LMFDB - Embedded newform 4527.2.a.k.1.7

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			1.7
Root			$1.07636$ 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)

$f(q)$	$=$	$q+1.37178 q^{2} -0.118218 q^{4} -1.17276 q^{5} +0.469303 q^{7} -2.90573 q^{8} +O(q^{10})$	\(q+1.37178 q^{2} -0.118218 q^{4} -1.17276 q^{5} +0.469303 q^{7} -2.90573 q^{8} -1.60876 q^{10} +5.74596 q^{11} -1.85873 q^{13} +0.643780 q^{14} -3.74959 q^{16} -5.22916 q^{17} +2.12602 q^{19} +0.138641 q^{20} +7.88220 q^{22} -0.171951 q^{23} -3.62464 q^{25} -2.54977 q^{26} -0.0554799 q^{28} +6.19149 q^{29} -0.396234 q^{31} +0.667846 q^{32} -7.17326 q^{34} -0.550377 q^{35} -8.17999 q^{37} +2.91644 q^{38} +3.40771 q^{40} +12.4282 q^{41} -4.97920 q^{43} -0.679275 q^{44} -0.235879 q^{46} +0.521599 q^{47} -6.77976 q^{49} -4.97222 q^{50} +0.219735 q^{52} -8.76106 q^{53} -6.73861 q^{55} -1.36367 q^{56} +8.49336 q^{58} -3.35297 q^{59} -5.38243 q^{61} -0.543546 q^{62} +8.41532 q^{64} +2.17983 q^{65} +8.42823 q^{67} +0.618179 q^{68} -0.754997 q^{70} -7.47643 q^{71} -4.60009 q^{73} -11.2212 q^{74} -0.251334 q^{76} +2.69660 q^{77} -17.1992 q^{79} +4.39735 q^{80} +17.0487 q^{82} -5.97721 q^{83} +6.13253 q^{85} -6.83038 q^{86} -16.6962 q^{88} +4.25595 q^{89} -0.872306 q^{91} +0.0203277 q^{92} +0.715520 q^{94} -2.49331 q^{95} -2.64532 q^{97} -9.30034 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.37178	0.969995	0.484998	−	0.874515i	$-0.338820\pi$
$2$	1.37178	0.969995	0.484998	+	0.874515i	$0.338820\pi$
$3$	0	0
$3$	0	0
$4$	−0.118218	−0.0591089
$5$	−1.17276	−0.524472	−0.262236	−	0.965004i	$-0.584460\pi$
$5$	−1.17276	−0.524472	−0.262236	+	0.965004i	$0.584460\pi$
$6$	0	0
$7$	0.469303	0.177380	0.0886899	−	0.996059i	$-0.471732\pi$
$7$	0.469303	0.177380	0.0886899	+	0.996059i	$0.471732\pi$
$8$	−2.90573	−1.02733
$9$	0	0
$10$	−1.60876	−0.508736
$11$	5.74596	1.73247	0.866237	−	0.499634i	$-0.166532\pi$
$11$	5.74596	1.73247	0.866237	+	0.499634i	$0.166532\pi$
$12$	0	0
$13$	−1.85873	−0.515518	−0.257759	−	0.966209i	$-0.582984\pi$
$13$	−1.85873	−0.515518	−0.257759	+	0.966209i	$0.582984\pi$
$14$	0.643780	0.172058
$15$	0	0
$16$	−3.74959	−0.937397
$17$	−5.22916	−1.26826	−0.634129	−	0.773228i	$-0.718641\pi$
$17$	−5.22916	−1.26826	−0.634129	+	0.773228i	$0.718641\pi$
$18$	0	0
$19$	2.12602	0.487743	0.243872	−	0.969808i	$-0.421582\pi$
$19$	2.12602	0.487743	0.243872	+	0.969808i	$0.421582\pi$
$20$	0.138641	0.0310010
$21$	0	0
$22$	7.88220	1.68049
$23$	−0.171951	−0.0358543	−0.0179271	−	0.999839i	$-0.505707\pi$
$23$	−0.171951	−0.0358543	−0.0179271	+	0.999839i	$0.505707\pi$
$24$	0	0
$25$	−3.62464	−0.724929
$26$	−2.54977	−0.500050
$27$	0	0
$28$	−0.0554799	−0.0104847
$29$	6.19149	1.14973	0.574865	−	0.818248i	$-0.305054\pi$
$29$	6.19149	1.14973	0.574865	+	0.818248i	$0.305054\pi$
$30$	0	0
$31$	−0.396234	−0.0711658	−0.0355829	−	0.999367i	$-0.511329\pi$
$31$	−0.396234	−0.0711658	−0.0355829	+	0.999367i	$0.511329\pi$
$32$	0.667846	0.118060
$33$	0	0
$34$	−7.17326	−1.23020
$35$	−0.550377	−0.0930307
$36$	0	0
$37$	−8.17999	−1.34478	−0.672391	−	0.740196i	$-0.734733\pi$
$37$	−8.17999	−1.34478	−0.672391	+	0.740196i	$0.734733\pi$
$38$	2.91644	0.473109
$39$	0	0
$40$	3.40771	0.538807
$41$	12.4282	1.94095	0.970477	−	0.241193i	$-0.0775388\pi$
$41$	12.4282	1.94095	0.970477	+	0.241193i	$0.0775388\pi$
$42$	0	0
$43$	−4.97920	−0.759321	−0.379661	−	0.925126i	$-0.623959\pi$
$43$	−4.97920	−0.759321	−0.379661	+	0.925126i	$0.623959\pi$
$44$	−0.679275	−0.102405
$45$	0	0
$46$	−0.235879	−0.0347785
$47$	0.521599	0.0760831	0.0380415	−	0.999276i	$-0.487888\pi$
$47$	0.521599	0.0760831	0.0380415	+	0.999276i	$0.487888\pi$
$48$	0	0
$49$	−6.77976	−0.968536
$50$	−4.97222	−0.703178
$51$	0	0
$52$	0.219735	0.0304717
$53$	−8.76106	−1.20342	−0.601712	−	0.798713i	$-0.705514\pi$
$53$	−8.76106	−1.20342	−0.601712	+	0.798713i	$0.705514\pi$
$54$	0	0
$55$	−6.73861	−0.908634
$56$	−1.36367	−0.182228
$57$	0	0
$58$	8.49336	1.11523
$59$	−3.35297	−0.436519	−0.218260	−	0.975891i	$-0.570038\pi$
$59$	−3.35297	−0.436519	−0.218260	+	0.975891i	$0.570038\pi$
$60$	0	0
$61$	−5.38243	−0.689150	−0.344575	−	0.938759i	$-0.611977\pi$
$61$	−5.38243	−0.689150	−0.344575	+	0.938759i	$0.611977\pi$
$62$	−0.543546	−0.0690305
$63$	0	0
$64$	8.41532	1.05191
$65$	2.17983	0.270375
$66$	0	0
$67$	8.42823	1.02967	0.514836	−	0.857289i	$-0.327853\pi$
$67$	8.42823	1.02967	0.514836	+	0.857289i	$0.327853\pi$
$68$	0.618179	0.0749653
$69$	0	0
$70$	−0.754997	−0.0902394
$71$	−7.47643	−0.887289	−0.443644	−	0.896203i	$-0.646315\pi$
$71$	−7.47643	−0.887289	−0.443644	+	0.896203i	$0.646315\pi$
$72$	0	0
$73$	−4.60009	−0.538400	−0.269200	−	0.963084i	$-0.586759\pi$
$73$	−4.60009	−0.538400	−0.269200	+	0.963084i	$0.586759\pi$
$74$	−11.2212	−1.30443
$75$	0	0
$76$	−0.251334	−0.0288299
$77$	2.69660	0.307306
$78$	0	0
$79$	−17.1992	−1.93506	−0.967530	−	0.252758i	$-0.918662\pi$
$79$	−17.1992	−1.93506	−0.967530	+	0.252758i	$0.918662\pi$
$80$	4.39735	0.491639
$81$	0	0
$82$	17.0487	1.88272
$83$	−5.97721	−0.656084	−0.328042	−	0.944663i	$-0.606389\pi$
$83$	−5.97721	−0.656084	−0.328042	+	0.944663i	$0.606389\pi$
$84$	0	0
$85$	6.13253	0.665166
$86$	−6.83038	−0.736538
$87$	0	0
$88$	−16.6962	−1.77982
$89$	4.25595	0.451130	0.225565	−	0.974228i	$-0.427577\pi$
$89$	4.25595	0.451130	0.225565	+	0.974228i	$0.427577\pi$
$90$	0	0
$91$	−0.872306	−0.0914425
$92$	0.0203277	0.00211930
$93$	0	0
$94$	0.715520	0.0738002
$95$	−2.49331	−0.255808
$96$	0	0
$97$	−2.64532	−0.268592	−0.134296	−	0.990941i	$-0.542877\pi$
$97$	−2.64532	−0.268592	−0.134296	+	0.990941i	$0.542877\pi$
$98$	−9.30034	−0.939476
$99$	0	0
$100$	0.428497	0.0428497
$101$	−6.39045	−0.635873	−0.317937	−	0.948112i	$-0.602990\pi$
$101$	−6.39045	−0.635873	−0.317937	+	0.948112i	$0.602990\pi$
$102$	0	0
$103$	−16.6014	−1.63578	−0.817890	−	0.575375i	$-0.804856\pi$
$103$	−16.6014	−1.63578	−0.817890	+	0.575375i	$0.804856\pi$
$104$	5.40096	0.529608
$105$	0	0
$106$	−12.0182	−1.16732
$107$	−9.15760	−0.885299	−0.442649	−	0.896695i	$-0.645961\pi$
$107$	−9.15760	−0.885299	−0.442649	+	0.896695i	$0.645961\pi$
$108$	0	0
$109$	6.51891	0.624398	0.312199	−	0.950017i	$-0.398934\pi$
$109$	6.51891	0.624398	0.312199	+	0.950017i	$0.398934\pi$
$110$	−9.24390	−0.881371
$111$	0	0
$112$	−1.75969	−0.166275
$113$	−1.36694	−0.128591	−0.0642954	−	0.997931i	$-0.520480\pi$
$113$	−1.36694	−0.128591	−0.0642954	+	0.997931i	$0.520480\pi$
$114$	0	0
$115$	0.201656	0.0188046
$116$	−0.731944	−0.0679593
$117$	0	0
$118$	−4.59954	−0.423422
$119$	−2.45406	−0.224963
$120$	0	0
$121$	22.0161	2.00146
$122$	−7.38352	−0.668473
$123$	0	0
$124$	0.0468419	0.00420653
$125$	10.1146	0.904677
$126$	0	0
$127$	6.77469	0.601157	0.300578	−	0.953757i	$-0.402820\pi$
$127$	6.77469	0.601157	0.300578	+	0.953757i	$0.402820\pi$
$128$	10.2083	0.902293
$129$	0	0
$130$	2.99025	0.262263
$131$	−16.6839	−1.45768	−0.728838	−	0.684687i	$-0.759939\pi$
$131$	−16.6839	−1.45768	−0.728838	+	0.684687i	$0.759939\pi$
$132$	0	0
$133$	0.997748	0.0865158
$134$	11.5617	0.998777
$135$	0	0
$136$	15.1945	1.30292
$137$	−13.0662	−1.11632	−0.558162	−	0.829732i	$-0.688493\pi$
$137$	−13.0662	−1.11632	−0.558162	+	0.829732i	$0.688493\pi$
$138$	0	0
$139$	−10.8468	−0.920017	−0.460008	−	0.887915i	$-0.652154\pi$
$139$	−10.8468	−0.920017	−0.460008	+	0.887915i	$0.652154\pi$
$140$	0.0650644	0.00549894
$141$	0	0
$142$	−10.2560	−0.860666
$143$	−10.6802	−0.893122
$144$	0	0
$145$	−7.26110	−0.603002
$146$	−6.31032	−0.522245
$147$	0	0
$148$	0.967020	0.0794886
$149$	−0.0210826	−0.00172716	−0.000863578	−	1.00000i	$-0.500275\pi$
$149$	−0.0210826	−0.00172716	−0.000863578		1.00000i	$0.500275\pi$
$150$	0	0
$151$	15.1704	1.23455	0.617276	−	0.786746i	$-0.288236\pi$
$151$	15.1704	1.23455	0.617276	+	0.786746i	$0.288236\pi$
$152$	−6.17765	−0.501074
$153$	0	0
$154$	3.69914	0.298085
$155$	0.464686	0.0373245
$156$	0	0
$157$	11.1437	0.889367	0.444684	−	0.895688i	$-0.353316\pi$
$157$	11.1437	0.889367	0.444684	+	0.895688i	$0.353316\pi$
$158$	−23.5935	−1.87700
$159$	0	0
$160$	−0.783221	−0.0619190
$161$	−0.0806970	−0.00635982
$162$	0	0
$163$	−10.3388	−0.809797	−0.404899	−	0.914362i	$-0.632693\pi$
$163$	−10.3388	−0.809797	−0.404899	+	0.914362i	$0.632693\pi$
$164$	−1.46923	−0.114728
$165$	0	0
$166$	−8.19942	−0.636398
$167$	15.6444	1.21060	0.605300	−	0.795998i	$-0.293053\pi$
$167$	15.6444	1.21060	0.605300	+	0.795998i	$0.293053\pi$
$168$	0	0
$169$	−9.54513	−0.734241
$170$	8.41248	0.645208
$171$	0	0
$172$	0.588630	0.0448826
$173$	5.69316	0.432843	0.216422	−	0.976300i	$-0.430561\pi$
$173$	5.69316	0.432843	0.216422	+	0.976300i	$0.430561\pi$
$174$	0	0
$175$	−1.70105	−0.128588
$176$	−21.5450	−1.62402
$177$	0	0
$178$	5.83823	0.437594
$179$	−10.8571	−0.811495	−0.405748	−	0.913985i	$-0.632989\pi$
$179$	−10.8571	−0.811495	−0.405748	+	0.913985i	$0.632989\pi$
$180$	0	0
$181$	6.26595	0.465744	0.232872	−	0.972507i	$-0.425188\pi$
$181$	6.26595	0.465744	0.232872	+	0.972507i	$0.425188\pi$
$182$	−1.19661	−0.0886988
$183$	0	0
$184$	0.499643	0.0368342
$185$	9.59313	0.705301
$186$	0	0
$187$	−30.0466	−2.19722
$188$	−0.0616623	−0.00449718
$189$	0	0
$190$	−3.42027	−0.248132
$191$	−8.26296	−0.597887	−0.298943	−	0.954271i	$-0.596634\pi$
$191$	−8.26296	−0.597887	−0.298943	+	0.954271i	$0.596634\pi$
$192$	0	0
$193$	−19.8469	−1.42861	−0.714306	−	0.699833i	$-0.753258\pi$
$193$	−19.8469	−1.42861	−0.714306	+	0.699833i	$0.753258\pi$
$194$	−3.62880	−0.260533
$195$	0	0
$196$	0.801487	0.0572491
$197$	4.24866	0.302704	0.151352	−	0.988480i	$-0.451637\pi$
$197$	4.24866	0.302704	0.151352	+	0.988480i	$0.451637\pi$
$198$	0	0
$199$	4.63286	0.328415	0.164207	−	0.986426i	$-0.447493\pi$
$199$	4.63286	0.328415	0.164207	+	0.986426i	$0.447493\pi$
$200$	10.5322	0.744742
$201$	0	0
$202$	−8.76630	−0.616794
$203$	2.90568	0.203939
$204$	0	0
$205$	−14.5752	−1.01798
$206$	−22.7734	−1.58670
$207$	0	0
$208$	6.96947	0.483246
$209$	12.2161	0.845002
$210$	0	0
$211$	10.2699	0.707007	0.353503	−	0.935433i	$-0.384990\pi$
$211$	10.2699	0.707007	0.353503	+	0.935433i	$0.384990\pi$
$212$	1.03571	0.0711330
$213$	0	0
$214$	−12.5622	−0.858736
$215$	5.83939	0.398243
$216$	0	0
$217$	−0.185954	−0.0126234
$218$	8.94251	0.605663
$219$	0	0
$220$	0.796624	0.0537083
$221$	9.71958	0.653810
$222$	0	0
$223$	14.7887	0.990326	0.495163	−	0.868800i	$-0.335108\pi$
$223$	14.7887	0.990326	0.495163	+	0.868800i	$0.335108\pi$
$224$	0.313422	0.0209414
$225$	0	0
$226$	−1.87514	−0.124732
$227$	−14.9154	−0.989969	−0.494984	−	0.868902i	$-0.664826\pi$
$227$	−14.9154	−0.989969	−0.494984	+	0.868902i	$0.664826\pi$
$228$	0	0
$229$	−18.2488	−1.20592	−0.602959	−	0.797772i	$-0.706012\pi$
$229$	−18.2488	−1.20592	−0.602959	+	0.797772i	$0.706012\pi$
$230$	0.276628	0.0182403
$231$	0	0
$232$	−17.9908	−1.18115
$233$	16.6741	1.09235	0.546177	−	0.837670i	$-0.316082\pi$
$233$	16.6741	1.09235	0.546177	+	0.837670i	$0.316082\pi$
$234$	0	0
$235$	−0.611708	−0.0399035
$236$	0.396380	0.0258022
$237$	0	0
$238$	−3.36643	−0.218213
$239$	−24.3691	−1.57631	−0.788153	−	0.615479i	$-0.788963\pi$
$239$	−24.3691	−1.57631	−0.788153	+	0.615479i	$0.788963\pi$
$240$	0	0
$241$	18.1071	1.16638	0.583190	−	0.812336i	$-0.301804\pi$
$241$	18.1071	1.16638	0.583190	+	0.812336i	$0.301804\pi$
$242$	30.2013	1.94141
$243$	0	0
$244$	0.636299	0.0407349
$245$	7.95100	0.507971
$246$	0	0
$247$	−3.95170	−0.251441
$248$	1.15135	0.0731108
$249$	0	0
$250$	13.8750	0.877533
$251$	−26.9687	−1.70225	−0.851124	−	0.524965i	$-0.824078\pi$
$251$	−26.9687	−1.70225	−0.851124	+	0.524965i	$0.824078\pi$
$252$	0	0
$253$	−0.988024	−0.0621165
$254$	9.29340	0.583119
$255$	0	0
$256$	−2.82712	−0.176695
$257$	−20.3022	−1.26642	−0.633208	−	0.773982i	$-0.718262\pi$
$257$	−20.3022	−1.26642	−0.633208	+	0.773982i	$0.718262\pi$
$258$	0	0
$259$	−3.83889	−0.238537
$260$	−0.257695	−0.0159816
$261$	0	0
$262$	−22.8866	−1.41394
$263$	12.0595	0.743622	0.371811	−	0.928309i	$-0.378737\pi$
$263$	12.0595	0.743622	0.371811	+	0.928309i	$0.378737\pi$
$264$	0	0
$265$	10.2746	0.631162
$266$	1.36869	0.0839199
$267$	0	0
$268$	−0.996366	−0.0608628
$269$	25.0120	1.52501	0.762506	−	0.646982i	$-0.223969\pi$
$269$	25.0120	1.52501	0.762506	+	0.646982i	$0.223969\pi$
$270$	0	0
$271$	−18.0354	−1.09557	−0.547787	−	0.836618i	$-0.684530\pi$
$271$	−18.0354	−1.09557	−0.547787	+	0.836618i	$0.684530\pi$
$272$	19.6072	1.18886
$273$	0	0
$274$	−17.9240	−1.08283
$275$	−20.8271	−1.25592
$276$	0	0
$277$	−29.3688	−1.76460	−0.882301	−	0.470686i	$-0.844006\pi$
$277$	−29.3688	−1.76460	−0.882301	+	0.470686i	$0.844006\pi$
$278$	−14.8795	−0.892412
$279$	0	0
$280$	1.59925	0.0955733
$281$	−12.0977	−0.721686	−0.360843	−	0.932626i	$-0.617511\pi$
$281$	−12.0977	−0.721686	−0.360843	+	0.932626i	$0.617511\pi$
$282$	0	0
$283$	−8.04149	−0.478017	−0.239009	−	0.971017i	$-0.576822\pi$
$283$	−8.04149	−0.478017	−0.239009	+	0.971017i	$0.576822\pi$
$284$	0.883846	0.0524466
$285$	0	0
$286$	−14.6509	−0.866324
$287$	5.83257	0.344286
$288$	0	0
$289$	10.3441	0.608477
$290$	−9.96064	−0.584909
$291$	0	0
$292$	0.543812	0.0318242
$293$	13.5236	0.790056	0.395028	−	0.918669i	$-0.370735\pi$
$293$	13.5236	0.790056	0.395028	+	0.918669i	$0.370735\pi$
$294$	0	0
$295$	3.93221	0.228942
$296$	23.7688	1.38154
$297$	0	0
$298$	−0.0289207	−0.00167533
$299$	0.319610	0.0184835
$300$	0	0
$301$	−2.33675	−0.134688
$302$	20.8105	1.19751
$303$	0	0
$304$	−7.97171	−0.457209
$305$	6.31228	0.361440
$306$	0	0
$307$	3.60632	0.205824	0.102912	−	0.994690i	$-0.467184\pi$
$307$	3.60632	0.205824	0.102912	+	0.994690i	$0.467184\pi$
$308$	−0.318785	−0.0181645
$309$	0	0
$310$	0.637447	0.0362046
$311$	27.5438	1.56187	0.780934	−	0.624613i	$-0.214743\pi$
$311$	27.5438	1.56187	0.780934	+	0.624613i	$0.214743\pi$
$312$	0	0
$313$	−30.2190	−1.70808	−0.854039	−	0.520209i	$-0.825854\pi$
$313$	−30.2190	−1.70808	−0.854039	+	0.520209i	$0.825854\pi$
$314$	15.2868	0.862682
$315$	0	0
$316$	2.03325	0.114379
$317$	5.23152	0.293831	0.146916	−	0.989149i	$-0.453065\pi$
$317$	5.23152	0.293831	0.146916	+	0.989149i	$0.453065\pi$
$318$	0	0
$319$	35.5761	1.99188
$320$	−9.86911	−0.551700
$321$	0	0
$322$	−0.110699	−0.00616899
$323$	−11.1173	−0.618584
$324$	0	0
$325$	6.73723	0.373714
$326$	−14.1826	−0.785500
$327$	0	0
$328$	−36.1129	−1.99400
$329$	0.244788	0.0134956
$330$	0	0
$331$	−5.56111	−0.305666	−0.152833	−	0.988252i	$-0.548840\pi$
$331$	−5.56111	−0.305666	−0.152833	+	0.988252i	$0.548840\pi$
$332$	0.706612	0.0387804
$333$	0	0
$334$	21.4607	1.17428
$335$	−9.88426	−0.540035
$336$	0	0
$337$	1.20314	0.0655392	0.0327696	−	0.999463i	$-0.489567\pi$
$337$	1.20314	0.0655392	0.0327696	+	0.999463i	$0.489567\pi$
$338$	−13.0938	−0.712210
$339$	0	0
$340$	−0.724973	−0.0393172
$341$	−2.27675	−0.123293
$342$	0	0
$343$	−6.46688	−0.349178
$344$	14.4682	0.780074
$345$	0	0
$346$	7.80977	0.419856
$347$	−18.8760	−1.01332	−0.506659	−	0.862147i	$-0.669120\pi$
$347$	−18.8760	−1.01332	−0.506659	+	0.862147i	$0.669120\pi$
$348$	0	0
$349$	18.3633	0.982966	0.491483	−	0.870887i	$-0.336455\pi$
$349$	18.3633	0.982966	0.491483	+	0.870887i	$0.336455\pi$
$350$	−2.33347	−0.124729
$351$	0	0
$352$	3.83742	0.204535
$353$	2.06519	0.109919	0.0549596	−	0.998489i	$-0.482497\pi$
$353$	2.06519	0.109919	0.0549596	+	0.998489i	$0.482497\pi$
$354$	0	0
$355$	8.76802	0.465358
$356$	−0.503129	−0.0266658
$357$	0	0
$358$	−14.8935	−0.787147
$359$	7.83605	0.413571	0.206786	−	0.978386i	$-0.433700\pi$
$359$	7.83605	0.413571	0.206786	+	0.978386i	$0.433700\pi$
$360$	0	0
$361$	−14.4800	−0.762107
$362$	8.59550	0.451770
$363$	0	0
$364$	0.103122	0.00540506
$365$	5.39478	0.282376
$366$	0	0
$367$	27.3479	1.42755	0.713775	−	0.700375i	$-0.246984\pi$
$367$	27.3479	1.42755	0.713775	+	0.700375i	$0.246984\pi$
$368$	0.644745	0.0336097
$369$	0	0
$370$	13.1597	0.684139
$371$	−4.11159	−0.213463
$372$	0	0
$373$	31.3381	1.62263	0.811313	−	0.584612i	$-0.198753\pi$
$373$	31.3381	1.62263	0.811313	+	0.584612i	$0.198753\pi$
$374$	−41.2173	−2.13130
$375$	0	0
$376$	−1.51563	−0.0781625
$377$	−11.5083	−0.592707
$378$	0	0
$379$	−1.97192	−0.101291	−0.0506454	−	0.998717i	$-0.516128\pi$
$379$	−1.97192	−0.101291	−0.0506454	+	0.998717i	$0.516128\pi$
$380$	0.294753	0.0151205
$381$	0	0
$382$	−11.3350	−0.579948
$383$	24.9455	1.27466	0.637328	−	0.770593i	$-0.280040\pi$
$383$	24.9455	1.27466	0.637328	+	0.770593i	$0.280040\pi$
$384$	0	0
$385$	−3.16245	−0.161173
$386$	−27.2256	−1.38575
$387$	0	0
$388$	0.312724	0.0158761
$389$	8.98152	0.455381	0.227691	−	0.973734i	$-0.426883\pi$
$389$	8.98152	0.455381	0.227691	+	0.973734i	$0.426883\pi$
$390$	0	0
$391$	0.899159	0.0454724
$392$	19.7001	0.995007
$393$	0	0
$394$	5.82822	0.293622
$395$	20.1704	1.01488
$396$	0	0
$397$	−10.3125	−0.517569	−0.258784	−	0.965935i	$-0.583322\pi$
$397$	−10.3125	−0.517569	−0.258784	+	0.965935i	$0.583322\pi$
$398$	6.35527	0.318561
$399$	0	0
$400$	13.5909	0.679546
$401$	22.2691	1.11207	0.556033	−	0.831160i	$-0.312323\pi$
$401$	22.2691	1.11207	0.556033	+	0.831160i	$0.312323\pi$
$402$	0	0
$403$	0.736492	0.0366873
$404$	0.755464	0.0375858
$405$	0	0
$406$	3.98596	0.197820
$407$	−47.0019	−2.32980
$408$	0	0
$409$	−9.36108	−0.462876	−0.231438	−	0.972850i	$-0.574343\pi$
$409$	−9.36108	−0.462876	−0.231438	+	0.972850i	$0.574343\pi$
$410$	−19.9940	−0.987433
$411$	0	0
$412$	1.96257	0.0966891
$413$	−1.57356	−0.0774297
$414$	0	0
$415$	7.00980	0.344098
$416$	−1.24134	−0.0608619
$417$	0	0
$418$	16.7578	0.819648
$419$	34.9797	1.70887	0.854436	−	0.519557i	$-0.173903\pi$
$419$	34.9797	1.70887	0.854436	+	0.519557i	$0.173903\pi$
$420$	0	0
$421$	15.9244	0.776107	0.388054	−	0.921637i	$-0.373147\pi$
$421$	15.9244	0.776107	0.388054	+	0.921637i	$0.373147\pi$
$422$	14.0880	0.685793
$423$	0	0
$424$	25.4573	1.23631
$425$	18.9538	0.919396
$426$	0	0
$427$	−2.52599	−0.122241
$428$	1.08259	0.0523290
$429$	0	0
$430$	8.01036	0.386294
$431$	37.5257	1.80755	0.903775	−	0.428008i	$-0.140785\pi$
$431$	37.5257	1.80755	0.903775	+	0.428008i	$0.140785\pi$
$432$	0	0
$433$	27.7776	1.33491	0.667454	−	0.744651i	$-0.267384\pi$
$433$	27.7776	1.33491	0.667454	+	0.744651i	$0.267384\pi$
$434$	−0.255088	−0.0122446
$435$	0	0
$436$	−0.770651	−0.0369075
$437$	−0.365572	−0.0174877
$438$	0	0
$439$	37.3507	1.78265	0.891327	−	0.453362i	$-0.149775\pi$
$439$	37.3507	1.78265	0.891327	+	0.453362i	$0.149775\pi$
$440$	19.5806	0.933468
$441$	0	0
$442$	13.3331	0.634193
$443$	−15.8245	−0.751844	−0.375922	−	0.926651i	$-0.622674\pi$
$443$	−15.8245	−0.751844	−0.375922	+	0.926651i	$0.622674\pi$
$444$	0	0
$445$	−4.99119	−0.236605
$446$	20.2869	0.960612
$447$	0	0
$448$	3.94933	0.186588
$449$	33.0395	1.55923	0.779616	−	0.626258i	$-0.215414\pi$
$449$	33.0395	1.55923	0.779616	+	0.626258i	$0.215414\pi$
$450$	0	0
$451$	71.4118	3.36265
$452$	0.161596	0.00760086
$453$	0	0
$454$	−20.4606	−0.960265
$455$	1.02300	0.0479591
$456$	0	0
$457$	−6.38585	−0.298717	−0.149359	−	0.988783i	$-0.547721\pi$
$457$	−6.38585	−0.298717	−0.149359	+	0.988783i	$0.547721\pi$
$458$	−25.0334	−1.16973
$459$	0	0
$460$	−0.0238394	−0.00111152
$461$	−4.10510	−0.191193	−0.0955967	−	0.995420i	$-0.530476\pi$
$461$	−4.10510	−0.191193	−0.0955967	+	0.995420i	$0.530476\pi$
$462$	0	0
$463$	−21.9482	−1.02002	−0.510011	−	0.860168i	$-0.670359\pi$
$463$	−21.9482	−1.02002	−0.510011	+	0.860168i	$0.670359\pi$
$464$	−23.2155	−1.07775
$465$	0	0
$466$	22.8732	1.05958
$467$	20.9539	0.969633	0.484816	−	0.874616i	$-0.338886\pi$
$467$	20.9539	0.969633	0.484816	+	0.874616i	$0.338886\pi$
$468$	0	0
$469$	3.95539	0.182643
$470$	−0.839130	−0.0387062
$471$	0	0
$472$	9.74282	0.448450
$473$	−28.6103	−1.31550
$474$	0	0
$475$	−7.70608	−0.353579
$476$	0.290113	0.0132973
$477$	0	0
$478$	−33.4291	−1.52901
$479$	−40.1617	−1.83503	−0.917517	−	0.397698i	$-0.869809\pi$
$479$	−40.1617	−1.83503	−0.917517	+	0.397698i	$0.869809\pi$
$480$	0	0
$481$	15.2044	0.693260
$482$	24.8389	1.13138
$483$	0	0
$484$	−2.60269	−0.118304
$485$	3.10232	0.140869
$486$	0	0
$487$	−1.56804	−0.0710547	−0.0355273	−	0.999369i	$-0.511311\pi$
$487$	−1.56804	−0.0710547	−0.0355273	+	0.999369i	$0.511311\pi$
$488$	15.6399	0.707985
$489$	0	0
$490$	10.9070	0.492729
$491$	−9.66243	−0.436059	−0.218030	−	0.975942i	$-0.569963\pi$
$491$	−9.66243	−0.436059	−0.218030	+	0.975942i	$0.569963\pi$
$492$	0	0
$493$	−32.3763	−1.45815
$494$	−5.42086	−0.243896
$495$	0	0
$496$	1.48572	0.0667106
$497$	−3.50871	−0.157387
$498$	0	0
$499$	−20.6363	−0.923810	−0.461905	−	0.886930i	$-0.652834\pi$
$499$	−20.6363	−0.923810	−0.461905	+	0.886930i	$0.652834\pi$
$500$	−1.19573	−0.0534745
$501$	0	0
$502$	−36.9951	−1.65117
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	7.49444	0.333498
$506$	−1.35535	−0.0602528
$507$	0	0
$508$	−0.800889	−0.0355337
$509$	23.3284	1.03401	0.517006	−	0.855982i	$-0.327047\pi$
$509$	23.3284	1.03401	0.517006	+	0.855982i	$0.327047\pi$
$510$	0	0
$511$	−2.15883	−0.0955012
$512$	−24.2947	−1.07369
$513$	0	0
$514$	−27.8501	−1.22842
$515$	19.4693	0.857921
$516$	0	0
$517$	2.99709	0.131812
$518$	−5.26612	−0.231380
$519$	0	0
$520$	−6.33401	−0.277765
$521$	43.2647	1.89546	0.947730	−	0.319073i	$-0.103371\pi$
$521$	43.2647	1.89546	0.947730	+	0.319073i	$0.103371\pi$
$522$	0	0
$523$	−43.4082	−1.89811	−0.949054	−	0.315112i	$-0.897958\pi$
$523$	−43.4082	−1.89811	−0.949054	+	0.315112i	$0.897958\pi$
$524$	1.97233	0.0861615
$525$	0	0
$526$	16.5430	0.721310
$527$	2.07197	0.0902565
$528$	0	0
$529$	−22.9704	−0.998714
$530$	14.0945	0.612225
$531$	0	0
$532$	−0.117952	−0.00511385
$533$	−23.1006	−1.00060
$534$	0	0
$535$	10.7396	0.464315
$536$	−24.4902	−1.05781
$537$	0	0
$538$	34.3110	1.47925
$539$	−38.9562	−1.67796
$540$	0	0
$541$	32.5657	1.40011	0.700054	−	0.714090i	$-0.253159\pi$
$541$	32.5657	1.40011	0.700054	+	0.714090i	$0.253159\pi$
$542$	−24.7406	−1.06270
$543$	0	0
$544$	−3.49228	−0.149730
$545$	−7.64509	−0.327480
$546$	0	0
$547$	27.8377	1.19025	0.595127	−	0.803631i	$-0.297102\pi$
$547$	27.8377	1.19025	0.595127	+	0.803631i	$0.297102\pi$
$548$	1.54466	0.0659847
$549$	0	0
$550$	−28.5702	−1.21824
$551$	13.1633	0.560773
$552$	0	0
$553$	−8.07162	−0.343240
$554$	−40.2876	−1.71166
$555$	0	0
$556$	1.28229	0.0543811
$557$	−15.7225	−0.666182	−0.333091	−	0.942895i	$-0.608092\pi$
$557$	−15.7225	−0.666182	−0.333091	+	0.942895i	$0.608092\pi$
$558$	0	0
$559$	9.25498	0.391444
$560$	2.06369	0.0872068
$561$	0	0
$562$	−16.5953	−0.700033
$563$	24.7817	1.04442	0.522212	−	0.852816i	$-0.325107\pi$
$563$	24.7817	1.04442	0.522212	+	0.852816i	$0.325107\pi$
$564$	0	0
$565$	1.60309	0.0674423
$566$	−11.0312	−0.463674
$567$	0	0
$568$	21.7245	0.911539
$569$	−2.95346	−0.123816	−0.0619078	−	0.998082i	$-0.519718\pi$
$569$	−2.95346	−0.123816	−0.0619078	+	0.998082i	$0.519718\pi$
$570$	0	0
$571$	−10.7926	−0.451656	−0.225828	−	0.974167i	$-0.572509\pi$
$571$	−10.7926	−0.451656	−0.225828	+	0.974167i	$0.572509\pi$
$572$	1.26259	0.0527914
$573$	0	0
$574$	8.00101	0.333956
$575$	0.623261	0.0259918
$576$	0	0
$577$	−5.92593	−0.246700	−0.123350	−	0.992363i	$-0.539364\pi$
$577$	−5.92593	−0.246700	−0.123350	+	0.992363i	$0.539364\pi$
$578$	14.1898	0.590220
$579$	0	0
$580$	0.858391	0.0356428
$581$	−2.80512	−0.116376
$582$	0	0
$583$	−50.3407	−2.08490
$584$	13.3666	0.553115
$585$	0	0
$586$	18.5514	0.766350
$587$	14.8090	0.611234	0.305617	−	0.952155i	$-0.401137\pi$
$587$	14.8090	0.611234	0.305617	+	0.952155i	$0.401137\pi$
$588$	0	0
$589$	−0.842403	−0.0347106
$590$	5.39414	0.222073
$591$	0	0
$592$	30.6716	1.26060
$593$	−33.4465	−1.37348	−0.686741	−	0.726902i	$-0.740959\pi$
$593$	−33.4465	−1.37348	−0.686741	+	0.726902i	$0.740959\pi$
$594$	0	0
$595$	2.87801	0.117987
$596$	0.00249234	0.000102090 0
$597$	0	0
$598$	0.438435	0.0179289
$599$	−9.35823	−0.382367	−0.191183	−	0.981554i	$-0.561233\pi$
$599$	−9.35823	−0.382367	−0.191183	+	0.981554i	$0.561233\pi$
$600$	0	0
$601$	−6.16182	−0.251346	−0.125673	−	0.992072i	$-0.540109\pi$
$601$	−6.16182	−0.251346	−0.125673	+	0.992072i	$0.540109\pi$
$602$	−3.20551	−0.130647
$603$	0	0
$604$	−1.79341	−0.0729730
$605$	−25.8195	−1.04971
$606$	0	0
$607$	35.7361	1.45048	0.725242	−	0.688494i	$-0.241728\pi$
$607$	35.7361	1.45048	0.725242	+	0.688494i	$0.241728\pi$
$608$	1.41986	0.0575828
$609$	0	0
$610$	8.65907	0.350595
$611$	−0.969511	−0.0392222
$612$	0	0
$613$	−1.96528	−0.0793771	−0.0396885	−	0.999212i	$-0.512637\pi$
$613$	−1.96528	−0.0793771	−0.0396885	+	0.999212i	$0.512637\pi$
$614$	4.94708	0.199648
$615$	0	0
$616$	−7.83558	−0.315705
$617$	35.4461	1.42701	0.713503	−	0.700652i	$-0.247107\pi$
$617$	35.4461	1.42701	0.713503	+	0.700652i	$0.247107\pi$
$618$	0	0
$619$	−34.4643	−1.38524	−0.692618	−	0.721304i	$-0.743543\pi$
$619$	−34.4643	−1.38524	−0.692618	+	0.721304i	$0.743543\pi$
$620$	−0.0549341	−0.00220621
$621$	0	0
$622$	37.7841	1.51501
$623$	1.99733	0.0800213
$624$	0	0
$625$	6.26126	0.250451
$626$	−41.4538	−1.65683
$627$	0	0
$628$	−1.31739	−0.0525695
$629$	42.7745	1.70553
$630$	0	0
$631$	−29.7560	−1.18457	−0.592284	−	0.805729i	$-0.701774\pi$
$631$	−29.7560	−1.18457	−0.592284	+	0.805729i	$0.701774\pi$
$632$	49.9762	1.98795
$633$	0	0
$634$	7.17650	0.285015
$635$	−7.94506	−0.315290
$636$	0	0
$637$	12.6017	0.499298
$638$	48.8026	1.93211
$639$	0	0
$640$	−11.9718	−0.473228
$641$	32.9327	1.30077	0.650383	−	0.759607i	$-0.274609\pi$
$641$	32.9327	1.30077	0.650383	+	0.759607i	$0.274609\pi$
$642$	0	0
$643$	−0.492253	−0.0194126	−0.00970628	−	0.999953i	$-0.503090\pi$
$643$	−0.492253	−0.0194126	−0.00970628	+	0.999953i	$0.503090\pi$
$644$	0.00953982	0.000375922 0
$645$	0	0
$646$	−15.2505	−0.600024
$647$	−3.76376	−0.147969	−0.0739844	−	0.997259i	$-0.523571\pi$
$647$	−3.76376	−0.147969	−0.0739844	+	0.997259i	$0.523571\pi$
$648$	0	0
$649$	−19.2660	−0.756258
$650$	9.24200	0.362501
$651$	0	0
$652$	1.22223	0.0478662
$653$	−18.5728	−0.726809	−0.363404	−	0.931631i	$-0.618386\pi$
$653$	−18.5728	−0.726809	−0.363404	+	0.931631i	$0.618386\pi$
$654$	0	0
$655$	19.5661	0.764510
$656$	−46.6005	−1.81945
$657$	0	0
$658$	0.335795	0.0130907
$659$	34.1930	1.33197	0.665985	−	0.745965i	$-0.268012\pi$
$659$	34.1930	1.33197	0.665985	+	0.745965i	$0.268012\pi$
$660$	0	0
$661$	29.5439	1.14912	0.574562	−	0.818461i	$-0.305173\pi$
$661$	29.5439	1.14912	0.574562	+	0.818461i	$0.305173\pi$
$662$	−7.62862	−0.296495
$663$	0	0
$664$	17.3682	0.674015
$665$	−1.17012	−0.0453751
$666$	0	0
$667$	−1.06463	−0.0412227
$668$	−1.84944	−0.0715572
$669$	0	0
$670$	−13.5590	−0.523831
$671$	−30.9273	−1.19393
$672$	0	0
$673$	−9.15185	−0.352778	−0.176389	−	0.984321i	$-0.556442\pi$
$673$	−9.15185	−0.352778	−0.176389	+	0.984321i	$0.556442\pi$
$674$	1.65045	0.0635728
$675$	0	0
$676$	1.12840	0.0434001
$677$	33.1256	1.27312	0.636561	−	0.771227i	$-0.280356\pi$
$677$	33.1256	1.27312	0.636561	+	0.771227i	$0.280356\pi$
$678$	0	0
$679$	−1.24146	−0.0476427
$680$	−17.8195	−0.683345
$681$	0	0
$682$	−3.12320	−0.119593
$683$	−49.1452	−1.88049	−0.940245	−	0.340498i	$-0.889404\pi$
$683$	−49.1452	−1.88049	−0.940245	+	0.340498i	$0.889404\pi$
$684$	0	0
$685$	15.3235	0.585481
$686$	−8.87113	−0.338701
$687$	0	0
$688$	18.6700	0.711786
$689$	16.2844	0.620387
$690$	0	0
$691$	34.1013	1.29727	0.648637	−	0.761098i	$-0.275339\pi$
$691$	34.1013	1.29727	0.648637	+	0.761098i	$0.275339\pi$
$692$	−0.673033	−0.0255849
$693$	0	0
$694$	−25.8938	−0.982914
$695$	12.7207	0.482523
$696$	0	0
$697$	−64.9889	−2.46163
$698$	25.1904	0.953473
$699$	0	0
$700$	0.201095	0.00760067
$701$	−26.2590	−0.991787	−0.495894	−	0.868383i	$-0.665159\pi$
$701$	−26.2590	−0.991787	−0.495894	+	0.868383i	$0.665159\pi$
$702$	0	0
$703$	−17.3909	−0.655909
$704$	48.3541	1.82241
$705$	0	0
$706$	2.83299	0.106621
$707$	−2.99905	−0.112791
$708$	0	0
$709$	28.4811	1.06963	0.534815	−	0.844969i	$-0.320381\pi$
$709$	28.4811	1.06963	0.534815	+	0.844969i	$0.320381\pi$
$710$	12.0278	0.451395
$711$	0	0
$712$	−12.3666	−0.463460
$713$	0.0681329	0.00255160
$714$	0	0
$715$	12.5252	0.468418
$716$	1.28350	0.0479666
$717$	0	0
$718$	10.7493	0.401162
$719$	−13.8368	−0.516024	−0.258012	−	0.966142i	$-0.583067\pi$
$719$	−13.8368	−0.516024	−0.258012	+	0.966142i	$0.583067\pi$
$720$	0	0
$721$	−7.79106	−0.290154
$722$	−19.8634	−0.739240
$723$	0	0
$724$	−0.740746	−0.0275296
$725$	−22.4419	−0.833473
$726$	0	0
$727$	9.61759	0.356697	0.178348	−	0.983967i	$-0.442925\pi$
$727$	9.61759	0.356697	0.178348	+	0.983967i	$0.442925\pi$
$728$	2.53469	0.0939417
$729$	0	0
$730$	7.40046	0.273903
$731$	26.0370	0.963015
$732$	0	0
$733$	−16.9530	−0.626175	−0.313087	−	0.949724i	$-0.601363\pi$
$733$	−16.9530	−0.626175	−0.313087	+	0.949724i	$0.601363\pi$
$734$	37.5153	1.38472
$735$	0	0
$736$	−0.114837	−0.00423294
$737$	48.4283	1.78388
$738$	0	0
$739$	9.82452	0.361401	0.180700	−	0.983538i	$-0.442164\pi$
$739$	9.82452	0.361401	0.180700	+	0.983538i	$0.442164\pi$
$740$	−1.13408	−0.0416895
$741$	0	0
$742$	−5.64019	−0.207058
$743$	−28.8650	−1.05896	−0.529478	−	0.848324i	$-0.677612\pi$
$743$	−28.8650	−1.05896	−0.529478	+	0.848324i	$0.677612\pi$
$744$	0	0
$745$	0.0247248	0.000905845 0
$746$	42.9890	1.57394
$747$	0	0
$748$	3.55204	0.129875
$749$	−4.29769	−0.157034
$750$	0	0
$751$	−9.70834	−0.354262	−0.177131	−	0.984187i	$-0.556682\pi$
$751$	−9.70834	−0.354262	−0.177131	+	0.984187i	$0.556682\pi$
$752$	−1.95578	−0.0713200
$753$	0	0
$754$	−15.7869	−0.574923
$755$	−17.7912	−0.647489
$756$	0	0
$757$	−5.99476	−0.217883	−0.108942	−	0.994048i	$-0.534746\pi$
$757$	−5.99476	−0.217883	−0.108942	+	0.994048i	$0.534746\pi$
$758$	−2.70505	−0.0982517
$759$	0	0
$760$	7.24488	0.262799
$761$	−15.4793	−0.561123	−0.280562	−	0.959836i	$-0.590521\pi$
$761$	−15.4793	−0.561123	−0.280562	+	0.959836i	$0.590521\pi$
$762$	0	0
$763$	3.05934	0.110756
$764$	0.976829	0.0353404
$765$	0	0
$766$	34.2198	1.23641
$767$	6.23226	0.225034
$768$	0	0
$769$	−27.9740	−1.00877	−0.504384	−	0.863480i	$-0.668280\pi$
$769$	−27.9740	−1.00877	−0.504384	+	0.863480i	$0.668280\pi$
$770$	−4.33819	−0.156337
$771$	0	0
$772$	2.34626	0.0844436
$773$	−31.7330	−1.14136	−0.570679	−	0.821173i	$-0.693320\pi$
$773$	−31.7330	−1.14136	−0.570679	+	0.821173i	$0.693320\pi$
$774$	0	0
$775$	1.43621	0.0515901
$776$	7.68659	0.275932
$777$	0	0
$778$	12.3207	0.441718
$779$	26.4226	0.946687
$780$	0	0
$781$	−42.9593	−1.53720
$782$	1.23345	0.0441080
$783$	0	0
$784$	25.4213	0.907903
$785$	−13.0689	−0.466448
$786$	0	0
$787$	19.1339	0.682051	0.341025	−	0.940054i	$-0.389226\pi$
$787$	19.1339	0.682051	0.341025	+	0.940054i	$0.389226\pi$
$788$	−0.502266	−0.0178925
$789$	0	0
$790$	27.6694	0.984434
$791$	−0.641508	−0.0228094
$792$	0	0
$793$	10.0045	0.355270
$794$	−14.1465	−0.502039
$795$	0	0
$796$	−0.547686	−0.0194122
$797$	2.45993	0.0871352	0.0435676	−	0.999050i	$-0.486128\pi$
$797$	2.45993	0.0871352	0.0435676	+	0.999050i	$0.486128\pi$
$798$	0	0
$799$	−2.72752	−0.0964929
$800$	−2.42071	−0.0855849
$801$	0	0
$802$	30.5483	1.07870
$803$	−26.4320	−0.932764
$804$	0	0
$805$	0.0946379	0.00333555
$806$	1.01031	0.0355865
$807$	0	0
$808$	18.5689	0.653252
$809$	−16.9350	−0.595404	−0.297702	−	0.954659i	$-0.596220\pi$
$809$	−16.9350	−0.595404	−0.297702	+	0.954659i	$0.596220\pi$
$810$	0	0
$811$	−23.7584	−0.834269	−0.417135	−	0.908845i	$-0.636966\pi$
$811$	−23.7584	−0.834269	−0.417135	+	0.908845i	$0.636966\pi$
$812$	−0.343503	−0.0120546
$813$	0	0
$814$	−64.4764	−2.25990
$815$	12.1249	0.424716
$816$	0	0
$817$	−10.5859	−0.370354
$818$	−12.8413	−0.448987
$819$	0	0
$820$	1.72305	0.0601714
$821$	−19.2765	−0.672755	−0.336378	−	0.941727i	$-0.609202\pi$
$821$	−19.2765	−0.672755	−0.336378	+	0.941727i	$0.609202\pi$
$822$	0	0
$823$	52.8192	1.84116	0.920580	−	0.390553i	$-0.127716\pi$
$823$	52.8192	1.84116	0.920580	+	0.390553i	$0.127716\pi$
$824$	48.2390	1.68049
$825$	0	0
$826$	−2.15858	−0.0751064
$827$	25.7144	0.894176	0.447088	−	0.894490i	$-0.352461\pi$
$827$	25.7144	0.894176	0.447088	+	0.894490i	$0.352461\pi$
$828$	0	0
$829$	23.3380	0.810562	0.405281	−	0.914192i	$-0.367174\pi$
$829$	23.3380	0.810562	0.405281	+	0.914192i	$0.367174\pi$
$830$	9.61591	0.333773
$831$	0	0
$832$	−15.6418	−0.542281
$833$	35.4524	1.22835
$834$	0	0
$835$	−18.3471	−0.634926
$836$	−1.44415	−0.0499471
$837$	0	0
$838$	47.9845	1.65760
$839$	−18.8028	−0.649144	−0.324572	−	0.945861i	$-0.605220\pi$
$839$	−18.8028	−0.649144	−0.324572	+	0.945861i	$0.605220\pi$
$840$	0	0
$841$	9.33453	0.321880
$842$	21.8448	0.752821
$843$	0	0
$844$	−1.21408	−0.0417904
$845$	11.1941	0.385089
$846$	0	0
$847$	10.3322	0.355019
$848$	32.8504	1.12809
$849$	0	0
$850$	26.0005	0.891810
$851$	1.40656	0.0482162
$852$	0	0
$853$	−45.0553	−1.54266	−0.771332	−	0.636433i	$-0.780409\pi$
$853$	−45.0553	−1.54266	−0.771332	+	0.636433i	$0.780409\pi$
$854$	−3.46511	−0.118573
$855$	0	0
$856$	26.6095	0.909495
$857$	33.4362	1.14216	0.571079	−	0.820895i	$-0.306525\pi$
$857$	33.4362	1.14216	0.571079	+	0.820895i	$0.306525\pi$
$858$	0	0
$859$	19.4483	0.663567	0.331784	−	0.943355i	$-0.392350\pi$
$859$	19.4483	0.663567	0.331784	+	0.943355i	$0.392350\pi$
$860$	−0.690319	−0.0235397
$861$	0	0
$862$	51.4770	1.75332
$863$	16.1628	0.550186	0.275093	−	0.961418i	$-0.411291\pi$
$863$	16.1628	0.550186	0.275093	+	0.961418i	$0.411291\pi$
$864$	0	0
$865$	−6.67669	−0.227014
$866$	38.1048	1.29485
$867$	0	0
$868$	0.0219830	0.000746153 0
$869$	−98.8259	−3.35244
$870$	0	0
$871$	−15.6658	−0.530815
$872$	−18.9422	−0.641463
$873$	0	0
$874$	−0.501484	−0.0169630
$875$	4.74681	0.160471
$876$	0	0
$877$	−22.3959	−0.756255	−0.378127	−	0.925754i	$-0.623432\pi$
$877$	−22.3959	−0.756255	−0.378127	+	0.925754i	$0.623432\pi$
$878$	51.2370	1.72917
$879$	0	0
$880$	25.2670	0.851751
$881$	−10.4074	−0.350633	−0.175316	−	0.984512i	$-0.556095\pi$
$881$	−10.4074	−0.350633	−0.175316	+	0.984512i	$0.556095\pi$
$882$	0	0
$883$	−18.3582	−0.617803	−0.308902	−	0.951094i	$-0.599961\pi$
$883$	−18.3582	−0.617803	−0.308902	+	0.951094i	$0.599961\pi$
$884$	−1.14903	−0.0386460
$885$	0	0
$886$	−21.7077	−0.729286
$887$	22.9758	0.771453	0.385726	−	0.922613i	$-0.373951\pi$
$887$	22.9758	0.771453	0.385726	+	0.922613i	$0.373951\pi$
$888$	0	0
$889$	3.17938	0.106633
$890$	−6.84682	−0.229506
$891$	0	0
$892$	−1.74829	−0.0585370
$893$	1.10893	0.0371090
$894$	0	0
$895$	12.7327	0.425607
$896$	4.79077	0.160048
$897$	0	0
$898$	45.3230	1.51245
$899$	−2.45328	−0.0818215
$900$	0	0
$901$	45.8130	1.52625
$902$	97.9614	3.26176
$903$	0	0
$904$	3.97196	0.132105
$905$	−7.34843	−0.244270
$906$	0	0
$907$	0.415835	0.0138076	0.00690378	−	0.999976i	$-0.497802\pi$
$907$	0.415835	0.0138076	0.00690378	+	0.999976i	$0.497802\pi$
$908$	1.76326	0.0585159
$909$	0	0
$910$	1.40333	0.0465201
$911$	13.6089	0.450881	0.225441	−	0.974257i	$-0.427618\pi$
$911$	13.6089	0.450881	0.225441	+	0.974257i	$0.427618\pi$
$912$	0	0
$913$	−34.3448	−1.13665
$914$	−8.75998	−0.289754
$915$	0	0
$916$	2.15734	0.0712804
$917$	−7.82978	−0.258562
$918$	0	0
$919$	−14.8913	−0.491218	−0.245609	−	0.969369i	$-0.578988\pi$
$919$	−14.8913	−0.491218	−0.245609	+	0.969369i	$0.578988\pi$
$920$	−0.585959	−0.0193185
$921$	0	0
$922$	−5.63129	−0.185457
$923$	13.8966	0.457414
$924$	0	0
$925$	29.6496	0.974871
$926$	−30.1082	−0.989416
$927$	0	0
$928$	4.13496	0.135737
$929$	19.4977	0.639699	0.319850	−	0.947468i	$-0.396368\pi$
$929$	19.4977	0.639699	0.319850	+	0.947468i	$0.396368\pi$
$930$	0	0
$931$	−14.4139	−0.472397
$932$	−1.97117	−0.0645678
$933$	0	0
$934$	28.7442	0.940539
$935$	35.2373	1.15238
$936$	0	0
$937$	12.5125	0.408764	0.204382	−	0.978891i	$-0.434481\pi$
$937$	12.5125	0.408764	0.204382	+	0.978891i	$0.434481\pi$
$938$	5.42593	0.177163
$939$	0	0
$940$	0.0723148	0.00235865
$941$	−4.08256	−0.133088	−0.0665439	−	0.997783i	$-0.521197\pi$
$941$	−4.08256	−0.133088	−0.0665439	+	0.997783i	$0.521197\pi$
$942$	0	0
$943$	−2.13704	−0.0695915
$944$	12.5723	0.409192
$945$	0	0
$946$	−39.2471	−1.27603
$947$	0.698574	0.0227006	0.0113503	−	0.999936i	$-0.496387\pi$
$947$	0.698574	0.0227006	0.0113503	+	0.999936i	$0.496387\pi$
$948$	0	0
$949$	8.55032	0.277555
$950$	−10.5710	−0.342970
$951$	0	0
$952$	7.13083	0.231112
$953$	−7.31223	−0.236866	−0.118433	−	0.992962i	$-0.537787\pi$
$953$	−7.31223	−0.236866	−0.118433	+	0.992962i	$0.537787\pi$
$954$	0	0
$955$	9.69044	0.313575
$956$	2.88086	0.0931737
$957$	0	0
$958$	−55.0930	−1.77997
$959$	−6.13202	−0.198013
$960$	0	0
$961$	−30.8430	−0.994935
$962$	20.8571	0.672459
$963$	0	0
$964$	−2.14058	−0.0689434
$965$	23.2756	0.749267
$966$	0	0
$967$	38.1107	1.22556	0.612778	−	0.790255i	$-0.290052\pi$
$967$	38.1107	1.22556	0.612778	+	0.790255i	$0.290052\pi$
$968$	−63.9729	−2.05617
$969$	0	0
$970$	4.25570	0.136642
$971$	51.0424	1.63803	0.819014	−	0.573773i	$-0.194521\pi$
$971$	51.0424	1.63803	0.819014	+	0.573773i	$0.194521\pi$
$972$	0	0
$973$	−5.09045	−0.163192
$974$	−2.15101	−0.0689227
$975$	0	0
$976$	20.1819	0.646007
$977$	−53.4134	−1.70885	−0.854424	−	0.519577i	$-0.826090\pi$
$977$	−53.4134	−1.70885	−0.854424	+	0.519577i	$0.826090\pi$
$978$	0	0
$979$	24.4546	0.781571
$980$	−0.939949	−0.0300256
$981$	0	0
$982$	−13.2547	−0.422976
$983$	−1.20932	−0.0385714	−0.0192857	−	0.999814i	$-0.506139\pi$
$983$	−1.20932	−0.0385714	−0.0192857	+	0.999814i	$0.506139\pi$
$984$	0	0
$985$	−4.98264	−0.158760
$986$	−44.4132	−1.41440
$987$	0	0
$988$	0.467161	0.0148624
$989$	0.856179	0.0272249
$990$	0	0
$991$	−13.3298	−0.423433	−0.211717	−	0.977331i	$-0.567905\pi$
$991$	−13.3298	−0.423433	−0.211717	+	0.977331i	$0.567905\pi$
$992$	−0.264624	−0.00840181
$993$	0	0
$994$	−4.81318	−0.152665
$995$	−5.43321	−0.172244
$996$	0	0
$997$	−39.2475	−1.24298	−0.621490	−	0.783422i	$-0.713472\pi$
$997$	−39.2475	−1.24298	−0.621490	+	0.783422i	$0.713472\pi$
$998$	−28.3085	−0.896091
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.7		10
3.2	odd	2		503.2.a.e.1.4	✓	10
12.11	even	2		8048.2.a.p.1.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.4	✓	10	3.2	odd	2
4527.2.a.k.1.7		10	1.1	even	1	trivial
8048.2.a.p.1.4		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)

\(f(q)\)	\(=\)	\(q+1.37178 q^{2} -0.118218 q^{4} -1.17276 q^{5} +0.469303 q^{7} -2.90573 q^{8} +O(q^{10})\)	\(q+1.37178 q^{2} -0.118218 q^{4} -1.17276 q^{5} +0.469303 q^{7} -2.90573 q^{8} -1.60876 q^{10} +5.74596 q^{11} -1.85873 q^{13} +0.643780 q^{14} -3.74959 q^{16} -5.22916 q^{17} +2.12602 q^{19} +0.138641 q^{20} +7.88220 q^{22} -0.171951 q^{23} -3.62464 q^{25} -2.54977 q^{26} -0.0554799 q^{28} +6.19149 q^{29} -0.396234 q^{31} +0.667846 q^{32} -7.17326 q^{34} -0.550377 q^{35} -8.17999 q^{37} +2.91644 q^{38} +3.40771 q^{40} +12.4282 q^{41} -4.97920 q^{43} -0.679275 q^{44} -0.235879 q^{46} +0.521599 q^{47} -6.77976 q^{49} -4.97222 q^{50} +0.219735 q^{52} -8.76106 q^{53} -6.73861 q^{55} -1.36367 q^{56} +8.49336 q^{58} -3.35297 q^{59} -5.38243 q^{61} -0.543546 q^{62} +8.41532 q^{64} +2.17983 q^{65} +8.42823 q^{67} +0.618179 q^{68} -0.754997 q^{70} -7.47643 q^{71} -4.60009 q^{73} -11.2212 q^{74} -0.251334 q^{76} +2.69660 q^{77} -17.1992 q^{79} +4.39735 q^{80} +17.0487 q^{82} -5.97721 q^{83} +6.13253 q^{85} -6.83038 q^{86} -16.6962 q^{88} +4.25595 q^{89} -0.872306 q^{91} +0.0203277 q^{92} +0.715520 q^{94} -2.49331 q^{95} -2.64532 q^{97} -9.30034 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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

L-functions

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