LMFDB - Embedded newform 7569.2.a.bl.1.6

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			1.6
Root			$0.915678$ 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)

$f(q)$	$=$	$q+0.915678 q^{2} -1.16153 q^{4} -0.285007 q^{5} +5.03918 q^{7} -2.89495 q^{8} +O(q^{10})$	\(q+0.915678 q^{2} -1.16153 q^{4} -0.285007 q^{5} +5.03918 q^{7} -2.89495 q^{8} -0.260975 q^{10} +4.18236 q^{11} +2.27527 q^{13} +4.61427 q^{14} -0.327776 q^{16} +3.40834 q^{17} +0.852557 q^{19} +0.331045 q^{20} +3.82970 q^{22} +7.24173 q^{23} -4.91877 q^{25} +2.08342 q^{26} -5.85318 q^{28} -3.27177 q^{31} +5.48976 q^{32} +3.12094 q^{34} -1.43620 q^{35} +0.380705 q^{37} +0.780668 q^{38} +0.825080 q^{40} +5.56008 q^{41} +7.90289 q^{43} -4.85795 q^{44} +6.63110 q^{46} -6.53585 q^{47} +18.3934 q^{49} -4.50401 q^{50} -2.64281 q^{52} -2.89257 q^{53} -1.19200 q^{55} -14.5882 q^{56} +4.68212 q^{59} -6.93927 q^{61} -2.99589 q^{62} +5.68240 q^{64} -0.648469 q^{65} -2.84060 q^{67} -3.95890 q^{68} -1.31510 q^{70} +11.4943 q^{71} -11.3234 q^{73} +0.348603 q^{74} -0.990273 q^{76} +21.0757 q^{77} -6.76944 q^{79} +0.0934183 q^{80} +5.09125 q^{82} -6.27986 q^{83} -0.971400 q^{85} +7.23650 q^{86} -12.1077 q^{88} -5.70609 q^{89} +11.4655 q^{91} -8.41151 q^{92} -5.98474 q^{94} -0.242985 q^{95} +0.502800 q^{97} +16.8424 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.915678	0.647482	0.323741	−	0.946146i	$-0.395059\pi$
$2$	0.915678	0.647482	0.323741	+	0.946146i	$0.395059\pi$
$3$	0	0
$3$	0	0
$4$	−1.16153	−0.580766
$5$	−0.285007	−0.127459	−0.0637295	−	0.997967i	$-0.520299\pi$
$5$	−0.285007	−0.127459	−0.0637295	+	0.997967i	$0.520299\pi$
$6$	0	0
$7$	5.03918	1.90463	0.952316	−	0.305113i	$-0.0986941\pi$
$7$	5.03918	1.90463	0.952316	+	0.305113i	$0.0986941\pi$
$8$	−2.89495	−1.02352
$9$	0	0
$10$	−0.260975	−0.0825275
$11$	4.18236	1.26103	0.630514	−	0.776178i	$-0.282844\pi$
$11$	4.18236	1.26103	0.630514	+	0.776178i	$0.282844\pi$
$12$	0	0
$13$	2.27527	0.631047	0.315524	−	0.948918i	$-0.397820\pi$
$13$	2.27527	0.631047	0.315524	+	0.948918i	$0.397820\pi$
$14$	4.61427	1.23322
$15$	0	0
$16$	−0.327776	−0.0819439
$17$	3.40834	0.826644	0.413322	−	0.910585i	$-0.364368\pi$
$17$	3.40834	0.826644	0.413322	+	0.910585i	$0.364368\pi$
$18$	0	0
$19$	0.852557	0.195590	0.0977950	−	0.995207i	$-0.468821\pi$
$19$	0.852557	0.195590	0.0977950	+	0.995207i	$0.468821\pi$
$20$	0.331045	0.0740239
$21$	0	0
$22$	3.82970	0.816494
$23$	7.24173	1.51001	0.755003	−	0.655721i	$-0.227635\pi$
$23$	7.24173	1.51001	0.755003	+	0.655721i	$0.227635\pi$
$24$	0	0
$25$	−4.91877	−0.983754
$26$	2.08342	0.408592
$27$	0	0
$28$	−5.85318	−1.10615
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−3.27177	−0.587627	−0.293813	−	0.955863i	$-0.594924\pi$
$31$	−3.27177	−0.587627	−0.293813	+	0.955863i	$0.594924\pi$
$32$	5.48976	0.970461
$33$	0	0
$34$	3.12094	0.535237
$35$	−1.43620	−0.242763
$36$	0	0
$37$	0.380705	0.0625875	0.0312937	−	0.999510i	$-0.490037\pi$
$37$	0.380705	0.0625875	0.0312937	+	0.999510i	$0.490037\pi$
$38$	0.780668	0.126641
$39$	0	0
$40$	0.825080	0.130457
$41$	5.56008	0.868339	0.434169	−	0.900831i	$-0.357042\pi$
$41$	5.56008	0.868339	0.434169	+	0.900831i	$0.357042\pi$
$42$	0	0
$43$	7.90289	1.20518	0.602589	−	0.798051i	$-0.294136\pi$
$43$	7.90289	1.20518	0.602589	+	0.798051i	$0.294136\pi$
$44$	−4.85795	−0.732363
$45$	0	0
$46$	6.63110	0.977702
$47$	−6.53585	−0.953352	−0.476676	−	0.879079i	$-0.658158\pi$
$47$	−6.53585	−0.953352	−0.476676	+	0.879079i	$0.658158\pi$
$48$	0	0
$49$	18.3934	2.62763
$50$	−4.50401	−0.636964
$51$	0	0
$52$	−2.64281	−0.366491
$53$	−2.89257	−0.397325	−0.198663	−	0.980068i	$-0.563660\pi$
$53$	−2.89257	−0.397325	−0.198663	+	0.980068i	$0.563660\pi$
$54$	0	0
$55$	−1.19200	−0.160729
$56$	−14.5882	−1.94943
$57$	0	0
$58$	0	0
$59$	4.68212	0.609560	0.304780	−	0.952423i	$-0.401417\pi$
$59$	4.68212	0.609560	0.304780	+	0.952423i	$0.401417\pi$
$60$	0	0
$61$	−6.93927	−0.888483	−0.444241	−	0.895907i	$-0.646527\pi$
$61$	−6.93927	−0.888483	−0.444241	+	0.895907i	$0.646527\pi$
$62$	−2.99589	−0.380478
$63$	0	0
$64$	5.68240	0.710301
$65$	−0.648469	−0.0804327
$66$	0	0
$67$	−2.84060	−0.347035	−0.173518	−	0.984831i	$-0.555513\pi$
$67$	−2.84060	−0.347035	−0.173518	+	0.984831i	$0.555513\pi$
$68$	−3.95890	−0.480087
$69$	0	0
$70$	−1.31510	−0.157184
$71$	11.4943	1.36412	0.682061	−	0.731296i	$-0.261084\pi$
$71$	11.4943	1.36412	0.682061	+	0.731296i	$0.261084\pi$
$72$	0	0
$73$	−11.3234	−1.32531	−0.662653	−	0.748926i	$-0.730570\pi$
$73$	−11.3234	−1.32531	−0.662653	+	0.748926i	$0.730570\pi$
$74$	0.348603	0.0405243
$75$	0	0
$76$	−0.990273	−0.113592
$77$	21.0757	2.40180
$78$	0	0
$79$	−6.76944	−0.761621	−0.380811	−	0.924653i	$-0.624355\pi$
$79$	−6.76944	−0.761621	−0.380811	+	0.924653i	$0.624355\pi$
$80$	0.0934183	0.0104445
$81$	0	0
$82$	5.09125	0.562234
$83$	−6.27986	−0.689304	−0.344652	−	0.938731i	$-0.612003\pi$
$83$	−6.27986	−0.689304	−0.344652	+	0.938731i	$0.612003\pi$
$84$	0	0
$85$	−0.971400	−0.105363
$86$	7.23650	0.780332
$87$	0	0
$88$	−12.1077	−1.29069
$89$	−5.70609	−0.604844	−0.302422	−	0.953174i	$-0.597795\pi$
$89$	−5.70609	−0.604844	−0.302422	+	0.953174i	$0.597795\pi$
$90$	0	0
$91$	11.4655	1.20191
$92$	−8.41151	−0.876961
$93$	0	0
$94$	−5.98474	−0.617278
$95$	−0.242985	−0.0249297
$96$	0	0
$97$	0.502800	0.0510516	0.0255258	−	0.999674i	$-0.491874\pi$
$97$	0.502800	0.0510516	0.0255258	+	0.999674i	$0.491874\pi$
$98$	16.8424	1.70134
$99$	0	0
$100$	5.71331	0.571331
$101$	4.50787	0.448550	0.224275	−	0.974526i	$-0.427999\pi$
$101$	4.50787	0.448550	0.224275	+	0.974526i	$0.427999\pi$
$102$	0	0
$103$	8.12388	0.800469	0.400235	−	0.916413i	$-0.368929\pi$
$103$	8.12388	0.800469	0.400235	+	0.916413i	$0.368929\pi$
$104$	−6.58680	−0.645889
$105$	0	0
$106$	−2.64866	−0.257261
$107$	−9.51920	−0.920256	−0.460128	−	0.887853i	$-0.652196\pi$
$107$	−9.51920	−0.920256	−0.460128	+	0.887853i	$0.652196\pi$
$108$	0	0
$109$	−7.43141	−0.711800	−0.355900	−	0.934524i	$-0.615826\pi$
$109$	−7.43141	−0.711800	−0.355900	+	0.934524i	$0.615826\pi$
$110$	−1.09149	−0.104069
$111$	0	0
$112$	−1.65172	−0.156073
$113$	−4.23804	−0.398682	−0.199341	−	0.979930i	$-0.563880\pi$
$113$	−4.23804	−0.398682	−0.199341	+	0.979930i	$0.563880\pi$
$114$	0	0
$115$	−2.06394	−0.192464
$116$	0	0
$117$	0	0
$118$	4.28732	0.394680
$119$	17.1752	1.57445
$120$	0	0
$121$	6.49212	0.590193
$122$	−6.35414	−0.575277
$123$	0	0
$124$	3.80027	0.341274
$125$	2.82692	0.252847
$126$	0	0
$127$	−16.9993	−1.50845	−0.754224	−	0.656618i	$-0.771987\pi$
$127$	−16.9993	−1.50845	−0.754224	+	0.656618i	$0.771987\pi$
$128$	−5.77626	−0.510554
$129$	0	0
$130$	−0.593789	−0.0520787
$131$	12.6952	1.10918	0.554591	−	0.832123i	$-0.312875\pi$
$131$	12.6952	1.10918	0.554591	+	0.832123i	$0.312875\pi$
$132$	0	0
$133$	4.29619	0.372527
$134$	−2.60108	−0.224699
$135$	0	0
$136$	−9.86696	−0.846085
$137$	−9.74328	−0.832425	−0.416212	−	0.909267i	$-0.636643\pi$
$137$	−9.74328	−0.832425	−0.416212	+	0.909267i	$0.636643\pi$
$138$	0	0
$139$	−12.4834	−1.05883	−0.529416	−	0.848362i	$-0.677589\pi$
$139$	−12.4834	−1.05883	−0.529416	+	0.848362i	$0.677589\pi$
$140$	1.66820	0.140988
$141$	0	0
$142$	10.5251	0.883245
$143$	9.51601	0.795769
$144$	0	0
$145$	0	0
$146$	−10.3686	−0.858113
$147$	0	0
$148$	−0.442201	−0.0363487
$149$	22.6007	1.85152	0.925760	−	0.378111i	$-0.123426\pi$
$149$	22.6007	1.85152	0.925760	+	0.378111i	$0.123426\pi$
$150$	0	0
$151$	−6.63564	−0.540001	−0.270000	−	0.962860i	$-0.587024\pi$
$151$	−6.63564	−0.540001	−0.270000	+	0.962860i	$0.587024\pi$
$152$	−2.46811	−0.200190
$153$	0	0
$154$	19.2985	1.55512
$155$	0.932477	0.0748983
$156$	0	0
$157$	−13.3793	−1.06778	−0.533890	−	0.845554i	$-0.679270\pi$
$157$	−13.3793	−1.06778	−0.533890	+	0.845554i	$0.679270\pi$
$158$	−6.19863	−0.493136
$159$	0	0
$160$	−1.56462	−0.123694
$161$	36.4924	2.87601
$162$	0	0
$163$	10.1562	0.795497	0.397748	−	0.917495i	$-0.369792\pi$
$163$	10.1562	0.795497	0.397748	+	0.917495i	$0.369792\pi$
$164$	−6.45822	−0.504302
$165$	0	0
$166$	−5.75033	−0.446312
$167$	−24.5296	−1.89816	−0.949078	−	0.315042i	$-0.897982\pi$
$167$	−24.5296	−1.89816	−0.949078	+	0.315042i	$0.897982\pi$
$168$	0	0
$169$	−7.82313	−0.601779
$170$	−0.889490	−0.0682208
$171$	0	0
$172$	−9.17946	−0.699927
$173$	22.0884	1.67935	0.839674	−	0.543091i	$-0.182746\pi$
$173$	22.0884	1.67935	0.839674	+	0.543091i	$0.182746\pi$
$174$	0	0
$175$	−24.7866	−1.87369
$176$	−1.37087	−0.103334
$177$	0	0
$178$	−5.22494	−0.391626
$179$	−0.735055	−0.0549406	−0.0274703	−	0.999623i	$-0.508745\pi$
$179$	−0.735055	−0.0549406	−0.0274703	+	0.999623i	$0.508745\pi$
$180$	0	0
$181$	13.6765	1.01656	0.508281	−	0.861191i	$-0.330281\pi$
$181$	13.6765	1.01656	0.508281	+	0.861191i	$0.330281\pi$
$182$	10.4987	0.778218
$183$	0	0
$184$	−20.9644	−1.54552
$185$	−0.108503	−0.00797733
$186$	0	0
$187$	14.2549	1.04242
$188$	7.59160	0.553675
$189$	0	0
$190$	−0.222496	−0.0161415
$191$	−1.16101	−0.0840075	−0.0420038	−	0.999117i	$-0.513374\pi$
$191$	−1.16101	−0.0840075	−0.0420038	+	0.999117i	$0.513374\pi$
$192$	0	0
$193$	20.6010	1.48289	0.741446	−	0.671013i	$-0.234141\pi$
$193$	20.6010	1.48289	0.741446	+	0.671013i	$0.234141\pi$
$194$	0.460403	0.0330550
$195$	0	0
$196$	−21.3645	−1.52604
$197$	−5.15819	−0.367506	−0.183753	−	0.982972i	$-0.558825\pi$
$197$	−5.15819	−0.367506	−0.183753	+	0.982972i	$0.558825\pi$
$198$	0	0
$199$	−2.80583	−0.198900	−0.0994500	−	0.995043i	$-0.531708\pi$
$199$	−2.80583	−0.198900	−0.0994500	+	0.995043i	$0.531708\pi$
$200$	14.2396	1.00689
$201$	0	0
$202$	4.12776	0.290428
$203$	0	0
$204$	0	0
$205$	−1.58466	−0.110678
$206$	7.43886	0.518290
$207$	0	0
$208$	−0.745779	−0.0517105
$209$	3.56570	0.246645
$210$	0	0
$211$	−4.84875	−0.333802	−0.166901	−	0.985974i	$-0.553376\pi$
$211$	−4.84875	−0.333802	−0.166901	+	0.985974i	$0.553376\pi$
$212$	3.35982	0.230753
$213$	0	0
$214$	−8.71653	−0.595850
$215$	−2.25238	−0.153611
$216$	0	0
$217$	−16.4870	−1.11921
$218$	−6.80478	−0.460878
$219$	0	0
$220$	1.38455	0.0933463
$221$	7.75490	0.521651
$222$	0	0
$223$	10.3956	0.696141	0.348070	−	0.937468i	$-0.386837\pi$
$223$	10.3956	0.696141	0.348070	+	0.937468i	$0.386837\pi$
$224$	27.6639	1.84837
$225$	0	0
$226$	−3.88069	−0.258139
$227$	27.9845	1.85739	0.928697	−	0.370838i	$-0.120930\pi$
$227$	27.9845	1.85739	0.928697	+	0.370838i	$0.120930\pi$
$228$	0	0
$229$	−3.49621	−0.231036	−0.115518	−	0.993305i	$-0.536853\pi$
$229$	−3.49621	−0.231036	−0.115518	+	0.993305i	$0.536853\pi$
$230$	−1.88991	−0.124617
$231$	0	0
$232$	0	0
$233$	3.35269	0.219642	0.109821	−	0.993951i	$-0.464972\pi$
$233$	3.35269	0.219642	0.109821	+	0.993951i	$0.464972\pi$
$234$	0	0
$235$	1.86276	0.121513
$236$	−5.43844	−0.354012
$237$	0	0
$238$	15.7270	1.01943
$239$	18.3933	1.18977	0.594883	−	0.803812i	$-0.297198\pi$
$239$	18.3933	1.18977	0.594883	+	0.803812i	$0.297198\pi$
$240$	0	0
$241$	5.62516	0.362348	0.181174	−	0.983451i	$-0.442010\pi$
$241$	5.62516	0.362348	0.181174	+	0.983451i	$0.442010\pi$
$242$	5.94470	0.382140
$243$	0	0
$244$	8.06019	0.516001
$245$	−5.24224	−0.334914
$246$	0	0
$247$	1.93980	0.123427
$248$	9.47160	0.601447
$249$	0	0
$250$	2.58855	0.163714
$251$	26.8112	1.69231	0.846153	−	0.532939i	$-0.178913\pi$
$251$	26.8112	1.69231	0.846153	+	0.532939i	$0.178913\pi$
$252$	0	0
$253$	30.2875	1.90416
$254$	−15.5659	−0.976693
$255$	0	0
$256$	−16.6540	−1.04088
$257$	3.70597	0.231172	0.115586	−	0.993297i	$-0.463125\pi$
$257$	3.70597	0.231172	0.115586	+	0.993297i	$0.463125\pi$
$258$	0	0
$259$	1.91844	0.119206
$260$	0.753218	0.0467126
$261$	0	0
$262$	11.6247	0.718176
$263$	−17.6520	−1.08847	−0.544235	−	0.838933i	$-0.683180\pi$
$263$	−17.6520	−1.08847	−0.544235	+	0.838933i	$0.683180\pi$
$264$	0	0
$265$	0.824403	0.0506427
$266$	3.93393	0.241205
$267$	0	0
$268$	3.29946	0.201546
$269$	14.3360	0.874082	0.437041	−	0.899442i	$-0.356027\pi$
$269$	14.3360	0.874082	0.437041	+	0.899442i	$0.356027\pi$
$270$	0	0
$271$	30.1902	1.83392	0.916961	−	0.398977i	$-0.130635\pi$
$271$	30.1902	1.83392	0.916961	+	0.398977i	$0.130635\pi$
$272$	−1.11717	−0.0677384
$273$	0	0
$274$	−8.92171	−0.538980
$275$	−20.5721	−1.24054
$276$	0	0
$277$	8.44567	0.507451	0.253726	−	0.967276i	$-0.418344\pi$
$277$	8.44567	0.507451	0.253726	+	0.967276i	$0.418344\pi$
$278$	−11.4308	−0.685575
$279$	0	0
$280$	4.15773	0.248472
$281$	12.6756	0.756163	0.378082	−	0.925772i	$-0.376584\pi$
$281$	12.6756	0.756163	0.378082	+	0.925772i	$0.376584\pi$
$282$	0	0
$283$	−24.6429	−1.46487	−0.732434	−	0.680839i	$-0.761616\pi$
$283$	−24.6429	−1.46487	−0.732434	+	0.680839i	$0.761616\pi$
$284$	−13.3510	−0.792236
$285$	0	0
$286$	8.71361	0.515246
$287$	28.0183	1.65387
$288$	0	0
$289$	−5.38323	−0.316661
$290$	0	0
$291$	0	0
$292$	13.1525	0.769694
$293$	4.60722	0.269157	0.134578	−	0.990903i	$-0.457032\pi$
$293$	4.60722	0.269157	0.134578	+	0.990903i	$0.457032\pi$
$294$	0	0
$295$	−1.33444	−0.0776939
$296$	−1.10212	−0.0640594
$297$	0	0
$298$	20.6950	1.19883
$299$	16.4769	0.952885
$300$	0	0
$301$	39.8241	2.29542
$302$	−6.07611	−0.349641
$303$	0	0
$304$	−0.279447	−0.0160274
$305$	1.97774	0.113245
$306$	0	0
$307$	2.80588	0.160140	0.0800700	−	0.996789i	$-0.474486\pi$
$307$	2.80588	0.160140	0.0800700	+	0.996789i	$0.474486\pi$
$308$	−24.4801	−1.39488
$309$	0	0
$310$	0.853849	0.0484954
$311$	−15.7203	−0.891418	−0.445709	−	0.895178i	$-0.647048\pi$
$311$	−15.7203	−0.891418	−0.445709	+	0.895178i	$0.647048\pi$
$312$	0	0
$313$	18.2248	1.03013	0.515063	−	0.857152i	$-0.327768\pi$
$313$	18.2248	1.03013	0.515063	+	0.857152i	$0.327768\pi$
$314$	−12.2511	−0.691369
$315$	0	0
$316$	7.86293	0.442324
$317$	−1.87997	−0.105589	−0.0527947	−	0.998605i	$-0.516813\pi$
$317$	−1.87997	−0.105589	−0.0527947	+	0.998605i	$0.516813\pi$
$318$	0	0
$319$	0	0
$320$	−1.61952	−0.0905342
$321$	0	0
$322$	33.4153	1.86216
$323$	2.90580	0.161683
$324$	0	0
$325$	−11.1916	−0.620796
$326$	9.29984	0.515070
$327$	0	0
$328$	−16.0961	−0.888761
$329$	−32.9353	−1.81578
$330$	0	0
$331$	6.14086	0.337532	0.168766	−	0.985656i	$-0.446022\pi$
$331$	6.14086	0.337532	0.168766	+	0.985656i	$0.446022\pi$
$332$	7.29426	0.400324
$333$	0	0
$334$	−22.4612	−1.22902
$335$	0.809592	0.0442327
$336$	0	0
$337$	−7.25198	−0.395040	−0.197520	−	0.980299i	$-0.563289\pi$
$337$	−7.25198	−0.395040	−0.197520	+	0.980299i	$0.563289\pi$
$338$	−7.16347	−0.389641
$339$	0	0
$340$	1.12831	0.0611914
$341$	−13.6837	−0.741014
$342$	0	0
$343$	57.4133	3.10003
$344$	−22.8784	−1.23352
$345$	0	0
$346$	20.2259	1.08735
$347$	1.50956	0.0810374	0.0405187	−	0.999179i	$-0.487099\pi$
$347$	1.50956	0.0810374	0.0405187	+	0.999179i	$0.487099\pi$
$348$	0	0
$349$	14.5718	0.780011	0.390006	−	0.920813i	$-0.372473\pi$
$349$	14.5718	0.780011	0.390006	+	0.920813i	$0.372473\pi$
$350$	−22.6965	−1.21318
$351$	0	0
$352$	22.9601	1.22378
$353$	21.8048	1.16055	0.580276	−	0.814420i	$-0.302945\pi$
$353$	21.8048	1.16055	0.580276	+	0.814420i	$0.302945\pi$
$354$	0	0
$355$	−3.27595	−0.173870
$356$	6.62781	0.351273
$357$	0	0
$358$	−0.673074	−0.0355731
$359$	−8.45420	−0.446196	−0.223098	−	0.974796i	$-0.571617\pi$
$359$	−8.45420	−0.446196	−0.223098	+	0.974796i	$0.571617\pi$
$360$	0	0
$361$	−18.2731	−0.961745
$362$	12.5232	0.658207
$363$	0	0
$364$	−13.3176	−0.698031
$365$	3.22725	0.168922
$366$	0	0
$367$	−6.88980	−0.359645	−0.179822	−	0.983699i	$-0.557552\pi$
$367$	−6.88980	−0.359645	−0.179822	+	0.983699i	$0.557552\pi$
$368$	−2.37366	−0.123736
$369$	0	0
$370$	−0.0993543	−0.00516518
$371$	−14.5762	−0.756758
$372$	0	0
$373$	−19.2607	−0.997281	−0.498641	−	0.866809i	$-0.666167\pi$
$373$	−19.2607	−0.997281	−0.498641	+	0.866809i	$0.666167\pi$
$374$	13.0529	0.674949
$375$	0	0
$376$	18.9209	0.975773
$377$	0	0
$378$	0	0
$379$	−34.9742	−1.79650	−0.898251	−	0.439483i	$-0.855162\pi$
$379$	−34.9742	−1.79650	−0.898251	+	0.439483i	$0.855162\pi$
$380$	0.282235	0.0144783
$381$	0	0
$382$	−1.06311	−0.0543934
$383$	20.4809	1.04652	0.523262	−	0.852172i	$-0.324715\pi$
$383$	20.4809	1.04652	0.523262	+	0.852172i	$0.324715\pi$
$384$	0	0
$385$	−6.00671	−0.306130
$386$	18.8639	0.960146
$387$	0	0
$388$	−0.584019	−0.0296491
$389$	7.13745	0.361883	0.180942	−	0.983494i	$-0.442086\pi$
$389$	7.13745	0.361883	0.180942	+	0.983494i	$0.442086\pi$
$390$	0	0
$391$	24.6823	1.24824
$392$	−53.2479	−2.68942
$393$	0	0
$394$	−4.72325	−0.237954
$395$	1.92934	0.0970755
$396$	0	0
$397$	−26.3181	−1.32087	−0.660435	−	0.750884i	$-0.729628\pi$
$397$	−26.3181	−1.32087	−0.660435	+	0.750884i	$0.729628\pi$
$398$	−2.56924	−0.128784
$399$	0	0
$400$	1.61225	0.0806126
$401$	5.05597	0.252483	0.126242	−	0.992000i	$-0.459709\pi$
$401$	5.05597	0.252483	0.126242	+	0.992000i	$0.459709\pi$
$402$	0	0
$403$	−7.44417	−0.370820
$404$	−5.23604	−0.260503
$405$	0	0
$406$	0	0
$407$	1.59224	0.0789246
$408$	0	0
$409$	6.45406	0.319133	0.159566	−	0.987187i	$-0.448990\pi$
$409$	6.45406	0.319133	0.159566	+	0.987187i	$0.448990\pi$
$410$	−1.45104	−0.0716618
$411$	0	0
$412$	−9.43615	−0.464886
$413$	23.5941	1.16099
$414$	0	0
$415$	1.78980	0.0878580
$416$	12.4907	0.612407
$417$	0	0
$418$	3.26503	0.159698
$419$	−1.23163	−0.0601693	−0.0300846	−	0.999547i	$-0.509578\pi$
$419$	−1.23163	−0.0601693	−0.0300846	+	0.999547i	$0.509578\pi$
$420$	0	0
$421$	8.84808	0.431229	0.215614	−	0.976479i	$-0.430825\pi$
$421$	8.84808	0.431229	0.215614	+	0.976479i	$0.430825\pi$
$422$	−4.43990	−0.216131
$423$	0	0
$424$	8.37384	0.406670
$425$	−16.7648	−0.813214
$426$	0	0
$427$	−34.9683	−1.69223
$428$	11.0569	0.534454
$429$	0	0
$430$	−2.06245	−0.0994603
$431$	9.86743	0.475297	0.237649	−	0.971351i	$-0.423623\pi$
$431$	9.86743	0.475297	0.237649	+	0.971351i	$0.423623\pi$
$432$	0	0
$433$	−33.0899	−1.59020	−0.795099	−	0.606480i	$-0.792581\pi$
$433$	−33.0899	−1.59020	−0.795099	+	0.606480i	$0.792581\pi$
$434$	−15.0968	−0.724671
$435$	0	0
$436$	8.63182	0.413389
$437$	6.17399	0.295342
$438$	0	0
$439$	−10.7748	−0.514253	−0.257126	−	0.966378i	$-0.582776\pi$
$439$	−10.7748	−0.514253	−0.257126	+	0.966378i	$0.582776\pi$
$440$	3.45078	0.164510
$441$	0	0
$442$	7.10100	0.337760
$443$	−34.0393	−1.61726	−0.808629	−	0.588319i	$-0.799790\pi$
$443$	−34.0393	−1.61726	−0.808629	+	0.588319i	$0.799790\pi$
$444$	0	0
$445$	1.62627	0.0770928
$446$	9.51903	0.450739
$447$	0	0
$448$	28.6347	1.35286
$449$	21.8826	1.03270	0.516351	−	0.856377i	$-0.327290\pi$
$449$	21.8826	1.03270	0.516351	+	0.856377i	$0.327290\pi$
$450$	0	0
$451$	23.2543	1.09500
$452$	4.92263	0.231541
$453$	0	0
$454$	25.6248	1.20263
$455$	−3.26775	−0.153195
$456$	0	0
$457$	29.0963	1.36107	0.680535	−	0.732716i	$-0.261748\pi$
$457$	29.0963	1.36107	0.680535	+	0.732716i	$0.261748\pi$
$458$	−3.20140	−0.149592
$459$	0	0
$460$	2.39734	0.111777
$461$	−18.7521	−0.873373	−0.436686	−	0.899614i	$-0.643848\pi$
$461$	−18.7521	−0.873373	−0.436686	+	0.899614i	$0.643848\pi$
$462$	0	0
$463$	22.8658	1.06266	0.531332	−	0.847164i	$-0.321692\pi$
$463$	22.8658	1.06266	0.531332	+	0.847164i	$0.321692\pi$
$464$	0	0
$465$	0	0
$466$	3.06998	0.142214
$467$	−19.6329	−0.908500	−0.454250	−	0.890874i	$-0.650093\pi$
$467$	−19.6329	−0.908500	−0.454250	+	0.890874i	$0.650093\pi$
$468$	0	0
$469$	−14.3143	−0.660974
$470$	1.70569	0.0786777
$471$	0	0
$472$	−13.5545	−0.623896
$473$	33.0527	1.51976
$474$	0	0
$475$	−4.19353	−0.192413
$476$	−19.9496	−0.914389
$477$	0	0
$478$	16.8424	0.770353
$479$	−22.5224	−1.02908	−0.514538	−	0.857468i	$-0.672037\pi$
$479$	−22.5224	−1.02908	−0.514538	+	0.857468i	$0.672037\pi$
$480$	0	0
$481$	0.866207	0.0394957
$482$	5.15084	0.234614
$483$	0	0
$484$	−7.54081	−0.342764
$485$	−0.143301	−0.00650698
$486$	0	0
$487$	1.72031	0.0779548	0.0389774	−	0.999240i	$-0.487590\pi$
$487$	1.72031	0.0779548	0.0389774	+	0.999240i	$0.487590\pi$
$488$	20.0888	0.909378
$489$	0	0
$490$	−4.80021	−0.216851
$491$	−29.4366	−1.32845	−0.664227	−	0.747531i	$-0.731239\pi$
$491$	−29.4366	−1.32845	−0.664227	+	0.747531i	$0.731239\pi$
$492$	0	0
$493$	0	0
$494$	1.77623	0.0799165
$495$	0	0
$496$	1.07241	0.0481524
$497$	57.9219	2.59815
$498$	0	0
$499$	−13.8879	−0.621709	−0.310855	−	0.950458i	$-0.600615\pi$
$499$	−13.8879	−0.621709	−0.310855	+	0.950458i	$0.600615\pi$
$500$	−3.28356	−0.146845
$501$	0	0
$502$	24.5504	1.09574
$503$	26.5111	1.18207	0.591035	−	0.806646i	$-0.298719\pi$
$503$	26.5111	1.18207	0.591035	+	0.806646i	$0.298719\pi$
$504$	0	0
$505$	−1.28477	−0.0571717
$506$	27.7336	1.23291
$507$	0	0
$508$	19.7453	0.876055
$509$	5.77185	0.255833	0.127916	−	0.991785i	$-0.459171\pi$
$509$	5.77185	0.255833	0.127916	+	0.991785i	$0.459171\pi$
$510$	0	0
$511$	−57.0608	−2.52422
$512$	−3.69719	−0.163394
$513$	0	0
$514$	3.39348	0.149680
$515$	−2.31536	−0.102027
$516$	0	0
$517$	−27.3353	−1.20220
$518$	1.75668	0.0771839
$519$	0	0
$520$	1.87728	0.0823243
$521$	−23.7796	−1.04180	−0.520901	−	0.853617i	$-0.674404\pi$
$521$	−23.7796	−1.04180	−0.520901	+	0.853617i	$0.674404\pi$
$522$	0	0
$523$	24.9044	1.08899	0.544497	−	0.838763i	$-0.316720\pi$
$523$	24.9044	1.08899	0.544497	+	0.838763i	$0.316720\pi$
$524$	−14.7459	−0.644176
$525$	0	0
$526$	−16.1636	−0.704765
$527$	−11.1513	−0.485758
$528$	0	0
$529$	29.4427	1.28012
$530$	0.754888	0.0327902
$531$	0	0
$532$	−4.99017	−0.216351
$533$	12.6507	0.547963
$534$	0	0
$535$	2.71304	0.117295
$536$	8.22340	0.355197
$537$	0	0
$538$	13.1272	0.565953
$539$	76.9277	3.31351
$540$	0	0
$541$	−20.8777	−0.897604	−0.448802	−	0.893631i	$-0.648149\pi$
$541$	−20.8777	−0.897604	−0.448802	+	0.893631i	$0.648149\pi$
$542$	27.6445	1.18743
$543$	0	0
$544$	18.7110	0.802226
$545$	2.11800	0.0907253
$546$	0	0
$547$	0.189502	0.00810253	0.00405126	−	0.999992i	$-0.498710\pi$
$547$	0.189502	0.00810253	0.00405126	+	0.999992i	$0.498710\pi$
$548$	11.3171	0.483444
$549$	0	0
$550$	−18.8374	−0.803229
$551$	0	0
$552$	0	0
$553$	−34.1124	−1.45061
$554$	7.73352	0.328566
$555$	0	0
$556$	14.4999	0.614934
$557$	24.6305	1.04363	0.521814	−	0.853059i	$-0.325255\pi$
$557$	24.6305	1.04363	0.521814	+	0.853059i	$0.325255\pi$
$558$	0	0
$559$	17.9812	0.760525
$560$	0.470752	0.0198929
$561$	0	0
$562$	11.6068	0.489602
$563$	24.8717	1.04822	0.524109	−	0.851651i	$-0.324398\pi$
$563$	24.8717	1.04822	0.524109	+	0.851651i	$0.324398\pi$
$564$	0	0
$565$	1.20787	0.0508156
$566$	−22.5650	−0.948476
$567$	0	0
$568$	−33.2754	−1.39620
$569$	14.2830	0.598775	0.299387	−	0.954132i	$-0.403218\pi$
$569$	14.2830	0.598775	0.299387	+	0.954132i	$0.403218\pi$
$570$	0	0
$571$	10.5361	0.440921	0.220460	−	0.975396i	$-0.429244\pi$
$571$	10.5361	0.440921	0.220460	+	0.975396i	$0.429244\pi$
$572$	−11.0532	−0.462156
$573$	0	0
$574$	25.6557	1.07085
$575$	−35.6204	−1.48547
$576$	0	0
$577$	8.44369	0.351515	0.175758	−	0.984433i	$-0.443762\pi$
$577$	8.44369	0.351515	0.175758	+	0.984433i	$0.443762\pi$
$578$	−4.92931	−0.205032
$579$	0	0
$580$	0	0
$581$	−31.6454	−1.31287
$582$	0	0
$583$	−12.0978	−0.501038
$584$	32.7807	1.35648
$585$	0	0
$586$	4.21874	0.174274
$587$	−9.35632	−0.386177	−0.193088	−	0.981181i	$-0.561850\pi$
$587$	−9.35632	−0.386177	−0.193088	+	0.981181i	$0.561850\pi$
$588$	0	0
$589$	−2.78937	−0.114934
$590$	−1.22192	−0.0503055
$591$	0	0
$592$	−0.124786	−0.00512866
$593$	5.06904	0.208161	0.104080	−	0.994569i	$-0.466810\pi$
$593$	5.06904	0.208161	0.104080	+	0.994569i	$0.466810\pi$
$594$	0	0
$595$	−4.89506	−0.200678
$596$	−26.2514	−1.07530
$597$	0	0
$598$	15.0876	0.616976
$599$	−23.2449	−0.949763	−0.474881	−	0.880050i	$-0.657509\pi$
$599$	−23.2449	−0.949763	−0.474881	+	0.880050i	$0.657509\pi$
$600$	0	0
$601$	−6.06774	−0.247508	−0.123754	−	0.992313i	$-0.539493\pi$
$601$	−6.06774	−0.247508	−0.123754	+	0.992313i	$0.539493\pi$
$602$	36.4661	1.48625
$603$	0	0
$604$	7.70751	0.313614
$605$	−1.85030	−0.0752254
$606$	0	0
$607$	−39.3130	−1.59566	−0.797832	−	0.602880i	$-0.794020\pi$
$607$	−39.3130	−1.59566	−0.797832	+	0.602880i	$0.794020\pi$
$608$	4.68033	0.189813
$609$	0	0
$610$	1.81097	0.0733242
$611$	−14.8708	−0.601610
$612$	0	0
$613$	24.6911	0.997266	0.498633	−	0.866813i	$-0.333836\pi$
$613$	24.6911	0.997266	0.498633	+	0.866813i	$0.333836\pi$
$614$	2.56928	0.103688
$615$	0	0
$616$	−61.0130	−2.45828
$617$	−18.9571	−0.763183	−0.381592	−	0.924331i	$-0.624624\pi$
$617$	−18.9571	−0.763183	−0.381592	+	0.924331i	$0.624624\pi$
$618$	0	0
$619$	−28.6122	−1.15002	−0.575010	−	0.818147i	$-0.695002\pi$
$619$	−28.6122	−1.15002	−0.575010	+	0.818147i	$0.695002\pi$
$620$	−1.08310	−0.0434984
$621$	0	0
$622$	−14.3948	−0.577178
$623$	−28.7540	−1.15201
$624$	0	0
$625$	23.7882	0.951527
$626$	16.6881	0.666989
$627$	0	0
$628$	15.5404	0.620131
$629$	1.29757	0.0517375
$630$	0	0
$631$	21.0131	0.836518	0.418259	−	0.908328i	$-0.362640\pi$
$631$	21.0131	0.836518	0.418259	+	0.908328i	$0.362640\pi$
$632$	19.5972	0.779534
$633$	0	0
$634$	−1.72144	−0.0683673
$635$	4.84493	0.192265
$636$	0	0
$637$	41.8500	1.65816
$638$	0	0
$639$	0	0
$640$	1.64627	0.0650747
$641$	−6.06906	−0.239713	−0.119857	−	0.992791i	$-0.538244\pi$
$641$	−6.06906	−0.239713	−0.119857	+	0.992791i	$0.538244\pi$
$642$	0	0
$643$	−45.2441	−1.78425	−0.892126	−	0.451787i	$-0.850787\pi$
$643$	−45.2441	−1.78425	−0.892126	+	0.451787i	$0.850787\pi$
$644$	−42.3872	−1.67029
$645$	0	0
$646$	2.66078	0.104687
$647$	11.5929	0.455762	0.227881	−	0.973689i	$-0.426820\pi$
$647$	11.5929	0.455762	0.227881	+	0.973689i	$0.426820\pi$
$648$	0	0
$649$	19.5823	0.768673
$650$	−10.2479	−0.401954
$651$	0	0
$652$	−11.7968	−0.461998
$653$	−12.6420	−0.494720	−0.247360	−	0.968924i	$-0.579563\pi$
$653$	−12.6420	−0.494720	−0.247360	+	0.968924i	$0.579563\pi$
$654$	0	0
$655$	−3.61821	−0.141375
$656$	−1.82246	−0.0711551
$657$	0	0
$658$	−30.1582	−1.17569
$659$	−46.2995	−1.80357	−0.901786	−	0.432183i	$-0.857744\pi$
$659$	−46.2995	−1.80357	−0.901786	+	0.432183i	$0.857744\pi$
$660$	0	0
$661$	−3.17597	−0.123531	−0.0617655	−	0.998091i	$-0.519673\pi$
$661$	−3.17597	−0.123531	−0.0617655	+	0.998091i	$0.519673\pi$
$662$	5.62306	0.218546
$663$	0	0
$664$	18.1799	0.705515
$665$	−1.22444	−0.0474819
$666$	0	0
$667$	0	0
$668$	28.4919	1.10239
$669$	0	0
$670$	0.741326	0.0286399
$671$	−29.0225	−1.12040
$672$	0	0
$673$	11.3207	0.436383	0.218191	−	0.975906i	$-0.429984\pi$
$673$	11.3207	0.436383	0.218191	+	0.975906i	$0.429984\pi$
$674$	−6.64048	−0.255782
$675$	0	0
$676$	9.08682	0.349493
$677$	−4.44930	−0.171000	−0.0855002	−	0.996338i	$-0.527249\pi$
$677$	−4.44930	−0.171000	−0.0855002	+	0.996338i	$0.527249\pi$
$678$	0	0
$679$	2.53370	0.0972345
$680$	2.81215	0.107841
$681$	0	0
$682$	−12.5299	−0.479794
$683$	−25.0062	−0.956835	−0.478418	−	0.878132i	$-0.658789\pi$
$683$	−25.0062	−0.956835	−0.478418	+	0.878132i	$0.658789\pi$
$684$	0	0
$685$	2.77690	0.106100
$686$	52.5721	2.00721
$687$	0	0
$688$	−2.59037	−0.0987570
$689$	−6.58139	−0.250731
$690$	0	0
$691$	−9.03178	−0.343585	−0.171793	−	0.985133i	$-0.554956\pi$
$691$	−9.03178	−0.343585	−0.171793	+	0.985133i	$0.554956\pi$
$692$	−25.6564	−0.975309
$693$	0	0
$694$	1.38227	0.0524703
$695$	3.55787	0.134958
$696$	0	0
$697$	18.9506	0.717807
$698$	13.3431	0.505044
$699$	0	0
$700$	28.7904	1.08818
$701$	26.8957	1.01584	0.507919	−	0.861405i	$-0.330415\pi$
$701$	26.8957	1.01584	0.507919	+	0.861405i	$0.330415\pi$
$702$	0	0
$703$	0.324572	0.0122415
$704$	23.7659	0.895709
$705$	0	0
$706$	19.9662	0.751437
$707$	22.7160	0.854322
$708$	0	0
$709$	−35.3956	−1.32931	−0.664655	−	0.747151i	$-0.731421\pi$
$709$	−35.3956	−1.32931	−0.664655	+	0.747151i	$0.731421\pi$
$710$	−2.99972	−0.112577
$711$	0	0
$712$	16.5188	0.619069
$713$	−23.6933	−0.887320
$714$	0	0
$715$	−2.71213	−0.101428
$716$	0.853791	0.0319077
$717$	0	0
$718$	−7.74133	−0.288904
$719$	−10.5778	−0.394486	−0.197243	−	0.980355i	$-0.563199\pi$
$719$	−10.5778	−0.394486	−0.197243	+	0.980355i	$0.563199\pi$
$720$	0	0
$721$	40.9377	1.52460
$722$	−16.7323	−0.622713
$723$	0	0
$724$	−15.8857	−0.590386
$725$	0	0
$726$	0	0
$727$	45.8396	1.70010	0.850048	−	0.526705i	$-0.176573\pi$
$727$	45.8396	1.70010	0.850048	+	0.526705i	$0.176573\pi$
$728$	−33.1921	−1.23018
$729$	0	0
$730$	2.95513	0.109374
$731$	26.9357	0.996253
$732$	0	0
$733$	−51.9120	−1.91741	−0.958707	−	0.284394i	$-0.908208\pi$
$733$	−51.9120	−1.91741	−0.958707	+	0.284394i	$0.908208\pi$
$734$	−6.30884	−0.232864
$735$	0	0
$736$	39.7554	1.46540
$737$	−11.8804	−0.437621
$738$	0	0
$739$	−16.2033	−0.596046	−0.298023	−	0.954559i	$-0.596327\pi$
$739$	−16.2033	−0.596046	−0.298023	+	0.954559i	$0.596327\pi$
$740$	0.126030	0.00463297
$741$	0	0
$742$	−13.3471	−0.489988
$743$	−36.1219	−1.32518	−0.662591	−	0.748981i	$-0.730543\pi$
$743$	−36.1219	−1.32518	−0.662591	+	0.748981i	$0.730543\pi$
$744$	0	0
$745$	−6.44135	−0.235993
$746$	−17.6366	−0.645722
$747$	0	0
$748$	−16.5575	−0.605403
$749$	−47.9690	−1.75275
$750$	0	0
$751$	41.6701	1.52056	0.760282	−	0.649593i	$-0.225061\pi$
$751$	41.6701	1.52056	0.760282	+	0.649593i	$0.225061\pi$
$752$	2.14229	0.0781213
$753$	0	0
$754$	0	0
$755$	1.89120	0.0688279
$756$	0	0
$757$	7.81689	0.284110	0.142055	−	0.989859i	$-0.454629\pi$
$757$	7.81689	0.284110	0.142055	+	0.989859i	$0.454629\pi$
$758$	−32.0251	−1.16320
$759$	0	0
$760$	0.703428	0.0255160
$761$	21.6344	0.784245	0.392123	−	0.919913i	$-0.371741\pi$
$761$	21.6344	0.784245	0.392123	+	0.919913i	$0.371741\pi$
$762$	0	0
$763$	−37.4482	−1.35572
$764$	1.34855	0.0487887
$765$	0	0
$766$	18.7539	0.677606
$767$	10.6531	0.384661
$768$	0	0
$769$	4.41685	0.159276	0.0796379	−	0.996824i	$-0.474624\pi$
$769$	4.41685	0.159276	0.0796379	+	0.996824i	$0.474624\pi$
$770$	−5.50022	−0.198214
$771$	0	0
$772$	−23.9287	−0.861213
$773$	−3.01602	−0.108479	−0.0542394	−	0.998528i	$-0.517273\pi$
$773$	−3.01602	−0.108479	−0.0542394	+	0.998528i	$0.517273\pi$
$774$	0	0
$775$	16.0931	0.578081
$776$	−1.45558	−0.0522523
$777$	0	0
$778$	6.53561	0.234313
$779$	4.74029	0.169838
$780$	0	0
$781$	48.0733	1.72020
$782$	22.6010	0.808211
$783$	0	0
$784$	−6.02890	−0.215318
$785$	3.81318	0.136098
$786$	0	0
$787$	−27.7911	−0.990646	−0.495323	−	0.868709i	$-0.664950\pi$
$787$	−27.7911	−0.990646	−0.495323	+	0.868709i	$0.664950\pi$
$788$	5.99141	0.213435
$789$	0	0
$790$	1.76665	0.0628547
$791$	−21.3563	−0.759342
$792$	0	0
$793$	−15.7887	−0.560675
$794$	−24.0989	−0.855240
$795$	0	0
$796$	3.25906	0.115514
$797$	−28.1611	−0.997519	−0.498759	−	0.866741i	$-0.666211\pi$
$797$	−28.1611	−0.997519	−0.498759	+	0.866741i	$0.666211\pi$
$798$	0	0
$799$	−22.2764	−0.788082
$800$	−27.0029	−0.954695
$801$	0	0
$802$	4.62965	0.163479
$803$	−47.3586	−1.67125
$804$	0	0
$805$	−10.4006	−0.366573
$806$	−6.81647	−0.240100
$807$	0	0
$808$	−13.0500	−0.459099
$809$	24.8636	0.874159	0.437080	−	0.899423i	$-0.356013\pi$
$809$	24.8636	0.874159	0.437080	+	0.899423i	$0.356013\pi$
$810$	0	0
$811$	−17.5911	−0.617705	−0.308853	−	0.951110i	$-0.599945\pi$
$811$	−17.5911	−0.617705	−0.308853	+	0.951110i	$0.599945\pi$
$812$	0	0
$813$	0	0
$814$	1.45798	0.0511023
$815$	−2.89459	−0.101393
$816$	0	0
$817$	6.73766	0.235721
$818$	5.90984	0.206633
$819$	0	0
$820$	1.84064	0.0642778
$821$	14.3123	0.499503	0.249752	−	0.968310i	$-0.419651\pi$
$821$	14.3123	0.499503	0.249752	+	0.968310i	$0.419651\pi$
$822$	0	0
$823$	15.1918	0.529551	0.264776	−	0.964310i	$-0.414702\pi$
$823$	15.1918	0.529551	0.264776	+	0.964310i	$0.414702\pi$
$824$	−23.5182	−0.819295
$825$	0	0
$826$	21.6046	0.751720
$827$	35.3653	1.22977	0.614887	−	0.788615i	$-0.289202\pi$
$827$	35.3653	1.22977	0.614887	+	0.788615i	$0.289202\pi$
$828$	0	0
$829$	19.3932	0.673555	0.336777	−	0.941584i	$-0.390663\pi$
$829$	19.3932	0.673555	0.336777	+	0.941584i	$0.390663\pi$
$830$	1.63888	0.0568865
$831$	0	0
$832$	12.9290	0.448233
$833$	62.6909	2.17211
$834$	0	0
$835$	6.99110	0.241937
$836$	−4.14168	−0.143243
$837$	0	0
$838$	−1.12778	−0.0389585
$839$	35.7024	1.23258	0.616292	−	0.787518i	$-0.288634\pi$
$839$	35.7024	1.23258	0.616292	+	0.787518i	$0.288634\pi$
$840$	0	0
$841$	0	0
$842$	8.10199	0.279213
$843$	0	0
$844$	5.63199	0.193861
$845$	2.22965	0.0767022
$846$	0	0
$847$	32.7150	1.12410
$848$	0.948114	0.0325584
$849$	0	0
$850$	−15.3512	−0.526542
$851$	2.75696	0.0945074
$852$	0	0
$853$	−2.67243	−0.0915024	−0.0457512	−	0.998953i	$-0.514568\pi$
$853$	−2.67243	−0.0915024	−0.0457512	+	0.998953i	$0.514568\pi$
$854$	−32.0197	−1.09569
$855$	0	0
$856$	27.5576	0.941899
$857$	32.1387	1.09784	0.548918	−	0.835876i	$-0.315040\pi$
$857$	32.1387	1.09784	0.548918	+	0.835876i	$0.315040\pi$
$858$	0	0
$859$	−11.7709	−0.401617	−0.200808	−	0.979631i	$-0.564357\pi$
$859$	−11.7709	−0.401617	−0.200808	+	0.979631i	$0.564357\pi$
$860$	2.61621	0.0892120
$861$	0	0
$862$	9.03539	0.307747
$863$	−31.9118	−1.08629	−0.543146	−	0.839638i	$-0.682767\pi$
$863$	−31.9118	−1.08629	−0.543146	+	0.839638i	$0.682767\pi$
$864$	0	0
$865$	−6.29534	−0.214048
$866$	−30.2997	−1.02963
$867$	0	0
$868$	19.1502	0.650002
$869$	−28.3122	−0.960426
$870$	0	0
$871$	−6.46315	−0.218996
$872$	21.5135	0.728540
$873$	0	0
$874$	5.65339	0.191229
$875$	14.2454	0.481581
$876$	0	0
$877$	8.86548	0.299366	0.149683	−	0.988734i	$-0.452175\pi$
$877$	8.86548	0.299366	0.149683	+	0.988734i	$0.452175\pi$
$878$	−9.86624	−0.332969
$879$	0	0
$880$	0.390709	0.0131708
$881$	−4.16892	−0.140454	−0.0702272	−	0.997531i	$-0.522372\pi$
$881$	−4.16892	−0.140454	−0.0702272	+	0.997531i	$0.522372\pi$
$882$	0	0
$883$	23.5054	0.791018	0.395509	−	0.918462i	$-0.370568\pi$
$883$	23.5054	0.791018	0.395509	+	0.918462i	$0.370568\pi$
$884$	−9.00757	−0.302958
$885$	0	0
$886$	−31.1691	−1.04715
$887$	−20.2426	−0.679680	−0.339840	−	0.940483i	$-0.610373\pi$
$887$	−20.2426	−0.679680	−0.339840	+	0.940483i	$0.610373\pi$
$888$	0	0
$889$	−85.6628	−2.87304
$890$	1.48914	0.0499162
$891$	0	0
$892$	−12.0748	−0.404295
$893$	−5.57218	−0.186466
$894$	0	0
$895$	0.209496	0.00700267
$896$	−29.1076	−0.972418
$897$	0	0
$898$	20.0374	0.668657
$899$	0	0
$900$	0	0
$901$	−9.85886	−0.328446
$902$	21.2934	0.708993
$903$	0	0
$904$	12.2689	0.408058
$905$	−3.89789	−0.129570
$906$	0	0
$907$	−2.27974	−0.0756974	−0.0378487	−	0.999283i	$-0.512050\pi$
$907$	−2.27974	−0.0756974	−0.0378487	+	0.999283i	$0.512050\pi$
$908$	−32.5049	−1.07871
$909$	0	0
$910$	−2.99221	−0.0991909
$911$	18.8815	0.625572	0.312786	−	0.949824i	$-0.398738\pi$
$911$	18.8815	0.625572	0.312786	+	0.949824i	$0.398738\pi$
$912$	0	0
$913$	−26.2646	−0.869232
$914$	26.6429	0.881268
$915$	0	0
$916$	4.06096	0.134178
$917$	63.9733	2.11258
$918$	0	0
$919$	−16.3791	−0.540296	−0.270148	−	0.962819i	$-0.587073\pi$
$919$	−16.3791	−0.540296	−0.270148	+	0.962819i	$0.587073\pi$
$920$	5.97501	0.196990
$921$	0	0
$922$	−17.1709	−0.565493
$923$	26.1527	0.860825
$924$	0	0
$925$	−1.87260	−0.0615707
$926$	20.9377	0.688056
$927$	0	0
$928$	0	0
$929$	−31.0737	−1.01950	−0.509748	−	0.860324i	$-0.670261\pi$
$929$	−31.0737	−1.01950	−0.509748	+	0.860324i	$0.670261\pi$
$930$	0	0
$931$	15.6814	0.513937
$932$	−3.89426	−0.127561
$933$	0	0
$934$	−17.9774	−0.588238
$935$	−4.06274	−0.132866
$936$	0	0
$937$	26.1466	0.854171	0.427085	−	0.904211i	$-0.359540\pi$
$937$	26.1466	0.854171	0.427085	+	0.904211i	$0.359540\pi$
$938$	−13.1073	−0.427969
$939$	0	0
$940$	−2.16366	−0.0705708
$941$	−40.0522	−1.30566	−0.652832	−	0.757503i	$-0.726419\pi$
$941$	−40.0522	−1.30566	−0.652832	+	0.757503i	$0.726419\pi$
$942$	0	0
$943$	40.2646	1.31120
$944$	−1.53468	−0.0499497
$945$	0	0
$946$	30.2656	0.984021
$947$	−41.8585	−1.36022	−0.680109	−	0.733111i	$-0.738068\pi$
$947$	−41.8585	−1.36022	−0.680109	+	0.733111i	$0.738068\pi$
$948$	0	0
$949$	−25.7639	−0.836331
$950$	−3.83993	−0.124584
$951$	0	0
$952$	−49.7214	−1.61148
$953$	30.7325	0.995522	0.497761	−	0.867314i	$-0.334156\pi$
$953$	30.7325	0.995522	0.497761	+	0.867314i	$0.334156\pi$
$954$	0	0
$955$	0.330895	0.0107075
$956$	−21.3645	−0.690976
$957$	0	0
$958$	−20.6233	−0.666309
$959$	−49.0982	−1.58546
$960$	0	0
$961$	−20.2955	−0.654695
$962$	0.793167	0.0255727
$963$	0	0
$964$	−6.53381	−0.210440
$965$	−5.87142	−0.189008
$966$	0	0
$967$	−11.4758	−0.369037	−0.184519	−	0.982829i	$-0.559073\pi$
$967$	−11.4758	−0.369037	−0.184519	+	0.982829i	$0.559073\pi$
$968$	−18.7944	−0.604074
$969$	0	0
$970$	−0.131218	−0.00421316
$971$	12.8983	0.413927	0.206963	−	0.978349i	$-0.433642\pi$
$971$	12.8983	0.413927	0.206963	+	0.978349i	$0.433642\pi$
$972$	0	0
$973$	−62.9064	−2.01669
$974$	1.57525	0.0504744
$975$	0	0
$976$	2.27452	0.0728057
$977$	62.0399	1.98483	0.992415	−	0.122929i	$-0.0392288\pi$
$977$	62.0399	1.98483	0.992415	+	0.122929i	$0.0392288\pi$
$978$	0	0
$979$	−23.8649	−0.762725
$980$	6.08903	0.194507
$981$	0	0
$982$	−26.9544	−0.860150
$983$	−8.89801	−0.283803	−0.141901	−	0.989881i	$-0.545322\pi$
$983$	−8.89801	−0.283803	−0.141901	+	0.989881i	$0.545322\pi$
$984$	0	0
$985$	1.47012	0.0468420
$986$	0	0
$987$	0	0
$988$	−2.25314	−0.0716820
$989$	57.2306	1.81983
$990$	0	0
$991$	2.83908	0.0901863	0.0450931	−	0.998983i	$-0.485642\pi$
$991$	2.83908	0.0901863	0.0450931	+	0.998983i	$0.485642\pi$
$992$	−17.9612	−0.570269
$993$	0	0
$994$	53.0378	1.68226
$995$	0.799681	0.0253516
$996$	0	0
$997$	−57.0290	−1.80613	−0.903063	−	0.429508i	$-0.858687\pi$
$997$	−57.0290	−1.80613	−0.903063	+	0.429508i	$0.858687\pi$
$998$	−12.7169	−0.402546
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.6		9
3.2	odd	2		2523.2.a.p.1.4		9
29.23	even	7		261.2.k.b.181.2		18
29.24	even	7		261.2.k.b.199.2		18
29.28	even	2		7569.2.a.bk.1.4		9
87.23	odd	14		87.2.g.b.7.2	✓	18
87.53	odd	14		87.2.g.b.25.2	yes	18
87.86	odd	2		2523.2.a.q.1.6		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.7.2	✓	18	87.23	odd	14
87.2.g.b.25.2	yes	18	87.53	odd	14
261.2.k.b.181.2		18	29.23	even	7
261.2.k.b.199.2		18	29.24	even	7
2523.2.a.p.1.4		9	3.2	odd	2
2523.2.a.q.1.6		9	87.86	odd	2
7569.2.a.bk.1.4		9	29.28	even	2
7569.2.a.bl.1.6		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)

\(f(q)\)	\(=\)	\(q+0.915678 q^{2} -1.16153 q^{4} -0.285007 q^{5} +5.03918 q^{7} -2.89495 q^{8} +O(q^{10})\)	\(q+0.915678 q^{2} -1.16153 q^{4} -0.285007 q^{5} +5.03918 q^{7} -2.89495 q^{8} -0.260975 q^{10} +4.18236 q^{11} +2.27527 q^{13} +4.61427 q^{14} -0.327776 q^{16} +3.40834 q^{17} +0.852557 q^{19} +0.331045 q^{20} +3.82970 q^{22} +7.24173 q^{23} -4.91877 q^{25} +2.08342 q^{26} -5.85318 q^{28} -3.27177 q^{31} +5.48976 q^{32} +3.12094 q^{34} -1.43620 q^{35} +0.380705 q^{37} +0.780668 q^{38} +0.825080 q^{40} +5.56008 q^{41} +7.90289 q^{43} -4.85795 q^{44} +6.63110 q^{46} -6.53585 q^{47} +18.3934 q^{49} -4.50401 q^{50} -2.64281 q^{52} -2.89257 q^{53} -1.19200 q^{55} -14.5882 q^{56} +4.68212 q^{59} -6.93927 q^{61} -2.99589 q^{62} +5.68240 q^{64} -0.648469 q^{65} -2.84060 q^{67} -3.95890 q^{68} -1.31510 q^{70} +11.4943 q^{71} -11.3234 q^{73} +0.348603 q^{74} -0.990273 q^{76} +21.0757 q^{77} -6.76944 q^{79} +0.0934183 q^{80} +5.09125 q^{82} -6.27986 q^{83} -0.971400 q^{85} +7.23650 q^{86} -12.1077 q^{88} -5.70609 q^{89} +11.4655 q^{91} -8.41151 q^{92} -5.98474 q^{94} -0.242985 q^{95} +0.502800 q^{97} +16.8424 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 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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