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Properties

Label 1000.2.bd.a.109.59
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 109.59
Character \(\chi\) \(=\) 1000.109
Dual form 1000.2.bd.a.789.59

$q$-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+(-0.472714 + 1.33287i) q^{2} +(0.698121 + 0.843883i) q^{3} +(-1.55308 - 1.26013i) q^{4} +(-2.03025 + 0.937060i) q^{5} +(-1.45480 + 0.531589i) q^{6} +(1.23613 - 0.401644i) q^{7} +(2.41376 - 1.47438i) q^{8} +(0.337378 - 1.76860i) q^{9} +(-0.289252 - 3.14902i) q^{10} +(0.219083 - 1.73422i) q^{11} +(-0.0208353 - 2.19035i) q^{12} +(-0.0419820 + 0.220077i) q^{13} +(-0.0489983 + 1.83747i) q^{14} +(-2.20813 - 1.05911i) q^{15} +(0.824134 + 3.91418i) q^{16} +(-0.945912 - 3.68408i) q^{17} +(2.19783 + 1.28572i) q^{18} +(4.99827 + 4.13493i) q^{19} +(4.33397 + 1.10305i) q^{20} +(1.20191 + 0.762756i) q^{21} +(2.20793 + 1.11180i) q^{22} +(-1.85158 + 1.97173i) q^{23} +(2.92929 + 1.00764i) q^{24} +(3.24384 - 3.80494i) q^{25} +(-0.273489 - 0.159990i) q^{26} +(4.60727 - 2.53287i) q^{27} +(-2.42594 - 0.933904i) q^{28} +(6.22807 - 0.391837i) q^{29} +(2.45547 - 2.44249i) q^{30} +(4.02911 - 1.03450i) q^{31} +(-5.60667 - 0.751824i) q^{32} +(1.61643 - 1.02582i) q^{33} +(5.35755 + 0.480740i) q^{34} +(-2.13329 + 1.97377i) q^{35} +(-2.75264 + 2.32164i) q^{36} +(3.21413 + 1.76699i) q^{37} +(-7.87407 + 4.70740i) q^{38} +(-0.215028 + 0.118213i) q^{39} +(-3.51895 + 5.25519i) q^{40} +(7.66491 - 7.19783i) q^{41} +(-1.58481 + 1.24142i) q^{42} +(-3.84943 + 2.79678i) q^{43} +(-2.52560 + 2.41732i) q^{44} +(0.972320 + 3.90684i) q^{45} +(-1.75279 - 3.39997i) q^{46} +(-1.13384 - 2.86375i) q^{47} +(-2.72777 + 3.42804i) q^{48} +(-4.29641 + 3.12153i) q^{49} +(3.53808 + 6.12226i) q^{50} +(2.44858 - 3.37017i) q^{51} +(0.342528 - 0.288895i) q^{52} +(-3.38258 + 5.33010i) q^{53} +(1.19806 + 7.33821i) q^{54} +(1.18028 + 3.72620i) q^{55} +(2.39155 - 2.79199i) q^{56} +7.10464i q^{57} +(-2.42183 + 8.48643i) q^{58} +(0.877787 - 0.413055i) q^{59} +(2.09479 + 4.42743i) q^{60} +(-5.80787 + 6.18476i) q^{61} +(-0.525764 + 5.85931i) q^{62} +(-0.293302 - 2.32173i) q^{63} +(3.65244 - 7.11756i) q^{64} +(-0.120992 - 0.486152i) q^{65} +(0.603171 + 2.63941i) q^{66} +(-0.0875745 + 1.39196i) q^{67} +(-3.17335 + 6.91366i) q^{68} +(-2.95653 - 0.186009i) q^{69} +(-1.62234 - 3.77643i) q^{70} +(8.36832 - 3.31325i) q^{71} +(-1.79323 - 4.76638i) q^{72} +(3.45272 + 1.62473i) q^{73} +(-3.87453 + 3.44874i) q^{74} +(5.47551 + 0.0811132i) q^{75} +(-2.55217 - 12.7204i) q^{76} +(-0.425724 - 2.23172i) q^{77} +(-0.0559153 - 0.342485i) q^{78} +(8.56130 + 10.3488i) q^{79} +(-5.34102 - 7.17450i) q^{80} +(0.331723 + 0.131339i) q^{81} +(5.97046 + 13.6188i) q^{82} +(3.77354 - 4.56142i) q^{83} +(-0.905494 - 2.69919i) q^{84} +(5.37265 + 6.59324i) q^{85} +(-1.90806 - 6.45286i) q^{86} +(4.67861 + 4.98222i) q^{87} +(-2.02808 - 4.50900i) q^{88} +(5.89626 - 12.5302i) q^{89} +(-5.66694 - 0.550840i) q^{90} +(0.0364974 + 0.288906i) q^{91} +(5.36029 - 0.729027i) q^{92} +(3.68581 + 2.67789i) q^{93} +(4.35298 - 0.157525i) q^{94} +(-14.0224 - 3.71126i) q^{95} +(-3.27968 - 5.25624i) q^{96} +(-1.94213 + 0.122189i) q^{97} +(-2.12961 - 7.20215i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.472714 + 1.33287i −0.334259 + 0.942481i
\(3\) 0.698121 + 0.843883i 0.403060 + 0.487216i 0.932520 − 0.361119i \(-0.117605\pi\)
−0.529459 + 0.848335i \(0.677605\pi\)
\(4\) −1.55308 − 1.26013i −0.776542 − 0.630066i
\(5\) −2.03025 + 0.937060i −0.907956 + 0.419066i
\(6\) −1.45480 + 0.531589i −0.593919 + 0.217020i
\(7\) 1.23613 − 0.401644i 0.467214 − 0.151807i −0.0659428 − 0.997823i \(-0.521006\pi\)
0.533157 + 0.846016i \(0.321006\pi\)
\(8\) 2.41376 − 1.47438i 0.853392 − 0.521270i
\(9\) 0.337378 − 1.76860i 0.112459 − 0.589532i
\(10\) −0.289252 − 3.14902i −0.0914694 − 0.995808i
\(11\) 0.219083 − 1.73422i 0.0660561 − 0.522888i −0.924019 − 0.382347i \(-0.875116\pi\)
0.990075 − 0.140541i \(-0.0448841\pi\)
\(12\) −0.0208353 − 2.19035i −0.00601465 − 0.632298i
\(13\) −0.0419820 + 0.220077i −0.0116437 + 0.0610384i −0.987635 − 0.156772i \(-0.949891\pi\)
0.975991 + 0.217810i \(0.0698913\pi\)
\(14\) −0.0489983 + 1.83747i −0.0130953 + 0.491083i
\(15\) −2.20813 − 1.05911i −0.570137 − 0.273462i
\(16\) 0.824134 + 3.91418i 0.206033 + 0.978545i
\(17\) −0.945912 − 3.68408i −0.229417 − 0.893521i −0.974103 − 0.226106i \(-0.927400\pi\)
0.744685 − 0.667416i \(-0.232600\pi\)
\(18\) 2.19783 + 1.28572i 0.518032 + 0.303047i
\(19\) 4.99827 + 4.13493i 1.14668 + 0.948618i 0.999182 − 0.0404442i \(-0.0128773\pi\)
0.147500 + 0.989062i \(0.452877\pi\)
\(20\) 4.33397 + 1.10305i 0.969105 + 0.246650i
\(21\) 1.20191 + 0.762756i 0.262278 + 0.166447i
\(22\) 2.20793 + 1.11180i 0.470732 + 0.237037i
\(23\) −1.85158 + 1.97173i −0.386080 + 0.411134i −0.892424 − 0.451198i \(-0.850997\pi\)
0.506343 + 0.862332i \(0.330997\pi\)
\(24\) 2.92929 + 1.00764i 0.597940 + 0.205683i
\(25\) 3.24384 − 3.80494i 0.648767 − 0.760987i
\(26\) −0.273489 − 0.159990i −0.0536356 − 0.0313766i
\(27\) 4.60727 − 2.53287i 0.886670 − 0.487451i
\(28\) −2.42594 − 0.933904i −0.458460 − 0.176491i
\(29\) 6.22807 − 0.391837i 1.15652 − 0.0727623i 0.527085 − 0.849812i \(-0.323285\pi\)
0.629439 + 0.777050i \(0.283285\pi\)
\(30\) 2.45547 − 2.44249i 0.448306 − 0.445936i
\(31\) 4.02911 − 1.03450i 0.723650 − 0.185802i 0.131139 − 0.991364i \(-0.458137\pi\)
0.592511 + 0.805562i \(0.298137\pi\)
\(32\) −5.60667 − 0.751824i −0.991129 − 0.132905i
\(33\) 1.61643 − 1.02582i 0.281384 − 0.178572i
\(34\) 5.35755 + 0.480740i 0.918812 + 0.0824463i
\(35\) −2.13329 + 1.97377i −0.360593 + 0.333628i
\(36\) −2.75264 + 2.32164i −0.458774 + 0.386939i
\(37\) 3.21413 + 1.76699i 0.528400 + 0.290491i 0.723537 − 0.690285i \(-0.242515\pi\)
−0.195137 + 0.980776i \(0.562515\pi\)
\(38\) −7.87407 + 4.70740i −1.27734 + 0.763642i
\(39\) −0.215028 + 0.118213i −0.0344320 + 0.0189292i
\(40\) −3.51895 + 5.25519i −0.556395 + 0.830918i
\(41\) 7.66491 − 7.19783i 1.19706 − 1.12411i 0.207763 − 0.978179i \(-0.433382\pi\)
0.989295 − 0.145933i \(-0.0466183\pi\)
\(42\) −1.58481 + 1.24142i −0.244542 + 0.191556i
\(43\) −3.84943 + 2.79678i −0.587033 + 0.426504i −0.841253 − 0.540642i \(-0.818181\pi\)
0.254220 + 0.967146i \(0.418181\pi\)
\(44\) −2.52560 + 2.41732i −0.380749 + 0.364424i
\(45\) 0.972320 + 3.90684i 0.144945 + 0.582397i
\(46\) −1.75279 − 3.39997i −0.258435 − 0.501299i
\(47\) −1.13384 − 2.86375i −0.165387 − 0.417721i 0.823321 − 0.567577i \(-0.192119\pi\)
−0.988708 + 0.149856i \(0.952119\pi\)
\(48\) −2.72777 + 3.42804i −0.393719 + 0.494795i
\(49\) −4.29641 + 3.12153i −0.613773 + 0.445932i
\(50\) 3.53808 + 6.12226i 0.500360 + 0.865818i
\(51\) 2.44858 − 3.37017i 0.342869 − 0.471919i
\(52\) 0.342528 − 0.288895i 0.0475001 − 0.0400626i
\(53\) −3.38258 + 5.33010i −0.464634 + 0.732146i −0.993159 − 0.116772i \(-0.962745\pi\)
0.528525 + 0.848918i \(0.322745\pi\)
\(54\) 1.19806 + 7.33821i 0.163036 + 0.998605i
\(55\) 1.18028 + 3.72620i 0.159149 + 0.502441i
\(56\) 2.39155 − 2.79199i 0.319584 − 0.373096i
\(57\) 7.10464i 0.941032i
\(58\) −2.42183 + 8.48643i −0.318002 + 1.11432i
\(59\) 0.877787 − 0.413055i 0.114278 − 0.0537752i −0.367790 − 0.929909i \(-0.619886\pi\)
0.482069 + 0.876134i \(0.339886\pi\)
\(60\) 2.09479 + 4.42743i 0.270436 + 0.571578i
\(61\) −5.80787 + 6.18476i −0.743622 + 0.791877i −0.983880 − 0.178832i \(-0.942768\pi\)
0.240258 + 0.970709i \(0.422768\pi\)
\(62\) −0.525764 + 5.85931i −0.0667721 + 0.744133i
\(63\) −0.293302 − 2.32173i −0.0369526 − 0.292510i
\(64\) 3.65244 − 7.11756i 0.456554 − 0.889695i
\(65\) −0.120992 − 0.486152i −0.0150072 − 0.0602997i
\(66\) 0.603171 + 2.63941i 0.0742453 + 0.324888i
\(67\) −0.0875745 + 1.39196i −0.0106989 + 0.170055i 0.989049 + 0.147590i \(0.0471515\pi\)
−0.999748 + 0.0224651i \(0.992849\pi\)
\(68\) −3.17335 + 6.91366i −0.384826 + 0.838405i
\(69\) −2.95653 − 0.186009i −0.355925 − 0.0223929i
\(70\) −1.62234 − 3.77643i −0.193906 − 0.451370i
\(71\) 8.36832 − 3.31325i 0.993137 − 0.393211i 0.185342 − 0.982674i \(-0.440661\pi\)
0.807795 + 0.589463i \(0.200661\pi\)
\(72\) −1.79323 − 4.76638i −0.211334 − 0.561724i
\(73\) 3.45272 + 1.62473i 0.404110 + 0.190160i 0.617083 − 0.786898i \(-0.288314\pi\)
−0.212972 + 0.977058i \(0.568314\pi\)
\(74\) −3.87453 + 3.44874i −0.450405 + 0.400908i
\(75\) 5.47551 + 0.0811132i 0.632257 + 0.00936615i
\(76\) −2.55217 − 12.7204i −0.292754 − 1.45913i
\(77\) −0.425724 − 2.23172i −0.0485157 − 0.254328i
\(78\) −0.0559153 − 0.342485i −0.00633116 − 0.0387788i
\(79\) 8.56130 + 10.3488i 0.963221 + 1.16433i 0.986360 + 0.164602i \(0.0526341\pi\)
−0.0231386 + 0.999732i \(0.507366\pi\)
\(80\) −5.34102 − 7.17450i −0.597144 − 0.802134i
\(81\) 0.331723 + 0.131339i 0.0368581 + 0.0145932i
\(82\) 5.97046 + 13.6188i 0.659327 + 1.50395i
\(83\) 3.77354 − 4.56142i 0.414200 − 0.500681i −0.521612 − 0.853183i \(-0.674669\pi\)
0.935811 + 0.352502i \(0.114669\pi\)
\(84\) −0.905494 − 2.69919i −0.0987975 − 0.294506i
\(85\) 5.37265 + 6.59324i 0.582745 + 0.715137i
\(86\) −1.90806 − 6.45286i −0.205751 − 0.695830i
\(87\) 4.67861 + 4.98222i 0.501600 + 0.534150i
\(88\) −2.02808 − 4.50900i −0.216194 − 0.480661i
\(89\) 5.89626 − 12.5302i 0.625002 − 1.32820i −0.302060 − 0.953289i \(-0.597674\pi\)
0.927062 − 0.374908i \(-0.122326\pi\)
\(90\) −5.66694 − 0.550840i −0.597347 − 0.0580637i
\(91\) 0.0364974 + 0.288906i 0.00382596 + 0.0302856i
\(92\) 5.36029 − 0.729027i 0.558849 − 0.0760064i
\(93\) 3.68581 + 2.67789i 0.382200 + 0.277685i
\(94\) 4.35298 − 0.157525i 0.448976 − 0.0162474i
\(95\) −14.0224 − 3.71126i −1.43867 − 0.380767i
\(96\) −3.27968 − 5.25624i −0.334731 − 0.536463i
\(97\) −1.94213 + 0.122189i −0.197194 + 0.0124064i −0.161080 − 0.986941i \(-0.551498\pi\)
−0.0361135 + 0.999348i \(0.511498\pi\)
\(98\) −2.12961 − 7.20215i −0.215124 − 0.727527i
\(99\) −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.109.59 ✓ 2960
8.5 even 2 inner 1000.2.bd.a.109.79 yes 2960
125.39 even 50 inner 1000.2.bd.a.789.79 yes 2960
1000.789 even 50 inner 1000.2.bd.a.789.59 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.109.59 ✓ 2960 1.1 even 1 trivial
1000.2.bd.a.109.79 yes 2960 8.5 even 2 inner
1000.2.bd.a.789.59 yes 2960 1000.789 even 50 inner
1000.2.bd.a.789.79 yes 2960 125.39 even 50 inner