LMFDB - Embedded newform 2075.4.a.l.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2075,4,Mod(1,2075)] 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("2075.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2075, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2075 = 5^{2} \cdot 83 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2075.a (trivial)

Newform invariants

comment:select newform

sage:traces = [47,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$122.428963262$
Analytic rank:	$0$
Dimension:	$47$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Character	$\chi$	$=$	2075.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.52319 q^{2} +6.73619 q^{3} +12.4592 q^{4} -30.4691 q^{6} +5.86549 q^{7} -20.1699 q^{8} +18.3763 q^{9} -65.4392 q^{11} +83.9277 q^{12} +56.9013 q^{13} -26.5307 q^{14} -8.44158 q^{16} +46.3849 q^{17} -83.1194 q^{18} -122.840 q^{19} +39.5111 q^{21} +295.994 q^{22} -81.3605 q^{23} -135.868 q^{24} -257.375 q^{26} -58.0909 q^{27} +73.0794 q^{28} +196.716 q^{29} +32.9330 q^{31} +199.542 q^{32} -440.811 q^{33} -209.808 q^{34} +228.954 q^{36} -141.312 q^{37} +555.626 q^{38} +383.298 q^{39} +360.968 q^{41} -178.716 q^{42} -296.066 q^{43} -815.322 q^{44} +368.009 q^{46} +453.133 q^{47} -56.8641 q^{48} -308.596 q^{49} +312.458 q^{51} +708.945 q^{52} -727.330 q^{53} +262.756 q^{54} -118.306 q^{56} -827.471 q^{57} -889.785 q^{58} -188.286 q^{59} +815.194 q^{61} -148.962 q^{62} +107.786 q^{63} -835.033 q^{64} +1993.87 q^{66} +1022.81 q^{67} +577.920 q^{68} -548.060 q^{69} -432.215 q^{71} -370.648 q^{72} +515.790 q^{73} +639.182 q^{74} -1530.48 q^{76} -383.833 q^{77} -1733.73 q^{78} +1025.94 q^{79} -887.472 q^{81} -1632.73 q^{82} -83.0000 q^{83} +492.277 q^{84} +1339.16 q^{86} +1325.12 q^{87} +1319.90 q^{88} +405.444 q^{89} +333.754 q^{91} -1013.69 q^{92} +221.843 q^{93} -2049.61 q^{94} +1344.15 q^{96} -260.768 q^{97} +1395.84 q^{98} -1202.53 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 47 q + q^{2} - 6 q^{3} + 229 q^{4} + 9 q^{6} - 6 q^{7} + 39 q^{8} + 531 q^{9} + 90 q^{11} + 6 q^{12} + 50 q^{13} + 115 q^{14} + 1149 q^{16} - 68 q^{17} + 8 q^{18} + 344 q^{19} + 625 q^{21} + 402 q^{22}+ \cdots + 13991 q^{99}+O(q^{100}) $ 47 * q + q^2 - 6 * q^3 + 229 * q^4 + 9 * q^6 - 6 * q^7 + 39 * q^8 + 531 * q^9 + 90 * q^11 + 6 * q^12 + 50 * q^13 + 115 * q^14 + 1149 * q^16 - 68 * q^17 + 8 * q^18 + 344 * q^19 + 625 * q^21 + 402 * q^22 - 469 * q^23 + 262 * q^24 + 1209 * q^26 + 12 * q^27 - 603 * q^28 + 480 * q^29 + 876 * q^31 + 1653 * q^32 - 757 * q^33 + 733 * q^34 + 4318 * q^36 + 634 * q^37 - 1190 * q^38 + 618 * q^39 + 1455 * q^41 + 2747 * q^42 - 652 * q^43 + 595 * q^44 + 1662 * q^46 + 446 * q^47 - 2145 * q^48 + 4099 * q^49 + 1681 * q^51 + 207 * q^52 - 126 * q^53 + 670 * q^54 + 3100 * q^56 + 650 * q^57 - 1082 * q^58 + 1144 * q^59 + 2762 * q^61 + 1916 * q^62 - 885 * q^63 + 6891 * q^64 + 2025 * q^66 - 100 * q^67 - 2052 * q^68 + 2556 * q^69 + 2546 * q^71 + 2940 * q^72 + 14 * q^73 + 1066 * q^74 + 4828 * q^76 - 210 * q^77 - 343 * q^78 + 1636 * q^79 + 8203 * q^81 + 480 * q^82 - 3901 * q^83 + 6209 * q^84 + 2706 * q^86 - 5479 * q^87 + 2178 * q^88 + 2558 * q^89 + 3342 * q^91 - 7501 * q^92 + 3353 * q^93 + 5769 * q^94 - 918 * q^96 - 3648 * q^97 + 8109 * q^98 + 13991 * q^99

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 $

$2$	−4.52319	−1.59919	−0.799594	−	0.600541i	$-0.794952\pi$
$2$	−4.52319	−1.59919	−0.799594	+	0.600541i	$0.794952\pi$
$3$	6.73619	1.29638	0.648191	−	0.761478i	$-0.275526\pi$
$3$	6.73619	1.29638	0.648191	+	0.761478i	$0.275526\pi$
$4$	12.4592	1.55740
$5$	0	0
$5$	0	0
$6$	−30.4691	−2.07316
$7$	5.86549	0.316707	0.158353	−	0.987383i	$-0.449381\pi$
$7$	5.86549	0.316707	0.158353	+	0.987383i	$0.449381\pi$
$8$	−20.1699	−0.891392
$9$	18.3763	0.680604
$10$	0	0
$11$	−65.4392	−1.79370	−0.896849	−	0.442338i	$-0.854149\pi$
$11$	−65.4392	−1.79370	−0.896849	+	0.442338i	$0.854149\pi$
$12$	83.9277	2.01899
$13$	56.9013	1.21397	0.606983	−	0.794714i	$-0.292379\pi$
$13$	56.9013	1.21397	0.606983	+	0.794714i	$0.292379\pi$
$14$	−26.5307	−0.506474
$15$	0	0
$16$	−8.44158	−0.131900
$17$	46.3849	0.661764	0.330882	−	0.943672i	$-0.392654\pi$
$17$	46.3849	0.661764	0.330882	+	0.943672i	$0.392654\pi$
$18$	−83.1194	−1.08841
$19$	−122.840	−1.48323	−0.741614	−	0.670827i	$-0.765939\pi$
$19$	−122.840	−1.48323	−0.741614	+	0.670827i	$0.765939\pi$
$20$	0	0
$21$	39.5111	0.410573
$22$	295.994	2.86846
$23$	−81.3605	−0.737602	−0.368801	−	0.929508i	$-0.620232\pi$
$23$	−81.3605	−0.737602	−0.368801	+	0.929508i	$0.620232\pi$
$24$	−135.868	−1.15558
$25$	0	0
$26$	−257.375	−1.94136
$27$	−58.0909	−0.414059
$28$	73.0794	0.493240
$29$	196.716	1.25963	0.629816	−	0.776744i	$-0.283130\pi$
$29$	196.716	1.25963	0.629816	+	0.776744i	$0.283130\pi$
$30$	0	0
$31$	32.9330	0.190805	0.0954024	−	0.995439i	$-0.469586\pi$
$31$	32.9330	0.190805	0.0954024	+	0.995439i	$0.469586\pi$
$32$	199.542	1.10232
$33$	−440.811	−2.32531
$34$	−209.808	−1.05829
$35$	0	0
$36$	228.954	1.05997
$37$	−141.312	−0.627881	−0.313940	−	0.949443i	$-0.601649\pi$
$37$	−141.312	−0.627881	−0.313940	+	0.949443i	$0.601649\pi$
$38$	555.626	2.37196
$39$	383.298	1.57376
$40$	0	0
$41$	360.968	1.37497	0.687485	−	0.726198i	$-0.258715\pi$
$41$	360.968	1.37497	0.687485	+	0.726198i	$0.258715\pi$
$42$	−178.716	−0.656583
$43$	−296.066	−1.04999	−0.524995	−	0.851105i	$-0.675933\pi$
$43$	−296.066	−1.04999	−0.524995	+	0.851105i	$0.675933\pi$
$44$	−815.322	−2.79351
$45$	0	0
$46$	368.009	1.17956
$47$	453.133	1.40630	0.703152	−	0.711040i	$-0.251775\pi$
$47$	453.133	1.40630	0.703152	+	0.711040i	$0.251775\pi$
$48$	−56.8641	−0.170992
$49$	−308.596	−0.899697
$50$	0	0
$51$	312.458	0.857898
$52$	708.945	1.89064
$53$	−727.330	−1.88503	−0.942514	−	0.334167i	$-0.891545\pi$
$53$	−727.330	−1.88503	−0.942514	+	0.334167i	$0.891545\pi$
$54$	262.756	0.662159
$55$	0	0
$56$	−118.306	−0.282310
$57$	−827.471	−1.92283
$58$	−889.785	−2.01439
$59$	−188.286	−0.415470	−0.207735	−	0.978185i	$-0.566609\pi$
$59$	−188.286	−0.415470	−0.207735	+	0.978185i	$0.566609\pi$
$60$	0	0
$61$	815.194	1.71106	0.855532	−	0.517750i	$-0.173230\pi$
$61$	815.194	1.71106	0.855532	+	0.517750i	$0.173230\pi$
$62$	−148.962	−0.305133
$63$	107.786	0.215552
$64$	−835.033	−1.63092
$65$	0	0
$66$	1993.87	3.71862
$67$	1022.81	1.86503	0.932513	−	0.361137i	$-0.117611\pi$
$67$	1022.81	1.86503	0.932513	+	0.361137i	$0.117611\pi$
$68$	577.920	1.03063
$69$	−548.060	−0.956213
$70$	0	0
$71$	−432.215	−0.722458	−0.361229	−	0.932477i	$-0.617643\pi$
$71$	−432.215	−0.722458	−0.361229	+	0.932477i	$0.617643\pi$
$72$	−370.648	−0.606684
$73$	515.790	0.826968	0.413484	−	0.910511i	$-0.364312\pi$
$73$	515.790	0.826968	0.413484	+	0.910511i	$0.364312\pi$
$74$	639.182	1.00410
$75$	0	0
$76$	−1530.48	−2.30998
$77$	−383.833	−0.568076
$78$	−1733.73	−2.51674
$79$	1025.94	1.46111	0.730554	−	0.682855i	$-0.239262\pi$
$79$	1025.94	1.46111	0.730554	+	0.682855i	$0.239262\pi$
$80$	0	0
$81$	−887.472	−1.21738
$82$	−1632.73	−2.19884
$83$	−83.0000	−0.109764
$83$	−83.0000	−0.109764
$84$	492.277	0.639427
$85$	0	0
$86$	1339.16	1.67913
$87$	1325.12	1.63296
$88$	1319.90	1.59889
$89$	405.444	0.482887	0.241444	−	0.970415i	$-0.422379\pi$
$89$	405.444	0.482887	0.241444	+	0.970415i	$0.422379\pi$
$90$	0	0
$91$	333.754	0.384472
$92$	−1013.69	−1.14874
$93$	221.843	0.247356
$94$	−2049.61	−2.24894
$95$	0	0
$96$	1344.15	1.42903
$97$	−260.768	−0.272959	−0.136479	−	0.990643i	$-0.543579\pi$
$97$	−260.768	−0.272959	−0.136479	+	0.990643i	$0.543579\pi$
$98$	1395.84	1.43878
$99$	−1202.53	−1.22080
$100$	0	0
$101$	−1207.14	−1.18926	−0.594628	−	0.804001i	$-0.702701\pi$
$101$	−1207.14	−1.18926	−0.594628	+	0.804001i	$0.702701\pi$
$102$	−1413.30	−1.37194
$103$	344.234	0.329305	0.164652	−	0.986352i	$-0.447350\pi$
$103$	344.234	0.329305	0.164652	+	0.986352i	$0.447350\pi$
$104$	−1147.69	−1.08212
$105$	0	0
$106$	3289.85	3.01451
$107$	−113.688	−0.102717	−0.0513583	−	0.998680i	$-0.516355\pi$
$107$	−113.688	−0.102717	−0.0513583	+	0.998680i	$0.516355\pi$
$108$	−723.767	−0.644857
$109$	1173.05	1.03081	0.515405	−	0.856947i	$-0.327642\pi$
$109$	1173.05	1.03081	0.515405	+	0.856947i	$0.327642\pi$
$110$	0	0
$111$	−951.907	−0.813973
$112$	−49.5140	−0.0417735
$113$	1704.59	1.41907	0.709534	−	0.704671i	$-0.248905\pi$
$113$	1704.59	1.41907	0.709534	+	0.704671i	$0.248905\pi$
$114$	3742.81	3.07496
$115$	0	0
$116$	2450.93	1.96175
$117$	1045.63	0.826230
$118$	851.651	0.664414
$119$	272.070	0.209585
$120$	0	0
$121$	2951.29	2.21735
$122$	−3687.27	−2.73631
$123$	2431.55	1.78249
$124$	410.320	0.297160
$125$	0	0
$126$	−487.536	−0.344708
$127$	1974.36	1.37949	0.689747	−	0.724050i	$-0.257722\pi$
$127$	1974.36	1.37949	0.689747	+	0.724050i	$0.257722\pi$
$128$	2180.67	1.50583
$129$	−1994.35	−1.36119
$130$	0	0
$131$	1244.80	0.830217	0.415108	−	0.909772i	$-0.363744\pi$
$131$	1244.80	0.830217	0.415108	+	0.909772i	$0.363744\pi$
$132$	−5492.16	−3.62145
$133$	−720.514	−0.469748
$134$	−4626.38	−2.98253
$135$	0	0
$136$	−935.578	−0.589891
$137$	−1369.41	−0.853992	−0.426996	−	0.904253i	$-0.640428\pi$
$137$	−1369.41	−0.853992	−0.426996	+	0.904253i	$0.640428\pi$
$138$	2478.98	1.52916
$139$	841.810	0.513679	0.256839	−	0.966454i	$-0.417319\pi$
$139$	841.810	0.513679	0.256839	+	0.966454i	$0.417319\pi$
$140$	0	0
$141$	3052.39	1.82311
$142$	1954.99	1.15535
$143$	−3723.57	−2.17749
$144$	−155.125	−0.0897714
$145$	0	0
$146$	−2333.01	−1.32248
$147$	−2078.76	−1.16635
$148$	−1760.64	−0.977863
$149$	2959.53	1.62721	0.813603	−	0.581420i	$-0.197503\pi$
$149$	2959.53	1.62721	0.813603	+	0.581420i	$0.197503\pi$
$150$	0	0
$151$	1852.29	0.998258	0.499129	−	0.866528i	$-0.333653\pi$
$151$	1852.29	0.998258	0.499129	+	0.866528i	$0.333653\pi$
$152$	2477.66	1.32214
$153$	852.383	0.450399
$154$	1736.15	0.908460
$155$	0	0
$156$	4775.59	2.45098
$157$	3242.63	1.64835	0.824173	−	0.566338i	$-0.191640\pi$
$157$	3242.63	1.64835	0.824173	+	0.566338i	$0.191640\pi$
$158$	−4640.53	−2.33659
$159$	−4899.44	−2.44371
$160$	0	0
$161$	−477.220	−0.233604
$162$	4014.20	1.94682
$163$	3916.00	1.88175	0.940873	−	0.338760i	$-0.110008\pi$
$163$	3916.00	1.88175	0.940873	+	0.338760i	$0.110008\pi$
$164$	4497.39	2.14138
$165$	0	0
$166$	375.425	0.175534
$167$	−995.910	−0.461472	−0.230736	−	0.973016i	$-0.574113\pi$
$167$	−995.910	−0.461472	−0.230736	+	0.973016i	$0.574113\pi$
$168$	−796.934	−0.365981
$169$	1040.75	0.473716
$170$	0	0
$171$	−2257.34	−1.00949
$172$	−3688.75	−1.63526
$173$	177.521	0.0780152	0.0390076	−	0.999239i	$-0.487580\pi$
$173$	177.521	0.0780152	0.0390076	+	0.999239i	$0.487580\pi$
$174$	−5993.77	−2.61142
$175$	0	0
$176$	552.410	0.236588
$177$	−1268.33	−0.538607
$178$	−1833.90	−0.772228
$179$	−2552.45	−1.06581	−0.532903	−	0.846176i	$-0.678899\pi$
$179$	−2552.45	−1.06581	−0.532903	+	0.846176i	$0.678899\pi$
$180$	0	0
$181$	−4589.91	−1.88489	−0.942446	−	0.334360i	$-0.891480\pi$
$181$	−4589.91	−1.88489	−0.942446	+	0.334360i	$0.891480\pi$
$182$	−1509.63	−0.614842
$183$	5491.30	2.21819
$184$	1641.03	0.657492
$185$	0	0
$186$	−1003.44	−0.395568
$187$	−3035.39	−1.18700
$188$	5645.69	2.19018
$189$	−340.732	−0.131135
$190$	0	0
$191$	1116.49	0.422964	0.211482	−	0.977382i	$-0.432171\pi$
$191$	1116.49	0.422964	0.211482	+	0.977382i	$0.432171\pi$
$192$	−5624.94	−2.11430
$193$	2539.65	0.947192	0.473596	−	0.880742i	$-0.342956\pi$
$193$	2539.65	0.947192	0.473596	+	0.880742i	$0.342956\pi$
$194$	1179.50	0.436512
$195$	0	0
$196$	−3844.87	−1.40119
$197$	−4980.87	−1.80138	−0.900690	−	0.434462i	$-0.856939\pi$
$197$	−4980.87	−1.80138	−0.900690	+	0.434462i	$0.856939\pi$
$198$	5439.27	1.95228
$199$	−226.366	−0.0806366	−0.0403183	−	0.999187i	$-0.512837\pi$
$199$	−226.366	−0.0806366	−0.0403183	+	0.999187i	$0.512837\pi$
$200$	0	0
$201$	6889.88	2.41778
$202$	5460.11	1.90184
$203$	1153.84	0.398934
$204$	3892.98	1.33609
$205$	0	0
$206$	−1557.04	−0.526620
$207$	−1495.11	−0.502015
$208$	−480.336	−0.160122
$209$	8038.52	2.66046
$210$	0	0
$211$	5290.15	1.72601	0.863007	−	0.505192i	$-0.168578\pi$
$211$	5290.15	1.72601	0.863007	+	0.505192i	$0.168578\pi$
$212$	−9061.97	−2.93575
$213$	−2911.49	−0.936581
$214$	514.234	0.164263
$215$	0	0
$216$	1171.69	0.369089
$217$	193.168	0.0604292
$218$	−5305.94	−1.64846
$219$	3474.46	1.07207
$220$	0	0
$221$	2639.36	0.803360
$222$	4305.65	1.30170
$223$	−6063.77	−1.82090	−0.910448	−	0.413624i	$-0.864263\pi$
$223$	−6063.77	−1.82090	−0.910448	+	0.413624i	$0.864263\pi$
$224$	1170.41	0.349113
$225$	0	0
$226$	−7710.20	−2.26936
$227$	−5046.09	−1.47542	−0.737711	−	0.675116i	$-0.764093\pi$
$227$	−5046.09	−1.47542	−0.737711	+	0.675116i	$0.764093\pi$
$228$	−10309.6	−2.99462
$229$	265.451	0.0766003	0.0383002	−	0.999266i	$-0.487806\pi$
$229$	265.451	0.0766003	0.0383002	+	0.999266i	$0.487806\pi$
$230$	0	0
$231$	−2585.57	−0.736443
$232$	−3967.75	−1.12283
$233$	−2146.70	−0.603585	−0.301792	−	0.953374i	$-0.597585\pi$
$233$	−2146.70	−0.603585	−0.301792	+	0.953374i	$0.597585\pi$
$234$	−4729.60	−1.32130
$235$	0	0
$236$	−2345.89	−0.647054
$237$	6910.94	1.89415
$238$	−1230.62	−0.335166
$239$	−1713.91	−0.463864	−0.231932	−	0.972732i	$-0.574505\pi$
$239$	−1713.91	−0.463864	−0.231932	+	0.972732i	$0.574505\pi$
$240$	0	0
$241$	5516.40	1.47445	0.737225	−	0.675647i	$-0.236136\pi$
$241$	5516.40	1.47445	0.737225	+	0.675647i	$0.236136\pi$
$242$	−13349.2	−3.54596
$243$	−4409.73	−1.16413
$244$	10156.7	2.66481
$245$	0	0
$246$	−10998.4	−2.85053
$247$	−6989.72	−1.80059
$248$	−664.256	−0.170082
$249$	−559.104	−0.142296
$250$	0	0
$251$	−3896.23	−0.979792	−0.489896	−	0.871781i	$-0.662965\pi$
$251$	−3896.23	−0.979792	−0.489896	+	0.871781i	$0.662965\pi$
$252$	1342.93	0.335701
$253$	5324.17	1.32303
$254$	−8930.39	−2.20607
$255$	0	0
$256$	−3183.34	−0.777182
$257$	5029.31	1.22070	0.610350	−	0.792132i	$-0.291029\pi$
$257$	5029.31	1.22070	0.610350	+	0.792132i	$0.291029\pi$
$258$	9020.84	2.17679
$259$	−828.866	−0.198854
$260$	0	0
$261$	3614.92	0.857310
$262$	−5630.45	−1.32767
$263$	−6106.31	−1.43168	−0.715839	−	0.698265i	$-0.753956\pi$
$263$	−6106.31	−1.43168	−0.715839	+	0.698265i	$0.753956\pi$
$264$	8891.11	2.07277
$265$	0	0
$266$	3259.02	0.751216
$267$	2731.15	0.626006
$268$	12743.5	2.90460
$269$	860.994	0.195151	0.0975757	−	0.995228i	$-0.468891\pi$
$269$	860.994	0.195151	0.0975757	+	0.995228i	$0.468891\pi$
$270$	0	0
$271$	−5692.97	−1.27610	−0.638050	−	0.769995i	$-0.720259\pi$
$271$	−5692.97	−1.27610	−0.638050	+	0.769995i	$0.720259\pi$
$272$	−391.562	−0.0872865
$273$	2248.23	0.498422
$274$	6194.12	1.36569
$275$	0	0
$276$	−6828.40	−1.48921
$277$	−555.413	−0.120475	−0.0602375	−	0.998184i	$-0.519186\pi$
$277$	−555.413	−0.120475	−0.0602375	+	0.998184i	$0.519186\pi$
$278$	−3807.66	−0.821469
$279$	605.187	0.129862
$280$	0	0
$281$	−1313.68	−0.278889	−0.139444	−	0.990230i	$-0.544532\pi$
$281$	−1313.68	−0.278889	−0.139444	+	0.990230i	$0.544532\pi$
$282$	−13806.5	−2.91549
$283$	3620.37	0.760456	0.380228	−	0.924893i	$-0.375846\pi$
$283$	3620.37	0.760456	0.380228	+	0.924893i	$0.375846\pi$
$284$	−5385.07	−1.12516
$285$	0	0
$286$	16842.4	3.48221
$287$	2117.26	0.435463
$288$	3666.84	0.750246
$289$	−2761.44	−0.562068
$290$	0	0
$291$	−1756.58	−0.353858
$292$	6426.34	1.28792
$293$	556.667	0.110993	0.0554963	−	0.998459i	$-0.482326\pi$
$293$	556.667	0.110993	0.0554963	+	0.998459i	$0.482326\pi$
$294$	9402.63	1.86521
$295$	0	0
$296$	2850.25	0.559688
$297$	3801.42	0.742697
$298$	−13386.5	−2.60221
$299$	−4629.52	−0.895424
$300$	0	0
$301$	−1736.57	−0.332539
$302$	−8378.24	−1.59640
$303$	−8131.52	−1.54173
$304$	1036.96	0.195637
$305$	0	0
$306$	−3855.49	−0.720273
$307$	−2266.06	−0.421273	−0.210636	−	0.977564i	$-0.567554\pi$
$307$	−2266.06	−0.421273	−0.210636	+	0.977564i	$0.567554\pi$
$308$	−4782.26	−0.884723
$309$	2318.83	0.426905
$310$	0	0
$311$	−68.4613	−0.0124826	−0.00624130	−	0.999981i	$-0.501987\pi$
$311$	−68.4613	−0.0124826	−0.00624130	+	0.999981i	$0.501987\pi$
$312$	−7731.08	−1.40284
$313$	−1158.53	−0.209214	−0.104607	−	0.994514i	$-0.533358\pi$
$313$	−1158.53	−0.209214	−0.104607	+	0.994514i	$0.533358\pi$
$314$	−14667.0	−2.63601
$315$	0	0
$316$	12782.4	2.27553
$317$	5008.05	0.887320	0.443660	−	0.896195i	$-0.353680\pi$
$317$	5008.05	0.887320	0.443660	+	0.896195i	$0.353680\pi$
$318$	22161.1	3.90796
$319$	−12873.0	−2.25940
$320$	0	0
$321$	−765.828	−0.133160
$322$	2158.55	0.373576
$323$	−5697.90	−0.981547
$324$	−11057.2	−1.89595
$325$	0	0
$326$	−17712.8	−3.00926
$327$	7901.92	1.33632
$328$	−7280.69	−1.22564
$329$	2657.85	0.445386
$330$	0	0
$331$	6572.56	1.09142	0.545710	−	0.837974i	$-0.316260\pi$
$331$	6572.56	1.09142	0.545710	+	0.837974i	$0.316260\pi$
$332$	−1034.12	−0.170947
$333$	−2596.80	−0.427338
$334$	4504.69	0.737981
$335$	0	0
$336$	−333.536	−0.0541544
$337$	10274.9	1.66086	0.830431	−	0.557122i	$-0.188094\pi$
$337$	10274.9	1.66086	0.830431	+	0.557122i	$0.188094\pi$
$338$	−4707.52	−0.757560
$339$	11482.5	1.83965
$340$	0	0
$341$	−2155.11	−0.342246
$342$	10210.4	1.61436
$343$	−3821.93	−0.601647
$344$	5971.61	0.935952
$345$	0	0
$346$	−802.959	−0.124761
$347$	12524.4	1.93759	0.968797	−	0.247855i	$-0.0797256\pi$
$347$	12524.4	1.93759	0.968797	+	0.247855i	$0.0797256\pi$
$348$	16510.0	2.54318
$349$	135.155	0.0207298	0.0103649	−	0.999946i	$-0.496701\pi$
$349$	135.155	0.0207298	0.0103649	+	0.999946i	$0.496701\pi$
$350$	0	0
$351$	−3305.45	−0.502654
$352$	−13057.9	−1.97724
$353$	−5754.61	−0.867669	−0.433835	−	0.900993i	$-0.642840\pi$
$353$	−5754.61	−0.867669	−0.433835	+	0.900993i	$0.642840\pi$
$354$	5736.89	0.861334
$355$	0	0
$356$	5051.52	0.752050
$357$	1832.72	0.271702
$358$	11545.2	1.70443
$359$	−820.184	−0.120578	−0.0602892	−	0.998181i	$-0.519202\pi$
$359$	−820.184	−0.120578	−0.0602892	+	0.998181i	$0.519202\pi$
$360$	0	0
$361$	8230.55	1.19996
$362$	20761.0	3.01430
$363$	19880.5	2.87453
$364$	4158.31	0.598777
$365$	0	0
$366$	−24838.2	−3.54730
$367$	2135.79	0.303780	0.151890	−	0.988397i	$-0.451464\pi$
$367$	2135.79	0.303780	0.151890	+	0.988397i	$0.451464\pi$
$368$	686.811	0.0972895
$369$	6633.26	0.935810
$370$	0	0
$371$	−4266.15	−0.597001
$372$	2763.99	0.385232
$373$	2760.94	0.383261	0.191630	−	0.981467i	$-0.438622\pi$
$373$	2760.94	0.383261	0.191630	+	0.981467i	$0.438622\pi$
$374$	13729.6	1.89824
$375$	0	0
$376$	−9139.65	−1.25357
$377$	11193.4	1.52915
$378$	1541.19	0.209710
$379$	8327.71	1.12867	0.564334	−	0.825546i	$-0.309133\pi$
$379$	8327.71	1.12867	0.564334	+	0.825546i	$0.309133\pi$
$380$	0	0
$381$	13299.6	1.78835
$382$	−5050.08	−0.676399
$383$	4449.55	0.593633	0.296816	−	0.954935i	$-0.404075\pi$
$383$	4449.55	0.593633	0.296816	+	0.954935i	$0.404075\pi$
$384$	14689.4	1.95213
$385$	0	0
$386$	−11487.3	−1.51474
$387$	−5440.59	−0.714627
$388$	−3248.97	−0.425106
$389$	−531.074	−0.0692199	−0.0346099	−	0.999401i	$-0.511019\pi$
$389$	−531.074	−0.0692199	−0.0346099	+	0.999401i	$0.511019\pi$
$390$	0	0
$391$	−3773.90	−0.488118
$392$	6224.35	0.801982
$393$	8385.19	1.07628
$394$	22529.4	2.88075
$395$	0	0
$396$	−14982.6	−1.90127
$397$	6459.58	0.816617	0.408309	−	0.912844i	$-0.366119\pi$
$397$	6459.58	0.816617	0.408309	+	0.912844i	$0.366119\pi$
$398$	1023.90	0.128953
$399$	−4853.52	−0.608973
$400$	0	0
$401$	6989.37	0.870405	0.435202	−	0.900333i	$-0.356677\pi$
$401$	6989.37	0.870405	0.435202	+	0.900333i	$0.356677\pi$
$402$	−31164.2	−3.86649
$403$	1873.93	0.231631
$404$	−15040.0	−1.85215
$405$	0	0
$406$	−5219.03	−0.637970
$407$	9247.36	1.12623
$408$	−6302.24	−0.764723
$409$	−1983.57	−0.239807	−0.119904	−	0.992786i	$-0.538259\pi$
$409$	−1983.57	−0.239807	−0.119904	+	0.992786i	$0.538259\pi$
$410$	0	0
$411$	−9224.64	−1.10710
$412$	4288.89	0.512860
$413$	−1104.39	−0.131582
$414$	6762.64	0.802816
$415$	0	0
$416$	11354.2	1.33818
$417$	5670.59	0.665924
$418$	−36359.7	−4.25458
$419$	6092.50	0.710353	0.355177	−	0.934799i	$-0.384421\pi$
$419$	6092.50	0.710353	0.355177	+	0.934799i	$0.384421\pi$
$420$	0	0
$421$	3594.15	0.416076	0.208038	−	0.978121i	$-0.433292\pi$
$421$	3594.15	0.416076	0.208038	+	0.978121i	$0.433292\pi$
$422$	−23928.3	−2.76022
$423$	8326.91	0.957136
$424$	14670.2	1.68030
$425$	0	0
$426$	13169.2	1.49777
$427$	4781.51	0.541905
$428$	−1416.47	−0.159971
$429$	−25082.7	−2.82286
$430$	0	0
$431$	6771.79	0.756812	0.378406	−	0.925640i	$-0.376472\pi$
$431$	6771.79	0.756812	0.378406	+	0.925640i	$0.376472\pi$
$432$	490.379	0.0546143
$433$	−7698.42	−0.854417	−0.427208	−	0.904153i	$-0.640503\pi$
$433$	−7698.42	−0.854417	−0.427208	+	0.904153i	$0.640503\pi$
$434$	−873.737	−0.0966376
$435$	0	0
$436$	14615.3	1.60539
$437$	9994.29	1.09403
$438$	−15715.6	−1.71443
$439$	7802.65	0.848291	0.424146	−	0.905594i	$-0.360574\pi$
$439$	7802.65	0.848291	0.424146	+	0.905594i	$0.360574\pi$
$440$	0	0
$441$	−5670.85	−0.612337
$442$	−11938.3	−1.28472
$443$	−11011.7	−1.18100	−0.590500	−	0.807038i	$-0.701069\pi$
$443$	−11011.7	−1.18100	−0.590500	+	0.807038i	$0.701069\pi$
$444$	−11860.0	−1.26768
$445$	0	0
$446$	27427.5	2.91195
$447$	19935.9	2.10948
$448$	−4897.88	−0.516525
$449$	7173.28	0.753960	0.376980	−	0.926221i	$-0.376963\pi$
$449$	7173.28	0.753960	0.376980	+	0.926221i	$0.376963\pi$
$450$	0	0
$451$	−23621.5	−2.46628
$452$	21237.9	2.21006
$453$	12477.4	1.29412
$454$	22824.4	2.35948
$455$	0	0
$456$	16690.0	1.71399
$457$	−7868.84	−0.805446	−0.402723	−	0.915322i	$-0.631936\pi$
$457$	−7868.84	−0.805446	−0.402723	+	0.915322i	$0.631936\pi$
$458$	−1200.68	−0.122498
$459$	−2694.54	−0.274010
$460$	0	0
$461$	9634.02	0.973321	0.486660	−	0.873591i	$-0.338215\pi$
$461$	9634.02	0.973321	0.486660	+	0.873591i	$0.338215\pi$
$462$	11695.0	1.17771
$463$	4248.25	0.426421	0.213211	−	0.977006i	$-0.431608\pi$
$463$	4248.25	0.426421	0.213211	+	0.977006i	$0.431608\pi$
$464$	−1660.60	−0.166145
$465$	0	0
$466$	9709.95	0.965246
$467$	−3925.28	−0.388951	−0.194476	−	0.980907i	$-0.562300\pi$
$467$	−3925.28	−0.388951	−0.194476	+	0.980907i	$0.562300\pi$
$468$	13027.8	1.28677
$469$	5999.31	0.590666
$470$	0	0
$471$	21843.0	2.13688
$472$	3797.70	0.370346
$473$	19374.3	1.88336
$474$	−31259.5	−3.02910
$475$	0	0
$476$	3389.78	0.326408
$477$	−13365.6	−1.28296
$478$	7752.33	0.741806
$479$	−13728.7	−1.30957	−0.654783	−	0.755817i	$-0.727240\pi$
$479$	−13728.7	−1.30957	−0.654783	+	0.755817i	$0.727240\pi$
$480$	0	0
$481$	−8040.85	−0.762227
$482$	−24951.7	−2.35792
$483$	−3214.64	−0.302839
$484$	36770.8	3.45331
$485$	0	0
$486$	19946.0	1.86167
$487$	−19.3960	−0.00180476	−0.000902379	−	1.00000i	$-0.500287\pi$
$487$	−19.3960	−0.00180476	−0.000902379		1.00000i	$0.500287\pi$
$488$	−16442.4	−1.52523
$489$	26378.9	2.43946
$490$	0	0
$491$	13917.1	1.27916	0.639580	−	0.768724i	$-0.279108\pi$
$491$	13917.1	1.27916	0.639580	+	0.768724i	$0.279108\pi$
$492$	30295.3	2.77605
$493$	9124.67	0.833579
$494$	31615.8	2.87948
$495$	0	0
$496$	−278.007	−0.0251671
$497$	−2535.16	−0.228807
$498$	2528.93	0.227559
$499$	−14102.7	−1.26518	−0.632591	−	0.774486i	$-0.718008\pi$
$499$	−14102.7	−1.26518	−0.632591	+	0.774486i	$0.718008\pi$
$500$	0	0
$501$	−6708.64	−0.598244
$502$	17623.4	1.56687
$503$	−13565.5	−1.20249	−0.601246	−	0.799064i	$-0.705329\pi$
$503$	−13565.5	−1.20249	−0.601246	+	0.799064i	$0.705329\pi$
$504$	−2174.03	−0.192141
$505$	0	0
$506$	−24082.2	−2.11578
$507$	7010.71	0.614116
$508$	24598.9	2.14843
$509$	18271.4	1.59109	0.795545	−	0.605895i	$-0.207185\pi$
$509$	18271.4	1.59109	0.795545	+	0.605895i	$0.207185\pi$
$510$	0	0
$511$	3025.36	0.261906
$512$	−3046.58	−0.262970
$513$	7135.86	0.614144
$514$	−22748.5	−1.95213
$515$	0	0
$516$	−24848.1	−2.11992
$517$	−29652.7	−2.52248
$518$	3749.11	0.318005
$519$	1195.81	0.101137
$520$	0	0
$521$	13617.3	1.14508	0.572539	−	0.819878i	$-0.305959\pi$
$521$	13617.3	1.14508	0.572539	+	0.819878i	$0.305959\pi$
$522$	−16351.0	−1.37100
$523$	7770.59	0.649683	0.324841	−	0.945769i	$-0.394689\pi$
$523$	7770.59	0.649683	0.324841	+	0.945769i	$0.394689\pi$
$524$	15509.2	1.29298
$525$	0	0
$526$	27620.0	2.28952
$527$	1527.59	0.126268
$528$	3721.14	0.306708
$529$	−5547.46	−0.455943
$530$	0	0
$531$	−3459.99	−0.282770
$532$	−8977.05	−0.731587
$533$	20539.6	1.66917
$534$	−12353.5	−1.00110
$535$	0	0
$536$	−20630.1	−1.66247
$537$	−17193.8	−1.38169
$538$	−3894.44	−0.312084
$539$	20194.3	1.61378
$540$	0	0
$541$	12848.5	1.02107	0.510536	−	0.859857i	$-0.329447\pi$
$541$	12848.5	1.02107	0.510536	+	0.859857i	$0.329447\pi$
$542$	25750.4	2.04072
$543$	−30918.5	−2.44354
$544$	9255.73	0.729478
$545$	0	0
$546$	−10169.2	−0.797070
$547$	2607.39	0.203810	0.101905	−	0.994794i	$-0.467506\pi$
$547$	2607.39	0.203810	0.101905	+	0.994794i	$0.467506\pi$
$548$	−17061.8	−1.33001
$549$	14980.2	1.16456
$550$	0	0
$551$	−24164.6	−1.86832
$552$	11054.3	0.852360
$553$	6017.65	0.462743
$554$	2512.24	0.192662
$555$	0	0
$556$	10488.3	0.800005
$557$	8799.57	0.669389	0.334695	−	0.942327i	$-0.391367\pi$
$557$	8799.57	0.669389	0.334695	+	0.942327i	$0.391367\pi$
$558$	−2737.38	−0.207674
$559$	−16846.5	−1.27465
$560$	0	0
$561$	−20447.0	−1.53881
$562$	5942.03	0.445995
$563$	11831.0	0.885647	0.442823	−	0.896609i	$-0.353977\pi$
$563$	11831.0	0.885647	0.442823	+	0.896609i	$0.353977\pi$
$564$	38030.4	2.83931
$565$	0	0
$566$	−16375.6	−1.21611
$567$	−5205.46	−0.385553
$568$	8717.74	0.643993
$569$	−1354.82	−0.0998193	−0.0499096	−	0.998754i	$-0.515893\pi$
$569$	−1354.82	−0.0998193	−0.0499096	+	0.998754i	$0.515893\pi$
$570$	0	0
$571$	15591.7	1.14272	0.571358	−	0.820701i	$-0.306417\pi$
$571$	15591.7	1.14272	0.571358	+	0.820701i	$0.306417\pi$
$572$	−46392.8	−3.39123
$573$	7520.87	0.548323
$574$	−9576.75	−0.696386
$575$	0	0
$576$	−15344.8	−1.11001
$577$	16372.8	1.18130	0.590649	−	0.806928i	$-0.298872\pi$
$577$	16372.8	1.18130	0.590649	+	0.806928i	$0.298872\pi$
$578$	12490.5	0.898853
$579$	17107.6	1.22792
$580$	0	0
$581$	−486.836	−0.0347631
$582$	7945.36	0.565886
$583$	47595.9	3.38117
$584$	−10403.4	−0.737152
$585$	0	0
$586$	−2517.91	−0.177498
$587$	25181.6	1.77063	0.885313	−	0.464996i	$-0.153944\pi$
$587$	25181.6	1.77063	0.885313	+	0.464996i	$0.153944\pi$
$588$	−25899.8	−1.81648
$589$	−4045.48	−0.283007
$590$	0	0
$591$	−33552.1	−2.33528
$592$	1192.90	0.0828173
$593$	−6008.36	−0.416077	−0.208039	−	0.978121i	$-0.566708\pi$
$593$	−6008.36	−0.416077	−0.208039	+	0.978121i	$0.566708\pi$
$594$	−17194.5	−1.18771
$595$	0	0
$596$	36873.4	2.53422
$597$	−1524.85	−0.104536
$598$	20940.2	1.43195
$599$	1039.51	0.0709069	0.0354535	−	0.999371i	$-0.488712\pi$
$599$	1039.51	0.0709069	0.0354535	+	0.999371i	$0.488712\pi$
$600$	0	0
$601$	−5984.93	−0.406207	−0.203103	−	0.979157i	$-0.565103\pi$
$601$	−5984.93	−0.406207	−0.203103	+	0.979157i	$0.565103\pi$
$602$	7854.83	0.531792
$603$	18795.5	1.26934
$604$	23078.1	1.55469
$605$	0	0
$606$	36780.4	2.46551
$607$	3563.84	0.238306	0.119153	−	0.992876i	$-0.461982\pi$
$607$	3563.84	0.238306	0.119153	+	0.992876i	$0.461982\pi$
$608$	−24511.6	−1.63500
$609$	7772.48	0.517170
$610$	0	0
$611$	25783.9	1.70721
$612$	10620.0	0.701453
$613$	−16062.7	−1.05835	−0.529174	−	0.848514i	$-0.677498\pi$
$613$	−16062.7	−1.05835	−0.529174	+	0.848514i	$0.677498\pi$
$614$	10249.8	0.673695
$615$	0	0
$616$	7741.87	0.506378
$617$	9083.93	0.592715	0.296357	−	0.955077i	$-0.404228\pi$
$617$	9083.93	0.592715	0.296357	+	0.955077i	$0.404228\pi$
$618$	−10488.5	−0.682701
$619$	−4218.77	−0.273936	−0.136968	−	0.990575i	$-0.543736\pi$
$619$	−4218.77	−0.273936	−0.136968	+	0.990575i	$0.543736\pi$
$620$	0	0
$621$	4726.31	0.305411
$622$	309.663	0.0199620
$623$	2378.13	0.152934
$624$	−3235.64	−0.207579
$625$	0	0
$626$	5240.24	0.334572
$627$	54149.0	3.44897
$628$	40400.7	2.56714
$629$	−6554.75	−0.415509
$630$	0	0
$631$	11752.6	0.741464	0.370732	−	0.928740i	$-0.379107\pi$
$631$	11752.6	0.741464	0.370732	+	0.928740i	$0.379107\pi$
$632$	−20693.1	−1.30242
$633$	35635.5	2.23757
$634$	−22652.4	−1.41899
$635$	0	0
$636$	−61043.2	−3.80585
$637$	−17559.5	−1.09220
$638$	58226.9	3.61320
$639$	−7942.52	−0.491708
$640$	0	0
$641$	−7409.24	−0.456548	−0.228274	−	0.973597i	$-0.573308\pi$
$641$	−7409.24	−0.456548	−0.228274	+	0.973597i	$0.573308\pi$
$642$	3463.98	0.212948
$643$	−17249.8	−1.05796	−0.528978	−	0.848635i	$-0.677425\pi$
$643$	−17249.8	−1.05796	−0.528978	+	0.848635i	$0.677425\pi$
$644$	−5945.78	−0.363815
$645$	0	0
$646$	25772.7	1.56968
$647$	6716.44	0.408115	0.204057	−	0.978959i	$-0.434587\pi$
$647$	6716.44	0.408115	0.204057	+	0.978959i	$0.434587\pi$
$648$	17900.2	1.08516
$649$	12321.3	0.745227
$650$	0	0
$651$	1301.22	0.0783392
$652$	48790.3	2.93064
$653$	−24217.5	−1.45130	−0.725652	−	0.688061i	$-0.758462\pi$
$653$	−24217.5	−1.45130	−0.725652	+	0.688061i	$0.758462\pi$
$654$	−35741.9	−2.13703
$655$	0	0
$656$	−3047.14	−0.181358
$657$	9478.31	0.562837
$658$	−12022.0	−0.712256
$659$	2281.13	0.134841	0.0674205	−	0.997725i	$-0.478523\pi$
$659$	2281.13	0.134841	0.0674205	+	0.997725i	$0.478523\pi$
$660$	0	0
$661$	18590.7	1.09394	0.546969	−	0.837153i	$-0.315782\pi$
$661$	18590.7	1.09394	0.546969	+	0.837153i	$0.315782\pi$
$662$	−29728.9	−1.74539
$663$	17779.2	1.04146
$664$	1674.10	0.0978429
$665$	0	0
$666$	11745.8	0.683394
$667$	−16005.0	−0.929107
$668$	−12408.3	−0.718698
$669$	−40846.7	−2.36057
$670$	0	0
$671$	−53345.7	−3.06913
$672$	7884.12	0.452584
$673$	−12090.5	−0.692500	−0.346250	−	0.938142i	$-0.612545\pi$
$673$	−12090.5	−0.692500	−0.346250	+	0.938142i	$0.612545\pi$
$674$	−46475.4	−2.65603
$675$	0	0
$676$	12967.0	0.737766
$677$	−4092.55	−0.232333	−0.116166	−	0.993230i	$-0.537061\pi$
$677$	−4092.55	−0.232333	−0.116166	+	0.993230i	$0.537061\pi$
$678$	−51937.4	−2.94195
$679$	−1529.53	−0.0864478
$680$	0	0
$681$	−33991.4	−1.91271
$682$	9747.97	0.547316
$683$	6612.31	0.370443	0.185222	−	0.982697i	$-0.440700\pi$
$683$	6612.31	0.370443	0.185222	+	0.982697i	$0.440700\pi$
$684$	−28124.7	−1.57218
$685$	0	0
$686$	17287.3	0.962146
$687$	1788.13	0.0993032
$688$	2499.26	0.138493
$689$	−41386.0	−2.28836
$690$	0	0
$691$	24193.9	1.33195	0.665975	−	0.745974i	$-0.268016\pi$
$691$	24193.9	1.33195	0.665975	+	0.745974i	$0.268016\pi$
$692$	2211.77	0.121501
$693$	−7053.43	−0.386635
$694$	−56650.2	−3.09858
$695$	0	0
$696$	−26727.5	−1.45561
$697$	16743.5	0.909906
$698$	−611.332	−0.0331508
$699$	−14460.6	−0.782476
$700$	0	0
$701$	−10638.9	−0.573220	−0.286610	−	0.958047i	$-0.592528\pi$
$701$	−10638.9	−0.573220	−0.286610	+	0.958047i	$0.592528\pi$
$702$	14951.1	0.803839
$703$	17358.7	0.931290
$704$	54643.9	2.92538
$705$	0	0
$706$	26029.2	1.38757
$707$	−7080.46	−0.376645
$708$	−15802.4	−0.838828
$709$	28206.6	1.49410	0.747052	−	0.664765i	$-0.231468\pi$
$709$	28206.6	1.49410	0.747052	+	0.664765i	$0.231468\pi$
$710$	0	0
$711$	18853.0	0.994435
$712$	−8177.76	−0.430442
$713$	−2679.45	−0.140738
$714$	−8289.72	−0.434503
$715$	0	0
$716$	−31801.6	−1.65989
$717$	−11545.2	−0.601345
$718$	3709.85	0.192828
$719$	−31730.6	−1.64583	−0.822914	−	0.568166i	$-0.807653\pi$
$719$	−31730.6	−1.64583	−0.822914	+	0.568166i	$0.807653\pi$
$720$	0	0
$721$	2019.10	0.104293
$722$	−37228.3	−1.91897
$723$	37159.5	1.91145
$724$	−57186.7	−2.93553
$725$	0	0
$726$	−89923.1	−4.59691
$727$	−20767.0	−1.05943	−0.529716	−	0.848175i	$-0.677701\pi$
$727$	−20767.0	−1.05943	−0.529716	+	0.848175i	$0.677701\pi$
$728$	−6731.78	−0.342715
$729$	−5743.03	−0.291776
$730$	0	0
$731$	−13733.0	−0.694846
$732$	68417.4	3.45462
$733$	17354.9	0.874511	0.437256	−	0.899337i	$-0.355951\pi$
$733$	17354.9	0.874511	0.437256	+	0.899337i	$0.355951\pi$
$734$	−9660.56	−0.485801
$735$	0	0
$736$	−16234.8	−0.813076
$737$	−66932.2	−3.34529
$738$	−30003.5	−1.49654
$739$	−30293.5	−1.50793	−0.753967	−	0.656912i	$-0.771862\pi$
$739$	−30293.5	−1.50793	−0.753967	+	0.656912i	$0.771862\pi$
$740$	0	0
$741$	−47084.1	−2.33425
$742$	19296.6	0.954717
$743$	−12672.7	−0.625731	−0.312865	−	0.949797i	$-0.601289\pi$
$743$	−12672.7	−0.625731	−0.312865	+	0.949797i	$0.601289\pi$
$744$	−4474.55	−0.220491
$745$	0	0
$746$	−12488.3	−0.612906
$747$	−1525.23	−0.0747060
$748$	−37818.6	−1.84864
$749$	−666.839	−0.0325311
$750$	0	0
$751$	−11849.2	−0.575745	−0.287872	−	0.957669i	$-0.592948\pi$
$751$	−11849.2	−0.575745	−0.287872	+	0.957669i	$0.592948\pi$
$752$	−3825.16	−0.185491
$753$	−26245.7	−1.27018
$754$	−50629.9	−2.44540
$755$	0	0
$756$	−4245.25	−0.204231
$757$	10176.3	0.488591	0.244295	−	0.969701i	$-0.421443\pi$
$757$	10176.3	0.488591	0.244295	+	0.969701i	$0.421443\pi$
$758$	−37667.8	−1.80495
$759$	35864.6	1.71516
$760$	0	0
$761$	−9435.81	−0.449472	−0.224736	−	0.974420i	$-0.572152\pi$
$761$	−9435.81	−0.449472	−0.224736	+	0.974420i	$0.572152\pi$
$762$	−60156.8	−2.85991
$763$	6880.54	0.326464
$764$	13910.6	0.658725
$765$	0	0
$766$	−20126.1	−0.949331
$767$	−10713.7	−0.504366
$768$	−21443.6	−1.00752
$769$	26037.0	1.22096	0.610480	−	0.792031i	$-0.290976\pi$
$769$	26037.0	1.22096	0.610480	+	0.792031i	$0.290976\pi$
$770$	0	0
$771$	33878.4	1.58249
$772$	31642.1	1.47516
$773$	603.106	0.0280624	0.0140312	−	0.999902i	$-0.495534\pi$
$773$	603.106	0.0280624	0.0140312	+	0.999902i	$0.495534\pi$
$774$	24608.8	1.14282
$775$	0	0
$776$	5259.66	0.243313
$777$	−5583.40	−0.257791
$778$	2402.15	0.110696
$779$	−44341.2	−2.03939
$780$	0	0
$781$	28283.8	1.29587
$782$	17070.1	0.780593
$783$	−11427.4	−0.521562
$784$	2605.04	0.118670
$785$	0	0
$786$	−37927.8	−1.72117
$787$	−36411.6	−1.64922	−0.824609	−	0.565704i	$-0.808605\pi$
$787$	−36411.6	−1.64922	−0.824609	+	0.565704i	$0.808605\pi$
$788$	−62057.7	−2.80547
$789$	−41133.3	−1.85600
$790$	0	0
$791$	9998.28	0.449429
$792$	24254.9	1.08821
$793$	46385.6	2.07717
$794$	−29217.9	−1.30592
$795$	0	0
$796$	−2820.35	−0.125584
$797$	17099.3	0.759962	0.379981	−	0.924994i	$-0.375931\pi$
$797$	17099.3	0.759962	0.379981	+	0.924994i	$0.375931\pi$
$798$	21953.4	0.973862
$799$	21018.5	0.930641
$800$	0	0
$801$	7450.56	0.328655
$802$	−31614.2	−1.39194
$803$	−33752.9	−1.48333
$804$	85842.5	3.76546
$805$	0	0
$806$	−8476.14	−0.370421
$807$	5799.82	0.252991
$808$	24347.9	1.06009
$809$	26965.2	1.17187	0.585936	−	0.810357i	$-0.300727\pi$
$809$	26965.2	1.17187	0.585936	+	0.810357i	$0.300727\pi$
$810$	0	0
$811$	−5732.60	−0.248211	−0.124105	−	0.992269i	$-0.539606\pi$
$811$	−5732.60	−0.248211	−0.124105	+	0.992269i	$0.539606\pi$
$812$	14375.9	0.621301
$813$	−38348.9	−1.65431
$814$	−41827.6	−1.80105
$815$	0	0
$816$	−2637.64	−0.113156
$817$	36368.6	1.55737
$818$	8972.05	0.383497
$819$	6133.16	0.261673
$820$	0	0
$821$	32257.0	1.37123	0.685613	−	0.727966i	$-0.259534\pi$
$821$	32257.0	1.37123	0.685613	+	0.727966i	$0.259534\pi$
$822$	41724.8	1.77046
$823$	22691.4	0.961085	0.480543	−	0.876971i	$-0.340440\pi$
$823$	22691.4	0.961085	0.480543	+	0.876971i	$0.340440\pi$
$824$	−6943.16	−0.293540
$825$	0	0
$826$	4995.35	0.210424
$827$	26040.8	1.09495	0.547476	−	0.836821i	$-0.315589\pi$
$827$	26040.8	1.09495	0.547476	+	0.836821i	$0.315589\pi$
$828$	−18627.9	−0.781839
$829$	−20949.9	−0.877706	−0.438853	−	0.898559i	$-0.644615\pi$
$829$	−20949.9	−0.877706	−0.438853	+	0.898559i	$0.644615\pi$
$830$	0	0
$831$	−3741.37	−0.156181
$832$	−47514.4	−1.97989
$833$	−14314.2	−0.595387
$834$	−25649.2	−1.06494
$835$	0	0
$836$	100154.	4.14341
$837$	−1913.11	−0.0790045
$838$	−27557.5	−1.13599
$839$	−19611.9	−0.807005	−0.403502	−	0.914979i	$-0.632207\pi$
$839$	−19611.9	−0.807005	−0.403502	+	0.914979i	$0.632207\pi$
$840$	0	0
$841$	14308.4	0.586673
$842$	−16257.0	−0.665384
$843$	−8849.22	−0.361546
$844$	65911.2	2.68810
$845$	0	0
$846$	−37664.2	−1.53064
$847$	17310.8	0.702249
$848$	6139.81	0.248635
$849$	24387.5	0.985840
$850$	0	0
$851$	11497.2	0.463126
$852$	−36274.9	−1.45863
$853$	3064.53	0.123010	0.0615050	−	0.998107i	$-0.480410\pi$
$853$	3064.53	0.123010	0.0615050	+	0.998107i	$0.480410\pi$
$854$	−21627.7	−0.866609
$855$	0	0
$856$	2293.08	0.0915608
$857$	2117.85	0.0844160	0.0422080	−	0.999109i	$-0.486561\pi$
$857$	2117.85	0.0844160	0.0422080	+	0.999109i	$0.486561\pi$
$858$	113454.	4.51428
$859$	−8459.87	−0.336027	−0.168013	−	0.985785i	$-0.553735\pi$
$859$	−8459.87	−0.336027	−0.168013	+	0.985785i	$0.553735\pi$
$860$	0	0
$861$	14262.3	0.564525
$862$	−30630.1	−1.21028
$863$	24613.9	0.970876	0.485438	−	0.874271i	$-0.338660\pi$
$863$	24613.9	0.970876	0.485438	+	0.874271i	$0.338660\pi$
$864$	−11591.6	−0.456427
$865$	0	0
$866$	34821.4	1.36637
$867$	−18601.6	−0.728655
$868$	2406.73	0.0941125
$869$	−67136.8	−2.62078
$870$	0	0
$871$	58199.4	2.26408
$872$	−23660.4	−0.918855
$873$	−4791.95	−0.185777
$874$	−45206.0	−1.74956
$875$	0	0
$876$	43289.1	1.66964
$877$	−15233.0	−0.586523	−0.293262	−	0.956032i	$-0.594741\pi$
$877$	−15233.0	−0.586523	−0.293262	+	0.956032i	$0.594741\pi$
$878$	−35292.8	−1.35658
$879$	3749.82	0.143889
$880$	0	0
$881$	3208.92	0.122714	0.0613571	−	0.998116i	$-0.480457\pi$
$881$	3208.92	0.122714	0.0613571	+	0.998116i	$0.480457\pi$
$882$	25650.3	0.979242
$883$	64.7286	0.00246692	0.00123346	−	0.999999i	$-0.499607\pi$
$883$	64.7286	0.00246692	0.00123346	+	0.999999i	$0.499607\pi$
$884$	32884.4	1.25115
$885$	0	0
$886$	49808.1	1.88864
$887$	7670.60	0.290365	0.145182	−	0.989405i	$-0.453623\pi$
$887$	7670.60	0.290365	0.145182	+	0.989405i	$0.453623\pi$
$888$	19199.9	0.725569
$889$	11580.6	0.436895
$890$	0	0
$891$	58075.5	2.18361
$892$	−75549.8	−2.83587
$893$	−55662.7	−2.08587
$894$	−90174.0	−3.37346
$895$	0	0
$896$	12790.7	0.476907
$897$	−31185.3	−1.16081
$898$	−32446.1	−1.20572
$899$	6478.47	0.240344
$900$	0	0
$901$	−33737.1	−1.24744
$902$	106844.	3.94405
$903$	−11697.9	−0.431097
$904$	−34381.5	−1.26495
$905$	0	0
$906$	−56437.5	−2.06955
$907$	−49822.3	−1.82395	−0.911974	−	0.410248i	$-0.865442\pi$
$907$	−49822.3	−1.82395	−0.911974	+	0.410248i	$0.865442\pi$
$908$	−62870.4	−2.29783
$909$	−22182.7	−0.809412
$910$	0	0
$911$	−37443.8	−1.36176	−0.680882	−	0.732393i	$-0.738403\pi$
$911$	−37443.8	−1.36176	−0.680882	+	0.732393i	$0.738403\pi$
$912$	6985.16	0.253620
$913$	5431.46	0.196884
$914$	35592.2	1.28806
$915$	0	0
$916$	3307.31	0.119298
$917$	7301.34	0.262935
$918$	12187.9	0.438193
$919$	21124.9	0.758265	0.379132	−	0.925342i	$-0.376222\pi$
$919$	21124.9	0.758265	0.379132	+	0.925342i	$0.376222\pi$
$920$	0	0
$921$	−15264.6	−0.546130
$922$	−43576.5	−1.55652
$923$	−24593.6	−0.877040
$924$	−32214.2	−1.14694
$925$	0	0
$926$	−19215.7	−0.681928
$927$	6325.75	0.224126
$928$	39253.2	1.38852
$929$	44899.6	1.58569	0.792846	−	0.609423i	$-0.208599\pi$
$929$	44899.6	1.58569	0.792846	+	0.609423i	$0.208599\pi$
$930$	0	0
$931$	37907.8	1.33446
$932$	−26746.3	−0.940025
$933$	−461.169	−0.0161822
$934$	17754.8	0.622006
$935$	0	0
$936$	−21090.3	−0.736495
$937$	−18219.9	−0.635237	−0.317618	−	0.948219i	$-0.602883\pi$
$937$	−18219.9	−0.635237	−0.317618	+	0.948219i	$0.602883\pi$
$938$	−27136.0	−0.944586
$939$	−7804.08	−0.271221
$940$	0	0
$941$	11980.0	0.415022	0.207511	−	0.978233i	$-0.433464\pi$
$941$	11980.0	0.415022	0.207511	+	0.978233i	$0.433464\pi$
$942$	−98800.0	−3.41728
$943$	−29368.6	−1.01418
$944$	1589.43	0.0548003
$945$	0	0
$946$	−87633.6	−3.01185
$947$	−7091.81	−0.243350	−0.121675	−	0.992570i	$-0.538827\pi$
$947$	−7091.81	−0.243350	−0.121675	+	0.992570i	$0.538827\pi$
$948$	86105.0	2.94996
$949$	29349.1	1.00391
$950$	0	0
$951$	33735.2	1.15030
$952$	−5487.63	−0.186822
$953$	−47185.3	−1.60386	−0.801932	−	0.597415i	$-0.796194\pi$
$953$	−47185.3	−1.60386	−0.801932	+	0.597415i	$0.796194\pi$
$954$	60455.3	2.05169
$955$	0	0
$956$	−21354.0	−0.722424
$957$	−86714.8	−2.92904
$958$	62097.7	2.09424
$959$	−8032.29	−0.270465
$960$	0	0
$961$	−28706.4	−0.963594
$962$	36370.2	1.21894
$963$	−2089.17	−0.0699093
$964$	68730.0	2.29631
$965$	0	0
$966$	14540.4	0.484297
$967$	−3853.22	−0.128140	−0.0640700	−	0.997945i	$-0.520408\pi$
$967$	−3853.22	−0.128140	−0.0640700	+	0.997945i	$0.520408\pi$
$968$	−59527.2	−1.97653
$969$	−38382.1	−1.27246
$970$	0	0
$971$	−3656.74	−0.120855	−0.0604276	−	0.998173i	$-0.519246\pi$
$971$	−3656.74	−0.120855	−0.0604276	+	0.998173i	$0.519246\pi$
$972$	−54941.8	−1.81302
$973$	4937.63	0.162686
$974$	87.7318	0.00288615
$975$	0	0
$976$	−6881.52	−0.225689
$977$	−36862.1	−1.20709	−0.603543	−	0.797330i	$-0.706245\pi$
$977$	−36862.1	−1.20709	−0.603543	+	0.797330i	$0.706245\pi$
$978$	−119317.	−3.90115
$979$	−26531.9	−0.866154
$980$	0	0
$981$	21556.4	0.701573
$982$	−62949.4	−2.04562
$983$	7076.78	0.229618	0.114809	−	0.993388i	$-0.463374\pi$
$983$	7076.78	0.229618	0.114809	+	0.993388i	$0.463374\pi$
$984$	−49044.2	−1.58889
$985$	0	0
$986$	−41272.6	−1.33305
$987$	17903.8	0.577390
$988$	−87086.5	−2.80424
$989$	24088.1	0.774475
$990$	0	0
$991$	−47739.0	−1.53025	−0.765125	−	0.643881i	$-0.777323\pi$
$991$	−47739.0	−1.53025	−0.765125	+	0.643881i	$0.777323\pi$
$992$	6571.52	0.210329
$993$	44274.0	1.41490
$994$	11467.0	0.365906
$995$	0	0
$996$	−6966.00	−0.221613
$997$	−5961.03	−0.189356	−0.0946779	−	0.995508i	$-0.530182\pi$
$997$	−5961.03	−0.189356	−0.0946779	+	0.995508i	$0.530182\pi$
$998$	63789.3	2.02326
$999$	8208.96	0.259980

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2075.4.a.l.1.7	yes	47
5.4	even	2		2075.4.a.k.1.41	✓	47

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2075.4.a.k.1.41	✓	47	5.4	even	2
2075.4.a.l.1.7	yes	47	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:	\( N \)	\(=\)	\( 2075 = 5^{2} \cdot 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2075.a (trivial)

\(f(q)\)	\(=\)	\(q-4.52319 q^{2} +6.73619 q^{3} +12.4592 q^{4} -30.4691 q^{6} +5.86549 q^{7} -20.1699 q^{8} +18.3763 q^{9} -65.4392 q^{11} +83.9277 q^{12} +56.9013 q^{13} -26.5307 q^{14} -8.44158 q^{16} +46.3849 q^{17} -83.1194 q^{18} -122.840 q^{19} +39.5111 q^{21} +295.994 q^{22} -81.3605 q^{23} -135.868 q^{24} -257.375 q^{26} -58.0909 q^{27} +73.0794 q^{28} +196.716 q^{29} +32.9330 q^{31} +199.542 q^{32} -440.811 q^{33} -209.808 q^{34} +228.954 q^{36} -141.312 q^{37} +555.626 q^{38} +383.298 q^{39} +360.968 q^{41} -178.716 q^{42} -296.066 q^{43} -815.322 q^{44} +368.009 q^{46} +453.133 q^{47} -56.8641 q^{48} -308.596 q^{49} +312.458 q^{51} +708.945 q^{52} -727.330 q^{53} +262.756 q^{54} -118.306 q^{56} -827.471 q^{57} -889.785 q^{58} -188.286 q^{59} +815.194 q^{61} -148.962 q^{62} +107.786 q^{63} -835.033 q^{64} +1993.87 q^{66} +1022.81 q^{67} +577.920 q^{68} -548.060 q^{69} -432.215 q^{71} -370.648 q^{72} +515.790 q^{73} +639.182 q^{74} -1530.48 q^{76} -383.833 q^{77} -1733.73 q^{78} +1025.94 q^{79} -887.472 q^{81} -1632.73 q^{82} -83.0000 q^{83} +492.277 q^{84} +1339.16 q^{86} +1325.12 q^{87} +1319.90 q^{88} +405.444 q^{89} +333.754 q^{91} -1013.69 q^{92} +221.843 q^{93} -2049.61 q^{94} +1344.15 q^{96} -260.768 q^{97} +1395.84 q^{98} -1202.53 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 47 q + q^{2} - 6 q^{3} + 229 q^{4} + 9 q^{6} - 6 q^{7} + 39 q^{8} + 531 q^{9} + 90 q^{11} + 6 q^{12} + 50 q^{13} + 115 q^{14} + 1149 q^{16} - 68 q^{17} + 8 q^{18} + 344 q^{19} + 625 q^{21} + 402 q^{22}+ \cdots + 13991 q^{99}+O(q^{100}) \) 47 * q + q^2 - 6 * q^3 + 229 * q^4 + 9 * q^6 - 6 * q^7 + 39 * q^8 + 531 * q^9 + 90 * q^11 + 6 * q^12 + 50 * q^13 + 115 * q^14 + 1149 * q^16 - 68 * q^17 + 8 * q^18 + 344 * q^19 + 625 * q^21 + 402 * q^22 - 469 * q^23 + 262 * q^24 + 1209 * q^26 + 12 * q^27 - 603 * q^28 + 480 * q^29 + 876 * q^31 + 1653 * q^32 - 757 * q^33 + 733 * q^34 + 4318 * q^36 + 634 * q^37 - 1190 * q^38 + 618 * q^39 + 1455 * q^41 + 2747 * q^42 - 652 * q^43 + 595 * q^44 + 1662 * q^46 + 446 * q^47 - 2145 * q^48 + 4099 * q^49 + 1681 * q^51 + 207 * q^52 - 126 * q^53 + 670 * q^54 + 3100 * q^56 + 650 * q^57 - 1082 * q^58 + 1144 * q^59 + 2762 * q^61 + 1916 * q^62 - 885 * q^63 + 6891 * q^64 + 2025 * q^66 - 100 * q^67 - 2052 * q^68 + 2556 * q^69 + 2546 * q^71 + 2940 * q^72 + 14 * q^73 + 1066 * q^74 + 4828 * q^76 - 210 * q^77 - 343 * q^78 + 1636 * q^79 + 8203 * q^81 + 480 * q^82 - 3901 * q^83 + 6209 * q^84 + 2706 * q^86 - 5479 * q^87 + 2178 * q^88 + 2558 * q^89 + 3342 * q^91 - 7501 * q^92 + 3353 * q^93 + 5769 * q^94 - 918 * q^96 - 3648 * q^97 + 8109 * q^98 + 13991 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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