LMFDB - Embedded newform 49.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [49,4,Mod(1,49)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(49, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("49.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 49 = 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	49.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$2.89109359028$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{65})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 35x^{2} + 36x + 194 $ x^4 - 2x^3 - 35x^2 + 36*x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 7 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.94534$ of defining polynomial
Character	$\chi$	$=$	49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.53113 q^{2} +7.82220 q^{3} +4.46887 q^{4} +2.07730 q^{5} -27.6212 q^{6} +12.4689 q^{8} +34.1868 q^{9} +O(q^{10})$	\(q-3.53113 q^{2} +7.82220 q^{3} +4.46887 q^{4} +2.07730 q^{5} -27.6212 q^{6} +12.4689 q^{8} +34.1868 q^{9} -7.33521 q^{10} +49.1868 q^{11} +34.9564 q^{12} +44.8559 q^{13} +16.2490 q^{15} -79.7802 q^{16} -26.5179 q^{17} -120.718 q^{18} -77.7350 q^{19} +9.28317 q^{20} -173.685 q^{22} +55.7510 q^{23} +97.5340 q^{24} -120.685 q^{25} -158.392 q^{26} +56.2164 q^{27} +121.436 q^{29} -57.3774 q^{30} -305.553 q^{31} +181.963 q^{32} +384.749 q^{33} +93.6380 q^{34} +152.776 q^{36} +77.1868 q^{37} +274.492 q^{38} +350.872 q^{39} +25.9016 q^{40} -248.720 q^{41} -147.179 q^{43} +219.809 q^{44} +71.0161 q^{45} -196.864 q^{46} -269.851 q^{47} -624.056 q^{48} +426.154 q^{50} -207.428 q^{51} +200.455 q^{52} -141.121 q^{53} -198.507 q^{54} +102.176 q^{55} -608.058 q^{57} -428.805 q^{58} +424.834 q^{59} +72.6148 q^{60} +587.996 q^{61} +1078.95 q^{62} -4.29373 q^{64} +93.1790 q^{65} -1358.60 q^{66} -179.634 q^{67} -118.505 q^{68} +436.095 q^{69} +674.872 q^{71} +426.270 q^{72} -237.489 q^{73} -272.556 q^{74} -944.021 q^{75} -347.388 q^{76} -1238.97 q^{78} +495.852 q^{79} -165.727 q^{80} -483.307 q^{81} +878.262 q^{82} +24.4406 q^{83} -55.0855 q^{85} +519.708 q^{86} +949.895 q^{87} +613.304 q^{88} -1072.29 q^{89} -250.767 q^{90} +249.144 q^{92} -2390.09 q^{93} +952.877 q^{94} -161.479 q^{95} +1423.35 q^{96} -1667.43 q^{97} +1681.54 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9}+O(q^{10}) $ 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9	$ 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9} + 100 q^{11} - 64 q^{15} - 174 q^{16} - 370 q^{18} - 340 q^{22} + 352 q^{23} - 128 q^{25} + 260 q^{29} - 552 q^{30} - 30 q^{32} - 50 q^{36} + 212 q^{37} + 952 q^{39} + 540 q^{43} + 460 q^{44} + 696 q^{46} + 1366 q^{50} + 428 q^{51} + 16 q^{53} - 1884 q^{57} - 780 q^{58} - 1064 q^{60} - 1678 q^{64} - 756 q^{65} - 1944 q^{67} + 2248 q^{71} + 270 q^{72} - 284 q^{74} - 1344 q^{78} - 1048 q^{79} - 1256 q^{81} - 3284 q^{85} + 4820 q^{86} + 1260 q^{88} + 3512 q^{92} - 5368 q^{93} + 2192 q^{95} + 3340 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9 + 100 * q^11 - 64 * q^15 - 174 * q^16 - 370 * q^18 - 340 * q^22 + 352 * q^23 - 128 * q^25 + 260 * q^29 - 552 * q^30 - 30 * q^32 - 50 * q^36 + 212 * q^37 + 952 * q^39 + 540 * q^43 + 460 * q^44 + 696 * q^46 + 1366 * q^50 + 428 * q^51 + 16 * q^53 - 1884 * q^57 - 780 * q^58 - 1064 * q^60 - 1678 * q^64 - 756 * q^65 - 1944 * q^67 + 2248 * q^71 + 270 * q^72 - 284 * q^74 - 1344 * q^78 - 1048 * q^79 - 1256 * q^81 - 3284 * q^85 + 4820 * q^86 + 1260 * q^88 + 3512 * q^92 - 5368 * q^93 + 2192 * q^95 + 3340 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	−3.53113	−1.24844	−0.624221	−	0.781248i	$-0.714584\pi$
$2$	−3.53113	−1.24844	−0.624221	+	0.781248i	$0.714584\pi$
$3$	7.82220	1.50538	0.752691	−	0.658374i	$-0.228755\pi$
$3$	7.82220	1.50538	0.752691	+	0.658374i	$0.228755\pi$
$4$	4.46887	0.558609
$5$	2.07730	0.185799	0.0928996	−	0.995675i	$-0.470386\pi$
$5$	2.07730	0.185799	0.0928996	+	0.995675i	$0.470386\pi$
$6$	−27.6212	−1.87938
$7$	0	0
$7$	0	0
$8$	12.4689	0.551051
$9$	34.1868	1.26618
$10$	−7.33521	−0.231960
$11$	49.1868	1.34822	0.674108	−	0.738633i	$-0.264528\pi$
$11$	49.1868	1.34822	0.674108	+	0.738633i	$0.264528\pi$
$12$	34.9564	0.840920
$13$	44.8559	0.956983	0.478492	−	0.878092i	$-0.341184\pi$
$13$	44.8559	0.956983	0.478492	+	0.878092i	$0.341184\pi$
$14$	0	0
$15$	16.2490	0.279699
$16$	−79.7802	−1.24656
$17$	−26.5179	−0.378325	−0.189163	−	0.981946i	$-0.560577\pi$
$17$	−26.5179	−0.378325	−0.189163	+	0.981946i	$0.560577\pi$
$18$	−120.718	−1.58075
$19$	−77.7350	−0.938612	−0.469306	−	0.883036i	$-0.655496\pi$
$19$	−77.7350	−0.938612	−0.469306	+	0.883036i	$0.655496\pi$
$20$	9.28317	0.103789
$21$	0	0
$22$	−173.685	−1.68317
$23$	55.7510	0.505430	0.252715	−	0.967541i	$-0.418677\pi$
$23$	55.7510	0.505430	0.252715	+	0.967541i	$0.418677\pi$
$24$	97.5340	0.829543
$25$	−120.685	−0.965479
$26$	−158.392	−1.19474
$27$	56.2164	0.400698
$28$	0	0
$29$	121.436	0.777588	0.388794	−	0.921325i	$-0.372892\pi$
$29$	121.436	0.777588	0.388794	+	0.921325i	$0.372892\pi$
$30$	−57.3774	−0.349188
$31$	−305.553	−1.77029	−0.885143	−	0.465319i	$-0.845940\pi$
$31$	−305.553	−1.77029	−0.885143	+	0.465319i	$0.845940\pi$
$32$	181.963	1.00521
$33$	384.749	2.02958
$34$	93.6380	0.472317
$35$	0	0
$36$	152.776	0.707298
$37$	77.1868	0.342957	0.171479	−	0.985188i	$-0.445146\pi$
$37$	77.1868	0.342957	0.171479	+	0.985188i	$0.445146\pi$
$38$	274.492	1.17180
$39$	350.872	1.44063
$40$	25.9016	0.102385
$41$	−248.720	−0.947403	−0.473702	−	0.880685i	$-0.657083\pi$
$41$	−248.720	−0.947403	−0.473702	+	0.880685i	$0.657083\pi$
$42$	0	0
$43$	−147.179	−0.521967	−0.260984	−	0.965343i	$-0.584047\pi$
$43$	−147.179	−0.521967	−0.260984	+	0.965343i	$0.584047\pi$
$44$	219.809	0.753125
$45$	71.0161	0.235255
$46$	−196.864	−0.631000
$47$	−269.851	−0.837484	−0.418742	−	0.908105i	$-0.637529\pi$
$47$	−269.851	−0.837484	−0.418742	+	0.908105i	$0.637529\pi$
$48$	−624.056	−1.87656
$49$	0	0
$50$	426.154	1.20534
$51$	−207.428	−0.569524
$52$	200.455	0.534579
$53$	−141.121	−0.365744	−0.182872	−	0.983137i	$-0.558539\pi$
$53$	−141.121	−0.365744	−0.182872	+	0.983137i	$0.558539\pi$
$54$	−198.507	−0.500248
$55$	102.176	0.250497
$56$	0	0
$57$	−608.058	−1.41297
$58$	−428.805	−0.970774
$59$	424.834	0.937434	0.468717	−	0.883348i	$-0.344716\pi$
$59$	424.834	0.937434	0.468717	+	0.883348i	$0.344716\pi$
$60$	72.6148	0.156242
$61$	587.996	1.23418	0.617092	−	0.786891i	$-0.288311\pi$
$61$	587.996	1.23418	0.617092	+	0.786891i	$0.288311\pi$
$62$	1078.95	2.21010
$63$	0	0
$64$	−4.29373	−0.00838618
$65$	93.1790	0.177807
$66$	−1358.60	−2.53381
$67$	−179.634	−0.327549	−0.163775	−	0.986498i	$-0.552367\pi$
$67$	−179.634	−0.327549	−0.163775	+	0.986498i	$0.552367\pi$
$68$	−118.505	−0.211336
$69$	436.095	0.760865
$70$	0	0
$71$	674.872	1.12806	0.564032	−	0.825753i	$-0.309250\pi$
$71$	674.872	1.12806	0.564032	+	0.825753i	$0.309250\pi$
$72$	426.270	0.697729
$73$	−237.489	−0.380767	−0.190383	−	0.981710i	$-0.560973\pi$
$73$	−237.489	−0.380767	−0.190383	+	0.981710i	$0.560973\pi$
$74$	−272.556	−0.428163
$75$	−944.021	−1.45341
$76$	−347.388	−0.524317
$77$	0	0
$78$	−1238.97	−1.79854
$79$	495.852	0.706174	0.353087	−	0.935591i	$-0.385132\pi$
$79$	495.852	0.706174	0.353087	+	0.935591i	$0.385132\pi$
$80$	−165.727	−0.231611
$81$	−483.307	−0.662973
$82$	878.262	1.18278
$83$	24.4406	0.0323217	0.0161609	−	0.999869i	$-0.494856\pi$
$83$	24.4406	0.0323217	0.0161609	+	0.999869i	$0.494856\pi$
$84$	0	0
$85$	−55.0855	−0.0702925
$86$	519.708	0.651646
$87$	949.895	1.17057
$88$	613.304	0.742936
$89$	−1072.29	−1.27710	−0.638552	−	0.769579i	$-0.720466\pi$
$89$	−1072.29	−1.27710	−0.638552	+	0.769579i	$0.720466\pi$
$90$	−250.767	−0.293702
$91$	0	0
$92$	249.144	0.282337
$93$	−2390.09	−2.66496
$94$	952.877	1.04555
$95$	−161.479	−0.174393
$96$	1423.35	1.51323
$97$	−1667.43	−1.74538	−0.872690	−	0.488275i	$-0.837626\pi$
$97$	−1667.43	−1.74538	−0.872690	+	0.488275i	$0.837626\pi$
$98$	0	0
$99$	1681.54	1.70708
$100$	−539.325	−0.539325
$101$	77.1187	0.0759762	0.0379881	−	0.999278i	$-0.487905\pi$
$101$	77.1187	0.0759762	0.0379881	+	0.999278i	$0.487905\pi$
$102$	732.455	0.711018
$103$	−164.693	−0.157550	−0.0787749	−	0.996892i	$-0.525101\pi$
$103$	−164.693	−0.157550	−0.0787749	+	0.996892i	$0.525101\pi$
$104$	559.302	0.527347
$105$	0	0
$106$	498.315	0.456610
$107$	1022.62	0.923931	0.461966	−	0.886898i	$-0.347144\pi$
$107$	1022.62	0.923931	0.461966	+	0.886898i	$0.347144\pi$
$108$	251.224	0.223833
$109$	1362.52	1.19730	0.598649	−	0.801011i	$-0.295704\pi$
$109$	1362.52	1.19730	0.598649	+	0.801011i	$0.295704\pi$
$110$	−360.795	−0.312731
$111$	603.770	0.516282
$112$	0	0
$113$	−1538.41	−1.28072	−0.640360	−	0.768075i	$-0.721215\pi$
$113$	−1538.41	−1.28072	−0.640360	+	0.768075i	$0.721215\pi$
$114$	2147.13	1.76401
$115$	115.811	0.0939084
$116$	542.681	0.434368
$117$	1533.48	1.21171
$118$	−1500.14	−1.17033
$119$	0	0
$120$	202.607	0.154128
$121$	1088.34	0.817685
$122$	−2076.29	−1.54081
$123$	−1945.54	−1.42620
$124$	−1365.48	−0.988898
$125$	−510.360	−0.365184
$126$	0	0
$127$	−170.358	−0.119030	−0.0595151	−	0.998227i	$-0.518955\pi$
$127$	−170.358	−0.119030	−0.0595151	+	0.998227i	$0.518955\pi$
$128$	−1440.54	−0.994744
$129$	−1151.26	−0.785760
$130$	−329.027	−0.221981
$131$	751.935	0.501503	0.250751	−	0.968051i	$-0.419322\pi$
$131$	751.935	0.501503	0.250751	+	0.968051i	$0.419322\pi$
$132$	1719.39	1.13374
$133$	0	0
$134$	634.312	0.408927
$135$	116.778	0.0744493
$136$	−330.648	−0.208477
$137$	518.623	0.323423	0.161711	−	0.986838i	$-0.448299\pi$
$137$	518.623	0.323423	0.161711	+	0.986838i	$0.448299\pi$
$138$	−1539.91	−0.949896
$139$	2975.72	1.81581	0.907905	−	0.419177i	$-0.137681\pi$
$139$	2975.72	1.81581	0.907905	+	0.419177i	$0.137681\pi$
$140$	0	0
$141$	−2110.83	−1.26073
$142$	−2383.06	−1.40832
$143$	2206.32	1.29022
$144$	−2727.43	−1.57837
$145$	252.258	0.144475
$146$	838.604	0.475365
$147$	0	0
$148$	344.938	0.191579
$149$	−2717.94	−1.49438	−0.747188	−	0.664612i	$-0.768597\pi$
$149$	−2717.94	−1.49438	−0.747188	+	0.664612i	$0.768597\pi$
$150$	3333.46	1.81450
$151$	707.650	0.381376	0.190688	−	0.981651i	$-0.438928\pi$
$151$	707.650	0.381376	0.190688	+	0.981651i	$0.438928\pi$
$152$	−969.267	−0.517224
$153$	−906.561	−0.479027
$154$	0	0
$155$	−634.724	−0.328918
$156$	1568.00	0.804747
$157$	−3117.91	−1.58495	−0.792473	−	0.609906i	$-0.791207\pi$
$157$	−3117.91	−1.58495	−0.792473	+	0.609906i	$0.791207\pi$
$158$	−1750.92	−0.881618
$159$	−1103.87	−0.550584
$160$	377.991	0.186768
$161$	0	0
$162$	1706.62	0.827684
$163$	1808.77	0.869167	0.434583	−	0.900632i	$-0.356896\pi$
$163$	1808.77	0.869167	0.434583	+	0.900632i	$0.356896\pi$
$164$	−1111.50	−0.529228
$165$	799.237	0.377094
$166$	−86.3028	−0.0403518
$167$	−3147.38	−1.45839	−0.729197	−	0.684303i	$-0.760106\pi$
$167$	−3147.38	−1.45839	−0.729197	+	0.684303i	$0.760106\pi$
$168$	0	0
$169$	−184.949	−0.0841827
$170$	194.514	0.0877562
$171$	−2657.51	−1.18845
$172$	−657.724	−0.291576
$173$	3284.36	1.44338	0.721691	−	0.692215i	$-0.243365\pi$
$173$	3284.36	1.44338	0.721691	+	0.692215i	$0.243365\pi$
$174$	−3354.20	−1.46139
$175$	0	0
$176$	−3924.13	−1.68064
$177$	3323.13	1.41120
$178$	3786.39	1.59439
$179$	2798.83	1.16868	0.584341	−	0.811508i	$-0.301353\pi$
$179$	2798.83	1.16868	0.584341	+	0.811508i	$0.301353\pi$
$180$	317.362	0.131415
$181$	3723.04	1.52890	0.764451	−	0.644682i	$-0.223010\pi$
$181$	3723.04	1.52890	0.764451	+	0.644682i	$0.223010\pi$
$182$	0	0
$183$	4599.42	1.85792
$184$	695.152	0.278518
$185$	160.340	0.0637212
$186$	8439.73	3.32705
$187$	−1304.33	−0.510064
$188$	−1205.93	−0.467826
$189$	0	0
$190$	570.202	0.217720
$191$	959.650	0.363549	0.181774	−	0.983340i	$-0.441816\pi$
$191$	959.650	0.363549	0.181774	+	0.983340i	$0.441816\pi$
$192$	−33.5864	−0.0126244
$193$	−3790.25	−1.41362	−0.706808	−	0.707406i	$-0.749865\pi$
$193$	−3790.25	−1.41362	−0.706808	+	0.707406i	$0.749865\pi$
$194$	5887.91	2.17901
$195$	728.865	0.267667
$196$	0	0
$197$	5117.99	1.85097	0.925487	−	0.378779i	$-0.123656\pi$
$197$	5117.99	1.85097	0.925487	+	0.378779i	$0.123656\pi$
$198$	−5937.72	−2.13119
$199$	−864.855	−0.308080	−0.154040	−	0.988065i	$-0.549228\pi$
$199$	−864.855	−0.308080	−0.154040	+	0.988065i	$0.549228\pi$
$200$	−1504.80	−0.532028
$201$	−1405.13	−0.493087
$202$	−272.316	−0.0948519
$203$	0	0
$204$	−926.969	−0.318141
$205$	−516.665	−0.176027
$206$	581.550	0.196692
$207$	1905.95	0.639963
$208$	−3578.61	−1.19294
$209$	−3823.53	−1.26545
$210$	0	0
$211$	−1344.61	−0.438707	−0.219353	−	0.975645i	$-0.570395\pi$
$211$	−1344.61	−0.438707	−0.219353	+	0.975645i	$0.570395\pi$
$212$	−630.650	−0.204308
$213$	5278.98	1.69817
$214$	−3611.01	−1.15348
$215$	−305.735	−0.0969811
$216$	700.955	0.220805
$217$	0	0
$218$	−4811.23	−1.49476
$219$	−1857.68	−0.573200
$220$	456.609	0.139930
$221$	−1189.48	−0.362051
$222$	−2131.99	−0.644549
$223$	864.916	0.259727	0.129863	−	0.991532i	$-0.458546\pi$
$223$	864.916	0.259727	0.129863	+	0.991532i	$0.458546\pi$
$224$	0	0
$225$	−4125.83	−1.22247
$226$	5432.32	1.59890
$227$	−1715.34	−0.501548	−0.250774	−	0.968046i	$-0.580685\pi$
$227$	−1715.34	−0.501548	−0.250774	+	0.968046i	$0.580685\pi$
$228$	−2717.33	−0.789298
$229$	−1045.46	−0.301685	−0.150842	−	0.988558i	$-0.548199\pi$
$229$	−1045.46	−0.301685	−0.150842	+	0.988558i	$0.548199\pi$
$230$	−408.945	−0.117239
$231$	0	0
$232$	1514.17	0.428491
$233$	1448.67	0.407320	0.203660	−	0.979042i	$-0.434716\pi$
$233$	1448.67	0.407320	0.203660	+	0.979042i	$0.434716\pi$
$234$	−5414.91	−1.51275
$235$	−560.560	−0.155604
$236$	1898.53	0.523659
$237$	3878.65	1.06306
$238$	0	0
$239$	−3153.12	−0.853383	−0.426691	−	0.904397i	$-0.640321\pi$
$239$	−3153.12	−0.853383	−0.426691	+	0.904397i	$0.640321\pi$
$240$	−1296.35	−0.348663
$241$	−381.012	−0.101839	−0.0509194	−	0.998703i	$-0.516215\pi$
$241$	−381.012	−0.101839	−0.0509194	+	0.998703i	$0.516215\pi$
$242$	−3843.06	−1.02083
$243$	−5298.37	−1.39873
$244$	2627.68	0.689426
$245$	0	0
$246$	6869.94	1.78053
$247$	−3486.87	−0.898236
$248$	−3809.90	−0.975519
$249$	191.179	0.0486565
$250$	1802.15	0.455912
$251$	3776.23	0.949617	0.474808	−	0.880089i	$-0.342517\pi$
$251$	3776.23	0.949617	0.474808	+	0.880089i	$0.342517\pi$
$252$	0	0
$253$	2742.21	0.681428
$254$	601.556	0.148602
$255$	−430.890	−0.105817
$256$	5121.09	1.25027
$257$	4258.42	1.03359	0.516795	−	0.856109i	$-0.327125\pi$
$257$	4258.42	1.03359	0.516795	+	0.856109i	$0.327125\pi$
$258$	4065.26	0.980977
$259$	0	0
$260$	416.405	0.0993244
$261$	4151.50	0.984564
$262$	−2655.18	−0.626098
$263$	4198.83	0.984451	0.492226	−	0.870468i	$-0.336184\pi$
$263$	4198.83	0.984451	0.492226	+	0.870468i	$0.336184\pi$
$264$	4797.38	1.11840
$265$	−293.150	−0.0679548
$266$	0	0
$267$	−8387.65	−1.92253
$268$	−802.762	−0.182972
$269$	3740.59	0.847835	0.423917	−	0.905701i	$-0.360655\pi$
$269$	3740.59	0.847835	0.423917	+	0.905701i	$0.360655\pi$
$270$	−412.359	−0.0929457
$271$	4356.30	0.976480	0.488240	−	0.872709i	$-0.337639\pi$
$271$	4356.30	0.976480	0.488240	+	0.872709i	$0.337639\pi$
$272$	2115.60	0.471607
$273$	0	0
$274$	−1831.32	−0.403775
$275$	−5936.10	−1.30167
$276$	1948.85	0.425026
$277$	−1344.30	−0.291593	−0.145797	−	0.989315i	$-0.546575\pi$
$277$	−1344.30	−0.291593	−0.145797	+	0.989315i	$0.546575\pi$
$278$	−10507.7	−2.26693
$279$	−10445.9	−2.24150
$280$	0	0
$281$	4205.54	0.892817	0.446408	−	0.894829i	$-0.352703\pi$
$281$	4205.54	0.892817	0.446408	+	0.894829i	$0.352703\pi$
$282$	7453.60	1.57395
$283$	4752.03	0.998159	0.499079	−	0.866556i	$-0.333672\pi$
$283$	4752.03	0.998159	0.499079	+	0.866556i	$0.333672\pi$
$284$	3015.91	0.630146
$285$	−1263.12	−0.262529
$286$	−7790.79	−1.61077
$287$	0	0
$288$	6220.73	1.27278
$289$	−4209.80	−0.856870
$290$	−890.757	−0.180369
$291$	−13043.0	−2.62746
$292$	−1061.31	−0.212700
$293$	−4961.17	−0.989196	−0.494598	−	0.869122i	$-0.664685\pi$
$293$	−4961.17	−0.989196	−0.494598	+	0.869122i	$0.664685\pi$
$294$	0	0
$295$	882.506	0.174174
$296$	962.432	0.188987
$297$	2765.10	0.540227
$298$	9597.39	1.86564
$299$	2500.76	0.483688
$300$	−4218.71	−0.811890
$301$	0	0
$302$	−2498.80	−0.476126
$303$	603.237	0.114373
$304$	6201.71	1.17004
$305$	1221.44	0.229310
$306$	3201.18	0.598037
$307$	−4234.00	−0.787124	−0.393562	−	0.919298i	$-0.628757\pi$
$307$	−4234.00	−0.787124	−0.393562	+	0.919298i	$0.628757\pi$
$308$	0	0
$309$	−1288.26	−0.237173
$310$	2241.29	0.410635
$311$	−684.700	−0.124842	−0.0624209	−	0.998050i	$-0.519882\pi$
$311$	−684.700	−0.124842	−0.0624209	+	0.998050i	$0.519882\pi$
$312$	4374.97	0.793859
$313$	−5944.07	−1.07341	−0.536707	−	0.843768i	$-0.680332\pi$
$313$	−5944.07	−1.07341	−0.536707	+	0.843768i	$0.680332\pi$
$314$	11009.8	1.97872
$315$	0	0
$316$	2215.90	0.394475
$317$	−2823.89	−0.500333	−0.250166	−	0.968203i	$-0.580485\pi$
$317$	−2823.89	−0.500333	−0.250166	+	0.968203i	$0.580485\pi$
$318$	3897.92	0.687373
$319$	5973.04	1.04836
$320$	−8.91935	−0.00155815
$321$	7999.16	1.39087
$322$	0	0
$323$	2061.37	0.355101
$324$	−2159.84	−0.370343
$325$	−5413.43	−0.923947
$326$	−6387.02	−1.08510
$327$	10657.9	1.80239
$328$	−3101.26	−0.522068
$329$	0	0
$330$	−2822.21	−0.470780
$331$	−2812.97	−0.467114	−0.233557	−	0.972343i	$-0.575037\pi$
$331$	−2812.97	−0.467114	−0.233557	+	0.972343i	$0.575037\pi$
$332$	109.222	0.0180552
$333$	2638.77	0.434245
$334$	11113.8	1.82072
$335$	−373.154	−0.0608584
$336$	0	0
$337$	4260.10	0.688612	0.344306	−	0.938857i	$-0.388114\pi$
$337$	4260.10	0.688612	0.344306	+	0.938857i	$0.388114\pi$
$338$	653.080	0.105097
$339$	−12033.7	−1.92797
$340$	−246.170	−0.0392660
$341$	−15029.2	−2.38673
$342$	9384.00	1.48371
$343$	0	0
$344$	−1835.16	−0.287631
$345$	905.899	0.141368
$346$	−11597.5	−1.80198
$347$	36.0584	0.00557843	0.00278922	−	0.999996i	$-0.499112\pi$
$347$	36.0584	0.00557843	0.00278922	+	0.999996i	$0.499112\pi$
$348$	4244.96	0.653890
$349$	−242.692	−0.0372236	−0.0186118	−	0.999827i	$-0.505925\pi$
$349$	−242.692	−0.0372236	−0.0186118	+	0.999827i	$0.505925\pi$
$350$	0	0
$351$	2521.63	0.383461
$352$	8950.18	1.35524
$353$	−109.990	−0.0165840	−0.00829201	−	0.999966i	$-0.502639\pi$
$353$	−109.990	−0.0165840	−0.00829201	+	0.999966i	$0.502639\pi$
$354$	−11734.4	−1.76180
$355$	1401.91	0.209593
$356$	−4791.92	−0.713402
$357$	0	0
$358$	−9883.01	−1.45903
$359$	12404.5	1.82363	0.911814	−	0.410604i	$-0.134682\pi$
$359$	12404.5	1.82363	0.911814	+	0.410604i	$0.134682\pi$
$360$	885.491	0.129637
$361$	−816.273	−0.119008
$362$	−13146.5	−1.90875
$363$	8513.20	1.23093
$364$	0	0
$365$	−493.335	−0.0707461
$366$	−16241.2	−2.31951
$367$	13859.6	1.97130	0.985649	−	0.168807i	$-0.0539913\pi$
$367$	13859.6	1.97130	0.985649	+	0.168807i	$0.0539913\pi$
$368$	−4447.82	−0.630051
$369$	−8502.93	−1.19958
$370$	−566.181	−0.0795523
$371$	0	0
$372$	−10681.0	−1.48867
$373$	4898.06	0.679925	0.339963	−	0.940439i	$-0.389586\pi$
$373$	4898.06	0.679925	0.339963	+	0.940439i	$0.389586\pi$
$374$	4605.75	0.636786
$375$	−3992.14	−0.549742
$376$	−3364.73	−0.461497
$377$	5447.11	0.744139
$378$	0	0
$379$	−9806.25	−1.32906	−0.664530	−	0.747262i	$-0.731368\pi$
$379$	−9806.25	−1.32906	−0.664530	+	0.747262i	$0.731368\pi$
$380$	−721.627	−0.0974176
$381$	−1332.57	−0.179186
$382$	−3388.65	−0.453870
$383$	10729.7	1.43149	0.715746	−	0.698361i	$-0.246087\pi$
$383$	10729.7	1.43149	0.715746	+	0.698361i	$0.246087\pi$
$384$	−11268.2	−1.49747
$385$	0	0
$386$	13383.8	1.76482
$387$	−5031.58	−0.660903
$388$	−7451.53	−0.974984
$389$	5264.05	0.686113	0.343057	−	0.939315i	$-0.388538\pi$
$389$	5264.05	0.686113	0.343057	+	0.939315i	$0.388538\pi$
$390$	−2573.72	−0.334167
$391$	−1478.40	−0.191217
$392$	0	0
$393$	5881.79	0.754954
$394$	−18072.3	−2.31083
$395$	1030.03	0.131206
$396$	7514.57	0.953590
$397$	1214.90	0.153587	0.0767935	−	0.997047i	$-0.475532\pi$
$397$	1214.90	0.153587	0.0767935	+	0.997047i	$0.475532\pi$
$398$	3053.91	0.384620
$399$	0	0
$400$	9628.26	1.20353
$401$	2295.45	0.285859	0.142929	−	0.989733i	$-0.454348\pi$
$401$	2295.45	0.285859	0.142929	+	0.989733i	$0.454348\pi$
$402$	4961.71	0.615591
$403$	−13705.8	−1.69413
$404$	344.633	0.0424410
$405$	−1003.97	−0.123180
$406$	0	0
$407$	3796.57	0.462381
$408$	−2586.39	−0.313837
$409$	4646.54	0.561753	0.280876	−	0.959744i	$-0.409375\pi$
$409$	4646.54	0.561753	0.280876	+	0.959744i	$0.409375\pi$
$410$	1824.41	0.219759
$411$	4056.77	0.486875
$412$	−735.990	−0.0880087
$413$	0	0
$414$	−6730.14	−0.798957
$415$	50.7703	0.00600535
$416$	8162.11	0.961973
$417$	23276.7	2.73349
$418$	13501.4	1.57984
$419$	7541.24	0.879269	0.439634	−	0.898177i	$-0.355108\pi$
$419$	7541.24	0.879269	0.439634	+	0.898177i	$0.355108\pi$
$420$	0	0
$421$	−6243.63	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$421$	−6243.63	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$422$	4748.01	0.547700
$423$	−9225.32	−1.06040
$424$	−1759.62	−0.201544
$425$	3200.31	0.365265
$426$	−18640.8	−2.12006
$427$	0	0
$428$	4569.97	0.516116
$429$	17258.2	1.94227
$430$	1079.59	0.121075
$431$	11465.8	1.28141	0.640706	−	0.767786i	$-0.278642\pi$
$431$	11465.8	1.28141	0.640706	+	0.767786i	$0.278642\pi$
$432$	−4484.95	−0.499496
$433$	5156.40	0.572289	0.286144	−	0.958187i	$-0.407626\pi$
$433$	5156.40	0.572289	0.286144	+	0.958187i	$0.407626\pi$
$434$	0	0
$435$	1973.21	0.217491
$436$	6088.92	0.668822
$437$	−4333.80	−0.474402
$438$	6559.72	0.715607
$439$	−5064.25	−0.550577	−0.275289	−	0.961362i	$-0.588773\pi$
$439$	−5064.25	−0.550577	−0.275289	+	0.961362i	$0.588773\pi$
$440$	1274.01	0.138037
$441$	0	0
$442$	4200.22	0.452000
$443$	12703.6	1.36246	0.681228	−	0.732071i	$-0.261446\pi$
$443$	12703.6	1.36246	0.681228	+	0.732071i	$0.261446\pi$
$444$	2698.17	0.288400
$445$	−2227.46	−0.237285
$446$	−3054.13	−0.324254
$447$	−21260.2	−2.24961
$448$	0	0
$449$	13942.2	1.46542	0.732709	−	0.680542i	$-0.238256\pi$
$449$	13942.2	1.46542	0.732709	+	0.680542i	$0.238256\pi$
$450$	14568.8	1.52618
$451$	−12233.7	−1.27730
$452$	−6874.95	−0.715421
$453$	5535.38	0.574116
$454$	6057.10	0.626154
$455$	0	0
$456$	−7581.80	−0.778619
$457$	−15214.0	−1.55729	−0.778646	−	0.627464i	$-0.784093\pi$
$457$	−15214.0	−1.55729	−0.778646	+	0.627464i	$0.784093\pi$
$458$	3691.65	0.376636
$459$	−1490.74	−0.151594
$460$	517.546	0.0524581
$461$	−11430.2	−1.15479	−0.577394	−	0.816465i	$-0.695930\pi$
$461$	−11430.2	−1.15479	−0.577394	+	0.816465i	$0.695930\pi$
$462$	0	0
$463$	−9347.88	−0.938300	−0.469150	−	0.883119i	$-0.655440\pi$
$463$	−9347.88	−0.938300	−0.469150	+	0.883119i	$0.655440\pi$
$464$	−9688.17	−0.969314
$465$	−4964.94	−0.495147
$466$	−5115.44	−0.508515
$467$	3630.84	0.359776	0.179888	−	0.983687i	$-0.442427\pi$
$467$	3630.84	0.359776	0.179888	+	0.983687i	$0.442427\pi$
$468$	6852.92	0.676872
$469$	0	0
$470$	1979.41	0.194262
$471$	−24388.9	−2.38595
$472$	5297.20	0.516575
$473$	−7239.26	−0.703724
$474$	−13696.0	−1.32717
$475$	9381.43	0.906210
$476$	0	0
$477$	−4824.46	−0.463096
$478$	11134.1	1.06540
$479$	6521.25	0.622053	0.311027	−	0.950401i	$-0.399327\pi$
$479$	6521.25	0.622053	0.311027	+	0.950401i	$0.399327\pi$
$480$	2956.72	0.281157
$481$	3462.28	0.328205
$482$	1345.40	0.127140
$483$	0	0
$484$	4863.65	0.456766
$485$	−3463.75	−0.324290
$486$	18709.2	1.74623
$487$	−3666.29	−0.341140	−0.170570	−	0.985346i	$-0.554561\pi$
$487$	−3666.29	−0.341140	−0.170570	+	0.985346i	$0.554561\pi$
$488$	7331.65	0.680099
$489$	14148.6	1.30843
$490$	0	0
$491$	−12470.7	−1.14623	−0.573113	−	0.819476i	$-0.694264\pi$
$491$	−12470.7	−1.14623	−0.573113	+	0.819476i	$0.694264\pi$
$492$	−8694.35	−0.796691
$493$	−3220.22	−0.294181
$494$	12312.6	1.12140
$495$	3493.05	0.317174
$496$	24377.0	2.20678
$497$	0	0
$498$	−675.078	−0.0607449
$499$	−2303.93	−0.206690	−0.103345	−	0.994646i	$-0.532955\pi$
$499$	−2303.93	−0.206690	−0.103345	+	0.994646i	$0.532955\pi$
$500$	−2280.74	−0.203995
$501$	−24619.5	−2.19544
$502$	−13334.4	−1.18554
$503$	−10520.4	−0.932570	−0.466285	−	0.884635i	$-0.654408\pi$
$503$	−10520.4	−0.932570	−0.466285	+	0.884635i	$0.654408\pi$
$504$	0	0
$505$	160.198	0.0141163
$506$	−9683.10	−0.850724
$507$	−1446.71	−0.126727
$508$	−761.308	−0.0664913
$509$	−9662.22	−0.841395	−0.420698	−	0.907201i	$-0.638215\pi$
$509$	−9662.22	−0.841395	−0.420698	+	0.907201i	$0.638215\pi$
$510$	1521.53	0.132107
$511$	0	0
$512$	−6558.89	−0.566142
$513$	−4369.98	−0.376100
$514$	−15037.0	−1.29038
$515$	−342.115	−0.0292726
$516$	−5144.85	−0.438933
$517$	−13273.1	−1.12911
$518$	0	0
$519$	25690.9	2.17284
$520$	1161.84	0.0979806
$521$	8607.81	0.723829	0.361914	−	0.932211i	$-0.382123\pi$
$521$	8607.81	0.723829	0.361914	+	0.932211i	$0.382123\pi$
$522$	−14659.5	−1.22917
$523$	−10482.7	−0.876439	−0.438219	−	0.898868i	$-0.644391\pi$
$523$	−10482.7	−0.876439	−0.438219	+	0.898868i	$0.644391\pi$
$524$	3360.30	0.280144
$525$	0	0
$526$	−14826.6	−1.22903
$527$	8102.61	0.669744
$528$	−30695.3	−2.53000
$529$	−9058.83	−0.744541
$530$	1035.15	0.0848377
$531$	14523.7	1.18696
$532$	0	0
$533$	−11156.6	−0.906649
$534$	29617.9	2.40017
$535$	2124.29	0.171666
$536$	−2239.84	−0.180497
$537$	21893.0	1.75931
$538$	−13208.5	−1.05847
$539$	0	0
$540$	521.866	0.0415881
$541$	20722.6	1.64683	0.823416	−	0.567438i	$-0.192066\pi$
$541$	20722.6	1.64683	0.823416	+	0.567438i	$0.192066\pi$
$542$	−15382.6	−1.21908
$543$	29122.3	2.30158
$544$	−4825.27	−0.380298
$545$	2830.35	0.222457
$546$	0	0
$547$	−4175.09	−0.326351	−0.163176	−	0.986597i	$-0.552174\pi$
$547$	−4175.09	−0.326351	−0.163176	+	0.986597i	$0.552174\pi$
$548$	2317.66	0.180667
$549$	20101.7	1.56270
$550$	20961.1	1.62506
$551$	−9439.81	−0.729854
$552$	5437.61	0.419276
$553$	0	0
$554$	4746.91	0.364038
$555$	1254.21	0.0959248
$556$	13298.1	1.01433
$557$	−10161.7	−0.773011	−0.386505	−	0.922287i	$-0.626318\pi$
$557$	−10161.7	−0.773011	−0.386505	+	0.922287i	$0.626318\pi$
$558$	36885.7	2.79838
$559$	−6601.85	−0.499514
$560$	0	0
$561$	−10202.7	−0.767841
$562$	−14850.3	−1.11463
$563$	−17104.4	−1.28040	−0.640201	−	0.768208i	$-0.721149\pi$
$563$	−17104.4	−1.28040	−0.640201	+	0.768208i	$0.721149\pi$
$564$	−9433.01	−0.704257
$565$	−3195.73	−0.237957
$566$	−16780.0	−1.24614
$567$	0	0
$568$	8414.89	0.621621
$569$	−18257.6	−1.34516	−0.672581	−	0.740023i	$-0.734815\pi$
$569$	−18257.6	−1.34516	−0.672581	+	0.740023i	$0.734815\pi$
$570$	4460.23	0.327752
$571$	13630.5	0.998982	0.499491	−	0.866319i	$-0.333520\pi$
$571$	13630.5	0.998982	0.499491	+	0.866319i	$0.333520\pi$
$572$	9859.74	0.720728
$573$	7506.57	0.547280
$574$	0	0
$575$	−6728.30	−0.487981
$576$	−146.789	−0.0106184
$577$	4442.08	0.320496	0.160248	−	0.987077i	$-0.448771\pi$
$577$	4442.08	0.320496	0.160248	+	0.987077i	$0.448771\pi$
$578$	14865.4	1.06975
$579$	−29648.0	−2.12803
$580$	1127.31	0.0807052
$581$	0	0
$582$	46056.4	3.28024
$583$	−6941.27	−0.493101
$584$	−2961.22	−0.209822
$585$	3185.49	0.225135
$586$	17518.5	1.23495
$587$	3103.38	0.218211	0.109106	−	0.994030i	$-0.465201\pi$
$587$	3103.38	0.218211	0.109106	+	0.994030i	$0.465201\pi$
$588$	0	0
$589$	23752.1	1.66161
$590$	−3116.24	−0.217447
$591$	40033.9	2.78642
$592$	−6157.97	−0.427519
$593$	5937.71	0.411185	0.205592	−	0.978638i	$-0.434088\pi$
$593$	5937.71	0.411185	0.205592	+	0.978638i	$0.434088\pi$
$594$	−9763.93	−0.674443
$595$	0	0
$596$	−12146.1	−0.834772
$597$	−6765.07	−0.463778
$598$	−8830.50	−0.603856
$599$	−2600.33	−0.177373	−0.0886866	−	0.996060i	$-0.528267\pi$
$599$	−2600.33	−0.177373	−0.0886866	+	0.996060i	$0.528267\pi$
$600$	−11770.9	−0.800906
$601$	−13881.4	−0.942156	−0.471078	−	0.882092i	$-0.656135\pi$
$601$	−13881.4	−0.942156	−0.471078	+	0.882092i	$0.656135\pi$
$602$	0	0
$603$	−6141.11	−0.414735
$604$	3162.40	0.213040
$605$	2260.80	0.151925
$606$	−2130.11	−0.142788
$607$	−12284.6	−0.821442	−0.410721	−	0.911761i	$-0.634723\pi$
$607$	−12284.6	−0.821442	−0.410721	+	0.911761i	$0.634723\pi$
$608$	−14144.9	−0.943505
$609$	0	0
$610$	−4313.07	−0.286281
$611$	−12104.4	−0.801459
$612$	−4051.30	−0.267589
$613$	22062.0	1.45363	0.726815	−	0.686833i	$-0.241000\pi$
$613$	22062.0	1.45363	0.726815	+	0.686833i	$0.241000\pi$
$614$	14950.8	0.982679
$615$	−4041.46	−0.264988
$616$	0	0
$617$	−12182.2	−0.794871	−0.397436	−	0.917630i	$-0.630100\pi$
$617$	−12182.2	−0.794871	−0.397436	+	0.917630i	$0.630100\pi$
$618$	4549.00	0.296097
$619$	23248.6	1.50960	0.754799	−	0.655956i	$-0.227734\pi$
$619$	23248.6	1.50960	0.754799	+	0.655956i	$0.227734\pi$
$620$	−2836.50	−0.183736
$621$	3134.12	0.202525
$622$	2417.76	0.155858
$623$	0	0
$624$	−27992.6	−1.79583
$625$	14025.4	0.897628
$626$	20989.3	1.34010
$627$	−29908.4	−1.90499
$628$	−13933.6	−0.885365
$629$	−2046.83	−0.129749
$630$	0	0
$631$	19184.4	1.21033	0.605165	−	0.796100i	$-0.293107\pi$
$631$	19184.4	1.21033	0.605165	+	0.796100i	$0.293107\pi$
$632$	6182.72	0.389138
$633$	−10517.8	−0.660421
$634$	9971.52	0.624637
$635$	−353.884	−0.0221157
$636$	−4933.07	−0.307561
$637$	0	0
$638$	−21091.6	−1.30881
$639$	23071.7	1.42833
$640$	−2992.44	−0.184823
$641$	−19433.4	−1.19746	−0.598730	−	0.800951i	$-0.704328\pi$
$641$	−19433.4	−1.19746	−0.598730	+	0.800951i	$0.704328\pi$
$642$	−28246.0	−1.73642
$643$	5777.47	0.354341	0.177170	−	0.984180i	$-0.443306\pi$
$643$	5777.47	0.354341	0.177170	+	0.984180i	$0.443306\pi$
$644$	0	0
$645$	−2391.52	−0.145994
$646$	−7278.95	−0.443323
$647$	−29231.5	−1.77621	−0.888106	−	0.459640i	$-0.847979\pi$
$647$	−29231.5	−1.77621	−0.888106	+	0.459640i	$0.847979\pi$
$648$	−6026.30	−0.365332
$649$	20896.2	1.26386
$650$	19115.5	1.15349
$651$	0	0
$652$	8083.18	0.485524
$653$	−7093.27	−0.425086	−0.212543	−	0.977152i	$-0.568174\pi$
$653$	−7093.27	−0.425086	−0.212543	+	0.977152i	$0.568174\pi$
$654$	−37634.4	−2.25018
$655$	1561.99	0.0931788
$656$	19842.9	1.18100
$657$	−8118.98	−0.482118
$658$	0	0
$659$	19014.2	1.12396	0.561980	−	0.827151i	$-0.310040\pi$
$659$	19014.2	1.12396	0.561980	+	0.827151i	$0.310040\pi$
$660$	3571.69	0.210648
$661$	−21058.4	−1.23915	−0.619573	−	0.784939i	$-0.712694\pi$
$661$	−21058.4	−1.23915	−0.619573	+	0.784939i	$0.712694\pi$
$662$	9932.96	0.583165
$663$	−9304.37	−0.545025
$664$	304.746	0.0178109
$665$	0	0
$666$	−9317.83	−0.542130
$667$	6770.16	0.393016
$668$	−14065.3	−0.814672
$669$	6765.54	0.390988
$670$	1317.65	0.0759782
$671$	28921.6	1.66395
$672$	0	0
$673$	9634.87	0.551853	0.275926	−	0.961179i	$-0.411015\pi$
$673$	9634.87	0.551853	0.275926	+	0.961179i	$0.411015\pi$
$674$	−15043.0	−0.859693
$675$	−6784.46	−0.386865
$676$	−826.515	−0.0470252
$677$	8371.31	0.475237	0.237619	−	0.971359i	$-0.423633\pi$
$677$	8371.31	0.475237	0.237619	+	0.971359i	$0.423633\pi$
$678$	42492.7	2.40696
$679$	0	0
$680$	−686.854	−0.0387348
$681$	−13417.8	−0.755022
$682$	53069.9	2.97969
$683$	−12068.8	−0.676137	−0.338069	−	0.941121i	$-0.609774\pi$
$683$	−12068.8	−0.676137	−0.338069	+	0.941121i	$0.609774\pi$
$684$	−11876.1	−0.663878
$685$	1077.33	0.0600917
$686$	0	0
$687$	−8177.78	−0.454151
$688$	11742.0	0.650666
$689$	−6330.09	−0.350011
$690$	−3198.85	−0.176490
$691$	2981.29	0.164130	0.0820648	−	0.996627i	$-0.473849\pi$
$691$	2981.29	0.164130	0.0820648	+	0.996627i	$0.473849\pi$
$692$	14677.4	0.806286
$693$	0	0
$694$	−127.327	−0.00696435
$695$	6181.46	0.337376
$696$	11844.1	0.645043
$697$	6595.53	0.358427
$698$	856.978	0.0464715
$699$	11331.8	0.613172
$700$	0	0
$701$	−28978.0	−1.56132	−0.780660	−	0.624956i	$-0.785117\pi$
$701$	−28978.0	−1.56132	−0.780660	+	0.624956i	$0.785117\pi$
$702$	−8904.22	−0.478729
$703$	−6000.11	−0.321904
$704$	−211.195	−0.0113064
$705$	−4384.81	−0.234243
$706$	388.388	0.0207042
$707$	0	0
$708$	14850.7	0.788307
$709$	−16372.4	−0.867249	−0.433625	−	0.901094i	$-0.642766\pi$
$709$	−16372.4	−0.867249	−0.433625	+	0.901094i	$0.642766\pi$
$710$	−4950.32	−0.261665
$711$	16951.6	0.894141
$712$	−13370.2	−0.703750
$713$	−17034.9	−0.894755
$714$	0	0
$715$	4583.18	0.239722
$716$	12507.6	0.652836
$717$	−24664.3	−1.28467
$718$	−43801.7	−2.27669
$719$	23010.5	1.19353	0.596765	−	0.802416i	$-0.296453\pi$
$719$	23010.5	1.19353	0.596765	+	0.802416i	$0.296453\pi$
$720$	−5665.68	−0.293260
$721$	0	0
$722$	2882.36	0.148574
$723$	−2980.35	−0.153306
$724$	16637.8	0.854058
$725$	−14655.5	−0.750745
$726$	−30061.2	−1.53674
$727$	−24636.8	−1.25685	−0.628423	−	0.777872i	$-0.716299\pi$
$727$	−24636.8	−1.25685	−0.628423	+	0.777872i	$0.716299\pi$
$728$	0	0
$729$	−28395.6	−1.44264
$730$	1742.03	0.0883225
$731$	3902.87	0.197473
$732$	20554.2	1.03785
$733$	6904.76	0.347931	0.173965	−	0.984752i	$-0.444342\pi$
$733$	6904.76	0.347931	0.173965	+	0.984752i	$0.444342\pi$
$734$	−48940.1	−2.46105
$735$	0	0
$736$	10144.6	0.508065
$737$	−8835.63	−0.441607
$738$	30025.0	1.49761
$739$	−9234.89	−0.459690	−0.229845	−	0.973227i	$-0.573822\pi$
$739$	−9234.89	−0.459690	−0.229845	+	0.973227i	$0.573822\pi$
$740$	716.538	0.0355952
$741$	−27275.0	−1.35219
$742$	0	0
$743$	20216.9	0.998232	0.499116	−	0.866535i	$-0.333658\pi$
$743$	20216.9	0.998232	0.499116	+	0.866535i	$0.333658\pi$
$744$	−29801.8	−1.46853
$745$	−5645.97	−0.277654
$746$	−17295.7	−0.848847
$747$	835.544	0.0409250
$748$	−5828.88	−0.284926
$749$	0	0
$750$	14096.8	0.686321
$751$	24054.9	1.16881	0.584405	−	0.811462i	$-0.301328\pi$
$751$	24054.9	1.16881	0.584405	+	0.811462i	$0.301328\pi$
$752$	21528.7	1.04398
$753$	29538.4	1.42954
$754$	−19234.5	−0.929015
$755$	1470.00	0.0708593
$756$	0	0
$757$	−30328.2	−1.45614	−0.728069	−	0.685504i	$-0.759582\pi$
$757$	−30328.2	−1.45614	−0.728069	+	0.685504i	$0.759582\pi$
$758$	34627.1	1.65925
$759$	21450.1	1.02581
$760$	−2013.46	−0.0960997
$761$	−33834.1	−1.61168	−0.805839	−	0.592135i	$-0.798285\pi$
$761$	−33834.1	−1.61168	−0.805839	+	0.592135i	$0.798285\pi$
$762$	4705.49	0.223703
$763$	0	0
$764$	4288.55	0.203082
$765$	−1883.20	−0.0890027
$766$	−37887.9	−1.78713
$767$	19056.3	0.897109
$768$	40058.2	1.88213
$769$	31738.1	1.48830	0.744151	−	0.668011i	$-0.232854\pi$
$769$	31738.1	1.48830	0.744151	+	0.668011i	$0.232854\pi$
$770$	0	0
$771$	33310.2	1.55595
$772$	−16938.1	−0.789658
$773$	27494.2	1.27930	0.639650	−	0.768667i	$-0.279080\pi$
$773$	27494.2	1.27930	0.639650	+	0.768667i	$0.279080\pi$
$774$	17767.1	0.825099
$775$	36875.6	1.70917
$776$	−20791.0	−0.961794
$777$	0	0
$778$	−18588.0	−0.856573
$779$	19334.2	0.889244
$780$	3257.20	0.149521
$781$	33194.8	1.52087
$782$	5220.41	0.238723
$783$	6826.68	0.311578
$784$	0	0
$785$	−6476.84	−0.294482
$786$	−20769.3	−0.942517
$787$	−468.356	−0.0212136	−0.0106068	−	0.999944i	$-0.503376\pi$
$787$	−468.356	−0.0212136	−0.0106068	+	0.999944i	$0.503376\pi$
$788$	22871.6	1.03397
$789$	32844.0	1.48198
$790$	−3637.18	−0.163804
$791$	0	0
$792$	20966.9	0.940688
$793$	26375.1	1.18109
$794$	−4289.97	−0.191745
$795$	−2293.07	−0.102298
$796$	−3864.93	−0.172096
$797$	37723.8	1.67659	0.838297	−	0.545214i	$-0.183551\pi$
$797$	37723.8	1.67659	0.838297	+	0.545214i	$0.183551\pi$
$798$	0	0
$799$	7155.87	0.316841
$800$	−21960.2	−0.970512
$801$	−36658.1	−1.61704
$802$	−8105.53	−0.356878
$803$	−11681.3	−0.513356
$804$	−6279.36	−0.275443
$805$	0	0
$806$	48397.1	2.11503
$807$	29259.6	1.27632
$808$	961.583	0.0418668
$809$	−7797.13	−0.338854	−0.169427	−	0.985543i	$-0.554192\pi$
$809$	−7797.13	−0.338854	−0.169427	+	0.985543i	$0.554192\pi$
$810$	3545.16	0.153783
$811$	16925.9	0.732860	0.366430	−	0.930446i	$-0.380580\pi$
$811$	16925.9	0.732860	0.366430	+	0.930446i	$0.380580\pi$
$812$	0	0
$813$	34075.8	1.46998
$814$	−13406.2	−0.577256
$815$	3757.36	0.161490
$816$	16548.6	0.709949
$817$	11441.0	0.489925
$818$	−16407.5	−0.701316
$819$	0	0
$820$	−2308.91	−0.0983301
$821$	−30009.3	−1.27568	−0.637840	−	0.770169i	$-0.720172\pi$
$821$	−30009.3	−1.27568	−0.637840	+	0.770169i	$0.720172\pi$
$822$	−14325.0	−0.607835
$823$	−23385.6	−0.990486	−0.495243	−	0.868754i	$-0.664921\pi$
$823$	−23385.6	−0.990486	−0.495243	+	0.868754i	$0.664921\pi$
$824$	−2053.53	−0.0868181
$825$	−46433.3	−1.95952
$826$	0	0
$827$	−37325.9	−1.56947	−0.784734	−	0.619833i	$-0.787200\pi$
$827$	−37325.9	−1.56947	−0.784734	+	0.619833i	$0.787200\pi$
$828$	8517.43	0.357489
$829$	−24671.3	−1.03362	−0.516809	−	0.856100i	$-0.672880\pi$
$829$	−24671.3	−1.03362	−0.516809	+	0.856100i	$0.672880\pi$
$830$	−179.277	−0.00749733
$831$	−10515.4	−0.438960
$832$	−192.599	−0.00802544
$833$	0	0
$834$	−82193.0	−3.41260
$835$	−6538.05	−0.270969
$836$	−17086.9	−0.706892
$837$	−17177.1	−0.709350
$838$	−26629.1	−1.09772
$839$	14147.4	0.582147	0.291074	−	0.956701i	$-0.405987\pi$
$839$	14147.4	0.582147	0.291074	+	0.956701i	$0.405987\pi$
$840$	0	0
$841$	−9642.35	−0.395356
$842$	22047.1	0.902366
$843$	32896.6	1.34403
$844$	−6008.91	−0.245065
$845$	−384.195	−0.0156411
$846$	32575.8	1.32385
$847$	0	0
$848$	11258.6	0.455923
$849$	37171.3	1.50261
$850$	−11300.7	−0.456012
$851$	4303.24	0.173341
$852$	23591.1	0.948611
$853$	27963.6	1.12246	0.561229	−	0.827661i	$-0.310329\pi$
$853$	27963.6	1.12246	0.561229	+	0.827661i	$0.310329\pi$
$854$	0	0
$855$	−5520.44	−0.220813
$856$	12750.9	0.509134
$857$	33855.8	1.34947	0.674734	−	0.738061i	$-0.264259\pi$
$857$	33855.8	1.34947	0.674734	+	0.738061i	$0.264259\pi$
$858$	−60941.1	−2.42482
$859$	−24282.7	−0.964511	−0.482255	−	0.876031i	$-0.660182\pi$
$859$	−24282.7	−0.964511	−0.482255	+	0.876031i	$0.660182\pi$
$860$	−1366.29	−0.0541745
$861$	0	0
$862$	−40487.2	−1.59977
$863$	−19667.0	−0.775750	−0.387875	−	0.921712i	$-0.626791\pi$
$863$	−19667.0	−0.775750	−0.387875	+	0.921712i	$0.626791\pi$
$864$	10229.3	0.402787
$865$	6822.59	0.268179
$866$	−18207.9	−0.714469
$867$	−32929.9	−1.28992
$868$	0	0
$869$	24389.4	0.952075
$870$	−6967.67	−0.271524
$871$	−8057.65	−0.313459
$872$	16989.1	0.659773
$873$	−57004.0	−2.20996
$874$	15303.2	0.592264
$875$	0	0
$876$	−8301.75	−0.320194
$877$	−36061.0	−1.38848	−0.694238	−	0.719745i	$-0.744259\pi$
$877$	−36061.0	−1.38848	−0.694238	+	0.719745i	$0.744259\pi$
$878$	17882.5	0.687364
$879$	−38807.2	−1.48912
$880$	−8151.58	−0.312261
$881$	−15889.7	−0.607646	−0.303823	−	0.952728i	$-0.598263\pi$
$881$	−15889.7	−0.607646	−0.303823	+	0.952728i	$0.598263\pi$
$882$	0	0
$883$	14861.3	0.566390	0.283195	−	0.959062i	$-0.408606\pi$
$883$	14861.3	0.566390	0.283195	+	0.959062i	$0.408606\pi$
$884$	−5315.65	−0.202245
$885$	6903.13	0.262199
$886$	−44858.2	−1.70095
$887$	−38189.9	−1.44565	−0.722824	−	0.691032i	$-0.757156\pi$
$887$	−38189.9	−1.44565	−0.722824	+	0.691032i	$0.757156\pi$
$888$	7528.33	0.284498
$889$	0	0
$890$	7865.45	0.296237
$891$	−23772.3	−0.893831
$892$	3865.20	0.145086
$893$	20976.8	0.786073
$894$	75072.7	2.80851
$895$	5813.99	0.217140
$896$	0	0
$897$	19561.4	0.728135
$898$	−49231.7	−1.82949
$899$	−37105.0	−1.37655
$900$	−18437.8	−0.682881
$901$	3742.22	0.138370
$902$	43198.9	1.59464
$903$	0	0
$904$	−19182.2	−0.705742
$905$	7733.86	0.284069
$906$	−19546.1	−0.716751
$907$	16865.5	0.617431	0.308715	−	0.951154i	$-0.400101\pi$
$907$	16865.5	0.617431	0.308715	+	0.951154i	$0.400101\pi$
$908$	−7665.65	−0.280169
$909$	2636.44	0.0961993
$910$	0	0
$911$	26754.1	0.973000	0.486500	−	0.873681i	$-0.338273\pi$
$911$	26754.1	0.973000	0.486500	+	0.873681i	$0.338273\pi$
$912$	48511.0	1.76136
$913$	1202.15	0.0435766
$914$	53722.7	1.94419
$915$	9554.37	0.345200
$916$	−4672.02	−0.168524
$917$	0	0
$918$	5263.99	0.189257
$919$	−41527.5	−1.49061	−0.745303	−	0.666726i	$-0.767695\pi$
$919$	−41527.5	−1.49061	−0.745303	+	0.666726i	$0.767695\pi$
$920$	1444.04	0.0517484
$921$	−33119.2	−1.18492
$922$	40361.5	1.44169
$923$	30272.0	1.07954
$924$	0	0
$925$	−9315.27	−0.331118
$926$	33008.6	1.17141
$927$	−5630.31	−0.199486
$928$	22096.8	0.781642
$929$	4584.68	0.161914	0.0809572	−	0.996718i	$-0.474202\pi$
$929$	4584.68	0.161914	0.0809572	+	0.996718i	$0.474202\pi$
$930$	17531.8	0.618163
$931$	0	0
$932$	6473.92	0.227532
$933$	−5355.86	−0.187935
$934$	−12821.0	−0.449159
$935$	−2709.48	−0.0947694
$936$	19120.7	0.667715
$937$	−6928.18	−0.241552	−0.120776	−	0.992680i	$-0.538538\pi$
$937$	−6928.18	−0.241552	−0.120776	+	0.992680i	$0.538538\pi$
$938$	0	0
$939$	−46495.7	−1.61590
$940$	−2505.07	−0.0869217
$941$	20944.9	0.725596	0.362798	−	0.931868i	$-0.381822\pi$
$941$	20944.9	0.725596	0.362798	+	0.931868i	$0.381822\pi$
$942$	86120.5	2.97872
$943$	−13866.4	−0.478846
$944$	−33893.3	−1.16857
$945$	0	0
$946$	25562.8	0.878559
$947$	29278.9	1.00468	0.502342	−	0.864669i	$-0.332472\pi$
$947$	29278.9	1.00468	0.502342	+	0.864669i	$0.332472\pi$
$948$	17333.2	0.593836
$949$	−10652.8	−0.364387
$950$	−33127.1	−1.13135
$951$	−22089.0	−0.753192
$952$	0	0
$953$	2136.81	0.0726316	0.0363158	−	0.999340i	$-0.488438\pi$
$953$	2136.81	0.0726316	0.0363158	+	0.999340i	$0.488438\pi$
$954$	17035.8	0.578149
$955$	1993.48	0.0675470
$956$	−14090.9	−0.476707
$957$	46722.3	1.57818
$958$	−23027.4	−0.776597
$959$	0	0
$960$	−69.7689	−0.00234561
$961$	63571.4	2.13391
$962$	−12225.8	−0.409745
$963$	34960.2	1.16986
$964$	−1702.70	−0.0568881
$965$	−7873.47	−0.262649
$966$	0	0
$967$	−3921.32	−0.130405	−0.0652023	−	0.997872i	$-0.520769\pi$
$967$	−3921.32	−0.130405	−0.0652023	+	0.997872i	$0.520769\pi$
$968$	13570.4	0.450586
$969$	16124.4	0.534562
$970$	12230.9	0.404857
$971$	47809.1	1.58009	0.790045	−	0.613049i	$-0.210057\pi$
$971$	47809.1	1.58009	0.790045	+	0.613049i	$0.210057\pi$
$972$	−23677.7	−0.781341
$973$	0	0
$974$	12946.1	0.425894
$975$	−42344.9	−1.39089
$976$	−46910.5	−1.53849
$977$	−51329.1	−1.68082	−0.840411	−	0.541949i	$-0.817686\pi$
$977$	−51329.1	−1.68082	−0.840411	+	0.541949i	$0.817686\pi$
$978$	−49960.5	−1.63350
$979$	−52742.4	−1.72181
$980$	0	0
$981$	46580.1	1.51599
$982$	44035.8	1.43100
$983$	−16326.3	−0.529734	−0.264867	−	0.964285i	$-0.585328\pi$
$983$	−16326.3	−0.529734	−0.264867	+	0.964285i	$0.585328\pi$
$984$	−24258.7	−0.785912
$985$	10631.6	0.343909
$986$	11371.0	0.367268
$987$	0	0
$988$	−15582.4	−0.501763
$989$	−8205.37	−0.263818
$990$	−12334.4	−0.395973
$991$	33770.0	1.08248	0.541242	−	0.840867i	$-0.317954\pi$
$991$	33770.0	1.08248	0.541242	+	0.840867i	$0.317954\pi$
$992$	−55599.3	−1.77952
$993$	−22003.6	−0.703185
$994$	0	0
$995$	−1796.56	−0.0572410
$996$	854.354	0.0271800
$997$	50695.4	1.61037	0.805185	−	0.593023i	$-0.202066\pi$
$997$	50695.4	1.61037	0.805185	+	0.593023i	$0.202066\pi$
$998$	8135.48	0.258040
$999$	4339.16	0.137422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	49.4.a.e.1.2	yes	4
3.2	odd	2		441.4.a.u.1.3		4
4.3	odd	2		784.4.a.bf.1.1		4
5.4	even	2		1225.4.a.bb.1.3		4
7.2	even	3		49.4.c.e.18.3		8
7.3	odd	6		49.4.c.e.30.4		8
7.4	even	3		49.4.c.e.30.3		8
7.5	odd	6		49.4.c.e.18.4		8
7.6	odd	2	inner	49.4.a.e.1.1	✓	4
21.2	odd	6		441.4.e.y.361.2		8
21.5	even	6		441.4.e.y.361.1		8
21.11	odd	6		441.4.e.y.226.2		8
21.17	even	6		441.4.e.y.226.1		8
21.20	even	2		441.4.a.u.1.4		4
28.27	even	2		784.4.a.bf.1.4		4
35.34	odd	2		1225.4.a.bb.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.a.e.1.1	✓	4	7.6	odd	2	inner
49.4.a.e.1.2	yes	4	1.1	even	1	trivial
49.4.c.e.18.3		8	7.2	even	3
49.4.c.e.18.4		8	7.5	odd	6
49.4.c.e.30.3		8	7.4	even	3
49.4.c.e.30.4		8	7.3	odd	6
441.4.a.u.1.3		4	3.2	odd	2
441.4.a.u.1.4		4	21.20	even	2
441.4.e.y.226.1		8	21.17	even	6
441.4.e.y.226.2		8	21.11	odd	6
441.4.e.y.361.1		8	21.5	even	6
441.4.e.y.361.2		8	21.2	odd	6
784.4.a.bf.1.1		4	4.3	odd	2
784.4.a.bf.1.4		4	28.27	even	2
1225.4.a.bb.1.3		4	5.4	even	2
1225.4.a.bb.1.4		4	35.34	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 49 = 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	49.a (trivial)

\(f(q)\)	\(=\)	\(q-3.53113 q^{2} +7.82220 q^{3} +4.46887 q^{4} +2.07730 q^{5} -27.6212 q^{6} +12.4689 q^{8} +34.1868 q^{9} +O(q^{10})\)	\(q-3.53113 q^{2} +7.82220 q^{3} +4.46887 q^{4} +2.07730 q^{5} -27.6212 q^{6} +12.4689 q^{8} +34.1868 q^{9} -7.33521 q^{10} +49.1868 q^{11} +34.9564 q^{12} +44.8559 q^{13} +16.2490 q^{15} -79.7802 q^{16} -26.5179 q^{17} -120.718 q^{18} -77.7350 q^{19} +9.28317 q^{20} -173.685 q^{22} +55.7510 q^{23} +97.5340 q^{24} -120.685 q^{25} -158.392 q^{26} +56.2164 q^{27} +121.436 q^{29} -57.3774 q^{30} -305.553 q^{31} +181.963 q^{32} +384.749 q^{33} +93.6380 q^{34} +152.776 q^{36} +77.1868 q^{37} +274.492 q^{38} +350.872 q^{39} +25.9016 q^{40} -248.720 q^{41} -147.179 q^{43} +219.809 q^{44} +71.0161 q^{45} -196.864 q^{46} -269.851 q^{47} -624.056 q^{48} +426.154 q^{50} -207.428 q^{51} +200.455 q^{52} -141.121 q^{53} -198.507 q^{54} +102.176 q^{55} -608.058 q^{57} -428.805 q^{58} +424.834 q^{59} +72.6148 q^{60} +587.996 q^{61} +1078.95 q^{62} -4.29373 q^{64} +93.1790 q^{65} -1358.60 q^{66} -179.634 q^{67} -118.505 q^{68} +436.095 q^{69} +674.872 q^{71} +426.270 q^{72} -237.489 q^{73} -272.556 q^{74} -944.021 q^{75} -347.388 q^{76} -1238.97 q^{78} +495.852 q^{79} -165.727 q^{80} -483.307 q^{81} +878.262 q^{82} +24.4406 q^{83} -55.0855 q^{85} +519.708 q^{86} +949.895 q^{87} +613.304 q^{88} -1072.29 q^{89} -250.767 q^{90} +249.144 q^{92} -2390.09 q^{93} +952.877 q^{94} -161.479 q^{95} +1423.35 q^{96} -1667.43 q^{97} +1681.54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9}+O(q^{10}) \) 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9	\( 4 q + 2 q^{2} + 34 q^{4} + 66 q^{8} + 40 q^{9} + 100 q^{11} - 64 q^{15} - 174 q^{16} - 370 q^{18} - 340 q^{22} + 352 q^{23} - 128 q^{25} + 260 q^{29} - 552 q^{30} - 30 q^{32} - 50 q^{36} + 212 q^{37} + 952 q^{39} + 540 q^{43} + 460 q^{44} + 696 q^{46} + 1366 q^{50} + 428 q^{51} + 16 q^{53} - 1884 q^{57} - 780 q^{58} - 1064 q^{60} - 1678 q^{64} - 756 q^{65} - 1944 q^{67} + 2248 q^{71} + 270 q^{72} - 284 q^{74} - 1344 q^{78} - 1048 q^{79} - 1256 q^{81} - 3284 q^{85} + 4820 q^{86} + 1260 q^{88} + 3512 q^{92} - 5368 q^{93} + 2192 q^{95} + 3340 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 34 * q^4 + 66 * q^8 + 40 * q^9 + 100 * q^11 - 64 * q^15 - 174 * q^16 - 370 * q^18 - 340 * q^22 + 352 * q^23 - 128 * q^25 + 260 * q^29 - 552 * q^30 - 30 * q^32 - 50 * q^36 + 212 * q^37 + 952 * q^39 + 540 * q^43 + 460 * q^44 + 696 * q^46 + 1366 * q^50 + 428 * q^51 + 16 * q^53 - 1884 * q^57 - 780 * q^58 - 1064 * q^60 - 1678 * q^64 - 756 * q^65 - 1944 * q^67 + 2248 * q^71 + 270 * q^72 - 284 * q^74 - 1344 * q^78 - 1048 * q^79 - 1256 * q^81 - 3284 * q^85 + 4820 * q^86 + 1260 * q^88 + 3512 * q^92 - 5368 * q^93 + 2192 * q^95 + 3340 * q^99

Introduction

L-functions

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