LMFDB - Embedded newform 1638.4.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,4,Mod(1,1638)] 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("1638.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-4,0,8,-5,0,-14,-16,0,10,32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$96.6451285894$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{105}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 26 $ x^2 - x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 546)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.62348$ of defining polynomial
Character	$\chi$	$=$	1638.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} +4.00000 q^{4} +2.62348 q^{5} -7.00000 q^{7} -8.00000 q^{8} -5.24695 q^{10} -4.49390 q^{11} -13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -12.2348 q^{17} +62.1052 q^{19} +10.4939 q^{20} +8.98780 q^{22} +58.1052 q^{23} -118.117 q^{25} +26.0000 q^{26} -28.0000 q^{28} +182.081 q^{29} -280.809 q^{31} -32.0000 q^{32} +24.4695 q^{34} -18.3643 q^{35} +159.198 q^{37} -124.210 q^{38} -20.9878 q^{40} +151.790 q^{41} +55.3521 q^{43} -17.9756 q^{44} -116.210 q^{46} +448.785 q^{47} +49.0000 q^{49} +236.235 q^{50} -52.0000 q^{52} -147.748 q^{53} -11.7896 q^{55} +56.0000 q^{56} -364.162 q^{58} +250.259 q^{59} -652.963 q^{61} +561.619 q^{62} +64.0000 q^{64} -34.1052 q^{65} -546.607 q^{67} -48.9390 q^{68} +36.7287 q^{70} -1095.42 q^{71} -715.117 q^{73} -318.396 q^{74} +248.421 q^{76} +31.4573 q^{77} -60.9710 q^{79} +41.9756 q^{80} -303.579 q^{82} +817.578 q^{83} -32.0976 q^{85} -110.704 q^{86} +35.9512 q^{88} +616.145 q^{89} +91.0000 q^{91} +232.421 q^{92} -897.570 q^{94} +162.931 q^{95} -1042.40 q^{97} -98.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{2} + 8 q^{4} - 5 q^{5} - 14 q^{7} - 16 q^{8} + 10 q^{10} + 32 q^{11} - 26 q^{13} + 28 q^{14} + 32 q^{16} + 78 q^{17} - 9 q^{19} - 20 q^{20} - 64 q^{22} - 17 q^{23} - 185 q^{25} + 52 q^{26} - 56 q^{28}+ \cdots - 196 q^{98}+O(q^{100}) $ 2 * q - 4 * q^2 + 8 * q^4 - 5 * q^5 - 14 * q^7 - 16 * q^8 + 10 * q^10 + 32 * q^11 - 26 * q^13 + 28 * q^14 + 32 * q^16 + 78 * q^17 - 9 * q^19 - 20 * q^20 - 64 * q^22 - 17 * q^23 - 185 * q^25 + 52 * q^26 - 56 * q^28 + 67 * q^29 - 121 * q^31 - 64 * q^32 - 156 * q^34 + 35 * q^35 - 30 * q^37 + 18 * q^38 + 40 * q^40 + 570 * q^41 - 43 * q^43 + 128 * q^44 + 34 * q^46 + 293 * q^47 + 98 * q^49 + 370 * q^50 - 104 * q^52 + 555 * q^53 - 290 * q^55 + 112 * q^56 - 134 * q^58 + 562 * q^59 - 1060 * q^61 + 242 * q^62 + 128 * q^64 + 65 * q^65 - 130 * q^67 + 312 * q^68 - 70 * q^70 + 2 * q^71 - 1379 * q^73 + 60 * q^74 - 36 * q^76 - 224 * q^77 + 913 * q^79 - 80 * q^80 - 1140 * q^82 - 363 * q^83 - 720 * q^85 + 86 * q^86 - 256 * q^88 - 315 * q^89 + 182 * q^91 - 68 * q^92 - 586 * q^94 + 705 * q^95 - 599 * q^97 - 196 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	2.62348	0.234651	0.117325	−	0.993094i	$-0.462568\pi$
$5$	2.62348	0.234651	0.117325	+	0.993094i	$0.462568\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−5.24695	−0.165923
$11$	−4.49390	−0.123178	−0.0615892	−	0.998102i	$-0.519617\pi$
$11$	−4.49390	−0.123178	−0.0615892	+	0.998102i	$0.519617\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−12.2348	−0.174551	−0.0872754	−	0.996184i	$-0.527816\pi$
$17$	−12.2348	−0.174551	−0.0872754	+	0.996184i	$0.527816\pi$
$18$	0	0
$19$	62.1052	0.749890	0.374945	−	0.927047i	$-0.377662\pi$
$19$	62.1052	0.749890	0.374945	+	0.927047i	$0.377662\pi$
$20$	10.4939	0.117325
$21$	0	0
$22$	8.98780	0.0871003
$23$	58.1052	0.526773	0.263386	−	0.964690i	$-0.415161\pi$
$23$	58.1052	0.526773	0.263386	+	0.964690i	$0.415161\pi$
$24$	0	0
$25$	−118.117	−0.944939
$26$	26.0000	0.196116
$27$	0	0
$28$	−28.0000	−0.188982
$29$	182.081	1.16592	0.582958	−	0.812502i	$-0.301895\pi$
$29$	182.081	1.16592	0.582958	+	0.812502i	$0.301895\pi$
$30$	0	0
$31$	−280.809	−1.62693	−0.813466	−	0.581613i	$-0.802422\pi$
$31$	−280.809	−1.62693	−0.813466	+	0.581613i	$0.802422\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	24.4695	0.123426
$35$	−18.3643	−0.0886897
$36$	0	0
$37$	159.198	0.707352	0.353676	−	0.935368i	$-0.384932\pi$
$37$	159.198	0.707352	0.353676	+	0.935368i	$0.384932\pi$
$38$	−124.210	−0.530252
$39$	0	0
$40$	−20.9878	−0.0829616
$41$	151.790	0.578184	0.289092	−	0.957301i	$-0.406647\pi$
$41$	151.790	0.578184	0.289092	+	0.957301i	$0.406647\pi$
$42$	0	0
$43$	55.3521	0.196305	0.0981526	−	0.995171i	$-0.468707\pi$
$43$	55.3521	0.196305	0.0981526	+	0.995171i	$0.468707\pi$
$44$	−17.9756	−0.0615892
$45$	0	0
$46$	−116.210	−0.372484
$47$	448.785	1.39281	0.696405	−	0.717649i	$-0.254782\pi$
$47$	448.785	1.39281	0.696405	+	0.717649i	$0.254782\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	236.235	0.668173
$51$	0	0
$52$	−52.0000	−0.138675
$53$	−147.748	−0.382921	−0.191460	−	0.981500i	$-0.561322\pi$
$53$	−147.748	−0.382921	−0.191460	+	0.981500i	$0.561322\pi$
$54$	0	0
$55$	−11.7896	−0.0289039
$56$	56.0000	0.133631
$57$	0	0
$58$	−364.162	−0.824427
$59$	250.259	0.552220	0.276110	−	0.961126i	$-0.410955\pi$
$59$	250.259	0.552220	0.276110	+	0.961126i	$0.410955\pi$
$60$	0	0
$61$	−652.963	−1.37055	−0.685274	−	0.728286i	$-0.740317\pi$
$61$	−652.963	−1.37055	−0.685274	+	0.728286i	$0.740317\pi$
$62$	561.619	1.15041
$63$	0	0
$64$	64.0000	0.125000
$65$	−34.1052	−0.0650804
$66$	0	0
$67$	−546.607	−0.996696	−0.498348	−	0.866977i	$-0.666060\pi$
$67$	−546.607	−0.996696	−0.498348	+	0.866977i	$0.666060\pi$
$68$	−48.9390	−0.0872754
$69$	0	0
$70$	36.7287	0.0627131
$71$	−1095.42	−1.83103	−0.915513	−	0.402288i	$-0.868215\pi$
$71$	−1095.42	−1.83103	−0.915513	+	0.402288i	$0.868215\pi$
$72$	0	0
$73$	−715.117	−1.14655	−0.573275	−	0.819363i	$-0.694327\pi$
$73$	−715.117	−1.14655	−0.573275	+	0.819363i	$0.694327\pi$
$74$	−318.396	−0.500173
$75$	0	0
$76$	248.421	0.374945
$77$	31.4573	0.0465571
$78$	0	0
$79$	−60.9710	−0.0868326	−0.0434163	−	0.999057i	$-0.513824\pi$
$79$	−60.9710	−0.0868326	−0.0434163	+	0.999057i	$0.513824\pi$
$80$	41.9756	0.0586627
$81$	0	0
$82$	−303.579	−0.408838
$83$	817.578	1.08121	0.540607	−	0.841275i	$-0.318195\pi$
$83$	817.578	1.08121	0.540607	+	0.841275i	$0.318195\pi$
$84$	0	0
$85$	−32.0976	−0.0409585
$86$	−110.704	−0.138809
$87$	0	0
$88$	35.9512	0.0435501
$89$	616.145	0.733834	0.366917	−	0.930254i	$-0.380413\pi$
$89$	616.145	0.733834	0.366917	+	0.930254i	$0.380413\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	232.421	0.263386
$93$	0	0
$94$	−897.570	−0.984865
$95$	162.931	0.175962
$96$	0	0
$97$	−1042.40	−1.09113	−0.545567	−	0.838067i	$-0.683686\pi$
$97$	−1042.40	−1.09113	−0.545567	+	0.838067i	$0.683686\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	−472.470	−0.472470
$101$	125.774	0.123911	0.0619556	−	0.998079i	$-0.480266\pi$
$101$	125.774	0.123911	0.0619556	+	0.998079i	$0.480266\pi$
$102$	0	0
$103$	866.622	0.829037	0.414518	−	0.910041i	$-0.363950\pi$
$103$	866.622	0.829037	0.414518	+	0.910041i	$0.363950\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	295.497	0.270766
$107$	−935.482	−0.845200	−0.422600	−	0.906316i	$-0.638883\pi$
$107$	−935.482	−0.845200	−0.422600	+	0.906316i	$0.638883\pi$
$108$	0	0
$109$	1660.10	1.45880	0.729399	−	0.684088i	$-0.239800\pi$
$109$	1660.10	1.45880	0.729399	+	0.684088i	$0.239800\pi$
$110$	23.5793	0.0204381
$111$	0	0
$112$	−112.000	−0.0944911
$113$	1142.10	0.950791	0.475395	−	0.879772i	$-0.342305\pi$
$113$	1142.10	0.950791	0.475395	+	0.879772i	$0.342305\pi$
$114$	0	0
$115$	152.438	0.123608
$116$	728.323	0.582958
$117$	0	0
$118$	−500.518	−0.390478
$119$	85.6433	0.0659740
$120$	0	0
$121$	−1310.80	−0.984827
$122$	1305.93	0.969123
$123$	0	0
$124$	−1123.24	−0.813466
$125$	−637.812	−0.456381
$126$	0	0
$127$	−690.299	−0.482316	−0.241158	−	0.970486i	$-0.577527\pi$
$127$	−690.299	−0.482316	−0.241158	+	0.970486i	$0.577527\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	68.2104	0.0460188
$131$	−1029.93	−0.686909	−0.343455	−	0.939169i	$-0.611597\pi$
$131$	−1029.93	−0.686909	−0.343455	+	0.939169i	$0.611597\pi$
$132$	0	0
$133$	−434.736	−0.283432
$134$	1093.21	0.704771
$135$	0	0
$136$	97.8780	0.0617130
$137$	−1420.23	−0.885679	−0.442840	−	0.896601i	$-0.646029\pi$
$137$	−1420.23	−0.885679	−0.442840	+	0.896601i	$0.646029\pi$
$138$	0	0
$139$	438.979	0.267868	0.133934	−	0.990990i	$-0.457239\pi$
$139$	438.979	0.267868	0.133934	+	0.990990i	$0.457239\pi$
$140$	−73.4573	−0.0443448
$141$	0	0
$142$	2190.85	1.29473
$143$	58.4207	0.0341635
$144$	0	0
$145$	477.684	0.273583
$146$	1430.23	0.810733
$147$	0	0
$148$	636.793	0.353676
$149$	1413.08	0.776941	0.388470	−	0.921461i	$-0.373004\pi$
$149$	1413.08	0.776941	0.388470	+	0.921461i	$0.373004\pi$
$150$	0	0
$151$	−961.116	−0.517977	−0.258988	−	0.965880i	$-0.583389\pi$
$151$	−961.116	−0.517977	−0.258988	+	0.965880i	$0.583389\pi$
$152$	−496.841	−0.265126
$153$	0	0
$154$	−62.9146	−0.0329208
$155$	−736.697	−0.381761
$156$	0	0
$157$	804.607	0.409010	0.204505	−	0.978865i	$-0.434441\pi$
$157$	804.607	0.409010	0.204505	+	0.978865i	$0.434441\pi$
$158$	121.942	0.0613999
$159$	0	0
$160$	−83.9512	−0.0414808
$161$	−406.736	−0.199101
$162$	0	0
$163$	1278.45	0.614329	0.307164	−	0.951657i	$-0.400620\pi$
$163$	1278.45	0.614329	0.307164	+	0.951657i	$0.400620\pi$
$164$	607.159	0.289092
$165$	0	0
$166$	−1635.16	−0.764534
$167$	−3859.36	−1.78830	−0.894150	−	0.447768i	$-0.852219\pi$
$167$	−3859.36	−1.78830	−0.894150	+	0.447768i	$0.852219\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	64.1952	0.0289620
$171$	0	0
$172$	221.409	0.0981526
$173$	−1601.97	−0.704019	−0.352009	−	0.935996i	$-0.614501\pi$
$173$	−1601.97	−0.704019	−0.352009	+	0.935996i	$0.614501\pi$
$174$	0	0
$175$	826.822	0.357153
$176$	−71.9024	−0.0307946
$177$	0	0
$178$	−1232.29	−0.518899
$179$	677.569	0.282927	0.141463	−	0.989944i	$-0.454819\pi$
$179$	677.569	0.282927	0.141463	+	0.989944i	$0.454819\pi$
$180$	0	0
$181$	−2581.37	−1.06006	−0.530032	−	0.847978i	$-0.677820\pi$
$181$	−2581.37	−1.06006	−0.530032	+	0.847978i	$0.677820\pi$
$182$	−182.000	−0.0741249
$183$	0	0
$184$	−464.841	−0.186242
$185$	417.652	0.165981
$186$	0	0
$187$	54.9818	0.0215009
$188$	1795.14	0.696405
$189$	0	0
$190$	−325.863	−0.124424
$191$	−2321.33	−0.879400	−0.439700	−	0.898145i	$-0.644915\pi$
$191$	−2321.33	−0.879400	−0.439700	+	0.898145i	$0.644915\pi$
$192$	0	0
$193$	3818.00	1.42397	0.711984	−	0.702196i	$-0.247797\pi$
$193$	3818.00	1.42397	0.711984	+	0.702196i	$0.247797\pi$
$194$	2084.81	0.771549
$195$	0	0
$196$	196.000	0.0714286
$197$	1040.79	0.376412	0.188206	−	0.982130i	$-0.439733\pi$
$197$	1040.79	0.376412	0.188206	+	0.982130i	$0.439733\pi$
$198$	0	0
$199$	2911.77	1.03724	0.518618	−	0.855006i	$-0.326447\pi$
$199$	2911.77	1.03724	0.518618	+	0.855006i	$0.326447\pi$
$200$	944.939	0.334086
$201$	0	0
$202$	−251.549	−0.0876184
$203$	−1274.57	−0.440675
$204$	0	0
$205$	398.216	0.135671
$206$	−1733.24	−0.586218
$207$	0	0
$208$	−208.000	−0.0693375
$209$	−279.095	−0.0923702
$210$	0	0
$211$	−2831.74	−0.923910	−0.461955	−	0.886903i	$-0.652852\pi$
$211$	−2831.74	−0.923910	−0.461955	+	0.886903i	$0.652852\pi$
$212$	−590.994	−0.191460
$213$	0	0
$214$	1870.96	0.597647
$215$	145.215	0.0460632
$216$	0	0
$217$	1965.67	0.614922
$218$	−3320.21	−1.03153
$219$	0	0
$220$	−47.1586	−0.0144520
$221$	159.052	0.0484117
$222$	0	0
$223$	−1888.75	−0.567174	−0.283587	−	0.958947i	$-0.591524\pi$
$223$	−1888.75	−0.567174	−0.283587	+	0.958947i	$0.591524\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	−2284.19	−0.672311
$227$	−935.649	−0.273574	−0.136787	−	0.990601i	$-0.543678\pi$
$227$	−935.649	−0.273574	−0.136787	+	0.990601i	$0.543678\pi$
$228$	0	0
$229$	186.677	0.0538688	0.0269344	−	0.999637i	$-0.491425\pi$
$229$	186.677	0.0538688	0.0269344	+	0.999637i	$0.491425\pi$
$230$	−304.875	−0.0874038
$231$	0	0
$232$	−1456.65	−0.412213
$233$	−6642.35	−1.86762	−0.933809	−	0.357772i	$-0.883536\pi$
$233$	−6642.35	−1.86762	−0.933809	+	0.357772i	$0.883536\pi$
$234$	0	0
$235$	1177.38	0.326824
$236$	1001.04	0.276110
$237$	0	0
$238$	−171.287	−0.0466507
$239$	−2650.59	−0.717375	−0.358688	−	0.933458i	$-0.616776\pi$
$239$	−2650.59	−0.717375	−0.358688	+	0.933458i	$0.616776\pi$
$240$	0	0
$241$	−6748.15	−1.80368	−0.901839	−	0.432072i	$-0.857783\pi$
$241$	−6748.15	−1.80368	−0.901839	+	0.432072i	$0.857783\pi$
$242$	2621.61	0.696378
$243$	0	0
$244$	−2611.85	−0.685274
$245$	128.550	0.0335215
$246$	0	0
$247$	−807.367	−0.207982
$248$	2246.48	0.575207
$249$	0	0
$250$	1275.62	0.322710
$251$	5782.67	1.45418	0.727089	−	0.686543i	$-0.240873\pi$
$251$	5782.67	1.45418	0.727089	+	0.686543i	$0.240873\pi$
$252$	0	0
$253$	−261.119	−0.0648870
$254$	1380.60	0.341049
$255$	0	0
$256$	256.000	0.0625000
$257$	−12.8658	−0.00312276	−0.00156138	−	0.999999i	$-0.500497\pi$
$257$	−12.8658	−0.00312276	−0.00156138	+	0.999999i	$0.500497\pi$
$258$	0	0
$259$	−1114.39	−0.267354
$260$	−136.421	−0.0325402
$261$	0	0
$262$	2059.85	0.485718
$263$	−4398.77	−1.03133	−0.515665	−	0.856790i	$-0.672455\pi$
$263$	−4398.77	−1.03133	−0.515665	+	0.856790i	$0.672455\pi$
$264$	0	0
$265$	−387.614	−0.0898527
$266$	869.473	0.200416
$267$	0	0
$268$	−2186.43	−0.498348
$269$	−3717.24	−0.842543	−0.421272	−	0.906934i	$-0.638416\pi$
$269$	−3717.24	−0.842543	−0.421272	+	0.906934i	$0.638416\pi$
$270$	0	0
$271$	4947.56	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$271$	4947.56	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$272$	−195.756	−0.0436377
$273$	0	0
$274$	2840.45	0.626270
$275$	530.808	0.116396
$276$	0	0
$277$	−5140.19	−1.11496	−0.557480	−	0.830190i	$-0.688232\pi$
$277$	−5140.19	−1.11496	−0.557480	+	0.830190i	$0.688232\pi$
$278$	−877.957	−0.189411
$279$	0	0
$280$	146.915	0.0313565
$281$	2453.75	0.520919	0.260460	−	0.965485i	$-0.416126\pi$
$281$	2453.75	0.520919	0.260460	+	0.965485i	$0.416126\pi$
$282$	0	0
$283$	−1893.29	−0.397684	−0.198842	−	0.980032i	$-0.563718\pi$
$283$	−1893.29	−0.397684	−0.198842	+	0.980032i	$0.563718\pi$
$284$	−4381.69	−0.915513
$285$	0	0
$286$	−116.841	−0.0241573
$287$	−1062.53	−0.218533
$288$	0	0
$289$	−4763.31	−0.969532
$290$	−955.369	−0.193452
$291$	0	0
$292$	−2860.47	−0.573275
$293$	−987.639	−0.196923	−0.0984615	−	0.995141i	$-0.531392\pi$
$293$	−987.639	−0.196923	−0.0984615	+	0.995141i	$0.531392\pi$
$294$	0	0
$295$	656.549	0.129579
$296$	−1273.59	−0.250087
$297$	0	0
$298$	−2826.16	−0.549380
$299$	−755.367	−0.146100
$300$	0	0
$301$	−387.465	−0.0741964
$302$	1922.23	0.366265
$303$	0	0
$304$	993.683	0.187472
$305$	−1713.03	−0.321600
$306$	0	0
$307$	−7723.40	−1.43582	−0.717911	−	0.696134i	$-0.754902\pi$
$307$	−7723.40	−1.43582	−0.717911	+	0.696134i	$0.754902\pi$
$308$	125.829	0.0232785
$309$	0	0
$310$	1473.39	0.269946
$311$	−4171.64	−0.760618	−0.380309	−	0.924859i	$-0.624182\pi$
$311$	−4171.64	−0.760618	−0.380309	+	0.924859i	$0.624182\pi$
$312$	0	0
$313$	−6476.71	−1.16960	−0.584801	−	0.811177i	$-0.698827\pi$
$313$	−6476.71	−1.16960	−0.584801	+	0.811177i	$0.698827\pi$
$314$	−1609.21	−0.289214
$315$	0	0
$316$	−243.884	−0.0434163
$317$	−6787.16	−1.20254	−0.601269	−	0.799046i	$-0.705338\pi$
$317$	−6787.16	−1.20254	−0.601269	+	0.799046i	$0.705338\pi$
$318$	0	0
$319$	−818.253	−0.143616
$320$	167.902	0.0293313
$321$	0	0
$322$	813.473	0.140786
$323$	−759.842	−0.130894
$324$	0	0
$325$	1535.53	0.262079
$326$	−2556.89	−0.434396
$327$	0	0
$328$	−1214.32	−0.204419
$329$	−3141.50	−0.526432
$330$	0	0
$331$	11544.4	1.91704	0.958518	−	0.285032i	$-0.0920043\pi$
$331$	11544.4	1.91704	0.958518	+	0.285032i	$0.0920043\pi$
$332$	3270.31	0.540607
$333$	0	0
$334$	7718.72	1.26452
$335$	−1434.01	−0.233875
$336$	0	0
$337$	−8042.44	−1.30000	−0.649999	−	0.759935i	$-0.725231\pi$
$337$	−8042.44	−1.30000	−0.649999	+	0.759935i	$0.725231\pi$
$338$	−338.000	−0.0543928
$339$	0	0
$340$	−128.390	−0.0204792
$341$	1261.93	0.200403
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−442.817	−0.0694043
$345$	0	0
$346$	3203.93	0.497816
$347$	−2433.15	−0.376422	−0.188211	−	0.982129i	$-0.560269\pi$
$347$	−2433.15	−0.376422	−0.188211	+	0.982129i	$0.560269\pi$
$348$	0	0
$349$	−7263.90	−1.11412	−0.557060	−	0.830472i	$-0.688071\pi$
$349$	−7263.90	−1.11412	−0.557060	+	0.830472i	$0.688071\pi$
$350$	−1653.64	−0.252546
$351$	0	0
$352$	143.805	0.0217751
$353$	−12209.7	−1.84096	−0.920479	−	0.390792i	$-0.872201\pi$
$353$	−12209.7	−1.84096	−0.920479	+	0.390792i	$0.872201\pi$
$354$	0	0
$355$	−2873.82	−0.429652
$356$	2464.58	0.366917
$357$	0	0
$358$	−1355.14	−0.200059
$359$	−1383.75	−0.203431	−0.101715	−	0.994814i	$-0.532433\pi$
$359$	−1383.75	−0.203431	−0.101715	+	0.994814i	$0.532433\pi$
$360$	0	0
$361$	−3001.95	−0.437665
$362$	5162.73	0.749578
$363$	0	0
$364$	364.000	0.0524142
$365$	−1876.09	−0.269039
$366$	0	0
$367$	5658.55	0.804834	0.402417	−	0.915457i	$-0.368170\pi$
$367$	5658.55	0.804834	0.402417	+	0.915457i	$0.368170\pi$
$368$	929.683	0.131693
$369$	0	0
$370$	−835.305	−0.117366
$371$	1034.24	0.144731
$372$	0	0
$373$	1654.27	0.229638	0.114819	−	0.993386i	$-0.463371\pi$
$373$	1654.27	0.229638	0.114819	+	0.993386i	$0.463371\pi$
$374$	−109.964	−0.0152034
$375$	0	0
$376$	−3590.28	−0.492432
$377$	−2367.05	−0.323367
$378$	0	0
$379$	−4751.42	−0.643968	−0.321984	−	0.946745i	$-0.604350\pi$
$379$	−4751.42	−0.643968	−0.321984	+	0.946745i	$0.604350\pi$
$380$	651.726	0.0879811
$381$	0	0
$382$	4642.66	0.621830
$383$	2901.77	0.387137	0.193569	−	0.981087i	$-0.437994\pi$
$383$	2901.77	0.387137	0.193569	+	0.981087i	$0.437994\pi$
$384$	0	0
$385$	82.5275	0.0109246
$386$	−7636.00	−1.00690
$387$	0	0
$388$	−4169.62	−0.545567
$389$	9036.49	1.17781	0.588905	−	0.808202i	$-0.299559\pi$
$389$	9036.49	1.17781	0.588905	+	0.808202i	$0.299559\pi$
$390$	0	0
$391$	−710.903	−0.0919486
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−2081.58	−0.266164
$395$	−159.956	−0.0203753
$396$	0	0
$397$	−3480.01	−0.439942	−0.219971	−	0.975506i	$-0.570596\pi$
$397$	−3480.01	−0.439942	−0.219971	+	0.975506i	$0.570596\pi$
$398$	−5823.54	−0.733436
$399$	0	0
$400$	−1889.88	−0.236235
$401$	−11592.8	−1.44368	−0.721841	−	0.692059i	$-0.756704\pi$
$401$	−11592.8	−1.44368	−0.721841	+	0.692059i	$0.756704\pi$
$402$	0	0
$403$	3650.52	0.451230
$404$	503.098	0.0619556
$405$	0	0
$406$	2549.13	0.311604
$407$	−715.421	−0.0871305
$408$	0	0
$409$	−6873.07	−0.830932	−0.415466	−	0.909609i	$-0.636382\pi$
$409$	−6873.07	−0.830932	−0.415466	+	0.909609i	$0.636382\pi$
$410$	−796.433	−0.0959342
$411$	0	0
$412$	3466.49	0.414518
$413$	−1751.81	−0.208719
$414$	0	0
$415$	2144.89	0.253708
$416$	416.000	0.0490290
$417$	0	0
$418$	558.189	0.0653156
$419$	4450.14	0.518863	0.259431	−	0.965762i	$-0.416465\pi$
$419$	4450.14	0.518863	0.259431	+	0.965762i	$0.416465\pi$
$420$	0	0
$421$	−10239.8	−1.18541	−0.592703	−	0.805421i	$-0.701939\pi$
$421$	−10239.8	−1.18541	−0.592703	+	0.805421i	$0.701939\pi$
$422$	5663.48	0.653303
$423$	0	0
$424$	1181.99	0.135383
$425$	1445.14	0.164940
$426$	0	0
$427$	4570.74	0.518018
$428$	−3741.93	−0.422600
$429$	0	0
$430$	−290.430	−0.0325716
$431$	23.7196	0.00265089	0.00132545	−	0.999999i	$-0.499578\pi$
$431$	23.7196	0.00265089	0.00132545	+	0.999999i	$0.499578\pi$
$432$	0	0
$433$	−338.101	−0.0375244	−0.0187622	−	0.999824i	$-0.505973\pi$
$433$	−338.101	−0.0375244	−0.0187622	+	0.999824i	$0.505973\pi$
$434$	−3931.33	−0.434816
$435$	0	0
$436$	6640.41	0.729399
$437$	3608.63	0.395021
$438$	0	0
$439$	1965.95	0.213735	0.106868	−	0.994273i	$-0.465918\pi$
$439$	1965.95	0.213735	0.106868	+	0.994273i	$0.465918\pi$
$440$	94.3171	0.0102191
$441$	0	0
$442$	−318.104	−0.0342322
$443$	−3840.86	−0.411929	−0.205965	−	0.978559i	$-0.566033\pi$
$443$	−3840.86	−0.411929	−0.205965	+	0.978559i	$0.566033\pi$
$444$	0	0
$445$	1616.44	0.172195
$446$	3777.49	0.401052
$447$	0	0
$448$	−448.000	−0.0472456
$449$	13122.5	1.37926	0.689631	−	0.724161i	$-0.257773\pi$
$449$	13122.5	1.37926	0.689631	+	0.724161i	$0.257773\pi$
$450$	0	0
$451$	−682.128	−0.0712198
$452$	4568.38	0.475395
$453$	0	0
$454$	1871.30	0.193446
$455$	238.736	0.0245981
$456$	0	0
$457$	−9095.17	−0.930972	−0.465486	−	0.885055i	$-0.654120\pi$
$457$	−9095.17	−0.930972	−0.465486	+	0.885055i	$0.654120\pi$
$458$	−373.353	−0.0380910
$459$	0	0
$460$	609.750	0.0618038
$461$	9244.86	0.934005	0.467002	−	0.884256i	$-0.345334\pi$
$461$	9244.86	0.934005	0.467002	+	0.884256i	$0.345334\pi$
$462$	0	0
$463$	−6415.22	−0.643932	−0.321966	−	0.946751i	$-0.604344\pi$
$463$	−6415.22	−0.643932	−0.321966	+	0.946751i	$0.604344\pi$
$464$	2913.29	0.291479
$465$	0	0
$466$	13284.7	1.32061
$467$	−3979.75	−0.394348	−0.197174	−	0.980368i	$-0.563176\pi$
$467$	−3979.75	−0.394348	−0.197174	+	0.980368i	$0.563176\pi$
$468$	0	0
$469$	3826.25	0.376716
$470$	−2354.75	−0.231099
$471$	0	0
$472$	−2002.07	−0.195239
$473$	−248.747	−0.0241806
$474$	0	0
$475$	−7335.70	−0.708600
$476$	342.573	0.0329870
$477$	0	0
$478$	5301.19	0.507261
$479$	−15594.1	−1.48750	−0.743752	−	0.668456i	$-0.766955\pi$
$479$	−15594.1	−1.48750	−0.743752	+	0.668456i	$0.766955\pi$
$480$	0	0
$481$	−2069.58	−0.196184
$482$	13496.3	1.27539
$483$	0	0
$484$	−5243.22	−0.492414
$485$	−2734.72	−0.256036
$486$	0	0
$487$	−1815.08	−0.168889	−0.0844445	−	0.996428i	$-0.526912\pi$
$487$	−1815.08	−0.168889	−0.0844445	+	0.996428i	$0.526912\pi$
$488$	5223.71	0.484562
$489$	0	0
$490$	−257.101	−0.0237033
$491$	7005.37	0.643886	0.321943	−	0.946759i	$-0.395664\pi$
$491$	7005.37	0.643886	0.321943	+	0.946759i	$0.395664\pi$
$492$	0	0
$493$	−2227.71	−0.203512
$494$	1614.73	0.147065
$495$	0	0
$496$	−4492.95	−0.406733
$497$	7667.97	0.692063
$498$	0	0
$499$	11854.7	1.06351	0.531755	−	0.846898i	$-0.321533\pi$
$499$	11854.7	1.06351	0.531755	+	0.846898i	$0.321533\pi$
$500$	−2551.25	−0.228191
$501$	0	0
$502$	−11565.3	−1.02826
$503$	−5098.54	−0.451954	−0.225977	−	0.974133i	$-0.572557\pi$
$503$	−5098.54	−0.451954	−0.225977	+	0.974133i	$0.572557\pi$
$504$	0	0
$505$	329.966	0.0290758
$506$	522.238	0.0458820
$507$	0	0
$508$	−2761.20	−0.241158
$509$	19337.8	1.68395	0.841977	−	0.539514i	$-0.181392\pi$
$509$	19337.8	1.68395	0.841977	+	0.539514i	$0.181392\pi$
$510$	0	0
$511$	5005.82	0.433355
$512$	−512.000	−0.0441942
$513$	0	0
$514$	25.7317	0.00220812
$515$	2273.56	0.194534
$516$	0	0
$517$	−2016.80	−0.171564
$518$	2228.77	0.189048
$519$	0	0
$520$	272.841	0.0230094
$521$	−5922.37	−0.498011	−0.249005	−	0.968502i	$-0.580104\pi$
$521$	−5922.37	−0.498011	−0.249005	+	0.968502i	$0.580104\pi$
$522$	0	0
$523$	−20591.8	−1.72164	−0.860821	−	0.508908i	$-0.830049\pi$
$523$	−20591.8	−1.72164	−0.860821	+	0.508908i	$0.830049\pi$
$524$	−4119.71	−0.343455
$525$	0	0
$526$	8797.54	0.729260
$527$	3435.63	0.283982
$528$	0	0
$529$	−8790.79	−0.722511
$530$	775.229	0.0635355
$531$	0	0
$532$	−1738.95	−0.141716
$533$	−1973.27	−0.160359
$534$	0	0
$535$	−2454.21	−0.198327
$536$	4372.85	0.352385
$537$	0	0
$538$	7434.48	0.595768
$539$	−220.201	−0.0175969
$540$	0	0
$541$	−3441.66	−0.273509	−0.136755	−	0.990605i	$-0.543667\pi$
$541$	−3441.66	−0.273509	−0.136755	+	0.990605i	$0.543667\pi$
$542$	−9895.12	−0.784192
$543$	0	0
$544$	391.512	0.0308565
$545$	4355.24	0.342308
$546$	0	0
$547$	−6202.33	−0.484812	−0.242406	−	0.970175i	$-0.577937\pi$
$547$	−6202.33	−0.484812	−0.242406	+	0.970175i	$0.577937\pi$
$548$	−5680.90	−0.442840
$549$	0	0
$550$	−1061.62	−0.0823044
$551$	11308.2	0.874308
$552$	0	0
$553$	426.797	0.0328196
$554$	10280.4	0.788396
$555$	0	0
$556$	1755.91	0.133934
$557$	21554.5	1.63967	0.819834	−	0.572601i	$-0.194066\pi$
$557$	21554.5	1.63967	0.819834	+	0.572601i	$0.194066\pi$
$558$	0	0
$559$	−719.578	−0.0544452
$560$	−293.829	−0.0221724
$561$	0	0
$562$	−4907.49	−0.368345
$563$	15131.2	1.13269	0.566346	−	0.824168i	$-0.308357\pi$
$563$	15131.2	1.13269	0.566346	+	0.824168i	$0.308357\pi$
$564$	0	0
$565$	2996.26	0.223104
$566$	3786.59	0.281205
$567$	0	0
$568$	8763.39	0.647366
$569$	6164.50	0.454182	0.227091	−	0.973874i	$-0.427079\pi$
$569$	6164.50	0.454182	0.227091	+	0.973874i	$0.427079\pi$
$570$	0	0
$571$	5559.06	0.407425	0.203712	−	0.979031i	$-0.434699\pi$
$571$	5559.06	0.407425	0.203712	+	0.979031i	$0.434699\pi$
$572$	233.683	0.0170818
$573$	0	0
$574$	2125.05	0.154526
$575$	−6863.23	−0.497768
$576$	0	0
$577$	20328.3	1.46669	0.733345	−	0.679857i	$-0.237958\pi$
$577$	20328.3	1.46669	0.733345	+	0.679857i	$0.237958\pi$
$578$	9526.62	0.685563
$579$	0	0
$580$	1910.74	0.136792
$581$	−5723.04	−0.408661
$582$	0	0
$583$	663.967	0.0471676
$584$	5720.94	0.405367
$585$	0	0
$586$	1975.28	0.139246
$587$	−18040.7	−1.26851	−0.634257	−	0.773122i	$-0.718694\pi$
$587$	−18040.7	−1.26851	−0.634257	+	0.773122i	$0.718694\pi$
$588$	0	0
$589$	−17439.7	−1.22002
$590$	−1313.10	−0.0916260
$591$	0	0
$592$	2547.17	0.176838
$593$	28571.8	1.97859	0.989295	−	0.145929i	$-0.0466171\pi$
$593$	28571.8	1.97859	0.989295	+	0.145929i	$0.0466171\pi$
$594$	0	0
$595$	224.683	0.0154809
$596$	5652.33	0.388470
$597$	0	0
$598$	1510.73	0.103309
$599$	9970.07	0.680077	0.340038	−	0.940412i	$-0.389560\pi$
$599$	9970.07	0.680077	0.340038	+	0.940412i	$0.389560\pi$
$600$	0	0
$601$	2444.33	0.165901	0.0829505	−	0.996554i	$-0.473566\pi$
$601$	2444.33	0.165901	0.0829505	+	0.996554i	$0.473566\pi$
$602$	774.930	0.0524648
$603$	0	0
$604$	−3844.46	−0.258988
$605$	−3438.86	−0.231090
$606$	0	0
$607$	−9292.75	−0.621385	−0.310693	−	0.950510i	$-0.600561\pi$
$607$	−9292.75	−0.621385	−0.310693	+	0.950510i	$0.600561\pi$
$608$	−1987.37	−0.132563
$609$	0	0
$610$	3426.07	0.227406
$611$	−5834.21	−0.386296
$612$	0	0
$613$	13698.9	0.902599	0.451300	−	0.892372i	$-0.350961\pi$
$613$	13698.9	0.902599	0.451300	+	0.892372i	$0.350961\pi$
$614$	15446.8	1.01528
$615$	0	0
$616$	−251.658	−0.0164604
$617$	9612.50	0.627204	0.313602	−	0.949555i	$-0.398464\pi$
$617$	9612.50	0.627204	0.313602	+	0.949555i	$0.398464\pi$
$618$	0	0
$619$	7038.12	0.457005	0.228502	−	0.973543i	$-0.426617\pi$
$619$	7038.12	0.457005	0.228502	+	0.973543i	$0.426617\pi$
$620$	−2946.79	−0.190880
$621$	0	0
$622$	8343.29	0.537838
$623$	−4313.01	−0.277363
$624$	0	0
$625$	13091.4	0.837849
$626$	12953.4	0.827034
$627$	0	0
$628$	3218.43	0.204505
$629$	−1947.75	−0.123469
$630$	0	0
$631$	−14423.9	−0.909992	−0.454996	−	0.890494i	$-0.650359\pi$
$631$	−14423.9	−0.909992	−0.454996	+	0.890494i	$0.650359\pi$
$632$	487.768	0.0307000
$633$	0	0
$634$	13574.3	0.850323
$635$	−1810.98	−0.113176
$636$	0	0
$637$	−637.000	−0.0396214
$638$	1636.51	0.101552
$639$	0	0
$640$	−335.805	−0.0207404
$641$	−18079.2	−1.11402	−0.557008	−	0.830507i	$-0.688051\pi$
$641$	−18079.2	−1.11402	−0.557008	+	0.830507i	$0.688051\pi$
$642$	0	0
$643$	15384.8	0.943575	0.471787	−	0.881712i	$-0.343609\pi$
$643$	15384.8	0.943575	0.471787	+	0.881712i	$0.343609\pi$
$644$	−1626.95	−0.0995506
$645$	0	0
$646$	1519.68	0.0925559
$647$	−24239.5	−1.47288	−0.736440	−	0.676503i	$-0.763495\pi$
$647$	−24239.5	−1.47288	−0.736440	+	0.676503i	$0.763495\pi$
$648$	0	0
$649$	−1124.64	−0.0680215
$650$	−3071.05	−0.185318
$651$	0	0
$652$	5113.78	0.307164
$653$	−11193.5	−0.670805	−0.335402	−	0.942075i	$-0.608872\pi$
$653$	−11193.5	−0.670805	−0.335402	+	0.942075i	$0.608872\pi$
$654$	0	0
$655$	−2701.99	−0.161184
$656$	2428.63	0.144546
$657$	0	0
$658$	6282.99	0.372244
$659$	1005.71	0.0594491	0.0297246	−	0.999558i	$-0.490537\pi$
$659$	1005.71	0.0594491	0.0297246	+	0.999558i	$0.490537\pi$
$660$	0	0
$661$	17523.5	1.03114	0.515572	−	0.856846i	$-0.327579\pi$
$661$	17523.5	1.03114	0.515572	+	0.856846i	$0.327579\pi$
$662$	−23088.8	−1.35555
$663$	0	0
$664$	−6540.62	−0.382267
$665$	−1140.52	−0.0665075
$666$	0	0
$667$	10579.8	0.614172
$668$	−15437.4	−0.894150
$669$	0	0
$670$	2868.02	0.165375
$671$	2934.35	0.168822
$672$	0	0
$673$	13120.0	0.751470	0.375735	−	0.926727i	$-0.377390\pi$
$673$	13120.0	0.751470	0.375735	+	0.926727i	$0.377390\pi$
$674$	16084.9	0.919238
$675$	0	0
$676$	676.000	0.0384615
$677$	−4710.83	−0.267433	−0.133716	−	0.991020i	$-0.542691\pi$
$677$	−4710.83	−0.267433	−0.133716	+	0.991020i	$0.542691\pi$
$678$	0	0
$679$	7296.83	0.412410
$680$	256.781	0.0144810
$681$	0	0
$682$	−2523.86	−0.141706
$683$	19643.8	1.10051	0.550256	−	0.834996i	$-0.314530\pi$
$683$	19643.8	1.10051	0.550256	+	0.834996i	$0.314530\pi$
$684$	0	0
$685$	−3725.93	−0.207825
$686$	686.000	0.0381802
$687$	0	0
$688$	885.634	0.0490763
$689$	1920.73	0.106203
$690$	0	0
$691$	7354.29	0.404878	0.202439	−	0.979295i	$-0.435113\pi$
$691$	7354.29	0.404878	0.202439	+	0.979295i	$0.435113\pi$
$692$	−6407.87	−0.352009
$693$	0	0
$694$	4866.31	0.266171
$695$	1151.65	0.0628555
$696$	0	0
$697$	−1857.11	−0.100923
$698$	14527.8	0.787802
$699$	0	0
$700$	3307.29	0.178577
$701$	−28333.8	−1.52661	−0.763304	−	0.646040i	$-0.776424\pi$
$701$	−28333.8	−1.52661	−0.763304	+	0.646040i	$0.776424\pi$
$702$	0	0
$703$	9887.03	0.530436
$704$	−287.610	−0.0153973
$705$	0	0
$706$	24419.5	1.30175
$707$	−880.421	−0.0468340
$708$	0	0
$709$	−23072.9	−1.22218	−0.611088	−	0.791563i	$-0.709268\pi$
$709$	−23072.9	−1.22218	−0.611088	+	0.791563i	$0.709268\pi$
$710$	5747.63	0.303810
$711$	0	0
$712$	−4929.16	−0.259449
$713$	−16316.5	−0.857023
$714$	0	0
$715$	153.265	0.00801650
$716$	2710.27	0.141463
$717$	0	0
$718$	2767.51	0.143847
$719$	24591.0	1.27551	0.637754	−	0.770240i	$-0.279864\pi$
$719$	24591.0	1.27551	0.637754	+	0.770240i	$0.279864\pi$
$720$	0	0
$721$	−6066.35	−0.313346
$722$	6003.89	0.309476
$723$	0	0
$724$	−10325.5	−0.530032
$725$	−21506.9	−1.10172
$726$	0	0
$727$	10868.5	0.554459	0.277229	−	0.960804i	$-0.410584\pi$
$727$	10868.5	0.554459	0.277229	+	0.960804i	$0.410584\pi$
$728$	−728.000	−0.0370625
$729$	0	0
$730$	3752.19	0.190239
$731$	−677.220	−0.0342652
$732$	0	0
$733$	23242.7	1.17120	0.585600	−	0.810600i	$-0.300859\pi$
$733$	23242.7	1.17120	0.585600	+	0.810600i	$0.300859\pi$
$734$	−11317.1	−0.569103
$735$	0	0
$736$	−1859.37	−0.0931211
$737$	2456.40	0.122771
$738$	0	0
$739$	33549.4	1.67000	0.835002	−	0.550247i	$-0.185466\pi$
$739$	33549.4	1.67000	0.835002	+	0.550247i	$0.185466\pi$
$740$	1670.61	0.0829903
$741$	0	0
$742$	−2068.48	−0.102340
$743$	−23365.8	−1.15371	−0.576856	−	0.816846i	$-0.695721\pi$
$743$	−23365.8	−1.15371	−0.576856	+	0.816846i	$0.695721\pi$
$744$	0	0
$745$	3707.19	0.182310
$746$	−3308.55	−0.162379
$747$	0	0
$748$	219.927	0.0107504
$749$	6548.37	0.319456
$750$	0	0
$751$	17437.5	0.847275	0.423637	−	0.905832i	$-0.360753\pi$
$751$	17437.5	0.847275	0.423637	+	0.905832i	$0.360753\pi$
$752$	7180.56	0.348202
$753$	0	0
$754$	4734.10	0.228655
$755$	−2521.46	−0.121544
$756$	0	0
$757$	−27482.3	−1.31950	−0.659750	−	0.751485i	$-0.729338\pi$
$757$	−27482.3	−1.31950	−0.659750	+	0.751485i	$0.729338\pi$
$758$	9502.84	0.455354
$759$	0	0
$760$	−1303.45	−0.0622120
$761$	19059.0	0.907867	0.453933	−	0.891036i	$-0.350020\pi$
$761$	19059.0	0.907867	0.453933	+	0.891036i	$0.350020\pi$
$762$	0	0
$763$	−11620.7	−0.551374
$764$	−9285.32	−0.439700
$765$	0	0
$766$	−5803.54	−0.273747
$767$	−3253.37	−0.153158
$768$	0	0
$769$	8103.85	0.380016	0.190008	−	0.981783i	$-0.439149\pi$
$769$	8103.85	0.380016	0.190008	+	0.981783i	$0.439149\pi$
$770$	−165.055	−0.00772489
$771$	0	0
$772$	15272.0	0.711984
$773$	−9898.23	−0.460562	−0.230281	−	0.973124i	$-0.573965\pi$
$773$	−9898.23	−0.460562	−0.230281	+	0.973124i	$0.573965\pi$
$774$	0	0
$775$	33168.5	1.53735
$776$	8339.23	0.385774
$777$	0	0
$778$	−18073.0	−0.832837
$779$	9426.92	0.433575
$780$	0	0
$781$	4922.73	0.225543
$782$	1421.81	0.0650175
$783$	0	0
$784$	784.000	0.0357143
$785$	2110.87	0.0959746
$786$	0	0
$787$	10468.3	0.474149	0.237075	−	0.971491i	$-0.423811\pi$
$787$	10468.3	0.474149	0.237075	+	0.971491i	$0.423811\pi$
$788$	4163.16	0.188206
$789$	0	0
$790$	319.912	0.0144075
$791$	−7994.67	−0.359365
$792$	0	0
$793$	8488.52	0.380121
$794$	6960.03	0.311086
$795$	0	0
$796$	11647.1	0.518618
$797$	381.927	0.0169743	0.00848717	−	0.999964i	$-0.497298\pi$
$797$	381.927	0.0169743	0.00848717	+	0.999964i	$0.497298\pi$
$798$	0	0
$799$	−5490.77	−0.243116
$800$	3779.76	0.167043
$801$	0	0
$802$	23185.6	1.02084
$803$	3213.67	0.141230
$804$	0	0
$805$	−1067.06	−0.0467193
$806$	−7301.05	−0.319067
$807$	0	0
$808$	−1006.20	−0.0438092
$809$	5687.15	0.247156	0.123578	−	0.992335i	$-0.460563\pi$
$809$	5687.15	0.247156	0.123578	+	0.992335i	$0.460563\pi$
$810$	0	0
$811$	37564.5	1.62647	0.813236	−	0.581935i	$-0.197704\pi$
$811$	37564.5	1.62647	0.813236	+	0.581935i	$0.197704\pi$
$812$	−5098.26	−0.220337
$813$	0	0
$814$	1430.84	0.0616105
$815$	3353.97	0.144153
$816$	0	0
$817$	3437.65	0.147207
$818$	13746.1	0.587558
$819$	0	0
$820$	1592.87	0.0678357
$821$	9003.37	0.382728	0.191364	−	0.981519i	$-0.438709\pi$
$821$	9003.37	0.382728	0.191364	+	0.981519i	$0.438709\pi$
$822$	0	0
$823$	19669.0	0.833071	0.416535	−	0.909119i	$-0.363244\pi$
$823$	19669.0	0.833071	0.416535	+	0.909119i	$0.363244\pi$
$824$	−6932.98	−0.293109
$825$	0	0
$826$	3503.63	0.147587
$827$	−19091.0	−0.802734	−0.401367	−	0.915917i	$-0.631465\pi$
$827$	−19091.0	−0.802734	−0.401367	+	0.915917i	$0.631465\pi$
$828$	0	0
$829$	−12674.4	−0.531000	−0.265500	−	0.964111i	$-0.585537\pi$
$829$	−12674.4	−0.531000	−0.265500	+	0.964111i	$0.585537\pi$
$830$	−4289.79	−0.179399
$831$	0	0
$832$	−832.000	−0.0346688
$833$	−599.503	−0.0249358
$834$	0	0
$835$	−10124.9	−0.419626
$836$	−1116.38	−0.0461851
$837$	0	0
$838$	−8900.27	−0.366891
$839$	28864.0	1.18772	0.593860	−	0.804569i	$-0.297603\pi$
$839$	28864.0	1.18772	0.593860	+	0.804569i	$0.297603\pi$
$840$	0	0
$841$	8764.41	0.359359
$842$	20479.6	0.838209
$843$	0	0
$844$	−11327.0	−0.461955
$845$	443.367	0.0180501
$846$	0	0
$847$	9175.63	0.372230
$848$	−2363.98	−0.0957302
$849$	0	0
$850$	−2890.27	−0.116630
$851$	9250.24	0.372613
$852$	0	0
$853$	−4468.74	−0.179375	−0.0896873	−	0.995970i	$-0.528587\pi$
$853$	−4468.74	−0.179375	−0.0896873	+	0.995970i	$0.528587\pi$
$854$	−9141.49	−0.366294
$855$	0	0
$856$	7483.85	0.298823
$857$	6709.51	0.267436	0.133718	−	0.991019i	$-0.457308\pi$
$857$	6709.51	0.267436	0.133718	+	0.991019i	$0.457308\pi$
$858$	0	0
$859$	13601.7	0.540260	0.270130	−	0.962824i	$-0.412933\pi$
$859$	13601.7	0.540260	0.270130	+	0.962824i	$0.412933\pi$
$860$	580.860	0.0230316
$861$	0	0
$862$	−47.4392	−0.00187446
$863$	−9919.12	−0.391252	−0.195626	−	0.980679i	$-0.562674\pi$
$863$	−9919.12	−0.391252	−0.195626	+	0.980679i	$0.562674\pi$
$864$	0	0
$865$	−4202.72	−0.165199
$866$	676.201	0.0265338
$867$	0	0
$868$	7862.66	0.307461
$869$	273.998	0.0106959
$870$	0	0
$871$	7105.89	0.276434
$872$	−13280.8	−0.515763
$873$	0	0
$874$	−7217.27	−0.279322
$875$	4464.69	0.172496
$876$	0	0
$877$	48217.7	1.85655	0.928276	−	0.371892i	$-0.121291\pi$
$877$	48217.7	1.85655	0.928276	+	0.371892i	$0.121291\pi$
$878$	−3931.91	−0.151134
$879$	0	0
$880$	−188.634	−0.00722598
$881$	−40625.3	−1.55358	−0.776788	−	0.629763i	$-0.783152\pi$
$881$	−40625.3	−1.55358	−0.776788	+	0.629763i	$0.783152\pi$
$882$	0	0
$883$	−35652.3	−1.35877	−0.679386	−	0.733781i	$-0.737754\pi$
$883$	−35652.3	−1.35877	−0.679386	+	0.733781i	$0.737754\pi$
$884$	636.207	0.0242058
$885$	0	0
$886$	7681.72	0.291278
$887$	28335.1	1.07260	0.536301	−	0.844027i	$-0.319821\pi$
$887$	28335.1	1.07260	0.536301	+	0.844027i	$0.319821\pi$
$888$	0	0
$889$	4832.09	0.182298
$890$	−3232.88	−0.121760
$891$	0	0
$892$	−7554.98	−0.283587
$893$	27871.9	1.04445
$894$	0	0
$895$	1777.58	0.0663889
$896$	896.000	0.0334077
$897$	0	0
$898$	−26245.0	−0.975286
$899$	−51130.0	−1.89686
$900$	0	0
$901$	1807.67	0.0668392
$902$	1364.26	0.0503600
$903$	0	0
$904$	−9136.77	−0.336155
$905$	−6772.15	−0.248745
$906$	0	0
$907$	−17124.4	−0.626910	−0.313455	−	0.949603i	$-0.601486\pi$
$907$	−17124.4	−0.626910	−0.313455	+	0.949603i	$0.601486\pi$
$908$	−3742.60	−0.136787
$909$	0	0
$910$	−477.473	−0.0173935
$911$	20610.5	0.749567	0.374784	−	0.927112i	$-0.377717\pi$
$911$	20610.5	0.749567	0.374784	+	0.927112i	$0.377717\pi$
$912$	0	0
$913$	−3674.11	−0.133182
$914$	18190.3	0.658297
$915$	0	0
$916$	746.707	0.0269344
$917$	7209.49	0.259627
$918$	0	0
$919$	−22868.1	−0.820836	−0.410418	−	0.911898i	$-0.634617\pi$
$919$	−22868.1	−0.820836	−0.410418	+	0.911898i	$0.634617\pi$
$920$	−1219.50	−0.0437019
$921$	0	0
$922$	−18489.7	−0.660441
$923$	14240.5	0.507835
$924$	0	0
$925$	−18804.1	−0.668404
$926$	12830.4	0.455329
$927$	0	0
$928$	−5826.59	−0.206107
$929$	−2643.43	−0.0933564	−0.0466782	−	0.998910i	$-0.514864\pi$
$929$	−2643.43	−0.0933564	−0.0466782	+	0.998910i	$0.514864\pi$
$930$	0	0
$931$	3043.15	0.107127
$932$	−26569.4	−0.933809
$933$	0	0
$934$	7959.49	0.278846
$935$	144.243	0.00504520
$936$	0	0
$937$	4443.82	0.154934	0.0774671	−	0.996995i	$-0.475317\pi$
$937$	4443.82	0.154934	0.0774671	+	0.996995i	$0.475317\pi$
$938$	−7652.49	−0.266378
$939$	0	0
$940$	4709.51	0.163412
$941$	−37681.1	−1.30539	−0.652694	−	0.757622i	$-0.726361\pi$
$941$	−37681.1	−1.30539	−0.652694	+	0.757622i	$0.726361\pi$
$942$	0	0
$943$	8819.76	0.304572
$944$	4004.15	0.138055
$945$	0	0
$946$	497.494	0.0170982
$947$	2077.40	0.0712844	0.0356422	−	0.999365i	$-0.488652\pi$
$947$	2077.40	0.0712844	0.0356422	+	0.999365i	$0.488652\pi$
$948$	0	0
$949$	9296.53	0.317996
$950$	14671.4	0.501056
$951$	0	0
$952$	−685.146	−0.0233253
$953$	25207.9	0.856836	0.428418	−	0.903581i	$-0.359071\pi$
$953$	25207.9	0.856836	0.428418	+	0.903581i	$0.359071\pi$
$954$	0	0
$955$	−6089.95	−0.206352
$956$	−10602.4	−0.358688
$957$	0	0
$958$	31188.3	1.05182
$959$	9941.58	0.334755
$960$	0	0
$961$	49062.9	1.64690
$962$	4139.15	0.138723
$963$	0	0
$964$	−26992.6	−0.901839
$965$	10016.4	0.334135
$966$	0	0
$967$	33306.4	1.10761	0.553806	−	0.832645i	$-0.313175\pi$
$967$	33306.4	1.10761	0.553806	+	0.832645i	$0.313175\pi$
$968$	10486.4	0.348189
$969$	0	0
$970$	5469.44	0.181045
$971$	20916.1	0.691277	0.345638	−	0.938368i	$-0.387662\pi$
$971$	20916.1	0.691277	0.345638	+	0.938368i	$0.387662\pi$
$972$	0	0
$973$	−3072.85	−0.101245
$974$	3630.15	0.119423
$975$	0	0
$976$	−10447.4	−0.342637
$977$	−28002.4	−0.916965	−0.458482	−	0.888703i	$-0.651607\pi$
$977$	−28002.4	−0.916965	−0.458482	+	0.888703i	$0.651607\pi$
$978$	0	0
$979$	−2768.89	−0.0903925
$980$	514.201	0.0167608
$981$	0	0
$982$	−14010.7	−0.455296
$983$	−22186.6	−0.719879	−0.359940	−	0.932976i	$-0.617203\pi$
$983$	−22186.6	−0.719879	−0.359940	+	0.932976i	$0.617203\pi$
$984$	0	0
$985$	2730.49	0.0883254
$986$	4455.43	0.143904
$987$	0	0
$988$	−3229.47	−0.103991
$989$	3216.25	0.103408
$990$	0	0
$991$	23038.4	0.738485	0.369243	−	0.929333i	$-0.379617\pi$
$991$	23038.4	0.738485	0.369243	+	0.929333i	$0.379617\pi$
$992$	8985.90	0.287604
$993$	0	0
$994$	−15335.9	−0.489362
$995$	7638.95	0.243388
$996$	0	0
$997$	−38018.8	−1.20769	−0.603845	−	0.797102i	$-0.706365\pi$
$997$	−38018.8	−1.20769	−0.603845	+	0.797102i	$0.706365\pi$
$998$	−23709.5	−0.752015
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.m.1.2		2
3.2	odd	2		546.4.a.j.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
546.4.a.j.1.1	✓	2	3.2	odd	2
1638.4.a.m.1.2		2	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +4.00000 q^{4} +2.62348 q^{5} -7.00000 q^{7} -8.00000 q^{8} -5.24695 q^{10} -4.49390 q^{11} -13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -12.2348 q^{17} +62.1052 q^{19} +10.4939 q^{20} +8.98780 q^{22} +58.1052 q^{23} -118.117 q^{25} +26.0000 q^{26} -28.0000 q^{28} +182.081 q^{29} -280.809 q^{31} -32.0000 q^{32} +24.4695 q^{34} -18.3643 q^{35} +159.198 q^{37} -124.210 q^{38} -20.9878 q^{40} +151.790 q^{41} +55.3521 q^{43} -17.9756 q^{44} -116.210 q^{46} +448.785 q^{47} +49.0000 q^{49} +236.235 q^{50} -52.0000 q^{52} -147.748 q^{53} -11.7896 q^{55} +56.0000 q^{56} -364.162 q^{58} +250.259 q^{59} -652.963 q^{61} +561.619 q^{62} +64.0000 q^{64} -34.1052 q^{65} -546.607 q^{67} -48.9390 q^{68} +36.7287 q^{70} -1095.42 q^{71} -715.117 q^{73} -318.396 q^{74} +248.421 q^{76} +31.4573 q^{77} -60.9710 q^{79} +41.9756 q^{80} -303.579 q^{82} +817.578 q^{83} -32.0976 q^{85} -110.704 q^{86} +35.9512 q^{88} +616.145 q^{89} +91.0000 q^{91} +232.421 q^{92} -897.570 q^{94} +162.931 q^{95} -1042.40 q^{97} -98.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{2} + 8 q^{4} - 5 q^{5} - 14 q^{7} - 16 q^{8} + 10 q^{10} + 32 q^{11} - 26 q^{13} + 28 q^{14} + 32 q^{16} + 78 q^{17} - 9 q^{19} - 20 q^{20} - 64 q^{22} - 17 q^{23} - 185 q^{25} + 52 q^{26} - 56 q^{28}+ \cdots - 196 q^{98}+O(q^{100}) \) 2 * q - 4 * q^2 + 8 * q^4 - 5 * q^5 - 14 * q^7 - 16 * q^8 + 10 * q^10 + 32 * q^11 - 26 * q^13 + 28 * q^14 + 32 * q^16 + 78 * q^17 - 9 * q^19 - 20 * q^20 - 64 * q^22 - 17 * q^23 - 185 * q^25 + 52 * q^26 - 56 * q^28 + 67 * q^29 - 121 * q^31 - 64 * q^32 - 156 * q^34 + 35 * q^35 - 30 * q^37 + 18 * q^38 + 40 * q^40 + 570 * q^41 - 43 * q^43 + 128 * q^44 + 34 * q^46 + 293 * q^47 + 98 * q^49 + 370 * q^50 - 104 * q^52 + 555 * q^53 - 290 * q^55 + 112 * q^56 - 134 * q^58 + 562 * q^59 - 1060 * q^61 + 242 * q^62 + 128 * q^64 + 65 * q^65 - 130 * q^67 + 312 * q^68 - 70 * q^70 + 2 * q^71 - 1379 * q^73 + 60 * q^74 - 36 * q^76 - 224 * q^77 + 913 * q^79 - 80 * q^80 - 1140 * q^82 - 363 * q^83 - 720 * q^85 + 86 * q^86 - 256 * q^88 - 315 * q^89 + 182 * q^91 - 68 * q^92 - 586 * q^94 + 705 * q^95 - 599 * q^97 - 196 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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