LMFDB - Embedded newform 8002.2.a.e.1.40

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8002,2,Mod(1,8002)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8002, 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("8002.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8002 = 2 \cdot 4001 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8002.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:	$63.8962916974$
Analytic rank:	$0$
Dimension:	$77$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.40
Character	$\chi$	$=$	8002.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +0.119243 q^{3} +1.00000 q^{4} -0.0709742 q^{5} -0.119243 q^{6} +1.96442 q^{7} -1.00000 q^{8} -2.98578 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +0.119243 q^{3} +1.00000 q^{4} -0.0709742 q^{5} -0.119243 q^{6} +1.96442 q^{7} -1.00000 q^{8} -2.98578 q^{9} +0.0709742 q^{10} +5.12920 q^{11} +0.119243 q^{12} +0.551579 q^{13} -1.96442 q^{14} -0.00846320 q^{15} +1.00000 q^{16} -2.75848 q^{17} +2.98578 q^{18} +2.88018 q^{19} -0.0709742 q^{20} +0.234245 q^{21} -5.12920 q^{22} +5.89411 q^{23} -0.119243 q^{24} -4.99496 q^{25} -0.551579 q^{26} -0.713765 q^{27} +1.96442 q^{28} +1.16607 q^{29} +0.00846320 q^{30} -2.13263 q^{31} -1.00000 q^{32} +0.611623 q^{33} +2.75848 q^{34} -0.139423 q^{35} -2.98578 q^{36} +8.50867 q^{37} -2.88018 q^{38} +0.0657721 q^{39} +0.0709742 q^{40} +4.41854 q^{41} -0.234245 q^{42} +7.25488 q^{43} +5.12920 q^{44} +0.211913 q^{45} -5.89411 q^{46} +7.67828 q^{47} +0.119243 q^{48} -3.14104 q^{49} +4.99496 q^{50} -0.328931 q^{51} +0.551579 q^{52} -7.67102 q^{53} +0.713765 q^{54} -0.364041 q^{55} -1.96442 q^{56} +0.343443 q^{57} -1.16607 q^{58} -9.61202 q^{59} -0.00846320 q^{60} -1.20342 q^{61} +2.13263 q^{62} -5.86534 q^{63} +1.00000 q^{64} -0.0391479 q^{65} -0.611623 q^{66} -8.40982 q^{67} -2.75848 q^{68} +0.702834 q^{69} +0.139423 q^{70} -6.85391 q^{71} +2.98578 q^{72} -4.85168 q^{73} -8.50867 q^{74} -0.595616 q^{75} +2.88018 q^{76} +10.0759 q^{77} -0.0657721 q^{78} +8.99305 q^{79} -0.0709742 q^{80} +8.87223 q^{81} -4.41854 q^{82} +5.67967 q^{83} +0.234245 q^{84} +0.195781 q^{85} -7.25488 q^{86} +0.139046 q^{87} -5.12920 q^{88} +14.3983 q^{89} -0.211913 q^{90} +1.08353 q^{91} +5.89411 q^{92} -0.254302 q^{93} -7.67828 q^{94} -0.204419 q^{95} -0.119243 q^{96} +17.2277 q^{97} +3.14104 q^{98} -15.3147 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) $ 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0.119243	0.0688452	0.0344226	−	0.999407i	$-0.489041\pi$
$3$	0.119243	0.0688452	0.0344226	+	0.999407i	$0.489041\pi$
$4$	1.00000	0.500000
$5$	−0.0709742	−0.0317406	−0.0158703	−	0.999874i	$-0.505052\pi$
$5$	−0.0709742	−0.0317406	−0.0158703	+	0.999874i	$0.505052\pi$
$6$	−0.119243	−0.0486809
$7$	1.96442	0.742482	0.371241	−	0.928536i	$-0.378932\pi$
$7$	1.96442	0.742482	0.371241	+	0.928536i	$0.378932\pi$
$8$	−1.00000	−0.353553
$9$	−2.98578	−0.995260
$10$	0.0709742	0.0224440
$11$	5.12920	1.54651	0.773256	−	0.634094i	$-0.218627\pi$
$11$	5.12920	1.54651	0.773256	+	0.634094i	$0.218627\pi$
$12$	0.119243	0.0344226
$13$	0.551579	0.152980	0.0764902	−	0.997070i	$-0.475629\pi$
$13$	0.551579	0.152980	0.0764902	+	0.997070i	$0.475629\pi$
$14$	−1.96442	−0.525014
$15$	−0.00846320	−0.00218519
$16$	1.00000	0.250000
$17$	−2.75848	−0.669030	−0.334515	−	0.942390i	$-0.608573\pi$
$17$	−2.75848	−0.669030	−0.334515	+	0.942390i	$0.608573\pi$
$18$	2.98578	0.703755
$19$	2.88018	0.660759	0.330380	−	0.943848i	$-0.392823\pi$
$19$	2.88018	0.660759	0.330380	+	0.943848i	$0.392823\pi$
$20$	−0.0709742	−0.0158703
$21$	0.234245	0.0511163
$22$	−5.12920	−1.09355
$23$	5.89411	1.22901	0.614504	−	0.788914i	$-0.289356\pi$
$23$	5.89411	1.22901	0.614504	+	0.788914i	$0.289356\pi$
$24$	−0.119243	−0.0243404
$25$	−4.99496	−0.998993
$26$	−0.551579	−0.108173
$27$	−0.713765	−0.137364
$28$	1.96442	0.371241
$29$	1.16607	0.216533	0.108267	−	0.994122i	$-0.465470\pi$
$29$	1.16607	0.216533	0.108267	+	0.994122i	$0.465470\pi$
$30$	0.00846320	0.00154516
$31$	−2.13263	−0.383032	−0.191516	−	0.981489i	$-0.561340\pi$
$31$	−2.13263	−0.383032	−0.191516	+	0.981489i	$0.561340\pi$
$32$	−1.00000	−0.176777
$33$	0.611623	0.106470
$34$	2.75848	0.473076
$35$	−0.139423	−0.0235669
$36$	−2.98578	−0.497630
$37$	8.50867	1.39882	0.699409	−	0.714722i	$-0.253447\pi$
$37$	8.50867	1.39882	0.699409	+	0.714722i	$0.253447\pi$
$38$	−2.88018	−0.467228
$39$	0.0657721	0.0105320
$40$	0.0709742	0.0112220
$41$	4.41854	0.690060	0.345030	−	0.938592i	$-0.387869\pi$
$41$	4.41854	0.690060	0.345030	+	0.938592i	$0.387869\pi$
$42$	−0.234245	−0.0361447
$43$	7.25488	1.10636	0.553180	−	0.833062i	$-0.313414\pi$
$43$	7.25488	1.10636	0.553180	+	0.833062i	$0.313414\pi$
$44$	5.12920	0.773256
$45$	0.211913	0.0315902
$46$	−5.89411	−0.869039
$47$	7.67828	1.11999	0.559996	−	0.828495i	$-0.310803\pi$
$47$	7.67828	1.11999	0.559996	+	0.828495i	$0.310803\pi$
$48$	0.119243	0.0172113
$49$	−3.14104	−0.448720
$50$	4.99496	0.706394
$51$	−0.328931	−0.0460595
$52$	0.551579	0.0764902
$53$	−7.67102	−1.05370	−0.526848	−	0.849959i	$-0.676626\pi$
$53$	−7.67102	−1.05370	−0.526848	+	0.849959i	$0.676626\pi$
$54$	0.713765	0.0971311
$55$	−0.364041	−0.0490873
$56$	−1.96442	−0.262507
$57$	0.343443	0.0454901
$58$	−1.16607	−0.153112
$59$	−9.61202	−1.25138	−0.625689	−	0.780072i	$-0.715182\pi$
$59$	−9.61202	−1.25138	−0.625689	+	0.780072i	$0.715182\pi$
$60$	−0.00846320	−0.00109259
$61$	−1.20342	−0.154083	−0.0770413	−	0.997028i	$-0.524547\pi$
$61$	−1.20342	−0.154083	−0.0770413	+	0.997028i	$0.524547\pi$
$62$	2.13263	0.270845
$63$	−5.86534	−0.738963
$64$	1.00000	0.125000
$65$	−0.0391479	−0.00485569
$66$	−0.611623	−0.0752856
$67$	−8.40982	−1.02742	−0.513712	−	0.857963i	$-0.671730\pi$
$67$	−8.40982	−1.02742	−0.513712	+	0.857963i	$0.671730\pi$
$68$	−2.75848	−0.334515
$69$	0.702834	0.0846112
$70$	0.139423	0.0166643
$71$	−6.85391	−0.813409	−0.406705	−	0.913560i	$-0.633322\pi$
$71$	−6.85391	−0.813409	−0.406705	+	0.913560i	$0.633322\pi$
$72$	2.98578	0.351878
$73$	−4.85168	−0.567846	−0.283923	−	0.958847i	$-0.591636\pi$
$73$	−4.85168	−0.567846	−0.283923	+	0.958847i	$0.591636\pi$
$74$	−8.50867	−0.989113
$75$	−0.595616	−0.0687758
$76$	2.88018	0.330380
$77$	10.0759	1.14826
$78$	−0.0657721	−0.00744722
$79$	8.99305	1.01180	0.505898	−	0.862593i	$-0.331161\pi$
$79$	8.99305	1.01180	0.505898	+	0.862593i	$0.331161\pi$
$80$	−0.0709742	−0.00793516
$81$	8.87223	0.985803
$82$	−4.41854	−0.487946
$83$	5.67967	0.623425	0.311712	−	0.950177i	$-0.399097\pi$
$83$	5.67967	0.623425	0.311712	+	0.950177i	$0.399097\pi$
$84$	0.234245	0.0255582
$85$	0.195781	0.0212354
$86$	−7.25488	−0.782314
$87$	0.139046	0.0149073
$88$	−5.12920	−0.546775
$89$	14.3983	1.52621	0.763107	−	0.646272i	$-0.223673\pi$
$89$	14.3983	1.52621	0.763107	+	0.646272i	$0.223673\pi$
$90$	−0.211913	−0.0223376
$91$	1.08353	0.113585
$92$	5.89411	0.614504
$93$	−0.254302	−0.0263699
$94$	−7.67828	−0.791954
$95$	−0.204419	−0.0209729
$96$	−0.119243	−0.0121702
$97$	17.2277	1.74921	0.874603	−	0.484840i	$-0.161122\pi$
$97$	17.2277	1.74921	0.874603	+	0.484840i	$0.161122\pi$
$98$	3.14104	0.317293
$99$	−15.3147	−1.53918
$100$	−4.99496	−0.499496
$101$	15.1030	1.50280	0.751400	−	0.659846i	$-0.229379\pi$
$101$	15.1030	1.50280	0.751400	+	0.659846i	$0.229379\pi$
$102$	0.328931	0.0325690
$103$	−5.96394	−0.587644	−0.293822	−	0.955860i	$-0.594927\pi$
$103$	−5.96394	−0.587644	−0.293822	+	0.955860i	$0.594927\pi$
$104$	−0.551579	−0.0540867
$105$	−0.0166253	−0.00162246
$106$	7.67102	0.745076
$107$	−5.88966	−0.569375	−0.284687	−	0.958620i	$-0.591890\pi$
$107$	−5.88966	−0.569375	−0.284687	+	0.958620i	$0.591890\pi$
$108$	−0.713765	−0.0686820
$109$	19.5546	1.87299	0.936494	−	0.350683i	$-0.114050\pi$
$109$	19.5546	1.87299	0.936494	+	0.350683i	$0.114050\pi$
$110$	0.364041	0.0347099
$111$	1.01460	0.0963018
$112$	1.96442	0.185621
$113$	−2.39291	−0.225106	−0.112553	−	0.993646i	$-0.535903\pi$
$113$	−2.39291	−0.225106	−0.112553	+	0.993646i	$0.535903\pi$
$114$	−0.343443	−0.0321664
$115$	−0.418330	−0.0390095
$116$	1.16607	0.108267
$117$	−1.64689	−0.152255
$118$	9.61202	0.884858
$119$	−5.41883	−0.496743
$120$	0.00846320	0.000772581 0
$121$	15.3087	1.39170
$122$	1.20342	0.108953
$123$	0.526881	0.0475073
$124$	−2.13263	−0.191516
$125$	0.709385	0.0634493
$126$	5.86534	0.522526
$127$	−13.7379	−1.21904	−0.609520	−	0.792770i	$-0.708638\pi$
$127$	−13.7379	−1.21904	−0.609520	+	0.792770i	$0.708638\pi$
$128$	−1.00000	−0.0883883
$129$	0.865097	0.0761675
$130$	0.0391479	0.00343349
$131$	−16.3354	−1.42723	−0.713614	−	0.700539i	$-0.752943\pi$
$131$	−16.3354	−1.42723	−0.713614	+	0.700539i	$0.752943\pi$
$132$	0.611623	0.0532350
$133$	5.65790	0.490602
$134$	8.40982	0.726498
$135$	0.0506589	0.00436002
$136$	2.75848	0.236538
$137$	1.76482	0.150778	0.0753892	−	0.997154i	$-0.475980\pi$
$137$	1.76482	0.150778	0.0753892	+	0.997154i	$0.475980\pi$
$138$	−0.702834	−0.0598292
$139$	0.977582	0.0829175	0.0414587	−	0.999140i	$-0.486799\pi$
$139$	0.977582	0.0829175	0.0414587	+	0.999140i	$0.486799\pi$
$140$	−0.139423	−0.0117834
$141$	0.915584	0.0771061
$142$	6.85391	0.575167
$143$	2.82916	0.236586
$144$	−2.98578	−0.248815
$145$	−0.0827607	−0.00687290
$146$	4.85168	0.401527
$147$	−0.374548	−0.0308922
$148$	8.50867	0.699409
$149$	−4.00607	−0.328190	−0.164095	−	0.986445i	$-0.552470\pi$
$149$	−4.00607	−0.328190	−0.164095	+	0.986445i	$0.552470\pi$
$150$	0.595616	0.0486319
$151$	−20.7180	−1.68601	−0.843005	−	0.537905i	$-0.819216\pi$
$151$	−20.7180	−1.68601	−0.843005	+	0.537905i	$0.819216\pi$
$152$	−2.88018	−0.233614
$153$	8.23622	0.665859
$154$	−10.0759	−0.811941
$155$	0.151362	0.0121577
$156$	0.0657721	0.00526598
$157$	−16.1265	−1.28704	−0.643518	−	0.765431i	$-0.722526\pi$
$157$	−16.1265	−1.28704	−0.643518	+	0.765431i	$0.722526\pi$
$158$	−8.99305	−0.715448
$159$	−0.914719	−0.0725419
$160$	0.0709742	0.00561100
$161$	11.5785	0.912516
$162$	−8.87223	−0.697068
$163$	12.4350	0.973984	0.486992	−	0.873406i	$-0.338094\pi$
$163$	12.4350	0.973984	0.486992	+	0.873406i	$0.338094\pi$
$164$	4.41854	0.345030
$165$	−0.0434095	−0.00337942
$166$	−5.67967	−0.440828
$167$	−4.47550	−0.346325	−0.173162	−	0.984893i	$-0.555399\pi$
$167$	−4.47550	−0.346325	−0.173162	+	0.984893i	$0.555399\pi$
$168$	−0.234245	−0.0180724
$169$	−12.6958	−0.976597
$170$	−0.195781	−0.0150157
$171$	−8.59960	−0.657628
$172$	7.25488	0.553180
$173$	−6.82431	−0.518843	−0.259421	−	0.965764i	$-0.583532\pi$
$173$	−6.82431	−0.518843	−0.259421	+	0.965764i	$0.583532\pi$
$174$	−0.139046	−0.0105410
$175$	−9.81222	−0.741734
$176$	5.12920	0.386628
$177$	−1.14617	−0.0861514
$178$	−14.3983	−1.07920
$179$	24.2127	1.80974	0.904872	−	0.425684i	$-0.139967\pi$
$179$	24.2127	1.80974	0.904872	+	0.425684i	$0.139967\pi$
$180$	0.211913	0.0157951
$181$	19.0511	1.41606	0.708028	−	0.706185i	$-0.249585\pi$
$181$	19.0511	1.41606	0.708028	+	0.706185i	$0.249585\pi$
$182$	−1.08353	−0.0803169
$183$	−0.143500	−0.0106078
$184$	−5.89411	−0.434520
$185$	−0.603896	−0.0443993
$186$	0.254302	0.0186464
$187$	−14.1488	−1.03466
$188$	7.67828	0.559996
$189$	−1.40214	−0.101990
$190$	0.204419	0.0148301
$191$	−17.1791	−1.24304	−0.621520	−	0.783399i	$-0.713484\pi$
$191$	−17.1791	−1.24304	−0.621520	+	0.783399i	$0.713484\pi$
$192$	0.119243	0.00860565
$193$	−2.92368	−0.210451	−0.105226	−	0.994448i	$-0.533556\pi$
$193$	−2.92368	−0.210451	−0.105226	+	0.994448i	$0.533556\pi$
$194$	−17.2277	−1.23688
$195$	−0.00466812	−0.000334291 0
$196$	−3.14104	−0.224360
$197$	11.0479	0.787129	0.393565	−	0.919297i	$-0.371242\pi$
$197$	11.0479	0.787129	0.393565	+	0.919297i	$0.371242\pi$
$198$	15.3147	1.08837
$199$	7.01744	0.497453	0.248727	−	0.968574i	$-0.419988\pi$
$199$	7.01744	0.497453	0.248727	+	0.968574i	$0.419988\pi$
$200$	4.99496	0.353197
$201$	−1.00282	−0.0707331
$202$	−15.1030	−1.06264
$203$	2.29065	0.160772
$204$	−0.328931	−0.0230298
$205$	−0.313602	−0.0219029
$206$	5.96394	0.415527
$207$	−17.5985	−1.22318
$208$	0.551579	0.0382451
$209$	14.7730	1.02187
$210$	0.0166253	0.00114726
$211$	12.7316	0.876477	0.438239	−	0.898859i	$-0.355602\pi$
$211$	12.7316	0.876477	0.438239	+	0.898859i	$0.355602\pi$
$212$	−7.67102	−0.526848
$213$	−0.817283	−0.0559993
$214$	5.88966	0.402609
$215$	−0.514910	−0.0351165
$216$	0.713765	0.0485655
$217$	−4.18940	−0.284395
$218$	−19.5546	−1.32440
$219$	−0.578530	−0.0390934
$220$	−0.364041	−0.0245436
$221$	−1.52152	−0.102348
$222$	−1.01460	−0.0680957
$223$	1.21606	0.0814334	0.0407167	−	0.999171i	$-0.487036\pi$
$223$	1.21606	0.0814334	0.0407167	+	0.999171i	$0.487036\pi$
$224$	−1.96442	−0.131254
$225$	14.9139	0.994258
$226$	2.39291	0.159174
$227$	−1.19008	−0.0789881	−0.0394941	−	0.999220i	$-0.512575\pi$
$227$	−1.19008	−0.0789881	−0.0394941	+	0.999220i	$0.512575\pi$
$228$	0.343443	0.0227451
$229$	−21.8149	−1.44157	−0.720783	−	0.693161i	$-0.756218\pi$
$229$	−21.8149	−1.44157	−0.720783	+	0.693161i	$0.756218\pi$
$230$	0.418330	0.0275839
$231$	1.20149	0.0790521
$232$	−1.16607	−0.0765561
$233$	6.37810	0.417843	0.208922	−	0.977932i	$-0.433005\pi$
$233$	6.37810	0.417843	0.208922	+	0.977932i	$0.433005\pi$
$234$	1.64689	0.107661
$235$	−0.544960	−0.0355493
$236$	−9.61202	−0.625689
$237$	1.07236	0.0696573
$238$	5.41883	0.351250
$239$	18.0909	1.17020	0.585101	−	0.810960i	$-0.301055\pi$
$239$	18.0909	1.17020	0.585101	+	0.810960i	$0.301055\pi$
$240$	−0.00846320	−0.000546297 0
$241$	7.60740	0.490036	0.245018	−	0.969519i	$-0.421206\pi$
$241$	7.60740	0.490036	0.245018	+	0.969519i	$0.421206\pi$
$242$	−15.3087	−0.984081
$243$	3.19925	0.205232
$244$	−1.20342	−0.0770413
$245$	0.222933	0.0142426
$246$	−0.526881	−0.0335927
$247$	1.58865	0.101083
$248$	2.13263	0.135422
$249$	0.677263	0.0429198
$250$	−0.709385	−0.0448654
$251$	23.2159	1.46537	0.732687	−	0.680565i	$-0.238266\pi$
$251$	23.2159	1.46537	0.732687	+	0.680565i	$0.238266\pi$
$252$	−5.86534	−0.369482
$253$	30.2321	1.90067
$254$	13.7379	0.861992
$255$	0.0233456	0.00146196
$256$	1.00000	0.0625000
$257$	19.7002	1.22887	0.614434	−	0.788968i	$-0.289384\pi$
$257$	19.7002	1.22887	0.614434	+	0.788968i	$0.289384\pi$
$258$	−0.865097	−0.0538586
$259$	16.7146	1.03860
$260$	−0.0391479	−0.00242785
$261$	−3.48162	−0.215507
$262$	16.3354	1.00920
$263$	8.10650	0.499868	0.249934	−	0.968263i	$-0.419591\pi$
$263$	8.10650	0.499868	0.249934	+	0.968263i	$0.419591\pi$
$264$	−0.611623	−0.0376428
$265$	0.544445	0.0334450
$266$	−5.65790	−0.346908
$267$	1.71690	0.105072
$268$	−8.40982	−0.513712
$269$	1.08307	0.0660358	0.0330179	−	0.999455i	$-0.489488\pi$
$269$	1.08307	0.0660358	0.0330179	+	0.999455i	$0.489488\pi$
$270$	−0.0506589	−0.00308300
$271$	−3.70028	−0.224776	−0.112388	−	0.993664i	$-0.535850\pi$
$271$	−3.70028	−0.224776	−0.112388	+	0.993664i	$0.535850\pi$
$272$	−2.75848	−0.167258
$273$	0.129204	0.00781980
$274$	−1.76482	−0.106616
$275$	−25.6202	−1.54495
$276$	0.702834	0.0423056
$277$	22.7710	1.36817	0.684087	−	0.729400i	$-0.260201\pi$
$277$	22.7710	1.36817	0.684087	+	0.729400i	$0.260201\pi$
$278$	−0.977582	−0.0586315
$279$	6.36758	0.381217
$280$	0.139423	0.00833214
$281$	−8.83444	−0.527018	−0.263509	−	0.964657i	$-0.584880\pi$
$281$	−8.83444	−0.527018	−0.263509	+	0.964657i	$0.584880\pi$
$282$	−0.915584	−0.0545222
$283$	0.868019	0.0515984	0.0257992	−	0.999667i	$-0.491787\pi$
$283$	0.868019	0.0515984	0.0257992	+	0.999667i	$0.491787\pi$
$284$	−6.85391	−0.406705
$285$	−0.0243756	−0.00144388
$286$	−2.82916	−0.167292
$287$	8.67988	0.512357
$288$	2.98578	0.175939
$289$	−9.39078	−0.552399
$290$	0.0827607	0.00485988
$291$	2.05429	0.120424
$292$	−4.85168	−0.283923
$293$	−11.6653	−0.681492	−0.340746	−	0.940155i	$-0.610680\pi$
$293$	−11.6653	−0.681492	−0.340746	+	0.940155i	$0.610680\pi$
$294$	0.374548	0.0218441
$295$	0.682206	0.0397196
$296$	−8.50867	−0.494557
$297$	−3.66104	−0.212435
$298$	4.00607	0.232065
$299$	3.25107	0.188014
$300$	−0.595616	−0.0343879
$301$	14.2517	0.821452
$302$	20.7180	1.19219
$303$	1.80093	0.103461
$304$	2.88018	0.165190
$305$	0.0854120	0.00489068
$306$	−8.23622	−0.470833
$307$	11.0978	0.633383	0.316692	−	0.948529i	$-0.397428\pi$
$307$	11.0978	0.633383	0.316692	+	0.948529i	$0.397428\pi$
$308$	10.0759	0.574129
$309$	−0.711160	−0.0404565
$310$	−0.151362	−0.00859678
$311$	−0.807940	−0.0458141	−0.0229070	−	0.999738i	$-0.507292\pi$
$311$	−0.807940	−0.0458141	−0.0229070	+	0.999738i	$0.507292\pi$
$312$	−0.0657721	−0.00372361
$313$	9.46612	0.535057	0.267528	−	0.963550i	$-0.413793\pi$
$313$	9.46612	0.535057	0.267528	+	0.963550i	$0.413793\pi$
$314$	16.1265	0.910072
$315$	0.416288	0.0234552
$316$	8.99305	0.505898
$317$	25.4678	1.43041	0.715206	−	0.698913i	$-0.246333\pi$
$317$	25.4678	1.43041	0.715206	+	0.698913i	$0.246333\pi$
$318$	0.914719	0.0512949
$319$	5.98100	0.334871
$320$	−0.0709742	−0.00396758
$321$	−0.702303	−0.0391987
$322$	−11.5785	−0.645246
$323$	−7.94493	−0.442068
$324$	8.87223	0.492902
$325$	−2.75511	−0.152826
$326$	−12.4350	−0.688711
$327$	2.33175	0.128946
$328$	−4.41854	−0.243973
$329$	15.0834	0.831575
$330$	0.0434095	0.00238961
$331$	27.5023	1.51166	0.755832	−	0.654766i	$-0.227233\pi$
$331$	27.5023	1.51166	0.755832	+	0.654766i	$0.227233\pi$
$332$	5.67967	0.311712
$333$	−25.4050	−1.39219
$334$	4.47550	0.244889
$335$	0.596880	0.0326111
$336$	0.234245	0.0127791
$337$	−21.4574	−1.16886	−0.584429	−	0.811445i	$-0.698681\pi$
$337$	−21.4574	−1.16886	−0.584429	+	0.811445i	$0.698681\pi$
$338$	12.6958	0.690558
$339$	−0.285339	−0.0154975
$340$	0.195781	0.0106177
$341$	−10.9387	−0.592364
$342$	8.59960	0.465013
$343$	−19.9213	−1.07565
$344$	−7.25488	−0.391157
$345$	−0.0498831	−0.00268561
$346$	6.82431	0.366877
$347$	12.8880	0.691862	0.345931	−	0.938260i	$-0.387563\pi$
$347$	12.8880	0.691862	0.345931	+	0.938260i	$0.387563\pi$
$348$	0.139046	0.00745364
$349$	−27.8506	−1.49081	−0.745405	−	0.666612i	$-0.767744\pi$
$349$	−27.8506	−1.49081	−0.745405	+	0.666612i	$0.767744\pi$
$350$	9.81222	0.524485
$351$	−0.393697	−0.0210140
$352$	−5.12920	−0.273387
$353$	32.1463	1.71098	0.855488	−	0.517822i	$-0.173257\pi$
$353$	32.1463	1.71098	0.855488	+	0.517822i	$0.173257\pi$
$354$	1.14617	0.0609182
$355$	0.486451	0.0258181
$356$	14.3983	0.763107
$357$	−0.646159	−0.0341984
$358$	−24.2127	−1.27968
$359$	8.43702	0.445289	0.222645	−	0.974900i	$-0.428531\pi$
$359$	8.43702	0.445289	0.222645	+	0.974900i	$0.428531\pi$
$360$	−0.211913	−0.0111688
$361$	−10.7045	−0.563397
$362$	−19.0511	−1.00130
$363$	1.82546	0.0958119
$364$	1.08353	0.0567926
$365$	0.344344	0.0180238
$366$	0.143500	0.00750088
$367$	−23.9924	−1.25239	−0.626197	−	0.779665i	$-0.715389\pi$
$367$	−23.9924	−1.25239	−0.626197	+	0.779665i	$0.715389\pi$
$368$	5.89411	0.307252
$369$	−13.1928	−0.686789
$370$	0.603896	0.0313951
$371$	−15.0691	−0.782351
$372$	−0.254302	−0.0131850
$373$	−25.9552	−1.34391	−0.671956	−	0.740591i	$-0.734546\pi$
$373$	−25.9552	−1.34391	−0.671956	+	0.740591i	$0.734546\pi$
$374$	14.1488	0.731617
$375$	0.0845894	0.00436818
$376$	−7.67828	−0.395977
$377$	0.643178	0.0331254
$378$	1.40214	0.0721181
$379$	0.00938018	0.000481827 0	0.000240914	−	1.00000i	$-0.499923\pi$
$379$	0.00938018	0.000481827 0	0.000240914		1.00000i	$0.499923\pi$
$380$	−0.204419	−0.0104865
$381$	−1.63815	−0.0839251
$382$	17.1791	0.878961
$383$	5.06068	0.258588	0.129294	−	0.991606i	$-0.458729\pi$
$383$	5.06068	0.258588	0.129294	+	0.991606i	$0.458729\pi$
$384$	−0.119243	−0.00608511
$385$	−0.715131	−0.0364464
$386$	2.92368	0.148811
$387$	−21.6615	−1.10112
$388$	17.2277	0.874603
$389$	−24.4811	−1.24124	−0.620620	−	0.784111i	$-0.713119\pi$
$389$	−24.4811	−1.24124	−0.620620	+	0.784111i	$0.713119\pi$
$390$	0.00466812	0.000236380 0
$391$	−16.2588	−0.822243
$392$	3.14104	0.158646
$393$	−1.94789	−0.0982578
$394$	−11.0479	−0.556584
$395$	−0.638274	−0.0321151
$396$	−15.3147	−0.769591
$397$	6.77666	0.340111	0.170055	−	0.985435i	$-0.445605\pi$
$397$	6.77666	0.340111	0.170055	+	0.985435i	$0.445605\pi$
$398$	−7.01744	−0.351753
$399$	0.674667	0.0337756
$400$	−4.99496	−0.249748
$401$	−8.57950	−0.428440	−0.214220	−	0.976785i	$-0.568721\pi$
$401$	−8.57950	−0.428440	−0.214220	+	0.976785i	$0.568721\pi$
$402$	1.00282	0.0500159
$403$	−1.17631	−0.0585964
$404$	15.1030	0.751400
$405$	−0.629700	−0.0312900
$406$	−2.29065	−0.113683
$407$	43.6427	2.16329
$408$	0.328931	0.0162845
$409$	−18.2104	−0.900444	−0.450222	−	0.892917i	$-0.648655\pi$
$409$	−18.2104	−0.900444	−0.450222	+	0.892917i	$0.648655\pi$
$410$	0.313602	0.0154877
$411$	0.210443	0.0103804
$412$	−5.96394	−0.293822
$413$	−18.8821	−0.929127
$414$	17.5985	0.864920
$415$	−0.403110	−0.0197879
$416$	−0.551579	−0.0270434
$417$	0.116570	0.00570847
$418$	−14.7730	−0.722573
$419$	15.8273	0.773212	0.386606	−	0.922245i	$-0.373647\pi$
$419$	15.8273	0.773212	0.386606	+	0.922245i	$0.373647\pi$
$420$	−0.0166253	−0.000811232 0
$421$	30.8170	1.50193	0.750964	−	0.660343i	$-0.229589\pi$
$421$	30.8170	1.50193	0.750964	+	0.660343i	$0.229589\pi$
$422$	−12.7316	−0.619763
$423$	−22.9257	−1.11468
$424$	7.67102	0.372538
$425$	13.7785	0.668356
$426$	0.817283	0.0395975
$427$	−2.36403	−0.114404
$428$	−5.88966	−0.284687
$429$	0.337358	0.0162878
$430$	0.514910	0.0248311
$431$	−0.161348	−0.00777188	−0.00388594	−	0.999992i	$-0.501237\pi$
$431$	−0.161348	−0.00777188	−0.00388594	+	0.999992i	$0.501237\pi$
$432$	−0.713765	−0.0343410
$433$	−20.6871	−0.994157	−0.497078	−	0.867706i	$-0.665594\pi$
$433$	−20.6871	−0.994157	−0.497078	+	0.867706i	$0.665594\pi$
$434$	4.18940	0.201097
$435$	−0.00986867	−0.000473166 0
$436$	19.5546	0.936494
$437$	16.9761	0.812078
$438$	0.578530	0.0276432
$439$	−20.7114	−0.988500	−0.494250	−	0.869320i	$-0.664557\pi$
$439$	−20.7114	−0.988500	−0.494250	+	0.869320i	$0.664557\pi$
$440$	0.364041	0.0173550
$441$	9.37845	0.446593
$442$	1.52152	0.0723713
$443$	−8.25046	−0.391991	−0.195995	−	0.980605i	$-0.562794\pi$
$443$	−8.25046	−0.391991	−0.195995	+	0.980605i	$0.562794\pi$
$444$	1.01460	0.0481509
$445$	−1.02191	−0.0484430
$446$	−1.21606	−0.0575821
$447$	−0.477697	−0.0225943
$448$	1.96442	0.0928103
$449$	−13.0499	−0.615863	−0.307932	−	0.951408i	$-0.599637\pi$
$449$	−13.0499	−0.615863	−0.307932	+	0.951408i	$0.599637\pi$
$450$	−14.9139	−0.703046
$451$	22.6636	1.06719
$452$	−2.39291	−0.112553
$453$	−2.47049	−0.116074
$454$	1.19008	0.0558530
$455$	−0.0769030	−0.00360527
$456$	−0.343443	−0.0160832
$457$	5.39405	0.252323	0.126161	−	0.992010i	$-0.459734\pi$
$457$	5.39405	0.252323	0.126161	+	0.992010i	$0.459734\pi$
$458$	21.8149	1.01934
$459$	1.96891	0.0919007
$460$	−0.418330	−0.0195047
$461$	31.9160	1.48648	0.743238	−	0.669027i	$-0.233289\pi$
$461$	31.9160	1.48648	0.743238	+	0.669027i	$0.233289\pi$
$462$	−1.20149	−0.0558982
$463$	−1.80650	−0.0839551	−0.0419776	−	0.999119i	$-0.513366\pi$
$463$	−1.80650	−0.0839551	−0.0419776	+	0.999119i	$0.513366\pi$
$464$	1.16607	0.0541333
$465$	0.0180489	0.000836998 0
$466$	−6.37810	−0.295460
$467$	34.4508	1.59419	0.797097	−	0.603851i	$-0.206368\pi$
$467$	34.4508	1.59419	0.797097	+	0.603851i	$0.206368\pi$
$468$	−1.64689	−0.0761277
$469$	−16.5205	−0.762844
$470$	0.544960	0.0251371
$471$	−1.92298	−0.0886063
$472$	9.61202	0.442429
$473$	37.2118	1.71100
$474$	−1.07236	−0.0492552
$475$	−14.3864	−0.660094
$476$	−5.41883	−0.248372
$477$	22.9040	1.04870
$478$	−18.0909	−0.827458
$479$	−9.45509	−0.432014	−0.216007	−	0.976392i	$-0.569303\pi$
$479$	−9.45509	−0.432014	−0.216007	+	0.976392i	$0.569303\pi$
$480$	0.00846320	0.000386291 0
$481$	4.69320	0.213992
$482$	−7.60740	−0.346508
$483$	1.38066	0.0628223
$484$	15.3087	0.695850
$485$	−1.22272	−0.0555209
$486$	−3.19925	−0.145121
$487$	33.2566	1.50700	0.753499	−	0.657449i	$-0.228364\pi$
$487$	33.2566	1.50700	0.753499	+	0.657449i	$0.228364\pi$
$488$	1.20342	0.0544764
$489$	1.48279	0.0670541
$490$	−0.222933	−0.0100711
$491$	−0.908954	−0.0410205	−0.0205103	−	0.999790i	$-0.506529\pi$
$491$	−0.908954	−0.0410205	−0.0205103	+	0.999790i	$0.506529\pi$
$492$	0.526881	0.0237536
$493$	−3.21658	−0.144867
$494$	−1.58865	−0.0714766
$495$	1.08695	0.0488546
$496$	−2.13263	−0.0957581
$497$	−13.4640	−0.603942
$498$	−0.677263	−0.0303489
$499$	−9.26109	−0.414584	−0.207292	−	0.978279i	$-0.566465\pi$
$499$	−9.26109	−0.414584	−0.207292	+	0.978279i	$0.566465\pi$
$500$	0.709385	0.0317246
$501$	−0.533674	−0.0238428
$502$	−23.2159	−1.03618
$503$	−15.6347	−0.697115	−0.348558	−	0.937287i	$-0.613328\pi$
$503$	−15.6347	−0.697115	−0.348558	+	0.937287i	$0.613328\pi$
$504$	5.86534	0.261263
$505$	−1.07192	−0.0476998
$506$	−30.2321	−1.34398
$507$	−1.51389	−0.0672340
$508$	−13.7379	−0.609520
$509$	0.584827	0.0259220	0.0129610	−	0.999916i	$-0.495874\pi$
$509$	0.584827	0.0259220	0.0129610	+	0.999916i	$0.495874\pi$
$510$	−0.0233456	−0.00103376
$511$	−9.53075	−0.421615
$512$	−1.00000	−0.0441942
$513$	−2.05577	−0.0907646
$514$	−19.7002	−0.868941
$515$	0.423286	0.0186522
$516$	0.865097	0.0380838
$517$	39.3834	1.73208
$518$	−16.7146	−0.734399
$519$	−0.813753	−0.0357198
$520$	0.0391479	0.00171675
$521$	27.0830	1.18653	0.593264	−	0.805008i	$-0.297839\pi$
$521$	27.0830	1.18653	0.593264	+	0.805008i	$0.297839\pi$
$522$	3.48162	0.152386
$523$	−22.3927	−0.979163	−0.489581	−	0.871958i	$-0.662850\pi$
$523$	−22.3927	−0.979163	−0.489581	+	0.871958i	$0.662850\pi$
$524$	−16.3354	−0.713614
$525$	−1.17004	−0.0510648
$526$	−8.10650	−0.353460
$527$	5.88283	0.256260
$528$	0.611623	0.0266175
$529$	11.7405	0.510458
$530$	−0.544445	−0.0236492
$531$	28.6994	1.24545
$532$	5.65790	0.245301
$533$	2.43717	0.105566
$534$	−1.71690	−0.0742975
$535$	0.418014	0.0180723
$536$	8.40982	0.363249
$537$	2.88721	0.124592
$538$	−1.08307	−0.0466943
$539$	−16.1110	−0.693951
$540$	0.0506589	0.00218001
$541$	45.7592	1.96734	0.983670	−	0.179981i	$-0.0576036\pi$
$541$	45.7592	1.96734	0.983670	+	0.179981i	$0.0576036\pi$
$542$	3.70028	0.158941
$543$	2.27171	0.0974886
$544$	2.75848	0.118269
$545$	−1.38787	−0.0594498
$546$	−0.129204	−0.00552943
$547$	34.0272	1.45490	0.727449	−	0.686162i	$-0.240706\pi$
$547$	34.0272	1.45490	0.727449	+	0.686162i	$0.240706\pi$
$548$	1.76482	0.0753892
$549$	3.59316	0.153352
$550$	25.6202	1.09245
$551$	3.35849	0.143076
$552$	−0.702834	−0.0299146
$553$	17.6662	0.751241
$554$	−22.7710	−0.967445
$555$	−0.0720106	−0.00305668
$556$	0.977582	0.0414587
$557$	−1.38435	−0.0586570	−0.0293285	−	0.999570i	$-0.509337\pi$
$557$	−1.38435	−0.0586570	−0.0293285	+	0.999570i	$0.509337\pi$
$558$	−6.36758	−0.269561
$559$	4.00164	0.169251
$560$	−0.139423	−0.00589171
$561$	−1.68715	−0.0712316
$562$	8.83444	0.372658
$563$	36.4878	1.53778	0.768888	−	0.639384i	$-0.220811\pi$
$563$	36.4878	1.53778	0.768888	+	0.639384i	$0.220811\pi$
$564$	0.915584	0.0385530
$565$	0.169835	0.00714501
$566$	−0.868019	−0.0364856
$567$	17.4288	0.731942
$568$	6.85391	0.287584
$569$	−26.1953	−1.09817	−0.549083	−	0.835768i	$-0.685023\pi$
$569$	−26.1953	−1.09817	−0.549083	+	0.835768i	$0.685023\pi$
$570$	0.0243756	0.00102098
$571$	40.0472	1.67592	0.837962	−	0.545728i	$-0.183747\pi$
$571$	40.0472	1.67592	0.837962	+	0.545728i	$0.183747\pi$
$572$	2.82916	0.118293
$573$	−2.04850	−0.0855773
$574$	−8.67988	−0.362291
$575$	−29.4409	−1.22777
$576$	−2.98578	−0.124408
$577$	−2.39783	−0.0998230	−0.0499115	−	0.998754i	$-0.515894\pi$
$577$	−2.39783	−0.0998230	−0.0499115	+	0.998754i	$0.515894\pi$
$578$	9.39078	0.390605
$579$	−0.348629	−0.0144885
$580$	−0.0827607	−0.00343645
$581$	11.1573	0.462882
$582$	−2.05429	−0.0851529
$583$	−39.3462	−1.62955
$584$	4.85168	0.200764
$585$	0.116887	0.00483268
$586$	11.6653	0.481888
$587$	−4.02611	−0.166175	−0.0830876	−	0.996542i	$-0.526478\pi$
$587$	−4.02611	−0.166175	−0.0830876	+	0.996542i	$0.526478\pi$
$588$	−0.374548	−0.0154461
$589$	−6.14238	−0.253092
$590$	−0.682206	−0.0280860
$591$	1.31739	0.0541901
$592$	8.50867	0.349704
$593$	35.2969	1.44947	0.724735	−	0.689028i	$-0.241962\pi$
$593$	35.2969	1.44947	0.724735	+	0.689028i	$0.241962\pi$
$594$	3.66104	0.150214
$595$	0.384597	0.0157669
$596$	−4.00607	−0.164095
$597$	0.836784	0.0342473
$598$	−3.25107	−0.132946
$599$	−11.7555	−0.480319	−0.240159	−	0.970733i	$-0.577200\pi$
$599$	−11.7555	−0.480319	−0.240159	+	0.970733i	$0.577200\pi$
$600$	0.595616	0.0243159
$601$	1.20169	0.0490180	0.0245090	−	0.999700i	$-0.492198\pi$
$601$	1.20169	0.0490180	0.0245090	+	0.999700i	$0.492198\pi$
$602$	−14.2517	−0.580855
$603$	25.1099	1.02255
$604$	−20.7180	−0.843005
$605$	−1.08652	−0.0441734
$606$	−1.80093	−0.0731577
$607$	−23.8145	−0.966601	−0.483301	−	0.875454i	$-0.660562\pi$
$607$	−23.8145	−0.966601	−0.483301	+	0.875454i	$0.660562\pi$
$608$	−2.88018	−0.116807
$609$	0.273145	0.0110684
$610$	−0.0854120	−0.00345823
$611$	4.23518	0.171337
$612$	8.23622	0.332930
$613$	−27.3705	−1.10548	−0.552741	−	0.833353i	$-0.686418\pi$
$613$	−27.3705	−1.10548	−0.552741	+	0.833353i	$0.686418\pi$
$614$	−11.0978	−0.447870
$615$	−0.0373950	−0.00150791
$616$	−10.0759	−0.405971
$617$	−17.2491	−0.694422	−0.347211	−	0.937787i	$-0.612871\pi$
$617$	−17.2491	−0.694422	−0.347211	+	0.937787i	$0.612871\pi$
$618$	0.711160	0.0286071
$619$	−5.30745	−0.213324	−0.106662	−	0.994295i	$-0.534016\pi$
$619$	−5.30745	−0.213324	−0.106662	+	0.994295i	$0.534016\pi$
$620$	0.151362	0.00607884
$621$	−4.20701	−0.168821
$622$	0.807940	0.0323954
$623$	28.2843	1.13319
$624$	0.0657721	0.00263299
$625$	24.9245	0.996979
$626$	−9.46612	−0.378342
$627$	1.76159	0.0703510
$628$	−16.1265	−0.643518
$629$	−23.4710	−0.935851
$630$	−0.416288	−0.0165853
$631$	23.4857	0.934950	0.467475	−	0.884006i	$-0.345164\pi$
$631$	23.4857	0.934950	0.467475	+	0.884006i	$0.345164\pi$
$632$	−8.99305	−0.357724
$633$	1.51815	0.0603412
$634$	−25.4678	−1.01145
$635$	0.975036	0.0386931
$636$	−0.914719	−0.0362710
$637$	−1.73253	−0.0686453
$638$	−5.98100	−0.236790
$639$	20.4643	0.809554
$640$	0.0709742	0.00280550
$641$	14.3258	0.565834	0.282917	−	0.959144i	$-0.408698\pi$
$641$	14.3258	0.565834	0.282917	+	0.959144i	$0.408698\pi$
$642$	0.702303	0.0277177
$643$	−3.50636	−0.138277	−0.0691386	−	0.997607i	$-0.522025\pi$
$643$	−3.50636	−0.138277	−0.0691386	+	0.997607i	$0.522025\pi$
$644$	11.5785	0.456258
$645$	−0.0613995	−0.00241760
$646$	7.94493	0.312589
$647$	−15.2117	−0.598033	−0.299017	−	0.954248i	$-0.596659\pi$
$647$	−15.2117	−0.598033	−0.299017	+	0.954248i	$0.596659\pi$
$648$	−8.87223	−0.348534
$649$	−49.3020	−1.93527
$650$	2.75511	0.108064
$651$	−0.499558	−0.0195792
$652$	12.4350	0.486992
$653$	46.2470	1.80979	0.904893	−	0.425640i	$-0.139951\pi$
$653$	46.2470	1.80979	0.904893	+	0.425640i	$0.139951\pi$
$654$	−2.33175	−0.0911788
$655$	1.15939	0.0453011
$656$	4.41854	0.172515
$657$	14.4860	0.565154
$658$	−15.0834	−0.588012
$659$	−30.0978	−1.17244	−0.586221	−	0.810151i	$-0.699385\pi$
$659$	−30.0978	−1.17244	−0.586221	+	0.810151i	$0.699385\pi$
$660$	−0.0434095	−0.00168971
$661$	11.5318	0.448535	0.224268	−	0.974528i	$-0.428001\pi$
$661$	11.5318	0.448535	0.224268	+	0.974528i	$0.428001\pi$
$662$	−27.5023	−1.06891
$663$	−0.181431	−0.00704620
$664$	−5.67967	−0.220414
$665$	−0.401565	−0.0155720
$666$	25.4050	0.984425
$667$	6.87293	0.266121
$668$	−4.47550	−0.173162
$669$	0.145007	0.00560630
$670$	−0.596880	−0.0230595
$671$	−6.17260	−0.238291
$672$	−0.234245	−0.00903618
$673$	−8.65441	−0.333603	−0.166801	−	0.985991i	$-0.553344\pi$
$673$	−8.65441	−0.333603	−0.166801	+	0.985991i	$0.553344\pi$
$674$	21.4574	0.826507
$675$	3.56523	0.137226
$676$	−12.6958	−0.488299
$677$	41.6483	1.60067	0.800337	−	0.599550i	$-0.204654\pi$
$677$	41.6483	1.60067	0.800337	+	0.599550i	$0.204654\pi$
$678$	0.285339	0.0109584
$679$	33.8425	1.29875
$680$	−0.195781	−0.00750786
$681$	−0.141909	−0.00543795
$682$	10.9387	0.418865
$683$	0.508286	0.0194490	0.00972451	−	0.999953i	$-0.496905\pi$
$683$	0.508286	0.0194490	0.00972451	+	0.999953i	$0.496905\pi$
$684$	−8.59960	−0.328814
$685$	−0.125256	−0.00478580
$686$	19.9213	0.760599
$687$	−2.60128	−0.0992449
$688$	7.25488	0.276590
$689$	−4.23117	−0.161195
$690$	0.0498831	0.00189902
$691$	−16.7359	−0.636664	−0.318332	−	0.947979i	$-0.603123\pi$
$691$	−16.7359	−0.636664	−0.318332	+	0.947979i	$0.603123\pi$
$692$	−6.82431	−0.259421
$693$	−30.0845	−1.14282
$694$	−12.8880	−0.489221
$695$	−0.0693831	−0.00263185
$696$	−0.139046	−0.00527052
$697$	−12.1885	−0.461671
$698$	27.8506	1.05416
$699$	0.760546	0.0287665
$700$	−9.81222	−0.370867
$701$	31.8736	1.20385	0.601926	−	0.798552i	$-0.294400\pi$
$701$	31.8736	1.20385	0.601926	+	0.798552i	$0.294400\pi$
$702$	0.393697	0.0148591
$703$	24.5065	0.924282
$704$	5.12920	0.193314
$705$	−0.0649828	−0.00244740
$706$	−32.1463	−1.20984
$707$	29.6686	1.11580
$708$	−1.14617	−0.0430757
$709$	1.75023	0.0657314	0.0328657	−	0.999460i	$-0.489537\pi$
$709$	1.75023	0.0657314	0.0328657	+	0.999460i	$0.489537\pi$
$710$	−0.486451	−0.0182562
$711$	−26.8513	−1.00700
$712$	−14.3983	−0.539598
$713$	−12.5700	−0.470749
$714$	0.646159	0.0241819
$715$	−0.200797	−0.00750939
$716$	24.2127	0.904872
$717$	2.15722	0.0805628
$718$	−8.43702	−0.314867
$719$	48.3390	1.80274	0.901370	−	0.433049i	$-0.142562\pi$
$719$	48.3390	1.80274	0.901370	+	0.433049i	$0.142562\pi$
$720$	0.211913	0.00789755
$721$	−11.7157	−0.436316
$722$	10.7045	0.398382
$723$	0.907132	0.0337366
$724$	19.0511	0.708028
$725$	−5.82446	−0.216315
$726$	−1.82546	−0.0677492
$727$	7.36251	0.273060	0.136530	−	0.990636i	$-0.456405\pi$
$727$	7.36251	0.273060	0.136530	+	0.990636i	$0.456405\pi$
$728$	−1.08353	−0.0401584
$729$	−26.2352	−0.971674
$730$	−0.344344	−0.0127447
$731$	−20.0125	−0.740188
$732$	−0.143500	−0.00530392
$733$	18.3521	0.677850	0.338925	−	0.940813i	$-0.389937\pi$
$733$	18.3521	0.677850	0.338925	+	0.940813i	$0.389937\pi$
$734$	23.9924	0.885576
$735$	0.0265832	0.000980538 0
$736$	−5.89411	−0.217260
$737$	−43.1357	−1.58892
$738$	13.1928	0.485633
$739$	−13.2031	−0.485682	−0.242841	−	0.970066i	$-0.578079\pi$
$739$	−13.2031	−0.485682	−0.242841	+	0.970066i	$0.578079\pi$
$740$	−0.603896	−0.0221997
$741$	0.189436	0.00695909
$742$	15.0691	0.553206
$743$	36.6743	1.34545	0.672725	−	0.739892i	$-0.265124\pi$
$743$	36.6743	1.34545	0.672725	+	0.739892i	$0.265124\pi$
$744$	0.254302	0.00932318
$745$	0.284328	0.0104170
$746$	25.9552	0.950289
$747$	−16.9582	−0.620470
$748$	−14.1488	−0.517332
$749$	−11.5698	−0.422751
$750$	−0.0845894	−0.00308877
$751$	−43.5598	−1.58952	−0.794760	−	0.606923i	$-0.792404\pi$
$751$	−43.5598	−1.58952	−0.794760	+	0.606923i	$0.792404\pi$
$752$	7.67828	0.279998
$753$	2.76834	0.100884
$754$	−0.643178	−0.0234232
$755$	1.47045	0.0535150
$756$	−1.40214	−0.0509952
$757$	41.6722	1.51460	0.757302	−	0.653065i	$-0.226517\pi$
$757$	41.6722	1.51460	0.757302	+	0.653065i	$0.226517\pi$
$758$	−0.00938018	−0.000340703 0
$759$	3.60498	0.130852
$760$	0.204419	0.00741505
$761$	12.3813	0.448821	0.224410	−	0.974495i	$-0.427954\pi$
$761$	12.3813	0.448821	0.224410	+	0.974495i	$0.427954\pi$
$762$	1.63815	0.0593440
$763$	38.4135	1.39066
$764$	−17.1791	−0.621520
$765$	−0.584559	−0.0211348
$766$	−5.06068	−0.182850
$767$	−5.30179	−0.191436
$768$	0.119243	0.00430282
$769$	−32.0545	−1.15592	−0.577958	−	0.816067i	$-0.696150\pi$
$769$	−32.0545	−1.15592	−0.577958	+	0.816067i	$0.696150\pi$
$770$	0.715131	0.0257715
$771$	2.34912	0.0846016
$772$	−2.92368	−0.105226
$773$	−22.0889	−0.794484	−0.397242	−	0.917714i	$-0.630033\pi$
$773$	−22.0889	−0.794484	−0.397242	+	0.917714i	$0.630033\pi$
$774$	21.6615	0.778606
$775$	10.6524	0.382646
$776$	−17.2277	−0.618438
$777$	1.99311	0.0715024
$778$	24.4811	0.877689
$779$	12.7262	0.455963
$780$	−0.00466812	−0.000167146 0
$781$	−35.1551	−1.25795
$782$	16.2588	0.581413
$783$	−0.832298	−0.0297439
$784$	−3.14104	−0.112180
$785$	1.14457	0.0408514
$786$	1.94789	0.0694788
$787$	37.8197	1.34813	0.674064	−	0.738673i	$-0.264547\pi$
$787$	37.8197	1.34813	0.674064	+	0.738673i	$0.264547\pi$
$788$	11.0479	0.393565
$789$	0.966647	0.0344135
$790$	0.638274	0.0227088
$791$	−4.70069	−0.167137
$792$	15.3147	0.544183
$793$	−0.663782	−0.0235716
$794$	−6.77666	−0.240495
$795$	0.0649214	0.00230253
$796$	7.01744	0.248727
$797$	30.9780	1.09730	0.548649	−	0.836053i	$-0.315142\pi$
$797$	30.9780	1.09730	0.548649	+	0.836053i	$0.315142\pi$
$798$	−0.674667	−0.0238830
$799$	−21.1804	−0.749309
$800$	4.99496	0.176599
$801$	−42.9901	−1.51898
$802$	8.57950	0.302953
$803$	−24.8852	−0.878180
$804$	−1.00282	−0.0353666
$805$	−0.821777	−0.0289638
$806$	1.17631	0.0414339
$807$	0.129149	0.00454625
$808$	−15.1030	−0.531320
$809$	−9.54858	−0.335710	−0.167855	−	0.985812i	$-0.553684\pi$
$809$	−9.54858	−0.335710	−0.167855	+	0.985812i	$0.553684\pi$
$810$	0.629700	0.0221254
$811$	25.7504	0.904218	0.452109	−	0.891963i	$-0.350672\pi$
$811$	25.7504	0.904218	0.452109	+	0.891963i	$0.350672\pi$
$812$	2.29065	0.0803861
$813$	−0.441234	−0.0154747
$814$	−43.6427	−1.52968
$815$	−0.882564	−0.0309149
$816$	−0.328931	−0.0115149
$817$	20.8954	0.731037
$818$	18.2104	0.636710
$819$	−3.23520	−0.113047
$820$	−0.313602	−0.0109515
$821$	−16.8853	−0.589302	−0.294651	−	0.955605i	$-0.595203\pi$
$821$	−16.8853	−0.589302	−0.294651	+	0.955605i	$0.595203\pi$
$822$	−0.210443	−0.00734003
$823$	−51.7405	−1.80356	−0.901780	−	0.432194i	$-0.857739\pi$
$823$	−51.7405	−1.80356	−0.901780	+	0.432194i	$0.857739\pi$
$824$	5.96394	0.207764
$825$	−3.05504	−0.106363
$826$	18.8821	0.656992
$827$	−0.978449	−0.0340240	−0.0170120	−	0.999855i	$-0.505415\pi$
$827$	−0.978449	−0.0340240	−0.0170120	+	0.999855i	$0.505415\pi$
$828$	−17.5985	−0.611591
$829$	24.0974	0.836938	0.418469	−	0.908231i	$-0.362567\pi$
$829$	24.0974	0.836938	0.418469	+	0.908231i	$0.362567\pi$
$830$	0.403110	0.0139922
$831$	2.71529	0.0941922
$832$	0.551579	0.0191225
$833$	8.66450	0.300207
$834$	−0.116570	−0.00403650
$835$	0.317645	0.0109926
$836$	14.7730	0.510936
$837$	1.52220	0.0526149
$838$	−15.8273	−0.546743
$839$	42.1722	1.45595	0.727974	−	0.685605i	$-0.240462\pi$
$839$	42.1722	1.45595	0.727974	+	0.685605i	$0.240462\pi$
$840$	0.0166253	0.000573628 0
$841$	−27.6403	−0.953113
$842$	−30.8170	−1.06202
$843$	−1.05345	−0.0362827
$844$	12.7316	0.438239
$845$	0.901072	0.0309978
$846$	22.9257	0.788201
$847$	30.0728	1.03331
$848$	−7.67102	−0.263424
$849$	0.103506	0.00355230
$850$	−13.7785	−0.472599
$851$	50.1511	1.71916
$852$	−0.817283	−0.0279997
$853$	15.4520	0.529067	0.264534	−	0.964376i	$-0.414782\pi$
$853$	15.4520	0.529067	0.264534	+	0.964376i	$0.414782\pi$
$854$	2.36403	0.0808955
$855$	0.610350	0.0208735
$856$	5.88966	0.201304
$857$	7.05816	0.241102	0.120551	−	0.992707i	$-0.461534\pi$
$857$	7.05816	0.241102	0.120551	+	0.992707i	$0.461534\pi$
$858$	−0.337358	−0.0115172
$859$	−1.25543	−0.0428347	−0.0214174	−	0.999771i	$-0.506818\pi$
$859$	−1.25543	−0.0428347	−0.0214174	+	0.999771i	$0.506818\pi$
$860$	−0.514910	−0.0175583
$861$	1.03502	0.0352733
$862$	0.161348	0.00549555
$863$	−5.42921	−0.184813	−0.0924063	−	0.995721i	$-0.529456\pi$
$863$	−5.42921	−0.184813	−0.0924063	+	0.995721i	$0.529456\pi$
$864$	0.713765	0.0242828
$865$	0.484350	0.0164684
$866$	20.6871	0.702975
$867$	−1.11979	−0.0380300
$868$	−4.18940	−0.142197
$869$	46.1271	1.56476
$870$	0.00986867	0.000334579 0
$871$	−4.63868	−0.157176
$872$	−19.5546	−0.662201
$873$	−51.4381	−1.74091
$874$	−16.9761	−0.574226
$875$	1.39353	0.0471100
$876$	−0.578530	−0.0195467
$877$	−41.0723	−1.38691	−0.693457	−	0.720498i	$-0.743913\pi$
$877$	−41.0723	−1.38691	−0.693457	+	0.720498i	$0.743913\pi$
$878$	20.7114	0.698975
$879$	−1.39101	−0.0469174
$880$	−0.364041	−0.0122718
$881$	53.0396	1.78695	0.893474	−	0.449114i	$-0.148260\pi$
$881$	53.0396	1.78695	0.893474	+	0.449114i	$0.148260\pi$
$882$	−9.37845	−0.315789
$883$	−12.4518	−0.419036	−0.209518	−	0.977805i	$-0.567189\pi$
$883$	−12.4518	−0.419036	−0.209518	+	0.977805i	$0.567189\pi$
$884$	−1.52152	−0.0511742
$885$	0.0813485	0.00273450
$886$	8.25046	0.277179
$887$	11.2707	0.378432	0.189216	−	0.981935i	$-0.439405\pi$
$887$	11.2707	0.378432	0.189216	+	0.981935i	$0.439405\pi$
$888$	−1.01460	−0.0340478
$889$	−26.9870	−0.905116
$890$	1.02191	0.0342544
$891$	45.5075	1.52456
$892$	1.21606	0.0407167
$893$	22.1149	0.740046
$894$	0.477697	0.0159766
$895$	−1.71848	−0.0574424
$896$	−1.96442	−0.0656268
$897$	0.387668	0.0129439
$898$	13.0499	0.435481
$899$	−2.48680	−0.0829393
$900$	14.9139	0.497129
$901$	21.1604	0.704954
$902$	−22.6636	−0.754614
$903$	1.69942	0.0565530
$904$	2.39291	0.0795871
$905$	−1.35214	−0.0449465
$906$	2.47049	0.0820765
$907$	−21.1268	−0.701505	−0.350753	−	0.936468i	$-0.614074\pi$
$907$	−21.1268	−0.701505	−0.350753	+	0.936468i	$0.614074\pi$
$908$	−1.19008	−0.0394941
$909$	−45.0941	−1.49568
$910$	0.0769030	0.00254931
$911$	36.4400	1.20731	0.603655	−	0.797246i	$-0.293711\pi$
$911$	36.4400	1.20731	0.603655	+	0.797246i	$0.293711\pi$
$912$	0.343443	0.0113725
$913$	29.1322	0.964134
$914$	−5.39405	−0.178419
$915$	0.0101848	0.000336700 0
$916$	−21.8149	−0.720783
$917$	−32.0896	−1.05969
$918$	−1.96891	−0.0649836
$919$	5.34988	0.176476	0.0882381	−	0.996099i	$-0.471876\pi$
$919$	5.34988	0.176476	0.0882381	+	0.996099i	$0.471876\pi$
$920$	0.418330	0.0137919
$921$	1.32334	0.0436054
$922$	−31.9160	−1.05110
$923$	−3.78047	−0.124436
$924$	1.20149	0.0395260
$925$	−42.5005	−1.39741
$926$	1.80650	0.0593652
$927$	17.8070	0.584859
$928$	−1.16607	−0.0382780
$929$	−43.6751	−1.43293	−0.716466	−	0.697622i	$-0.754242\pi$
$929$	−43.6751	−1.43293	−0.716466	+	0.697622i	$0.754242\pi$
$930$	−0.0180489	−0.000591847 0
$931$	−9.04677	−0.296496
$932$	6.37810	0.208922
$933$	−0.0963414	−0.00315408
$934$	−34.4508	−1.12727
$935$	1.00420	0.0328409
$936$	1.64689	0.0538304
$937$	3.45358	0.112823	0.0564117	−	0.998408i	$-0.482034\pi$
$937$	3.45358	0.112823	0.0564117	+	0.998408i	$0.482034\pi$
$938$	16.5205	0.539412
$939$	1.12877	0.0368361
$940$	−0.544960	−0.0177746
$941$	−21.4969	−0.700779	−0.350389	−	0.936604i	$-0.613951\pi$
$941$	−21.4969	−0.700779	−0.350389	+	0.936604i	$0.613951\pi$
$942$	1.92298	0.0626541
$943$	26.0434	0.848088
$944$	−9.61202	−0.312845
$945$	0.0995155	0.00323724
$946$	−37.2118	−1.20986
$947$	0.889700	0.0289114	0.0144557	−	0.999896i	$-0.495398\pi$
$947$	0.889700	0.0289114	0.0144557	+	0.999896i	$0.495398\pi$
$948$	1.07236	0.0348287
$949$	−2.67608	−0.0868692
$950$	14.3864	0.466757
$951$	3.03686	0.0984771
$952$	5.41883	0.175625
$953$	26.8855	0.870908	0.435454	−	0.900211i	$-0.356588\pi$
$953$	26.8855	0.870908	0.435454	+	0.900211i	$0.356588\pi$
$954$	−22.9040	−0.741544
$955$	1.21928	0.0394548
$956$	18.0909	0.585101
$957$	0.713194	0.0230543
$958$	9.45509	0.305480
$959$	3.46685	0.111950
$960$	−0.00846320	−0.000273149 0
$961$	−26.4519	−0.853286
$962$	−4.69320	−0.151315
$963$	17.5852	0.566676
$964$	7.60740	0.245018
$965$	0.207506	0.00667985
$966$	−1.38066	−0.0444221
$967$	3.67291	0.118113	0.0590565	−	0.998255i	$-0.481191\pi$
$967$	3.67291	0.118113	0.0590565	+	0.998255i	$0.481191\pi$
$968$	−15.3087	−0.492040
$969$	−0.947381	−0.0304343
$970$	1.22272	0.0392592
$971$	58.1925	1.86749	0.933743	−	0.357945i	$-0.116522\pi$
$971$	58.1925	1.86749	0.933743	+	0.357945i	$0.116522\pi$
$972$	3.19925	0.102616
$973$	1.92039	0.0615648
$974$	−33.2566	−1.06561
$975$	−0.328529	−0.0105214
$976$	−1.20342	−0.0385206
$977$	2.92256	0.0935010	0.0467505	−	0.998907i	$-0.485113\pi$
$977$	2.92256	0.0935010	0.0467505	+	0.998907i	$0.485113\pi$
$978$	−1.48279	−0.0474144
$979$	73.8516	2.36031
$980$	0.222933	0.00712132
$981$	−58.3857	−1.86411
$982$	0.908954	0.0290059
$983$	−12.1006	−0.385949	−0.192975	−	0.981204i	$-0.561814\pi$
$983$	−12.1006	−0.385949	−0.192975	+	0.981204i	$0.561814\pi$
$984$	−0.526881	−0.0167964
$985$	−0.784115	−0.0249840
$986$	3.21658	0.102437
$987$	1.79860	0.0572499
$988$	1.58865	0.0505416
$989$	42.7611	1.35972
$990$	−1.08695	−0.0345454
$991$	4.86896	0.154668	0.0773339	−	0.997005i	$-0.475359\pi$
$991$	4.86896	0.154668	0.0773339	+	0.997005i	$0.475359\pi$
$992$	2.13263	0.0677112
$993$	3.27947	0.104071
$994$	13.4640	0.427052
$995$	−0.498057	−0.0157895
$996$	0.677263	0.0214599
$997$	38.4162	1.21665	0.608327	−	0.793687i	$-0.291841\pi$
$997$	38.4162	1.21665	0.608327	+	0.793687i	$0.291841\pi$
$998$	9.26109	0.293155
$999$	−6.07319	−0.192147

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8002.2.a.e.1.40	✓	77

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8002.2.a.e.1.40	✓	77	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:	\( N \)	\(=\)	\( 8002 = 2 \cdot 4001 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8002.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +0.119243 q^{3} +1.00000 q^{4} -0.0709742 q^{5} -0.119243 q^{6} +1.96442 q^{7} -1.00000 q^{8} -2.98578 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +0.119243 q^{3} +1.00000 q^{4} -0.0709742 q^{5} -0.119243 q^{6} +1.96442 q^{7} -1.00000 q^{8} -2.98578 q^{9} +0.0709742 q^{10} +5.12920 q^{11} +0.119243 q^{12} +0.551579 q^{13} -1.96442 q^{14} -0.00846320 q^{15} +1.00000 q^{16} -2.75848 q^{17} +2.98578 q^{18} +2.88018 q^{19} -0.0709742 q^{20} +0.234245 q^{21} -5.12920 q^{22} +5.89411 q^{23} -0.119243 q^{24} -4.99496 q^{25} -0.551579 q^{26} -0.713765 q^{27} +1.96442 q^{28} +1.16607 q^{29} +0.00846320 q^{30} -2.13263 q^{31} -1.00000 q^{32} +0.611623 q^{33} +2.75848 q^{34} -0.139423 q^{35} -2.98578 q^{36} +8.50867 q^{37} -2.88018 q^{38} +0.0657721 q^{39} +0.0709742 q^{40} +4.41854 q^{41} -0.234245 q^{42} +7.25488 q^{43} +5.12920 q^{44} +0.211913 q^{45} -5.89411 q^{46} +7.67828 q^{47} +0.119243 q^{48} -3.14104 q^{49} +4.99496 q^{50} -0.328931 q^{51} +0.551579 q^{52} -7.67102 q^{53} +0.713765 q^{54} -0.364041 q^{55} -1.96442 q^{56} +0.343443 q^{57} -1.16607 q^{58} -9.61202 q^{59} -0.00846320 q^{60} -1.20342 q^{61} +2.13263 q^{62} -5.86534 q^{63} +1.00000 q^{64} -0.0391479 q^{65} -0.611623 q^{66} -8.40982 q^{67} -2.75848 q^{68} +0.702834 q^{69} +0.139423 q^{70} -6.85391 q^{71} +2.98578 q^{72} -4.85168 q^{73} -8.50867 q^{74} -0.595616 q^{75} +2.88018 q^{76} +10.0759 q^{77} -0.0657721 q^{78} +8.99305 q^{79} -0.0709742 q^{80} +8.87223 q^{81} -4.41854 q^{82} +5.67967 q^{83} +0.234245 q^{84} +0.195781 q^{85} -7.25488 q^{86} +0.139046 q^{87} -5.12920 q^{88} +14.3983 q^{89} -0.211913 q^{90} +1.08353 q^{91} +5.89411 q^{92} -0.254302 q^{93} -7.67828 q^{94} -0.204419 q^{95} -0.119243 q^{96} +17.2277 q^{97} +3.14104 q^{98} -15.3147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9}+O(q^{10}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9	\( 77 q - 77 q^{2} + 10 q^{3} + 77 q^{4} + 18 q^{5} - 10 q^{6} + 21 q^{7} - 77 q^{8} + 71 q^{9} - 18 q^{10} + 30 q^{11} + 10 q^{12} - 2 q^{13} - 21 q^{14} + 21 q^{15} + 77 q^{16} + 60 q^{17} - 71 q^{18} - 3 q^{19} + 18 q^{20} + 10 q^{21} - 30 q^{22} + 53 q^{23} - 10 q^{24} + 59 q^{25} + 2 q^{26} + 43 q^{27} + 21 q^{28} + 30 q^{29} - 21 q^{30} + 22 q^{31} - 77 q^{32} + 31 q^{33} - 60 q^{34} + 41 q^{35} + 71 q^{36} - 3 q^{37} + 3 q^{38} + 44 q^{39} - 18 q^{40} + 48 q^{41} - 10 q^{42} + 21 q^{43} + 30 q^{44} + 33 q^{45} - 53 q^{46} + 107 q^{47} + 10 q^{48} + 24 q^{49} - 59 q^{50} + 18 q^{51} - 2 q^{52} + 42 q^{53} - 43 q^{54} + 49 q^{55} - 21 q^{56} + 32 q^{57} - 30 q^{58} + 42 q^{59} + 21 q^{60} - 31 q^{61} - 22 q^{62} + 109 q^{63} + 77 q^{64} + 39 q^{65} - 31 q^{66} - q^{67} + 60 q^{68} - 33 q^{69} - 41 q^{70} + 58 q^{71} - 71 q^{72} + 35 q^{73} + 3 q^{74} + 34 q^{75} - 3 q^{76} + 86 q^{77} - 44 q^{78} + 25 q^{79} + 18 q^{80} + 53 q^{81} - 48 q^{82} + 107 q^{83} + 10 q^{84} + 21 q^{85} - 21 q^{86} + 100 q^{87} - 30 q^{88} + 34 q^{89} - 33 q^{90} - 51 q^{91} + 53 q^{92} + 48 q^{93} - 107 q^{94} + 118 q^{95} - 10 q^{96} - 13 q^{97} - 24 q^{98} + 63 q^{99}+O(q^{100}) \) 77 * q - 77 * q^2 + 10 * q^3 + 77 * q^4 + 18 * q^5 - 10 * q^6 + 21 * q^7 - 77 * q^8 + 71 * q^9 - 18 * q^10 + 30 * q^11 + 10 * q^12 - 2 * q^13 - 21 * q^14 + 21 * q^15 + 77 * q^16 + 60 * q^17 - 71 * q^18 - 3 * q^19 + 18 * q^20 + 10 * q^21 - 30 * q^22 + 53 * q^23 - 10 * q^24 + 59 * q^25 + 2 * q^26 + 43 * q^27 + 21 * q^28 + 30 * q^29 - 21 * q^30 + 22 * q^31 - 77 * q^32 + 31 * q^33 - 60 * q^34 + 41 * q^35 + 71 * q^36 - 3 * q^37 + 3 * q^38 + 44 * q^39 - 18 * q^40 + 48 * q^41 - 10 * q^42 + 21 * q^43 + 30 * q^44 + 33 * q^45 - 53 * q^46 + 107 * q^47 + 10 * q^48 + 24 * q^49 - 59 * q^50 + 18 * q^51 - 2 * q^52 + 42 * q^53 - 43 * q^54 + 49 * q^55 - 21 * q^56 + 32 * q^57 - 30 * q^58 + 42 * q^59 + 21 * q^60 - 31 * q^61 - 22 * q^62 + 109 * q^63 + 77 * q^64 + 39 * q^65 - 31 * q^66 - q^67 + 60 * q^68 - 33 * q^69 - 41 * q^70 + 58 * q^71 - 71 * q^72 + 35 * q^73 + 3 * q^74 + 34 * q^75 - 3 * q^76 + 86 * q^77 - 44 * q^78 + 25 * q^79 + 18 * q^80 + 53 * q^81 - 48 * q^82 + 107 * q^83 + 10 * q^84 + 21 * q^85 - 21 * q^86 + 100 * q^87 - 30 * q^88 + 34 * q^89 - 33 * q^90 - 51 * q^91 + 53 * q^92 + 48 * q^93 - 107 * q^94 + 118 * q^95 - 10 * q^96 - 13 * q^97 - 24 * q^98 + 63 * q^99

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