LMFDB - Embedded newform 7569.2.a.bm.1.3

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} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 $ x^9 - 4x^8 - 6x^7 + 33x^6 + 6x^5 - 90x^4 + 21x^3 + 84x^2 - 36x - 8
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			1.3
Root			$1.69494$ 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)

$f(q)$	$=$	$q-0.694939 q^{2} -1.51706 q^{4} +2.01780 q^{5} -2.39488 q^{7} +2.44414 q^{8} -1.40225 q^{10} -0.957887 q^{11} +6.53321 q^{13} +1.66430 q^{14} +1.33559 q^{16} -3.81642 q^{17} +4.31846 q^{19} -3.06113 q^{20} +0.665673 q^{22} +5.38804 q^{23} -0.928470 q^{25} -4.54018 q^{26} +3.63318 q^{28} -2.73463 q^{31} -5.81644 q^{32} +2.65217 q^{34} -4.83240 q^{35} +1.86068 q^{37} -3.00106 q^{38} +4.93180 q^{40} +11.8282 q^{41} +3.70437 q^{43} +1.45317 q^{44} -3.74436 q^{46} -3.23270 q^{47} -1.26453 q^{49} +0.645230 q^{50} -9.91127 q^{52} -2.26772 q^{53} -1.93283 q^{55} -5.85343 q^{56} +9.30726 q^{59} -3.13852 q^{61} +1.90040 q^{62} +1.37088 q^{64} +13.1827 q^{65} +12.1012 q^{67} +5.78973 q^{68} +3.35822 q^{70} +4.60343 q^{71} -9.09135 q^{73} -1.29306 q^{74} -6.55136 q^{76} +2.29403 q^{77} -4.28781 q^{79} +2.69496 q^{80} -8.21984 q^{82} -16.1813 q^{83} -7.70078 q^{85} -2.57431 q^{86} -2.34121 q^{88} +15.4430 q^{89} -15.6463 q^{91} -8.17399 q^{92} +2.24653 q^{94} +8.71380 q^{95} -15.7632 q^{97} +0.878774 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * 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 $

$2$	−0.694939	−0.491396	−0.245698	−	0.969346i	$-0.579017\pi$
$2$	−0.694939	−0.491396	−0.245698	+	0.969346i	$0.579017\pi$
$3$	0	0
$3$	0	0
$4$	−1.51706	−0.758530
$5$	2.01780	0.902389	0.451195	−	0.892426i	$-0.350998\pi$
$5$	2.01780	0.902389	0.451195	+	0.892426i	$0.350998\pi$
$6$	0	0
$7$	−2.39488	−0.905181	−0.452590	−	0.891719i	$-0.649500\pi$
$7$	−2.39488	−0.905181	−0.452590	+	0.891719i	$0.649500\pi$
$8$	2.44414	0.864134
$9$	0	0
$10$	−1.40225	−0.443430
$11$	−0.957887	−0.288814	−0.144407	−	0.989518i	$-0.546127\pi$
$11$	−0.957887	−0.288814	−0.144407	+	0.989518i	$0.546127\pi$
$12$	0	0
$13$	6.53321	1.81199	0.905993	−	0.423294i	$-0.139126\pi$
$13$	6.53321	1.81199	0.905993	+	0.423294i	$0.139126\pi$
$14$	1.66430	0.444802
$15$	0	0
$16$	1.33559	0.333898
$17$	−3.81642	−0.925617	−0.462808	−	0.886458i	$-0.653158\pi$
$17$	−3.81642	−0.925617	−0.462808	+	0.886458i	$0.653158\pi$
$18$	0	0
$19$	4.31846	0.990722	0.495361	−	0.868687i	$-0.335036\pi$
$19$	4.31846	0.990722	0.495361	+	0.868687i	$0.335036\pi$
$20$	−3.06113	−0.684489
$21$	0	0
$22$	0.665673	0.141922
$23$	5.38804	1.12348	0.561742	−	0.827312i	$-0.310131\pi$
$23$	5.38804	1.12348	0.561742	+	0.827312i	$0.310131\pi$
$24$	0	0
$25$	−0.928470	−0.185694
$26$	−4.54018	−0.890402
$27$	0	0
$28$	3.63318	0.686607
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−2.73463	−0.491155	−0.245577	−	0.969377i	$-0.578978\pi$
$31$	−2.73463	−0.491155	−0.245577	+	0.969377i	$0.578978\pi$
$32$	−5.81644	−1.02821
$33$	0	0
$34$	2.65217	0.454844
$35$	−4.83240	−0.816825
$36$	0	0
$37$	1.86068	0.305894	0.152947	−	0.988234i	$-0.451124\pi$
$37$	1.86068	0.305894	0.152947	+	0.988234i	$0.451124\pi$
$38$	−3.00106	−0.486837
$39$	0	0
$40$	4.93180	0.779785
$41$	11.8282	1.84725	0.923624	−	0.383300i	$-0.125212\pi$
$41$	11.8282	1.84725	0.923624	+	0.383300i	$0.125212\pi$
$42$	0	0
$43$	3.70437	0.564911	0.282455	−	0.959280i	$-0.408851\pi$
$43$	3.70437	0.564911	0.282455	+	0.959280i	$0.408851\pi$
$44$	1.45317	0.219074
$45$	0	0
$46$	−3.74436	−0.552075
$47$	−3.23270	−0.471538	−0.235769	−	0.971809i	$-0.575761\pi$
$47$	−3.23270	−0.471538	−0.235769	+	0.971809i	$0.575761\pi$
$48$	0	0
$49$	−1.26453	−0.180648
$50$	0.645230	0.0912493
$51$	0	0
$52$	−9.91127	−1.37445
$53$	−2.26772	−0.311495	−0.155748	−	0.987797i	$-0.549779\pi$
$53$	−2.26772	−0.311495	−0.155748	+	0.987797i	$0.549779\pi$
$54$	0	0
$55$	−1.93283	−0.260622
$56$	−5.85343	−0.782198
$57$	0	0
$58$	0	0
$59$	9.30726	1.21170	0.605851	−	0.795578i	$-0.292833\pi$
$59$	9.30726	1.21170	0.605851	+	0.795578i	$0.292833\pi$
$60$	0	0
$61$	−3.13852	−0.401847	−0.200923	−	0.979607i	$-0.564394\pi$
$61$	−3.13852	−0.401847	−0.200923	+	0.979607i	$0.564394\pi$
$62$	1.90040	0.241351
$63$	0	0
$64$	1.37088	0.171360
$65$	13.1827	1.63512
$66$	0	0
$67$	12.1012	1.47840	0.739200	−	0.673487i	$-0.235204\pi$
$67$	12.1012	1.47840	0.739200	+	0.673487i	$0.235204\pi$
$68$	5.78973	0.702108
$69$	0	0
$70$	3.35822	0.401384
$71$	4.60343	0.546327	0.273163	−	0.961968i	$-0.411930\pi$
$71$	4.60343	0.546327	0.273163	+	0.961968i	$0.411930\pi$
$72$	0	0
$73$	−9.09135	−1.06406	−0.532031	−	0.846725i	$-0.678571\pi$
$73$	−9.09135	−1.06406	−0.532031	+	0.846725i	$0.678571\pi$
$74$	−1.29306	−0.150315
$75$	0	0
$76$	−6.55136	−0.751492
$77$	2.29403	0.261429
$78$	0	0
$79$	−4.28781	−0.482416	−0.241208	−	0.970473i	$-0.577544\pi$
$79$	−4.28781	−0.482416	−0.241208	+	0.970473i	$0.577544\pi$
$80$	2.69496	0.301306
$81$	0	0
$82$	−8.21984	−0.907730
$83$	−16.1813	−1.77612	−0.888062	−	0.459725i	$-0.847948\pi$
$83$	−16.1813	−1.77612	−0.888062	+	0.459725i	$0.847948\pi$
$84$	0	0
$85$	−7.70078	−0.835266
$86$	−2.57431	−0.277595
$87$	0	0
$88$	−2.34121	−0.249574
$89$	15.4430	1.63696	0.818478	−	0.574538i	$-0.194818\pi$
$89$	15.4430	1.63696	0.818478	+	0.574538i	$0.194818\pi$
$90$	0	0
$91$	−15.6463	−1.64017
$92$	−8.17399	−0.852197
$93$	0	0
$94$	2.24653	0.231712
$95$	8.71380	0.894017
$96$	0	0
$97$	−15.7632	−1.60051	−0.800253	−	0.599662i	$-0.795302\pi$
$97$	−15.7632	−1.60051	−0.800253	+	0.599662i	$0.795302\pi$
$98$	0.878774	0.0887696
$99$	0	0
$100$	1.40855	0.140855
$101$	9.25649	0.921055	0.460528	−	0.887645i	$-0.347660\pi$
$101$	9.25649	0.921055	0.460528	+	0.887645i	$0.347660\pi$
$102$	0	0
$103$	10.5946	1.04391	0.521956	−	0.852972i	$-0.325202\pi$
$103$	10.5946	1.04391	0.521956	+	0.852972i	$0.325202\pi$
$104$	15.9681	1.56580
$105$	0	0
$106$	1.57593	0.153068
$107$	0.125704	0.0121523	0.00607614	−	0.999982i	$-0.498066\pi$
$107$	0.125704	0.0121523	0.00607614	+	0.999982i	$0.498066\pi$
$108$	0	0
$109$	−9.64659	−0.923976	−0.461988	−	0.886886i	$-0.652864\pi$
$109$	−9.64659	−0.923976	−0.461988	+	0.886886i	$0.652864\pi$
$110$	1.34320	0.128069
$111$	0	0
$112$	−3.19859	−0.302238
$113$	3.10847	0.292420	0.146210	−	0.989254i	$-0.453292\pi$
$113$	3.10847	0.292420	0.146210	+	0.989254i	$0.453292\pi$
$114$	0	0
$115$	10.8720	1.01382
$116$	0	0
$117$	0	0
$118$	−6.46797	−0.595425
$119$	9.13987	0.837850
$120$	0	0
$121$	−10.0825	−0.916587
$122$	2.18108	0.197466
$123$	0	0
$124$	4.14861	0.372556
$125$	−11.9625	−1.06996
$126$	0	0
$127$	3.92002	0.347846	0.173923	−	0.984759i	$-0.444356\pi$
$127$	3.92002	0.347846	0.173923	+	0.984759i	$0.444356\pi$
$128$	10.6802	0.944005
$129$	0	0
$130$	−9.16118	−0.803489
$131$	−16.4949	−1.44117	−0.720585	−	0.693367i	$-0.756127\pi$
$131$	−16.4949	−1.44117	−0.720585	+	0.693367i	$0.756127\pi$
$132$	0	0
$133$	−10.3422	−0.896782
$134$	−8.40960	−0.726479
$135$	0	0
$136$	−9.32786	−0.799857
$137$	−3.04505	−0.260156	−0.130078	−	0.991504i	$-0.541523\pi$
$137$	−3.04505	−0.260156	−0.130078	+	0.991504i	$0.541523\pi$
$138$	0	0
$139$	11.2805	0.956803	0.478401	−	0.878141i	$-0.341216\pi$
$139$	11.2805	0.956803	0.478401	+	0.878141i	$0.341216\pi$
$140$	7.33105	0.619587
$141$	0	0
$142$	−3.19910	−0.268463
$143$	−6.25807	−0.523326
$144$	0	0
$145$	0	0
$146$	6.31793	0.522875
$147$	0	0
$148$	−2.82277	−0.232030
$149$	−4.89777	−0.401241	−0.200621	−	0.979669i	$-0.564296\pi$
$149$	−4.89777	−0.401241	−0.200621	+	0.979669i	$0.564296\pi$
$150$	0	0
$151$	17.0267	1.38561	0.692805	−	0.721125i	$-0.256375\pi$
$151$	17.0267	1.38561	0.692805	+	0.721125i	$0.256375\pi$
$152$	10.5549	0.856117
$153$	0	0
$154$	−1.59421	−0.128465
$155$	−5.51795	−0.443213
$156$	0	0
$157$	−20.3897	−1.62728	−0.813639	−	0.581371i	$-0.802517\pi$
$157$	−20.3897	−1.62728	−0.813639	+	0.581371i	$0.802517\pi$
$158$	2.97976	0.237057
$159$	0	0
$160$	−11.7364	−0.927846
$161$	−12.9037	−1.01696
$162$	0	0
$163$	−7.63026	−0.597648	−0.298824	−	0.954308i	$-0.596594\pi$
$163$	−7.63026	−0.597648	−0.298824	+	0.954308i	$0.596594\pi$
$164$	−17.9440	−1.40119
$165$	0	0
$166$	11.2450	0.872779
$167$	21.7308	1.68158	0.840791	−	0.541360i	$-0.182090\pi$
$167$	21.7308	1.68158	0.840791	+	0.541360i	$0.182090\pi$
$168$	0	0
$169$	29.6828	2.28329
$170$	5.35157	0.410446
$171$	0	0
$172$	−5.61975	−0.428502
$173$	−4.01358	−0.305147	−0.152573	−	0.988292i	$-0.548756\pi$
$173$	−4.01358	−0.305147	−0.152573	+	0.988292i	$0.548756\pi$
$174$	0	0
$175$	2.22358	0.168087
$176$	−1.27935	−0.0964344
$177$	0	0
$178$	−10.7319	−0.804393
$179$	3.86773	0.289088	0.144544	−	0.989498i	$-0.453829\pi$
$179$	3.86773	0.289088	0.144544	+	0.989498i	$0.453829\pi$
$180$	0	0
$181$	16.6298	1.23608	0.618042	−	0.786145i	$-0.287926\pi$
$181$	16.6298	1.23608	0.618042	+	0.786145i	$0.287926\pi$
$182$	10.8732	0.805975
$183$	0	0
$184$	13.1691	0.970841
$185$	3.75449	0.276036
$186$	0	0
$187$	3.65569	0.267331
$188$	4.90421	0.357676
$189$	0	0
$190$	−6.05555	−0.439316
$191$	−2.64112	−0.191105	−0.0955525	−	0.995424i	$-0.530462\pi$
$191$	−2.64112	−0.191105	−0.0955525	+	0.995424i	$0.530462\pi$
$192$	0	0
$193$	−6.54111	−0.470839	−0.235420	−	0.971894i	$-0.575646\pi$
$193$	−6.54111	−0.470839	−0.235420	+	0.971894i	$0.575646\pi$
$194$	10.9544	0.786482
$195$	0	0
$196$	1.91838	0.137027
$197$	9.82434	0.699956	0.349978	−	0.936758i	$-0.386189\pi$
$197$	9.82434	0.699956	0.349978	+	0.936758i	$0.386189\pi$
$198$	0	0
$199$	0.524712	0.0371958	0.0185979	−	0.999827i	$-0.494080\pi$
$199$	0.524712	0.0371958	0.0185979	+	0.999827i	$0.494080\pi$
$200$	−2.26931	−0.160465
$201$	0	0
$202$	−6.43269	−0.452603
$203$	0	0
$204$	0	0
$205$	23.8669	1.66694
$206$	−7.36256	−0.512974
$207$	0	0
$208$	8.72570	0.605019
$209$	−4.13659	−0.286134
$210$	0	0
$211$	−16.5476	−1.13919	−0.569593	−	0.821927i	$-0.692899\pi$
$211$	−16.5476	−1.13919	−0.569593	+	0.821927i	$0.692899\pi$
$212$	3.44027	0.236279
$213$	0	0
$214$	−0.0873567	−0.00597158
$215$	7.47469	0.509769
$216$	0	0
$217$	6.54913	0.444584
$218$	6.70379	0.454038
$219$	0	0
$220$	2.93222	0.197690
$221$	−24.9334	−1.67720
$222$	0	0
$223$	−11.7450	−0.786504	−0.393252	−	0.919431i	$-0.628650\pi$
$223$	−11.7450	−0.786504	−0.393252	+	0.919431i	$0.628650\pi$
$224$	13.9297	0.930716
$225$	0	0
$226$	−2.16019	−0.143694
$227$	21.3071	1.41420	0.707100	−	0.707114i	$-0.250003\pi$
$227$	21.3071	1.41420	0.707100	+	0.707114i	$0.250003\pi$
$228$	0	0
$229$	−0.183313	−0.0121137	−0.00605683	−	0.999982i	$-0.501928\pi$
$229$	−0.183313	−0.0121137	−0.00605683	+	0.999982i	$0.501928\pi$
$230$	−7.55538	−0.498187
$231$	0	0
$232$	0	0
$233$	23.8810	1.56449	0.782247	−	0.622969i	$-0.214074\pi$
$233$	23.8810	1.56449	0.782247	+	0.622969i	$0.214074\pi$
$234$	0	0
$235$	−6.52296	−0.425511
$236$	−14.1197	−0.919113
$237$	0	0
$238$	−6.35165	−0.411716
$239$	8.40937	0.543957	0.271979	−	0.962303i	$-0.412322\pi$
$239$	8.40937	0.543957	0.271979	+	0.962303i	$0.412322\pi$
$240$	0	0
$241$	−0.538363	−0.0346790	−0.0173395	−	0.999850i	$-0.505520\pi$
$241$	−0.538363	−0.0346790	−0.0173395	+	0.999850i	$0.505520\pi$
$242$	7.00668	0.450407
$243$	0	0
$244$	4.76133	0.304813
$245$	−2.55158	−0.163015
$246$	0	0
$247$	28.2134	1.79517
$248$	−6.68383	−0.424424
$249$	0	0
$250$	8.31319	0.525772
$251$	−1.71008	−0.107939	−0.0539695	−	0.998543i	$-0.517187\pi$
$251$	−1.71008	−0.107939	−0.0539695	+	0.998543i	$0.517187\pi$
$252$	0	0
$253$	−5.16114	−0.324478
$254$	−2.72417	−0.170930
$255$	0	0
$256$	−10.1638	−0.635240
$257$	18.0843	1.12807	0.564034	−	0.825752i	$-0.309249\pi$
$257$	18.0843	1.12807	0.564034	+	0.825752i	$0.309249\pi$
$258$	0	0
$259$	−4.45612	−0.276890
$260$	−19.9990	−1.24028
$261$	0	0
$262$	11.4630	0.708185
$263$	0.965486	0.0595344	0.0297672	−	0.999557i	$-0.490523\pi$
$263$	0.965486	0.0595344	0.0297672	+	0.999557i	$0.490523\pi$
$264$	0	0
$265$	−4.57581	−0.281090
$266$	7.18719	0.440675
$267$	0	0
$268$	−18.3583	−1.12141
$269$	9.70803	0.591909	0.295955	−	0.955202i	$-0.404362\pi$
$269$	9.70803	0.591909	0.295955	+	0.955202i	$0.404362\pi$
$270$	0	0
$271$	−16.9509	−1.02969	−0.514845	−	0.857283i	$-0.672151\pi$
$271$	−16.9509	−1.02969	−0.514845	+	0.857283i	$0.672151\pi$
$272$	−5.09718	−0.309062
$273$	0	0
$274$	2.11612	0.127840
$275$	0.889370	0.0536310
$276$	0	0
$277$	−22.0209	−1.32311	−0.661554	−	0.749897i	$-0.730103\pi$
$277$	−22.0209	−1.32311	−0.661554	+	0.749897i	$0.730103\pi$
$278$	−7.83928	−0.470169
$279$	0	0
$280$	−11.8111	−0.705847
$281$	−10.7284	−0.640003	−0.320001	−	0.947417i	$-0.603683\pi$
$281$	−10.7284	−0.640003	−0.320001	+	0.947417i	$0.603683\pi$
$282$	0	0
$283$	15.2187	0.904656	0.452328	−	0.891852i	$-0.350594\pi$
$283$	15.2187	0.904656	0.452328	+	0.891852i	$0.350594\pi$
$284$	−6.98368	−0.414405
$285$	0	0
$286$	4.34898	0.257160
$287$	−28.3271	−1.67209
$288$	0	0
$289$	−2.43497	−0.143234
$290$	0	0
$291$	0	0
$292$	13.7921	0.807123
$293$	23.1861	1.35455	0.677274	−	0.735731i	$-0.263161\pi$
$293$	23.1861	1.35455	0.677274	+	0.735731i	$0.263161\pi$
$294$	0	0
$295$	18.7802	1.09343
$296$	4.54777	0.264334
$297$	0	0
$298$	3.40365	0.197168
$299$	35.2012	2.03574
$300$	0	0
$301$	−8.87153	−0.511346
$302$	−11.8325	−0.680883
$303$	0	0
$304$	5.76770	0.330800
$305$	−6.33292	−0.362622
$306$	0	0
$307$	18.8555	1.07614	0.538070	−	0.842900i	$-0.319154\pi$
$307$	18.8555	1.07614	0.538070	+	0.842900i	$0.319154\pi$
$308$	−3.48018	−0.198302
$309$	0	0
$310$	3.83464	0.217793
$311$	27.1467	1.53935	0.769675	−	0.638436i	$-0.220418\pi$
$311$	27.1467	1.53935	0.769675	+	0.638436i	$0.220418\pi$
$312$	0	0
$313$	7.41189	0.418945	0.209472	−	0.977815i	$-0.432825\pi$
$313$	7.41189	0.418945	0.209472	+	0.977815i	$0.432825\pi$
$314$	14.1696	0.799637
$315$	0	0
$316$	6.50487	0.365927
$317$	2.23194	0.125358	0.0626790	−	0.998034i	$-0.480036\pi$
$317$	2.23194	0.125358	0.0626790	+	0.998034i	$0.480036\pi$
$318$	0	0
$319$	0	0
$320$	2.76617	0.154633
$321$	0	0
$322$	8.96730	0.499728
$323$	−16.4810	−0.917029
$324$	0	0
$325$	−6.06589	−0.336475
$326$	5.30256	0.293682
$327$	0	0
$328$	28.9097	1.59627
$329$	7.74194	0.426827
$330$	0	0
$331$	23.2235	1.27648	0.638241	−	0.769837i	$-0.279663\pi$
$331$	23.2235	1.27648	0.638241	+	0.769837i	$0.279663\pi$
$332$	24.5479	1.34724
$333$	0	0
$334$	−15.1016	−0.826322
$335$	24.4179	1.33409
$336$	0	0
$337$	−27.0723	−1.47472	−0.737362	−	0.675498i	$-0.763929\pi$
$337$	−27.0723	−1.47472	−0.737362	+	0.675498i	$0.763929\pi$
$338$	−20.6277	−1.12200
$339$	0	0
$340$	11.6825	0.633575
$341$	2.61947	0.141852
$342$	0	0
$343$	19.7926	1.06870
$344$	9.05400	0.488159
$345$	0	0
$346$	2.78919	0.149948
$347$	33.0378	1.77356	0.886780	−	0.462191i	$-0.152937\pi$
$347$	33.0378	1.77356	0.886780	+	0.462191i	$0.152937\pi$
$348$	0	0
$349$	6.36434	0.340675	0.170338	−	0.985386i	$-0.445514\pi$
$349$	6.36434	0.340675	0.170338	+	0.985386i	$0.445514\pi$
$350$	−1.54525	−0.0825971
$351$	0	0
$352$	5.57149	0.296961
$353$	−17.5725	−0.935292	−0.467646	−	0.883916i	$-0.654898\pi$
$353$	−17.5725	−0.935292	−0.467646	+	0.883916i	$0.654898\pi$
$354$	0	0
$355$	9.28882	0.492999
$356$	−23.4280	−1.24168
$357$	0	0
$358$	−2.68783	−0.142056
$359$	7.81990	0.412719	0.206359	−	0.978476i	$-0.433838\pi$
$359$	7.81990	0.412719	0.206359	+	0.978476i	$0.433838\pi$
$360$	0	0
$361$	−0.350934	−0.0184702
$362$	−11.5567	−0.607407
$363$	0	0
$364$	23.7363	1.24412
$365$	−18.3446	−0.960198
$366$	0	0
$367$	−29.1659	−1.52245	−0.761224	−	0.648489i	$-0.775401\pi$
$367$	−29.1659	−1.52245	−0.761224	+	0.648489i	$0.775401\pi$
$368$	7.19623	0.375130
$369$	0	0
$370$	−2.60914	−0.135643
$371$	5.43093	0.281960
$372$	0	0
$373$	−5.85615	−0.303220	−0.151610	−	0.988440i	$-0.548446\pi$
$373$	−5.85615	−0.303220	−0.151610	+	0.988440i	$0.548446\pi$
$374$	−2.54048	−0.131365
$375$	0	0
$376$	−7.90118	−0.407472
$377$	0	0
$378$	0	0
$379$	2.31759	0.119046	0.0595232	−	0.998227i	$-0.481042\pi$
$379$	2.31759	0.119046	0.0595232	+	0.998227i	$0.481042\pi$
$380$	−13.2194	−0.678139
$381$	0	0
$382$	1.83542	0.0939082
$383$	18.2436	0.932206	0.466103	−	0.884730i	$-0.345658\pi$
$383$	18.2436	0.932206	0.466103	+	0.884730i	$0.345658\pi$
$384$	0	0
$385$	4.62890	0.235910
$386$	4.54567	0.231368
$387$	0	0
$388$	23.9137	1.21403
$389$	29.6576	1.50370	0.751850	−	0.659335i	$-0.229162\pi$
$389$	29.6576	1.50370	0.751850	+	0.659335i	$0.229162\pi$
$390$	0	0
$391$	−20.5630	−1.03992
$392$	−3.09070	−0.156104
$393$	0	0
$394$	−6.82732	−0.343955
$395$	−8.65195	−0.435327
$396$	0	0
$397$	2.03228	0.101997	0.0509985	−	0.998699i	$-0.483760\pi$
$397$	2.03228	0.101997	0.0509985	+	0.998699i	$0.483760\pi$
$398$	−0.364643	−0.0182779
$399$	0	0
$400$	−1.24006	−0.0620029
$401$	0.142599	0.00712106	0.00356053	−	0.999994i	$-0.498867\pi$
$401$	0.142599	0.00712106	0.00356053	+	0.999994i	$0.498867\pi$
$402$	0	0
$403$	−17.8659	−0.889965
$404$	−14.0427	−0.698648
$405$	0	0
$406$	0	0
$407$	−1.78232	−0.0883465
$408$	0	0
$409$	20.7433	1.02569	0.512845	−	0.858481i	$-0.328592\pi$
$409$	20.7433	1.02569	0.512845	+	0.858481i	$0.328592\pi$
$410$	−16.5860	−0.819125
$411$	0	0
$412$	−16.0726	−0.791839
$413$	−22.2898	−1.09681
$414$	0	0
$415$	−32.6506	−1.60275
$416$	−38.0000	−1.86310
$417$	0	0
$418$	2.87468	0.140605
$419$	−6.55154	−0.320064	−0.160032	−	0.987112i	$-0.551160\pi$
$419$	−6.55154	−0.320064	−0.160032	+	0.987112i	$0.551160\pi$
$420$	0	0
$421$	−8.62850	−0.420527	−0.210264	−	0.977645i	$-0.567432\pi$
$421$	−8.62850	−0.420527	−0.210264	+	0.977645i	$0.567432\pi$
$422$	11.4996	0.559791
$423$	0	0
$424$	−5.54263	−0.269174
$425$	3.54343	0.171882
$426$	0	0
$427$	7.51640	0.363744
$428$	−0.190701	−0.00921788
$429$	0	0
$430$	−5.19445	−0.250499
$431$	−31.5733	−1.52083	−0.760417	−	0.649435i	$-0.775006\pi$
$431$	−31.5733	−1.52083	−0.760417	+	0.649435i	$0.775006\pi$
$432$	0	0
$433$	3.15030	0.151394	0.0756969	−	0.997131i	$-0.475882\pi$
$433$	3.15030	0.151394	0.0756969	+	0.997131i	$0.475882\pi$
$434$	−4.55124	−0.218467
$435$	0	0
$436$	14.6345	0.700863
$437$	23.2680	1.11306
$438$	0	0
$439$	22.8086	1.08859	0.544296	−	0.838893i	$-0.316797\pi$
$439$	22.8086	1.08859	0.544296	+	0.838893i	$0.316797\pi$
$440$	−4.72410	−0.225213
$441$	0	0
$442$	17.3272	0.824171
$443$	19.1260	0.908705	0.454353	−	0.890822i	$-0.349871\pi$
$443$	19.1260	0.908705	0.454353	+	0.890822i	$0.349871\pi$
$444$	0	0
$445$	31.1610	1.47717
$446$	8.16206	0.386485
$447$	0	0
$448$	−3.28310	−0.155112
$449$	−39.7898	−1.87780	−0.938898	−	0.344196i	$-0.888152\pi$
$449$	−39.7898	−1.87780	−0.938898	+	0.344196i	$0.888152\pi$
$450$	0	0
$451$	−11.3300	−0.533511
$452$	−4.71573	−0.221809
$453$	0	0
$454$	−14.8071	−0.694932
$455$	−31.5711	−1.48008
$456$	0	0
$457$	29.2169	1.36671	0.683355	−	0.730086i	$-0.260520\pi$
$457$	29.2169	1.36671	0.683355	+	0.730086i	$0.260520\pi$
$458$	0.127391	0.00595260
$459$	0	0
$460$	−16.4935	−0.769013
$461$	0.982645	0.0457663	0.0228832	−	0.999738i	$-0.492715\pi$
$461$	0.982645	0.0457663	0.0228832	+	0.999738i	$0.492715\pi$
$462$	0	0
$463$	−10.1420	−0.471340	−0.235670	−	0.971833i	$-0.575728\pi$
$463$	−10.1420	−0.471340	−0.235670	+	0.971833i	$0.575728\pi$
$464$	0	0
$465$	0	0
$466$	−16.5958	−0.768785
$467$	17.9779	0.831917	0.415959	−	0.909384i	$-0.363446\pi$
$467$	17.9779	0.831917	0.415959	+	0.909384i	$0.363446\pi$
$468$	0	0
$469$	−28.9810	−1.33822
$470$	4.53305	0.209094
$471$	0	0
$472$	22.7483	1.04707
$473$	−3.54837	−0.163154
$474$	0	0
$475$	−4.00956	−0.183971
$476$	−13.8657	−0.635535
$477$	0	0
$478$	−5.84400	−0.267298
$479$	20.5926	0.940902	0.470451	−	0.882426i	$-0.344091\pi$
$479$	20.5926	0.940902	0.470451	+	0.882426i	$0.344091\pi$
$480$	0	0
$481$	12.1562	0.554276
$482$	0.374129	0.0170411
$483$	0	0
$484$	15.2957	0.695259
$485$	−31.8070	−1.44428
$486$	0	0
$487$	15.9915	0.724643	0.362321	−	0.932053i	$-0.381984\pi$
$487$	15.9915	0.724643	0.362321	+	0.932053i	$0.381984\pi$
$488$	−7.67099	−0.347250
$489$	0	0
$490$	1.77319	0.0801047
$491$	−10.2005	−0.460341	−0.230171	−	0.973150i	$-0.573928\pi$
$491$	−10.2005	−0.460341	−0.230171	+	0.973150i	$0.573928\pi$
$492$	0	0
$493$	0	0
$494$	−19.6066	−0.882141
$495$	0	0
$496$	−3.65236	−0.163996
$497$	−11.0247	−0.494525
$498$	0	0
$499$	−6.15200	−0.275401	−0.137701	−	0.990474i	$-0.543971\pi$
$499$	−6.15200	−0.275401	−0.137701	+	0.990474i	$0.543971\pi$
$500$	18.1478	0.811595
$501$	0	0
$502$	1.18840	0.0530408
$503$	−20.2354	−0.902253	−0.451126	−	0.892460i	$-0.648978\pi$
$503$	−20.2354	−0.902253	−0.451126	+	0.892460i	$0.648978\pi$
$504$	0	0
$505$	18.6778	0.831150
$506$	3.58667	0.159447
$507$	0	0
$508$	−5.94691	−0.263851
$509$	4.05922	0.179922	0.0899608	−	0.995945i	$-0.471326\pi$
$509$	4.05922	0.179922	0.0899608	+	0.995945i	$0.471326\pi$
$510$	0	0
$511$	21.7727	0.963168
$512$	−14.2971	−0.631851
$513$	0	0
$514$	−12.5675	−0.554328
$515$	21.3777	0.942015
$516$	0	0
$517$	3.09656	0.136187
$518$	3.09673	0.136062
$519$	0	0
$520$	32.2204	1.41296
$521$	−10.9066	−0.477828	−0.238914	−	0.971041i	$-0.576791\pi$
$521$	−10.9066	−0.477828	−0.238914	+	0.971041i	$0.576791\pi$
$522$	0	0
$523$	3.65147	0.159668	0.0798339	−	0.996808i	$-0.474561\pi$
$523$	3.65147	0.159668	0.0798339	+	0.996808i	$0.474561\pi$
$524$	25.0238	1.09317
$525$	0	0
$526$	−0.670954	−0.0292550
$527$	10.4365	0.454621
$528$	0	0
$529$	6.03100	0.262217
$530$	3.17991	0.138126
$531$	0	0
$532$	15.6897	0.680237
$533$	77.2758	3.34719
$534$	0	0
$535$	0.253646	0.0109661
$536$	29.5771	1.27754
$537$	0	0
$538$	−6.74649	−0.290862
$539$	1.21128	0.0521736
$540$	0	0
$541$	33.9307	1.45879	0.729397	−	0.684091i	$-0.239801\pi$
$541$	33.9307	1.45879	0.729397	+	0.684091i	$0.239801\pi$
$542$	11.7798	0.505986
$543$	0	0
$544$	22.1979	0.951729
$545$	−19.4649	−0.833785
$546$	0	0
$547$	19.6642	0.840779	0.420389	−	0.907344i	$-0.361893\pi$
$547$	19.6642	0.840779	0.420389	+	0.907344i	$0.361893\pi$
$548$	4.61953	0.197336
$549$	0	0
$550$	−0.618057	−0.0263541
$551$	0	0
$552$	0	0
$553$	10.2688	0.436674
$554$	15.3032	0.650170
$555$	0	0
$556$	−17.1133	−0.725764
$557$	38.2463	1.62055	0.810275	−	0.586050i	$-0.199318\pi$
$557$	38.2463	1.62055	0.810275	+	0.586050i	$0.199318\pi$
$558$	0	0
$559$	24.2014	1.02361
$560$	−6.45412	−0.272737
$561$	0	0
$562$	7.45558	0.314495
$563$	−23.4745	−0.989330	−0.494665	−	0.869084i	$-0.664709\pi$
$563$	−23.4745	−0.989330	−0.494665	+	0.869084i	$0.664709\pi$
$564$	0	0
$565$	6.27228	0.263877
$566$	−10.5760	−0.444544
$567$	0	0
$568$	11.2514	0.472100
$569$	−3.40949	−0.142933	−0.0714665	−	0.997443i	$-0.522768\pi$
$569$	−3.40949	−0.142933	−0.0714665	+	0.997443i	$0.522768\pi$
$570$	0	0
$571$	23.2044	0.971073	0.485536	−	0.874216i	$-0.338624\pi$
$571$	23.2044	0.971073	0.485536	+	0.874216i	$0.338624\pi$
$572$	9.49387	0.396959
$573$	0	0
$574$	19.6856	0.821660
$575$	−5.00264	−0.208624
$576$	0	0
$577$	0.293784	0.0122304	0.00611519	−	0.999981i	$-0.498053\pi$
$577$	0.293784	0.0122304	0.00611519	+	0.999981i	$0.498053\pi$
$578$	1.69216	0.0703844
$579$	0	0
$580$	0	0
$581$	38.7522	1.60771
$582$	0	0
$583$	2.17222	0.0899642
$584$	−22.2205	−0.919492
$585$	0	0
$586$	−16.1129	−0.665619
$587$	2.75456	0.113693	0.0568465	−	0.998383i	$-0.481895\pi$
$587$	2.75456	0.113693	0.0568465	+	0.998383i	$0.481895\pi$
$588$	0	0
$589$	−11.8094	−0.486598
$590$	−13.0511	−0.537305
$591$	0	0
$592$	2.48512	0.102138
$593$	20.7991	0.854116	0.427058	−	0.904224i	$-0.359550\pi$
$593$	20.7991	0.854116	0.427058	+	0.904224i	$0.359550\pi$
$594$	0	0
$595$	18.4425	0.756067
$596$	7.43022	0.304354
$597$	0	0
$598$	−24.4627	−1.00035
$599$	34.9339	1.42736	0.713681	−	0.700471i	$-0.247027\pi$
$599$	34.9339	1.42736	0.713681	+	0.700471i	$0.247027\pi$
$600$	0	0
$601$	−2.88126	−0.117529	−0.0587646	−	0.998272i	$-0.518716\pi$
$601$	−2.88126	−0.117529	−0.0587646	+	0.998272i	$0.518716\pi$
$602$	6.16517	0.251273
$603$	0	0
$604$	−25.8305	−1.05103
$605$	−20.3444	−0.827118
$606$	0	0
$607$	−22.2741	−0.904076	−0.452038	−	0.891999i	$-0.649303\pi$
$607$	−22.2741	−0.904076	−0.452038	+	0.891999i	$0.649303\pi$
$608$	−25.1180	−1.01867
$609$	0	0
$610$	4.40099	0.178191
$611$	−21.1199	−0.854420
$612$	0	0
$613$	1.67163	0.0675166	0.0337583	−	0.999430i	$-0.489252\pi$
$613$	1.67163	0.0675166	0.0337583	+	0.999430i	$0.489252\pi$
$614$	−13.1034	−0.528810
$615$	0	0
$616$	5.60693	0.225909
$617$	14.4007	0.579752	0.289876	−	0.957064i	$-0.406386\pi$
$617$	14.4007	0.579752	0.289876	+	0.957064i	$0.406386\pi$
$618$	0	0
$619$	2.74833	0.110465	0.0552324	−	0.998474i	$-0.482410\pi$
$619$	2.74833	0.110465	0.0552324	+	0.998474i	$0.482410\pi$
$620$	8.37107	0.336190
$621$	0	0
$622$	−18.8653	−0.756430
$623$	−36.9842	−1.48174
$624$	0	0
$625$	−19.4956	−0.779824
$626$	−5.15081	−0.205868
$627$	0	0
$628$	30.9324	1.23434
$629$	−7.10114	−0.283141
$630$	0	0
$631$	−9.07172	−0.361139	−0.180570	−	0.983562i	$-0.557794\pi$
$631$	−9.07172	−0.361139	−0.180570	+	0.983562i	$0.557794\pi$
$632$	−10.4800	−0.416872
$633$	0	0
$634$	−1.55106	−0.0616004
$635$	7.90983	0.313892
$636$	0	0
$637$	−8.26146	−0.327331
$638$	0	0
$639$	0	0
$640$	21.5505	0.851860
$641$	−11.3851	−0.449686	−0.224843	−	0.974395i	$-0.572187\pi$
$641$	−11.3851	−0.449686	−0.224843	+	0.974395i	$0.572187\pi$
$642$	0	0
$643$	−10.8211	−0.426742	−0.213371	−	0.976971i	$-0.568444\pi$
$643$	−10.8211	−0.426742	−0.213371	+	0.976971i	$0.568444\pi$
$644$	19.5757	0.771392
$645$	0	0
$646$	11.4533	0.450624
$647$	22.0900	0.868448	0.434224	−	0.900805i	$-0.357023\pi$
$647$	22.0900	0.868448	0.434224	+	0.900805i	$0.357023\pi$
$648$	0	0
$649$	−8.91530	−0.349956
$650$	4.21542	0.165342
$651$	0	0
$652$	11.5756	0.453334
$653$	19.3795	0.758377	0.379189	−	0.925319i	$-0.376203\pi$
$653$	19.3795	0.758377	0.379189	+	0.925319i	$0.376203\pi$
$654$	0	0
$655$	−33.2836	−1.30050
$656$	15.7976	0.616793
$657$	0	0
$658$	−5.38018	−0.209741
$659$	6.69003	0.260607	0.130303	−	0.991474i	$-0.458405\pi$
$659$	6.69003	0.260607	0.130303	+	0.991474i	$0.458405\pi$
$660$	0	0
$661$	−44.4774	−1.72997	−0.864985	−	0.501798i	$-0.832672\pi$
$661$	−44.4774	−1.72997	−0.864985	+	0.501798i	$0.832672\pi$
$662$	−16.1389	−0.627257
$663$	0	0
$664$	−39.5493	−1.53481
$665$	−20.8685	−0.809247
$666$	0	0
$667$	0	0
$668$	−32.9670	−1.27553
$669$	0	0
$670$	−16.9689	−0.655567
$671$	3.00635	0.116059
$672$	0	0
$673$	21.0755	0.812402	0.406201	−	0.913784i	$-0.366853\pi$
$673$	21.0755	0.812402	0.406201	+	0.913784i	$0.366853\pi$
$674$	18.8136	0.724673
$675$	0	0
$676$	−45.0306	−1.73194
$677$	3.58338	0.137720	0.0688602	−	0.997626i	$-0.478064\pi$
$677$	3.58338	0.137720	0.0688602	+	0.997626i	$0.478064\pi$
$678$	0	0
$679$	37.7509	1.44875
$680$	−18.8218	−0.721782
$681$	0	0
$682$	−1.82037	−0.0697056
$683$	13.4710	0.515454	0.257727	−	0.966218i	$-0.417027\pi$
$683$	13.4710	0.515454	0.257727	+	0.966218i	$0.417027\pi$
$684$	0	0
$685$	−6.14432	−0.234762
$686$	−13.7546	−0.525154
$687$	0	0
$688$	4.94753	0.188623
$689$	−14.8155	−0.564425
$690$	0	0
$691$	17.4813	0.665021	0.332511	−	0.943100i	$-0.392104\pi$
$691$	17.4813	0.665021	0.332511	+	0.943100i	$0.392104\pi$
$692$	6.08884	0.231463
$693$	0	0
$694$	−22.9592	−0.871520
$695$	22.7619	0.863408
$696$	0	0
$697$	−45.1412	−1.70984
$698$	−4.42283	−0.167406
$699$	0	0
$700$	−3.37330	−0.127499
$701$	29.5015	1.11426	0.557128	−	0.830427i	$-0.311903\pi$
$701$	29.5015	1.11426	0.557128	+	0.830427i	$0.311903\pi$
$702$	0	0
$703$	8.03528	0.303056
$704$	−1.31315	−0.0494911
$705$	0	0
$706$	12.2118	0.459598
$707$	−22.1682	−0.833722
$708$	0	0
$709$	24.6468	0.925630	0.462815	−	0.886455i	$-0.346839\pi$
$709$	24.6468	0.925630	0.462815	+	0.886455i	$0.346839\pi$
$710$	−6.45516	−0.242258
$711$	0	0
$712$	37.7449	1.41455
$713$	−14.7343	−0.551805
$714$	0	0
$715$	−12.6276	−0.472244
$716$	−5.86758	−0.219282
$717$	0	0
$718$	−5.43435	−0.202808
$719$	1.38271	0.0515665	0.0257833	−	0.999668i	$-0.491792\pi$
$719$	1.38271	0.0515665	0.0257833	+	0.999668i	$0.491792\pi$
$720$	0	0
$721$	−25.3727	−0.944929
$722$	0.243877	0.00907617
$723$	0	0
$724$	−25.2284	−0.937607
$725$	0	0
$726$	0	0
$727$	−37.7570	−1.40033	−0.700164	−	0.713982i	$-0.746890\pi$
$727$	−37.7570	−1.40033	−0.700164	+	0.713982i	$0.746890\pi$
$728$	−38.2417	−1.41733
$729$	0	0
$730$	12.7483	0.471837
$731$	−14.1374	−0.522891
$732$	0	0
$733$	32.5735	1.20313	0.601564	−	0.798824i	$-0.294544\pi$
$733$	32.5735	1.20313	0.601564	+	0.798824i	$0.294544\pi$
$734$	20.2685	0.748125
$735$	0	0
$736$	−31.3392	−1.15518
$737$	−11.5916	−0.426982
$738$	0	0
$739$	37.9628	1.39648	0.698241	−	0.715863i	$-0.253966\pi$
$739$	37.9628	1.39648	0.698241	+	0.715863i	$0.253966\pi$
$740$	−5.69579	−0.209381
$741$	0	0
$742$	−3.77416	−0.138554
$743$	−13.1194	−0.481306	−0.240653	−	0.970611i	$-0.577362\pi$
$743$	−13.1194	−0.481306	−0.240653	+	0.970611i	$0.577362\pi$
$744$	0	0
$745$	−9.88275	−0.362076
$746$	4.06966	0.149001
$747$	0	0
$748$	−5.54591	−0.202779
$749$	−0.301047	−0.0110000
$750$	0	0
$751$	28.1170	1.02601	0.513003	−	0.858387i	$-0.328533\pi$
$751$	28.1170	1.02601	0.513003	+	0.858387i	$0.328533\pi$
$752$	−4.31758	−0.157446
$753$	0	0
$754$	0	0
$755$	34.3565	1.25036
$756$	0	0
$757$	19.0677	0.693028	0.346514	−	0.938045i	$-0.387365\pi$
$757$	19.0677	0.693028	0.346514	+	0.938045i	$0.387365\pi$
$758$	−1.61058	−0.0584989
$759$	0	0
$760$	21.2977	0.772550
$761$	−8.84457	−0.320615	−0.160308	−	0.987067i	$-0.551249\pi$
$761$	−8.84457	−0.320615	−0.160308	+	0.987067i	$0.551249\pi$
$762$	0	0
$763$	23.1025	0.836365
$764$	4.00674	0.144959
$765$	0	0
$766$	−12.6782	−0.458082
$767$	60.8062	2.19559
$768$	0	0
$769$	14.4007	0.519303	0.259651	−	0.965702i	$-0.416392\pi$
$769$	14.4007	0.519303	0.259651	+	0.965702i	$0.416392\pi$
$770$	−3.21680	−0.115925
$771$	0	0
$772$	9.92326	0.357146
$773$	−5.71792	−0.205659	−0.102830	−	0.994699i	$-0.532790\pi$
$773$	−5.71792	−0.205659	−0.102830	+	0.994699i	$0.532790\pi$
$774$	0	0
$775$	2.53903	0.0912045
$776$	−38.5274	−1.38305
$777$	0	0
$778$	−20.6102	−0.738911
$779$	51.0794	1.83011
$780$	0	0
$781$	−4.40957	−0.157787
$782$	14.2900	0.511010
$783$	0	0
$784$	−1.68890	−0.0603180
$785$	−41.1425	−1.46844
$786$	0	0
$787$	38.7631	1.38176	0.690878	−	0.722972i	$-0.257224\pi$
$787$	38.7631	1.38176	0.690878	+	0.722972i	$0.257224\pi$
$788$	−14.9041	−0.530937
$789$	0	0
$790$	6.01258	0.213918
$791$	−7.44442	−0.264693
$792$	0	0
$793$	−20.5046	−0.728140
$794$	−1.41231	−0.0501209
$795$	0	0
$796$	−0.796020	−0.0282142
$797$	3.16247	0.112021	0.0560103	−	0.998430i	$-0.482162\pi$
$797$	3.16247	0.112021	0.0560103	+	0.998430i	$0.482162\pi$
$798$	0	0
$799$	12.3373	0.436464
$800$	5.40039	0.190933
$801$	0	0
$802$	−0.0990976	−0.00349926
$803$	8.70848	0.307316
$804$	0	0
$805$	−26.0372	−0.917690
$806$	12.4157	0.437325
$807$	0	0
$808$	22.6242	0.795916
$809$	11.8313	0.415965	0.207983	−	0.978133i	$-0.433310\pi$
$809$	11.8313	0.415965	0.207983	+	0.978133i	$0.433310\pi$
$810$	0	0
$811$	−11.2027	−0.393381	−0.196691	−	0.980466i	$-0.563019\pi$
$811$	−11.2027	−0.393381	−0.196691	+	0.980466i	$0.563019\pi$
$812$	0	0
$813$	0	0
$814$	1.23861	0.0434131
$815$	−15.3964	−0.539311
$816$	0	0
$817$	15.9972	0.559670
$818$	−14.4153	−0.504020
$819$	0	0
$820$	−36.2075	−1.26442
$821$	−14.8331	−0.517678	−0.258839	−	0.965920i	$-0.583340\pi$
$821$	−14.8331	−0.517678	−0.258839	+	0.965920i	$0.583340\pi$
$822$	0	0
$823$	−43.8191	−1.52744	−0.763719	−	0.645548i	$-0.776629\pi$
$823$	−43.8191	−1.52744	−0.763719	+	0.645548i	$0.776629\pi$
$824$	25.8946	0.902080
$825$	0	0
$826$	15.4900	0.538967
$827$	−7.62477	−0.265139	−0.132570	−	0.991174i	$-0.542323\pi$
$827$	−7.62477	−0.265139	−0.132570	+	0.991174i	$0.542323\pi$
$828$	0	0
$829$	20.4341	0.709707	0.354854	−	0.934922i	$-0.384531\pi$
$829$	20.4341	0.709707	0.354854	+	0.934922i	$0.384531\pi$
$830$	22.6901	0.787587
$831$	0	0
$832$	8.95624	0.310502
$833$	4.82599	0.167211
$834$	0	0
$835$	43.8485	1.51744
$836$	6.27546	0.217041
$837$	0	0
$838$	4.55292	0.157278
$839$	16.9391	0.584802	0.292401	−	0.956296i	$-0.405546\pi$
$839$	16.9391	0.584802	0.292401	+	0.956296i	$0.405546\pi$
$840$	0	0
$841$	0	0
$842$	5.99628	0.206645
$843$	0	0
$844$	25.1038	0.864107
$845$	59.8940	2.06042
$846$	0	0
$847$	24.1463	0.829677
$848$	−3.02875	−0.104008
$849$	0	0
$850$	−2.46247	−0.0844619
$851$	10.0254	0.343668
$852$	0	0
$853$	9.95085	0.340711	0.170355	−	0.985383i	$-0.445508\pi$
$853$	9.95085	0.340711	0.170355	+	0.985383i	$0.445508\pi$
$854$	−5.22343	−0.178742
$855$	0	0
$856$	0.307239	0.0105012
$857$	−32.4424	−1.10821	−0.554106	−	0.832446i	$-0.686940\pi$
$857$	−32.4424	−1.10821	−0.554106	+	0.832446i	$0.686940\pi$
$858$	0	0
$859$	51.9501	1.77252	0.886258	−	0.463192i	$-0.153296\pi$
$859$	51.9501	1.77252	0.886258	+	0.463192i	$0.153296\pi$
$860$	−11.3396	−0.386675
$861$	0	0
$862$	21.9415	0.747332
$863$	−20.8719	−0.710489	−0.355245	−	0.934773i	$-0.615602\pi$
$863$	−20.8719	−0.710489	−0.355245	+	0.934773i	$0.615602\pi$
$864$	0	0
$865$	−8.09862	−0.275361
$866$	−2.18926	−0.0743942
$867$	0	0
$868$	−9.93543	−0.337230
$869$	4.10724	0.139328
$870$	0	0
$871$	79.0597	2.67884
$872$	−23.5776	−0.798439
$873$	0	0
$874$	−16.1698	−0.546953
$875$	28.6488	0.968505
$876$	0	0
$877$	−15.0698	−0.508873	−0.254436	−	0.967090i	$-0.581890\pi$
$877$	−15.0698	−0.508873	−0.254436	+	0.967090i	$0.581890\pi$
$878$	−15.8505	−0.534930
$879$	0	0
$880$	−2.58147	−0.0870214
$881$	−15.0028	−0.505456	−0.252728	−	0.967537i	$-0.581328\pi$
$881$	−15.0028	−0.505456	−0.252728	+	0.967537i	$0.581328\pi$
$882$	0	0
$883$	−4.09571	−0.137832	−0.0689158	−	0.997622i	$-0.521954\pi$
$883$	−4.09571	−0.137832	−0.0689158	+	0.997622i	$0.521954\pi$
$884$	37.8255	1.27221
$885$	0	0
$886$	−13.2914	−0.446534
$887$	16.1647	0.542756	0.271378	−	0.962473i	$-0.412521\pi$
$887$	16.1647	0.542756	0.271378	+	0.962473i	$0.412521\pi$
$888$	0	0
$889$	−9.38799	−0.314863
$890$	−21.6549	−0.725875
$891$	0	0
$892$	17.8179	0.596587
$893$	−13.9603	−0.467163
$894$	0	0
$895$	7.80431	0.260869
$896$	−25.5778	−0.854495
$897$	0	0
$898$	27.6514	0.922741
$899$	0	0
$900$	0	0
$901$	8.65457	0.288325
$902$	7.87368	0.262165
$903$	0	0
$904$	7.59753	0.252690
$905$	33.5557	1.11543
$906$	0	0
$907$	−41.3224	−1.37209	−0.686044	−	0.727560i	$-0.740654\pi$
$907$	−41.3224	−1.37209	−0.686044	+	0.727560i	$0.740654\pi$
$908$	−32.3241	−1.07271
$909$	0	0
$910$	21.9400	0.727303
$911$	31.3566	1.03889	0.519446	−	0.854504i	$-0.326138\pi$
$911$	31.3566	1.03889	0.519446	+	0.854504i	$0.326138\pi$
$912$	0	0
$913$	15.4998	0.512969
$914$	−20.3040	−0.671596
$915$	0	0
$916$	0.278097	0.00918858
$917$	39.5035	1.30452
$918$	0	0
$919$	29.9401	0.987633	0.493817	−	0.869566i	$-0.335601\pi$
$919$	29.9401	0.987633	0.493817	+	0.869566i	$0.335601\pi$
$920$	26.5727	0.876077
$921$	0	0
$922$	−0.682878	−0.0224894
$923$	30.0752	0.989936
$924$	0	0
$925$	−1.72759	−0.0568028
$926$	7.04809	0.231615
$927$	0	0
$928$	0	0
$929$	1.38920	0.0455781	0.0227891	−	0.999740i	$-0.492745\pi$
$929$	1.38920	0.0455781	0.0227891	+	0.999740i	$0.492745\pi$
$930$	0	0
$931$	−5.46084	−0.178972
$932$	−36.2289	−1.18672
$933$	0	0
$934$	−12.4935	−0.408801
$935$	7.37647	0.241236
$936$	0	0
$937$	−7.08484	−0.231451	−0.115726	−	0.993281i	$-0.536919\pi$
$937$	−7.08484	−0.231451	−0.115726	+	0.993281i	$0.536919\pi$
$938$	20.1400	0.657595
$939$	0	0
$940$	9.89572	0.322763
$941$	−56.3655	−1.83746	−0.918731	−	0.394883i	$-0.870785\pi$
$941$	−56.3655	−1.83746	−0.918731	+	0.394883i	$0.870785\pi$
$942$	0	0
$943$	63.7306	2.07535
$944$	12.4307	0.404585
$945$	0	0
$946$	2.46590	0.0801732
$947$	23.7926	0.773155	0.386578	−	0.922257i	$-0.373657\pi$
$947$	23.7926	0.773155	0.386578	+	0.922257i	$0.373657\pi$
$948$	0	0
$949$	−59.3956	−1.92806
$950$	2.78640	0.0904027
$951$	0	0
$952$	22.3391	0.724015
$953$	−21.7614	−0.704920	−0.352460	−	0.935827i	$-0.614655\pi$
$953$	−21.7614	−0.704920	−0.352460	+	0.935827i	$0.614655\pi$
$954$	0	0
$955$	−5.32927	−0.172451
$956$	−12.7575	−0.412608
$957$	0	0
$958$	−14.3106	−0.462355
$959$	7.29255	0.235489
$960$	0	0
$961$	−23.5218	−0.758767
$962$	−8.44783	−0.272369
$963$	0	0
$964$	0.816729	0.0263051
$965$	−13.1987	−0.424880
$966$	0	0
$967$	−24.8725	−0.799846	−0.399923	−	0.916549i	$-0.630963\pi$
$967$	−24.8725	−0.799846	−0.399923	+	0.916549i	$0.630963\pi$
$968$	−24.6429	−0.792054
$969$	0	0
$970$	22.1039	0.709713
$971$	59.9657	1.92439	0.962196	−	0.272358i	$-0.0878036\pi$
$971$	59.9657	1.92439	0.962196	+	0.272358i	$0.0878036\pi$
$972$	0	0
$973$	−27.0156	−0.866080
$974$	−11.1131	−0.356086
$975$	0	0
$976$	−4.19179	−0.134176
$977$	46.1910	1.47778	0.738891	−	0.673825i	$-0.235350\pi$
$977$	46.1910	1.47778	0.738891	+	0.673825i	$0.235350\pi$
$978$	0	0
$979$	−14.7927	−0.472775
$980$	3.87090	0.123651
$981$	0	0
$982$	7.08871	0.226210
$983$	−0.667578	−0.0212924	−0.0106462	−	0.999943i	$-0.503389\pi$
$983$	−0.667578	−0.0212924	−0.0106462	+	0.999943i	$0.503389\pi$
$984$	0	0
$985$	19.8236	0.631632
$986$	0	0
$987$	0	0
$988$	−42.8014	−1.36169
$989$	19.9593	0.634669
$990$	0	0
$991$	−44.8159	−1.42362	−0.711811	−	0.702371i	$-0.752125\pi$
$991$	−44.8159	−1.42362	−0.711811	+	0.702371i	$0.752125\pi$
$992$	15.9058	0.505011
$993$	0	0
$994$	7.66148	0.243007
$995$	1.05877	0.0335651
$996$	0	0
$997$	11.1717	0.353812	0.176906	−	0.984228i	$-0.443391\pi$
$997$	11.1717	0.353812	0.176906	+	0.984228i	$0.443391\pi$
$998$	4.27526	0.135331
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bm.1.3		9
3.2	odd	2		2523.2.a.o.1.7		9
29.4	even	14		261.2.k.c.190.1		18
29.22	even	14		261.2.k.c.136.1		18
29.28	even	2		7569.2.a.bj.1.7		9
87.62	odd	14		87.2.g.a.16.3	✓	18
87.80	odd	14		87.2.g.a.49.3	yes	18
87.86	odd	2		2523.2.a.r.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.16.3	✓	18	87.62	odd	14
87.2.g.a.49.3	yes	18	87.80	odd	14
261.2.k.c.136.1		18	29.22	even	14
261.2.k.c.190.1		18	29.4	even	14
2523.2.a.o.1.7		9	3.2	odd	2
2523.2.a.r.1.3		9	87.86	odd	2
7569.2.a.bj.1.7		9	29.28	even	2
7569.2.a.bm.1.3		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)

\(f(q)\)	\(=\)	\(q-0.694939 q^{2} -1.51706 q^{4} +2.01780 q^{5} -2.39488 q^{7} +2.44414 q^{8} -1.40225 q^{10} -0.957887 q^{11} +6.53321 q^{13} +1.66430 q^{14} +1.33559 q^{16} -3.81642 q^{17} +4.31846 q^{19} -3.06113 q^{20} +0.665673 q^{22} +5.38804 q^{23} -0.928470 q^{25} -4.54018 q^{26} +3.63318 q^{28} -2.73463 q^{31} -5.81644 q^{32} +2.65217 q^{34} -4.83240 q^{35} +1.86068 q^{37} -3.00106 q^{38} +4.93180 q^{40} +11.8282 q^{41} +3.70437 q^{43} +1.45317 q^{44} -3.74436 q^{46} -3.23270 q^{47} -1.26453 q^{49} +0.645230 q^{50} -9.91127 q^{52} -2.26772 q^{53} -1.93283 q^{55} -5.85343 q^{56} +9.30726 q^{59} -3.13852 q^{61} +1.90040 q^{62} +1.37088 q^{64} +13.1827 q^{65} +12.1012 q^{67} +5.78973 q^{68} +3.35822 q^{70} +4.60343 q^{71} -9.09135 q^{73} -1.29306 q^{74} -6.55136 q^{76} +2.29403 q^{77} -4.28781 q^{79} +2.69496 q^{80} -8.21984 q^{82} -16.1813 q^{83} -7.70078 q^{85} -2.57431 q^{86} -2.34121 q^{88} +15.4430 q^{89} -15.6463 q^{91} -8.17399 q^{92} +2.24653 q^{94} +8.71380 q^{95} -15.7632 q^{97} +0.878774 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * q^98

Introduction

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