LMFDB - Embedded newform 99.4.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [99,4,Mod(1,99)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(99, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("99.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 99 = 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	99.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:	$5.84118909057$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	99.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.46410 q^{2} +11.9282 q^{4} -6.53590 q^{5} +9.32051 q^{7} -17.5359 q^{8} +O(q^{10})$	\(q-4.46410 q^{2} +11.9282 q^{4} -6.53590 q^{5} +9.32051 q^{7} -17.5359 q^{8} +29.1769 q^{10} +11.0000 q^{11} +1.89488 q^{13} -41.6077 q^{14} -17.1436 q^{16} -61.2154 q^{17} -121.923 q^{19} -77.9615 q^{20} -49.1051 q^{22} -117.177 q^{23} -82.2820 q^{25} -8.45895 q^{26} +111.177 q^{28} +139.990 q^{29} -314.067 q^{31} +216.818 q^{32} +273.272 q^{34} -60.9179 q^{35} +88.6410 q^{37} +544.277 q^{38} +114.613 q^{40} -92.1436 q^{41} +396.697 q^{43} +131.210 q^{44} +523.090 q^{46} +82.7461 q^{47} -256.128 q^{49} +367.315 q^{50} +22.6025 q^{52} -581.787 q^{53} -71.8949 q^{55} -163.443 q^{56} -624.928 q^{58} -697.338 q^{59} +184.095 q^{61} +1402.03 q^{62} -830.749 q^{64} -12.3848 q^{65} -219.405 q^{67} -730.190 q^{68} +271.944 q^{70} +886.628 q^{71} +338.708 q^{73} -395.703 q^{74} -1454.32 q^{76} +102.526 q^{77} +208.536 q^{79} +112.049 q^{80} +411.338 q^{82} -463.923 q^{83} +400.098 q^{85} -1770.90 q^{86} -192.895 q^{88} -1137.68 q^{89} +17.6613 q^{91} -1397.71 q^{92} -369.387 q^{94} +796.877 q^{95} +1522.49 q^{97} +1143.38 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 10 q^{4} - 20 q^{5} - 16 q^{7} - 42 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 10 * q^4 - 20 * q^5 - 16 * q^7 - 42 * q^8	$ 2 q - 2 q^{2} + 10 q^{4} - 20 q^{5} - 16 q^{7} - 42 q^{8} - 4 q^{10} + 22 q^{11} + 80 q^{13} - 104 q^{14} - 62 q^{16} - 164 q^{17} - 36 q^{19} - 52 q^{20} - 22 q^{22} - 172 q^{23} - 26 q^{25} + 184 q^{26} + 160 q^{28} - 108 q^{29} - 448 q^{31} + 302 q^{32} + 20 q^{34} + 280 q^{35} + 108 q^{37} + 756 q^{38} + 444 q^{40} - 212 q^{41} + 156 q^{43} + 110 q^{44} + 388 q^{46} + 20 q^{47} + 42 q^{49} + 506 q^{50} - 128 q^{52} + 132 q^{53} - 220 q^{55} + 456 q^{56} - 1236 q^{58} - 688 q^{59} - 96 q^{61} + 1072 q^{62} - 262 q^{64} - 1064 q^{65} + 448 q^{67} - 532 q^{68} + 1112 q^{70} - 132 q^{71} + 428 q^{73} - 348 q^{74} - 1620 q^{76} - 176 q^{77} + 424 q^{79} + 716 q^{80} + 116 q^{82} - 720 q^{83} + 1784 q^{85} - 2364 q^{86} - 462 q^{88} - 1056 q^{89} - 1960 q^{91} - 1292 q^{92} - 524 q^{94} - 360 q^{95} + 52 q^{97} + 1878 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 10 * q^4 - 20 * q^5 - 16 * q^7 - 42 * q^8 - 4 * q^10 + 22 * q^11 + 80 * q^13 - 104 * q^14 - 62 * q^16 - 164 * q^17 - 36 * q^19 - 52 * q^20 - 22 * q^22 - 172 * q^23 - 26 * q^25 + 184 * q^26 + 160 * q^28 - 108 * q^29 - 448 * q^31 + 302 * q^32 + 20 * q^34 + 280 * q^35 + 108 * q^37 + 756 * q^38 + 444 * q^40 - 212 * q^41 + 156 * q^43 + 110 * q^44 + 388 * q^46 + 20 * q^47 + 42 * q^49 + 506 * q^50 - 128 * q^52 + 132 * q^53 - 220 * q^55 + 456 * q^56 - 1236 * q^58 - 688 * q^59 - 96 * q^61 + 1072 * q^62 - 262 * q^64 - 1064 * q^65 + 448 * q^67 - 532 * q^68 + 1112 * q^70 - 132 * q^71 + 428 * q^73 - 348 * q^74 - 1620 * q^76 - 176 * q^77 + 424 * q^79 + 716 * q^80 + 116 * q^82 - 720 * q^83 + 1784 * q^85 - 2364 * q^86 - 462 * q^88 - 1056 * q^89 - 1960 * q^91 - 1292 * q^92 - 524 * q^94 - 360 * q^95 + 52 * q^97 + 1878 * q^98

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 $

$2$	−4.46410	−1.57830	−0.789149	−	0.614202i	$-0.789478\pi$
$2$	−4.46410	−1.57830	−0.789149	+	0.614202i	$0.789478\pi$
$3$	0	0
$3$	0	0
$4$	11.9282	1.49103
$5$	−6.53590	−0.584589	−0.292294	−	0.956328i	$-0.594419\pi$
$5$	−6.53590	−0.584589	−0.292294	+	0.956328i	$0.594419\pi$
$6$	0	0
$7$	9.32051	0.503260	0.251630	−	0.967823i	$-0.419033\pi$
$7$	9.32051	0.503260	0.251630	+	0.967823i	$0.419033\pi$
$8$	−17.5359	−0.774985
$9$	0	0
$10$	29.1769	0.922655
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	1.89488	0.0404266	0.0202133	−	0.999796i	$-0.493565\pi$
$13$	1.89488	0.0404266	0.0202133	+	0.999796i	$0.493565\pi$
$14$	−41.6077	−0.794295
$15$	0	0
$16$	−17.1436	−0.267869
$17$	−61.2154	−0.873348	−0.436674	−	0.899620i	$-0.643844\pi$
$17$	−61.2154	−0.873348	−0.436674	+	0.899620i	$0.643844\pi$
$18$	0	0
$19$	−121.923	−1.47216	−0.736081	−	0.676894i	$-0.763326\pi$
$19$	−121.923	−1.47216	−0.736081	+	0.676894i	$0.763326\pi$
$20$	−77.9615	−0.871636
$21$	0	0
$22$	−49.1051	−0.475875
$23$	−117.177	−1.06231	−0.531154	−	0.847275i	$-0.678241\pi$
$23$	−117.177	−1.06231	−0.531154	+	0.847275i	$0.678241\pi$
$24$	0	0
$25$	−82.2820	−0.658256
$26$	−8.45895	−0.0638052
$27$	0	0
$28$	111.177	0.750374
$29$	139.990	0.896394	0.448197	−	0.893935i	$-0.352066\pi$
$29$	139.990	0.896394	0.448197	+	0.893935i	$0.352066\pi$
$30$	0	0
$31$	−314.067	−1.81961	−0.909807	−	0.415032i	$-0.863771\pi$
$31$	−314.067	−1.81961	−0.909807	+	0.415032i	$0.863771\pi$
$32$	216.818	1.19776
$33$	0	0
$34$	273.272	1.37840
$35$	−60.9179	−0.294200
$36$	0	0
$37$	88.6410	0.393851	0.196926	−	0.980418i	$-0.436904\pi$
$37$	88.6410	0.393851	0.196926	+	0.980418i	$0.436904\pi$
$38$	544.277	2.32351
$39$	0	0
$40$	114.613	0.453047
$41$	−92.1436	−0.350986	−0.175493	−	0.984481i	$-0.556152\pi$
$41$	−92.1436	−0.350986	−0.175493	+	0.984481i	$0.556152\pi$
$42$	0	0
$43$	396.697	1.40688	0.703439	−	0.710755i	$-0.251647\pi$
$43$	396.697	1.40688	0.703439	+	0.710755i	$0.251647\pi$
$44$	131.210	0.449561
$45$	0	0
$46$	523.090	1.67664
$47$	82.7461	0.256803	0.128402	−	0.991722i	$-0.459015\pi$
$47$	82.7461	0.256803	0.128402	+	0.991722i	$0.459015\pi$
$48$	0	0
$49$	−256.128	−0.746729
$50$	367.315	1.03892
$51$	0	0
$52$	22.6025	0.0602771
$53$	−581.787	−1.50782	−0.753911	−	0.656976i	$-0.771835\pi$
$53$	−581.787	−1.50782	−0.753911	+	0.656976i	$0.771835\pi$
$54$	0	0
$55$	−71.8949	−0.176260
$56$	−163.443	−0.390019
$57$	0	0
$58$	−624.928	−1.41478
$59$	−697.338	−1.53874	−0.769371	−	0.638803i	$-0.779430\pi$
$59$	−697.338	−1.53874	−0.769371	+	0.638803i	$0.779430\pi$
$60$	0	0
$61$	184.095	0.386409	0.193204	−	0.981159i	$-0.438112\pi$
$61$	184.095	0.386409	0.193204	+	0.981159i	$0.438112\pi$
$62$	1402.03	2.87189
$63$	0	0
$64$	−830.749	−1.62256
$65$	−12.3848	−0.0236329
$66$	0	0
$67$	−219.405	−0.400068	−0.200034	−	0.979789i	$-0.564105\pi$
$67$	−219.405	−0.400068	−0.200034	+	0.979789i	$0.564105\pi$
$68$	−730.190	−1.30218
$69$	0	0
$70$	271.944	0.464336
$71$	886.628	1.48202	0.741010	−	0.671494i	$-0.234347\pi$
$71$	886.628	1.48202	0.741010	+	0.671494i	$0.234347\pi$
$72$	0	0
$73$	338.708	0.543051	0.271526	−	0.962431i	$-0.412472\pi$
$73$	338.708	0.543051	0.271526	+	0.962431i	$0.412472\pi$
$74$	−395.703	−0.621615
$75$	0	0
$76$	−1454.32	−2.19503
$77$	102.526	0.151739
$78$	0	0
$79$	208.536	0.296989	0.148494	−	0.988913i	$-0.452557\pi$
$79$	208.536	0.296989	0.148494	+	0.988913i	$0.452557\pi$
$80$	112.049	0.156593
$81$	0	0
$82$	411.338	0.553960
$83$	−463.923	−0.613520	−0.306760	−	0.951787i	$-0.599245\pi$
$83$	−463.923	−0.613520	−0.306760	+	0.951787i	$0.599245\pi$
$84$	0	0
$85$	400.098	0.510549
$86$	−1770.90	−2.22047
$87$	0	0
$88$	−192.895	−0.233667
$89$	−1137.68	−1.35499	−0.677495	−	0.735528i	$-0.736934\pi$
$89$	−1137.68	−1.35499	−0.677495	+	0.735528i	$0.736934\pi$
$90$	0	0
$91$	17.6613	0.0203451
$92$	−1397.71	−1.58393
$93$	0	0
$94$	−369.387	−0.405313
$95$	796.877	0.860609
$96$	0	0
$97$	1522.49	1.59367	0.796833	−	0.604199i	$-0.206507\pi$
$97$	1522.49	1.59367	0.796833	+	0.604199i	$0.206507\pi$
$98$	1143.38	1.17856
$99$	0	0
$100$	−981.477	−0.981477
$101$	−1819.34	−1.79239	−0.896193	−	0.443665i	$-0.853678\pi$
$101$	−1819.34	−1.79239	−0.896193	+	0.443665i	$0.853678\pi$
$102$	0	0
$103$	1142.24	1.09270	0.546349	−	0.837558i	$-0.316017\pi$
$103$	1142.24	1.09270	0.546349	+	0.837558i	$0.316017\pi$
$104$	−33.2285	−0.0313300
$105$	0	0
$106$	2597.16	2.37979
$107$	1056.59	0.954625	0.477313	−	0.878734i	$-0.341611\pi$
$107$	1056.59	0.954625	0.477313	+	0.878734i	$0.341611\pi$
$108$	0	0
$109$	−1382.68	−1.21502	−0.607508	−	0.794314i	$-0.707831\pi$
$109$	−1382.68	−1.21502	−0.607508	+	0.794314i	$0.707831\pi$
$110$	320.946	0.278191
$111$	0	0
$112$	−159.787	−0.134808
$113$	263.405	0.219284	0.109642	−	0.993971i	$-0.465030\pi$
$113$	263.405	0.219284	0.109642	+	0.993971i	$0.465030\pi$
$114$	0	0
$115$	765.856	0.621013
$116$	1669.83	1.33655
$117$	0	0
$118$	3112.99	2.42859
$119$	−570.559	−0.439521
$120$	0	0
$121$	121.000	0.0909091
$122$	−821.818	−0.609868
$123$	0	0
$124$	−3746.25	−2.71309
$125$	1354.77	0.969398
$126$	0	0
$127$	−1217.68	−0.850800	−0.425400	−	0.905006i	$-0.639867\pi$
$127$	−1217.68	−0.850800	−0.425400	+	0.905006i	$0.639867\pi$
$128$	1974.00	1.36312
$129$	0	0
$130$	55.2868	0.0372998
$131$	925.626	0.617346	0.308673	−	0.951168i	$-0.400115\pi$
$131$	925.626	0.617346	0.308673	+	0.951168i	$0.400115\pi$
$132$	0	0
$133$	−1136.38	−0.740880
$134$	979.446	0.631427
$135$	0	0
$136$	1073.47	0.676831
$137$	1116.32	0.696160	0.348080	−	0.937465i	$-0.386834\pi$
$137$	1116.32	0.696160	0.348080	+	0.937465i	$0.386834\pi$
$138$	0	0
$139$	−576.846	−0.351996	−0.175998	−	0.984391i	$-0.556315\pi$
$139$	−576.846	−0.351996	−0.175998	+	0.984391i	$0.556315\pi$
$140$	−726.641	−0.438660
$141$	0	0
$142$	−3958.00	−2.33907
$143$	20.8437	0.0121891
$144$	0	0
$145$	−914.958	−0.524022
$146$	−1512.03	−0.857097
$147$	0	0
$148$	1057.33	0.587242
$149$	−1630.36	−0.896404	−0.448202	−	0.893932i	$-0.647935\pi$
$149$	−1630.36	−0.896404	−0.448202	+	0.893932i	$0.647935\pi$
$150$	0	0
$151$	1348.67	0.726842	0.363421	−	0.931625i	$-0.381609\pi$
$151$	1348.67	0.726842	0.363421	+	0.931625i	$0.381609\pi$
$152$	2138.03	1.14090
$153$	0	0
$154$	−457.685	−0.239489
$155$	2052.71	1.06373
$156$	0	0
$157$	−1357.54	−0.690088	−0.345044	−	0.938587i	$-0.612136\pi$
$157$	−1357.54	−0.690088	−0.345044	+	0.938587i	$0.612136\pi$
$158$	−930.925	−0.468737
$159$	0	0
$160$	−1417.10	−0.700197
$161$	−1092.15	−0.534617
$162$	0	0
$163$	−817.507	−0.392835	−0.196418	−	0.980520i	$-0.562931\pi$
$163$	−817.507	−0.392835	−0.196418	+	0.980520i	$0.562931\pi$
$164$	−1099.11	−0.523329
$165$	0	0
$166$	2071.00	0.968318
$167$	1665.68	0.771822	0.385911	−	0.922536i	$-0.373887\pi$
$167$	1665.68	0.771822	0.385911	+	0.922536i	$0.373887\pi$
$168$	0	0
$169$	−2193.41	−0.998366
$170$	−1786.08	−0.805799
$171$	0	0
$172$	4731.89	2.09769
$173$	731.907	0.321652	0.160826	−	0.986983i	$-0.448584\pi$
$173$	731.907	0.321652	0.160826	+	0.986983i	$0.448584\pi$
$174$	0	0
$175$	−766.910	−0.331274
$176$	−188.580	−0.0807654
$177$	0	0
$178$	5078.73	2.13858
$179$	4272.97	1.78423	0.892114	−	0.451811i	$-0.149222\pi$
$179$	4272.97	1.78423	0.892114	+	0.451811i	$0.149222\pi$
$180$	0	0
$181$	1049.04	0.430797	0.215398	−	0.976526i	$-0.430895\pi$
$181$	1049.04	0.430797	0.215398	+	0.976526i	$0.430895\pi$
$182$	−78.8417	−0.0321106
$183$	0	0
$184$	2054.80	0.823272
$185$	−579.349	−0.230241
$186$	0	0
$187$	−673.369	−0.263324
$188$	987.013	0.382901
$189$	0	0
$190$	−3557.34	−1.35830
$191$	2369.05	0.897480	0.448740	−	0.893662i	$-0.351873\pi$
$191$	2369.05	0.897480	0.448740	+	0.893662i	$0.351873\pi$
$192$	0	0
$193$	4519.52	1.68561	0.842804	−	0.538221i	$-0.180903\pi$
$193$	4519.52	1.68561	0.842804	+	0.538221i	$0.180903\pi$
$194$	−6796.56	−2.51528
$195$	0	0
$196$	−3055.15	−1.11339
$197$	−3624.62	−1.31088	−0.655440	−	0.755247i	$-0.727517\pi$
$197$	−3624.62	−1.31088	−0.655440	+	0.755247i	$0.727517\pi$
$198$	0	0
$199$	−1015.00	−0.361565	−0.180782	−	0.983523i	$-0.557863\pi$
$199$	−1015.00	−0.361565	−0.180782	+	0.983523i	$0.557863\pi$
$200$	1442.89	0.510138
$201$	0	0
$202$	8121.71	2.82892
$203$	1304.78	0.451119
$204$	0	0
$205$	602.241	0.205182
$206$	−5099.06	−1.72460
$207$	0	0
$208$	−32.4851	−0.0108290
$209$	−1341.15	−0.443873
$210$	0	0
$211$	610.912	0.199322	0.0996610	−	0.995021i	$-0.468224\pi$
$211$	610.912	0.199322	0.0996610	+	0.995021i	$0.468224\pi$
$212$	−6939.67	−2.24820
$213$	0	0
$214$	−4716.75	−1.50668
$215$	−2592.77	−0.822445
$216$	0	0
$217$	−2927.26	−0.915739
$218$	6172.42	1.91766
$219$	0	0
$220$	−857.577	−0.262808
$221$	−115.996	−0.0353065
$222$	0	0
$223$	−444.565	−0.133499	−0.0667495	−	0.997770i	$-0.521263\pi$
$223$	−444.565	−0.133499	−0.0667495	+	0.997770i	$0.521263\pi$
$224$	2020.85	0.602785
$225$	0	0
$226$	−1175.87	−0.346095
$227$	−5816.84	−1.70078	−0.850390	−	0.526152i	$-0.823634\pi$
$227$	−5816.84	−1.70078	−0.850390	+	0.526152i	$0.823634\pi$
$228$	0	0
$229$	−5815.83	−1.67826	−0.839129	−	0.543933i	$-0.816935\pi$
$229$	−5815.83	−1.67826	−0.839129	+	0.543933i	$0.816935\pi$
$230$	−3418.86	−0.980144
$231$	0	0
$232$	−2454.84	−0.694692
$233$	2408.26	0.677125	0.338562	−	0.940944i	$-0.390059\pi$
$233$	2408.26	0.677125	0.338562	+	0.940944i	$0.390059\pi$
$234$	0	0
$235$	−540.820	−0.150124
$236$	−8317.99	−2.29430
$237$	0	0
$238$	2547.03	0.693695
$239$	6298.09	1.70456	0.852280	−	0.523086i	$-0.175220\pi$
$239$	6298.09	1.70456	0.852280	+	0.523086i	$0.175220\pi$
$240$	0	0
$241$	4040.05	1.07984	0.539922	−	0.841715i	$-0.318454\pi$
$241$	4040.05	1.07984	0.539922	+	0.841715i	$0.318454\pi$
$242$	−540.156	−0.143482
$243$	0	0
$244$	2195.92	0.576145
$245$	1674.03	0.436529
$246$	0	0
$247$	−231.030	−0.0595145
$248$	5507.44	1.41017
$249$	0	0
$250$	−6047.85	−1.53000
$251$	2394.38	0.602120	0.301060	−	0.953605i	$-0.402660\pi$
$251$	2394.38	0.602120	0.301060	+	0.953605i	$0.402660\pi$
$252$	0	0
$253$	−1288.95	−0.320298
$254$	5435.84	1.34282
$255$	0	0
$256$	−2166.16	−0.528847
$257$	2181.73	0.529544	0.264772	−	0.964311i	$-0.414703\pi$
$257$	2181.73	0.529544	0.264772	+	0.964311i	$0.414703\pi$
$258$	0	0
$259$	826.179	0.198210
$260$	−147.728	−0.0352373
$261$	0	0
$262$	−4132.09	−0.974356
$263$	3883.01	0.910406	0.455203	−	0.890388i	$-0.349567\pi$
$263$	3883.01	0.910406	0.455203	+	0.890388i	$0.349567\pi$
$264$	0	0
$265$	3802.50	0.881456
$266$	5072.94	1.16933
$267$	0	0
$268$	−2617.11	−0.596512
$269$	−3612.57	−0.818818	−0.409409	−	0.912351i	$-0.634265\pi$
$269$	−3612.57	−0.818818	−0.409409	+	0.912351i	$0.634265\pi$
$270$	0	0
$271$	−716.290	−0.160559	−0.0802795	−	0.996772i	$-0.525581\pi$
$271$	−716.290	−0.160559	−0.0802795	+	0.996772i	$0.525581\pi$
$272$	1049.45	0.233943
$273$	0	0
$274$	−4983.38	−1.09875
$275$	−905.102	−0.198472
$276$	0	0
$277$	6995.36	1.51737	0.758683	−	0.651460i	$-0.225843\pi$
$277$	6995.36	1.51737	0.758683	+	0.651460i	$0.225843\pi$
$278$	2575.10	0.555554
$279$	0	0
$280$	1068.25	0.228001
$281$	−8787.00	−1.86544	−0.932720	−	0.360603i	$-0.882571\pi$
$281$	−8787.00	−1.86544	−0.932720	+	0.360603i	$0.882571\pi$
$282$	0	0
$283$	3572.22	0.750342	0.375171	−	0.926956i	$-0.377584\pi$
$283$	3572.22	0.750342	0.375171	+	0.926956i	$0.377584\pi$
$284$	10575.9	2.20973
$285$	0	0
$286$	−93.0484	−0.0192380
$287$	−858.825	−0.176637
$288$	0	0
$289$	−1165.68	−0.237264
$290$	4084.47	0.827063
$291$	0	0
$292$	4040.17	0.809703
$293$	−4840.00	−0.965038	−0.482519	−	0.875886i	$-0.660278\pi$
$293$	−4840.00	−0.965038	−0.482519	+	0.875886i	$0.660278\pi$
$294$	0	0
$295$	4557.73	0.899530
$296$	−1554.40	−0.305229
$297$	0	0
$298$	7278.09	1.41479
$299$	−222.036	−0.0429455
$300$	0	0
$301$	3697.42	0.708026
$302$	−6020.60	−1.14717
$303$	0	0
$304$	2090.20	0.394346
$305$	−1203.22	−0.225890
$306$	0	0
$307$	4917.94	0.914273	0.457136	−	0.889397i	$-0.348875\pi$
$307$	4917.94	0.914273	0.457136	+	0.889397i	$0.348875\pi$
$308$	1222.95	0.226246
$309$	0	0
$310$	−9163.50	−1.67888
$311$	4875.93	0.889031	0.444516	−	0.895771i	$-0.353376\pi$
$311$	4875.93	0.889031	0.444516	+	0.895771i	$0.353376\pi$
$312$	0	0
$313$	4283.96	0.773622	0.386811	−	0.922159i	$-0.373577\pi$
$313$	4283.96	0.773622	0.386811	+	0.922159i	$0.373577\pi$
$314$	6060.21	1.08916
$315$	0	0
$316$	2487.46	0.442818
$317$	−8879.08	−1.57318	−0.786592	−	0.617474i	$-0.788156\pi$
$317$	−8879.08	−1.57318	−0.786592	+	0.617474i	$0.788156\pi$
$318$	0	0
$319$	1539.89	0.270273
$320$	5429.69	0.948527
$321$	0	0
$322$	4875.46	0.843785
$323$	7463.57	1.28571
$324$	0	0
$325$	−155.915	−0.0266111
$326$	3649.44	0.620011
$327$	0	0
$328$	1615.82	0.272008
$329$	771.236	0.129239
$330$	0	0
$331$	−7084.95	−1.17651	−0.588254	−	0.808676i	$-0.700185\pi$
$331$	−7084.95	−1.17651	−0.588254	+	0.808676i	$0.700185\pi$
$332$	−5533.77	−0.914774
$333$	0	0
$334$	−7435.77	−1.21817
$335$	1434.01	0.233875
$336$	0	0
$337$	−11557.2	−1.86813	−0.934066	−	0.357099i	$-0.883766\pi$
$337$	−11557.2	−1.86813	−0.934066	+	0.357099i	$0.883766\pi$
$338$	9791.60	1.57572
$339$	0	0
$340$	4772.45	0.761242
$341$	−3454.73	−0.548634
$342$	0	0
$343$	−5584.18	−0.879059
$344$	−6956.44	−1.09031
$345$	0	0
$346$	−3267.31	−0.507664
$347$	−3384.55	−0.523608	−0.261804	−	0.965121i	$-0.584317\pi$
$347$	−3384.55	−0.523608	−0.261804	+	0.965121i	$0.584317\pi$
$348$	0	0
$349$	8069.34	1.23766	0.618828	−	0.785527i	$-0.287608\pi$
$349$	8069.34	1.23766	0.618828	+	0.785527i	$0.287608\pi$
$350$	3423.57	0.522849
$351$	0	0
$352$	2385.00	0.361139
$353$	498.421	0.0751509	0.0375755	−	0.999294i	$-0.488037\pi$
$353$	498.421	0.0751509	0.0375755	+	0.999294i	$0.488037\pi$
$354$	0	0
$355$	−5794.91	−0.866372
$356$	−13570.5	−2.02032
$357$	0	0
$358$	−19075.0	−2.81604
$359$	−488.919	−0.0718779	−0.0359390	−	0.999354i	$-0.511442\pi$
$359$	−488.919	−0.0718779	−0.0359390	+	0.999354i	$0.511442\pi$
$360$	0	0
$361$	8006.23	1.16726
$362$	−4683.00	−0.679926
$363$	0	0
$364$	210.667	0.0303351
$365$	−2213.76	−0.317461
$366$	0	0
$367$	−4204.61	−0.598034	−0.299017	−	0.954248i	$-0.596659\pi$
$367$	−4204.61	−0.598034	−0.299017	+	0.954248i	$0.596659\pi$
$368$	2008.83	0.284559
$369$	0	0
$370$	2586.27	0.363389
$371$	−5422.55	−0.758827
$372$	0	0
$373$	−8499.45	−1.17985	−0.589926	−	0.807457i	$-0.700843\pi$
$373$	−8499.45	−1.17985	−0.589926	+	0.807457i	$0.700843\pi$
$374$	3005.99	0.415604
$375$	0	0
$376$	−1451.03	−0.199019
$377$	265.264	0.0362382
$378$	0	0
$379$	−3902.19	−0.528871	−0.264436	−	0.964403i	$-0.585186\pi$
$379$	−3902.19	−0.528871	−0.264436	+	0.964403i	$0.585186\pi$
$380$	9505.31	1.28319
$381$	0	0
$382$	−10575.7	−1.41649
$383$	−11681.3	−1.55845	−0.779226	−	0.626743i	$-0.784387\pi$
$383$	−11681.3	−1.55845	−0.779226	+	0.626743i	$0.784387\pi$
$384$	0	0
$385$	−670.097	−0.0887047
$386$	−20175.6	−2.66039
$387$	0	0
$388$	18160.6	2.37620
$389$	−8914.04	−1.16185	−0.580925	−	0.813957i	$-0.697309\pi$
$389$	−8914.04	−1.16185	−0.580925	+	0.813957i	$0.697309\pi$
$390$	0	0
$391$	7173.03	0.927764
$392$	4491.44	0.578704
$393$	0	0
$394$	16180.7	2.06896
$395$	−1362.97	−0.173616
$396$	0	0
$397$	−13445.5	−1.69977	−0.849885	−	0.526967i	$-0.823329\pi$
$397$	−13445.5	−1.69977	−0.849885	+	0.526967i	$0.823329\pi$
$398$	4531.06	0.570657
$399$	0	0
$400$	1410.61	0.176326
$401$	11815.1	1.47137	0.735683	−	0.677326i	$-0.236861\pi$
$401$	11815.1	1.47137	0.735683	+	0.677326i	$0.236861\pi$
$402$	0	0
$403$	−595.119	−0.0735608
$404$	−21701.4	−2.67249
$405$	0	0
$406$	−5824.65	−0.712001
$407$	975.051	0.118751
$408$	0	0
$409$	−14699.5	−1.77713	−0.888564	−	0.458752i	$-0.848297\pi$
$409$	−14699.5	−1.77713	−0.888564	+	0.458752i	$0.848297\pi$
$410$	−2688.47	−0.323839
$411$	0	0
$412$	13624.8	1.62924
$413$	−6499.55	−0.774387
$414$	0	0
$415$	3032.15	0.358657
$416$	410.844	0.0484214
$417$	0	0
$418$	5987.05	0.700565
$419$	6376.77	0.743498	0.371749	−	0.928333i	$-0.378758\pi$
$419$	6376.77	0.743498	0.371749	+	0.928333i	$0.378758\pi$
$420$	0	0
$421$	6745.79	0.780925	0.390463	−	0.920619i	$-0.372315\pi$
$421$	6745.79	0.780925	0.390463	+	0.920619i	$0.372315\pi$
$422$	−2727.17	−0.314590
$423$	0	0
$424$	10202.2	1.16854
$425$	5036.93	0.574887
$426$	0	0
$427$	1715.86	0.194464
$428$	12603.3	1.42337
$429$	0	0
$430$	11574.4	1.29806
$431$	−13592.8	−1.51913	−0.759563	−	0.650434i	$-0.774587\pi$
$431$	−13592.8	−1.51913	−0.759563	+	0.650434i	$0.774587\pi$
$432$	0	0
$433$	−2511.11	−0.278698	−0.139349	−	0.990243i	$-0.544501\pi$
$433$	−2511.11	−0.278698	−0.139349	+	0.990243i	$0.544501\pi$
$434$	13067.6	1.44531
$435$	0	0
$436$	−16492.9	−1.81162
$437$	14286.6	1.56389
$438$	0	0
$439$	9721.66	1.05692	0.528462	−	0.848957i	$-0.322769\pi$
$439$	9721.66	1.05692	0.528462	+	0.848957i	$0.322769\pi$
$440$	1260.74	0.136599
$441$	0	0
$442$	517.818	0.0557242
$443$	11323.6	1.21445	0.607226	−	0.794529i	$-0.292282\pi$
$443$	11323.6	1.21445	0.607226	+	0.794529i	$0.292282\pi$
$444$	0	0
$445$	7435.77	0.792111
$446$	1984.58	0.210701
$447$	0	0
$448$	−7743.00	−0.816568
$449$	−10422.4	−1.09546	−0.547732	−	0.836654i	$-0.684509\pi$
$449$	−10422.4	−1.09546	−0.547732	+	0.836654i	$0.684509\pi$
$450$	0	0
$451$	−1013.58	−0.105826
$452$	3141.95	0.326958
$453$	0	0
$454$	25967.0	2.68434
$455$	−115.432	−0.0118935
$456$	0	0
$457$	−6168.96	−0.631448	−0.315724	−	0.948851i	$-0.602247\pi$
$457$	−6168.96	−0.631448	−0.315724	+	0.948851i	$0.602247\pi$
$458$	25962.5	2.64879
$459$	0	0
$460$	9135.29	0.925946
$461$	2860.00	0.288945	0.144472	−	0.989509i	$-0.453851\pi$
$461$	2860.00	0.288945	0.144472	+	0.989509i	$0.453851\pi$
$462$	0	0
$463$	−10508.4	−1.05478	−0.527392	−	0.849622i	$-0.676830\pi$
$463$	−10508.4	−1.05478	−0.527392	+	0.849622i	$0.676830\pi$
$464$	−2399.93	−0.240116
$465$	0	0
$466$	−10750.7	−1.06871
$467$	9373.42	0.928802	0.464401	−	0.885625i	$-0.346270\pi$
$467$	9373.42	0.928802	0.464401	+	0.885625i	$0.346270\pi$
$468$	0	0
$469$	−2044.97	−0.201339
$470$	2414.28	0.236941
$471$	0	0
$472$	12228.5	1.19250
$473$	4363.67	0.424190
$474$	0	0
$475$	10032.1	0.969059
$476$	−6805.74	−0.655337
$477$	0	0
$478$	−28115.3	−2.69030
$479$	−556.775	−0.0531100	−0.0265550	−	0.999647i	$-0.508454\pi$
$479$	−556.775	−0.0531100	−0.0265550	+	0.999647i	$0.508454\pi$
$480$	0	0
$481$	167.964	0.0159221
$482$	−18035.2	−1.70431
$483$	0	0
$484$	1443.31	0.135548
$485$	−9950.85	−0.931639
$486$	0	0
$487$	−6462.85	−0.601355	−0.300677	−	0.953726i	$-0.597213\pi$
$487$	−6462.85	−0.601355	−0.300677	+	0.953726i	$0.597213\pi$
$488$	−3228.27	−0.299461
$489$	0	0
$490$	−7473.03	−0.688974
$491$	3106.60	0.285538	0.142769	−	0.989756i	$-0.454399\pi$
$491$	3106.60	0.285538	0.142769	+	0.989756i	$0.454399\pi$
$492$	0	0
$493$	−8569.52	−0.782864
$494$	1031.34	0.0939316
$495$	0	0
$496$	5384.23	0.487418
$497$	8263.82	0.745841
$498$	0	0
$499$	223.024	0.0200079	0.0100039	−	0.999950i	$-0.496816\pi$
$499$	223.024	0.0200079	0.0100039	+	0.999950i	$0.496816\pi$
$500$	16160.0	1.44540
$501$	0	0
$502$	−10688.8	−0.950325
$503$	−12042.2	−1.06747	−0.533734	−	0.845653i	$-0.679212\pi$
$503$	−12042.2	−1.06747	−0.533734	+	0.845653i	$0.679212\pi$
$504$	0	0
$505$	11891.0	1.04781
$506$	5753.99	0.505525
$507$	0	0
$508$	−14524.7	−1.26856
$509$	−5229.03	−0.455349	−0.227674	−	0.973737i	$-0.573112\pi$
$509$	−5229.03	−0.455349	−0.227674	+	0.973737i	$0.573112\pi$
$510$	0	0
$511$	3156.93	0.273296
$512$	−6122.06	−0.528437
$513$	0	0
$514$	−9739.48	−0.835778
$515$	−7465.54	−0.638778
$516$	0	0
$517$	910.207	0.0774292
$518$	−3688.15	−0.312834
$519$	0	0
$520$	217.178	0.0183152
$521$	21307.2	1.79171	0.895857	−	0.444342i	$-0.146562\pi$
$521$	21307.2	1.79171	0.895857	+	0.444342i	$0.146562\pi$
$522$	0	0
$523$	12533.6	1.04791	0.523956	−	0.851746i	$-0.324456\pi$
$523$	12533.6	1.04791	0.523956	+	0.851746i	$0.324456\pi$
$524$	11041.0	0.920478
$525$	0	0
$526$	−17334.2	−1.43689
$527$	19225.7	1.58916
$528$	0	0
$529$	1563.43	0.128498
$530$	−16974.7	−1.39120
$531$	0	0
$532$	−13555.0	−1.10467
$533$	−174.601	−0.0141892
$534$	0	0
$535$	−6905.80	−0.558063
$536$	3847.46	0.310047
$537$	0	0
$538$	16126.9	1.29234
$539$	−2817.41	−0.225147
$540$	0	0
$541$	1194.10	0.0948952	0.0474476	−	0.998874i	$-0.484891\pi$
$541$	1194.10	0.0948952	0.0474476	+	0.998874i	$0.484891\pi$
$542$	3197.59	0.253410
$543$	0	0
$544$	−13272.6	−1.04606
$545$	9037.05	0.710284
$546$	0	0
$547$	−17077.5	−1.33488	−0.667441	−	0.744662i	$-0.732611\pi$
$547$	−17077.5	−1.33488	−0.667441	+	0.744662i	$0.732611\pi$
$548$	13315.7	1.03799
$549$	0	0
$550$	4040.47	0.313248
$551$	−17068.0	−1.31964
$552$	0	0
$553$	1943.66	0.149463
$554$	−31228.0	−2.39485
$555$	0	0
$556$	−6880.73	−0.524835
$557$	14523.5	1.10481	0.552407	−	0.833574i	$-0.313709\pi$
$557$	14523.5	1.10481	0.552407	+	0.833574i	$0.313709\pi$
$558$	0	0
$559$	751.695	0.0568753
$560$	1044.35	0.0788070
$561$	0	0
$562$	39226.1	2.94422
$563$	5179.82	0.387751	0.193875	−	0.981026i	$-0.437894\pi$
$563$	5179.82	0.387751	0.193875	+	0.981026i	$0.437894\pi$
$564$	0	0
$565$	−1721.59	−0.128191
$566$	−15946.8	−1.18426
$567$	0	0
$568$	−15547.8	−1.14854
$569$	10532.1	0.775971	0.387985	−	0.921666i	$-0.373171\pi$
$569$	10532.1	0.775971	0.387985	+	0.921666i	$0.373171\pi$
$570$	0	0
$571$	16946.5	1.24201	0.621007	−	0.783805i	$-0.286724\pi$
$571$	16946.5	1.24201	0.621007	+	0.783805i	$0.286724\pi$
$572$	248.628	0.0181742
$573$	0	0
$574$	3833.88	0.278786
$575$	9641.55	0.699271
$576$	0	0
$577$	13375.8	0.965060	0.482530	−	0.875879i	$-0.339718\pi$
$577$	13375.8	0.965060	0.482530	+	0.875879i	$0.339718\pi$
$578$	5203.70	0.374473
$579$	0	0
$580$	−10913.8	−0.781330
$581$	−4324.00	−0.308760
$582$	0	0
$583$	−6399.66	−0.454626
$584$	−5939.54	−0.420856
$585$	0	0
$586$	21606.3	1.52312
$587$	−22817.6	−1.60440	−0.802201	−	0.597054i	$-0.796338\pi$
$587$	−22817.6	−1.60440	−0.802201	+	0.597054i	$0.796338\pi$
$588$	0	0
$589$	38292.0	2.67877
$590$	−20346.2	−1.41973
$591$	0	0
$592$	−1519.63	−0.105500
$593$	1325.54	0.0917934	0.0458967	−	0.998946i	$-0.485385\pi$
$593$	1325.54	0.0917934	0.0458967	+	0.998946i	$0.485385\pi$
$594$	0	0
$595$	3729.11	0.256939
$596$	−19447.2	−1.33656
$597$	0	0
$598$	991.193	0.0677808
$599$	−1757.35	−0.119872	−0.0599361	−	0.998202i	$-0.519090\pi$
$599$	−1757.35	−0.119872	−0.0599361	+	0.998202i	$0.519090\pi$
$600$	0	0
$601$	−12632.4	−0.857382	−0.428691	−	0.903451i	$-0.641025\pi$
$601$	−12632.4	−0.857382	−0.428691	+	0.903451i	$0.641025\pi$
$602$	−16505.7	−1.11748
$603$	0	0
$604$	16087.2	1.08374
$605$	−790.844	−0.0531444
$606$	0	0
$607$	13468.3	0.900598	0.450299	−	0.892878i	$-0.351317\pi$
$607$	13468.3	0.900598	0.450299	+	0.892878i	$0.351317\pi$
$608$	−26435.1	−1.76330
$609$	0	0
$610$	5371.32	0.356522
$611$	156.794	0.0103817
$612$	0	0
$613$	2901.94	0.191205	0.0956023	−	0.995420i	$-0.469522\pi$
$613$	2901.94	0.191205	0.0956023	+	0.995420i	$0.469522\pi$
$614$	−21954.2	−1.44300
$615$	0	0
$616$	−1797.88	−0.117595
$617$	22969.7	1.49874	0.749372	−	0.662149i	$-0.230355\pi$
$617$	22969.7	1.49874	0.749372	+	0.662149i	$0.230355\pi$
$618$	0	0
$619$	2859.62	0.185683	0.0928416	−	0.995681i	$-0.470405\pi$
$619$	2859.62	0.185683	0.0928416	+	0.995681i	$0.470405\pi$
$620$	24485.1	1.58604
$621$	0	0
$622$	−21766.6	−1.40316
$623$	−10603.8	−0.681912
$624$	0	0
$625$	1430.59	0.0915576
$626$	−19124.0	−1.22101
$627$	0	0
$628$	−16193.1	−1.02894
$629$	−5426.19	−0.343969
$630$	0	0
$631$	−990.450	−0.0624869	−0.0312434	−	0.999512i	$-0.509947\pi$
$631$	−990.450	−0.0624869	−0.0312434	+	0.999512i	$0.509947\pi$
$632$	−3656.86	−0.230162
$633$	0	0
$634$	39637.1	2.48295
$635$	7958.63	0.497368
$636$	0	0
$637$	−485.333	−0.0301877
$638$	−6874.21	−0.426571
$639$	0	0
$640$	−12901.9	−0.796862
$641$	2721.01	0.167665	0.0838327	−	0.996480i	$-0.473284\pi$
$641$	2721.01	0.167665	0.0838327	+	0.996480i	$0.473284\pi$
$642$	0	0
$643$	1014.22	0.0622033	0.0311017	−	0.999516i	$-0.490098\pi$
$643$	1014.22	0.0622033	0.0311017	+	0.999516i	$0.490098\pi$
$644$	−13027.4	−0.797128
$645$	0	0
$646$	−33318.1	−2.02923
$647$	25988.5	1.57916	0.789578	−	0.613650i	$-0.210299\pi$
$647$	25988.5	1.57916	0.789578	+	0.613650i	$0.210299\pi$
$648$	0	0
$649$	−7670.72	−0.463948
$650$	696.019	0.0420002
$651$	0	0
$652$	−9751.39	−0.585727
$653$	−1127.80	−0.0675870	−0.0337935	−	0.999429i	$-0.510759\pi$
$653$	−1127.80	−0.0675870	−0.0337935	+	0.999429i	$0.510759\pi$
$654$	0	0
$655$	−6049.79	−0.360893
$656$	1579.67	0.0940181
$657$	0	0
$658$	−3442.88	−0.203978
$659$	−28673.0	−1.69490	−0.847452	−	0.530873i	$-0.821864\pi$
$659$	−28673.0	−1.69490	−0.847452	+	0.530873i	$0.821864\pi$
$660$	0	0
$661$	−14066.2	−0.827705	−0.413852	−	0.910344i	$-0.635817\pi$
$661$	−14066.2	−0.827705	−0.413852	+	0.910344i	$0.635817\pi$
$662$	31628.0	1.85688
$663$	0	0
$664$	8135.31	0.475469
$665$	7427.30	0.433110
$666$	0	0
$667$	−16403.6	−0.952246
$668$	19868.6	1.15081
$669$	0	0
$670$	−6401.56	−0.369125
$671$	2025.04	0.116507
$672$	0	0
$673$	−11713.0	−0.670879	−0.335439	−	0.942062i	$-0.608885\pi$
$673$	−11713.0	−0.670879	−0.335439	+	0.942062i	$0.608885\pi$
$674$	51592.5	2.94847
$675$	0	0
$676$	−26163.4	−1.48859
$677$	3194.15	0.181331	0.0906655	−	0.995881i	$-0.471101\pi$
$677$	3194.15	0.181331	0.0906655	+	0.995881i	$0.471101\pi$
$678$	0	0
$679$	14190.4	0.802029
$680$	−7016.07	−0.395668
$681$	0	0
$682$	15422.3	0.865909
$683$	−1138.53	−0.0637845	−0.0318923	−	0.999491i	$-0.510153\pi$
$683$	−1138.53	−0.0637845	−0.0318923	+	0.999491i	$0.510153\pi$
$684$	0	0
$685$	−7296.17	−0.406967
$686$	24928.3	1.38742
$687$	0	0
$688$	−6800.82	−0.376859
$689$	−1102.42	−0.0609561
$690$	0	0
$691$	4538.68	0.249869	0.124934	−	0.992165i	$-0.460128\pi$
$691$	4538.68	0.249869	0.124934	+	0.992165i	$0.460128\pi$
$692$	8730.34	0.479592
$693$	0	0
$694$	15109.0	0.826410
$695$	3770.21	0.205773
$696$	0	0
$697$	5640.61	0.306533
$698$	−36022.4	−1.95339
$699$	0	0
$700$	−9147.86	−0.493938
$701$	−18226.5	−0.982036	−0.491018	−	0.871150i	$-0.663375\pi$
$701$	−18226.5	−0.982036	−0.491018	+	0.871150i	$0.663375\pi$
$702$	0	0
$703$	−10807.4	−0.579812
$704$	−9138.23	−0.489219
$705$	0	0
$706$	−2225.00	−0.118611
$707$	−16957.2	−0.902036
$708$	0	0
$709$	−9626.62	−0.509923	−0.254962	−	0.966951i	$-0.582063\pi$
$709$	−9626.62	−0.509923	−0.254962	+	0.966951i	$0.582063\pi$
$710$	25869.1	1.36739
$711$	0	0
$712$	19950.3	1.05010
$713$	36801.4	1.93299
$714$	0	0
$715$	−136.232	−0.00712560
$716$	50968.8	2.66033
$717$	0	0
$718$	2182.59	0.113445
$719$	217.065	0.0112589	0.00562947	−	0.999984i	$-0.498208\pi$
$719$	217.065	0.0112589	0.00562947	+	0.999984i	$0.498208\pi$
$720$	0	0
$721$	10646.2	0.549911
$722$	−35740.6	−1.84228
$723$	0	0
$724$	12513.1	0.642329
$725$	−11518.6	−0.590057
$726$	0	0
$727$	−28648.2	−1.46149	−0.730746	−	0.682650i	$-0.760828\pi$
$727$	−28648.2	−1.46149	−0.730746	+	0.682650i	$0.760828\pi$
$728$	−309.706	−0.0157671
$729$	0	0
$730$	9882.44	0.501049
$731$	−24284.0	−1.22869
$732$	0	0
$733$	−5760.02	−0.290247	−0.145123	−	0.989414i	$-0.546358\pi$
$733$	−5760.02	−0.290247	−0.145123	+	0.989414i	$0.546358\pi$
$734$	18769.8	0.943877
$735$	0	0
$736$	−25406.1	−1.27239
$737$	−2413.46	−0.120625
$738$	0	0
$739$	−17937.2	−0.892870	−0.446435	−	0.894816i	$-0.647307\pi$
$739$	−17937.2	−0.892870	−0.446435	+	0.894816i	$0.647307\pi$
$740$	−6910.59	−0.343295
$741$	0	0
$742$	24206.8	1.19766
$743$	−20217.9	−0.998279	−0.499140	−	0.866522i	$-0.666351\pi$
$743$	−20217.9	−0.998279	−0.499140	+	0.866522i	$0.666351\pi$
$744$	0	0
$745$	10655.9	0.524027
$746$	37942.4	1.86216
$747$	0	0
$748$	−8032.09	−0.392623
$749$	9848.00	0.480425
$750$	0	0
$751$	6649.29	0.323084	0.161542	−	0.986866i	$-0.448353\pi$
$751$	6649.29	0.323084	0.161542	+	0.986866i	$0.448353\pi$
$752$	−1418.57	−0.0687896
$753$	0	0
$754$	−1184.17	−0.0571946
$755$	−8814.76	−0.424903
$756$	0	0
$757$	9493.38	0.455803	0.227901	−	0.973684i	$-0.426814\pi$
$757$	9493.38	0.455803	0.227901	+	0.973684i	$0.426814\pi$
$758$	17419.8	0.834717
$759$	0	0
$760$	−13973.9	−0.666958
$761$	18781.3	0.894642	0.447321	−	0.894374i	$-0.352378\pi$
$761$	18781.3	0.894642	0.447321	+	0.894374i	$0.352378\pi$
$762$	0	0
$763$	−12887.3	−0.611469
$764$	28258.5	1.33817
$765$	0	0
$766$	52146.5	2.45970
$767$	−1321.37	−0.0622061
$768$	0	0
$769$	−11023.8	−0.516942	−0.258471	−	0.966019i	$-0.583219\pi$
$769$	−11023.8	−0.516942	−0.258471	+	0.966019i	$0.583219\pi$
$770$	2991.38	0.140002
$771$	0	0
$772$	53909.8	2.51328
$773$	−14027.8	−0.652710	−0.326355	−	0.945247i	$-0.605820\pi$
$773$	−14027.8	−0.652710	−0.326355	+	0.945247i	$0.605820\pi$
$774$	0	0
$775$	25842.0	1.19777
$776$	−26698.3	−1.23507
$777$	0	0
$778$	39793.2	1.83375
$779$	11234.4	0.516708
$780$	0	0
$781$	9752.91	0.446846
$782$	−32021.1	−1.46429
$783$	0	0
$784$	4390.96	0.200025
$785$	8872.77	0.403417
$786$	0	0
$787$	−30940.0	−1.40139	−0.700693	−	0.713463i	$-0.747126\pi$
$787$	−30940.0	−1.40139	−0.700693	+	0.713463i	$0.747126\pi$
$788$	−43235.2	−1.95456
$789$	0	0
$790$	6084.43	0.274018
$791$	2455.07	0.110357
$792$	0	0
$793$	348.838	0.0156212
$794$	60022.0	2.68275
$795$	0	0
$796$	−12107.1	−0.539102
$797$	−33897.3	−1.50653	−0.753265	−	0.657717i	$-0.771522\pi$
$797$	−33897.3	−1.50653	−0.753265	+	0.657717i	$0.771522\pi$
$798$	0	0
$799$	−5065.34	−0.224279
$800$	−17840.2	−0.788434
$801$	0	0
$802$	−52743.8	−2.32226
$803$	3725.78	0.163736
$804$	0	0
$805$	7138.17	0.312531
$806$	2656.67	0.116101
$807$	0	0
$808$	31903.7	1.38907
$809$	34155.0	1.48433	0.742166	−	0.670216i	$-0.233798\pi$
$809$	34155.0	1.48433	0.742166	+	0.670216i	$0.233798\pi$
$810$	0	0
$811$	42765.4	1.85166	0.925830	−	0.377941i	$-0.123368\pi$
$811$	42765.4	1.85166	0.925830	+	0.377941i	$0.123368\pi$
$812$	15563.6	0.672631
$813$	0	0
$814$	−4352.73	−0.187424
$815$	5343.15	0.229647
$816$	0	0
$817$	−48366.6	−2.07115
$818$	65620.3	2.80484
$819$	0	0
$820$	7183.66	0.305932
$821$	−15670.6	−0.666148	−0.333074	−	0.942901i	$-0.608086\pi$
$821$	−15670.6	−0.666148	−0.333074	+	0.942901i	$0.608086\pi$
$822$	0	0
$823$	16111.0	0.682373	0.341186	−	0.939996i	$-0.389171\pi$
$823$	16111.0	0.682373	0.341186	+	0.939996i	$0.389171\pi$
$824$	−20030.1	−0.846824
$825$	0	0
$826$	29014.6	1.22221
$827$	−12505.1	−0.525810	−0.262905	−	0.964822i	$-0.584681\pi$
$827$	−12505.1	−0.525810	−0.262905	+	0.964822i	$0.584681\pi$
$828$	0	0
$829$	−5981.63	−0.250604	−0.125302	−	0.992119i	$-0.539990\pi$
$829$	−5981.63	−0.250604	−0.125302	+	0.992119i	$0.539990\pi$
$830$	−13535.8	−0.566067
$831$	0	0
$832$	−1574.17	−0.0655944
$833$	15679.0	0.652154
$834$	0	0
$835$	−10886.7	−0.451198
$836$	−15997.6	−0.661826
$837$	0	0
$838$	−28466.6	−1.17346
$839$	−42174.9	−1.73545	−0.867724	−	0.497047i	$-0.834418\pi$
$839$	−42174.9	−1.73545	−0.867724	+	0.497047i	$0.834418\pi$
$840$	0	0
$841$	−4791.89	−0.196477
$842$	−30113.9	−1.23253
$843$	0	0
$844$	7287.09	0.297194
$845$	14335.9	0.583633
$846$	0	0
$847$	1127.78	0.0457509
$848$	9973.92	0.403898
$849$	0	0
$850$	−22485.4	−0.907343
$851$	−10386.7	−0.418391
$852$	0	0
$853$	20426.0	0.819898	0.409949	−	0.912108i	$-0.365546\pi$
$853$	20426.0	0.819898	0.409949	+	0.912108i	$0.365546\pi$
$854$	−7659.76	−0.306922
$855$	0	0
$856$	−18528.3	−0.739820
$857$	−4325.90	−0.172427	−0.0862136	−	0.996277i	$-0.527477\pi$
$857$	−4325.90	−0.172427	−0.0862136	+	0.996277i	$0.527477\pi$
$858$	0	0
$859$	−37355.7	−1.48377	−0.741887	−	0.670525i	$-0.766069\pi$
$859$	−37355.7	−1.48377	−0.741887	+	0.670525i	$0.766069\pi$
$860$	−30927.1	−1.22629
$861$	0	0
$862$	60679.7	2.39763
$863$	−38444.8	−1.51643	−0.758214	−	0.652006i	$-0.773928\pi$
$863$	−38444.8	−1.51643	−0.758214	+	0.652006i	$0.773928\pi$
$864$	0	0
$865$	−4783.67	−0.188034
$866$	11209.8	0.439868
$867$	0	0
$868$	−34917.0	−1.36539
$869$	2293.89	0.0895455
$870$	0	0
$871$	−415.747	−0.0161734
$872$	24246.5	0.941618
$873$	0	0
$874$	−63776.7	−2.46828
$875$	12627.2	0.487859
$876$	0	0
$877$	−21521.3	−0.828645	−0.414323	−	0.910130i	$-0.635982\pi$
$877$	−21521.3	−0.828645	−0.414323	+	0.910130i	$0.635982\pi$
$878$	−43398.5	−1.66814
$879$	0	0
$880$	1232.54	0.0472145
$881$	−24114.4	−0.922173	−0.461086	−	0.887355i	$-0.652540\pi$
$881$	−24114.4	−0.922173	−0.461086	+	0.887355i	$0.652540\pi$
$882$	0	0
$883$	5203.36	0.198309	0.0991546	−	0.995072i	$-0.468386\pi$
$883$	5203.36	0.198309	0.0991546	+	0.995072i	$0.468386\pi$
$884$	−1383.62	−0.0526429
$885$	0	0
$886$	−50549.9	−1.91677
$887$	−25749.8	−0.974738	−0.487369	−	0.873196i	$-0.662043\pi$
$887$	−25749.8	−0.974738	−0.487369	+	0.873196i	$0.662043\pi$
$888$	0	0
$889$	−11349.4	−0.428174
$890$	−33194.0	−1.25019
$891$	0	0
$892$	−5302.86	−0.199050
$893$	−10088.7	−0.378056
$894$	0	0
$895$	−27927.7	−1.04304
$896$	18398.7	0.686002
$897$	0	0
$898$	46526.6	1.72897
$899$	−43966.1	−1.63109
$900$	0	0
$901$	35614.3	1.31685
$902$	4524.72	0.167025
$903$	0	0
$904$	−4619.04	−0.169941
$905$	−6856.39	−0.251839
$906$	0	0
$907$	41969.7	1.53647	0.768237	−	0.640166i	$-0.221134\pi$
$907$	41969.7	1.53647	0.768237	+	0.640166i	$0.221134\pi$
$908$	−69384.4	−2.53591
$909$	0	0
$910$	515.301	0.0187715
$911$	5980.10	0.217486	0.108743	−	0.994070i	$-0.465317\pi$
$911$	5980.10	0.217486	0.108743	+	0.994070i	$0.465317\pi$
$912$	0	0
$913$	−5103.15	−0.184983
$914$	27538.9	0.996613
$915$	0	0
$916$	−69372.4	−2.50232
$917$	8627.30	0.310685
$918$	0	0
$919$	8614.03	0.309195	0.154598	−	0.987978i	$-0.450592\pi$
$919$	8614.03	0.309195	0.154598	+	0.987978i	$0.450592\pi$
$920$	−13430.0	−0.481275
$921$	0	0
$922$	−12767.3	−0.456041
$923$	1680.06	0.0599130
$924$	0	0
$925$	−7293.56	−0.259255
$926$	46910.4	1.66476
$927$	0	0
$928$	30352.3	1.07367
$929$	29940.6	1.05740	0.528698	−	0.848810i	$-0.322681\pi$
$929$	29940.6	1.05740	0.528698	+	0.848810i	$0.322681\pi$
$930$	0	0
$931$	31227.9	1.09931
$932$	28726.2	1.00961
$933$	0	0
$934$	−41843.9	−1.46593
$935$	4401.07	0.153936
$936$	0	0
$937$	7361.97	0.256676	0.128338	−	0.991731i	$-0.459036\pi$
$937$	7361.97	0.256676	0.128338	+	0.991731i	$0.459036\pi$
$938$	9128.94	0.317772
$939$	0	0
$940$	−6451.01	−0.223839
$941$	13364.9	0.463001	0.231501	−	0.972835i	$-0.425636\pi$
$941$	13364.9	0.463001	0.231501	+	0.972835i	$0.425636\pi$
$942$	0	0
$943$	10797.1	0.372855
$944$	11954.9	0.412180
$945$	0	0
$946$	−19479.9	−0.669498
$947$	11892.5	0.408081	0.204041	−	0.978962i	$-0.434593\pi$
$947$	11892.5	0.408081	0.204041	+	0.978962i	$0.434593\pi$
$948$	0	0
$949$	641.811	0.0219537
$950$	−44784.2	−1.52946
$951$	0	0
$952$	10005.3	0.340622
$953$	39943.0	1.35769	0.678846	−	0.734280i	$-0.262480\pi$
$953$	39943.0	1.35769	0.678846	+	0.734280i	$0.262480\pi$
$954$	0	0
$955$	−15483.9	−0.524656
$956$	75124.9	2.54154
$957$	0	0
$958$	2485.50	0.0838235
$959$	10404.7	0.350350
$960$	0	0
$961$	68846.9	2.31100
$962$	−749.810	−0.0251298
$963$	0	0
$964$	48190.5	1.61007
$965$	−29539.1	−0.985387
$966$	0	0
$967$	51655.2	1.71781	0.858903	−	0.512138i	$-0.171146\pi$
$967$	51655.2	1.71781	0.858903	+	0.512138i	$0.171146\pi$
$968$	−2121.84	−0.0704531
$969$	0	0
$970$	44421.6	1.47040
$971$	11420.4	0.377445	0.188722	−	0.982030i	$-0.439565\pi$
$971$	11420.4	0.377445	0.188722	+	0.982030i	$0.439565\pi$
$972$	0	0
$973$	−5376.50	−0.177145
$974$	28850.8	0.949117
$975$	0	0
$976$	−3156.05	−0.103507
$977$	31161.4	1.02041	0.510205	−	0.860053i	$-0.329570\pi$
$977$	31161.4	1.02041	0.510205	+	0.860053i	$0.329570\pi$
$978$	0	0
$979$	−12514.5	−0.408545
$980$	19968.1	0.650876
$981$	0	0
$982$	−13868.2	−0.450664
$983$	−23732.3	−0.770032	−0.385016	−	0.922910i	$-0.625804\pi$
$983$	−23732.3	−0.770032	−0.385016	+	0.922910i	$0.625804\pi$
$984$	0	0
$985$	23690.2	0.766326
$986$	38255.2	1.23559
$987$	0	0
$988$	−2755.77	−0.0887376
$989$	−46483.8	−1.49454
$990$	0	0
$991$	25272.0	0.810081	0.405041	−	0.914299i	$-0.367257\pi$
$991$	25272.0	0.810081	0.405041	+	0.914299i	$0.367257\pi$
$992$	−68095.3	−2.17946
$993$	0	0
$994$	−36890.5	−1.17716
$995$	6633.93	0.211367
$996$	0	0
$997$	−31125.7	−0.988726	−0.494363	−	0.869256i	$-0.664599\pi$
$997$	−31125.7	−0.988726	−0.494363	+	0.869256i	$0.664599\pi$
$998$	−995.603	−0.0315784
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	99.4.a.d.1.1	✓	2
3.2	odd	2		99.4.a.g.1.2	yes	2
4.3	odd	2		1584.4.a.w.1.2		2
5.4	even	2		2475.4.a.r.1.2		2
11.10	odd	2		1089.4.a.w.1.2		2
12.11	even	2		1584.4.a.bk.1.1		2
15.14	odd	2		2475.4.a.m.1.1		2
33.32	even	2		1089.4.a.l.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.4.a.d.1.1	✓	2	1.1	even	1	trivial
99.4.a.g.1.2	yes	2	3.2	odd	2
1089.4.a.l.1.1		2	33.32	even	2
1089.4.a.w.1.2		2	11.10	odd	2
1584.4.a.w.1.2		2	4.3	odd	2
1584.4.a.bk.1.1		2	12.11	even	2
2475.4.a.m.1.1		2	15.14	odd	2
2475.4.a.r.1.2		2	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:	\( N \)	\(=\)	\( 99 = 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	99.a (trivial)

\(f(q)\)	\(=\)	\(q-4.46410 q^{2} +11.9282 q^{4} -6.53590 q^{5} +9.32051 q^{7} -17.5359 q^{8} +O(q^{10})\)	\(q-4.46410 q^{2} +11.9282 q^{4} -6.53590 q^{5} +9.32051 q^{7} -17.5359 q^{8} +29.1769 q^{10} +11.0000 q^{11} +1.89488 q^{13} -41.6077 q^{14} -17.1436 q^{16} -61.2154 q^{17} -121.923 q^{19} -77.9615 q^{20} -49.1051 q^{22} -117.177 q^{23} -82.2820 q^{25} -8.45895 q^{26} +111.177 q^{28} +139.990 q^{29} -314.067 q^{31} +216.818 q^{32} +273.272 q^{34} -60.9179 q^{35} +88.6410 q^{37} +544.277 q^{38} +114.613 q^{40} -92.1436 q^{41} +396.697 q^{43} +131.210 q^{44} +523.090 q^{46} +82.7461 q^{47} -256.128 q^{49} +367.315 q^{50} +22.6025 q^{52} -581.787 q^{53} -71.8949 q^{55} -163.443 q^{56} -624.928 q^{58} -697.338 q^{59} +184.095 q^{61} +1402.03 q^{62} -830.749 q^{64} -12.3848 q^{65} -219.405 q^{67} -730.190 q^{68} +271.944 q^{70} +886.628 q^{71} +338.708 q^{73} -395.703 q^{74} -1454.32 q^{76} +102.526 q^{77} +208.536 q^{79} +112.049 q^{80} +411.338 q^{82} -463.923 q^{83} +400.098 q^{85} -1770.90 q^{86} -192.895 q^{88} -1137.68 q^{89} +17.6613 q^{91} -1397.71 q^{92} -369.387 q^{94} +796.877 q^{95} +1522.49 q^{97} +1143.38 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 10 q^{4} - 20 q^{5} - 16 q^{7} - 42 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 10 * q^4 - 20 * q^5 - 16 * q^7 - 42 * q^8	\( 2 q - 2 q^{2} + 10 q^{4} - 20 q^{5} - 16 q^{7} - 42 q^{8} - 4 q^{10} + 22 q^{11} + 80 q^{13} - 104 q^{14} - 62 q^{16} - 164 q^{17} - 36 q^{19} - 52 q^{20} - 22 q^{22} - 172 q^{23} - 26 q^{25} + 184 q^{26} + 160 q^{28} - 108 q^{29} - 448 q^{31} + 302 q^{32} + 20 q^{34} + 280 q^{35} + 108 q^{37} + 756 q^{38} + 444 q^{40} - 212 q^{41} + 156 q^{43} + 110 q^{44} + 388 q^{46} + 20 q^{47} + 42 q^{49} + 506 q^{50} - 128 q^{52} + 132 q^{53} - 220 q^{55} + 456 q^{56} - 1236 q^{58} - 688 q^{59} - 96 q^{61} + 1072 q^{62} - 262 q^{64} - 1064 q^{65} + 448 q^{67} - 532 q^{68} + 1112 q^{70} - 132 q^{71} + 428 q^{73} - 348 q^{74} - 1620 q^{76} - 176 q^{77} + 424 q^{79} + 716 q^{80} + 116 q^{82} - 720 q^{83} + 1784 q^{85} - 2364 q^{86} - 462 q^{88} - 1056 q^{89} - 1960 q^{91} - 1292 q^{92} - 524 q^{94} - 360 q^{95} + 52 q^{97} + 1878 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 10 * q^4 - 20 * q^5 - 16 * q^7 - 42 * q^8 - 4 * q^10 + 22 * q^11 + 80 * q^13 - 104 * q^14 - 62 * q^16 - 164 * q^17 - 36 * q^19 - 52 * q^20 - 22 * q^22 - 172 * q^23 - 26 * q^25 + 184 * q^26 + 160 * q^28 - 108 * q^29 - 448 * q^31 + 302 * q^32 + 20 * q^34 + 280 * q^35 + 108 * q^37 + 756 * q^38 + 444 * q^40 - 212 * q^41 + 156 * q^43 + 110 * q^44 + 388 * q^46 + 20 * q^47 + 42 * q^49 + 506 * q^50 - 128 * q^52 + 132 * q^53 - 220 * q^55 + 456 * q^56 - 1236 * q^58 - 688 * q^59 - 96 * q^61 + 1072 * q^62 - 262 * q^64 - 1064 * q^65 + 448 * q^67 - 532 * q^68 + 1112 * q^70 - 132 * q^71 + 428 * q^73 - 348 * q^74 - 1620 * q^76 - 176 * q^77 + 424 * q^79 + 716 * q^80 + 116 * q^82 - 720 * q^83 + 1784 * q^85 - 2364 * q^86 - 462 * q^88 - 1056 * q^89 - 1960 * q^91 - 1292 * q^92 - 524 * q^94 - 360 * q^95 + 52 * q^97 + 1878 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more