LMFDB - Embedded newform 847.4.a.k.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [847,4,Mod(1,847)] 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("847.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(847, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	847.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.9746177749$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 45x^{6} + 73x^{5} + 584x^{4} - 787x^{3} - 2076x^{2} + 1820x + 1488 $ x^8 - 2x^7 - 45x^6 + 73x^5 + 584x^4 - 787x^3 - 2076x^2 + 1820*x + 1488
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.00760$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.00760 q^{2} +6.42839 q^{3} +8.06082 q^{4} -10.5476 q^{5} -25.7624 q^{6} +7.00000 q^{7} -0.243753 q^{8} +14.3241 q^{9} +42.2704 q^{10} +51.8181 q^{12} +60.3406 q^{13} -28.0532 q^{14} -67.8038 q^{15} -63.5097 q^{16} +33.3802 q^{17} -57.4054 q^{18} +69.2475 q^{19} -85.0221 q^{20} +44.9987 q^{21} +159.000 q^{23} -1.56694 q^{24} -13.7489 q^{25} -241.821 q^{26} -81.4853 q^{27} +56.4258 q^{28} -106.351 q^{29} +271.730 q^{30} -133.567 q^{31} +256.471 q^{32} -133.774 q^{34} -73.8330 q^{35} +115.464 q^{36} -264.019 q^{37} -277.516 q^{38} +387.892 q^{39} +2.57100 q^{40} +59.5186 q^{41} -180.337 q^{42} -260.611 q^{43} -151.085 q^{45} -637.207 q^{46} +212.453 q^{47} -408.265 q^{48} +49.0000 q^{49} +55.0999 q^{50} +214.581 q^{51} +486.395 q^{52} +559.528 q^{53} +326.560 q^{54} -1.70627 q^{56} +445.150 q^{57} +426.210 q^{58} +837.450 q^{59} -546.555 q^{60} +492.234 q^{61} +535.281 q^{62} +100.269 q^{63} -519.755 q^{64} -636.446 q^{65} -402.391 q^{67} +269.071 q^{68} +1022.11 q^{69} +295.893 q^{70} -856.461 q^{71} -3.49155 q^{72} -105.552 q^{73} +1058.08 q^{74} -88.3831 q^{75} +558.192 q^{76} -1554.52 q^{78} +446.524 q^{79} +669.873 q^{80} -910.571 q^{81} -238.526 q^{82} +971.903 q^{83} +362.727 q^{84} -352.079 q^{85} +1044.42 q^{86} -683.662 q^{87} +779.257 q^{89} +605.487 q^{90} +422.384 q^{91} +1281.67 q^{92} -858.618 q^{93} -851.425 q^{94} -730.393 q^{95} +1648.70 q^{96} -1766.59 q^{97} -196.372 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{2} + 2 q^{3} + 30 q^{4} + 2 q^{5} + 53 q^{6} + 56 q^{7} + 27 q^{8} + 120 q^{9} + 66 q^{10} + 36 q^{12} + 124 q^{13} + 14 q^{14} + 78 q^{15} + 122 q^{16} + 192 q^{17} - 162 q^{18} + 96 q^{19}+ \cdots + 98 q^{98}+O(q^{100}) $ 8 * q + 2 * q^2 + 2 * q^3 + 30 * q^4 + 2 * q^5 + 53 * q^6 + 56 * q^7 + 27 * q^8 + 120 * q^9 + 66 * q^10 + 36 * q^12 + 124 * q^13 + 14 * q^14 + 78 * q^15 + 122 * q^16 + 192 * q^17 - 162 * q^18 + 96 * q^19 + 340 * q^20 + 14 * q^21 + 172 * q^23 + 17 * q^24 + 406 * q^25 - 347 * q^26 - 220 * q^27 + 210 * q^28 + 294 * q^29 + 205 * q^30 + 394 * q^31 + 788 * q^32 - 767 * q^34 + 14 * q^35 + 87 * q^36 - 252 * q^37 + 81 * q^38 - 208 * q^39 + 345 * q^40 + 1120 * q^41 + 371 * q^42 + 384 * q^43 - 1458 * q^45 - 40 * q^46 - 368 * q^47 - 523 * q^48 + 392 * q^49 - 1306 * q^50 + 960 * q^51 + 183 * q^52 + 602 * q^53 + 2515 * q^54 + 189 * q^56 + 1324 * q^57 + 1727 * q^58 + 66 * q^59 - 2460 * q^60 - 2388 * q^61 + 1595 * q^62 + 840 * q^63 + 741 * q^64 + 762 * q^65 - 894 * q^67 + 1352 * q^68 + 1470 * q^69 + 462 * q^70 - 2104 * q^71 - 1010 * q^72 + 1644 * q^73 - 2533 * q^74 - 2002 * q^75 + 5765 * q^76 - 2068 * q^78 + 678 * q^79 + 3846 * q^80 + 2464 * q^81 - 4764 * q^82 + 474 * q^83 + 252 * q^84 + 1494 * q^85 + 3471 * q^86 + 3460 * q^87 + 88 * q^89 + 2014 * q^90 + 868 * q^91 + 3669 * q^92 - 4170 * q^93 - 888 * q^94 + 4994 * q^95 + 214 * q^96 - 392 * q^97 + 98 * 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 $

$2$	−4.00760	−1.41690	−0.708450	−	0.705761i	$-0.750605\pi$
$2$	−4.00760	−1.41690	−0.708450	+	0.705761i	$0.750605\pi$
$3$	6.42839	1.23714	0.618572	−	0.785728i	$-0.287712\pi$
$3$	6.42839	1.23714	0.618572	+	0.785728i	$0.287712\pi$
$4$	8.06082	1.00760
$5$	−10.5476	−0.943403	−0.471701	−	0.881758i	$-0.656360\pi$
$5$	−10.5476	−0.943403	−0.471701	+	0.881758i	$0.656360\pi$
$6$	−25.7624	−1.75291
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−0.243753	−0.0107725
$9$	14.3241	0.530524
$10$	42.2704	1.33671
$11$	0	0
$11$	0	0
$12$	51.8181	1.24655
$13$	60.3406	1.28734	0.643672	−	0.765302i	$-0.277410\pi$
$13$	60.3406	1.28734	0.643672	+	0.765302i	$0.277410\pi$
$14$	−28.0532	−0.535537
$15$	−67.8038	−1.16712
$16$	−63.5097	−0.992339
$17$	33.3802	0.476228	0.238114	−	0.971237i	$-0.423471\pi$
$17$	33.3802	0.476228	0.238114	+	0.971237i	$0.423471\pi$
$18$	−57.4054	−0.751699
$19$	69.2475	0.836130	0.418065	−	0.908417i	$-0.362708\pi$
$19$	69.2475	0.836130	0.418065	+	0.908417i	$0.362708\pi$
$20$	−85.0221	−0.950575
$21$	44.9987	0.467596
$22$	0	0
$23$	159.000	1.44147	0.720734	−	0.693212i	$-0.243805\pi$
$23$	159.000	1.44147	0.720734	+	0.693212i	$0.243805\pi$
$24$	−1.56694	−0.0133271
$25$	−13.7489	−0.109991
$26$	−241.821	−1.82404
$27$	−81.4853	−0.580809
$28$	56.4258	0.380838
$29$	−106.351	−0.680993	−0.340496	−	0.940246i	$-0.610595\pi$
$29$	−106.351	−0.680993	−0.340496	+	0.940246i	$0.610595\pi$
$30$	271.730	1.65370
$31$	−133.567	−0.773847	−0.386924	−	0.922112i	$-0.626462\pi$
$31$	−133.567	−0.773847	−0.386924	+	0.922112i	$0.626462\pi$
$32$	256.471	1.41682
$33$	0	0
$34$	−133.774	−0.674767
$35$	−73.8330	−0.356573
$36$	115.464	0.534557
$37$	−264.019	−1.17309	−0.586547	−	0.809916i	$-0.699513\pi$
$37$	−264.019	−1.17309	−0.586547	+	0.809916i	$0.699513\pi$
$38$	−277.516	−1.18471
$39$	387.892	1.59263
$40$	2.57100	0.0101628
$41$	59.5186	0.226713	0.113357	−	0.993554i	$-0.463840\pi$
$41$	59.5186	0.226713	0.113357	+	0.993554i	$0.463840\pi$
$42$	−180.337	−0.662537
$43$	−260.611	−0.924250	−0.462125	−	0.886815i	$-0.652913\pi$
$43$	−260.611	−0.924250	−0.462125	+	0.886815i	$0.652913\pi$
$44$	0	0
$45$	−151.085	−0.500498
$46$	−637.207	−2.04241
$47$	212.453	0.659350	0.329675	−	0.944095i	$-0.393061\pi$
$47$	212.453	0.659350	0.329675	+	0.944095i	$0.393061\pi$
$48$	−408.265	−1.22767
$49$	49.0000	0.142857
$50$	55.0999	0.155846
$51$	214.581	0.589162
$52$	486.395	1.29713
$53$	559.528	1.45013	0.725067	−	0.688679i	$-0.241809\pi$
$53$	559.528	1.45013	0.725067	+	0.688679i	$0.241809\pi$
$54$	326.560	0.822948
$55$	0	0
$56$	−1.70627	−0.00407161
$57$	445.150	1.03441
$58$	426.210	0.964898
$59$	837.450	1.84791	0.923955	−	0.382500i	$-0.124937\pi$
$59$	837.450	1.84791	0.923955	+	0.382500i	$0.124937\pi$
$60$	−546.555	−1.17600
$61$	492.234	1.03318	0.516591	−	0.856232i	$-0.327201\pi$
$61$	492.234	1.03318	0.516591	+	0.856232i	$0.327201\pi$
$62$	535.281	1.09646
$63$	100.269	0.200519
$64$	−519.755	−1.01515
$65$	−636.446	−1.21448
$66$	0	0
$67$	−402.391	−0.733729	−0.366864	−	0.930274i	$-0.619569\pi$
$67$	−402.391	−0.733729	−0.366864	+	0.930274i	$0.619569\pi$
$68$	269.071	0.479849
$69$	1022.11	1.78330
$70$	295.893	0.505228
$71$	−856.461	−1.43160	−0.715798	−	0.698308i	$-0.753937\pi$
$71$	−856.461	−1.43160	−0.715798	+	0.698308i	$0.753937\pi$
$72$	−3.49155	−0.00571505
$73$	−105.552	−0.169233	−0.0846163	−	0.996414i	$-0.526966\pi$
$73$	−105.552	−0.169233	−0.0846163	+	0.996414i	$0.526966\pi$
$74$	1058.08	1.66215
$75$	−88.3831	−0.136075
$76$	558.192	0.842487
$77$	0	0
$78$	−1554.52	−2.25659
$79$	446.524	0.635923	0.317961	−	0.948104i	$-0.397002\pi$
$79$	446.524	0.635923	0.317961	+	0.948104i	$0.397002\pi$
$80$	669.873	0.936176
$81$	−910.571	−1.24907
$82$	−238.526	−0.321230
$83$	971.903	1.28530	0.642652	−	0.766158i	$-0.277834\pi$
$83$	971.903	1.28530	0.642652	+	0.766158i	$0.277834\pi$
$84$	362.727	0.471151
$85$	−352.079	−0.449275
$86$	1044.42	1.30957
$87$	−683.662	−0.842486
$88$	0	0
$89$	779.257	0.928102	0.464051	−	0.885808i	$-0.346395\pi$
$89$	779.257	0.928102	0.464051	+	0.885808i	$0.346395\pi$
$90$	605.487	0.709155
$91$	422.384	0.486570
$92$	1281.67	1.45243
$93$	−858.618	−0.957360
$94$	−851.425	−0.934232
$95$	−730.393	−0.788808
$96$	1648.70	1.75281
$97$	−1766.59	−1.84917	−0.924586	−	0.380972i	$-0.875589\pi$
$97$	−1766.59	−1.84917	−0.924586	+	0.380972i	$0.875589\pi$
$98$	−196.372	−0.202414
$99$	0	0
$100$	−110.827	−0.110827
$101$	−339.383	−0.334355	−0.167178	−	0.985927i	$-0.553465\pi$
$101$	−339.383	−0.334355	−0.167178	+	0.985927i	$0.553465\pi$
$102$	−859.952	−0.834784
$103$	1045.45	1.00011	0.500056	−	0.865993i	$-0.333313\pi$
$103$	1045.45	1.00011	0.500056	+	0.865993i	$0.333313\pi$
$104$	−14.7082	−0.0138678
$105$	−474.627	−0.441132
$106$	−2242.36	−2.05469
$107$	−1090.57	−0.985319	−0.492659	−	0.870222i	$-0.663975\pi$
$107$	−1090.57	−0.985319	−0.492659	+	0.870222i	$0.663975\pi$
$108$	−656.838	−0.585225
$109$	1551.05	1.36297	0.681486	−	0.731831i	$-0.261334\pi$
$109$	1551.05	1.36297	0.681486	+	0.731831i	$0.261334\pi$
$110$	0	0
$111$	−1697.22	−1.45128
$112$	−444.568	−0.375069
$113$	1292.60	1.07609	0.538044	−	0.842917i	$-0.319163\pi$
$113$	1292.60	1.07609	0.538044	+	0.842917i	$0.319163\pi$
$114$	−1783.98	−1.46566
$115$	−1677.06	−1.35988
$116$	−857.273	−0.686170
$117$	864.327	0.682966
$118$	−3356.16	−2.61830
$119$	233.661	0.179997
$120$	16.5274	0.0125728
$121$	0	0
$122$	−1972.67	−1.46391
$123$	382.608	0.280477
$124$	−1076.66	−0.779731
$125$	1463.46	1.04717
$126$	−401.838	−0.284115
$127$	−995.997	−0.695909	−0.347955	−	0.937511i	$-0.613124\pi$
$127$	−995.997	−0.695909	−0.347955	+	0.937511i	$0.613124\pi$
$128$	31.1995	0.0215443
$129$	−1675.30	−1.14343
$130$	2550.62	1.72080
$131$	2610.47	1.74105	0.870527	−	0.492121i	$-0.163778\pi$
$131$	2610.47	1.74105	0.870527	+	0.492121i	$0.163778\pi$
$132$	0	0
$133$	484.733	0.316027
$134$	1612.62	1.03962
$135$	859.471	0.547937
$136$	−8.13651	−0.00513014
$137$	117.638	0.0733613	0.0366806	−	0.999327i	$-0.488322\pi$
$137$	117.638	0.0733613	0.0366806	+	0.999327i	$0.488322\pi$
$138$	−4096.21	−2.52676
$139$	985.960	0.601640	0.300820	−	0.953681i	$-0.402740\pi$
$139$	985.960	0.601640	0.300820	+	0.953681i	$0.402740\pi$
$140$	−595.154	−0.359284
$141$	1365.73	0.815710
$142$	3432.35	2.02843
$143$	0	0
$144$	−909.723	−0.526460
$145$	1121.74	0.642451
$146$	423.011	0.239785
$147$	314.991	0.176735
$148$	−2128.21	−1.18201
$149$	523.599	0.287885	0.143943	−	0.989586i	$-0.454022\pi$
$149$	523.599	0.287885	0.143943	+	0.989586i	$0.454022\pi$
$150$	354.204	0.192804
$151$	1368.70	0.737636	0.368818	−	0.929502i	$-0.379763\pi$
$151$	1368.70	0.737636	0.368818	+	0.929502i	$0.379763\pi$
$152$	−16.8793	−0.00900717
$153$	478.142	0.252650
$154$	0	0
$155$	1408.80	0.730050
$156$	3126.73	1.60474
$157$	−67.3334	−0.0342279	−0.0171140	−	0.999854i	$-0.505448\pi$
$157$	−67.3334	−0.0342279	−0.0171140	+	0.999854i	$0.505448\pi$
$158$	−1789.49	−0.901038
$159$	3596.86	1.79402
$160$	−2705.15	−1.33663
$161$	1113.00	0.544823
$162$	3649.20	1.76980
$163$	3404.71	1.63606	0.818029	−	0.575177i	$-0.195067\pi$
$163$	3404.71	1.63606	0.818029	+	0.575177i	$0.195067\pi$
$164$	479.769	0.228437
$165$	0	0
$166$	−3894.99	−1.82115
$167$	−2865.57	−1.32781	−0.663905	−	0.747817i	$-0.731102\pi$
$167$	−2865.57	−1.32781	−0.663905	+	0.747817i	$0.731102\pi$
$168$	−10.9686	−0.00503716
$169$	1443.98	0.657253
$170$	1410.99	0.636577
$171$	991.912	0.443587
$172$	−2100.74	−0.931276
$173$	3254.33	1.43018	0.715092	−	0.699030i	$-0.246385\pi$
$173$	3254.33	1.43018	0.715092	+	0.699030i	$0.246385\pi$
$174$	2739.84	1.19372
$175$	−96.2421	−0.0415727
$176$	0	0
$177$	5383.45	2.28613
$178$	−3122.95	−1.31503
$179$	−3319.65	−1.38616	−0.693080	−	0.720861i	$-0.743747\pi$
$179$	−3319.65	−1.38616	−0.693080	+	0.720861i	$0.743747\pi$
$180$	−1217.87	−0.504303
$181$	2471.00	1.01474	0.507369	−	0.861729i	$-0.330618\pi$
$181$	2471.00	1.01474	0.507369	+	0.861729i	$0.330618\pi$
$182$	−1692.74	−0.689421
$183$	3164.27	1.27819
$184$	−38.7566	−0.0155281
$185$	2784.76	1.10670
$186$	3440.99	1.35648
$187$	0	0
$188$	1712.55	0.664363
$189$	−570.397	−0.219525
$190$	2927.12	1.11766
$191$	4638.29	1.75715	0.878573	−	0.477608i	$-0.158496\pi$
$191$	4638.29	1.75715	0.878573	+	0.477608i	$0.158496\pi$
$192$	−3341.19	−1.25588
$193$	4849.58	1.80871	0.904353	−	0.426786i	$-0.140354\pi$
$193$	4849.58	1.80871	0.904353	+	0.426786i	$0.140354\pi$
$194$	7079.77	2.62009
$195$	−4091.32	−1.50249
$196$	394.980	0.143943
$197$	−3193.21	−1.15486	−0.577428	−	0.816442i	$-0.695944\pi$
$197$	−3193.21	−1.15486	−0.577428	+	0.816442i	$0.695944\pi$
$198$	0	0
$199$	−3060.65	−1.09027	−0.545135	−	0.838348i	$-0.683522\pi$
$199$	−3060.65	−1.09027	−0.545135	+	0.838348i	$0.683522\pi$
$200$	3.35133	0.00118487
$201$	−2586.72	−0.907728
$202$	1360.11	0.473748
$203$	−744.454	−0.257391
$204$	1729.70	0.593642
$205$	−627.776	−0.213882
$206$	−4189.75	−1.41706
$207$	2277.54	0.764733
$208$	−3832.21	−1.27748
$209$	0	0
$210$	1902.11	0.625039
$211$	−1855.21	−0.605298	−0.302649	−	0.953102i	$-0.597871\pi$
$211$	−1855.21	−0.605298	−0.302649	+	0.953102i	$0.597871\pi$
$212$	4510.26	1.46116
$213$	−5505.66	−1.77109
$214$	4370.55	1.39610
$215$	2748.81	0.871940
$216$	19.8623	0.00625674
$217$	−934.966	−0.292487
$218$	−6215.99	−1.93119
$219$	−678.532	−0.209365
$220$	0	0
$221$	2014.18	0.613069
$222$	6801.75	2.05632
$223$	636.378	0.191099	0.0955494	−	0.995425i	$-0.469539\pi$
$223$	636.378	0.191099	0.0955494	+	0.995425i	$0.469539\pi$
$224$	1795.30	0.535507
$225$	−196.941	−0.0583529
$226$	−5180.24	−1.52471
$227$	5015.66	1.46652	0.733262	−	0.679947i	$-0.237997\pi$
$227$	5015.66	1.46652	0.733262	+	0.679947i	$0.237997\pi$
$228$	3588.27	1.04228
$229$	2583.85	0.745614	0.372807	−	0.927909i	$-0.378395\pi$
$229$	2583.85	0.745614	0.372807	+	0.927909i	$0.378395\pi$
$230$	6720.98	1.92682
$231$	0	0
$232$	25.9232	0.00733597
$233$	−1824.49	−0.512989	−0.256494	−	0.966546i	$-0.582568\pi$
$233$	−1824.49	−0.512989	−0.256494	+	0.966546i	$0.582568\pi$
$234$	−3463.87	−0.967694
$235$	−2240.86	−0.622033
$236$	6750.54	1.86196
$237$	2870.43	0.786728
$238$	−936.419	−0.255038
$239$	5329.10	1.44230	0.721152	−	0.692777i	$-0.243613\pi$
$239$	5329.10	1.44230	0.721152	+	0.692777i	$0.243613\pi$
$240$	4306.20	1.15818
$241$	2159.12	0.577101	0.288550	−	0.957465i	$-0.406827\pi$
$241$	2159.12	0.577101	0.288550	+	0.957465i	$0.406827\pi$
$242$	0	0
$243$	−3653.40	−0.964467
$244$	3967.81	1.04104
$245$	−516.831	−0.134772
$246$	−1533.34	−0.397407
$247$	4178.44	1.07639
$248$	32.5572	0.00833624
$249$	6247.77	1.59011
$250$	−5864.97	−1.48373
$251$	−4627.53	−1.16369	−0.581847	−	0.813299i	$-0.697670\pi$
$251$	−4627.53	−1.16369	−0.581847	+	0.813299i	$0.697670\pi$
$252$	808.251	0.202044
$253$	0	0
$254$	3991.55	0.986033
$255$	−2263.30	−0.555817
$256$	4033.01	0.984621
$257$	5548.11	1.34662	0.673311	−	0.739360i	$-0.264872\pi$
$257$	5548.11	1.34662	0.673311	+	0.739360i	$0.264872\pi$
$258$	6713.94	1.62012
$259$	−1848.13	−0.443388
$260$	−5130.28	−1.22372
$261$	−1523.38	−0.361283
$262$	−10461.7	−2.46690
$263$	559.458	0.131170	0.0655849	−	0.997847i	$-0.479109\pi$
$263$	559.458	0.131170	0.0655849	+	0.997847i	$0.479109\pi$
$264$	0	0
$265$	−5901.66	−1.36806
$266$	−1942.61	−0.447779
$267$	5009.37	1.14820
$268$	−3243.60	−0.739307
$269$	−3900.98	−0.884189	−0.442094	−	0.896969i	$-0.645764\pi$
$269$	−3900.98	−0.884189	−0.442094	+	0.896969i	$0.645764\pi$
$270$	−3444.41	−0.776372
$271$	5066.32	1.13563	0.567817	−	0.823155i	$-0.307788\pi$
$271$	5066.32	1.13563	0.567817	+	0.823155i	$0.307788\pi$
$272$	−2119.96	−0.472580
$273$	2715.25	0.601957
$274$	−471.445	−0.103945
$275$	0	0
$276$	8239.06	1.79686
$277$	−6248.30	−1.35532	−0.677661	−	0.735375i	$-0.737006\pi$
$277$	−6248.30	−1.35532	−0.677661	+	0.735375i	$0.737006\pi$
$278$	−3951.33	−0.852464
$279$	−1913.23	−0.410545
$280$	17.9970	0.00384116
$281$	6130.56	1.30149	0.650744	−	0.759297i	$-0.274457\pi$
$281$	6130.56	1.30149	0.650744	+	0.759297i	$0.274457\pi$
$282$	−5473.29	−1.15578
$283$	4823.70	1.01321	0.506606	−	0.862178i	$-0.330900\pi$
$283$	4823.70	1.01321	0.506606	+	0.862178i	$0.330900\pi$
$284$	−6903.78	−1.44248
$285$	−4695.25	−0.975868
$286$	0	0
$287$	416.630	0.0856895
$288$	3673.73	0.751655
$289$	−3798.77	−0.773207
$290$	−4495.48	−0.910288
$291$	−11356.3	−2.28769
$292$	−850.839	−0.170519
$293$	2977.52	0.593681	0.296840	−	0.954927i	$-0.404067\pi$
$293$	2977.52	0.593681	0.296840	+	0.954927i	$0.404067\pi$
$294$	−1262.36	−0.250415
$295$	−8833.06	−1.74332
$296$	64.3554	0.0126371
$297$	0	0
$298$	−2098.37	−0.407904
$299$	9594.14	1.85566
$300$	−712.440	−0.137109
$301$	−1824.27	−0.349333
$302$	−5485.19	−1.04516
$303$	−2181.68	−0.413645
$304$	−4397.89	−0.829725
$305$	−5191.87	−0.974707
$306$	−1916.20	−0.357980
$307$	613.140	0.113986	0.0569930	−	0.998375i	$-0.481849\pi$
$307$	613.140	0.113986	0.0569930	+	0.998375i	$0.481849\pi$
$308$	0	0
$309$	6720.57	1.23728
$310$	−5645.91	−1.03441
$311$	3663.37	0.667944	0.333972	−	0.942583i	$-0.391611\pi$
$311$	3663.37	0.667944	0.333972	+	0.942583i	$0.391611\pi$
$312$	−94.5499	−0.0171565
$313$	3746.40	0.676546	0.338273	−	0.941048i	$-0.390157\pi$
$313$	3746.40	0.676546	0.338273	+	0.941048i	$0.390157\pi$
$314$	269.845	0.0484975
$315$	−1057.59	−0.189170
$316$	3599.35	0.640758
$317$	−5156.84	−0.913681	−0.456840	−	0.889549i	$-0.651019\pi$
$317$	−5156.84	−0.913681	−0.456840	+	0.889549i	$0.651019\pi$
$318$	−14414.8	−2.54195
$319$	0	0
$320$	5482.15	0.957693
$321$	−7010.59	−1.21898
$322$	−4460.45	−0.771960
$323$	2311.49	0.398189
$324$	−7339.95	−1.25856
$325$	−829.615	−0.141596
$326$	−13644.7	−2.31813
$327$	9970.76	1.68619
$328$	−14.5078	−0.00244226
$329$	1487.17	0.249211
$330$	0	0
$331$	1602.92	0.266177	0.133088	−	0.991104i	$-0.457511\pi$
$331$	1602.92	0.266177	0.133088	+	0.991104i	$0.457511\pi$
$332$	7834.34	1.29508
$333$	−3781.85	−0.622354
$334$	11484.0	1.88137
$335$	4244.24	0.692202
$336$	−2857.85	−0.464014
$337$	−6375.79	−1.03060	−0.515299	−	0.857010i	$-0.672319\pi$
$337$	−6375.79	−1.03060	−0.515299	+	0.857010i	$0.672319\pi$
$338$	−5786.90	−0.931261
$339$	8309.36	1.33128
$340$	−2838.05	−0.452691
$341$	0	0
$342$	−3975.18	−0.628518
$343$	343.000	0.0539949
$344$	63.5245	0.00995644
$345$	−10780.8	−1.68237
$346$	−13042.0	−2.02643
$347$	1069.17	0.165406	0.0827031	−	0.996574i	$-0.473645\pi$
$347$	1069.17	0.165406	0.0827031	+	0.996574i	$0.473645\pi$
$348$	−5510.88	−0.848891
$349$	−4621.76	−0.708874	−0.354437	−	0.935080i	$-0.615327\pi$
$349$	−4621.76	−0.708874	−0.354437	+	0.935080i	$0.615327\pi$
$350$	385.700	0.0589043
$351$	−4916.87	−0.747701
$352$	0	0
$353$	1369.54	0.206496	0.103248	−	0.994656i	$-0.467076\pi$
$353$	1369.54	0.206496	0.103248	+	0.994656i	$0.467076\pi$
$354$	−21574.7	−3.23922
$355$	9033.58	1.35057
$356$	6281.46	0.935159
$357$	1502.06	0.222682
$358$	13303.8	1.96405
$359$	4486.12	0.659522	0.329761	−	0.944064i	$-0.393032\pi$
$359$	4486.12	0.659522	0.329761	+	0.944064i	$0.393032\pi$
$360$	36.8274	0.00539159
$361$	−2063.78	−0.300886
$362$	−9902.75	−1.43778
$363$	0	0
$364$	3404.76	0.490269
$365$	1113.32	0.159654
$366$	−12681.1	−1.81107
$367$	−5726.82	−0.814544	−0.407272	−	0.913307i	$-0.633520\pi$
$367$	−5726.82	−0.814544	−0.407272	+	0.913307i	$0.633520\pi$
$368$	−10098.0	−1.43042
$369$	852.552	0.120277
$370$	−11160.2	−1.56808
$371$	3916.70	0.548099
$372$	−6921.16	−0.964639
$373$	−13410.0	−1.86151	−0.930754	−	0.365645i	$-0.880848\pi$
$373$	−13410.0	−1.86151	−0.930754	+	0.365645i	$0.880848\pi$
$374$	0	0
$375$	9407.70	1.29550
$376$	−51.7860	−0.00710282
$377$	−6417.25	−0.876672
$378$	2285.92	0.311045
$379$	11796.0	1.59873	0.799367	−	0.600843i	$-0.205168\pi$
$379$	11796.0	1.59873	0.799367	+	0.600843i	$0.205168\pi$
$380$	−5887.57	−0.794805
$381$	−6402.65	−0.860939
$382$	−18588.4	−2.48970
$383$	−10664.1	−1.42274	−0.711368	−	0.702820i	$-0.751924\pi$
$383$	−10664.1	−1.42274	−0.711368	+	0.702820i	$0.751924\pi$
$384$	200.562	0.0266534
$385$	0	0
$386$	−19435.1	−2.56275
$387$	−3733.02	−0.490337
$388$	−14240.1	−1.86323
$389$	2931.68	0.382113	0.191056	−	0.981579i	$-0.438809\pi$
$389$	2931.68	0.382113	0.191056	+	0.981579i	$0.438809\pi$
$390$	16396.4	2.12888
$391$	5307.44	0.686467
$392$	−11.9439	−0.00153892
$393$	16781.1	2.15393
$394$	12797.1	1.63631
$395$	−4709.74	−0.599931
$396$	0	0
$397$	−8477.41	−1.07171	−0.535855	−	0.844310i	$-0.680011\pi$
$397$	−8477.41	−1.07171	−0.535855	+	0.844310i	$0.680011\pi$
$398$	12265.8	1.54480
$399$	3116.05	0.390971
$400$	873.187	0.109148
$401$	−11982.2	−1.49218	−0.746088	−	0.665848i	$-0.768070\pi$
$401$	−11982.2	−1.49218	−0.746088	+	0.665848i	$0.768070\pi$
$402$	10366.5	1.28616
$403$	−8059.48	−0.996207
$404$	−2735.71	−0.336897
$405$	9604.30	1.17837
$406$	2983.47	0.364697
$407$	0	0
$408$	−52.3046	−0.00634672
$409$	−7195.61	−0.869926	−0.434963	−	0.900448i	$-0.643239\pi$
$409$	−7195.61	−0.869926	−0.434963	+	0.900448i	$0.643239\pi$
$410$	2515.87	0.303049
$411$	756.222	0.0907584
$412$	8427.20	1.00771
$413$	5862.15	0.698445
$414$	−9127.44	−1.08355
$415$	−10251.2	−1.21256
$416$	15475.6	1.82393
$417$	6338.13	0.744315
$418$	0	0
$419$	−52.9772	−0.00617686	−0.00308843	−	0.999995i	$-0.500983\pi$
$419$	−52.9772	−0.00617686	−0.00308843	+	0.999995i	$0.500983\pi$
$420$	−3825.88	−0.444486
$421$	2660.28	0.307967	0.153984	−	0.988073i	$-0.450790\pi$
$421$	2660.28	0.307967	0.153984	+	0.988073i	$0.450790\pi$
$422$	7434.93	0.857646
$423$	3043.21	0.349801
$424$	−136.387	−0.0156215
$425$	−458.940	−0.0523808
$426$	22064.5	2.50945
$427$	3445.64	0.390506
$428$	−8790.87	−0.992810
$429$	0	0
$430$	−11016.1	−1.23545
$431$	−2862.89	−0.319955	−0.159978	−	0.987121i	$-0.551142\pi$
$431$	−2862.89	−0.319955	−0.159978	+	0.987121i	$0.551142\pi$
$432$	5175.11	0.576360
$433$	−8159.52	−0.905592	−0.452796	−	0.891614i	$-0.649573\pi$
$433$	−8159.52	−0.905592	−0.452796	+	0.891614i	$0.649573\pi$
$434$	3746.97	0.414424
$435$	7210.97	0.794804
$436$	12502.8	1.37333
$437$	11010.3	1.20525
$438$	2719.28	0.296649
$439$	7349.43	0.799018	0.399509	−	0.916729i	$-0.369181\pi$
$439$	7349.43	0.799018	0.399509	+	0.916729i	$0.369181\pi$
$440$	0	0
$441$	701.883	0.0757891
$442$	−8072.01	−0.868657
$443$	−1637.08	−0.175576	−0.0877881	−	0.996139i	$-0.527980\pi$
$443$	−1637.08	−0.175576	−0.0877881	+	0.996139i	$0.527980\pi$
$444$	−13681.0	−1.46232
$445$	−8219.27	−0.875575
$446$	−2550.35	−0.270768
$447$	3365.89	0.356155
$448$	−3638.29	−0.383690
$449$	6409.20	0.673650	0.336825	−	0.941567i	$-0.390647\pi$
$449$	6409.20	0.673650	0.336825	+	0.941567i	$0.390647\pi$
$450$	789.260	0.0826801
$451$	0	0
$452$	10419.5	1.08427
$453$	8798.52	0.912562
$454$	−20100.7	−2.07792
$455$	−4455.12	−0.459032
$456$	−108.507	−0.0111432
$457$	7465.07	0.764116	0.382058	−	0.924138i	$-0.375215\pi$
$457$	7465.07	0.764116	0.382058	+	0.924138i	$0.375215\pi$
$458$	−10355.0	−1.05646
$459$	−2719.99	−0.276598
$460$	−13518.5	−1.37022
$461$	−13900.8	−1.40440	−0.702198	−	0.711982i	$-0.747798\pi$
$461$	−13900.8	−1.40440	−0.702198	+	0.711982i	$0.747798\pi$
$462$	0	0
$463$	−9514.60	−0.955034	−0.477517	−	0.878622i	$-0.658463\pi$
$463$	−9514.60	−0.955034	−0.477517	+	0.878622i	$0.658463\pi$
$464$	6754.29	0.675776
$465$	9056.33	0.903176
$466$	7311.82	0.726853
$467$	8032.77	0.795957	0.397979	−	0.917395i	$-0.369712\pi$
$467$	8032.77	0.795957	0.397979	+	0.917395i	$0.369712\pi$
$468$	6967.19	0.688159
$469$	−2816.73	−0.277323
$470$	8980.46	0.881357
$471$	−432.845	−0.0423449
$472$	−204.131	−0.0199065
$473$	0	0
$474$	−11503.5	−1.11471
$475$	−952.076	−0.0919668
$476$	1883.50	0.181366
$477$	8014.76	0.769331
$478$	−21356.9	−2.04360
$479$	−4584.42	−0.437301	−0.218651	−	0.975803i	$-0.570166\pi$
$479$	−4584.42	−0.437301	−0.218651	+	0.975803i	$0.570166\pi$
$480$	−17389.7	−1.65360
$481$	−15931.1	−1.51017
$482$	−8652.89	−0.817693
$483$	7154.78	0.674025
$484$	0	0
$485$	18633.2	1.74452
$486$	14641.3	1.36655
$487$	16788.0	1.56209	0.781045	−	0.624474i	$-0.214687\pi$
$487$	16788.0	1.56209	0.781045	+	0.624474i	$0.214687\pi$
$488$	−119.983	−0.0111299
$489$	21886.8	2.02404
$490$	2071.25	0.190958
$491$	−10364.3	−0.952614	−0.476307	−	0.879279i	$-0.658025\pi$
$491$	−10364.3	−0.952614	−0.476307	+	0.879279i	$0.658025\pi$
$492$	3084.14	0.282609
$493$	−3550.00	−0.324308
$494$	−16745.5	−1.52513
$495$	0	0
$496$	8482.78	0.767919
$497$	−5995.23	−0.541092
$498$	−25038.5	−2.25302
$499$	−18075.0	−1.62154	−0.810768	−	0.585368i	$-0.800950\pi$
$499$	−18075.0	−1.62154	−0.810768	+	0.585368i	$0.800950\pi$
$500$	11796.7	1.05513
$501$	−18421.0	−1.64269
$502$	18545.3	1.64884
$503$	13824.0	1.22541	0.612704	−	0.790312i	$-0.290082\pi$
$503$	13824.0	1.22541	0.612704	+	0.790312i	$0.290082\pi$
$504$	−24.4409	−0.00216008
$505$	3579.66	0.315432
$506$	0	0
$507$	9282.49	0.813116
$508$	−8028.56	−0.701200
$509$	−1438.23	−0.125242	−0.0626212	−	0.998037i	$-0.519946\pi$
$509$	−1438.23	−0.125242	−0.0626212	+	0.998037i	$0.519946\pi$
$510$	9070.40	0.787537
$511$	−738.867	−0.0639639
$512$	−16412.3	−1.41665
$513$	−5642.65	−0.485632
$514$	−22234.6	−1.90803
$515$	−11027.0	−0.943508
$516$	−13504.3	−1.15212
$517$	0	0
$518$	7406.57	0.628235
$519$	20920.1	1.76934
$520$	155.136	0.0130830
$521$	−7030.47	−0.591191	−0.295596	−	0.955313i	$-0.595518\pi$
$521$	−7030.47	−0.591191	−0.295596	+	0.955313i	$0.595518\pi$
$522$	6105.09	0.511902
$523$	12311.3	1.02932	0.514662	−	0.857393i	$-0.327918\pi$
$523$	12311.3	1.02932	0.514662	+	0.857393i	$0.327918\pi$
$524$	21042.6	1.75429
$525$	−618.682	−0.0514314
$526$	−2242.08	−0.185854
$527$	−4458.47	−0.368528
$528$	0	0
$529$	13113.9	1.07783
$530$	23651.5	1.93840
$531$	11995.8	0.980361
$532$	3907.34	0.318430
$533$	3591.38	0.291858
$534$	−20075.5	−1.62688
$535$	11502.8	0.929553
$536$	98.0838	0.00790406
$537$	−21340.0	−1.71488
$538$	15633.5	1.25281
$539$	0	0
$540$	6928.05	0.552103
$541$	−21377.6	−1.69888	−0.849442	−	0.527681i	$-0.823062\pi$
$541$	−21377.6	−1.69888	−0.849442	+	0.527681i	$0.823062\pi$
$542$	−20303.8	−1.60908
$543$	15884.5	1.25538
$544$	8561.05	0.674728
$545$	−16359.8	−1.28583
$546$	−10881.6	−0.852912
$547$	6871.32	0.537105	0.268552	−	0.963265i	$-0.413455\pi$
$547$	6871.32	0.537105	0.268552	+	0.963265i	$0.413455\pi$
$548$	948.259	0.0739190
$549$	7050.83	0.548128
$550$	0	0
$551$	−7364.51	−0.569399
$552$	−249.143	−0.0192105
$553$	3125.67	0.240356
$554$	25040.7	1.92035
$555$	17901.5	1.36915
$556$	7947.65	0.606214
$557$	5440.65	0.413874	0.206937	−	0.978354i	$-0.433650\pi$
$557$	5440.65	0.413874	0.206937	+	0.978354i	$0.433650\pi$
$558$	7667.44	0.581700
$559$	−15725.4	−1.18983
$560$	4689.11	0.353841
$561$	0	0
$562$	−24568.8	−1.84408
$563$	4349.17	0.325570	0.162785	−	0.986662i	$-0.447952\pi$
$563$	4349.17	0.325570	0.162785	+	0.986662i	$0.447952\pi$
$564$	11008.9	0.821912
$565$	−13633.8	−1.01518
$566$	−19331.4	−1.43562
$567$	−6374.00	−0.472103
$568$	208.765	0.0154218
$569$	−15711.4	−1.15757	−0.578785	−	0.815480i	$-0.696473\pi$
$569$	−15711.4	−1.15757	−0.578785	+	0.815480i	$0.696473\pi$
$570$	18816.7	1.38271
$571$	23775.8	1.74253	0.871266	−	0.490812i	$-0.163300\pi$
$571$	23775.8	1.74253	0.871266	+	0.490812i	$0.163300\pi$
$572$	0	0
$573$	29816.7	2.17384
$574$	−1669.68	−0.121413
$575$	−2186.07	−0.158548
$576$	−7445.05	−0.538560
$577$	−9031.16	−0.651598	−0.325799	−	0.945439i	$-0.605633\pi$
$577$	−9031.16	−0.651598	−0.325799	+	0.945439i	$0.605633\pi$
$578$	15223.9	1.09556
$579$	31175.0	2.23763
$580$	9042.14	0.647335
$581$	6803.32	0.485799
$582$	45511.5	3.24143
$583$	0	0
$584$	25.7287	0.00182305
$585$	−9116.55	−0.644312
$586$	−11932.7	−0.841185
$587$	13460.1	0.946439	0.473220	−	0.880945i	$-0.343092\pi$
$587$	13460.1	0.946439	0.473220	+	0.880945i	$0.343092\pi$
$588$	2539.09	0.178078
$589$	−9249.16	−0.647037
$590$	35399.3	2.47011
$591$	−20527.2	−1.42872
$592$	16767.8	1.16411
$593$	8573.25	0.593695	0.296848	−	0.954925i	$-0.404065\pi$
$593$	8573.25	0.593695	0.296848	+	0.954925i	$0.404065\pi$
$594$	0	0
$595$	−2464.56	−0.169810
$596$	4220.64	0.290074
$597$	−19675.0	−1.34882
$598$	−38449.4	−2.62929
$599$	7719.48	0.526560	0.263280	−	0.964720i	$-0.415196\pi$
$599$	7719.48	0.526560	0.263280	+	0.964720i	$0.415196\pi$
$600$	21.5436	0.00146586
$601$	11571.5	0.785378	0.392689	−	0.919671i	$-0.371545\pi$
$601$	11571.5	0.785378	0.392689	+	0.919671i	$0.371545\pi$
$602$	7310.95	0.494970
$603$	−5763.90	−0.389261
$604$	11032.8	0.743244
$605$	0	0
$606$	8743.31	0.586094
$607$	15871.8	1.06131	0.530654	−	0.847588i	$-0.321946\pi$
$607$	15871.8	1.06131	0.530654	+	0.847588i	$0.321946\pi$
$608$	17760.0	1.18464
$609$	−4785.64	−0.318430
$610$	20806.9	1.38106
$611$	12819.5	0.848810
$612$	3854.22	0.254571
$613$	−22045.9	−1.45257	−0.726286	−	0.687393i	$-0.758755\pi$
$613$	−22045.9	−1.45257	−0.726286	+	0.687393i	$0.758755\pi$
$614$	−2457.22	−0.161507
$615$	−4035.59	−0.264602
$616$	0	0
$617$	−7518.67	−0.490584	−0.245292	−	0.969449i	$-0.578884\pi$
$617$	−7518.67	−0.490584	−0.245292	+	0.969449i	$0.578884\pi$
$618$	−26933.3	−1.75310
$619$	14304.5	0.928829	0.464415	−	0.885618i	$-0.346265\pi$
$619$	14304.5	0.928829	0.464415	+	0.885618i	$0.346265\pi$
$620$	11356.1	0.735600
$621$	−12956.1	−0.837218
$622$	−14681.3	−0.946409
$623$	5454.80	0.350790
$624$	−24634.9	−1.58043
$625$	−13717.4	−0.877911
$626$	−15014.0	−0.958597
$627$	0	0
$628$	−542.762	−0.0344882
$629$	−8812.99	−0.558660
$630$	4238.41	0.268035
$631$	−17226.2	−1.08679	−0.543396	−	0.839476i	$-0.682862\pi$
$631$	−17226.2	−1.08679	−0.543396	+	0.839476i	$0.682862\pi$
$632$	−108.842	−0.00685045
$633$	−11926.0	−0.748840
$634$	20666.5	1.29459
$635$	10505.3	0.656523
$636$	28993.7	1.80766
$637$	2956.69	0.183906
$638$	0	0
$639$	−12268.1	−0.759496
$640$	−329.078	−0.0203249
$641$	21684.0	1.33614	0.668071	−	0.744098i	$-0.267120\pi$
$641$	21684.0	1.33614	0.668071	+	0.744098i	$0.267120\pi$
$642$	28095.6	1.72717
$643$	−18945.1	−1.16193	−0.580966	−	0.813928i	$-0.697325\pi$
$643$	−18945.1	−1.16193	−0.580966	+	0.813928i	$0.697325\pi$
$644$	8971.68	0.548966
$645$	17670.4	1.07871
$646$	−9263.53	−0.564193
$647$	14479.4	0.879819	0.439909	−	0.898042i	$-0.355011\pi$
$647$	14479.4	0.879819	0.439909	+	0.898042i	$0.355011\pi$
$648$	221.954	0.0134555
$649$	0	0
$650$	3324.76	0.200627
$651$	−6010.32	−0.361848
$652$	27444.8	1.64850
$653$	−5549.22	−0.332554	−0.166277	−	0.986079i	$-0.553175\pi$
$653$	−5549.22	−0.332554	−0.166277	+	0.986079i	$0.553175\pi$
$654$	−39958.8	−2.38916
$655$	−27534.1	−1.64251
$656$	−3780.01	−0.224976
$657$	−1511.95	−0.0897819
$658$	−5959.98	−0.353107
$659$	−18756.9	−1.10875	−0.554375	−	0.832267i	$-0.687043\pi$
$659$	−18756.9	−1.10875	−0.554375	+	0.832267i	$0.687043\pi$
$660$	0	0
$661$	14240.3	0.837945	0.418973	−	0.907999i	$-0.362390\pi$
$661$	14240.3	0.837945	0.418973	+	0.907999i	$0.362390\pi$
$662$	−6423.86	−0.377146
$663$	12947.9	0.758454
$664$	−236.904	−0.0138459
$665$	−5112.75	−0.298141
$666$	15156.1	0.881813
$667$	−16909.7	−0.981629
$668$	−23098.8	−1.33790
$669$	4090.88	0.236417
$670$	−17009.2	−0.980780
$671$	0	0
$672$	11540.9	0.662498
$673$	−30202.7	−1.72991	−0.864955	−	0.501849i	$-0.832653\pi$
$673$	−30202.7	−1.72991	−0.864955	+	0.501849i	$0.832653\pi$
$674$	25551.6	1.46025
$675$	1120.33	0.0638838
$676$	11639.7	0.662250
$677$	9296.11	0.527738	0.263869	−	0.964559i	$-0.415001\pi$
$677$	9296.11	0.527738	0.263869	+	0.964559i	$0.415001\pi$
$678$	−33300.6	−1.88628
$679$	−12366.1	−0.698922
$680$	85.8203	0.00483979
$681$	32242.6	1.81430
$682$	0	0
$683$	−5266.10	−0.295024	−0.147512	−	0.989060i	$-0.547127\pi$
$683$	−5266.10	−0.295024	−0.147512	+	0.989060i	$0.547127\pi$
$684$	7995.62	0.446960
$685$	−1240.79	−0.0692092
$686$	−1374.61	−0.0765054
$687$	16610.0	0.922432
$688$	16551.3	0.917169
$689$	33762.2	1.86682
$690$	43205.1	2.38375
$691$	1655.42	0.0911360	0.0455680	−	0.998961i	$-0.485490\pi$
$691$	1655.42	0.0911360	0.0455680	+	0.998961i	$0.485490\pi$
$692$	26232.5	1.44106
$693$	0	0
$694$	−4284.79	−0.234364
$695$	−10399.5	−0.567589
$696$	166.645	0.00907564
$697$	1986.74	0.107967
$698$	18522.1	1.00440
$699$	−11728.5	−0.634641
$700$	−775.791	−0.0418888
$701$	−1649.62	−0.0888803	−0.0444402	−	0.999012i	$-0.514150\pi$
$701$	−1649.62	−0.0888803	−0.0444402	+	0.999012i	$0.514150\pi$
$702$	19704.8	1.05942
$703$	−18282.7	−0.980859
$704$	0	0
$705$	−14405.1	−0.769544
$706$	−5488.55	−0.292584
$707$	−2375.68	−0.126374
$708$	43395.1	2.30351
$709$	6209.02	0.328893	0.164446	−	0.986386i	$-0.447416\pi$
$709$	6209.02	0.328893	0.164446	+	0.986386i	$0.447416\pi$
$710$	−36202.9	−1.91362
$711$	6396.08	0.337372
$712$	−189.946	−0.00999794
$713$	−21237.1	−1.11548
$714$	−6019.66	−0.315519
$715$	0	0
$716$	−26759.1	−1.39670
$717$	34257.5	1.78434
$718$	−17978.6	−0.934476
$719$	−28127.4	−1.45894	−0.729469	−	0.684014i	$-0.760233\pi$
$719$	−28127.4	−1.45894	−0.729469	+	0.684014i	$0.760233\pi$
$720$	9595.36	0.496664
$721$	7318.16	0.378007
$722$	8270.79	0.426326
$723$	13879.7	0.713956
$724$	19918.3	1.02245
$725$	1462.20	0.0749031
$726$	0	0
$727$	−24136.1	−1.23130	−0.615652	−	0.788018i	$-0.711107\pi$
$727$	−24136.1	−1.23130	−0.615652	+	0.788018i	$0.711107\pi$
$728$	−102.957	−0.00524155
$729$	1099.96	0.0558837
$730$	−4461.74	−0.226214
$731$	−8699.22	−0.440154
$732$	25506.6	1.28791
$733$	−21669.1	−1.09191	−0.545953	−	0.837816i	$-0.683832\pi$
$733$	−21669.1	−1.09191	−0.545953	+	0.837816i	$0.683832\pi$
$734$	22950.8	1.15413
$735$	−3322.39	−0.166732
$736$	40778.9	2.04230
$737$	0	0
$738$	−3416.69	−0.170420
$739$	35191.4	1.75174	0.875870	−	0.482548i	$-0.160288\pi$
$739$	35191.4	1.75174	0.875870	+	0.482548i	$0.160288\pi$
$740$	22447.4	1.11511
$741$	26860.6	1.33164
$742$	−15696.5	−0.776601
$743$	−22771.0	−1.12434	−0.562172	−	0.827020i	$-0.690034\pi$
$743$	−22771.0	−1.12434	−0.562172	+	0.827020i	$0.690034\pi$
$744$	209.290	0.0103131
$745$	−5522.69	−0.271592
$746$	53741.8	2.63757
$747$	13921.7	0.681884
$748$	0	0
$749$	−7633.97	−0.372416
$750$	−37702.3	−1.83559
$751$	−31234.7	−1.51767	−0.758834	−	0.651284i	$-0.774231\pi$
$751$	−31234.7	−1.51767	−0.758834	+	0.651284i	$0.774231\pi$
$752$	−13492.8	−0.654299
$753$	−29747.5	−1.43966
$754$	25717.7	1.24216
$755$	−14436.4	−0.695888
$756$	−4597.87	−0.221194
$757$	−3601.63	−0.172924	−0.0864619	−	0.996255i	$-0.527556\pi$
$757$	−3601.63	−0.172924	−0.0864619	+	0.996255i	$0.527556\pi$
$758$	−47273.6	−2.26524
$759$	0	0
$760$	178.035	0.00849739
$761$	25457.1	1.21264	0.606319	−	0.795221i	$-0.292645\pi$
$761$	25457.1	1.21264	0.606319	+	0.795221i	$0.292645\pi$
$762$	25659.2	1.21986
$763$	10857.4	0.515155
$764$	37388.4	1.77051
$765$	−5043.24	−0.238351
$766$	42737.2	2.01587
$767$	50532.2	2.37890
$768$	25925.7	1.21812
$769$	−10381.4	−0.486816	−0.243408	−	0.969924i	$-0.578265\pi$
$769$	−10381.4	−0.486816	−0.243408	+	0.969924i	$0.578265\pi$
$770$	0	0
$771$	35665.4	1.66596
$772$	39091.6	1.82246
$773$	−14476.9	−0.673607	−0.336804	−	0.941575i	$-0.609346\pi$
$773$	−14476.9	−0.673607	−0.336804	+	0.941575i	$0.609346\pi$
$774$	14960.4	0.694757
$775$	1836.39	0.0851163
$776$	430.611	0.0199201
$777$	−11880.5	−0.548534
$778$	−11749.0	−0.541415
$779$	4121.51	0.189562
$780$	−32979.4	−1.51391
$781$	0	0
$782$	−21270.1	−0.972655
$783$	8666.00	0.395527
$784$	−3111.98	−0.141763
$785$	710.203	0.0322907
$786$	−67251.9	−3.05191
$787$	11803.9	0.534645	0.267322	−	0.963607i	$-0.413861\pi$
$787$	11803.9	0.534645	0.267322	+	0.963607i	$0.413861\pi$
$788$	−25739.9	−1.16364
$789$	3596.41	0.162276
$790$	18874.7	0.850042
$791$	9048.23	0.406723
$792$	0	0
$793$	29701.7	1.33006
$794$	33974.0	1.51851
$795$	−37938.1	−1.69249
$796$	−24671.4	−1.09856
$797$	−5079.93	−0.225772	−0.112886	−	0.993608i	$-0.536010\pi$
$797$	−5079.93	−0.225772	−0.112886	+	0.993608i	$0.536010\pi$
$798$	−12487.9	−0.553967
$799$	7091.71	0.314001
$800$	−3526.19	−0.155837
$801$	11162.2	0.492381
$802$	48019.8	2.11426
$803$	0	0
$804$	−20851.1	−0.914629
$805$	−11739.4	−0.513988
$806$	32299.2	1.41153
$807$	−25077.0	−1.09387
$808$	82.7256	0.00360183
$809$	−8699.51	−0.378070	−0.189035	−	0.981970i	$-0.560536\pi$
$809$	−8699.51	−0.378070	−0.189035	+	0.981970i	$0.560536\pi$
$810$	−38490.2	−1.66964
$811$	15480.1	0.670259	0.335130	−	0.942172i	$-0.391220\pi$
$811$	15480.1	0.670259	0.335130	+	0.942172i	$0.391220\pi$
$812$	−6000.91	−0.259348
$813$	32568.2	1.40494
$814$	0	0
$815$	−35911.4	−1.54346
$816$	−13627.9	−0.584649
$817$	−18046.6	−0.772793
$818$	28837.1	1.23260
$819$	6050.29	0.258137
$820$	−5060.39	−0.215508
$821$	−15593.0	−0.662849	−0.331425	−	0.943482i	$-0.607529\pi$
$821$	−15593.0	−0.662849	−0.331425	+	0.943482i	$0.607529\pi$
$822$	−3030.63	−0.128595
$823$	−36816.1	−1.55933	−0.779665	−	0.626197i	$-0.784611\pi$
$823$	−36816.1	−1.55933	−0.779665	+	0.626197i	$0.784611\pi$
$824$	−254.832	−0.0107737
$825$	0	0
$826$	−23493.1	−0.989626
$827$	12108.9	0.509149	0.254575	−	0.967053i	$-0.418065\pi$
$827$	12108.9	0.509149	0.254575	+	0.967053i	$0.418065\pi$
$828$	18358.8	0.770547
$829$	5851.08	0.245134	0.122567	−	0.992460i	$-0.460887\pi$
$829$	5851.08	0.245134	0.122567	+	0.992460i	$0.460887\pi$
$830$	41082.7	1.71807
$831$	−40166.5	−1.67673
$832$	−31362.3	−1.30684
$833$	1635.63	0.0680326
$834$	−25400.7	−1.05462
$835$	30224.8	1.25266
$836$	0	0
$837$	10883.7	0.449458
$838$	212.311	0.00875199
$839$	22806.8	0.938473	0.469237	−	0.883072i	$-0.344529\pi$
$839$	22806.8	0.938473	0.469237	+	0.883072i	$0.344529\pi$
$840$	115.692	0.00475207
$841$	−13078.6	−0.536249
$842$	−10661.3	−0.436359
$843$	39409.6	1.61013
$844$	−14954.5	−0.609900
$845$	−15230.5	−0.620054
$846$	−12195.9	−0.495633
$847$	0	0
$848$	−35535.5	−1.43902
$849$	31008.6	1.25349
$850$	1839.24	0.0742183
$851$	−41979.0	−1.69098
$852$	−44380.2	−1.78455
$853$	42190.7	1.69353	0.846765	−	0.531968i	$-0.178547\pi$
$853$	42190.7	1.69353	0.846765	+	0.531968i	$0.178547\pi$
$854$	−13808.7	−0.553308
$855$	−10462.3	−0.418481
$856$	265.829	0.0106143
$857$	7162.44	0.285489	0.142745	−	0.989760i	$-0.454407\pi$
$857$	7162.44	0.285489	0.142745	+	0.989760i	$0.454407\pi$
$858$	0	0
$859$	−38407.9	−1.52557	−0.762783	−	0.646654i	$-0.776168\pi$
$859$	−38407.9	−1.52557	−0.762783	+	0.646654i	$0.776168\pi$
$860$	22157.6	0.878569
$861$	2678.26	0.106010
$862$	11473.3	0.453344
$863$	10042.3	0.396112	0.198056	−	0.980191i	$-0.436537\pi$
$863$	10042.3	0.396112	0.198056	+	0.980191i	$0.436537\pi$
$864$	−20898.6	−0.822900
$865$	−34325.2	−1.34924
$866$	32700.0	1.28313
$867$	−24419.9	−0.956568
$868$	−7536.60	−0.294711
$869$	0	0
$870$	−28898.7	−1.12616
$871$	−24280.5	−0.944561
$872$	−378.073	−0.0146825
$873$	−25304.9	−0.981031
$874$	−44125.0	−1.70772
$875$	10244.2	0.395793
$876$	−5469.52	−0.210957
$877$	49108.6	1.89085	0.945427	−	0.325835i	$-0.105645\pi$
$877$	49108.6	1.89085	0.945427	+	0.325835i	$0.105645\pi$
$878$	−29453.5	−1.13213
$879$	19140.6	0.734468
$880$	0	0
$881$	17395.9	0.665245	0.332623	−	0.943060i	$-0.392066\pi$
$881$	17395.9	0.665245	0.332623	+	0.943060i	$0.392066\pi$
$882$	−2812.86	−0.107386
$883$	−27861.5	−1.06185	−0.530925	−	0.847419i	$-0.678155\pi$
$883$	−27861.5	−1.06185	−0.530925	+	0.847419i	$0.678155\pi$
$884$	16235.9	0.617730
$885$	−56782.3	−2.15674
$886$	6560.77	0.248774
$887$	−30594.6	−1.15813	−0.579067	−	0.815280i	$-0.696583\pi$
$887$	−30594.6	−1.15813	−0.579067	+	0.815280i	$0.696583\pi$
$888$	413.701	0.0156339
$889$	−6971.98	−0.263029
$890$	32939.5	1.24060
$891$	0	0
$892$	5129.73	0.192552
$893$	14711.8	0.551302
$894$	−13489.1	−0.504636
$895$	35014.3	1.30771
$896$	218.396	0.00814297
$897$	61674.8	2.29572
$898$	−25685.5	−0.954494
$899$	14204.9	0.526985
$900$	−1587.51	−0.0587965
$901$	18677.1	0.690594
$902$	0	0
$903$	−11727.1	−0.432176
$904$	−315.076	−0.0115921
$905$	−26063.0	−0.957307
$906$	−35260.9	−1.29301
$907$	16597.9	0.607634	0.303817	−	0.952730i	$-0.401739\pi$
$907$	16597.9	0.607634	0.303817	+	0.952730i	$0.401739\pi$
$908$	40430.3	1.47767
$909$	−4861.37	−0.177383
$910$	17854.3	0.650401
$911$	242.944	0.00883543	0.00441772	−	0.999990i	$-0.498594\pi$
$911$	242.944	0.00883543	0.00441772	+	0.999990i	$0.498594\pi$
$912$	−28271.3	−1.02649
$913$	0	0
$914$	−29917.0	−1.08268
$915$	−33375.3	−1.20585
$916$	20828.0	0.751283
$917$	18273.3	0.658056
$918$	10900.6	0.391911
$919$	−1920.43	−0.0689327	−0.0344663	−	0.999406i	$-0.510973\pi$
$919$	−1920.43	−0.0689327	−0.0344663	+	0.999406i	$0.510973\pi$
$920$	408.788	0.0146493
$921$	3941.50	0.141017
$922$	55708.9	1.98989
$923$	−51679.4	−1.84295
$924$	0	0
$925$	3629.96	0.129030
$926$	38130.7	1.35319
$927$	14975.2	0.530583
$928$	−27275.9	−0.964842
$929$	−49689.0	−1.75484	−0.877419	−	0.479725i	$-0.840736\pi$
$929$	−49689.0	−1.75484	−0.877419	+	0.479725i	$0.840736\pi$
$930$	−36294.1	−1.27971
$931$	3393.13	0.119447
$932$	−14706.9	−0.516889
$933$	23549.5	0.826342
$934$	−32192.1	−1.12779
$935$	0	0
$936$	−210.682	−0.00735722
$937$	−38642.1	−1.34726	−0.673629	−	0.739069i	$-0.735266\pi$
$937$	−38642.1	−1.34726	−0.673629	+	0.739069i	$0.735266\pi$
$938$	11288.3	0.392939
$939$	24083.3	0.836984
$940$	−18063.2	−0.626762
$941$	−16537.8	−0.572921	−0.286460	−	0.958092i	$-0.592479\pi$
$941$	−16537.8	−0.572921	−0.286460	+	0.958092i	$0.592479\pi$
$942$	1734.67	0.0599984
$943$	9463.44	0.326800
$944$	−53186.2	−1.83375
$945$	6016.30	0.207101
$946$	0	0
$947$	−16166.1	−0.554729	−0.277365	−	0.960765i	$-0.589461\pi$
$947$	−16166.1	−0.554729	−0.277365	+	0.960765i	$0.589461\pi$
$948$	23138.0	0.792709
$949$	−6369.09	−0.217860
$950$	3815.53	0.130308
$951$	−33150.1	−1.13035
$952$	−56.9555	−0.00193901
$953$	15004.7	0.510020	0.255010	−	0.966938i	$-0.417921\pi$
$953$	15004.7	0.510020	0.255010	+	0.966938i	$0.417921\pi$
$954$	−32119.9	−1.09006
$955$	−48922.7	−1.65770
$956$	42956.9	1.45327
$957$	0	0
$958$	18372.5	0.619612
$959$	823.466	0.0277279
$960$	35241.4	1.18480
$961$	−11951.0	−0.401160
$962$	63845.2	2.13976
$963$	−15621.4	−0.522735
$964$	17404.3	0.581488
$965$	−51151.2	−1.70634
$966$	−28673.5	−0.955025
$967$	−23983.8	−0.797589	−0.398794	−	0.917040i	$-0.630571\pi$
$967$	−23983.8	−0.797589	−0.398794	+	0.917040i	$0.630571\pi$
$968$	0	0
$969$	14859.2	0.492616
$970$	−74674.3	−2.47180
$971$	−31184.5	−1.03065	−0.515323	−	0.856996i	$-0.672328\pi$
$971$	−31184.5	−1.03065	−0.515323	+	0.856996i	$0.672328\pi$
$972$	−29449.4	−0.971800
$973$	6901.72	0.227399
$974$	−67279.6	−2.21333
$975$	−5333.09	−0.175175
$976$	−31261.6	−1.02527
$977$	−3320.82	−0.108744	−0.0543718	−	0.998521i	$-0.517316\pi$
$977$	−3320.82	−0.108744	−0.0543718	+	0.998521i	$0.517316\pi$
$978$	−87713.4	−2.86786
$979$	0	0
$980$	−4166.08	−0.135796
$981$	22217.5	0.723089
$982$	41535.9	1.34976
$983$	8000.50	0.259589	0.129795	−	0.991541i	$-0.458568\pi$
$983$	8000.50	0.259589	0.129795	+	0.991541i	$0.458568\pi$
$984$	−93.2618	−0.00302142
$985$	33680.6	1.08949
$986$	14227.0	0.459512
$987$	9560.11	0.308310
$988$	33681.6	1.08457
$989$	−41437.0	−1.33228
$990$	0	0
$991$	37294.6	1.19546	0.597731	−	0.801697i	$-0.296069\pi$
$991$	37294.6	1.19546	0.597731	+	0.801697i	$0.296069\pi$
$992$	−34256.0	−1.09640
$993$	10304.2	0.329299
$994$	24026.5	0.766673
$995$	32282.4	1.02856
$996$	50362.1	1.60219
$997$	−16065.0	−0.510315	−0.255158	−	0.966899i	$-0.582127\pi$
$997$	−16065.0	−0.510315	−0.255158	+	0.966899i	$0.582127\pi$
$998$	72437.1	2.29755
$999$	21513.7	0.681343

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.4.a.k.1.2	yes	8
11.10	odd	2		847.4.a.j.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.4.a.j.1.7	✓	8	11.10	odd	2
847.4.a.k.1.2	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-4.00760 q^{2} +6.42839 q^{3} +8.06082 q^{4} -10.5476 q^{5} -25.7624 q^{6} +7.00000 q^{7} -0.243753 q^{8} +14.3241 q^{9} +42.2704 q^{10} +51.8181 q^{12} +60.3406 q^{13} -28.0532 q^{14} -67.8038 q^{15} -63.5097 q^{16} +33.3802 q^{17} -57.4054 q^{18} +69.2475 q^{19} -85.0221 q^{20} +44.9987 q^{21} +159.000 q^{23} -1.56694 q^{24} -13.7489 q^{25} -241.821 q^{26} -81.4853 q^{27} +56.4258 q^{28} -106.351 q^{29} +271.730 q^{30} -133.567 q^{31} +256.471 q^{32} -133.774 q^{34} -73.8330 q^{35} +115.464 q^{36} -264.019 q^{37} -277.516 q^{38} +387.892 q^{39} +2.57100 q^{40} +59.5186 q^{41} -180.337 q^{42} -260.611 q^{43} -151.085 q^{45} -637.207 q^{46} +212.453 q^{47} -408.265 q^{48} +49.0000 q^{49} +55.0999 q^{50} +214.581 q^{51} +486.395 q^{52} +559.528 q^{53} +326.560 q^{54} -1.70627 q^{56} +445.150 q^{57} +426.210 q^{58} +837.450 q^{59} -546.555 q^{60} +492.234 q^{61} +535.281 q^{62} +100.269 q^{63} -519.755 q^{64} -636.446 q^{65} -402.391 q^{67} +269.071 q^{68} +1022.11 q^{69} +295.893 q^{70} -856.461 q^{71} -3.49155 q^{72} -105.552 q^{73} +1058.08 q^{74} -88.3831 q^{75} +558.192 q^{76} -1554.52 q^{78} +446.524 q^{79} +669.873 q^{80} -910.571 q^{81} -238.526 q^{82} +971.903 q^{83} +362.727 q^{84} -352.079 q^{85} +1044.42 q^{86} -683.662 q^{87} +779.257 q^{89} +605.487 q^{90} +422.384 q^{91} +1281.67 q^{92} -858.618 q^{93} -851.425 q^{94} -730.393 q^{95} +1648.70 q^{96} -1766.59 q^{97} -196.372 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{2} + 2 q^{3} + 30 q^{4} + 2 q^{5} + 53 q^{6} + 56 q^{7} + 27 q^{8} + 120 q^{9} + 66 q^{10} + 36 q^{12} + 124 q^{13} + 14 q^{14} + 78 q^{15} + 122 q^{16} + 192 q^{17} - 162 q^{18} + 96 q^{19}+ \cdots + 98 q^{98}+O(q^{100}) \) 8 * q + 2 * q^2 + 2 * q^3 + 30 * q^4 + 2 * q^5 + 53 * q^6 + 56 * q^7 + 27 * q^8 + 120 * q^9 + 66 * q^10 + 36 * q^12 + 124 * q^13 + 14 * q^14 + 78 * q^15 + 122 * q^16 + 192 * q^17 - 162 * q^18 + 96 * q^19 + 340 * q^20 + 14 * q^21 + 172 * q^23 + 17 * q^24 + 406 * q^25 - 347 * q^26 - 220 * q^27 + 210 * q^28 + 294 * q^29 + 205 * q^30 + 394 * q^31 + 788 * q^32 - 767 * q^34 + 14 * q^35 + 87 * q^36 - 252 * q^37 + 81 * q^38 - 208 * q^39 + 345 * q^40 + 1120 * q^41 + 371 * q^42 + 384 * q^43 - 1458 * q^45 - 40 * q^46 - 368 * q^47 - 523 * q^48 + 392 * q^49 - 1306 * q^50 + 960 * q^51 + 183 * q^52 + 602 * q^53 + 2515 * q^54 + 189 * q^56 + 1324 * q^57 + 1727 * q^58 + 66 * q^59 - 2460 * q^60 - 2388 * q^61 + 1595 * q^62 + 840 * q^63 + 741 * q^64 + 762 * q^65 - 894 * q^67 + 1352 * q^68 + 1470 * q^69 + 462 * q^70 - 2104 * q^71 - 1010 * q^72 + 1644 * q^73 - 2533 * q^74 - 2002 * q^75 + 5765 * q^76 - 2068 * q^78 + 678 * q^79 + 3846 * q^80 + 2464 * q^81 - 4764 * q^82 + 474 * q^83 + 252 * q^84 + 1494 * q^85 + 3471 * q^86 + 3460 * q^87 + 88 * q^89 + 2014 * q^90 + 868 * q^91 + 3669 * q^92 - 4170 * q^93 - 888 * q^94 + 4994 * q^95 + 214 * q^96 - 392 * q^97 + 98 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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