LMFDB - Embedded newform 4009.2.a.d.1.5

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			1.5
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)

$f(q)$	$=$	$q-2.62780 q^{2} +3.38775 q^{3} +4.90531 q^{4} -4.01450 q^{5} -8.90232 q^{6} +2.03245 q^{7} -7.63457 q^{8} +8.47685 q^{9} +O(q^{10})$	\(q-2.62780 q^{2} +3.38775 q^{3} +4.90531 q^{4} -4.01450 q^{5} -8.90232 q^{6} +2.03245 q^{7} -7.63457 q^{8} +8.47685 q^{9} +10.5493 q^{10} -1.08340 q^{11} +16.6180 q^{12} -3.24731 q^{13} -5.34087 q^{14} -13.6001 q^{15} +10.2515 q^{16} -4.85879 q^{17} -22.2754 q^{18} -1.00000 q^{19} -19.6924 q^{20} +6.88543 q^{21} +2.84696 q^{22} -4.43727 q^{23} -25.8640 q^{24} +11.1162 q^{25} +8.53326 q^{26} +18.5542 q^{27} +9.96981 q^{28} +0.720392 q^{29} +35.7383 q^{30} +5.61339 q^{31} -11.6697 q^{32} -3.67030 q^{33} +12.7679 q^{34} -8.15927 q^{35} +41.5816 q^{36} -5.04997 q^{37} +2.62780 q^{38} -11.0011 q^{39} +30.6490 q^{40} -0.00862364 q^{41} -18.0935 q^{42} +7.54006 q^{43} -5.31443 q^{44} -34.0303 q^{45} +11.6602 q^{46} -0.470506 q^{47} +34.7294 q^{48} -2.86914 q^{49} -29.2111 q^{50} -16.4604 q^{51} -15.9291 q^{52} +2.26278 q^{53} -48.7566 q^{54} +4.34932 q^{55} -15.5169 q^{56} -3.38775 q^{57} -1.89304 q^{58} -5.81858 q^{59} -66.7128 q^{60} +4.83244 q^{61} -14.7508 q^{62} +17.2288 q^{63} +10.1625 q^{64} +13.0363 q^{65} +9.64479 q^{66} -9.51968 q^{67} -23.8339 q^{68} -15.0324 q^{69} +21.4409 q^{70} -0.556975 q^{71} -64.7171 q^{72} -16.2479 q^{73} +13.2703 q^{74} +37.6589 q^{75} -4.90531 q^{76} -2.20196 q^{77} +28.9085 q^{78} -16.7386 q^{79} -41.1545 q^{80} +37.4264 q^{81} +0.0226612 q^{82} +13.3319 q^{83} +33.7752 q^{84} +19.5056 q^{85} -19.8137 q^{86} +2.44051 q^{87} +8.27132 q^{88} -8.64655 q^{89} +89.4247 q^{90} -6.59999 q^{91} -21.7662 q^{92} +19.0168 q^{93} +1.23639 q^{94} +4.01450 q^{95} -39.5339 q^{96} +10.7704 q^{97} +7.53953 q^{98} -9.18384 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.62780	−1.85813	−0.929066	−	0.369913i	$-0.879387\pi$
$2$	−2.62780	−1.85813	−0.929066	+	0.369913i	$0.879387\pi$
$3$	3.38775	1.95592	0.977959	−	0.208797i	$-0.0669546\pi$
$3$	3.38775	1.95592	0.977959	+	0.208797i	$0.0669546\pi$
$4$	4.90531	2.45266
$5$	−4.01450	−1.79534	−0.897669	−	0.440671i	$-0.854741\pi$
$5$	−4.01450	−1.79534	−0.897669	+	0.440671i	$0.854741\pi$
$6$	−8.90232	−3.63436
$7$	2.03245	0.768194	0.384097	−	0.923293i	$-0.374513\pi$
$7$	2.03245	0.768194	0.384097	+	0.923293i	$0.374513\pi$
$8$	−7.63457	−2.69923
$9$	8.47685	2.82562
$10$	10.5493	3.33598
$11$	−1.08340	−0.326658	−0.163329	−	0.986572i	$-0.552223\pi$
$11$	−1.08340	−0.326658	−0.163329	+	0.986572i	$0.552223\pi$
$12$	16.6180	4.79720
$13$	−3.24731	−0.900641	−0.450320	−	0.892867i	$-0.648690\pi$
$13$	−3.24731	−0.900641	−0.450320	+	0.892867i	$0.648690\pi$
$14$	−5.34087	−1.42741
$15$	−13.6001	−3.51153
$16$	10.2515	2.56287
$17$	−4.85879	−1.17843	−0.589215	−	0.807976i	$-0.700563\pi$
$17$	−4.85879	−1.17843	−0.589215	+	0.807976i	$0.700563\pi$
$18$	−22.2754	−5.25037
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−19.6924	−4.40335
$21$	6.88543	1.50252
$22$	2.84696	0.606974
$23$	−4.43727	−0.925234	−0.462617	−	0.886558i	$-0.653090\pi$
$23$	−4.43727	−0.925234	−0.462617	+	0.886558i	$0.653090\pi$
$24$	−25.8640	−5.27947
$25$	11.1162	2.22324
$26$	8.53326	1.67351
$27$	18.5542	3.57076
$28$	9.96981	1.88412
$29$	0.720392	0.133773	0.0668867	−	0.997761i	$-0.478693\pi$
$29$	0.720392	0.133773	0.0668867	+	0.997761i	$0.478693\pi$
$30$	35.7383	6.52490
$31$	5.61339	1.00819	0.504097	−	0.863647i	$-0.331825\pi$
$31$	5.61339	1.00819	0.504097	+	0.863647i	$0.331825\pi$
$32$	−11.6697	−2.06292
$33$	−3.67030	−0.638917
$34$	12.7679	2.18968
$35$	−8.15927	−1.37917
$36$	41.5816	6.93027
$37$	−5.04997	−0.830210	−0.415105	−	0.909773i	$-0.636255\pi$
$37$	−5.04997	−0.830210	−0.415105	+	0.909773i	$0.636255\pi$
$38$	2.62780	0.426285
$39$	−11.0011	−1.76158
$40$	30.6490	4.84603
$41$	−0.00862364	−0.00134679	−0.000673393	−	1.00000i	$-0.500214\pi$
$41$	−0.00862364	−0.00134679	−0.000673393		1.00000i	$0.500214\pi$
$42$	−18.0935	−2.79189
$43$	7.54006	1.14985	0.574924	−	0.818207i	$-0.305032\pi$
$43$	7.54006	1.14985	0.574924	+	0.818207i	$0.305032\pi$
$44$	−5.31443	−0.801180
$45$	−34.0303	−5.07293
$46$	11.6602	1.71921
$47$	−0.470506	−0.0686303	−0.0343151	−	0.999411i	$-0.510925\pi$
$47$	−0.470506	−0.0686303	−0.0343151	+	0.999411i	$0.510925\pi$
$48$	34.7294	5.01276
$49$	−2.86914	−0.409878
$50$	−29.2111	−4.13107
$51$	−16.4604	−2.30491
$52$	−15.9291	−2.20896
$53$	2.26278	0.310817	0.155408	−	0.987850i	$-0.450331\pi$
$53$	2.26278	0.310817	0.155408	+	0.987850i	$0.450331\pi$
$54$	−48.7566	−6.63494
$55$	4.34932	0.586462
$56$	−15.5169	−2.07353
$57$	−3.38775	−0.448718
$58$	−1.89304	−0.248569
$59$	−5.81858	−0.757514	−0.378757	−	0.925496i	$-0.623648\pi$
$59$	−5.81858	−0.757514	−0.378757	+	0.925496i	$0.623648\pi$
$60$	−66.7128	−8.61259
$61$	4.83244	0.618731	0.309365	−	0.950943i	$-0.399883\pi$
$61$	4.83244	0.618731	0.309365	+	0.950943i	$0.399883\pi$
$62$	−14.7508	−1.87336
$63$	17.2288	2.17062
$64$	10.1625	1.27031
$65$	13.0363	1.61695
$66$	9.64479	1.18719
$67$	−9.51968	−1.16301	−0.581507	−	0.813541i	$-0.697537\pi$
$67$	−9.51968	−1.16301	−0.581507	+	0.813541i	$0.697537\pi$
$68$	−23.8339	−2.89028
$69$	−15.0324	−1.80968
$70$	21.4409	2.56268
$71$	−0.556975	−0.0661008	−0.0330504	−	0.999454i	$-0.510522\pi$
$71$	−0.556975	−0.0661008	−0.0330504	+	0.999454i	$0.510522\pi$
$72$	−64.7171	−7.62699
$73$	−16.2479	−1.90167	−0.950835	−	0.309697i	$-0.899772\pi$
$73$	−16.2479	−1.90167	−0.950835	+	0.309697i	$0.899772\pi$
$74$	13.2703	1.54264
$75$	37.6589	4.34847
$76$	−4.90531	−0.562678
$77$	−2.20196	−0.250937
$78$	28.9085	3.27325
$79$	−16.7386	−1.88324	−0.941619	−	0.336680i	$-0.890696\pi$
$79$	−16.7386	−1.88324	−0.941619	+	0.336680i	$0.890696\pi$
$80$	−41.1545	−4.60122
$81$	37.4264	4.15849
$82$	0.0226612	0.00250251
$83$	13.3319	1.46337	0.731684	−	0.681644i	$-0.238735\pi$
$83$	13.3319	1.46337	0.731684	+	0.681644i	$0.238735\pi$
$84$	33.7752	3.68518
$85$	19.5056	2.11568
$86$	−19.8137	−2.13657
$87$	2.44051	0.261650
$88$	8.27132	0.881725
$89$	−8.64655	−0.916532	−0.458266	−	0.888815i	$-0.651529\pi$
$89$	−8.64655	−0.916532	−0.458266	+	0.888815i	$0.651529\pi$
$90$	89.4247	9.42619
$91$	−6.59999	−0.691867
$92$	−21.7662	−2.26928
$93$	19.0168	1.97195
$94$	1.23639	0.127524
$95$	4.01450	0.411879
$96$	−39.5339	−4.03491
$97$	10.7704	1.09357	0.546785	−	0.837273i	$-0.315851\pi$
$97$	10.7704	1.09357	0.546785	+	0.837273i	$0.315851\pi$
$98$	7.53953	0.761607
$99$	−9.18384	−0.923010
$100$	54.5284	5.45284
$101$	−18.7150	−1.86221	−0.931106	−	0.364750i	$-0.881155\pi$
$101$	−18.7150	−1.86221	−0.931106	+	0.364750i	$0.881155\pi$
$102$	43.2545	4.28283
$103$	−5.49725	−0.541660	−0.270830	−	0.962627i	$-0.587298\pi$
$103$	−5.49725	−0.541660	−0.270830	+	0.962627i	$0.587298\pi$
$104$	24.7918	2.43104
$105$	−27.6416	−2.69754
$106$	−5.94612	−0.577538
$107$	8.45415	0.817294	0.408647	−	0.912693i	$-0.366001\pi$
$107$	8.45415	0.817294	0.408647	+	0.912693i	$0.366001\pi$
$108$	91.0141	8.75784
$109$	−18.4322	−1.76549	−0.882743	−	0.469855i	$-0.844306\pi$
$109$	−18.4322	−1.76549	−0.882743	+	0.469855i	$0.844306\pi$
$110$	−11.4291	−1.08972
$111$	−17.1080	−1.62382
$112$	20.8356	1.96878
$113$	6.79595	0.639309	0.319655	−	0.947534i	$-0.396433\pi$
$113$	6.79595	0.639309	0.319655	+	0.947534i	$0.396433\pi$
$114$	8.90232	0.833778
$115$	17.8134	1.66111
$116$	3.53375	0.328100
$117$	−27.5269	−2.54486
$118$	15.2900	1.40756
$119$	−9.87525	−0.905263
$120$	103.831	9.47844
$121$	−9.82624	−0.893294
$122$	−12.6987	−1.14968
$123$	−0.0292147	−0.00263420
$124$	27.5354	2.47275
$125$	−24.5534	−2.19612
$126$	−45.2737	−4.03330
$127$	8.53461	0.757325	0.378662	−	0.925535i	$-0.376384\pi$
$127$	8.53461	0.757325	0.378662	+	0.925535i	$0.376384\pi$
$128$	−3.36570	−0.297489
$129$	25.5438	2.24901
$130$	−34.2567	−3.00452
$131$	−15.8890	−1.38822	−0.694112	−	0.719867i	$-0.744203\pi$
$131$	−15.8890	−1.38822	−0.694112	+	0.719867i	$0.744203\pi$
$132$	−18.0040	−1.56704
$133$	−2.03245	−0.176236
$134$	25.0158	2.16103
$135$	−74.4857	−6.41071
$136$	37.0948	3.18085
$137$	6.66418	0.569359	0.284680	−	0.958623i	$-0.408113\pi$
$137$	6.66418	0.569359	0.284680	+	0.958623i	$0.408113\pi$
$138$	39.5020	3.36263
$139$	−15.5512	−1.31904	−0.659519	−	0.751688i	$-0.729240\pi$
$139$	−15.5512	−1.31904	−0.659519	+	0.751688i	$0.729240\pi$
$140$	−40.0238	−3.38263
$141$	−1.59396	−0.134235
$142$	1.46362	0.122824
$143$	3.51814	0.294202
$144$	86.9002	7.24169
$145$	−2.89201	−0.240169
$146$	42.6961	3.53356
$147$	−9.71994	−0.801687
$148$	−24.7717	−2.03622
$149$	7.03324	0.576186	0.288093	−	0.957602i	$-0.406979\pi$
$149$	7.03324	0.576186	0.288093	+	0.957602i	$0.406979\pi$
$150$	−98.9598	−8.08004
$151$	−22.1170	−1.79985	−0.899926	−	0.436042i	$-0.856380\pi$
$151$	−22.1170	−1.79985	−0.899926	+	0.436042i	$0.856380\pi$
$152$	7.63457	0.619246
$153$	−41.1872	−3.32979
$154$	5.78631	0.466274
$155$	−22.5349	−1.81005
$156$	−53.9637	−4.32055
$157$	−2.18932	−0.174726	−0.0873632	−	0.996177i	$-0.527844\pi$
$157$	−2.18932	−0.174726	−0.0873632	+	0.996177i	$0.527844\pi$
$158$	43.9856	3.49931
$159$	7.66573	0.607932
$160$	46.8478	3.70364
$161$	−9.01853	−0.710760
$162$	−98.3490	−7.72703
$163$	−24.7994	−1.94244	−0.971219	−	0.238190i	$-0.923446\pi$
$163$	−24.7994	−1.94244	−0.971219	+	0.238190i	$0.923446\pi$
$164$	−0.0423017	−0.00330320
$165$	14.7344	1.14707
$166$	−35.0336	−2.71913
$167$	−1.79944	−0.139245	−0.0696225	−	0.997573i	$-0.522179\pi$
$167$	−1.79944	−0.139245	−0.0696225	+	0.997573i	$0.522179\pi$
$168$	−52.5674	−4.05566
$169$	−2.45500	−0.188846
$170$	−51.2568	−3.93121
$171$	−8.47685	−0.648241
$172$	36.9863	2.82018
$173$	10.8316	0.823508	0.411754	−	0.911295i	$-0.364916\pi$
$173$	10.8316	0.823508	0.411754	+	0.911295i	$0.364916\pi$
$174$	−6.41316	−0.486180
$175$	22.5931	1.70788
$176$	−11.1065	−0.837182
$177$	−19.7119	−1.48164
$178$	22.7214	1.70304
$179$	21.4368	1.60226	0.801129	−	0.598491i	$-0.204233\pi$
$179$	21.4368	1.60226	0.801129	+	0.598491i	$0.204233\pi$
$180$	−166.929	−12.4422
$181$	1.57913	0.117376	0.0586879	−	0.998276i	$-0.481308\pi$
$181$	1.57913	0.117376	0.0586879	+	0.998276i	$0.481308\pi$
$182$	17.3434	1.28558
$183$	16.3711	1.21019
$184$	33.8766	2.49742
$185$	20.2731	1.49051
$186$	−49.9722	−3.66414
$187$	5.26403	0.384944
$188$	−2.30798	−0.168327
$189$	37.7105	2.74303
$190$	−10.5493	−0.765325
$191$	−0.982671	−0.0711036	−0.0355518	−	0.999368i	$-0.511319\pi$
$191$	−0.982671	−0.0711036	−0.0355518	+	0.999368i	$0.511319\pi$
$192$	34.4280	2.48463
$193$	−6.55718	−0.471996	−0.235998	−	0.971754i	$-0.575836\pi$
$193$	−6.55718	−0.471996	−0.235998	+	0.971754i	$0.575836\pi$
$194$	−28.3025	−2.03200
$195$	44.1637	3.16263
$196$	−14.0741	−1.00529
$197$	0.370937	0.0264281	0.0132141	−	0.999913i	$-0.495794\pi$
$197$	0.370937	0.0264281	0.0132141	+	0.999913i	$0.495794\pi$
$198$	24.1333	1.71508
$199$	12.7345	0.902726	0.451363	−	0.892341i	$-0.350938\pi$
$199$	12.7345	0.902726	0.451363	+	0.892341i	$0.350938\pi$
$200$	−84.8674	−6.00103
$201$	−32.2503	−2.27476
$202$	49.1792	3.46024
$203$	1.46416	0.102764
$204$	−80.7433	−5.65316
$205$	0.0346196	0.00241794
$206$	14.4456	1.00648
$207$	−37.6140	−2.61436
$208$	−33.2897	−2.30822
$209$	1.08340	0.0749405
$210$	72.6364	5.01239
$211$	−1.00000	−0.0688428
$211$	−1.00000	−0.0688428
$212$	11.0996	0.762327
$213$	−1.88689	−0.129288
$214$	−22.2158	−1.51864
$215$	−30.2695	−2.06437
$216$	−141.653	−9.63829
$217$	11.4089	0.774489
$218$	48.4361	3.28051
$219$	−55.0438	−3.71951
$220$	21.3348	1.43839
$221$	15.7780	1.06134
$222$	44.9565	3.01728
$223$	−10.7524	−0.720034	−0.360017	−	0.932946i	$-0.617229\pi$
$223$	−10.7524	−0.720034	−0.360017	+	0.932946i	$0.617229\pi$
$224$	−23.7180	−1.58472
$225$	94.2302	6.28202
$226$	−17.8584	−1.18792
$227$	10.5660	0.701287	0.350644	−	0.936509i	$-0.385963\pi$
$227$	10.5660	0.701287	0.350644	+	0.936509i	$0.385963\pi$
$228$	−16.6180	−1.10055
$229$	9.59543	0.634084	0.317042	−	0.948412i	$-0.397310\pi$
$229$	9.59543	0.634084	0.317042	+	0.948412i	$0.397310\pi$
$230$	−46.8100	−3.08656
$231$	−7.45970	−0.490812
$232$	−5.49989	−0.361085
$233$	17.2452	1.12977	0.564886	−	0.825169i	$-0.308920\pi$
$233$	17.2452	1.12977	0.564886	+	0.825169i	$0.308920\pi$
$234$	72.3351	4.72870
$235$	1.88884	0.123215
$236$	−28.5419	−1.85792
$237$	−56.7061	−3.68346
$238$	25.9502	1.68210
$239$	−21.9275	−1.41837	−0.709187	−	0.705021i	$-0.750938\pi$
$239$	−21.9275	−1.41837	−0.709187	+	0.705021i	$0.750938\pi$
$240$	−139.421	−8.99960
$241$	7.86492	0.506624	0.253312	−	0.967385i	$-0.418480\pi$
$241$	7.86492	0.506624	0.253312	+	0.967385i	$0.418480\pi$
$242$	25.8214	1.65986
$243$	71.1287	4.56291
$244$	23.7047	1.51753
$245$	11.5182	0.735869
$246$	0.0767704	0.00489470
$247$	3.24731	0.206621
$248$	−42.8558	−2.72135
$249$	45.1652	2.86223
$250$	64.5214	4.08069
$251$	16.7412	1.05670	0.528349	−	0.849027i	$-0.322811\pi$
$251$	16.7412	1.05670	0.528349	+	0.849027i	$0.322811\pi$
$252$	84.5126	5.32379
$253$	4.80735	0.302235
$254$	−22.4272	−1.40721
$255$	66.0801	4.13810
$256$	−11.4806	−0.717540
$257$	−22.9497	−1.43156	−0.715782	−	0.698323i	$-0.753930\pi$
$257$	−22.9497	−1.43156	−0.715782	+	0.698323i	$0.753930\pi$
$258$	−67.1240	−4.17896
$259$	−10.2638	−0.637763
$260$	63.9471	3.96583
$261$	6.10665	0.377992
$262$	41.7530	2.57951
$263$	−10.5379	−0.649794	−0.324897	−	0.945749i	$-0.605330\pi$
$263$	−10.5379	−0.649794	−0.324897	+	0.945749i	$0.605330\pi$
$264$	28.0211	1.72458
$265$	−9.08392	−0.558021
$266$	5.34087	0.327470
$267$	−29.2923	−1.79266
$268$	−46.6970	−2.85247
$269$	−18.6399	−1.13650	−0.568249	−	0.822857i	$-0.692379\pi$
$269$	−18.6399	−1.13650	−0.568249	+	0.822857i	$0.692379\pi$
$270$	195.733	11.9120
$271$	−16.6859	−1.01359	−0.506797	−	0.862066i	$-0.669171\pi$
$271$	−16.6859	−1.01359	−0.506797	+	0.862066i	$0.669171\pi$
$272$	−49.8098	−3.02016
$273$	−22.3591	−1.35323
$274$	−17.5121	−1.05795
$275$	−12.0433	−0.726239
$276$	−73.7384	−4.43853
$277$	12.2464	0.735814	0.367907	−	0.929863i	$-0.380074\pi$
$277$	12.2464	0.735814	0.367907	+	0.929863i	$0.380074\pi$
$278$	40.8654	2.45095
$279$	47.5838	2.84877
$280$	62.2925	3.72269
$281$	−23.5148	−1.40277	−0.701387	−	0.712780i	$-0.747436\pi$
$281$	−23.5148	−1.40277	−0.701387	+	0.712780i	$0.747436\pi$
$282$	4.18859	0.249427
$283$	20.1354	1.19692	0.598462	−	0.801151i	$-0.295779\pi$
$283$	20.1354	1.19692	0.598462	+	0.801151i	$0.295779\pi$
$284$	−2.73214	−0.162123
$285$	13.6001	0.805601
$286$	−9.24495	−0.546666
$287$	−0.0175271	−0.00103459
$288$	−98.9219	−5.82903
$289$	6.60785	0.388697
$290$	7.59962	0.446265
$291$	36.4875	2.13893
$292$	−79.7010	−4.66415
$293$	−24.8629	−1.45251	−0.726254	−	0.687427i	$-0.758740\pi$
$293$	−24.8629	−1.45251	−0.726254	+	0.687427i	$0.758740\pi$
$294$	25.5420	1.48964
$295$	23.3587	1.35999
$296$	38.5544	2.24093
$297$	−20.1017	−1.16642
$298$	−18.4819	−1.07063
$299$	14.4092	0.833304
$300$	184.729	10.6653
$301$	15.3248	0.883306
$302$	58.1189	3.34436
$303$	−63.4017	−3.64233
$304$	−10.2515	−0.587963
$305$	−19.3998	−1.11083
$306$	108.232	6.18719
$307$	−6.47609	−0.369610	−0.184805	−	0.982775i	$-0.559165\pi$
$307$	−6.47609	−0.369610	−0.184805	+	0.982775i	$0.559165\pi$
$308$	−10.8013	−0.615462
$309$	−18.6233	−1.05944
$310$	59.2172	3.36331
$311$	−11.0194	−0.624851	−0.312425	−	0.949942i	$-0.601141\pi$
$311$	−11.0194	−0.624851	−0.312425	+	0.949942i	$0.601141\pi$
$312$	83.9884	4.75491
$313$	16.8213	0.950796	0.475398	−	0.879771i	$-0.342304\pi$
$313$	16.8213	0.950796	0.475398	+	0.879771i	$0.342304\pi$
$314$	5.75308	0.324665
$315$	−69.1649	−3.89700
$316$	−82.1080	−4.61894
$317$	23.8739	1.34089	0.670445	−	0.741959i	$-0.266103\pi$
$317$	23.8739	1.34089	0.670445	+	0.741959i	$0.266103\pi$
$318$	−20.1440	−1.12962
$319$	−0.780474	−0.0436982
$320$	−40.7974	−2.28064
$321$	28.6406	1.59856
$322$	23.6989	1.32069
$323$	4.85879	0.270350
$324$	183.588	10.1993
$325$	−36.0977	−2.00234
$326$	65.1677	3.60931
$327$	−62.4438	−3.45315
$328$	0.0658378	0.00363528
$329$	−0.956279	−0.0527214
$330$	−38.7190	−2.13141
$331$	18.7184	1.02886	0.514428	−	0.857534i	$-0.328004\pi$
$331$	18.7184	1.02886	0.514428	+	0.857534i	$0.328004\pi$
$332$	65.3972	3.58914
$333$	−42.8079	−2.34586
$334$	4.72857	0.258736
$335$	38.2167	2.08800
$336$	70.5859	3.85078
$337$	8.39608	0.457364	0.228682	−	0.973501i	$-0.426558\pi$
$337$	8.39608	0.457364	0.228682	+	0.973501i	$0.426558\pi$
$338$	6.45125	0.350902
$339$	23.0230	1.25044
$340$	95.6811	5.18904
$341$	−6.08156	−0.329335
$342$	22.2754	1.20452
$343$	−20.0585	−1.08306
$344$	−57.5651	−3.10370
$345$	60.3473	3.24899
$346$	−28.4631	−1.53019
$347$	−10.6394	−0.571155	−0.285577	−	0.958356i	$-0.592185\pi$
$347$	−10.6394	−0.571155	−0.285577	+	0.958356i	$0.592185\pi$
$348$	11.9715	0.641738
$349$	6.58753	0.352622	0.176311	−	0.984334i	$-0.443583\pi$
$349$	6.58753	0.352622	0.176311	+	0.984334i	$0.443583\pi$
$350$	−59.3701	−3.17346
$351$	−60.2511	−3.21597
$352$	12.6429	0.673870
$353$	21.0663	1.12125	0.560624	−	0.828071i	$-0.310562\pi$
$353$	21.0663	1.12125	0.560624	+	0.828071i	$0.310562\pi$
$354$	51.7988	2.75308
$355$	2.23598	0.118673
$356$	−42.4140	−2.24794
$357$	−33.4549	−1.77062
$358$	−56.3314	−2.97721
$359$	−11.9346	−0.629881	−0.314941	−	0.949111i	$-0.601985\pi$
$359$	−11.9346	−0.629881	−0.314941	+	0.949111i	$0.601985\pi$
$360$	259.807	13.6930
$361$	1.00000	0.0526316
$362$	−4.14963	−0.218100
$363$	−33.2888	−1.74721
$364$	−32.3750	−1.69691
$365$	65.2271	3.41414
$366$	−43.0199	−2.24869
$367$	17.9906	0.939101	0.469550	−	0.882906i	$-0.344416\pi$
$367$	17.9906	0.939101	0.469550	+	0.882906i	$0.344416\pi$
$368$	−45.4886	−2.37125
$369$	−0.0731013	−0.00380550
$370$	−53.2736	−2.76956
$371$	4.59899	0.238767
$372$	93.2832	4.83651
$373$	−24.5506	−1.27118	−0.635590	−	0.772027i	$-0.719243\pi$
$373$	−24.5506	−1.27118	−0.635590	+	0.772027i	$0.719243\pi$
$374$	−13.8328	−0.715277
$375$	−83.1808	−4.29544
$376$	3.59211	0.185249
$377$	−2.33933	−0.120482
$378$	−99.0955	−5.09692
$379$	22.0063	1.13039	0.565195	−	0.824957i	$-0.308801\pi$
$379$	22.0063	1.13039	0.565195	+	0.824957i	$0.308801\pi$
$380$	19.6924	1.01020
$381$	28.9131	1.48126
$382$	2.58226	0.132120
$383$	−20.4747	−1.04621	−0.523104	−	0.852269i	$-0.675226\pi$
$383$	−20.4747	−1.04621	−0.523104	+	0.852269i	$0.675226\pi$
$384$	−11.4022	−0.581864
$385$	8.83977	0.450516
$386$	17.2309	0.877031
$387$	63.9159	3.24903
$388$	52.8323	2.68215
$389$	−5.29022	−0.268225	−0.134112	−	0.990966i	$-0.542818\pi$
$389$	−5.29022	−0.268225	−0.134112	+	0.990966i	$0.542818\pi$
$390$	−116.053	−5.87659
$391$	21.5598	1.09032
$392$	21.9047	1.10635
$393$	−53.8278	−2.71525
$394$	−0.974746	−0.0491070
$395$	67.1970	3.38105
$396$	−45.0496	−2.26383
$397$	−6.88793	−0.345696	−0.172848	−	0.984949i	$-0.555297\pi$
$397$	−6.88793	−0.345696	−0.172848	+	0.984949i	$0.555297\pi$
$398$	−33.4637	−1.67738
$399$	−6.88543	−0.344703
$400$	113.957	5.69787
$401$	11.8518	0.591853	0.295927	−	0.955211i	$-0.404372\pi$
$401$	11.8518	0.591853	0.295927	+	0.955211i	$0.404372\pi$
$402$	84.7472	4.22681
$403$	−18.2284	−0.908021
$404$	−91.8029	−4.56737
$405$	−150.248	−7.46589
$406$	−3.84752	−0.190949
$407$	5.47115	0.271195
$408$	125.668	6.22149
$409$	40.1754	1.98655	0.993273	−	0.115798i	$-0.0369424\pi$
$409$	40.1754	1.98655	0.993273	+	0.115798i	$0.0369424\pi$
$410$	−0.0909732	−0.00449285
$411$	22.5766	1.11362
$412$	−26.9657	−1.32851
$413$	−11.8260	−0.581918
$414$	98.8420	4.85782
$415$	−53.5209	−2.62724
$416$	37.8949	1.85795
$417$	−52.6836	−2.57993
$418$	−2.84696	−0.139249
$419$	−20.6681	−1.00970	−0.504851	−	0.863207i	$-0.668453\pi$
$419$	−20.6681	−1.00970	−0.504851	+	0.863207i	$0.668453\pi$
$420$	−135.591	−6.61614
$421$	−32.0340	−1.56124	−0.780620	−	0.625006i	$-0.785097\pi$
$421$	−32.0340	−1.56124	−0.780620	+	0.625006i	$0.785097\pi$
$422$	2.62780	0.127919
$423$	−3.98840	−0.193923
$424$	−17.2754	−0.838965
$425$	−54.0112	−2.61993
$426$	4.95837	0.240234
$427$	9.82170	0.475305
$428$	41.4703	2.00454
$429$	11.9186	0.575434
$430$	79.5422	3.83586
$431$	−38.6760	−1.86296	−0.931479	−	0.363795i	$-0.881481\pi$
$431$	−38.6760	−1.86296	−0.931479	+	0.363795i	$0.881481\pi$
$432$	190.208	9.15138
$433$	−1.81673	−0.0873065	−0.0436532	−	0.999047i	$-0.513900\pi$
$433$	−1.81673	−0.0873065	−0.0436532	+	0.999047i	$0.513900\pi$
$434$	−29.9804	−1.43910
$435$	−9.79741	−0.469750
$436$	−90.4158	−4.33013
$437$	4.43727	0.212263
$438$	144.644	6.91135
$439$	−0.870965	−0.0415689	−0.0207844	−	0.999784i	$-0.506616\pi$
$439$	−0.870965	−0.0415689	−0.0207844	+	0.999784i	$0.506616\pi$
$440$	−33.2052	−1.58299
$441$	−24.3213	−1.15816
$442$	−41.4613	−1.97211
$443$	1.26016	0.0598721	0.0299360	−	0.999552i	$-0.490470\pi$
$443$	1.26016	0.0598721	0.0299360	+	0.999552i	$0.490470\pi$
$444$	−83.9203	−3.98268
$445$	34.7116	1.64549
$446$	28.2551	1.33792
$447$	23.8269	1.12697
$448$	20.6548	0.975847
$449$	−11.1479	−0.526104	−0.263052	−	0.964782i	$-0.584729\pi$
$449$	−11.1479	−0.526104	−0.263052	+	0.964782i	$0.584729\pi$
$450$	−247.618	−11.6728
$451$	0.00934287	0.000439939 0
$452$	33.3363	1.56801
$453$	−74.9267	−3.52036
$454$	−27.7652	−1.30308
$455$	26.4956	1.24213
$456$	25.8640	1.21119
$457$	−10.3412	−0.483739	−0.241869	−	0.970309i	$-0.577761\pi$
$457$	−10.3412	−0.483739	−0.241869	+	0.970309i	$0.577761\pi$
$458$	−25.2148	−1.17821
$459$	−90.1509	−4.20789
$460$	87.3803	4.07413
$461$	−3.12820	−0.145695	−0.0728473	−	0.997343i	$-0.523209\pi$
$461$	−3.12820	−0.145695	−0.0728473	+	0.997343i	$0.523209\pi$
$462$	19.6026	0.911994
$463$	1.35329	0.0628929	0.0314464	−	0.999505i	$-0.489989\pi$
$463$	1.35329	0.0628929	0.0314464	+	0.999505i	$0.489989\pi$
$464$	7.38508	0.342844
$465$	−76.3427	−3.54031
$466$	−45.3169	−2.09927
$467$	2.56343	0.118621	0.0593106	−	0.998240i	$-0.481110\pi$
$467$	2.56343	0.118621	0.0593106	+	0.998240i	$0.481110\pi$
$468$	−135.028	−6.24168
$469$	−19.3483	−0.893421
$470$	−4.96350	−0.228949
$471$	−7.41686	−0.341751
$472$	44.4224	2.04470
$473$	−8.16892	−0.375607
$474$	149.012	6.84436
$475$	−11.1162	−0.510046
$476$	−48.4412	−2.22030
$477$	19.1812	0.878248
$478$	57.6210	2.63553
$479$	19.5467	0.893114	0.446557	−	0.894755i	$-0.352650\pi$
$479$	19.5467	0.893114	0.446557	+	0.894755i	$0.352650\pi$
$480$	158.709	7.24402
$481$	16.3988	0.747721
$482$	−20.6674	−0.941375
$483$	−30.5525	−1.39019
$484$	−48.2008	−2.19095
$485$	−43.2378	−1.96333
$486$	−186.912	−8.47849
$487$	36.2669	1.64341	0.821704	−	0.569914i	$-0.193024\pi$
$487$	36.2669	1.64341	0.821704	+	0.569914i	$0.193024\pi$
$488$	−36.8936	−1.67010
$489$	−84.0141	−3.79925
$490$	−30.2674	−1.36734
$491$	−33.1590	−1.49644	−0.748222	−	0.663449i	$-0.769092\pi$
$491$	−33.1590	−1.49644	−0.748222	+	0.663449i	$0.769092\pi$
$492$	−0.143307	−0.00646080
$493$	−3.50023	−0.157643
$494$	−8.53326	−0.383929
$495$	36.8685	1.65712
$496$	57.5455	2.58387
$497$	−1.13202	−0.0507782
$498$	−118.685	−5.31840
$499$	1.21074	0.0542002	0.0271001	−	0.999633i	$-0.491373\pi$
$499$	1.21074	0.0542002	0.0271001	+	0.999633i	$0.491373\pi$
$500$	−120.442	−5.38634
$501$	−6.09606	−0.272352
$502$	−43.9926	−1.96349
$503$	38.3537	1.71011	0.855054	−	0.518539i	$-0.173524\pi$
$503$	38.3537	1.71011	0.855054	+	0.518539i	$0.173524\pi$
$504$	−131.534	−5.85901
$505$	75.1313	3.34330
$506$	−12.6327	−0.561593
$507$	−8.31694	−0.369368
$508$	41.8650	1.85746
$509$	27.2903	1.20962	0.604811	−	0.796369i	$-0.293249\pi$
$509$	27.2903	1.20962	0.604811	+	0.796369i	$0.293249\pi$
$510$	−173.645	−7.68913
$511$	−33.0230	−1.46085
$512$	36.9002	1.63077
$513$	−18.5542	−0.819188
$514$	60.3072	2.66004
$515$	22.0687	0.972462
$516$	125.300	5.51605
$517$	0.509747	0.0224186
$518$	26.9712	1.18505
$519$	36.6946	1.61071
$520$	−99.5266	−4.36453
$521$	11.9701	0.524418	0.262209	−	0.965011i	$-0.415549\pi$
$521$	11.9701	0.524418	0.262209	+	0.965011i	$0.415549\pi$
$522$	−16.0470	−0.702360
$523$	14.1263	0.617700	0.308850	−	0.951111i	$-0.400056\pi$
$523$	14.1263	0.617700	0.308850	+	0.951111i	$0.400056\pi$
$524$	−77.9403	−3.40484
$525$	76.5398	3.34047
$526$	27.6914	1.20740
$527$	−27.2743	−1.18809
$528$	−37.6260	−1.63746
$529$	−3.31066	−0.143942
$530$	23.8707	1.03688
$531$	−49.3232	−2.14044
$532$	−9.96981	−0.432246
$533$	0.0280036	0.00121297
$534$	76.9743	3.33100
$535$	−33.9392	−1.46732
$536$	72.6787	3.13924
$537$	72.6224	3.13389
$538$	48.9820	2.11176
$539$	3.10844	0.133890
$540$	−365.376	−15.7233
$541$	25.4447	1.09395	0.546976	−	0.837148i	$-0.315779\pi$
$541$	25.4447	1.09395	0.546976	+	0.837148i	$0.315779\pi$
$542$	43.8470	1.88339
$543$	5.34970	0.229577
$544$	56.7004	2.43101
$545$	73.9961	3.16965
$546$	58.7552	2.51449
$547$	39.8941	1.70575	0.852873	−	0.522118i	$-0.174858\pi$
$547$	39.8941	1.70575	0.852873	+	0.522118i	$0.174858\pi$
$548$	32.6899	1.39644
$549$	40.9639	1.74830
$550$	31.6473	1.34945
$551$	−0.720392	−0.0306897
$552$	114.766	4.88475
$553$	−34.0204	−1.44669
$554$	−32.1810	−1.36724
$555$	68.6802	2.91531
$556$	−76.2836	−3.23515
$557$	−37.8960	−1.60571	−0.802853	−	0.596177i	$-0.796686\pi$
$557$	−37.8960	−1.60571	−0.802853	+	0.596177i	$0.796686\pi$
$558$	−125.041	−5.29339
$559$	−24.4849	−1.03560
$560$	−83.6446	−3.53463
$561$	17.8332	0.752918
$562$	61.7921	2.60654
$563$	20.2490	0.853393	0.426697	−	0.904395i	$-0.359677\pi$
$563$	20.2490	0.853393	0.426697	+	0.904395i	$0.359677\pi$
$564$	−7.81885	−0.329233
$565$	−27.2823	−1.14778
$566$	−52.9117	−2.22404
$567$	76.0673	3.19453
$568$	4.25227	0.178421
$569$	−9.50978	−0.398671	−0.199335	−	0.979931i	$-0.563878\pi$
$569$	−9.50978	−0.398671	−0.199335	+	0.979931i	$0.563878\pi$
$570$	−35.7383	−1.49691
$571$	30.0794	1.25879	0.629393	−	0.777087i	$-0.283304\pi$
$571$	30.0794	1.25879	0.629393	+	0.777087i	$0.283304\pi$
$572$	17.2576	0.721576
$573$	−3.32904	−0.139073
$574$	0.0460577	0.00192241
$575$	−49.3255	−2.05702
$576$	86.1460	3.58942
$577$	−3.23596	−0.134715	−0.0673574	−	0.997729i	$-0.521457\pi$
$577$	−3.23596	−0.134715	−0.0673574	+	0.997729i	$0.521457\pi$
$578$	−17.3641	−0.722251
$579$	−22.2141	−0.923186
$580$	−14.1862	−0.589051
$581$	27.0965	1.12415
$582$	−95.8817	−3.97442
$583$	−2.45150	−0.101531
$584$	124.046	5.13305
$585$	110.507	4.56889
$586$	65.3347	2.69895
$587$	−3.18885	−0.131618	−0.0658090	−	0.997832i	$-0.520963\pi$
$587$	−3.18885	−0.131618	−0.0658090	+	0.997832i	$0.520963\pi$
$588$	−47.6794	−1.96626
$589$	−5.61339	−0.231296
$590$	−61.3818	−2.52705
$591$	1.25664	0.0516913
$592$	−51.7697	−2.12772
$593$	−42.1586	−1.73125	−0.865624	−	0.500694i	$-0.833078\pi$
$593$	−42.1586	−1.73125	−0.865624	+	0.500694i	$0.833078\pi$
$594$	52.8231	2.16736
$595$	39.6442	1.62525
$596$	34.5003	1.41319
$597$	43.1413	1.76566
$598$	−37.8644	−1.54839
$599$	−35.5775	−1.45366	−0.726829	−	0.686819i	$-0.759007\pi$
$599$	−35.5775	−1.45366	−0.726829	+	0.686819i	$0.759007\pi$
$600$	−287.509	−11.7375
$601$	−28.4291	−1.15965	−0.579823	−	0.814742i	$-0.696878\pi$
$601$	−28.4291	−1.15965	−0.579823	+	0.814742i	$0.696878\pi$
$602$	−40.2704	−1.64130
$603$	−80.6969	−3.28623
$604$	−108.491	−4.41442
$605$	39.4474	1.60377
$606$	166.607	6.76794
$607$	−34.1711	−1.38696	−0.693482	−	0.720474i	$-0.743924\pi$
$607$	−34.1711	−1.38696	−0.693482	+	0.720474i	$0.743924\pi$
$608$	11.6697	0.473267
$609$	4.96021	0.200998
$610$	50.9788	2.06407
$611$	1.52788	0.0618112
$612$	−202.036	−8.16683
$613$	−9.42365	−0.380618	−0.190309	−	0.981724i	$-0.560949\pi$
$613$	−9.42365	−0.380618	−0.190309	+	0.981724i	$0.560949\pi$
$614$	17.0179	0.686785
$615$	0.117282	0.00472928
$616$	16.8110	0.677336
$617$	10.6051	0.426945	0.213473	−	0.976949i	$-0.431523\pi$
$617$	10.6051	0.426945	0.213473	+	0.976949i	$0.431523\pi$
$618$	48.9382	1.96858
$619$	9.18847	0.369316	0.184658	−	0.982803i	$-0.440882\pi$
$619$	9.18847	0.369316	0.184658	+	0.982803i	$0.440882\pi$
$620$	−110.541	−4.43943
$621$	−82.3299	−3.30379
$622$	28.9567	1.16106
$623$	−17.5737	−0.704075
$624$	−112.777	−4.51470
$625$	42.9887	1.71955
$626$	−44.2030	−1.76671
$627$	3.67030	0.146578
$628$	−10.7393	−0.428544
$629$	24.5368	0.978345
$630$	181.751	7.24114
$631$	−7.50937	−0.298943	−0.149472	−	0.988766i	$-0.547757\pi$
$631$	−7.50937	−0.298943	−0.149472	+	0.988766i	$0.547757\pi$
$632$	127.792	5.08329
$633$	−3.38775	−0.134651
$634$	−62.7357	−2.49155
$635$	−34.2622	−1.35965
$636$	37.6028	1.49105
$637$	9.31699	0.369153
$638$	2.05093	0.0811970
$639$	−4.72139	−0.186775
$640$	13.5116	0.534093
$641$	−28.9594	−1.14383	−0.571913	−	0.820314i	$-0.693799\pi$
$641$	−28.9594	−1.14383	−0.571913	+	0.820314i	$0.693799\pi$
$642$	−75.2615	−2.97034
$643$	41.7695	1.64723	0.823614	−	0.567151i	$-0.191954\pi$
$643$	41.7695	1.64723	0.823614	+	0.567151i	$0.191954\pi$
$644$	−44.2387	−1.74325
$645$	−102.546	−4.03773
$646$	−12.7679	−0.502347
$647$	22.1377	0.870322	0.435161	−	0.900353i	$-0.356691\pi$
$647$	22.1377	0.870322	0.435161	+	0.900353i	$0.356691\pi$
$648$	−285.735	−11.2247
$649$	6.30386	0.247448
$650$	94.8573	3.72061
$651$	38.6506	1.51484
$652$	−121.649	−4.76413
$653$	−26.3599	−1.03154	−0.515772	−	0.856726i	$-0.672495\pi$
$653$	−26.3599	−1.03154	−0.515772	+	0.856726i	$0.672495\pi$
$654$	164.089	6.41641
$655$	63.7862	2.49233
$656$	−0.0884050	−0.00345164
$657$	−137.731	−5.37339
$658$	2.51291	0.0979633
$659$	47.7289	1.85925	0.929627	−	0.368502i	$-0.120129\pi$
$659$	47.7289	1.85925	0.929627	+	0.368502i	$0.120129\pi$
$660$	72.2768	2.81337
$661$	24.9030	0.968615	0.484308	−	0.874898i	$-0.339072\pi$
$661$	24.9030	0.968615	0.484308	+	0.874898i	$0.339072\pi$
$662$	−49.1881	−1.91175
$663$	53.4519	2.07590
$664$	−101.784	−3.94997
$665$	8.15927	0.316403
$666$	112.490	4.35891
$667$	−3.19657	−0.123772
$668$	−8.82683	−0.341520
$669$	−36.4264	−1.40833
$670$	−100.426	−3.87979
$671$	−5.23548	−0.202113
$672$	−80.3506	−3.09959
$673$	19.2144	0.740662	0.370331	−	0.928900i	$-0.379244\pi$
$673$	19.2144	0.740662	0.370331	+	0.928900i	$0.379244\pi$
$674$	−22.0632	−0.849842
$675$	206.252	7.93864
$676$	−12.0426	−0.463176
$677$	11.7148	0.450237	0.225119	−	0.974331i	$-0.427723\pi$
$677$	11.7148	0.450237	0.225119	+	0.974331i	$0.427723\pi$
$678$	−60.4997	−2.32348
$679$	21.8903	0.840074
$680$	−148.917	−5.71071
$681$	35.7948	1.37166
$682$	15.9811	0.611948
$683$	−39.8912	−1.52639	−0.763197	−	0.646166i	$-0.776371\pi$
$683$	−39.8912	−1.52639	−0.763197	+	0.646166i	$0.776371\pi$
$684$	−41.5816	−1.58991
$685$	−26.7533	−1.02219
$686$	52.7098	2.01247
$687$	32.5069	1.24022
$688$	77.2967	2.94691
$689$	−7.34794	−0.279934
$690$	−158.581	−6.03706
$691$	24.0852	0.916242	0.458121	−	0.888890i	$-0.348523\pi$
$691$	24.0852	0.916242	0.458121	+	0.888890i	$0.348523\pi$
$692$	53.1322	2.01978
$693$	−18.6657	−0.709051
$694$	27.9583	1.06128
$695$	62.4303	2.36812
$696$	−18.6322	−0.706253
$697$	0.0419005	0.00158709
$698$	−17.3107	−0.655219
$699$	58.4225	2.20974
$700$	110.826	4.18884
$701$	−37.4396	−1.41408	−0.707038	−	0.707176i	$-0.749969\pi$
$701$	−37.4396	−1.41408	−0.707038	+	0.707176i	$0.749969\pi$
$702$	158.328	5.97569
$703$	5.04997	0.190463
$704$	−11.0101	−0.414958
$705$	6.39893	0.240998
$706$	−55.3580	−2.08343
$707$	−38.0373	−1.43054
$708$	−96.6930	−3.63394
$709$	−10.1462	−0.381049	−0.190524	−	0.981682i	$-0.561019\pi$
$709$	−10.1462	−0.381049	−0.190524	+	0.981682i	$0.561019\pi$
$710$	−5.87569	−0.220511
$711$	−141.890	−5.32131
$712$	66.0127	2.47393
$713$	−24.9081	−0.932816
$714$	87.9126	3.29005
$715$	−14.1236	−0.528191
$716$	105.154	3.92979
$717$	−74.2849	−2.77422
$718$	31.3616	1.17040
$719$	24.4884	0.913263	0.456631	−	0.889656i	$-0.349056\pi$
$719$	24.4884	0.913263	0.456631	+	0.889656i	$0.349056\pi$
$720$	−348.861	−13.0013
$721$	−11.1729	−0.416100
$722$	−2.62780	−0.0977965
$723$	26.6444	0.990915
$724$	7.74613	0.287883
$725$	8.00801	0.297410
$726$	87.4763	3.24655
$727$	16.3003	0.604546	0.302273	−	0.953221i	$-0.402255\pi$
$727$	16.3003	0.604546	0.302273	+	0.953221i	$0.402255\pi$
$728$	50.3881	1.86751
$729$	128.687	4.76619
$730$	−171.403	−6.34393
$731$	−36.6356	−1.35502
$732$	80.3054	2.96817
$733$	−5.01348	−0.185177	−0.0925886	−	0.995704i	$-0.529514\pi$
$733$	−5.01348	−0.185177	−0.0925886	+	0.995704i	$0.529514\pi$
$734$	−47.2756	−1.74497
$735$	39.0207	1.43930
$736$	51.7814	1.90869
$737$	10.3136	0.379908
$738$	0.192095	0.00707112
$739$	8.26318	0.303966	0.151983	−	0.988383i	$-0.451434\pi$
$739$	8.26318	0.303966	0.151983	+	0.988383i	$0.451434\pi$
$740$	99.4459	3.65570
$741$	11.0011	0.404134
$742$	−12.0852	−0.443662
$743$	10.1696	0.373088	0.186544	−	0.982447i	$-0.440271\pi$
$743$	10.1696	0.373088	0.186544	+	0.982447i	$0.440271\pi$
$744$	−145.185	−5.32273
$745$	−28.2349	−1.03445
$746$	64.5139	2.36202
$747$	113.013	4.13492
$748$	25.8217	0.944135
$749$	17.1826	0.627840
$750$	218.582	7.98150
$751$	16.8876	0.616237	0.308118	−	0.951348i	$-0.400301\pi$
$751$	16.8876	0.616237	0.308118	+	0.951348i	$0.400301\pi$
$752$	−4.82338	−0.175890
$753$	56.7152	2.06682
$754$	6.14729	0.223871
$755$	88.7885	3.23134
$756$	184.982	6.72772
$757$	−30.1863	−1.09714	−0.548570	−	0.836104i	$-0.684828\pi$
$757$	−30.1863	−1.09714	−0.548570	+	0.836104i	$0.684828\pi$
$758$	−57.8282	−2.10041
$759$	16.2861	0.591147
$760$	−30.6490	−1.11176
$761$	2.86643	0.103908	0.0519539	−	0.998649i	$-0.483455\pi$
$761$	2.86643	0.103908	0.0519539	+	0.998649i	$0.483455\pi$
$762$	−75.9778	−2.75239
$763$	−37.4626	−1.35624
$764$	−4.82031	−0.174393
$765$	165.346	5.97810
$766$	53.8034	1.94399
$767$	18.8947	0.682248
$768$	−38.8935	−1.40345
$769$	−16.3324	−0.588962	−0.294481	−	0.955657i	$-0.595147\pi$
$769$	−16.3324	−0.588962	−0.294481	+	0.955657i	$0.595147\pi$
$770$	−23.2291	−0.837119
$771$	−77.7479	−2.80002
$772$	−32.1650	−1.15764
$773$	−31.5413	−1.13446	−0.567231	−	0.823559i	$-0.691985\pi$
$773$	−31.5413	−1.13446	−0.567231	+	0.823559i	$0.691985\pi$
$774$	−167.958	−6.03713
$775$	62.3995	2.24146
$776$	−82.2276	−2.95180
$777$	−34.7713	−1.24741
$778$	13.9016	0.498397
$779$	0.00862364	0.000308974 0
$780$	216.637	7.75685
$781$	0.603428	0.0215924
$782$	−56.6547	−2.02597
$783$	13.3663	0.477672
$784$	−29.4130	−1.05046
$785$	8.78900	0.313693
$786$	141.449	5.04530
$787$	2.52873	0.0901395	0.0450698	−	0.998984i	$-0.485649\pi$
$787$	2.52873	0.0901395	0.0450698	+	0.998984i	$0.485649\pi$
$788$	1.81956	0.0648192
$789$	−35.6997	−1.27094
$790$	−176.580	−6.28244
$791$	13.8124	0.491113
$792$	70.1147	2.49142
$793$	−15.6924	−0.557254
$794$	18.1001	0.642348
$795$	−30.7740	−1.09144
$796$	62.4668	2.21408
$797$	7.84016	0.277713	0.138856	−	0.990313i	$-0.455657\pi$
$797$	7.84016	0.277713	0.138856	+	0.990313i	$0.455657\pi$
$798$	18.0935	0.640504
$799$	2.28609	0.0808760
$800$	−129.722	−4.58637
$801$	−73.2955	−2.58977
$802$	−31.1442	−1.09974
$803$	17.6030	0.621196
$804$	−158.198	−5.57921
$805$	36.2049	1.27605
$806$	47.9005	1.68722
$807$	−63.1475	−2.22290
$808$	142.881	5.02654
$809$	−35.8976	−1.26209	−0.631047	−	0.775745i	$-0.717374\pi$
$809$	−35.8976	−1.26209	−0.631047	+	0.775745i	$0.717374\pi$
$810$	394.822	13.8726
$811$	−36.7260	−1.28962	−0.644812	−	0.764341i	$-0.723064\pi$
$811$	−36.7260	−1.28962	−0.644812	+	0.764341i	$0.723064\pi$
$812$	7.18217	0.252045
$813$	−56.5275	−1.98251
$814$	−14.3771	−0.503916
$815$	99.5570	3.48733
$816$	−168.743	−5.90719
$817$	−7.54006	−0.263793
$818$	−105.573	−3.69127
$819$	−55.9471	−1.95495
$820$	0.169820	0.00593037
$821$	−17.6110	−0.614626	−0.307313	−	0.951608i	$-0.599430\pi$
$821$	−17.6110	−0.614626	−0.307313	+	0.951608i	$0.599430\pi$
$822$	−59.3266	−2.06925
$823$	32.9207	1.14754	0.573771	−	0.819016i	$-0.305480\pi$
$823$	32.9207	1.14754	0.573771	+	0.819016i	$0.305480\pi$
$824$	41.9691	1.46206
$825$	−40.7997	−1.42046
$826$	31.0762	1.08128
$827$	21.0601	0.732330	0.366165	−	0.930550i	$-0.380671\pi$
$827$	21.0601	0.732330	0.366165	+	0.930550i	$0.380671\pi$
$828$	−184.509	−6.41212
$829$	26.2384	0.911296	0.455648	−	0.890160i	$-0.349407\pi$
$829$	26.2384	0.911296	0.455648	+	0.890160i	$0.349407\pi$
$830$	140.642	4.88176
$831$	41.4877	1.43919
$832$	−33.0008	−1.14410
$833$	13.9406	0.483012
$834$	138.442	4.79385
$835$	7.22385	0.249992
$836$	5.31443	0.183803
$837$	104.152	3.60002
$838$	54.3115	1.87616
$839$	29.0276	1.00215	0.501073	−	0.865405i	$-0.332939\pi$
$839$	29.0276	1.00215	0.501073	+	0.865405i	$0.332939\pi$
$840$	211.032	7.28128
$841$	−28.4810	−0.982105
$842$	84.1788	2.90099
$843$	−79.6622	−2.74371
$844$	−4.90531	−0.168848
$845$	9.85561	0.339043
$846$	10.4807	0.360334
$847$	−19.9713	−0.686224
$848$	23.1968	0.796582
$849$	68.2137	2.34109
$850$	141.931	4.86818
$851$	22.4081	0.768139
$852$	−9.25580	−0.317099
$853$	4.01959	0.137628	0.0688140	−	0.997630i	$-0.478078\pi$
$853$	4.01959	0.137628	0.0688140	+	0.997630i	$0.478078\pi$
$854$	−25.8094	−0.883181
$855$	34.0303	1.16381
$856$	−64.5439	−2.20606
$857$	13.7857	0.470910	0.235455	−	0.971885i	$-0.424342\pi$
$857$	13.7857	0.470910	0.235455	+	0.971885i	$0.424342\pi$
$858$	−31.3196	−1.06923
$859$	2.42563	0.0827615	0.0413808	−	0.999143i	$-0.486824\pi$
$859$	2.42563	0.0827615	0.0413808	+	0.999143i	$0.486824\pi$
$860$	−148.482	−5.06318
$861$	−0.0593775	−0.00202358
$862$	101.633	3.46162
$863$	49.7920	1.69494	0.847469	−	0.530844i	$-0.178125\pi$
$863$	49.7920	1.69494	0.847469	+	0.530844i	$0.178125\pi$
$864$	−216.521	−7.36619
$865$	−43.4832	−1.47847
$866$	4.77400	0.162227
$867$	22.3858	0.760260
$868$	55.9644	1.89956
$869$	18.1346	0.615175
$870$	25.7456	0.872858
$871$	30.9133	1.04746
$872$	140.722	4.76545
$873$	91.2992	3.09001
$874$	−11.6602	−0.394413
$875$	−49.9036	−1.68705
$876$	−270.007	−9.12269
$877$	−17.8249	−0.601905	−0.300952	−	0.953639i	$-0.597305\pi$
$877$	−17.8249	−0.601905	−0.300952	+	0.953639i	$0.597305\pi$
$878$	2.28872	0.0772405
$879$	−84.2294	−2.84099
$880$	44.5869	1.50302
$881$	−12.4497	−0.419441	−0.209720	−	0.977761i	$-0.567255\pi$
$881$	−12.4497	−0.419441	−0.209720	+	0.977761i	$0.567255\pi$
$882$	63.9114	2.15201
$883$	−41.1992	−1.38646	−0.693232	−	0.720714i	$-0.743814\pi$
$883$	−41.1992	−1.38646	−0.693232	+	0.720714i	$0.743814\pi$
$884$	77.3960	2.60311
$885$	79.1333	2.66004
$886$	−3.31145	−0.111250
$887$	1.45448	0.0488367	0.0244183	−	0.999702i	$-0.492227\pi$
$887$	1.45448	0.0488367	0.0244183	+	0.999702i	$0.492227\pi$
$888$	130.613	4.38307
$889$	17.3462	0.581772
$890$	−91.2149	−3.05753
$891$	−40.5479	−1.35840
$892$	−52.7439	−1.76600
$893$	0.470506	0.0157449
$894$	−62.6121	−2.09406
$895$	−86.0578	−2.87660
$896$	−6.84063	−0.228529
$897$	48.8146	1.62987
$898$	29.2945	0.977571
$899$	4.04384	0.134870
$900$	462.229	15.4076
$901$	−10.9944	−0.366276
$902$	−0.0245512	−0.000817464 0
$903$	51.9166	1.72768
$904$	−51.8842	−1.72564
$905$	−6.33941	−0.210729
$906$	196.892	6.54130
$907$	−55.5778	−1.84543	−0.922715	−	0.385482i	$-0.874035\pi$
$907$	−55.5778	−1.84543	−0.922715	+	0.385482i	$0.874035\pi$
$908$	51.8293	1.72002
$909$	−158.644	−5.26189
$910$	−69.6251	−2.30805
$911$	13.6113	0.450963	0.225481	−	0.974247i	$-0.427604\pi$
$911$	13.6113	0.450963	0.225481	+	0.974247i	$0.427604\pi$
$912$	−34.7294	−1.15001
$913$	−14.4438	−0.478021
$914$	27.1744	0.898851
$915$	−65.7218	−2.17269
$916$	47.0686	1.55519
$917$	−32.2935	−1.06643
$918$	236.898	7.81881
$919$	−35.9614	−1.18626	−0.593128	−	0.805108i	$-0.702107\pi$
$919$	−35.9614	−1.18626	−0.593128	+	0.805108i	$0.702107\pi$
$920$	−135.998	−4.48371
$921$	−21.9394	−0.722927
$922$	8.22027	0.270720
$923$	1.80867	0.0595331
$924$	−36.5922	−1.20379
$925$	−56.1364	−1.84575
$926$	−3.55618	−0.116863
$927$	−46.5993	−1.53052
$928$	−8.40672	−0.275964
$929$	2.77699	0.0911102	0.0455551	−	0.998962i	$-0.485494\pi$
$929$	2.77699	0.0911102	0.0455551	+	0.998962i	$0.485494\pi$
$930$	200.613	6.57836
$931$	2.86914	0.0940324
$932$	84.5932	2.77094
$933$	−37.3309	−1.22216
$934$	−6.73616	−0.220414
$935$	−21.1324	−0.691104
$936$	210.156	6.86917
$937$	−19.3909	−0.633472	−0.316736	−	0.948514i	$-0.602587\pi$
$937$	−19.3909	−0.633472	−0.316736	+	0.948514i	$0.602587\pi$
$938$	50.8433	1.66009
$939$	56.9864	1.85968
$940$	9.26537	0.302203
$941$	25.5358	0.832444	0.416222	−	0.909263i	$-0.363354\pi$
$941$	25.5358	0.832444	0.416222	+	0.909263i	$0.363354\pi$
$942$	19.4900	0.635018
$943$	0.0382654	0.00124609
$944$	−59.6490	−1.94141
$945$	−151.389	−4.92467
$946$	21.4662	0.697928
$947$	19.5951	0.636757	0.318378	−	0.947964i	$-0.396862\pi$
$947$	19.5951	0.636757	0.318378	+	0.947964i	$0.396862\pi$
$948$	−278.161	−9.03426
$949$	52.7618	1.71272
$950$	29.2111	0.947732
$951$	80.8787	2.62267
$952$	75.3934	2.44351
$953$	−23.6811	−0.767108	−0.383554	−	0.923519i	$-0.625300\pi$
$953$	−23.6811	−0.767108	−0.383554	+	0.923519i	$0.625300\pi$
$954$	−50.4044	−1.63190
$955$	3.94493	0.127655
$956$	−107.561	−3.47878
$957$	−2.64405	−0.0854701
$958$	−51.3649	−1.65952
$959$	13.5446	0.437378
$960$	−138.211	−4.46075
$961$	0.510127	0.0164557
$962$	−43.0927	−1.38937
$963$	71.6646	2.30936
$964$	38.5799	1.24258
$965$	26.3238	0.847392
$966$	80.2858	2.58315
$967$	34.2050	1.09996	0.549979	−	0.835178i	$-0.314636\pi$
$967$	34.2050	1.09996	0.549979	+	0.835178i	$0.314636\pi$
$968$	75.0192	2.41121
$969$	16.4604	0.528783
$970$	113.620	3.64812
$971$	−26.3655	−0.846108	−0.423054	−	0.906104i	$-0.639042\pi$
$971$	−26.3655	−0.846108	−0.423054	+	0.906104i	$0.639042\pi$
$972$	348.909	11.1913
$973$	−31.6071	−1.01328
$974$	−95.3019	−3.05367
$975$	−122.290	−3.91641
$976$	49.5397	1.58573
$977$	−8.32698	−0.266404	−0.133202	−	0.991089i	$-0.542526\pi$
$977$	−8.32698	−0.266404	−0.133202	+	0.991089i	$0.542526\pi$
$978$	220.772	7.05951
$979$	9.36769	0.299393
$980$	56.5003	1.80483
$981$	−156.247	−4.98859
$982$	87.1350	2.78059
$983$	47.6393	1.51946	0.759729	−	0.650240i	$-0.225332\pi$
$983$	47.6393	1.51946	0.759729	+	0.650240i	$0.225332\pi$
$984$	0.223042	0.00711032
$985$	−1.48912	−0.0474474
$986$	9.19790	0.292921
$987$	−3.23963	−0.103119
$988$	15.9291	0.506771
$989$	−33.4572	−1.06388
$990$	−96.8829	−3.07914
$991$	51.8802	1.64803	0.824015	−	0.566568i	$-0.191729\pi$
$991$	51.8802	1.64803	0.824015	+	0.566568i	$0.191729\pi$
$992$	−65.5063	−2.07983
$993$	63.4132	2.01236
$994$	2.97473	0.0943527
$995$	−51.1227	−1.62070
$996$	221.549	7.02006
$997$	8.50370	0.269315	0.134657	−	0.990892i	$-0.457007\pi$
$997$	8.50370	0.269315	0.134657	+	0.990892i	$0.457007\pi$
$998$	−3.18158	−0.100711
$999$	−93.6982	−2.96448

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.d.1.5	✓	75

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.d.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-2.62780 q^{2} +3.38775 q^{3} +4.90531 q^{4} -4.01450 q^{5} -8.90232 q^{6} +2.03245 q^{7} -7.63457 q^{8} +8.47685 q^{9} +O(q^{10})\)	\(q-2.62780 q^{2} +3.38775 q^{3} +4.90531 q^{4} -4.01450 q^{5} -8.90232 q^{6} +2.03245 q^{7} -7.63457 q^{8} +8.47685 q^{9} +10.5493 q^{10} -1.08340 q^{11} +16.6180 q^{12} -3.24731 q^{13} -5.34087 q^{14} -13.6001 q^{15} +10.2515 q^{16} -4.85879 q^{17} -22.2754 q^{18} -1.00000 q^{19} -19.6924 q^{20} +6.88543 q^{21} +2.84696 q^{22} -4.43727 q^{23} -25.8640 q^{24} +11.1162 q^{25} +8.53326 q^{26} +18.5542 q^{27} +9.96981 q^{28} +0.720392 q^{29} +35.7383 q^{30} +5.61339 q^{31} -11.6697 q^{32} -3.67030 q^{33} +12.7679 q^{34} -8.15927 q^{35} +41.5816 q^{36} -5.04997 q^{37} +2.62780 q^{38} -11.0011 q^{39} +30.6490 q^{40} -0.00862364 q^{41} -18.0935 q^{42} +7.54006 q^{43} -5.31443 q^{44} -34.0303 q^{45} +11.6602 q^{46} -0.470506 q^{47} +34.7294 q^{48} -2.86914 q^{49} -29.2111 q^{50} -16.4604 q^{51} -15.9291 q^{52} +2.26278 q^{53} -48.7566 q^{54} +4.34932 q^{55} -15.5169 q^{56} -3.38775 q^{57} -1.89304 q^{58} -5.81858 q^{59} -66.7128 q^{60} +4.83244 q^{61} -14.7508 q^{62} +17.2288 q^{63} +10.1625 q^{64} +13.0363 q^{65} +9.64479 q^{66} -9.51968 q^{67} -23.8339 q^{68} -15.0324 q^{69} +21.4409 q^{70} -0.556975 q^{71} -64.7171 q^{72} -16.2479 q^{73} +13.2703 q^{74} +37.6589 q^{75} -4.90531 q^{76} -2.20196 q^{77} +28.9085 q^{78} -16.7386 q^{79} -41.1545 q^{80} +37.4264 q^{81} +0.0226612 q^{82} +13.3319 q^{83} +33.7752 q^{84} +19.5056 q^{85} -19.8137 q^{86} +2.44051 q^{87} +8.27132 q^{88} -8.64655 q^{89} +89.4247 q^{90} -6.59999 q^{91} -21.7662 q^{92} +19.0168 q^{93} +1.23639 q^{94} +4.01450 q^{95} -39.5339 q^{96} +10.7704 q^{97} +7.53953 q^{98} -9.18384 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 75 q - 11 q^{2} - 4 q^{3} + 67 q^{4} - 18 q^{5} - 15 q^{6} - 19 q^{7} - 30 q^{8} + 57 q^{9}+O(q^{10}) \) 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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