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Properties

Label 1000.2.bd.a.69.103
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 69.103
Character \(\chi\) \(=\) 1000.69
Dual form 1000.2.bd.a.29.103

$q$-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+(0.840959 - 1.13701i) q^{2} +(2.72371 + 1.49737i) q^{3} +(-0.585577 - 1.91235i) q^{4} +(1.47591 - 1.67979i) q^{5} +(3.99305 - 1.83765i) q^{6} +(2.25056 + 3.09763i) q^{7} +(-2.66681 - 0.942404i) q^{8} +(3.56898 + 5.62382i) q^{9} +(-0.668753 - 3.09076i) q^{10} +(-1.75746 - 4.43883i) q^{11} +(1.26856 - 6.08552i) q^{12} +(-1.09412 - 1.72406i) q^{13} +(5.41466 + 0.0460722i) q^{14} +(6.53522 - 2.36526i) q^{15} +(-3.31420 + 2.23966i) q^{16} +(0.691074 - 0.735919i) q^{17} +(9.39570 + 0.671433i) q^{18} +(-0.453349 - 0.824639i) q^{19} +(-4.07661 - 1.83882i) q^{20} +(1.49157 + 11.8070i) q^{21} +(-6.52494 - 1.73463i) q^{22} +(-6.82667 + 5.64751i) q^{23} +(-5.85248 - 6.56004i) q^{24} +(-0.643373 - 4.95843i) q^{25} +(-2.88038 - 0.205837i) q^{26} +(0.714436 + 11.3556i) q^{27} +(4.60589 - 6.11777i) q^{28} +(0.370210 + 0.0706213i) q^{29} +(2.80652 - 9.41969i) q^{30} +(3.10102 + 2.91205i) q^{31} +(-0.240587 + 5.65174i) q^{32} +(1.85978 - 14.7217i) q^{33} +(-0.255582 - 1.40464i) q^{34} +(8.52498 + 0.791363i) q^{35} +(8.66482 - 10.1183i) q^{36} +(-0.592926 + 9.42429i) q^{37} +(-1.31887 - 0.178025i) q^{38} +(-0.398510 - 6.33414i) q^{39} +(-5.51901 + 3.08877i) q^{40} +(-3.74836 + 4.53099i) q^{41} +(14.6790 + 8.23324i) q^{42} +(2.62524 - 8.07965i) q^{43} +(-7.45950 + 5.96017i) q^{44} +(14.7143 + 2.30512i) q^{45} +(0.680324 + 12.5113i) q^{46} +(8.79283 - 4.13759i) q^{47} +(-12.3805 + 1.13760i) q^{48} +(-2.36717 + 7.28540i) q^{49} +(-6.17883 - 3.43832i) q^{50} +(2.98423 - 0.969635i) q^{51} +(-2.65632 + 3.10192i) q^{52} +(-3.92047 + 0.495270i) q^{53} +(13.5123 + 8.73730i) q^{54} +(-10.0502 - 3.59917i) q^{55} +(-3.08260 - 10.3817i) q^{56} -2.92491i q^{57} +(0.391628 - 0.361542i) q^{58} +(-2.02076 - 7.87034i) q^{59} +(-8.35010 - 11.1126i) q^{60} +(-9.80860 + 8.11438i) q^{61} +(5.91886 - 1.07697i) q^{62} +(-9.38830 + 23.7121i) q^{63} +(6.22375 + 5.02642i) q^{64} +(-4.51088 - 0.706666i) q^{65} +(-15.1747 - 14.4949i) q^{66} +(0.277842 + 1.45650i) q^{67} +(-1.81202 - 0.890641i) q^{68} +(-27.0503 + 5.16012i) q^{69} +(8.06894 - 9.02748i) q^{70} +(-2.96228 - 6.29516i) q^{71} +(-4.21789 - 18.3611i) q^{72} +(-3.21628 + 12.5266i) q^{73} +(10.2169 + 8.59960i) q^{74} +(5.67226 - 14.4687i) q^{75} +(-1.31153 + 1.34985i) q^{76} +(9.79460 - 15.4338i) q^{77} +(-7.53710 - 4.87364i) q^{78} +(-4.64460 - 2.55339i) q^{79} +(-1.12930 + 8.87269i) q^{80} +(-6.54971 + 13.9189i) q^{81} +(1.99956 + 8.07229i) q^{82} +(10.5097 - 5.77775i) q^{83} +(21.7057 - 9.76629i) q^{84} +(-0.216224 - 2.24701i) q^{85} +(-6.97891 - 9.77956i) q^{86} +(0.902597 + 0.746693i) q^{87} +(0.503633 + 13.4938i) q^{88} +(-7.88738 - 2.02514i) q^{89} +(14.9951 - 14.7918i) q^{90} +(2.87811 - 7.26927i) q^{91} +(14.7976 + 9.74795i) q^{92} +(4.08585 + 12.5750i) q^{93} +(2.68993 - 13.4771i) q^{94} +(-2.05432 - 0.455563i) q^{95} +(-9.11803 + 15.0334i) q^{96} +(12.1774 + 2.32297i) q^{97} +(6.29287 + 8.81821i) q^{98} +(18.6909 - 25.7258i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.840959 − 1.13701i 0.594647 − 0.803987i
\(3\) 2.72371 + 1.49737i 1.57253 + 0.864508i 0.998971 + 0.0453457i \(0.0144389\pi\)
0.573562 + 0.819162i \(0.305561\pi\)
\(4\) −0.585577 − 1.91235i −0.292789 − 0.956177i
\(5\) 1.47591 − 1.67979i 0.660047 − 0.751224i
\(6\) 3.99305 − 1.83765i 1.63016 − 0.750219i
\(7\) 2.25056 + 3.09763i 0.850631 + 1.17079i 0.983723 + 0.179690i \(0.0575095\pi\)
−0.133092 + 0.991104i \(0.542491\pi\)
\(8\) −2.66681 − 0.942404i −0.942860 − 0.333190i
\(9\) 3.56898 + 5.62382i 1.18966 + 1.87461i
\(10\) −0.668753 − 3.09076i −0.211478 − 0.977383i
\(11\) −1.75746 − 4.43883i −0.529894 − 1.33836i −0.910457 − 0.413604i \(-0.864270\pi\)
0.380563 − 0.924755i \(-0.375730\pi\)
\(12\) 1.26856 − 6.08552i 0.366202 − 1.75674i
\(13\) −1.09412 − 1.72406i −0.303454 − 0.478168i 0.658043 − 0.752980i \(-0.271384\pi\)
−0.961498 + 0.274812i \(0.911384\pi\)
\(14\) 5.41466 + 0.0460722i 1.44713 + 0.0123133i
\(15\) 6.53522 − 2.36526i 1.68739 − 0.610709i
\(16\) −3.31420 + 2.23966i −0.828550 + 0.559916i
\(17\) 0.691074 − 0.735919i 0.167610 − 0.178487i −0.639680 − 0.768641i \(-0.720933\pi\)
0.807290 + 0.590155i \(0.200933\pi\)
\(18\) 9.39570 + 0.671433i 2.21459 + 0.158258i
\(19\) −0.453349 − 0.824639i −0.104005 − 0.189185i 0.819184 − 0.573531i \(-0.194427\pi\)
−0.923189 + 0.384346i \(0.874427\pi\)
\(20\) −4.07661 − 1.83882i −0.911558 − 0.411172i
\(21\) 1.49157 + 11.8070i 0.325486 + 2.57649i
\(22\) −6.52494 − 1.73463i −1.39112 − 0.369824i
\(23\) −6.82667 + 5.64751i −1.42346 + 1.17759i −0.467534 + 0.883975i \(0.654858\pi\)
−0.955924 + 0.293613i \(0.905142\pi\)
\(24\) −5.85248 − 6.56004i −1.19463 − 1.33906i
\(25\) −0.643373 − 4.95843i −0.128675 − 0.991687i
\(26\) −2.88038 − 0.205837i −0.564889 − 0.0403679i
\(27\) 0.714436 + 11.3556i 0.137493 + 2.18539i
\(28\) 4.60589 − 6.11777i 0.870431 − 1.15615i
\(29\) 0.370210 + 0.0706213i 0.0687462 + 0.0131140i 0.221640 − 0.975129i \(-0.428859\pi\)
−0.152893 + 0.988243i \(0.548859\pi\)
\(30\) 2.80652 − 9.41969i 0.512398 − 1.71979i
\(31\) 3.10102 + 2.91205i 0.556960 + 0.523020i 0.911107 − 0.412170i \(-0.135229\pi\)
−0.354148 + 0.935190i \(0.615229\pi\)
\(32\) −0.240587 + 5.65174i −0.0425301 + 0.999095i
\(33\) 1.85978 − 14.7217i 0.323746 − 2.56271i
\(34\) −0.255582 − 1.40464i −0.0438319 − 0.240893i
\(35\) 8.52498 + 0.791363i 1.44099 + 0.133765i
\(36\) 8.66482 − 10.1183i 1.44414 − 1.68639i
\(37\) −0.592926 + 9.42429i −0.0974764 + 1.54934i 0.580500 + 0.814260i \(0.302857\pi\)
−0.677977 + 0.735083i \(0.737143\pi\)
\(38\) −1.31887 − 0.178025i −0.213949 − 0.0288795i
\(39\) −0.398510 − 6.33414i −0.0638127 − 1.01427i
\(40\) −5.51901 + 3.08877i −0.872633 + 0.488377i
\(41\) −3.74836 + 4.53099i −0.585396 + 0.707622i −0.977175 − 0.212438i \(-0.931860\pi\)
0.391779 + 0.920059i \(0.371860\pi\)
\(42\) 14.6790 + 8.23324i 2.26501 + 1.27042i
\(43\) 2.62524 − 8.07965i 0.400345 − 1.23213i −0.524376 − 0.851487i \(-0.675701\pi\)
0.924720 − 0.380647i \(-0.124299\pi\)
\(44\) −7.45950 + 5.96017i −1.12456 + 0.898529i
\(45\) 14.7143 + 2.30512i 2.19348 + 0.343627i
\(46\) 0.680324 + 12.5113i 0.100308 + 1.84469i
\(47\) 8.79283 − 4.13759i 1.28257 − 0.603530i 0.340906 − 0.940097i \(-0.389266\pi\)
0.941659 + 0.336568i \(0.109266\pi\)
\(48\) −12.3805 + 1.13760i −1.78697 + 0.164199i
\(49\) −2.36717 + 7.28540i −0.338167 + 1.04077i
\(50\) −6.17883 − 3.43832i −0.873819 − 0.486251i
\(51\) 2.98423 − 0.969635i 0.417876 − 0.135776i
\(52\) −2.65632 + 3.10192i −0.368365 + 0.430158i
\(53\) −3.92047 + 0.495270i −0.538518 + 0.0680306i −0.389888 − 0.920862i \(-0.627486\pi\)
−0.148630 + 0.988893i \(0.547486\pi\)
\(54\) 13.5123 + 8.73730i 1.83879 + 1.18900i
\(55\) −10.0502 − 3.59917i −1.35516 − 0.485312i
\(56\) −3.08260 − 10.3817i −0.411929 − 1.38732i
\(57\) − 2.92491i − 0.387413i
\(58\) 0.391628 − 0.361542i 0.0514233 − 0.0474728i
\(59\) −2.02076 − 7.87034i −0.263081 − 1.02463i −0.953542 − 0.301262i \(-0.902592\pi\)
0.690461 − 0.723370i \(-0.257408\pi\)
\(60\) −8.35010 − 11.1126i −1.07799 − 1.43463i
\(61\) −9.80860 + 8.11438i −1.25586 + 1.03894i −0.258526 + 0.966004i \(0.583237\pi\)
−0.997337 + 0.0729352i \(0.976763\pi\)
\(62\) 5.91886 − 1.07697i 0.751695 − 0.136776i
\(63\) −9.38830 + 23.7121i −1.18281 + 2.98745i
\(64\) 6.22375 + 5.02642i 0.777969 + 0.628303i
\(65\) −4.51088 − 0.706666i −0.559505 − 0.0876511i
\(66\) −15.1747 − 14.4949i −1.86787 − 1.78420i
\(67\) 0.277842 + 1.45650i 0.0339438 + 0.177940i 0.995222 − 0.0976412i \(-0.0311298\pi\)
−0.961278 + 0.275581i \(0.911130\pi\)
\(68\) −1.81202 − 0.890641i −0.219739 − 0.108006i
\(69\) −27.0503 + 5.16012i −3.25647 + 0.621205i
\(70\) 8.06894 − 9.02748i 0.964423 − 1.07899i
\(71\) −2.96228 − 6.29516i −0.351558 − 0.747098i 0.648341 − 0.761350i \(-0.275463\pi\)
−0.999898 + 0.0142522i \(0.995463\pi\)
\(72\) −4.21789 − 18.3611i −0.497083 − 2.16387i
\(73\) −3.21628 + 12.5266i −0.376438 + 1.46613i 0.444846 + 0.895607i \(0.353258\pi\)
−0.821283 + 0.570520i \(0.806742\pi\)
\(74\) 10.2169 + 8.59960i 1.18769 + 0.999683i
\(75\) 5.67226 − 14.4687i 0.654976 − 1.67070i
\(76\) −1.31153 + 1.34985i −0.150443 + 0.154839i
\(77\) 9.79460 − 15.4338i 1.11620 − 1.75885i
\(78\) −7.53710 − 4.87364i −0.853408 − 0.551831i
\(79\) −4.64460 − 2.55339i −0.522559 − 0.287279i 0.198562 − 0.980088i \(-0.436373\pi\)
−0.721121 + 0.692809i \(0.756373\pi\)
\(80\) −1.12930 + 8.87269i −0.126260 + 0.991997i
\(81\) −6.54971 + 13.9189i −0.727746 + 1.54654i
\(82\) 1.99956 + 8.07229i 0.220814 + 0.891436i
\(83\) 10.5097 − 5.77775i 1.15359 − 0.634190i 0.214113 − 0.976809i \(-0.431314\pi\)
0.939474 + 0.342619i \(0.111314\pi\)
\(84\) 21.7057 − 9.76629i 2.36828 − 1.06559i
\(85\) −0.216224 − 2.24701i −0.0234528 − 0.243722i
\(86\) −6.97891 − 9.77956i −0.752556 − 1.05456i
\(87\) 0.902597 + 0.746693i 0.0967685 + 0.0800539i
\(88\) 0.503633 + 13.4938i 0.0536875 + 1.43844i
\(89\) −7.88738 − 2.02514i −0.836061 − 0.214664i −0.193695 − 0.981062i \(-0.562047\pi\)
−0.642366 + 0.766398i \(0.722047\pi\)
\(90\) 14.9951 − 14.7918i 1.58062 − 1.55919i
\(91\) 2.87811 − 7.26927i 0.301708 − 0.762027i
\(92\) 14.7976 + 9.74795i 1.54276 + 1.01629i
\(93\) 4.08585 + 12.5750i 0.423683 + 1.30396i
\(94\) 2.68993 − 13.4771i 0.277445 − 1.39005i
\(95\) −2.05432 − 0.455563i −0.210769 − 0.0467398i
\(96\) −9.11803 + 15.0334i −0.930606 + 1.53434i
\(97\) 12.1774 + 2.32297i 1.23643 + 0.235862i 0.763832 − 0.645415i \(-0.223316\pi\)
0.472600 + 0.881277i \(0.343316\pi\)
\(98\) 6.29287 + 8.81821i 0.635676 + 0.890774i
\(99\) 18.6909 − 25.7258i 1.87850 − 2.58554i
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.69.103 yes 2960
8.5 even 2 inner 1000.2.bd.a.69.10 yes 2960
125.29 even 50 inner 1000.2.bd.a.29.10 ✓ 2960
1000.29 even 50 inner 1000.2.bd.a.29.103 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.29.10 ✓ 2960 125.29 even 50 inner
1000.2.bd.a.29.103 yes 2960 1000.29 even 50 inner
1000.2.bd.a.69.10 yes 2960 8.5 even 2 inner
1000.2.bd.a.69.103 yes 2960 1.1 even 1 trivial