LMFDB - Embedded newform 8039.2.a.a.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8039,2,Mod(1,8039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8039, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8039 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8039.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:	$64.1917381849$
Analytic rank:	$1$
Dimension:	$279$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	8039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.62410 q^{2} +1.81984 q^{3} +4.88588 q^{4} -0.731341 q^{5} -4.77543 q^{6} -1.89048 q^{7} -7.57282 q^{8} +0.311808 q^{9} +O(q^{10})$	\(q-2.62410 q^{2} +1.81984 q^{3} +4.88588 q^{4} -0.731341 q^{5} -4.77543 q^{6} -1.89048 q^{7} -7.57282 q^{8} +0.311808 q^{9} +1.91911 q^{10} +6.30107 q^{11} +8.89151 q^{12} +3.74858 q^{13} +4.96081 q^{14} -1.33092 q^{15} +10.1001 q^{16} -0.632565 q^{17} -0.818213 q^{18} -0.230636 q^{19} -3.57325 q^{20} -3.44037 q^{21} -16.5346 q^{22} +1.40654 q^{23} -13.7813 q^{24} -4.46514 q^{25} -9.83663 q^{26} -4.89207 q^{27} -9.23667 q^{28} +0.508013 q^{29} +3.49247 q^{30} +4.87473 q^{31} -11.3579 q^{32} +11.4669 q^{33} +1.65991 q^{34} +1.38259 q^{35} +1.52345 q^{36} +3.88252 q^{37} +0.605212 q^{38} +6.82180 q^{39} +5.53832 q^{40} -10.4430 q^{41} +9.02786 q^{42} -6.07192 q^{43} +30.7862 q^{44} -0.228038 q^{45} -3.69089 q^{46} -5.89107 q^{47} +18.3805 q^{48} -3.42608 q^{49} +11.7170 q^{50} -1.15117 q^{51} +18.3151 q^{52} -3.34001 q^{53} +12.8373 q^{54} -4.60823 q^{55} +14.3163 q^{56} -0.419721 q^{57} -1.33308 q^{58} -0.445697 q^{59} -6.50273 q^{60} -12.7613 q^{61} -12.7918 q^{62} -0.589467 q^{63} +9.60404 q^{64} -2.74149 q^{65} -30.0903 q^{66} -1.60690 q^{67} -3.09064 q^{68} +2.55967 q^{69} -3.62804 q^{70} -11.1687 q^{71} -2.36126 q^{72} -12.0801 q^{73} -10.1881 q^{74} -8.12583 q^{75} -1.12686 q^{76} -11.9121 q^{77} -17.9011 q^{78} -4.92954 q^{79} -7.38659 q^{80} -9.83820 q^{81} +27.4035 q^{82} +13.2847 q^{83} -16.8092 q^{84} +0.462621 q^{85} +15.9333 q^{86} +0.924501 q^{87} -47.7169 q^{88} +8.49096 q^{89} +0.598393 q^{90} -7.08662 q^{91} +6.87218 q^{92} +8.87121 q^{93} +15.4587 q^{94} +0.168674 q^{95} -20.6695 q^{96} +5.53376 q^{97} +8.99036 q^{98} +1.96472 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) $ 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * 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 $

$2$	−2.62410	−1.85552	−0.927758	−	0.373182i	$-0.878267\pi$
$2$	−2.62410	−1.85552	−0.927758	+	0.373182i	$0.878267\pi$
$3$	1.81984	1.05068	0.525342	−	0.850891i	$-0.323937\pi$
$3$	1.81984	1.05068	0.525342	+	0.850891i	$0.323937\pi$
$4$	4.88588	2.44294
$5$	−0.731341	−0.327066	−0.163533	−	0.986538i	$-0.552289\pi$
$5$	−0.731341	−0.327066	−0.163533	+	0.986538i	$0.552289\pi$
$6$	−4.77543	−1.94956
$7$	−1.89048	−0.714535	−0.357268	−	0.934002i	$-0.616292\pi$
$7$	−1.89048	−0.714535	−0.357268	+	0.934002i	$0.616292\pi$
$8$	−7.57282	−2.67740
$9$	0.311808	0.103936
$10$	1.91911	0.606876
$11$	6.30107	1.89984	0.949921	−	0.312489i	$-0.101163\pi$
$11$	6.30107	1.89984	0.949921	+	0.312489i	$0.101163\pi$
$12$	8.89151	2.56676
$13$	3.74858	1.03967	0.519834	−	0.854267i	$-0.325994\pi$
$13$	3.74858	1.03967	0.519834	+	0.854267i	$0.325994\pi$
$14$	4.96081	1.32583
$15$	−1.33092	−0.343643
$16$	10.1001	2.52501
$17$	−0.632565	−0.153420	−0.0767098	−	0.997053i	$-0.524442\pi$
$17$	−0.632565	−0.153420	−0.0767098	+	0.997053i	$0.524442\pi$
$18$	−0.818213	−0.192855
$19$	−0.230636	−0.0529116	−0.0264558	−	0.999650i	$-0.508422\pi$
$19$	−0.230636	−0.0529116	−0.0264558	+	0.999650i	$0.508422\pi$
$20$	−3.57325	−0.799002
$21$	−3.44037	−0.750750
$22$	−16.5346	−3.52519
$23$	1.40654	0.293284	0.146642	−	0.989190i	$-0.453154\pi$
$23$	1.40654	0.293284	0.146642	+	0.989190i	$0.453154\pi$
$24$	−13.7813	−2.81310
$25$	−4.46514	−0.893028
$26$	−9.83663	−1.92912
$27$	−4.89207	−0.941480
$28$	−9.23667	−1.74557
$29$	0.508013	0.0943357	0.0471678	−	0.998887i	$-0.484980\pi$
$29$	0.508013	0.0943357	0.0471678	+	0.998887i	$0.484980\pi$
$30$	3.49247	0.637634
$31$	4.87473	0.875527	0.437764	−	0.899090i	$-0.355771\pi$
$31$	4.87473	0.875527	0.437764	+	0.899090i	$0.355771\pi$
$32$	−11.3579	−2.00781
$33$	11.4669	1.99613
$34$	1.65991	0.284673
$35$	1.38259	0.233700
$36$	1.52345	0.253909
$37$	3.88252	0.638282	0.319141	−	0.947707i	$-0.396606\pi$
$37$	3.88252	0.638282	0.319141	+	0.947707i	$0.396606\pi$
$38$	0.605212	0.0981784
$39$	6.82180	1.09236
$40$	5.53832	0.875685
$41$	−10.4430	−1.63093	−0.815464	−	0.578808i	$-0.803518\pi$
$41$	−10.4430	−1.63093	−0.815464	+	0.578808i	$0.803518\pi$
$42$	9.02786	1.39303
$43$	−6.07192	−0.925958	−0.462979	−	0.886369i	$-0.653220\pi$
$43$	−6.07192	−0.925958	−0.462979	+	0.886369i	$0.653220\pi$
$44$	30.7862	4.64120
$45$	−0.228038	−0.0339939
$46$	−3.69089	−0.544193
$47$	−5.89107	−0.859301	−0.429650	−	0.902995i	$-0.641363\pi$
$47$	−5.89107	−0.859301	−0.429650	+	0.902995i	$0.641363\pi$
$48$	18.3805	2.65299
$49$	−3.42608	−0.489440
$50$	11.7170	1.65703
$51$	−1.15117	−0.161196
$52$	18.3151	2.53985
$53$	−3.34001	−0.458785	−0.229393	−	0.973334i	$-0.573674\pi$
$53$	−3.34001	−0.458785	−0.229393	+	0.973334i	$0.573674\pi$
$54$	12.8373	1.74693
$55$	−4.60823	−0.621374
$56$	14.3163	1.91309
$57$	−0.419721	−0.0555934
$58$	−1.33308	−0.175041
$59$	−0.445697	−0.0580248	−0.0290124	−	0.999579i	$-0.509236\pi$
$59$	−0.445697	−0.0580248	−0.0290124	+	0.999579i	$0.509236\pi$
$60$	−6.50273	−0.839498
$61$	−12.7613	−1.63392	−0.816961	−	0.576693i	$-0.804343\pi$
$61$	−12.7613	−1.63392	−0.816961	+	0.576693i	$0.804343\pi$
$62$	−12.7918	−1.62455
$63$	−0.589467	−0.0742658
$64$	9.60404	1.20050
$65$	−2.74149	−0.340040
$66$	−30.0903	−3.70386
$67$	−1.60690	−0.196314	−0.0981568	−	0.995171i	$-0.531295\pi$
$67$	−1.60690	−0.196314	−0.0981568	+	0.995171i	$0.531295\pi$
$68$	−3.09064	−0.374795
$69$	2.55967	0.308148
$70$	−3.62804	−0.433634
$71$	−11.1687	−1.32548	−0.662740	−	0.748850i	$-0.730606\pi$
$71$	−11.1687	−1.32548	−0.662740	+	0.748850i	$0.730606\pi$
$72$	−2.36126	−0.278278
$73$	−12.0801	−1.41386	−0.706932	−	0.707282i	$-0.749921\pi$
$73$	−12.0801	−1.41386	−0.706932	+	0.707282i	$0.749921\pi$
$74$	−10.1881	−1.18434
$75$	−8.12583	−0.938290
$76$	−1.12686	−0.129260
$77$	−11.9121	−1.35750
$78$	−17.9011	−2.02690
$79$	−4.92954	−0.554617	−0.277308	−	0.960781i	$-0.589442\pi$
$79$	−4.92954	−0.554617	−0.277308	+	0.960781i	$0.589442\pi$
$80$	−7.38659	−0.825846
$81$	−9.83820	−1.09313
$82$	27.4035	3.02621
$83$	13.2847	1.45819	0.729095	−	0.684412i	$-0.239941\pi$
$83$	13.2847	1.45819	0.729095	+	0.684412i	$0.239941\pi$
$84$	−16.8092	−1.83404
$85$	0.462621	0.0501783
$86$	15.9333	1.71813
$87$	0.924501	0.0991169
$88$	−47.7169	−5.08664
$89$	8.49096	0.900040	0.450020	−	0.893018i	$-0.351417\pi$
$89$	8.49096	0.900040	0.450020	+	0.893018i	$0.351417\pi$
$90$	0.598393	0.0630762
$91$	−7.08662	−0.742880
$92$	6.87218	0.716474
$93$	8.87121	0.919902
$94$	15.4587	1.59445
$95$	0.168674	0.0173056
$96$	−20.6695	−2.10957
$97$	5.53376	0.561868	0.280934	−	0.959727i	$-0.409356\pi$
$97$	5.53376	0.561868	0.280934	+	0.959727i	$0.409356\pi$
$98$	8.99036	0.908163
$99$	1.96472	0.197462
$100$	−21.8161	−2.18161
$101$	−14.4589	−1.43871	−0.719357	−	0.694641i	$-0.755563\pi$
$101$	−14.4589	−1.43871	−0.719357	+	0.694641i	$0.755563\pi$
$102$	3.02077	0.299101
$103$	−18.2975	−1.80291	−0.901455	−	0.432874i	$-0.857500\pi$
$103$	−18.2975	−1.80291	−0.901455	+	0.432874i	$0.857500\pi$
$104$	−28.3873	−2.78361
$105$	2.51608	0.245545
$106$	8.76450	0.851283
$107$	7.39818	0.715209	0.357605	−	0.933873i	$-0.383594\pi$
$107$	7.39818	0.715209	0.357605	+	0.933873i	$0.383594\pi$
$108$	−23.9021	−2.29998
$109$	−9.42074	−0.902343	−0.451172	−	0.892437i	$-0.648994\pi$
$109$	−9.42074	−0.902343	−0.451172	+	0.892437i	$0.648994\pi$
$110$	12.0924	1.15297
$111$	7.06555	0.670632
$112$	−19.0940	−1.80421
$113$	7.90971	0.744083	0.372042	−	0.928216i	$-0.378658\pi$
$113$	7.90971	0.744083	0.372042	+	0.928216i	$0.378658\pi$
$114$	1.10139	0.103154
$115$	−1.02866	−0.0959231
$116$	2.48209	0.230456
$117$	1.16884	0.108059
$118$	1.16955	0.107666
$119$	1.19585	0.109624
$120$	10.0788	0.920068
$121$	28.7034	2.60940
$122$	33.4870	3.03177
$123$	−19.0046	−1.71359
$124$	23.8173	2.13886
$125$	6.92225	0.619145
$126$	1.54682	0.137801
$127$	11.2168	0.995328	0.497664	−	0.867370i	$-0.334191\pi$
$127$	11.2168	0.995328	0.497664	+	0.867370i	$0.334191\pi$
$128$	−2.48616	−0.219748
$129$	−11.0499	−0.972889
$130$	7.19393	0.630950
$131$	−18.4718	−1.61388	−0.806942	−	0.590630i	$-0.798879\pi$
$131$	−18.4718	−1.61388	−0.806942	+	0.590630i	$0.798879\pi$
$132$	56.0260	4.87643
$133$	0.436014	0.0378072
$134$	4.21665	0.364263
$135$	3.57778	0.307926
$136$	4.79031	0.410765
$137$	19.3916	1.65674	0.828368	−	0.560185i	$-0.189270\pi$
$137$	19.3916	1.65674	0.828368	+	0.560185i	$0.189270\pi$
$138$	−6.71683	−0.571774
$139$	9.18273	0.778869	0.389434	−	0.921054i	$-0.372670\pi$
$139$	9.18273	0.778869	0.389434	+	0.921054i	$0.372670\pi$
$140$	6.75516	0.570915
$141$	−10.7208	−0.902853
$142$	29.3077	2.45945
$143$	23.6200	1.97521
$144$	3.14928	0.262440
$145$	−0.371531	−0.0308540
$146$	31.6992	2.62345
$147$	−6.23490	−0.514246
$148$	18.9695	1.55928
$149$	−2.35433	−0.192874	−0.0964372	−	0.995339i	$-0.530745\pi$
$149$	−2.35433	−0.192874	−0.0964372	+	0.995339i	$0.530745\pi$
$150$	21.3230	1.74101
$151$	−9.96550	−0.810981	−0.405490	−	0.914099i	$-0.632899\pi$
$151$	−9.96550	−0.810981	−0.405490	+	0.914099i	$0.632899\pi$
$152$	1.74657	0.141665
$153$	−0.197239	−0.0159458
$154$	31.2584	2.51887
$155$	−3.56509	−0.286355
$156$	33.3305	2.66858
$157$	10.3665	0.827336	0.413668	−	0.910428i	$-0.364248\pi$
$157$	10.3665	0.827336	0.413668	+	0.910428i	$0.364248\pi$
$158$	12.9356	1.02910
$159$	−6.07827	−0.482038
$160$	8.30648	0.656685
$161$	−2.65904	−0.209561
$162$	25.8164	2.02833
$163$	−4.61886	−0.361777	−0.180889	−	0.983504i	$-0.557897\pi$
$163$	−4.61886	−0.361777	−0.180889	+	0.983504i	$0.557897\pi$
$164$	−51.0234	−3.98426
$165$	−8.38623	−0.652867
$166$	−34.8605	−2.70570
$167$	−1.48695	−0.115064	−0.0575319	−	0.998344i	$-0.518323\pi$
$167$	−1.48695	−0.115064	−0.0575319	+	0.998344i	$0.518323\pi$
$168$	26.0533	2.01006
$169$	1.05184	0.0809106
$170$	−1.21396	−0.0931067
$171$	−0.0719142	−0.00549942
$172$	−29.6666	−2.26206
$173$	21.1441	1.60755	0.803777	−	0.594931i	$-0.202821\pi$
$173$	21.1441	1.60755	0.803777	+	0.594931i	$0.202821\pi$
$174$	−2.42598	−0.183913
$175$	8.44127	0.638100
$176$	63.6411	4.79713
$177$	−0.811097	−0.0609657
$178$	−22.2811	−1.67004
$179$	−1.01922	−0.0761804	−0.0380902	−	0.999274i	$-0.512127\pi$
$179$	−1.01922	−0.0761804	−0.0380902	+	0.999274i	$0.512127\pi$
$180$	−1.11417	−0.0830450
$181$	6.74671	0.501479	0.250739	−	0.968055i	$-0.419326\pi$
$181$	6.74671	0.501479	0.250739	+	0.968055i	$0.419326\pi$
$182$	18.5960	1.37843
$183$	−23.2236	−1.71673
$184$	−10.6515	−0.785237
$185$	−2.83945	−0.208760
$186$	−23.2789	−1.70689
$187$	−3.98584	−0.291473
$188$	−28.7830	−2.09922
$189$	9.24838	0.672720
$190$	−0.442617	−0.0321108
$191$	11.9775	0.866658	0.433329	−	0.901236i	$-0.357339\pi$
$191$	11.9775	0.866658	0.433329	+	0.901236i	$0.357339\pi$
$192$	17.4778	1.26135
$193$	−9.59093	−0.690370	−0.345185	−	0.938535i	$-0.612184\pi$
$193$	−9.59093	−0.690370	−0.345185	+	0.938535i	$0.612184\pi$
$194$	−14.5211	−1.04255
$195$	−4.98907	−0.357274
$196$	−16.7394	−1.19567
$197$	14.7606	1.05165	0.525824	−	0.850593i	$-0.323757\pi$
$197$	14.7606	1.05165	0.525824	+	0.850593i	$0.323757\pi$
$198$	−5.15562	−0.366394
$199$	−0.571938	−0.0405436	−0.0202718	−	0.999795i	$-0.506453\pi$
$199$	−0.571938	−0.0405436	−0.0202718	+	0.999795i	$0.506453\pi$
$200$	33.8137	2.39099
$201$	−2.92429	−0.206263
$202$	37.9415	2.66956
$203$	−0.960390	−0.0674062
$204$	−5.62446	−0.393791
$205$	7.63742	0.533420
$206$	48.0145	3.34533
$207$	0.438570	0.0304827
$208$	37.8609	2.62518
$209$	−1.45326	−0.100524
$210$	−6.60245	−0.455612
$211$	0.687260	0.0473129	0.0236565	−	0.999720i	$-0.492469\pi$
$211$	0.687260	0.0473129	0.0236565	+	0.999720i	$0.492469\pi$
$212$	−16.3189	−1.12078
$213$	−20.3252	−1.39266
$214$	−19.4135	−1.32708
$215$	4.44064	0.302849
$216$	37.0468	2.52072
$217$	−9.21559	−0.625595
$218$	24.7209	1.67431
$219$	−21.9837	−1.48552
$220$	−22.5153	−1.51798
$221$	−2.37122	−0.159506
$222$	−18.5407	−1.24437
$223$	−7.66230	−0.513106	−0.256553	−	0.966530i	$-0.582587\pi$
$223$	−7.66230	−0.513106	−0.256553	+	0.966530i	$0.582587\pi$
$224$	21.4719	1.43465
$225$	−1.39226	−0.0928177
$226$	−20.7558	−1.38066
$227$	−6.77022	−0.449355	−0.224678	−	0.974433i	$-0.572133\pi$
$227$	−6.77022	−0.449355	−0.224678	+	0.974433i	$0.572133\pi$
$228$	−2.05071	−0.135811
$229$	−1.09823	−0.0725728	−0.0362864	−	0.999341i	$-0.511553\pi$
$229$	−1.09823	−0.0725728	−0.0362864	+	0.999341i	$0.511553\pi$
$230$	2.69930	0.177987
$231$	−21.6780	−1.42631
$232$	−3.84709	−0.252574
$233$	−12.4459	−0.815356	−0.407678	−	0.913126i	$-0.633661\pi$
$233$	−12.4459	−0.815356	−0.407678	+	0.913126i	$0.633661\pi$
$234$	−3.06714	−0.200505
$235$	4.30838	0.281048
$236$	−2.17762	−0.141751
$237$	−8.97097	−0.582727
$238$	−3.13803	−0.203409
$239$	−6.33676	−0.409891	−0.204945	−	0.978773i	$-0.565702\pi$
$239$	−6.33676	−0.409891	−0.204945	+	0.978773i	$0.565702\pi$
$240$	−13.4424	−0.867703
$241$	−22.3658	−1.44071	−0.720354	−	0.693607i	$-0.756021\pi$
$241$	−22.3658	−1.44071	−0.720354	+	0.693607i	$0.756021\pi$
$242$	−75.3206	−4.84179
$243$	−3.22770	−0.207057
$244$	−62.3503	−3.99157
$245$	2.50563	0.160079
$246$	49.8699	3.17959
$247$	−0.864559	−0.0550106
$248$	−36.9155	−2.34413
$249$	24.1761	1.53210
$250$	−18.1646	−1.14883
$251$	17.6261	1.11255	0.556274	−	0.830999i	$-0.312231\pi$
$251$	17.6261	1.11255	0.556274	+	0.830999i	$0.312231\pi$
$252$	−2.88006	−0.181427
$253$	8.86270	0.557193
$254$	−29.4339	−1.84685
$255$	0.841895	0.0527215
$256$	−12.6841	−0.792759
$257$	−6.44430	−0.401985	−0.200992	−	0.979593i	$-0.564417\pi$
$257$	−6.44430	−0.401985	−0.200992	+	0.979593i	$0.564417\pi$
$258$	28.9960	1.80521
$259$	−7.33983	−0.456075
$260$	−13.3946	−0.830697
$261$	0.158402	0.00980486
$262$	48.4717	2.99459
$263$	−17.5312	−1.08102	−0.540509	−	0.841338i	$-0.681768\pi$
$263$	−17.5312	−1.08102	−0.540509	+	0.841338i	$0.681768\pi$
$264$	−86.8369	−5.34444
$265$	2.44269	0.150053
$266$	−1.14414	−0.0701519
$267$	15.4522	0.945657
$268$	−7.85110	−0.479582
$269$	−1.07497	−0.0655419	−0.0327710	−	0.999463i	$-0.510433\pi$
$269$	−1.07497	−0.0655419	−0.0327710	+	0.999463i	$0.510433\pi$
$270$	−9.38842	−0.571361
$271$	3.04673	0.185076	0.0925378	−	0.995709i	$-0.470502\pi$
$271$	3.04673	0.185076	0.0925378	+	0.995709i	$0.470502\pi$
$272$	−6.38895	−0.387387
$273$	−12.8965	−0.780531
$274$	−50.8854	−3.07410
$275$	−28.1351	−1.69661
$276$	12.5062	0.752788
$277$	−3.64905	−0.219250	−0.109625	−	0.993973i	$-0.534965\pi$
$277$	−3.64905	−0.219250	−0.109625	+	0.993973i	$0.534965\pi$
$278$	−24.0964	−1.44520
$279$	1.51998	0.0909987
$280$	−10.4701	−0.625708
$281$	−17.4377	−1.04025	−0.520124	−	0.854091i	$-0.674114\pi$
$281$	−17.4377	−1.04025	−0.520124	+	0.854091i	$0.674114\pi$
$282$	28.1324	1.67526
$283$	12.5930	0.748576	0.374288	−	0.927312i	$-0.377887\pi$
$283$	12.5930	0.748576	0.374288	+	0.927312i	$0.377887\pi$
$284$	−54.5689	−3.23807
$285$	0.306959	0.0181827
$286$	−61.9812	−3.66503
$287$	19.7424	1.16535
$288$	−3.54147	−0.208683
$289$	−16.5999	−0.976462
$290$	0.974933	0.0572500
$291$	10.0705	0.590345
$292$	−59.0217	−3.45398
$293$	5.31736	0.310643	0.155322	−	0.987864i	$-0.450359\pi$
$293$	5.31736	0.310643	0.155322	+	0.987864i	$0.450359\pi$
$294$	16.3610	0.954192
$295$	0.325957	0.0189779
$296$	−29.4016	−1.70893
$297$	−30.8253	−1.78866
$298$	6.17800	0.357882
$299$	5.27252	0.304918
$300$	−39.7018	−2.29219
$301$	11.4788	0.661630
$302$	26.1504	1.50479
$303$	−26.3128	−1.51163
$304$	−2.32944	−0.133603
$305$	9.33289	0.534400
$306$	0.517573	0.0295877
$307$	−22.7938	−1.30091	−0.650455	−	0.759545i	$-0.725422\pi$
$307$	−22.7938	−1.30091	−0.650455	+	0.759545i	$0.725422\pi$
$308$	−58.2009	−3.31630
$309$	−33.2985	−1.89429
$310$	9.35514	0.531336
$311$	−17.8313	−1.01112	−0.505561	−	0.862791i	$-0.668714\pi$
$311$	−17.8313	−1.01112	−0.505561	+	0.862791i	$0.668714\pi$
$312$	−51.6603	−2.92469
$313$	−8.19125	−0.462997	−0.231498	−	0.972835i	$-0.574363\pi$
$313$	−8.19125	−0.462997	−0.231498	+	0.972835i	$0.574363\pi$
$314$	−27.2026	−1.53513
$315$	0.431102	0.0242898
$316$	−24.0852	−1.35490
$317$	17.4711	0.981272	0.490636	−	0.871365i	$-0.336764\pi$
$317$	17.4711	0.981272	0.490636	+	0.871365i	$0.336764\pi$
$318$	15.9500	0.894429
$319$	3.20102	0.179223
$320$	−7.02383	−0.392644
$321$	13.4635	0.751459
$322$	6.97757	0.388845
$323$	0.145893	0.00811768
$324$	−48.0683	−2.67046
$325$	−16.7379	−0.928453
$326$	12.1203	0.671283
$327$	−17.1442	−0.948077
$328$	79.0832	4.36664
$329$	11.1370	0.614000
$330$	22.0063	1.21141
$331$	−10.4178	−0.572612	−0.286306	−	0.958138i	$-0.592427\pi$
$331$	−10.4178	−0.572612	−0.286306	+	0.958138i	$0.592427\pi$
$332$	64.9077	3.56227
$333$	1.21060	0.0663404
$334$	3.90190	0.213503
$335$	1.17519	0.0642074
$336$	−34.7479	−1.89566
$337$	14.1918	0.773075	0.386537	−	0.922274i	$-0.373671\pi$
$337$	14.1918	0.773075	0.386537	+	0.922274i	$0.373671\pi$
$338$	−2.76012	−0.150131
$339$	14.3944	0.781796
$340$	2.26031	0.122583
$341$	30.7160	1.66336
$342$	0.188710	0.0102043
$343$	19.7103	1.06426
$344$	45.9815	2.47916
$345$	−1.87199	−0.100785
$346$	−55.4841	−2.98284
$347$	25.0772	1.34622	0.673108	−	0.739545i	$-0.264959\pi$
$347$	25.0772	1.34622	0.673108	+	0.739545i	$0.264959\pi$
$348$	4.51700	0.242137
$349$	9.16236	0.490450	0.245225	−	0.969466i	$-0.421138\pi$
$349$	9.16236	0.490450	0.245225	+	0.969466i	$0.421138\pi$
$350$	−22.1507	−1.18400
$351$	−18.3383	−0.978827
$352$	−71.5667	−3.81452
$353$	28.9181	1.53916	0.769578	−	0.638552i	$-0.220466\pi$
$353$	28.9181	1.53916	0.769578	+	0.638552i	$0.220466\pi$
$354$	2.12840	0.113123
$355$	8.16812	0.433519
$356$	41.4858	2.19874
$357$	2.17626	0.115180
$358$	2.67454	0.141354
$359$	−21.6771	−1.14407	−0.572037	−	0.820228i	$-0.693847\pi$
$359$	−21.6771	−1.14407	−0.572037	+	0.820228i	$0.693847\pi$
$360$	1.72689	0.0910151
$361$	−18.9468	−0.997200
$362$	−17.7040	−0.930502
$363$	52.2356	2.74166
$364$	−34.6244	−1.81481
$365$	8.83465	0.462427
$366$	60.9408	3.18543
$367$	−8.66027	−0.452062	−0.226031	−	0.974120i	$-0.572575\pi$
$367$	−8.66027	−0.452062	−0.226031	+	0.974120i	$0.572575\pi$
$368$	14.2061	0.740546
$369$	−3.25622	−0.169512
$370$	7.45098	0.387358
$371$	6.31422	0.327818
$372$	43.3437	2.24727
$373$	21.6556	1.12129	0.560643	−	0.828058i	$-0.310554\pi$
$373$	21.6556	1.12129	0.560643	+	0.828058i	$0.310554\pi$
$374$	10.4592	0.540833
$375$	12.5974	0.650525
$376$	44.6120	2.30069
$377$	1.90433	0.0980778
$378$	−24.2686	−1.24824
$379$	12.1621	0.624727	0.312364	−	0.949963i	$-0.398879\pi$
$379$	12.1621	0.624727	0.312364	+	0.949963i	$0.398879\pi$
$380$	0.824121	0.0422765
$381$	20.4127	1.04578
$382$	−31.4300	−1.60810
$383$	−31.3611	−1.60248	−0.801240	−	0.598343i	$-0.795826\pi$
$383$	−31.3611	−1.60248	−0.801240	+	0.598343i	$0.795826\pi$
$384$	−4.52441	−0.230885
$385$	8.71178	0.443993
$386$	25.1675	1.28099
$387$	−1.89327	−0.0962403
$388$	27.0373	1.37261
$389$	7.77970	0.394447	0.197223	−	0.980359i	$-0.436808\pi$
$389$	7.77970	0.394447	0.197223	+	0.980359i	$0.436808\pi$
$390$	13.0918	0.662928
$391$	−0.889728	−0.0449955
$392$	25.9451	1.31042
$393$	−33.6156	−1.69568
$394$	−38.7332	−1.95135
$395$	3.60518	0.181396
$396$	9.59939	0.482387
$397$	−28.9317	−1.45204	−0.726019	−	0.687674i	$-0.758632\pi$
$397$	−28.9317	−1.45204	−0.726019	+	0.687674i	$0.758632\pi$
$398$	1.50082	0.0752293
$399$	0.793475	0.0397234
$400$	−45.0982	−2.25491
$401$	−25.3695	−1.26689	−0.633445	−	0.773788i	$-0.718360\pi$
$401$	−25.3695	−1.26689	−0.633445	+	0.773788i	$0.718360\pi$
$402$	7.67361	0.382725
$403$	18.2733	0.910258
$404$	−70.6444	−3.51469
$405$	7.19508	0.357526
$406$	2.52016	0.125073
$407$	24.4640	1.21264
$408$	8.71758	0.431585
$409$	−28.4155	−1.40506	−0.702529	−	0.711655i	$-0.747946\pi$
$409$	−28.4155	−1.40506	−0.702529	+	0.711655i	$0.747946\pi$
$410$	−20.0413	−0.989770
$411$	35.2895	1.74070
$412$	−89.3995	−4.40440
$413$	0.842583	0.0414608
$414$	−1.15085	−0.0565611
$415$	−9.71569	−0.476924
$416$	−42.5759	−2.08745
$417$	16.7111	0.818345
$418$	3.81348	0.186524
$419$	16.5979	0.810862	0.405431	−	0.914126i	$-0.367121\pi$
$419$	16.5979	0.810862	0.405431	+	0.914126i	$0.367121\pi$
$420$	12.2933	0.599851
$421$	−27.8704	−1.35832	−0.679160	−	0.733990i	$-0.737656\pi$
$421$	−27.8704	−1.35832	−0.679160	+	0.733990i	$0.737656\pi$
$422$	−1.80344	−0.0877899
$423$	−1.83688	−0.0893122
$424$	25.2933	1.22835
$425$	2.82449	0.137008
$426$	53.3353	2.58410
$427$	24.1251	1.16749
$428$	36.1466	1.74721
$429$	42.9846	2.07532
$430$	−11.6527	−0.561942
$431$	−12.5869	−0.606290	−0.303145	−	0.952945i	$-0.598037\pi$
$431$	−12.5869	−0.606290	−0.303145	+	0.952945i	$0.598037\pi$
$432$	−49.4102	−2.37725
$433$	−1.86750	−0.0897461	−0.0448731	−	0.998993i	$-0.514288\pi$
$433$	−1.86750	−0.0897461	−0.0448731	+	0.998993i	$0.514288\pi$
$434$	24.1826	1.16080
$435$	−0.676126	−0.0324178
$436$	−46.0286	−2.20437
$437$	−0.324399	−0.0155181
$438$	57.6874	2.75641
$439$	3.70650	0.176902	0.0884508	−	0.996081i	$-0.471808\pi$
$439$	3.70650	0.176902	0.0884508	+	0.996081i	$0.471808\pi$
$440$	34.8973	1.66366
$441$	−1.06828	−0.0508703
$442$	6.22231	0.295965
$443$	−39.7199	−1.88715	−0.943575	−	0.331160i	$-0.892560\pi$
$443$	−39.7199	−1.88715	−0.943575	+	0.331160i	$0.892560\pi$
$444$	34.5214	1.63831
$445$	−6.20979	−0.294372
$446$	20.1066	0.952076
$447$	−4.28450	−0.202650
$448$	−18.1563	−0.857803
$449$	−4.17730	−0.197139	−0.0985695	−	0.995130i	$-0.531427\pi$
$449$	−4.17730	−0.197139	−0.0985695	+	0.995130i	$0.531427\pi$
$450$	3.65344	0.172225
$451$	−65.8022	−3.09851
$452$	38.6459	1.81775
$453$	−18.1356	−0.852084
$454$	17.7657	0.833786
$455$	5.18274	0.242971
$456$	3.17847	0.148846
$457$	0.261156	0.0122164	0.00610818	−	0.999981i	$-0.498056\pi$
$457$	0.261156	0.0122164	0.00610818	+	0.999981i	$0.498056\pi$
$458$	2.88185	0.134660
$459$	3.09456	0.144442
$460$	−5.02591	−0.234334
$461$	−1.18377	−0.0551337	−0.0275668	−	0.999620i	$-0.508776\pi$
$461$	−1.18377	−0.0551337	−0.0275668	+	0.999620i	$0.508776\pi$
$462$	56.8851	2.64654
$463$	37.9762	1.76490	0.882451	−	0.470404i	$-0.155892\pi$
$463$	37.9762	1.76490	0.882451	+	0.470404i	$0.155892\pi$
$464$	5.13096	0.238199
$465$	−6.48789	−0.300868
$466$	32.6591	1.51291
$467$	−10.9144	−0.505059	−0.252530	−	0.967589i	$-0.581263\pi$
$467$	−10.9144	−0.505059	−0.252530	+	0.967589i	$0.581263\pi$
$468$	5.71079	0.263981
$469$	3.03781	0.140273
$470$	−11.3056	−0.521489
$471$	18.8653	0.869268
$472$	3.37519	0.155356
$473$	−38.2595	−1.75918
$474$	23.5407	1.08126
$475$	1.02982	0.0472516
$476$	5.84280	0.267804
$477$	−1.04144	−0.0476843
$478$	16.6283	0.760559
$479$	−7.03791	−0.321571	−0.160785	−	0.986989i	$-0.551403\pi$
$479$	−7.03791	−0.321571	−0.160785	+	0.986989i	$0.551403\pi$
$480$	15.1164	0.689968
$481$	14.5539	0.663602
$482$	58.6900	2.67326
$483$	−4.83901	−0.220183
$484$	140.242	6.37461
$485$	−4.04707	−0.183768
$486$	8.46980	0.384198
$487$	−35.3661	−1.60259	−0.801295	−	0.598269i	$-0.795855\pi$
$487$	−35.3661	−1.60259	−0.801295	+	0.598269i	$0.795855\pi$
$488$	96.6394	4.37466
$489$	−8.40558	−0.380113
$490$	−6.57502	−0.297029
$491$	−18.0257	−0.813489	−0.406744	−	0.913542i	$-0.633336\pi$
$491$	−18.0257	−0.813489	−0.406744	+	0.913542i	$0.633336\pi$
$492$	−92.8542	−4.18619
$493$	−0.321352	−0.0144729
$494$	2.26869	0.102073
$495$	−1.43688	−0.0645830
$496$	49.2351	2.21072
$497$	21.1142	0.947102
$498$	−63.4404	−2.84283
$499$	−33.1520	−1.48409	−0.742044	−	0.670352i	$-0.766143\pi$
$499$	−33.1520	−1.48409	−0.742044	+	0.670352i	$0.766143\pi$
$500$	33.8213	1.51253
$501$	−2.70601	−0.120896
$502$	−46.2525	−2.06435
$503$	26.1373	1.16540	0.582701	−	0.812686i	$-0.301996\pi$
$503$	26.1373	1.16540	0.582701	+	0.812686i	$0.301996\pi$
$504$	4.46393	0.198839
$505$	10.5744	0.470554
$506$	−23.2566	−1.03388
$507$	1.91417	0.0850115
$508$	54.8038	2.43153
$509$	44.3080	1.96392	0.981959	−	0.189095i	$-0.0605555\pi$
$509$	44.3080	1.96392	0.981959	+	0.189095i	$0.0605555\pi$
$510$	−2.20921	−0.0978257
$511$	22.8371	1.01026
$512$	38.2567	1.69073
$513$	1.12829	0.0498152
$514$	16.9105	0.745889
$515$	13.3817	0.589670
$516$	−53.9885	−2.37671
$517$	−37.1200	−1.63254
$518$	19.2604	0.846254
$519$	38.4788	1.68903
$520$	20.7608	0.910422
$521$	26.0412	1.14088	0.570442	−	0.821338i	$-0.306772\pi$
$521$	26.0412	1.14088	0.570442	+	0.821338i	$0.306772\pi$
$522$	−0.415663	−0.0181931
$523$	2.74939	0.120222	0.0601112	−	0.998192i	$-0.480854\pi$
$523$	2.74939	0.120222	0.0601112	+	0.998192i	$0.480854\pi$
$524$	−90.2508	−3.94262
$525$	15.3617	0.670441
$526$	46.0034	2.00584
$527$	−3.08359	−0.134323
$528$	115.817	5.04027
$529$	−21.0216	−0.913985
$530$	−6.40984	−0.278426
$531$	−0.138972	−0.00603086
$532$	2.13031	0.0923608
$533$	−39.1465	−1.69562
$534$	−40.5480	−1.75468
$535$	−5.41060	−0.233921
$536$	12.1687	0.525609
$537$	−1.85482	−0.0800415
$538$	2.82082	0.121614
$539$	−21.5879	−0.929858
$540$	17.4806	0.752244
$541$	17.2817	0.743000	0.371500	−	0.928433i	$-0.378844\pi$
$541$	17.2817	0.743000	0.371500	+	0.928433i	$0.378844\pi$
$542$	−7.99491	−0.343411
$543$	12.2779	0.526895
$544$	7.18460	0.308037
$545$	6.88978	0.295126
$546$	33.8416	1.44829
$547$	36.6080	1.56524	0.782622	−	0.622498i	$-0.213882\pi$
$547$	36.6080	1.56524	0.782622	+	0.622498i	$0.213882\pi$
$548$	94.7450	4.04730
$549$	−3.97908	−0.169823
$550$	73.8293	3.14809
$551$	−0.117166	−0.00499145
$552$	−19.3839	−0.825036
$553$	9.31921	0.396293
$554$	9.57545	0.406822
$555$	−5.16733	−0.219341
$556$	44.8657	1.90273
$557$	−3.65941	−0.155054	−0.0775271	−	0.996990i	$-0.524702\pi$
$557$	−3.65941	−0.155054	−0.0775271	+	0.996990i	$0.524702\pi$
$558$	−3.98857	−0.168850
$559$	−22.7610	−0.962690
$560$	13.9642	0.590096
$561$	−7.25357	−0.306246
$562$	45.7583	1.93020
$563$	−21.3222	−0.898623	−0.449311	−	0.893375i	$-0.648331\pi$
$563$	−21.3222	−0.898623	−0.449311	+	0.893375i	$0.648331\pi$
$564$	−52.3805	−2.20562
$565$	−5.78470	−0.243364
$566$	−33.0452	−1.38900
$567$	18.5989	0.781082
$568$	84.5785	3.54884
$569$	10.1287	0.424616	0.212308	−	0.977203i	$-0.431902\pi$
$569$	10.1287	0.424616	0.212308	+	0.977203i	$0.431902\pi$
$570$	−0.805490	−0.0337383
$571$	4.48604	0.187735	0.0938675	−	0.995585i	$-0.470077\pi$
$571$	4.48604	0.187735	0.0938675	+	0.995585i	$0.470077\pi$
$572$	115.405	4.82531
$573$	21.7970	0.910584
$574$	−51.8058	−2.16233
$575$	−6.28039	−0.261911
$576$	2.99461	0.124776
$577$	−33.6482	−1.40079	−0.700397	−	0.713754i	$-0.746994\pi$
$577$	−33.6482	−1.40079	−0.700397	+	0.713754i	$0.746994\pi$
$578$	43.5596	1.81184
$579$	−17.4539	−0.725361
$580$	−1.81526	−0.0753744
$581$	−25.1146	−1.04193
$582$	−26.4261	−1.09540
$583$	−21.0456	−0.871620
$584$	91.4802	3.78548
$585$	−0.854818	−0.0353424
$586$	−13.9533	−0.576404
$587$	−12.9354	−0.533902	−0.266951	−	0.963710i	$-0.586016\pi$
$587$	−12.9354	−0.533902	−0.266951	+	0.963710i	$0.586016\pi$
$588$	−30.4630	−1.25627
$589$	−1.12429	−0.0463256
$590$	−0.855342	−0.0352139
$591$	26.8619	1.10495
$592$	39.2137	1.61167
$593$	−4.97155	−0.204157	−0.102079	−	0.994776i	$-0.532549\pi$
$593$	−4.97155	−0.204157	−0.102079	+	0.994776i	$0.532549\pi$
$594$	80.8885	3.31889
$595$	−0.874577	−0.0358542
$596$	−11.5030	−0.471181
$597$	−1.04083	−0.0425985
$598$	−13.8356	−0.565780
$599$	1.01851	0.0416150	0.0208075	−	0.999784i	$-0.493376\pi$
$599$	1.01851	0.0416150	0.0208075	+	0.999784i	$0.493376\pi$
$600$	61.5355	2.51217
$601$	2.71352	0.110687	0.0553434	−	0.998467i	$-0.482375\pi$
$601$	2.71352	0.110687	0.0553434	+	0.998467i	$0.482375\pi$
$602$	−30.1216	−1.22766
$603$	−0.501042	−0.0204040
$604$	−48.6902	−1.98118
$605$	−20.9920	−0.853447
$606$	69.0474	2.80486
$607$	−3.11913	−0.126602	−0.0633009	−	0.997994i	$-0.520163\pi$
$607$	−3.11913	−0.126602	−0.0633009	+	0.997994i	$0.520163\pi$
$608$	2.61954	0.106236
$609$	−1.74775	−0.0708225
$610$	−24.4904	−0.991588
$611$	−22.0831	−0.893388
$612$	−0.963685	−0.0389547
$613$	−34.6368	−1.39897	−0.699483	−	0.714649i	$-0.746586\pi$
$613$	−34.6368	−1.39897	−0.699483	+	0.714649i	$0.746586\pi$
$614$	59.8131	2.41386
$615$	13.8989	0.560456
$616$	90.2079	3.63458
$617$	20.7948	0.837166	0.418583	−	0.908179i	$-0.362527\pi$
$617$	20.7948	0.837166	0.418583	+	0.908179i	$0.362527\pi$
$618$	87.3785	3.51488
$619$	−20.3937	−0.819693	−0.409847	−	0.912154i	$-0.634418\pi$
$619$	−20.3937	−0.819693	−0.409847	+	0.912154i	$0.634418\pi$
$620$	−17.4186	−0.699548
$621$	−6.88089	−0.276121
$622$	46.7911	1.87615
$623$	−16.0520	−0.643110
$624$	68.9006	2.75823
$625$	17.2632	0.690527
$626$	21.4946	0.859098
$627$	−2.64469	−0.105619
$628$	50.6494	2.02113
$629$	−2.45595	−0.0979250
$630$	−1.13125	−0.0450701
$631$	−35.9338	−1.43050	−0.715251	−	0.698868i	$-0.753688\pi$
$631$	−35.9338	−1.43050	−0.715251	+	0.698868i	$0.753688\pi$
$632$	37.3306	1.48493
$633$	1.25070	0.0497109
$634$	−45.8457	−1.82077
$635$	−8.20330	−0.325538
$636$	−29.6977	−1.17759
$637$	−12.8429	−0.508855
$638$	−8.39980	−0.332551
$639$	−3.48248	−0.137765
$640$	1.81823	0.0718719
$641$	26.1124	1.03138	0.515688	−	0.856776i	$-0.327536\pi$
$641$	26.1124	1.03138	0.515688	+	0.856776i	$0.327536\pi$
$642$	−35.3295	−1.39434
$643$	−3.86134	−0.152277	−0.0761383	−	0.997097i	$-0.524259\pi$
$643$	−3.86134	−0.152277	−0.0761383	+	0.997097i	$0.524259\pi$
$644$	−12.9917	−0.511946
$645$	8.08125	0.318199
$646$	−0.382836	−0.0150625
$647$	37.3707	1.46919	0.734596	−	0.678505i	$-0.237372\pi$
$647$	37.3707	1.46919	0.734596	+	0.678505i	$0.237372\pi$
$648$	74.5030	2.92675
$649$	−2.80837	−0.110238
$650$	43.9219	1.72276
$651$	−16.7709	−0.657302
$652$	−22.5672	−0.883800
$653$	−18.2258	−0.713229	−0.356614	−	0.934252i	$-0.616069\pi$
$653$	−18.2258	−0.713229	−0.356614	+	0.934252i	$0.616069\pi$
$654$	44.9881	1.75917
$655$	13.5092	0.527847
$656$	−105.475	−4.11811
$657$	−3.76666	−0.146951
$658$	−29.2244	−1.13929
$659$	11.2746	0.439197	0.219599	−	0.975590i	$-0.429525\pi$
$659$	11.2746	0.439197	0.219599	+	0.975590i	$0.429525\pi$
$660$	−40.9741	−1.59491
$661$	−9.87750	−0.384190	−0.192095	−	0.981376i	$-0.561528\pi$
$661$	−9.87750	−0.384190	−0.192095	+	0.981376i	$0.561528\pi$
$662$	27.3372	1.06249
$663$	−4.31524	−0.167590
$664$	−100.603	−3.90416
$665$	−0.318875	−0.0123654
$666$	−3.17673	−0.123096
$667$	0.714540	0.0276671
$668$	−7.26506	−0.281094
$669$	−13.9441	−0.539112
$670$	−3.08381	−0.119138
$671$	−80.4100	−3.10419
$672$	39.0753	1.50736
$673$	−42.1163	−1.62346	−0.811732	−	0.584030i	$-0.801475\pi$
$673$	−42.1163	−1.62346	−0.811732	+	0.584030i	$0.801475\pi$
$674$	−37.2406	−1.43445
$675$	21.8438	0.840768
$676$	5.13915	0.197660
$677$	−13.8249	−0.531333	−0.265667	−	0.964065i	$-0.585592\pi$
$677$	−13.8249	−0.531333	−0.265667	+	0.964065i	$0.585592\pi$
$678$	−37.7723	−1.45063
$679$	−10.4615	−0.401474
$680$	−3.50335	−0.134347
$681$	−12.3207	−0.472130
$682$	−80.6017	−3.08640
$683$	−27.6543	−1.05816	−0.529082	−	0.848571i	$-0.677463\pi$
$683$	−27.6543	−1.05816	−0.529082	+	0.848571i	$0.677463\pi$
$684$	−0.351364	−0.0134347
$685$	−14.1819	−0.541861
$686$	−51.7218	−1.97475
$687$	−1.99859	−0.0762510
$688$	−61.3267	−2.33806
$689$	−12.5203	−0.476985
$690$	4.91229	0.187008
$691$	−1.27951	−0.0486748	−0.0243374	−	0.999704i	$-0.507748\pi$
$691$	−1.27951	−0.0486748	−0.0243374	+	0.999704i	$0.507748\pi$
$692$	103.307	3.92716
$693$	−3.71427	−0.141093
$694$	−65.8050	−2.49792
$695$	−6.71571	−0.254741
$696$	−7.00109	−0.265376
$697$	6.60590	0.250216
$698$	−24.0429	−0.910038
$699$	−22.6495	−0.856681
$700$	41.2430	1.55884
$701$	11.0095	0.415823	0.207912	−	0.978148i	$-0.433333\pi$
$701$	11.0095	0.415823	0.207912	+	0.978148i	$0.433333\pi$
$702$	48.1215	1.81623
$703$	−0.895450	−0.0337725
$704$	60.5157	2.28077
$705$	7.84055	0.295292
$706$	−75.8840	−2.85593
$707$	27.3343	1.02801
$708$	−3.96292	−0.148936
$709$	19.8095	0.743961	0.371980	−	0.928241i	$-0.378679\pi$
$709$	19.8095	0.743961	0.371980	+	0.928241i	$0.378679\pi$
$710$	−21.4339	−0.804401
$711$	−1.53707	−0.0576446
$712$	−64.3005	−2.40976
$713$	6.85650	0.256778
$714$	−5.71071	−0.213718
$715$	−17.2743	−0.646023
$716$	−4.97981	−0.186104
$717$	−11.5319	−0.430666
$718$	56.8828	2.12285
$719$	45.3355	1.69073	0.845364	−	0.534191i	$-0.179384\pi$
$719$	45.3355	1.69073	0.845364	+	0.534191i	$0.179384\pi$
$720$	−2.30320	−0.0858350
$721$	34.5912	1.28824
$722$	49.7182	1.85032
$723$	−40.7021	−1.51373
$724$	32.9636	1.22508
$725$	−2.26835	−0.0842444
$726$	−137.071	−5.08719
$727$	27.8573	1.03317	0.516586	−	0.856236i	$-0.327203\pi$
$727$	27.8573	1.03317	0.516586	+	0.856236i	$0.327203\pi$
$728$	53.6657	1.98898
$729$	23.6407	0.875582
$730$	−23.1830	−0.858040
$731$	3.84088	0.142060
$732$	−113.467	−4.19388
$733$	−25.6822	−0.948594	−0.474297	−	0.880365i	$-0.657298\pi$
$733$	−25.6822	−0.948594	−0.474297	+	0.880365i	$0.657298\pi$
$734$	22.7254	0.838809
$735$	4.55984	0.168192
$736$	−15.9753	−0.588857
$737$	−10.1252	−0.372965
$738$	8.54462	0.314532
$739$	−45.9036	−1.68859	−0.844296	−	0.535877i	$-0.819981\pi$
$739$	−45.9036	−1.68859	−0.844296	+	0.535877i	$0.819981\pi$
$740$	−13.8732	−0.509989
$741$	−1.57336	−0.0577987
$742$	−16.5691	−0.608272
$743$	29.4301	1.07968	0.539842	−	0.841766i	$-0.318484\pi$
$743$	29.4301	1.07968	0.539842	+	0.841766i	$0.318484\pi$
$744$	−67.1801	−2.46294
$745$	1.72182	0.0630826
$746$	−56.8264	−2.08056
$747$	4.14229	0.151558
$748$	−19.4743	−0.712052
$749$	−13.9861	−0.511042
$750$	−33.0567	−1.20706
$751$	36.4645	1.33061	0.665304	−	0.746573i	$-0.268302\pi$
$751$	36.4645	1.33061	0.665304	+	0.746573i	$0.268302\pi$
$752$	−59.5001	−2.16975
$753$	32.0766	1.16893
$754$	−4.99714	−0.181985
$755$	7.28818	0.265244
$756$	45.1864	1.64342
$757$	33.5601	1.21976	0.609880	−	0.792493i	$-0.291217\pi$
$757$	33.5601	1.21976	0.609880	+	0.792493i	$0.291217\pi$
$758$	−31.9146	−1.15919
$759$	16.1287	0.585433
$760$	−1.27734	−0.0463339
$761$	49.9981	1.81243	0.906215	−	0.422818i	$-0.138959\pi$
$761$	49.9981	1.81243	0.906215	+	0.422818i	$0.138959\pi$
$762$	−53.5649	−1.94045
$763$	17.8097	0.644756
$764$	58.5204	2.11719
$765$	0.144249	0.00521533
$766$	82.2947	2.97343
$767$	−1.67073	−0.0603266
$768$	−23.0831	−0.832939
$769$	−12.8839	−0.464607	−0.232304	−	0.972643i	$-0.574626\pi$
$769$	−12.8839	−0.464607	−0.232304	+	0.972643i	$0.574626\pi$
$770$	−22.8605	−0.823837
$771$	−11.7276	−0.422359
$772$	−46.8601	−1.68653
$773$	35.1053	1.26265	0.631326	−	0.775518i	$-0.282511\pi$
$773$	35.1053	1.26265	0.631326	+	0.775518i	$0.282511\pi$
$774$	4.96812	0.178575
$775$	−21.7663	−0.781870
$776$	−41.9062	−1.50434
$777$	−13.3573	−0.479190
$778$	−20.4147	−0.731902
$779$	2.40854	0.0862950
$780$	−24.3760	−0.872800
$781$	−70.3746	−2.51820
$782$	2.33473	0.0834898
$783$	−2.48524	−0.0888151
$784$	−34.6036	−1.23584
$785$	−7.58144	−0.270593
$786$	88.2105	3.14637
$787$	−42.4933	−1.51472	−0.757361	−	0.652997i	$-0.773512\pi$
$787$	−42.4933	−1.51472	−0.757361	+	0.652997i	$0.773512\pi$
$788$	72.1185	2.56911
$789$	−31.9038	−1.13581
$790$	−9.46034	−0.336584
$791$	−14.9532	−0.531673
$792$	−14.8785	−0.528684
$793$	−47.8369	−1.69874
$794$	75.9195	2.69428
$795$	4.44529	0.157658
$796$	−2.79442	−0.0990456
$797$	31.3525	1.11056	0.555281	−	0.831663i	$-0.312611\pi$
$797$	31.3525	1.11056	0.555281	+	0.831663i	$0.312611\pi$
$798$	−2.08215	−0.0737074
$799$	3.72649	0.131834
$800$	50.7145	1.79303
$801$	2.64755	0.0935465
$802$	66.5719	2.35074
$803$	−76.1173	−2.68612
$804$	−14.2877	−0.503889
$805$	1.94466	0.0685404
$806$	−47.9509	−1.68900
$807$	−1.95627	−0.0688638
$808$	109.495	3.85201
$809$	−43.6902	−1.53607	−0.768033	−	0.640411i	$-0.778764\pi$
$809$	−43.6902	−1.53607	−0.768033	+	0.640411i	$0.778764\pi$
$810$	−18.8806	−0.663396
$811$	−34.8213	−1.22274	−0.611370	−	0.791345i	$-0.709381\pi$
$811$	−34.8213	−1.22274	−0.611370	+	0.791345i	$0.709381\pi$
$812$	−4.69235	−0.164669
$813$	5.54455	0.194456
$814$	−64.1959	−2.25006
$815$	3.37796	0.118325
$816$	−11.6268	−0.407021
$817$	1.40041	0.0489940
$818$	74.5651	2.60711
$819$	−2.20966	−0.0772119
$820$	37.3155	1.30311
$821$	−14.5400	−0.507451	−0.253725	−	0.967276i	$-0.581656\pi$
$821$	−14.5400	−0.507451	−0.253725	+	0.967276i	$0.581656\pi$
$822$	−92.6031	−3.22990
$823$	5.02820	0.175272	0.0876360	−	0.996153i	$-0.472069\pi$
$823$	5.02820	0.175272	0.0876360	+	0.996153i	$0.472069\pi$
$824$	138.564	4.82711
$825$	−51.2014	−1.78260
$826$	−2.21102	−0.0769312
$827$	−0.434996	−0.0151263	−0.00756315	−	0.999971i	$-0.502407\pi$
$827$	−0.434996	−0.0151263	−0.00756315	+	0.999971i	$0.502407\pi$
$828$	2.14280	0.0744674
$829$	−3.90717	−0.135701	−0.0678507	−	0.997695i	$-0.521614\pi$
$829$	−3.90717	−0.135701	−0.0678507	+	0.997695i	$0.521614\pi$
$830$	25.4949	0.884941
$831$	−6.64067	−0.230362
$832$	36.0015	1.24813
$833$	2.16722	0.0750897
$834$	−43.8515	−1.51845
$835$	1.08747	0.0376334
$836$	−7.10043	−0.245574
$837$	−23.8475	−0.824291
$838$	−43.5546	−1.50457
$839$	−4.24154	−0.146434	−0.0732171	−	0.997316i	$-0.523327\pi$
$839$	−4.24154	−0.146434	−0.0732171	+	0.997316i	$0.523327\pi$
$840$	−19.0539	−0.657421
$841$	−28.7419	−0.991101
$842$	73.1346	2.52038
$843$	−31.7339	−1.09297
$844$	3.35787	0.115583
$845$	−0.769253	−0.0264631
$846$	4.82015	0.165720
$847$	−54.2633	−1.86451
$848$	−33.7343	−1.15844
$849$	22.9172	0.786517
$850$	−7.41174	−0.254221
$851$	5.46091	0.187198
$852$	−99.3065	−3.40218
$853$	5.74964	0.196864	0.0984320	−	0.995144i	$-0.468617\pi$
$853$	5.74964	0.196864	0.0984320	+	0.995144i	$0.468617\pi$
$854$	−63.3065	−2.16630
$855$	0.0525938	0.00179867
$856$	−56.0251	−1.91490
$857$	−10.0568	−0.343535	−0.171768	−	0.985137i	$-0.554948\pi$
$857$	−10.0568	−0.343535	−0.171768	+	0.985137i	$0.554948\pi$
$858$	−112.796	−3.85078
$859$	47.4722	1.61973	0.809866	−	0.586615i	$-0.199540\pi$
$859$	47.4722	1.61973	0.809866	+	0.586615i	$0.199540\pi$
$860$	21.6964	0.739843
$861$	35.9279	1.22442
$862$	33.0292	1.12498
$863$	−47.8985	−1.63048	−0.815242	−	0.579120i	$-0.803396\pi$
$863$	−47.8985	−1.63048	−0.815242	+	0.579120i	$0.803396\pi$
$864$	55.5635	1.89031
$865$	−15.4635	−0.525776
$866$	4.90049	0.166525
$867$	−30.2090	−1.02595
$868$	−45.0263	−1.52829
$869$	−31.0614	−1.05369
$870$	1.77422	0.0601517
$871$	−6.02357	−0.204101
$872$	71.3416	2.41593
$873$	1.72547	0.0583982
$874$	0.851255	0.0287941
$875$	−13.0864	−0.442401
$876$	−107.410	−3.62904
$877$	−3.70451	−0.125092	−0.0625462	−	0.998042i	$-0.519922\pi$
$877$	−3.70451	−0.125092	−0.0625462	+	0.998042i	$0.519922\pi$
$878$	−9.72622	−0.328244
$879$	9.67672	0.326388
$880$	−46.5434	−1.56898
$881$	39.0765	1.31652	0.658260	−	0.752791i	$-0.271293\pi$
$881$	39.0765	1.31652	0.658260	+	0.752791i	$0.271293\pi$
$882$	2.80326	0.0943907
$883$	32.2817	1.08636	0.543182	−	0.839615i	$-0.317219\pi$
$883$	32.2817	1.08636	0.543182	+	0.839615i	$0.317219\pi$
$884$	−11.5855	−0.389663
$885$	0.593188	0.0199398
$886$	104.229	3.50164
$887$	−13.1365	−0.441082	−0.220541	−	0.975378i	$-0.570782\pi$
$887$	−13.1365	−0.441082	−0.220541	+	0.975378i	$0.570782\pi$
$888$	−53.5062	−1.79555
$889$	−21.2051	−0.711197
$890$	16.2951	0.546212
$891$	−61.9911	−2.07678
$892$	−37.4371	−1.25349
$893$	1.35869	0.0454670
$894$	11.2429	0.376020
$895$	0.745401	0.0249160
$896$	4.70004	0.157017
$897$	9.59513	0.320372
$898$	10.9616	0.365795
$899$	2.47643	0.0825935
$900$	−6.80244	−0.226748
$901$	2.11277	0.0703867
$902$	172.671	5.74933
$903$	20.8896	0.695163
$904$	−59.8989	−1.99221
$905$	−4.93415	−0.164017
$906$	47.5895	1.58106
$907$	−41.7801	−1.38729	−0.693643	−	0.720319i	$-0.743995\pi$
$907$	−41.7801	−1.38729	−0.693643	+	0.720319i	$0.743995\pi$
$908$	−33.0785	−1.09775
$909$	−4.50840	−0.149534
$910$	−13.6000	−0.450836
$911$	46.2722	1.53306	0.766532	−	0.642206i	$-0.221980\pi$
$911$	46.2722	1.53306	0.766532	+	0.642206i	$0.221980\pi$
$912$	−4.23921	−0.140374
$913$	83.7081	2.77033
$914$	−0.685298	−0.0226676
$915$	16.9843	0.561485
$916$	−5.36580	−0.177291
$917$	34.9205	1.15318
$918$	−8.12041	−0.268014
$919$	54.8904	1.81067	0.905333	−	0.424703i	$-0.139621\pi$
$919$	54.8904	1.81067	0.905333	+	0.424703i	$0.139621\pi$
$920$	7.78986	0.256824
$921$	−41.4810	−1.36684
$922$	3.10633	0.102301
$923$	−41.8667	−1.37806
$924$	−105.916	−3.48438
$925$	−17.3360	−0.570004
$926$	−99.6531	−3.27480
$927$	−5.70531	−0.187387
$928$	−5.76995	−0.189408
$929$	39.0087	1.27983	0.639917	−	0.768444i	$-0.278969\pi$
$929$	39.0087	1.27983	0.639917	+	0.768444i	$0.278969\pi$
$930$	17.0248	0.558266
$931$	0.790178	0.0258970
$932$	−60.8090	−1.99186
$933$	−32.4501	−1.06237
$934$	28.6405	0.937146
$935$	2.91501	0.0953309
$936$	−8.85139	−0.289317
$937$	21.7649	0.711029	0.355515	−	0.934671i	$-0.384306\pi$
$937$	21.7649	0.711029	0.355515	+	0.934671i	$0.384306\pi$
$938$	−7.97150	−0.260279
$939$	−14.9067	−0.486463
$940$	21.0502	0.686583
$941$	17.9682	0.585746	0.292873	−	0.956151i	$-0.405389\pi$
$941$	17.9682	0.585746	0.292873	+	0.956151i	$0.405389\pi$
$942$	−49.5044	−1.61294
$943$	−14.6885	−0.478324
$944$	−4.50157	−0.146514
$945$	−6.76372	−0.220024
$946$	100.397	3.26418
$947$	−21.6724	−0.704258	−0.352129	−	0.935952i	$-0.614542\pi$
$947$	−21.6724	−0.704258	−0.352129	+	0.935952i	$0.614542\pi$
$948$	−43.8311	−1.42357
$949$	−45.2830	−1.46995
$950$	−2.70236	−0.0876760
$951$	31.7945	1.03101
$952$	−9.05599	−0.293506
$953$	−18.6648	−0.604612	−0.302306	−	0.953211i	$-0.597756\pi$
$953$	−18.6648	−0.604612	−0.302306	+	0.953211i	$0.597756\pi$
$954$	2.73284	0.0884789
$955$	−8.75961	−0.283454
$956$	−30.9606	−1.00134
$957$	5.82534	0.188307
$958$	18.4682	0.596679
$959$	−36.6594	−1.18380
$960$	−12.7822	−0.412545
$961$	−7.23702	−0.233452
$962$	−38.1909	−1.23132
$963$	2.30681	0.0743359
$964$	−109.277	−3.51956
$965$	7.01424	0.225797
$966$	12.6980	0.408553
$967$	−12.3341	−0.396639	−0.198320	−	0.980137i	$-0.563548\pi$
$967$	−12.3341	−0.396639	−0.198320	+	0.980137i	$0.563548\pi$
$968$	−217.366	−6.98641
$969$	0.265501	0.00852912
$970$	10.6199	0.340984
$971$	−0.830081	−0.0266386	−0.0133193	−	0.999911i	$-0.504240\pi$
$971$	−0.830081	−0.0266386	−0.0133193	+	0.999911i	$0.504240\pi$
$972$	−15.7702	−0.505828
$973$	−17.3598	−0.556529
$974$	92.8040	2.97363
$975$	−30.4603	−0.975510
$976$	−128.890	−4.12568
$977$	53.7143	1.71847	0.859237	−	0.511579i	$-0.170939\pi$
$977$	53.7143	1.71847	0.859237	+	0.511579i	$0.170939\pi$
$978$	22.0570	0.705306
$979$	53.5021	1.70993
$980$	12.2422	0.391063
$981$	−2.93746	−0.0937859
$982$	47.3012	1.50944
$983$	−43.1927	−1.37763	−0.688816	−	0.724936i	$-0.741869\pi$
$983$	−43.1927	−1.37763	−0.688816	+	0.724936i	$0.741869\pi$
$984$	143.919	4.58796
$985$	−10.7950	−0.343958
$986$	0.843257	0.0268548
$987$	20.2675	0.645120
$988$	−4.22413	−0.134387
$989$	−8.54039	−0.271568
$990$	3.77052	0.119835
$991$	−6.64149	−0.210974	−0.105487	−	0.994421i	$-0.533640\pi$
$991$	−6.64149	−0.210974	−0.105487	+	0.994421i	$0.533640\pi$
$992$	−55.3666	−1.75789
$993$	−18.9586	−0.601633
$994$	−55.4057	−1.75736
$995$	0.418282	0.0132604
$996$	118.121	3.74282
$997$	24.1376	0.764445	0.382222	−	0.924070i	$-0.375159\pi$
$997$	24.1376	0.764445	0.382222	+	0.924070i	$0.375159\pi$
$998$	86.9940	2.75375
$999$	−18.9936	−0.600930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8039.2.a.a.1.11	✓	279

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8039.2.a.a.1.11	✓	279	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:	\( N \)	\(=\)	\( 8039 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8039.a (trivial)

\(f(q)\)	\(=\)	\(q-2.62410 q^{2} +1.81984 q^{3} +4.88588 q^{4} -0.731341 q^{5} -4.77543 q^{6} -1.89048 q^{7} -7.57282 q^{8} +0.311808 q^{9} +O(q^{10})\)	\(q-2.62410 q^{2} +1.81984 q^{3} +4.88588 q^{4} -0.731341 q^{5} -4.77543 q^{6} -1.89048 q^{7} -7.57282 q^{8} +0.311808 q^{9} +1.91911 q^{10} +6.30107 q^{11} +8.89151 q^{12} +3.74858 q^{13} +4.96081 q^{14} -1.33092 q^{15} +10.1001 q^{16} -0.632565 q^{17} -0.818213 q^{18} -0.230636 q^{19} -3.57325 q^{20} -3.44037 q^{21} -16.5346 q^{22} +1.40654 q^{23} -13.7813 q^{24} -4.46514 q^{25} -9.83663 q^{26} -4.89207 q^{27} -9.23667 q^{28} +0.508013 q^{29} +3.49247 q^{30} +4.87473 q^{31} -11.3579 q^{32} +11.4669 q^{33} +1.65991 q^{34} +1.38259 q^{35} +1.52345 q^{36} +3.88252 q^{37} +0.605212 q^{38} +6.82180 q^{39} +5.53832 q^{40} -10.4430 q^{41} +9.02786 q^{42} -6.07192 q^{43} +30.7862 q^{44} -0.228038 q^{45} -3.69089 q^{46} -5.89107 q^{47} +18.3805 q^{48} -3.42608 q^{49} +11.7170 q^{50} -1.15117 q^{51} +18.3151 q^{52} -3.34001 q^{53} +12.8373 q^{54} -4.60823 q^{55} +14.3163 q^{56} -0.419721 q^{57} -1.33308 q^{58} -0.445697 q^{59} -6.50273 q^{60} -12.7613 q^{61} -12.7918 q^{62} -0.589467 q^{63} +9.60404 q^{64} -2.74149 q^{65} -30.0903 q^{66} -1.60690 q^{67} -3.09064 q^{68} +2.55967 q^{69} -3.62804 q^{70} -11.1687 q^{71} -2.36126 q^{72} -12.0801 q^{73} -10.1881 q^{74} -8.12583 q^{75} -1.12686 q^{76} -11.9121 q^{77} -17.9011 q^{78} -4.92954 q^{79} -7.38659 q^{80} -9.83820 q^{81} +27.4035 q^{82} +13.2847 q^{83} -16.8092 q^{84} +0.462621 q^{85} +15.9333 q^{86} +0.924501 q^{87} -47.7169 q^{88} +8.49096 q^{89} +0.598393 q^{90} -7.08662 q^{91} +6.87218 q^{92} +8.87121 q^{93} +15.4587 q^{94} +0.168674 q^{95} -20.6695 q^{96} +5.53376 q^{97} +8.99036 q^{98} +1.96472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9}+O(q^{10}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9	\( 279 q - 13 q^{2} - 12 q^{3} + 227 q^{4} - 20 q^{5} - 40 q^{6} - 57 q^{7} - 39 q^{8} + 175 q^{9} - 42 q^{10} - 53 q^{11} - 36 q^{12} - 75 q^{13} - 31 q^{14} - 60 q^{15} + 127 q^{16} - 55 q^{17} - 57 q^{18} - 113 q^{19} - 43 q^{20} - 103 q^{21} - 73 q^{22} - 30 q^{23} - 106 q^{24} + 75 q^{25} - 42 q^{26} - 45 q^{27} - 146 q^{28} - 92 q^{29} - 76 q^{30} - 84 q^{31} - 71 q^{32} - 117 q^{33} - 106 q^{34} - 49 q^{35} + 67 q^{36} - 123 q^{37} - 21 q^{38} - 92 q^{39} - 97 q^{40} - 116 q^{41} - 19 q^{42} - 126 q^{43} - 131 q^{44} - 85 q^{45} - 183 q^{46} - 42 q^{47} - 47 q^{48} - 22 q^{49} - 64 q^{50} - 90 q^{51} - 158 q^{52} - 60 q^{53} - 117 q^{54} - 99 q^{55} - 65 q^{56} - 182 q^{57} - 93 q^{58} - 58 q^{59} - 141 q^{60} - 217 q^{61} - 16 q^{62} - 141 q^{63} - 47 q^{64} - 197 q^{65} - 53 q^{66} - 147 q^{67} - 90 q^{68} - 103 q^{69} - 118 q^{70} - 78 q^{71} - 135 q^{72} - 282 q^{73} - 98 q^{74} - 53 q^{75} - 296 q^{76} - 53 q^{77} - 27 q^{78} - 153 q^{79} - 52 q^{80} - 89 q^{81} - 81 q^{82} - 54 q^{83} - 164 q^{84} - 303 q^{85} - 82 q^{86} - 29 q^{87} - 203 q^{88} - 185 q^{89} - 56 q^{90} - 163 q^{91} - 66 q^{92} - 156 q^{93} - 134 q^{94} - 69 q^{95} - 189 q^{96} - 212 q^{97} - 13 q^{98} - 181 q^{99}+O(q^{100}) \) 279 * q - 13 * q^2 - 12 * q^3 + 227 * q^4 - 20 * q^5 - 40 * q^6 - 57 * q^7 - 39 * q^8 + 175 * q^9 - 42 * q^10 - 53 * q^11 - 36 * q^12 - 75 * q^13 - 31 * q^14 - 60 * q^15 + 127 * q^16 - 55 * q^17 - 57 * q^18 - 113 * q^19 - 43 * q^20 - 103 * q^21 - 73 * q^22 - 30 * q^23 - 106 * q^24 + 75 * q^25 - 42 * q^26 - 45 * q^27 - 146 * q^28 - 92 * q^29 - 76 * q^30 - 84 * q^31 - 71 * q^32 - 117 * q^33 - 106 * q^34 - 49 * q^35 + 67 * q^36 - 123 * q^37 - 21 * q^38 - 92 * q^39 - 97 * q^40 - 116 * q^41 - 19 * q^42 - 126 * q^43 - 131 * q^44 - 85 * q^45 - 183 * q^46 - 42 * q^47 - 47 * q^48 - 22 * q^49 - 64 * q^50 - 90 * q^51 - 158 * q^52 - 60 * q^53 - 117 * q^54 - 99 * q^55 - 65 * q^56 - 182 * q^57 - 93 * q^58 - 58 * q^59 - 141 * q^60 - 217 * q^61 - 16 * q^62 - 141 * q^63 - 47 * q^64 - 197 * q^65 - 53 * q^66 - 147 * q^67 - 90 * q^68 - 103 * q^69 - 118 * q^70 - 78 * q^71 - 135 * q^72 - 282 * q^73 - 98 * q^74 - 53 * q^75 - 296 * q^76 - 53 * q^77 - 27 * q^78 - 153 * q^79 - 52 * q^80 - 89 * q^81 - 81 * q^82 - 54 * q^83 - 164 * q^84 - 303 * q^85 - 82 * q^86 - 29 * q^87 - 203 * q^88 - 185 * q^89 - 56 * q^90 - 163 * q^91 - 66 * q^92 - 156 * q^93 - 134 * q^94 - 69 * q^95 - 189 * q^96 - 212 * q^97 - 13 * q^98 - 181 * q^99

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