LMFDB - Embedded newform 2075.4.a.g.1.14

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 = [21,5] 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:	$21$
Twist minimal:	no (minimal twist has level 415)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
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)

$f(q)$	$=$	\(q+1.52351 q^{2} -2.27996 q^{3} -5.67892 q^{4} -3.47354 q^{6} -9.53696 q^{7} -20.8400 q^{8} -21.8018 q^{9} +58.3417 q^{11} +12.9477 q^{12} +68.7822 q^{13} -14.5297 q^{14} +13.6814 q^{16} -87.2391 q^{17} -33.2153 q^{18} -122.038 q^{19} +21.7439 q^{21} +88.8842 q^{22} -181.049 q^{23} +47.5142 q^{24} +104.790 q^{26} +111.266 q^{27} +54.1596 q^{28} +39.0467 q^{29} -159.515 q^{31} +187.564 q^{32} -133.016 q^{33} -132.910 q^{34} +123.811 q^{36} +174.925 q^{37} -185.926 q^{38} -156.820 q^{39} -232.187 q^{41} +33.1270 q^{42} +268.245 q^{43} -331.318 q^{44} -275.830 q^{46} -373.579 q^{47} -31.1930 q^{48} -252.046 q^{49} +198.901 q^{51} -390.608 q^{52} +532.530 q^{53} +169.515 q^{54} +198.750 q^{56} +278.241 q^{57} +59.4880 q^{58} -492.537 q^{59} +324.629 q^{61} -243.024 q^{62} +207.923 q^{63} +176.304 q^{64} -202.652 q^{66} -583.030 q^{67} +495.424 q^{68} +412.783 q^{69} -1048.99 q^{71} +454.349 q^{72} +510.729 q^{73} +266.500 q^{74} +693.043 q^{76} -556.403 q^{77} -238.917 q^{78} -1197.20 q^{79} +334.967 q^{81} -353.739 q^{82} -83.0000 q^{83} -123.481 q^{84} +408.673 q^{86} -89.0247 q^{87} -1215.84 q^{88} -170.431 q^{89} -655.973 q^{91} +1028.16 q^{92} +363.688 q^{93} -569.151 q^{94} -427.636 q^{96} -464.590 q^{97} -383.995 q^{98} -1271.95 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 5 q^{2} + 12 q^{3} + 87 q^{4} - 7 q^{6} + 11 q^{7} + 84 q^{8} + 153 q^{9} - 30 q^{11} + 244 q^{12} + 89 q^{13} - 191 q^{14} + 583 q^{16} + 357 q^{17} + 281 q^{18} - 175 q^{19} - 41 q^{21} + 122 q^{22}+ \cdots - 5369 q^{99}+O(q^{100}) $ 21 * q + 5 * q^2 + 12 * q^3 + 87 * q^4 - 7 * q^6 + 11 * q^7 + 84 * q^8 + 153 * q^9 - 30 * q^11 + 244 * q^12 + 89 * q^13 - 191 * q^14 + 583 * q^16 + 357 * q^17 + 281 * q^18 - 175 * q^19 - 41 * q^21 + 122 * q^22 + 554 * q^23 - 136 * q^24 - 933 * q^26 + 555 * q^27 + 773 * q^28 - 96 * q^29 + 99 * q^31 + 1307 * q^32 + 1097 * q^33 + 393 * q^34 - 746 * q^36 + 756 * q^37 + 1192 * q^38 + 40 * q^39 - 1436 * q^41 + 1112 * q^42 + 1003 * q^43 - 706 * q^44 - 775 * q^46 + 858 * q^47 + 2837 * q^48 + 1384 * q^49 - 279 * q^51 + 1092 * q^52 + 1625 * q^53 - 269 * q^54 - 3806 * q^56 + 1740 * q^57 + 1087 * q^58 - 1224 * q^59 - 794 * q^61 + 2979 * q^62 + 1236 * q^63 + 2570 * q^64 - 1094 * q^66 + 571 * q^67 + 4064 * q^68 + 774 * q^69 - 1078 * q^71 + 2610 * q^72 + 1986 * q^73 - 1872 * q^74 - 2854 * q^76 + 2519 * q^77 + 2449 * q^78 - 432 * q^79 - 219 * q^81 + 2143 * q^82 - 1743 * q^83 - 377 * q^84 + 1556 * q^86 + 5681 * q^87 + 3158 * q^88 - 6252 * q^89 + 684 * q^91 + 7623 * q^92 + 3314 * q^93 - 3877 * q^94 - 954 * q^96 + 4880 * q^97 + 5072 * q^98 - 5369 * 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 $

$2$	1.52351	0.538642	0.269321	−	0.963050i	$-0.413201\pi$
$2$	1.52351	0.538642	0.269321	+	0.963050i	$0.413201\pi$
$3$	−2.27996	−0.438778	−0.219389	−	0.975638i	$-0.570406\pi$
$3$	−2.27996	−0.438778	−0.219389	+	0.975638i	$0.570406\pi$
$4$	−5.67892	−0.709864
$5$	0	0
$5$	0	0
$6$	−3.47354	−0.236344
$7$	−9.53696	−0.514948	−0.257474	−	0.966285i	$-0.582890\pi$
$7$	−9.53696	−0.514948	−0.257474	+	0.966285i	$0.582890\pi$
$8$	−20.8400	−0.921005
$9$	−21.8018	−0.807474
$10$	0	0
$11$	58.3417	1.59915	0.799577	−	0.600564i	$-0.205057\pi$
$11$	58.3417	1.59915	0.799577	+	0.600564i	$0.205057\pi$
$12$	12.9477	0.311473
$13$	68.7822	1.46744	0.733721	−	0.679451i	$-0.237782\pi$
$13$	68.7822	1.46744	0.733721	+	0.679451i	$0.237782\pi$
$14$	−14.5297	−0.277373
$15$	0	0
$16$	13.6814	0.213772
$17$	−87.2391	−1.24462	−0.622312	−	0.782770i	$-0.713806\pi$
$17$	−87.2391	−1.24462	−0.622312	+	0.782770i	$0.713806\pi$
$18$	−33.2153	−0.434940
$19$	−122.038	−1.47355	−0.736774	−	0.676139i	$-0.763652\pi$
$19$	−122.038	−1.47355	−0.736774	+	0.676139i	$0.763652\pi$
$20$	0	0
$21$	21.7439	0.225947
$22$	88.8842	0.861372
$23$	−181.049	−1.64136	−0.820679	−	0.571389i	$-0.806405\pi$
$23$	−181.049	−1.64136	−0.820679	+	0.571389i	$0.806405\pi$
$24$	47.5142	0.404117
$25$	0	0
$26$	104.790	0.790427
$27$	111.266	0.793079
$28$	54.1596	0.365543
$29$	39.0467	0.250027	0.125014	−	0.992155i	$-0.460103\pi$
$29$	39.0467	0.250027	0.125014	+	0.992155i	$0.460103\pi$
$30$	0	0
$31$	−159.515	−0.924188	−0.462094	−	0.886831i	$-0.652902\pi$
$31$	−159.515	−0.924188	−0.462094	+	0.886831i	$0.652902\pi$
$32$	187.564	1.03615
$33$	−133.016	−0.701673
$34$	−132.910	−0.670407
$35$	0	0
$36$	123.811	0.573197
$37$	174.925	0.777230	0.388615	−	0.921400i	$-0.372954\pi$
$37$	174.925	0.777230	0.388615	+	0.921400i	$0.372954\pi$
$38$	−185.926	−0.793716
$39$	−156.820	−0.643881
$40$	0	0
$41$	−232.187	−0.884427	−0.442213	−	0.896910i	$-0.645807\pi$
$41$	−232.187	−0.884427	−0.442213	+	0.896910i	$0.645807\pi$
$42$	33.1270	0.121705
$43$	268.245	0.951324	0.475662	−	0.879628i	$-0.342209\pi$
$43$	268.245	0.951324	0.475662	+	0.879628i	$0.342209\pi$
$44$	−331.318	−1.13518
$45$	0	0
$46$	−275.830	−0.884105
$47$	−373.579	−1.15941	−0.579703	−	0.814828i	$-0.696831\pi$
$47$	−373.579	−1.15941	−0.579703	+	0.814828i	$0.696831\pi$
$48$	−31.1930	−0.0937983
$49$	−252.046	−0.734829
$50$	0	0
$51$	198.901	0.546113
$52$	−390.608	−1.04169
$53$	532.530	1.38016	0.690081	−	0.723732i	$-0.257575\pi$
$53$	532.530	1.38016	0.690081	+	0.723732i	$0.257575\pi$
$54$	169.515	0.427186
$55$	0	0
$56$	198.750	0.474270
$57$	278.241	0.646560
$58$	59.4880	0.134675
$59$	−492.537	−1.08683	−0.543414	−	0.839465i	$-0.682869\pi$
$59$	−492.537	−1.08683	−0.543414	+	0.839465i	$0.682869\pi$
$60$	0	0
$61$	324.629	0.681386	0.340693	−	0.940175i	$-0.389338\pi$
$61$	324.629	0.681386	0.340693	+	0.940175i	$0.389338\pi$
$62$	−243.024	−0.497807
$63$	207.923	0.415807
$64$	176.304	0.344344
$65$	0	0
$66$	−202.652	−0.377951
$67$	−583.030	−1.06311	−0.531556	−	0.847023i	$-0.678392\pi$
$67$	−583.030	−1.06311	−0.531556	+	0.847023i	$0.678392\pi$
$68$	495.424	0.883514
$69$	412.783	0.720191
$70$	0	0
$71$	−1048.99	−1.75340	−0.876702	−	0.481034i	$-0.840261\pi$
$71$	−1048.99	−1.75340	−0.876702	+	0.481034i	$0.840261\pi$
$72$	454.349	0.743688
$73$	510.729	0.818854	0.409427	−	0.912343i	$-0.365729\pi$
$73$	510.729	0.818854	0.409427	+	0.912343i	$0.365729\pi$
$74$	266.500	0.418649
$75$	0	0
$76$	693.043	1.04602
$77$	−556.403	−0.823480
$78$	−238.917	−0.346821
$79$	−1197.20	−1.70501	−0.852505	−	0.522719i	$-0.824918\pi$
$79$	−1197.20	−1.70501	−0.852505	+	0.522719i	$0.824918\pi$
$80$	0	0
$81$	334.967	0.459489
$82$	−353.739	−0.476390
$83$	−83.0000	−0.109764
$83$	−83.0000	−0.109764
$84$	−123.481	−0.160392
$85$	0	0
$86$	408.673	0.512423
$87$	−89.0247	−0.109706
$88$	−1215.84	−1.47283
$89$	−170.431	−0.202985	−0.101493	−	0.994836i	$-0.532362\pi$
$89$	−170.431	−0.202985	−0.101493	+	0.994836i	$0.532362\pi$
$90$	0	0
$91$	−655.973	−0.755656
$92$	1028.16	1.16514
$93$	363.688	0.405513
$94$	−569.151	−0.624505
$95$	0	0
$96$	−427.636	−0.454640
$97$	−464.590	−0.486309	−0.243155	−	0.969988i	$-0.578182\pi$
$97$	−464.590	−0.486309	−0.243155	+	0.969988i	$0.578182\pi$
$98$	−383.995	−0.395810
$99$	−1271.95	−1.29128
$100$	0	0
$101$	−1050.82	−1.03525	−0.517624	−	0.855608i	$-0.673183\pi$
$101$	−1050.82	−1.03525	−0.517624	+	0.855608i	$0.673183\pi$
$102$	303.028	0.294159
$103$	1435.65	1.37339	0.686694	−	0.726947i	$-0.259061\pi$
$103$	1435.65	1.37339	0.686694	+	0.726947i	$0.259061\pi$
$104$	−1433.42	−1.35152
$105$	0	0
$106$	811.315	0.743414
$107$	636.749	0.575297	0.287649	−	0.957736i	$-0.407126\pi$
$107$	636.749	0.575297	0.287649	+	0.957736i	$0.407126\pi$
$108$	−631.870	−0.562979
$109$	2104.83	1.84960	0.924799	−	0.380456i	$-0.124233\pi$
$109$	2104.83	1.84960	0.924799	+	0.380456i	$0.124233\pi$
$110$	0	0
$111$	−398.821	−0.341031
$112$	−130.479	−0.110081
$113$	156.174	0.130015	0.0650073	−	0.997885i	$-0.479293\pi$
$113$	156.174	0.130015	0.0650073	+	0.997885i	$0.479293\pi$
$114$	423.903	0.348265
$115$	0	0
$116$	−221.743	−0.177485
$117$	−1499.58	−1.18492
$118$	−750.385	−0.585411
$119$	831.996	0.640916
$120$	0	0
$121$	2072.76	1.55729
$122$	494.576	0.367023
$123$	529.376	0.388067
$124$	905.875	0.656048
$125$	0	0
$126$	316.773	0.223971
$127$	2480.31	1.73301	0.866505	−	0.499168i	$-0.166361\pi$
$127$	2480.31	1.73301	0.866505	+	0.499168i	$0.166361\pi$
$128$	−1231.91	−0.850674
$129$	−611.586	−0.417419
$130$	0	0
$131$	124.301	0.0829024	0.0414512	−	0.999141i	$-0.486802\pi$
$131$	124.301	0.0829024	0.0414512	+	0.999141i	$0.486802\pi$
$132$	755.389	0.498092
$133$	1163.87	0.758800
$134$	−888.252	−0.572637
$135$	0	0
$136$	1818.06	1.14630
$137$	1442.09	0.899317	0.449659	−	0.893200i	$-0.351546\pi$
$137$	1442.09	0.899317	0.449659	+	0.893200i	$0.351546\pi$
$138$	628.879	0.387926
$139$	−434.791	−0.265313	−0.132656	−	0.991162i	$-0.542351\pi$
$139$	−434.791	−0.265313	−0.132656	+	0.991162i	$0.542351\pi$
$140$	0	0
$141$	851.743	0.508721
$142$	−1598.14	−0.944458
$143$	4012.87	2.34667
$144$	−298.279	−0.172615
$145$	0	0
$146$	778.102	0.441069
$147$	574.654	0.322426
$148$	−993.385	−0.551728
$149$	2831.58	1.55686	0.778429	−	0.627732i	$-0.216017\pi$
$149$	2831.58	1.55686	0.778429	+	0.627732i	$0.216017\pi$
$150$	0	0
$151$	1287.43	0.693840	0.346920	−	0.937895i	$-0.387227\pi$
$151$	1287.43	0.693840	0.346920	+	0.937895i	$0.387227\pi$
$152$	2543.27	1.35715
$153$	1901.97	1.00500
$154$	−847.686	−0.443561
$155$	0	0
$156$	890.569	0.457068
$157$	1550.95	0.788403	0.394201	−	0.919024i	$-0.371021\pi$
$157$	1550.95	0.788403	0.394201	+	0.919024i	$0.371021\pi$
$158$	−1823.95	−0.918391
$159$	−1214.14	−0.605584
$160$	0	0
$161$	1726.65	0.845214
$162$	510.326	0.247500
$163$	1681.15	0.807840	0.403920	−	0.914794i	$-0.367647\pi$
$163$	1681.15	0.807840	0.403920	+	0.914794i	$0.367647\pi$
$164$	1318.57	0.627823
$165$	0	0
$166$	−126.451	−0.0591237
$167$	−311.253	−0.144224	−0.0721122	−	0.997397i	$-0.522974\pi$
$167$	−311.253	−0.144224	−0.0721122	+	0.997397i	$0.522974\pi$
$168$	−453.141	−0.208099
$169$	2533.99	1.15339
$170$	0	0
$171$	2660.65	1.18985
$172$	−1523.34	−0.675311
$173$	−3487.62	−1.53271	−0.766355	−	0.642418i	$-0.777931\pi$
$173$	−3487.62	−1.53271	−0.766355	+	0.642418i	$0.777931\pi$
$174$	−135.630	−0.0590924
$175$	0	0
$176$	798.196	0.341854
$177$	1122.96	0.476876
$178$	−259.654	−0.109336
$179$	2671.46	1.11550	0.557749	−	0.830010i	$-0.311665\pi$
$179$	2671.46	1.11550	0.557749	+	0.830010i	$0.311665\pi$
$180$	0	0
$181$	2311.46	0.949222	0.474611	−	0.880196i	$-0.342589\pi$
$181$	2311.46	0.949222	0.474611	+	0.880196i	$0.342589\pi$
$182$	−999.383	−0.407028
$183$	−740.140	−0.298977
$184$	3773.05	1.51170
$185$	0	0
$186$	554.083	0.218426
$187$	−5089.68	−1.99034
$188$	2121.52	0.823021
$189$	−1061.14	−0.408394
$190$	0	0
$191$	2619.04	0.992184	0.496092	−	0.868270i	$-0.334768\pi$
$191$	2619.04	0.992184	0.496092	+	0.868270i	$0.334768\pi$
$192$	−401.965	−0.151090
$193$	1752.12	0.653474	0.326737	−	0.945115i	$-0.394051\pi$
$193$	1752.12	0.653474	0.326737	+	0.945115i	$0.394051\pi$
$194$	−707.809	−0.261947
$195$	0	0
$196$	1431.35	0.521629
$197$	−3723.06	−1.34648	−0.673241	−	0.739423i	$-0.735098\pi$
$197$	−3723.06	−1.34648	−0.673241	+	0.739423i	$0.735098\pi$
$198$	−1937.84	−0.695536
$199$	−3665.93	−1.30589	−0.652943	−	0.757407i	$-0.726466\pi$
$199$	−3665.93	−1.30589	−0.652943	+	0.757407i	$0.726466\pi$
$200$	0	0
$201$	1329.28	0.466469
$202$	−1600.93	−0.557628
$203$	−372.387	−0.128751
$204$	−1129.54	−0.387666
$205$	0	0
$206$	2187.23	0.739765
$207$	3947.19	1.32535
$208$	941.037	0.313698
$209$	−7119.90	−2.35643
$210$	0	0
$211$	5551.32	1.81123	0.905613	−	0.424104i	$-0.139411\pi$
$211$	5551.32	1.81123	0.905613	+	0.424104i	$0.139411\pi$
$212$	−3024.19	−0.979728
$213$	2391.64	0.769354
$214$	970.094	0.309880
$215$	0	0
$216$	−2318.78	−0.730430
$217$	1521.29	0.475908
$218$	3206.73	0.996272
$219$	−1164.44	−0.359295
$220$	0	0
$221$	−6000.50	−1.82641
$222$	−607.609	−0.183694
$223$	345.414	0.103725	0.0518623	−	0.998654i	$-0.483484\pi$
$223$	345.414	0.103725	0.0518623	+	0.998654i	$0.483484\pi$
$224$	−1788.79	−0.533564
$225$	0	0
$226$	237.933	0.0700314
$227$	3965.94	1.15960	0.579798	−	0.814760i	$-0.303131\pi$
$227$	3965.94	1.15960	0.579798	+	0.814760i	$0.303131\pi$
$228$	−1580.11	−0.458970
$229$	775.056	0.223656	0.111828	−	0.993728i	$-0.464329\pi$
$229$	775.056	0.223656	0.111828	+	0.993728i	$0.464329\pi$
$230$	0	0
$231$	1268.57	0.361325
$232$	−813.732	−0.230276
$233$	−1470.86	−0.413559	−0.206780	−	0.978388i	$-0.566298\pi$
$233$	−1470.86	−0.413559	−0.206780	+	0.978388i	$0.566298\pi$
$234$	−2284.62	−0.638249
$235$	0	0
$236$	2797.07	0.771500
$237$	2729.57	0.748120
$238$	1267.56	0.345224
$239$	11.3141	0.00306214	0.00153107	−	0.999999i	$-0.499513\pi$
$239$	11.3141	0.00306214	0.00153107	+	0.999999i	$0.499513\pi$
$240$	0	0
$241$	−2035.05	−0.543939	−0.271969	−	0.962306i	$-0.587675\pi$
$241$	−2035.05	−0.543939	−0.271969	+	0.962306i	$0.587675\pi$
$242$	3157.86	0.838823
$243$	−3767.89	−0.994693
$244$	−1843.54	−0.483692
$245$	0	0
$246$	806.510	0.209029
$247$	−8394.04	−2.16235
$248$	3324.30	0.851182
$249$	189.236	0.0481621
$250$	0	0
$251$	392.845	0.0987894	0.0493947	−	0.998779i	$-0.484271\pi$
$251$	392.845	0.0987894	0.0493947	+	0.998779i	$0.484271\pi$
$252$	−1180.78	−0.295167
$253$	−10562.7	−2.62478
$254$	3778.79	0.933473
$255$	0	0
$256$	−3287.26	−0.802553
$257$	1734.89	0.421087	0.210543	−	0.977585i	$-0.432477\pi$
$257$	1734.89	0.421087	0.210543	+	0.977585i	$0.432477\pi$
$258$	−931.757	−0.224840
$259$	−1668.25	−0.400233
$260$	0	0
$261$	−851.288	−0.201890
$262$	189.374	0.0446547
$263$	−191.388	−0.0448725	−0.0224363	−	0.999748i	$-0.507142\pi$
$263$	−191.388	−0.0448725	−0.0224363	+	0.999748i	$0.507142\pi$
$264$	2772.06	0.646244
$265$	0	0
$266$	1773.17	0.408722
$267$	388.576	0.0890653
$268$	3310.98	0.754665
$269$	3106.75	0.704171	0.352085	−	0.935968i	$-0.385473\pi$
$269$	3106.75	0.704171	0.352085	+	0.935968i	$0.385473\pi$
$270$	0	0
$271$	−8143.14	−1.82532	−0.912658	−	0.408723i	$-0.865974\pi$
$271$	−8143.14	−1.82532	−0.912658	+	0.408723i	$0.865974\pi$
$272$	−1193.55	−0.266065
$273$	1495.59	0.331565
$274$	2197.05	0.484410
$275$	0	0
$276$	−2344.16	−0.511238
$277$	6428.64	1.39444	0.697219	−	0.716858i	$-0.254421\pi$
$277$	6428.64	1.39444	0.697219	+	0.716858i	$0.254421\pi$
$278$	−662.409	−0.142909
$279$	3477.73	0.746258
$280$	0	0
$281$	−6379.48	−1.35433	−0.677167	−	0.735830i	$-0.736792\pi$
$281$	−6379.48	−1.35433	−0.677167	+	0.735830i	$0.736792\pi$
$282$	1297.64	0.274019
$283$	1477.39	0.310325	0.155163	−	0.987889i	$-0.450410\pi$
$283$	1477.39	0.310325	0.155163	+	0.987889i	$0.450410\pi$
$284$	5957.10	1.24468
$285$	0	0
$286$	6113.65	1.26401
$287$	2214.36	0.455434
$288$	−4089.22	−0.836666
$289$	2697.66	0.549087
$290$	0	0
$291$	1059.25	0.213382
$292$	−2900.39	−0.581275
$293$	−3391.39	−0.676202	−0.338101	−	0.941110i	$-0.609785\pi$
$293$	−3391.39	−0.676202	−0.338101	+	0.941110i	$0.609785\pi$
$294$	875.492	0.173673
$295$	0	0
$296$	−3645.43	−0.715833
$297$	6491.44	1.26826
$298$	4313.94	0.838590
$299$	−12452.9	−2.40860
$300$	0	0
$301$	−2558.24	−0.489882
$302$	1961.42	0.373732
$303$	2395.81	0.454244
$304$	−1669.65	−0.315003
$305$	0	0
$306$	2897.67	0.541336
$307$	7178.81	1.33458	0.667291	−	0.744798i	$-0.267454\pi$
$307$	7178.81	1.33458	0.667291	+	0.744798i	$0.267454\pi$
$308$	3159.76	0.584559
$309$	−3273.22	−0.602612
$310$	0	0
$311$	4757.32	0.867404	0.433702	−	0.901056i	$-0.357207\pi$
$311$	4757.32	0.867404	0.433702	+	0.901056i	$0.357207\pi$
$312$	3268.13	0.593018
$313$	1566.01	0.282799	0.141399	−	0.989953i	$-0.454840\pi$
$313$	1566.01	0.282799	0.141399	+	0.989953i	$0.454840\pi$
$314$	2362.89	0.424667
$315$	0	0
$316$	6798.81	1.21033
$317$	9079.93	1.60877	0.804384	−	0.594109i	$-0.202495\pi$
$317$	9079.93	1.60877	0.804384	+	0.594109i	$0.202495\pi$
$318$	−1849.76	−0.326193
$319$	2278.05	0.399832
$320$	0	0
$321$	−1451.76	−0.252428
$322$	2630.58	0.455268
$323$	10646.5	1.83401
$324$	−1902.25	−0.326175
$325$	0	0
$326$	2561.25	0.435137
$327$	−4798.92	−0.811562
$328$	4838.77	0.814562
$329$	3562.81	0.597033
$330$	0	0
$331$	5363.00	0.890565	0.445283	−	0.895390i	$-0.353103\pi$
$331$	5363.00	0.890565	0.445283	+	0.895390i	$0.353103\pi$
$332$	471.350	0.0779177
$333$	−3813.68	−0.627593
$334$	−474.197	−0.0776854
$335$	0	0
$336$	297.486	0.0483012
$337$	10899.9	1.76188	0.880940	−	0.473228i	$-0.156911\pi$
$337$	10899.9	1.76188	0.880940	+	0.473228i	$0.156911\pi$
$338$	3860.56	0.621263
$339$	−356.071	−0.0570475
$340$	0	0
$341$	−9306.41	−1.47792
$342$	4053.52	0.640905
$343$	5674.94	0.893346
$344$	−5590.21	−0.876174
$345$	0	0
$346$	−5313.42	−0.825582
$347$	−4599.17	−0.711517	−0.355758	−	0.934578i	$-0.615777\pi$
$347$	−4599.17	−0.711517	−0.355758	+	0.934578i	$0.615777\pi$
$348$	505.563	0.0778766
$349$	−2707.96	−0.415341	−0.207670	−	0.978199i	$-0.566588\pi$
$349$	−2707.96	−0.415341	−0.207670	+	0.978199i	$0.566588\pi$
$350$	0	0
$351$	7653.12	1.16380
$352$	10942.8	1.65697
$353$	5693.22	0.858413	0.429206	−	0.903206i	$-0.358793\pi$
$353$	5693.22	0.858413	0.429206	+	0.903206i	$0.358793\pi$
$354$	1710.84	0.256865
$355$	0	0
$356$	967.865	0.144092
$357$	−1896.91	−0.281219
$358$	4069.99	0.600854
$359$	11778.5	1.73160	0.865800	−	0.500389i	$-0.166810\pi$
$359$	11778.5	1.73160	0.865800	+	0.500389i	$0.166810\pi$
$360$	0	0
$361$	8034.26	1.17135
$362$	3521.53	0.511291
$363$	−4725.79	−0.683305
$364$	3725.22	0.536413
$365$	0	0
$366$	−1127.61	−0.161042
$367$	−2945.24	−0.418911	−0.209455	−	0.977818i	$-0.567169\pi$
$367$	−2945.24	−0.418911	−0.209455	+	0.977818i	$0.567169\pi$
$368$	−2477.00	−0.350876
$369$	5062.09	0.714152
$370$	0	0
$371$	−5078.72	−0.710711
$372$	−2065.35	−0.287859
$373$	1706.46	0.236882	0.118441	−	0.992961i	$-0.462210\pi$
$373$	1706.46	0.236882	0.118441	+	0.992961i	$0.462210\pi$
$374$	−7754.18	−1.07208
$375$	0	0
$376$	7785.37	1.06782
$377$	2685.72	0.366900
$378$	−1616.66	−0.219978
$379$	−11508.4	−1.55976	−0.779879	−	0.625930i	$-0.784719\pi$
$379$	−11508.4	−1.55976	−0.779879	+	0.625930i	$0.784719\pi$
$380$	0	0
$381$	−5655.01	−0.760406
$382$	3990.14	0.534432
$383$	−11376.4	−1.51777	−0.758885	−	0.651225i	$-0.774255\pi$
$383$	−11376.4	−1.51777	−0.758885	+	0.651225i	$0.774255\pi$
$384$	2808.69	0.373257
$385$	0	0
$386$	2669.37	0.351989
$387$	−5848.22	−0.768169
$388$	2638.37	0.345214
$389$	−1455.14	−0.189663	−0.0948313	−	0.995493i	$-0.530231\pi$
$389$	−1455.14	−0.189663	−0.0948313	+	0.995493i	$0.530231\pi$
$390$	0	0
$391$	15794.5	2.04287
$392$	5252.64	0.676781
$393$	−283.400	−0.0363757
$394$	−5672.12	−0.725272
$395$	0	0
$396$	7223.32	0.916630
$397$	−10435.0	−1.31919	−0.659594	−	0.751622i	$-0.729272\pi$
$397$	−10435.0	−1.31919	−0.659594	+	0.751622i	$0.729272\pi$
$398$	−5585.09	−0.703405
$399$	−2653.57	−0.332945
$400$	0	0
$401$	−4464.91	−0.556027	−0.278014	−	0.960577i	$-0.589676\pi$
$401$	−4464.91	−0.556027	−0.278014	+	0.960577i	$0.589676\pi$
$402$	2025.18	0.251260
$403$	−10971.8	−1.35619
$404$	5967.49	0.734886
$405$	0	0
$406$	−567.335	−0.0693507
$407$	10205.4	1.24291
$408$	−4145.10	−0.502973
$409$	−1305.92	−0.157882	−0.0789408	−	0.996879i	$-0.525154\pi$
$409$	−1305.92	−0.157882	−0.0789408	+	0.996879i	$0.525154\pi$
$410$	0	0
$411$	−3287.91	−0.394600
$412$	−8152.94	−0.974919
$413$	4697.31	0.559659
$414$	6013.58	0.713892
$415$	0	0
$416$	12901.0	1.52049
$417$	991.304	0.116413
$418$	−10847.2	−1.26927
$419$	−9560.35	−1.11469	−0.557343	−	0.830282i	$-0.688179\pi$
$419$	−9560.35	−1.11469	−0.557343	+	0.830282i	$0.688179\pi$
$420$	0	0
$421$	12725.5	1.47316	0.736581	−	0.676349i	$-0.236439\pi$
$421$	12725.5	1.47316	0.736581	+	0.676349i	$0.236439\pi$
$422$	8457.50	0.975603
$423$	8144.69	0.936190
$424$	−11097.9	−1.27114
$425$	0	0
$426$	3643.69	0.414407
$427$	−3095.98	−0.350878
$428$	−3616.04	−0.408383
$429$	−9149.17	−1.02966
$430$	0	0
$431$	−3802.40	−0.424954	−0.212477	−	0.977166i	$-0.568153\pi$
$431$	−3802.40	−0.424954	−0.212477	+	0.977166i	$0.568153\pi$
$432$	1522.27	0.169538
$433$	−5008.23	−0.555844	−0.277922	−	0.960604i	$-0.589646\pi$
$433$	−5008.23	−0.555844	−0.277922	+	0.960604i	$0.589646\pi$
$434$	2317.71	0.256344
$435$	0	0
$436$	−11953.2	−1.31296
$437$	22094.8	2.41862
$438$	−1774.04	−0.193531
$439$	759.014	0.0825188	0.0412594	−	0.999148i	$-0.486863\pi$
$439$	759.014	0.0825188	0.0412594	+	0.999148i	$0.486863\pi$
$440$	0	0
$441$	5495.06	0.593355
$442$	−9141.82	−0.983783
$443$	−5220.82	−0.559930	−0.279965	−	0.960010i	$-0.590323\pi$
$443$	−5220.82	−0.559930	−0.279965	+	0.960010i	$0.590323\pi$
$444$	2264.87	0.242086
$445$	0	0
$446$	526.241	0.0558705
$447$	−6455.87	−0.683115
$448$	−1681.40	−0.177319
$449$	144.140	0.0151501	0.00757506	−	0.999971i	$-0.497589\pi$
$449$	144.140	0.0151501	0.00757506	+	0.999971i	$0.497589\pi$
$450$	0	0
$451$	−13546.2	−1.41433
$452$	−886.902	−0.0922928
$453$	−2935.29	−0.304442
$454$	6042.15	0.624608
$455$	0	0
$456$	−5798.54	−0.595485
$457$	−4176.41	−0.427493	−0.213747	−	0.976889i	$-0.568567\pi$
$457$	−4176.41	−0.427493	−0.213747	+	0.976889i	$0.568567\pi$
$458$	1180.81	0.120470
$459$	−9706.74	−0.987085
$460$	0	0
$461$	10280.5	1.03864	0.519318	−	0.854581i	$-0.326186\pi$
$461$	10280.5	1.03864	0.519318	+	0.854581i	$0.326186\pi$
$462$	1932.69	0.194625
$463$	63.5675	0.00638063	0.00319032	−	0.999995i	$-0.498984\pi$
$463$	63.5675	0.00638063	0.00319032	+	0.999995i	$0.498984\pi$
$464$	534.213	0.0534487
$465$	0	0
$466$	−2240.87	−0.222760
$467$	13276.1	1.31551	0.657754	−	0.753232i	$-0.271506\pi$
$467$	13276.1	1.31551	0.657754	+	0.753232i	$0.271506\pi$
$468$	8515.97	0.841134
$469$	5560.34	0.547447
$470$	0	0
$471$	−3536.09	−0.345933
$472$	10264.5	1.00097
$473$	15649.8	1.52131
$474$	4158.53	0.402969
$475$	0	0
$476$	−4724.84	−0.454963
$477$	−11610.1	−1.11445
$478$	17.2372	0.00164940
$479$	362.208	0.0345505	0.0172752	−	0.999851i	$-0.494501\pi$
$479$	362.208	0.0345505	0.0172752	+	0.999851i	$0.494501\pi$
$480$	0	0
$481$	12031.7	1.14054
$482$	−3100.42	−0.292989
$483$	−3936.69	−0.370861
$484$	−11771.0	−1.10547
$485$	0	0
$486$	−5740.42	−0.535784
$487$	−4517.95	−0.420386	−0.210193	−	0.977660i	$-0.567409\pi$
$487$	−4517.95	−0.420386	−0.210193	+	0.977660i	$0.567409\pi$
$488$	−6765.27	−0.627560
$489$	−3832.95	−0.354462
$490$	0	0
$491$	13557.8	1.24614	0.623070	−	0.782166i	$-0.285885\pi$
$491$	13557.8	1.24614	0.623070	+	0.782166i	$0.285885\pi$
$492$	−3006.28	−0.275475
$493$	−3406.40	−0.311189
$494$	−12788.4	−1.16473
$495$	0	0
$496$	−2182.39	−0.197565
$497$	10004.1	0.902911
$498$	288.303	0.0259421
$499$	−9368.11	−0.840429	−0.420215	−	0.907425i	$-0.638045\pi$
$499$	−9368.11	−0.840429	−0.420215	+	0.907425i	$0.638045\pi$
$500$	0	0
$501$	709.643	0.0632825
$502$	598.503	0.0532121
$503$	14154.3	1.25469	0.627345	−	0.778741i	$-0.284142\pi$
$503$	14154.3	1.25469	0.627345	+	0.778741i	$0.284142\pi$
$504$	−4333.11	−0.382960
$505$	0	0
$506$	−16092.4	−1.41382
$507$	−5777.39	−0.506080
$508$	−14085.5	−1.23020
$509$	−20074.0	−1.74806	−0.874032	−	0.485868i	$-0.838504\pi$
$509$	−20074.0	−1.74806	−0.874032	+	0.485868i	$0.838504\pi$
$510$	0	0
$511$	−4870.81	−0.421667
$512$	4847.09	0.418385
$513$	−13578.7	−1.16864
$514$	2643.12	0.226815
$515$	0	0
$516$	3473.14	0.296311
$517$	−21795.2	−1.85407
$518$	−2541.60	−0.215582
$519$	7951.61	0.672519
$520$	0	0
$521$	−9883.02	−0.831061	−0.415531	−	0.909579i	$-0.636404\pi$
$521$	−9883.02	−0.831061	−0.415531	+	0.909579i	$0.636404\pi$
$522$	−1296.95	−0.108747
$523$	−11943.5	−0.998573	−0.499286	−	0.866437i	$-0.666404\pi$
$523$	−11943.5	−0.998573	−0.499286	+	0.866437i	$0.666404\pi$
$524$	−705.894	−0.0588495
$525$	0	0
$526$	−291.581	−0.0241702
$527$	13916.0	1.15027
$528$	−1819.85	−0.149998
$529$	20611.6	1.69406
$530$	0	0
$531$	10738.2	0.877585
$532$	−6609.53	−0.538645
$533$	−15970.3	−1.29785
$534$	591.999	0.0479744
$535$	0	0
$536$	12150.3	0.979131
$537$	−6090.80	−0.489455
$538$	4733.17	0.379296
$539$	−14704.8	−1.17510
$540$	0	0
$541$	−12252.3	−0.973690	−0.486845	−	0.873488i	$-0.661852\pi$
$541$	−12252.3	−0.973690	−0.486845	+	0.873488i	$0.661852\pi$
$542$	−12406.2	−0.983193
$543$	−5270.02	−0.416497
$544$	−16362.9	−1.28962
$545$	0	0
$546$	2278.55	0.178595
$547$	−6227.09	−0.486748	−0.243374	−	0.969933i	$-0.578254\pi$
$547$	−6227.09	−0.486748	−0.243374	+	0.969933i	$0.578254\pi$
$548$	−8189.53	−0.638393
$549$	−7077.51	−0.550202
$550$	0	0
$551$	−4765.17	−0.368427
$552$	−8602.38	−0.663300
$553$	11417.7	0.877991
$554$	9794.09	0.751103
$555$	0	0
$556$	2469.14	0.188336
$557$	−4186.97	−0.318505	−0.159253	−	0.987238i	$-0.550908\pi$
$557$	−4186.97	−0.318505	−0.159253	+	0.987238i	$0.550908\pi$
$558$	5298.35	0.401966
$559$	18450.5	1.39601
$560$	0	0
$561$	11604.2	0.873318
$562$	−9719.20	−0.729501
$563$	11298.2	0.845757	0.422878	−	0.906186i	$-0.361020\pi$
$563$	11298.2	0.845757	0.422878	+	0.906186i	$0.361020\pi$
$564$	−4836.97	−0.361123
$565$	0	0
$566$	2250.83	0.167154
$567$	−3194.57	−0.236613
$568$	21860.8	1.61489
$569$	13663.2	1.00667	0.503333	−	0.864093i	$-0.332107\pi$
$569$	13663.2	1.00667	0.503333	+	0.864093i	$0.332107\pi$
$570$	0	0
$571$	−9967.49	−0.730519	−0.365260	−	0.930906i	$-0.619020\pi$
$571$	−9967.49	−0.730519	−0.365260	+	0.930906i	$0.619020\pi$
$572$	−22788.8	−1.66581
$573$	−5971.30	−0.435348
$574$	3373.60	0.245316
$575$	0	0
$576$	−3843.74	−0.278049
$577$	−9794.28	−0.706657	−0.353329	−	0.935499i	$-0.614950\pi$
$577$	−9794.28	−0.706657	−0.353329	+	0.935499i	$0.614950\pi$
$578$	4109.92	0.295761
$579$	−3994.76	−0.286730
$580$	0	0
$581$	791.568	0.0565228
$582$	1613.77	0.114936
$583$	31068.7	2.20709
$584$	−10643.6	−0.754169
$585$	0	0
$586$	−5166.82	−0.364231
$587$	7683.46	0.540256	0.270128	−	0.962824i	$-0.412934\pi$
$587$	7683.46	0.540256	0.270128	+	0.962824i	$0.412934\pi$
$588$	−3263.41	−0.228879
$589$	19466.9	1.36184
$590$	0	0
$591$	8488.41	0.590806
$592$	2393.22	0.166150
$593$	9643.48	0.667808	0.333904	−	0.942607i	$-0.391634\pi$
$593$	9643.48	0.667808	0.333904	+	0.942607i	$0.391634\pi$
$594$	9889.79	0.683136
$595$	0	0
$596$	−16080.3	−1.10516
$597$	8358.16	0.572993
$598$	−18972.2	−1.29737
$599$	7857.22	0.535955	0.267978	−	0.963425i	$-0.413645\pi$
$599$	7857.22	0.535955	0.267978	+	0.963425i	$0.413645\pi$
$600$	0	0
$601$	−22538.6	−1.52973	−0.764864	−	0.644191i	$-0.777194\pi$
$601$	−22538.6	−1.52973	−0.764864	+	0.644191i	$0.777194\pi$
$602$	−3897.50	−0.263871
$603$	12711.1	0.858435
$604$	−7311.23	−0.492533
$605$	0	0
$606$	3650.05	0.244675
$607$	25082.9	1.67724	0.838620	−	0.544717i	$-0.183363\pi$
$607$	25082.9	1.67724	0.838620	+	0.544717i	$0.183363\pi$
$608$	−22889.9	−1.52682
$609$	849.025	0.0564930
$610$	0	0
$611$	−25695.6	−1.70136
$612$	−10801.1	−0.713415
$613$	−7745.54	−0.510342	−0.255171	−	0.966896i	$-0.582132\pi$
$613$	−7745.54	−0.510342	−0.255171	+	0.966896i	$0.582132\pi$
$614$	10937.0	0.718862
$615$	0	0
$616$	11595.4	0.758430
$617$	1569.47	0.102406	0.0512031	−	0.998688i	$-0.483694\pi$
$617$	1569.47	0.102406	0.0512031	+	0.998688i	$0.483694\pi$
$618$	−4986.79	−0.324592
$619$	21425.6	1.39122	0.695612	−	0.718418i	$-0.255133\pi$
$619$	21425.6	1.39122	0.695612	+	0.718418i	$0.255133\pi$
$620$	0	0
$621$	−20144.5	−1.30173
$622$	7247.83	0.467221
$623$	1625.40	0.104527
$624$	−2145.52	−0.137644
$625$	0	0
$626$	2385.83	0.152327
$627$	16233.1	1.03395
$628$	−8807.71	−0.559659
$629$	−15260.3	−0.967359
$630$	0	0
$631$	11830.4	0.746371	0.373186	−	0.927757i	$-0.378265\pi$
$631$	11830.4	0.746371	0.373186	+	0.927757i	$0.378265\pi$
$632$	24949.7	1.57032
$633$	−12656.8	−0.794726
$634$	13833.4	0.866551
$635$	0	0
$636$	6895.02	0.429883
$637$	−17336.3	−1.07832
$638$	3470.63	0.215366
$639$	22869.8	1.41583
$640$	0	0
$641$	29383.1	1.81055	0.905274	−	0.424828i	$-0.139665\pi$
$641$	29383.1	1.81055	0.905274	+	0.424828i	$0.139665\pi$
$642$	−2211.77	−0.135968
$643$	−21889.3	−1.34250	−0.671251	−	0.741230i	$-0.734243\pi$
$643$	−21889.3	−1.34250	−0.671251	+	0.741230i	$0.734243\pi$
$644$	−9805.52	−0.599987
$645$	0	0
$646$	16220.0	0.987877
$647$	−9421.28	−0.572471	−0.286235	−	0.958159i	$-0.592404\pi$
$647$	−9421.28	−0.572471	−0.286235	+	0.958159i	$0.592404\pi$
$648$	−6980.71	−0.423192
$649$	−28735.4	−1.73800
$650$	0	0
$651$	−3468.48	−0.208818
$652$	−9547.11	−0.573457
$653$	1168.63	0.0700335	0.0350168	−	0.999387i	$-0.488852\pi$
$653$	1168.63	0.0700335	0.0350168	+	0.999387i	$0.488852\pi$
$654$	−7311.20	−0.437142
$655$	0	0
$656$	−3176.64	−0.189066
$657$	−11134.8	−0.661204
$658$	5427.97	0.321587
$659$	13876.8	0.820277	0.410139	−	0.912023i	$-0.365480\pi$
$659$	13876.8	0.820277	0.410139	+	0.912023i	$0.365480\pi$
$660$	0	0
$661$	−6084.50	−0.358033	−0.179016	−	0.983846i	$-0.557291\pi$
$661$	−6084.50	−0.358033	−0.179016	+	0.983846i	$0.557291\pi$
$662$	8170.59	0.479696
$663$	13680.9	0.801389
$664$	1729.72	0.101093
$665$	0	0
$666$	−5810.19	−0.338048
$667$	−7069.35	−0.410384
$668$	1767.58	0.102380
$669$	−787.527	−0.0455121
$670$	0	0
$671$	18939.4	1.08964
$672$	4078.35	0.234116
$673$	7481.24	0.428500	0.214250	−	0.976779i	$-0.431269\pi$
$673$	7481.24	0.428500	0.214250	+	0.976779i	$0.431269\pi$
$674$	16606.1	0.949023
$675$	0	0
$676$	−14390.3	−0.818748
$677$	−20122.6	−1.14235	−0.571176	−	0.820827i	$-0.693513\pi$
$677$	−20122.6	−1.14235	−0.571176	+	0.820827i	$0.693513\pi$
$678$	−542.478	−0.0307282
$679$	4430.78	0.250424
$680$	0	0
$681$	−9042.16	−0.508805
$682$	−14178.4	−0.796069
$683$	−21064.6	−1.18011	−0.590054	−	0.807364i	$-0.700894\pi$
$683$	−21064.6	−1.18011	−0.590054	+	0.807364i	$0.700894\pi$
$684$	−15109.6	−0.844634
$685$	0	0
$686$	8645.82	0.481194
$687$	−1767.09	−0.0981351
$688$	3669.96	0.203366
$689$	36628.6	2.02531
$690$	0	0
$691$	11210.0	0.617145	0.308573	−	0.951201i	$-0.400149\pi$
$691$	11210.0	0.617145	0.308573	+	0.951201i	$0.400149\pi$
$692$	19805.9	1.08802
$693$	12130.6	0.664939
$694$	−7006.88	−0.383253
$695$	0	0
$696$	1855.27	0.101040
$697$	20255.8	1.10078
$698$	−4125.61	−0.223720
$699$	3353.50	0.181460
$700$	0	0
$701$	10860.9	0.585178	0.292589	−	0.956238i	$-0.405483\pi$
$701$	10860.9	0.585178	0.292589	+	0.956238i	$0.405483\pi$
$702$	11659.6	0.626871
$703$	−21347.5	−1.14529
$704$	10285.9	0.550658
$705$	0	0
$706$	8673.69	0.462378
$707$	10021.6	0.533098
$708$	−6377.20	−0.338517
$709$	−14059.1	−0.744709	−0.372355	−	0.928090i	$-0.621450\pi$
$709$	−14059.1	−0.744709	−0.372355	+	0.928090i	$0.621450\pi$
$710$	0	0
$711$	26101.2	1.37675
$712$	3551.78	0.186950
$713$	28880.1	1.51692
$714$	−2889.97	−0.151477
$715$	0	0
$716$	−15171.0	−0.791852
$717$	−25.7957	−0.00134360
$718$	17944.6	0.932714
$719$	−12419.6	−0.644191	−0.322096	−	0.946707i	$-0.604387\pi$
$719$	−12419.6	−0.644191	−0.322096	+	0.946707i	$0.604387\pi$
$720$	0	0
$721$	−13691.8	−0.707223
$722$	12240.3	0.630936
$723$	4639.83	0.238668
$724$	−13126.6	−0.673819
$725$	0	0
$726$	−7199.79	−0.368057
$727$	−15819.2	−0.807019	−0.403509	−	0.914975i	$-0.632210\pi$
$727$	−15819.2	−0.807019	−0.403509	+	0.914975i	$0.632210\pi$
$728$	13670.5	0.695963
$729$	−453.498	−0.0230401
$730$	0	0
$731$	−23401.4	−1.18404
$732$	4203.20	0.212233
$733$	171.656	0.00864974	0.00432487	−	0.999991i	$-0.498623\pi$
$733$	171.656	0.00864974	0.00432487	+	0.999991i	$0.498623\pi$
$734$	−4487.10	−0.225643
$735$	0	0
$736$	−33958.1	−1.70070
$737$	−34015.0	−1.70008
$738$	7712.15	0.384673
$739$	21681.8	1.07926	0.539632	−	0.841901i	$-0.318563\pi$
$739$	21681.8	1.07926	0.539632	+	0.841901i	$0.318563\pi$
$740$	0	0
$741$	19138.0	0.948790
$742$	−7737.48	−0.382819
$743$	−34876.3	−1.72205	−0.861027	−	0.508559i	$-0.830179\pi$
$743$	−34876.3	−1.72205	−0.861027	+	0.508559i	$0.830179\pi$
$744$	−7579.25	−0.373480
$745$	0	0
$746$	2599.81	0.127595
$747$	1809.55	0.0886318
$748$	28903.9	1.41287
$749$	−6072.65	−0.296248
$750$	0	0
$751$	−15466.9	−0.751524	−0.375762	−	0.926716i	$-0.622619\pi$
$751$	−15466.9	−0.751524	−0.375762	+	0.926716i	$0.622619\pi$
$752$	−5111.08	−0.247848
$753$	−895.668	−0.0433466
$754$	4091.72	0.197628
$755$	0	0
$756$	6026.12	0.289905
$757$	24842.6	1.19276	0.596380	−	0.802702i	$-0.296605\pi$
$757$	24842.6	1.19276	0.596380	+	0.802702i	$0.296605\pi$
$758$	−17533.2	−0.840152
$759$	24082.5	1.15170
$760$	0	0
$761$	−830.614	−0.0395660	−0.0197830	−	0.999804i	$-0.506298\pi$
$761$	−830.614	−0.0395660	−0.0197830	+	0.999804i	$0.506298\pi$
$762$	−8615.46	−0.409587
$763$	−20073.7	−0.952446
$764$	−14873.3	−0.704316
$765$	0	0
$766$	−17332.0	−0.817535
$767$	−33877.8	−1.59486
$768$	7494.79	0.352142
$769$	22797.1	1.06903	0.534514	−	0.845159i	$-0.320495\pi$
$769$	22797.1	1.06903	0.534514	+	0.845159i	$0.320495\pi$
$770$	0	0
$771$	−3955.46	−0.184763
$772$	−9950.14	−0.463878
$773$	11780.8	0.548157	0.274079	−	0.961707i	$-0.411627\pi$
$773$	11780.8	0.548157	0.274079	+	0.961707i	$0.411627\pi$
$774$	−8909.82	−0.413769
$775$	0	0
$776$	9682.05	0.447894
$777$	3803.55	0.175613
$778$	−2216.93	−0.102160
$779$	28335.6	1.30325
$780$	0	0
$781$	−61199.6	−2.80396
$782$	24063.1	1.10038
$783$	4344.56	0.198291
$784$	−3448.35	−0.157086
$785$	0	0
$786$	−431.763	−0.0195935
$787$	3803.40	0.172270	0.0861350	−	0.996283i	$-0.472548\pi$
$787$	3803.40	0.172270	0.0861350	+	0.996283i	$0.472548\pi$
$788$	21142.9	0.955819
$789$	436.356	0.0196891
$790$	0	0
$791$	−1489.43	−0.0669508
$792$	26507.5	1.18927
$793$	22328.7	0.999894
$794$	−15897.8	−0.710570
$795$	0	0
$796$	20818.5	0.927001
$797$	−28395.0	−1.26199	−0.630993	−	0.775789i	$-0.717352\pi$
$797$	−28395.0	−1.26199	−0.630993	+	0.775789i	$0.717352\pi$
$798$	−4042.75	−0.179338
$799$	32590.7	1.44302
$800$	0	0
$801$	3715.71	0.163905
$802$	−6802.34	−0.299500
$803$	29796.8	1.30947
$804$	−7548.88	−0.331130
$805$	0	0
$806$	−16715.7	−0.730503
$807$	−7083.25	−0.308974
$808$	21899.0	0.953469
$809$	−21087.0	−0.916416	−0.458208	−	0.888845i	$-0.651508\pi$
$809$	−21087.0	−0.916416	−0.458208	+	0.888845i	$0.651508\pi$
$810$	0	0
$811$	5516.13	0.238838	0.119419	−	0.992844i	$-0.461897\pi$
$811$	5516.13	0.238838	0.119419	+	0.992844i	$0.461897\pi$
$812$	2114.75	0.0913956
$813$	18566.0	0.800908
$814$	15548.1	0.669484
$815$	0	0
$816$	2721.25	0.116744
$817$	−32736.0	−1.40182
$818$	−1989.58	−0.0850417
$819$	14301.4	0.610173
$820$	0	0
$821$	−9186.35	−0.390507	−0.195253	−	0.980753i	$-0.562553\pi$
$821$	−9186.35	−0.390507	−0.195253	+	0.980753i	$0.562553\pi$
$822$	−5009.17	−0.212548
$823$	−30176.7	−1.27812	−0.639060	−	0.769157i	$-0.720676\pi$
$823$	−30176.7	−1.27812	−0.639060	+	0.769157i	$0.720676\pi$
$824$	−29918.9	−1.26490
$825$	0	0
$826$	7156.39	0.301456
$827$	4173.79	0.175498	0.0877491	−	0.996143i	$-0.472033\pi$
$827$	4173.79	0.175498	0.0877491	+	0.996143i	$0.472033\pi$
$828$	−22415.7	−0.940822
$829$	18419.9	0.771714	0.385857	−	0.922559i	$-0.373906\pi$
$829$	18419.9	0.771714	0.385857	+	0.922559i	$0.373906\pi$
$830$	0	0
$831$	−14657.0	−0.611848
$832$	12126.6	0.505304
$833$	21988.3	0.914585
$834$	1510.26	0.0627051
$835$	0	0
$836$	40433.3	1.67275
$837$	−17748.6	−0.732954
$838$	−14565.3	−0.600418
$839$	21442.5	0.882335	0.441167	−	0.897425i	$-0.354565\pi$
$839$	21442.5	0.882335	0.441167	+	0.897425i	$0.354565\pi$
$840$	0	0
$841$	−22864.4	−0.937486
$842$	19387.4	0.793508
$843$	14544.9	0.594251
$844$	−31525.5	−1.28573
$845$	0	0
$846$	12408.5	0.504272
$847$	−19767.8	−0.801924
$848$	7285.75	0.295040
$849$	−3368.39	−0.136164
$850$	0	0
$851$	−31670.0	−1.27571
$852$	−13581.9	−0.546137
$853$	27609.3	1.10823	0.554117	−	0.832439i	$-0.313056\pi$
$853$	27609.3	1.10823	0.554117	+	0.832439i	$0.313056\pi$
$854$	−4716.76	−0.188998
$855$	0	0
$856$	−13269.8	−0.529852
$857$	3202.65	0.127655	0.0638275	−	0.997961i	$-0.479669\pi$
$857$	3202.65	0.127655	0.0638275	+	0.997961i	$0.479669\pi$
$858$	−13938.9	−0.554621
$859$	−18828.1	−0.747855	−0.373928	−	0.927458i	$-0.621989\pi$
$859$	−18828.1	−0.747855	−0.373928	+	0.927458i	$0.621989\pi$
$860$	0	0
$861$	−5048.64	−0.199834
$862$	−5793.00	−0.228898
$863$	17394.9	0.686129	0.343064	−	0.939312i	$-0.388535\pi$
$863$	17394.9	0.686129	0.343064	+	0.939312i	$0.388535\pi$
$864$	20869.4	0.821751
$865$	0	0
$866$	−7630.10	−0.299401
$867$	−6150.55	−0.240927
$868$	−8639.30	−0.337830
$869$	−69846.8	−2.72657
$870$	0	0
$871$	−40102.1	−1.56005
$872$	−43864.6	−1.70349
$873$	10128.9	0.392682
$874$	33661.7	1.30277
$875$	0	0
$876$	6612.76	0.255051
$877$	26088.8	1.00451	0.502255	−	0.864719i	$-0.332504\pi$
$877$	26088.8	1.00451	0.502255	+	0.864719i	$0.332504\pi$
$878$	1156.37	0.0444481
$879$	7732.22	0.296702
$880$	0	0
$881$	42096.4	1.60983	0.804917	−	0.593388i	$-0.202210\pi$
$881$	42096.4	1.60983	0.804917	+	0.593388i	$0.202210\pi$
$882$	8371.79	0.319606
$883$	−9252.21	−0.352618	−0.176309	−	0.984335i	$-0.556416\pi$
$883$	−9252.21	−0.352618	−0.176309	+	0.984335i	$0.556416\pi$
$884$	34076.3	1.29651
$885$	0	0
$886$	−7953.98	−0.301602
$887$	16116.1	0.610062	0.305031	−	0.952342i	$-0.401333\pi$
$887$	16116.1	0.610062	0.305031	+	0.952342i	$0.401333\pi$
$888$	8311.43	0.314092
$889$	−23654.7	−0.892410
$890$	0	0
$891$	19542.6	0.734793
$892$	−1961.57	−0.0736305
$893$	45590.8	1.70844
$894$	−9835.59	−0.367955
$895$	0	0
$896$	11748.7	0.438053
$897$	28392.1	1.05684
$898$	219.599	0.00816050
$899$	−6228.55	−0.231072
$900$	0	0
$901$	−46457.4	−1.71778
$902$	−20637.8	−0.761820
$903$	5832.67	0.214949
$904$	−3254.67	−0.119744
$905$	0	0
$906$	−4471.95	−0.163985
$907$	−26185.5	−0.958628	−0.479314	−	0.877643i	$-0.659114\pi$
$907$	−26185.5	−0.958628	−0.479314	+	0.877643i	$0.659114\pi$
$908$	−22522.2	−0.823157
$909$	22909.7	0.835936
$910$	0	0
$911$	−27823.4	−1.01189	−0.505945	−	0.862566i	$-0.668856\pi$
$911$	−27823.4	−1.01189	−0.505945	+	0.862566i	$0.668856\pi$
$912$	3806.73	0.138216
$913$	−4842.36	−0.175530
$914$	−6362.81	−0.230266
$915$	0	0
$916$	−4401.48	−0.158765
$917$	−1185.45	−0.0426904
$918$	−14788.3	−0.531686
$919$	47772.2	1.71476	0.857378	−	0.514688i	$-0.172092\pi$
$919$	47772.2	1.71476	0.857378	+	0.514688i	$0.172092\pi$
$920$	0	0
$921$	−16367.4	−0.585584
$922$	15662.5	0.559454
$923$	−72151.5	−2.57302
$924$	−7204.12	−0.256492
$925$	0	0
$926$	96.8458	0.00343688
$927$	−31299.8	−1.10897
$928$	7323.73	0.259066
$929$	1947.77	0.0687883	0.0343942	−	0.999408i	$-0.489050\pi$
$929$	1947.77	0.0687883	0.0343942	+	0.999408i	$0.489050\pi$
$930$	0	0
$931$	30759.2	1.08281
$932$	8352.89	0.293571
$933$	−10846.5	−0.380598
$934$	20226.2	0.708589
$935$	0	0
$936$	31251.1	1.09132
$937$	19185.4	0.668899	0.334449	−	0.942414i	$-0.391450\pi$
$937$	19185.4	0.668899	0.334449	+	0.942414i	$0.391450\pi$
$938$	8471.23	0.294878
$939$	−3570.43	−0.124086
$940$	0	0
$941$	45452.2	1.57460	0.787300	−	0.616570i	$-0.211478\pi$
$941$	45452.2	1.57460	0.787300	+	0.616570i	$0.211478\pi$
$942$	−5387.28	−0.186334
$943$	42037.1	1.45166
$944$	−6738.59	−0.232333
$945$	0	0
$946$	23842.7	0.819443
$947$	−7896.17	−0.270952	−0.135476	−	0.990781i	$-0.543256\pi$
$947$	−7896.17	−0.270952	−0.135476	+	0.990781i	$0.543256\pi$
$948$	−15501.0	−0.531064
$949$	35129.1	1.20162
$950$	0	0
$951$	−20701.8	−0.705892
$952$	−17338.8	−0.590287
$953$	−18152.5	−0.617017	−0.308508	−	0.951222i	$-0.599830\pi$
$953$	−18152.5	−0.617017	−0.308508	+	0.951222i	$0.599830\pi$
$954$	−17688.1	−0.600287
$955$	0	0
$956$	−64.2520	−0.00217370
$957$	−5193.85	−0.175437
$958$	551.827	0.0186104
$959$	−13753.2	−0.463101
$960$	0	0
$961$	−4345.81	−0.145876
$962$	18330.5	0.614343
$963$	−13882.3	−0.464538
$964$	11556.9	0.386123
$965$	0	0
$966$	−5997.60	−0.199761
$967$	23418.6	0.778791	0.389396	−	0.921071i	$-0.372684\pi$
$967$	23418.6	0.778791	0.389396	+	0.921071i	$0.372684\pi$
$968$	−43196.2	−1.43427
$969$	−24273.5	−0.804724
$970$	0	0
$971$	43994.5	1.45402	0.727009	−	0.686628i	$-0.240910\pi$
$971$	43994.5	1.45402	0.727009	+	0.686628i	$0.240910\pi$
$972$	21397.5	0.706097
$973$	4146.59	0.136622
$974$	−6883.15	−0.226438
$975$	0	0
$976$	4441.38	0.145661
$977$	40451.4	1.32462	0.662310	−	0.749230i	$-0.269576\pi$
$977$	40451.4	1.32462	0.662310	+	0.749230i	$0.269576\pi$
$978$	−5839.54	−0.190928
$979$	−9943.25	−0.324604
$980$	0	0
$981$	−45889.1	−1.49350
$982$	20655.4	0.671224
$983$	20180.4	0.654785	0.327393	−	0.944888i	$-0.393830\pi$
$983$	20180.4	0.654785	0.327393	+	0.944888i	$0.393830\pi$
$984$	−11032.2	−0.357412
$985$	0	0
$986$	−5189.68	−0.167620
$987$	−8123.04	−0.261965
$988$	47669.0	1.53497
$989$	−48565.3	−1.56146
$990$	0	0
$991$	−37145.5	−1.19068	−0.595341	−	0.803473i	$-0.702983\pi$
$991$	−37145.5	−1.19068	−0.595341	+	0.803473i	$0.702983\pi$
$992$	−29919.3	−0.957599
$993$	−12227.4	−0.390760
$994$	15241.4	0.486346
$995$	0	0
$996$	−1074.66	−0.0341886
$997$	−6005.07	−0.190755	−0.0953774	−	0.995441i	$-0.530406\pi$
$997$	−6005.07	−0.190755	−0.0953774	+	0.995441i	$0.530406\pi$
$998$	−14272.4	−0.452691
$999$	19463.2	0.616405

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2075.4.a.g.1.14		21
5.4	even	2		415.4.a.c.1.8	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
415.4.a.c.1.8	✓	21	5.4	even	2
2075.4.a.g.1.14		21	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)

\(f(q)\)	\(=\)	\(q+1.52351 q^{2} -2.27996 q^{3} -5.67892 q^{4} -3.47354 q^{6} -9.53696 q^{7} -20.8400 q^{8} -21.8018 q^{9} +58.3417 q^{11} +12.9477 q^{12} +68.7822 q^{13} -14.5297 q^{14} +13.6814 q^{16} -87.2391 q^{17} -33.2153 q^{18} -122.038 q^{19} +21.7439 q^{21} +88.8842 q^{22} -181.049 q^{23} +47.5142 q^{24} +104.790 q^{26} +111.266 q^{27} +54.1596 q^{28} +39.0467 q^{29} -159.515 q^{31} +187.564 q^{32} -133.016 q^{33} -132.910 q^{34} +123.811 q^{36} +174.925 q^{37} -185.926 q^{38} -156.820 q^{39} -232.187 q^{41} +33.1270 q^{42} +268.245 q^{43} -331.318 q^{44} -275.830 q^{46} -373.579 q^{47} -31.1930 q^{48} -252.046 q^{49} +198.901 q^{51} -390.608 q^{52} +532.530 q^{53} +169.515 q^{54} +198.750 q^{56} +278.241 q^{57} +59.4880 q^{58} -492.537 q^{59} +324.629 q^{61} -243.024 q^{62} +207.923 q^{63} +176.304 q^{64} -202.652 q^{66} -583.030 q^{67} +495.424 q^{68} +412.783 q^{69} -1048.99 q^{71} +454.349 q^{72} +510.729 q^{73} +266.500 q^{74} +693.043 q^{76} -556.403 q^{77} -238.917 q^{78} -1197.20 q^{79} +334.967 q^{81} -353.739 q^{82} -83.0000 q^{83} -123.481 q^{84} +408.673 q^{86} -89.0247 q^{87} -1215.84 q^{88} -170.431 q^{89} -655.973 q^{91} +1028.16 q^{92} +363.688 q^{93} -569.151 q^{94} -427.636 q^{96} -464.590 q^{97} -383.995 q^{98} -1271.95 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 5 q^{2} + 12 q^{3} + 87 q^{4} - 7 q^{6} + 11 q^{7} + 84 q^{8} + 153 q^{9} - 30 q^{11} + 244 q^{12} + 89 q^{13} - 191 q^{14} + 583 q^{16} + 357 q^{17} + 281 q^{18} - 175 q^{19} - 41 q^{21} + 122 q^{22}+ \cdots - 5369 q^{99}+O(q^{100}) \) 21 * q + 5 * q^2 + 12 * q^3 + 87 * q^4 - 7 * q^6 + 11 * q^7 + 84 * q^8 + 153 * q^9 - 30 * q^11 + 244 * q^12 + 89 * q^13 - 191 * q^14 + 583 * q^16 + 357 * q^17 + 281 * q^18 - 175 * q^19 - 41 * q^21 + 122 * q^22 + 554 * q^23 - 136 * q^24 - 933 * q^26 + 555 * q^27 + 773 * q^28 - 96 * q^29 + 99 * q^31 + 1307 * q^32 + 1097 * q^33 + 393 * q^34 - 746 * q^36 + 756 * q^37 + 1192 * q^38 + 40 * q^39 - 1436 * q^41 + 1112 * q^42 + 1003 * q^43 - 706 * q^44 - 775 * q^46 + 858 * q^47 + 2837 * q^48 + 1384 * q^49 - 279 * q^51 + 1092 * q^52 + 1625 * q^53 - 269 * q^54 - 3806 * q^56 + 1740 * q^57 + 1087 * q^58 - 1224 * q^59 - 794 * q^61 + 2979 * q^62 + 1236 * q^63 + 2570 * q^64 - 1094 * q^66 + 571 * q^67 + 4064 * q^68 + 774 * q^69 - 1078 * q^71 + 2610 * q^72 + 1986 * q^73 - 1872 * q^74 - 2854 * q^76 + 2519 * q^77 + 2449 * q^78 - 432 * q^79 - 219 * q^81 + 2143 * q^82 - 1743 * q^83 - 377 * q^84 + 1556 * q^86 + 5681 * q^87 + 3158 * q^88 - 6252 * q^89 + 684 * q^91 + 7623 * q^92 + 3314 * q^93 - 3877 * q^94 - 954 * q^96 + 4880 * q^97 + 5072 * q^98 - 5369 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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