LMFDB - Embedded newform 1922.4.a.x.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1922,4,Mod(1,1922)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1922, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("1922.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 1922 = 2 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1922.a (trivial)

Newform invariants

comment:select newform

sage:traces = [32,-64,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$113.401671031$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	1922.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.00000 q^{2} -0.915774 q^{3} +4.00000 q^{4} +5.16716 q^{5} +1.83155 q^{6} -2.76909 q^{7} -8.00000 q^{8} -26.1614 q^{9} -10.3343 q^{10} -48.5280 q^{11} -3.66310 q^{12} +36.7499 q^{13} +5.53817 q^{14} -4.73195 q^{15} +16.0000 q^{16} +109.156 q^{17} +52.3227 q^{18} -109.456 q^{19} +20.6687 q^{20} +2.53586 q^{21} +97.0560 q^{22} +5.46425 q^{23} +7.32619 q^{24} -98.3004 q^{25} -73.4997 q^{26} +48.6838 q^{27} -11.0763 q^{28} -11.8771 q^{29} +9.46391 q^{30} -32.0000 q^{32} +44.4407 q^{33} -218.312 q^{34} -14.3083 q^{35} -104.645 q^{36} -332.952 q^{37} +218.911 q^{38} -33.6546 q^{39} -41.3373 q^{40} +247.094 q^{41} -5.07172 q^{42} +17.3041 q^{43} -194.112 q^{44} -135.180 q^{45} -10.9285 q^{46} +81.6399 q^{47} -14.6524 q^{48} -335.332 q^{49} +196.601 q^{50} -99.9620 q^{51} +146.999 q^{52} -182.822 q^{53} -97.3676 q^{54} -250.752 q^{55} +22.1527 q^{56} +100.237 q^{57} +23.7541 q^{58} +221.015 q^{59} -18.9278 q^{60} -14.4425 q^{61} +72.4431 q^{63} +64.0000 q^{64} +189.893 q^{65} -88.8814 q^{66} -611.554 q^{67} +436.623 q^{68} -5.00402 q^{69} +28.6167 q^{70} +1005.57 q^{71} +209.291 q^{72} -464.909 q^{73} +665.904 q^{74} +90.0210 q^{75} -437.822 q^{76} +134.378 q^{77} +67.3091 q^{78} +711.985 q^{79} +82.6746 q^{80} +661.773 q^{81} -494.188 q^{82} -21.3995 q^{83} +10.1434 q^{84} +564.026 q^{85} -34.6083 q^{86} +10.8767 q^{87} +388.224 q^{88} -1211.40 q^{89} +270.360 q^{90} -101.764 q^{91} +21.8570 q^{92} -163.280 q^{94} -565.575 q^{95} +29.3048 q^{96} +1500.39 q^{97} +670.664 q^{98} +1269.56 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 64 q^{2} + 128 q^{4} + 112 q^{7} - 256 q^{8} + 288 q^{9} - 224 q^{14} + 512 q^{16} - 576 q^{18} + 304 q^{19} + 1200 q^{25} + 448 q^{28} - 1024 q^{32} - 272 q^{33} + 1152 q^{36} - 608 q^{38} + 1616 q^{39}+ \cdots - 6176 q^{98}+O(q^{100}) $ 32 * q - 64 * q^2 + 128 * q^4 + 112 * q^7 - 256 * q^8 + 288 * q^9 - 224 * q^14 + 512 * q^16 - 576 * q^18 + 304 * q^19 + 1200 * q^25 + 448 * q^28 - 1024 * q^32 - 272 * q^33 + 1152 * q^36 - 608 * q^38 + 1616 * q^39 + 368 * q^41 + 1360 * q^45 + 784 * q^47 + 3088 * q^49 - 2400 * q^50 - 464 * q^51 - 896 * q^56 - 848 * q^59 + 6992 * q^63 + 2048 * q^64 + 544 * q^66 + 3216 * q^67 - 1664 * q^69 + 2688 * q^71 - 2304 * q^72 + 1216 * q^76 - 3232 * q^78 + 3584 * q^81 - 736 * q^82 + 4960 * q^87 - 2720 * q^90 - 1568 * q^94 - 6816 * q^95 + 7504 * q^97 - 6176 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	−0.915774	−0.176241	−0.0881204	−	0.996110i	$-0.528086\pi$
$3$	−0.915774	−0.176241	−0.0881204	+	0.996110i	$0.528086\pi$
$4$	4.00000	0.500000
$5$	5.16716	0.462165	0.231083	−	0.972934i	$-0.425773\pi$
$5$	5.16716	0.462165	0.231083	+	0.972934i	$0.425773\pi$
$6$	1.83155	0.124621
$7$	−2.76909	−0.149517	−0.0747583	−	0.997202i	$-0.523819\pi$
$7$	−2.76909	−0.149517	−0.0747583	+	0.997202i	$0.523819\pi$
$8$	−8.00000	−0.353553
$9$	−26.1614	−0.968939
$10$	−10.3343	−0.326800
$11$	−48.5280	−1.33016	−0.665080	−	0.746773i	$-0.731602\pi$
$11$	−48.5280	−1.33016	−0.665080	+	0.746773i	$0.731602\pi$
$12$	−3.66310	−0.0881204
$13$	36.7499	0.784044	0.392022	−	0.919956i	$-0.371776\pi$
$13$	36.7499	0.784044	0.392022	+	0.919956i	$0.371776\pi$
$14$	5.53817	0.105724
$15$	−4.73195	−0.0814524
$16$	16.0000	0.250000
$17$	109.156	1.55730	0.778652	−	0.627456i	$-0.215904\pi$
$17$	109.156	1.55730	0.778652	+	0.627456i	$0.215904\pi$
$18$	52.3227	0.685143
$19$	−109.456	−1.32162	−0.660811	−	0.750552i	$-0.729788\pi$
$19$	−109.456	−1.32162	−0.660811	+	0.750552i	$0.729788\pi$
$20$	20.6687	0.231083
$21$	2.53586	0.0263509
$22$	97.0560	0.940564
$23$	5.46425	0.0495380	0.0247690	−	0.999693i	$-0.492115\pi$
$23$	5.46425	0.0495380	0.0247690	+	0.999693i	$0.492115\pi$
$24$	7.32619	0.0623105
$25$	−98.3004	−0.786403
$26$	−73.4997	−0.554403
$27$	48.6838	0.347007
$28$	−11.0763	−0.0747583
$29$	−11.8771	−0.0760523	−0.0380261	−	0.999277i	$-0.512107\pi$
$29$	−11.8771	−0.0760523	−0.0380261	+	0.999277i	$0.512107\pi$
$30$	9.46391	0.0575955
$31$	0	0
$31$	0	0
$32$	−32.0000	−0.176777
$33$	44.4407	0.234428
$34$	−218.312	−1.10118
$35$	−14.3083	−0.0691014
$36$	−104.645	−0.484470
$37$	−332.952	−1.47938	−0.739688	−	0.672949i	$-0.765027\pi$
$37$	−332.952	−1.47938	−0.739688	+	0.672949i	$0.765027\pi$
$38$	218.911	0.934528
$39$	−33.6546	−0.138181
$40$	−41.3373	−0.163400
$41$	247.094	0.941211	0.470605	−	0.882344i	$-0.344036\pi$
$41$	247.094	0.941211	0.470605	+	0.882344i	$0.344036\pi$
$42$	−5.07172	−0.0186329
$43$	17.3041	0.0613688	0.0306844	−	0.999529i	$-0.490231\pi$
$43$	17.3041	0.0613688	0.0306844	+	0.999529i	$0.490231\pi$
$44$	−194.112	−0.665080
$45$	−135.180	−0.447810
$46$	−10.9285	−0.0350287
$47$	81.6399	0.253370	0.126685	−	0.991943i	$-0.459566\pi$
$47$	81.6399	0.253370	0.126685	+	0.991943i	$0.459566\pi$
$48$	−14.6524	−0.0440602
$49$	−335.332	−0.977645
$50$	196.601	0.556071
$51$	−99.9620	−0.274460
$52$	146.999	0.392022
$53$	−182.822	−0.473822	−0.236911	−	0.971531i	$-0.576135\pi$
$53$	−182.822	−0.473822	−0.236911	+	0.971531i	$0.576135\pi$
$54$	−97.3676	−0.245371
$55$	−250.752	−0.614753
$56$	22.1527	0.0528621
$57$	100.237	0.232924
$58$	23.7541	0.0537771
$59$	221.015	0.487690	0.243845	−	0.969814i	$-0.421591\pi$
$59$	221.015	0.487690	0.243845	+	0.969814i	$0.421591\pi$
$60$	−18.9278	−0.0407262
$61$	−14.4425	−0.0303144	−0.0151572	−	0.999885i	$-0.504825\pi$
$61$	−14.4425	−0.0303144	−0.0151572	+	0.999885i	$0.504825\pi$
$62$	0	0
$63$	72.4431	0.144873
$64$	64.0000	0.125000
$65$	189.893	0.362358
$66$	−88.8814	−0.165766
$67$	−611.554	−1.11512	−0.557562	−	0.830136i	$-0.688263\pi$
$67$	−611.554	−1.11512	−0.557562	+	0.830136i	$0.688263\pi$
$68$	436.623	0.778652
$69$	−5.00402	−0.00873062
$70$	28.6167	0.0488621
$71$	1005.57	1.68083	0.840416	−	0.541943i	$-0.182311\pi$
$71$	1005.57	1.68083	0.840416	+	0.541943i	$0.182311\pi$
$72$	209.291	0.342572
$73$	−464.909	−0.745391	−0.372695	−	0.927954i	$-0.621566\pi$
$73$	−464.909	−0.745391	−0.372695	+	0.927954i	$0.621566\pi$
$74$	665.904	1.04608
$75$	90.0210	0.138596
$76$	−437.822	−0.660811
$77$	134.378	0.198881
$78$	67.3091	0.0977085
$79$	711.985	1.01398	0.506991	−	0.861951i	$-0.330758\pi$
$79$	711.985	1.01398	0.506991	+	0.861951i	$0.330758\pi$
$80$	82.6746	0.115541
$81$	661.773	0.907782
$82$	−494.188	−0.665536
$83$	−21.3995	−0.0283001	−0.0141500	−	0.999900i	$-0.504504\pi$
$83$	−21.3995	−0.0283001	−0.0141500	+	0.999900i	$0.504504\pi$
$84$	10.1434	0.0131755
$85$	564.026	0.719732
$86$	−34.6083	−0.0433943
$87$	10.8767	0.0134035
$88$	388.224	0.470282
$89$	−1211.40	−1.44278	−0.721392	−	0.692527i	$-0.756497\pi$
$89$	−1211.40	−1.44278	−0.721392	+	0.692527i	$0.756497\pi$
$90$	270.360	0.316649
$91$	−101.764	−0.117228
$92$	21.8570	0.0247690
$93$	0	0
$94$	−163.280	−0.179160
$95$	−565.575	−0.610808
$96$	29.3048	0.0311553
$97$	1500.39	1.57053	0.785267	−	0.619158i	$-0.212526\pi$
$97$	1500.39	1.57053	0.785267	+	0.619158i	$0.212526\pi$
$98$	670.664	0.691299
$99$	1269.56	1.28884
$100$	−393.202	−0.393202
$101$	−175.935	−0.173329	−0.0866644	−	0.996238i	$-0.527621\pi$
$101$	−175.935	−0.173329	−0.0866644	+	0.996238i	$0.527621\pi$
$102$	199.924	0.194073
$103$	934.861	0.894316	0.447158	−	0.894455i	$-0.352436\pi$
$103$	934.861	0.894316	0.447158	+	0.894455i	$0.352436\pi$
$104$	−293.999	−0.277202
$105$	13.1032	0.0121785
$106$	365.645	0.335043
$107$	1399.38	1.26433	0.632164	−	0.774835i	$-0.282167\pi$
$107$	1399.38	1.26433	0.632164	+	0.774835i	$0.282167\pi$
$108$	194.735	0.173504
$109$	1002.49	0.880929	0.440464	−	0.897770i	$-0.354814\pi$
$109$	1002.49	0.880929	0.440464	+	0.897770i	$0.354814\pi$
$110$	501.505	0.434696
$111$	304.909	0.260727
$112$	−44.3054	−0.0373792
$113$	−1164.13	−0.969133	−0.484566	−	0.874755i	$-0.661023\pi$
$113$	−1164.13	−0.969133	−0.484566	+	0.874755i	$0.661023\pi$
$114$	−200.473	−0.164702
$115$	28.2347	0.0228948
$116$	−47.5083	−0.0380261
$117$	−961.426	−0.759691
$118$	−442.030	−0.344849
$119$	−302.262	−0.232843
$120$	37.8556	0.0287978
$121$	1023.97	0.769323
$122$	28.8850	0.0214355
$123$	−226.282	−0.165880
$124$	0	0
$125$	−1153.83	−0.825613
$126$	−144.886	−0.102440
$127$	−2195.23	−1.53382	−0.766910	−	0.641754i	$-0.778207\pi$
$127$	−2195.23	−1.53382	−0.766910	+	0.641754i	$0.778207\pi$
$128$	−128.000	−0.0883883
$129$	−15.8467	−0.0108157
$130$	−379.785	−0.256226
$131$	1051.91	0.701572	0.350786	−	0.936456i	$-0.385914\pi$
$131$	1051.91	0.701572	0.350786	+	0.936456i	$0.385914\pi$
$132$	177.763	0.117214
$133$	303.092	0.197604
$134$	1223.11	0.788511
$135$	251.557	0.160375
$136$	−873.246	−0.550590
$137$	−1749.63	−1.09110	−0.545551	−	0.838077i	$-0.683680\pi$
$137$	−1749.63	−1.09110	−0.545551	+	0.838077i	$0.683680\pi$
$138$	10.0080	0.00617348
$139$	1786.32	1.09003	0.545013	−	0.838428i	$-0.316525\pi$
$139$	1786.32	1.09003	0.545013	+	0.838428i	$0.316525\pi$
$140$	−57.2333	−0.0345507
$141$	−74.7637	−0.0446542
$142$	−2011.14	−1.18853
$143$	−1783.40	−1.04290
$144$	−418.582	−0.242235
$145$	−61.3708	−0.0351487
$146$	929.819	0.527071
$147$	307.088	0.172301
$148$	−1331.81	−0.739688
$149$	−2487.93	−1.36792	−0.683958	−	0.729522i	$-0.739743\pi$
$149$	−2487.93	−1.36792	−0.683958	+	0.729522i	$0.739743\pi$
$150$	−180.042	−0.0980024
$151$	3235.02	1.74346	0.871729	−	0.489989i	$-0.162999\pi$
$151$	3235.02	1.74346	0.871729	+	0.489989i	$0.162999\pi$
$152$	875.644	0.467264
$153$	−2855.66	−1.50893
$154$	−268.757	−0.140630
$155$	0	0
$156$	−134.618	−0.0690903
$157$	1075.13	0.546527	0.273263	−	0.961939i	$-0.411897\pi$
$157$	1075.13	0.546527	0.273263	+	0.961939i	$0.411897\pi$
$158$	−1423.97	−0.716993
$159$	167.424	0.0835068
$160$	−165.349	−0.0817000
$161$	−15.1310	−0.00740676
$162$	−1323.55	−0.641899
$163$	−913.394	−0.438911	−0.219456	−	0.975622i	$-0.570428\pi$
$163$	−913.394	−0.438911	−0.219456	+	0.975622i	$0.570428\pi$
$164$	988.377	0.470605
$165$	229.632	0.108345
$166$	42.7991	0.0200112
$167$	3516.67	1.62951	0.814754	−	0.579806i	$-0.196872\pi$
$167$	3516.67	1.62951	0.814754	+	0.579806i	$0.196872\pi$
$168$	−20.2869	−0.00931646
$169$	−846.447	−0.385274
$170$	−1128.05	−0.508927
$171$	2863.51	1.28057
$172$	69.2166	0.0306844
$173$	−2899.75	−1.27436	−0.637178	−	0.770716i	$-0.719899\pi$
$173$	−2899.75	−1.27436	−0.637178	+	0.770716i	$0.719899\pi$
$174$	−21.7534	−0.00947772
$175$	272.202	0.117580
$176$	−776.448	−0.332540
$177$	−202.400	−0.0859508
$178$	2422.79	1.02020
$179$	−3206.74	−1.33901	−0.669506	−	0.742806i	$-0.733494\pi$
$179$	−3206.74	−1.33901	−0.669506	+	0.742806i	$0.733494\pi$
$180$	−540.720	−0.223905
$181$	−333.405	−0.136916	−0.0684579	−	0.997654i	$-0.521808\pi$
$181$	−333.405	−0.136916	−0.0684579	+	0.997654i	$0.521808\pi$
$182$	203.527	0.0828925
$183$	13.2261	0.00534263
$184$	−43.7140	−0.0175143
$185$	−1720.42	−0.683717
$186$	0	0
$187$	−5297.11	−2.07146
$188$	326.560	0.126685
$189$	−134.810	−0.0518834
$190$	1131.15	0.431906
$191$	1894.64	0.717755	0.358877	−	0.933385i	$-0.383160\pi$
$191$	1894.64	0.717755	0.358877	+	0.933385i	$0.383160\pi$
$192$	−58.6095	−0.0220301
$193$	2460.46	0.917657	0.458828	−	0.888525i	$-0.348269\pi$
$193$	2460.46	0.917657	0.458828	+	0.888525i	$0.348269\pi$
$194$	−3000.78	−1.11053
$195$	−173.899	−0.0638623
$196$	−1341.33	−0.488822
$197$	−4239.92	−1.53341	−0.766705	−	0.642000i	$-0.778105\pi$
$197$	−4239.92	−1.53341	−0.766705	+	0.642000i	$0.778105\pi$
$198$	−2539.12	−0.911350
$199$	2536.76	0.903647	0.451824	−	0.892107i	$-0.350774\pi$
$199$	2536.76	0.903647	0.451824	+	0.892107i	$0.350774\pi$
$200$	786.403	0.278036
$201$	560.046	0.196530
$202$	351.871	0.122562
$203$	32.8886	0.0113711
$204$	−399.848	−0.137230
$205$	1276.78	0.434995
$206$	−1869.72	−0.632377
$207$	−142.952	−0.0479994
$208$	587.998	0.196011
$209$	5311.66	1.75797
$210$	−26.2064	−0.00861149
$211$	−1333.05	−0.434933	−0.217467	−	0.976068i	$-0.569779\pi$
$211$	−1333.05	−0.434933	−0.217467	+	0.976068i	$0.569779\pi$
$212$	−731.289	−0.236911
$213$	−920.873	−0.296231
$214$	−2798.76	−0.894014
$215$	89.4134	0.0283625
$216$	−389.470	−0.122686
$217$	0	0
$218$	−2004.98	−0.622911
$219$	425.752	0.131368
$220$	−1003.01	−0.307377
$221$	4011.46	1.22100
$222$	−609.817	−0.184362
$223$	6074.25	1.82404	0.912022	−	0.410142i	$-0.134521\pi$
$223$	6074.25	1.82404	0.912022	+	0.410142i	$0.134521\pi$
$224$	88.6108	0.0264311
$225$	2571.67	0.761977
$226$	2328.26	0.685280
$227$	4249.70	1.24257	0.621284	−	0.783586i	$-0.286611\pi$
$227$	4249.70	1.24257	0.621284	+	0.783586i	$0.286611\pi$
$228$	400.946	0.116462
$229$	1395.51	0.402697	0.201348	−	0.979520i	$-0.435468\pi$
$229$	1395.51	0.402697	0.201348	+	0.979520i	$0.435468\pi$
$230$	−56.4694	−0.0161890
$231$	−123.060	−0.0350509
$232$	95.0165	0.0268885
$233$	−1622.55	−0.456210	−0.228105	−	0.973637i	$-0.573253\pi$
$233$	−1622.55	−0.456210	−0.228105	+	0.973637i	$0.573253\pi$
$234$	1922.85	0.537183
$235$	421.847	0.117099
$236$	884.060	0.243845
$237$	−652.017	−0.178705
$238$	604.524	0.164645
$239$	834.389	0.225825	0.112912	−	0.993605i	$-0.463982\pi$
$239$	834.389	0.225825	0.112912	+	0.993605i	$0.463982\pi$
$240$	−75.7113	−0.0203631
$241$	−1582.61	−0.423008	−0.211504	−	0.977377i	$-0.567836\pi$
$241$	−1582.61	−0.423008	−0.211504	+	0.977377i	$0.567836\pi$
$242$	−2047.94	−0.543994
$243$	−1920.50	−0.506996
$244$	−57.7701	−0.0151572
$245$	−1732.72	−0.451833
$246$	452.565	0.117295
$247$	−4022.48	−1.03621
$248$	0	0
$249$	19.5971	0.00498763
$250$	2307.66	0.583797
$251$	1711.78	0.430464	0.215232	−	0.976563i	$-0.430949\pi$
$251$	1711.78	0.430464	0.215232	+	0.976563i	$0.430949\pi$
$252$	289.772	0.0724363
$253$	−265.169	−0.0658935
$254$	4390.46	1.08458
$255$	−516.520	−0.126846
$256$	256.000	0.0625000
$257$	7420.43	1.80106	0.900532	−	0.434790i	$-0.143177\pi$
$257$	7420.43	1.80106	0.900532	+	0.434790i	$0.143177\pi$
$258$	31.6934	0.00764784
$259$	921.973	0.221191
$260$	759.570	0.181179
$261$	310.720	0.0736900
$262$	−2103.82	−0.496087
$263$	4558.38	1.06875	0.534376	−	0.845247i	$-0.320547\pi$
$263$	4558.38	1.06875	0.534376	+	0.845247i	$0.320547\pi$
$264$	−355.526	−0.0828829
$265$	−944.673	−0.218984
$266$	−606.184	−0.139727
$267$	1109.37	0.254277
$268$	−2446.22	−0.557562
$269$	−4893.07	−1.10905	−0.554527	−	0.832166i	$-0.687101\pi$
$269$	−4893.07	−1.10905	−0.554527	+	0.832166i	$0.687101\pi$
$270$	−503.114	−0.113402
$271$	7527.82	1.68739	0.843694	−	0.536824i	$-0.180376\pi$
$271$	7527.82	1.68739	0.843694	+	0.536824i	$0.180376\pi$
$272$	1746.49	0.389326
$273$	93.1924	0.0206603
$274$	3499.26	0.771526
$275$	4770.32	1.04604
$276$	−20.0161	−0.00436531
$277$	−4365.22	−0.946861	−0.473430	−	0.880831i	$-0.656984\pi$
$277$	−4365.22	−0.946861	−0.473430	+	0.880831i	$0.656984\pi$
$278$	−3572.64	−0.770765
$279$	0	0
$280$	114.467	0.0244310
$281$	5895.33	1.25155	0.625775	−	0.780003i	$-0.284783\pi$
$281$	5895.33	1.25155	0.625775	+	0.780003i	$0.284783\pi$
$282$	149.527	0.0315753
$283$	5152.63	1.08230	0.541152	−	0.840925i	$-0.317988\pi$
$283$	5152.63	1.08230	0.541152	+	0.840925i	$0.317988\pi$
$284$	4022.27	0.840416
$285$	517.939	0.107649
$286$	3566.80	0.737444
$287$	−684.225	−0.140727
$288$	837.163	0.171286
$289$	7001.99	1.42520
$290$	122.742	0.0248539
$291$	−1374.02	−0.276792
$292$	−1859.64	−0.372695
$293$	3843.62	0.766371	0.383186	−	0.923671i	$-0.374827\pi$
$293$	3843.62	0.766371	0.383186	+	0.923671i	$0.374827\pi$
$294$	−614.177	−0.121835
$295$	1142.02	0.225393
$296$	2663.61	0.523039
$297$	−2362.53	−0.461575
$298$	4975.87	0.967262
$299$	200.810	0.0388400
$300$	360.084	0.0692982
$301$	−47.9167	−0.00917565
$302$	−6470.04	−1.23281
$303$	161.117	0.0305476
$304$	−1751.29	−0.330406
$305$	−74.6269	−0.0140102
$306$	5711.33	1.06698
$307$	10347.6	1.92368	0.961839	−	0.273618i	$-0.0882203\pi$
$307$	10347.6	1.92368	0.961839	+	0.273618i	$0.0882203\pi$
$308$	537.513	0.0994405
$309$	−856.121	−0.157615
$310$	0	0
$311$	3853.49	0.702609	0.351305	−	0.936261i	$-0.385738\pi$
$311$	3853.49	0.702609	0.351305	+	0.936261i	$0.385738\pi$
$312$	269.237	0.0488542
$313$	−35.3480	−0.00638334	−0.00319167	−	0.999995i	$-0.501016\pi$
$313$	−35.3480	−0.00638334	−0.00319167	+	0.999995i	$0.501016\pi$
$314$	−2150.26	−0.386453
$315$	374.325	0.0669550
$316$	2847.94	0.506991
$317$	−432.260	−0.0765873	−0.0382936	−	0.999267i	$-0.512192\pi$
$317$	−432.260	−0.0765873	−0.0382936	+	0.999267i	$0.512192\pi$
$318$	−334.848	−0.0590482
$319$	576.371	0.101162
$320$	330.699	0.0577707
$321$	−1281.51	−0.222826
$322$	30.2620	0.00523737
$323$	−11947.7	−2.05817
$324$	2647.09	0.453891
$325$	−3612.53	−0.616575
$326$	1826.79	0.310357
$327$	−918.055	−0.155256
$328$	−1976.75	−0.332768
$329$	−226.068	−0.0378831
$330$	−459.265	−0.0766112
$331$	−10140.0	−1.68383	−0.841913	−	0.539613i	$-0.818571\pi$
$331$	−10140.0	−1.68383	−0.841913	+	0.539613i	$0.818571\pi$
$332$	−85.5982	−0.0141500
$333$	8710.47	1.43343
$334$	−7033.34	−1.15224
$335$	−3160.00	−0.515371
$336$	40.5737	0.00658773
$337$	8818.30	1.42541	0.712705	−	0.701464i	$-0.247470\pi$
$337$	8818.30	1.42541	0.712705	+	0.701464i	$0.247470\pi$
$338$	1692.89	0.272430
$339$	1066.08	0.170801
$340$	2256.10	0.359866
$341$	0	0
$342$	−5727.01	−0.905501
$343$	1878.36	0.295691
$344$	−138.433	−0.0216971
$345$	−25.8566	−0.00403499
$346$	5799.50	0.901106
$347$	−1729.22	−0.267520	−0.133760	−	0.991014i	$-0.542705\pi$
$347$	−1729.22	−0.267520	−0.133760	+	0.991014i	$0.542705\pi$
$348$	43.5068	0.00670176
$349$	8253.61	1.26592	0.632959	−	0.774185i	$-0.281840\pi$
$349$	8253.61	1.26592	0.632959	+	0.774185i	$0.281840\pi$
$350$	−544.405	−0.0831419
$351$	1789.12	0.272069
$352$	1552.90	0.235141
$353$	24.4799	0.00369103	0.00184552	−	0.999998i	$-0.499413\pi$
$353$	24.4799	0.00369103	0.00184552	+	0.999998i	$0.499413\pi$
$354$	404.800	0.0607764
$355$	5195.94	0.776822
$356$	−4845.59	−0.721392
$357$	276.804	0.0410364
$358$	6413.49	0.946825
$359$	807.552	0.118721	0.0593607	−	0.998237i	$-0.481094\pi$
$359$	807.552	0.118721	0.0593607	+	0.998237i	$0.481094\pi$
$360$	1081.44	0.158325
$361$	5121.51	0.746685
$362$	666.809	0.0968141
$363$	−937.724	−0.135586
$364$	−407.054	−0.0586138
$365$	−2402.26	−0.344494
$366$	−26.4522	−0.00377781
$367$	−8656.53	−1.23125	−0.615623	−	0.788041i	$-0.711096\pi$
$367$	−8656.53	−1.23125	−0.615623	+	0.788041i	$0.711096\pi$
$368$	87.4280	0.0123845
$369$	−6464.32	−0.911976
$370$	3440.83	0.483461
$371$	506.251	0.0708443
$372$	0	0
$373$	7263.93	1.00834	0.504171	−	0.863604i	$-0.331798\pi$
$373$	7263.93	1.00834	0.504171	+	0.863604i	$0.331798\pi$
$374$	10594.2	1.46474
$375$	1056.65	0.145507
$376$	−653.119	−0.0895800
$377$	−436.481	−0.0596284
$378$	269.619	0.0366871
$379$	3293.79	0.446413	0.223206	−	0.974771i	$-0.428348\pi$
$379$	3293.79	0.446413	0.223206	+	0.974771i	$0.428348\pi$
$380$	−2262.30	−0.305404
$381$	2010.34	0.270322
$382$	−3789.27	−0.507529
$383$	−12397.7	−1.65403	−0.827013	−	0.562183i	$-0.809962\pi$
$383$	−12397.7	−1.65403	−0.827013	+	0.562183i	$0.809962\pi$
$384$	117.219	0.0155776
$385$	694.355	0.0919158
$386$	−4920.92	−0.648881
$387$	−452.700	−0.0594626
$388$	6001.57	0.785267
$389$	8920.39	1.16268	0.581339	−	0.813662i	$-0.302529\pi$
$389$	8920.39	1.16268	0.581339	+	0.813662i	$0.302529\pi$
$390$	347.797	0.0451574
$391$	596.455	0.0771458
$392$	2682.66	0.345650
$393$	−963.314	−0.123646
$394$	8479.84	1.08428
$395$	3678.94	0.468627
$396$	5078.24	0.644422
$397$	−13226.6	−1.67211	−0.836053	−	0.548649i	$-0.815142\pi$
$397$	−13226.6	−1.67211	−0.836053	+	0.548649i	$0.815142\pi$
$398$	−5073.51	−0.638975
$399$	−277.564	−0.0348260
$400$	−1572.81	−0.196601
$401$	3260.02	0.405979	0.202990	−	0.979181i	$-0.434934\pi$
$401$	3260.02	0.405979	0.202990	+	0.979181i	$0.434934\pi$
$402$	−1120.09	−0.138968
$403$	0	0
$404$	−703.741	−0.0866644
$405$	3419.49	0.419545
$406$	−65.7773	−0.00804057
$407$	16157.5	1.96781
$408$	799.696	0.0970364
$409$	−682.917	−0.0825625	−0.0412813	−	0.999148i	$-0.513144\pi$
$409$	−682.917	−0.0825625	−0.0412813	+	0.999148i	$0.513144\pi$
$410$	−2553.55	−0.307588
$411$	1602.27	0.192297
$412$	3739.44	0.447158
$413$	−612.010	−0.0729177
$414$	285.904	0.0339407
$415$	−110.575	−0.0130793
$416$	−1176.00	−0.138601
$417$	−1635.86	−0.192107
$418$	−10623.3	−1.24307
$419$	5479.25	0.638852	0.319426	−	0.947611i	$-0.396510\pi$
$419$	5479.25	0.638852	0.319426	+	0.947611i	$0.396510\pi$
$420$	52.4128	0.00608924
$421$	13823.8	1.60031	0.800157	−	0.599790i	$-0.204749\pi$
$421$	13823.8	1.60031	0.800157	+	0.599790i	$0.204749\pi$
$422$	2666.10	0.307544
$423$	−2135.81	−0.245500
$424$	1462.58	0.167521
$425$	−10730.1	−1.22467
$426$	1841.75	0.209467
$427$	39.9926	0.00453250
$428$	5597.51	0.632164
$429$	1633.19	0.183802
$430$	−178.827	−0.0200553
$431$	−11057.2	−1.23575	−0.617875	−	0.786277i	$-0.712006\pi$
$431$	−11057.2	−1.23575	−0.617875	+	0.786277i	$0.712006\pi$
$432$	778.941	0.0867518
$433$	−2806.32	−0.311462	−0.155731	−	0.987800i	$-0.549773\pi$
$433$	−2806.32	−0.311462	−0.155731	+	0.987800i	$0.549773\pi$
$434$	0	0
$435$	56.2017	0.00619464
$436$	4009.96	0.440464
$437$	−598.092	−0.0654706
$438$	−851.504	−0.0928914
$439$	12658.5	1.37621	0.688106	−	0.725610i	$-0.258442\pi$
$439$	12658.5	1.37621	0.688106	+	0.725610i	$0.258442\pi$
$440$	2006.02	0.217348
$441$	8772.74	0.947278
$442$	−8022.92	−0.863374
$443$	−6579.42	−0.705638	−0.352819	−	0.935692i	$-0.614777\pi$
$443$	−6579.42	−0.705638	−0.352819	+	0.935692i	$0.614777\pi$
$444$	1219.63	0.130363
$445$	−6259.48	−0.666805
$446$	−12148.5	−1.28979
$447$	2278.38	0.241082
$448$	−177.222	−0.0186896
$449$	18640.8	1.95928	0.979638	−	0.200774i	$-0.0643457\pi$
$449$	18640.8	1.95928	0.979638	+	0.200774i	$0.0643457\pi$
$450$	−5143.34	−0.538799
$451$	−11991.0	−1.25196
$452$	−4656.51	−0.484566
$453$	−2962.55	−0.307268
$454$	−8499.41	−0.878628
$455$	−525.829	−0.0541786
$456$	−801.892	−0.0823510
$457$	5850.71	0.598872	0.299436	−	0.954116i	$-0.403202\pi$
$457$	5850.71	0.598872	0.299436	+	0.954116i	$0.403202\pi$
$458$	−2791.01	−0.284750
$459$	5314.12	0.540396
$460$	112.939	0.0114474
$461$	14207.6	1.43539	0.717693	−	0.696360i	$-0.245198\pi$
$461$	14207.6	1.43539	0.717693	+	0.696360i	$0.245198\pi$
$462$	246.120	0.0247847
$463$	8035.18	0.806536	0.403268	−	0.915082i	$-0.367874\pi$
$463$	8035.18	0.806536	0.403268	+	0.915082i	$0.367874\pi$
$464$	−190.033	−0.0190131
$465$	0	0
$466$	3245.10	0.322589
$467$	2373.97	0.235234	0.117617	−	0.993059i	$-0.462474\pi$
$467$	2373.97	0.235234	0.117617	+	0.993059i	$0.462474\pi$
$468$	−3845.71	−0.379846
$469$	1693.45	0.166729
$470$	−843.694	−0.0828015
$471$	−984.576	−0.0963203
$472$	−1768.12	−0.172424
$473$	−839.736	−0.0816302
$474$	1304.03	0.126363
$475$	10759.5	1.03933
$476$	−1209.05	−0.116421
$477$	4782.88	0.459105
$478$	−1668.78	−0.159682
$479$	−20610.0	−1.96596	−0.982980	−	0.183713i	$-0.941188\pi$
$479$	−20610.0	−1.96596	−0.982980	+	0.183713i	$0.941188\pi$
$480$	151.423	0.0143989
$481$	−12235.9	−1.15990
$482$	3165.22	0.299112
$483$	13.8566	0.00130537
$484$	4095.88	0.384662
$485$	7752.77	0.725846
$486$	3840.99	0.358500
$487$	−6337.51	−0.589692	−0.294846	−	0.955545i	$-0.595268\pi$
$487$	−6337.51	−0.589692	−0.294846	+	0.955545i	$0.595268\pi$
$488$	115.540	0.0107177
$489$	836.463	0.0773541
$490$	3465.43	0.319494
$491$	−6928.02	−0.636776	−0.318388	−	0.947960i	$-0.603142\pi$
$491$	−6928.02	−0.636776	−0.318388	+	0.947960i	$0.603142\pi$
$492$	−905.130	−0.0829398
$493$	−1296.45	−0.118437
$494$	8044.95	0.732711
$495$	6560.02	0.595658
$496$	0	0
$497$	−2784.51	−0.251312
$498$	−39.1943	−0.00352678
$499$	11924.2	1.06974	0.534870	−	0.844935i	$-0.320361\pi$
$499$	11924.2	1.06974	0.534870	+	0.844935i	$0.320361\pi$
$500$	−4615.32	−0.412807
$501$	−3220.47	−0.287186
$502$	−3423.56	−0.304384
$503$	7551.92	0.669430	0.334715	−	0.942319i	$-0.391360\pi$
$503$	7551.92	0.669430	0.334715	+	0.942319i	$0.391360\pi$
$504$	−579.545	−0.0512202
$505$	−909.087	−0.0801066
$506$	530.339	0.0465937
$507$	775.155	0.0679010
$508$	−8780.92	−0.766910
$509$	−2076.02	−0.180782	−0.0903908	−	0.995906i	$-0.528812\pi$
$509$	−2076.02	−0.180782	−0.0903908	+	0.995906i	$0.528812\pi$
$510$	1033.04	0.0896937
$511$	1287.37	0.111448
$512$	−512.000	−0.0441942
$513$	−5328.71	−0.458613
$514$	−14840.9	−1.27354
$515$	4830.58	0.413322
$516$	−63.3867	−0.00540784
$517$	−3961.82	−0.337023
$518$	−1843.95	−0.156406
$519$	2655.51	0.224594
$520$	−1519.14	−0.128113
$521$	14969.7	1.25880	0.629399	−	0.777082i	$-0.283301\pi$
$521$	14969.7	1.25880	0.629399	+	0.777082i	$0.283301\pi$
$522$	−621.440	−0.0521067
$523$	1892.57	0.158234	0.0791169	−	0.996865i	$-0.474790\pi$
$523$	1892.57	0.158234	0.0791169	+	0.996865i	$0.474790\pi$
$524$	4207.65	0.350786
$525$	−249.276	−0.0207225
$526$	−9116.76	−0.755721
$527$	0	0
$528$	711.051	0.0586071
$529$	−12137.1	−0.997546
$530$	1889.35	0.154845
$531$	−5782.05	−0.472542
$532$	1212.37	0.0988022
$533$	9080.68	0.737951
$534$	−2218.73	−0.179801
$535$	7230.82	0.584328
$536$	4892.43	0.394256
$537$	2936.65	0.235989
$538$	9786.13	0.784220
$539$	16273.0	1.30042
$540$	1006.23	0.0801874
$541$	24138.1	1.91826	0.959131	−	0.282962i	$-0.0913170\pi$
$541$	24138.1	1.91826	0.959131	+	0.282962i	$0.0913170\pi$
$542$	−15055.6	−1.19316
$543$	305.323	0.0241302
$544$	−3492.99	−0.275295
$545$	5180.03	0.407135
$546$	−186.385	−0.0146090
$547$	14517.6	1.13478	0.567392	−	0.823448i	$-0.307953\pi$
$547$	14517.6	1.13478	0.567392	+	0.823448i	$0.307953\pi$
$548$	−6998.52	−0.545551
$549$	377.836	0.0293728
$550$	−9540.65	−0.739663
$551$	1300.01	0.100512
$552$	40.0322	0.00308674
$553$	−1971.55	−0.151607
$554$	8730.43	0.669532
$555$	1575.51	0.120499
$556$	7145.28	0.545013
$557$	689.679	0.0524644	0.0262322	−	0.999656i	$-0.491649\pi$
$557$	689.679	0.0524644	0.0262322	+	0.999656i	$0.491649\pi$
$558$	0	0
$559$	635.925	0.0481158
$560$	−228.933	−0.0172753
$561$	4850.96	0.365076
$562$	−11790.7	−0.884980
$563$	−16924.1	−1.26690	−0.633451	−	0.773783i	$-0.718362\pi$
$563$	−16924.1	−1.26690	−0.633451	+	0.773783i	$0.718362\pi$
$564$	−299.055	−0.0223271
$565$	−6015.24	−0.447899
$566$	−10305.3	−0.765304
$567$	−1832.51	−0.135729
$568$	−8044.55	−0.594264
$569$	19315.8	1.42313	0.711566	−	0.702619i	$-0.247986\pi$
$569$	19315.8	1.42313	0.711566	+	0.702619i	$0.247986\pi$
$570$	−1035.88	−0.0761195
$571$	4131.46	0.302795	0.151398	−	0.988473i	$-0.451623\pi$
$571$	4131.46	0.302795	0.151398	+	0.988473i	$0.451623\pi$
$572$	−7133.59	−0.521452
$573$	−1735.06	−0.126498
$574$	1368.45	0.0995088
$575$	−537.138	−0.0389569
$576$	−1674.33	−0.121117
$577$	3387.46	0.244405	0.122203	−	0.992505i	$-0.461004\pi$
$577$	3387.46	0.244405	0.122203	+	0.992505i	$0.461004\pi$
$578$	−14004.0	−1.00777
$579$	−2253.22	−0.161729
$580$	−245.483	−0.0175744
$581$	59.2572	0.00423133
$582$	2748.04	0.195722
$583$	8872.01	0.630259
$584$	3719.27	0.263535
$585$	−4967.85	−0.351103
$586$	−7687.24	−0.541906
$587$	−9312.37	−0.654792	−0.327396	−	0.944887i	$-0.606171\pi$
$587$	−9312.37	−0.654792	−0.327396	+	0.944887i	$0.606171\pi$
$588$	1228.35	0.0861504
$589$	0	0
$590$	−2284.04	−0.159377
$591$	3882.81	0.270249
$592$	−5327.23	−0.369844
$593$	7254.28	0.502357	0.251178	−	0.967941i	$-0.419182\pi$
$593$	7254.28	0.502357	0.251178	+	0.967941i	$0.419182\pi$
$594$	4725.06	0.326383
$595$	−1561.84	−0.107612
$596$	−9951.73	−0.683958
$597$	−2323.09	−0.159260
$598$	−401.621	−0.0274640
$599$	−4162.23	−0.283914	−0.141957	−	0.989873i	$-0.545339\pi$
$599$	−4162.23	−0.283914	−0.141957	+	0.989873i	$0.545339\pi$
$600$	−720.168	−0.0490012
$601$	−5369.44	−0.364432	−0.182216	−	0.983258i	$-0.558327\pi$
$601$	−5369.44	−0.364432	−0.182216	+	0.983258i	$0.558327\pi$
$602$	95.8334	0.00648817
$603$	15999.1	1.08049
$604$	12940.1	0.871729
$605$	5291.02	0.355554
$606$	−322.234	−0.0216004
$607$	15058.5	1.00693	0.503464	−	0.864016i	$-0.332059\pi$
$607$	15058.5	1.00693	0.503464	+	0.864016i	$0.332059\pi$
$608$	3502.58	0.233632
$609$	−30.1186	−0.00200405
$610$	149.254	0.00990673
$611$	3000.26	0.198654
$612$	−11422.7	−0.754466
$613$	615.520	0.0405557	0.0202778	−	0.999794i	$-0.493545\pi$
$613$	615.520	0.0405557	0.0202778	+	0.999794i	$0.493545\pi$
$614$	−20695.2	−1.36025
$615$	−1169.24	−0.0766638
$616$	−1075.03	−0.0703150
$617$	15501.7	1.01147	0.505733	−	0.862690i	$-0.331222\pi$
$617$	15501.7	1.01147	0.505733	+	0.862690i	$0.331222\pi$
$618$	1712.24	0.111451
$619$	702.135	0.0455916	0.0227958	−	0.999740i	$-0.492743\pi$
$619$	702.135	0.0455916	0.0227958	+	0.999740i	$0.492743\pi$
$620$	0	0
$621$	266.020	0.0171901
$622$	−7706.98	−0.496820
$623$	3354.46	0.215720
$624$	−538.473	−0.0345452
$625$	6325.52	0.404834
$626$	70.6960	0.00451371
$627$	−4864.28	−0.309826
$628$	4300.52	0.273263
$629$	−36343.6	−2.30384
$630$	−748.650	−0.0473444
$631$	−20880.4	−1.31733	−0.658665	−	0.752436i	$-0.728879\pi$
$631$	−20880.4	−1.31733	−0.658665	+	0.752436i	$0.728879\pi$
$632$	−5695.88	−0.358497
$633$	1220.77	0.0766529
$634$	864.521	0.0541554
$635$	−11343.1	−0.708879
$636$	669.696	0.0417534
$637$	−12323.4	−0.766517
$638$	−1152.74	−0.0715321
$639$	−26307.0	−1.62862
$640$	−661.397	−0.0408500
$641$	1854.24	0.114256	0.0571280	−	0.998367i	$-0.481806\pi$
$641$	1854.24	0.114256	0.0571280	+	0.998367i	$0.481806\pi$
$642$	2563.03	0.157562
$643$	−8596.20	−0.527218	−0.263609	−	0.964630i	$-0.584913\pi$
$643$	−8596.20	−0.527218	−0.263609	+	0.964630i	$0.584913\pi$
$644$	−60.5239	−0.00370338
$645$	−81.8824	−0.00499863
$646$	23895.4	1.45534
$647$	−30338.2	−1.84346	−0.921728	−	0.387837i	$-0.873222\pi$
$647$	−30338.2	−1.84346	−0.921728	+	0.387837i	$0.873222\pi$
$648$	−5294.19	−0.320950
$649$	−10725.4	−0.648705
$650$	7225.05	0.435985
$651$	0	0
$652$	−3653.58	−0.219456
$653$	−25490.7	−1.52761	−0.763804	−	0.645448i	$-0.776671\pi$
$653$	−25490.7	−1.52761	−0.763804	+	0.645448i	$0.776671\pi$
$654$	1836.11	0.109782
$655$	5435.40	0.324242
$656$	3953.51	0.235303
$657$	12162.7	0.722238
$658$	452.136	0.0267874
$659$	28789.8	1.70181	0.850905	−	0.525319i	$-0.176054\pi$
$659$	28789.8	1.70181	0.850905	+	0.525319i	$0.176054\pi$
$660$	918.530	0.0541723
$661$	−27723.9	−1.63137	−0.815686	−	0.578496i	$-0.803640\pi$
$661$	−27723.9	−1.63137	−0.815686	+	0.578496i	$0.803640\pi$
$662$	20280.1	1.19065
$663$	−3673.59	−0.215189
$664$	171.196	0.0100056
$665$	1566.13	0.0913259
$666$	−17420.9	−1.01359
$667$	−64.8993	−0.00376748
$668$	14066.7	0.814754
$669$	−5562.64	−0.321471
$670$	6320.00	0.364422
$671$	700.867	0.0403229
$672$	−81.1474	−0.00465823
$673$	3358.20	0.192346	0.0961732	−	0.995365i	$-0.469340\pi$
$673$	3358.20	0.192346	0.0961732	+	0.995365i	$0.469340\pi$
$674$	−17636.6	−1.00792
$675$	−4785.64	−0.272888
$676$	−3385.79	−0.192637
$677$	−22710.6	−1.28928	−0.644638	−	0.764488i	$-0.722992\pi$
$677$	−22710.6	−1.28928	−0.644638	+	0.764488i	$0.722992\pi$
$678$	−2132.16	−0.120774
$679$	−4154.72	−0.234821
$680$	−4512.21	−0.254464
$681$	−3891.77	−0.218991
$682$	0	0
$683$	33669.2	1.88626	0.943130	−	0.332424i	$-0.107867\pi$
$683$	33669.2	1.88626	0.943130	+	0.332424i	$0.107867\pi$
$684$	11454.0	0.640286
$685$	−9040.63	−0.504270
$686$	−3756.72	−0.209085
$687$	−1277.97	−0.0709716
$688$	276.866	0.0153422
$689$	−6718.70	−0.371498
$690$	51.7132	0.00285317
$691$	20114.3	1.10736	0.553680	−	0.832730i	$-0.313223\pi$
$691$	20114.3	1.10736	0.553680	+	0.832730i	$0.313223\pi$
$692$	−11599.0	−0.637178
$693$	−3515.52	−0.192704
$694$	3458.44	0.189165
$695$	9230.21	0.503772
$696$	−87.0137	−0.00473886
$697$	26971.8	1.46575
$698$	−16507.2	−0.895140
$699$	1485.89	0.0804028
$700$	1088.81	0.0587902
$701$	−29681.9	−1.59924	−0.799621	−	0.600505i	$-0.794966\pi$
$701$	−29681.9	−1.59924	−0.799621	+	0.600505i	$0.794966\pi$
$702$	−3578.25	−0.192382
$703$	36443.4	1.95518
$704$	−3105.79	−0.166270
$705$	−386.316	−0.0206376
$706$	−48.9598	−0.00260995
$707$	487.180	0.0259156
$708$	−809.599	−0.0429754
$709$	−20451.8	−1.08334	−0.541668	−	0.840593i	$-0.682207\pi$
$709$	−20451.8	−1.08334	−0.541668	+	0.840593i	$0.682207\pi$
$710$	−10391.9	−0.549296
$711$	−18626.5	−0.982486
$712$	9691.17	0.510101
$713$	0	0
$714$	−553.607	−0.0290171
$715$	−9215.11	−0.481994
$716$	−12827.0	−0.669506
$717$	−764.112	−0.0397996
$718$	−1615.10	−0.0839487
$719$	−27476.2	−1.42516	−0.712578	−	0.701593i	$-0.752473\pi$
$719$	−27476.2	−1.42516	−0.712578	+	0.701593i	$0.752473\pi$
$720$	−2162.88	−0.111952
$721$	−2588.71	−0.133715
$722$	−10243.0	−0.527986
$723$	1449.31	0.0745513
$724$	−1333.62	−0.0684579
$725$	1167.52	0.0598078
$726$	1875.45	0.0958739
$727$	−18123.2	−0.924555	−0.462278	−	0.886735i	$-0.652968\pi$
$727$	−18123.2	−0.924555	−0.462278	+	0.886735i	$0.652968\pi$
$728$	814.109	0.0414462
$729$	−16109.1	−0.818429
$730$	4804.53	0.243594
$731$	1888.85	0.0955698
$732$	52.9043	0.00267131
$733$	18487.3	0.931576	0.465788	−	0.884896i	$-0.345771\pi$
$733$	18487.3	0.931576	0.465788	+	0.884896i	$0.345771\pi$
$734$	17313.1	0.870622
$735$	1586.78	0.0796315
$736$	−174.856	−0.00875717
$737$	29677.5	1.48329
$738$	12928.6	0.644864
$739$	29885.3	1.48762	0.743809	−	0.668392i	$-0.233017\pi$
$739$	29885.3	1.48762	0.743809	+	0.668392i	$0.233017\pi$
$740$	−6881.67	−0.341858
$741$	3683.68	0.182623
$742$	−1012.50	−0.0500945
$743$	14694.6	0.725563	0.362781	−	0.931874i	$-0.381827\pi$
$743$	14694.6	0.725563	0.362781	+	0.931874i	$0.381827\pi$
$744$	0	0
$745$	−12855.6	−0.632203
$746$	−14527.9	−0.713006
$747$	559.841	0.0274210
$748$	−21188.5	−1.03573
$749$	−3875.00	−0.189038
$750$	−2113.29	−0.102889
$751$	6746.22	0.327794	0.163897	−	0.986477i	$-0.447594\pi$
$751$	6746.22	0.327794	0.163897	+	0.986477i	$0.447594\pi$
$752$	1306.24	0.0633426
$753$	−1567.60	−0.0758654
$754$	872.961	0.0421636
$755$	16715.9	0.805765
$756$	−539.239	−0.0259417
$757$	16302.4	0.782720	0.391360	−	0.920238i	$-0.372005\pi$
$757$	16302.4	0.782720	0.391360	+	0.920238i	$0.372005\pi$
$758$	−6587.57	−0.315661
$759$	242.835	0.0116131
$760$	4524.60	0.215953
$761$	−19843.3	−0.945227	−0.472613	−	0.881270i	$-0.656689\pi$
$761$	−19843.3	−0.945227	−0.472613	+	0.881270i	$0.656689\pi$
$762$	−4020.67	−0.191146
$763$	−2775.98	−0.131713
$764$	7578.55	0.358877
$765$	−14755.7	−0.697376
$766$	24795.4	1.16957
$767$	8122.27	0.382371
$768$	−234.438	−0.0110150
$769$	7233.85	0.339219	0.169609	−	0.985511i	$-0.445749\pi$
$769$	7233.85	0.339219	0.169609	+	0.985511i	$0.445749\pi$
$770$	−1388.71	−0.0649943
$771$	−6795.43	−0.317421
$772$	9841.84	0.458828
$773$	−14721.8	−0.685003	−0.342502	−	0.939517i	$-0.611274\pi$
$773$	−14721.8	−0.685003	−0.342502	+	0.939517i	$0.611274\pi$
$774$	905.400	0.0420464
$775$	0	0
$776$	−12003.1	−0.555267
$777$	−844.319	−0.0389830
$778$	−17840.8	−0.822137
$779$	−27045.8	−1.24392
$780$	−695.595	−0.0319311
$781$	−48798.2	−2.23577
$782$	−1192.91	−0.0545503
$783$	−578.221	−0.0263907
$784$	−5365.31	−0.244411
$785$	5555.37	0.252586
$786$	1926.63	0.0874307
$787$	18989.6	0.860108	0.430054	−	0.902803i	$-0.358495\pi$
$787$	18989.6	0.860108	0.430054	+	0.902803i	$0.358495\pi$
$788$	−16959.7	−0.766705
$789$	−4174.44	−0.188358
$790$	−7357.88	−0.331369
$791$	3223.57	0.144901
$792$	−10156.5	−0.455675
$793$	−530.761	−0.0237678
$794$	26453.3	1.18236
$795$	865.107	0.0385939
$796$	10147.0	0.451824
$797$	−1756.41	−0.0780618	−0.0390309	−	0.999238i	$-0.512427\pi$
$797$	−1756.41	−0.0780618	−0.0390309	+	0.999238i	$0.512427\pi$
$798$	555.127	0.0246257
$799$	8911.47	0.394575
$800$	3145.61	0.139018
$801$	31691.8	1.39797
$802$	−6520.04	−0.287071
$803$	22561.1	0.991488
$804$	2240.18	0.0982651
$805$	−78.1843	−0.00342315
$806$	0	0
$807$	4480.94	0.195461
$808$	1407.48	0.0612810
$809$	31426.4	1.36575	0.682876	−	0.730534i	$-0.260729\pi$
$809$	31426.4	1.36575	0.682876	+	0.730534i	$0.260729\pi$
$810$	−6838.98	−0.296663
$811$	−40043.1	−1.73379	−0.866896	−	0.498489i	$-0.833888\pi$
$811$	−40043.1	−1.73379	−0.866896	+	0.498489i	$0.833888\pi$
$812$	131.555	0.00568554
$813$	−6893.78	−0.297387
$814$	−32315.0	−1.39145
$815$	−4719.66	−0.202850
$816$	−1599.39	−0.0686151
$817$	−1894.03	−0.0811063
$818$	1365.83	0.0583805
$819$	2662.27	0.113587
$820$	5107.11	0.217497
$821$	25691.7	1.09214	0.546069	−	0.837740i	$-0.316124\pi$
$821$	25691.7	1.09214	0.546069	+	0.837740i	$0.316124\pi$
$822$	−3204.53	−0.135974
$823$	−15468.1	−0.655146	−0.327573	−	0.944826i	$-0.606231\pi$
$823$	−15468.1	−0.655146	−0.327573	+	0.944826i	$0.606231\pi$
$824$	−7478.89	−0.316189
$825$	−4368.54	−0.184355
$826$	1224.02	0.0515606
$827$	17847.9	0.750461	0.375231	−	0.926932i	$-0.377564\pi$
$827$	17847.9	0.750461	0.375231	+	0.926932i	$0.377564\pi$
$828$	−571.809	−0.0239997
$829$	−22101.4	−0.925950	−0.462975	−	0.886371i	$-0.653218\pi$
$829$	−22101.4	−0.925950	−0.462975	+	0.886371i	$0.653218\pi$
$830$	221.150	0.00924846
$831$	3997.55	0.166875
$832$	2351.99	0.0980056
$833$	−36603.4	−1.52249
$834$	3271.73	0.135840
$835$	18171.2	0.753102
$836$	21246.6	0.878984
$837$	0	0
$838$	−10958.5	−0.451736
$839$	−17592.5	−0.723910	−0.361955	−	0.932196i	$-0.617891\pi$
$839$	−17592.5	−0.723910	−0.361955	+	0.932196i	$0.617891\pi$
$840$	−104.826	−0.00430574
$841$	−24247.9	−0.994216
$842$	−27647.7	−1.13159
$843$	−5398.79	−0.220574
$844$	−5332.20	−0.217467
$845$	−4373.73	−0.178060
$846$	4271.62	0.173595
$847$	−2835.46	−0.115027
$848$	−2925.16	−0.118456
$849$	−4718.64	−0.190746
$850$	21460.1	0.865972
$851$	−1819.33	−0.0732854
$852$	−3683.49	−0.148116
$853$	3048.40	0.122362	0.0611812	−	0.998127i	$-0.480513\pi$
$853$	3048.40	0.122362	0.0611812	+	0.998127i	$0.480513\pi$
$854$	−79.9852	−0.00320496
$855$	14796.2	0.591836
$856$	−11195.0	−0.447007
$857$	−12355.4	−0.492475	−0.246238	−	0.969210i	$-0.579194\pi$
$857$	−12355.4	−0.492475	−0.246238	+	0.969210i	$0.579194\pi$
$858$	−3266.38	−0.129968
$859$	−32228.4	−1.28012	−0.640058	−	0.768327i	$-0.721090\pi$
$859$	−32228.4	−1.28012	−0.640058	+	0.768327i	$0.721090\pi$
$860$	357.653	0.0141813
$861$	626.596	0.0248018
$862$	22114.5	0.873807
$863$	−7764.22	−0.306254	−0.153127	−	0.988207i	$-0.548934\pi$
$863$	−7764.22	−0.306254	−0.153127	+	0.988207i	$0.548934\pi$
$864$	−1557.88	−0.0613428
$865$	−14983.5	−0.588963
$866$	5612.63	0.220237
$867$	−6412.24	−0.251178
$868$	0	0
$869$	−34551.2	−1.34876
$870$	−112.403	−0.00438027
$871$	−22474.5	−0.874306
$872$	−8019.93	−0.311455
$873$	−39252.3	−1.52175
$874$	1196.18	0.0462947
$875$	3195.06	0.123443
$876$	1703.01	0.0656841
$877$	−33655.6	−1.29586	−0.647931	−	0.761699i	$-0.724365\pi$
$877$	−33655.6	−1.29586	−0.647931	+	0.761699i	$0.724365\pi$
$878$	−25317.0	−0.973129
$879$	−3519.89	−0.135066
$880$	−4012.04	−0.153688
$881$	23192.1	0.886902	0.443451	−	0.896299i	$-0.353754\pi$
$881$	23192.1	0.886902	0.443451	+	0.896299i	$0.353754\pi$
$882$	−17545.5	−0.669827
$883$	5748.29	0.219078	0.109539	−	0.993983i	$-0.465063\pi$
$883$	5748.29	0.219078	0.109539	+	0.993983i	$0.465063\pi$
$884$	16045.8	0.610498
$885$	−1045.83	−0.0397235
$886$	13158.8	0.498962
$887$	13125.7	0.496862	0.248431	−	0.968650i	$-0.420085\pi$
$887$	13125.7	0.496862	0.248431	+	0.968650i	$0.420085\pi$
$888$	−2439.27	−0.0921808
$889$	6078.79	0.229332
$890$	12519.0	0.471502
$891$	−32114.6	−1.20749
$892$	24297.0	0.912022
$893$	−8935.94	−0.334860
$894$	−4556.77	−0.170471
$895$	−16569.8	−0.618845
$896$	354.443	0.0132155
$897$	−183.897	−0.00684520
$898$	−37281.6	−1.38542
$899$	0	0
$900$	10286.7	0.380988
$901$	−19956.1	−0.737885
$902$	23982.0	0.885269
$903$	43.8808	0.00161712
$904$	9313.03	0.342640
$905$	−1722.76	−0.0632777
$906$	5925.09	0.217271
$907$	−14616.4	−0.535092	−0.267546	−	0.963545i	$-0.586213\pi$
$907$	−14616.4	−0.535092	−0.267546	+	0.963545i	$0.586213\pi$
$908$	16998.8	0.621284
$909$	4602.71	0.167945
$910$	1051.66	0.0383100
$911$	21826.9	0.793805	0.396902	−	0.917861i	$-0.370085\pi$
$911$	21826.9	0.793805	0.396902	+	0.917861i	$0.370085\pi$
$912$	1603.78	0.0582309
$913$	1038.48	0.0376436
$914$	−11701.4	−0.423466
$915$	68.3414	0.00246918
$916$	5582.02	0.201348
$917$	−2912.84	−0.104897
$918$	−10628.2	−0.382118
$919$	16021.5	0.575083	0.287541	−	0.957768i	$-0.407162\pi$
$919$	16021.5	0.575083	0.287541	+	0.957768i	$0.407162\pi$
$920$	−225.877	−0.00809452
$921$	−9476.07	−0.339030
$922$	−28415.1	−1.01497
$923$	36954.5	1.31785
$924$	−492.241	−0.0175255
$925$	32729.3	1.16339
$926$	−16070.4	−0.570307
$927$	−24457.2	−0.866538
$928$	380.066	0.0134443
$929$	18139.9	0.640636	0.320318	−	0.947310i	$-0.396210\pi$
$929$	18139.9	0.640636	0.320318	+	0.947310i	$0.396210\pi$
$930$	0	0
$931$	36704.0	1.29208
$932$	−6490.21	−0.228105
$933$	−3528.93	−0.123828
$934$	−4747.95	−0.166336
$935$	−27371.1	−0.957358
$936$	7691.41	0.268591
$937$	−7809.38	−0.272275	−0.136137	−	0.990690i	$-0.543469\pi$
$937$	−7809.38	−0.272275	−0.136137	+	0.990690i	$0.543469\pi$
$938$	−3386.89	−0.117896
$939$	32.3708	0.00112501
$940$	1687.39	0.0585495
$941$	30853.2	1.06885	0.534423	−	0.845217i	$-0.320529\pi$
$941$	30853.2	1.06885	0.534423	+	0.845217i	$0.320529\pi$
$942$	1969.15	0.0681087
$943$	1350.18	0.0466257
$944$	3536.24	0.121922
$945$	−696.584	−0.0239787
$946$	1679.47	0.0577213
$947$	50075.1	1.71829	0.859146	−	0.511730i	$-0.170995\pi$
$947$	50075.1	1.71829	0.859146	+	0.511730i	$0.170995\pi$
$948$	−2608.07	−0.0893524
$949$	−17085.4	−0.584420
$950$	−21519.0	−0.734916
$951$	395.853	0.0134978
$952$	2418.09	0.0823224
$953$	−17737.3	−0.602905	−0.301453	−	0.953481i	$-0.597472\pi$
$953$	−17737.3	−0.602905	−0.301453	+	0.953481i	$0.597472\pi$
$954$	−9565.76	−0.324636
$955$	9789.90	0.331721
$956$	3337.56	0.112912
$957$	−527.825	−0.0178288
$958$	41220.0	1.39014
$959$	4844.88	0.163138
$960$	−302.845	−0.0101815
$961$	0	0
$962$	24471.9	0.820171
$963$	−36609.6	−1.22506
$964$	−6330.45	−0.211504
$965$	12713.6	0.424109
$966$	−27.7131	−0.000923039 0
$967$	−27015.5	−0.898408	−0.449204	−	0.893429i	$-0.648292\pi$
$967$	−27015.5	−0.898408	−0.449204	+	0.893429i	$0.648292\pi$
$968$	−8191.75	−0.271997
$969$	10941.4	0.362733
$970$	−15505.5	−0.513251
$971$	6966.75	0.230251	0.115125	−	0.993351i	$-0.463273\pi$
$971$	6966.75	0.230251	0.115125	+	0.993351i	$0.463273\pi$
$972$	−7681.99	−0.253498
$973$	−4946.47	−0.162977
$974$	12675.0	0.416975
$975$	3308.26	0.108666
$976$	−231.080	−0.00757859
$977$	−27076.0	−0.886631	−0.443316	−	0.896366i	$-0.646198\pi$
$977$	−27076.0	−0.886631	−0.443316	+	0.896366i	$0.646198\pi$
$978$	−1672.93	−0.0546976
$979$	58786.7	1.91913
$980$	−6930.87	−0.225917
$981$	−26226.5	−0.853566
$982$	13856.0	0.450269
$983$	34139.2	1.10770	0.553851	−	0.832616i	$-0.313158\pi$
$983$	34139.2	1.10770	0.553851	+	0.832616i	$0.313158\pi$
$984$	1810.26	0.0586473
$985$	−21908.4	−0.708689
$986$	2592.90	0.0837473
$987$	207.027	0.00667654
$988$	−16089.9	−0.518105
$989$	94.5542	0.00304009
$990$	−13120.0	−0.421194
$991$	51547.9	1.65234	0.826172	−	0.563418i	$-0.190514\pi$
$991$	51547.9	1.65234	0.826172	+	0.563418i	$0.190514\pi$
$992$	0	0
$993$	9285.98	0.296759
$994$	5569.01	0.177705
$995$	13107.8	0.417634
$996$	78.3886	0.00249381
$997$	50209.2	1.59493	0.797463	−	0.603368i	$-0.206175\pi$
$997$	50209.2	1.59493	0.797463	+	0.603368i	$0.206175\pi$
$998$	−23848.4	−0.756420
$999$	−16209.4	−0.513355

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1922.4.a.x.1.15	✓	32
31.30	odd	2	inner	1922.4.a.x.1.18	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1922.4.a.x.1.15	✓	32	1.1	even	1	trivial
1922.4.a.x.1.18	yes	32	31.30	odd	2	inner

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:	\( N \)	\(=\)	\( 1922 = 2 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1922.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} -0.915774 q^{3} +4.00000 q^{4} +5.16716 q^{5} +1.83155 q^{6} -2.76909 q^{7} -8.00000 q^{8} -26.1614 q^{9} -10.3343 q^{10} -48.5280 q^{11} -3.66310 q^{12} +36.7499 q^{13} +5.53817 q^{14} -4.73195 q^{15} +16.0000 q^{16} +109.156 q^{17} +52.3227 q^{18} -109.456 q^{19} +20.6687 q^{20} +2.53586 q^{21} +97.0560 q^{22} +5.46425 q^{23} +7.32619 q^{24} -98.3004 q^{25} -73.4997 q^{26} +48.6838 q^{27} -11.0763 q^{28} -11.8771 q^{29} +9.46391 q^{30} -32.0000 q^{32} +44.4407 q^{33} -218.312 q^{34} -14.3083 q^{35} -104.645 q^{36} -332.952 q^{37} +218.911 q^{38} -33.6546 q^{39} -41.3373 q^{40} +247.094 q^{41} -5.07172 q^{42} +17.3041 q^{43} -194.112 q^{44} -135.180 q^{45} -10.9285 q^{46} +81.6399 q^{47} -14.6524 q^{48} -335.332 q^{49} +196.601 q^{50} -99.9620 q^{51} +146.999 q^{52} -182.822 q^{53} -97.3676 q^{54} -250.752 q^{55} +22.1527 q^{56} +100.237 q^{57} +23.7541 q^{58} +221.015 q^{59} -18.9278 q^{60} -14.4425 q^{61} +72.4431 q^{63} +64.0000 q^{64} +189.893 q^{65} -88.8814 q^{66} -611.554 q^{67} +436.623 q^{68} -5.00402 q^{69} +28.6167 q^{70} +1005.57 q^{71} +209.291 q^{72} -464.909 q^{73} +665.904 q^{74} +90.0210 q^{75} -437.822 q^{76} +134.378 q^{77} +67.3091 q^{78} +711.985 q^{79} +82.6746 q^{80} +661.773 q^{81} -494.188 q^{82} -21.3995 q^{83} +10.1434 q^{84} +564.026 q^{85} -34.6083 q^{86} +10.8767 q^{87} +388.224 q^{88} -1211.40 q^{89} +270.360 q^{90} -101.764 q^{91} +21.8570 q^{92} -163.280 q^{94} -565.575 q^{95} +29.3048 q^{96} +1500.39 q^{97} +670.664 q^{98} +1269.56 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 64 q^{2} + 128 q^{4} + 112 q^{7} - 256 q^{8} + 288 q^{9} - 224 q^{14} + 512 q^{16} - 576 q^{18} + 304 q^{19} + 1200 q^{25} + 448 q^{28} - 1024 q^{32} - 272 q^{33} + 1152 q^{36} - 608 q^{38} + 1616 q^{39}+ \cdots - 6176 q^{98}+O(q^{100}) \) 32 * q - 64 * q^2 + 128 * q^4 + 112 * q^7 - 256 * q^8 + 288 * q^9 - 224 * q^14 + 512 * q^16 - 576 * q^18 + 304 * q^19 + 1200 * q^25 + 448 * q^28 - 1024 * q^32 - 272 * q^33 + 1152 * q^36 - 608 * q^38 + 1616 * q^39 + 368 * q^41 + 1360 * q^45 + 784 * q^47 + 3088 * q^49 - 2400 * q^50 - 464 * q^51 - 896 * q^56 - 848 * q^59 + 6992 * q^63 + 2048 * q^64 + 544 * q^66 + 3216 * q^67 - 1664 * q^69 + 2688 * q^71 - 2304 * q^72 + 1216 * q^76 - 3232 * q^78 + 3584 * q^81 - 736 * q^82 + 4960 * q^87 - 2720 * q^90 - 1568 * q^94 - 6816 * q^95 + 7504 * q^97 - 6176 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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