LMFDB - Embedded newform 7569.2.a.bl.1.2

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			1.2
Root			$-1.83470$ 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)

$f(q)$	$=$	$q-1.83470 q^{2} +1.36614 q^{4} +2.25851 q^{5} -0.744275 q^{7} +1.16295 q^{8} -4.14369 q^{10} -6.30880 q^{11} -1.64589 q^{13} +1.36552 q^{14} -4.86594 q^{16} -4.16595 q^{17} +1.04086 q^{19} +3.08543 q^{20} +11.5748 q^{22} +4.57353 q^{23} +0.100848 q^{25} +3.01971 q^{26} -1.01678 q^{28} -3.53151 q^{31} +6.60167 q^{32} +7.64329 q^{34} -1.68095 q^{35} -1.07555 q^{37} -1.90967 q^{38} +2.62652 q^{40} +1.20065 q^{41} -12.8267 q^{43} -8.61870 q^{44} -8.39108 q^{46} +7.88593 q^{47} -6.44605 q^{49} -0.185026 q^{50} -2.24851 q^{52} +13.1918 q^{53} -14.2485 q^{55} -0.865553 q^{56} -11.0585 q^{59} -6.44864 q^{61} +6.47928 q^{62} -2.38023 q^{64} -3.71724 q^{65} -6.87762 q^{67} -5.69127 q^{68} +3.08405 q^{70} +3.27762 q^{71} +7.09043 q^{73} +1.97331 q^{74} +1.42196 q^{76} +4.69549 q^{77} +6.38938 q^{79} -10.9898 q^{80} -2.20283 q^{82} +5.15394 q^{83} -9.40883 q^{85} +23.5333 q^{86} -7.33681 q^{88} -9.31849 q^{89} +1.22499 q^{91} +6.24808 q^{92} -14.4684 q^{94} +2.35079 q^{95} -15.1681 q^{97} +11.8266 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

Coefficient data

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

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

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

$2$	−1.83470	−1.29733	−0.648666	−	0.761073i	$-0.724673\pi$
$2$	−1.83470	−1.29733	−0.648666	+	0.761073i	$0.724673\pi$
$3$	0	0
$3$	0	0
$4$	1.36614	0.683070
$5$	2.25851	1.01003	0.505017	−	0.863109i	$-0.331486\pi$
$5$	2.25851	1.01003	0.505017	+	0.863109i	$0.331486\pi$
$6$	0	0
$7$	−0.744275	−0.281310	−0.140655	−	0.990059i	$-0.544921\pi$
$7$	−0.744275	−0.281310	−0.140655	+	0.990059i	$0.544921\pi$
$8$	1.16295	0.411164
$9$	0	0
$10$	−4.14369	−1.31035
$11$	−6.30880	−1.90218	−0.951088	−	0.308920i	$-0.900032\pi$
$11$	−6.30880	−1.90218	−0.951088	+	0.308920i	$0.900032\pi$
$12$	0	0
$13$	−1.64589	−0.456486	−0.228243	−	0.973604i	$-0.573298\pi$
$13$	−1.64589	−0.456486	−0.228243	+	0.973604i	$0.573298\pi$
$14$	1.36552	0.364952
$15$	0	0
$16$	−4.86594	−1.21649
$17$	−4.16595	−1.01039	−0.505196	−	0.863005i	$-0.668580\pi$
$17$	−4.16595	−1.01039	−0.505196	+	0.863005i	$0.668580\pi$
$18$	0	0
$19$	1.04086	0.238789	0.119395	−	0.992847i	$-0.461905\pi$
$19$	1.04086	0.238789	0.119395	+	0.992847i	$0.461905\pi$
$20$	3.08543	0.689924
$21$	0	0
$22$	11.5748	2.46775
$23$	4.57353	0.953648	0.476824	−	0.878999i	$-0.341788\pi$
$23$	4.57353	0.953648	0.476824	+	0.878999i	$0.341788\pi$
$24$	0	0
$25$	0.100848	0.0201696
$26$	3.01971	0.592214
$27$	0	0
$28$	−1.01678	−0.192154
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−3.53151	−0.634278	−0.317139	−	0.948379i	$-0.602722\pi$
$31$	−3.53151	−0.634278	−0.317139	+	0.948379i	$0.602722\pi$
$32$	6.60167	1.16702
$33$	0	0
$34$	7.64329	1.31081
$35$	−1.68095	−0.284132
$36$	0	0
$37$	−1.07555	−0.176819	−0.0884094	−	0.996084i	$-0.528178\pi$
$37$	−1.07555	−0.176819	−0.0884094	+	0.996084i	$0.528178\pi$
$38$	−1.90967	−0.309789
$39$	0	0
$40$	2.62652	0.415290
$41$	1.20065	0.187510	0.0937548	−	0.995595i	$-0.470113\pi$
$41$	1.20065	0.187510	0.0937548	+	0.995595i	$0.470113\pi$
$42$	0	0
$43$	−12.8267	−1.95606	−0.978029	−	0.208468i	$-0.933152\pi$
$43$	−12.8267	−1.95606	−0.978029	+	0.208468i	$0.933152\pi$
$44$	−8.61870	−1.29932
$45$	0	0
$46$	−8.39108	−1.23720
$47$	7.88593	1.15028	0.575141	−	0.818055i	$-0.304947\pi$
$47$	7.88593	1.15028	0.575141	+	0.818055i	$0.304947\pi$
$48$	0	0
$49$	−6.44605	−0.920865
$50$	−0.185026	−0.0261667
$51$	0	0
$52$	−2.24851	−0.311812
$53$	13.1918	1.81203	0.906016	−	0.423244i	$-0.139109\pi$
$53$	13.1918	1.81203	0.906016	+	0.423244i	$0.139109\pi$
$54$	0	0
$55$	−14.2485	−1.92126
$56$	−0.865553	−0.115664
$57$	0	0
$58$	0	0
$59$	−11.0585	−1.43970	−0.719849	−	0.694131i	$-0.755789\pi$
$59$	−11.0585	−1.43970	−0.719849	+	0.694131i	$0.755789\pi$
$60$	0	0
$61$	−6.44864	−0.825664	−0.412832	−	0.910807i	$-0.635460\pi$
$61$	−6.44864	−0.825664	−0.412832	+	0.910807i	$0.635460\pi$
$62$	6.47928	0.822869
$63$	0	0
$64$	−2.38023	−0.297528
$65$	−3.71724	−0.461067
$66$	0	0
$67$	−6.87762	−0.840235	−0.420118	−	0.907470i	$-0.638011\pi$
$67$	−6.87762	−0.840235	−0.420118	+	0.907470i	$0.638011\pi$
$68$	−5.69127	−0.690168
$69$	0	0
$70$	3.08405	0.368614
$71$	3.27762	0.388982	0.194491	−	0.980904i	$-0.437694\pi$
$71$	3.27762	0.388982	0.194491	+	0.980904i	$0.437694\pi$
$72$	0	0
$73$	7.09043	0.829872	0.414936	−	0.909851i	$-0.363804\pi$
$73$	7.09043	0.829872	0.414936	+	0.909851i	$0.363804\pi$
$74$	1.97331	0.229393
$75$	0	0
$76$	1.42196	0.163110
$77$	4.69549	0.535100
$78$	0	0
$79$	6.38938	0.718861	0.359431	−	0.933172i	$-0.382971\pi$
$79$	6.38938	0.718861	0.359431	+	0.933172i	$0.382971\pi$
$80$	−10.9898	−1.22869
$81$	0	0
$82$	−2.20283	−0.243262
$83$	5.15394	0.565718	0.282859	−	0.959161i	$-0.408717\pi$
$83$	5.15394	0.565718	0.282859	+	0.959161i	$0.408717\pi$
$84$	0	0
$85$	−9.40883	−1.02053
$86$	23.5333	2.53766
$87$	0	0
$88$	−7.33681	−0.782106
$89$	−9.31849	−0.987758	−0.493879	−	0.869531i	$-0.664421\pi$
$89$	−9.31849	−0.987758	−0.493879	+	0.869531i	$0.664421\pi$
$90$	0	0
$91$	1.22499	0.128414
$92$	6.24808	0.651408
$93$	0	0
$94$	−14.4684	−1.49230
$95$	2.35079	0.241186
$96$	0	0
$97$	−15.1681	−1.54009	−0.770043	−	0.637992i	$-0.779765\pi$
$97$	−15.1681	−1.54009	−0.770043	+	0.637992i	$0.779765\pi$
$98$	11.8266	1.19467
$99$	0	0
$100$	0.137772	0.0137772
$101$	−4.44158	−0.441954	−0.220977	−	0.975279i	$-0.570925\pi$
$101$	−4.44158	−0.441954	−0.220977	+	0.975279i	$0.570925\pi$
$102$	0	0
$103$	15.3937	1.51678	0.758392	−	0.651799i	$-0.225986\pi$
$103$	15.3937	1.51678	0.758392	+	0.651799i	$0.225986\pi$
$104$	−1.91408	−0.187691
$105$	0	0
$106$	−24.2030	−2.35081
$107$	7.91673	0.765339	0.382670	−	0.923885i	$-0.375005\pi$
$107$	7.91673	0.765339	0.382670	+	0.923885i	$0.375005\pi$
$108$	0	0
$109$	4.18903	0.401236	0.200618	−	0.979669i	$-0.435705\pi$
$109$	4.18903	0.401236	0.200618	+	0.979669i	$0.435705\pi$
$110$	26.1417	2.49252
$111$	0	0
$112$	3.62160	0.342209
$113$	10.8248	1.01831	0.509155	−	0.860675i	$-0.329958\pi$
$113$	10.8248	1.01831	0.509155	+	0.860675i	$0.329958\pi$
$114$	0	0
$115$	10.3294	0.963217
$116$	0	0
$117$	0	0
$118$	20.2891	1.86776
$119$	3.10062	0.284233
$120$	0	0
$121$	28.8010	2.61827
$122$	11.8314	1.07116
$123$	0	0
$124$	−4.82454	−0.433256
$125$	−11.0648	−0.989662
$126$	0	0
$127$	8.06968	0.716068	0.358034	−	0.933709i	$-0.383447\pi$
$127$	8.06968	0.716068	0.358034	+	0.933709i	$0.383447\pi$
$128$	−8.83633	−0.781029
$129$	0	0
$130$	6.82004	0.598157
$131$	18.7048	1.63424	0.817122	−	0.576465i	$-0.195568\pi$
$131$	18.7048	1.63424	0.817122	+	0.576465i	$0.195568\pi$
$132$	0	0
$133$	−0.774686	−0.0671738
$134$	12.6184	1.09006
$135$	0	0
$136$	−4.84478	−0.415437
$137$	3.73212	0.318856	0.159428	−	0.987210i	$-0.449035\pi$
$137$	3.73212	0.318856	0.159428	+	0.987210i	$0.449035\pi$
$138$	0	0
$139$	13.5448	1.14886	0.574430	−	0.818554i	$-0.305224\pi$
$139$	13.5448	1.14886	0.574430	+	0.818554i	$0.305224\pi$
$140$	−2.29641	−0.194082
$141$	0	0
$142$	−6.01347	−0.504639
$143$	10.3836	0.868317
$144$	0	0
$145$	0	0
$146$	−13.0088	−1.07662
$147$	0	0
$148$	−1.46935	−0.120780
$149$	2.05954	0.168724	0.0843621	−	0.996435i	$-0.473115\pi$
$149$	2.05954	0.168724	0.0843621	+	0.996435i	$0.473115\pi$
$150$	0	0
$151$	−12.4698	−1.01478	−0.507390	−	0.861717i	$-0.669389\pi$
$151$	−12.4698	−1.01478	−0.507390	+	0.861717i	$0.669389\pi$
$152$	1.21046	0.0981816
$153$	0	0
$154$	−8.61483	−0.694203
$155$	−7.97594	−0.640643
$156$	0	0
$157$	−13.1146	−1.04666	−0.523329	−	0.852131i	$-0.675310\pi$
$157$	−13.1146	−1.04666	−0.523329	+	0.852131i	$0.675310\pi$
$158$	−11.7226	−0.932601
$159$	0	0
$160$	14.9099	1.17873
$161$	−3.40397	−0.268270
$162$	0	0
$163$	10.8794	0.852144	0.426072	−	0.904689i	$-0.359897\pi$
$163$	10.8794	0.852144	0.426072	+	0.904689i	$0.359897\pi$
$164$	1.64025	0.128082
$165$	0	0
$166$	−9.45596	−0.733925
$167$	20.2789	1.56923	0.784614	−	0.619984i	$-0.212861\pi$
$167$	20.2789	1.56923	0.784614	+	0.619984i	$0.212861\pi$
$168$	0	0
$169$	−10.2911	−0.791620
$170$	17.2624	1.32397
$171$	0	0
$172$	−17.5231	−1.33612
$173$	0.158953	0.0120850	0.00604249	−	0.999982i	$-0.498077\pi$
$173$	0.158953	0.0120850	0.00604249	+	0.999982i	$0.498077\pi$
$174$	0	0
$175$	−0.0750587	−0.00567390
$176$	30.6983	2.31397
$177$	0	0
$178$	17.0967	1.28145
$179$	14.3983	1.07618	0.538090	−	0.842887i	$-0.319146\pi$
$179$	14.3983	1.07618	0.538090	+	0.842887i	$0.319146\pi$
$180$	0	0
$181$	12.9365	0.961561	0.480781	−	0.876841i	$-0.340353\pi$
$181$	12.9365	0.961561	0.480781	+	0.876841i	$0.340353\pi$
$182$	−2.24750	−0.166596
$183$	0	0
$184$	5.31878	0.392106
$185$	−2.42913	−0.178593
$186$	0	0
$187$	26.2822	1.92194
$188$	10.7733	0.785722
$189$	0	0
$190$	−4.31300	−0.312898
$191$	5.61297	0.406140	0.203070	−	0.979164i	$-0.434908\pi$
$191$	5.61297	0.406140	0.203070	+	0.979164i	$0.434908\pi$
$192$	0	0
$193$	9.06736	0.652683	0.326341	−	0.945252i	$-0.394184\pi$
$193$	9.06736	0.652683	0.326341	+	0.945252i	$0.394184\pi$
$194$	27.8289	1.99800
$195$	0	0
$196$	−8.80621	−0.629015
$197$	−9.64306	−0.687039	−0.343520	−	0.939145i	$-0.611619\pi$
$197$	−9.64306	−0.687039	−0.343520	+	0.939145i	$0.611619\pi$
$198$	0	0
$199$	11.9070	0.844068	0.422034	−	0.906580i	$-0.361316\pi$
$199$	11.9070	0.844068	0.422034	+	0.906580i	$0.361316\pi$
$200$	0.117281	0.00829301
$201$	0	0
$202$	8.14899	0.573361
$203$	0	0
$204$	0	0
$205$	2.71167	0.189391
$206$	−28.2428	−1.96777
$207$	0	0
$208$	8.00878	0.555309
$209$	−6.56657	−0.454219
$210$	0	0
$211$	−5.79422	−0.398891	−0.199445	−	0.979909i	$-0.563914\pi$
$211$	−5.79422	−0.398891	−0.199445	+	0.979909i	$0.563914\pi$
$212$	18.0218	1.23774
$213$	0	0
$214$	−14.5249	−0.992899
$215$	−28.9692	−1.97569
$216$	0	0
$217$	2.62842	0.178429
$218$	−7.68563	−0.520537
$219$	0	0
$220$	−19.4654	−1.31236
$221$	6.85668	0.461230
$222$	0	0
$223$	−10.3821	−0.695235	−0.347618	−	0.937636i	$-0.613009\pi$
$223$	−10.3821	−0.695235	−0.347618	+	0.937636i	$0.613009\pi$
$224$	−4.91346	−0.328294
$225$	0	0
$226$	−19.8603	−1.32109
$227$	−1.39818	−0.0928007	−0.0464003	−	0.998923i	$-0.514775\pi$
$227$	−1.39818	−0.0928007	−0.0464003	+	0.998923i	$0.514775\pi$
$228$	0	0
$229$	−6.89122	−0.455384	−0.227692	−	0.973733i	$-0.573118\pi$
$229$	−6.89122	−0.455384	−0.227692	+	0.973733i	$0.573118\pi$
$230$	−18.9513	−1.24961
$231$	0	0
$232$	0	0
$233$	0.133957	0.00877579	0.00438789	−	0.999990i	$-0.498603\pi$
$233$	0.133957	0.00877579	0.00438789	+	0.999990i	$0.498603\pi$
$234$	0	0
$235$	17.8104	1.16182
$236$	−15.1075	−0.983413
$237$	0	0
$238$	−5.68871	−0.368744
$239$	−10.3770	−0.671234	−0.335617	−	0.941998i	$-0.608945\pi$
$239$	−10.3770	−0.671234	−0.335617	+	0.941998i	$0.608945\pi$
$240$	0	0
$241$	8.46604	0.545345	0.272673	−	0.962107i	$-0.412092\pi$
$241$	8.46604	0.545345	0.272673	+	0.962107i	$0.412092\pi$
$242$	−52.8413	−3.39677
$243$	0	0
$244$	−8.80974	−0.563986
$245$	−14.5585	−0.930105
$246$	0	0
$247$	−1.71313	−0.109004
$248$	−4.10696	−0.260792
$249$	0	0
$250$	20.3006	1.28392
$251$	−7.55152	−0.476647	−0.238324	−	0.971186i	$-0.576598\pi$
$251$	−7.55152	−0.476647	−0.238324	+	0.971186i	$0.576598\pi$
$252$	0	0
$253$	−28.8535	−1.81401
$254$	−14.8055	−0.928978
$255$	0	0
$256$	20.9725	1.31078
$257$	−21.3098	−1.32927	−0.664633	−	0.747170i	$-0.731412\pi$
$257$	−21.3098	−1.32927	−0.664633	+	0.747170i	$0.731412\pi$
$258$	0	0
$259$	0.800503	0.0497408
$260$	−5.07827	−0.314941
$261$	0	0
$262$	−34.3177	−2.12016
$263$	25.8179	1.59200	0.796001	−	0.605296i	$-0.206945\pi$
$263$	25.8179	1.59200	0.796001	+	0.605296i	$0.206945\pi$
$264$	0	0
$265$	29.7937	1.83021
$266$	1.42132	0.0871466
$267$	0	0
$268$	−9.39579	−0.573939
$269$	−19.1133	−1.16536	−0.582678	−	0.812703i	$-0.697995\pi$
$269$	−19.1133	−1.16536	−0.582678	+	0.812703i	$0.697995\pi$
$270$	0	0
$271$	−3.48307	−0.211581	−0.105791	−	0.994388i	$-0.533737\pi$
$271$	−3.48307	−0.211581	−0.105791	+	0.994388i	$0.533737\pi$
$272$	20.2713	1.22913
$273$	0	0
$274$	−6.84734	−0.413663
$275$	−0.636230	−0.0383661
$276$	0	0
$277$	−10.1669	−0.610870	−0.305435	−	0.952213i	$-0.598802\pi$
$277$	−10.1669	−0.610870	−0.305435	+	0.952213i	$0.598802\pi$
$278$	−24.8508	−1.49045
$279$	0	0
$280$	−1.95486	−0.116825
$281$	1.07622	0.0642017	0.0321009	−	0.999485i	$-0.489780\pi$
$281$	1.07622	0.0642017	0.0321009	+	0.999485i	$0.489780\pi$
$282$	0	0
$283$	9.12794	0.542600	0.271300	−	0.962495i	$-0.412546\pi$
$283$	9.12794	0.542600	0.271300	+	0.962495i	$0.412546\pi$
$284$	4.47769	0.265702
$285$	0	0
$286$	−19.0508	−1.12650
$287$	−0.893612	−0.0527483
$288$	0	0
$289$	0.355162	0.0208919
$290$	0	0
$291$	0	0
$292$	9.68651	0.566860
$293$	13.7721	0.804575	0.402287	−	0.915513i	$-0.368215\pi$
$293$	13.7721	0.804575	0.402287	+	0.915513i	$0.368215\pi$
$294$	0	0
$295$	−24.9757	−1.45414
$296$	−1.25080	−0.0727015
$297$	0	0
$298$	−3.77865	−0.218891
$299$	−7.52751	−0.435327
$300$	0	0
$301$	9.54662	0.550258
$302$	22.8784	1.31651
$303$	0	0
$304$	−5.06476	−0.290484
$305$	−14.5643	−0.833949
$306$	0	0
$307$	3.43747	0.196187	0.0980934	−	0.995177i	$-0.468726\pi$
$307$	3.43747	0.196187	0.0980934	+	0.995177i	$0.468726\pi$
$308$	6.41469	0.365511
$309$	0	0
$310$	14.6335	0.831126
$311$	29.9213	1.69668	0.848341	−	0.529450i	$-0.177602\pi$
$311$	29.9213	1.69668	0.848341	+	0.529450i	$0.177602\pi$
$312$	0	0
$313$	30.6301	1.73132	0.865658	−	0.500637i	$-0.166901\pi$
$313$	30.6301	1.73132	0.865658	+	0.500637i	$0.166901\pi$
$314$	24.0614	1.35786
$315$	0	0
$316$	8.72878	0.491032
$317$	−4.43441	−0.249061	−0.124531	−	0.992216i	$-0.539743\pi$
$317$	−4.43441	−0.249061	−0.124531	+	0.992216i	$0.539743\pi$
$318$	0	0
$319$	0	0
$320$	−5.37575	−0.300514
$321$	0	0
$322$	6.24527	0.348036
$323$	−4.33617	−0.241271
$324$	0	0
$325$	−0.165984	−0.00920715
$326$	−19.9606	−1.10551
$327$	0	0
$328$	1.39629	0.0770972
$329$	−5.86930	−0.323585
$330$	0	0
$331$	−12.6242	−0.693891	−0.346946	−	0.937885i	$-0.612781\pi$
$331$	−12.6242	−0.693891	−0.346946	+	0.937885i	$0.612781\pi$
$332$	7.04100	0.386425
$333$	0	0
$334$	−37.2058	−2.03581
$335$	−15.5331	−0.848666
$336$	0	0
$337$	−17.0903	−0.930968	−0.465484	−	0.885056i	$-0.654120\pi$
$337$	−17.0903	−0.930968	−0.465484	+	0.885056i	$0.654120\pi$
$338$	18.8811	1.02699
$339$	0	0
$340$	−12.8538	−0.697093
$341$	22.2796	1.20651
$342$	0	0
$343$	10.0076	0.540358
$344$	−14.9168	−0.804261
$345$	0	0
$346$	−0.291632	−0.0156782
$347$	−5.48266	−0.294325	−0.147162	−	0.989112i	$-0.547014\pi$
$347$	−5.48266	−0.294325	−0.147162	+	0.989112i	$0.547014\pi$
$348$	0	0
$349$	−20.4824	−1.09640	−0.548198	−	0.836349i	$-0.684686\pi$
$349$	−20.4824	−1.09640	−0.548198	+	0.836349i	$0.684686\pi$
$350$	0.137710	0.00736093
$351$	0	0
$352$	−41.6486	−2.21988
$353$	18.8850	1.00515	0.502574	−	0.864534i	$-0.332387\pi$
$353$	18.8850	1.00515	0.502574	+	0.864534i	$0.332387\pi$
$354$	0	0
$355$	7.40253	0.392885
$356$	−12.7304	−0.674707
$357$	0	0
$358$	−26.4166	−1.39616
$359$	4.29579	0.226723	0.113361	−	0.993554i	$-0.463838\pi$
$359$	4.29579	0.226723	0.113361	+	0.993554i	$0.463838\pi$
$360$	0	0
$361$	−17.9166	−0.942980
$362$	−23.7346	−1.24746
$363$	0	0
$364$	1.67351	0.0877157
$365$	16.0138	0.838199
$366$	0	0
$367$	13.6676	0.713445	0.356723	−	0.934210i	$-0.383894\pi$
$367$	13.6676	0.713445	0.356723	+	0.934210i	$0.383894\pi$
$368$	−22.2546	−1.16010
$369$	0	0
$370$	4.45673	0.231695
$371$	−9.81832	−0.509742
$372$	0	0
$373$	12.5156	0.648034	0.324017	−	0.946051i	$-0.394966\pi$
$373$	12.5156	0.648034	0.324017	+	0.946051i	$0.394966\pi$
$374$	−48.2200	−2.49340
$375$	0	0
$376$	9.17092	0.472954
$377$	0	0
$378$	0	0
$379$	11.1109	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$379$	11.1109	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$380$	3.21150	0.164746
$381$	0	0
$382$	−10.2981	−0.526899
$383$	−15.5503	−0.794582	−0.397291	−	0.917693i	$-0.630050\pi$
$383$	−15.5503	−0.794582	−0.397291	+	0.917693i	$0.630050\pi$
$384$	0	0
$385$	10.6048	0.540470
$386$	−16.6359	−0.846746
$387$	0	0
$388$	−20.7217	−1.05199
$389$	0.482991	0.0244886	0.0122443	−	0.999925i	$-0.496102\pi$
$389$	0.482991	0.0244886	0.0122443	+	0.999925i	$0.496102\pi$
$390$	0	0
$391$	−19.0531	−0.963558
$392$	−7.49642	−0.378626
$393$	0	0
$394$	17.6922	0.891318
$395$	14.4304	0.726075
$396$	0	0
$397$	−7.77402	−0.390167	−0.195083	−	0.980787i	$-0.562498\pi$
$397$	−7.77402	−0.390167	−0.195083	+	0.980787i	$0.562498\pi$
$398$	−21.8459	−1.09504
$399$	0	0
$400$	−0.490721	−0.0245360
$401$	1.55128	0.0774671	0.0387335	−	0.999250i	$-0.487668\pi$
$401$	1.55128	0.0774671	0.0387335	+	0.999250i	$0.487668\pi$
$402$	0	0
$403$	5.81246	0.289539
$404$	−6.06782	−0.301885
$405$	0	0
$406$	0	0
$407$	6.78542	0.336341
$408$	0	0
$409$	10.1682	0.502788	0.251394	−	0.967885i	$-0.419111\pi$
$409$	10.1682	0.502788	0.251394	+	0.967885i	$0.419111\pi$
$410$	−4.97511	−0.245703
$411$	0	0
$412$	21.0299	1.03607
$413$	8.23059	0.405001
$414$	0	0
$415$	11.6402	0.571395
$416$	−10.8656	−0.532729
$417$	0	0
$418$	12.0477	0.589273
$419$	15.0196	0.733755	0.366878	−	0.930269i	$-0.380427\pi$
$419$	15.0196	0.733755	0.366878	+	0.930269i	$0.380427\pi$
$420$	0	0
$421$	−8.57843	−0.418087	−0.209044	−	0.977906i	$-0.567035\pi$
$421$	−8.57843	−0.418087	−0.209044	+	0.977906i	$0.567035\pi$
$422$	10.6307	0.517494
$423$	0	0
$424$	15.3414	0.745042
$425$	−0.420128	−0.0203792
$426$	0	0
$427$	4.79957	0.232267
$428$	10.8154	0.522780
$429$	0	0
$430$	53.1500	2.56312
$431$	9.39451	0.452518	0.226259	−	0.974067i	$-0.427350\pi$
$431$	9.39451	0.452518	0.226259	+	0.974067i	$0.427350\pi$
$432$	0	0
$433$	22.2308	1.06835	0.534173	−	0.845375i	$-0.320623\pi$
$433$	22.2308	1.06835	0.534173	+	0.845375i	$0.320623\pi$
$434$	−4.82237	−0.231481
$435$	0	0
$436$	5.72280	0.274072
$437$	4.76040	0.227721
$438$	0	0
$439$	−2.81005	−0.134116	−0.0670581	−	0.997749i	$-0.521361\pi$
$439$	−2.81005	−0.134116	−0.0670581	+	0.997749i	$0.521361\pi$
$440$	−16.5702	−0.789954
$441$	0	0
$442$	−12.5800	−0.598369
$443$	4.92058	0.233784	0.116892	−	0.993145i	$-0.462707\pi$
$443$	4.92058	0.233784	0.116892	+	0.993145i	$0.462707\pi$
$444$	0	0
$445$	−21.0459	−0.997670
$446$	19.0480	0.901951
$447$	0	0
$448$	1.77154	0.0836975
$449$	37.9915	1.79293	0.896464	−	0.443116i	$-0.146127\pi$
$449$	37.9915	1.79293	0.896464	+	0.443116i	$0.146127\pi$
$450$	0	0
$451$	−7.57465	−0.356676
$452$	14.7882	0.695576
$453$	0	0
$454$	2.56525	0.120393
$455$	2.76665	0.129703
$456$	0	0
$457$	−4.65673	−0.217833	−0.108916	−	0.994051i	$-0.534738\pi$
$457$	−4.65673	−0.217833	−0.108916	+	0.994051i	$0.534738\pi$
$458$	12.6433	0.590785
$459$	0	0
$460$	14.1113	0.657944
$461$	19.9667	0.929940	0.464970	−	0.885326i	$-0.346065\pi$
$461$	19.9667	0.929940	0.464970	+	0.885326i	$0.346065\pi$
$462$	0	0
$463$	−19.4998	−0.906230	−0.453115	−	0.891452i	$-0.649687\pi$
$463$	−19.4998	−0.906230	−0.453115	+	0.891452i	$0.649687\pi$
$464$	0	0
$465$	0	0
$466$	−0.245771	−0.0113851
$467$	−34.5823	−1.60028	−0.800139	−	0.599814i	$-0.795241\pi$
$467$	−34.5823	−1.60028	−0.800139	+	0.599814i	$0.795241\pi$
$468$	0	0
$469$	5.11884	0.236366
$470$	−32.6769	−1.50727
$471$	0	0
$472$	−12.8605	−0.591952
$473$	80.9213	3.72077
$474$	0	0
$475$	0.104969	0.00481629
$476$	4.23587	0.194151
$477$	0	0
$478$	19.0388	0.870813
$479$	−6.16844	−0.281843	−0.140922	−	0.990021i	$-0.545007\pi$
$479$	−6.16844	−0.281843	−0.140922	+	0.990021i	$0.545007\pi$
$480$	0	0
$481$	1.77023	0.0807154
$482$	−15.5327	−0.707494
$483$	0	0
$484$	39.3462	1.78846
$485$	−34.2572	−1.55554
$486$	0	0
$487$	6.54624	0.296638	0.148319	−	0.988940i	$-0.452614\pi$
$487$	6.54624	0.296638	0.148319	+	0.988940i	$0.452614\pi$
$488$	−7.49943	−0.339483
$489$	0	0
$490$	26.7104	1.20666
$491$	−18.2086	−0.821744	−0.410872	−	0.911693i	$-0.634776\pi$
$491$	−18.2086	−0.821744	−0.410872	+	0.911693i	$0.634776\pi$
$492$	0	0
$493$	0	0
$494$	3.14309	0.141414
$495$	0	0
$496$	17.1841	0.771590
$497$	−2.43945	−0.109424
$498$	0	0
$499$	16.7536	0.749995	0.374997	−	0.927026i	$-0.377644\pi$
$499$	16.7536	0.749995	0.374997	+	0.927026i	$0.377644\pi$
$500$	−15.1160	−0.676008
$501$	0	0
$502$	13.8548	0.618370
$503$	−3.32173	−0.148109	−0.0740544	−	0.997254i	$-0.523594\pi$
$503$	−3.32173	−0.148109	−0.0740544	+	0.997254i	$0.523594\pi$
$504$	0	0
$505$	−10.0313	−0.446389
$506$	52.9377	2.35337
$507$	0	0
$508$	11.0243	0.489124
$509$	−18.8251	−0.834409	−0.417205	−	0.908813i	$-0.636990\pi$
$509$	−18.8251	−0.834409	−0.417205	+	0.908813i	$0.636990\pi$
$510$	0	0
$511$	−5.27723	−0.233451
$512$	−20.8057	−0.919490
$513$	0	0
$514$	39.0971	1.72450
$515$	34.7667	1.53200
$516$	0	0
$517$	−49.7508	−2.18804
$518$	−1.46869	−0.0645304
$519$	0	0
$520$	−4.32295	−0.189574
$521$	3.09335	0.135522	0.0677611	−	0.997702i	$-0.478414\pi$
$521$	3.09335	0.135522	0.0677611	+	0.997702i	$0.478414\pi$
$522$	0	0
$523$	34.6039	1.51312	0.756561	−	0.653924i	$-0.226878\pi$
$523$	34.6039	1.51312	0.756561	+	0.653924i	$0.226878\pi$
$524$	25.5533	1.11630
$525$	0	0
$526$	−47.3683	−2.06535
$527$	14.7121	0.640870
$528$	0	0
$529$	−2.08278	−0.0905556
$530$	−54.6627	−2.37439
$531$	0	0
$532$	−1.05833	−0.0458843
$533$	−1.97613	−0.0855956
$534$	0	0
$535$	17.8800	0.773019
$536$	−7.99831	−0.345474
$537$	0	0
$538$	35.0672	1.51185
$539$	40.6669	1.75165
$540$	0	0
$541$	27.7601	1.19350	0.596751	−	0.802427i	$-0.296458\pi$
$541$	27.7601	1.19350	0.596751	+	0.802427i	$0.296458\pi$
$542$	6.39040	0.274491
$543$	0	0
$544$	−27.5022	−1.17915
$545$	9.46095	0.405263
$546$	0	0
$547$	−32.6717	−1.39694	−0.698471	−	0.715639i	$-0.746136\pi$
$547$	−32.6717	−1.39694	−0.698471	+	0.715639i	$0.746136\pi$
$548$	5.09859	0.217801
$549$	0	0
$550$	1.16729	0.0497736
$551$	0	0
$552$	0	0
$553$	−4.75546	−0.202223
$554$	18.6533	0.792501
$555$	0	0
$556$	18.5041	0.784751
$557$	−43.4204	−1.83978	−0.919891	−	0.392175i	$-0.871723\pi$
$557$	−43.4204	−1.83978	−0.919891	+	0.392175i	$0.871723\pi$
$558$	0	0
$559$	21.1113	0.892914
$560$	8.17940	0.345643
$561$	0	0
$562$	−1.97454	−0.0832909
$563$	0.551999	0.0232640	0.0116320	−	0.999932i	$-0.496297\pi$
$563$	0.551999	0.0232640	0.0116320	+	0.999932i	$0.496297\pi$
$564$	0	0
$565$	24.4478	1.02853
$566$	−16.7471	−0.703932
$567$	0	0
$568$	3.81170	0.159935
$569$	27.9559	1.17197	0.585986	−	0.810322i	$-0.300708\pi$
$569$	27.9559	1.17197	0.585986	+	0.810322i	$0.300708\pi$
$570$	0	0
$571$	33.2897	1.39313	0.696566	−	0.717493i	$-0.254710\pi$
$571$	33.2897	1.39313	0.696566	+	0.717493i	$0.254710\pi$
$572$	14.1854	0.593121
$573$	0	0
$574$	1.63951	0.0684320
$575$	0.461232	0.0192347
$576$	0	0
$577$	19.3682	0.806309	0.403154	−	0.915132i	$-0.367914\pi$
$577$	19.3682	0.806309	0.403154	+	0.915132i	$0.367914\pi$
$578$	−0.651618	−0.0271037
$579$	0	0
$580$	0	0
$581$	−3.83595	−0.159142
$582$	0	0
$583$	−83.2244	−3.44680
$584$	8.24580	0.341214
$585$	0	0
$586$	−25.2677	−1.04380
$587$	−5.82917	−0.240596	−0.120298	−	0.992738i	$-0.538385\pi$
$587$	−5.82917	−0.240596	−0.120298	+	0.992738i	$0.538385\pi$
$588$	0	0
$589$	−3.67581	−0.151459
$590$	45.8231	1.88651
$591$	0	0
$592$	5.23355	0.215098
$593$	2.27254	0.0933219	0.0466609	−	0.998911i	$-0.485142\pi$
$593$	2.27254	0.0933219	0.0466609	+	0.998911i	$0.485142\pi$
$594$	0	0
$595$	7.00276	0.287085
$596$	2.81362	0.115250
$597$	0	0
$598$	13.8108	0.564764
$599$	−30.3666	−1.24074	−0.620372	−	0.784307i	$-0.713019\pi$
$599$	−30.3666	−1.24074	−0.620372	+	0.784307i	$0.713019\pi$
$600$	0	0
$601$	19.9450	0.813572	0.406786	−	0.913524i	$-0.366650\pi$
$601$	19.9450	0.813572	0.406786	+	0.913524i	$0.366650\pi$
$602$	−17.5152	−0.713867
$603$	0	0
$604$	−17.0355	−0.693165
$605$	65.0472	2.64455
$606$	0	0
$607$	33.9427	1.37769	0.688846	−	0.724907i	$-0.258117\pi$
$607$	33.9427	1.37769	0.688846	+	0.724907i	$0.258117\pi$
$608$	6.87141	0.278672
$609$	0	0
$610$	26.7212	1.08191
$611$	−12.9793	−0.525088
$612$	0	0
$613$	26.5569	1.07262	0.536311	−	0.844021i	$-0.319818\pi$
$613$	26.5569	1.07262	0.536311	+	0.844021i	$0.319818\pi$
$614$	−6.30674	−0.254519
$615$	0	0
$616$	5.46060	0.220014
$617$	0.847640	0.0341247	0.0170624	−	0.999854i	$-0.494569\pi$
$617$	0.847640	0.0341247	0.0170624	+	0.999854i	$0.494569\pi$
$618$	0	0
$619$	44.1556	1.77476	0.887381	−	0.461036i	$-0.152522\pi$
$619$	44.1556	1.77476	0.887381	+	0.461036i	$0.152522\pi$
$620$	−10.8962	−0.437604
$621$	0	0
$622$	−54.8968	−2.20116
$623$	6.93552	0.277866
$624$	0	0
$625$	−25.4941	−1.01976
$626$	−56.1972	−2.24609
$627$	0	0
$628$	−17.9163	−0.714940
$629$	4.48068	0.178656
$630$	0	0
$631$	−4.70583	−0.187336	−0.0936681	−	0.995603i	$-0.529859\pi$
$631$	−4.70583	−0.187336	−0.0936681	+	0.995603i	$0.529859\pi$
$632$	7.43051	0.295570
$633$	0	0
$634$	8.13583	0.323115
$635$	18.2254	0.723253
$636$	0	0
$637$	10.6095	0.420362
$638$	0	0
$639$	0	0
$640$	−19.9569	−0.788866
$641$	−26.2799	−1.03799	−0.518996	−	0.854776i	$-0.673694\pi$
$641$	−26.2799	−1.03799	−0.518996	+	0.854776i	$0.673694\pi$
$642$	0	0
$643$	−21.4377	−0.845421	−0.422710	−	0.906265i	$-0.638921\pi$
$643$	−21.4377	−0.845421	−0.422710	+	0.906265i	$0.638921\pi$
$644$	−4.65029	−0.183247
$645$	0	0
$646$	7.95559	0.313008
$647$	−31.4103	−1.23487	−0.617433	−	0.786623i	$-0.711827\pi$
$647$	−31.4103	−1.23487	−0.617433	+	0.786623i	$0.711827\pi$
$648$	0	0
$649$	69.7661	2.73856
$650$	0.304532	0.0119447
$651$	0	0
$652$	14.8628	0.582073
$653$	34.4884	1.34963	0.674817	−	0.737985i	$-0.264223\pi$
$653$	34.4884	1.34963	0.674817	+	0.737985i	$0.264223\pi$
$654$	0	0
$655$	42.2448	1.65064
$656$	−5.84228	−0.228103
$657$	0	0
$658$	10.7684	0.419797
$659$	−23.6867	−0.922702	−0.461351	−	0.887218i	$-0.652635\pi$
$659$	−23.6867	−0.922702	−0.461351	+	0.887218i	$0.652635\pi$
$660$	0	0
$661$	−32.0028	−1.24477	−0.622383	−	0.782713i	$-0.713835\pi$
$661$	−32.0028	−1.24477	−0.622383	+	0.782713i	$0.713835\pi$
$662$	23.1618	0.900207
$663$	0	0
$664$	5.99376	0.232603
$665$	−1.74963	−0.0678478
$666$	0	0
$667$	0	0
$668$	27.7038	1.07189
$669$	0	0
$670$	28.4987	1.10100
$671$	40.6832	1.57056
$672$	0	0
$673$	11.9640	0.461180	0.230590	−	0.973051i	$-0.425934\pi$
$673$	11.9640	0.461180	0.230590	+	0.973051i	$0.425934\pi$
$674$	31.3557	1.20777
$675$	0	0
$676$	−14.0590	−0.540732
$677$	40.0569	1.53951	0.769755	−	0.638339i	$-0.220378\pi$
$677$	40.0569	1.53951	0.769755	+	0.638339i	$0.220378\pi$
$678$	0	0
$679$	11.2892	0.433241
$680$	−10.9420	−0.419605
$681$	0	0
$682$	−40.8765	−1.56524
$683$	20.7589	0.794318	0.397159	−	0.917750i	$-0.369996\pi$
$683$	20.7589	0.794318	0.397159	+	0.917750i	$0.369996\pi$
$684$	0	0
$685$	8.42901	0.322056
$686$	−18.3609	−0.701023
$687$	0	0
$688$	62.4141	2.37952
$689$	−21.7122	−0.827168
$690$	0	0
$691$	3.33679	0.126937	0.0634687	−	0.997984i	$-0.479784\pi$
$691$	3.33679	0.126937	0.0634687	+	0.997984i	$0.479784\pi$
$692$	0.217152	0.00825489
$693$	0	0
$694$	10.0591	0.381837
$695$	30.5911	1.16039
$696$	0	0
$697$	−5.00184	−0.189458
$698$	37.5791	1.42239
$699$	0	0
$700$	−0.102541	−0.00387567
$701$	−24.0330	−0.907714	−0.453857	−	0.891074i	$-0.649952\pi$
$701$	−24.0330	−0.907714	−0.453857	+	0.891074i	$0.649952\pi$
$702$	0	0
$703$	−1.11949	−0.0422225
$704$	15.0164	0.565951
$705$	0	0
$706$	−34.6484	−1.30401
$707$	3.30576	0.124326
$708$	0	0
$709$	−47.3439	−1.77804	−0.889019	−	0.457871i	$-0.848612\pi$
$709$	−47.3439	−1.77804	−0.889019	+	0.457871i	$0.848612\pi$
$710$	−13.5814	−0.509703
$711$	0	0
$712$	−10.8369	−0.406130
$713$	−16.1515	−0.604878
$714$	0	0
$715$	23.4513	0.877030
$716$	19.6701	0.735106
$717$	0	0
$718$	−7.88150	−0.294135
$719$	−28.9066	−1.07803	−0.539016	−	0.842295i	$-0.681204\pi$
$719$	−28.9066	−1.07803	−0.539016	+	0.842295i	$0.681204\pi$
$720$	0	0
$721$	−11.4571	−0.426686
$722$	32.8717	1.22336
$723$	0	0
$724$	17.6730	0.656813
$725$	0	0
$726$	0	0
$727$	3.66717	0.136008	0.0680038	−	0.997685i	$-0.478337\pi$
$727$	3.66717	0.136008	0.0680038	+	0.997685i	$0.478337\pi$
$728$	1.42460	0.0527992
$729$	0	0
$730$	−29.3805	−1.08742
$731$	53.4356	1.97639
$732$	0	0
$733$	−41.1371	−1.51943	−0.759716	−	0.650255i	$-0.774662\pi$
$733$	−41.1371	−1.51943	−0.759716	+	0.650255i	$0.774662\pi$
$734$	−25.0761	−0.925575
$735$	0	0
$736$	30.1930	1.11293
$737$	43.3896	1.59827
$738$	0	0
$739$	42.9540	1.58009	0.790043	−	0.613051i	$-0.210058\pi$
$739$	42.9540	1.58009	0.790043	+	0.613051i	$0.210058\pi$
$740$	−3.31853	−0.121992
$741$	0	0
$742$	18.0137	0.661304
$743$	−28.2979	−1.03815	−0.519074	−	0.854729i	$-0.673723\pi$
$743$	−28.2979	−1.03815	−0.519074	+	0.854729i	$0.673723\pi$
$744$	0	0
$745$	4.65149	0.170417
$746$	−22.9625	−0.840715
$747$	0	0
$748$	35.9051	1.31282
$749$	−5.89222	−0.215297
$750$	0	0
$751$	52.1447	1.90279	0.951393	−	0.307978i	$-0.0996523\pi$
$751$	52.1447	1.90279	0.951393	+	0.307978i	$0.0996523\pi$
$752$	−38.3725	−1.39930
$753$	0	0
$754$	0	0
$755$	−28.1631	−1.02496
$756$	0	0
$757$	−40.8852	−1.48600	−0.742999	−	0.669292i	$-0.766598\pi$
$757$	−40.8852	−1.48600	−0.742999	+	0.669292i	$0.766598\pi$
$758$	−20.3853	−0.740427
$759$	0	0
$760$	2.73384	0.0991668
$761$	13.1093	0.475210	0.237605	−	0.971362i	$-0.423638\pi$
$761$	13.1093	0.475210	0.237605	+	0.971362i	$0.423638\pi$
$762$	0	0
$763$	−3.11779	−0.112872
$764$	7.66810	0.277422
$765$	0	0
$766$	28.5302	1.03084
$767$	18.2011	0.657202
$768$	0	0
$769$	35.8690	1.29347	0.646734	−	0.762716i	$-0.276134\pi$
$769$	35.8690	1.29347	0.646734	+	0.762716i	$0.276134\pi$
$770$	−19.4566	−0.701169
$771$	0	0
$772$	12.3873	0.445828
$773$	38.0995	1.37034	0.685172	−	0.728381i	$-0.259727\pi$
$773$	38.0995	1.37034	0.685172	+	0.728381i	$0.259727\pi$
$774$	0	0
$775$	−0.356146	−0.0127931
$776$	−17.6397	−0.633228
$777$	0	0
$778$	−0.886146	−0.0317699
$779$	1.24970	0.0447753
$780$	0	0
$781$	−20.6779	−0.739912
$782$	34.9569	1.25005
$783$	0	0
$784$	31.3661	1.12022
$785$	−29.6194	−1.05716
$786$	0	0
$787$	−16.2121	−0.577901	−0.288950	−	0.957344i	$-0.593306\pi$
$787$	−16.2121	−0.577901	−0.288950	+	0.957344i	$0.593306\pi$
$788$	−13.1738	−0.469296
$789$	0	0
$790$	−26.4756	−0.941960
$791$	−8.05662	−0.286460
$792$	0	0
$793$	10.6137	0.376904
$794$	14.2630	0.506176
$795$	0	0
$796$	16.2667	0.576557
$797$	28.8327	1.02131	0.510653	−	0.859787i	$-0.329404\pi$
$797$	28.8327	1.02131	0.510653	+	0.859787i	$0.329404\pi$
$798$	0	0
$799$	−32.8524	−1.16224
$800$	0.665765	0.0235384
$801$	0	0
$802$	−2.84613	−0.100500
$803$	−44.7321	−1.57856
$804$	0	0
$805$	−7.68788	−0.270962
$806$	−10.6641	−0.375629
$807$	0	0
$808$	−5.16533	−0.181716
$809$	12.7510	0.448301	0.224151	−	0.974555i	$-0.428039\pi$
$809$	12.7510	0.448301	0.224151	+	0.974555i	$0.428039\pi$
$810$	0	0
$811$	11.6687	0.409742	0.204871	−	0.978789i	$-0.434323\pi$
$811$	11.6687	0.409742	0.204871	+	0.978789i	$0.434323\pi$
$812$	0	0
$813$	0	0
$814$	−12.4492	−0.436345
$815$	24.5713	0.860695
$816$	0	0
$817$	−13.3508	−0.467086
$818$	−18.6557	−0.652282
$819$	0	0
$820$	3.70452	0.129367
$821$	45.4267	1.58540	0.792701	−	0.609611i	$-0.208674\pi$
$821$	45.4267	1.58540	0.792701	+	0.609611i	$0.208674\pi$
$822$	0	0
$823$	−11.1415	−0.388370	−0.194185	−	0.980965i	$-0.562206\pi$
$823$	−11.1415	−0.388370	−0.194185	+	0.980965i	$0.562206\pi$
$824$	17.9020	0.623646
$825$	0	0
$826$	−15.1007	−0.525420
$827$	−8.81985	−0.306696	−0.153348	−	0.988172i	$-0.549006\pi$
$827$	−8.81985	−0.306696	−0.153348	+	0.988172i	$0.549006\pi$
$828$	0	0
$829$	−40.1793	−1.39548	−0.697742	−	0.716349i	$-0.745812\pi$
$829$	−40.1793	−1.39548	−0.697742	+	0.716349i	$0.745812\pi$
$830$	−21.3563	−0.741289
$831$	0	0
$832$	3.91758	0.135818
$833$	26.8540	0.930435
$834$	0	0
$835$	45.8000	1.58497
$836$	−8.97085	−0.310263
$837$	0	0
$838$	−27.5565	−0.951924
$839$	31.9033	1.10142	0.550712	−	0.834695i	$-0.314356\pi$
$839$	31.9033	1.10142	0.550712	+	0.834695i	$0.314356\pi$
$840$	0	0
$841$	0	0
$842$	15.7389	0.542398
$843$	0	0
$844$	−7.91571	−0.272470
$845$	−23.2424	−0.799564
$846$	0	0
$847$	−21.4359	−0.736545
$848$	−64.1905	−2.20431
$849$	0	0
$850$	0.770811	0.0264386
$851$	−4.91905	−0.168623
$852$	0	0
$853$	7.50388	0.256928	0.128464	−	0.991714i	$-0.458995\pi$
$853$	7.50388	0.256928	0.128464	+	0.991714i	$0.458995\pi$
$854$	−8.80578	−0.301328
$855$	0	0
$856$	9.20674	0.314680
$857$	−10.9514	−0.374092	−0.187046	−	0.982351i	$-0.559891\pi$
$857$	−10.9514	−0.374092	−0.187046	+	0.982351i	$0.559891\pi$
$858$	0	0
$859$	−11.4776	−0.391610	−0.195805	−	0.980643i	$-0.562732\pi$
$859$	−11.4776	−0.391610	−0.195805	+	0.980643i	$0.562732\pi$
$860$	−39.5760	−1.34953
$861$	0	0
$862$	−17.2361	−0.587066
$863$	7.52988	0.256320	0.128160	−	0.991753i	$-0.459093\pi$
$863$	7.52988	0.256320	0.128160	+	0.991753i	$0.459093\pi$
$864$	0	0
$865$	0.358997	0.0122063
$866$	−40.7870	−1.38600
$867$	0	0
$868$	3.59078	0.121879
$869$	−40.3093	−1.36740
$870$	0	0
$871$	11.3198	0.383556
$872$	4.87162	0.164974
$873$	0	0
$874$	−8.73393	−0.295430
$875$	8.23523	0.278402
$876$	0	0
$877$	27.3723	0.924297	0.462149	−	0.886803i	$-0.347079\pi$
$877$	27.3723	0.924297	0.462149	+	0.886803i	$0.347079\pi$
$878$	5.15560	0.173993
$879$	0	0
$880$	69.3322	2.33719
$881$	−26.2414	−0.884094	−0.442047	−	0.896992i	$-0.645748\pi$
$881$	−26.2414	−0.884094	−0.442047	+	0.896992i	$0.645748\pi$
$882$	0	0
$883$	30.3863	1.02258	0.511290	−	0.859408i	$-0.329168\pi$
$883$	30.3863	1.02258	0.511290	+	0.859408i	$0.329168\pi$
$884$	9.36718	0.315052
$885$	0	0
$886$	−9.02781	−0.303295
$887$	−8.08209	−0.271370	−0.135685	−	0.990752i	$-0.543324\pi$
$887$	−8.08209	−0.271370	−0.135685	+	0.990752i	$0.543324\pi$
$888$	0	0
$889$	−6.00606	−0.201437
$890$	38.6129	1.29431
$891$	0	0
$892$	−14.1834	−0.474894
$893$	8.20814	0.274675
$894$	0	0
$895$	32.5187	1.08698
$896$	6.57666	0.219711
$897$	0	0
$898$	−69.7031	−2.32602
$899$	0	0
$900$	0	0
$901$	−54.9564	−1.83086
$902$	13.8972	0.462728
$903$	0	0
$904$	12.5886	0.418692
$905$	29.2171	0.971210
$906$	0	0
$907$	7.94924	0.263950	0.131975	−	0.991253i	$-0.457868\pi$
$907$	7.94924	0.263950	0.131975	+	0.991253i	$0.457868\pi$
$908$	−1.91011	−0.0633893
$909$	0	0
$910$	−5.07598	−0.168267
$911$	5.66777	0.187782	0.0938908	−	0.995583i	$-0.470070\pi$
$911$	5.66777	0.187782	0.0938908	+	0.995583i	$0.470070\pi$
$912$	0	0
$913$	−32.5152	−1.07610
$914$	8.54372	0.282601
$915$	0	0
$916$	−9.41436	−0.311059
$917$	−13.9215	−0.459728
$918$	0	0
$919$	26.8324	0.885118	0.442559	−	0.896739i	$-0.354071\pi$
$919$	26.8324	0.885118	0.442559	+	0.896739i	$0.354071\pi$
$920$	12.0125	0.396040
$921$	0	0
$922$	−36.6329	−1.20644
$923$	−5.39459	−0.177565
$924$	0	0
$925$	−0.108467	−0.00356637
$926$	35.7763	1.17568
$927$	0	0
$928$	0	0
$929$	24.4580	0.802441	0.401220	−	0.915982i	$-0.368586\pi$
$929$	24.4580	0.802441	0.401220	+	0.915982i	$0.368586\pi$
$930$	0	0
$931$	−6.70943	−0.219893
$932$	0.183003	0.00599447
$933$	0	0
$934$	63.4483	2.07609
$935$	59.3584	1.94123
$936$	0	0
$937$	34.2397	1.11856	0.559281	−	0.828978i	$-0.311077\pi$
$937$	34.2397	1.11856	0.559281	+	0.828978i	$0.311077\pi$
$938$	−9.39156	−0.306645
$939$	0	0
$940$	24.3315	0.793606
$941$	38.0807	1.24139	0.620697	−	0.784050i	$-0.286850\pi$
$941$	38.0807	1.24139	0.620697	+	0.784050i	$0.286850\pi$
$942$	0	0
$943$	5.49120	0.178818
$944$	53.8101	1.75137
$945$	0	0
$946$	−148.467	−4.82707
$947$	49.7451	1.61650	0.808250	−	0.588839i	$-0.200415\pi$
$947$	49.7451	1.61650	0.808250	+	0.588839i	$0.200415\pi$
$948$	0	0
$949$	−11.6700	−0.378825
$950$	−0.192586	−0.00624832
$951$	0	0
$952$	3.60585	0.116866
$953$	9.58035	0.310338	0.155169	−	0.987888i	$-0.450408\pi$
$953$	9.58035	0.310338	0.155169	+	0.987888i	$0.450408\pi$
$954$	0	0
$955$	12.6769	0.410216
$956$	−14.1765	−0.458500
$957$	0	0
$958$	11.3173	0.365644
$959$	−2.77772	−0.0896974
$960$	0	0
$961$	−18.5284	−0.597691
$962$	−3.24784	−0.104715
$963$	0	0
$964$	11.5658	0.372509
$965$	20.4787	0.659232
$966$	0	0
$967$	45.1604	1.45226	0.726131	−	0.687557i	$-0.241317\pi$
$967$	45.1604	1.45226	0.726131	+	0.687557i	$0.241317\pi$
$968$	33.4940	1.07654
$969$	0	0
$970$	62.8518	2.01805
$971$	−18.1666	−0.582994	−0.291497	−	0.956572i	$-0.594153\pi$
$971$	−18.1666	−0.582994	−0.291497	+	0.956572i	$0.594153\pi$
$972$	0	0
$973$	−10.0811	−0.323185
$974$	−12.0104	−0.384838
$975$	0	0
$976$	31.3787	1.00441
$977$	−1.43021	−0.0457566	−0.0228783	−	0.999738i	$-0.507283\pi$
$977$	−1.43021	−0.0457566	−0.0228783	+	0.999738i	$0.507283\pi$
$978$	0	0
$979$	58.7885	1.87889
$980$	−19.8889	−0.635327
$981$	0	0
$982$	33.4074	1.06607
$983$	−41.1614	−1.31285	−0.656423	−	0.754393i	$-0.727931\pi$
$983$	−41.1614	−1.31285	−0.656423	+	0.754393i	$0.727931\pi$
$984$	0	0
$985$	−21.7789	−0.693933
$986$	0	0
$987$	0	0
$988$	−2.34038	−0.0744574
$989$	−58.6635	−1.86539
$990$	0	0
$991$	−3.93474	−0.124991	−0.0624956	−	0.998045i	$-0.519906\pi$
$991$	−3.93474	−0.124991	−0.0624956	+	0.998045i	$0.519906\pi$
$992$	−23.3139	−0.740216
$993$	0	0
$994$	4.47567	0.141960
$995$	26.8921	0.852538
$996$	0	0
$997$	−46.8707	−1.48441	−0.742206	−	0.670172i	$-0.766220\pi$
$997$	−46.8707	−1.48441	−0.742206	+	0.670172i	$0.766220\pi$
$998$	−30.7379	−0.972992
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bl.1.2		9
3.2	odd	2		2523.2.a.p.1.8		9
29.16	even	7		261.2.k.b.82.1		18
29.20	even	7		261.2.k.b.226.1		18
29.28	even	2		7569.2.a.bk.1.8		9
87.20	odd	14		87.2.g.b.52.3	✓	18
87.74	odd	14		87.2.g.b.82.3	yes	18
87.86	odd	2		2523.2.a.q.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.52.3	✓	18	87.20	odd	14
87.2.g.b.82.3	yes	18	87.74	odd	14
261.2.k.b.82.1		18	29.16	even	7
261.2.k.b.226.1		18	29.20	even	7
2523.2.a.p.1.8		9	3.2	odd	2
2523.2.a.q.1.2		9	87.86	odd	2
7569.2.a.bk.1.8		9	29.28	even	2
7569.2.a.bl.1.2		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)

\(f(q)\)	\(=\)	\(q-1.83470 q^{2} +1.36614 q^{4} +2.25851 q^{5} -0.744275 q^{7} +1.16295 q^{8} -4.14369 q^{10} -6.30880 q^{11} -1.64589 q^{13} +1.36552 q^{14} -4.86594 q^{16} -4.16595 q^{17} +1.04086 q^{19} +3.08543 q^{20} +11.5748 q^{22} +4.57353 q^{23} +0.100848 q^{25} +3.01971 q^{26} -1.01678 q^{28} -3.53151 q^{31} +6.60167 q^{32} +7.64329 q^{34} -1.68095 q^{35} -1.07555 q^{37} -1.90967 q^{38} +2.62652 q^{40} +1.20065 q^{41} -12.8267 q^{43} -8.61870 q^{44} -8.39108 q^{46} +7.88593 q^{47} -6.44605 q^{49} -0.185026 q^{50} -2.24851 q^{52} +13.1918 q^{53} -14.2485 q^{55} -0.865553 q^{56} -11.0585 q^{59} -6.44864 q^{61} +6.47928 q^{62} -2.38023 q^{64} -3.71724 q^{65} -6.87762 q^{67} -5.69127 q^{68} +3.08405 q^{70} +3.27762 q^{71} +7.09043 q^{73} +1.97331 q^{74} +1.42196 q^{76} +4.69549 q^{77} +6.38938 q^{79} -10.9898 q^{80} -2.20283 q^{82} +5.15394 q^{83} -9.40883 q^{85} +23.5333 q^{86} -7.33681 q^{88} -9.31849 q^{89} +1.22499 q^{91} +6.24808 q^{92} -14.4684 q^{94} +2.35079 q^{95} -15.1681 q^{97} +11.8266 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31}+ \cdots - 12 q^{98}+O(q^{100}) \) 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

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