LMFDB - Embedded newform 4009.2.a.d.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$1$
Dimension:	$75$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.17778 q^{2} -1.25453 q^{3} +2.74274 q^{4} +2.27371 q^{5} +2.73208 q^{6} -1.62232 q^{7} -1.61752 q^{8} -1.42616 q^{9} +O(q^{10})$	\(q-2.17778 q^{2} -1.25453 q^{3} +2.74274 q^{4} +2.27371 q^{5} +2.73208 q^{6} -1.62232 q^{7} -1.61752 q^{8} -1.42616 q^{9} -4.95164 q^{10} +0.712705 q^{11} -3.44083 q^{12} -6.45412 q^{13} +3.53307 q^{14} -2.85242 q^{15} -1.96287 q^{16} -3.12354 q^{17} +3.10588 q^{18} -1.00000 q^{19} +6.23618 q^{20} +2.03525 q^{21} -1.55212 q^{22} +7.14256 q^{23} +2.02922 q^{24} +0.169744 q^{25} +14.0557 q^{26} +5.55274 q^{27} -4.44961 q^{28} +8.26282 q^{29} +6.21196 q^{30} +2.05885 q^{31} +7.50974 q^{32} -0.894107 q^{33} +6.80238 q^{34} -3.68869 q^{35} -3.91159 q^{36} +7.42228 q^{37} +2.17778 q^{38} +8.09686 q^{39} -3.67777 q^{40} -2.54806 q^{41} -4.43233 q^{42} -8.68556 q^{43} +1.95476 q^{44} -3.24268 q^{45} -15.5549 q^{46} +10.6920 q^{47} +2.46247 q^{48} -4.36806 q^{49} -0.369666 q^{50} +3.91856 q^{51} -17.7019 q^{52} +6.48311 q^{53} -12.0927 q^{54} +1.62048 q^{55} +2.62414 q^{56} +1.25453 q^{57} -17.9946 q^{58} +7.84811 q^{59} -7.82345 q^{60} -6.34020 q^{61} -4.48374 q^{62} +2.31370 q^{63} -12.4288 q^{64} -14.6748 q^{65} +1.94717 q^{66} +4.48368 q^{67} -8.56704 q^{68} -8.96053 q^{69} +8.03317 q^{70} -12.7709 q^{71} +2.30685 q^{72} +16.5987 q^{73} -16.1641 q^{74} -0.212949 q^{75} -2.74274 q^{76} -1.15624 q^{77} -17.6332 q^{78} -7.83514 q^{79} -4.46299 q^{80} -2.68756 q^{81} +5.54912 q^{82} -0.482513 q^{83} +5.58215 q^{84} -7.10201 q^{85} +18.9153 q^{86} -10.3659 q^{87} -1.15281 q^{88} -7.50702 q^{89} +7.06185 q^{90} +10.4707 q^{91} +19.5902 q^{92} -2.58289 q^{93} -23.2849 q^{94} -2.27371 q^{95} -9.42116 q^{96} -9.95412 q^{97} +9.51269 q^{98} -1.01643 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) $ 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * 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 $

$2$	−2.17778	−1.53992	−0.769962	−	0.638089i	$-0.779725\pi$
$2$	−2.17778	−1.53992	−0.769962	+	0.638089i	$0.779725\pi$
$3$	−1.25453	−0.724301	−0.362150	−	0.932120i	$-0.617957\pi$
$3$	−1.25453	−0.724301	−0.362150	+	0.932120i	$0.617957\pi$
$4$	2.74274	1.37137
$5$	2.27371	1.01683	0.508416	−	0.861111i	$-0.330231\pi$
$5$	2.27371	1.01683	0.508416	+	0.861111i	$0.330231\pi$
$6$	2.73208	1.11537
$7$	−1.62232	−0.613181	−0.306591	−	0.951841i	$-0.599188\pi$
$7$	−1.62232	−0.613181	−0.306591	+	0.951841i	$0.599188\pi$
$8$	−1.61752	−0.571880
$9$	−1.42616	−0.475388
$10$	−4.95164	−1.56585
$11$	0.712705	0.214889	0.107444	−	0.994211i	$-0.465733\pi$
$11$	0.712705	0.214889	0.107444	+	0.994211i	$0.465733\pi$
$12$	−3.44083	−0.993283
$13$	−6.45412	−1.79005	−0.895025	−	0.446016i	$-0.852842\pi$
$13$	−6.45412	−1.79005	−0.895025	+	0.446016i	$0.852842\pi$
$14$	3.53307	0.944253
$15$	−2.85242	−0.736493
$16$	−1.96287	−0.490717
$17$	−3.12354	−0.757569	−0.378785	−	0.925485i	$-0.623658\pi$
$17$	−3.12354	−0.757569	−0.378785	+	0.925485i	$0.623658\pi$
$18$	3.10588	0.732062
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	6.23618	1.39445
$21$	2.03525	0.444128
$22$	−1.55212	−0.330912
$23$	7.14256	1.48933	0.744664	−	0.667440i	$-0.232610\pi$
$23$	7.14256	1.48933	0.744664	+	0.667440i	$0.232610\pi$
$24$	2.02922	0.414213
$25$	0.169744	0.0339488
$26$	14.0557	2.75654
$27$	5.55274	1.06863
$28$	−4.44961	−0.840897
$29$	8.26282	1.53437	0.767184	−	0.641427i	$-0.221657\pi$
$29$	8.26282	1.53437	0.767184	+	0.641427i	$0.221657\pi$
$30$	6.21196	1.13414
$31$	2.05885	0.369781	0.184891	−	0.982759i	$-0.440807\pi$
$31$	2.05885	0.369781	0.184891	+	0.982759i	$0.440807\pi$
$32$	7.50974	1.32755
$33$	−0.894107	−0.155644
$34$	6.80238	1.16660
$35$	−3.68869	−0.623503
$36$	−3.91159	−0.651932
$37$	7.42228	1.22021	0.610107	−	0.792319i	$-0.291126\pi$
$37$	7.42228	1.22021	0.610107	+	0.792319i	$0.291126\pi$
$38$	2.17778	0.353283
$39$	8.09686	1.29653
$40$	−3.67777	−0.581506
$41$	−2.54806	−0.397940	−0.198970	−	0.980006i	$-0.563760\pi$
$41$	−2.54806	−0.397940	−0.198970	+	0.980006i	$0.563760\pi$
$42$	−4.43233	−0.683923
$43$	−8.68556	−1.32454	−0.662268	−	0.749267i	$-0.730406\pi$
$43$	−8.68556	−1.32454	−0.662268	+	0.749267i	$0.730406\pi$
$44$	1.95476	0.294692
$45$	−3.24268	−0.483390
$46$	−15.5549	−2.29345
$47$	10.6920	1.55959	0.779794	−	0.626036i	$-0.215324\pi$
$47$	10.6920	1.55959	0.779794	+	0.626036i	$0.215324\pi$
$48$	2.46247	0.355427
$49$	−4.36806	−0.624009
$50$	−0.369666	−0.0522787
$51$	3.91856	0.548708
$52$	−17.7019	−2.45482
$53$	6.48311	0.890523	0.445262	−	0.895401i	$-0.353111\pi$
$53$	6.48311	0.890523	0.445262	+	0.895401i	$0.353111\pi$
$54$	−12.0927	−1.64560
$55$	1.62048	0.218506
$56$	2.62414	0.350666
$57$	1.25453	0.166166
$58$	−17.9946	−2.36281
$59$	7.84811	1.02174	0.510868	−	0.859659i	$-0.329324\pi$
$59$	7.84811	1.02174	0.510868	+	0.859659i	$0.329324\pi$
$60$	−7.82345	−1.01000
$61$	−6.34020	−0.811780	−0.405890	−	0.913922i	$-0.633038\pi$
$61$	−6.34020	−0.811780	−0.405890	+	0.913922i	$0.633038\pi$
$62$	−4.48374	−0.569435
$63$	2.31370	0.291499
$64$	−12.4288	−1.55361
$65$	−14.6748	−1.82018
$66$	1.94717	0.239680
$67$	4.48368	0.547768	0.273884	−	0.961763i	$-0.411692\pi$
$67$	4.48368	0.547768	0.273884	+	0.961763i	$0.411692\pi$
$68$	−8.56704	−1.03891
$69$	−8.96053	−1.07872
$70$	8.03317	0.960147
$71$	−12.7709	−1.51563	−0.757815	−	0.652469i	$-0.773733\pi$
$71$	−12.7709	−1.51563	−0.757815	+	0.652469i	$0.773733\pi$
$72$	2.30685	0.271865
$73$	16.5987	1.94273	0.971367	−	0.237582i	$-0.0763549\pi$
$73$	16.5987	1.94273	0.971367	+	0.237582i	$0.0763549\pi$
$74$	−16.1641	−1.87904
$75$	−0.212949	−0.0245892
$76$	−2.74274	−0.314614
$77$	−1.15624	−0.131766
$78$	−17.6332	−1.99657
$79$	−7.83514	−0.881522	−0.440761	−	0.897625i	$-0.645291\pi$
$79$	−7.83514	−0.881522	−0.440761	+	0.897625i	$0.645291\pi$
$80$	−4.46299	−0.498977
$81$	−2.68756	−0.298618
$82$	5.54912	0.612798
$83$	−0.482513	−0.0529627	−0.0264813	−	0.999649i	$-0.508430\pi$
$83$	−0.482513	−0.0529627	−0.0264813	+	0.999649i	$0.508430\pi$
$84$	5.58215	0.609063
$85$	−7.10201	−0.770321
$86$	18.9153	2.03969
$87$	−10.3659	−1.11134
$88$	−1.15281	−0.122890
$89$	−7.50702	−0.795743	−0.397871	−	0.917441i	$-0.630251\pi$
$89$	−7.50702	−0.795743	−0.397871	+	0.917441i	$0.630251\pi$
$90$	7.06185	0.744385
$91$	10.4707	1.09762
$92$	19.5902	2.04242
$93$	−2.58289	−0.267833
$94$	−23.2849	−2.40165
$95$	−2.27371	−0.233277
$96$	−9.42116	−0.961543
$97$	−9.95412	−1.01069	−0.505344	−	0.862918i	$-0.668634\pi$
$97$	−9.95412	−1.01069	−0.505344	+	0.862918i	$0.668634\pi$
$98$	9.51269	0.960927
$99$	−1.01643	−0.102156
$100$	0.465564	0.0465564
$101$	−10.5230	−1.04708	−0.523539	−	0.852002i	$-0.675388\pi$
$101$	−10.5230	−1.04708	−0.523539	+	0.852002i	$0.675388\pi$
$102$	−8.53377	−0.844969
$103$	10.0283	0.988114	0.494057	−	0.869430i	$-0.335513\pi$
$103$	10.0283	0.988114	0.494057	+	0.869430i	$0.335513\pi$
$104$	10.4397	1.02369
$105$	4.62756	0.451604
$106$	−14.1188	−1.37134
$107$	−4.85447	−0.469299	−0.234650	−	0.972080i	$-0.575394\pi$
$107$	−4.85447	−0.469299	−0.234650	+	0.972080i	$0.575394\pi$
$108$	15.2297	1.46548
$109$	11.7741	1.12775	0.563875	−	0.825860i	$-0.309310\pi$
$109$	11.7741	1.12775	0.563875	+	0.825860i	$0.309310\pi$
$110$	−3.52906	−0.336483
$111$	−9.31144	−0.883803
$112$	3.18441	0.300898
$113$	12.9644	1.21959	0.609793	−	0.792561i	$-0.291253\pi$
$113$	12.9644	1.21959	0.609793	+	0.792561i	$0.291253\pi$
$114$	−2.73208	−0.255883
$115$	16.2401	1.51440
$116$	22.6628	2.10418
$117$	9.20463	0.850968
$118$	−17.0915	−1.57340
$119$	5.06739	0.464527
$120$	4.61385	0.421185
$121$	−10.4921	−0.953823
$122$	13.8076	1.25008
$123$	3.19661	0.288228
$124$	5.64690	0.507106
$125$	−10.9826	−0.982312
$126$	−5.03874	−0.448887
$127$	−1.66599	−0.147833	−0.0739164	−	0.997264i	$-0.523550\pi$
$127$	−1.66599	−0.147833	−0.0739164	+	0.997264i	$0.523550\pi$
$128$	12.0478	1.06489
$129$	10.8963	0.959363
$130$	31.9585	2.80294
$131$	−5.43444	−0.474809	−0.237405	−	0.971411i	$-0.576297\pi$
$131$	−5.43444	−0.474809	−0.237405	+	0.971411i	$0.576297\pi$
$132$	−2.45230	−0.213445
$133$	1.62232	0.140673
$134$	−9.76447	−0.843522
$135$	12.6253	1.08661
$136$	5.05238	0.433238
$137$	16.0015	1.36710	0.683549	−	0.729905i	$-0.260436\pi$
$137$	16.0015	1.36710	0.683549	+	0.729905i	$0.260436\pi$
$138$	19.5141	1.66115
$139$	−15.9860	−1.35591	−0.677956	−	0.735102i	$-0.737134\pi$
$139$	−15.9860	−1.35591	−0.677956	+	0.735102i	$0.737134\pi$
$140$	−10.1171	−0.855052
$141$	−13.4134	−1.12961
$142$	27.8123	2.33396
$143$	−4.59988	−0.384661
$144$	2.79937	0.233281
$145$	18.7872	1.56020
$146$	−36.1484	−2.99167
$147$	5.47985	0.451970
$148$	20.3574	1.67336
$149$	−1.84763	−0.151364	−0.0756818	−	0.997132i	$-0.524113\pi$
$149$	−1.84763	−0.151364	−0.0756818	+	0.997132i	$0.524113\pi$
$150$	0.463756	0.0378655
$151$	−4.04319	−0.329030	−0.164515	−	0.986375i	$-0.552606\pi$
$151$	−4.04319	−0.329030	−0.164515	+	0.986375i	$0.552606\pi$
$152$	1.61752	0.131198
$153$	4.45468	0.360139
$154$	2.51804	0.202909
$155$	4.68123	0.376006
$156$	22.2075	1.77803
$157$	−12.1727	−0.971490	−0.485745	−	0.874100i	$-0.661452\pi$
$157$	−12.1727	−0.971490	−0.485745	+	0.874100i	$0.661452\pi$
$158$	17.0632	1.35748
$159$	−8.13323	−0.645007
$160$	17.0749	1.34989
$161$	−11.5876	−0.913227
$162$	5.85292	0.459849
$163$	−21.1981	−1.66037	−0.830183	−	0.557491i	$-0.811764\pi$
$163$	−21.1981	−1.66037	−0.830183	+	0.557491i	$0.811764\pi$
$164$	−6.98866	−0.545722
$165$	−2.03294	−0.158264
$166$	1.05081	0.0815585
$167$	12.8006	0.990543	0.495272	−	0.868738i	$-0.335069\pi$
$167$	12.8006	0.990543	0.495272	+	0.868738i	$0.335069\pi$
$168$	−3.29205	−0.253988
$169$	28.6556	2.20428
$170$	15.4666	1.18624
$171$	1.42616	0.109062
$172$	−23.8222	−1.81643
$173$	−15.1651	−1.15298	−0.576490	−	0.817104i	$-0.695578\pi$
$173$	−15.1651	−1.15298	−0.576490	+	0.817104i	$0.695578\pi$
$174$	22.5747	1.71139
$175$	−0.275380	−0.0208168
$176$	−1.39895	−0.105450
$177$	−9.84566	−0.740045
$178$	16.3487	1.22538
$179$	−0.483723	−0.0361551	−0.0180776	−	0.999837i	$-0.505755\pi$
$179$	−0.483723	−0.0361551	−0.0180776	+	0.999837i	$0.505755\pi$
$180$	−8.89382	−0.662906
$181$	18.6568	1.38675	0.693374	−	0.720578i	$-0.256123\pi$
$181$	18.6568	1.38675	0.693374	+	0.720578i	$0.256123\pi$
$182$	−22.8028	−1.69026
$183$	7.95395	0.587973
$184$	−11.5532	−0.851716
$185$	16.8761	1.24075
$186$	5.62497	0.412443
$187$	−2.22616	−0.162793
$188$	29.3254	2.13877
$189$	−9.00835	−0.655261
$190$	4.95164	0.359230
$191$	−25.9827	−1.88004	−0.940021	−	0.341116i	$-0.889195\pi$
$191$	−25.9827	−1.88004	−0.940021	+	0.341116i	$0.889195\pi$
$192$	15.5923	1.12528
$193$	−22.3774	−1.61076	−0.805379	−	0.592760i	$-0.798038\pi$
$193$	−22.3774	−1.61076	−0.805379	+	0.592760i	$0.798038\pi$
$194$	21.6779	1.55638
$195$	18.4099	1.31836
$196$	−11.9804	−0.855746
$197$	8.25198	0.587929	0.293964	−	0.955816i	$-0.405025\pi$
$197$	8.25198	0.587929	0.293964	+	0.955816i	$0.405025\pi$
$198$	2.21357	0.157312
$199$	−14.3758	−1.01907	−0.509536	−	0.860449i	$-0.670183\pi$
$199$	−14.3758	−1.01907	−0.509536	+	0.860449i	$0.670183\pi$
$200$	−0.274565	−0.0194147
$201$	−5.62489	−0.396749
$202$	22.9168	1.61242
$203$	−13.4050	−0.940845
$204$	10.7476	0.752481
$205$	−5.79354	−0.404638
$206$	−21.8394	−1.52162
$207$	−10.1865	−0.708009
$208$	12.6686	0.878408
$209$	−0.712705	−0.0492989
$210$	−10.0778	−0.695436
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	17.7815	1.22124
$213$	16.0215	1.09777
$214$	10.5720	0.722685
$215$	−19.7484	−1.34683
$216$	−8.98166	−0.611125
$217$	−3.34013	−0.226743
$218$	−25.6413	−1.73665
$219$	−20.8235	−1.40712
$220$	4.44456	0.299652
$221$	20.1597	1.35609
$222$	20.2783	1.36099
$223$	27.2686	1.82604	0.913020	−	0.407914i	$-0.133744\pi$
$223$	27.2686	1.82604	0.913020	+	0.407914i	$0.133744\pi$
$224$	−12.1832	−0.814027
$225$	−0.242083	−0.0161389
$226$	−28.2336	−1.87807
$227$	−13.0188	−0.864086	−0.432043	−	0.901853i	$-0.642207\pi$
$227$	−13.0188	−0.864086	−0.432043	+	0.901853i	$0.642207\pi$
$228$	3.44083	0.227875
$229$	10.6244	0.702078	0.351039	−	0.936361i	$-0.385828\pi$
$229$	10.6244	0.702078	0.351039	+	0.936361i	$0.385828\pi$
$230$	−35.3674	−2.33206
$231$	1.45053	0.0954380
$232$	−13.3653	−0.877474
$233$	−0.717334	−0.0469941	−0.0234970	−	0.999724i	$-0.507480\pi$
$233$	−0.717334	−0.0469941	−0.0234970	+	0.999724i	$0.507480\pi$
$234$	−20.0457	−1.31043
$235$	24.3105	1.58584
$236$	21.5253	1.40118
$237$	9.82938	0.638487
$238$	−11.0357	−0.715337
$239$	−22.1898	−1.43534	−0.717671	−	0.696382i	$-0.754792\pi$
$239$	−22.1898	−1.43534	−0.717671	+	0.696382i	$0.754792\pi$
$240$	5.59893	0.361410
$241$	−21.2455	−1.36855	−0.684273	−	0.729226i	$-0.739880\pi$
$241$	−21.2455	−1.36855	−0.684273	+	0.729226i	$0.739880\pi$
$242$	22.8494	1.46882
$243$	−13.2866	−0.852336
$244$	−17.3895	−1.11325
$245$	−9.93169	−0.634513
$246$	−6.96151	−0.443850
$247$	6.45412	0.410666
$248$	−3.33024	−0.211470
$249$	0.605325	0.0383609
$250$	23.9177	1.51269
$251$	14.2507	0.899493	0.449747	−	0.893156i	$-0.351514\pi$
$251$	14.2507	0.899493	0.449747	+	0.893156i	$0.351514\pi$
$252$	6.34588	0.399753
$253$	5.09054	0.320040
$254$	3.62816	0.227651
$255$	8.90965	0.557944
$256$	−1.37989	−0.0862431
$257$	−17.1217	−1.06802	−0.534012	−	0.845477i	$-0.679316\pi$
$257$	−17.1217	−1.06802	−0.534012	+	0.845477i	$0.679316\pi$
$258$	−23.7297	−1.47735
$259$	−12.0413	−0.748213
$260$	−40.2490	−2.49614
$261$	−11.7841	−0.729420
$262$	11.8350	0.731170
$263$	0.262179	0.0161667	0.00808333	−	0.999967i	$-0.497427\pi$
$263$	0.262179	0.0161667	0.00808333	+	0.999967i	$0.497427\pi$
$264$	1.44624	0.0890097
$265$	14.7407	0.905513
$266$	−3.53307	−0.216626
$267$	9.41776	0.576357
$268$	12.2975	0.751192
$269$	18.7365	1.14239	0.571193	−	0.820816i	$-0.306481\pi$
$269$	18.7365	1.14239	0.571193	+	0.820816i	$0.306481\pi$
$270$	−27.4952	−1.67330
$271$	31.1309	1.89107	0.945533	−	0.325526i	$-0.105542\pi$
$271$	31.1309	1.89107	0.945533	+	0.325526i	$0.105542\pi$
$272$	6.13109	0.371752
$273$	−13.1357	−0.795011
$274$	−34.8477	−2.10523
$275$	0.120978	0.00729522
$276$	−24.5764	−1.47932
$277$	−7.70318	−0.462839	−0.231420	−	0.972854i	$-0.574337\pi$
$277$	−7.70318	−0.462839	−0.231420	+	0.972854i	$0.574337\pi$
$278$	34.8140	2.08800
$279$	−2.93627	−0.175790
$280$	5.96653	0.356568
$281$	8.57365	0.511461	0.255731	−	0.966748i	$-0.417684\pi$
$281$	8.57365	0.511461	0.255731	+	0.966748i	$0.417684\pi$
$282$	29.2115	1.73952
$283$	−28.0967	−1.67018	−0.835088	−	0.550116i	$-0.814584\pi$
$283$	−28.0967	−1.67018	−0.835088	+	0.550116i	$0.814584\pi$
$284$	−35.0273	−2.07849
$285$	2.85242	0.168963
$286$	10.0175	0.592350
$287$	4.13378	0.244009
$288$	−10.7101	−0.631100
$289$	−7.24352	−0.426089
$290$	−40.9145	−2.40258
$291$	12.4877	0.732042
$292$	45.5260	2.66421
$293$	23.7454	1.38722	0.693612	−	0.720349i	$-0.256018\pi$
$293$	23.7454	1.38722	0.693612	+	0.720349i	$0.256018\pi$
$294$	−11.9339	−0.696000
$295$	17.8443	1.03894
$296$	−12.0057	−0.697816
$297$	3.95747	0.229635
$298$	4.02373	0.233089
$299$	−46.0989	−2.66597
$300$	−0.584062	−0.0337208
$301$	14.0908	0.812180
$302$	8.80518	0.506681
$303$	13.2014	0.758399
$304$	1.96287	0.112578
$305$	−14.4158	−0.825444
$306$	−9.70132	−0.554588
$307$	−23.4846	−1.34034	−0.670169	−	0.742209i	$-0.733778\pi$
$307$	−23.4846	−1.34034	−0.670169	+	0.742209i	$0.733778\pi$
$308$	−3.17126	−0.180699
$309$	−12.5807	−0.715692
$310$	−10.1947	−0.579020
$311$	−26.2750	−1.48992	−0.744959	−	0.667110i	$-0.767531\pi$
$311$	−26.2750	−1.48992	−0.744959	+	0.667110i	$0.767531\pi$
$312$	−13.0968	−0.741462
$313$	24.8220	1.40302	0.701510	−	0.712660i	$-0.252510\pi$
$313$	24.8220	1.40302	0.701510	+	0.712660i	$0.252510\pi$
$314$	26.5096	1.49602
$315$	5.26068	0.296406
$316$	−21.4897	−1.20889
$317$	−12.7207	−0.714466	−0.357233	−	0.934015i	$-0.616280\pi$
$317$	−12.7207	−0.714466	−0.357233	+	0.934015i	$0.616280\pi$
$318$	17.7124	0.993262
$319$	5.88896	0.329718
$320$	−28.2595	−1.57976
$321$	6.09006	0.339914
$322$	25.2352	1.40630
$323$	3.12354	0.173798
$324$	−7.37127	−0.409515
$325$	−1.09555	−0.0607701
$326$	46.1649	2.55684
$327$	−14.7709	−0.816831
$328$	4.12154	0.227574
$329$	−17.3459	−0.956311
$330$	4.42730	0.243715
$331$	10.8931	0.598741	0.299370	−	0.954137i	$-0.403223\pi$
$331$	10.8931	0.598741	0.299370	+	0.954137i	$0.403223\pi$
$332$	−1.32341	−0.0726313
$333$	−10.5854	−0.580076
$334$	−27.8770	−1.52536
$335$	10.1946	0.556989
$336$	−3.99492	−0.217941
$337$	−20.2631	−1.10380	−0.551900	−	0.833910i	$-0.686097\pi$
$337$	−20.2631	−1.10380	−0.551900	+	0.833910i	$0.686097\pi$
$338$	−62.4057	−3.39442
$339$	−16.2641	−0.883347
$340$	−19.4789	−1.05639
$341$	1.46736	0.0794618
$342$	−3.10588	−0.167947
$343$	18.4427	0.995812
$344$	14.0491	0.757475
$345$	−20.3736	−1.09688
$346$	33.0263	1.77550
$347$	13.4010	0.719404	0.359702	−	0.933067i	$-0.382878\pi$
$347$	13.4010	0.719404	0.359702	+	0.933067i	$0.382878\pi$
$348$	−28.4310	−1.52406
$349$	−17.4114	−0.932011	−0.466006	−	0.884782i	$-0.654307\pi$
$349$	−17.4114	−0.932011	−0.466006	+	0.884782i	$0.654307\pi$
$350$	0.599718	0.0320563
$351$	−35.8380	−1.91289
$352$	5.35223	0.285275
$353$	−0.910640	−0.0484685	−0.0242342	−	0.999706i	$-0.507715\pi$
$353$	−0.910640	−0.0484685	−0.0242342	+	0.999706i	$0.507715\pi$
$354$	21.4417	1.13961
$355$	−29.0373	−1.54114
$356$	−20.5898	−1.09126
$357$	−6.35718	−0.336457
$358$	1.05344	0.0556762
$359$	2.56165	0.135199	0.0675993	−	0.997713i	$-0.478466\pi$
$359$	2.56165	0.135199	0.0675993	+	0.997713i	$0.478466\pi$
$360$	5.24510	0.276441
$361$	1.00000	0.0526316
$362$	−40.6304	−2.13549
$363$	13.1626	0.690855
$364$	28.7183	1.50525
$365$	37.7407	1.97544
$366$	−17.3220	−0.905434
$367$	−16.8641	−0.880300	−0.440150	−	0.897924i	$-0.645075\pi$
$367$	−16.8641	−0.880300	−0.440150	+	0.897924i	$0.645075\pi$
$368$	−14.0199	−0.730838
$369$	3.63395	0.189176
$370$	−36.7524	−1.91067
$371$	−10.5177	−0.546052
$372$	−7.08418	−0.367298
$373$	25.4816	1.31939	0.659694	−	0.751534i	$-0.270686\pi$
$373$	25.4816	1.31939	0.659694	+	0.751534i	$0.270686\pi$
$374$	4.84810	0.250689
$375$	13.7779	0.711490
$376$	−17.2945	−0.891897
$377$	−53.3292	−2.74659
$378$	19.6182	1.00905
$379$	−22.5136	−1.15645	−0.578224	−	0.815878i	$-0.696254\pi$
$379$	−22.5136	−1.15645	−0.578224	+	0.815878i	$0.696254\pi$
$380$	−6.23618	−0.319909
$381$	2.09003	0.107075
$382$	56.5847	2.89512
$383$	0.555086	0.0283636	0.0141818	−	0.999899i	$-0.495486\pi$
$383$	0.555086	0.0283636	0.0141818	+	0.999899i	$0.495486\pi$
$384$	−15.1143	−0.771300
$385$	−2.62895	−0.133984
$386$	48.7330	2.48045
$387$	12.3870	0.629669
$388$	−27.3015	−1.38603
$389$	−33.6053	−1.70385	−0.851927	−	0.523660i	$-0.824566\pi$
$389$	−33.6053	−1.70385	−0.851927	+	0.523660i	$0.824566\pi$
$390$	−40.0927	−2.03017
$391$	−22.3101	−1.12827
$392$	7.06543	0.356858
$393$	6.81764	0.343905
$394$	−17.9710	−0.905366
$395$	−17.8148	−0.896360
$396$	−2.78781	−0.140093
$397$	−3.11457	−0.156316	−0.0781579	−	0.996941i	$-0.524904\pi$
$397$	−3.11457	−0.156316	−0.0781579	+	0.996941i	$0.524904\pi$
$398$	31.3073	1.56929
$399$	−2.03525	−0.101890
$400$	−0.333185	−0.0166593
$401$	−17.4089	−0.869360	−0.434680	−	0.900585i	$-0.643139\pi$
$401$	−17.4089	−0.869360	−0.434680	+	0.900585i	$0.643139\pi$
$402$	12.2498	0.610964
$403$	−13.2881	−0.661927
$404$	−28.8618	−1.43593
$405$	−6.11073	−0.303644
$406$	29.1931	1.44883
$407$	5.28990	0.262210
$408$	−6.33835	−0.313795
$409$	21.9092	1.08334	0.541670	−	0.840591i	$-0.317792\pi$
$409$	21.9092	1.08334	0.541670	+	0.840591i	$0.317792\pi$
$410$	12.6171	0.623113
$411$	−20.0743	−0.990190
$412$	27.5049	1.35507
$413$	−12.7322	−0.626510
$414$	22.1839	1.09028
$415$	−1.09709	−0.0538542
$416$	−48.4687	−2.37637
$417$	20.0548	0.982088
$418$	1.55212	0.0759165
$419$	−29.1254	−1.42287	−0.711434	−	0.702753i	$-0.751954\pi$
$419$	−29.1254	−1.42287	−0.711434	+	0.702753i	$0.751954\pi$
$420$	12.6922	0.619315
$421$	4.18309	0.203871	0.101936	−	0.994791i	$-0.467496\pi$
$421$	4.18309	0.203871	0.101936	+	0.994791i	$0.467496\pi$
$422$	2.17778	0.106013
$423$	−15.2486	−0.741410
$424$	−10.4866	−0.509272
$425$	−0.530202	−0.0257186
$426$	−34.8913	−1.69049
$427$	10.2859	0.497768
$428$	−13.3145	−0.643582
$429$	5.77067	0.278611
$430$	43.0078	2.07402
$431$	16.3175	0.785987	0.392994	−	0.919541i	$-0.371439\pi$
$431$	16.3175	0.785987	0.392994	+	0.919541i	$0.371439\pi$
$432$	−10.8993	−0.524392
$433$	−14.8070	−0.711577	−0.355789	−	0.934566i	$-0.615788\pi$
$433$	−14.8070	−0.711577	−0.355789	+	0.934566i	$0.615788\pi$
$434$	7.27408	0.349167
$435$	−23.5691	−1.13005
$436$	32.2931	1.54656
$437$	−7.14256	−0.341675
$438$	45.3492	2.16687
$439$	−8.60942	−0.410905	−0.205453	−	0.978667i	$-0.565867\pi$
$439$	−8.60942	−0.410905	−0.205453	+	0.978667i	$0.565867\pi$
$440$	−2.62116	−0.124959
$441$	6.22957	0.296646
$442$	−43.9034	−2.08827
$443$	−10.1847	−0.483890	−0.241945	−	0.970290i	$-0.577785\pi$
$443$	−10.1847	−0.483890	−0.241945	+	0.970290i	$0.577785\pi$
$444$	−25.5388	−1.21202
$445$	−17.0688	−0.809137
$446$	−59.3851	−2.81197
$447$	2.31790	0.109633
$448$	20.1636	0.952642
$449$	−3.23149	−0.152503	−0.0762516	−	0.997089i	$-0.524295\pi$
$449$	−3.23149	−0.152503	−0.0762516	+	0.997089i	$0.524295\pi$
$450$	0.527205	0.0248527
$451$	−1.81602	−0.0855128
$452$	35.5579	1.67250
$453$	5.07228	0.238317
$454$	28.3520	1.33063
$455$	23.8072	1.11610
$456$	−2.02922	−0.0950270
$457$	−41.1592	−1.92535	−0.962674	−	0.270664i	$-0.912757\pi$
$457$	−41.1592	−1.92535	−0.962674	+	0.270664i	$0.912757\pi$
$458$	−23.1376	−1.08115
$459$	−17.3442	−0.809557
$460$	44.5423	2.07680
$461$	27.3481	1.27373	0.636864	−	0.770976i	$-0.280231\pi$
$461$	27.3481	1.27373	0.636864	+	0.770976i	$0.280231\pi$
$462$	−3.15894	−0.146967
$463$	−11.6025	−0.539215	−0.269607	−	0.962970i	$-0.586894\pi$
$463$	−11.6025	−0.539215	−0.269607	+	0.962970i	$0.586894\pi$
$464$	−16.2188	−0.752940
$465$	−5.87273	−0.272341
$466$	1.56220	0.0723674
$467$	−9.98224	−0.461923	−0.230962	−	0.972963i	$-0.574187\pi$
$467$	−9.98224	−0.461923	−0.230962	+	0.972963i	$0.574187\pi$
$468$	25.2459	1.16699
$469$	−7.27398	−0.335881
$470$	−52.9429	−2.44208
$471$	15.2710	0.703651
$472$	−12.6945	−0.584310
$473$	−6.19025	−0.284628
$474$	−21.4063	−0.983222
$475$	−0.169744	−0.00778840
$476$	13.8985	0.637038
$477$	−9.24598	−0.423344
$478$	48.3247	2.21032
$479$	1.33788	0.0611292	0.0305646	−	0.999533i	$-0.490269\pi$
$479$	1.33788	0.0611292	0.0305646	+	0.999533i	$0.490269\pi$
$480$	−21.4210	−0.977729
$481$	−47.9042	−2.18425
$482$	46.2682	2.10746
$483$	14.5369	0.661452
$484$	−28.7769	−1.30804
$485$	−22.6328	−1.02770
$486$	28.9353	1.31253
$487$	−13.0425	−0.591010	−0.295505	−	0.955341i	$-0.595488\pi$
$487$	−13.0425	−0.591010	−0.295505	+	0.955341i	$0.595488\pi$
$488$	10.2554	0.464240
$489$	26.5936	1.20260
$490$	21.6291	0.977102
$491$	5.72160	0.258212	0.129106	−	0.991631i	$-0.458789\pi$
$491$	5.72160	0.258212	0.129106	+	0.991631i	$0.458789\pi$
$492$	8.76745	0.395267
$493$	−25.8092	−1.16239
$494$	−14.0557	−0.632394
$495$	−2.31108	−0.103875
$496$	−4.04126	−0.181458
$497$	20.7186	0.929356
$498$	−1.31827	−0.0590729
$499$	−25.0032	−1.11930	−0.559649	−	0.828730i	$-0.689064\pi$
$499$	−25.0032	−1.11930	−0.559649	+	0.828730i	$0.689064\pi$
$500$	−30.1223	−1.34711
$501$	−16.0587	−0.717451
$502$	−31.0348	−1.38515
$503$	27.5813	1.22979	0.614894	−	0.788610i	$-0.289199\pi$
$503$	27.5813	1.22979	0.614894	+	0.788610i	$0.289199\pi$
$504$	−3.74246	−0.166702
$505$	−23.9262	−1.06470
$506$	−11.0861	−0.492837
$507$	−35.9492	−1.59656
$508$	−4.56937	−0.202733
$509$	27.1055	1.20143	0.600715	−	0.799463i	$-0.294883\pi$
$509$	27.1055	1.20143	0.600715	+	0.799463i	$0.294883\pi$
$510$	−19.4033	−0.859192
$511$	−26.9285	−1.19125
$512$	−21.0906	−0.932081
$513$	−5.55274	−0.245159
$514$	37.2874	1.64468
$515$	22.8013	1.00475
$516$	29.8856	1.31564
$517$	7.62025	0.335138
$518$	26.2234	1.15219
$519$	19.0250	0.835105
$520$	23.7367	1.04092
$521$	35.4205	1.55180	0.775900	−	0.630856i	$-0.217296\pi$
$521$	35.4205	1.55180	0.775900	+	0.630856i	$0.217296\pi$
$522$	25.6633	1.12325
$523$	18.9029	0.826567	0.413283	−	0.910603i	$-0.364382\pi$
$523$	18.9029	0.826567	0.413283	+	0.910603i	$0.364382\pi$
$524$	−14.9052	−0.651138
$525$	0.345472	0.0150776
$526$	−0.570969	−0.0248954
$527$	−6.43091	−0.280135
$528$	1.75501	0.0763772
$529$	28.0162	1.21810
$530$	−32.1020	−1.39442
$531$	−11.1927	−0.485722
$532$	4.44961	0.192915
$533$	16.4455	0.712332
$534$	−20.5098	−0.887547
$535$	−11.0376	−0.477199
$536$	−7.25243	−0.313257
$537$	0.606843	0.0261872
$538$	−40.8041	−1.75919
$539$	−3.11314	−0.134092
$540$	34.6279	1.49015
$541$	4.66919	0.200744	0.100372	−	0.994950i	$-0.467997\pi$
$541$	4.66919	0.200744	0.100372	+	0.994950i	$0.467997\pi$
$542$	−67.7963	−2.91210
$543$	−23.4054	−1.00442
$544$	−23.4569	−1.00571
$545$	26.7708	1.14673
$546$	28.6068	1.22426
$547$	−26.6989	−1.14156	−0.570782	−	0.821102i	$-0.693360\pi$
$547$	−26.6989	−1.14156	−0.570782	+	0.821102i	$0.693360\pi$
$548$	43.8878	1.87480
$549$	9.04217	0.385911
$550$	−0.263463	−0.0112341
$551$	−8.26282	−0.352008
$552$	14.4938	0.616899
$553$	12.7111	0.540533
$554$	16.7759	0.712738
$555$	−21.1715	−0.898680
$556$	−43.8453	−1.85946
$557$	2.55208	0.108135	0.0540675	−	0.998537i	$-0.482781\pi$
$557$	2.55208	0.108135	0.0540675	+	0.998537i	$0.482781\pi$
$558$	6.39455	0.270703
$559$	56.0576	2.37099
$560$	7.24041	0.305963
$561$	2.79278	0.117911
$562$	−18.6715	−0.787612
$563$	−32.5993	−1.37390	−0.686949	−	0.726706i	$-0.741050\pi$
$563$	−32.5993	−1.37390	−0.686949	+	0.726706i	$0.741050\pi$
$564$	−36.7894	−1.54911
$565$	29.4772	1.24011
$566$	61.1886	2.57195
$567$	4.36010	0.183107
$568$	20.6572	0.866758
$569$	−30.6223	−1.28375	−0.641876	−	0.766808i	$-0.721844\pi$
$569$	−30.6223	−1.28375	−0.641876	+	0.766808i	$0.721844\pi$
$570$	−6.21196	−0.260190
$571$	−3.76798	−0.157685	−0.0788425	−	0.996887i	$-0.525122\pi$
$571$	−3.76798	−0.157685	−0.0788425	+	0.996887i	$0.525122\pi$
$572$	−12.6163	−0.527513
$573$	32.5960	1.36172
$574$	−9.00247	−0.375756
$575$	1.21241	0.0505609
$576$	17.7256	0.738566
$577$	−10.3599	−0.431287	−0.215644	−	0.976472i	$-0.569185\pi$
$577$	−10.3599	−0.431287	−0.215644	+	0.976472i	$0.569185\pi$
$578$	15.7748	0.656145
$579$	28.0730	1.16667
$580$	51.5285	2.13960
$581$	0.782793	0.0324757
$582$	−27.1955	−1.12729
$583$	4.62054	0.191363
$584$	−26.8488	−1.11101
$585$	20.9286	0.865293
$586$	−51.7124	−2.13622
$587$	−23.2242	−0.958564	−0.479282	−	0.877661i	$-0.659103\pi$
$587$	−23.2242	−0.958564	−0.479282	+	0.877661i	$0.659103\pi$
$588$	15.0298	0.619818
$589$	−2.05885	−0.0848336
$590$	−38.8610	−1.59988
$591$	−10.3523	−0.425837
$592$	−14.5689	−0.598780
$593$	14.6206	0.600394	0.300197	−	0.953877i	$-0.402948\pi$
$593$	14.6206	0.600394	0.300197	+	0.953877i	$0.402948\pi$
$594$	−8.61850	−0.353621
$595$	11.5218	0.472346
$596$	−5.06756	−0.207575
$597$	18.0348	0.738115
$598$	100.393	4.10539
$599$	−14.0129	−0.572553	−0.286276	−	0.958147i	$-0.592418\pi$
$599$	−14.0129	−0.572553	−0.286276	+	0.958147i	$0.592418\pi$
$600$	0.344448	0.0140621
$601$	−2.64546	−0.107911	−0.0539553	−	0.998543i	$-0.517183\pi$
$601$	−2.64546	−0.107911	−0.0539553	+	0.998543i	$0.517183\pi$
$602$	−30.6867	−1.25070
$603$	−6.39446	−0.260403
$604$	−11.0894	−0.451221
$605$	−23.8559	−0.969878
$606$	−28.7497	−1.16788
$607$	9.50925	0.385968	0.192984	−	0.981202i	$-0.438183\pi$
$607$	9.50925	0.385968	0.192984	+	0.981202i	$0.438183\pi$
$608$	−7.50974	−0.304560
$609$	16.8169	0.681455
$610$	31.3944	1.27112
$611$	−69.0074	−2.79174
$612$	12.2180	0.493884
$613$	−17.9974	−0.726907	−0.363454	−	0.931612i	$-0.618402\pi$
$613$	−17.9974	−0.726907	−0.363454	+	0.931612i	$0.618402\pi$
$614$	51.1444	2.06402
$615$	7.26815	0.293080
$616$	1.87024	0.0753541
$617$	11.8842	0.478440	0.239220	−	0.970965i	$-0.423108\pi$
$617$	11.8842	0.478440	0.239220	+	0.970965i	$0.423108\pi$
$618$	27.3981	1.10211
$619$	43.0835	1.73167	0.865836	−	0.500329i	$-0.166787\pi$
$619$	43.0835	1.73167	0.865836	+	0.500329i	$0.166787\pi$
$620$	12.8394	0.515642
$621$	39.6608	1.59153
$622$	57.2212	2.29436
$623$	12.1788	0.487935
$624$	−15.8931	−0.636232
$625$	−25.8199	−1.03280
$626$	−54.0568	−2.16055
$627$	0.894107	0.0357072
$628$	−33.3866	−1.33227
$629$	−23.1838	−0.924397
$630$	−11.4566	−0.456443
$631$	42.9134	1.70835	0.854177	−	0.519982i	$-0.174061\pi$
$631$	42.9134	1.70835	0.854177	+	0.519982i	$0.174061\pi$
$632$	12.6735	0.504124
$633$	1.25453	0.0498629
$634$	27.7029	1.10022
$635$	−3.78797	−0.150321
$636$	−22.3073	−0.884542
$637$	28.1920	1.11701
$638$	−12.8249	−0.507741
$639$	18.2134	0.720513
$640$	27.3933	1.08281
$641$	−21.2655	−0.839938	−0.419969	−	0.907538i	$-0.637959\pi$
$641$	−21.2655	−0.839938	−0.419969	+	0.907538i	$0.637959\pi$
$642$	−13.2628	−0.523442
$643$	−38.0529	−1.50066	−0.750330	−	0.661063i	$-0.770105\pi$
$643$	−38.0529	−1.50066	−0.750330	+	0.661063i	$0.770105\pi$
$644$	−31.7816	−1.25237
$645$	24.7749	0.975511
$646$	−6.80238	−0.267636
$647$	19.0452	0.748744	0.374372	−	0.927279i	$-0.377858\pi$
$647$	19.0452	0.748744	0.374372	+	0.927279i	$0.377858\pi$
$648$	4.34718	0.170773
$649$	5.59339	0.219560
$650$	2.38587	0.0935814
$651$	4.19028	0.164230
$652$	−58.1409	−2.27697
$653$	−1.97433	−0.0772615	−0.0386308	−	0.999254i	$-0.512300\pi$
$653$	−1.97433	−0.0772615	−0.0386308	+	0.999254i	$0.512300\pi$
$654$	32.1677	1.25786
$655$	−12.3563	−0.482801
$656$	5.00150	0.195276
$657$	−23.6725	−0.923553
$658$	37.7756	1.47265
$659$	−20.2403	−0.788449	−0.394224	−	0.919014i	$-0.628987\pi$
$659$	−20.2403	−0.788449	−0.394224	+	0.919014i	$0.628987\pi$
$660$	−5.57581	−0.217038
$661$	−19.7257	−0.767240	−0.383620	−	0.923491i	$-0.625323\pi$
$661$	−19.7257	−0.767240	−0.383620	+	0.923491i	$0.625323\pi$
$662$	−23.7229	−0.922015
$663$	−25.2908	−0.982215
$664$	0.780474	0.0302883
$665$	3.68869	0.143041
$666$	23.0527	0.893273
$667$	59.0177	2.28518
$668$	35.1088	1.35840
$669$	−34.2092	−1.32260
$670$	−22.2015	−0.857721
$671$	−4.51870	−0.174442
$672$	15.2842	0.589600
$673$	22.4766	0.866408	0.433204	−	0.901296i	$-0.357383\pi$
$673$	22.4766	0.866408	0.433204	+	0.901296i	$0.357383\pi$
$674$	44.1285	1.69977
$675$	0.942545	0.0362786
$676$	78.5948	3.02288
$677$	18.0876	0.695165	0.347582	−	0.937649i	$-0.387003\pi$
$677$	18.0876	0.695165	0.347582	+	0.937649i	$0.387003\pi$
$678$	35.4198	1.36029
$679$	16.1488	0.619735
$680$	11.4876	0.440531
$681$	16.3324	0.625858
$682$	−3.19558	−0.122365
$683$	−5.27378	−0.201796	−0.100898	−	0.994897i	$-0.532171\pi$
$683$	−5.27378	−0.201796	−0.100898	+	0.994897i	$0.532171\pi$
$684$	3.91159	0.149564
$685$	36.3827	1.39011
$686$	−40.1642	−1.53348
$687$	−13.3286	−0.508516
$688$	17.0486	0.649972
$689$	−41.8427	−1.59408
$690$	44.3693	1.68911
$691$	42.0872	1.60107	0.800536	−	0.599284i	$-0.204548\pi$
$691$	42.0872	1.60107	0.800536	+	0.599284i	$0.204548\pi$
$692$	−41.5938	−1.58116
$693$	1.64899	0.0626399
$694$	−29.1845	−1.10783
$695$	−36.3474	−1.37874
$696$	16.7671	0.635555
$697$	7.95896	0.301467
$698$	37.9183	1.43523
$699$	0.899914	0.0340379
$700$	−0.755296	−0.0285475
$701$	−29.2419	−1.10445	−0.552225	−	0.833695i	$-0.686221\pi$
$701$	−29.2419	−1.10445	−0.552225	+	0.833695i	$0.686221\pi$
$702$	78.0474	2.94571
$703$	−7.42228	−0.279936
$704$	−8.85810	−0.333852
$705$	−30.4981	−1.14863
$706$	1.98318	0.0746378
$707$	17.0717	0.642048
$708$	−27.0040	−1.01487
$709$	31.2570	1.17388	0.586941	−	0.809630i	$-0.300332\pi$
$709$	31.2570	1.17388	0.586941	+	0.809630i	$0.300332\pi$
$710$	63.2370	2.37324
$711$	11.1742	0.419065
$712$	12.1428	0.455069
$713$	14.7055	0.550725
$714$	13.8445	0.518119
$715$	−10.4588	−0.391136
$716$	−1.32672	−0.0495820
$717$	27.8377	1.03962
$718$	−5.57871	−0.208196
$719$	−16.1608	−0.602698	−0.301349	−	0.953514i	$-0.597437\pi$
$719$	−16.1608	−0.602698	−0.301349	+	0.953514i	$0.597437\pi$
$720$	6.36495	0.237208
$721$	−16.2691	−0.605893
$722$	−2.17778	−0.0810487
$723$	26.6531	0.991239
$724$	51.1707	1.90174
$725$	1.40257	0.0520900
$726$	−28.6652	−1.06386
$727$	8.97529	0.332875	0.166438	−	0.986052i	$-0.446774\pi$
$727$	8.97529	0.332875	0.166438	+	0.986052i	$0.446774\pi$
$728$	−16.9365	−0.627709
$729$	24.7311	0.915966
$730$	−82.1910	−3.04202
$731$	27.1297	1.00343
$732$	21.8156	0.806328
$733$	−12.3542	−0.456312	−0.228156	−	0.973625i	$-0.573270\pi$
$733$	−12.3542	−0.456312	−0.228156	+	0.973625i	$0.573270\pi$
$734$	36.7264	1.35560
$735$	12.4596	0.459578
$736$	53.6388	1.97715
$737$	3.19554	0.117709
$738$	−7.91396	−0.291317
$739$	−39.8909	−1.46741	−0.733704	−	0.679469i	$-0.762210\pi$
$739$	−39.8909	−1.46741	−0.733704	+	0.679469i	$0.762210\pi$
$740$	46.2867	1.70153
$741$	−8.09686	−0.297445
$742$	22.9053	0.840879
$743$	−29.9849	−1.10004	−0.550020	−	0.835152i	$-0.685380\pi$
$743$	−29.9849	−1.10004	−0.550020	+	0.835152i	$0.685380\pi$
$744$	4.17787	0.153168
$745$	−4.20096	−0.153911
$746$	−55.4934	−2.03176
$747$	0.688143	0.0251778
$748$	−6.10578	−0.223249
$749$	7.87552	0.287765
$750$	−30.0054	−1.09564
$751$	21.1038	0.770087	0.385044	−	0.922898i	$-0.374186\pi$
$751$	21.1038	0.770087	0.385044	+	0.922898i	$0.374186\pi$
$752$	−20.9870	−0.765317
$753$	−17.8778	−0.651504
$754$	116.139	4.22955
$755$	−9.19302	−0.334568
$756$	−24.7075	−0.898604
$757$	49.6105	1.80313	0.901563	−	0.432649i	$-0.142421\pi$
$757$	49.6105	1.80313	0.901563	+	0.432649i	$0.142421\pi$
$758$	49.0298	1.78084
$759$	−6.38622	−0.231805
$760$	3.67777	0.133407
$761$	−15.0888	−0.546967	−0.273483	−	0.961877i	$-0.588176\pi$
$761$	−15.0888	−0.546967	−0.273483	+	0.961877i	$0.588176\pi$
$762$	−4.55163	−0.164888
$763$	−19.1014	−0.691515
$764$	−71.2637	−2.57823
$765$	10.1286	0.366201
$766$	−1.20886	−0.0436778
$767$	−50.6526	−1.82896
$768$	1.73111	0.0624660
$769$	26.3814	0.951337	0.475668	−	0.879625i	$-0.342206\pi$
$769$	26.3814	0.951337	0.475668	+	0.879625i	$0.342206\pi$
$770$	5.72528	0.206325
$771$	21.4797	0.773571
$772$	−61.3752	−2.20894
$773$	43.4230	1.56182	0.780908	−	0.624646i	$-0.214757\pi$
$773$	43.4230	1.56182	0.780908	+	0.624646i	$0.214757\pi$
$774$	−26.9763	−0.969642
$775$	0.349479	0.0125536
$776$	16.1010	0.577992
$777$	15.1062	0.541931
$778$	73.1849	2.62381
$779$	2.54806	0.0912937
$780$	50.4935	1.80796
$781$	−9.10191	−0.325692
$782$	48.5865	1.73745
$783$	45.8813	1.63966
$784$	8.57393	0.306212
$785$	−27.6772	−0.987843
$786$	−14.8473	−0.529587
$787$	12.2713	0.437424	0.218712	−	0.975790i	$-0.429815\pi$
$787$	12.2713	0.437424	0.218712	+	0.975790i	$0.429815\pi$
$788$	22.6330	0.806267
$789$	−0.328910	−0.0117095
$790$	38.7968	1.38033
$791$	−21.0324	−0.747827
$792$	1.64410	0.0584207
$793$	40.9204	1.45313
$794$	6.78286	0.240715
$795$	−18.4926	−0.655864
$796$	−39.4290	−1.39752
$797$	−17.6721	−0.625979	−0.312989	−	0.949757i	$-0.601330\pi$
$797$	−17.6721	−0.625979	−0.312989	+	0.949757i	$0.601330\pi$
$798$	4.43233	0.156903
$799$	−33.3969	−1.18150
$800$	1.27473	0.0450687
$801$	10.7063	0.378287
$802$	37.9129	1.33875
$803$	11.8300	0.417472
$804$	−15.4276	−0.544089
$805$	−26.3467	−0.928600
$806$	28.9386	1.01932
$807$	−23.5055	−0.827431
$808$	17.0212	0.598802
$809$	−44.5671	−1.56690	−0.783448	−	0.621457i	$-0.786541\pi$
$809$	−44.5671	−1.56690	−0.783448	+	0.621457i	$0.786541\pi$
$810$	13.3078	0.467590
$811$	7.96283	0.279613	0.139806	−	0.990179i	$-0.455352\pi$
$811$	7.96283	0.279613	0.139806	+	0.990179i	$0.455352\pi$
$812$	−36.7663	−1.29025
$813$	−39.0545	−1.36970
$814$	−11.5202	−0.403784
$815$	−48.1983	−1.68831
$816$	−7.69161	−0.269260
$817$	8.68556	0.303869
$818$	−47.7135	−1.66826
$819$	−14.9329	−0.521798
$820$	−15.8902	−0.554908
$821$	6.66857	0.232735	0.116367	−	0.993206i	$-0.462875\pi$
$821$	6.66857	0.232735	0.116367	+	0.993206i	$0.462875\pi$
$822$	43.7174	1.52482
$823$	−49.6409	−1.73037	−0.865187	−	0.501449i	$-0.832800\pi$
$823$	−49.6409	−1.73037	−0.865187	+	0.501449i	$0.832800\pi$
$824$	−16.2209	−0.565082
$825$	−0.151770	−0.00528394
$826$	27.7279	0.964778
$827$	−37.6406	−1.30889	−0.654446	−	0.756109i	$-0.727098\pi$
$827$	−37.6406	−1.30889	−0.654446	+	0.756109i	$0.727098\pi$
$828$	−27.9388	−0.970941
$829$	−19.3013	−0.670363	−0.335182	−	0.942154i	$-0.608798\pi$
$829$	−19.3013	−0.670363	−0.335182	+	0.942154i	$0.608798\pi$
$830$	2.38923	0.0829314
$831$	9.66384	0.335235
$832$	80.2172	2.78103
$833$	13.6438	0.472730
$834$	−43.6750	−1.51234
$835$	29.1049	1.00722
$836$	−1.95476	−0.0676069
$837$	11.4323	0.395157
$838$	63.4287	2.19111
$839$	−46.6157	−1.60935	−0.804676	−	0.593714i	$-0.797661\pi$
$839$	−46.6157	−1.60935	−0.804676	+	0.593714i	$0.797661\pi$
$840$	−7.48517	−0.258263
$841$	39.2742	1.35428
$842$	−9.10986	−0.313946
$843$	−10.7559	−0.370452
$844$	−2.74274	−0.0944089
$845$	65.1545	2.24138
$846$	33.2080	1.14172
$847$	17.0215	0.584866
$848$	−12.7255	−0.436995
$849$	35.2481	1.20971
$850$	1.15467	0.0396047
$851$	53.0141	1.81730
$852$	43.9427	1.50545
$853$	30.6139	1.04820	0.524100	−	0.851657i	$-0.324402\pi$
$853$	30.6139	1.04820	0.524100	+	0.851657i	$0.324402\pi$
$854$	−22.4004	−0.766526
$855$	3.24268	0.110897
$856$	7.85220	0.268383
$857$	−21.4588	−0.733020	−0.366510	−	0.930414i	$-0.619447\pi$
$857$	−21.4588	−0.733020	−0.366510	+	0.930414i	$0.619447\pi$
$858$	−12.5673	−0.429039
$859$	−8.95932	−0.305688	−0.152844	−	0.988250i	$-0.548843\pi$
$859$	−8.95932	−0.305688	−0.152844	+	0.988250i	$0.548843\pi$
$860$	−54.1647	−1.84700
$861$	−5.18593	−0.176736
$862$	−35.5360	−1.21036
$863$	16.1485	0.549702	0.274851	−	0.961487i	$-0.411371\pi$
$863$	16.1485	0.549702	0.274851	+	0.961487i	$0.411371\pi$
$864$	41.6996	1.41865
$865$	−34.4810	−1.17239
$866$	32.2463	1.09578
$867$	9.08718	0.308617
$868$	−9.16110	−0.310948
$869$	−5.58414	−0.189429
$870$	51.3283	1.74019
$871$	−28.9382	−0.980532
$872$	−19.0448	−0.644937
$873$	14.1962	0.480469
$874$	15.5549	0.526154
$875$	17.8173	0.602336
$876$	−57.1135	−1.92969
$877$	40.2224	1.35821	0.679107	−	0.734040i	$-0.262367\pi$
$877$	40.2224	1.35821	0.679107	+	0.734040i	$0.262367\pi$
$878$	18.7494	0.632763
$879$	−29.7893	−1.00477
$880$	−3.18079	−0.107225
$881$	14.7861	0.498158	0.249079	−	0.968483i	$-0.419872\pi$
$881$	14.7861	0.498158	0.249079	+	0.968483i	$0.419872\pi$
$882$	−13.5667	−0.456813
$883$	−4.64374	−0.156274	−0.0781371	−	0.996943i	$-0.524897\pi$
$883$	−4.64374	−0.156274	−0.0781371	+	0.996943i	$0.524897\pi$
$884$	55.2927	1.85969
$885$	−22.3861	−0.752502
$886$	22.1801	0.745155
$887$	45.2760	1.52022	0.760110	−	0.649794i	$-0.225145\pi$
$887$	45.2760	1.52022	0.760110	+	0.649794i	$0.225145\pi$
$888$	15.0614	0.505429
$889$	2.70278	0.0906482
$890$	37.1721	1.24601
$891$	−1.91544	−0.0641696
$892$	74.7906	2.50418
$893$	−10.6920	−0.357794
$894$	−5.04788	−0.168826
$895$	−1.09984	−0.0367637
$896$	−19.5455	−0.652970
$897$	57.8323	1.93096
$898$	7.03747	0.234844
$899$	17.0120	0.567380
$900$	−0.663971	−0.0221324
$901$	−20.2502	−0.674633
$902$	3.95489	0.131683
$903$	−17.6773	−0.588263
$904$	−20.9701	−0.697456
$905$	42.4201	1.41009
$906$	−11.0463	−0.366990
$907$	−16.4443	−0.546023	−0.273012	−	0.962011i	$-0.588020\pi$
$907$	−16.4443	−0.546023	−0.273012	+	0.962011i	$0.588020\pi$
$908$	−35.7071	−1.18498
$909$	15.0075	0.497768
$910$	−51.8470	−1.71871
$911$	−35.3486	−1.17115	−0.585576	−	0.810618i	$-0.699132\pi$
$911$	−35.3486	−1.17115	−0.585576	+	0.810618i	$0.699132\pi$
$912$	−2.46247	−0.0815405
$913$	−0.343889	−0.0113811
$914$	89.6359	2.96489
$915$	18.0850	0.597870
$916$	29.1399	0.962808
$917$	8.81642	0.291144
$918$	37.7719	1.24666
$919$	−12.4219	−0.409761	−0.204880	−	0.978787i	$-0.565681\pi$
$919$	−12.4219	−0.409761	−0.204880	+	0.978787i	$0.565681\pi$
$920$	−26.2687	−0.866053
$921$	29.4621	0.970808
$922$	−59.5582	−1.96145
$923$	82.4250	2.71305
$924$	3.97843	0.130881
$925$	1.25989	0.0414249
$926$	25.2678	0.830350
$927$	−14.3019	−0.469738
$928$	62.0516	2.03694
$929$	53.7588	1.76377	0.881885	−	0.471465i	$-0.156275\pi$
$929$	53.7588	1.76377	0.881885	+	0.471465i	$0.156275\pi$
$930$	12.7895	0.419385
$931$	4.36806	0.143157
$932$	−1.96746	−0.0644462
$933$	32.9627	1.07915
$934$	21.7392	0.711327
$935$	−5.06164	−0.165533
$936$	−14.8887	−0.486651
$937$	−1.50118	−0.0490414	−0.0245207	−	0.999699i	$-0.507806\pi$
$937$	−1.50118	−0.0490414	−0.0245207	+	0.999699i	$0.507806\pi$
$938$	15.8411	0.517232
$939$	−31.1398	−1.01621
$940$	66.6773	2.17477
$941$	58.6855	1.91309	0.956547	−	0.291578i	$-0.0941804\pi$
$941$	58.6855	1.91309	0.956547	+	0.291578i	$0.0941804\pi$
$942$	−33.2570	−1.08357
$943$	−18.1997	−0.592663
$944$	−15.4048	−0.501384
$945$	−20.4823	−0.666291
$946$	13.4810	0.438305
$947$	−41.5938	−1.35162	−0.675809	−	0.737077i	$-0.736206\pi$
$947$	−41.5938	−1.35162	−0.675809	+	0.737077i	$0.736206\pi$
$948$	26.9594	0.875601
$949$	−107.130	−3.47759
$950$	0.369666	0.0119936
$951$	15.9585	0.517488
$952$	−8.19661	−0.265654
$953$	4.64545	0.150481	0.0752404	−	0.997165i	$-0.476028\pi$
$953$	4.64545	0.150481	0.0752404	+	0.997165i	$0.476028\pi$
$954$	20.1357	0.651918
$955$	−59.0770	−1.91169
$956$	−60.8609	−1.96838
$957$	−7.38785	−0.238815
$958$	−2.91360	−0.0941343
$959$	−25.9596	−0.838279
$960$	35.4523	1.14422
$961$	−26.7611	−0.863262
$962$	104.325	3.36357
$963$	6.92327	0.223099
$964$	−58.2709	−1.87678
$965$	−50.8796	−1.63787
$966$	−31.6582	−1.01859
$967$	32.0649	1.03114	0.515568	−	0.856849i	$-0.327581\pi$
$967$	32.0649	1.03114	0.515568	+	0.856849i	$0.327581\pi$
$968$	16.9711	0.545472
$969$	−3.91856	−0.125882
$970$	49.2892	1.58258
$971$	−29.7655	−0.955222	−0.477611	−	0.878571i	$-0.658497\pi$
$971$	−29.7655	−0.955222	−0.477611	+	0.878571i	$0.658497\pi$
$972$	−36.4416	−1.16887
$973$	25.9344	0.831420
$974$	28.4036	0.910111
$975$	1.37439	0.0440159
$976$	12.4450	0.398354
$977$	−15.3194	−0.490110	−0.245055	−	0.969509i	$-0.578806\pi$
$977$	−15.3194	−0.490110	−0.245055	+	0.969509i	$0.578806\pi$
$978$	−57.9151	−1.85192
$979$	−5.35030	−0.170996
$980$	−27.2400	−0.870151
$981$	−16.7917	−0.536119
$982$	−12.4604	−0.397627
$983$	−15.6221	−0.498266	−0.249133	−	0.968469i	$-0.580146\pi$
$983$	−15.6221	−0.498266	−0.249133	+	0.968469i	$0.580146\pi$
$984$	−5.17057	−0.164832
$985$	18.7626	0.597825
$986$	56.2069	1.78999
$987$	21.7609	0.692657
$988$	17.7019	0.563174
$989$	−62.0372	−1.97267
$990$	5.03302	0.159960
$991$	29.5564	0.938891	0.469445	−	0.882962i	$-0.344454\pi$
$991$	29.5564	0.938891	0.469445	+	0.882962i	$0.344454\pi$
$992$	15.4615	0.490902
$993$	−13.6657	−0.433668
$994$	−45.1206	−1.43114
$995$	−32.6863	−1.03623
$996$	1.66025	0.0526069
$997$	−20.7357	−0.656707	−0.328354	−	0.944555i	$-0.606494\pi$
$997$	−20.7357	−0.656707	−0.328354	+	0.944555i	$0.606494\pi$
$998$	54.4516	1.72364
$999$	41.2140	1.30395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.13	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.13	✓	75	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.17778 q^{2} -1.25453 q^{3} +2.74274 q^{4} +2.27371 q^{5} +2.73208 q^{6} -1.62232 q^{7} -1.61752 q^{8} -1.42616 q^{9} +O(q^{10})\)	\(q-2.17778 q^{2} -1.25453 q^{3} +2.74274 q^{4} +2.27371 q^{5} +2.73208 q^{6} -1.62232 q^{7} -1.61752 q^{8} -1.42616 q^{9} -4.95164 q^{10} +0.712705 q^{11} -3.44083 q^{12} -6.45412 q^{13} +3.53307 q^{14} -2.85242 q^{15} -1.96287 q^{16} -3.12354 q^{17} +3.10588 q^{18} -1.00000 q^{19} +6.23618 q^{20} +2.03525 q^{21} -1.55212 q^{22} +7.14256 q^{23} +2.02922 q^{24} +0.169744 q^{25} +14.0557 q^{26} +5.55274 q^{27} -4.44961 q^{28} +8.26282 q^{29} +6.21196 q^{30} +2.05885 q^{31} +7.50974 q^{32} -0.894107 q^{33} +6.80238 q^{34} -3.68869 q^{35} -3.91159 q^{36} +7.42228 q^{37} +2.17778 q^{38} +8.09686 q^{39} -3.67777 q^{40} -2.54806 q^{41} -4.43233 q^{42} -8.68556 q^{43} +1.95476 q^{44} -3.24268 q^{45} -15.5549 q^{46} +10.6920 q^{47} +2.46247 q^{48} -4.36806 q^{49} -0.369666 q^{50} +3.91856 q^{51} -17.7019 q^{52} +6.48311 q^{53} -12.0927 q^{54} +1.62048 q^{55} +2.62414 q^{56} +1.25453 q^{57} -17.9946 q^{58} +7.84811 q^{59} -7.82345 q^{60} -6.34020 q^{61} -4.48374 q^{62} +2.31370 q^{63} -12.4288 q^{64} -14.6748 q^{65} +1.94717 q^{66} +4.48368 q^{67} -8.56704 q^{68} -8.96053 q^{69} +8.03317 q^{70} -12.7709 q^{71} +2.30685 q^{72} +16.5987 q^{73} -16.1641 q^{74} -0.212949 q^{75} -2.74274 q^{76} -1.15624 q^{77} -17.6332 q^{78} -7.83514 q^{79} -4.46299 q^{80} -2.68756 q^{81} +5.54912 q^{82} -0.482513 q^{83} +5.58215 q^{84} -7.10201 q^{85} +18.9153 q^{86} -10.3659 q^{87} -1.15281 q^{88} -7.50702 q^{89} +7.06185 q^{90} +10.4707 q^{91} +19.5902 q^{92} -2.58289 q^{93} -23.2849 q^{94} -2.27371 q^{95} -9.42116 q^{96} -9.95412 q^{97} +9.51269 q^{98} -1.01643 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9} - 48 q^{11} - 14 q^{12} - 3 q^{13} - 4 q^{14} - 39 q^{15} + 59 q^{16} - 23 q^{17} - 24 q^{18} - 75 q^{19} - 62 q^{20} - 3 q^{21} - 6 q^{22} - 73 q^{23} - 64 q^{24} + 57 q^{25} - 46 q^{26} - 22 q^{27} - 26 q^{28} - 39 q^{29} - 14 q^{30} - 44 q^{31} - 71 q^{32} - 3 q^{33} - 9 q^{34} - 49 q^{35} + 20 q^{36} - 12 q^{37} + 11 q^{38} - 90 q^{39} - 8 q^{40} - 42 q^{41} - 45 q^{42} - 24 q^{43} - 120 q^{44} - 63 q^{45} - 39 q^{46} - 59 q^{47} - 4 q^{48} + 48 q^{49} - 100 q^{50} - 55 q^{51} + 2 q^{52} + 13 q^{53} - 87 q^{54} - 36 q^{55} - 12 q^{56} + 4 q^{57} - 17 q^{58} - 47 q^{59} - 45 q^{60} - 35 q^{61} - 40 q^{62} - 69 q^{63} + 26 q^{64} - 44 q^{65} + 33 q^{66} - 39 q^{67} - 63 q^{68} + 42 q^{69} + 40 q^{70} - 154 q^{71} - 51 q^{72} - 29 q^{73} - 95 q^{74} + 37 q^{75} - 67 q^{76} - 24 q^{77} - 19 q^{78} - 95 q^{79} - 146 q^{80} + 23 q^{81} + 7 q^{82} - 52 q^{83} - 72 q^{84} - 36 q^{85} - 44 q^{86} - 103 q^{87} + 67 q^{88} + q^{89} - 2 q^{90} - 64 q^{91} - 183 q^{92} - 49 q^{93} + 5 q^{94} + 18 q^{95} - 69 q^{96} - 7 q^{97} - 23 q^{98} - 100 q^{99}+O(q^{100}) \) 75 * q - 11 * q^2 - 4 * q^3 + 67 * q^4 - 18 * q^5 - 15 * q^6 - 19 * q^7 - 30 * q^8 + 57 * q^9 - 48 * q^11 - 14 * q^12 - 3 * q^13 - 4 * q^14 - 39 * q^15 + 59 * q^16 - 23 * q^17 - 24 * q^18 - 75 * q^19 - 62 * q^20 - 3 * q^21 - 6 * q^22 - 73 * q^23 - 64 * q^24 + 57 * q^25 - 46 * q^26 - 22 * q^27 - 26 * q^28 - 39 * q^29 - 14 * q^30 - 44 * q^31 - 71 * q^32 - 3 * q^33 - 9 * q^34 - 49 * q^35 + 20 * q^36 - 12 * q^37 + 11 * q^38 - 90 * q^39 - 8 * q^40 - 42 * q^41 - 45 * q^42 - 24 * q^43 - 120 * q^44 - 63 * q^45 - 39 * q^46 - 59 * q^47 - 4 * q^48 + 48 * q^49 - 100 * q^50 - 55 * q^51 + 2 * q^52 + 13 * q^53 - 87 * q^54 - 36 * q^55 - 12 * q^56 + 4 * q^57 - 17 * q^58 - 47 * q^59 - 45 * q^60 - 35 * q^61 - 40 * q^62 - 69 * q^63 + 26 * q^64 - 44 * q^65 + 33 * q^66 - 39 * q^67 - 63 * q^68 + 42 * q^69 + 40 * q^70 - 154 * q^71 - 51 * q^72 - 29 * q^73 - 95 * q^74 + 37 * q^75 - 67 * q^76 - 24 * q^77 - 19 * q^78 - 95 * q^79 - 146 * q^80 + 23 * q^81 + 7 * q^82 - 52 * q^83 - 72 * q^84 - 36 * q^85 - 44 * q^86 - 103 * q^87 + 67 * q^88 + q^89 - 2 * q^90 - 64 * q^91 - 183 * q^92 - 49 * q^93 + 5 * q^94 + 18 * q^95 - 69 * q^96 - 7 * q^97 - 23 * q^98 - 100 * q^99

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