LMFDB - Embedded newform 1503.2.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,2,Mod(1,1503)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, 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("1503.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1503.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:	$12.0015154238$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.506287$ of defining polynomial
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.842403 q^{2} -1.29036 q^{4} -1.21255 q^{5} -3.40756 q^{7} -2.77181 q^{8} +O(q^{10})$	\(q+0.842403 q^{2} -1.29036 q^{4} -1.21255 q^{5} -3.40756 q^{7} -2.77181 q^{8} -1.02146 q^{10} +4.19997 q^{11} -2.51125 q^{13} -2.87054 q^{14} +0.245734 q^{16} +4.79075 q^{17} +3.24068 q^{19} +1.56462 q^{20} +3.53807 q^{22} +0.142037 q^{23} -3.52972 q^{25} -2.11549 q^{26} +4.39696 q^{28} -0.264204 q^{29} -10.0831 q^{31} +5.75062 q^{32} +4.03574 q^{34} +4.13183 q^{35} +6.54621 q^{37} +2.72996 q^{38} +3.36095 q^{40} +9.58639 q^{41} +10.2791 q^{43} -5.41947 q^{44} +0.119653 q^{46} +7.14732 q^{47} +4.61145 q^{49} -2.97345 q^{50} +3.24041 q^{52} +0.687009 q^{53} -5.09268 q^{55} +9.44509 q^{56} -0.222567 q^{58} +14.3473 q^{59} +10.7130 q^{61} -8.49403 q^{62} +4.35287 q^{64} +3.04502 q^{65} -1.19870 q^{67} -6.18177 q^{68} +3.48067 q^{70} -3.15248 q^{71} +0.153683 q^{73} +5.51455 q^{74} -4.18164 q^{76} -14.3117 q^{77} -3.65659 q^{79} -0.297965 q^{80} +8.07561 q^{82} +3.96483 q^{83} -5.80902 q^{85} +8.65919 q^{86} -11.6415 q^{88} -8.14055 q^{89} +8.55724 q^{91} -0.183279 q^{92} +6.02092 q^{94} -3.92949 q^{95} +3.44602 q^{97} +3.88470 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) $ 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8	$ 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8} + 5 q^{10} + 15 q^{11} + 3 q^{14} + 10 q^{16} + 11 q^{17} - 16 q^{19} + 9 q^{20} + 7 q^{22} + 9 q^{23} + 6 q^{25} - 7 q^{26} + 9 q^{28} + q^{29} - 18 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 7 q^{37} - 20 q^{38} + 32 q^{40} + 10 q^{41} + 6 q^{43} + 5 q^{44} + 13 q^{46} + 7 q^{47} + 11 q^{49} - 3 q^{50} - 6 q^{52} + 9 q^{53} + 17 q^{55} + 15 q^{56} - 19 q^{58} + 37 q^{59} - 2 q^{61} - 18 q^{62} + 14 q^{64} + 16 q^{65} - 26 q^{68} + 7 q^{70} - 13 q^{71} - 6 q^{73} - 13 q^{74} - 50 q^{76} + 2 q^{77} + 14 q^{79} + 10 q^{80} + 39 q^{82} + 5 q^{83} + 29 q^{85} + 37 q^{86} + 48 q^{88} + 30 q^{89} - 33 q^{91} + 31 q^{92} + 34 q^{94} - 43 q^{95} - 9 q^{97} - 36 q^{98}+O(q^{100}) $ 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8 + 5 * q^10 + 15 * q^11 + 3 * q^14 + 10 * q^16 + 11 * q^17 - 16 * q^19 + 9 * q^20 + 7 * q^22 + 9 * q^23 + 6 * q^25 - 7 * q^26 + 9 * q^28 + q^29 - 18 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 7 * q^37 - 20 * q^38 + 32 * q^40 + 10 * q^41 + 6 * q^43 + 5 * q^44 + 13 * q^46 + 7 * q^47 + 11 * q^49 - 3 * q^50 - 6 * q^52 + 9 * q^53 + 17 * q^55 + 15 * q^56 - 19 * q^58 + 37 * q^59 - 2 * q^61 - 18 * q^62 + 14 * q^64 + 16 * q^65 - 26 * q^68 + 7 * q^70 - 13 * q^71 - 6 * q^73 - 13 * q^74 - 50 * q^76 + 2 * q^77 + 14 * q^79 + 10 * q^80 + 39 * q^82 + 5 * q^83 + 29 * q^85 + 37 * q^86 + 48 * q^88 + 30 * q^89 - 33 * q^91 + 31 * q^92 + 34 * q^94 - 43 * q^95 - 9 * q^97 - 36 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0.842403	0.595669	0.297835	−	0.954618i	$-0.403736\pi$
$2$	0.842403	0.595669	0.297835	+	0.954618i	$0.403736\pi$
$3$	0	0
$3$	0	0
$4$	−1.29036	−0.645178
$5$	−1.21255	−0.542268	−0.271134	−	0.962542i	$-0.587399\pi$
$5$	−1.21255	−0.542268	−0.271134	+	0.962542i	$0.587399\pi$
$6$	0	0
$7$	−3.40756	−1.28794	−0.643968	−	0.765053i	$-0.722713\pi$
$7$	−3.40756	−1.28794	−0.643968	+	0.765053i	$0.722713\pi$
$8$	−2.77181	−0.979982
$9$	0	0
$10$	−1.02146	−0.323013
$11$	4.19997	1.26634	0.633170	−	0.774013i	$-0.281753\pi$
$11$	4.19997	1.26634	0.633170	+	0.774013i	$0.281753\pi$
$12$	0	0
$13$	−2.51125	−0.696497	−0.348248	−	0.937402i	$-0.613223\pi$
$13$	−2.51125	−0.696497	−0.348248	+	0.937402i	$0.613223\pi$
$14$	−2.87054	−0.767183
$15$	0	0
$16$	0.245734	0.0614335
$17$	4.79075	1.16193	0.580964	−	0.813930i	$-0.302676\pi$
$17$	4.79075	1.16193	0.580964	+	0.813930i	$0.302676\pi$
$18$	0	0
$19$	3.24068	0.743464	0.371732	−	0.928340i	$-0.378764\pi$
$19$	3.24068	0.743464	0.371732	+	0.928340i	$0.378764\pi$
$20$	1.56462	0.349860
$21$	0	0
$22$	3.53807	0.754320
$23$	0.142037	0.0296168	0.0148084	−	0.999890i	$-0.495286\pi$
$23$	0.142037	0.0296168	0.0148084	+	0.999890i	$0.495286\pi$
$24$	0	0
$25$	−3.52972	−0.705945
$26$	−2.11549	−0.414881
$27$	0	0
$28$	4.39696	0.830948
$29$	−0.264204	−0.0490615	−0.0245308	−	0.999699i	$-0.507809\pi$
$29$	−0.264204	−0.0490615	−0.0245308	+	0.999699i	$0.507809\pi$
$30$	0	0
$31$	−10.0831	−1.81098	−0.905488	−	0.424372i	$-0.860495\pi$
$31$	−10.0831	−1.81098	−0.905488	+	0.424372i	$0.860495\pi$
$32$	5.75062	1.01658
$33$	0	0
$34$	4.03574	0.692124
$35$	4.13183	0.698407
$36$	0	0
$37$	6.54621	1.07619	0.538095	−	0.842884i	$-0.319144\pi$
$37$	6.54621	1.07619	0.538095	+	0.842884i	$0.319144\pi$
$38$	2.72996	0.442858
$39$	0	0
$40$	3.36095	0.531413
$41$	9.58639	1.49714	0.748571	−	0.663054i	$-0.230740\pi$
$41$	9.58639	1.49714	0.748571	+	0.663054i	$0.230740\pi$
$42$	0	0
$43$	10.2791	1.56756	0.783778	−	0.621042i	$-0.213290\pi$
$43$	10.2791	1.56756	0.783778	+	0.621042i	$0.213290\pi$
$44$	−5.41947	−0.817015
$45$	0	0
$46$	0.119653	0.0176418
$47$	7.14732	1.04254	0.521272	−	0.853391i	$-0.325458\pi$
$47$	7.14732	1.04254	0.521272	+	0.853391i	$0.325458\pi$
$48$	0	0
$49$	4.61145	0.658778
$50$	−2.97345	−0.420510
$51$	0	0
$52$	3.24041	0.449364
$53$	0.687009	0.0943679	0.0471840	−	0.998886i	$-0.484975\pi$
$53$	0.687009	0.0943679	0.0471840	+	0.998886i	$0.484975\pi$
$54$	0	0
$55$	−5.09268	−0.686696
$56$	9.44509	1.26215
$57$	0	0
$58$	−0.222567	−0.0292244
$59$	14.3473	1.86786	0.933929	−	0.357458i	$-0.116357\pi$
$59$	14.3473	1.86786	0.933929	+	0.357458i	$0.116357\pi$
$60$	0	0
$61$	10.7130	1.37166	0.685830	−	0.727762i	$-0.259439\pi$
$61$	10.7130	1.37166	0.685830	+	0.727762i	$0.259439\pi$
$62$	−8.49403	−1.07874
$63$	0	0
$64$	4.35287	0.544109
$65$	3.04502	0.377688
$66$	0	0
$67$	−1.19870	−0.146445	−0.0732224	−	0.997316i	$-0.523328\pi$
$67$	−1.19870	−0.146445	−0.0732224	+	0.997316i	$0.523328\pi$
$68$	−6.18177	−0.749650
$69$	0	0
$70$	3.48067	0.416019
$71$	−3.15248	−0.374131	−0.187065	−	0.982347i	$-0.559898\pi$
$71$	−3.15248	−0.374131	−0.187065	+	0.982347i	$0.559898\pi$
$72$	0	0
$73$	0.153683	0.0179872	0.00899359	−	0.999960i	$-0.497137\pi$
$73$	0.153683	0.0179872	0.00899359	+	0.999960i	$0.497137\pi$
$74$	5.51455	0.641054
$75$	0	0
$76$	−4.18164	−0.479667
$77$	−14.3117	−1.63096
$78$	0	0
$79$	−3.65659	−0.411398	−0.205699	−	0.978615i	$-0.565947\pi$
$79$	−3.65659	−0.411398	−0.205699	+	0.978615i	$0.565947\pi$
$80$	−0.297965	−0.0333134
$81$	0	0
$82$	8.07561	0.891801
$83$	3.96483	0.435197	0.217599	−	0.976038i	$-0.430178\pi$
$83$	3.96483	0.435197	0.217599	+	0.976038i	$0.430178\pi$
$84$	0	0
$85$	−5.80902	−0.630076
$86$	8.65919	0.933744
$87$	0	0
$88$	−11.6415	−1.24099
$89$	−8.14055	−0.862897	−0.431448	−	0.902138i	$-0.641997\pi$
$89$	−8.14055	−0.862897	−0.431448	+	0.902138i	$0.641997\pi$
$90$	0	0
$91$	8.55724	0.897043
$92$	−0.183279	−0.0191081
$93$	0	0
$94$	6.02092	0.621011
$95$	−3.92949	−0.403157
$96$	0	0
$97$	3.44602	0.349890	0.174945	−	0.984578i	$-0.444025\pi$
$97$	3.44602	0.349890	0.174945	+	0.984578i	$0.444025\pi$
$98$	3.88470	0.392414
$99$	0	0
$100$	4.55460	0.455460
$101$	−13.3447	−1.32785	−0.663923	−	0.747801i	$-0.731110\pi$
$101$	−13.3447	−1.32785	−0.663923	+	0.747801i	$0.731110\pi$
$102$	0	0
$103$	−15.8434	−1.56109	−0.780547	−	0.625097i	$-0.785059\pi$
$103$	−15.8434	−1.56109	−0.780547	+	0.625097i	$0.785059\pi$
$104$	6.96071	0.682554
$105$	0	0
$106$	0.578738	0.0562120
$107$	−0.0656293	−0.00634462	−0.00317231	−	0.999995i	$-0.501010\pi$
$107$	−0.0656293	−0.00634462	−0.00317231	+	0.999995i	$0.501010\pi$
$108$	0	0
$109$	17.7446	1.69963	0.849814	−	0.527082i	$-0.176714\pi$
$109$	17.7446	1.69963	0.849814	+	0.527082i	$0.176714\pi$
$110$	−4.29009	−0.409044
$111$	0	0
$112$	−0.837353	−0.0791224
$113$	−7.48673	−0.704292	−0.352146	−	0.935945i	$-0.614548\pi$
$113$	−7.48673	−0.704292	−0.352146	+	0.935945i	$0.614548\pi$
$114$	0	0
$115$	−0.172227	−0.0160603
$116$	0.340918	0.0316534
$117$	0	0
$118$	12.0862	1.11263
$119$	−16.3247	−1.49649
$120$	0	0
$121$	6.63979	0.603617
$122$	9.02467	0.817055
$123$	0	0
$124$	13.0108	1.16840
$125$	10.3427	0.925080
$126$	0	0
$127$	20.4833	1.81760	0.908802	−	0.417228i	$-0.136999\pi$
$127$	20.4833	1.81760	0.908802	+	0.417228i	$0.136999\pi$
$128$	−7.83437	−0.692467
$129$	0	0
$130$	2.56513	0.224977
$131$	−10.4722	−0.914963	−0.457482	−	0.889219i	$-0.651248\pi$
$131$	−10.4722	−0.914963	−0.457482	+	0.889219i	$0.651248\pi$
$132$	0	0
$133$	−11.0428	−0.957533
$134$	−1.00979	−0.0872326
$135$	0	0
$136$	−13.2790	−1.13867
$137$	0.157548	0.0134602	0.00673010	−	0.999977i	$-0.497858\pi$
$137$	0.157548	0.0134602	0.00673010	+	0.999977i	$0.497858\pi$
$138$	0	0
$139$	−8.11580	−0.688373	−0.344187	−	0.938901i	$-0.611845\pi$
$139$	−8.11580	−0.688373	−0.344187	+	0.938901i	$0.611845\pi$
$140$	−5.33154	−0.450597
$141$	0	0
$142$	−2.65566	−0.222858
$143$	−10.5472	−0.882001
$144$	0	0
$145$	0.320361	0.0266045
$146$	0.129463	0.0107144
$147$	0	0
$148$	−8.44695	−0.694335
$149$	−16.7804	−1.37470	−0.687351	−	0.726325i	$-0.741227\pi$
$149$	−16.7804	−1.37470	−0.687351	+	0.726325i	$0.741227\pi$
$150$	0	0
$151$	−13.1837	−1.07288	−0.536438	−	0.843940i	$-0.680230\pi$
$151$	−13.1837	−1.07288	−0.536438	+	0.843940i	$0.680230\pi$
$152$	−8.98255	−0.728581
$153$	0	0
$154$	−12.0562	−0.971515
$155$	12.2262	0.982035
$156$	0	0
$157$	17.2372	1.37568	0.687840	−	0.725862i	$-0.258559\pi$
$157$	17.2372	1.37568	0.687840	+	0.725862i	$0.258559\pi$
$158$	−3.08032	−0.245057
$159$	0	0
$160$	−6.97291	−0.551257
$161$	−0.484000	−0.0381445
$162$	0	0
$163$	1.08125	0.0846901	0.0423451	−	0.999103i	$-0.486517\pi$
$163$	1.08125	0.0846901	0.0423451	+	0.999103i	$0.486517\pi$
$164$	−12.3699	−0.965924
$165$	0	0
$166$	3.33999	0.259233
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−6.69360	−0.514893
$170$	−4.89354	−0.375317
$171$	0	0
$172$	−13.2638	−1.01135
$173$	15.9953	1.21610	0.608049	−	0.793899i	$-0.291952\pi$
$173$	15.9953	1.21610	0.608049	+	0.793899i	$0.291952\pi$
$174$	0	0
$175$	12.0277	0.909212
$176$	1.03208	0.0777957
$177$	0	0
$178$	−6.85763	−0.514001
$179$	19.0256	1.42204	0.711018	−	0.703173i	$-0.248234\pi$
$179$	19.0256	1.42204	0.711018	+	0.703173i	$0.248234\pi$
$180$	0	0
$181$	−20.3346	−1.51146	−0.755731	−	0.654883i	$-0.772718\pi$
$181$	−20.3346	−1.51146	−0.755731	+	0.654883i	$0.772718\pi$
$182$	7.20865	0.534341
$183$	0	0
$184$	−0.393700	−0.0290239
$185$	−7.93761	−0.583584
$186$	0	0
$187$	20.1210	1.47139
$188$	−9.22259	−0.672626
$189$	0	0
$190$	−3.31021	−0.240148
$191$	16.4252	1.18849	0.594243	−	0.804286i	$-0.297452\pi$
$191$	16.4252	1.18849	0.594243	+	0.804286i	$0.297452\pi$
$192$	0	0
$193$	6.63951	0.477922	0.238961	−	0.971029i	$-0.423193\pi$
$193$	6.63951	0.477922	0.238961	+	0.971029i	$0.423193\pi$
$194$	2.90294	0.208419
$195$	0	0
$196$	−5.95041	−0.425029
$197$	−7.07078	−0.503772	−0.251886	−	0.967757i	$-0.581051\pi$
$197$	−7.07078	−0.503772	−0.251886	+	0.967757i	$0.581051\pi$
$198$	0	0
$199$	−0.717075	−0.0508321	−0.0254161	−	0.999677i	$-0.508091\pi$
$199$	−0.717075	−0.0508321	−0.0254161	+	0.999677i	$0.508091\pi$
$200$	9.78372	0.691813
$201$	0	0
$202$	−11.2416	−0.790957
$203$	0.900292	0.0631881
$204$	0	0
$205$	−11.6240	−0.811853
$206$	−13.3465	−0.929895
$207$	0	0
$208$	−0.617100	−0.0427882
$209$	13.6108	0.941478
$210$	0	0
$211$	12.1736	0.838066	0.419033	−	0.907971i	$-0.362369\pi$
$211$	12.1736	0.838066	0.419033	+	0.907971i	$0.362369\pi$
$212$	−0.886486	−0.0608841
$213$	0	0
$214$	−0.0552863	−0.00377930
$215$	−12.4640	−0.850036
$216$	0	0
$217$	34.3587	2.33242
$218$	14.9481	1.01242
$219$	0	0
$220$	6.57137	0.443042
$221$	−12.0308	−0.809278
$222$	0	0
$223$	6.72478	0.450325	0.225162	−	0.974321i	$-0.427709\pi$
$223$	6.72478	0.450325	0.225162	+	0.974321i	$0.427709\pi$
$224$	−19.5956	−1.30928
$225$	0	0
$226$	−6.30685	−0.419525
$227$	11.2701	0.748023	0.374011	−	0.927424i	$-0.377982\pi$
$227$	11.2701	0.748023	0.374011	+	0.927424i	$0.377982\pi$
$228$	0	0
$229$	−6.52089	−0.430912	−0.215456	−	0.976514i	$-0.569124\pi$
$229$	−6.52089	−0.430912	−0.215456	+	0.976514i	$0.569124\pi$
$230$	−0.145085	−0.00956660
$231$	0	0
$232$	0.732324	0.0480794
$233$	22.0249	1.44290	0.721451	−	0.692466i	$-0.243476\pi$
$233$	22.0249	1.44290	0.721451	+	0.692466i	$0.243476\pi$
$234$	0	0
$235$	−8.66647	−0.565338
$236$	−18.5131	−1.20510
$237$	0	0
$238$	−13.7520	−0.891411
$239$	−4.88322	−0.315869	−0.157934	−	0.987450i	$-0.550483\pi$
$239$	−4.88322	−0.315869	−0.157934	+	0.987450i	$0.550483\pi$
$240$	0	0
$241$	−14.3294	−0.923039	−0.461519	−	0.887130i	$-0.652696\pi$
$241$	−14.3294	−0.923039	−0.461519	+	0.887130i	$0.652696\pi$
$242$	5.59338	0.359556
$243$	0	0
$244$	−13.8236	−0.884965
$245$	−5.59161	−0.357235
$246$	0	0
$247$	−8.13818	−0.517820
$248$	27.9484	1.77472
$249$	0	0
$250$	8.71273	0.551042
$251$	−2.59502	−0.163796	−0.0818981	−	0.996641i	$-0.526098\pi$
$251$	−2.59502	−0.163796	−0.0818981	+	0.996641i	$0.526098\pi$
$252$	0	0
$253$	0.596553	0.0375049
$254$	17.2552	1.08269
$255$	0	0
$256$	−15.3054	−0.956590
$257$	29.6878	1.85187	0.925937	−	0.377679i	$-0.123278\pi$
$257$	29.6878	1.85187	0.925937	+	0.377679i	$0.123278\pi$
$258$	0	0
$259$	−22.3066	−1.38606
$260$	−3.92916	−0.243676
$261$	0	0
$262$	−8.82185	−0.545015
$263$	4.24468	0.261738	0.130869	−	0.991400i	$-0.458223\pi$
$263$	4.24468	0.261738	0.130869	+	0.991400i	$0.458223\pi$
$264$	0	0
$265$	−0.833032	−0.0511727
$266$	−9.30250	−0.570373
$267$	0	0
$268$	1.54675	0.0944830
$269$	29.8482	1.81988	0.909938	−	0.414744i	$-0.136129\pi$
$269$	29.8482	1.81988	0.909938	+	0.414744i	$0.136129\pi$
$270$	0	0
$271$	−31.6214	−1.92086	−0.960432	−	0.278516i	$-0.910158\pi$
$271$	−31.6214	−1.92086	−0.960432	+	0.278516i	$0.910158\pi$
$272$	1.17725	0.0713812
$273$	0	0
$274$	0.132719	0.00801782
$275$	−14.8248	−0.893966
$276$	0	0
$277$	20.4671	1.22975	0.614874	−	0.788625i	$-0.289207\pi$
$277$	20.4671	1.22975	0.614874	+	0.788625i	$0.289207\pi$
$278$	−6.83678	−0.410043
$279$	0	0
$280$	−11.4526	−0.684426
$281$	22.8045	1.36040	0.680202	−	0.733025i	$-0.261892\pi$
$281$	22.8045	1.36040	0.680202	+	0.733025i	$0.261892\pi$
$282$	0	0
$283$	−6.36109	−0.378128	−0.189064	−	0.981965i	$-0.560545\pi$
$283$	−6.36109	−0.378128	−0.189064	+	0.981965i	$0.560545\pi$
$284$	4.06783	0.241381
$285$	0	0
$286$	−8.88500	−0.525381
$287$	−32.6662	−1.92822
$288$	0	0
$289$	5.95127	0.350075
$290$	0.269873	0.0158475
$291$	0	0
$292$	−0.198305	−0.0116049
$293$	3.49011	0.203894	0.101947	−	0.994790i	$-0.467493\pi$
$293$	3.49011	0.203894	0.101947	+	0.994790i	$0.467493\pi$
$294$	0	0
$295$	−17.3968	−1.01288
$296$	−18.1448	−1.05465
$297$	0	0
$298$	−14.1358	−0.818868
$299$	−0.356691	−0.0206280
$300$	0	0
$301$	−35.0268	−2.01891
$302$	−11.1060	−0.639079
$303$	0	0
$304$	0.796346	0.0456736
$305$	−12.9900	−0.743808
$306$	0	0
$307$	−3.64448	−0.208002	−0.104001	−	0.994577i	$-0.533164\pi$
$307$	−3.64448	−0.208002	−0.104001	+	0.994577i	$0.533164\pi$
$308$	18.4671	1.05226
$309$	0	0
$310$	10.2994	0.584968
$311$	−10.9171	−0.619049	−0.309525	−	0.950891i	$-0.600170\pi$
$311$	−10.9171	−0.619049	−0.309525	+	0.950891i	$0.600170\pi$
$312$	0	0
$313$	−2.69366	−0.152254	−0.0761272	−	0.997098i	$-0.524256\pi$
$313$	−2.69366	−0.152254	−0.0761272	+	0.997098i	$0.524256\pi$
$314$	14.5207	0.819450
$315$	0	0
$316$	4.71830	0.265425
$317$	11.6786	0.655937	0.327968	−	0.944689i	$-0.393636\pi$
$317$	11.6786	0.655937	0.327968	+	0.944689i	$0.393636\pi$
$318$	0	0
$319$	−1.10965	−0.0621286
$320$	−5.27807	−0.295053
$321$	0	0
$322$	−0.407723	−0.0227215
$323$	15.5253	0.863851
$324$	0	0
$325$	8.86403	0.491688
$326$	0.910850	0.0504473
$327$	0	0
$328$	−26.5716	−1.46717
$329$	−24.3549	−1.34273
$330$	0	0
$331$	−4.93784	−0.271408	−0.135704	−	0.990749i	$-0.543330\pi$
$331$	−4.93784	−0.271408	−0.135704	+	0.990749i	$0.543330\pi$
$332$	−5.11605	−0.280780
$333$	0	0
$334$	−0.842403	−0.0460943
$335$	1.45348	0.0794124
$336$	0	0
$337$	1.78104	0.0970192	0.0485096	−	0.998823i	$-0.484553\pi$
$337$	1.78104	0.0970192	0.0485096	+	0.998823i	$0.484553\pi$
$338$	−5.63871	−0.306706
$339$	0	0
$340$	7.49571	0.406512
$341$	−42.3487	−2.29331
$342$	0	0
$343$	8.13913	0.439472
$344$	−28.4918	−1.53618
$345$	0	0
$346$	13.4745	0.724392
$347$	−0.767188	−0.0411848	−0.0205924	−	0.999788i	$-0.506555\pi$
$347$	−0.767188	−0.0411848	−0.0205924	+	0.999788i	$0.506555\pi$
$348$	0	0
$349$	21.8125	1.16760	0.583798	−	0.811899i	$-0.301566\pi$
$349$	21.8125	1.16760	0.583798	+	0.811899i	$0.301566\pi$
$350$	10.1322	0.541589
$351$	0	0
$352$	24.1525	1.28733
$353$	−22.7534	−1.21104	−0.605521	−	0.795829i	$-0.707035\pi$
$353$	−22.7534	−1.21104	−0.605521	+	0.795829i	$0.707035\pi$
$354$	0	0
$355$	3.82254	0.202879
$356$	10.5042	0.556722
$357$	0	0
$358$	16.0272	0.847063
$359$	30.8595	1.62870	0.814350	−	0.580374i	$-0.197093\pi$
$359$	30.8595	1.62870	0.814350	+	0.580374i	$0.197093\pi$
$360$	0	0
$361$	−8.49797	−0.447262
$362$	−17.1300	−0.900331
$363$	0	0
$364$	−11.0419	−0.578753
$365$	−0.186348	−0.00975388
$366$	0	0
$367$	5.53204	0.288770	0.144385	−	0.989522i	$-0.453880\pi$
$367$	5.53204	0.288770	0.144385	+	0.989522i	$0.453880\pi$
$368$	0.0349034	0.00181946
$369$	0	0
$370$	−6.68666	−0.347623
$371$	−2.34102	−0.121540
$372$	0	0
$373$	32.5894	1.68742	0.843708	−	0.536802i	$-0.180368\pi$
$373$	32.5894	1.68742	0.843708	+	0.536802i	$0.180368\pi$
$374$	16.9500	0.876464
$375$	0	0
$376$	−19.8110	−1.02167
$377$	0.663484	0.0341712
$378$	0	0
$379$	−34.8665	−1.79097	−0.895487	−	0.445088i	$-0.853172\pi$
$379$	−34.8665	−1.79097	−0.895487	+	0.445088i	$0.853172\pi$
$380$	5.07044	0.260108
$381$	0	0
$382$	13.8366	0.707944
$383$	−2.30034	−0.117542	−0.0587709	−	0.998271i	$-0.518718\pi$
$383$	−2.30034	−0.117542	−0.0587709	+	0.998271i	$0.518718\pi$
$384$	0	0
$385$	17.3536	0.884421
$386$	5.59314	0.284683
$387$	0	0
$388$	−4.44660	−0.225742
$389$	−0.332692	−0.0168682	−0.00843408	−	0.999964i	$-0.502685\pi$
$389$	−0.332692	−0.0168682	−0.00843408	+	0.999964i	$0.502685\pi$
$390$	0	0
$391$	0.680464	0.0344126
$392$	−12.7820	−0.645591
$393$	0	0
$394$	−5.95645	−0.300082
$395$	4.43379	0.223088
$396$	0	0
$397$	27.8441	1.39745	0.698727	−	0.715388i	$-0.253750\pi$
$397$	27.8441	1.39745	0.698727	+	0.715388i	$0.253750\pi$
$398$	−0.604067	−0.0302791
$399$	0	0
$400$	−0.867373	−0.0433687
$401$	9.63219	0.481009	0.240504	−	0.970648i	$-0.422687\pi$
$401$	9.63219	0.481009	0.240504	+	0.970648i	$0.422687\pi$
$402$	0	0
$403$	25.3212	1.26134
$404$	17.2194	0.856698
$405$	0	0
$406$	0.758409	0.0376392
$407$	27.4939	1.36282
$408$	0	0
$409$	37.0060	1.82983	0.914915	−	0.403647i	$-0.132258\pi$
$409$	37.0060	1.82983	0.914915	+	0.403647i	$0.132258\pi$
$410$	−9.79207	−0.483596
$411$	0	0
$412$	20.4436	1.00718
$413$	−48.8892	−2.40568
$414$	0	0
$415$	−4.80756	−0.235994
$416$	−14.4413	−0.708042
$417$	0	0
$418$	11.4658	0.560809
$419$	6.04230	0.295186	0.147593	−	0.989048i	$-0.452848\pi$
$419$	6.04230	0.295186	0.147593	+	0.989048i	$0.452848\pi$
$420$	0	0
$421$	14.4831	0.705862	0.352931	−	0.935649i	$-0.385185\pi$
$421$	14.4831	0.705862	0.352931	+	0.935649i	$0.385185\pi$
$422$	10.2551	0.499210
$423$	0	0
$424$	−1.90426	−0.0924788
$425$	−16.9100	−0.820256
$426$	0	0
$427$	−36.5052	−1.76661
$428$	0.0846852	0.00409341
$429$	0	0
$430$	−10.4997	−0.506340
$431$	3.16758	0.152577	0.0762885	−	0.997086i	$-0.475693\pi$
$431$	3.16758	0.152577	0.0762885	+	0.997086i	$0.475693\pi$
$432$	0	0
$433$	26.8759	1.29157	0.645787	−	0.763518i	$-0.276529\pi$
$433$	26.8759	1.29157	0.645787	+	0.763518i	$0.276529\pi$
$434$	28.9439	1.38935
$435$	0	0
$436$	−22.8969	−1.09656
$437$	0.460298	0.0220190
$438$	0	0
$439$	14.6647	0.699909	0.349955	−	0.936767i	$-0.386197\pi$
$439$	14.6647	0.699909	0.349955	+	0.936767i	$0.386197\pi$
$440$	14.1159	0.672950
$441$	0	0
$442$	−10.1348	−0.482062
$443$	−2.62513	−0.124724	−0.0623618	−	0.998054i	$-0.519863\pi$
$443$	−2.62513	−0.124724	−0.0623618	+	0.998054i	$0.519863\pi$
$444$	0	0
$445$	9.87082	0.467922
$446$	5.66498	0.268244
$447$	0	0
$448$	−14.8327	−0.700778
$449$	−2.26746	−0.107008	−0.0535041	−	0.998568i	$-0.517039\pi$
$449$	−2.26746	−0.107008	−0.0535041	+	0.998568i	$0.517039\pi$
$450$	0	0
$451$	40.2626	1.89589
$452$	9.66055	0.454394
$453$	0	0
$454$	9.49397	0.445574
$455$	−10.3761	−0.486438
$456$	0	0
$457$	−13.2244	−0.618610	−0.309305	−	0.950963i	$-0.600096\pi$
$457$	−13.2244	−0.618610	−0.309305	+	0.950963i	$0.600096\pi$
$458$	−5.49322	−0.256681
$459$	0	0
$460$	0.222234	0.0103617
$461$	33.5434	1.56227	0.781135	−	0.624362i	$-0.214641\pi$
$461$	33.5434	1.56227	0.781135	+	0.624362i	$0.214641\pi$
$462$	0	0
$463$	−22.8174	−1.06042	−0.530208	−	0.847867i	$-0.677886\pi$
$463$	−22.8174	−1.06042	−0.530208	+	0.847867i	$0.677886\pi$
$464$	−0.0649240	−0.00301402
$465$	0	0
$466$	18.5539	0.859492
$467$	−12.1148	−0.560605	−0.280303	−	0.959912i	$-0.590435\pi$
$467$	−12.1148	−0.560605	−0.280303	+	0.959912i	$0.590435\pi$
$468$	0	0
$469$	4.08464	0.188611
$470$	−7.30066	−0.336755
$471$	0	0
$472$	−39.7679	−1.83047
$473$	43.1722	1.98506
$474$	0	0
$475$	−11.4387	−0.524844
$476$	21.0647	0.965501
$477$	0	0
$478$	−4.11364	−0.188153
$479$	2.35149	0.107442	0.0537211	−	0.998556i	$-0.482892\pi$
$479$	2.35149	0.107442	0.0537211	+	0.998556i	$0.482892\pi$
$480$	0	0
$481$	−16.4392	−0.749563
$482$	−12.0711	−0.549826
$483$	0	0
$484$	−8.56769	−0.389441
$485$	−4.17847	−0.189735
$486$	0	0
$487$	−15.3635	−0.696187	−0.348094	−	0.937460i	$-0.613171\pi$
$487$	−15.3635	−0.696187	−0.348094	+	0.937460i	$0.613171\pi$
$488$	−29.6944	−1.34420
$489$	0	0
$490$	−4.71039	−0.212794
$491$	20.0376	0.904283	0.452141	−	0.891946i	$-0.350660\pi$
$491$	20.0376	0.904283	0.452141	+	0.891946i	$0.350660\pi$
$492$	0	0
$493$	−1.26574	−0.0570059
$494$	−6.85563	−0.308449
$495$	0	0
$496$	−2.47776	−0.111255
$497$	10.7423	0.481857
$498$	0	0
$499$	−1.87902	−0.0841166	−0.0420583	−	0.999115i	$-0.513392\pi$
$499$	−1.87902	−0.0841166	−0.0420583	+	0.999115i	$0.513392\pi$
$500$	−13.3458	−0.596842
$501$	0	0
$502$	−2.18605	−0.0975683
$503$	−18.7896	−0.837787	−0.418893	−	0.908035i	$-0.637582\pi$
$503$	−18.7896	−0.837787	−0.418893	+	0.908035i	$0.637582\pi$
$504$	0	0
$505$	16.1811	0.720049
$506$	0.502538	0.0223405
$507$	0	0
$508$	−26.4308	−1.17268
$509$	−40.2594	−1.78447	−0.892233	−	0.451576i	$-0.850862\pi$
$509$	−40.2594	−1.78447	−0.892233	+	0.451576i	$0.850862\pi$
$510$	0	0
$511$	−0.523682	−0.0231663
$512$	2.77538	0.122656
$513$	0	0
$514$	25.0091	1.10310
$515$	19.2109	0.846532
$516$	0	0
$517$	30.0185	1.32021
$518$	−18.7911	−0.825636
$519$	0	0
$520$	−8.44021	−0.370128
$521$	−10.1422	−0.444337	−0.222168	−	0.975008i	$-0.571313\pi$
$521$	−10.1422	−0.444337	−0.222168	+	0.975008i	$0.571313\pi$
$522$	0	0
$523$	−32.5975	−1.42539	−0.712694	−	0.701475i	$-0.752525\pi$
$523$	−32.5975	−1.42539	−0.712694	+	0.701475i	$0.752525\pi$
$524$	13.5129	0.590315
$525$	0	0
$526$	3.57573	0.155909
$527$	−48.3055	−2.10422
$528$	0	0
$529$	−22.9798	−0.999123
$530$	−0.701749	−0.0304820
$531$	0	0
$532$	14.2492	0.617780
$533$	−24.0739	−1.04275
$534$	0	0
$535$	0.0795787	0.00344049
$536$	3.32257	0.143513
$537$	0	0
$538$	25.1442	1.08404
$539$	19.3680	0.834237
$540$	0	0
$541$	−15.3949	−0.661880	−0.330940	−	0.943652i	$-0.607366\pi$
$541$	−15.3949	−0.661880	−0.330940	+	0.943652i	$0.607366\pi$
$542$	−26.6380	−1.14420
$543$	0	0
$544$	27.5498	1.18119
$545$	−21.5163	−0.921655
$546$	0	0
$547$	41.5584	1.77691	0.888455	−	0.458964i	$-0.151779\pi$
$547$	41.5584	1.77691	0.888455	+	0.458964i	$0.151779\pi$
$548$	−0.203292	−0.00868422
$549$	0	0
$550$	−12.4884	−0.532508
$551$	−0.856203	−0.0364755
$552$	0	0
$553$	12.4600	0.529855
$554$	17.2415	0.732523
$555$	0	0
$556$	10.4723	0.444124
$557$	−18.5224	−0.784817	−0.392409	−	0.919791i	$-0.628358\pi$
$557$	−18.5224	−0.784817	−0.392409	+	0.919791i	$0.628358\pi$
$558$	0	0
$559$	−25.8135	−1.09180
$560$	1.01533	0.0429056
$561$	0	0
$562$	19.2106	0.810350
$563$	−31.9142	−1.34502	−0.672511	−	0.740087i	$-0.734784\pi$
$563$	−31.9142	−1.34502	−0.672511	+	0.740087i	$0.734784\pi$
$564$	0	0
$565$	9.07803	0.381916
$566$	−5.35860	−0.225239
$567$	0	0
$568$	8.73808	0.366642
$569$	27.6777	1.16031	0.580156	−	0.814506i	$-0.302992\pi$
$569$	27.6777	1.16031	0.580156	+	0.814506i	$0.302992\pi$
$570$	0	0
$571$	−32.6675	−1.36709	−0.683546	−	0.729907i	$-0.739563\pi$
$571$	−32.6675	−1.36709	−0.683546	+	0.729907i	$0.739563\pi$
$572$	13.6097	0.569048
$573$	0	0
$574$	−27.5181	−1.14858
$575$	−0.501352	−0.0209078
$576$	0	0
$577$	−32.7241	−1.36232	−0.681162	−	0.732133i	$-0.738525\pi$
$577$	−32.7241	−1.36232	−0.681162	+	0.732133i	$0.738525\pi$
$578$	5.01337	0.208529
$579$	0	0
$580$	−0.413380	−0.0171647
$581$	−13.5104	−0.560506
$582$	0	0
$583$	2.88542	0.119502
$584$	−0.425978	−0.0176271
$585$	0	0
$586$	2.94008	0.121454
$587$	17.2455	0.711798	0.355899	−	0.934524i	$-0.384175\pi$
$587$	17.2455	0.711798	0.355899	+	0.934524i	$0.384175\pi$
$588$	0	0
$589$	−32.6761	−1.34639
$590$	−14.6551	−0.603342
$591$	0	0
$592$	1.60863	0.0661142
$593$	44.0912	1.81061	0.905305	−	0.424762i	$-0.139642\pi$
$593$	44.0912	1.81061	0.905305	+	0.424762i	$0.139642\pi$
$594$	0	0
$595$	19.7946	0.811498
$596$	21.6527	0.886928
$597$	0	0
$598$	−0.300478	−0.0122875
$599$	−4.74123	−0.193721	−0.0968606	−	0.995298i	$-0.530880\pi$
$599$	−4.74123	−0.193721	−0.0968606	+	0.995298i	$0.530880\pi$
$600$	0	0
$601$	3.28382	0.133950	0.0669750	−	0.997755i	$-0.478665\pi$
$601$	3.28382	0.133950	0.0669750	+	0.997755i	$0.478665\pi$
$602$	−29.5067	−1.20260
$603$	0	0
$604$	17.0117	0.692196
$605$	−8.05107	−0.327323
$606$	0	0
$607$	−14.4426	−0.586206	−0.293103	−	0.956081i	$-0.594688\pi$
$607$	−14.4426	−0.586206	−0.293103	+	0.956081i	$0.594688\pi$
$608$	18.6359	0.755787
$609$	0	0
$610$	−10.9429	−0.443063
$611$	−17.9487	−0.726128
$612$	0	0
$613$	1.58209	0.0639000	0.0319500	−	0.999489i	$-0.489828\pi$
$613$	1.58209	0.0639000	0.0319500	+	0.999489i	$0.489828\pi$
$614$	−3.07013	−0.123900
$615$	0	0
$616$	39.6691	1.59832
$617$	11.2773	0.454008	0.227004	−	0.973894i	$-0.427107\pi$
$617$	11.2773	0.454008	0.227004	+	0.973894i	$0.427107\pi$
$618$	0	0
$619$	−28.9029	−1.16171	−0.580854	−	0.814008i	$-0.697281\pi$
$619$	−28.9029	−1.16171	−0.580854	+	0.814008i	$0.697281\pi$
$620$	−15.7762	−0.633588
$621$	0	0
$622$	−9.19656	−0.368748
$623$	27.7394	1.11136
$624$	0	0
$625$	5.10758	0.204303
$626$	−2.26914	−0.0906933
$627$	0	0
$628$	−22.2422	−0.887559
$629$	31.3613	1.25046
$630$	0	0
$631$	−3.50507	−0.139534	−0.0697672	−	0.997563i	$-0.522226\pi$
$631$	−3.50507	−0.139534	−0.0697672	+	0.997563i	$0.522226\pi$
$632$	10.1354	0.403163
$633$	0	0
$634$	9.83811	0.390721
$635$	−24.8371	−0.985629
$636$	0	0
$637$	−11.5805	−0.458837
$638$	−0.934774	−0.0370081
$639$	0	0
$640$	9.49956	0.375503
$641$	4.89788	0.193454	0.0967272	−	0.995311i	$-0.469163\pi$
$641$	4.89788	0.193454	0.0967272	+	0.995311i	$0.469163\pi$
$642$	0	0
$643$	31.2912	1.23400	0.617001	−	0.786962i	$-0.288347\pi$
$643$	31.2912	1.23400	0.617001	+	0.786962i	$0.288347\pi$
$644$	0.624533	0.0246100
$645$	0	0
$646$	13.0786	0.514569
$647$	−30.8557	−1.21306	−0.606532	−	0.795059i	$-0.707440\pi$
$647$	−30.8557	−1.21306	−0.606532	+	0.795059i	$0.707440\pi$
$648$	0	0
$649$	60.2583	2.36534
$650$	7.46709	0.292883
$651$	0	0
$652$	−1.39520	−0.0546402
$653$	−29.1220	−1.13963	−0.569815	−	0.821773i	$-0.692985\pi$
$653$	−29.1220	−1.13963	−0.569815	+	0.821773i	$0.692985\pi$
$654$	0	0
$655$	12.6981	0.496156
$656$	2.35570	0.0919747
$657$	0	0
$658$	−20.5166	−0.799822
$659$	−21.0879	−0.821469	−0.410734	−	0.911755i	$-0.634728\pi$
$659$	−21.0879	−0.821469	−0.410734	+	0.911755i	$0.634728\pi$
$660$	0	0
$661$	−17.9637	−0.698709	−0.349354	−	0.936991i	$-0.613599\pi$
$661$	−17.9637	−0.698709	−0.349354	+	0.936991i	$0.613599\pi$
$662$	−4.15965	−0.161669
$663$	0	0
$664$	−10.9898	−0.426485
$665$	13.3900	0.519240
$666$	0	0
$667$	−0.0375269	−0.00145305
$668$	1.29036	0.0499254
$669$	0	0
$670$	1.22442	0.0473035
$671$	44.9943	1.73699
$672$	0	0
$673$	−16.9700	−0.654146	−0.327073	−	0.944999i	$-0.606062\pi$
$673$	−16.9700	−0.654146	−0.327073	+	0.944999i	$0.606062\pi$
$674$	1.50035	0.0577913
$675$	0	0
$676$	8.63714	0.332198
$677$	21.1696	0.813614	0.406807	−	0.913514i	$-0.366642\pi$
$677$	21.1696	0.813614	0.406807	+	0.913514i	$0.366642\pi$
$678$	0	0
$679$	−11.7425	−0.450636
$680$	16.1015	0.617463
$681$	0	0
$682$	−35.6747	−1.36605
$683$	23.8834	0.913874	0.456937	−	0.889499i	$-0.348946\pi$
$683$	23.8834	0.913874	0.456937	+	0.889499i	$0.348946\pi$
$684$	0	0
$685$	−0.191034	−0.00729904
$686$	6.85643	0.261780
$687$	0	0
$688$	2.52594	0.0963004
$689$	−1.72525	−0.0657269
$690$	0	0
$691$	−7.70109	−0.292963	−0.146482	−	0.989213i	$-0.546795\pi$
$691$	−7.70109	−0.292963	−0.146482	+	0.989213i	$0.546795\pi$
$692$	−20.6396	−0.784601
$693$	0	0
$694$	−0.646281	−0.0245325
$695$	9.84081	0.373283
$696$	0	0
$697$	45.9260	1.73957
$698$	18.3749	0.695500
$699$	0	0
$700$	−15.5201	−0.586604
$701$	27.4380	1.03632	0.518159	−	0.855284i	$-0.326618\pi$
$701$	27.4380	1.03632	0.518159	+	0.855284i	$0.326618\pi$
$702$	0	0
$703$	21.2142	0.800109
$704$	18.2820	0.689027
$705$	0	0
$706$	−19.1675	−0.721380
$707$	45.4728	1.71018
$708$	0	0
$709$	13.6483	0.512572	0.256286	−	0.966601i	$-0.417501\pi$
$709$	13.6483	0.512572	0.256286	+	0.966601i	$0.417501\pi$
$710$	3.22012	0.120849
$711$	0	0
$712$	22.5640	0.845623
$713$	−1.43217	−0.0536353
$714$	0	0
$715$	12.7890	0.478282
$716$	−24.5498	−0.917467
$717$	0	0
$718$	25.9961	0.970167
$719$	−12.2274	−0.456007	−0.228003	−	0.973660i	$-0.573220\pi$
$719$	−12.2274	−0.456007	−0.228003	+	0.973660i	$0.573220\pi$
$720$	0	0
$721$	53.9872	2.01059
$722$	−7.15872	−0.266420
$723$	0	0
$724$	26.2389	0.975162
$725$	0.932569	0.0346347
$726$	0	0
$727$	−16.9710	−0.629420	−0.314710	−	0.949188i	$-0.601907\pi$
$727$	−16.9710	−0.629420	−0.314710	+	0.949188i	$0.601907\pi$
$728$	−23.7190	−0.879086
$729$	0	0
$730$	−0.156980	−0.00581009
$731$	49.2448	1.82138
$732$	0	0
$733$	−24.4232	−0.902092	−0.451046	−	0.892501i	$-0.648949\pi$
$733$	−24.4232	−0.902092	−0.451046	+	0.892501i	$0.648949\pi$
$734$	4.66021	0.172012
$735$	0	0
$736$	0.816802	0.0301077
$737$	−5.03452	−0.185449
$738$	0	0
$739$	5.37262	0.197635	0.0988175	−	0.995106i	$-0.468494\pi$
$739$	5.37262	0.197635	0.0988175	+	0.995106i	$0.468494\pi$
$740$	10.2423	0.376516
$741$	0	0
$742$	−1.97208	−0.0723975
$743$	16.3799	0.600921	0.300461	−	0.953794i	$-0.402860\pi$
$743$	16.3799	0.600921	0.300461	+	0.953794i	$0.402860\pi$
$744$	0	0
$745$	20.3470	0.745458
$746$	27.4534	1.00514
$747$	0	0
$748$	−25.9633	−0.949312
$749$	0.223636	0.00817146
$750$	0	0
$751$	5.00958	0.182802	0.0914011	−	0.995814i	$-0.470865\pi$
$751$	5.00958	0.182802	0.0914011	+	0.995814i	$0.470865\pi$
$752$	1.75634	0.0640471
$753$	0	0
$754$	0.558921	0.0203547
$755$	15.9859	0.581787
$756$	0	0
$757$	−19.0233	−0.691415	−0.345708	−	0.938342i	$-0.612361\pi$
$757$	−19.0233	−0.691415	−0.345708	+	0.938342i	$0.612361\pi$
$758$	−29.3717	−1.06683
$759$	0	0
$760$	10.8918	0.395087
$761$	−22.5846	−0.818690	−0.409345	−	0.912380i	$-0.634243\pi$
$761$	−22.5846	−0.818690	−0.409345	+	0.912380i	$0.634243\pi$
$762$	0	0
$763$	−60.4659	−2.18901
$764$	−21.1944	−0.766785
$765$	0	0
$766$	−1.93781	−0.0700160
$767$	−36.0297	−1.30096
$768$	0	0
$769$	−4.96145	−0.178915	−0.0894573	−	0.995991i	$-0.528513\pi$
$769$	−4.96145	−0.178915	−0.0894573	+	0.995991i	$0.528513\pi$
$770$	14.6187	0.526822
$771$	0	0
$772$	−8.56733	−0.308345
$773$	−50.0966	−1.80185	−0.900925	−	0.433974i	$-0.857111\pi$
$773$	−50.0966	−1.80185	−0.900925	+	0.433974i	$0.857111\pi$
$774$	0	0
$775$	35.5905	1.27845
$776$	−9.55171	−0.342886
$777$	0	0
$778$	−0.280261	−0.0100478
$779$	31.0664	1.11307
$780$	0	0
$781$	−13.2404	−0.473777
$782$	0.573225	0.0204985
$783$	0	0
$784$	1.13319	0.0404710
$785$	−20.9010	−0.745988
$786$	0	0
$787$	6.00849	0.214180	0.107090	−	0.994249i	$-0.465847\pi$
$787$	6.00849	0.214180	0.107090	+	0.994249i	$0.465847\pi$
$788$	9.12383	0.325023
$789$	0	0
$790$	3.73504	0.132887
$791$	25.5115	0.907083
$792$	0	0
$793$	−26.9031	−0.955356
$794$	23.4559	0.832420
$795$	0	0
$796$	0.925283	0.0327958
$797$	18.3258	0.649134	0.324567	−	0.945863i	$-0.394781\pi$
$797$	18.3258	0.649134	0.324567	+	0.945863i	$0.394781\pi$
$798$	0	0
$799$	34.2410	1.21136
$800$	−20.2981	−0.717647
$801$	0	0
$802$	8.11419	0.286522
$803$	0.645463	0.0227779
$804$	0	0
$805$	0.586874	0.0206846
$806$	21.3307	0.751340
$807$	0	0
$808$	36.9889	1.30127
$809$	−10.5235	−0.369988	−0.184994	−	0.982740i	$-0.559227\pi$
$809$	−10.5235	−0.369988	−0.184994	+	0.982740i	$0.559227\pi$
$810$	0	0
$811$	2.86669	0.100663	0.0503315	−	0.998733i	$-0.483972\pi$
$811$	2.86669	0.100663	0.0503315	+	0.998733i	$0.483972\pi$
$812$	−1.16170	−0.0407676
$813$	0	0
$814$	23.1610	0.811792
$815$	−1.31107	−0.0459248
$816$	0	0
$817$	33.3115	1.16542
$818$	31.1740	1.08997
$819$	0	0
$820$	14.9991	0.523790
$821$	24.3763	0.850738	0.425369	−	0.905020i	$-0.360144\pi$
$821$	24.3763	0.850738	0.425369	+	0.905020i	$0.360144\pi$
$822$	0	0
$823$	43.9211	1.53099	0.765497	−	0.643440i	$-0.222493\pi$
$823$	43.9211	1.53099	0.765497	+	0.643440i	$0.222493\pi$
$824$	43.9148	1.52984
$825$	0	0
$826$	−41.1844	−1.43299
$827$	−45.4139	−1.57920	−0.789598	−	0.613625i	$-0.789711\pi$
$827$	−45.4139	−1.57920	−0.789598	+	0.613625i	$0.789711\pi$
$828$	0	0
$829$	28.6971	0.996691	0.498346	−	0.866978i	$-0.333941\pi$
$829$	28.6971	0.996691	0.498346	+	0.866978i	$0.333941\pi$
$830$	−4.04990	−0.140574
$831$	0	0
$832$	−10.9312	−0.378970
$833$	22.0923	0.765452
$834$	0	0
$835$	1.21255	0.0419620
$836$	−17.5628	−0.607421
$837$	0	0
$838$	5.09005	0.175833
$839$	−55.5225	−1.91685	−0.958425	−	0.285344i	$-0.907892\pi$
$839$	−55.5225	−1.91685	−0.958425	+	0.285344i	$0.907892\pi$
$840$	0	0
$841$	−28.9302	−0.997593
$842$	12.2006	0.420460
$843$	0	0
$844$	−15.7083	−0.540702
$845$	8.11632	0.279210
$846$	0	0
$847$	−22.6255	−0.777420
$848$	0.168821	0.00579735
$849$	0	0
$850$	−14.2451	−0.488601
$851$	0.929806	0.0318733
$852$	0	0
$853$	−25.5294	−0.874109	−0.437054	−	0.899435i	$-0.643978\pi$
$853$	−25.5294	−0.874109	−0.437054	+	0.899435i	$0.643978\pi$
$854$	−30.7521	−1.05231
$855$	0	0
$856$	0.181912	0.00621761
$857$	28.2575	0.965258	0.482629	−	0.875825i	$-0.339682\pi$
$857$	28.2575	0.965258	0.482629	+	0.875825i	$0.339682\pi$
$858$	0	0
$859$	−38.5020	−1.31367	−0.656835	−	0.754034i	$-0.728105\pi$
$859$	−38.5020	−1.31367	−0.656835	+	0.754034i	$0.728105\pi$
$860$	16.0830	0.548425
$861$	0	0
$862$	2.66838	0.0908854
$863$	10.3942	0.353824	0.176912	−	0.984227i	$-0.443389\pi$
$863$	10.3942	0.353824	0.176912	+	0.984227i	$0.443389\pi$
$864$	0	0
$865$	−19.3951	−0.659452
$866$	22.6404	0.769351
$867$	0	0
$868$	−44.3350	−1.50483
$869$	−15.3576	−0.520970
$870$	0	0
$871$	3.01024	0.101998
$872$	−49.1847	−1.66561
$873$	0	0
$874$	0.387756	0.0131160
$875$	−35.2434	−1.19144
$876$	0	0
$877$	36.3335	1.22689	0.613447	−	0.789736i	$-0.289782\pi$
$877$	36.3335	1.22689	0.613447	+	0.789736i	$0.289782\pi$
$878$	12.3536	0.416914
$879$	0	0
$880$	−1.25144	−0.0421862
$881$	−21.9157	−0.738360	−0.369180	−	0.929358i	$-0.620361\pi$
$881$	−21.9157	−0.738360	−0.369180	+	0.929358i	$0.620361\pi$
$882$	0	0
$883$	22.7689	0.766233	0.383117	−	0.923700i	$-0.374851\pi$
$883$	22.7689	0.766233	0.383117	+	0.923700i	$0.374851\pi$
$884$	15.5240	0.522129
$885$	0	0
$886$	−2.21142	−0.0742940
$887$	−3.73121	−0.125282	−0.0626409	−	0.998036i	$-0.519952\pi$
$887$	−3.73121	−0.125282	−0.0626409	+	0.998036i	$0.519952\pi$
$888$	0	0
$889$	−69.7982	−2.34096
$890$	8.31521	0.278727
$891$	0	0
$892$	−8.67737	−0.290540
$893$	23.1622	0.775093
$894$	0	0
$895$	−23.0694	−0.771126
$896$	26.6961	0.891853
$897$	0	0
$898$	−1.91012	−0.0637415
$899$	2.66400	0.0888492
$900$	0	0
$901$	3.29129	0.109649
$902$	33.9173	1.12932
$903$	0	0
$904$	20.7518	0.690194
$905$	24.6567	0.819618
$906$	0	0
$907$	57.8699	1.92154	0.960770	−	0.277347i	$-0.0894551\pi$
$907$	57.8699	1.92154	0.960770	+	0.277347i	$0.0894551\pi$
$908$	−14.5424	−0.482608
$909$	0	0
$910$	−8.74084	−0.289756
$911$	−47.6341	−1.57819	−0.789095	−	0.614272i	$-0.789450\pi$
$911$	−47.6341	−1.57819	−0.789095	+	0.614272i	$0.789450\pi$
$912$	0	0
$913$	16.6522	0.551107
$914$	−11.1403	−0.368487
$915$	0	0
$916$	8.41427	0.278015
$917$	35.6847	1.17841
$918$	0	0
$919$	51.2555	1.69076	0.845381	−	0.534164i	$-0.179374\pi$
$919$	51.2555	1.69076	0.845381	+	0.534164i	$0.179374\pi$
$920$	0.477380	0.0157388
$921$	0	0
$922$	28.2571	0.930596
$923$	7.91669	0.260581
$924$	0	0
$925$	−23.1063	−0.759731
$926$	−19.2215	−0.631657
$927$	0	0
$928$	−1.51934	−0.0498748
$929$	−14.4121	−0.472847	−0.236424	−	0.971650i	$-0.575975\pi$
$929$	−14.4121	−0.472847	−0.236424	+	0.971650i	$0.575975\pi$
$930$	0	0
$931$	14.9442	0.489778
$932$	−28.4200	−0.930929
$933$	0	0
$934$	−10.2055	−0.333935
$935$	−24.3977	−0.797891
$936$	0	0
$937$	3.65491	0.119401	0.0597004	−	0.998216i	$-0.480985\pi$
$937$	3.65491	0.119401	0.0597004	+	0.998216i	$0.480985\pi$
$938$	3.44092	0.112350
$939$	0	0
$940$	11.1828	0.364744
$941$	31.3761	1.02283	0.511416	−	0.859333i	$-0.329121\pi$
$941$	31.3761	1.02283	0.511416	+	0.859333i	$0.329121\pi$
$942$	0	0
$943$	1.36162	0.0443406
$944$	3.52562	0.114749
$945$	0	0
$946$	36.3684	1.18244
$947$	2.52840	0.0821620	0.0410810	−	0.999156i	$-0.486920\pi$
$947$	2.52840	0.0821620	0.0410810	+	0.999156i	$0.486920\pi$
$948$	0	0
$949$	−0.385936	−0.0125280
$950$	−9.63601	−0.312634
$951$	0	0
$952$	45.2491	1.46653
$953$	−25.2734	−0.818687	−0.409344	−	0.912380i	$-0.634242\pi$
$953$	−25.2734	−0.818687	−0.409344	+	0.912380i	$0.634242\pi$
$954$	0	0
$955$	−19.9164	−0.644478
$956$	6.30109	0.203792
$957$	0	0
$958$	1.98090	0.0640000
$959$	−0.536852	−0.0173359
$960$	0	0
$961$	70.6686	2.27963
$962$	−13.8484	−0.446492
$963$	0	0
$964$	18.4901	0.595525
$965$	−8.05073	−0.259162
$966$	0	0
$967$	0.518625	0.0166779	0.00833893	−	0.999965i	$-0.497346\pi$
$967$	0.518625	0.0166779	0.00833893	+	0.999965i	$0.497346\pi$
$968$	−18.4042	−0.591534
$969$	0	0
$970$	−3.51996	−0.113019
$971$	47.8549	1.53574	0.767869	−	0.640607i	$-0.221317\pi$
$971$	47.8549	1.53574	0.767869	+	0.640607i	$0.221317\pi$
$972$	0	0
$973$	27.6551	0.886580
$974$	−12.9423	−0.414697
$975$	0	0
$976$	2.63255	0.0842658
$977$	45.8694	1.46749	0.733746	−	0.679423i	$-0.237770\pi$
$977$	45.8694	1.46749	0.733746	+	0.679423i	$0.237770\pi$
$978$	0	0
$979$	−34.1901	−1.09272
$980$	7.21517	0.230480
$981$	0	0
$982$	16.8797	0.538653
$983$	27.4603	0.875848	0.437924	−	0.899012i	$-0.355714\pi$
$983$	27.4603	0.875848	0.437924	+	0.899012i	$0.355714\pi$
$984$	0	0
$985$	8.57367	0.273180
$986$	−1.06626	−0.0339567
$987$	0	0
$988$	10.5012	0.334086
$989$	1.46002	0.0464260
$990$	0	0
$991$	−8.77212	−0.278656	−0.139328	−	0.990246i	$-0.544494\pi$
$991$	−8.77212	−0.278656	−0.139328	+	0.990246i	$0.544494\pi$
$992$	−57.9840	−1.84099
$993$	0	0
$994$	9.04932	0.287027
$995$	0.869489	0.0275647
$996$	0	0
$997$	−7.87335	−0.249351	−0.124676	−	0.992198i	$-0.539789\pi$
$997$	−7.87335	−0.249351	−0.124676	+	0.992198i	$0.539789\pi$
$998$	−1.58289	−0.0501056
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.2.a.d.1.3		5
3.2	odd	2		501.2.a.b.1.3	✓	5
12.11	even	2		8016.2.a.p.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.b.1.3	✓	5	3.2	odd	2
1503.2.a.d.1.3		5	1.1	even	1	trivial
8016.2.a.p.1.5		5	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:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q+0.842403 q^{2} -1.29036 q^{4} -1.21255 q^{5} -3.40756 q^{7} -2.77181 q^{8} +O(q^{10})\)	\(q+0.842403 q^{2} -1.29036 q^{4} -1.21255 q^{5} -3.40756 q^{7} -2.77181 q^{8} -1.02146 q^{10} +4.19997 q^{11} -2.51125 q^{13} -2.87054 q^{14} +0.245734 q^{16} +4.79075 q^{17} +3.24068 q^{19} +1.56462 q^{20} +3.53807 q^{22} +0.142037 q^{23} -3.52972 q^{25} -2.11549 q^{26} +4.39696 q^{28} -0.264204 q^{29} -10.0831 q^{31} +5.75062 q^{32} +4.03574 q^{34} +4.13183 q^{35} +6.54621 q^{37} +2.72996 q^{38} +3.36095 q^{40} +9.58639 q^{41} +10.2791 q^{43} -5.41947 q^{44} +0.119653 q^{46} +7.14732 q^{47} +4.61145 q^{49} -2.97345 q^{50} +3.24041 q^{52} +0.687009 q^{53} -5.09268 q^{55} +9.44509 q^{56} -0.222567 q^{58} +14.3473 q^{59} +10.7130 q^{61} -8.49403 q^{62} +4.35287 q^{64} +3.04502 q^{65} -1.19870 q^{67} -6.18177 q^{68} +3.48067 q^{70} -3.15248 q^{71} +0.153683 q^{73} +5.51455 q^{74} -4.18164 q^{76} -14.3117 q^{77} -3.65659 q^{79} -0.297965 q^{80} +8.07561 q^{82} +3.96483 q^{83} -5.80902 q^{85} +8.65919 q^{86} -11.6415 q^{88} -8.14055 q^{89} +8.55724 q^{91} -0.183279 q^{92} +6.02092 q^{94} -3.92949 q^{95} +3.44602 q^{97} +3.88470 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8}+O(q^{10}) \) 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8	\( 5 q + 4 q^{2} + 4 q^{4} + 9 q^{5} - 4 q^{7} + 12 q^{8} + 5 q^{10} + 15 q^{11} + 3 q^{14} + 10 q^{16} + 11 q^{17} - 16 q^{19} + 9 q^{20} + 7 q^{22} + 9 q^{23} + 6 q^{25} - 7 q^{26} + 9 q^{28} + q^{29} - 18 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 7 q^{37} - 20 q^{38} + 32 q^{40} + 10 q^{41} + 6 q^{43} + 5 q^{44} + 13 q^{46} + 7 q^{47} + 11 q^{49} - 3 q^{50} - 6 q^{52} + 9 q^{53} + 17 q^{55} + 15 q^{56} - 19 q^{58} + 37 q^{59} - 2 q^{61} - 18 q^{62} + 14 q^{64} + 16 q^{65} - 26 q^{68} + 7 q^{70} - 13 q^{71} - 6 q^{73} - 13 q^{74} - 50 q^{76} + 2 q^{77} + 14 q^{79} + 10 q^{80} + 39 q^{82} + 5 q^{83} + 29 q^{85} + 37 q^{86} + 48 q^{88} + 30 q^{89} - 33 q^{91} + 31 q^{92} + 34 q^{94} - 43 q^{95} - 9 q^{97} - 36 q^{98}+O(q^{100}) \) 5 * q + 4 * q^2 + 4 * q^4 + 9 * q^5 - 4 * q^7 + 12 * q^8 + 5 * q^10 + 15 * q^11 + 3 * q^14 + 10 * q^16 + 11 * q^17 - 16 * q^19 + 9 * q^20 + 7 * q^22 + 9 * q^23 + 6 * q^25 - 7 * q^26 + 9 * q^28 + q^29 - 18 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 7 * q^37 - 20 * q^38 + 32 * q^40 + 10 * q^41 + 6 * q^43 + 5 * q^44 + 13 * q^46 + 7 * q^47 + 11 * q^49 - 3 * q^50 - 6 * q^52 + 9 * q^53 + 17 * q^55 + 15 * q^56 - 19 * q^58 + 37 * q^59 - 2 * q^61 - 18 * q^62 + 14 * q^64 + 16 * q^65 - 26 * q^68 + 7 * q^70 - 13 * q^71 - 6 * q^73 - 13 * q^74 - 50 * q^76 + 2 * q^77 + 14 * q^79 + 10 * q^80 + 39 * q^82 + 5 * q^83 + 29 * q^85 + 37 * q^86 + 48 * q^88 + 30 * q^89 - 33 * q^91 + 31 * q^92 + 34 * q^94 - 43 * q^95 - 9 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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