LMFDB - Embedded newform 1629.4.a.b.1.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1629,4,Mod(1,1629)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1629.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1629, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1629 = 3^{2} \cdot 181 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1629.a (trivial)

Newform invariants

comment:select newform

sage:traces = [17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$96.1141113994$
Analytic rank:	$1$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 5 x^{16} - 71 x^{15} + 350 x^{14} + 1993 x^{13} - 9702 x^{12} - 28280 x^{11} + 136606 x^{10} + \cdots - 19616 $ x^17 - 5x^16 - 71x^15 + 350x^14 + 1993x^13 - 9702x^12 - 28280x^11 + 136606x^10 + 213561x^9 - 1035619x^8 - 816085x^7 + 4074332x^6 + 1288101x^5 - 7036810x^4 - 384244x^3 + 2665504x^2 + 597216x - 19616
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 543)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$2.22090$ of defining polynomial
Character	$\chi$	$=$	1629.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.22090 q^{2} -3.06759 q^{4} -12.5951 q^{5} +24.1343 q^{7} -24.5800 q^{8} -27.9724 q^{10} +41.6126 q^{11} -69.2885 q^{13} +53.5999 q^{14} -30.0492 q^{16} +124.194 q^{17} -114.386 q^{19} +38.6365 q^{20} +92.4176 q^{22} +69.0499 q^{23} +33.6357 q^{25} -153.883 q^{26} -74.0341 q^{28} -138.456 q^{29} +194.464 q^{31} +129.904 q^{32} +275.822 q^{34} -303.973 q^{35} +58.7545 q^{37} -254.040 q^{38} +309.587 q^{40} +207.753 q^{41} -304.683 q^{43} -127.651 q^{44} +153.353 q^{46} +354.608 q^{47} +239.464 q^{49} +74.7016 q^{50} +212.549 q^{52} -57.2922 q^{53} -524.114 q^{55} -593.222 q^{56} -307.497 q^{58} +67.1797 q^{59} -512.348 q^{61} +431.887 q^{62} +528.898 q^{64} +872.694 q^{65} +106.780 q^{67} -380.976 q^{68} -675.094 q^{70} -372.183 q^{71} -451.820 q^{73} +130.488 q^{74} +350.889 q^{76} +1004.29 q^{77} -661.578 q^{79} +378.471 q^{80} +461.399 q^{82} -235.552 q^{83} -1564.23 q^{85} -676.671 q^{86} -1022.84 q^{88} -1352.43 q^{89} -1672.23 q^{91} -211.817 q^{92} +787.549 q^{94} +1440.70 q^{95} -131.453 q^{97} +531.825 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 5 q^{2} + 31 q^{4} + 21 q^{5} - 62 q^{7} + 60 q^{8} - 127 q^{10} + 65 q^{11} - 246 q^{13} + 121 q^{14} - 85 q^{16} + 433 q^{17} - 459 q^{19} + 208 q^{20} - 270 q^{22} + 165 q^{23} - 146 q^{25} + 89 q^{26}+ \cdots - 1803 q^{98}+O(q^{100}) $ 17 * q + 5 * q^2 + 31 * q^4 + 21 * q^5 - 62 * q^7 + 60 * q^8 - 127 * q^10 + 65 * q^11 - 246 * q^13 + 121 * q^14 - 85 * q^16 + 433 * q^17 - 459 * q^19 + 208 * q^20 - 270 * q^22 + 165 * q^23 - 146 * q^25 + 89 * q^26 - 625 * q^28 + 271 * q^29 - 328 * q^31 + 690 * q^32 - 699 * q^34 + 662 * q^35 - 1068 * q^37 + 571 * q^38 - 1038 * q^40 + 837 * q^41 - 1808 * q^43 + 48 * q^44 - 808 * q^46 + 707 * q^47 - 1085 * q^49 - 981 * q^50 - 1551 * q^52 + 154 * q^53 - 1914 * q^55 - 1098 * q^56 + 1104 * q^58 + 223 * q^59 - 1459 * q^61 - 1631 * q^62 + 718 * q^64 + 280 * q^65 - 1860 * q^67 - 684 * q^68 + 400 * q^70 + 307 * q^71 - 3177 * q^73 - 1138 * q^74 - 1674 * q^76 + 714 * q^77 - 3487 * q^79 - 1636 * q^80 - 1497 * q^82 + 1556 * q^83 - 1810 * q^85 - 3881 * q^86 - 3632 * q^88 + 715 * q^89 - 1776 * q^91 - 1033 * q^92 - 1979 * q^94 + 100 * q^95 - 3254 * q^97 - 1803 * q^98

Coefficient data

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

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

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

$2$	2.22090	0.785208	0.392604	−	0.919708i	$-0.371574\pi$
$2$	2.22090	0.785208	0.392604	+	0.919708i	$0.371574\pi$
$3$	0	0
$3$	0	0
$4$	−3.06759	−0.383449
$5$	−12.5951	−1.12654	−0.563268	−	0.826274i	$-0.690456\pi$
$5$	−12.5951	−1.12654	−0.563268	+	0.826274i	$0.690456\pi$
$6$	0	0
$7$	24.1343	1.30313	0.651564	−	0.758593i	$-0.274113\pi$
$7$	24.1343	1.30313	0.651564	+	0.758593i	$0.274113\pi$
$8$	−24.5800	−1.08629
$9$	0	0
$10$	−27.9724	−0.884566
$11$	41.6126	1.14061	0.570304	−	0.821434i	$-0.306826\pi$
$11$	41.6126	1.14061	0.570304	+	0.821434i	$0.306826\pi$
$12$	0	0
$13$	−69.2885	−1.47824	−0.739122	−	0.673571i	$-0.764760\pi$
$13$	−69.2885	−1.47824	−0.739122	+	0.673571i	$0.764760\pi$
$14$	53.5999	1.02323
$15$	0	0
$16$	−30.0492	−0.469518
$17$	124.194	1.77185	0.885924	−	0.463830i	$-0.153525\pi$
$17$	124.194	1.77185	0.885924	+	0.463830i	$0.153525\pi$
$18$	0	0
$19$	−114.386	−1.38116	−0.690578	−	0.723258i	$-0.742644\pi$
$19$	−114.386	−1.38116	−0.690578	+	0.723258i	$0.742644\pi$
$20$	38.6365	0.431969
$21$	0	0
$22$	92.4176	0.895613
$23$	69.0499	0.625996	0.312998	−	0.949754i	$-0.398667\pi$
$23$	69.0499	0.625996	0.312998	+	0.949754i	$0.398667\pi$
$24$	0	0
$25$	33.6357	0.269086
$26$	−153.883	−1.16073
$27$	0	0
$28$	−74.0341	−0.499683
$29$	−138.456	−0.886572	−0.443286	−	0.896380i	$-0.646187\pi$
$29$	−138.456	−0.886572	−0.443286	+	0.896380i	$0.646187\pi$
$30$	0	0
$31$	194.464	1.12667	0.563336	−	0.826228i	$-0.309518\pi$
$31$	194.464	1.12667	0.563336	+	0.826228i	$0.309518\pi$
$32$	129.904	0.717625
$33$	0	0
$34$	275.822	1.39127
$35$	−303.973	−1.46802
$36$	0	0
$37$	58.7545	0.261059	0.130529	−	0.991444i	$-0.458332\pi$
$37$	58.7545	0.261059	0.130529	+	0.991444i	$0.458332\pi$
$38$	−254.040	−1.08449
$39$	0	0
$40$	309.587	1.22375
$41$	207.753	0.791355	0.395678	−	0.918389i	$-0.370510\pi$
$41$	207.753	0.791355	0.395678	+	0.918389i	$0.370510\pi$
$42$	0	0
$43$	−304.683	−1.08055	−0.540275	−	0.841488i	$-0.681680\pi$
$43$	−304.683	−1.08055	−0.540275	+	0.841488i	$0.681680\pi$
$44$	−127.651	−0.437365
$45$	0	0
$46$	153.353	0.491537
$47$	354.608	1.10053	0.550264	−	0.834990i	$-0.314527\pi$
$47$	354.608	1.10053	0.550264	+	0.834990i	$0.314527\pi$
$48$	0	0
$49$	239.464	0.698145
$50$	74.7016	0.211288
$51$	0	0
$52$	212.549	0.566831
$53$	−57.2922	−0.148485	−0.0742423	−	0.997240i	$-0.523654\pi$
$53$	−57.2922	−0.148485	−0.0742423	+	0.997240i	$0.523654\pi$
$54$	0	0
$55$	−524.114	−1.28494
$56$	−593.222	−1.41558
$57$	0	0
$58$	−307.497	−0.696143
$59$	67.1797	0.148238	0.0741190	−	0.997249i	$-0.476386\pi$
$59$	67.1797	0.148238	0.0741190	+	0.997249i	$0.476386\pi$
$60$	0	0
$61$	−512.348	−1.07540	−0.537701	−	0.843136i	$-0.680707\pi$
$61$	−512.348	−1.07540	−0.537701	+	0.843136i	$0.680707\pi$
$62$	431.887	0.884672
$63$	0	0
$64$	528.898	1.03300
$65$	872.694	1.66530
$66$	0	0
$67$	106.780	0.194705	0.0973527	−	0.995250i	$-0.468963\pi$
$67$	106.780	0.194705	0.0973527	+	0.995250i	$0.468963\pi$
$68$	−380.976	−0.679413
$69$	0	0
$70$	−675.094	−1.15270
$71$	−372.183	−0.622113	−0.311056	−	0.950391i	$-0.600683\pi$
$71$	−372.183	−0.622113	−0.311056	+	0.950391i	$0.600683\pi$
$72$	0	0
$73$	−451.820	−0.724404	−0.362202	−	0.932100i	$-0.617975\pi$
$73$	−451.820	−0.724404	−0.362202	+	0.932100i	$0.617975\pi$
$74$	130.488	0.204986
$75$	0	0
$76$	350.889	0.529602
$77$	1004.29	1.48636
$78$	0	0
$79$	−661.578	−0.942194	−0.471097	−	0.882081i	$-0.656142\pi$
$79$	−661.578	−0.942194	−0.471097	+	0.882081i	$0.656142\pi$
$80$	378.471	0.528930
$81$	0	0
$82$	461.399	0.621378
$83$	−235.552	−0.311509	−0.155754	−	0.987796i	$-0.549781\pi$
$83$	−235.552	−0.311509	−0.155754	+	0.987796i	$0.549781\pi$
$84$	0	0
$85$	−1564.23	−1.99605
$86$	−676.671	−0.848457
$87$	0	0
$88$	−1022.84	−1.23904
$89$	−1352.43	−1.61076	−0.805380	−	0.592759i	$-0.798039\pi$
$89$	−1352.43	−1.61076	−0.805380	+	0.592759i	$0.798039\pi$
$90$	0	0
$91$	−1672.23	−1.92634
$92$	−211.817	−0.240037
$93$	0	0
$94$	787.549	0.864144
$95$	1440.70	1.55592
$96$	0	0
$97$	−131.453	−0.137599	−0.0687994	−	0.997631i	$-0.521917\pi$
$97$	−131.453	−0.137599	−0.0687994	+	0.997631i	$0.521917\pi$
$98$	531.825	0.548188
$99$	0	0
$100$	−103.181	−0.103181
$101$	−1264.62	−1.24589	−0.622944	−	0.782266i	$-0.714064\pi$
$101$	−1264.62	−1.24589	−0.622944	+	0.782266i	$0.714064\pi$
$102$	0	0
$103$	843.745	0.807152	0.403576	−	0.914946i	$-0.367767\pi$
$103$	843.745	0.807152	0.403576	+	0.914946i	$0.367767\pi$
$104$	1703.12	1.60581
$105$	0	0
$106$	−127.240	−0.116591
$107$	−205.666	−0.185817	−0.0929087	−	0.995675i	$-0.529616\pi$
$107$	−205.666	−0.185817	−0.0929087	+	0.995675i	$0.529616\pi$
$108$	0	0
$109$	1416.75	1.24495	0.622476	−	0.782639i	$-0.286127\pi$
$109$	1416.75	1.24495	0.622476	+	0.782639i	$0.286127\pi$
$110$	−1164.01	−1.00894
$111$	0	0
$112$	−725.215	−0.611843
$113$	−2107.97	−1.75488	−0.877440	−	0.479686i	$-0.840751\pi$
$113$	−2107.97	−1.75488	−0.877440	+	0.479686i	$0.840751\pi$
$114$	0	0
$115$	−869.688	−0.705208
$116$	424.726	0.339955
$117$	0	0
$118$	149.200	0.116398
$119$	2997.33	2.30895
$120$	0	0
$121$	400.611	0.300985
$122$	−1137.88	−0.844413
$123$	0	0
$124$	−596.537	−0.432021
$125$	1150.74	0.823402
$126$	0	0
$127$	−2481.52	−1.73386	−0.866928	−	0.498433i	$-0.833909\pi$
$127$	−2481.52	−1.73386	−0.866928	+	0.498433i	$0.833909\pi$
$128$	135.397	0.0934966
$129$	0	0
$130$	1938.17	1.30760
$131$	−636.102	−0.424248	−0.212124	−	0.977243i	$-0.568038\pi$
$131$	−636.102	−0.424248	−0.212124	+	0.977243i	$0.568038\pi$
$132$	0	0
$133$	−2760.62	−1.79982
$134$	237.148	0.152884
$135$	0	0
$136$	−3052.69	−1.92475
$137$	251.472	0.156823	0.0784114	−	0.996921i	$-0.475015\pi$
$137$	251.472	0.156823	0.0784114	+	0.996921i	$0.475015\pi$
$138$	0	0
$139$	−2121.90	−1.29480	−0.647400	−	0.762151i	$-0.724144\pi$
$139$	−2121.90	−1.29480	−0.647400	+	0.762151i	$0.724144\pi$
$140$	932.464	0.562912
$141$	0	0
$142$	−826.582	−0.488488
$143$	−2883.28	−1.68610
$144$	0	0
$145$	1743.86	0.998756
$146$	−1003.45	−0.568808
$147$	0	0
$148$	−180.235	−0.100103
$149$	1929.37	1.06081	0.530404	−	0.847745i	$-0.322040\pi$
$149$	1929.37	1.06081	0.530404	+	0.847745i	$0.322040\pi$
$150$	0	0
$151$	−1892.02	−1.01967	−0.509837	−	0.860271i	$-0.670294\pi$
$151$	−1892.02	−1.01967	−0.509837	+	0.860271i	$0.670294\pi$
$152$	2811.61	1.50034
$153$	0	0
$154$	2230.43	1.16710
$155$	−2449.29	−1.26924
$156$	0	0
$157$	−2989.77	−1.51981	−0.759904	−	0.650036i	$-0.774754\pi$
$157$	−2989.77	−1.51981	−0.759904	+	0.650036i	$0.774754\pi$
$158$	−1469.30	−0.739818
$159$	0	0
$160$	−1636.15	−0.808432
$161$	1666.47	0.815753
$162$	0	0
$163$	2488.85	1.19596	0.597980	−	0.801511i	$-0.295970\pi$
$163$	2488.85	1.19596	0.597980	+	0.801511i	$0.295970\pi$
$164$	−637.301	−0.303444
$165$	0	0
$166$	−523.139	−0.244599
$167$	−2212.25	−1.02509	−0.512543	−	0.858661i	$-0.671297\pi$
$167$	−2212.25	−1.02509	−0.512543	+	0.858661i	$0.671297\pi$
$168$	0	0
$169$	2603.90	1.18521
$170$	−3474.00	−1.56732
$171$	0	0
$172$	934.642	0.414336
$173$	1293.44	0.568432	0.284216	−	0.958760i	$-0.408267\pi$
$173$	1293.44	0.568432	0.284216	+	0.958760i	$0.408267\pi$
$174$	0	0
$175$	811.774	0.350653
$176$	−1250.42	−0.535536
$177$	0	0
$178$	−3003.62	−1.26478
$179$	1516.30	0.633149	0.316574	−	0.948568i	$-0.397467\pi$
$179$	1516.30	0.633149	0.316574	+	0.948568i	$0.397467\pi$
$180$	0	0
$181$	−181.000	−0.0743294
$181$	−181.000	−0.0743294
$182$	−3713.86	−1.51258
$183$	0	0
$184$	−1697.25	−0.680016
$185$	−740.017	−0.294093
$186$	0	0
$187$	5168.03	2.02098
$188$	−1087.79	−0.421997
$189$	0	0
$190$	3199.65	1.22172
$191$	−4170.02	−1.57975	−0.789875	−	0.613268i	$-0.789854\pi$
$191$	−4170.02	−1.57975	−0.789875	+	0.613268i	$0.789854\pi$
$192$	0	0
$193$	−4053.18	−1.51168	−0.755840	−	0.654757i	$-0.772771\pi$
$193$	−4053.18	−1.51168	−0.755840	+	0.654757i	$0.772771\pi$
$194$	−291.945	−0.108044
$195$	0	0
$196$	−734.576	−0.267703
$197$	−4980.20	−1.80114	−0.900571	−	0.434710i	$-0.856851\pi$
$197$	−4980.20	−1.80114	−0.900571	+	0.434710i	$0.856851\pi$
$198$	0	0
$199$	−775.235	−0.276155	−0.138078	−	0.990421i	$-0.544092\pi$
$199$	−775.235	−0.276155	−0.138078	+	0.990421i	$0.544092\pi$
$200$	−826.767	−0.292306
$201$	0	0
$202$	−2808.60	−0.978281
$203$	−3341.53	−1.15532
$204$	0	0
$205$	−2616.66	−0.891491
$206$	1873.87	0.633782
$207$	0	0
$208$	2082.06	0.694063
$209$	−4759.90	−1.57536
$210$	0	0
$211$	2014.44	0.657252	0.328626	−	0.944460i	$-0.393415\pi$
$211$	2014.44	0.657252	0.328626	+	0.944460i	$0.393415\pi$
$212$	175.749	0.0569363
$213$	0	0
$214$	−456.764	−0.145905
$215$	3837.50	1.21728
$216$	0	0
$217$	4693.26	1.46820
$218$	3146.46	0.977546
$219$	0	0
$220$	1607.77	0.492707
$221$	−8605.21	−2.61923
$222$	0	0
$223$	1581.37	0.474870	0.237435	−	0.971403i	$-0.423693\pi$
$223$	1581.37	0.474870	0.237435	+	0.971403i	$0.423693\pi$
$224$	3135.14	0.935158
$225$	0	0
$226$	−4681.60	−1.37795
$227$	−3284.88	−0.960464	−0.480232	−	0.877142i	$-0.659447\pi$
$227$	−3284.88	−0.960464	−0.480232	+	0.877142i	$0.659447\pi$
$228$	0	0
$229$	−2869.66	−0.828089	−0.414045	−	0.910257i	$-0.635884\pi$
$229$	−2869.66	−0.828089	−0.414045	+	0.910257i	$0.635884\pi$
$230$	−1931.49	−0.553734
$231$	0	0
$232$	3403.25	0.963078
$233$	1453.15	0.408581	0.204290	−	0.978910i	$-0.434511\pi$
$233$	1453.15	0.408581	0.204290	+	0.978910i	$0.434511\pi$
$234$	0	0
$235$	−4466.31	−1.23979
$236$	−206.080	−0.0568417
$237$	0	0
$238$	6656.77	1.81300
$239$	6263.29	1.69514	0.847570	−	0.530683i	$-0.178065\pi$
$239$	6263.29	1.69514	0.847570	+	0.530683i	$0.178065\pi$
$240$	0	0
$241$	−5448.22	−1.45623	−0.728114	−	0.685456i	$-0.759603\pi$
$241$	−5448.22	−1.45623	−0.728114	+	0.685456i	$0.759603\pi$
$242$	889.717	0.236335
$243$	0	0
$244$	1571.68	0.412361
$245$	−3016.06	−0.786486
$246$	0	0
$247$	7925.64	2.04169
$248$	−4779.94	−1.22390
$249$	0	0
$250$	2555.68	0.646542
$251$	6442.28	1.62005	0.810026	−	0.586394i	$-0.199453\pi$
$251$	6442.28	1.62005	0.810026	+	0.586394i	$0.199453\pi$
$252$	0	0
$253$	2873.35	0.714015
$254$	−5511.23	−1.36144
$255$	0	0
$256$	−3930.48	−0.959589
$257$	−1661.83	−0.403356	−0.201678	−	0.979452i	$-0.564639\pi$
$257$	−1661.83	−0.403356	−0.201678	+	0.979452i	$0.564639\pi$
$258$	0	0
$259$	1418.00	0.340193
$260$	−2677.07	−0.638556
$261$	0	0
$262$	−1412.72	−0.333123
$263$	2563.49	0.601033	0.300517	−	0.953777i	$-0.402841\pi$
$263$	2563.49	0.601033	0.300517	+	0.953777i	$0.402841\pi$
$264$	0	0
$265$	721.599	0.167273
$266$	−6131.08	−1.41324
$267$	0	0
$268$	−327.558	−0.0746596
$269$	−3782.12	−0.857249	−0.428624	−	0.903483i	$-0.641002\pi$
$269$	−3782.12	−0.857249	−0.428624	+	0.903483i	$0.641002\pi$
$270$	0	0
$271$	3065.79	0.687209	0.343605	−	0.939114i	$-0.388352\pi$
$271$	3065.79	0.687209	0.343605	+	0.939114i	$0.388352\pi$
$272$	−3731.92	−0.831915
$273$	0	0
$274$	558.495	0.123138
$275$	1399.67	0.306921
$276$	0	0
$277$	−3315.35	−0.719133	−0.359567	−	0.933119i	$-0.617075\pi$
$277$	−3315.35	−0.719133	−0.359567	+	0.933119i	$0.617075\pi$
$278$	−4712.53	−1.01669
$279$	0	0
$280$	7471.67	1.59471
$281$	−3047.80	−0.647034	−0.323517	−	0.946222i	$-0.604865\pi$
$281$	−3047.80	−0.647034	−0.323517	+	0.946222i	$0.604865\pi$
$282$	0	0
$283$	636.812	0.133762	0.0668808	−	0.997761i	$-0.478695\pi$
$283$	636.812	0.133762	0.0668808	+	0.997761i	$0.478695\pi$
$284$	1141.71	0.238548
$285$	0	0
$286$	−6403.48	−1.32394
$287$	5013.97	1.03124
$288$	0	0
$289$	10511.1	2.13945
$290$	3872.94	0.784231
$291$	0	0
$292$	1386.00	0.277772
$293$	8294.83	1.65389	0.826944	−	0.562284i	$-0.190077\pi$
$293$	8294.83	1.65389	0.826944	+	0.562284i	$0.190077\pi$
$294$	0	0
$295$	−846.132	−0.166996
$296$	−1444.19	−0.283587
$297$	0	0
$298$	4284.95	0.832955
$299$	−4784.37	−0.925375
$300$	0	0
$301$	−7353.30	−1.40810
$302$	−4202.00	−0.800656
$303$	0	0
$304$	3437.20	0.648477
$305$	6453.06	1.21148
$306$	0	0
$307$	−2003.72	−0.372502	−0.186251	−	0.982502i	$-0.559634\pi$
$307$	−2003.72	−0.372502	−0.186251	+	0.982502i	$0.559634\pi$
$308$	−3080.75	−0.569942
$309$	0	0
$310$	−5439.64	−0.996615
$311$	8950.67	1.63198	0.815990	−	0.578065i	$-0.196192\pi$
$311$	8950.67	1.63198	0.815990	+	0.578065i	$0.196192\pi$
$312$	0	0
$313$	−6502.45	−1.17425	−0.587125	−	0.809497i	$-0.699740\pi$
$313$	−6502.45	−1.17425	−0.587125	+	0.809497i	$0.699740\pi$
$314$	−6639.99	−1.19336
$315$	0	0
$316$	2029.45	0.361283
$317$	−7060.10	−1.25090	−0.625449	−	0.780265i	$-0.715084\pi$
$317$	−7060.10	−1.25090	−0.625449	+	0.780265i	$0.715084\pi$
$318$	0	0
$319$	−5761.51	−1.01123
$320$	−6661.50	−1.16372
$321$	0	0
$322$	3701.07	0.640536
$323$	−14206.0	−2.44720
$324$	0	0
$325$	−2330.57	−0.397774
$326$	5527.49	0.939078
$327$	0	0
$328$	−5106.58	−0.859645
$329$	8558.20	1.43413
$330$	0	0
$331$	9067.92	1.50579	0.752897	−	0.658138i	$-0.228656\pi$
$331$	9067.92	1.50579	0.752897	+	0.658138i	$0.228656\pi$
$332$	722.578	0.119448
$333$	0	0
$334$	−4913.20	−0.804906
$335$	−1344.90	−0.219343
$336$	0	0
$337$	11310.3	1.82822	0.914108	−	0.405470i	$-0.132892\pi$
$337$	11310.3	1.82822	0.914108	+	0.405470i	$0.132892\pi$
$338$	5783.01	0.930634
$339$	0	0
$340$	4798.41	0.765384
$341$	8092.17	1.28509
$342$	0	0
$343$	−2498.78	−0.393356
$344$	7489.12	1.17380
$345$	0	0
$346$	2872.61	0.446337
$347$	−3949.05	−0.610940	−0.305470	−	0.952202i	$-0.598814\pi$
$347$	−3949.05	−0.610940	−0.305470	+	0.952202i	$0.598814\pi$
$348$	0	0
$349$	1396.69	0.214221	0.107110	−	0.994247i	$-0.465840\pi$
$349$	1396.69	0.214221	0.107110	+	0.994247i	$0.465840\pi$
$350$	1802.87	0.275336
$351$	0	0
$352$	5405.65	0.818529
$353$	3218.35	0.485256	0.242628	−	0.970119i	$-0.421991\pi$
$353$	3218.35	0.485256	0.242628	+	0.970119i	$0.421991\pi$
$354$	0	0
$355$	4687.67	0.700833
$356$	4148.71	0.617644
$357$	0	0
$358$	3367.56	0.497153
$359$	2851.63	0.419230	0.209615	−	0.977784i	$-0.432779\pi$
$359$	2851.63	0.419230	0.209615	+	0.977784i	$0.432779\pi$
$360$	0	0
$361$	6225.16	0.907590
$362$	−401.983	−0.0583640
$363$	0	0
$364$	5129.71	0.738654
$365$	5690.70	0.816068
$366$	0	0
$367$	6627.24	0.942613	0.471307	−	0.881969i	$-0.343783\pi$
$367$	6627.24	0.942613	0.471307	+	0.881969i	$0.343783\pi$
$368$	−2074.89	−0.293916
$369$	0	0
$370$	−1643.51	−0.230924
$371$	−1382.71	−0.193495
$372$	0	0
$373$	4853.81	0.673782	0.336891	−	0.941544i	$-0.390625\pi$
$373$	4853.81	0.673782	0.336891	+	0.941544i	$0.390625\pi$
$374$	11477.7	1.58689
$375$	0	0
$376$	−8716.27	−1.19550
$377$	9593.40	1.31057
$378$	0	0
$379$	9236.82	1.25188	0.625941	−	0.779870i	$-0.284715\pi$
$379$	9236.82	1.25188	0.625941	+	0.779870i	$0.284715\pi$
$380$	−4419.48	−0.596617
$381$	0	0
$382$	−9261.21	−1.24043
$383$	6565.85	0.875977	0.437988	−	0.898981i	$-0.355691\pi$
$383$	6565.85	0.875977	0.437988	+	0.898981i	$0.355691\pi$
$384$	0	0
$385$	−12649.1	−1.67444
$386$	−9001.71	−1.18698
$387$	0	0
$388$	403.245	0.0527621
$389$	−4912.64	−0.640310	−0.320155	−	0.947365i	$-0.603735\pi$
$389$	−4912.64	−0.640310	−0.320155	+	0.947365i	$0.603735\pi$
$390$	0	0
$391$	8575.57	1.10917
$392$	−5886.03	−0.758391
$393$	0	0
$394$	−11060.5	−1.41427
$395$	8332.62	1.06142
$396$	0	0
$397$	−7872.23	−0.995204	−0.497602	−	0.867405i	$-0.665786\pi$
$397$	−7872.23	−0.995204	−0.497602	+	0.867405i	$0.665786\pi$
$398$	−1721.72	−0.216839
$399$	0	0
$400$	−1010.72	−0.126341
$401$	−1393.53	−0.173540	−0.0867698	−	0.996228i	$-0.527654\pi$
$401$	−1393.53	−0.173540	−0.0867698	+	0.996228i	$0.527654\pi$
$402$	0	0
$403$	−13474.2	−1.66550
$404$	3879.35	0.477734
$405$	0	0
$406$	−7421.21	−0.907164
$407$	2444.93	0.297766
$408$	0	0
$409$	−8216.36	−0.993332	−0.496666	−	0.867942i	$-0.665443\pi$
$409$	−8216.36	−0.993332	−0.496666	+	0.867942i	$0.665443\pi$
$410$	−5811.35	−0.700006
$411$	0	0
$412$	−2588.26	−0.309501
$413$	1621.33	0.193173
$414$	0	0
$415$	2966.80	0.350926
$416$	−9000.86	−1.06083
$417$	0	0
$418$	−10571.3	−1.23698
$419$	8188.66	0.954755	0.477377	−	0.878698i	$-0.341587\pi$
$419$	8188.66	0.954755	0.477377	+	0.878698i	$0.341587\pi$
$420$	0	0
$421$	−9284.71	−1.07484	−0.537422	−	0.843314i	$-0.680602\pi$
$421$	−9284.71	−1.07484	−0.537422	+	0.843314i	$0.680602\pi$
$422$	4473.88	0.516079
$423$	0	0
$424$	1408.24	0.161298
$425$	4177.34	0.476779
$426$	0	0
$427$	−12365.2	−1.40139
$428$	630.899	0.0712515
$429$	0	0
$430$	8522.71	0.955818
$431$	245.242	0.0274081	0.0137041	−	0.999906i	$-0.495638\pi$
$431$	245.242	0.0274081	0.0137041	+	0.999906i	$0.495638\pi$
$432$	0	0
$433$	−13239.2	−1.46937	−0.734683	−	0.678411i	$-0.762669\pi$
$433$	−13239.2	−1.46937	−0.734683	+	0.678411i	$0.762669\pi$
$434$	10423.3	1.15284
$435$	0	0
$436$	−4346.00	−0.477376
$437$	−7898.35	−0.864598
$438$	0	0
$439$	10250.7	1.11444	0.557218	−	0.830366i	$-0.311869\pi$
$439$	10250.7	1.11444	0.557218	+	0.830366i	$0.311869\pi$
$440$	12882.7	1.39582
$441$	0	0
$442$	−19111.3	−2.05664
$443$	−13273.9	−1.42362	−0.711811	−	0.702371i	$-0.752125\pi$
$443$	−13273.9	−1.42362	−0.711811	+	0.702371i	$0.752125\pi$
$444$	0	0
$445$	17034.0	1.81458
$446$	3512.06	0.372872
$447$	0	0
$448$	12764.6	1.34614
$449$	13798.5	1.45032	0.725160	−	0.688581i	$-0.241766\pi$
$449$	13798.5	1.45032	0.725160	+	0.688581i	$0.241766\pi$
$450$	0	0
$451$	8645.15	0.902625
$452$	6466.40	0.672907
$453$	0	0
$454$	−7295.40	−0.754164
$455$	21061.8	2.17010
$456$	0	0
$457$	6261.74	0.640945	0.320472	−	0.947258i	$-0.396158\pi$
$457$	6261.74	0.640945	0.320472	+	0.947258i	$0.396158\pi$
$458$	−6373.23	−0.650222
$459$	0	0
$460$	2667.85	0.270411
$461$	−11419.4	−1.15370	−0.576850	−	0.816850i	$-0.695718\pi$
$461$	−11419.4	−1.15370	−0.576850	+	0.816850i	$0.695718\pi$
$462$	0	0
$463$	1294.61	0.129948	0.0649739	−	0.997887i	$-0.479304\pi$
$463$	1294.61	0.129948	0.0649739	+	0.997887i	$0.479304\pi$
$464$	4160.48	0.416262
$465$	0	0
$466$	3227.31	0.320821
$467$	6534.30	0.647476	0.323738	−	0.946147i	$-0.395060\pi$
$467$	6534.30	0.647476	0.323738	+	0.946147i	$0.395060\pi$
$468$	0	0
$469$	2577.06	0.253726
$470$	−9919.24	−0.973490
$471$	0	0
$472$	−1651.28	−0.161030
$473$	−12678.7	−1.23248
$474$	0	0
$475$	−3847.45	−0.371649
$476$	−9194.58	−0.885363
$477$	0	0
$478$	13910.2	1.33104
$479$	−2316.45	−0.220963	−0.110482	−	0.993878i	$-0.535239\pi$
$479$	−2316.45	−0.220963	−0.110482	+	0.993878i	$0.535239\pi$
$480$	0	0
$481$	−4071.01	−0.385909
$482$	−12100.0	−1.14344
$483$	0	0
$484$	−1228.91	−0.115412
$485$	1655.66	0.155010
$486$	0	0
$487$	−6271.75	−0.583573	−0.291787	−	0.956483i	$-0.594250\pi$
$487$	−6271.75	−0.583573	−0.291787	+	0.956483i	$0.594250\pi$
$488$	12593.5	1.16820
$489$	0	0
$490$	−6698.38	−0.617555
$491$	−21736.4	−1.99786	−0.998932	−	0.0461948i	$-0.985291\pi$
$491$	−21736.4	−1.99786	−0.998932	+	0.0461948i	$0.985291\pi$
$492$	0	0
$493$	−17195.3	−1.57087
$494$	17602.1	1.60315
$495$	0	0
$496$	−5843.49	−0.528993
$497$	−8982.37	−0.810693
$498$	0	0
$499$	3086.65	0.276909	0.138454	−	0.990369i	$-0.455787\pi$
$499$	3086.65	0.276909	0.138454	+	0.990369i	$0.455787\pi$
$500$	−3530.00	−0.315733
$501$	0	0
$502$	14307.7	1.27208
$503$	9728.59	0.862378	0.431189	−	0.902262i	$-0.358094\pi$
$503$	9728.59	0.862378	0.431189	+	0.902262i	$0.358094\pi$
$504$	0	0
$505$	15928.0	1.40354
$506$	6381.43	0.560650
$507$	0	0
$508$	7612.30	0.664845
$509$	−14650.3	−1.27576	−0.637881	−	0.770135i	$-0.720189\pi$
$509$	−14650.3	−1.27576	−0.637881	+	0.770135i	$0.720189\pi$
$510$	0	0
$511$	−10904.3	−0.943992
$512$	−9812.39	−0.846973
$513$	0	0
$514$	−3690.77	−0.316718
$515$	−10627.0	−0.909286
$516$	0	0
$517$	14756.2	1.25527
$518$	3149.24	0.267123
$519$	0	0
$520$	−21450.9	−1.80900
$521$	−9999.74	−0.840876	−0.420438	−	0.907321i	$-0.638124\pi$
$521$	−9999.74	−0.840876	−0.420438	+	0.907321i	$0.638124\pi$
$522$	0	0
$523$	1356.47	0.113412	0.0567060	−	0.998391i	$-0.481940\pi$
$523$	1356.47	0.113412	0.0567060	+	0.998391i	$0.481940\pi$
$524$	1951.30	0.162677
$525$	0	0
$526$	5693.27	0.471936
$527$	24151.3	1.99629
$528$	0	0
$529$	−7399.11	−0.608129
$530$	1602.60	0.131344
$531$	0	0
$532$	8468.47	0.690140
$533$	−14394.9	−1.16982
$534$	0	0
$535$	2590.38	0.209330
$536$	−2624.66	−0.211508
$537$	0	0
$538$	−8399.72	−0.673118
$539$	9964.71	0.796309
$540$	0	0
$541$	−19207.0	−1.52638	−0.763190	−	0.646174i	$-0.776368\pi$
$541$	−19207.0	−1.52638	−0.763190	+	0.646174i	$0.776368\pi$
$542$	6808.83	0.539602
$543$	0	0
$544$	16133.3	1.27152
$545$	−17844.0	−1.40248
$546$	0	0
$547$	7518.52	0.587694	0.293847	−	0.955852i	$-0.405064\pi$
$547$	7518.52	0.587694	0.293847	+	0.955852i	$0.405064\pi$
$548$	−771.414	−0.0601335
$549$	0	0
$550$	3108.53	0.240997
$551$	15837.4	1.22449
$552$	0	0
$553$	−15966.7	−1.22780
$554$	−7363.06	−0.564669
$555$	0	0
$556$	6509.12	0.496489
$557$	−6622.61	−0.503787	−0.251893	−	0.967755i	$-0.581053\pi$
$557$	−6622.61	−0.503787	−0.251893	+	0.967755i	$0.581053\pi$
$558$	0	0
$559$	21111.0	1.59732
$560$	9134.13	0.689263
$561$	0	0
$562$	−6768.87	−0.508056
$563$	−2100.74	−0.157257	−0.0786286	−	0.996904i	$-0.525054\pi$
$563$	−2100.74	−0.157257	−0.0786286	+	0.996904i	$0.525054\pi$
$564$	0	0
$565$	26550.1	1.97694
$566$	1414.30	0.105031
$567$	0	0
$568$	9148.28	0.675798
$569$	−1053.41	−0.0776120	−0.0388060	−	0.999247i	$-0.512355\pi$
$569$	−1053.41	−0.0776120	−0.0388060	+	0.999247i	$0.512355\pi$
$570$	0	0
$571$	9008.40	0.660227	0.330114	−	0.943941i	$-0.392913\pi$
$571$	9008.40	0.660227	0.330114	+	0.943941i	$0.392913\pi$
$572$	8844.72	0.646532
$573$	0	0
$574$	11135.5	0.809736
$575$	2322.54	0.168446
$576$	0	0
$577$	−22068.6	−1.59225	−0.796123	−	0.605135i	$-0.793119\pi$
$577$	−22068.6	−1.59225	−0.796123	+	0.605135i	$0.793119\pi$
$578$	23344.1	1.67991
$579$	0	0
$580$	−5349.45	−0.382972
$581$	−5684.89	−0.405936
$582$	0	0
$583$	−2384.08	−0.169363
$584$	11105.8	0.786916
$585$	0	0
$586$	18422.0	1.29865
$587$	12450.1	0.875417	0.437709	−	0.899117i	$-0.355790\pi$
$587$	12450.1	0.875417	0.437709	+	0.899117i	$0.355790\pi$
$588$	0	0
$589$	−22244.0	−1.55611
$590$	−1879.18	−0.131126
$591$	0	0
$592$	−1765.52	−0.122572
$593$	−4923.34	−0.340940	−0.170470	−	0.985363i	$-0.554529\pi$
$593$	−4923.34	−0.340940	−0.170470	+	0.985363i	$0.554529\pi$
$594$	0	0
$595$	−37751.5	−2.60111
$596$	−5918.53	−0.406766
$597$	0	0
$598$	−10625.6	−0.726612
$599$	−6790.19	−0.463171	−0.231586	−	0.972815i	$-0.574391\pi$
$599$	−6790.19	−0.463171	−0.231586	+	0.972815i	$0.574391\pi$
$600$	0	0
$601$	12782.4	0.867562	0.433781	−	0.901018i	$-0.357179\pi$
$601$	12782.4	0.867562	0.433781	+	0.901018i	$0.357179\pi$
$602$	−16331.0	−1.10565
$603$	0	0
$604$	5803.96	0.390993
$605$	−5045.72	−0.339070
$606$	0	0
$607$	23914.8	1.59913	0.799565	−	0.600580i	$-0.205064\pi$
$607$	23914.8	1.59913	0.799565	+	0.600580i	$0.205064\pi$
$608$	−14859.2	−0.991152
$609$	0	0
$610$	14331.6	0.951263
$611$	−24570.3	−1.62685
$612$	0	0
$613$	−24235.4	−1.59683	−0.798417	−	0.602105i	$-0.794329\pi$
$613$	−24235.4	−1.59683	−0.798417	+	0.602105i	$0.794329\pi$
$614$	−4450.06	−0.292492
$615$	0	0
$616$	−24685.5	−1.61462
$617$	−12764.3	−0.832855	−0.416427	−	0.909169i	$-0.636718\pi$
$617$	−12764.3	−0.832855	−0.416427	+	0.909169i	$0.636718\pi$
$618$	0	0
$619$	−19695.8	−1.27890	−0.639451	−	0.768832i	$-0.720838\pi$
$619$	−19695.8	−1.27890	−0.639451	+	0.768832i	$0.720838\pi$
$620$	7513.43	0.486688
$621$	0	0
$622$	19878.6	1.28144
$623$	−32640.0	−2.09903
$624$	0	0
$625$	−18698.1	−1.19668
$626$	−14441.3	−0.922029
$627$	0	0
$628$	9171.39	0.582768
$629$	7296.95	0.462557
$630$	0	0
$631$	−15589.6	−0.983539	−0.491769	−	0.870725i	$-0.663650\pi$
$631$	−15589.6	−0.983539	−0.491769	+	0.870725i	$0.663650\pi$
$632$	16261.6	1.02350
$633$	0	0
$634$	−15679.8	−0.982215
$635$	31255.0	1.95325
$636$	0	0
$637$	−16592.1	−1.03203
$638$	−12795.7	−0.794026
$639$	0	0
$640$	−1705.34	−0.105327
$641$	7419.16	0.457159	0.228580	−	0.973525i	$-0.426592\pi$
$641$	7419.16	0.457159	0.228580	+	0.973525i	$0.426592\pi$
$642$	0	0
$643$	20440.9	1.25367	0.626837	−	0.779151i	$-0.284349\pi$
$643$	20440.9	1.25367	0.626837	+	0.779151i	$0.284349\pi$
$644$	−5112.05	−0.312800
$645$	0	0
$646$	−31550.2	−1.92156
$647$	12837.7	0.780066	0.390033	−	0.920801i	$-0.372464\pi$
$647$	12837.7	0.780066	0.390033	+	0.920801i	$0.372464\pi$
$648$	0	0
$649$	2795.52	0.169081
$650$	−5175.97	−0.312336
$651$	0	0
$652$	−7634.77	−0.458590
$653$	8034.18	0.481473	0.240736	−	0.970591i	$-0.422611\pi$
$653$	8034.18	0.481473	0.240736	+	0.970591i	$0.422611\pi$
$654$	0	0
$655$	8011.74	0.477931
$656$	−6242.80	−0.371556
$657$	0	0
$658$	19006.9	1.12609
$659$	2980.52	0.176183	0.0880914	−	0.996112i	$-0.471923\pi$
$659$	2980.52	0.176183	0.0880914	+	0.996112i	$0.471923\pi$
$660$	0	0
$661$	30120.5	1.77239	0.886195	−	0.463312i	$-0.153339\pi$
$661$	30120.5	1.77239	0.886195	+	0.463312i	$0.153339\pi$
$662$	20139.0	1.18236
$663$	0	0
$664$	5789.89	0.338390
$665$	34770.2	2.02757
$666$	0	0
$667$	−9560.36	−0.554990
$668$	6786.29	0.393068
$669$	0	0
$670$	−2986.90	−0.172230
$671$	−21320.2	−1.22661
$672$	0	0
$673$	2403.47	0.137662	0.0688312	−	0.997628i	$-0.478073\pi$
$673$	2403.47	0.137662	0.0688312	+	0.997628i	$0.478073\pi$
$674$	25119.0	1.43553
$675$	0	0
$676$	−7987.70	−0.454467
$677$	−8398.45	−0.476778	−0.238389	−	0.971170i	$-0.576619\pi$
$677$	−8398.45	−0.476778	−0.238389	+	0.971170i	$0.576619\pi$
$678$	0	0
$679$	−3172.53	−0.179309
$680$	38448.8	2.16830
$681$	0	0
$682$	17971.9	1.00906
$683$	−7666.40	−0.429497	−0.214749	−	0.976669i	$-0.568893\pi$
$683$	−7666.40	−0.429497	−0.214749	+	0.976669i	$0.568893\pi$
$684$	0	0
$685$	−3167.31	−0.176667
$686$	−5549.54	−0.308867
$687$	0	0
$688$	9155.46	0.507338
$689$	3969.69	0.219497
$690$	0	0
$691$	22625.9	1.24563	0.622814	−	0.782370i	$-0.285989\pi$
$691$	22625.9	1.24563	0.622814	+	0.782370i	$0.285989\pi$
$692$	−3967.76	−0.217965
$693$	0	0
$694$	−8770.45	−0.479715
$695$	26725.5	1.45864
$696$	0	0
$697$	25801.6	1.40216
$698$	3101.91	0.168208
$699$	0	0
$700$	−2490.19	−0.134458
$701$	22649.0	1.22031	0.610156	−	0.792281i	$-0.291107\pi$
$701$	22649.0	1.22031	0.610156	+	0.792281i	$0.291107\pi$
$702$	0	0
$703$	−6720.69	−0.360563
$704$	22008.8	1.17825
$705$	0	0
$706$	7147.64	0.381027
$707$	−30520.8	−1.62355
$708$	0	0
$709$	5919.34	0.313548	0.156774	−	0.987634i	$-0.449891\pi$
$709$	5919.34	0.313548	0.156774	+	0.987634i	$0.449891\pi$
$710$	10410.9	0.550300
$711$	0	0
$712$	33242.9	1.74976
$713$	13427.8	0.705292
$714$	0	0
$715$	36315.1	1.89945
$716$	−4651.39	−0.242780
$717$	0	0
$718$	6333.20	0.329183
$719$	29546.8	1.53256	0.766279	−	0.642508i	$-0.222106\pi$
$719$	29546.8	1.53256	0.766279	+	0.642508i	$0.222106\pi$
$720$	0	0
$721$	20363.2	1.05182
$722$	13825.5	0.712647
$723$	0	0
$724$	555.234	0.0285015
$725$	−4657.06	−0.238564
$726$	0	0
$727$	31490.3	1.60648	0.803240	−	0.595656i	$-0.203108\pi$
$727$	31490.3	1.60648	0.803240	+	0.595656i	$0.203108\pi$
$728$	41103.5	2.09258
$729$	0	0
$730$	12638.5	0.640783
$731$	−37839.7	−1.91457
$732$	0	0
$733$	−27583.9	−1.38995	−0.694975	−	0.719034i	$-0.744585\pi$
$733$	−27583.9	−1.38995	−0.694975	+	0.719034i	$0.744585\pi$
$734$	14718.5	0.740147
$735$	0	0
$736$	8969.87	0.449231
$737$	4443.40	0.222082
$738$	0	0
$739$	−35392.5	−1.76175	−0.880876	−	0.473348i	$-0.843045\pi$
$739$	−35392.5	−1.76175	−0.880876	+	0.473348i	$0.843045\pi$
$740$	2270.07	0.112769
$741$	0	0
$742$	−3070.85	−0.151933
$743$	35369.8	1.74643	0.873213	−	0.487340i	$-0.162033\pi$
$743$	35369.8	1.74643	0.873213	+	0.487340i	$0.162033\pi$
$744$	0	0
$745$	−24300.6	−1.19504
$746$	10779.8	0.529059
$747$	0	0
$748$	−15853.4	−0.774943
$749$	−4963.60	−0.242144
$750$	0	0
$751$	1628.37	0.0791210	0.0395605	−	0.999217i	$-0.487404\pi$
$751$	1628.37	0.0791210	0.0395605	+	0.999217i	$0.487404\pi$
$752$	−10655.7	−0.516718
$753$	0	0
$754$	21306.0	1.02907
$755$	23830.2	1.14870
$756$	0	0
$757$	−4093.04	−0.196518	−0.0982590	−	0.995161i	$-0.531327\pi$
$757$	−4093.04	−0.196518	−0.0982590	+	0.995161i	$0.531327\pi$
$758$	20514.1	0.982988
$759$	0	0
$760$	−35412.5	−1.69019
$761$	−27568.6	−1.31322	−0.656610	−	0.754230i	$-0.728010\pi$
$761$	−27568.6	−1.31322	−0.656610	+	0.754230i	$0.728010\pi$
$762$	0	0
$763$	34192.2	1.62233
$764$	12791.9	0.605753
$765$	0	0
$766$	14582.1	0.687824
$767$	−4654.78	−0.219132
$768$	0	0
$769$	−19012.2	−0.891545	−0.445773	−	0.895146i	$-0.647071\pi$
$769$	−19012.2	−0.891545	−0.445773	+	0.895146i	$0.647071\pi$
$770$	−28092.4	−1.31478
$771$	0	0
$772$	12433.5	0.579652
$773$	−18694.2	−0.869836	−0.434918	−	0.900470i	$-0.643223\pi$
$773$	−18694.2	−0.869836	−0.434918	+	0.900470i	$0.643223\pi$
$774$	0	0
$775$	6540.95	0.303171
$776$	3231.13	0.149473
$777$	0	0
$778$	−10910.5	−0.502776
$779$	−23764.0	−1.09298
$780$	0	0
$781$	−15487.5	−0.709586
$782$	19045.5	0.870928
$783$	0	0
$784$	−7195.68	−0.327792
$785$	37656.4	1.71212
$786$	0	0
$787$	17033.3	0.771502	0.385751	−	0.922603i	$-0.373942\pi$
$787$	17033.3	0.771502	0.385751	+	0.922603i	$0.373942\pi$
$788$	15277.2	0.690646
$789$	0	0
$790$	18505.9	0.833433
$791$	−50874.4	−2.28684
$792$	0	0
$793$	35499.9	1.58971
$794$	−17483.5	−0.781442
$795$	0	0
$796$	2378.10	0.105891
$797$	−4978.07	−0.221245	−0.110622	−	0.993863i	$-0.535284\pi$
$797$	−4978.07	−0.221245	−0.110622	+	0.993863i	$0.535284\pi$
$798$	0	0
$799$	44040.1	1.94997
$800$	4369.42	0.193103
$801$	0	0
$802$	−3094.89	−0.136265
$803$	−18801.4	−0.826261
$804$	0	0
$805$	−20989.3	−0.918976
$806$	−29924.8	−1.30776
$807$	0	0
$808$	31084.5	1.35340
$809$	−13743.3	−0.597266	−0.298633	−	0.954368i	$-0.596531\pi$
$809$	−13743.3	−0.597266	−0.298633	+	0.954368i	$0.596531\pi$
$810$	0	0
$811$	−14258.2	−0.617352	−0.308676	−	0.951167i	$-0.599886\pi$
$811$	−14258.2	−0.617352	−0.308676	+	0.951167i	$0.599886\pi$
$812$	10250.4	0.443005
$813$	0	0
$814$	5429.95	0.233808
$815$	−31347.2	−1.34729
$816$	0	0
$817$	34851.5	1.49241
$818$	−18247.7	−0.779972
$819$	0	0
$820$	8026.85	0.341841
$821$	−39920.0	−1.69698	−0.848489	−	0.529213i	$-0.822487\pi$
$821$	−39920.0	−1.69698	−0.848489	+	0.529213i	$0.822487\pi$
$822$	0	0
$823$	−18426.0	−0.780424	−0.390212	−	0.920725i	$-0.627598\pi$
$823$	−18426.0	−0.780424	−0.390212	+	0.920725i	$0.627598\pi$
$824$	−20739.3	−0.876805
$825$	0	0
$826$	3600.82	0.151681
$827$	−5145.34	−0.216350	−0.108175	−	0.994132i	$-0.534501\pi$
$827$	−5145.34	−0.216350	−0.108175	+	0.994132i	$0.534501\pi$
$828$	0	0
$829$	−6858.20	−0.287328	−0.143664	−	0.989626i	$-0.545889\pi$
$829$	−6858.20	−0.287328	−0.143664	+	0.989626i	$0.545889\pi$
$830$	6588.97	0.275550
$831$	0	0
$832$	−36646.5	−1.52703
$833$	29739.9	1.23701
$834$	0	0
$835$	27863.5	1.15480
$836$	14601.4	0.604068
$837$	0	0
$838$	18186.2	0.749681
$839$	−13863.3	−0.570457	−0.285228	−	0.958460i	$-0.592069\pi$
$839$	−13863.3	−0.570457	−0.285228	+	0.958460i	$0.592069\pi$
$840$	0	0
$841$	−5219.01	−0.213990
$842$	−20620.4	−0.843976
$843$	0	0
$844$	−6179.49	−0.252022
$845$	−32796.3	−1.33518
$846$	0	0
$847$	9668.45	0.392222
$848$	1721.58	0.0697162
$849$	0	0
$850$	9277.48	0.374370
$851$	4057.00	0.163422
$852$	0	0
$853$	−43369.7	−1.74085	−0.870427	−	0.492297i	$-0.836157\pi$
$853$	−43369.7	−1.74085	−0.870427	+	0.492297i	$0.836157\pi$
$854$	−27461.8	−1.10038
$855$	0	0
$856$	5055.28	0.201853
$857$	31753.5	1.26567	0.632834	−	0.774287i	$-0.281891\pi$
$857$	31753.5	1.26567	0.632834	+	0.774287i	$0.281891\pi$
$858$	0	0
$859$	−17071.5	−0.678083	−0.339041	−	0.940771i	$-0.610103\pi$
$859$	−17071.5	−0.678083	−0.339041	+	0.940771i	$0.610103\pi$
$860$	−11771.9	−0.466765
$861$	0	0
$862$	544.659	0.0215211
$863$	14310.1	0.564452	0.282226	−	0.959348i	$-0.408927\pi$
$863$	14310.1	0.564452	0.282226	+	0.959348i	$0.408927\pi$
$864$	0	0
$865$	−16291.0	−0.640359
$866$	−29403.0	−1.15376
$867$	0	0
$868$	−14397.0	−0.562979
$869$	−27530.0	−1.07467
$870$	0	0
$871$	−7398.64	−0.287822
$872$	−34823.7	−1.35239
$873$	0	0
$874$	−17541.5	−0.678889
$875$	27772.3	1.07300
$876$	0	0
$877$	−33188.1	−1.27786	−0.638930	−	0.769265i	$-0.720623\pi$
$877$	−33188.1	−1.27786	−0.638930	+	0.769265i	$0.720623\pi$
$878$	22765.7	0.875064
$879$	0	0
$880$	15749.2	0.603301
$881$	29862.3	1.14198	0.570990	−	0.820957i	$-0.306559\pi$
$881$	29862.3	1.14198	0.570990	+	0.820957i	$0.306559\pi$
$882$	0	0
$883$	−19945.6	−0.760160	−0.380080	−	0.924954i	$-0.624104\pi$
$883$	−19945.6	−0.760160	−0.380080	+	0.924954i	$0.624104\pi$
$884$	26397.2	1.00434
$885$	0	0
$886$	−29480.1	−1.11784
$887$	44470.7	1.68340	0.841702	−	0.539943i	$-0.181554\pi$
$887$	44470.7	1.68340	0.841702	+	0.539943i	$0.181554\pi$
$888$	0	0
$889$	−59889.8	−2.25944
$890$	37830.8	1.42482
$891$	0	0
$892$	−4850.98	−0.182088
$893$	−40562.2	−1.52000
$894$	0	0
$895$	−19097.9	−0.713266
$896$	3267.72	0.121838
$897$	0	0
$898$	30645.2	1.13880
$899$	−26924.7	−0.998876
$900$	0	0
$901$	−7115.33	−0.263092
$902$	19200.0	0.708748
$903$	0	0
$904$	51814.1	1.90632
$905$	2279.71	0.0837348
$906$	0	0
$907$	−12086.1	−0.442462	−0.221231	−	0.975221i	$-0.571008\pi$
$907$	−12086.1	−0.442462	−0.221231	+	0.975221i	$0.571008\pi$
$908$	10076.7	0.368289
$909$	0	0
$910$	46776.3	1.70398
$911$	−12341.1	−0.448825	−0.224413	−	0.974494i	$-0.572046\pi$
$911$	−12341.1	−0.448825	−0.224413	+	0.974494i	$0.572046\pi$
$912$	0	0
$913$	−9801.95	−0.355309
$914$	13906.7	0.503275
$915$	0	0
$916$	8802.94	0.317530
$917$	−15351.9	−0.552850
$918$	0	0
$919$	27933.9	1.00267	0.501335	−	0.865253i	$-0.332843\pi$
$919$	27933.9	1.00267	0.501335	+	0.865253i	$0.332843\pi$
$920$	21377.0	0.766063
$921$	0	0
$922$	−25361.4	−0.905894
$923$	25788.0	0.919635
$924$	0	0
$925$	1976.25	0.0702472
$926$	2875.21	0.102036
$927$	0	0
$928$	−17986.0	−0.636227
$929$	9624.47	0.339901	0.169951	−	0.985453i	$-0.445639\pi$
$929$	9624.47	0.339901	0.169951	+	0.985453i	$0.445639\pi$
$930$	0	0
$931$	−27391.3	−0.964246
$932$	−4457.68	−0.156670
$933$	0	0
$934$	14512.0	0.508403
$935$	−65091.7	−2.27671
$936$	0	0
$937$	9150.38	0.319029	0.159514	−	0.987196i	$-0.449007\pi$
$937$	9150.38	0.319029	0.159514	+	0.987196i	$0.449007\pi$
$938$	5723.40	0.199228
$939$	0	0
$940$	13700.8	0.475395
$941$	27348.1	0.947421	0.473710	−	0.880681i	$-0.342914\pi$
$941$	27348.1	0.947421	0.473710	+	0.880681i	$0.342914\pi$
$942$	0	0
$943$	14345.3	0.495385
$944$	−2018.69	−0.0696005
$945$	0	0
$946$	−28158.0	−0.967756
$947$	18536.5	0.636067	0.318034	−	0.948079i	$-0.396978\pi$
$947$	18536.5	0.636067	0.318034	+	0.948079i	$0.396978\pi$
$948$	0	0
$949$	31305.9	1.07085
$950$	−8544.82	−0.291822
$951$	0	0
$952$	−73674.4	−2.50820
$953$	−30118.8	−1.02376	−0.511880	−	0.859057i	$-0.671051\pi$
$953$	−30118.8	−1.02376	−0.511880	+	0.859057i	$0.671051\pi$
$954$	0	0
$955$	52521.7	1.77965
$956$	−19213.2	−0.650000
$957$	0	0
$958$	−5144.62	−0.173502
$959$	6069.10	0.204360
$960$	0	0
$961$	8025.40	0.269390
$962$	−9041.33	−0.303019
$963$	0	0
$964$	16712.9	0.558389
$965$	51050.0	1.70296
$966$	0	0
$967$	−42667.4	−1.41891	−0.709457	−	0.704749i	$-0.751060\pi$
$967$	−42667.4	−1.41891	−0.709457	+	0.704749i	$0.751060\pi$
$968$	−9847.02	−0.326958
$969$	0	0
$970$	3677.07	0.121715
$971$	−20101.0	−0.664337	−0.332169	−	0.943220i	$-0.607780\pi$
$971$	−20101.0	−0.664337	−0.332169	+	0.943220i	$0.607780\pi$
$972$	0	0
$973$	−51210.5	−1.68729
$974$	−13928.9	−0.458226
$975$	0	0
$976$	15395.6	0.504920
$977$	44958.6	1.47221	0.736107	−	0.676865i	$-0.236662\pi$
$977$	44958.6	1.47221	0.736107	+	0.676865i	$0.236662\pi$
$978$	0	0
$979$	−56278.3	−1.83724
$980$	9252.04	0.301577
$981$	0	0
$982$	−48274.5	−1.56874
$983$	−21743.8	−0.705515	−0.352757	−	0.935715i	$-0.614756\pi$
$983$	−21743.8	−0.705515	−0.352757	+	0.935715i	$0.614756\pi$
$984$	0	0
$985$	62726.0	2.02905
$986$	−38189.2	−1.23346
$987$	0	0
$988$	−24312.6	−0.782882
$989$	−21038.3	−0.676420
$990$	0	0
$991$	36395.9	1.16665	0.583327	−	0.812237i	$-0.301750\pi$
$991$	36395.9	1.16665	0.583327	+	0.812237i	$0.301750\pi$
$992$	25261.7	0.808529
$993$	0	0
$994$	−19949.0	−0.636562
$995$	9764.13	0.311099
$996$	0	0
$997$	36605.3	1.16279	0.581395	−	0.813622i	$-0.302507\pi$
$997$	36605.3	1.16279	0.581395	+	0.813622i	$0.302507\pi$
$998$	6855.15	0.217431
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1629.4.a.b.1.11		17
3.2	odd	2		543.4.a.a.1.7	✓	17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
543.4.a.a.1.7	✓	17	3.2	odd	2
1629.4.a.b.1.11		17	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1629 = 3^{2} \cdot 181 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1629.a (trivial)

\(f(q)\)	\(=\)	\(q+2.22090 q^{2} -3.06759 q^{4} -12.5951 q^{5} +24.1343 q^{7} -24.5800 q^{8} -27.9724 q^{10} +41.6126 q^{11} -69.2885 q^{13} +53.5999 q^{14} -30.0492 q^{16} +124.194 q^{17} -114.386 q^{19} +38.6365 q^{20} +92.4176 q^{22} +69.0499 q^{23} +33.6357 q^{25} -153.883 q^{26} -74.0341 q^{28} -138.456 q^{29} +194.464 q^{31} +129.904 q^{32} +275.822 q^{34} -303.973 q^{35} +58.7545 q^{37} -254.040 q^{38} +309.587 q^{40} +207.753 q^{41} -304.683 q^{43} -127.651 q^{44} +153.353 q^{46} +354.608 q^{47} +239.464 q^{49} +74.7016 q^{50} +212.549 q^{52} -57.2922 q^{53} -524.114 q^{55} -593.222 q^{56} -307.497 q^{58} +67.1797 q^{59} -512.348 q^{61} +431.887 q^{62} +528.898 q^{64} +872.694 q^{65} +106.780 q^{67} -380.976 q^{68} -675.094 q^{70} -372.183 q^{71} -451.820 q^{73} +130.488 q^{74} +350.889 q^{76} +1004.29 q^{77} -661.578 q^{79} +378.471 q^{80} +461.399 q^{82} -235.552 q^{83} -1564.23 q^{85} -676.671 q^{86} -1022.84 q^{88} -1352.43 q^{89} -1672.23 q^{91} -211.817 q^{92} +787.549 q^{94} +1440.70 q^{95} -131.453 q^{97} +531.825 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 5 q^{2} + 31 q^{4} + 21 q^{5} - 62 q^{7} + 60 q^{8} - 127 q^{10} + 65 q^{11} - 246 q^{13} + 121 q^{14} - 85 q^{16} + 433 q^{17} - 459 q^{19} + 208 q^{20} - 270 q^{22} + 165 q^{23} - 146 q^{25} + 89 q^{26}+ \cdots - 1803 q^{98}+O(q^{100}) \) 17 * q + 5 * q^2 + 31 * q^4 + 21 * q^5 - 62 * q^7 + 60 * q^8 - 127 * q^10 + 65 * q^11 - 246 * q^13 + 121 * q^14 - 85 * q^16 + 433 * q^17 - 459 * q^19 + 208 * q^20 - 270 * q^22 + 165 * q^23 - 146 * q^25 + 89 * q^26 - 625 * q^28 + 271 * q^29 - 328 * q^31 + 690 * q^32 - 699 * q^34 + 662 * q^35 - 1068 * q^37 + 571 * q^38 - 1038 * q^40 + 837 * q^41 - 1808 * q^43 + 48 * q^44 - 808 * q^46 + 707 * q^47 - 1085 * q^49 - 981 * q^50 - 1551 * q^52 + 154 * q^53 - 1914 * q^55 - 1098 * q^56 + 1104 * q^58 + 223 * q^59 - 1459 * q^61 - 1631 * q^62 + 718 * q^64 + 280 * q^65 - 1860 * q^67 - 684 * q^68 + 400 * q^70 + 307 * q^71 - 3177 * q^73 - 1138 * q^74 - 1674 * q^76 + 714 * q^77 - 3487 * q^79 - 1636 * q^80 - 1497 * q^82 + 1556 * q^83 - 1810 * q^85 - 3881 * q^86 - 3632 * q^88 + 715 * q^89 - 1776 * q^91 - 1033 * q^92 - 1979 * q^94 + 100 * q^95 - 3254 * q^97 - 1803 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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