LMFDB - Embedded newform 4003.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$1$
Dimension:	$152$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71344 q^{2} -2.64650 q^{3} +5.36273 q^{4} -1.98460 q^{5} +7.18110 q^{6} -4.86510 q^{7} -9.12456 q^{8} +4.00395 q^{9} +O(q^{10})$	\(q-2.71344 q^{2} -2.64650 q^{3} +5.36273 q^{4} -1.98460 q^{5} +7.18110 q^{6} -4.86510 q^{7} -9.12456 q^{8} +4.00395 q^{9} +5.38509 q^{10} +4.74904 q^{11} -14.1925 q^{12} -0.0343281 q^{13} +13.2011 q^{14} +5.25224 q^{15} +14.0334 q^{16} +3.60088 q^{17} -10.8645 q^{18} -3.71179 q^{19} -10.6429 q^{20} +12.8755 q^{21} -12.8862 q^{22} -5.09332 q^{23} +24.1481 q^{24} -1.06136 q^{25} +0.0931470 q^{26} -2.65695 q^{27} -26.0902 q^{28} -10.4557 q^{29} -14.2516 q^{30} -3.93498 q^{31} -19.8297 q^{32} -12.5683 q^{33} -9.77077 q^{34} +9.65528 q^{35} +21.4721 q^{36} +0.369241 q^{37} +10.0717 q^{38} +0.0908491 q^{39} +18.1086 q^{40} +2.32546 q^{41} -34.9367 q^{42} +7.81321 q^{43} +25.4678 q^{44} -7.94624 q^{45} +13.8204 q^{46} -3.83967 q^{47} -37.1394 q^{48} +16.6692 q^{49} +2.87993 q^{50} -9.52973 q^{51} -0.184092 q^{52} -12.6157 q^{53} +7.20945 q^{54} -9.42495 q^{55} +44.3919 q^{56} +9.82324 q^{57} +28.3709 q^{58} -1.72956 q^{59} +28.1664 q^{60} +2.62444 q^{61} +10.6773 q^{62} -19.4796 q^{63} +25.7398 q^{64} +0.0681275 q^{65} +34.1033 q^{66} +9.42755 q^{67} +19.3106 q^{68} +13.4794 q^{69} -26.1990 q^{70} +14.5835 q^{71} -36.5343 q^{72} -9.64534 q^{73} -1.00191 q^{74} +2.80888 q^{75} -19.9053 q^{76} -23.1045 q^{77} -0.246513 q^{78} +3.62057 q^{79} -27.8508 q^{80} -4.98024 q^{81} -6.31000 q^{82} +11.9062 q^{83} +69.0477 q^{84} -7.14632 q^{85} -21.2006 q^{86} +27.6710 q^{87} -43.3329 q^{88} +3.03326 q^{89} +21.5616 q^{90} +0.167009 q^{91} -27.3141 q^{92} +10.4139 q^{93} +10.4187 q^{94} +7.36642 q^{95} +52.4793 q^{96} +18.1865 q^{97} -45.2307 q^{98} +19.0149 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) $ 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * 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 $

$2$	−2.71344	−1.91869	−0.959344	−	0.282238i	$-0.908923\pi$
$2$	−2.71344	−1.91869	−0.959344	+	0.282238i	$0.908923\pi$
$3$	−2.64650	−1.52796	−0.763978	−	0.645242i	$-0.776756\pi$
$3$	−2.64650	−1.52796	−0.763978	+	0.645242i	$0.776756\pi$
$4$	5.36273	2.68137
$5$	−1.98460	−0.887541	−0.443770	−	0.896141i	$-0.646359\pi$
$5$	−1.98460	−0.887541	−0.443770	+	0.896141i	$0.646359\pi$
$6$	7.18110	2.93167
$7$	−4.86510	−1.83883	−0.919417	−	0.393284i	$-0.871339\pi$
$7$	−4.86510	−1.83883	−0.919417	+	0.393284i	$0.871339\pi$
$8$	−9.12456	−3.22602
$9$	4.00395	1.33465
$10$	5.38509	1.70291
$11$	4.74904	1.43189	0.715944	−	0.698157i	$-0.245996\pi$
$11$	4.74904	1.43189	0.715944	+	0.698157i	$0.245996\pi$
$12$	−14.1925	−4.09701
$13$	−0.0343281	−0.00952089	−0.00476044	−	0.999989i	$-0.501515\pi$
$13$	−0.0343281	−0.00952089	−0.00476044	+	0.999989i	$0.501515\pi$
$14$	13.2011	3.52815
$15$	5.25224	1.35612
$16$	14.0334	3.50836
$17$	3.60088	0.873343	0.436671	−	0.899621i	$-0.356157\pi$
$17$	3.60088	0.873343	0.436671	+	0.899621i	$0.356157\pi$
$18$	−10.8645	−2.56078
$19$	−3.71179	−0.851543	−0.425771	−	0.904831i	$-0.639997\pi$
$19$	−3.71179	−0.851543	−0.425771	+	0.904831i	$0.639997\pi$
$20$	−10.6429	−2.37982
$21$	12.8755	2.80966
$22$	−12.8862	−2.74735
$23$	−5.09332	−1.06203	−0.531015	−	0.847362i	$-0.678189\pi$
$23$	−5.09332	−1.06203	−0.531015	+	0.847362i	$0.678189\pi$
$24$	24.1481	4.92921
$25$	−1.06136	−0.212272
$26$	0.0931470	0.0182676
$27$	−2.65695	−0.511329
$28$	−26.0902	−4.93059
$29$	−10.4557	−1.94158	−0.970789	−	0.239937i	$-0.922873\pi$
$29$	−10.4557	−1.94158	−0.970789	+	0.239937i	$0.922873\pi$
$30$	−14.2516	−2.60198
$31$	−3.93498	−0.706744	−0.353372	−	0.935483i	$-0.614965\pi$
$31$	−3.93498	−0.706744	−0.353372	+	0.935483i	$0.614965\pi$
$32$	−19.8297	−3.50543
$33$	−12.5683	−2.18786
$34$	−9.77077	−1.67567
$35$	9.65528	1.63204
$36$	21.4721	3.57868
$37$	0.369241	0.0607029	0.0303514	−	0.999539i	$-0.490337\pi$
$37$	0.369241	0.0607029	0.0303514	+	0.999539i	$0.490337\pi$
$38$	10.0717	1.63385
$39$	0.0908491	0.0145475
$40$	18.1086	2.86322
$41$	2.32546	0.363176	0.181588	−	0.983375i	$-0.441876\pi$
$41$	2.32546	0.363176	0.181588	+	0.983375i	$0.441876\pi$
$42$	−34.9367	−5.39086
$43$	7.81321	1.19150	0.595751	−	0.803169i	$-0.296854\pi$
$43$	7.81321	1.19150	0.595751	+	0.803169i	$0.296854\pi$
$44$	25.4678	3.83942
$45$	−7.94624	−1.18456
$46$	13.8204	2.03770
$47$	−3.83967	−0.560073	−0.280037	−	0.959989i	$-0.590347\pi$
$47$	−3.83967	−0.560073	−0.280037	+	0.959989i	$0.590347\pi$
$48$	−37.1394	−5.36062
$49$	16.6692	2.38131
$50$	2.87993	0.407283
$51$	−9.52973	−1.33443
$52$	−0.184092	−0.0255290
$53$	−12.6157	−1.73290	−0.866452	−	0.499261i	$-0.833605\pi$
$53$	−12.6157	−1.73290	−0.866452	+	0.499261i	$0.833605\pi$
$54$	7.20945	0.981082
$55$	−9.42495	−1.27086
$56$	44.3919	5.93211
$57$	9.82324	1.30112
$58$	28.3709	3.72528
$59$	−1.72956	−0.225170	−0.112585	−	0.993642i	$-0.535913\pi$
$59$	−1.72956	−0.225170	−0.112585	+	0.993642i	$0.535913\pi$
$60$	28.1664	3.63626
$61$	2.62444	0.336025	0.168013	−	0.985785i	$-0.446265\pi$
$61$	2.62444	0.336025	0.168013	+	0.985785i	$0.446265\pi$
$62$	10.6773	1.35602
$63$	−19.4796	−2.45420
$64$	25.7398	3.21747
$65$	0.0681275	0.00845018
$66$	34.1033	4.19783
$67$	9.42755	1.15176	0.575879	−	0.817535i	$-0.304660\pi$
$67$	9.42755	1.15176	0.575879	+	0.817535i	$0.304660\pi$
$68$	19.3106	2.34175
$69$	13.4794	1.62274
$70$	−26.1990	−3.13138
$71$	14.5835	1.73075	0.865374	−	0.501127i	$-0.167081\pi$
$71$	14.5835	1.73075	0.865374	+	0.501127i	$0.167081\pi$
$72$	−36.5343	−4.30560
$73$	−9.64534	−1.12890	−0.564451	−	0.825467i	$-0.690912\pi$
$73$	−9.64534	−1.12890	−0.564451	+	0.825467i	$0.690912\pi$
$74$	−1.00191	−0.116470
$75$	2.80888	0.324342
$76$	−19.9053	−2.28330
$77$	−23.1045	−2.63301
$78$	−0.246513	−0.0279121
$79$	3.62057	0.407346	0.203673	−	0.979039i	$-0.434712\pi$
$79$	3.62057	0.407346	0.203673	+	0.979039i	$0.434712\pi$
$80$	−27.8508	−3.11381
$81$	−4.98024	−0.553360
$82$	−6.31000	−0.696822
$83$	11.9062	1.30688	0.653439	−	0.756979i	$-0.273326\pi$
$83$	11.9062	1.30688	0.653439	+	0.756979i	$0.273326\pi$
$84$	69.0477	7.53372
$85$	−7.14632	−0.775127
$86$	−21.2006	−2.28612
$87$	27.6710	2.96664
$88$	−43.3329	−4.61930
$89$	3.03326	0.321525	0.160763	−	0.986993i	$-0.448605\pi$
$89$	3.03326	0.321525	0.160763	+	0.986993i	$0.448605\pi$
$90$	21.5616	2.27279
$91$	0.167009	0.0175073
$92$	−27.3141	−2.84769
$93$	10.4139	1.07987
$94$	10.4187	1.07461
$95$	7.36642	0.755779
$96$	52.4793	5.35614
$97$	18.1865	1.84656	0.923280	−	0.384127i	$-0.125498\pi$
$97$	18.1865	1.84656	0.923280	+	0.384127i	$0.125498\pi$
$98$	−45.2307	−4.56899
$99$	19.0149	1.91107
$100$	−5.69178	−0.569178
$101$	−15.8696	−1.57909	−0.789543	−	0.613696i	$-0.789682\pi$
$101$	−15.8696	−1.57909	−0.789543	+	0.613696i	$0.789682\pi$
$102$	25.8583	2.56035
$103$	8.79965	0.867055	0.433528	−	0.901140i	$-0.357269\pi$
$103$	8.79965	0.867055	0.433528	+	0.901140i	$0.357269\pi$
$104$	0.313228	0.0307146
$105$	−25.5527	−2.49369
$106$	34.2320	3.32490
$107$	−15.5394	−1.50225	−0.751127	−	0.660157i	$-0.770490\pi$
$107$	−15.5394	−1.50225	−0.751127	+	0.660157i	$0.770490\pi$
$108$	−14.2485	−1.37106
$109$	8.64675	0.828209	0.414104	−	0.910229i	$-0.364095\pi$
$109$	8.64675	0.828209	0.414104	+	0.910229i	$0.364095\pi$
$110$	25.5740	2.43838
$111$	−0.977196	−0.0927513
$112$	−68.2740	−6.45129
$113$	−6.44293	−0.606100	−0.303050	−	0.952975i	$-0.598005\pi$
$113$	−6.44293	−0.606100	−0.303050	+	0.952975i	$0.598005\pi$
$114$	−26.6547	−2.49644
$115$	10.1082	0.942595
$116$	−56.0712	−5.20608
$117$	−0.137448	−0.0127070
$118$	4.69306	0.432031
$119$	−17.5186	−1.60593
$120$	−47.9244	−4.37488
$121$	11.5534	1.05031
$122$	−7.12125	−0.644727
$123$	−6.15433	−0.554918
$124$	−21.1023	−1.89504
$125$	12.0294	1.07594
$126$	52.8566	4.70884
$127$	20.1579	1.78873	0.894363	−	0.447343i	$-0.147630\pi$
$127$	20.1579	1.78873	0.894363	+	0.447343i	$0.147630\pi$
$128$	−30.1837	−2.66789
$129$	−20.6776	−1.82056
$130$	−0.184860	−0.0162133
$131$	14.5061	1.26741	0.633704	−	0.773576i	$-0.281534\pi$
$131$	14.5061	1.26741	0.633704	+	0.773576i	$0.281534\pi$
$132$	−67.4005	−5.86646
$133$	18.0582	1.56585
$134$	−25.5810	−2.20987
$135$	5.27298	0.453826
$136$	−32.8565	−2.81742
$137$	−10.7896	−0.921814	−0.460907	−	0.887448i	$-0.652476\pi$
$137$	−10.7896	−0.921814	−0.460907	+	0.887448i	$0.652476\pi$
$138$	−36.5756	−3.11352
$139$	4.76264	0.403962	0.201981	−	0.979389i	$-0.435262\pi$
$139$	4.76264	0.403962	0.201981	+	0.979389i	$0.435262\pi$
$140$	51.7787	4.37610
$141$	10.1617	0.855767
$142$	−39.5715	−3.32077
$143$	−0.163025	−0.0136329
$144$	56.1891	4.68243
$145$	20.7504	1.72323
$146$	26.1720	2.16601
$147$	−44.1149	−3.63854
$148$	1.98014	0.162767
$149$	−18.5520	−1.51984	−0.759921	−	0.650015i	$-0.774762\pi$
$149$	−18.5520	−1.51984	−0.759921	+	0.650015i	$0.774762\pi$
$150$	−7.62171	−0.622310
$151$	14.6621	1.19319	0.596594	−	0.802543i	$-0.296520\pi$
$151$	14.6621	1.19319	0.596594	+	0.802543i	$0.296520\pi$
$152$	33.8684	2.74709
$153$	14.4178	1.16561
$154$	62.6927	5.05192
$155$	7.80938	0.627264
$156$	0.487199	0.0390072
$157$	0.170936	0.0136422	0.00682109	−	0.999977i	$-0.497829\pi$
$157$	0.170936	0.0136422	0.00682109	+	0.999977i	$0.497829\pi$
$158$	−9.82418	−0.781569
$159$	33.3875	2.64780
$160$	39.3541	3.11121
$161$	24.7795	1.95290
$162$	13.5136	1.06173
$163$	2.94393	0.230587	0.115293	−	0.993331i	$-0.463219\pi$
$163$	2.94393	0.230587	0.115293	+	0.993331i	$0.463219\pi$
$164$	12.4708	0.973809
$165$	24.9431	1.94182
$166$	−32.3068	−2.50749
$167$	16.9104	1.30856	0.654282	−	0.756251i	$-0.272971\pi$
$167$	16.9104	1.30856	0.654282	+	0.756251i	$0.272971\pi$
$168$	−117.483	−9.06401
$169$	−12.9988	−0.999909
$170$	19.3911	1.48723
$171$	−14.8618	−1.13651
$172$	41.9001	3.19486
$173$	19.4805	1.48108	0.740538	−	0.672015i	$-0.234571\pi$
$173$	19.4805	1.48108	0.740538	+	0.672015i	$0.234571\pi$
$174$	−75.0835	−5.69207
$175$	5.16361	0.390332
$176$	66.6453	5.02358
$177$	4.57729	0.344050
$178$	−8.23056	−0.616907
$179$	−5.95394	−0.445019	−0.222509	−	0.974931i	$-0.571425\pi$
$179$	−5.95394	−0.445019	−0.222509	+	0.974931i	$0.571425\pi$
$180$	−42.6136	−3.17623
$181$	4.71521	0.350479	0.175239	−	0.984526i	$-0.443930\pi$
$181$	4.71521	0.350479	0.175239	+	0.984526i	$0.443930\pi$
$182$	−0.453169	−0.0335911
$183$	−6.94557	−0.513432
$184$	46.4743	3.42613
$185$	−0.732797	−0.0538763
$186$	−28.2575	−2.07194
$187$	17.1007	1.25053
$188$	−20.5911	−1.50176
$189$	12.9263	0.940250
$190$	−19.9883	−1.45010
$191$	−2.28622	−0.165425	−0.0827124	−	0.996573i	$-0.526358\pi$
$191$	−2.28622	−0.165425	−0.0827124	+	0.996573i	$0.526358\pi$
$192$	−68.1202	−4.91615
$193$	−2.13543	−0.153712	−0.0768559	−	0.997042i	$-0.524488\pi$
$193$	−2.13543	−0.153712	−0.0768559	+	0.997042i	$0.524488\pi$
$194$	−49.3479	−3.54297
$195$	−0.180299	−0.0129115
$196$	89.3923	6.38516
$197$	3.40347	0.242487	0.121244	−	0.992623i	$-0.461312\pi$
$197$	3.40347	0.242487	0.121244	+	0.992623i	$0.461312\pi$
$198$	−51.5957	−3.66675
$199$	−17.1412	−1.21510	−0.607552	−	0.794280i	$-0.707849\pi$
$199$	−17.1412	−1.21510	−0.607552	+	0.794280i	$0.707849\pi$
$200$	9.68442	0.684792
$201$	−24.9500	−1.75984
$202$	43.0612	3.02977
$203$	50.8681	3.57024
$204$	−51.1054	−3.57809
$205$	−4.61512	−0.322334
$206$	−23.8773	−1.66361
$207$	−20.3934	−1.41744
$208$	−0.481740	−0.0334027
$209$	−17.6274	−1.21931
$210$	69.3355	4.78461
$211$	28.0241	1.92926	0.964630	−	0.263609i	$-0.0849130\pi$
$211$	28.0241	1.92926	0.964630	+	0.263609i	$0.0849130\pi$
$212$	−67.6548	−4.64655
$213$	−38.5953	−2.64451
$214$	42.1653	2.88236
$215$	−15.5061	−1.05751
$216$	24.2435	1.64956
$217$	19.1441	1.29959
$218$	−23.4624	−1.58907
$219$	25.5264	1.72491
$220$	−50.5435	−3.40764
$221$	−0.123611	−0.00831500
$222$	2.65156	0.177961
$223$	−17.2363	−1.15423	−0.577114	−	0.816663i	$-0.695821\pi$
$223$	−17.2363	−1.15423	−0.577114	+	0.816663i	$0.695821\pi$
$224$	96.4734	6.44590
$225$	−4.24962	−0.283308
$226$	17.4825	1.16292
$227$	2.96262	0.196636	0.0983180	−	0.995155i	$-0.468654\pi$
$227$	2.96262	0.196636	0.0983180	+	0.995155i	$0.468654\pi$
$228$	52.6794	3.48878
$229$	7.21530	0.476801	0.238400	−	0.971167i	$-0.423377\pi$
$229$	7.21530	0.476801	0.238400	+	0.971167i	$0.423377\pi$
$230$	−27.4280	−1.80855
$231$	61.1461	4.02312
$232$	95.4038	6.26356
$233$	11.0720	0.725353	0.362676	−	0.931915i	$-0.381863\pi$
$233$	11.0720	0.725353	0.362676	+	0.931915i	$0.381863\pi$
$234$	0.372956	0.0243809
$235$	7.62021	0.497088
$236$	−9.27519	−0.603764
$237$	−9.58182	−0.622406
$238$	47.5357	3.08128
$239$	9.33487	0.603823	0.301911	−	0.953336i	$-0.402375\pi$
$239$	9.33487	0.603823	0.301911	+	0.953336i	$0.402375\pi$
$240$	73.7070	4.75777
$241$	−11.4957	−0.740506	−0.370253	−	0.928931i	$-0.620729\pi$
$241$	−11.4957	−0.740506	−0.370253	+	0.928931i	$0.620729\pi$
$242$	−31.3493	−2.01521
$243$	21.1510	1.35684
$244$	14.0742	0.901006
$245$	−33.0817	−2.11351
$246$	16.6994	1.06471
$247$	0.127418	0.00810744
$248$	35.9050	2.27997
$249$	−31.5098	−1.99685
$250$	−32.6409	−2.06439
$251$	5.70283	0.359959	0.179980	−	0.983670i	$-0.442397\pi$
$251$	5.70283	0.359959	0.179980	+	0.983670i	$0.442397\pi$
$252$	−104.464	−6.58060
$253$	−24.1884	−1.52071
$254$	−54.6972	−3.43201
$255$	18.9127	1.18436
$256$	30.4222	1.90138
$257$	0.891949	0.0556383	0.0278191	−	0.999613i	$-0.491144\pi$
$257$	0.891949	0.0556383	0.0278191	+	0.999613i	$0.491144\pi$
$258$	56.1074	3.49310
$259$	−1.79639	−0.111623
$260$	0.365350	0.0226580
$261$	−41.8641	−2.59132
$262$	−39.3615	−2.43176
$263$	−15.0509	−0.928079	−0.464039	−	0.885815i	$-0.653600\pi$
$263$	−15.0509	−0.928079	−0.464039	+	0.885815i	$0.653600\pi$
$264$	114.680	7.05809
$265$	25.0372	1.53802
$266$	−48.9998	−3.00437
$267$	−8.02752	−0.491276
$268$	50.5574	3.08829
$269$	0.133043	0.00811179	0.00405589	−	0.999992i	$-0.498709\pi$
$269$	0.133043	0.00811179	0.00405589	+	0.999992i	$0.498709\pi$
$270$	−14.3079	−0.870750
$271$	10.9188	0.663268	0.331634	−	0.943408i	$-0.392400\pi$
$271$	10.9188	0.663268	0.331634	+	0.943408i	$0.392400\pi$
$272$	50.5328	3.06400
$273$	−0.441990	−0.0267504
$274$	29.2768	1.76867
$275$	−5.04043	−0.303949
$276$	72.2867	4.35115
$277$	12.6256	0.758597	0.379298	−	0.925274i	$-0.376165\pi$
$277$	12.6256	0.758597	0.379298	+	0.925274i	$0.376165\pi$
$278$	−12.9231	−0.775078
$279$	−15.7555	−0.943256
$280$	−88.1001	−5.26499
$281$	11.5896	0.691376	0.345688	−	0.938349i	$-0.387646\pi$
$281$	11.5896	0.691376	0.345688	+	0.938349i	$0.387646\pi$
$282$	−27.5730	−1.64195
$283$	−25.5451	−1.51850	−0.759250	−	0.650799i	$-0.774434\pi$
$283$	−25.5451	−1.51850	−0.759250	+	0.650799i	$0.774434\pi$
$284$	78.2076	4.64077
$285$	−19.4952	−1.15480
$286$	0.442358	0.0261572
$287$	−11.3136	−0.667821
$288$	−79.3971	−4.67852
$289$	−4.03364	−0.237273
$290$	−56.3049	−3.30634
$291$	−48.1306	−2.82146
$292$	−51.7254	−3.02700
$293$	−1.72183	−0.100590	−0.0502952	−	0.998734i	$-0.516016\pi$
$293$	−1.72183	−0.100590	−0.0502952	+	0.998734i	$0.516016\pi$
$294$	119.703	6.98122
$295$	3.43250	0.199848
$296$	−3.36916	−0.195829
$297$	−12.6179	−0.732167
$298$	50.3398	2.91610
$299$	0.174844	0.0101115
$300$	15.0633	0.869678
$301$	−38.0120	−2.19098
$302$	−39.7848	−2.28936
$303$	41.9989	2.41277
$304$	−52.0891	−2.98752
$305$	−5.20847	−0.298236
$306$	−39.1216	−2.23644
$307$	−14.9800	−0.854954	−0.427477	−	0.904026i	$-0.640597\pi$
$307$	−14.9800	−0.854954	−0.427477	+	0.904026i	$0.640597\pi$
$308$	−123.903	−7.06005
$309$	−23.2883	−1.32482
$310$	−21.1902	−1.20352
$311$	−30.3754	−1.72243	−0.861216	−	0.508240i	$-0.830296\pi$
$311$	−30.3754	−1.72243	−0.861216	+	0.508240i	$0.830296\pi$
$312$	−0.828958	−0.0469305
$313$	8.96448	0.506702	0.253351	−	0.967374i	$-0.418467\pi$
$313$	8.96448	0.506702	0.253351	+	0.967374i	$0.418467\pi$
$314$	−0.463824	−0.0261751
$315$	38.6592	2.17820
$316$	19.4161	1.09224
$317$	−10.8579	−0.609842	−0.304921	−	0.952378i	$-0.598630\pi$
$317$	−10.8579	−0.609842	−0.304921	+	0.952378i	$0.598630\pi$
$318$	−90.5948	−5.08030
$319$	−49.6546	−2.78012
$320$	−51.0831	−2.85563
$321$	41.1251	2.29538
$322$	−67.2375	−3.74700
$323$	−13.3657	−0.743689
$324$	−26.7077	−1.48376
$325$	0.0364343	0.00202101
$326$	−7.98817	−0.442424
$327$	−22.8836	−1.26547
$328$	−21.2188	−1.17161
$329$	18.6804	1.02988
$330$	−67.6815	−3.72574
$331$	27.7984	1.52794	0.763969	−	0.645253i	$-0.223248\pi$
$331$	27.7984	1.52794	0.763969	+	0.645253i	$0.223248\pi$
$332$	63.8499	3.50422
$333$	1.47842	0.0810171
$334$	−45.8852	−2.51073
$335$	−18.7099	−1.02223
$336$	180.687	9.85728
$337$	5.20485	0.283526	0.141763	−	0.989901i	$-0.454723\pi$
$337$	5.20485	0.283526	0.141763	+	0.989901i	$0.454723\pi$
$338$	35.2715	1.91851
$339$	17.0512	0.926094
$340$	−38.3238	−2.07840
$341$	−18.6874	−1.01198
$342$	40.3266	2.18061
$343$	−47.0415	−2.54000
$344$	−71.2921	−3.84381
$345$	−26.7513	−1.44024
$346$	−52.8591	−2.84172
$347$	19.0644	1.02343	0.511714	−	0.859156i	$-0.329011\pi$
$347$	19.0644	1.02343	0.511714	+	0.859156i	$0.329011\pi$
$348$	148.392	7.95466
$349$	25.9284	1.38791	0.693957	−	0.720016i	$-0.255866\pi$
$349$	25.9284	1.38791	0.693957	+	0.720016i	$0.255866\pi$
$350$	−14.0111	−0.748926
$351$	0.0912078	0.00486831
$352$	−94.1720	−5.01939
$353$	−12.0698	−0.642409	−0.321205	−	0.947010i	$-0.604088\pi$
$353$	−12.0698	−0.642409	−0.321205	+	0.947010i	$0.604088\pi$
$354$	−12.4202	−0.660125
$355$	−28.9425	−1.53611
$356$	16.2666	0.862127
$357$	46.3631	2.45379
$358$	16.1556	0.853852
$359$	−21.2454	−1.12129	−0.560644	−	0.828057i	$-0.689446\pi$
$359$	−21.2454	−1.12129	−0.560644	+	0.828057i	$0.689446\pi$
$360$	72.5059	3.82140
$361$	−5.22262	−0.274875
$362$	−12.7944	−0.672460
$363$	−30.5760	−1.60482
$364$	0.895626	0.0469436
$365$	19.1422	1.00195
$366$	18.8464	0.985115
$367$	−8.83887	−0.461385	−0.230693	−	0.973027i	$-0.574099\pi$
$367$	−8.83887	−0.461385	−0.230693	+	0.973027i	$0.574099\pi$
$368$	−71.4767	−3.72598
$369$	9.31104	0.484713
$370$	1.98840	0.103372
$371$	61.3767	3.18652
$372$	55.8471	2.89554
$373$	−8.19249	−0.424191	−0.212096	−	0.977249i	$-0.568029\pi$
$373$	−8.19249	−0.424191	−0.212096	+	0.977249i	$0.568029\pi$
$374$	−46.4017	−2.39938
$375$	−31.8357	−1.64399
$376$	35.0353	1.80681
$377$	0.358924	0.0184855
$378$	−35.0747	−1.80405
$379$	−1.08109	−0.0555316	−0.0277658	−	0.999614i	$-0.508839\pi$
$379$	−1.08109	−0.0555316	−0.0277658	+	0.999614i	$0.508839\pi$
$380$	39.5041	2.02652
$381$	−53.3479	−2.73309
$382$	6.20350	0.317399
$383$	33.7632	1.72522	0.862611	−	0.505868i	$-0.168828\pi$
$383$	33.7632	1.72522	0.862611	+	0.505868i	$0.168828\pi$
$384$	79.8812	4.07642
$385$	45.8533	2.33690
$386$	5.79436	0.294925
$387$	31.2837	1.59024
$388$	97.5294	4.95131
$389$	12.9279	0.655469	0.327734	−	0.944770i	$-0.393715\pi$
$389$	12.9279	0.655469	0.327734	+	0.944770i	$0.393715\pi$
$390$	0.489230	0.0247731
$391$	−18.3404	−0.927516
$392$	−152.099	−7.68215
$393$	−38.3905	−1.93654
$394$	−9.23509	−0.465257
$395$	−7.18538	−0.361536
$396$	101.972	5.12428
$397$	29.3298	1.47202	0.736010	−	0.676970i	$-0.236707\pi$
$397$	29.3298	1.47202	0.736010	+	0.676970i	$0.236707\pi$
$398$	46.5114	2.33141
$399$	−47.7910	−2.39254
$400$	−14.8945	−0.744725
$401$	−16.6130	−0.829613	−0.414806	−	0.909910i	$-0.636151\pi$
$401$	−16.6130	−0.829613	−0.414806	+	0.909910i	$0.636151\pi$
$402$	67.7001	3.37658
$403$	0.135080	0.00672883
$404$	−85.1045	−4.23411
$405$	9.88380	0.491130
$406$	−138.027	−6.85017
$407$	1.75354	0.0869198
$408$	86.9546	4.30489
$409$	−5.85432	−0.289477	−0.144739	−	0.989470i	$-0.546234\pi$
$409$	−5.85432	−0.289477	−0.144739	+	0.989470i	$0.546234\pi$
$410$	12.5228	0.618458
$411$	28.5545	1.40849
$412$	47.1902	2.32489
$413$	8.41450	0.414051
$414$	55.3361	2.71962
$415$	−23.6291	−1.15991
$416$	0.680715	0.0333748
$417$	−12.6043	−0.617236
$418$	47.8309	2.33949
$419$	11.1072	0.542620	0.271310	−	0.962492i	$-0.412543\pi$
$419$	11.1072	0.542620	0.271310	+	0.962492i	$0.412543\pi$
$420$	−137.032	−6.68648
$421$	0.733952	0.0357706	0.0178853	−	0.999840i	$-0.494307\pi$
$421$	0.733952	0.0357706	0.0178853	+	0.999840i	$0.494307\pi$
$422$	−76.0416	−3.70165
$423$	−15.3738	−0.747501
$424$	115.113	5.59038
$425$	−3.82182	−0.185386
$426$	104.726	5.07398
$427$	−12.7682	−0.617894
$428$	−83.3339	−4.02809
$429$	0.431446	0.0208304
$430$	42.0748	2.02903
$431$	22.0473	1.06198	0.530990	−	0.847378i	$-0.321820\pi$
$431$	22.0473	1.06198	0.530990	+	0.847378i	$0.321820\pi$
$432$	−37.2861	−1.79393
$433$	17.6984	0.850532	0.425266	−	0.905068i	$-0.360180\pi$
$433$	17.6984	0.850532	0.425266	+	0.905068i	$0.360180\pi$
$434$	−51.9462	−2.49350
$435$	−54.9159	−2.63302
$436$	46.3702	2.22073
$437$	18.9053	0.904364
$438$	−69.2642	−3.30957
$439$	19.2876	0.920547	0.460274	−	0.887777i	$-0.347751\pi$
$439$	19.2876	0.920547	0.460274	+	0.887777i	$0.347751\pi$
$440$	85.9985	4.09982
$441$	66.7425	3.17821
$442$	0.335411	0.0159539
$443$	−21.0561	−1.00041	−0.500203	−	0.865908i	$-0.666741\pi$
$443$	−21.0561	−1.00041	−0.500203	+	0.865908i	$0.666741\pi$
$444$	−5.24044	−0.248700
$445$	−6.01982	−0.285367
$446$	46.7696	2.21461
$447$	49.0979	2.32225
$448$	−125.226	−5.91639
$449$	−5.91283	−0.279044	−0.139522	−	0.990219i	$-0.544557\pi$
$449$	−5.91283	−0.279044	−0.139522	+	0.990219i	$0.544557\pi$
$450$	11.5311	0.543580
$451$	11.0437	0.520028
$452$	−34.5517	−1.62518
$453$	−38.8033	−1.82314
$454$	−8.03888	−0.377283
$455$	−0.331447	−0.0155385
$456$	−89.6327	−4.19744
$457$	−6.28400	−0.293953	−0.146977	−	0.989140i	$-0.546954\pi$
$457$	−6.28400	−0.293953	−0.146977	+	0.989140i	$0.546954\pi$
$458$	−19.5783	−0.914832
$459$	−9.56735	−0.446566
$460$	54.2076	2.52744
$461$	−15.2872	−0.711998	−0.355999	−	0.934486i	$-0.615859\pi$
$461$	−15.2872	−0.711998	−0.355999	+	0.934486i	$0.615859\pi$
$462$	−165.916	−7.71911
$463$	−10.1649	−0.472401	−0.236200	−	0.971704i	$-0.575902\pi$
$463$	−10.1649	−0.472401	−0.236200	+	0.971704i	$0.575902\pi$
$464$	−146.730	−6.81175
$465$	−20.6675	−0.958432
$466$	−30.0432	−1.39173
$467$	10.3905	0.480817	0.240409	−	0.970672i	$-0.422719\pi$
$467$	10.3905	0.480817	0.240409	+	0.970672i	$0.422719\pi$
$468$	−0.737095	−0.0340722
$469$	−45.8659	−2.11789
$470$	−20.6769	−0.953756
$471$	−0.452382	−0.0208447
$472$	15.7815	0.726403
$473$	37.1052	1.70610
$474$	25.9997	1.19420
$475$	3.93954	0.180758
$476$	−93.9478	−4.30609
$477$	−50.5127	−2.31282
$478$	−25.3296	−1.15855
$479$	−37.0241	−1.69168	−0.845838	−	0.533440i	$-0.820899\pi$
$479$	−37.0241	−1.69168	−0.845838	+	0.533440i	$0.820899\pi$
$480$	−104.150	−4.75379
$481$	−0.0126753	−0.000577945 0
$482$	31.1930	1.42080
$483$	−65.5788	−2.98394
$484$	61.9576	2.81626
$485$	−36.0930	−1.63890
$486$	−57.3920	−2.60335
$487$	−30.3561	−1.37557	−0.687784	−	0.725915i	$-0.741416\pi$
$487$	−30.3561	−1.37557	−0.687784	+	0.725915i	$0.741416\pi$
$488$	−23.9469	−1.08402
$489$	−7.79111	−0.352326
$490$	89.7650	4.05517
$491$	−38.3828	−1.73219	−0.866096	−	0.499878i	$-0.833378\pi$
$491$	−38.3828	−1.73219	−0.866096	+	0.499878i	$0.833378\pi$
$492$	−33.0040	−1.48794
$493$	−37.6498	−1.69566
$494$	−0.345742	−0.0155557
$495$	−37.7370	−1.69615
$496$	−55.2213	−2.47951
$497$	−70.9503	−3.18256
$498$	85.4998	3.83134
$499$	16.5331	0.740122	0.370061	−	0.929007i	$-0.379337\pi$
$499$	16.5331	0.740122	0.370061	+	0.929007i	$0.379337\pi$
$500$	64.5103	2.88499
$501$	−44.7533	−1.99943
$502$	−15.4743	−0.690650
$503$	6.51841	0.290642	0.145321	−	0.989385i	$-0.453579\pi$
$503$	6.51841	0.290642	0.145321	+	0.989385i	$0.453579\pi$
$504$	177.743	7.91729
$505$	31.4948	1.40150
$506$	65.6336	2.91777
$507$	34.4013	1.52782
$508$	108.101	4.79623
$509$	1.01563	0.0450172	0.0225086	−	0.999747i	$-0.492835\pi$
$509$	1.01563	0.0450172	0.0225086	+	0.999747i	$0.492835\pi$
$510$	−51.3184	−2.27242
$511$	46.9255	2.07586
$512$	−22.1811	−0.980274
$513$	9.86202	0.435419
$514$	−2.42025	−0.106752
$515$	−17.4638	−0.769547
$516$	−110.889	−4.88160
$517$	−18.2347	−0.801962
$518$	4.87440	0.214169
$519$	−51.5551	−2.26302
$520$	−0.621633	−0.0272604
$521$	2.06206	0.0903407	0.0451704	−	0.998979i	$-0.485617\pi$
$521$	2.06206	0.0903407	0.0451704	+	0.998979i	$0.485617\pi$
$522$	113.596	4.97195
$523$	−31.9386	−1.39658	−0.698288	−	0.715817i	$-0.746055\pi$
$523$	−31.9386	−1.39658	−0.698288	+	0.715817i	$0.746055\pi$
$524$	77.7925	3.39838
$525$	−13.6655	−0.596410
$526$	40.8397	1.78069
$527$	−14.1694	−0.617230
$528$	−176.377	−7.67581
$529$	2.94188	0.127908
$530$	−67.9368	−2.95099
$531$	−6.92509	−0.300523
$532$	96.8414	4.19861
$533$	−0.0798286	−0.00345776
$534$	21.7822	0.942606
$535$	30.8396	1.33331
$536$	−86.0222	−3.71559
$537$	15.7571	0.679969
$538$	−0.361004	−0.0155640
$539$	79.1625	3.40977
$540$	28.2776	1.21687
$541$	−44.0977	−1.89591	−0.947953	−	0.318410i	$-0.896851\pi$
$541$	−44.0977	−1.89591	−0.947953	+	0.318410i	$0.896851\pi$
$542$	−29.6274	−1.27261
$543$	−12.4788	−0.535516
$544$	−71.4044	−3.06144
$545$	−17.1604	−0.735069
$546$	1.19931	0.0513258
$547$	19.6921	0.841975	0.420987	−	0.907067i	$-0.361684\pi$
$547$	19.6921	0.841975	0.420987	+	0.907067i	$0.361684\pi$
$548$	−57.8615	−2.47172
$549$	10.5081	0.448476
$550$	13.6769	0.583184
$551$	38.8094	1.65334
$552$	−122.994	−5.23497
$553$	−17.6144	−0.749041
$554$	−34.2587	−1.45551
$555$	1.93934	0.0823206
$556$	25.5408	1.08317
$557$	−22.7118	−0.962331	−0.481166	−	0.876630i	$-0.659786\pi$
$557$	−22.7118	−0.962331	−0.481166	+	0.876630i	$0.659786\pi$
$558$	42.7515	1.80981
$559$	−0.268212	−0.0113442
$560$	135.497	5.72578
$561$	−45.2570	−1.91075
$562$	−31.4476	−1.32654
$563$	12.9127	0.544206	0.272103	−	0.962268i	$-0.412281\pi$
$563$	12.9127	0.544206	0.272103	+	0.962268i	$0.412281\pi$
$564$	54.4943	2.29462
$565$	12.7867	0.537939
$566$	69.3151	2.91353
$567$	24.2294	1.01754
$568$	−133.068	−5.58342
$569$	−10.2081	−0.427945	−0.213973	−	0.976840i	$-0.568640\pi$
$569$	−10.2081	−0.427945	−0.213973	+	0.976840i	$0.568640\pi$
$570$	52.8990	2.21570
$571$	−22.7597	−0.952462	−0.476231	−	0.879320i	$-0.657997\pi$
$571$	−22.7597	−0.952462	−0.476231	+	0.879320i	$0.657997\pi$
$572$	−0.874261	−0.0365547
$573$	6.05046	0.252762
$574$	30.6987	1.28134
$575$	5.40583	0.225439
$576$	103.061	4.29419
$577$	3.75852	0.156469	0.0782347	−	0.996935i	$-0.475072\pi$
$577$	3.75852	0.156469	0.0782347	+	0.996935i	$0.475072\pi$
$578$	10.9450	0.455253
$579$	5.65142	0.234865
$580$	111.279	4.62061
$581$	−57.9250	−2.40313
$582$	130.599	5.41351
$583$	−59.9126	−2.48133
$584$	88.0095	3.64186
$585$	0.272779	0.0112780
$586$	4.67207	0.193002
$587$	−5.02212	−0.207285	−0.103642	−	0.994615i	$-0.533050\pi$
$587$	−5.02212	−0.207285	−0.103642	+	0.994615i	$0.533050\pi$
$588$	−236.576	−9.75625
$589$	14.6058	0.601823
$590$	−9.31386	−0.383445
$591$	−9.00727	−0.370510
$592$	5.18172	0.212968
$593$	6.96580	0.286051	0.143026	−	0.989719i	$-0.454317\pi$
$593$	6.96580	0.286051	0.143026	+	0.989719i	$0.454317\pi$
$594$	34.2380	1.40480
$595$	34.7675	1.42533
$596$	−99.4896	−4.07525
$597$	45.3640	1.85663
$598$	−0.474427	−0.0194008
$599$	27.3644	1.11808	0.559040	−	0.829141i	$-0.311170\pi$
$599$	27.3644	1.11808	0.559040	+	0.829141i	$0.311170\pi$
$600$	−25.6298	−1.04633
$601$	−4.02471	−0.164171	−0.0820857	−	0.996625i	$-0.526158\pi$
$601$	−4.02471	−0.164171	−0.0820857	+	0.996625i	$0.526158\pi$
$602$	103.143	4.20380
$603$	37.7474	1.53719
$604$	78.6291	3.19937
$605$	−22.9288	−0.932189
$606$	−113.961	−4.62936
$607$	36.7703	1.49246	0.746230	−	0.665688i	$-0.231862\pi$
$607$	36.7703	1.49246	0.746230	+	0.665688i	$0.231862\pi$
$608$	73.6037	2.98502
$609$	−134.622	−5.45517
$610$	14.1328	0.572222
$611$	0.131808	0.00533239
$612$	77.3185	3.12542
$613$	−6.94466	−0.280492	−0.140246	−	0.990117i	$-0.544789\pi$
$613$	−6.94466	−0.280492	−0.140246	+	0.990117i	$0.544789\pi$
$614$	40.6473	1.64039
$615$	12.2139	0.492512
$616$	210.819	8.49413
$617$	−1.46402	−0.0589392	−0.0294696	−	0.999566i	$-0.509382\pi$
$617$	−1.46402	−0.0589392	−0.0294696	+	0.999566i	$0.509382\pi$
$618$	63.1912	2.54192
$619$	13.4341	0.539961	0.269980	−	0.962866i	$-0.412983\pi$
$619$	13.4341	0.539961	0.269980	+	0.962866i	$0.412983\pi$
$620$	41.8796	1.68192
$621$	13.5327	0.543047
$622$	82.4217	3.30481
$623$	−14.7571	−0.591231
$624$	1.27492	0.0510378
$625$	−18.5667	−0.742669
$626$	−24.3245	−0.972204
$627$	46.6509	1.86306
$628$	0.916684	0.0365797
$629$	1.32959	0.0530144
$630$	−104.899	−4.17929
$631$	−6.97152	−0.277532	−0.138766	−	0.990325i	$-0.544314\pi$
$631$	−6.97152	−0.277532	−0.138766	+	0.990325i	$0.544314\pi$
$632$	−33.0361	−1.31410
$633$	−74.1657	−2.94782
$634$	29.4623	1.17010
$635$	−40.0054	−1.58757
$636$	179.048	7.09972
$637$	−0.572220	−0.0226722
$638$	134.735	5.33419
$639$	58.3917	2.30994
$640$	59.9027	2.36786
$641$	−6.22642	−0.245929	−0.122964	−	0.992411i	$-0.539240\pi$
$641$	−6.22642	−0.245929	−0.122964	+	0.992411i	$0.539240\pi$
$642$	−111.590	−4.40412
$643$	24.6830	0.973401	0.486701	−	0.873569i	$-0.338200\pi$
$643$	24.6830	0.973401	0.486701	+	0.873569i	$0.338200\pi$
$644$	132.886	5.23643
$645$	41.0369	1.61582
$646$	36.2670	1.42691
$647$	20.1731	0.793088	0.396544	−	0.918016i	$-0.370209\pi$
$647$	20.1731	0.793088	0.396544	+	0.918016i	$0.370209\pi$
$648$	45.4425	1.78515
$649$	−8.21377	−0.322419
$650$	−0.0988622	−0.00387770
$651$	−50.6648	−1.98571
$652$	15.7875	0.618287
$653$	−26.2964	−1.02906	−0.514529	−	0.857473i	$-0.672033\pi$
$653$	−26.2964	−1.02906	−0.514529	+	0.857473i	$0.672033\pi$
$654$	62.0932	2.42804
$655$	−28.7889	−1.12488
$656$	32.6342	1.27415
$657$	−38.6194	−1.50669
$658$	−50.6879	−1.97602
$659$	−33.5966	−1.30874	−0.654369	−	0.756176i	$-0.727066\pi$
$659$	−33.5966	−1.30874	−0.654369	+	0.756176i	$0.727066\pi$
$660$	133.763	5.20672
$661$	43.8683	1.70628	0.853140	−	0.521683i	$-0.174695\pi$
$661$	43.8683	1.70628	0.853140	+	0.521683i	$0.174695\pi$
$662$	−75.4291	−2.93164
$663$	0.327137	0.0127049
$664$	−108.639	−4.21601
$665$	−35.8384	−1.38975
$666$	−4.01161	−0.155447
$667$	53.2543	2.06201
$668$	90.6858	3.50874
$669$	45.6159	1.76361
$670$	50.7682	1.96135
$671$	12.4636	0.481151
$672$	−255.317	−9.84905
$673$	28.8287	1.11126	0.555631	−	0.831429i	$-0.312477\pi$
$673$	28.8287	1.11126	0.555631	+	0.831429i	$0.312477\pi$
$674$	−14.1230	−0.543998
$675$	2.81997	0.108541
$676$	−69.7092	−2.68112
$677$	−0.337843	−0.0129844	−0.00649218	−	0.999979i	$-0.502067\pi$
$677$	−0.337843	−0.0129844	−0.00649218	+	0.999979i	$0.502067\pi$
$678$	−46.2673	−1.77689
$679$	−88.4792	−3.39552
$680$	65.2070	2.50057
$681$	−7.84057	−0.300451
$682$	50.7070	1.94167
$683$	−7.15032	−0.273600	−0.136800	−	0.990599i	$-0.543682\pi$
$683$	−7.15032	−0.273600	−0.136800	+	0.990599i	$0.543682\pi$
$684$	−79.6999	−3.04740
$685$	21.4130	0.818147
$686$	127.644	4.87347
$687$	−19.0953	−0.728531
$688$	109.646	4.18022
$689$	0.433073	0.0164988
$690$	72.5880	2.76338
$691$	−41.9893	−1.59735	−0.798674	−	0.601764i	$-0.794465\pi$
$691$	−41.9893	−1.59735	−0.798674	+	0.601764i	$0.794465\pi$
$692$	104.469	3.97131
$693$	−92.5094	−3.51414
$694$	−51.7299	−1.96364
$695$	−9.45195	−0.358533
$696$	−252.486	−9.57045
$697$	8.37372	0.317177
$698$	−70.3550	−2.66297
$699$	−29.3021	−1.10831
$700$	27.6910	1.04662
$701$	22.9393	0.866407	0.433203	−	0.901296i	$-0.357383\pi$
$701$	22.9393	0.866407	0.433203	+	0.901296i	$0.357383\pi$
$702$	−0.247486	−0.00934077
$703$	−1.37055	−0.0516911
$704$	122.239	4.60706
$705$	−20.1669	−0.759528
$706$	32.7506	1.23258
$707$	77.2072	2.90368
$708$	24.5468	0.922524
$709$	−20.3350	−0.763697	−0.381849	−	0.924225i	$-0.624712\pi$
$709$	−20.3350	−0.763697	−0.381849	+	0.924225i	$0.624712\pi$
$710$	78.5336	2.94732
$711$	14.4966	0.543664
$712$	−27.6772	−1.03725
$713$	20.0421	0.750583
$714$	−125.803	−4.70807
$715$	0.323540	0.0120997
$716$	−31.9294	−1.19326
$717$	−24.7047	−0.922614
$718$	57.6479	2.15140
$719$	−0.890809	−0.0332216	−0.0166108	−	0.999862i	$-0.505288\pi$
$719$	−0.890809	−0.0332216	−0.0166108	+	0.999862i	$0.505288\pi$
$720$	−111.513	−4.15585
$721$	−42.8112	−1.59437
$722$	14.1712	0.527399
$723$	30.4235	1.13146
$724$	25.2864	0.939763
$725$	11.0972	0.412142
$726$	82.9659	3.07915
$727$	−10.1021	−0.374666	−0.187333	−	0.982296i	$-0.559984\pi$
$727$	−10.1021	−0.374666	−0.187333	+	0.982296i	$0.559984\pi$
$728$	−1.52389	−0.0564790
$729$	−41.0354	−1.51983
$730$	−51.9410	−1.92242
$731$	28.1344	1.04059
$732$	−37.2472	−1.37670
$733$	3.04749	0.112561	0.0562807	−	0.998415i	$-0.482076\pi$
$733$	3.04749	0.112561	0.0562807	+	0.998415i	$0.482076\pi$
$734$	23.9837	0.885255
$735$	87.5505	3.22935
$736$	100.999	3.72287
$737$	44.7718	1.64919
$738$	−25.2649	−0.930014
$739$	21.4491	0.789019	0.394510	−	0.918892i	$-0.370914\pi$
$739$	21.4491	0.789019	0.394510	+	0.918892i	$0.370914\pi$
$740$	−3.92979	−0.144462
$741$	−0.337213	−0.0123878
$742$	−166.542	−6.11394
$743$	−28.0663	−1.02965	−0.514826	−	0.857295i	$-0.672143\pi$
$743$	−28.0663	−1.02965	−0.514826	+	0.857295i	$0.672143\pi$
$744$	−95.0225	−3.48369
$745$	36.8184	1.34892
$746$	22.2298	0.813891
$747$	47.6719	1.74422
$748$	91.7067	3.35313
$749$	75.6009	2.76240
$750$	86.3842	3.15430
$751$	−27.7716	−1.01340	−0.506700	−	0.862122i	$-0.669135\pi$
$751$	−27.7716	−1.01340	−0.506700	+	0.862122i	$0.669135\pi$
$752$	−53.8837	−1.96494
$753$	−15.0925	−0.550002
$754$	−0.973918	−0.0354680
$755$	−29.0985	−1.05900
$756$	69.3203	2.52115
$757$	−6.74776	−0.245251	−0.122626	−	0.992453i	$-0.539131\pi$
$757$	−6.74776	−0.245251	−0.122626	+	0.992453i	$0.539131\pi$
$758$	2.93346	0.106548
$759$	64.0144	2.32358
$760$	−67.2153	−2.43816
$761$	32.2845	1.17031	0.585156	−	0.810920i	$-0.301033\pi$
$761$	32.2845	1.17031	0.585156	+	0.810920i	$0.301033\pi$
$762$	144.756	5.24396
$763$	−42.0673	−1.52294
$764$	−12.2604	−0.443564
$765$	−28.6135	−1.03452
$766$	−91.6144	−3.31016
$767$	0.0593726	0.00214382
$768$	−80.5122	−2.90523
$769$	41.6877	1.50330	0.751649	−	0.659563i	$-0.229259\pi$
$769$	41.6877	1.50330	0.751649	+	0.659563i	$0.229259\pi$
$770$	−124.420	−4.48378
$771$	−2.36054	−0.0850128
$772$	−11.4518	−0.412158
$773$	−14.6025	−0.525217	−0.262608	−	0.964903i	$-0.584583\pi$
$773$	−14.6025	−0.525217	−0.262608	+	0.964903i	$0.584583\pi$
$774$	−84.8862	−3.05117
$775$	4.17643	0.150022
$776$	−165.944	−5.95704
$777$	4.75415	0.170554
$778$	−35.0789	−1.25764
$779$	−8.63163	−0.309260
$780$	−0.966897	−0.0346204
$781$	69.2578	2.47824
$782$	49.7656	1.77961
$783$	27.7803	0.992786
$784$	233.926	8.35449
$785$	−0.339240	−0.0121080
$786$	104.170	3.71562
$787$	42.2699	1.50676	0.753379	−	0.657587i	$-0.228423\pi$
$787$	42.2699	1.50676	0.753379	+	0.657587i	$0.228423\pi$
$788$	18.2519	0.650197
$789$	39.8322	1.41806
$790$	19.4971	0.693675
$791$	31.3455	1.11452
$792$	−173.503	−6.16515
$793$	−0.0900919	−0.00319926
$794$	−79.5845	−2.82435
$795$	−66.2609	−2.35003
$796$	−91.9235	−3.25814
$797$	−33.8714	−1.19979	−0.599894	−	0.800080i	$-0.704791\pi$
$797$	−33.8714	−1.19979	−0.599894	+	0.800080i	$0.704791\pi$
$798$	129.678	4.59055
$799$	−13.8262	−0.489136
$800$	21.0464	0.744103
$801$	12.1450	0.429123
$802$	45.0783	1.59177
$803$	−45.8061	−1.61646
$804$	−133.800	−4.71876
$805$	−49.1774	−1.73328
$806$	−0.366532	−0.0129105
$807$	−0.352099	−0.0123945
$808$	144.803	5.09416
$809$	−27.9630	−0.983128	−0.491564	−	0.870842i	$-0.663575\pi$
$809$	−27.9630	−0.983128	−0.491564	+	0.870842i	$0.663575\pi$
$810$	−26.8191	−0.942325
$811$	−40.0624	−1.40678	−0.703390	−	0.710804i	$-0.748331\pi$
$811$	−40.0624	−1.40678	−0.703390	+	0.710804i	$0.748331\pi$
$812$	272.792	9.57312
$813$	−28.8965	−1.01344
$814$	−4.75812	−0.166772
$815$	−5.84253	−0.204655
$816$	−133.735	−4.68165
$817$	−29.0010	−1.01462
$818$	15.8853	0.555417
$819$	0.668697	0.0233661
$820$	−24.7496	−0.864295
$821$	53.0461	1.85132	0.925661	−	0.378353i	$-0.123510\pi$
$821$	53.0461	1.85132	0.925661	+	0.378353i	$0.123510\pi$
$822$	−77.4809	−2.70246
$823$	−13.2914	−0.463308	−0.231654	−	0.972798i	$-0.574414\pi$
$823$	−13.2914	−0.463308	−0.231654	+	0.972798i	$0.574414\pi$
$824$	−80.2929	−2.79714
$825$	13.3395	0.464421
$826$	−22.8322	−0.794434
$827$	38.3575	1.33382	0.666910	−	0.745138i	$-0.267616\pi$
$827$	38.3575	1.33382	0.666910	+	0.745138i	$0.267616\pi$
$828$	−109.364	−3.80067
$829$	−39.3863	−1.36794	−0.683972	−	0.729509i	$-0.739749\pi$
$829$	−39.3863	−1.36794	−0.683972	+	0.729509i	$0.739749\pi$
$830$	64.1161	2.22550
$831$	−33.4135	−1.15910
$832$	−0.883595	−0.0306332
$833$	60.0237	2.07970
$834$	34.2010	1.18428
$835$	−33.5604	−1.16140
$836$	−94.5312	−3.26943
$837$	10.4550	0.361379
$838$	−30.1386	−1.04112
$839$	43.9661	1.51788	0.758939	−	0.651162i	$-0.225718\pi$
$839$	43.9661	1.51788	0.758939	+	0.651162i	$0.225718\pi$
$840$	233.157	8.04467
$841$	80.3219	2.76972
$842$	−1.99153	−0.0686327
$843$	−30.6718	−1.05639
$844$	150.286	5.17305
$845$	25.7975	0.887460
$846$	41.7159	1.43422
$847$	−56.2083	−1.93134
$848$	−177.042	−6.07965
$849$	67.6051	2.32020
$850$	10.3703	0.355698
$851$	−1.88066	−0.0644683
$852$	−206.976	−7.09089
$853$	19.5038	0.667798	0.333899	−	0.942609i	$-0.391636\pi$
$853$	19.5038	0.667798	0.333899	+	0.942609i	$0.391636\pi$
$854$	34.6456	1.18555
$855$	29.4948	1.00870
$856$	141.791	4.84630
$857$	−1.14295	−0.0390423	−0.0195211	−	0.999809i	$-0.506214\pi$
$857$	−1.14295	−0.0390423	−0.0195211	+	0.999809i	$0.506214\pi$
$858$	−1.17070	−0.0399671
$859$	−41.5952	−1.41921	−0.709605	−	0.704599i	$-0.751127\pi$
$859$	−41.5952	−1.41921	−0.709605	+	0.704599i	$0.751127\pi$
$860$	−83.1551	−2.83556
$861$	29.9414	1.02040
$862$	−59.8239	−2.03761
$863$	−0.738241	−0.0251300	−0.0125650	−	0.999921i	$-0.504000\pi$
$863$	−0.738241	−0.0251300	−0.0125650	+	0.999921i	$0.504000\pi$
$864$	52.6864	1.79243
$865$	−38.6610	−1.31452
$866$	−48.0236	−1.63191
$867$	10.6750	0.362542
$868$	102.665	3.48466
$869$	17.1942	0.583274
$870$	149.011	5.05194
$871$	−0.323629	−0.0109658
$872$	−78.8978	−2.67182
$873$	72.8179	2.46451
$874$	−51.2984	−1.73519
$875$	−58.5241	−1.97848
$876$	136.891	4.62512
$877$	−51.8481	−1.75079	−0.875393	−	0.483413i	$-0.839397\pi$
$877$	−51.8481	−1.75079	−0.875393	+	0.483413i	$0.839397\pi$
$878$	−52.3357	−1.76624
$879$	4.55682	0.153698
$880$	−132.264	−4.45863
$881$	−56.1590	−1.89204	−0.946022	−	0.324103i	$-0.894938\pi$
$881$	−56.1590	−1.89204	−0.946022	+	0.324103i	$0.894938\pi$
$882$	−181.101	−6.09800
$883$	−36.9204	−1.24247	−0.621236	−	0.783624i	$-0.713369\pi$
$883$	−36.9204	−1.24247	−0.621236	+	0.783624i	$0.713369\pi$
$884$	−0.662894	−0.0222956
$885$	−9.08409	−0.305358
$886$	57.1344	1.91947
$887$	29.7015	0.997277	0.498639	−	0.866810i	$-0.333833\pi$
$887$	29.7015	0.997277	0.498639	+	0.866810i	$0.333833\pi$
$888$	8.91648	0.299218
$889$	−98.0702	−3.28917
$890$	16.3344	0.547530
$891$	−23.6514	−0.792351
$892$	−92.4337	−3.09491
$893$	14.2520	0.476926
$894$	−133.224	−4.45568
$895$	11.8162	0.394972
$896$	146.847	4.90581
$897$	−0.462723	−0.0154499
$898$	16.0441	0.535398
$899$	41.1431	1.37220
$900$	−22.7896	−0.759653
$901$	−45.4278	−1.51342
$902$	−29.9664	−0.997772
$903$	100.599	3.34771
$904$	58.7889	1.95529
$905$	−9.35782	−0.311064
$906$	105.290	3.49803
$907$	−34.3483	−1.14052	−0.570259	−	0.821465i	$-0.693157\pi$
$907$	−34.3483	−1.14052	−0.570259	+	0.821465i	$0.693157\pi$
$908$	15.8877	0.527253
$909$	−63.5411	−2.10752
$910$	0.899360	0.0298135
$911$	−47.1301	−1.56149	−0.780745	−	0.624850i	$-0.785160\pi$
$911$	−47.1301	−1.56149	−0.780745	+	0.624850i	$0.785160\pi$
$912$	137.854	4.56480
$913$	56.5431	1.87130
$914$	17.0512	0.564004
$915$	13.7842	0.455691
$916$	38.6937	1.27848
$917$	−70.5738	−2.33055
$918$	25.9604	0.856821
$919$	−16.1997	−0.534380	−0.267190	−	0.963644i	$-0.586095\pi$
$919$	−16.1997	−0.534380	−0.267190	+	0.963644i	$0.586095\pi$
$920$	−92.2329	−3.04083
$921$	39.6445	1.30633
$922$	41.4810	1.36610
$923$	−0.500625	−0.0164783
$924$	327.910	10.7875
$925$	−0.391897	−0.0128855
$926$	27.5817	0.906390
$927$	35.2333	1.15721
$928$	207.334	6.80606
$929$	39.8133	1.30623	0.653117	−	0.757257i	$-0.273461\pi$
$929$	39.8133	1.30623	0.653117	+	0.757257i	$0.273461\pi$
$930$	56.0799	1.83893
$931$	−61.8725	−2.02779
$932$	59.3763	1.94494
$933$	80.3884	2.63180
$934$	−28.1941	−0.922539
$935$	−33.9381	−1.10990
$936$	1.25415	0.0409932
$937$	29.4900	0.963397	0.481699	−	0.876337i	$-0.340020\pi$
$937$	29.4900	0.963397	0.481699	+	0.876337i	$0.340020\pi$
$938$	124.454	4.06357
$939$	−23.7245	−0.774219
$940$	40.8651	1.33287
$941$	−52.2897	−1.70459	−0.852297	−	0.523058i	$-0.824791\pi$
$941$	−52.2897	−1.70459	−0.852297	+	0.523058i	$0.824791\pi$
$942$	1.22751	0.0399944
$943$	−11.8443	−0.385704
$944$	−24.2717	−0.789978
$945$	−25.6536	−0.834510
$946$	−100.683	−3.27347
$947$	26.2321	0.852429	0.426215	−	0.904622i	$-0.359847\pi$
$947$	26.2321	0.852429	0.426215	+	0.904622i	$0.359847\pi$
$948$	−51.3847	−1.66890
$949$	0.331106	0.0107481
$950$	−10.6897	−0.346819
$951$	28.7355	0.931811
$952$	159.850	5.18077
$953$	−21.9584	−0.711302	−0.355651	−	0.934619i	$-0.615741\pi$
$953$	−21.9584	−0.711302	−0.355651	+	0.934619i	$0.615741\pi$
$954$	137.063	4.43758
$955$	4.53723	0.146821
$956$	50.0604	1.61907
$957$	131.411	4.24791
$958$	100.463	3.24580
$959$	52.4922	1.69506
$960$	135.191	4.36328
$961$	−15.5159	−0.500513
$962$	0.0343937	0.00110890
$963$	−62.2191	−2.00498
$964$	−61.6486	−1.98557
$965$	4.23798	0.136426
$966$	177.944	5.72525
$967$	−36.9220	−1.18733	−0.593665	−	0.804712i	$-0.702320\pi$
$967$	−36.9220	−1.18733	−0.593665	+	0.804712i	$0.702320\pi$
$968$	−105.419	−3.38831
$969$	35.3723	1.13632
$970$	97.9360	3.14453
$971$	−19.5418	−0.627127	−0.313564	−	0.949567i	$-0.601523\pi$
$971$	−19.5418	−0.627127	−0.313564	+	0.949567i	$0.601523\pi$
$972$	113.427	3.63818
$973$	−23.1707	−0.742819
$974$	82.3694	2.63929
$975$	−0.0964234	−0.00308802
$976$	36.8299	1.17890
$977$	−20.2680	−0.648429	−0.324215	−	0.945984i	$-0.605100\pi$
$977$	−20.2680	−0.648429	−0.324215	+	0.945984i	$0.605100\pi$
$978$	21.1407	0.676004
$979$	14.4051	0.460388
$980$	−177.408	−5.66709
$981$	34.6211	1.10537
$982$	104.149	3.32354
$983$	53.4834	1.70586	0.852928	−	0.522028i	$-0.174825\pi$
$983$	53.4834	1.70586	0.852928	+	0.522028i	$0.174825\pi$
$984$	56.1556	1.79017
$985$	−6.75453	−0.215217
$986$	102.160	3.25345
$987$	−49.4375	−1.57361
$988$	0.683311	0.0217390
$989$	−39.7951	−1.26541
$990$	102.397	3.25439
$991$	34.3746	1.09194	0.545972	−	0.837803i	$-0.316160\pi$
$991$	34.3746	1.09194	0.545972	+	0.837803i	$0.316160\pi$
$992$	78.0296	2.47744
$993$	−73.5684	−2.33462
$994$	192.519	6.10634
$995$	34.0184	1.07846
$996$	−168.979	−5.35429
$997$	−16.2246	−0.513838	−0.256919	−	0.966433i	$-0.582707\pi$
$997$	−16.2246	−0.513838	−0.256919	+	0.966433i	$0.582707\pi$
$998$	−44.8614	−1.42006
$999$	−0.981054	−0.0310392

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.b.1.5	✓	152

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.b.1.5	✓	152	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71344 q^{2} -2.64650 q^{3} +5.36273 q^{4} -1.98460 q^{5} +7.18110 q^{6} -4.86510 q^{7} -9.12456 q^{8} +4.00395 q^{9} +O(q^{10})\)	\(q-2.71344 q^{2} -2.64650 q^{3} +5.36273 q^{4} -1.98460 q^{5} +7.18110 q^{6} -4.86510 q^{7} -9.12456 q^{8} +4.00395 q^{9} +5.38509 q^{10} +4.74904 q^{11} -14.1925 q^{12} -0.0343281 q^{13} +13.2011 q^{14} +5.25224 q^{15} +14.0334 q^{16} +3.60088 q^{17} -10.8645 q^{18} -3.71179 q^{19} -10.6429 q^{20} +12.8755 q^{21} -12.8862 q^{22} -5.09332 q^{23} +24.1481 q^{24} -1.06136 q^{25} +0.0931470 q^{26} -2.65695 q^{27} -26.0902 q^{28} -10.4557 q^{29} -14.2516 q^{30} -3.93498 q^{31} -19.8297 q^{32} -12.5683 q^{33} -9.77077 q^{34} +9.65528 q^{35} +21.4721 q^{36} +0.369241 q^{37} +10.0717 q^{38} +0.0908491 q^{39} +18.1086 q^{40} +2.32546 q^{41} -34.9367 q^{42} +7.81321 q^{43} +25.4678 q^{44} -7.94624 q^{45} +13.8204 q^{46} -3.83967 q^{47} -37.1394 q^{48} +16.6692 q^{49} +2.87993 q^{50} -9.52973 q^{51} -0.184092 q^{52} -12.6157 q^{53} +7.20945 q^{54} -9.42495 q^{55} +44.3919 q^{56} +9.82324 q^{57} +28.3709 q^{58} -1.72956 q^{59} +28.1664 q^{60} +2.62444 q^{61} +10.6773 q^{62} -19.4796 q^{63} +25.7398 q^{64} +0.0681275 q^{65} +34.1033 q^{66} +9.42755 q^{67} +19.3106 q^{68} +13.4794 q^{69} -26.1990 q^{70} +14.5835 q^{71} -36.5343 q^{72} -9.64534 q^{73} -1.00191 q^{74} +2.80888 q^{75} -19.9053 q^{76} -23.1045 q^{77} -0.246513 q^{78} +3.62057 q^{79} -27.8508 q^{80} -4.98024 q^{81} -6.31000 q^{82} +11.9062 q^{83} +69.0477 q^{84} -7.14632 q^{85} -21.2006 q^{86} +27.6710 q^{87} -43.3329 q^{88} +3.03326 q^{89} +21.5616 q^{90} +0.167009 q^{91} -27.3141 q^{92} +10.4139 q^{93} +10.4187 q^{94} +7.36642 q^{95} +52.4793 q^{96} +18.1865 q^{97} -45.2307 q^{98} +19.0149 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * q^99

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