LMFDB - Embedded newform 2023.4.a.v.1.20

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2023,4,Mod(1,2023)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2023, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2023.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2023 = 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2023.a (trivial)

Newform invariants

comment:select newform

sage:traces = [56,8,24,240,80,68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$119.360863942$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	2023.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.91072 q^{2} -3.61256 q^{3} -4.34916 q^{4} -15.8569 q^{5} +6.90258 q^{6} +7.00000 q^{7} +23.5957 q^{8} -13.9494 q^{9} +30.2981 q^{10} -28.2171 q^{11} +15.7116 q^{12} +57.6130 q^{13} -13.3750 q^{14} +57.2841 q^{15} -10.2915 q^{16} +26.6533 q^{18} -82.0772 q^{19} +68.9644 q^{20} -25.2879 q^{21} +53.9148 q^{22} +170.815 q^{23} -85.2411 q^{24} +126.442 q^{25} -110.082 q^{26} +147.932 q^{27} -30.4441 q^{28} -110.697 q^{29} -109.454 q^{30} -30.1681 q^{31} -169.102 q^{32} +101.936 q^{33} -110.998 q^{35} +60.6682 q^{36} +36.2332 q^{37} +156.826 q^{38} -208.131 q^{39} -374.156 q^{40} +519.242 q^{41} +48.3181 q^{42} -346.930 q^{43} +122.721 q^{44} +221.195 q^{45} -326.378 q^{46} +90.0869 q^{47} +37.1785 q^{48} +49.0000 q^{49} -241.595 q^{50} -250.568 q^{52} -637.790 q^{53} -282.656 q^{54} +447.436 q^{55} +165.170 q^{56} +296.509 q^{57} +211.511 q^{58} -216.045 q^{59} -249.138 q^{60} -669.329 q^{61} +57.6426 q^{62} -97.6458 q^{63} +405.437 q^{64} -913.565 q^{65} -194.771 q^{66} +48.5739 q^{67} -617.078 q^{69} +212.087 q^{70} -814.570 q^{71} -329.146 q^{72} +677.396 q^{73} -69.2314 q^{74} -456.780 q^{75} +356.967 q^{76} -197.520 q^{77} +397.678 q^{78} +6.39511 q^{79} +163.191 q^{80} -157.781 q^{81} -992.124 q^{82} -427.264 q^{83} +109.981 q^{84} +662.885 q^{86} +399.901 q^{87} -665.803 q^{88} +522.938 q^{89} -422.640 q^{90} +403.291 q^{91} -742.901 q^{92} +108.984 q^{93} -172.130 q^{94} +1301.49 q^{95} +610.891 q^{96} +172.218 q^{97} -93.6251 q^{98} +393.611 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{2} + 24 q^{3} + 240 q^{4} + 80 q^{5} + 68 q^{6} + 392 q^{7} + 96 q^{8} + 576 q^{9} + 80 q^{10} + 176 q^{11} + 288 q^{12} - 96 q^{13} + 56 q^{14} + 192 q^{15} + 1088 q^{16} + 216 q^{18} + 48 q^{19}+ \cdots + 6576 q^{99}+O(q^{100}) $ 56 * q + 8 * q^2 + 24 * q^3 + 240 * q^4 + 80 * q^5 + 68 * q^6 + 392 * q^7 + 96 * q^8 + 576 * q^9 + 80 * q^10 + 176 * q^11 + 288 * q^12 - 96 * q^13 + 56 * q^14 + 192 * q^15 + 1088 * q^16 + 216 * q^18 + 48 * q^19 + 1660 * q^20 + 168 * q^21 + 352 * q^22 + 608 * q^23 + 736 * q^24 + 1792 * q^25 + 864 * q^27 + 1680 * q^28 + 928 * q^29 - 100 * q^30 + 496 * q^31 + 436 * q^32 + 384 * q^33 + 560 * q^35 + 2804 * q^36 + 1392 * q^37 + 624 * q^39 - 1020 * q^40 + 2088 * q^41 + 476 * q^42 + 1312 * q^43 + 2112 * q^44 + 4840 * q^45 + 736 * q^46 + 5228 * q^48 + 2744 * q^49 + 2020 * q^50 - 1536 * q^52 + 1272 * q^53 + 2268 * q^54 - 336 * q^55 + 672 * q^56 + 2448 * q^57 + 1376 * q^58 + 2372 * q^60 + 976 * q^61 + 3776 * q^62 + 4032 * q^63 + 5376 * q^64 + 6080 * q^65 + 976 * q^67 + 1248 * q^69 + 560 * q^70 + 3216 * q^71 + 4532 * q^72 + 4088 * q^73 - 2456 * q^74 - 760 * q^75 - 2640 * q^76 + 1232 * q^77 + 3744 * q^78 + 1792 * q^79 + 13920 * q^80 + 4184 * q^81 + 8204 * q^82 - 2832 * q^83 + 2016 * q^84 + 4908 * q^86 + 832 * q^87 + 2424 * q^88 - 2768 * q^89 + 8660 * q^90 - 672 * q^91 + 6784 * q^92 + 784 * q^93 - 928 * q^94 + 4560 * q^95 + 6656 * q^96 + 9464 * q^97 + 392 * q^98 + 6576 * 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 $

$2$	−1.91072	−0.675540	−0.337770	−	0.941229i	$-0.609673\pi$
$2$	−1.91072	−0.675540	−0.337770	+	0.941229i	$0.609673\pi$
$3$	−3.61256	−0.695238	−0.347619	−	0.937636i	$-0.613010\pi$
$3$	−3.61256	−0.695238	−0.347619	+	0.937636i	$0.613010\pi$
$4$	−4.34916	−0.543646
$5$	−15.8569	−1.41829	−0.709143	−	0.705064i	$-0.750918\pi$
$5$	−15.8569	−1.41829	−0.709143	+	0.705064i	$0.750918\pi$
$6$	6.90258	0.469661
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	23.5957	1.04279
$9$	−13.9494	−0.516644
$10$	30.2981	0.958110
$11$	−28.2171	−0.773434	−0.386717	−	0.922198i	$-0.626391\pi$
$11$	−28.2171	−0.773434	−0.386717	+	0.922198i	$0.626391\pi$
$12$	15.7116	0.377963
$13$	57.6130	1.22915	0.614576	−	0.788858i	$-0.289327\pi$
$13$	57.6130	1.22915	0.614576	+	0.788858i	$0.289327\pi$
$14$	−13.3750	−0.255330
$15$	57.2841	0.986047
$16$	−10.2915	−0.160804
$17$	0	0
$17$	0	0
$18$	26.6533	0.349014
$19$	−82.0772	−0.991042	−0.495521	−	0.868596i	$-0.665023\pi$
$19$	−82.0772	−0.991042	−0.495521	+	0.868596i	$0.665023\pi$
$20$	68.9644	0.771045
$21$	−25.2879	−0.262775
$22$	53.9148	0.522486
$23$	170.815	1.54858	0.774289	−	0.632832i	$-0.218108\pi$
$23$	170.815	1.54858	0.774289	+	0.632832i	$0.218108\pi$
$24$	−85.2411	−0.724990
$25$	126.442	1.01154
$26$	−110.082	−0.830341
$27$	147.932	1.05443
$28$	−30.4441	−0.205479
$29$	−110.697	−0.708826	−0.354413	−	0.935089i	$-0.615319\pi$
$29$	−110.697	−0.708826	−0.354413	+	0.935089i	$0.615319\pi$
$30$	−109.454	−0.666114
$31$	−30.1681	−0.174785	−0.0873927	−	0.996174i	$-0.527853\pi$
$31$	−30.1681	−0.174785	−0.0873927	+	0.996174i	$0.527853\pi$
$32$	−169.102	−0.934165
$33$	101.936	0.537721
$34$	0	0
$35$	−110.998	−0.536062
$36$	60.6682	0.280871
$37$	36.2332	0.160992	0.0804960	−	0.996755i	$-0.474350\pi$
$37$	36.2332	0.160992	0.0804960	+	0.996755i	$0.474350\pi$
$38$	156.826	0.669489
$39$	−208.131	−0.854553
$40$	−374.156	−1.47898
$41$	519.242	1.97785	0.988927	−	0.148404i	$-0.0474135\pi$
$41$	519.242	1.97785	0.988927	+	0.148404i	$0.0474135\pi$
$42$	48.3181	0.177515
$43$	−346.930	−1.23038	−0.615190	−	0.788379i	$-0.710921\pi$
$43$	−346.930	−1.23038	−0.615190	+	0.788379i	$0.710921\pi$
$44$	122.721	0.420474
$45$	221.195	0.732750
$46$	−326.378	−1.04613
$47$	90.0869	0.279586	0.139793	−	0.990181i	$-0.455356\pi$
$47$	90.0869	0.279586	0.139793	+	0.990181i	$0.455356\pi$
$48$	37.1785	0.111797
$49$	49.0000	0.142857
$50$	−241.595	−0.683334
$51$	0	0
$52$	−250.568	−0.668223
$53$	−637.790	−1.65297	−0.826483	−	0.562962i	$-0.809662\pi$
$53$	−637.790	−1.65297	−0.826483	+	0.562962i	$0.809662\pi$
$54$	−282.656	−0.712309
$55$	447.436	1.09695
$56$	165.170	0.394139
$57$	296.509	0.689010
$58$	211.511	0.478841
$59$	−216.045	−0.476722	−0.238361	−	0.971177i	$-0.576610\pi$
$59$	−216.045	−0.476722	−0.238361	+	0.971177i	$0.576610\pi$
$60$	−249.138	−0.536060
$61$	−669.329	−1.40490	−0.702449	−	0.711734i	$-0.747910\pi$
$61$	−669.329	−1.40490	−0.702449	+	0.711734i	$0.747910\pi$
$62$	57.6426	0.118075
$63$	−97.6458	−0.195273
$64$	405.437	0.791870
$65$	−913.565	−1.74329
$66$	−194.771	−0.363252
$67$	48.5739	0.0885708	0.0442854	−	0.999019i	$-0.485899\pi$
$67$	48.5739	0.0885708	0.0442854	+	0.999019i	$0.485899\pi$
$68$	0	0
$69$	−617.078	−1.07663
$70$	212.087	0.362131
$71$	−814.570	−1.36157	−0.680786	−	0.732482i	$-0.738362\pi$
$71$	−814.570	−1.36157	−0.680786	+	0.732482i	$0.738362\pi$
$72$	−329.146	−0.538754
$73$	677.396	1.08607	0.543036	−	0.839709i	$-0.317275\pi$
$73$	677.396	1.08607	0.543036	+	0.839709i	$0.317275\pi$
$74$	−69.2314	−0.108757
$75$	−456.780	−0.703259
$76$	356.967	0.538775
$77$	−197.520	−0.292330
$78$	397.678	0.577285
$79$	6.39511	0.00910767	0.00455384	−	0.999990i	$-0.498550\pi$
$79$	6.39511	0.00910767	0.00455384	+	0.999990i	$0.498550\pi$
$80$	163.191	0.228066
$81$	−157.781	−0.216435
$82$	−992.124	−1.33612
$83$	−427.264	−0.565040	−0.282520	−	0.959261i	$-0.591170\pi$
$83$	−427.264	−0.565040	−0.282520	+	0.959261i	$0.591170\pi$
$84$	109.981	0.142857
$85$	0	0
$86$	662.885	0.831172
$87$	399.901	0.492803
$88$	−665.803	−0.806532
$89$	522.938	0.622824	0.311412	−	0.950275i	$-0.399198\pi$
$89$	522.938	0.622824	0.311412	+	0.950275i	$0.399198\pi$
$90$	−422.640	−0.495002
$91$	403.291	0.464576
$92$	−742.901	−0.841878
$93$	108.984	0.121517
$94$	−172.130	−0.188871
$95$	1301.49	1.40558
$96$	610.891	0.649467
$97$	172.218	0.180269	0.0901345	−	0.995930i	$-0.471270\pi$
$97$	172.218	0.180269	0.0901345	+	0.995930i	$0.471270\pi$
$98$	−93.6251	−0.0965057
$99$	393.611	0.399590
$100$	−549.918	−0.549918
$101$	−1909.23	−1.88095	−0.940474	−	0.339865i	$-0.889619\pi$
$101$	−1909.23	−1.88095	−0.940474	+	0.339865i	$0.889619\pi$
$102$	0	0
$103$	−1592.64	−1.52357	−0.761784	−	0.647831i	$-0.775676\pi$
$103$	−1592.64	−1.52357	−0.761784	+	0.647831i	$0.775676\pi$
$104$	1359.42	1.28175
$105$	400.989	0.372691
$106$	1218.64	1.11664
$107$	−235.424	−0.212704	−0.106352	−	0.994329i	$-0.533917\pi$
$107$	−235.424	−0.212704	−0.106352	+	0.994329i	$0.533917\pi$
$108$	−643.382	−0.573235
$109$	−1546.55	−1.35901	−0.679506	−	0.733670i	$-0.737806\pi$
$109$	−1546.55	−1.35901	−0.679506	+	0.733670i	$0.737806\pi$
$110$	−854.924	−0.741034
$111$	−130.895	−0.111928
$112$	−72.0402	−0.0607782
$113$	315.525	0.262673	0.131337	−	0.991338i	$-0.458073\pi$
$113$	315.525	0.262673	0.131337	+	0.991338i	$0.458073\pi$
$114$	−566.544	−0.465454
$115$	−2708.59	−2.19633
$116$	481.441	0.385350
$117$	−803.666	−0.635034
$118$	412.800	0.322045
$119$	0	0
$120$	1351.66	1.02824
$121$	−534.796	−0.401800
$122$	1278.90	0.949066
$123$	−1875.79	−1.37508
$124$	131.206	0.0950213
$125$	−22.8682	−0.0163631
$126$	186.573	0.131915
$127$	966.820	0.675523	0.337761	−	0.941232i	$-0.390330\pi$
$127$	966.820	0.675523	0.337761	+	0.941232i	$0.390330\pi$
$128$	578.140	0.399225
$129$	1253.31	0.855407
$130$	1745.56	1.17766
$131$	−532.805	−0.355354	−0.177677	−	0.984089i	$-0.556858\pi$
$131$	−532.805	−0.355354	−0.177677	+	0.984089i	$0.556858\pi$
$132$	−443.336	−0.292329
$133$	−574.540	−0.374579
$134$	−92.8109	−0.0598331
$135$	−2345.75	−1.49548
$136$	0	0
$137$	−1849.21	−1.15320	−0.576600	−	0.817027i	$-0.695621\pi$
$137$	−1849.21	−1.15320	−0.576600	+	0.817027i	$0.695621\pi$
$138$	1179.06	0.727307
$139$	−102.170	−0.0623451	−0.0311726	−	0.999514i	$-0.509924\pi$
$139$	−102.170	−0.0623451	−0.0311726	+	0.999514i	$0.509924\pi$
$140$	482.751	0.291428
$141$	−325.444	−0.194379
$142$	1556.41	0.919797
$143$	−1625.67	−0.950667
$144$	143.560	0.0830785
$145$	1755.32	1.00532
$146$	−1294.31	−0.733685
$147$	−177.016	−0.0993197
$148$	−157.584	−0.0875226
$149$	3088.45	1.69809	0.849046	−	0.528318i	$-0.177177\pi$
$149$	3088.45	1.69809	0.849046	+	0.528318i	$0.177177\pi$
$150$	872.777	0.475080
$151$	−516.149	−0.278170	−0.139085	−	0.990280i	$-0.544416\pi$
$151$	−516.149	−0.278170	−0.139085	+	0.990280i	$0.544416\pi$
$152$	−1936.67	−1.03345
$153$	0	0
$154$	377.404	0.197481
$155$	478.373	0.247896
$156$	905.194	0.464574
$157$	−2952.38	−1.50080	−0.750399	−	0.660985i	$-0.770139\pi$
$157$	−2952.38	−1.50080	−0.750399	+	0.660985i	$0.770139\pi$
$158$	−12.2192	−0.00615260
$159$	2304.06	1.14920
$160$	2681.44	1.32491
$161$	1195.70	0.585308
$162$	301.474	0.146210
$163$	−2728.24	−1.31099	−0.655497	−	0.755198i	$-0.727541\pi$
$163$	−2728.24	−1.31099	−0.655497	+	0.755198i	$0.727541\pi$
$164$	−2258.27	−1.07525
$165$	−1616.39	−0.762642
$166$	816.380	0.381707
$167$	−304.263	−0.140985	−0.0704926	−	0.997512i	$-0.522457\pi$
$167$	−304.263	−0.140985	−0.0704926	+	0.997512i	$0.522457\pi$
$168$	−596.688	−0.274021
$169$	1122.26	0.510814
$170$	0	0
$171$	1144.93	0.512016
$172$	1508.86	0.668891
$173$	1038.31	0.456310	0.228155	−	0.973625i	$-0.426731\pi$
$173$	1038.31	0.456310	0.228155	+	0.973625i	$0.426731\pi$
$174$	−764.097	−0.332908
$175$	885.095	0.382325
$176$	290.395	0.124371
$177$	780.475	0.331435
$178$	−999.186	−0.420742
$179$	3656.98	1.52701	0.763507	−	0.645799i	$-0.223476\pi$
$179$	3656.98	1.52701	0.763507	+	0.645799i	$0.223476\pi$
$180$	−962.011	−0.398356
$181$	586.824	0.240985	0.120492	−	0.992714i	$-0.461553\pi$
$181$	586.824	0.240985	0.120492	+	0.992714i	$0.461553\pi$
$182$	−770.575	−0.313839
$183$	2417.99	0.976739
$184$	4030.50	1.61485
$185$	−574.547	−0.228333
$186$	−208.238	−0.0820899
$187$	0	0
$188$	−391.803	−0.151995
$189$	1035.53	0.398537
$190$	−2486.78	−0.949527
$191$	1014.66	0.384388	0.192194	−	0.981357i	$-0.438440\pi$
$191$	1014.66	0.384388	0.192194	+	0.981357i	$0.438440\pi$
$192$	−1464.67	−0.550538
$193$	3309.22	1.23421	0.617106	−	0.786880i	$-0.288305\pi$
$193$	3309.22	1.23421	0.617106	+	0.786880i	$0.288305\pi$
$194$	−329.060	−0.121779
$195$	3300.31	1.21200
$196$	−213.109	−0.0776636
$197$	3087.91	1.11677	0.558386	−	0.829581i	$-0.311421\pi$
$197$	3087.91	1.11677	0.558386	+	0.829581i	$0.311421\pi$
$198$	−752.079	−0.269939
$199$	893.733	0.318367	0.159183	−	0.987249i	$-0.449114\pi$
$199$	893.733	0.318367	0.159183	+	0.987249i	$0.449114\pi$
$200$	2983.50	1.05483
$201$	−175.476	−0.0615778
$202$	3648.00	1.27066
$203$	−774.881	−0.267911
$204$	0	0
$205$	−8233.59	−2.80516
$206$	3043.08	1.02923
$207$	−2382.76	−0.800064
$208$	−592.922	−0.197653
$209$	2315.98	0.766505
$210$	−766.176	−0.251767
$211$	−3136.57	−1.02337	−0.511683	−	0.859174i	$-0.670978\pi$
$211$	−3136.57	−1.02337	−0.511683	+	0.859174i	$0.670978\pi$
$212$	2773.85	0.898627
$213$	2942.68	0.946617
$214$	449.829	0.143690
$215$	5501.25	1.74503
$216$	3490.57	1.09955
$217$	−211.177	−0.0660627
$218$	2955.01	0.918068
$219$	−2447.14	−0.755078
$220$	−1945.97	−0.596352
$221$	0	0
$222$	250.103	0.0756117
$223$	40.7664	0.0122418	0.00612090	−	0.999981i	$-0.498052\pi$
$223$	40.7664	0.0122418	0.00612090	+	0.999981i	$0.498052\pi$
$224$	−1183.71	−0.353081
$225$	−1763.79	−0.522605
$226$	−602.879	−0.177446
$227$	−5794.92	−1.69437	−0.847185	−	0.531297i	$-0.821705\pi$
$227$	−5794.92	−1.69437	−0.847185	+	0.531297i	$0.821705\pi$
$228$	−1289.57	−0.374577
$229$	−1405.79	−0.405665	−0.202832	−	0.979213i	$-0.565015\pi$
$229$	−1405.79	−0.405665	−0.202832	+	0.979213i	$0.565015\pi$
$230$	5175.36	1.48371
$231$	713.552	0.203239
$232$	−2611.98	−0.739160
$233$	4818.67	1.35486	0.677428	−	0.735589i	$-0.263094\pi$
$233$	4818.67	1.35486	0.677428	+	0.735589i	$0.263094\pi$
$234$	1535.58	0.428991
$235$	−1428.50	−0.396533
$236$	939.613	0.259168
$237$	−23.1027	−0.00633200
$238$	0	0
$239$	4767.10	1.29020	0.645101	−	0.764098i	$-0.276815\pi$
$239$	4767.10	1.29020	0.645101	+	0.764098i	$0.276815\pi$
$240$	−589.537	−0.158560
$241$	−1385.95	−0.370443	−0.185222	−	0.982697i	$-0.559300\pi$
$241$	−1385.95	−0.370443	−0.185222	+	0.982697i	$0.559300\pi$
$242$	1021.84	0.271432
$243$	−3424.18	−0.903955
$244$	2911.02	0.763767
$245$	−776.989	−0.202612
$246$	3584.11	0.928921
$247$	−4728.71	−1.21814
$248$	−711.838	−0.182265
$249$	1543.52	0.392837
$250$	43.6946	0.0110540
$251$	6681.73	1.68027	0.840133	−	0.542380i	$-0.182477\pi$
$251$	6681.73	1.68027	0.840133	+	0.542380i	$0.182477\pi$
$252$	424.677	0.106159
$253$	−4819.89	−1.19772
$254$	−1847.32	−0.456343
$255$	0	0
$256$	−4348.16	−1.06156
$257$	3582.54	0.869542	0.434771	−	0.900541i	$-0.356829\pi$
$257$	3582.54	0.869542	0.434771	+	0.900541i	$0.356829\pi$
$258$	−2394.71	−0.577862
$259$	253.632	0.0608493
$260$	3973.24	0.947731
$261$	1544.16	0.366211
$262$	1018.04	0.240056
$263$	−6028.82	−1.41351	−0.706754	−	0.707459i	$-0.749841\pi$
$263$	−6028.82	−1.41351	−0.706754	+	0.707459i	$0.749841\pi$
$264$	2405.26	0.560732
$265$	10113.4	2.34438
$266$	1097.78	0.253043
$267$	−1889.15	−0.433011
$268$	−211.256	−0.0481511
$269$	2082.33	0.471978	0.235989	−	0.971756i	$-0.424167\pi$
$269$	2082.33	0.471978	0.235989	+	0.971756i	$0.424167\pi$
$270$	4482.06	1.01026
$271$	−4984.04	−1.11719	−0.558596	−	0.829440i	$-0.688660\pi$
$271$	−4984.04	−1.11719	−0.558596	+	0.829440i	$0.688660\pi$
$272$	0	0
$273$	−1456.91	−0.322991
$274$	3533.31	0.779033
$275$	−3567.83	−0.782357
$276$	2683.78	0.585305
$277$	1678.03	0.363983	0.181991	−	0.983300i	$-0.441746\pi$
$277$	1678.03	0.363983	0.181991	+	0.983300i	$0.441746\pi$
$278$	195.219	0.0421166
$279$	420.826	0.0903019
$280$	−2619.09	−0.559003
$281$	−2950.83	−0.626448	−0.313224	−	0.949679i	$-0.601409\pi$
$281$	−2950.83	−0.626448	−0.313224	+	0.949679i	$0.601409\pi$
$282$	621.832	0.131310
$283$	6387.17	1.34162	0.670808	−	0.741631i	$-0.265947\pi$
$283$	6387.17	1.34162	0.670808	+	0.741631i	$0.265947\pi$
$284$	3542.70	0.740213
$285$	−4701.72	−0.977214
$286$	3106.20	0.642214
$287$	3634.70	0.747558
$288$	2358.87	0.482631
$289$	0	0
$290$	−3353.92	−0.679133
$291$	−622.148	−0.125330
$292$	−2946.11	−0.590438
$293$	−6966.22	−1.38898	−0.694490	−	0.719503i	$-0.744370\pi$
$293$	−6966.22	−1.38898	−0.694490	+	0.719503i	$0.744370\pi$
$294$	338.226	0.0670944
$295$	3425.80	0.676129
$296$	854.950	0.167882
$297$	−4174.22	−0.815531
$298$	−5901.15	−1.14713
$299$	9841.14	1.90344
$300$	1986.61	0.382324
$301$	−2428.51	−0.465040
$302$	986.214	0.187915
$303$	6897.22	1.30771
$304$	844.694	0.159364
$305$	10613.5	1.99255
$306$	0	0
$307$	−1423.61	−0.264656	−0.132328	−	0.991206i	$-0.542245\pi$
$307$	−1423.61	−0.264656	−0.132328	+	0.991206i	$0.542245\pi$
$308$	859.045	0.158924
$309$	5753.51	1.05924
$310$	−914.035	−0.167464
$311$	2640.30	0.481407	0.240703	−	0.970599i	$-0.422622\pi$
$311$	2640.30	0.481407	0.240703	+	0.970599i	$0.422622\pi$
$312$	−4911.00	−0.891123
$313$	1834.80	0.331340	0.165670	−	0.986181i	$-0.447021\pi$
$313$	1834.80	0.331340	0.165670	+	0.986181i	$0.447021\pi$
$314$	5641.16	1.01385
$315$	1548.36	0.276953
$316$	−27.8134	−0.00495135
$317$	819.489	0.145196	0.0725980	−	0.997361i	$-0.476871\pi$
$317$	819.489	0.145196	0.0725980	+	0.997361i	$0.476871\pi$
$318$	−4402.40	−0.776334
$319$	3123.55	0.548230
$320$	−6428.99	−1.12310
$321$	850.484	0.147880
$322$	−2284.65	−0.395399
$323$	0	0
$324$	686.215	0.117664
$325$	7284.71	1.24333
$326$	5212.89	0.885629
$327$	5587.00	0.944837
$328$	12251.9	2.06249
$329$	630.608	0.105673
$330$	3088.47	0.515195
$331$	3179.76	0.528022	0.264011	−	0.964520i	$-0.414954\pi$
$331$	3179.76	0.528022	0.264011	+	0.964520i	$0.414954\pi$
$332$	1858.24	0.307181
$333$	−505.431	−0.0831756
$334$	581.359	0.0952412
$335$	−770.232	−0.125619
$336$	260.250	0.0422553
$337$	−9833.46	−1.58950	−0.794752	−	0.606934i	$-0.792399\pi$
$337$	−9833.46	−1.58950	−0.794752	+	0.606934i	$0.792399\pi$
$338$	−2144.32	−0.345075
$339$	−1139.85	−0.182621
$340$	0	0
$341$	851.255	0.135185
$342$	−2187.63	−0.345887
$343$	343.000	0.0539949
$344$	−8186.08	−1.28303
$345$	9784.97	1.52697
$346$	−1983.93	−0.308256
$347$	−5312.81	−0.821922	−0.410961	−	0.911653i	$-0.634807\pi$
$347$	−5312.81	−0.821922	−0.410961	+	0.911653i	$0.634807\pi$
$348$	−1739.23	−0.267910
$349$	−223.175	−0.0342300	−0.0171150	−	0.999854i	$-0.505448\pi$
$349$	−223.175	−0.0342300	−0.0171150	+	0.999854i	$0.505448\pi$
$350$	−1691.17	−0.258276
$351$	8522.82	1.29605
$352$	4771.56	0.722515
$353$	−4317.81	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$353$	−4317.81	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$354$	−1491.27	−0.223898
$355$	12916.6	1.93110
$356$	−2274.34	−0.338595
$357$	0	0
$358$	−6987.45	−1.03156
$359$	−9910.55	−1.45699	−0.728494	−	0.685052i	$-0.759779\pi$
$359$	−9910.55	−1.45699	−0.728494	+	0.685052i	$0.759779\pi$
$360$	5219.25	0.764107
$361$	−122.337	−0.0178359
$362$	−1121.25	−0.162795
$363$	1931.98	0.279347
$364$	−1753.98	−0.252564
$365$	−10741.4	−1.54036
$366$	−4620.10	−0.659826
$367$	11737.6	1.66947	0.834736	−	0.550651i	$-0.185620\pi$
$367$	11737.6	1.66947	0.834736	+	0.550651i	$0.185620\pi$
$368$	−1757.93	−0.249018
$369$	−7243.11	−1.02185
$370$	1097.80	0.154248
$371$	−4464.53	−0.624762
$372$	−473.990	−0.0660624
$373$	−6427.28	−0.892204	−0.446102	−	0.894982i	$-0.647188\pi$
$373$	−6427.28	−0.892204	−0.446102	+	0.894982i	$0.647188\pi$
$374$	0	0
$375$	82.6127	0.0113763
$376$	2125.67	0.291550
$377$	−6377.60	−0.871255
$378$	−1978.60	−0.269227
$379$	7003.12	0.949145	0.474572	−	0.880216i	$-0.342603\pi$
$379$	7003.12	0.949145	0.474572	+	0.880216i	$0.342603\pi$
$380$	−5660.40	−0.764138
$381$	−3492.70	−0.469649
$382$	−1938.73	−0.259670
$383$	5814.35	0.775717	0.387858	−	0.921719i	$-0.373215\pi$
$383$	5814.35	0.775717	0.387858	+	0.921719i	$0.373215\pi$
$384$	−2088.57	−0.277556
$385$	3132.05	0.414608
$386$	−6322.99	−0.833760
$387$	4839.47	0.635669
$388$	−749.005	−0.0980024
$389$	8003.88	1.04322	0.521610	−	0.853184i	$-0.325332\pi$
$389$	8003.88	1.04322	0.521610	+	0.853184i	$0.325332\pi$
$390$	−6305.96	−0.818755
$391$	0	0
$392$	1156.19	0.148971
$393$	1924.79	0.247056
$394$	−5900.11	−0.754425
$395$	−101.407	−0.0129173
$396$	−1711.88	−0.217235
$397$	−9991.89	−1.26317	−0.631585	−	0.775306i	$-0.717595\pi$
$397$	−9991.89	−1.26317	−0.631585	+	0.775306i	$0.717595\pi$
$398$	−1707.67	−0.215070
$399$	2075.56	0.260421
$400$	−1301.27	−0.162659
$401$	8968.88	1.11692	0.558460	−	0.829532i	$-0.311393\pi$
$401$	8968.88	1.11692	0.558460	+	0.829532i	$0.311393\pi$
$402$	335.285	0.0415983
$403$	−1738.07	−0.214838
$404$	8303.57	1.02257
$405$	2501.92	0.306966
$406$	1480.58	0.180985
$407$	−1022.40	−0.124517
$408$	0	0
$409$	−9120.04	−1.10258	−0.551292	−	0.834312i	$-0.685865\pi$
$409$	−9120.04	−1.10258	−0.551292	+	0.834312i	$0.685865\pi$
$410$	15732.0	1.89500
$411$	6680.38	0.801749
$412$	6926.66	0.828281
$413$	−1512.31	−0.180184
$414$	4552.78	0.540475
$415$	6775.09	0.801388
$416$	−9742.47	−1.14823
$417$	369.097	0.0433447
$418$	−4425.18	−0.517805
$419$	−6426.52	−0.749298	−0.374649	−	0.927167i	$-0.622237\pi$
$419$	−6426.52	−0.749298	−0.374649	+	0.927167i	$0.622237\pi$
$420$	−1743.97	−0.202612
$421$	5456.72	0.631697	0.315848	−	0.948810i	$-0.397711\pi$
$421$	5456.72	0.631697	0.315848	+	0.948810i	$0.397711\pi$
$422$	5993.09	0.691325
$423$	−1256.66	−0.144446
$424$	−15049.1	−1.72370
$425$	0	0
$426$	−5622.64	−0.639478
$427$	−4685.30	−0.531002
$428$	1023.90	0.115635
$429$	5872.84	0.660940
$430$	−10511.3	−1.17884
$431$	−8847.65	−0.988809	−0.494404	−	0.869232i	$-0.664614\pi$
$431$	−8847.65	−0.988809	−0.494404	+	0.869232i	$0.664614\pi$
$432$	−1522.44	−0.169556
$433$	−1394.80	−0.154804	−0.0774018	−	0.997000i	$-0.524662\pi$
$433$	−1394.80	−0.154804	−0.0774018	+	0.997000i	$0.524662\pi$
$434$	403.498	0.0446280
$435$	−6341.20	−0.698936
$436$	6726.19	0.738821
$437$	−14020.0	−1.53471
$438$	4675.78	0.510086
$439$	9388.35	1.02069	0.510344	−	0.859971i	$-0.329518\pi$
$439$	9388.35	1.02069	0.510344	+	0.859971i	$0.329518\pi$
$440$	10557.6	1.14389
$441$	−683.520	−0.0738063
$442$	0	0
$443$	−3142.98	−0.337083	−0.168541	−	0.985695i	$-0.553906\pi$
$443$	−3142.98	−0.337083	−0.168541	+	0.985695i	$0.553906\pi$
$444$	569.283	0.0608490
$445$	−8292.19	−0.883342
$446$	−77.8930	−0.00826982
$447$	−11157.2	−1.18058
$448$	2838.06	0.299299
$449$	15910.6	1.67231	0.836157	−	0.548491i	$-0.184797\pi$
$449$	15910.6	1.67231	0.836157	+	0.548491i	$0.184797\pi$
$450$	3370.10	0.353041
$451$	−14651.5	−1.52974
$452$	−1372.27	−0.142801
$453$	1864.62	0.193394
$454$	11072.4	1.14462
$455$	−6394.96	−0.658901
$456$	6996.35	0.718496
$457$	−15685.8	−1.60558	−0.802789	−	0.596263i	$-0.796652\pi$
$457$	−15685.8	−1.60558	−0.802789	+	0.596263i	$0.796652\pi$
$458$	2686.07	0.274043
$459$	0	0
$460$	11780.1	1.19402
$461$	−6739.63	−0.680902	−0.340451	−	0.940262i	$-0.610580\pi$
$461$	−6739.63	−0.680902	−0.340451	+	0.940262i	$0.610580\pi$
$462$	−1363.40	−0.137296
$463$	7122.10	0.714886	0.357443	−	0.933935i	$-0.383649\pi$
$463$	7122.10	0.714886	0.357443	+	0.933935i	$0.383649\pi$
$464$	1139.24	0.113982
$465$	−1728.15	−0.172347
$466$	−9207.11	−0.915260
$467$	−11169.5	−1.10677	−0.553387	−	0.832924i	$-0.686665\pi$
$467$	−11169.5	−1.10677	−0.553387	+	0.832924i	$0.686665\pi$
$468$	3495.28	0.345233
$469$	340.017	0.0334766
$470$	2729.46	0.267874
$471$	10665.6	1.04341
$472$	−5097.73	−0.497123
$473$	9789.36	0.951618
$474$	44.1428	0.00427752
$475$	−10378.0	−1.00248
$476$	0	0
$477$	8896.78	0.853995
$478$	−9108.58	−0.871583
$479$	18417.2	1.75679	0.878397	−	0.477932i	$-0.158613\pi$
$479$	18417.2	1.75679	0.878397	+	0.477932i	$0.158613\pi$
$480$	−9686.86	−0.921130
$481$	2087.50	0.197884
$482$	2648.16	0.250249
$483$	−4319.55	−0.406928
$484$	2325.92	0.218437
$485$	−2730.85	−0.255673
$486$	6542.63	0.610658
$487$	4059.30	0.377709	0.188855	−	0.982005i	$-0.439522\pi$
$487$	4059.30	0.377709	0.188855	+	0.982005i	$0.439522\pi$
$488$	−15793.3	−1.46502
$489$	9855.93	0.911453
$490$	1484.61	0.136873
$491$	5705.87	0.524444	0.262222	−	0.965008i	$-0.415545\pi$
$491$	5705.87	0.524444	0.262222	+	0.965008i	$0.415545\pi$
$492$	8158.14	0.747556
$493$	0	0
$494$	9035.23	0.822903
$495$	−6241.46	−0.566733
$496$	310.474	0.0281062
$497$	−5701.99	−0.514626
$498$	−2949.22	−0.265377
$499$	15753.1	1.41323	0.706617	−	0.707596i	$-0.250220\pi$
$499$	15753.1	1.41323	0.706617	+	0.707596i	$0.250220\pi$
$500$	99.4574	0.00889574
$501$	1099.17	0.0980183
$502$	−12766.9	−1.13509
$503$	−6741.05	−0.597552	−0.298776	−	0.954323i	$-0.596578\pi$
$503$	−6741.05	−0.597552	−0.298776	+	0.954323i	$0.596578\pi$
$504$	−2304.02	−0.203630
$505$	30274.6	2.66772
$506$	9209.44	0.809110
$507$	−4054.22	−0.355137
$508$	−4204.86	−0.367245
$509$	−6496.37	−0.565710	−0.282855	−	0.959163i	$-0.591282\pi$
$509$	−6496.37	−0.565710	−0.282855	+	0.959163i	$0.591282\pi$
$510$	0	0
$511$	4741.78	0.410497
$512$	3682.98	0.317903
$513$	−12141.9	−1.04498
$514$	−6845.21	−0.587411
$515$	25254.4	2.16086
$516$	−5450.84	−0.465038
$517$	−2541.99	−0.216241
$518$	−484.620	−0.0411061
$519$	−3750.98	−0.317244
$520$	−21556.3	−1.81789
$521$	9641.97	0.810792	0.405396	−	0.914141i	$-0.367134\pi$
$521$	9641.97	0.810792	0.405396	+	0.914141i	$0.367134\pi$
$522$	−2950.45	−0.247390
$523$	1071.46	0.0895823	0.0447911	−	0.998996i	$-0.485738\pi$
$523$	1071.46	0.0895823	0.0447911	+	0.998996i	$0.485738\pi$
$524$	2317.26	0.193187
$525$	−3197.46	−0.265807
$526$	11519.4	0.954882
$527$	0	0
$528$	−1049.07	−0.0864676
$529$	17010.6	1.39810
$530$	−19323.8	−1.58372
$531$	3013.69	0.246296
$532$	2498.77	0.203638
$533$	29915.1	2.43108
$534$	3609.62	0.292516
$535$	3733.10	0.301675
$536$	1146.14	0.0923611
$537$	−13211.1	−1.06164
$538$	−3978.75	−0.318840
$539$	−1382.64	−0.110491
$540$	10202.1	0.813012
$541$	5025.45	0.399373	0.199687	−	0.979860i	$-0.436008\pi$
$541$	5025.45	0.399373	0.199687	+	0.979860i	$0.436008\pi$
$542$	9523.08	0.754708
$543$	−2119.94	−0.167542
$544$	0	0
$545$	24523.5	1.92747
$546$	2783.75	0.218193
$547$	17977.4	1.40522	0.702611	−	0.711574i	$-0.252017\pi$
$547$	17977.4	1.40522	0.702611	+	0.711574i	$0.252017\pi$
$548$	8042.51	0.626932
$549$	9336.74	0.725833
$550$	6817.11	0.528514
$551$	9085.72	0.702477
$552$	−14560.4	−1.12270
$553$	44.7658	0.00344238
$554$	−3206.24	−0.245885
$555$	2075.59	0.158746
$556$	444.356	0.0338937
$557$	−16741.4	−1.27353	−0.636766	−	0.771057i	$-0.719728\pi$
$557$	−16741.4	−1.27353	−0.636766	+	0.771057i	$0.719728\pi$
$558$	−804.080	−0.0610025
$559$	−19987.7	−1.51232
$560$	1142.34	0.0862009
$561$	0	0
$562$	5638.21	0.423191
$563$	12030.5	0.900574	0.450287	−	0.892884i	$-0.351322\pi$
$563$	12030.5	0.900574	0.450287	+	0.892884i	$0.351322\pi$
$564$	1415.41	0.105673
$565$	−5003.26	−0.372546
$566$	−12204.1	−0.906316
$567$	−1104.47	−0.0818046
$568$	−19220.4	−1.41984
$569$	21215.1	1.56306	0.781531	−	0.623866i	$-0.214439\pi$
$569$	21215.1	1.56306	0.781531	+	0.623866i	$0.214439\pi$
$570$	8983.65	0.660147
$571$	−5362.53	−0.393021	−0.196511	−	0.980502i	$-0.562961\pi$
$571$	−5362.53	−0.393021	−0.196511	+	0.980502i	$0.562961\pi$
$572$	7070.31	0.516826
$573$	−3665.52	−0.267241
$574$	−6944.87	−0.505006
$575$	21598.2	1.56644
$576$	−5655.61	−0.409115
$577$	−59.6556	−0.00430415	−0.00215208	−	0.999998i	$-0.500685\pi$
$577$	−59.6556	−0.00430415	−0.00215208	+	0.999998i	$0.500685\pi$
$578$	0	0
$579$	−11954.8	−0.858072
$580$	−7634.17	−0.546537
$581$	−2990.85	−0.213565
$582$	1188.75	0.0846653
$583$	17996.6	1.27846
$584$	15983.7	1.13255
$585$	12743.7	0.900660
$586$	13310.5	0.938311
$587$	7136.33	0.501785	0.250893	−	0.968015i	$-0.419276\pi$
$587$	7136.33	0.501785	0.250893	+	0.968015i	$0.419276\pi$
$588$	769.870	0.0539947
$589$	2476.11	0.173220
$590$	−6545.74	−0.456752
$591$	−11155.3	−0.776423
$592$	−372.893	−0.0258882
$593$	−5224.41	−0.361789	−0.180894	−	0.983503i	$-0.557899\pi$
$593$	−5224.41	−0.361789	−0.180894	+	0.983503i	$0.557899\pi$
$594$	7975.74	0.550924
$595$	0	0
$596$	−13432.2	−0.923160
$597$	−3228.67	−0.221341
$598$	−18803.6	−1.28585
$599$	11073.2	0.755321	0.377661	−	0.925944i	$-0.376729\pi$
$599$	11073.2	0.755321	0.377661	+	0.925944i	$0.376729\pi$
$600$	−10778.1	−0.733355
$601$	−11840.1	−0.803604	−0.401802	−	0.915726i	$-0.631616\pi$
$601$	−11840.1	−0.803604	−0.401802	+	0.915726i	$0.631616\pi$
$602$	4640.20	0.314153
$603$	−677.576	−0.0457596
$604$	2244.82	0.151226
$605$	8480.22	0.569868
$606$	−13178.6	−0.883408
$607$	−22565.9	−1.50893	−0.754464	−	0.656341i	$-0.772103\pi$
$607$	−22565.9	−1.50893	−0.754464	+	0.656341i	$0.772103\pi$
$608$	13879.4	0.925797
$609$	2799.31	0.186262
$610$	−20279.4	−1.34605
$611$	5190.18	0.343653
$612$	0	0
$613$	−2171.69	−0.143089	−0.0715445	−	0.997437i	$-0.522793\pi$
$613$	−2171.69	−0.143089	−0.0715445	+	0.997437i	$0.522793\pi$
$614$	2720.11	0.178786
$615$	29744.3	1.95026
$616$	−4660.62	−0.304841
$617$	−20959.7	−1.36759	−0.683796	−	0.729673i	$-0.739672\pi$
$617$	−20959.7	−1.36759	−0.683796	+	0.729673i	$0.739672\pi$
$618$	−10993.3	−0.715561
$619$	−21463.0	−1.39365	−0.696826	−	0.717240i	$-0.745405\pi$
$619$	−21463.0	−1.39365	−0.696826	+	0.717240i	$0.745405\pi$
$620$	−2080.52	−0.134767
$621$	25269.0	1.63287
$622$	−5044.86	−0.325210
$623$	3660.56	0.235405
$624$	2141.97	0.137416
$625$	−15442.7	−0.988330
$626$	−3505.79	−0.223833
$627$	−8366.62	−0.532904
$628$	12840.4	0.815902
$629$	0	0
$630$	−2958.48	−0.187093
$631$	−3171.61	−0.200095	−0.100047	−	0.994983i	$-0.531899\pi$
$631$	−3171.61	−0.200095	−0.100047	+	0.994983i	$0.531899\pi$
$632$	150.897	0.00949743
$633$	11331.0	0.711483
$634$	−1565.81	−0.0980857
$635$	−15330.8	−0.958085
$636$	−10020.7	−0.624760
$637$	2823.04	0.175593
$638$	−5968.23	−0.370352
$639$	11362.8	0.703449
$640$	−9167.52	−0.566216
$641$	7114.09	0.438362	0.219181	−	0.975684i	$-0.429662\pi$
$641$	7114.09	0.438362	0.219181	+	0.975684i	$0.429662\pi$
$642$	−1625.03	−0.0998987
$643$	−5505.92	−0.337686	−0.168843	−	0.985643i	$-0.554003\pi$
$643$	−5505.92	−0.337686	−0.168843	+	0.985643i	$0.554003\pi$
$644$	−5200.31	−0.318200
$645$	−19873.6	−1.21321
$646$	0	0
$647$	11481.3	0.697644	0.348822	−	0.937189i	$-0.386582\pi$
$647$	11481.3	0.697644	0.348822	+	0.937189i	$0.386582\pi$
$648$	−3722.96	−0.225697
$649$	6096.15	0.368713
$650$	−13919.0	−0.839921
$651$	762.888	0.0459293
$652$	11865.6	0.712716
$653$	28543.4	1.71055	0.855275	−	0.518175i	$-0.173388\pi$
$653$	28543.4	1.71055	0.855275	+	0.518175i	$0.173388\pi$
$654$	−10675.2	−0.638276
$655$	8448.66	0.503994
$656$	−5343.76	−0.318047
$657$	−9449.27	−0.561113
$658$	−1204.91	−0.0713866
$659$	−9752.35	−0.576476	−0.288238	−	0.957559i	$-0.593069\pi$
$659$	−9752.35	−0.576476	−0.288238	+	0.957559i	$0.593069\pi$
$660$	7029.95	0.414607
$661$	22309.4	1.31276	0.656380	−	0.754431i	$-0.272087\pi$
$661$	22309.4	1.31276	0.656380	+	0.754431i	$0.272087\pi$
$662$	−6075.62	−0.356700
$663$	0	0
$664$	−10081.6	−0.589220
$665$	9110.44	0.531260
$666$	965.736	0.0561884
$667$	−18908.7	−1.09767
$668$	1323.29	0.0766460
$669$	−147.271	−0.00851096
$670$	1471.70	0.0848605
$671$	18886.5	1.08660
$672$	4276.24	0.245475
$673$	7760.76	0.444510	0.222255	−	0.974989i	$-0.428658\pi$
$673$	7760.76	0.444510	0.222255	+	0.974989i	$0.428658\pi$
$674$	18789.0	1.07377
$675$	18704.9	1.06659
$676$	−4880.88	−0.277701
$677$	−22366.2	−1.26972	−0.634862	−	0.772625i	$-0.718943\pi$
$677$	−22366.2	−1.26972	−0.634862	+	0.772625i	$0.718943\pi$
$678$	2177.94	0.123368
$679$	1205.53	0.0681353
$680$	0	0
$681$	20934.5	1.17799
$682$	−1626.51	−0.0913228
$683$	8110.06	0.454353	0.227176	−	0.973854i	$-0.427051\pi$
$683$	8110.06	0.454353	0.227176	+	0.973854i	$0.427051\pi$
$684$	−4979.47	−0.278355
$685$	29322.7	1.63557
$686$	−655.376	−0.0364757
$687$	5078.51	0.282034
$688$	3570.42	0.197850
$689$	−36745.0	−2.03174
$690$	−18696.3	−1.03153
$691$	−7867.18	−0.433114	−0.216557	−	0.976270i	$-0.569483\pi$
$691$	−7867.18	−0.433114	−0.216557	+	0.976270i	$0.569483\pi$
$692$	−4515.80	−0.248071
$693$	2755.28	0.151031
$694$	10151.3	0.555241
$695$	1620.11	0.0884233
$696$	9435.96	0.513892
$697$	0	0
$698$	426.424	0.0231237
$699$	−17407.8	−0.941948
$700$	−3849.42	−0.207849
$701$	8047.92	0.433617	0.216809	−	0.976214i	$-0.430435\pi$
$701$	8047.92	0.433617	0.216809	+	0.976214i	$0.430435\pi$
$702$	−16284.7	−0.875536
$703$	−2973.92	−0.159550
$704$	−11440.3	−0.612459
$705$	5160.55	0.275684
$706$	8250.11	0.439798
$707$	−13364.6	−0.710932
$708$	−3394.41	−0.180183
$709$	−15725.6	−0.832988	−0.416494	−	0.909139i	$-0.636741\pi$
$709$	−15725.6	−0.832988	−0.416494	+	0.909139i	$0.636741\pi$
$710$	−24679.9	−1.30454
$711$	−89.2079	−0.00470543
$712$	12339.1	0.649477
$713$	−5153.15	−0.270669
$714$	0	0
$715$	25778.1	1.34832
$716$	−15904.8	−0.830155
$717$	−17221.4	−0.896997
$718$	18936.2	0.984254
$719$	−17506.4	−0.908036	−0.454018	−	0.890993i	$-0.650010\pi$
$719$	−17506.4	−0.908036	−0.454018	+	0.890993i	$0.650010\pi$
$720$	−2276.41	−0.117829
$721$	−11148.5	−0.575855
$722$	233.750	0.0120489
$723$	5006.83	0.257546
$724$	−2552.19	−0.131010
$725$	−13996.8	−0.717004
$726$	−3691.47	−0.188710
$727$	14020.8	0.715271	0.357636	−	0.933861i	$-0.383583\pi$
$727$	14020.8	0.715271	0.357636	+	0.933861i	$0.383583\pi$
$728$	9515.95	0.484457
$729$	16630.1	0.844898
$730$	20523.8	1.04058
$731$	0	0
$732$	−10516.3	−0.531000
$733$	−23109.6	−1.16449	−0.582246	−	0.813012i	$-0.697826\pi$
$733$	−23109.6	−1.16449	−0.582246	+	0.813012i	$0.697826\pi$
$734$	−22427.2	−1.12779
$735$	2806.92	0.140864
$736$	−28885.1	−1.44663
$737$	−1370.61	−0.0685036
$738$	13839.5	0.690298
$739$	10006.8	0.498113	0.249057	−	0.968489i	$-0.419880\pi$
$739$	10006.8	0.498113	0.249057	+	0.968489i	$0.419880\pi$
$740$	2498.80	0.124132
$741$	17082.8	0.846898
$742$	8530.45	0.422052
$743$	−23020.2	−1.13665	−0.568324	−	0.822805i	$-0.692408\pi$
$743$	−23020.2	−1.13665	−0.568324	+	0.822805i	$0.692408\pi$
$744$	2571.56	0.126718
$745$	−48973.3	−2.40838
$746$	12280.7	0.602719
$747$	5960.07	0.291924
$748$	0	0
$749$	−1647.97	−0.0803945
$750$	−157.849	−0.00768513
$751$	−7372.82	−0.358240	−0.179120	−	0.983827i	$-0.557325\pi$
$751$	−7372.82	−0.358240	−0.179120	+	0.983827i	$0.557325\pi$
$752$	−927.125	−0.0449585
$753$	−24138.2	−1.16818
$754$	12185.8	0.588568
$755$	8184.54	0.394524
$756$	−4503.67	−0.216663
$757$	8747.05	0.419970	0.209985	−	0.977705i	$-0.432659\pi$
$757$	8747.05	0.419970	0.209985	+	0.977705i	$0.432659\pi$
$758$	−13381.0	−0.641185
$759$	17412.2	0.832702
$760$	30709.7	1.46573
$761$	10210.9	0.486395	0.243197	−	0.969977i	$-0.421804\pi$
$761$	10210.9	0.486395	0.243197	+	0.969977i	$0.421804\pi$
$762$	6673.55	0.317267
$763$	−10825.8	−0.513659
$764$	−4412.92	−0.208971
$765$	0	0
$766$	−11109.6	−0.524028
$767$	−12447.0	−0.585964
$768$	15708.0	0.738039
$769$	−890.499	−0.0417584	−0.0208792	−	0.999782i	$-0.506647\pi$
$769$	−890.499	−0.0417584	−0.0208792	+	0.999782i	$0.506647\pi$
$770$	−5984.47	−0.280085
$771$	−12942.1	−0.604539
$772$	−14392.4	−0.670974
$773$	−6290.28	−0.292685	−0.146343	−	0.989234i	$-0.546750\pi$
$773$	−6290.28	−0.292685	−0.146343	+	0.989234i	$0.546750\pi$
$774$	−9246.85	−0.429420
$775$	−3814.52	−0.176802
$776$	4063.61	0.187984
$777$	−916.263	−0.0423047
$778$	−15293.1	−0.704737
$779$	−42617.9	−1.96014
$780$	−14353.6	−0.658899
$781$	22984.8	1.05309
$782$	0	0
$783$	−16375.7	−0.747407
$784$	−504.281	−0.0229720
$785$	46815.6	2.12856
$786$	−3677.73	−0.166896
$787$	11597.6	0.525298	0.262649	−	0.964891i	$-0.415404\pi$
$787$	11597.6	0.525298	0.262649	+	0.964891i	$0.415404\pi$
$788$	−13429.8	−0.607128
$789$	21779.5	0.982725
$790$	193.760	0.00872615
$791$	2208.68	0.0992812
$792$	9287.55	0.416690
$793$	−38562.1	−1.72683
$794$	19091.7	0.853322
$795$	−36535.2	−1.62990
$796$	−3886.99	−0.173079
$797$	231.692	0.0102973	0.00514866	−	0.999987i	$-0.498361\pi$
$797$	231.692	0.0102973	0.00514866	+	0.999987i	$0.498361\pi$
$798$	−3965.81	−0.175925
$799$	0	0
$800$	−21381.6	−0.944943
$801$	−7294.66	−0.321778
$802$	−17137.0	−0.754524
$803$	−19114.2	−0.840005
$804$	763.175	0.0334765
$805$	−18960.2	−0.830134
$806$	3320.96	0.145132
$807$	−7522.56	−0.328137
$808$	−45049.8	−1.96144
$809$	11846.2	0.514821	0.257410	−	0.966302i	$-0.417131\pi$
$809$	11846.2	0.514821	0.257410	+	0.966302i	$0.417131\pi$
$810$	−4780.46	−0.207368
$811$	−25322.4	−1.09641	−0.548206	−	0.836343i	$-0.684689\pi$
$811$	−25322.4	−1.09641	−0.548206	+	0.836343i	$0.684689\pi$
$812$	3370.08	0.145649
$813$	18005.2	0.776714
$814$	1953.51	0.0841160
$815$	43261.5	1.85937
$816$	0	0
$817$	28475.1	1.21936
$818$	17425.8	0.744840
$819$	−5625.66	−0.240020
$820$	35809.2	1.52501
$821$	2529.16	0.107513	0.0537565	−	0.998554i	$-0.482881\pi$
$821$	2529.16	0.107513	0.0537565	+	0.998554i	$0.482881\pi$
$822$	−12764.3	−0.541613
$823$	−24803.7	−1.05055	−0.525275	−	0.850932i	$-0.676038\pi$
$823$	−24803.7	−1.05055	−0.525275	+	0.850932i	$0.676038\pi$
$824$	−37579.5	−1.58877
$825$	12889.0	0.543924
$826$	2889.60	0.121722
$827$	−30192.9	−1.26954	−0.634770	−	0.772701i	$-0.718905\pi$
$827$	−30192.9	−1.26954	−0.634770	+	0.772701i	$0.718905\pi$
$828$	10363.0	0.434951
$829$	12774.3	0.535187	0.267594	−	0.963532i	$-0.413772\pi$
$829$	12774.3	0.535187	0.267594	+	0.963532i	$0.413772\pi$
$830$	−12945.3	−0.541370
$831$	−6062.00	−0.253055
$832$	23358.5	0.973328
$833$	0	0
$834$	−705.239	−0.0292811
$835$	4824.67	0.199958
$836$	−10072.6	−0.416707
$837$	−4462.83	−0.184299
$838$	12279.3	0.506181
$839$	−15261.8	−0.628006	−0.314003	−	0.949422i	$-0.601670\pi$
$839$	−15261.8	−0.628006	−0.314003	+	0.949422i	$0.601670\pi$
$840$	9461.63	0.388640
$841$	−12135.1	−0.497565
$842$	−10426.2	−0.426736
$843$	10660.1	0.435531
$844$	13641.4	0.556348
$845$	−17795.6	−0.724480
$846$	2401.12	0.0975792
$847$	−3743.57	−0.151866
$848$	6563.79	0.265803
$849$	−23074.0	−0.932743
$850$	0	0
$851$	6189.16	0.249309
$852$	−12798.2	−0.514624
$853$	−12238.3	−0.491242	−0.245621	−	0.969366i	$-0.578992\pi$
$853$	−12238.3	−0.491242	−0.245621	+	0.969366i	$0.578992\pi$
$854$	8952.29	0.358713
$855$	−18155.0	−0.726186
$856$	−5555.01	−0.221806
$857$	15895.1	0.633567	0.316784	−	0.948498i	$-0.397397\pi$
$857$	15895.1	0.633567	0.316784	+	0.948498i	$0.397397\pi$
$858$	−11221.3	−0.446492
$859$	20083.0	0.797698	0.398849	−	0.917017i	$-0.369410\pi$
$859$	20083.0	0.797698	0.398849	+	0.917017i	$0.369410\pi$
$860$	−23925.8	−0.948679
$861$	−13130.6	−0.519731
$862$	16905.4	0.667980
$863$	−31483.2	−1.24183	−0.620915	−	0.783878i	$-0.713239\pi$
$863$	−31483.2	−1.24183	−0.620915	+	0.783878i	$0.713239\pi$
$864$	−25015.6	−0.985010
$865$	−16464.5	−0.647178
$866$	2665.07	0.104576
$867$	0	0
$868$	918.441	0.0359147
$869$	−180.451	−0.00704418
$870$	12116.2	0.472159
$871$	2798.49	0.108867
$872$	−36491.9	−1.41717
$873$	−2402.34	−0.0931349
$874$	26788.2	1.03676
$875$	−160.077	−0.00618468
$876$	10643.0	0.410495
$877$	10154.6	0.390989	0.195495	−	0.980705i	$-0.437369\pi$
$877$	10154.6	0.390989	0.195495	+	0.980705i	$0.437369\pi$
$878$	−17938.5	−0.689515
$879$	25165.9	0.965671
$880$	−4604.77	−0.176394
$881$	−18592.9	−0.711022	−0.355511	−	0.934672i	$-0.615693\pi$
$881$	−18592.9	−0.711022	−0.355511	+	0.934672i	$0.615693\pi$
$882$	1306.01	0.0498591
$883$	29151.1	1.11100	0.555500	−	0.831517i	$-0.312527\pi$
$883$	29151.1	1.11100	0.555500	+	0.831517i	$0.312527\pi$
$884$	0	0
$885$	−12375.9	−0.470070
$886$	6005.35	0.227713
$887$	30725.1	1.16308	0.581538	−	0.813519i	$-0.302451\pi$
$887$	30725.1	1.16308	0.581538	+	0.813519i	$0.302451\pi$
$888$	−3088.56	−0.116718
$889$	6767.74	0.255324
$890$	15844.0	0.596733
$891$	4452.12	0.167398
$892$	−177.300	−0.00665520
$893$	−7394.08	−0.277081
$894$	21318.3	0.797528
$895$	−57988.5	−2.16574
$896$	4046.98	0.150893
$897$	−35551.7	−1.32334
$898$	−30400.7	−1.12971
$899$	3339.52	0.123893
$900$	7671.02	0.284112
$901$	0	0
$902$	27994.9	1.03340
$903$	8773.15	0.323314
$904$	7445.05	0.273914
$905$	−9305.22	−0.341786
$906$	−3562.76	−0.130645
$907$	35086.2	1.28448	0.642238	−	0.766506i	$-0.278006\pi$
$907$	35086.2	1.28448	0.642238	+	0.766506i	$0.278006\pi$
$908$	25203.0	0.921137
$909$	26632.6	0.971781
$910$	12218.9	0.445114
$911$	−8601.50	−0.312821	−0.156411	−	0.987692i	$-0.549992\pi$
$911$	−8601.50	−0.312821	−0.156411	+	0.987692i	$0.549992\pi$
$912$	−3051.51	−0.110796
$913$	12056.1	0.437021
$914$	29971.0	1.08463
$915$	−38342.0	−1.38530
$916$	6114.01	0.220538
$917$	−3729.64	−0.134311
$918$	0	0
$919$	−51964.6	−1.86524	−0.932619	−	0.360862i	$-0.882483\pi$
$919$	−51964.6	−1.86524	−0.932619	+	0.360862i	$0.882483\pi$
$920$	−63911.3	−2.29032
$921$	5142.86	0.183999
$922$	12877.5	0.459976
$923$	−46929.8	−1.67358
$924$	−3103.35	−0.110490
$925$	4581.41	0.162849
$926$	−13608.3	−0.482934
$927$	22216.4	0.787143
$928$	18719.1	0.662161
$929$	−13563.7	−0.479020	−0.239510	−	0.970894i	$-0.576987\pi$
$929$	−13563.7	−0.479020	−0.239510	+	0.970894i	$0.576987\pi$
$930$	3302.01	0.116427
$931$	−4021.78	−0.141577
$932$	−20957.2	−0.736562
$933$	−9538.24	−0.334692
$934$	21341.8	0.747670
$935$	0	0
$936$	−18963.1	−0.662210
$937$	36010.7	1.25552	0.627758	−	0.778409i	$-0.283973\pi$
$937$	36010.7	1.25552	0.627758	+	0.778409i	$0.283973\pi$
$938$	−649.676	−0.0226148
$939$	−6628.35	−0.230360
$940$	6212.79	0.215573
$941$	−14082.1	−0.487847	−0.243923	−	0.969795i	$-0.578435\pi$
$941$	−14082.1	−0.487847	−0.243923	+	0.969795i	$0.578435\pi$
$942$	−20379.0	−0.704867
$943$	88694.1	3.06286
$944$	2223.41	0.0766589
$945$	−16420.3	−0.565239
$946$	−18704.7	−0.642856
$947$	37827.2	1.29801	0.649007	−	0.760783i	$-0.275185\pi$
$947$	37827.2	1.29801	0.649007	+	0.760783i	$0.275185\pi$
$948$	100.478	0.00344236
$949$	39026.8	1.33495
$950$	19829.4	0.677213
$951$	−2960.46	−0.100946
$952$	0	0
$953$	35976.8	1.22288	0.611438	−	0.791292i	$-0.290591\pi$
$953$	35976.8	1.22288	0.611438	+	0.791292i	$0.290591\pi$
$954$	−16999.2	−0.576908
$955$	−16089.4	−0.545173
$956$	−20732.9	−0.701412
$957$	−11284.0	−0.381151
$958$	−35190.1	−1.18678
$959$	−12944.4	−0.435869
$960$	23225.1	0.780821
$961$	−28880.9	−0.969450
$962$	−3988.63	−0.133678
$963$	3284.02	0.109892
$964$	6027.72	0.201390
$965$	−52474.1	−1.75047
$966$	8253.43	0.274896
$967$	−41254.5	−1.37193	−0.685965	−	0.727635i	$-0.740620\pi$
$967$	−41254.5	−1.37193	−0.685965	+	0.727635i	$0.740620\pi$
$968$	−12618.9	−0.418995
$969$	0	0
$970$	5217.88	0.172717
$971$	−39640.6	−1.31012	−0.655061	−	0.755576i	$-0.727357\pi$
$971$	−39640.6	−1.31012	−0.655061	+	0.755576i	$0.727357\pi$
$972$	14892.3	0.491431
$973$	−715.192	−0.0235642
$974$	−7756.17	−0.255158
$975$	−26316.5	−0.864412
$976$	6888.37	0.225913
$977$	23299.1	0.762953	0.381477	−	0.924378i	$-0.375416\pi$
$977$	23299.1	0.762953	0.381477	+	0.924378i	$0.375416\pi$
$978$	−18831.9	−0.615723
$979$	−14755.8	−0.481713
$980$	3379.25	0.110149
$981$	21573.4	0.702126
$982$	−10902.3	−0.354283
$983$	10453.2	0.339171	0.169586	−	0.985515i	$-0.445757\pi$
$983$	10453.2	0.339171	0.169586	+	0.985515i	$0.445757\pi$
$984$	−44260.8	−1.43392
$985$	−48964.7	−1.58390
$986$	0	0
$987$	−2278.11	−0.0734682
$988$	20565.9	0.662237
$989$	−59260.8	−1.90534
$990$	11925.7	0.382851
$991$	10554.9	0.338332	0.169166	−	0.985588i	$-0.445893\pi$
$991$	10554.9	0.338332	0.169166	+	0.985588i	$0.445893\pi$
$992$	5101.48	0.163278
$993$	−11487.1	−0.367101
$994$	10894.9	0.347651
$995$	−14171.9	−0.451536
$996$	−6713.01	−0.213564
$997$	−51756.4	−1.64407	−0.822037	−	0.569434i	$-0.807163\pi$
$997$	−51756.4	−1.64407	−0.822037	+	0.569434i	$0.807163\pi$
$998$	−30099.6	−0.954697
$999$	5360.06	0.169755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2023.4.a.v.1.20		56
17.5	odd	16		119.4.k.a.8.19	✓	112
17.7	odd	16		119.4.k.a.15.19	yes	112
17.16	even	2		2023.4.a.u.1.20		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.4.k.a.8.19	✓	112	17.5	odd	16
119.4.k.a.15.19	yes	112	17.7	odd	16
2023.4.a.u.1.20		56	17.16	even	2
2023.4.a.v.1.20		56	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:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2023.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91072 q^{2} -3.61256 q^{3} -4.34916 q^{4} -15.8569 q^{5} +6.90258 q^{6} +7.00000 q^{7} +23.5957 q^{8} -13.9494 q^{9} +30.2981 q^{10} -28.2171 q^{11} +15.7116 q^{12} +57.6130 q^{13} -13.3750 q^{14} +57.2841 q^{15} -10.2915 q^{16} +26.6533 q^{18} -82.0772 q^{19} +68.9644 q^{20} -25.2879 q^{21} +53.9148 q^{22} +170.815 q^{23} -85.2411 q^{24} +126.442 q^{25} -110.082 q^{26} +147.932 q^{27} -30.4441 q^{28} -110.697 q^{29} -109.454 q^{30} -30.1681 q^{31} -169.102 q^{32} +101.936 q^{33} -110.998 q^{35} +60.6682 q^{36} +36.2332 q^{37} +156.826 q^{38} -208.131 q^{39} -374.156 q^{40} +519.242 q^{41} +48.3181 q^{42} -346.930 q^{43} +122.721 q^{44} +221.195 q^{45} -326.378 q^{46} +90.0869 q^{47} +37.1785 q^{48} +49.0000 q^{49} -241.595 q^{50} -250.568 q^{52} -637.790 q^{53} -282.656 q^{54} +447.436 q^{55} +165.170 q^{56} +296.509 q^{57} +211.511 q^{58} -216.045 q^{59} -249.138 q^{60} -669.329 q^{61} +57.6426 q^{62} -97.6458 q^{63} +405.437 q^{64} -913.565 q^{65} -194.771 q^{66} +48.5739 q^{67} -617.078 q^{69} +212.087 q^{70} -814.570 q^{71} -329.146 q^{72} +677.396 q^{73} -69.2314 q^{74} -456.780 q^{75} +356.967 q^{76} -197.520 q^{77} +397.678 q^{78} +6.39511 q^{79} +163.191 q^{80} -157.781 q^{81} -992.124 q^{82} -427.264 q^{83} +109.981 q^{84} +662.885 q^{86} +399.901 q^{87} -665.803 q^{88} +522.938 q^{89} -422.640 q^{90} +403.291 q^{91} -742.901 q^{92} +108.984 q^{93} -172.130 q^{94} +1301.49 q^{95} +610.891 q^{96} +172.218 q^{97} -93.6251 q^{98} +393.611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{2} + 24 q^{3} + 240 q^{4} + 80 q^{5} + 68 q^{6} + 392 q^{7} + 96 q^{8} + 576 q^{9} + 80 q^{10} + 176 q^{11} + 288 q^{12} - 96 q^{13} + 56 q^{14} + 192 q^{15} + 1088 q^{16} + 216 q^{18} + 48 q^{19}+ \cdots + 6576 q^{99}+O(q^{100}) \) 56 * q + 8 * q^2 + 24 * q^3 + 240 * q^4 + 80 * q^5 + 68 * q^6 + 392 * q^7 + 96 * q^8 + 576 * q^9 + 80 * q^10 + 176 * q^11 + 288 * q^12 - 96 * q^13 + 56 * q^14 + 192 * q^15 + 1088 * q^16 + 216 * q^18 + 48 * q^19 + 1660 * q^20 + 168 * q^21 + 352 * q^22 + 608 * q^23 + 736 * q^24 + 1792 * q^25 + 864 * q^27 + 1680 * q^28 + 928 * q^29 - 100 * q^30 + 496 * q^31 + 436 * q^32 + 384 * q^33 + 560 * q^35 + 2804 * q^36 + 1392 * q^37 + 624 * q^39 - 1020 * q^40 + 2088 * q^41 + 476 * q^42 + 1312 * q^43 + 2112 * q^44 + 4840 * q^45 + 736 * q^46 + 5228 * q^48 + 2744 * q^49 + 2020 * q^50 - 1536 * q^52 + 1272 * q^53 + 2268 * q^54 - 336 * q^55 + 672 * q^56 + 2448 * q^57 + 1376 * q^58 + 2372 * q^60 + 976 * q^61 + 3776 * q^62 + 4032 * q^63 + 5376 * q^64 + 6080 * q^65 + 976 * q^67 + 1248 * q^69 + 560 * q^70 + 3216 * q^71 + 4532 * q^72 + 4088 * q^73 - 2456 * q^74 - 760 * q^75 - 2640 * q^76 + 1232 * q^77 + 3744 * q^78 + 1792 * q^79 + 13920 * q^80 + 4184 * q^81 + 8204 * q^82 - 2832 * q^83 + 2016 * q^84 + 4908 * q^86 + 832 * q^87 + 2424 * q^88 - 2768 * q^89 + 8660 * q^90 - 672 * q^91 + 6784 * q^92 + 784 * q^93 - 928 * q^94 + 4560 * q^95 + 6656 * q^96 + 9464 * q^97 + 392 * q^98 + 6576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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