LMFDB - Embedded newform 10000.2.a.be.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10000,2,Mod(1,10000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10000, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("10000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 10000 = 2^{4} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	10000.a (trivial)

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:	yes
Analytic conductor:	$79.8504020213$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.6152203125.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 8x^{6} + 20x^{5} + 26x^{4} - 35x^{3} - 27x^{2} + 16x + 1 $ x^8 - 3x^7 - 8x^6 + 20x^5 + 26x^4 - 35x^3 - 27x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 625)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.499011$ of defining polynomial
Character	$\chi$	$=$	10000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30231 q^{3} -3.59425 q^{7} +2.30065 q^{9} +O(q^{10})$	$q+2.30231 q^{3} -3.59425 q^{7} +2.30065 q^{9} +0.497788 q^{11} +2.64789 q^{13} +5.10719 q^{17} +0.987277 q^{19} -8.27508 q^{21} -6.41382 q^{23} -1.61013 q^{27} -5.57001 q^{29} -6.05507 q^{31} +1.14606 q^{33} -4.59612 q^{37} +6.09627 q^{39} -2.87475 q^{41} +9.48858 q^{43} -5.36834 q^{47} +5.91861 q^{49} +11.7583 q^{51} -0.307600 q^{53} +2.27302 q^{57} +1.26645 q^{59} -6.22625 q^{61} -8.26910 q^{63} +5.28626 q^{67} -14.7666 q^{69} +0.151963 q^{71} -14.8741 q^{73} -1.78917 q^{77} +16.5886 q^{79} -10.6090 q^{81} -14.5960 q^{83} -12.8239 q^{87} +11.3822 q^{89} -9.51717 q^{91} -13.9407 q^{93} +0.849192 q^{97} +1.14523 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	$ 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) $ 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * q^99

Display 100 coefficients 10 coefficients

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
$3$	2.30231	1.32924	0.664621	−	0.747181i	$-0.268593\pi$
$3$	2.30231	1.32924	0.664621	+	0.747181i	$0.268593\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.59425	−1.35850	−0.679249	−	0.733908i	$-0.737694\pi$
$7$	−3.59425	−1.35850	−0.679249	+	0.733908i	$0.737694\pi$
$8$	0	0
$9$	2.30065	0.766883
$10$	0	0
$11$	0.497788	0.150089	0.0750443	−	0.997180i	$-0.476090\pi$
$11$	0.497788	0.150089	0.0750443	+	0.997180i	$0.476090\pi$
$12$	0	0
$13$	2.64789	0.734392	0.367196	−	0.930143i	$-0.380318\pi$
$13$	2.64789	0.734392	0.367196	+	0.930143i	$0.380318\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.10719	1.23867	0.619337	−	0.785125i	$-0.287401\pi$
$17$	5.10719	1.23867	0.619337	+	0.785125i	$0.287401\pi$
$18$	0	0
$19$	0.987277	0.226497	0.113248	−	0.993567i	$-0.463874\pi$
$19$	0.987277	0.226497	0.113248	+	0.993567i	$0.463874\pi$
$20$	0	0
$21$	−8.27508	−1.80577
$22$	0	0
$23$	−6.41382	−1.33737	−0.668687	−	0.743544i	$-0.733143\pi$
$23$	−6.41382	−1.33737	−0.668687	+	0.743544i	$0.733143\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.61013	−0.309869
$28$	0	0
$29$	−5.57001	−1.03432	−0.517162	−	0.855887i	$-0.673012\pi$
$29$	−5.57001	−1.03432	−0.517162	+	0.855887i	$0.673012\pi$
$30$	0	0
$31$	−6.05507	−1.08752	−0.543762	−	0.839240i	$-0.683000\pi$
$31$	−6.05507	−1.08752	−0.543762	+	0.839240i	$0.683000\pi$
$32$	0	0
$33$	1.14606	0.199504
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−4.59612	−0.755598	−0.377799	−	0.925888i	$-0.623319\pi$
$37$	−4.59612	−0.755598	−0.377799	+	0.925888i	$0.623319\pi$
$38$	0	0
$39$	6.09627	0.976185
$40$	0	0
$41$	−2.87475	−0.448960	−0.224480	−	0.974479i	$-0.572068\pi$
$41$	−2.87475	−0.448960	−0.224480	+	0.974479i	$0.572068\pi$
$42$	0	0
$43$	9.48858	1.44700	0.723498	−	0.690327i	$-0.242533\pi$
$43$	9.48858	1.44700	0.723498	+	0.690327i	$0.242533\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.36834	−0.783053	−0.391527	−	0.920167i	$-0.628053\pi$
$47$	−5.36834	−0.783053	−0.391527	+	0.920167i	$0.628053\pi$
$48$	0	0
$49$	5.91861	0.845515
$50$	0	0
$51$	11.7583	1.64650
$52$	0	0
$53$	−0.307600	−0.0422521	−0.0211261	−	0.999777i	$-0.506725\pi$
$53$	−0.307600	−0.0422521	−0.0211261	+	0.999777i	$0.506725\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.27302	0.301069
$58$	0	0
$59$	1.26645	0.164878	0.0824389	−	0.996596i	$-0.473729\pi$
$59$	1.26645	0.164878	0.0824389	+	0.996596i	$0.473729\pi$
$60$	0	0
$61$	−6.22625	−0.797190	−0.398595	−	0.917127i	$-0.630502\pi$
$61$	−6.22625	−0.797190	−0.398595	+	0.917127i	$0.630502\pi$
$62$	0	0
$63$	−8.26910	−1.04181
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.28626	0.645819	0.322910	−	0.946430i	$-0.395339\pi$
$67$	5.28626	0.645819	0.322910	+	0.946430i	$0.395339\pi$
$68$	0	0
$69$	−14.7666	−1.77769
$70$	0	0
$71$	0.151963	0.0180347	0.00901734	−	0.999959i	$-0.497130\pi$
$71$	0.151963	0.0180347	0.00901734	+	0.999959i	$0.497130\pi$
$72$	0	0
$73$	−14.8741	−1.74088	−0.870439	−	0.492276i	$-0.836165\pi$
$73$	−14.8741	−1.74088	−0.870439	+	0.492276i	$0.836165\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.78917	−0.203895
$78$	0	0
$79$	16.5886	1.86636	0.933181	−	0.359406i	$-0.117021\pi$
$79$	16.5886	1.86636	0.933181	+	0.359406i	$0.117021\pi$
$80$	0	0
$81$	−10.6090	−1.17877
$82$	0	0
$83$	−14.5960	−1.60212	−0.801058	−	0.598587i	$-0.795729\pi$
$83$	−14.5960	−1.60212	−0.801058	+	0.598587i	$0.795729\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−12.8239	−1.37487
$88$	0	0
$89$	11.3822	1.20652	0.603258	−	0.797546i	$-0.293869\pi$
$89$	11.3822	1.20652	0.603258	+	0.797546i	$0.293869\pi$
$90$	0	0
$91$	−9.51717	−0.997670
$92$	0	0
$93$	−13.9407	−1.44558
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.849192	0.0862223	0.0431112	−	0.999070i	$-0.486273\pi$
$97$	0.849192	0.0862223	0.0431112	+	0.999070i	$0.486273\pi$
$98$	0	0
$99$	1.14523	0.115100
$100$	0	0
$101$	−13.2498	−1.31841	−0.659203	−	0.751965i	$-0.729106\pi$
$101$	−13.2498	−1.31841	−0.659203	+	0.751965i	$0.729106\pi$
$102$	0	0
$103$	−0.830909	−0.0818719	−0.0409360	−	0.999162i	$-0.513034\pi$
$103$	−0.830909	−0.0818719	−0.0409360	+	0.999162i	$0.513034\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.0722844	0.00698800	0.00349400	−	0.999994i	$-0.498888\pi$
$107$	0.0722844	0.00698800	0.00349400	+	0.999994i	$0.498888\pi$
$108$	0	0
$109$	5.59621	0.536020	0.268010	−	0.963416i	$-0.413634\pi$
$109$	5.59621	0.536020	0.268010	+	0.963416i	$0.413634\pi$
$110$	0	0
$111$	−10.5817	−1.00437
$112$	0	0
$113$	−2.60239	−0.244812	−0.122406	−	0.992480i	$-0.539061\pi$
$113$	−2.60239	−0.244812	−0.122406	+	0.992480i	$0.539061\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	6.09186	0.563193
$118$	0	0
$119$	−18.3565	−1.68274
$120$	0	0
$121$	−10.7522	−0.977473
$122$	0	0
$123$	−6.61857	−0.596777
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.94483	−0.438783	−0.219391	−	0.975637i	$-0.570407\pi$
$127$	−4.94483	−0.438783	−0.219391	+	0.975637i	$0.570407\pi$
$128$	0	0
$129$	21.8457	1.92341
$130$	0	0
$131$	−2.70342	−0.236199	−0.118099	−	0.993002i	$-0.537680\pi$
$131$	−2.70342	−0.236199	−0.118099	+	0.993002i	$0.537680\pi$
$132$	0	0
$133$	−3.54852	−0.307695
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.24838	−0.192092	−0.0960459	−	0.995377i	$-0.530620\pi$
$137$	−2.24838	−0.192092	−0.0960459	+	0.995377i	$0.530620\pi$
$138$	0	0
$139$	10.8032	0.916317	0.458159	−	0.888870i	$-0.348509\pi$
$139$	10.8032	0.916317	0.458159	+	0.888870i	$0.348509\pi$
$140$	0	0
$141$	−12.3596	−1.04087
$142$	0	0
$143$	1.31809	0.110224
$144$	0	0
$145$	0	0
$146$	0	0
$147$	13.6265	1.12389
$148$	0	0
$149$	12.1878	0.998460	0.499230	−	0.866469i	$-0.333616\pi$
$149$	12.1878	0.998460	0.499230	+	0.866469i	$0.333616\pi$
$150$	0	0
$151$	−17.0860	−1.39044	−0.695220	−	0.718797i	$-0.744693\pi$
$151$	−17.0860	−1.39044	−0.695220	+	0.718797i	$0.744693\pi$
$152$	0	0
$153$	11.7498	0.949918
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−7.49835	−0.598433	−0.299217	−	0.954185i	$-0.596725\pi$
$157$	−7.49835	−0.598433	−0.299217	+	0.954185i	$0.596725\pi$
$158$	0	0
$159$	−0.708192	−0.0561633
$160$	0	0
$161$	23.0529	1.81682
$162$	0	0
$163$	−1.95259	−0.152939	−0.0764693	−	0.997072i	$-0.524365\pi$
$163$	−1.95259	−0.152939	−0.0764693	+	0.997072i	$0.524365\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	0.356578	0.0275928	0.0137964	−	0.999905i	$-0.495608\pi$
$167$	0.356578	0.0275928	0.0137964	+	0.999905i	$0.495608\pi$
$168$	0	0
$169$	−5.98868	−0.460668
$170$	0	0
$171$	2.27138	0.173697
$172$	0	0
$173$	−9.95032	−0.756509	−0.378254	−	0.925702i	$-0.623476\pi$
$173$	−9.95032	−0.756509	−0.378254	+	0.925702i	$0.623476\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.91577	0.219162
$178$	0	0
$179$	−15.3824	−1.14973	−0.574867	−	0.818247i	$-0.694946\pi$
$179$	−15.3824	−1.14973	−0.574867	+	0.818247i	$0.694946\pi$
$180$	0	0
$181$	8.91917	0.662957	0.331478	−	0.943463i	$-0.392453\pi$
$181$	8.91917	0.662957	0.331478	+	0.943463i	$0.392453\pi$
$182$	0	0
$183$	−14.3348	−1.05966
$184$	0	0
$185$	0	0
$186$	0	0
$187$	2.54230	0.185911
$188$	0	0
$189$	5.78719	0.420956
$190$	0	0
$191$	11.3887	0.824056	0.412028	−	0.911171i	$-0.364821\pi$
$191$	11.3887	0.824056	0.412028	+	0.911171i	$0.364821\pi$
$192$	0	0
$193$	17.3321	1.24759	0.623795	−	0.781588i	$-0.285590\pi$
$193$	17.3321	1.24759	0.623795	+	0.781588i	$0.285590\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.9609	1.84964	0.924820	−	0.380405i	$-0.124215\pi$
$197$	25.9609	1.84964	0.924820	+	0.380405i	$0.124215\pi$
$198$	0	0
$199$	8.38571	0.594447	0.297223	−	0.954808i	$-0.403939\pi$
$199$	8.38571	0.594447	0.297223	+	0.954808i	$0.403939\pi$
$200$	0	0
$201$	12.1706	0.858450
$202$	0	0
$203$	20.0200	1.40513
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−14.7559	−1.02561
$208$	0	0
$209$	0.491454	0.0339946
$210$	0	0
$211$	−15.9135	−1.09553	−0.547765	−	0.836632i	$-0.684521\pi$
$211$	−15.9135	−1.09553	−0.547765	+	0.836632i	$0.684521\pi$
$212$	0	0
$213$	0.349866	0.0239724
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21.7634	1.47740
$218$	0	0
$219$	−34.2448	−2.31405
$220$	0	0
$221$	13.5233	0.909674
$222$	0	0
$223$	0.379706	0.0254270	0.0127135	−	0.999919i	$-0.495953\pi$
$223$	0.379706	0.0254270	0.0127135	+	0.999919i	$0.495953\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.3009	−1.34742	−0.673710	−	0.738996i	$-0.735300\pi$
$227$	−20.3009	−1.34742	−0.673710	+	0.738996i	$0.735300\pi$
$228$	0	0
$229$	−11.9107	−0.787078	−0.393539	−	0.919308i	$-0.628749\pi$
$229$	−11.9107	−0.787078	−0.393539	+	0.919308i	$0.628749\pi$
$230$	0	0
$231$	−4.11923	−0.271026
$232$	0	0
$233$	−19.0610	−1.24873	−0.624364	−	0.781133i	$-0.714642\pi$
$233$	−19.0610	−1.24873	−0.624364	+	0.781133i	$0.714642\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	38.1922	2.48085
$238$	0	0
$239$	−4.12493	−0.266819	−0.133410	−	0.991061i	$-0.542593\pi$
$239$	−4.12493	−0.266819	−0.133410	+	0.991061i	$0.542593\pi$
$240$	0	0
$241$	16.1858	1.04262	0.521310	−	0.853367i	$-0.325444\pi$
$241$	16.1858	1.04262	0.521310	+	0.853367i	$0.325444\pi$
$242$	0	0
$243$	−19.5948	−1.25701
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.61420	0.166338
$248$	0	0
$249$	−33.6045	−2.12960
$250$	0	0
$251$	−11.8718	−0.749344	−0.374672	−	0.927157i	$-0.622245\pi$
$251$	−11.8718	−0.749344	−0.374672	+	0.927157i	$0.622245\pi$
$252$	0	0
$253$	−3.19272	−0.200725
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.9164	−1.17997	−0.589986	−	0.807414i	$-0.700867\pi$
$257$	−18.9164	−1.17997	−0.589986	+	0.807414i	$0.700867\pi$
$258$	0	0
$259$	16.5196	1.02648
$260$	0	0
$261$	−12.8146	−0.793206
$262$	0	0
$263$	6.74703	0.416040	0.208020	−	0.978125i	$-0.433298\pi$
$263$	6.74703	0.416040	0.208020	+	0.978125i	$0.433298\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	26.2055	1.60375
$268$	0	0
$269$	−25.3329	−1.54457	−0.772286	−	0.635275i	$-0.780887\pi$
$269$	−25.3329	−1.54457	−0.772286	+	0.635275i	$0.780887\pi$
$270$	0	0
$271$	−9.40735	−0.571456	−0.285728	−	0.958311i	$-0.592235\pi$
$271$	−9.40735	−0.571456	−0.285728	+	0.958311i	$0.592235\pi$
$272$	0	0
$273$	−21.9115	−1.32614
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5.96736	−0.358544	−0.179272	−	0.983800i	$-0.557374\pi$
$277$	−5.96736	−0.358544	−0.179272	+	0.983800i	$0.557374\pi$
$278$	0	0
$279$	−13.9306	−0.834003
$280$	0	0
$281$	11.0723	0.660516	0.330258	−	0.943891i	$-0.392864\pi$
$281$	11.0723	0.660516	0.330258	+	0.943891i	$0.392864\pi$
$282$	0	0
$283$	3.02406	0.179762	0.0898810	−	0.995953i	$-0.471351\pi$
$283$	3.02406	0.179762	0.0898810	+	0.995953i	$0.471351\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	10.3326	0.609911
$288$	0	0
$289$	9.08337	0.534316
$290$	0	0
$291$	1.95511	0.114610
$292$	0	0
$293$	6.26426	0.365962	0.182981	−	0.983116i	$-0.441425\pi$
$293$	6.26426	0.365962	0.182981	+	0.983116i	$0.441425\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−0.801501	−0.0465078
$298$	0	0
$299$	−16.9831	−0.982158
$300$	0	0
$301$	−34.1043	−1.96574
$302$	0	0
$303$	−30.5052	−1.75248
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−25.8734	−1.47667	−0.738337	−	0.674432i	$-0.764388\pi$
$307$	−25.8734	−1.47667	−0.738337	+	0.674432i	$0.764388\pi$
$308$	0	0
$309$	−1.91301	−0.108828
$310$	0	0
$311$	9.00277	0.510500	0.255250	−	0.966875i	$-0.417842\pi$
$311$	9.00277	0.510500	0.255250	+	0.966875i	$0.417842\pi$
$312$	0	0
$313$	−33.2274	−1.87812	−0.939062	−	0.343748i	$-0.888304\pi$
$313$	−33.2274	−1.87812	−0.939062	+	0.343748i	$0.888304\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.9693	−1.45858	−0.729290	−	0.684204i	$-0.760150\pi$
$317$	−25.9693	−1.45858	−0.729290	+	0.684204i	$0.760150\pi$
$318$	0	0
$319$	−2.77268	−0.155240
$320$	0	0
$321$	0.166421	0.00928874
$322$	0	0
$323$	5.04221	0.280556
$324$	0	0
$325$	0	0
$326$	0	0
$327$	12.8842	0.712500
$328$	0	0
$329$	19.2951	1.06378
$330$	0	0
$331$	−12.1284	−0.666636	−0.333318	−	0.942814i	$-0.608168\pi$
$331$	−12.1284	−0.666636	−0.333318	+	0.942814i	$0.608168\pi$
$332$	0	0
$333$	−10.5741	−0.579455
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−30.2519	−1.64793	−0.823963	−	0.566644i	$-0.808241\pi$
$337$	−30.2519	−1.64793	−0.823963	+	0.566644i	$0.808241\pi$
$338$	0	0
$339$	−5.99152	−0.325415
$340$	0	0
$341$	−3.01414	−0.163225
$342$	0	0
$343$	3.88680	0.209867
$344$	0	0
$345$	0	0
$346$	0	0
$347$	13.3283	0.715502	0.357751	−	0.933817i	$-0.383544\pi$
$347$	13.3283	0.715502	0.357751	+	0.933817i	$0.383544\pi$
$348$	0	0
$349$	−27.4444	−1.46906	−0.734532	−	0.678574i	$-0.762598\pi$
$349$	−27.4444	−1.46906	−0.734532	+	0.678574i	$0.762598\pi$
$350$	0	0
$351$	−4.26344	−0.227566
$352$	0	0
$353$	15.5564	0.827983	0.413992	−	0.910281i	$-0.364134\pi$
$353$	15.5564	0.827983	0.413992	+	0.910281i	$0.364134\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−42.2624	−2.23676
$358$	0	0
$359$	21.9829	1.16021	0.580107	−	0.814540i	$-0.303011\pi$
$359$	21.9829	1.16021	0.580107	+	0.814540i	$0.303011\pi$
$360$	0	0
$361$	−18.0253	−0.948699
$362$	0	0
$363$	−24.7550	−1.29930
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−12.1103	−0.632155	−0.316078	−	0.948733i	$-0.602366\pi$
$367$	−12.1103	−0.632155	−0.316078	+	0.948733i	$0.602366\pi$
$368$	0	0
$369$	−6.61379	−0.344300
$370$	0	0
$371$	1.10559	0.0573994
$372$	0	0
$373$	7.08604	0.366901	0.183451	−	0.983029i	$-0.441273\pi$
$373$	7.08604	0.366901	0.183451	+	0.983029i	$0.441273\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−14.7488	−0.759600
$378$	0	0
$379$	21.5030	1.10453	0.552266	−	0.833668i	$-0.313763\pi$
$379$	21.5030	1.10453	0.552266	+	0.833668i	$0.313763\pi$
$380$	0	0
$381$	−11.3845	−0.583248
$382$	0	0
$383$	−24.8816	−1.27139	−0.635696	−	0.771940i	$-0.719287\pi$
$383$	−24.8816	−1.27139	−0.635696	+	0.771940i	$0.719287\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	21.8299	1.10968
$388$	0	0
$389$	−35.4142	−1.79557	−0.897785	−	0.440434i	$-0.854825\pi$
$389$	−35.4142	−1.79557	−0.897785	+	0.440434i	$0.854825\pi$
$390$	0	0
$391$	−32.7566	−1.65657
$392$	0	0
$393$	−6.22412	−0.313965
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.49705	0.275889	0.137944	−	0.990440i	$-0.455950\pi$
$397$	5.49705	0.275889	0.137944	+	0.990440i	$0.455950\pi$
$398$	0	0
$399$	−8.16980	−0.409001
$400$	0	0
$401$	24.8463	1.24077	0.620383	−	0.784299i	$-0.286977\pi$
$401$	24.8463	1.24077	0.620383	+	0.784299i	$0.286977\pi$
$402$	0	0
$403$	−16.0332	−0.798669
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.28789	−0.113407
$408$	0	0
$409$	3.93157	0.194404	0.0972019	−	0.995265i	$-0.469011\pi$
$409$	3.93157	0.194404	0.0972019	+	0.995265i	$0.469011\pi$
$410$	0	0
$411$	−5.17647	−0.255336
$412$	0	0
$413$	−4.55194	−0.223986
$414$	0	0
$415$	0	0
$416$	0	0
$417$	24.8724	1.21801
$418$	0	0
$419$	4.76604	0.232836	0.116418	−	0.993200i	$-0.462859\pi$
$419$	4.76604	0.232836	0.116418	+	0.993200i	$0.462859\pi$
$420$	0	0
$421$	16.0581	0.782625	0.391312	−	0.920258i	$-0.372021\pi$
$421$	16.0581	0.782625	0.391312	+	0.920258i	$0.372021\pi$
$422$	0	0
$423$	−12.3507	−0.600510
$424$	0	0
$425$	0	0
$426$	0	0
$427$	22.3787	1.08298
$428$	0	0
$429$	3.03465	0.146514
$430$	0	0
$431$	41.3693	1.99269	0.996344	−	0.0854360i	$-0.0272283\pi$
$431$	41.3693	1.99269	0.996344	+	0.0854360i	$0.0272283\pi$
$432$	0	0
$433$	15.1854	0.729763	0.364881	−	0.931054i	$-0.381110\pi$
$433$	15.1854	0.729763	0.364881	+	0.931054i	$0.381110\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.33222	−0.302911
$438$	0	0
$439$	−17.1194	−0.817063	−0.408532	−	0.912744i	$-0.633959\pi$
$439$	−17.1194	−0.817063	−0.408532	+	0.912744i	$0.633959\pi$
$440$	0	0
$441$	13.6166	0.648411
$442$	0	0
$443$	−35.3909	−1.68147	−0.840736	−	0.541445i	$-0.817877\pi$
$443$	−35.3909	−1.68147	−0.840736	+	0.541445i	$0.817877\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	28.0601	1.32719
$448$	0	0
$449$	20.6830	0.976090	0.488045	−	0.872818i	$-0.337710\pi$
$449$	20.6830	0.976090	0.488045	+	0.872818i	$0.337710\pi$
$450$	0	0
$451$	−1.43101	−0.0673838
$452$	0	0
$453$	−39.3374	−1.84823
$454$	0	0
$455$	0	0
$456$	0	0
$457$	41.7664	1.95375	0.976876	−	0.213807i	$-0.0685863\pi$
$457$	41.7664	1.95375	0.976876	+	0.213807i	$0.0685863\pi$
$458$	0	0
$459$	−8.22322	−0.383827
$460$	0	0
$461$	−10.6783	−0.497340	−0.248670	−	0.968588i	$-0.579993\pi$
$461$	−10.6783	−0.497340	−0.248670	+	0.968588i	$0.579993\pi$
$462$	0	0
$463$	7.83033	0.363906	0.181953	−	0.983307i	$-0.441758\pi$
$463$	7.83033	0.363906	0.181953	+	0.983307i	$0.441758\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.98100	−0.230493	−0.115246	−	0.993337i	$-0.536766\pi$
$467$	−4.98100	−0.230493	−0.115246	+	0.993337i	$0.536766\pi$
$468$	0	0
$469$	−19.0001	−0.877344
$470$	0	0
$471$	−17.2635	−0.795462
$472$	0	0
$473$	4.72330	0.217178
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−0.707679	−0.0324024
$478$	0	0
$479$	16.3981	0.749247	0.374623	−	0.927177i	$-0.377772\pi$
$479$	16.3981	0.749247	0.374623	+	0.927177i	$0.377772\pi$
$480$	0	0
$481$	−12.1700	−0.554906
$482$	0	0
$483$	53.0749	2.41499
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−0.876508	−0.0397184	−0.0198592	−	0.999803i	$-0.506322\pi$
$487$	−0.876508	−0.0397184	−0.0198592	+	0.999803i	$0.506322\pi$
$488$	0	0
$489$	−4.49547	−0.203292
$490$	0	0
$491$	−12.2118	−0.551112	−0.275556	−	0.961285i	$-0.588862\pi$
$491$	−12.2118	−0.551112	−0.275556	+	0.961285i	$0.588862\pi$
$492$	0	0
$493$	−28.4471	−1.28119
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.546192	−0.0245001
$498$	0	0
$499$	3.34603	0.149789	0.0748945	−	0.997191i	$-0.476138\pi$
$499$	3.34603	0.149789	0.0748945	+	0.997191i	$0.476138\pi$
$500$	0	0
$501$	0.820954	0.0366775
$502$	0	0
$503$	−11.8820	−0.529792	−0.264896	−	0.964277i	$-0.585338\pi$
$503$	−11.8820	−0.529792	−0.264896	+	0.964277i	$0.585338\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−13.7878	−0.612339
$508$	0	0
$509$	36.3183	1.60978	0.804890	−	0.593424i	$-0.202224\pi$
$509$	36.3183	1.60978	0.804890	+	0.593424i	$0.202224\pi$
$510$	0	0
$511$	53.4611	2.36498
$512$	0	0
$513$	−1.58964	−0.0701844
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.67229	−0.117527
$518$	0	0
$519$	−22.9088	−1.00558
$520$	0	0
$521$	0.204224	0.00894722	0.00447361	−	0.999990i	$-0.498576\pi$
$521$	0.204224	0.00894722	0.00447361	+	0.999990i	$0.498576\pi$
$522$	0	0
$523$	−28.0312	−1.22572	−0.612859	−	0.790192i	$-0.709981\pi$
$523$	−28.0312	−1.22572	−0.612859	+	0.790192i	$0.709981\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−30.9244	−1.34709
$528$	0	0
$529$	18.1371	0.788570
$530$	0	0
$531$	2.91366	0.126442
$532$	0	0
$533$	−7.61202	−0.329713
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−35.4151	−1.52827
$538$	0	0
$539$	2.94621	0.126902
$540$	0	0
$541$	−9.32216	−0.400791	−0.200396	−	0.979715i	$-0.564223\pi$
$541$	−9.32216	−0.400791	−0.200396	+	0.979715i	$0.564223\pi$
$542$	0	0
$543$	20.5347	0.881230
$544$	0	0
$545$	0	0
$546$	0	0
$547$	20.0378	0.856755	0.428378	−	0.903600i	$-0.359085\pi$
$547$	20.0378	0.856755	0.428378	+	0.903600i	$0.359085\pi$
$548$	0	0
$549$	−14.3244	−0.611351
$550$	0	0
$551$	−5.49914	−0.234271
$552$	0	0
$553$	−59.6235	−2.53545
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.9839	−1.22809	−0.614043	−	0.789272i	$-0.710458\pi$
$557$	−28.9839	−1.22809	−0.614043	+	0.789272i	$0.710458\pi$
$558$	0	0
$559$	25.1247	1.06266
$560$	0	0
$561$	5.85316	0.247121
$562$	0	0
$563$	−13.7025	−0.577492	−0.288746	−	0.957406i	$-0.593238\pi$
$563$	−13.7025	−0.577492	−0.288746	+	0.957406i	$0.593238\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	38.1312	1.60136
$568$	0	0
$569$	34.6152	1.45115	0.725573	−	0.688146i	$-0.241575\pi$
$569$	34.6152	1.45115	0.725573	+	0.688146i	$0.241575\pi$
$570$	0	0
$571$	42.4197	1.77521	0.887604	−	0.460607i	$-0.152368\pi$
$571$	42.4197	1.77521	0.887604	+	0.460607i	$0.152368\pi$
$572$	0	0
$573$	26.2203	1.09537
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−34.3049	−1.42813	−0.714067	−	0.700078i	$-0.753149\pi$
$577$	−34.3049	−1.42813	−0.714067	+	0.700078i	$0.753149\pi$
$578$	0	0
$579$	39.9039	1.65835
$580$	0	0
$581$	52.4615	2.17647
$582$	0	0
$583$	−0.153120	−0.00634156
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−29.7521	−1.22800	−0.614001	−	0.789305i	$-0.710441\pi$
$587$	−29.7521	−1.22800	−0.614001	+	0.789305i	$0.710441\pi$
$588$	0	0
$589$	−5.97803	−0.246321
$590$	0	0
$591$	59.7702	2.45862
$592$	0	0
$593$	−14.8105	−0.608195	−0.304098	−	0.952641i	$-0.598355\pi$
$593$	−14.8105	−0.608195	−0.304098	+	0.952641i	$0.598355\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.3065	0.790163
$598$	0	0
$599$	−27.2394	−1.11297	−0.556486	−	0.830857i	$-0.687851\pi$
$599$	−27.2394	−1.11297	−0.556486	+	0.830857i	$0.687851\pi$
$600$	0	0
$601$	33.1682	1.35296	0.676480	−	0.736461i	$-0.263505\pi$
$601$	33.1682	1.35296	0.676480	+	0.736461i	$0.263505\pi$
$602$	0	0
$603$	12.1618	0.495268
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−5.79849	−0.235353	−0.117677	−	0.993052i	$-0.537545\pi$
$607$	−5.79849	−0.235353	−0.117677	+	0.993052i	$0.537545\pi$
$608$	0	0
$609$	46.0923	1.86775
$610$	0	0
$611$	−14.2148	−0.575068
$612$	0	0
$613$	4.04653	0.163438	0.0817189	−	0.996655i	$-0.473959\pi$
$613$	4.04653	0.163438	0.0817189	+	0.996655i	$0.473959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.1749	1.49661	0.748303	−	0.663357i	$-0.230869\pi$
$617$	37.1749	1.49661	0.748303	+	0.663357i	$0.230869\pi$
$618$	0	0
$619$	37.7118	1.51576	0.757882	−	0.652391i	$-0.226234\pi$
$619$	37.7118	1.51576	0.757882	+	0.652391i	$0.226234\pi$
$620$	0	0
$621$	10.3271	0.414411
$622$	0	0
$623$	−40.9106	−1.63905
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.13148	0.0451870
$628$	0	0
$629$	−23.4733	−0.935940
$630$	0	0
$631$	−18.5225	−0.737368	−0.368684	−	0.929555i	$-0.620192\pi$
$631$	−18.5225	−0.737368	−0.368684	+	0.929555i	$0.620192\pi$
$632$	0	0
$633$	−36.6379	−1.45622
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15.6718	0.620940
$638$	0	0
$639$	0.349613	0.0138305
$640$	0	0
$641$	−19.7116	−0.778560	−0.389280	−	0.921120i	$-0.627276\pi$
$641$	−19.7116	−0.778560	−0.389280	+	0.921120i	$0.627276\pi$
$642$	0	0
$643$	−42.5897	−1.67957	−0.839787	−	0.542916i	$-0.817320\pi$
$643$	−42.5897	−1.67957	−0.839787	+	0.542916i	$0.817320\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27.4018	−1.07728	−0.538638	−	0.842537i	$-0.681061\pi$
$647$	−27.4018	−1.07728	−0.538638	+	0.842537i	$0.681061\pi$
$648$	0	0
$649$	0.630424	0.0247463
$650$	0	0
$651$	50.1062	1.96382
$652$	0	0
$653$	21.4414	0.839068	0.419534	−	0.907740i	$-0.362193\pi$
$653$	21.4414	0.839068	0.419534	+	0.907740i	$0.362193\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−34.2200	−1.33505
$658$	0	0
$659$	36.7342	1.43096	0.715480	−	0.698633i	$-0.246208\pi$
$659$	36.7342	1.43096	0.715480	+	0.698633i	$0.246208\pi$
$660$	0	0
$661$	−15.7515	−0.612664	−0.306332	−	0.951925i	$-0.599102\pi$
$661$	−15.7515	−0.612664	−0.306332	+	0.951925i	$0.599102\pi$
$662$	0	0
$663$	31.1348	1.20918
$664$	0	0
$665$	0	0
$666$	0	0
$667$	35.7250	1.38328
$668$	0	0
$669$	0.874201	0.0337986
$670$	0	0
$671$	−3.09935	−0.119649
$672$	0	0
$673$	31.0107	1.19537	0.597687	−	0.801730i	$-0.296087\pi$
$673$	31.0107	1.19537	0.597687	+	0.801730i	$0.296087\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−9.59882	−0.368912	−0.184456	−	0.982841i	$-0.559052\pi$
$677$	−9.59882	−0.368912	−0.184456	+	0.982841i	$0.559052\pi$
$678$	0	0
$679$	−3.05220	−0.117133
$680$	0	0
$681$	−46.7391	−1.79105
$682$	0	0
$683$	0.868350	0.0332265	0.0166132	−	0.999862i	$-0.494712\pi$
$683$	0.868350	0.0332265	0.0166132	+	0.999862i	$0.494712\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−27.4221	−1.04622
$688$	0	0
$689$	−0.814491	−0.0310296
$690$	0	0
$691$	13.1375	0.499772	0.249886	−	0.968275i	$-0.419607\pi$
$691$	13.1375	0.499772	0.249886	+	0.968275i	$0.419607\pi$
$692$	0	0
$693$	−4.11625	−0.156364
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−14.6819	−0.556116
$698$	0	0
$699$	−43.8844	−1.65986
$700$	0	0
$701$	7.13602	0.269524	0.134762	−	0.990878i	$-0.456973\pi$
$701$	7.13602	0.269524	0.134762	+	0.990878i	$0.456973\pi$
$702$	0	0
$703$	−4.53765	−0.171141
$704$	0	0
$705$	0	0
$706$	0	0
$707$	47.6231	1.79105
$708$	0	0
$709$	−8.00513	−0.300639	−0.150319	−	0.988637i	$-0.548030\pi$
$709$	−8.00513	−0.300639	−0.150319	+	0.988637i	$0.548030\pi$
$710$	0	0
$711$	38.1645	1.43128
$712$	0	0
$713$	38.8362	1.45443
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.49688	−0.354667
$718$	0	0
$719$	41.2567	1.53861	0.769307	−	0.638879i	$-0.220602\pi$
$719$	41.2567	1.53861	0.769307	+	0.638879i	$0.220602\pi$
$720$	0	0
$721$	2.98649	0.111223
$722$	0	0
$723$	37.2648	1.38589
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−39.2602	−1.45608	−0.728040	−	0.685534i	$-0.759569\pi$
$727$	−39.2602	−1.45608	−0.728040	+	0.685534i	$0.759569\pi$
$728$	0	0
$729$	−13.2864	−0.492090
$730$	0	0
$731$	48.4600	1.79236
$732$	0	0
$733$	0.912055	0.0336875	0.0168438	−	0.999858i	$-0.494638\pi$
$733$	0.912055	0.0336875	0.0168438	+	0.999858i	$0.494638\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.63143	0.0969301
$738$	0	0
$739$	11.6941	0.430174	0.215087	−	0.976595i	$-0.430997\pi$
$739$	11.6941	0.430174	0.215087	+	0.976595i	$0.430997\pi$
$740$	0	0
$741$	6.01871	0.221103
$742$	0	0
$743$	−11.5631	−0.424211	−0.212105	−	0.977247i	$-0.568032\pi$
$743$	−11.5631	−0.424211	−0.212105	+	0.977247i	$0.568032\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−33.5802	−1.22863
$748$	0	0
$749$	−0.259808	−0.00949318
$750$	0	0
$751$	27.2430	0.994112	0.497056	−	0.867718i	$-0.334414\pi$
$751$	27.2430	0.994112	0.497056	+	0.867718i	$0.334414\pi$
$752$	0	0
$753$	−27.3327	−0.996059
$754$	0	0
$755$	0	0
$756$	0	0
$757$	17.1080	0.621800	0.310900	−	0.950443i	$-0.399370\pi$
$757$	17.1080	0.621800	0.310900	+	0.950443i	$0.399370\pi$
$758$	0	0
$759$	−7.35065	−0.266812
$760$	0	0
$761$	37.7173	1.36725	0.683625	−	0.729833i	$-0.260402\pi$
$761$	37.7173	1.36725	0.683625	+	0.729833i	$0.260402\pi$
$762$	0	0
$763$	−20.1142	−0.728182
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.35342	0.121085
$768$	0	0
$769$	−5.93347	−0.213966	−0.106983	−	0.994261i	$-0.534119\pi$
$769$	−5.93347	−0.213966	−0.106983	+	0.994261i	$0.534119\pi$
$770$	0	0
$771$	−43.5514	−1.56847
$772$	0	0
$773$	23.5908	0.848503	0.424252	−	0.905544i	$-0.360537\pi$
$773$	23.5908	0.848503	0.424252	+	0.905544i	$0.360537\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	38.0333	1.36444
$778$	0	0
$779$	−2.83817	−0.101688
$780$	0	0
$781$	0.0756453	0.00270680
$782$	0	0
$783$	8.96842	0.320505
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18.2307	0.649854	0.324927	−	0.945739i	$-0.394660\pi$
$787$	18.2307	0.649854	0.324927	+	0.945739i	$0.394660\pi$
$788$	0	0
$789$	15.5338	0.553017
$790$	0	0
$791$	9.35364	0.332577
$792$	0	0
$793$	−16.4864	−0.585451
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.8296	−0.383603	−0.191801	−	0.981434i	$-0.561433\pi$
$797$	−10.8296	−0.383603	−0.191801	+	0.981434i	$0.561433\pi$
$798$	0	0
$799$	−27.4171	−0.969948
$800$	0	0
$801$	26.1865	0.925256
$802$	0	0
$803$	−7.40413	−0.261286
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−58.3242	−2.05311
$808$	0	0
$809$	44.8177	1.57571	0.787854	−	0.615863i	$-0.211192\pi$
$809$	44.8177	1.57571	0.787854	+	0.615863i	$0.211192\pi$
$810$	0	0
$811$	3.06296	0.107555	0.0537775	−	0.998553i	$-0.482874\pi$
$811$	3.06296	0.107555	0.0537775	+	0.998553i	$0.482874\pi$
$812$	0	0
$813$	−21.6587	−0.759602
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.36786	0.327740
$818$	0	0
$819$	−21.8957	−0.765096
$820$	0	0
$821$	20.3210	0.709208	0.354604	−	0.935017i	$-0.384616\pi$
$821$	20.3210	0.709208	0.354604	+	0.935017i	$0.384616\pi$
$822$	0	0
$823$	50.0369	1.74418	0.872089	−	0.489348i	$-0.162765\pi$
$823$	50.0369	1.74418	0.872089	+	0.489348i	$0.162765\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.6293	1.09986	0.549929	−	0.835211i	$-0.314655\pi$
$827$	31.6293	1.09986	0.549929	+	0.835211i	$0.314655\pi$
$828$	0	0
$829$	33.1369	1.15089	0.575446	−	0.817840i	$-0.304829\pi$
$829$	33.1369	1.15089	0.575446	+	0.817840i	$0.304829\pi$
$830$	0	0
$831$	−13.7387	−0.476591
$832$	0	0
$833$	30.2274	1.04732
$834$	0	0
$835$	0	0
$836$	0	0
$837$	9.74944	0.336990
$838$	0	0
$839$	−10.3854	−0.358543	−0.179271	−	0.983800i	$-0.557374\pi$
$839$	−10.3854	−0.358543	−0.179271	+	0.983800i	$0.557374\pi$
$840$	0	0
$841$	2.02500	0.0698275
$842$	0	0
$843$	25.4918	0.877986
$844$	0	0
$845$	0	0
$846$	0	0
$847$	38.6461	1.32790
$848$	0	0
$849$	6.96235	0.238947
$850$	0	0
$851$	29.4787	1.01052
$852$	0	0
$853$	−9.31456	−0.318924	−0.159462	−	0.987204i	$-0.550976\pi$
$853$	−9.31456	−0.318924	−0.159462	+	0.987204i	$0.550976\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−34.0314	−1.16249	−0.581245	−	0.813729i	$-0.697434\pi$
$857$	−34.0314	−1.16249	−0.581245	+	0.813729i	$0.697434\pi$
$858$	0	0
$859$	−33.4243	−1.14042	−0.570211	−	0.821498i	$-0.693139\pi$
$859$	−33.4243	−1.14042	−0.570211	+	0.821498i	$0.693139\pi$
$860$	0	0
$861$	23.7888	0.810719
$862$	0	0
$863$	−16.5900	−0.564730	−0.282365	−	0.959307i	$-0.591119\pi$
$863$	−16.5900	−0.564730	−0.282365	+	0.959307i	$0.591119\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20.9128	0.710235
$868$	0	0
$869$	8.25760	0.280120
$870$	0	0
$871$	13.9974	0.474285
$872$	0	0
$873$	1.95369	0.0661224
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−32.9538	−1.11277	−0.556385	−	0.830925i	$-0.687812\pi$
$877$	−32.9538	−1.11277	−0.556385	+	0.830925i	$0.687812\pi$
$878$	0	0
$879$	14.4223	0.486452
$880$	0	0
$881$	−50.0755	−1.68709	−0.843544	−	0.537061i	$-0.819535\pi$
$881$	−50.0755	−1.68709	−0.843544	+	0.537061i	$0.819535\pi$
$882$	0	0
$883$	−12.5335	−0.421786	−0.210893	−	0.977509i	$-0.567637\pi$
$883$	−12.5335	−0.421786	−0.210893	+	0.977509i	$0.567637\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−26.9358	−0.904415	−0.452207	−	0.891913i	$-0.649363\pi$
$887$	−26.9358	−0.904415	−0.452207	+	0.891913i	$0.649363\pi$
$888$	0	0
$889$	17.7729	0.596085
$890$	0	0
$891$	−5.28101	−0.176921
$892$	0	0
$893$	−5.30004	−0.177359
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−39.1004	−1.30552
$898$	0	0
$899$	33.7268	1.12485
$900$	0	0
$901$	−1.57097	−0.0523366
$902$	0	0
$903$	−78.5188	−2.61294
$904$	0	0
$905$	0	0
$906$	0	0
$907$	28.6510	0.951342	0.475671	−	0.879623i	$-0.342205\pi$
$907$	28.6510	0.951342	0.475671	+	0.879623i	$0.342205\pi$
$908$	0	0
$909$	−30.4832	−1.01106
$910$	0	0
$911$	8.90071	0.294894	0.147447	−	0.989070i	$-0.452894\pi$
$911$	8.90071	0.294894	0.147447	+	0.989070i	$0.452894\pi$
$912$	0	0
$913$	−7.26569	−0.240459
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.71676	0.320876
$918$	0	0
$919$	−17.1901	−0.567049	−0.283525	−	0.958965i	$-0.591504\pi$
$919$	−17.1901	−0.567049	−0.283525	+	0.958965i	$0.591504\pi$
$920$	0	0
$921$	−59.5687	−1.96286
$922$	0	0
$923$	0.402381	0.0132445
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.91163	−0.0627861
$928$	0	0
$929$	34.7789	1.14106	0.570530	−	0.821277i	$-0.306738\pi$
$929$	34.7789	1.14106	0.570530	+	0.821277i	$0.306738\pi$
$930$	0	0
$931$	5.84330	0.191507
$932$	0	0
$933$	20.7272	0.678578
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.5386	−0.540292	−0.270146	−	0.962819i	$-0.587072\pi$
$937$	−16.5386	−0.540292	−0.270146	+	0.962819i	$0.587072\pi$
$938$	0	0
$939$	−76.4999	−2.49648
$940$	0	0
$941$	41.1982	1.34302	0.671512	−	0.740994i	$-0.265645\pi$
$941$	41.1982	1.34302	0.671512	+	0.740994i	$0.265645\pi$
$942$	0	0
$943$	18.4381	0.600428
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.1110	1.20594	0.602972	−	0.797762i	$-0.293983\pi$
$947$	37.1110	1.20594	0.602972	+	0.797762i	$0.293983\pi$
$948$	0	0
$949$	−39.3849	−1.27849
$950$	0	0
$951$	−59.7895	−1.93881
$952$	0	0
$953$	20.8962	0.676895	0.338447	−	0.940985i	$-0.390098\pi$
$953$	20.8962	0.676895	0.338447	+	0.940985i	$0.390098\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−6.38358	−0.206352
$958$	0	0
$959$	8.08122	0.260956
$960$	0	0
$961$	5.66391	0.182707
$962$	0	0
$963$	0.166301	0.00535897
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−8.28057	−0.266285	−0.133143	−	0.991097i	$-0.542507\pi$
$967$	−8.28057	−0.266285	−0.133143	+	0.991097i	$0.542507\pi$
$968$	0	0
$969$	11.6087	0.372927
$970$	0	0
$971$	26.4077	0.847463	0.423731	−	0.905788i	$-0.360720\pi$
$971$	26.4077	0.847463	0.423731	+	0.905788i	$0.360720\pi$
$972$	0	0
$973$	−38.8294	−1.24481
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.3280	0.810314	0.405157	−	0.914247i	$-0.367217\pi$
$977$	25.3280	0.810314	0.405157	+	0.914247i	$0.367217\pi$
$978$	0	0
$979$	5.66594	0.181084
$980$	0	0
$981$	12.8749	0.411065
$982$	0	0
$983$	5.12811	0.163561	0.0817807	−	0.996650i	$-0.473939\pi$
$983$	5.12811	0.163561	0.0817807	+	0.996650i	$0.473939\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	44.4235	1.41401
$988$	0	0
$989$	−60.8581	−1.93517
$990$	0	0
$991$	26.5396	0.843059	0.421530	−	0.906815i	$-0.361493\pi$
$991$	26.5396	0.843059	0.421530	+	0.906815i	$0.361493\pi$
$992$	0	0
$993$	−27.9233	−0.886120
$994$	0	0
$995$	0	0
$996$	0	0
$997$	44.4973	1.40924	0.704622	−	0.709583i	$-0.251117\pi$
$997$	44.4973	1.40924	0.704622	+	0.709583i	$0.251117\pi$
$998$	0	0
$999$	7.40034	0.234136

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	10000.2.a.be.1.8		8
4.3	odd	2		625.2.a.g.1.6	yes	8
5.4	even	2		10000.2.a.bn.1.1		8
12.11	even	2		5625.2.a.s.1.3		8
20.3	even	4		625.2.b.d.624.3		16
20.7	even	4		625.2.b.d.624.14		16
20.19	odd	2		625.2.a.e.1.3	✓	8
60.59	even	2		5625.2.a.be.1.6		8
100.3	even	20		625.2.e.j.249.2		32
100.11	odd	10		625.2.d.n.501.3		16
100.19	odd	10		625.2.d.q.251.3		16
100.23	even	20		625.2.e.k.124.2		32
100.27	even	20		625.2.e.k.124.7		32
100.31	odd	10		625.2.d.m.251.2		16
100.39	odd	10		625.2.d.p.501.2		16
100.47	even	20		625.2.e.j.249.7		32
100.59	odd	10		625.2.d.p.126.2		16
100.63	even	20		625.2.e.k.499.7		32
100.67	even	20		625.2.e.j.374.2		32
100.71	odd	10		625.2.d.m.376.2		16
100.79	odd	10		625.2.d.q.376.3		16
100.83	even	20		625.2.e.j.374.7		32
100.87	even	20		625.2.e.k.499.2		32
100.91	odd	10		625.2.d.n.126.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
625.2.a.e.1.3	✓	8	20.19	odd	2
625.2.a.g.1.6	yes	8	4.3	odd	2
625.2.b.d.624.3		16	20.3	even	4
625.2.b.d.624.14		16	20.7	even	4
625.2.d.m.251.2		16	100.31	odd	10
625.2.d.m.376.2		16	100.71	odd	10
625.2.d.n.126.3		16	100.91	odd	10
625.2.d.n.501.3		16	100.11	odd	10
625.2.d.p.126.2		16	100.59	odd	10
625.2.d.p.501.2		16	100.39	odd	10
625.2.d.q.251.3		16	100.19	odd	10
625.2.d.q.376.3		16	100.79	odd	10
625.2.e.j.249.2		32	100.3	even	20
625.2.e.j.249.7		32	100.47	even	20
625.2.e.j.374.2		32	100.67	even	20
625.2.e.j.374.7		32	100.83	even	20
625.2.e.k.124.2		32	100.23	even	20
625.2.e.k.124.7		32	100.27	even	20
625.2.e.k.499.2		32	100.87	even	20
625.2.e.k.499.7		32	100.63	even	20
5625.2.a.s.1.3		8	12.11	even	2
5625.2.a.be.1.6		8	60.59	even	2
10000.2.a.be.1.8		8	1.1	even	1	trivial
10000.2.a.bn.1.1		8	5.4	even	2

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:	\( N \)	\(=\)	\( 10000 = 2^{4} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	10000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30231 q^{3} -3.59425 q^{7} +2.30065 q^{9} +O(q^{10})\)	\(q+2.30231 q^{3} -3.59425 q^{7} +2.30065 q^{9} +0.497788 q^{11} +2.64789 q^{13} +5.10719 q^{17} +0.987277 q^{19} -8.27508 q^{21} -6.41382 q^{23} -1.61013 q^{27} -5.57001 q^{29} -6.05507 q^{31} +1.14606 q^{33} -4.59612 q^{37} +6.09627 q^{39} -2.87475 q^{41} +9.48858 q^{43} -5.36834 q^{47} +5.91861 q^{49} +11.7583 q^{51} -0.307600 q^{53} +2.27302 q^{57} +1.26645 q^{59} -6.22625 q^{61} -8.26910 q^{63} +5.28626 q^{67} -14.7666 q^{69} +0.151963 q^{71} -14.8741 q^{73} -1.78917 q^{77} +16.5886 q^{79} -10.6090 q^{81} -14.5960 q^{83} -12.8239 q^{87} +11.3822 q^{89} -9.51717 q^{91} -13.9407 q^{93} +0.849192 q^{97} +1.14523 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9	\( 8 q - 5 q^{3} - 10 q^{7} + 9 q^{9} - q^{11} + 10 q^{13} + 15 q^{17} + 10 q^{19} - 14 q^{21} - 30 q^{23} - 20 q^{27} + 10 q^{29} + 9 q^{31} + 5 q^{33} - 10 q^{37} - 8 q^{39} - 4 q^{41} - 30 q^{47} - 4 q^{49} + 14 q^{51} + 10 q^{53} - 10 q^{57} + 5 q^{59} + 6 q^{61} - 10 q^{67} + 3 q^{69} + 9 q^{71} + 5 q^{77} + 20 q^{79} + 8 q^{81} - 40 q^{83} - 40 q^{87} - 5 q^{89} - 6 q^{91} - 40 q^{93} + 22 q^{99}+O(q^{100}) \) 8 * q - 5 * q^3 - 10 * q^7 + 9 * q^9 - q^11 + 10 * q^13 + 15 * q^17 + 10 * q^19 - 14 * q^21 - 30 * q^23 - 20 * q^27 + 10 * q^29 + 9 * q^31 + 5 * q^33 - 10 * q^37 - 8 * q^39 - 4 * q^41 - 30 * q^47 - 4 * q^49 + 14 * q^51 + 10 * q^53 - 10 * q^57 + 5 * q^59 + 6 * q^61 - 10 * q^67 + 3 * q^69 + 9 * q^71 + 5 * q^77 + 20 * q^79 + 8 * q^81 - 40 * q^83 - 40 * q^87 - 5 * q^89 - 6 * q^91 - 40 * q^93 + 22 * q^99

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