LMFDB - Embedded newform 522.4.a.j.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [522,4,Mod(1,522)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(522, 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("522.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 522 = 2 \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	522.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:	$30.7989970230$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	522.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} +4.00000 q^{4} -9.69694 q^{5} -27.5959 q^{7} +8.00000 q^{8} +O(q^{10})$	\(q+2.00000 q^{2} +4.00000 q^{4} -9.69694 q^{5} -27.5959 q^{7} +8.00000 q^{8} -19.3939 q^{10} +52.3485 q^{11} -5.40408 q^{13} -55.1918 q^{14} +16.0000 q^{16} -17.1918 q^{17} +44.2020 q^{19} -38.7878 q^{20} +104.697 q^{22} +205.060 q^{23} -30.9694 q^{25} -10.8082 q^{26} -110.384 q^{28} -29.0000 q^{29} +299.994 q^{31} +32.0000 q^{32} -34.3837 q^{34} +267.596 q^{35} +29.7980 q^{37} +88.4041 q^{38} -77.5755 q^{40} +43.9592 q^{41} +64.8230 q^{43} +209.394 q^{44} +410.120 q^{46} +499.499 q^{47} +418.535 q^{49} -61.9388 q^{50} -21.6163 q^{52} +351.627 q^{53} -507.620 q^{55} -220.767 q^{56} -58.0000 q^{58} -522.372 q^{59} +484.606 q^{61} +599.989 q^{62} +64.0000 q^{64} +52.4031 q^{65} -504.990 q^{67} -68.7673 q^{68} +535.192 q^{70} -481.283 q^{71} +3.11019 q^{73} +59.5959 q^{74} +176.808 q^{76} -1444.60 q^{77} -1043.27 q^{79} -155.151 q^{80} +87.9184 q^{82} +1007.08 q^{83} +166.708 q^{85} +129.646 q^{86} +418.788 q^{88} +295.637 q^{89} +149.131 q^{91} +820.241 q^{92} +998.999 q^{94} -428.624 q^{95} +428.949 q^{97} +837.069 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8}+O(q^{10}) $ 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 16 * q^7 + 16 * q^8	$ 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8} + 20 q^{10} + 90 q^{11} - 50 q^{13} - 32 q^{14} + 32 q^{16} + 44 q^{17} + 108 q^{19} + 40 q^{20} + 180 q^{22} + 28 q^{23} + 232 q^{25} - 100 q^{26} - 64 q^{28} - 58 q^{29} + 66 q^{31} + 64 q^{32} + 88 q^{34} + 496 q^{35} + 40 q^{37} + 216 q^{38} + 80 q^{40} - 304 q^{41} - 130 q^{43} + 360 q^{44} + 56 q^{46} + 514 q^{47} + 210 q^{49} + 464 q^{50} - 200 q^{52} + 958 q^{53} + 234 q^{55} - 128 q^{56} - 116 q^{58} + 180 q^{59} + 1028 q^{61} + 132 q^{62} + 128 q^{64} - 826 q^{65} - 912 q^{67} + 176 q^{68} + 992 q^{70} - 796 q^{71} - 856 q^{73} + 80 q^{74} + 432 q^{76} - 1008 q^{77} - 318 q^{79} + 160 q^{80} - 608 q^{82} + 1828 q^{83} + 1372 q^{85} - 260 q^{86} + 720 q^{88} + 944 q^{89} - 368 q^{91} + 112 q^{92} + 1028 q^{94} + 828 q^{95} + 368 q^{97} + 420 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 16 * q^7 + 16 * q^8 + 20 * q^10 + 90 * q^11 - 50 * q^13 - 32 * q^14 + 32 * q^16 + 44 * q^17 + 108 * q^19 + 40 * q^20 + 180 * q^22 + 28 * q^23 + 232 * q^25 - 100 * q^26 - 64 * q^28 - 58 * q^29 + 66 * q^31 + 64 * q^32 + 88 * q^34 + 496 * q^35 + 40 * q^37 + 216 * q^38 + 80 * q^40 - 304 * q^41 - 130 * q^43 + 360 * q^44 + 56 * q^46 + 514 * q^47 + 210 * q^49 + 464 * q^50 - 200 * q^52 + 958 * q^53 + 234 * q^55 - 128 * q^56 - 116 * q^58 + 180 * q^59 + 1028 * q^61 + 132 * q^62 + 128 * q^64 - 826 * q^65 - 912 * q^67 + 176 * q^68 + 992 * q^70 - 796 * q^71 - 856 * q^73 + 80 * q^74 + 432 * q^76 - 1008 * q^77 - 318 * q^79 + 160 * q^80 - 608 * q^82 + 1828 * q^83 + 1372 * q^85 - 260 * q^86 + 720 * q^88 + 944 * q^89 - 368 * q^91 + 112 * q^92 + 1028 * q^94 + 828 * q^95 + 368 * q^97 + 420 * 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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	−9.69694	−0.867321	−0.433660	−	0.901076i	$-0.642778\pi$
$5$	−9.69694	−0.867321	−0.433660	+	0.901076i	$0.642778\pi$
$6$	0	0
$7$	−27.5959	−1.49004	−0.745020	−	0.667042i	$-0.767560\pi$
$7$	−27.5959	−1.49004	−0.745020	+	0.667042i	$0.767560\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	−19.3939	−0.613288
$11$	52.3485	1.43488	0.717439	−	0.696621i	$-0.245314\pi$
$11$	52.3485	1.43488	0.717439	+	0.696621i	$0.245314\pi$
$12$	0	0
$13$	−5.40408	−0.115294	−0.0576470	−	0.998337i	$-0.518360\pi$
$13$	−5.40408	−0.115294	−0.0576470	+	0.998337i	$0.518360\pi$
$14$	−55.1918	−1.05362
$15$	0	0
$16$	16.0000	0.250000
$17$	−17.1918	−0.245273	−0.122636	−	0.992452i	$-0.539135\pi$
$17$	−17.1918	−0.245273	−0.122636	+	0.992452i	$0.539135\pi$
$18$	0	0
$19$	44.2020	0.533718	0.266859	−	0.963736i	$-0.414014\pi$
$19$	44.2020	0.533718	0.266859	+	0.963736i	$0.414014\pi$
$20$	−38.7878	−0.433660
$21$	0	0
$22$	104.697	1.01461
$23$	205.060	1.85904	0.929522	−	0.368767i	$-0.120220\pi$
$23$	205.060	1.85904	0.929522	+	0.368767i	$0.120220\pi$
$24$	0	0
$25$	−30.9694	−0.247755
$26$	−10.8082	−0.0815252
$27$	0	0
$28$	−110.384	−0.745020
$29$	−29.0000	−0.185695
$29$	−29.0000	−0.185695
$30$	0	0
$31$	299.994	1.73808	0.869042	−	0.494739i	$-0.164736\pi$
$31$	299.994	1.73808	0.869042	+	0.494739i	$0.164736\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−34.3837	−0.173434
$35$	267.596	1.29234
$36$	0	0
$37$	29.7980	0.132399	0.0661994	−	0.997806i	$-0.478913\pi$
$37$	29.7980	0.132399	0.0661994	+	0.997806i	$0.478913\pi$
$38$	88.4041	0.377396
$39$	0	0
$40$	−77.5755	−0.306644
$41$	43.9592	0.167446	0.0837228	−	0.996489i	$-0.473319\pi$
$41$	43.9592	0.167446	0.0837228	+	0.996489i	$0.473319\pi$
$42$	0	0
$43$	64.8230	0.229893	0.114947	−	0.993372i	$-0.463330\pi$
$43$	64.8230	0.229893	0.114947	+	0.993372i	$0.463330\pi$
$44$	209.394	0.717439
$45$	0	0
$46$	410.120	1.31454
$47$	499.499	1.55020	0.775101	−	0.631837i	$-0.217699\pi$
$47$	499.499	1.55020	0.775101	+	0.631837i	$0.217699\pi$
$48$	0	0
$49$	418.535	1.22022
$50$	−61.9388	−0.175189
$51$	0	0
$52$	−21.6163	−0.0576470
$53$	351.627	0.911314	0.455657	−	0.890156i	$-0.349404\pi$
$53$	351.627	0.911314	0.455657	+	0.890156i	$0.349404\pi$
$54$	0	0
$55$	−507.620	−1.24450
$56$	−220.767	−0.526809
$57$	0	0
$58$	−58.0000	−0.131306
$59$	−522.372	−1.15266	−0.576331	−	0.817216i	$-0.695516\pi$
$59$	−522.372	−1.15266	−0.576331	+	0.817216i	$0.695516\pi$
$60$	0	0
$61$	484.606	1.01717	0.508586	−	0.861011i	$-0.330168\pi$
$61$	484.606	1.01717	0.508586	+	0.861011i	$0.330168\pi$
$62$	599.989	1.22901
$63$	0	0
$64$	64.0000	0.125000
$65$	52.4031	0.0999969
$66$	0	0
$67$	−504.990	−0.920811	−0.460405	−	0.887709i	$-0.652296\pi$
$67$	−504.990	−0.920811	−0.460405	+	0.887709i	$0.652296\pi$
$68$	−68.7673	−0.122636
$69$	0	0
$70$	535.192	0.913824
$71$	−481.283	−0.804475	−0.402238	−	0.915535i	$-0.631767\pi$
$71$	−481.283	−0.804475	−0.402238	+	0.915535i	$0.631767\pi$
$72$	0	0
$73$	3.11019	0.00498659	0.00249329	−	0.999997i	$-0.499206\pi$
$73$	3.11019	0.00498659	0.00249329	+	0.999997i	$0.499206\pi$
$74$	59.5959	0.0936201
$75$	0	0
$76$	176.808	0.266859
$77$	−1444.60	−2.13802
$78$	0	0
$79$	−1043.27	−1.48578	−0.742890	−	0.669414i	$-0.766545\pi$
$79$	−1043.27	−1.48578	−0.742890	+	0.669414i	$0.766545\pi$
$80$	−155.151	−0.216830
$81$	0	0
$82$	87.9184	0.118402
$83$	1007.08	1.33182	0.665912	−	0.746030i	$-0.268042\pi$
$83$	1007.08	1.33182	0.665912	+	0.746030i	$0.268042\pi$
$84$	0	0
$85$	166.708	0.212730
$86$	129.646	0.162559
$87$	0	0
$88$	418.788	0.507306
$89$	295.637	0.352106	0.176053	−	0.984381i	$-0.443667\pi$
$89$	295.637	0.352106	0.176053	+	0.984381i	$0.443667\pi$
$90$	0	0
$91$	149.131	0.171793
$92$	820.241	0.929522
$93$	0	0
$94$	998.999	1.09616
$95$	−428.624	−0.462905
$96$	0	0
$97$	428.949	0.449002	0.224501	−	0.974474i	$-0.427925\pi$
$97$	428.949	0.449002	0.224501	+	0.974474i	$0.427925\pi$
$98$	837.069	0.862824
$99$	0	0
$100$	−123.878	−0.123878
$101$	1212.16	1.19420	0.597101	−	0.802166i	$-0.296319\pi$
$101$	1212.16	1.19420	0.597101	+	0.802166i	$0.296319\pi$
$102$	0	0
$103$	1032.07	0.987310	0.493655	−	0.869658i	$-0.335661\pi$
$103$	1032.07	0.987310	0.493655	+	0.869658i	$0.335661\pi$
$104$	−43.2327	−0.0407626
$105$	0	0
$106$	703.253	0.644396
$107$	1176.06	1.06256	0.531280	−	0.847196i	$-0.321711\pi$
$107$	1176.06	1.06256	0.531280	+	0.847196i	$0.321711\pi$
$108$	0	0
$109$	−2167.86	−1.90498	−0.952491	−	0.304568i	$-0.901488\pi$
$109$	−2167.86	−1.90498	−0.952491	+	0.304568i	$0.901488\pi$
$110$	−1015.24	−0.879994
$111$	0	0
$112$	−441.535	−0.372510
$113$	−1623.65	−1.35169	−0.675843	−	0.737046i	$-0.736220\pi$
$113$	−1623.65	−1.35169	−0.675843	+	0.737046i	$0.736220\pi$
$114$	0	0
$115$	−1988.46	−1.61239
$116$	−116.000	−0.0928477
$117$	0	0
$118$	−1044.74	−0.815056
$119$	474.424	0.365466
$120$	0	0
$121$	1409.36	1.05887
$122$	969.212	0.719249
$123$	0	0
$124$	1199.98	0.869042
$125$	1512.43	1.08220
$126$	0	0
$127$	333.473	0.233000	0.116500	−	0.993191i	$-0.462833\pi$
$127$	333.473	0.233000	0.116500	+	0.993191i	$0.462833\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	104.806	0.0707085
$131$	933.492	0.622592	0.311296	−	0.950313i	$-0.399237\pi$
$131$	933.492	0.622592	0.311296	+	0.950313i	$0.399237\pi$
$132$	0	0
$133$	−1219.80	−0.795261
$134$	−1009.98	−0.651112
$135$	0	0
$136$	−137.535	−0.0867169
$137$	1090.32	0.679944	0.339972	−	0.940436i	$-0.389582\pi$
$137$	1090.32	0.679944	0.339972	+	0.940436i	$0.389582\pi$
$138$	0	0
$139$	−1924.60	−1.17440	−0.587202	−	0.809440i	$-0.699771\pi$
$139$	−1924.60	−1.17440	−0.587202	+	0.809440i	$0.699771\pi$
$140$	1070.38	0.646171
$141$	0	0
$142$	−962.565	−0.568850
$143$	−282.895	−0.165433
$144$	0	0
$145$	281.211	0.161057
$146$	6.22039	0.00352605
$147$	0	0
$148$	119.192	0.0661994
$149$	1703.62	0.936687	0.468343	−	0.883546i	$-0.344851\pi$
$149$	1703.62	0.936687	0.468343	+	0.883546i	$0.344851\pi$
$150$	0	0
$151$	1082.81	0.583560	0.291780	−	0.956486i	$-0.405753\pi$
$151$	1082.81	0.583560	0.291780	+	0.956486i	$0.405753\pi$
$152$	353.616	0.188698
$153$	0	0
$154$	−2889.21	−1.51181
$155$	−2909.03	−1.50748
$156$	0	0
$157$	2407.90	1.22402	0.612010	−	0.790850i	$-0.290361\pi$
$157$	2407.90	1.22402	0.612010	+	0.790850i	$0.290361\pi$
$158$	−2086.53	−1.05060
$159$	0	0
$160$	−310.302	−0.153322
$161$	−5658.82	−2.77005
$162$	0	0
$163$	1061.59	0.510123	0.255061	−	0.966925i	$-0.417904\pi$
$163$	1061.59	0.510123	0.255061	+	0.966925i	$0.417904\pi$
$164$	175.837	0.0837228
$165$	0	0
$166$	2014.16	0.941742
$167$	−1315.63	−0.609618	−0.304809	−	0.952414i	$-0.598593\pi$
$167$	−1315.63	−0.609618	−0.304809	+	0.952414i	$0.598593\pi$
$168$	0	0
$169$	−2167.80	−0.986707
$170$	333.416	0.150423
$171$	0	0
$172$	259.292	0.114947
$173$	653.800	0.287327	0.143663	−	0.989627i	$-0.454112\pi$
$173$	653.800	0.287327	0.143663	+	0.989627i	$0.454112\pi$
$174$	0	0
$175$	854.629	0.369165
$176$	837.576	0.358719
$177$	0	0
$178$	591.273	0.248977
$179$	−2055.28	−0.858208	−0.429104	−	0.903255i	$-0.641171\pi$
$179$	−2055.28	−0.858208	−0.429104	+	0.903255i	$0.641171\pi$
$180$	0	0
$181$	3398.36	1.39557	0.697785	−	0.716307i	$-0.254169\pi$
$181$	3398.36	1.39557	0.697785	+	0.716307i	$0.254169\pi$
$182$	298.261	0.121476
$183$	0	0
$184$	1640.48	0.657271
$185$	−288.949	−0.114832
$186$	0	0
$187$	−899.966	−0.351936
$188$	1998.00	0.775101
$189$	0	0
$190$	−857.249	−0.327323
$191$	−4853.92	−1.83883	−0.919417	−	0.393285i	$-0.871339\pi$
$191$	−4853.92	−1.83883	−0.919417	+	0.393285i	$0.871339\pi$
$192$	0	0
$193$	−4877.24	−1.81902	−0.909512	−	0.415677i	$-0.863545\pi$
$193$	−4877.24	−1.81902	−0.909512	+	0.415677i	$0.863545\pi$
$194$	857.898	0.317492
$195$	0	0
$196$	1674.14	0.610109
$197$	1031.42	0.373022	0.186511	−	0.982453i	$-0.440282\pi$
$197$	1031.42	0.373022	0.186511	+	0.982453i	$0.440282\pi$
$198$	0	0
$199$	−1167.70	−0.415960	−0.207980	−	0.978133i	$-0.566689\pi$
$199$	−1167.70	−0.415960	−0.207980	+	0.978133i	$0.566689\pi$
$200$	−247.755	−0.0875946
$201$	0	0
$202$	2424.32	0.844428
$203$	800.282	0.276693
$204$	0	0
$205$	−426.269	−0.145229
$206$	2064.14	0.698134
$207$	0	0
$208$	−86.4653	−0.0288235
$209$	2313.91	0.765820
$210$	0	0
$211$	794.341	0.259169	0.129585	−	0.991568i	$-0.458636\pi$
$211$	794.341	0.259169	0.129585	+	0.991568i	$0.458636\pi$
$212$	1406.51	0.455657
$213$	0	0
$214$	2352.12	0.751343
$215$	−628.584	−0.199391
$216$	0	0
$217$	−8278.62	−2.58981
$218$	−4335.71	−1.34703
$219$	0	0
$220$	−2030.48	−0.622250
$221$	92.9061	0.0282785
$222$	0	0
$223$	5136.08	1.54232	0.771161	−	0.636641i	$-0.219676\pi$
$223$	5136.08	1.54232	0.771161	+	0.636641i	$0.219676\pi$
$224$	−883.069	−0.263404
$225$	0	0
$226$	−3247.31	−0.955786
$227$	5032.84	1.47155	0.735774	−	0.677228i	$-0.236819\pi$
$227$	5032.84	1.47155	0.735774	+	0.677228i	$0.236819\pi$
$228$	0	0
$229$	−6213.12	−1.79290	−0.896451	−	0.443142i	$-0.853864\pi$
$229$	−6213.12	−1.79290	−0.896451	+	0.443142i	$0.853864\pi$
$230$	−3976.91	−1.14013
$231$	0	0
$232$	−232.000	−0.0656532
$233$	3117.59	0.876566	0.438283	−	0.898837i	$-0.355587\pi$
$233$	3117.59	0.876566	0.438283	+	0.898837i	$0.355587\pi$
$234$	0	0
$235$	−4843.62	−1.34452
$236$	−2089.49	−0.576331
$237$	0	0
$238$	948.849	0.258423
$239$	−7193.09	−1.94679	−0.973394	−	0.229136i	$-0.926410\pi$
$239$	−7193.09	−1.94679	−0.973394	+	0.229136i	$0.926410\pi$
$240$	0	0
$241$	−837.347	−0.223810	−0.111905	−	0.993719i	$-0.535695\pi$
$241$	−837.347	−0.223810	−0.111905	+	0.993719i	$0.535695\pi$
$242$	2818.72	0.748737
$243$	0	0
$244$	1938.42	0.508586
$245$	−4058.51	−1.05832
$246$	0	0
$247$	−238.871	−0.0615345
$248$	2399.96	0.614505
$249$	0	0
$250$	3024.85	0.765234
$251$	−4306.56	−1.08298	−0.541489	−	0.840708i	$-0.682139\pi$
$251$	−4306.56	−1.08298	−0.541489	+	0.840708i	$0.682139\pi$
$252$	0	0
$253$	10734.6	2.66750
$254$	666.947	0.164756
$255$	0	0
$256$	256.000	0.0625000
$257$	−401.238	−0.0973873	−0.0486936	−	0.998814i	$-0.515506\pi$
$257$	−401.238	−0.0973873	−0.0486936	+	0.998814i	$0.515506\pi$
$258$	0	0
$259$	−822.302	−0.197279
$260$	209.612	0.0499985
$261$	0	0
$262$	1866.98	0.440239
$263$	2682.72	0.628988	0.314494	−	0.949260i	$-0.398165\pi$
$263$	2682.72	0.628988	0.314494	+	0.949260i	$0.398165\pi$
$264$	0	0
$265$	−3409.70	−0.790401
$266$	−2439.59	−0.562334
$267$	0	0
$268$	−2019.96	−0.460405
$269$	4732.38	1.07263	0.536316	−	0.844017i	$-0.319816\pi$
$269$	4732.38	1.07263	0.536316	+	0.844017i	$0.319816\pi$
$270$	0	0
$271$	4529.66	1.01534	0.507670	−	0.861552i	$-0.330507\pi$
$271$	4529.66	1.01534	0.507670	+	0.861552i	$0.330507\pi$
$272$	−275.069	−0.0613181
$273$	0	0
$274$	2180.64	0.480793
$275$	−1621.20	−0.355498
$276$	0	0
$277$	7158.69	1.55279	0.776397	−	0.630245i	$-0.217045\pi$
$277$	7158.69	1.55279	0.776397	+	0.630245i	$0.217045\pi$
$278$	−3849.19	−0.830429
$279$	0	0
$280$	2140.77	0.456912
$281$	5769.75	1.22489	0.612446	−	0.790512i	$-0.290186\pi$
$281$	5769.75	1.22489	0.612446	+	0.790512i	$0.290186\pi$
$282$	0	0
$283$	−3815.25	−0.801389	−0.400694	−	0.916212i	$-0.631231\pi$
$283$	−3815.25	−0.801389	−0.400694	+	0.916212i	$0.631231\pi$
$284$	−1925.13	−0.402238
$285$	0	0
$286$	−565.791	−0.116979
$287$	−1213.09	−0.249501
$288$	0	0
$289$	−4617.44	−0.939841
$290$	562.422	0.113885
$291$	0	0
$292$	12.4408	0.00249329
$293$	−5416.66	−1.08002	−0.540008	−	0.841660i	$-0.681579\pi$
$293$	−5416.66	−1.08002	−0.540008	+	0.841660i	$0.681579\pi$
$294$	0	0
$295$	5065.41	0.999728
$296$	238.384	0.0468100
$297$	0	0
$298$	3407.25	0.662338
$299$	−1108.16	−0.214337
$300$	0	0
$301$	−1788.85	−0.342550
$302$	2165.61	0.412639
$303$	0	0
$304$	707.233	0.133430
$305$	−4699.20	−0.882214
$306$	0	0
$307$	4929.06	0.916339	0.458169	−	0.888865i	$-0.348505\pi$
$307$	4929.06	0.916339	0.458169	+	0.888865i	$0.348505\pi$
$308$	−5778.42	−1.06901
$309$	0	0
$310$	−5818.05	−1.06595
$311$	−1117.93	−0.203832	−0.101916	−	0.994793i	$-0.532497\pi$
$311$	−1117.93	−0.203832	−0.101916	+	0.994793i	$0.532497\pi$
$312$	0	0
$313$	2032.21	0.366988	0.183494	−	0.983021i	$-0.441259\pi$
$313$	2032.21	0.366988	0.183494	+	0.983021i	$0.441259\pi$
$314$	4815.79	0.865512
$315$	0	0
$316$	−4173.06	−0.742890
$317$	−2051.75	−0.363526	−0.181763	−	0.983342i	$-0.558180\pi$
$317$	−2051.75	−0.363526	−0.181763	+	0.983342i	$0.558180\pi$
$318$	0	0
$319$	−1518.11	−0.266450
$320$	−620.604	−0.108415
$321$	0	0
$322$	−11317.6	−1.95872
$323$	−759.914	−0.130906
$324$	0	0
$325$	167.361	0.0285647
$326$	2123.18	0.360711
$327$	0	0
$328$	351.673	0.0592010
$329$	−13784.1	−2.30986
$330$	0	0
$331$	−9652.92	−1.60294	−0.801469	−	0.598036i	$-0.795948\pi$
$331$	−9652.92	−1.60294	−0.801469	+	0.598036i	$0.795948\pi$
$332$	4028.32	0.665912
$333$	0	0
$334$	−2631.25	−0.431065
$335$	4896.85	0.798638
$336$	0	0
$337$	4277.57	0.691437	0.345719	−	0.938338i	$-0.387635\pi$
$337$	4277.57	0.691437	0.345719	+	0.938338i	$0.387635\pi$
$338$	−4335.59	−0.697707
$339$	0	0
$340$	666.833	0.106365
$341$	15704.2	2.49394
$342$	0	0
$343$	−2084.45	−0.328133
$344$	518.584	0.0812795
$345$	0	0
$346$	1307.60	0.203171
$347$	9475.83	1.46596	0.732982	−	0.680248i	$-0.238128\pi$
$347$	9475.83	1.46596	0.732982	+	0.680248i	$0.238128\pi$
$348$	0	0
$349$	2968.57	0.455313	0.227656	−	0.973742i	$-0.426894\pi$
$349$	2968.57	0.455313	0.227656	+	0.973742i	$0.426894\pi$
$350$	1709.26	0.261039
$351$	0	0
$352$	1675.15	0.253653
$353$	−3969.16	−0.598461	−0.299231	−	0.954181i	$-0.596730\pi$
$353$	−3969.16	−0.598461	−0.299231	+	0.954181i	$0.596730\pi$
$354$	0	0
$355$	4666.97	0.697738
$356$	1182.55	0.176053
$357$	0	0
$358$	−4110.57	−0.606845
$359$	3503.03	0.514994	0.257497	−	0.966279i	$-0.417102\pi$
$359$	3503.03	0.514994	0.257497	+	0.966279i	$0.417102\pi$
$360$	0	0
$361$	−4905.18	−0.715145
$362$	6796.72	0.986817
$363$	0	0
$364$	596.522	0.0858963
$365$	−30.1594	−0.00432497
$366$	0	0
$367$	−8253.53	−1.17393	−0.586963	−	0.809614i	$-0.699677\pi$
$367$	−8253.53	−1.17393	−0.586963	+	0.809614i	$0.699677\pi$
$368$	3280.96	0.464761
$369$	0	0
$370$	−577.898	−0.0811986
$371$	−9703.46	−1.35789
$372$	0	0
$373$	1253.95	0.174067	0.0870335	−	0.996205i	$-0.472261\pi$
$373$	1253.95	0.174067	0.0870335	+	0.996205i	$0.472261\pi$
$374$	−1799.93	−0.248856
$375$	0	0
$376$	3996.00	0.548079
$377$	156.718	0.0214096
$378$	0	0
$379$	−6966.52	−0.944185	−0.472092	−	0.881549i	$-0.656501\pi$
$379$	−6966.52	−0.944185	−0.472092	+	0.881549i	$0.656501\pi$
$380$	−1714.50	−0.231452
$381$	0	0
$382$	−9707.84	−1.30025
$383$	9821.84	1.31037	0.655186	−	0.755467i	$-0.272590\pi$
$383$	9821.84	1.31037	0.655186	+	0.755467i	$0.272590\pi$
$384$	0	0
$385$	14008.2	1.85435
$386$	−9754.49	−1.28624
$387$	0	0
$388$	1715.80	0.224501
$389$	−3942.96	−0.513923	−0.256962	−	0.966422i	$-0.582721\pi$
$389$	−3942.96	−0.513923	−0.256962	+	0.966422i	$0.582721\pi$
$390$	0	0
$391$	−3525.36	−0.455972
$392$	3348.28	0.431412
$393$	0	0
$394$	2062.83	0.263767
$395$	10116.5	1.28865
$396$	0	0
$397$	8725.50	1.10307	0.551537	−	0.834150i	$-0.314041\pi$
$397$	8725.50	1.10307	0.551537	+	0.834150i	$0.314041\pi$
$398$	−2335.40	−0.294128
$399$	0	0
$400$	−495.510	−0.0619388
$401$	−3986.86	−0.496494	−0.248247	−	0.968697i	$-0.579854\pi$
$401$	−3986.86	−0.496494	−0.248247	+	0.968697i	$0.579854\pi$
$402$	0	0
$403$	−1621.19	−0.200391
$404$	4848.64	0.597101
$405$	0	0
$406$	1600.56	0.195652
$407$	1559.88	0.189976
$408$	0	0
$409$	−704.847	−0.0852138	−0.0426069	−	0.999092i	$-0.513566\pi$
$409$	−704.847	−0.0852138	−0.0426069	+	0.999092i	$0.513566\pi$
$410$	−852.539	−0.102692
$411$	0	0
$412$	4128.28	0.493655
$413$	14415.3	1.71751
$414$	0	0
$415$	−9765.60	−1.15512
$416$	−172.931	−0.0203813
$417$	0	0
$418$	4627.82	0.541517
$419$	−6705.33	−0.781807	−0.390903	−	0.920432i	$-0.627837\pi$
$419$	−6705.33	−0.781807	−0.390903	+	0.920432i	$0.627837\pi$
$420$	0	0
$421$	8996.55	1.04148	0.520742	−	0.853714i	$-0.325655\pi$
$421$	8996.55	1.04148	0.520742	+	0.853714i	$0.325655\pi$
$422$	1588.68	0.183260
$423$	0	0
$424$	2813.01	0.322198
$425$	532.421	0.0607675
$426$	0	0
$427$	−13373.2	−1.51563
$428$	4704.24	0.531280
$429$	0	0
$430$	−1257.17	−0.140991
$431$	−5383.16	−0.601619	−0.300810	−	0.953684i	$-0.597257\pi$
$431$	−5383.16	−0.601619	−0.300810	+	0.953684i	$0.597257\pi$
$432$	0	0
$433$	−6901.64	−0.765985	−0.382993	−	0.923751i	$-0.625107\pi$
$433$	−6901.64	−0.765985	−0.382993	+	0.923751i	$0.625107\pi$
$434$	−16557.2	−1.83127
$435$	0	0
$436$	−8671.42	−0.952491
$437$	9064.08	0.992205
$438$	0	0
$439$	12119.4	1.31760	0.658802	−	0.752316i	$-0.271063\pi$
$439$	12119.4	1.31760	0.658802	+	0.752316i	$0.271063\pi$
$440$	−4060.96	−0.439997
$441$	0	0
$442$	185.812	0.0199959
$443$	−5229.40	−0.560849	−0.280425	−	0.959876i	$-0.590475\pi$
$443$	−5229.40	−0.560849	−0.280425	+	0.959876i	$0.590475\pi$
$444$	0	0
$445$	−2866.77	−0.305389
$446$	10272.2	1.09059
$447$	0	0
$448$	−1766.14	−0.186255
$449$	10220.3	1.07422	0.537112	−	0.843511i	$-0.319515\pi$
$449$	10220.3	1.07422	0.537112	+	0.843511i	$0.319515\pi$
$450$	0	0
$451$	2301.20	0.240264
$452$	−6494.61	−0.675843
$453$	0	0
$454$	10065.7	1.04054
$455$	−1446.11	−0.148999
$456$	0	0
$457$	12030.6	1.23143	0.615717	−	0.787967i	$-0.288866\pi$
$457$	12030.6	1.23143	0.615717	+	0.787967i	$0.288866\pi$
$458$	−12426.2	−1.26777
$459$	0	0
$460$	−7953.82	−0.806193
$461$	−1063.24	−0.107418	−0.0537092	−	0.998557i	$-0.517104\pi$
$461$	−1063.24	−0.107418	−0.0537092	+	0.998557i	$0.517104\pi$
$462$	0	0
$463$	15498.8	1.55570	0.777851	−	0.628449i	$-0.216310\pi$
$463$	15498.8	1.55570	0.777851	+	0.628449i	$0.216310\pi$
$464$	−464.000	−0.0464238
$465$	0	0
$466$	6235.17	0.619826
$467$	−13215.6	−1.30952	−0.654759	−	0.755838i	$-0.727230\pi$
$467$	−13215.6	−1.30952	−0.654759	+	0.755838i	$0.727230\pi$
$468$	0	0
$469$	13935.7	1.37204
$470$	−9687.23	−0.950721
$471$	0	0
$472$	−4178.98	−0.407528
$473$	3393.38	0.329869
$474$	0	0
$475$	−1368.91	−0.132231
$476$	1897.70	0.182733
$477$	0	0
$478$	−14386.2	−1.37659
$479$	19058.5	1.81796	0.908981	−	0.416837i	$-0.136861\pi$
$479$	19058.5	1.81796	0.908981	+	0.416837i	$0.136861\pi$
$480$	0	0
$481$	−161.031	−0.0152648
$482$	−1674.69	−0.158258
$483$	0	0
$484$	5637.45	0.529437
$485$	−4159.49	−0.389428
$486$	0	0
$487$	−487.214	−0.0453343	−0.0226671	−	0.999743i	$-0.507216\pi$
$487$	−487.214	−0.0453343	−0.0226671	+	0.999743i	$0.507216\pi$
$488$	3876.85	0.359624
$489$	0	0
$490$	−8117.01	−0.748345
$491$	6462.89	0.594025	0.297013	−	0.954874i	$-0.404010\pi$
$491$	6462.89	0.594025	0.297013	+	0.954874i	$0.404010\pi$
$492$	0	0
$493$	498.563	0.0455460
$494$	−477.743	−0.0435115
$495$	0	0
$496$	4799.91	0.434521
$497$	13281.4	1.19870
$498$	0	0
$499$	−7947.79	−0.713010	−0.356505	−	0.934293i	$-0.616032\pi$
$499$	−7947.79	−0.713010	−0.356505	+	0.934293i	$0.616032\pi$
$500$	6049.70	0.541102
$501$	0	0
$502$	−8613.12	−0.765781
$503$	2307.81	0.204573	0.102287	−	0.994755i	$-0.467384\pi$
$503$	2307.81	0.204573	0.102287	+	0.994755i	$0.467384\pi$
$504$	0	0
$505$	−11754.2	−1.03576
$506$	21469.2	1.88621
$507$	0	0
$508$	1333.89	0.116500
$509$	16496.2	1.43651	0.718253	−	0.695782i	$-0.244942\pi$
$509$	16496.2	1.43651	0.718253	+	0.695782i	$0.244942\pi$
$510$	0	0
$511$	−85.8287	−0.00743021
$512$	512.000	0.0441942
$513$	0	0
$514$	−802.476	−0.0688632
$515$	−10007.9	−0.856314
$516$	0	0
$517$	26148.0	2.22435
$518$	−1644.60	−0.139498
$519$	0	0
$520$	419.224	0.0353542
$521$	10386.5	0.873398	0.436699	−	0.899608i	$-0.356147\pi$
$521$	10386.5	0.873398	0.436699	+	0.899608i	$0.356147\pi$
$522$	0	0
$523$	−918.174	−0.0767666	−0.0383833	−	0.999263i	$-0.512221\pi$
$523$	−918.174	−0.0767666	−0.0383833	+	0.999263i	$0.512221\pi$
$524$	3733.97	0.311296
$525$	0	0
$526$	5365.44	0.444761
$527$	−5157.45	−0.426304
$528$	0	0
$529$	29882.7	2.45604
$530$	−6819.40	−0.558898
$531$	0	0
$532$	−4879.18	−0.397631
$533$	−237.559	−0.0193055
$534$	0	0
$535$	−11404.2	−0.921580
$536$	−4039.92	−0.325556
$537$	0	0
$538$	9464.75	0.758465
$539$	21909.7	1.75086
$540$	0	0
$541$	−12860.2	−1.02200	−0.511001	−	0.859580i	$-0.670725\pi$
$541$	−12860.2	−1.02200	−0.511001	+	0.859580i	$0.670725\pi$
$542$	9059.32	0.717954
$543$	0	0
$544$	−550.139	−0.0433585
$545$	21021.6	1.65223
$546$	0	0
$547$	4471.84	0.349547	0.174774	−	0.984609i	$-0.444081\pi$
$547$	4471.84	0.349547	0.174774	+	0.984609i	$0.444081\pi$
$548$	4361.28	0.339972
$549$	0	0
$550$	−3242.40	−0.251375
$551$	−1281.86	−0.0991090
$552$	0	0
$553$	28789.9	2.21387
$554$	14317.4	1.09799
$555$	0	0
$556$	−7698.39	−0.587202
$557$	−2649.40	−0.201542	−0.100771	−	0.994910i	$-0.532131\pi$
$557$	−2649.40	−0.201542	−0.100771	+	0.994910i	$0.532131\pi$
$558$	0	0
$559$	−350.309	−0.0265053
$560$	4281.53	0.323085
$561$	0	0
$562$	11539.5	0.866130
$563$	−12615.8	−0.944395	−0.472198	−	0.881493i	$-0.656539\pi$
$563$	−12615.8	−0.944395	−0.472198	+	0.881493i	$0.656539\pi$
$564$	0	0
$565$	15744.5	1.17234
$566$	−7630.50	−0.566667
$567$	0	0
$568$	−3850.26	−0.284425
$569$	−617.443	−0.0454913	−0.0227456	−	0.999741i	$-0.507241\pi$
$569$	−617.443	−0.0454913	−0.0227456	+	0.999741i	$0.507241\pi$
$570$	0	0
$571$	8005.50	0.586725	0.293362	−	0.956001i	$-0.405226\pi$
$571$	8005.50	0.586725	0.293362	+	0.956001i	$0.405226\pi$
$572$	−1131.58	−0.0827164
$573$	0	0
$574$	−2426.19	−0.176424
$575$	−6350.59	−0.460588
$576$	0	0
$577$	11761.1	0.848565	0.424282	−	0.905530i	$-0.360526\pi$
$577$	11761.1	0.848565	0.424282	+	0.905530i	$0.360526\pi$
$578$	−9234.88	−0.664568
$579$	0	0
$580$	1124.84	0.0805287
$581$	−27791.3	−1.98447
$582$	0	0
$583$	18407.1	1.30762
$584$	24.8816	0.00176302
$585$	0	0
$586$	−10833.3	−0.763686
$587$	1211.03	0.0851528	0.0425764	−	0.999093i	$-0.486443\pi$
$587$	1211.03	0.0851528	0.0425764	+	0.999093i	$0.486443\pi$
$588$	0	0
$589$	13260.4	0.927646
$590$	10130.8	0.706914
$591$	0	0
$592$	476.767	0.0330997
$593$	−459.376	−0.0318117	−0.0159058	−	0.999873i	$-0.505063\pi$
$593$	−459.376	−0.0318117	−0.0159058	+	0.999873i	$0.505063\pi$
$594$	0	0
$595$	−4600.47	−0.316976
$596$	6814.50	0.468343
$597$	0	0
$598$	−2216.32	−0.151559
$599$	−16686.2	−1.13820	−0.569099	−	0.822269i	$-0.692708\pi$
$599$	−16686.2	−1.13820	−0.569099	+	0.822269i	$0.692708\pi$
$600$	0	0
$601$	11625.9	0.789072	0.394536	−	0.918881i	$-0.370905\pi$
$601$	11625.9	0.789072	0.394536	+	0.918881i	$0.370905\pi$
$602$	−3577.70	−0.242219
$603$	0	0
$604$	4331.22	0.291780
$605$	−13666.5	−0.918384
$606$	0	0
$607$	−21556.9	−1.44146	−0.720731	−	0.693215i	$-0.756194\pi$
$607$	−21556.9	−1.44146	−0.720731	+	0.693215i	$0.756194\pi$
$608$	1414.47	0.0943489
$609$	0	0
$610$	−9398.39	−0.623819
$611$	−2699.34	−0.178729
$612$	0	0
$613$	−19273.2	−1.26988	−0.634940	−	0.772561i	$-0.718975\pi$
$613$	−19273.2	−1.26988	−0.634940	+	0.772561i	$0.718975\pi$
$614$	9858.11	0.647949
$615$	0	0
$616$	−11556.8	−0.755906
$617$	3270.70	0.213409	0.106704	−	0.994291i	$-0.465970\pi$
$617$	3270.70	0.213409	0.106704	+	0.994291i	$0.465970\pi$
$618$	0	0
$619$	75.8068	0.00492234	0.00246117	−	0.999997i	$-0.499217\pi$
$619$	75.8068	0.00492234	0.00246117	+	0.999997i	$0.499217\pi$
$620$	−11636.1	−0.753738
$621$	0	0
$622$	−2235.85	−0.144131
$623$	−8158.37	−0.524652
$624$	0	0
$625$	−10794.7	−0.690862
$626$	4064.41	0.259499
$627$	0	0
$628$	9631.58	0.612010
$629$	−512.282	−0.0324738
$630$	0	0
$631$	−4915.71	−0.310129	−0.155064	−	0.987904i	$-0.549559\pi$
$631$	−4915.71	−0.310129	−0.155064	+	0.987904i	$0.549559\pi$
$632$	−8346.13	−0.525302
$633$	0	0
$634$	−4103.50	−0.257052
$635$	−3233.67	−0.202086
$636$	0	0
$637$	−2261.80	−0.140684
$638$	−3036.21	−0.188409
$639$	0	0
$640$	−1241.21	−0.0766610
$641$	−3317.63	−0.204428	−0.102214	−	0.994762i	$-0.532593\pi$
$641$	−3317.63	−0.204428	−0.102214	+	0.994762i	$0.532593\pi$
$642$	0	0
$643$	−27222.7	−1.66961	−0.834805	−	0.550546i	$-0.814420\pi$
$643$	−27222.7	−1.66961	−0.834805	+	0.550546i	$0.814420\pi$
$644$	−22635.3	−1.38502
$645$	0	0
$646$	−1519.83	−0.0925648
$647$	−24433.6	−1.48468	−0.742338	−	0.670025i	$-0.766283\pi$
$647$	−24433.6	−1.48468	−0.742338	+	0.670025i	$0.766283\pi$
$648$	0	0
$649$	−27345.4	−1.65393
$650$	334.722	0.0201983
$651$	0	0
$652$	4246.35	0.255061
$653$	−5755.36	−0.344907	−0.172454	−	0.985018i	$-0.555170\pi$
$653$	−5755.36	−0.344907	−0.172454	+	0.985018i	$0.555170\pi$
$654$	0	0
$655$	−9052.01	−0.539987
$656$	703.347	0.0418614
$657$	0	0
$658$	−27568.3	−1.63332
$659$	−965.300	−0.0570603	−0.0285301	−	0.999593i	$-0.509083\pi$
$659$	−965.300	−0.0570603	−0.0285301	+	0.999593i	$0.509083\pi$
$660$	0	0
$661$	−22791.0	−1.34110	−0.670549	−	0.741866i	$-0.733941\pi$
$661$	−22791.0	−1.34110	−0.670549	+	0.741866i	$0.733941\pi$
$662$	−19305.8	−1.13345
$663$	0	0
$664$	8056.64	0.470871
$665$	11828.3	0.689746
$666$	0	0
$667$	−5946.75	−0.345216
$668$	−5262.50	−0.304809
$669$	0	0
$670$	9793.71	0.564722
$671$	25368.4	1.45952
$672$	0	0
$673$	12412.1	0.710925	0.355463	−	0.934690i	$-0.384323\pi$
$673$	12412.1	0.710925	0.355463	+	0.934690i	$0.384323\pi$
$674$	8555.15	0.488920
$675$	0	0
$676$	−8671.18	−0.493354
$677$	−8998.03	−0.510816	−0.255408	−	0.966833i	$-0.582210\pi$
$677$	−8998.03	−0.510816	−0.255408	+	0.966833i	$0.582210\pi$
$678$	0	0
$679$	−11837.2	−0.669030
$680$	1333.67	0.0752114
$681$	0	0
$682$	31408.5	1.76348
$683$	−13439.1	−0.752904	−0.376452	−	0.926436i	$-0.622856\pi$
$683$	−13439.1	−0.752904	−0.376452	+	0.926436i	$0.622856\pi$
$684$	0	0
$685$	−10572.8	−0.589730
$686$	−4168.90	−0.232025
$687$	0	0
$688$	1037.17	0.0574733
$689$	−1900.22	−0.105069
$690$	0	0
$691$	−16220.4	−0.892983	−0.446492	−	0.894788i	$-0.647327\pi$
$691$	−16220.4	−0.892983	−0.446492	+	0.894788i	$0.647327\pi$
$692$	2615.20	0.143663
$693$	0	0
$694$	18951.7	1.03659
$695$	18662.7	1.01858
$696$	0	0
$697$	−755.739	−0.0410698
$698$	5937.15	0.321955
$699$	0	0
$700$	3418.51	0.184582
$701$	18919.6	1.01938	0.509689	−	0.860359i	$-0.329761\pi$
$701$	18919.6	1.01938	0.509689	+	0.860359i	$0.329761\pi$
$702$	0	0
$703$	1317.13	0.0706636
$704$	3350.30	0.179360
$705$	0	0
$706$	−7938.31	−0.423176
$707$	−33450.6	−1.77941
$708$	0	0
$709$	−4781.99	−0.253302	−0.126651	−	0.991947i	$-0.540423\pi$
$709$	−4781.99	−0.253302	−0.126651	+	0.991947i	$0.540423\pi$
$710$	9333.94	0.493375
$711$	0	0
$712$	2365.09	0.124488
$713$	61516.9	3.23117
$714$	0	0
$715$	2743.22	0.143483
$716$	−8221.14	−0.429104
$717$	0	0
$718$	7006.06	0.364156
$719$	−13020.1	−0.675336	−0.337668	−	0.941265i	$-0.609638\pi$
$719$	−13020.1	−0.675336	−0.337668	+	0.941265i	$0.609638\pi$
$720$	0	0
$721$	−28480.9	−1.47113
$722$	−9810.36	−0.505684
$723$	0	0
$724$	13593.4	0.697785
$725$	898.112	0.0460070
$726$	0	0
$727$	−8738.68	−0.445804	−0.222902	−	0.974841i	$-0.571553\pi$
$727$	−8738.68	−0.445804	−0.222902	+	0.974841i	$0.571553\pi$
$728$	1193.04	0.0607379
$729$	0	0
$730$	−60.3187	−0.00305821
$731$	−1114.43	−0.0563865
$732$	0	0
$733$	−12775.2	−0.643744	−0.321872	−	0.946783i	$-0.604312\pi$
$733$	−12775.2	−0.643744	−0.321872	+	0.946783i	$0.604312\pi$
$734$	−16507.1	−0.830091
$735$	0	0
$736$	6561.93	0.328636
$737$	−26435.4	−1.32125
$738$	0	0
$739$	5397.68	0.268683	0.134342	−	0.990935i	$-0.457108\pi$
$739$	5397.68	0.268683	0.134342	+	0.990935i	$0.457108\pi$
$740$	−1155.80	−0.0574161
$741$	0	0
$742$	−19406.9	−0.960176
$743$	37984.2	1.87551	0.937757	−	0.347293i	$-0.112899\pi$
$743$	37984.2	1.87551	0.937757	+	0.347293i	$0.112899\pi$
$744$	0	0
$745$	−16519.9	−0.812408
$746$	2507.90	0.123084
$747$	0	0
$748$	−3599.87	−0.175968
$749$	−32454.4	−1.58326
$750$	0	0
$751$	26384.8	1.28202	0.641009	−	0.767534i	$-0.278516\pi$
$751$	26384.8	1.28202	0.641009	+	0.767534i	$0.278516\pi$
$752$	7991.99	0.387550
$753$	0	0
$754$	313.437	0.0151389
$755$	−10499.9	−0.506133
$756$	0	0
$757$	6889.95	0.330805	0.165403	−	0.986226i	$-0.447108\pi$
$757$	6889.95	0.330805	0.165403	+	0.986226i	$0.447108\pi$
$758$	−13933.0	−0.667639
$759$	0	0
$760$	−3429.00	−0.163662
$761$	−31579.3	−1.50427	−0.752134	−	0.659010i	$-0.770975\pi$
$761$	−31579.3	−1.50427	−0.752134	+	0.659010i	$0.770975\pi$
$762$	0	0
$763$	59824.0	2.83850
$764$	−19415.7	−0.919417
$765$	0	0
$766$	19643.7	0.926574
$767$	2822.94	0.132895
$768$	0	0
$769$	−12232.7	−0.573632	−0.286816	−	0.957986i	$-0.592597\pi$
$769$	−12232.7	−0.573632	−0.286816	+	0.957986i	$0.592597\pi$
$770$	28016.5	1.31123
$771$	0	0
$772$	−19509.0	−0.909512
$773$	−9958.46	−0.463365	−0.231682	−	0.972792i	$-0.574423\pi$
$773$	−9958.46	−0.463365	−0.231682	+	0.972792i	$0.574423\pi$
$774$	0	0
$775$	−9290.64	−0.430619
$776$	3431.59	0.158746
$777$	0	0
$778$	−7885.92	−0.363398
$779$	1943.09	0.0893688
$780$	0	0
$781$	−25194.4	−1.15432
$782$	−7050.72	−0.322421
$783$	0	0
$784$	6696.55	0.305054
$785$	−23349.2	−1.06162
$786$	0	0
$787$	103.911	0.00470653	0.00235326	−	0.999997i	$-0.499251\pi$
$787$	103.911	0.00470653	0.00235326	+	0.999997i	$0.499251\pi$
$788$	4125.67	0.186511
$789$	0	0
$790$	20233.0	0.911211
$791$	44806.2	2.01406
$792$	0	0
$793$	−2618.85	−0.117274
$794$	17451.0	0.779991
$795$	0	0
$796$	−4670.80	−0.207980
$797$	−30624.4	−1.36107	−0.680534	−	0.732717i	$-0.738252\pi$
$797$	−30624.4	−1.36107	−0.680534	+	0.732717i	$0.738252\pi$
$798$	0	0
$799$	−8587.31	−0.380222
$800$	−991.020	−0.0437973
$801$	0	0
$802$	−7973.71	−0.351074
$803$	162.814	0.00715514
$804$	0	0
$805$	54873.3	2.40252
$806$	−3242.39	−0.141698
$807$	0	0
$808$	9697.27	0.422214
$809$	7737.50	0.336262	0.168131	−	0.985765i	$-0.446227\pi$
$809$	7737.50	0.336262	0.168131	+	0.985765i	$0.446227\pi$
$810$	0	0
$811$	−12974.1	−0.561756	−0.280878	−	0.959744i	$-0.590626\pi$
$811$	−12974.1	−0.561756	−0.280878	+	0.959744i	$0.590626\pi$
$812$	3201.13	0.138347
$813$	0	0
$814$	3119.76	0.134333
$815$	−10294.2	−0.442440
$816$	0	0
$817$	2865.31	0.122698
$818$	−1409.69	−0.0602552
$819$	0	0
$820$	−1705.08	−0.0726145
$821$	−3264.26	−0.138762	−0.0693809	−	0.997590i	$-0.522102\pi$
$821$	−3264.26	−0.138762	−0.0693809	+	0.997590i	$0.522102\pi$
$822$	0	0
$823$	23509.7	0.995745	0.497872	−	0.867250i	$-0.334115\pi$
$823$	23509.7	0.995745	0.497872	+	0.867250i	$0.334115\pi$
$824$	8256.56	0.349067
$825$	0	0
$826$	28830.7	1.21447
$827$	−27116.5	−1.14019	−0.570093	−	0.821580i	$-0.693093\pi$
$827$	−27116.5	−1.14019	−0.570093	+	0.821580i	$0.693093\pi$
$828$	0	0
$829$	15393.6	0.644923	0.322462	−	0.946583i	$-0.395490\pi$
$829$	15393.6	0.644923	0.322462	+	0.946583i	$0.395490\pi$
$830$	−19531.2	−0.816792
$831$	0	0
$832$	−345.861	−0.0144118
$833$	−7195.38	−0.299286
$834$	0	0
$835$	12757.5	0.528734
$836$	9255.64	0.382910
$837$	0	0
$838$	−13410.7	−0.552821
$839$	7906.51	0.325343	0.162672	−	0.986680i	$-0.447989\pi$
$839$	7906.51	0.325343	0.162672	+	0.986680i	$0.447989\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	17993.1	0.736441
$843$	0	0
$844$	3177.37	0.129585
$845$	21021.0	0.855791
$846$	0	0
$847$	−38892.6	−1.57777
$848$	5626.02	0.227828
$849$	0	0
$850$	1064.84	0.0429691
$851$	6110.38	0.246135
$852$	0	0
$853$	−11887.1	−0.477146	−0.238573	−	0.971125i	$-0.576680\pi$
$853$	−11887.1	−0.477146	−0.238573	+	0.971125i	$0.576680\pi$
$854$	−26746.3	−1.07171
$855$	0	0
$856$	9408.47	0.375672
$857$	655.328	0.0261208	0.0130604	−	0.999915i	$-0.495843\pi$
$857$	655.328	0.0261208	0.0130604	+	0.999915i	$0.495843\pi$
$858$	0	0
$859$	−48758.4	−1.93669	−0.968345	−	0.249617i	$-0.919695\pi$
$859$	−48758.4	−1.93669	−0.968345	+	0.249617i	$0.919695\pi$
$860$	−2514.34	−0.0996956
$861$	0	0
$862$	−10766.3	−0.425409
$863$	10398.2	0.410150	0.205075	−	0.978746i	$-0.434256\pi$
$863$	10398.2	0.410150	0.205075	+	0.978746i	$0.434256\pi$
$864$	0	0
$865$	−6339.86	−0.249204
$866$	−13803.3	−0.541633
$867$	0	0
$868$	−33114.5	−1.29491
$869$	−54613.4	−2.13191
$870$	0	0
$871$	2729.01	0.106164
$872$	−17342.8	−0.673513
$873$	0	0
$874$	18128.2	0.701595
$875$	−41736.8	−1.61253
$876$	0	0
$877$	−31024.2	−1.19454	−0.597272	−	0.802039i	$-0.703749\pi$
$877$	−31024.2	−1.19454	−0.597272	+	0.802039i	$0.703749\pi$
$878$	24238.8	0.931687
$879$	0	0
$880$	−8121.92	−0.311125
$881$	1510.50	0.0577641	0.0288821	−	0.999583i	$-0.490805\pi$
$881$	1510.50	0.0577641	0.0288821	+	0.999583i	$0.490805\pi$
$882$	0	0
$883$	−32376.4	−1.23392	−0.616961	−	0.786993i	$-0.711637\pi$
$883$	−32376.4	−1.23392	−0.616961	+	0.786993i	$0.711637\pi$
$884$	371.624	0.0141392
$885$	0	0
$886$	−10458.8	−0.396580
$887$	−10474.6	−0.396510	−0.198255	−	0.980150i	$-0.563527\pi$
$887$	−10474.6	−0.396510	−0.198255	+	0.980150i	$0.563527\pi$
$888$	0	0
$889$	−9202.51	−0.347179
$890$	−5733.54	−0.215942
$891$	0	0
$892$	20544.3	0.771161
$893$	22078.9	0.827371
$894$	0	0
$895$	19930.0	0.744341
$896$	−3532.28	−0.131702
$897$	0	0
$898$	20440.6	0.759591
$899$	−8699.84	−0.322754
$900$	0	0
$901$	−6045.11	−0.223520
$902$	4602.39	0.169892
$903$	0	0
$904$	−12989.2	−0.477893
$905$	−32953.7	−1.21041
$906$	0	0
$907$	−34634.3	−1.26793	−0.633965	−	0.773362i	$-0.718574\pi$
$907$	−34634.3	−1.26793	−0.633965	+	0.773362i	$0.718574\pi$
$908$	20131.4	0.735774
$909$	0	0
$910$	−2892.22	−0.105358
$911$	−1186.14	−0.0431377	−0.0215688	−	0.999767i	$-0.506866\pi$
$911$	−1186.14	−0.0431377	−0.0215688	+	0.999767i	$0.506866\pi$
$912$	0	0
$913$	52719.1	1.91101
$914$	24061.1	0.870755
$915$	0	0
$916$	−24852.5	−0.896451
$917$	−25760.6	−0.927687
$918$	0	0
$919$	−5685.58	−0.204080	−0.102040	−	0.994780i	$-0.532537\pi$
$919$	−5685.58	−0.204080	−0.102040	+	0.994780i	$0.532537\pi$
$920$	−15907.6	−0.570065
$921$	0	0
$922$	−2126.47	−0.0759563
$923$	2600.89	0.0927512
$924$	0	0
$925$	−922.824	−0.0328025
$926$	30997.6	1.10005
$927$	0	0
$928$	−928.000	−0.0328266
$929$	19143.7	0.676087	0.338044	−	0.941130i	$-0.390235\pi$
$929$	19143.7	0.676087	0.338044	+	0.941130i	$0.390235\pi$
$930$	0	0
$931$	18500.1	0.651252
$932$	12470.3	0.438283
$933$	0	0
$934$	−26431.2	−0.925969
$935$	8726.92	0.305241
$936$	0	0
$937$	−53623.5	−1.86959	−0.934794	−	0.355191i	$-0.884416\pi$
$937$	−53623.5	−1.86959	−0.934794	+	0.355191i	$0.884416\pi$
$938$	27871.3	0.970182
$939$	0	0
$940$	−19374.5	−0.672261
$941$	7947.57	0.275328	0.137664	−	0.990479i	$-0.456041\pi$
$941$	7947.57	0.275328	0.137664	+	0.990479i	$0.456041\pi$
$942$	0	0
$943$	9014.28	0.311289
$944$	−8357.96	−0.288166
$945$	0	0
$946$	6786.77	0.233252
$947$	41047.8	1.40853	0.704263	−	0.709939i	$-0.251277\pi$
$947$	41047.8	1.40853	0.704263	+	0.709939i	$0.251277\pi$
$948$	0	0
$949$	−16.8077	−0.000574924 0
$950$	−2737.82	−0.0935017
$951$	0	0
$952$	3795.40	0.129212
$953$	12000.8	0.407915	0.203957	−	0.978980i	$-0.434620\pi$
$953$	12000.8	0.407915	0.203957	+	0.978980i	$0.434620\pi$
$954$	0	0
$955$	47068.1	1.59486
$956$	−28772.4	−0.973394
$957$	0	0
$958$	38117.0	1.28549
$959$	−30088.4	−1.01314
$960$	0	0
$961$	60205.6	2.02093
$962$	−322.061	−0.0107938
$963$	0	0
$964$	−3349.39	−0.111905
$965$	47294.3	1.57768
$966$	0	0
$967$	−55391.0	−1.84204	−0.921021	−	0.389514i	$-0.872643\pi$
$967$	−55391.0	−1.84204	−0.921021	+	0.389514i	$0.872643\pi$
$968$	11274.9	0.374369
$969$	0	0
$970$	−8318.98	−0.275367
$971$	23937.0	0.791117	0.395558	−	0.918441i	$-0.370551\pi$
$971$	23937.0	0.791117	0.395558	+	0.918441i	$0.370551\pi$
$972$	0	0
$973$	53111.0	1.74991
$974$	−974.429	−0.0320562
$975$	0	0
$976$	7753.70	0.254293
$977$	43982.3	1.44024	0.720122	−	0.693847i	$-0.244086\pi$
$977$	43982.3	1.44024	0.720122	+	0.693847i	$0.244086\pi$
$978$	0	0
$979$	15476.1	0.505229
$980$	−16234.0	−0.529160
$981$	0	0
$982$	12925.8	0.420039
$983$	−2724.78	−0.0884100	−0.0442050	−	0.999022i	$-0.514075\pi$
$983$	−2724.78	−0.0884100	−0.0442050	+	0.999022i	$0.514075\pi$
$984$	0	0
$985$	−10001.6	−0.323530
$986$	997.126	0.0322059
$987$	0	0
$988$	−955.486	−0.0307673
$989$	13292.6	0.427382
$990$	0	0
$991$	−23748.6	−0.761250	−0.380625	−	0.924729i	$-0.624291\pi$
$991$	−23748.6	−0.761250	−0.380625	+	0.924729i	$0.624291\pi$
$992$	9599.82	0.307253
$993$	0	0
$994$	26562.9	0.847609
$995$	11323.1	0.360771
$996$	0	0
$997$	24718.0	0.785182	0.392591	−	0.919713i	$-0.371579\pi$
$997$	24718.0	0.785182	0.392591	+	0.919713i	$0.371579\pi$
$998$	−15895.6	−0.504174
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	522.4.a.j.1.1		2
3.2	odd	2		58.4.a.c.1.1	✓	2
12.11	even	2		464.4.a.e.1.2		2
15.14	odd	2		1450.4.a.g.1.2		2
24.5	odd	2		1856.4.a.l.1.2		2
24.11	even	2		1856.4.a.i.1.1		2
87.86	odd	2		1682.4.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.4.a.c.1.1	✓	2	3.2	odd	2
464.4.a.e.1.2		2	12.11	even	2
522.4.a.j.1.1		2	1.1	even	1	trivial
1450.4.a.g.1.2		2	15.14	odd	2
1682.4.a.c.1.2		2	87.86	odd	2
1856.4.a.i.1.1		2	24.11	even	2
1856.4.a.l.1.2		2	24.5	odd	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:	\( N \)	\(=\)	\( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	522.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +4.00000 q^{4} -9.69694 q^{5} -27.5959 q^{7} +8.00000 q^{8} +O(q^{10})\)	\(q+2.00000 q^{2} +4.00000 q^{4} -9.69694 q^{5} -27.5959 q^{7} +8.00000 q^{8} -19.3939 q^{10} +52.3485 q^{11} -5.40408 q^{13} -55.1918 q^{14} +16.0000 q^{16} -17.1918 q^{17} +44.2020 q^{19} -38.7878 q^{20} +104.697 q^{22} +205.060 q^{23} -30.9694 q^{25} -10.8082 q^{26} -110.384 q^{28} -29.0000 q^{29} +299.994 q^{31} +32.0000 q^{32} -34.3837 q^{34} +267.596 q^{35} +29.7980 q^{37} +88.4041 q^{38} -77.5755 q^{40} +43.9592 q^{41} +64.8230 q^{43} +209.394 q^{44} +410.120 q^{46} +499.499 q^{47} +418.535 q^{49} -61.9388 q^{50} -21.6163 q^{52} +351.627 q^{53} -507.620 q^{55} -220.767 q^{56} -58.0000 q^{58} -522.372 q^{59} +484.606 q^{61} +599.989 q^{62} +64.0000 q^{64} +52.4031 q^{65} -504.990 q^{67} -68.7673 q^{68} +535.192 q^{70} -481.283 q^{71} +3.11019 q^{73} +59.5959 q^{74} +176.808 q^{76} -1444.60 q^{77} -1043.27 q^{79} -155.151 q^{80} +87.9184 q^{82} +1007.08 q^{83} +166.708 q^{85} +129.646 q^{86} +418.788 q^{88} +295.637 q^{89} +149.131 q^{91} +820.241 q^{92} +998.999 q^{94} -428.624 q^{95} +428.949 q^{97} +837.069 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8}+O(q^{10}) \) 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 16 * q^7 + 16 * q^8	\( 2 q + 4 q^{2} + 8 q^{4} + 10 q^{5} - 16 q^{7} + 16 q^{8} + 20 q^{10} + 90 q^{11} - 50 q^{13} - 32 q^{14} + 32 q^{16} + 44 q^{17} + 108 q^{19} + 40 q^{20} + 180 q^{22} + 28 q^{23} + 232 q^{25} - 100 q^{26} - 64 q^{28} - 58 q^{29} + 66 q^{31} + 64 q^{32} + 88 q^{34} + 496 q^{35} + 40 q^{37} + 216 q^{38} + 80 q^{40} - 304 q^{41} - 130 q^{43} + 360 q^{44} + 56 q^{46} + 514 q^{47} + 210 q^{49} + 464 q^{50} - 200 q^{52} + 958 q^{53} + 234 q^{55} - 128 q^{56} - 116 q^{58} + 180 q^{59} + 1028 q^{61} + 132 q^{62} + 128 q^{64} - 826 q^{65} - 912 q^{67} + 176 q^{68} + 992 q^{70} - 796 q^{71} - 856 q^{73} + 80 q^{74} + 432 q^{76} - 1008 q^{77} - 318 q^{79} + 160 q^{80} - 608 q^{82} + 1828 q^{83} + 1372 q^{85} - 260 q^{86} + 720 q^{88} + 944 q^{89} - 368 q^{91} + 112 q^{92} + 1028 q^{94} + 828 q^{95} + 368 q^{97} + 420 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 + 8 * q^4 + 10 * q^5 - 16 * q^7 + 16 * q^8 + 20 * q^10 + 90 * q^11 - 50 * q^13 - 32 * q^14 + 32 * q^16 + 44 * q^17 + 108 * q^19 + 40 * q^20 + 180 * q^22 + 28 * q^23 + 232 * q^25 - 100 * q^26 - 64 * q^28 - 58 * q^29 + 66 * q^31 + 64 * q^32 + 88 * q^34 + 496 * q^35 + 40 * q^37 + 216 * q^38 + 80 * q^40 - 304 * q^41 - 130 * q^43 + 360 * q^44 + 56 * q^46 + 514 * q^47 + 210 * q^49 + 464 * q^50 - 200 * q^52 + 958 * q^53 + 234 * q^55 - 128 * q^56 - 116 * q^58 + 180 * q^59 + 1028 * q^61 + 132 * q^62 + 128 * q^64 - 826 * q^65 - 912 * q^67 + 176 * q^68 + 992 * q^70 - 796 * q^71 - 856 * q^73 + 80 * q^74 + 432 * q^76 - 1008 * q^77 - 318 * q^79 + 160 * q^80 - 608 * q^82 + 1828 * q^83 + 1372 * q^85 - 260 * q^86 + 720 * q^88 + 944 * q^89 - 368 * q^91 + 112 * q^92 + 1028 * q^94 + 828 * q^95 + 368 * q^97 + 420 * q^98

Introduction

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