LMFDB - Embedded newform 1728.3.g.l.703.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(703,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.703");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.207360000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 6x^{6} + 32x^{4} + 24x^{2} + 16 $ x^8 + 6x^6 + 32x^4 + 24*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{4} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.4
Root			$-0.437016 - 0.756934i$ of defining polynomial
Character	$\chi$	$=$	1728.703
Dual form			1728.3.g.l.703.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.08036 q^{5} +6.01392i q^{7} +O(q^{10})$	$q-1.08036 q^{5} +6.01392i q^{7} +17.7247i q^{11} -12.4164 q^{13} -26.3786 q^{17} -5.19615i q^{19} -29.8356i q^{23} -23.8328 q^{25} -4.32145 q^{29} -44.8403i q^{31} -6.49721i q^{35} +20.4164 q^{37} +59.2393 q^{41} -19.1491i q^{43} +41.0631i q^{47} +12.8328 q^{49} +70.0430 q^{53} -19.1491i q^{55} -28.9521i q^{59} -18.4164 q^{61} +13.4142 q^{65} -94.8767i q^{67} +83.8931i q^{71} +55.8328 q^{73} -106.595 q^{77} -41.0410i q^{79} -35.4493i q^{83} +28.4984 q^{85} -26.3786 q^{89} -74.6712i q^{91} +5.61373i q^{95} +76.3313 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q + 8 q^{13} + 24 q^{25} + 56 q^{37} - 112 q^{49} - 40 q^{61} + 232 q^{73} - 416 q^{85} - 248 q^{97}+O(q^{100}) $ 8 * q + 8 * q^13 + 24 * q^25 + 56 * q^37 - 112 * q^49 - 40 * q^61 + 232 * q^73 - 416 * q^85 - 248 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.08036	−0.216073	−0.108036	−	0.994147i	$-0.534456\pi$
$5$	−1.08036	−0.216073	−0.108036	+	0.994147i	$0.534456\pi$
$6$	0	0
$7$	6.01392i	0.859131i	0.903036	+	0.429565i	$0.141333\pi$
$7$	6.01392i	0.859131i	−0.903036	+	0.429565i	$0.858667\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	17.7247i	1.61133i	0.592369	+	0.805667i	$0.298193\pi$
$11$	17.7247i	1.61133i	−0.592369	+	0.805667i	$0.701807\pi$
$12$	0	0
$13$	−12.4164	−0.955108	−0.477554	−	0.878602i	$-0.658477\pi$
$13$	−12.4164	−0.955108	−0.477554	+	0.878602i	$0.658477\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.3786	−1.55168	−0.775841	−	0.630929i	$-0.782674\pi$
$17$	−26.3786	−1.55168	−0.775841	+	0.630929i	$0.782674\pi$
$18$	0	0
$19$	− 5.19615i	− 0.273482i	−0.990607	−	0.136741i	$-0.956337\pi$
$19$	− 5.19615i	− 0.273482i	0.990607	−	0.136741i	$-0.0436628\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 29.8356i	− 1.29720i	−0.761129	−	0.648600i	$-0.775355\pi$
$23$	− 29.8356i	− 1.29720i	0.761129	−	0.648600i	$-0.224645\pi$
$24$	0	0
$25$	−23.8328	−0.953313
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.32145	−0.149016	−0.0745078	−	0.997220i	$-0.523739\pi$
$29$	−4.32145	−0.149016	−0.0745078	+	0.997220i	$0.523739\pi$
$30$	0	0
$31$	− 44.8403i	− 1.44646i	−0.690607	−	0.723230i	$-0.742657\pi$
$31$	− 44.8403i	− 1.44646i	0.690607	−	0.723230i	$-0.257343\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.49721i	− 0.185635i
$36$	0	0
$37$	20.4164	0.551795	0.275897	−	0.961187i	$-0.411025\pi$
$37$	20.4164	0.551795	0.275897	+	0.961187i	$0.411025\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	59.2393	1.44486	0.722431	−	0.691443i	$-0.243025\pi$
$41$	59.2393	1.44486	0.722431	+	0.691443i	$0.243025\pi$
$42$	0	0
$43$	− 19.1491i	− 0.445328i	−0.974895	−	0.222664i	$-0.928525\pi$
$43$	− 19.1491i	− 0.445328i	0.974895	−	0.222664i	$-0.0714752\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	41.0631i	0.873683i	0.899539	+	0.436841i	$0.143903\pi$
$47$	41.0631i	0.873683i	−0.899539	+	0.436841i	$0.856097\pi$
$48$	0	0
$49$	12.8328	0.261894
$50$	0	0
$51$	0	0
$52$	0	0
$53$	70.0430	1.32157	0.660783	−	0.750577i	$-0.270224\pi$
$53$	70.0430	1.32157	0.660783	+	0.750577i	$0.270224\pi$
$54$	0	0
$55$	− 19.1491i	− 0.348165i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 28.9521i	− 0.490714i	−0.969433	−	0.245357i	$-0.921095\pi$
$59$	− 28.9521i	− 0.490714i	0.969433	−	0.245357i	$-0.0789052\pi$
$60$	0	0
$61$	−18.4164	−0.301908	−0.150954	−	0.988541i	$-0.548235\pi$
$61$	−18.4164	−0.301908	−0.150954	+	0.988541i	$0.548235\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.4142	0.206373
$66$	0	0
$67$	− 94.8767i	− 1.41607i	−0.706177	−	0.708035i	$-0.749582\pi$
$67$	− 94.8767i	− 1.41607i	0.706177	−	0.708035i	$-0.250418\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	83.8931i	1.18159i	0.806821	+	0.590797i	$0.201186\pi$
$71$	83.8931i	1.18159i	−0.806821	+	0.590797i	$0.798814\pi$
$72$	0	0
$73$	55.8328	0.764833	0.382417	−	0.923990i	$-0.375092\pi$
$73$	55.8328	0.764833	0.382417	+	0.923990i	$0.375092\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−106.595	−1.38435
$78$	0	0
$79$	− 41.0410i	− 0.519507i	−0.965675	−	0.259753i	$-0.916359\pi$
$79$	− 41.0410i	− 0.519507i	0.965675	−	0.259753i	$-0.0836413\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 35.4493i	− 0.427101i	−0.976932	−	0.213550i	$-0.931497\pi$
$83$	− 35.4493i	− 0.427101i	0.976932	−	0.213550i	$-0.0685027\pi$
$84$	0	0
$85$	28.4984	0.335276
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−26.3786	−0.296389	−0.148194	−	0.988958i	$-0.547346\pi$
$89$	−26.3786	−0.296389	−0.148194	+	0.988958i	$0.547346\pi$
$90$	0	0
$91$	− 74.6712i	− 0.820563i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.61373i	0.0590919i
$96$	0	0
$97$	76.3313	0.786920	0.393460	−	0.919342i	$-0.371278\pi$
$97$	76.3313	0.786920	0.393460	+	0.919342i	$0.371278\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−72.2037	−0.714888	−0.357444	−	0.933935i	$-0.616352\pi$
$101$	−72.2037	−0.714888	−0.357444	+	0.933935i	$0.616352\pi$
$102$	0	0
$103$	− 57.9754i	− 0.562868i	−0.959581	−	0.281434i	$-0.909190\pi$
$103$	− 57.9754i	− 0.562868i	0.959581	−	0.281434i	$-0.0908101\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 64.4015i	− 0.601883i	−0.953643	−	0.300942i	$-0.902699\pi$
$107$	− 64.4015i	− 0.601883i	0.953643	−	0.300942i	$-0.0973009\pi$
$108$	0	0
$109$	43.1672	0.396029	0.198015	−	0.980199i	$-0.436551\pi$
$109$	43.1672	0.396029	0.198015	+	0.980199i	$0.436551\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−81.2965	−0.719438	−0.359719	−	0.933061i	$-0.617127\pi$
$113$	−81.2965	−0.719438	−0.359719	+	0.933061i	$0.617127\pi$
$114$	0	0
$115$	32.2333i	0.280290i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 158.639i	− 1.33310i
$120$	0	0
$121$	−193.164	−1.59640
$122$	0	0
$123$	0	0
$124$	0	0
$125$	52.7572	0.422057
$126$	0	0
$127$	− 191.968i	− 1.51156i	−0.654826	−	0.755780i	$-0.727258\pi$
$127$	− 191.968i	− 1.51156i	0.654826	−	0.755780i	$-0.272742\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	141.797i	1.08242i	0.840887	+	0.541211i	$0.182034\pi$
$131$	141.797i	1.08242i	−0.840887	+	0.541211i	$0.817966\pi$
$132$	0	0
$133$	31.2492	0.234957
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−87.7787	−0.640720	−0.320360	−	0.947296i	$-0.603804\pi$
$137$	−87.7787	−0.640720	−0.320360	+	0.947296i	$0.603804\pi$
$138$	0	0
$139$	− 183.501i	− 1.32015i	−0.751200	−	0.660075i	$-0.770524\pi$
$139$	− 183.501i	− 1.32015i	0.751200	−	0.660075i	$-0.229476\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 220.077i	− 1.53900i
$144$	0	0
$145$	4.66874	0.0321982
$146$	0	0
$147$	0	0
$148$	0	0
$149$	153.950	1.03322	0.516611	−	0.856220i	$-0.327193\pi$
$149$	153.950	1.03322	0.516611	+	0.856220i	$0.327193\pi$
$150$	0	0
$151$	− 185.375i	− 1.22765i	−0.789442	−	0.613825i	$-0.789630\pi$
$151$	− 185.375i	− 1.22765i	0.789442	−	0.613825i	$-0.210370\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	48.4438i	0.312540i
$156$	0	0
$157$	−196.164	−1.24945	−0.624726	−	0.780844i	$-0.714789\pi$
$157$	−196.164	−1.24945	−0.624726	+	0.780844i	$0.714789\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	179.429	1.11447
$162$	0	0
$163$	− 129.325i	− 0.793403i	−0.917948	−	0.396701i	$-0.870155\pi$
$163$	− 129.325i	− 0.793403i	0.917948	−	0.396701i	$-0.129845\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	137.951i	0.826052i	0.910719	+	0.413026i	$0.135528\pi$
$167$	137.951i	0.826052i	−0.910719	+	0.413026i	$0.864472\pi$
$168$	0	0
$169$	−14.8328	−0.0877681
$170$	0	0
$171$	0	0
$172$	0	0
$173$	160.432	0.927354	0.463677	−	0.886004i	$-0.346530\pi$
$173$	160.432	0.927354	0.463677	+	0.886004i	$0.346530\pi$
$174$	0	0
$175$	− 143.329i	− 0.819020i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	201.469i	1.12552i	0.826619	+	0.562762i	$0.190261\pi$
$179$	201.469i	1.12552i	−0.826619	+	0.562762i	$0.809739\pi$
$180$	0	0
$181$	48.2523	0.266587	0.133294	−	0.991077i	$-0.457445\pi$
$181$	48.2523	0.266587	0.133294	+	0.991077i	$0.457445\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−22.0571	−0.119228
$186$	0	0
$187$	− 467.552i	− 2.50028i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	147.411i	0.771786i	0.922544	+	0.385893i	$0.126107\pi$
$191$	147.411i	0.771786i	−0.922544	+	0.385893i	$0.873893\pi$
$192$	0	0
$193$	25.1641	0.130384	0.0651919	−	0.997873i	$-0.479234\pi$
$193$	25.1641	0.130384	0.0651919	+	0.997873i	$0.479234\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−385.506	−1.95688	−0.978441	−	0.206528i	$-0.933784\pi$
$197$	−385.506	−1.95688	−0.978441	+	0.206528i	$0.933784\pi$
$198$	0	0
$199$	121.386i	0.609978i	0.952356	+	0.304989i	$0.0986528\pi$
$199$	121.386i	0.609978i	−0.952356	+	0.304989i	$0.901347\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 25.9888i	− 0.128024i
$204$	0	0
$205$	−64.0000	−0.312195
$206$	0	0
$207$	0	0
$208$	0	0
$209$	92.1001	0.440670
$210$	0	0
$211$	261.631i	1.23996i	0.784619	+	0.619978i	$0.212859\pi$
$211$	261.631i	1.23996i	−0.784619	+	0.619978i	$0.787141\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.6880i	0.0962231i
$216$	0	0
$217$	269.666	1.24270
$218$	0	0
$219$	0	0
$220$	0	0
$221$	327.527	1.48202
$222$	0	0
$223$	− 70.0542i	− 0.314144i	−0.987587	−	0.157072i	$-0.949794\pi$
$223$	− 70.0542i	− 0.314144i	0.987587	−	0.157072i	$-0.0502056\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 298.356i	− 1.31434i	−0.753740	−	0.657172i	$-0.771752\pi$
$227$	− 298.356i	− 1.31434i	0.753740	−	0.657172i	$-0.228248\pi$
$228$	0	0
$229$	−259.666	−1.13391	−0.566956	−	0.823748i	$-0.691879\pi$
$229$	−259.666	−1.13391	−0.566956	+	0.823748i	$0.691879\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	7.38192	0.0316821	0.0158410	−	0.999875i	$-0.494957\pi$
$233$	7.38192	0.0316821	0.0158410	+	0.999875i	$0.494957\pi$
$234$	0	0
$235$	− 44.3630i	− 0.188779i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	82.1262i	0.343624i	0.985130	+	0.171812i	$0.0549622\pi$
$239$	82.1262i	0.343624i	−0.985130	+	0.171812i	$0.945038\pi$
$240$	0	0
$241$	−415.827	−1.72542	−0.862711	−	0.505698i	$-0.831235\pi$
$241$	−415.827	−1.72542	−0.862711	+	0.505698i	$0.831235\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−13.8641	−0.0565882
$246$	0	0
$247$	64.5175i	0.261205i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 140.030i	− 0.557890i	−0.960307	−	0.278945i	$-0.910015\pi$
$251$	− 140.030i	− 0.557890i	0.960307	−	0.278945i	$-0.0899847\pi$
$252$	0	0
$253$	528.827	2.09022
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−66.9825	−0.260632	−0.130316	−	0.991472i	$-0.541599\pi$
$257$	−66.9825	−0.260632	−0.130316	+	0.991472i	$0.541599\pi$
$258$	0	0
$259$	122.783i	0.474064i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 37.2163i	− 0.141507i	−0.997494	−	0.0707534i	$-0.977460\pi$
$263$	− 37.2163i	− 0.141507i	0.997494	−	0.0707534i	$-0.0225404\pi$
$264$	0	0
$265$	−75.6718	−0.285554
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−279.092	−1.03752	−0.518758	−	0.854921i	$-0.673605\pi$
$269$	−279.092	−1.03752	−0.518758	+	0.854921i	$0.673605\pi$
$270$	0	0
$271$	− 406.305i	− 1.49928i	−0.661845	−	0.749641i	$-0.730226\pi$
$271$	− 406.305i	− 1.49928i	0.661845	−	0.749641i	$-0.269774\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 422.429i	− 1.53611i
$276$	0	0
$277$	−4.83282	−0.0174470	−0.00872349	−	0.999962i	$-0.502777\pi$
$277$	−4.83282	−0.0174470	−0.00872349	+	0.999962i	$0.502777\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.32145	0.0153788	0.00768942	−	0.999970i	$-0.497552\pi$
$281$	4.32145	0.0153788	0.00768942	+	0.999970i	$0.497552\pi$
$282$	0	0
$283$	− 44.3630i	− 0.156760i	−0.996924	−	0.0783799i	$-0.975025\pi$
$283$	− 44.3630i	− 0.156760i	0.996924	−	0.0783799i	$-0.0249747\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	356.260i	1.24133i
$288$	0	0
$289$	406.830	1.40772
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−388.927	−1.32740	−0.663699	−	0.748000i	$-0.731014\pi$
$293$	−388.927	−1.32740	−0.663699	+	0.748000i	$0.731014\pi$
$294$	0	0
$295$	31.2788i	0.106030i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	370.451i	1.23897i
$300$	0	0
$301$	115.161	0.382595
$302$	0	0
$303$	0	0
$304$	0	0
$305$	19.8964	0.0652341
$306$	0	0
$307$	− 96.6999i	− 0.314983i	−0.987520	−	0.157492i	$-0.949659\pi$
$307$	− 96.6999i	− 0.314983i	0.987520	−	0.157492i	$-0.0503407\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 136.184i	− 0.437890i	−0.975737	−	0.218945i	$-0.929739\pi$
$311$	− 136.184i	− 0.437890i	0.975737	−	0.218945i	$-0.0702615\pi$
$312$	0	0
$313$	282.161	0.901473	0.450736	−	0.892657i	$-0.351161\pi$
$313$	282.161	0.901473	0.450736	+	0.892657i	$0.351161\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−275.489	−0.869051	−0.434526	−	0.900659i	$-0.643084\pi$
$317$	−275.489	−0.869051	−0.434526	+	0.900659i	$0.643084\pi$
$318$	0	0
$319$	− 76.5963i	− 0.240114i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	137.067i	0.424356i
$324$	0	0
$325$	295.918	0.910517
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−246.950	−0.750608
$330$	0	0
$331$	− 55.5221i	− 0.167741i	−0.996477	−	0.0838703i	$-0.973272\pi$
$331$	− 55.5221i	− 0.167741i	0.996477	−	0.0838703i	$-0.0267281\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	102.501i	0.305974i
$336$	0	0
$337$	−430.659	−1.27792	−0.638961	−	0.769239i	$-0.720635\pi$
$337$	−430.659	−1.27792	−0.638961	+	0.769239i	$0.720635\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	794.779	2.33073
$342$	0	0
$343$	371.857i	1.08413i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	135.871i	0.391559i	0.980648	+	0.195779i	$0.0627236\pi$
$347$	135.871i	0.391559i	−0.980648	+	0.195779i	$0.937276\pi$
$348$	0	0
$349$	−288.082	−0.825450	−0.412725	−	0.910856i	$-0.635423\pi$
$349$	−288.082	−0.825450	−0.412725	+	0.910856i	$0.635423\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−198.964	−0.563638	−0.281819	−	0.959468i	$-0.590938\pi$
$353$	−198.964	−0.563638	−0.281819	+	0.959468i	$0.590938\pi$
$354$	0	0
$355$	− 90.6350i	− 0.255310i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 324.658i	− 0.904339i	−0.891932	−	0.452170i	$-0.850650\pi$
$359$	− 324.658i	− 0.904339i	0.891932	−	0.452170i	$-0.149350\pi$
$360$	0	0
$361$	334.000	0.925208
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−60.3197	−0.165259
$366$	0	0
$367$	− 176.141i	− 0.479948i	−0.970779	−	0.239974i	$-0.922861\pi$
$367$	− 176.141i	− 0.479948i	0.970779	−	0.239974i	$-0.0771390\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	421.233i	1.13540i
$372$	0	0
$373$	−596.580	−1.59941	−0.799706	−	0.600392i	$-0.795011\pi$
$373$	−596.580	−1.59941	−0.799706	+	0.600392i	$0.795011\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	53.6569	0.142326
$378$	0	0
$379$	30.3082i	0.0799689i	0.999200	+	0.0399844i	$0.0127308\pi$
$379$	30.3082i	0.0799689i	−0.999200	+	0.0399844i	$0.987269\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 707.220i	− 1.84653i	−0.384167	−	0.923264i	$-0.625511\pi$
$383$	− 707.220i	− 1.84653i	0.384167	−	0.923264i	$-0.374489\pi$
$384$	0	0
$385$	115.161	0.299119
$386$	0	0
$387$	0	0
$388$	0	0
$389$	577.088	1.48352	0.741758	−	0.670668i	$-0.233992\pi$
$389$	577.088	1.48352	0.741758	+	0.670668i	$0.233992\pi$
$390$	0	0
$391$	787.021i	2.01284i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	44.3392i	0.112251i
$396$	0	0
$397$	−26.8266	−0.0675733	−0.0337867	−	0.999429i	$-0.510757\pi$
$397$	−26.8266	−0.0675733	−0.0337867	+	0.999429i	$0.510757\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	505.065	1.25951	0.629756	−	0.776793i	$-0.283155\pi$
$401$	505.065	1.25951	0.629756	+	0.776793i	$0.283155\pi$
$402$	0	0
$403$	556.755i	1.38153i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	361.874i	0.889126i
$408$	0	0
$409$	−683.328	−1.67073	−0.835364	−	0.549696i	$-0.814743\pi$
$409$	−683.328	−1.67073	−0.835364	+	0.549696i	$0.814743\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	174.116	0.421588
$414$	0	0
$415$	38.2982i	0.0922847i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	306.620i	0.731791i	0.930656	+	0.365895i	$0.119237\pi$
$419$	306.620i	0.731791i	−0.930656	+	0.365895i	$0.880763\pi$
$420$	0	0
$421$	18.5867	0.0441489	0.0220745	−	0.999756i	$-0.492973\pi$
$421$	18.5867	0.0441489	0.0220745	+	0.999756i	$0.492973\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	628.676	1.47924
$426$	0	0
$427$	− 110.755i	− 0.259379i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 308.130i	− 0.714918i	−0.933929	−	0.357459i	$-0.883643\pi$
$431$	− 308.130i	− 0.714918i	0.933929	−	0.357459i	$-0.116357\pi$
$432$	0	0
$433$	174.839	0.403785	0.201893	−	0.979408i	$-0.435291\pi$
$433$	174.839	0.403785	0.201893	+	0.979408i	$0.435291\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−155.030	−0.354761
$438$	0	0
$439$	480.159i	1.09376i	0.837212	+	0.546878i	$0.184184\pi$
$439$	480.159i	1.09376i	−0.837212	+	0.546878i	$0.815816\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 555.962i	− 1.25499i	−0.778619	−	0.627497i	$-0.784080\pi$
$443$	− 555.962i	− 1.25499i	0.778619	−	0.627497i	$-0.215920\pi$
$444$	0	0
$445$	28.4984	0.0640415
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−801.711	−1.78555	−0.892774	−	0.450504i	$-0.851244\pi$
$449$	−801.711	−1.78555	−0.892774	+	0.450504i	$0.851244\pi$
$450$	0	0
$451$	1050.00i	2.32816i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	80.6720i	0.177301i
$456$	0	0
$457$	−137.830	−0.301597	−0.150798	−	0.988565i	$-0.548184\pi$
$457$	−137.830	−0.301597	−0.150798	+	0.988565i	$0.548184\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−188.341	−0.408549	−0.204274	−	0.978914i	$-0.565483\pi$
$461$	−188.341	−0.408549	−0.204274	+	0.978914i	$0.565483\pi$
$462$	0	0
$463$	− 548.425i	− 1.18450i	−0.805753	−	0.592251i	$-0.798239\pi$
$463$	− 548.425i	− 1.18450i	0.805753	−	0.592251i	$-0.201761\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 618.597i	− 1.32462i	−0.749231	−	0.662309i	$-0.769577\pi$
$467$	− 618.597i	− 1.32462i	0.749231	−	0.662309i	$-0.230423\pi$
$468$	0	0
$469$	570.580	1.21659
$470$	0	0
$471$	0	0
$472$	0	0
$473$	339.411	0.717571
$474$	0	0
$475$	123.839i	0.260714i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 770.425i	− 1.60840i	−0.594357	−	0.804202i	$-0.702593\pi$
$479$	− 770.425i	− 1.60840i	0.594357	−	0.804202i	$-0.297407\pi$
$480$	0	0
$481$	−253.498	−0.527024
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−82.4655	−0.170032
$486$	0	0
$487$	549.208i	1.12774i	0.825865	+	0.563868i	$0.190687\pi$
$487$	549.208i	1.12774i	−0.825865	+	0.563868i	$0.809313\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 23.0256i	− 0.0468952i	−0.999725	−	0.0234476i	$-0.992536\pi$
$491$	− 23.0256i	− 0.0468952i	0.999725	−	0.0234476i	$-0.00746429\pi$
$492$	0	0
$493$	113.994	0.231225
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−504.526	−1.01514
$498$	0	0
$499$	626.809i	1.25613i	0.778160	+	0.628065i	$0.216153\pi$
$499$	626.809i	1.25613i	−0.778160	+	0.628065i	$0.783847\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	732.896i	1.45705i	0.685019	+	0.728525i	$0.259794\pi$
$503$	732.896i	1.45705i	−0.685019	+	0.728525i	$0.740206\pi$
$504$	0	0
$505$	78.0062	0.154468
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−437.363	−0.859259	−0.429630	−	0.903005i	$-0.641356\pi$
$509$	−437.363	−0.859259	−0.429630	+	0.903005i	$0.641356\pi$
$510$	0	0
$511$	335.774i	0.657092i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	62.6345i	0.121620i
$516$	0	0
$517$	−727.830	−1.40779
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−385.236	−0.739417	−0.369709	−	0.929148i	$-0.620542\pi$
$521$	−385.236	−0.739417	−0.369709	+	0.929148i	$0.620542\pi$
$522$	0	0
$523$	− 418.572i	− 0.800328i	−0.916443	−	0.400164i	$-0.868953\pi$
$523$	− 418.572i	− 0.800328i	0.916443	−	0.400164i	$-0.131047\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1182.82i	2.24445i
$528$	0	0
$529$	−361.164	−0.682730
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−735.540	−1.38000
$534$	0	0
$535$	69.5770i	0.130050i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	227.457i	0.421999i
$540$	0	0
$541$	23.4257	0.0433008	0.0216504	−	0.999766i	$-0.493108\pi$
$541$	23.4257	0.0433008	0.0216504	+	0.999766i	$0.493108\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−46.6362	−0.0855711
$546$	0	0
$547$	722.163i	1.32023i	0.751167	+	0.660113i	$0.229491\pi$
$547$	722.163i	1.32023i	−0.751167	+	0.660113i	$0.770509\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	22.4549i	0.0407530i
$552$	0	0
$553$	246.817	0.446324
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.34135	0.00420349	0.00210175	−	0.999998i	$-0.499331\pi$
$557$	2.34135	0.00420349	0.00210175	+	0.999998i	$0.499331\pi$
$558$	0	0
$559$	237.763i	0.425336i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	359.794i	0.639066i	0.947575	+	0.319533i	$0.103526\pi$
$563$	359.794i	0.639066i	−0.947575	+	0.319533i	$0.896474\pi$
$564$	0	0
$565$	87.8297	0.155451
$566$	0	0
$567$	0	0
$568$	0	0
$569$	18.6354	0.0327512	0.0163756	−	0.999866i	$-0.494787\pi$
$569$	18.6354	0.0327512	0.0163756	+	0.999866i	$0.494787\pi$
$570$	0	0
$571$	− 284.528i	− 0.498298i	−0.968465	−	0.249149i	$-0.919849\pi$
$571$	− 284.528i	− 0.498298i	0.968465	−	0.249149i	$-0.0801509\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	711.067i	1.23664i
$576$	0	0
$577$	664.823	1.15221	0.576104	−	0.817377i	$-0.304572\pi$
$577$	664.823	1.15221	0.576104	+	0.817377i	$0.304572\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	213.189	0.366935
$582$	0	0
$583$	1241.49i	2.12948i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 754.467i	− 1.28529i	−0.766162	−	0.642647i	$-0.777836\pi$
$587$	− 754.467i	− 1.28529i	0.766162	−	0.642647i	$-0.222164\pi$
$588$	0	0
$589$	−232.997	−0.395580
$590$	0	0
$591$	0	0
$592$	0	0
$593$	550.801	0.928838	0.464419	−	0.885615i	$-0.346263\pi$
$593$	550.801	0.928838	0.464419	+	0.885615i	$0.346263\pi$
$594$	0	0
$595$	171.387i	0.288046i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.9888i	0.0433871i	0.999765	+	0.0216935i	$0.00690581\pi$
$599$	25.9888i	0.0433871i	−0.999765	+	0.0216935i	$0.993094\pi$
$600$	0	0
$601$	186.170	0.309768	0.154884	−	0.987933i	$-0.450500\pi$
$601$	186.170	0.309768	0.154884	+	0.987933i	$0.450500\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	208.687	0.344938
$606$	0	0
$607$	− 99.5447i	− 0.163994i	−0.996633	−	0.0819972i	$-0.973870\pi$
$607$	− 99.5447i	− 0.163994i	0.996633	−	0.0819972i	$-0.0261299\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 509.856i	− 0.834461i
$612$	0	0
$613$	960.234	1.56645	0.783225	−	0.621739i	$-0.213573\pi$
$613$	960.234	1.56645	0.783225	+	0.621739i	$0.213573\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	354.625	0.574757	0.287378	−	0.957817i	$-0.407216\pi$
$617$	354.625	0.574757	0.287378	+	0.957817i	$0.407216\pi$
$618$	0	0
$619$	− 360.647i	− 0.582629i	−0.956627	−	0.291314i	$-0.905907\pi$
$619$	− 360.647i	− 0.582629i	0.956627	−	0.291314i	$-0.0940926\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 158.639i	− 0.254637i
$624$	0	0
$625$	538.823	0.862118
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−538.556	−0.856210
$630$	0	0
$631$	82.7121i	0.131081i	0.997850	+	0.0655405i	$0.0208772\pi$
$631$	82.7121i	0.131081i	−0.997850	+	0.0655405i	$0.979123\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	207.395i	0.326607i
$636$	0	0
$637$	−159.337	−0.250137
$638$	0	0
$639$	0	0
$640$	0	0
$641$	760.569	1.18653	0.593267	−	0.805005i	$-0.297838\pi$
$641$	760.569	1.18653	0.593267	+	0.805005i	$0.297838\pi$
$642$	0	0
$643$	787.021i	1.22398i	0.790864	+	0.611992i	$0.209631\pi$
$643$	787.021i	1.22398i	−0.790864	+	0.611992i	$0.790369\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 978.962i	− 1.51308i	−0.653948	−	0.756539i	$-0.726889\pi$
$647$	− 978.962i	− 1.51308i	0.653948	−	0.756539i	$-0.273111\pi$
$648$	0	0
$649$	513.167	0.790704
$650$	0	0
$651$	0	0
$652$	0	0
$653$	939.548	1.43882	0.719409	−	0.694587i	$-0.244413\pi$
$653$	939.548	1.43882	0.719409	+	0.694587i	$0.244413\pi$
$654$	0	0
$655$	− 153.193i	− 0.233882i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	149.491i	0.226845i	0.993547	+	0.113423i	$0.0361814\pi$
$659$	149.491i	0.226845i	−0.993547	+	0.113423i	$0.963819\pi$
$660$	0	0
$661$	−337.420	−0.510468	−0.255234	−	0.966879i	$-0.582153\pi$
$661$	−337.420	−0.510468	−0.255234	+	0.966879i	$0.582153\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−33.7605	−0.0507677
$666$	0	0
$667$	128.933i	0.193303i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 326.425i	− 0.486475i
$672$	0	0
$673$	−321.341	−0.477475	−0.238737	−	0.971084i	$-0.576734\pi$
$673$	−321.341	−0.477475	−0.238737	+	0.971084i	$0.576734\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	741.841	1.09578	0.547889	−	0.836551i	$-0.315432\pi$
$677$	741.841	1.09578	0.547889	+	0.836551i	$0.315432\pi$
$678$	0	0
$679$	459.050i	0.676067i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1259.02i	1.84337i	0.387938	+	0.921686i	$0.373188\pi$
$683$	1259.02i	1.84337i	−0.387938	+	0.921686i	$0.626812\pi$
$684$	0	0
$685$	94.8328	0.138442
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−869.682	−1.26224
$690$	0	0
$691$	222.770i	0.322387i	0.986923	+	0.161194i	$0.0515344\pi$
$691$	222.770i	0.322387i	−0.986923	+	0.161194i	$0.948466\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	198.248i	0.285248i
$696$	0	0
$697$	−1562.65	−2.24197
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.34135	0.00334001	0.00167000	−	0.999999i	$-0.499468\pi$
$701$	2.34135	0.00334001	0.00167000	+	0.999999i	$0.499468\pi$
$702$	0	0
$703$	− 106.087i	− 0.150906i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 434.227i	− 0.614182i
$708$	0	0
$709$	495.085	0.698287	0.349143	−	0.937069i	$-0.386473\pi$
$709$	495.085	0.698287	0.349143	+	0.937069i	$0.386473\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1337.84	−1.87635
$714$	0	0
$715$	237.763i	0.332535i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 962.433i	− 1.33857i	−0.743005	−	0.669286i	$-0.766600\pi$
$719$	− 962.433i	− 1.33857i	0.743005	−	0.669286i	$-0.233400\pi$
$720$	0	0
$721$	348.659	0.483578
$722$	0	0
$723$	0	0
$724$	0	0
$725$	102.992	0.142058
$726$	0	0
$727$	− 544.625i	− 0.749141i	−0.927198	−	0.374570i	$-0.877790\pi$
$727$	− 544.625i	− 0.749141i	0.927198	−	0.374570i	$-0.122210\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	505.126i	0.691006i
$732$	0	0
$733$	146.170	0.199414	0.0997069	−	0.995017i	$-0.468209\pi$
$733$	146.170	0.199414	0.0997069	+	0.995017i	$0.468209\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1681.66	2.28176
$738$	0	0
$739$	1172.87i	1.58710i	0.608505	+	0.793550i	$0.291769\pi$
$739$	1172.87i	1.58710i	−0.608505	+	0.793550i	$0.708231\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 494.837i	− 0.665998i	−0.942927	−	0.332999i	$-0.891939\pi$
$743$	− 494.837i	− 0.665998i	0.942927	−	0.332999i	$-0.108061\pi$
$744$	0	0
$745$	−166.322	−0.223251
$746$	0	0
$747$	0	0
$748$	0	0
$749$	387.305	0.517096
$750$	0	0
$751$	927.250i	1.23469i	0.786693	+	0.617344i	$0.211791\pi$
$751$	927.250i	1.23469i	−0.786693	+	0.617344i	$0.788209\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	200.272i	0.265261i
$756$	0	0
$757$	757.748	1.00099	0.500494	−	0.865740i	$-0.333152\pi$
$757$	757.748	1.00099	0.500494	+	0.865740i	$0.333152\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−953.139	−1.25248	−0.626241	−	0.779629i	$-0.715408\pi$
$761$	−953.139	−1.25248	−0.626241	+	0.779629i	$0.715408\pi$
$762$	0	0
$763$	259.604i	0.340241i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	359.482i	0.468685i
$768$	0	0
$769$	145.675	0.189434	0.0947171	−	0.995504i	$-0.469805\pi$
$769$	145.675	0.189434	0.0947171	+	0.995504i	$0.469805\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	60.1391	0.0777996	0.0388998	−	0.999243i	$-0.487615\pi$
$773$	60.1391	0.0777996	0.0388998	+	0.999243i	$0.487615\pi$
$774$	0	0
$775$	1068.67i	1.37893i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 307.817i	− 0.395143i
$780$	0	0
$781$	−1486.98	−1.90394
$782$	0	0
$783$	0	0
$784$	0	0
$785$	211.928	0.269973
$786$	0	0
$787$	328.312i	0.417169i	0.978004	+	0.208585i	$0.0668856\pi$
$787$	328.312i	0.417169i	−0.978004	+	0.208585i	$0.933114\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 488.910i	− 0.618091i
$792$	0	0
$793$	228.666	0.288355
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−446.725	−0.560508	−0.280254	−	0.959926i	$-0.590419\pi$
$797$	−446.725	−0.560508	−0.280254	+	0.959926i	$0.590419\pi$
$798$	0	0
$799$	− 1083.19i	− 1.35568i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	989.618i	1.23240i
$804$	0	0
$805$	−193.848	−0.240805
$806$	0	0
$807$	0	0
$808$	0	0
$809$	793.341	0.980644	0.490322	−	0.871541i	$-0.336879\pi$
$809$	793.341	0.980644	0.490322	+	0.871541i	$0.336879\pi$
$810$	0	0
$811$	− 788.930i	− 0.972787i	−0.873740	−	0.486394i	$-0.838312\pi$
$811$	− 788.930i	− 0.972787i	0.873740	−	0.486394i	$-0.161688\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	139.718i	0.171433i
$816$	0	0
$817$	−99.5016	−0.121789
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1448.75	−1.76462	−0.882310	−	0.470668i	$-0.844013\pi$
$821$	−1448.75	−1.76462	−0.882310	+	0.470668i	$0.844013\pi$
$822$	0	0
$823$	129.939i	0.157884i	0.996879	+	0.0789421i	$0.0251542\pi$
$823$	129.939i	0.157884i	−0.996879	+	0.0789421i	$0.974846\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 350.389i	− 0.423687i	−0.977304	−	0.211843i	$-0.932053\pi$
$827$	− 350.389i	− 0.423687i	0.977304	−	0.211843i	$-0.0679467\pi$
$828$	0	0
$829$	−167.748	−0.202349	−0.101175	−	0.994869i	$-0.532260\pi$
$829$	−167.748	−0.202349	−0.101175	+	0.994869i	$0.532260\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−338.512	−0.406376
$834$	0	0
$835$	− 149.037i	− 0.178487i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	661.684i	0.788658i	0.918969	+	0.394329i	$0.129023\pi$
$839$	661.684i	0.788658i	−0.918969	+	0.394329i	$0.870977\pi$
$840$	0	0
$841$	−822.325	−0.977794
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.0248	0.0189643
$846$	0	0
$847$	− 1161.67i	− 1.37151i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 609.136i	− 0.715789i
$852$	0	0
$853$	−17.0789	−0.0200222	−0.0100111	−	0.999950i	$-0.503187\pi$
$853$	−17.0789	−0.0200222	−0.0100111	+	0.999950i	$0.503187\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−422.419	−0.492904	−0.246452	−	0.969155i	$-0.579265\pi$
$857$	−422.419	−0.492904	−0.246452	+	0.969155i	$0.579265\pi$
$858$	0	0
$859$	− 1323.18i	− 1.54037i	−0.637819	−	0.770186i	$-0.720163\pi$
$859$	− 1323.18i	− 1.54037i	0.637819	−	0.770186i	$-0.279837\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1455.50i	1.68656i	0.537473	+	0.843281i	$0.319379\pi$
$863$	1455.50i	1.68656i	−0.537473	+	0.843281i	$0.680621\pi$
$864$	0	0
$865$	−173.325	−0.200376
$866$	0	0
$867$	0	0
$868$	0	0
$869$	727.439	0.837099
$870$	0	0
$871$	1178.03i	1.35250i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	317.277i	0.362602i
$876$	0	0
$877$	753.085	0.858706	0.429353	−	0.903137i	$-0.358742\pi$
$877$	753.085	0.858706	0.429353	+	0.903137i	$0.358742\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−996.177	−1.13073	−0.565367	−	0.824839i	$-0.691265\pi$
$881$	−996.177	−1.13073	−0.565367	+	0.824839i	$0.691265\pi$
$882$	0	0
$883$	− 950.011i	− 1.07589i	−0.842980	−	0.537945i	$-0.819201\pi$
$883$	− 950.011i	− 1.07589i	0.842980	−	0.537945i	$-0.180799\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 433.086i	− 0.488259i	−0.969743	−	0.244129i	$-0.921498\pi$
$887$	− 433.086i	− 0.488259i	0.969743	−	0.244129i	$-0.0785022\pi$
$888$	0	0
$889$	1154.48	1.29863
$890$	0	0
$891$	0	0
$892$	0	0
$893$	213.370	0.238936
$894$	0	0
$895$	− 217.659i	− 0.243195i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	193.775i	0.215545i
$900$	0	0
$901$	−1847.63	−2.05065
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−52.1300	−0.0576022
$906$	0	0
$907$	− 685.704i	− 0.756014i	−0.925803	−	0.378007i	$-0.876610\pi$
$907$	− 685.704i	− 0.756014i	0.925803	−	0.378007i	$-0.123390\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	36.0750i	0.0395994i	0.999804	+	0.0197997i	$0.00630285\pi$
$911$	36.0750i	0.0395994i	−0.999804	+	0.0197997i	$0.993697\pi$
$912$	0	0
$913$	628.328	0.688202
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−852.758	−0.929943
$918$	0	0
$919$	− 953.775i	− 1.03784i	−0.854823	−	0.518920i	$-0.826334\pi$
$919$	− 953.775i	− 1.03784i	0.854823	−	0.518920i	$-0.173666\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1041.65i	− 1.12855i
$924$	0	0
$925$	−486.580	−0.526033
$926$	0	0
$927$	0	0
$928$	0	0
$929$	149.629	0.161064	0.0805321	−	0.996752i	$-0.474338\pi$
$929$	149.629	0.161064	0.0805321	+	0.996752i	$0.474338\pi$
$930$	0	0
$931$	− 66.6813i	− 0.0716233i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	505.126i	0.540241i
$936$	0	0
$937$	−661.158	−0.705611	−0.352806	−	0.935697i	$-0.614772\pi$
$937$	−661.158	−0.705611	−0.352806	+	0.935697i	$0.614772\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−466.713	−0.495976	−0.247988	−	0.968763i	$-0.579769\pi$
$941$	−466.713	−0.495976	−0.247988	+	0.968763i	$0.579769\pi$
$942$	0	0
$943$	− 1767.44i	− 1.87428i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1302.74i	− 1.37565i	−0.725879	−	0.687823i	$-0.758567\pi$
$947$	− 1302.74i	− 1.37565i	0.725879	−	0.687823i	$-0.241433\pi$
$948$	0	0
$949$	−693.243	−0.730498
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.6959	0.0227659	0.0113829	−	0.999935i	$-0.496377\pi$
$953$	21.6959	0.0227659	0.0113829	+	0.999935i	$0.496377\pi$
$954$	0	0
$955$	− 159.258i	− 0.166762i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 527.893i	− 0.550462i
$960$	0	0
$961$	−1049.65	−1.09225
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−27.1863	−0.0281724
$966$	0	0
$967$	640.899i	0.662770i	0.943496	+	0.331385i	$0.107516\pi$
$967$	640.899i	0.662770i	−0.943496	+	0.331385i	$0.892484\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1396.09i	− 1.43779i	−0.695121	−	0.718893i	$-0.744649\pi$
$971$	− 1396.09i	− 1.43779i	0.695121	−	0.718893i	$-0.255351\pi$
$972$	0	0
$973$	1103.56	1.13418
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1009.95	1.03373	0.516864	−	0.856068i	$-0.327099\pi$
$977$	1009.95	1.03373	0.516864	+	0.856068i	$0.327099\pi$
$978$	0	0
$979$	− 467.552i	− 0.477581i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 485.376i	− 0.493770i	−0.969045	−	0.246885i	$-0.920593\pi$
$983$	− 485.376i	− 0.493770i	0.969045	−	0.246885i	$-0.0794071\pi$
$984$	0	0
$985$	416.486	0.422828
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−571.325	−0.577679
$990$	0	0
$991$	265.720i	0.268133i	0.990972	+	0.134066i	$0.0428035\pi$
$991$	265.720i	0.268133i	−0.990972	+	0.134066i	$0.957196\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 131.141i	− 0.131800i
$996$	0	0
$997$	1715.81	1.72097	0.860487	−	0.509472i	$-0.170159\pi$
$997$	1715.81	1.72097	0.860487	+	0.509472i	$0.170159\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.g.l.703.4		8
3.2	odd	2	inner	1728.3.g.l.703.6		8
4.3	odd	2	inner	1728.3.g.l.703.3		8
8.3	odd	2		108.3.d.d.55.7	yes	8
8.5	even	2		108.3.d.d.55.8	yes	8
12.11	even	2	inner	1728.3.g.l.703.5		8
24.5	odd	2		108.3.d.d.55.1	✓	8
24.11	even	2		108.3.d.d.55.2	yes	8
72.5	odd	6		324.3.f.p.55.3		8
72.11	even	6		324.3.f.p.271.4		8
72.13	even	6		324.3.f.p.55.2		8
72.29	odd	6		324.3.f.o.271.3		8
72.43	odd	6		324.3.f.p.271.1		8
72.59	even	6		324.3.f.o.55.3		8
72.61	even	6		324.3.f.o.271.2		8
72.67	odd	6		324.3.f.o.55.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.3.d.d.55.1	✓	8	24.5	odd	2
108.3.d.d.55.2	yes	8	24.11	even	2
108.3.d.d.55.7	yes	8	8.3	odd	2
108.3.d.d.55.8	yes	8	8.5	even	2
324.3.f.o.55.2		8	72.67	odd	6
324.3.f.o.55.3		8	72.59	even	6
324.3.f.o.271.2		8	72.61	even	6
324.3.f.o.271.3		8	72.29	odd	6
324.3.f.p.55.2		8	72.13	even	6
324.3.f.p.55.3		8	72.5	odd	6
324.3.f.p.271.1		8	72.43	odd	6
324.3.f.p.271.4		8	72.11	even	6
1728.3.g.l.703.3		8	4.3	odd	2	inner
1728.3.g.l.703.4		8	1.1	even	1	trivial
1728.3.g.l.703.5		8	12.11	even	2	inner
1728.3.g.l.703.6		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.08036 q^{5} +6.01392i q^{7} +O(q^{10})\)	\(q-1.08036 q^{5} +6.01392i q^{7} +17.7247i q^{11} -12.4164 q^{13} -26.3786 q^{17} -5.19615i q^{19} -29.8356i q^{23} -23.8328 q^{25} -4.32145 q^{29} -44.8403i q^{31} -6.49721i q^{35} +20.4164 q^{37} +59.2393 q^{41} -19.1491i q^{43} +41.0631i q^{47} +12.8328 q^{49} +70.0430 q^{53} -19.1491i q^{55} -28.9521i q^{59} -18.4164 q^{61} +13.4142 q^{65} -94.8767i q^{67} +83.8931i q^{71} +55.8328 q^{73} -106.595 q^{77} -41.0410i q^{79} -35.4493i q^{83} +28.4984 q^{85} -26.3786 q^{89} -74.6712i q^{91} +5.61373i q^{95} +76.3313 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q + 8 q^{13} + 24 q^{25} + 56 q^{37} - 112 q^{49} - 40 q^{61} + 232 q^{73} - 416 q^{85} - 248 q^{97}+O(q^{100}) \) 8 * q + 8 * q^13 + 24 * q^25 + 56 * q^37 - 112 * q^49 - 40 * q^61 + 232 * q^73 - 416 * q^85 - 248 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more