0.149856i \(0.952119\pi\)
\(48\) −3.79816 − 2.18316i −0.548217 − 0.315112i
\(49\) −4.29641 + 3.12153i −0.613773 + 0.445932i
\(50\) 5.76945 + 4.08821i 0.815924 + 0.578160i
\(51\) −2.44858 + 3.37017i −0.342869 + 0.471919i
\(52\) −0.00426221 + 0.448071i −0.000591063 + 0.0621363i
\(53\) 3.38258 − 5.33010i 0.464634 − 0.732146i −0.528525 − 0.848918i \(-0.677255\pi\)
0.993159 + 0.116772i \(0.0372547\pi\)
\(54\) −4.15000 − 6.16946i −0.564744 − 0.839557i
\(55\) 1.18028 + 3.72620i 0.159149 + 0.502441i
\(56\) −2.02276 − 3.06972i −0.270303 − 0.410208i
\(57\) 7.10464i 0.941032i
\(58\) −1.33910 − 8.72305i −0.175832 − 1.14539i
\(59\) −0.877787 + 0.413055i −0.114278 + 0.0537752i −0.482069 − 0.876134i \(-0.660114\pi\)
0.367790 + 0.929909i \(0.380114\pi\)
\(60\) 4.72282 + 1.29816i 0.609713 + 0.167592i
\(61\) 5.80787 − 6.18476i 0.743622 − 0.791877i −0.240258 − 0.970709i \(-0.577232\pi\)
0.983880 + 0.178832i \(0.0572319\pi\)
\(62\) 1.96640 + 5.54447i 0.249733 + 0.704149i
\(63\) −0.293302 − 2.32173i −0.0369526 − 0.292510i
\(64\) −6.87418 + 4.09214i −0.859273 + 0.511518i
\(65\) −0.120992 − 0.486152i −0.0150072 − 0.0602997i
\(66\) 1.64922 + 2.14717i 0.203005 + 0.264299i
\(67\) 0.0875745 − 1.39196i 0.0106989 − 0.170055i −0.989049 − 0.147590i \(-0.952849\pi\)
0.999748 − 0.0224651i \(-0.00715147\pi\)
\(68\) 3.17335 + 6.91366i 0.384826 + 0.838405i
\(69\) 2.95653 + 0.186009i 0.355925 + 0.0223929i
\(70\) 3.04979 + 2.75539i 0.364520 + 0.329332i
\(71\) 8.36832 − 3.31325i 0.993137 − 0.393211i 0.185342 − 0.982674i \(-0.440661\pi\)
0.807795 + 0.589463i \(0.200661\pi\)
\(72\) −5.07627 + 0.406866i −0.598244 + 0.0479497i
\(73\) 3.45272 + 1.62473i 0.404110 + 0.190160i 0.617083 − 0.786898i \(-0.288314\pi\)
−0.212972 + 0.977058i \(0.568314\pi\)
\(74\) 2.08266 − 4.75062i 0.242104 − 0.552248i
\(75\) −5.47551 − 0.0811132i −0.632257 − 0.00936615i
\(76\) 11.3091 + 6.35807i 1.29725 + 0.729321i
\(77\) 0.425724 + 2.23172i 0.0485157 + 0.254328i
\(78\) −0.193687 − 0.287937i −0.0219307 − 0.0326025i
\(79\) 8.56130 + 10.3488i 0.963221 + 1.16433i 0.986360 + 0.164602i \(0.0526341\pi\)
−0.0231386 + 0.999732i \(0.507366\pi\)
\(80\) 6.29295 − 6.35601i 0.703573 − 0.710623i
\(81\) 0.331723 + 0.131339i 0.0368581 + 0.0145932i
\(82\) 11.1073 + 9.88669i 1.22660 + 1.09180i
\(83\) −3.77354 + 4.56142i −0.414200 + 0.500681i −0.935811 − 0.352502i \(-0.885331\pi\)
0.521612 + 0.853183i \(0.325331\pi\)
\(84\) 2.63701 + 1.07319i 0.287721 + 0.117095i
\(85\) −5.37265 − 6.59324i −0.582745 − 0.715137i
\(86\) 4.42595 + 5.06864i 0.477262 + 0.546566i
\(87\) 4.67861 + 4.98222i 0.501600 + 0.534150i
\(88\) 4.89378 − 0.703678i 0.521678 − 0.0750123i
\(89\) 5.89626 − 12.5302i 0.625002 − 1.32820i −0.302060 − 0.953289i \(-0.597674\pi\)
0.927062 − 0.374908i \(-0.122326\pi\)
\(90\) 5.38010 − 1.86336i 0.567112 − 0.196415i
\(91\) −0.0364974 − 0.288906i −0.00382596 − 0.0302856i
\(92\) 2.94194 − 4.53973i 0.306719 − 0.473300i
\(93\) −3.68581 − 2.67789i −0.382200 − 0.277685i
\(94\) 3.89044 − 1.95903i 0.401268 − 0.202058i
\(95\) −14.0224 − 3.71126i −1.43867 − 0.380767i
\(96\) 2.59504 − 5.62585i 0.264855 − 0.574186i
\(97\) −1.94213 + 0.122189i −0.197194 + 0.0124064i −0.161080 − 0.986941i \(-0.551498\pi\)
−0.0361135 + 0.999348i \(0.511498\pi\)
\(98\) −4.93987 − 5.65720i −0.499002 − 0.571463i
\(99\) 2.99323 + 0.972558i 0.300831 + 0.0977458i
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.109.79 yes 2960
8.5 even 2 inner 1000.2.bd.a.109.59 ✓ 2960
125.39 even 50 inner 1000.2.bd.a.789.59 yes 2960
1000.789 even 50 inner 1000.2.bd.a.789.79 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.109.59 ✓ 2960 8.5 even 2 inner
1000.2.bd.a.109.79 yes 2960 1.1 even 1 trivial
1000.2.bd.a.789.59 yes 2960 125.39 even 50 inner
1000.2.bd.a.789.79 yes 2960 1000.789 even 50 inner