LMFDB - Embedded newform 4527.2.a.k.1.9

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			1.9
Root			$2.78533$ 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)

$f(q)$	$=$	$q+2.03947 q^{2} +2.15945 q^{4} +0.701114 q^{5} -2.02991 q^{7} +0.325186 q^{8} +O(q^{10})$	\(q+2.03947 q^{2} +2.15945 q^{4} +0.701114 q^{5} -2.02991 q^{7} +0.325186 q^{8} +1.42990 q^{10} +0.626970 q^{11} -2.93743 q^{13} -4.13995 q^{14} -3.65568 q^{16} +2.71003 q^{17} -1.11114 q^{19} +1.51402 q^{20} +1.27869 q^{22} +0.412395 q^{23} -4.50844 q^{25} -5.99081 q^{26} -4.38349 q^{28} -6.46349 q^{29} -4.14074 q^{31} -8.10604 q^{32} +5.52704 q^{34} -1.42320 q^{35} +2.98634 q^{37} -2.26613 q^{38} +0.227992 q^{40} +0.135430 q^{41} -0.861393 q^{43} +1.35391 q^{44} +0.841067 q^{46} -1.67307 q^{47} -2.87945 q^{49} -9.19483 q^{50} -6.34323 q^{52} +8.51721 q^{53} +0.439578 q^{55} -0.660099 q^{56} -13.1821 q^{58} +0.406685 q^{59} -8.53132 q^{61} -8.44492 q^{62} -9.22067 q^{64} -2.05948 q^{65} -13.0293 q^{67} +5.85217 q^{68} -2.90258 q^{70} -2.78095 q^{71} -2.11734 q^{73} +6.09055 q^{74} -2.39944 q^{76} -1.27270 q^{77} -0.317368 q^{79} -2.56305 q^{80} +0.276205 q^{82} +5.05120 q^{83} +1.90004 q^{85} -1.75679 q^{86} +0.203882 q^{88} -16.4699 q^{89} +5.96273 q^{91} +0.890544 q^{92} -3.41218 q^{94} -0.779033 q^{95} +7.68873 q^{97} -5.87256 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.03947	1.44212	0.721062	−	0.692870i	$-0.243654\pi$
$2$	2.03947	1.44212	0.721062	+	0.692870i	$0.243654\pi$
$3$	0	0
$3$	0	0
$4$	2.15945	1.07972
$5$	0.701114	0.313548	0.156774	−	0.987635i	$-0.449891\pi$
$5$	0.701114	0.313548	0.156774	+	0.987635i	$0.449891\pi$
$6$	0	0
$7$	−2.02991	−0.767235	−0.383618	−	0.923492i	$-0.625322\pi$
$7$	−2.02991	−0.767235	−0.383618	+	0.923492i	$0.625322\pi$
$8$	0.325186	0.114971
$9$	0	0
$10$	1.42990	0.452175
$11$	0.626970	0.189039	0.0945194	−	0.995523i	$-0.469869\pi$
$11$	0.626970	0.189039	0.0945194	+	0.995523i	$0.469869\pi$
$12$	0	0
$13$	−2.93743	−0.814697	−0.407349	−	0.913273i	$-0.633547\pi$
$13$	−2.93743	−0.814697	−0.407349	+	0.913273i	$0.633547\pi$
$14$	−4.13995	−1.10645
$15$	0	0
$16$	−3.65568	−0.913921
$17$	2.71003	0.657280	0.328640	−	0.944455i	$-0.393410\pi$
$17$	2.71003	0.657280	0.328640	+	0.944455i	$0.393410\pi$
$18$	0	0
$19$	−1.11114	−0.254912	−0.127456	−	0.991844i	$-0.540681\pi$
$19$	−1.11114	−0.254912	−0.127456	+	0.991844i	$0.540681\pi$
$20$	1.51402	0.338545
$21$	0	0
$22$	1.27869	0.272617
$23$	0.412395	0.0859902	0.0429951	−	0.999075i	$-0.486310\pi$
$23$	0.412395	0.0859902	0.0429951	+	0.999075i	$0.486310\pi$
$24$	0	0
$25$	−4.50844	−0.901688
$26$	−5.99081	−1.17489
$27$	0	0
$28$	−4.38349	−0.828402
$29$	−6.46349	−1.20024	−0.600120	−	0.799910i	$-0.704880\pi$
$29$	−6.46349	−1.20024	−0.600120	+	0.799910i	$0.704880\pi$
$30$	0	0
$31$	−4.14074	−0.743698	−0.371849	−	0.928293i	$-0.621276\pi$
$31$	−4.14074	−0.743698	−0.371849	+	0.928293i	$0.621276\pi$
$32$	−8.10604	−1.43296
$33$	0	0
$34$	5.52704	0.947879
$35$	−1.42320	−0.240565
$36$	0	0
$37$	2.98634	0.490951	0.245475	−	0.969403i	$-0.421056\pi$
$37$	2.98634	0.490951	0.245475	+	0.969403i	$0.421056\pi$
$38$	−2.26613	−0.367615
$39$	0	0
$40$	0.227992	0.0360488
$41$	0.135430	0.0211505	0.0105753	−	0.999944i	$-0.496634\pi$
$41$	0.135430	0.0211505	0.0105753	+	0.999944i	$0.496634\pi$
$42$	0	0
$43$	−0.861393	−0.131361	−0.0656806	−	0.997841i	$-0.520922\pi$
$43$	−0.861393	−0.131361	−0.0656806	+	0.997841i	$0.520922\pi$
$44$	1.35391	0.204109
$45$	0	0
$46$	0.841067	0.124009
$47$	−1.67307	−0.244043	−0.122021	−	0.992527i	$-0.538938\pi$
$47$	−1.67307	−0.244043	−0.122021	+	0.992527i	$0.538938\pi$
$48$	0	0
$49$	−2.87945	−0.411350
$50$	−9.19483	−1.30035
$51$	0	0
$52$	−6.34323	−0.879647
$53$	8.51721	1.16993	0.584964	−	0.811059i	$-0.301108\pi$
$53$	8.51721	1.16993	0.584964	+	0.811059i	$0.301108\pi$
$54$	0	0
$55$	0.439578	0.0592727
$56$	−0.660099	−0.0882095
$57$	0	0
$58$	−13.1821	−1.73089
$59$	0.406685	0.0529459	0.0264729	−	0.999650i	$-0.491572\pi$
$59$	0.406685	0.0529459	0.0264729	+	0.999650i	$0.491572\pi$
$60$	0	0
$61$	−8.53132	−1.09232	−0.546162	−	0.837680i	$-0.683912\pi$
$61$	−8.53132	−1.09232	−0.546162	+	0.837680i	$0.683912\pi$
$62$	−8.44492	−1.07251
$63$	0	0
$64$	−9.22067	−1.15258
$65$	−2.05948	−0.255446
$66$	0	0
$67$	−13.0293	−1.59178	−0.795889	−	0.605443i	$-0.792996\pi$
$67$	−13.0293	−1.59178	−0.795889	+	0.605443i	$0.792996\pi$
$68$	5.85217	0.709680
$69$	0	0
$70$	−2.90258	−0.346925
$71$	−2.78095	−0.330039	−0.165019	−	0.986290i	$-0.552769\pi$
$71$	−2.78095	−0.330039	−0.165019	+	0.986290i	$0.552769\pi$
$72$	0	0
$73$	−2.11734	−0.247816	−0.123908	−	0.992294i	$-0.539543\pi$
$73$	−2.11734	−0.247816	−0.123908	+	0.992294i	$0.539543\pi$
$74$	6.09055	0.708012
$75$	0	0
$76$	−2.39944	−0.275234
$77$	−1.27270	−0.145037
$78$	0	0
$79$	−0.317368	−0.0357067	−0.0178534	−	0.999841i	$-0.505683\pi$
$79$	−0.317368	−0.0357067	−0.0178534	+	0.999841i	$0.505683\pi$
$80$	−2.56305	−0.286558
$81$	0	0
$82$	0.276205	0.0305017
$83$	5.05120	0.554442	0.277221	−	0.960806i	$-0.410587\pi$
$83$	5.05120	0.554442	0.277221	+	0.960806i	$0.410587\pi$
$84$	0	0
$85$	1.90004	0.206089
$86$	−1.75679	−0.189439
$87$	0	0
$88$	0.203882	0.0217339
$89$	−16.4699	−1.74580	−0.872901	−	0.487897i	$-0.837764\pi$
$89$	−16.4699	−1.74580	−0.872901	+	0.487897i	$0.837764\pi$
$90$	0	0
$91$	5.96273	0.625064
$92$	0.890544	0.0928456
$93$	0	0
$94$	−3.41218	−0.351940
$95$	−0.779033	−0.0799271
$96$	0	0
$97$	7.68873	0.780673	0.390336	−	0.920672i	$-0.372359\pi$
$97$	7.68873	0.780673	0.390336	+	0.920672i	$0.372359\pi$
$98$	−5.87256	−0.593218
$99$	0	0
$100$	−9.73573	−0.973573
$101$	0.549020	0.0546296	0.0273148	−	0.999627i	$-0.491304\pi$
$101$	0.549020	0.0546296	0.0273148	+	0.999627i	$0.491304\pi$
$102$	0	0
$103$	1.32758	0.130810	0.0654051	−	0.997859i	$-0.479166\pi$
$103$	1.32758	0.130810	0.0654051	+	0.997859i	$0.479166\pi$
$104$	−0.955211	−0.0936662
$105$	0	0
$106$	17.3706	1.68718
$107$	−2.57528	−0.248961	−0.124481	−	0.992222i	$-0.539726\pi$
$107$	−2.57528	−0.248961	−0.124481	+	0.992222i	$0.539726\pi$
$108$	0	0
$109$	8.89011	0.851518	0.425759	−	0.904837i	$-0.360007\pi$
$109$	8.89011	0.851518	0.425759	+	0.904837i	$0.360007\pi$
$110$	0.896507	0.0854786
$111$	0	0
$112$	7.42072	0.701192
$113$	1.91987	0.180606	0.0903032	−	0.995914i	$-0.471216\pi$
$113$	1.91987	0.180606	0.0903032	+	0.995914i	$0.471216\pi$
$114$	0	0
$115$	0.289136	0.0269621
$116$	−13.9576	−1.29593
$117$	0	0
$118$	0.829423	0.0763546
$119$	−5.50114	−0.504288
$120$	0	0
$121$	−10.6069	−0.964264
$122$	−17.3994	−1.57527
$123$	0	0
$124$	−8.94170	−0.802988
$125$	−6.66650	−0.596270
$126$	0	0
$127$	21.8010	1.93452	0.967261	−	0.253782i	$-0.0816746\pi$
$127$	21.8010	1.93452	0.967261	+	0.253782i	$0.0816746\pi$
$128$	−2.59322	−0.229210
$129$	0	0
$130$	−4.20024	−0.368386
$131$	−8.78145	−0.767239	−0.383620	−	0.923491i	$-0.625323\pi$
$131$	−8.78145	−0.767239	−0.383620	+	0.923491i	$0.625323\pi$
$132$	0	0
$133$	2.25551	0.195577
$134$	−26.5728	−2.29554
$135$	0	0
$136$	0.881265	0.0755678
$137$	4.26127	0.364065	0.182032	−	0.983293i	$-0.441732\pi$
$137$	4.26127	0.364065	0.182032	+	0.983293i	$0.441732\pi$
$138$	0	0
$139$	13.4951	1.14464	0.572321	−	0.820030i	$-0.306043\pi$
$139$	13.4951	1.14464	0.572321	+	0.820030i	$0.306043\pi$
$140$	−3.07333	−0.259743
$141$	0	0
$142$	−5.67168	−0.475957
$143$	−1.84168	−0.154009
$144$	0	0
$145$	−4.53164	−0.376332
$146$	−4.31826	−0.357382
$147$	0	0
$148$	6.44884	0.530091
$149$	−11.2453	−0.921253	−0.460626	−	0.887594i	$-0.652375\pi$
$149$	−11.2453	−0.921253	−0.460626	+	0.887594i	$0.652375\pi$
$150$	0	0
$151$	−2.36107	−0.192141	−0.0960705	−	0.995375i	$-0.530627\pi$
$151$	−2.36107	−0.192141	−0.0960705	+	0.995375i	$0.530627\pi$
$152$	−0.361325	−0.0293074
$153$	0	0
$154$	−2.59563	−0.209162
$155$	−2.90313	−0.233185
$156$	0	0
$157$	−17.5242	−1.39859	−0.699293	−	0.714835i	$-0.746502\pi$
$157$	−17.5242	−1.39859	−0.699293	+	0.714835i	$0.746502\pi$
$158$	−0.647264	−0.0514936
$159$	0	0
$160$	−5.68326	−0.449301
$161$	−0.837126	−0.0659747
$162$	0	0
$163$	−7.12153	−0.557801	−0.278901	−	0.960320i	$-0.589970\pi$
$163$	−7.12153	−0.557801	−0.278901	+	0.960320i	$0.589970\pi$
$164$	0.292453	0.0228367
$165$	0	0
$166$	10.3018	0.799574
$167$	−4.56080	−0.352925	−0.176463	−	0.984307i	$-0.556465\pi$
$167$	−4.56080	−0.352925	−0.176463	+	0.984307i	$0.556465\pi$
$168$	0	0
$169$	−4.37149	−0.336269
$170$	3.87509	0.297206
$171$	0	0
$172$	−1.86013	−0.141834
$173$	−2.91357	−0.221514	−0.110757	−	0.993847i	$-0.535328\pi$
$173$	−2.91357	−0.221514	−0.110757	+	0.993847i	$0.535328\pi$
$174$	0	0
$175$	9.15174	0.691807
$176$	−2.29201	−0.172766
$177$	0	0
$178$	−33.5898	−2.51766
$179$	9.01141	0.673544	0.336772	−	0.941586i	$-0.390665\pi$
$179$	9.01141	0.673544	0.336772	+	0.941586i	$0.390665\pi$
$180$	0	0
$181$	14.4065	1.07082	0.535412	−	0.844591i	$-0.320156\pi$
$181$	14.4065	1.07082	0.535412	+	0.844591i	$0.320156\pi$
$182$	12.1608	0.901420
$183$	0	0
$184$	0.134105	0.00988635
$185$	2.09376	0.153937
$186$	0	0
$187$	1.69911	0.124251
$188$	−3.61291	−0.263499
$189$	0	0
$190$	−1.58882	−0.115265
$191$	−4.40730	−0.318901	−0.159450	−	0.987206i	$-0.550972\pi$
$191$	−4.40730	−0.318901	−0.159450	+	0.987206i	$0.550972\pi$
$192$	0	0
$193$	−2.90027	−0.208766	−0.104383	−	0.994537i	$-0.533287\pi$
$193$	−2.90027	−0.208766	−0.104383	+	0.994537i	$0.533287\pi$
$194$	15.6810	1.12583
$195$	0	0
$196$	−6.21802	−0.444144
$197$	−5.91510	−0.421433	−0.210717	−	0.977547i	$-0.567580\pi$
$197$	−5.91510	−0.421433	−0.210717	+	0.977547i	$0.567580\pi$
$198$	0	0
$199$	−4.73473	−0.335636	−0.167818	−	0.985818i	$-0.553672\pi$
$199$	−4.73473	−0.335636	−0.167818	+	0.985818i	$0.553672\pi$
$200$	−1.46608	−0.103668
$201$	0	0
$202$	1.11971	0.0787827
$203$	13.1203	0.920866
$204$	0	0
$205$	0.0949516	0.00663171
$206$	2.70756	0.188644
$207$	0	0
$208$	10.7383	0.744569
$209$	−0.696649	−0.0481882
$210$	0	0
$211$	−1.50671	−0.103726	−0.0518631	−	0.998654i	$-0.516516\pi$
$211$	−1.50671	−0.103726	−0.0518631	+	0.998654i	$0.516516\pi$
$212$	18.3925	1.26320
$213$	0	0
$214$	−5.25220	−0.359033
$215$	−0.603935	−0.0411880
$216$	0	0
$217$	8.40534	0.570591
$218$	18.1311	1.22799
$219$	0	0
$220$	0.949245	0.0639981
$221$	−7.96054	−0.535484
$222$	0	0
$223$	−9.79220	−0.655734	−0.327867	−	0.944724i	$-0.606330\pi$
$223$	−9.79220	−0.655734	−0.327867	+	0.944724i	$0.606330\pi$
$224$	16.4546	1.09942
$225$	0	0
$226$	3.91552	0.260457
$227$	−6.88512	−0.456981	−0.228491	−	0.973546i	$-0.573379\pi$
$227$	−6.88512	−0.456981	−0.228491	+	0.973546i	$0.573379\pi$
$228$	0	0
$229$	12.1880	0.805405	0.402703	−	0.915331i	$-0.368071\pi$
$229$	12.1880	0.805405	0.402703	+	0.915331i	$0.368071\pi$
$230$	0.589684	0.0388826
$231$	0	0
$232$	−2.10183	−0.137992
$233$	21.0547	1.37934	0.689669	−	0.724125i	$-0.257756\pi$
$233$	21.0547	1.37934	0.689669	+	0.724125i	$0.257756\pi$
$234$	0	0
$235$	−1.17301	−0.0765190
$236$	0.878215	0.0571669
$237$	0	0
$238$	−11.2194	−0.727247
$239$	−9.11944	−0.589888	−0.294944	−	0.955515i	$-0.595301\pi$
$239$	−9.11944	−0.589888	−0.294944	+	0.955515i	$0.595301\pi$
$240$	0	0
$241$	22.7352	1.46450	0.732252	−	0.681034i	$-0.238469\pi$
$241$	22.7352	1.46450	0.732252	+	0.681034i	$0.238469\pi$
$242$	−21.6325	−1.39059
$243$	0	0
$244$	−18.4229	−1.17941
$245$	−2.01882	−0.128978
$246$	0	0
$247$	3.26388	0.207676
$248$	−1.34651	−0.0855034
$249$	0	0
$250$	−13.5961	−0.859896
$251$	11.6824	0.737385	0.368692	−	0.929551i	$-0.379806\pi$
$251$	11.6824	0.737385	0.368692	+	0.929551i	$0.379806\pi$
$252$	0	0
$253$	0.258559	0.0162555
$254$	44.4625	2.78982
$255$	0	0
$256$	13.1525	0.822034
$257$	16.3010	1.01683	0.508415	−	0.861112i	$-0.330232\pi$
$257$	16.3010	1.01683	0.508415	+	0.861112i	$0.330232\pi$
$258$	0	0
$259$	−6.06201	−0.376675
$260$	−4.44733	−0.275811
$261$	0	0
$262$	−17.9095	−1.10645
$263$	15.7955	0.973993	0.486996	−	0.873404i	$-0.338093\pi$
$263$	15.7955	0.973993	0.486996	+	0.873404i	$0.338093\pi$
$264$	0	0
$265$	5.97154	0.366829
$266$	4.60005	0.282047
$267$	0	0
$268$	−28.1360	−1.71868
$269$	−11.2401	−0.685323	−0.342662	−	0.939459i	$-0.611328\pi$
$269$	−11.2401	−0.685323	−0.342662	+	0.939459i	$0.611328\pi$
$270$	0	0
$271$	22.7114	1.37962	0.689809	−	0.723991i	$-0.257694\pi$
$271$	22.7114	1.37962	0.689809	+	0.723991i	$0.257694\pi$
$272$	−9.90703	−0.600702
$273$	0	0
$274$	8.69074	0.525027
$275$	−2.82666	−0.170454
$276$	0	0
$277$	−3.42609	−0.205854	−0.102927	−	0.994689i	$-0.532821\pi$
$277$	−3.42609	−0.205854	−0.102927	+	0.994689i	$0.532821\pi$
$278$	27.5229	1.65072
$279$	0	0
$280$	−0.462805	−0.0276579
$281$	6.41168	0.382488	0.191244	−	0.981542i	$-0.438748\pi$
$281$	6.41168	0.382488	0.191244	+	0.981542i	$0.438748\pi$
$282$	0	0
$283$	−22.3462	−1.32834	−0.664172	−	0.747580i	$-0.731216\pi$
$283$	−22.3462	−1.32834	−0.664172	+	0.747580i	$0.731216\pi$
$284$	−6.00532	−0.356350
$285$	0	0
$286$	−3.75606	−0.222101
$287$	−0.274910	−0.0162274
$288$	0	0
$289$	−9.65571	−0.567983
$290$	−9.24216	−0.542718
$291$	0	0
$292$	−4.57228	−0.267573
$293$	−4.95366	−0.289396	−0.144698	−	0.989476i	$-0.546221\pi$
$293$	−4.95366	−0.289396	−0.144698	+	0.989476i	$0.546221\pi$
$294$	0	0
$295$	0.285133	0.0166011
$296$	0.971115	0.0564449
$297$	0	0
$298$	−22.9345	−1.32856
$299$	−1.21138	−0.0700560
$300$	0	0
$301$	1.74855	0.100785
$302$	−4.81533	−0.277091
$303$	0	0
$304$	4.06196	0.232969
$305$	−5.98143	−0.342496
$306$	0	0
$307$	−1.86937	−0.106690	−0.0533452	−	0.998576i	$-0.516988\pi$
$307$	−1.86937	−0.106690	−0.0533452	+	0.998576i	$0.516988\pi$
$308$	−2.74832	−0.156600
$309$	0	0
$310$	−5.92085	−0.336282
$311$	4.29662	0.243639	0.121819	−	0.992552i	$-0.461127\pi$
$311$	4.29662	0.243639	0.121819	+	0.992552i	$0.461127\pi$
$312$	0	0
$313$	−24.2378	−1.37000	−0.684999	−	0.728544i	$-0.740198\pi$
$313$	−24.2378	−1.37000	−0.684999	+	0.728544i	$0.740198\pi$
$314$	−35.7402	−2.01693
$315$	0	0
$316$	−0.685340	−0.0385534
$317$	28.1902	1.58332	0.791661	−	0.610961i	$-0.209217\pi$
$317$	28.1902	1.58332	0.791661	+	0.610961i	$0.209217\pi$
$318$	0	0
$319$	−4.05242	−0.226892
$320$	−6.46474	−0.361390
$321$	0	0
$322$	−1.70729	−0.0951438
$323$	−3.01121	−0.167548
$324$	0	0
$325$	13.2432	0.734602
$326$	−14.5242	−0.804419
$327$	0	0
$328$	0.0440398	0.00243169
$329$	3.39619	0.187238
$330$	0	0
$331$	13.5185	0.743046	0.371523	−	0.928424i	$-0.378836\pi$
$331$	13.5185	0.743046	0.371523	+	0.928424i	$0.378836\pi$
$332$	10.9078	0.598644
$333$	0	0
$334$	−9.30162	−0.508962
$335$	−9.13500	−0.499098
$336$	0	0
$337$	−1.73199	−0.0943475	−0.0471738	−	0.998887i	$-0.515021\pi$
$337$	−1.73199	−0.0943475	−0.0471738	+	0.998887i	$0.515021\pi$
$338$	−8.91554	−0.484941
$339$	0	0
$340$	4.10304	0.222519
$341$	−2.59612	−0.140588
$342$	0	0
$343$	20.0544	1.08284
$344$	−0.280113	−0.0151027
$345$	0	0
$346$	−5.94214	−0.319451
$347$	27.7262	1.48842	0.744210	−	0.667946i	$-0.232826\pi$
$347$	27.7262	1.48842	0.744210	+	0.667946i	$0.232826\pi$
$348$	0	0
$349$	−16.0612	−0.859737	−0.429869	−	0.902891i	$-0.641440\pi$
$349$	−16.0612	−0.859737	−0.429869	+	0.902891i	$0.641440\pi$
$350$	18.6647	0.997671
$351$	0	0
$352$	−5.08225	−0.270885
$353$	16.0011	0.851652	0.425826	−	0.904805i	$-0.359984\pi$
$353$	16.0011	0.851652	0.425826	+	0.904805i	$0.359984\pi$
$354$	0	0
$355$	−1.94977	−0.103483
$356$	−35.5658	−1.88498
$357$	0	0
$358$	18.3785	0.971335
$359$	−29.2698	−1.54480	−0.772400	−	0.635136i	$-0.780944\pi$
$359$	−29.2698	−1.54480	−0.772400	+	0.635136i	$0.780944\pi$
$360$	0	0
$361$	−17.7654	−0.935020
$362$	29.3816	1.54426
$363$	0	0
$364$	12.8762	0.674896
$365$	−1.48450	−0.0777022
$366$	0	0
$367$	19.5331	1.01962	0.509810	−	0.860287i	$-0.329716\pi$
$367$	19.5331	1.01962	0.509810	+	0.860287i	$0.329716\pi$
$368$	−1.50759	−0.0785883
$369$	0	0
$370$	4.27017	0.221996
$371$	−17.2892	−0.897611
$372$	0	0
$373$	−22.0592	−1.14218	−0.571091	−	0.820887i	$-0.693480\pi$
$373$	−22.0592	−1.14218	−0.571091	+	0.820887i	$0.693480\pi$
$374$	3.46529	0.179186
$375$	0	0
$376$	−0.544059	−0.0280577
$377$	18.9861	0.977832
$378$	0	0
$379$	10.7429	0.551828	0.275914	−	0.961182i	$-0.411020\pi$
$379$	10.7429	0.551828	0.275914	+	0.961182i	$0.411020\pi$
$380$	−1.68228	−0.0862991
$381$	0	0
$382$	−8.98856	−0.459895
$383$	−7.81876	−0.399520	−0.199760	−	0.979845i	$-0.564016\pi$
$383$	−7.81876	−0.399520	−0.199760	+	0.979845i	$0.564016\pi$
$384$	0	0
$385$	−0.892305	−0.0454761
$386$	−5.91502	−0.301067
$387$	0	0
$388$	16.6034	0.842910
$389$	35.0312	1.77615	0.888075	−	0.459698i	$-0.152043\pi$
$389$	35.0312	1.77615	0.888075	+	0.459698i	$0.152043\pi$
$390$	0	0
$391$	1.11760	0.0565197
$392$	−0.936357	−0.0472932
$393$	0	0
$394$	−12.0637	−0.607759
$395$	−0.222512	−0.0111958
$396$	0	0
$397$	−0.245438	−0.0123182	−0.00615909	−	0.999981i	$-0.501961\pi$
$397$	−0.245438	−0.0123182	−0.00615909	+	0.999981i	$0.501961\pi$
$398$	−9.65635	−0.484029
$399$	0	0
$400$	16.4814	0.824072
$401$	16.3434	0.816151	0.408075	−	0.912948i	$-0.366200\pi$
$401$	16.3434	0.816151	0.408075	+	0.912948i	$0.366200\pi$
$402$	0	0
$403$	12.1631	0.605889
$404$	1.18558	0.0589848
$405$	0	0
$406$	26.7585	1.32800
$407$	1.87235	0.0928087
$408$	0	0
$409$	4.60893	0.227897	0.113948	−	0.993487i	$-0.463650\pi$
$409$	4.60893	0.227897	0.113948	+	0.993487i	$0.463650\pi$
$410$	0.193651	0.00956375
$411$	0	0
$412$	2.86683	0.141239
$413$	−0.825536	−0.0406220
$414$	0	0
$415$	3.54147	0.173844
$416$	23.8109	1.16743
$417$	0	0
$418$	−1.42080	−0.0694934
$419$	−11.4509	−0.559414	−0.279707	−	0.960085i	$-0.590237\pi$
$419$	−11.4509	−0.559414	−0.279707	+	0.960085i	$0.590237\pi$
$420$	0	0
$421$	−22.2626	−1.08501	−0.542505	−	0.840052i	$-0.682524\pi$
$421$	−22.2626	−1.08501	−0.542505	+	0.840052i	$0.682524\pi$
$422$	−3.07289	−0.149586
$423$	0	0
$424$	2.76968	0.134507
$425$	−12.2180	−0.592661
$426$	0	0
$427$	17.3178	0.838069
$428$	−5.56117	−0.268809
$429$	0	0
$430$	−1.23171	−0.0593982
$431$	1.13299	0.0545741	0.0272871	−	0.999628i	$-0.491313\pi$
$431$	1.13299	0.0545741	0.0272871	+	0.999628i	$0.491313\pi$
$432$	0	0
$433$	−2.65692	−0.127683	−0.0638417	−	0.997960i	$-0.520335\pi$
$433$	−2.65692	−0.127683	−0.0638417	+	0.997960i	$0.520335\pi$
$434$	17.1425	0.822864
$435$	0	0
$436$	19.1977	0.919403
$437$	−0.458226	−0.0219199
$438$	0	0
$439$	6.57228	0.313678	0.156839	−	0.987624i	$-0.449870\pi$
$439$	6.57228	0.313678	0.156839	+	0.987624i	$0.449870\pi$
$440$	0.142945	0.00681461
$441$	0	0
$442$	−16.2353	−0.772235
$443$	−1.34406	−0.0638583	−0.0319291	−	0.999490i	$-0.510165\pi$
$443$	−1.34406	−0.0638583	−0.0319291	+	0.999490i	$0.510165\pi$
$444$	0	0
$445$	−11.5473	−0.547393
$446$	−19.9709	−0.945650
$447$	0	0
$448$	18.7172	0.884303
$449$	−26.3410	−1.24311	−0.621555	−	0.783371i	$-0.713499\pi$
$449$	−26.3410	−1.24311	−0.621555	+	0.783371i	$0.713499\pi$
$450$	0	0
$451$	0.0849103	0.00399827
$452$	4.14586	0.195005
$453$	0	0
$454$	−14.0420	−0.659024
$455$	4.18056	0.195988
$456$	0	0
$457$	−30.0693	−1.40658	−0.703290	−	0.710903i	$-0.748287\pi$
$457$	−30.0693	−1.40658	−0.703290	+	0.710903i	$0.748287\pi$
$458$	24.8571	1.16149
$459$	0	0
$460$	0.624373	0.0291115
$461$	7.72477	0.359778	0.179889	−	0.983687i	$-0.442426\pi$
$461$	7.72477	0.359778	0.179889	+	0.983687i	$0.442426\pi$
$462$	0	0
$463$	7.82961	0.363873	0.181936	−	0.983310i	$-0.441764\pi$
$463$	7.82961	0.363873	0.181936	+	0.983310i	$0.441764\pi$
$464$	23.6285	1.09692
$465$	0	0
$466$	42.9404	1.98918
$467$	7.19623	0.333002	0.166501	−	0.986041i	$-0.446753\pi$
$467$	7.19623	0.333002	0.166501	+	0.986041i	$0.446753\pi$
$468$	0	0
$469$	26.4483	1.22127
$470$	−2.39233	−0.110350
$471$	0	0
$472$	0.132248	0.00608722
$473$	−0.540068	−0.0248323
$474$	0	0
$475$	5.00948	0.229851
$476$	−11.8794	−0.544492
$477$	0	0
$478$	−18.5988	−0.850691
$479$	12.1955	0.557229	0.278614	−	0.960403i	$-0.410125\pi$
$479$	12.1955	0.557229	0.278614	+	0.960403i	$0.410125\pi$
$480$	0	0
$481$	−8.77217	−0.399976
$482$	46.3678	2.11200
$483$	0	0
$484$	−22.9050	−1.04114
$485$	5.39068	0.244778
$486$	0	0
$487$	35.9684	1.62988	0.814942	−	0.579543i	$-0.196769\pi$
$487$	35.9684	1.62988	0.814942	+	0.579543i	$0.196769\pi$
$488$	−2.77426	−0.125585
$489$	0	0
$490$	−4.11734	−0.186002
$491$	17.3618	0.783527	0.391763	−	0.920066i	$-0.371865\pi$
$491$	17.3618	0.783527	0.391763	+	0.920066i	$0.371865\pi$
$492$	0	0
$493$	−17.5163	−0.788893
$494$	6.65660	0.299495
$495$	0	0
$496$	15.1372	0.679682
$497$	5.64510	0.253217
$498$	0	0
$499$	−30.0817	−1.34664	−0.673320	−	0.739351i	$-0.735133\pi$
$499$	−30.0817	−1.34664	−0.673320	+	0.739351i	$0.735133\pi$
$500$	−14.3960	−0.643806
$501$	0	0
$502$	23.8259	1.06340
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	0.384926	0.0171290
$506$	0.527324	0.0234424
$507$	0	0
$508$	47.0780	2.08875
$509$	32.5495	1.44273	0.721366	−	0.692554i	$-0.243515\pi$
$509$	32.5495	1.44273	0.721366	+	0.692554i	$0.243515\pi$
$510$	0	0
$511$	4.29802	0.190133
$512$	32.0107	1.41469
$513$	0	0
$514$	33.2455	1.46639
$515$	0.930784	0.0410152
$516$	0	0
$517$	−1.04897	−0.0461335
$518$	−12.3633	−0.543212
$519$	0	0
$520$	−0.669712	−0.0293688
$521$	12.6474	0.554094	0.277047	−	0.960856i	$-0.410644\pi$
$521$	12.6474	0.554094	0.277047	+	0.960856i	$0.410644\pi$
$522$	0	0
$523$	14.7294	0.644073	0.322036	−	0.946727i	$-0.395633\pi$
$523$	14.7294	0.644073	0.322036	+	0.946727i	$0.395633\pi$
$524$	−18.9631	−0.828406
$525$	0	0
$526$	32.2145	1.40462
$527$	−11.2215	−0.488818
$528$	0	0
$529$	−22.8299	−0.992606
$530$	12.1788	0.529013
$531$	0	0
$532$	4.87065	0.211169
$533$	−0.397815	−0.0172313
$534$	0	0
$535$	−1.80556	−0.0780613
$536$	−4.23693	−0.183008
$537$	0	0
$538$	−22.9239	−0.988321
$539$	−1.80533	−0.0777611
$540$	0	0
$541$	−27.4172	−1.17876	−0.589379	−	0.807857i	$-0.700627\pi$
$541$	−27.4172	−1.17876	−0.589379	+	0.807857i	$0.700627\pi$
$542$	46.3192	1.98958
$543$	0	0
$544$	−21.9676	−0.941855
$545$	6.23298	0.266992
$546$	0	0
$547$	−16.1884	−0.692164	−0.346082	−	0.938204i	$-0.612488\pi$
$547$	−16.1884	−0.692164	−0.346082	+	0.938204i	$0.612488\pi$
$548$	9.20198	0.393089
$549$	0	0
$550$	−5.76489	−0.245816
$551$	7.18181	0.305955
$552$	0	0
$553$	0.644231	0.0273955
$554$	−6.98742	−0.296867
$555$	0	0
$556$	29.1420	1.23590
$557$	13.3858	0.567176	0.283588	−	0.958946i	$-0.408475\pi$
$557$	13.3858	0.567176	0.283588	+	0.958946i	$0.408475\pi$
$558$	0	0
$559$	2.53028	0.107020
$560$	5.20277	0.219857
$561$	0	0
$562$	13.0764	0.551596
$563$	−23.3845	−0.985538	−0.492769	−	0.870160i	$-0.664015\pi$
$563$	−23.3845	−0.985538	−0.492769	+	0.870160i	$0.664015\pi$
$564$	0	0
$565$	1.34605	0.0566287
$566$	−45.5745	−1.91564
$567$	0	0
$568$	−0.904327	−0.0379447
$569$	−43.0171	−1.80337	−0.901685	−	0.432393i	$-0.857669\pi$
$569$	−43.0171	−1.80337	−0.901685	+	0.432393i	$0.857669\pi$
$570$	0	0
$571$	−1.24920	−0.0522772	−0.0261386	−	0.999658i	$-0.508321\pi$
$571$	−1.24920	−0.0522772	−0.0261386	+	0.999658i	$0.508321\pi$
$572$	−3.97702	−0.166287
$573$	0	0
$574$	−0.560672	−0.0234020
$575$	−1.85926	−0.0775364
$576$	0	0
$577$	−38.6392	−1.60857	−0.804284	−	0.594244i	$-0.797451\pi$
$577$	−38.6392	−1.60857	−0.804284	+	0.594244i	$0.797451\pi$
$578$	−19.6926	−0.819102
$579$	0	0
$580$	−9.78584	−0.406335
$581$	−10.2535	−0.425387
$582$	0	0
$583$	5.34004	0.221162
$584$	−0.688529	−0.0284915
$585$	0	0
$586$	−10.1029	−0.417345
$587$	13.5179	0.557943	0.278972	−	0.960299i	$-0.410006\pi$
$587$	13.5179	0.557943	0.278972	+	0.960299i	$0.410006\pi$
$588$	0	0
$589$	4.60092	0.189578
$590$	0.581520	0.0239408
$591$	0	0
$592$	−10.9171	−0.448690
$593$	6.72274	0.276070	0.138035	−	0.990427i	$-0.455921\pi$
$593$	6.72274	0.276070	0.138035	+	0.990427i	$0.455921\pi$
$594$	0	0
$595$	−3.85692	−0.158118
$596$	−24.2837	−0.994698
$597$	0	0
$598$	−2.47058	−0.101029
$599$	−4.15139	−0.169621	−0.0848106	−	0.996397i	$-0.527029\pi$
$599$	−4.15139	−0.169621	−0.0848106	+	0.996397i	$0.527029\pi$
$600$	0	0
$601$	−39.8754	−1.62655	−0.813276	−	0.581878i	$-0.802318\pi$
$601$	−39.8754	−1.62655	−0.813276	+	0.581878i	$0.802318\pi$
$602$	3.56612	0.145344
$603$	0	0
$604$	−5.09860	−0.207459
$605$	−7.43665	−0.302343
$606$	0	0
$607$	−27.9713	−1.13532	−0.567660	−	0.823263i	$-0.692151\pi$
$607$	−27.9713	−1.13532	−0.567660	+	0.823263i	$0.692151\pi$
$608$	9.00690	0.365278
$609$	0	0
$610$	−12.1990	−0.493921
$611$	4.91454	0.198821
$612$	0	0
$613$	−30.7521	−1.24207	−0.621033	−	0.783785i	$-0.713287\pi$
$613$	−30.7521	−1.24207	−0.621033	+	0.783785i	$0.713287\pi$
$614$	−3.81252	−0.153861
$615$	0	0
$616$	−0.413863	−0.0166750
$617$	24.4669	0.984999	0.492500	−	0.870313i	$-0.336083\pi$
$617$	24.4669	0.984999	0.492500	+	0.870313i	$0.336083\pi$
$618$	0	0
$619$	37.1492	1.49315	0.746575	−	0.665301i	$-0.231697\pi$
$619$	37.1492	1.49315	0.746575	+	0.665301i	$0.231697\pi$
$620$	−6.26915	−0.251775
$621$	0	0
$622$	8.76284	0.351358
$623$	33.4324	1.33944
$624$	0	0
$625$	17.8682	0.714729
$626$	−49.4322	−1.97571
$627$	0	0
$628$	−37.8426	−1.51009
$629$	8.09308	0.322692
$630$	0	0
$631$	−15.7642	−0.627563	−0.313781	−	0.949495i	$-0.601596\pi$
$631$	−15.7642	−0.627563	−0.313781	+	0.949495i	$0.601596\pi$
$632$	−0.103204	−0.00410522
$633$	0	0
$634$	57.4932	2.28335
$635$	15.2850	0.606565
$636$	0	0
$637$	8.45819	0.335126
$638$	−8.26479	−0.327206
$639$	0	0
$640$	−1.81814	−0.0718684
$641$	−48.3528	−1.90982	−0.954911	−	0.296892i	$-0.904050\pi$
$641$	−48.3528	−1.90982	−0.954911	+	0.296892i	$0.904050\pi$
$642$	0	0
$643$	3.30378	0.130288	0.0651441	−	0.997876i	$-0.479249\pi$
$643$	3.30378	0.130288	0.0651441	+	0.997876i	$0.479249\pi$
$644$	−1.80773	−0.0712344
$645$	0	0
$646$	−6.14129	−0.241626
$647$	−34.3017	−1.34854	−0.674269	−	0.738486i	$-0.735541\pi$
$647$	−34.3017	−1.34854	−0.674269	+	0.738486i	$0.735541\pi$
$648$	0	0
$649$	0.254980	0.0100088
$650$	27.0092	1.05939
$651$	0	0
$652$	−15.3786	−0.602271
$653$	13.5211	0.529120	0.264560	−	0.964369i	$-0.414773\pi$
$653$	13.5211	0.529120	0.264560	+	0.964369i	$0.414773\pi$
$654$	0	0
$655$	−6.15680	−0.240566
$656$	−0.495088	−0.0193299
$657$	0	0
$658$	6.92644	0.270021
$659$	−29.2605	−1.13983	−0.569914	−	0.821704i	$-0.693023\pi$
$659$	−29.2605	−1.13983	−0.569914	+	0.821704i	$0.693023\pi$
$660$	0	0
$661$	36.0006	1.40026	0.700131	−	0.714015i	$-0.253125\pi$
$661$	36.0006	1.40026	0.700131	+	0.714015i	$0.253125\pi$
$662$	27.5707	1.07157
$663$	0	0
$664$	1.64258	0.0637445
$665$	1.58137	0.0613229
$666$	0	0
$667$	−2.66551	−0.103209
$668$	−9.84880	−0.381061
$669$	0	0
$670$	−18.6306	−0.719762
$671$	−5.34889	−0.206491
$672$	0	0
$673$	14.3874	0.554595	0.277298	−	0.960784i	$-0.410561\pi$
$673$	14.3874	0.554595	0.277298	+	0.960784i	$0.410561\pi$
$674$	−3.53235	−0.136061
$675$	0	0
$676$	−9.44000	−0.363077
$677$	−25.3712	−0.975094	−0.487547	−	0.873097i	$-0.662108\pi$
$677$	−25.3712	−0.975094	−0.487547	+	0.873097i	$0.662108\pi$
$678$	0	0
$679$	−15.6075	−0.598960
$680$	0.617867	0.0236941
$681$	0	0
$682$	−5.29471	−0.202745
$683$	−31.2549	−1.19594	−0.597968	−	0.801520i	$-0.704025\pi$
$683$	−31.2549	−1.19594	−0.597968	+	0.801520i	$0.704025\pi$
$684$	0	0
$685$	2.98764	0.114152
$686$	40.9005	1.56159
$687$	0	0
$688$	3.14898	0.120054
$689$	−25.0187	−0.953138
$690$	0	0
$691$	−1.92143	−0.0730946	−0.0365473	−	0.999332i	$-0.511636\pi$
$691$	−1.92143	−0.0730946	−0.0365473	+	0.999332i	$0.511636\pi$
$692$	−6.29169	−0.239174
$693$	0	0
$694$	56.5468	2.14649
$695$	9.46162	0.358900
$696$	0	0
$697$	0.367019	0.0139018
$698$	−32.7564	−1.23985
$699$	0	0
$700$	19.7627	0.746960
$701$	−5.22129	−0.197205	−0.0986027	−	0.995127i	$-0.531437\pi$
$701$	−5.22129	−0.197205	−0.0986027	+	0.995127i	$0.531437\pi$
$702$	0	0
$703$	−3.31823	−0.125149
$704$	−5.78109	−0.217883
$705$	0	0
$706$	32.6338	1.22819
$707$	−1.11446	−0.0419137
$708$	0	0
$709$	1.68414	0.0632493	0.0316247	−	0.999500i	$-0.489932\pi$
$709$	1.68414	0.0632493	0.0316247	+	0.999500i	$0.489932\pi$
$710$	−3.97649	−0.149235
$711$	0	0
$712$	−5.35577	−0.200716
$713$	−1.70762	−0.0639508
$714$	0	0
$715$	−1.29123	−0.0482893
$716$	19.4596	0.727241
$717$	0	0
$718$	−59.6949	−2.22779
$719$	38.7868	1.44651	0.723253	−	0.690583i	$-0.242646\pi$
$719$	38.7868	1.44651	0.723253	+	0.690583i	$0.242646\pi$
$720$	0	0
$721$	−2.69487	−0.100362
$722$	−36.2320	−1.34842
$723$	0	0
$724$	31.1100	1.15619
$725$	29.1402	1.08224
$726$	0	0
$727$	−18.0472	−0.669333	−0.334666	−	0.942337i	$-0.608624\pi$
$727$	−18.0472	−0.669333	−0.334666	+	0.942337i	$0.608624\pi$
$728$	1.93900	0.0718640
$729$	0	0
$730$	−3.02759	−0.112056
$731$	−2.33440	−0.0863410
$732$	0	0
$733$	2.36945	0.0875175	0.0437587	−	0.999042i	$-0.486067\pi$
$733$	2.36945	0.0875175	0.0437587	+	0.999042i	$0.486067\pi$
$734$	39.8373	1.47042
$735$	0	0
$736$	−3.34289	−0.123220
$737$	−8.16896	−0.300907
$738$	0	0
$739$	23.0868	0.849260	0.424630	−	0.905367i	$-0.360404\pi$
$739$	23.0868	0.849260	0.424630	+	0.905367i	$0.360404\pi$
$740$	4.52137	0.166209
$741$	0	0
$742$	−35.2608	−1.29447
$743$	−2.24456	−0.0823449	−0.0411725	−	0.999152i	$-0.513109\pi$
$743$	−2.24456	−0.0823449	−0.0411725	+	0.999152i	$0.513109\pi$
$744$	0	0
$745$	−7.88426	−0.288857
$746$	−44.9891	−1.64717
$747$	0	0
$748$	3.66914	0.134157
$749$	5.22759	0.191012
$750$	0	0
$751$	54.7798	1.99894	0.999472	−	0.0324826i	$-0.0103413\pi$
$751$	54.7798	1.99894	0.999472	+	0.0324826i	$0.0103413\pi$
$752$	6.11622	0.223036
$753$	0	0
$754$	38.7215	1.41015
$755$	−1.65538	−0.0602454
$756$	0	0
$757$	−5.96003	−0.216621	−0.108311	−	0.994117i	$-0.534544\pi$
$757$	−5.96003	−0.216621	−0.108311	+	0.994117i	$0.534544\pi$
$758$	21.9099	0.795804
$759$	0	0
$760$	−0.253330	−0.00918926
$761$	33.5210	1.21513	0.607567	−	0.794268i	$-0.292146\pi$
$761$	33.5210	1.21513	0.607567	+	0.794268i	$0.292146\pi$
$762$	0	0
$763$	−18.0461	−0.653314
$764$	−9.51732	−0.344325
$765$	0	0
$766$	−15.9461	−0.576158
$767$	−1.19461	−0.0431349
$768$	0	0
$769$	−41.9889	−1.51416	−0.757080	−	0.653322i	$-0.773375\pi$
$769$	−41.9889	−1.51416	−0.757080	+	0.653322i	$0.773375\pi$
$770$	−1.81983	−0.0655822
$771$	0	0
$772$	−6.26298	−0.225409
$773$	−11.9695	−0.430512	−0.215256	−	0.976558i	$-0.569059\pi$
$773$	−11.9695	−0.430512	−0.215256	+	0.976558i	$0.569059\pi$
$774$	0	0
$775$	18.6683	0.670584
$776$	2.50027	0.0897544
$777$	0	0
$778$	71.4451	2.56143
$779$	−0.150481	−0.00539153
$780$	0	0
$781$	−1.74358	−0.0623901
$782$	2.27932	0.0815084
$783$	0	0
$784$	10.5264	0.375942
$785$	−12.2865	−0.438523
$786$	0	0
$787$	−14.5883	−0.520017	−0.260008	−	0.965606i	$-0.583725\pi$
$787$	−14.5883	−0.520017	−0.260008	+	0.965606i	$0.583725\pi$
$788$	−12.7733	−0.455031
$789$	0	0
$790$	−0.453806	−0.0161457
$791$	−3.89717	−0.138568
$792$	0	0
$793$	25.0602	0.889913
$794$	−0.500564	−0.0177643
$795$	0	0
$796$	−10.2244	−0.362394
$797$	−13.0225	−0.461281	−0.230641	−	0.973039i	$-0.574082\pi$
$797$	−13.0225	−0.461281	−0.230641	+	0.973039i	$0.574082\pi$
$798$	0	0
$799$	−4.53408	−0.160404
$800$	36.5456	1.29208
$801$	0	0
$802$	33.3319	1.17699
$803$	−1.32751	−0.0468468
$804$	0	0
$805$	−0.586921	−0.0206862
$806$	24.8064	0.873767
$807$	0	0
$808$	0.178534	0.00628079
$809$	−23.5857	−0.829230	−0.414615	−	0.909997i	$-0.636084\pi$
$809$	−23.5857	−0.829230	−0.414615	+	0.909997i	$0.636084\pi$
$810$	0	0
$811$	44.3230	1.55639	0.778195	−	0.628023i	$-0.216136\pi$
$811$	44.3230	1.55639	0.778195	+	0.628023i	$0.216136\pi$
$812$	28.3326	0.994280
$813$	0	0
$814$	3.81860	0.133842
$815$	−4.99300	−0.174897
$816$	0	0
$817$	0.957124	0.0334855
$818$	9.39977	0.328655
$819$	0	0
$820$	0.205043	0.00716041
$821$	11.0481	0.385581	0.192791	−	0.981240i	$-0.438246\pi$
$821$	11.0481	0.385581	0.192791	+	0.981240i	$0.438246\pi$
$822$	0	0
$823$	8.97374	0.312805	0.156403	−	0.987693i	$-0.450010\pi$
$823$	8.97374	0.312805	0.156403	+	0.987693i	$0.450010\pi$
$824$	0.431710	0.0150393
$825$	0	0
$826$	−1.68366	−0.0585819
$827$	−36.0996	−1.25531	−0.627654	−	0.778493i	$-0.715985\pi$
$827$	−36.0996	−1.25531	−0.627654	+	0.778493i	$0.715985\pi$
$828$	0	0
$829$	−5.63971	−0.195875	−0.0979375	−	0.995193i	$-0.531225\pi$
$829$	−5.63971	−0.195875	−0.0979375	+	0.995193i	$0.531225\pi$
$830$	7.22273	0.250705
$831$	0	0
$832$	27.0851	0.939007
$833$	−7.80341	−0.270372
$834$	0	0
$835$	−3.19764	−0.110659
$836$	−1.50438	−0.0520299
$837$	0	0
$838$	−23.3538	−0.806745
$839$	19.2580	0.664859	0.332429	−	0.943128i	$-0.392132\pi$
$839$	19.2580	0.664859	0.332429	+	0.943128i	$0.392132\pi$
$840$	0	0
$841$	12.7767	0.440575
$842$	−45.4039	−1.56472
$843$	0	0
$844$	−3.25366	−0.111996
$845$	−3.06492	−0.105436
$846$	0	0
$847$	21.5311	0.739818
$848$	−31.1362	−1.06922
$849$	0	0
$850$	−24.9183	−0.854691
$851$	1.23155	0.0422170
$852$	0	0
$853$	29.1126	0.996796	0.498398	−	0.866948i	$-0.333922\pi$
$853$	29.1126	0.996796	0.498398	+	0.866948i	$0.333922\pi$
$854$	35.3193	1.20860
$855$	0	0
$856$	−0.837443	−0.0286232
$857$	18.6286	0.636342	0.318171	−	0.948033i	$-0.396931\pi$
$857$	18.6286	0.636342	0.318171	+	0.948033i	$0.396931\pi$
$858$	0	0
$859$	18.5140	0.631690	0.315845	−	0.948811i	$-0.397712\pi$
$859$	18.5140	0.631690	0.315845	+	0.948811i	$0.397712\pi$
$860$	−1.30416	−0.0444716
$861$	0	0
$862$	2.31070	0.0787027
$863$	43.5150	1.48127	0.740635	−	0.671908i	$-0.234525\pi$
$863$	43.5150	1.48127	0.740635	+	0.671908i	$0.234525\pi$
$864$	0	0
$865$	−2.04274	−0.0694553
$866$	−5.41872	−0.184135
$867$	0	0
$868$	18.1509	0.616081
$869$	−0.198981	−0.00674996
$870$	0	0
$871$	38.2726	1.29682
$872$	2.89094	0.0978995
$873$	0	0
$874$	−0.934540	−0.0316113
$875$	13.5324	0.457479
$876$	0	0
$877$	11.9837	0.404662	0.202331	−	0.979317i	$-0.435148\pi$
$877$	11.9837	0.404662	0.202331	+	0.979317i	$0.435148\pi$
$878$	13.4040	0.452363
$879$	0	0
$880$	−1.60696	−0.0541705
$881$	8.33770	0.280904	0.140452	−	0.990087i	$-0.455144\pi$
$881$	8.33770	0.280904	0.140452	+	0.990087i	$0.455144\pi$
$882$	0	0
$883$	14.7773	0.497296	0.248648	−	0.968594i	$-0.420014\pi$
$883$	14.7773	0.497296	0.248648	+	0.968594i	$0.420014\pi$
$884$	−17.1904	−0.578174
$885$	0	0
$886$	−2.74118	−0.0920916
$887$	26.6636	0.895278	0.447639	−	0.894214i	$-0.352265\pi$
$887$	26.6636	0.895278	0.447639	+	0.894214i	$0.352265\pi$
$888$	0	0
$889$	−44.2541	−1.48423
$890$	−23.5503	−0.789408
$891$	0	0
$892$	−21.1457	−0.708011
$893$	1.85901	0.0622094
$894$	0	0
$895$	6.31803	0.211188
$896$	5.26401	0.175858
$897$	0	0
$898$	−53.7218	−1.79272
$899$	26.7636	0.892616
$900$	0	0
$901$	23.0819	0.768971
$902$	0.173172	0.00576601
$903$	0	0
$904$	0.624315	0.0207644
$905$	10.1006	0.335754
$906$	0	0
$907$	−41.3358	−1.37253	−0.686266	−	0.727351i	$-0.740751\pi$
$907$	−41.3358	−1.37253	−0.686266	+	0.727351i	$0.740751\pi$
$908$	−14.8680	−0.493413
$909$	0	0
$910$	8.52613	0.282638
$911$	32.2613	1.06887	0.534433	−	0.845211i	$-0.320525\pi$
$911$	32.2613	1.06887	0.534433	+	0.845211i	$0.320525\pi$
$912$	0	0
$913$	3.16696	0.104811
$914$	−61.3254	−2.02846
$915$	0	0
$916$	26.3193	0.869615
$917$	17.8256	0.588653
$918$	0	0
$919$	−48.5538	−1.60164	−0.800821	−	0.598903i	$-0.795603\pi$
$919$	−48.5538	−1.60164	−0.800821	+	0.598903i	$0.795603\pi$
$920$	0.0940229	0.00309984
$921$	0	0
$922$	15.7545	0.518845
$923$	8.16886	0.268881
$924$	0	0
$925$	−13.4637	−0.442684
$926$	15.9683	0.524750
$927$	0	0
$928$	52.3933	1.71989
$929$	−28.1849	−0.924717	−0.462358	−	0.886693i	$-0.652997\pi$
$929$	−28.1849	−0.924717	−0.462358	+	0.886693i	$0.652997\pi$
$930$	0	0
$931$	3.19946	0.104858
$932$	45.4664	1.48930
$933$	0	0
$934$	14.6765	0.480230
$935$	1.19127	0.0389587
$936$	0	0
$937$	−59.4487	−1.94211	−0.971053	−	0.238865i	$-0.923225\pi$
$937$	−59.4487	−1.94211	−0.971053	+	0.238865i	$0.923225\pi$
$938$	53.9405	1.76122
$939$	0	0
$940$	−2.53306	−0.0826194
$941$	−24.0890	−0.785277	−0.392639	−	0.919693i	$-0.628438\pi$
$941$	−24.0890	−0.785277	−0.392639	+	0.919693i	$0.628438\pi$
$942$	0	0
$943$	0.0558504	0.00181874
$944$	−1.48671	−0.0483884
$945$	0	0
$946$	−1.10145	−0.0358113
$947$	−40.0415	−1.30117	−0.650586	−	0.759432i	$-0.725477\pi$
$947$	−40.0415	−1.30117	−0.650586	+	0.759432i	$0.725477\pi$
$948$	0	0
$949$	6.21955	0.201895
$950$	10.2167	0.331474
$951$	0	0
$952$	−1.78889	−0.0579783
$953$	24.4039	0.790521	0.395261	−	0.918569i	$-0.370654\pi$
$953$	24.4039	0.790521	0.395261	+	0.918569i	$0.370654\pi$
$954$	0	0
$955$	−3.09002	−0.0999907
$956$	−19.6929	−0.636915
$957$	0	0
$958$	24.8725	0.803593
$959$	−8.65001	−0.279323
$960$	0	0
$961$	−13.8543	−0.446913
$962$	−17.8906	−0.576816
$963$	0	0
$964$	49.0955	1.58126
$965$	−2.03342	−0.0654581
$966$	0	0
$967$	−16.2701	−0.523212	−0.261606	−	0.965175i	$-0.584252\pi$
$967$	−16.2701	−0.523212	−0.261606	+	0.965175i	$0.584252\pi$
$968$	−3.44922	−0.110862
$969$	0	0
$970$	10.9941	0.353001
$971$	53.0256	1.70167	0.850837	−	0.525430i	$-0.176096\pi$
$971$	53.0256	1.70167	0.850837	+	0.525430i	$0.176096\pi$
$972$	0	0
$973$	−27.3939	−0.878209
$974$	73.3565	2.35050
$975$	0	0
$976$	31.1878	0.998298
$977$	47.0435	1.50505	0.752527	−	0.658561i	$-0.228835\pi$
$977$	47.0435	1.50505	0.752527	+	0.658561i	$0.228835\pi$
$978$	0	0
$979$	−10.3261	−0.330024
$980$	−4.35954	−0.139260
$981$	0	0
$982$	35.4089	1.12994
$983$	10.9007	0.347678	0.173839	−	0.984774i	$-0.444383\pi$
$983$	10.9007	0.347678	0.173839	+	0.984774i	$0.444383\pi$
$984$	0	0
$985$	−4.14716	−0.132139
$986$	−35.7239	−1.13768
$987$	0	0
$988$	7.04818	0.224233
$989$	−0.355234	−0.0112958
$990$	0	0
$991$	−48.5839	−1.54332	−0.771659	−	0.636036i	$-0.780573\pi$
$991$	−48.5839	−1.54332	−0.771659	+	0.636036i	$0.780573\pi$
$992$	33.5650	1.06569
$993$	0	0
$994$	11.5130	0.365171
$995$	−3.31959	−0.105238
$996$	0	0
$997$	20.3630	0.644902	0.322451	−	0.946586i	$-0.395493\pi$
$997$	20.3630	0.644902	0.322451	+	0.946586i	$0.395493\pi$
$998$	−61.3507	−1.94202
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.9		10
3.2	odd	2		503.2.a.e.1.2	✓	10
12.11	even	2		8048.2.a.p.1.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.2	✓	10	3.2	odd	2
4527.2.a.k.1.9		10	1.1	even	1	trivial
8048.2.a.p.1.1		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)

\(f(q)\)	\(=\)	\(q+2.03947 q^{2} +2.15945 q^{4} +0.701114 q^{5} -2.02991 q^{7} +0.325186 q^{8} +O(q^{10})\)	\(q+2.03947 q^{2} +2.15945 q^{4} +0.701114 q^{5} -2.02991 q^{7} +0.325186 q^{8} +1.42990 q^{10} +0.626970 q^{11} -2.93743 q^{13} -4.13995 q^{14} -3.65568 q^{16} +2.71003 q^{17} -1.11114 q^{19} +1.51402 q^{20} +1.27869 q^{22} +0.412395 q^{23} -4.50844 q^{25} -5.99081 q^{26} -4.38349 q^{28} -6.46349 q^{29} -4.14074 q^{31} -8.10604 q^{32} +5.52704 q^{34} -1.42320 q^{35} +2.98634 q^{37} -2.26613 q^{38} +0.227992 q^{40} +0.135430 q^{41} -0.861393 q^{43} +1.35391 q^{44} +0.841067 q^{46} -1.67307 q^{47} -2.87945 q^{49} -9.19483 q^{50} -6.34323 q^{52} +8.51721 q^{53} +0.439578 q^{55} -0.660099 q^{56} -13.1821 q^{58} +0.406685 q^{59} -8.53132 q^{61} -8.44492 q^{62} -9.22067 q^{64} -2.05948 q^{65} -13.0293 q^{67} +5.85217 q^{68} -2.90258 q^{70} -2.78095 q^{71} -2.11734 q^{73} +6.09055 q^{74} -2.39944 q^{76} -1.27270 q^{77} -0.317368 q^{79} -2.56305 q^{80} +0.276205 q^{82} +5.05120 q^{83} +1.90004 q^{85} -1.75679 q^{86} +0.203882 q^{88} -16.4699 q^{89} +5.96273 q^{91} +0.890544 q^{92} -3.41218 q^{94} -0.779033 q^{95} +7.68873 q^{97} -5.87256 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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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