LMFDB - Embedded newform 471.4.a.c.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [471,4,Mod(1,471)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(471, 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("471.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 471 = 3 \cdot 157 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	471.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:	$27.7898996127$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	471.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.43477 q^{2} -3.00000 q^{3} -2.07188 q^{4} -12.7413 q^{5} -7.30432 q^{6} -29.4801 q^{7} -24.5227 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.43477 q^{2} -3.00000 q^{3} -2.07188 q^{4} -12.7413 q^{5} -7.30432 q^{6} -29.4801 q^{7} -24.5227 q^{8} +9.00000 q^{9} -31.0221 q^{10} -62.2952 q^{11} +6.21565 q^{12} +72.0794 q^{13} -71.7773 q^{14} +38.2238 q^{15} -43.1322 q^{16} +84.0694 q^{17} +21.9130 q^{18} -68.6221 q^{19} +26.3984 q^{20} +88.4402 q^{21} -151.675 q^{22} +180.815 q^{23} +73.5682 q^{24} +37.3399 q^{25} +175.497 q^{26} -27.0000 q^{27} +61.0793 q^{28} -82.5432 q^{29} +93.0663 q^{30} -128.042 q^{31} +91.1648 q^{32} +186.885 q^{33} +204.690 q^{34} +375.613 q^{35} -18.6469 q^{36} +195.069 q^{37} -167.079 q^{38} -216.238 q^{39} +312.451 q^{40} -138.169 q^{41} +215.332 q^{42} -433.228 q^{43} +129.068 q^{44} -114.671 q^{45} +440.243 q^{46} -579.648 q^{47} +129.397 q^{48} +526.074 q^{49} +90.9143 q^{50} -252.208 q^{51} -149.340 q^{52} +459.505 q^{53} -65.7389 q^{54} +793.719 q^{55} +722.932 q^{56} +205.866 q^{57} -200.974 q^{58} -710.579 q^{59} -79.1953 q^{60} -48.8133 q^{61} -311.754 q^{62} -265.321 q^{63} +567.023 q^{64} -918.383 q^{65} +455.024 q^{66} -138.593 q^{67} -174.182 q^{68} -542.445 q^{69} +914.533 q^{70} +596.513 q^{71} -220.705 q^{72} +284.148 q^{73} +474.949 q^{74} -112.020 q^{75} +142.177 q^{76} +1836.47 q^{77} -526.491 q^{78} -423.604 q^{79} +549.559 q^{80} +81.0000 q^{81} -336.409 q^{82} +1032.20 q^{83} -183.238 q^{84} -1071.15 q^{85} -1054.81 q^{86} +247.630 q^{87} +1527.65 q^{88} -86.9284 q^{89} -279.199 q^{90} -2124.90 q^{91} -374.627 q^{92} +384.127 q^{93} -1411.31 q^{94} +874.333 q^{95} -273.494 q^{96} +1056.41 q^{97} +1280.87 q^{98} -560.656 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) $ 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * 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 $

$2$	2.43477	0.860822	0.430411	−	0.902633i	$-0.358369\pi$
$2$	2.43477	0.860822	0.430411	+	0.902633i	$0.358369\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	−2.07188	−0.258985
$5$	−12.7413	−1.13961	−0.569807	−	0.821779i	$-0.692982\pi$
$5$	−12.7413	−1.13961	−0.569807	+	0.821779i	$0.692982\pi$
$6$	−7.30432	−0.496996
$7$	−29.4801	−1.59177	−0.795887	−	0.605445i	$-0.792995\pi$
$7$	−29.4801	−1.59177	−0.795887	+	0.605445i	$0.792995\pi$
$8$	−24.5227	−1.08376
$9$	9.00000	0.333333
$10$	−31.0221	−0.981005
$11$	−62.2952	−1.70752	−0.853759	−	0.520669i	$-0.825683\pi$
$11$	−62.2952	−1.70752	−0.853759	+	0.520669i	$0.825683\pi$
$12$	6.21565	0.149525
$13$	72.0794	1.53779	0.768893	−	0.639377i	$-0.220808\pi$
$13$	72.0794	1.53779	0.768893	+	0.639377i	$0.220808\pi$
$14$	−71.7773	−1.37023
$15$	38.2238	0.657956
$16$	−43.1322	−0.673941
$17$	84.0694	1.19940	0.599700	−	0.800225i	$-0.295286\pi$
$17$	84.0694	1.19940	0.599700	+	0.800225i	$0.295286\pi$
$18$	21.9130	0.286941
$19$	−68.6221	−0.828578	−0.414289	−	0.910145i	$-0.635970\pi$
$19$	−68.6221	−0.828578	−0.414289	+	0.910145i	$0.635970\pi$
$20$	26.3984	0.295143
$21$	88.4402	0.919011
$22$	−151.675	−1.46987
$23$	180.815	1.63924	0.819620	−	0.572908i	$-0.194185\pi$
$23$	180.815	1.63924	0.819620	+	0.572908i	$0.194185\pi$
$24$	73.5682	0.625711
$25$	37.3399	0.298720
$26$	175.497	1.32376
$27$	−27.0000	−0.192450
$28$	61.0793	0.412246
$29$	−82.5432	−0.528548	−0.264274	−	0.964448i	$-0.585132\pi$
$29$	−82.5432	−0.528548	−0.264274	+	0.964448i	$0.585132\pi$
$30$	93.0663	0.566383
$31$	−128.042	−0.741841	−0.370921	−	0.928665i	$-0.620958\pi$
$31$	−128.042	−0.741841	−0.370921	+	0.928665i	$0.620958\pi$
$32$	91.1648	0.503619
$33$	186.885	0.985836
$34$	204.690	1.03247
$35$	375.613	1.81401
$36$	−18.6469	−0.0863285
$37$	195.069	0.866734	0.433367	−	0.901218i	$-0.357325\pi$
$37$	195.069	0.866734	0.433367	+	0.901218i	$0.357325\pi$
$38$	−167.079	−0.713259
$39$	−216.238	−0.887841
$40$	312.451	1.23507
$41$	−138.169	−0.526300	−0.263150	−	0.964755i	$-0.584761\pi$
$41$	−138.169	−0.526300	−0.263150	+	0.964755i	$0.584761\pi$
$42$	215.332	0.791105
$43$	−433.228	−1.53644	−0.768218	−	0.640189i	$-0.778856\pi$
$43$	−433.228	−1.53644	−0.768218	+	0.640189i	$0.778856\pi$
$44$	129.068	0.442222
$45$	−114.671	−0.379871
$46$	440.243	1.41109
$47$	−579.648	−1.79894	−0.899472	−	0.436978i	$-0.856049\pi$
$47$	−579.648	−1.79894	−0.899472	+	0.436978i	$0.856049\pi$
$48$	129.397	0.389100
$49$	526.074	1.53374
$50$	90.9143	0.257144
$51$	−252.208	−0.692474
$52$	−149.340	−0.398264
$53$	459.505	1.19090	0.595451	−	0.803392i	$-0.296973\pi$
$53$	459.505	1.19090	0.595451	+	0.803392i	$0.296973\pi$
$54$	−65.7389	−0.165665
$55$	793.719	1.94591
$56$	722.932	1.72510
$57$	205.866	0.478380
$58$	−200.974	−0.454986
$59$	−710.579	−1.56796	−0.783979	−	0.620787i	$-0.786813\pi$
$59$	−710.579	−1.56796	−0.783979	+	0.620787i	$0.786813\pi$
$60$	−79.1953	−0.170401
$61$	−48.8133	−0.102457	−0.0512287	−	0.998687i	$-0.516314\pi$
$61$	−48.8133	−0.102457	−0.0512287	+	0.998687i	$0.516314\pi$
$62$	−311.754	−0.638593
$63$	−265.321	−0.530591
$64$	567.023	1.10747
$65$	−918.383	−1.75248
$66$	455.024	0.848629
$67$	−138.593	−0.252713	−0.126357	−	0.991985i	$-0.540328\pi$
$67$	−138.593	−0.252713	−0.126357	+	0.991985i	$0.540328\pi$
$68$	−174.182	−0.310627
$69$	−542.445	−0.946416
$70$	914.533	1.56154
$71$	596.513	0.997086	0.498543	−	0.866865i	$-0.333869\pi$
$71$	596.513	0.997086	0.498543	+	0.866865i	$0.333869\pi$
$72$	−220.705	−0.361254
$73$	284.148	0.455576	0.227788	−	0.973711i	$-0.426851\pi$
$73$	284.148	0.455576	0.227788	+	0.973711i	$0.426851\pi$
$74$	474.949	0.746104
$75$	−112.020	−0.172466
$76$	142.177	0.214590
$77$	1836.47	2.71798
$78$	−526.491	−0.764273
$79$	−423.604	−0.603281	−0.301640	−	0.953422i	$-0.597534\pi$
$79$	−423.604	−0.603281	−0.301640	+	0.953422i	$0.597534\pi$
$80$	549.559	0.768033
$81$	81.0000	0.111111
$82$	−336.409	−0.453051
$83$	1032.20	1.36505	0.682525	−	0.730862i	$-0.260882\pi$
$83$	1032.20	1.36505	0.682525	+	0.730862i	$0.260882\pi$
$84$	−183.238	−0.238010
$85$	−1071.15	−1.36685
$86$	−1054.81	−1.32260
$87$	247.630	0.305157
$88$	1527.65	1.85054
$89$	−86.9284	−0.103532	−0.0517662	−	0.998659i	$-0.516485\pi$
$89$	−86.9284	−0.103532	−0.0517662	+	0.998659i	$0.516485\pi$
$90$	−279.199	−0.327002
$91$	−2124.90	−2.44781
$92$	−374.627	−0.424539
$93$	384.127	0.428302
$94$	−1411.31	−1.54857
$95$	874.333	0.944259
$96$	−273.494	−0.290765
$97$	1056.41	1.10580	0.552898	−	0.833249i	$-0.313522\pi$
$97$	1056.41	1.10580	0.552898	+	0.833249i	$0.313522\pi$
$98$	1280.87	1.32028
$99$	−560.656	−0.569173
$100$	−77.3640	−0.0773640
$101$	−757.968	−0.746739	−0.373369	−	0.927683i	$-0.621798\pi$
$101$	−757.968	−0.746739	−0.373369	+	0.927683i	$0.621798\pi$
$102$	−614.069	−0.596097
$103$	781.918	0.748006	0.374003	−	0.927427i	$-0.377985\pi$
$103$	781.918	0.748006	0.374003	+	0.927427i	$0.377985\pi$
$104$	−1767.58	−1.66659
$105$	−1126.84	−1.04732
$106$	1118.79	1.02515
$107$	732.858	0.662131	0.331066	−	0.943608i	$-0.392592\pi$
$107$	732.858	0.662131	0.331066	+	0.943608i	$0.392592\pi$
$108$	55.9408	0.0498418
$109$	−208.451	−0.183175	−0.0915873	−	0.995797i	$-0.529194\pi$
$109$	−208.451	−0.183175	−0.0915873	+	0.995797i	$0.529194\pi$
$110$	1932.53	1.67508
$111$	−585.207	−0.500409
$112$	1271.54	1.07276
$113$	1889.19	1.57274	0.786371	−	0.617754i	$-0.211957\pi$
$113$	1889.19	1.57274	0.786371	+	0.617754i	$0.211957\pi$
$114$	501.238	0.411800
$115$	−2303.81	−1.86810
$116$	171.020	0.136886
$117$	648.714	0.512595
$118$	−1730.10	−1.34973
$119$	−2478.37	−1.90918
$120$	−937.353	−0.713068
$121$	2549.69	1.91562
$122$	−118.849	−0.0881975
$123$	414.506	0.303859
$124$	265.289	0.192126
$125$	1116.90	0.799189
$126$	−645.995	−0.456745
$127$	250.527	0.175045	0.0875223	−	0.996163i	$-0.472105\pi$
$127$	250.527	0.175045	0.0875223	+	0.996163i	$0.472105\pi$
$128$	651.255	0.449713
$129$	1299.69	0.887061
$130$	−2236.05	−1.50858
$131$	1410.99	0.941057	0.470528	−	0.882385i	$-0.344063\pi$
$131$	1410.99	0.941057	0.470528	+	0.882385i	$0.344063\pi$
$132$	−387.205	−0.255317
$133$	2022.98	1.31891
$134$	−337.442	−0.217541
$135$	344.014	0.219319
$136$	−2061.61	−1.29987
$137$	−2370.55	−1.47832	−0.739158	−	0.673532i	$-0.764776\pi$
$137$	−2370.55	−1.47832	−0.739158	+	0.673532i	$0.764776\pi$
$138$	−1320.73	−0.814695
$139$	−1379.10	−0.841536	−0.420768	−	0.907168i	$-0.638239\pi$
$139$	−1379.10	−0.841536	−0.420768	+	0.907168i	$0.638239\pi$
$140$	−778.227	−0.469802
$141$	1738.95	1.03862
$142$	1452.37	0.858313
$143$	−4490.20	−2.62580
$144$	−388.190	−0.224647
$145$	1051.71	0.602341
$146$	691.836	0.392170
$147$	−1578.22	−0.885508
$148$	−404.160	−0.224471
$149$	−2438.75	−1.34088	−0.670438	−	0.741966i	$-0.733894\pi$
$149$	−2438.75	−1.34088	−0.670438	+	0.741966i	$0.733894\pi$
$150$	−272.743	−0.148462
$151$	−269.446	−0.145213	−0.0726066	−	0.997361i	$-0.523132\pi$
$151$	−269.446	−0.145213	−0.0726066	+	0.997361i	$0.523132\pi$
$152$	1682.80	0.897982
$153$	756.624	0.399800
$154$	4471.38	2.33970
$155$	1631.42	0.845412
$156$	448.020	0.229938
$157$	−157.000	−0.0798087
$157$	−157.000	−0.0798087
$158$	−1031.38	−0.519317
$159$	−1378.51	−0.687568
$160$	−1161.55	−0.573931
$161$	−5330.44	−2.60930
$162$	197.217	0.0956469
$163$	370.432	0.178003	0.0890015	−	0.996031i	$-0.471632\pi$
$163$	370.432	0.178003	0.0890015	+	0.996031i	$0.471632\pi$
$164$	286.269	0.136304
$165$	−2381.16	−1.12347
$166$	2513.18	1.17507
$167$	3772.11	1.74787	0.873937	−	0.486039i	$-0.161559\pi$
$167$	3772.11	1.74787	0.873937	+	0.486039i	$0.161559\pi$
$168$	−2168.80	−0.995990
$169$	2998.44	1.36479
$170$	−2608.01	−1.17662
$171$	−617.599	−0.276193
$172$	897.599	0.397914
$173$	4391.91	1.93012	0.965061	−	0.262027i	$-0.0843910\pi$
$173$	4391.91	1.93012	0.965061	+	0.262027i	$0.0843910\pi$
$174$	602.922	0.262686
$175$	−1100.78	−0.475494
$176$	2686.93	1.15077
$177$	2131.74	0.905261
$178$	−211.651	−0.0891230
$179$	1502.22	0.627271	0.313635	−	0.949544i	$-0.398453\pi$
$179$	1502.22	0.627271	0.313635	+	0.949544i	$0.398453\pi$
$180$	237.586	0.0983811
$181$	−2985.24	−1.22592	−0.612960	−	0.790114i	$-0.710021\pi$
$181$	−2985.24	−1.22592	−0.612960	+	0.790114i	$0.710021\pi$
$182$	−5173.66	−2.10713
$183$	146.440	0.0591538
$184$	−4434.08	−1.77655
$185$	−2485.43	−0.987742
$186$	935.262	0.368692
$187$	−5237.11	−2.04800
$188$	1200.96	0.465900
$189$	795.962	0.306337
$190$	2128.80	0.812839
$191$	−229.894	−0.0870917	−0.0435459	−	0.999051i	$-0.513865\pi$
$191$	−229.894	−0.0870917	−0.0435459	+	0.999051i	$0.513865\pi$
$192$	−1701.07	−0.639397
$193$	−4793.72	−1.78787	−0.893937	−	0.448193i	$-0.852068\pi$
$193$	−4793.72	−1.78787	−0.893937	+	0.448193i	$0.852068\pi$
$194$	2572.12	0.951894
$195$	2755.15	1.01180
$196$	−1089.96	−0.397218
$197$	−304.699	−0.110197	−0.0550987	−	0.998481i	$-0.517547\pi$
$197$	−304.699	−0.110197	−0.0550987	+	0.998481i	$0.517547\pi$
$198$	−1365.07	−0.489956
$199$	3672.29	1.30815	0.654075	−	0.756430i	$-0.273058\pi$
$199$	3672.29	1.30815	0.654075	+	0.756430i	$0.273058\pi$
$200$	−915.678	−0.323741
$201$	415.778	0.145904
$202$	−1845.48	−0.642809
$203$	2433.38	0.841329
$204$	522.546	0.179341
$205$	1760.44	0.599779
$206$	1903.79	0.643900
$207$	1627.33	0.546413
$208$	−3108.94	−1.03638
$209$	4274.82	1.41481
$210$	−2743.60	−0.901554
$211$	859.712	0.280498	0.140249	−	0.990116i	$-0.455210\pi$
$211$	859.712	0.280498	0.140249	+	0.990116i	$0.455210\pi$
$212$	−952.040	−0.308426
$213$	−1789.54	−0.575668
$214$	1784.34	0.569977
$215$	5519.88	1.75094
$216$	662.114	0.208570
$217$	3774.70	1.18084
$218$	−507.532	−0.157681
$219$	−852.445	−0.263027
$220$	−1644.49	−0.503962
$221$	6059.67	1.84442
$222$	−1424.85	−0.430763
$223$	−1614.10	−0.484700	−0.242350	−	0.970189i	$-0.577918\pi$
$223$	−1614.10	−0.484700	−0.242350	+	0.970189i	$0.577918\pi$
$224$	−2687.54	−0.801648
$225$	336.059	0.0995732
$226$	4599.74	1.35385
$227$	1988.15	0.581313	0.290657	−	0.956827i	$-0.406126\pi$
$227$	1988.15	0.581313	0.290657	+	0.956827i	$0.406126\pi$
$228$	−426.531	−0.123893
$229$	−6901.42	−1.99152	−0.995761	−	0.0919778i	$-0.970681\pi$
$229$	−6901.42	−1.99152	−0.995761	+	0.0919778i	$0.970681\pi$
$230$	−5609.26	−1.60810
$231$	−5509.40	−1.56923
$232$	2024.19	0.572820
$233$	2495.85	0.701755	0.350877	−	0.936421i	$-0.385883\pi$
$233$	2495.85	0.701755	0.350877	+	0.936421i	$0.385883\pi$
$234$	1579.47	0.441253
$235$	7385.46	2.05010
$236$	1472.24	0.406078
$237$	1270.81	0.348304
$238$	−6034.27	−1.64346
$239$	4920.71	1.33177	0.665887	−	0.746052i	$-0.268053\pi$
$239$	4920.71	1.33177	0.665887	+	0.746052i	$0.268053\pi$
$240$	−1648.68	−0.443424
$241$	−2918.74	−0.780136	−0.390068	−	0.920786i	$-0.627549\pi$
$241$	−2918.74	−0.780136	−0.390068	+	0.920786i	$0.627549\pi$
$242$	6207.91	1.64901
$243$	−243.000	−0.0641500
$244$	101.135	0.0265350
$245$	−6702.86	−1.74788
$246$	1009.23	0.261569
$247$	−4946.24	−1.27418
$248$	3139.95	0.803980
$249$	−3096.61	−0.788112
$250$	2719.40	0.687959
$251$	−6297.09	−1.58354	−0.791770	−	0.610819i	$-0.790840\pi$
$251$	−6297.09	−1.58354	−0.791770	+	0.610819i	$0.790840\pi$
$252$	549.713	0.137415
$253$	−11263.9	−2.79903
$254$	609.976	0.150682
$255$	3213.45	0.789153
$256$	−2950.53	−0.720344
$257$	1558.87	0.378364	0.189182	−	0.981942i	$-0.439416\pi$
$257$	1558.87	0.378364	0.189182	+	0.981942i	$0.439416\pi$
$258$	3164.44	0.763602
$259$	−5750.65	−1.37964
$260$	1902.78	0.453867
$261$	−742.889	−0.176183
$262$	3435.43	0.810082
$263$	2276.79	0.533814	0.266907	−	0.963722i	$-0.413998\pi$
$263$	2276.79	0.533814	0.266907	+	0.963722i	$0.413998\pi$
$264$	−4582.94	−1.06841
$265$	−5854.67	−1.35717
$266$	4925.51	1.13535
$267$	260.785	0.0597745
$268$	287.148	0.0654491
$269$	503.020	0.114014	0.0570069	−	0.998374i	$-0.481844\pi$
$269$	503.020	0.114014	0.0570069	+	0.998374i	$0.481844\pi$
$270$	837.596	0.188794
$271$	1871.81	0.419573	0.209787	−	0.977747i	$-0.432723\pi$
$271$	1871.81	0.419573	0.209787	+	0.977747i	$0.432723\pi$
$272$	−3626.10	−0.808326
$273$	6374.71	1.41324
$274$	−5771.74	−1.27257
$275$	−2326.10	−0.510069
$276$	1123.88	0.245108
$277$	8784.15	1.90537	0.952686	−	0.303956i	$-0.0983075\pi$
$277$	8784.15	1.90537	0.952686	+	0.303956i	$0.0983075\pi$
$278$	−3357.79	−0.724413
$279$	−1152.38	−0.247280
$280$	−9211.07	−1.96595
$281$	−4636.83	−0.984378	−0.492189	−	0.870488i	$-0.663803\pi$
$281$	−4636.83	−0.984378	−0.492189	+	0.870488i	$0.663803\pi$
$282$	4233.94	0.894068
$283$	−503.695	−0.105801	−0.0529003	−	0.998600i	$-0.516847\pi$
$283$	−503.695	−0.105801	−0.0529003	+	0.998600i	$0.516847\pi$
$284$	−1235.91	−0.258231
$285$	−2623.00	−0.545168
$286$	−10932.6	−2.26034
$287$	4073.22	0.837751
$288$	820.483	0.167873
$289$	2154.66	0.438563
$290$	2560.66	0.518508
$291$	−3169.23	−0.638432
$292$	−588.722	−0.117988
$293$	−5742.45	−1.14497	−0.572487	−	0.819914i	$-0.694021\pi$
$293$	−5742.45	−1.14497	−0.572487	+	0.819914i	$0.694021\pi$
$294$	−3842.61	−0.762265
$295$	9053.68	1.78687
$296$	−4783.63	−0.939334
$297$	1681.97	0.328612
$298$	−5937.81	−1.15426
$299$	13033.0	2.52080
$300$	232.092	0.0446661
$301$	12771.6	2.44566
$302$	−656.040	−0.125003
$303$	2273.90	0.431130
$304$	2959.82	0.558413
$305$	621.943	0.116762
$306$	1842.21	0.344157
$307$	1263.96	0.234977	0.117488	−	0.993074i	$-0.462516\pi$
$307$	1263.96	0.234977	0.117488	+	0.993074i	$0.462516\pi$
$308$	−3804.94	−0.703918
$309$	−2345.75	−0.431862
$310$	3972.14	0.727750
$311$	4847.26	0.883804	0.441902	−	0.897063i	$-0.354304\pi$
$311$	4847.26	0.883804	0.441902	+	0.897063i	$0.354304\pi$
$312$	5302.75	0.962209
$313$	−8946.56	−1.61562	−0.807811	−	0.589442i	$-0.799348\pi$
$313$	−8946.56	−1.61562	−0.807811	+	0.589442i	$0.799348\pi$
$314$	−382.259	−0.0687011
$315$	3380.52	0.604669
$316$	877.658	0.156241
$317$	1333.22	0.236218	0.118109	−	0.993001i	$-0.462317\pi$
$317$	1333.22	0.236218	0.118109	+	0.993001i	$0.462317\pi$
$318$	−3356.37	−0.591873
$319$	5142.04	0.902505
$320$	−7224.60	−1.26209
$321$	−2198.57	−0.382282
$322$	−12978.4	−2.24614
$323$	−5769.02	−0.993798
$324$	−167.823	−0.0287762
$325$	2691.44	0.459367
$326$	901.918	0.153229
$327$	625.354	0.105756
$328$	3388.27	0.570384
$329$	17088.1	2.86351
$330$	−5797.58	−0.967110
$331$	−2583.26	−0.428970	−0.214485	−	0.976727i	$-0.568807\pi$
$331$	−2583.26	−0.428970	−0.214485	+	0.976727i	$0.568807\pi$
$332$	−2138.61	−0.353528
$333$	1755.62	0.288911
$334$	9184.24	1.50461
$335$	1765.85	0.287996
$336$	−3814.62	−0.619359
$337$	−2202.06	−0.355945	−0.177973	−	0.984035i	$-0.556954\pi$
$337$	−2202.06	−0.355945	−0.177973	+	0.984035i	$0.556954\pi$
$338$	7300.51	1.17484
$339$	−5667.56	−0.908023
$340$	2219.30	0.353995
$341$	7976.41	1.26671
$342$	−1503.71	−0.237753
$343$	−5397.05	−0.849601
$344$	10624.0	1.66513
$345$	6911.44	1.07855
$346$	10693.3	1.66149
$347$	−10041.4	−1.55347	−0.776733	−	0.629830i	$-0.783125\pi$
$347$	−10041.4	−1.55347	−0.776733	+	0.629830i	$0.783125\pi$
$348$	−513.060	−0.0790313
$349$	−5456.07	−0.836838	−0.418419	−	0.908254i	$-0.637416\pi$
$349$	−5456.07	−0.836838	−0.418419	+	0.908254i	$0.637416\pi$
$350$	−2680.16	−0.409316
$351$	−1946.14	−0.295947
$352$	−5679.12	−0.859938
$353$	4638.97	0.699455	0.349728	−	0.936851i	$-0.386274\pi$
$353$	4638.97	0.699455	0.349728	+	0.936851i	$0.386274\pi$
$354$	5190.30	0.779269
$355$	−7600.33	−1.13629
$356$	180.105	0.0268134
$357$	7435.11	1.10226
$358$	3657.57	0.539968
$359$	716.412	0.105322	0.0526612	−	0.998612i	$-0.483230\pi$
$359$	716.412	0.105322	0.0526612	+	0.998612i	$0.483230\pi$
$360$	2812.06	0.411690
$361$	−2150.01	−0.313458
$362$	−7268.39	−1.05530
$363$	−7649.06	−1.10598
$364$	4402.55	0.633947
$365$	−3620.41	−0.519181
$366$	356.548	0.0509209
$367$	632.258	0.0899281	0.0449640	−	0.998989i	$-0.485683\pi$
$367$	632.258	0.0899281	0.0449640	+	0.998989i	$0.485683\pi$
$368$	−7798.95	−1.10475
$369$	−1243.52	−0.175433
$370$	−6051.45	−0.850270
$371$	−13546.2	−1.89565
$372$	−795.866	−0.110924
$373$	−5222.42	−0.724951	−0.362475	−	0.931993i	$-0.618068\pi$
$373$	−5222.42	−0.724951	−0.362475	+	0.931993i	$0.618068\pi$
$374$	−12751.2	−1.76296
$375$	−3350.70	−0.461412
$376$	14214.6	1.94963
$377$	−5949.66	−0.812794
$378$	1937.99	0.263702
$379$	7679.44	1.04081	0.520404	−	0.853920i	$-0.325781\pi$
$379$	7679.44	1.04081	0.520404	+	0.853920i	$0.325781\pi$
$380$	−1811.52	−0.244549
$381$	−751.580	−0.101062
$382$	−559.739	−0.0749705
$383$	14666.6	1.95673	0.978364	−	0.206893i	$-0.0663351\pi$
$383$	14666.6	1.95673	0.978364	+	0.206893i	$0.0663351\pi$
$384$	−1953.76	−0.259642
$385$	−23398.9	−3.09745
$386$	−11671.6	−1.53904
$387$	−3899.06	−0.512145
$388$	−2188.76	−0.286385
$389$	2844.05	0.370692	0.185346	−	0.982673i	$-0.440659\pi$
$389$	2844.05	0.370692	0.185346	+	0.982673i	$0.440659\pi$
$390$	6708.16	0.870976
$391$	15201.0	1.96611
$392$	−12900.8	−1.66221
$393$	−4232.96	−0.543319
$394$	−741.872	−0.0948603
$395$	5397.25	0.687507
$396$	1161.61	0.147407
$397$	3027.64	0.382752	0.191376	−	0.981517i	$-0.438705\pi$
$397$	3027.64	0.382752	0.191376	+	0.981517i	$0.438705\pi$
$398$	8941.19	1.12608
$399$	−6068.95	−0.761473
$400$	−1610.56	−0.201319
$401$	−1671.64	−0.208173	−0.104087	−	0.994568i	$-0.533192\pi$
$401$	−1671.64	−0.208173	−0.104087	+	0.994568i	$0.533192\pi$
$402$	1012.33	0.125598
$403$	−9229.21	−1.14079
$404$	1570.42	0.193394
$405$	−1032.04	−0.126624
$406$	5924.73	0.724235
$407$	−12151.9	−1.47996
$408$	6184.83	0.750478
$409$	1276.97	0.154381	0.0771907	−	0.997016i	$-0.475405\pi$
$409$	1276.97	0.154381	0.0771907	+	0.997016i	$0.475405\pi$
$410$	4286.28	0.516303
$411$	7111.64	0.853506
$412$	−1620.04	−0.193723
$413$	20947.9	2.49584
$414$	3962.19	0.470365
$415$	−13151.6	−1.55563
$416$	6571.10	0.774458
$417$	4137.29	0.485861
$418$	10408.2	1.21790
$419$	15141.2	1.76539	0.882694	−	0.469948i	$-0.155728\pi$
$419$	15141.2	1.76539	0.882694	+	0.469948i	$0.155728\pi$
$420$	2334.68	0.271240
$421$	−4494.65	−0.520322	−0.260161	−	0.965565i	$-0.583776\pi$
$421$	−4494.65	−0.520322	−0.260161	+	0.965565i	$0.583776\pi$
$422$	2093.20	0.241459
$423$	−5216.84	−0.599648
$424$	−11268.3	−1.29066
$425$	3139.15	0.358284
$426$	−4357.12	−0.495547
$427$	1439.02	0.163089
$428$	−1518.40	−0.171482
$429$	13470.6	1.51600
$430$	13439.7	1.50725
$431$	4641.09	0.518686	0.259343	−	0.965785i	$-0.416494\pi$
$431$	4641.09	0.518686	0.259343	+	0.965785i	$0.416494\pi$
$432$	1164.57	0.129700
$433$	15216.1	1.68877	0.844385	−	0.535737i	$-0.179966\pi$
$433$	15216.1	1.68877	0.844385	+	0.535737i	$0.179966\pi$
$434$	9190.53	1.01650
$435$	−3155.12	−0.347761
$436$	431.887	0.0474395
$437$	−12407.9	−1.35824
$438$	−2075.51	−0.226419
$439$	5195.01	0.564793	0.282397	−	0.959298i	$-0.408871\pi$
$439$	5195.01	0.564793	0.282397	+	0.959298i	$0.408871\pi$
$440$	−19464.2	−2.10890
$441$	4734.67	0.511248
$442$	14753.9	1.58772
$443$	10890.7	1.16802	0.584012	−	0.811745i	$-0.301482\pi$
$443$	10890.7	1.16802	0.584012	+	0.811745i	$0.301482\pi$
$444$	1212.48	0.129599
$445$	1107.58	0.117987
$446$	−3929.96	−0.417240
$447$	7316.26	0.774155
$448$	−16715.9	−1.76284
$449$	−524.696	−0.0551491	−0.0275746	−	0.999620i	$-0.508778\pi$
$449$	−524.696	−0.0551491	−0.0275746	+	0.999620i	$0.508778\pi$
$450$	818.228	0.0857148
$451$	8607.23	0.898667
$452$	−3914.18	−0.407317
$453$	808.338	0.0838389
$454$	4840.69	0.500407
$455$	27074.0	2.78956
$456$	−5048.41	−0.518450
$457$	4433.31	0.453789	0.226894	−	0.973919i	$-0.427143\pi$
$457$	4433.31	0.453789	0.226894	+	0.973919i	$0.427143\pi$
$458$	−16803.4	−1.71435
$459$	−2269.87	−0.230825
$460$	4773.23	0.483811
$461$	8038.71	0.812147	0.406074	−	0.913840i	$-0.366898\pi$
$461$	8038.71	0.812147	0.406074	+	0.913840i	$0.366898\pi$
$462$	−13414.1	−1.35083
$463$	2678.60	0.268866	0.134433	−	0.990923i	$-0.457079\pi$
$463$	2678.60	0.268866	0.134433	+	0.990923i	$0.457079\pi$
$464$	3560.27	0.356210
$465$	−4894.26	−0.488099
$466$	6076.84	0.604086
$467$	11851.9	1.17439	0.587197	−	0.809444i	$-0.300232\pi$
$467$	11851.9	1.17439	0.587197	+	0.809444i	$0.300232\pi$
$468$	−1344.06	−0.132755
$469$	4085.72	0.402263
$470$	17981.9	1.76477
$471$	471.000	0.0460776
$472$	17425.4	1.69929
$473$	26988.0	2.62349
$474$	3094.14	0.299828
$475$	−2562.35	−0.247513
$476$	5134.89	0.494449
$477$	4135.54	0.396967
$478$	11980.8	1.14642
$479$	648.708	0.0618793	0.0309397	−	0.999521i	$-0.490150\pi$
$479$	648.708	0.0618793	0.0309397	+	0.999521i	$0.490150\pi$
$480$	3484.66	0.331359
$481$	14060.5	1.33285
$482$	−7106.48	−0.671559
$483$	15991.3	1.50648
$484$	−5282.65	−0.496117
$485$	−13460.0	−1.26018
$486$	−591.650	−0.0552218
$487$	−3493.55	−0.325067	−0.162534	−	0.986703i	$-0.551967\pi$
$487$	−3493.55	−0.325067	−0.162534	+	0.986703i	$0.551967\pi$
$488$	1197.03	0.111039
$489$	−1111.30	−0.102770
$490$	−16319.9	−1.50461
$491$	11420.8	1.04972	0.524859	−	0.851189i	$-0.324118\pi$
$491$	11420.8	1.04972	0.524859	+	0.851189i	$0.324118\pi$
$492$	−858.807	−0.0786952
$493$	−6939.36	−0.633941
$494$	−12043.0	−1.09684
$495$	7143.47	0.648637
$496$	5522.75	0.499957
$497$	−17585.2	−1.58714
$498$	−7539.55	−0.678425
$499$	3932.15	0.352760	0.176380	−	0.984322i	$-0.443561\pi$
$499$	3932.15	0.352760	0.176380	+	0.984322i	$0.443561\pi$
$500$	−2314.09	−0.206978
$501$	−11316.3	−1.00914
$502$	−15332.0	−1.36315
$503$	6810.53	0.603711	0.301856	−	0.953354i	$-0.402394\pi$
$503$	6810.53	0.603711	0.301856	+	0.953354i	$0.402394\pi$
$504$	6506.39	0.575035
$505$	9657.47	0.850994
$506$	−27425.0	−2.40947
$507$	−8995.31	−0.787960
$508$	−519.062	−0.0453340
$509$	8428.39	0.733952	0.366976	−	0.930230i	$-0.380393\pi$
$509$	8428.39	0.733952	0.366976	+	0.930230i	$0.380393\pi$
$510$	7824.02	0.679321
$511$	−8376.71	−0.725174
$512$	−12393.9	−1.06980
$513$	1852.80	0.159460
$514$	3795.49	0.325704
$515$	−9962.63	−0.852438
$516$	−2692.80	−0.229736
$517$	36109.3	3.07173
$518$	−14001.5	−1.18763
$519$	−13175.7	−1.11436
$520$	22521.3	1.89927
$521$	−17383.0	−1.46173	−0.730867	−	0.682520i	$-0.760884\pi$
$521$	−17383.0	−1.46173	−0.730867	+	0.682520i	$0.760884\pi$
$522$	−1808.77	−0.151662
$523$	936.723	0.0783175	0.0391587	−	0.999233i	$-0.487532\pi$
$523$	936.723	0.0783175	0.0391587	+	0.999233i	$0.487532\pi$
$524$	−2923.40	−0.243720
$525$	3302.35	0.274527
$526$	5543.47	0.459519
$527$	−10764.4	−0.889765
$528$	−8060.79	−0.664395
$529$	20527.0	1.68711
$530$	−14254.8	−1.16828
$531$	−6395.21	−0.522653
$532$	−4191.39	−0.341578
$533$	−9959.10	−0.809337
$534$	634.953	0.0514552
$535$	−9337.54	−0.754574
$536$	3398.68	0.273881
$537$	−4506.67	−0.362155
$538$	1224.74	0.0981455
$539$	−32771.9	−2.61890
$540$	−712.757	−0.0568004
$541$	11519.5	0.915456	0.457728	−	0.889092i	$-0.348663\pi$
$541$	11519.5	0.915456	0.457728	+	0.889092i	$0.348663\pi$
$542$	4557.43	0.361178
$543$	8955.73	0.707785
$544$	7664.16	0.604041
$545$	2655.94	0.208748
$546$	15521.0	1.21655
$547$	−17222.7	−1.34623	−0.673117	−	0.739536i	$-0.735045\pi$
$547$	−17222.7	−1.34623	−0.673117	+	0.739536i	$0.735045\pi$
$548$	4911.49	0.382862
$549$	−439.319	−0.0341524
$550$	−5663.52	−0.439079
$551$	5664.29	0.437943
$552$	13302.2	1.02569
$553$	12487.9	0.960286
$554$	21387.4	1.64019
$555$	7456.28	0.570273
$556$	2857.33	0.217946
$557$	10493.6	0.798254	0.399127	−	0.916896i	$-0.369313\pi$
$557$	10493.6	0.798254	0.399127	+	0.916896i	$0.369313\pi$
$558$	−2805.78	−0.212864
$559$	−31226.8	−2.36271
$560$	−16201.0	−1.22253
$561$	15711.3	1.18241
$562$	−11289.6	−0.847374
$563$	−8803.10	−0.658981	−0.329491	−	0.944159i	$-0.606877\pi$
$563$	−8803.10	−0.658981	−0.329491	+	0.944159i	$0.606877\pi$
$564$	−3602.89	−0.268988
$565$	−24070.7	−1.79232
$566$	−1226.38	−0.0910755
$567$	−2387.89	−0.176864
$568$	−14628.1	−1.08060
$569$	8334.29	0.614045	0.307023	−	0.951702i	$-0.400667\pi$
$569$	8334.29	0.614045	0.307023	+	0.951702i	$0.400667\pi$
$570$	−6386.40	−0.469293
$571$	−10829.6	−0.793701	−0.396850	−	0.917883i	$-0.629897\pi$
$571$	−10829.6	−0.793701	−0.396850	+	0.917883i	$0.629897\pi$
$572$	9303.16	0.680043
$573$	689.681	0.0502824
$574$	9917.36	0.721154
$575$	6751.62	0.489673
$576$	5103.21	0.369156
$577$	1539.51	0.111076	0.0555379	−	0.998457i	$-0.482313\pi$
$577$	1539.51	0.111076	0.0555379	+	0.998457i	$0.482313\pi$
$578$	5246.10	0.377524
$579$	14381.2	1.03223
$580$	−2179.01	−0.155997
$581$	−30429.5	−2.17285
$582$	−7716.36	−0.549577
$583$	−28624.9	−2.03349
$584$	−6968.10	−0.493736
$585$	−8265.44	−0.584161
$586$	−13981.5	−0.985618
$587$	12347.1	0.868174	0.434087	−	0.900871i	$-0.357071\pi$
$587$	12347.1	0.868174	0.434087	+	0.900871i	$0.357071\pi$
$588$	3269.89	0.229334
$589$	8786.53	0.614674
$590$	22043.7	1.53817
$591$	914.096	0.0636225
$592$	−8413.76	−0.584128
$593$	20674.3	1.43169	0.715844	−	0.698260i	$-0.246042\pi$
$593$	20674.3	1.43169	0.715844	+	0.698260i	$0.246042\pi$
$594$	4095.21	0.282876
$595$	31577.6	2.17572
$596$	5052.81	0.347267
$597$	−11016.9	−0.755261
$598$	31732.5	2.16996
$599$	1456.74	0.0993666	0.0496833	−	0.998765i	$-0.484179\pi$
$599$	1456.74	0.0993666	0.0496833	+	0.998765i	$0.484179\pi$
$600$	2747.03	0.186912
$601$	−2571.27	−0.174516	−0.0872581	−	0.996186i	$-0.527810\pi$
$601$	−2571.27	−0.174516	−0.0872581	+	0.996186i	$0.527810\pi$
$602$	31096.0	2.10528
$603$	−1247.34	−0.0842378
$604$	558.260	0.0376081
$605$	−32486.2	−2.18306
$606$	5536.44	0.371126
$607$	−19235.6	−1.28624	−0.643121	−	0.765765i	$-0.722361\pi$
$607$	−19235.6	−1.28624	−0.643121	+	0.765765i	$0.722361\pi$
$608$	−6255.92	−0.417288
$609$	−7300.14	−0.485741
$610$	1514.29	0.100511
$611$	−41780.7	−2.76639
$612$	−1567.64	−0.103542
$613$	−9413.60	−0.620248	−0.310124	−	0.950696i	$-0.600371\pi$
$613$	−9413.60	−0.620248	−0.310124	+	0.950696i	$0.600371\pi$
$614$	3077.45	0.202273
$615$	−5281.33	−0.346282
$616$	−45035.2	−2.94565
$617$	−4447.93	−0.290222	−0.145111	−	0.989415i	$-0.546354\pi$
$617$	−4447.93	−0.290222	−0.145111	+	0.989415i	$0.546354\pi$
$618$	−5711.38	−0.371756
$619$	21607.6	1.40304	0.701521	−	0.712648i	$-0.252505\pi$
$619$	21607.6	1.40304	0.701521	+	0.712648i	$0.252505\pi$
$620$	−3380.11	−0.218949
$621$	−4882.00	−0.315472
$622$	11802.0	0.760798
$623$	2562.65	0.164800
$624$	9326.83	0.598353
$625$	−18898.2	−1.20949
$626$	−21782.8	−1.39076
$627$	−12824.5	−0.816842
$628$	325.286	0.0206693
$629$	16399.3	1.03956
$630$	8230.80	0.520513
$631$	−23503.7	−1.48283	−0.741417	−	0.671045i	$-0.765846\pi$
$631$	−23503.7	−1.48283	−0.741417	+	0.671045i	$0.765846\pi$
$632$	10387.9	0.653813
$633$	−2579.14	−0.161945
$634$	3246.09	0.203342
$635$	−3192.03	−0.199483
$636$	2856.12	0.178070
$637$	37919.1	2.35857
$638$	12519.7	0.776896
$639$	5368.62	0.332362
$640$	−8297.81	−0.512500
$641$	−24920.6	−1.53558	−0.767789	−	0.640703i	$-0.778643\pi$
$641$	−24920.6	−1.53558	−0.767789	+	0.640703i	$0.778643\pi$
$642$	−5353.02	−0.329076
$643$	−18558.0	−1.13819	−0.569095	−	0.822272i	$-0.692706\pi$
$643$	−18558.0	−1.13819	−0.569095	+	0.822272i	$0.692706\pi$
$644$	11044.0	0.675771
$645$	−16559.6	−1.01091
$646$	−14046.2	−0.855483
$647$	−10022.1	−0.608982	−0.304491	−	0.952515i	$-0.598486\pi$
$647$	−10022.1	−0.608982	−0.304491	+	0.952515i	$0.598486\pi$
$648$	−1986.34	−0.120418
$649$	44265.6	2.67732
$650$	6553.04	0.395433
$651$	−11324.1	−0.681760
$652$	−767.493	−0.0461002
$653$	−14741.4	−0.883422	−0.441711	−	0.897157i	$-0.645628\pi$
$653$	−14741.4	−0.883422	−0.441711	+	0.897157i	$0.645628\pi$
$654$	1522.60	0.0910370
$655$	−17977.8	−1.07244
$656$	5959.52	0.354695
$657$	2557.33	0.151859
$658$	41605.6	2.46498
$659$	−9028.93	−0.533713	−0.266857	−	0.963736i	$-0.585985\pi$
$659$	−9028.93	−0.533713	−0.266857	+	0.963736i	$0.585985\pi$
$660$	4933.48	0.290963
$661$	−28921.0	−1.70181	−0.850907	−	0.525317i	$-0.823947\pi$
$661$	−28921.0	−1.70181	−0.850907	+	0.525317i	$0.823947\pi$
$662$	−6289.66	−0.369267
$663$	−18179.0	−1.06488
$664$	−25312.5	−1.47939
$665$	−25775.4	−1.50305
$666$	4274.54	0.248701
$667$	−14925.0	−0.866417
$668$	−7815.38	−0.452674
$669$	4842.30	0.279842
$670$	4299.44	0.247913
$671$	3040.83	0.174948
$672$	8062.63	0.462831
$673$	12079.8	0.691893	0.345946	−	0.938254i	$-0.387558\pi$
$673$	12079.8	0.691893	0.345946	+	0.938254i	$0.387558\pi$
$674$	−5361.50	−0.306406
$675$	−1008.18	−0.0574886
$676$	−6212.41	−0.353460
$677$	15168.5	0.861114	0.430557	−	0.902563i	$-0.358317\pi$
$677$	15168.5	0.861114	0.430557	+	0.902563i	$0.358317\pi$
$678$	−13799.2	−0.781646
$679$	−31143.1	−1.76018
$680$	26267.5	1.48134
$681$	−5964.45	−0.335621
$682$	19420.8	1.09041
$683$	−6462.49	−0.362050	−0.181025	−	0.983478i	$-0.557942\pi$
$683$	−6462.49	−0.362050	−0.181025	+	0.983478i	$0.557942\pi$
$684$	1279.59	0.0715299
$685$	30203.8	1.68471
$686$	−13140.6	−0.731355
$687$	20704.3	1.14981
$688$	18686.1	1.03547
$689$	33120.8	1.83135
$690$	16827.8	0.928438
$691$	3884.21	0.213838	0.106919	−	0.994268i	$-0.465901\pi$
$691$	3884.21	0.213838	0.106919	+	0.994268i	$0.465901\pi$
$692$	−9099.53	−0.499873
$693$	16528.2	0.905994
$694$	−24448.6	−1.33726
$695$	17571.5	0.959026
$696$	−6072.56	−0.330718
$697$	−11615.7	−0.631245
$698$	−13284.3	−0.720369
$699$	−7487.56	−0.405158
$700$	2280.70	0.123146
$701$	−33442.8	−1.80188	−0.900939	−	0.433946i	$-0.857121\pi$
$701$	−33442.8	−1.80188	−0.900939	+	0.433946i	$0.857121\pi$
$702$	−4738.42	−0.254758
$703$	−13386.0	−0.718157
$704$	−35322.8	−1.89102
$705$	−22156.4	−1.18363
$706$	11294.8	0.602106
$707$	22344.9	1.18864
$708$	−4416.71	−0.234449
$709$	−3456.93	−0.183114	−0.0915569	−	0.995800i	$-0.529184\pi$
$709$	−3456.93	−0.183114	−0.0915569	+	0.995800i	$0.529184\pi$
$710$	−18505.1	−0.978146
$711$	−3812.44	−0.201094
$712$	2131.72	0.112205
$713$	−23152.0	−1.21606
$714$	18102.8	0.948852
$715$	57210.8	2.99239
$716$	−3112.43	−0.162454
$717$	−14762.1	−0.768900
$718$	1744.30	0.0906639
$719$	−21200.2	−1.09963	−0.549815	−	0.835287i	$-0.685302\pi$
$719$	−21200.2	−1.09963	−0.549815	+	0.835287i	$0.685302\pi$
$720$	4946.03	0.256011
$721$	−23051.0	−1.19066
$722$	−5234.78	−0.269831
$723$	8756.23	0.450412
$724$	6185.08	0.317495
$725$	−3082.16	−0.157888
$726$	−18623.7	−0.952054
$727$	−28851.2	−1.47185	−0.735924	−	0.677064i	$-0.763252\pi$
$727$	−28851.2	−1.47185	−0.735924	+	0.677064i	$0.763252\pi$
$728$	52108.5	2.65284
$729$	729.000	0.0370370
$730$	−8814.87	−0.446922
$731$	−36421.2	−1.84280
$732$	−303.406	−0.0153200
$733$	−10043.7	−0.506101	−0.253051	−	0.967453i	$-0.581434\pi$
$733$	−10043.7	−0.506101	−0.253051	+	0.967453i	$0.581434\pi$
$734$	1539.40	0.0774121
$735$	20108.6	1.00914
$736$	16484.0	0.825552
$737$	8633.66	0.431513
$738$	−3027.68	−0.151017
$739$	−5595.67	−0.278539	−0.139269	−	0.990255i	$-0.544475\pi$
$739$	−5595.67	−0.278539	−0.139269	+	0.990255i	$0.544475\pi$
$740$	5149.51	0.255811
$741$	14838.7	0.735646
$742$	−32982.0	−1.63182
$743$	−11495.8	−0.567619	−0.283810	−	0.958881i	$-0.591598\pi$
$743$	−11495.8	−0.567619	−0.283810	+	0.958881i	$0.591598\pi$
$744$	−9419.85	−0.464178
$745$	31072.8	1.52808
$746$	−12715.4	−0.624054
$747$	9289.84	0.455017
$748$	10850.7	0.530402
$749$	−21604.7	−1.05396
$750$	−8158.20	−0.397194
$751$	16846.3	0.818551	0.409276	−	0.912411i	$-0.365781\pi$
$751$	16846.3	0.818551	0.409276	+	0.912411i	$0.365781\pi$
$752$	25001.5	1.21238
$753$	18891.3	0.914258
$754$	−14486.1	−0.699671
$755$	3433.08	0.165487
$756$	−1649.14	−0.0793368
$757$	16510.5	0.792712	0.396356	−	0.918097i	$-0.370275\pi$
$757$	16510.5	0.792712	0.396356	+	0.918097i	$0.370275\pi$
$758$	18697.7	0.895951
$759$	33791.7	1.61602
$760$	−21441.0	−1.02335
$761$	35795.9	1.70513	0.852563	−	0.522625i	$-0.175047\pi$
$761$	35795.9	1.70513	0.852563	+	0.522625i	$0.175047\pi$
$762$	−1829.93	−0.0869964
$763$	6145.16	0.291572
$764$	476.313	0.0225555
$765$	−9640.35	−0.455618
$766$	35709.8	1.68439
$767$	−51218.1	−2.41119
$768$	8851.59	0.415891
$769$	30356.4	1.42351	0.711755	−	0.702428i	$-0.247901\pi$
$769$	30356.4	1.42351	0.711755	+	0.702428i	$0.247901\pi$
$770$	−56971.0	−2.66635
$771$	−4676.60	−0.218448
$772$	9932.03	0.463033
$773$	−18454.6	−0.858686	−0.429343	−	0.903142i	$-0.641255\pi$
$773$	−18454.6	−0.858686	−0.429343	+	0.903142i	$0.641255\pi$
$774$	−9493.32	−0.440866
$775$	−4781.09	−0.221602
$776$	−25906.1	−1.19842
$777$	17251.9	0.796538
$778$	6924.62	0.319100
$779$	9481.42	0.436081
$780$	−5708.35	−0.262040
$781$	−37159.9	−1.70254
$782$	37011.0	1.69247
$783$	2228.67	0.101719
$784$	−22690.8	−1.03365
$785$	2000.38	0.0909511
$786$	−10306.3	−0.467701
$787$	37751.3	1.70989	0.854947	−	0.518715i	$-0.173589\pi$
$787$	37751.3	1.70989	0.854947	+	0.518715i	$0.173589\pi$
$788$	631.300	0.0285395
$789$	−6830.38	−0.308198
$790$	13141.1	0.591821
$791$	−55693.4	−2.50345
$792$	13748.8	0.616848
$793$	−3518.43	−0.157557
$794$	7371.60	0.329482
$795$	17564.0	0.783562
$796$	−7608.56	−0.338792
$797$	2781.41	0.123617	0.0618084	−	0.998088i	$-0.480313\pi$
$797$	2781.41	0.123617	0.0618084	+	0.998088i	$0.480313\pi$
$798$	−14776.5	−0.655493
$799$	−48730.7	−2.15766
$800$	3404.09	0.150441
$801$	−782.356	−0.0345108
$802$	−4070.05	−0.179200
$803$	−17701.1	−0.777904
$804$	−861.444	−0.0377871
$805$	67916.5	2.97359
$806$	−22471.0	−0.982020
$807$	−1509.06	−0.0658259
$808$	18587.4	0.809287
$809$	−14803.1	−0.643325	−0.321662	−	0.946854i	$-0.604242\pi$
$809$	−14803.1	−0.643325	−0.321662	+	0.946854i	$0.604242\pi$
$810$	−2512.79	−0.109001
$811$	32864.0	1.42295	0.711473	−	0.702713i	$-0.248028\pi$
$811$	32864.0	1.42295	0.711473	+	0.702713i	$0.248028\pi$
$812$	−5041.68	−0.217892
$813$	−5615.43	−0.242241
$814$	−29587.0	−1.27399
$815$	−4719.78	−0.202855
$816$	10878.3	0.466687
$817$	29729.0	1.27306
$818$	3109.12	0.132895
$819$	−19124.1	−0.815936
$820$	−3647.43	−0.155334
$821$	2870.66	0.122030	0.0610152	−	0.998137i	$-0.480566\pi$
$821$	2870.66	0.122030	0.0610152	+	0.998137i	$0.480566\pi$
$822$	17315.2	0.734717
$823$	8661.45	0.366852	0.183426	−	0.983034i	$-0.441281\pi$
$823$	8661.45	0.366852	0.183426	+	0.983034i	$0.441281\pi$
$824$	−19174.8	−0.810661
$825$	6978.29	0.294488
$826$	51003.4	2.14847
$827$	34845.3	1.46516	0.732581	−	0.680680i	$-0.238316\pi$
$827$	34845.3	1.46516	0.732581	+	0.680680i	$0.238316\pi$
$828$	−3371.65	−0.141513
$829$	1575.92	0.0660239	0.0330120	−	0.999455i	$-0.489490\pi$
$829$	1575.92	0.0660239	0.0330120	+	0.999455i	$0.489490\pi$
$830$	−32021.2	−1.33912
$831$	−26352.4	−1.10007
$832$	40870.7	1.70305
$833$	44226.7	1.83958
$834$	10073.4	0.418240
$835$	−48061.5	−1.99190
$836$	−8856.94	−0.366416
$837$	3457.14	0.142767
$838$	36865.4	1.51968
$839$	9066.80	0.373088	0.186544	−	0.982447i	$-0.440271\pi$
$839$	9066.80	0.373088	0.186544	+	0.982447i	$0.440271\pi$
$840$	27633.2	1.13504
$841$	−17575.6	−0.720637
$842$	−10943.4	−0.447905
$843$	13910.5	0.568331
$844$	−1781.22	−0.0726448
$845$	−38203.9	−1.55533
$846$	−12701.8	−0.516190
$847$	−75164.9	−3.04923
$848$	−19819.5	−0.802598
$849$	1511.09	0.0610840
$850$	7643.10	0.308419
$851$	35271.4	1.42078
$852$	3707.72	0.149090
$853$	16980.5	0.681595	0.340797	−	0.940137i	$-0.389303\pi$
$853$	16980.5	0.681595	0.340797	+	0.940137i	$0.389303\pi$
$854$	3503.68	0.140391
$855$	7868.99	0.314753
$856$	−17971.7	−0.717593
$857$	−2443.34	−0.0973897	−0.0486949	−	0.998814i	$-0.515506\pi$
$857$	−2443.34	−0.0973897	−0.0486949	+	0.998814i	$0.515506\pi$
$858$	32797.8	1.30501
$859$	49148.0	1.95216	0.976082	−	0.217401i	$-0.0697578\pi$
$859$	49148.0	1.95216	0.976082	+	0.217401i	$0.0697578\pi$
$860$	−11436.5	−0.453469
$861$	−12219.7	−0.483676
$862$	11300.0	0.446496
$863$	3197.70	0.126131	0.0630655	−	0.998009i	$-0.479912\pi$
$863$	3197.70	0.126131	0.0630655	+	0.998009i	$0.479912\pi$
$864$	−2461.45	−0.0969215
$865$	−55958.5	−2.19959
$866$	37047.7	1.45373
$867$	−6463.97	−0.253204
$868$	−7820.73	−0.305821
$869$	26388.5	1.03011
$870$	−7681.99	−0.299361
$871$	−9989.68	−0.388619
$872$	5111.80	0.198518
$873$	9507.70	0.368599
$874$	−30210.4	−1.16920
$875$	−32926.3	−1.27213
$876$	1766.17	0.0681201
$877$	39753.5	1.53065	0.765326	−	0.643643i	$-0.222578\pi$
$877$	39753.5	1.53065	0.765326	+	0.643643i	$0.222578\pi$
$878$	12648.7	0.486186
$879$	17227.3	0.661051
$880$	−34234.9	−1.31143
$881$	31706.8	1.21252	0.606259	−	0.795267i	$-0.292669\pi$
$881$	31706.8	1.21252	0.606259	+	0.795267i	$0.292669\pi$
$882$	11527.8	0.440094
$883$	−36079.4	−1.37505	−0.687524	−	0.726162i	$-0.741302\pi$
$883$	−36079.4	−1.37505	−0.687524	+	0.726162i	$0.741302\pi$
$884$	−12554.9	−0.477678
$885$	−27161.0	−1.03165
$886$	26516.5	1.00546
$887$	−20239.6	−0.766156	−0.383078	−	0.923716i	$-0.625136\pi$
$887$	−20239.6	−0.766156	−0.383078	+	0.923716i	$0.625136\pi$
$888$	14350.9	0.542324
$889$	−7385.55	−0.278631
$890$	2696.70	0.101566
$891$	−5045.91	−0.189724
$892$	3344.22	0.125530
$893$	39776.7	1.49057
$894$	17813.4	0.666410
$895$	−19140.2	−0.714846
$896$	−19199.0	−0.715842
$897$	−39099.1	−1.45538
$898$	−1277.52	−0.0474736
$899$	10569.0	0.392099
$900$	−696.276	−0.0257880
$901$	38630.3	1.42837
$902$	20956.6	0.773592
$903$	−38314.8	−1.41200
$904$	−46328.1	−1.70448
$905$	38035.8	1.39707
$906$	1968.12	0.0721704
$907$	25279.5	0.925461	0.462730	−	0.886499i	$-0.346870\pi$
$907$	25279.5	0.925461	0.462730	+	0.886499i	$0.346870\pi$
$908$	−4119.21	−0.150552
$909$	−6821.71	−0.248913
$910$	65919.0	2.40131
$911$	−9750.13	−0.354595	−0.177298	−	0.984157i	$-0.556735\pi$
$911$	−9750.13	−0.354595	−0.177298	+	0.984157i	$0.556735\pi$
$912$	−8879.47	−0.322400
$913$	−64301.4	−2.33085
$914$	10794.1	0.390631
$915$	−1865.83	−0.0674124
$916$	14298.9	0.515775
$917$	−41596.0	−1.49795
$918$	−5526.62	−0.198699
$919$	5781.34	0.207518	0.103759	−	0.994602i	$-0.466913\pi$
$919$	5781.34	0.207518	0.103759	+	0.994602i	$0.466913\pi$
$920$	56495.8	2.02458
$921$	−3791.87	−0.135664
$922$	19572.4	0.699114
$923$	42996.3	1.53330
$924$	11414.8	0.406407
$925$	7283.87	0.258910
$926$	6521.78	0.231446
$927$	7037.26	0.249335
$928$	−7525.03	−0.266187
$929$	52931.7	1.86936	0.934678	−	0.355494i	$-0.115688\pi$
$929$	52931.7	1.86936	0.934678	+	0.355494i	$0.115688\pi$
$930$	−11916.4	−0.420166
$931$	−36100.3	−1.27083
$932$	−5171.12	−0.181744
$933$	−14541.8	−0.510264
$934$	28856.7	1.01094
$935$	66727.5	2.33393
$936$	−15908.3	−0.555532
$937$	−14911.8	−0.519901	−0.259951	−	0.965622i	$-0.583706\pi$
$937$	−14911.8	−0.519901	−0.259951	+	0.965622i	$0.583706\pi$
$938$	9947.81	0.346277
$939$	26839.7	0.932779
$940$	−15301.8	−0.530947
$941$	−14139.4	−0.489832	−0.244916	−	0.969544i	$-0.578760\pi$
$941$	−14139.4	−0.489832	−0.244916	+	0.969544i	$0.578760\pi$
$942$	1146.78	0.0396646
$943$	−24982.9	−0.862732
$944$	30648.9	1.05671
$945$	−10141.6	−0.349106
$946$	65709.7	2.25836
$947$	−29079.8	−0.997853	−0.498926	−	0.866644i	$-0.666272\pi$
$947$	−29079.8	−0.997853	−0.498926	+	0.866644i	$0.666272\pi$
$948$	−2632.97	−0.0902057
$949$	20481.2	0.700578
$950$	−6238.73	−0.213064
$951$	−3999.67	−0.136381
$952$	60776.4	2.06909
$953$	−18445.5	−0.626978	−0.313489	−	0.949592i	$-0.601498\pi$
$953$	−18445.5	−0.626978	−0.313489	+	0.949592i	$0.601498\pi$
$954$	10069.1	0.341718
$955$	2929.14	0.0992509
$956$	−10195.1	−0.344910
$957$	−15426.1	−0.521062
$958$	1579.46	0.0532671
$959$	69883.8	2.35315
$960$	21673.8	0.728665
$961$	−13396.2	−0.449672
$962$	34234.0	1.14735
$963$	6595.72	0.220710
$964$	6047.30	0.202044
$965$	61078.1	2.03749
$966$	38935.2	1.29681
$967$	10725.9	0.356692	0.178346	−	0.983968i	$-0.442925\pi$
$967$	10725.9	0.356692	0.178346	+	0.983968i	$0.442925\pi$
$968$	−62525.3	−2.07607
$969$	17307.0	0.573769
$970$	−32772.1	−1.08479
$971$	5019.20	0.165884	0.0829422	−	0.996554i	$-0.473568\pi$
$971$	5019.20	0.165884	0.0829422	+	0.996554i	$0.473568\pi$
$972$	503.468	0.0166139
$973$	40655.9	1.33954
$974$	−8506.00	−0.279825
$975$	−8074.32	−0.265216
$976$	2105.42	0.0690502
$977$	−22654.0	−0.741827	−0.370914	−	0.928667i	$-0.620955\pi$
$977$	−22654.0	−0.741827	−0.370914	+	0.928667i	$0.620955\pi$
$978$	−2705.76	−0.0884668
$979$	5415.22	0.176784
$980$	13887.5	0.452675
$981$	−1876.06	−0.0610582
$982$	27806.9	0.903621
$983$	10322.6	0.334933	0.167467	−	0.985878i	$-0.446441\pi$
$983$	10322.6	0.334933	0.167467	+	0.985878i	$0.446441\pi$
$984$	−10164.8	−0.329311
$985$	3882.25	0.125582
$986$	−16895.8	−0.545710
$987$	−51264.2	−1.65325
$988$	10248.0	0.329993
$989$	−78334.2	−2.51859
$990$	17392.7	0.558361
$991$	2881.16	0.0923543	0.0461771	−	0.998933i	$-0.485296\pi$
$991$	2881.16	0.0923543	0.0461771	+	0.998933i	$0.485296\pi$
$992$	−11672.9	−0.373605
$993$	7749.79	0.247666
$994$	−42816.1	−1.36624
$995$	−46789.6	−1.49079
$996$	6415.82	0.204110
$997$	−22170.1	−0.704246	−0.352123	−	0.935954i	$-0.614540\pi$
$997$	−22170.1	−0.704246	−0.352123	+	0.935954i	$0.614540\pi$
$998$	9573.89	0.303664
$999$	−5266.86	−0.166803

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	471.4.a.c.1.16	✓	22
3.2	odd	2		1413.4.a.e.1.7		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
471.4.a.c.1.16	✓	22	1.1	even	1	trivial
1413.4.a.e.1.7		22	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 471 = 3 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	471.a (trivial)

\(f(q)\)	\(=\)	\(q+2.43477 q^{2} -3.00000 q^{3} -2.07188 q^{4} -12.7413 q^{5} -7.30432 q^{6} -29.4801 q^{7} -24.5227 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.43477 q^{2} -3.00000 q^{3} -2.07188 q^{4} -12.7413 q^{5} -7.30432 q^{6} -29.4801 q^{7} -24.5227 q^{8} +9.00000 q^{9} -31.0221 q^{10} -62.2952 q^{11} +6.21565 q^{12} +72.0794 q^{13} -71.7773 q^{14} +38.2238 q^{15} -43.1322 q^{16} +84.0694 q^{17} +21.9130 q^{18} -68.6221 q^{19} +26.3984 q^{20} +88.4402 q^{21} -151.675 q^{22} +180.815 q^{23} +73.5682 q^{24} +37.3399 q^{25} +175.497 q^{26} -27.0000 q^{27} +61.0793 q^{28} -82.5432 q^{29} +93.0663 q^{30} -128.042 q^{31} +91.1648 q^{32} +186.885 q^{33} +204.690 q^{34} +375.613 q^{35} -18.6469 q^{36} +195.069 q^{37} -167.079 q^{38} -216.238 q^{39} +312.451 q^{40} -138.169 q^{41} +215.332 q^{42} -433.228 q^{43} +129.068 q^{44} -114.671 q^{45} +440.243 q^{46} -579.648 q^{47} +129.397 q^{48} +526.074 q^{49} +90.9143 q^{50} -252.208 q^{51} -149.340 q^{52} +459.505 q^{53} -65.7389 q^{54} +793.719 q^{55} +722.932 q^{56} +205.866 q^{57} -200.974 q^{58} -710.579 q^{59} -79.1953 q^{60} -48.8133 q^{61} -311.754 q^{62} -265.321 q^{63} +567.023 q^{64} -918.383 q^{65} +455.024 q^{66} -138.593 q^{67} -174.182 q^{68} -542.445 q^{69} +914.533 q^{70} +596.513 q^{71} -220.705 q^{72} +284.148 q^{73} +474.949 q^{74} -112.020 q^{75} +142.177 q^{76} +1836.47 q^{77} -526.491 q^{78} -423.604 q^{79} +549.559 q^{80} +81.0000 q^{81} -336.409 q^{82} +1032.20 q^{83} -183.238 q^{84} -1071.15 q^{85} -1054.81 q^{86} +247.630 q^{87} +1527.65 q^{88} -86.9284 q^{89} -279.199 q^{90} -2124.90 q^{91} -374.627 q^{92} +384.127 q^{93} -1411.31 q^{94} +874.333 q^{95} -273.494 q^{96} +1056.41 q^{97} +1280.87 q^{98} -560.656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9}+O(q^{10}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9	\( 22 q + 4 q^{2} - 66 q^{3} + 90 q^{4} + 32 q^{5} - 12 q^{6} - 4 q^{7} + 27 q^{8} + 198 q^{9} + 13 q^{10} + 61 q^{11} - 270 q^{12} + 4 q^{13} + 133 q^{14} - 96 q^{15} + 342 q^{16} + 308 q^{17} + 36 q^{18} + 32 q^{19} + 407 q^{20} + 12 q^{21} - 166 q^{22} + 53 q^{23} - 81 q^{24} + 746 q^{25} + 467 q^{26} - 594 q^{27} + 85 q^{28} + 634 q^{29} - 39 q^{30} - 163 q^{31} + 150 q^{32} - 183 q^{33} + 37 q^{34} + 782 q^{35} + 810 q^{36} - 2 q^{37} + 584 q^{38} - 12 q^{39} + 864 q^{40} + 1593 q^{41} - 399 q^{42} - 891 q^{43} + 2093 q^{44} + 288 q^{45} + 108 q^{46} + 1200 q^{47} - 1026 q^{48} + 2816 q^{49} + 4703 q^{50} - 924 q^{51} + 1866 q^{52} + 1182 q^{53} - 108 q^{54} + 970 q^{55} + 5362 q^{56} - 96 q^{57} + 1814 q^{58} + 2802 q^{59} - 1221 q^{60} + 2629 q^{61} + 2378 q^{62} - 36 q^{63} + 625 q^{64} + 2264 q^{65} + 498 q^{66} - 1074 q^{67} + 4383 q^{68} - 159 q^{69} + 4009 q^{70} + 3920 q^{71} + 243 q^{72} + 1086 q^{73} + 4904 q^{74} - 2238 q^{75} + 3750 q^{76} + 2966 q^{77} - 1401 q^{78} - 30 q^{79} + 7777 q^{80} + 1782 q^{81} + 2932 q^{82} + 1900 q^{83} - 255 q^{84} + 524 q^{85} + 3209 q^{86} - 1902 q^{87} - 100 q^{88} + 4488 q^{89} + 117 q^{90} - 818 q^{91} + 6210 q^{92} + 489 q^{93} + 3220 q^{94} + 3500 q^{95} - 450 q^{96} + 2178 q^{97} + 7629 q^{98} + 549 q^{99}+O(q^{100}) \) 22 * q + 4 * q^2 - 66 * q^3 + 90 * q^4 + 32 * q^5 - 12 * q^6 - 4 * q^7 + 27 * q^8 + 198 * q^9 + 13 * q^10 + 61 * q^11 - 270 * q^12 + 4 * q^13 + 133 * q^14 - 96 * q^15 + 342 * q^16 + 308 * q^17 + 36 * q^18 + 32 * q^19 + 407 * q^20 + 12 * q^21 - 166 * q^22 + 53 * q^23 - 81 * q^24 + 746 * q^25 + 467 * q^26 - 594 * q^27 + 85 * q^28 + 634 * q^29 - 39 * q^30 - 163 * q^31 + 150 * q^32 - 183 * q^33 + 37 * q^34 + 782 * q^35 + 810 * q^36 - 2 * q^37 + 584 * q^38 - 12 * q^39 + 864 * q^40 + 1593 * q^41 - 399 * q^42 - 891 * q^43 + 2093 * q^44 + 288 * q^45 + 108 * q^46 + 1200 * q^47 - 1026 * q^48 + 2816 * q^49 + 4703 * q^50 - 924 * q^51 + 1866 * q^52 + 1182 * q^53 - 108 * q^54 + 970 * q^55 + 5362 * q^56 - 96 * q^57 + 1814 * q^58 + 2802 * q^59 - 1221 * q^60 + 2629 * q^61 + 2378 * q^62 - 36 * q^63 + 625 * q^64 + 2264 * q^65 + 498 * q^66 - 1074 * q^67 + 4383 * q^68 - 159 * q^69 + 4009 * q^70 + 3920 * q^71 + 243 * q^72 + 1086 * q^73 + 4904 * q^74 - 2238 * q^75 + 3750 * q^76 + 2966 * q^77 - 1401 * q^78 - 30 * q^79 + 7777 * q^80 + 1782 * q^81 + 2932 * q^82 + 1900 * q^83 - 255 * q^84 + 524 * q^85 + 3209 * q^86 - 1902 * q^87 - 100 * q^88 + 4488 * q^89 + 117 * q^90 - 818 * q^91 + 6210 * q^92 + 489 * q^93 + 3220 * q^94 + 3500 * q^95 - 450 * q^96 + 2178 * q^97 + 7629 * q^98 + 549 * q^99

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