LMFDB - Embedded newform 4009.2.a.c.1.18

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:	$71$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
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)

$f(q)$	$=$	$q-1.89239 q^{2} -0.431235 q^{3} +1.58115 q^{4} +3.97235 q^{5} +0.816065 q^{6} -2.66937 q^{7} +0.792636 q^{8} -2.81404 q^{9} +O(q^{10})$	\(q-1.89239 q^{2} -0.431235 q^{3} +1.58115 q^{4} +3.97235 q^{5} +0.816065 q^{6} -2.66937 q^{7} +0.792636 q^{8} -2.81404 q^{9} -7.51724 q^{10} -2.92001 q^{11} -0.681845 q^{12} +1.55018 q^{13} +5.05149 q^{14} -1.71301 q^{15} -4.66227 q^{16} -1.96101 q^{17} +5.32526 q^{18} +1.00000 q^{19} +6.28087 q^{20} +1.15112 q^{21} +5.52581 q^{22} -6.26871 q^{23} -0.341812 q^{24} +10.7796 q^{25} -2.93355 q^{26} +2.50721 q^{27} -4.22066 q^{28} +8.06876 q^{29} +3.24170 q^{30} -4.58316 q^{31} +7.23757 q^{32} +1.25921 q^{33} +3.71099 q^{34} -10.6037 q^{35} -4.44940 q^{36} +8.58707 q^{37} -1.89239 q^{38} -0.668492 q^{39} +3.14863 q^{40} +8.34308 q^{41} -2.17838 q^{42} +6.30592 q^{43} -4.61697 q^{44} -11.1783 q^{45} +11.8628 q^{46} +3.37259 q^{47} +2.01053 q^{48} +0.125532 q^{49} -20.3992 q^{50} +0.845653 q^{51} +2.45106 q^{52} -3.53695 q^{53} -4.74463 q^{54} -11.5993 q^{55} -2.11584 q^{56} -0.431235 q^{57} -15.2693 q^{58} +1.15247 q^{59} -2.70853 q^{60} -2.16887 q^{61} +8.67313 q^{62} +7.51170 q^{63} -4.37177 q^{64} +6.15787 q^{65} -2.38292 q^{66} -7.38999 q^{67} -3.10064 q^{68} +2.70328 q^{69} +20.0663 q^{70} -3.50612 q^{71} -2.23051 q^{72} -12.4025 q^{73} -16.2501 q^{74} -4.64852 q^{75} +1.58115 q^{76} +7.79459 q^{77} +1.26505 q^{78} -10.7182 q^{79} -18.5202 q^{80} +7.36091 q^{81} -15.7884 q^{82} +11.4521 q^{83} +1.82010 q^{84} -7.78980 q^{85} -11.9333 q^{86} -3.47953 q^{87} -2.31451 q^{88} +2.08592 q^{89} +21.1538 q^{90} -4.13801 q^{91} -9.91174 q^{92} +1.97642 q^{93} -6.38226 q^{94} +3.97235 q^{95} -3.12109 q^{96} -3.23881 q^{97} -0.237556 q^{98} +8.21702 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) $ 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * 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 $

$2$	−1.89239	−1.33812	−0.669061	−	0.743207i	$-0.733304\pi$
$2$	−1.89239	−1.33812	−0.669061	+	0.743207i	$0.733304\pi$
$3$	−0.431235	−0.248973	−0.124487	−	0.992221i	$-0.539728\pi$
$3$	−0.431235	−0.248973	−0.124487	+	0.992221i	$0.539728\pi$
$4$	1.58115	0.790573
$5$	3.97235	1.77649	0.888245	−	0.459371i	$-0.151925\pi$
$5$	3.97235	1.77649	0.888245	+	0.459371i	$0.151925\pi$
$6$	0.816065	0.333157
$7$	−2.66937	−1.00893	−0.504463	−	0.863433i	$-0.668310\pi$
$7$	−2.66937	−1.00893	−0.504463	+	0.863433i	$0.668310\pi$
$8$	0.792636	0.280239
$9$	−2.81404	−0.938012
$10$	−7.51724	−2.37716
$11$	−2.92001	−0.880417	−0.440208	−	0.897896i	$-0.645095\pi$
$11$	−2.92001	−0.880417	−0.440208	+	0.897896i	$0.645095\pi$
$12$	−0.681845	−0.196832
$13$	1.55018	0.429943	0.214972	−	0.976620i	$-0.431034\pi$
$13$	1.55018	0.429943	0.214972	+	0.976620i	$0.431034\pi$
$14$	5.05149	1.35007
$15$	−1.71301	−0.442299
$16$	−4.66227	−1.16557
$17$	−1.96101	−0.475614	−0.237807	−	0.971312i	$-0.576429\pi$
$17$	−1.96101	−0.475614	−0.237807	+	0.971312i	$0.576429\pi$
$18$	5.32526	1.25518
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	6.28087	1.40444
$21$	1.15112	0.251196
$22$	5.52581	1.17811
$23$	−6.26871	−1.30712	−0.653558	−	0.756877i	$-0.726724\pi$
$23$	−6.26871	−1.30712	−0.653558	+	0.756877i	$0.726724\pi$
$24$	−0.341812	−0.0697721
$25$	10.7796	2.15591
$26$	−2.93355	−0.575317
$27$	2.50721	0.482513
$28$	−4.22066	−0.797630
$29$	8.06876	1.49833	0.749166	−	0.662383i	$-0.230455\pi$
$29$	8.06876	1.49833	0.749166	+	0.662383i	$0.230455\pi$
$30$	3.24170	0.591850
$31$	−4.58316	−0.823159	−0.411580	−	0.911374i	$-0.635023\pi$
$31$	−4.58316	−0.823159	−0.411580	+	0.911374i	$0.635023\pi$
$32$	7.23757	1.27943
$33$	1.25921	0.219200
$34$	3.71099	0.636430
$35$	−10.6037	−1.79235
$36$	−4.44940	−0.741567
$37$	8.58707	1.41171	0.705853	−	0.708358i	$-0.250564\pi$
$37$	8.58707	1.41171	0.705853	+	0.708358i	$0.250564\pi$
$38$	−1.89239	−0.306986
$39$	−0.668492	−0.107044
$40$	3.14863	0.497842
$41$	8.34308	1.30297	0.651485	−	0.758662i	$-0.274146\pi$
$41$	8.34308	1.30297	0.651485	+	0.758662i	$0.274146\pi$
$42$	−2.17838	−0.336131
$43$	6.30592	0.961643	0.480822	−	0.876818i	$-0.340338\pi$
$43$	6.30592	0.961643	0.480822	+	0.876818i	$0.340338\pi$
$44$	−4.61697	−0.696034
$45$	−11.1783	−1.66637
$46$	11.8628	1.74908
$47$	3.37259	0.491943	0.245971	−	0.969277i	$-0.420893\pi$
$47$	3.37259	0.491943	0.245971	+	0.969277i	$0.420893\pi$
$48$	2.01053	0.290195
$49$	0.125532	0.0179331
$50$	−20.3992	−2.88488
$51$	0.845653	0.118415
$52$	2.45106	0.339901
$53$	−3.53695	−0.485837	−0.242919	−	0.970047i	$-0.578105\pi$
$53$	−3.53695	−0.485837	−0.242919	+	0.970047i	$0.578105\pi$
$54$	−4.74463	−0.645662
$55$	−11.5993	−1.56405
$56$	−2.11584	−0.282741
$57$	−0.431235	−0.0571184
$58$	−15.2693	−2.00495
$59$	1.15247	0.150039	0.0750194	−	0.997182i	$-0.476098\pi$
$59$	1.15247	0.150039	0.0750194	+	0.997182i	$0.476098\pi$
$60$	−2.70853	−0.349669
$61$	−2.16887	−0.277695	−0.138848	−	0.990314i	$-0.544340\pi$
$61$	−2.16887	−0.277695	−0.138848	+	0.990314i	$0.544340\pi$
$62$	8.67313	1.10149
$63$	7.51170	0.946386
$64$	−4.37177	−0.546472
$65$	6.15787	0.763789
$66$	−2.38292	−0.293317
$67$	−7.38999	−0.902831	−0.451416	−	0.892314i	$-0.649081\pi$
$67$	−7.38999	−0.902831	−0.451416	+	0.892314i	$0.649081\pi$
$68$	−3.10064	−0.376007
$69$	2.70328	0.325437
$70$	20.0663	2.39838
$71$	−3.50612	−0.416100	−0.208050	−	0.978118i	$-0.566712\pi$
$71$	−3.50612	−0.416100	−0.208050	+	0.978118i	$0.566712\pi$
$72$	−2.23051	−0.262868
$73$	−12.4025	−1.45160	−0.725802	−	0.687904i	$-0.758531\pi$
$73$	−12.4025	−1.45160	−0.725802	+	0.687904i	$0.758531\pi$
$74$	−16.2501	−1.88904
$75$	−4.64852	−0.536765
$76$	1.58115	0.181370
$77$	7.79459	0.888276
$78$	1.26505	0.143239
$79$	−10.7182	−1.20589	−0.602944	−	0.797784i	$-0.706006\pi$
$79$	−10.7182	−1.20589	−0.602944	+	0.797784i	$0.706006\pi$
$80$	−18.5202	−2.07062
$81$	7.36091	0.817879
$82$	−15.7884	−1.74353
$83$	11.4521	1.25704	0.628518	−	0.777795i	$-0.283662\pi$
$83$	11.4521	1.25704	0.628518	+	0.777795i	$0.283662\pi$
$84$	1.82010	0.198589
$85$	−7.78980	−0.844923
$86$	−11.9333	−1.28680
$87$	−3.47953	−0.373045
$88$	−2.31451	−0.246727
$89$	2.08592	0.221107	0.110554	−	0.993870i	$-0.464738\pi$
$89$	2.08592	0.221107	0.110554	+	0.993870i	$0.464738\pi$
$90$	21.1538	2.22981
$91$	−4.13801	−0.433781
$92$	−9.91174	−1.03337
$93$	1.97642	0.204945
$94$	−6.38226	−0.658280
$95$	3.97235	0.407555
$96$	−3.12109	−0.318545
$97$	−3.23881	−0.328852	−0.164426	−	0.986389i	$-0.552577\pi$
$97$	−3.23881	−0.328852	−0.164426	+	0.986389i	$0.552577\pi$
$98$	−0.237556	−0.0239967
$99$	8.21702	0.825842
$100$	17.0441	1.70441
$101$	−8.66946	−0.862643	−0.431322	−	0.902198i	$-0.641953\pi$
$101$	−8.66946	−0.862643	−0.431322	+	0.902198i	$0.641953\pi$
$102$	−1.60031	−0.158454
$103$	−17.3840	−1.71289	−0.856447	−	0.516234i	$-0.827333\pi$
$103$	−17.3840	−1.71289	−0.856447	+	0.516234i	$0.827333\pi$
$104$	1.22873	0.120487
$105$	4.57267	0.446247
$106$	6.69329	0.650110
$107$	−13.7047	−1.32488	−0.662440	−	0.749115i	$-0.730479\pi$
$107$	−13.7047	−1.32488	−0.662440	+	0.749115i	$0.730479\pi$
$108$	3.96427	0.381462
$109$	8.52053	0.816118	0.408059	−	0.912955i	$-0.366206\pi$
$109$	8.52053	0.816118	0.408059	+	0.912955i	$0.366206\pi$
$110$	21.9504	2.09289
$111$	−3.70304	−0.351477
$112$	12.4453	1.17597
$113$	−7.74204	−0.728310	−0.364155	−	0.931338i	$-0.618642\pi$
$113$	−7.74204	−0.728310	−0.364155	+	0.931338i	$0.618642\pi$
$114$	0.816065	0.0764315
$115$	−24.9015	−2.32208
$116$	12.7579	1.18454
$117$	−4.36227	−0.403292
$118$	−2.18093	−0.200770
$119$	5.23465	0.479859
$120$	−1.35780	−0.123949
$121$	−2.47353	−0.224866
$122$	4.10435	0.371591
$123$	−3.59782	−0.324405
$124$	−7.24664	−0.650768
$125$	22.9585	2.05347
$126$	−14.2151	−1.26638
$127$	−0.746775	−0.0662656	−0.0331328	−	0.999451i	$-0.510548\pi$
$127$	−0.746775	−0.0662656	−0.0331328	+	0.999451i	$0.510548\pi$
$128$	−6.20203	−0.548187
$129$	−2.71933	−0.239424
$130$	−11.6531	−1.02204
$131$	−0.288153	−0.0251761	−0.0125880	−	0.999921i	$-0.504007\pi$
$131$	−0.288153	−0.0251761	−0.0125880	+	0.999921i	$0.504007\pi$
$132$	1.99100	0.173294
$133$	−2.66937	−0.231464
$134$	13.9848	1.20810
$135$	9.95953	0.857180
$136$	−1.55436	−0.133286
$137$	−8.17721	−0.698627	−0.349313	−	0.937006i	$-0.613585\pi$
$137$	−8.17721	−0.698627	−0.349313	+	0.937006i	$0.613585\pi$
$138$	−5.11567	−0.435475
$139$	−5.55725	−0.471360	−0.235680	−	0.971831i	$-0.575732\pi$
$139$	−5.55725	−0.471360	−0.235680	+	0.971831i	$0.575732\pi$
$140$	−16.7660	−1.41698
$141$	−1.45438	−0.122481
$142$	6.63496	0.556793
$143$	−4.52655	−0.378529
$144$	13.1198	1.09332
$145$	32.0519	2.66177
$146$	23.4704	1.94242
$147$	−0.0541337	−0.00446487
$148$	13.5774	1.11606
$149$	−2.53650	−0.207798	−0.103899	−	0.994588i	$-0.533132\pi$
$149$	−2.53650	−0.207798	−0.103899	+	0.994588i	$0.533132\pi$
$150$	8.79683	0.718258
$151$	−0.252628	−0.0205586	−0.0102793	−	0.999947i	$-0.503272\pi$
$151$	−0.252628	−0.0205586	−0.0102793	+	0.999947i	$0.503272\pi$
$152$	0.792636	0.0642913
$153$	5.51834	0.446132
$154$	−14.7504	−1.18862
$155$	−18.2059	−1.46233
$156$	−1.05698	−0.0846264
$157$	−16.8917	−1.34810	−0.674050	−	0.738685i	$-0.735447\pi$
$157$	−16.8917	−1.34810	−0.674050	+	0.738685i	$0.735447\pi$
$158$	20.2830	1.61363
$159$	1.52525	0.120961
$160$	28.7502	2.27290
$161$	16.7335	1.31878
$162$	−13.9297	−1.09442
$163$	18.6982	1.46456	0.732279	−	0.681004i	$-0.238456\pi$
$163$	18.6982	1.46456	0.732279	+	0.681004i	$0.238456\pi$
$164$	13.1916	1.03009
$165$	5.00203	0.389407
$166$	−21.6719	−1.68207
$167$	14.3754	1.11240	0.556202	−	0.831047i	$-0.312258\pi$
$167$	14.3754	1.11240	0.556202	+	0.831047i	$0.312258\pi$
$168$	0.912423	0.0703949
$169$	−10.5969	−0.815149
$170$	14.7414	1.13061
$171$	−2.81404	−0.215195
$172$	9.97057	0.760249
$173$	−16.8972	−1.28467	−0.642336	−	0.766423i	$-0.722035\pi$
$173$	−16.8972	−1.28467	−0.642336	+	0.766423i	$0.722035\pi$
$174$	6.58463	0.499180
$175$	−28.7747	−2.17516
$176$	13.6139	1.02619
$177$	−0.496985	−0.0373557
$178$	−3.94738	−0.295869
$179$	−10.4442	−0.780637	−0.390319	−	0.920680i	$-0.627635\pi$
$179$	−10.4442	−0.780637	−0.390319	+	0.920680i	$0.627635\pi$
$180$	−17.6746	−1.31739
$181$	8.68371	0.645455	0.322727	−	0.946492i	$-0.395400\pi$
$181$	8.68371	0.645455	0.322727	+	0.946492i	$0.395400\pi$
$182$	7.83073	0.580452
$183$	0.935292	0.0691388
$184$	−4.96880	−0.366305
$185$	34.1109	2.50788
$186$	−3.74015	−0.274241
$187$	5.72616	0.418738
$188$	5.33255	0.388917
$189$	−6.69268	−0.486821
$190$	−7.51724	−0.545358
$191$	−20.7391	−1.50063	−0.750316	−	0.661080i	$-0.770099\pi$
$191$	−20.7391	−1.50063	−0.750316	+	0.661080i	$0.770099\pi$
$192$	1.88526	0.136057
$193$	−2.83556	−0.204108	−0.102054	−	0.994779i	$-0.532541\pi$
$193$	−2.83556	−0.204108	−0.102054	+	0.994779i	$0.532541\pi$
$194$	6.12910	0.440044
$195$	−2.65549	−0.190163
$196$	0.198484	0.0141775
$197$	0.931748	0.0663843	0.0331921	−	0.999449i	$-0.489433\pi$
$197$	0.931748	0.0663843	0.0331921	+	0.999449i	$0.489433\pi$
$198$	−15.5498	−1.10508
$199$	−4.76979	−0.338122	−0.169061	−	0.985606i	$-0.554073\pi$
$199$	−4.76979	−0.338122	−0.169061	+	0.985606i	$0.554073\pi$
$200$	8.54428	0.604172
$201$	3.18682	0.224781
$202$	16.4060	1.15432
$203$	−21.5385	−1.51171
$204$	1.33710	0.0936158
$205$	33.1416	2.31471
$206$	32.8973	2.29206
$207$	17.6404	1.22609
$208$	−7.22737	−0.501128
$209$	−2.92001	−0.201982
$210$	−8.65328	−0.597133
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	−5.59243	−0.384090
$213$	1.51196	0.103598
$214$	25.9346	1.77285
$215$	25.0493	1.70835
$216$	1.98731	0.135219
$217$	12.2341	0.830508
$218$	−16.1242	−1.09207
$219$	5.34839	0.361411
$220$	−18.3402	−1.23650
$221$	−3.03992	−0.204487
$222$	7.00761	0.470320
$223$	−21.2792	−1.42496	−0.712481	−	0.701691i	$-0.752429\pi$
$223$	−21.2792	−1.42496	−0.712481	+	0.701691i	$0.752429\pi$
$224$	−19.3197	−1.29085
$225$	−30.3341	−2.02227
$226$	14.6510	0.974568
$227$	5.41960	0.359712	0.179856	−	0.983693i	$-0.442437\pi$
$227$	5.41960	0.359712	0.179856	+	0.983693i	$0.442437\pi$
$228$	−0.681845	−0.0451563
$229$	22.6413	1.49618	0.748089	−	0.663598i	$-0.230972\pi$
$229$	22.6413	1.49618	0.748089	+	0.663598i	$0.230972\pi$
$230$	47.1234	3.10722
$231$	−3.36130	−0.221157
$232$	6.39559	0.419891
$233$	26.0017	1.70343	0.851715	−	0.524006i	$-0.175563\pi$
$233$	26.0017	1.70343	0.851715	+	0.524006i	$0.175563\pi$
$234$	8.25512	0.539654
$235$	13.3971	0.873931
$236$	1.82222	0.118617
$237$	4.62204	0.300234
$238$	−9.90600	−0.642111
$239$	2.91530	0.188575	0.0942875	−	0.995545i	$-0.469943\pi$
$239$	2.91530	0.188575	0.0942875	+	0.995545i	$0.469943\pi$
$240$	7.98654	0.515529
$241$	−18.3673	−1.18314	−0.591570	−	0.806254i	$-0.701491\pi$
$241$	−18.3673	−1.18314	−0.591570	+	0.806254i	$0.701491\pi$
$242$	4.68088	0.300898
$243$	−10.6959	−0.686144
$244$	−3.42930	−0.219538
$245$	0.498657	0.0318580
$246$	6.80849	0.434093
$247$	1.55018	0.0986357
$248$	−3.63278	−0.230682
$249$	−4.93856	−0.312969
$250$	−43.4464	−2.74779
$251$	−1.77480	−0.112025	−0.0560123	−	0.998430i	$-0.517839\pi$
$251$	−1.77480	−0.112025	−0.0560123	+	0.998430i	$0.517839\pi$
$252$	11.8771	0.748187
$253$	18.3047	1.15081
$254$	1.41319	0.0886715
$255$	3.35923	0.210363
$256$	20.4802	1.28001
$257$	29.6639	1.85038	0.925191	−	0.379502i	$-0.123905\pi$
$257$	29.6639	1.85038	0.925191	+	0.379502i	$0.123905\pi$
$258$	5.14603	0.320378
$259$	−22.9221	−1.42431
$260$	9.73649	0.603831
$261$	−22.7058	−1.40545
$262$	0.545299	0.0336887
$263$	−18.5302	−1.14262	−0.571311	−	0.820733i	$-0.693565\pi$
$263$	−18.5302	−1.14262	−0.571311	+	0.820733i	$0.693565\pi$
$264$	0.998096	0.0614285
$265$	−14.0500	−0.863085
$266$	5.05149	0.309727
$267$	−0.899522	−0.0550499
$268$	−11.6847	−0.713754
$269$	−12.7956	−0.780161	−0.390080	−	0.920781i	$-0.627553\pi$
$269$	−12.7956	−0.780161	−0.390080	+	0.920781i	$0.627553\pi$
$270$	−18.8473	−1.14701
$271$	−18.6582	−1.13340	−0.566702	−	0.823923i	$-0.691781\pi$
$271$	−18.6582	−1.13340	−0.566702	+	0.823923i	$0.691781\pi$
$272$	9.14274	0.554360
$273$	1.78445	0.108000
$274$	15.4745	0.934848
$275$	−31.4765	−1.89810
$276$	4.27428	0.257282
$277$	−9.93806	−0.597120	−0.298560	−	0.954391i	$-0.596506\pi$
$277$	−9.93806	−0.597120	−0.298560	+	0.954391i	$0.596506\pi$
$278$	10.5165	0.630737
$279$	12.8972	0.772134
$280$	−8.40485	−0.502286
$281$	−2.85464	−0.170294	−0.0851468	−	0.996368i	$-0.527136\pi$
$281$	−2.85464	−0.170294	−0.0851468	+	0.996368i	$0.527136\pi$
$282$	2.75225	0.163894
$283$	7.89406	0.469253	0.234627	−	0.972086i	$-0.424613\pi$
$283$	7.89406	0.469253	0.234627	+	0.972086i	$0.424613\pi$
$284$	−5.54369	−0.328958
$285$	−1.71301	−0.101470
$286$	8.56601	0.506519
$287$	−22.2708	−1.31460
$288$	−20.3668	−1.20012
$289$	−13.1545	−0.773792
$290$	−60.6548	−3.56177
$291$	1.39669	0.0818753
$292$	−19.6102	−1.14760
$293$	−29.3435	−1.71427	−0.857133	−	0.515096i	$-0.827756\pi$
$293$	−29.3435	−1.71427	−0.857133	+	0.515096i	$0.827756\pi$
$294$	0.102442	0.00597455
$295$	4.57802	0.266542
$296$	6.80642	0.395615
$297$	−7.32110	−0.424813
$298$	4.80004	0.278059
$299$	−9.71764	−0.561985
$300$	−7.34999	−0.424352
$301$	−16.8328	−0.970227
$302$	0.478071	0.0275099
$303$	3.73857	0.214775
$304$	−4.66227	−0.267399
$305$	−8.61551	−0.493323
$306$	−10.4429	−0.596979
$307$	21.8870	1.24916	0.624579	−	0.780962i	$-0.285271\pi$
$307$	21.8870	1.24916	0.624579	+	0.780962i	$0.285271\pi$
$308$	12.3244	0.702247
$309$	7.49658	0.426465
$310$	34.4527	1.95678
$311$	6.66403	0.377882	0.188941	−	0.981988i	$-0.439494\pi$
$311$	6.66403	0.377882	0.188941	+	0.981988i	$0.439494\pi$
$312$	−0.529871	−0.0299980
$313$	9.76104	0.551726	0.275863	−	0.961197i	$-0.411036\pi$
$313$	9.76104	0.551726	0.275863	+	0.961197i	$0.411036\pi$
$314$	31.9656	1.80392
$315$	29.8391	1.68124
$316$	−16.9470	−0.953342
$317$	−7.69015	−0.431922	−0.215961	−	0.976402i	$-0.569288\pi$
$317$	−7.69015	−0.431922	−0.215961	+	0.976402i	$0.569288\pi$
$318$	−2.88638	−0.161860
$319$	−23.5609	−1.31916
$320$	−17.3662	−0.970801
$321$	5.90993	0.329860
$322$	−31.6663	−1.76469
$323$	−1.96101	−0.109113
$324$	11.6387	0.646593
$325$	16.7103	0.926921
$326$	−35.3844	−1.95976
$327$	−3.67435	−0.203192
$328$	6.61302	0.365143
$329$	−9.00268	−0.496334
$330$	−9.46579	−0.521075
$331$	−21.2827	−1.16980	−0.584901	−	0.811104i	$-0.698867\pi$
$331$	−21.2827	−1.16980	−0.584901	+	0.811104i	$0.698867\pi$
$332$	18.1075	0.993779
$333$	−24.1643	−1.32420
$334$	−27.2039	−1.48853
$335$	−29.3556	−1.60387
$336$	−5.36685	−0.292786
$337$	−6.59326	−0.359158	−0.179579	−	0.983744i	$-0.557473\pi$
$337$	−6.59326	−0.359158	−0.179579	+	0.983744i	$0.557473\pi$
$338$	20.0536	1.09077
$339$	3.33864	0.181330
$340$	−12.3168	−0.667973
$341$	13.3829	0.724724
$342$	5.32526	0.287957
$343$	18.3505	0.990834
$344$	4.99830	0.269490
$345$	10.7384	0.578135
$346$	31.9762	1.71905
$347$	−34.8761	−1.87224	−0.936122	−	0.351675i	$-0.885612\pi$
$347$	−34.8761	−1.87224	−0.936122	+	0.351675i	$0.885612\pi$
$348$	−5.50164	−0.294919
$349$	−28.0182	−1.49978	−0.749891	−	0.661561i	$-0.769894\pi$
$349$	−28.0182	−1.49978	−0.749891	+	0.661561i	$0.769894\pi$
$350$	54.4529	2.91063
$351$	3.88664	0.207453
$352$	−21.1338	−1.12643
$353$	14.3987	0.766365	0.383183	−	0.923673i	$-0.374828\pi$
$353$	14.3987	0.766365	0.383183	+	0.923673i	$0.374828\pi$
$354$	0.940490	0.0499865
$355$	−13.9275	−0.739198
$356$	3.29815	0.174802
$357$	−2.25736	−0.119472
$358$	19.7645	1.04459
$359$	−31.1781	−1.64552	−0.822759	−	0.568390i	$-0.807567\pi$
$359$	−31.1781	−1.64552	−0.822759	+	0.568390i	$0.807567\pi$
$360$	−8.86036	−0.466982
$361$	1.00000	0.0526316
$362$	−16.4330	−0.863698
$363$	1.06667	0.0559856
$364$	−6.54279	−0.342936
$365$	−49.2671	−2.57876
$366$	−1.76994	−0.0925162
$367$	23.0055	1.20088	0.600438	−	0.799671i	$-0.294993\pi$
$367$	23.0055	1.20088	0.600438	+	0.799671i	$0.294993\pi$
$368$	29.2264	1.52353
$369$	−23.4777	−1.22220
$370$	−64.5511	−3.35585
$371$	9.44142	0.490174
$372$	3.12500	0.162024
$373$	−9.84297	−0.509650	−0.254825	−	0.966987i	$-0.582018\pi$
$373$	−9.84297	−0.509650	−0.254825	+	0.966987i	$0.582018\pi$
$374$	−10.8361	−0.560324
$375$	−9.90049	−0.511259
$376$	2.67324	0.137862
$377$	12.5080	0.644197
$378$	12.6652	0.651426
$379$	19.9138	1.02290	0.511452	−	0.859312i	$-0.329108\pi$
$379$	19.9138	1.02290	0.511452	+	0.859312i	$0.329108\pi$
$380$	6.28087	0.322202
$381$	0.322035	0.0164984
$382$	39.2466	2.00803
$383$	−13.4333	−0.686410	−0.343205	−	0.939260i	$-0.611513\pi$
$383$	−13.4333	−0.686410	−0.343205	+	0.939260i	$0.611513\pi$
$384$	2.67453	0.136484
$385$	30.9629	1.57801
$386$	5.36598	0.273121
$387$	−17.7451	−0.902033
$388$	−5.12104	−0.259981
$389$	−6.83937	−0.346770	−0.173385	−	0.984854i	$-0.555470\pi$
$389$	−6.83937	−0.346770	−0.173385	+	0.984854i	$0.555470\pi$
$390$	5.02522	0.254462
$391$	12.2930	0.621682
$392$	0.0995011	0.00502557
$393$	0.124262	0.00626817
$394$	−1.76323	−0.0888303
$395$	−42.5763	−2.14225
$396$	12.9923	0.652888
$397$	18.3971	0.923324	0.461662	−	0.887056i	$-0.347253\pi$
$397$	18.3971	0.923324	0.461662	+	0.887056i	$0.347253\pi$
$398$	9.02631	0.452448
$399$	1.15112	0.0576283
$400$	−50.2573	−2.51286
$401$	−8.37620	−0.418288	−0.209144	−	0.977885i	$-0.567068\pi$
$401$	−8.37620	−0.418288	−0.209144	+	0.977885i	$0.567068\pi$
$402$	−6.03071	−0.300785
$403$	−7.10473	−0.353912
$404$	−13.7077	−0.681983
$405$	29.2401	1.45295
$406$	40.7593	2.02285
$407$	−25.0744	−1.24289
$408$	0.670295	0.0331846
$409$	−24.9165	−1.23204	−0.616020	−	0.787731i	$-0.711256\pi$
$409$	−24.9165	−1.23204	−0.616020	+	0.787731i	$0.711256\pi$
$410$	−62.7169	−3.09737
$411$	3.52630	0.173939
$412$	−27.4866	−1.35417
$413$	−3.07637	−0.151378
$414$	−33.3825	−1.64066
$415$	45.4919	2.23311
$416$	11.2195	0.550084
$417$	2.39648	0.117356
$418$	5.52581	0.270276
$419$	−3.30379	−0.161401	−0.0807005	−	0.996738i	$-0.525716\pi$
$419$	−3.30379	−0.161401	−0.0807005	+	0.996738i	$0.525716\pi$
$420$	7.23006	0.352791
$421$	−17.4464	−0.850287	−0.425144	−	0.905126i	$-0.639776\pi$
$421$	−17.4464	−0.850287	−0.425144	+	0.905126i	$0.639776\pi$
$422$	−1.89239	−0.0921202
$423$	−9.49059	−0.461448
$424$	−2.80351	−0.136151
$425$	−21.1388	−1.02538
$426$	−2.86122	−0.138627
$427$	5.78952	0.280174
$428$	−21.6691	−1.04741
$429$	1.95201	0.0942437
$430$	−47.4031	−2.28598
$431$	29.6915	1.43019	0.715094	−	0.699028i	$-0.246384\pi$
$431$	29.6915	1.43019	0.715094	+	0.699028i	$0.246384\pi$
$432$	−11.6893	−0.562402
$433$	−1.25365	−0.0602466	−0.0301233	−	0.999546i	$-0.509590\pi$
$433$	−1.25365	−0.0602466	−0.0301233	+	0.999546i	$0.509590\pi$
$434$	−23.1518	−1.11132
$435$	−13.8219	−0.662710
$436$	13.4722	0.645201
$437$	−6.26871	−0.299873
$438$	−10.1212	−0.483612
$439$	−31.5486	−1.50573	−0.752866	−	0.658173i	$-0.771329\pi$
$439$	−31.5486	−1.50573	−0.752866	+	0.658173i	$0.771329\pi$
$440$	−9.19404	−0.438308
$441$	−0.353251	−0.0168215
$442$	5.75271	0.273629
$443$	17.8480	0.847983	0.423992	−	0.905666i	$-0.360629\pi$
$443$	17.8480	0.847983	0.423992	+	0.905666i	$0.360629\pi$
$444$	−5.85505	−0.277868
$445$	8.28602	0.392795
$446$	40.2686	1.90678
$447$	1.09382	0.0517361
$448$	11.6699	0.551350
$449$	−30.3609	−1.43282	−0.716411	−	0.697679i	$-0.754216\pi$
$449$	−30.3609	−1.43282	−0.716411	+	0.697679i	$0.754216\pi$
$450$	57.4040	2.70605
$451$	−24.3619	−1.14716
$452$	−12.2413	−0.575782
$453$	0.108942	0.00511854
$454$	−10.2560	−0.481339
$455$	−16.4376	−0.770608
$456$	−0.341812	−0.0160068
$457$	7.24655	0.338979	0.169490	−	0.985532i	$-0.445788\pi$
$457$	7.24655	0.338979	0.169490	+	0.985532i	$0.445788\pi$
$458$	−42.8462	−2.00207
$459$	−4.91666	−0.229490
$460$	−39.3729	−1.83577
$461$	−0.629689	−0.0293275	−0.0146638	−	0.999892i	$-0.504668\pi$
$461$	−0.629689	−0.0293275	−0.0146638	+	0.999892i	$0.504668\pi$
$462$	6.36089	0.295935
$463$	26.8896	1.24966	0.624832	−	0.780759i	$-0.285167\pi$
$463$	26.8896	1.24966	0.624832	+	0.780759i	$0.285167\pi$
$464$	−37.6187	−1.74641
$465$	7.85102	0.364082
$466$	−49.2054	−2.27940
$467$	−25.0649	−1.15987	−0.579933	−	0.814664i	$-0.696921\pi$
$467$	−25.0649	−1.15987	−0.579933	+	0.814664i	$0.696921\pi$
$468$	−6.89738	−0.318832
$469$	19.7266	0.910891
$470$	−25.3526	−1.16943
$471$	7.28427	0.335641
$472$	0.913490	0.0420468
$473$	−18.4134	−0.846647
$474$	−8.74671	−0.401750
$475$	10.7796	0.494601
$476$	8.27674	0.379364
$477$	9.95310	0.455721
$478$	−5.51689	−0.252337
$479$	−23.3100	−1.06506	−0.532530	−	0.846411i	$-0.678759\pi$
$479$	−23.3100	−1.06506	−0.532530	+	0.846411i	$0.678759\pi$
$480$	−12.3981	−0.565891
$481$	13.3115	0.606953
$482$	34.7580	1.58319
$483$	−7.21606	−0.328342
$484$	−3.91100	−0.177773
$485$	−12.8657	−0.584202
$486$	20.2409	0.918145
$487$	18.4438	0.835767	0.417883	−	0.908501i	$-0.362772\pi$
$487$	18.4438	0.835767	0.417883	+	0.908501i	$0.362772\pi$
$488$	−1.71912	−0.0778211
$489$	−8.06333	−0.364636
$490$	−0.943654	−0.0426299
$491$	−9.24161	−0.417068	−0.208534	−	0.978015i	$-0.566869\pi$
$491$	−9.24161	−0.417068	−0.208534	+	0.978015i	$0.566869\pi$
$492$	−5.68868	−0.256466
$493$	−15.8229	−0.712627
$494$	−2.93355	−0.131987
$495$	32.6409	1.46710
$496$	21.3679	0.959448
$497$	9.35914	0.419815
$498$	9.34569	0.418790
$499$	29.0119	1.29875	0.649376	−	0.760468i	$-0.275030\pi$
$499$	29.0119	1.29875	0.649376	+	0.760468i	$0.275030\pi$
$500$	36.3007	1.62342
$501$	−6.19918	−0.276959
$502$	3.35862	0.149903
$503$	−0.439864	−0.0196126	−0.00980629	−	0.999952i	$-0.503121\pi$
$503$	−0.439864	−0.0196126	−0.00980629	+	0.999952i	$0.503121\pi$
$504$	5.95405	0.265214
$505$	−34.4381	−1.53248
$506$	−34.6397	−1.53992
$507$	4.56976	0.202950
$508$	−1.18076	−0.0523878
$509$	−40.4136	−1.79130	−0.895651	−	0.444758i	$-0.853290\pi$
$509$	−40.4136	−1.79130	−0.895651	+	0.444758i	$0.853290\pi$
$510$	−6.35698	−0.281492
$511$	33.1069	1.46456
$512$	−26.3525	−1.16463
$513$	2.50721	0.110696
$514$	−56.1357	−2.47604
$515$	−69.0553	−3.04294
$516$	−4.29966	−0.189282
$517$	−9.84800	−0.433115
$518$	43.3775	1.90590
$519$	7.28667	0.319849
$520$	4.88095	0.214044
$521$	19.4698	0.852987	0.426493	−	0.904491i	$-0.359749\pi$
$521$	19.4698	0.852987	0.426493	+	0.904491i	$0.359749\pi$
$522$	42.9682	1.88067
$523$	19.7990	0.865748	0.432874	−	0.901454i	$-0.357499\pi$
$523$	19.7990	0.865748	0.432874	+	0.901454i	$0.357499\pi$
$524$	−0.455612	−0.0199035
$525$	12.4086	0.541557
$526$	35.0664	1.52897
$527$	8.98760	0.391506
$528$	−5.87078	−0.255493
$529$	16.2967	0.708551
$530$	26.5881	1.15491
$531$	−3.24309	−0.140738
$532$	−4.22066	−0.182989
$533$	12.9333	0.560203
$534$	1.70225	0.0736635
$535$	−54.4398	−2.35364
$536$	−5.85757	−0.253009
$537$	4.50391	0.194358
$538$	24.2143	1.04395
$539$	−0.366555	−0.0157886
$540$	15.7475	0.677663
$541$	14.4034	0.619251	0.309625	−	0.950859i	$-0.399796\pi$
$541$	14.4034	0.619251	0.309625	+	0.950859i	$0.399796\pi$
$542$	35.3086	1.51663
$543$	−3.74471	−0.160701
$544$	−14.1929	−0.608516
$545$	33.8465	1.44983
$546$	−3.37688	−0.144517
$547$	5.80940	0.248392	0.124196	−	0.992258i	$-0.460365\pi$
$547$	5.80940	0.248392	0.124196	+	0.992258i	$0.460365\pi$
$548$	−12.9294	−0.552315
$549$	6.10328	0.260482
$550$	59.5658	2.53990
$551$	8.06876	0.343741
$552$	2.14272	0.0912002
$553$	28.6107	1.21665
$554$	18.8067	0.799020
$555$	−14.7098	−0.624395
$556$	−8.78682	−0.372644
$557$	−23.5050	−0.995938	−0.497969	−	0.867195i	$-0.665921\pi$
$557$	−23.5050	−0.995938	−0.497969	+	0.867195i	$0.665921\pi$
$558$	−24.4065	−1.03321
$559$	9.77532	0.413452
$560$	49.4372	2.08910
$561$	−2.46932	−0.104255
$562$	5.40210	0.227874
$563$	6.15833	0.259543	0.129771	−	0.991544i	$-0.458576\pi$
$563$	6.15833	0.259543	0.129771	+	0.991544i	$0.458576\pi$
$564$	−2.29958	−0.0968299
$565$	−30.7541	−1.29383
$566$	−14.9387	−0.627919
$567$	−19.6490	−0.825180
$568$	−2.77908	−0.116608
$569$	−4.45139	−0.186612	−0.0933060	−	0.995637i	$-0.529743\pi$
$569$	−4.45139	−0.186612	−0.0933060	+	0.995637i	$0.529743\pi$
$570$	3.24170	0.135780
$571$	−17.1721	−0.718631	−0.359315	−	0.933216i	$-0.616990\pi$
$571$	−17.1721	−0.718631	−0.359315	+	0.933216i	$0.616990\pi$
$572$	−7.15714	−0.299255
$573$	8.94343	0.373617
$574$	42.1450	1.75910
$575$	−67.5740	−2.81803
$576$	12.3023	0.512597
$577$	7.52251	0.313166	0.156583	−	0.987665i	$-0.449952\pi$
$577$	7.52251	0.313166	0.156583	+	0.987665i	$0.449952\pi$
$578$	24.8934	1.03543
$579$	1.22279	0.0508174
$580$	50.6788	2.10432
$581$	−30.5700	−1.26826
$582$	−2.64308	−0.109559
$583$	10.3279	0.427739
$584$	−9.83067	−0.406796
$585$	−17.3285	−0.716444
$586$	55.5294	2.29390
$587$	41.1449	1.69823	0.849117	−	0.528205i	$-0.177135\pi$
$587$	41.1449	1.69823	0.849117	+	0.528205i	$0.177135\pi$
$588$	−0.0855933	−0.00352981
$589$	−4.58316	−0.188846
$590$	−8.66340	−0.356667
$591$	−0.401802	−0.0165279
$592$	−40.0352	−1.64544
$593$	−44.7769	−1.83877	−0.919384	−	0.393361i	$-0.871312\pi$
$593$	−44.7769	−1.83877	−0.919384	+	0.393361i	$0.871312\pi$
$594$	13.8544	0.568452
$595$	20.7939	0.852465
$596$	−4.01057	−0.164279
$597$	2.05690	0.0841833
$598$	18.3896	0.752005
$599$	25.3682	1.03651	0.518257	−	0.855225i	$-0.326581\pi$
$599$	25.3682	1.03651	0.518257	+	0.855225i	$0.326581\pi$
$600$	−3.68459	−0.150423
$601$	14.2376	0.580763	0.290381	−	0.956911i	$-0.406218\pi$
$601$	14.2376	0.580763	0.290381	+	0.956911i	$0.406218\pi$
$602$	31.8543	1.29828
$603$	20.7957	0.846867
$604$	−0.399441	−0.0162530
$605$	−9.82571	−0.399472
$606$	−7.07484	−0.287396
$607$	22.4003	0.909201	0.454601	−	0.890695i	$-0.349782\pi$
$607$	22.4003	0.909201	0.454601	+	0.890695i	$0.349782\pi$
$608$	7.23757	0.293522
$609$	9.28815	0.376375
$610$	16.3039	0.660127
$611$	5.22813	0.211507
$612$	8.72530	0.352700
$613$	37.4461	1.51244	0.756218	−	0.654320i	$-0.227045\pi$
$613$	37.4461	1.51244	0.756218	+	0.654320i	$0.227045\pi$
$614$	−41.4188	−1.67153
$615$	−14.2918	−0.576302
$616$	6.17827	0.248930
$617$	−12.0902	−0.486732	−0.243366	−	0.969935i	$-0.578252\pi$
$617$	−12.0902	−0.486732	−0.243366	+	0.969935i	$0.578252\pi$
$618$	−14.1865	−0.570663
$619$	−4.32161	−0.173700	−0.0868501	−	0.996221i	$-0.527680\pi$
$619$	−4.32161	−0.173700	−0.0868501	+	0.996221i	$0.527680\pi$
$620$	−28.7862	−1.15608
$621$	−15.7170	−0.630701
$622$	−12.6109	−0.505653
$623$	−5.56810	−0.223081
$624$	3.11669	0.124767
$625$	37.3013	1.49205
$626$	−18.4717	−0.738278
$627$	1.25921	0.0502880
$628$	−26.7082	−1.06577
$629$	−16.8393	−0.671427
$630$	−56.4673	−2.24971
$631$	−8.99677	−0.358156	−0.179078	−	0.983835i	$-0.557311\pi$
$631$	−8.99677	−0.358156	−0.179078	+	0.983835i	$0.557311\pi$
$632$	−8.49560	−0.337937
$633$	−0.431235	−0.0171400
$634$	14.5528	0.577965
$635$	−2.96645	−0.117720
$636$	2.41165	0.0956281
$637$	0.194597	0.00771023
$638$	44.5864	1.76519
$639$	9.86636	0.390307
$640$	−24.6366	−0.973849
$641$	7.61792	0.300890	0.150445	−	0.988618i	$-0.451929\pi$
$641$	7.61792	0.300890	0.150445	+	0.988618i	$0.451929\pi$
$642$	−11.1839	−0.441393
$643$	24.0500	0.948441	0.474220	−	0.880406i	$-0.342730\pi$
$643$	24.0500	0.948441	0.474220	+	0.880406i	$0.342730\pi$
$644$	26.4581	1.04259
$645$	−10.8021	−0.425333
$646$	3.71099	0.146007
$647$	19.1015	0.750956	0.375478	−	0.926831i	$-0.377479\pi$
$647$	19.1015	0.750956	0.375478	+	0.926831i	$0.377479\pi$
$648$	5.83453	0.229202
$649$	−3.36523	−0.132097
$650$	−31.6224	−1.24033
$651$	−5.27578	−0.206774
$652$	29.5646	1.15784
$653$	−25.6284	−1.00292	−0.501458	−	0.865182i	$-0.667203\pi$
$653$	−25.6284	−1.00292	−0.501458	+	0.865182i	$0.667203\pi$
$654$	6.95330	0.271896
$655$	−1.14465	−0.0447250
$656$	−38.8977	−1.51870
$657$	34.9011	1.36162
$658$	17.0366	0.664156
$659$	−22.5265	−0.877509	−0.438754	−	0.898607i	$-0.644580\pi$
$659$	−22.5265	−0.877509	−0.438754	+	0.898607i	$0.644580\pi$
$660$	7.90893	0.307855
$661$	50.5326	1.96549	0.982745	−	0.184964i	$-0.0592170\pi$
$661$	50.5326	1.96549	0.982745	+	0.184964i	$0.0592170\pi$
$662$	40.2752	1.56534
$663$	1.31092	0.0509118
$664$	9.07738	0.352271
$665$	−10.6037	−0.411193
$666$	45.7284	1.77194
$667$	−50.5807	−1.95849
$668$	22.7296	0.879436
$669$	9.17634	0.354778
$670$	55.5524	2.14618
$671$	6.33313	0.244488
$672$	8.33134	0.321388
$673$	1.03113	0.0397471	0.0198736	−	0.999803i	$-0.493674\pi$
$673$	1.03113	0.0397471	0.0198736	+	0.999803i	$0.493674\pi$
$674$	12.4770	0.480597
$675$	27.0267	1.04026
$676$	−16.7553	−0.644435
$677$	−11.9146	−0.457917	−0.228958	−	0.973436i	$-0.573532\pi$
$677$	−11.9146	−0.457917	−0.228958	+	0.973436i	$0.573532\pi$
$678$	−6.31801	−0.242642
$679$	8.64559	0.331787
$680$	−6.17448	−0.236780
$681$	−2.33712	−0.0895587
$682$	−25.3256	−0.969769
$683$	44.0075	1.68390	0.841950	−	0.539555i	$-0.181408\pi$
$683$	44.0075	1.68390	0.841950	+	0.539555i	$0.181408\pi$
$684$	−4.44940	−0.170127
$685$	−32.4828	−1.24110
$686$	−34.7263	−1.32586
$687$	−9.76370	−0.372509
$688$	−29.3999	−1.12086
$689$	−5.48291	−0.208882
$690$	−20.3212	−0.773616
$691$	21.2501	0.808393	0.404196	−	0.914672i	$-0.367551\pi$
$691$	21.2501	0.808393	0.404196	+	0.914672i	$0.367551\pi$
$692$	−26.7170	−1.01563
$693$	−21.9343	−0.833214
$694$	65.9991	2.50529
$695$	−22.0754	−0.837366
$696$	−2.75800	−0.104542
$697$	−16.3608	−0.619710
$698$	53.0215	2.00689
$699$	−11.2128	−0.424109
$700$	−45.4969	−1.71962
$701$	−13.6685	−0.516250	−0.258125	−	0.966111i	$-0.583105\pi$
$701$	−13.6685	−0.516250	−0.258125	+	0.966111i	$0.583105\pi$
$702$	−7.35504	−0.277598
$703$	8.58707	0.323868
$704$	12.7656	0.481123
$705$	−5.77730	−0.217586
$706$	−27.2480	−1.02549
$707$	23.1420	0.870344
$708$	−0.785806	−0.0295324
$709$	4.88290	0.183381	0.0916906	−	0.995788i	$-0.470773\pi$
$709$	4.88290	0.183381	0.0916906	+	0.995788i	$0.470773\pi$
$710$	26.3564	0.989137
$711$	30.1613	1.13114
$712$	1.65338	0.0619629
$713$	28.7305	1.07596
$714$	4.27181	0.159869
$715$	−17.9811	−0.672453
$716$	−16.5138	−0.617151
$717$	−1.25718	−0.0469502
$718$	59.0012	2.20191
$719$	20.7888	0.775291	0.387646	−	0.921808i	$-0.373288\pi$
$719$	20.7888	0.775291	0.387646	+	0.921808i	$0.373288\pi$
$720$	52.1164	1.94227
$721$	46.4043	1.72819
$722$	−1.89239	−0.0704275
$723$	7.92060	0.294570
$724$	13.7302	0.510279
$725$	86.9778	3.23027
$726$	−2.01856	−0.0749157
$727$	43.3897	1.60923	0.804617	−	0.593794i	$-0.202371\pi$
$727$	43.3897	1.60923	0.804617	+	0.593794i	$0.202371\pi$
$728$	−3.27993	−0.121562
$729$	−17.4703	−0.647048
$730$	93.2327	3.45070
$731$	−12.3659	−0.457371
$732$	1.47883	0.0546592
$733$	50.5500	1.86711	0.933553	−	0.358439i	$-0.116691\pi$
$733$	50.5500	1.86711	0.933553	+	0.358439i	$0.116691\pi$
$734$	−43.5354	−1.60692
$735$	−0.215038	−0.00793180
$736$	−45.3702	−1.67237
$737$	21.5789	0.794868
$738$	44.4291	1.63546
$739$	−32.1331	−1.18203	−0.591017	−	0.806659i	$-0.701273\pi$
$739$	−32.1331	−1.18203	−0.591017	+	0.806659i	$0.701273\pi$
$740$	53.9342	1.98266
$741$	−0.668492	−0.0245577
$742$	−17.8669	−0.655913
$743$	34.7049	1.27320	0.636599	−	0.771195i	$-0.280341\pi$
$743$	34.7049	1.27320	0.636599	+	0.771195i	$0.280341\pi$
$744$	1.56658	0.0574336
$745$	−10.0758	−0.369151
$746$	18.6268	0.681974
$747$	−32.2268	−1.17912
$748$	9.05390	0.331043
$749$	36.5828	1.33671
$750$	18.7356	0.684128
$751$	−23.1989	−0.846540	−0.423270	−	0.906003i	$-0.639118\pi$
$751$	−23.1989	−0.846540	−0.423270	+	0.906003i	$0.639118\pi$
$752$	−15.7239	−0.573392
$753$	0.765357	0.0278911
$754$	−23.6701	−0.862015
$755$	−1.00353	−0.0365221
$756$	−10.5821	−0.384867
$757$	−26.6271	−0.967777	−0.483888	−	0.875130i	$-0.660776\pi$
$757$	−26.6271	−0.967777	−0.483888	+	0.875130i	$0.660776\pi$
$758$	−37.6847	−1.36877
$759$	−7.89362	−0.286520
$760$	3.14863	0.114213
$761$	48.4487	1.75626	0.878131	−	0.478420i	$-0.158790\pi$
$761$	48.4487	1.75626	0.878131	+	0.478420i	$0.158790\pi$
$762$	−0.609417	−0.0220768
$763$	−22.7444	−0.823404
$764$	−32.7916	−1.18636
$765$	21.9208	0.792548
$766$	25.4211	0.918501
$767$	1.78654	0.0645082
$768$	−8.83178	−0.318689
$769$	7.31530	0.263796	0.131898	−	0.991263i	$-0.457893\pi$
$769$	7.31530	0.263796	0.131898	+	0.991263i	$0.457893\pi$
$770$	−58.5938	−2.11158
$771$	−12.7921	−0.460696
$772$	−4.48343	−0.161362
$773$	−28.3452	−1.01951	−0.509754	−	0.860320i	$-0.670263\pi$
$773$	−28.3452	−1.01951	−0.509754	+	0.860320i	$0.670263\pi$
$774$	33.5806	1.20703
$775$	−49.4045	−1.77466
$776$	−2.56720	−0.0921571
$777$	9.88479	0.354615
$778$	12.9428	0.464021
$779$	8.34308	0.298922
$780$	−4.19871	−0.150338
$781$	10.2379	0.366342
$782$	−23.2631	−0.831887
$783$	20.2301	0.722965
$784$	−0.585264	−0.0209023
$785$	−67.0996	−2.39489
$786$	−0.235152	−0.00838759
$787$	10.4549	0.372676	0.186338	−	0.982486i	$-0.440338\pi$
$787$	10.4549	0.372676	0.186338	+	0.982486i	$0.440338\pi$
$788$	1.47323	0.0524816
$789$	7.99087	0.284483
$790$	80.5710	2.86659
$791$	20.6664	0.734811
$792$	6.51311	0.231433
$793$	−3.36214	−0.119393
$794$	−34.8145	−1.23552
$795$	6.05885	0.214885
$796$	−7.54174	−0.267310
$797$	−13.7938	−0.488603	−0.244302	−	0.969699i	$-0.578559\pi$
$797$	−13.7938	−0.488603	−0.244302	+	0.969699i	$0.578559\pi$
$798$	−2.17838	−0.0771137
$799$	−6.61367	−0.233975
$800$	78.0179	2.75835
$801$	−5.86986	−0.207401
$802$	15.8511	0.559720
$803$	36.2155	1.27802
$804$	5.03883	0.177706
$805$	66.4713	2.34281
$806$	13.4449	0.473577
$807$	5.51790	0.194239
$808$	−6.87173	−0.241746
$809$	−1.78829	−0.0628729	−0.0314365	−	0.999506i	$-0.510008\pi$
$809$	−1.78829	−0.0628729	−0.0314365	+	0.999506i	$0.510008\pi$
$810$	−55.3338	−1.94423
$811$	−17.3843	−0.610445	−0.305222	−	0.952281i	$-0.598731\pi$
$811$	−17.3843	−0.610445	−0.305222	+	0.952281i	$0.598731\pi$
$812$	−34.0555	−1.19511
$813$	8.04606	0.282188
$814$	47.4505	1.66314
$815$	74.2760	2.60177
$816$	−3.94266	−0.138021
$817$	6.30592	0.220616
$818$	47.1517	1.64862
$819$	11.6445	0.406892
$820$	52.4018	1.82995
$821$	11.4029	0.397966	0.198983	−	0.980003i	$-0.436236\pi$
$821$	11.4029	0.397966	0.198983	+	0.980003i	$0.436236\pi$
$822$	−6.67314	−0.232752
$823$	−19.2968	−0.672644	−0.336322	−	0.941747i	$-0.609183\pi$
$823$	−19.2968	−0.672644	−0.336322	+	0.941747i	$0.609183\pi$
$824$	−13.7792	−0.480020
$825$	13.5737	0.472577
$826$	5.82170	0.202563
$827$	13.5234	0.470254	0.235127	−	0.971965i	$-0.424449\pi$
$827$	13.5234	0.470254	0.235127	+	0.971965i	$0.424449\pi$
$828$	27.8920	0.969314
$829$	33.5736	1.16606	0.583029	−	0.812451i	$-0.301867\pi$
$829$	33.5736	1.16606	0.583029	+	0.812451i	$0.301867\pi$
$830$	−86.0886	−2.98818
$831$	4.28564	0.148667
$832$	−6.77704	−0.234952
$833$	−0.246169	−0.00852925
$834$	−4.53508	−0.157037
$835$	57.1042	1.97617
$836$	−4.61697	−0.159681
$837$	−11.4910	−0.397186
$838$	6.25207	0.215974
$839$	−41.0171	−1.41607	−0.708034	−	0.706179i	$-0.750417\pi$
$839$	−41.0171	−1.41607	−0.708034	+	0.706179i	$0.750417\pi$
$840$	3.62446	0.125056
$841$	36.1049	1.24500
$842$	33.0155	1.13779
$843$	1.23102	0.0423986
$844$	1.58115	0.0544253
$845$	−42.0947	−1.44810
$846$	17.9599	0.617474
$847$	6.60275	0.226873
$848$	16.4902	0.566276
$849$	−3.40419	−0.116832
$850$	40.0029	1.37209
$851$	−53.8298	−1.84526
$852$	2.39063	0.0819017
$853$	10.3099	0.353005	0.176503	−	0.984300i	$-0.443522\pi$
$853$	10.3099	0.353005	0.176503	+	0.984300i	$0.443522\pi$
$854$	−10.9560	−0.374908
$855$	−11.1783	−0.382291
$856$	−10.8628	−0.371283
$857$	−47.1413	−1.61032	−0.805158	−	0.593060i	$-0.797920\pi$
$857$	−47.1413	−1.61032	−0.805158	+	0.593060i	$0.797920\pi$
$858$	−3.69396	−0.126110
$859$	−56.7810	−1.93734	−0.968672	−	0.248345i	$-0.920113\pi$
$859$	−56.7810	−1.93734	−0.968672	+	0.248345i	$0.920113\pi$
$860$	39.6066	1.35057
$861$	9.60392	0.327301
$862$	−56.1879	−1.91377
$863$	−38.6274	−1.31489	−0.657446	−	0.753502i	$-0.728363\pi$
$863$	−38.6274	−1.31489	−0.657446	+	0.753502i	$0.728363\pi$
$864$	18.1461	0.617344
$865$	−67.1217	−2.28221
$866$	2.37240	0.0806174
$867$	5.67266	0.192654
$868$	19.3440	0.656577
$869$	31.2972	1.06168
$870$	26.1565	0.886787
$871$	−11.4558	−0.388166
$872$	6.75368	0.228708
$873$	9.11414	0.308467
$874$	11.8628	0.401267
$875$	−61.2847	−2.07180
$876$	8.45658	0.285722
$877$	−5.82626	−0.196739	−0.0983695	−	0.995150i	$-0.531363\pi$
$877$	−5.82626	−0.196739	−0.0983695	+	0.995150i	$0.531363\pi$
$878$	59.7023	2.01486
$879$	12.6539	0.426806
$880$	54.0791	1.82301
$881$	11.7275	0.395111	0.197555	−	0.980292i	$-0.436700\pi$
$881$	11.7275	0.395111	0.197555	+	0.980292i	$0.436700\pi$
$882$	0.668490	0.0225092
$883$	−29.8526	−1.00462	−0.502310	−	0.864688i	$-0.667516\pi$
$883$	−29.8526	−1.00462	−0.502310	+	0.864688i	$0.667516\pi$
$884$	−4.80655	−0.161662
$885$	−1.97420	−0.0663620
$886$	−33.7754	−1.13471
$887$	−56.0411	−1.88168	−0.940838	−	0.338856i	$-0.889960\pi$
$887$	−56.0411	−1.88168	−0.940838	+	0.338856i	$0.889960\pi$
$888$	−2.93516	−0.0984977
$889$	1.99342	0.0668571
$890$	−15.6804	−0.525608
$891$	−21.4940	−0.720075
$892$	−33.6456	−1.12654
$893$	3.37259	0.112859
$894$	−2.06994	−0.0692293
$895$	−41.4881	−1.38679
$896$	16.5555	0.553081
$897$	4.19058	0.139919
$898$	57.4548	1.91729
$899$	−36.9804	−1.23337
$900$	−47.9627	−1.59876
$901$	6.93598	0.231071
$902$	46.1022	1.53504
$903$	7.25889	0.241561
$904$	−6.13662	−0.204101
$905$	34.4947	1.14664
$906$	−0.206161	−0.00684923
$907$	−45.6727	−1.51654	−0.758268	−	0.651943i	$-0.773954\pi$
$907$	−45.6727	−1.51654	−0.758268	+	0.651943i	$0.773954\pi$
$908$	8.56918	0.284378
$909$	24.3962	0.809170
$910$	31.1064	1.03117
$911$	41.6338	1.37939	0.689695	−	0.724100i	$-0.257745\pi$
$911$	41.6338	1.37939	0.689695	+	0.724100i	$0.257745\pi$
$912$	2.01053	0.0665754
$913$	−33.4404	−1.10672
$914$	−13.7133	−0.453596
$915$	3.71531	0.122824
$916$	35.7992	1.18284
$917$	0.769188	0.0254008
$918$	9.30425	0.307086
$919$	−46.5275	−1.53480	−0.767400	−	0.641169i	$-0.778450\pi$
$919$	−46.5275	−1.53480	−0.767400	+	0.641169i	$0.778450\pi$
$920$	−19.7378	−0.650737
$921$	−9.43843	−0.311007
$922$	1.19162	0.0392438
$923$	−5.43513	−0.178899
$924$	−5.31470	−0.174841
$925$	92.5649	3.04352
$926$	−50.8856	−1.67220
$927$	48.9192	1.60672
$928$	58.3982	1.91701
$929$	4.90083	0.160791	0.0803955	−	0.996763i	$-0.474382\pi$
$929$	4.90083	0.160791	0.0803955	+	0.996763i	$0.474382\pi$
$930$	−14.8572	−0.487187
$931$	0.125532	0.00411414
$932$	41.1125	1.34668
$933$	−2.87376	−0.0940826
$934$	47.4326	1.55204
$935$	22.7463	0.743884
$936$	−3.45769	−0.113018
$937$	−17.7886	−0.581128	−0.290564	−	0.956856i	$-0.593843\pi$
$937$	−17.7886	−0.581128	−0.290564	+	0.956856i	$0.593843\pi$
$938$	−37.3305	−1.21888
$939$	−4.20930	−0.137365
$940$	21.1828	0.690906
$941$	29.5405	0.962992	0.481496	−	0.876448i	$-0.340094\pi$
$941$	29.5405	0.962992	0.481496	+	0.876448i	$0.340094\pi$
$942$	−13.7847	−0.449129
$943$	−52.3003	−1.70313
$944$	−5.37313	−0.174880
$945$	−26.5857	−0.864832
$946$	34.8453	1.13292
$947$	22.5425	0.732534	0.366267	−	0.930510i	$-0.380636\pi$
$947$	22.5425	0.732534	0.366267	+	0.930510i	$0.380636\pi$
$948$	7.30812	0.237357
$949$	−19.2261	−0.624107
$950$	−20.3992	−0.661836
$951$	3.31626	0.107537
$952$	4.14917	0.134475
$953$	−2.55275	−0.0826918	−0.0413459	−	0.999145i	$-0.513165\pi$
$953$	−2.55275	−0.0826918	−0.0413459	+	0.999145i	$0.513165\pi$
$954$	−18.8352	−0.609811
$955$	−82.3831	−2.66586
$956$	4.60951	0.149082
$957$	10.1603	0.328435
$958$	44.1116	1.42518
$959$	21.8280	0.704863
$960$	7.48891	0.241704
$961$	−9.99466	−0.322409
$962$	−25.1906	−0.812178
$963$	38.5655	1.24275
$964$	−29.0413	−0.935358
$965$	−11.2638	−0.362595
$966$	13.6556	0.439362
$967$	28.2237	0.907613	0.453806	−	0.891100i	$-0.350066\pi$
$967$	28.2237	0.907613	0.453806	+	0.891100i	$0.350066\pi$
$968$	−1.96061	−0.0630162
$969$	0.845653	0.0271663
$970$	24.3470	0.781734
$971$	−27.1250	−0.870483	−0.435241	−	0.900314i	$-0.643337\pi$
$971$	−27.1250	−0.870483	−0.435241	+	0.900314i	$0.643337\pi$
$972$	−16.9118	−0.542447
$973$	14.8344	0.475568
$974$	−34.9028	−1.11836
$975$	−7.20606	−0.230779
$976$	10.1119	0.323673
$977$	−32.2918	−1.03311	−0.516553	−	0.856255i	$-0.672785\pi$
$977$	−32.2918	−1.03311	−0.516553	+	0.856255i	$0.672785\pi$
$978$	15.2590	0.487928
$979$	−6.09092	−0.194667
$980$	0.788449	0.0251861
$981$	−23.9771	−0.765529
$982$	17.4887	0.558088
$983$	7.83824	0.250001	0.125001	−	0.992157i	$-0.460107\pi$
$983$	7.83824	0.250001	0.125001	+	0.992157i	$0.460107\pi$
$984$	−2.85176	−0.0909109
$985$	3.70123	0.117931
$986$	29.9431	0.953582
$987$	3.88227	0.123574
$988$	2.45106	0.0779787
$989$	−39.5299	−1.25698
$990$	−61.7694	−1.96316
$991$	56.6231	1.79869	0.899346	−	0.437239i	$-0.144043\pi$
$991$	56.6231	1.79869	0.899346	+	0.437239i	$0.144043\pi$
$992$	−33.1709	−1.05318
$993$	9.17783	0.291250
$994$	−17.7111	−0.561764
$995$	−18.9473	−0.600669
$996$	−7.80858	−0.247424
$997$	−40.2125	−1.27354	−0.636771	−	0.771053i	$-0.719730\pi$
$997$	−40.2125	−1.27354	−0.636771	+	0.771053i	$0.719730\pi$
$998$	−54.9019	−1.73789
$999$	21.5296	0.681167

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.c.1.18	✓	71

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.c.1.18	✓	71	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)

\(f(q)\)	\(=\)	\(q-1.89239 q^{2} -0.431235 q^{3} +1.58115 q^{4} +3.97235 q^{5} +0.816065 q^{6} -2.66937 q^{7} +0.792636 q^{8} -2.81404 q^{9} +O(q^{10})\)	\(q-1.89239 q^{2} -0.431235 q^{3} +1.58115 q^{4} +3.97235 q^{5} +0.816065 q^{6} -2.66937 q^{7} +0.792636 q^{8} -2.81404 q^{9} -7.51724 q^{10} -2.92001 q^{11} -0.681845 q^{12} +1.55018 q^{13} +5.05149 q^{14} -1.71301 q^{15} -4.66227 q^{16} -1.96101 q^{17} +5.32526 q^{18} +1.00000 q^{19} +6.28087 q^{20} +1.15112 q^{21} +5.52581 q^{22} -6.26871 q^{23} -0.341812 q^{24} +10.7796 q^{25} -2.93355 q^{26} +2.50721 q^{27} -4.22066 q^{28} +8.06876 q^{29} +3.24170 q^{30} -4.58316 q^{31} +7.23757 q^{32} +1.25921 q^{33} +3.71099 q^{34} -10.6037 q^{35} -4.44940 q^{36} +8.58707 q^{37} -1.89239 q^{38} -0.668492 q^{39} +3.14863 q^{40} +8.34308 q^{41} -2.17838 q^{42} +6.30592 q^{43} -4.61697 q^{44} -11.1783 q^{45} +11.8628 q^{46} +3.37259 q^{47} +2.01053 q^{48} +0.125532 q^{49} -20.3992 q^{50} +0.845653 q^{51} +2.45106 q^{52} -3.53695 q^{53} -4.74463 q^{54} -11.5993 q^{55} -2.11584 q^{56} -0.431235 q^{57} -15.2693 q^{58} +1.15247 q^{59} -2.70853 q^{60} -2.16887 q^{61} +8.67313 q^{62} +7.51170 q^{63} -4.37177 q^{64} +6.15787 q^{65} -2.38292 q^{66} -7.38999 q^{67} -3.10064 q^{68} +2.70328 q^{69} +20.0663 q^{70} -3.50612 q^{71} -2.23051 q^{72} -12.4025 q^{73} -16.2501 q^{74} -4.64852 q^{75} +1.58115 q^{76} +7.79459 q^{77} +1.26505 q^{78} -10.7182 q^{79} -18.5202 q^{80} +7.36091 q^{81} -15.7884 q^{82} +11.4521 q^{83} +1.82010 q^{84} -7.78980 q^{85} -11.9333 q^{86} -3.47953 q^{87} -2.31451 q^{88} +2.08592 q^{89} +21.1538 q^{90} -4.13801 q^{91} -9.91174 q^{92} +1.97642 q^{93} -6.38226 q^{94} +3.97235 q^{95} -3.12109 q^{96} -3.23881 q^{97} -0.237556 q^{98} +8.21702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9}+O(q^{10}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9	\( 71 q - 15 q^{2} - 8 q^{3} + 69 q^{4} - 18 q^{5} - 9 q^{6} - 19 q^{7} - 39 q^{8} + 63 q^{9} - 10 q^{10} - 52 q^{11} - 9 q^{12} - 15 q^{13} - 53 q^{14} - 33 q^{15} + 53 q^{16} - 10 q^{17} - 35 q^{18} + 71 q^{19} - 33 q^{20} - 38 q^{21} - 6 q^{22} - 65 q^{23} - 30 q^{24} + 51 q^{25} - 4 q^{26} - 23 q^{27} - 29 q^{28} - 97 q^{29} - 27 q^{30} - 53 q^{31} - 78 q^{32} - 17 q^{33} - 24 q^{34} - 38 q^{35} + 24 q^{36} - 33 q^{37} - 15 q^{38} - 86 q^{39} + 25 q^{40} - 69 q^{41} + 64 q^{42} - 10 q^{43} - 94 q^{44} - 34 q^{45} - 6 q^{46} - 37 q^{47} - q^{48} + 74 q^{49} - 41 q^{50} - 46 q^{51} - 30 q^{52} - 50 q^{53} - 17 q^{54} - 30 q^{55} - 116 q^{56} - 8 q^{57} + 11 q^{58} - 93 q^{59} - 56 q^{60} - 18 q^{61} - q^{62} - 84 q^{63} + 93 q^{64} - 78 q^{65} - 53 q^{66} - 5 q^{67} - 9 q^{68} - 69 q^{69} - 10 q^{70} - 221 q^{71} - 73 q^{72} - 34 q^{73} - 58 q^{74} - 70 q^{75} + 69 q^{76} - 2 q^{77} + 7 q^{78} - 68 q^{79} - 71 q^{80} + 39 q^{81} + 26 q^{82} - 45 q^{83} - 10 q^{84} - 44 q^{85} - 80 q^{86} - 7 q^{87} - 46 q^{88} - 143 q^{89} + 41 q^{90} - 30 q^{91} - 46 q^{92} + 32 q^{93} + 41 q^{94} - 18 q^{95} - 140 q^{96} - 18 q^{97} - 97 q^{98} - 142 q^{99}+O(q^{100}) \) 71 * q - 15 * q^2 - 8 * q^3 + 69 * q^4 - 18 * q^5 - 9 * q^6 - 19 * q^7 - 39 * q^8 + 63 * q^9 - 10 * q^10 - 52 * q^11 - 9 * q^12 - 15 * q^13 - 53 * q^14 - 33 * q^15 + 53 * q^16 - 10 * q^17 - 35 * q^18 + 71 * q^19 - 33 * q^20 - 38 * q^21 - 6 * q^22 - 65 * q^23 - 30 * q^24 + 51 * q^25 - 4 * q^26 - 23 * q^27 - 29 * q^28 - 97 * q^29 - 27 * q^30 - 53 * q^31 - 78 * q^32 - 17 * q^33 - 24 * q^34 - 38 * q^35 + 24 * q^36 - 33 * q^37 - 15 * q^38 - 86 * q^39 + 25 * q^40 - 69 * q^41 + 64 * q^42 - 10 * q^43 - 94 * q^44 - 34 * q^45 - 6 * q^46 - 37 * q^47 - q^48 + 74 * q^49 - 41 * q^50 - 46 * q^51 - 30 * q^52 - 50 * q^53 - 17 * q^54 - 30 * q^55 - 116 * q^56 - 8 * q^57 + 11 * q^58 - 93 * q^59 - 56 * q^60 - 18 * q^61 - q^62 - 84 * q^63 + 93 * q^64 - 78 * q^65 - 53 * q^66 - 5 * q^67 - 9 * q^68 - 69 * q^69 - 10 * q^70 - 221 * q^71 - 73 * q^72 - 34 * q^73 - 58 * q^74 - 70 * q^75 + 69 * q^76 - 2 * q^77 + 7 * q^78 - 68 * q^79 - 71 * q^80 + 39 * q^81 + 26 * q^82 - 45 * q^83 - 10 * q^84 - 44 * q^85 - 80 * q^86 - 7 * q^87 - 46 * q^88 - 143 * q^89 + 41 * q^90 - 30 * q^91 - 46 * q^92 + 32 * q^93 + 41 * q^94 - 18 * q^95 - 140 * q^96 - 18 * q^97 - 97 * q^98 - 142 * q^99

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