LMFDB - Embedded newform 7569.2.a.bl.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.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:	$60.4387692899$
Analytic rank:	$0$
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} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 $ x^9 - x^8 - 14x^7 + 9x^6 + 70x^5 - 23x^4 - 141x^3 + 14x^2 + 84*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.19659$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.19659 q^{2} +2.82501 q^{4} +4.07769 q^{5} +3.64733 q^{7} +1.81221 q^{8} +O(q^{10})$	$q+2.19659 q^{2} +2.82501 q^{4} +4.07769 q^{5} +3.64733 q^{7} +1.81221 q^{8} +8.95700 q^{10} +3.52776 q^{11} +0.926540 q^{13} +8.01169 q^{14} -1.66934 q^{16} -1.61861 q^{17} -3.98608 q^{19} +11.5195 q^{20} +7.74905 q^{22} -1.54577 q^{23} +11.6275 q^{25} +2.03523 q^{26} +10.3037 q^{28} -1.60775 q^{31} -7.29128 q^{32} -3.55543 q^{34} +14.8727 q^{35} -5.30404 q^{37} -8.75579 q^{38} +7.38960 q^{40} +6.71590 q^{41} +4.42936 q^{43} +9.96596 q^{44} -3.39542 q^{46} -3.33787 q^{47} +6.30302 q^{49} +25.5409 q^{50} +2.61748 q^{52} -2.03823 q^{53} +14.3851 q^{55} +6.60971 q^{56} -7.29049 q^{59} -6.00554 q^{61} -3.53156 q^{62} -12.6773 q^{64} +3.77814 q^{65} +9.48235 q^{67} -4.57260 q^{68} +32.6692 q^{70} -8.44759 q^{71} +7.56757 q^{73} -11.6508 q^{74} -11.2607 q^{76} +12.8669 q^{77} +0.287887 q^{79} -6.80706 q^{80} +14.7521 q^{82} +6.47754 q^{83} -6.60020 q^{85} +9.72949 q^{86} +6.39303 q^{88} -16.5093 q^{89} +3.37940 q^{91} -4.36681 q^{92} -7.33194 q^{94} -16.2540 q^{95} +10.0522 q^{97} +13.8451 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	$ 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) $ 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * 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 $

$2$	2.19659	1.55322	0.776612	−	0.629979i	$-0.216937\pi$
$2$	2.19659	1.55322	0.776612	+	0.629979i	$0.216937\pi$
$3$	0	0
$3$	0	0
$4$	2.82501	1.41250
$5$	4.07769	1.82360	0.911798	−	0.410638i	$-0.134694\pi$
$5$	4.07769	1.82360	0.911798	+	0.410638i	$0.134694\pi$
$6$	0	0
$7$	3.64733	1.37856	0.689281	−	0.724495i	$-0.257927\pi$
$7$	3.64733	1.37856	0.689281	+	0.724495i	$0.257927\pi$
$8$	1.81221	0.640711
$9$	0	0
$10$	8.95700	2.83245
$11$	3.52776	1.06366	0.531830	−	0.846851i	$-0.321504\pi$
$11$	3.52776	1.06366	0.531830	+	0.846851i	$0.321504\pi$
$12$	0	0
$13$	0.926540	0.256976	0.128488	−	0.991711i	$-0.458988\pi$
$13$	0.926540	0.256976	0.128488	+	0.991711i	$0.458988\pi$
$14$	8.01169	2.14121
$15$	0	0
$16$	−1.66934	−0.417336
$17$	−1.61861	−0.392571	−0.196286	−	0.980547i	$-0.562888\pi$
$17$	−1.61861	−0.392571	−0.196286	+	0.980547i	$0.562888\pi$
$18$	0	0
$19$	−3.98608	−0.914470	−0.457235	−	0.889346i	$-0.651160\pi$
$19$	−3.98608	−0.914470	−0.457235	+	0.889346i	$0.651160\pi$
$20$	11.5195	2.57584
$21$	0	0
$22$	7.74905	1.65210
$23$	−1.54577	−0.322315	−0.161158	−	0.986929i	$-0.551523\pi$
$23$	−1.54577	−0.322315	−0.161158	+	0.986929i	$0.551523\pi$
$24$	0	0
$25$	11.6275	2.32550
$26$	2.03523	0.399141
$27$	0	0
$28$	10.3037	1.94722
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−1.60775	−0.288760	−0.144380	−	0.989522i	$-0.546119\pi$
$31$	−1.60775	−0.288760	−0.144380	+	0.989522i	$0.546119\pi$
$32$	−7.29128	−1.28893
$33$	0	0
$34$	−3.55543	−0.609751
$35$	14.8727	2.51394
$36$	0	0
$37$	−5.30404	−0.871979	−0.435990	−	0.899952i	$-0.643602\pi$
$37$	−5.30404	−0.871979	−0.435990	+	0.899952i	$0.643602\pi$
$38$	−8.75579	−1.42038
$39$	0	0
$40$	7.38960	1.16840
$41$	6.71590	1.04885	0.524423	−	0.851458i	$-0.324281\pi$
$41$	6.71590	1.04885	0.524423	+	0.851458i	$0.324281\pi$
$42$	0	0
$43$	4.42936	0.675471	0.337736	−	0.941241i	$-0.390339\pi$
$43$	4.42936	0.675471	0.337736	+	0.941241i	$0.390339\pi$
$44$	9.96596	1.50242
$45$	0	0
$46$	−3.39542	−0.500628
$47$	−3.33787	−0.486879	−0.243439	−	0.969916i	$-0.578276\pi$
$47$	−3.33787	−0.486879	−0.243439	+	0.969916i	$0.578276\pi$
$48$	0	0
$49$	6.30302	0.900431
$50$	25.5409	3.61203
$51$	0	0
$52$	2.61748	0.362980
$53$	−2.03823	−0.279972	−0.139986	−	0.990153i	$-0.544706\pi$
$53$	−2.03823	−0.279972	−0.139986	+	0.990153i	$0.544706\pi$
$54$	0	0
$55$	14.3851	1.93969
$56$	6.60971	0.883260
$57$	0	0
$58$	0	0
$59$	−7.29049	−0.949141	−0.474571	−	0.880217i	$-0.657397\pi$
$59$	−7.29049	−0.949141	−0.474571	+	0.880217i	$0.657397\pi$
$60$	0	0
$61$	−6.00554	−0.768930	−0.384465	−	0.923139i	$-0.625614\pi$
$61$	−6.00554	−0.768930	−0.384465	+	0.923139i	$0.625614\pi$
$62$	−3.53156	−0.448509
$63$	0	0
$64$	−12.6773	−1.58466
$65$	3.77814	0.468620
$66$	0	0
$67$	9.48235	1.15845	0.579227	−	0.815167i	$-0.303355\pi$
$67$	9.48235	1.15845	0.579227	+	0.815167i	$0.303355\pi$
$68$	−4.57260	−0.554509
$69$	0	0
$70$	32.6692	3.90471
$71$	−8.44759	−1.00255	−0.501273	−	0.865289i	$-0.667135\pi$
$71$	−8.44759	−1.00255	−0.501273	+	0.865289i	$0.667135\pi$
$72$	0	0
$73$	7.56757	0.885718	0.442859	−	0.896591i	$-0.353964\pi$
$73$	7.56757	0.885718	0.442859	+	0.896591i	$0.353964\pi$
$74$	−11.6508	−1.35438
$75$	0	0
$76$	−11.2607	−1.29169
$77$	12.8669	1.46632
$78$	0	0
$79$	0.287887	0.0323898	0.0161949	−	0.999869i	$-0.494845\pi$
$79$	0.287887	0.0323898	0.0161949	+	0.999869i	$0.494845\pi$
$80$	−6.80706	−0.761053
$81$	0	0
$82$	14.7521	1.62909
$83$	6.47754	0.711003	0.355501	−	0.934676i	$-0.384310\pi$
$83$	6.47754	0.711003	0.355501	+	0.934676i	$0.384310\pi$
$84$	0	0
$85$	−6.60020	−0.715892
$86$	9.72949	1.04916
$87$	0	0
$88$	6.39303	0.681499
$89$	−16.5093	−1.74999	−0.874993	−	0.484136i	$-0.839134\pi$
$89$	−16.5093	−1.74999	−0.874993	+	0.484136i	$0.839134\pi$
$90$	0	0
$91$	3.37940	0.354257
$92$	−4.36681	−0.455272
$93$	0	0
$94$	−7.33194	−0.756232
$95$	−16.2540	−1.66762
$96$	0	0
$97$	10.0522	1.02064	0.510322	−	0.859983i	$-0.329526\pi$
$97$	10.0522	1.02064	0.510322	+	0.859983i	$0.329526\pi$
$98$	13.8451	1.39857
$99$	0	0
$100$	32.8478	3.28478
$101$	−8.23922	−0.819833	−0.409917	−	0.912123i	$-0.634442\pi$
$101$	−8.23922	−0.819833	−0.409917	+	0.912123i	$0.634442\pi$
$102$	0	0
$103$	7.62704	0.751515	0.375757	−	0.926718i	$-0.377383\pi$
$103$	7.62704	0.751515	0.375757	+	0.926718i	$0.377383\pi$
$104$	1.67908	0.164647
$105$	0	0
$106$	−4.47715	−0.434859
$107$	16.7576	1.62002	0.810011	−	0.586414i	$-0.199461\pi$
$107$	16.7576	1.62002	0.810011	+	0.586414i	$0.199461\pi$
$108$	0	0
$109$	−12.7116	−1.21755	−0.608775	−	0.793343i	$-0.708339\pi$
$109$	−12.7116	−1.21755	−0.608775	+	0.793343i	$0.708339\pi$
$110$	31.5982	3.01277
$111$	0	0
$112$	−6.08865	−0.575323
$113$	8.46482	0.796303	0.398152	−	0.917320i	$-0.369652\pi$
$113$	8.46482	0.796303	0.398152	+	0.917320i	$0.369652\pi$
$114$	0	0
$115$	−6.30316	−0.587773
$116$	0	0
$117$	0	0
$118$	−16.0142	−1.47423
$119$	−5.90362	−0.541184
$120$	0	0
$121$	1.44511	0.131374
$122$	−13.1917	−1.19432
$123$	0	0
$124$	−4.54190	−0.407875
$125$	27.0250	2.41719
$126$	0	0
$127$	−17.8052	−1.57995	−0.789977	−	0.613137i	$-0.789907\pi$
$127$	−17.8052	−1.57995	−0.789977	+	0.613137i	$0.789907\pi$
$128$	−13.2642	−1.17240
$129$	0	0
$130$	8.29902	0.727872
$131$	−11.3299	−0.989897	−0.494949	−	0.868922i	$-0.664813\pi$
$131$	−11.3299	−0.989897	−0.494949	+	0.868922i	$0.664813\pi$
$132$	0	0
$133$	−14.5386	−1.26065
$134$	20.8288	1.79934
$135$	0	0
$136$	−2.93326	−0.251525
$137$	−8.31246	−0.710182	−0.355091	−	0.934832i	$-0.615550\pi$
$137$	−8.31246	−0.710182	−0.355091	+	0.934832i	$0.615550\pi$
$138$	0	0
$139$	−3.11275	−0.264020	−0.132010	−	0.991248i	$-0.542143\pi$
$139$	−3.11275	−0.264020	−0.132010	+	0.991248i	$0.542143\pi$
$140$	42.0154	3.55095
$141$	0	0
$142$	−18.5559	−1.55718
$143$	3.26861	0.273335
$144$	0	0
$145$	0	0
$146$	16.6229	1.37572
$147$	0	0
$148$	−14.9840	−1.23167
$149$	−15.1558	−1.24161	−0.620807	−	0.783963i	$-0.713195\pi$
$149$	−15.1558	−1.24161	−0.620807	+	0.783963i	$0.713195\pi$
$150$	0	0
$151$	0.678890	0.0552473	0.0276237	−	0.999618i	$-0.491206\pi$
$151$	0.678890	0.0552473	0.0276237	+	0.999618i	$0.491206\pi$
$152$	−7.22360	−0.585911
$153$	0	0
$154$	28.2633	2.27752
$155$	−6.55589	−0.526582
$156$	0	0
$157$	−6.23584	−0.497674	−0.248837	−	0.968545i	$-0.580048\pi$
$157$	−6.23584	−0.497674	−0.248837	+	0.968545i	$0.580048\pi$
$158$	0.632370	0.0503086
$159$	0	0
$160$	−29.7315	−2.35048
$161$	−5.63793	−0.444331
$162$	0	0
$163$	−15.3163	−1.19967	−0.599834	−	0.800124i	$-0.704767\pi$
$163$	−15.3163	−1.19967	−0.599834	+	0.800124i	$0.704767\pi$
$164$	18.9725	1.48150
$165$	0	0
$166$	14.2285	1.10435
$167$	15.7861	1.22156	0.610781	−	0.791799i	$-0.290855\pi$
$167$	15.7861	1.22156	0.610781	+	0.791799i	$0.290855\pi$
$168$	0	0
$169$	−12.1415	−0.933963
$170$	−14.4979	−1.11194
$171$	0	0
$172$	12.5130	0.954106
$173$	3.44502	0.261920	0.130960	−	0.991388i	$-0.458194\pi$
$173$	3.44502	0.261920	0.130960	+	0.991388i	$0.458194\pi$
$174$	0	0
$175$	42.4094	3.20585
$176$	−5.88905	−0.443904
$177$	0	0
$178$	−36.2642	−2.71812
$179$	12.0378	0.899744	0.449872	−	0.893093i	$-0.351470\pi$
$179$	12.0378	0.899744	0.449872	+	0.893093i	$0.351470\pi$
$180$	0	0
$181$	2.41926	0.179823	0.0899113	−	0.995950i	$-0.471342\pi$
$181$	2.41926	0.179823	0.0899113	+	0.995950i	$0.471342\pi$
$182$	7.42315	0.550241
$183$	0	0
$184$	−2.80125	−0.206511
$185$	−21.6282	−1.59014
$186$	0	0
$187$	−5.71008	−0.417563
$188$	−9.42952	−0.687719
$189$	0	0
$190$	−35.7034	−2.59019
$191$	−5.83662	−0.422323	−0.211162	−	0.977451i	$-0.567725\pi$
$191$	−5.83662	−0.422323	−0.211162	+	0.977451i	$0.567725\pi$
$192$	0	0
$193$	19.9903	1.43894	0.719468	−	0.694525i	$-0.244386\pi$
$193$	19.9903	1.43894	0.719468	+	0.694525i	$0.244386\pi$
$194$	22.0805	1.58529
$195$	0	0
$196$	17.8061	1.27186
$197$	11.7126	0.834486	0.417243	−	0.908795i	$-0.362996\pi$
$197$	11.7126	0.834486	0.417243	+	0.908795i	$0.362996\pi$
$198$	0	0
$199$	−16.6484	−1.18017	−0.590086	−	0.807340i	$-0.700906\pi$
$199$	−16.6484	−1.18017	−0.590086	+	0.807340i	$0.700906\pi$
$200$	21.0715	1.48998
$201$	0	0
$202$	−18.0982	−1.27338
$203$	0	0
$204$	0	0
$205$	27.3853	1.91267
$206$	16.7535	1.16727
$207$	0	0
$208$	−1.54671	−0.107245
$209$	−14.0620	−0.972686
$210$	0	0
$211$	−24.2435	−1.66899	−0.834496	−	0.551014i	$-0.814241\pi$
$211$	−24.2435	−1.66899	−0.834496	+	0.551014i	$0.814241\pi$
$212$	−5.75801	−0.395462
$213$	0	0
$214$	36.8097	2.51626
$215$	18.0615	1.23179
$216$	0	0
$217$	−5.86399	−0.398073
$218$	−27.9222	−1.89113
$219$	0	0
$220$	40.6381	2.73982
$221$	−1.49971	−0.100881
$222$	0	0
$223$	28.3179	1.89631	0.948153	−	0.317814i	$-0.102949\pi$
$223$	28.3179	1.89631	0.948153	+	0.317814i	$0.102949\pi$
$224$	−26.5937	−1.77687
$225$	0	0
$226$	18.5937	1.23684
$227$	−9.86418	−0.654709	−0.327354	−	0.944902i	$-0.606157\pi$
$227$	−9.86418	−0.654709	−0.327354	+	0.944902i	$0.606157\pi$
$228$	0	0
$229$	15.6570	1.03464	0.517322	−	0.855791i	$-0.326929\pi$
$229$	15.6570	1.03464	0.517322	+	0.855791i	$0.326929\pi$
$230$	−13.8455	−0.912943
$231$	0	0
$232$	0	0
$233$	19.7879	1.29635	0.648176	−	0.761491i	$-0.275532\pi$
$233$	19.7879	1.29635	0.648176	+	0.761491i	$0.275532\pi$
$234$	0	0
$235$	−13.6108	−0.887871
$236$	−20.5957	−1.34067
$237$	0	0
$238$	−12.9678	−0.840579
$239$	9.33298	0.603700	0.301850	−	0.953355i	$-0.402396\pi$
$239$	9.33298	0.603700	0.301850	+	0.953355i	$0.402396\pi$
$240$	0	0
$241$	−24.5737	−1.58293	−0.791464	−	0.611215i	$-0.790681\pi$
$241$	−24.5737	−1.58293	−0.791464	+	0.611215i	$0.790681\pi$
$242$	3.17432	0.204053
$243$	0	0
$244$	−16.9657	−1.08612
$245$	25.7017	1.64202
$246$	0	0
$247$	−3.69327	−0.234997
$248$	−2.91357	−0.185012
$249$	0	0
$250$	59.3627	3.75443
$251$	−4.66182	−0.294252	−0.147126	−	0.989118i	$-0.547002\pi$
$251$	−4.66182	−0.294252	−0.147126	+	0.989118i	$0.547002\pi$
$252$	0	0
$253$	−5.45311	−0.342834
$254$	−39.1107	−2.45402
$255$	0	0
$256$	−3.78146	−0.236341
$257$	28.3543	1.76869	0.884345	−	0.466834i	$-0.154605\pi$
$257$	28.3543	1.76869	0.884345	+	0.466834i	$0.154605\pi$
$258$	0	0
$259$	−19.3456	−1.20208
$260$	10.6733	0.661928
$261$	0	0
$262$	−24.8871	−1.53753
$263$	26.7811	1.65139	0.825697	−	0.564114i	$-0.190782\pi$
$263$	26.7811	1.65139	0.825697	+	0.564114i	$0.190782\pi$
$264$	0	0
$265$	−8.31125	−0.510556
$266$	−31.9353	−1.95808
$267$	0	0
$268$	26.7877	1.63632
$269$	0.785551	0.0478959	0.0239479	−	0.999713i	$-0.492376\pi$
$269$	0.785551	0.0478959	0.0239479	+	0.999713i	$0.492376\pi$
$270$	0	0
$271$	−12.1514	−0.738148	−0.369074	−	0.929400i	$-0.620325\pi$
$271$	−12.1514	−0.738148	−0.369074	+	0.929400i	$0.620325\pi$
$272$	2.70202	0.163834
$273$	0	0
$274$	−18.2591	−1.10307
$275$	41.0191	2.47355
$276$	0	0
$277$	−15.7837	−0.948352	−0.474176	−	0.880430i	$-0.657254\pi$
$277$	−15.7837	−0.948352	−0.474176	+	0.880430i	$0.657254\pi$
$278$	−6.83744	−0.410083
$279$	0	0
$280$	26.9523	1.61071
$281$	−20.0723	−1.19741	−0.598705	−	0.800969i	$-0.704318\pi$
$281$	−20.0723	−1.19741	−0.598705	+	0.800969i	$0.704318\pi$
$282$	0	0
$283$	−1.95557	−0.116246	−0.0581232	−	0.998309i	$-0.518512\pi$
$283$	−1.95557	−0.116246	−0.0581232	+	0.998309i	$0.518512\pi$
$284$	−23.8645	−1.41610
$285$	0	0
$286$	7.17980	0.424551
$287$	24.4951	1.44590
$288$	0	0
$289$	−14.3801	−0.845888
$290$	0	0
$291$	0	0
$292$	21.3785	1.25108
$293$	2.41420	0.141039	0.0705194	−	0.997510i	$-0.477534\pi$
$293$	2.41420	0.141039	0.0705194	+	0.997510i	$0.477534\pi$
$294$	0	0
$295$	−29.7283	−1.73085
$296$	−9.61201	−0.558687
$297$	0	0
$298$	−33.2912	−1.92851
$299$	−1.43222	−0.0828273
$300$	0	0
$301$	16.1553	0.931178
$302$	1.49124	0.0858114
$303$	0	0
$304$	6.65415	0.381641
$305$	−24.4887	−1.40222
$306$	0	0
$307$	−1.11367	−0.0635606	−0.0317803	−	0.999495i	$-0.510118\pi$
$307$	−1.11367	−0.0635606	−0.0317803	+	0.999495i	$0.510118\pi$
$308$	36.3491	2.07118
$309$	0	0
$310$	−14.4006	−0.817899
$311$	−4.42989	−0.251196	−0.125598	−	0.992081i	$-0.540085\pi$
$311$	−4.42989	−0.251196	−0.125598	+	0.992081i	$0.540085\pi$
$312$	0	0
$313$	−30.1002	−1.70136	−0.850682	−	0.525680i	$-0.823811\pi$
$313$	−30.1002	−1.70136	−0.850682	+	0.525680i	$0.823811\pi$
$314$	−13.6976	−0.772999
$315$	0	0
$316$	0.813283	0.0457507
$317$	6.21069	0.348827	0.174414	−	0.984672i	$-0.444197\pi$
$317$	6.21069	0.348827	0.174414	+	0.984672i	$0.444197\pi$
$318$	0	0
$319$	0	0
$320$	−51.6939	−2.88978
$321$	0	0
$322$	−12.3842	−0.690146
$323$	6.45193	0.358995
$324$	0	0
$325$	10.7734	0.597599
$326$	−33.6437	−1.86335
$327$	0	0
$328$	12.1706	0.672008
$329$	−12.1743	−0.671192
$330$	0	0
$331$	10.0058	0.549967	0.274984	−	0.961449i	$-0.411328\pi$
$331$	10.0058	0.549967	0.274984	+	0.961449i	$0.411328\pi$
$332$	18.2991	1.00429
$333$	0	0
$334$	34.6755	1.89736
$335$	38.6660	2.11255
$336$	0	0
$337$	−27.2955	−1.48688	−0.743440	−	0.668803i	$-0.766807\pi$
$337$	−27.2955	−1.48688	−0.743440	+	0.668803i	$0.766807\pi$
$338$	−26.6700	−1.45065
$339$	0	0
$340$	−18.6456	−1.01120
$341$	−5.67175	−0.307143
$342$	0	0
$343$	−2.54213	−0.137262
$344$	8.02691	0.432782
$345$	0	0
$346$	7.56729	0.406820
$347$	−1.19493	−0.0641473	−0.0320736	−	0.999486i	$-0.510211\pi$
$347$	−1.19493	−0.0641473	−0.0320736	+	0.999486i	$0.510211\pi$
$348$	0	0
$349$	7.25120	0.388148	0.194074	−	0.980987i	$-0.437830\pi$
$349$	7.25120	0.388148	0.194074	+	0.980987i	$0.437830\pi$
$350$	93.1561	4.97940
$351$	0	0
$352$	−25.7219	−1.37098
$353$	5.26353	0.280150	0.140075	−	0.990141i	$-0.455266\pi$
$353$	5.26353	0.280150	0.140075	+	0.990141i	$0.455266\pi$
$354$	0	0
$355$	−34.4466	−1.82824
$356$	−46.6390	−2.47186
$357$	0	0
$358$	26.4420	1.39750
$359$	14.6286	0.772067	0.386034	−	0.922485i	$-0.373845\pi$
$359$	14.6286	0.772067	0.386034	+	0.922485i	$0.373845\pi$
$360$	0	0
$361$	−3.11114	−0.163744
$362$	5.31413	0.279305
$363$	0	0
$364$	9.54682	0.500390
$365$	30.8582	1.61519
$366$	0	0
$367$	5.55318	0.289874	0.144937	−	0.989441i	$-0.453702\pi$
$367$	5.55318	0.289874	0.144937	+	0.989441i	$0.453702\pi$
$368$	2.58042	0.134514
$369$	0	0
$370$	−47.5083	−2.46984
$371$	−7.43409	−0.385959
$372$	0	0
$373$	29.7459	1.54019	0.770093	−	0.637931i	$-0.220210\pi$
$373$	29.7459	1.54019	0.770093	+	0.637931i	$0.220210\pi$
$374$	−12.5427	−0.648568
$375$	0	0
$376$	−6.04891	−0.311949
$377$	0	0
$378$	0	0
$379$	−1.96231	−0.100797	−0.0503986	−	0.998729i	$-0.516049\pi$
$379$	−1.96231	−0.100797	−0.0503986	+	0.998729i	$0.516049\pi$
$380$	−45.9177	−2.35553
$381$	0	0
$382$	−12.8207	−0.655962
$383$	11.8165	0.603796	0.301898	−	0.953340i	$-0.402380\pi$
$383$	11.8165	0.603796	0.301898	+	0.953340i	$0.402380\pi$
$384$	0	0
$385$	52.4672	2.67398
$386$	43.9106	2.23499
$387$	0	0
$388$	28.3975	1.44166
$389$	−11.9013	−0.603422	−0.301711	−	0.953399i	$-0.597558\pi$
$389$	−11.9013	−0.603422	−0.301711	+	0.953399i	$0.597558\pi$
$390$	0	0
$391$	2.50200	0.126532
$392$	11.4224	0.576916
$393$	0	0
$394$	25.7277	1.29614
$395$	1.17391	0.0590660
$396$	0	0
$397$	19.6778	0.987603	0.493801	−	0.869575i	$-0.335607\pi$
$397$	19.6778	0.987603	0.493801	+	0.869575i	$0.335607\pi$
$398$	−36.5697	−1.83307
$399$	0	0
$400$	−19.4103	−0.970517
$401$	20.5264	1.02504	0.512520	−	0.858675i	$-0.328712\pi$
$401$	20.5264	1.02504	0.512520	+	0.858675i	$0.328712\pi$
$402$	0	0
$403$	−1.48964	−0.0742044
$404$	−23.2759	−1.15802
$405$	0	0
$406$	0	0
$407$	−18.7114	−0.927490
$408$	0	0
$409$	32.1820	1.59130	0.795648	−	0.605760i	$-0.207131\pi$
$409$	32.1820	1.59130	0.795648	+	0.605760i	$0.207131\pi$
$410$	60.1543	2.97081
$411$	0	0
$412$	21.5464	1.06152
$413$	−26.5908	−1.30845
$414$	0	0
$415$	26.4134	1.29658
$416$	−6.75566	−0.331223
$417$	0	0
$418$	−30.8884	−1.51080
$419$	−1.58695	−0.0775275	−0.0387638	−	0.999248i	$-0.512342\pi$
$419$	−1.58695	−0.0775275	−0.0387638	+	0.999248i	$0.512342\pi$
$420$	0	0
$421$	5.86317	0.285753	0.142877	−	0.989741i	$-0.454365\pi$
$421$	5.86317	0.285753	0.142877	+	0.989741i	$0.454365\pi$
$422$	−53.2530	−2.59232
$423$	0	0
$424$	−3.69369	−0.179381
$425$	−18.8205	−0.912926
$426$	0	0
$427$	−21.9042	−1.06002
$428$	47.3405	2.28829
$429$	0	0
$430$	39.6738	1.91324
$431$	8.46235	0.407617	0.203809	−	0.979011i	$-0.434668\pi$
$431$	8.46235	0.407617	0.203809	+	0.979011i	$0.434668\pi$
$432$	0	0
$433$	12.7185	0.611212	0.305606	−	0.952158i	$-0.401141\pi$
$433$	12.7185	0.611212	0.305606	+	0.952158i	$0.401141\pi$
$434$	−12.8808	−0.618297
$435$	0	0
$436$	−35.9104	−1.71980
$437$	6.16157	0.294748
$438$	0	0
$439$	−7.50539	−0.358213	−0.179106	−	0.983830i	$-0.557321\pi$
$439$	−7.50539	−0.358213	−0.179106	+	0.983830i	$0.557321\pi$
$440$	26.0688	1.24278
$441$	0	0
$442$	−3.29425	−0.156691
$443$	11.6583	0.553902	0.276951	−	0.960884i	$-0.410676\pi$
$443$	11.6583	0.553902	0.276951	+	0.960884i	$0.410676\pi$
$444$	0	0
$445$	−67.3199	−3.19127
$446$	62.2028	2.94539
$447$	0	0
$448$	−46.2381	−2.18455
$449$	19.4512	0.917957	0.458978	−	0.888447i	$-0.348216\pi$
$449$	19.4512	0.917957	0.458978	+	0.888447i	$0.348216\pi$
$450$	0	0
$451$	23.6921	1.11562
$452$	23.9132	1.12478
$453$	0	0
$454$	−21.6676	−1.01691
$455$	13.7801	0.646022
$456$	0	0
$457$	17.1125	0.800489	0.400245	−	0.916408i	$-0.368925\pi$
$457$	17.1125	0.800489	0.400245	+	0.916408i	$0.368925\pi$
$458$	34.3920	1.60703
$459$	0	0
$460$	−17.8065	−0.830232
$461$	−8.53020	−0.397291	−0.198646	−	0.980071i	$-0.563654\pi$
$461$	−8.53020	−0.397291	−0.198646	+	0.980071i	$0.563654\pi$
$462$	0	0
$463$	1.49772	0.0696050	0.0348025	−	0.999394i	$-0.488920\pi$
$463$	1.49772	0.0696050	0.0348025	+	0.999394i	$0.488920\pi$
$464$	0	0
$465$	0	0
$466$	43.4660	2.01352
$467$	21.4307	0.991695	0.495848	−	0.868410i	$-0.334857\pi$
$467$	21.4307	0.991695	0.495848	+	0.868410i	$0.334857\pi$
$468$	0	0
$469$	34.5853	1.59700
$470$	−29.8974	−1.37906
$471$	0	0
$472$	−13.2119	−0.608126
$473$	15.6257	0.718472
$474$	0	0
$475$	−46.3483	−2.12660
$476$	−16.6778	−0.764424
$477$	0	0
$478$	20.5007	0.937682
$479$	28.5538	1.30465	0.652327	−	0.757937i	$-0.273793\pi$
$479$	28.5538	1.30465	0.652327	+	0.757937i	$0.273793\pi$
$480$	0	0
$481$	−4.91441	−0.224078
$482$	−53.9783	−2.45864
$483$	0	0
$484$	4.08245	0.185566
$485$	40.9896	1.86124
$486$	0	0
$487$	25.1555	1.13991	0.569953	−	0.821677i	$-0.306961\pi$
$487$	25.1555	1.13991	0.569953	+	0.821677i	$0.306961\pi$
$488$	−10.8833	−0.492662
$489$	0	0
$490$	56.4561	2.55043
$491$	21.6846	0.978614	0.489307	−	0.872112i	$-0.337250\pi$
$491$	21.6846	0.978614	0.489307	+	0.872112i	$0.337250\pi$
$492$	0	0
$493$	0	0
$494$	−8.11259	−0.365003
$495$	0	0
$496$	2.68388	0.120510
$497$	−30.8112	−1.38207
$498$	0	0
$499$	5.68927	0.254687	0.127343	−	0.991859i	$-0.459355\pi$
$499$	5.68927	0.254687	0.127343	+	0.991859i	$0.459355\pi$
$500$	76.3457	3.41428
$501$	0	0
$502$	−10.2401	−0.457039
$503$	9.46973	0.422234	0.211117	−	0.977461i	$-0.432290\pi$
$503$	9.46973	0.422234	0.211117	+	0.977461i	$0.432290\pi$
$504$	0	0
$505$	−33.5970	−1.49505
$506$	−11.9782	−0.532498
$507$	0	0
$508$	−50.2998	−2.23169
$509$	30.5939	1.35605	0.678024	−	0.735039i	$-0.262836\pi$
$509$	30.5939	1.35605	0.678024	+	0.735039i	$0.262836\pi$
$510$	0	0
$511$	27.6014	1.22102
$512$	18.2220	0.805308
$513$	0	0
$514$	62.2827	2.74717
$515$	31.1007	1.37046
$516$	0	0
$517$	−11.7752	−0.517874
$518$	−42.4943	−1.86709
$519$	0	0
$520$	6.84676	0.300250
$521$	−8.99098	−0.393902	−0.196951	−	0.980413i	$-0.563104\pi$
$521$	−8.99098	−0.393902	−0.196951	+	0.980413i	$0.563104\pi$
$522$	0	0
$523$	−1.43930	−0.0629362	−0.0314681	−	0.999505i	$-0.510018\pi$
$523$	−1.43930	−0.0629362	−0.0314681	+	0.999505i	$0.510018\pi$
$524$	−32.0070	−1.39823
$525$	0	0
$526$	58.8271	2.56498
$527$	2.60232	0.113359
$528$	0	0
$529$	−20.6106	−0.896113
$530$	−18.2564	−0.793008
$531$	0	0
$532$	−41.0715	−1.78068
$533$	6.22255	0.269528
$534$	0	0
$535$	68.3324	2.95427
$536$	17.1840	0.742234
$537$	0	0
$538$	1.72553	0.0743930
$539$	22.2355	0.957753
$540$	0	0
$541$	−27.0344	−1.16230	−0.581150	−	0.813796i	$-0.697397\pi$
$541$	−27.0344	−1.16230	−0.581150	+	0.813796i	$0.697397\pi$
$542$	−26.6917	−1.14651
$543$	0	0
$544$	11.8018	0.505996
$545$	−51.8339	−2.22032
$546$	0	0
$547$	−37.2739	−1.59372	−0.796858	−	0.604166i	$-0.793506\pi$
$547$	−37.2739	−1.59372	−0.796858	+	0.604166i	$0.793506\pi$
$548$	−23.4828	−1.00313
$549$	0	0
$550$	90.1022	3.84197
$551$	0	0
$552$	0	0
$553$	1.05002	0.0446513
$554$	−34.6704	−1.47300
$555$	0	0
$556$	−8.79356	−0.372930
$557$	−31.7679	−1.34605	−0.673026	−	0.739619i	$-0.735006\pi$
$557$	−31.7679	−1.34605	−0.673026	+	0.739619i	$0.735006\pi$
$558$	0	0
$559$	4.10398	0.173580
$560$	−24.8276	−1.04916
$561$	0	0
$562$	−44.0905	−1.85985
$563$	−13.4735	−0.567842	−0.283921	−	0.958848i	$-0.591635\pi$
$563$	−13.4735	−0.567842	−0.283921	+	0.958848i	$0.591635\pi$
$564$	0	0
$565$	34.5169	1.45214
$566$	−4.29558	−0.180557
$567$	0	0
$568$	−15.3088	−0.642342
$569$	5.53839	0.232181	0.116091	−	0.993239i	$-0.462964\pi$
$569$	5.53839	0.232181	0.116091	+	0.993239i	$0.462964\pi$
$570$	0	0
$571$	41.3036	1.72850	0.864250	−	0.503062i	$-0.167793\pi$
$571$	41.3036	1.72850	0.864250	+	0.503062i	$0.167793\pi$
$572$	9.23386	0.386087
$573$	0	0
$574$	53.8057	2.24581
$575$	−17.9735	−0.749546
$576$	0	0
$577$	−39.7871	−1.65636	−0.828180	−	0.560463i	$-0.810623\pi$
$577$	−39.7871	−1.65636	−0.828180	+	0.560463i	$0.810623\pi$
$578$	−31.5872	−1.31385
$579$	0	0
$580$	0	0
$581$	23.6257	0.980161
$582$	0	0
$583$	−7.19039	−0.297795
$584$	13.7140	0.567489
$585$	0	0
$586$	5.30300	0.219065
$587$	−38.7105	−1.59775	−0.798877	−	0.601495i	$-0.794572\pi$
$587$	−38.7105	−1.59775	−0.798877	+	0.601495i	$0.794572\pi$
$588$	0	0
$589$	6.40861	0.264062
$590$	−65.3010	−2.68840
$591$	0	0
$592$	8.85427	0.363908
$593$	1.40712	0.0577833	0.0288916	−	0.999583i	$-0.490802\pi$
$593$	1.40712	0.0577833	0.0288916	+	0.999583i	$0.490802\pi$
$594$	0	0
$595$	−24.0731	−0.986901
$596$	−42.8154	−1.75379
$597$	0	0
$598$	−3.14599	−0.128649
$599$	−30.7455	−1.25623	−0.628113	−	0.778122i	$-0.716173\pi$
$599$	−30.7455	−1.25623	−0.628113	+	0.778122i	$0.716173\pi$
$600$	0	0
$601$	−13.2534	−0.540616	−0.270308	−	0.962774i	$-0.587125\pi$
$601$	−13.2534	−0.540616	−0.270308	+	0.962774i	$0.587125\pi$
$602$	35.4867	1.44633
$603$	0	0
$604$	1.91787	0.0780371
$605$	5.89271	0.239573
$606$	0	0
$607$	9.85282	0.399914	0.199957	−	0.979805i	$-0.435920\pi$
$607$	9.85282	0.399914	0.199957	+	0.979805i	$0.435920\pi$
$608$	29.0636	1.17869
$609$	0	0
$610$	−53.7916	−2.17796
$611$	−3.09267	−0.125116
$612$	0	0
$613$	−23.0646	−0.931571	−0.465785	−	0.884898i	$-0.654228\pi$
$613$	−23.0646	−0.931571	−0.465785	+	0.884898i	$0.654228\pi$
$614$	−2.44628	−0.0987238
$615$	0	0
$616$	23.3175	0.939488
$617$	2.14430	0.0863262	0.0431631	−	0.999068i	$-0.486256\pi$
$617$	2.14430	0.0863262	0.0431631	+	0.999068i	$0.486256\pi$
$618$	0	0
$619$	29.7463	1.19560	0.597802	−	0.801644i	$-0.296041\pi$
$619$	29.7463	1.19560	0.597802	+	0.801644i	$0.296041\pi$
$620$	−18.5204	−0.743799
$621$	0	0
$622$	−9.73065	−0.390163
$623$	−60.2150	−2.41246
$624$	0	0
$625$	52.0617	2.08247
$626$	−66.1178	−2.64260
$627$	0	0
$628$	−17.6163	−0.702966
$629$	8.58519	0.342314
$630$	0	0
$631$	−4.52395	−0.180096	−0.0900478	−	0.995937i	$-0.528702\pi$
$631$	−4.52395	−0.180096	−0.0900478	+	0.995937i	$0.528702\pi$
$632$	0.521710	0.0207525
$633$	0	0
$634$	13.6423	0.541807
$635$	−72.6039	−2.88120
$636$	0	0
$637$	5.84000	0.231389
$638$	0	0
$639$	0	0
$640$	−54.0872	−2.13798
$641$	30.7540	1.21471	0.607354	−	0.794431i	$-0.292231\pi$
$641$	30.7540	1.21471	0.607354	+	0.794431i	$0.292231\pi$
$642$	0	0
$643$	16.8712	0.665337	0.332668	−	0.943044i	$-0.392051\pi$
$643$	16.8712	0.665337	0.332668	+	0.943044i	$0.392051\pi$
$644$	−15.9272	−0.627620
$645$	0	0
$646$	14.1722	0.557599
$647$	−14.1310	−0.555548	−0.277774	−	0.960646i	$-0.589597\pi$
$647$	−14.1310	−0.555548	−0.277774	+	0.960646i	$0.589597\pi$
$648$	0	0
$649$	−25.7191	−1.00956
$650$	23.6647	0.928205
$651$	0	0
$652$	−43.2688	−1.69454
$653$	38.7238	1.51538	0.757689	−	0.652615i	$-0.226328\pi$
$653$	38.7238	1.51538	0.757689	+	0.652615i	$0.226328\pi$
$654$	0	0
$655$	−46.1997	−1.80517
$656$	−11.2111	−0.437722
$657$	0	0
$658$	−26.7420	−1.04251
$659$	−17.3822	−0.677116	−0.338558	−	0.940945i	$-0.609939\pi$
$659$	−17.3822	−0.677116	−0.338558	+	0.940945i	$0.609939\pi$
$660$	0	0
$661$	0.959081	0.0373039	0.0186520	−	0.999826i	$-0.494063\pi$
$661$	0.959081	0.0373039	0.0186520	+	0.999826i	$0.494063\pi$
$662$	21.9786	0.854222
$663$	0	0
$664$	11.7386	0.455547
$665$	−59.2837	−2.29892
$666$	0	0
$667$	0	0
$668$	44.5958	1.72546
$669$	0	0
$670$	84.9334	3.28127
$671$	−21.1861	−0.817881
$672$	0	0
$673$	−30.1107	−1.16068	−0.580341	−	0.814374i	$-0.697081\pi$
$673$	−30.1107	−1.16068	−0.580341	+	0.814374i	$0.697081\pi$
$674$	−59.9570	−2.30946
$675$	0	0
$676$	−34.2999	−1.31923
$677$	−41.7661	−1.60520	−0.802602	−	0.596515i	$-0.796552\pi$
$677$	−41.7661	−1.60520	−0.802602	+	0.596515i	$0.796552\pi$
$678$	0	0
$679$	36.6636	1.40702
$680$	−11.9609	−0.458680
$681$	0	0
$682$	−12.4585	−0.477061
$683$	−40.2674	−1.54079	−0.770396	−	0.637566i	$-0.779941\pi$
$683$	−40.2674	−1.54079	−0.770396	+	0.637566i	$0.779941\pi$
$684$	0	0
$685$	−33.8956	−1.29508
$686$	−5.58402	−0.213199
$687$	0	0
$688$	−7.39413	−0.281899
$689$	−1.88850	−0.0719461
$690$	0	0
$691$	41.3791	1.57414	0.787069	−	0.616866i	$-0.211598\pi$
$691$	41.3791	1.57414	0.787069	+	0.616866i	$0.211598\pi$
$692$	9.73221	0.369963
$693$	0	0
$694$	−2.62477	−0.0996351
$695$	−12.6928	−0.481467
$696$	0	0
$697$	−10.8704	−0.411747
$698$	15.9279	0.602881
$699$	0	0
$700$	119.807	4.52828
$701$	−2.28878	−0.0864459	−0.0432230	−	0.999065i	$-0.513763\pi$
$701$	−2.28878	−0.0864459	−0.0432230	+	0.999065i	$0.513763\pi$
$702$	0	0
$703$	21.1423	0.797399
$704$	−44.7224	−1.68554
$705$	0	0
$706$	11.5618	0.435135
$707$	−30.0512	−1.13019
$708$	0	0
$709$	−27.3080	−1.02557	−0.512786	−	0.858516i	$-0.671387\pi$
$709$	−27.3080	−1.02557	−0.512786	+	0.858516i	$0.671387\pi$
$710$	−75.6651	−2.83966
$711$	0	0
$712$	−29.9183	−1.12124
$713$	2.48521	0.0930717
$714$	0	0
$715$	13.3284	0.498453
$716$	34.0068	1.27089
$717$	0	0
$718$	32.1330	1.19919
$719$	19.0751	0.711383	0.355691	−	0.934604i	$-0.384245\pi$
$719$	19.0751	0.711383	0.355691	+	0.934604i	$0.384245\pi$
$720$	0	0
$721$	27.8183	1.03601
$722$	−6.83391	−0.254332
$723$	0	0
$724$	6.83444	0.254000
$725$	0	0
$726$	0	0
$727$	−29.2574	−1.08510	−0.542548	−	0.840024i	$-0.682541\pi$
$727$	−29.2574	−1.08510	−0.542548	+	0.840024i	$0.682541\pi$
$728$	6.12416	0.226976
$729$	0	0
$730$	67.7828	2.50875
$731$	−7.16942	−0.265171
$732$	0	0
$733$	20.9488	0.773760	0.386880	−	0.922130i	$-0.373553\pi$
$733$	20.9488	0.773760	0.386880	+	0.922130i	$0.373553\pi$
$734$	12.1981	0.450239
$735$	0	0
$736$	11.2706	0.415441
$737$	33.4515	1.23220
$738$	0	0
$739$	−33.7623	−1.24197	−0.620983	−	0.783824i	$-0.713266\pi$
$739$	−33.7623	−1.24197	−0.620983	+	0.783824i	$0.713266\pi$
$740$	−61.0999	−2.24608
$741$	0	0
$742$	−16.3297	−0.599480
$743$	44.2943	1.62500	0.812500	−	0.582962i	$-0.198106\pi$
$743$	44.2943	1.62500	0.812500	+	0.582962i	$0.198106\pi$
$744$	0	0
$745$	−61.8007	−2.26420
$746$	65.3396	2.39225
$747$	0	0
$748$	−16.1310	−0.589809
$749$	61.1207	2.23330
$750$	0	0
$751$	−7.65239	−0.279240	−0.139620	−	0.990205i	$-0.544588\pi$
$751$	−7.65239	−0.279240	−0.139620	+	0.990205i	$0.544588\pi$
$752$	5.57206	0.203192
$753$	0	0
$754$	0	0
$755$	2.76830	0.100749
$756$	0	0
$757$	16.2856	0.591911	0.295955	−	0.955202i	$-0.404362\pi$
$757$	16.2856	0.591911	0.295955	+	0.955202i	$0.404362\pi$
$758$	−4.31040	−0.156561
$759$	0	0
$760$	−29.4556	−1.06847
$761$	28.5492	1.03491	0.517453	−	0.855711i	$-0.326880\pi$
$761$	28.5492	1.03491	0.517453	+	0.855711i	$0.326880\pi$
$762$	0	0
$763$	−46.3634	−1.67847
$764$	−16.4885	−0.596533
$765$	0	0
$766$	25.9560	0.937830
$767$	−6.75493	−0.243907
$768$	0	0
$769$	49.8716	1.79842	0.899208	−	0.437521i	$-0.144143\pi$
$769$	49.8716	1.79842	0.899208	+	0.437521i	$0.144143\pi$
$770$	115.249	4.15329
$771$	0	0
$772$	56.4729	2.03250
$773$	−4.99992	−0.179835	−0.0899173	−	0.995949i	$-0.528660\pi$
$773$	−4.99992	−0.179835	−0.0899173	+	0.995949i	$0.528660\pi$
$774$	0	0
$775$	−18.6941	−0.671513
$776$	18.2166	0.653938
$777$	0	0
$778$	−26.1424	−0.937249
$779$	−26.7701	−0.959139
$780$	0	0
$781$	−29.8011	−1.06637
$782$	5.49588	0.196532
$783$	0	0
$784$	−10.5219	−0.375782
$785$	−25.4278	−0.907556
$786$	0	0
$787$	19.1052	0.681027	0.340513	−	0.940240i	$-0.389399\pi$
$787$	19.1052	0.681027	0.340513	+	0.940240i	$0.389399\pi$
$788$	33.0881	1.17872
$789$	0	0
$790$	2.57860	0.0917426
$791$	30.8740	1.09775
$792$	0	0
$793$	−5.56437	−0.197597
$794$	43.2242	1.53397
$795$	0	0
$796$	−47.0318	−1.66700
$797$	23.2432	0.823318	0.411659	−	0.911338i	$-0.364950\pi$
$797$	23.2432	0.823318	0.411659	+	0.911338i	$0.364950\pi$
$798$	0	0
$799$	5.40273	0.191135
$800$	−84.7795	−2.99741
$801$	0	0
$802$	45.0881	1.59212
$803$	26.6966	0.942103
$804$	0	0
$805$	−22.9897	−0.810281
$806$	−3.27213	−0.115256
$807$	0	0
$808$	−14.9312	−0.525276
$809$	7.53860	0.265043	0.132521	−	0.991180i	$-0.457693\pi$
$809$	7.53860	0.265043	0.132521	+	0.991180i	$0.457693\pi$
$810$	0	0
$811$	−48.2745	−1.69515	−0.847573	−	0.530679i	$-0.821937\pi$
$811$	−48.2745	−1.69515	−0.847573	+	0.530679i	$0.821937\pi$
$812$	0	0
$813$	0	0
$814$	−41.1013	−1.44060
$815$	−62.4552	−2.18771
$816$	0	0
$817$	−17.6558	−0.617698
$818$	70.6906	2.47164
$819$	0	0
$820$	77.3637	2.70166
$821$	−6.07390	−0.211980	−0.105990	−	0.994367i	$-0.533801\pi$
$821$	−6.07390	−0.211980	−0.105990	+	0.994367i	$0.533801\pi$
$822$	0	0
$823$	40.0147	1.39483	0.697413	−	0.716670i	$-0.254334\pi$
$823$	40.0147	1.39483	0.697413	+	0.716670i	$0.254334\pi$
$824$	13.8218	0.481504
$825$	0	0
$826$	−58.4092	−2.03231
$827$	28.8632	1.00367	0.501837	−	0.864962i	$-0.332658\pi$
$827$	28.8632	1.00367	0.501837	+	0.864962i	$0.332658\pi$
$828$	0	0
$829$	−41.9301	−1.45629	−0.728147	−	0.685421i	$-0.759618\pi$
$829$	−41.9301	−1.45629	−0.728147	+	0.685421i	$0.759618\pi$
$830$	58.0194	2.01388
$831$	0	0
$832$	−11.7460	−0.407219
$833$	−10.2021	−0.353483
$834$	0	0
$835$	64.3706	2.22764
$836$	−39.7251	−1.37392
$837$	0	0
$838$	−3.48588	−0.120418
$839$	30.7540	1.06175	0.530873	−	0.847451i	$-0.321864\pi$
$839$	30.7540	1.06175	0.530873	+	0.847451i	$0.321864\pi$
$840$	0	0
$841$	0	0
$842$	12.8790	0.443839
$843$	0	0
$844$	−68.4881	−2.35746
$845$	−49.5093	−1.70317
$846$	0	0
$847$	5.27079	0.181107
$848$	3.40251	0.116843
$849$	0	0
$850$	−41.3408	−1.41798
$851$	8.19883	0.281052
$852$	0	0
$853$	−14.3499	−0.491330	−0.245665	−	0.969355i	$-0.579006\pi$
$853$	−14.3499	−0.491330	−0.245665	+	0.969355i	$0.579006\pi$
$854$	−48.1145	−1.64644
$855$	0	0
$856$	30.3683	1.03797
$857$	−12.0892	−0.412959	−0.206479	−	0.978451i	$-0.566201\pi$
$857$	−12.0892	−0.412959	−0.206479	+	0.978451i	$0.566201\pi$
$858$	0	0
$859$	22.5722	0.770154	0.385077	−	0.922885i	$-0.374175\pi$
$859$	22.5722	0.770154	0.385077	+	0.922885i	$0.374175\pi$
$860$	51.0240	1.73990
$861$	0	0
$862$	18.5883	0.633121
$863$	20.0243	0.681637	0.340818	−	0.940129i	$-0.389296\pi$
$863$	20.0243	0.681637	0.340818	+	0.940129i	$0.389296\pi$
$864$	0	0
$865$	14.0477	0.477636
$866$	27.9373	0.949348
$867$	0	0
$868$	−16.5658	−0.562280
$869$	1.01560	0.0344518
$870$	0	0
$871$	8.78578	0.297695
$872$	−23.0360	−0.780098
$873$	0	0
$874$	13.5344	0.457809
$875$	98.5689	3.33224
$876$	0	0
$877$	−42.0059	−1.41844	−0.709219	−	0.704988i	$-0.750952\pi$
$877$	−42.0059	−1.41844	−0.709219	+	0.704988i	$0.750952\pi$
$878$	−16.4863	−0.556384
$879$	0	0
$880$	−24.0137	−0.809502
$881$	−56.1769	−1.89265	−0.946323	−	0.323221i	$-0.895234\pi$
$881$	−56.1769	−1.89265	−0.946323	+	0.323221i	$0.895234\pi$
$882$	0	0
$883$	−54.5727	−1.83652	−0.918259	−	0.395980i	$-0.870405\pi$
$883$	−54.5727	−1.83652	−0.918259	+	0.395980i	$0.870405\pi$
$884$	−4.23669	−0.142495
$885$	0	0
$886$	25.6085	0.860334
$887$	−39.2007	−1.31623	−0.658115	−	0.752917i	$-0.728646\pi$
$887$	−39.2007	−1.31623	−0.658115	+	0.752917i	$0.728646\pi$
$888$	0	0
$889$	−64.9413	−2.17806
$890$	−147.874	−4.95675
$891$	0	0
$892$	79.9983	2.67854
$893$	13.3050	0.445236
$894$	0	0
$895$	49.0862	1.64077
$896$	−48.3789	−1.61622
$897$	0	0
$898$	42.7262	1.42579
$899$	0	0
$900$	0	0
$901$	3.29910	0.109909
$902$	52.0418	1.73280
$903$	0	0
$904$	15.3400	0.510200
$905$	9.86500	0.327924
$906$	0	0
$907$	3.36661	0.111786	0.0558932	−	0.998437i	$-0.482199\pi$
$907$	3.36661	0.111786	0.0558932	+	0.998437i	$0.482199\pi$
$908$	−27.8664	−0.924779
$909$	0	0
$910$	30.2693	1.00342
$911$	−54.9714	−1.82128	−0.910641	−	0.413198i	$-0.864412\pi$
$911$	−54.9714	−1.82128	−0.910641	+	0.413198i	$0.864412\pi$
$912$	0	0
$913$	22.8512	0.756265
$914$	37.5892	1.24334
$915$	0	0
$916$	44.2312	1.46144
$917$	−41.3238	−1.36463
$918$	0	0
$919$	26.6983	0.880696	0.440348	−	0.897827i	$-0.354855\pi$
$919$	26.6983	0.880696	0.440348	+	0.897827i	$0.354855\pi$
$920$	−11.4226	−0.376593
$921$	0	0
$922$	−18.7374	−0.617082
$923$	−7.82703	−0.257630
$924$	0	0
$925$	−61.6729	−2.02779
$926$	3.28988	0.108112
$927$	0	0
$928$	0	0
$929$	−34.4983	−1.13185	−0.565926	−	0.824456i	$-0.691481\pi$
$929$	−34.4983	−1.13185	−0.565926	+	0.824456i	$0.691481\pi$
$930$	0	0
$931$	−25.1243	−0.823417
$932$	55.9011	1.83110
$933$	0	0
$934$	47.0745	1.54032
$935$	−23.2839	−0.761466
$936$	0	0
$937$	−36.3860	−1.18868	−0.594340	−	0.804214i	$-0.702587\pi$
$937$	−36.3860	−1.18868	−0.594340	+	0.804214i	$0.702587\pi$
$938$	75.9696	2.48050
$939$	0	0
$940$	−38.4506	−1.25412
$941$	−10.4980	−0.342226	−0.171113	−	0.985251i	$-0.554736\pi$
$941$	−10.4980	−0.342226	−0.171113	+	0.985251i	$0.554736\pi$
$942$	0	0
$943$	−10.3812	−0.338059
$944$	12.1703	0.396111
$945$	0	0
$946$	34.3233	1.11595
$947$	8.01774	0.260542	0.130271	−	0.991478i	$-0.458415\pi$
$947$	8.01774	0.260542	0.130271	+	0.991478i	$0.458415\pi$
$948$	0	0
$949$	7.01166	0.227608
$950$	−101.808	−3.30309
$951$	0	0
$952$	−10.6986	−0.346742
$953$	23.8505	0.772594	0.386297	−	0.922374i	$-0.373754\pi$
$953$	23.8505	0.772594	0.386297	+	0.922374i	$0.373754\pi$
$954$	0	0
$955$	−23.7999	−0.770147
$956$	26.3658	0.852729
$957$	0	0
$958$	62.7209	2.02642
$959$	−30.3183	−0.979029
$960$	0	0
$961$	−28.4151	−0.916618
$962$	−10.7949	−0.348043
$963$	0	0
$964$	−69.4208	−2.23589
$965$	81.5143	2.62404
$966$	0	0
$967$	−1.65641	−0.0532667	−0.0266333	−	0.999645i	$-0.508479\pi$
$967$	−1.65641	−0.0532667	−0.0266333	+	0.999645i	$0.508479\pi$
$968$	2.61884	0.0841726
$969$	0	0
$970$	90.0374	2.89093
$971$	30.7444	0.986635	0.493317	−	0.869849i	$-0.335784\pi$
$971$	30.7444	0.986635	0.493317	+	0.869849i	$0.335784\pi$
$972$	0	0
$973$	−11.3532	−0.363968
$974$	55.2564	1.77053
$975$	0	0
$976$	10.0253	0.320902
$977$	45.2764	1.44852	0.724260	−	0.689527i	$-0.242181\pi$
$977$	45.2764	1.44852	0.724260	+	0.689527i	$0.242181\pi$
$978$	0	0
$979$	−58.2410	−1.86139
$980$	72.6076	2.31936
$981$	0	0
$982$	47.6323	1.52001
$983$	54.0341	1.72342	0.861711	−	0.507400i	$-0.169393\pi$
$983$	54.0341	1.72342	0.861711	+	0.507400i	$0.169393\pi$
$984$	0	0
$985$	47.7602	1.52177
$986$	0	0
$987$	0	0
$988$	−10.4335	−0.331934
$989$	−6.84677	−0.217715
$990$	0	0
$991$	2.45456	0.0779718	0.0389859	−	0.999240i	$-0.487587\pi$
$991$	2.45456	0.0779718	0.0389859	+	0.999240i	$0.487587\pi$
$992$	11.7225	0.372191
$993$	0	0
$994$	−67.6795	−2.14666
$995$	−67.8868	−2.15216
$996$	0	0
$997$	−18.5499	−0.587480	−0.293740	−	0.955885i	$-0.594900\pi$
$997$	−18.5499	−0.587480	−0.293740	+	0.955885i	$0.594900\pi$
$998$	12.4970	0.395586
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.8		9
3.2	odd	2		2523.2.a.p.1.2		9
29.16	even	7		261.2.k.b.82.3		18
29.20	even	7		261.2.k.b.226.3		18
29.28	even	2		7569.2.a.bk.1.2		9
87.20	odd	14		87.2.g.b.52.1	✓	18
87.74	odd	14		87.2.g.b.82.1	yes	18
87.86	odd	2		2523.2.a.q.1.8		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.52.1	✓	18	87.20	odd	14
87.2.g.b.82.1	yes	18	87.74	odd	14
261.2.k.b.82.3		18	29.16	even	7
261.2.k.b.226.3		18	29.20	even	7
2523.2.a.p.1.2		9	3.2	odd	2
2523.2.a.q.1.8		9	87.86	odd	2
7569.2.a.bk.1.2		9	29.28	even	2
7569.2.a.bl.1.8		9	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+2.19659 q^{2} +2.82501 q^{4} +4.07769 q^{5} +3.64733 q^{7} +1.81221 q^{8} +O(q^{10})\)	\(q+2.19659 q^{2} +2.82501 q^{4} +4.07769 q^{5} +3.64733 q^{7} +1.81221 q^{8} +8.95700 q^{10} +3.52776 q^{11} +0.926540 q^{13} +8.01169 q^{14} -1.66934 q^{16} -1.61861 q^{17} -3.98608 q^{19} +11.5195 q^{20} +7.74905 q^{22} -1.54577 q^{23} +11.6275 q^{25} +2.03523 q^{26} +10.3037 q^{28} -1.60775 q^{31} -7.29128 q^{32} -3.55543 q^{34} +14.8727 q^{35} -5.30404 q^{37} -8.75579 q^{38} +7.38960 q^{40} +6.71590 q^{41} +4.42936 q^{43} +9.96596 q^{44} -3.39542 q^{46} -3.33787 q^{47} +6.30302 q^{49} +25.5409 q^{50} +2.61748 q^{52} -2.03823 q^{53} +14.3851 q^{55} +6.60971 q^{56} -7.29049 q^{59} -6.00554 q^{61} -3.53156 q^{62} -12.6773 q^{64} +3.77814 q^{65} +9.48235 q^{67} -4.57260 q^{68} +32.6692 q^{70} -8.44759 q^{71} +7.56757 q^{73} -11.6508 q^{74} -11.2607 q^{76} +12.8669 q^{77} +0.287887 q^{79} -6.80706 q^{80} +14.7521 q^{82} +6.47754 q^{83} -6.60020 q^{85} +9.72949 q^{86} +6.39303 q^{88} -16.5093 q^{89} +3.37940 q^{91} -4.36681 q^{92} -7.33194 q^{94} -16.2540 q^{95} +10.0522 q^{97} +13.8451 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) \) 9 * q + q^2 + 11 * q^4 + 5 * q^7 + 12 * q^8 - 4 * q^10 + 3 * q^11 + 5 * q^13 + 15 * q^14 - 5 * q^16 + 16 * q^17 + q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 + 12 * q^26 - 24 * q^28 + 4 * q^31 + 25 * q^32 + 24 * q^34 + 44 * q^35 - 25 * q^37 + 10 * q^38 - 5 * q^40 + 34 * q^41 - 12 * q^43 - 23 * q^44 - 6 * q^46 + 8 * q^47 + 26 * q^49 + 27 * q^50 - 23 * q^52 + 32 * q^53 + 5 * q^55 - 14 * q^56 - 10 * q^59 - 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 + 11 * q^68 + 14 * q^70 - 7 * q^71 - 17 * q^73 + 62 * q^74 + 6 * q^76 + 64 * q^77 - 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 - 42 * q^85 + 70 * q^86 - 29 * q^88 - 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 - 20 * q^95 - 16 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

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