LMFDB - Embedded newform 531.4.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [531,4,Mod(1,531)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(531, 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("531.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 531 = 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	531.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:	$31.3300142130$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 49x^{6} + 89x^{5} + 648x^{4} - 1023x^{3} - 1476x^{2} + 1940x - 384 $ x^8 - 2x^7 - 49x^6 + 89x^5 + 648x^4 - 1023x^3 - 1476x^2 + 1940*x - 384
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17127$ of defining polynomial
Character	$\chi$	$=$	531.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17127 q^{2} -3.28558 q^{4} -9.58086 q^{5} +14.1591 q^{7} +24.5041 q^{8} +O(q^{10})$	\(q-2.17127 q^{2} -3.28558 q^{4} -9.58086 q^{5} +14.1591 q^{7} +24.5041 q^{8} +20.8027 q^{10} -19.1455 q^{11} +15.5364 q^{13} -30.7433 q^{14} -26.9204 q^{16} -100.737 q^{17} +74.3408 q^{19} +31.4787 q^{20} +41.5701 q^{22} -98.9130 q^{23} -33.2071 q^{25} -33.7338 q^{26} -46.5208 q^{28} -194.021 q^{29} +52.9293 q^{31} -137.581 q^{32} +218.728 q^{34} -135.656 q^{35} -212.802 q^{37} -161.414 q^{38} -234.770 q^{40} +395.318 q^{41} +305.894 q^{43} +62.9041 q^{44} +214.767 q^{46} +630.864 q^{47} -142.520 q^{49} +72.1018 q^{50} -51.0461 q^{52} +109.277 q^{53} +183.431 q^{55} +346.956 q^{56} +421.272 q^{58} +59.0000 q^{59} +240.030 q^{61} -114.924 q^{62} +514.089 q^{64} -148.852 q^{65} -100.162 q^{67} +330.980 q^{68} +294.547 q^{70} -263.939 q^{71} -296.815 q^{73} +462.051 q^{74} -244.253 q^{76} -271.083 q^{77} +626.260 q^{79} +257.920 q^{80} -858.343 q^{82} +7.08512 q^{83} +965.148 q^{85} -664.179 q^{86} -469.143 q^{88} +132.516 q^{89} +219.982 q^{91} +324.986 q^{92} -1369.78 q^{94} -712.249 q^{95} +1476.73 q^{97} +309.449 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	$ 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.17127	−0.767661	−0.383830	−	0.923404i	$-0.625395\pi$
$2$	−2.17127	−0.767661	−0.383830	+	0.923404i	$0.625395\pi$
$3$	0	0
$3$	0	0
$4$	−3.28558	−0.410697
$5$	−9.58086	−0.856938	−0.428469	−	0.903556i	$-0.640947\pi$
$5$	−9.58086	−0.856938	−0.428469	+	0.903556i	$0.640947\pi$
$6$	0	0
$7$	14.1591	0.764520	0.382260	−	0.924055i	$-0.375146\pi$
$7$	14.1591	0.764520	0.382260	+	0.924055i	$0.375146\pi$
$8$	24.5041	1.08294
$9$	0	0
$10$	20.8027	0.657838
$11$	−19.1455	−0.524781	−0.262391	−	0.964962i	$-0.584511\pi$
$11$	−19.1455	−0.524781	−0.262391	+	0.964962i	$0.584511\pi$
$12$	0	0
$13$	15.5364	0.331464	0.165732	−	0.986171i	$-0.447001\pi$
$13$	15.5364	0.331464	0.165732	+	0.986171i	$0.447001\pi$
$14$	−30.7433	−0.586892
$15$	0	0
$16$	−26.9204	−0.420631
$17$	−100.737	−1.43720	−0.718598	−	0.695426i	$-0.755216\pi$
$17$	−100.737	−1.43720	−0.718598	+	0.695426i	$0.755216\pi$
$18$	0	0
$19$	74.3408	0.897629	0.448815	−	0.893625i	$-0.351846\pi$
$19$	74.3408	0.897629	0.448815	+	0.893625i	$0.351846\pi$
$20$	31.4787	0.351942
$21$	0	0
$22$	41.5701	0.402854
$23$	−98.9130	−0.896730	−0.448365	−	0.893851i	$-0.647994\pi$
$23$	−98.9130	−0.896730	−0.448365	+	0.893851i	$0.647994\pi$
$24$	0	0
$25$	−33.2071	−0.265657
$26$	−33.7338	−0.254452
$27$	0	0
$28$	−46.5208	−0.313986
$29$	−194.021	−1.24237	−0.621185	−	0.783664i	$-0.713348\pi$
$29$	−194.021	−1.24237	−0.621185	+	0.783664i	$0.713348\pi$
$30$	0	0
$31$	52.9293	0.306658	0.153329	−	0.988175i	$-0.451001\pi$
$31$	52.9293	0.306658	0.153329	+	0.988175i	$0.451001\pi$
$32$	−137.581	−0.760035
$33$	0	0
$34$	218.728	1.10328
$35$	−135.656	−0.655146
$36$	0	0
$37$	−212.802	−0.945525	−0.472762	−	0.881190i	$-0.656743\pi$
$37$	−212.802	−0.945525	−0.472762	+	0.881190i	$0.656743\pi$
$38$	−161.414	−0.689075
$39$	0	0
$40$	−234.770	−0.928010
$41$	395.318	1.50581	0.752906	−	0.658128i	$-0.228652\pi$
$41$	395.318	1.50581	0.752906	+	0.658128i	$0.228652\pi$
$42$	0	0
$43$	305.894	1.08485	0.542423	−	0.840106i	$-0.317507\pi$
$43$	305.894	1.08485	0.542423	+	0.840106i	$0.317507\pi$
$44$	62.9041	0.215526
$45$	0	0
$46$	214.767	0.688384
$47$	630.864	1.95789	0.978947	−	0.204116i	$-0.0654320\pi$
$47$	630.864	1.95789	0.978947	+	0.204116i	$0.0654320\pi$
$48$	0	0
$49$	−142.520	−0.415510
$50$	72.1018	0.203935
$51$	0	0
$52$	−51.0461	−0.136131
$53$	109.277	0.283214	0.141607	−	0.989923i	$-0.454773\pi$
$53$	109.277	0.283214	0.141607	+	0.989923i	$0.454773\pi$
$54$	0	0
$55$	183.431	0.449705
$56$	346.956	0.827926
$57$	0	0
$58$	421.272	0.953718
$59$	59.0000	0.130189
$59$	59.0000	0.130189
$60$	0	0
$61$	240.030	0.503815	0.251907	−	0.967751i	$-0.418942\pi$
$61$	240.030	0.503815	0.251907	+	0.967751i	$0.418942\pi$
$62$	−114.924	−0.235409
$63$	0	0
$64$	514.089	1.00408
$65$	−148.852	−0.284044
$66$	0	0
$67$	−100.162	−0.182638	−0.0913191	−	0.995822i	$-0.529108\pi$
$67$	−100.162	−0.182638	−0.0913191	+	0.995822i	$0.529108\pi$
$68$	330.980	0.590252
$69$	0	0
$70$	294.547	0.502930
$71$	−263.939	−0.441180	−0.220590	−	0.975367i	$-0.570798\pi$
$71$	−263.939	−0.441180	−0.220590	+	0.975367i	$0.570798\pi$
$72$	0	0
$73$	−296.815	−0.475884	−0.237942	−	0.971279i	$-0.576473\pi$
$73$	−296.815	−0.475884	−0.237942	+	0.971279i	$0.576473\pi$
$74$	462.051	0.725842
$75$	0	0
$76$	−244.253	−0.368654
$77$	−271.083	−0.401205
$78$	0	0
$79$	626.260	0.891896	0.445948	−	0.895059i	$-0.352867\pi$
$79$	626.260	0.891896	0.445948	+	0.895059i	$0.352867\pi$
$80$	257.920	0.360454
$81$	0	0
$82$	−858.343	−1.15595
$83$	7.08512	0.00936980	0.00468490	−	0.999989i	$-0.498509\pi$
$83$	7.08512	0.00936980	0.00468490	+	0.999989i	$0.498509\pi$
$84$	0	0
$85$	965.148	1.23159
$86$	−664.179	−0.832793
$87$	0	0
$88$	−469.143	−0.568305
$89$	132.516	0.157828	0.0789138	−	0.996881i	$-0.474855\pi$
$89$	132.516	0.157828	0.0789138	+	0.996881i	$0.474855\pi$
$90$	0	0
$91$	219.982	0.253411
$92$	324.986	0.368284
$93$	0	0
$94$	−1369.78	−1.50300
$95$	−712.249	−0.769213
$96$	0	0
$97$	1476.73	1.54577	0.772885	−	0.634546i	$-0.218813\pi$
$97$	1476.73	1.54577	0.772885	+	0.634546i	$0.218813\pi$
$98$	309.449	0.318970
$99$	0	0
$100$	109.105	0.109105
$101$	1349.75	1.32975	0.664875	−	0.746955i	$-0.268485\pi$
$101$	1349.75	1.32975	0.664875	+	0.746955i	$0.268485\pi$
$102$	0	0
$103$	1258.15	1.20359	0.601794	−	0.798651i	$-0.294453\pi$
$103$	1258.15	1.20359	0.601794	+	0.798651i	$0.294453\pi$
$104$	380.705	0.358954
$105$	0	0
$106$	−237.270	−0.217412
$107$	1690.18	1.52706	0.763532	−	0.645771i	$-0.223464\pi$
$107$	1690.18	1.52706	0.763532	+	0.645771i	$0.223464\pi$
$108$	0	0
$109$	−1293.50	−1.13665	−0.568324	−	0.822805i	$-0.692408\pi$
$109$	−1293.50	−1.13665	−0.568324	+	0.822805i	$0.692408\pi$
$110$	−398.278	−0.345221
$111$	0	0
$112$	−381.168	−0.321580
$113$	−1086.11	−0.904185	−0.452093	−	0.891971i	$-0.649322\pi$
$113$	−1086.11	−0.904185	−0.452093	+	0.891971i	$0.649322\pi$
$114$	0	0
$115$	947.672	0.768442
$116$	637.470	0.510238
$117$	0	0
$118$	−128.105	−0.0999409
$119$	−1426.35	−1.09876
$120$	0	0
$121$	−964.449	−0.724605
$122$	−521.171	−0.386759
$123$	0	0
$124$	−173.903	−0.125943
$125$	1515.76	1.08459
$126$	0	0
$127$	−681.878	−0.476432	−0.238216	−	0.971212i	$-0.576563\pi$
$127$	−681.878	−0.476432	−0.238216	+	0.971212i	$0.576563\pi$
$128$	−15.5784	−0.0107574
$129$	0	0
$130$	323.199	0.218049
$131$	1373.82	0.916267	0.458134	−	0.888883i	$-0.348518\pi$
$131$	1373.82	0.916267	0.458134	+	0.888883i	$0.348518\pi$
$132$	0	0
$133$	1052.60	0.686255
$134$	217.479	0.140204
$135$	0	0
$136$	−2468.47	−1.55639
$137$	920.396	0.573976	0.286988	−	0.957934i	$-0.407346\pi$
$137$	920.396	0.573976	0.286988	+	0.957934i	$0.407346\pi$
$138$	0	0
$139$	2435.03	1.48587	0.742936	−	0.669362i	$-0.233433\pi$
$139$	2435.03	1.48587	0.742936	+	0.669362i	$0.233433\pi$
$140$	445.709	0.269067
$141$	0	0
$142$	573.083	0.338676
$143$	−297.453	−0.173946
$144$	0	0
$145$	1858.88	1.06463
$146$	644.465	0.365317
$147$	0	0
$148$	699.177	0.388324
$149$	−1240.27	−0.681925	−0.340963	−	0.940077i	$-0.610753\pi$
$149$	−1240.27	−0.681925	−0.340963	+	0.940077i	$0.610753\pi$
$150$	0	0
$151$	3384.25	1.82389	0.911943	−	0.410318i	$-0.134582\pi$
$151$	3384.25	1.82389	0.911943	+	0.410318i	$0.134582\pi$
$152$	1821.65	0.972076
$153$	0	0
$154$	588.596	0.307990
$155$	−507.108	−0.262787
$156$	0	0
$157$	195.675	0.0994685	0.0497342	−	0.998762i	$-0.484163\pi$
$157$	195.675	0.0994685	0.0497342	+	0.998762i	$0.484163\pi$
$158$	−1359.78	−0.684674
$159$	0	0
$160$	1318.14	0.651303
$161$	−1400.52	−0.685568
$162$	0	0
$163$	135.992	0.0653479	0.0326739	−	0.999466i	$-0.489598\pi$
$163$	135.992	0.0653479	0.0326739	+	0.999466i	$0.489598\pi$
$164$	−1298.85	−0.618433
$165$	0	0
$166$	−15.3837	−0.00719283
$167$	627.557	0.290789	0.145395	−	0.989374i	$-0.453555\pi$
$167$	627.557	0.290789	0.145395	+	0.989374i	$0.453555\pi$
$168$	0	0
$169$	−1955.62	−0.890132
$170$	−2095.60	−0.945442
$171$	0	0
$172$	−1005.04	−0.445543
$173$	2869.43	1.26103	0.630515	−	0.776177i	$-0.282844\pi$
$173$	2869.43	1.26103	0.630515	+	0.776177i	$0.282844\pi$
$174$	0	0
$175$	−470.183	−0.203100
$176$	515.404	0.220739
$177$	0	0
$178$	−287.728	−0.121158
$179$	−2266.38	−0.946353	−0.473176	−	0.880968i	$-0.656893\pi$
$179$	−2266.38	−0.946353	−0.473176	+	0.880968i	$0.656893\pi$
$180$	0	0
$181$	449.089	0.184423	0.0922114	−	0.995739i	$-0.470606\pi$
$181$	449.089	0.184423	0.0922114	+	0.995739i	$0.470606\pi$
$182$	−477.640	−0.194533
$183$	0	0
$184$	−2423.77	−0.971102
$185$	2038.82	0.810256
$186$	0	0
$187$	1928.66	0.754213
$188$	−2072.75	−0.804101
$189$	0	0
$190$	1546.49	0.590494
$191$	−495.207	−0.187602	−0.0938009	−	0.995591i	$-0.529902\pi$
$191$	−495.207	−0.187602	−0.0938009	+	0.995591i	$0.529902\pi$
$192$	0	0
$193$	−2118.19	−0.790002	−0.395001	−	0.918681i	$-0.629256\pi$
$193$	−2118.19	−0.790002	−0.395001	+	0.918681i	$0.629256\pi$
$194$	−3206.39	−1.18663
$195$	0	0
$196$	468.260	0.170649
$197$	−372.268	−0.134635	−0.0673173	−	0.997732i	$-0.521444\pi$
$197$	−372.268	−0.134635	−0.0673173	+	0.997732i	$0.521444\pi$
$198$	0	0
$199$	989.744	0.352568	0.176284	−	0.984339i	$-0.443592\pi$
$199$	989.744	0.352568	0.176284	+	0.984339i	$0.443592\pi$
$200$	−813.710	−0.287690
$201$	0	0
$202$	−2930.66	−1.02080
$203$	−2747.16	−0.949816
$204$	0	0
$205$	−3787.48	−1.29039
$206$	−2731.80	−0.923948
$207$	0	0
$208$	−418.246	−0.139424
$209$	−1423.29	−0.471059
$210$	0	0
$211$	−1268.28	−0.413802	−0.206901	−	0.978362i	$-0.566338\pi$
$211$	−1268.28	−0.413802	−0.206901	+	0.978362i	$0.566338\pi$
$212$	−359.037	−0.116315
$213$	0	0
$214$	−3669.84	−1.17227
$215$	−2930.72	−0.929645
$216$	0	0
$217$	749.432	0.234446
$218$	2808.54	0.872560
$219$	0	0
$220$	−602.675	−0.184692
$221$	−1565.09	−0.476378
$222$	0	0
$223$	1129.78	0.339262	0.169631	−	0.985508i	$-0.445742\pi$
$223$	1129.78	0.339262	0.169631	+	0.985508i	$0.445742\pi$
$224$	−1948.02	−0.581062
$225$	0	0
$226$	2358.25	0.694108
$227$	161.210	0.0471359	0.0235680	−	0.999722i	$-0.492497\pi$
$227$	161.210	0.0471359	0.0235680	+	0.999722i	$0.492497\pi$
$228$	0	0
$229$	118.980	0.0343338	0.0171669	−	0.999853i	$-0.494535\pi$
$229$	118.980	0.0343338	0.0171669	+	0.999853i	$0.494535\pi$
$230$	−2057.65	−0.589903
$231$	0	0
$232$	−4754.29	−1.34541
$233$	4469.51	1.25668	0.628342	−	0.777937i	$-0.283734\pi$
$233$	4469.51	1.25668	0.628342	+	0.777937i	$0.283734\pi$
$234$	0	0
$235$	−6044.22	−1.67779
$236$	−193.849	−0.0534682
$237$	0	0
$238$	3096.99	0.843479
$239$	4682.67	1.26735	0.633675	−	0.773600i	$-0.281546\pi$
$239$	4682.67	1.26735	0.633675	+	0.773600i	$0.281546\pi$
$240$	0	0
$241$	−3836.61	−1.02547	−0.512735	−	0.858547i	$-0.671367\pi$
$241$	−3836.61	−1.02547	−0.512735	+	0.858547i	$0.671367\pi$
$242$	2094.08	0.556251
$243$	0	0
$244$	−788.638	−0.206915
$245$	1365.46	0.356066
$246$	0	0
$247$	1154.99	0.297531
$248$	1296.98	0.332091
$249$	0	0
$250$	−3291.13	−0.832597
$251$	331.121	0.0832677	0.0416338	−	0.999133i	$-0.486744\pi$
$251$	331.121	0.0832677	0.0416338	+	0.999133i	$0.486744\pi$
$252$	0	0
$253$	1893.74	0.470587
$254$	1480.54	0.365738
$255$	0	0
$256$	−4078.89	−0.995822
$257$	−1423.82	−0.345585	−0.172792	−	0.984958i	$-0.555279\pi$
$257$	−1423.82	−0.345585	−0.172792	+	0.984958i	$0.555279\pi$
$258$	0	0
$259$	−3013.08	−0.722872
$260$	489.066	0.116656
$261$	0	0
$262$	−2982.93	−0.703382
$263$	5205.39	1.22045	0.610225	−	0.792228i	$-0.291079\pi$
$263$	5205.39	1.22045	0.610225	+	0.792228i	$0.291079\pi$
$264$	0	0
$265$	−1046.97	−0.242697
$266$	−2285.48	−0.526811
$267$	0	0
$268$	329.091	0.0750090
$269$	−349.224	−0.0791544	−0.0395772	−	0.999217i	$-0.512601\pi$
$269$	−349.224	−0.0791544	−0.0395772	+	0.999217i	$0.512601\pi$
$270$	0	0
$271$	−8231.66	−1.84516	−0.922579	−	0.385808i	$-0.873923\pi$
$271$	−8231.66	−1.84516	−0.922579	+	0.385808i	$0.873923\pi$
$272$	2711.88	0.604529
$273$	0	0
$274$	−1998.43	−0.440619
$275$	635.768	0.139412
$276$	0	0
$277$	−7242.28	−1.57093	−0.785463	−	0.618908i	$-0.787575\pi$
$277$	−7242.28	−1.57093	−0.785463	+	0.618908i	$0.787575\pi$
$278$	−5287.11	−1.14065
$279$	0	0
$280$	−3324.13	−0.709482
$281$	−1769.76	−0.375713	−0.187856	−	0.982197i	$-0.560154\pi$
$281$	−1769.76	−0.375713	−0.187856	+	0.982197i	$0.560154\pi$
$282$	0	0
$283$	5303.35	1.11396	0.556981	−	0.830525i	$-0.311960\pi$
$283$	5303.35	1.11396	0.556981	+	0.830525i	$0.311960\pi$
$284$	867.191	0.181191
$285$	0	0
$286$	645.851	0.133531
$287$	5597.35	1.15122
$288$	0	0
$289$	5234.96	1.06553
$290$	−4036.14	−0.817278
$291$	0	0
$292$	975.207	0.195444
$293$	−138.305	−0.0275764	−0.0137882	−	0.999905i	$-0.504389\pi$
$293$	−138.305	−0.0275764	−0.0137882	+	0.999905i	$0.504389\pi$
$294$	0	0
$295$	−565.271	−0.111564
$296$	−5214.51	−1.02394
$297$	0	0
$298$	2692.96	0.523487
$299$	−1536.75	−0.297233
$300$	0	0
$301$	4331.18	0.829386
$302$	−7348.14	−1.40012
$303$	0	0
$304$	−2001.28	−0.377570
$305$	−2299.70	−0.431738
$306$	0	0
$307$	2464.57	0.458178	0.229089	−	0.973406i	$-0.426425\pi$
$307$	2464.57	0.458178	0.229089	+	0.973406i	$0.426425\pi$
$308$	890.665	0.164774
$309$	0	0
$310$	1101.07	0.201731
$311$	9275.20	1.69115	0.845576	−	0.533855i	$-0.179257\pi$
$311$	9275.20	1.69115	0.845576	+	0.533855i	$0.179257\pi$
$312$	0	0
$313$	5699.12	1.02918	0.514590	−	0.857437i	$-0.327944\pi$
$313$	5699.12	1.02918	0.514590	+	0.857437i	$0.327944\pi$
$314$	−424.863	−0.0763580
$315$	0	0
$316$	−2057.63	−0.366299
$317$	859.960	0.152366	0.0761832	−	0.997094i	$-0.475727\pi$
$317$	859.960	0.152366	0.0761832	+	0.997094i	$0.475727\pi$
$318$	0	0
$319$	3714.63	0.651972
$320$	−4925.41	−0.860434
$321$	0	0
$322$	3040.91	0.526283
$323$	−7488.88	−1.29007
$324$	0	0
$325$	−515.920	−0.0880557
$326$	−295.275	−0.0501650
$327$	0	0
$328$	9686.89	1.63070
$329$	8932.47	1.49685
$330$	0	0
$331$	10291.1	1.70891	0.854454	−	0.519527i	$-0.173892\pi$
$331$	10291.1	1.70891	0.854454	+	0.519527i	$0.173892\pi$
$332$	−23.2787	−0.00384815
$333$	0	0
$334$	−1362.60	−0.223227
$335$	959.640	0.156510
$336$	0	0
$337$	1861.32	0.300869	0.150434	−	0.988620i	$-0.451933\pi$
$337$	1861.32	0.300869	0.150434	+	0.988620i	$0.451933\pi$
$338$	4246.18	0.683319
$339$	0	0
$340$	−3171.07	−0.505810
$341$	−1013.36	−0.160928
$342$	0	0
$343$	−6874.52	−1.08218
$344$	7495.64	1.17482
$345$	0	0
$346$	−6230.30	−0.968044
$347$	2342.82	0.362447	0.181224	−	0.983442i	$-0.441994\pi$
$347$	2342.82	0.362447	0.181224	+	0.983442i	$0.441994\pi$
$348$	0	0
$349$	−9507.00	−1.45816	−0.729081	−	0.684428i	$-0.760052\pi$
$349$	−9507.00	−1.45816	−0.729081	+	0.684428i	$0.760052\pi$
$350$	1020.90	0.155912
$351$	0	0
$352$	2634.06	0.398852
$353$	−7366.56	−1.11071	−0.555357	−	0.831612i	$-0.687419\pi$
$353$	−7366.56	−1.11071	−0.555357	+	0.831612i	$0.687419\pi$
$354$	0	0
$355$	2528.76	0.378064
$356$	−435.391	−0.0648193
$357$	0	0
$358$	4920.93	0.726478
$359$	−7674.15	−1.12821	−0.564103	−	0.825704i	$-0.690778\pi$
$359$	−7674.15	−1.12821	−0.564103	+	0.825704i	$0.690778\pi$
$360$	0	0
$361$	−1332.44	−0.194262
$362$	−975.095	−0.141574
$363$	0	0
$364$	−722.767	−0.104075
$365$	2843.74	0.407803
$366$	0	0
$367$	4845.83	0.689238	0.344619	−	0.938743i	$-0.388008\pi$
$367$	4845.83	0.689238	0.344619	+	0.938743i	$0.388008\pi$
$368$	2662.77	0.377192
$369$	0	0
$370$	−4426.84	−0.622002
$371$	1547.26	0.216522
$372$	0	0
$373$	−3677.62	−0.510510	−0.255255	−	0.966874i	$-0.582159\pi$
$373$	−3677.62	−0.510510	−0.255255	+	0.966874i	$0.582159\pi$
$374$	−4187.65	−0.578980
$375$	0	0
$376$	15458.7	2.12027
$377$	−3014.39	−0.411800
$378$	0	0
$379$	2448.51	0.331851	0.165925	−	0.986138i	$-0.446939\pi$
$379$	2448.51	0.331851	0.165925	+	0.986138i	$0.446939\pi$
$380$	2340.15	0.315913
$381$	0	0
$382$	1075.23	0.144014
$383$	−7777.55	−1.03764	−0.518818	−	0.854885i	$-0.673628\pi$
$383$	−7777.55	−1.03764	−0.518818	+	0.854885i	$0.673628\pi$
$384$	0	0
$385$	2597.21	0.343808
$386$	4599.16	0.606453
$387$	0	0
$388$	−4851.93	−0.634843
$389$	13158.8	1.71512	0.857558	−	0.514387i	$-0.171981\pi$
$389$	13158.8	1.71512	0.857558	+	0.514387i	$0.171981\pi$
$390$	0	0
$391$	9964.21	1.28878
$392$	−3492.31	−0.449971
$393$	0	0
$394$	808.296	0.103354
$395$	−6000.11	−0.764300
$396$	0	0
$397$	−7496.45	−0.947698	−0.473849	−	0.880606i	$-0.657136\pi$
$397$	−7496.45	−0.947698	−0.473849	+	0.880606i	$0.657136\pi$
$398$	−2149.00	−0.270653
$399$	0	0
$400$	893.949	0.111744
$401$	4346.75	0.541313	0.270657	−	0.962676i	$-0.412759\pi$
$401$	4346.75	0.541313	0.270657	+	0.962676i	$0.412759\pi$
$402$	0	0
$403$	822.332	0.101646
$404$	−4434.69	−0.546124
$405$	0	0
$406$	5964.83	0.729136
$407$	4074.20	0.496193
$408$	0	0
$409$	14635.7	1.76941	0.884706	−	0.466149i	$-0.154359\pi$
$409$	14635.7	1.76941	0.884706	+	0.466149i	$0.154359\pi$
$410$	8223.66	0.990580
$411$	0	0
$412$	−4133.76	−0.494311
$413$	835.387	0.0995320
$414$	0	0
$415$	−67.8816	−0.00802934
$416$	−2137.52	−0.251924
$417$	0	0
$418$	3090.36	0.361613
$419$	6292.05	0.733620	0.366810	−	0.930296i	$-0.380450\pi$
$419$	6292.05	0.733620	0.366810	+	0.930296i	$0.380450\pi$
$420$	0	0
$421$	−10784.3	−1.24845	−0.624223	−	0.781246i	$-0.714584\pi$
$421$	−10784.3	−1.24845	−0.624223	+	0.781246i	$0.714584\pi$
$422$	2753.79	0.317660
$423$	0	0
$424$	2677.73	0.306702
$425$	3345.19	0.381802
$426$	0	0
$427$	3398.61	0.385176
$428$	−5553.21	−0.627160
$429$	0	0
$430$	6363.40	0.713652
$431$	−5954.37	−0.665457	−0.332728	−	0.943023i	$-0.607969\pi$
$431$	−5954.37	−0.665457	−0.332728	+	0.943023i	$0.607969\pi$
$432$	0	0
$433$	4561.77	0.506292	0.253146	−	0.967428i	$-0.418535\pi$
$433$	4561.77	0.506292	0.253146	+	0.967428i	$0.418535\pi$
$434$	−1627.22	−0.179975
$435$	0	0
$436$	4249.89	0.466818
$437$	−7353.28	−0.804931
$438$	0	0
$439$	−7301.09	−0.793763	−0.396882	−	0.917870i	$-0.629908\pi$
$439$	−7301.09	−0.793763	−0.396882	+	0.917870i	$0.629908\pi$
$440$	4494.79	0.487002
$441$	0	0
$442$	3398.25	0.365697
$443$	15813.1	1.69595	0.847973	−	0.530040i	$-0.177823\pi$
$443$	15813.1	1.69595	0.847973	+	0.530040i	$0.177823\pi$
$444$	0	0
$445$	−1269.62	−0.135248
$446$	−2453.05	−0.260438
$447$	0	0
$448$	7279.04	0.767639
$449$	−1604.95	−0.168691	−0.0843456	−	0.996437i	$-0.526880\pi$
$449$	−1604.95	−0.168691	−0.0843456	+	0.996437i	$0.526880\pi$
$450$	0	0
$451$	−7568.57	−0.790221
$452$	3568.51	0.371346
$453$	0	0
$454$	−350.030	−0.0361844
$455$	−2107.61	−0.217157
$456$	0	0
$457$	9914.58	1.01485	0.507423	−	0.861697i	$-0.330598\pi$
$457$	9914.58	1.01485	0.507423	+	0.861697i	$0.330598\pi$
$458$	−258.339	−0.0263567
$459$	0	0
$460$	−3113.65	−0.315597
$461$	5717.04	0.577590	0.288795	−	0.957391i	$-0.406745\pi$
$461$	5717.04	0.577590	0.288795	+	0.957391i	$0.406745\pi$
$462$	0	0
$463$	15831.2	1.58906	0.794532	−	0.607222i	$-0.207716\pi$
$463$	15831.2	1.58906	0.794532	+	0.607222i	$0.207716\pi$
$464$	5223.11	0.522579
$465$	0	0
$466$	−9704.53	−0.964707
$467$	−375.189	−0.0371770	−0.0185885	−	0.999827i	$-0.505917\pi$
$467$	−375.189	−0.0371770	−0.0185885	+	0.999827i	$0.505917\pi$
$468$	0	0
$469$	−1418.21	−0.139630
$470$	13123.6	1.28798
$471$	0	0
$472$	1445.74	0.140986
$473$	−5856.49	−0.569306
$474$	0	0
$475$	−2468.65	−0.238462
$476$	4686.37	0.451260
$477$	0	0
$478$	−10167.3	−0.972894
$479$	3601.59	0.343551	0.171775	−	0.985136i	$-0.445050\pi$
$479$	3601.59	0.343551	0.171775	+	0.985136i	$0.445050\pi$
$480$	0	0
$481$	−3306.18	−0.313407
$482$	8330.33	0.787212
$483$	0	0
$484$	3168.77	0.297593
$485$	−14148.4	−1.32463
$486$	0	0
$487$	−14132.2	−1.31498	−0.657488	−	0.753465i	$-0.728381\pi$
$487$	−14132.2	−1.31498	−0.657488	+	0.753465i	$0.728381\pi$
$488$	5881.71	0.545600
$489$	0	0
$490$	−2964.79	−0.273338
$491$	293.022	0.0269326	0.0134663	−	0.999909i	$-0.495713\pi$
$491$	293.022	0.0269326	0.0134663	+	0.999909i	$0.495713\pi$
$492$	0	0
$493$	19545.1	1.78553
$494$	−2507.80	−0.228403
$495$	0	0
$496$	−1424.88	−0.128990
$497$	−3737.14	−0.337291
$498$	0	0
$499$	15571.2	1.39692	0.698459	−	0.715650i	$-0.253869\pi$
$499$	15571.2	1.39692	0.698459	+	0.715650i	$0.253869\pi$
$500$	−4980.15	−0.445438
$501$	0	0
$502$	−718.954	−0.0639213
$503$	−12697.7	−1.12557	−0.562784	−	0.826604i	$-0.690270\pi$
$503$	−12697.7	−1.12557	−0.562784	+	0.826604i	$0.690270\pi$
$504$	0	0
$505$	−12931.7	−1.13951
$506$	−4111.83	−0.361251
$507$	0	0
$508$	2240.36	0.195669
$509$	4294.84	0.373999	0.187000	−	0.982360i	$-0.440124\pi$
$509$	4294.84	0.373999	0.187000	+	0.982360i	$0.440124\pi$
$510$	0	0
$511$	−4202.63	−0.363822
$512$	8981.00	0.775210
$513$	0	0
$514$	3091.49	0.265292
$515$	−12054.2	−1.03140
$516$	0	0
$517$	−12078.2	−1.02747
$518$	6542.22	0.554921
$519$	0	0
$520$	−3647.48	−0.307601
$521$	22351.1	1.87950	0.939751	−	0.341860i	$-0.111057\pi$
$521$	22351.1	1.87950	0.939751	+	0.341860i	$0.111057\pi$
$522$	0	0
$523$	12800.7	1.07024	0.535118	−	0.844777i	$-0.320267\pi$
$523$	12800.7	1.07024	0.535118	+	0.844777i	$0.320267\pi$
$524$	−4513.78	−0.376308
$525$	0	0
$526$	−11302.3	−0.936891
$527$	−5331.95	−0.440727
$528$	0	0
$529$	−2383.22	−0.195875
$530$	2273.25	0.186309
$531$	0	0
$532$	−3458.40	−0.281843
$533$	6141.83	0.499122
$534$	0	0
$535$	−16193.4	−1.30860
$536$	−2454.38	−0.197786
$537$	0	0
$538$	758.260	0.0607637
$539$	2728.62	0.218052
$540$	0	0
$541$	18834.0	1.49674	0.748372	−	0.663280i	$-0.230836\pi$
$541$	18834.0	1.49674	0.748372	+	0.663280i	$0.230836\pi$
$542$	17873.2	1.41646
$543$	0	0
$544$	13859.5	1.09232
$545$	12392.8	0.974037
$546$	0	0
$547$	−16024.5	−1.25258	−0.626289	−	0.779591i	$-0.715427\pi$
$547$	−16024.5	−1.25258	−0.626289	+	0.779591i	$0.715427\pi$
$548$	−3024.03	−0.235730
$549$	0	0
$550$	−1380.43	−0.107021
$551$	−14423.7	−1.11519
$552$	0	0
$553$	8867.29	0.681872
$554$	15725.0	1.20594
$555$	0	0
$556$	−8000.47	−0.610244
$557$	−12161.8	−0.925157	−0.462578	−	0.886578i	$-0.653076\pi$
$557$	−12161.8	−0.925157	−0.462578	+	0.886578i	$0.653076\pi$
$558$	0	0
$559$	4752.49	0.359587
$560$	3651.92	0.275575
$561$	0	0
$562$	3842.64	0.288420
$563$	−20165.9	−1.50958	−0.754790	−	0.655967i	$-0.772261\pi$
$563$	−20165.9	−1.50958	−0.754790	+	0.655967i	$0.772261\pi$
$564$	0	0
$565$	10405.9	0.774831
$566$	−11515.0	−0.855144
$567$	0	0
$568$	−6467.57	−0.477770
$569$	18351.0	1.35205	0.676024	−	0.736880i	$-0.263702\pi$
$569$	18351.0	1.35205	0.676024	+	0.736880i	$0.263702\pi$
$570$	0	0
$571$	−19894.1	−1.45804	−0.729021	−	0.684492i	$-0.760024\pi$
$571$	−19894.1	−1.45804	−0.729021	+	0.684492i	$0.760024\pi$
$572$	977.304	0.0714391
$573$	0	0
$574$	−12153.4	−0.883748
$575$	3284.62	0.238223
$576$	0	0
$577$	916.177	0.0661022	0.0330511	−	0.999454i	$-0.489478\pi$
$577$	916.177	0.0661022	0.0330511	+	0.999454i	$0.489478\pi$
$578$	−11366.5	−0.817968
$579$	0	0
$580$	−6107.51	−0.437242
$581$	100.319	0.00716340
$582$	0	0
$583$	−2092.16	−0.148625
$584$	−7273.16	−0.515352
$585$	0	0
$586$	300.298	0.0211693
$587$	1121.91	0.0788862	0.0394431	−	0.999222i	$-0.487442\pi$
$587$	1121.91	0.0788862	0.0394431	+	0.999222i	$0.487442\pi$
$588$	0	0
$589$	3934.81	0.275265
$590$	1227.36	0.0856432
$591$	0	0
$592$	5728.70	0.397717
$593$	−16327.9	−1.13070	−0.565352	−	0.824850i	$-0.691260\pi$
$593$	−16327.9	−1.13070	−0.565352	+	0.824850i	$0.691260\pi$
$594$	0	0
$595$	13665.6	0.941573
$596$	4075.00	0.280065
$597$	0	0
$598$	3336.71	0.228174
$599$	−24940.5	−1.70124	−0.850620	−	0.525780i	$-0.823773\pi$
$599$	−24940.5	−1.70124	−0.850620	+	0.525780i	$0.823773\pi$
$600$	0	0
$601$	17458.9	1.18496	0.592482	−	0.805584i	$-0.298148\pi$
$601$	17458.9	1.18496	0.592482	+	0.805584i	$0.298148\pi$
$602$	−9404.17	−0.636687
$603$	0	0
$604$	−11119.2	−0.749064
$605$	9240.25	0.620941
$606$	0	0
$607$	19831.7	1.32610	0.663052	−	0.748573i	$-0.269261\pi$
$607$	19831.7	1.32610	0.663052	+	0.748573i	$0.269261\pi$
$608$	−10227.9	−0.682230
$609$	0	0
$610$	4993.26	0.331428
$611$	9801.37	0.648971
$612$	0	0
$613$	4099.73	0.270125	0.135062	−	0.990837i	$-0.456877\pi$
$613$	4099.73	0.270125	0.135062	+	0.990837i	$0.456877\pi$
$614$	−5351.26	−0.351725
$615$	0	0
$616$	−6642.64	−0.434480
$617$	−19906.4	−1.29887	−0.649434	−	0.760418i	$-0.724994\pi$
$617$	−19906.4	−1.29887	−0.649434	+	0.760418i	$0.724994\pi$
$618$	0	0
$619$	−11172.3	−0.725450	−0.362725	−	0.931896i	$-0.618154\pi$
$619$	−11172.3	−0.725450	−0.362725	+	0.931896i	$0.618154\pi$
$620$	1666.14	0.107926
$621$	0	0
$622$	−20139.0	−1.29823
$623$	1876.31	0.120662
$624$	0	0
$625$	−10371.4	−0.663769
$626$	−12374.3	−0.790060
$627$	0	0
$628$	−642.905	−0.0408514
$629$	21437.0	1.35890
$630$	0	0
$631$	23430.1	1.47819	0.739096	−	0.673600i	$-0.235253\pi$
$631$	23430.1	1.47819	0.739096	+	0.673600i	$0.235253\pi$
$632$	15345.9	0.965867
$633$	0	0
$634$	−1867.21	−0.116966
$635$	6532.97	0.408273
$636$	0	0
$637$	−2214.25	−0.137726
$638$	−8065.46	−0.500493
$639$	0	0
$640$	149.254	0.00921844
$641$	16450.0	1.01363	0.506816	−	0.862055i	$-0.330823\pi$
$641$	16450.0	1.01363	0.506816	+	0.862055i	$0.330823\pi$
$642$	0	0
$643$	−11207.2	−0.687355	−0.343677	−	0.939088i	$-0.611673\pi$
$643$	−11207.2	−0.687355	−0.343677	+	0.939088i	$0.611673\pi$
$644$	4601.52	0.281561
$645$	0	0
$646$	16260.4	0.990335
$647$	−28416.4	−1.72668	−0.863341	−	0.504621i	$-0.831632\pi$
$647$	−28416.4	−1.72668	−0.863341	+	0.504621i	$0.831632\pi$
$648$	0	0
$649$	−1129.59	−0.0683207
$650$	1120.20	0.0675969
$651$	0	0
$652$	−446.812	−0.0268382
$653$	16811.3	1.00747	0.503734	−	0.863859i	$-0.331959\pi$
$653$	16811.3	1.00747	0.503734	+	0.863859i	$0.331959\pi$
$654$	0	0
$655$	−13162.4	−0.785184
$656$	−10642.1	−0.633391
$657$	0	0
$658$	−19394.8	−1.14907
$659$	7979.14	0.471659	0.235829	−	0.971794i	$-0.424219\pi$
$659$	7979.14	0.471659	0.235829	+	0.971794i	$0.424219\pi$
$660$	0	0
$661$	19673.4	1.15765	0.578824	−	0.815453i	$-0.303512\pi$
$661$	19673.4	1.15765	0.578824	+	0.815453i	$0.303512\pi$
$662$	−22344.7	−1.31186
$663$	0	0
$664$	173.614	0.0101469
$665$	−10084.8	−0.588078
$666$	0	0
$667$	19191.2	1.11407
$668$	−2061.89	−0.119426
$669$	0	0
$670$	−2083.64	−0.120146
$671$	−4595.50	−0.264393
$672$	0	0
$673$	−12964.9	−0.742588	−0.371294	−	0.928515i	$-0.621086\pi$
$673$	−12964.9	−0.742588	−0.371294	+	0.928515i	$0.621086\pi$
$674$	−4041.44	−0.230965
$675$	0	0
$676$	6425.34	0.365575
$677$	−26269.4	−1.49131	−0.745655	−	0.666332i	$-0.767863\pi$
$677$	−26269.4	−1.49131	−0.745655	+	0.666332i	$0.767863\pi$
$678$	0	0
$679$	20909.2	1.18177
$680$	23650.0	1.33373
$681$	0	0
$682$	2200.28	0.123538
$683$	8043.98	0.450650	0.225325	−	0.974284i	$-0.427656\pi$
$683$	8043.98	0.450650	0.225325	+	0.974284i	$0.427656\pi$
$684$	0	0
$685$	−8818.18	−0.491862
$686$	14926.5	0.830751
$687$	0	0
$688$	−8234.77	−0.456319
$689$	1697.77	0.0938751
$690$	0	0
$691$	3316.01	0.182557	0.0912785	−	0.995825i	$-0.470905\pi$
$691$	3316.01	0.182557	0.0912785	+	0.995825i	$0.470905\pi$
$692$	−9427.72	−0.517902
$693$	0	0
$694$	−5086.90	−0.278236
$695$	−23329.6	−1.27330
$696$	0	0
$697$	−39823.2	−2.16415
$698$	20642.3	1.11937
$699$	0	0
$700$	1544.82	0.0834127
$701$	13786.8	0.742824	0.371412	−	0.928468i	$-0.378874\pi$
$701$	13786.8	0.742824	0.371412	+	0.928468i	$0.378874\pi$
$702$	0	0
$703$	−15819.9	−0.848731
$704$	−9842.50	−0.526922
$705$	0	0
$706$	15994.8	0.852652
$707$	19111.2	1.01662
$708$	0	0
$709$	−5117.29	−0.271063	−0.135532	−	0.990773i	$-0.543274\pi$
$709$	−5117.29	−0.271063	−0.135532	+	0.990773i	$0.543274\pi$
$710$	−5490.63	−0.290225
$711$	0	0
$712$	3247.18	0.170917
$713$	−5235.40	−0.274989
$714$	0	0
$715$	2849.85	0.149061
$716$	7446.36	0.388664
$717$	0	0
$718$	16662.7	0.866079
$719$	5501.98	0.285382	0.142691	−	0.989767i	$-0.454425\pi$
$719$	5501.98	0.285382	0.142691	+	0.989767i	$0.454425\pi$
$720$	0	0
$721$	17814.3	0.920167
$722$	2893.09	0.149127
$723$	0	0
$724$	−1475.52	−0.0757420
$725$	6442.87	0.330044
$726$	0	0
$727$	−23570.5	−1.20245	−0.601226	−	0.799079i	$-0.705321\pi$
$727$	−23570.5	−1.20245	−0.601226	+	0.799079i	$0.705321\pi$
$728$	5390.45	0.274428
$729$	0	0
$730$	−6174.53	−0.313054
$731$	−30814.8	−1.55914
$732$	0	0
$733$	30142.7	1.51889	0.759446	−	0.650571i	$-0.225470\pi$
$733$	30142.7	1.51889	0.759446	+	0.650571i	$0.225470\pi$
$734$	−10521.6	−0.529101
$735$	0	0
$736$	13608.6	0.681546
$737$	1917.66	0.0958450
$738$	0	0
$739$	−17267.8	−0.859548	−0.429774	−	0.902937i	$-0.641407\pi$
$739$	−17267.8	−0.859548	−0.429774	+	0.902937i	$0.641407\pi$
$740$	−6698.72	−0.332770
$741$	0	0
$742$	−3359.53	−0.166216
$743$	−36299.4	−1.79232	−0.896161	−	0.443728i	$-0.853656\pi$
$743$	−36299.4	−1.79232	−0.896161	+	0.443728i	$0.853656\pi$
$744$	0	0
$745$	11882.8	0.584368
$746$	7985.12	0.391898
$747$	0	0
$748$	−6336.78	−0.309753
$749$	23931.4	1.16747
$750$	0	0
$751$	−19115.5	−0.928807	−0.464404	−	0.885624i	$-0.653731\pi$
$751$	−19115.5	−0.928807	−0.464404	+	0.885624i	$0.653731\pi$
$752$	−16983.1	−0.823550
$753$	0	0
$754$	6545.05	0.316123
$755$	−32424.1	−1.56296
$756$	0	0
$757$	7724.96	0.370896	0.185448	−	0.982654i	$-0.440626\pi$
$757$	7724.96	0.370896	0.185448	+	0.982654i	$0.440626\pi$
$758$	−5316.38	−0.254749
$759$	0	0
$760$	−17453.0	−0.833009
$761$	−15791.5	−0.752223	−0.376112	−	0.926574i	$-0.622739\pi$
$761$	−15791.5	−0.752223	−0.376112	+	0.926574i	$0.622739\pi$
$762$	0	0
$763$	−18314.8	−0.868990
$764$	1627.04	0.0770475
$765$	0	0
$766$	16887.2	0.796552
$767$	916.649	0.0431529
$768$	0	0
$769$	−29233.6	−1.37086	−0.685430	−	0.728139i	$-0.740386\pi$
$769$	−29233.6	−1.37086	−0.685430	+	0.728139i	$0.740386\pi$
$770$	−5639.25	−0.263928
$771$	0	0
$772$	6959.46	0.324451
$773$	37733.7	1.75574	0.877869	−	0.478901i	$-0.158965\pi$
$773$	37733.7	1.75574	0.877869	+	0.478901i	$0.158965\pi$
$774$	0	0
$775$	−1757.63	−0.0814658
$776$	36186.0	1.67397
$777$	0	0
$778$	−28571.4	−1.31663
$779$	29388.3	1.35166
$780$	0	0
$781$	5053.25	0.231523
$782$	−21635.0	−0.989343
$783$	0	0
$784$	3836.69	0.174776
$785$	−1874.73	−0.0852383
$786$	0	0
$787$	−13387.8	−0.606384	−0.303192	−	0.952929i	$-0.598052\pi$
$787$	−13387.8	−0.606384	−0.303192	+	0.952929i	$0.598052\pi$
$788$	1223.12	0.0552941
$789$	0	0
$790$	13027.9	0.586723
$791$	−15378.4	−0.691268
$792$	0	0
$793$	3729.21	0.166996
$794$	16276.8	0.727510
$795$	0	0
$796$	−3251.88	−0.144799
$797$	−38966.4	−1.73182	−0.865911	−	0.500199i	$-0.833260\pi$
$797$	−38966.4	−1.73182	−0.865911	+	0.500199i	$0.833260\pi$
$798$	0	0
$799$	−63551.4	−2.81388
$800$	4568.67	0.201909
$801$	0	0
$802$	−9437.99	−0.415545
$803$	5682.67	0.249735
$804$	0	0
$805$	13418.2	0.587489
$806$	−1785.51	−0.0780295
$807$	0	0
$808$	33074.2	1.44003
$809$	22024.2	0.957145	0.478572	−	0.878048i	$-0.341154\pi$
$809$	22024.2	0.957145	0.478572	+	0.878048i	$0.341154\pi$
$810$	0	0
$811$	4588.64	0.198679	0.0993397	−	0.995054i	$-0.468327\pi$
$811$	4588.64	0.198679	0.0993397	+	0.995054i	$0.468327\pi$
$812$	9026.00	0.390087
$813$	0	0
$814$	−8846.20	−0.380908
$815$	−1302.92	−0.0559991
$816$	0	0
$817$	22740.4	0.973789
$818$	−31778.1	−1.35831
$819$	0	0
$820$	12444.1	0.529958
$821$	45052.8	1.91517	0.957584	−	0.288154i	$-0.0930414\pi$
$821$	45052.8	1.91517	0.957584	+	0.288154i	$0.0930414\pi$
$822$	0	0
$823$	−17701.8	−0.749752	−0.374876	−	0.927075i	$-0.622315\pi$
$823$	−17701.8	−0.749752	−0.374876	+	0.927075i	$0.622315\pi$
$824$	30829.9	1.30341
$825$	0	0
$826$	−1813.85	−0.0764068
$827$	−10882.0	−0.457562	−0.228781	−	0.973478i	$-0.573474\pi$
$827$	−10882.0	−0.457562	−0.228781	+	0.973478i	$0.573474\pi$
$828$	0	0
$829$	26577.8	1.11349	0.556746	−	0.830683i	$-0.312050\pi$
$829$	26577.8	1.11349	0.556746	+	0.830683i	$0.312050\pi$
$830$	147.389	0.00616381
$831$	0	0
$832$	7987.10	0.332816
$833$	14357.0	0.597169
$834$	0	0
$835$	−6012.53	−0.249188
$836$	4676.34	0.193463
$837$	0	0
$838$	−13661.8	−0.563171
$839$	13400.2	0.551402	0.275701	−	0.961243i	$-0.411090\pi$
$839$	13400.2	0.551402	0.275701	+	0.961243i	$0.411090\pi$
$840$	0	0
$841$	13255.0	0.543483
$842$	23415.7	0.958383
$843$	0	0
$844$	4167.05	0.169947
$845$	18736.5	0.762788
$846$	0	0
$847$	−13655.7	−0.553975
$848$	−2941.77	−0.119128
$849$	0	0
$850$	−7263.32	−0.293094
$851$	21048.9	0.847880
$852$	0	0
$853$	−3838.21	−0.154065	−0.0770327	−	0.997029i	$-0.524545\pi$
$853$	−3838.21	−0.154065	−0.0770327	+	0.997029i	$0.524545\pi$
$854$	−7379.31	−0.295685
$855$	0	0
$856$	41416.2	1.65371
$857$	−44265.5	−1.76439	−0.882194	−	0.470886i	$-0.843934\pi$
$857$	−44265.5	−1.76439	−0.882194	+	0.470886i	$0.843934\pi$
$858$	0	0
$859$	−26682.8	−1.05984	−0.529921	−	0.848047i	$-0.677779\pi$
$859$	−26682.8	−1.05984	−0.529921	+	0.848047i	$0.677779\pi$
$860$	9629.12	0.381803
$861$	0	0
$862$	12928.6	0.510845
$863$	3259.02	0.128550	0.0642748	−	0.997932i	$-0.479527\pi$
$863$	3259.02	0.128550	0.0642748	+	0.997932i	$0.479527\pi$
$864$	0	0
$865$	−27491.6	−1.08063
$866$	−9904.84	−0.388661
$867$	0	0
$868$	−2462.32	−0.0962862
$869$	−11990.1	−0.468050
$870$	0	0
$871$	−1556.16	−0.0605379
$872$	−31695.9	−1.23092
$873$	0	0
$874$	15966.0	0.617914
$875$	21461.8	0.829190
$876$	0	0
$877$	−37049.0	−1.42652	−0.713259	−	0.700900i	$-0.752782\pi$
$877$	−37049.0	−1.42652	−0.713259	+	0.700900i	$0.752782\pi$
$878$	15852.7	0.609341
$879$	0	0
$880$	−4938.02	−0.189160
$881$	−7446.48	−0.284765	−0.142383	−	0.989812i	$-0.545476\pi$
$881$	−7446.48	−0.284765	−0.142383	+	0.989812i	$0.545476\pi$
$882$	0	0
$883$	23801.2	0.907106	0.453553	−	0.891229i	$-0.350156\pi$
$883$	23801.2	0.907106	0.453553	+	0.891229i	$0.350156\pi$
$884$	5142.24	0.195647
$885$	0	0
$886$	−34334.6	−1.30191
$887$	−20503.7	−0.776154	−0.388077	−	0.921627i	$-0.626860\pi$
$887$	−20503.7	−0.776154	−0.388077	+	0.921627i	$0.626860\pi$
$888$	0	0
$889$	−9654.77	−0.364242
$890$	2756.68	0.103825
$891$	0	0
$892$	−3711.97	−0.139334
$893$	46899.0	1.75746
$894$	0	0
$895$	21713.9	0.810966
$896$	−220.576	−0.00822426
$897$	0	0
$898$	3484.79	0.129498
$899$	−10269.4	−0.380982
$900$	0	0
$901$	−11008.2	−0.407034
$902$	16433.4	0.606622
$903$	0	0
$904$	−26614.2	−0.979176
$905$	−4302.66	−0.158039
$906$	0	0
$907$	−10620.2	−0.388797	−0.194399	−	0.980923i	$-0.562276\pi$
$907$	−10620.2	−0.388797	−0.194399	+	0.980923i	$0.562276\pi$
$908$	−529.666	−0.0193586
$909$	0	0
$910$	4576.20	0.166703
$911$	6121.02	0.222611	0.111305	−	0.993786i	$-0.464497\pi$
$911$	6121.02	0.222611	0.111305	+	0.993786i	$0.464497\pi$
$912$	0	0
$913$	−135.648	−0.00491709
$914$	−21527.2	−0.779057
$915$	0	0
$916$	−390.919	−0.0141008
$917$	19452.0	0.700504
$918$	0	0
$919$	−50071.6	−1.79729	−0.898645	−	0.438677i	$-0.855447\pi$
$919$	−50071.6	−1.79729	−0.898645	+	0.438677i	$0.855447\pi$
$920$	23221.8	0.832174
$921$	0	0
$922$	−12413.2	−0.443393
$923$	−4100.66	−0.146235
$924$	0	0
$925$	7066.54	0.251185
$926$	−34373.8	−1.21986
$927$	0	0
$928$	26693.6	0.944245
$929$	−39484.7	−1.39446	−0.697228	−	0.716849i	$-0.745584\pi$
$929$	−39484.7	−1.39446	−0.697228	+	0.716849i	$0.745584\pi$
$930$	0	0
$931$	−10595.0	−0.372974
$932$	−14684.9	−0.516117
$933$	0	0
$934$	814.637	0.0285393
$935$	−18478.3	−0.646314
$936$	0	0
$937$	−50370.5	−1.75617	−0.878085	−	0.478504i	$-0.841179\pi$
$937$	−50370.5	−1.75617	−0.878085	+	0.478504i	$0.841179\pi$
$938$	3079.31	0.107189
$939$	0	0
$940$	19858.8	0.689065
$941$	18333.3	0.635119	0.317560	−	0.948238i	$-0.397137\pi$
$941$	18333.3	0.635119	0.317560	+	0.948238i	$0.397137\pi$
$942$	0	0
$943$	−39102.1	−1.35031
$944$	−1588.30	−0.0547615
$945$	0	0
$946$	12716.0	0.437034
$947$	12078.7	0.414473	0.207236	−	0.978291i	$-0.433553\pi$
$947$	12078.7	0.414473	0.207236	+	0.978291i	$0.433553\pi$
$948$	0	0
$949$	−4611.44	−0.157738
$950$	5360.10	0.183058
$951$	0	0
$952$	−34951.3	−1.18989
$953$	−40916.7	−1.39079	−0.695395	−	0.718628i	$-0.744770\pi$
$953$	−40916.7	−1.39079	−0.695395	+	0.718628i	$0.744770\pi$
$954$	0	0
$955$	4744.51	0.160763
$956$	−15385.3	−0.520497
$957$	0	0
$958$	−7820.03	−0.263730
$959$	13032.0	0.438816
$960$	0	0
$961$	−26989.5	−0.905961
$962$	7178.62	0.240590
$963$	0	0
$964$	12605.5	0.421157
$965$	20294.0	0.676983
$966$	0	0
$967$	27358.4	0.909812	0.454906	−	0.890539i	$-0.349673\pi$
$967$	27358.4	0.909812	0.454906	+	0.890539i	$0.349673\pi$
$968$	−23632.9	−0.784701
$969$	0	0
$970$	30720.0	1.01687
$971$	25783.7	0.852151	0.426076	−	0.904688i	$-0.359896\pi$
$971$	25783.7	0.852151	0.426076	+	0.904688i	$0.359896\pi$
$972$	0	0
$973$	34477.8	1.13598
$974$	30684.9	1.00945
$975$	0	0
$976$	−6461.70	−0.211920
$977$	−17329.6	−0.567475	−0.283738	−	0.958902i	$-0.591575\pi$
$977$	−17329.6	−0.567475	−0.283738	+	0.958902i	$0.591575\pi$
$978$	0	0
$979$	−2537.09	−0.0828249
$980$	−4486.33	−0.146235
$981$	0	0
$982$	−636.231	−0.0206751
$983$	23593.4	0.765527	0.382764	−	0.923846i	$-0.374972\pi$
$983$	23593.4	0.765527	0.382764	+	0.923846i	$0.374972\pi$
$984$	0	0
$985$	3566.65	0.115374
$986$	−42437.7	−1.37068
$987$	0	0
$988$	−3794.81	−0.122195
$989$	−30256.9	−0.972813
$990$	0	0
$991$	27698.5	0.887863	0.443931	−	0.896061i	$-0.353583\pi$
$991$	27698.5	0.887863	0.443931	+	0.896061i	$0.353583\pi$
$992$	−7282.07	−0.233071
$993$	0	0
$994$	8114.34	0.258925
$995$	−9482.60	−0.302129
$996$	0	0
$997$	21554.5	0.684691	0.342346	−	0.939574i	$-0.388779\pi$
$997$	21554.5	0.684691	0.342346	+	0.939574i	$0.388779\pi$
$998$	−33809.3	−1.07236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	531.4.a.f.1.3		8
3.2	odd	2		177.4.a.c.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.4.a.c.1.6	✓	8	3.2	odd	2
531.4.a.f.1.3		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 531 = 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	531.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17127 q^{2} -3.28558 q^{4} -9.58086 q^{5} +14.1591 q^{7} +24.5041 q^{8} +O(q^{10})\)	\(q-2.17127 q^{2} -3.28558 q^{4} -9.58086 q^{5} +14.1591 q^{7} +24.5041 q^{8} +20.8027 q^{10} -19.1455 q^{11} +15.5364 q^{13} -30.7433 q^{14} -26.9204 q^{16} -100.737 q^{17} +74.3408 q^{19} +31.4787 q^{20} +41.5701 q^{22} -98.9130 q^{23} -33.2071 q^{25} -33.7338 q^{26} -46.5208 q^{28} -194.021 q^{29} +52.9293 q^{31} -137.581 q^{32} +218.728 q^{34} -135.656 q^{35} -212.802 q^{37} -161.414 q^{38} -234.770 q^{40} +395.318 q^{41} +305.894 q^{43} +62.9041 q^{44} +214.767 q^{46} +630.864 q^{47} -142.520 q^{49} +72.1018 q^{50} -51.0461 q^{52} +109.277 q^{53} +183.431 q^{55} +346.956 q^{56} +421.272 q^{58} +59.0000 q^{59} +240.030 q^{61} -114.924 q^{62} +514.089 q^{64} -148.852 q^{65} -100.162 q^{67} +330.980 q^{68} +294.547 q^{70} -263.939 q^{71} -296.815 q^{73} +462.051 q^{74} -244.253 q^{76} -271.083 q^{77} +626.260 q^{79} +257.920 q^{80} -858.343 q^{82} +7.08512 q^{83} +965.148 q^{85} -664.179 q^{86} -469.143 q^{88} +132.516 q^{89} +219.982 q^{91} +324.986 q^{92} -1369.78 q^{94} -712.249 q^{95} +1476.73 q^{97} +309.449 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8}+O(q^{10}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8	\( 8 q - 2 q^{2} + 38 q^{4} + 12 q^{5} + 53 q^{7} - 3 q^{8} + 29 q^{10} + 27 q^{11} + 89 q^{13} + 37 q^{14} + 362 q^{16} - 79 q^{17} + 288 q^{19} - 457 q^{20} + 596 q^{22} - 202 q^{23} + 264 q^{25} - 270 q^{26} + 702 q^{28} + 114 q^{29} + 538 q^{31} - 316 q^{32} + 498 q^{34} + 196 q^{35} + 395 q^{37} - 397 q^{38} + 918 q^{40} + 39 q^{41} + 527 q^{43} - 64 q^{44} - 539 q^{46} - 860 q^{47} + 347 q^{49} + 591 q^{50} - 644 q^{52} + 812 q^{53} + 536 q^{55} + 2218 q^{56} - 1154 q^{58} + 472 q^{59} - 460 q^{61} + 2014 q^{62} - 451 q^{64} + 986 q^{65} + 1934 q^{67} + 69 q^{68} - 1028 q^{70} + 1687 q^{71} + 1980 q^{73} + 2400 q^{74} - 940 q^{76} + 821 q^{77} + 3319 q^{79} + 2119 q^{80} + 429 q^{82} - 2057 q^{83} + 566 q^{85} + 6690 q^{86} + 1189 q^{88} - 1668 q^{89} + 2427 q^{91} + 980 q^{92} + 332 q^{94} - 2146 q^{95} + 1956 q^{97} + 2026 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 38 * q^4 + 12 * q^5 + 53 * q^7 - 3 * q^8 + 29 * q^10 + 27 * q^11 + 89 * q^13 + 37 * q^14 + 362 * q^16 - 79 * q^17 + 288 * q^19 - 457 * q^20 + 596 * q^22 - 202 * q^23 + 264 * q^25 - 270 * q^26 + 702 * q^28 + 114 * q^29 + 538 * q^31 - 316 * q^32 + 498 * q^34 + 196 * q^35 + 395 * q^37 - 397 * q^38 + 918 * q^40 + 39 * q^41 + 527 * q^43 - 64 * q^44 - 539 * q^46 - 860 * q^47 + 347 * q^49 + 591 * q^50 - 644 * q^52 + 812 * q^53 + 536 * q^55 + 2218 * q^56 - 1154 * q^58 + 472 * q^59 - 460 * q^61 + 2014 * q^62 - 451 * q^64 + 986 * q^65 + 1934 * q^67 + 69 * q^68 - 1028 * q^70 + 1687 * q^71 + 1980 * q^73 + 2400 * q^74 - 940 * q^76 + 821 * q^77 + 3319 * q^79 + 2119 * q^80 + 429 * q^82 - 2057 * q^83 + 566 * q^85 + 6690 * q^86 + 1189 * q^88 - 1668 * q^89 + 2427 * q^91 + 980 * q^92 + 332 * q^94 - 2146 * q^95 + 1956 * q^97 + 2026 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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