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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 301.3
Character \(\chi\) \(=\) 800.301
Dual form 800.2.y.e.101.3

$q$-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+(-1.26838 - 0.625467i) q^{2} +(-0.986257 + 2.38104i) q^{3} +(1.21758 + 1.58666i) q^{4} +(2.74021 - 2.40319i) q^{6} +(-2.66071 + 2.66071i) q^{7} +(-0.551955 - 2.77405i) q^{8} +(-2.57531 - 2.57531i) q^{9} +(-2.06648 - 4.98892i) q^{11} +(-4.97875 + 1.33425i) q^{12} +(3.28494 + 1.36067i) q^{13} +(5.03897 - 1.71061i) q^{14} +(-1.03499 + 3.86378i) q^{16} -6.47754i q^{17} +(1.65570 + 4.87724i) q^{18} +(-1.25082 - 0.518105i) q^{19} +(-3.71110 - 8.95938i) q^{21} +(-0.499321 + 7.62037i) q^{22} +(-3.93037 - 3.93037i) q^{23} +(7.14948 + 1.42170i) q^{24} +(-3.31550 - 3.78046i) q^{26} +(1.52870 - 0.633209i) q^{27} +(-7.46127 - 0.982010i) q^{28} +(-0.515400 + 1.24429i) q^{29} +4.32793 q^{31} +(3.72942 - 4.25340i) q^{32} +13.9169 q^{33} +(-4.05149 + 8.21599i) q^{34} +(0.950490 - 7.22178i) q^{36} +(2.65068 - 1.09795i) q^{37} +(1.26245 + 1.43950i) q^{38} +(-6.47958 + 6.47958i) q^{39} +(6.83045 + 6.83045i) q^{41} +(-0.896709 + 13.6851i) q^{42} +(-0.124633 - 0.300891i) q^{43} +(5.39962 - 9.35322i) q^{44} +(2.52689 + 7.44352i) q^{46} +7.79266i q^{47} +(-8.17904 - 6.27502i) q^{48} -7.15871i q^{49} +(15.4232 + 6.38852i) q^{51} +(1.84077 + 6.86880i) q^{52} +(-2.36819 - 5.71731i) q^{53} +(-2.33503 - 0.153002i) q^{54} +(8.84952 + 5.91234i) q^{56} +(2.46725 - 2.46725i) q^{57} +(1.43198 - 1.25586i) q^{58} +(-8.46858 + 3.50780i) q^{59} +(1.27530 - 3.07884i) q^{61} +(-5.48947 - 2.70698i) q^{62} +13.7043 q^{63} +(-7.39069 + 3.06230i) q^{64} +(-17.6519 - 8.70454i) q^{66} +(3.86267 - 9.32531i) q^{67} +(10.2777 - 7.88694i) q^{68} +(13.2347 - 5.48199i) q^{69} +(9.55305 - 9.55305i) q^{71} +(-5.72257 + 8.56548i) q^{72} +(-1.93135 - 1.93135i) q^{73} +(-4.04880 - 0.265296i) q^{74} +(-0.700914 - 2.61546i) q^{76} +(18.7723 + 7.77576i) q^{77} +(12.2713 - 4.16582i) q^{78} -6.31686i q^{79} -6.66170i q^{81} +(-4.39140 - 12.9358i) q^{82} +(-11.6422 - 4.82236i) q^{83} +(9.69693 - 16.7970i) q^{84} +(-0.0301150 + 0.459598i) q^{86} +(-2.45437 - 2.45437i) q^{87} +(-12.6989 + 8.48617i) q^{88} +(2.03773 - 2.03773i) q^{89} +(-12.3606 + 5.11992i) q^{91} +(1.45061 - 11.0217i) q^{92} +(-4.26845 + 10.3050i) q^{93} +(4.87405 - 9.88407i) q^{94} +(6.44932 + 13.0748i) q^{96} +1.02663 q^{97} +(-4.47754 + 9.07998i) q^{98} +(-7.52618 + 18.1698i) 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 \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(e\left(\frac{7}{8}\right)\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.26838 − 0.625467i −0.896881 − 0.442272i
\(3\) −0.986257 + 2.38104i −0.569416 + 1.37469i 0.332633 + 0.943057i \(0.392063\pi\)
−0.902048 + 0.431635i \(0.857937\pi\)
\(4\) 1.21758 + 1.58666i 0.608791 + 0.793330i
\(5\) 0 0
\(6\) 2.74021 − 2.40319i 1.11869 − 0.981098i
\(7\) −2.66071 + 2.66071i −1.00565 + 1.00565i −0.00566848 + 0.999984i \(0.501804\pi\)
−0.999984 + 0.00566848i \(0.998196\pi\)
\(8\) −0.551955 − 2.77405i −0.195146 − 0.980774i
\(9\) −2.57531 − 2.57531i −0.858435 − 0.858435i
\(10\) 0 0
\(11\) −2.06648 − 4.98892i −0.623067 − 1.50422i −0.848084 − 0.529861i \(-0.822244\pi\)
0.225018 − 0.974355i \(-0.427756\pi\)
\(12\) −4.97875 + 1.33425i −1.43724 + 0.385165i
\(13\) 3.28494 + 1.36067i 0.911077 + 0.377381i 0.788469 − 0.615075i \(-0.210874\pi\)
0.122608 + 0.992455i \(0.460874\pi\)
\(14\) 5.03897 − 1.71061i 1.34672 − 0.457179i
\(15\) 0 0
\(16\) −1.03499 + 3.86378i −0.258747 + 0.965945i
\(17\) − 6.47754i − 1.57103i −0.618840 − 0.785517i \(-0.712397\pi\)
0.618840 − 0.785517i \(-0.287603\pi\)
\(18\) 1.65570 + 4.87724i 0.390253 + 1.14958i
\(19\) −1.25082 − 0.518105i −0.286957 − 0.118861i 0.234562 − 0.972101i \(-0.424634\pi\)
−0.521519 + 0.853240i \(0.674634\pi\)
\(20\) 0 0
\(21\) −3.71110 − 8.95938i −0.809827 − 1.95510i
\(22\) −0.499321 + 7.62037i −0.106456 + 1.62467i
\(23\) −3.93037 − 3.93037i −0.819538 − 0.819538i 0.166503 − 0.986041i \(-0.446753\pi\)
−0.986041 + 0.166503i \(0.946753\pi\)
\(24\) 7.14948 + 1.42170i 1.45938 + 0.290203i
\(25\) 0 0
\(26\) −3.31550 − 3.78046i −0.650223 − 0.741409i
\(27\) 1.52870 − 0.633209i 0.294199 − 0.121861i
\(28\) −7.46127 − 0.982010i −1.41005 − 0.185582i
\(29\) −0.515400 + 1.24429i −0.0957074 + 0.231058i −0.964481 − 0.264151i \(-0.914908\pi\)
0.868774 + 0.495209i \(0.164908\pi\)
\(30\) 0 0
\(31\) 4.32793 0.777319 0.388660 − 0.921381i \(-0.372938\pi\)
0.388660 + 0.921381i \(0.372938\pi\)
\(32\) 3.72942 − 4.25340i 0.659275 − 0.751902i
\(33\) 13.9169 2.42262
\(34\) −4.05149 + 8.21599i −0.694824 + 1.40903i
\(35\) 0 0
\(36\) 0.950490 − 7.22178i 0.158415 − 1.20363i
\(37\) 2.65068 − 1.09795i 0.435769 − 0.180502i −0.154004 − 0.988070i \(-0.549217\pi\)
0.589774 + 0.807569i \(0.299217\pi\)
\(38\) 1.26245 + 1.43950i 0.204797 + 0.233517i
\(39\) −6.47958 + 6.47958i −1.03756 + 1.03756i
\(40\) 0 0
\(41\) 6.83045 + 6.83045i 1.06674 + 1.06674i 0.997608 + 0.0691302i \(0.0220224\pi\)
0.0691302 + 0.997608i \(0.477978\pi\)
\(42\) −0.896709 + 13.6851i −0.138365 + 2.11165i
\(43\) −0.124633 − 0.300891i −0.0190064 − 0.0458854i 0.914091 − 0.405508i \(-0.132905\pi\)
−0.933098 + 0.359623i \(0.882905\pi\)
\(44\) 5.39962 − 9.35322i 0.814023 − 1.41005i
\(45\) 0 0
\(46\) 2.52689 + 7.44352i 0.372570 + 1.09749i
\(47\) 7.79266i 1.13668i 0.822795 + 0.568338i \(0.192414\pi\)
−0.822795 + 0.568338i \(0.807586\pi\)
\(48\) −8.17904 − 6.27502i −1.18054 − 0.905721i
\(49\) − 7.15871i − 1.02267i
\(50\) 0 0
\(51\) 15.4232 + 6.38852i 2.15969 + 0.894571i
\(52\) 1.84077 + 6.86880i 0.255268 + 0.952531i
\(53\) −2.36819 − 5.71731i −0.325296 − 0.785334i −0.998929 − 0.0462678i \(-0.985267\pi\)
0.673633 − 0.739066i \(-0.264733\pi\)
\(54\) −2.33503 − 0.153002i −0.317757 − 0.0208209i
\(55\) 0 0
\(56\) 8.84952 + 5.91234i 1.18257 + 0.790069i
\(57\) 2.46725 − 2.46725i 0.326795 − 0.326795i
\(58\) 1.43198 − 1.25586i 0.188029 − 0.164903i
\(59\) −8.46858 + 3.50780i −1.10252 + 0.456677i −0.858354 − 0.513058i \(-0.828513\pi\)
−0.244162 + 0.969735i \(0.578513\pi\)
\(60\) 0 0
\(61\) 1.27530 − 3.07884i 0.163285 − 0.394205i −0.820967 − 0.570976i \(-0.806565\pi\)
0.984252 + 0.176770i \(0.0565650\pi\)
\(62\) −5.48947 − 2.70698i −0.697163 − 0.343786i
\(63\) 13.7043 1.72658
\(64\) −7.39069 + 3.06230i −0.923836 + 0.382788i
\(65\) 0 0
\(66\) −17.6519 − 8.70454i −2.17280 − 1.07146i
\(67\) 3.86267 − 9.32531i 0.471900 − 1.13927i −0.491422 − 0.870922i \(-0.663523\pi\)
0.963322 − 0.268347i \(-0.0864773\pi\)
\(68\) 10.2777 − 7.88694i 1.24635 − 0.956432i
\(69\) 13.2347 − 5.48199i 1.59327 − 0.659954i
\(70\) 0 0
\(71\) 9.55305 − 9.55305i 1.13374 − 1.13374i 0.144188 − 0.989550i \(-0.453943\pi\)
0.989550 − 0.144188i \(-0.0460571\pi\)
\(72\) −5.72257 + 8.56548i −0.674411 + 1.00945i
\(73\) −1.93135 − 1.93135i −0.226047 − 0.226047i 0.584992 − 0.811039i \(-0.301098\pi\)
−0.811039 + 0.584992i \(0.801098\pi\)
\(74\) −4.04880 − 0.265296i −0.470664 − 0.0308401i
\(75\) 0 0
\(76\) −0.700914 − 2.61546i −0.0804004 − 0.300013i
\(77\) 18.7723 + 7.77576i 2.13931 + 0.886130i
\(78\) 12.2713 − 4.16582i 1.38946 − 0.471686i
\(79\) − 6.31686i − 0.710702i −0.934733 − 0.355351i \(-0.884361\pi\)
0.934733 − 0.355351i \(-0.115639\pi\)
\(80\) 0 0
\(81\) − 6.66170i − 0.740188i
\(82\) −4.39140 − 12.9358i −0.484949 − 1.42853i
\(83\) −11.6422 − 4.82236i −1.27790 − 0.529323i −0.362542 − 0.931967i \(-0.618091\pi\)
−0.915357 + 0.402644i \(0.868091\pi\)
\(84\) 9.69693 − 16.7970i 1.05802 − 1.83271i
\(85\) 0 0
\(86\) −0.0301150 + 0.459598i −0.00324738 + 0.0495598i
\(87\) −2.45437 − 2.45437i −0.263136 − 0.263136i
\(88\) −12.6989 + 8.48617i −1.35371 + 0.904629i
\(89\) 2.03773 − 2.03773i 0.215999 − 0.215999i −0.590811 − 0.806810i \(-0.701192\pi\)
0.806810 + 0.590811i \(0.201192\pi\)
\(90\) 0 0
\(91\) −12.3606 + 5.11992i −1.29574 + 0.536713i
\(92\) 1.45061 − 11.0217i 0.151237 − 1.14909i
\(93\) −4.26845 + 10.3050i −0.442618 + 1.06857i
\(94\) 4.87405 − 9.88407i 0.502720 − 1.01946i
\(95\) 0 0
\(96\) 6.44932 + 13.0748i 0.658231 + 1.33444i
\(97\) 1.02663 0.104238 0.0521190 − 0.998641i \(-0.483402\pi\)
0.0521190 + 0.998641i \(0.483402\pi\)
\(98\) −4.47754 + 9.07998i −0.452300 + 0.917217i
\(99\) −7.52618 + 18.1698i −0.756410 + 1.82613i
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.301.3 yes 64
5.2 odd 4 800.2.ba.h.749.12 64
5.3 odd 4 800.2.ba.f.749.5 64
5.4 even 2 800.2.y.d.301.14 yes 64
32.5 even 8 inner 800.2.y.e.101.3 yes 64
160.37 odd 8 800.2.ba.f.549.5 64
160.69 even 8 800.2.y.d.101.14 ✓ 64
160.133 odd 8 800.2.ba.h.549.12 64
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
800.2.y.d.101.14 ✓ 64 160.69 even 8
800.2.y.d.301.14 yes 64 5.4 even 2
800.2.y.e.101.3 yes 64 32.5 even 8 inner
800.2.y.e.301.3 yes 64 1.1 even 1 trivial
800.2.ba.f.549.5 64 160.37 odd 8
800.2.ba.f.749.5 64 5.3 odd 4
800.2.ba.h.549.12 64 160.133 odd 8
800.2.ba.h.749.12 64 5.2 odd 4