LMFDB - Embedded newform 966.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [966,4,Mod(1,966)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(966, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("966.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 966 = 2 \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	966.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:	$56.9958450655$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 267x^{3} + 1502x^{2} + 1857x + 198 $ x^5 - 2x^4 - 267x^3 + 1502x^2 + 1857x + 198
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$12.7990$ of defining polynomial
Character	$\chi$	$=$	966.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -8.30883 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -8.30883 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +16.6177 q^{10} -14.5035 q^{11} +12.0000 q^{12} -22.1543 q^{13} +14.0000 q^{14} -24.9265 q^{15} +16.0000 q^{16} -72.5456 q^{17} -18.0000 q^{18} +97.6211 q^{19} -33.2353 q^{20} -21.0000 q^{21} +29.0070 q^{22} +23.0000 q^{23} -24.0000 q^{24} -55.9633 q^{25} +44.3085 q^{26} +27.0000 q^{27} -28.0000 q^{28} -169.610 q^{29} +49.8530 q^{30} +212.661 q^{31} -32.0000 q^{32} -43.5105 q^{33} +145.091 q^{34} +58.1618 q^{35} +36.0000 q^{36} -269.643 q^{37} -195.242 q^{38} -66.4628 q^{39} +66.4707 q^{40} +270.944 q^{41} +42.0000 q^{42} +155.803 q^{43} -58.0140 q^{44} -74.7795 q^{45} -46.0000 q^{46} +92.6401 q^{47} +48.0000 q^{48} +49.0000 q^{49} +111.927 q^{50} -217.637 q^{51} -88.6171 q^{52} -307.965 q^{53} -54.0000 q^{54} +120.507 q^{55} +56.0000 q^{56} +292.863 q^{57} +339.220 q^{58} +347.269 q^{59} -99.7060 q^{60} +391.103 q^{61} -425.322 q^{62} -63.0000 q^{63} +64.0000 q^{64} +184.076 q^{65} +87.0210 q^{66} +271.676 q^{67} -290.182 q^{68} +69.0000 q^{69} -116.324 q^{70} +230.501 q^{71} -72.0000 q^{72} -766.459 q^{73} +539.287 q^{74} -167.890 q^{75} +390.484 q^{76} +101.524 q^{77} +132.926 q^{78} +1166.26 q^{79} -132.941 q^{80} +81.0000 q^{81} -541.889 q^{82} +986.304 q^{83} -84.0000 q^{84} +602.769 q^{85} -311.606 q^{86} -508.830 q^{87} +116.028 q^{88} -566.945 q^{89} +149.559 q^{90} +155.080 q^{91} +92.0000 q^{92} +637.983 q^{93} -185.280 q^{94} -811.117 q^{95} -96.0000 q^{96} +153.496 q^{97} -98.0000 q^{98} -130.531 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) $ 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9	\( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9} + 18 q^{11} + 60 q^{12} - 22 q^{13} + 70 q^{14} + 80 q^{16} + 44 q^{17} - 90 q^{18} - 14 q^{19} - 105 q^{21} - 36 q^{22} + 115 q^{23} - 120 q^{24} + 43 q^{25} + 44 q^{26} + 135 q^{27} - 140 q^{28} - 39 q^{29} - 100 q^{31} - 160 q^{32} + 54 q^{33} - 88 q^{34} + 180 q^{36} + 255 q^{37} + 28 q^{38} - 66 q^{39} + 69 q^{41} + 210 q^{42} + 912 q^{43} + 72 q^{44} - 230 q^{46} - 319 q^{47} + 240 q^{48} + 245 q^{49} - 86 q^{50} + 132 q^{51} - 88 q^{52} + 745 q^{53} - 270 q^{54} + 2199 q^{55} + 280 q^{56} - 42 q^{57} + 78 q^{58} - 315 q^{59} + 1091 q^{61} + 200 q^{62} - 315 q^{63} + 320 q^{64} + 533 q^{65} - 108 q^{66} + 991 q^{67} + 176 q^{68} + 345 q^{69} + 923 q^{71} - 360 q^{72} + 1144 q^{73} - 510 q^{74} + 129 q^{75} - 56 q^{76} - 126 q^{77} + 132 q^{78} - 110 q^{79} + 405 q^{81} - 138 q^{82} + 218 q^{83} - 420 q^{84} + 2973 q^{85} - 1824 q^{86} - 117 q^{87} - 144 q^{88} - 13 q^{89} + 154 q^{91} + 460 q^{92} - 300 q^{93} + 638 q^{94} - 347 q^{95} - 480 q^{96} + 761 q^{97} - 490 q^{98} + 162 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9 + 18 * q^11 + 60 * q^12 - 22 * q^13 + 70 * q^14 + 80 * q^16 + 44 * q^17 - 90 * q^18 - 14 * q^19 - 105 * q^21 - 36 * q^22 + 115 * q^23 - 120 * q^24 + 43 * q^25 + 44 * q^26 + 135 * q^27 - 140 * q^28 - 39 * q^29 - 100 * q^31 - 160 * q^32 + 54 * q^33 - 88 * q^34 + 180 * q^36 + 255 * q^37 + 28 * q^38 - 66 * q^39 + 69 * q^41 + 210 * q^42 + 912 * q^43 + 72 * q^44 - 230 * q^46 - 319 * q^47 + 240 * q^48 + 245 * q^49 - 86 * q^50 + 132 * q^51 - 88 * q^52 + 745 * q^53 - 270 * q^54 + 2199 * q^55 + 280 * q^56 - 42 * q^57 + 78 * q^58 - 315 * q^59 + 1091 * q^61 + 200 * q^62 - 315 * q^63 + 320 * q^64 + 533 * q^65 - 108 * q^66 + 991 * q^67 + 176 * q^68 + 345 * q^69 + 923 * q^71 - 360 * q^72 + 1144 * q^73 - 510 * q^74 + 129 * q^75 - 56 * q^76 - 126 * q^77 + 132 * q^78 - 110 * q^79 + 405 * q^81 - 138 * q^82 + 218 * q^83 - 420 * q^84 + 2973 * q^85 - 1824 * q^86 - 117 * q^87 - 144 * q^88 - 13 * q^89 + 154 * q^91 + 460 * q^92 - 300 * q^93 + 638 * q^94 - 347 * q^95 - 480 * q^96 + 761 * q^97 - 490 * q^98 + 162 * 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 $

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	−8.30883	−0.743165	−0.371582	−	0.928400i	$-0.621185\pi$
$5$	−8.30883	−0.743165	−0.371582	+	0.928400i	$0.621185\pi$
$6$	−6.00000	−0.408248
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	9.00000	0.333333
$10$	16.6177	0.525497
$11$	−14.5035	−0.397543	−0.198771	−	0.980046i	$-0.563695\pi$
$11$	−14.5035	−0.397543	−0.198771	+	0.980046i	$0.563695\pi$
$12$	12.0000	0.288675
$13$	−22.1543	−0.472653	−0.236326	−	0.971674i	$-0.575943\pi$
$13$	−22.1543	−0.472653	−0.236326	+	0.971674i	$0.575943\pi$
$14$	14.0000	0.267261
$15$	−24.9265	−0.429066
$16$	16.0000	0.250000
$17$	−72.5456	−1.03499	−0.517497	−	0.855685i	$-0.673136\pi$
$17$	−72.5456	−1.03499	−0.517497	+	0.855685i	$0.673136\pi$
$18$	−18.0000	−0.235702
$19$	97.6211	1.17873	0.589363	−	0.807868i	$-0.299379\pi$
$19$	97.6211	1.17873	0.589363	+	0.807868i	$0.299379\pi$
$20$	−33.2353	−0.371582
$21$	−21.0000	−0.218218
$22$	29.0070	0.281105
$23$	23.0000	0.208514
$23$	23.0000	0.208514
$24$	−24.0000	−0.204124
$25$	−55.9633	−0.447706
$26$	44.3085	0.334216
$27$	27.0000	0.192450
$28$	−28.0000	−0.188982
$29$	−169.610	−1.08606	−0.543031	−	0.839713i	$-0.682723\pi$
$29$	−169.610	−1.08606	−0.543031	+	0.839713i	$0.682723\pi$
$30$	49.8530	0.303396
$31$	212.661	1.23210	0.616049	−	0.787708i	$-0.288732\pi$
$31$	212.661	1.23210	0.616049	+	0.787708i	$0.288732\pi$
$32$	−32.0000	−0.176777
$33$	−43.5105	−0.229521
$34$	145.091	0.731851
$35$	58.1618	0.280890
$36$	36.0000	0.166667
$37$	−269.643	−1.19808	−0.599042	−	0.800718i	$-0.704452\pi$
$37$	−269.643	−1.19808	−0.599042	+	0.800718i	$0.704452\pi$
$38$	−195.242	−0.833486
$39$	−66.4628	−0.272886
$40$	66.4707	0.262748
$41$	270.944	1.03206	0.516029	−	0.856571i	$-0.327410\pi$
$41$	270.944	1.03206	0.516029	+	0.856571i	$0.327410\pi$
$42$	42.0000	0.154303
$43$	155.803	0.552552	0.276276	−	0.961078i	$-0.410900\pi$
$43$	155.803	0.552552	0.276276	+	0.961078i	$0.410900\pi$
$44$	−58.0140	−0.198771
$45$	−74.7795	−0.247722
$46$	−46.0000	−0.147442
$47$	92.6401	0.287510	0.143755	−	0.989613i	$-0.454082\pi$
$47$	92.6401	0.287510	0.143755	+	0.989613i	$0.454082\pi$
$48$	48.0000	0.144338
$49$	49.0000	0.142857
$50$	111.927	0.316576
$51$	−217.637	−0.597554
$52$	−88.6171	−0.236326
$53$	−307.965	−0.798156	−0.399078	−	0.916917i	$-0.630670\pi$
$53$	−307.965	−0.798156	−0.399078	+	0.916917i	$0.630670\pi$
$54$	−54.0000	−0.136083
$55$	120.507	0.295440
$56$	56.0000	0.133631
$57$	292.863	0.680538
$58$	339.220	0.767961
$59$	347.269	0.766280	0.383140	−	0.923690i	$-0.374843\pi$
$59$	347.269	0.766280	0.383140	+	0.923690i	$0.374843\pi$
$60$	−99.7060	−0.214533
$61$	391.103	0.820911	0.410456	−	0.911881i	$-0.365370\pi$
$61$	391.103	0.820911	0.410456	+	0.911881i	$0.365370\pi$
$62$	−425.322	−0.871225
$63$	−63.0000	−0.125988
$64$	64.0000	0.125000
$65$	184.076	0.351259
$66$	87.0210	0.162296
$67$	271.676	0.495381	0.247691	−	0.968839i	$-0.420328\pi$
$67$	271.676	0.495381	0.247691	+	0.968839i	$0.420328\pi$
$68$	−290.182	−0.517497
$69$	69.0000	0.120386
$70$	−116.324	−0.198619
$71$	230.501	0.385288	0.192644	−	0.981269i	$-0.438294\pi$
$71$	230.501	0.385288	0.192644	+	0.981269i	$0.438294\pi$
$72$	−72.0000	−0.117851
$73$	−766.459	−1.22887	−0.614433	−	0.788969i	$-0.710615\pi$
$73$	−766.459	−1.22887	−0.614433	+	0.788969i	$0.710615\pi$
$74$	539.287	0.847173
$75$	−167.890	−0.258483
$76$	390.484	0.589363
$77$	101.524	0.150257
$78$	132.926	0.192960
$79$	1166.26	1.66094	0.830472	−	0.557061i	$-0.188071\pi$
$79$	1166.26	1.66094	0.830472	+	0.557061i	$0.188071\pi$
$80$	−132.941	−0.185791
$81$	81.0000	0.111111
$82$	−541.889	−0.729776
$83$	986.304	1.30435	0.652175	−	0.758069i	$-0.273857\pi$
$83$	986.304	1.30435	0.652175	+	0.758069i	$0.273857\pi$
$84$	−84.0000	−0.109109
$85$	602.769	0.769170
$86$	−311.606	−0.390713
$87$	−508.830	−0.627038
$88$	116.028	0.140553
$89$	−566.945	−0.675236	−0.337618	−	0.941283i	$-0.609621\pi$
$89$	−566.945	−0.675236	−0.337618	+	0.941283i	$0.609621\pi$
$90$	149.559	0.175166
$91$	155.080	0.178646
$92$	92.0000	0.104257
$93$	637.983	0.711352
$94$	−185.280	−0.203300
$95$	−811.117	−0.875988
$96$	−96.0000	−0.102062
$97$	153.496	0.160672	0.0803360	−	0.996768i	$-0.474401\pi$
$97$	153.496	0.160672	0.0803360	+	0.996768i	$0.474401\pi$
$98$	−98.0000	−0.101015
$99$	−130.531	−0.132514
$100$	−223.853	−0.223853
$101$	636.895	0.627459	0.313730	−	0.949512i	$-0.398421\pi$
$101$	636.895	0.627459	0.313730	+	0.949512i	$0.398421\pi$
$102$	435.273	0.422534
$103$	1199.75	1.14771	0.573857	−	0.818956i	$-0.305447\pi$
$103$	1199.75	1.14771	0.573857	+	0.818956i	$0.305447\pi$
$104$	177.234	0.167108
$105$	174.485	0.162172
$106$	615.931	0.564382
$107$	−1000.85	−0.904258	−0.452129	−	0.891953i	$-0.649335\pi$
$107$	−1000.85	−0.904258	−0.452129	+	0.891953i	$0.649335\pi$
$108$	108.000	0.0962250
$109$	2103.23	1.84819	0.924097	−	0.382157i	$-0.124819\pi$
$109$	2103.23	1.84819	0.924097	+	0.382157i	$0.124819\pi$
$110$	−241.014	−0.208907
$111$	−808.930	−0.691714
$112$	−112.000	−0.0944911
$113$	−1288.14	−1.07237	−0.536187	−	0.844099i	$-0.680136\pi$
$113$	−1288.14	−1.07237	−0.536187	+	0.844099i	$0.680136\pi$
$114$	−585.726	−0.481213
$115$	−191.103	−0.154961
$116$	−678.440	−0.543031
$117$	−199.388	−0.157551
$118$	−694.537	−0.541842
$119$	507.819	0.391191
$120$	199.412	0.151698
$121$	−1120.65	−0.841960
$122$	−782.206	−0.580472
$123$	812.833	0.595859
$124$	850.644	0.616049
$125$	1503.59	1.07588
$126$	126.000	0.0890871
$127$	198.668	0.138810	0.0694051	−	0.997589i	$-0.477890\pi$
$127$	198.668	0.138810	0.0694051	+	0.997589i	$0.477890\pi$
$128$	−128.000	−0.0883883
$129$	467.409	0.319016
$130$	−368.152	−0.248378
$131$	−1801.35	−1.20141	−0.600703	−	0.799472i	$-0.705113\pi$
$131$	−1801.35	−1.20141	−0.600703	+	0.799472i	$0.705113\pi$
$132$	−174.042	−0.114761
$133$	−683.347	−0.445517
$134$	−543.353	−0.350288
$135$	−224.338	−0.143022
$136$	580.365	0.365925
$137$	592.328	0.369387	0.184693	−	0.982796i	$-0.440871\pi$
$137$	592.328	0.369387	0.184693	+	0.982796i	$0.440871\pi$
$138$	−138.000	−0.0851257
$139$	−1905.39	−1.16268	−0.581341	−	0.813660i	$-0.697472\pi$
$139$	−1905.39	−1.16268	−0.581341	+	0.813660i	$0.697472\pi$
$140$	232.647	0.140445
$141$	277.920	0.165994
$142$	−461.002	−0.272440
$143$	321.314	0.187900
$144$	144.000	0.0833333
$145$	1409.26	0.807122
$146$	1532.92	0.868940
$147$	147.000	0.0824786
$148$	−1078.57	−0.599042
$149$	725.774	0.399045	0.199523	−	0.979893i	$-0.436061\pi$
$149$	725.774	0.399045	0.199523	+	0.979893i	$0.436061\pi$
$150$	335.780	0.182775
$151$	2168.98	1.16894	0.584468	−	0.811417i	$-0.301303\pi$
$151$	2168.98	1.16894	0.584468	+	0.811417i	$0.301303\pi$
$152$	−780.969	−0.416743
$153$	−652.910	−0.344998
$154$	−203.049	−0.106248
$155$	−1766.96	−0.915651
$156$	−265.851	−0.136443
$157$	3166.00	1.60939	0.804696	−	0.593687i	$-0.202328\pi$
$157$	3166.00	1.60939	0.804696	+	0.593687i	$0.202328\pi$
$158$	−2332.52	−1.17446
$159$	−923.896	−0.460816
$160$	265.883	0.131374
$161$	−161.000	−0.0788110
$162$	−162.000	−0.0785674
$163$	1850.98	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$163$	1850.98	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$164$	1083.78	0.516029
$165$	361.521	0.170572
$166$	−1972.61	−0.922314
$167$	−2134.03	−0.988841	−0.494421	−	0.869223i	$-0.664620\pi$
$167$	−2134.03	−0.988841	−0.494421	+	0.869223i	$0.664620\pi$
$168$	168.000	0.0771517
$169$	−1706.19	−0.776599
$170$	−1205.54	−0.543886
$171$	878.590	0.392909
$172$	623.212	0.276276
$173$	−2283.38	−1.00348	−0.501741	−	0.865018i	$-0.667307\pi$
$173$	−2283.38	−1.00348	−0.501741	+	0.865018i	$0.667307\pi$
$174$	1017.66	0.443383
$175$	391.743	0.169217
$176$	−232.056	−0.0993856
$177$	1041.81	0.442412
$178$	1133.89	0.477464
$179$	3629.67	1.51561	0.757805	−	0.652482i	$-0.226272\pi$
$179$	3629.67	1.51561	0.757805	+	0.652482i	$0.226272\pi$
$180$	−299.118	−0.123861
$181$	4028.04	1.65415	0.827076	−	0.562089i	$-0.190002\pi$
$181$	4028.04	1.65415	0.827076	+	0.562089i	$0.190002\pi$
$182$	−310.160	−0.126322
$183$	1173.31	0.473953
$184$	−184.000	−0.0737210
$185$	2240.42	0.890374
$186$	−1275.97	−0.503002
$187$	1052.16	0.411454
$188$	370.560	0.143755
$189$	−189.000	−0.0727393
$190$	1622.23	0.619417
$191$	3578.19	1.35554	0.677772	−	0.735272i	$-0.262946\pi$
$191$	3578.19	1.35554	0.677772	+	0.735272i	$0.262946\pi$
$192$	192.000	0.0721688
$193$	3504.46	1.30703	0.653515	−	0.756913i	$-0.273294\pi$
$193$	3504.46	1.30703	0.653515	+	0.756913i	$0.273294\pi$
$194$	−306.992	−0.113612
$195$	552.228	0.202799
$196$	196.000	0.0714286
$197$	2692.23	0.973673	0.486837	−	0.873493i	$-0.338151\pi$
$197$	2692.23	0.973673	0.486837	+	0.873493i	$0.338151\pi$
$198$	261.063	0.0937017
$199$	1414.69	0.503945	0.251972	−	0.967734i	$-0.418921\pi$
$199$	1414.69	0.503945	0.251972	+	0.967734i	$0.418921\pi$
$200$	447.706	0.158288
$201$	815.029	0.286009
$202$	−1273.79	−0.443681
$203$	1187.27	0.410493
$204$	−870.547	−0.298777
$205$	−2251.23	−0.766990
$206$	−2399.49	−0.811556
$207$	207.000	0.0695048
$208$	−354.468	−0.118163
$209$	−1415.85	−0.468594
$210$	−348.971	−0.114673
$211$	4057.01	1.32368	0.661839	−	0.749646i	$-0.269776\pi$
$211$	4057.01	1.32368	0.661839	+	0.749646i	$0.269776\pi$
$212$	−1231.86	−0.399078
$213$	691.503	0.222446
$214$	2001.70	0.639407
$215$	−1294.54	−0.410637
$216$	−216.000	−0.0680414
$217$	−1488.63	−0.465689
$218$	−4206.47	−1.30687
$219$	−2299.38	−0.709487
$220$	482.028	0.147720
$221$	1607.19	0.489193
$222$	1617.86	0.489116
$223$	2144.14	0.643867	0.321934	−	0.946762i	$-0.395667\pi$
$223$	2144.14	0.643867	0.321934	+	0.946762i	$0.395667\pi$
$224$	224.000	0.0668153
$225$	−503.670	−0.149235
$226$	2576.29	0.758283
$227$	3257.99	0.952601	0.476301	−	0.879283i	$-0.341977\pi$
$227$	3257.99	0.952601	0.476301	+	0.879283i	$0.341977\pi$
$228$	1171.45	0.340269
$229$	−4284.29	−1.23631	−0.618153	−	0.786058i	$-0.712119\pi$
$229$	−4284.29	−1.23631	−0.618153	+	0.786058i	$0.712119\pi$
$230$	382.206	0.109574
$231$	304.573	0.0867509
$232$	1356.88	0.383981
$233$	−1777.85	−0.499875	−0.249938	−	0.968262i	$-0.580410\pi$
$233$	−1777.85	−0.499875	−0.249938	+	0.968262i	$0.580410\pi$
$234$	398.777	0.111405
$235$	−769.731	−0.213667
$236$	1389.07	0.383140
$237$	3498.78	0.958946
$238$	−1015.64	−0.276614
$239$	4467.17	1.20903	0.604513	−	0.796595i	$-0.293368\pi$
$239$	4467.17	1.20903	0.604513	+	0.796595i	$0.293368\pi$
$240$	−398.824	−0.107267
$241$	−1419.44	−0.379395	−0.189698	−	0.981843i	$-0.560751\pi$
$241$	−1419.44	−0.379395	−0.189698	+	0.981843i	$0.560751\pi$
$242$	2241.30	0.595356
$243$	243.000	0.0641500
$244$	1564.41	0.410456
$245$	−407.133	−0.106166
$246$	−1625.67	−0.421336
$247$	−2162.72	−0.557129
$248$	−1701.29	−0.435612
$249$	2958.91	0.753066
$250$	−3007.19	−0.760765
$251$	2706.45	0.680596	0.340298	−	0.940318i	$-0.389472\pi$
$251$	2706.45	0.680596	0.340298	+	0.940318i	$0.389472\pi$
$252$	−252.000	−0.0629941
$253$	−333.580	−0.0828933
$254$	−397.335	−0.0981537
$255$	1808.31	0.444081
$256$	256.000	0.0625000
$257$	4598.01	1.11602	0.558008	−	0.829836i	$-0.311566\pi$
$257$	4598.01	1.11602	0.558008	+	0.829836i	$0.311566\pi$
$258$	−934.818	−0.225578
$259$	1887.50	0.452833
$260$	736.304	0.175629
$261$	−1526.49	−0.362020
$262$	3602.69	0.849523
$263$	−1253.69	−0.293939	−0.146969	−	0.989141i	$-0.546952\pi$
$263$	−1253.69	−0.293939	−0.146969	+	0.989141i	$0.546952\pi$
$264$	348.084	0.0811480
$265$	2558.83	0.593161
$266$	1366.69	0.315028
$267$	−1700.83	−0.389848
$268$	1086.71	0.247691
$269$	−6818.93	−1.54557	−0.772783	−	0.634670i	$-0.781136\pi$
$269$	−6818.93	−1.54557	−0.772783	+	0.634670i	$0.781136\pi$
$270$	448.677	0.101132
$271$	5139.23	1.15198	0.575989	−	0.817457i	$-0.304617\pi$
$271$	5139.23	1.15198	0.575989	+	0.817457i	$0.304617\pi$
$272$	−1160.73	−0.258748
$273$	465.240	0.103141
$274$	−1184.66	−0.261196
$275$	811.663	0.177982
$276$	276.000	0.0601929
$277$	6604.65	1.43262	0.716308	−	0.697784i	$-0.245830\pi$
$277$	6604.65	1.43262	0.716308	+	0.697784i	$0.245830\pi$
$278$	3810.78	0.822141
$279$	1913.95	0.410699
$280$	−465.295	−0.0993095
$281$	3673.35	0.779835	0.389918	−	0.920850i	$-0.372503\pi$
$281$	3673.35	0.779835	0.389918	+	0.920850i	$0.372503\pi$
$282$	−555.841	−0.117375
$283$	−640.568	−0.134551	−0.0672753	−	0.997734i	$-0.521431\pi$
$283$	−640.568	−0.134551	−0.0672753	+	0.997734i	$0.521431\pi$
$284$	922.004	0.192644
$285$	−2433.35	−0.505752
$286$	−642.629	−0.132865
$287$	−1896.61	−0.390082
$288$	−288.000	−0.0589256
$289$	349.861	0.0712113
$290$	−2818.52	−0.570722
$291$	460.489	0.0927640
$292$	−3065.84	−0.614433
$293$	−8154.50	−1.62591	−0.812954	−	0.582328i	$-0.802142\pi$
$293$	−8154.50	−1.62591	−0.812954	+	0.582328i	$0.802142\pi$
$294$	−294.000	−0.0583212
$295$	−2885.40	−0.569472
$296$	2157.15	0.423587
$297$	−391.594	−0.0765071
$298$	−1451.55	−0.282168
$299$	−509.548	−0.0985550
$300$	−671.560	−0.129242
$301$	−1090.62	−0.208845
$302$	−4337.97	−0.826563
$303$	1910.68	0.362264
$304$	1561.94	0.294682
$305$	−3249.61	−0.610072
$306$	1305.82	0.243950
$307$	−10499.1	−1.95184	−0.975920	−	0.218128i	$-0.930005\pi$
$307$	−10499.1	−1.95184	−0.975920	+	0.218128i	$0.930005\pi$
$308$	406.098	0.0751285
$309$	3599.24	0.662633
$310$	3533.93	0.647463
$311$	−1395.30	−0.254405	−0.127203	−	0.991877i	$-0.540600\pi$
$311$	−1395.30	−0.254405	−0.127203	+	0.991877i	$0.540600\pi$
$312$	531.702	0.0964799
$313$	3926.14	0.709005	0.354503	−	0.935055i	$-0.384650\pi$
$313$	3926.14	0.709005	0.354503	+	0.935055i	$0.384650\pi$
$314$	−6332.01	−1.13801
$315$	523.456	0.0936299
$316$	4665.04	0.830472
$317$	−5256.41	−0.931323	−0.465661	−	0.884963i	$-0.654183\pi$
$317$	−5256.41	−0.931323	−0.465661	+	0.884963i	$0.654183\pi$
$318$	1847.79	0.325846
$319$	2459.94	0.431756
$320$	−531.765	−0.0928956
$321$	−3002.54	−0.522074
$322$	322.000	0.0557278
$323$	−7081.98	−1.21997
$324$	324.000	0.0555556
$325$	1239.83	0.211610
$326$	−3701.97	−0.628936
$327$	6309.70	1.06706
$328$	−2167.56	−0.364888
$329$	−648.481	−0.108668
$330$	−723.043	−0.120613
$331$	−3818.72	−0.634127	−0.317063	−	0.948404i	$-0.602697\pi$
$331$	−3818.72	−0.634127	−0.317063	+	0.948404i	$0.602697\pi$
$332$	3945.22	0.652175
$333$	−2426.79	−0.399361
$334$	4268.07	0.699216
$335$	−2257.31	−0.368150
$336$	−336.000	−0.0545545
$337$	1472.21	0.237972	0.118986	−	0.992896i	$-0.462036\pi$
$337$	1472.21	0.237972	0.118986	+	0.992896i	$0.462036\pi$
$338$	3412.38	0.549139
$339$	−3864.43	−0.619136
$340$	2411.08	0.384585
$341$	−3084.33	−0.489811
$342$	−1757.18	−0.277829
$343$	−343.000	−0.0539949
$344$	−1246.42	−0.195357
$345$	−573.309	−0.0894665
$346$	4566.77	0.709569
$347$	6812.47	1.05393	0.526964	−	0.849888i	$-0.323330\pi$
$347$	6812.47	1.05393	0.526964	+	0.849888i	$0.323330\pi$
$348$	−2035.32	−0.313519
$349$	−6220.80	−0.954130	−0.477065	−	0.878868i	$-0.658299\pi$
$349$	−6220.80	−0.954130	−0.477065	+	0.878868i	$0.658299\pi$
$350$	−783.486	−0.119655
$351$	−598.165	−0.0909621
$352$	464.112	0.0702763
$353$	−7551.64	−1.13862	−0.569311	−	0.822123i	$-0.692790\pi$
$353$	−7551.64	−1.13862	−0.569311	+	0.822123i	$0.692790\pi$
$354$	−2083.61	−0.312832
$355$	−1915.19	−0.286332
$356$	−2267.78	−0.337618
$357$	1523.46	0.225854
$358$	−7259.33	−1.07170
$359$	−3493.70	−0.513623	−0.256811	−	0.966462i	$-0.582672\pi$
$359$	−3493.70	−0.513623	−0.256811	+	0.966462i	$0.582672\pi$
$360$	598.236	0.0875828
$361$	2670.87	0.389397
$362$	−8056.08	−1.16966
$363$	−3361.95	−0.486106
$364$	620.320	0.0893230
$365$	6368.38	0.913250
$366$	−2346.62	−0.335136
$367$	9120.11	1.29718	0.648591	−	0.761137i	$-0.275359\pi$
$367$	9120.11	1.29718	0.648591	+	0.761137i	$0.275359\pi$
$368$	368.000	0.0521286
$369$	2438.50	0.344020
$370$	−4480.84	−0.629589
$371$	2155.76	0.301675
$372$	2551.93	0.355676
$373$	−10976.5	−1.52371	−0.761855	−	0.647748i	$-0.775711\pi$
$373$	−10976.5	−1.52371	−0.761855	+	0.647748i	$0.775711\pi$
$374$	−2104.33	−0.290942
$375$	4510.78	0.621162
$376$	−741.121	−0.101650
$377$	3757.59	0.513330
$378$	378.000	0.0514344
$379$	13270.1	1.79853	0.899263	−	0.437409i	$-0.144104\pi$
$379$	13270.1	1.79853	0.899263	+	0.437409i	$0.144104\pi$
$380$	−3244.47	−0.437994
$381$	596.003	0.0801421
$382$	−7156.39	−0.958515
$383$	−3359.14	−0.448157	−0.224079	−	0.974571i	$-0.571937\pi$
$383$	−3359.14	−0.448157	−0.224079	+	0.974571i	$0.571937\pi$
$384$	−384.000	−0.0510310
$385$	−843.550	−0.111666
$386$	−7008.93	−0.924210
$387$	1402.23	0.184184
$388$	613.985	0.0803360
$389$	−911.871	−0.118853	−0.0594263	−	0.998233i	$-0.518927\pi$
$389$	−911.871	−0.118853	−0.0594263	+	0.998233i	$0.518927\pi$
$390$	−1104.46	−0.143401
$391$	−1668.55	−0.215811
$392$	−392.000	−0.0505076
$393$	−5404.04	−0.693633
$394$	−5384.46	−0.688491
$395$	−9690.26	−1.23435
$396$	−522.126	−0.0662571
$397$	7469.67	0.944312	0.472156	−	0.881515i	$-0.343476\pi$
$397$	7469.67	0.944312	0.472156	+	0.881515i	$0.343476\pi$
$398$	−2829.39	−0.356343
$399$	−2050.04	−0.257219
$400$	−895.413	−0.111927
$401$	−12650.4	−1.57539	−0.787697	−	0.616063i	$-0.788727\pi$
$401$	−12650.4	−1.57539	−0.787697	+	0.616063i	$0.788727\pi$
$402$	−1630.06	−0.202239
$403$	−4711.35	−0.582355
$404$	2547.58	0.313730
$405$	−673.015	−0.0825738
$406$	−2374.54	−0.290262
$407$	3910.77	0.476289
$408$	1741.09	0.211267
$409$	−7038.74	−0.850962	−0.425481	−	0.904967i	$-0.639895\pi$
$409$	−7038.74	−0.850962	−0.425481	+	0.904967i	$0.639895\pi$
$410$	4502.46	0.542343
$411$	1776.98	0.213266
$412$	4798.98	0.573857
$413$	−2430.88	−0.289627
$414$	−414.000	−0.0491473
$415$	−8195.04	−0.969346
$416$	708.937	0.0835540
$417$	−5716.16	−0.671275
$418$	2831.69	0.331346
$419$	878.152	0.102388	0.0511939	−	0.998689i	$-0.483697\pi$
$419$	878.152	0.102388	0.0511939	+	0.998689i	$0.483697\pi$
$420$	697.942	0.0810859
$421$	−14980.2	−1.73418	−0.867089	−	0.498152i	$-0.834012\pi$
$421$	−14980.2	−1.73418	−0.867089	+	0.498152i	$0.834012\pi$
$422$	−8114.02	−0.935981
$423$	833.761	0.0958365
$424$	2463.72	0.282191
$425$	4059.89	0.463373
$426$	−1383.01	−0.157293
$427$	−2737.72	−0.310275
$428$	−4003.39	−0.452129
$429$	963.943	0.108484
$430$	2589.08	0.290364
$431$	8205.26	0.917015	0.458507	−	0.888691i	$-0.348384\pi$
$431$	8205.26	0.917015	0.458507	+	0.888691i	$0.348384\pi$
$432$	432.000	0.0481125
$433$	5787.39	0.642320	0.321160	−	0.947025i	$-0.395927\pi$
$433$	5787.39	0.642320	0.321160	+	0.947025i	$0.395927\pi$
$434$	2977.25	0.329292
$435$	4227.78	0.465992
$436$	8412.93	0.924097
$437$	2245.28	0.245782
$438$	4598.76	0.501683
$439$	−12480.2	−1.35682	−0.678412	−	0.734682i	$-0.737331\pi$
$439$	−12480.2	−1.35682	−0.678412	+	0.734682i	$0.737331\pi$
$440$	−964.057	−0.104454
$441$	441.000	0.0476190
$442$	−3214.39	−0.345911
$443$	395.620	0.0424299	0.0212150	−	0.999775i	$-0.493247\pi$
$443$	395.620	0.0424299	0.0212150	+	0.999775i	$0.493247\pi$
$444$	−3235.72	−0.345857
$445$	4710.65	0.501811
$446$	−4288.28	−0.455283
$447$	2177.32	0.230389
$448$	−448.000	−0.0472456
$449$	−11165.8	−1.17360	−0.586800	−	0.809732i	$-0.699613\pi$
$449$	−11165.8	−1.17360	−0.586800	+	0.809732i	$0.699613\pi$
$450$	1007.34	0.105525
$451$	−3929.64	−0.410287
$452$	−5152.57	−0.536187
$453$	6506.95	0.674886
$454$	−6515.98	−0.673591
$455$	−1288.53	−0.132763
$456$	−2342.91	−0.240607
$457$	216.987	0.0222106	0.0111053	−	0.999938i	$-0.496465\pi$
$457$	216.987	0.0222106	0.0111053	+	0.999938i	$0.496465\pi$
$458$	8568.59	0.874200
$459$	−1958.73	−0.199185
$460$	−764.413	−0.0774803
$461$	9388.00	0.948466	0.474233	−	0.880399i	$-0.342725\pi$
$461$	9388.00	0.948466	0.474233	+	0.880399i	$0.342725\pi$
$462$	−609.147	−0.0613421
$463$	−17584.3	−1.76504	−0.882518	−	0.470278i	$-0.844154\pi$
$463$	−17584.3	−1.76504	−0.882518	+	0.470278i	$0.844154\pi$
$464$	−2713.76	−0.271515
$465$	−5300.89	−0.528652
$466$	3555.70	0.353465
$467$	−9387.19	−0.930165	−0.465083	−	0.885267i	$-0.653975\pi$
$467$	−9387.19	−0.930165	−0.465083	+	0.885267i	$0.653975\pi$
$468$	−797.554	−0.0787755
$469$	−1901.73	−0.187237
$470$	1539.46	0.151085
$471$	9498.01	0.929183
$472$	−2778.15	−0.270921
$473$	−2259.69	−0.219663
$474$	−6997.56	−0.678077
$475$	−5463.20	−0.527724
$476$	2031.28	0.195595
$477$	−2771.69	−0.266052
$478$	−8934.34	−0.854910
$479$	2273.49	0.216865	0.108433	−	0.994104i	$-0.465417\pi$
$479$	2273.49	0.216865	0.108433	+	0.994104i	$0.465417\pi$
$480$	797.648	0.0758489
$481$	5973.75	0.566278
$482$	2838.88	0.268273
$483$	−483.000	−0.0455016
$484$	−4482.59	−0.420980
$485$	−1275.37	−0.119406
$486$	−486.000	−0.0453609
$487$	1878.52	0.174793	0.0873964	−	0.996174i	$-0.472145\pi$
$487$	1878.52	0.174793	0.0873964	+	0.996174i	$0.472145\pi$
$488$	−3128.82	−0.290236
$489$	5552.95	0.513524
$490$	814.266	0.0750710
$491$	5633.46	0.517789	0.258895	−	0.965906i	$-0.416642\pi$
$491$	5633.46	0.517789	0.258895	+	0.965906i	$0.416642\pi$
$492$	3251.33	0.297930
$493$	12304.5	1.12407
$494$	4325.45	0.393950
$495$	1084.56	0.0984798
$496$	3402.58	0.308024
$497$	−1613.51	−0.145625
$498$	−5917.83	−0.532498
$499$	−14298.3	−1.28273	−0.641365	−	0.767236i	$-0.721631\pi$
$499$	−14298.3	−1.28273	−0.641365	+	0.767236i	$0.721631\pi$
$500$	6014.38	0.537942
$501$	−6402.10	−0.570908
$502$	−5412.90	−0.481254
$503$	−2513.24	−0.222783	−0.111392	−	0.993777i	$-0.535531\pi$
$503$	−2513.24	−0.222783	−0.111392	+	0.993777i	$0.535531\pi$
$504$	504.000	0.0445435
$505$	−5291.85	−0.466306
$506$	667.161	0.0586144
$507$	−5118.57	−0.448370
$508$	794.671	0.0694051
$509$	5630.22	0.490285	0.245143	−	0.969487i	$-0.421165\pi$
$509$	5630.22	0.490285	0.245143	+	0.969487i	$0.421165\pi$
$510$	−3616.61	−0.314012
$511$	5365.22	0.464468
$512$	−512.000	−0.0441942
$513$	2635.77	0.226846
$514$	−9196.02	−0.789142
$515$	−9968.49	−0.852940
$516$	1869.64	0.159508
$517$	−1343.61	−0.114297
$518$	−3775.01	−0.320201
$519$	−6850.15	−0.579361
$520$	−1472.61	−0.124189
$521$	−10388.1	−0.873538	−0.436769	−	0.899574i	$-0.643877\pi$
$521$	−10388.1	−0.873538	−0.436769	+	0.899574i	$0.643877\pi$
$522$	3052.98	0.255987
$523$	11591.4	0.969137	0.484568	−	0.874753i	$-0.338977\pi$
$523$	11591.4	0.969137	0.484568	+	0.874753i	$0.338977\pi$
$524$	−7205.38	−0.600703
$525$	1175.23	0.0976976
$526$	2507.38	0.207846
$527$	−15427.6	−1.27521
$528$	−696.168	−0.0573803
$529$	529.000	0.0434783
$530$	−5117.66	−0.419428
$531$	3125.42	0.255427
$532$	−2733.39	−0.222758
$533$	−6002.58	−0.487806
$534$	3401.67	0.275664
$535$	8315.88	0.672013
$536$	−2173.41	−0.175144
$537$	10889.0	0.875037
$538$	13637.9	1.09288
$539$	−710.671	−0.0567918
$540$	−897.354	−0.0715110
$541$	−5848.10	−0.464750	−0.232375	−	0.972626i	$-0.574650\pi$
$541$	−5848.10	−0.464750	−0.232375	+	0.972626i	$0.574650\pi$
$542$	−10278.5	−0.814572
$543$	12084.1	0.955026
$544$	2321.46	0.182963
$545$	−17475.4	−1.37351
$546$	−930.479	−0.0729319
$547$	18885.5	1.47621	0.738103	−	0.674688i	$-0.235722\pi$
$547$	18885.5	1.47621	0.738103	+	0.674688i	$0.235722\pi$
$548$	2369.31	0.184693
$549$	3519.93	0.273637
$550$	−1623.33	−0.125853
$551$	−16557.5	−1.28017
$552$	−552.000	−0.0425628
$553$	−8163.82	−0.627778
$554$	−13209.3	−1.01301
$555$	6721.27	0.514057
$556$	−7621.55	−0.581341
$557$	5881.61	0.447418	0.223709	−	0.974656i	$-0.428183\pi$
$557$	5881.61	0.447418	0.223709	+	0.974656i	$0.428183\pi$
$558$	−3827.90	−0.290408
$559$	−3451.70	−0.261165
$560$	930.589	0.0702225
$561$	3156.49	0.237553
$562$	−7346.70	−0.551427
$563$	−1310.11	−0.0980719	−0.0490360	−	0.998797i	$-0.515615\pi$
$563$	−1310.11	−0.0980719	−0.0490360	+	0.998797i	$0.515615\pi$
$564$	1111.68	0.0829969
$565$	10703.0	0.796951
$566$	1281.14	0.0951417
$567$	−567.000	−0.0419961
$568$	−1844.01	−0.136220
$569$	−1541.80	−0.113595	−0.0567975	−	0.998386i	$-0.518089\pi$
$569$	−1541.80	−0.113595	−0.0567975	+	0.998386i	$0.518089\pi$
$570$	4866.70	0.357621
$571$	19602.9	1.43670	0.718349	−	0.695683i	$-0.244898\pi$
$571$	19602.9	1.43670	0.718349	+	0.695683i	$0.244898\pi$
$572$	1285.26	0.0939498
$573$	10734.6	0.782624
$574$	3793.22	0.275829
$575$	−1287.16	−0.0933532
$576$	576.000	0.0416667
$577$	13220.7	0.953874	0.476937	−	0.878937i	$-0.341747\pi$
$577$	13220.7	0.953874	0.476937	+	0.878937i	$0.341747\pi$
$578$	−699.722	−0.0503540
$579$	10513.4	0.754614
$580$	5637.04	0.403561
$581$	−6904.13	−0.492998
$582$	−920.977	−0.0655940
$583$	4466.57	0.317301
$584$	6131.68	0.434470
$585$	1656.68	0.117086
$586$	16309.0	1.14969
$587$	−9448.20	−0.664342	−0.332171	−	0.943219i	$-0.607781\pi$
$587$	−9448.20	−0.664342	−0.332171	+	0.943219i	$0.607781\pi$
$588$	588.000	0.0412393
$589$	20760.2	1.45231
$590$	5770.79	0.402678
$591$	8076.70	0.562150
$592$	−4314.29	−0.299521
$593$	5273.96	0.365220	0.182610	−	0.983185i	$-0.441545\pi$
$593$	5273.96	0.365220	0.182610	+	0.983185i	$0.441545\pi$
$594$	783.189	0.0540987
$595$	−4219.38	−0.290719
$596$	2903.10	0.199523
$597$	4244.08	0.290953
$598$	1019.10	0.0696889
$599$	17549.3	1.19707	0.598536	−	0.801096i	$-0.295749\pi$
$599$	17549.3	1.19707	0.598536	+	0.801096i	$0.295749\pi$
$600$	1343.12	0.0913877
$601$	−895.932	−0.0608084	−0.0304042	−	0.999538i	$-0.509679\pi$
$601$	−895.932	−0.0608084	−0.0304042	+	0.999538i	$0.509679\pi$
$602$	2181.24	0.147676
$603$	2445.09	0.165127
$604$	8675.93	0.584468
$605$	9311.28	0.625715
$606$	−3821.37	−0.256159
$607$	4998.74	0.334255	0.167127	−	0.985935i	$-0.446551\pi$
$607$	4998.74	0.334255	0.167127	+	0.985935i	$0.446551\pi$
$608$	−3123.87	−0.208371
$609$	3561.81	0.236998
$610$	6499.22	0.431386
$611$	−2052.37	−0.135892
$612$	−2611.64	−0.172499
$613$	3867.39	0.254817	0.127408	−	0.991850i	$-0.459334\pi$
$613$	3867.39	0.254817	0.127408	+	0.991850i	$0.459334\pi$
$614$	20998.2	1.38016
$615$	−6753.70	−0.442822
$616$	−812.196	−0.0531239
$617$	−16805.5	−1.09654	−0.548270	−	0.836301i	$-0.684713\pi$
$617$	−16805.5	−1.09654	−0.548270	+	0.836301i	$0.684713\pi$
$618$	−7198.47	−0.468552
$619$	−27427.0	−1.78091	−0.890456	−	0.455070i	$-0.849614\pi$
$619$	−27427.0	−1.78091	−0.890456	+	0.455070i	$0.849614\pi$
$620$	−7067.86	−0.457826
$621$	621.000	0.0401286
$622$	2790.59	0.179892
$623$	3968.61	0.255215
$624$	−1063.40	−0.0682216
$625$	−5497.70	−0.351853
$626$	−7852.29	−0.501343
$627$	−4247.54	−0.270543
$628$	12664.0	0.804696
$629$	19561.4	1.24001
$630$	−1046.91	−0.0662064
$631$	−18481.8	−1.16601	−0.583003	−	0.812470i	$-0.698122\pi$
$631$	−18481.8	−1.16601	−0.583003	+	0.812470i	$0.698122\pi$
$632$	−9330.08	−0.587232
$633$	12171.0	0.764226
$634$	10512.8	0.658544
$635$	−1650.70	−0.103159
$636$	−3695.58	−0.230408
$637$	−1085.56	−0.0675219
$638$	−4919.87	−0.305297
$639$	2074.51	0.128429
$640$	1063.53	0.0656871
$641$	5671.20	0.349452	0.174726	−	0.984617i	$-0.444096\pi$
$641$	5671.20	0.349452	0.174726	+	0.984617i	$0.444096\pi$
$642$	6005.09	0.369162
$643$	−1071.38	−0.0657092	−0.0328546	−	0.999460i	$-0.510460\pi$
$643$	−1071.38	−0.0657092	−0.0328546	+	0.999460i	$0.510460\pi$
$644$	−644.000	−0.0394055
$645$	−3883.62	−0.237081
$646$	14164.0	0.862652
$647$	−18753.0	−1.13950	−0.569750	−	0.821818i	$-0.692960\pi$
$647$	−18753.0	−1.13950	−0.569750	+	0.821818i	$0.692960\pi$
$648$	−648.000	−0.0392837
$649$	−5036.61	−0.304629
$650$	−2479.65	−0.149631
$651$	−4465.88	−0.268866
$652$	7403.94	0.444725
$653$	−14041.2	−0.841463	−0.420732	−	0.907185i	$-0.638227\pi$
$653$	−14041.2	−0.841463	−0.420732	+	0.907185i	$0.638227\pi$
$654$	−12619.4	−0.754522
$655$	14967.1	0.892843
$656$	4335.11	0.258015
$657$	−6898.13	−0.409622
$658$	1296.96	0.0768402
$659$	−22843.8	−1.35033	−0.675166	−	0.737665i	$-0.735928\pi$
$659$	−22843.8	−1.35033	−0.675166	+	0.737665i	$0.735928\pi$
$660$	1446.09	0.0852860
$661$	5950.92	0.350173	0.175086	−	0.984553i	$-0.443980\pi$
$661$	5950.92	0.350173	0.175086	+	0.984553i	$0.443980\pi$
$662$	7637.45	0.448395
$663$	4821.58	0.282436
$664$	−7890.44	−0.461157
$665$	5677.82	0.331092
$666$	4853.58	0.282391
$667$	−3901.03	−0.226459
$668$	−8536.14	−0.494421
$669$	6432.43	0.371737
$670$	4514.63	0.260321
$671$	−5672.36	−0.326347
$672$	672.000	0.0385758
$673$	15124.6	0.866288	0.433144	−	0.901325i	$-0.357404\pi$
$673$	15124.6	0.866288	0.433144	+	0.901325i	$0.357404\pi$
$674$	−2944.42	−0.168272
$675$	−1511.01	−0.0861611
$676$	−6824.75	−0.388300
$677$	20532.9	1.16565	0.582824	−	0.812598i	$-0.301948\pi$
$677$	20532.9	1.16565	0.582824	+	0.812598i	$0.301948\pi$
$678$	7728.86	0.437795
$679$	−1074.47	−0.0607283
$680$	−4822.15	−0.271943
$681$	9773.97	0.549985
$682$	6168.65	0.346349
$683$	18650.6	1.04487	0.522434	−	0.852680i	$-0.325024\pi$
$683$	18650.6	1.04487	0.522434	+	0.852680i	$0.325024\pi$
$684$	3514.36	0.196454
$685$	−4921.55	−0.274515
$686$	686.000	0.0381802
$687$	−12852.9	−0.713782
$688$	2492.85	0.138138
$689$	6822.75	0.377251
$690$	1146.62	0.0632624
$691$	−6696.25	−0.368650	−0.184325	−	0.982865i	$-0.559010\pi$
$691$	−6696.25	−0.368650	−0.184325	+	0.982865i	$0.559010\pi$
$692$	−9133.53	−0.501741
$693$	913.720	0.0500857
$694$	−13624.9	−0.745239
$695$	15831.5	0.864064
$696$	4070.64	0.221691
$697$	−19655.8	−1.06817
$698$	12441.6	0.674672
$699$	−5333.55	−0.288603
$700$	1566.97	0.0846086
$701$	30931.0	1.66655	0.833273	−	0.552861i	$-0.186464\pi$
$701$	30931.0	1.66655	0.833273	+	0.552861i	$0.186464\pi$
$702$	1196.33	0.0643199
$703$	−26322.9	−1.41221
$704$	−928.224	−0.0496928
$705$	−2309.19	−0.123361
$706$	15103.3	0.805127
$707$	−4458.26	−0.237157
$708$	4167.22	0.221206
$709$	21650.5	1.14683	0.573414	−	0.819266i	$-0.305619\pi$
$709$	21650.5	1.14683	0.573414	+	0.819266i	$0.305619\pi$
$710$	3830.39	0.202468
$711$	10496.3	0.553648
$712$	4535.56	0.238732
$713$	4891.20	0.256910
$714$	−3046.91	−0.159703
$715$	−2669.75	−0.139640
$716$	14518.7	0.757805
$717$	13401.5	0.698031
$718$	6987.41	0.363186
$719$	29412.9	1.52562	0.762808	−	0.646625i	$-0.223820\pi$
$719$	29412.9	1.52562	0.762808	+	0.646625i	$0.223820\pi$
$720$	−1196.47	−0.0619304
$721$	−8398.22	−0.433795
$722$	−5341.75	−0.275345
$723$	−4258.32	−0.219044
$724$	16112.2	0.827076
$725$	9491.93	0.486237
$726$	6723.89	0.343729
$727$	−10968.5	−0.559559	−0.279780	−	0.960064i	$-0.590261\pi$
$727$	−10968.5	−0.559559	−0.279780	+	0.960064i	$0.590261\pi$
$728$	−1240.64	−0.0631609
$729$	729.000	0.0370370
$730$	−12736.8	−0.645766
$731$	−11302.8	−0.571888
$732$	4693.23	0.236977
$733$	−1031.75	−0.0519899	−0.0259949	−	0.999662i	$-0.508275\pi$
$733$	−1031.75	−0.0519899	−0.0259949	+	0.999662i	$0.508275\pi$
$734$	−18240.2	−0.917246
$735$	−1221.40	−0.0612952
$736$	−736.000	−0.0368605
$737$	−3940.26	−0.196935
$738$	−4877.00	−0.243259
$739$	−37494.0	−1.86636	−0.933179	−	0.359413i	$-0.882977\pi$
$739$	−37494.0	−1.86636	−0.933179	+	0.359413i	$0.882977\pi$
$740$	8961.69	0.445187
$741$	−6488.17	−0.321658
$742$	−4311.51	−0.213316
$743$	−2363.98	−0.116724	−0.0583620	−	0.998295i	$-0.518588\pi$
$743$	−2363.98	−0.116724	−0.0583620	+	0.998295i	$0.518588\pi$
$744$	−5103.86	−0.251501
$745$	−6030.34	−0.296556
$746$	21953.1	1.07743
$747$	8876.74	0.434783
$748$	4208.66	0.205727
$749$	7005.94	0.341777
$750$	−9021.56	−0.439228
$751$	−19164.4	−0.931184	−0.465592	−	0.884999i	$-0.654159\pi$
$751$	−19164.4	−0.931184	−0.465592	+	0.884999i	$0.654159\pi$
$752$	1482.24	0.0718774
$753$	8119.35	0.392942
$754$	−7515.17	−0.362979
$755$	−18021.7	−0.868712
$756$	−756.000	−0.0363696
$757$	3686.58	0.177002	0.0885012	−	0.996076i	$-0.471792\pi$
$757$	3686.58	0.177002	0.0885012	+	0.996076i	$0.471792\pi$
$758$	−26540.3	−1.27175
$759$	−1000.74	−0.0478585
$760$	6488.94	0.309709
$761$	−7136.48	−0.339944	−0.169972	−	0.985449i	$-0.554368\pi$
$761$	−7136.48	−0.339944	−0.169972	+	0.985449i	$0.554368\pi$
$762$	−1192.01	−0.0566690
$763$	−14722.6	−0.698552
$764$	14312.8	0.677772
$765$	5424.92	0.256390
$766$	6718.28	0.316895
$767$	−7693.48	−0.362184
$768$	768.000	0.0360844
$769$	21984.9	1.03095	0.515473	−	0.856906i	$-0.327616\pi$
$769$	21984.9	1.03095	0.515473	+	0.856906i	$0.327616\pi$
$770$	1687.10	0.0789595
$771$	13794.0	0.644332
$772$	14017.9	0.653515
$773$	14038.5	0.653209	0.326604	−	0.945161i	$-0.394096\pi$
$773$	14038.5	0.653209	0.326604	+	0.945161i	$0.394096\pi$
$774$	−2804.46	−0.130238
$775$	−11901.2	−0.551618
$776$	−1227.97	−0.0568061
$777$	5662.51	0.261443
$778$	1823.74	0.0840415
$779$	26449.9	1.21652
$780$	2208.91	0.101400
$781$	−3343.07	−0.153168
$782$	3337.10	0.152601
$783$	−4579.47	−0.209013
$784$	784.000	0.0357143
$785$	−26305.8	−1.19604
$786$	10808.1	0.490472
$787$	8222.69	0.372436	0.186218	−	0.982508i	$-0.440377\pi$
$787$	8222.69	0.372436	0.186218	+	0.982508i	$0.440377\pi$
$788$	10768.9	0.486837
$789$	−3761.07	−0.169705
$790$	19380.5	0.872820
$791$	9017.00	0.405319
$792$	1044.25	0.0468508
$793$	−8664.60	−0.388006
$794$	−14939.3	−0.667729
$795$	7676.50	0.342462
$796$	5658.78	0.251972
$797$	−30260.1	−1.34488	−0.672438	−	0.740153i	$-0.734753\pi$
$797$	−30260.1	−1.34488	−0.672438	+	0.740153i	$0.734753\pi$
$798$	4100.08	0.181881
$799$	−6720.63	−0.297571
$800$	1790.83	0.0791441
$801$	−5102.50	−0.225079
$802$	25300.9	1.11397
$803$	11116.3	0.488527
$804$	3260.12	0.143004
$805$	1337.72	0.0585696
$806$	9422.70	0.411787
$807$	−20456.8	−0.892333
$808$	−5095.16	−0.221840
$809$	16996.1	0.738631	0.369316	−	0.929304i	$-0.379592\pi$
$809$	16996.1	0.738631	0.369316	+	0.929304i	$0.379592\pi$
$810$	1346.03	0.0583885
$811$	−22534.8	−0.975713	−0.487857	−	0.872924i	$-0.662221\pi$
$811$	−22534.8	−0.975713	−0.487857	+	0.872924i	$0.662221\pi$
$812$	4749.08	0.205246
$813$	15417.7	0.665095
$814$	−7821.54	−0.336787
$815$	−15379.5	−0.661007
$816$	−3482.19	−0.149388
$817$	15209.7	0.651308
$818$	14077.5	0.601721
$819$	1395.72	0.0595487
$820$	−9004.93	−0.383495
$821$	28687.1	1.21947	0.609736	−	0.792605i	$-0.291275\pi$
$821$	28687.1	1.21947	0.609736	+	0.792605i	$0.291275\pi$
$822$	−3553.97	−0.150802
$823$	−37057.9	−1.56957	−0.784786	−	0.619767i	$-0.787227\pi$
$823$	−37057.9	−1.56957	−0.784786	+	0.619767i	$0.787227\pi$
$824$	−9597.97	−0.405778
$825$	2434.99	0.102758
$826$	4861.76	0.204797
$827$	24957.4	1.04940	0.524699	−	0.851288i	$-0.324178\pi$
$827$	24957.4	1.04940	0.524699	+	0.851288i	$0.324178\pi$
$828$	828.000	0.0347524
$829$	19930.2	0.834988	0.417494	−	0.908680i	$-0.362909\pi$
$829$	19930.2	0.834988	0.417494	+	0.908680i	$0.362909\pi$
$830$	16390.1	0.685431
$831$	19813.9	0.827122
$832$	−1417.87	−0.0590816
$833$	−3554.73	−0.147856
$834$	11432.3	0.474663
$835$	17731.3	0.734872
$836$	−5663.39	−0.234297
$837$	5741.85	0.237117
$838$	−1756.30	−0.0723992
$839$	29030.3	1.19456	0.597282	−	0.802032i	$-0.296247\pi$
$839$	29030.3	1.19456	0.597282	+	0.802032i	$0.296247\pi$
$840$	−1395.88	−0.0573364
$841$	4378.54	0.179529
$842$	29960.4	1.22625
$843$	11020.1	0.450238
$844$	16228.0	0.661839
$845$	14176.4	0.577141
$846$	−1667.52	−0.0677667
$847$	7844.54	0.318231
$848$	−4927.44	−0.199539
$849$	−1921.70	−0.0776828
$850$	−8119.78	−0.327654
$851$	−6201.80	−0.249818
$852$	2766.01	0.111223
$853$	−14721.5	−0.590919	−0.295459	−	0.955355i	$-0.595473\pi$
$853$	−14721.5	−0.590919	−0.295459	+	0.955355i	$0.595473\pi$
$854$	5475.44	0.219398
$855$	−7300.05	−0.291996
$856$	8006.78	0.319704
$857$	−660.025	−0.0263081	−0.0131540	−	0.999913i	$-0.504187\pi$
$857$	−660.025	−0.0263081	−0.0131540	+	0.999913i	$0.504187\pi$
$858$	−1927.89	−0.0767097
$859$	−21141.3	−0.839735	−0.419868	−	0.907585i	$-0.637924\pi$
$859$	−21141.3	−0.839735	−0.419868	+	0.907585i	$0.637924\pi$
$860$	−5178.17	−0.205319
$861$	−5689.83	−0.225214
$862$	−16410.5	−0.648427
$863$	25517.4	1.00651	0.503257	−	0.864137i	$-0.332135\pi$
$863$	25517.4	1.00651	0.503257	+	0.864137i	$0.332135\pi$
$864$	−864.000	−0.0340207
$865$	18972.2	0.745752
$866$	−11574.8	−0.454189
$867$	1049.58	0.0411138
$868$	−5954.51	−0.232845
$869$	−16914.8	−0.660296
$870$	−8455.57	−0.329506
$871$	−6018.79	−0.234143
$872$	−16825.9	−0.653435
$873$	1381.47	0.0535573
$874$	−4490.57	−0.173794
$875$	−10525.2	−0.406646
$876$	−9197.51	−0.354743
$877$	−24757.2	−0.953240	−0.476620	−	0.879109i	$-0.658138\pi$
$877$	−24757.2	−0.953240	−0.476620	+	0.879109i	$0.658138\pi$
$878$	24960.3	0.959419
$879$	−24463.5	−0.938718
$880$	1928.11	0.0738599
$881$	4304.38	0.164607	0.0823033	−	0.996607i	$-0.473772\pi$
$881$	4304.38	0.164607	0.0823033	+	0.996607i	$0.473772\pi$
$882$	−882.000	−0.0336718
$883$	11491.6	0.437966	0.218983	−	0.975729i	$-0.429726\pi$
$883$	11491.6	0.437966	0.218983	+	0.975729i	$0.429726\pi$
$884$	6428.78	0.244596
$885$	−8656.19	−0.328785
$886$	−791.239	−0.0300025
$887$	14538.9	0.550358	0.275179	−	0.961393i	$-0.411263\pi$
$887$	14538.9	0.550358	0.275179	+	0.961393i	$0.411263\pi$
$888$	6471.44	0.244558
$889$	−1390.67	−0.0524653
$890$	−9421.29	−0.354834
$891$	−1174.78	−0.0441714
$892$	8576.57	0.321934
$893$	9043.63	0.338895
$894$	−4354.65	−0.162910
$895$	−30158.3	−1.12635
$896$	896.000	0.0334077
$897$	−1528.64	−0.0569007
$898$	22331.6	0.829860
$899$	−36069.4	−1.33813
$900$	−2014.68	−0.0746177
$901$	22341.5	0.826086
$902$	7859.28	0.290117
$903$	−3271.86	−0.120577
$904$	10305.1	0.379142
$905$	−33468.3	−1.22931
$906$	−13013.9	−0.477216
$907$	43139.4	1.57930	0.789648	−	0.613560i	$-0.210263\pi$
$907$	43139.4	1.57930	0.789648	+	0.613560i	$0.210263\pi$
$908$	13032.0	0.476301
$909$	5732.05	0.209153
$910$	2577.07	0.0938779
$911$	30133.3	1.09590	0.547948	−	0.836513i	$-0.315409\pi$
$911$	30133.3	1.09590	0.547948	+	0.836513i	$0.315409\pi$
$912$	4685.81	0.170135
$913$	−14304.9	−0.518534
$914$	−433.974	−0.0157052
$915$	−9748.82	−0.352225
$916$	−17137.2	−0.618153
$917$	12609.4	0.454089
$918$	3917.46	0.140845
$919$	−30662.3	−1.10061	−0.550303	−	0.834965i	$-0.685488\pi$
$919$	−30662.3	−1.10061	−0.550303	+	0.834965i	$0.685488\pi$
$920$	1528.83	0.0547868
$921$	−31497.3	−1.12690
$922$	−18776.0	−0.670667
$923$	−5106.58	−0.182108
$924$	1218.29	0.0433754
$925$	15090.1	0.536390
$926$	35168.6	1.24807
$927$	10797.7	0.382571
$928$	5427.52	0.191990
$929$	−21557.4	−0.761329	−0.380664	−	0.924713i	$-0.624305\pi$
$929$	−21557.4	−0.761329	−0.380664	+	0.924713i	$0.624305\pi$
$930$	10601.8	0.373813
$931$	4783.43	0.168390
$932$	−7111.40	−0.249938
$933$	−4185.89	−0.146881
$934$	18774.4	0.657726
$935$	−8742.26	−0.305778
$936$	1595.11	0.0557027
$937$	21141.7	0.737106	0.368553	−	0.929607i	$-0.379853\pi$
$937$	21141.7	0.737106	0.368553	+	0.929607i	$0.379853\pi$
$938$	3803.47	0.132396
$939$	11778.4	0.409345
$940$	−3078.92	−0.106833
$941$	10324.0	0.357656	0.178828	−	0.983880i	$-0.442769\pi$
$941$	10324.0	0.357656	0.178828	+	0.983880i	$0.442769\pi$
$942$	−18996.0	−0.657032
$943$	6231.72	0.215199
$944$	5556.30	0.191570
$945$	1570.37	0.0540573
$946$	4519.38	0.155325
$947$	28630.4	0.982431	0.491215	−	0.871038i	$-0.336553\pi$
$947$	28630.4	0.982431	0.491215	+	0.871038i	$0.336553\pi$
$948$	13995.1	0.479473
$949$	16980.3	0.580828
$950$	10926.4	0.373157
$951$	−15769.2	−0.537699
$952$	−4062.55	−0.138307
$953$	27602.9	0.938243	0.469121	−	0.883134i	$-0.344571\pi$
$953$	27602.9	0.938243	0.469121	+	0.883134i	$0.344571\pi$
$954$	5543.37	0.188127
$955$	−29730.6	−1.00739
$956$	17868.7	0.604513
$957$	7379.81	0.249274
$958$	−4546.98	−0.153347
$959$	−4146.30	−0.139615
$960$	−1595.30	−0.0536333
$961$	15433.7	0.518065
$962$	−11947.5	−0.400419
$963$	−9007.63	−0.301419
$964$	−5677.77	−0.189698
$965$	−29118.0	−0.971339
$966$	966.000	0.0321745
$967$	−18995.6	−0.631705	−0.315852	−	0.948808i	$-0.602290\pi$
$967$	−18995.6	−0.631705	−0.315852	+	0.948808i	$0.602290\pi$
$968$	8965.19	0.297678
$969$	−21245.9	−0.704353
$970$	2550.75	0.0844326
$971$	−5118.27	−0.169159	−0.0845793	−	0.996417i	$-0.526955\pi$
$971$	−5118.27	−0.169159	−0.0845793	+	0.996417i	$0.526955\pi$
$972$	972.000	0.0320750
$973$	13337.7	0.439453
$974$	−3757.05	−0.123597
$975$	3719.48	0.122173
$976$	6257.65	0.205228
$977$	35484.8	1.16198	0.580992	−	0.813909i	$-0.302665\pi$
$977$	35484.8	1.16198	0.580992	+	0.813909i	$0.302665\pi$
$978$	−11105.9	−0.363116
$979$	8222.68	0.268435
$980$	−1628.53	−0.0530832
$981$	18929.1	0.616065
$982$	−11266.9	−0.366132
$983$	16072.4	0.521494	0.260747	−	0.965407i	$-0.416031\pi$
$983$	16072.4	0.521494	0.260747	+	0.965407i	$0.416031\pi$
$984$	−6502.67	−0.210668
$985$	−22369.3	−0.723599
$986$	−24608.9	−0.794835
$987$	−1945.44	−0.0627397
$988$	−8650.89	−0.278564
$989$	3583.47	0.115215
$990$	−2169.13	−0.0696358
$991$	48573.5	1.55700	0.778500	−	0.627644i	$-0.215981\pi$
$991$	48573.5	1.55700	0.778500	+	0.627644i	$0.215981\pi$
$992$	−6805.15	−0.217806
$993$	−11456.2	−0.366113
$994$	3227.02	0.102973
$995$	−11754.5	−0.374514
$996$	11835.7	0.376533
$997$	−30100.7	−0.956167	−0.478083	−	0.878314i	$-0.658668\pi$
$997$	−30100.7	−0.956167	−0.478083	+	0.878314i	$0.658668\pi$
$998$	28596.7	0.907027
$999$	−7280.37	−0.230571

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	966.4.a.n.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.4.a.n.1.2	✓	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 966 = 2 \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	966.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -8.30883 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} -8.30883 q^{5} -6.00000 q^{6} -7.00000 q^{7} -8.00000 q^{8} +9.00000 q^{9} +16.6177 q^{10} -14.5035 q^{11} +12.0000 q^{12} -22.1543 q^{13} +14.0000 q^{14} -24.9265 q^{15} +16.0000 q^{16} -72.5456 q^{17} -18.0000 q^{18} +97.6211 q^{19} -33.2353 q^{20} -21.0000 q^{21} +29.0070 q^{22} +23.0000 q^{23} -24.0000 q^{24} -55.9633 q^{25} +44.3085 q^{26} +27.0000 q^{27} -28.0000 q^{28} -169.610 q^{29} +49.8530 q^{30} +212.661 q^{31} -32.0000 q^{32} -43.5105 q^{33} +145.091 q^{34} +58.1618 q^{35} +36.0000 q^{36} -269.643 q^{37} -195.242 q^{38} -66.4628 q^{39} +66.4707 q^{40} +270.944 q^{41} +42.0000 q^{42} +155.803 q^{43} -58.0140 q^{44} -74.7795 q^{45} -46.0000 q^{46} +92.6401 q^{47} +48.0000 q^{48} +49.0000 q^{49} +111.927 q^{50} -217.637 q^{51} -88.6171 q^{52} -307.965 q^{53} -54.0000 q^{54} +120.507 q^{55} +56.0000 q^{56} +292.863 q^{57} +339.220 q^{58} +347.269 q^{59} -99.7060 q^{60} +391.103 q^{61} -425.322 q^{62} -63.0000 q^{63} +64.0000 q^{64} +184.076 q^{65} +87.0210 q^{66} +271.676 q^{67} -290.182 q^{68} +69.0000 q^{69} -116.324 q^{70} +230.501 q^{71} -72.0000 q^{72} -766.459 q^{73} +539.287 q^{74} -167.890 q^{75} +390.484 q^{76} +101.524 q^{77} +132.926 q^{78} +1166.26 q^{79} -132.941 q^{80} +81.0000 q^{81} -541.889 q^{82} +986.304 q^{83} -84.0000 q^{84} +602.769 q^{85} -311.606 q^{86} -508.830 q^{87} +116.028 q^{88} -566.945 q^{89} +149.559 q^{90} +155.080 q^{91} +92.0000 q^{92} +637.983 q^{93} -185.280 q^{94} -811.117 q^{95} -96.0000 q^{96} +153.496 q^{97} -98.0000 q^{98} -130.531 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9}+O(q^{10}) \) 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9	\( 5 q - 10 q^{2} + 15 q^{3} + 20 q^{4} - 30 q^{6} - 35 q^{7} - 40 q^{8} + 45 q^{9} + 18 q^{11} + 60 q^{12} - 22 q^{13} + 70 q^{14} + 80 q^{16} + 44 q^{17} - 90 q^{18} - 14 q^{19} - 105 q^{21} - 36 q^{22} + 115 q^{23} - 120 q^{24} + 43 q^{25} + 44 q^{26} + 135 q^{27} - 140 q^{28} - 39 q^{29} - 100 q^{31} - 160 q^{32} + 54 q^{33} - 88 q^{34} + 180 q^{36} + 255 q^{37} + 28 q^{38} - 66 q^{39} + 69 q^{41} + 210 q^{42} + 912 q^{43} + 72 q^{44} - 230 q^{46} - 319 q^{47} + 240 q^{48} + 245 q^{49} - 86 q^{50} + 132 q^{51} - 88 q^{52} + 745 q^{53} - 270 q^{54} + 2199 q^{55} + 280 q^{56} - 42 q^{57} + 78 q^{58} - 315 q^{59} + 1091 q^{61} + 200 q^{62} - 315 q^{63} + 320 q^{64} + 533 q^{65} - 108 q^{66} + 991 q^{67} + 176 q^{68} + 345 q^{69} + 923 q^{71} - 360 q^{72} + 1144 q^{73} - 510 q^{74} + 129 q^{75} - 56 q^{76} - 126 q^{77} + 132 q^{78} - 110 q^{79} + 405 q^{81} - 138 q^{82} + 218 q^{83} - 420 q^{84} + 2973 q^{85} - 1824 q^{86} - 117 q^{87} - 144 q^{88} - 13 q^{89} + 154 q^{91} + 460 q^{92} - 300 q^{93} + 638 q^{94} - 347 q^{95} - 480 q^{96} + 761 q^{97} - 490 q^{98} + 162 q^{99}+O(q^{100}) \) 5 * q - 10 * q^2 + 15 * q^3 + 20 * q^4 - 30 * q^6 - 35 * q^7 - 40 * q^8 + 45 * q^9 + 18 * q^11 + 60 * q^12 - 22 * q^13 + 70 * q^14 + 80 * q^16 + 44 * q^17 - 90 * q^18 - 14 * q^19 - 105 * q^21 - 36 * q^22 + 115 * q^23 - 120 * q^24 + 43 * q^25 + 44 * q^26 + 135 * q^27 - 140 * q^28 - 39 * q^29 - 100 * q^31 - 160 * q^32 + 54 * q^33 - 88 * q^34 + 180 * q^36 + 255 * q^37 + 28 * q^38 - 66 * q^39 + 69 * q^41 + 210 * q^42 + 912 * q^43 + 72 * q^44 - 230 * q^46 - 319 * q^47 + 240 * q^48 + 245 * q^49 - 86 * q^50 + 132 * q^51 - 88 * q^52 + 745 * q^53 - 270 * q^54 + 2199 * q^55 + 280 * q^56 - 42 * q^57 + 78 * q^58 - 315 * q^59 + 1091 * q^61 + 200 * q^62 - 315 * q^63 + 320 * q^64 + 533 * q^65 - 108 * q^66 + 991 * q^67 + 176 * q^68 + 345 * q^69 + 923 * q^71 - 360 * q^72 + 1144 * q^73 - 510 * q^74 + 129 * q^75 - 56 * q^76 - 126 * q^77 + 132 * q^78 - 110 * q^79 + 405 * q^81 - 138 * q^82 + 218 * q^83 - 420 * q^84 + 2973 * q^85 - 1824 * q^86 - 117 * q^87 - 144 * q^88 - 13 * q^89 + 154 * q^91 + 460 * q^92 - 300 * q^93 + 638 * q^94 - 347 * q^95 - 480 * q^96 + 761 * q^97 - 490 * q^98 + 162 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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