LMFDB - Embedded newform 2000.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2000,2,Mod(1,2000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2000, 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("2000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2000 = 2^{4} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2000.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:	$15.9700804043$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1000)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.54336$ of defining polynomial
Character	$\chi$	$=$	2000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54336 q^{3} +2.87918 q^{7} +3.46869 q^{9} +O(q^{10})$	$q+2.54336 q^{3} +2.87918 q^{7} +3.46869 q^{9} -4.99442 q^{11} -0.149344 q^{13} +6.23049 q^{17} +1.90770 q^{19} +7.32279 q^{21} +4.35689 q^{23} +1.19205 q^{27} -8.93525 q^{29} +8.99442 q^{31} -12.7026 q^{33} +2.56444 q^{37} -0.379836 q^{39} +7.99787 q^{41} +4.54336 q^{43} +9.68713 q^{47} +1.28967 q^{49} +15.8464 q^{51} -1.52786 q^{53} +4.85197 q^{57} -11.4666 q^{59} -3.55541 q^{61} +9.98698 q^{63} +4.08115 q^{67} +11.0811 q^{69} +8.56444 q^{71} -15.2671 q^{73} -14.3798 q^{77} +15.0867 q^{79} -7.37426 q^{81} -8.96033 q^{83} -22.7256 q^{87} -9.26362 q^{89} -0.429988 q^{91} +22.8761 q^{93} +3.66606 q^{97} -17.3241 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 6 * q^7 + 6 * q^9	$ 4 q + 4 q^{3} + 6 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} + 8 q^{21} + 14 q^{23} + 22 q^{27} + 10 q^{29} + 16 q^{31} - 4 q^{33} + 24 q^{39} + 2 q^{41} + 12 q^{43} + 16 q^{47} + 2 q^{49} - 24 q^{53} - 24 q^{57} - 8 q^{59} + 6 q^{61} + 18 q^{63} - 16 q^{67} + 12 q^{69} + 24 q^{71} - 4 q^{73} - 32 q^{77} + 48 q^{79} + 16 q^{81} + 2 q^{83} - 22 q^{87} + 10 q^{89} + 8 q^{91} + 20 q^{93} - 4 q^{97} - 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 6 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 + 8 * q^21 + 14 * q^23 + 22 * q^27 + 10 * q^29 + 16 * q^31 - 4 * q^33 + 24 * q^39 + 2 * q^41 + 12 * q^43 + 16 * q^47 + 2 * q^49 - 24 * q^53 - 24 * q^57 - 8 * q^59 + 6 * q^61 + 18 * q^63 - 16 * q^67 + 12 * q^69 + 24 * q^71 - 4 * q^73 - 32 * q^77 + 48 * q^79 + 16 * q^81 + 2 * q^83 - 22 * q^87 + 10 * q^89 + 8 * q^91 + 20 * q^93 - 4 * q^97 - 8 * 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
$3$	2.54336	1.46841	0.734205	−	0.678927i	$-0.237555\pi$
$3$	2.54336	1.46841	0.734205	+	0.678927i	$0.237555\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.87918	1.08823	0.544114	−	0.839012i	$-0.316866\pi$
$7$	2.87918	1.08823	0.544114	+	0.839012i	$0.316866\pi$
$8$	0	0
$9$	3.46869	1.15623
$10$	0	0
$11$	−4.99442	−1.50588	−0.752938	−	0.658092i	$-0.771364\pi$
$11$	−4.99442	−1.50588	−0.752938	+	0.658092i	$0.771364\pi$
$12$	0	0
$13$	−0.149344	−0.0414206	−0.0207103	−	0.999786i	$-0.506593\pi$
$13$	−0.149344	−0.0414206	−0.0207103	+	0.999786i	$0.506593\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.23049	1.51112	0.755558	−	0.655082i	$-0.227366\pi$
$17$	6.23049	1.51112	0.755558	+	0.655082i	$0.227366\pi$
$18$	0	0
$19$	1.90770	0.437656	0.218828	−	0.975763i	$-0.429777\pi$
$19$	1.90770	0.437656	0.218828	+	0.975763i	$0.429777\pi$
$20$	0	0
$21$	7.32279	1.59796
$22$	0	0
$23$	4.35689	0.908474	0.454237	−	0.890881i	$-0.349912\pi$
$23$	4.35689	0.908474	0.454237	+	0.890881i	$0.349912\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.19205	0.229410
$28$	0	0
$29$	−8.93525	−1.65923	−0.829617	−	0.558333i	$-0.811441\pi$
$29$	−8.93525	−1.65923	−0.829617	+	0.558333i	$0.811441\pi$
$30$	0	0
$31$	8.99442	1.61545	0.807723	−	0.589562i	$-0.200700\pi$
$31$	8.99442	1.61545	0.807723	+	0.589562i	$0.200700\pi$
$32$	0	0
$33$	−12.7026	−2.21124
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.56444	0.421591	0.210795	−	0.977530i	$-0.432395\pi$
$37$	2.56444	0.421591	0.210795	+	0.977530i	$0.432395\pi$
$38$	0	0
$39$	−0.379836	−0.0608225
$40$	0	0
$41$	7.99787	1.24906	0.624529	−	0.781002i	$-0.285291\pi$
$41$	7.99787	1.24906	0.624529	+	0.781002i	$0.285291\pi$
$42$	0	0
$43$	4.54336	0.692856	0.346428	−	0.938077i	$-0.387394\pi$
$43$	4.54336	0.692856	0.346428	+	0.938077i	$0.387394\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.68713	1.41301	0.706507	−	0.707706i	$-0.250270\pi$
$47$	9.68713	1.41301	0.706507	+	0.707706i	$0.250270\pi$
$48$	0	0
$49$	1.28967	0.184238
$50$	0	0
$51$	15.8464	2.21894
$52$	0	0
$53$	−1.52786	−0.209868	−0.104934	−	0.994479i	$-0.533463\pi$
$53$	−1.52786	−0.209868	−0.104934	+	0.994479i	$0.533463\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4.85197	0.642659
$58$	0	0
$59$	−11.4666	−1.49282	−0.746409	−	0.665487i	$-0.768224\pi$
$59$	−11.4666	−1.49282	−0.746409	+	0.665487i	$0.768224\pi$
$60$	0	0
$61$	−3.55541	−0.455224	−0.227612	−	0.973752i	$-0.573092\pi$
$61$	−3.55541	−0.455224	−0.227612	+	0.973752i	$0.573092\pi$
$62$	0	0
$63$	9.98698	1.25824
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.08115	0.498592	0.249296	−	0.968427i	$-0.419801\pi$
$67$	4.08115	0.498592	0.249296	+	0.968427i	$0.419801\pi$
$68$	0	0
$69$	11.0811	1.33401
$70$	0	0
$71$	8.56444	1.01641	0.508206	−	0.861236i	$-0.330309\pi$
$71$	8.56444	1.01641	0.508206	+	0.861236i	$0.330309\pi$
$72$	0	0
$73$	−15.2671	−1.78687	−0.893437	−	0.449188i	$-0.851713\pi$
$73$	−15.2671	−1.78687	−0.893437	+	0.449188i	$0.851713\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−14.3798	−1.63873
$78$	0	0
$79$	15.0867	1.69739	0.848695	−	0.528883i	$-0.177389\pi$
$79$	15.0867	1.69739	0.848695	+	0.528883i	$0.177389\pi$
$80$	0	0
$81$	−7.37426	−0.819362
$82$	0	0
$83$	−8.96033	−0.983524	−0.491762	−	0.870730i	$-0.663647\pi$
$83$	−8.96033	−0.983524	−0.491762	+	0.870730i	$0.663647\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−22.7256	−2.43644
$88$	0	0
$89$	−9.26362	−0.981942	−0.490971	−	0.871176i	$-0.663358\pi$
$89$	−9.26362	−0.981942	−0.490971	+	0.871176i	$0.663358\pi$
$90$	0	0
$91$	−0.429988	−0.0450750
$92$	0	0
$93$	22.8761	2.37214
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.66606	0.372232	0.186116	−	0.982528i	$-0.440410\pi$
$97$	3.66606	0.372232	0.186116	+	0.982528i	$0.440410\pi$
$98$	0	0
$99$	−17.3241	−1.74114
$100$	0	0
$101$	−2.28409	−0.227275	−0.113638	−	0.993522i	$-0.536250\pi$
$101$	−2.28409	−0.227275	−0.113638	+	0.993522i	$0.536250\pi$
$102$	0	0
$103$	−14.7379	−1.45217	−0.726083	−	0.687607i	$-0.758661\pi$
$103$	−14.7379	−1.45217	−0.726083	+	0.687607i	$0.758661\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.71681	−0.262644	−0.131322	−	0.991340i	$-0.541922\pi$
$107$	−2.71681	−0.262644	−0.131322	+	0.991340i	$0.541922\pi$
$108$	0	0
$109$	0.381966	0.0365857	0.0182929	−	0.999833i	$-0.494177\pi$
$109$	0.381966	0.0365857	0.0182929	+	0.999833i	$0.494177\pi$
$110$	0	0
$111$	6.52229	0.619068
$112$	0	0
$113$	−17.8395	−1.67820	−0.839100	−	0.543978i	$-0.816918\pi$
$113$	−17.8395	−1.67820	−0.839100	+	0.543978i	$0.816918\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−0.518029	−0.0478918
$118$	0	0
$119$	17.9387	1.64444
$120$	0	0
$121$	13.9443	1.26766
$122$	0	0
$123$	20.3415	1.83413
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.77385	−0.246140	−0.123070	−	0.992398i	$-0.539274\pi$
$127$	−2.77385	−0.246140	−0.123070	+	0.992398i	$0.539274\pi$
$128$	0	0
$129$	11.5554	1.01740
$130$	0	0
$131$	3.08672	0.269688	0.134844	−	0.990867i	$-0.456947\pi$
$131$	3.08672	0.269688	0.134844	+	0.990867i	$0.456947\pi$
$132$	0	0
$133$	5.49261	0.476270
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−18.5774	−1.58717	−0.793587	−	0.608457i	$-0.791789\pi$
$137$	−18.5774	−1.58717	−0.793587	+	0.608457i	$0.791789\pi$
$138$	0	0
$139$	10.6034	0.899372	0.449686	−	0.893187i	$-0.351536\pi$
$139$	10.6034	0.899372	0.449686	+	0.893187i	$0.351536\pi$
$140$	0	0
$141$	24.6379	2.07488
$142$	0	0
$143$	0.745888	0.0623743
$144$	0	0
$145$	0	0
$146$	0	0
$147$	3.28009	0.270537
$148$	0	0
$149$	−2.91672	−0.238947	−0.119474	−	0.992837i	$-0.538121\pi$
$149$	−2.91672	−0.238947	−0.119474	+	0.992837i	$0.538121\pi$
$150$	0	0
$151$	−10.0198	−0.815403	−0.407702	−	0.913115i	$-0.633670\pi$
$151$	−10.0198	−0.815403	−0.407702	+	0.913115i	$0.633670\pi$
$152$	0	0
$153$	21.6116	1.74720
$154$	0	0
$155$	0	0
$156$	0	0
$157$	10.2305	0.816482	0.408241	−	0.912874i	$-0.366142\pi$
$157$	10.2305	0.816482	0.408241	+	0.912874i	$0.366142\pi$
$158$	0	0
$159$	−3.88591	−0.308173
$160$	0	0
$161$	12.5443	0.988626
$162$	0	0
$163$	−8.49508	−0.665386	−0.332693	−	0.943035i	$-0.607957\pi$
$163$	−8.49508	−0.665386	−0.332693	+	0.943035i	$0.607957\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.35745	−0.414572	−0.207286	−	0.978280i	$-0.566463\pi$
$167$	−5.35745	−0.414572	−0.207286	+	0.978280i	$0.566463\pi$
$168$	0	0
$169$	−12.9777	−0.998284
$170$	0	0
$171$	6.61722	0.506031
$172$	0	0
$173$	16.2193	1.23313	0.616567	−	0.787303i	$-0.288523\pi$
$173$	16.2193	1.23313	0.616567	+	0.787303i	$0.288523\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−29.1636	−2.19207
$178$	0	0
$179$	−3.65116	−0.272900	−0.136450	−	0.990647i	$-0.543569\pi$
$179$	−3.65116	−0.272900	−0.136450	+	0.990647i	$0.543569\pi$
$180$	0	0
$181$	5.33424	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$181$	5.33424	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$182$	0	0
$183$	−9.04270	−0.668456
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−31.1177	−2.27555
$188$	0	0
$189$	3.43212	0.249650
$190$	0	0
$191$	−2.09230	−0.151393	−0.0756967	−	0.997131i	$-0.524118\pi$
$191$	−2.09230	−0.151393	−0.0756967	+	0.997131i	$0.524118\pi$
$192$	0	0
$193$	17.0366	1.22632	0.613160	−	0.789959i	$-0.289898\pi$
$193$	17.0366	1.22632	0.613160	+	0.789959i	$0.289898\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.8315	−1.55543	−0.777715	−	0.628617i	$-0.783621\pi$
$197$	−21.8315	−1.55543	−0.777715	+	0.628617i	$0.783621\pi$
$198$	0	0
$199$	−5.31032	−0.376439	−0.188219	−	0.982127i	$-0.560272\pi$
$199$	−5.31032	−0.376439	−0.188219	+	0.982127i	$0.560272\pi$
$200$	0	0
$201$	10.3798	0.732137
$202$	0	0
$203$	−25.7262	−1.80562
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.1127	1.05041
$208$	0	0
$209$	−9.52786	−0.659056
$210$	0	0
$211$	−5.21196	−0.358806	−0.179403	−	0.983776i	$-0.557417\pi$
$211$	−5.21196	−0.358806	−0.179403	+	0.983776i	$0.557417\pi$
$212$	0	0
$213$	21.7825	1.49251
$214$	0	0
$215$	0	0
$216$	0	0
$217$	25.8965	1.75797
$218$	0	0
$219$	−38.8297	−2.62387
$220$	0	0
$221$	−0.930487	−0.0625914
$222$	0	0
$223$	3.72928	0.249731	0.124865	−	0.992174i	$-0.460150\pi$
$223$	3.72928	0.249731	0.124865	+	0.992174i	$0.460150\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.5725	1.23270	0.616350	−	0.787473i	$-0.288611\pi$
$227$	18.5725	1.23270	0.616350	+	0.787473i	$0.288611\pi$
$228$	0	0
$229$	25.7005	1.69834	0.849168	−	0.528122i	$-0.177104\pi$
$229$	25.7005	1.69834	0.849168	+	0.528122i	$0.177104\pi$
$230$	0	0
$231$	−36.5731	−2.40634
$232$	0	0
$233$	−21.5626	−1.41261	−0.706307	−	0.707906i	$-0.749640\pi$
$233$	−21.5626	−1.41261	−0.706307	+	0.707906i	$0.749640\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	38.3710	2.49246
$238$	0	0
$239$	4.91328	0.317813	0.158907	−	0.987294i	$-0.449203\pi$
$239$	4.91328	0.317813	0.158907	+	0.987294i	$0.449203\pi$
$240$	0	0
$241$	−27.2172	−1.75321	−0.876607	−	0.481206i	$-0.840199\pi$
$241$	−27.2172	−1.75321	−0.876607	+	0.481206i	$0.840199\pi$
$242$	0	0
$243$	−22.3316	−1.43257
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−0.284904	−0.0181280
$248$	0	0
$249$	−22.7894	−1.44422
$250$	0	0
$251$	11.1177	0.701744	0.350872	−	0.936423i	$-0.385885\pi$
$251$	11.1177	0.701744	0.350872	+	0.936423i	$0.385885\pi$
$252$	0	0
$253$	−21.7602	−1.36805
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.52786	−0.344819	−0.172409	−	0.985025i	$-0.555155\pi$
$257$	−5.52786	−0.344819	−0.172409	+	0.985025i	$0.555155\pi$
$258$	0	0
$259$	7.38347	0.458786
$260$	0	0
$261$	−30.9936	−1.91846
$262$	0	0
$263$	−12.3327	−0.760468	−0.380234	−	0.924890i	$-0.624156\pi$
$263$	−12.3327	−0.760468	−0.380234	+	0.924890i	$0.624156\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−23.5607	−1.44189
$268$	0	0
$269$	2.94427	0.179515	0.0897577	−	0.995964i	$-0.471391\pi$
$269$	2.94427	0.179515	0.0897577	+	0.995964i	$0.471391\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	0	0
$273$	−1.09362	−0.0661887
$274$	0	0
$275$	0	0
$276$	0	0
$277$	7.17476	0.431090	0.215545	−	0.976494i	$-0.430847\pi$
$277$	7.17476	0.431090	0.215545	+	0.976494i	$0.430847\pi$
$278$	0	0
$279$	31.1989	1.86783
$280$	0	0
$281$	−21.1143	−1.25957	−0.629786	−	0.776769i	$-0.716857\pi$
$281$	−21.1143	−1.25957	−0.629786	+	0.776769i	$0.716857\pi$
$282$	0	0
$283$	−1.44672	−0.0859983	−0.0429992	−	0.999075i	$-0.513691\pi$
$283$	−1.44672	−0.0859983	−0.0429992	+	0.999075i	$0.513691\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	23.0273	1.35926
$288$	0	0
$289$	21.8190	1.28347
$290$	0	0
$291$	9.32411	0.546589
$292$	0	0
$293$	11.5658	0.675678	0.337839	−	0.941204i	$-0.390304\pi$
$293$	11.5658	0.675678	0.337839	+	0.941204i	$0.390304\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.95359	−0.345462
$298$	0	0
$299$	−0.650676	−0.0376296
$300$	0	0
$301$	13.0811	0.753985
$302$	0	0
$303$	−5.80927	−0.333734
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.6574	1.63556	0.817781	−	0.575529i	$-0.195204\pi$
$307$	28.6574	1.63556	0.817781	+	0.575529i	$0.195204\pi$
$308$	0	0
$309$	−37.4838	−2.13238
$310$	0	0
$311$	−2.60658	−0.147806	−0.0739029	−	0.997265i	$-0.523545\pi$
$311$	−2.60658	−0.147806	−0.0739029	+	0.997265i	$0.523545\pi$
$312$	0	0
$313$	−0.298688	−0.0168829	−0.00844143	−	0.999964i	$-0.502687\pi$
$313$	−0.298688	−0.0168829	−0.00844143	+	0.999964i	$0.502687\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−15.2100	−0.854280	−0.427140	−	0.904186i	$-0.640479\pi$
$317$	−15.2100	−0.854280	−0.427140	+	0.904186i	$0.640479\pi$
$318$	0	0
$319$	44.6264	2.49860
$320$	0	0
$321$	−6.90983	−0.385669
$322$	0	0
$323$	11.8859	0.661350
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0.971478	0.0537228
$328$	0	0
$329$	27.8910	1.53768
$330$	0	0
$331$	−13.4586	−0.739749	−0.369875	−	0.929082i	$-0.620599\pi$
$331$	−13.4586	−0.739749	−0.369875	+	0.929082i	$0.620599\pi$
$332$	0	0
$333$	8.89523	0.487456
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.50508	0.463301	0.231651	−	0.972799i	$-0.425587\pi$
$337$	8.50508	0.463301	0.231651	+	0.972799i	$0.425587\pi$
$338$	0	0
$339$	−45.3723	−2.46429
$340$	0	0
$341$	−44.9220	−2.43266
$342$	0	0
$343$	−16.4411	−0.887734
$344$	0	0
$345$	0	0
$346$	0	0
$347$	16.6797	0.895412	0.447706	−	0.894181i	$-0.352241\pi$
$347$	16.6797	0.895412	0.447706	+	0.894181i	$0.352241\pi$
$348$	0	0
$349$	−20.1657	−1.07945	−0.539724	−	0.841842i	$-0.681471\pi$
$349$	−20.1657	−1.07945	−0.539724	+	0.841842i	$0.681471\pi$
$350$	0	0
$351$	−0.178025	−0.00950229
$352$	0	0
$353$	23.2448	1.23719	0.618597	−	0.785709i	$-0.287701\pi$
$353$	23.2448	1.23719	0.618597	+	0.785709i	$0.287701\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	45.6246	2.41471
$358$	0	0
$359$	3.35247	0.176937	0.0884683	−	0.996079i	$-0.471803\pi$
$359$	3.35247	0.176937	0.0884683	+	0.996079i	$0.471803\pi$
$360$	0	0
$361$	−15.3607	−0.808457
$362$	0	0
$363$	35.4653	1.86145
$364$	0	0
$365$	0	0
$366$	0	0
$367$	7.15982	0.373740	0.186870	−	0.982385i	$-0.440166\pi$
$367$	7.15982	0.373740	0.186870	+	0.982385i	$0.440166\pi$
$368$	0	0
$369$	27.7421	1.44420
$370$	0	0
$371$	−4.39899	−0.228384
$372$	0	0
$373$	−16.8520	−0.872562	−0.436281	−	0.899810i	$-0.643705\pi$
$373$	−16.8520	−0.872562	−0.436281	+	0.899810i	$0.643705\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.33443	0.0687265
$378$	0	0
$379$	−13.3915	−0.687874	−0.343937	−	0.938993i	$-0.611761\pi$
$379$	−13.3915	−0.687874	−0.343937	+	0.938993i	$0.611761\pi$
$380$	0	0
$381$	−7.05491	−0.361434
$382$	0	0
$383$	11.1431	0.569383	0.284692	−	0.958619i	$-0.408109\pi$
$383$	11.1431	0.569383	0.284692	+	0.958619i	$0.408109\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.7595	0.801101
$388$	0	0
$389$	−4.96130	−0.251548	−0.125774	−	0.992059i	$-0.540141\pi$
$389$	−4.96130	−0.251548	−0.125774	+	0.992059i	$0.540141\pi$
$390$	0	0
$391$	27.1456	1.37281
$392$	0	0
$393$	7.85066	0.396013
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−13.5408	−0.679594	−0.339797	−	0.940499i	$-0.610358\pi$
$397$	−13.5408	−0.679594	−0.339797	+	0.940499i	$0.610358\pi$
$398$	0	0
$399$	13.9697	0.699359
$400$	0	0
$401$	11.6515	0.581846	0.290923	−	0.956746i	$-0.406038\pi$
$401$	11.6515	0.581846	0.290923	+	0.956746i	$0.406038\pi$
$402$	0	0
$403$	−1.34326	−0.0669128
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−12.8079	−0.634863
$408$	0	0
$409$	19.2172	0.950230	0.475115	−	0.879924i	$-0.342406\pi$
$409$	19.2172	0.950230	0.475115	+	0.879924i	$0.342406\pi$
$410$	0	0
$411$	−47.2490	−2.33062
$412$	0	0
$413$	−33.0143	−1.62453
$414$	0	0
$415$	0	0
$416$	0	0
$417$	26.9684	1.32065
$418$	0	0
$419$	−8.48329	−0.414436	−0.207218	−	0.978295i	$-0.566441\pi$
$419$	−8.48329	−0.414436	−0.207218	+	0.978295i	$0.566441\pi$
$420$	0	0
$421$	11.7803	0.574138	0.287069	−	0.957910i	$-0.407319\pi$
$421$	11.7803	0.574138	0.287069	+	0.957910i	$0.407319\pi$
$422$	0	0
$423$	33.6016	1.63377
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−10.2367	−0.495387
$428$	0	0
$429$	1.89706	0.0915911
$430$	0	0
$431$	−22.8018	−1.09833	−0.549163	−	0.835716i	$-0.685053\pi$
$431$	−22.8018	−1.09833	−0.549163	+	0.835716i	$0.685053\pi$
$432$	0	0
$433$	19.9559	0.959020	0.479510	−	0.877536i	$-0.340814\pi$
$433$	19.9559	0.959020	0.479510	+	0.877536i	$0.340814\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.31164	0.397600
$438$	0	0
$439$	1.74540	0.0833036	0.0416518	−	0.999132i	$-0.486738\pi$
$439$	1.74540	0.0833036	0.0416518	+	0.999132i	$0.486738\pi$
$440$	0	0
$441$	4.47345	0.213022
$442$	0	0
$443$	0.198339	0.00942337	0.00471169	−	0.999989i	$-0.498500\pi$
$443$	0.198339	0.00942337	0.00471169	+	0.999989i	$0.498500\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−7.41828	−0.350873
$448$	0	0
$449$	−34.5751	−1.63170	−0.815849	−	0.578265i	$-0.803730\pi$
$449$	−34.5751	−1.63170	−0.815849	+	0.578265i	$0.803730\pi$
$450$	0	0
$451$	−39.9448	−1.88093
$452$	0	0
$453$	−25.4841	−1.19735
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−20.7597	−0.971097	−0.485548	−	0.874210i	$-0.661380\pi$
$457$	−20.7597	−0.971097	−0.485548	+	0.874210i	$0.661380\pi$
$458$	0	0
$459$	7.42705	0.346665
$460$	0	0
$461$	7.52016	0.350249	0.175124	−	0.984546i	$-0.443967\pi$
$461$	7.52016	0.350249	0.175124	+	0.984546i	$0.443967\pi$
$462$	0	0
$463$	8.05820	0.374496	0.187248	−	0.982313i	$-0.440043\pi$
$463$	8.05820	0.374496	0.187248	+	0.982313i	$0.440043\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.2993	1.21699	0.608494	−	0.793559i	$-0.291774\pi$
$467$	26.2993	1.21699	0.608494	+	0.793559i	$0.291774\pi$
$468$	0	0
$469$	11.7504	0.542581
$470$	0	0
$471$	26.0198	1.19893
$472$	0	0
$473$	−22.6915	−1.04336
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−5.29969	−0.242656
$478$	0	0
$479$	−37.0533	−1.69301	−0.846504	−	0.532383i	$-0.821297\pi$
$479$	−37.0533	−1.69301	−0.846504	+	0.532383i	$0.821297\pi$
$480$	0	0
$481$	−0.382983	−0.0174625
$482$	0	0
$483$	31.9046	1.45171
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−40.6512	−1.84208	−0.921042	−	0.389464i	$-0.872660\pi$
$487$	−40.6512	−1.84208	−0.921042	+	0.389464i	$0.872660\pi$
$488$	0	0
$489$	−21.6061	−0.977060
$490$	0	0
$491$	3.15066	0.142187	0.0710937	−	0.997470i	$-0.477351\pi$
$491$	3.15066	0.142187	0.0710937	+	0.997470i	$0.477351\pi$
$492$	0	0
$493$	−55.6710	−2.50730
$494$	0	0
$495$	0	0
$496$	0	0
$497$	24.6585	1.10609
$498$	0	0
$499$	−32.7267	−1.46505	−0.732525	−	0.680740i	$-0.761658\pi$
$499$	−32.7267	−1.46505	−0.732525	+	0.680740i	$0.761658\pi$
$500$	0	0
$501$	−13.6259	−0.608761
$502$	0	0
$503$	23.1097	1.03041	0.515205	−	0.857067i	$-0.327716\pi$
$503$	23.1097	1.03041	0.515205	+	0.857067i	$0.327716\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−33.0070	−1.46589
$508$	0	0
$509$	3.87476	0.171746	0.0858728	−	0.996306i	$-0.472632\pi$
$509$	3.87476	0.171746	0.0858728	+	0.996306i	$0.472632\pi$
$510$	0	0
$511$	−43.9566	−1.94453
$512$	0	0
$513$	2.27407	0.100403
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−48.3816	−2.12782
$518$	0	0
$519$	41.2517	1.81075
$520$	0	0
$521$	32.8020	1.43708	0.718541	−	0.695485i	$-0.244810\pi$
$521$	32.8020	1.43708	0.718541	+	0.695485i	$0.244810\pi$
$522$	0	0
$523$	−21.8927	−0.957300	−0.478650	−	0.878006i	$-0.658874\pi$
$523$	−21.8927	−0.957300	−0.478650	+	0.878006i	$0.658874\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	56.0397	2.44113
$528$	0	0
$529$	−4.01751	−0.174674
$530$	0	0
$531$	−39.7739	−1.72604
$532$	0	0
$533$	−1.19444	−0.0517367
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.28622	−0.400730
$538$	0	0
$539$	−6.44114	−0.277440
$540$	0	0
$541$	18.6477	0.801728	0.400864	−	0.916138i	$-0.368710\pi$
$541$	18.6477	0.801728	0.400864	+	0.916138i	$0.368710\pi$
$542$	0	0
$543$	13.5669	0.582212
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.73673	−0.117014	−0.0585070	−	0.998287i	$-0.518634\pi$
$547$	−2.73673	−0.117014	−0.0585070	+	0.998287i	$0.518634\pi$
$548$	0	0
$549$	−12.3326	−0.526344
$550$	0	0
$551$	−17.0458	−0.726175
$552$	0	0
$553$	43.4374	1.84714
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.3816	−1.54154	−0.770770	−	0.637114i	$-0.780128\pi$
$557$	−36.3816	−1.54154	−0.770770	+	0.637114i	$0.780128\pi$
$558$	0	0
$559$	−0.678524	−0.0286985
$560$	0	0
$561$	−79.1436	−3.34145
$562$	0	0
$563$	−17.7713	−0.748971	−0.374486	−	0.927233i	$-0.622181\pi$
$563$	−17.7713	−0.748971	−0.374486	+	0.927233i	$0.622181\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.2318	−0.891652
$568$	0	0
$569$	16.1644	0.677648	0.338824	−	0.940850i	$-0.389971\pi$
$569$	16.1644	0.677648	0.338824	+	0.940850i	$0.389971\pi$
$570$	0	0
$571$	3.10394	0.129896	0.0649478	−	0.997889i	$-0.479312\pi$
$571$	3.10394	0.129896	0.0649478	+	0.997889i	$0.479312\pi$
$572$	0	0
$573$	−5.32148	−0.222308
$574$	0	0
$575$	0	0
$576$	0	0
$577$	17.6122	0.733204	0.366602	−	0.930378i	$-0.380521\pi$
$577$	17.6122	0.733204	0.366602	+	0.930378i	$0.380521\pi$
$578$	0	0
$579$	43.3302	1.80074
$580$	0	0
$581$	−25.7984	−1.07030
$582$	0	0
$583$	7.63080	0.316035
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.16181	0.378148	0.189074	−	0.981963i	$-0.439451\pi$
$587$	9.16181	0.378148	0.189074	+	0.981963i	$0.439451\pi$
$588$	0	0
$589$	17.1587	0.707010
$590$	0	0
$591$	−55.5254	−2.28401
$592$	0	0
$593$	5.25096	0.215631	0.107816	−	0.994171i	$-0.465614\pi$
$593$	5.25096	0.215631	0.107816	+	0.994171i	$0.465614\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.5061	−0.552767
$598$	0	0
$599$	−2.14245	−0.0875382	−0.0437691	−	0.999042i	$-0.513937\pi$
$599$	−2.14245	−0.0875382	−0.0437691	+	0.999042i	$0.513937\pi$
$600$	0	0
$601$	−9.13849	−0.372767	−0.186383	−	0.982477i	$-0.559677\pi$
$601$	−9.13849	−0.372767	−0.186383	+	0.982477i	$0.559677\pi$
$602$	0	0
$603$	14.1562	0.576487
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−45.1034	−1.83069	−0.915346	−	0.402669i	$-0.868083\pi$
$607$	−45.1034	−1.83069	−0.915346	+	0.402669i	$0.868083\pi$
$608$	0	0
$609$	−65.4310	−2.65140
$610$	0	0
$611$	−1.44672	−0.0585279
$612$	0	0
$613$	−25.6902	−1.03762	−0.518808	−	0.854891i	$-0.673624\pi$
$613$	−25.6902	−1.03762	−0.518808	+	0.854891i	$0.673624\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−11.2974	−0.454815	−0.227408	−	0.973800i	$-0.573025\pi$
$617$	−11.2974	−0.454815	−0.227408	+	0.973800i	$0.573025\pi$
$618$	0	0
$619$	6.36868	0.255979	0.127990	−	0.991776i	$-0.459148\pi$
$619$	6.36868	0.255979	0.127990	+	0.991776i	$0.459148\pi$
$620$	0	0
$621$	5.19362	0.208413
$622$	0	0
$623$	−26.6716	−1.06858
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−24.2328	−0.967765
$628$	0	0
$629$	15.9777	0.637072
$630$	0	0
$631$	35.5148	1.41382	0.706910	−	0.707303i	$-0.250088\pi$
$631$	35.5148	1.41382	0.706910	+	0.707303i	$0.250088\pi$
$632$	0	0
$633$	−13.2559	−0.526875
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.192604	−0.00763125
$638$	0	0
$639$	29.7074	1.17521
$640$	0	0
$641$	15.5628	0.614693	0.307347	−	0.951598i	$-0.400559\pi$
$641$	15.5628	0.614693	0.307347	+	0.951598i	$0.400559\pi$
$642$	0	0
$643$	39.8252	1.57055	0.785277	−	0.619144i	$-0.212520\pi$
$643$	39.8252	1.57055	0.785277	+	0.619144i	$0.212520\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−16.0949	−0.632757	−0.316379	−	0.948633i	$-0.602467\pi$
$647$	−16.0949	−0.632757	−0.316379	+	0.948633i	$0.602467\pi$
$648$	0	0
$649$	57.2689	2.24800
$650$	0	0
$651$	65.8643	2.58143
$652$	0	0
$653$	−44.0700	−1.72459	−0.862296	−	0.506404i	$-0.830974\pi$
$653$	−44.0700	−1.72459	−0.862296	+	0.506404i	$0.830974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−52.9567	−2.06604
$658$	0	0
$659$	−23.4133	−0.912051	−0.456026	−	0.889967i	$-0.650727\pi$
$659$	−23.4133	−0.912051	−0.456026	+	0.889967i	$0.650727\pi$
$660$	0	0
$661$	−14.8679	−0.578294	−0.289147	−	0.957285i	$-0.593372\pi$
$661$	−14.8679	−0.578294	−0.289147	+	0.957285i	$0.593372\pi$
$662$	0	0
$663$	−2.36657	−0.0919098
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−38.9299	−1.50737
$668$	0	0
$669$	9.48490	0.366708
$670$	0	0
$671$	17.7572	0.685511
$672$	0	0
$673$	−22.8979	−0.882648	−0.441324	−	0.897348i	$-0.645491\pi$
$673$	−22.8979	−0.882648	−0.441324	+	0.897348i	$0.645491\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.9295	0.612220	0.306110	−	0.951996i	$-0.400972\pi$
$677$	15.9295	0.612220	0.306110	+	0.951996i	$0.400972\pi$
$678$	0	0
$679$	10.5552	0.405073
$680$	0	0
$681$	47.2366	1.81011
$682$	0	0
$683$	12.8309	0.490961	0.245480	−	0.969402i	$-0.421054\pi$
$683$	12.8309	0.490961	0.245480	+	0.969402i	$0.421054\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	65.3657	2.49386
$688$	0	0
$689$	0.228178	0.00869287
$690$	0	0
$691$	33.0756	1.25825	0.629127	−	0.777303i	$-0.283413\pi$
$691$	33.0756	1.25825	0.629127	+	0.777303i	$0.283413\pi$
$692$	0	0
$693$	−49.8792	−1.89475
$694$	0	0
$695$	0	0
$696$	0	0
$697$	49.8307	1.88747
$698$	0	0
$699$	−54.8415	−2.07430
$700$	0	0
$701$	26.3469	0.995109	0.497554	−	0.867433i	$-0.334232\pi$
$701$	26.3469	0.995109	0.497554	+	0.867433i	$0.334232\pi$
$702$	0	0
$703$	4.89217	0.184512
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.57630	−0.247327
$708$	0	0
$709$	49.1651	1.84643	0.923217	−	0.384278i	$-0.125550\pi$
$709$	49.1651	1.84643	0.923217	+	0.384278i	$0.125550\pi$
$710$	0	0
$711$	52.3312	1.96257
$712$	0	0
$713$	39.1877	1.46759
$714$	0	0
$715$	0	0
$716$	0	0
$717$	12.4962	0.466681
$718$	0	0
$719$	−20.5785	−0.767449	−0.383724	−	0.923448i	$-0.625359\pi$
$719$	−20.5785	−0.767449	−0.383724	+	0.923448i	$0.625359\pi$
$720$	0	0
$721$	−42.4330	−1.58029
$722$	0	0
$723$	−69.2232	−2.57444
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−18.9611	−0.703228	−0.351614	−	0.936145i	$-0.614367\pi$
$727$	−18.9611	−0.703228	−0.351614	+	0.936145i	$0.614367\pi$
$728$	0	0
$729$	−34.6744	−1.28424
$730$	0	0
$731$	28.3074	1.04699
$732$	0	0
$733$	40.6241	1.50049	0.750243	−	0.661162i	$-0.229937\pi$
$733$	40.6241	1.50049	0.750243	+	0.661162i	$0.229937\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−20.3830	−0.750817
$738$	0	0
$739$	36.5625	1.34497	0.672486	−	0.740109i	$-0.265226\pi$
$739$	36.5625	1.34497	0.672486	+	0.740109i	$0.265226\pi$
$740$	0	0
$741$	−0.724614	−0.0266193
$742$	0	0
$743$	−36.8890	−1.35333	−0.676664	−	0.736292i	$-0.736575\pi$
$743$	−36.8890	−1.35333	−0.676664	+	0.736292i	$0.736575\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−31.0806	−1.13718
$748$	0	0
$749$	−7.82218	−0.285816
$750$	0	0
$751$	29.0051	1.05841	0.529205	−	0.848494i	$-0.322490\pi$
$751$	29.0051	1.05841	0.529205	+	0.848494i	$0.322490\pi$
$752$	0	0
$753$	28.2764	1.03045
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.7976	−1.11936	−0.559678	−	0.828710i	$-0.689075\pi$
$757$	−30.7976	−1.11936	−0.559678	+	0.828710i	$0.689075\pi$
$758$	0	0
$759$	−55.3440	−2.00886
$760$	0	0
$761$	17.9315	0.650017	0.325008	−	0.945711i	$-0.394633\pi$
$761$	17.9315	0.650017	0.325008	+	0.945711i	$0.394633\pi$
$762$	0	0
$763$	1.09975	0.0398136
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.71246	0.0618335
$768$	0	0
$769$	0.274769	0.00990844	0.00495422	−	0.999988i	$-0.498423\pi$
$769$	0.274769	0.00990844	0.00495422	+	0.999988i	$0.498423\pi$
$770$	0	0
$771$	−14.0594	−0.506335
$772$	0	0
$773$	40.0620	1.44093	0.720465	−	0.693491i	$-0.243928\pi$
$773$	40.0620	1.44093	0.720465	+	0.693491i	$0.243928\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	18.7788	0.673687
$778$	0	0
$779$	15.2575	0.546658
$780$	0	0
$781$	−42.7744	−1.53059
$782$	0	0
$783$	−10.6512	−0.380644
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.0911	0.431000	0.215500	−	0.976504i	$-0.430862\pi$
$787$	12.0911	0.431000	0.215500	+	0.976504i	$0.430862\pi$
$788$	0	0
$789$	−31.3666	−1.11668
$790$	0	0
$791$	−51.3631	−1.82626
$792$	0	0
$793$	0.530980	0.0188557
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−37.4182	−1.32542	−0.662710	−	0.748876i	$-0.730594\pi$
$797$	−37.4182	−1.32542	−0.662710	+	0.748876i	$0.730594\pi$
$798$	0	0
$799$	60.3556	2.13523
$800$	0	0
$801$	−32.1326	−1.13535
$802$	0	0
$803$	76.2502	2.69081
$804$	0	0
$805$	0	0
$806$	0	0
$807$	7.48835	0.263602
$808$	0	0
$809$	1.80589	0.0634919	0.0317459	−	0.999496i	$-0.489893\pi$
$809$	1.80589	0.0634919	0.0317459	+	0.999496i	$0.489893\pi$
$810$	0	0
$811$	−13.6623	−0.479749	−0.239874	−	0.970804i	$-0.577106\pi$
$811$	−13.6623	−0.479749	−0.239874	+	0.970804i	$0.577106\pi$
$812$	0	0
$813$	40.6938	1.42719
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.66737	0.303233
$818$	0	0
$819$	−1.49150	−0.0521171
$820$	0	0
$821$	8.50658	0.296882	0.148441	−	0.988921i	$-0.452575\pi$
$821$	8.50658	0.296882	0.148441	+	0.988921i	$0.452575\pi$
$822$	0	0
$823$	0.897547	0.0312865	0.0156433	−	0.999878i	$-0.495020\pi$
$823$	0.897547	0.0312865	0.0156433	+	0.999878i	$0.495020\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.48640	0.329875	0.164937	−	0.986304i	$-0.447258\pi$
$827$	9.48640	0.329875	0.164937	+	0.986304i	$0.447258\pi$
$828$	0	0
$829$	36.5443	1.26923	0.634617	−	0.772827i	$-0.281158\pi$
$829$	36.5443	1.26923	0.634617	+	0.772827i	$0.281158\pi$
$830$	0	0
$831$	18.2480	0.633017
$832$	0	0
$833$	8.03526	0.278405
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10.7218	0.370599
$838$	0	0
$839$	4.24539	0.146567	0.0732836	−	0.997311i	$-0.476652\pi$
$839$	4.24539	0.146567	0.0732836	+	0.997311i	$0.476652\pi$
$840$	0	0
$841$	50.8387	1.75306
$842$	0	0
$843$	−53.7012	−1.84957
$844$	0	0
$845$	0	0
$846$	0	0
$847$	40.1480	1.37950
$848$	0	0
$849$	−3.67952	−0.126281
$850$	0	0
$851$	11.1730	0.383004
$852$	0	0
$853$	16.4472	0.563141	0.281571	−	0.959540i	$-0.409145\pi$
$853$	16.4472	0.563141	0.281571	+	0.959540i	$0.409145\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−19.5917	−0.669238	−0.334619	−	0.942353i	$-0.608608\pi$
$857$	−19.5917	−0.669238	−0.334619	+	0.942353i	$0.608608\pi$
$858$	0	0
$859$	8.49707	0.289916	0.144958	−	0.989438i	$-0.453695\pi$
$859$	8.49707	0.289916	0.144958	+	0.989438i	$0.453695\pi$
$860$	0	0
$861$	58.5667	1.99595
$862$	0	0
$863$	−27.1736	−0.925000	−0.462500	−	0.886619i	$-0.653048\pi$
$863$	−27.1736	−0.925000	−0.462500	+	0.886619i	$0.653048\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	55.4937	1.88466
$868$	0	0
$869$	−75.3495	−2.55606
$870$	0	0
$871$	−0.609496	−0.0206520
$872$	0	0
$873$	12.7164	0.430385
$874$	0	0
$875$	0	0
$876$	0	0
$877$	35.7008	1.20553	0.602765	−	0.797919i	$-0.294066\pi$
$877$	35.7008	1.20553	0.602765	+	0.797919i	$0.294066\pi$
$878$	0	0
$879$	29.4159	0.992173
$880$	0	0
$881$	−56.1477	−1.89166	−0.945832	−	0.324656i	$-0.894751\pi$
$881$	−56.1477	−1.89166	−0.945832	+	0.324656i	$0.894751\pi$
$882$	0	0
$883$	−26.2876	−0.884648	−0.442324	−	0.896855i	$-0.645846\pi$
$883$	−26.2876	−0.884648	−0.442324	+	0.896855i	$0.645846\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−6.03327	−0.202577	−0.101289	−	0.994857i	$-0.532297\pi$
$887$	−6.03327	−0.202577	−0.101289	+	0.994857i	$0.532297\pi$
$888$	0	0
$889$	−7.98642	−0.267856
$890$	0	0
$891$	36.8302	1.23386
$892$	0	0
$893$	18.4801	0.618414
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.65490	−0.0552557
$898$	0	0
$899$	−80.3674	−2.68040
$900$	0	0
$901$	−9.51934	−0.317135
$902$	0	0
$903$	33.2701	1.10716
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.46094	−0.148123	−0.0740615	−	0.997254i	$-0.523596\pi$
$907$	−4.46094	−0.148123	−0.0740615	+	0.997254i	$0.523596\pi$
$908$	0	0
$909$	−7.92280	−0.262783
$910$	0	0
$911$	−6.61407	−0.219134	−0.109567	−	0.993979i	$-0.534946\pi$
$911$	−6.61407	−0.219134	−0.109567	+	0.993979i	$0.534946\pi$
$912$	0	0
$913$	44.7517	1.48106
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.88723	0.293482
$918$	0	0
$919$	−1.77131	−0.0584301	−0.0292150	−	0.999573i	$-0.509301\pi$
$919$	−1.77131	−0.0584301	−0.0292150	+	0.999573i	$0.509301\pi$
$920$	0	0
$921$	72.8861	2.40168
$922$	0	0
$923$	−1.27905	−0.0421004
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−51.1211	−1.67904
$928$	0	0
$929$	−14.5785	−0.478306	−0.239153	−	0.970982i	$-0.576870\pi$
$929$	−14.5785	−0.478306	−0.239153	+	0.970982i	$0.576870\pi$
$930$	0	0
$931$	2.46030	0.0806330
$932$	0	0
$933$	−6.62948	−0.217040
$934$	0	0
$935$	0	0
$936$	0	0
$937$	14.6344	0.478086	0.239043	−	0.971009i	$-0.423166\pi$
$937$	14.6344	0.478086	0.239043	+	0.971009i	$0.423166\pi$
$938$	0	0
$939$	−0.759672	−0.0247910
$940$	0	0
$941$	25.7856	0.840587	0.420293	−	0.907388i	$-0.361927\pi$
$941$	25.7856	0.840587	0.420293	+	0.907388i	$0.361927\pi$
$942$	0	0
$943$	34.8458	1.13474
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.45301	0.209695	0.104847	−	0.994488i	$-0.466565\pi$
$947$	6.45301	0.209695	0.104847	+	0.994488i	$0.466565\pi$
$948$	0	0
$949$	2.28005	0.0740134
$950$	0	0
$951$	−38.6846	−1.25443
$952$	0	0
$953$	6.84133	0.221613	0.110806	−	0.993842i	$-0.464657\pi$
$953$	6.84133	0.221613	0.110806	+	0.993842i	$0.464657\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	113.501	3.66897
$958$	0	0
$959$	−53.4876	−1.72720
$960$	0	0
$961$	49.8997	1.60967
$962$	0	0
$963$	−9.42377	−0.303677
$964$	0	0
$965$	0	0
$966$	0	0
$967$	43.1953	1.38907	0.694534	−	0.719460i	$-0.255611\pi$
$967$	43.1953	1.38907	0.694534	+	0.719460i	$0.255611\pi$
$968$	0	0
$969$	30.2302	0.971133
$970$	0	0
$971$	−32.3074	−1.03679	−0.518397	−	0.855140i	$-0.673471\pi$
$971$	−32.3074	−1.03679	−0.518397	+	0.855140i	$0.673471\pi$
$972$	0	0
$973$	30.5292	0.978721
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.9354	1.43761	0.718806	−	0.695211i	$-0.244689\pi$
$977$	44.9354	1.43761	0.718806	+	0.695211i	$0.244689\pi$
$978$	0	0
$979$	46.2664	1.47868
$980$	0	0
$981$	1.32492	0.0423015
$982$	0	0
$983$	31.9585	1.01932	0.509660	−	0.860376i	$-0.329771\pi$
$983$	31.9585	1.01932	0.509660	+	0.860376i	$0.329771\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	70.9368	2.25794
$988$	0	0
$989$	19.7949	0.629442
$990$	0	0
$991$	−18.4498	−0.586078	−0.293039	−	0.956100i	$-0.594667\pi$
$991$	−18.4498	−0.586078	−0.293039	+	0.956100i	$0.594667\pi$
$992$	0	0
$993$	−34.2300	−1.08626
$994$	0	0
$995$	0	0
$996$	0	0
$997$	21.0718	0.667351	0.333676	−	0.942688i	$-0.391711\pi$
$997$	21.0718	0.667351	0.333676	+	0.942688i	$0.391711\pi$
$998$	0	0
$999$	3.05693	0.0967170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2000.2.a.r.1.3		4
4.3	odd	2		1000.2.a.e.1.2	✓	4
5.2	odd	4		2000.2.c.j.1249.2		8
5.3	odd	4		2000.2.c.j.1249.7		8
5.4	even	2		2000.2.a.m.1.2		4
8.3	odd	2		8000.2.a.br.1.3		4
8.5	even	2		8000.2.a.bb.1.2		4
12.11	even	2		9000.2.a.r.1.2		4
20.3	even	4		1000.2.c.d.249.2		8
20.7	even	4		1000.2.c.d.249.7		8
20.19	odd	2		1000.2.a.h.1.3	yes	4
40.19	odd	2		8000.2.a.ba.1.2		4
40.29	even	2		8000.2.a.bq.1.3		4
60.59	even	2		9000.2.a.ba.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1000.2.a.e.1.2	✓	4	4.3	odd	2
1000.2.a.h.1.3	yes	4	20.19	odd	2
1000.2.c.d.249.2		8	20.3	even	4
1000.2.c.d.249.7		8	20.7	even	4
2000.2.a.m.1.2		4	5.4	even	2
2000.2.a.r.1.3		4	1.1	even	1	trivial
2000.2.c.j.1249.2		8	5.2	odd	4
2000.2.c.j.1249.7		8	5.3	odd	4
8000.2.a.ba.1.2		4	40.19	odd	2
8000.2.a.bb.1.2		4	8.5	even	2
8000.2.a.bq.1.3		4	40.29	even	2
8000.2.a.br.1.3		4	8.3	odd	2
9000.2.a.r.1.2		4	12.11	even	2
9000.2.a.ba.1.3		4	60.59	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:	\( N \)	\(=\)	\( 2000 = 2^{4} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2000.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54336 q^{3} +2.87918 q^{7} +3.46869 q^{9} +O(q^{10})\)	\(q+2.54336 q^{3} +2.87918 q^{7} +3.46869 q^{9} -4.99442 q^{11} -0.149344 q^{13} +6.23049 q^{17} +1.90770 q^{19} +7.32279 q^{21} +4.35689 q^{23} +1.19205 q^{27} -8.93525 q^{29} +8.99442 q^{31} -12.7026 q^{33} +2.56444 q^{37} -0.379836 q^{39} +7.99787 q^{41} +4.54336 q^{43} +9.68713 q^{47} +1.28967 q^{49} +15.8464 q^{51} -1.52786 q^{53} +4.85197 q^{57} -11.4666 q^{59} -3.55541 q^{61} +9.98698 q^{63} +4.08115 q^{67} +11.0811 q^{69} +8.56444 q^{71} -15.2671 q^{73} -14.3798 q^{77} +15.0867 q^{79} -7.37426 q^{81} -8.96033 q^{83} -22.7256 q^{87} -9.26362 q^{89} -0.429988 q^{91} +22.8761 q^{93} +3.66606 q^{97} -17.3241 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 6 * q^7 + 6 * q^9	\( 4 q + 4 q^{3} + 6 q^{7} + 6 q^{9} - 4 q^{13} - 4 q^{17} + 8 q^{21} + 14 q^{23} + 22 q^{27} + 10 q^{29} + 16 q^{31} - 4 q^{33} + 24 q^{39} + 2 q^{41} + 12 q^{43} + 16 q^{47} + 2 q^{49} - 24 q^{53} - 24 q^{57} - 8 q^{59} + 6 q^{61} + 18 q^{63} - 16 q^{67} + 12 q^{69} + 24 q^{71} - 4 q^{73} - 32 q^{77} + 48 q^{79} + 16 q^{81} + 2 q^{83} - 22 q^{87} + 10 q^{89} + 8 q^{91} + 20 q^{93} - 4 q^{97} - 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 6 * q^7 + 6 * q^9 - 4 * q^13 - 4 * q^17 + 8 * q^21 + 14 * q^23 + 22 * q^27 + 10 * q^29 + 16 * q^31 - 4 * q^33 + 24 * q^39 + 2 * q^41 + 12 * q^43 + 16 * q^47 + 2 * q^49 - 24 * q^53 - 24 * q^57 - 8 * q^59 + 6 * q^61 + 18 * q^63 - 16 * q^67 + 12 * q^69 + 24 * q^71 - 4 * q^73 - 32 * q^77 + 48 * q^79 + 16 * q^81 + 2 * q^83 - 22 * q^87 + 10 * q^89 + 8 * q^91 + 20 * q^93 - 4 * q^97 - 8 * q^99

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