LMFDB - Embedded newform 4527.2.a.k.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4527,2,Mod(1,4527)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4527, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4527.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4527 = 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4527.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:	$36.1482769950$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.858231$ of defining polynomial
Character	$\chi$	$=$	4527.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.58686 q^{2} +4.69185 q^{4} -1.44291 q^{5} -1.96509 q^{7} +6.96343 q^{8} +O(q^{10})$	\(q+2.58686 q^{2} +4.69185 q^{4} -1.44291 q^{5} -1.96509 q^{7} +6.96343 q^{8} -3.73261 q^{10} -2.85614 q^{11} -3.84427 q^{13} -5.08341 q^{14} +8.62972 q^{16} -1.30329 q^{17} +3.53196 q^{19} -6.76991 q^{20} -7.38845 q^{22} -4.20650 q^{23} -2.91801 q^{25} -9.94458 q^{26} -9.21989 q^{28} +1.10402 q^{29} -3.53201 q^{31} +8.39703 q^{32} -3.37144 q^{34} +2.83545 q^{35} -6.61903 q^{37} +9.13668 q^{38} -10.0476 q^{40} -0.671095 q^{41} -4.32289 q^{43} -13.4006 q^{44} -10.8816 q^{46} +0.378524 q^{47} -3.13843 q^{49} -7.54849 q^{50} -18.0367 q^{52} +7.13700 q^{53} +4.12116 q^{55} -13.6838 q^{56} +2.85594 q^{58} +13.8806 q^{59} -6.51374 q^{61} -9.13682 q^{62} +4.46249 q^{64} +5.54693 q^{65} -7.55189 q^{67} -6.11485 q^{68} +7.33490 q^{70} -0.0744967 q^{71} +3.29052 q^{73} -17.1225 q^{74} +16.5714 q^{76} +5.61258 q^{77} +8.57578 q^{79} -12.4519 q^{80} -1.73603 q^{82} -14.8165 q^{83} +1.88053 q^{85} -11.1827 q^{86} -19.8886 q^{88} +17.5415 q^{89} +7.55432 q^{91} -19.7362 q^{92} +0.979188 q^{94} -5.09630 q^{95} -8.01702 q^{97} -8.11867 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * 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 $

$2$	2.58686	1.82919	0.914593	−	0.404375i	$-0.132511\pi$
$2$	2.58686	1.82919	0.914593	+	0.404375i	$0.132511\pi$
$3$	0	0
$3$	0	0
$4$	4.69185	2.34592
$5$	−1.44291	−0.645289	−0.322644	−	0.946520i	$-0.604572\pi$
$5$	−1.44291	−0.645289	−0.322644	+	0.946520i	$0.604572\pi$
$6$	0	0
$7$	−1.96509	−0.742734	−0.371367	−	0.928486i	$-0.621111\pi$
$7$	−1.96509	−0.742734	−0.371367	+	0.928486i	$0.621111\pi$
$8$	6.96343	2.46194
$9$	0	0
$10$	−3.73261	−1.18035
$11$	−2.85614	−0.861160	−0.430580	−	0.902552i	$-0.641691\pi$
$11$	−2.85614	−0.861160	−0.430580	+	0.902552i	$0.641691\pi$
$12$	0	0
$13$	−3.84427	−1.06621	−0.533104	−	0.846050i	$-0.678974\pi$
$13$	−3.84427	−1.06621	−0.533104	+	0.846050i	$0.678974\pi$
$14$	−5.08341	−1.35860
$15$	0	0
$16$	8.62972	2.15743
$17$	−1.30329	−0.316095	−0.158047	−	0.987432i	$-0.550520\pi$
$17$	−1.30329	−0.316095	−0.158047	+	0.987432i	$0.550520\pi$
$18$	0	0
$19$	3.53196	0.810287	0.405143	−	0.914253i	$-0.367222\pi$
$19$	3.53196	0.810287	0.405143	+	0.914253i	$0.367222\pi$
$20$	−6.76991	−1.51380
$21$	0	0
$22$	−7.38845	−1.57522
$23$	−4.20650	−0.877115	−0.438558	−	0.898703i	$-0.644510\pi$
$23$	−4.20650	−0.877115	−0.438558	+	0.898703i	$0.644510\pi$
$24$	0	0
$25$	−2.91801	−0.583602
$26$	−9.94458	−1.95029
$27$	0	0
$28$	−9.21989	−1.74240
$29$	1.10402	0.205011	0.102506	−	0.994732i	$-0.467314\pi$
$29$	1.10402	0.205011	0.102506	+	0.994732i	$0.467314\pi$
$30$	0	0
$31$	−3.53201	−0.634368	−0.317184	−	0.948364i	$-0.602737\pi$
$31$	−3.53201	−0.634368	−0.317184	+	0.948364i	$0.602737\pi$
$32$	8.39703	1.48440
$33$	0	0
$34$	−3.37144	−0.578196
$35$	2.83545	0.479278
$36$	0	0
$37$	−6.61903	−1.08816	−0.544081	−	0.839033i	$-0.683121\pi$
$37$	−6.61903	−1.08816	−0.544081	+	0.839033i	$0.683121\pi$
$38$	9.13668	1.48217
$39$	0	0
$40$	−10.0476	−1.58866
$41$	−0.671095	−0.104807	−0.0524037	−	0.998626i	$-0.516688\pi$
$41$	−0.671095	−0.104807	−0.0524037	+	0.998626i	$0.516688\pi$
$42$	0	0
$43$	−4.32289	−0.659235	−0.329617	−	0.944115i	$-0.606920\pi$
$43$	−4.32289	−0.659235	−0.329617	+	0.944115i	$0.606920\pi$
$44$	−13.4006	−2.02021
$45$	0	0
$46$	−10.8816	−1.60441
$47$	0.378524	0.0552134	0.0276067	−	0.999619i	$-0.491211\pi$
$47$	0.378524	0.0552134	0.0276067	+	0.999619i	$0.491211\pi$
$48$	0	0
$49$	−3.13843	−0.448347
$50$	−7.54849	−1.06752
$51$	0	0
$52$	−18.0367	−2.50124
$53$	7.13700	0.980342	0.490171	−	0.871626i	$-0.336934\pi$
$53$	7.13700	0.980342	0.490171	+	0.871626i	$0.336934\pi$
$54$	0	0
$55$	4.12116	0.555697
$56$	−13.6838	−1.82857
$57$	0	0
$58$	2.85594	0.375004
$59$	13.8806	1.80709	0.903547	−	0.428489i	$-0.140954\pi$
$59$	13.8806	1.80709	0.903547	+	0.428489i	$0.140954\pi$
$60$	0	0
$61$	−6.51374	−0.833999	−0.417000	−	0.908907i	$-0.636918\pi$
$61$	−6.51374	−0.833999	−0.417000	+	0.908907i	$0.636918\pi$
$62$	−9.13682	−1.16038
$63$	0	0
$64$	4.46249	0.557812
$65$	5.54693	0.688012
$66$	0	0
$67$	−7.55189	−0.922610	−0.461305	−	0.887242i	$-0.652619\pi$
$67$	−7.55189	−0.922610	−0.461305	+	0.887242i	$0.652619\pi$
$68$	−6.11485	−0.741534
$69$	0	0
$70$	7.33490	0.876688
$71$	−0.0744967	−0.00884113	−0.00442057	−	0.999990i	$-0.501407\pi$
$71$	−0.0744967	−0.00884113	−0.00442057	+	0.999990i	$0.501407\pi$
$72$	0	0
$73$	3.29052	0.385126	0.192563	−	0.981285i	$-0.438320\pi$
$73$	3.29052	0.385126	0.192563	+	0.981285i	$0.438320\pi$
$74$	−17.1225	−1.99045
$75$	0	0
$76$	16.5714	1.90087
$77$	5.61258	0.639612
$78$	0	0
$79$	8.57578	0.964850	0.482425	−	0.875937i	$-0.339756\pi$
$79$	8.57578	0.964850	0.482425	+	0.875937i	$0.339756\pi$
$80$	−12.4519	−1.39217
$81$	0	0
$82$	−1.73603	−0.191712
$83$	−14.8165	−1.62632	−0.813158	−	0.582042i	$-0.802254\pi$
$83$	−14.8165	−1.62632	−0.813158	+	0.582042i	$0.802254\pi$
$84$	0	0
$85$	1.88053	0.203973
$86$	−11.1827	−1.20586
$87$	0	0
$88$	−19.8886	−2.12013
$89$	17.5415	1.85940	0.929700	−	0.368318i	$-0.120066\pi$
$89$	17.5415	1.85940	0.929700	+	0.368318i	$0.120066\pi$
$90$	0	0
$91$	7.55432	0.791908
$92$	−19.7362	−2.05764
$93$	0	0
$94$	0.979188	0.100996
$95$	−5.09630	−0.522869
$96$	0	0
$97$	−8.01702	−0.814005	−0.407003	−	0.913427i	$-0.633426\pi$
$97$	−8.01702	−0.814005	−0.407003	+	0.913427i	$0.633426\pi$
$98$	−8.11867	−0.820110
$99$	0	0
$100$	−13.6909	−1.36909
$101$	−9.11148	−0.906626	−0.453313	−	0.891351i	$-0.649758\pi$
$101$	−9.11148	−0.906626	−0.453313	+	0.891351i	$0.649758\pi$
$102$	0	0
$103$	−15.2943	−1.50699	−0.753494	−	0.657455i	$-0.771633\pi$
$103$	−15.2943	−1.50699	−0.753494	+	0.657455i	$0.771633\pi$
$104$	−26.7693	−2.62494
$105$	0	0
$106$	18.4624	1.79323
$107$	−3.50420	−0.338764	−0.169382	−	0.985550i	$-0.554177\pi$
$107$	−3.50420	−0.338764	−0.169382	+	0.985550i	$0.554177\pi$
$108$	0	0
$109$	2.49301	0.238787	0.119393	−	0.992847i	$-0.461905\pi$
$109$	2.49301	0.238787	0.119393	+	0.992847i	$0.461905\pi$
$110$	10.6609	1.01647
$111$	0	0
$112$	−16.9582	−1.60240
$113$	14.5343	1.36727	0.683634	−	0.729825i	$-0.260399\pi$
$113$	14.5343	1.36727	0.683634	+	0.729825i	$0.260399\pi$
$114$	0	0
$115$	6.06959	0.565993
$116$	5.17989	0.480941
$117$	0	0
$118$	35.9070	3.30551
$119$	2.56109	0.234774
$120$	0	0
$121$	−2.84244	−0.258404
$122$	−16.8501	−1.52554
$123$	0	0
$124$	−16.5716	−1.48818
$125$	11.4250	1.02188
$126$	0	0
$127$	5.52702	0.490443	0.245222	−	0.969467i	$-0.421139\pi$
$127$	5.52702	0.490443	0.245222	+	0.969467i	$0.421139\pi$
$128$	−5.25021	−0.464057
$129$	0	0
$130$	14.3491	1.25850
$131$	21.2174	1.85377	0.926885	−	0.375346i	$-0.122476\pi$
$131$	21.2174	1.85377	0.926885	+	0.375346i	$0.122476\pi$
$132$	0	0
$133$	−6.94061	−0.601827
$134$	−19.5357	−1.68763
$135$	0	0
$136$	−9.07538	−0.778208
$137$	−11.6794	−0.997835	−0.498917	−	0.866650i	$-0.666269\pi$
$137$	−11.6794	−0.997835	−0.498917	+	0.866650i	$0.666269\pi$
$138$	0	0
$139$	−10.8676	−0.921778	−0.460889	−	0.887458i	$-0.652469\pi$
$139$	−10.8676	−0.921778	−0.460889	+	0.887458i	$0.652469\pi$
$140$	13.3035	1.12435
$141$	0	0
$142$	−0.192713	−0.0161721
$143$	10.9798	0.918175
$144$	0	0
$145$	−1.59300	−0.132292
$146$	8.51212	0.704468
$147$	0	0
$148$	−31.0555	−2.55274
$149$	7.40209	0.606403	0.303202	−	0.952926i	$-0.401944\pi$
$149$	7.40209	0.606403	0.303202	+	0.952926i	$0.401944\pi$
$150$	0	0
$151$	6.09188	0.495750	0.247875	−	0.968792i	$-0.420268\pi$
$151$	6.09188	0.495750	0.247875	+	0.968792i	$0.420268\pi$
$152$	24.5945	1.99488
$153$	0	0
$154$	14.5189	1.16997
$155$	5.09637	0.409350
$156$	0	0
$157$	2.22790	0.177806	0.0889028	−	0.996040i	$-0.471664\pi$
$157$	2.22790	0.177806	0.0889028	+	0.996040i	$0.471664\pi$
$158$	22.1843	1.76489
$159$	0	0
$160$	−12.1162	−0.957866
$161$	8.26614	0.651463
$162$	0	0
$163$	−18.7370	−1.46759	−0.733797	−	0.679368i	$-0.762254\pi$
$163$	−18.7370	−1.46759	−0.733797	+	0.679368i	$0.762254\pi$
$164$	−3.14867	−0.245870
$165$	0	0
$166$	−38.3281	−2.97484
$167$	−22.7237	−1.75841	−0.879205	−	0.476443i	$-0.841926\pi$
$167$	−22.7237	−1.75841	−0.879205	+	0.476443i	$0.841926\pi$
$168$	0	0
$169$	1.77838	0.136798
$170$	4.86468	0.373104
$171$	0	0
$172$	−20.2823	−1.54651
$173$	2.25141	0.171172	0.0855859	−	0.996331i	$-0.472724\pi$
$173$	2.25141	0.171172	0.0855859	+	0.996331i	$0.472724\pi$
$174$	0	0
$175$	5.73415	0.433461
$176$	−24.6477	−1.85789
$177$	0	0
$178$	45.3775	3.40119
$179$	−15.8986	−1.18832	−0.594159	−	0.804348i	$-0.702515\pi$
$179$	−15.8986	−1.18832	−0.594159	+	0.804348i	$0.702515\pi$
$180$	0	0
$181$	16.5942	1.23344	0.616718	−	0.787184i	$-0.288462\pi$
$181$	16.5942	1.23344	0.616718	+	0.787184i	$0.288462\pi$
$182$	19.5420	1.44855
$183$	0	0
$184$	−29.2916	−2.15941
$185$	9.55066	0.702179
$186$	0	0
$187$	3.72239	0.272208
$188$	1.77598	0.129526
$189$	0	0
$190$	−13.1834	−0.956425
$191$	2.03640	0.147348	0.0736742	−	0.997282i	$-0.476527\pi$
$191$	2.03640	0.147348	0.0736742	+	0.997282i	$0.476527\pi$
$192$	0	0
$193$	−15.1384	−1.08968	−0.544841	−	0.838539i	$-0.683410\pi$
$193$	−15.1384	−1.08968	−0.544841	+	0.838539i	$0.683410\pi$
$194$	−20.7389	−1.48897
$195$	0	0
$196$	−14.7250	−1.05179
$197$	6.01285	0.428398	0.214199	−	0.976790i	$-0.431286\pi$
$197$	6.01285	0.428398	0.214199	+	0.976790i	$0.431286\pi$
$198$	0	0
$199$	23.5513	1.66951	0.834753	−	0.550624i	$-0.185610\pi$
$199$	23.5513	1.66951	0.834753	+	0.550624i	$0.185610\pi$
$200$	−20.3194	−1.43680
$201$	0	0
$202$	−23.5701	−1.65839
$203$	−2.16950	−0.152269
$204$	0	0
$205$	0.968329	0.0676311
$206$	−39.5641	−2.75656
$207$	0	0
$208$	−33.1749	−2.30027
$209$	−10.0878	−0.697786
$210$	0	0
$211$	2.35555	0.162163	0.0810814	−	0.996707i	$-0.474163\pi$
$211$	2.35555	0.162163	0.0810814	+	0.996707i	$0.474163\pi$
$212$	33.4857	2.29981
$213$	0	0
$214$	−9.06488	−0.619662
$215$	6.23754	0.425397
$216$	0	0
$217$	6.94071	0.471166
$218$	6.44906	0.436785
$219$	0	0
$220$	19.3358	1.30362
$221$	5.01020	0.337023
$222$	0	0
$223$	13.6268	0.912519	0.456260	−	0.889847i	$-0.349189\pi$
$223$	13.6268	0.912519	0.456260	+	0.889847i	$0.349189\pi$
$224$	−16.5009	−1.10251
$225$	0	0
$226$	37.5981	2.50099
$227$	−11.9812	−0.795218	−0.397609	−	0.917555i	$-0.630160\pi$
$227$	−11.9812	−0.795218	−0.397609	+	0.917555i	$0.630160\pi$
$228$	0	0
$229$	27.2066	1.79786	0.898932	−	0.438087i	$-0.144344\pi$
$229$	27.2066	1.79786	0.898932	+	0.438087i	$0.144344\pi$
$230$	15.7012	1.03531
$231$	0	0
$232$	7.68776	0.504726
$233$	−14.4504	−0.946679	−0.473339	−	0.880880i	$-0.656952\pi$
$233$	−14.4504	−0.946679	−0.473339	+	0.880880i	$0.656952\pi$
$234$	0	0
$235$	−0.546176	−0.0356286
$236$	65.1254	4.23930
$237$	0	0
$238$	6.62517	0.429446
$239$	23.9361	1.54830	0.774148	−	0.633005i	$-0.218179\pi$
$239$	23.9361	1.54830	0.774148	+	0.633005i	$0.218179\pi$
$240$	0	0
$241$	−13.7734	−0.887224	−0.443612	−	0.896219i	$-0.646303\pi$
$241$	−13.7734	−0.887224	−0.443612	+	0.896219i	$0.646303\pi$
$242$	−7.35300	−0.472668
$243$	0	0
$244$	−30.5615	−1.95650
$245$	4.52847	0.289313
$246$	0	0
$247$	−13.5778	−0.863934
$248$	−24.5949	−1.56178
$249$	0	0
$250$	29.5548	1.86921
$251$	8.26975	0.521982	0.260991	−	0.965341i	$-0.415951\pi$
$251$	8.26975	0.521982	0.260991	+	0.965341i	$0.415951\pi$
$252$	0	0
$253$	12.0144	0.755336
$254$	14.2976	0.897112
$255$	0	0
$256$	−22.5065	−1.40666
$257$	20.1486	1.25684	0.628418	−	0.777876i	$-0.283703\pi$
$257$	20.1486	1.25684	0.628418	+	0.777876i	$0.283703\pi$
$258$	0	0
$259$	13.0070	0.808214
$260$	26.0253	1.61402
$261$	0	0
$262$	54.8863	3.39089
$263$	27.9229	1.72180	0.860900	−	0.508774i	$-0.169901\pi$
$263$	27.9229	1.72180	0.860900	+	0.508774i	$0.169901\pi$
$264$	0	0
$265$	−10.2980	−0.632604
$266$	−17.9544	−1.10085
$267$	0	0
$268$	−35.4323	−2.16437
$269$	−11.6165	−0.708269	−0.354135	−	0.935194i	$-0.615225\pi$
$269$	−11.6165	−0.708269	−0.354135	+	0.935194i	$0.615225\pi$
$270$	0	0
$271$	−3.27787	−0.199117	−0.0995583	−	0.995032i	$-0.531743\pi$
$271$	−3.27787	−0.199117	−0.0995583	+	0.995032i	$0.531743\pi$
$272$	−11.2471	−0.681953
$273$	0	0
$274$	−30.2129	−1.82523
$275$	8.33426	0.502575
$276$	0	0
$277$	10.1157	0.607795	0.303897	−	0.952705i	$-0.401712\pi$
$277$	10.1157	0.607795	0.303897	+	0.952705i	$0.401712\pi$
$278$	−28.1130	−1.68610
$279$	0	0
$280$	19.7444	1.17995
$281$	−16.7656	−1.00015	−0.500077	−	0.865981i	$-0.666695\pi$
$281$	−16.7656	−1.00015	−0.500077	+	0.865981i	$0.666695\pi$
$282$	0	0
$283$	15.4382	0.917705	0.458852	−	0.888513i	$-0.348261\pi$
$283$	15.4382	0.917705	0.458852	+	0.888513i	$0.348261\pi$
$284$	−0.349527	−0.0207406
$285$	0	0
$286$	28.4031	1.67951
$287$	1.31876	0.0778440
$288$	0	0
$289$	−15.3014	−0.900084
$290$	−4.12087	−0.241986
$291$	0	0
$292$	15.4386	0.903477
$293$	7.52887	0.439841	0.219921	−	0.975518i	$-0.429420\pi$
$293$	7.52887	0.439841	0.219921	+	0.975518i	$0.429420\pi$
$294$	0	0
$295$	−20.0284	−1.16610
$296$	−46.0911	−2.67899
$297$	0	0
$298$	19.1482	1.10922
$299$	16.1709	0.935187
$300$	0	0
$301$	8.49487	0.489636
$302$	15.7588	0.906819
$303$	0	0
$304$	30.4798	1.74814
$305$	9.39874	0.538170
$306$	0	0
$307$	17.1932	0.981265	0.490632	−	0.871367i	$-0.336766\pi$
$307$	17.1932	0.981265	0.490632	+	0.871367i	$0.336766\pi$
$308$	26.3333	1.50048
$309$	0	0
$310$	13.1836	0.748778
$311$	−10.1595	−0.576094	−0.288047	−	0.957616i	$-0.593006\pi$
$311$	−10.1595	−0.576094	−0.288047	+	0.957616i	$0.593006\pi$
$312$	0	0
$313$	26.6909	1.50866	0.754329	−	0.656496i	$-0.227962\pi$
$313$	26.6909	1.50866	0.754329	+	0.656496i	$0.227962\pi$
$314$	5.76326	0.325240
$315$	0	0
$316$	40.2362	2.26346
$317$	−19.3855	−1.08880	−0.544398	−	0.838827i	$-0.683242\pi$
$317$	−19.3855	−1.08880	−0.544398	+	0.838827i	$0.683242\pi$
$318$	0	0
$319$	−3.15324	−0.176548
$320$	−6.43898	−0.359950
$321$	0	0
$322$	21.3833	1.19165
$323$	−4.60317	−0.256127
$324$	0	0
$325$	11.2176	0.622241
$326$	−48.4700	−2.68450
$327$	0	0
$328$	−4.67312	−0.258030
$329$	−0.743833	−0.0410088
$330$	0	0
$331$	−26.0579	−1.43227	−0.716135	−	0.697962i	$-0.754090\pi$
$331$	−26.0579	−1.43227	−0.716135	+	0.697962i	$0.754090\pi$
$332$	−69.5165	−3.81521
$333$	0	0
$334$	−58.7830	−3.21646
$335$	10.8967	0.595350
$336$	0	0
$337$	−11.6240	−0.633199	−0.316600	−	0.948559i	$-0.602541\pi$
$337$	−11.6240	−0.633199	−0.316600	+	0.948559i	$0.602541\pi$
$338$	4.60042	0.250230
$339$	0	0
$340$	8.82317	0.478504
$341$	10.0879	0.546292
$342$	0	0
$343$	19.9229	1.07574
$344$	−30.1021	−1.62300
$345$	0	0
$346$	5.82409	0.313105
$347$	−24.5102	−1.31578	−0.657888	−	0.753115i	$-0.728550\pi$
$347$	−24.5102	−1.31578	−0.657888	+	0.753115i	$0.728550\pi$
$348$	0	0
$349$	10.6510	0.570133	0.285066	−	0.958508i	$-0.407984\pi$
$349$	10.6510	0.570133	0.285066	+	0.958508i	$0.407984\pi$
$350$	14.8334	0.792881
$351$	0	0
$352$	−23.9831	−1.27830
$353$	−1.04651	−0.0557003	−0.0278501	−	0.999612i	$-0.508866\pi$
$353$	−1.04651	−0.0557003	−0.0278501	+	0.999612i	$0.508866\pi$
$354$	0	0
$355$	0.107492	0.00570508
$356$	82.3022	4.36201
$357$	0	0
$358$	−41.1275	−2.17365
$359$	24.0194	1.26769	0.633847	−	0.773459i	$-0.281475\pi$
$359$	24.0194	1.26769	0.633847	+	0.773459i	$0.281475\pi$
$360$	0	0
$361$	−6.52528	−0.343436
$362$	42.9268	2.25618
$363$	0	0
$364$	35.4437	1.85776
$365$	−4.74793	−0.248518
$366$	0	0
$367$	−18.6963	−0.975937	−0.487969	−	0.872861i	$-0.662262\pi$
$367$	−18.6963	−0.975937	−0.487969	+	0.872861i	$0.662262\pi$
$368$	−36.3009	−1.89231
$369$	0	0
$370$	24.7062	1.28442
$371$	−14.0248	−0.728133
$372$	0	0
$373$	−23.3008	−1.20647	−0.603235	−	0.797564i	$-0.706122\pi$
$373$	−23.3008	−1.20647	−0.603235	+	0.797564i	$0.706122\pi$
$374$	9.62931	0.497920
$375$	0	0
$376$	2.63582	0.135932
$377$	−4.24415	−0.218585
$378$	0	0
$379$	−9.79951	−0.503367	−0.251683	−	0.967810i	$-0.580984\pi$
$379$	−9.79951	−0.503367	−0.251683	+	0.967810i	$0.580984\pi$
$380$	−23.9110	−1.22661
$381$	0	0
$382$	5.26787	0.269528
$383$	1.97295	0.100813	0.0504066	−	0.998729i	$-0.483948\pi$
$383$	1.97295	0.100813	0.0504066	+	0.998729i	$0.483948\pi$
$384$	0	0
$385$	−8.09844	−0.412735
$386$	−39.1608	−1.99323
$387$	0	0
$388$	−37.6146	−1.90959
$389$	6.61202	0.335243	0.167621	−	0.985851i	$-0.446391\pi$
$389$	6.61202	0.335243	0.167621	+	0.985851i	$0.446391\pi$
$390$	0	0
$391$	5.48230	0.277252
$392$	−21.8542	−1.10380
$393$	0	0
$394$	15.5544	0.783619
$395$	−12.3741	−0.622607
$396$	0	0
$397$	−7.61750	−0.382311	−0.191156	−	0.981560i	$-0.561224\pi$
$397$	−7.61750	−0.382311	−0.191156	+	0.981560i	$0.561224\pi$
$398$	60.9239	3.05384
$399$	0	0
$400$	−25.1816	−1.25908
$401$	−25.8162	−1.28920	−0.644601	−	0.764519i	$-0.722976\pi$
$401$	−25.8162	−1.28920	−0.644601	+	0.764519i	$0.722976\pi$
$402$	0	0
$403$	13.5780	0.676368
$404$	−42.7497	−2.12687
$405$	0	0
$406$	−5.61218	−0.278528
$407$	18.9049	0.937081
$408$	0	0
$409$	−13.6924	−0.677044	−0.338522	−	0.940958i	$-0.609927\pi$
$409$	−13.6924	−0.677044	−0.338522	+	0.940958i	$0.609927\pi$
$410$	2.50493	0.123710
$411$	0	0
$412$	−71.7583	−3.53528
$413$	−27.2765	−1.34219
$414$	0	0
$415$	21.3788	1.04944
$416$	−32.2804	−1.58268
$417$	0	0
$418$	−26.0957	−1.27638
$419$	−24.0027	−1.17261	−0.586305	−	0.810091i	$-0.699418\pi$
$419$	−24.0027	−1.17261	−0.586305	+	0.810091i	$0.699418\pi$
$420$	0	0
$421$	31.1293	1.51715	0.758573	−	0.651588i	$-0.225897\pi$
$421$	31.1293	1.51715	0.758573	+	0.651588i	$0.225897\pi$
$422$	6.09348	0.296626
$423$	0	0
$424$	49.6979	2.41355
$425$	3.80302	0.184474
$426$	0	0
$427$	12.8001	0.619439
$428$	−16.4412	−0.794714
$429$	0	0
$430$	16.1357	0.778130
$431$	−28.0585	−1.35153	−0.675766	−	0.737116i	$-0.736187\pi$
$431$	−28.0585	−1.35153	−0.675766	+	0.737116i	$0.736187\pi$
$432$	0	0
$433$	−20.6011	−0.990026	−0.495013	−	0.868886i	$-0.664837\pi$
$433$	−20.6011	−0.990026	−0.495013	+	0.868886i	$0.664837\pi$
$434$	17.9547	0.861851
$435$	0	0
$436$	11.6968	0.560175
$437$	−14.8572	−0.710715
$438$	0	0
$439$	−36.4050	−1.73752	−0.868758	−	0.495238i	$-0.835081\pi$
$439$	−36.4050	−1.73752	−0.868758	+	0.495238i	$0.835081\pi$
$440$	28.6974	1.36809
$441$	0	0
$442$	12.9607	0.616477
$443$	28.1192	1.33598	0.667992	−	0.744169i	$-0.267154\pi$
$443$	28.1192	1.33598	0.667992	+	0.744169i	$0.267154\pi$
$444$	0	0
$445$	−25.3109	−1.19985
$446$	35.2507	1.66917
$447$	0	0
$448$	−8.76920	−0.414306
$449$	12.9493	0.611116	0.305558	−	0.952173i	$-0.401157\pi$
$449$	12.9493	0.611116	0.305558	+	0.952173i	$0.401157\pi$
$450$	0	0
$451$	1.91674	0.0902559
$452$	68.1925	3.20750
$453$	0	0
$454$	−30.9936	−1.45460
$455$	−10.9002	−0.511010
$456$	0	0
$457$	0.149626	0.00699919	0.00349960	−	0.999994i	$-0.498886\pi$
$457$	0.149626	0.00699919	0.00349960	+	0.999994i	$0.498886\pi$
$458$	70.3797	3.28863
$459$	0	0
$460$	28.4776	1.32777
$461$	−38.9917	−1.81603	−0.908013	−	0.418942i	$-0.862401\pi$
$461$	−38.9917	−1.81603	−0.908013	+	0.418942i	$0.862401\pi$
$462$	0	0
$463$	−20.1386	−0.935918	−0.467959	−	0.883750i	$-0.655011\pi$
$463$	−20.1386	−0.935918	−0.467959	+	0.883750i	$0.655011\pi$
$464$	9.52738	0.442298
$465$	0	0
$466$	−37.3812	−1.73165
$467$	−14.8664	−0.687936	−0.343968	−	0.938981i	$-0.611771\pi$
$467$	−14.8664	−0.687936	−0.343968	+	0.938981i	$0.611771\pi$
$468$	0	0
$469$	14.8401	0.685254
$470$	−1.41288	−0.0651713
$471$	0	0
$472$	96.6562	4.44896
$473$	12.3468	0.567707
$474$	0	0
$475$	−10.3063	−0.472885
$476$	12.0162	0.550762
$477$	0	0
$478$	61.9193	2.83212
$479$	35.2515	1.61068	0.805342	−	0.592810i	$-0.201982\pi$
$479$	35.2515	1.61068	0.805342	+	0.592810i	$0.201982\pi$
$480$	0	0
$481$	25.4453	1.16021
$482$	−35.6299	−1.62290
$483$	0	0
$484$	−13.3363	−0.606195
$485$	11.5678	0.525268
$486$	0	0
$487$	27.4362	1.24325	0.621626	−	0.783314i	$-0.286472\pi$
$487$	27.4362	1.24325	0.621626	+	0.783314i	$0.286472\pi$
$488$	−45.3580	−2.05326
$489$	0	0
$490$	11.7145	0.529208
$491$	4.83937	0.218398	0.109199	−	0.994020i	$-0.465171\pi$
$491$	4.83937	0.218398	0.109199	+	0.994020i	$0.465171\pi$
$492$	0	0
$493$	−1.43886	−0.0648030
$494$	−35.1238	−1.58030
$495$	0	0
$496$	−30.4803	−1.36860
$497$	0.146393	0.00656661
$498$	0	0
$499$	16.7739	0.750904	0.375452	−	0.926842i	$-0.377488\pi$
$499$	16.7739	0.750904	0.375452	+	0.926842i	$0.377488\pi$
$500$	53.6042	2.39725
$501$	0	0
$502$	21.3927	0.954803
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	13.1470	0.585036
$506$	31.0795	1.38165
$507$	0	0
$508$	25.9319	1.15054
$509$	27.5401	1.22069	0.610347	−	0.792134i	$-0.291030\pi$
$509$	27.5401	1.22069	0.610347	+	0.792134i	$0.291030\pi$
$510$	0	0
$511$	−6.46617	−0.286046
$512$	−47.7209	−2.10898
$513$	0	0
$514$	52.1216	2.29899
$515$	22.0682	0.972443
$516$	0	0
$517$	−1.08112	−0.0475475
$518$	33.6472	1.47837
$519$	0	0
$520$	38.6256	1.69385
$521$	−25.7250	−1.12703	−0.563515	−	0.826106i	$-0.690551\pi$
$521$	−25.7250	−1.12703	−0.563515	+	0.826106i	$0.690551\pi$
$522$	0	0
$523$	−29.5746	−1.29321	−0.646605	−	0.762825i	$-0.723812\pi$
$523$	−29.5746	−1.29321	−0.646605	+	0.762825i	$0.723812\pi$
$524$	99.5486	4.34880
$525$	0	0
$526$	72.2327	3.14949
$527$	4.60324	0.200520
$528$	0	0
$529$	−5.30539	−0.230669
$530$	−26.6396	−1.15715
$531$	0	0
$532$	−32.5643	−1.41184
$533$	2.57987	0.111746
$534$	0	0
$535$	5.05624	0.218601
$536$	−52.5870	−2.27141
$537$	0	0
$538$	−30.0502	−1.29556
$539$	8.96380	0.386098
$540$	0	0
$541$	−13.1889	−0.567035	−0.283517	−	0.958967i	$-0.591501\pi$
$541$	−13.1889	−0.567035	−0.283517	+	0.958967i	$0.591501\pi$
$542$	−8.47940	−0.364222
$543$	0	0
$544$	−10.9438	−0.469211
$545$	−3.59718	−0.154086
$546$	0	0
$547$	−17.9543	−0.767673	−0.383836	−	0.923401i	$-0.625397\pi$
$547$	−17.9543	−0.767673	−0.383836	+	0.923401i	$0.625397\pi$
$548$	−54.7977	−2.34084
$549$	0	0
$550$	21.5596	0.919303
$551$	3.89935	0.166118
$552$	0	0
$553$	−16.8522	−0.716627
$554$	26.1680	1.11177
$555$	0	0
$556$	−50.9891	−2.16242
$557$	−9.47412	−0.401431	−0.200716	−	0.979650i	$-0.564327\pi$
$557$	−9.47412	−0.401431	−0.200716	+	0.979650i	$0.564327\pi$
$558$	0	0
$559$	16.6183	0.702881
$560$	24.4691	1.03401
$561$	0	0
$562$	−43.3703	−1.82947
$563$	33.2995	1.40341	0.701704	−	0.712469i	$-0.252423\pi$
$563$	33.2995	1.40341	0.701704	+	0.712469i	$0.252423\pi$
$564$	0	0
$565$	−20.9716	−0.882283
$566$	39.9364	1.67865
$567$	0	0
$568$	−0.518752	−0.0217664
$569$	28.8525	1.20956	0.604780	−	0.796393i	$-0.293261\pi$
$569$	28.8525	1.20956	0.604780	+	0.796393i	$0.293261\pi$
$570$	0	0
$571$	22.7379	0.951550	0.475775	−	0.879567i	$-0.342168\pi$
$571$	22.7379	0.951550	0.475775	+	0.879567i	$0.342168\pi$
$572$	51.5154	2.15397
$573$	0	0
$574$	3.41145	0.142391
$575$	12.2746	0.511886
$576$	0	0
$577$	−4.60031	−0.191513	−0.0957567	−	0.995405i	$-0.530527\pi$
$577$	−4.60031	−0.191513	−0.0957567	+	0.995405i	$0.530527\pi$
$578$	−39.5827	−1.64642
$579$	0	0
$580$	−7.47411	−0.310346
$581$	29.1156	1.20792
$582$	0	0
$583$	−20.3843	−0.844231
$584$	22.9133	0.948159
$585$	0	0
$586$	19.4761	0.804552
$587$	−32.6537	−1.34776	−0.673881	−	0.738839i	$-0.735374\pi$
$587$	−32.6537	−1.34776	−0.673881	+	0.738839i	$0.735374\pi$
$588$	0	0
$589$	−12.4749	−0.514020
$590$	−51.8106	−2.13301
$591$	0	0
$592$	−57.1204	−2.34763
$593$	36.4077	1.49509	0.747543	−	0.664214i	$-0.231233\pi$
$593$	36.4077	1.49509	0.747543	+	0.664214i	$0.231233\pi$
$594$	0	0
$595$	−3.69542	−0.151497
$596$	34.7295	1.42257
$597$	0	0
$598$	41.8318	1.71063
$599$	40.2375	1.64406	0.822031	−	0.569443i	$-0.192841\pi$
$599$	40.2375	1.64406	0.822031	+	0.569443i	$0.192841\pi$
$600$	0	0
$601$	8.80257	0.359065	0.179532	−	0.983752i	$-0.442542\pi$
$601$	8.80257	0.359065	0.179532	+	0.983752i	$0.442542\pi$
$602$	21.9750	0.895635
$603$	0	0
$604$	28.5822	1.16299
$605$	4.10139	0.166745
$606$	0	0
$607$	13.3824	0.543176	0.271588	−	0.962414i	$-0.412451\pi$
$607$	13.3824	0.543176	0.271588	+	0.962414i	$0.412451\pi$
$608$	29.6579	1.20279
$609$	0	0
$610$	24.3132	0.984414
$611$	−1.45515	−0.0588689
$612$	0	0
$613$	−15.6160	−0.630725	−0.315363	−	0.948971i	$-0.602126\pi$
$613$	−15.6160	−0.630725	−0.315363	+	0.948971i	$0.602126\pi$
$614$	44.4763	1.79492
$615$	0	0
$616$	39.0828	1.57469
$617$	−4.29634	−0.172964	−0.0864820	−	0.996253i	$-0.527563\pi$
$617$	−4.29634	−0.172964	−0.0864820	+	0.996253i	$0.527563\pi$
$618$	0	0
$619$	−10.8380	−0.435614	−0.217807	−	0.975992i	$-0.569890\pi$
$619$	−10.8380	−0.435614	−0.217807	+	0.975992i	$0.569890\pi$
$620$	23.9114	0.960305
$621$	0	0
$622$	−26.2813	−1.05378
$623$	−34.4707	−1.38104
$624$	0	0
$625$	−1.89515	−0.0758062
$626$	69.0456	2.75962
$627$	0	0
$628$	10.4529	0.417118
$629$	8.62653	0.343962
$630$	0	0
$631$	29.5073	1.17467	0.587334	−	0.809345i	$-0.300178\pi$
$631$	29.5073	1.17467	0.587334	+	0.809345i	$0.300178\pi$
$632$	59.7168	2.37541
$633$	0	0
$634$	−50.1475	−1.99161
$635$	−7.97499	−0.316478
$636$	0	0
$637$	12.0649	0.478031
$638$	−8.15699	−0.322938
$639$	0	0
$640$	7.57558	0.299451
$641$	−7.72732	−0.305211	−0.152605	−	0.988287i	$-0.548766\pi$
$641$	−7.72732	−0.305211	−0.152605	+	0.988287i	$0.548766\pi$
$642$	0	0
$643$	10.0167	0.395021	0.197511	−	0.980301i	$-0.436714\pi$
$643$	10.0167	0.395021	0.197511	+	0.980301i	$0.436714\pi$
$644$	38.7834	1.52828
$645$	0	0
$646$	−11.9078	−0.468505
$647$	13.6884	0.538147	0.269074	−	0.963120i	$-0.413282\pi$
$647$	13.6884	0.538147	0.269074	+	0.963120i	$0.413282\pi$
$648$	0	0
$649$	−39.6449	−1.55620
$650$	29.0184	1.13819
$651$	0	0
$652$	−87.9111	−3.44286
$653$	−38.1399	−1.49253	−0.746264	−	0.665650i	$-0.768155\pi$
$653$	−38.1399	−1.49253	−0.746264	+	0.665650i	$0.768155\pi$
$654$	0	0
$655$	−30.6147	−1.19622
$656$	−5.79136	−0.226115
$657$	0	0
$658$	−1.92419	−0.0750128
$659$	−31.5253	−1.22805	−0.614026	−	0.789286i	$-0.710451\pi$
$659$	−31.5253	−1.22805	−0.614026	+	0.789286i	$0.710451\pi$
$660$	0	0
$661$	−30.1122	−1.17123	−0.585614	−	0.810590i	$-0.699146\pi$
$661$	−30.1122	−1.17123	−0.585614	+	0.810590i	$0.699146\pi$
$662$	−67.4080	−2.61989
$663$	0	0
$664$	−103.173	−4.00390
$665$	10.0147	0.388352
$666$	0	0
$667$	−4.64405	−0.179818
$668$	−106.616	−4.12510
$669$	0	0
$670$	28.1882	1.08901
$671$	18.6042	0.718207
$672$	0	0
$673$	−40.7982	−1.57265	−0.786327	−	0.617810i	$-0.788020\pi$
$673$	−40.7982	−1.57265	−0.786327	+	0.617810i	$0.788020\pi$
$674$	−30.0697	−1.15824
$675$	0	0
$676$	8.34388	0.320918
$677$	−3.18994	−0.122599	−0.0612997	−	0.998119i	$-0.519525\pi$
$677$	−3.18994	−0.122599	−0.0612997	+	0.998119i	$0.519525\pi$
$678$	0	0
$679$	15.7542	0.604589
$680$	13.0950	0.502169
$681$	0	0
$682$	26.0961	0.999270
$683$	−25.0239	−0.957512	−0.478756	−	0.877948i	$-0.658912\pi$
$683$	−25.0239	−0.957512	−0.478756	+	0.877948i	$0.658912\pi$
$684$	0	0
$685$	16.8523	0.643892
$686$	51.5378	1.96772
$687$	0	0
$688$	−37.3054	−1.42225
$689$	−27.4365	−1.04525
$690$	0	0
$691$	36.5349	1.38985	0.694927	−	0.719081i	$-0.255437\pi$
$691$	36.5349	1.38985	0.694927	+	0.719081i	$0.255437\pi$
$692$	10.5633	0.401556
$693$	0	0
$694$	−63.4045	−2.40680
$695$	15.6810	0.594813
$696$	0	0
$697$	0.874633	0.0331291
$698$	27.5525	1.04288
$699$	0	0
$700$	26.9037	1.01687
$701$	−40.6252	−1.53439	−0.767196	−	0.641413i	$-0.778348\pi$
$701$	−40.6252	−1.53439	−0.767196	+	0.641413i	$0.778348\pi$
$702$	0	0
$703$	−23.3781	−0.881723
$704$	−12.7455	−0.480365
$705$	0	0
$706$	−2.70718	−0.101886
$707$	17.9049	0.673382
$708$	0	0
$709$	−29.3401	−1.10189	−0.550945	−	0.834542i	$-0.685733\pi$
$709$	−29.3401	−1.10189	−0.550945	+	0.834542i	$0.685733\pi$
$710$	0.278067	0.0104357
$711$	0	0
$712$	122.149	4.57774
$713$	14.8574	0.556414
$714$	0	0
$715$	−15.8428	−0.592488
$716$	−74.5938	−2.78770
$717$	0	0
$718$	62.1347	2.31885
$719$	5.69273	0.212303	0.106152	−	0.994350i	$-0.466147\pi$
$719$	5.69273	0.212303	0.106152	+	0.994350i	$0.466147\pi$
$720$	0	0
$721$	30.0546	1.11929
$722$	−16.8800	−0.628208
$723$	0	0
$724$	77.8574	2.89355
$725$	−3.22154	−0.119645
$726$	0	0
$727$	31.4072	1.16483	0.582414	−	0.812892i	$-0.302108\pi$
$727$	31.4072	1.16483	0.582414	+	0.812892i	$0.302108\pi$
$728$	52.6040	1.94963
$729$	0	0
$730$	−12.2822	−0.454585
$731$	5.63399	0.208381
$732$	0	0
$733$	47.6365	1.75949	0.879747	−	0.475441i	$-0.157712\pi$
$733$	47.6365	1.75949	0.879747	+	0.475441i	$0.157712\pi$
$734$	−48.3646	−1.78517
$735$	0	0
$736$	−35.3221	−1.30199
$737$	21.5693	0.794515
$738$	0	0
$739$	9.88794	0.363734	0.181867	−	0.983323i	$-0.441786\pi$
$739$	9.88794	0.363734	0.181867	+	0.983323i	$0.441786\pi$
$740$	44.8102	1.64726
$741$	0	0
$742$	−36.2803	−1.33189
$743$	−17.7935	−0.652782	−0.326391	−	0.945235i	$-0.605833\pi$
$743$	−17.7935	−0.652782	−0.326391	+	0.945235i	$0.605833\pi$
$744$	0	0
$745$	−10.6806	−0.391305
$746$	−60.2759	−2.20686
$747$	0	0
$748$	17.4649	0.638579
$749$	6.88606	0.251611
$750$	0	0
$751$	−43.7934	−1.59804	−0.799022	−	0.601302i	$-0.794649\pi$
$751$	−43.7934	−1.59804	−0.799022	+	0.601302i	$0.794649\pi$
$752$	3.26656	0.119119
$753$	0	0
$754$	−10.9790	−0.399832
$755$	−8.79003	−0.319902
$756$	0	0
$757$	0.950190	0.0345353	0.0172676	−	0.999851i	$-0.494503\pi$
$757$	0.950190	0.0345353	0.0172676	+	0.999851i	$0.494503\pi$
$758$	−25.3499	−0.920752
$759$	0	0
$760$	−35.4877	−1.28727
$761$	5.00434	0.181407	0.0907035	−	0.995878i	$-0.471088\pi$
$761$	5.00434	0.181407	0.0907035	+	0.995878i	$0.471088\pi$
$762$	0	0
$763$	−4.89898	−0.177355
$764$	9.55445	0.345668
$765$	0	0
$766$	5.10375	0.184406
$767$	−53.3605	−1.92674
$768$	0	0
$769$	16.5794	0.597870	0.298935	−	0.954274i	$-0.403369\pi$
$769$	16.5794	0.597870	0.298935	+	0.954274i	$0.403369\pi$
$770$	−20.9495	−0.754969
$771$	0	0
$772$	−71.0268	−2.55631
$773$	30.4696	1.09592	0.547958	−	0.836506i	$-0.315405\pi$
$773$	30.4696	1.09592	0.547958	+	0.836506i	$0.315405\pi$
$774$	0	0
$775$	10.3064	0.370218
$776$	−55.8259	−2.00403
$777$	0	0
$778$	17.1044	0.613221
$779$	−2.37028	−0.0849240
$780$	0	0
$781$	0.212773	0.00761363
$782$	14.1819	0.507145
$783$	0	0
$784$	−27.0837	−0.967277
$785$	−3.21465	−0.114736
$786$	0	0
$787$	29.7903	1.06191	0.530954	−	0.847400i	$-0.321834\pi$
$787$	29.7903	1.06191	0.530954	+	0.847400i	$0.321834\pi$
$788$	28.2114	1.00499
$789$	0	0
$790$	−32.0100	−1.13886
$791$	−28.5611	−1.01552
$792$	0	0
$793$	25.0406	0.889216
$794$	−19.7054	−0.699318
$795$	0	0
$796$	110.499	3.91653
$797$	−0.503125	−0.0178216	−0.00891081	−	0.999960i	$-0.502836\pi$
$797$	−0.503125	−0.0178216	−0.00891081	+	0.999960i	$0.502836\pi$
$798$	0	0
$799$	−0.493327	−0.0174527
$800$	−24.5026	−0.866299
$801$	0	0
$802$	−66.7830	−2.35819
$803$	−9.39820	−0.331655
$804$	0	0
$805$	−11.9273	−0.420382
$806$	35.1244	1.23720
$807$	0	0
$808$	−63.4471	−2.23206
$809$	21.1371	0.743142	0.371571	−	0.928405i	$-0.378819\pi$
$809$	21.1371	0.743142	0.371571	+	0.928405i	$0.378819\pi$
$810$	0	0
$811$	15.7238	0.552139	0.276069	−	0.961138i	$-0.410968\pi$
$811$	15.7238	0.552139	0.276069	+	0.961138i	$0.410968\pi$
$812$	−10.1789	−0.357211
$813$	0	0
$814$	48.9043	1.71410
$815$	27.0358	0.947023
$816$	0	0
$817$	−15.2683	−0.534169
$818$	−35.4202	−1.23844
$819$	0	0
$820$	4.54325	0.158657
$821$	42.7588	1.49229	0.746147	−	0.665781i	$-0.231902\pi$
$821$	42.7588	1.49229	0.746147	+	0.665781i	$0.231902\pi$
$822$	0	0
$823$	−33.2920	−1.16049	−0.580243	−	0.814444i	$-0.697042\pi$
$823$	−33.2920	−1.16049	−0.580243	+	0.814444i	$0.697042\pi$
$824$	−106.500	−3.71012
$825$	0	0
$826$	−70.5605	−2.45511
$827$	−14.2343	−0.494975	−0.247487	−	0.968891i	$-0.579605\pi$
$827$	−14.2343	−0.494975	−0.247487	+	0.968891i	$0.579605\pi$
$828$	0	0
$829$	−12.3766	−0.429855	−0.214928	−	0.976630i	$-0.568952\pi$
$829$	−12.3766	−0.429855	−0.214928	+	0.976630i	$0.568952\pi$
$830$	55.3040	1.91963
$831$	0	0
$832$	−17.1550	−0.594743
$833$	4.09029	0.141720
$834$	0	0
$835$	32.7882	1.13468
$836$	−47.3303	−1.63695
$837$	0	0
$838$	−62.0917	−2.14492
$839$	−31.6864	−1.09394	−0.546969	−	0.837153i	$-0.684218\pi$
$839$	−31.6864	−1.09394	−0.546969	+	0.837153i	$0.684218\pi$
$840$	0	0
$841$	−27.7811	−0.957970
$842$	80.5270	2.77514
$843$	0	0
$844$	11.0519	0.380421
$845$	−2.56604	−0.0882745
$846$	0	0
$847$	5.58565	0.191925
$848$	61.5903	2.11502
$849$	0	0
$850$	9.83789	0.337437
$851$	27.8429	0.954443
$852$	0	0
$853$	30.2234	1.03483	0.517416	−	0.855734i	$-0.326894\pi$
$853$	30.2234	1.03483	0.517416	+	0.855734i	$0.326894\pi$
$854$	33.1120	1.13307
$855$	0	0
$856$	−24.4012	−0.834017
$857$	31.4206	1.07331	0.536653	−	0.843803i	$-0.319688\pi$
$857$	31.4206	1.07331	0.536653	+	0.843803i	$0.319688\pi$
$858$	0	0
$859$	27.5866	0.941244	0.470622	−	0.882335i	$-0.344030\pi$
$859$	27.5866	0.941244	0.470622	+	0.882335i	$0.344030\pi$
$860$	29.2656	0.997948
$861$	0	0
$862$	−72.5835	−2.47220
$863$	−40.3690	−1.37418	−0.687088	−	0.726574i	$-0.741111\pi$
$863$	−40.3690	−1.37418	−0.687088	+	0.726574i	$0.741111\pi$
$864$	0	0
$865$	−3.24859	−0.110455
$866$	−53.2922	−1.81094
$867$	0	0
$868$	32.5648	1.10532
$869$	−24.4937	−0.830890
$870$	0	0
$871$	29.0315	0.983694
$872$	17.3599	0.587879
$873$	0	0
$874$	−38.4334	−1.30003
$875$	−22.4511	−0.758985
$876$	0	0
$877$	−0.330616	−0.0111641	−0.00558205	−	0.999984i	$-0.501777\pi$
$877$	−0.330616	−0.0111641	−0.00558205	+	0.999984i	$0.501777\pi$
$878$	−94.1746	−3.17824
$879$	0	0
$880$	35.5644	1.19888
$881$	4.51734	0.152193	0.0760965	−	0.997100i	$-0.475754\pi$
$881$	4.51734	0.152193	0.0760965	+	0.997100i	$0.475754\pi$
$882$	0	0
$883$	−46.7445	−1.57308	−0.786539	−	0.617540i	$-0.788129\pi$
$883$	−46.7445	−1.57308	−0.786539	+	0.617540i	$0.788129\pi$
$884$	23.5071	0.790629
$885$	0	0
$886$	72.7404	2.44376
$887$	9.29057	0.311947	0.155973	−	0.987761i	$-0.450149\pi$
$887$	9.29057	0.311947	0.155973	+	0.987761i	$0.450149\pi$
$888$	0	0
$889$	−10.8611	−0.364269
$890$	−65.4757	−2.19475
$891$	0	0
$892$	63.9349	2.14070
$893$	1.33693	0.0447387
$894$	0	0
$895$	22.9403	0.766808
$896$	10.3171	0.344671
$897$	0	0
$898$	33.4981	1.11785
$899$	−3.89941	−0.130053
$900$	0	0
$901$	−9.30159	−0.309881
$902$	4.95835	0.165095
$903$	0	0
$904$	101.208	3.36614
$905$	−23.9439	−0.795923
$906$	0	0
$907$	−24.0423	−0.798313	−0.399156	−	0.916883i	$-0.630697\pi$
$907$	−24.0423	−0.798313	−0.399156	+	0.916883i	$0.630697\pi$
$908$	−56.2138	−1.86552
$909$	0	0
$910$	−28.1973	−0.934732
$911$	−6.04735	−0.200358	−0.100179	−	0.994969i	$-0.531941\pi$
$911$	−6.04735	−0.200358	−0.100179	+	0.994969i	$0.531941\pi$
$912$	0	0
$913$	42.3179	1.40052
$914$	0.387061	0.0128028
$915$	0	0
$916$	127.649	4.21765
$917$	−41.6940	−1.37686
$918$	0	0
$919$	−34.8990	−1.15121	−0.575606	−	0.817727i	$-0.695234\pi$
$919$	−34.8990	−1.15121	−0.575606	+	0.817727i	$0.695234\pi$
$920$	42.2652	1.39344
$921$	0	0
$922$	−100.866	−3.32185
$923$	0.286385	0.00942648
$924$	0	0
$925$	19.3144	0.635054
$926$	−52.0956	−1.71197
$927$	0	0
$928$	9.27048	0.304319
$929$	−39.8343	−1.30692	−0.653460	−	0.756961i	$-0.726683\pi$
$929$	−39.8343	−1.30692	−0.653460	+	0.756961i	$0.726683\pi$
$930$	0	0
$931$	−11.0848	−0.363289
$932$	−67.7992	−2.22084
$933$	0	0
$934$	−38.4573	−1.25836
$935$	−5.37108	−0.175653
$936$	0	0
$937$	10.6822	0.348971	0.174485	−	0.984660i	$-0.444174\pi$
$937$	10.6822	0.348971	0.174485	+	0.984660i	$0.444174\pi$
$938$	38.3894	1.25346
$939$	0	0
$940$	−2.56257	−0.0835819
$941$	1.40329	0.0457459	0.0228729	−	0.999738i	$-0.492719\pi$
$941$	1.40329	0.0457459	0.0228729	+	0.999738i	$0.492719\pi$
$942$	0	0
$943$	2.82296	0.0919281
$944$	119.785	3.89868
$945$	0	0
$946$	31.9395	1.03844
$947$	−9.93625	−0.322885	−0.161442	−	0.986882i	$-0.551615\pi$
$947$	−9.93625	−0.322885	−0.161442	+	0.986882i	$0.551615\pi$
$948$	0	0
$949$	−12.6496	−0.410625
$950$	−26.6609	−0.864995
$951$	0	0
$952$	17.8339	0.578001
$953$	−11.4560	−0.371097	−0.185548	−	0.982635i	$-0.559406\pi$
$953$	−11.4560	−0.371097	−0.185548	+	0.982635i	$0.559406\pi$
$954$	0	0
$955$	−2.93834	−0.0950823
$956$	112.304	3.63218
$957$	0	0
$958$	91.1908	2.94624
$959$	22.9510	0.741126
$960$	0	0
$961$	−18.5249	−0.597578
$962$	65.8234	2.12223
$963$	0	0
$964$	−64.6227	−2.08136
$965$	21.8433	0.703160
$966$	0	0
$967$	−32.0337	−1.03014	−0.515068	−	0.857150i	$-0.672233\pi$
$967$	−32.0337	−1.03014	−0.515068	+	0.857150i	$0.672233\pi$
$968$	−19.7931	−0.636175
$969$	0	0
$970$	29.9244	0.960814
$971$	21.7771	0.698859	0.349429	−	0.936963i	$-0.386375\pi$
$971$	21.7771	0.698859	0.349429	+	0.936963i	$0.386375\pi$
$972$	0	0
$973$	21.3558	0.684636
$974$	70.9736	2.27414
$975$	0	0
$976$	−56.2118	−1.79930
$977$	−48.8640	−1.56330	−0.781649	−	0.623719i	$-0.785621\pi$
$977$	−48.8640	−1.56330	−0.781649	+	0.623719i	$0.785621\pi$
$978$	0	0
$979$	−50.1012	−1.60124
$980$	21.2469	0.678706
$981$	0	0
$982$	12.5188	0.399490
$983$	−43.9288	−1.40111	−0.700555	−	0.713598i	$-0.747064\pi$
$983$	−43.9288	−1.40111	−0.700555	+	0.713598i	$0.747064\pi$
$984$	0	0
$985$	−8.67600	−0.276440
$986$	−3.72213	−0.118537
$987$	0	0
$988$	−63.7049	−2.02672
$989$	18.1842	0.578225
$990$	0	0
$991$	18.0715	0.574060	0.287030	−	0.957922i	$-0.407332\pi$
$991$	18.0715	0.574060	0.287030	+	0.957922i	$0.407332\pi$
$992$	−29.6584	−0.941655
$993$	0	0
$994$	0.378697	0.0120115
$995$	−33.9824	−1.07731
$996$	0	0
$997$	−36.5966	−1.15903	−0.579514	−	0.814963i	$-0.696757\pi$
$997$	−36.5966	−1.15903	−0.579514	+	0.814963i	$0.696757\pi$
$998$	43.3918	1.37354
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.10		10
3.2	odd	2		503.2.a.e.1.1	✓	10
12.11	even	2		8048.2.a.p.1.8		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.1	✓	10	3.2	odd	2
4527.2.a.k.1.10		10	1.1	even	1	trivial
8048.2.a.p.1.8		10	12.11	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:	\( N \)	\(=\)	\( 4527 = 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4527.a (trivial)

\(f(q)\)	\(=\)	\(q+2.58686 q^{2} +4.69185 q^{4} -1.44291 q^{5} -1.96509 q^{7} +6.96343 q^{8} +O(q^{10})\)	\(q+2.58686 q^{2} +4.69185 q^{4} -1.44291 q^{5} -1.96509 q^{7} +6.96343 q^{8} -3.73261 q^{10} -2.85614 q^{11} -3.84427 q^{13} -5.08341 q^{14} +8.62972 q^{16} -1.30329 q^{17} +3.53196 q^{19} -6.76991 q^{20} -7.38845 q^{22} -4.20650 q^{23} -2.91801 q^{25} -9.94458 q^{26} -9.21989 q^{28} +1.10402 q^{29} -3.53201 q^{31} +8.39703 q^{32} -3.37144 q^{34} +2.83545 q^{35} -6.61903 q^{37} +9.13668 q^{38} -10.0476 q^{40} -0.671095 q^{41} -4.32289 q^{43} -13.4006 q^{44} -10.8816 q^{46} +0.378524 q^{47} -3.13843 q^{49} -7.54849 q^{50} -18.0367 q^{52} +7.13700 q^{53} +4.12116 q^{55} -13.6838 q^{56} +2.85594 q^{58} +13.8806 q^{59} -6.51374 q^{61} -9.13682 q^{62} +4.46249 q^{64} +5.54693 q^{65} -7.55189 q^{67} -6.11485 q^{68} +7.33490 q^{70} -0.0744967 q^{71} +3.29052 q^{73} -17.1225 q^{74} +16.5714 q^{76} +5.61258 q^{77} +8.57578 q^{79} -12.4519 q^{80} -1.73603 q^{82} -14.8165 q^{83} +1.88053 q^{85} -11.1827 q^{86} -19.8886 q^{88} +17.5415 q^{89} +7.55432 q^{91} -19.7362 q^{92} +0.979188 q^{94} -5.09630 q^{95} -8.01702 q^{97} -8.11867 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * q^98

Introduction

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