LMFDB - Embedded newform 4923.2.a.l.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4923,2,Mod(1,4923)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4923, 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("4923.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4923 = 3^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4923.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:	$39.3103529151$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$2.72204$ of defining polynomial
Character	$\chi$	$=$	4923.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.72204 q^{2} +5.40952 q^{4} -0.469688 q^{5} +1.03831 q^{7} +9.28087 q^{8} +O(q^{10})$	\(q+2.72204 q^{2} +5.40952 q^{4} -0.469688 q^{5} +1.03831 q^{7} +9.28087 q^{8} -1.27851 q^{10} +1.84194 q^{11} -0.700934 q^{13} +2.82633 q^{14} +14.4439 q^{16} -0.793975 q^{17} +3.65845 q^{19} -2.54079 q^{20} +5.01384 q^{22} +3.65159 q^{23} -4.77939 q^{25} -1.90797 q^{26} +5.61677 q^{28} -5.52848 q^{29} +6.13605 q^{31} +20.7552 q^{32} -2.16124 q^{34} -0.487683 q^{35} -2.73114 q^{37} +9.95846 q^{38} -4.35911 q^{40} +9.69220 q^{41} -4.25340 q^{43} +9.96402 q^{44} +9.93978 q^{46} +5.21084 q^{47} -5.92191 q^{49} -13.0097 q^{50} -3.79172 q^{52} +7.01107 q^{53} -0.865138 q^{55} +9.63643 q^{56} -15.0488 q^{58} -6.98691 q^{59} -3.87686 q^{61} +16.7026 q^{62} +27.6087 q^{64} +0.329220 q^{65} -5.10753 q^{67} -4.29503 q^{68} -1.32749 q^{70} -1.33192 q^{71} -7.83377 q^{73} -7.43429 q^{74} +19.7905 q^{76} +1.91251 q^{77} +4.36166 q^{79} -6.78412 q^{80} +26.3826 q^{82} +13.3348 q^{83} +0.372921 q^{85} -11.5779 q^{86} +17.0948 q^{88} +0.208861 q^{89} -0.727788 q^{91} +19.7533 q^{92} +14.1841 q^{94} -1.71833 q^{95} +3.08809 q^{97} -16.1197 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * 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 $

$2$	2.72204	1.92478	0.962388	−	0.271679i	$-0.0875790\pi$
$2$	2.72204	1.92478	0.962388	+	0.271679i	$0.0875790\pi$
$3$	0	0
$3$	0	0
$4$	5.40952	2.70476
$5$	−0.469688	−0.210051	−0.105025	−	0.994470i	$-0.533492\pi$
$5$	−0.469688	−0.210051	−0.105025	+	0.994470i	$0.533492\pi$
$6$	0	0
$7$	1.03831	0.392445	0.196222	−	0.980559i	$-0.437133\pi$
$7$	1.03831	0.392445	0.196222	+	0.980559i	$0.437133\pi$
$8$	9.28087	3.28128
$9$	0	0
$10$	−1.27851	−0.404301
$11$	1.84194	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$11$	1.84194	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$12$	0	0
$13$	−0.700934	−0.194404	−0.0972021	−	0.995265i	$-0.530989\pi$
$13$	−0.700934	−0.194404	−0.0972021	+	0.995265i	$0.530989\pi$
$14$	2.82633	0.755368
$15$	0	0
$16$	14.4439	3.61097
$17$	−0.793975	−0.192567	−0.0962837	−	0.995354i	$-0.530696\pi$
$17$	−0.793975	−0.192567	−0.0962837	+	0.995354i	$0.530696\pi$
$18$	0	0
$19$	3.65845	0.839306	0.419653	−	0.907685i	$-0.362152\pi$
$19$	3.65845	0.839306	0.419653	+	0.907685i	$0.362152\pi$
$20$	−2.54079	−0.568138
$21$	0	0
$22$	5.01384	1.06895
$23$	3.65159	0.761409	0.380704	−	0.924697i	$-0.375682\pi$
$23$	3.65159	0.761409	0.380704	+	0.924697i	$0.375682\pi$
$24$	0	0
$25$	−4.77939	−0.955879
$26$	−1.90797	−0.374184
$27$	0	0
$28$	5.61677	1.06147
$29$	−5.52848	−1.02661	−0.513307	−	0.858205i	$-0.671580\pi$
$29$	−5.52848	−1.02661	−0.513307	+	0.858205i	$0.671580\pi$
$30$	0	0
$31$	6.13605	1.10207	0.551034	−	0.834483i	$-0.314234\pi$
$31$	6.13605	1.10207	0.551034	+	0.834483i	$0.314234\pi$
$32$	20.7552	3.66903
$33$	0	0
$34$	−2.16124	−0.370649
$35$	−0.487683	−0.0824334
$36$	0	0
$37$	−2.73114	−0.448997	−0.224498	−	0.974474i	$-0.572074\pi$
$37$	−2.73114	−0.448997	−0.224498	+	0.974474i	$0.572074\pi$
$38$	9.95846	1.61547
$39$	0	0
$40$	−4.35911	−0.689237
$41$	9.69220	1.51367	0.756833	−	0.653608i	$-0.226745\pi$
$41$	9.69220	1.51367	0.756833	+	0.653608i	$0.226745\pi$
$42$	0	0
$43$	−4.25340	−0.648638	−0.324319	−	0.945948i	$-0.605135\pi$
$43$	−4.25340	−0.648638	−0.324319	+	0.945948i	$0.605135\pi$
$44$	9.96402	1.50213
$45$	0	0
$46$	9.93978	1.46554
$47$	5.21084	0.760078	0.380039	−	0.924970i	$-0.375910\pi$
$47$	5.21084	0.760078	0.380039	+	0.924970i	$0.375910\pi$
$48$	0	0
$49$	−5.92191	−0.845987
$50$	−13.0097	−1.83985
$51$	0	0
$52$	−3.79172	−0.525817
$53$	7.01107	0.963044	0.481522	−	0.876434i	$-0.340084\pi$
$53$	7.01107	0.963044	0.481522	+	0.876434i	$0.340084\pi$
$54$	0	0
$55$	−0.865138	−0.116655
$56$	9.63643	1.28772
$57$	0	0
$58$	−15.0488	−1.97600
$59$	−6.98691	−0.909619	−0.454809	−	0.890589i	$-0.650293\pi$
$59$	−6.98691	−0.909619	−0.454809	+	0.890589i	$0.650293\pi$
$60$	0	0
$61$	−3.87686	−0.496381	−0.248191	−	0.968711i	$-0.579836\pi$
$61$	−3.87686	−0.496381	−0.248191	+	0.968711i	$0.579836\pi$
$62$	16.7026	2.12123
$63$	0	0
$64$	27.6087	3.45109
$65$	0.329220	0.0408348
$66$	0	0
$67$	−5.10753	−0.623984	−0.311992	−	0.950085i	$-0.600996\pi$
$67$	−5.10753	−0.623984	−0.311992	+	0.950085i	$0.600996\pi$
$68$	−4.29503	−0.520849
$69$	0	0
$70$	−1.32749	−0.158666
$71$	−1.33192	−0.158070	−0.0790350	−	0.996872i	$-0.525184\pi$
$71$	−1.33192	−0.158070	−0.0790350	+	0.996872i	$0.525184\pi$
$72$	0	0
$73$	−7.83377	−0.916874	−0.458437	−	0.888727i	$-0.651591\pi$
$73$	−7.83377	−0.916874	−0.458437	+	0.888727i	$0.651591\pi$
$74$	−7.43429	−0.864218
$75$	0	0
$76$	19.7905	2.27012
$77$	1.91251	0.217950
$78$	0	0
$79$	4.36166	0.490726	0.245363	−	0.969431i	$-0.421093\pi$
$79$	4.36166	0.490726	0.245363	+	0.969431i	$0.421093\pi$
$80$	−6.78412	−0.758488
$81$	0	0
$82$	26.3826	2.91347
$83$	13.3348	1.46368	0.731842	−	0.681475i	$-0.238661\pi$
$83$	13.3348	1.46368	0.731842	+	0.681475i	$0.238661\pi$
$84$	0	0
$85$	0.372921	0.0404489
$86$	−11.5779	−1.24848
$87$	0	0
$88$	17.0948	1.82231
$89$	0.208861	0.0221393	0.0110696	−	0.999939i	$-0.496476\pi$
$89$	0.208861	0.0221393	0.0110696	+	0.999939i	$0.496476\pi$
$90$	0	0
$91$	−0.727788	−0.0762929
$92$	19.7533	2.05943
$93$	0	0
$94$	14.1841	1.46298
$95$	−1.71833	−0.176297
$96$	0	0
$97$	3.08809	0.313549	0.156774	−	0.987634i	$-0.449890\pi$
$97$	3.08809	0.313549	0.156774	+	0.987634i	$0.449890\pi$
$98$	−16.1197	−1.62834
$99$	0	0
$100$	−25.8542	−2.58542
$101$	8.00261	0.796290	0.398145	−	0.917323i	$-0.369654\pi$
$101$	8.00261	0.796290	0.398145	+	0.917323i	$0.369654\pi$
$102$	0	0
$103$	−3.26470	−0.321681	−0.160840	−	0.986980i	$-0.551420\pi$
$103$	−3.26470	−0.321681	−0.160840	+	0.986980i	$0.551420\pi$
$104$	−6.50528	−0.637895
$105$	0	0
$106$	19.0844	1.85364
$107$	18.4884	1.78735	0.893673	−	0.448719i	$-0.148120\pi$
$107$	18.4884	1.78735	0.893673	+	0.448719i	$0.148120\pi$
$108$	0	0
$109$	−18.6769	−1.78893	−0.894463	−	0.447141i	$-0.852442\pi$
$109$	−18.6769	−1.78893	−0.894463	+	0.447141i	$0.852442\pi$
$110$	−2.35494	−0.224535
$111$	0	0
$112$	14.9973	1.41711
$113$	−7.85874	−0.739288	−0.369644	−	0.929173i	$-0.620520\pi$
$113$	−7.85874	−0.739288	−0.369644	+	0.929173i	$0.620520\pi$
$114$	0	0
$115$	−1.71511	−0.159935
$116$	−29.9064	−2.77674
$117$	0	0
$118$	−19.0187	−1.75081
$119$	−0.824394	−0.0755720
$120$	0	0
$121$	−7.60726	−0.691569
$122$	−10.5530	−0.955422
$123$	0	0
$124$	33.1931	2.98083
$125$	4.59326	0.410834
$126$	0	0
$127$	9.84052	0.873205	0.436602	−	0.899655i	$-0.356182\pi$
$127$	9.84052	0.873205	0.436602	+	0.899655i	$0.356182\pi$
$128$	33.6417	2.97354
$129$	0	0
$130$	0.896152	0.0785978
$131$	−8.02091	−0.700791	−0.350395	−	0.936602i	$-0.613953\pi$
$131$	−8.02091	−0.700791	−0.350395	+	0.936602i	$0.613953\pi$
$132$	0	0
$133$	3.79861	0.329381
$134$	−13.9029	−1.20103
$135$	0	0
$136$	−7.36878	−0.631868
$137$	−16.8434	−1.43903	−0.719516	−	0.694476i	$-0.755636\pi$
$137$	−16.8434	−1.43903	−0.719516	+	0.694476i	$0.755636\pi$
$138$	0	0
$139$	11.2506	0.954260	0.477130	−	0.878833i	$-0.341677\pi$
$139$	11.2506	0.954260	0.477130	+	0.878833i	$0.341677\pi$
$140$	−2.63813	−0.222963
$141$	0	0
$142$	−3.62555	−0.304249
$143$	−1.29108	−0.107965
$144$	0	0
$145$	2.59666	0.215641
$146$	−21.3239	−1.76478
$147$	0	0
$148$	−14.7742	−1.21443
$149$	18.8662	1.54558	0.772788	−	0.634664i	$-0.218862\pi$
$149$	18.8662	1.54558	0.772788	+	0.634664i	$0.218862\pi$
$150$	0	0
$151$	−6.40292	−0.521062	−0.260531	−	0.965465i	$-0.583898\pi$
$151$	−6.40292	−0.521062	−0.260531	+	0.965465i	$0.583898\pi$
$152$	33.9536	2.75400
$153$	0	0
$154$	5.20593	0.419506
$155$	−2.88203	−0.231490
$156$	0	0
$157$	−14.9254	−1.19118	−0.595589	−	0.803290i	$-0.703081\pi$
$157$	−14.9254	−1.19118	−0.595589	+	0.803290i	$0.703081\pi$
$158$	11.8726	0.944537
$159$	0	0
$160$	−9.74845	−0.770683
$161$	3.79148	0.298811
$162$	0	0
$163$	−20.3834	−1.59655	−0.798274	−	0.602294i	$-0.794253\pi$
$163$	−20.3834	−1.59655	−0.798274	+	0.602294i	$0.794253\pi$
$164$	52.4302	4.09411
$165$	0	0
$166$	36.2979	2.81726
$167$	−15.8853	−1.22924	−0.614622	−	0.788822i	$-0.710691\pi$
$167$	−15.8853	−1.22924	−0.614622	+	0.788822i	$0.710691\pi$
$168$	0	0
$169$	−12.5087	−0.962207
$170$	1.01511	0.0778551
$171$	0	0
$172$	−23.0089	−1.75441
$173$	−16.4110	−1.24771	−0.623854	−	0.781541i	$-0.714434\pi$
$173$	−16.4110	−1.24771	−0.623854	+	0.781541i	$0.714434\pi$
$174$	0	0
$175$	−4.96250	−0.375130
$176$	26.6048	2.00541
$177$	0	0
$178$	0.568530	0.0426131
$179$	20.4269	1.52678	0.763390	−	0.645938i	$-0.223534\pi$
$179$	20.4269	1.52678	0.763390	+	0.645938i	$0.223534\pi$
$180$	0	0
$181$	−10.3683	−0.770666	−0.385333	−	0.922778i	$-0.625913\pi$
$181$	−10.3683	−0.770666	−0.385333	+	0.922778i	$0.625913\pi$
$182$	−1.98107	−0.146847
$183$	0	0
$184$	33.8899	2.49840
$185$	1.28279	0.0943122
$186$	0	0
$187$	−1.46246	−0.106945
$188$	28.1881	2.05583
$189$	0	0
$190$	−4.67737	−0.339332
$191$	7.95483	0.575591	0.287796	−	0.957692i	$-0.407078\pi$
$191$	7.95483	0.575591	0.287796	+	0.957692i	$0.407078\pi$
$192$	0	0
$193$	23.9566	1.72443	0.862217	−	0.506539i	$-0.169075\pi$
$193$	23.9566	1.72443	0.862217	+	0.506539i	$0.169075\pi$
$194$	8.40593	0.603511
$195$	0	0
$196$	−32.0347	−2.28819
$197$	−5.15127	−0.367013	−0.183506	−	0.983019i	$-0.558745\pi$
$197$	−5.15127	−0.367013	−0.183506	+	0.983019i	$0.558745\pi$
$198$	0	0
$199$	1.54311	0.109388	0.0546940	−	0.998503i	$-0.482582\pi$
$199$	1.54311	0.109388	0.0546940	+	0.998503i	$0.482582\pi$
$200$	−44.3569	−3.13651
$201$	0	0
$202$	21.7835	1.53268
$203$	−5.74028	−0.402889
$204$	0	0
$205$	−4.55231	−0.317947
$206$	−8.88666	−0.619163
$207$	0	0
$208$	−10.1242	−0.701988
$209$	6.73864	0.466122
$210$	0	0
$211$	8.81178	0.606628	0.303314	−	0.952891i	$-0.401907\pi$
$211$	8.81178	0.606628	0.303314	+	0.952891i	$0.401907\pi$
$212$	37.9265	2.60480
$213$	0	0
$214$	50.3264	3.44024
$215$	1.99777	0.136247
$216$	0	0
$217$	6.37113	0.432501
$218$	−50.8395	−3.44328
$219$	0	0
$220$	−4.67998	−0.315524
$221$	0.556524	0.0374359
$222$	0	0
$223$	24.2050	1.62089	0.810445	−	0.585815i	$-0.199226\pi$
$223$	24.2050	1.62089	0.810445	+	0.585815i	$0.199226\pi$
$224$	21.5503	1.43989
$225$	0	0
$226$	−21.3918	−1.42296
$227$	3.55162	0.235729	0.117865	−	0.993030i	$-0.462395\pi$
$227$	3.55162	0.235729	0.117865	+	0.993030i	$0.462395\pi$
$228$	0	0
$229$	−18.8164	−1.24342	−0.621711	−	0.783247i	$-0.713562\pi$
$229$	−18.8164	−1.24342	−0.621711	+	0.783247i	$0.713562\pi$
$230$	−4.66860	−0.307838
$231$	0	0
$232$	−51.3091	−3.36861
$233$	−12.1897	−0.798575	−0.399288	−	0.916826i	$-0.630742\pi$
$233$	−12.1897	−0.798575	−0.399288	+	0.916826i	$0.630742\pi$
$234$	0	0
$235$	−2.44747	−0.159655
$236$	−37.7959	−2.46030
$237$	0	0
$238$	−2.24404	−0.145459
$239$	−30.6423	−1.98208	−0.991042	−	0.133553i	$-0.957361\pi$
$239$	−30.6423	−1.98208	−0.991042	+	0.133553i	$0.957361\pi$
$240$	0	0
$241$	−7.26294	−0.467847	−0.233924	−	0.972255i	$-0.575157\pi$
$241$	−7.26294	−0.467847	−0.233924	+	0.972255i	$0.575157\pi$
$242$	−20.7073	−1.33111
$243$	0	0
$244$	−20.9720	−1.34259
$245$	2.78145	0.177700
$246$	0	0
$247$	−2.56433	−0.163164
$248$	56.9479	3.61620
$249$	0	0
$250$	12.5031	0.790764
$251$	25.2813	1.59574	0.797871	−	0.602828i	$-0.205959\pi$
$251$	25.2813	1.59574	0.797871	+	0.602828i	$0.205959\pi$
$252$	0	0
$253$	6.72601	0.422860
$254$	26.7863	1.68072
$255$	0	0
$256$	36.3569	2.27230
$257$	26.5017	1.65313	0.826564	−	0.562842i	$-0.190292\pi$
$257$	26.5017	1.65313	0.826564	+	0.562842i	$0.190292\pi$
$258$	0	0
$259$	−2.83578	−0.176207
$260$	1.78093	0.110448
$261$	0	0
$262$	−21.8333	−1.34886
$263$	−12.5983	−0.776845	−0.388422	−	0.921481i	$-0.626980\pi$
$263$	−12.5983	−0.776845	−0.388422	+	0.921481i	$0.626980\pi$
$264$	0	0
$265$	−3.29301	−0.202288
$266$	10.3400	0.633985
$267$	0	0
$268$	−27.6293	−1.68773
$269$	−1.45914	−0.0889652	−0.0444826	−	0.999010i	$-0.514164\pi$
$269$	−1.45914	−0.0889652	−0.0444826	+	0.999010i	$0.514164\pi$
$270$	0	0
$271$	5.16205	0.313572	0.156786	−	0.987633i	$-0.449887\pi$
$271$	5.16205	0.313572	0.156786	+	0.987633i	$0.449887\pi$
$272$	−11.4681	−0.695355
$273$	0	0
$274$	−45.8486	−2.76981
$275$	−8.80336	−0.530862
$276$	0	0
$277$	14.0110	0.841837	0.420918	−	0.907098i	$-0.361708\pi$
$277$	14.0110	0.841837	0.420918	+	0.907098i	$0.361708\pi$
$278$	30.6245	1.83674
$279$	0	0
$280$	−4.52612	−0.270487
$281$	−16.3617	−0.976059	−0.488030	−	0.872827i	$-0.662284\pi$
$281$	−16.3617	−0.976059	−0.488030	+	0.872827i	$0.662284\pi$
$282$	0	0
$283$	18.5182	1.10079	0.550397	−	0.834903i	$-0.314476\pi$
$283$	18.5182	1.10079	0.550397	+	0.834903i	$0.314476\pi$
$284$	−7.20506	−0.427542
$285$	0	0
$286$	−3.51437	−0.207809
$287$	10.0635	0.594031
$288$	0	0
$289$	−16.3696	−0.962918
$290$	7.06823	0.415061
$291$	0	0
$292$	−42.3770	−2.47993
$293$	−30.5239	−1.78323	−0.891613	−	0.452798i	$-0.850426\pi$
$293$	−30.5239	−1.78323	−0.891613	+	0.452798i	$0.850426\pi$
$294$	0	0
$295$	3.28167	0.191066
$296$	−25.3474	−1.47329
$297$	0	0
$298$	51.3545	2.97489
$299$	−2.55952	−0.148021
$300$	0	0
$301$	−4.41636	−0.254555
$302$	−17.4290	−1.00293
$303$	0	0
$304$	52.8422	3.03071
$305$	1.82092	0.104265
$306$	0	0
$307$	3.36035	0.191785	0.0958926	−	0.995392i	$-0.469429\pi$
$307$	3.36035	0.191785	0.0958926	+	0.995392i	$0.469429\pi$
$308$	10.3458	0.589504
$309$	0	0
$310$	−7.84501	−0.445567
$311$	−26.0631	−1.47791	−0.738953	−	0.673757i	$-0.764679\pi$
$311$	−26.0631	−1.47791	−0.738953	+	0.673757i	$0.764679\pi$
$312$	0	0
$313$	−28.4960	−1.61069	−0.805345	−	0.592807i	$-0.798020\pi$
$313$	−28.4960	−1.61069	−0.805345	+	0.592807i	$0.798020\pi$
$314$	−40.6276	−2.29275
$315$	0	0
$316$	23.5945	1.32730
$317$	8.26530	0.464225	0.232113	−	0.972689i	$-0.425436\pi$
$317$	8.26530	0.464225	0.232113	+	0.972689i	$0.425436\pi$
$318$	0	0
$319$	−10.1831	−0.570146
$320$	−12.9675	−0.724904
$321$	0	0
$322$	10.3206	0.575144
$323$	−2.90472	−0.161623
$324$	0	0
$325$	3.35004	0.185827
$326$	−55.4844	−3.07300
$327$	0	0
$328$	89.9520	4.96677
$329$	5.41047	0.298289
$330$	0	0
$331$	12.5162	0.687951	0.343976	−	0.938979i	$-0.388226\pi$
$331$	12.5162	0.687951	0.343976	+	0.938979i	$0.388226\pi$
$332$	72.1348	3.95891
$333$	0	0
$334$	−43.2406	−2.36602
$335$	2.39895	0.131068
$336$	0	0
$337$	−20.0461	−1.09198	−0.545990	−	0.837792i	$-0.683846\pi$
$337$	−20.0461	−1.09198	−0.545990	+	0.837792i	$0.683846\pi$
$338$	−34.0492	−1.85203
$339$	0	0
$340$	2.01732	0.109405
$341$	11.3022	0.612051
$342$	0	0
$343$	−13.4170	−0.724448
$344$	−39.4753	−2.12836
$345$	0	0
$346$	−44.6716	−2.40156
$347$	−0.748255	−0.0401684	−0.0200842	−	0.999798i	$-0.506393\pi$
$347$	−0.748255	−0.0401684	−0.0200842	+	0.999798i	$0.506393\pi$
$348$	0	0
$349$	4.16941	0.223183	0.111592	−	0.993754i	$-0.464405\pi$
$349$	4.16941	0.223183	0.111592	+	0.993754i	$0.464405\pi$
$350$	−13.5081	−0.722040
$351$	0	0
$352$	38.2298	2.03765
$353$	−28.9002	−1.53820	−0.769100	−	0.639128i	$-0.779295\pi$
$353$	−28.9002	−1.53820	−0.769100	+	0.639128i	$0.779295\pi$
$354$	0	0
$355$	0.625588	0.0332028
$356$	1.12984	0.0598814
$357$	0	0
$358$	55.6030	2.93871
$359$	−22.5238	−1.18876	−0.594380	−	0.804185i	$-0.702602\pi$
$359$	−22.5238	−1.18876	−0.594380	+	0.804185i	$0.702602\pi$
$360$	0	0
$361$	−5.61576	−0.295566
$362$	−28.2228	−1.48336
$363$	0	0
$364$	−3.93698	−0.206354
$365$	3.67943	0.192590
$366$	0	0
$367$	−3.07935	−0.160741	−0.0803705	−	0.996765i	$-0.525610\pi$
$367$	−3.07935	−0.160741	−0.0803705	+	0.996765i	$0.525610\pi$
$368$	52.7431	2.74943
$369$	0	0
$370$	3.49180	0.181530
$371$	7.27967	0.377942
$372$	0	0
$373$	−23.8756	−1.23623	−0.618115	−	0.786088i	$-0.712103\pi$
$373$	−23.8756	−1.23623	−0.618115	+	0.786088i	$0.712103\pi$
$374$	−3.98087	−0.205846
$375$	0	0
$376$	48.3611	2.49403
$377$	3.87510	0.199578
$378$	0	0
$379$	3.60479	0.185166	0.0925828	−	0.995705i	$-0.470488\pi$
$379$	3.60479	0.185166	0.0925828	+	0.995705i	$0.470488\pi$
$380$	−9.29534	−0.476841
$381$	0	0
$382$	21.6534	1.10788
$383$	3.63594	0.185788	0.0928939	−	0.995676i	$-0.470388\pi$
$383$	3.63594	0.185788	0.0928939	+	0.995676i	$0.470388\pi$
$384$	0	0
$385$	−0.898282	−0.0457807
$386$	65.2109	3.31915
$387$	0	0
$388$	16.7051	0.848074
$389$	−21.6735	−1.09889	−0.549445	−	0.835530i	$-0.685161\pi$
$389$	−21.6735	−1.09889	−0.549445	+	0.835530i	$0.685161\pi$
$390$	0	0
$391$	−2.89927	−0.146622
$392$	−54.9605	−2.77592
$393$	0	0
$394$	−14.0220	−0.706417
$395$	−2.04862	−0.103077
$396$	0	0
$397$	6.59119	0.330802	0.165401	−	0.986226i	$-0.447108\pi$
$397$	6.59119	0.330802	0.165401	+	0.986226i	$0.447108\pi$
$398$	4.20041	0.210547
$399$	0	0
$400$	−69.0330	−3.45165
$401$	4.35340	0.217398	0.108699	−	0.994075i	$-0.465332\pi$
$401$	4.35340	0.217398	0.108699	+	0.994075i	$0.465332\pi$
$402$	0	0
$403$	−4.30097	−0.214246
$404$	43.2903	2.15377
$405$	0	0
$406$	−15.6253	−0.775471
$407$	−5.03060	−0.249358
$408$	0	0
$409$	31.8412	1.57445	0.787223	−	0.616668i	$-0.211518\pi$
$409$	31.8412	1.57445	0.787223	+	0.616668i	$0.211518\pi$
$410$	−12.3916	−0.611977
$411$	0	0
$412$	−17.6605	−0.870069
$413$	−7.25459	−0.356975
$414$	0	0
$415$	−6.26319	−0.307448
$416$	−14.5480	−0.713274
$417$	0	0
$418$	18.3429	0.897180
$419$	29.8225	1.45692	0.728462	−	0.685087i	$-0.240236\pi$
$419$	29.8225	1.45692	0.728462	+	0.685087i	$0.240236\pi$
$420$	0	0
$421$	−7.93092	−0.386530	−0.193265	−	0.981147i	$-0.561908\pi$
$421$	−7.93092	−0.386530	−0.193265	+	0.981147i	$0.561908\pi$
$422$	23.9860	1.16762
$423$	0	0
$424$	65.0688	3.16002
$425$	3.79472	0.184071
$426$	0	0
$427$	−4.02539	−0.194802
$428$	100.014	4.83434
$429$	0	0
$430$	5.43803	0.262245
$431$	24.9537	1.20198	0.600988	−	0.799258i	$-0.294774\pi$
$431$	24.9537	1.20198	0.600988	+	0.799258i	$0.294774\pi$
$432$	0	0
$433$	23.1505	1.11254	0.556272	−	0.831000i	$-0.312231\pi$
$433$	23.1505	1.11254	0.556272	+	0.831000i	$0.312231\pi$
$434$	17.3425	0.832467
$435$	0	0
$436$	−101.033	−4.83862
$437$	13.3591	0.639054
$438$	0	0
$439$	18.9654	0.905167	0.452584	−	0.891722i	$-0.350502\pi$
$439$	18.9654	0.905167	0.452584	+	0.891722i	$0.350502\pi$
$440$	−8.02923	−0.382778
$441$	0	0
$442$	1.51488	0.0720557
$443$	−21.6203	−1.02721	−0.513606	−	0.858026i	$-0.671691\pi$
$443$	−21.6203	−1.02721	−0.513606	+	0.858026i	$0.671691\pi$
$444$	0	0
$445$	−0.0980997	−0.00465037
$446$	65.8872	3.11985
$447$	0	0
$448$	28.6664	1.35436
$449$	−15.1299	−0.714023	−0.357011	−	0.934100i	$-0.616204\pi$
$449$	−15.1299	−0.714023	−0.357011	+	0.934100i	$0.616204\pi$
$450$	0	0
$451$	17.8524	0.840639
$452$	−42.5121	−1.99960
$453$	0	0
$454$	9.66767	0.453726
$455$	0.341833	0.0160254
$456$	0	0
$457$	−7.87882	−0.368556	−0.184278	−	0.982874i	$-0.558995\pi$
$457$	−7.87882	−0.368556	−0.184278	+	0.982874i	$0.558995\pi$
$458$	−51.2190	−2.39331
$459$	0	0
$460$	−9.27791	−0.432585
$461$	33.2729	1.54967	0.774836	−	0.632162i	$-0.217832\pi$
$461$	33.2729	1.54967	0.774836	+	0.632162i	$0.217832\pi$
$462$	0	0
$463$	31.9493	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$463$	31.9493	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$464$	−79.8528	−3.70707
$465$	0	0
$466$	−33.1810	−1.53708
$467$	−6.03218	−0.279136	−0.139568	−	0.990213i	$-0.544571\pi$
$467$	−6.03218	−0.279136	−0.139568	+	0.990213i	$0.544571\pi$
$468$	0	0
$469$	−5.30320	−0.244879
$470$	−6.66211	−0.307300
$471$	0	0
$472$	−64.8446	−2.98472
$473$	−7.83451	−0.360231
$474$	0	0
$475$	−17.4852	−0.802274
$476$	−4.45958	−0.204404
$477$	0	0
$478$	−83.4096	−3.81507
$479$	−18.1368	−0.828693	−0.414347	−	0.910119i	$-0.635990\pi$
$479$	−18.1368	−0.828693	−0.414347	+	0.910119i	$0.635990\pi$
$480$	0	0
$481$	1.91435	0.0872869
$482$	−19.7700	−0.900501
$483$	0	0
$484$	−41.1516	−1.87053
$485$	−1.45044	−0.0658612
$486$	0	0
$487$	−23.3591	−1.05850	−0.529251	−	0.848465i	$-0.677527\pi$
$487$	−23.3591	−1.05850	−0.529251	+	0.848465i	$0.677527\pi$
$488$	−35.9806	−1.62877
$489$	0	0
$490$	7.57123	0.342033
$491$	4.28889	0.193555	0.0967775	−	0.995306i	$-0.469146\pi$
$491$	4.28889	0.193555	0.0967775	+	0.995306i	$0.469146\pi$
$492$	0	0
$493$	4.38948	0.197692
$494$	−6.98022	−0.314055
$495$	0	0
$496$	88.6285	3.97954
$497$	−1.38295	−0.0620338
$498$	0	0
$499$	−12.0079	−0.537549	−0.268775	−	0.963203i	$-0.586619\pi$
$499$	−12.0079	−0.537549	−0.268775	+	0.963203i	$0.586619\pi$
$500$	24.8474	1.11121
$501$	0	0
$502$	68.8169	3.07145
$503$	22.0435	0.982872	0.491436	−	0.870914i	$-0.336472\pi$
$503$	22.0435	0.982872	0.491436	+	0.870914i	$0.336472\pi$
$504$	0	0
$505$	−3.75873	−0.167261
$506$	18.3085	0.813911
$507$	0	0
$508$	53.2325	2.36181
$509$	11.9642	0.530302	0.265151	−	0.964207i	$-0.414578\pi$
$509$	11.9642	0.530302	0.265151	+	0.964207i	$0.414578\pi$
$510$	0	0
$511$	−8.13390	−0.359822
$512$	31.6816	1.40014
$513$	0	0
$514$	72.1387	3.18190
$515$	1.53339	0.0675693
$516$	0	0
$517$	9.59805	0.422122
$518$	−7.71911	−0.339158
$519$	0	0
$520$	3.05545	0.133990
$521$	5.47217	0.239740	0.119870	−	0.992790i	$-0.461752\pi$
$521$	5.47217	0.239740	0.119870	+	0.992790i	$0.461752\pi$
$522$	0	0
$523$	−31.1363	−1.36149	−0.680747	−	0.732519i	$-0.738345\pi$
$523$	−31.1363	−1.36149	−0.680747	+	0.732519i	$0.738345\pi$
$524$	−43.3893	−1.89547
$525$	0	0
$526$	−34.2931	−1.49525
$527$	−4.87187	−0.212222
$528$	0	0
$529$	−9.66591	−0.420257
$530$	−8.96373	−0.389360
$531$	0	0
$532$	20.5487	0.890897
$533$	−6.79359	−0.294263
$534$	0	0
$535$	−8.68380	−0.375434
$536$	−47.4023	−2.04747
$537$	0	0
$538$	−3.97183	−0.171238
$539$	−10.9078	−0.469832
$540$	0	0
$541$	−15.0578	−0.647387	−0.323694	−	0.946162i	$-0.604925\pi$
$541$	−15.0578	−0.647387	−0.323694	+	0.946162i	$0.604925\pi$
$542$	14.0513	0.603556
$543$	0	0
$544$	−16.4791	−0.706535
$545$	8.77234	0.375766
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−91.1150	−3.89224
$549$	0	0
$550$	−23.9631	−1.02179
$551$	−20.2257	−0.861642
$552$	0	0
$553$	4.52877	0.192583
$554$	38.1384	1.62035
$555$	0	0
$556$	60.8602	2.58105
$557$	−23.5488	−0.997796	−0.498898	−	0.866661i	$-0.666262\pi$
$557$	−23.5488	−0.997796	−0.498898	+	0.866661i	$0.666262\pi$
$558$	0	0
$559$	2.98136	0.126098
$560$	−7.04403	−0.297665
$561$	0	0
$562$	−44.5374	−1.87869
$563$	29.0653	1.22496	0.612478	−	0.790488i	$-0.290173\pi$
$563$	29.0653	1.22496	0.612478	+	0.790488i	$0.290173\pi$
$564$	0	0
$565$	3.69116	0.155288
$566$	50.4074	2.11878
$567$	0	0
$568$	−12.3614	−0.518673
$569$	0.227367	0.00953174	0.00476587	−	0.999989i	$-0.498483\pi$
$569$	0.227367	0.00953174	0.00476587	+	0.999989i	$0.498483\pi$
$570$	0	0
$571$	0.874566	0.0365994	0.0182997	−	0.999833i	$-0.494175\pi$
$571$	0.874566	0.0365994	0.0182997	+	0.999833i	$0.494175\pi$
$572$	−6.98412	−0.292021
$573$	0	0
$574$	27.3933	1.14338
$575$	−17.4524	−0.727814
$576$	0	0
$577$	−29.2997	−1.21976	−0.609882	−	0.792492i	$-0.708783\pi$
$577$	−29.2997	−1.21976	−0.609882	+	0.792492i	$0.708783\pi$
$578$	−44.5588	−1.85340
$579$	0	0
$580$	14.0467	0.583257
$581$	13.8457	0.574415
$582$	0	0
$583$	12.9140	0.534842
$584$	−72.7042	−3.00852
$585$	0	0
$586$	−83.0875	−3.43231
$587$	10.9789	0.453149	0.226574	−	0.973994i	$-0.427247\pi$
$587$	10.9789	0.453149	0.226574	+	0.973994i	$0.427247\pi$
$588$	0	0
$589$	22.4484	0.924971
$590$	8.93285	0.367760
$591$	0	0
$592$	−39.4483	−1.62132
$593$	42.7926	1.75728	0.878641	−	0.477483i	$-0.158451\pi$
$593$	42.7926	1.75728	0.878641	+	0.477483i	$0.158451\pi$
$594$	0	0
$595$	0.387208	0.0158740
$596$	102.057	4.18042
$597$	0	0
$598$	−6.96713	−0.284907
$599$	−0.841737	−0.0343925	−0.0171962	−	0.999852i	$-0.505474\pi$
$599$	−0.841737	−0.0343925	−0.0171962	+	0.999852i	$0.505474\pi$
$600$	0	0
$601$	−19.7479	−0.805536	−0.402768	−	0.915302i	$-0.631952\pi$
$601$	−19.7479	−0.805536	−0.402768	+	0.915302i	$0.631952\pi$
$602$	−12.0215	−0.489960
$603$	0	0
$604$	−34.6367	−1.40935
$605$	3.57304	0.145265
$606$	0	0
$607$	−0.928802	−0.0376989	−0.0188495	−	0.999822i	$-0.506000\pi$
$607$	−0.928802	−0.0376989	−0.0188495	+	0.999822i	$0.506000\pi$
$608$	75.9317	3.07944
$609$	0	0
$610$	4.95661	0.200687
$611$	−3.65245	−0.147762
$612$	0	0
$613$	20.1162	0.812485	0.406242	−	0.913765i	$-0.366839\pi$
$613$	20.1162	0.812485	0.406242	+	0.913765i	$0.366839\pi$
$614$	9.14702	0.369144
$615$	0	0
$616$	17.7497	0.715157
$617$	13.2513	0.533479	0.266739	−	0.963769i	$-0.414054\pi$
$617$	13.2513	0.533479	0.266739	+	0.963769i	$0.414054\pi$
$618$	0	0
$619$	−29.2870	−1.17714	−0.588572	−	0.808445i	$-0.700310\pi$
$619$	−29.2870	−1.17714	−0.588572	+	0.808445i	$0.700310\pi$
$620$	−15.5904	−0.626126
$621$	0	0
$622$	−70.9450	−2.84464
$623$	0.216863	0.00868844
$624$	0	0
$625$	21.7396	0.869583
$626$	−77.5674	−3.10022
$627$	0	0
$628$	−80.7393	−3.22185
$629$	2.16846	0.0864621
$630$	0	0
$631$	−46.5766	−1.85418	−0.927092	−	0.374834i	$-0.877700\pi$
$631$	−46.5766	−1.85418	−0.927092	+	0.374834i	$0.877700\pi$
$632$	40.4800	1.61021
$633$	0	0
$634$	22.4985	0.893530
$635$	−4.62197	−0.183417
$636$	0	0
$637$	4.15087	0.164463
$638$	−27.7189	−1.09740
$639$	0	0
$640$	−15.8011	−0.624594
$641$	5.11165	0.201898	0.100949	−	0.994892i	$-0.467812\pi$
$641$	5.11165	0.201898	0.100949	+	0.994892i	$0.467812\pi$
$642$	0	0
$643$	−4.32994	−0.170756	−0.0853781	−	0.996349i	$-0.527210\pi$
$643$	−4.32994	−0.170756	−0.0853781	+	0.996349i	$0.527210\pi$
$644$	20.5101	0.808212
$645$	0	0
$646$	−7.90677	−0.311088
$647$	12.0401	0.473343	0.236672	−	0.971590i	$-0.423943\pi$
$647$	12.0401	0.473343	0.236672	+	0.971590i	$0.423943\pi$
$648$	0	0
$649$	−12.8695	−0.505171
$650$	9.11895	0.357675
$651$	0	0
$652$	−110.264	−4.31828
$653$	−39.4366	−1.54327	−0.771636	−	0.636064i	$-0.780561\pi$
$653$	−39.4366	−1.54327	−0.771636	+	0.636064i	$0.780561\pi$
$654$	0	0
$655$	3.76733	0.147202
$656$	139.993	5.46581
$657$	0	0
$658$	14.7275	0.574139
$659$	−43.8625	−1.70864	−0.854320	−	0.519748i	$-0.826026\pi$
$659$	−43.8625	−1.70864	−0.854320	+	0.519748i	$0.826026\pi$
$660$	0	0
$661$	−5.18941	−0.201845	−0.100922	−	0.994894i	$-0.532179\pi$
$661$	−5.18941	−0.201845	−0.100922	+	0.994894i	$0.532179\pi$
$662$	34.0696	1.32415
$663$	0	0
$664$	123.758	4.80276
$665$	−1.78416	−0.0691868
$666$	0	0
$667$	−20.1877	−0.781672
$668$	−85.9321	−3.32481
$669$	0	0
$670$	6.53003	0.252277
$671$	−7.14094	−0.275673
$672$	0	0
$673$	37.7606	1.45557	0.727783	−	0.685808i	$-0.240551\pi$
$673$	37.7606	1.45557	0.727783	+	0.685808i	$0.240551\pi$
$674$	−54.5663	−2.10182
$675$	0	0
$676$	−67.6661	−2.60254
$677$	36.6368	1.40806	0.704032	−	0.710168i	$-0.251381\pi$
$677$	36.6368	1.40806	0.704032	+	0.710168i	$0.251381\pi$
$678$	0	0
$679$	3.20640	0.123050
$680$	3.46103	0.132724
$681$	0	0
$682$	30.7652	1.17806
$683$	−31.5096	−1.20568	−0.602841	−	0.797861i	$-0.705965\pi$
$683$	−31.5096	−1.20568	−0.602841	+	0.797861i	$0.705965\pi$
$684$	0	0
$685$	7.91116	0.302270
$686$	−36.5216	−1.39440
$687$	0	0
$688$	−61.4357	−2.34221
$689$	−4.91430	−0.187220
$690$	0	0
$691$	18.9407	0.720540	0.360270	−	0.932848i	$-0.382685\pi$
$691$	18.9407	0.720540	0.360270	+	0.932848i	$0.382685\pi$
$692$	−88.7759	−3.37475
$693$	0	0
$694$	−2.03678	−0.0773153
$695$	−5.28426	−0.200443
$696$	0	0
$697$	−7.69537	−0.291483
$698$	11.3493	0.429578
$699$	0	0
$700$	−26.8447	−1.01464
$701$	−42.8730	−1.61929	−0.809646	−	0.586919i	$-0.800341\pi$
$701$	−42.8730	−1.61929	−0.809646	+	0.586919i	$0.800341\pi$
$702$	0	0
$703$	−9.99174	−0.376846
$704$	50.8535	1.91662
$705$	0	0
$706$	−78.6675	−2.96069
$707$	8.30921	0.312500
$708$	0	0
$709$	35.4811	1.33252	0.666260	−	0.745719i	$-0.267894\pi$
$709$	35.4811	1.33252	0.666260	+	0.745719i	$0.267894\pi$
$710$	1.70288	0.0639079
$711$	0	0
$712$	1.93842	0.0726452
$713$	22.4063	0.839124
$714$	0	0
$715$	0.606404	0.0226782
$716$	110.500	4.12957
$717$	0	0
$718$	−61.3107	−2.28810
$719$	24.4790	0.912913	0.456456	−	0.889746i	$-0.349118\pi$
$719$	24.4790	0.912913	0.456456	+	0.889746i	$0.349118\pi$
$720$	0	0
$721$	−3.38978	−0.126242
$722$	−15.2863	−0.568899
$723$	0	0
$724$	−56.0873	−2.08447
$725$	26.4228	0.981317
$726$	0	0
$727$	−28.7796	−1.06737	−0.533687	−	0.845682i	$-0.679194\pi$
$727$	−28.7796	−1.06737	−0.533687	+	0.845682i	$0.679194\pi$
$728$	−6.75450	−0.250339
$729$	0	0
$730$	10.0156	0.370693
$731$	3.37710	0.124906
$732$	0	0
$733$	23.0494	0.851350	0.425675	−	0.904876i	$-0.360037\pi$
$733$	23.0494	0.851350	0.425675	+	0.904876i	$0.360037\pi$
$734$	−8.38213	−0.309390
$735$	0	0
$736$	75.7893	2.79363
$737$	−9.40776	−0.346539
$738$	0	0
$739$	−9.02906	−0.332139	−0.166070	−	0.986114i	$-0.553108\pi$
$739$	−9.02906	−0.332139	−0.166070	+	0.986114i	$0.553108\pi$
$740$	6.93925	0.255092
$741$	0	0
$742$	19.8156	0.727453
$743$	47.5028	1.74271	0.871355	−	0.490653i	$-0.163242\pi$
$743$	47.5028	1.74271	0.871355	+	0.490653i	$0.163242\pi$
$744$	0	0
$745$	−8.86121	−0.324650
$746$	−64.9903	−2.37947
$747$	0	0
$748$	−7.91118	−0.289262
$749$	19.1968	0.701434
$750$	0	0
$751$	−20.5388	−0.749473	−0.374737	−	0.927131i	$-0.622267\pi$
$751$	−20.5388	−0.749473	−0.374737	+	0.927131i	$0.622267\pi$
$752$	75.2647	2.74462
$753$	0	0
$754$	10.5482	0.384143
$755$	3.00737	0.109450
$756$	0	0
$757$	36.3077	1.31962	0.659812	−	0.751430i	$-0.270636\pi$
$757$	36.3077	1.31962	0.659812	+	0.751430i	$0.270636\pi$
$758$	9.81240	0.356402
$759$	0	0
$760$	−15.9476	−0.578480
$761$	−23.0597	−0.835915	−0.417957	−	0.908467i	$-0.637254\pi$
$761$	−23.0597	−0.835915	−0.417957	+	0.908467i	$0.637254\pi$
$762$	0	0
$763$	−19.3925	−0.702055
$764$	43.0318	1.55684
$765$	0	0
$766$	9.89719	0.357600
$767$	4.89737	0.176834
$768$	0	0
$769$	4.40403	0.158813	0.0794066	−	0.996842i	$-0.474697\pi$
$769$	4.40403	0.158813	0.0794066	+	0.996842i	$0.474697\pi$
$770$	−2.44516	−0.0881176
$771$	0	0
$772$	129.594	4.66418
$773$	25.6259	0.921699	0.460849	−	0.887478i	$-0.347545\pi$
$773$	25.6259	0.921699	0.460849	+	0.887478i	$0.347545\pi$
$774$	0	0
$775$	−29.3266	−1.05344
$776$	28.6602	1.02884
$777$	0	0
$778$	−58.9962	−2.11512
$779$	35.4584	1.27043
$780$	0	0
$781$	−2.45332	−0.0877867
$782$	−7.89194	−0.282215
$783$	0	0
$784$	−85.5354	−3.05484
$785$	7.01029	0.250208
$786$	0	0
$787$	46.4038	1.65412	0.827059	−	0.562115i	$-0.190012\pi$
$787$	46.4038	1.65412	0.827059	+	0.562115i	$0.190012\pi$
$788$	−27.8659	−0.992681
$789$	0	0
$790$	−5.57644	−0.198401
$791$	−8.15982	−0.290130
$792$	0	0
$793$	2.71742	0.0964985
$794$	17.9415	0.636720
$795$	0	0
$796$	8.34748	0.295868
$797$	33.6692	1.19262	0.596312	−	0.802753i	$-0.296632\pi$
$797$	33.6692	1.19262	0.596312	+	0.802753i	$0.296632\pi$
$798$	0	0
$799$	−4.13727	−0.146366
$800$	−99.1971	−3.50715
$801$	0	0
$802$	11.8501	0.418443
$803$	−14.4293	−0.509200
$804$	0	0
$805$	−1.78082	−0.0627655
$806$	−11.7074	−0.412376
$807$	0	0
$808$	74.2712	2.61285
$809$	−32.2939	−1.13539	−0.567696	−	0.823238i	$-0.692165\pi$
$809$	−32.2939	−1.13539	−0.567696	+	0.823238i	$0.692165\pi$
$810$	0	0
$811$	27.4278	0.963119	0.481559	−	0.876413i	$-0.340071\pi$
$811$	27.4278	0.963119	0.481559	+	0.876413i	$0.340071\pi$
$812$	−31.0522	−1.08972
$813$	0	0
$814$	−13.6935	−0.479957
$815$	9.57382	0.335356
$816$	0	0
$817$	−15.5609	−0.544405
$818$	86.6732	3.03046
$819$	0	0
$820$	−24.6258	−0.859971
$821$	17.5646	0.613008	0.306504	−	0.951869i	$-0.400841\pi$
$821$	17.5646	0.613008	0.306504	+	0.951869i	$0.400841\pi$
$822$	0	0
$823$	−7.94705	−0.277017	−0.138508	−	0.990361i	$-0.544231\pi$
$823$	−7.94705	−0.277017	−0.138508	+	0.990361i	$0.544231\pi$
$824$	−30.2993	−1.05552
$825$	0	0
$826$	−19.7473	−0.687097
$827$	5.16367	0.179558	0.0897792	−	0.995962i	$-0.471384\pi$
$827$	5.16367	0.179558	0.0897792	+	0.995962i	$0.471384\pi$
$828$	0	0
$829$	−20.0606	−0.696734	−0.348367	−	0.937358i	$-0.613264\pi$
$829$	−20.0606	−0.696734	−0.348367	+	0.937358i	$0.613264\pi$
$830$	−17.0487	−0.591768
$831$	0	0
$832$	−19.3519	−0.670905
$833$	4.70185	0.162909
$834$	0	0
$835$	7.46115	0.258204
$836$	36.4528	1.26075
$837$	0	0
$838$	81.1781	2.80425
$839$	−39.3019	−1.35685	−0.678426	−	0.734669i	$-0.737338\pi$
$839$	−39.3019	−1.35685	−0.678426	+	0.734669i	$0.737338\pi$
$840$	0	0
$841$	1.56409	0.0539343
$842$	−21.5883	−0.743983
$843$	0	0
$844$	47.6675	1.64078
$845$	5.87518	0.202112
$846$	0	0
$847$	−7.89870	−0.271403
$848$	101.267	3.47753
$849$	0	0
$850$	10.3294	0.354295
$851$	−9.97300	−0.341870
$852$	0	0
$853$	41.2623	1.41280	0.706398	−	0.707815i	$-0.250319\pi$
$853$	41.2623	1.41280	0.706398	+	0.707815i	$0.250319\pi$
$854$	−10.9573	−0.374950
$855$	0	0
$856$	171.589	5.86479
$857$	−4.93880	−0.168706	−0.0843532	−	0.996436i	$-0.526882\pi$
$857$	−4.93880	−0.168706	−0.0843532	+	0.996436i	$0.526882\pi$
$858$	0	0
$859$	−38.1044	−1.30011	−0.650053	−	0.759889i	$-0.725253\pi$
$859$	−38.1044	−1.30011	−0.650053	+	0.759889i	$0.725253\pi$
$860$	10.8070	0.368516
$861$	0	0
$862$	67.9250	2.31353
$863$	1.71273	0.0583019	0.0291510	−	0.999575i	$-0.490720\pi$
$863$	1.71273	0.0583019	0.0291510	+	0.999575i	$0.490720\pi$
$864$	0	0
$865$	7.70808	0.262082
$866$	63.0168	2.14140
$867$	0	0
$868$	34.4648	1.16981
$869$	8.03393	0.272532
$870$	0	0
$871$	3.58004	0.121305
$872$	−173.338	−5.86997
$873$	0	0
$874$	36.3642	1.23004
$875$	4.76924	0.161230
$876$	0	0
$877$	19.2114	0.648724	0.324362	−	0.945933i	$-0.394850\pi$
$877$	19.2114	0.648724	0.324362	+	0.945933i	$0.394850\pi$
$878$	51.6245	1.74224
$879$	0	0
$880$	−12.4960	−0.421238
$881$	−0.0458679	−0.00154533	−0.000772664	−	1.00000i	$-0.500246\pi$
$881$	−0.0458679	−0.00154533	−0.000772664		1.00000i	$0.500246\pi$
$882$	0	0
$883$	42.1763	1.41935	0.709674	−	0.704531i	$-0.248842\pi$
$883$	42.1763	1.41935	0.709674	+	0.704531i	$0.248842\pi$
$884$	3.01053	0.101255
$885$	0	0
$886$	−58.8515	−1.97715
$887$	−36.1371	−1.21337	−0.606683	−	0.794944i	$-0.707500\pi$
$887$	−36.1371	−1.21337	−0.606683	+	0.794944i	$0.707500\pi$
$888$	0	0
$889$	10.2175	0.342685
$890$	−0.267032	−0.00895093
$891$	0	0
$892$	130.938	4.38412
$893$	19.0636	0.637938
$894$	0	0
$895$	−9.59428	−0.320701
$896$	34.9306	1.16695
$897$	0	0
$898$	−41.1842	−1.37433
$899$	−33.9230	−1.13140
$900$	0	0
$901$	−5.56661	−0.185451
$902$	48.5951	1.61804
$903$	0	0
$904$	−72.9360	−2.42581
$905$	4.86985	0.161879
$906$	0	0
$907$	−21.2700	−0.706259	−0.353129	−	0.935574i	$-0.614882\pi$
$907$	−21.2700	−0.706259	−0.353129	+	0.935574i	$0.614882\pi$
$908$	19.2126	0.637592
$909$	0	0
$910$	0.930485	0.0308453
$911$	10.2529	0.339694	0.169847	−	0.985470i	$-0.445673\pi$
$911$	10.2529	0.339694	0.169847	+	0.985470i	$0.445673\pi$
$912$	0	0
$913$	24.5619	0.812880
$914$	−21.4465	−0.709387
$915$	0	0
$916$	−101.788	−3.36316
$917$	−8.32821	−0.275022
$918$	0	0
$919$	−39.3681	−1.29864	−0.649318	−	0.760517i	$-0.724945\pi$
$919$	−39.3681	−1.29864	−0.649318	+	0.760517i	$0.724945\pi$
$920$	−15.9177	−0.524791
$921$	0	0
$922$	90.5703	2.98277
$923$	0.933589	0.0307295
$924$	0	0
$925$	13.0532	0.429187
$926$	86.9674	2.85793
$927$	0	0
$928$	−114.745	−3.76667
$929$	−48.6956	−1.59765	−0.798826	−	0.601562i	$-0.794545\pi$
$929$	−48.6956	−1.59765	−0.798826	+	0.601562i	$0.794545\pi$
$930$	0	0
$931$	−21.6650	−0.710042
$932$	−65.9406	−2.15996
$933$	0	0
$934$	−16.4198	−0.537274
$935$	0.686898	0.0224640
$936$	0	0
$937$	−12.4806	−0.407723	−0.203862	−	0.979000i	$-0.565349\pi$
$937$	−12.4806	−0.407723	−0.203862	+	0.979000i	$0.565349\pi$
$938$	−14.4356	−0.471338
$939$	0	0
$940$	−13.2396	−0.431829
$941$	44.4531	1.44913	0.724565	−	0.689207i	$-0.242041\pi$
$941$	44.4531	1.44913	0.724565	+	0.689207i	$0.242041\pi$
$942$	0	0
$943$	35.3919	1.15252
$944$	−100.918	−3.28461
$945$	0	0
$946$	−21.3259	−0.693365
$947$	7.49160	0.243444	0.121722	−	0.992564i	$-0.461158\pi$
$947$	7.49160	0.243444	0.121722	+	0.992564i	$0.461158\pi$
$948$	0	0
$949$	5.49096	0.178244
$950$	−47.5954	−1.54420
$951$	0	0
$952$	−7.65109	−0.247973
$953$	14.6341	0.474045	0.237023	−	0.971504i	$-0.423828\pi$
$953$	14.6341	0.474045	0.237023	+	0.971504i	$0.423828\pi$
$954$	0	0
$955$	−3.73629	−0.120903
$956$	−165.760	−5.36106
$957$	0	0
$958$	−49.3693	−1.59505
$959$	−17.4887	−0.564741
$960$	0	0
$961$	6.65114	0.214553
$962$	5.21095	0.168008
$963$	0	0
$964$	−39.2891	−1.26542
$965$	−11.2521	−0.362219
$966$	0	0
$967$	1.24602	0.0400692	0.0200346	−	0.999799i	$-0.493622\pi$
$967$	1.24602	0.0400692	0.0200346	+	0.999799i	$0.493622\pi$
$968$	−70.6020	−2.26923
$969$	0	0
$970$	−3.94817	−0.126768
$971$	−15.4693	−0.496434	−0.248217	−	0.968704i	$-0.579845\pi$
$971$	−15.4693	−0.496434	−0.248217	+	0.968704i	$0.579845\pi$
$972$	0	0
$973$	11.6816	0.374494
$974$	−63.5846	−2.03738
$975$	0	0
$976$	−55.9969	−1.79242
$977$	−29.2177	−0.934757	−0.467379	−	0.884057i	$-0.654801\pi$
$977$	−29.2177	−0.934757	−0.467379	+	0.884057i	$0.654801\pi$
$978$	0	0
$979$	0.384710	0.0122954
$980$	15.0463	0.480637
$981$	0	0
$982$	11.6745	0.372550
$983$	2.39930	0.0765259	0.0382629	−	0.999268i	$-0.487818\pi$
$983$	2.39930	0.0765259	0.0382629	+	0.999268i	$0.487818\pi$
$984$	0	0
$985$	2.41949	0.0770913
$986$	11.9483	0.380513
$987$	0	0
$988$	−13.8718	−0.441321
$989$	−15.5317	−0.493878
$990$	0	0
$991$	51.4773	1.63523	0.817614	−	0.575766i	$-0.195296\pi$
$991$	51.4773	1.63523	0.817614	+	0.575766i	$0.195296\pi$
$992$	127.355	4.04352
$993$	0	0
$994$	−3.76445	−0.119401
$995$	−0.724779	−0.0229771
$996$	0	0
$997$	−41.2171	−1.30536	−0.652680	−	0.757634i	$-0.726355\pi$
$997$	−41.2171	−1.30536	−0.652680	+	0.757634i	$0.726355\pi$
$998$	−32.6862	−1.03466
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.18		18
3.2	odd	2		547.2.a.b.1.1	✓	18
12.11	even	2		8752.2.a.s.1.13		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.1	✓	18	3.2	odd	2
4923.2.a.l.1.18		18	1.1	even	1	trivial
8752.2.a.s.1.13		18	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:	\( N \)	\(=\)	\( 4923 = 3^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4923.a (trivial)

\(f(q)\)	\(=\)	\(q+2.72204 q^{2} +5.40952 q^{4} -0.469688 q^{5} +1.03831 q^{7} +9.28087 q^{8} +O(q^{10})\)	\(q+2.72204 q^{2} +5.40952 q^{4} -0.469688 q^{5} +1.03831 q^{7} +9.28087 q^{8} -1.27851 q^{10} +1.84194 q^{11} -0.700934 q^{13} +2.82633 q^{14} +14.4439 q^{16} -0.793975 q^{17} +3.65845 q^{19} -2.54079 q^{20} +5.01384 q^{22} +3.65159 q^{23} -4.77939 q^{25} -1.90797 q^{26} +5.61677 q^{28} -5.52848 q^{29} +6.13605 q^{31} +20.7552 q^{32} -2.16124 q^{34} -0.487683 q^{35} -2.73114 q^{37} +9.95846 q^{38} -4.35911 q^{40} +9.69220 q^{41} -4.25340 q^{43} +9.96402 q^{44} +9.93978 q^{46} +5.21084 q^{47} -5.92191 q^{49} -13.0097 q^{50} -3.79172 q^{52} +7.01107 q^{53} -0.865138 q^{55} +9.63643 q^{56} -15.0488 q^{58} -6.98691 q^{59} -3.87686 q^{61} +16.7026 q^{62} +27.6087 q^{64} +0.329220 q^{65} -5.10753 q^{67} -4.29503 q^{68} -1.32749 q^{70} -1.33192 q^{71} -7.83377 q^{73} -7.43429 q^{74} +19.7905 q^{76} +1.91251 q^{77} +4.36166 q^{79} -6.78412 q^{80} +26.3826 q^{82} +13.3348 q^{83} +0.372921 q^{85} -11.5779 q^{86} +17.0948 q^{88} +0.208861 q^{89} -0.727788 q^{91} +19.7533 q^{92} +14.1841 q^{94} -1.71833 q^{95} +3.08809 q^{97} -16.1197 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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