LMFDB - Embedded newform 4527.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4527,2,Mod(1,4527)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4527, 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("4527.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4527 = 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4527.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:	$36.1482769950$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.510671$ of defining polynomial
Character	$\chi$	$=$	4527.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.15783 q^{2} +2.65622 q^{4} +2.23445 q^{5} -3.60329 q^{7} -1.41602 q^{8} +O(q^{10})$	\(q-2.15783 q^{2} +2.65622 q^{4} +2.23445 q^{5} -3.60329 q^{7} -1.41602 q^{8} -4.82156 q^{10} +1.50306 q^{11} +0.00459548 q^{13} +7.77529 q^{14} -2.25692 q^{16} -1.11927 q^{17} +2.49963 q^{19} +5.93520 q^{20} -3.24334 q^{22} -1.72480 q^{23} -0.00724425 q^{25} -0.00991627 q^{26} -9.57116 q^{28} +0.572061 q^{29} +3.25375 q^{31} +7.70209 q^{32} +2.41520 q^{34} -8.05137 q^{35} -6.13809 q^{37} -5.39377 q^{38} -3.16402 q^{40} +5.89549 q^{41} -7.66800 q^{43} +3.99246 q^{44} +3.72183 q^{46} -6.44840 q^{47} +5.98373 q^{49} +0.0156318 q^{50} +0.0122066 q^{52} +7.82500 q^{53} +3.35851 q^{55} +5.10234 q^{56} -1.23441 q^{58} -0.253382 q^{59} +7.08244 q^{61} -7.02103 q^{62} -12.1059 q^{64} +0.0102684 q^{65} +6.16046 q^{67} -2.97304 q^{68} +17.3735 q^{70} +9.32311 q^{71} -15.2654 q^{73} +13.2450 q^{74} +6.63958 q^{76} -5.41596 q^{77} -1.99661 q^{79} -5.04297 q^{80} -12.7215 q^{82} -6.91660 q^{83} -2.50096 q^{85} +16.5462 q^{86} -2.12836 q^{88} -3.37422 q^{89} -0.0165589 q^{91} -4.58146 q^{92} +13.9146 q^{94} +5.58529 q^{95} -14.8842 q^{97} -12.9119 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.15783	−1.52582	−0.762908	−	0.646507i	$-0.776229\pi$
$2$	−2.15783	−1.52582	−0.762908	+	0.646507i	$0.776229\pi$
$3$	0	0
$3$	0	0
$4$	2.65622	1.32811
$5$	2.23445	0.999275	0.499638	−	0.866235i	$-0.333466\pi$
$5$	2.23445	0.999275	0.499638	+	0.866235i	$0.333466\pi$
$6$	0	0
$7$	−3.60329	−1.36192	−0.680959	−	0.732322i	$-0.738437\pi$
$7$	−3.60329	−1.36192	−0.680959	+	0.732322i	$0.738437\pi$
$8$	−1.41602	−0.500639
$9$	0	0
$10$	−4.82156	−1.52471
$11$	1.50306	0.453189	0.226595	−	0.973989i	$-0.427241\pi$
$11$	1.50306	0.453189	0.226595	+	0.973989i	$0.427241\pi$
$12$	0	0
$13$	0.00459548	0.00127456	0.000637279	−	1.00000i	$-0.499797\pi$
$13$	0.00459548	0.00127456	0.000637279		1.00000i	$0.499797\pi$
$14$	7.77529	2.07803
$15$	0	0
$16$	−2.25692	−0.564230
$17$	−1.11927	−0.271464	−0.135732	−	0.990746i	$-0.543339\pi$
$17$	−1.11927	−0.271464	−0.135732	+	0.990746i	$0.543339\pi$
$18$	0	0
$19$	2.49963	0.573454	0.286727	−	0.958012i	$-0.407433\pi$
$19$	2.49963	0.573454	0.286727	+	0.958012i	$0.407433\pi$
$20$	5.93520	1.32715
$21$	0	0
$22$	−3.24334	−0.691483
$23$	−1.72480	−0.359646	−0.179823	−	0.983699i	$-0.557553\pi$
$23$	−1.72480	−0.359646	−0.179823	+	0.983699i	$0.557553\pi$
$24$	0	0
$25$	−0.00724425	−0.00144885
$26$	−0.00991627	−0.00194474
$27$	0	0
$28$	−9.57116	−1.80878
$29$	0.572061	0.106229	0.0531145	−	0.998588i	$-0.483085\pi$
$29$	0.572061	0.106229	0.0531145	+	0.998588i	$0.483085\pi$
$30$	0	0
$31$	3.25375	0.584390	0.292195	−	0.956359i	$-0.405614\pi$
$31$	3.25375	0.584390	0.292195	+	0.956359i	$0.405614\pi$
$32$	7.70209	1.36155
$33$	0	0
$34$	2.41520	0.414203
$35$	−8.05137	−1.36093
$36$	0	0
$37$	−6.13809	−1.00910	−0.504548	−	0.863384i	$-0.668341\pi$
$37$	−6.13809	−1.00910	−0.504548	+	0.863384i	$0.668341\pi$
$38$	−5.39377	−0.874986
$39$	0	0
$40$	−3.16402	−0.500276
$41$	5.89549	0.920721	0.460360	−	0.887732i	$-0.347720\pi$
$41$	5.89549	0.920721	0.460360	+	0.887732i	$0.347720\pi$
$42$	0	0
$43$	−7.66800	−1.16936	−0.584680	−	0.811264i	$-0.698780\pi$
$43$	−7.66800	−1.16936	−0.584680	+	0.811264i	$0.698780\pi$
$44$	3.99246	0.601886
$45$	0	0
$46$	3.72183	0.548754
$47$	−6.44840	−0.940596	−0.470298	−	0.882508i	$-0.655854\pi$
$47$	−6.44840	−0.940596	−0.470298	+	0.882508i	$0.655854\pi$
$48$	0	0
$49$	5.98373	0.854819
$50$	0.0156318	0.00221068
$51$	0	0
$52$	0.0122066	0.00169276
$53$	7.82500	1.07485	0.537423	−	0.843313i	$-0.319398\pi$
$53$	7.82500	1.07485	0.537423	+	0.843313i	$0.319398\pi$
$54$	0	0
$55$	3.35851	0.452861
$56$	5.10234	0.681829
$57$	0	0
$58$	−1.23441	−0.162086
$59$	−0.253382	−0.0329875	−0.0164938	−	0.999864i	$-0.505250\pi$
$59$	−0.253382	−0.0329875	−0.0164938	+	0.999864i	$0.505250\pi$
$60$	0	0
$61$	7.08244	0.906814	0.453407	−	0.891304i	$-0.350208\pi$
$61$	7.08244	0.906814	0.453407	+	0.891304i	$0.350208\pi$
$62$	−7.02103	−0.891672
$63$	0	0
$64$	−12.1059	−1.51324
$65$	0.0102684	0.00127363
$66$	0	0
$67$	6.16046	0.752621	0.376310	−	0.926494i	$-0.377193\pi$
$67$	6.16046	0.752621	0.376310	+	0.926494i	$0.377193\pi$
$68$	−2.97304	−0.360534
$69$	0	0
$70$	17.3735	2.07653
$71$	9.32311	1.10645	0.553225	−	0.833032i	$-0.313397\pi$
$71$	9.32311	1.10645	0.553225	+	0.833032i	$0.313397\pi$
$72$	0	0
$73$	−15.2654	−1.78668	−0.893338	−	0.449385i	$-0.851643\pi$
$73$	−15.2654	−1.78668	−0.893338	+	0.449385i	$0.851643\pi$
$74$	13.2450	1.53969
$75$	0	0
$76$	6.63958	0.761612
$77$	−5.41596	−0.617206
$78$	0	0
$79$	−1.99661	−0.224636	−0.112318	−	0.993672i	$-0.535828\pi$
$79$	−1.99661	−0.224636	−0.112318	+	0.993672i	$0.535828\pi$
$80$	−5.04297	−0.563821
$81$	0	0
$82$	−12.7215	−1.40485
$83$	−6.91660	−0.759196	−0.379598	−	0.925152i	$-0.623938\pi$
$83$	−6.91660	−0.759196	−0.379598	+	0.925152i	$0.623938\pi$
$84$	0	0
$85$	−2.50096	−0.271267
$86$	16.5462	1.78423
$87$	0	0
$88$	−2.12836	−0.226884
$89$	−3.37422	−0.357666	−0.178833	−	0.983879i	$-0.557232\pi$
$89$	−3.37422	−0.357666	−0.178833	+	0.983879i	$0.557232\pi$
$90$	0	0
$91$	−0.0165589	−0.00173584
$92$	−4.58146	−0.477651
$93$	0	0
$94$	13.9146	1.43518
$95$	5.58529	0.573039
$96$	0	0
$97$	−14.8842	−1.51126	−0.755631	−	0.654997i	$-0.772670\pi$
$97$	−14.8842	−1.51126	−0.755631	+	0.654997i	$0.772670\pi$
$98$	−12.9119	−1.30430
$99$	0	0
$100$	−0.0192424	−0.00192424
$101$	−11.2591	−1.12032	−0.560159	−	0.828385i	$-0.689260\pi$
$101$	−11.2591	−1.12032	−0.560159	+	0.828385i	$0.689260\pi$
$102$	0	0
$103$	4.65004	0.458182	0.229091	−	0.973405i	$-0.426425\pi$
$103$	4.65004	0.458182	0.229091	+	0.973405i	$0.426425\pi$
$104$	−0.00650730	−0.000638093 0
$105$	0	0
$106$	−16.8850	−1.64002
$107$	8.08323	0.781435	0.390718	−	0.920511i	$-0.372227\pi$
$107$	8.08323	0.781435	0.390718	+	0.920511i	$0.372227\pi$
$108$	0	0
$109$	−1.20991	−0.115889	−0.0579444	−	0.998320i	$-0.518455\pi$
$109$	−1.20991	−0.115889	−0.0579444	+	0.998320i	$0.518455\pi$
$110$	−7.24708	−0.690982
$111$	0	0
$112$	8.13234	0.768434
$113$	−7.68412	−0.722861	−0.361431	−	0.932399i	$-0.617712\pi$
$113$	−7.68412	−0.722861	−0.361431	+	0.932399i	$0.617712\pi$
$114$	0	0
$115$	−3.85398	−0.359386
$116$	1.51952	0.141084
$117$	0	0
$118$	0.546755	0.0503329
$119$	4.03307	0.369711
$120$	0	0
$121$	−8.74081	−0.794619
$122$	−15.2827	−1.38363
$123$	0	0
$124$	8.64268	0.776136
$125$	−11.1884	−1.00072
$126$	0	0
$127$	8.83928	0.784360	0.392180	−	0.919889i	$-0.371721\pi$
$127$	8.83928	0.784360	0.392180	+	0.919889i	$0.371721\pi$
$128$	10.7184	0.947380
$129$	0	0
$130$	−0.0221574	−0.00194333
$131$	12.9102	1.12797	0.563984	−	0.825786i	$-0.309268\pi$
$131$	12.9102	1.12797	0.563984	+	0.825786i	$0.309268\pi$
$132$	0	0
$133$	−9.00690	−0.780998
$134$	−13.2932	−1.14836
$135$	0	0
$136$	1.58491	0.135905
$137$	−3.83616	−0.327745	−0.163873	−	0.986482i	$-0.552399\pi$
$137$	−3.83616	−0.327745	−0.163873	+	0.986482i	$0.552399\pi$
$138$	0	0
$139$	9.22921	0.782811	0.391406	−	0.920218i	$-0.371989\pi$
$139$	9.22921	0.782811	0.391406	+	0.920218i	$0.371989\pi$
$140$	−21.3863	−1.80747
$141$	0	0
$142$	−20.1177	−1.68824
$143$	0.00690728	0.000577616 0
$144$	0	0
$145$	1.27824	0.106152
$146$	32.9401	2.72614
$147$	0	0
$148$	−16.3042	−1.34019
$149$	−22.6884	−1.85871	−0.929353	−	0.369191i	$-0.879635\pi$
$149$	−22.6884	−1.85871	−0.929353	+	0.369191i	$0.879635\pi$
$150$	0	0
$151$	−2.22521	−0.181085	−0.0905427	−	0.995893i	$-0.528860\pi$
$151$	−2.22521	−0.181085	−0.0905427	+	0.995893i	$0.528860\pi$
$152$	−3.53953	−0.287094
$153$	0	0
$154$	11.6867	0.941743
$155$	7.27033	0.583967
$156$	0	0
$157$	−22.3177	−1.78114	−0.890571	−	0.454844i	$-0.849695\pi$
$157$	−22.3177	−1.78114	−0.890571	+	0.454844i	$0.849695\pi$
$158$	4.30834	0.342753
$159$	0	0
$160$	17.2099	1.36056
$161$	6.21497	0.489808
$162$	0	0
$163$	7.22321	0.565765	0.282883	−	0.959155i	$-0.408709\pi$
$163$	7.22321	0.565765	0.282883	+	0.959155i	$0.408709\pi$
$164$	15.6597	1.22282
$165$	0	0
$166$	14.9248	1.15839
$167$	11.4485	0.885915	0.442958	−	0.896543i	$-0.353929\pi$
$167$	11.4485	0.885915	0.442958	+	0.896543i	$0.353929\pi$
$168$	0	0
$169$	−13.0000	−0.999998
$170$	5.39664	0.413903
$171$	0	0
$172$	−20.3679	−1.55304
$173$	12.5821	0.956599	0.478300	−	0.878197i	$-0.341253\pi$
$173$	12.5821	0.956599	0.478300	+	0.878197i	$0.341253\pi$
$174$	0	0
$175$	0.0261032	0.00197321
$176$	−3.39228	−0.255703
$177$	0	0
$178$	7.28098	0.545732
$179$	−1.23407	−0.0922389	−0.0461194	−	0.998936i	$-0.514685\pi$
$179$	−1.23407	−0.0922389	−0.0461194	+	0.998936i	$0.514685\pi$
$180$	0	0
$181$	−17.0030	−1.26382	−0.631911	−	0.775041i	$-0.717729\pi$
$181$	−17.0030	−1.26382	−0.631911	+	0.775041i	$0.717729\pi$
$182$	0.0357312	0.00264858
$183$	0	0
$184$	2.44236	0.180053
$185$	−13.7152	−1.00836
$186$	0	0
$187$	−1.68233	−0.123024
$188$	−17.1284	−1.24922
$189$	0	0
$190$	−12.0521	−0.874352
$191$	−0.213652	−0.0154593	−0.00772966	−	0.999970i	$-0.502460\pi$
$191$	−0.213652	−0.0154593	−0.00772966	+	0.999970i	$0.502460\pi$
$192$	0	0
$193$	−3.51336	−0.252897	−0.126449	−	0.991973i	$-0.540358\pi$
$193$	−3.51336	−0.252897	−0.126449	+	0.991973i	$0.540358\pi$
$194$	32.1176	2.30591
$195$	0	0
$196$	15.8941	1.13530
$197$	11.5456	0.822587	0.411294	−	0.911503i	$-0.365077\pi$
$197$	11.5456	0.822587	0.411294	+	0.911503i	$0.365077\pi$
$198$	0	0
$199$	−4.63698	−0.328707	−0.164353	−	0.986402i	$-0.552554\pi$
$199$	−4.63698	−0.328707	−0.164353	+	0.986402i	$0.552554\pi$
$200$	0.0102580	0.000725351 0
$201$	0	0
$202$	24.2951	1.70940
$203$	−2.06130	−0.144675
$204$	0	0
$205$	13.1732	0.920054
$206$	−10.0340	−0.699101
$207$	0	0
$208$	−0.0103716	−0.000719144 0
$209$	3.75709	0.259883
$210$	0	0
$211$	−0.622411	−0.0428485	−0.0214243	−	0.999770i	$-0.506820\pi$
$211$	−0.622411	−0.0428485	−0.0214243	+	0.999770i	$0.506820\pi$
$212$	20.7850	1.42752
$213$	0	0
$214$	−17.4422	−1.19233
$215$	−17.1338	−1.16851
$216$	0	0
$217$	−11.7242	−0.795891
$218$	2.61079	0.176825
$219$	0	0
$220$	8.92095	0.601450
$221$	−0.00514360	−0.000345996 0
$222$	0	0
$223$	0.0281489	0.00188499	0.000942495	−	1.00000i	$-0.499700\pi$
$223$	0.0281489	0.00188499	0.000942495		1.00000i	$0.499700\pi$
$224$	−27.7529	−1.85432
$225$	0	0
$226$	16.5810	1.10295
$227$	8.12837	0.539499	0.269749	−	0.962931i	$-0.413059\pi$
$227$	8.12837	0.539499	0.269749	+	0.962931i	$0.413059\pi$
$228$	0	0
$229$	23.9995	1.58593	0.792966	−	0.609266i	$-0.208536\pi$
$229$	23.9995	1.58593	0.792966	+	0.609266i	$0.208536\pi$
$230$	8.31623	0.548356
$231$	0	0
$232$	−0.810050	−0.0531824
$233$	12.5903	0.824819	0.412410	−	0.910999i	$-0.364687\pi$
$233$	12.5903	0.824819	0.412410	+	0.910999i	$0.364687\pi$
$234$	0	0
$235$	−14.4086	−0.939915
$236$	−0.673040	−0.0438112
$237$	0	0
$238$	−8.70268	−0.564111
$239$	−24.1351	−1.56117	−0.780586	−	0.625049i	$-0.785079\pi$
$239$	−24.1351	−1.56117	−0.780586	+	0.625049i	$0.785079\pi$
$240$	0	0
$241$	25.4713	1.64075	0.820374	−	0.571827i	$-0.193765\pi$
$241$	25.4713	1.64075	0.820374	+	0.571827i	$0.193765\pi$
$242$	18.8612	1.21244
$243$	0	0
$244$	18.8126	1.20435
$245$	13.3703	0.854199
$246$	0	0
$247$	0.0114870	0.000730901 0
$248$	−4.60737	−0.292569
$249$	0	0
$250$	24.1427	1.52692
$251$	−4.12389	−0.260298	−0.130149	−	0.991494i	$-0.541546\pi$
$251$	−4.12389	−0.260298	−0.130149	+	0.991494i	$0.541546\pi$
$252$	0	0
$253$	−2.59248	−0.162988
$254$	−19.0737	−1.19679
$255$	0	0
$256$	1.08345	0.0677158
$257$	16.7266	1.04337	0.521687	−	0.853137i	$-0.325303\pi$
$257$	16.7266	1.04337	0.521687	+	0.853137i	$0.325303\pi$
$258$	0	0
$259$	22.1174	1.37431
$260$	0.0272751	0.00169153
$261$	0	0
$262$	−27.8579	−1.72107
$263$	−28.7329	−1.77174	−0.885872	−	0.463930i	$-0.846439\pi$
$263$	−28.7329	−1.77174	−0.885872	+	0.463930i	$0.846439\pi$
$264$	0	0
$265$	17.4845	1.07407
$266$	19.4354	1.19166
$267$	0	0
$268$	16.3636	0.999565
$269$	−28.6099	−1.74438	−0.872189	−	0.489169i	$-0.837300\pi$
$269$	−28.6099	−1.74438	−0.872189	+	0.489169i	$0.837300\pi$
$270$	0	0
$271$	−13.8852	−0.843467	−0.421733	−	0.906720i	$-0.638578\pi$
$271$	−13.8852	−0.843467	−0.421733	+	0.906720i	$0.638578\pi$
$272$	2.52611	0.153168
$273$	0	0
$274$	8.27778	0.500079
$275$	−0.0108885	−0.000656603 0
$276$	0	0
$277$	−14.4664	−0.869200	−0.434600	−	0.900624i	$-0.643110\pi$
$277$	−14.4664	−0.869200	−0.434600	+	0.900624i	$0.643110\pi$
$278$	−19.9150	−1.19443
$279$	0	0
$280$	11.4009	0.681335
$281$	15.4967	0.924456	0.462228	−	0.886761i	$-0.347050\pi$
$281$	15.4967	0.924456	0.462228	+	0.886761i	$0.347050\pi$
$282$	0	0
$283$	10.4295	0.619970	0.309985	−	0.950741i	$-0.399676\pi$
$283$	10.4295	0.619970	0.309985	+	0.950741i	$0.399676\pi$
$284$	24.7643	1.46949
$285$	0	0
$286$	−0.0149047	−0.000881336 0
$287$	−21.2432	−1.25395
$288$	0	0
$289$	−15.7472	−0.926307
$290$	−2.75822	−0.161968
$291$	0	0
$292$	−40.5483	−2.37291
$293$	1.90244	0.111142	0.0555708	−	0.998455i	$-0.482302\pi$
$293$	1.90244	0.111142	0.0555708	+	0.998455i	$0.482302\pi$
$294$	0	0
$295$	−0.566169	−0.0329636
$296$	8.69167	0.505193
$297$	0	0
$298$	48.9577	2.83604
$299$	−0.00792630	−0.000458390 0
$300$	0	0
$301$	27.6301	1.59257
$302$	4.80163	0.276303
$303$	0	0
$304$	−5.64146	−0.323560
$305$	15.8253	0.906157
$306$	0	0
$307$	−17.6876	−1.00949	−0.504744	−	0.863269i	$-0.668413\pi$
$307$	−17.6876	−1.00949	−0.504744	+	0.863269i	$0.668413\pi$
$308$	−14.3860	−0.819720
$309$	0	0
$310$	−15.6881	−0.891025
$311$	28.0074	1.58815	0.794076	−	0.607818i	$-0.207955\pi$
$311$	28.0074	1.58815	0.794076	+	0.607818i	$0.207955\pi$
$312$	0	0
$313$	5.02123	0.283817	0.141908	−	0.989880i	$-0.454676\pi$
$313$	5.02123	0.283817	0.141908	+	0.989880i	$0.454676\pi$
$314$	48.1577	2.71769
$315$	0	0
$316$	−5.30344	−0.298342
$317$	−0.356974	−0.0200496	−0.0100248	−	0.999950i	$-0.503191\pi$
$317$	−0.356974	−0.0200496	−0.0100248	+	0.999950i	$0.503191\pi$
$318$	0	0
$319$	0.859841	0.0481418
$320$	−27.0501	−1.51215
$321$	0	0
$322$	−13.4108	−0.747357
$323$	−2.79777	−0.155672
$324$	0	0
$325$	−3.32908e−5 0	−1.84664e−6 0
$326$	−15.5864	−0.863253
$327$	0	0
$328$	−8.34814	−0.460949
$329$	23.2355	1.28101
$330$	0	0
$331$	16.2490	0.893123	0.446562	−	0.894753i	$-0.352648\pi$
$331$	16.2490	0.893123	0.446562	+	0.894753i	$0.352648\pi$
$332$	−18.3720	−1.00830
$333$	0	0
$334$	−24.7040	−1.35174
$335$	13.7652	0.752075
$336$	0	0
$337$	−24.0444	−1.30978	−0.654892	−	0.755722i	$-0.727286\pi$
$337$	−24.0444	−1.30978	−0.654892	+	0.755722i	$0.727286\pi$
$338$	28.0517	1.52581
$339$	0	0
$340$	−6.64310	−0.360273
$341$	4.89057	0.264839
$342$	0	0
$343$	3.66192	0.197725
$344$	10.8581	0.585427
$345$	0	0
$346$	−27.1500	−1.45959
$347$	10.7861	0.579028	0.289514	−	0.957174i	$-0.406506\pi$
$347$	10.7861	0.579028	0.289514	+	0.957174i	$0.406506\pi$
$348$	0	0
$349$	17.2044	0.920933	0.460466	−	0.887677i	$-0.347682\pi$
$349$	17.2044	0.920933	0.460466	+	0.887677i	$0.347682\pi$
$350$	−0.0563262	−0.00301076
$351$	0	0
$352$	11.5767	0.617040
$353$	2.83276	0.150773	0.0753863	−	0.997154i	$-0.475981\pi$
$353$	2.83276	0.150773	0.0753863	+	0.997154i	$0.475981\pi$
$354$	0	0
$355$	20.8320	1.10565
$356$	−8.96268	−0.475021
$357$	0	0
$358$	2.66292	0.140739
$359$	−23.3759	−1.23373	−0.616867	−	0.787068i	$-0.711598\pi$
$359$	−23.3759	−1.23373	−0.616867	+	0.787068i	$0.711598\pi$
$360$	0	0
$361$	−12.7518	−0.671150
$362$	36.6895	1.92836
$363$	0	0
$364$	−0.0439841	−0.00230539
$365$	−34.1097	−1.78538
$366$	0	0
$367$	−21.8406	−1.14007	−0.570034	−	0.821621i	$-0.693070\pi$
$367$	−21.8406	−1.14007	−0.570034	+	0.821621i	$0.693070\pi$
$368$	3.89274	0.202923
$369$	0	0
$370$	29.5951	1.53858
$371$	−28.1958	−1.46385
$372$	0	0
$373$	0.594294	0.0307714	0.0153857	−	0.999882i	$-0.495102\pi$
$373$	0.594294	0.0307714	0.0153857	+	0.999882i	$0.495102\pi$
$374$	3.63019	0.187713
$375$	0	0
$376$	9.13108	0.470899
$377$	0.00262890	0.000135395 0
$378$	0	0
$379$	−16.0631	−0.825105	−0.412553	−	0.910934i	$-0.635363\pi$
$379$	−16.0631	−0.825105	−0.412553	+	0.910934i	$0.635363\pi$
$380$	14.8358	0.761060
$381$	0	0
$382$	0.461025	0.0235881
$383$	−15.0745	−0.770271	−0.385135	−	0.922860i	$-0.625845\pi$
$383$	−15.0745	−0.770271	−0.385135	+	0.922860i	$0.625845\pi$
$384$	0	0
$385$	−12.1017	−0.616759
$386$	7.58123	0.385875
$387$	0	0
$388$	−39.5358	−2.00713
$389$	−8.06793	−0.409060	−0.204530	−	0.978860i	$-0.565567\pi$
$389$	−8.06793	−0.409060	−0.204530	+	0.978860i	$0.565567\pi$
$390$	0	0
$391$	1.93053	0.0976309
$392$	−8.47309	−0.427956
$393$	0	0
$394$	−24.9133	−1.25512
$395$	−4.46132	−0.224473
$396$	0	0
$397$	7.64285	0.383583	0.191792	−	0.981436i	$-0.438570\pi$
$397$	7.64285	0.383583	0.191792	+	0.981436i	$0.438570\pi$
$398$	10.0058	0.501546
$399$	0	0
$400$	0.0163497	0.000817484 0
$401$	−37.0556	−1.85047	−0.925234	−	0.379397i	$-0.876132\pi$
$401$	−37.0556	−1.85047	−0.925234	+	0.379397i	$0.876132\pi$
$402$	0	0
$403$	0.0149525	0.000744839 0
$404$	−29.9066	−1.48791
$405$	0	0
$406$	4.44794	0.220747
$407$	−9.22591	−0.457312
$408$	0	0
$409$	−29.1516	−1.44145	−0.720726	−	0.693220i	$-0.756191\pi$
$409$	−29.1516	−1.44145	−0.720726	+	0.693220i	$0.756191\pi$
$410$	−28.4254	−1.40383
$411$	0	0
$412$	12.3515	0.608517
$413$	0.913010	0.0449263
$414$	0	0
$415$	−15.4548	−0.758645
$416$	0.0353948	0.00173537
$417$	0	0
$418$	−8.10716	−0.396534
$419$	−31.8690	−1.55690	−0.778451	−	0.627705i	$-0.783994\pi$
$419$	−31.8690	−1.55690	−0.778451	+	0.627705i	$0.783994\pi$
$420$	0	0
$421$	−26.0459	−1.26940	−0.634699	−	0.772759i	$-0.718876\pi$
$421$	−26.0459	−1.26940	−0.634699	+	0.772759i	$0.718876\pi$
$422$	1.34306	0.0653790
$423$	0	0
$424$	−11.0804	−0.538110
$425$	0.00810829	0.000393310 0
$426$	0	0
$427$	−25.5201	−1.23501
$428$	21.4709	1.03783
$429$	0	0
$430$	36.9717	1.78293
$431$	22.6196	1.08955	0.544774	−	0.838583i	$-0.316616\pi$
$431$	22.6196	1.08955	0.544774	+	0.838583i	$0.316616\pi$
$432$	0	0
$433$	−22.7272	−1.09220	−0.546099	−	0.837720i	$-0.683888\pi$
$433$	−22.7272	−1.09220	−0.546099	+	0.837720i	$0.683888\pi$
$434$	25.2988	1.21438
$435$	0	0
$436$	−3.21380	−0.153913
$437$	−4.31137	−0.206241
$438$	0	0
$439$	24.5936	1.17379	0.586895	−	0.809663i	$-0.300350\pi$
$439$	24.5936	1.17379	0.586895	+	0.809663i	$0.300350\pi$
$440$	−4.75572	−0.226720
$441$	0	0
$442$	0.0110990	0.000527926 0
$443$	−23.4765	−1.11540	−0.557702	−	0.830041i	$-0.688317\pi$
$443$	−23.4765	−1.11540	−0.557702	+	0.830041i	$0.688317\pi$
$444$	0	0
$445$	−7.53951	−0.357407
$446$	−0.0607405	−0.00287615
$447$	0	0
$448$	43.6213	2.06091
$449$	13.6452	0.643958	0.321979	−	0.946747i	$-0.395652\pi$
$449$	13.6452	0.643958	0.321979	+	0.946747i	$0.395652\pi$
$450$	0	0
$451$	8.86127	0.417261
$452$	−20.4107	−0.960041
$453$	0	0
$454$	−17.5396	−0.823176
$455$	−0.0370000	−0.00173458
$456$	0	0
$457$	−2.17967	−0.101960	−0.0509802	−	0.998700i	$-0.516235\pi$
$457$	−2.17967	−0.101960	−0.0509802	+	0.998700i	$0.516235\pi$
$458$	−51.7868	−2.41984
$459$	0	0
$460$	−10.2370	−0.477304
$461$	−12.2229	−0.569276	−0.284638	−	0.958635i	$-0.591873\pi$
$461$	−12.2229	−0.569276	−0.284638	+	0.958635i	$0.591873\pi$
$462$	0	0
$463$	−1.99436	−0.0926860	−0.0463430	−	0.998926i	$-0.514757\pi$
$463$	−1.99436	−0.0926860	−0.0463430	+	0.998926i	$0.514757\pi$
$464$	−1.29109	−0.0599375
$465$	0	0
$466$	−27.1677	−1.25852
$467$	−41.2601	−1.90929	−0.954645	−	0.297747i	$-0.903765\pi$
$467$	−41.2601	−1.90929	−0.954645	+	0.297747i	$0.903765\pi$
$468$	0	0
$469$	−22.1980	−1.02501
$470$	31.0913	1.43414
$471$	0	0
$472$	0.358794	0.0165148
$473$	−11.5255	−0.529941
$474$	0	0
$475$	−0.0181079	−0.000830849 0
$476$	10.7127	0.491018
$477$	0	0
$478$	52.0795	2.38206
$479$	−23.4921	−1.07338	−0.536691	−	0.843779i	$-0.680326\pi$
$479$	−23.4921	−1.07338	−0.536691	+	0.843779i	$0.680326\pi$
$480$	0	0
$481$	−0.0282075	−0.00128615
$482$	−54.9626	−2.50348
$483$	0	0
$484$	−23.2176	−1.05534
$485$	−33.2580	−1.51017
$486$	0	0
$487$	6.09387	0.276139	0.138070	−	0.990423i	$-0.455910\pi$
$487$	6.09387	0.276139	0.138070	+	0.990423i	$0.455910\pi$
$488$	−10.0289	−0.453986
$489$	0	0
$490$	−28.8509	−1.30335
$491$	7.88983	0.356063	0.178032	−	0.984025i	$-0.443027\pi$
$491$	7.88983	0.356063	0.178032	+	0.984025i	$0.443027\pi$
$492$	0	0
$493$	−0.640292	−0.0288373
$494$	−0.0247870	−0.00111522
$495$	0	0
$496$	−7.34344	−0.329730
$497$	−33.5939	−1.50689
$498$	0	0
$499$	−0.289120	−0.0129428	−0.00647139	−	0.999979i	$-0.502060\pi$
$499$	−0.289120	−0.0129428	−0.00647139	+	0.999979i	$0.502060\pi$
$500$	−29.7190	−1.32907
$501$	0	0
$502$	8.89865	0.397166
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−25.1578	−1.11951
$506$	5.59413	0.248689
$507$	0	0
$508$	23.4791	1.04172
$509$	−27.0889	−1.20069	−0.600346	−	0.799740i	$-0.704970\pi$
$509$	−27.0889	−1.20069	−0.600346	+	0.799740i	$0.704970\pi$
$510$	0	0
$511$	55.0056	2.43331
$512$	−23.7747	−1.05070
$513$	0	0
$514$	−36.0931	−1.59200
$515$	10.3903	0.457850
$516$	0	0
$517$	−9.69233	−0.426268
$518$	−47.7255	−2.09694
$519$	0	0
$520$	−0.0145402	−0.000637631 0
$521$	−21.1689	−0.927424	−0.463712	−	0.885986i	$-0.653483\pi$
$521$	−21.1689	−0.927424	−0.463712	+	0.885986i	$0.653483\pi$
$522$	0	0
$523$	−0.868369	−0.0379711	−0.0189856	−	0.999820i	$-0.506044\pi$
$523$	−0.868369	−0.0379711	−0.0189856	+	0.999820i	$0.506044\pi$
$524$	34.2923	1.49807
$525$	0	0
$526$	62.0006	2.70335
$527$	−3.64183	−0.158641
$528$	0	0
$529$	−20.0251	−0.870655
$530$	−37.7287	−1.63883
$531$	0	0
$532$	−23.9244	−1.03725
$533$	0.0270926	0.00117351
$534$	0	0
$535$	18.0616	0.780869
$536$	−8.72335	−0.376791
$537$	0	0
$538$	61.7353	2.66160
$539$	8.99390	0.387395
$540$	0	0
$541$	14.8513	0.638508	0.319254	−	0.947669i	$-0.396568\pi$
$541$	14.8513	0.638508	0.319254	+	0.947669i	$0.396568\pi$
$542$	29.9619	1.28697
$543$	0	0
$544$	−8.62074	−0.369611
$545$	−2.70349	−0.115805
$546$	0	0
$547$	−18.8256	−0.804925	−0.402463	−	0.915436i	$-0.631846\pi$
$547$	−18.8256	−0.804925	−0.402463	+	0.915436i	$0.631846\pi$
$548$	−10.1897	−0.435283
$549$	0	0
$550$	0.0234956	0.00100186
$551$	1.42994	0.0609175
$552$	0	0
$553$	7.19437	0.305936
$554$	31.2160	1.32624
$555$	0	0
$556$	24.5149	1.03966
$557$	−11.9201	−0.505071	−0.252535	−	0.967588i	$-0.581264\pi$
$557$	−11.9201	−0.505071	−0.252535	+	0.967588i	$0.581264\pi$
$558$	0	0
$559$	−0.0352382	−0.00149042
$560$	18.1713	0.767877
$561$	0	0
$562$	−33.4392	−1.41055
$563$	22.7172	0.957414	0.478707	−	0.877975i	$-0.341106\pi$
$563$	22.7172	0.957414	0.478707	+	0.877975i	$0.341106\pi$
$564$	0	0
$565$	−17.1698	−0.722337
$566$	−22.5051	−0.945960
$567$	0	0
$568$	−13.2017	−0.553932
$569$	16.9403	0.710176	0.355088	−	0.934833i	$-0.384451\pi$
$569$	16.9403	0.710176	0.355088	+	0.934833i	$0.384451\pi$
$570$	0	0
$571$	16.2940	0.681883	0.340941	−	0.940085i	$-0.389254\pi$
$571$	16.2940	0.681883	0.340941	+	0.940085i	$0.389254\pi$
$572$	0.0183473	0.000767139 0
$573$	0	0
$574$	45.8392	1.91329
$575$	0.0124949	0.000521073 0
$576$	0	0
$577$	−44.3876	−1.84788	−0.923941	−	0.382535i	$-0.875051\pi$
$577$	−44.3876	−1.84788	−0.923941	+	0.382535i	$0.875051\pi$
$578$	33.9798	1.41337
$579$	0	0
$580$	3.39529	0.140982
$581$	24.9226	1.03396
$582$	0	0
$583$	11.7614	0.487109
$584$	21.6161	0.894480
$585$	0	0
$586$	−4.10513	−0.169582
$587$	−33.6221	−1.38773	−0.693865	−	0.720105i	$-0.744094\pi$
$587$	−33.6221	−1.38773	−0.693865	+	0.720105i	$0.744094\pi$
$588$	0	0
$589$	8.13316	0.335121
$590$	1.22170	0.0502964
$591$	0	0
$592$	13.8532	0.569362
$593$	−14.3513	−0.589336	−0.294668	−	0.955600i	$-0.595209\pi$
$593$	−14.3513	−0.589336	−0.294668	+	0.955600i	$0.595209\pi$
$594$	0	0
$595$	9.01169	0.369443
$596$	−60.2655	−2.46857
$597$	0	0
$598$	0.0171036	0.000699418 0
$599$	4.26954	0.174449	0.0872243	−	0.996189i	$-0.472200\pi$
$599$	4.26954	0.174449	0.0872243	+	0.996189i	$0.472200\pi$
$600$	0	0
$601$	18.2764	0.745510	0.372755	−	0.927930i	$-0.378413\pi$
$601$	18.2764	0.745510	0.372755	+	0.927930i	$0.378413\pi$
$602$	−59.6210	−2.42997
$603$	0	0
$604$	−5.91067	−0.240502
$605$	−19.5309	−0.794044
$606$	0	0
$607$	−29.2048	−1.18539	−0.592694	−	0.805428i	$-0.701936\pi$
$607$	−29.2048	−1.18539	−0.592694	+	0.805428i	$0.701936\pi$
$608$	19.2524	0.780787
$609$	0	0
$610$	−34.1484	−1.38263
$611$	−0.0296335	−0.00119884
$612$	0	0
$613$	−24.0622	−0.971861	−0.485931	−	0.873997i	$-0.661519\pi$
$613$	−24.0622	−0.971861	−0.485931	+	0.873997i	$0.661519\pi$
$614$	38.1669	1.54029
$615$	0	0
$616$	7.66912	0.308998
$617$	20.7734	0.836307	0.418154	−	0.908376i	$-0.362677\pi$
$617$	20.7734	0.836307	0.418154	+	0.908376i	$0.362677\pi$
$618$	0	0
$619$	−6.43218	−0.258531	−0.129266	−	0.991610i	$-0.541262\pi$
$619$	−6.43218	−0.258531	−0.129266	+	0.991610i	$0.541262\pi$
$620$	19.3116	0.775574
$621$	0	0
$622$	−60.4351	−2.42323
$623$	12.1583	0.487112
$624$	0	0
$625$	−24.9637	−0.998549
$626$	−10.8350	−0.433052
$627$	0	0
$628$	−59.2807	−2.36556
$629$	6.87020	0.273933
$630$	0	0
$631$	−43.2199	−1.72056	−0.860279	−	0.509824i	$-0.829710\pi$
$631$	−43.2199	−1.72056	−0.860279	+	0.509824i	$0.829710\pi$
$632$	2.82724	0.112462
$633$	0	0
$634$	0.770288	0.0305920
$635$	19.7509	0.783791
$636$	0	0
$637$	0.0274981	0.00108952
$638$	−1.85539	−0.0734556
$639$	0	0
$640$	23.9497	0.946694
$641$	49.6276	1.96017	0.980086	−	0.198574i	$-0.0636309\pi$
$641$	49.6276	1.96017	0.980086	+	0.198574i	$0.0636309\pi$
$642$	0	0
$643$	−45.1377	−1.78006	−0.890028	−	0.455906i	$-0.849315\pi$
$643$	−45.1377	−1.78006	−0.890028	+	0.455906i	$0.849315\pi$
$644$	16.5084	0.650521
$645$	0	0
$646$	6.03711	0.237527
$647$	29.7930	1.17128	0.585642	−	0.810570i	$-0.300843\pi$
$647$	29.7930	1.17128	0.585642	+	0.810570i	$0.300843\pi$
$648$	0	0
$649$	−0.380848	−0.0149496
$650$	7.18359e−5 0	2.81764e−6 0
$651$	0	0
$652$	19.1865	0.751400
$653$	−29.1127	−1.13927	−0.569635	−	0.821898i	$-0.692915\pi$
$653$	−29.1127	−1.13927	−0.569635	+	0.821898i	$0.692915\pi$
$654$	0	0
$655$	28.8471	1.12715
$656$	−13.3056	−0.519498
$657$	0	0
$658$	−50.1382	−1.95459
$659$	−17.1327	−0.667395	−0.333697	−	0.942680i	$-0.608296\pi$
$659$	−17.1327	−0.667395	−0.333697	+	0.942680i	$0.608296\pi$
$660$	0	0
$661$	−0.266846	−0.0103791	−0.00518956	−	0.999987i	$-0.501652\pi$
$661$	−0.266846	−0.0103791	−0.00518956	+	0.999987i	$0.501652\pi$
$662$	−35.0625	−1.36274
$663$	0	0
$664$	9.79405	0.380083
$665$	−20.1255	−0.780432
$666$	0	0
$667$	−0.986691	−0.0382048
$668$	30.4099	1.17660
$669$	0	0
$670$	−29.7030	−1.14753
$671$	10.6453	0.410958
$672$	0	0
$673$	41.4814	1.59899	0.799495	−	0.600673i	$-0.205101\pi$
$673$	41.4814	1.59899	0.799495	+	0.600673i	$0.205101\pi$
$674$	51.8838	1.99849
$675$	0	0
$676$	−34.5309	−1.32811
$677$	−33.0242	−1.26922	−0.634611	−	0.772832i	$-0.718840\pi$
$677$	−33.0242	−1.26922	−0.634611	+	0.772832i	$0.718840\pi$
$678$	0	0
$679$	53.6322	2.05822
$680$	3.54141	0.135807
$681$	0	0
$682$	−10.5530	−0.404096
$683$	−0.471704	−0.0180492	−0.00902462	−	0.999959i	$-0.502873\pi$
$683$	−0.471704	−0.0180492	−0.00902462	+	0.999959i	$0.502873\pi$
$684$	0	0
$685$	−8.57170	−0.327508
$686$	−7.90179	−0.301692
$687$	0	0
$688$	17.3061	0.659788
$689$	0.0359597	0.00136995
$690$	0	0
$691$	−12.1972	−0.464003	−0.232002	−	0.972715i	$-0.574527\pi$
$691$	−12.1972	−0.464003	−0.232002	+	0.972715i	$0.574527\pi$
$692$	33.4209	1.27047
$693$	0	0
$694$	−23.2745	−0.883489
$695$	20.6222	0.782244
$696$	0	0
$697$	−6.59866	−0.249942
$698$	−37.1242	−1.40517
$699$	0	0
$700$	0.0693359	0.00262065
$701$	47.0220	1.77600	0.887999	−	0.459846i	$-0.152095\pi$
$701$	47.0220	1.77600	0.887999	+	0.459846i	$0.152095\pi$
$702$	0	0
$703$	−15.3430	−0.578671
$704$	−18.1960	−0.685786
$705$	0	0
$706$	−6.11261	−0.230051
$707$	40.5697	1.52578
$708$	0	0
$709$	26.6796	1.00197	0.500986	−	0.865455i	$-0.332971\pi$
$709$	26.6796	1.00197	0.500986	+	0.865455i	$0.332971\pi$
$710$	−44.9519	−1.68701
$711$	0	0
$712$	4.77796	0.179062
$713$	−5.61207	−0.210174
$714$	0	0
$715$	0.0154340	0.000577198 0
$716$	−3.27797	−0.122504
$717$	0	0
$718$	50.4412	1.88245
$719$	−12.7457	−0.475336	−0.237668	−	0.971346i	$-0.576383\pi$
$719$	−12.7457	−0.475336	−0.237668	+	0.971346i	$0.576383\pi$
$720$	0	0
$721$	−16.7555	−0.624006
$722$	27.5163	1.02405
$723$	0	0
$724$	−45.1637	−1.67850
$725$	−0.00414415	−0.000153910 0
$726$	0	0
$727$	1.14219	0.0423614	0.0211807	−	0.999776i	$-0.493257\pi$
$727$	1.14219	0.0423614	0.0211807	+	0.999776i	$0.493257\pi$
$728$	0.0234477	0.000869031 0
$729$	0	0
$730$	73.6028	2.72416
$731$	8.58259	0.317439
$732$	0	0
$733$	−20.6111	−0.761289	−0.380644	−	0.924721i	$-0.624298\pi$
$733$	−20.6111	−0.761289	−0.380644	+	0.924721i	$0.624298\pi$
$734$	47.1282	1.73953
$735$	0	0
$736$	−13.2846	−0.489676
$737$	9.25954	0.341080
$738$	0	0
$739$	−40.1054	−1.47530	−0.737651	−	0.675182i	$-0.764065\pi$
$739$	−40.1054	−1.47530	−0.737651	+	0.675182i	$0.764065\pi$
$740$	−36.4308	−1.33922
$741$	0	0
$742$	60.8417	2.23357
$743$	31.7658	1.16538	0.582688	−	0.812696i	$-0.302001\pi$
$743$	31.7658	1.16538	0.582688	+	0.812696i	$0.302001\pi$
$744$	0	0
$745$	−50.6961	−1.85736
$746$	−1.28238	−0.0469514
$747$	0	0
$748$	−4.46866	−0.163390
$749$	−29.1263	−1.06425
$750$	0	0
$751$	3.19661	0.116646	0.0583230	−	0.998298i	$-0.481425\pi$
$751$	3.19661	0.116646	0.0583230	+	0.998298i	$0.481425\pi$
$752$	14.5535	0.530713
$753$	0	0
$754$	−0.00567271	−0.000206588 0
$755$	−4.97212	−0.180954
$756$	0	0
$757$	45.6783	1.66021	0.830103	−	0.557610i	$-0.188281\pi$
$757$	45.6783	1.66021	0.830103	+	0.557610i	$0.188281\pi$
$758$	34.6614	1.25896
$759$	0	0
$760$	−7.90889	−0.286886
$761$	−37.7902	−1.36989	−0.684946	−	0.728594i	$-0.740174\pi$
$761$	−37.7902	−1.36989	−0.684946	+	0.728594i	$0.740174\pi$
$762$	0	0
$763$	4.35968	0.157831
$764$	−0.567508	−0.0205317
$765$	0	0
$766$	32.5282	1.17529
$767$	−0.00116441	−4.20445e−5 0
$768$	0	0
$769$	11.6770	0.421084	0.210542	−	0.977585i	$-0.432477\pi$
$769$	11.6770	0.421084	0.210542	+	0.977585i	$0.432477\pi$
$770$	26.1134	0.941061
$771$	0	0
$772$	−9.33228	−0.335876
$773$	43.2250	1.55470	0.777348	−	0.629070i	$-0.216564\pi$
$773$	43.2250	1.55470	0.777348	+	0.629070i	$0.216564\pi$
$774$	0	0
$775$	−0.0235710	−0.000846694 0
$776$	21.0764	0.756597
$777$	0	0
$778$	17.4092	0.624150
$779$	14.7365	0.527992
$780$	0	0
$781$	14.0132	0.501431
$782$	−4.16574	−0.148967
$783$	0	0
$784$	−13.5048	−0.482314
$785$	−49.8676	−1.77985
$786$	0	0
$787$	41.6501	1.48467	0.742334	−	0.670030i	$-0.233719\pi$
$787$	41.6501	1.48467	0.742334	+	0.670030i	$0.233719\pi$
$788$	30.6676	1.09249
$789$	0	0
$790$	9.62676	0.342505
$791$	27.6881	0.984477
$792$	0	0
$793$	0.0325473	0.00115579
$794$	−16.4920	−0.585278
$795$	0	0
$796$	−12.3169	−0.436560
$797$	−20.0955	−0.711819	−0.355910	−	0.934520i	$-0.615829\pi$
$797$	−20.0955	−0.711819	−0.355910	+	0.934520i	$0.615829\pi$
$798$	0	0
$799$	7.21753	0.255338
$800$	−0.0557958	−0.00197268
$801$	0	0
$802$	79.9596	2.82347
$803$	−22.9448	−0.809703
$804$	0	0
$805$	13.8870	0.489453
$806$	−0.0322650	−0.00113649
$807$	0	0
$808$	15.9431	0.560875
$809$	10.5903	0.372334	0.186167	−	0.982518i	$-0.440393\pi$
$809$	10.5903	0.372334	0.186167	+	0.982518i	$0.440393\pi$
$810$	0	0
$811$	29.9332	1.05110	0.525549	−	0.850764i	$-0.323860\pi$
$811$	29.9332	1.05110	0.525549	+	0.850764i	$0.323860\pi$
$812$	−5.47528	−0.192145
$813$	0	0
$814$	19.9079	0.697773
$815$	16.1399	0.565355
$816$	0	0
$817$	−19.1672	−0.670575
$818$	62.9041	2.19939
$819$	0	0
$820$	34.9909	1.22193
$821$	−1.47970	−0.0516419	−0.0258209	−	0.999667i	$-0.508220\pi$
$821$	−1.47970	−0.0516419	−0.0258209	+	0.999667i	$0.508220\pi$
$822$	0	0
$823$	−20.9277	−0.729494	−0.364747	−	0.931107i	$-0.618845\pi$
$823$	−20.9277	−0.729494	−0.364747	+	0.931107i	$0.618845\pi$
$824$	−6.58455	−0.229384
$825$	0	0
$826$	−1.97012	−0.0685492
$827$	−14.8863	−0.517646	−0.258823	−	0.965925i	$-0.583335\pi$
$827$	−14.8863	−0.517646	−0.258823	+	0.965925i	$0.583335\pi$
$828$	0	0
$829$	54.9732	1.90930	0.954649	−	0.297733i	$-0.0962304\pi$
$829$	54.9732	1.90930	0.954649	+	0.297733i	$0.0962304\pi$
$830$	33.3488	1.15755
$831$	0	0
$832$	−0.0556327	−0.00192872
$833$	−6.69743	−0.232052
$834$	0	0
$835$	25.5812	0.885273
$836$	9.97968	0.345154
$837$	0	0
$838$	68.7678	2.37554
$839$	40.8460	1.41016	0.705080	−	0.709128i	$-0.250911\pi$
$839$	40.8460	1.41016	0.705080	+	0.709128i	$0.250911\pi$
$840$	0	0
$841$	−28.6727	−0.988715
$842$	56.2026	1.93687
$843$	0	0
$844$	−1.65326	−0.0569077
$845$	−29.0478	−0.999274
$846$	0	0
$847$	31.4957	1.08221
$848$	−17.6604	−0.606460
$849$	0	0
$850$	−0.0174963	−0.000600118 0
$851$	10.5870	0.362918
$852$	0	0
$853$	−8.90994	−0.305071	−0.152535	−	0.988298i	$-0.548744\pi$
$853$	−8.90994	−0.305071	−0.152535	+	0.988298i	$0.548744\pi$
$854$	55.0681	1.88439
$855$	0	0
$856$	−11.4460	−0.391217
$857$	−32.1479	−1.09815	−0.549076	−	0.835773i	$-0.685020\pi$
$857$	−32.1479	−1.09815	−0.549076	+	0.835773i	$0.685020\pi$
$858$	0	0
$859$	−25.4577	−0.868606	−0.434303	−	0.900767i	$-0.643005\pi$
$859$	−25.4577	−0.868606	−0.434303	+	0.900767i	$0.643005\pi$
$860$	−45.5111	−1.55192
$861$	0	0
$862$	−48.8092	−1.66245
$863$	42.3500	1.44161	0.720805	−	0.693138i	$-0.243772\pi$
$863$	42.3500	1.44161	0.720805	+	0.693138i	$0.243772\pi$
$864$	0	0
$865$	28.1140	0.955906
$866$	49.0414	1.66649
$867$	0	0
$868$	−31.1421	−1.05703
$869$	−3.00102	−0.101803
$870$	0	0
$871$	0.0283103	0.000959259 0
$872$	1.71326	0.0580184
$873$	0	0
$874$	9.30319	0.314685
$875$	40.3152	1.36290
$876$	0	0
$877$	−54.7498	−1.84877	−0.924385	−	0.381461i	$-0.875421\pi$
$877$	−54.7498	−1.84877	−0.924385	+	0.381461i	$0.875421\pi$
$878$	−53.0688	−1.79099
$879$	0	0
$880$	−7.57988	−0.255518
$881$	42.7752	1.44113	0.720566	−	0.693386i	$-0.243882\pi$
$881$	42.7752	1.44113	0.720566	+	0.693386i	$0.243882\pi$
$882$	0	0
$883$	−51.0453	−1.71781	−0.858906	−	0.512133i	$-0.828856\pi$
$883$	−51.0453	−1.71781	−0.858906	+	0.512133i	$0.828856\pi$
$884$	−0.0136626	−0.000459522 0
$885$	0	0
$886$	50.6584	1.70190
$887$	15.6713	0.526191	0.263096	−	0.964770i	$-0.415256\pi$
$887$	15.6713	0.526191	0.263096	+	0.964770i	$0.415256\pi$
$888$	0	0
$889$	−31.8505	−1.06823
$890$	16.2690	0.545337
$891$	0	0
$892$	0.0747698	0.00250348
$893$	−16.1186	−0.539389
$894$	0	0
$895$	−2.75747	−0.0921720
$896$	−38.6215	−1.29025
$897$	0	0
$898$	−29.4441	−0.982561
$899$	1.86134	0.0620792
$900$	0	0
$901$	−8.75831	−0.291782
$902$	−19.1211	−0.636663
$903$	0	0
$904$	10.8809	0.361892
$905$	−37.9923	−1.26291
$906$	0	0
$907$	50.8029	1.68688	0.843442	−	0.537221i	$-0.180526\pi$
$907$	50.8029	1.68688	0.843442	+	0.537221i	$0.180526\pi$
$908$	21.5908	0.716515
$909$	0	0
$910$	0.0798396	0.00264666
$911$	47.4669	1.57265	0.786324	−	0.617815i	$-0.211982\pi$
$911$	47.4669	1.57265	0.786324	+	0.617815i	$0.211982\pi$
$912$	0	0
$913$	−10.3961	−0.344059
$914$	4.70335	0.155573
$915$	0	0
$916$	63.7481	2.10630
$917$	−46.5192	−1.53620
$918$	0	0
$919$	18.1424	0.598462	0.299231	−	0.954181i	$-0.403270\pi$
$919$	18.1424	0.598462	0.299231	+	0.954181i	$0.403270\pi$
$920$	5.45732	0.179922
$921$	0	0
$922$	26.3749	0.868610
$923$	0.0428442	0.00141023
$924$	0	0
$925$	0.0444659	0.00146203
$926$	4.30350	0.141422
$927$	0	0
$928$	4.40606	0.144636
$929$	36.7632	1.20616	0.603081	−	0.797680i	$-0.293940\pi$
$929$	36.7632	1.20616	0.603081	+	0.797680i	$0.293940\pi$
$930$	0	0
$931$	14.9571	0.490200
$932$	33.4427	1.09545
$933$	0	0
$934$	89.0322	2.91322
$935$	−3.75909	−0.122935
$936$	0	0
$937$	−29.1597	−0.952607	−0.476304	−	0.879281i	$-0.658024\pi$
$937$	−29.1597	−0.952607	−0.476304	+	0.879281i	$0.658024\pi$
$938$	47.8994	1.56397
$939$	0	0
$940$	−38.2725	−1.24831
$941$	11.1041	0.361983	0.180991	−	0.983485i	$-0.442069\pi$
$941$	11.1041	0.361983	0.180991	+	0.983485i	$0.442069\pi$
$942$	0	0
$943$	−10.1686	−0.331134
$944$	0.571863	0.0186125
$945$	0	0
$946$	24.8700	0.808593
$947$	−7.44473	−0.241921	−0.120961	−	0.992657i	$-0.538597\pi$
$947$	−7.44473	−0.241921	−0.120961	+	0.992657i	$0.538597\pi$
$948$	0	0
$949$	−0.0701518	−0.00227722
$950$	0.0390738	0.00126772
$951$	0	0
$952$	−5.71091	−0.185092
$953$	40.8632	1.32369	0.661844	−	0.749641i	$-0.269774\pi$
$953$	40.8632	1.32369	0.661844	+	0.749641i	$0.269774\pi$
$954$	0	0
$955$	−0.477394	−0.0154481
$956$	−64.1083	−2.07341
$957$	0	0
$958$	50.6919	1.63778
$959$	13.8228	0.446362
$960$	0	0
$961$	−20.4131	−0.658488
$962$	0.0608670	0.00196243
$963$	0	0
$964$	67.6574	2.17910
$965$	−7.85042	−0.252714
$966$	0	0
$967$	2.29088	0.0736697	0.0368349	−	0.999321i	$-0.488272\pi$
$967$	2.29088	0.0736697	0.0368349	+	0.999321i	$0.488272\pi$
$968$	12.3772	0.397817
$969$	0	0
$970$	71.7651	2.30424
$971$	27.8481	0.893689	0.446845	−	0.894612i	$-0.352548\pi$
$971$	27.8481	0.893689	0.446845	+	0.894612i	$0.352548\pi$
$972$	0	0
$973$	−33.2556	−1.06612
$974$	−13.1495	−0.421338
$975$	0	0
$976$	−15.9845	−0.511651
$977$	−2.65190	−0.0848419	−0.0424209	−	0.999100i	$-0.513507\pi$
$977$	−2.65190	−0.0848419	−0.0424209	+	0.999100i	$0.513507\pi$
$978$	0	0
$979$	−5.07165	−0.162090
$980$	35.5146	1.13447
$981$	0	0
$982$	−17.0249	−0.543287
$983$	−17.5940	−0.561162	−0.280581	−	0.959830i	$-0.590527\pi$
$983$	−17.5940	−0.561162	−0.280581	+	0.959830i	$0.590527\pi$
$984$	0	0
$985$	25.7980	0.821991
$986$	1.38164	0.0440004
$987$	0	0
$988$	0.0305121	0.000970719 0
$989$	13.2258	0.420556
$990$	0	0
$991$	−40.1464	−1.27529	−0.637646	−	0.770329i	$-0.720092\pi$
$991$	−40.1464	−1.27529	−0.637646	+	0.770329i	$0.720092\pi$
$992$	25.0606	0.795676
$993$	0	0
$994$	72.4899	2.29924
$995$	−10.3611	−0.328469
$996$	0	0
$997$	−52.3474	−1.65786	−0.828930	−	0.559353i	$-0.811050\pi$
$997$	−52.3474	−1.65786	−0.828930	+	0.559353i	$0.811050\pi$
$998$	0.623871	0.0197483
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.1		10
3.2	odd	2		503.2.a.e.1.10	✓	10
12.11	even	2		8048.2.a.p.1.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.10	✓	10	3.2	odd	2
4527.2.a.k.1.1		10	1.1	even	1	trivial
8048.2.a.p.1.7		10	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4527 = 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4527.a (trivial)

\(f(q)\)	\(=\)	\(q-2.15783 q^{2} +2.65622 q^{4} +2.23445 q^{5} -3.60329 q^{7} -1.41602 q^{8} +O(q^{10})\)	\(q-2.15783 q^{2} +2.65622 q^{4} +2.23445 q^{5} -3.60329 q^{7} -1.41602 q^{8} -4.82156 q^{10} +1.50306 q^{11} +0.00459548 q^{13} +7.77529 q^{14} -2.25692 q^{16} -1.11927 q^{17} +2.49963 q^{19} +5.93520 q^{20} -3.24334 q^{22} -1.72480 q^{23} -0.00724425 q^{25} -0.00991627 q^{26} -9.57116 q^{28} +0.572061 q^{29} +3.25375 q^{31} +7.70209 q^{32} +2.41520 q^{34} -8.05137 q^{35} -6.13809 q^{37} -5.39377 q^{38} -3.16402 q^{40} +5.89549 q^{41} -7.66800 q^{43} +3.99246 q^{44} +3.72183 q^{46} -6.44840 q^{47} +5.98373 q^{49} +0.0156318 q^{50} +0.0122066 q^{52} +7.82500 q^{53} +3.35851 q^{55} +5.10234 q^{56} -1.23441 q^{58} -0.253382 q^{59} +7.08244 q^{61} -7.02103 q^{62} -12.1059 q^{64} +0.0102684 q^{65} +6.16046 q^{67} -2.97304 q^{68} +17.3735 q^{70} +9.32311 q^{71} -15.2654 q^{73} +13.2450 q^{74} +6.63958 q^{76} -5.41596 q^{77} -1.99661 q^{79} -5.04297 q^{80} -12.7215 q^{82} -6.91660 q^{83} -2.50096 q^{85} +16.5462 q^{86} -2.12836 q^{88} -3.37422 q^{89} -0.0165589 q^{91} -4.58146 q^{92} +13.9146 q^{94} +5.58529 q^{95} -14.8842 q^{97} -12.9119 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more