LMFDB - Embedded newform 1521.4.a.be.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,4,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1521.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:	$89.7419051187$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 48x^{7} + 29x^{6} + 772x^{5} - 150x^{4} - 4745x^{3} - 966x^{2} + 9428x + 5144 $ x^9 - x^8 - 48x^7 + 29x^6 + 772x^5 - 150x^4 - 4745x^3 - 966x^2 + 9428*x + 5144
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.614643$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61464 q^{2} -5.39293 q^{4} -1.20859 q^{5} +28.2769 q^{7} +21.6248 q^{8} +O(q^{10})$	$q-1.61464 q^{2} -5.39293 q^{4} -1.20859 q^{5} +28.2769 q^{7} +21.6248 q^{8} +1.95145 q^{10} +31.7768 q^{11} -45.6572 q^{14} +8.22709 q^{16} +16.0963 q^{17} -58.5273 q^{19} +6.51785 q^{20} -51.3081 q^{22} -152.860 q^{23} -123.539 q^{25} -152.496 q^{28} -265.310 q^{29} -56.9241 q^{31} -186.282 q^{32} -25.9899 q^{34} -34.1753 q^{35} +444.864 q^{37} +94.5007 q^{38} -26.1356 q^{40} +189.276 q^{41} -132.752 q^{43} -171.370 q^{44} +246.814 q^{46} -113.693 q^{47} +456.586 q^{49} +199.472 q^{50} -300.506 q^{53} -38.4051 q^{55} +611.483 q^{56} +428.381 q^{58} -513.561 q^{59} +619.902 q^{61} +91.9121 q^{62} +234.963 q^{64} +597.308 q^{67} -86.8064 q^{68} +55.1809 q^{70} -826.673 q^{71} -332.560 q^{73} -718.297 q^{74} +315.633 q^{76} +898.549 q^{77} +679.621 q^{79} -9.94319 q^{80} -305.614 q^{82} -88.2682 q^{83} -19.4539 q^{85} +214.347 q^{86} +687.166 q^{88} -1484.39 q^{89} +824.363 q^{92} +183.574 q^{94} +70.7356 q^{95} +154.995 q^{97} -737.223 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8}+O(q^{10}) $ 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8	$ 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8} + 198 q^{10} - 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} - 72 q^{19} - 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} + 132 q^{28} + 547 q^{29} - 521 q^{31} - 721 q^{32} - 100 q^{34} - 138 q^{35} + 584 q^{37} + 416 q^{38} + 1342 q^{40} - 482 q^{41} + 158 q^{43} - 1453 q^{44} + 1537 q^{46} - 1500 q^{47} + 642 q^{49} - 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} + 1455 q^{58} - 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} + 252 q^{67} + 1579 q^{68} + 2492 q^{70} - 2352 q^{71} + 903 q^{73} - 1037 q^{74} - 485 q^{76} + 1686 q^{77} - 115 q^{79} - 5701 q^{80} - 5147 q^{82} - 1207 q^{83} + 4308 q^{85} - 5691 q^{86} - 484 q^{88} - 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} + 2155 q^{97} - 5593 q^{98}+O(q^{100}) $ 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8 + 198 * q^10 - 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 - 72 * q^19 - 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 + 132 * q^28 + 547 * q^29 - 521 * q^31 - 721 * q^32 - 100 * q^34 - 138 * q^35 + 584 * q^37 + 416 * q^38 + 1342 * q^40 - 482 * q^41 + 158 * q^43 - 1453 * q^44 + 1537 * q^46 - 1500 * q^47 + 642 * q^49 - 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 + 1455 * q^58 - 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 + 252 * q^67 + 1579 * q^68 + 2492 * q^70 - 2352 * q^71 + 903 * q^73 - 1037 * q^74 - 485 * q^76 + 1686 * q^77 - 115 * q^79 - 5701 * q^80 - 5147 * q^82 - 1207 * q^83 + 4308 * q^85 - 5691 * q^86 - 484 * q^88 - 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 + 2155 * q^97 - 5593 * 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 $

$2$	−1.61464	−0.570863	−0.285431	−	0.958399i	$-0.592137\pi$
$2$	−1.61464	−0.570863	−0.285431	+	0.958399i	$0.592137\pi$
$3$	0	0
$3$	0	0
$4$	−5.39293	−0.674116
$5$	−1.20859	−0.108100	−0.0540499	−	0.998538i	$-0.517213\pi$
$5$	−1.20859	−0.108100	−0.0540499	+	0.998538i	$0.517213\pi$
$6$	0	0
$7$	28.2769	1.52681	0.763406	−	0.645919i	$-0.223526\pi$
$7$	28.2769	1.52681	0.763406	+	0.645919i	$0.223526\pi$
$8$	21.6248	0.955690
$9$	0	0
$10$	1.95145	0.0617101
$11$	31.7768	0.871005	0.435502	−	0.900188i	$-0.356571\pi$
$11$	31.7768	0.871005	0.435502	+	0.900188i	$0.356571\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−45.6572	−0.871600
$15$	0	0
$16$	8.22709	0.128548
$17$	16.0963	0.229643	0.114822	−	0.993386i	$-0.463370\pi$
$17$	16.0963	0.229643	0.114822	+	0.993386i	$0.463370\pi$
$18$	0	0
$19$	−58.5273	−0.706688	−0.353344	−	0.935493i	$-0.614956\pi$
$19$	−58.5273	−0.706688	−0.353344	+	0.935493i	$0.614956\pi$
$20$	6.51785	0.0728718
$21$	0	0
$22$	−51.3081	−0.497224
$23$	−152.860	−1.38580	−0.692902	−	0.721031i	$-0.743668\pi$
$23$	−152.860	−1.38580	−0.692902	+	0.721031i	$0.743668\pi$
$24$	0	0
$25$	−123.539	−0.988314
$26$	0	0
$27$	0	0
$28$	−152.496	−1.02925
$29$	−265.310	−1.69886	−0.849429	−	0.527703i	$-0.823053\pi$
$29$	−265.310	−1.69886	−0.849429	+	0.527703i	$0.823053\pi$
$30$	0	0
$31$	−56.9241	−0.329802	−0.164901	−	0.986310i	$-0.552731\pi$
$31$	−56.9241	−0.329802	−0.164901	+	0.986310i	$0.552731\pi$
$32$	−186.282	−1.02907
$33$	0	0
$34$	−25.9899	−0.131095
$35$	−34.1753	−0.165048
$36$	0	0
$37$	444.864	1.97663	0.988314	−	0.152434i	$-0.0487111\pi$
$37$	444.864	1.97663	0.988314	+	0.152434i	$0.0487111\pi$
$38$	94.5007	0.403422
$39$	0	0
$40$	−26.1356	−0.103310
$41$	189.276	0.720976	0.360488	−	0.932764i	$-0.382610\pi$
$41$	189.276	0.720976	0.360488	+	0.932764i	$0.382610\pi$
$42$	0	0
$43$	−132.752	−0.470802	−0.235401	−	0.971898i	$-0.575640\pi$
$43$	−132.752	−0.470802	−0.235401	+	0.971898i	$0.575640\pi$
$44$	−171.370	−0.587158
$45$	0	0
$46$	246.814	0.791104
$47$	−113.693	−0.352847	−0.176424	−	0.984314i	$-0.556453\pi$
$47$	−113.693	−0.352847	−0.176424	+	0.984314i	$0.556453\pi$
$48$	0	0
$49$	456.586	1.33115
$50$	199.472	0.564192
$51$	0	0
$52$	0	0
$53$	−300.506	−0.778824	−0.389412	−	0.921064i	$-0.627322\pi$
$53$	−300.506	−0.778824	−0.389412	+	0.921064i	$0.627322\pi$
$54$	0	0
$55$	−38.4051	−0.0941554
$56$	611.483	1.45916
$57$	0	0
$58$	428.381	0.969814
$59$	−513.561	−1.13322	−0.566610	−	0.823986i	$-0.691745\pi$
$59$	−513.561	−1.13322	−0.566610	+	0.823986i	$0.691745\pi$
$60$	0	0
$61$	619.902	1.30115	0.650577	−	0.759441i	$-0.274527\pi$
$61$	619.902	1.30115	0.650577	+	0.759441i	$0.274527\pi$
$62$	91.9121	0.188272
$63$	0	0
$64$	234.963	0.458911
$65$	0	0
$66$	0	0
$67$	597.308	1.08915	0.544573	−	0.838713i	$-0.316692\pi$
$67$	597.308	1.08915	0.544573	+	0.838713i	$0.316692\pi$
$68$	−86.8064	−0.154806
$69$	0	0
$70$	55.1809	0.0942197
$71$	−826.673	−1.38180	−0.690902	−	0.722949i	$-0.742786\pi$
$71$	−826.673	−1.38180	−0.690902	+	0.722949i	$0.742786\pi$
$72$	0	0
$73$	−332.560	−0.533194	−0.266597	−	0.963808i	$-0.585899\pi$
$73$	−332.560	−0.533194	−0.266597	+	0.963808i	$0.585899\pi$
$74$	−718.297	−1.12838
$75$	0	0
$76$	315.633	0.476390
$77$	898.549	1.32986
$78$	0	0
$79$	679.621	0.967891	0.483945	−	0.875098i	$-0.339203\pi$
$79$	679.621	0.967891	0.483945	+	0.875098i	$0.339203\pi$
$80$	−9.94319	−0.0138960
$81$	0	0
$82$	−305.614	−0.411578
$83$	−88.2682	−0.116731	−0.0583656	−	0.998295i	$-0.518589\pi$
$83$	−88.2682	−0.116731	−0.0583656	+	0.998295i	$0.518589\pi$
$84$	0	0
$85$	−19.4539	−0.0248244
$86$	214.347	0.268763
$87$	0	0
$88$	687.166	0.832411
$89$	−1484.39	−1.76792	−0.883959	−	0.467564i	$-0.845132\pi$
$89$	−1484.39	−1.76792	−0.883959	+	0.467564i	$0.845132\pi$
$90$	0	0
$91$	0	0
$92$	824.363	0.934193
$93$	0	0
$94$	183.574	0.201427
$95$	70.7356	0.0763929
$96$	0	0
$97$	154.995	0.162241	0.0811206	−	0.996704i	$-0.474150\pi$
$97$	154.995	0.162241	0.0811206	+	0.996704i	$0.474150\pi$
$98$	−737.223	−0.759906
$99$	0	0
$100$	666.238	0.666238
$101$	1963.48	1.93440	0.967198	−	0.254025i	$-0.0817544\pi$
$101$	1963.48	1.93440	0.967198	+	0.254025i	$0.0817544\pi$
$102$	0	0
$103$	−1620.75	−1.55046	−0.775228	−	0.631682i	$-0.782365\pi$
$103$	−1620.75	−1.55046	−0.775228	+	0.631682i	$0.782365\pi$
$104$	0	0
$105$	0	0
$106$	485.210	0.444601
$107$	−321.126	−0.290135	−0.145068	−	0.989422i	$-0.546340\pi$
$107$	−321.126	−0.290135	−0.145068	+	0.989422i	$0.546340\pi$
$108$	0	0
$109$	−1305.27	−1.14699	−0.573495	−	0.819209i	$-0.694413\pi$
$109$	−1305.27	−1.14699	−0.573495	+	0.819209i	$0.694413\pi$
$110$	62.0106	0.0537498
$111$	0	0
$112$	232.637	0.196269
$113$	1086.88	0.904820	0.452410	−	0.891810i	$-0.350564\pi$
$113$	1086.88	0.904820	0.452410	+	0.891810i	$0.350564\pi$
$114$	0	0
$115$	184.745	0.149805
$116$	1430.80	1.14523
$117$	0	0
$118$	829.217	0.646912
$119$	455.155	0.350622
$120$	0	0
$121$	−321.238	−0.241351
$122$	−1000.92	−0.742780
$123$	0	0
$124$	306.988	0.222325
$125$	300.383	0.214936
$126$	0	0
$127$	1326.41	0.926774	0.463387	−	0.886156i	$-0.346634\pi$
$127$	1326.41	0.926774	0.463387	+	0.886156i	$0.346634\pi$
$128$	1110.88	0.767098
$129$	0	0
$130$	0	0
$131$	−2379.32	−1.58689	−0.793443	−	0.608644i	$-0.791714\pi$
$131$	−2379.32	−1.58689	−0.793443	+	0.608644i	$0.791714\pi$
$132$	0	0
$133$	−1654.97	−1.07898
$134$	−964.440	−0.621753
$135$	0	0
$136$	348.080	0.219468
$137$	−1633.55	−1.01871	−0.509355	−	0.860556i	$-0.670116\pi$
$137$	−1633.55	−1.01871	−0.509355	+	0.860556i	$0.670116\pi$
$138$	0	0
$139$	−1254.97	−0.765791	−0.382895	−	0.923792i	$-0.625073\pi$
$139$	−1254.97	−0.765791	−0.382895	+	0.923792i	$0.625073\pi$
$140$	184.305	0.111261
$141$	0	0
$142$	1334.78	0.788820
$143$	0	0
$144$	0	0
$145$	320.652	0.183646
$146$	536.965	0.304380
$147$	0	0
$148$	−2399.12	−1.33248
$149$	−461.974	−0.254002	−0.127001	−	0.991903i	$-0.540535\pi$
$149$	−461.974	−0.254002	−0.127001	+	0.991903i	$0.540535\pi$
$150$	0	0
$151$	−766.065	−0.412857	−0.206429	−	0.978462i	$-0.566184\pi$
$151$	−766.065	−0.412857	−0.206429	+	0.978462i	$0.566184\pi$
$152$	−1265.64	−0.675375
$153$	0	0
$154$	−1450.84	−0.759167
$155$	68.7980	0.0356516
$156$	0	0
$157$	−3424.00	−1.74054	−0.870270	−	0.492576i	$-0.836055\pi$
$157$	−3424.00	−1.74054	−0.870270	+	0.492576i	$0.836055\pi$
$158$	−1097.35	−0.552533
$159$	0	0
$160$	225.139	0.111243
$161$	−4322.41	−2.11586
$162$	0	0
$163$	−1722.36	−0.827640	−0.413820	−	0.910359i	$-0.635806\pi$
$163$	−1722.36	−0.827640	−0.413820	+	0.910359i	$0.635806\pi$
$164$	−1020.75	−0.486021
$165$	0	0
$166$	142.522	0.0666375
$167$	2360.22	1.09365	0.546824	−	0.837247i	$-0.315837\pi$
$167$	2360.22	1.09365	0.546824	+	0.837247i	$0.315837\pi$
$168$	0	0
$169$	0	0
$170$	31.4111	0.0141713
$171$	0	0
$172$	715.921	0.317375
$173$	−4240.41	−1.86354	−0.931769	−	0.363052i	$-0.881735\pi$
$173$	−4240.41	−1.86354	−0.931769	+	0.363052i	$0.881735\pi$
$174$	0	0
$175$	−3493.31	−1.50897
$176$	261.430	0.111966
$177$	0	0
$178$	2396.76	1.00924
$179$	145.366	0.0606991	0.0303496	−	0.999539i	$-0.490338\pi$
$179$	145.366	0.0606991	0.0303496	+	0.999539i	$0.490338\pi$
$180$	0	0
$181$	1447.40	0.594388	0.297194	−	0.954817i	$-0.403949\pi$
$181$	1447.40	0.594388	0.297194	+	0.954817i	$0.403949\pi$
$182$	0	0
$183$	0	0
$184$	−3305.57	−1.32440
$185$	−537.659	−0.213673
$186$	0	0
$187$	511.490	0.200020
$188$	613.138	0.237860
$189$	0	0
$190$	−114.213	−0.0436098
$191$	3772.56	1.42918	0.714589	−	0.699544i	$-0.246614\pi$
$191$	3772.56	1.42918	0.714589	+	0.699544i	$0.246614\pi$
$192$	0	0
$193$	4396.27	1.63964	0.819819	−	0.572622i	$-0.194074\pi$
$193$	4396.27	1.63964	0.819819	+	0.572622i	$0.194074\pi$
$194$	−250.262	−0.0926175
$195$	0	0
$196$	−2462.33	−0.897352
$197$	−1220.50	−0.441405	−0.220703	−	0.975341i	$-0.570835\pi$
$197$	−1220.50	−0.441405	−0.220703	+	0.975341i	$0.570835\pi$
$198$	0	0
$199$	1851.42	0.659516	0.329758	−	0.944066i	$-0.393033\pi$
$199$	1851.42	0.659516	0.329758	+	0.944066i	$0.393033\pi$
$200$	−2671.51	−0.944522
$201$	0	0
$202$	−3170.33	−1.10427
$203$	−7502.16	−2.59384
$204$	0	0
$205$	−228.758	−0.0779373
$206$	2616.93	0.885097
$207$	0	0
$208$	0	0
$209$	−1859.81	−0.615529
$210$	0	0
$211$	565.901	0.184636	0.0923180	−	0.995730i	$-0.470572\pi$
$211$	565.901	0.184636	0.0923180	+	0.995730i	$0.470572\pi$
$212$	1620.61	0.525017
$213$	0	0
$214$	518.505	0.165627
$215$	160.443	0.0508936
$216$	0	0
$217$	−1609.64	−0.503546
$218$	2107.54	0.654774
$219$	0	0
$220$	207.116	0.0634717
$221$	0	0
$222$	0	0
$223$	1073.79	0.322448	0.161224	−	0.986918i	$-0.448456\pi$
$223$	1073.79	0.322448	0.161224	+	0.986918i	$0.448456\pi$
$224$	−5267.49	−1.57120
$225$	0	0
$226$	−1754.92	−0.516528
$227$	5756.24	1.68306	0.841531	−	0.540208i	$-0.181655\pi$
$227$	5756.24	1.68306	0.841531	+	0.540208i	$0.181655\pi$
$228$	0	0
$229$	2577.30	0.743725	0.371862	−	0.928288i	$-0.378719\pi$
$229$	2577.30	0.743725	0.371862	+	0.928288i	$0.378719\pi$
$230$	−298.298	−0.0855182
$231$	0	0
$232$	−5737.28	−1.62358
$233$	347.566	0.0977246	0.0488623	−	0.998806i	$-0.484440\pi$
$233$	347.566	0.0977246	0.0488623	+	0.998806i	$0.484440\pi$
$234$	0	0
$235$	137.408	0.0381427
$236$	2769.60	0.763921
$237$	0	0
$238$	−734.914	−0.200157
$239$	−2201.35	−0.595788	−0.297894	−	0.954599i	$-0.596284\pi$
$239$	−2201.35	−0.595788	−0.297894	+	0.954599i	$0.596284\pi$
$240$	0	0
$241$	699.561	0.186982	0.0934910	−	0.995620i	$-0.470197\pi$
$241$	699.561	0.186982	0.0934910	+	0.995620i	$0.470197\pi$
$242$	518.685	0.137778
$243$	0	0
$244$	−3343.09	−0.877128
$245$	−551.826	−0.143897
$246$	0	0
$247$	0	0
$248$	−1230.97	−0.315189
$249$	0	0
$250$	−485.011	−0.122699
$251$	−6033.42	−1.51724	−0.758618	−	0.651536i	$-0.774125\pi$
$251$	−6033.42	−1.51724	−0.758618	+	0.651536i	$0.774125\pi$
$252$	0	0
$253$	−4857.39	−1.20704
$254$	−2141.69	−0.529061
$255$	0	0
$256$	−3673.37	−0.896819
$257$	3501.49	0.849872	0.424936	−	0.905223i	$-0.360297\pi$
$257$	3501.49	0.849872	0.424936	+	0.905223i	$0.360297\pi$
$258$	0	0
$259$	12579.4	3.01794
$260$	0	0
$261$	0	0
$262$	3841.75	0.905894
$263$	66.3098	0.0155469	0.00777346	−	0.999970i	$-0.497526\pi$
$263$	66.3098	0.0155469	0.00777346	+	0.999970i	$0.497526\pi$
$264$	0	0
$265$	363.189	0.0841907
$266$	2672.19	0.615949
$267$	0	0
$268$	−3221.24	−0.734211
$269$	−3128.48	−0.709096	−0.354548	−	0.935038i	$-0.615365\pi$
$269$	−3128.48	−0.709096	−0.354548	+	0.935038i	$0.615365\pi$
$270$	0	0
$271$	−1260.57	−0.282561	−0.141280	−	0.989970i	$-0.545122\pi$
$271$	−1260.57	−0.282561	−0.141280	+	0.989970i	$0.545122\pi$
$272$	132.426	0.0295202
$273$	0	0
$274$	2637.60	0.581544
$275$	−3925.68	−0.860826
$276$	0	0
$277$	−1534.66	−0.332884	−0.166442	−	0.986051i	$-0.553228\pi$
$277$	−1534.66	−0.332884	−0.166442	+	0.986051i	$0.553228\pi$
$278$	2026.32	0.437161
$279$	0	0
$280$	−739.034	−0.157735
$281$	−652.474	−0.138517	−0.0692586	−	0.997599i	$-0.522063\pi$
$281$	−652.474	−0.138517	−0.0692586	+	0.997599i	$0.522063\pi$
$282$	0	0
$283$	922.235	0.193714	0.0968572	−	0.995298i	$-0.469121\pi$
$283$	922.235	0.193714	0.0968572	+	0.995298i	$0.469121\pi$
$284$	4458.19	0.931496
$285$	0	0
$286$	0	0
$287$	5352.16	1.10079
$288$	0	0
$289$	−4653.91	−0.947264
$290$	−517.739	−0.104837
$291$	0	0
$292$	1793.47	0.359434
$293$	−5570.98	−1.11078	−0.555392	−	0.831588i	$-0.687432\pi$
$293$	−5570.98	−1.11078	−0.555392	+	0.831588i	$0.687432\pi$
$294$	0	0
$295$	620.686	0.122501
$296$	9620.10	1.88904
$297$	0	0
$298$	745.923	0.145000
$299$	0	0
$300$	0	0
$301$	−3753.82	−0.718825
$302$	1236.92	0.235685
$303$	0	0
$304$	−481.509	−0.0908435
$305$	−749.209	−0.140654
$306$	0	0
$307$	−7069.91	−1.31434	−0.657168	−	0.753744i	$-0.728246\pi$
$307$	−7069.91	−1.31434	−0.657168	+	0.753744i	$0.728246\pi$
$308$	−4845.81	−0.896480
$309$	0	0
$310$	−111.084	−0.0203521
$311$	6857.82	1.25039	0.625195	−	0.780468i	$-0.285019\pi$
$311$	6857.82	1.25039	0.625195	+	0.780468i	$0.285019\pi$
$312$	0	0
$313$	3820.23	0.689879	0.344940	−	0.938625i	$-0.387899\pi$
$313$	3820.23	0.689879	0.344940	+	0.938625i	$0.387899\pi$
$314$	5528.53	0.993609
$315$	0	0
$316$	−3665.15	−0.652471
$317$	−6324.21	−1.12051	−0.560257	−	0.828319i	$-0.689298\pi$
$317$	−6324.21	−1.12051	−0.560257	+	0.828319i	$0.689298\pi$
$318$	0	0
$319$	−8430.70	−1.47971
$320$	−283.974	−0.0496082
$321$	0	0
$322$	6979.16	1.20787
$323$	−942.075	−0.162286
$324$	0	0
$325$	0	0
$326$	2780.99	0.472469
$327$	0	0
$328$	4093.06	0.689030
$329$	−3214.89	−0.538731
$330$	0	0
$331$	7928.22	1.31654	0.658269	−	0.752783i	$-0.271289\pi$
$331$	7928.22	1.31654	0.658269	+	0.752783i	$0.271289\pi$
$332$	476.024	0.0786904
$333$	0	0
$334$	−3810.91	−0.624323
$335$	−721.902	−0.117737
$336$	0	0
$337$	−9305.11	−1.50410	−0.752050	−	0.659106i	$-0.770935\pi$
$337$	−9305.11	−1.50410	−0.752050	+	0.659106i	$0.770935\pi$
$338$	0	0
$339$	0	0
$340$	104.914	0.0167345
$341$	−1808.86	−0.287259
$342$	0	0
$343$	3211.86	0.505609
$344$	−2870.73	−0.449940
$345$	0	0
$346$	6846.74	1.06382
$347$	2063.17	0.319184	0.159592	−	0.987183i	$-0.448982\pi$
$347$	2063.17	0.319184	0.159592	+	0.987183i	$0.448982\pi$
$348$	0	0
$349$	3148.44	0.482900	0.241450	−	0.970413i	$-0.422377\pi$
$349$	3148.44	0.482900	0.241450	+	0.970413i	$0.422377\pi$
$350$	5640.46	0.861414
$351$	0	0
$352$	−5919.44	−0.896328
$353$	−4543.87	−0.685116	−0.342558	−	0.939497i	$-0.611293\pi$
$353$	−4543.87	−0.685116	−0.342558	+	0.939497i	$0.611293\pi$
$354$	0	0
$355$	999.111	0.149373
$356$	8005.19	1.19178
$357$	0	0
$358$	−234.714	−0.0346509
$359$	−21.5084	−0.00316204	−0.00158102	−	0.999999i	$-0.500503\pi$
$359$	−21.5084	−0.00316204	−0.00158102	+	0.999999i	$0.500503\pi$
$360$	0	0
$361$	−3433.56	−0.500591
$362$	−2337.03	−0.339314
$363$	0	0
$364$	0	0
$365$	401.929	0.0576381
$366$	0	0
$367$	−13050.8	−1.85626	−0.928129	−	0.372258i	$-0.878584\pi$
$367$	−13050.8	−1.85626	−0.928129	+	0.372258i	$0.878584\pi$
$368$	−1257.59	−0.178143
$369$	0	0
$370$	868.128	0.121978
$371$	−8497.39	−1.18912
$372$	0	0
$373$	−10150.6	−1.40905	−0.704527	−	0.709677i	$-0.748841\pi$
$373$	−10150.6	−1.40905	−0.704527	+	0.709677i	$0.748841\pi$
$374$	−825.873	−0.114184
$375$	0	0
$376$	−2458.59	−0.337213
$377$	0	0
$378$	0	0
$379$	−3443.65	−0.466724	−0.233362	−	0.972390i	$-0.574973\pi$
$379$	−3443.65	−0.466724	−0.233362	+	0.972390i	$0.574973\pi$
$380$	−381.472	−0.0514977
$381$	0	0
$382$	−6091.34	−0.815865
$383$	−5784.45	−0.771728	−0.385864	−	0.922556i	$-0.626097\pi$
$383$	−5784.45	−0.771728	−0.385864	+	0.922556i	$0.626097\pi$
$384$	0	0
$385$	−1085.98	−0.143758
$386$	−7098.40	−0.936008
$387$	0	0
$388$	−835.879	−0.109369
$389$	2235.07	0.291318	0.145659	−	0.989335i	$-0.453470\pi$
$389$	2235.07	0.291318	0.145659	+	0.989335i	$0.453470\pi$
$390$	0	0
$391$	−2460.49	−0.318241
$392$	9873.57	1.27217
$393$	0	0
$394$	1970.67	0.251982
$395$	−821.385	−0.104629
$396$	0	0
$397$	3585.93	0.453332	0.226666	−	0.973973i	$-0.427217\pi$
$397$	3585.93	0.453332	0.226666	+	0.973973i	$0.427217\pi$
$398$	−2989.38	−0.376493
$399$	0	0
$400$	−1016.37	−0.127046
$401$	4130.00	0.514321	0.257160	−	0.966369i	$-0.417213\pi$
$401$	4130.00	0.514321	0.257160	+	0.966369i	$0.417213\pi$
$402$	0	0
$403$	0	0
$404$	−10588.9	−1.30401
$405$	0	0
$406$	12113.3	1.48072
$407$	14136.3	1.72165
$408$	0	0
$409$	10133.6	1.22512	0.612558	−	0.790425i	$-0.290140\pi$
$409$	10133.6	1.22512	0.612558	+	0.790425i	$0.290140\pi$
$410$	369.363	0.0444915
$411$	0	0
$412$	8740.57	1.04519
$413$	−14521.9	−1.73021
$414$	0	0
$415$	106.680	0.0126186
$416$	0	0
$417$	0	0
$418$	3002.92	0.351382
$419$	7801.05	0.909561	0.454781	−	0.890604i	$-0.349718\pi$
$419$	7801.05	0.909561	0.454781	+	0.890604i	$0.349718\pi$
$420$	0	0
$421$	12154.2	1.40704	0.703518	−	0.710678i	$-0.251612\pi$
$421$	12154.2	1.40704	0.703518	+	0.710678i	$0.251612\pi$
$422$	−913.728	−0.105402
$423$	0	0
$424$	−6498.38	−0.744314
$425$	−1988.53	−0.226960
$426$	0	0
$427$	17528.9	1.98662
$428$	1731.81	0.195585
$429$	0	0
$430$	−259.058	−0.0290532
$431$	−13124.4	−1.46678	−0.733390	−	0.679808i	$-0.762063\pi$
$431$	−13124.4	−1.46678	−0.733390	+	0.679808i	$0.762063\pi$
$432$	0	0
$433$	−4457.79	−0.494752	−0.247376	−	0.968920i	$-0.579568\pi$
$433$	−4457.79	−0.494752	−0.247376	+	0.968920i	$0.579568\pi$
$434$	2598.99	0.287456
$435$	0	0
$436$	7039.21	0.773204
$437$	8946.48	0.979332
$438$	0	0
$439$	−4830.11	−0.525122	−0.262561	−	0.964915i	$-0.584567\pi$
$439$	−4830.11	−0.525122	−0.262561	+	0.964915i	$0.584567\pi$
$440$	−830.503	−0.0899834
$441$	0	0
$442$	0	0
$443$	−9154.26	−0.981788	−0.490894	−	0.871219i	$-0.663330\pi$
$443$	−9154.26	−0.981788	−0.490894	+	0.871219i	$0.663330\pi$
$444$	0	0
$445$	1794.02	0.191112
$446$	−1733.78	−0.184074
$447$	0	0
$448$	6644.02	0.700671
$449$	2576.20	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$449$	2576.20	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$450$	0	0
$451$	6014.59	0.627973
$452$	−5861.44	−0.609954
$453$	0	0
$454$	−9294.28	−0.960797
$455$	0	0
$456$	0	0
$457$	−1489.35	−0.152448	−0.0762240	−	0.997091i	$-0.524286\pi$
$457$	−1489.35	−0.152448	−0.0762240	+	0.997091i	$0.524286\pi$
$458$	−4161.42	−0.424565
$459$	0	0
$460$	−996.319	−0.100986
$461$	−13869.1	−1.40118	−0.700592	−	0.713562i	$-0.747081\pi$
$461$	−13869.1	−1.40118	−0.700592	+	0.713562i	$0.747081\pi$
$462$	0	0
$463$	−10538.6	−1.05782	−0.528908	−	0.848679i	$-0.677398\pi$
$463$	−10538.6	−1.05782	−0.528908	+	0.848679i	$0.677398\pi$
$464$	−2182.73	−0.218385
$465$	0	0
$466$	−561.196	−0.0557873
$467$	−12356.9	−1.22443	−0.612214	−	0.790692i	$-0.709721\pi$
$467$	−12356.9	−1.22443	−0.612214	+	0.790692i	$0.709721\pi$
$468$	0	0
$469$	16890.1	1.66292
$470$	−221.866	−0.0217742
$471$	0	0
$472$	−11105.6	−1.08301
$473$	−4218.42	−0.410070
$474$	0	0
$475$	7230.42	0.698430
$476$	−2454.62	−0.236360
$477$	0	0
$478$	3554.39	0.340113
$479$	−4204.24	−0.401037	−0.200518	−	0.979690i	$-0.564263\pi$
$479$	−4204.24	−0.401037	−0.200518	+	0.979690i	$0.564263\pi$
$480$	0	0
$481$	0	0
$482$	−1129.54	−0.106741
$483$	0	0
$484$	1732.41	0.162699
$485$	−187.326	−0.0175382
$486$	0	0
$487$	−12581.8	−1.17071	−0.585355	−	0.810777i	$-0.699045\pi$
$487$	−12581.8	−1.17071	−0.585355	+	0.810777i	$0.699045\pi$
$488$	13405.3	1.24350
$489$	0	0
$490$	891.002	0.0821457
$491$	−3085.05	−0.283557	−0.141778	−	0.989898i	$-0.545282\pi$
$491$	−3085.05	−0.283557	−0.141778	+	0.989898i	$0.545282\pi$
$492$	0	0
$493$	−4270.53	−0.390131
$494$	0	0
$495$	0	0
$496$	−468.319	−0.0423955
$497$	−23375.8	−2.10975
$498$	0	0
$499$	−6275.13	−0.562953	−0.281477	−	0.959568i	$-0.590824\pi$
$499$	−6275.13	−0.562953	−0.281477	+	0.959568i	$0.590824\pi$
$500$	−1619.94	−0.144892
$501$	0	0
$502$	9741.83	0.866133
$503$	17843.6	1.58173	0.790863	−	0.611994i	$-0.209632\pi$
$503$	17843.6	1.58173	0.790863	+	0.611994i	$0.209632\pi$
$504$	0	0
$505$	−2373.05	−0.209108
$506$	7842.96	0.689055
$507$	0	0
$508$	−7153.26	−0.624753
$509$	−10033.0	−0.873684	−0.436842	−	0.899538i	$-0.643903\pi$
$509$	−10033.0	−0.873684	−0.436842	+	0.899538i	$0.643903\pi$
$510$	0	0
$511$	−9403.77	−0.814086
$512$	−2955.83	−0.255138
$513$	0	0
$514$	−5653.66	−0.485160
$515$	1958.82	0.167604
$516$	0	0
$517$	−3612.79	−0.307332
$518$	−20311.2	−1.72283
$519$	0	0
$520$	0	0
$521$	2406.44	0.202357	0.101179	−	0.994868i	$-0.467739\pi$
$521$	2406.44	0.202357	0.101179	+	0.994868i	$0.467739\pi$
$522$	0	0
$523$	1987.60	0.166179	0.0830895	−	0.996542i	$-0.473521\pi$
$523$	1987.60	0.166179	0.0830895	+	0.996542i	$0.473521\pi$
$524$	12831.5	1.06975
$525$	0	0
$526$	−107.067	−0.00887515
$527$	−916.270	−0.0757369
$528$	0	0
$529$	11199.2	0.920455
$530$	−586.421	−0.0480613
$531$	0	0
$532$	8925.15	0.727358
$533$	0	0
$534$	0	0
$535$	388.111	0.0313635
$536$	12916.7	1.04089
$537$	0	0
$538$	5051.38	0.404797
$539$	14508.8	1.15944
$540$	0	0
$541$	9255.22	0.735514	0.367757	−	0.929922i	$-0.380126\pi$
$541$	9255.22	0.735514	0.367757	+	0.929922i	$0.380126\pi$
$542$	2035.37	0.161303
$543$	0	0
$544$	−2998.46	−0.236320
$545$	1577.54	0.123989
$546$	0	0
$547$	−7539.31	−0.589319	−0.294660	−	0.955602i	$-0.595206\pi$
$547$	−7539.31	−0.589319	−0.294660	+	0.955602i	$0.595206\pi$
$548$	8809.60	0.686729
$549$	0	0
$550$	6338.57	0.491414
$551$	15527.9	1.20056
$552$	0	0
$553$	19217.6	1.47779
$554$	2477.93	0.190031
$555$	0	0
$556$	6767.94	0.516232
$557$	−13615.2	−1.03572	−0.517860	−	0.855465i	$-0.673271\pi$
$557$	−13615.2	−1.03572	−0.517860	+	0.855465i	$0.673271\pi$
$558$	0	0
$559$	0	0
$560$	−281.163	−0.0212166
$561$	0	0
$562$	1053.51	0.0790743
$563$	4751.68	0.355700	0.177850	−	0.984058i	$-0.443086\pi$
$563$	4751.68	0.355700	0.177850	+	0.984058i	$0.443086\pi$
$564$	0	0
$565$	−1313.59	−0.0978109
$566$	−1489.08	−0.110584
$567$	0	0
$568$	−17876.6	−1.32058
$569$	−17296.9	−1.27438	−0.637191	−	0.770706i	$-0.719904\pi$
$569$	−17296.9	−1.27438	−0.637191	+	0.770706i	$0.719904\pi$
$570$	0	0
$571$	3685.46	0.270108	0.135054	−	0.990838i	$-0.456879\pi$
$571$	3685.46	0.270108	0.135054	+	0.990838i	$0.456879\pi$
$572$	0	0
$573$	0	0
$574$	−8641.83	−0.628402
$575$	18884.2	1.36961
$576$	0	0
$577$	−10066.3	−0.726287	−0.363143	−	0.931733i	$-0.618296\pi$
$577$	−10066.3	−0.726287	−0.363143	+	0.931733i	$0.618296\pi$
$578$	7514.40	0.540758
$579$	0	0
$580$	−1729.25	−0.123799
$581$	−2495.95	−0.178227
$582$	0	0
$583$	−9549.10	−0.678359
$584$	−7191.53	−0.509568
$585$	0	0
$586$	8995.14	0.634106
$587$	19809.2	1.39286	0.696432	−	0.717622i	$-0.254770\pi$
$587$	19809.2	1.39286	0.696432	+	0.717622i	$0.254770\pi$
$588$	0	0
$589$	3331.61	0.233067
$590$	−1002.19	−0.0699311
$591$	0	0
$592$	3659.94	0.254092
$593$	5085.44	0.352165	0.176082	−	0.984375i	$-0.443657\pi$
$593$	5085.44	0.352165	0.176082	+	0.984375i	$0.443657\pi$
$594$	0	0
$595$	−550.097	−0.0379022
$596$	2491.39	0.171227
$597$	0	0
$598$	0	0
$599$	20473.7	1.39655	0.698273	−	0.715832i	$-0.253952\pi$
$599$	20473.7	1.39655	0.698273	+	0.715832i	$0.253952\pi$
$600$	0	0
$601$	6131.90	0.416182	0.208091	−	0.978109i	$-0.433275\pi$
$601$	6131.90	0.416182	0.208091	+	0.978109i	$0.433275\pi$
$602$	6061.07	0.410350
$603$	0	0
$604$	4131.33	0.278314
$605$	388.246	0.0260900
$606$	0	0
$607$	27464.1	1.83647	0.918233	−	0.396041i	$-0.129616\pi$
$607$	27464.1	1.83647	0.918233	+	0.396041i	$0.129616\pi$
$608$	10902.6	0.727234
$609$	0	0
$610$	1209.71	0.0802943
$611$	0	0
$612$	0	0
$613$	13563.4	0.893669	0.446835	−	0.894617i	$-0.352551\pi$
$613$	13563.4	0.893669	0.446835	+	0.894617i	$0.352551\pi$
$614$	11415.4	0.750305
$615$	0	0
$616$	19431.0	1.27093
$617$	−4983.73	−0.325182	−0.162591	−	0.986694i	$-0.551985\pi$
$617$	−4983.73	−0.325182	−0.162591	+	0.986694i	$0.551985\pi$
$618$	0	0
$619$	10088.7	0.655085	0.327543	−	0.944836i	$-0.393779\pi$
$619$	10088.7	0.655085	0.327543	+	0.944836i	$0.393779\pi$
$620$	−371.023	−0.0240333
$621$	0	0
$622$	−11072.9	−0.713801
$623$	−41973.9	−2.69928
$624$	0	0
$625$	15079.4	0.965080
$626$	−6168.31	−0.393826
$627$	0	0
$628$	18465.4	1.17333
$629$	7160.69	0.453919
$630$	0	0
$631$	−6909.71	−0.435929	−0.217964	−	0.975957i	$-0.569942\pi$
$631$	−6909.71	−0.435929	−0.217964	+	0.975957i	$0.569942\pi$
$632$	14696.7	0.925004
$633$	0	0
$634$	10211.3	0.639660
$635$	−1603.10	−0.100184
$636$	0	0
$637$	0	0
$638$	13612.6	0.844713
$639$	0	0
$640$	−1342.60	−0.0829232
$641$	−24827.6	−1.52984	−0.764922	−	0.644123i	$-0.777222\pi$
$641$	−24827.6	−1.52984	−0.764922	+	0.644123i	$0.777222\pi$
$642$	0	0
$643$	−9712.36	−0.595674	−0.297837	−	0.954617i	$-0.596265\pi$
$643$	−9712.36	−0.595674	−0.297837	+	0.954617i	$0.596265\pi$
$644$	23310.5	1.42634
$645$	0	0
$646$	1521.12	0.0926432
$647$	−422.401	−0.0256666	−0.0128333	−	0.999918i	$-0.504085\pi$
$647$	−422.401	−0.0256666	−0.0128333	+	0.999918i	$0.504085\pi$
$648$	0	0
$649$	−16319.3	−0.987039
$650$	0	0
$651$	0	0
$652$	9288.54	0.557925
$653$	5691.05	0.341053	0.170527	−	0.985353i	$-0.445453\pi$
$653$	5691.05	0.341053	0.170527	+	0.985353i	$0.445453\pi$
$654$	0	0
$655$	2875.63	0.171542
$656$	1557.19	0.0926802
$657$	0	0
$658$	5190.90	0.307542
$659$	11497.8	0.679655	0.339827	−	0.940488i	$-0.389631\pi$
$659$	11497.8	0.679655	0.339827	+	0.940488i	$0.389631\pi$
$660$	0	0
$661$	−22058.8	−1.29802	−0.649008	−	0.760781i	$-0.724816\pi$
$661$	−22058.8	−1.29802	−0.649008	+	0.760781i	$0.724816\pi$
$662$	−12801.2	−0.751562
$663$	0	0
$664$	−1908.78	−0.111559
$665$	2000.19	0.116638
$666$	0	0
$667$	40555.3	2.35429
$668$	−12728.5	−0.737246
$669$	0	0
$670$	1165.61	0.0672114
$671$	19698.5	1.13331
$672$	0	0
$673$	−4767.64	−0.273074	−0.136537	−	0.990635i	$-0.543597\pi$
$673$	−4767.64	−0.273074	−0.136537	+	0.990635i	$0.543597\pi$
$674$	15024.4	0.858634
$675$	0	0
$676$	0	0
$677$	14024.5	0.796165	0.398082	−	0.917350i	$-0.369676\pi$
$677$	14024.5	0.796165	0.398082	+	0.917350i	$0.369676\pi$
$678$	0	0
$679$	4382.80	0.247712
$680$	−420.687	−0.0237244
$681$	0	0
$682$	2920.67	0.163986
$683$	4652.81	0.260666	0.130333	−	0.991470i	$-0.458395\pi$
$683$	4652.81	0.260666	0.130333	+	0.991470i	$0.458395\pi$
$684$	0	0
$685$	1974.29	0.110122
$686$	−5186.00	−0.288633
$687$	0	0
$688$	−1092.16	−0.0605207
$689$	0	0
$690$	0	0
$691$	−10978.9	−0.604426	−0.302213	−	0.953240i	$-0.597725\pi$
$691$	−10978.9	−0.604426	−0.302213	+	0.953240i	$0.597725\pi$
$692$	22868.2	1.25624
$693$	0	0
$694$	−3331.28	−0.182210
$695$	1516.74	0.0827818
$696$	0	0
$697$	3046.66	0.165567
$698$	−5083.60	−0.275669
$699$	0	0
$700$	18839.2	1.01722
$701$	4451.47	0.239843	0.119921	−	0.992783i	$-0.461736\pi$
$701$	4451.47	0.239843	0.119921	+	0.992783i	$0.461736\pi$
$702$	0	0
$703$	−26036.7	−1.39686
$704$	7466.35	0.399714
$705$	0	0
$706$	7336.74	0.391107
$707$	55521.3	2.95346
$708$	0	0
$709$	−18370.9	−0.973107	−0.486554	−	0.873651i	$-0.661746\pi$
$709$	−18370.9	−0.973107	−0.486554	+	0.873651i	$0.661746\pi$
$710$	−1613.21	−0.0852713
$711$	0	0
$712$	−32099.6	−1.68958
$713$	8701.42	0.457042
$714$	0	0
$715$	0	0
$716$	−783.947	−0.0409183
$717$	0	0
$718$	34.7284	0.00180509
$719$	31594.0	1.63874	0.819372	−	0.573263i	$-0.194323\pi$
$719$	31594.0	1.63874	0.819372	+	0.573263i	$0.194323\pi$
$720$	0	0
$721$	−45829.8	−2.36725
$722$	5543.97	0.285769
$723$	0	0
$724$	−7805.71	−0.400687
$725$	32776.2	1.67901
$726$	0	0
$727$	−21370.5	−1.09022	−0.545109	−	0.838365i	$-0.683512\pi$
$727$	−21370.5	−1.09022	−0.545109	+	0.838365i	$0.683512\pi$
$728$	0	0
$729$	0	0
$730$	−648.972	−0.0329035
$731$	−2136.82	−0.108116
$732$	0	0
$733$	13371.4	0.673785	0.336893	−	0.941543i	$-0.390624\pi$
$733$	13371.4	0.673785	0.336893	+	0.941543i	$0.390624\pi$
$734$	21072.4	1.05967
$735$	0	0
$736$	28475.1	1.42610
$737$	18980.5	0.948652
$738$	0	0
$739$	−6426.29	−0.319885	−0.159942	−	0.987126i	$-0.551131\pi$
$739$	−6426.29	−0.319885	−0.159942	+	0.987126i	$0.551131\pi$
$740$	2899.56	0.144040
$741$	0	0
$742$	13720.2	0.678822
$743$	−25071.8	−1.23795	−0.618975	−	0.785411i	$-0.712452\pi$
$743$	−25071.8	−1.23795	−0.618975	+	0.785411i	$0.712452\pi$
$744$	0	0
$745$	558.338	0.0274576
$746$	16389.6	0.804376
$747$	0	0
$748$	−2758.43	−0.134837
$749$	−9080.47	−0.442982
$750$	0	0
$751$	5426.94	0.263691	0.131846	−	0.991270i	$-0.457910\pi$
$751$	5426.94	0.263691	0.131846	+	0.991270i	$0.457910\pi$
$752$	−935.362	−0.0453579
$753$	0	0
$754$	0	0
$755$	925.860	0.0446298
$756$	0	0
$757$	−2227.69	−0.106957	−0.0534786	−	0.998569i	$-0.517031\pi$
$757$	−2227.69	−0.106957	−0.0534786	+	0.998569i	$0.517031\pi$
$758$	5560.26	0.266435
$759$	0	0
$760$	1529.64	0.0730079
$761$	30769.6	1.46570	0.732851	−	0.680390i	$-0.238189\pi$
$761$	30769.6	1.46570	0.732851	+	0.680390i	$0.238189\pi$
$762$	0	0
$763$	−36909.0	−1.75124
$764$	−20345.2	−0.963432
$765$	0	0
$766$	9339.83	0.440551
$767$	0	0
$768$	0	0
$769$	−1346.49	−0.0631415	−0.0315707	−	0.999502i	$-0.510051\pi$
$769$	−1346.49	−0.0631415	−0.0315707	+	0.999502i	$0.510051\pi$
$770$	1753.47	0.0820658
$771$	0	0
$772$	−23708.8	−1.10531
$773$	−18487.6	−0.860226	−0.430113	−	0.902775i	$-0.641526\pi$
$773$	−18487.6	−0.860226	−0.430113	+	0.902775i	$0.641526\pi$
$774$	0	0
$775$	7032.36	0.325948
$776$	3351.75	0.155052
$777$	0	0
$778$	−3608.84	−0.166302
$779$	−11077.8	−0.509505
$780$	0	0
$781$	−26269.0	−1.20356
$782$	3972.81	0.181672
$783$	0	0
$784$	3756.37	0.171117
$785$	4138.22	0.188152
$786$	0	0
$787$	1388.74	0.0629010	0.0314505	−	0.999505i	$-0.489987\pi$
$787$	1388.74	0.0629010	0.0314505	+	0.999505i	$0.489987\pi$
$788$	6582.05	0.297558
$789$	0	0
$790$	1326.24	0.0597287
$791$	30733.5	1.38149
$792$	0	0
$793$	0	0
$794$	−5790.00	−0.258790
$795$	0	0
$796$	−9984.57	−0.444590
$797$	−27516.6	−1.22295	−0.611473	−	0.791265i	$-0.709423\pi$
$797$	−27516.6	−1.22295	−0.611473	+	0.791265i	$0.709423\pi$
$798$	0	0
$799$	−1830.04	−0.0810290
$800$	23013.2	1.01705
$801$	0	0
$802$	−6668.48	−0.293606
$803$	−10567.7	−0.464414
$804$	0	0
$805$	5224.04	0.228724
$806$	0	0
$807$	0	0
$808$	42459.9	1.84868
$809$	−31550.8	−1.37116	−0.685580	−	0.727998i	$-0.740451\pi$
$809$	−31550.8	−1.37116	−0.685580	+	0.727998i	$0.740451\pi$
$810$	0	0
$811$	−16536.1	−0.715980	−0.357990	−	0.933725i	$-0.616538\pi$
$811$	−16536.1	−0.715980	−0.357990	+	0.933725i	$0.616538\pi$
$812$	40458.6	1.74855
$813$	0	0
$814$	−22825.1	−0.982826
$815$	2081.63	0.0894677
$816$	0	0
$817$	7769.61	0.332710
$818$	−16362.1	−0.699373
$819$	0	0
$820$	1233.68	0.0525388
$821$	3643.64	0.154889	0.0774445	−	0.996997i	$-0.475324\pi$
$821$	3643.64	0.154889	0.0774445	+	0.996997i	$0.475324\pi$
$822$	0	0
$823$	−26439.9	−1.11985	−0.559925	−	0.828543i	$-0.689170\pi$
$823$	−26439.9	−1.11985	−0.559925	+	0.828543i	$0.689170\pi$
$824$	−35048.3	−1.48175
$825$	0	0
$826$	23447.7	0.987713
$827$	−13793.7	−0.579995	−0.289997	−	0.957027i	$-0.593654\pi$
$827$	−13793.7	−0.579995	−0.289997	+	0.957027i	$0.593654\pi$
$828$	0	0
$829$	−34019.6	−1.42527	−0.712636	−	0.701534i	$-0.752499\pi$
$829$	−34019.6	−1.42527	−0.712636	+	0.701534i	$0.752499\pi$
$830$	−172.251	−0.00720350
$831$	0	0
$832$	0	0
$833$	7349.36	0.305691
$834$	0	0
$835$	−2852.54	−0.118223
$836$	10029.8	0.414938
$837$	0	0
$838$	−12595.9	−0.519234
$839$	26565.2	1.09312	0.546562	−	0.837418i	$-0.315936\pi$
$839$	26565.2	1.09312	0.546562	+	0.837418i	$0.315936\pi$
$840$	0	0
$841$	46000.5	1.88612
$842$	−19624.8	−0.803224
$843$	0	0
$844$	−3051.86	−0.124466
$845$	0	0
$846$	0	0
$847$	−9083.63	−0.368497
$848$	−2472.29	−0.100116
$849$	0	0
$850$	3210.77	0.129563
$851$	−68001.9	−2.73922
$852$	0	0
$853$	15220.2	0.610939	0.305470	−	0.952202i	$-0.401187\pi$
$853$	15220.2	0.610939	0.305470	+	0.952202i	$0.401187\pi$
$854$	−28303.0	−1.13408
$855$	0	0
$856$	−6944.29	−0.277279
$857$	−39559.1	−1.57680	−0.788398	−	0.615165i	$-0.789089\pi$
$857$	−39559.1	−1.57680	−0.788398	+	0.615165i	$0.789089\pi$
$858$	0	0
$859$	41510.0	1.64878	0.824390	−	0.566022i	$-0.191518\pi$
$859$	41510.0	1.64878	0.824390	+	0.566022i	$0.191518\pi$
$860$	−865.257	−0.0343082
$861$	0	0
$862$	21191.3	0.837330
$863$	34778.6	1.37182	0.685908	−	0.727688i	$-0.259405\pi$
$863$	34778.6	1.37182	0.685908	+	0.727688i	$0.259405\pi$
$864$	0	0
$865$	5124.92	0.201448
$866$	7197.74	0.282435
$867$	0	0
$868$	8680.67	0.339448
$869$	21596.2	0.843037
$870$	0	0
$871$	0	0
$872$	−28226.1	−1.09617
$873$	0	0
$874$	−14445.4	−0.559064
$875$	8493.91	0.328167
$876$	0	0
$877$	28204.1	1.08596	0.542979	−	0.839746i	$-0.317296\pi$
$877$	28204.1	1.08596	0.542979	+	0.839746i	$0.317296\pi$
$878$	7798.90	0.299772
$879$	0	0
$880$	−315.962	−0.0121035
$881$	1006.27	0.0384813	0.0192406	−	0.999815i	$-0.493875\pi$
$881$	1006.27	0.0384813	0.0192406	+	0.999815i	$0.493875\pi$
$882$	0	0
$883$	20823.5	0.793620	0.396810	−	0.917901i	$-0.370117\pi$
$883$	20823.5	0.793620	0.396810	+	0.917901i	$0.370117\pi$
$884$	0	0
$885$	0	0
$886$	14780.9	0.560466
$887$	−17301.0	−0.654917	−0.327458	−	0.944866i	$-0.606192\pi$
$887$	−17301.0	−0.654917	−0.327458	+	0.944866i	$0.606192\pi$
$888$	0	0
$889$	37507.0	1.41501
$890$	−2896.70	−0.109098
$891$	0	0
$892$	−5790.85	−0.217368
$893$	6654.14	0.249353
$894$	0	0
$895$	−175.688	−0.00656156
$896$	31412.2	1.17121
$897$	0	0
$898$	−4159.64	−0.154576
$899$	15102.6	0.560287
$900$	0	0
$901$	−4837.04	−0.178852
$902$	−9711.42	−0.358486
$903$	0	0
$904$	23503.5	0.864728
$905$	−1749.31	−0.0642532
$906$	0	0
$907$	1878.58	0.0687732	0.0343866	−	0.999409i	$-0.489052\pi$
$907$	1878.58	0.0687732	0.0343866	+	0.999409i	$0.489052\pi$
$908$	−31043.0	−1.13458
$909$	0	0
$910$	0	0
$911$	29940.8	1.08890	0.544448	−	0.838795i	$-0.316739\pi$
$911$	29940.8	1.08890	0.544448	+	0.838795i	$0.316739\pi$
$912$	0	0
$913$	−2804.88	−0.101673
$914$	2404.77	0.0870269
$915$	0	0
$916$	−13899.2	−0.501357
$917$	−67279.9	−2.42288
$918$	0	0
$919$	10648.8	0.382233	0.191116	−	0.981567i	$-0.438789\pi$
$919$	10648.8	0.382233	0.191116	+	0.981567i	$0.438789\pi$
$920$	3995.08	0.143167
$921$	0	0
$922$	22393.6	0.799884
$923$	0	0
$924$	0	0
$925$	−54958.2	−1.95353
$926$	17016.0	0.603867
$927$	0	0
$928$	49422.6	1.74825
$929$	22670.9	0.800654	0.400327	−	0.916372i	$-0.368897\pi$
$929$	22670.9	0.800654	0.400327	+	0.916372i	$0.368897\pi$
$930$	0	0
$931$	−26722.7	−0.940711
$932$	−1874.40	−0.0658777
$933$	0	0
$934$	19952.0	0.698980
$935$	−618.182	−0.0216222
$936$	0	0
$937$	−18160.3	−0.633161	−0.316580	−	0.948566i	$-0.602535\pi$
$937$	−18160.3	−0.633161	−0.316580	+	0.948566i	$0.602535\pi$
$938$	−27271.4	−0.949300
$939$	0	0
$940$	−741.034	−0.0257126
$941$	17564.9	0.608503	0.304251	−	0.952592i	$-0.401594\pi$
$941$	17564.9	0.608503	0.304251	+	0.952592i	$0.401594\pi$
$942$	0	0
$943$	−28932.8	−0.999132
$944$	−4225.11	−0.145673
$945$	0	0
$946$	6811.25	0.234094
$947$	−40490.9	−1.38942	−0.694708	−	0.719292i	$-0.744467\pi$
$947$	−40490.9	−1.38942	−0.694708	+	0.719292i	$0.744467\pi$
$948$	0	0
$949$	0	0
$950$	−11674.5	−0.398708
$951$	0	0
$952$	9842.65	0.335086
$953$	11971.4	0.406917	0.203458	−	0.979084i	$-0.434782\pi$
$953$	11971.4	0.406917	0.203458	+	0.979084i	$0.434782\pi$
$954$	0	0
$955$	−4559.49	−0.154494
$956$	11871.7	0.401630
$957$	0	0
$958$	6788.35	0.228937
$959$	−46191.7	−1.55538
$960$	0	0
$961$	−26550.6	−0.891230
$962$	0	0
$963$	0	0
$964$	−3772.68	−0.126048
$965$	−5313.30	−0.177245
$966$	0	0
$967$	−341.617	−0.0113605	−0.00568027	−	0.999984i	$-0.501808\pi$
$967$	−341.617	−0.0113605	−0.00568027	+	0.999984i	$0.501808\pi$
$968$	−6946.71	−0.230657
$969$	0	0
$970$	302.465	0.0100119
$971$	2743.85	0.0906843	0.0453422	−	0.998972i	$-0.485562\pi$
$971$	2743.85	0.0906843	0.0453422	+	0.998972i	$0.485562\pi$
$972$	0	0
$973$	−35486.6	−1.16922
$974$	20315.1	0.668314
$975$	0	0
$976$	5099.99	0.167261
$977$	19743.5	0.646520	0.323260	−	0.946310i	$-0.395221\pi$
$977$	19743.5	0.646520	0.323260	+	0.946310i	$0.395221\pi$
$978$	0	0
$979$	−47169.0	−1.53986
$980$	2975.96	0.0970035
$981$	0	0
$982$	4981.26	0.161872
$983$	−17848.8	−0.579135	−0.289567	−	0.957158i	$-0.593511\pi$
$983$	−17848.8	−0.579135	−0.289567	+	0.957158i	$0.593511\pi$
$984$	0	0
$985$	1475.08	0.0477158
$986$	6895.37	0.222711
$987$	0	0
$988$	0	0
$989$	20292.4	0.652439
$990$	0	0
$991$	38391.0	1.23061	0.615303	−	0.788291i	$-0.289034\pi$
$991$	38391.0	1.23061	0.615303	+	0.788291i	$0.289034\pi$
$992$	10603.9	0.339391
$993$	0	0
$994$	37743.6	1.20438
$995$	−2237.61	−0.0712935
$996$	0	0
$997$	−7060.87	−0.224293	−0.112146	−	0.993692i	$-0.535773\pi$
$997$	−7060.87	−0.224293	−0.112146	+	0.993692i	$0.535773\pi$
$998$	10132.1	0.321369
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.4.a.be.1.5		9
3.2	odd	2		507.4.a.q.1.5	yes	9
13.12	even	2		1521.4.a.bj.1.5		9
39.5	even	4		507.4.b.j.337.8		18
39.8	even	4		507.4.b.j.337.11		18
39.38	odd	2		507.4.a.n.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.n.1.5	✓	9	39.38	odd	2
507.4.a.q.1.5	yes	9	3.2	odd	2
507.4.b.j.337.8		18	39.5	even	4
507.4.b.j.337.11		18	39.8	even	4
1521.4.a.be.1.5		9	1.1	even	1	trivial
1521.4.a.bj.1.5		9	13.12	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61464 q^{2} -5.39293 q^{4} -1.20859 q^{5} +28.2769 q^{7} +21.6248 q^{8} +O(q^{10})\)	\(q-1.61464 q^{2} -5.39293 q^{4} -1.20859 q^{5} +28.2769 q^{7} +21.6248 q^{8} +1.95145 q^{10} +31.7768 q^{11} -45.6572 q^{14} +8.22709 q^{16} +16.0963 q^{17} -58.5273 q^{19} +6.51785 q^{20} -51.3081 q^{22} -152.860 q^{23} -123.539 q^{25} -152.496 q^{28} -265.310 q^{29} -56.9241 q^{31} -186.282 q^{32} -25.9899 q^{34} -34.1753 q^{35} +444.864 q^{37} +94.5007 q^{38} -26.1356 q^{40} +189.276 q^{41} -132.752 q^{43} -171.370 q^{44} +246.814 q^{46} -113.693 q^{47} +456.586 q^{49} +199.472 q^{50} -300.506 q^{53} -38.4051 q^{55} +611.483 q^{56} +428.381 q^{58} -513.561 q^{59} +619.902 q^{61} +91.9121 q^{62} +234.963 q^{64} +597.308 q^{67} -86.8064 q^{68} +55.1809 q^{70} -826.673 q^{71} -332.560 q^{73} -718.297 q^{74} +315.633 q^{76} +898.549 q^{77} +679.621 q^{79} -9.94319 q^{80} -305.614 q^{82} -88.2682 q^{83} -19.4539 q^{85} +214.347 q^{86} +687.166 q^{88} -1484.39 q^{89} +824.363 q^{92} +183.574 q^{94} +70.7356 q^{95} +154.995 q^{97} -737.223 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8}+O(q^{10}) \) 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8	\( 9 q - 8 q^{2} + 32 q^{4} - 41 q^{5} + q^{7} - 111 q^{8} + 198 q^{10} - 37 q^{11} - 98 q^{14} + 32 q^{16} + 134 q^{17} - 72 q^{19} - 356 q^{20} + 274 q^{22} - 226 q^{23} + 612 q^{25} + 132 q^{28} + 547 q^{29} - 521 q^{31} - 721 q^{32} - 100 q^{34} - 138 q^{35} + 584 q^{37} + 416 q^{38} + 1342 q^{40} - 482 q^{41} + 158 q^{43} - 1453 q^{44} + 1537 q^{46} - 1500 q^{47} + 642 q^{49} - 2777 q^{50} - 1399 q^{53} - 1408 q^{55} + 616 q^{56} + 1455 q^{58} - 1541 q^{59} + 2092 q^{61} + 293 q^{62} + 2481 q^{64} + 252 q^{67} + 1579 q^{68} + 2492 q^{70} - 2352 q^{71} + 903 q^{73} - 1037 q^{74} - 485 q^{76} + 1686 q^{77} - 115 q^{79} - 5701 q^{80} - 5147 q^{82} - 1207 q^{83} + 4308 q^{85} - 5691 q^{86} - 484 q^{88} - 2336 q^{89} - 2087 q^{92} - 468 q^{94} + 222 q^{95} + 2155 q^{97} - 5593 q^{98}+O(q^{100}) \) 9 * q - 8 * q^2 + 32 * q^4 - 41 * q^5 + q^7 - 111 * q^8 + 198 * q^10 - 37 * q^11 - 98 * q^14 + 32 * q^16 + 134 * q^17 - 72 * q^19 - 356 * q^20 + 274 * q^22 - 226 * q^23 + 612 * q^25 + 132 * q^28 + 547 * q^29 - 521 * q^31 - 721 * q^32 - 100 * q^34 - 138 * q^35 + 584 * q^37 + 416 * q^38 + 1342 * q^40 - 482 * q^41 + 158 * q^43 - 1453 * q^44 + 1537 * q^46 - 1500 * q^47 + 642 * q^49 - 2777 * q^50 - 1399 * q^53 - 1408 * q^55 + 616 * q^56 + 1455 * q^58 - 1541 * q^59 + 2092 * q^61 + 293 * q^62 + 2481 * q^64 + 252 * q^67 + 1579 * q^68 + 2492 * q^70 - 2352 * q^71 + 903 * q^73 - 1037 * q^74 - 485 * q^76 + 1686 * q^77 - 115 * q^79 - 5701 * q^80 - 5147 * q^82 - 1207 * q^83 + 4308 * q^85 - 5691 * q^86 - 484 * q^88 - 2336 * q^89 - 2087 * q^92 - 468 * q^94 + 222 * q^95 + 2155 * q^97 - 5593 * q^98

Introduction

L-functions

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