LMFDB - Embedded newform 1617.4.a.be.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1617,4,Mod(1,1617)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1617.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1617, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1617 = 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1617.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,4,-48,72,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$95.4060884793$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 92 x^{14} + 346 x^{13} + 3385 x^{12} - 11756 x^{11} - 63875 x^{10} + 199850 x^{9} + \cdots + 5479424 $ x^16 - 4x^15 - 92x^14 + 346x^13 + 3385x^12 - 11756x^11 - 63875x^10 + 199850x^9 + 660294x^8 - 1796196x^7 - 3723569x^6 + 8337264x^5 + 10859648x^4 - 17983904x^3 - 14390128x^2 + 13240576*x + 5479424
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{4}\cdot 7^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.31445$ of defining polynomial
Character	$\chi$	$=$	1617.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.31445 q^{2} -3.00000 q^{3} +10.6145 q^{4} -14.6492 q^{5} +12.9433 q^{6} -11.2800 q^{8} +9.00000 q^{9} +63.2031 q^{10} +11.0000 q^{11} -31.8434 q^{12} -24.6988 q^{13} +43.9475 q^{15} -36.2488 q^{16} +61.9469 q^{17} -38.8300 q^{18} -16.6166 q^{19} -155.493 q^{20} -47.4589 q^{22} -50.6775 q^{23} +33.8400 q^{24} +89.5985 q^{25} +106.562 q^{26} -27.0000 q^{27} -100.342 q^{29} -189.609 q^{30} -113.542 q^{31} +246.634 q^{32} -33.0000 q^{33} -267.267 q^{34} +95.5302 q^{36} -1.84796 q^{37} +71.6914 q^{38} +74.0963 q^{39} +165.243 q^{40} +62.0499 q^{41} -302.869 q^{43} +116.759 q^{44} -131.843 q^{45} +218.645 q^{46} -41.4573 q^{47} +108.746 q^{48} -386.568 q^{50} -185.841 q^{51} -262.164 q^{52} +678.402 q^{53} +116.490 q^{54} -161.141 q^{55} +49.8498 q^{57} +432.921 q^{58} -144.369 q^{59} +466.480 q^{60} -228.420 q^{61} +489.871 q^{62} -774.097 q^{64} +361.817 q^{65} +142.377 q^{66} +482.768 q^{67} +657.534 q^{68} +152.032 q^{69} +559.130 q^{71} -101.520 q^{72} -218.369 q^{73} +7.97293 q^{74} -268.796 q^{75} -176.376 q^{76} -319.685 q^{78} +1175.69 q^{79} +531.015 q^{80} +81.0000 q^{81} -267.711 q^{82} -187.873 q^{83} -907.471 q^{85} +1306.71 q^{86} +301.027 q^{87} -124.080 q^{88} +34.7130 q^{89} +568.828 q^{90} -537.915 q^{92} +340.626 q^{93} +178.865 q^{94} +243.419 q^{95} -739.901 q^{96} -712.441 q^{97} +99.0000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20}+ \cdots + 1584 q^{99}+O(q^{100}) $ 16 * q + 4 * q^2 - 48 * q^3 + 72 * q^4 - 40 * q^5 - 12 * q^6 + 66 * q^8 + 144 * q^9 - 178 * q^10 + 176 * q^11 - 216 * q^12 - 104 * q^13 + 120 * q^15 + 220 * q^16 - 180 * q^17 + 36 * q^18 - 152 * q^19 - 298 * q^20 + 44 * q^22 + 4 * q^23 - 198 * q^24 + 588 * q^25 - 406 * q^26 - 432 * q^27 + 412 * q^29 + 534 * q^30 - 628 * q^31 + 592 * q^32 - 528 * q^33 + 88 * q^34 + 648 * q^36 + 148 * q^37 - 446 * q^38 + 312 * q^39 - 1376 * q^40 - 596 * q^41 - 260 * q^43 + 792 * q^44 - 360 * q^45 - 148 * q^46 - 2220 * q^47 - 660 * q^48 + 82 * q^50 + 540 * q^51 - 1046 * q^52 + 168 * q^53 - 108 * q^54 - 440 * q^55 + 456 * q^57 + 538 * q^58 + 48 * q^59 + 894 * q^60 - 1504 * q^61 - 1276 * q^62 + 630 * q^64 + 1224 * q^65 - 132 * q^66 + 116 * q^67 - 356 * q^68 - 12 * q^69 + 320 * q^71 + 594 * q^72 - 652 * q^73 + 1062 * q^74 - 1764 * q^75 + 594 * q^76 + 1218 * q^78 + 1136 * q^79 - 1970 * q^80 + 1296 * q^81 + 416 * q^82 - 3300 * q^83 - 1148 * q^85 + 1864 * q^86 - 1236 * q^87 + 726 * q^88 - 2416 * q^89 - 1602 * q^90 - 1064 * q^92 + 1884 * q^93 - 914 * q^94 + 1120 * q^95 - 1776 * q^96 - 3616 * q^97 + 1584 * q^99

Coefficient data

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

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

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

$2$	−4.31445	−1.52539	−0.762694	−	0.646759i	$-0.776124\pi$
$2$	−4.31445	−1.52539	−0.762694	+	0.646759i	$0.776124\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	10.6145	1.32681
$5$	−14.6492	−1.31026	−0.655131	−	0.755515i	$-0.727387\pi$
$5$	−14.6492	−1.31026	−0.655131	+	0.755515i	$0.727387\pi$
$6$	12.9433	0.880683
$7$	0	0
$7$	0	0
$8$	−11.2800	−0.498510
$9$	9.00000	0.333333
$10$	63.2031	1.99866
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−31.8434	−0.766033
$13$	−24.6988	−0.526939	−0.263469	−	0.964668i	$-0.584867\pi$
$13$	−24.6988	−0.526939	−0.263469	+	0.964668i	$0.584867\pi$
$14$	0	0
$15$	43.9475	0.756480
$16$	−36.2488	−0.566387
$17$	61.9469	0.883784	0.441892	−	0.897068i	$-0.354307\pi$
$17$	61.9469	0.883784	0.441892	+	0.897068i	$0.354307\pi$
$18$	−38.8300	−0.508463
$19$	−16.6166	−0.200637	−0.100319	−	0.994955i	$-0.531986\pi$
$19$	−16.6166	−0.200637	−0.100319	+	0.994955i	$0.531986\pi$
$20$	−155.493	−1.73847
$21$	0	0
$22$	−47.4589	−0.459922
$23$	−50.6775	−0.459434	−0.229717	−	0.973257i	$-0.573780\pi$
$23$	−50.6775	−0.459434	−0.229717	+	0.973257i	$0.573780\pi$
$24$	33.8400	0.287815
$25$	89.5985	0.716788
$26$	106.562	0.803786
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−100.342	−0.642520	−0.321260	−	0.946991i	$-0.604106\pi$
$29$	−100.342	−0.642520	−0.321260	+	0.946991i	$0.604106\pi$
$30$	−189.609	−1.15393
$31$	−113.542	−0.657830	−0.328915	−	0.944359i	$-0.606683\pi$
$31$	−113.542	−0.657830	−0.328915	+	0.944359i	$0.606683\pi$
$32$	246.634	1.36247
$33$	−33.0000	−0.174078
$34$	−267.267	−1.34811
$35$	0	0
$36$	95.5302	0.442270
$37$	−1.84796	−0.00821088	−0.00410544	−	0.999992i	$-0.501307\pi$
$37$	−1.84796	−0.00821088	−0.00410544	+	0.999992i	$0.501307\pi$
$38$	71.6914	0.306050
$39$	74.0963	0.304228
$40$	165.243	0.653179
$41$	62.0499	0.236355	0.118178	−	0.992992i	$-0.462295\pi$
$41$	62.0499	0.236355	0.118178	+	0.992992i	$0.462295\pi$
$42$	0	0
$43$	−302.869	−1.07412	−0.537059	−	0.843545i	$-0.680465\pi$
$43$	−302.869	−1.07412	−0.537059	+	0.843545i	$0.680465\pi$
$44$	116.759	0.400048
$45$	−131.843	−0.436754
$46$	218.645	0.700815
$47$	−41.4573	−0.128663	−0.0643316	−	0.997929i	$-0.520492\pi$
$47$	−41.4573	−0.128663	−0.0643316	+	0.997929i	$0.520492\pi$
$48$	108.746	0.327004
$49$	0	0
$50$	−386.568	−1.09338
$51$	−185.841	−0.510253
$52$	−262.164	−0.699147
$53$	678.402	1.75822	0.879110	−	0.476619i	$-0.158138\pi$
$53$	678.402	1.75822	0.879110	+	0.476619i	$0.158138\pi$
$54$	116.490	0.293561
$55$	−161.141	−0.395059
$56$	0	0
$57$	49.8498	0.115838
$58$	432.921	0.980092
$59$	−144.369	−0.318563	−0.159281	−	0.987233i	$-0.550918\pi$
$59$	−144.369	−0.318563	−0.159281	+	0.987233i	$0.550918\pi$
$60$	466.480	1.00370
$61$	−228.420	−0.479446	−0.239723	−	0.970841i	$-0.577057\pi$
$61$	−228.420	−0.479446	−0.239723	+	0.970841i	$0.577057\pi$
$62$	489.871	1.00345
$63$	0	0
$64$	−774.097	−1.51191
$65$	361.817	0.690428
$66$	142.377	0.265536
$67$	482.768	0.880291	0.440146	−	0.897926i	$-0.354927\pi$
$67$	482.768	0.880291	0.440146	+	0.897926i	$0.354927\pi$
$68$	657.534	1.17261
$69$	152.032	0.265254
$70$	0	0
$71$	559.130	0.934599	0.467299	−	0.884099i	$-0.345227\pi$
$71$	559.130	0.934599	0.467299	+	0.884099i	$0.345227\pi$
$72$	−101.520	−0.166170
$73$	−218.369	−0.350112	−0.175056	−	0.984558i	$-0.556011\pi$
$73$	−218.369	−0.350112	−0.175056	+	0.984558i	$0.556011\pi$
$74$	7.97293	0.0125248
$75$	−268.796	−0.413838
$76$	−176.376	−0.266207
$77$	0	0
$78$	−319.685	−0.464066
$79$	1175.69	1.67438	0.837189	−	0.546914i	$-0.184197\pi$
$79$	1175.69	1.67438	0.837189	+	0.546914i	$0.184197\pi$
$80$	531.015	0.742116
$81$	81.0000	0.111111
$82$	−267.711	−0.360534
$83$	−187.873	−0.248455	−0.124227	−	0.992254i	$-0.539645\pi$
$83$	−187.873	−0.248455	−0.124227	+	0.992254i	$0.539645\pi$
$84$	0	0
$85$	−907.471	−1.15799
$86$	1306.71	1.63845
$87$	301.027	0.370959
$88$	−124.080	−0.150306
$89$	34.7130	0.0413435	0.0206718	−	0.999786i	$-0.493420\pi$
$89$	34.7130	0.0413435	0.0206718	+	0.999786i	$0.493420\pi$
$90$	568.828	0.666220
$91$	0	0
$92$	−537.915	−0.609581
$93$	340.626	0.379798
$94$	178.865	0.196261
$95$	243.419	0.262887
$96$	−739.901	−0.786623
$97$	−712.441	−0.745747	−0.372873	−	0.927882i	$-0.621627\pi$
$97$	−712.441	−0.745747	−0.372873	+	0.927882i	$0.621627\pi$
$98$	0	0
$99$	99.0000	0.100504
$100$	951.041	0.951041
$101$	89.9994	0.0886661	0.0443331	−	0.999017i	$-0.485884\pi$
$101$	89.9994	0.0886661	0.0443331	+	0.999017i	$0.485884\pi$
$102$	801.800	0.778334
$103$	−1109.04	−1.06094	−0.530469	−	0.847705i	$-0.677984\pi$
$103$	−1109.04	−1.06094	−0.530469	+	0.847705i	$0.677984\pi$
$104$	278.602	0.262684
$105$	0	0
$106$	−2926.93	−2.68197
$107$	950.441	0.858716	0.429358	−	0.903134i	$-0.358740\pi$
$107$	950.441	0.858716	0.429358	+	0.903134i	$0.358740\pi$
$108$	−286.591	−0.255344
$109$	1574.34	1.38343	0.691716	−	0.722169i	$-0.256855\pi$
$109$	1574.34	1.38343	0.691716	+	0.722169i	$0.256855\pi$
$110$	695.235	0.602618
$111$	5.54388	0.00474056
$112$	0	0
$113$	1593.80	1.32683	0.663417	−	0.748250i	$-0.269105\pi$
$113$	1593.80	1.32683	0.663417	+	0.748250i	$0.269105\pi$
$114$	−215.074	−0.176698
$115$	742.384	0.601979
$116$	−1065.08	−0.852501
$117$	−222.289	−0.175646
$118$	622.872	0.485932
$119$	0	0
$120$	−495.728	−0.377113
$121$	121.000	0.0909091
$122$	985.508	0.731342
$123$	−186.150	−0.136460
$124$	−1205.19	−0.872815
$125$	518.603	0.371082
$126$	0	0
$127$	−2056.40	−1.43682	−0.718409	−	0.695621i	$-0.755130\pi$
$127$	−2056.40	−1.43682	−0.718409	+	0.695621i	$0.755130\pi$
$128$	1366.74	0.943777
$129$	908.607	0.620142
$130$	−1561.04	−1.05317
$131$	−95.2553	−0.0635305	−0.0317652	−	0.999495i	$-0.510113\pi$
$131$	−95.2553	−0.0635305	−0.0317652	+	0.999495i	$0.510113\pi$
$132$	−350.278	−0.230968
$133$	0	0
$134$	−2082.88	−1.34279
$135$	395.528	0.252160
$136$	−698.761	−0.440575
$137$	1468.93	0.916052	0.458026	−	0.888939i	$-0.348557\pi$
$137$	1468.93	0.916052	0.458026	+	0.888939i	$0.348557\pi$
$138$	−655.936	−0.404616
$139$	1864.27	1.13759	0.568795	−	0.822479i	$-0.307410\pi$
$139$	1864.27	1.13759	0.568795	+	0.822479i	$0.307410\pi$
$140$	0	0
$141$	124.372	0.0742837
$142$	−2412.34	−1.42563
$143$	−271.686	−0.158878
$144$	−326.239	−0.188796
$145$	1469.93	0.841870
$146$	942.142	0.534056
$147$	0	0
$148$	−19.6151	−0.0108943
$149$	485.799	0.267102	0.133551	−	0.991042i	$-0.457362\pi$
$149$	485.799	0.267102	0.133551	+	0.991042i	$0.457362\pi$
$150$	1159.70	0.631263
$151$	−2444.09	−1.31720	−0.658600	−	0.752494i	$-0.728851\pi$
$151$	−2444.09	−1.31720	−0.658600	+	0.752494i	$0.728851\pi$
$152$	187.435	0.100020
$153$	557.522	0.294595
$154$	0	0
$155$	1663.30	0.861930
$156$	786.493	0.403653
$157$	3473.99	1.76595	0.882977	−	0.469417i	$-0.155536\pi$
$157$	3473.99	1.76595	0.882977	+	0.469417i	$0.155536\pi$
$158$	−5072.47	−2.55408
$159$	−2035.21	−1.01511
$160$	−3612.98	−1.78519
$161$	0	0
$162$	−349.470	−0.169488
$163$	−1383.85	−0.664977	−0.332488	−	0.943107i	$-0.607888\pi$
$163$	−1383.85	−0.664977	−0.332488	+	0.943107i	$0.607888\pi$
$164$	658.627	0.313598
$165$	483.423	0.228087
$166$	810.569	0.378990
$167$	2609.95	1.20936	0.604682	−	0.796467i	$-0.293300\pi$
$167$	2609.95	1.20936	0.604682	+	0.796467i	$0.293300\pi$
$168$	0	0
$169$	−1586.97	−0.722335
$170$	3915.24	1.76638
$171$	−149.549	−0.0668791
$172$	−3214.79	−1.42515
$173$	−1836.78	−0.807213	−0.403607	−	0.914933i	$-0.632244\pi$
$173$	−1836.78	−0.807213	−0.403607	+	0.914933i	$0.632244\pi$
$174$	−1298.76	−0.565857
$175$	0	0
$176$	−398.737	−0.170772
$177$	433.106	0.183922
$178$	−149.768	−0.0630649
$179$	3474.43	1.45079	0.725394	−	0.688334i	$-0.241657\pi$
$179$	3474.43	1.45079	0.725394	+	0.688334i	$0.241657\pi$
$180$	−1399.44	−0.579489
$181$	1930.00	0.792574	0.396287	−	0.918127i	$-0.370298\pi$
$181$	1930.00	0.792574	0.396287	+	0.918127i	$0.370298\pi$
$182$	0	0
$183$	685.261	0.276808
$184$	571.642	0.229033
$185$	27.0711	0.0107584
$186$	−1469.61	−0.579340
$187$	681.416	0.266471
$188$	−440.047	−0.170711
$189$	0	0
$190$	−1050.22	−0.401005
$191$	4739.40	1.79545	0.897726	−	0.440555i	$-0.145218\pi$
$191$	4739.40	1.79545	0.897726	+	0.440555i	$0.145218\pi$
$192$	2322.29	0.872901
$193$	616.631	0.229980	0.114990	−	0.993367i	$-0.463316\pi$
$193$	616.631	0.229980	0.114990	+	0.993367i	$0.463316\pi$
$194$	3073.79	1.13755
$195$	−1085.45	−0.398619
$196$	0	0
$197$	266.547	0.0963996	0.0481998	−	0.998838i	$-0.484652\pi$
$197$	266.547	0.0963996	0.0481998	+	0.998838i	$0.484652\pi$
$198$	−427.130	−0.153307
$199$	−1763.43	−0.628173	−0.314086	−	0.949394i	$-0.601698\pi$
$199$	−1763.43	−0.628173	−0.314086	+	0.949394i	$0.601698\pi$
$200$	−1010.67	−0.357326
$201$	−1448.30	−0.508236
$202$	−388.298	−0.135250
$203$	0	0
$204$	−1972.60	−0.677008
$205$	−908.981	−0.309688
$206$	4784.88	1.61834
$207$	−456.097	−0.153145
$208$	895.300	0.298451
$209$	−182.783	−0.0604944
$210$	0	0
$211$	−1939.41	−0.632769	−0.316384	−	0.948631i	$-0.602469\pi$
$211$	−1939.41	−0.632769	−0.316384	+	0.948631i	$0.602469\pi$
$212$	7200.87	2.33282
$213$	−1677.39	−0.539591
$214$	−4100.63	−1.30988
$215$	4436.78	1.40738
$216$	304.560	0.0959383
$217$	0	0
$218$	−6792.40	−2.11027
$219$	655.107	0.202137
$220$	−1710.43	−0.524168
$221$	−1530.01	−0.465700
$222$	−23.9188	−0.00723119
$223$	−1505.23	−0.452007	−0.226004	−	0.974126i	$-0.572566\pi$
$223$	−1505.23	−0.452007	−0.226004	+	0.974126i	$0.572566\pi$
$224$	0	0
$225$	806.387	0.238929
$226$	−6876.38	−2.02394
$227$	−5793.53	−1.69397	−0.846983	−	0.531620i	$-0.821583\pi$
$227$	−5793.53	−1.69397	−0.846983	+	0.531620i	$0.821583\pi$
$228$	529.129	0.153695
$229$	5867.31	1.69311	0.846557	−	0.532299i	$-0.178672\pi$
$229$	5867.31	1.69311	0.846557	+	0.532299i	$0.178672\pi$
$230$	−3202.98	−0.918252
$231$	0	0
$232$	1131.86	0.320303
$233$	1959.31	0.550894	0.275447	−	0.961316i	$-0.411174\pi$
$233$	1959.31	0.550894	0.275447	+	0.961316i	$0.411174\pi$
$234$	959.054	0.267929
$235$	607.316	0.168583
$236$	−1532.40	−0.422672
$237$	−3527.08	−0.966703
$238$	0	0
$239$	407.911	0.110400	0.0552000	−	0.998475i	$-0.482420\pi$
$239$	407.911	0.110400	0.0552000	+	0.998475i	$0.482420\pi$
$240$	−1593.05	−0.428461
$241$	−3637.51	−0.972251	−0.486125	−	0.873889i	$-0.661590\pi$
$241$	−3637.51	−0.972251	−0.486125	+	0.873889i	$0.661590\pi$
$242$	−522.048	−0.138672
$243$	−243.000	−0.0641500
$244$	−2424.56	−0.636134
$245$	0	0
$246$	803.134	0.208154
$247$	410.409	0.105724
$248$	1280.75	0.327935
$249$	563.619	0.143445
$250$	−2237.48	−0.566044
$251$	603.356	0.151727	0.0758635	−	0.997118i	$-0.475829\pi$
$251$	603.356	0.151727	0.0758635	+	0.997118i	$0.475829\pi$
$252$	0	0
$253$	−557.452	−0.138525
$254$	8872.23	2.19171
$255$	2722.41	0.668566
$256$	296.068	0.0722823
$257$	−3594.38	−0.872418	−0.436209	−	0.899845i	$-0.643679\pi$
$257$	−3594.38	−0.872418	−0.436209	+	0.899845i	$0.643679\pi$
$258$	−3920.14	−0.945958
$259$	0	0
$260$	3840.49	0.916066
$261$	−903.080	−0.214173
$262$	410.974	0.0969087
$263$	−3805.24	−0.892173	−0.446086	−	0.894990i	$-0.647183\pi$
$263$	−3805.24	−0.892173	−0.446086	+	0.894990i	$0.647183\pi$
$264$	372.240	0.0867795
$265$	−9938.03	−2.30373
$266$	0	0
$267$	−104.139	−0.0238697
$268$	5124.33	1.16798
$269$	−6927.58	−1.57019	−0.785097	−	0.619373i	$-0.787387\pi$
$269$	−6927.58	−1.57019	−0.785097	+	0.619373i	$0.787387\pi$
$270$	−1706.48	−0.384642
$271$	853.216	0.191251	0.0956257	−	0.995417i	$-0.469515\pi$
$271$	853.216	0.191251	0.0956257	+	0.995417i	$0.469515\pi$
$272$	−2245.50	−0.500564
$273$	0	0
$274$	−6337.62	−1.39734
$275$	985.584	0.216120
$276$	1613.74	0.351942
$277$	5412.54	1.17404	0.587018	−	0.809574i	$-0.300302\pi$
$277$	5412.54	1.17404	0.587018	+	0.809574i	$0.300302\pi$
$278$	−8043.29	−1.73527
$279$	−1021.88	−0.219277
$280$	0	0
$281$	−659.873	−0.140088	−0.0700440	−	0.997544i	$-0.522314\pi$
$281$	−659.873	−0.140088	−0.0700440	+	0.997544i	$0.522314\pi$
$282$	−536.596	−0.113312
$283$	5841.18	1.22693	0.613467	−	0.789721i	$-0.289774\pi$
$283$	5841.18	1.22693	0.613467	+	0.789721i	$0.289774\pi$
$284$	5934.87	1.24003
$285$	−730.258	−0.151778
$286$	1172.18	0.242351
$287$	0	0
$288$	2219.70	0.454157
$289$	−1075.58	−0.218925
$290$	−6341.94	−1.28418
$291$	2137.32	0.430557
$292$	−2317.87	−0.464531
$293$	−3035.44	−0.605230	−0.302615	−	0.953113i	$-0.597860\pi$
$293$	−3035.44	−0.605230	−0.302615	+	0.953113i	$0.597860\pi$
$294$	0	0
$295$	2114.88	0.417401
$296$	20.8450	0.00409321
$297$	−297.000	−0.0580259
$298$	−2095.95	−0.407434
$299$	1251.67	0.242094
$300$	−2853.12	−0.549084
$301$	0	0
$302$	10544.9	2.00924
$303$	−269.998	−0.0511914
$304$	602.331	0.113638
$305$	3346.17	0.628201
$306$	−2405.40	−0.449371
$307$	4354.54	0.809534	0.404767	−	0.914420i	$-0.367353\pi$
$307$	4354.54	0.809534	0.404767	+	0.914420i	$0.367353\pi$
$308$	0	0
$309$	3327.11	0.612532
$310$	−7176.21	−1.31478
$311$	−3364.84	−0.613513	−0.306756	−	0.951788i	$-0.599244\pi$
$311$	−3364.84	−0.613513	−0.306756	+	0.951788i	$0.599244\pi$
$312$	−835.806	−0.151661
$313$	−7739.96	−1.39773	−0.698863	−	0.715255i	$-0.746310\pi$
$313$	−7739.96	−1.39773	−0.698863	+	0.715255i	$0.746310\pi$
$314$	−14988.4	−2.69376
$315$	0	0
$316$	12479.4	2.22158
$317$	−10892.4	−1.92990	−0.964949	−	0.262438i	$-0.915474\pi$
$317$	−10892.4	−1.92990	−0.964949	+	0.262438i	$0.915474\pi$
$318$	8780.79	1.54843
$319$	−1103.76	−0.193727
$320$	11339.9	1.98100
$321$	−2851.32	−0.495780
$322$	0	0
$323$	−1029.35	−0.177320
$324$	859.772	0.147423
$325$	−2212.97	−0.377704
$326$	5970.53	1.01435
$327$	−4723.01	−0.798725
$328$	−699.923	−0.117826
$329$	0	0
$330$	−2085.70	−0.347922
$331$	4314.69	0.716486	0.358243	−	0.933628i	$-0.383376\pi$
$331$	4314.69	0.716486	0.358243	+	0.933628i	$0.383376\pi$
$332$	−1994.17	−0.329652
$333$	−16.6316	−0.00273696
$334$	−11260.5	−1.84475
$335$	−7072.16	−1.15341
$336$	0	0
$337$	−5671.76	−0.916797	−0.458398	−	0.888747i	$-0.651577\pi$
$337$	−5671.76	−0.916797	−0.458398	+	0.888747i	$0.651577\pi$
$338$	6846.91	1.10184
$339$	−4781.41	−0.766048
$340$	−9632.33	−1.53643
$341$	−1248.96	−0.198343
$342$	645.223	0.102017
$343$	0	0
$344$	3416.36	0.535459
$345$	−2227.15	−0.347553
$346$	7924.70	1.23131
$347$	−3069.37	−0.474848	−0.237424	−	0.971406i	$-0.576303\pi$
$347$	−3069.37	−0.474848	−0.237424	+	0.971406i	$0.576303\pi$
$348$	3195.24	0.492192
$349$	−81.8995	−0.0125615	−0.00628077	−	0.999980i	$-0.501999\pi$
$349$	−81.8995	−0.0125615	−0.00628077	+	0.999980i	$0.501999\pi$
$350$	0	0
$351$	666.867	0.101409
$352$	2712.97	0.410800
$353$	−11726.5	−1.76809	−0.884047	−	0.467399i	$-0.845191\pi$
$353$	−11726.5	−1.76809	−0.884047	+	0.467399i	$0.845191\pi$
$354$	−1868.61	−0.280553
$355$	−8190.80	−1.22457
$356$	368.460	0.0548550
$357$	0	0
$358$	−14990.3	−2.21302
$359$	−9275.28	−1.36359	−0.681797	−	0.731541i	$-0.738801\pi$
$359$	−9275.28	−1.36359	−0.681797	+	0.731541i	$0.738801\pi$
$360$	1487.18	0.217726
$361$	−6582.89	−0.959745
$362$	−8326.90	−1.20898
$363$	−363.000	−0.0524864
$364$	0	0
$365$	3198.93	0.458739
$366$	−2956.52	−0.422240
$367$	−3892.49	−0.553641	−0.276820	−	0.960922i	$-0.589281\pi$
$367$	−3892.49	−0.553641	−0.276820	+	0.960922i	$0.589281\pi$
$368$	1837.00	0.260218
$369$	558.449	0.0787851
$370$	−116.797	−0.0164108
$371$	0	0
$372$	3615.56	0.503920
$373$	−11469.4	−1.59213	−0.796063	−	0.605213i	$-0.793088\pi$
$373$	−11469.4	−1.59213	−0.796063	+	0.605213i	$0.793088\pi$
$374$	−2939.93	−0.406472
$375$	−1555.81	−0.214244
$376$	467.638	0.0641399
$377$	2478.33	0.338569
$378$	0	0
$379$	6852.41	0.928719	0.464360	−	0.885647i	$-0.346284\pi$
$379$	6852.41	0.928719	0.464360	+	0.885647i	$0.346284\pi$
$380$	2583.77	0.348801
$381$	6169.20	0.829548
$382$	−20447.9	−2.73876
$383$	−6017.45	−0.802813	−0.401406	−	0.915900i	$-0.631478\pi$
$383$	−6017.45	−0.802813	−0.401406	+	0.915900i	$0.631478\pi$
$384$	−4100.21	−0.544890
$385$	0	0
$386$	−2660.42	−0.350808
$387$	−2725.82	−0.358039
$388$	−7562.18	−0.989463
$389$	9876.82	1.28734	0.643669	−	0.765304i	$-0.277411\pi$
$389$	9876.82	1.28734	0.643669	+	0.765304i	$0.277411\pi$
$390$	4683.12	0.608049
$391$	−3139.31	−0.406041
$392$	0	0
$393$	285.766	0.0366794
$394$	−1150.01	−0.147047
$395$	−17223.0	−2.19388
$396$	1050.83	0.133349
$397$	−7729.30	−0.977135	−0.488567	−	0.872526i	$-0.662480\pi$
$397$	−7729.30	−0.977135	−0.488567	+	0.872526i	$0.662480\pi$
$398$	7608.24	0.958207
$399$	0	0
$400$	−3247.84	−0.405980
$401$	−5461.32	−0.680113	−0.340056	−	0.940405i	$-0.610446\pi$
$401$	−5461.32	−0.680113	−0.340056	+	0.940405i	$0.610446\pi$
$402$	6248.63	0.775258
$403$	2804.35	0.346636
$404$	955.296	0.117643
$405$	−1186.58	−0.145585
$406$	0	0
$407$	−20.3276	−0.00247567
$408$	2096.28	0.254366
$409$	−184.816	−0.0223436	−0.0111718	−	0.999938i	$-0.503556\pi$
$409$	−184.816	−0.0223436	−0.0111718	+	0.999938i	$0.503556\pi$
$410$	3921.75	0.472394
$411$	−4406.79	−0.528883
$412$	−11771.8	−1.40766
$413$	0	0
$414$	1967.81	0.233605
$415$	2752.19	0.325541
$416$	−6091.54	−0.717939
$417$	−5592.80	−0.656788
$418$	788.606	0.0922774
$419$	5175.78	0.603469	0.301735	−	0.953392i	$-0.402434\pi$
$419$	5175.78	0.603469	0.301735	+	0.953392i	$0.402434\pi$
$420$	0	0
$421$	1782.44	0.206345	0.103172	−	0.994664i	$-0.467101\pi$
$421$	1782.44	0.206345	0.103172	+	0.994664i	$0.467101\pi$
$422$	8367.46	0.965218
$423$	−373.116	−0.0428877
$424$	−7652.37	−0.876490
$425$	5550.35	0.633486
$426$	7237.01	0.823086
$427$	0	0
$428$	10088.4	1.13935
$429$	815.059	0.0917283
$430$	−19142.3	−2.14680
$431$	11768.2	1.31521	0.657603	−	0.753365i	$-0.271571\pi$
$431$	11768.2	1.31521	0.657603	+	0.753365i	$0.271571\pi$
$432$	978.717	0.109001
$433$	−3809.95	−0.422851	−0.211426	−	0.977394i	$-0.567811\pi$
$433$	−3809.95	−0.422851	−0.211426	+	0.977394i	$0.567811\pi$
$434$	0	0
$435$	−4409.79	−0.486054
$436$	16710.8	1.83555
$437$	842.087	0.0921796
$438$	−2826.43	−0.308338
$439$	−16810.6	−1.82762	−0.913809	−	0.406144i	$-0.866873\pi$
$439$	−16810.6	−1.82762	−0.913809	+	0.406144i	$0.866873\pi$
$440$	1817.67	0.196941
$441$	0	0
$442$	6601.16	0.710374
$443$	−2512.40	−0.269453	−0.134726	−	0.990883i	$-0.543016\pi$
$443$	−2512.40	−0.269453	−0.134726	+	0.990883i	$0.543016\pi$
$444$	58.8453	0.00628981
$445$	−508.518	−0.0541709
$446$	6494.23	0.689486
$447$	−1457.40	−0.154211
$448$	0	0
$449$	3925.62	0.412609	0.206305	−	0.978488i	$-0.433856\pi$
$449$	3925.62	0.412609	0.206305	+	0.978488i	$0.433856\pi$
$450$	−3479.11	−0.364460
$451$	682.549	0.0712638
$452$	16917.4	1.76046
$453$	7332.26	0.760485
$454$	24995.9	2.58395
$455$	0	0
$456$	−562.305	−0.0577464
$457$	6761.59	0.692109	0.346055	−	0.938214i	$-0.387521\pi$
$457$	6761.59	0.692109	0.346055	+	0.938214i	$0.387521\pi$
$458$	−25314.2	−2.58265
$459$	−1672.57	−0.170084
$460$	7880.01	0.798712
$461$	11942.7	1.20656	0.603282	−	0.797528i	$-0.293859\pi$
$461$	11942.7	1.20656	0.603282	+	0.797528i	$0.293859\pi$
$462$	0	0
$463$	2063.81	0.207156	0.103578	−	0.994621i	$-0.466971\pi$
$463$	2063.81	0.207156	0.103578	+	0.994621i	$0.466971\pi$
$464$	3637.28	0.363915
$465$	−4989.89	−0.497636
$466$	−8453.32	−0.840328
$467$	−9098.55	−0.901565	−0.450782	−	0.892634i	$-0.648855\pi$
$467$	−9098.55	−0.901565	−0.450782	+	0.892634i	$0.648855\pi$
$468$	−2359.48	−0.233049
$469$	0	0
$470$	−2620.23	−0.257154
$471$	−10422.0	−1.01957
$472$	1628.48	0.158807
$473$	−3331.56	−0.323859
$474$	15217.4	1.47460
$475$	−1488.82	−0.143814
$476$	0	0
$477$	6105.62	0.586073
$478$	−1759.91	−0.168403
$479$	1993.08	0.190117	0.0950586	−	0.995472i	$-0.469696\pi$
$479$	1993.08	0.190117	0.0950586	+	0.995472i	$0.469696\pi$
$480$	10838.9	1.03068
$481$	45.6423	0.00432663
$482$	15693.8	1.48306
$483$	0	0
$484$	1284.35	0.120619
$485$	10436.7	0.977124
$486$	1048.41	0.0978537
$487$	8513.77	0.792189	0.396094	−	0.918210i	$-0.370365\pi$
$487$	8513.77	0.792189	0.396094	+	0.918210i	$0.370365\pi$
$488$	2576.58	0.239009
$489$	4151.54	0.383924
$490$	0	0
$491$	12336.6	1.13390	0.566950	−	0.823752i	$-0.308123\pi$
$491$	12336.6	1.13390	0.566950	+	0.823752i	$0.308123\pi$
$492$	−1975.88	−0.181056
$493$	−6215.89	−0.567849
$494$	−1770.69	−0.161269
$495$	−1450.27	−0.131686
$496$	4115.76	0.372587
$497$	0	0
$498$	−2431.71	−0.218810
$499$	11809.5	1.05945	0.529726	−	0.848169i	$-0.322295\pi$
$499$	11809.5	1.05945	0.529726	+	0.848169i	$0.322295\pi$
$500$	5504.69	0.492355
$501$	−7829.84	−0.698226
$502$	−2603.15	−0.231442
$503$	7808.72	0.692194	0.346097	−	0.938199i	$-0.387507\pi$
$503$	7808.72	0.692194	0.346097	+	0.938199i	$0.387507\pi$
$504$	0	0
$505$	−1318.42	−0.116176
$506$	2405.10	0.211304
$507$	4760.91	0.417041
$508$	−21827.6	−1.90638
$509$	14748.0	1.28427	0.642137	−	0.766590i	$-0.278048\pi$
$509$	14748.0	1.28427	0.642137	+	0.766590i	$0.278048\pi$
$510$	−11745.7	−1.01982
$511$	0	0
$512$	−12211.3	−1.05404
$513$	448.648	0.0386127
$514$	15507.8	1.33078
$515$	16246.5	1.39011
$516$	9644.38	0.822810
$517$	−456.030	−0.0387934
$518$	0	0
$519$	5510.34	0.466045
$520$	−4081.29	−0.344185
$521$	−1652.60	−0.138967	−0.0694835	−	0.997583i	$-0.522135\pi$
$521$	−1652.60	−0.138967	−0.0694835	+	0.997583i	$0.522135\pi$
$522$	3896.29	0.326697
$523$	−17241.3	−1.44151	−0.720754	−	0.693191i	$-0.756204\pi$
$523$	−17241.3	−1.44151	−0.720754	+	0.693191i	$0.756204\pi$
$524$	−1011.08	−0.0842928
$525$	0	0
$526$	16417.5	1.36091
$527$	−7033.57	−0.581380
$528$	1196.21	0.0985954
$529$	−9598.79	−0.788920
$530$	42877.1	3.51408
$531$	−1299.32	−0.106188
$532$	0	0
$533$	−1532.56	−0.124545
$534$	449.303	0.0364106
$535$	−13923.2	−1.12514
$536$	−5445.62	−0.438834
$537$	−10423.3	−0.837613
$538$	29888.7	2.39515
$539$	0	0
$540$	4198.32	0.334568
$541$	−18725.5	−1.48812	−0.744061	−	0.668112i	$-0.767103\pi$
$541$	−18725.5	−1.48812	−0.744061	+	0.668112i	$0.767103\pi$
$542$	−3681.16	−0.291733
$543$	−5790.01	−0.457593
$544$	15278.2	1.20413
$545$	−23062.8	−1.81266
$546$	0	0
$547$	5629.03	0.440000	0.220000	−	0.975500i	$-0.429394\pi$
$547$	5629.03	0.440000	0.220000	+	0.975500i	$0.429394\pi$
$548$	15591.9	1.21543
$549$	−2055.78	−0.159815
$550$	−4252.25	−0.329666
$551$	1667.35	0.128913
$552$	−1714.93	−0.132232
$553$	0	0
$554$	−23352.1	−1.79086
$555$	−81.2133	−0.00621137
$556$	19788.2	1.50937
$557$	1374.50	0.104559	0.0522797	−	0.998632i	$-0.483351\pi$
$557$	1374.50	0.104559	0.0522797	+	0.998632i	$0.483351\pi$
$558$	4408.84	0.334482
$559$	7480.49	0.565994
$560$	0	0
$561$	−2044.25	−0.153847
$562$	2846.99	0.213689
$563$	16202.9	1.21292	0.606458	−	0.795116i	$-0.292590\pi$
$563$	16202.9	1.21292	0.606458	+	0.795116i	$0.292590\pi$
$564$	1320.14	0.0985603
$565$	−23347.9	−1.73850
$566$	−25201.5	−1.87155
$567$	0	0
$568$	−6306.98	−0.465907
$569$	−1798.25	−0.132490	−0.0662448	−	0.997803i	$-0.521102\pi$
$569$	−1798.25	−0.132490	−0.0662448	+	0.997803i	$0.521102\pi$
$570$	3150.66	0.231521
$571$	−3775.13	−0.276680	−0.138340	−	0.990385i	$-0.544177\pi$
$571$	−3775.13	−0.276680	−0.138340	+	0.990385i	$0.544177\pi$
$572$	−2883.81	−0.210801
$573$	−14218.2	−1.03660
$574$	0	0
$575$	−4540.63	−0.329317
$576$	−6966.88	−0.503970
$577$	15695.5	1.13243	0.566216	−	0.824257i	$-0.308407\pi$
$577$	15695.5	1.13243	0.566216	+	0.824257i	$0.308407\pi$
$578$	4640.54	0.333946
$579$	−1849.89	−0.132779
$580$	15602.5	1.11700
$581$	0	0
$582$	−9221.37	−0.656767
$583$	7462.42	0.530123
$584$	2463.20	0.174534
$585$	3256.35	0.230143
$586$	13096.2	0.923210
$587$	−9710.67	−0.682798	−0.341399	−	0.939918i	$-0.610901\pi$
$587$	−9710.67	−0.682798	−0.341399	+	0.939918i	$0.610901\pi$
$588$	0	0
$589$	1886.68	0.131985
$590$	−9124.56	−0.636699
$591$	−799.642	−0.0556563
$592$	66.9863	0.00465054
$593$	18925.8	1.31061	0.655304	−	0.755365i	$-0.272540\pi$
$593$	18925.8	1.31061	0.655304	+	0.755365i	$0.272540\pi$
$594$	1281.39	0.0885120
$595$	0	0
$596$	5156.49	0.354393
$597$	5290.30	0.362676
$598$	−5400.27	−0.369287
$599$	−9502.00	−0.648149	−0.324075	−	0.946032i	$-0.605053\pi$
$599$	−9502.00	−0.648149	−0.324075	+	0.946032i	$0.605053\pi$
$600$	3032.01	0.206302
$601$	25373.8	1.72216	0.861082	−	0.508467i	$-0.169788\pi$
$601$	25373.8	1.72216	0.861082	+	0.508467i	$0.169788\pi$
$602$	0	0
$603$	4344.91	0.293430
$604$	−25942.7	−1.74767
$605$	−1772.55	−0.119115
$606$	1164.89	0.0780867
$607$	−11459.1	−0.766247	−0.383123	−	0.923697i	$-0.625152\pi$
$607$	−11459.1	−0.766247	−0.383123	+	0.923697i	$0.625152\pi$
$608$	−4098.21	−0.273362
$609$	0	0
$610$	−14436.9	−0.958250
$611$	1023.94	0.0677976
$612$	5917.80	0.390871
$613$	−27701.7	−1.82522	−0.912611	−	0.408829i	$-0.865937\pi$
$613$	−27701.7	−1.82522	−0.912611	+	0.408829i	$0.865937\pi$
$614$	−18787.4	−1.23485
$615$	2726.94	0.178798
$616$	0	0
$617$	−19793.9	−1.29153	−0.645764	−	0.763537i	$-0.723461\pi$
$617$	−19793.9	−1.29153	−0.645764	+	0.763537i	$0.723461\pi$
$618$	−14354.6	−0.934350
$619$	20932.3	1.35919	0.679597	−	0.733586i	$-0.262155\pi$
$619$	20932.3	1.35919	0.679597	+	0.733586i	$0.262155\pi$
$620$	17655.0	1.14362
$621$	1368.29	0.0884181
$622$	14517.4	0.935845
$623$	0	0
$624$	−2685.90	−0.172311
$625$	−18796.9	−1.20300
$626$	33393.7	2.13207
$627$	548.348	0.0349265
$628$	36874.6	2.34308
$629$	−114.475	−0.00725665
$630$	0	0
$631$	−17090.6	−1.07823	−0.539117	−	0.842231i	$-0.681242\pi$
$631$	−17090.6	−1.07823	−0.539117	+	0.842231i	$0.681242\pi$
$632$	−13261.8	−0.834694
$633$	5818.22	0.365329
$634$	46994.7	2.94384
$635$	30124.6	1.88261
$636$	−21602.6	−1.34685
$637$	0	0
$638$	4762.14	0.295509
$639$	5032.17	0.311533
$640$	−20021.6	−1.23660
$641$	24542.3	1.51226	0.756132	−	0.654419i	$-0.227087\pi$
$641$	24542.3	1.51226	0.756132	+	0.654419i	$0.227087\pi$
$642$	12301.9	0.756257
$643$	14247.9	0.873846	0.436923	−	0.899499i	$-0.356068\pi$
$643$	14247.9	0.873846	0.436923	+	0.899499i	$0.356068\pi$
$644$	0	0
$645$	−13310.3	−0.812549
$646$	4441.06	0.270482
$647$	17073.8	1.03747	0.518733	−	0.854936i	$-0.326404\pi$
$647$	17073.8	1.03747	0.518733	+	0.854936i	$0.326404\pi$
$648$	−913.680	−0.0553900
$649$	−1588.06	−0.0960503
$650$	9547.76	0.576144
$651$	0	0
$652$	−14688.8	−0.882297
$653$	−9800.30	−0.587313	−0.293657	−	0.955911i	$-0.594872\pi$
$653$	−9800.30	−0.587313	−0.293657	+	0.955911i	$0.594872\pi$
$654$	20377.2	1.21837
$655$	1395.41	0.0832416
$656$	−2249.24	−0.133869
$657$	−1965.32	−0.116704
$658$	0	0
$659$	7476.99	0.441976	0.220988	−	0.975277i	$-0.429072\pi$
$659$	7476.99	0.441976	0.220988	+	0.975277i	$0.429072\pi$
$660$	5131.28	0.302628
$661$	8361.23	0.492004	0.246002	−	0.969269i	$-0.420883\pi$
$661$	8361.23	0.492004	0.246002	+	0.969269i	$0.420883\pi$
$662$	−18615.5	−1.09292
$663$	4590.04	0.268872
$664$	2119.21	0.123857
$665$	0	0
$666$	71.7564	0.00417493
$667$	5085.09	0.295196
$668$	27703.2	1.60459
$669$	4515.69	0.260966
$670$	30512.5	1.75940
$671$	−2512.62	−0.144558
$672$	0	0
$673$	30985.6	1.77475	0.887375	−	0.461049i	$-0.152527\pi$
$673$	30985.6	1.77475	0.887375	+	0.461049i	$0.152527\pi$
$674$	24470.5	1.39847
$675$	−2419.16	−0.137946
$676$	−16844.9	−0.958401
$677$	−15795.6	−0.896711	−0.448356	−	0.893855i	$-0.647990\pi$
$677$	−15795.6	−0.896711	−0.448356	+	0.893855i	$0.647990\pi$
$678$	20629.1	1.16852
$679$	0	0
$680$	10236.3	0.577269
$681$	17380.6	0.978011
$682$	5388.58	0.302550
$683$	−14968.4	−0.838582	−0.419291	−	0.907852i	$-0.637721\pi$
$683$	−14968.4	−0.838582	−0.419291	+	0.907852i	$0.637721\pi$
$684$	−1587.39	−0.0887357
$685$	−21518.6	−1.20027
$686$	0	0
$687$	−17601.9	−0.977519
$688$	10978.6	0.608367
$689$	−16755.7	−0.926474
$690$	9608.93	0.530153
$691$	−27858.1	−1.53368	−0.766840	−	0.641838i	$-0.778172\pi$
$691$	−27858.1	−1.53368	−0.766840	+	0.641838i	$0.778172\pi$
$692$	−19496.5	−1.07102
$693$	0	0
$694$	13242.6	0.724327
$695$	−27310.0	−1.49054
$696$	−3395.58	−0.184927
$697$	3843.80	0.208887
$698$	353.351	0.0191612
$699$	−5877.92	−0.318059
$700$	0	0
$701$	−1192.90	−0.0642729	−0.0321365	−	0.999483i	$-0.510231\pi$
$701$	−1192.90	−0.0642729	−0.0321365	+	0.999483i	$0.510231\pi$
$702$	−2877.16	−0.154689
$703$	30.7068	0.00164741
$704$	−8515.07	−0.455858
$705$	−1821.95	−0.0973312
$706$	50593.2	2.69703
$707$	0	0
$708$	4597.19	0.244030
$709$	−13888.3	−0.735664	−0.367832	−	0.929892i	$-0.619900\pi$
$709$	−13888.3	−0.735664	−0.367832	+	0.929892i	$0.619900\pi$
$710$	35338.8	1.86794
$711$	10581.2	0.558126
$712$	−391.563	−0.0206102
$713$	5754.02	0.302230
$714$	0	0
$715$	3979.98	0.208172
$716$	36879.2	1.92492
$717$	−1223.73	−0.0637394
$718$	40017.7	2.08001
$719$	−31737.7	−1.64620	−0.823098	−	0.567899i	$-0.807756\pi$
$719$	−31737.7	−1.64620	−0.823098	+	0.567899i	$0.807756\pi$
$720$	4779.14	0.247372
$721$	0	0
$722$	28401.5	1.46398
$723$	10912.5	0.561329
$724$	20486.0	1.05159
$725$	−8990.51	−0.460551
$726$	1566.14	0.0800621
$727$	14730.4	0.751472	0.375736	−	0.926727i	$-0.377390\pi$
$727$	14730.4	0.751472	0.375736	+	0.926727i	$0.377390\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−13801.6	−0.699754
$731$	−18761.8	−0.949288
$732$	7273.68	0.367272
$733$	6008.51	0.302769	0.151384	−	0.988475i	$-0.451627\pi$
$733$	6008.51	0.302769	0.151384	+	0.988475i	$0.451627\pi$
$734$	16793.9	0.844517
$735$	0	0
$736$	−12498.8	−0.625966
$737$	5310.45	0.265418
$738$	−2409.40	−0.120178
$739$	8723.11	0.434215	0.217108	−	0.976148i	$-0.430338\pi$
$739$	8723.11	0.434215	0.217108	+	0.976148i	$0.430338\pi$
$740$	287.345	0.0142744
$741$	−1231.23	−0.0610395
$742$	0	0
$743$	28162.9	1.39057	0.695286	−	0.718733i	$-0.255278\pi$
$743$	28162.9	1.39057	0.695286	+	0.718733i	$0.255278\pi$
$744$	−3842.26	−0.189333
$745$	−7116.55	−0.349974
$746$	49484.2	2.42861
$747$	−1690.86	−0.0828183
$748$	7232.87	0.353556
$749$	0	0
$750$	6712.45	0.326806
$751$	−29044.8	−1.41126	−0.705632	−	0.708578i	$-0.749337\pi$
$751$	−29044.8	−1.41126	−0.705632	+	0.708578i	$0.749337\pi$
$752$	1502.78	0.0728732
$753$	−1810.07	−0.0875996
$754$	−10692.6	−0.516449
$755$	35803.9	1.72588
$756$	0	0
$757$	13181.3	0.632871	0.316435	−	0.948614i	$-0.397514\pi$
$757$	13181.3	0.632871	0.316435	+	0.948614i	$0.397514\pi$
$758$	−29564.4	−1.41666
$759$	1672.36	0.0799772
$760$	−2745.77	−0.131052
$761$	32209.1	1.53427	0.767134	−	0.641487i	$-0.221682\pi$
$761$	32209.1	1.53427	0.767134	+	0.641487i	$0.221682\pi$
$762$	−26616.7	−1.26538
$763$	0	0
$764$	50306.2	2.38222
$765$	−8167.24	−0.385996
$766$	25962.0	1.22460
$767$	3565.73	0.167863
$768$	−888.205	−0.0417322
$769$	−11714.3	−0.549320	−0.274660	−	0.961541i	$-0.588565\pi$
$769$	−11714.3	−0.549320	−0.274660	+	0.961541i	$0.588565\pi$
$770$	0	0
$771$	10783.1	0.503691
$772$	6545.21	0.305139
$773$	−22457.9	−1.04496	−0.522481	−	0.852651i	$-0.674993\pi$
$773$	−22457.9	−1.04496	−0.522481	+	0.852651i	$0.674993\pi$
$774$	11760.4	0.546149
$775$	−10173.2	−0.471525
$776$	8036.33	0.371762
$777$	0	0
$778$	−42613.0	−1.96369
$779$	−1031.06	−0.0474217
$780$	−11521.5	−0.528891
$781$	6150.43	0.281792
$782$	13544.4	0.619370
$783$	2709.24	0.123653
$784$	0	0
$785$	−50891.1	−2.31386
$786$	−1232.92	−0.0559502
$787$	−12982.2	−0.588010	−0.294005	−	0.955804i	$-0.594988\pi$
$787$	−12982.2	−0.588010	−0.294005	+	0.955804i	$0.594988\pi$
$788$	2829.26	0.127904
$789$	11415.7	0.515096
$790$	74307.5	3.34651
$791$	0	0
$792$	−1116.72	−0.0501022
$793$	5641.70	0.252639
$794$	33347.7	1.49051
$795$	29814.1	1.33006
$796$	−18717.9	−0.833465
$797$	−18858.7	−0.838154	−0.419077	−	0.907951i	$-0.637646\pi$
$797$	−18858.7	−0.838154	−0.419077	+	0.907951i	$0.637646\pi$
$798$	0	0
$799$	−2568.15	−0.113710
$800$	22098.0	0.976603
$801$	312.417	0.0137812
$802$	23562.6	1.03744
$803$	−2402.06	−0.105563
$804$	−15373.0	−0.674332
$805$	0	0
$806$	−12099.2	−0.528755
$807$	20782.7	0.906552
$808$	−1015.19	−0.0442009
$809$	21479.6	0.933478	0.466739	−	0.884395i	$-0.345429\pi$
$809$	21479.6	0.933478	0.466739	+	0.884395i	$0.345429\pi$
$810$	5119.45	0.222073
$811$	4450.75	0.192709	0.0963545	−	0.995347i	$-0.469282\pi$
$811$	4450.75	0.192709	0.0963545	+	0.995347i	$0.469282\pi$
$812$	0	0
$813$	−2559.65	−0.110419
$814$	87.7022	0.00377636
$815$	20272.2	0.871294
$816$	6736.50	0.289001
$817$	5032.65	0.215508
$818$	797.377	0.0340827
$819$	0	0
$820$	−9648.35	−0.410896
$821$	−32636.1	−1.38734	−0.693670	−	0.720293i	$-0.744007\pi$
$821$	−32636.1	−1.38734	−0.693670	+	0.720293i	$0.744007\pi$
$822$	19012.9	0.806752
$823$	13553.8	0.574065	0.287033	−	0.957921i	$-0.407331\pi$
$823$	13553.8	0.574065	0.287033	+	0.957921i	$0.407331\pi$
$824$	12509.9	0.528888
$825$	−2956.75	−0.124777
$826$	0	0
$827$	−14489.8	−0.609263	−0.304631	−	0.952470i	$-0.598533\pi$
$827$	−14489.8	−0.609263	−0.304631	+	0.952470i	$0.598533\pi$
$828$	−4841.23	−0.203194
$829$	−26927.9	−1.12816	−0.564079	−	0.825721i	$-0.690769\pi$
$829$	−26927.9	−1.12816	−0.564079	+	0.825721i	$0.690769\pi$
$830$	−11874.2	−0.496576
$831$	−16237.6	−0.677830
$832$	19119.2	0.796684
$833$	0	0
$834$	24129.9	1.00186
$835$	−38233.6	−1.58458
$836$	−1940.14	−0.0802645
$837$	3065.63	0.126599
$838$	−22330.6	−0.920524
$839$	−34465.9	−1.41823	−0.709114	−	0.705094i	$-0.750905\pi$
$839$	−34465.9	−1.41823	−0.709114	+	0.705094i	$0.750905\pi$
$840$	0	0
$841$	−14320.4	−0.587168
$842$	−7690.27	−0.314755
$843$	1979.62	0.0808799
$844$	−20585.8	−0.839563
$845$	23247.8	0.946449
$846$	1609.79	0.0654204
$847$	0	0
$848$	−24591.2	−0.995833
$849$	−17523.5	−0.708370
$850$	−23946.7	−0.966312
$851$	93.6500	0.00377236
$852$	−17804.6	−0.715934
$853$	−951.037	−0.0381745	−0.0190873	−	0.999818i	$-0.506076\pi$
$853$	−951.037	−0.0381745	−0.0190873	+	0.999818i	$0.506076\pi$
$854$	0	0
$855$	2190.78	0.0876292
$856$	−10721.0	−0.428079
$857$	4628.31	0.184481	0.0922404	−	0.995737i	$-0.470597\pi$
$857$	4628.31	0.184481	0.0922404	+	0.995737i	$0.470597\pi$
$858$	−3516.53	−0.139921
$859$	30205.9	1.19978	0.599891	−	0.800082i	$-0.295211\pi$
$859$	30205.9	1.19978	0.599891	+	0.800082i	$0.295211\pi$
$860$	47094.1	1.86732
$861$	0	0
$862$	−50773.2	−2.00620
$863$	−30606.9	−1.20727	−0.603634	−	0.797262i	$-0.706281\pi$
$863$	−30606.9	−1.20727	−0.603634	+	0.797262i	$0.706281\pi$
$864$	−6659.11	−0.262208
$865$	26907.3	1.05766
$866$	16437.8	0.645012
$867$	3226.74	0.126397
$868$	0	0
$869$	12932.6	0.504844
$870$	19025.8	0.741421
$871$	−11923.8	−0.463860
$872$	−17758.5	−0.689655
$873$	−6411.97	−0.248582
$874$	−3633.14	−0.140610
$875$	0	0
$876$	6953.62	0.268197
$877$	20525.2	0.790291	0.395146	−	0.918619i	$-0.370694\pi$
$877$	20525.2	0.790291	0.395146	+	0.918619i	$0.370694\pi$
$878$	72528.3	2.78783
$879$	9106.32	0.349429
$880$	5841.17	0.223756
$881$	34579.5	1.32237	0.661187	−	0.750221i	$-0.270053\pi$
$881$	34579.5	1.32237	0.661187	+	0.750221i	$0.270053\pi$
$882$	0	0
$883$	11174.0	0.425861	0.212930	−	0.977067i	$-0.431699\pi$
$883$	11174.0	0.425861	0.212930	+	0.977067i	$0.431699\pi$
$884$	−16240.3	−0.617895
$885$	−6344.65	−0.240987
$886$	10839.6	0.411020
$887$	−28892.2	−1.09369	−0.546847	−	0.837233i	$-0.684172\pi$
$887$	−28892.2	−1.09369	−0.546847	+	0.837233i	$0.684172\pi$
$888$	−62.5349	−0.00236322
$889$	0	0
$890$	2193.97	0.0826316
$891$	891.000	0.0335013
$892$	−15977.2	−0.599727
$893$	688.879	0.0258146
$894$	6287.86	0.235232
$895$	−50897.6	−1.90091
$896$	0	0
$897$	−3755.01	−0.139773
$898$	−16936.9	−0.629389
$899$	11393.1	0.422669
$900$	8559.37	0.317014
$901$	42024.9	1.55389
$902$	−2944.82	−0.108705
$903$	0	0
$904$	−17978.1	−0.661440
$905$	−28273.0	−1.03848
$906$	−31634.7	−1.16004
$907$	−13080.3	−0.478859	−0.239429	−	0.970914i	$-0.576960\pi$
$907$	−13080.3	−0.478859	−0.239429	+	0.970914i	$0.576960\pi$
$908$	−61495.3	−2.24757
$909$	809.995	0.0295554
$910$	0	0
$911$	−13848.9	−0.503659	−0.251829	−	0.967772i	$-0.581032\pi$
$911$	−13848.9	−0.503659	−0.251829	+	0.967772i	$0.581032\pi$
$912$	−1806.99	−0.0656092
$913$	−2066.60	−0.0749119
$914$	−29172.6	−1.05574
$915$	−10038.5	−0.362692
$916$	62278.4	2.24644
$917$	0	0
$918$	7216.20	0.259445
$919$	41613.6	1.49370	0.746848	−	0.664995i	$-0.231566\pi$
$919$	41613.6	1.49370	0.746848	+	0.664995i	$0.231566\pi$
$920$	−8374.08	−0.300093
$921$	−13063.6	−0.467384
$922$	−51526.1	−1.84048
$923$	−13809.8	−0.492476
$924$	0	0
$925$	−165.574	−0.00588546
$926$	−8904.21	−0.315994
$927$	−9981.32	−0.353646
$928$	−24747.8	−0.875415
$929$	13583.7	0.479728	0.239864	−	0.970806i	$-0.422897\pi$
$929$	13583.7	0.479728	0.239864	+	0.970806i	$0.422897\pi$
$930$	21528.6	0.759088
$931$	0	0
$932$	20797.0	0.730931
$933$	10094.5	0.354212
$934$	39255.2	1.37524
$935$	−9982.19	−0.349147
$936$	2507.42	0.0875614
$937$	−47931.8	−1.67114	−0.835572	−	0.549381i	$-0.814864\pi$
$937$	−47931.8	−1.67114	−0.835572	+	0.549381i	$0.814864\pi$
$938$	0	0
$939$	23219.9	0.806978
$940$	6446.33	0.223677
$941$	30613.0	1.06053	0.530263	−	0.847833i	$-0.322093\pi$
$941$	30613.0	1.06053	0.530263	+	0.847833i	$0.322093\pi$
$942$	44965.1	1.55525
$943$	−3144.54	−0.108590
$944$	5233.19	0.180430
$945$	0	0
$946$	14373.8	0.494010
$947$	−6345.46	−0.217740	−0.108870	−	0.994056i	$-0.534723\pi$
$947$	−6345.46	−0.217740	−0.108870	+	0.994056i	$0.534723\pi$
$948$	−37438.1	−1.28263
$949$	5393.45	0.184488
$950$	6423.45	0.219373
$951$	32677.2	1.11423
$952$	0	0
$953$	−2674.07	−0.0908936	−0.0454468	−	0.998967i	$-0.514471\pi$
$953$	−2674.07	−0.0908936	−0.0454468	+	0.998967i	$0.514471\pi$
$954$	−26342.4	−0.893989
$955$	−69428.4	−2.35251
$956$	4329.76	0.146480
$957$	3311.29	0.111848
$958$	−8599.04	−0.290003
$959$	0	0
$960$	−34019.7	−1.14373
$961$	−16899.2	−0.567259
$962$	−196.921	−0.00659980
$963$	8553.97	0.286239
$964$	−38610.2	−1.28999
$965$	−9033.14	−0.301334
$966$	0	0
$967$	−20187.8	−0.671351	−0.335676	−	0.941978i	$-0.608965\pi$
$967$	−20187.8	−0.671351	−0.335676	+	0.941978i	$0.608965\pi$
$968$	−1364.88	−0.0453191
$969$	3088.04	0.102376
$970$	−45028.5	−1.49049
$971$	8015.21	0.264902	0.132451	−	0.991190i	$-0.457715\pi$
$971$	8015.21	0.264902	0.132451	+	0.991190i	$0.457715\pi$
$972$	−2579.32	−0.0851148
$973$	0	0
$974$	−36732.2	−1.20840
$975$	6638.92	0.218067
$976$	8279.96	0.271552
$977$	−10285.8	−0.336819	−0.168410	−	0.985717i	$-0.553863\pi$
$977$	−10285.8	−0.336819	−0.168410	+	0.985717i	$0.553863\pi$
$978$	−17911.6	−0.585634
$979$	381.843	0.0124655
$980$	0	0
$981$	14169.0	0.461144
$982$	−53225.8	−1.72964
$983$	−40809.2	−1.32412	−0.662061	−	0.749450i	$-0.730318\pi$
$983$	−40809.2	−1.32412	−0.662061	+	0.749450i	$0.730318\pi$
$984$	2099.77	0.0680266
$985$	−3904.70	−0.126309
$986$	26818.1	0.866190
$987$	0	0
$988$	4356.28	0.140275
$989$	15348.6	0.493486
$990$	6257.11	0.200873
$991$	−61319.6	−1.96557	−0.982786	−	0.184749i	$-0.940853\pi$
$991$	−61319.6	−1.96557	−0.982786	+	0.184749i	$0.940853\pi$
$992$	−28003.3	−0.896274
$993$	−12944.1	−0.413663
$994$	0	0
$995$	25832.8	0.823071
$996$	5982.52	0.190325
$997$	−22436.4	−0.712705	−0.356353	−	0.934352i	$-0.615980\pi$
$997$	−22436.4	−0.712705	−0.356353	+	0.934352i	$0.615980\pi$
$998$	−50951.6	−1.61608
$999$	49.8949	0.00158019

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1617.4.a.be.1.3	✓	16
7.6	odd	2		1617.4.a.bf.1.3	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1617.4.a.be.1.3	✓	16	1.1	even	1	trivial
1617.4.a.bf.1.3	yes	16	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1617.a (trivial)

\(f(q)\)	\(=\)	\(q-4.31445 q^{2} -3.00000 q^{3} +10.6145 q^{4} -14.6492 q^{5} +12.9433 q^{6} -11.2800 q^{8} +9.00000 q^{9} +63.2031 q^{10} +11.0000 q^{11} -31.8434 q^{12} -24.6988 q^{13} +43.9475 q^{15} -36.2488 q^{16} +61.9469 q^{17} -38.8300 q^{18} -16.6166 q^{19} -155.493 q^{20} -47.4589 q^{22} -50.6775 q^{23} +33.8400 q^{24} +89.5985 q^{25} +106.562 q^{26} -27.0000 q^{27} -100.342 q^{29} -189.609 q^{30} -113.542 q^{31} +246.634 q^{32} -33.0000 q^{33} -267.267 q^{34} +95.5302 q^{36} -1.84796 q^{37} +71.6914 q^{38} +74.0963 q^{39} +165.243 q^{40} +62.0499 q^{41} -302.869 q^{43} +116.759 q^{44} -131.843 q^{45} +218.645 q^{46} -41.4573 q^{47} +108.746 q^{48} -386.568 q^{50} -185.841 q^{51} -262.164 q^{52} +678.402 q^{53} +116.490 q^{54} -161.141 q^{55} +49.8498 q^{57} +432.921 q^{58} -144.369 q^{59} +466.480 q^{60} -228.420 q^{61} +489.871 q^{62} -774.097 q^{64} +361.817 q^{65} +142.377 q^{66} +482.768 q^{67} +657.534 q^{68} +152.032 q^{69} +559.130 q^{71} -101.520 q^{72} -218.369 q^{73} +7.97293 q^{74} -268.796 q^{75} -176.376 q^{76} -319.685 q^{78} +1175.69 q^{79} +531.015 q^{80} +81.0000 q^{81} -267.711 q^{82} -187.873 q^{83} -907.471 q^{85} +1306.71 q^{86} +301.027 q^{87} -124.080 q^{88} +34.7130 q^{89} +568.828 q^{90} -537.915 q^{92} +340.626 q^{93} +178.865 q^{94} +243.419 q^{95} -739.901 q^{96} -712.441 q^{97} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{2} - 48 q^{3} + 72 q^{4} - 40 q^{5} - 12 q^{6} + 66 q^{8} + 144 q^{9} - 178 q^{10} + 176 q^{11} - 216 q^{12} - 104 q^{13} + 120 q^{15} + 220 q^{16} - 180 q^{17} + 36 q^{18} - 152 q^{19} - 298 q^{20}+ \cdots + 1584 q^{99}+O(q^{100}) \) 16 * q + 4 * q^2 - 48 * q^3 + 72 * q^4 - 40 * q^5 - 12 * q^6 + 66 * q^8 + 144 * q^9 - 178 * q^10 + 176 * q^11 - 216 * q^12 - 104 * q^13 + 120 * q^15 + 220 * q^16 - 180 * q^17 + 36 * q^18 - 152 * q^19 - 298 * q^20 + 44 * q^22 + 4 * q^23 - 198 * q^24 + 588 * q^25 - 406 * q^26 - 432 * q^27 + 412 * q^29 + 534 * q^30 - 628 * q^31 + 592 * q^32 - 528 * q^33 + 88 * q^34 + 648 * q^36 + 148 * q^37 - 446 * q^38 + 312 * q^39 - 1376 * q^40 - 596 * q^41 - 260 * q^43 + 792 * q^44 - 360 * q^45 - 148 * q^46 - 2220 * q^47 - 660 * q^48 + 82 * q^50 + 540 * q^51 - 1046 * q^52 + 168 * q^53 - 108 * q^54 - 440 * q^55 + 456 * q^57 + 538 * q^58 + 48 * q^59 + 894 * q^60 - 1504 * q^61 - 1276 * q^62 + 630 * q^64 + 1224 * q^65 - 132 * q^66 + 116 * q^67 - 356 * q^68 - 12 * q^69 + 320 * q^71 + 594 * q^72 - 652 * q^73 + 1062 * q^74 - 1764 * q^75 + 594 * q^76 + 1218 * q^78 + 1136 * q^79 - 1970 * q^80 + 1296 * q^81 + 416 * q^82 - 3300 * q^83 - 1148 * q^85 + 1864 * q^86 - 1236 * q^87 + 726 * q^88 - 2416 * q^89 - 1602 * q^90 - 1064 * q^92 + 1884 * q^93 - 914 * q^94 + 1120 * q^95 - 1776 * q^96 - 3616 * q^97 + 1584 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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