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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [800,2,Mod(101,800)] 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("800.101"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.y (of order \(8\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [64,0,0,0,0,0,0,0,0,0,0,0,0,-16,0,20,0,20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(18)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{8})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 101.1
Character \(\chi\) \(=\) 800.101
Dual form 800.2.y.e.301.1

$q$-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+(-1.41064 + 0.100408i) q^{2} +(0.972675 + 2.34825i) q^{3} +(1.97984 - 0.283281i) q^{4} +(-1.60788 - 3.21487i) q^{6} +(-3.40884 - 3.40884i) q^{7} +(-2.76440 + 0.598400i) q^{8} +(-2.44684 + 2.44684i) q^{9} +(0.440065 - 1.06241i) q^{11} +(2.59095 + 4.37360i) q^{12} +(-3.33003 + 1.37934i) q^{13} +(5.15094 + 4.46639i) q^{14} +(3.83950 - 1.12170i) q^{16} -3.76004i q^{17} +(3.20594 - 3.69730i) q^{18} +(2.26019 - 0.936201i) q^{19} +(4.68910 - 11.3205i) q^{21} +(-0.514100 + 1.54287i) q^{22} +(2.96497 - 2.96497i) q^{23} +(-4.09406 - 5.90944i) q^{24} +(4.55899 - 2.28012i) q^{26} +(-1.08102 - 0.447773i) q^{27} +(-7.71461 - 5.78329i) q^{28} +(-3.13708 - 7.57358i) q^{29} -8.55745 q^{31} +(-5.30355 + 1.96784i) q^{32} +2.92284 q^{33} +(0.377539 + 5.30408i) q^{34} +(-4.15120 + 5.53748i) q^{36} +(4.63838 + 1.92128i) q^{37} +(-3.09432 + 1.54759i) q^{38} +(-6.47807 - 6.47807i) q^{39} +(4.22263 - 4.22263i) q^{41} +(-5.47799 + 16.4400i) q^{42} +(4.26503 - 10.2967i) q^{43} +(0.570296 - 2.22806i) q^{44} +(-3.88481 + 4.48023i) q^{46} +2.54081i q^{47} +(6.36861 + 7.92505i) q^{48} +16.2404i q^{49} +(8.82950 - 3.65730i) q^{51} +(-6.20216 + 3.67420i) q^{52} +(1.07959 - 2.60637i) q^{53} +(1.56990 + 0.523106i) q^{54} +(11.4633 + 7.38356i) q^{56} +(4.39686 + 4.39686i) q^{57} +(5.18576 + 10.3686i) q^{58} +(-5.77500 - 2.39208i) q^{59} +(1.40373 + 3.38890i) q^{61} +(12.0715 - 0.859239i) q^{62} +16.6818 q^{63} +(7.28383 - 3.30844i) q^{64} +(-4.12309 + 0.293477i) q^{66} +(-0.417645 - 1.00828i) q^{67} +(-1.06515 - 7.44427i) q^{68} +(9.84643 + 4.07853i) q^{69} +(-11.6383 - 11.6383i) q^{71} +(5.29986 - 8.22823i) q^{72} +(-10.6611 + 10.6611i) q^{73} +(-6.73602 - 2.24451i) q^{74} +(4.20960 - 2.49379i) q^{76} +(-5.12170 + 2.12148i) q^{77} +(9.78870 + 8.48780i) q^{78} -4.57691i q^{79} +7.40702i q^{81} +(-5.53265 + 6.38062i) q^{82} +(-3.71809 + 1.54008i) q^{83} +(6.07678 - 23.7411i) q^{84} +(-4.98257 + 14.9532i) q^{86} +(14.7333 - 14.7333i) q^{87} +(-0.580769 + 3.20026i) q^{88} +(3.12250 + 3.12250i) q^{89} +(16.0535 + 6.64957i) q^{91} +(5.03024 - 6.71008i) q^{92} +(-8.32362 - 20.0950i) q^{93} +(-0.255119 - 3.58418i) q^{94} +(-9.77959 - 10.5400i) q^{96} +0.0931883 q^{97} +(-1.63067 - 22.9095i) q^{98} +(1.52278 + 3.67631i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 64 q - 16 q^{14} + 20 q^{16} + 20 q^{18} - 4 q^{22} + 8 q^{23} - 28 q^{24} + 24 q^{27} - 20 q^{28} - 20 q^{32} - 20 q^{34} + 12 q^{36} - 20 q^{38} + 24 q^{39} - 100 q^{42} + 8 q^{43} + 40 q^{44} + 32 q^{46}+ \cdots - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(e\left(\frac{1}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41064 + 0.100408i −0.997476 + 0.0709994i
\(3\) 0.972675 + 2.34825i 0.561574 + 1.35576i 0.908507 + 0.417870i \(0.137223\pi\)
−0.346933 + 0.937890i \(0.612777\pi\)
\(4\) 1.97984 − 0.283281i 0.989918 − 0.141640i
\(5\) 0 0
\(6\) −1.60788 − 3.21487i −0.656415 − 1.31247i
\(7\) −3.40884 − 3.40884i −1.28842 − 1.28842i −0.935745 − 0.352676i \(-0.885272\pi\)
−0.352676 − 0.935745i \(-0.614728\pi\)
\(8\) −2.76440 + 0.598400i −0.977364 + 0.211566i
\(9\) −2.44684 + 2.44684i −0.815613 + 0.815613i
\(10\) 0 0
\(11\) 0.440065 − 1.06241i 0.132684 − 0.320329i −0.843548 − 0.537053i \(-0.819537\pi\)
0.976233 + 0.216725i \(0.0695374\pi\)
\(12\) 2.59095 + 4.37360i 0.747943 + 1.26255i
\(13\) −3.33003 + 1.37934i −0.923583 + 0.382561i −0.793240 − 0.608908i \(-0.791608\pi\)
−0.130342 + 0.991469i \(0.541608\pi\)
\(14\) 5.15094 + 4.46639i 1.37665 + 1.19369i
\(15\) 0 0
\(16\) 3.83950 − 1.12170i 0.959876 − 0.280425i
\(17\) − 3.76004i − 0.911944i −0.889994 − 0.455972i \(-0.849292\pi\)
0.889994 − 0.455972i \(-0.150708\pi\)
\(18\) 3.20594 − 3.69730i 0.755647 − 0.871463i
\(19\) 2.26019 − 0.936201i 0.518523 − 0.214779i −0.108045 − 0.994146i \(-0.534459\pi\)
0.626568 + 0.779367i \(0.284459\pi\)
\(20\) 0 0
\(21\) 4.68910 − 11.3205i 1.02325 − 2.47033i
\(22\) −0.514100 + 1.54287i −0.109607 + 0.328941i
\(23\) 2.96497 − 2.96497i 0.618239 − 0.618239i −0.326840 − 0.945080i \(-0.605984\pi\)
0.945080 + 0.326840i \(0.105984\pi\)
\(24\) −4.09406 − 5.90944i −0.835696 − 1.20626i
\(25\) 0 0
\(26\) 4.55899 − 2.28012i 0.894091 − 0.447169i
\(27\) −1.08102 − 0.447773i −0.208042 − 0.0861740i
\(28\) −7.71461 − 5.78329i −1.45792 − 1.09294i
\(29\) −3.13708 − 7.57358i −0.582541 − 1.40638i −0.890502 − 0.454980i \(-0.849646\pi\)
0.307960 − 0.951399i \(-0.400354\pi\)
\(30\) 0 0
\(31\) −8.55745 −1.53696 −0.768482 − 0.639872i \(-0.778988\pi\)
−0.768482 + 0.639872i \(0.778988\pi\)
\(32\) −5.30355 + 1.96784i −0.937544 + 0.347868i
\(33\) 2.92284 0.508801
\(34\) 0.377539 + 5.30408i 0.0647474 + 0.909643i
\(35\) 0 0
\(36\) −4.15120 + 5.53748i −0.691866 + 0.922914i
\(37\) 4.63838 + 1.92128i 0.762545 + 0.315857i 0.729849 − 0.683609i \(-0.239591\pi\)
0.0326965 + 0.999465i \(0.489591\pi\)
\(38\) −3.09432 + 1.54759i −0.501965 + 0.251052i
\(39\) −6.47807 − 6.47807i −1.03732 − 1.03732i
\(40\) 0 0
\(41\) 4.22263 − 4.22263i 0.659464 − 0.659464i −0.295789 − 0.955253i \(-0.595582\pi\)
0.955253 + 0.295789i \(0.0955825\pi\)
\(42\) −5.47799 + 16.4400i −0.845272 + 2.53675i
\(43\) 4.26503 − 10.2967i 0.650412 − 1.57023i −0.161770 − 0.986828i \(-0.551720\pi\)
0.812182 − 0.583404i \(-0.198280\pi\)
\(44\) 0.570296 − 2.22806i 0.0859753 − 0.335893i
\(45\) 0 0
\(46\) −3.88481 + 4.48023i −0.572785 + 0.660574i
\(47\) 2.54081i 0.370616i 0.982681 + 0.185308i \(0.0593282\pi\)
−0.982681 + 0.185308i \(0.940672\pi\)
\(48\) 6.36861 + 7.92505i 0.919230 + 1.14388i
\(49\) 16.2404i 2.32006i
\(50\) 0 0
\(51\) 8.82950 − 3.65730i 1.23638 − 0.512124i
\(52\) −6.20216 + 3.67420i −0.860085 + 0.509520i
\(53\) 1.07959 − 2.60637i 0.148294 − 0.358013i −0.832225 − 0.554438i \(-0.812933\pi\)
0.980519 + 0.196425i \(0.0629333\pi\)
\(54\) 1.56990 + 0.523106i 0.213636 + 0.0711857i
\(55\) 0 0
\(56\) 11.4633 + 7.38356i 1.53184 + 0.986670i
\(57\) 4.39686 + 4.39686i 0.582378 + 0.582378i
\(58\) 5.18576 + 10.3686i 0.680923 + 1.36147i
\(59\) −5.77500 − 2.39208i −0.751841 − 0.311423i −0.0263484 − 0.999653i \(-0.508388\pi\)
−0.725492 + 0.688230i \(0.758388\pi\)
\(60\) 0 0
\(61\) 1.40373 + 3.38890i 0.179729 + 0.433904i 0.987910 − 0.155030i \(-0.0495476\pi\)
−0.808181 + 0.588935i \(0.799548\pi\)
\(62\) 12.0715 − 0.859239i 1.53308 − 0.109123i
\(63\) 16.6818 2.10171
\(64\) 7.28383 − 3.30844i 0.910479 − 0.413555i
\(65\) 0 0
\(66\) −4.12309 + 0.293477i −0.507517 + 0.0361245i
\(67\) −0.417645 − 1.00828i −0.0510235 − 0.123182i 0.896313 − 0.443423i \(-0.146236\pi\)
−0.947336 + 0.320241i \(0.896236\pi\)
\(68\) −1.06515 − 7.44427i −0.129168 − 0.902750i
\(69\) 9.84643 + 4.07853i 1.18537 + 0.490997i
\(70\) 0 0
\(71\) −11.6383 − 11.6383i −1.38121 − 1.38121i −0.842474 − 0.538738i \(-0.818901\pi\)
−0.538738 − 0.842474i \(-0.681099\pi\)
\(72\) 5.29986 − 8.22823i 0.624594 − 0.969707i
\(73\) −10.6611 + 10.6611i −1.24779 + 1.24779i −0.291094 + 0.956694i \(0.594019\pi\)
−0.956694 + 0.291094i \(0.905981\pi\)
\(74\) −6.73602 − 2.24451i −0.783046 − 0.260919i
\(75\) 0 0
\(76\) 4.20960 − 2.49379i 0.482874 − 0.286058i
\(77\) −5.12170 + 2.12148i −0.583672 + 0.241765i
\(78\) 9.78870 + 8.48780i 1.10835 + 0.961054i
\(79\) − 4.57691i − 0.514942i −0.966286 − 0.257471i \(-0.917111\pi\)
0.966286 − 0.257471i \(-0.0828893\pi\)
\(80\) 0 0
\(81\) 7.40702i 0.823002i
\(82\) −5.53265 + 6.38062i −0.610979 + 0.704622i
\(83\) −3.71809 + 1.54008i −0.408114 + 0.169046i −0.577289 − 0.816540i \(-0.695890\pi\)
0.169176 + 0.985586i \(0.445890\pi\)
\(84\) 6.07678 − 23.7411i 0.663031 − 2.59036i
\(85\) 0 0
\(86\) −4.98257 + 14.9532i −0.537285 + 1.61245i
\(87\) 14.7333 − 14.7333i 1.57957 − 1.57957i
\(88\) −0.580769 + 3.20026i −0.0619102 + 0.341149i
\(89\) 3.12250 + 3.12250i 0.330985 + 0.330985i 0.852960 − 0.521976i \(-0.174805\pi\)
−0.521976 + 0.852960i \(0.674805\pi\)
\(90\) 0 0
\(91\) 16.0535 + 6.64957i 1.68286 + 0.697065i
\(92\) 5.03024 − 6.71008i 0.524439 − 0.699574i
\(93\) −8.32362 − 20.0950i −0.863119 − 2.08375i
\(94\) −0.255119 − 3.58418i −0.0263135 − 0.369680i
\(95\) 0 0
\(96\) −9.77959 − 10.5400i −0.998125 − 1.07573i
\(97\) 0.0931883 0.00946184 0.00473092 − 0.999989i \(-0.498494\pi\)
0.00473092 + 0.999989i \(0.498494\pi\)
\(98\) −1.63067 − 22.9095i −0.164723 − 2.31421i
\(99\) 1.52278 + 3.67631i 0.153045 + 0.369483i
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 800.2.y.e.101.1 yes 64
5.2 odd 4 800.2.ba.f.549.9 64
5.3 odd 4 800.2.ba.h.549.8 64
5.4 even 2 800.2.y.d.101.16 ✓ 64
32.13 even 8 inner 800.2.y.e.301.1 yes 64
160.13 odd 8 800.2.ba.f.749.9 64
160.77 odd 8 800.2.ba.h.749.8 64
160.109 even 8 800.2.y.d.301.16 yes 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.y.d.101.16 ✓ 64 5.4 even 2
800.2.y.d.301.16 yes 64 160.109 even 8
800.2.y.e.101.1 yes 64 1.1 even 1 trivial
800.2.y.e.301.1 yes 64 32.13 even 8 inner
800.2.ba.f.549.9 64 5.2 odd 4
800.2.ba.f.749.9 64 160.13 odd 8
800.2.ba.h.549.8 64 5.3 odd 4
800.2.ba.h.749.8 64 160.77 odd 8