LMFDB - Embedded newform 49.4.a.e.1.1

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			1.1
Root			$-2.11692$ 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	$-0.771245\pi$
$3$	−7.82220	−1.50538	−0.752691	+	0.658374i	$0.771245\pi$
$4$	4.46887	0.558609
$5$	−2.07730	−0.185799	−0.0928996	−	0.995675i	$-0.529614\pi$
$5$	−2.07730	−0.185799	−0.0928996	+	0.995675i	$0.529614\pi$
$6$	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	$-0.658816\pi$
$13$	−44.8559	−0.956983	−0.478492	+	0.878092i	$0.658816\pi$
$14$	0	0
$15$	16.2490	0.279699
$16$	−79.7802	−1.24656
$17$	26.5179	0.378325	0.189163	−	0.981946i	$-0.439423\pi$
$17$	26.5179	0.378325	0.189163	+	0.981946i	$0.439423\pi$
$18$	−120.718	−1.58075
$19$	77.7350	0.938612	0.469306	−	0.883036i	$-0.344504\pi$
$19$	77.7350	0.938612	0.469306	+	0.883036i	$0.344504\pi$
$20$	−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	$-0.154060\pi$
$31$	305.553	1.77029	0.885143	+	0.465319i	$0.154060\pi$
$32$	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	$-0.342917\pi$
$41$	248.720	0.947403	0.473702	+	0.880685i	$0.342917\pi$
$42$	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	$-0.362471\pi$
$47$	269.851	0.837484	0.418742	+	0.908105i	$0.362471\pi$
$48$	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	$-0.655284\pi$
$59$	−424.834	−0.937434	−0.468717	+	0.883348i	$0.655284\pi$
$60$	72.6148	0.156242
$61$	−587.996	−1.23418	−0.617092	−	0.786891i	$-0.711689\pi$
$61$	−587.996	−1.23418	−0.617092	+	0.786891i	$0.711689\pi$
$62$	−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	$-0.439027\pi$
$73$	237.489	0.380767	0.190383	+	0.981710i	$0.439027\pi$
$74$	−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	$-0.505144\pi$
$83$	−24.4406	−0.0323217	−0.0161609	+	0.999869i	$0.505144\pi$
$84$	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	$-0.279534\pi$
$89$	1072.29	1.27710	0.638552	+	0.769579i	$0.279534\pi$
$90$	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	$-0.162374\pi$
$97$	1667.43	1.74538	0.872690	+	0.488275i	$0.162374\pi$
$98$	0	0
$99$	1681.54	1.70708
$100$	−539.325	−0.539325
$101$	−77.1187	−0.0759762	−0.0379881	−	0.999278i	$-0.512095\pi$
$101$	−77.1187	−0.0759762	−0.0379881	+	0.999278i	$0.512095\pi$
$102$	732.455	0.711018
$103$	164.693	0.157550	0.0787749	−	0.996892i	$-0.474899\pi$
$103$	164.693	0.157550	0.0787749	+	0.996892i	$0.474899\pi$
$104$	−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	$-0.580678\pi$
$131$	−751.935	−0.501503	−0.250751	+	0.968051i	$0.580678\pi$
$132$	−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	$-0.862319\pi$
$139$	−2975.72	−1.81581	−0.907905	+	0.419177i	$0.862319\pi$
$140$	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	$-0.208793\pi$
$157$	3117.91	1.58495	0.792473	+	0.609906i	$0.208793\pi$
$158$	−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	$-0.239894\pi$
$167$	3147.38	1.45839	0.729197	+	0.684303i	$0.239894\pi$
$168$	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	$-0.756635\pi$
$173$	−3284.36	−1.44338	−0.721691	+	0.692215i	$0.756635\pi$
$174$	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	$-0.776990\pi$
$181$	−3723.04	−1.52890	−0.764451	+	0.644682i	$0.776990\pi$
$182$	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	$-0.450772\pi$
$199$	864.855	0.308080	0.154040	+	0.988065i	$0.450772\pi$
$200$	−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	$-0.541454\pi$
$223$	−864.916	−0.259727	−0.129863	+	0.991532i	$0.541454\pi$
$224$	0	0
$225$	−4125.83	−1.22247
$226$	5432.32	1.59890
$227$	1715.34	0.501548	0.250774	−	0.968046i	$-0.419315\pi$
$227$	1715.34	0.501548	0.250774	+	0.968046i	$0.419315\pi$
$228$	−2717.33	−0.789298
$229$	1045.46	0.301685	0.150842	−	0.988558i	$-0.451801\pi$
$229$	1045.46	0.301685	0.150842	+	0.988558i	$0.451801\pi$
$230$	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	$-0.483785\pi$
$241$	381.012	0.101839	0.0509194	+	0.998703i	$0.483785\pi$
$242$	−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	$-0.657483\pi$
$251$	−3776.23	−0.949617	−0.474808	+	0.880089i	$0.657483\pi$
$252$	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	$-0.672875\pi$
$257$	−4258.42	−1.03359	−0.516795	+	0.856109i	$0.672875\pi$
$258$	−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	$-0.639345\pi$
$269$	−3740.59	−0.847835	−0.423917	+	0.905701i	$0.639345\pi$
$270$	−412.359	−0.0929457
$271$	−4356.30	−0.976480	−0.488240	−	0.872709i	$-0.662361\pi$
$271$	−4356.30	−0.976480	−0.488240	+	0.872709i	$0.662361\pi$
$272$	−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	$-0.666328\pi$
$283$	−4752.03	−0.998159	−0.499079	+	0.866556i	$0.666328\pi$
$284$	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	$-0.335315\pi$
$293$	4961.17	0.989196	0.494598	+	0.869122i	$0.335315\pi$
$294$	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	$-0.371243\pi$
$307$	4234.00	0.787124	0.393562	+	0.919298i	$0.371243\pi$
$308$	0	0
$309$	−1288.26	−0.237173
$310$	2241.29	0.410635
$311$	684.700	0.124842	0.0624209	−	0.998050i	$-0.480118\pi$
$311$	684.700	0.124842	0.0624209	+	0.998050i	$0.480118\pi$
$312$	4374.97	0.793859
$313$	5944.07	1.07341	0.536707	−	0.843768i	$-0.319668\pi$
$313$	5944.07	1.07341	0.536707	+	0.843768i	$0.319668\pi$
$314$	−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	$-0.494075\pi$
$349$	242.692	0.0372236	0.0186118	+	0.999827i	$0.494075\pi$
$350$	0	0
$351$	2521.63	0.383461
$352$	8950.18	1.35524
$353$	109.990	0.0165840	0.00829201	−	0.999966i	$-0.497361\pi$
$353$	109.990	0.0165840	0.00829201	+	0.999966i	$0.497361\pi$
$354$	−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	$-0.946009\pi$
$367$	−13859.6	−1.97130	−0.985649	+	0.168807i	$0.946009\pi$
$368$	−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	$-0.753913\pi$
$383$	−10729.7	−1.43149	−0.715746	+	0.698361i	$0.753913\pi$
$384$	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	$-0.524468\pi$
$397$	−1214.90	−0.153587	−0.0767935	+	0.997047i	$0.524468\pi$
$398$	−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	$-0.590625\pi$
$409$	−4646.54	−0.561753	−0.280876	+	0.959744i	$0.590625\pi$
$410$	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	$-0.644892\pi$
$419$	−7541.24	−0.879269	−0.439634	+	0.898177i	$0.644892\pi$
$420$	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	$-0.592374\pi$
$433$	−5156.40	−0.572289	−0.286144	+	0.958187i	$0.592374\pi$
$434$	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	$-0.411227\pi$
$439$	5064.25	0.550577	0.275289	+	0.961362i	$0.411227\pi$
$440$	−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	$-0.304070\pi$
$461$	11430.2	1.15479	0.577394	+	0.816465i	$0.304070\pi$
$462$	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	$-0.557573\pi$
$467$	−3630.84	−0.359776	−0.179888	+	0.983687i	$0.557573\pi$
$468$	−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	$-0.600673\pi$
$479$	−6521.25	−0.622053	−0.311027	+	0.950401i	$0.600673\pi$
$480$	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	$-0.345592\pi$
$503$	10520.4	0.932570	0.466285	+	0.884635i	$0.345592\pi$
$504$	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	$-0.361785\pi$
$509$	9662.22	0.841395	0.420698	+	0.907201i	$0.361785\pi$
$510$	−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	$-0.617877\pi$
$521$	−8607.81	−0.723829	−0.361914	+	0.932211i	$0.617877\pi$
$522$	−14659.5	−1.22917
$523$	10482.7	0.876439	0.438219	−	0.898868i	$-0.355609\pi$
$523$	10482.7	0.876439	0.438219	+	0.898868i	$0.355609\pi$
$524$	−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	$-0.278851\pi$
$563$	17104.4	1.28040	0.640201	+	0.768208i	$0.278851\pi$
$564$	−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	$-0.551229\pi$
$577$	−4442.08	−0.320496	−0.160248	+	0.987077i	$0.551229\pi$
$578$	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	$-0.534799\pi$
$587$	−3103.38	−0.218211	−0.109106	+	0.994030i	$0.534799\pi$
$588$	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	$-0.565912\pi$
$593$	−5937.71	−0.411185	−0.205592	+	0.978638i	$0.565912\pi$
$594$	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	$-0.343865\pi$
$601$	13881.4	0.942156	0.471078	+	0.882092i	$0.343865\pi$
$602$	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	$-0.365277\pi$
$607$	12284.6	0.821442	0.410721	+	0.911761i	$0.365277\pi$
$608$	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	$-0.772266\pi$
$619$	−23248.6	−1.50960	−0.754799	+	0.655956i	$0.772266\pi$
$620$	−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	$-0.556694\pi$
$643$	−5777.47	−0.354341	−0.177170	+	0.984180i	$0.556694\pi$
$644$	0	0
$645$	−2391.52	−0.145994
$646$	−7278.95	−0.443323
$647$	29231.5	1.77621	0.888106	−	0.459640i	$-0.152021\pi$
$647$	29231.5	1.77621	0.888106	+	0.459640i	$0.152021\pi$
$648$	−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	$-0.287306\pi$
$661$	21058.4	1.23915	0.619573	+	0.784939i	$0.287306\pi$
$662$	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	$-0.576367\pi$
$677$	−8371.31	−0.475237	−0.237619	+	0.971359i	$0.576367\pi$
$678$	−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	$-0.526151\pi$
$691$	−2981.29	−0.164130	−0.0820648	+	0.996627i	$0.526151\pi$
$692$	−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	$-0.703547\pi$
$719$	−23010.5	−1.19353	−0.596765	+	0.802416i	$0.703547\pi$
$720$	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	$-0.283701\pi$
$727$	24636.8	1.25685	0.628423	+	0.777872i	$0.283701\pi$
$728$	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	$-0.555658\pi$
$733$	−6904.76	−0.347931	−0.173965	+	0.984752i	$0.555658\pi$
$734$	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	$-0.201715\pi$
$761$	33834.1	1.61168	0.805839	+	0.592135i	$0.201715\pi$
$762$	−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	$-0.767146\pi$
$769$	−31738.1	−1.48830	−0.744151	+	0.668011i	$0.767146\pi$
$770$	0	0
$771$	33310.2	1.55595
$772$	−16938.1	−0.789658
$773$	−27494.2	−1.27930	−0.639650	−	0.768667i	$-0.720920\pi$
$773$	−27494.2	−1.27930	−0.639650	+	0.768667i	$0.720920\pi$
$774$	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	$-0.496624\pi$
$787$	468.356	0.0212136	0.0106068	+	0.999944i	$0.496624\pi$
$788$	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	$-0.816449\pi$
$797$	−37723.8	−1.67659	−0.838297	+	0.545214i	$0.816449\pi$
$798$	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	$-0.619420\pi$
$811$	−16925.9	−0.732860	−0.366430	+	0.930446i	$0.619420\pi$
$812$	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	$-0.327120\pi$
$829$	24671.3	1.03362	0.516809	+	0.856100i	$0.327120\pi$
$830$	−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	$-0.594013\pi$
$839$	−14147.4	−0.582147	−0.291074	+	0.956701i	$0.594013\pi$
$840$	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	$-0.689671\pi$
$853$	−27963.6	−1.12246	−0.561229	+	0.827661i	$0.689671\pi$
$854$	0	0
$855$	−5520.44	−0.220813
$856$	12750.9	0.509134
$857$	−33855.8	−1.34947	−0.674734	−	0.738061i	$-0.735741\pi$
$857$	−33855.8	−1.34947	−0.674734	+	0.738061i	$0.735741\pi$
$858$	−60941.1	−2.42482
$859$	24282.7	0.964511	0.482255	−	0.876031i	$-0.339818\pi$
$859$	24282.7	0.964511	0.482255	+	0.876031i	$0.339818\pi$
$860$	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	$-0.401737\pi$
$881$	15889.7	0.607646	0.303823	+	0.952728i	$0.401737\pi$
$882$	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	$-0.242844\pi$
$887$	38189.9	1.44565	0.722824	+	0.691032i	$0.242844\pi$
$888$	−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	$-0.525798\pi$
$929$	−4584.68	−0.161914	−0.0809572	+	0.996718i	$0.525798\pi$
$930$	−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	$-0.461462\pi$
$937$	6928.18	0.241552	0.120776	+	0.992680i	$0.461462\pi$
$938$	0	0
$939$	−46495.7	−1.61590
$940$	−2505.07	−0.0869217
$941$	−20944.9	−0.725596	−0.362798	−	0.931868i	$-0.618178\pi$
$941$	−20944.9	−0.725596	−0.362798	+	0.931868i	$0.618178\pi$
$942$	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	$-0.789943\pi$
$971$	−47809.1	−1.58009	−0.790045	+	0.613049i	$0.789943\pi$
$972$	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	$-0.414672\pi$
$983$	16326.3	0.529734	0.264867	+	0.964285i	$0.414672\pi$
$984$	−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	$-0.797934\pi$
$997$	−50695.4	−1.61037	−0.805185	+	0.593023i	$0.797934\pi$
$998$	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.1.1	✓	4
3.2	odd	2		441.4.a.u.1.4		4
4.3	odd	2		784.4.a.bf.1.4		4
5.4	even	2		1225.4.a.bb.1.4		4
7.2	even	3		49.4.c.e.18.4		8
7.3	odd	6		49.4.c.e.30.3		8
7.4	even	3		49.4.c.e.30.4		8
7.5	odd	6		49.4.c.e.18.3		8
7.6	odd	2	inner	49.4.a.e.1.2	yes	4
21.2	odd	6		441.4.e.y.361.1		8
21.5	even	6		441.4.e.y.361.2		8
21.11	odd	6		441.4.e.y.226.1		8
21.17	even	6		441.4.e.y.226.2		8
21.20	even	2		441.4.a.u.1.3		4
28.27	even	2		784.4.a.bf.1.1		4
35.34	odd	2		1225.4.a.bb.1.3		4

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