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Properties

Label 1000.2.bd.a.29.10
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 29.10
Character \(\chi\) \(=\) 1000.29
Dual form 1000.2.bd.a.69.10

$q$-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+(-1.39137 + 0.253168i) q^{2} +(-2.72371 + 1.49737i) q^{3} +(1.87181 - 0.704500i) q^{4} +(-1.47591 - 1.67979i) q^{5} +(3.41059 - 2.77295i) q^{6} +(2.25056 - 3.09763i) q^{7} +(-2.42602 + 1.45410i) q^{8} +(3.56898 - 5.62382i) q^{9} +(2.47880 + 1.96355i) q^{10} +(1.75746 - 4.43883i) q^{11} +(-4.04337 + 4.72165i) q^{12} +(1.09412 - 1.72406i) q^{13} +(-2.34714 + 4.87971i) q^{14} +(6.53522 + 2.36526i) q^{15} +(3.00736 - 2.63738i) q^{16} +(0.691074 + 0.735919i) q^{17} +(-3.54200 + 8.72836i) q^{18} +(0.453349 - 0.824639i) q^{19} +(-3.94604 - 2.10447i) q^{20} +(-1.49157 + 11.8070i) q^{21} +(-1.32150 + 6.62099i) q^{22} +(-6.82667 - 5.64751i) q^{23} +(4.43045 - 7.59320i) q^{24} +(-0.643373 + 4.95843i) q^{25} +(-1.08585 + 2.67580i) q^{26} +(-0.714436 + 11.3556i) q^{27} +(2.03035 - 7.38370i) q^{28} +(-0.370210 + 0.0706213i) q^{29} +(-9.69170 - 1.63645i) q^{30} +(3.10102 - 2.91205i) q^{31} +(-3.51665 + 4.43094i) q^{32} +(1.85978 + 14.7217i) q^{33} +(-1.14785 - 0.848977i) q^{34} +(-8.52498 + 0.791363i) q^{35} +(2.71849 - 13.0411i) q^{36} +(0.592926 + 9.42429i) q^{37} +(-0.422003 + 1.26215i) q^{38} +(-0.398510 + 6.33414i) q^{39} +(6.02318 + 1.92908i) q^{40} +(-3.74836 - 4.53099i) q^{41} +(-0.913826 - 16.8054i) q^{42} +(-2.62524 - 8.07965i) q^{43} +(0.162475 - 9.54679i) q^{44} +(-14.7143 + 2.30512i) q^{45} +(10.9282 + 6.12948i) q^{46} +(8.79283 + 4.13759i) q^{47} +(-4.24203 + 11.6866i) q^{48} +(-2.36717 - 7.28540i) q^{49} +(-0.360147 - 7.06189i) q^{50} +(-2.98423 - 0.969635i) q^{51} +(0.833390 - 3.99792i) q^{52} +(3.92047 + 0.495270i) q^{53} +(-1.88084 - 15.9807i) q^{54} +(-10.0502 + 3.59917i) q^{55} +(-0.955643 + 10.7875i) q^{56} +2.92491i q^{57} +(0.497219 - 0.191985i) q^{58} +(2.02076 - 7.87034i) q^{59} +(13.8990 - 0.176727i) q^{60} +(9.80860 + 8.11438i) q^{61} +(-3.57742 + 4.83681i) q^{62} +(-9.38830 - 23.7121i) q^{63} +(3.77118 - 7.05537i) q^{64} +(-4.51088 + 0.706666i) q^{65} +(-6.31469 - 20.0124i) q^{66} +(-0.277842 + 1.45650i) q^{67} +(1.81202 + 0.890641i) q^{68} +(27.0503 + 5.16012i) q^{69} +(11.6610 - 3.25933i) q^{70} +(-2.96228 + 6.29516i) q^{71} +(-0.480836 + 18.8332i) q^{72} +(-3.21628 - 12.5266i) q^{73} +(-3.21091 - 12.9625i) q^{74} +(-5.67226 - 14.4687i) q^{75} +(0.267626 - 1.86295i) q^{76} +(-9.79460 - 15.4338i) q^{77} +(-1.04913 - 8.91401i) q^{78} +(-4.64460 + 2.55339i) q^{79} +(-8.86884 - 1.15919i) q^{80} +(-6.54971 - 13.9189i) q^{81} +(6.36245 + 5.35531i) q^{82} +(-10.5097 - 5.77775i) q^{83} +(5.52607 + 23.1512i) q^{84} +(0.216224 - 2.24701i) q^{85} +(5.69818 + 10.5771i) q^{86} +(0.902597 - 0.746693i) q^{87} +(2.19088 + 13.3242i) q^{88} +(-7.88738 + 2.02514i) q^{89} +(19.8895 - 6.93247i) q^{90} +(-2.87811 - 7.26927i) q^{91} +(-16.7569 - 5.76169i) q^{92} +(-4.08585 + 12.5750i) q^{93} +(-13.2816 - 3.53085i) q^{94} +(-2.05432 + 0.455563i) q^{95} +(2.94356 - 17.3343i) q^{96} +(12.1774 - 2.32297i) q^{97} +(5.13803 + 9.53738i) 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 \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39137 + 0.253168i −0.983846 + 0.179017i
\(3\) −2.72371 + 1.49737i −1.57253 + 0.864508i −0.573562 + 0.819162i \(0.694439\pi\)
−0.998971 + 0.0453457i \(0.985561\pi\)
\(4\) 1.87181 − 0.704500i 0.935906 − 0.352250i
\(5\) −1.47591 − 1.67979i −0.660047 − 0.751224i
\(6\) 3.41059 − 2.77295i 1.39237 − 1.13205i
\(7\) 2.25056 − 3.09763i 0.850631 − 1.17079i −0.133092 − 0.991104i \(-0.542491\pi\)
0.983723 − 0.179690i \(-0.0575095\pi\)
\(8\) −2.42602 + 1.45410i −0.857729 + 0.514102i
\(9\) 3.56898 − 5.62382i 1.18966 − 1.87461i
\(10\) 2.47880 + 1.96355i 0.783867 + 0.620929i
\(11\) 1.75746 − 4.43883i 0.529894 − 1.33836i −0.380563 − 0.924755i \(-0.624270\pi\)
0.910457 − 0.413604i \(-0.135730\pi\)
\(12\) −4.04337 + 4.72165i −1.16722 + 1.36302i
\(13\) 1.09412 − 1.72406i 0.303454 − 0.478168i −0.658043 − 0.752980i \(-0.728616\pi\)
0.961498 + 0.274812i \(0.0886157\pi\)
\(14\) −2.34714 + 4.87971i −0.627299 + 1.30416i
\(15\) 6.53522 + 2.36526i 1.68739 + 0.610709i
\(16\) 3.00736 − 2.63738i 0.751840 − 0.659345i
\(17\) 0.691074 + 0.735919i 0.167610 + 0.178487i 0.807290 − 0.590155i \(-0.200933\pi\)
−0.639680 + 0.768641i \(0.720933\pi\)
\(18\) −3.54200 + 8.72836i −0.834858 + 2.05729i
\(19\) 0.453349 − 0.824639i 0.104005 − 0.189185i −0.819184 − 0.573531i \(-0.805573\pi\)
0.923189 + 0.384346i \(0.125573\pi\)
\(20\) −3.94604 − 2.10447i −0.882361 − 0.470573i
\(21\) −1.49157 + 11.8070i −0.325486 + 2.57649i
\(22\) −1.32150 + 6.62099i −0.281745 + 1.41160i
\(23\) −6.82667 − 5.64751i −1.42346 − 1.17759i −0.955924 − 0.293613i \(-0.905142\pi\)
−0.467534 − 0.883975i \(-0.654858\pi\)
\(24\) 4.43045 − 7.59320i 0.904362 − 1.54996i
\(25\) −0.643373 + 4.95843i −0.128675 + 0.991687i
\(26\) −1.08585 + 2.67580i −0.212952 + 0.524767i
\(27\) −0.714436 + 11.3556i −0.137493 + 2.18539i
\(28\) 2.03035 − 7.38370i 0.383699 − 1.39539i
\(29\) −0.370210 + 0.0706213i −0.0687462 + 0.0131140i −0.221640 − 0.975129i \(-0.571141\pi\)
0.152893 + 0.988243i \(0.451141\pi\)
\(30\) −9.69170 − 1.63645i −1.76945 − 0.298773i
\(31\) 3.10102 − 2.91205i 0.556960 − 0.523020i −0.354148 − 0.935190i \(-0.615229\pi\)
0.911107 + 0.412170i \(0.135229\pi\)
\(32\) −3.51665 + 4.43094i −0.621661 + 0.783286i
\(33\) 1.85978 + 14.7217i 0.323746 + 2.56271i
\(34\) −1.14785 − 0.848977i −0.196855 − 0.145598i
\(35\) −8.52498 + 0.791363i −1.44099 + 0.133765i
\(36\) 2.71849 − 13.0411i 0.453082 − 2.17351i
\(37\) 0.592926 + 9.42429i 0.0974764 + 1.54934i 0.677977 + 0.735083i \(0.262857\pi\)
−0.580500 + 0.814260i \(0.697143\pi\)
\(38\) −0.422003 + 1.26215i −0.0684580 + 0.204748i
\(39\) −0.398510 + 6.33414i −0.0638127 + 1.01427i
\(40\) 6.02318 + 1.92908i 0.952348 + 0.305014i
\(41\) −3.74836 − 4.53099i −0.585396 − 0.707622i 0.391779 − 0.920059i \(-0.371860\pi\)
−0.977175 + 0.212438i \(0.931860\pi\)
\(42\) −0.913826 − 16.8054i −0.141006 − 2.59314i
\(43\) −2.62524 − 8.07965i −0.400345 − 1.23213i −0.924720 − 0.380647i \(-0.875701\pi\)
0.524376 − 0.851487i \(-0.324299\pi\)
\(44\) 0.162475 − 9.54679i 0.0244941 − 1.43923i
\(45\) −14.7143 + 2.30512i −2.19348 + 0.343627i
\(46\) 10.9282 + 6.12948i 1.61127 + 0.903742i
\(47\) 8.79283 + 4.13759i 1.28257 + 0.603530i 0.941659 − 0.336568i \(-0.109266\pi\)
0.340906 + 0.940097i \(0.389266\pi\)
\(48\) −4.24203 + 11.6866i −0.612285 + 1.68681i
\(49\) −2.36717 − 7.28540i −0.338167 − 1.04077i
\(50\) −0.360147 − 7.06189i −0.0509325 − 0.998702i
\(51\) −2.98423 − 0.969635i −0.417876 − 0.135776i
\(52\) 0.833390 − 3.99792i 0.115570 − 0.554412i
\(53\) 3.92047 + 0.495270i 0.538518 + 0.0680306i 0.389888 − 0.920862i \(-0.372514\pi\)
0.148630 + 0.988893i \(0.452514\pi\)
\(54\) −1.88084 − 15.9807i −0.255950 − 2.17470i
\(55\) −10.0502 + 3.59917i −1.35516 + 0.485312i
\(56\) −0.955643 + 10.7875i −0.127703 + 1.44154i
\(57\) 2.92491i 0.387413i
\(58\) 0.497219 − 0.191985i 0.0652881 − 0.0252089i
\(59\) 2.02076 − 7.87034i 0.263081 − 1.02463i −0.690461 − 0.723370i \(-0.742592\pi\)
0.953542 − 0.301262i \(-0.0974078\pi\)
\(60\) 13.8990 − 0.176727i 1.79436 − 0.0228154i
\(61\) 9.80860 + 8.11438i 1.25586 + 1.03894i 0.997337 + 0.0729352i \(0.0232366\pi\)
0.258526 + 0.966004i \(0.416763\pi\)
\(62\) −3.57742 + 4.83681i −0.454333 + 0.614276i
\(63\) −9.38830 − 23.7121i −1.18281 − 2.98745i
\(64\) 3.77118 − 7.05537i 0.471397 − 0.881921i
\(65\) −4.51088 + 0.706666i −0.559505 + 0.0876511i
\(66\) −6.31469 − 20.0124i −0.777284 − 2.46336i
\(67\) −0.277842 + 1.45650i −0.0339438 + 0.177940i −0.995222 − 0.0976412i \(-0.968870\pi\)
0.961278 + 0.275581i \(0.0888702\pi\)
\(68\) 1.81202 + 0.890641i 0.219739 + 0.108006i
\(69\) 27.0503 + 5.16012i 3.25647 + 0.621205i
\(70\) 11.6610 − 3.25933i 1.39376 − 0.389564i
\(71\) −2.96228 + 6.29516i −0.351558 + 0.747098i −0.999898 − 0.0142522i \(-0.995463\pi\)
0.648341 + 0.761350i \(0.275463\pi\)
\(72\) −0.480836 + 18.8332i −0.0566671 + 2.21951i
\(73\) −3.21628 − 12.5266i −0.376438 − 1.46613i −0.821283 − 0.570520i \(-0.806742\pi\)
0.444846 − 0.895607i \(-0.353258\pi\)
\(74\) −3.21091 − 12.9625i −0.373260 − 1.50687i
\(75\) −5.67226 − 14.4687i −0.654976 − 1.67070i
\(76\) 0.267626 − 1.86295i 0.0306988 − 0.213695i
\(77\) −9.79460 − 15.4338i −1.11620 − 1.75885i
\(78\) −1.04913 − 8.91401i −0.118790 − 1.00931i
\(79\) −4.64460 + 2.55339i −0.522559 + 0.287279i −0.721121 − 0.692809i \(-0.756373\pi\)
0.198562 + 0.980088i \(0.436373\pi\)
\(80\) −8.86884 − 1.15919i −0.991566 − 0.129601i
\(81\) −6.54971 − 13.9189i −0.727746 − 1.54654i
\(82\) 6.36245 + 5.35531i 0.702615 + 0.591395i
\(83\) −10.5097 − 5.77775i −1.15359 − 0.634190i −0.214113 − 0.976809i \(-0.568686\pi\)
−0.939474 + 0.342619i \(0.888686\pi\)
\(84\) 5.52607 + 23.1512i 0.602943 + 2.52600i
\(85\) 0.216224 − 2.24701i 0.0234528 − 0.243722i
\(86\) 5.69818 + 10.5771i 0.614450 + 1.14056i
\(87\) 0.902597 − 0.746693i 0.0967685 − 0.0800539i
\(88\) 2.19088 + 13.3242i 0.233548 + 1.42037i
\(89\) −7.88738 + 2.02514i −0.836061 + 0.214664i −0.642366 − 0.766398i \(-0.722047\pi\)
−0.193695 + 0.981062i \(0.562047\pi\)
\(90\) 19.8895 − 6.93247i 2.09653 − 0.730746i
\(91\) −2.87811 − 7.26927i −0.301708 − 0.762027i
\(92\) −16.7569 − 5.76169i −1.74703 − 0.600698i
\(93\) −4.08585 + 12.5750i −0.423683 + 1.30396i
\(94\) −13.2816 − 3.53085i −1.36989 − 0.364179i
\(95\) −2.05432 + 0.455563i −0.210769 + 0.0467398i
\(96\) 2.94356 − 17.3343i 0.300426 − 1.76917i
\(97\) 12.1774 − 2.32297i 1.23643 − 0.235862i 0.472600 − 0.881277i \(-0.343316\pi\)
0.763832 + 0.645415i \(0.223316\pi\)
\(98\) 5.13803 + 9.53738i 0.519020 + 0.963421i
\(99\) −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.29.10 ✓ 2960
8.5 even 2 inner 1000.2.bd.a.29.103 yes 2960
125.69 even 50 inner 1000.2.bd.a.69.103 yes 2960
1000.69 even 50 inner 1000.2.bd.a.69.10 yes 2960
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1000.2.bd.a.29.10 ✓ 2960 1.1 even 1 trivial
1000.2.bd.a.29.103 yes 2960 8.5 even 2 inner
1000.2.bd.a.69.10 yes 2960 1000.69 even 50 inner
1000.2.bd.a.69.103 yes 2960 125.69 even 50 inner