LMFDB - Embedded newform 693.4.a.u.1.3

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			1.3
Root			$2.94273$ 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)

$f(q)$	$=$	$q-1.94273 q^{2} -4.22579 q^{4} -4.10121 q^{5} +7.00000 q^{7} +23.7514 q^{8} +O(q^{10})$	\(q-1.94273 q^{2} -4.22579 q^{4} -4.10121 q^{5} +7.00000 q^{7} +23.7514 q^{8} +7.96756 q^{10} +11.0000 q^{11} +71.7277 q^{13} -13.5991 q^{14} -12.3363 q^{16} +79.3194 q^{17} -159.920 q^{19} +17.3309 q^{20} -21.3701 q^{22} -137.959 q^{23} -108.180 q^{25} -139.348 q^{26} -29.5805 q^{28} +72.2721 q^{29} +235.611 q^{31} -166.045 q^{32} -154.096 q^{34} -28.7085 q^{35} -236.396 q^{37} +310.681 q^{38} -97.4097 q^{40} -147.737 q^{41} -147.130 q^{43} -46.4837 q^{44} +268.018 q^{46} +538.615 q^{47} +49.0000 q^{49} +210.165 q^{50} -303.106 q^{52} +571.743 q^{53} -45.1133 q^{55} +166.260 q^{56} -140.405 q^{58} +749.305 q^{59} -758.200 q^{61} -457.729 q^{62} +421.272 q^{64} -294.171 q^{65} -40.4747 q^{67} -335.187 q^{68} +55.7729 q^{70} -131.133 q^{71} +238.274 q^{73} +459.255 q^{74} +675.787 q^{76} +77.0000 q^{77} +588.655 q^{79} +50.5940 q^{80} +287.013 q^{82} +481.542 q^{83} -325.306 q^{85} +285.833 q^{86} +261.266 q^{88} +1368.29 q^{89} +502.094 q^{91} +582.988 q^{92} -1046.39 q^{94} +655.865 q^{95} -667.821 q^{97} -95.1939 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.94273	−0.686860	−0.343430	−	0.939178i	$-0.611589\pi$
$2$	−1.94273	−0.686860	−0.343430	+	0.939178i	$0.611589\pi$
$3$	0	0
$3$	0	0
$4$	−4.22579	−0.528224
$5$	−4.10121	−0.366824	−0.183412	−	0.983036i	$-0.558714\pi$
$5$	−4.10121	−0.366824	−0.183412	+	0.983036i	$0.558714\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	23.7514	1.04968
$9$	0	0
$10$	7.96756	0.251956
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	71.7277	1.53028	0.765142	−	0.643862i	$-0.222669\pi$
$13$	71.7277	1.53028	0.765142	+	0.643862i	$0.222669\pi$
$14$	−13.5991	−0.259608
$15$	0	0
$16$	−12.3363	−0.192755
$17$	79.3194	1.13163	0.565817	−	0.824531i	$-0.308561\pi$
$17$	79.3194	1.13163	0.565817	+	0.824531i	$0.308561\pi$
$18$	0	0
$19$	−159.920	−1.93095	−0.965476	−	0.260492i	$-0.916115\pi$
$19$	−159.920	−1.93095	−0.965476	+	0.260492i	$0.916115\pi$
$20$	17.3309	0.193765
$21$	0	0
$22$	−21.3701	−0.207096
$23$	−137.959	−1.25072	−0.625360	−	0.780337i	$-0.715048\pi$
$23$	−137.959	−1.25072	−0.625360	+	0.780337i	$0.715048\pi$
$24$	0	0
$25$	−108.180	−0.865440
$26$	−139.348	−1.05109
$27$	0	0
$28$	−29.5805	−0.199650
$29$	72.2721	0.462779	0.231389	−	0.972861i	$-0.425673\pi$
$29$	72.2721	0.462779	0.231389	+	0.972861i	$0.425673\pi$
$30$	0	0
$31$	235.611	1.36506	0.682532	−	0.730855i	$-0.260879\pi$
$31$	235.611	1.36506	0.682532	+	0.730855i	$0.260879\pi$
$32$	−166.045	−0.917279
$33$	0	0
$34$	−154.096	−0.777273
$35$	−28.7085	−0.138646
$36$	0	0
$37$	−236.396	−1.05036	−0.525180	−	0.850991i	$-0.676002\pi$
$37$	−236.396	−1.05036	−0.525180	+	0.850991i	$0.676002\pi$
$38$	310.681	1.32629
$39$	0	0
$40$	−97.4097	−0.385046
$41$	−147.737	−0.562747	−0.281373	−	0.959598i	$-0.590790\pi$
$41$	−147.737	−0.562747	−0.281373	+	0.959598i	$0.590790\pi$
$42$	0	0
$43$	−147.130	−0.521792	−0.260896	−	0.965367i	$-0.584018\pi$
$43$	−147.130	−0.521792	−0.260896	+	0.965367i	$0.584018\pi$
$44$	−46.4837	−0.159266
$45$	0	0
$46$	268.018	0.859068
$47$	538.615	1.67160	0.835799	−	0.549035i	$-0.185005\pi$
$47$	538.615	1.67160	0.835799	+	0.549035i	$0.185005\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	210.165	0.594436
$51$	0	0
$52$	−303.106	−0.808333
$53$	571.743	1.48179	0.740896	−	0.671620i	$-0.234401\pi$
$53$	571.743	1.48179	0.740896	+	0.671620i	$0.234401\pi$
$54$	0	0
$55$	−45.1133	−0.110601
$56$	166.260	0.396740
$57$	0	0
$58$	−140.405	−0.317864
$59$	749.305	1.65341	0.826705	−	0.562636i	$-0.190213\pi$
$59$	749.305	1.65341	0.826705	+	0.562636i	$0.190213\pi$
$60$	0	0
$61$	−758.200	−1.59144	−0.795718	−	0.605668i	$-0.792906\pi$
$61$	−758.200	−1.59144	−0.795718	+	0.605668i	$0.792906\pi$
$62$	−457.729	−0.937608
$63$	0	0
$64$	421.272	0.822797
$65$	−294.171	−0.561344
$66$	0	0
$67$	−40.4747	−0.0738026	−0.0369013	−	0.999319i	$-0.511749\pi$
$67$	−40.4747	−0.0738026	−0.0369013	+	0.999319i	$0.511749\pi$
$68$	−335.187	−0.597756
$69$	0	0
$70$	55.7729	0.0952305
$71$	−131.133	−0.219192	−0.109596	−	0.993976i	$-0.534956\pi$
$71$	−131.133	−0.219192	−0.109596	+	0.993976i	$0.534956\pi$
$72$	0	0
$73$	238.274	0.382026	0.191013	−	0.981588i	$-0.438823\pi$
$73$	238.274	0.382026	0.191013	+	0.981588i	$0.438823\pi$
$74$	459.255	0.721450
$75$	0	0
$76$	675.787	1.01998
$77$	77.0000	0.113961
$78$	0	0
$79$	588.655	0.838339	0.419170	−	0.907908i	$-0.362321\pi$
$79$	588.655	0.838339	0.419170	+	0.907908i	$0.362321\pi$
$80$	50.5940	0.0707072
$81$	0	0
$82$	287.013	0.386528
$83$	481.542	0.636820	0.318410	−	0.947953i	$-0.396851\pi$
$83$	481.542	0.636820	0.318410	+	0.947953i	$0.396851\pi$
$84$	0	0
$85$	−325.306	−0.415110
$86$	285.833	0.358398
$87$	0	0
$88$	261.266	0.316489
$89$	1368.29	1.62965	0.814823	−	0.579710i	$-0.196834\pi$
$89$	1368.29	1.62965	0.814823	+	0.579710i	$0.196834\pi$
$90$	0	0
$91$	502.094	0.578393
$92$	582.988	0.660660
$93$	0	0
$94$	−1046.39	−1.14815
$95$	655.865	0.708319
$96$	0	0
$97$	−667.821	−0.699040	−0.349520	−	0.936929i	$-0.613655\pi$
$97$	−667.821	−0.699040	−0.349520	+	0.936929i	$0.613655\pi$
$98$	−95.1939	−0.0981228
$99$	0	0
$100$	457.146	0.457146
$101$	−801.092	−0.789224	−0.394612	−	0.918848i	$-0.629121\pi$
$101$	−801.092	−0.789224	−0.394612	+	0.918848i	$0.629121\pi$
$102$	0	0
$103$	−301.925	−0.288831	−0.144415	−	0.989517i	$-0.546130\pi$
$103$	−301.925	−0.288831	−0.144415	+	0.989517i	$0.546130\pi$
$104$	1703.64	1.60630
$105$	0	0
$106$	−1110.74	−1.01778
$107$	1637.95	1.47988	0.739938	−	0.672675i	$-0.234855\pi$
$107$	1637.95	1.47988	0.739938	+	0.672675i	$0.234855\pi$
$108$	0	0
$109$	459.430	0.403719	0.201860	−	0.979414i	$-0.435302\pi$
$109$	459.430	0.403719	0.201860	+	0.979414i	$0.435302\pi$
$110$	87.6431	0.0759677
$111$	0	0
$112$	−86.3544	−0.0728547
$113$	−1598.23	−1.33052	−0.665262	−	0.746610i	$-0.731680\pi$
$113$	−1598.23	−1.33052	−0.665262	+	0.746610i	$0.731680\pi$
$114$	0	0
$115$	565.801	0.458793
$116$	−305.407	−0.244451
$117$	0	0
$118$	−1455.70	−1.13566
$119$	555.236	0.427717
$120$	0	0
$121$	121.000	0.0909091
$122$	1472.98	1.09309
$123$	0	0
$124$	−995.644	−0.721060
$125$	956.321	0.684288
$126$	0	0
$127$	39.2165	0.0274008	0.0137004	−	0.999906i	$-0.495639\pi$
$127$	39.2165	0.0274008	0.0137004	+	0.999906i	$0.495639\pi$
$128$	509.943	0.352133
$129$	0	0
$130$	571.495	0.385565
$131$	822.089	0.548292	0.274146	−	0.961688i	$-0.411605\pi$
$131$	822.089	0.548292	0.274146	+	0.961688i	$0.411605\pi$
$132$	0	0
$133$	−1119.44	−0.729831
$134$	78.6315	0.0506920
$135$	0	0
$136$	1883.95	1.18785
$137$	−2913.27	−1.81677	−0.908384	−	0.418137i	$-0.862683\pi$
$137$	−2913.27	−1.81677	−0.908384	+	0.418137i	$0.862683\pi$
$138$	0	0
$139$	772.680	0.471496	0.235748	−	0.971814i	$-0.424246\pi$
$139$	772.680	0.471496	0.235748	+	0.971814i	$0.424246\pi$
$140$	121.316	0.0732363
$141$	0	0
$142$	254.756	0.150554
$143$	789.005	0.461398
$144$	0	0
$145$	−296.403	−0.169758
$146$	−462.903	−0.262398
$147$	0	0
$148$	998.961	0.554825
$149$	2396.19	1.31747	0.658736	−	0.752374i	$-0.271092\pi$
$149$	2396.19	1.31747	0.658736	+	0.752374i	$0.271092\pi$
$150$	0	0
$151$	1679.18	0.904968	0.452484	−	0.891773i	$-0.350538\pi$
$151$	1679.18	0.904968	0.452484	+	0.891773i	$0.350538\pi$
$152$	−3798.32	−2.02687
$153$	0	0
$154$	−149.590	−0.0782749
$155$	−966.291	−0.500738
$156$	0	0
$157$	2960.94	1.50515	0.752576	−	0.658505i	$-0.228811\pi$
$157$	2960.94	1.50515	0.752576	+	0.658505i	$0.228811\pi$
$158$	−1143.60	−0.575821
$159$	0	0
$160$	680.987	0.336480
$161$	−965.716	−0.472727
$162$	0	0
$163$	−731.153	−0.351339	−0.175670	−	0.984449i	$-0.556209\pi$
$163$	−731.153	−0.351339	−0.175670	+	0.984449i	$0.556209\pi$
$164$	624.305	0.297256
$165$	0	0
$166$	−935.507	−0.437406
$167$	−425.782	−0.197293	−0.0986466	−	0.995123i	$-0.531451\pi$
$167$	−425.782	−0.197293	−0.0986466	+	0.995123i	$0.531451\pi$
$168$	0	0
$169$	2947.87	1.34177
$170$	631.982	0.285122
$171$	0	0
$172$	621.739	0.275623
$173$	−492.101	−0.216264	−0.108132	−	0.994137i	$-0.534487\pi$
$173$	−492.101	−0.216264	−0.108132	+	0.994137i	$0.534487\pi$
$174$	0	0
$175$	−757.260	−0.327106
$176$	−135.700	−0.0581179
$177$	0	0
$178$	−2658.22	−1.11934
$179$	4034.94	1.68483	0.842417	−	0.538826i	$-0.181132\pi$
$179$	4034.94	1.68483	0.842417	+	0.538826i	$0.181132\pi$
$180$	0	0
$181$	1468.55	0.603076	0.301538	−	0.953454i	$-0.402500\pi$
$181$	1468.55	0.603076	0.301538	+	0.953454i	$0.402500\pi$
$182$	−975.434	−0.397275
$183$	0	0
$184$	−3276.74	−1.31285
$185$	969.511	0.385297
$186$	0	0
$187$	872.513	0.341200
$188$	−2276.08	−0.882979
$189$	0	0
$190$	−1274.17	−0.486515
$191$	4914.52	1.86179	0.930895	−	0.365286i	$-0.119029\pi$
$191$	4914.52	1.86179	0.930895	+	0.365286i	$0.119029\pi$
$192$	0	0
$193$	4625.39	1.72509	0.862547	−	0.505977i	$-0.168868\pi$
$193$	4625.39	1.72509	0.862547	+	0.505977i	$0.168868\pi$
$194$	1297.40	0.480143
$195$	0	0
$196$	−207.064	−0.0754606
$197$	2997.37	1.08403	0.542015	−	0.840369i	$-0.317662\pi$
$197$	2997.37	1.08403	0.542015	+	0.840369i	$0.317662\pi$
$198$	0	0
$199$	1324.74	0.471902	0.235951	−	0.971765i	$-0.424179\pi$
$199$	1324.74	0.471902	0.235951	+	0.971765i	$0.424179\pi$
$200$	−2569.43	−0.908431
$201$	0	0
$202$	1556.31	0.542086
$203$	505.904	0.174914
$204$	0	0
$205$	605.900	0.206429
$206$	586.560	0.198386
$207$	0	0
$208$	−884.858	−0.294970
$209$	−1759.12	−0.582204
$210$	0	0
$211$	−4781.25	−1.55998	−0.779988	−	0.625795i	$-0.784775\pi$
$211$	−4781.25	−1.55998	−0.779988	+	0.625795i	$0.784775\pi$
$212$	−2416.07	−0.782718
$213$	0	0
$214$	−3182.10	−1.01647
$215$	603.409	0.191405
$216$	0	0
$217$	1649.28	0.515946
$218$	−892.549	−0.277298
$219$	0	0
$220$	190.640	0.0584224
$221$	5689.40	1.73172
$222$	0	0
$223$	4897.49	1.47067	0.735337	−	0.677702i	$-0.237024\pi$
$223$	4897.49	1.47067	0.735337	+	0.677702i	$0.237024\pi$
$224$	−1162.32	−0.346699
$225$	0	0
$226$	3104.94	0.913883
$227$	−4710.24	−1.37722	−0.688611	−	0.725131i	$-0.741779\pi$
$227$	−4710.24	−1.37722	−0.688611	+	0.725131i	$0.741779\pi$
$228$	0	0
$229$	4740.56	1.36797	0.683985	−	0.729496i	$-0.260245\pi$
$229$	4740.56	1.36797	0.683985	+	0.729496i	$0.260245\pi$
$230$	−1099.20	−0.315127
$231$	0	0
$232$	1716.57	0.485767
$233$	−2907.39	−0.817465	−0.408733	−	0.912654i	$-0.634029\pi$
$233$	−2907.39	−0.817465	−0.408733	+	0.912654i	$0.634029\pi$
$234$	0	0
$235$	−2208.98	−0.613182
$236$	−3166.41	−0.873371
$237$	0	0
$238$	−1078.67	−0.293782
$239$	−2771.48	−0.750092	−0.375046	−	0.927006i	$-0.622373\pi$
$239$	−2771.48	−0.750092	−0.375046	+	0.927006i	$0.622373\pi$
$240$	0	0
$241$	−2109.36	−0.563801	−0.281901	−	0.959444i	$-0.590965\pi$
$241$	−2109.36	−0.563801	−0.281901	+	0.959444i	$0.590965\pi$
$242$	−235.071	−0.0624418
$243$	0	0
$244$	3204.00	0.840634
$245$	−200.959	−0.0524034
$246$	0	0
$247$	−11470.7	−2.95490
$248$	5596.10	1.43287
$249$	0	0
$250$	−1857.88	−0.470009
$251$	−2244.35	−0.564391	−0.282195	−	0.959357i	$-0.591063\pi$
$251$	−2244.35	−0.564391	−0.282195	+	0.959357i	$0.591063\pi$
$252$	0	0
$253$	−1517.55	−0.377106
$254$	−76.1872	−0.0188205
$255$	0	0
$256$	−4360.86	−1.06466
$257$	−3583.91	−0.869877	−0.434938	−	0.900460i	$-0.643230\pi$
$257$	−3583.91	−0.869877	−0.434938	+	0.900460i	$0.643230\pi$
$258$	0	0
$259$	−1654.77	−0.396999
$260$	1243.10	0.296515
$261$	0	0
$262$	−1597.10	−0.376600
$263$	−1865.86	−0.437467	−0.218734	−	0.975785i	$-0.570193\pi$
$263$	−1865.86	−0.437467	−0.218734	+	0.975785i	$0.570193\pi$
$264$	0	0
$265$	−2344.84	−0.543556
$266$	2174.77	0.501292
$267$	0	0
$268$	171.038	0.0389843
$269$	1277.50	0.289555	0.144778	−	0.989464i	$-0.453753\pi$
$269$	1277.50	0.289555	0.144778	+	0.989464i	$0.453753\pi$
$270$	0	0
$271$	1314.40	0.294628	0.147314	−	0.989090i	$-0.452937\pi$
$271$	1314.40	0.294628	0.147314	+	0.989090i	$0.452937\pi$
$272$	−978.511	−0.218129
$273$	0	0
$274$	5659.70	1.24786
$275$	−1189.98	−0.260940
$276$	0	0
$277$	−5015.84	−1.08799	−0.543994	−	0.839089i	$-0.683089\pi$
$277$	−5015.84	−1.08799	−0.543994	+	0.839089i	$0.683089\pi$
$278$	−1501.11	−0.323851
$279$	0	0
$280$	−681.868	−0.145534
$281$	−318.428	−0.0676009	−0.0338004	−	0.999429i	$-0.510761\pi$
$281$	−318.428	−0.0676009	−0.0338004	+	0.999429i	$0.510761\pi$
$282$	0	0
$283$	4911.16	1.03158	0.515792	−	0.856714i	$-0.327498\pi$
$283$	4911.16	1.03158	0.515792	+	0.856714i	$0.327498\pi$
$284$	554.140	0.115782
$285$	0	0
$286$	−1532.83	−0.316916
$287$	−1034.16	−0.212698
$288$	0	0
$289$	1378.56	0.280595
$290$	575.832	0.116600
$291$	0	0
$292$	−1006.90	−0.201795
$293$	3488.46	0.695557	0.347778	−	0.937577i	$-0.386936\pi$
$293$	3488.46	0.695557	0.347778	+	0.937577i	$0.386936\pi$
$294$	0	0
$295$	−3073.06	−0.606510
$296$	−5614.75	−1.10254
$297$	0	0
$298$	−4655.15	−0.904918
$299$	−9895.52	−1.91396
$300$	0	0
$301$	−1029.91	−0.197219
$302$	−3262.21	−0.621586
$303$	0	0
$304$	1972.82	0.372201
$305$	3109.54	0.583776
$306$	0	0
$307$	4511.48	0.838710	0.419355	−	0.907822i	$-0.362256\pi$
$307$	4511.48	0.838710	0.419355	+	0.907822i	$0.362256\pi$
$308$	−325.386	−0.0601967
$309$	0	0
$310$	1877.25	0.343937
$311$	562.319	0.102528	0.0512640	−	0.998685i	$-0.483675\pi$
$311$	562.319	0.102528	0.0512640	+	0.998685i	$0.483675\pi$
$312$	0	0
$313$	−3380.05	−0.610388	−0.305194	−	0.952290i	$-0.598721\pi$
$313$	−3380.05	−0.610388	−0.305194	+	0.952290i	$0.598721\pi$
$314$	−5752.32	−1.03383
$315$	0	0
$316$	−2487.53	−0.442831
$317$	4074.85	0.721975	0.360988	−	0.932571i	$-0.382440\pi$
$317$	4074.85	0.721975	0.360988	+	0.932571i	$0.382440\pi$
$318$	0	0
$319$	794.993	0.139533
$320$	−1727.73	−0.301822
$321$	0	0
$322$	1876.13	0.324697
$323$	−12684.7	−2.18513
$324$	0	0
$325$	−7759.51	−1.32437
$326$	1420.43	0.241321
$327$	0	0
$328$	−3508.96	−0.590701
$329$	3770.31	0.631805
$330$	0	0
$331$	−5539.37	−0.919853	−0.459926	−	0.887957i	$-0.652124\pi$
$331$	−5539.37	−0.919853	−0.459926	+	0.887957i	$0.652124\pi$
$332$	−2034.90	−0.336384
$333$	0	0
$334$	827.180	0.135513
$335$	165.995	0.0270725
$336$	0	0
$337$	−749.792	−0.121198	−0.0605990	−	0.998162i	$-0.519301\pi$
$337$	−749.792	−0.121198	−0.0605990	+	0.998162i	$0.519301\pi$
$338$	−5726.91	−0.921607
$339$	0	0
$340$	1374.67	0.219271
$341$	2591.72	0.411583
$342$	0	0
$343$	343.000	0.0539949
$344$	−3494.54	−0.547712
$345$	0	0
$346$	956.020	0.148543
$347$	−6660.32	−1.03039	−0.515194	−	0.857074i	$-0.672280\pi$
$347$	−6660.32	−1.03039	−0.515194	+	0.857074i	$0.672280\pi$
$348$	0	0
$349$	12085.7	1.85368	0.926841	−	0.375455i	$-0.122513\pi$
$349$	12085.7	1.85368	0.926841	+	0.375455i	$0.122513\pi$
$350$	1471.15	0.224676
$351$	0	0
$352$	−1826.50	−0.276570
$353$	3173.90	0.478554	0.239277	−	0.970951i	$-0.423090\pi$
$353$	3173.90	0.478554	0.239277	+	0.970951i	$0.423090\pi$
$354$	0	0
$355$	537.804	0.0804047
$356$	−5782.11	−0.860818
$357$	0	0
$358$	−7838.80	−1.15724
$359$	10519.4	1.54649	0.773246	−	0.634107i	$-0.218632\pi$
$359$	10519.4	1.54649	0.773246	+	0.634107i	$0.218632\pi$
$360$	0	0
$361$	18715.3	2.72858
$362$	−2853.01	−0.414228
$363$	0	0
$364$	−2121.74	−0.305521
$365$	−977.213	−0.140136
$366$	0	0
$367$	−1117.72	−0.158977	−0.0794884	−	0.996836i	$-0.525329\pi$
$367$	−1117.72	−0.158977	−0.0794884	+	0.996836i	$0.525329\pi$
$368$	1701.92	0.241083
$369$	0	0
$370$	−1883.50	−0.264645
$371$	4002.20	0.560064
$372$	0	0
$373$	5063.48	0.702887	0.351444	−	0.936209i	$-0.385691\pi$
$373$	5063.48	0.702887	0.351444	+	0.936209i	$0.385691\pi$
$374$	−1695.06	−0.234357
$375$	0	0
$376$	12792.9	1.75464
$377$	5183.91	0.708183
$378$	0	0
$379$	7008.77	0.949912	0.474956	−	0.880010i	$-0.342464\pi$
$379$	7008.77	0.949912	0.474956	+	0.880010i	$0.342464\pi$
$380$	−2771.55	−0.374151
$381$	0	0
$382$	−9547.59	−1.27879
$383$	3726.73	0.497199	0.248599	−	0.968606i	$-0.420030\pi$
$383$	3726.73	0.497199	0.248599	+	0.968606i	$0.420030\pi$
$384$	0	0
$385$	−315.793	−0.0418034
$386$	−8985.90	−1.18490
$387$	0	0
$388$	2822.07	0.369250
$389$	−5041.12	−0.657056	−0.328528	−	0.944494i	$-0.606553\pi$
$389$	−5041.12	−0.657056	−0.328528	+	0.944494i	$0.606553\pi$
$390$	0	0
$391$	−10942.9	−1.41536
$392$	1163.82	0.149954
$393$	0	0
$394$	−5823.09	−0.744576
$395$	−2414.20	−0.307523
$396$	0	0
$397$	−12535.1	−1.58469	−0.792343	−	0.610076i	$-0.791139\pi$
$397$	−12535.1	−1.58469	−0.792343	+	0.610076i	$0.791139\pi$
$398$	−2573.62	−0.324131
$399$	0	0
$400$	1334.55	0.166818
$401$	−14848.8	−1.84916	−0.924582	−	0.380984i	$-0.875585\pi$
$401$	−14848.8	−1.84916	−0.924582	+	0.380984i	$0.875585\pi$
$402$	0	0
$403$	16899.8	2.08894
$404$	3385.25	0.416887
$405$	0	0
$406$	−982.837	−0.120141
$407$	−2600.36	−0.316695
$408$	0	0
$409$	−10835.0	−1.30991	−0.654957	−	0.755666i	$-0.727313\pi$
$409$	−10835.0	−1.30991	−0.654957	+	0.755666i	$0.727313\pi$
$410$	−1177.10	−0.141788
$411$	0	0
$412$	1275.87	0.152567
$413$	5245.13	0.624930
$414$	0	0
$415$	−1974.91	−0.233601
$416$	−11910.0	−1.40370
$417$	0	0
$418$	3417.49	0.399892
$419$	−9072.77	−1.05784	−0.528918	−	0.848673i	$-0.677402\pi$
$419$	−9072.77	−1.05784	−0.528918	+	0.848673i	$0.677402\pi$
$420$	0	0
$421$	−15709.6	−1.81862	−0.909311	−	0.416117i	$-0.863391\pi$
$421$	−15709.6	−1.81862	−0.909311	+	0.416117i	$0.863391\pi$
$422$	9288.69	1.07148
$423$	0	0
$424$	13579.7	1.55540
$425$	−8580.77	−0.979362
$426$	0	0
$427$	−5307.40	−0.601506
$428$	−6921.64	−0.781706
$429$	0	0
$430$	−1172.26	−0.131469
$431$	15198.4	1.69856	0.849282	−	0.527940i	$-0.177035\pi$
$431$	15198.4	1.69856	0.849282	+	0.527940i	$0.177035\pi$
$432$	0	0
$433$	14668.5	1.62800	0.813999	−	0.580867i	$-0.197286\pi$
$433$	14668.5	1.62800	0.813999	+	0.580867i	$0.197286\pi$
$434$	−3204.11	−0.354382
$435$	0	0
$436$	−1941.45	−0.213254
$437$	22062.4	2.41508
$438$	0	0
$439$	−5452.78	−0.592817	−0.296409	−	0.955061i	$-0.595789\pi$
$439$	−5452.78	−0.592817	−0.296409	+	0.955061i	$0.595789\pi$
$440$	−1071.51	−0.116096
$441$	0	0
$442$	−11053.0	−1.18945
$443$	17927.7	1.92273	0.961365	−	0.275278i	$-0.0887700\pi$
$443$	17927.7	1.92273	0.961365	+	0.275278i	$0.0887700\pi$
$444$	0	0
$445$	−5611.65	−0.597793
$446$	−9514.52	−1.01015
$447$	0	0
$448$	2948.91	0.310988
$449$	9146.60	0.961369	0.480685	−	0.876893i	$-0.340388\pi$
$449$	9146.60	0.961369	0.480685	+	0.876893i	$0.340388\pi$
$450$	0	0
$451$	−1625.10	−0.169674
$452$	6753.80	0.702814
$453$	0	0
$454$	9150.73	0.945958
$455$	−2059.19	−0.212168
$456$	0	0
$457$	−401.495	−0.0410966	−0.0205483	−	0.999789i	$-0.506541\pi$
$457$	−401.495	−0.0410966	−0.0205483	+	0.999789i	$0.506541\pi$
$458$	−9209.64	−0.939603
$459$	0	0
$460$	−2390.96	−0.242346
$461$	1090.23	0.110146	0.0550729	−	0.998482i	$-0.482461\pi$
$461$	1090.23	0.110146	0.0550729	+	0.998482i	$0.482461\pi$
$462$	0	0
$463$	7462.97	0.749101	0.374550	−	0.927207i	$-0.377797\pi$
$463$	7462.97	0.749101	0.374550	+	0.927207i	$0.377797\pi$
$464$	−891.573	−0.0892031
$465$	0	0
$466$	5648.28	0.561484
$467$	−5634.38	−0.558304	−0.279152	−	0.960247i	$-0.590053\pi$
$467$	−5634.38	−0.558304	−0.279152	+	0.960247i	$0.590053\pi$
$468$	0	0
$469$	−283.323	−0.0278947
$470$	4291.45	0.421170
$471$	0	0
$472$	17797.1	1.73554
$473$	−1618.42	−0.157326
$474$	0	0
$475$	17300.1	1.67112
$476$	−2346.31	−0.225931
$477$	0	0
$478$	5384.24	0.515208
$479$	8134.97	0.775984	0.387992	−	0.921663i	$-0.373169\pi$
$479$	8134.97	0.775984	0.387992	+	0.921663i	$0.373169\pi$
$480$	0	0
$481$	−16956.2	−1.60735
$482$	4097.93	0.387252
$483$	0	0
$484$	−511.321	−0.0480204
$485$	2738.87	0.256425
$486$	0	0
$487$	−18635.3	−1.73397	−0.866986	−	0.498333i	$-0.833946\pi$
$487$	−18635.3	−1.73397	−0.866986	+	0.498333i	$0.833946\pi$
$488$	−18008.3	−1.67049
$489$	0	0
$490$	390.410	0.0359938
$491$	17849.3	1.64058	0.820291	−	0.571946i	$-0.193811\pi$
$491$	17849.3	1.64058	0.820291	+	0.571946i	$0.193811\pi$
$492$	0	0
$493$	5732.57	0.523696
$494$	22284.4	2.02960
$495$	0	0
$496$	−2906.58	−0.263124
$497$	−917.930	−0.0828467
$498$	0	0
$499$	5667.26	0.508419	0.254210	−	0.967149i	$-0.418185\pi$
$499$	5667.26	0.508419	0.254210	+	0.967149i	$0.418185\pi$
$500$	−4041.21	−0.361457
$501$	0	0
$502$	4360.17	0.387657
$503$	16813.6	1.49042	0.745209	−	0.666831i	$-0.232350\pi$
$503$	16813.6	1.49042	0.745209	+	0.666831i	$0.232350\pi$
$504$	0	0
$505$	3285.45	0.289506
$506$	2948.20	0.259019
$507$	0	0
$508$	−165.721	−0.0144738
$509$	−16268.7	−1.41670	−0.708348	−	0.705864i	$-0.750559\pi$
$509$	−16268.7	−1.41670	−0.708348	+	0.705864i	$0.750559\pi$
$510$	0	0
$511$	1667.92	0.144392
$512$	4392.44	0.379141
$513$	0	0
$514$	6962.58	0.597483
$515$	1238.26	0.105950
$516$	0	0
$517$	5924.77	0.504006
$518$	3214.78	0.272682
$519$	0	0
$520$	−6986.98	−0.589229
$521$	3638.19	0.305935	0.152967	−	0.988231i	$-0.451117\pi$
$521$	3638.19	0.305935	0.152967	+	0.988231i	$0.451117\pi$
$522$	0	0
$523$	16134.9	1.34901	0.674504	−	0.738271i	$-0.264357\pi$
$523$	16134.9	1.34901	0.674504	+	0.738271i	$0.264357\pi$
$524$	−3473.98	−0.289621
$525$	0	0
$526$	3624.87	0.300479
$527$	18688.5	1.54475
$528$	0	0
$529$	6865.82	0.564299
$530$	4555.39	0.373347
$531$	0	0
$532$	4730.51	0.385514
$533$	−10596.8	−0.861162
$534$	0	0
$535$	−6717.58	−0.542853
$536$	−961.332	−0.0774687
$537$	0	0
$538$	−2481.84	−0.198884
$539$	539.000	0.0430730
$540$	0	0
$541$	3076.95	0.244525	0.122263	−	0.992498i	$-0.460985\pi$
$541$	3076.95	0.244525	0.122263	+	0.992498i	$0.460985\pi$
$542$	−2553.53	−0.202368
$543$	0	0
$544$	−13170.6	−1.03802
$545$	−1884.22	−0.148094
$546$	0	0
$547$	3248.03	0.253886	0.126943	−	0.991910i	$-0.459483\pi$
$547$	3248.03	0.253886	0.126943	+	0.991910i	$0.459483\pi$
$548$	12310.9	0.959660
$549$	0	0
$550$	2311.81	0.179229
$551$	−11557.7	−0.893603
$552$	0	0
$553$	4120.58	0.316863
$554$	9744.44	0.747295
$555$	0	0
$556$	−3265.19	−0.249055
$557$	8464.19	0.643877	0.321938	−	0.946761i	$-0.395666\pi$
$557$	8464.19	0.643877	0.321938	+	0.946761i	$0.395666\pi$
$558$	0	0
$559$	−10553.3	−0.798489
$560$	354.158	0.0267248
$561$	0	0
$562$	618.621	0.0464323
$563$	−2750.50	−0.205897	−0.102948	−	0.994687i	$-0.532828\pi$
$563$	−2750.50	−0.205897	−0.102948	+	0.994687i	$0.532828\pi$
$564$	0	0
$565$	6554.70	0.488067
$566$	−9541.07	−0.708553
$567$	0	0
$568$	−3114.60	−0.230080
$569$	−20832.3	−1.53486	−0.767429	−	0.641134i	$-0.778464\pi$
$569$	−20832.3	−1.53486	−0.767429	+	0.641134i	$0.778464\pi$
$570$	0	0
$571$	5708.91	0.418407	0.209204	−	0.977872i	$-0.432913\pi$
$571$	5708.91	0.418407	0.209204	+	0.977872i	$0.432913\pi$
$572$	−3334.17	−0.243721
$573$	0	0
$574$	2009.09	0.146094
$575$	14924.5	1.08242
$576$	0	0
$577$	11087.7	0.799974	0.399987	−	0.916521i	$-0.369015\pi$
$577$	11087.7	0.799974	0.399987	+	0.916521i	$0.369015\pi$
$578$	−2678.18	−0.192729
$579$	0	0
$580$	1252.54	0.0896703
$581$	3370.79	0.240695
$582$	0	0
$583$	6289.17	0.446777
$584$	5659.35	0.401003
$585$	0	0
$586$	−6777.15	−0.477750
$587$	8468.55	0.595459	0.297730	−	0.954650i	$-0.403771\pi$
$587$	8468.55	0.595459	0.297730	+	0.954650i	$0.403771\pi$
$588$	0	0
$589$	−37678.9	−2.63587
$590$	5970.13	0.416587
$591$	0	0
$592$	2916.27	0.202463
$593$	−22765.6	−1.57651	−0.788256	−	0.615347i	$-0.789016\pi$
$593$	−22765.6	−1.57651	−0.788256	+	0.615347i	$0.789016\pi$
$594$	0	0
$595$	−2277.14	−0.156897
$596$	−10125.8	−0.695920
$597$	0	0
$598$	19224.3	1.31462
$599$	14725.7	1.00447	0.502233	−	0.864733i	$-0.332512\pi$
$599$	14725.7	1.00447	0.502233	+	0.864733i	$0.332512\pi$
$600$	0	0
$601$	−9283.77	−0.630105	−0.315052	−	0.949074i	$-0.602022\pi$
$601$	−9283.77	−0.630105	−0.315052	+	0.949074i	$0.602022\pi$
$602$	2000.83	0.135462
$603$	0	0
$604$	−7095.88	−0.478026
$605$	−496.247	−0.0333476
$606$	0	0
$607$	−445.560	−0.0297936	−0.0148968	−	0.999889i	$-0.504742\pi$
$607$	−445.560	−0.0297936	−0.0148968	+	0.999889i	$0.504742\pi$
$608$	26553.9	1.77122
$609$	0	0
$610$	−6041.00	−0.400972
$611$	38633.7	2.55802
$612$	0	0
$613$	1623.92	0.106997	0.0534987	−	0.998568i	$-0.482963\pi$
$613$	1623.92	0.106997	0.0534987	+	0.998568i	$0.482963\pi$
$614$	−8764.61	−0.576076
$615$	0	0
$616$	1828.86	0.119622
$617$	9598.21	0.626271	0.313135	−	0.949708i	$-0.398621\pi$
$617$	9598.21	0.626271	0.313135	+	0.949708i	$0.398621\pi$
$618$	0	0
$619$	−27233.7	−1.76836	−0.884179	−	0.467149i	$-0.845281\pi$
$619$	−27233.7	−1.76836	−0.884179	+	0.467149i	$0.845281\pi$
$620$	4083.35	0.264502
$621$	0	0
$622$	−1092.44	−0.0704223
$623$	9578.03	0.615948
$624$	0	0
$625$	9600.43	0.614428
$626$	6566.52	0.419251
$627$	0	0
$628$	−12512.3	−0.795057
$629$	−18750.8	−1.18862
$630$	0	0
$631$	18470.5	1.16529	0.582647	−	0.812725i	$-0.302017\pi$
$631$	18470.5	1.16529	0.582647	+	0.812725i	$0.302017\pi$
$632$	13981.4	0.879984
$633$	0	0
$634$	−7916.34	−0.495896
$635$	−160.835	−0.0100513
$636$	0	0
$637$	3514.66	0.218612
$638$	−1544.46	−0.0958396
$639$	0	0
$640$	−2091.39	−0.129171
$641$	24286.3	1.49649	0.748247	−	0.663420i	$-0.230896\pi$
$641$	24286.3	1.49649	0.748247	+	0.663420i	$0.230896\pi$
$642$	0	0
$643$	−16390.9	−1.00528	−0.502640	−	0.864496i	$-0.667638\pi$
$643$	−16390.9	−1.00528	−0.502640	+	0.864496i	$0.667638\pi$
$644$	4080.92	0.249706
$645$	0	0
$646$	24643.0	1.50088
$647$	−12590.9	−0.765067	−0.382534	−	0.923942i	$-0.624948\pi$
$647$	−12590.9	−0.765067	−0.382534	+	0.923942i	$0.624948\pi$
$648$	0	0
$649$	8242.35	0.498522
$650$	15074.6	0.909656
$651$	0	0
$652$	3089.70	0.185586
$653$	−9552.86	−0.572484	−0.286242	−	0.958157i	$-0.592406\pi$
$653$	−9552.86	−0.572484	−0.286242	+	0.958157i	$0.592406\pi$
$654$	0	0
$655$	−3371.56	−0.201126
$656$	1822.53	0.108472
$657$	0	0
$658$	−7324.70	−0.433961
$659$	546.125	0.0322823	0.0161411	−	0.999870i	$-0.494862\pi$
$659$	546.125	0.0322823	0.0161411	+	0.999870i	$0.494862\pi$
$660$	0	0
$661$	1618.43	0.0952338	0.0476169	−	0.998866i	$-0.484837\pi$
$661$	1618.43	0.0952338	0.0476169	+	0.998866i	$0.484837\pi$
$662$	10761.5	0.631809
$663$	0	0
$664$	11437.3	0.668454
$665$	4591.05	0.267719
$666$	0	0
$667$	−9970.62	−0.578806
$668$	1799.26	0.104215
$669$	0	0
$670$	−322.484	−0.0185950
$671$	−8340.20	−0.479836
$672$	0	0
$673$	−10901.5	−0.624403	−0.312202	−	0.950016i	$-0.601066\pi$
$673$	−10901.5	−0.624403	−0.312202	+	0.950016i	$0.601066\pi$
$674$	1456.64	0.0832461
$675$	0	0
$676$	−12457.1	−0.708754
$677$	9491.82	0.538848	0.269424	−	0.963022i	$-0.413167\pi$
$677$	9491.82	0.538848	0.269424	+	0.963022i	$0.413167\pi$
$678$	0	0
$679$	−4674.74	−0.264212
$680$	−7726.48	−0.435731
$681$	0	0
$682$	−5035.02	−0.282699
$683$	−1485.73	−0.0832355	−0.0416178	−	0.999134i	$-0.513251\pi$
$683$	−1485.73	−0.0832355	−0.0416178	+	0.999134i	$0.513251\pi$
$684$	0	0
$685$	11947.9	0.666433
$686$	−666.357	−0.0370869
$687$	0	0
$688$	1815.04	0.100578
$689$	41009.8	2.26756
$690$	0	0
$691$	−31472.9	−1.73268	−0.866341	−	0.499452i	$-0.833535\pi$
$691$	−31472.9	−1.73268	−0.866341	+	0.499452i	$0.833535\pi$
$692$	2079.51	0.114236
$693$	0	0
$694$	12939.2	0.707732
$695$	−3168.93	−0.172956
$696$	0	0
$697$	−11718.4	−0.636823
$698$	−23479.4	−1.27322
$699$	0	0
$700$	3200.02	0.172785
$701$	−18134.1	−0.977057	−0.488528	−	0.872548i	$-0.662466\pi$
$701$	−18134.1	−0.977057	−0.488528	+	0.872548i	$0.662466\pi$
$702$	0	0
$703$	37804.4	2.02819
$704$	4634.00	0.248083
$705$	0	0
$706$	−6166.04	−0.328699
$707$	−5607.64	−0.298299
$708$	0	0
$709$	−18224.7	−0.965363	−0.482682	−	0.875796i	$-0.660337\pi$
$709$	−18224.7	−0.965363	−0.482682	+	0.875796i	$0.660337\pi$
$710$	−1044.81	−0.0552267
$711$	0	0
$712$	32498.9	1.71060
$713$	−32504.8	−1.70731
$714$	0	0
$715$	−3235.88	−0.169252
$716$	−17050.8	−0.889970
$717$	0	0
$718$	−20436.3	−1.06222
$719$	4787.72	0.248333	0.124167	−	0.992261i	$-0.460374\pi$
$719$	4787.72	0.248333	0.124167	+	0.992261i	$0.460374\pi$
$720$	0	0
$721$	−2113.48	−0.109168
$722$	−36358.8	−1.87415
$723$	0	0
$724$	−6205.80	−0.318559
$725$	−7818.39	−0.400507
$726$	0	0
$727$	−18469.6	−0.942228	−0.471114	−	0.882072i	$-0.656148\pi$
$727$	−18469.6	−0.942228	−0.471114	+	0.882072i	$0.656148\pi$
$728$	11925.5	0.607125
$729$	0	0
$730$	1898.46	0.0962538
$731$	−11670.2	−0.590477
$732$	0	0
$733$	3108.79	0.156652	0.0783260	−	0.996928i	$-0.475043\pi$
$733$	3108.79	0.156652	0.0783260	+	0.996928i	$0.475043\pi$
$734$	2171.43	0.109195
$735$	0	0
$736$	22907.5	1.14726
$737$	−445.222	−0.0222523
$738$	0	0
$739$	11836.0	0.589165	0.294583	−	0.955626i	$-0.404819\pi$
$739$	11836.0	0.589165	0.294583	+	0.955626i	$0.404819\pi$
$740$	−4096.95	−0.203523
$741$	0	0
$742$	−7775.20	−0.384686
$743$	−426.050	−0.0210367	−0.0105183	−	0.999945i	$-0.503348\pi$
$743$	−426.050	−0.0210367	−0.0105183	+	0.999945i	$0.503348\pi$
$744$	0	0
$745$	−9827.27	−0.483280
$746$	−9836.98	−0.482785
$747$	0	0
$748$	−3687.06	−0.180230
$749$	11465.7	0.559340
$750$	0	0
$751$	5776.10	0.280656	0.140328	−	0.990105i	$-0.455184\pi$
$751$	5776.10	0.280656	0.140328	+	0.990105i	$0.455184\pi$
$752$	−6644.55	−0.322210
$753$	0	0
$754$	−10070.9	−0.486422
$755$	−6886.69	−0.331963
$756$	0	0
$757$	21009.3	1.00871	0.504356	−	0.863496i	$-0.331730\pi$
$757$	21009.3	1.00871	0.504356	+	0.863496i	$0.331730\pi$
$758$	−13616.2	−0.652456
$759$	0	0
$760$	15577.7	0.743505
$761$	2037.17	0.0970401	0.0485201	−	0.998822i	$-0.484550\pi$
$761$	2037.17	0.0970401	0.0485201	+	0.998822i	$0.484550\pi$
$762$	0	0
$763$	3216.01	0.152591
$764$	−20767.7	−0.983443
$765$	0	0
$766$	−7240.04	−0.341506
$767$	53745.9	2.53019
$768$	0	0
$769$	−7297.29	−0.342194	−0.171097	−	0.985254i	$-0.554731\pi$
$769$	−7297.29	−0.342194	−0.171097	+	0.985254i	$0.554731\pi$
$770$	613.502	0.0287131
$771$	0	0
$772$	−19546.0	−0.911236
$773$	21086.9	0.981166	0.490583	−	0.871394i	$-0.336784\pi$
$773$	21086.9	0.981166	0.490583	+	0.871394i	$0.336784\pi$
$774$	0	0
$775$	−25488.4	−1.18138
$776$	−15861.7	−0.733765
$777$	0	0
$778$	9793.54	0.451305
$779$	23626.0	1.08664
$780$	0	0
$781$	−1442.46	−0.0660888
$782$	21259.0	0.972151
$783$	0	0
$784$	−604.481	−0.0275365
$785$	−12143.4	−0.552125
$786$	0	0
$787$	−16997.6	−0.769886	−0.384943	−	0.922940i	$-0.625779\pi$
$787$	−16997.6	−0.769886	−0.384943	+	0.922940i	$0.625779\pi$
$788$	−12666.3	−0.572611
$789$	0	0
$790$	4690.14	0.211225
$791$	−11187.6	−0.502891
$792$	0	0
$793$	−54384.0	−2.43535
$794$	24352.4	1.08846
$795$	0	0
$796$	−5598.09	−0.249270
$797$	6632.97	0.294796	0.147398	−	0.989077i	$-0.452910\pi$
$797$	6632.97	0.294796	0.147398	+	0.989077i	$0.452910\pi$
$798$	0	0
$799$	42722.6	1.89164
$800$	17962.8	0.793851
$801$	0	0
$802$	28847.3	1.27012
$803$	2621.02	0.115185
$804$	0	0
$805$	3960.61	0.173408
$806$	−32831.9	−1.43481
$807$	0	0
$808$	−19027.1	−0.828429
$809$	31469.8	1.36764	0.683818	−	0.729653i	$-0.260318\pi$
$809$	31469.8	1.36764	0.683818	+	0.729653i	$0.260318\pi$
$810$	0	0
$811$	10571.5	0.457727	0.228864	−	0.973458i	$-0.426499\pi$
$811$	10571.5	0.457727	0.228864	+	0.973458i	$0.426499\pi$
$812$	−2137.85	−0.0923937
$813$	0	0
$814$	5051.80	0.217525
$815$	2998.61	0.128880
$816$	0	0
$817$	23528.9	1.00755
$818$	21049.4	0.899727
$819$	0	0
$820$	−2560.41	−0.109041
$821$	1867.28	0.0793770	0.0396885	−	0.999212i	$-0.487363\pi$
$821$	1867.28	0.0793770	0.0396885	+	0.999212i	$0.487363\pi$
$822$	0	0
$823$	−24030.7	−1.01781	−0.508904	−	0.860823i	$-0.669949\pi$
$823$	−24030.7	−1.01781	−0.508904	+	0.860823i	$0.669949\pi$
$824$	−7171.16	−0.303179
$825$	0	0
$826$	−10189.9	−0.429239
$827$	−7606.03	−0.319816	−0.159908	−	0.987132i	$-0.551120\pi$
$827$	−7606.03	−0.319816	−0.159908	+	0.987132i	$0.551120\pi$
$828$	0	0
$829$	−17718.4	−0.742321	−0.371161	−	0.928569i	$-0.621040\pi$
$829$	−17718.4	−0.742321	−0.371161	+	0.928569i	$0.621040\pi$
$830$	3836.71	0.160451
$831$	0	0
$832$	30216.9	1.25911
$833$	3886.65	0.161662
$834$	0	0
$835$	1746.22	0.0723718
$836$	7433.66	0.307534
$837$	0	0
$838$	17626.0	0.726585
$839$	−8104.23	−0.333479	−0.166740	−	0.986001i	$-0.553324\pi$
$839$	−8104.23	−0.333479	−0.166740	+	0.986001i	$0.553324\pi$
$840$	0	0
$841$	−19165.8	−0.785836
$842$	30519.6	1.24914
$843$	0	0
$844$	20204.6	0.824017
$845$	−12089.8	−0.492192
$846$	0	0
$847$	847.000	0.0343604
$848$	−7053.22	−0.285623
$849$	0	0
$850$	16670.1	0.672684
$851$	32613.1	1.31371
$852$	0	0
$853$	−11568.1	−0.464341	−0.232171	−	0.972675i	$-0.574583\pi$
$853$	−11568.1	−0.464341	−0.232171	+	0.972675i	$0.574583\pi$
$854$	10310.9	0.413150
$855$	0	0
$856$	38903.7	1.55339
$857$	−26485.6	−1.05570	−0.527848	−	0.849339i	$-0.677001\pi$
$857$	−26485.6	−1.05570	−0.527848	+	0.849339i	$0.677001\pi$
$858$	0	0
$859$	32310.6	1.28338	0.641689	−	0.766965i	$-0.278234\pi$
$859$	32310.6	1.28338	0.641689	+	0.766965i	$0.278234\pi$
$860$	−2549.88	−0.101105
$861$	0	0
$862$	−29526.4	−1.16667
$863$	−874.086	−0.0344777	−0.0172388	−	0.999851i	$-0.505488\pi$
$863$	−874.086	−0.0344777	−0.0172388	+	0.999851i	$0.505488\pi$
$864$	0	0
$865$	2018.21	0.0793308
$866$	−28497.0	−1.11821
$867$	0	0
$868$	−6969.51	−0.272535
$869$	6475.20	0.252769
$870$	0	0
$871$	−2903.16	−0.112939
$872$	10912.1	0.423774
$873$	0	0
$874$	−42861.4	−1.65882
$875$	6694.25	0.258636
$876$	0	0
$877$	12152.6	0.467919	0.233959	−	0.972246i	$-0.424832\pi$
$877$	12152.6	0.467919	0.233959	+	0.972246i	$0.424832\pi$
$878$	10593.3	0.407182
$879$	0	0
$880$	556.534	0.0213190
$881$	−21220.4	−0.811500	−0.405750	−	0.913984i	$-0.632990\pi$
$881$	−21220.4	−0.811500	−0.405750	+	0.913984i	$0.632990\pi$
$882$	0	0
$883$	−32328.9	−1.23211	−0.616055	−	0.787703i	$-0.711270\pi$
$883$	−32328.9	−1.23211	−0.616055	+	0.787703i	$0.711270\pi$
$884$	−24042.2	−0.914737
$885$	0	0
$886$	−34828.6	−1.32064
$887$	−46075.5	−1.74415	−0.872076	−	0.489370i	$-0.837227\pi$
$887$	−46075.5	−1.74415	−0.872076	+	0.489370i	$0.837227\pi$
$888$	0	0
$889$	274.516	0.0103565
$890$	10901.9	0.410600
$891$	0	0
$892$	−20695.8	−0.776845
$893$	−86135.2	−3.22778
$894$	0	0
$895$	−16548.1	−0.618037
$896$	3569.60	0.133094
$897$	0	0
$898$	−17769.4	−0.660326
$899$	17028.1	0.631723
$900$	0	0
$901$	45350.3	1.67684
$902$	3157.14	0.116543
$903$	0	0
$904$	−37960.3	−1.39662
$905$	−6022.85	−0.221222
$906$	0	0
$907$	2154.27	0.0788658	0.0394329	−	0.999222i	$-0.487445\pi$
$907$	2154.27	0.0788658	0.0394329	+	0.999222i	$0.487445\pi$
$908$	19904.5	0.727482
$909$	0	0
$910$	4000.46	0.145730
$911$	6282.73	0.228492	0.114246	−	0.993452i	$-0.463555\pi$
$911$	6282.73	0.228492	0.114246	+	0.993452i	$0.463555\pi$
$912$	0	0
$913$	5296.96	0.192009
$914$	779.997	0.0282276
$915$	0	0
$916$	−20032.6	−0.722595
$917$	5754.62	0.207235
$918$	0	0
$919$	−2441.49	−0.0876358	−0.0438179	−	0.999040i	$-0.513952\pi$
$919$	−2441.49	−0.0876358	−0.0438179	+	0.999040i	$0.513952\pi$
$920$	13438.6	0.481584
$921$	0	0
$922$	−2118.03	−0.0756546
$923$	−9405.86	−0.335426
$924$	0	0
$925$	25573.4	0.909024
$926$	−14498.6	−0.514527
$927$	0	0
$928$	−12000.4	−0.424497
$929$	−43741.5	−1.54479	−0.772396	−	0.635141i	$-0.780942\pi$
$929$	−43741.5	−1.54479	−0.772396	+	0.635141i	$0.780942\pi$
$930$	0	0
$931$	−7836.06	−0.275850
$932$	12286.0	0.431805
$933$	0	0
$934$	10946.1	0.383476
$935$	−3578.36	−0.125160
$936$	0	0
$937$	−5530.26	−0.192813	−0.0964064	−	0.995342i	$-0.530735\pi$
$937$	−5530.26	−0.192813	−0.0964064	+	0.995342i	$0.530735\pi$
$938$	550.420	0.0191598
$939$	0	0
$940$	9334.67	0.323897
$941$	−53251.1	−1.84478	−0.922389	−	0.386262i	$-0.873766\pi$
$941$	−53251.1	−1.84478	−0.922389	+	0.386262i	$0.873766\pi$
$942$	0	0
$943$	20381.7	0.703838
$944$	−9243.68	−0.318704
$945$	0	0
$946$	3144.17	0.108061
$947$	−14734.1	−0.505590	−0.252795	−	0.967520i	$-0.581350\pi$
$947$	−14734.1	−0.505590	−0.252795	+	0.967520i	$0.581350\pi$
$948$	0	0
$949$	17090.9	0.584608
$950$	−33609.5	−1.14783
$951$	0	0
$952$	13187.6	0.448964
$953$	14754.5	0.501517	0.250758	−	0.968050i	$-0.419320\pi$
$953$	14754.5	0.501517	0.250758	+	0.968050i	$0.419320\pi$
$954$	0	0
$955$	−20155.5	−0.682949
$956$	11711.7	0.396216
$957$	0	0
$958$	−15804.1	−0.532992
$959$	−20392.9	−0.686674
$960$	0	0
$961$	25721.6	0.863402
$962$	32941.3	1.10402
$963$	0	0
$964$	8913.73	0.297813
$965$	−18969.7	−0.632805
$966$	0	0
$967$	45826.3	1.52397	0.761983	−	0.647597i	$-0.224226\pi$
$967$	45826.3	1.52397	0.761983	+	0.647597i	$0.224226\pi$
$968$	2873.92	0.0954250
$969$	0	0
$970$	−5320.90	−0.176128
$971$	−13645.4	−0.450980	−0.225490	−	0.974245i	$-0.572398\pi$
$971$	−13645.4	−0.450980	−0.225490	+	0.974245i	$0.572398\pi$
$972$	0	0
$973$	5408.76	0.178209
$974$	36203.3	1.19099
$975$	0	0
$976$	9353.42	0.306758
$977$	−7591.69	−0.248597	−0.124299	−	0.992245i	$-0.539668\pi$
$977$	−7591.69	−0.248597	−0.124299	+	0.992245i	$0.539668\pi$
$978$	0	0
$979$	15051.2	0.491357
$980$	849.213	0.0276807
$981$	0	0
$982$	−34676.3	−1.12685
$983$	9428.03	0.305908	0.152954	−	0.988233i	$-0.451121\pi$
$983$	9428.03	0.305908	0.152954	+	0.988233i	$0.451121\pi$
$984$	0	0
$985$	−12292.9	−0.397648
$986$	−11136.9	−0.359706
$987$	0	0
$988$	48472.7	1.56085
$989$	20297.9	0.652615
$990$	0	0
$991$	−47149.3	−1.51135	−0.755675	−	0.654947i	$-0.772691\pi$
$991$	−47149.3	−1.51135	−0.755675	+	0.654947i	$0.772691\pi$
$992$	−39122.1	−1.25215
$993$	0	0
$994$	1783.29	0.0569040
$995$	−5433.05	−0.173105
$996$	0	0
$997$	28331.5	0.899968	0.449984	−	0.893037i	$-0.351430\pi$
$997$	28331.5	0.899968	0.449984	+	0.893037i	$0.351430\pi$
$998$	−11010.0	−0.349213
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.4.a.u.1.3	yes	8
3.2	odd	2		693.4.a.r.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.4.a.r.1.6	✓	8	3.2	odd	2
693.4.a.u.1.3	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)

\(f(q)\)	\(=\)	\(q-1.94273 q^{2} -4.22579 q^{4} -4.10121 q^{5} +7.00000 q^{7} +23.7514 q^{8} +O(q^{10})\)	\(q-1.94273 q^{2} -4.22579 q^{4} -4.10121 q^{5} +7.00000 q^{7} +23.7514 q^{8} +7.96756 q^{10} +11.0000 q^{11} +71.7277 q^{13} -13.5991 q^{14} -12.3363 q^{16} +79.3194 q^{17} -159.920 q^{19} +17.3309 q^{20} -21.3701 q^{22} -137.959 q^{23} -108.180 q^{25} -139.348 q^{26} -29.5805 q^{28} +72.2721 q^{29} +235.611 q^{31} -166.045 q^{32} -154.096 q^{34} -28.7085 q^{35} -236.396 q^{37} +310.681 q^{38} -97.4097 q^{40} -147.737 q^{41} -147.130 q^{43} -46.4837 q^{44} +268.018 q^{46} +538.615 q^{47} +49.0000 q^{49} +210.165 q^{50} -303.106 q^{52} +571.743 q^{53} -45.1133 q^{55} +166.260 q^{56} -140.405 q^{58} +749.305 q^{59} -758.200 q^{61} -457.729 q^{62} +421.272 q^{64} -294.171 q^{65} -40.4747 q^{67} -335.187 q^{68} +55.7729 q^{70} -131.133 q^{71} +238.274 q^{73} +459.255 q^{74} +675.787 q^{76} +77.0000 q^{77} +588.655 q^{79} +50.5940 q^{80} +287.013 q^{82} +481.542 q^{83} -325.306 q^{85} +285.833 q^{86} +261.266 q^{88} +1368.29 q^{89} +502.094 q^{91} +582.988 q^{92} -1046.39 q^{94} +655.865 q^{95} -667.821 q^{97} -95.1939 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{2} + 30 q^{4} + 10 q^{5} + 56 q^{7} + 81 q^{8}+O(q^{10}) \) 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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