LMFDB - Embedded newform 8045.2.a.b.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8045,2,Mod(1,8045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8045, 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("8045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8045 = 5 \cdot 1609 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8045.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:	$64.2396484261$
Analytic rank:	$1$
Dimension:	$126$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57758 q^{2} +3.01279 q^{3} +4.64389 q^{4} -1.00000 q^{5} -7.76569 q^{6} -3.99315 q^{7} -6.81483 q^{8} +6.07689 q^{9} +O(q^{10})$	\(q-2.57758 q^{2} +3.01279 q^{3} +4.64389 q^{4} -1.00000 q^{5} -7.76569 q^{6} -3.99315 q^{7} -6.81483 q^{8} +6.07689 q^{9} +2.57758 q^{10} -0.240356 q^{11} +13.9911 q^{12} +2.34990 q^{13} +10.2927 q^{14} -3.01279 q^{15} +8.27796 q^{16} -3.26499 q^{17} -15.6636 q^{18} -0.497976 q^{19} -4.64389 q^{20} -12.0305 q^{21} +0.619536 q^{22} +8.52360 q^{23} -20.5316 q^{24} +1.00000 q^{25} -6.05703 q^{26} +9.27002 q^{27} -18.5438 q^{28} -8.64370 q^{29} +7.76569 q^{30} +2.72610 q^{31} -7.70739 q^{32} -0.724142 q^{33} +8.41575 q^{34} +3.99315 q^{35} +28.2204 q^{36} -10.1746 q^{37} +1.28357 q^{38} +7.07974 q^{39} +6.81483 q^{40} -3.03998 q^{41} +31.0096 q^{42} +3.25111 q^{43} -1.11619 q^{44} -6.07689 q^{45} -21.9702 q^{46} +9.83322 q^{47} +24.9397 q^{48} +8.94527 q^{49} -2.57758 q^{50} -9.83672 q^{51} +10.9127 q^{52} +6.68259 q^{53} -23.8942 q^{54} +0.240356 q^{55} +27.2127 q^{56} -1.50030 q^{57} +22.2798 q^{58} +7.90583 q^{59} -13.9911 q^{60} -8.36120 q^{61} -7.02674 q^{62} -24.2660 q^{63} +3.31047 q^{64} -2.34990 q^{65} +1.86653 q^{66} -13.3648 q^{67} -15.1623 q^{68} +25.6798 q^{69} -10.2927 q^{70} -5.17378 q^{71} -41.4130 q^{72} -5.72996 q^{73} +26.2259 q^{74} +3.01279 q^{75} -2.31255 q^{76} +0.959778 q^{77} -18.2486 q^{78} +1.13925 q^{79} -8.27796 q^{80} +9.69794 q^{81} +7.83577 q^{82} -7.96542 q^{83} -55.8685 q^{84} +3.26499 q^{85} -8.37997 q^{86} -26.0416 q^{87} +1.63799 q^{88} -18.1631 q^{89} +15.6636 q^{90} -9.38349 q^{91} +39.5827 q^{92} +8.21317 q^{93} -25.3459 q^{94} +0.497976 q^{95} -23.2207 q^{96} +2.21873 q^{97} -23.0571 q^{98} -1.46062 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) $ 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * 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 $

$2$	−2.57758	−1.82262	−0.911310	−	0.411720i	$-0.864928\pi$
$2$	−2.57758	−1.82262	−0.911310	+	0.411720i	$0.864928\pi$
$3$	3.01279	1.73943	0.869717	−	0.493551i	$-0.164301\pi$
$3$	3.01279	1.73943	0.869717	+	0.493551i	$0.164301\pi$
$4$	4.64389	2.32195
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−7.76569	−3.17033
$7$	−3.99315	−1.50927	−0.754635	−	0.656145i	$-0.772186\pi$
$7$	−3.99315	−1.50927	−0.754635	+	0.656145i	$0.772186\pi$
$8$	−6.81483	−2.40941
$9$	6.07689	2.02563
$10$	2.57758	0.815101
$11$	−0.240356	−0.0724701	−0.0362350	−	0.999343i	$-0.511536\pi$
$11$	−0.240356	−0.0724701	−0.0362350	+	0.999343i	$0.511536\pi$
$12$	13.9911	4.03887
$13$	2.34990	0.651744	0.325872	−	0.945414i	$-0.394342\pi$
$13$	2.34990	0.651744	0.325872	+	0.945414i	$0.394342\pi$
$14$	10.2927	2.75083
$15$	−3.01279	−0.777899
$16$	8.27796	2.06949
$17$	−3.26499	−0.791876	−0.395938	−	0.918277i	$-0.629581\pi$
$17$	−3.26499	−0.791876	−0.395938	+	0.918277i	$0.629581\pi$
$18$	−15.6636	−3.69196
$19$	−0.497976	−0.114244	−0.0571218	−	0.998367i	$-0.518192\pi$
$19$	−0.497976	−0.114244	−0.0571218	+	0.998367i	$0.518192\pi$
$20$	−4.64389	−1.03841
$21$	−12.0305	−2.62528
$22$	0.619536	0.132085
$23$	8.52360	1.77729	0.888647	−	0.458593i	$-0.151646\pi$
$23$	8.52360	1.77729	0.888647	+	0.458593i	$0.151646\pi$
$24$	−20.5316	−4.19101
$25$	1.00000	0.200000
$26$	−6.05703	−1.18788
$27$	9.27002	1.78402
$28$	−18.5438	−3.50444
$29$	−8.64370	−1.60510	−0.802548	−	0.596588i	$-0.796523\pi$
$29$	−8.64370	−1.60510	−0.802548	+	0.596588i	$0.796523\pi$
$30$	7.76569	1.41781
$31$	2.72610	0.489623	0.244811	−	0.969571i	$-0.421274\pi$
$31$	2.72610	0.489623	0.244811	+	0.969571i	$0.421274\pi$
$32$	−7.70739	−1.36249
$33$	−0.724142	−0.126057
$34$	8.41575	1.44329
$35$	3.99315	0.674966
$36$	28.2204	4.70341
$37$	−10.1746	−1.67270	−0.836350	−	0.548195i	$-0.815315\pi$
$37$	−10.1746	−1.67270	−0.836350	+	0.548195i	$0.815315\pi$
$38$	1.28357	0.208223
$39$	7.07974	1.13367
$40$	6.81483	1.07752
$41$	−3.03998	−0.474764	−0.237382	−	0.971416i	$-0.576289\pi$
$41$	−3.03998	−0.474764	−0.237382	+	0.971416i	$0.576289\pi$
$42$	31.0096	4.78488
$43$	3.25111	0.495789	0.247894	−	0.968787i	$-0.420261\pi$
$43$	3.25111	0.495789	0.247894	+	0.968787i	$0.420261\pi$
$44$	−1.11619	−0.168272
$45$	−6.07689	−0.905890
$46$	−21.9702	−3.23933
$47$	9.83322	1.43432	0.717161	−	0.696907i	$-0.245441\pi$
$47$	9.83322	1.43432	0.717161	+	0.696907i	$0.245441\pi$
$48$	24.9397	3.59974
$49$	8.94527	1.27790
$50$	−2.57758	−0.364524
$51$	−9.83672	−1.37742
$52$	10.9127	1.51331
$53$	6.68259	0.917924	0.458962	−	0.888456i	$-0.348221\pi$
$53$	6.68259	0.917924	0.458962	+	0.888456i	$0.348221\pi$
$54$	−23.8942	−3.25159
$55$	0.240356	0.0324096
$56$	27.2127	3.63645
$57$	−1.50030	−0.198719
$58$	22.2798	2.92548
$59$	7.90583	1.02925	0.514626	−	0.857415i	$-0.327931\pi$
$59$	7.90583	1.02925	0.514626	+	0.857415i	$0.327931\pi$
$60$	−13.9911	−1.80624
$61$	−8.36120	−1.07054	−0.535271	−	0.844680i	$-0.679791\pi$
$61$	−8.36120	−1.07054	−0.535271	+	0.844680i	$0.679791\pi$
$62$	−7.02674	−0.892396
$63$	−24.2660	−3.05722
$64$	3.31047	0.413808
$65$	−2.34990	−0.291469
$66$	1.86653	0.229754
$67$	−13.3648	−1.63277	−0.816384	−	0.577509i	$-0.804025\pi$
$67$	−13.3648	−1.63277	−0.816384	+	0.577509i	$0.804025\pi$
$68$	−15.1623	−1.83869
$69$	25.6798	3.09148
$70$	−10.2927	−1.23021
$71$	−5.17378	−0.614015	−0.307007	−	0.951707i	$-0.599328\pi$
$71$	−5.17378	−0.614015	−0.307007	+	0.951707i	$0.599328\pi$
$72$	−41.4130	−4.88057
$73$	−5.72996	−0.670641	−0.335320	−	0.942104i	$-0.608845\pi$
$73$	−5.72996	−0.670641	−0.335320	+	0.942104i	$0.608845\pi$
$74$	26.2259	3.04870
$75$	3.01279	0.347887
$76$	−2.31255	−0.265267
$77$	0.959778	0.109377
$78$	−18.2486	−2.06624
$79$	1.13925	0.128176	0.0640878	−	0.997944i	$-0.479586\pi$
$79$	1.13925	0.128176	0.0640878	+	0.997944i	$0.479586\pi$
$80$	−8.27796	−0.925504
$81$	9.69794	1.07755
$82$	7.83577	0.865316
$83$	−7.96542	−0.874318	−0.437159	−	0.899384i	$-0.644015\pi$
$83$	−7.96542	−0.874318	−0.437159	+	0.899384i	$0.644015\pi$
$84$	−55.8685	−6.09575
$85$	3.26499	0.354138
$86$	−8.37997	−0.903635
$87$	−26.0416	−2.79196
$88$	1.63799	0.174610
$89$	−18.1631	−1.92528	−0.962640	−	0.270783i	$-0.912717\pi$
$89$	−18.1631	−1.92528	−0.962640	+	0.270783i	$0.912717\pi$
$90$	15.6636	1.65109
$91$	−9.38349	−0.983657
$92$	39.5827	4.12678
$93$	8.21317	0.851666
$94$	−25.3459	−2.61423
$95$	0.497976	0.0510913
$96$	−23.2207	−2.36996
$97$	2.21873	0.225278	0.112639	−	0.993636i	$-0.464070\pi$
$97$	2.21873	0.225278	0.112639	+	0.993636i	$0.464070\pi$
$98$	−23.0571	−2.32912
$99$	−1.46062	−0.146798
$100$	4.64389	0.464389
$101$	4.34445	0.432289	0.216145	−	0.976361i	$-0.430652\pi$
$101$	4.34445	0.432289	0.216145	+	0.976361i	$0.430652\pi$
$102$	25.3549	2.51051
$103$	12.1373	1.19592	0.597960	−	0.801526i	$-0.295978\pi$
$103$	12.1373	1.19592	0.597960	+	0.801526i	$0.295978\pi$
$104$	−16.0141	−1.57032
$105$	12.0305	1.17406
$106$	−17.2249	−1.67303
$107$	17.3196	1.67435	0.837174	−	0.546936i	$-0.184206\pi$
$107$	17.3196	1.67435	0.837174	+	0.546936i	$0.184206\pi$
$108$	43.0490	4.14239
$109$	−5.65447	−0.541600	−0.270800	−	0.962636i	$-0.587288\pi$
$109$	−5.65447	−0.541600	−0.270800	+	0.962636i	$0.587288\pi$
$110$	−0.619536	−0.0590704
$111$	−30.6540	−2.90955
$112$	−33.0552	−3.12342
$113$	−16.1558	−1.51981	−0.759903	−	0.650036i	$-0.774754\pi$
$113$	−16.1558	−1.51981	−0.759903	+	0.650036i	$0.774754\pi$
$114$	3.86713	0.362190
$115$	−8.52360	−0.794830
$116$	−40.1404	−3.72695
$117$	14.2801	1.32019
$118$	−20.3779	−1.87594
$119$	13.0376	1.19515
$120$	20.5316	1.87427
$121$	−10.9422	−0.994748
$122$	21.5516	1.95119
$123$	−9.15880	−0.825821
$124$	12.6597	1.13688
$125$	−1.00000	−0.0894427
$126$	62.5473	5.57216
$127$	−4.54153	−0.402995	−0.201498	−	0.979489i	$-0.564581\pi$
$127$	−4.54153	−0.402995	−0.201498	+	0.979489i	$0.564581\pi$
$128$	6.88181	0.608272
$129$	9.79489	0.862392
$130$	6.05703	0.531237
$131$	1.96531	0.171710	0.0858548	−	0.996308i	$-0.472638\pi$
$131$	1.96531	0.171710	0.0858548	+	0.996308i	$0.472638\pi$
$132$	−3.36284	−0.292697
$133$	1.98849	0.172424
$134$	34.4487	2.97592
$135$	−9.27002	−0.797837
$136$	22.2504	1.90795
$137$	19.4013	1.65756	0.828781	−	0.559574i	$-0.189035\pi$
$137$	19.4013	1.65756	0.828781	+	0.559574i	$0.189035\pi$
$138$	−66.1916	−5.63460
$139$	11.0335	0.935850	0.467925	−	0.883768i	$-0.345002\pi$
$139$	11.0335	0.935850	0.467925	+	0.883768i	$0.345002\pi$
$140$	18.5438	1.56724
$141$	29.6254	2.49491
$142$	13.3358	1.11912
$143$	−0.564812	−0.0472319
$144$	50.3043	4.19202
$145$	8.64370	0.717821
$146$	14.7694	1.22232
$147$	26.9502	2.22282
$148$	−47.2499	−3.88392
$149$	−13.6644	−1.11943	−0.559717	−	0.828684i	$-0.689090\pi$
$149$	−13.6644	−1.11943	−0.559717	+	0.828684i	$0.689090\pi$
$150$	−7.76569	−0.634066
$151$	−9.82144	−0.799257	−0.399629	−	0.916677i	$-0.630861\pi$
$151$	−9.82144	−0.799257	−0.399629	+	0.916677i	$0.630861\pi$
$152$	3.39362	0.275259
$153$	−19.8410	−1.60405
$154$	−2.47390	−0.199353
$155$	−2.72610	−0.218966
$156$	32.8776	2.63231
$157$	13.1191	1.04702	0.523511	−	0.852019i	$-0.324622\pi$
$157$	13.1191	1.04702	0.523511	+	0.852019i	$0.324622\pi$
$158$	−2.93650	−0.233615
$159$	20.1332	1.59667
$160$	7.70739	0.609323
$161$	−34.0360	−2.68242
$162$	−24.9972	−1.96396
$163$	3.28657	0.257424	0.128712	−	0.991682i	$-0.458916\pi$
$163$	3.28657	0.257424	0.128712	+	0.991682i	$0.458916\pi$
$164$	−14.1173	−1.10238
$165$	0.724142	0.0563744
$166$	20.5315	1.59355
$167$	22.4390	1.73638	0.868192	−	0.496228i	$-0.165282\pi$
$167$	22.4390	1.73638	0.868192	+	0.496228i	$0.165282\pi$
$168$	81.9860	6.32536
$169$	−7.47799	−0.575230
$170$	−8.41575	−0.645459
$171$	−3.02615	−0.231415
$172$	15.0978	1.15120
$173$	−14.0921	−1.07140	−0.535700	−	0.844408i	$-0.679952\pi$
$173$	−14.0921	−1.07140	−0.535700	+	0.844408i	$0.679952\pi$
$174$	67.1243	5.08868
$175$	−3.99315	−0.301854
$176$	−1.98966	−0.149976
$177$	23.8186	1.79032
$178$	46.8167	3.50906
$179$	−7.80616	−0.583459	−0.291730	−	0.956501i	$-0.594231\pi$
$179$	−7.80616	−0.583459	−0.291730	+	0.956501i	$0.594231\pi$
$180$	−28.2204	−2.10343
$181$	−3.78428	−0.281284	−0.140642	−	0.990061i	$-0.544917\pi$
$181$	−3.78428	−0.281284	−0.140642	+	0.990061i	$0.544917\pi$
$182$	24.1867	1.79283
$183$	−25.1905	−1.86214
$184$	−58.0869	−4.28222
$185$	10.1746	0.748054
$186$	−21.1701	−1.55226
$187$	0.784760	0.0573873
$188$	45.6644	3.33042
$189$	−37.0166	−2.69256
$190$	−1.28357	−0.0931200
$191$	−0.273835	−0.0198140	−0.00990700	−	0.999951i	$-0.503154\pi$
$191$	−0.273835	−0.0198140	−0.00990700	+	0.999951i	$0.503154\pi$
$192$	9.97373	0.719792
$193$	−15.2027	−1.09432	−0.547158	−	0.837030i	$-0.684290\pi$
$193$	−15.2027	−1.09432	−0.547158	+	0.837030i	$0.684290\pi$
$194$	−5.71895	−0.410597
$195$	−7.07974	−0.506991
$196$	41.5409	2.96721
$197$	16.2555	1.15816	0.579079	−	0.815272i	$-0.303412\pi$
$197$	16.2555	1.15816	0.579079	+	0.815272i	$0.303412\pi$
$198$	3.76485	0.267556
$199$	−3.86764	−0.274170	−0.137085	−	0.990559i	$-0.543773\pi$
$199$	−3.86764	−0.274170	−0.137085	+	0.990559i	$0.543773\pi$
$200$	−6.81483	−0.481881
$201$	−40.2653	−2.84009
$202$	−11.1982	−0.787899
$203$	34.5156	2.42252
$204$	−45.6807	−3.19829
$205$	3.03998	0.212321
$206$	−31.2847	−2.17971
$207$	51.7970	3.60014
$208$	19.4523	1.34878
$209$	0.119692	0.00827924
$210$	−31.0096	−2.13986
$211$	7.71898	0.531396	0.265698	−	0.964056i	$-0.414398\pi$
$211$	7.71898	0.531396	0.265698	+	0.964056i	$0.414398\pi$
$212$	31.0332	2.13137
$213$	−15.5875	−1.06804
$214$	−44.6426	−3.05170
$215$	−3.25111	−0.221724
$216$	−63.1737	−4.29842
$217$	−10.8857	−0.738973
$218$	14.5748	0.987131
$219$	−17.2631	−1.16653
$220$	1.11619	0.0752534
$221$	−7.67238	−0.516100
$222$	79.0131	5.30301
$223$	14.8784	0.996332	0.498166	−	0.867082i	$-0.334007\pi$
$223$	14.8784	0.996332	0.498166	+	0.867082i	$0.334007\pi$
$224$	30.7768	2.05636
$225$	6.07689	0.405126
$226$	41.6427	2.77003
$227$	−21.3315	−1.41582	−0.707912	−	0.706301i	$-0.750362\pi$
$227$	−21.3315	−1.41582	−0.707912	+	0.706301i	$0.750362\pi$
$228$	−6.96722	−0.461415
$229$	−15.6076	−1.03138	−0.515688	−	0.856776i	$-0.672464\pi$
$229$	−15.6076	−1.03138	−0.515688	+	0.856776i	$0.672464\pi$
$230$	21.9702	1.44867
$231$	2.89161	0.190254
$232$	58.9054	3.86733
$233$	−5.98641	−0.392183	−0.196091	−	0.980586i	$-0.562825\pi$
$233$	−5.98641	−0.392183	−0.196091	+	0.980586i	$0.562825\pi$
$234$	−36.8079	−2.40621
$235$	−9.83322	−0.641449
$236$	36.7138	2.38987
$237$	3.43232	0.222953
$238$	−33.6054	−2.17831
$239$	21.8642	1.41428	0.707138	−	0.707076i	$-0.249986\pi$
$239$	21.8642	1.41428	0.707138	+	0.707076i	$0.249986\pi$
$240$	−24.9397	−1.60985
$241$	−1.97038	−0.126924	−0.0634618	−	0.997984i	$-0.520214\pi$
$241$	−1.97038	−0.126924	−0.0634618	+	0.997984i	$0.520214\pi$
$242$	28.2044	1.81305
$243$	1.40776	0.0903081
$244$	−38.8285	−2.48574
$245$	−8.94527	−0.571492
$246$	23.6075	1.50516
$247$	−1.17019	−0.0744575
$248$	−18.5779	−1.17970
$249$	−23.9981	−1.52082
$250$	2.57758	0.163020
$251$	3.79513	0.239547	0.119773	−	0.992801i	$-0.461783\pi$
$251$	3.79513	0.239547	0.119773	+	0.992801i	$0.461783\pi$
$252$	−112.689	−7.09871
$253$	−2.04870	−0.128801
$254$	11.7061	0.734508
$255$	9.83672	0.615999
$256$	−24.3593	−1.52246
$257$	−28.0664	−1.75073	−0.875367	−	0.483459i	$-0.839380\pi$
$257$	−28.0664	−1.75073	−0.875367	+	0.483459i	$0.839380\pi$
$258$	−25.2471	−1.57181
$259$	40.6289	2.52456
$260$	−10.9127	−0.676775
$261$	−52.5269	−3.25133
$262$	−5.06573	−0.312962
$263$	12.1836	0.751274	0.375637	−	0.926767i	$-0.377424\pi$
$263$	12.1836	0.751274	0.375637	+	0.926767i	$0.377424\pi$
$264$	4.93491	0.303722
$265$	−6.68259	−0.410508
$266$	−5.12549	−0.314264
$267$	−54.7215	−3.34890
$268$	−62.0646	−3.79120
$269$	−8.70140	−0.530534	−0.265267	−	0.964175i	$-0.585460\pi$
$269$	−8.70140	−0.530534	−0.265267	+	0.964175i	$0.585460\pi$
$270$	23.8942	1.45415
$271$	−19.0702	−1.15843	−0.579217	−	0.815174i	$-0.696641\pi$
$271$	−19.0702	−1.15843	−0.579217	+	0.815174i	$0.696641\pi$
$272$	−27.0274	−1.63878
$273$	−28.2705	−1.71101
$274$	−50.0082	−3.02111
$275$	−0.240356	−0.0144940
$276$	119.254	7.17826
$277$	−25.4817	−1.53104	−0.765522	−	0.643409i	$-0.777519\pi$
$277$	−25.4817	−1.53104	−0.765522	+	0.643409i	$0.777519\pi$
$278$	−28.4397	−1.70570
$279$	16.5662	0.991795
$280$	−27.2127	−1.62627
$281$	−8.87894	−0.529673	−0.264837	−	0.964293i	$-0.585318\pi$
$281$	−8.87894	−0.529673	−0.264837	+	0.964293i	$0.585318\pi$
$282$	−76.3617	−4.54727
$283$	−17.5897	−1.04560	−0.522800	−	0.852456i	$-0.675112\pi$
$283$	−17.5897	−1.04560	−0.522800	+	0.852456i	$0.675112\pi$
$284$	−24.0265	−1.42571
$285$	1.50030	0.0888699
$286$	1.45584	0.0860859
$287$	12.1391	0.716548
$288$	−46.8370	−2.75990
$289$	−6.33985	−0.372932
$290$	−22.2798	−1.30831
$291$	6.68458	0.391857
$292$	−26.6093	−1.55719
$293$	9.13388	0.533607	0.266803	−	0.963751i	$-0.414033\pi$
$293$	9.13388	0.533607	0.266803	+	0.963751i	$0.414033\pi$
$294$	−69.4662	−4.05135
$295$	−7.90583	−0.460295
$296$	69.3385	4.03022
$297$	−2.22811	−0.129288
$298$	35.2211	2.04030
$299$	20.0296	1.15834
$300$	13.9911	0.807775
$301$	−12.9822	−0.748279
$302$	25.3155	1.45674
$303$	13.0889	0.751938
$304$	−4.12222	−0.236426
$305$	8.36120	0.478761
$306$	51.1416	2.92357
$307$	−9.47729	−0.540897	−0.270449	−	0.962734i	$-0.587172\pi$
$307$	−9.47729	−0.540897	−0.270449	+	0.962734i	$0.587172\pi$
$308$	4.45711	0.253967
$309$	36.5670	2.08022
$310$	7.02674	0.399092
$311$	−8.29199	−0.470196	−0.235098	−	0.971972i	$-0.575541\pi$
$311$	−8.29199	−0.470196	−0.235098	+	0.971972i	$0.575541\pi$
$312$	−48.2472	−2.73146
$313$	−20.6675	−1.16820	−0.584099	−	0.811682i	$-0.698552\pi$
$313$	−20.6675	−1.16820	−0.584099	+	0.811682i	$0.698552\pi$
$314$	−33.8156	−1.90832
$315$	24.2660	1.36723
$316$	5.29055	0.297617
$317$	−2.17977	−0.122428	−0.0612140	−	0.998125i	$-0.519497\pi$
$317$	−2.17977	−0.122428	−0.0612140	+	0.998125i	$0.519497\pi$
$318$	−51.8949	−2.91012
$319$	2.07757	0.116321
$320$	−3.31047	−0.185061
$321$	52.1803	2.91242
$322$	87.7304	4.88903
$323$	1.62589	0.0904667
$324$	45.0362	2.50201
$325$	2.34990	0.130349
$326$	−8.47139	−0.469187
$327$	−17.0357	−0.942077
$328$	20.7169	1.14390
$329$	−39.2656	−2.16478
$330$	−1.86653	−0.102749
$331$	−0.326641	−0.0179538	−0.00897691	−	0.999960i	$-0.502857\pi$
$331$	−0.326641	−0.0179538	−0.00897691	+	0.999960i	$0.502857\pi$
$332$	−36.9906	−2.03012
$333$	−61.8302	−3.38827
$334$	−57.8383	−3.16477
$335$	13.3648	0.730196
$336$	−99.5882	−5.43298
$337$	−22.5823	−1.23014	−0.615068	−	0.788474i	$-0.710872\pi$
$337$	−22.5823	−1.23014	−0.615068	+	0.788474i	$0.710872\pi$
$338$	19.2751	1.04843
$339$	−48.6739	−2.64360
$340$	15.1623	0.822289
$341$	−0.655235	−0.0354830
$342$	7.80012	0.421782
$343$	−7.76776	−0.419420
$344$	−22.1557	−1.19456
$345$	−25.6798	−1.38255
$346$	36.3234	1.95276
$347$	32.7526	1.75825	0.879127	−	0.476588i	$-0.158127\pi$
$347$	32.7526	1.75825	0.879127	+	0.476588i	$0.158127\pi$
$348$	−120.935	−6.48278
$349$	1.88652	0.100983	0.0504916	−	0.998724i	$-0.483921\pi$
$349$	1.88652	0.100983	0.0504916	+	0.998724i	$0.483921\pi$
$350$	10.2927	0.550165
$351$	21.7836	1.16272
$352$	1.85252	0.0987396
$353$	12.1793	0.648241	0.324120	−	0.946016i	$-0.394932\pi$
$353$	12.1793	0.648241	0.324120	+	0.946016i	$0.394932\pi$
$354$	−61.3942	−3.26307
$355$	5.17378	0.274596
$356$	−84.3473	−4.47040
$357$	39.2795	2.07889
$358$	20.1210	1.06343
$359$	−28.5411	−1.50634	−0.753171	−	0.657825i	$-0.771477\pi$
$359$	−28.5411	−1.50634	−0.753171	+	0.657825i	$0.771477\pi$
$360$	41.4130	2.18266
$361$	−18.7520	−0.986948
$362$	9.75428	0.512673
$363$	−32.9666	−1.73030
$364$	−43.5759	−2.28400
$365$	5.72996	0.299920
$366$	64.9305	3.39397
$367$	1.14379	0.0597051	0.0298526	−	0.999554i	$-0.490496\pi$
$367$	1.14379	0.0597051	0.0298526	+	0.999554i	$0.490496\pi$
$368$	70.5580	3.67809
$369$	−18.4736	−0.961697
$370$	−26.2259	−1.36342
$371$	−26.6846	−1.38540
$372$	38.1411	1.97752
$373$	−35.2826	−1.82687	−0.913433	−	0.406989i	$-0.866579\pi$
$373$	−35.2826	−1.82687	−0.913433	+	0.406989i	$0.866579\pi$
$374$	−2.02278	−0.104595
$375$	−3.01279	−0.155580
$376$	−67.0118	−3.45587
$377$	−20.3118	−1.04611
$378$	95.4131	4.90752
$379$	−15.9848	−0.821087	−0.410543	−	0.911841i	$-0.634661\pi$
$379$	−15.9848	−0.821087	−0.410543	+	0.911841i	$0.634661\pi$
$380$	2.31255	0.118631
$381$	−13.6827	−0.700984
$382$	0.705830	0.0361134
$383$	−21.9768	−1.12296	−0.561481	−	0.827490i	$-0.689768\pi$
$383$	−21.9768	−1.12296	−0.561481	+	0.827490i	$0.689768\pi$
$384$	20.7334	1.05805
$385$	−0.959778	−0.0489148
$386$	39.1861	1.99452
$387$	19.7566	1.00429
$388$	10.3036	0.523084
$389$	13.9080	0.705163	0.352581	−	0.935781i	$-0.385304\pi$
$389$	13.9080	0.705163	0.352581	+	0.935781i	$0.385304\pi$
$390$	18.2486	0.924052
$391$	−27.8295	−1.40740
$392$	−60.9605	−3.07897
$393$	5.92105	0.298678
$394$	−41.8998	−2.11088
$395$	−1.13925	−0.0573218
$396$	−6.78295	−0.340856
$397$	6.33726	0.318058	0.159029	−	0.987274i	$-0.449164\pi$
$397$	6.33726	0.318058	0.159029	+	0.987274i	$0.449164\pi$
$398$	9.96913	0.499707
$399$	5.99091	0.299921
$400$	8.27796	0.413898
$401$	−21.4939	−1.07335	−0.536677	−	0.843788i	$-0.680321\pi$
$401$	−21.4939	−1.07335	−0.536677	+	0.843788i	$0.680321\pi$
$402$	103.787	5.17641
$403$	6.40606	0.319109
$404$	20.1752	1.00375
$405$	−9.69794	−0.481894
$406$	−88.9666	−4.41534
$407$	2.44554	0.121221
$408$	67.0356	3.31876
$409$	4.56383	0.225667	0.112833	−	0.993614i	$-0.464007\pi$
$409$	4.56383	0.225667	0.112833	+	0.993614i	$0.464007\pi$
$410$	−7.83577	−0.386981
$411$	58.4519	2.88322
$412$	56.3642	2.77686
$413$	−31.5692	−1.55342
$414$	−133.511	−6.56169
$415$	7.96542	0.391007
$416$	−18.1116	−0.887993
$417$	33.2416	1.62785
$418$	−0.308514	−0.0150899
$419$	7.17368	0.350457	0.175229	−	0.984528i	$-0.443934\pi$
$419$	7.17368	0.350457	0.175229	+	0.984528i	$0.443934\pi$
$420$	55.8685	2.72610
$421$	15.1029	0.736069	0.368035	−	0.929812i	$-0.380031\pi$
$421$	15.1029	0.736069	0.368035	+	0.929812i	$0.380031\pi$
$422$	−19.8962	−0.968534
$423$	59.7554	2.90541
$424$	−45.5407	−2.21165
$425$	−3.26499	−0.158375
$426$	40.1780	1.94663
$427$	33.3876	1.61574
$428$	80.4304	3.88775
$429$	−1.70166	−0.0821568
$430$	8.37997	0.404118
$431$	−7.01244	−0.337778	−0.168889	−	0.985635i	$-0.554018\pi$
$431$	−7.01244	−0.337778	−0.168889	+	0.985635i	$0.554018\pi$
$432$	76.7369	3.69200
$433$	21.3385	1.02547	0.512733	−	0.858548i	$-0.328633\pi$
$433$	21.3385	1.02547	0.512733	+	0.858548i	$0.328633\pi$
$434$	28.0588	1.34687
$435$	26.0416	1.24860
$436$	−26.2587	−1.25757
$437$	−4.24455	−0.203044
$438$	44.4970	2.12615
$439$	−4.42737	−0.211307	−0.105653	−	0.994403i	$-0.533693\pi$
$439$	−4.42737	−0.211307	−0.105653	+	0.994403i	$0.533693\pi$
$440$	−1.63799	−0.0780879
$441$	54.3594	2.58854
$442$	19.7761	0.940655
$443$	−15.9698	−0.758748	−0.379374	−	0.925243i	$-0.623861\pi$
$443$	−15.9698	−0.758748	−0.379374	+	0.925243i	$0.623861\pi$
$444$	−142.354	−6.75583
$445$	18.1631	0.861012
$446$	−38.3502	−1.81594
$447$	−41.1681	−1.94718
$448$	−13.2192	−0.624548
$449$	1.82119	0.0859472	0.0429736	−	0.999076i	$-0.486317\pi$
$449$	1.82119	0.0859472	0.0429736	+	0.999076i	$0.486317\pi$
$450$	−15.6636	−0.738391
$451$	0.730677	0.0344062
$452$	−75.0257	−3.52891
$453$	−29.5899	−1.39026
$454$	54.9836	2.58051
$455$	9.38349	0.439905
$456$	10.2243	0.478795
$457$	−22.6484	−1.05945	−0.529723	−	0.848171i	$-0.677704\pi$
$457$	−22.6484	−1.05945	−0.529723	+	0.848171i	$0.677704\pi$
$458$	40.2296	1.87981
$459$	−30.2665	−1.41272
$460$	−39.5827	−1.84555
$461$	17.4422	0.812366	0.406183	−	0.913792i	$-0.366859\pi$
$461$	17.4422	0.812366	0.406183	+	0.913792i	$0.366859\pi$
$462$	−7.45334	−0.346761
$463$	−39.9694	−1.85753	−0.928767	−	0.370663i	$-0.879130\pi$
$463$	−39.9694	−1.85753	−0.928767	+	0.370663i	$0.879130\pi$
$464$	−71.5522	−3.32173
$465$	−8.21317	−0.380877
$466$	15.4304	0.714800
$467$	18.8262	0.871171	0.435585	−	0.900147i	$-0.356541\pi$
$467$	18.8262	0.871171	0.435585	+	0.900147i	$0.356541\pi$
$468$	66.3151	3.06542
$469$	53.3676	2.46429
$470$	25.3459	1.16912
$471$	39.5252	1.82122
$472$	−53.8769	−2.47989
$473$	−0.781423	−0.0359299
$474$	−8.84705	−0.406358
$475$	−0.497976	−0.0228487
$476$	60.5452	2.77509
$477$	40.6094	1.85938
$478$	−56.3566	−2.57769
$479$	14.3326	0.654875	0.327438	−	0.944873i	$-0.393815\pi$
$479$	14.3326	0.654875	0.327438	+	0.944873i	$0.393815\pi$
$480$	23.2207	1.05988
$481$	−23.9093	−1.09017
$482$	5.07881	0.231334
$483$	−102.543	−4.66588
$484$	−50.8145	−2.30975
$485$	−2.21873	−0.100748
$486$	−3.62862	−0.164597
$487$	35.7830	1.62148	0.810741	−	0.585405i	$-0.199064\pi$
$487$	35.7830	1.62148	0.810741	+	0.585405i	$0.199064\pi$
$488$	56.9802	2.57937
$489$	9.90175	0.447772
$490$	23.0571	1.04161
$491$	−8.99806	−0.406077	−0.203038	−	0.979171i	$-0.565082\pi$
$491$	−8.99806	−0.406077	−0.203038	+	0.979171i	$0.565082\pi$
$492$	−42.5325	−1.91751
$493$	28.2216	1.27104
$494$	3.01626	0.135708
$495$	1.46062	0.0656499
$496$	22.5666	1.01327
$497$	20.6597	0.926714
$498$	61.8570	2.77188
$499$	19.6495	0.879631	0.439815	−	0.898088i	$-0.355044\pi$
$499$	19.6495	0.879631	0.439815	+	0.898088i	$0.355044\pi$
$500$	−4.64389	−0.207681
$501$	67.6040	3.02033
$502$	−9.78224	−0.436603
$503$	−26.2325	−1.16965	−0.584824	−	0.811160i	$-0.698836\pi$
$503$	−26.2325	−1.16965	−0.584824	+	0.811160i	$0.698836\pi$
$504$	165.368	7.36610
$505$	−4.34445	−0.193326
$506$	5.28067	0.234755
$507$	−22.5296	−1.00057
$508$	−21.0904	−0.935734
$509$	−36.9629	−1.63835	−0.819175	−	0.573544i	$-0.805568\pi$
$509$	−36.9629	−1.63835	−0.819175	+	0.573544i	$0.805568\pi$
$510$	−25.3549	−1.12273
$511$	22.8806	1.01218
$512$	49.0243	2.16659
$513$	−4.61625	−0.203812
$514$	72.3433	3.19092
$515$	−12.1373	−0.534832
$516$	45.4864	2.00243
$517$	−2.36347	−0.103945
$518$	−104.724	−4.60131
$519$	−42.4564	−1.86363
$520$	16.0141	0.702267
$521$	7.87356	0.344947	0.172473	−	0.985014i	$-0.444824\pi$
$521$	7.87356	0.344947	0.172473	+	0.985014i	$0.444824\pi$
$522$	135.392	5.92594
$523$	−29.8560	−1.30551	−0.652757	−	0.757567i	$-0.726388\pi$
$523$	−29.8560	−1.30551	−0.652757	+	0.757567i	$0.726388\pi$
$524$	9.12668	0.398701
$525$	−12.0305	−0.525055
$526$	−31.4042	−1.36929
$527$	−8.90070	−0.387721
$528$	−5.99442	−0.260873
$529$	49.6517	2.15877
$530$	17.2249	0.748201
$531$	48.0429	2.08488
$532$	9.23436	0.400360
$533$	−7.14363	−0.309425
$534$	141.049	6.10377
$535$	−17.3196	−0.748791
$536$	91.0788	3.93400
$537$	−23.5183	−1.01489
$538$	22.4285	0.966962
$539$	−2.15005	−0.0926092
$540$	−43.0490	−1.85253
$541$	−2.91554	−0.125349	−0.0626745	−	0.998034i	$-0.519963\pi$
$541$	−2.91554	−0.125349	−0.0626745	+	0.998034i	$0.519963\pi$
$542$	49.1549	2.11138
$543$	−11.4012	−0.489274
$544$	25.1646	1.07892
$545$	5.65447	0.242211
$546$	72.8693	3.11852
$547$	25.7415	1.10063	0.550313	−	0.834959i	$-0.314509\pi$
$547$	25.7415	1.10063	0.550313	+	0.834959i	$0.314509\pi$
$548$	90.0974	3.84877
$549$	−50.8101	−2.16852
$550$	0.619536	0.0264171
$551$	4.30436	0.183372
$552$	−175.004	−7.44865
$553$	−4.54919	−0.193451
$554$	65.6809	2.79051
$555$	30.6540	1.30119
$556$	51.2384	2.17299
$557$	11.9451	0.506128	0.253064	−	0.967450i	$-0.418562\pi$
$557$	11.9451	0.506128	0.253064	+	0.967450i	$0.418562\pi$
$558$	−42.7007	−1.80767
$559$	7.63976	0.323127
$560$	33.0552	1.39684
$561$	2.36432	0.0998215
$562$	22.8861	0.965393
$563$	16.5052	0.695609	0.347805	−	0.937567i	$-0.386927\pi$
$563$	16.5052	0.695609	0.347805	+	0.937567i	$0.386927\pi$
$564$	137.577	5.79305
$565$	16.1558	0.679678
$566$	45.3388	1.90573
$567$	−38.7254	−1.62631
$568$	35.2585	1.47941
$569$	18.6182	0.780515	0.390258	−	0.920706i	$-0.372386\pi$
$569$	18.6182	0.780515	0.390258	+	0.920706i	$0.372386\pi$
$570$	−3.86713	−0.161976
$571$	−36.8115	−1.54051	−0.770257	−	0.637734i	$-0.779872\pi$
$571$	−36.8115	−1.54051	−0.770257	+	0.637734i	$0.779872\pi$
$572$	−2.62293	−0.109670
$573$	−0.825006	−0.0344651
$574$	−31.2894	−1.30599
$575$	8.52360	0.355459
$576$	20.1173	0.838222
$577$	−32.4206	−1.34969	−0.674844	−	0.737960i	$-0.735789\pi$
$577$	−32.4206	−1.34969	−0.674844	+	0.737960i	$0.735789\pi$
$578$	16.3414	0.679714
$579$	−45.8026	−1.90349
$580$	40.1404	1.66674
$581$	31.8071	1.31958
$582$	−17.2300	−0.714206
$583$	−1.60620	−0.0665220
$584$	39.0487	1.61585
$585$	−14.2801	−0.590408
$586$	−23.5433	−0.972563
$587$	−25.6717	−1.05959	−0.529793	−	0.848127i	$-0.677730\pi$
$587$	−25.6717	−1.05959	−0.529793	+	0.848127i	$0.677730\pi$
$588$	125.154	5.16126
$589$	−1.35753	−0.0559362
$590$	20.3779	0.838944
$591$	48.9744	2.01454
$592$	−84.2253	−3.46164
$593$	−37.1709	−1.52643	−0.763213	−	0.646147i	$-0.776379\pi$
$593$	−37.1709	−1.52643	−0.763213	+	0.646147i	$0.776379\pi$
$594$	5.74311	0.235643
$595$	−13.0376	−0.534490
$596$	−63.4562	−2.59927
$597$	−11.6524	−0.476900
$598$	−51.6277	−2.11121
$599$	−48.5737	−1.98467	−0.992334	−	0.123583i	$-0.960562\pi$
$599$	−48.5737	−1.98467	−0.992334	+	0.123583i	$0.960562\pi$
$600$	−20.5316	−0.838201
$601$	7.68144	0.313332	0.156666	−	0.987652i	$-0.449925\pi$
$601$	7.68144	0.313332	0.156666	+	0.987652i	$0.449925\pi$
$602$	33.4625	1.36383
$603$	−81.2163	−3.30738
$604$	−45.6097	−1.85583
$605$	10.9422	0.444865
$606$	−33.7377	−1.37050
$607$	38.9986	1.58291	0.791453	−	0.611230i	$-0.209325\pi$
$607$	38.9986	1.58291	0.791453	+	0.611230i	$0.209325\pi$
$608$	3.83810	0.155655
$609$	103.988	4.21382
$610$	−21.5516	−0.872600
$611$	23.1070	0.934811
$612$	−92.1394	−3.72452
$613$	8.89036	0.359078	0.179539	−	0.983751i	$-0.442539\pi$
$613$	8.89036	0.359078	0.179539	+	0.983751i	$0.442539\pi$
$614$	24.4284	0.985851
$615$	9.15880	0.369319
$616$	−6.54073	−0.263534
$617$	−22.3139	−0.898322	−0.449161	−	0.893451i	$-0.648277\pi$
$617$	−22.3139	−0.898322	−0.449161	+	0.893451i	$0.648277\pi$
$618$	−94.2542	−3.79146
$619$	−36.3477	−1.46094	−0.730468	−	0.682946i	$-0.760698\pi$
$619$	−36.3477	−1.46094	−0.730468	+	0.682946i	$0.760698\pi$
$620$	−12.6597	−0.508427
$621$	79.0140	3.17072
$622$	21.3732	0.856989
$623$	72.5279	2.90577
$624$	58.6058	2.34611
$625$	1.00000	0.0400000
$626$	53.2722	2.12918
$627$	0.360605	0.0144012
$628$	60.9239	2.43113
$629$	33.2201	1.32457
$630$	−62.5473	−2.49195
$631$	2.67504	0.106492	0.0532458	−	0.998581i	$-0.483043\pi$
$631$	2.67504	0.106492	0.0532458	+	0.998581i	$0.483043\pi$
$632$	−7.76379	−0.308827
$633$	23.2556	0.924329
$634$	5.61852	0.223140
$635$	4.54153	0.180225
$636$	93.4965	3.70738
$637$	21.0205	0.832861
$638$	−5.35508	−0.212010
$639$	−31.4405	−1.24377
$640$	−6.88181	−0.272027
$641$	1.68325	0.0664844	0.0332422	−	0.999447i	$-0.489417\pi$
$641$	1.68325	0.0664844	0.0332422	+	0.999447i	$0.489417\pi$
$642$	−134.499	−5.30824
$643$	−32.9427	−1.29914	−0.649568	−	0.760304i	$-0.725050\pi$
$643$	−32.9427	−1.29914	−0.649568	+	0.760304i	$0.725050\pi$
$644$	−158.060	−6.22842
$645$	−9.79489	−0.385673
$646$	−4.19084	−0.164887
$647$	27.3315	1.07451	0.537255	−	0.843420i	$-0.319461\pi$
$647$	27.3315	1.07451	0.537255	+	0.843420i	$0.319461\pi$
$648$	−66.0898	−2.59625
$649$	−1.90021	−0.0745899
$650$	−6.05703	−0.237576
$651$	−32.7965	−1.28539
$652$	15.2625	0.597725
$653$	−11.8858	−0.465128	−0.232564	−	0.972581i	$-0.574712\pi$
$653$	−11.8858	−0.465128	−0.232564	+	0.972581i	$0.574712\pi$
$654$	43.9108	1.71705
$655$	−1.96531	−0.0767909
$656$	−25.1648	−0.982520
$657$	−34.8203	−1.35847
$658$	101.210	3.94557
$659$	32.0413	1.24815	0.624076	−	0.781364i	$-0.285476\pi$
$659$	32.0413	1.24815	0.624076	+	0.781364i	$0.285476\pi$
$660$	3.36284	0.130898
$661$	−15.4885	−0.602434	−0.301217	−	0.953556i	$-0.597393\pi$
$661$	−15.4885	−0.602434	−0.301217	+	0.953556i	$0.597393\pi$
$662$	0.841942	0.0327230
$663$	−23.1153	−0.897723
$664$	54.2830	2.10659
$665$	−1.98849	−0.0771105
$666$	159.372	6.17554
$667$	−73.6755	−2.85273
$668$	104.204	4.03179
$669$	44.8255	1.73305
$670$	−34.4487	−1.33087
$671$	2.00967	0.0775823
$672$	92.7240	3.57690
$673$	46.4803	1.79168	0.895842	−	0.444372i	$-0.146573\pi$
$673$	46.4803	1.79168	0.895842	+	0.444372i	$0.146573\pi$
$674$	58.2076	2.24207
$675$	9.27002	0.356803
$676$	−34.7270	−1.33565
$677$	−23.4815	−0.902468	−0.451234	−	0.892406i	$-0.649016\pi$
$677$	−23.4815	−0.902468	−0.451234	+	0.892406i	$0.649016\pi$
$678$	125.461	4.81829
$679$	−8.85975	−0.340006
$680$	−22.2504	−0.853262
$681$	−64.2674	−2.46273
$682$	1.68892	0.0646720
$683$	−15.5946	−0.596711	−0.298356	−	0.954455i	$-0.596438\pi$
$683$	−15.5946	−0.596711	−0.298356	+	0.954455i	$0.596438\pi$
$684$	−14.0531	−0.537334
$685$	−19.4013	−0.741284
$686$	20.0220	0.764443
$687$	−47.0223	−1.79401
$688$	26.9125	1.02603
$689$	15.7034	0.598251
$690$	66.1916	2.51987
$691$	−23.0658	−0.877466	−0.438733	−	0.898617i	$-0.644573\pi$
$691$	−23.0658	−0.877466	−0.438733	+	0.898617i	$0.644573\pi$
$692$	−65.4420	−2.48773
$693$	5.83247	0.221557
$694$	−84.4224	−3.20463
$695$	−11.0335	−0.418525
$696$	177.469	6.72696
$697$	9.92549	0.375955
$698$	−4.86265	−0.184054
$699$	−18.0358	−0.682176
$700$	−18.5438	−0.700889
$701$	−43.9476	−1.65988	−0.829939	−	0.557855i	$-0.811625\pi$
$701$	−43.9476	−1.65988	−0.829939	+	0.557855i	$0.811625\pi$
$702$	−56.1488	−2.11920
$703$	5.06673	0.191095
$704$	−0.795690	−0.0299887
$705$	−29.6254	−1.11576
$706$	−31.3932	−1.18150
$707$	−17.3481	−0.652441
$708$	110.611	4.15702
$709$	21.2026	0.796279	0.398140	−	0.917325i	$-0.369656\pi$
$709$	21.2026	0.796279	0.398140	+	0.917325i	$0.369656\pi$
$710$	−13.3358	−0.500484
$711$	6.92309	0.259636
$712$	123.778	4.63879
$713$	23.2362	0.870203
$714$	−101.246	−3.78903
$715$	0.564812	0.0211228
$716$	−36.2510	−1.35476
$717$	65.8721	2.46004
$718$	73.5668	2.74549
$719$	17.5453	0.654328	0.327164	−	0.944968i	$-0.393907\pi$
$719$	17.5453	0.654328	0.327164	+	0.944968i	$0.393907\pi$
$720$	−50.3043	−1.87473
$721$	−48.4660	−1.80497
$722$	48.3347	1.79883
$723$	−5.93635	−0.220775
$724$	−17.5738	−0.653126
$725$	−8.64370	−0.321019
$726$	84.9739	3.15368
$727$	−40.6285	−1.50683	−0.753414	−	0.657547i	$-0.771594\pi$
$727$	−40.6285	−1.50683	−0.753414	+	0.657547i	$0.771594\pi$
$728$	63.9469	2.37003
$729$	−24.8525	−0.920464
$730$	−14.7694	−0.546640
$731$	−10.6148	−0.392603
$732$	−116.982	−4.32378
$733$	24.7541	0.914315	0.457157	−	0.889386i	$-0.348868\pi$
$733$	24.7541	0.914315	0.457157	+	0.889386i	$0.348868\pi$
$734$	−2.94819	−0.108820
$735$	−26.9502	−0.994073
$736$	−65.6947	−2.42154
$737$	3.21231	0.118327
$738$	47.6171	1.75281
$739$	−40.1899	−1.47841	−0.739204	−	0.673481i	$-0.764798\pi$
$739$	−40.1899	−1.47841	−0.739204	+	0.673481i	$0.764798\pi$
$740$	47.2499	1.73694
$741$	−3.52554	−0.129514
$742$	68.7815	2.52505
$743$	24.5455	0.900489	0.450244	−	0.892905i	$-0.351337\pi$
$743$	24.5455	0.900489	0.450244	+	0.892905i	$0.351337\pi$
$744$	−55.9714	−2.05201
$745$	13.6644	0.500626
$746$	90.9437	3.32968
$747$	−48.4050	−1.77105
$748$	3.64434	0.133250
$749$	−69.1598	−2.52704
$750$	7.76569	0.283563
$751$	26.1073	0.952668	0.476334	−	0.879264i	$-0.341965\pi$
$751$	26.1073	0.952668	0.476334	+	0.879264i	$0.341965\pi$
$752$	81.3990	2.96832
$753$	11.4339	0.416676
$754$	52.3552	1.90666
$755$	9.82144	0.357439
$756$	−171.901	−6.25199
$757$	−54.7903	−1.99139	−0.995694	−	0.0926956i	$-0.970452\pi$
$757$	−54.7903	−1.99139	−0.995694	+	0.0926956i	$0.970452\pi$
$758$	41.2021	1.49653
$759$	−6.17229	−0.224040
$760$	−3.39362	−0.123100
$761$	7.65409	0.277461	0.138730	−	0.990330i	$-0.455698\pi$
$761$	7.65409	0.277461	0.138730	+	0.990330i	$0.455698\pi$
$762$	35.2681	1.27763
$763$	22.5792	0.817420
$764$	−1.27166	−0.0460070
$765$	19.8410	0.717352
$766$	56.6469	2.04673
$767$	18.5779	0.670808
$768$	−73.3895	−2.64821
$769$	46.7016	1.68410	0.842051	−	0.539398i	$-0.181348\pi$
$769$	46.7016	1.68410	0.842051	+	0.539398i	$0.181348\pi$
$770$	2.47390	0.0891532
$771$	−84.5581	−3.04529
$772$	−70.5998	−2.54094
$773$	9.80710	0.352737	0.176368	−	0.984324i	$-0.443565\pi$
$773$	9.80710	0.352737	0.176368	+	0.984324i	$0.443565\pi$
$774$	−50.9242	−1.83043
$775$	2.72610	0.0979245
$776$	−15.1203	−0.542787
$777$	122.406	4.39130
$778$	−35.8489	−1.28524
$779$	1.51383	0.0542388
$780$	−32.8776	−1.17721
$781$	1.24355	0.0444977
$782$	71.7325	2.56515
$783$	−80.1273	−2.86352
$784$	74.0486	2.64459
$785$	−13.1191	−0.468242
$786$	−15.2620	−0.544376
$787$	−7.26128	−0.258837	−0.129418	−	0.991590i	$-0.541311\pi$
$787$	−7.26128	−0.258837	−0.129418	+	0.991590i	$0.541311\pi$
$788$	75.4889	2.68918
$789$	36.7067	1.30679
$790$	2.93650	0.104476
$791$	64.5125	2.29380
$792$	9.95387	0.353695
$793$	−19.6480	−0.697719
$794$	−16.3348	−0.579699
$795$	−20.1332	−0.714052
$796$	−17.9609	−0.636607
$797$	14.1109	0.499834	0.249917	−	0.968267i	$-0.419597\pi$
$797$	14.1109	0.499834	0.249917	+	0.968267i	$0.419597\pi$
$798$	−15.4420	−0.546642
$799$	−32.1054	−1.13581
$800$	−7.70739	−0.272497
$801$	−110.375	−3.89991
$802$	55.4021	1.95632
$803$	1.37723	0.0486014
$804$	−186.988	−6.59454
$805$	34.0360	1.19961
$806$	−16.5121	−0.581614
$807$	−26.2155	−0.922828
$808$	−29.6067	−1.04156
$809$	18.8275	0.661940	0.330970	−	0.943641i	$-0.392624\pi$
$809$	18.8275	0.661940	0.330970	+	0.943641i	$0.392624\pi$
$810$	24.9972	0.878311
$811$	−4.26956	−0.149925	−0.0749623	−	0.997186i	$-0.523884\pi$
$811$	−4.26956	−0.149925	−0.0749623	+	0.997186i	$0.523884\pi$
$812$	160.287	5.62497
$813$	−57.4545	−2.01502
$814$	−6.30355	−0.220939
$815$	−3.28657	−0.115124
$816$	−81.4280	−2.85055
$817$	−1.61897	−0.0566407
$818$	−11.7636	−0.411305
$819$	−57.0225	−1.99253
$820$	14.1173	0.492998
$821$	−2.35501	−0.0821904	−0.0410952	−	0.999155i	$-0.513085\pi$
$821$	−2.35501	−0.0821904	−0.0410952	+	0.999155i	$0.513085\pi$
$822$	−150.664	−5.25501
$823$	23.5420	0.820624	0.410312	−	0.911945i	$-0.365420\pi$
$823$	23.5420	0.820624	0.410312	+	0.911945i	$0.365420\pi$
$824$	−82.7135	−2.88146
$825$	−0.724142	−0.0252114
$826$	81.3720	2.83129
$827$	41.8496	1.45525	0.727627	−	0.685973i	$-0.240623\pi$
$827$	41.8496	1.45525	0.727627	+	0.685973i	$0.240623\pi$
$828$	240.540	8.35933
$829$	26.4075	0.917169	0.458584	−	0.888651i	$-0.348357\pi$
$829$	26.4075	0.917169	0.458584	+	0.888651i	$0.348357\pi$
$830$	−20.5315	−0.712658
$831$	−76.7708	−2.66315
$832$	7.77925	0.269697
$833$	−29.2062	−1.01194
$834$	−85.6828	−2.96695
$835$	−22.4390	−0.776535
$836$	0.555835	0.0192239
$837$	25.2710	0.873495
$838$	−18.4907	−0.638751
$839$	9.64131	0.332855	0.166427	−	0.986054i	$-0.446777\pi$
$839$	9.64131	0.332855	0.166427	+	0.986054i	$0.446777\pi$
$840$	−81.9860	−2.82879
$841$	45.7136	1.57633
$842$	−38.9288	−1.34157
$843$	−26.7504	−0.921331
$844$	35.8461	1.23387
$845$	7.47799	0.257251
$846$	−154.024	−5.29546
$847$	43.6940	1.50134
$848$	55.3182	1.89963
$849$	−52.9940	−1.81875
$850$	8.41575	0.288658
$851$	−86.7246	−2.97288
$852$	−72.3867	−2.47993
$853$	45.0847	1.54367	0.771836	−	0.635821i	$-0.219339\pi$
$853$	45.0847	1.54367	0.771836	+	0.635821i	$0.219339\pi$
$854$	−86.0590	−2.94488
$855$	3.02615	0.103492
$856$	−118.030	−4.03419
$857$	−44.7103	−1.52728	−0.763638	−	0.645645i	$-0.776589\pi$
$857$	−44.7103	−1.52728	−0.763638	+	0.645645i	$0.776589\pi$
$858$	4.38615	0.149741
$859$	15.0979	0.515134	0.257567	−	0.966260i	$-0.417079\pi$
$859$	15.0979	0.515134	0.257567	+	0.966260i	$0.417079\pi$
$860$	−15.0978	−0.514830
$861$	36.5725	1.24639
$862$	18.0751	0.615640
$863$	−4.38562	−0.149288	−0.0746442	−	0.997210i	$-0.523782\pi$
$863$	−4.38562	−0.149288	−0.0746442	+	0.997210i	$0.523782\pi$
$864$	−71.4477	−2.43070
$865$	14.0921	0.479145
$866$	−55.0017	−1.86903
$867$	−19.1006	−0.648691
$868$	−50.5523	−1.71586
$869$	−0.273825	−0.00928889
$870$	−67.1243	−2.27573
$871$	−31.4058	−1.06415
$872$	38.5342	1.30493
$873$	13.4830	0.456331
$874$	10.9406	0.370073
$875$	3.99315	0.134993
$876$	−80.1682	−2.70863
$877$	−24.9893	−0.843828	−0.421914	−	0.906636i	$-0.638642\pi$
$877$	−24.9893	−0.843828	−0.421914	+	0.906636i	$0.638642\pi$
$878$	11.4119	0.385132
$879$	27.5184	0.928174
$880$	1.98966	0.0670713
$881$	37.9542	1.27871	0.639355	−	0.768912i	$-0.279202\pi$
$881$	37.9542	1.27871	0.639355	+	0.768912i	$0.279202\pi$
$882$	−140.116	−4.71794
$883$	−28.3775	−0.954979	−0.477489	−	0.878637i	$-0.658453\pi$
$883$	−28.3775	−0.954979	−0.477489	+	0.878637i	$0.658453\pi$
$884$	−35.6297	−1.19836
$885$	−23.8186	−0.800653
$886$	41.1634	1.38291
$887$	51.6034	1.73267	0.866337	−	0.499460i	$-0.166468\pi$
$887$	51.6034	1.73267	0.866337	+	0.499460i	$0.166468\pi$
$888$	208.902	7.01030
$889$	18.1350	0.608229
$890$	−46.8167	−1.56930
$891$	−2.33096	−0.0780900
$892$	69.0937	2.31343
$893$	−4.89671	−0.163862
$894$	106.114	3.54897
$895$	7.80616	0.260931
$896$	−27.4801	−0.918047
$897$	60.3449	2.01486
$898$	−4.69425	−0.156649
$899$	−23.5636	−0.785891
$900$	28.2204	0.940681
$901$	−21.8186	−0.726882
$902$	−1.88337	−0.0627095
$903$	−39.1125	−1.30158
$904$	110.099	3.66183
$905$	3.78428	0.125794
$906$	76.2702	2.53391
$907$	37.4691	1.24414	0.622071	−	0.782961i	$-0.286292\pi$
$907$	37.4691	1.24414	0.622071	+	0.782961i	$0.286292\pi$
$908$	−99.0613	−3.28747
$909$	26.4008	0.875658
$910$	−24.1867	−0.801780
$911$	−29.9082	−0.990902	−0.495451	−	0.868636i	$-0.664997\pi$
$911$	−29.9082	−0.990902	−0.495451	+	0.868636i	$0.664997\pi$
$912$	−12.4194	−0.411247
$913$	1.91454	0.0633619
$914$	58.3779	1.93097
$915$	25.1905	0.832773
$916$	−72.4798	−2.39480
$917$	−7.84777	−0.259156
$918$	78.0142	2.57485
$919$	11.0570	0.364735	0.182368	−	0.983230i	$-0.441624\pi$
$919$	11.0570	0.364735	0.182368	+	0.983230i	$0.441624\pi$
$920$	58.0869	1.91507
$921$	−28.5531	−0.940855
$922$	−44.9587	−1.48064
$923$	−12.1578	−0.400180
$924$	13.4283	0.441759
$925$	−10.1746	−0.334540
$926$	103.024	3.38558
$927$	73.7569	2.42249
$928$	66.6204	2.18692
$929$	23.1418	0.759257	0.379629	−	0.925139i	$-0.376052\pi$
$929$	23.1418	0.759257	0.379629	+	0.925139i	$0.376052\pi$
$930$	21.1701	0.694194
$931$	−4.45453	−0.145991
$932$	−27.8002	−0.910627
$933$	−24.9820	−0.817875
$934$	−48.5258	−1.58781
$935$	−0.784760	−0.0256644
$936$	−97.3163	−3.18088
$937$	−44.8418	−1.46492	−0.732458	−	0.680812i	$-0.761627\pi$
$937$	−44.8418	−1.46492	−0.732458	+	0.680812i	$0.761627\pi$
$938$	−137.559	−4.49146
$939$	−62.2669	−2.03200
$940$	−45.6644	−1.48941
$941$	−17.6650	−0.575861	−0.287931	−	0.957651i	$-0.592967\pi$
$941$	−17.6650	−0.575861	−0.287931	+	0.957651i	$0.592967\pi$
$942$	−101.879	−3.31940
$943$	−25.9115	−0.843796
$944$	65.4441	2.13003
$945$	37.0166	1.20415
$946$	2.01418	0.0654865
$947$	27.9782	0.909170	0.454585	−	0.890703i	$-0.349788\pi$
$947$	27.9782	0.909170	0.454585	+	0.890703i	$0.349788\pi$
$948$	15.9393	0.517685
$949$	−13.4648	−0.437086
$950$	1.28357	0.0416445
$951$	−6.56718	−0.212955
$952$	−88.8491	−2.87962
$953$	35.2559	1.14205	0.571025	−	0.820933i	$-0.306546\pi$
$953$	35.2559	1.14205	0.571025	+	0.820933i	$0.306546\pi$
$954$	−104.674	−3.38894
$955$	0.273835	0.00886109
$956$	101.535	3.28387
$957$	6.25927	0.202333
$958$	−36.9435	−1.19359
$959$	−77.4722	−2.50171
$960$	−9.97373	−0.321901
$961$	−23.5684	−0.760270
$962$	61.6281	1.98697
$963$	105.249	3.39161
$964$	−9.15025	−0.294710
$965$	15.2027	0.489393
$966$	264.313	8.50414
$967$	1.38138	0.0444223	0.0222111	−	0.999753i	$-0.492929\pi$
$967$	1.38138	0.0444223	0.0222111	+	0.999753i	$0.492929\pi$
$968$	74.5695	2.39675
$969$	4.89845	0.157361
$970$	5.71895	0.183625
$971$	1.24139	0.0398382	0.0199191	−	0.999802i	$-0.493659\pi$
$971$	1.24139	0.0398382	0.0199191	+	0.999802i	$0.493659\pi$
$972$	6.53751	0.209691
$973$	−44.0585	−1.41245
$974$	−92.2334	−2.95535
$975$	7.07974	0.226733
$976$	−69.2137	−2.21548
$977$	−1.95008	−0.0623886	−0.0311943	−	0.999513i	$-0.509931\pi$
$977$	−1.95008	−0.0623886	−0.0311943	+	0.999513i	$0.509931\pi$
$978$	−25.5225	−0.816119
$979$	4.36560	0.139525
$980$	−41.5409	−1.32697
$981$	−34.3616	−1.09708
$982$	23.1932	0.740124
$983$	9.92419	0.316532	0.158266	−	0.987396i	$-0.449410\pi$
$983$	9.92419	0.316532	0.158266	+	0.987396i	$0.449410\pi$
$984$	62.4157	1.98974
$985$	−16.2555	−0.517944
$986$	−72.7433	−2.31662
$987$	−118.299	−3.76549
$988$	−5.43425	−0.172886
$989$	27.7111	0.881162
$990$	−3.76485	−0.119655
$991$	5.19434	0.165004	0.0825018	−	0.996591i	$-0.473709\pi$
$991$	5.19434	0.165004	0.0825018	+	0.996591i	$0.473709\pi$
$992$	−21.0111	−0.667105
$993$	−0.984101	−0.0312295
$994$	−53.2519	−1.68905
$995$	3.86764	0.122612
$996$	−111.445	−3.53126
$997$	24.1922	0.766174	0.383087	−	0.923712i	$-0.374861\pi$
$997$	24.1922	0.766174	0.383087	+	0.923712i	$0.374861\pi$
$998$	−50.6480	−1.60323
$999$	−94.3192	−2.98413

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8045.2.a.b.1.5	✓	126

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8045.2.a.b.1.5	✓	126	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:	\( N \)	\(=\)	\( 8045 = 5 \cdot 1609 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57758 q^{2} +3.01279 q^{3} +4.64389 q^{4} -1.00000 q^{5} -7.76569 q^{6} -3.99315 q^{7} -6.81483 q^{8} +6.07689 q^{9} +O(q^{10})\)	\(q-2.57758 q^{2} +3.01279 q^{3} +4.64389 q^{4} -1.00000 q^{5} -7.76569 q^{6} -3.99315 q^{7} -6.81483 q^{8} +6.07689 q^{9} +2.57758 q^{10} -0.240356 q^{11} +13.9911 q^{12} +2.34990 q^{13} +10.2927 q^{14} -3.01279 q^{15} +8.27796 q^{16} -3.26499 q^{17} -15.6636 q^{18} -0.497976 q^{19} -4.64389 q^{20} -12.0305 q^{21} +0.619536 q^{22} +8.52360 q^{23} -20.5316 q^{24} +1.00000 q^{25} -6.05703 q^{26} +9.27002 q^{27} -18.5438 q^{28} -8.64370 q^{29} +7.76569 q^{30} +2.72610 q^{31} -7.70739 q^{32} -0.724142 q^{33} +8.41575 q^{34} +3.99315 q^{35} +28.2204 q^{36} -10.1746 q^{37} +1.28357 q^{38} +7.07974 q^{39} +6.81483 q^{40} -3.03998 q^{41} +31.0096 q^{42} +3.25111 q^{43} -1.11619 q^{44} -6.07689 q^{45} -21.9702 q^{46} +9.83322 q^{47} +24.9397 q^{48} +8.94527 q^{49} -2.57758 q^{50} -9.83672 q^{51} +10.9127 q^{52} +6.68259 q^{53} -23.8942 q^{54} +0.240356 q^{55} +27.2127 q^{56} -1.50030 q^{57} +22.2798 q^{58} +7.90583 q^{59} -13.9911 q^{60} -8.36120 q^{61} -7.02674 q^{62} -24.2660 q^{63} +3.31047 q^{64} -2.34990 q^{65} +1.86653 q^{66} -13.3648 q^{67} -15.1623 q^{68} +25.6798 q^{69} -10.2927 q^{70} -5.17378 q^{71} -41.4130 q^{72} -5.72996 q^{73} +26.2259 q^{74} +3.01279 q^{75} -2.31255 q^{76} +0.959778 q^{77} -18.2486 q^{78} +1.13925 q^{79} -8.27796 q^{80} +9.69794 q^{81} +7.83577 q^{82} -7.96542 q^{83} -55.8685 q^{84} +3.26499 q^{85} -8.37997 q^{86} -26.0416 q^{87} +1.63799 q^{88} -18.1631 q^{89} +15.6636 q^{90} -9.38349 q^{91} +39.5827 q^{92} +8.21317 q^{93} -25.3459 q^{94} +0.497976 q^{95} -23.2207 q^{96} +2.21873 q^{97} -23.0571 q^{98} -1.46062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9}+O(q^{10}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9	\( 126 q + 5 q^{2} - 9 q^{3} + 109 q^{4} - 126 q^{5} - 21 q^{6} - 23 q^{7} + 12 q^{8} + 109 q^{9} - 5 q^{10} - 44 q^{11} - 11 q^{12} - 35 q^{13} - 14 q^{14} + 9 q^{15} + 75 q^{16} + 11 q^{17} - 15 q^{18} - 130 q^{19} - 109 q^{20} - 44 q^{21} - 14 q^{22} + 75 q^{23} - 63 q^{24} + 126 q^{25} - 43 q^{26} - 42 q^{27} - 77 q^{28} - 24 q^{29} + 21 q^{30} - 78 q^{31} + 24 q^{32} - 29 q^{33} - 57 q^{34} + 23 q^{35} + 50 q^{36} - 31 q^{37} - 3 q^{38} - 57 q^{39} - 12 q^{40} - 38 q^{41} - 10 q^{42} - 100 q^{43} - 90 q^{44} - 109 q^{45} - 96 q^{46} + 12 q^{47} - 22 q^{48} + 65 q^{49} + 5 q^{50} - 74 q^{51} - 112 q^{52} + 20 q^{53} - 90 q^{54} + 44 q^{55} - 57 q^{56} + 6 q^{57} - 35 q^{58} - 97 q^{59} + 11 q^{60} - 102 q^{61} - 16 q^{62} - 15 q^{63} + 4 q^{64} + 35 q^{65} - 83 q^{66} - 121 q^{67} + 41 q^{68} - 71 q^{69} + 14 q^{70} - 32 q^{71} - 32 q^{72} - 85 q^{73} - 42 q^{74} - 9 q^{75} - 275 q^{76} + 13 q^{77} + 10 q^{78} - 97 q^{79} - 75 q^{80} + 86 q^{81} - 55 q^{82} - 73 q^{83} - 111 q^{84} - 11 q^{85} - 56 q^{86} - q^{87} - 37 q^{88} - 67 q^{89} + 15 q^{90} - 180 q^{91} + 98 q^{92} - 44 q^{93} - 86 q^{94} + 130 q^{95} - 179 q^{96} - 50 q^{97} + 18 q^{98} - 217 q^{99}+O(q^{100}) \) 126 * q + 5 * q^2 - 9 * q^3 + 109 * q^4 - 126 * q^5 - 21 * q^6 - 23 * q^7 + 12 * q^8 + 109 * q^9 - 5 * q^10 - 44 * q^11 - 11 * q^12 - 35 * q^13 - 14 * q^14 + 9 * q^15 + 75 * q^16 + 11 * q^17 - 15 * q^18 - 130 * q^19 - 109 * q^20 - 44 * q^21 - 14 * q^22 + 75 * q^23 - 63 * q^24 + 126 * q^25 - 43 * q^26 - 42 * q^27 - 77 * q^28 - 24 * q^29 + 21 * q^30 - 78 * q^31 + 24 * q^32 - 29 * q^33 - 57 * q^34 + 23 * q^35 + 50 * q^36 - 31 * q^37 - 3 * q^38 - 57 * q^39 - 12 * q^40 - 38 * q^41 - 10 * q^42 - 100 * q^43 - 90 * q^44 - 109 * q^45 - 96 * q^46 + 12 * q^47 - 22 * q^48 + 65 * q^49 + 5 * q^50 - 74 * q^51 - 112 * q^52 + 20 * q^53 - 90 * q^54 + 44 * q^55 - 57 * q^56 + 6 * q^57 - 35 * q^58 - 97 * q^59 + 11 * q^60 - 102 * q^61 - 16 * q^62 - 15 * q^63 + 4 * q^64 + 35 * q^65 - 83 * q^66 - 121 * q^67 + 41 * q^68 - 71 * q^69 + 14 * q^70 - 32 * q^71 - 32 * q^72 - 85 * q^73 - 42 * q^74 - 9 * q^75 - 275 * q^76 + 13 * q^77 + 10 * q^78 - 97 * q^79 - 75 * q^80 + 86 * q^81 - 55 * q^82 - 73 * q^83 - 111 * q^84 - 11 * q^85 - 56 * q^86 - q^87 - 37 * q^88 - 67 * q^89 + 15 * q^90 - 180 * q^91 + 98 * q^92 - 44 * q^93 - 86 * q^94 + 130 * q^95 - 179 * q^96 - 50 * q^97 + 18 * q^98 - 217 * q^99

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