LMFDB - Embedded newform 693.4.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,4,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("693.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	693.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:	$40.8883236340$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 43x^{6} + 57x^{5} + 560x^{4} - 439x^{3} - 2246x^{2} + 384x + 1056 $ x^8 - 2x^7 - 43x^6 + 57x^5 + 560x^4 - 439x^3 - 2246x^2 + 384*x + 1056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$5.45684$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.45684 q^{2} +11.8634 q^{4} +12.1911 q^{5} +7.00000 q^{7} -17.2186 q^{8} +O(q^{10})$	\(q-4.45684 q^{2} +11.8634 q^{4} +12.1911 q^{5} +7.00000 q^{7} -17.2186 q^{8} -54.3338 q^{10} +11.0000 q^{11} -15.6475 q^{13} -31.1979 q^{14} -18.1668 q^{16} +43.3250 q^{17} +51.7758 q^{19} +144.628 q^{20} -49.0252 q^{22} +121.245 q^{23} +23.6233 q^{25} +69.7383 q^{26} +83.0439 q^{28} +187.274 q^{29} +91.9030 q^{31} +218.715 q^{32} -193.093 q^{34} +85.3378 q^{35} -226.675 q^{37} -230.756 q^{38} -209.914 q^{40} -11.2971 q^{41} -98.0582 q^{43} +130.497 q^{44} -540.371 q^{46} +186.239 q^{47} +49.0000 q^{49} -105.285 q^{50} -185.632 q^{52} +487.456 q^{53} +134.102 q^{55} -120.530 q^{56} -834.652 q^{58} -697.760 q^{59} +486.332 q^{61} -409.597 q^{62} -829.444 q^{64} -190.760 q^{65} -436.739 q^{67} +513.982 q^{68} -380.337 q^{70} -715.665 q^{71} -860.777 q^{73} +1010.25 q^{74} +614.237 q^{76} +77.0000 q^{77} +157.939 q^{79} -221.474 q^{80} +50.3492 q^{82} +397.064 q^{83} +528.180 q^{85} +437.030 q^{86} -189.404 q^{88} -237.813 q^{89} -109.532 q^{91} +1438.38 q^{92} -830.036 q^{94} +631.204 q^{95} +590.116 q^{97} -218.385 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8}+O(q^{10}) $ 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8	$ 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8} + 9 q^{10} + 88 q^{11} + 16 q^{13} + 42 q^{14} + 122 q^{16} + 90 q^{17} - 42 q^{19} + 291 q^{20} + 66 q^{22} + 338 q^{23} + 244 q^{25} + 209 q^{26} + 210 q^{28} + 496 q^{29} - 8 q^{31} + 524 q^{32} - 302 q^{34} + 70 q^{35} - 360 q^{37} + 45 q^{38} - 6 q^{40} + 242 q^{41} - 66 q^{43} + 330 q^{44} + 344 q^{46} + 540 q^{47} + 392 q^{49} + 1171 q^{50} + 465 q^{52} + 906 q^{53} + 110 q^{55} + 567 q^{56} + 977 q^{58} + 1242 q^{59} - 318 q^{61} - 110 q^{62} + 525 q^{64} + 1258 q^{65} + 522 q^{67} + 678 q^{68} + 63 q^{70} + 858 q^{71} - 78 q^{73} + 1651 q^{74} + 1775 q^{76} + 616 q^{77} + 516 q^{79} + 567 q^{80} - 1212 q^{82} + 3192 q^{83} + 720 q^{85} + 1322 q^{86} + 891 q^{88} + 2356 q^{89} + 112 q^{91} + 4504 q^{92} - 423 q^{94} + 3308 q^{95} - 1556 q^{97} + 294 q^{98}+O(q^{100}) $ 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8 + 9 * q^10 + 88 * q^11 + 16 * q^13 + 42 * q^14 + 122 * q^16 + 90 * q^17 - 42 * q^19 + 291 * q^20 + 66 * q^22 + 338 * q^23 + 244 * q^25 + 209 * q^26 + 210 * q^28 + 496 * q^29 - 8 * q^31 + 524 * q^32 - 302 * q^34 + 70 * q^35 - 360 * q^37 + 45 * q^38 - 6 * q^40 + 242 * q^41 - 66 * q^43 + 330 * q^44 + 344 * q^46 + 540 * q^47 + 392 * q^49 + 1171 * q^50 + 465 * q^52 + 906 * q^53 + 110 * q^55 + 567 * q^56 + 977 * q^58 + 1242 * q^59 - 318 * q^61 - 110 * q^62 + 525 * q^64 + 1258 * q^65 + 522 * q^67 + 678 * q^68 + 63 * q^70 + 858 * q^71 - 78 * q^73 + 1651 * q^74 + 1775 * q^76 + 616 * q^77 + 516 * q^79 + 567 * q^80 - 1212 * q^82 + 3192 * q^83 + 720 * q^85 + 1322 * q^86 + 891 * q^88 + 2356 * q^89 + 112 * q^91 + 4504 * q^92 - 423 * q^94 + 3308 * q^95 - 1556 * q^97 + 294 * q^98

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 $

$2$	−4.45684	−1.57573	−0.787865	−	0.615848i	$-0.788814\pi$
$2$	−4.45684	−1.57573	−0.787865	+	0.615848i	$0.788814\pi$
$3$	0	0
$3$	0	0
$4$	11.8634	1.48293
$5$	12.1911	1.09041	0.545203	−	0.838304i	$-0.316452\pi$
$5$	12.1911	1.09041	0.545203	+	0.838304i	$0.316452\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−17.2186	−0.760961
$9$	0	0
$10$	−54.3338	−1.71819
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−15.6475	−0.333833	−0.166917	−	0.985971i	$-0.553381\pi$
$13$	−15.6475	−0.333833	−0.166917	+	0.985971i	$0.553381\pi$
$14$	−31.1979	−0.595570
$15$	0	0
$16$	−18.1668	−0.283857
$17$	43.3250	0.618109	0.309055	−	0.951044i	$-0.399987\pi$
$17$	43.3250	0.618109	0.309055	+	0.951044i	$0.399987\pi$
$18$	0	0
$19$	51.7758	0.625167	0.312584	−	0.949890i	$-0.398806\pi$
$19$	51.7758	0.625167	0.312584	+	0.949890i	$0.398806\pi$
$20$	144.628	1.61699
$21$	0	0
$22$	−49.0252	−0.475101
$23$	121.245	1.09919	0.549595	−	0.835431i	$-0.314782\pi$
$23$	121.245	1.09919	0.549595	+	0.835431i	$0.314782\pi$
$24$	0	0
$25$	23.6233	0.188986
$26$	69.7383	0.526031
$27$	0	0
$28$	83.0439	0.560493
$29$	187.274	1.19917	0.599586	−	0.800310i	$-0.295332\pi$
$29$	187.274	1.19917	0.599586	+	0.800310i	$0.295332\pi$
$30$	0	0
$31$	91.9030	0.532460	0.266230	−	0.963910i	$-0.414222\pi$
$31$	91.9030	0.532460	0.266230	+	0.963910i	$0.414222\pi$
$32$	218.715	1.20824
$33$	0	0
$34$	−193.093	−0.973974
$35$	85.3378	0.412135
$36$	0	0
$37$	−226.675	−1.00717	−0.503583	−	0.863947i	$-0.667985\pi$
$37$	−226.675	−1.00717	−0.503583	+	0.863947i	$0.667985\pi$
$38$	−230.756	−0.985095
$39$	0	0
$40$	−209.914	−0.829757
$41$	−11.2971	−0.0430318	−0.0215159	−	0.999769i	$-0.506849\pi$
$41$	−11.2971	−0.0430318	−0.0215159	+	0.999769i	$0.506849\pi$
$42$	0	0
$43$	−98.0582	−0.347761	−0.173881	−	0.984767i	$-0.555631\pi$
$43$	−98.0582	−0.347761	−0.173881	+	0.984767i	$0.555631\pi$
$44$	130.497	0.447119
$45$	0	0
$46$	−540.371	−1.73203
$47$	186.239	0.577994	0.288997	−	0.957330i	$-0.406678\pi$
$47$	186.239	0.577994	0.288997	+	0.957330i	$0.406678\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−105.285	−0.297791
$51$	0	0
$52$	−185.632	−0.495050
$53$	487.456	1.26334	0.631671	−	0.775236i	$-0.282369\pi$
$53$	487.456	1.26334	0.631671	+	0.775236i	$0.282369\pi$
$54$	0	0
$55$	134.102	0.328770
$56$	−120.530	−0.287616
$57$	0	0
$58$	−834.652	−1.88957
$59$	−697.760	−1.53967	−0.769835	−	0.638243i	$-0.779662\pi$
$59$	−697.760	−1.53967	−0.769835	+	0.638243i	$0.779662\pi$
$60$	0	0
$61$	486.332	1.02079	0.510397	−	0.859939i	$-0.329498\pi$
$61$	486.332	1.02079	0.510397	+	0.859939i	$0.329498\pi$
$62$	−409.597	−0.839013
$63$	0	0
$64$	−829.444	−1.62001
$65$	−190.760	−0.364014
$66$	0	0
$67$	−436.739	−0.796361	−0.398181	−	0.917307i	$-0.630358\pi$
$67$	−436.739	−0.796361	−0.398181	+	0.917307i	$0.630358\pi$
$68$	513.982	0.916610
$69$	0	0
$70$	−380.337	−0.649413
$71$	−715.665	−1.19625	−0.598125	−	0.801403i	$-0.704087\pi$
$71$	−715.665	−1.19625	−0.598125	+	0.801403i	$0.704087\pi$
$72$	0	0
$73$	−860.777	−1.38009	−0.690043	−	0.723768i	$-0.742408\pi$
$73$	−860.777	−1.38009	−0.690043	+	0.723768i	$0.742408\pi$
$74$	1010.25	1.58702
$75$	0	0
$76$	614.237	0.927076
$77$	77.0000	0.113961
$78$	0	0
$79$	157.939	0.224930	0.112465	−	0.993656i	$-0.464125\pi$
$79$	157.939	0.224930	0.112465	+	0.993656i	$0.464125\pi$
$80$	−221.474	−0.309519
$81$	0	0
$82$	50.3492	0.0678066
$83$	397.064	0.525102	0.262551	−	0.964918i	$-0.415436\pi$
$83$	397.064	0.525102	0.262551	+	0.964918i	$0.415436\pi$
$84$	0	0
$85$	528.180	0.673990
$86$	437.030	0.547978
$87$	0	0
$88$	−189.404	−0.229438
$89$	−237.813	−0.283238	−0.141619	−	0.989921i	$-0.545231\pi$
$89$	−237.813	−0.283238	−0.141619	+	0.989921i	$0.545231\pi$
$90$	0	0
$91$	−109.532	−0.126177
$92$	1438.38	1.63002
$93$	0	0
$94$	−830.036	−0.910763
$95$	631.204	0.681686
$96$	0	0
$97$	590.116	0.617703	0.308851	−	0.951110i	$-0.400055\pi$
$97$	590.116	0.617703	0.308851	+	0.951110i	$0.400055\pi$
$98$	−218.385	−0.225104
$99$	0	0
$100$	280.252	0.280252
$101$	858.491	0.845773	0.422886	−	0.906183i	$-0.361017\pi$
$101$	858.491	0.845773	0.422886	+	0.906183i	$0.361017\pi$
$102$	0	0
$103$	18.6313	0.0178233	0.00891164	−	0.999960i	$-0.497163\pi$
$103$	18.6313	0.0178233	0.00891164	+	0.999960i	$0.497163\pi$
$104$	269.427	0.254034
$105$	0	0
$106$	−2172.51	−1.99069
$107$	−202.961	−0.183373	−0.0916866	−	0.995788i	$-0.529226\pi$
$107$	−202.961	−0.183373	−0.0916866	+	0.995788i	$0.529226\pi$
$108$	0	0
$109$	1023.62	0.899495	0.449748	−	0.893156i	$-0.351514\pi$
$109$	1023.62	0.899495	0.449748	+	0.893156i	$0.351514\pi$
$110$	−597.672	−0.518053
$111$	0	0
$112$	−127.168	−0.107288
$113$	1011.17	0.841792	0.420896	−	0.907109i	$-0.361716\pi$
$113$	1011.17	0.841792	0.420896	+	0.907109i	$0.361716\pi$
$114$	0	0
$115$	1478.11	1.19856
$116$	2221.71	1.77828
$117$	0	0
$118$	3109.80	2.42611
$119$	303.275	0.233623
$120$	0	0
$121$	121.000	0.0909091
$122$	−2167.50	−1.60850
$123$	0	0
$124$	1090.28	0.789599
$125$	−1235.90	−0.884335
$126$	0	0
$127$	2232.29	1.55971	0.779855	−	0.625960i	$-0.215293\pi$
$127$	2232.29	1.55971	0.779855	+	0.625960i	$0.215293\pi$
$128$	1946.98	1.34445
$129$	0	0
$130$	850.188	0.573588
$131$	−553.340	−0.369050	−0.184525	−	0.982828i	$-0.559075\pi$
$131$	−553.340	−0.369050	−0.184525	+	0.982828i	$0.559075\pi$
$132$	0	0
$133$	362.430	0.236291
$134$	1946.48	1.25485
$135$	0	0
$136$	−745.995	−0.470357
$137$	778.408	0.485430	0.242715	−	0.970098i	$-0.421962\pi$
$137$	778.408	0.485430	0.242715	+	0.970098i	$0.421962\pi$
$138$	0	0
$139$	890.735	0.543533	0.271767	−	0.962363i	$-0.412392\pi$
$139$	890.735	0.543533	0.271767	+	0.962363i	$0.412392\pi$
$140$	1012.40	0.611166
$141$	0	0
$142$	3189.60	1.88497
$143$	−172.122	−0.100654
$144$	0	0
$145$	2283.08	1.30759
$146$	3836.34	2.17464
$147$	0	0
$148$	−2689.14	−1.49355
$149$	−1810.56	−0.995481	−0.497740	−	0.867326i	$-0.665837\pi$
$149$	−1810.56	−0.995481	−0.497740	+	0.867326i	$0.665837\pi$
$150$	0	0
$151$	3043.06	1.64001	0.820003	−	0.572359i	$-0.193972\pi$
$151$	3043.06	1.64001	0.820003	+	0.572359i	$0.193972\pi$
$152$	−891.505	−0.475728
$153$	0	0
$154$	−343.177	−0.179571
$155$	1120.40	0.580598
$156$	0	0
$157$	−1299.44	−0.660551	−0.330276	−	0.943885i	$-0.607142\pi$
$157$	−1299.44	−0.660551	−0.330276	+	0.943885i	$0.607142\pi$
$158$	−703.908	−0.354430
$159$	0	0
$160$	2666.38	1.31748
$161$	848.717	0.415455
$162$	0	0
$163$	−1550.80	−0.745202	−0.372601	−	0.927992i	$-0.621534\pi$
$163$	−1550.80	−0.745202	−0.372601	+	0.927992i	$0.621534\pi$
$164$	−134.022	−0.0638130
$165$	0	0
$166$	−1769.65	−0.827420
$167$	1346.23	0.623799	0.311900	−	0.950115i	$-0.399035\pi$
$167$	1346.23	0.623799	0.311900	+	0.950115i	$0.399035\pi$
$168$	0	0
$169$	−1952.16	−0.888555
$170$	−2354.01	−1.06203
$171$	0	0
$172$	−1163.30	−0.515704
$173$	3058.10	1.34395	0.671975	−	0.740574i	$-0.265446\pi$
$173$	3058.10	1.34395	0.671975	+	0.740574i	$0.265446\pi$
$174$	0	0
$175$	165.363	0.0714301
$176$	−199.835	−0.0855860
$177$	0	0
$178$	1059.90	0.446307
$179$	1139.10	0.475645	0.237822	−	0.971309i	$-0.423566\pi$
$179$	1139.10	0.475645	0.237822	+	0.971309i	$0.423566\pi$
$180$	0	0
$181$	3156.25	1.29615	0.648073	−	0.761578i	$-0.275575\pi$
$181$	3156.25	1.29615	0.648073	+	0.761578i	$0.275575\pi$
$182$	488.168	0.198821
$183$	0	0
$184$	−2087.67	−0.836441
$185$	−2763.42	−1.09822
$186$	0	0
$187$	476.575	0.186367
$188$	2209.43	0.857122
$189$	0	0
$190$	−2813.18	−1.07415
$191$	−2195.74	−0.831822	−0.415911	−	0.909405i	$-0.636537\pi$
$191$	−2195.74	−0.831822	−0.415911	+	0.909405i	$0.636537\pi$
$192$	0	0
$193$	−1224.61	−0.456733	−0.228366	−	0.973575i	$-0.573338\pi$
$193$	−1224.61	−0.456733	−0.228366	+	0.973575i	$0.573338\pi$
$194$	−2630.05	−0.973333
$195$	0	0
$196$	581.307	0.211847
$197$	4502.49	1.62837	0.814186	−	0.580605i	$-0.197184\pi$
$197$	4502.49	1.62837	0.814186	+	0.580605i	$0.197184\pi$
$198$	0	0
$199$	5332.05	1.89939	0.949696	−	0.313174i	$-0.101392\pi$
$199$	5332.05	1.89939	0.949696	+	0.313174i	$0.101392\pi$
$200$	−406.759	−0.143811
$201$	0	0
$202$	−3826.15	−1.33271
$203$	1310.92	0.453245
$204$	0	0
$205$	−137.724	−0.0469222
$206$	−83.0367	−0.0280847
$207$	0	0
$208$	284.265	0.0947607
$209$	569.533	0.188495
$210$	0	0
$211$	−1925.26	−0.628152	−0.314076	−	0.949398i	$-0.601695\pi$
$211$	−1925.26	−0.628152	−0.314076	+	0.949398i	$0.601695\pi$
$212$	5782.88	1.87344
$213$	0	0
$214$	904.562	0.288947
$215$	−1195.44	−0.379201
$216$	0	0
$217$	643.321	0.201251
$218$	−4562.11	−1.41736
$219$	0	0
$220$	1590.91	0.487541
$221$	−677.927	−0.206345
$222$	0	0
$223$	1794.01	0.538724	0.269362	−	0.963039i	$-0.413187\pi$
$223$	1794.01	0.538724	0.269362	+	0.963039i	$0.413187\pi$
$224$	1531.01	0.456673
$225$	0	0
$226$	−4506.60	−1.32644
$227$	2809.22	0.821385	0.410692	−	0.911774i	$-0.365287\pi$
$227$	2809.22	0.821385	0.410692	+	0.911774i	$0.365287\pi$
$228$	0	0
$229$	−2445.83	−0.705785	−0.352893	−	0.935664i	$-0.614802\pi$
$229$	−2445.83	−0.705785	−0.352893	+	0.935664i	$0.614802\pi$
$230$	−6587.72	−1.88861
$231$	0	0
$232$	−3224.60	−0.912524
$233$	−3143.08	−0.883735	−0.441868	−	0.897080i	$-0.645684\pi$
$233$	−3143.08	−0.883735	−0.441868	+	0.897080i	$0.645684\pi$
$234$	0	0
$235$	2270.46	0.630248
$236$	−8277.81	−2.28322
$237$	0	0
$238$	−1351.65	−0.368127
$239$	−1011.71	−0.273817	−0.136909	−	0.990584i	$-0.543717\pi$
$239$	−1011.71	−0.273817	−0.136909	+	0.990584i	$0.543717\pi$
$240$	0	0
$241$	−2449.72	−0.654773	−0.327387	−	0.944890i	$-0.606168\pi$
$241$	−2449.72	−0.654773	−0.327387	+	0.944890i	$0.606168\pi$
$242$	−539.277	−0.143248
$243$	0	0
$244$	5769.56	1.51376
$245$	597.365	0.155772
$246$	0	0
$247$	−810.160	−0.208701
$248$	−1582.44	−0.405181
$249$	0	0
$250$	5508.19	1.39347
$251$	2605.03	0.655092	0.327546	−	0.944835i	$-0.393778\pi$
$251$	2605.03	0.655092	0.327546	+	0.944835i	$0.393778\pi$
$252$	0	0
$253$	1333.70	0.331418
$254$	−9948.93	−2.45768
$255$	0	0
$256$	−2041.80	−0.498487
$257$	−2323.64	−0.563987	−0.281994	−	0.959416i	$-0.590996\pi$
$257$	−2323.64	−0.563987	−0.281994	+	0.959416i	$0.590996\pi$
$258$	0	0
$259$	−1586.73	−0.380673
$260$	−2263.07	−0.539805
$261$	0	0
$262$	2466.15	0.581523
$263$	−2382.09	−0.558502	−0.279251	−	0.960218i	$-0.590086\pi$
$263$	−2382.09	−0.558502	−0.279251	+	0.960218i	$0.590086\pi$
$264$	0	0
$265$	5942.63	1.37756
$266$	−1615.29	−0.372331
$267$	0	0
$268$	−5181.22	−1.18095
$269$	−4768.03	−1.08071	−0.540357	−	0.841436i	$-0.681711\pi$
$269$	−4768.03	−1.08071	−0.540357	+	0.841436i	$0.681711\pi$
$270$	0	0
$271$	−6275.87	−1.40676	−0.703380	−	0.710814i	$-0.748327\pi$
$271$	−6275.87	−1.40676	−0.703380	+	0.710814i	$0.748327\pi$
$272$	−787.078	−0.175454
$273$	0	0
$274$	−3469.24	−0.764906
$275$	259.856	0.0569815
$276$	0	0
$277$	1833.05	0.397607	0.198803	−	0.980039i	$-0.436295\pi$
$277$	1833.05	0.397607	0.198803	+	0.980039i	$0.436295\pi$
$278$	−3969.86	−0.856462
$279$	0	0
$280$	−1469.40	−0.313619
$281$	7948.23	1.68737	0.843686	−	0.536837i	$-0.180381\pi$
$281$	7948.23	1.68737	0.843686	+	0.536837i	$0.180381\pi$
$282$	0	0
$283$	3141.50	0.659869	0.329935	−	0.944004i	$-0.392973\pi$
$283$	3141.50	0.659869	0.329935	+	0.944004i	$0.392973\pi$
$284$	−8490.22	−1.77395
$285$	0	0
$286$	767.121	0.158604
$287$	−79.0795	−0.0162645
$288$	0	0
$289$	−3035.94	−0.617941
$290$	−10175.3	−2.06040
$291$	0	0
$292$	−10211.7	−2.04657
$293$	3083.66	0.614845	0.307422	−	0.951573i	$-0.400534\pi$
$293$	3083.66	0.614845	0.307422	+	0.951573i	$0.400534\pi$
$294$	0	0
$295$	−8506.47	−1.67887
$296$	3903.02	0.766414
$297$	0	0
$298$	8069.36	1.56861
$299$	−1897.18	−0.366946
$300$	0	0
$301$	−686.408	−0.131441
$302$	−13562.4	−2.58421
$303$	0	0
$304$	−940.601	−0.177458
$305$	5928.93	1.11308
$306$	0	0
$307$	8216.45	1.52748	0.763742	−	0.645521i	$-0.223360\pi$
$307$	8216.45	1.52748	0.763742	+	0.645521i	$0.223360\pi$
$308$	913.482	0.168995
$309$	0	0
$310$	−4993.44	−0.914866
$311$	3241.70	0.591062	0.295531	−	0.955333i	$-0.404503\pi$
$311$	3241.70	0.591062	0.295531	+	0.955333i	$0.404503\pi$
$312$	0	0
$313$	−10645.0	−1.92233	−0.961167	−	0.275969i	$-0.911001\pi$
$313$	−10645.0	−1.92233	−0.961167	+	0.275969i	$0.911001\pi$
$314$	5791.39	1.04085
$315$	0	0
$316$	1873.69	0.333555
$317$	741.966	0.131460	0.0657302	−	0.997837i	$-0.479062\pi$
$317$	741.966	0.131460	0.0657302	+	0.997837i	$0.479062\pi$
$318$	0	0
$319$	2060.02	0.361564
$320$	−10111.8	−1.76647
$321$	0	0
$322$	−3782.59	−0.654645
$323$	2243.19	0.386422
$324$	0	0
$325$	−369.645	−0.0630898
$326$	6911.66	1.17424
$327$	0	0
$328$	194.519	0.0327455
$329$	1303.67	0.218461
$330$	0	0
$331$	3842.77	0.638120	0.319060	−	0.947735i	$-0.396633\pi$
$331$	3842.77	0.638120	0.319060	+	0.947735i	$0.396633\pi$
$332$	4710.54	0.778688
$333$	0	0
$334$	−5999.93	−0.982939
$335$	−5324.34	−0.868358
$336$	0	0
$337$	−1251.97	−0.202371	−0.101186	−	0.994868i	$-0.532264\pi$
$337$	−1251.97	−0.202371	−0.101186	+	0.994868i	$0.532264\pi$
$338$	8700.44	1.40012
$339$	0	0
$340$	6266.02	0.999478
$341$	1010.93	0.160543
$342$	0	0
$343$	343.000	0.0539949
$344$	1688.42	0.264633
$345$	0	0
$346$	−13629.5	−2.11770
$347$	12167.3	1.88234	0.941172	−	0.337927i	$-0.109726\pi$
$347$	12167.3	1.88234	0.941172	+	0.337927i	$0.109726\pi$
$348$	0	0
$349$	−2862.41	−0.439029	−0.219514	−	0.975609i	$-0.570447\pi$
$349$	−2862.41	−0.439029	−0.219514	+	0.975609i	$0.570447\pi$
$350$	−736.996	−0.112555
$351$	0	0
$352$	2405.87	0.364299
$353$	8633.06	1.30167	0.650837	−	0.759217i	$-0.274418\pi$
$353$	8633.06	1.30167	0.650837	+	0.759217i	$0.274418\pi$
$354$	0	0
$355$	−8724.75	−1.30440
$356$	−2821.28	−0.420021
$357$	0	0
$358$	−5076.79	−0.749488
$359$	−1013.23	−0.148958	−0.0744791	−	0.997223i	$-0.523729\pi$
$359$	−1013.23	−0.148958	−0.0744791	+	0.997223i	$0.523729\pi$
$360$	0	0
$361$	−4178.27	−0.609166
$362$	−14066.9	−2.04238
$363$	0	0
$364$	−1299.43	−0.187111
$365$	−10493.8	−1.50486
$366$	0	0
$367$	5282.35	0.751325	0.375662	−	0.926757i	$-0.377415\pi$
$367$	5282.35	0.751325	0.375662	+	0.926757i	$0.377415\pi$
$368$	−2202.64	−0.312013
$369$	0	0
$370$	12316.1	1.73050
$371$	3412.19	0.477499
$372$	0	0
$373$	−8770.95	−1.21754	−0.608770	−	0.793347i	$-0.708337\pi$
$373$	−8770.95	−1.21754	−0.608770	+	0.793347i	$0.708337\pi$
$374$	−2124.02	−0.293664
$375$	0	0
$376$	−3206.77	−0.439831
$377$	−2930.37	−0.400323
$378$	0	0
$379$	8431.99	1.14280	0.571401	−	0.820671i	$-0.306400\pi$
$379$	8431.99	1.14280	0.571401	+	0.820671i	$0.306400\pi$
$380$	7488.23	1.01089
$381$	0	0
$382$	9786.04	1.31073
$383$	11598.9	1.54745	0.773727	−	0.633519i	$-0.218390\pi$
$383$	11598.9	1.54745	0.773727	+	0.633519i	$0.218390\pi$
$384$	0	0
$385$	938.716	0.124263
$386$	5457.89	0.719688
$387$	0	0
$388$	7000.79	0.916008
$389$	1396.44	0.182011	0.0910053	−	0.995850i	$-0.470992\pi$
$389$	1396.44	0.182011	0.0910053	+	0.995850i	$0.470992\pi$
$390$	0	0
$391$	5252.95	0.679420
$392$	−843.711	−0.108709
$393$	0	0
$394$	−20066.9	−2.56587
$395$	1925.45	0.245266
$396$	0	0
$397$	5773.49	0.729882	0.364941	−	0.931031i	$-0.381089\pi$
$397$	5773.49	0.729882	0.364941	+	0.931031i	$0.381089\pi$
$398$	−23764.1	−2.99293
$399$	0	0
$400$	−429.160	−0.0536450
$401$	−505.794	−0.0629880	−0.0314940	−	0.999504i	$-0.510027\pi$
$401$	−505.794	−0.0629880	−0.0314940	+	0.999504i	$0.510027\pi$
$402$	0	0
$403$	−1438.05	−0.177753
$404$	10184.6	1.25422
$405$	0	0
$406$	−5842.57	−0.714191
$407$	−2493.43	−0.303672
$408$	0	0
$409$	1865.46	0.225528	0.112764	−	0.993622i	$-0.464030\pi$
$409$	1865.46	0.225528	0.112764	+	0.993622i	$0.464030\pi$
$410$	613.813	0.0739367
$411$	0	0
$412$	221.031	0.0264306
$413$	−4884.32	−0.581941
$414$	0	0
$415$	4840.66	0.572575
$416$	−3422.34	−0.403351
$417$	0	0
$418$	−2538.32	−0.297017
$419$	−7706.74	−0.898566	−0.449283	−	0.893390i	$-0.648320\pi$
$419$	−7706.74	−0.898566	−0.449283	+	0.893390i	$0.648320\pi$
$420$	0	0
$421$	−4534.79	−0.524969	−0.262485	−	0.964936i	$-0.584542\pi$
$421$	−4534.79	−0.524969	−0.262485	+	0.964936i	$0.584542\pi$
$422$	8580.56	0.989798
$423$	0	0
$424$	−8393.29	−0.961355
$425$	1023.48	0.116814
$426$	0	0
$427$	3404.33	0.385824
$428$	−2407.80	−0.271929
$429$	0	0
$430$	5327.88	0.597519
$431$	700.548	0.0782928	0.0391464	−	0.999233i	$-0.487536\pi$
$431$	700.548	0.0782928	0.0391464	+	0.999233i	$0.487536\pi$
$432$	0	0
$433$	13305.9	1.47676	0.738382	−	0.674382i	$-0.235590\pi$
$433$	13305.9	1.47676	0.738382	+	0.674382i	$0.235590\pi$
$434$	−2867.18	−0.317117
$435$	0	0
$436$	12143.6	1.33389
$437$	6277.57	0.687178
$438$	0	0
$439$	992.131	0.107863	0.0539315	−	0.998545i	$-0.482825\pi$
$439$	992.131	0.107863	0.0539315	+	0.998545i	$0.482825\pi$
$440$	−2309.05	−0.250181
$441$	0	0
$442$	3021.41	0.325145
$443$	7180.05	0.770055	0.385027	−	0.922905i	$-0.374192\pi$
$443$	7180.05	0.770055	0.385027	+	0.922905i	$0.374192\pi$
$444$	0	0
$445$	−2899.21	−0.308844
$446$	−7995.59	−0.848884
$447$	0	0
$448$	−5806.11	−0.612305
$449$	−17736.0	−1.86417	−0.932087	−	0.362236i	$-0.882014\pi$
$449$	−17736.0	−1.86417	−0.932087	+	0.362236i	$0.882014\pi$
$450$	0	0
$451$	−124.268	−0.0129746
$452$	11995.9	1.24832
$453$	0	0
$454$	−12520.2	−1.29428
$455$	−1335.32	−0.137584
$456$	0	0
$457$	−5926.25	−0.606604	−0.303302	−	0.952894i	$-0.598089\pi$
$457$	−5926.25	−0.606604	−0.303302	+	0.952894i	$0.598089\pi$
$458$	10900.7	1.11213
$459$	0	0
$460$	17535.5	1.77738
$461$	−11831.2	−1.19531	−0.597653	−	0.801755i	$-0.703900\pi$
$461$	−11831.2	−1.19531	−0.597653	+	0.801755i	$0.703900\pi$
$462$	0	0
$463$	2756.08	0.276643	0.138322	−	0.990387i	$-0.455829\pi$
$463$	2756.08	0.276643	0.138322	+	0.990387i	$0.455829\pi$
$464$	−3402.18	−0.340393
$465$	0	0
$466$	14008.2	1.39253
$467$	−7795.11	−0.772408	−0.386204	−	0.922413i	$-0.626214\pi$
$467$	−7795.11	−0.772408	−0.386204	+	0.922413i	$0.626214\pi$
$468$	0	0
$469$	−3057.18	−0.300996
$470$	−10119.1	−0.993101
$471$	0	0
$472$	12014.4	1.17163
$473$	−1078.64	−0.104854
$474$	0	0
$475$	1223.11	0.118148
$476$	3597.88	0.346446
$477$	0	0
$478$	4509.04	0.431462
$479$	13277.4	1.26651	0.633257	−	0.773942i	$-0.281718\pi$
$479$	13277.4	1.26651	0.633257	+	0.773942i	$0.281718\pi$
$480$	0	0
$481$	3546.89	0.336226
$482$	10918.0	1.03175
$483$	0	0
$484$	1435.47	0.134811
$485$	7194.17	0.673547
$486$	0	0
$487$	10601.3	0.986429	0.493214	−	0.869908i	$-0.335822\pi$
$487$	10601.3	0.986429	0.493214	+	0.869908i	$0.335822\pi$
$488$	−8373.95	−0.776785
$489$	0	0
$490$	−2662.36	−0.245455
$491$	−6095.15	−0.560225	−0.280112	−	0.959967i	$-0.590372\pi$
$491$	−6095.15	−0.560225	−0.280112	+	0.959967i	$0.590372\pi$
$492$	0	0
$493$	8113.67	0.741220
$494$	3610.75	0.328857
$495$	0	0
$496$	−1669.58	−0.151142
$497$	−5009.65	−0.452140
$498$	0	0
$499$	−18897.8	−1.69535	−0.847676	−	0.530515i	$-0.821999\pi$
$499$	−18897.8	−1.69535	−0.847676	+	0.530515i	$0.821999\pi$
$500$	−14661.9	−1.31140
$501$	0	0
$502$	−11610.2	−1.03225
$503$	12952.6	1.14817	0.574085	−	0.818796i	$-0.305358\pi$
$503$	12952.6	1.14817	0.574085	+	0.818796i	$0.305358\pi$
$504$	0	0
$505$	10466.0	0.922236
$506$	−5944.08	−0.522226
$507$	0	0
$508$	26482.5	2.31294
$509$	13187.2	1.14835	0.574176	−	0.818732i	$-0.305323\pi$
$509$	13187.2	1.14835	0.574176	+	0.818732i	$0.305323\pi$
$510$	0	0
$511$	−6025.44	−0.521624
$512$	−6475.82	−0.558971
$513$	0	0
$514$	10356.1	0.888691
$515$	227.136	0.0194346
$516$	0	0
$517$	2048.63	0.174272
$518$	7071.78	0.599838
$519$	0	0
$520$	3284.62	0.277000
$521$	2254.72	0.189599	0.0947995	−	0.995496i	$-0.469779\pi$
$521$	2254.72	0.189599	0.0947995	+	0.995496i	$0.469779\pi$
$522$	0	0
$523$	10786.8	0.901859	0.450929	−	0.892560i	$-0.351093\pi$
$523$	10786.8	0.901859	0.450929	+	0.892560i	$0.351093\pi$
$524$	−6564.50	−0.547274
$525$	0	0
$526$	10616.6	0.880049
$527$	3981.70	0.329119
$528$	0	0
$529$	2533.42	0.208220
$530$	−26485.3	−2.17066
$531$	0	0
$532$	4299.66	0.350402
$533$	176.771	0.0143655
$534$	0	0
$535$	−2474.32	−0.199951
$536$	7520.03	0.606000
$537$	0	0
$538$	21250.3	1.70291
$539$	539.000	0.0430730
$540$	0	0
$541$	−19113.2	−1.51893	−0.759464	−	0.650549i	$-0.774539\pi$
$541$	−19113.2	−1.51893	−0.759464	+	0.650549i	$0.774539\pi$
$542$	27970.5	2.21668
$543$	0	0
$544$	9475.84	0.746826
$545$	12479.1	0.980816
$546$	0	0
$547$	2707.11	0.211604	0.105802	−	0.994387i	$-0.466259\pi$
$547$	2707.11	0.211604	0.105802	+	0.994387i	$0.466259\pi$
$548$	9234.57	0.719856
$549$	0	0
$550$	−1158.14	−0.0897874
$551$	9696.28	0.749683
$552$	0	0
$553$	1105.57	0.0850157
$554$	−8169.59	−0.626521
$555$	0	0
$556$	10567.1	0.806019
$557$	−1527.87	−0.116226	−0.0581131	−	0.998310i	$-0.518508\pi$
$557$	−1527.87	−0.116226	−0.0581131	+	0.998310i	$0.518508\pi$
$558$	0	0
$559$	1534.36	0.116094
$560$	−1550.32	−0.116987
$561$	0	0
$562$	−35424.0	−2.65884
$563$	5678.01	0.425043	0.212522	−	0.977156i	$-0.431832\pi$
$563$	5678.01	0.425043	0.212522	+	0.977156i	$0.431832\pi$
$564$	0	0
$565$	12327.2	0.917896
$566$	−14001.2	−1.03978
$567$	0	0
$568$	12322.7	0.910300
$569$	19704.3	1.45175	0.725877	−	0.687824i	$-0.241434\pi$
$569$	19704.3	1.45175	0.725877	+	0.687824i	$0.241434\pi$
$570$	0	0
$571$	−19744.0	−1.44704	−0.723519	−	0.690304i	$-0.757477\pi$
$571$	−19744.0	−1.44704	−0.723519	+	0.690304i	$0.757477\pi$
$572$	−2041.96	−0.149263
$573$	0	0
$574$	352.444	0.0256285
$575$	2864.21	0.207732
$576$	0	0
$577$	−16525.4	−1.19231	−0.596153	−	0.802871i	$-0.703305\pi$
$577$	−16525.4	−1.19231	−0.596153	+	0.802871i	$0.703305\pi$
$578$	13530.7	0.973708
$579$	0	0
$580$	27085.2	1.93905
$581$	2779.45	0.198470
$582$	0	0
$583$	5362.01	0.380912
$584$	14821.4	1.05019
$585$	0	0
$586$	−13743.4	−0.968830
$587$	−15341.2	−1.07870	−0.539351	−	0.842081i	$-0.681330\pi$
$587$	−15341.2	−1.07870	−0.539351	+	0.842081i	$0.681330\pi$
$588$	0	0
$589$	4758.35	0.332876
$590$	37911.9	2.64544
$591$	0	0
$592$	4117.97	0.285891
$593$	−3742.99	−0.259201	−0.129600	−	0.991566i	$-0.541369\pi$
$593$	−3742.99	−0.259201	−0.129600	+	0.991566i	$0.541369\pi$
$594$	0	0
$595$	3697.26	0.254744
$596$	−21479.4	−1.47622
$597$	0	0
$598$	8455.44	0.578208
$599$	−11362.5	−0.775057	−0.387529	−	0.921858i	$-0.626671\pi$
$599$	−11362.5	−0.775057	−0.387529	+	0.921858i	$0.626671\pi$
$600$	0	0
$601$	6080.27	0.412678	0.206339	−	0.978481i	$-0.433845\pi$
$601$	6080.27	0.412678	0.206339	+	0.978481i	$0.433845\pi$
$602$	3059.21	0.207116
$603$	0	0
$604$	36101.1	2.43201
$605$	1475.12	0.0991279
$606$	0	0
$607$	−21318.7	−1.42553	−0.712766	−	0.701402i	$-0.752558\pi$
$607$	−21318.7	−1.42553	−0.712766	+	0.701402i	$0.752558\pi$
$608$	11324.1	0.755353
$609$	0	0
$610$	−26424.3	−1.75391
$611$	−2914.17	−0.192954
$612$	0	0
$613$	−7505.66	−0.494536	−0.247268	−	0.968947i	$-0.579533\pi$
$613$	−7505.66	−0.494536	−0.247268	+	0.968947i	$0.579533\pi$
$614$	−36619.4	−2.40690
$615$	0	0
$616$	−1325.83	−0.0867196
$617$	4837.21	0.315622	0.157811	−	0.987469i	$-0.449556\pi$
$617$	4837.21	0.315622	0.157811	+	0.987469i	$0.449556\pi$
$618$	0	0
$619$	−11483.4	−0.745649	−0.372824	−	0.927902i	$-0.621611\pi$
$619$	−11483.4	−0.745649	−0.372824	+	0.927902i	$0.621611\pi$
$620$	13291.8	0.860984
$621$	0	0
$622$	−14447.8	−0.931354
$623$	−1664.69	−0.107054
$624$	0	0
$625$	−18019.8	−1.15327
$626$	47443.0	3.02908
$627$	0	0
$628$	−15415.8	−0.979548
$629$	−9820.70	−0.622539
$630$	0	0
$631$	6720.61	0.423999	0.211999	−	0.977270i	$-0.432003\pi$
$631$	6720.61	0.423999	0.211999	+	0.977270i	$0.432003\pi$
$632$	−2719.48	−0.171163
$633$	0	0
$634$	−3306.82	−0.207146
$635$	27214.0	1.70072
$636$	0	0
$637$	−766.727	−0.0476904
$638$	−9181.17	−0.569728
$639$	0	0
$640$	23735.8	1.46600
$641$	−24827.3	−1.52983	−0.764915	−	0.644131i	$-0.777219\pi$
$641$	−24827.3	−1.52983	−0.764915	+	0.644131i	$0.777219\pi$
$642$	0	0
$643$	15682.9	0.961859	0.480929	−	0.876759i	$-0.340299\pi$
$643$	15682.9	0.961859	0.480929	+	0.876759i	$0.340299\pi$
$644$	10068.7	0.616089
$645$	0	0
$646$	−9997.52	−0.608896
$647$	22025.9	1.33838	0.669188	−	0.743093i	$-0.266642\pi$
$647$	22025.9	1.33838	0.669188	+	0.743093i	$0.266642\pi$
$648$	0	0
$649$	−7675.36	−0.464228
$650$	1647.45	0.0994126
$651$	0	0
$652$	−18397.8	−1.10508
$653$	14258.3	0.854469	0.427235	−	0.904141i	$-0.359488\pi$
$653$	14258.3	0.854469	0.427235	+	0.904141i	$0.359488\pi$
$654$	0	0
$655$	−6745.83	−0.402414
$656$	205.232	0.0122149
$657$	0	0
$658$	−5810.25	−0.344236
$659$	11822.2	0.698829	0.349414	−	0.936968i	$-0.386381\pi$
$659$	11822.2	0.698829	0.349414	+	0.936968i	$0.386381\pi$
$660$	0	0
$661$	−27196.3	−1.60033	−0.800163	−	0.599783i	$-0.795254\pi$
$661$	−27196.3	−1.60033	−0.800163	+	0.599783i	$0.795254\pi$
$662$	−17126.6	−1.00550
$663$	0	0
$664$	−6836.89	−0.399582
$665$	4418.43	0.257653
$666$	0	0
$667$	22706.1	1.31812
$668$	15970.9	0.925048
$669$	0	0
$670$	23729.7	1.36830
$671$	5349.65	0.307781
$672$	0	0
$673$	−2020.00	−0.115699	−0.0578495	−	0.998325i	$-0.518424\pi$
$673$	−2020.00	−0.115699	−0.0578495	+	0.998325i	$0.518424\pi$
$674$	5579.82	0.318882
$675$	0	0
$676$	−23159.2	−1.31766
$677$	−16276.1	−0.923989	−0.461994	−	0.886883i	$-0.652866\pi$
$677$	−16276.1	−0.923989	−0.461994	+	0.886883i	$0.652866\pi$
$678$	0	0
$679$	4130.81	0.233470
$680$	−9094.52	−0.512880
$681$	0	0
$682$	−4505.56	−0.252972
$683$	9926.97	0.556142	0.278071	−	0.960561i	$-0.410305\pi$
$683$	9926.97	0.556142	0.278071	+	0.960561i	$0.410305\pi$
$684$	0	0
$685$	9489.66	0.529316
$686$	−1528.70	−0.0850814
$687$	0	0
$688$	1781.41	0.0987144
$689$	−7627.45	−0.421746
$690$	0	0
$691$	−13631.7	−0.750471	−0.375236	−	0.926930i	$-0.622438\pi$
$691$	−13631.7	−0.750471	−0.375236	+	0.926930i	$0.622438\pi$
$692$	36279.5	1.99298
$693$	0	0
$694$	−54227.6	−2.96607
$695$	10859.0	0.592672
$696$	0	0
$697$	−489.446	−0.0265984
$698$	12757.3	0.691791
$699$	0	0
$700$	1961.77	0.105925
$701$	27909.6	1.50375	0.751875	−	0.659305i	$-0.229149\pi$
$701$	27909.6	1.50375	0.751875	+	0.659305i	$0.229149\pi$
$702$	0	0
$703$	−11736.3	−0.629647
$704$	−9123.88	−0.488451
$705$	0	0
$706$	−38476.1	−2.05109
$707$	6009.44	0.319672
$708$	0	0
$709$	12864.8	0.681452	0.340726	−	0.940163i	$-0.389327\pi$
$709$	12864.8	0.681452	0.340726	+	0.940163i	$0.389327\pi$
$710$	38884.8	2.05538
$711$	0	0
$712$	4094.81	0.215533
$713$	11142.8	0.585275
$714$	0	0
$715$	−2098.36	−0.109754
$716$	13513.6	0.705346
$717$	0	0
$718$	4515.78	0.234718
$719$	−27741.1	−1.43890	−0.719451	−	0.694544i	$-0.755606\pi$
$719$	−27741.1	−1.43890	−0.719451	+	0.694544i	$0.755606\pi$
$720$	0	0
$721$	130.419	0.00673656
$722$	18621.9	0.959882
$723$	0	0
$724$	37443.9	1.92209
$725$	4424.04	0.226627
$726$	0	0
$727$	−33694.0	−1.71890	−0.859450	−	0.511220i	$-0.829194\pi$
$727$	−33694.0	−1.71890	−0.859450	+	0.511220i	$0.829194\pi$
$728$	1885.99	0.0960158
$729$	0	0
$730$	46769.3	2.37125
$731$	−4248.37	−0.214955
$732$	0	0
$733$	−38697.7	−1.94998	−0.974988	−	0.222256i	$-0.928658\pi$
$733$	−38697.7	−1.94998	−0.974988	+	0.222256i	$0.928658\pi$
$734$	−23542.6	−1.18389
$735$	0	0
$736$	26518.2	1.32809
$737$	−4804.13	−0.240112
$738$	0	0
$739$	−22086.4	−1.09941	−0.549704	−	0.835359i	$-0.685260\pi$
$739$	−22086.4	−1.09941	−0.549704	+	0.835359i	$0.685260\pi$
$740$	−32783.6	−1.62858
$741$	0	0
$742$	−15207.6	−0.752409
$743$	−181.397	−0.00895666	−0.00447833	−	0.999990i	$-0.501426\pi$
$743$	−181.397	−0.00895666	−0.00447833	+	0.999990i	$0.501426\pi$
$744$	0	0
$745$	−22072.7	−1.08548
$746$	39090.7	1.91852
$747$	0	0
$748$	5653.81	0.276368
$749$	−1420.72	−0.0693086
$750$	0	0
$751$	−7735.78	−0.375876	−0.187938	−	0.982181i	$-0.560180\pi$
$751$	−7735.78	−0.375876	−0.187938	+	0.982181i	$0.560180\pi$
$752$	−3383.37	−0.164067
$753$	0	0
$754$	13060.2	0.630802
$755$	37098.3	1.78827
$756$	0	0
$757$	−17569.4	−0.843552	−0.421776	−	0.906700i	$-0.638593\pi$
$757$	−17569.4	−0.843552	−0.421776	+	0.906700i	$0.638593\pi$
$758$	−37580.0	−1.80075
$759$	0	0
$760$	−10868.4	−0.518737
$761$	−20086.4	−0.956807	−0.478404	−	0.878140i	$-0.658784\pi$
$761$	−20086.4	−0.956807	−0.478404	+	0.878140i	$0.658784\pi$
$762$	0	0
$763$	7165.34	0.339977
$764$	−26048.9	−1.23353
$765$	0	0
$766$	−51694.3	−2.43837
$767$	10918.2	0.513993
$768$	0	0
$769$	−7772.56	−0.364481	−0.182240	−	0.983254i	$-0.558335\pi$
$769$	−7772.56	−0.364481	−0.182240	+	0.983254i	$0.558335\pi$
$770$	−4183.70	−0.195806
$771$	0	0
$772$	−14528.1	−0.677301
$773$	−1055.68	−0.0491204	−0.0245602	−	0.999698i	$-0.507819\pi$
$773$	−1055.68	−0.0491204	−0.0245602	+	0.999698i	$0.507819\pi$
$774$	0	0
$775$	2171.05	0.100628
$776$	−10161.0	−0.470048
$777$	0	0
$778$	−6223.69	−0.286800
$779$	−584.914	−0.0269021
$780$	0	0
$781$	−7872.31	−0.360683
$782$	−23411.6	−1.07058
$783$	0	0
$784$	−890.174	−0.0405509
$785$	−15841.6	−0.720269
$786$	0	0
$787$	−34605.2	−1.56740	−0.783700	−	0.621140i	$-0.786670\pi$
$787$	−34605.2	−1.56740	−0.783700	+	0.621140i	$0.786670\pi$
$788$	53414.9	2.41475
$789$	0	0
$790$	−8581.42	−0.386472
$791$	7078.16	0.318168
$792$	0	0
$793$	−7609.87	−0.340775
$794$	−25731.5	−1.15010
$795$	0	0
$796$	63256.3	2.81666
$797$	−34013.8	−1.51171	−0.755853	−	0.654741i	$-0.772778\pi$
$797$	−34013.8	−1.51171	−0.755853	+	0.654741i	$0.772778\pi$
$798$	0	0
$799$	8068.80	0.357264
$800$	5166.77	0.228341
$801$	0	0
$802$	2254.24	0.0992520
$803$	−9468.55	−0.416112
$804$	0	0
$805$	10346.8	0.453015
$806$	6409.16	0.280090
$807$	0	0
$808$	−14782.0	−0.643600
$809$	−19962.0	−0.867525	−0.433763	−	0.901027i	$-0.642814\pi$
$809$	−19962.0	−0.867525	−0.433763	+	0.901027i	$0.642814\pi$
$810$	0	0
$811$	−31886.3	−1.38061	−0.690307	−	0.723517i	$-0.742525\pi$
$811$	−31886.3	−1.38061	−0.690307	+	0.723517i	$0.742525\pi$
$812$	15552.0	0.672128
$813$	0	0
$814$	11112.8	0.478505
$815$	−18906.0	−0.812573
$816$	0	0
$817$	−5077.04	−0.217409
$818$	−8314.04	−0.355371
$819$	0	0
$820$	−1633.87	−0.0695821
$821$	−6951.19	−0.295491	−0.147745	−	0.989025i	$-0.547202\pi$
$821$	−6951.19	−0.295491	−0.147745	+	0.989025i	$0.547202\pi$
$822$	0	0
$823$	36694.6	1.55418	0.777091	−	0.629388i	$-0.216694\pi$
$823$	36694.6	1.55418	0.777091	+	0.629388i	$0.216694\pi$
$824$	−320.805	−0.0135628
$825$	0	0
$826$	21768.6	0.916982
$827$	−22074.4	−0.928176	−0.464088	−	0.885789i	$-0.653618\pi$
$827$	−22074.4	−0.928176	−0.464088	+	0.885789i	$0.653618\pi$
$828$	0	0
$829$	23458.4	0.982802	0.491401	−	0.870933i	$-0.336485\pi$
$829$	23458.4	0.982802	0.491401	+	0.870933i	$0.336485\pi$
$830$	−21574.0	−0.902224
$831$	0	0
$832$	12978.7	0.540812
$833$	2122.93	0.0883013
$834$	0	0
$835$	16412.1	0.680195
$836$	6756.61	0.279524
$837$	0	0
$838$	34347.7	1.41590
$839$	−37736.8	−1.55282	−0.776412	−	0.630225i	$-0.782963\pi$
$839$	−37736.8	−1.55282	−0.776412	+	0.630225i	$0.782963\pi$
$840$	0	0
$841$	10682.7	0.438015
$842$	20210.8	0.827210
$843$	0	0
$844$	−22840.1	−0.931503
$845$	−23799.0	−0.968887
$846$	0	0
$847$	847.000	0.0343604
$848$	−8855.52	−0.358608
$849$	0	0
$850$	−4561.48	−0.184068
$851$	−27483.3	−1.10707
$852$	0	0
$853$	26345.9	1.05752	0.528762	−	0.848770i	$-0.322656\pi$
$853$	26345.9	1.05752	0.528762	+	0.848770i	$0.322656\pi$
$854$	−15172.5	−0.607955
$855$	0	0
$856$	3494.69	0.139540
$857$	−47972.7	−1.91215	−0.956077	−	0.293116i	$-0.905308\pi$
$857$	−47972.7	−1.91215	−0.956077	+	0.293116i	$0.905308\pi$
$858$	0	0
$859$	−30428.2	−1.20861	−0.604306	−	0.796752i	$-0.706549\pi$
$859$	−30428.2	−1.20861	−0.604306	+	0.796752i	$0.706549\pi$
$860$	−14182.0	−0.562327
$861$	0	0
$862$	−3122.23	−0.123368
$863$	8351.59	0.329422	0.164711	−	0.986342i	$-0.447331\pi$
$863$	8351.59	0.329422	0.164711	+	0.986342i	$0.447331\pi$
$864$	0	0
$865$	37281.7	1.46545
$866$	−59302.1	−2.32698
$867$	0	0
$868$	7631.98	0.298440
$869$	1737.33	0.0678191
$870$	0	0
$871$	6833.87	0.265852
$872$	−17625.3	−0.684481
$873$	0	0
$874$	−27978.1	−1.08281
$875$	−8651.27	−0.334247
$876$	0	0
$877$	−24609.5	−0.947552	−0.473776	−	0.880645i	$-0.657109\pi$
$877$	−24609.5	−0.947552	−0.473776	+	0.880645i	$0.657109\pi$
$878$	−4421.77	−0.169963
$879$	0	0
$880$	−2436.21	−0.0933235
$881$	17386.9	0.664901	0.332451	−	0.943121i	$-0.392125\pi$
$881$	17386.9	0.664901	0.332451	+	0.943121i	$0.392125\pi$
$882$	0	0
$883$	35566.6	1.35551	0.677753	−	0.735290i	$-0.262954\pi$
$883$	35566.6	1.35551	0.677753	+	0.735290i	$0.262954\pi$
$884$	−8042.53	−0.305995
$885$	0	0
$886$	−32000.3	−1.21340
$887$	21087.4	0.798248	0.399124	−	0.916897i	$-0.369314\pi$
$887$	21087.4	0.798248	0.399124	+	0.916897i	$0.369314\pi$
$888$	0	0
$889$	15626.0	0.589515
$890$	12921.3	0.486655
$891$	0	0
$892$	21283.0	0.798888
$893$	9642.65	0.361343
$894$	0	0
$895$	13886.9	0.518646
$896$	13628.8	0.508155
$897$	0	0
$898$	79046.5	2.93743
$899$	17211.1	0.638511
$900$	0	0
$901$	21119.0	0.780884
$902$	553.841	0.0204444
$903$	0	0
$904$	−17410.8	−0.640571
$905$	38478.2	1.41333
$906$	0	0
$907$	−46282.0	−1.69434	−0.847171	−	0.531321i	$-0.821696\pi$
$907$	−46282.0	−1.69434	−0.847171	+	0.531321i	$0.821696\pi$
$908$	33326.9	1.21805
$909$	0	0
$910$	5951.31	0.216796
$911$	14001.3	0.509203	0.254601	−	0.967046i	$-0.418056\pi$
$911$	14001.3	0.509203	0.254601	+	0.967046i	$0.418056\pi$
$912$	0	0
$913$	4367.71	0.158324
$914$	26412.3	0.955845
$915$	0	0
$916$	−29015.8	−1.04663
$917$	−3873.38	−0.139488
$918$	0	0
$919$	−7587.94	−0.272364	−0.136182	−	0.990684i	$-0.543483\pi$
$919$	−7587.94	−0.272364	−0.136182	+	0.990684i	$0.543483\pi$
$920$	−25451.0	−0.912061
$921$	0	0
$922$	52730.0	1.88348
$923$	11198.4	0.399348
$924$	0	0
$925$	−5354.81	−0.190341
$926$	−12283.4	−0.435915
$927$	0	0
$928$	40959.8	1.44889
$929$	−33002.7	−1.16554	−0.582768	−	0.812638i	$-0.698030\pi$
$929$	−33002.7	−1.16554	−0.582768	+	0.812638i	$0.698030\pi$
$930$	0	0
$931$	2537.01	0.0893096
$932$	−37287.7	−1.31051
$933$	0	0
$934$	34741.5	1.21711
$935$	5809.98	0.203216
$936$	0	0
$937$	−15187.7	−0.529519	−0.264760	−	0.964314i	$-0.585293\pi$
$937$	−15187.7	−0.529519	−0.264760	+	0.964314i	$0.585293\pi$
$938$	13625.3	0.474289
$939$	0	0
$940$	26935.4	0.934612
$941$	−32513.8	−1.12638	−0.563188	−	0.826328i	$-0.690425\pi$
$941$	−32513.8	−1.12638	−0.563188	+	0.826328i	$0.690425\pi$
$942$	0	0
$943$	−1369.72	−0.0473002
$944$	12676.1	0.437046
$945$	0	0
$946$	4807.33	0.165222
$947$	14865.8	0.510108	0.255054	−	0.966927i	$-0.417907\pi$
$947$	14865.8	0.510108	0.255054	+	0.966927i	$0.417907\pi$
$948$	0	0
$949$	13469.0	0.460719
$950$	−5451.22	−0.186169
$951$	0	0
$952$	−5221.97	−0.177778
$953$	−19708.9	−0.669920	−0.334960	−	0.942232i	$-0.608723\pi$
$953$	−19708.9	−0.669920	−0.334960	+	0.942232i	$0.608723\pi$
$954$	0	0
$955$	−26768.5	−0.907024
$956$	−12002.4	−0.406050
$957$	0	0
$958$	−59175.2	−1.99568
$959$	5448.85	0.183475
$960$	0	0
$961$	−21344.8	−0.716486
$962$	−15807.9	−0.529801
$963$	0	0
$964$	−29062.0	−0.970981
$965$	−14929.4	−0.498024
$966$	0	0
$967$	3670.99	0.122080	0.0610398	−	0.998135i	$-0.480558\pi$
$967$	3670.99	0.122080	0.0610398	+	0.998135i	$0.480558\pi$
$968$	−2083.45	−0.0691783
$969$	0	0
$970$	−32063.3	−1.06133
$971$	12572.5	0.415521	0.207761	−	0.978180i	$-0.433383\pi$
$971$	12572.5	0.415521	0.207761	+	0.978180i	$0.433383\pi$
$972$	0	0
$973$	6235.14	0.205436
$974$	−47248.3	−1.55435
$975$	0	0
$976$	−8835.11	−0.289759
$977$	14573.7	0.477232	0.238616	−	0.971114i	$-0.423306\pi$
$977$	14573.7	0.477232	0.238616	+	0.971114i	$0.423306\pi$
$978$	0	0
$979$	−2615.95	−0.0853994
$980$	7086.78	0.230999
$981$	0	0
$982$	27165.1	0.882763
$983$	−26717.6	−0.866896	−0.433448	−	0.901179i	$-0.642703\pi$
$983$	−26717.6	−0.866896	−0.433448	+	0.901179i	$0.642703\pi$
$984$	0	0
$985$	54890.4	1.77559
$986$	−36161.3	−1.16796
$987$	0	0
$988$	−9611.26	−0.309489
$989$	−11889.1	−0.382256
$990$	0	0
$991$	22048.7	0.706761	0.353380	−	0.935480i	$-0.385032\pi$
$991$	22048.7	0.706761	0.353380	+	0.935480i	$0.385032\pi$
$992$	20100.6	0.643341
$993$	0	0
$994$	22327.2	0.712451
$995$	65003.6	2.07111
$996$	0	0
$997$	29571.1	0.939345	0.469673	−	0.882841i	$-0.344372\pi$
$997$	29571.1	0.939345	0.469673	+	0.882841i	$0.344372\pi$
$998$	84224.3	2.67142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.u.1.1	yes	8
3.2	odd	2		693.4.a.r.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.r.1.8	✓	8	3.2	odd	2
693.4.a.u.1.1	yes	8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-4.45684 q^{2} +11.8634 q^{4} +12.1911 q^{5} +7.00000 q^{7} -17.2186 q^{8} +O(q^{10})\)	\(q-4.45684 q^{2} +11.8634 q^{4} +12.1911 q^{5} +7.00000 q^{7} -17.2186 q^{8} -54.3338 q^{10} +11.0000 q^{11} -15.6475 q^{13} -31.1979 q^{14} -18.1668 q^{16} +43.3250 q^{17} +51.7758 q^{19} +144.628 q^{20} -49.0252 q^{22} +121.245 q^{23} +23.6233 q^{25} +69.7383 q^{26} +83.0439 q^{28} +187.274 q^{29} +91.9030 q^{31} +218.715 q^{32} -193.093 q^{34} +85.3378 q^{35} -226.675 q^{37} -230.756 q^{38} -209.914 q^{40} -11.2971 q^{41} -98.0582 q^{43} +130.497 q^{44} -540.371 q^{46} +186.239 q^{47} +49.0000 q^{49} -105.285 q^{50} -185.632 q^{52} +487.456 q^{53} +134.102 q^{55} -120.530 q^{56} -834.652 q^{58} -697.760 q^{59} +486.332 q^{61} -409.597 q^{62} -829.444 q^{64} -190.760 q^{65} -436.739 q^{67} +513.982 q^{68} -380.337 q^{70} -715.665 q^{71} -860.777 q^{73} +1010.25 q^{74} +614.237 q^{76} +77.0000 q^{77} +157.939 q^{79} -221.474 q^{80} +50.3492 q^{82} +397.064 q^{83} +528.180 q^{85} +437.030 q^{86} -189.404 q^{88} -237.813 q^{89} -109.532 q^{91} +1438.38 q^{92} -830.036 q^{94} +631.204 q^{95} +590.116 q^{97} -218.385 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8}+O(q^{10}) \) 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8	\( 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8} + 9 q^{10} + 88 q^{11} + 16 q^{13} + 42 q^{14} + 122 q^{16} + 90 q^{17} - 42 q^{19} + 291 q^{20} + 66 q^{22} + 338 q^{23} + 244 q^{25} + 209 q^{26} + 210 q^{28} + 496 q^{29} - 8 q^{31} + 524 q^{32} - 302 q^{34} + 70 q^{35} - 360 q^{37} + 45 q^{38} - 6 q^{40} + 242 q^{41} - 66 q^{43} + 330 q^{44} + 344 q^{46} + 540 q^{47} + 392 q^{49} + 1171 q^{50} + 465 q^{52} + 906 q^{53} + 110 q^{55} + 567 q^{56} + 977 q^{58} + 1242 q^{59} - 318 q^{61} - 110 q^{62} + 525 q^{64} + 1258 q^{65} + 522 q^{67} + 678 q^{68} + 63 q^{70} + 858 q^{71} - 78 q^{73} + 1651 q^{74} + 1775 q^{76} + 616 q^{77} + 516 q^{79} + 567 q^{80} - 1212 q^{82} + 3192 q^{83} + 720 q^{85} + 1322 q^{86} + 891 q^{88} + 2356 q^{89} + 112 q^{91} + 4504 q^{92} - 423 q^{94} + 3308 q^{95} - 1556 q^{97} + 294 q^{98}+O(q^{100}) \) 8 * q + 6 * q^2 + 30 * q^4 + 10 * q^5 + 56 * q^7 + 81 * q^8 + 9 * q^10 + 88 * q^11 + 16 * q^13 + 42 * q^14 + 122 * q^16 + 90 * q^17 - 42 * q^19 + 291 * q^20 + 66 * q^22 + 338 * q^23 + 244 * q^25 + 209 * q^26 + 210 * q^28 + 496 * q^29 - 8 * q^31 + 524 * q^32 - 302 * q^34 + 70 * q^35 - 360 * q^37 + 45 * q^38 - 6 * q^40 + 242 * q^41 - 66 * q^43 + 330 * q^44 + 344 * q^46 + 540 * q^47 + 392 * q^49 + 1171 * q^50 + 465 * q^52 + 906 * q^53 + 110 * q^55 + 567 * q^56 + 977 * q^58 + 1242 * q^59 - 318 * q^61 - 110 * q^62 + 525 * q^64 + 1258 * q^65 + 522 * q^67 + 678 * q^68 + 63 * q^70 + 858 * q^71 - 78 * q^73 + 1651 * q^74 + 1775 * q^76 + 616 * q^77 + 516 * q^79 + 567 * q^80 - 1212 * q^82 + 3192 * q^83 + 720 * q^85 + 1322 * q^86 + 891 * q^88 + 2356 * q^89 + 112 * q^91 + 4504 * q^92 - 423 * q^94 + 3308 * q^95 - 1556 * q^97 + 294 * q^98

Introduction

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