LMFDB - Embedded newform 4527.2.a.k.1.8

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			1.8
Root			$-0.489003$ 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)

$f(q)$	$=$	$q+1.62786 q^{2} +0.649933 q^{4} +1.79865 q^{5} +0.552233 q^{7} -2.19772 q^{8} +O(q^{10})$	\(q+1.62786 q^{2} +0.649933 q^{4} +1.79865 q^{5} +0.552233 q^{7} -2.19772 q^{8} +2.92795 q^{10} -4.33718 q^{11} +2.54873 q^{13} +0.898958 q^{14} -4.87745 q^{16} -2.52304 q^{17} -5.11893 q^{19} +1.16900 q^{20} -7.06033 q^{22} +3.78409 q^{23} -1.76486 q^{25} +4.14898 q^{26} +0.358914 q^{28} -0.907287 q^{29} +0.380820 q^{31} -3.54438 q^{32} -4.10717 q^{34} +0.993273 q^{35} -5.43266 q^{37} -8.33291 q^{38} -3.95293 q^{40} -5.72459 q^{41} -9.21553 q^{43} -2.81888 q^{44} +6.15997 q^{46} +8.81077 q^{47} -6.69504 q^{49} -2.87295 q^{50} +1.65651 q^{52} +6.46357 q^{53} -7.80106 q^{55} -1.21365 q^{56} -1.47694 q^{58} -3.40568 q^{59} -1.06109 q^{61} +0.619922 q^{62} +3.98515 q^{64} +4.58427 q^{65} -0.253929 q^{67} -1.63981 q^{68} +1.61691 q^{70} -11.7311 q^{71} -16.1271 q^{73} -8.84362 q^{74} -3.32696 q^{76} -2.39513 q^{77} +7.76447 q^{79} -8.77283 q^{80} -9.31884 q^{82} +16.3846 q^{83} -4.53807 q^{85} -15.0016 q^{86} +9.53191 q^{88} +3.09992 q^{89} +1.40749 q^{91} +2.45940 q^{92} +14.3427 q^{94} -9.20715 q^{95} -4.28605 q^{97} -10.8986 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.62786	1.15107	0.575536	−	0.817776i	$-0.304793\pi$
$2$	1.62786	1.15107	0.575536	+	0.817776i	$0.304793\pi$
$3$	0	0
$3$	0	0
$4$	0.649933	0.324967
$5$	1.79865	0.804380	0.402190	−	0.915556i	$-0.368249\pi$
$5$	1.79865	0.804380	0.402190	+	0.915556i	$0.368249\pi$
$6$	0	0
$7$	0.552233	0.208724	0.104362	−	0.994539i	$-0.466720\pi$
$7$	0.552233	0.208724	0.104362	+	0.994539i	$0.466720\pi$
$8$	−2.19772	−0.777012
$9$	0	0
$10$	2.92795	0.925900
$11$	−4.33718	−1.30771	−0.653854	−	0.756620i	$-0.726849\pi$
$11$	−4.33718	−1.30771	−0.653854	+	0.756620i	$0.726849\pi$
$12$	0	0
$13$	2.54873	0.706891	0.353445	−	0.935455i	$-0.385010\pi$
$13$	2.54873	0.706891	0.353445	+	0.935455i	$0.385010\pi$
$14$	0.898958	0.240257
$15$	0	0
$16$	−4.87745	−1.21936
$17$	−2.52304	−0.611928	−0.305964	−	0.952043i	$-0.598979\pi$
$17$	−2.52304	−0.611928	−0.305964	+	0.952043i	$0.598979\pi$
$18$	0	0
$19$	−5.11893	−1.17436	−0.587181	−	0.809455i	$-0.699762\pi$
$19$	−5.11893	−1.17436	−0.587181	+	0.809455i	$0.699762\pi$
$20$	1.16900	0.261397
$21$	0	0
$22$	−7.06033	−1.50527
$23$	3.78409	0.789036	0.394518	−	0.918888i	$-0.370912\pi$
$23$	3.78409	0.789036	0.394518	+	0.918888i	$0.370912\pi$
$24$	0	0
$25$	−1.76486	−0.352972
$26$	4.14898	0.813682
$27$	0	0
$28$	0.358914	0.0678284
$29$	−0.907287	−0.168479	−0.0842395	−	0.996446i	$-0.526846\pi$
$29$	−0.907287	−0.168479	−0.0842395	+	0.996446i	$0.526846\pi$
$30$	0	0
$31$	0.380820	0.0683972	0.0341986	−	0.999415i	$-0.489112\pi$
$31$	0.380820	0.0683972	0.0341986	+	0.999415i	$0.489112\pi$
$32$	−3.54438	−0.626563
$33$	0	0
$34$	−4.10717	−0.704373
$35$	0.993273	0.167894
$36$	0	0
$37$	−5.43266	−0.893124	−0.446562	−	0.894753i	$-0.647352\pi$
$37$	−5.43266	−0.893124	−0.446562	+	0.894753i	$0.647352\pi$
$38$	−8.33291	−1.35178
$39$	0	0
$40$	−3.95293	−0.625013
$41$	−5.72459	−0.894031	−0.447015	−	0.894526i	$-0.647513\pi$
$41$	−5.72459	−0.894031	−0.447015	+	0.894526i	$0.647513\pi$
$42$	0	0
$43$	−9.21553	−1.40536	−0.702678	−	0.711508i	$-0.748012\pi$
$43$	−9.21553	−1.40536	−0.702678	+	0.711508i	$0.748012\pi$
$44$	−2.81888	−0.424962
$45$	0	0
$46$	6.15997	0.908238
$47$	8.81077	1.28518	0.642592	−	0.766209i	$-0.277859\pi$
$47$	8.81077	1.28518	0.642592	+	0.766209i	$0.277859\pi$
$48$	0	0
$49$	−6.69504	−0.956434
$50$	−2.87295	−0.406297
$51$	0	0
$52$	1.65651	0.229716
$53$	6.46357	0.887840	0.443920	−	0.896066i	$-0.353587\pi$
$53$	6.46357	0.887840	0.443920	+	0.896066i	$0.353587\pi$
$54$	0	0
$55$	−7.80106	−1.05190
$56$	−1.21365	−0.162181
$57$	0	0
$58$	−1.47694	−0.193932
$59$	−3.40568	−0.443381	−0.221691	−	0.975117i	$-0.571157\pi$
$59$	−3.40568	−0.443381	−0.221691	+	0.975117i	$0.571157\pi$
$60$	0	0
$61$	−1.06109	−0.135859	−0.0679294	−	0.997690i	$-0.521639\pi$
$61$	−1.06109	−0.135859	−0.0679294	+	0.997690i	$0.521639\pi$
$62$	0.619922	0.0787301
$63$	0	0
$64$	3.98515	0.498144
$65$	4.58427	0.568609
$66$	0	0
$67$	−0.253929	−0.0310223	−0.0155112	−	0.999880i	$-0.504938\pi$
$67$	−0.253929	−0.0310223	−0.0155112	+	0.999880i	$0.504938\pi$
$68$	−1.63981	−0.198856
$69$	0	0
$70$	1.61691	0.193258
$71$	−11.7311	−1.39222	−0.696112	−	0.717933i	$-0.745088\pi$
$71$	−11.7311	−1.39222	−0.696112	+	0.717933i	$0.745088\pi$
$72$	0	0
$73$	−16.1271	−1.88753	−0.943765	−	0.330617i	$-0.892743\pi$
$73$	−16.1271	−1.88753	−0.943765	+	0.330617i	$0.892743\pi$
$74$	−8.84362	−1.02805
$75$	0	0
$76$	−3.32696	−0.381629
$77$	−2.39513	−0.272951
$78$	0	0
$79$	7.76447	0.873571	0.436785	−	0.899566i	$-0.356117\pi$
$79$	7.76447	0.873571	0.436785	+	0.899566i	$0.356117\pi$
$80$	−8.77283	−0.980832
$81$	0	0
$82$	−9.31884	−1.02909
$83$	16.3846	1.79844	0.899222	−	0.437492i	$-0.144133\pi$
$83$	16.3846	1.79844	0.899222	+	0.437492i	$0.144133\pi$
$84$	0	0
$85$	−4.53807	−0.492223
$86$	−15.0016	−1.61767
$87$	0	0
$88$	9.53191	1.01611
$89$	3.09992	0.328590	0.164295	−	0.986411i	$-0.447465\pi$
$89$	3.09992	0.328590	0.164295	+	0.986411i	$0.447465\pi$
$90$	0	0
$91$	1.40749	0.147545
$92$	2.45940	0.256410
$93$	0	0
$94$	14.3427	1.47934
$95$	−9.20715	−0.944634
$96$	0	0
$97$	−4.28605	−0.435183	−0.217591	−	0.976040i	$-0.569820\pi$
$97$	−4.28605	−0.435183	−0.217591	+	0.976040i	$0.569820\pi$
$98$	−10.8986	−1.10092
$99$	0	0
$100$	−1.14704	−0.114704
$101$	3.73446	0.371592	0.185796	−	0.982588i	$-0.440514\pi$
$101$	3.73446	0.371592	0.185796	+	0.982588i	$0.440514\pi$
$102$	0	0
$103$	2.83220	0.279065	0.139533	−	0.990217i	$-0.455440\pi$
$103$	2.83220	0.279065	0.139533	+	0.990217i	$0.455440\pi$
$104$	−5.60140	−0.549263
$105$	0	0
$106$	10.5218	1.02197
$107$	−10.0994	−0.976343	−0.488172	−	0.872748i	$-0.662336\pi$
$107$	−10.0994	−0.976343	−0.488172	+	0.872748i	$0.662336\pi$
$108$	0	0
$109$	−6.86764	−0.657800	−0.328900	−	0.944365i	$-0.606678\pi$
$109$	−6.86764	−0.657800	−0.328900	+	0.944365i	$0.606678\pi$
$110$	−12.6991	−1.21081
$111$	0	0
$112$	−2.69349	−0.254511
$113$	−2.80558	−0.263927	−0.131963	−	0.991255i	$-0.542128\pi$
$113$	−2.80558	−0.263927	−0.131963	+	0.991255i	$0.542128\pi$
$114$	0	0
$115$	6.80624	0.634685
$116$	−0.589676	−0.0547501
$117$	0	0
$118$	−5.54397	−0.510364
$119$	−1.39331	−0.127724
$120$	0	0
$121$	7.81112	0.710102
$122$	−1.72731	−0.156383
$123$	0	0
$124$	0.247507	0.0222268
$125$	−12.1676	−1.08830
$126$	0	0
$127$	−14.9147	−1.32347	−0.661734	−	0.749739i	$-0.730179\pi$
$127$	−14.9147	−1.32347	−0.661734	+	0.749739i	$0.730179\pi$
$128$	13.5760	1.19996
$129$	0	0
$130$	7.46256	0.654510
$131$	−13.5207	−1.18131	−0.590654	−	0.806925i	$-0.701130\pi$
$131$	−13.5207	−1.18131	−0.590654	+	0.806925i	$0.701130\pi$
$132$	0	0
$133$	−2.82684	−0.245118
$134$	−0.413361	−0.0357090
$135$	0	0
$136$	5.54495	0.475476
$137$	12.4479	1.06350	0.531750	−	0.846901i	$-0.321535\pi$
$137$	12.4479	1.06350	0.531750	+	0.846901i	$0.321535\pi$
$138$	0	0
$139$	2.15443	0.182736	0.0913681	−	0.995817i	$-0.470876\pi$
$139$	2.15443	0.182736	0.0913681	+	0.995817i	$0.470876\pi$
$140$	0.645561	0.0545599
$141$	0	0
$142$	−19.0966	−1.60255
$143$	−11.0543	−0.924407
$144$	0	0
$145$	−1.63189	−0.135521
$146$	−26.2526	−2.17268
$147$	0	0
$148$	−3.53087	−0.290236
$149$	8.59253	0.703927	0.351964	−	0.936014i	$-0.385514\pi$
$149$	8.59253	0.703927	0.351964	+	0.936014i	$0.385514\pi$
$150$	0	0
$151$	0.527862	0.0429568	0.0214784	−	0.999769i	$-0.493163\pi$
$151$	0.527862	0.0429568	0.0214784	+	0.999769i	$0.493163\pi$
$152$	11.2500	0.912494
$153$	0	0
$154$	−3.89894	−0.314186
$155$	0.684961	0.0550174
$156$	0	0
$157$	11.1711	0.891554	0.445777	−	0.895144i	$-0.352927\pi$
$157$	11.1711	0.891554	0.445777	+	0.895144i	$0.352927\pi$
$158$	12.6395	1.00554
$159$	0	0
$160$	−6.37509	−0.503995
$161$	2.08970	0.164691
$162$	0	0
$163$	20.0109	1.56738	0.783689	−	0.621153i	$-0.213336\pi$
$163$	20.0109	1.56738	0.783689	+	0.621153i	$0.213336\pi$
$164$	−3.72060	−0.290530
$165$	0	0
$166$	26.6719	2.07014
$167$	11.9520	0.924877	0.462439	−	0.886651i	$-0.346975\pi$
$167$	11.9520	0.924877	0.462439	+	0.886651i	$0.346975\pi$
$168$	0	0
$169$	−6.50397	−0.500305
$170$	−7.38735	−0.566584
$171$	0	0
$172$	−5.98948	−0.456694
$173$	−3.14561	−0.239156	−0.119578	−	0.992825i	$-0.538154\pi$
$173$	−3.14561	−0.239156	−0.119578	+	0.992825i	$0.538154\pi$
$174$	0	0
$175$	−0.974615	−0.0736739
$176$	21.1544	1.59457
$177$	0	0
$178$	5.04623	0.378231
$179$	6.02096	0.450028	0.225014	−	0.974356i	$-0.427757\pi$
$179$	6.02096	0.450028	0.225014	+	0.974356i	$0.427757\pi$
$180$	0	0
$181$	−23.0475	−1.71311	−0.856553	−	0.516059i	$-0.827398\pi$
$181$	−23.0475	−1.71311	−0.856553	+	0.516059i	$0.827398\pi$
$182$	2.29120	0.169835
$183$	0	0
$184$	−8.31637	−0.613091
$185$	−9.77145	−0.718412
$186$	0	0
$187$	10.9429	0.800224
$188$	5.72641	0.417642
$189$	0	0
$190$	−14.9880	−1.08734
$191$	17.9774	1.30080	0.650398	−	0.759594i	$-0.274602\pi$
$191$	17.9774	1.30080	0.650398	+	0.759594i	$0.274602\pi$
$192$	0	0
$193$	−10.4429	−0.751697	−0.375848	−	0.926681i	$-0.622649\pi$
$193$	−10.4429	−0.751697	−0.375848	+	0.926681i	$0.622649\pi$
$194$	−6.97710	−0.500927
$195$	0	0
$196$	−4.35133	−0.310809
$197$	4.36026	0.310656	0.155328	−	0.987863i	$-0.450357\pi$
$197$	4.36026	0.310656	0.155328	+	0.987863i	$0.450357\pi$
$198$	0	0
$199$	14.0205	0.993888	0.496944	−	0.867783i	$-0.334455\pi$
$199$	14.0205	0.993888	0.496944	+	0.867783i	$0.334455\pi$
$200$	3.87868	0.274264
$201$	0	0
$202$	6.07918	0.427729
$203$	−0.501034	−0.0351657
$204$	0	0
$205$	−10.2965	−0.719141
$206$	4.61043	0.321224
$207$	0	0
$208$	−12.4313	−0.861957
$209$	22.2017	1.53572
$210$	0	0
$211$	−11.0580	−0.761265	−0.380633	−	0.924726i	$-0.624294\pi$
$211$	−11.0580	−0.761265	−0.380633	+	0.924726i	$0.624294\pi$
$212$	4.20089	0.288518
$213$	0	0
$214$	−16.4404	−1.12384
$215$	−16.5755	−1.13044
$216$	0	0
$217$	0.210301	0.0142762
$218$	−11.1796	−0.757175
$219$	0	0
$220$	−5.07017	−0.341831
$221$	−6.43056	−0.432566
$222$	0	0
$223$	−15.5502	−1.04132	−0.520658	−	0.853765i	$-0.674313\pi$
$223$	−15.5502	−1.04132	−0.520658	+	0.853765i	$0.674313\pi$
$224$	−1.95732	−0.130779
$225$	0	0
$226$	−4.56710	−0.303799
$227$	−1.27577	−0.0846759	−0.0423379	−	0.999103i	$-0.513481\pi$
$227$	−1.27577	−0.0846759	−0.0423379	+	0.999103i	$0.513481\pi$
$228$	0	0
$229$	−15.5232	−1.02580	−0.512901	−	0.858448i	$-0.671429\pi$
$229$	−15.5232	−1.02580	−0.512901	+	0.858448i	$0.671429\pi$
$230$	11.0796	0.730568
$231$	0	0
$232$	1.99397	0.130910
$233$	27.8010	1.82130	0.910651	−	0.413177i	$-0.135581\pi$
$233$	27.8010	1.82130	0.910651	+	0.413177i	$0.135581\pi$
$234$	0	0
$235$	15.8475	1.03378
$236$	−2.21346	−0.144084
$237$	0	0
$238$	−2.26811	−0.147020
$239$	−1.20369	−0.0778601	−0.0389301	−	0.999242i	$-0.512395\pi$
$239$	−1.20369	−0.0778601	−0.0389301	+	0.999242i	$0.512395\pi$
$240$	0	0
$241$	−25.6025	−1.64920	−0.824602	−	0.565713i	$-0.808601\pi$
$241$	−25.6025	−1.64920	−0.824602	+	0.565713i	$0.808601\pi$
$242$	12.7154	0.817379
$243$	0	0
$244$	−0.689638	−0.0441496
$245$	−12.0420	−0.769337
$246$	0	0
$247$	−13.0468	−0.830146
$248$	−0.836936	−0.0531455
$249$	0	0
$250$	−19.8072	−1.25272
$251$	−13.2638	−0.837205	−0.418603	−	0.908170i	$-0.637480\pi$
$251$	−13.2638	−0.837205	−0.418603	+	0.908170i	$0.637480\pi$
$252$	0	0
$253$	−16.4123	−1.03183
$254$	−24.2791	−1.52341
$255$	0	0
$256$	14.1296	0.883099
$257$	16.1006	1.00433	0.502163	−	0.864773i	$-0.332538\pi$
$257$	16.1006	1.00433	0.502163	+	0.864773i	$0.332538\pi$
$258$	0	0
$259$	−3.00009	−0.186417
$260$	2.97947	0.184779
$261$	0	0
$262$	−22.0098	−1.35977
$263$	−6.77336	−0.417663	−0.208832	−	0.977952i	$-0.566966\pi$
$263$	−6.77336	−0.417663	−0.208832	+	0.977952i	$0.566966\pi$
$264$	0	0
$265$	11.6257	0.714161
$266$	−4.60170	−0.282148
$267$	0	0
$268$	−0.165037	−0.0100812
$269$	−23.4853	−1.43192	−0.715962	−	0.698139i	$-0.754012\pi$
$269$	−23.4853	−1.43192	−0.715962	+	0.698139i	$0.754012\pi$
$270$	0	0
$271$	17.5886	1.06843	0.534216	−	0.845348i	$-0.320607\pi$
$271$	17.5886	1.06843	0.534216	+	0.845348i	$0.320607\pi$
$272$	12.3060	0.746163
$273$	0	0
$274$	20.2635	1.22417
$275$	7.65452	0.461585
$276$	0	0
$277$	−11.3448	−0.681645	−0.340823	−	0.940128i	$-0.610706\pi$
$277$	−11.3448	−0.681645	−0.340823	+	0.940128i	$0.610706\pi$
$278$	3.50711	0.210343
$279$	0	0
$280$	−2.18294	−0.130455
$281$	9.36752	0.558820	0.279410	−	0.960172i	$-0.409861\pi$
$281$	9.36752	0.558820	0.279410	+	0.960172i	$0.409861\pi$
$282$	0	0
$283$	26.8789	1.59778	0.798892	−	0.601474i	$-0.205420\pi$
$283$	26.8789	1.59778	0.798892	+	0.601474i	$0.205420\pi$
$284$	−7.62443	−0.452426
$285$	0	0
$286$	−17.9949	−1.06406
$287$	−3.16131	−0.186606
$288$	0	0
$289$	−10.6342	−0.625544
$290$	−2.65649	−0.155995
$291$	0	0
$292$	−10.4815	−0.613384
$293$	−17.2365	−1.00697	−0.503485	−	0.864004i	$-0.667949\pi$
$293$	−17.2365	−1.00697	−0.503485	+	0.864004i	$0.667949\pi$
$294$	0	0
$295$	−6.12561	−0.356647
$296$	11.9395	0.693968
$297$	0	0
$298$	13.9874	0.810271
$299$	9.64462	0.557763
$300$	0	0
$301$	−5.08912	−0.293332
$302$	0.859286	0.0494464
$303$	0	0
$304$	24.9673	1.43197
$305$	−1.90853	−0.109282
$306$	0	0
$307$	29.0001	1.65512	0.827561	−	0.561376i	$-0.189728\pi$
$307$	29.0001	1.65512	0.827561	+	0.561376i	$0.189728\pi$
$308$	−1.55668	−0.0886998
$309$	0	0
$310$	1.11502	0.0633290
$311$	−27.2312	−1.54414	−0.772071	−	0.635536i	$-0.780779\pi$
$311$	−27.2312	−1.54414	−0.772071	+	0.635536i	$0.780779\pi$
$312$	0	0
$313$	5.53059	0.312607	0.156304	−	0.987709i	$-0.450042\pi$
$313$	5.53059	0.312607	0.156304	+	0.987709i	$0.450042\pi$
$314$	18.1851	1.02624
$315$	0	0
$316$	5.04639	0.283881
$317$	−31.2030	−1.75253	−0.876267	−	0.481826i	$-0.839974\pi$
$317$	−31.2030	−1.75253	−0.876267	+	0.481826i	$0.839974\pi$
$318$	0	0
$319$	3.93507	0.220322
$320$	7.16789	0.400697
$321$	0	0
$322$	3.40173	0.189571
$323$	12.9153	0.718626
$324$	0	0
$325$	−4.49816	−0.249513
$326$	32.5751	1.80417
$327$	0	0
$328$	12.5811	0.694673
$329$	4.86560	0.268249
$330$	0	0
$331$	16.8025	0.923551	0.461775	−	0.886997i	$-0.347213\pi$
$331$	16.8025	0.923551	0.461775	+	0.886997i	$0.347213\pi$
$332$	10.6489	0.584434
$333$	0	0
$334$	19.4563	1.06460
$335$	−0.456729	−0.0249538
$336$	0	0
$337$	7.57943	0.412878	0.206439	−	0.978459i	$-0.433813\pi$
$337$	7.57943	0.412878	0.206439	+	0.978459i	$0.433813\pi$
$338$	−10.5876	−0.575887
$339$	0	0
$340$	−2.94944	−0.159956
$341$	−1.65168	−0.0894437
$342$	0	0
$343$	−7.56285	−0.408355
$344$	20.2532	1.09198
$345$	0	0
$346$	−5.12062	−0.275286
$347$	28.0855	1.50771	0.753853	−	0.657043i	$-0.228193\pi$
$347$	28.0855	1.50771	0.753853	+	0.657043i	$0.228193\pi$
$348$	0	0
$349$	11.6230	0.622164	0.311082	−	0.950383i	$-0.399309\pi$
$349$	11.6230	0.622164	0.311082	+	0.950383i	$0.399309\pi$
$350$	−1.58654	−0.0848040
$351$	0	0
$352$	15.3726	0.819362
$353$	6.16556	0.328160	0.164080	−	0.986447i	$-0.447535\pi$
$353$	6.16556	0.328160	0.164080	+	0.986447i	$0.447535\pi$
$354$	0	0
$355$	−21.1001	−1.11988
$356$	2.01474	0.106781
$357$	0	0
$358$	9.80129	0.518014
$359$	10.3896	0.548344	0.274172	−	0.961681i	$-0.411596\pi$
$359$	10.3896	0.548344	0.274172	+	0.961681i	$0.411596\pi$
$360$	0	0
$361$	7.20342	0.379127
$362$	−37.5181	−1.97191
$363$	0	0
$364$	0.914776	0.0479473
$365$	−29.0069	−1.51829
$366$	0	0
$367$	0.571294	0.0298213	0.0149107	−	0.999889i	$-0.495254\pi$
$367$	0.571294	0.0298213	0.0149107	+	0.999889i	$0.495254\pi$
$368$	−18.4567	−0.962122
$369$	0	0
$370$	−15.9066	−0.826943
$371$	3.56940	0.185314
$372$	0	0
$373$	2.95332	0.152917	0.0764586	−	0.997073i	$-0.475639\pi$
$373$	2.95332	0.152917	0.0764586	+	0.997073i	$0.475639\pi$
$374$	17.8135	0.921115
$375$	0	0
$376$	−19.3636	−0.998603
$377$	−2.31243	−0.119096
$378$	0	0
$379$	−26.2705	−1.34942	−0.674711	−	0.738082i	$-0.735732\pi$
$379$	−26.2705	−1.34942	−0.674711	+	0.738082i	$0.735732\pi$
$380$	−5.98404	−0.306975
$381$	0	0
$382$	29.2646	1.49731
$383$	−3.44503	−0.176033	−0.0880164	−	0.996119i	$-0.528053\pi$
$383$	−3.44503	−0.176033	−0.0880164	+	0.996119i	$0.528053\pi$
$384$	0	0
$385$	−4.30800	−0.219556
$386$	−16.9996	−0.865257
$387$	0	0
$388$	−2.78565	−0.141420
$389$	7.95150	0.403157	0.201578	−	0.979472i	$-0.435393\pi$
$389$	7.95150	0.403157	0.201578	+	0.979472i	$0.435393\pi$
$390$	0	0
$391$	−9.54742	−0.482834
$392$	14.7138	0.743161
$393$	0	0
$394$	7.09790	0.357587
$395$	13.9656	0.702683
$396$	0	0
$397$	21.7255	1.09037	0.545186	−	0.838315i	$-0.316459\pi$
$397$	21.7255	1.09037	0.545186	+	0.838315i	$0.316459\pi$
$398$	22.8235	1.14404
$399$	0	0
$400$	8.60803	0.430402
$401$	−14.1072	−0.704481	−0.352241	−	0.935909i	$-0.614580\pi$
$401$	−14.1072	−0.704481	−0.352241	+	0.935909i	$0.614580\pi$
$402$	0	0
$403$	0.970607	0.0483494
$404$	2.42715	0.120755
$405$	0	0
$406$	−0.815614	−0.0404782
$407$	23.5624	1.16795
$408$	0	0
$409$	−10.2905	−0.508834	−0.254417	−	0.967095i	$-0.581884\pi$
$409$	−10.2905	−0.508834	−0.254417	+	0.967095i	$0.581884\pi$
$410$	−16.7613	−0.827783
$411$	0	0
$412$	1.84074	0.0906869
$413$	−1.88072	−0.0925444
$414$	0	0
$415$	29.4702	1.44663
$416$	−9.03366	−0.442912
$417$	0	0
$418$	36.1413	1.76773
$419$	−19.0965	−0.932925	−0.466463	−	0.884541i	$-0.654472\pi$
$419$	−19.0965	−0.932925	−0.466463	+	0.884541i	$0.654472\pi$
$420$	0	0
$421$	26.3025	1.28191	0.640953	−	0.767580i	$-0.278539\pi$
$421$	26.3025	1.28191	0.640953	+	0.767580i	$0.278539\pi$
$422$	−18.0009	−0.876271
$423$	0	0
$424$	−14.2051	−0.689862
$425$	4.45283	0.215994
$426$	0	0
$427$	−0.585969	−0.0283570
$428$	−6.56392	−0.317279
$429$	0	0
$430$	−26.9826	−1.30122
$431$	16.5246	0.795963	0.397982	−	0.917393i	$-0.369711\pi$
$431$	16.5246	0.795963	0.397982	+	0.917393i	$0.369711\pi$
$432$	0	0
$433$	4.52467	0.217442	0.108721	−	0.994072i	$-0.465325\pi$
$433$	4.52467	0.217442	0.108721	+	0.994072i	$0.465325\pi$
$434$	0.342341	0.0164329
$435$	0	0
$436$	−4.46351	−0.213763
$437$	−19.3705	−0.926615
$438$	0	0
$439$	−9.06783	−0.432784	−0.216392	−	0.976307i	$-0.569429\pi$
$439$	−9.06783	−0.432784	−0.216392	+	0.976307i	$0.569429\pi$
$440$	17.1446	0.817335
$441$	0	0
$442$	−10.4681	−0.497915
$443$	12.6542	0.601219	0.300609	−	0.953747i	$-0.402810\pi$
$443$	12.6542	0.601219	0.300609	+	0.953747i	$0.402810\pi$
$444$	0	0
$445$	5.57566	0.264312
$446$	−25.3135	−1.19863
$447$	0	0
$448$	2.20073	0.103975
$449$	−34.8336	−1.64390	−0.821950	−	0.569560i	$-0.807114\pi$
$449$	−34.8336	−1.64390	−0.821950	+	0.569560i	$0.807114\pi$
$450$	0	0
$451$	24.8286	1.16913
$452$	−1.82344	−0.0857674
$453$	0	0
$454$	−2.07678	−0.0974680
$455$	2.53158	0.118683
$456$	0	0
$457$	−14.3718	−0.672283	−0.336141	−	0.941812i	$-0.609122\pi$
$457$	−14.3718	−0.672283	−0.336141	+	0.941812i	$0.609122\pi$
$458$	−25.2696	−1.18077
$459$	0	0
$460$	4.42360	0.206252
$461$	20.9323	0.974914	0.487457	−	0.873147i	$-0.337925\pi$
$461$	20.9323	0.974914	0.487457	+	0.873147i	$0.337925\pi$
$462$	0	0
$463$	19.4204	0.902541	0.451271	−	0.892387i	$-0.350971\pi$
$463$	19.4204	0.902541	0.451271	+	0.892387i	$0.350971\pi$
$464$	4.42525	0.205437
$465$	0	0
$466$	45.2561	2.09645
$467$	−22.0847	−1.02196	−0.510980	−	0.859593i	$-0.670717\pi$
$467$	−22.0847	−1.02196	−0.510980	+	0.859593i	$0.670717\pi$
$468$	0	0
$469$	−0.140228	−0.00647512
$470$	25.7975	1.18995
$471$	0	0
$472$	7.48473	0.344512
$473$	39.9694	1.83780
$474$	0	0
$475$	9.03420	0.414518
$476$	−0.905557	−0.0415061
$477$	0	0
$478$	−1.95944	−0.0896226
$479$	−25.4118	−1.16109	−0.580547	−	0.814227i	$-0.697161\pi$
$479$	−25.4118	−1.16109	−0.580547	+	0.814227i	$0.697161\pi$
$480$	0	0
$481$	−13.8464	−0.631342
$482$	−41.6774	−1.89835
$483$	0	0
$484$	5.07671	0.230760
$485$	−7.70911	−0.350053
$486$	0	0
$487$	−36.7671	−1.66608	−0.833039	−	0.553214i	$-0.813401\pi$
$487$	−36.7671	−1.66608	−0.833039	+	0.553214i	$0.813401\pi$
$488$	2.33198	0.105564
$489$	0	0
$490$	−19.6027	−0.885562
$491$	12.6572	0.571212	0.285606	−	0.958347i	$-0.407805\pi$
$491$	12.6572	0.571212	0.285606	+	0.958347i	$0.407805\pi$
$492$	0	0
$493$	2.28913	0.103097
$494$	−21.2383	−0.955558
$495$	0	0
$496$	−1.85743	−0.0834011
$497$	−6.47829	−0.290591
$498$	0	0
$499$	2.39500	0.107215	0.0536074	−	0.998562i	$-0.482928\pi$
$499$	2.39500	0.107215	0.0536074	+	0.998562i	$0.482928\pi$
$500$	−7.90814	−0.353663
$501$	0	0
$502$	−21.5917	−0.963683
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	6.71697	0.298901
$506$	−26.7169	−1.18771
$507$	0	0
$508$	−9.69357	−0.430083
$509$	−6.46329	−0.286480	−0.143240	−	0.989688i	$-0.545752\pi$
$509$	−6.46329	−0.286480	−0.143240	+	0.989688i	$0.545752\pi$
$510$	0	0
$511$	−8.90589	−0.393973
$512$	−4.15104	−0.183452
$513$	0	0
$514$	26.2095	1.15605
$515$	5.09414	0.224475
$516$	0	0
$517$	−38.2139	−1.68065
$518$	−4.88374	−0.214579
$519$	0	0
$520$	−10.0750	−0.441816
$521$	−34.3036	−1.50287	−0.751434	−	0.659808i	$-0.770637\pi$
$521$	−34.3036	−1.50287	−0.751434	+	0.659808i	$0.770637\pi$
$522$	0	0
$523$	30.3662	1.32782	0.663910	−	0.747812i	$-0.268896\pi$
$523$	30.3662	1.32782	0.663910	+	0.747812i	$0.268896\pi$
$524$	−8.78754	−0.383885
$525$	0	0
$526$	−11.0261	−0.480761
$527$	−0.960825	−0.0418542
$528$	0	0
$529$	−8.68070	−0.377422
$530$	18.9250	0.822051
$531$	0	0
$532$	−1.83726	−0.0796552
$533$	−14.5904	−0.631982
$534$	0	0
$535$	−18.1652	−0.785351
$536$	0.558065	0.0241047
$537$	0	0
$538$	−38.2308	−1.64825
$539$	29.0376	1.25074
$540$	0	0
$541$	39.1371	1.68263	0.841317	−	0.540542i	$-0.181781\pi$
$541$	39.1371	1.68263	0.841317	+	0.540542i	$0.181781\pi$
$542$	28.6318	1.22984
$543$	0	0
$544$	8.94262	0.383412
$545$	−12.3525	−0.529121
$546$	0	0
$547$	24.0487	1.02825	0.514125	−	0.857715i	$-0.328117\pi$
$547$	24.0487	1.02825	0.514125	+	0.857715i	$0.328117\pi$
$548$	8.09034	0.345602
$549$	0	0
$550$	12.4605	0.531318
$551$	4.64434	0.197855
$552$	0	0
$553$	4.28779	0.182335
$554$	−18.4678	−0.784623
$555$	0	0
$556$	1.40023	0.0593832
$557$	17.2987	0.732971	0.366486	−	0.930424i	$-0.380561\pi$
$557$	17.2987	0.732971	0.366486	+	0.930424i	$0.380561\pi$
$558$	0	0
$559$	−23.4879	−0.993433
$560$	−4.84464	−0.204723
$561$	0	0
$562$	15.2490	0.643242
$563$	−5.63944	−0.237674	−0.118837	−	0.992914i	$-0.537917\pi$
$563$	−5.63944	−0.237674	−0.118837	+	0.992914i	$0.537917\pi$
$564$	0	0
$565$	−5.04626	−0.212298
$566$	43.7551	1.83917
$567$	0	0
$568$	25.7817	1.08177
$569$	12.0350	0.504535	0.252268	−	0.967658i	$-0.418824\pi$
$569$	12.0350	0.504535	0.252268	+	0.967658i	$0.418824\pi$
$570$	0	0
$571$	29.9404	1.25297	0.626484	−	0.779434i	$-0.284493\pi$
$571$	29.9404	1.25297	0.626484	+	0.779434i	$0.284493\pi$
$572$	−7.18456	−0.300402
$573$	0	0
$574$	−5.14617	−0.214797
$575$	−6.67839	−0.278508
$576$	0	0
$577$	17.6080	0.733032	0.366516	−	0.930412i	$-0.380551\pi$
$577$	17.6080	0.733032	0.366516	+	0.930412i	$0.380551\pi$
$578$	−17.3111	−0.720046
$579$	0	0
$580$	−1.06062	−0.0440399
$581$	9.04811	0.375379
$582$	0	0
$583$	−28.0337	−1.16104
$584$	35.4428	1.46663
$585$	0	0
$586$	−28.0587	−1.15909
$587$	21.2268	0.876125	0.438062	−	0.898945i	$-0.355665\pi$
$587$	21.2268	0.876125	0.438062	+	0.898945i	$0.355665\pi$
$588$	0	0
$589$	−1.94939	−0.0803231
$590$	−9.97165	−0.410526
$591$	0	0
$592$	26.4976	1.08904
$593$	−1.78783	−0.0734175	−0.0367087	−	0.999326i	$-0.511687\pi$
$593$	−1.78783	−0.0734175	−0.0367087	+	0.999326i	$0.511687\pi$
$594$	0	0
$595$	−2.50607	−0.102739
$596$	5.58457	0.228753
$597$	0	0
$598$	15.7001	0.642025
$599$	4.15199	0.169646	0.0848228	−	0.996396i	$-0.472968\pi$
$599$	4.15199	0.169646	0.0848228	+	0.996396i	$0.472968\pi$
$600$	0	0
$601$	−4.87221	−0.198742	−0.0993708	−	0.995050i	$-0.531683\pi$
$601$	−4.87221	−0.198742	−0.0993708	+	0.995050i	$0.531683\pi$
$602$	−8.28438	−0.337646
$603$	0	0
$604$	0.343075	0.0139595
$605$	14.0495	0.571192
$606$	0	0
$607$	−22.0579	−0.895303	−0.447652	−	0.894208i	$-0.647739\pi$
$607$	−22.0579	−0.895303	−0.447652	+	0.894208i	$0.647739\pi$
$608$	18.1434	0.735812
$609$	0	0
$610$	−3.10682	−0.125792
$611$	22.4563	0.908484
$612$	0	0
$613$	−19.0369	−0.768895	−0.384447	−	0.923147i	$-0.625608\pi$
$613$	−19.0369	−0.768895	−0.384447	+	0.923147i	$0.625608\pi$
$614$	47.2081	1.90516
$615$	0	0
$616$	5.26383	0.212086
$617$	22.0697	0.888494	0.444247	−	0.895904i	$-0.353471\pi$
$617$	22.0697	0.888494	0.444247	+	0.895904i	$0.353471\pi$
$618$	0	0
$619$	−5.69372	−0.228850	−0.114425	−	0.993432i	$-0.536503\pi$
$619$	−5.69372	−0.228850	−0.114425	+	0.993432i	$0.536503\pi$
$620$	0.445179	0.0178788
$621$	0	0
$622$	−44.3287	−1.77742
$623$	1.71187	0.0685848
$624$	0	0
$625$	−13.0609	−0.522438
$626$	9.00303	0.359833
$627$	0	0
$628$	7.26050	0.289725
$629$	13.7069	0.546528
$630$	0	0
$631$	37.6327	1.49813	0.749067	−	0.662494i	$-0.230502\pi$
$631$	37.6327	1.49813	0.749067	+	0.662494i	$0.230502\pi$
$632$	−17.0641	−0.678775
$633$	0	0
$634$	−50.7941	−2.01729
$635$	−26.8263	−1.06457
$636$	0	0
$637$	−17.0639	−0.676095
$638$	6.40575	0.253606
$639$	0	0
$640$	24.4185	0.965226
$641$	−6.63550	−0.262086	−0.131043	−	0.991377i	$-0.541833\pi$
$641$	−6.63550	−0.262086	−0.131043	+	0.991377i	$0.541833\pi$
$642$	0	0
$643$	−29.4970	−1.16325	−0.581625	−	0.813457i	$-0.697583\pi$
$643$	−29.4970	−1.16325	−0.581625	+	0.813457i	$0.697583\pi$
$644$	1.35816	0.0535191
$645$	0	0
$646$	21.0243	0.827190
$647$	45.6726	1.79558	0.897788	−	0.440427i	$-0.145173\pi$
$647$	45.6726	1.79558	0.897788	+	0.440427i	$0.145173\pi$
$648$	0	0
$649$	14.7710	0.579813
$650$	−7.32238	−0.287207
$651$	0	0
$652$	13.0058	0.509346
$653$	31.6691	1.23931	0.619655	−	0.784875i	$-0.287273\pi$
$653$	31.6691	1.23931	0.619655	+	0.784875i	$0.287273\pi$
$654$	0	0
$655$	−24.3190	−0.950220
$656$	27.9214	1.09015
$657$	0	0
$658$	7.92052	0.308774
$659$	−9.73186	−0.379099	−0.189550	−	0.981871i	$-0.560703\pi$
$659$	−9.73186	−0.379099	−0.189550	+	0.981871i	$0.560703\pi$
$660$	0	0
$661$	−14.5544	−0.566100	−0.283050	−	0.959105i	$-0.591346\pi$
$661$	−14.5544	−0.566100	−0.283050	+	0.959105i	$0.591346\pi$
$662$	27.3522	1.06307
$663$	0	0
$664$	−36.0088	−1.39741
$665$	−5.08449	−0.197168
$666$	0	0
$667$	−3.43325	−0.132936
$668$	7.76803	0.300554
$669$	0	0
$670$	−0.743491	−0.0287236
$671$	4.60214	0.177664
$672$	0	0
$673$	−46.1914	−1.78055	−0.890273	−	0.455427i	$-0.849486\pi$
$673$	−46.1914	−1.78055	−0.890273	+	0.455427i	$0.849486\pi$
$674$	12.3383	0.475252
$675$	0	0
$676$	−4.22715	−0.162583
$677$	10.6087	0.407725	0.203863	−	0.978999i	$-0.434650\pi$
$677$	10.6087	0.407725	0.203863	+	0.978999i	$0.434650\pi$
$678$	0	0
$679$	−2.36690	−0.0908333
$680$	9.97342	0.382463
$681$	0	0
$682$	−2.68871	−0.102956
$683$	35.9985	1.37745	0.688723	−	0.725025i	$-0.258172\pi$
$683$	35.9985	1.37745	0.688723	+	0.725025i	$0.258172\pi$
$684$	0	0
$685$	22.3895	0.855458
$686$	−12.3113	−0.470046
$687$	0	0
$688$	44.9483	1.71364
$689$	16.4739	0.627606
$690$	0	0
$691$	24.7115	0.940070	0.470035	−	0.882648i	$-0.344241\pi$
$691$	24.7115	0.940070	0.470035	+	0.882648i	$0.344241\pi$
$692$	−2.04444	−0.0777178
$693$	0	0
$694$	45.7192	1.73548
$695$	3.87506	0.146989
$696$	0	0
$697$	14.4434	0.547083
$698$	18.9206	0.716155
$699$	0	0
$700$	−0.633434	−0.0239416
$701$	−28.8712	−1.09045	−0.545225	−	0.838290i	$-0.683556\pi$
$701$	−28.8712	−1.09045	−0.545225	+	0.838290i	$0.683556\pi$
$702$	0	0
$703$	27.8094	1.04885
$704$	−17.2843	−0.651428
$705$	0	0
$706$	10.0367	0.377735
$707$	2.06229	0.0775603
$708$	0	0
$709$	−31.6656	−1.18923	−0.594614	−	0.804012i	$-0.702695\pi$
$709$	−31.6656	−1.18923	−0.594614	+	0.804012i	$0.702695\pi$
$710$	−34.3481	−1.28906
$711$	0	0
$712$	−6.81275	−0.255319
$713$	1.44105	0.0539679
$714$	0	0
$715$	−19.8828	−0.743575
$716$	3.91322	0.146244
$717$	0	0
$718$	16.9129	0.631183
$719$	−30.1524	−1.12449	−0.562247	−	0.826969i	$-0.690063\pi$
$719$	−30.1524	−1.12449	−0.562247	+	0.826969i	$0.690063\pi$
$720$	0	0
$721$	1.56403	0.0582477
$722$	11.7262	0.436403
$723$	0	0
$724$	−14.9793	−0.556702
$725$	1.60124	0.0594685
$726$	0	0
$727$	15.1905	0.563384	0.281692	−	0.959505i	$-0.409104\pi$
$727$	15.1905	0.563384	0.281692	+	0.959505i	$0.409104\pi$
$728$	−3.09328	−0.114644
$729$	0	0
$730$	−47.2193	−1.74766
$731$	23.2512	0.859977
$732$	0	0
$733$	39.8270	1.47104	0.735522	−	0.677501i	$-0.236937\pi$
$733$	39.8270	1.47104	0.735522	+	0.677501i	$0.236937\pi$
$734$	0.929988	0.0343265
$735$	0	0
$736$	−13.4122	−0.494381
$737$	1.10134	0.0405682
$738$	0	0
$739$	−26.4367	−0.972489	−0.486245	−	0.873823i	$-0.661634\pi$
$739$	−26.4367	−0.972489	−0.486245	+	0.873823i	$0.661634\pi$
$740$	−6.35079	−0.233460
$741$	0	0
$742$	5.81048	0.213310
$743$	26.5745	0.974923	0.487462	−	0.873144i	$-0.337923\pi$
$743$	26.5745	0.974923	0.487462	+	0.873144i	$0.337923\pi$
$744$	0	0
$745$	15.4549	0.566225
$746$	4.80760	0.176019
$747$	0	0
$748$	7.11215	0.260046
$749$	−5.57720	−0.203787
$750$	0	0
$751$	−30.1977	−1.10193	−0.550964	−	0.834529i	$-0.685740\pi$
$751$	−30.1977	−1.10193	−0.550964	+	0.834529i	$0.685740\pi$
$752$	−42.9741	−1.56711
$753$	0	0
$754$	−3.76432	−0.137088
$755$	0.949439	0.0345536
$756$	0	0
$757$	31.3606	1.13982	0.569911	−	0.821706i	$-0.306978\pi$
$757$	31.3606	1.13982	0.569911	+	0.821706i	$0.306978\pi$
$758$	−42.7647	−1.55328
$759$	0	0
$760$	20.2348	0.733992
$761$	−40.3426	−1.46242	−0.731210	−	0.682153i	$-0.761044\pi$
$761$	−40.3426	−1.46242	−0.731210	+	0.682153i	$0.761044\pi$
$762$	0	0
$763$	−3.79253	−0.137299
$764$	11.6841	0.422715
$765$	0	0
$766$	−5.60803	−0.202626
$767$	−8.68015	−0.313422
$768$	0	0
$769$	−20.4304	−0.736739	−0.368369	−	0.929680i	$-0.620084\pi$
$769$	−20.4304	−0.736739	−0.368369	+	0.929680i	$0.620084\pi$
$770$	−7.01283	−0.252725
$771$	0	0
$772$	−6.78719	−0.244276
$773$	−45.6859	−1.64321	−0.821605	−	0.570058i	$-0.806921\pi$
$773$	−45.6859	−1.64321	−0.821605	+	0.570058i	$0.806921\pi$
$774$	0	0
$775$	−0.672094	−0.0241423
$776$	9.41956	0.338142
$777$	0	0
$778$	12.9439	0.464063
$779$	29.3038	1.04992
$780$	0	0
$781$	50.8798	1.82062
$782$	−15.5419	−0.555776
$783$	0	0
$784$	32.6547	1.16624
$785$	20.0930	0.717149
$786$	0	0
$787$	15.1038	0.538392	0.269196	−	0.963085i	$-0.413242\pi$
$787$	15.1038	0.538392	0.269196	+	0.963085i	$0.413242\pi$
$788$	2.83388	0.100953
$789$	0	0
$790$	22.7340	0.808839
$791$	−1.54933	−0.0550880
$792$	0	0
$793$	−2.70444	−0.0960373
$794$	35.3661	1.25510
$795$	0	0
$796$	9.11240	0.322980
$797$	−13.0697	−0.462954	−0.231477	−	0.972840i	$-0.574356\pi$
$797$	−13.0697	−0.462954	−0.231477	+	0.972840i	$0.574356\pi$
$798$	0	0
$799$	−22.2300	−0.786440
$800$	6.25533	0.221159
$801$	0	0
$802$	−22.9646	−0.810909
$803$	69.9460	2.46834
$804$	0	0
$805$	3.75863	0.132474
$806$	1.58001	0.0556536
$807$	0	0
$808$	−8.20729	−0.288732
$809$	19.0282	0.668996	0.334498	−	0.942396i	$-0.391433\pi$
$809$	19.0282	0.668996	0.334498	+	0.942396i	$0.391433\pi$
$810$	0	0
$811$	21.0749	0.740042	0.370021	−	0.929023i	$-0.379351\pi$
$811$	21.0749	0.740042	0.370021	+	0.929023i	$0.379351\pi$
$812$	−0.325638	−0.0114277
$813$	0	0
$814$	38.3564	1.34439
$815$	35.9927	1.26077
$816$	0	0
$817$	47.1736	1.65040
$818$	−16.7516	−0.585705
$819$	0	0
$820$	−6.69206	−0.233697
$821$	−10.5704	−0.368909	−0.184454	−	0.982841i	$-0.559052\pi$
$821$	−10.5704	−0.368909	−0.184454	+	0.982841i	$0.559052\pi$
$822$	0	0
$823$	40.5293	1.41276	0.706381	−	0.707832i	$-0.250327\pi$
$823$	40.5293	1.41276	0.706381	+	0.707832i	$0.250327\pi$
$824$	−6.22439	−0.216837
$825$	0	0
$826$	−3.06156	−0.106525
$827$	47.1057	1.63803	0.819013	−	0.573775i	$-0.194522\pi$
$827$	47.1057	1.63803	0.819013	+	0.573775i	$0.194522\pi$
$828$	0	0
$829$	−9.69487	−0.336717	−0.168358	−	0.985726i	$-0.553847\pi$
$829$	−9.69487	−0.336717	−0.168358	+	0.985726i	$0.553847\pi$
$830$	47.9733	1.66518
$831$	0	0
$832$	10.1571	0.352134
$833$	16.8919	0.585269
$834$	0	0
$835$	21.4975	0.743953
$836$	14.4296	0.499059
$837$	0	0
$838$	−31.0865	−1.07386
$839$	−29.2517	−1.00988	−0.504940	−	0.863154i	$-0.668485\pi$
$839$	−29.2517	−1.00988	−0.504940	+	0.863154i	$0.668485\pi$
$840$	0	0
$841$	−28.1768	−0.971615
$842$	42.8168	1.47557
$843$	0	0
$844$	−7.18697	−0.247386
$845$	−11.6984	−0.402436
$846$	0	0
$847$	4.31356	0.148216
$848$	−31.5258	−1.08260
$849$	0	0
$850$	7.24858	0.248624
$851$	−20.5577	−0.704708
$852$	0	0
$853$	−31.0610	−1.06351	−0.531754	−	0.846899i	$-0.678467\pi$
$853$	−31.0610	−1.06351	−0.531754	+	0.846899i	$0.678467\pi$
$854$	−0.953876	−0.0326410
$855$	0	0
$856$	22.1956	0.758630
$857$	27.3567	0.934486	0.467243	−	0.884129i	$-0.345247\pi$
$857$	27.3567	0.934486	0.467243	+	0.884129i	$0.345247\pi$
$858$	0	0
$859$	−37.0124	−1.26285	−0.631424	−	0.775438i	$-0.717529\pi$
$859$	−37.0124	−1.26285	−0.631424	+	0.775438i	$0.717529\pi$
$860$	−10.7730	−0.367355
$861$	0	0
$862$	26.8998	0.916211
$863$	37.6253	1.28078	0.640389	−	0.768051i	$-0.278773\pi$
$863$	37.6253	1.28078	0.640389	+	0.768051i	$0.278773\pi$
$864$	0	0
$865$	−5.65785	−0.192373
$866$	7.36554	0.250291
$867$	0	0
$868$	0.136682	0.00463928
$869$	−33.6759	−1.14238
$870$	0	0
$871$	−0.647196	−0.0219294
$872$	15.0932	0.511119
$873$	0	0
$874$	−31.5324	−1.06660
$875$	−6.71935	−0.227156
$876$	0	0
$877$	35.6727	1.20458	0.602291	−	0.798277i	$-0.294255\pi$
$877$	35.6727	1.20458	0.602291	+	0.798277i	$0.294255\pi$
$878$	−14.7612	−0.498165
$879$	0	0
$880$	38.0493	1.28264
$881$	30.3930	1.02397	0.511984	−	0.858995i	$-0.328911\pi$
$881$	30.3930	1.02397	0.511984	+	0.858995i	$0.328911\pi$
$882$	0	0
$883$	23.6017	0.794260	0.397130	−	0.917762i	$-0.370006\pi$
$883$	23.6017	0.794260	0.397130	+	0.917762i	$0.370006\pi$
$884$	−4.17944	−0.140570
$885$	0	0
$886$	20.5993	0.692046
$887$	18.1804	0.610437	0.305219	−	0.952282i	$-0.401270\pi$
$887$	18.1804	0.610437	0.305219	+	0.952282i	$0.401270\pi$
$888$	0	0
$889$	−8.23639	−0.276240
$890$	9.07640	0.304242
$891$	0	0
$892$	−10.1066	−0.338393
$893$	−45.1017	−1.50927
$894$	0	0
$895$	10.8296	0.361993
$896$	7.49713	0.250461
$897$	0	0
$898$	−56.7043	−1.89225
$899$	−0.345513	−0.0115235
$900$	0	0
$901$	−16.3079	−0.543294
$902$	40.4175	1.34576
$903$	0	0
$904$	6.16589	0.205074
$905$	−41.4543	−1.37799
$906$	0	0
$907$	−26.8615	−0.891921	−0.445961	−	0.895053i	$-0.647138\pi$
$907$	−26.8615	−0.891921	−0.445961	+	0.895053i	$0.647138\pi$
$908$	−0.829166	−0.0275168
$909$	0	0
$910$	4.12107	0.136612
$911$	−42.7019	−1.41478	−0.707388	−	0.706825i	$-0.750127\pi$
$911$	−42.7019	−1.41478	−0.707388	+	0.706825i	$0.750127\pi$
$912$	0	0
$913$	−71.0630	−2.35184
$914$	−23.3952	−0.773846
$915$	0	0
$916$	−10.0891	−0.333352
$917$	−7.46656	−0.246568
$918$	0	0
$919$	−7.59767	−0.250624	−0.125312	−	0.992117i	$-0.539993\pi$
$919$	−7.59767	−0.250624	−0.125312	+	0.992117i	$0.539993\pi$
$920$	−14.9582	−0.493158
$921$	0	0
$922$	34.0749	1.12220
$923$	−29.8994	−0.984151
$924$	0	0
$925$	9.58790	0.315248
$926$	31.6137	1.03889
$927$	0	0
$928$	3.21577	0.105563
$929$	34.5765	1.13442	0.567210	−	0.823574i	$-0.308023\pi$
$929$	34.5765	1.13442	0.567210	+	0.823574i	$0.308023\pi$
$930$	0	0
$931$	34.2714	1.12320
$932$	18.0688	0.591862
$933$	0	0
$934$	−35.9509	−1.17635
$935$	19.6824	0.643684
$936$	0	0
$937$	40.4353	1.32096	0.660482	−	0.750842i	$-0.270352\pi$
$937$	40.4353	1.32096	0.660482	+	0.750842i	$0.270352\pi$
$938$	−0.228271	−0.00745333
$939$	0	0
$940$	10.2998	0.335943
$941$	22.3193	0.727587	0.363794	−	0.931480i	$-0.381481\pi$
$941$	22.3193	0.727587	0.363794	+	0.931480i	$0.381481\pi$
$942$	0	0
$943$	−21.6623	−0.705423
$944$	16.6110	0.540643
$945$	0	0
$946$	65.0647	2.11543
$947$	40.0277	1.30073	0.650363	−	0.759623i	$-0.274617\pi$
$947$	40.0277	1.30073	0.650363	+	0.759623i	$0.274617\pi$
$948$	0	0
$949$	−41.1036	−1.33428
$950$	14.7064	0.477140
$951$	0	0
$952$	3.06210	0.0992433
$953$	6.33099	0.205081	0.102540	−	0.994729i	$-0.467303\pi$
$953$	6.33099	0.205081	0.102540	+	0.994729i	$0.467303\pi$
$954$	0	0
$955$	32.3349	1.04633
$956$	−0.782317	−0.0253019
$957$	0	0
$958$	−41.3669	−1.33650
$959$	6.87416	0.221978
$960$	0	0
$961$	−30.8550	−0.995322
$962$	−22.5400	−0.726720
$963$	0	0
$964$	−16.6399	−0.535936
$965$	−18.7831	−0.604650
$966$	0	0
$967$	−12.8388	−0.412867	−0.206433	−	0.978461i	$-0.566186\pi$
$967$	−12.8388	−0.412867	−0.206433	+	0.978461i	$0.566186\pi$
$968$	−17.1667	−0.551758
$969$	0	0
$970$	−12.5494	−0.402936
$971$	−22.1505	−0.710842	−0.355421	−	0.934706i	$-0.615662\pi$
$971$	−22.1505	−0.710842	−0.355421	+	0.934706i	$0.615662\pi$
$972$	0	0
$973$	1.18975	0.0381415
$974$	−59.8518	−1.91778
$975$	0	0
$976$	5.17542	0.165661
$977$	−41.0297	−1.31266	−0.656328	−	0.754475i	$-0.727891\pi$
$977$	−41.0297	−1.31266	−0.656328	+	0.754475i	$0.727891\pi$
$978$	0	0
$979$	−13.4449	−0.429701
$980$	−7.82651	−0.250009
$981$	0	0
$982$	20.6042	0.657506
$983$	−43.9691	−1.40240	−0.701199	−	0.712966i	$-0.747351\pi$
$983$	−43.9691	−1.40240	−0.701199	+	0.712966i	$0.747351\pi$
$984$	0	0
$985$	7.84258	0.249885
$986$	3.72638	0.118672
$987$	0	0
$988$	−8.47953	−0.269770
$989$	−34.8724	−1.10888
$990$	0	0
$991$	20.6965	0.657446	0.328723	−	0.944426i	$-0.393382\pi$
$991$	20.6965	0.657446	0.328723	+	0.944426i	$0.393382\pi$
$992$	−1.34977	−0.0428552
$993$	0	0
$994$	−10.5458	−0.334491
$995$	25.2180	0.799464
$996$	0	0
$997$	52.9018	1.67542	0.837709	−	0.546117i	$-0.183895\pi$
$997$	52.9018	1.67542	0.837709	+	0.546117i	$0.183895\pi$
$998$	3.89873	0.123412
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.8		10
3.2	odd	2		503.2.a.e.1.3	✓	10
12.11	even	2		8048.2.a.p.1.6		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.3	✓	10	3.2	odd	2
4527.2.a.k.1.8		10	1.1	even	1	trivial
8048.2.a.p.1.6		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)

\(f(q)\)	\(=\)	\(q+1.62786 q^{2} +0.649933 q^{4} +1.79865 q^{5} +0.552233 q^{7} -2.19772 q^{8} +O(q^{10})\)	\(q+1.62786 q^{2} +0.649933 q^{4} +1.79865 q^{5} +0.552233 q^{7} -2.19772 q^{8} +2.92795 q^{10} -4.33718 q^{11} +2.54873 q^{13} +0.898958 q^{14} -4.87745 q^{16} -2.52304 q^{17} -5.11893 q^{19} +1.16900 q^{20} -7.06033 q^{22} +3.78409 q^{23} -1.76486 q^{25} +4.14898 q^{26} +0.358914 q^{28} -0.907287 q^{29} +0.380820 q^{31} -3.54438 q^{32} -4.10717 q^{34} +0.993273 q^{35} -5.43266 q^{37} -8.33291 q^{38} -3.95293 q^{40} -5.72459 q^{41} -9.21553 q^{43} -2.81888 q^{44} +6.15997 q^{46} +8.81077 q^{47} -6.69504 q^{49} -2.87295 q^{50} +1.65651 q^{52} +6.46357 q^{53} -7.80106 q^{55} -1.21365 q^{56} -1.47694 q^{58} -3.40568 q^{59} -1.06109 q^{61} +0.619922 q^{62} +3.98515 q^{64} +4.58427 q^{65} -0.253929 q^{67} -1.63981 q^{68} +1.61691 q^{70} -11.7311 q^{71} -16.1271 q^{73} -8.84362 q^{74} -3.32696 q^{76} -2.39513 q^{77} +7.76447 q^{79} -8.77283 q^{80} -9.31884 q^{82} +16.3846 q^{83} -4.53807 q^{85} -15.0016 q^{86} +9.53191 q^{88} +3.09992 q^{89} +1.40749 q^{91} +2.45940 q^{92} +14.3427 q^{94} -9.20715 q^{95} -4.28605 q^{97} -10.8986 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 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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