LMFDB - Embedded newform 2075.4.a.k.1.40

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			1.40
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)

$f(q)$	$=$	\(q+4.47908 q^{2} -1.96805 q^{3} +12.0621 q^{4} -8.81505 q^{6} -27.9268 q^{7} +18.1946 q^{8} -23.1268 q^{9} +1.10633 q^{11} -23.7389 q^{12} -53.8543 q^{13} -125.086 q^{14} -15.0020 q^{16} +35.5100 q^{17} -103.587 q^{18} +150.926 q^{19} +54.9613 q^{21} +4.95532 q^{22} +35.5907 q^{23} -35.8079 q^{24} -241.218 q^{26} +98.6520 q^{27} -336.856 q^{28} -200.577 q^{29} +248.494 q^{31} -212.752 q^{32} -2.17730 q^{33} +159.052 q^{34} -278.958 q^{36} -349.834 q^{37} +676.008 q^{38} +105.988 q^{39} -69.5204 q^{41} +246.176 q^{42} +405.042 q^{43} +13.3446 q^{44} +159.414 q^{46} +307.844 q^{47} +29.5247 q^{48} +436.904 q^{49} -69.8855 q^{51} -649.598 q^{52} -19.7924 q^{53} +441.870 q^{54} -508.117 q^{56} -297.029 q^{57} -898.399 q^{58} +393.465 q^{59} +477.990 q^{61} +1113.03 q^{62} +645.856 q^{63} -832.917 q^{64} -9.75231 q^{66} -743.743 q^{67} +428.327 q^{68} -70.0444 q^{69} +527.495 q^{71} -420.783 q^{72} +171.209 q^{73} -1566.93 q^{74} +1820.49 q^{76} -30.8961 q^{77} +474.728 q^{78} +442.894 q^{79} +430.271 q^{81} -311.387 q^{82} +83.0000 q^{83} +662.950 q^{84} +1814.21 q^{86} +394.745 q^{87} +20.1292 q^{88} -673.015 q^{89} +1503.98 q^{91} +429.300 q^{92} -489.049 q^{93} +1378.86 q^{94} +418.707 q^{96} +1098.95 q^{97} +1956.93 q^{98} -25.5857 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.47908	1.58359	0.791796	−	0.610785i	$-0.209146\pi$
$2$	4.47908	1.58359	0.791796	+	0.610785i	$0.209146\pi$
$3$	−1.96805	−0.378751	−0.189376	−	0.981905i	$-0.560646\pi$
$3$	−1.96805	−0.378751	−0.189376	+	0.981905i	$0.560646\pi$
$4$	12.0621	1.50777
$5$	0	0
$5$	0	0
$6$	−8.81505	−0.599788
$7$	−27.9268	−1.50790	−0.753952	−	0.656930i	$-0.771855\pi$
$7$	−27.9268	−1.50790	−0.753952	+	0.656930i	$0.771855\pi$
$8$	18.1946	0.804096
$9$	−23.1268	−0.856547
$10$	0	0
$11$	1.10633	0.0303245	0.0151623	−	0.999885i	$-0.495174\pi$
$11$	1.10633	0.0303245	0.0151623	+	0.999885i	$0.495174\pi$
$12$	−23.7389	−0.571069
$13$	−53.8543	−1.14896	−0.574481	−	0.818518i	$-0.694796\pi$
$13$	−53.8543	−1.14896	−0.574481	+	0.818518i	$0.694796\pi$
$14$	−125.086	−2.38791
$15$	0	0
$16$	−15.0020	−0.234406
$17$	35.5100	0.506615	0.253307	−	0.967386i	$-0.418482\pi$
$17$	35.5100	0.506615	0.253307	+	0.967386i	$0.418482\pi$
$18$	−103.587	−1.35642
$19$	150.926	1.82235	0.911177	−	0.412014i	$-0.135175\pi$
$19$	150.926	1.82235	0.911177	+	0.412014i	$0.135175\pi$
$20$	0	0
$21$	54.9613	0.571121
$22$	4.95532	0.0480217
$23$	35.5907	0.322660	0.161330	−	0.986901i	$-0.448422\pi$
$23$	35.5907	0.322660	0.161330	+	0.986901i	$0.448422\pi$
$24$	−35.8079	−0.304552
$25$	0	0
$26$	−241.218	−1.81949
$27$	98.6520	0.703170
$28$	−336.856	−2.27357
$29$	−200.577	−1.28435	−0.642176	−	0.766557i	$-0.721968\pi$
$29$	−200.577	−1.28435	−0.642176	+	0.766557i	$0.721968\pi$
$30$	0	0
$31$	248.494	1.43971	0.719853	−	0.694126i	$-0.244209\pi$
$31$	248.494	1.43971	0.719853	+	0.694126i	$0.244209\pi$
$32$	−212.752	−1.17530
$33$	−2.17730	−0.0114855
$34$	159.052	0.802272
$35$	0	0
$36$	−278.958	−1.29147
$37$	−349.834	−1.55439	−0.777194	−	0.629261i	$-0.783358\pi$
$37$	−349.834	−1.55439	−0.777194	+	0.629261i	$0.783358\pi$
$38$	676.008	2.88587
$39$	105.988	0.435171
$40$	0	0
$41$	−69.5204	−0.264811	−0.132406	−	0.991196i	$-0.542270\pi$
$41$	−69.5204	−0.264811	−0.132406	+	0.991196i	$0.542270\pi$
$42$	246.176	0.904423
$43$	405.042	1.43647	0.718236	−	0.695799i	$-0.244950\pi$
$43$	405.042	1.43647	0.718236	+	0.695799i	$0.244950\pi$
$44$	13.3446	0.0457223
$45$	0	0
$46$	159.414	0.510962
$47$	307.844	0.955399	0.477699	−	0.878523i	$-0.341471\pi$
$47$	307.844	0.955399	0.477699	+	0.878523i	$0.341471\pi$
$48$	29.5247	0.0887817
$49$	436.904	1.27377
$50$	0	0
$51$	−69.8855	−0.191881
$52$	−649.598	−1.73237
$53$	−19.7924	−0.0512961	−0.0256480	−	0.999671i	$-0.508165\pi$
$53$	−19.7924	−0.0512961	−0.0256480	+	0.999671i	$0.508165\pi$
$54$	441.870	1.11353
$55$	0	0
$56$	−508.117	−1.21250
$57$	−297.029	−0.690219
$58$	−898.399	−2.03389
$59$	393.465	0.868216	0.434108	−	0.900861i	$-0.357064\pi$
$59$	393.465	0.868216	0.434108	+	0.900861i	$0.357064\pi$
$60$	0	0
$61$	477.990	1.00329	0.501643	−	0.865075i	$-0.332729\pi$
$61$	477.990	1.00329	0.501643	+	0.865075i	$0.332729\pi$
$62$	1113.03	2.27991
$63$	645.856	1.29159
$64$	−832.917	−1.62679
$65$	0	0
$66$	−9.75231	−0.0181883
$67$	−743.743	−1.35616	−0.678079	−	0.734989i	$-0.737187\pi$
$67$	−743.743	−1.35616	−0.678079	+	0.734989i	$0.737187\pi$
$68$	428.327	0.763857
$69$	−70.0444	−0.122208
$70$	0	0
$71$	527.495	0.881721	0.440860	−	0.897576i	$-0.354673\pi$
$71$	527.495	0.881721	0.440860	+	0.897576i	$0.354673\pi$
$72$	−420.783	−0.688746
$73$	171.209	0.274500	0.137250	−	0.990536i	$-0.456174\pi$
$73$	171.209	0.274500	0.137250	+	0.990536i	$0.456174\pi$
$74$	−1566.93	−2.46152
$75$	0	0
$76$	1820.49	2.74769
$77$	−30.8961	−0.0457265
$78$	474.728	0.689133
$79$	442.894	0.630753	0.315377	−	0.948967i	$-0.397869\pi$
$79$	442.894	0.630753	0.315377	+	0.948967i	$0.397869\pi$
$80$	0	0
$81$	430.271	0.590221
$82$	−311.387	−0.419354
$83$	83.0000	0.109764
$83$	83.0000	0.109764
$84$	662.950	0.861117
$85$	0	0
$86$	1814.21	2.27479
$87$	394.745	0.486450
$88$	20.1292	0.0243838
$89$	−673.015	−0.801567	−0.400784	−	0.916173i	$-0.631262\pi$
$89$	−673.015	−0.801567	−0.400784	+	0.916173i	$0.631262\pi$
$90$	0	0
$91$	1503.98	1.73252
$92$	429.300	0.486496
$93$	−489.049	−0.545291
$94$	1378.86	1.51296
$95$	0	0
$96$	418.707	0.445146
$97$	1098.95	1.15032	0.575162	−	0.818039i	$-0.304939\pi$
$97$	1098.95	1.15032	0.575162	+	0.818039i	$0.304939\pi$
$98$	1956.93	2.01714
$99$	−25.5857	−0.0259744
$100$	0	0
$101$	1447.30	1.42586	0.712930	−	0.701235i	$-0.247368\pi$
$101$	1447.30	1.42586	0.712930	+	0.701235i	$0.247368\pi$
$102$	−313.023	−0.303861
$103$	−228.980	−0.219049	−0.109525	−	0.993984i	$-0.534933\pi$
$103$	−228.980	−0.219049	−0.109525	+	0.993984i	$0.534933\pi$
$104$	−979.858	−0.923875
$105$	0	0
$106$	−88.6516	−0.0812321
$107$	1002.78	0.906000	0.453000	−	0.891511i	$-0.350354\pi$
$107$	1002.78	0.906000	0.453000	+	0.891511i	$0.350354\pi$
$108$	1189.95	1.06022
$109$	1153.38	1.01352	0.506762	−	0.862086i	$-0.330842\pi$
$109$	1153.38	1.01352	0.506762	+	0.862086i	$0.330842\pi$
$110$	0	0
$111$	688.491	0.588727
$112$	418.958	0.353462
$113$	−992.531	−0.826278	−0.413139	−	0.910668i	$-0.635568\pi$
$113$	−992.531	−0.826278	−0.413139	+	0.910668i	$0.635568\pi$
$114$	−1330.42	−1.09303
$115$	0	0
$116$	−2419.38	−1.93650
$117$	1245.48	0.984140
$118$	1762.36	1.37490
$119$	−991.681	−0.763926
$120$	0	0
$121$	−1329.78	−0.999080
$122$	2140.96	1.58880
$123$	136.820	0.100298
$124$	2997.37	2.17074
$125$	0	0
$126$	2892.84	2.04535
$127$	1474.40	1.03017	0.515087	−	0.857138i	$-0.327759\pi$
$127$	1474.40	1.03017	0.515087	+	0.857138i	$0.327759\pi$
$128$	−2028.68	−1.40087
$129$	−797.143	−0.544066
$130$	0	0
$131$	2398.88	1.59993	0.799967	−	0.600045i	$-0.204851\pi$
$131$	2398.88	1.59993	0.799967	+	0.600045i	$0.204851\pi$
$132$	−26.2629	−0.0173174
$133$	−4214.87	−2.74794
$134$	−3331.28	−2.14760
$135$	0	0
$136$	646.091	0.407367
$137$	−3130.43	−1.95220	−0.976099	−	0.217327i	$-0.930266\pi$
$137$	−3130.43	−1.95220	−0.976099	+	0.217327i	$0.930266\pi$
$138$	−313.734	−0.193528
$139$	−379.847	−0.231785	−0.115893	−	0.993262i	$-0.536973\pi$
$139$	−379.847	−0.231785	−0.115893	+	0.993262i	$0.536973\pi$
$140$	0	0
$141$	−605.853	−0.361859
$142$	2362.69	1.39629
$143$	−59.5804	−0.0348417
$144$	346.948	0.200780
$145$	0	0
$146$	766.858	0.434696
$147$	−859.850	−0.482444
$148$	−4219.75	−2.34366
$149$	−2901.73	−1.59543	−0.797715	−	0.603035i	$-0.793958\pi$
$149$	−2901.73	−1.59543	−0.797715	+	0.603035i	$0.793958\pi$
$150$	0	0
$151$	1624.86	0.875688	0.437844	−	0.899051i	$-0.355742\pi$
$151$	1624.86	0.875688	0.437844	+	0.899051i	$0.355742\pi$
$152$	2746.03	1.46535
$153$	−821.233	−0.433940
$154$	−138.386	−0.0724121
$155$	0	0
$156$	1278.44	0.656136
$157$	3252.16	1.65319	0.826594	−	0.562799i	$-0.190276\pi$
$157$	3252.16	1.65319	0.826594	+	0.562799i	$0.190276\pi$
$158$	1983.76	0.998857
$159$	38.9524	0.0194285
$160$	0	0
$161$	−993.934	−0.486540
$162$	1927.22	0.934670
$163$	−516.314	−0.248103	−0.124052	−	0.992276i	$-0.539589\pi$
$163$	−516.314	−0.248103	−0.124052	+	0.992276i	$0.539589\pi$
$164$	−838.565	−0.399274
$165$	0	0
$166$	371.763	0.173822
$167$	−1451.00	−0.672345	−0.336173	−	0.941800i	$-0.609133\pi$
$167$	−1451.00	−0.672345	−0.336173	+	0.941800i	$0.609133\pi$
$168$	999.999	0.459236
$169$	703.288	0.320113
$170$	0	0
$171$	−3490.43	−1.56093
$172$	4885.67	2.16587
$173$	−1569.44	−0.689723	−0.344861	−	0.938654i	$-0.612074\pi$
$173$	−1569.44	−0.689723	−0.344861	+	0.938654i	$0.612074\pi$
$174$	1768.09	0.770339
$175$	0	0
$176$	−16.5971	−0.00710826
$177$	−774.358	−0.328838
$178$	−3014.49	−1.26936
$179$	−1767.40	−0.737998	−0.368999	−	0.929430i	$-0.620299\pi$
$179$	−1767.40	−0.737998	−0.368999	+	0.929430i	$0.620299\pi$
$180$	0	0
$181$	−2154.03	−0.884573	−0.442286	−	0.896874i	$-0.645833\pi$
$181$	−2154.03	−0.884573	−0.442286	+	0.896874i	$0.645833\pi$
$182$	6736.43	2.74361
$183$	−940.709	−0.379996
$184$	647.560	0.259450
$185$	0	0
$186$	−2190.49	−0.863519
$187$	39.2857	0.0153629
$188$	3713.26	1.44052
$189$	−2755.03	−1.06031
$190$	0	0
$191$	−337.348	−0.127799	−0.0638996	−	0.997956i	$-0.520354\pi$
$191$	−337.348	−0.127799	−0.0638996	+	0.997956i	$0.520354\pi$
$192$	1639.22	0.616149
$193$	4783.20	1.78395	0.891975	−	0.452086i	$-0.149320\pi$
$193$	4783.20	1.78395	0.891975	+	0.452086i	$0.149320\pi$
$194$	4922.28	1.82165
$195$	0	0
$196$	5270.00	1.92055
$197$	276.353	0.0999458	0.0499729	−	0.998751i	$-0.484087\pi$
$197$	276.353	0.0999458	0.0499729	+	0.998751i	$0.484087\pi$
$198$	−114.601	−0.0411329
$199$	3127.48	1.11408	0.557038	−	0.830487i	$-0.311938\pi$
$199$	3127.48	1.11408	0.557038	+	0.830487i	$0.311938\pi$
$200$	0	0
$201$	1463.72	0.513647
$202$	6482.57	2.25798
$203$	5601.46	1.93668
$204$	−842.969	−0.289312
$205$	0	0
$206$	−1025.62	−0.346885
$207$	−823.099	−0.276374
$208$	807.923	0.269324
$209$	166.973	0.0552620
$210$	0	0
$211$	1355.25	0.442175	0.221088	−	0.975254i	$-0.429039\pi$
$211$	1355.25	0.442175	0.221088	+	0.975254i	$0.429039\pi$
$212$	−238.738	−0.0773425
$213$	−1038.14	−0.333953
$214$	4491.51	1.43473
$215$	0	0
$216$	1794.93	0.565416
$217$	−6939.64	−2.17094
$218$	5166.09	1.60501
$219$	−336.948	−0.103967
$220$	0	0
$221$	−1912.37	−0.582081
$222$	3083.80	0.932304
$223$	399.899	0.120086	0.0600430	−	0.998196i	$-0.480876\pi$
$223$	399.899	0.120086	0.0600430	+	0.998196i	$0.480876\pi$
$224$	5941.48	1.77224
$225$	0	0
$226$	−4445.62	−1.30849
$227$	1781.32	0.520839	0.260420	−	0.965496i	$-0.416139\pi$
$227$	1781.32	0.520839	0.260420	+	0.965496i	$0.416139\pi$
$228$	−3582.81	−1.04069
$229$	−4648.46	−1.34139	−0.670696	−	0.741733i	$-0.734004\pi$
$229$	−4648.46	−1.34139	−0.670696	+	0.741733i	$0.734004\pi$
$230$	0	0
$231$	60.8051	0.0173190
$232$	−3649.42	−1.03274
$233$	−4757.51	−1.33766	−0.668830	−	0.743415i	$-0.733205\pi$
$233$	−4757.51	−1.33766	−0.668830	+	0.743415i	$0.733205\pi$
$234$	5578.59	1.55848
$235$	0	0
$236$	4746.02	1.30907
$237$	−871.638	−0.238899
$238$	−4441.82	−1.20975
$239$	−3553.60	−0.961771	−0.480886	−	0.876783i	$-0.659685\pi$
$239$	−3553.60	−0.961771	−0.480886	+	0.876783i	$0.659685\pi$
$240$	0	0
$241$	−1989.57	−0.531783	−0.265891	−	0.964003i	$-0.585666\pi$
$241$	−1989.57	−0.531783	−0.265891	+	0.964003i	$0.585666\pi$
$242$	−5956.17	−1.58214
$243$	−3510.40	−0.926717
$244$	5765.58	1.51272
$245$	0	0
$246$	612.826	0.158831
$247$	−8128.00	−2.09382
$248$	4521.26	1.15766
$249$	−163.348	−0.0415734
$250$	0	0
$251$	3266.88	0.821528	0.410764	−	0.911742i	$-0.365262\pi$
$251$	3266.88	0.821528	0.410764	+	0.911742i	$0.365262\pi$
$252$	7790.40	1.94742
$253$	39.3750	0.00978451
$254$	6603.97	1.63138
$255$	0	0
$256$	−2423.29	−0.591624
$257$	5789.14	1.40512	0.702562	−	0.711623i	$-0.252040\pi$
$257$	5789.14	1.40512	0.702562	+	0.711623i	$0.252040\pi$
$258$	−3570.46	−0.861579
$259$	9769.74	2.34387
$260$	0	0
$261$	4638.70	1.10011
$262$	10744.8	2.53364
$263$	−1086.39	−0.254714	−0.127357	−	0.991857i	$-0.540649\pi$
$263$	−1086.39	−0.254714	−0.127357	+	0.991857i	$0.540649\pi$
$264$	−39.6152	−0.00923540
$265$	0	0
$266$	−18878.7	−4.35161
$267$	1324.53	0.303595
$268$	−8971.12	−2.04477
$269$	−6745.63	−1.52895	−0.764477	−	0.644651i	$-0.777003\pi$
$269$	−6745.63	−1.52895	−0.764477	+	0.644651i	$0.777003\pi$
$270$	0	0
$271$	8169.30	1.83118	0.915589	−	0.402115i	$-0.131725\pi$
$271$	8169.30	1.83118	0.915589	+	0.402115i	$0.131725\pi$
$272$	−532.722	−0.118754
$273$	−2959.90	−0.656196
$274$	−14021.5	−3.09149
$275$	0	0
$276$	−844.884	−0.184261
$277$	4871.18	1.05661	0.528305	−	0.849055i	$-0.322828\pi$
$277$	4871.18	1.05661	0.528305	+	0.849055i	$0.322828\pi$
$278$	−1701.36	−0.367054
$279$	−5746.87	−1.23318
$280$	0	0
$281$	7426.68	1.57665	0.788325	−	0.615260i	$-0.210949\pi$
$281$	7426.68	1.57665	0.788325	+	0.615260i	$0.210949\pi$
$282$	−2713.66	−0.573037
$283$	5125.86	1.07668	0.538340	−	0.842727i	$-0.319051\pi$
$283$	5125.86	1.07668	0.538340	+	0.842727i	$0.319051\pi$
$284$	6362.72	1.32943
$285$	0	0
$286$	−266.865	−0.0551751
$287$	1941.48	0.399310
$288$	4920.27	1.00670
$289$	−3652.04	−0.743341
$290$	0	0
$291$	−2162.79	−0.435687
$292$	2065.14	0.413882
$293$	4090.92	0.815679	0.407840	−	0.913054i	$-0.366282\pi$
$293$	4090.92	0.815679	0.407840	+	0.913054i	$0.366282\pi$
$294$	−3851.33	−0.763994
$295$	0	0
$296$	−6365.10	−1.24988
$297$	109.141	0.0213233
$298$	−12997.1	−2.52651
$299$	−1916.72	−0.370724
$300$	0	0
$301$	−11311.5	−2.16606
$302$	7277.86	1.38673
$303$	−2848.36	−0.540046
$304$	−2264.19	−0.427172
$305$	0	0
$306$	−3678.37	−0.687184
$307$	−4673.43	−0.868818	−0.434409	−	0.900716i	$-0.643043\pi$
$307$	−4673.43	−0.868818	−0.434409	+	0.900716i	$0.643043\pi$
$308$	−372.673	−0.0689448
$309$	450.645	0.0829653
$310$	0	0
$311$	−59.5176	−0.0108519	−0.00542594	−	0.999985i	$-0.501727\pi$
$311$	−59.5176	−0.0108519	−0.00542594	+	0.999985i	$0.501727\pi$
$312$	1928.41	0.349919
$313$	−6891.26	−1.24446	−0.622231	−	0.782833i	$-0.713774\pi$
$313$	−6891.26	−1.24446	−0.622231	+	0.782833i	$0.713774\pi$
$314$	14566.7	2.61798
$315$	0	0
$316$	5342.25	0.951029
$317$	−2115.68	−0.374853	−0.187426	−	0.982279i	$-0.560015\pi$
$317$	−2115.68	−0.374853	−0.187426	+	0.982279i	$0.560015\pi$
$318$	174.471	0.0307668
$319$	−221.903	−0.0389473
$320$	0	0
$321$	−1973.51	−0.343149
$322$	−4451.91	−0.770482
$323$	5359.38	0.923232
$324$	5189.99	0.889915
$325$	0	0
$326$	−2312.61	−0.392895
$327$	−2269.91	−0.383873
$328$	−1264.90	−0.212934
$329$	−8597.10	−1.44065
$330$	0	0
$331$	−432.484	−0.0718172	−0.0359086	−	0.999355i	$-0.511433\pi$
$331$	−432.484	−0.0718172	−0.0359086	+	0.999355i	$0.511433\pi$
$332$	1001.16	0.165499
$333$	8090.54	1.33141
$334$	−6499.14	−1.06472
$335$	0	0
$336$	−824.529	−0.133874
$337$	8042.09	1.29994	0.649971	−	0.759959i	$-0.274781\pi$
$337$	8042.09	1.29994	0.649971	+	0.759959i	$0.274781\pi$
$338$	3150.08	0.506929
$339$	1953.35	0.312954
$340$	0	0
$341$	274.916	0.0436584
$342$	−15633.9	−2.47188
$343$	−2622.45	−0.412825
$344$	7369.58	1.15506
$345$	0	0
$346$	−7029.63	−1.09224
$347$	−9375.09	−1.45038	−0.725189	−	0.688550i	$-0.758248\pi$
$347$	−9375.09	−1.45038	−0.725189	+	0.688550i	$0.758248\pi$
$348$	4761.47	0.733453
$349$	10570.2	1.62124	0.810619	−	0.585574i	$-0.199131\pi$
$349$	10570.2	1.62124	0.810619	+	0.585574i	$0.199131\pi$
$350$	0	0
$351$	−5312.84	−0.807915
$352$	−235.373	−0.0356404
$353$	323.155	0.0487246	0.0243623	−	0.999703i	$-0.492244\pi$
$353$	323.155	0.0487246	0.0243623	+	0.999703i	$0.492244\pi$
$354$	−3468.41	−0.520745
$355$	0	0
$356$	−8118.00	−1.20858
$357$	1951.68	0.289338
$358$	−7916.32	−1.16869
$359$	8509.95	1.25108	0.625540	−	0.780192i	$-0.284879\pi$
$359$	8509.95	1.25108	0.625540	+	0.780192i	$0.284879\pi$
$360$	0	0
$361$	15919.6	2.32098
$362$	−9648.06	−1.40080
$363$	2617.07	0.378403
$364$	18141.2	2.61224
$365$	0	0
$366$	−4213.51	−0.601759
$367$	−5850.12	−0.832081	−0.416041	−	0.909346i	$-0.636583\pi$
$367$	−5850.12	−0.832081	−0.416041	+	0.909346i	$0.636583\pi$
$368$	−533.933	−0.0756336
$369$	1607.78	0.226824
$370$	0	0
$371$	552.737	0.0773496
$372$	−5898.98	−0.822171
$373$	−2306.84	−0.320224	−0.160112	−	0.987099i	$-0.551186\pi$
$373$	−2306.84	−0.320224	−0.160112	+	0.987099i	$0.551186\pi$
$374$	175.964	0.0243285
$375$	0	0
$376$	5601.11	0.768232
$377$	10801.9	1.47567
$378$	−12340.0	−1.67910
$379$	−7465.12	−1.01176	−0.505881	−	0.862604i	$-0.668832\pi$
$379$	−7465.12	−1.01176	−0.505881	+	0.862604i	$0.668832\pi$
$380$	0	0
$381$	−2901.70	−0.390180
$382$	−1511.01	−0.202382
$383$	−5064.60	−0.675689	−0.337845	−	0.941202i	$-0.609698\pi$
$383$	−5064.60	−0.675689	−0.337845	+	0.941202i	$0.609698\pi$
$384$	3992.55	0.530583
$385$	0	0
$386$	21424.3	2.82505
$387$	−9367.32	−1.23041
$388$	13255.7	1.73442
$389$	14112.6	1.83942	0.919711	−	0.392595i	$-0.128423\pi$
$389$	14112.6	1.83942	0.919711	+	0.392595i	$0.128423\pi$
$390$	0	0
$391$	1263.83	0.163464
$392$	7949.31	1.02424
$393$	−4721.12	−0.605977
$394$	1237.81	0.158274
$395$	0	0
$396$	−308.619	−0.0391633
$397$	12519.1	1.58265	0.791327	−	0.611393i	$-0.209391\pi$
$397$	12519.1	1.58265	0.791327	+	0.611393i	$0.209391\pi$
$398$	14008.2	1.76424
$399$	8295.07	1.04078
$400$	0	0
$401$	5320.42	0.662566	0.331283	−	0.943531i	$-0.392519\pi$
$401$	5320.42	0.662566	0.331283	+	0.943531i	$0.392519\pi$
$402$	6556.13	0.813408
$403$	−13382.5	−1.65417
$404$	17457.5	2.14986
$405$	0	0
$406$	25089.4	3.06691
$407$	−387.030	−0.0471361
$408$	−1271.54	−0.154291
$409$	−2656.13	−0.321118	−0.160559	−	0.987026i	$-0.551330\pi$
$409$	−2656.13	−0.321118	−0.160559	+	0.987026i	$0.551330\pi$
$410$	0	0
$411$	6160.85	0.739397
$412$	−2761.99	−0.330275
$413$	−10988.2	−1.30919
$414$	−3686.73	−0.437663
$415$	0	0
$416$	11457.6	1.35037
$417$	747.557	0.0877891
$418$	747.885	0.0875126
$419$	7531.18	0.878096	0.439048	−	0.898464i	$-0.355316\pi$
$419$	7531.18	0.878096	0.439048	+	0.898464i	$0.355316\pi$
$420$	0	0
$421$	2962.90	0.343000	0.171500	−	0.985184i	$-0.445139\pi$
$421$	2962.90	0.343000	0.171500	+	0.985184i	$0.445139\pi$
$422$	6070.25	0.700225
$423$	−7119.45	−0.818344
$424$	−360.115	−0.0412470
$425$	0	0
$426$	−4649.90	−0.528846
$427$	−13348.7	−1.51286
$428$	12095.6	1.36604
$429$	117.257	0.0131963
$430$	0	0
$431$	6991.70	0.781388	0.390694	−	0.920521i	$-0.372235\pi$
$431$	6991.70	0.781388	0.390694	+	0.920521i	$0.372235\pi$
$432$	−1479.98	−0.164827
$433$	−10191.5	−1.13111	−0.565557	−	0.824709i	$-0.691339\pi$
$433$	−10191.5	−1.13111	−0.565557	+	0.824709i	$0.691339\pi$
$434$	−31083.2	−3.43788
$435$	0	0
$436$	13912.3	1.52816
$437$	5371.56	0.588001
$438$	−1509.21	−0.164642
$439$	9491.43	1.03189	0.515947	−	0.856621i	$-0.327440\pi$
$439$	9491.43	1.03189	0.515947	+	0.856621i	$0.327440\pi$
$440$	0	0
$441$	−10104.2	−1.09105
$442$	−8565.65	−0.921779
$443$	8017.52	0.859873	0.429936	−	0.902859i	$-0.358536\pi$
$443$	8017.52	0.859873	0.429936	+	0.902859i	$0.358536\pi$
$444$	8304.67	0.887663
$445$	0	0
$446$	1791.18	0.190167
$447$	5710.75	0.604271
$448$	23260.7	2.45304
$449$	−15361.7	−1.61462	−0.807310	−	0.590128i	$-0.799077\pi$
$449$	−15361.7	−1.61462	−0.807310	+	0.590128i	$0.799077\pi$
$450$	0	0
$451$	−76.9122	−0.00803028
$452$	−11972.0	−1.24583
$453$	−3197.80	−0.331668
$454$	7978.68	0.824797
$455$	0	0
$456$	−5404.33	−0.555002
$457$	−11444.8	−1.17148	−0.585741	−	0.810499i	$-0.699196\pi$
$457$	−11444.8	−1.17148	−0.585741	+	0.810499i	$0.699196\pi$
$458$	−20820.8	−2.12422
$459$	3503.14	0.356236
$460$	0	0
$461$	14701.8	1.48531	0.742656	−	0.669673i	$-0.233566\pi$
$461$	14701.8	1.48531	0.742656	+	0.669673i	$0.233566\pi$
$462$	272.351	0.0274262
$463$	21.0020	0.00210809	0.00105405	−	0.999999i	$-0.499664\pi$
$463$	21.0020	0.00210809	0.00105405	+	0.999999i	$0.499664\pi$
$464$	3009.06	0.301060
$465$	0	0
$466$	−21309.2	−2.11831
$467$	−14991.4	−1.48548	−0.742741	−	0.669579i	$-0.766475\pi$
$467$	−14991.4	−1.48548	−0.742741	+	0.669579i	$0.766475\pi$
$468$	15023.1	1.48385
$469$	20770.3	2.04496
$470$	0	0
$471$	−6400.41	−0.626147
$472$	7158.93	0.698128
$473$	448.108	0.0435603
$474$	−3904.14	−0.378318
$475$	0	0
$476$	−11961.8	−1.15182
$477$	457.734	0.0439375
$478$	−15916.9	−1.52305
$479$	18888.8	1.80177	0.900887	−	0.434053i	$-0.142917\pi$
$479$	18888.8	1.80177	0.900887	+	0.434053i	$0.142917\pi$
$480$	0	0
$481$	18840.1	1.78593
$482$	−8911.45	−0.842128
$483$	1956.11	0.184278
$484$	−16039.9	−1.50638
$485$	0	0
$486$	−15723.3	−1.46754
$487$	−6533.58	−0.607936	−0.303968	−	0.952682i	$-0.598312\pi$
$487$	−6533.58	−0.607936	−0.303968	+	0.952682i	$0.598312\pi$
$488$	8696.85	0.806737
$489$	1016.13	0.0939695
$490$	0	0
$491$	6971.17	0.640742	0.320371	−	0.947292i	$-0.396192\pi$
$491$	6971.17	0.640742	0.320371	+	0.947292i	$0.396192\pi$
$492$	1650.34	0.151226
$493$	−7122.49	−0.650671
$494$	−36406.0	−3.31575
$495$	0	0
$496$	−3727.91	−0.337476
$497$	−14731.2	−1.32955
$498$	−731.649	−0.0658353
$499$	1492.98	0.133938	0.0669689	−	0.997755i	$-0.478667\pi$
$499$	1492.98	0.133938	0.0669689	+	0.997755i	$0.478667\pi$
$500$	0	0
$501$	2855.64	0.254652
$502$	14632.6	1.30097
$503$	−4277.49	−0.379172	−0.189586	−	0.981864i	$-0.560715\pi$
$503$	−4277.49	−0.379172	−0.189586	+	0.981864i	$0.560715\pi$
$504$	11751.1	1.03856
$505$	0	0
$506$	176.363	0.0154947
$507$	−1384.11	−0.121243
$508$	17784.5	1.55326
$509$	−7020.29	−0.611334	−0.305667	−	0.952138i	$-0.598879\pi$
$509$	−7020.29	−0.611334	−0.305667	+	0.952138i	$0.598879\pi$
$510$	0	0
$511$	−4781.31	−0.413919
$512$	5375.35	0.463983
$513$	14889.1	1.28142
$514$	25930.0	2.22514
$515$	0	0
$516$	−9615.24	−0.820325
$517$	340.576	0.0289720
$518$	43759.4	3.71173
$519$	3088.73	0.261233
$520$	0	0
$521$	−4143.27	−0.348407	−0.174203	−	0.984710i	$-0.555735\pi$
$521$	−4143.27	−0.348407	−0.174203	+	0.984710i	$0.555735\pi$
$522$	20777.1	1.74212
$523$	6946.59	0.580790	0.290395	−	0.956907i	$-0.406213\pi$
$523$	6946.59	0.580790	0.290395	+	0.956907i	$0.406213\pi$
$524$	28935.6	2.41233
$525$	0	0
$526$	−4866.03	−0.403363
$527$	8824.04	0.729377
$528$	32.6639	0.00269226
$529$	−10900.3	−0.895890
$530$	0	0
$531$	−9099.57	−0.743668
$532$	−50840.3	−4.14325
$533$	3743.98	0.304258
$534$	5932.66	0.480770
$535$	0	0
$536$	−13532.1	−1.09048
$537$	3478.33	0.279518
$538$	−30214.2	−2.42124
$539$	483.359	0.0386266
$540$	0	0
$541$	7949.54	0.631751	0.315875	−	0.948801i	$-0.397702\pi$
$541$	7949.54	0.631751	0.315875	+	0.948801i	$0.397702\pi$
$542$	36590.9	2.89984
$543$	4239.24	0.335033
$544$	−7554.83	−0.595424
$545$	0	0
$546$	−13257.6	−1.03915
$547$	−15833.7	−1.23766	−0.618828	−	0.785526i	$-0.712392\pi$
$547$	−15833.7	−1.23766	−0.618828	+	0.785526i	$0.712392\pi$
$548$	−37759.7	−2.94346
$549$	−11054.4	−0.859361
$550$	0	0
$551$	−30272.2	−2.34054
$552$	−1274.43	−0.0982669
$553$	−12368.6	−0.951115
$554$	21818.4	1.67324
$555$	0	0
$556$	−4581.76	−0.349478
$557$	16832.3	1.28045	0.640224	−	0.768189i	$-0.278842\pi$
$557$	16832.3	1.28045	0.640224	+	0.768189i	$0.278842\pi$
$558$	−25740.7	−1.95285
$559$	−21813.3	−1.65045
$560$	0	0
$561$	−77.3162	−0.00581870
$562$	33264.7	2.49677
$563$	−22369.3	−1.67452	−0.837258	−	0.546808i	$-0.815843\pi$
$563$	−22369.3	−1.67452	−0.837258	+	0.546808i	$0.815843\pi$
$564$	−7307.88	−0.545598
$565$	0	0
$566$	22959.1	1.70502
$567$	−12016.1	−0.889996
$568$	9597.57	0.708988
$569$	18669.1	1.37548	0.687740	−	0.725957i	$-0.258603\pi$
$569$	18669.1	1.37548	0.687740	+	0.725957i	$0.258603\pi$
$570$	0	0
$571$	4298.30	0.315023	0.157512	−	0.987517i	$-0.449653\pi$
$571$	4298.30	0.315023	0.157512	+	0.987517i	$0.449653\pi$
$572$	−718.667	−0.0525332
$573$	663.917	0.0484041
$574$	8696.05	0.632345
$575$	0	0
$576$	19262.7	1.39342
$577$	−19951.5	−1.43950	−0.719752	−	0.694231i	$-0.755744\pi$
$577$	−19951.5	−1.43950	−0.719752	+	0.694231i	$0.755744\pi$
$578$	−16357.8	−1.17715
$579$	−9413.57	−0.675673
$580$	0	0
$581$	−2317.92	−0.165514
$582$	−9687.29	−0.689951
$583$	−21.8968	−0.00155553
$584$	3115.08	0.220724
$585$	0	0
$586$	18323.5	1.29170
$587$	17760.2	1.24880	0.624398	−	0.781106i	$-0.285344\pi$
$587$	17760.2	1.24880	0.624398	+	0.781106i	$0.285344\pi$
$588$	−10371.6	−0.727412
$589$	37504.2	2.62366
$590$	0	0
$591$	−543.876	−0.0378546
$592$	5248.21	0.364359
$593$	13990.5	0.968838	0.484419	−	0.874836i	$-0.339031\pi$
$593$	13990.5	0.968838	0.484419	+	0.874836i	$0.339031\pi$
$594$	488.852	0.0337674
$595$	0	0
$596$	−35001.0	−2.40553
$597$	−6155.04	−0.421958
$598$	−8585.12	−0.587076
$599$	23269.8	1.58728	0.793638	−	0.608391i	$-0.208185\pi$
$599$	23269.8	1.58728	0.793638	+	0.608391i	$0.208185\pi$
$600$	0	0
$601$	−13497.9	−0.916123	−0.458062	−	0.888921i	$-0.651456\pi$
$601$	−13497.9	−0.916123	−0.458062	+	0.888921i	$0.651456\pi$
$602$	−50665.1	−3.43016
$603$	17200.4	1.16161
$604$	19599.2	1.32033
$605$	0	0
$606$	−12758.0	−0.855214
$607$	8412.24	0.562508	0.281254	−	0.959633i	$-0.409250\pi$
$607$	8412.24	0.562508	0.281254	+	0.959633i	$0.409250\pi$
$608$	−32109.8	−2.14181
$609$	−11024.0	−0.733520
$610$	0	0
$611$	−16578.8	−1.09772
$612$	−9905.82	−0.654280
$613$	−19815.1	−1.30558	−0.652792	−	0.757537i	$-0.726403\pi$
$613$	−19815.1	−1.30558	−0.652792	+	0.757537i	$0.726403\pi$
$614$	−20932.7	−1.37585
$615$	0	0
$616$	−562.142	−0.0367684
$617$	−22031.0	−1.43750	−0.718749	−	0.695270i	$-0.755285\pi$
$617$	−22031.0	−1.43750	−0.718749	+	0.695270i	$0.755285\pi$
$618$	2018.47	0.131383
$619$	6870.20	0.446101	0.223051	−	0.974807i	$-0.428398\pi$
$619$	6870.20	0.446101	0.223051	+	0.974807i	$0.428398\pi$
$620$	0	0
$621$	3511.10	0.226885
$622$	−266.584	−0.0171850
$623$	18795.1	1.20869
$624$	−1590.03	−0.102007
$625$	0	0
$626$	−30866.5	−1.97072
$627$	−328.611	−0.0209306
$628$	39228.0	2.49262
$629$	−12422.6	−0.787476
$630$	0	0
$631$	−6443.69	−0.406528	−0.203264	−	0.979124i	$-0.565155\pi$
$631$	−6443.69	−0.406528	−0.203264	+	0.979124i	$0.565155\pi$
$632$	8058.29	0.507186
$633$	−2667.19	−0.167474
$634$	−9476.29	−0.593615
$635$	0	0
$636$	469.849	0.0292936
$637$	−23529.2	−1.46352
$638$	−993.922	−0.0616767
$639$	−12199.3	−0.755236
$640$	0	0
$641$	1436.28	0.0885017	0.0442508	−	0.999020i	$-0.485910\pi$
$641$	1436.28	0.0885017	0.0442508	+	0.999020i	$0.485910\pi$
$642$	−8839.51	−0.543408
$643$	−8864.97	−0.543702	−0.271851	−	0.962339i	$-0.587636\pi$
$643$	−8864.97	−0.543702	−0.271851	+	0.962339i	$0.587636\pi$
$644$	−11989.0	−0.733589
$645$	0	0
$646$	24005.1	1.46202
$647$	20844.0	1.26656	0.633279	−	0.773923i	$-0.281708\pi$
$647$	20844.0	1.26656	0.633279	+	0.773923i	$0.281708\pi$
$648$	7828.61	0.474594
$649$	435.300	0.0263282
$650$	0	0
$651$	13657.6	0.822246
$652$	−6227.85	−0.374082
$653$	4654.38	0.278928	0.139464	−	0.990227i	$-0.455462\pi$
$653$	4654.38	0.278928	0.139464	+	0.990227i	$0.455462\pi$
$654$	−10167.1	−0.607899
$655$	0	0
$656$	1042.95	0.0620735
$657$	−3959.51	−0.235122
$658$	−38507.1	−2.28140
$659$	6789.17	0.401318	0.200659	−	0.979661i	$-0.435692\pi$
$659$	6789.17	0.401318	0.200659	+	0.979661i	$0.435692\pi$
$660$	0	0
$661$	4486.83	0.264020	0.132010	−	0.991248i	$-0.457857\pi$
$661$	4486.83	0.264020	0.132010	+	0.991248i	$0.457857\pi$
$662$	−1937.13	−0.113729
$663$	3763.64	0.220464
$664$	1510.15	0.0882610
$665$	0	0
$666$	36238.1	2.10841
$667$	−7138.68	−0.414409
$668$	−17502.1	−1.01374
$669$	−787.021	−0.0454828
$670$	0	0
$671$	528.813	0.0304241
$672$	−11693.1	−0.671238
$673$	12403.8	0.710450	0.355225	−	0.934781i	$-0.384404\pi$
$673$	12403.8	0.710450	0.355225	+	0.934781i	$0.384404\pi$
$674$	36021.1	2.05858
$675$	0	0
$676$	8483.16	0.482656
$677$	20788.2	1.18014	0.590070	−	0.807352i	$-0.299100\pi$
$677$	20788.2	1.18014	0.590070	+	0.807352i	$0.299100\pi$
$678$	8749.21	0.495592
$679$	−30690.1	−1.73458
$680$	0	0
$681$	−3505.73	−0.197269
$682$	1231.37	0.0691371
$683$	−4355.19	−0.243992	−0.121996	−	0.992531i	$-0.538930\pi$
$683$	−4355.19	−0.243992	−0.121996	+	0.992531i	$0.538930\pi$
$684$	−42102.0	−2.35352
$685$	0	0
$686$	−11746.1	−0.653746
$687$	9148.39	0.508054
$688$	−6076.44	−0.336718
$689$	1065.91	0.0589372
$690$	0	0
$691$	7425.03	0.408772	0.204386	−	0.978890i	$-0.434480\pi$
$691$	7425.03	0.408772	0.204386	+	0.978890i	$0.434480\pi$
$692$	−18930.7	−1.03994
$693$	714.527	0.0391669
$694$	−41991.8	−2.29681
$695$	0	0
$696$	7182.24	0.391152
$697$	−2468.67	−0.134157
$698$	47344.9	2.56738
$699$	9363.01	0.506640
$700$	0	0
$701$	30391.1	1.63746	0.818728	−	0.574181i	$-0.194680\pi$
$701$	30391.1	1.63746	0.818728	+	0.574181i	$0.194680\pi$
$702$	−23796.6	−1.27941
$703$	−52799.0	−2.83265
$704$	−921.477	−0.0493316
$705$	0	0
$706$	1447.43	0.0771600
$707$	−40418.5	−2.15006
$708$	−9340.41	−0.495811
$709$	−9033.93	−0.478528	−0.239264	−	0.970955i	$-0.576906\pi$
$709$	−9033.93	−0.478528	−0.239264	+	0.970955i	$0.576906\pi$
$710$	0	0
$711$	−10242.7	−0.540270
$712$	−12245.3	−0.644537
$713$	8844.10	0.464536
$714$	8741.71	0.458194
$715$	0	0
$716$	−21318.6	−1.11273
$717$	6993.66	0.364272
$718$	38116.7	1.98120
$719$	−15255.8	−0.791301	−0.395651	−	0.918401i	$-0.629481\pi$
$719$	−15255.8	−0.791301	−0.395651	+	0.918401i	$0.629481\pi$
$720$	0	0
$721$	6394.68	0.330306
$722$	71305.0	3.67548
$723$	3915.58	0.201413
$724$	−25982.2	−1.33373
$725$	0	0
$726$	11722.0	0.599236
$727$	−31425.7	−1.60319	−0.801593	−	0.597870i	$-0.796014\pi$
$727$	−31425.7	−1.60319	−0.801593	+	0.597870i	$0.796014\pi$
$728$	27364.3	1.39311
$729$	−4708.68	−0.239226
$730$	0	0
$731$	14383.1	0.727738
$732$	−11347.0	−0.572945
$733$	20741.7	1.04517	0.522587	−	0.852586i	$-0.324967\pi$
$733$	20741.7	1.04517	0.522587	+	0.852586i	$0.324967\pi$
$734$	−26203.1	−1.31768
$735$	0	0
$736$	−7572.00	−0.379222
$737$	−822.821	−0.0411249
$738$	7201.39	0.359196
$739$	−19085.6	−0.950035	−0.475017	−	0.879976i	$-0.657558\pi$
$739$	−19085.6	−0.950035	−0.475017	+	0.879976i	$0.657558\pi$
$740$	0	0
$741$	15996.3	0.793036
$742$	2475.75	0.122490
$743$	17225.3	0.850517	0.425259	−	0.905072i	$-0.360183\pi$
$743$	17225.3	0.850517	0.425259	+	0.905072i	$0.360183\pi$
$744$	−8898.06	−0.438466
$745$	0	0
$746$	−10332.5	−0.507104
$747$	−1919.52	−0.0940183
$748$	473.869	0.0231636
$749$	−28004.3	−1.36616
$750$	0	0
$751$	−9934.87	−0.482728	−0.241364	−	0.970435i	$-0.577595\pi$
$751$	−9934.87	−0.482728	−0.241364	+	0.970435i	$0.577595\pi$
$752$	−4618.29	−0.223952
$753$	−6429.38	−0.311155
$754$	48382.7	2.33686
$755$	0	0
$756$	−33231.6	−1.59870
$757$	−68.9021	−0.00330818	−0.00165409	−	0.999999i	$-0.500527\pi$
$757$	−68.9021	−0.00330818	−0.00165409	+	0.999999i	$0.500527\pi$
$758$	−33436.9	−1.60222
$759$	−77.4919	−0.00370590
$760$	0	0
$761$	−9900.08	−0.471587	−0.235794	−	0.971803i	$-0.575769\pi$
$761$	−9900.08	−0.471587	−0.235794	+	0.971803i	$0.575769\pi$
$762$	−12996.9	−0.617886
$763$	−32210.2	−1.52830
$764$	−4069.13	−0.192691
$765$	0	0
$766$	−22684.7	−1.07002
$767$	−21189.8	−0.997546
$768$	4769.16	0.224078
$769$	9776.56	0.458455	0.229227	−	0.973373i	$-0.426380\pi$
$769$	9776.56	0.458455	0.229227	+	0.973373i	$0.426380\pi$
$770$	0	0
$771$	−11393.3	−0.532192
$772$	57695.6	2.68978
$773$	21990.2	1.02320	0.511599	−	0.859224i	$-0.329053\pi$
$773$	21990.2	1.02320	0.511599	+	0.859224i	$0.329053\pi$
$774$	−41956.9	−1.94846
$775$	0	0
$776$	19995.0	0.924971
$777$	−19227.3	−0.887743
$778$	63211.3	2.91290
$779$	−10492.4	−0.482580
$780$	0	0
$781$	583.582	0.0267378
$782$	5660.79	0.258861
$783$	−19787.3	−0.903117
$784$	−6554.44	−0.298581
$785$	0	0
$786$	−21146.3	−0.959621
$787$	13279.1	0.601461	0.300731	−	0.953709i	$-0.402769\pi$
$787$	13279.1	0.601461	0.300731	+	0.953709i	$0.402769\pi$
$788$	3333.41	0.150695
$789$	2138.07	0.0964732
$790$	0	0
$791$	27718.2	1.24595
$792$	−465.523	−0.0208859
$793$	−25741.9	−1.15274
$794$	56073.8	2.50628
$795$	0	0
$796$	37724.1	1.67977
$797$	21493.8	0.955269	0.477635	−	0.878559i	$-0.341494\pi$
$797$	21493.8	0.955269	0.477635	+	0.878559i	$0.341494\pi$
$798$	37154.3	1.64818
$799$	10931.6	0.484019
$800$	0	0
$801$	15564.7	0.686580
$802$	23830.6	1.04924
$803$	189.413	0.00832407
$804$	17655.6	0.774460
$805$	0	0
$806$	−59941.2	−2.61953
$807$	13275.7	0.579093
$808$	26333.1	1.14653
$809$	−29958.7	−1.30197	−0.650985	−	0.759091i	$-0.725644\pi$
$809$	−29958.7	−1.30197	−0.650985	+	0.759091i	$0.725644\pi$
$810$	0	0
$811$	26589.1	1.15126	0.575629	−	0.817711i	$-0.304757\pi$
$811$	26589.1	1.15126	0.575629	+	0.817711i	$0.304757\pi$
$812$	67565.6	2.92006
$813$	−16077.6	−0.693561
$814$	−1733.54	−0.0746444
$815$	0	0
$816$	1048.42	0.0449781
$817$	61131.3	2.61776
$818$	−11897.0	−0.508520
$819$	−34782.2	−1.48399
$820$	0	0
$821$	−32354.5	−1.37537	−0.687686	−	0.726009i	$-0.741373\pi$
$821$	−32354.5	−1.37537	−0.687686	+	0.726009i	$0.741373\pi$
$822$	27594.9	1.17090
$823$	−912.733	−0.0386584	−0.0193292	−	0.999813i	$-0.506153\pi$
$823$	−912.733	−0.0386584	−0.0193292	+	0.999813i	$0.506153\pi$
$824$	−4166.21	−0.176137
$825$	0	0
$826$	−49217.0	−2.07322
$827$	−17231.2	−0.724531	−0.362265	−	0.932075i	$-0.617997\pi$
$827$	−17231.2	−0.724531	−0.362265	+	0.932075i	$0.617997\pi$
$828$	−9928.33	−0.416707
$829$	−31991.9	−1.34032	−0.670160	−	0.742217i	$-0.733775\pi$
$829$	−31991.9	−1.34032	−0.670160	+	0.742217i	$0.733775\pi$
$830$	0	0
$831$	−9586.73	−0.400192
$832$	44856.2	1.86912
$833$	15514.5	0.645313
$834$	3348.37	0.139022
$835$	0	0
$836$	2014.05	0.0833223
$837$	24514.5	1.01236
$838$	33732.7	1.39055
$839$	11161.1	0.459267	0.229634	−	0.973277i	$-0.426247\pi$
$839$	11161.1	0.459267	0.229634	+	0.973277i	$0.426247\pi$
$840$	0	0
$841$	15842.1	0.649558
$842$	13271.1	0.543173
$843$	−14616.1	−0.597158
$844$	16347.2	0.666697
$845$	0	0
$846$	−31888.6	−1.29592
$847$	37136.3	1.50652
$848$	296.925	0.0120241
$849$	−10087.9	−0.407794
$850$	0	0
$851$	−12450.9	−0.501539
$852$	−12522.2	−0.503523
$853$	−15256.3	−0.612388	−0.306194	−	0.951969i	$-0.599056\pi$
$853$	−15256.3	−0.612388	−0.306194	+	0.951969i	$0.599056\pi$
$854$	−59790.0	−2.39575
$855$	0	0
$856$	18245.1	0.728511
$857$	−38889.4	−1.55010	−0.775051	−	0.631898i	$-0.782276\pi$
$857$	−38889.4	−1.55010	−0.775051	+	0.631898i	$0.782276\pi$
$858$	525.204	0.0208976
$859$	5149.06	0.204521	0.102261	−	0.994758i	$-0.467392\pi$
$859$	5149.06	0.204521	0.102261	+	0.994758i	$0.467392\pi$
$860$	0	0
$861$	−3820.93	−0.151239
$862$	31316.4	1.23740
$863$	−6731.55	−0.265521	−0.132761	−	0.991148i	$-0.542384\pi$
$863$	−6731.55	−0.265521	−0.132761	+	0.991148i	$0.542384\pi$
$864$	−20988.4	−0.826436
$865$	0	0
$866$	−45648.5	−1.79122
$867$	7187.39	0.281542
$868$	−83706.9	−3.27327
$869$	489.985	0.0191273
$870$	0	0
$871$	40053.8	1.55817
$872$	20985.3	0.814969
$873$	−25415.2	−0.985307
$874$	24059.6	0.931155
$875$	0	0
$876$	−4064.31	−0.156758
$877$	8176.68	0.314831	0.157415	−	0.987532i	$-0.449684\pi$
$877$	8176.68	0.314831	0.157415	+	0.987532i	$0.449684\pi$
$878$	42512.9	1.63410
$879$	−8051.13	−0.308940
$880$	0	0
$881$	−32106.5	−1.22780	−0.613901	−	0.789383i	$-0.710401\pi$
$881$	−32106.5	−1.22780	−0.613901	+	0.789383i	$0.710401\pi$
$882$	−45257.5	−1.72778
$883$	34983.7	1.33329	0.666645	−	0.745375i	$-0.267730\pi$
$883$	34983.7	1.33329	0.666645	+	0.745375i	$0.267730\pi$
$884$	−23067.3	−0.877642
$885$	0	0
$886$	35911.1	1.36169
$887$	10990.6	0.416040	0.208020	−	0.978125i	$-0.433298\pi$
$887$	10990.6	0.416040	0.208020	+	0.978125i	$0.433298\pi$
$888$	12526.8	0.473393
$889$	−41175.3	−1.55340
$890$	0	0
$891$	476.020	0.0178982
$892$	4823.63	0.181062
$893$	46461.7	1.74108
$894$	25578.9	0.956919
$895$	0	0
$896$	56654.5	2.11238
$897$	3772.19	0.140412
$898$	−68806.3	−2.55690
$899$	−49842.2	−1.84909
$900$	0	0
$901$	−702.828	−0.0259874
$902$	−344.496	−0.0127167
$903$	22261.6	0.820399
$904$	−18058.7	−0.664407
$905$	0	0
$906$	−14323.2	−0.525227
$907$	8587.67	0.314387	0.157193	−	0.987568i	$-0.449755\pi$
$907$	8587.67	0.314387	0.157193	+	0.987568i	$0.449755\pi$
$908$	21486.5	0.785304
$909$	−33471.4	−1.22132
$910$	0	0
$911$	16350.0	0.594620	0.297310	−	0.954781i	$-0.403911\pi$
$911$	16350.0	0.594620	0.297310	+	0.954781i	$0.403911\pi$
$912$	4456.04	0.161792
$913$	91.8250	0.00332855
$914$	−51262.3	−1.85515
$915$	0	0
$916$	−56070.3	−2.02251
$917$	−66993.0	−2.41255
$918$	15690.8	0.564133
$919$	−2057.65	−0.0738583	−0.0369291	−	0.999318i	$-0.511758\pi$
$919$	−2057.65	−0.0738583	−0.0369291	+	0.999318i	$0.511758\pi$
$920$	0	0
$921$	9197.55	0.329066
$922$	65850.3	2.35213
$923$	−28407.9	−1.01306
$924$	733.439	0.0261129
$925$	0	0
$926$	94.0698	0.00333836
$927$	5295.58	0.187626
$928$	42673.1	1.50950
$929$	21010.5	0.742016	0.371008	−	0.928630i	$-0.379012\pi$
$929$	21010.5	0.742016	0.371008	+	0.928630i	$0.379012\pi$
$930$	0	0
$931$	65940.1	2.32127
$932$	−57385.7	−2.01688
$933$	117.134	0.00411016
$934$	−67147.7	−2.35240
$935$	0	0
$936$	22661.0	0.791343
$937$	−6872.87	−0.239623	−0.119812	−	0.992797i	$-0.538229\pi$
$937$	−6872.87	−0.239623	−0.119812	+	0.992797i	$0.538229\pi$
$938$	93031.9	3.23838
$939$	13562.3	0.471342
$940$	0	0
$941$	23175.3	0.802861	0.401431	−	0.915889i	$-0.368513\pi$
$941$	23175.3	0.802861	0.401431	+	0.915889i	$0.368513\pi$
$942$	−28667.9	−0.991562
$943$	−2474.28	−0.0854441
$944$	−5902.76	−0.203515
$945$	0	0
$946$	2007.11	0.0689819
$947$	−39802.8	−1.36581	−0.682903	−	0.730510i	$-0.739283\pi$
$947$	−39802.8	−1.36581	−0.682903	+	0.730510i	$0.739283\pi$
$948$	−10513.8	−0.360203
$949$	−9220.34	−0.315390
$950$	0	0
$951$	4163.76	0.141976
$952$	−18043.2	−0.614270
$953$	12561.3	0.426968	0.213484	−	0.976947i	$-0.431519\pi$
$953$	12561.3	0.426968	0.213484	+	0.976947i	$0.431519\pi$
$954$	2050.23	0.0695792
$955$	0	0
$956$	−42864.0	−1.45013
$957$	436.717	0.0147514
$958$	84604.3	2.85328
$959$	87422.9	2.94373
$960$	0	0
$961$	31958.4	1.07275
$962$	84386.2	2.82819
$963$	−23191.0	−0.776032
$964$	−23998.5	−0.801804
$965$	0	0
$966$	8761.58	0.291821
$967$	37183.7	1.23655	0.618276	−	0.785961i	$-0.287831\pi$
$967$	37183.7	1.23655	0.618276	+	0.785961i	$0.287831\pi$
$968$	−24194.8	−0.803356
$969$	−10547.5	−0.349675
$970$	0	0
$971$	17128.5	0.566096	0.283048	−	0.959106i	$-0.408654\pi$
$971$	17128.5	0.566096	0.283048	+	0.959106i	$0.408654\pi$
$972$	−42342.9	−1.39727
$973$	10607.9	0.349510
$974$	−29264.4	−0.962723
$975$	0	0
$976$	−7170.82	−0.235176
$977$	23254.7	0.761497	0.380749	−	0.924679i	$-0.375666\pi$
$977$	23254.7	0.761497	0.380749	+	0.924679i	$0.375666\pi$
$978$	4551.33	0.148809
$979$	−744.574	−0.0243071
$980$	0	0
$981$	−26674.0	−0.868130
$982$	31224.4	1.01467
$983$	1776.47	0.0576406	0.0288203	−	0.999585i	$-0.490825\pi$
$983$	1776.47	0.0576406	0.0288203	+	0.999585i	$0.490825\pi$
$984$	2489.38	0.0806489
$985$	0	0
$986$	−31902.2	−1.03040
$987$	16919.5	0.545648
$988$	−98041.1	−3.15699
$989$	14415.7	0.463492
$990$	0	0
$991$	12681.8	0.406508	0.203254	−	0.979126i	$-0.434848\pi$
$991$	12681.8	0.406508	0.203254	+	0.979126i	$0.434848\pi$
$992$	−52867.7	−1.69209
$993$	851.150	0.0272008
$994$	−65982.4	−2.10547
$995$	0	0
$996$	−1970.33	−0.0626829
$997$	19332.0	0.614094	0.307047	−	0.951694i	$-0.400659\pi$
$997$	19332.0	0.614094	0.307047	+	0.951694i	$0.400659\pi$
$998$	6687.18	0.212103
$999$	−34511.8	−1.09300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2075.4.a.k.1.40	✓	47
5.4	even	2		2075.4.a.l.1.8	yes	47

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2075.4.a.k.1.40	✓	47	1.1	even	1	trivial
2075.4.a.l.1.8	yes	47	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+4.47908 q^{2} -1.96805 q^{3} +12.0621 q^{4} -8.81505 q^{6} -27.9268 q^{7} +18.1946 q^{8} -23.1268 q^{9} +1.10633 q^{11} -23.7389 q^{12} -53.8543 q^{13} -125.086 q^{14} -15.0020 q^{16} +35.5100 q^{17} -103.587 q^{18} +150.926 q^{19} +54.9613 q^{21} +4.95532 q^{22} +35.5907 q^{23} -35.8079 q^{24} -241.218 q^{26} +98.6520 q^{27} -336.856 q^{28} -200.577 q^{29} +248.494 q^{31} -212.752 q^{32} -2.17730 q^{33} +159.052 q^{34} -278.958 q^{36} -349.834 q^{37} +676.008 q^{38} +105.988 q^{39} -69.5204 q^{41} +246.176 q^{42} +405.042 q^{43} +13.3446 q^{44} +159.414 q^{46} +307.844 q^{47} +29.5247 q^{48} +436.904 q^{49} -69.8855 q^{51} -649.598 q^{52} -19.7924 q^{53} +441.870 q^{54} -508.117 q^{56} -297.029 q^{57} -898.399 q^{58} +393.465 q^{59} +477.990 q^{61} +1113.03 q^{62} +645.856 q^{63} -832.917 q^{64} -9.75231 q^{66} -743.743 q^{67} +428.327 q^{68} -70.0444 q^{69} +527.495 q^{71} -420.783 q^{72} +171.209 q^{73} -1566.93 q^{74} +1820.49 q^{76} -30.8961 q^{77} +474.728 q^{78} +442.894 q^{79} +430.271 q^{81} -311.387 q^{82} +83.0000 q^{83} +662.950 q^{84} +1814.21 q^{86} +394.745 q^{87} +20.1292 q^{88} -673.015 q^{89} +1503.98 q^{91} +429.300 q^{92} -489.049 q^{93} +1378.86 q^{94} +418.707 q^{96} +1098.95 q^{97} +1956.93 q^{98} -25.5857 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 47 q - q^{2} + 6 q^{3} + 229 q^{4} + 9 q^{6} + 6 q^{7} - 39 q^{8} + 531 q^{9} + 90 q^{11} - 6 q^{12} - 50 q^{13} + 115 q^{14} + 1149 q^{16} + 68 q^{17} - 8 q^{18} + 344 q^{19} + 625 q^{21} - 402 q^{22}+ \cdots + 13991 q^{99}+O(q^{100}) \) 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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