LMFDB - Embedded newform 4527.2.a.k.1.4

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			1.4
Root			$1.31567$ 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)

$f(q)$	$=$	$q-0.0830530 q^{2} -1.99310 q^{4} -2.25024 q^{5} -3.20647 q^{7} +0.331639 q^{8} +O(q^{10})$	\(q-0.0830530 q^{2} -1.99310 q^{4} -2.25024 q^{5} -3.20647 q^{7} +0.331639 q^{8} +0.186890 q^{10} +0.218022 q^{11} -4.17856 q^{13} +0.266307 q^{14} +3.95866 q^{16} +4.68939 q^{17} +3.43926 q^{19} +4.48497 q^{20} -0.0181074 q^{22} +3.99816 q^{23} +0.0635992 q^{25} +0.347042 q^{26} +6.39082 q^{28} +0.712153 q^{29} +1.04933 q^{31} -0.992057 q^{32} -0.389468 q^{34} +7.21534 q^{35} +1.70998 q^{37} -0.285641 q^{38} -0.746269 q^{40} -3.18460 q^{41} -6.35890 q^{43} -0.434541 q^{44} -0.332059 q^{46} +3.87861 q^{47} +3.28144 q^{49} -0.00528210 q^{50} +8.32830 q^{52} +10.8877 q^{53} -0.490604 q^{55} -1.06339 q^{56} -0.0591465 q^{58} +2.63793 q^{59} +11.1544 q^{61} -0.0871503 q^{62} -7.83493 q^{64} +9.40279 q^{65} +6.09170 q^{67} -9.34643 q^{68} -0.599256 q^{70} -9.40515 q^{71} -2.78532 q^{73} -0.142019 q^{74} -6.85479 q^{76} -0.699082 q^{77} +5.16993 q^{79} -8.90795 q^{80} +0.264491 q^{82} -1.83793 q^{83} -10.5523 q^{85} +0.528126 q^{86} +0.0723048 q^{88} +8.08786 q^{89} +13.3984 q^{91} -7.96873 q^{92} -0.322130 q^{94} -7.73917 q^{95} -12.9058 q^{97} -0.272533 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.0830530	−0.0587274	−0.0293637	−	0.999569i	$-0.509348\pi$
$2$	−0.0830530	−0.0587274	−0.0293637	+	0.999569i	$0.509348\pi$
$3$	0	0
$3$	0	0
$4$	−1.99310	−0.996551
$5$	−2.25024	−1.00634	−0.503170	−	0.864188i	$-0.667833\pi$
$5$	−2.25024	−1.00634	−0.503170	+	0.864188i	$0.667833\pi$
$6$	0	0
$7$	−3.20647	−1.21193	−0.605966	−	0.795491i	$-0.707213\pi$
$7$	−3.20647	−1.21193	−0.605966	+	0.795491i	$0.707213\pi$
$8$	0.331639	0.117252
$9$	0	0
$10$	0.186890	0.0590997
$11$	0.218022	0.0657362	0.0328681	−	0.999460i	$-0.489536\pi$
$11$	0.218022	0.0657362	0.0328681	+	0.999460i	$0.489536\pi$
$12$	0	0
$13$	−4.17856	−1.15892	−0.579462	−	0.814999i	$-0.696737\pi$
$13$	−4.17856	−1.15892	−0.579462	+	0.814999i	$0.696737\pi$
$14$	0.266307	0.0711735
$15$	0	0
$16$	3.95866	0.989665
$17$	4.68939	1.13734	0.568672	−	0.822564i	$-0.307457\pi$
$17$	4.68939	1.13734	0.568672	+	0.822564i	$0.307457\pi$
$18$	0	0
$19$	3.43926	0.789020	0.394510	−	0.918892i	$-0.370914\pi$
$19$	3.43926	0.789020	0.394510	+	0.918892i	$0.370914\pi$
$20$	4.48497	1.00287
$21$	0	0
$22$	−0.0181074	−0.00386052
$23$	3.99816	0.833673	0.416837	−	0.908981i	$-0.363139\pi$
$23$	3.99816	0.833673	0.416837	+	0.908981i	$0.363139\pi$
$24$	0	0
$25$	0.0635992	0.0127198
$26$	0.347042	0.0680606
$27$	0	0
$28$	6.39082	1.20775
$29$	0.712153	0.132243	0.0661217	−	0.997812i	$-0.478937\pi$
$29$	0.712153	0.132243	0.0661217	+	0.997812i	$0.478937\pi$
$30$	0	0
$31$	1.04933	0.188466	0.0942329	−	0.995550i	$-0.469960\pi$
$31$	1.04933	0.188466	0.0942329	+	0.995550i	$0.469960\pi$
$32$	−0.992057	−0.175373
$33$	0	0
$34$	−0.389468	−0.0667932
$35$	7.21534	1.21961
$36$	0	0
$37$	1.70998	0.281119	0.140560	−	0.990072i	$-0.455110\pi$
$37$	1.70998	0.281119	0.140560	+	0.990072i	$0.455110\pi$
$38$	−0.285641	−0.0463371
$39$	0	0
$40$	−0.746269	−0.117996
$41$	−3.18460	−0.497351	−0.248675	−	0.968587i	$-0.579995\pi$
$41$	−3.18460	−0.497351	−0.248675	+	0.968587i	$0.579995\pi$
$42$	0	0
$43$	−6.35890	−0.969723	−0.484861	−	0.874591i	$-0.661130\pi$
$43$	−6.35890	−0.969723	−0.484861	+	0.874591i	$0.661130\pi$
$44$	−0.434541	−0.0655095
$45$	0	0
$46$	−0.332059	−0.0489594
$47$	3.87861	0.565754	0.282877	−	0.959156i	$-0.408711\pi$
$47$	3.87861	0.565754	0.282877	+	0.959156i	$0.408711\pi$
$48$	0	0
$49$	3.28144	0.468777
$50$	−0.00528210	−0.000747002 0
$51$	0	0
$52$	8.32830	1.15493
$53$	10.8877	1.49555	0.747773	−	0.663954i	$-0.231123\pi$
$53$	10.8877	1.49555	0.747773	+	0.663954i	$0.231123\pi$
$54$	0	0
$55$	−0.490604	−0.0661530
$56$	−1.06339	−0.142102
$57$	0	0
$58$	−0.0591465	−0.00776631
$59$	2.63793	0.343429	0.171715	−	0.985147i	$-0.445069\pi$
$59$	2.63793	0.343429	0.171715	+	0.985147i	$0.445069\pi$
$60$	0	0
$61$	11.1544	1.42817	0.714084	−	0.700060i	$-0.246843\pi$
$61$	11.1544	1.42817	0.714084	+	0.700060i	$0.246843\pi$
$62$	−0.0871503	−0.0110681
$63$	0	0
$64$	−7.83493	−0.979366
$65$	9.40279	1.16627
$66$	0	0
$67$	6.09170	0.744219	0.372110	−	0.928189i	$-0.378635\pi$
$67$	6.09170	0.744219	0.372110	+	0.928189i	$0.378635\pi$
$68$	−9.34643	−1.13342
$69$	0	0
$70$	−0.599256	−0.0716247
$71$	−9.40515	−1.11619	−0.558093	−	0.829778i	$-0.688467\pi$
$71$	−9.40515	−1.11619	−0.558093	+	0.829778i	$0.688467\pi$
$72$	0	0
$73$	−2.78532	−0.325997	−0.162998	−	0.986626i	$-0.552117\pi$
$73$	−2.78532	−0.325997	−0.162998	+	0.986626i	$0.552117\pi$
$74$	−0.142019	−0.0165094
$75$	0	0
$76$	−6.85479	−0.786299
$77$	−0.699082	−0.0796678
$78$	0	0
$79$	5.16993	0.581662	0.290831	−	0.956774i	$-0.406068\pi$
$79$	5.16993	0.581662	0.290831	+	0.956774i	$0.406068\pi$
$80$	−8.90795	−0.995939
$81$	0	0
$82$	0.264491	0.0292081
$83$	−1.83793	−0.201739	−0.100870	−	0.994900i	$-0.532162\pi$
$83$	−1.83793	−0.201739	−0.100870	+	0.994900i	$0.532162\pi$
$84$	0	0
$85$	−10.5523	−1.14455
$86$	0.528126	0.0569493
$87$	0	0
$88$	0.0723048	0.00770772
$89$	8.08786	0.857311	0.428656	−	0.903468i	$-0.358987\pi$
$89$	8.08786	0.857311	0.428656	+	0.903468i	$0.358987\pi$
$90$	0	0
$91$	13.3984	1.40454
$92$	−7.96873	−0.830798
$93$	0	0
$94$	−0.322130	−0.0332252
$95$	−7.73917	−0.794022
$96$	0	0
$97$	−12.9058	−1.31039	−0.655193	−	0.755462i	$-0.727413\pi$
$97$	−12.9058	−1.31039	−0.655193	+	0.755462i	$0.727413\pi$
$98$	−0.272533	−0.0275300
$99$	0	0
$100$	−0.126760	−0.0126760
$101$	−2.17744	−0.216663	−0.108331	−	0.994115i	$-0.534551\pi$
$101$	−2.17744	−0.216663	−0.108331	+	0.994115i	$0.534551\pi$
$102$	0	0
$103$	14.7612	1.45447	0.727234	−	0.686390i	$-0.240806\pi$
$103$	14.7612	1.45447	0.727234	+	0.686390i	$0.240806\pi$
$104$	−1.38578	−0.135886
$105$	0	0
$106$	−0.904260	−0.0878295
$107$	−10.4459	−1.00984	−0.504920	−	0.863166i	$-0.668478\pi$
$107$	−10.4459	−1.00984	−0.504920	+	0.863166i	$0.668478\pi$
$108$	0	0
$109$	−5.69014	−0.545017	−0.272508	−	0.962153i	$-0.587853\pi$
$109$	−5.69014	−0.545017	−0.272508	+	0.962153i	$0.587853\pi$
$110$	0.0407461	0.00388499
$111$	0	0
$112$	−12.6933	−1.19941
$113$	−20.9980	−1.97533	−0.987665	−	0.156584i	$-0.949952\pi$
$113$	−20.9980	−1.97533	−0.987665	+	0.156584i	$0.949952\pi$
$114$	0	0
$115$	−8.99683	−0.838958
$116$	−1.41939	−0.131787
$117$	0	0
$118$	−0.219088	−0.0201687
$119$	−15.0364	−1.37838
$120$	0	0
$121$	−10.9525	−0.995679
$122$	−0.926403	−0.0838726
$123$	0	0
$124$	−2.09143	−0.187816
$125$	11.1081	0.993539
$126$	0	0
$127$	−0.0956906	−0.00849117	−0.00424558	−	0.999991i	$-0.501351\pi$
$127$	−0.0956906	−0.00849117	−0.00424558	+	0.999991i	$0.501351\pi$
$128$	2.63483	0.232888
$129$	0	0
$130$	−0.780930	−0.0684921
$131$	−14.5638	−1.27244	−0.636221	−	0.771507i	$-0.719504\pi$
$131$	−14.5638	−1.27244	−0.636221	+	0.771507i	$0.719504\pi$
$132$	0	0
$133$	−11.0279	−0.956238
$134$	−0.505934	−0.0437060
$135$	0	0
$136$	1.55519	0.133356
$137$	17.1079	1.46162	0.730811	−	0.682579i	$-0.239142\pi$
$137$	17.1079	1.46162	0.730811	+	0.682579i	$0.239142\pi$
$138$	0	0
$139$	−17.7270	−1.50358	−0.751791	−	0.659402i	$-0.770810\pi$
$139$	−17.7270	−1.50358	−0.751791	+	0.659402i	$0.770810\pi$
$140$	−14.3809	−1.21541
$141$	0	0
$142$	0.781126	0.0655507
$143$	−0.911020	−0.0761833
$144$	0	0
$145$	−1.60252	−0.133082
$146$	0.231329	0.0191449
$147$	0	0
$148$	−3.40817	−0.280150
$149$	3.37396	0.276406	0.138203	−	0.990404i	$-0.455867\pi$
$149$	3.37396	0.276406	0.138203	+	0.990404i	$0.455867\pi$
$150$	0	0
$151$	2.76168	0.224742	0.112371	−	0.993666i	$-0.464155\pi$
$151$	2.76168	0.224742	0.112371	+	0.993666i	$0.464155\pi$
$152$	1.14059	0.0925143
$153$	0	0
$154$	0.0580609	0.00467868
$155$	−2.36126	−0.189661
$156$	0	0
$157$	−8.93566	−0.713144	−0.356572	−	0.934268i	$-0.616054\pi$
$157$	−8.93566	−0.713144	−0.356572	+	0.934268i	$0.616054\pi$
$158$	−0.429378	−0.0341595
$159$	0	0
$160$	2.23237	0.176484
$161$	−12.8200	−1.01035
$162$	0	0
$163$	−8.75660	−0.685870	−0.342935	−	0.939359i	$-0.611421\pi$
$163$	−8.75660	−0.685870	−0.342935	+	0.939359i	$0.611421\pi$
$164$	6.34723	0.495635
$165$	0	0
$166$	0.152646	0.0118476
$167$	−14.3605	−1.11125	−0.555623	−	0.831434i	$-0.687520\pi$
$167$	−14.3605	−1.11125	−0.555623	+	0.831434i	$0.687520\pi$
$168$	0	0
$169$	4.46038	0.343106
$170$	0.876398	0.0672167
$171$	0	0
$172$	12.6739	0.966378
$173$	11.0498	0.840098	0.420049	−	0.907501i	$-0.362013\pi$
$173$	11.0498	0.840098	0.420049	+	0.907501i	$0.362013\pi$
$174$	0	0
$175$	−0.203929	−0.0154156
$176$	0.863077	0.0650569
$177$	0	0
$178$	−0.671721	−0.0503476
$179$	6.55060	0.489615	0.244808	−	0.969572i	$-0.421275\pi$
$179$	6.55060	0.489615	0.244808	+	0.969572i	$0.421275\pi$
$180$	0	0
$181$	−13.9694	−1.03834	−0.519169	−	0.854672i	$-0.673758\pi$
$181$	−13.9694	−1.03834	−0.519169	+	0.854672i	$0.673758\pi$
$182$	−1.11278	−0.0824848
$183$	0	0
$184$	1.32595	0.0977500
$185$	−3.84787	−0.282901
$186$	0	0
$187$	1.02239	0.0747647
$188$	−7.73047	−0.563802
$189$	0	0
$190$	0.642762	0.0466308
$191$	−2.23996	−0.162078	−0.0810390	−	0.996711i	$-0.525824\pi$
$191$	−2.23996	−0.162078	−0.0810390	+	0.996711i	$0.525824\pi$
$192$	0	0
$193$	22.1924	1.59744	0.798722	−	0.601700i	$-0.205510\pi$
$193$	22.1924	1.59744	0.798722	+	0.601700i	$0.205510\pi$
$194$	1.07187	0.0769555
$195$	0	0
$196$	−6.54024	−0.467160
$197$	11.0264	0.785595	0.392797	−	0.919625i	$-0.371507\pi$
$197$	11.0264	0.785595	0.392797	+	0.919625i	$0.371507\pi$
$198$	0	0
$199$	14.6196	1.03636	0.518179	−	0.855272i	$-0.326610\pi$
$199$	14.6196	1.03636	0.518179	+	0.855272i	$0.326610\pi$
$200$	0.0210920	0.00149143
$201$	0	0
$202$	0.180843	0.0127240
$203$	−2.28350	−0.160270
$204$	0	0
$205$	7.16612	0.500504
$206$	−1.22597	−0.0854170
$207$	0	0
$208$	−16.5415	−1.14695
$209$	0.749835	0.0518672
$210$	0	0
$211$	1.54449	0.106327	0.0531635	−	0.998586i	$-0.483070\pi$
$211$	1.54449	0.106327	0.0531635	+	0.998586i	$0.483070\pi$
$212$	−21.7004	−1.49039
$213$	0	0
$214$	0.867562	0.0593053
$215$	14.3091	0.975871
$216$	0	0
$217$	−3.36465	−0.228408
$218$	0.472584	0.0320074
$219$	0	0
$220$	0.977823	0.0659248
$221$	−19.5949	−1.31810
$222$	0	0
$223$	−26.6878	−1.78715	−0.893573	−	0.448918i	$-0.851810\pi$
$223$	−26.6878	−1.78715	−0.893573	+	0.448918i	$0.851810\pi$
$224$	3.18100	0.212540
$225$	0	0
$226$	1.74395	0.116006
$227$	6.41989	0.426103	0.213052	−	0.977041i	$-0.431660\pi$
$227$	6.41989	0.426103	0.213052	+	0.977041i	$0.431660\pi$
$228$	0	0
$229$	−27.8670	−1.84150	−0.920752	−	0.390149i	$-0.872423\pi$
$229$	−27.8670	−1.84150	−0.920752	+	0.390149i	$0.872423\pi$
$230$	0.747214	0.0492698
$231$	0	0
$232$	0.236178	0.0155058
$233$	−0.647350	−0.0424093	−0.0212047	−	0.999775i	$-0.506750\pi$
$233$	−0.647350	−0.0424093	−0.0212047	+	0.999775i	$0.506750\pi$
$234$	0	0
$235$	−8.72782	−0.569340
$236$	−5.25766	−0.342245
$237$	0	0
$238$	1.24882	0.0809488
$239$	12.3635	0.799727	0.399864	−	0.916575i	$-0.369057\pi$
$239$	12.3635	0.799727	0.399864	+	0.916575i	$0.369057\pi$
$240$	0	0
$241$	−12.7531	−0.821497	−0.410748	−	0.911749i	$-0.634732\pi$
$241$	−12.7531	−0.821497	−0.410748	+	0.911749i	$0.634732\pi$
$242$	0.909636	0.0584736
$243$	0	0
$244$	−22.2318	−1.42324
$245$	−7.38404	−0.471749
$246$	0	0
$247$	−14.3712	−0.914415
$248$	0.348000	0.0220980
$249$	0	0
$250$	−0.922562	−0.0583480
$251$	13.4917	0.851588	0.425794	−	0.904820i	$-0.359995\pi$
$251$	13.4917	0.851588	0.425794	+	0.904820i	$0.359995\pi$
$252$	0	0
$253$	0.871688	0.0548025
$254$	0.00794739	0.000498664 0
$255$	0	0
$256$	15.4510	0.965689
$257$	4.56318	0.284643	0.142322	−	0.989820i	$-0.454543\pi$
$257$	4.56318	0.284643	0.142322	+	0.989820i	$0.454543\pi$
$258$	0	0
$259$	−5.48300	−0.340697
$260$	−18.7407	−1.16225
$261$	0	0
$262$	1.20957	0.0747272
$263$	−16.0985	−0.992674	−0.496337	−	0.868130i	$-0.665322\pi$
$263$	−16.0985	−0.992674	−0.496337	+	0.868130i	$0.665322\pi$
$264$	0	0
$265$	−24.5001	−1.50503
$266$	0.915898	0.0561573
$267$	0	0
$268$	−12.1414	−0.741652
$269$	7.49840	0.457186	0.228593	−	0.973522i	$-0.426588\pi$
$269$	7.49840	0.457186	0.228593	+	0.973522i	$0.426588\pi$
$270$	0	0
$271$	6.47424	0.393282	0.196641	−	0.980476i	$-0.436997\pi$
$271$	6.47424	0.393282	0.196641	+	0.980476i	$0.436997\pi$
$272$	18.5637	1.12559
$273$	0	0
$274$	−1.42086	−0.0858373
$275$	0.0138660	0.000836154 0
$276$	0	0
$277$	8.32050	0.499930	0.249965	−	0.968255i	$-0.419581\pi$
$277$	8.32050	0.499930	0.249965	+	0.968255i	$0.419581\pi$
$278$	1.47228	0.0883014
$279$	0	0
$280$	2.39289	0.143002
$281$	26.3798	1.57369	0.786844	−	0.617152i	$-0.211714\pi$
$281$	26.3798	1.57369	0.786844	+	0.617152i	$0.211714\pi$
$282$	0	0
$283$	−23.4048	−1.39127	−0.695637	−	0.718394i	$-0.744878\pi$
$283$	−23.4048	−1.39127	−0.695637	+	0.718394i	$0.744878\pi$
$284$	18.7454	1.11234
$285$	0	0
$286$	0.0756630	0.00447405
$287$	10.2113	0.602755
$288$	0	0
$289$	4.99036	0.293551
$290$	0.133094	0.00781555
$291$	0	0
$292$	5.55142	0.324872
$293$	3.57088	0.208613	0.104307	−	0.994545i	$-0.466738\pi$
$293$	3.57088	0.208613	0.104307	+	0.994545i	$0.466738\pi$
$294$	0	0
$295$	−5.93598	−0.345606
$296$	0.567097	0.0329618
$297$	0	0
$298$	−0.280218	−0.0162326
$299$	−16.7065	−0.966164
$300$	0	0
$301$	20.3896	1.17524
$302$	−0.229366	−0.0131985
$303$	0	0
$304$	13.6149	0.780866
$305$	−25.1000	−1.43722
$306$	0	0
$307$	−2.46828	−0.140872	−0.0704362	−	0.997516i	$-0.522439\pi$
$307$	−2.46828	−0.140872	−0.0704362	+	0.997516i	$0.522439\pi$
$308$	1.39334	0.0793930
$309$	0	0
$310$	0.196110	0.0111383
$311$	19.8949	1.12814	0.564069	−	0.825727i	$-0.309235\pi$
$311$	19.8949	1.12814	0.564069	+	0.825727i	$0.309235\pi$
$312$	0	0
$313$	−10.0179	−0.566246	−0.283123	−	0.959084i	$-0.591370\pi$
$313$	−10.0179	−0.566246	−0.283123	+	0.959084i	$0.591370\pi$
$314$	0.742134	0.0418810
$315$	0	0
$316$	−10.3042	−0.579656
$317$	11.2440	0.631529	0.315764	−	0.948838i	$-0.397739\pi$
$317$	11.2440	0.631529	0.315764	+	0.948838i	$0.397739\pi$
$318$	0	0
$319$	0.155265	0.00869319
$320$	17.6305	0.985575
$321$	0	0
$322$	1.06474	0.0593354
$323$	16.1280	0.897387
$324$	0	0
$325$	−0.265753	−0.0147413
$326$	0.727262	0.0402793
$327$	0	0
$328$	−1.05614	−0.0583155
$329$	−12.4366	−0.685654
$330$	0	0
$331$	−30.0728	−1.65295	−0.826475	−	0.562974i	$-0.809657\pi$
$331$	−30.0728	−1.65295	−0.826475	+	0.562974i	$0.809657\pi$
$332$	3.66319	0.201043
$333$	0	0
$334$	1.19268	0.0652606
$335$	−13.7078	−0.748937
$336$	0	0
$337$	−17.3224	−0.943613	−0.471807	−	0.881702i	$-0.656398\pi$
$337$	−17.3224	−0.943613	−0.471807	+	0.881702i	$0.656398\pi$
$338$	−0.370448	−0.0201497
$339$	0	0
$340$	21.0317	1.14061
$341$	0.228778	0.0123890
$342$	0	0
$343$	11.9234	0.643806
$344$	−2.10886	−0.113702
$345$	0	0
$346$	−0.917716	−0.0493367
$347$	−21.2412	−1.14029	−0.570143	−	0.821545i	$-0.693112\pi$
$347$	−21.2412	−1.14029	−0.570143	+	0.821545i	$0.693112\pi$
$348$	0	0
$349$	11.6817	0.625308	0.312654	−	0.949867i	$-0.398782\pi$
$349$	11.6817	0.625308	0.312654	+	0.949867i	$0.398782\pi$
$350$	0.0169369	0.000905315 0
$351$	0	0
$352$	−0.216291	−0.0115283
$353$	−14.0775	−0.749271	−0.374636	−	0.927172i	$-0.622232\pi$
$353$	−14.0775	−0.749271	−0.374636	+	0.927172i	$0.622232\pi$
$354$	0	0
$355$	21.1639	1.12326
$356$	−16.1199	−0.854355
$357$	0	0
$358$	−0.544048	−0.0287538
$359$	−7.85706	−0.414680	−0.207340	−	0.978269i	$-0.566481\pi$
$359$	−7.85706	−0.414680	−0.207340	+	0.978269i	$0.566481\pi$
$360$	0	0
$361$	−7.17150	−0.377448
$362$	1.16020	0.0609788
$363$	0	0
$364$	−26.7044	−1.39969
$365$	6.26764	0.328063
$366$	0	0
$367$	−19.7726	−1.03212	−0.516062	−	0.856551i	$-0.672602\pi$
$367$	−19.7726	−1.03212	−0.516062	+	0.856551i	$0.672602\pi$
$368$	15.8273	0.825057
$369$	0	0
$370$	0.319578	0.0166141
$371$	−34.9112	−1.81250
$372$	0	0
$373$	7.49088	0.387863	0.193931	−	0.981015i	$-0.437876\pi$
$373$	7.49088	0.387863	0.193931	+	0.981015i	$0.437876\pi$
$374$	−0.0849127	−0.00439073
$375$	0	0
$376$	1.28630	0.0663358
$377$	−2.97577	−0.153260
$378$	0	0
$379$	27.5064	1.41291	0.706455	−	0.707758i	$-0.250293\pi$
$379$	27.5064	1.41291	0.706455	+	0.707758i	$0.250293\pi$
$380$	15.4250	0.791284
$381$	0	0
$382$	0.186036	0.00951842
$383$	−4.92412	−0.251611	−0.125805	−	0.992055i	$-0.540151\pi$
$383$	−4.92412	−0.251611	−0.125805	+	0.992055i	$0.540151\pi$
$384$	0	0
$385$	1.57311	0.0801729
$386$	−1.84315	−0.0938137
$387$	0	0
$388$	25.7226	1.30587
$389$	−38.9916	−1.97696	−0.988478	−	0.151368i	$-0.951632\pi$
$389$	−38.9916	−1.97696	−0.988478	+	0.151368i	$0.951632\pi$
$390$	0	0
$391$	18.7489	0.948173
$392$	1.08825	0.0549651
$393$	0	0
$394$	−0.915772	−0.0461359
$395$	−11.6336	−0.585350
$396$	0	0
$397$	−12.6661	−0.635694	−0.317847	−	0.948142i	$-0.602960\pi$
$397$	−12.6661	−0.635694	−0.317847	+	0.948142i	$0.602960\pi$
$398$	−1.21420	−0.0608626
$399$	0	0
$400$	0.251767	0.0125884
$401$	−26.7959	−1.33812	−0.669062	−	0.743207i	$-0.733304\pi$
$401$	−26.7959	−1.33812	−0.669062	+	0.743207i	$0.733304\pi$
$402$	0	0
$403$	−4.38471	−0.218418
$404$	4.33985	0.215916
$405$	0	0
$406$	0.189651	0.00941223
$407$	0.372814	0.0184797
$408$	0	0
$409$	35.3472	1.74781	0.873903	−	0.486101i	$-0.161581\pi$
$409$	35.3472	1.74781	0.873903	+	0.486101i	$0.161581\pi$
$410$	−0.595168	−0.0293933
$411$	0	0
$412$	−29.4206	−1.44945
$413$	−8.45844	−0.416212
$414$	0	0
$415$	4.13580	0.203018
$416$	4.14537	0.203244
$417$	0	0
$418$	−0.0622761	−0.00304602
$419$	2.49358	0.121819	0.0609096	−	0.998143i	$-0.480600\pi$
$419$	2.49358	0.121819	0.0609096	+	0.998143i	$0.480600\pi$
$420$	0	0
$421$	17.2671	0.841547	0.420773	−	0.907166i	$-0.361759\pi$
$421$	17.2671	0.841547	0.420773	+	0.907166i	$0.361759\pi$
$422$	−0.128274	−0.00624430
$423$	0	0
$424$	3.61080	0.175356
$425$	0.298241	0.0144668
$426$	0	0
$427$	−35.7661	−1.73084
$428$	20.8197	1.00636
$429$	0	0
$430$	−1.18841	−0.0573103
$431$	−31.0699	−1.49659	−0.748293	−	0.663368i	$-0.769126\pi$
$431$	−31.0699	−1.49659	−0.748293	+	0.663368i	$0.769126\pi$
$432$	0	0
$433$	16.4828	0.792112	0.396056	−	0.918226i	$-0.370379\pi$
$433$	16.4828	0.792112	0.396056	+	0.918226i	$0.370379\pi$
$434$	0.279445	0.0134138
$435$	0	0
$436$	11.3410	0.543137
$437$	13.7507	0.657785
$438$	0	0
$439$	33.0499	1.57739	0.788693	−	0.614787i	$-0.210758\pi$
$439$	33.0499	1.57739	0.788693	+	0.614787i	$0.210758\pi$
$440$	−0.162703	−0.00775658
$441$	0	0
$442$	1.62742	0.0774083
$443$	24.8330	1.17985	0.589925	−	0.807458i	$-0.299157\pi$
$443$	24.8330	1.17985	0.589925	+	0.807458i	$0.299157\pi$
$444$	0	0
$445$	−18.1997	−0.862747
$446$	2.21650	0.104954
$447$	0	0
$448$	25.1224	1.18692
$449$	18.6217	0.878811	0.439405	−	0.898289i	$-0.355189\pi$
$449$	18.6217	0.878811	0.439405	+	0.898289i	$0.355189\pi$
$450$	0	0
$451$	−0.694314	−0.0326940
$452$	41.8512	1.96852
$453$	0	0
$454$	−0.533192	−0.0250239
$455$	−30.1497	−1.41344
$456$	0	0
$457$	−34.2762	−1.60337	−0.801686	−	0.597745i	$-0.796063\pi$
$457$	−34.2762	−1.60337	−0.801686	+	0.597745i	$0.796063\pi$
$458$	2.31444	0.108147
$459$	0	0
$460$	17.9316	0.836065
$461$	−11.2220	−0.522659	−0.261329	−	0.965250i	$-0.584161\pi$
$461$	−11.2220	−0.522659	−0.261329	+	0.965250i	$0.584161\pi$
$462$	0	0
$463$	−26.9770	−1.25373	−0.626863	−	0.779129i	$-0.715662\pi$
$463$	−26.9770	−1.25373	−0.626863	+	0.779129i	$0.715662\pi$
$464$	2.81917	0.130877
$465$	0	0
$466$	0.0537644	0.00249059
$467$	−11.5336	−0.533711	−0.266856	−	0.963737i	$-0.585985\pi$
$467$	−11.5336	−0.533711	−0.266856	+	0.963737i	$0.585985\pi$
$468$	0	0
$469$	−19.5328	−0.901942
$470$	0.724872	0.0334359
$471$	0	0
$472$	0.874841	0.0402678
$473$	−1.38638	−0.0637459
$474$	0	0
$475$	0.218734	0.0100362
$476$	29.9690	1.37363
$477$	0	0
$478$	−1.02682	−0.0469659
$479$	28.8893	1.31998	0.659992	−	0.751272i	$-0.270559\pi$
$479$	28.8893	1.31998	0.659992	+	0.751272i	$0.270559\pi$
$480$	0	0
$481$	−7.14526	−0.325796
$482$	1.05918	0.0482443
$483$	0	0
$484$	21.8294	0.992245
$485$	29.0412	1.31869
$486$	0	0
$487$	1.63083	0.0739001	0.0369501	−	0.999317i	$-0.488236\pi$
$487$	1.63083	0.0739001	0.0369501	+	0.999317i	$0.488236\pi$
$488$	3.69922	0.167456
$489$	0	0
$490$	0.613267	0.0277046
$491$	6.45592	0.291352	0.145676	−	0.989332i	$-0.453464\pi$
$491$	6.45592	0.291352	0.145676	+	0.989332i	$0.453464\pi$
$492$	0	0
$493$	3.33956	0.150406
$494$	1.19357	0.0537012
$495$	0	0
$496$	4.15396	0.186518
$497$	30.1573	1.35274
$498$	0	0
$499$	29.7187	1.33039	0.665196	−	0.746669i	$-0.268348\pi$
$499$	29.7187	1.33039	0.665196	+	0.746669i	$0.268348\pi$
$500$	−22.1396	−0.990113
$501$	0	0
$502$	−1.12053	−0.0500115
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	4.89976	0.218037
$506$	−0.0723963	−0.00321841
$507$	0	0
$508$	0.190721	0.00846188
$509$	−7.29140	−0.323186	−0.161593	−	0.986858i	$-0.551663\pi$
$509$	−7.29140	−0.323186	−0.161593	+	0.986858i	$0.551663\pi$
$510$	0	0
$511$	8.93103	0.395086
$512$	−6.55291	−0.289601
$513$	0	0
$514$	−0.378986	−0.0167163
$515$	−33.2164	−1.46369
$516$	0	0
$517$	0.845624	0.0371905
$518$	0.455380	0.0200082
$519$	0	0
$520$	3.11833	0.136748
$521$	−1.73107	−0.0758394	−0.0379197	−	0.999281i	$-0.512073\pi$
$521$	−1.73107	−0.0758394	−0.0379197	+	0.999281i	$0.512073\pi$
$522$	0	0
$523$	15.8653	0.693742	0.346871	−	0.937913i	$-0.387244\pi$
$523$	15.8653	0.693742	0.346871	+	0.937913i	$0.387244\pi$
$524$	29.0271	1.26805
$525$	0	0
$526$	1.33703	0.0582971
$527$	4.92073	0.214350
$528$	0	0
$529$	−7.01475	−0.304989
$530$	2.03481	0.0883863
$531$	0	0
$532$	21.9797	0.952940
$533$	13.3070	0.576392
$534$	0	0
$535$	23.5058	1.01624
$536$	2.02025	0.0872613
$537$	0	0
$538$	−0.622765	−0.0268493
$539$	0.715427	0.0308156
$540$	0	0
$541$	0.384069	0.0165124	0.00825621	−	0.999966i	$-0.497372\pi$
$541$	0.384069	0.0165124	0.00825621	+	0.999966i	$0.497372\pi$
$542$	−0.537705	−0.0230964
$543$	0	0
$544$	−4.65214	−0.199459
$545$	12.8042	0.548472
$546$	0	0
$547$	38.3367	1.63916	0.819579	−	0.572967i	$-0.194208\pi$
$547$	38.3367	1.63916	0.819579	+	0.572967i	$0.194208\pi$
$548$	−34.0977	−1.45658
$549$	0	0
$550$	−0.00115162	−4.91051e−5 0
$551$	2.44928	0.104343
$552$	0	0
$553$	−16.5772	−0.704935
$554$	−0.691043	−0.0293596
$555$	0	0
$556$	35.3317	1.49840
$557$	−25.2935	−1.07172	−0.535860	−	0.844307i	$-0.680013\pi$
$557$	−25.2935	−1.07172	−0.535860	+	0.844307i	$0.680013\pi$
$558$	0	0
$559$	26.5710	1.12384
$560$	28.5631	1.20701
$561$	0	0
$562$	−2.19092	−0.0924185
$563$	−32.5764	−1.37293	−0.686465	−	0.727163i	$-0.740839\pi$
$563$	−32.5764	−1.37293	−0.686465	+	0.727163i	$0.740839\pi$
$564$	0	0
$565$	47.2507	1.98785
$566$	1.94384	0.0817058
$567$	0	0
$568$	−3.11912	−0.130875
$569$	36.9548	1.54923	0.774614	−	0.632435i	$-0.217944\pi$
$569$	36.9548	1.54923	0.774614	+	0.632435i	$0.217944\pi$
$570$	0	0
$571$	−32.4941	−1.35984	−0.679919	−	0.733288i	$-0.737985\pi$
$571$	−32.4941	−1.35984	−0.679919	+	0.733288i	$0.737985\pi$
$572$	1.81576	0.0759206
$573$	0	0
$574$	−0.848081	−0.0353982
$575$	0.254279	0.0106042
$576$	0	0
$577$	−3.08889	−0.128592	−0.0642960	−	0.997931i	$-0.520480\pi$
$577$	−3.08889	−0.128592	−0.0642960	+	0.997931i	$0.520480\pi$
$578$	−0.414465	−0.0172395
$579$	0	0
$580$	3.19398	0.132623
$581$	5.89327	0.244494
$582$	0	0
$583$	2.37377	0.0983116
$584$	−0.923721	−0.0382238
$585$	0	0
$586$	−0.296572	−0.0122513
$587$	39.5581	1.63274	0.816369	−	0.577530i	$-0.195983\pi$
$587$	39.5581	1.63274	0.816369	+	0.577530i	$0.195983\pi$
$588$	0	0
$589$	3.60893	0.148703
$590$	0.493002	0.0202966
$591$	0	0
$592$	6.76923	0.278214
$593$	−37.0809	−1.52273	−0.761366	−	0.648323i	$-0.775471\pi$
$593$	−37.0809	−1.52273	−0.761366	+	0.648323i	$0.775471\pi$
$594$	0	0
$595$	33.8355	1.38712
$596$	−6.72465	−0.275453
$597$	0	0
$598$	1.38753	0.0567403
$599$	7.21058	0.294616	0.147308	−	0.989091i	$-0.452939\pi$
$599$	7.21058	0.294616	0.147308	+	0.989091i	$0.452939\pi$
$600$	0	0
$601$	−15.6443	−0.638145	−0.319072	−	0.947730i	$-0.603371\pi$
$601$	−15.6443	−0.638145	−0.319072	+	0.947730i	$0.603371\pi$
$602$	−1.69342	−0.0690186
$603$	0	0
$604$	−5.50431	−0.223967
$605$	24.6457	1.00199
$606$	0	0
$607$	−5.03996	−0.204566	−0.102283	−	0.994755i	$-0.532615\pi$
$607$	−5.03996	−0.204566	−0.102283	+	0.994755i	$0.532615\pi$
$608$	−3.41194	−0.138372
$609$	0	0
$610$	2.08463	0.0844043
$611$	−16.2070	−0.655666
$612$	0	0
$613$	−41.4393	−1.67372	−0.836858	−	0.547420i	$-0.815610\pi$
$613$	−41.4393	−1.67372	−0.836858	+	0.547420i	$0.815610\pi$
$614$	0.204998	0.00827306
$615$	0	0
$616$	−0.231843	−0.00934122
$617$	−17.1685	−0.691178	−0.345589	−	0.938386i	$-0.612321\pi$
$617$	−17.1685	−0.691178	−0.345589	+	0.938386i	$0.612321\pi$
$618$	0	0
$619$	−16.9228	−0.680186	−0.340093	−	0.940392i	$-0.610459\pi$
$619$	−16.9228	−0.680186	−0.340093	+	0.940392i	$0.610459\pi$
$620$	4.70623	0.189007
$621$	0	0
$622$	−1.65234	−0.0662526
$623$	−25.9335	−1.03900
$624$	0	0
$625$	−25.3140	−1.01256
$626$	0.832018	0.0332541
$627$	0	0
$628$	17.8097	0.710684
$629$	8.01876	0.319729
$630$	0	0
$631$	15.9640	0.635518	0.317759	−	0.948171i	$-0.397070\pi$
$631$	15.9640	0.635518	0.317759	+	0.948171i	$0.397070\pi$
$632$	1.71455	0.0682012
$633$	0	0
$634$	−0.933852	−0.0370880
$635$	0.215327	0.00854500
$636$	0	0
$637$	−13.7117	−0.543277
$638$	−0.0128953	−0.000510528 0
$639$	0	0
$640$	−5.92901	−0.234365
$641$	−34.1774	−1.34993	−0.674964	−	0.737851i	$-0.735841\pi$
$641$	−34.1774	−1.34993	−0.674964	+	0.737851i	$0.735841\pi$
$642$	0	0
$643$	−1.27460	−0.0502653	−0.0251326	−	0.999684i	$-0.508001\pi$
$643$	−1.27460	−0.0502653	−0.0251326	+	0.999684i	$0.508001\pi$
$644$	25.5515	1.00687
$645$	0	0
$646$	−1.33948	−0.0527012
$647$	−16.4300	−0.645930	−0.322965	−	0.946411i	$-0.604680\pi$
$647$	−16.4300	−0.645930	−0.322965	+	0.946411i	$0.604680\pi$
$648$	0	0
$649$	0.575128	0.0225757
$650$	0.0220716	0.000865719 0
$651$	0	0
$652$	17.4528	0.683504
$653$	−46.5637	−1.82218	−0.911088	−	0.412211i	$-0.864757\pi$
$653$	−46.5637	−1.82218	−0.911088	+	0.412211i	$0.864757\pi$
$654$	0	0
$655$	32.7720	1.28051
$656$	−12.6067	−0.492211
$657$	0	0
$658$	1.03290	0.0402667
$659$	21.2263	0.826861	0.413431	−	0.910536i	$-0.364330\pi$
$659$	21.2263	0.826861	0.413431	+	0.910536i	$0.364330\pi$
$660$	0	0
$661$	31.5049	1.22540	0.612700	−	0.790316i	$-0.290084\pi$
$661$	31.5049	1.22540	0.612700	+	0.790316i	$0.290084\pi$
$662$	2.49764	0.0970734
$663$	0	0
$664$	−0.609530	−0.0236544
$665$	24.8154	0.962300
$666$	0	0
$667$	2.84730	0.110248
$668$	28.6219	1.10741
$669$	0	0
$670$	1.13847	0.0439831
$671$	2.43190	0.0938824
$672$	0	0
$673$	−46.0920	−1.77672	−0.888359	−	0.459150i	$-0.848154\pi$
$673$	−46.0920	−1.77672	−0.888359	+	0.459150i	$0.848154\pi$
$674$	1.43868	0.0554159
$675$	0	0
$676$	−8.89000	−0.341923
$677$	−5.22605	−0.200853	−0.100427	−	0.994944i	$-0.532021\pi$
$677$	−5.22605	−0.200853	−0.100427	+	0.994944i	$0.532021\pi$
$678$	0	0
$679$	41.3820	1.58810
$680$	−3.49955	−0.134201
$681$	0	0
$682$	−0.0190007	−0.000727575 0
$683$	26.7163	1.02227	0.511135	−	0.859500i	$-0.329225\pi$
$683$	26.7163	1.02227	0.511135	+	0.859500i	$0.329225\pi$
$684$	0	0
$685$	−38.4969	−1.47089
$686$	−0.990279	−0.0378090
$687$	0	0
$688$	−25.1727	−0.959701
$689$	−45.4951	−1.73323
$690$	0	0
$691$	−20.7575	−0.789653	−0.394826	−	0.918756i	$-0.629195\pi$
$691$	−20.7575	−0.789653	−0.394826	+	0.918756i	$0.629195\pi$
$692$	−22.0233	−0.837200
$693$	0	0
$694$	1.76414	0.0669660
$695$	39.8900	1.51311
$696$	0	0
$697$	−14.9338	−0.565659
$698$	−0.970202	−0.0367227
$699$	0	0
$700$	0.406451	0.0153624
$701$	48.5428	1.83344	0.916719	−	0.399533i	$-0.130828\pi$
$701$	48.5428	1.83344	0.916719	+	0.399533i	$0.130828\pi$
$702$	0	0
$703$	5.88106	0.221809
$704$	−1.70819	−0.0643798
$705$	0	0
$706$	1.16918	0.0440027
$707$	6.98188	0.262581
$708$	0	0
$709$	−21.9935	−0.825981	−0.412991	−	0.910735i	$-0.635516\pi$
$709$	−21.9935	−0.825981	−0.412991	+	0.910735i	$0.635516\pi$
$710$	−1.75772	−0.0659662
$711$	0	0
$712$	2.68225	0.100522
$713$	4.19540	0.157119
$714$	0	0
$715$	2.05002	0.0766663
$716$	−13.0560	−0.487926
$717$	0	0
$718$	0.652552	0.0243530
$719$	−9.78228	−0.364817	−0.182409	−	0.983223i	$-0.558389\pi$
$719$	−9.78228	−0.364817	−0.182409	+	0.983223i	$0.558389\pi$
$720$	0	0
$721$	−47.3314	−1.76271
$722$	0.595615	0.0221665
$723$	0	0
$724$	27.8425	1.03476
$725$	0.0452923	0.00168211
$726$	0	0
$727$	−41.0223	−1.52143	−0.760716	−	0.649085i	$-0.775152\pi$
$727$	−41.0223	−1.52143	−0.760716	+	0.649085i	$0.775152\pi$
$728$	4.44344	0.164685
$729$	0	0
$730$	−0.520547	−0.0192663
$731$	−29.8193	−1.10291
$732$	0	0
$733$	19.7580	0.729780	0.364890	−	0.931051i	$-0.381107\pi$
$733$	19.7580	0.729780	0.364890	+	0.931051i	$0.381107\pi$
$734$	1.64218	0.0606139
$735$	0	0
$736$	−3.96640	−0.146203
$737$	1.32813	0.0489222
$738$	0	0
$739$	8.08268	0.297326	0.148663	−	0.988888i	$-0.452503\pi$
$739$	8.08268	0.297326	0.148663	+	0.988888i	$0.452503\pi$
$740$	7.66921	0.281926
$741$	0	0
$742$	2.89948	0.106443
$743$	−10.5995	−0.388859	−0.194429	−	0.980917i	$-0.562286\pi$
$743$	−10.5995	−0.388859	−0.194429	+	0.980917i	$0.562286\pi$
$744$	0	0
$745$	−7.59224	−0.278158
$746$	−0.622140	−0.0227782
$747$	0	0
$748$	−2.03773	−0.0745068
$749$	33.4944	1.22386
$750$	0	0
$751$	14.0003	0.510877	0.255438	−	0.966825i	$-0.417780\pi$
$751$	14.0003	0.510877	0.255438	+	0.966825i	$0.417780\pi$
$752$	15.3541	0.559907
$753$	0	0
$754$	0.247147	0.00900057
$755$	−6.21445	−0.226167
$756$	0	0
$757$	−28.8413	−1.04826	−0.524128	−	0.851640i	$-0.675609\pi$
$757$	−28.8413	−1.04826	−0.524128	+	0.851640i	$0.675609\pi$
$758$	−2.28449	−0.0829765
$759$	0	0
$760$	−2.56661	−0.0931008
$761$	14.2005	0.514767	0.257383	−	0.966309i	$-0.417140\pi$
$761$	14.2005	0.514767	0.257383	+	0.966309i	$0.417140\pi$
$762$	0	0
$763$	18.2453	0.660523
$764$	4.46448	0.161519
$765$	0	0
$766$	0.408963	0.0147764
$767$	−11.0228	−0.398008
$768$	0	0
$769$	19.1077	0.689043	0.344521	−	0.938778i	$-0.388041\pi$
$769$	19.1077	0.689043	0.344521	+	0.938778i	$0.388041\pi$
$770$	−0.130651	−0.00470834
$771$	0	0
$772$	−44.2317	−1.59193
$773$	29.5204	1.06178	0.530888	−	0.847442i	$-0.321859\pi$
$773$	29.5204	1.06178	0.530888	+	0.847442i	$0.321859\pi$
$774$	0	0
$775$	0.0667367	0.00239725
$776$	−4.28007	−0.153646
$777$	0	0
$778$	3.23837	0.116101
$779$	−10.9527	−0.392420
$780$	0	0
$781$	−2.05053	−0.0733739
$782$	−1.55715	−0.0556837
$783$	0	0
$784$	12.9901	0.463932
$785$	20.1074	0.717665
$786$	0	0
$787$	−20.3547	−0.725568	−0.362784	−	0.931873i	$-0.618174\pi$
$787$	−20.3547	−0.725568	−0.362784	+	0.931873i	$0.618174\pi$
$788$	−21.9766	−0.782886
$789$	0	0
$790$	0.966206	0.0343761
$791$	67.3295	2.39396
$792$	0	0
$793$	−46.6092	−1.65514
$794$	1.05196	0.0373326
$795$	0	0
$796$	−29.1384	−1.03278
$797$	−51.2687	−1.81603	−0.908016	−	0.418935i	$-0.862403\pi$
$797$	−51.2687	−1.81603	−0.908016	+	0.418935i	$0.862403\pi$
$798$	0	0
$799$	18.1883	0.643456
$800$	−0.0630940	−0.00223071
$801$	0	0
$802$	2.22548	0.0785845
$803$	−0.607262	−0.0214298
$804$	0	0
$805$	28.8480	1.01676
$806$	0.364163	0.0128271
$807$	0	0
$808$	−0.722123	−0.0254042
$809$	−53.5568	−1.88296	−0.941478	−	0.337073i	$-0.890563\pi$
$809$	−53.5568	−1.88296	−0.941478	+	0.337073i	$0.890563\pi$
$810$	0	0
$811$	42.8963	1.50629	0.753146	−	0.657854i	$-0.228535\pi$
$811$	42.8963	1.50629	0.753146	+	0.657854i	$0.228535\pi$
$812$	4.55124	0.159717
$813$	0	0
$814$	−0.0309633	−0.00108526
$815$	19.7045	0.690218
$816$	0	0
$817$	−21.8699	−0.765130
$818$	−2.93569	−0.102644
$819$	0	0
$820$	−14.2828	−0.498778
$821$	−5.52686	−0.192889	−0.0964444	−	0.995338i	$-0.530747\pi$
$821$	−5.52686	−0.192889	−0.0964444	+	0.995338i	$0.530747\pi$
$822$	0	0
$823$	−19.0939	−0.665572	−0.332786	−	0.943002i	$-0.607989\pi$
$823$	−19.0939	−0.665572	−0.332786	+	0.943002i	$0.607989\pi$
$824$	4.89540	0.170539
$825$	0	0
$826$	0.702499	0.0244431
$827$	47.8774	1.66486	0.832430	−	0.554130i	$-0.186949\pi$
$827$	47.8774	1.66486	0.832430	+	0.554130i	$0.186949\pi$
$828$	0	0
$829$	13.1461	0.456583	0.228292	−	0.973593i	$-0.426686\pi$
$829$	13.1461	0.456583	0.228292	+	0.973593i	$0.426686\pi$
$830$	−0.343490	−0.0119227
$831$	0	0
$832$	32.7387	1.13501
$833$	15.3879	0.533161
$834$	0	0
$835$	32.3146	1.11829
$836$	−1.49450	−0.0516883
$837$	0	0
$838$	−0.207099	−0.00715412
$839$	−33.7580	−1.16545	−0.582727	−	0.812668i	$-0.698014\pi$
$839$	−33.7580	−1.16545	−0.582727	+	0.812668i	$0.698014\pi$
$840$	0	0
$841$	−28.4928	−0.982512
$842$	−1.43408	−0.0494218
$843$	0	0
$844$	−3.07832	−0.105960
$845$	−10.0370	−0.345282
$846$	0	0
$847$	35.1187	1.20669
$848$	43.1009	1.48009
$849$	0	0
$850$	−0.0247698	−0.000849598 0
$851$	6.83677	0.234361
$852$	0	0
$853$	16.1494	0.552946	0.276473	−	0.961022i	$-0.410834\pi$
$853$	16.1494	0.552946	0.276473	+	0.961022i	$0.410834\pi$
$854$	2.97048	0.101648
$855$	0	0
$856$	−3.46426	−0.118406
$857$	−21.1749	−0.723322	−0.361661	−	0.932310i	$-0.617790\pi$
$857$	−21.1749	−0.723322	−0.361661	+	0.932310i	$0.617790\pi$
$858$	0	0
$859$	−21.2136	−0.723798	−0.361899	−	0.932217i	$-0.617871\pi$
$859$	−21.2136	−0.723798	−0.361899	+	0.932217i	$0.617871\pi$
$860$	−28.5194	−0.972505
$861$	0	0
$862$	2.58045	0.0878906
$863$	45.3619	1.54414	0.772068	−	0.635539i	$-0.219222\pi$
$863$	45.3619	1.54414	0.772068	+	0.635539i	$0.219222\pi$
$864$	0	0
$865$	−24.8647	−0.845424
$866$	−1.36895	−0.0465186
$867$	0	0
$868$	6.70610	0.227620
$869$	1.12716	0.0382363
$870$	0	0
$871$	−25.4545	−0.862494
$872$	−1.88707	−0.0639044
$873$	0	0
$874$	−1.14204	−0.0386300
$875$	−35.6178	−1.20410
$876$	0	0
$877$	12.1962	0.411837	0.205918	−	0.978569i	$-0.433982\pi$
$877$	12.1962	0.411837	0.205918	+	0.978569i	$0.433982\pi$
$878$	−2.74490	−0.0926358
$879$	0	0
$880$	−1.94213	−0.0654693
$881$	−8.50736	−0.286620	−0.143310	−	0.989678i	$-0.545775\pi$
$881$	−8.50736	−0.286620	−0.143310	+	0.989678i	$0.545775\pi$
$882$	0	0
$883$	−26.5027	−0.891888	−0.445944	−	0.895061i	$-0.647132\pi$
$883$	−26.5027	−0.891888	−0.445944	+	0.895061i	$0.647132\pi$
$884$	39.0546	1.31355
$885$	0	0
$886$	−2.06245	−0.0692895
$887$	−33.1863	−1.11429	−0.557144	−	0.830416i	$-0.688103\pi$
$887$	−33.1863	−1.11429	−0.557144	+	0.830416i	$0.688103\pi$
$888$	0	0
$889$	0.306829	0.0102907
$890$	1.51154	0.0506668
$891$	0	0
$892$	53.1915	1.78098
$893$	13.3395	0.446391
$894$	0	0
$895$	−14.7405	−0.492719
$896$	−8.44850	−0.282244
$897$	0	0
$898$	−1.54659	−0.0516102
$899$	0.747286	0.0249234
$900$	0	0
$901$	51.0568	1.70095
$902$	0.0576649	0.00192003
$903$	0	0
$904$	−6.96377	−0.231612
$905$	31.4346	1.04492
$906$	0	0
$907$	6.31486	0.209682	0.104841	−	0.994489i	$-0.466567\pi$
$907$	6.31486	0.209682	0.104841	+	0.994489i	$0.466567\pi$
$908$	−12.7955	−0.424634
$909$	0	0
$910$	2.50403	0.0830077
$911$	−43.0949	−1.42780	−0.713898	−	0.700249i	$-0.753072\pi$
$911$	−43.0949	−1.42780	−0.713898	+	0.700249i	$0.753072\pi$
$912$	0	0
$913$	−0.400710	−0.0132616
$914$	2.84674	0.0941618
$915$	0	0
$916$	55.5418	1.83515
$917$	46.6983	1.54211
$918$	0	0
$919$	−19.4088	−0.640236	−0.320118	−	0.947378i	$-0.603723\pi$
$919$	−19.4088	−0.640236	−0.320118	+	0.947378i	$0.603723\pi$
$920$	−2.98370	−0.0983697
$921$	0	0
$922$	0.932018	0.0306944
$923$	39.3000	1.29358
$924$	0	0
$925$	0.108753	0.00357579
$926$	2.24052	0.0736281
$927$	0	0
$928$	−0.706496	−0.0231919
$929$	28.3914	0.931493	0.465747	−	0.884918i	$-0.345786\pi$
$929$	28.3914	0.931493	0.465747	+	0.884918i	$0.345786\pi$
$930$	0	0
$931$	11.2857	0.369874
$932$	1.29024	0.0422631
$933$	0	0
$934$	0.957900	0.0313434
$935$	−2.30063	−0.0752387
$936$	0	0
$937$	−4.00280	−0.130766	−0.0653829	−	0.997860i	$-0.520827\pi$
$937$	−4.00280	−0.130766	−0.0653829	+	0.997860i	$0.520827\pi$
$938$	1.62226	0.0529687
$939$	0	0
$940$	17.3954	0.567377
$941$	−13.9675	−0.455328	−0.227664	−	0.973740i	$-0.573109\pi$
$941$	−13.9675	−0.455328	−0.227664	+	0.973740i	$0.573109\pi$
$942$	0	0
$943$	−12.7325	−0.414628
$944$	10.4427	0.339880
$945$	0	0
$946$	0.115143	0.00374363
$947$	−57.2963	−1.86188	−0.930940	−	0.365171i	$-0.881010\pi$
$947$	−57.2963	−1.86188	−0.930940	+	0.365171i	$0.881010\pi$
$948$	0	0
$949$	11.6386	0.377806
$950$	−0.0181665	−0.000589400 0
$951$	0	0
$952$	−4.98665	−0.161618
$953$	−5.29452	−0.171506	−0.0857532	−	0.996316i	$-0.527330\pi$
$953$	−5.29452	−0.171506	−0.0857532	+	0.996316i	$0.527330\pi$
$954$	0	0
$955$	5.04047	0.163106
$956$	−24.6417	−0.796969
$957$	0	0
$958$	−2.39934	−0.0775192
$959$	−54.8558	−1.77139
$960$	0	0
$961$	−29.8989	−0.964481
$962$	0.593436	0.0191331
$963$	0	0
$964$	25.4181	0.818663
$965$	−49.9383	−1.60757
$966$	0	0
$967$	44.3615	1.42657	0.713284	−	0.700875i	$-0.247207\pi$
$967$	44.3615	1.42657	0.713284	+	0.700875i	$0.247207\pi$
$968$	−3.63227	−0.116746
$969$	0	0
$970$	−2.41196	−0.0774434
$971$	−51.4789	−1.65204	−0.826018	−	0.563644i	$-0.809399\pi$
$971$	−51.4789	−1.65204	−0.826018	+	0.563644i	$0.809399\pi$
$972$	0	0
$973$	56.8410	1.82224
$974$	−0.135446	−0.00433996
$975$	0	0
$976$	44.1563	1.41341
$977$	0.892938	0.0285676	0.0142838	−	0.999898i	$-0.495453\pi$
$977$	0.892938	0.0285676	0.0142838	+	0.999898i	$0.495453\pi$
$978$	0	0
$979$	1.76333	0.0563564
$980$	14.7171	0.470122
$981$	0	0
$982$	−0.536184	−0.0171103
$983$	−36.2679	−1.15677	−0.578384	−	0.815765i	$-0.696316\pi$
$983$	−36.2679	−1.15677	−0.578384	+	0.815765i	$0.696316\pi$
$984$	0	0
$985$	−24.8120	−0.790576
$986$	−0.277361	−0.00883296
$987$	0	0
$988$	28.6432	0.911261
$989$	−25.4239	−0.808432
$990$	0	0
$991$	29.3407	0.932036	0.466018	−	0.884775i	$-0.345688\pi$
$991$	29.3407	0.932036	0.466018	+	0.884775i	$0.345688\pi$
$992$	−1.04100	−0.0330518
$993$	0	0
$994$	−2.50466	−0.0794429
$995$	−32.8977	−1.04293
$996$	0	0
$997$	18.6388	0.590296	0.295148	−	0.955452i	$-0.404631\pi$
$997$	18.6388	0.590296	0.295148	+	0.955452i	$0.404631\pi$
$998$	−2.46823	−0.0781304
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.4		10
3.2	odd	2		503.2.a.e.1.7	✓	10
12.11	even	2		8048.2.a.p.1.3		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.7	✓	10	3.2	odd	2
4527.2.a.k.1.4		10	1.1	even	1	trivial
8048.2.a.p.1.3		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)

\(f(q)\)	\(=\)	\(q-0.0830530 q^{2} -1.99310 q^{4} -2.25024 q^{5} -3.20647 q^{7} +0.331639 q^{8} +O(q^{10})\)	\(q-0.0830530 q^{2} -1.99310 q^{4} -2.25024 q^{5} -3.20647 q^{7} +0.331639 q^{8} +0.186890 q^{10} +0.218022 q^{11} -4.17856 q^{13} +0.266307 q^{14} +3.95866 q^{16} +4.68939 q^{17} +3.43926 q^{19} +4.48497 q^{20} -0.0181074 q^{22} +3.99816 q^{23} +0.0635992 q^{25} +0.347042 q^{26} +6.39082 q^{28} +0.712153 q^{29} +1.04933 q^{31} -0.992057 q^{32} -0.389468 q^{34} +7.21534 q^{35} +1.70998 q^{37} -0.285641 q^{38} -0.746269 q^{40} -3.18460 q^{41} -6.35890 q^{43} -0.434541 q^{44} -0.332059 q^{46} +3.87861 q^{47} +3.28144 q^{49} -0.00528210 q^{50} +8.32830 q^{52} +10.8877 q^{53} -0.490604 q^{55} -1.06339 q^{56} -0.0591465 q^{58} +2.63793 q^{59} +11.1544 q^{61} -0.0871503 q^{62} -7.83493 q^{64} +9.40279 q^{65} +6.09170 q^{67} -9.34643 q^{68} -0.599256 q^{70} -9.40515 q^{71} -2.78532 q^{73} -0.142019 q^{74} -6.85479 q^{76} -0.699082 q^{77} +5.16993 q^{79} -8.90795 q^{80} +0.264491 q^{82} -1.83793 q^{83} -10.5523 q^{85} +0.528126 q^{86} +0.0723048 q^{88} +8.08786 q^{89} +13.3984 q^{91} -7.96873 q^{92} -0.322130 q^{94} -7.73917 q^{95} -12.9058 q^{97} -0.272533 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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

Modular forms

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Database

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