LMFDB - Embedded newform 8042.2.a.a.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8042,2,Mod(1,8042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8042, 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("8042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8042 = 2 \cdot 4021 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8042.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:	$64.2156933055$
Analytic rank:	$1$
Dimension:	$67$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	8042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -3.01737 q^{3} +1.00000 q^{4} -1.22094 q^{5} -3.01737 q^{6} +2.53637 q^{7} +1.00000 q^{8} +6.10451 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -3.01737 q^{3} +1.00000 q^{4} -1.22094 q^{5} -3.01737 q^{6} +2.53637 q^{7} +1.00000 q^{8} +6.10451 q^{9} -1.22094 q^{10} +0.137455 q^{11} -3.01737 q^{12} -0.806951 q^{13} +2.53637 q^{14} +3.68404 q^{15} +1.00000 q^{16} +2.15852 q^{17} +6.10451 q^{18} +1.07604 q^{19} -1.22094 q^{20} -7.65317 q^{21} +0.137455 q^{22} -5.61126 q^{23} -3.01737 q^{24} -3.50930 q^{25} -0.806951 q^{26} -9.36747 q^{27} +2.53637 q^{28} +3.70442 q^{29} +3.68404 q^{30} -1.71836 q^{31} +1.00000 q^{32} -0.414753 q^{33} +2.15852 q^{34} -3.09677 q^{35} +6.10451 q^{36} -3.29300 q^{37} +1.07604 q^{38} +2.43487 q^{39} -1.22094 q^{40} -7.64932 q^{41} -7.65317 q^{42} +5.20520 q^{43} +0.137455 q^{44} -7.45327 q^{45} -5.61126 q^{46} +4.92416 q^{47} -3.01737 q^{48} -0.566819 q^{49} -3.50930 q^{50} -6.51306 q^{51} -0.806951 q^{52} -1.93351 q^{53} -9.36747 q^{54} -0.167825 q^{55} +2.53637 q^{56} -3.24682 q^{57} +3.70442 q^{58} -11.7080 q^{59} +3.68404 q^{60} -0.669111 q^{61} -1.71836 q^{62} +15.4833 q^{63} +1.00000 q^{64} +0.985242 q^{65} -0.414753 q^{66} +5.09018 q^{67} +2.15852 q^{68} +16.9312 q^{69} -3.09677 q^{70} -7.94119 q^{71} +6.10451 q^{72} +2.48803 q^{73} -3.29300 q^{74} +10.5888 q^{75} +1.07604 q^{76} +0.348637 q^{77} +2.43487 q^{78} -13.0996 q^{79} -1.22094 q^{80} +9.95156 q^{81} -7.64932 q^{82} +4.45477 q^{83} -7.65317 q^{84} -2.63543 q^{85} +5.20520 q^{86} -11.1776 q^{87} +0.137455 q^{88} +12.1306 q^{89} -7.45327 q^{90} -2.04673 q^{91} -5.61126 q^{92} +5.18491 q^{93} +4.92416 q^{94} -1.31379 q^{95} -3.01737 q^{96} +9.54246 q^{97} -0.566819 q^{98} +0.839097 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) $ 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * 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
$3$	−3.01737	−1.74208	−0.871039	−	0.491213i	$-0.836554\pi$
$3$	−3.01737	−1.74208	−0.871039	+	0.491213i	$0.836554\pi$
$4$	1.00000	0.500000
$5$	−1.22094	−0.546023	−0.273011	−	0.962011i	$-0.588020\pi$
$5$	−1.22094	−0.546023	−0.273011	+	0.962011i	$0.588020\pi$
$6$	−3.01737	−1.23184
$7$	2.53637	0.958658	0.479329	−	0.877635i	$-0.340880\pi$
$7$	2.53637	0.958658	0.479329	+	0.877635i	$0.340880\pi$
$8$	1.00000	0.353553
$9$	6.10451	2.03484
$10$	−1.22094	−0.386096
$11$	0.137455	0.0414443	0.0207221	−	0.999785i	$-0.493403\pi$
$11$	0.137455	0.0414443	0.0207221	+	0.999785i	$0.493403\pi$
$12$	−3.01737	−0.871039
$13$	−0.806951	−0.223808	−0.111904	−	0.993719i	$-0.535695\pi$
$13$	−0.806951	−0.223808	−0.111904	+	0.993719i	$0.535695\pi$
$14$	2.53637	0.677874
$15$	3.68404	0.951214
$16$	1.00000	0.250000
$17$	2.15852	0.523519	0.261759	−	0.965133i	$-0.415697\pi$
$17$	2.15852	0.523519	0.261759	+	0.965133i	$0.415697\pi$
$18$	6.10451	1.43885
$19$	1.07604	0.246861	0.123431	−	0.992353i	$-0.460610\pi$
$19$	1.07604	0.246861	0.123431	+	0.992353i	$0.460610\pi$
$20$	−1.22094	−0.273011
$21$	−7.65317	−1.67006
$22$	0.137455	0.0293055
$23$	−5.61126	−1.17003	−0.585014	−	0.811023i	$-0.698911\pi$
$23$	−5.61126	−1.17003	−0.585014	+	0.811023i	$0.698911\pi$
$24$	−3.01737	−0.615918
$25$	−3.50930	−0.701859
$26$	−0.806951	−0.158256
$27$	−9.36747	−1.80277
$28$	2.53637	0.479329
$29$	3.70442	0.687894	0.343947	−	0.938989i	$-0.388236\pi$
$29$	3.70442	0.687894	0.343947	+	0.938989i	$0.388236\pi$
$30$	3.68404	0.672610
$31$	−1.71836	−0.308626	−0.154313	−	0.988022i	$-0.549316\pi$
$31$	−1.71836	−0.308626	−0.154313	+	0.988022i	$0.549316\pi$
$32$	1.00000	0.176777
$33$	−0.414753	−0.0721992
$34$	2.15852	0.370184
$35$	−3.09677	−0.523449
$36$	6.10451	1.01742
$37$	−3.29300	−0.541366	−0.270683	−	0.962669i	$-0.587250\pi$
$37$	−3.29300	−0.541366	−0.270683	+	0.962669i	$0.587250\pi$
$38$	1.07604	0.174557
$39$	2.43487	0.389891
$40$	−1.22094	−0.193048
$41$	−7.64932	−1.19462	−0.597312	−	0.802009i	$-0.703764\pi$
$41$	−7.64932	−1.19462	−0.597312	+	0.802009i	$0.703764\pi$
$42$	−7.65317	−1.18091
$43$	5.20520	0.793785	0.396893	−	0.917865i	$-0.370089\pi$
$43$	5.20520	0.793785	0.396893	+	0.917865i	$0.370089\pi$
$44$	0.137455	0.0207221
$45$	−7.45327	−1.11107
$46$	−5.61126	−0.827335
$47$	4.92416	0.718263	0.359132	−	0.933287i	$-0.383073\pi$
$47$	4.92416	0.718263	0.359132	+	0.933287i	$0.383073\pi$
$48$	−3.01737	−0.435520
$49$	−0.566819	−0.0809741
$50$	−3.50930	−0.496290
$51$	−6.51306	−0.912011
$52$	−0.806951	−0.111904
$53$	−1.93351	−0.265588	−0.132794	−	0.991144i	$-0.542395\pi$
$53$	−1.93351	−0.265588	−0.132794	+	0.991144i	$0.542395\pi$
$54$	−9.36747	−1.27475
$55$	−0.167825	−0.0226295
$56$	2.53637	0.338937
$57$	−3.24682	−0.430051
$58$	3.70442	0.486414
$59$	−11.7080	−1.52425	−0.762124	−	0.647431i	$-0.775843\pi$
$59$	−11.7080	−1.52425	−0.762124	+	0.647431i	$0.775843\pi$
$60$	3.68404	0.475607
$61$	−0.669111	−0.0856708	−0.0428354	−	0.999082i	$-0.513639\pi$
$61$	−0.669111	−0.0856708	−0.0428354	+	0.999082i	$0.513639\pi$
$62$	−1.71836	−0.218231
$63$	15.4833	1.95071
$64$	1.00000	0.125000
$65$	0.985242	0.122204
$66$	−0.414753	−0.0510525
$67$	5.09018	0.621864	0.310932	−	0.950432i	$-0.399359\pi$
$67$	5.09018	0.621864	0.310932	+	0.950432i	$0.399359\pi$
$68$	2.15852	0.261759
$69$	16.9312	2.03828
$70$	−3.09677	−0.370134
$71$	−7.94119	−0.942446	−0.471223	−	0.882014i	$-0.656187\pi$
$71$	−7.94119	−0.942446	−0.471223	+	0.882014i	$0.656187\pi$
$72$	6.10451	0.719424
$73$	2.48803	0.291202	0.145601	−	0.989343i	$-0.453488\pi$
$73$	2.48803	0.291202	0.145601	+	0.989343i	$0.453488\pi$
$74$	−3.29300	−0.382804
$75$	10.5888	1.22269
$76$	1.07604	0.123431
$77$	0.348637	0.0397309
$78$	2.43487	0.275695
$79$	−13.0996	−1.47382	−0.736912	−	0.675988i	$-0.763717\pi$
$79$	−13.0996	−1.47382	−0.736912	+	0.675988i	$0.763717\pi$
$80$	−1.22094	−0.136506
$81$	9.95156	1.10573
$82$	−7.64932	−0.844726
$83$	4.45477	0.488975	0.244487	−	0.969652i	$-0.421380\pi$
$83$	4.45477	0.488975	0.244487	+	0.969652i	$0.421380\pi$
$84$	−7.65317	−0.835029
$85$	−2.63543	−0.285853
$86$	5.20520	0.561291
$87$	−11.1776	−1.19837
$88$	0.137455	0.0146528
$89$	12.1306	1.28584	0.642921	−	0.765933i	$-0.277722\pi$
$89$	12.1306	1.28584	0.642921	+	0.765933i	$0.277722\pi$
$90$	−7.45327	−0.785643
$91$	−2.04673	−0.214555
$92$	−5.61126	−0.585014
$93$	5.18491	0.537650
$94$	4.92416	0.507889
$95$	−1.31379	−0.134792
$96$	−3.01737	−0.307959
$97$	9.54246	0.968890	0.484445	−	0.874822i	$-0.339022\pi$
$97$	9.54246	0.968890	0.484445	+	0.874822i	$0.339022\pi$
$98$	−0.566819	−0.0572573
$99$	0.839097	0.0843324
$100$	−3.50930	−0.350930
$101$	4.64189	0.461886	0.230943	−	0.972967i	$-0.425819\pi$
$101$	4.64189	0.461886	0.230943	+	0.972967i	$0.425819\pi$
$102$	−6.51306	−0.644889
$103$	14.4225	1.42109	0.710546	−	0.703651i	$-0.248448\pi$
$103$	14.4225	1.42109	0.710546	+	0.703651i	$0.248448\pi$
$104$	−0.806951	−0.0791281
$105$	9.34409	0.911890
$106$	−1.93351	−0.187799
$107$	12.9553	1.25243	0.626216	−	0.779650i	$-0.284603\pi$
$107$	12.9553	1.25243	0.626216	+	0.779650i	$0.284603\pi$
$108$	−9.36747	−0.901385
$109$	−16.0244	−1.53486	−0.767431	−	0.641132i	$-0.778465\pi$
$109$	−16.0244	−1.53486	−0.767431	+	0.641132i	$0.778465\pi$
$110$	−0.167825	−0.0160015
$111$	9.93620	0.943103
$112$	2.53637	0.239665
$113$	2.36333	0.222323	0.111161	−	0.993802i	$-0.464543\pi$
$113$	2.36333	0.222323	0.111161	+	0.993802i	$0.464543\pi$
$114$	−3.24682	−0.304092
$115$	6.85103	0.638862
$116$	3.70442	0.343947
$117$	−4.92605	−0.455413
$118$	−11.7080	−1.07781
$119$	5.47482	0.501876
$120$	3.68404	0.336305
$121$	−10.9811	−0.998282
$122$	−0.669111	−0.0605784
$123$	23.0808	2.08113
$124$	−1.71836	−0.154313
$125$	10.3894	0.929254
$126$	15.4833	1.37936
$127$	−11.3836	−1.01013	−0.505065	−	0.863081i	$-0.668532\pi$
$127$	−11.3836	−1.01013	−0.505065	+	0.863081i	$0.668532\pi$
$128$	1.00000	0.0883883
$129$	−15.7060	−1.38284
$130$	0.985242	0.0864115
$131$	16.1468	1.41075	0.705376	−	0.708833i	$-0.250778\pi$
$131$	16.1468	1.41075	0.705376	+	0.708833i	$0.250778\pi$
$132$	−0.414753	−0.0360996
$133$	2.72924	0.236655
$134$	5.09018	0.439724
$135$	11.4371	0.984353
$136$	2.15852	0.185092
$137$	−21.5338	−1.83976	−0.919880	−	0.392200i	$-0.871714\pi$
$137$	−21.5338	−1.83976	−0.919880	+	0.392200i	$0.871714\pi$
$138$	16.9312	1.44128
$139$	0.441567	0.0374532	0.0187266	−	0.999825i	$-0.494039\pi$
$139$	0.441567	0.0374532	0.0187266	+	0.999825i	$0.494039\pi$
$140$	−3.09677	−0.261725
$141$	−14.8580	−1.25127
$142$	−7.94119	−0.666410
$143$	−0.110920	−0.00927556
$144$	6.10451	0.508710
$145$	−4.52289	−0.375606
$146$	2.48803	0.205911
$147$	1.71030	0.141063
$148$	−3.29300	−0.270683
$149$	14.4044	1.18005	0.590027	−	0.807383i	$-0.299117\pi$
$149$	14.4044	1.18005	0.590027	+	0.807383i	$0.299117\pi$
$150$	10.5888	0.864575
$151$	−13.4602	−1.09538	−0.547688	−	0.836683i	$-0.684492\pi$
$151$	−13.4602	−1.09538	−0.547688	+	0.836683i	$0.684492\pi$
$152$	1.07604	0.0872786
$153$	13.1767	1.06528
$154$	0.348637	0.0280940
$155$	2.09801	0.168517
$156$	2.43487	0.194946
$157$	−17.3613	−1.38558	−0.692791	−	0.721138i	$-0.743619\pi$
$157$	−17.3613	−1.38558	−0.692791	+	0.721138i	$0.743619\pi$
$158$	−13.0996	−1.04215
$159$	5.83411	0.462675
$160$	−1.22094	−0.0965241
$161$	−14.2322	−1.12166
$162$	9.95156	0.781868
$163$	2.59472	0.203234	0.101617	−	0.994824i	$-0.467598\pi$
$163$	2.59472	0.203234	0.101617	+	0.994824i	$0.467598\pi$
$164$	−7.64932	−0.597312
$165$	0.506390	0.0394224
$166$	4.45477	0.345757
$167$	−20.0621	−1.55245	−0.776226	−	0.630454i	$-0.782869\pi$
$167$	−20.0621	−1.55245	−0.776226	+	0.630454i	$0.782869\pi$
$168$	−7.65317	−0.590455
$169$	−12.3488	−0.949910
$170$	−2.63543	−0.202129
$171$	6.56872	0.502322
$172$	5.20520	0.396893
$173$	13.8284	1.05135	0.525676	−	0.850685i	$-0.323813\pi$
$173$	13.8284	1.05135	0.525676	+	0.850685i	$0.323813\pi$
$174$	−11.1776	−0.847372
$175$	−8.90088	−0.672843
$176$	0.137455	0.0103611
$177$	35.3273	2.65536
$178$	12.1306	0.909227
$179$	10.1664	0.759870	0.379935	−	0.925013i	$-0.375946\pi$
$179$	10.1664	0.759870	0.379935	+	0.925013i	$0.375946\pi$
$180$	−7.45327	−0.555534
$181$	−0.529361	−0.0393471	−0.0196736	−	0.999806i	$-0.506263\pi$
$181$	−0.529361	−0.0393471	−0.0196736	+	0.999806i	$0.506263\pi$
$182$	−2.04673	−0.151714
$183$	2.01895	0.149245
$184$	−5.61126	−0.413668
$185$	4.02057	0.295598
$186$	5.18491	0.380176
$187$	0.296700	0.0216969
$188$	4.92416	0.359132
$189$	−23.7594	−1.72824
$190$	−1.31379	−0.0953121
$191$	−17.2259	−1.24642	−0.623211	−	0.782054i	$-0.714172\pi$
$191$	−17.2259	−1.24642	−0.623211	+	0.782054i	$0.714172\pi$
$192$	−3.01737	−0.217760
$193$	−5.40764	−0.389250	−0.194625	−	0.980878i	$-0.562349\pi$
$193$	−5.40764	−0.389250	−0.194625	+	0.980878i	$0.562349\pi$
$194$	9.54246	0.685109
$195$	−2.97284	−0.212889
$196$	−0.566819	−0.0404870
$197$	0.799118	0.0569348	0.0284674	−	0.999595i	$-0.490937\pi$
$197$	0.799118	0.0569348	0.0284674	+	0.999595i	$0.490937\pi$
$198$	0.839097	0.0596320
$199$	23.0955	1.63720	0.818600	−	0.574365i	$-0.194751\pi$
$199$	23.0955	1.63720	0.818600	+	0.574365i	$0.194751\pi$
$200$	−3.50930	−0.248145
$201$	−15.3589	−1.08334
$202$	4.64189	0.326602
$203$	9.39579	0.659455
$204$	−6.51306	−0.456005
$205$	9.33939	0.652291
$206$	14.4225	1.00486
$207$	−34.2540	−2.38082
$208$	−0.806951	−0.0559520
$209$	0.147907	0.0102310
$210$	9.34409	0.644803
$211$	13.5050	0.929724	0.464862	−	0.885383i	$-0.346104\pi$
$211$	13.5050	0.929724	0.464862	+	0.885383i	$0.346104\pi$
$212$	−1.93351	−0.132794
$213$	23.9615	1.64181
$214$	12.9553	0.885603
$215$	−6.35525	−0.433425
$216$	−9.36747	−0.637375
$217$	−4.35839	−0.295867
$218$	−16.0244	−1.08531
$219$	−7.50731	−0.507297
$220$	−0.167825	−0.0113148
$221$	−1.74182	−0.117168
$222$	9.93620	0.666874
$223$	−23.0827	−1.54573	−0.772866	−	0.634570i	$-0.781177\pi$
$223$	−23.0827	−1.54573	−0.772866	+	0.634570i	$0.781177\pi$
$224$	2.53637	0.169468
$225$	−21.4226	−1.42817
$226$	2.36333	0.157206
$227$	−2.95811	−0.196337	−0.0981684	−	0.995170i	$-0.531298\pi$
$227$	−2.95811	−0.196337	−0.0981684	+	0.995170i	$0.531298\pi$
$228$	−3.24682	−0.215026
$229$	−28.5272	−1.88513	−0.942566	−	0.334020i	$-0.891595\pi$
$229$	−28.5272	−1.88513	−0.942566	+	0.334020i	$0.891595\pi$
$230$	6.85103	0.451744
$231$	−1.05197	−0.0692143
$232$	3.70442	0.243207
$233$	−3.91475	−0.256464	−0.128232	−	0.991744i	$-0.540930\pi$
$233$	−3.91475	−0.256464	−0.128232	+	0.991744i	$0.540930\pi$
$234$	−4.92605	−0.322026
$235$	−6.01213	−0.392188
$236$	−11.7080	−0.762124
$237$	39.5264	2.56752
$238$	5.47482	0.354880
$239$	27.0971	1.75277	0.876384	−	0.481612i	$-0.159949\pi$
$239$	27.0971	1.75277	0.876384	+	0.481612i	$0.159949\pi$
$240$	3.68404	0.237804
$241$	26.8742	1.73112	0.865561	−	0.500803i	$-0.166962\pi$
$241$	26.8742	1.73112	0.865561	+	0.500803i	$0.166962\pi$
$242$	−10.9811	−0.705892
$243$	−1.92512	−0.123497
$244$	−0.669111	−0.0428354
$245$	0.692053	0.0442137
$246$	23.0808	1.47158
$247$	−0.868314	−0.0552495
$248$	−1.71836	−0.109116
$249$	−13.4417	−0.851832
$250$	10.3894	0.657082
$251$	−19.1225	−1.20700	−0.603501	−	0.797363i	$-0.706228\pi$
$251$	−19.1225	−1.20700	−0.603501	+	0.797363i	$0.706228\pi$
$252$	15.4833	0.975357
$253$	−0.771296	−0.0484910
$254$	−11.3836	−0.714270
$255$	7.95208	0.497978
$256$	1.00000	0.0625000
$257$	−14.1320	−0.881533	−0.440766	−	0.897622i	$-0.645293\pi$
$257$	−14.1320	−0.881533	−0.440766	+	0.897622i	$0.645293\pi$
$258$	−15.7060	−0.977813
$259$	−8.35228	−0.518985
$260$	0.985242	0.0611021
$261$	22.6137	1.39975
$262$	16.1468	0.997553
$263$	−20.0912	−1.23888	−0.619438	−	0.785045i	$-0.712640\pi$
$263$	−20.0912	−1.23888	−0.619438	+	0.785045i	$0.712640\pi$
$264$	−0.414753	−0.0255263
$265$	2.36071	0.145017
$266$	2.72924	0.167341
$267$	−36.6025	−2.24004
$268$	5.09018	0.310932
$269$	29.9173	1.82409	0.912046	−	0.410089i	$-0.134502\pi$
$269$	29.9173	1.82409	0.912046	+	0.410089i	$0.134502\pi$
$270$	11.4371	0.696043
$271$	18.7229	1.13734	0.568668	−	0.822567i	$-0.307459\pi$
$271$	18.7229	1.13734	0.568668	+	0.822567i	$0.307459\pi$
$272$	2.15852	0.130880
$273$	6.17574	0.373773
$274$	−21.5338	−1.30091
$275$	−0.482371	−0.0290880
$276$	16.9312	1.01914
$277$	−6.84176	−0.411081	−0.205541	−	0.978649i	$-0.565895\pi$
$277$	−6.84176	−0.411081	−0.205541	+	0.978649i	$0.565895\pi$
$278$	0.441567	0.0264834
$279$	−10.4897	−0.628003
$280$	−3.09677	−0.185067
$281$	−1.59746	−0.0952965	−0.0476482	−	0.998864i	$-0.515173\pi$
$281$	−1.59746	−0.0952965	−0.0476482	+	0.998864i	$0.515173\pi$
$282$	−14.8580	−0.884782
$283$	−11.6118	−0.690253	−0.345126	−	0.938556i	$-0.612164\pi$
$283$	−11.6118	−0.690253	−0.345126	+	0.938556i	$0.612164\pi$
$284$	−7.94119	−0.471223
$285$	3.96418	0.234818
$286$	−0.110920	−0.00655881
$287$	−19.4015	−1.14524
$288$	6.10451	0.359712
$289$	−12.3408	−0.725928
$290$	−4.52289	−0.265593
$291$	−28.7931	−1.68788
$292$	2.48803	0.145601
$293$	−4.40276	−0.257212	−0.128606	−	0.991696i	$-0.541050\pi$
$293$	−4.40276	−0.257212	−0.128606	+	0.991696i	$0.541050\pi$
$294$	1.71030	0.0997468
$295$	14.2948	0.832273
$296$	−3.29300	−0.191402
$297$	−1.28761	−0.0747145
$298$	14.4044	0.834425
$299$	4.52802	0.261862
$300$	10.5888	0.611347
$301$	13.2023	0.760969
$302$	−13.4602	−0.774548
$303$	−14.0063	−0.804641
$304$	1.07604	0.0617153
$305$	0.816946	0.0467782
$306$	13.1767	0.753264
$307$	−14.2102	−0.811018	−0.405509	−	0.914091i	$-0.632906\pi$
$307$	−14.2102	−0.811018	−0.405509	+	0.914091i	$0.632906\pi$
$308$	0.348637	0.0198654
$309$	−43.5180	−2.47565
$310$	2.09801	0.119159
$311$	−21.5820	−1.22380	−0.611900	−	0.790935i	$-0.709595\pi$
$311$	−21.5820	−1.22380	−0.611900	+	0.790935i	$0.709595\pi$
$312$	2.43487	0.137847
$313$	−5.62071	−0.317701	−0.158851	−	0.987303i	$-0.550779\pi$
$313$	−5.62071	−0.317701	−0.158851	+	0.987303i	$0.550779\pi$
$314$	−17.3613	−0.979755
$315$	−18.9043	−1.06513
$316$	−13.0996	−0.736912
$317$	−29.4976	−1.65675	−0.828375	−	0.560174i	$-0.810734\pi$
$317$	−29.4976	−1.65675	−0.828375	+	0.560174i	$0.810734\pi$
$318$	5.83411	0.327161
$319$	0.509192	0.0285093
$320$	−1.22094	−0.0682528
$321$	−39.0908	−2.18184
$322$	−14.2322	−0.793132
$323$	2.32266	0.129236
$324$	9.95156	0.552864
$325$	2.83183	0.157082
$326$	2.59472	0.143708
$327$	48.3516	2.67385
$328$	−7.64932	−0.422363
$329$	12.4895	0.688569
$330$	0.506390	0.0278758
$331$	−11.8249	−0.649958	−0.324979	−	0.945721i	$-0.605357\pi$
$331$	−11.8249	−0.649958	−0.324979	+	0.945721i	$0.605357\pi$
$332$	4.45477	0.244487
$333$	−20.1022	−1.10159
$334$	−20.0621	−1.09775
$335$	−6.21482	−0.339552
$336$	−7.65317	−0.417515
$337$	−31.5618	−1.71928	−0.859642	−	0.510898i	$-0.829313\pi$
$337$	−31.5618	−1.71928	−0.859642	+	0.510898i	$0.829313\pi$
$338$	−12.3488	−0.671688
$339$	−7.13103	−0.387304
$340$	−2.63543	−0.142927
$341$	−0.236197	−0.0127908
$342$	6.56872	0.355195
$343$	−19.1923	−1.03628
$344$	5.20520	0.280645
$345$	−20.6721	−1.11295
$346$	13.8284	0.743418
$347$	8.09057	0.434325	0.217162	−	0.976136i	$-0.430320\pi$
$347$	8.09057	0.434325	0.217162	+	0.976136i	$0.430320\pi$
$348$	−11.1776	−0.599183
$349$	−36.9391	−1.97730	−0.988652	−	0.150224i	$-0.952000\pi$
$349$	−36.9391	−1.97730	−0.988652	+	0.150224i	$0.952000\pi$
$350$	−8.90088	−0.475772
$351$	7.55909	0.403474
$352$	0.137455	0.00732638
$353$	−22.5658	−1.20106	−0.600528	−	0.799604i	$-0.705043\pi$
$353$	−22.5658	−1.20106	−0.600528	+	0.799604i	$0.705043\pi$
$354$	35.3273	1.87762
$355$	9.69574	0.514597
$356$	12.1306	0.642921
$357$	−16.5195	−0.874307
$358$	10.1664	0.537309
$359$	−20.2189	−1.06711	−0.533556	−	0.845765i	$-0.679145\pi$
$359$	−20.2189	−1.06711	−0.533556	+	0.845765i	$0.679145\pi$
$360$	−7.45327	−0.392822
$361$	−17.8421	−0.939060
$362$	−0.529361	−0.0278226
$363$	33.1340	1.73909
$364$	−2.04673	−0.107278
$365$	−3.03775	−0.159003
$366$	2.01895	0.105532
$367$	33.4484	1.74599	0.872996	−	0.487727i	$-0.162174\pi$
$367$	33.4484	1.74599	0.872996	+	0.487727i	$0.162174\pi$
$368$	−5.61126	−0.292507
$369$	−46.6954	−2.43087
$370$	4.02057	0.209019
$371$	−4.90410	−0.254608
$372$	5.18491	0.268825
$373$	−19.7495	−1.02259	−0.511295	−	0.859405i	$-0.670834\pi$
$373$	−19.7495	−1.02259	−0.511295	+	0.859405i	$0.670834\pi$
$374$	0.296700	0.0153420
$375$	−31.3486	−1.61883
$376$	4.92416	0.253944
$377$	−2.98929	−0.153956
$378$	−23.7594	−1.22205
$379$	18.5701	0.953881	0.476941	−	0.878936i	$-0.341746\pi$
$379$	18.5701	0.953881	0.476941	+	0.878936i	$0.341746\pi$
$380$	−1.31379	−0.0673958
$381$	34.3485	1.75973
$382$	−17.2259	−0.881353
$383$	−23.9812	−1.22538	−0.612690	−	0.790323i	$-0.709913\pi$
$383$	−23.9812	−1.22538	−0.612690	+	0.790323i	$0.709913\pi$
$384$	−3.01737	−0.153979
$385$	−0.425666	−0.0216940
$386$	−5.40764	−0.275241
$387$	31.7752	1.61522
$388$	9.54246	0.484445
$389$	−12.6126	−0.639483	−0.319742	−	0.947505i	$-0.603596\pi$
$389$	−12.6126	−0.639483	−0.319742	+	0.947505i	$0.603596\pi$
$390$	−2.97284	−0.150536
$391$	−12.1120	−0.612532
$392$	−0.566819	−0.0286287
$393$	−48.7209	−2.45764
$394$	0.799118	0.0402590
$395$	15.9939	0.804741
$396$	0.839097	0.0421662
$397$	−5.43803	−0.272927	−0.136464	−	0.990645i	$-0.543574\pi$
$397$	−5.43803	−0.272927	−0.136464	+	0.990645i	$0.543574\pi$
$398$	23.0955	1.15767
$399$	−8.23513	−0.412272
$400$	−3.50930	−0.175465
$401$	−12.0188	−0.600191	−0.300096	−	0.953909i	$-0.597019\pi$
$401$	−12.0188	−0.600191	−0.300096	+	0.953909i	$0.597019\pi$
$402$	−15.3589	−0.766035
$403$	1.38663	0.0690729
$404$	4.64189	0.230943
$405$	−12.1503	−0.603753
$406$	9.39579	0.466305
$407$	−0.452640	−0.0224365
$408$	−6.51306	−0.322445
$409$	−12.3241	−0.609386	−0.304693	−	0.952451i	$-0.598554\pi$
$409$	−12.3241	−0.609386	−0.304693	+	0.952451i	$0.598554\pi$
$410$	9.33939	0.461240
$411$	64.9755	3.20501
$412$	14.4225	0.710546
$413$	−29.6958	−1.46123
$414$	−34.2540	−1.68349
$415$	−5.43902	−0.266991
$416$	−0.806951	−0.0395641
$417$	−1.33237	−0.0652464
$418$	0.147907	0.00723439
$419$	13.0762	0.638814	0.319407	−	0.947618i	$-0.396516\pi$
$419$	13.0762	0.638814	0.319407	+	0.947618i	$0.396516\pi$
$420$	9.34409	0.455945
$421$	20.1913	0.984061	0.492031	−	0.870578i	$-0.336255\pi$
$421$	20.1913	0.984061	0.492031	+	0.870578i	$0.336255\pi$
$422$	13.5050	0.657414
$423$	30.0596	1.46155
$424$	−1.93351	−0.0938995
$425$	−7.57490	−0.367437
$426$	23.9615	1.16094
$427$	−1.69711	−0.0821291
$428$	12.9553	0.626216
$429$	0.334685	0.0161588
$430$	−6.35525	−0.306478
$431$	4.48938	0.216246	0.108123	−	0.994138i	$-0.465516\pi$
$431$	4.48938	0.216246	0.108123	+	0.994138i	$0.465516\pi$
$432$	−9.36747	−0.450692
$433$	0.826963	0.0397413	0.0198707	−	0.999803i	$-0.493675\pi$
$433$	0.826963	0.0397413	0.0198707	+	0.999803i	$0.493675\pi$
$434$	−4.35839	−0.209209
$435$	13.6472	0.654335
$436$	−16.0244	−0.767431
$437$	−6.03795	−0.288835
$438$	−7.50731	−0.358713
$439$	−3.58838	−0.171264	−0.0856321	−	0.996327i	$-0.527291\pi$
$439$	−3.58838	−0.171264	−0.0856321	+	0.996327i	$0.527291\pi$
$440$	−0.167825	−0.00800074
$441$	−3.46015	−0.164769
$442$	−1.74182	−0.0828501
$443$	−16.1827	−0.768864	−0.384432	−	0.923153i	$-0.625603\pi$
$443$	−16.1827	−0.768864	−0.384432	+	0.923153i	$0.625603\pi$
$444$	9.93620	0.471551
$445$	−14.8108	−0.702099
$446$	−23.0827	−1.09300
$447$	−43.4634	−2.05575
$448$	2.53637	0.119832
$449$	27.3050	1.28860	0.644300	−	0.764773i	$-0.277149\pi$
$449$	27.3050	1.28860	0.644300	+	0.764773i	$0.277149\pi$
$450$	−21.4226	−1.00987
$451$	−1.05144	−0.0495103
$452$	2.36333	0.111161
$453$	40.6144	1.90823
$454$	−2.95811	−0.138831
$455$	2.49894	0.117152
$456$	−3.24682	−0.152046
$457$	−10.5724	−0.494554	−0.247277	−	0.968945i	$-0.579536\pi$
$457$	−10.5724	−0.494554	−0.247277	+	0.968945i	$0.579536\pi$
$458$	−28.5272	−1.33299
$459$	−20.2199	−0.943784
$460$	6.85103	0.319431
$461$	34.4944	1.60657	0.803283	−	0.595598i	$-0.203085\pi$
$461$	34.4944	1.60657	0.803283	+	0.595598i	$0.203085\pi$
$462$	−1.05197	−0.0489419
$463$	−7.64417	−0.355255	−0.177627	−	0.984098i	$-0.556842\pi$
$463$	−7.64417	−0.355255	−0.177627	+	0.984098i	$0.556842\pi$
$464$	3.70442	0.171973
$465$	−6.33048	−0.293569
$466$	−3.91475	−0.181348
$467$	−2.34336	−0.108438	−0.0542189	−	0.998529i	$-0.517267\pi$
$467$	−2.34336	−0.108438	−0.0542189	+	0.998529i	$0.517267\pi$
$468$	−4.92605	−0.227707
$469$	12.9106	0.596155
$470$	−6.01213	−0.277319
$471$	52.3854	2.41379
$472$	−11.7080	−0.538903
$473$	0.715481	0.0328979
$474$	39.5264	1.81551
$475$	−3.77615	−0.173262
$476$	5.47482	0.250938
$477$	−11.8031	−0.540429
$478$	27.0971	1.23939
$479$	−0.134962	−0.00616657	−0.00308329	−	0.999995i	$-0.500981\pi$
$479$	−0.134962	−0.00616657	−0.00308329	+	0.999995i	$0.500981\pi$
$480$	3.68404	0.168153
$481$	2.65729	0.121162
$482$	26.8742	1.22409
$483$	42.9439	1.95402
$484$	−10.9811	−0.499141
$485$	−11.6508	−0.529036
$486$	−1.92512	−0.0873253
$487$	−27.9801	−1.26790	−0.633950	−	0.773374i	$-0.718567\pi$
$487$	−27.9801	−1.26790	−0.633950	+	0.773374i	$0.718567\pi$
$488$	−0.669111	−0.0302892
$489$	−7.82923	−0.354050
$490$	0.692053	0.0312638
$491$	35.1814	1.58771	0.793857	−	0.608104i	$-0.208070\pi$
$491$	35.1814	1.58771	0.793857	+	0.608104i	$0.208070\pi$
$492$	23.0808	1.04056
$493$	7.99608	0.360125
$494$	−0.868314	−0.0390673
$495$	−1.02449	−0.0460474
$496$	−1.71836	−0.0771564
$497$	−20.1418	−0.903484
$498$	−13.4417	−0.602336
$499$	12.4147	0.555759	0.277880	−	0.960616i	$-0.410368\pi$
$499$	12.4147	0.555759	0.277880	+	0.960616i	$0.410368\pi$
$500$	10.3894	0.464627
$501$	60.5348	2.70450
$502$	−19.1225	−0.853479
$503$	23.2076	1.03478	0.517388	−	0.855751i	$-0.326904\pi$
$503$	23.2076	1.03478	0.517388	+	0.855751i	$0.326904\pi$
$504$	15.4833	0.689682
$505$	−5.66749	−0.252200
$506$	−0.771296	−0.0342883
$507$	37.2610	1.65482
$508$	−11.3836	−0.505065
$509$	−12.7492	−0.565100	−0.282550	−	0.959253i	$-0.591180\pi$
$509$	−12.7492	−0.565100	−0.282550	+	0.959253i	$0.591180\pi$
$510$	7.95208	0.352124
$511$	6.31058	0.279163
$512$	1.00000	0.0441942
$513$	−10.0798	−0.445034
$514$	−14.1320	−0.623338
$515$	−17.6091	−0.775948
$516$	−15.7060	−0.691418
$517$	0.676851	0.0297679
$518$	−8.35228	−0.366978
$519$	−41.7253	−1.83154
$520$	0.985242	0.0432057
$521$	3.93655	0.172463	0.0862317	−	0.996275i	$-0.472517\pi$
$521$	3.93655	0.172463	0.0862317	+	0.996275i	$0.472517\pi$
$522$	22.6137	0.989775
$523$	20.9972	0.918143	0.459072	−	0.888399i	$-0.348182\pi$
$523$	20.9972	0.918143	0.459072	+	0.888399i	$0.348182\pi$
$524$	16.1468	0.705376
$525$	26.8572	1.17215
$526$	−20.0912	−0.876018
$527$	−3.70911	−0.161571
$528$	−0.414753	−0.0180498
$529$	8.48625	0.368967
$530$	2.36071	0.102543
$531$	−71.4715	−3.10160
$532$	2.72924	0.118328
$533$	6.17263	0.267366
$534$	−36.6025	−1.58395
$535$	−15.8176	−0.683856
$536$	5.09018	0.219862
$537$	−30.6757	−1.32375
$538$	29.9173	1.28983
$539$	−0.0779121	−0.00335591
$540$	11.4371	0.492176
$541$	2.69617	0.115917	0.0579587	−	0.998319i	$-0.481541\pi$
$541$	2.69617	0.115917	0.0579587	+	0.998319i	$0.481541\pi$
$542$	18.7229	0.804218
$543$	1.59728	0.0685457
$544$	2.15852	0.0925459
$545$	19.5649	0.838069
$546$	6.17574	0.264297
$547$	−14.7569	−0.630959	−0.315480	−	0.948932i	$-0.602165\pi$
$547$	−14.7569	−0.630959	−0.315480	+	0.948932i	$0.602165\pi$
$548$	−21.5338	−0.919880
$549$	−4.08460	−0.174326
$550$	−0.482371	−0.0205684
$551$	3.98612	0.169814
$552$	16.9312	0.720642
$553$	−33.2256	−1.41289
$554$	−6.84176	−0.290678
$555$	−12.1315	−0.514955
$556$	0.441567	0.0187266
$557$	−5.78948	−0.245308	−0.122654	−	0.992449i	$-0.539141\pi$
$557$	−5.78948	−0.245308	−0.122654	+	0.992449i	$0.539141\pi$
$558$	−10.4897	−0.444065
$559$	−4.20034	−0.177656
$560$	−3.09677	−0.130862
$561$	−0.895253	−0.0377976
$562$	−1.59746	−0.0673848
$563$	17.2791	0.728225	0.364113	−	0.931355i	$-0.381372\pi$
$563$	17.2791	0.728225	0.364113	+	0.931355i	$0.381372\pi$
$564$	−14.8580	−0.625635
$565$	−2.88549	−0.121393
$566$	−11.6118	−0.488082
$567$	25.2409	1.06002
$568$	−7.94119	−0.333205
$569$	11.3445	0.475587	0.237793	−	0.971316i	$-0.423576\pi$
$569$	11.3445	0.475587	0.237793	+	0.971316i	$0.423576\pi$
$570$	3.96418	0.166041
$571$	−24.6047	−1.02968	−0.514838	−	0.857287i	$-0.672148\pi$
$571$	−24.6047	−1.02968	−0.514838	+	0.857287i	$0.672148\pi$
$572$	−0.110920	−0.00463778
$573$	51.9769	2.17136
$574$	−19.4015	−0.809804
$575$	19.6916	0.821196
$576$	6.10451	0.254355
$577$	−5.98598	−0.249199	−0.124600	−	0.992207i	$-0.539765\pi$
$577$	−5.98598	−0.249199	−0.124600	+	0.992207i	$0.539765\pi$
$578$	−12.3408	−0.513309
$579$	16.3168	0.678105
$580$	−4.52289	−0.187803
$581$	11.2990	0.468760
$582$	−28.7931	−1.19351
$583$	−0.265771	−0.0110071
$584$	2.48803	0.102956
$585$	6.01443	0.248666
$586$	−4.40276	−0.181877
$587$	−24.4051	−1.00731	−0.503654	−	0.863906i	$-0.668011\pi$
$587$	−24.4051	−1.00731	−0.503654	+	0.863906i	$0.668011\pi$
$588$	1.71030	0.0705316
$589$	−1.84902	−0.0761877
$590$	14.2948	0.588506
$591$	−2.41123	−0.0991849
$592$	−3.29300	−0.135342
$593$	−25.8698	−1.06235	−0.531173	−	0.847263i	$-0.678249\pi$
$593$	−25.8698	−1.06235	−0.531173	+	0.847263i	$0.678249\pi$
$594$	−1.28761	−0.0528311
$595$	−6.68444	−0.274035
$596$	14.4044	0.590027
$597$	−69.6878	−2.85213
$598$	4.52802	0.185164
$599$	−9.41724	−0.384778	−0.192389	−	0.981319i	$-0.561624\pi$
$599$	−9.41724	−0.384778	−0.192389	+	0.981319i	$0.561624\pi$
$600$	10.5888	0.432288
$601$	−15.6127	−0.636853	−0.318427	−	0.947947i	$-0.603154\pi$
$601$	−15.6127	−0.636853	−0.318427	+	0.947947i	$0.603154\pi$
$602$	13.2023	0.538086
$603$	31.0731	1.26539
$604$	−13.4602	−0.547688
$605$	13.4073	0.545085
$606$	−14.0063	−0.568967
$607$	9.72071	0.394552	0.197276	−	0.980348i	$-0.436791\pi$
$607$	9.72071	0.394552	0.197276	+	0.980348i	$0.436791\pi$
$608$	1.07604	0.0436393
$609$	−28.3506	−1.14882
$610$	0.816946	0.0330772
$611$	−3.97356	−0.160753
$612$	13.1767	0.532638
$613$	8.20603	0.331438	0.165719	−	0.986173i	$-0.447005\pi$
$613$	8.20603	0.331438	0.165719	+	0.986173i	$0.447005\pi$
$614$	−14.2102	−0.573477
$615$	−28.1804	−1.13634
$616$	0.348637	0.0140470
$617$	−20.1399	−0.810801	−0.405400	−	0.914139i	$-0.632868\pi$
$617$	−20.1399	−0.810801	−0.405400	+	0.914139i	$0.632868\pi$
$618$	−43.5180	−1.75055
$619$	−34.1228	−1.37151	−0.685755	−	0.727833i	$-0.740528\pi$
$619$	−34.1228	−1.37151	−0.685755	+	0.727833i	$0.740528\pi$
$620$	2.09801	0.0842583
$621$	52.5633	2.10929
$622$	−21.5820	−0.865358
$623$	30.7677	1.23268
$624$	2.43487	0.0974728
$625$	4.86165	0.194466
$626$	−5.62071	−0.224649
$627$	−0.446291	−0.0178232
$628$	−17.3613	−0.692791
$629$	−7.10802	−0.283415
$630$	−18.9043	−0.753164
$631$	25.6941	1.02286	0.511432	−	0.859323i	$-0.329115\pi$
$631$	25.6941	1.02286	0.511432	+	0.859323i	$0.329115\pi$
$632$	−13.0996	−0.521076
$633$	−40.7496	−1.61965
$634$	−29.4976	−1.17150
$635$	13.8987	0.551554
$636$	5.83411	0.231338
$637$	0.457395	0.0181227
$638$	0.509192	0.0201591
$639$	−48.4771	−1.91772
$640$	−1.22094	−0.0482620
$641$	−23.5666	−0.930824	−0.465412	−	0.885094i	$-0.654094\pi$
$641$	−23.5666	−0.930824	−0.465412	+	0.885094i	$0.654094\pi$
$642$	−39.0908	−1.54279
$643$	−12.3993	−0.488979	−0.244489	−	0.969652i	$-0.578620\pi$
$643$	−12.3993	−0.488979	−0.244489	+	0.969652i	$0.578620\pi$
$644$	−14.2322	−0.560829
$645$	19.1761	0.755060
$646$	2.32266	0.0913839
$647$	7.93563	0.311982	0.155991	−	0.987758i	$-0.450143\pi$
$647$	7.93563	0.311982	0.155991	+	0.987758i	$0.450143\pi$
$648$	9.95156	0.390934
$649$	−1.60932	−0.0631713
$650$	2.83183	0.111074
$651$	13.1509	0.515423
$652$	2.59472	0.101617
$653$	2.23194	0.0873424	0.0436712	−	0.999046i	$-0.486095\pi$
$653$	2.23194	0.0873424	0.0436712	+	0.999046i	$0.486095\pi$
$654$	48.3516	1.89070
$655$	−19.7143	−0.770303
$656$	−7.64932	−0.298656
$657$	15.1882	0.592549
$658$	12.4895	0.486892
$659$	−32.6108	−1.27033	−0.635167	−	0.772374i	$-0.719069\pi$
$659$	−32.6108	−1.27033	−0.635167	+	0.772374i	$0.719069\pi$
$660$	0.506390	0.0197112
$661$	2.99151	0.116356	0.0581781	−	0.998306i	$-0.481471\pi$
$661$	2.99151	0.116356	0.0581781	+	0.998306i	$0.481471\pi$
$662$	−11.8249	−0.459589
$663$	5.25572	0.204115
$664$	4.45477	0.172879
$665$	−3.33225	−0.129219
$666$	−20.1022	−0.778944
$667$	−20.7865	−0.804856
$668$	−20.0621	−0.776226
$669$	69.6490	2.69279
$670$	−6.21482	−0.240099
$671$	−0.0919727	−0.00355056
$672$	−7.65317	−0.295227
$673$	12.8025	0.493500	0.246750	−	0.969079i	$-0.420637\pi$
$673$	12.8025	0.493500	0.246750	+	0.969079i	$0.420637\pi$
$674$	−31.5618	−1.21572
$675$	32.8732	1.26529
$676$	−12.3488	−0.474955
$677$	−37.2355	−1.43108	−0.715539	−	0.698573i	$-0.753819\pi$
$677$	−37.2355	−1.43108	−0.715539	+	0.698573i	$0.753819\pi$
$678$	−7.13103	−0.273865
$679$	24.2032	0.928835
$680$	−2.63543	−0.101064
$681$	8.92572	0.342034
$682$	−0.236197	−0.00904444
$683$	34.9802	1.33848	0.669241	−	0.743046i	$-0.266620\pi$
$683$	34.9802	1.33848	0.669241	+	0.743046i	$0.266620\pi$
$684$	6.56872	0.251161
$685$	26.2916	1.00455
$686$	−19.1923	−0.732764
$687$	86.0771	3.28405
$688$	5.20520	0.198446
$689$	1.56025	0.0594407
$690$	−20.6721	−0.786973
$691$	45.2095	1.71985	0.859926	−	0.510419i	$-0.170510\pi$
$691$	45.2095	1.71985	0.859926	+	0.510419i	$0.170510\pi$
$692$	13.8284	0.525676
$693$	2.12826	0.0808460
$694$	8.09057	0.307114
$695$	−0.539128	−0.0204503
$696$	−11.1776	−0.423686
$697$	−16.5112	−0.625408
$698$	−36.9391	−1.39816
$699$	11.8123	0.446781
$700$	−8.90088	−0.336422
$701$	−0.514658	−0.0194384	−0.00971919	−	0.999953i	$-0.503094\pi$
$701$	−0.514658	−0.0194384	−0.00971919	+	0.999953i	$0.503094\pi$
$702$	7.55909	0.285300
$703$	−3.54341	−0.133642
$704$	0.137455	0.00518053
$705$	18.1408	0.683222
$706$	−22.5658	−0.849274
$707$	11.7736	0.442791
$708$	35.3273	1.32768
$709$	−7.66312	−0.287795	−0.143897	−	0.989593i	$-0.545964\pi$
$709$	−7.66312	−0.287795	−0.143897	+	0.989593i	$0.545964\pi$
$710$	9.69574	0.363875
$711$	−79.9669	−2.99899
$712$	12.1306	0.454614
$713$	9.64214	0.361101
$714$	−16.5195	−0.618228
$715$	0.135427	0.00506467
$716$	10.1664	0.379935
$717$	−81.7621	−3.05346
$718$	−20.2189	−0.754563
$719$	39.9815	1.49106	0.745529	−	0.666473i	$-0.232197\pi$
$719$	39.9815	1.49106	0.745529	+	0.666473i	$0.232197\pi$
$720$	−7.45327	−0.277767
$721$	36.5809	1.36234
$722$	−17.8421	−0.664015
$723$	−81.0895	−3.01575
$724$	−0.529361	−0.0196736
$725$	−12.9999	−0.482805
$726$	33.1340	1.22972
$727$	8.90783	0.330373	0.165187	−	0.986262i	$-0.447177\pi$
$727$	8.90783	0.330373	0.165187	+	0.986262i	$0.447177\pi$
$728$	−2.04673	−0.0758568
$729$	−24.0459	−0.890588
$730$	−3.03775	−0.112432
$731$	11.2355	0.415561
$732$	2.01895	0.0746227
$733$	−41.0364	−1.51571	−0.757857	−	0.652420i	$-0.773754\pi$
$733$	−41.0364	−1.51571	−0.757857	+	0.652420i	$0.773754\pi$
$734$	33.4484	1.23460
$735$	−2.08818	−0.0770237
$736$	−5.61126	−0.206834
$737$	0.699671	0.0257727
$738$	−46.6954	−1.71888
$739$	−27.4996	−1.01159	−0.505794	−	0.862654i	$-0.668800\pi$
$739$	−27.4996	−1.01159	−0.505794	+	0.862654i	$0.668800\pi$
$740$	4.02057	0.147799
$741$	2.62002	0.0962490
$742$	−4.90410	−0.180035
$743$	−41.4321	−1.52000	−0.759998	−	0.649925i	$-0.774800\pi$
$743$	−41.4321	−1.52000	−0.759998	+	0.649925i	$0.774800\pi$
$744$	5.18491	0.190088
$745$	−17.5870	−0.644336
$746$	−19.7495	−0.723080
$747$	27.1942	0.994984
$748$	0.296700	0.0108484
$749$	32.8593	1.20065
$750$	−31.3486	−1.14469
$751$	32.2074	1.17526	0.587632	−	0.809128i	$-0.300060\pi$
$751$	32.2074	1.17526	0.587632	+	0.809128i	$0.300060\pi$
$752$	4.92416	0.179566
$753$	57.6996	2.10269
$754$	−2.98929	−0.108863
$755$	16.4342	0.598100
$756$	−23.7594	−0.864120
$757$	51.0877	1.85682	0.928408	−	0.371563i	$-0.121178\pi$
$757$	51.0877	1.85682	0.928408	+	0.371563i	$0.121178\pi$
$758$	18.5701	0.674496
$759$	2.32729	0.0844751
$760$	−1.31379	−0.0476561
$761$	36.2040	1.31239	0.656197	−	0.754590i	$-0.272164\pi$
$761$	36.2040	1.31239	0.656197	+	0.754590i	$0.272164\pi$
$762$	34.3485	1.24431
$763$	−40.6439	−1.47141
$764$	−17.2259	−0.623211
$765$	−16.0880	−0.581665
$766$	−23.9812	−0.866475
$767$	9.44776	0.341139
$768$	−3.01737	−0.108880
$769$	−28.7434	−1.03651	−0.518257	−	0.855225i	$-0.673419\pi$
$769$	−28.7434	−1.03651	−0.518257	+	0.855225i	$0.673419\pi$
$770$	−0.425666	−0.0153399
$771$	42.6416	1.53570
$772$	−5.40764	−0.194625
$773$	42.0247	1.51152	0.755761	−	0.654847i	$-0.227267\pi$
$773$	42.0247	1.51152	0.755761	+	0.654847i	$0.227267\pi$
$774$	31.7752	1.14214
$775$	6.03022	0.216612
$776$	9.54246	0.342554
$777$	25.2019	0.904113
$778$	−12.6126	−0.452183
$779$	−8.23100	−0.294906
$780$	−2.97284	−0.106445
$781$	−1.09156	−0.0390590
$782$	−12.1120	−0.433125
$783$	−34.7011	−1.24011
$784$	−0.566819	−0.0202435
$785$	21.1972	0.756559
$786$	−48.7209	−1.73782
$787$	−35.7797	−1.27541	−0.637705	−	0.770281i	$-0.720116\pi$
$787$	−35.7797	−1.27541	−0.637705	+	0.770281i	$0.720116\pi$
$788$	0.799118	0.0284674
$789$	60.6226	2.15822
$790$	15.9939	0.569038
$791$	5.99427	0.213132
$792$	0.839097	0.0298160
$793$	0.539940	0.0191738
$794$	−5.43803	−0.192989
$795$	−7.12312	−0.252631
$796$	23.0955	0.818600
$797$	−37.1514	−1.31597	−0.657985	−	0.753031i	$-0.728591\pi$
$797$	−37.1514	−1.31597	−0.657985	+	0.753031i	$0.728591\pi$
$798$	−8.23513	−0.291521
$799$	10.6289	0.376024
$800$	−3.50930	−0.124072
$801$	74.0515	2.61648
$802$	−12.0188	−0.424399
$803$	0.341993	0.0120687
$804$	−15.3589	−0.541668
$805$	17.3768	0.612450
$806$	1.38663	0.0488419
$807$	−90.2716	−3.17771
$808$	4.64189	0.163301
$809$	−2.55407	−0.0897963	−0.0448981	−	0.998992i	$-0.514296\pi$
$809$	−2.55407	−0.0897963	−0.0448981	+	0.998992i	$0.514296\pi$
$810$	−12.1503	−0.426918
$811$	45.3194	1.59138	0.795690	−	0.605704i	$-0.207108\pi$
$811$	45.3194	1.59138	0.795690	+	0.605704i	$0.207108\pi$
$812$	9.39579	0.329728
$813$	−56.4940	−1.98133
$814$	−0.452640	−0.0158650
$815$	−3.16801	−0.110970
$816$	−6.51306	−0.228003
$817$	5.60101	0.195955
$818$	−12.3241	−0.430901
$819$	−12.4943	−0.436586
$820$	9.33939	0.326146
$821$	5.30166	0.185029	0.0925146	−	0.995711i	$-0.470510\pi$
$821$	5.30166	0.185029	0.0925146	+	0.995711i	$0.470510\pi$
$822$	64.9755	2.26628
$823$	23.6127	0.823085	0.411543	−	0.911390i	$-0.364990\pi$
$823$	23.6127	0.823085	0.411543	+	0.911390i	$0.364990\pi$
$824$	14.4225	0.502432
$825$	1.45549	0.0506737
$826$	−29.6958	−1.03325
$827$	21.4154	0.744688	0.372344	−	0.928095i	$-0.378554\pi$
$827$	21.4154	0.744688	0.372344	+	0.928095i	$0.378554\pi$
$828$	−34.2540	−1.19041
$829$	−47.1120	−1.63627	−0.818134	−	0.575028i	$-0.804991\pi$
$829$	−47.1120	−1.63627	−0.818134	+	0.575028i	$0.804991\pi$
$830$	−5.43902	−0.188791
$831$	20.6441	0.716136
$832$	−0.806951	−0.0279760
$833$	−1.22349	−0.0423914
$834$	−1.33237	−0.0461362
$835$	24.4947	0.847674
$836$	0.147907	0.00511549
$837$	16.0966	0.556381
$838$	13.0762	0.451710
$839$	−23.6781	−0.817459	−0.408729	−	0.912656i	$-0.634028\pi$
$839$	−23.6781	−0.817459	−0.408729	+	0.912656i	$0.634028\pi$
$840$	9.34409	0.322402
$841$	−15.2773	−0.526802
$842$	20.1913	0.695836
$843$	4.82012	0.166014
$844$	13.5050	0.464862
$845$	15.0772	0.518672
$846$	30.0596	1.03347
$847$	−27.8522	−0.957012
$848$	−1.93351	−0.0663970
$849$	35.0372	1.20247
$850$	−7.57490	−0.259817
$851$	18.4779	0.633414
$852$	23.9615	0.820907
$853$	−55.6344	−1.90489	−0.952443	−	0.304718i	$-0.901438\pi$
$853$	−55.6344	−1.90489	−0.952443	+	0.304718i	$0.901438\pi$
$854$	−1.69711	−0.0580740
$855$	−8.02003	−0.274279
$856$	12.9553	0.442802
$857$	−16.6179	−0.567658	−0.283829	−	0.958875i	$-0.591605\pi$
$857$	−16.6179	−0.567658	−0.283829	+	0.958875i	$0.591605\pi$
$858$	0.334685	0.0114260
$859$	25.9876	0.886684	0.443342	−	0.896353i	$-0.353793\pi$
$859$	25.9876	0.886684	0.443342	+	0.896353i	$0.353793\pi$
$860$	−6.35525	−0.216712
$861$	58.5416	1.99509
$862$	4.48938	0.152909
$863$	−38.7168	−1.31793	−0.658967	−	0.752172i	$-0.729006\pi$
$863$	−38.7168	−1.31793	−0.658967	+	0.752172i	$0.729006\pi$
$864$	−9.36747	−0.318688
$865$	−16.8837	−0.574061
$866$	0.826963	0.0281014
$867$	37.2367	1.26462
$868$	−4.35839	−0.147933
$869$	−1.80061	−0.0610816
$870$	13.6472	0.462684
$871$	−4.10753	−0.139178
$872$	−16.0244	−0.542655
$873$	58.2521	1.97153
$874$	−6.03795	−0.204237
$875$	26.3513	0.890837
$876$	−7.50731	−0.253649
$877$	−47.0422	−1.58850	−0.794251	−	0.607590i	$-0.792137\pi$
$877$	−47.0422	−1.58850	−0.794251	+	0.607590i	$0.792137\pi$
$878$	−3.58838	−0.121102
$879$	13.2848	0.448084
$880$	−0.167825	−0.00565738
$881$	42.9098	1.44567	0.722833	−	0.691023i	$-0.242840\pi$
$881$	42.9098	1.44567	0.722833	+	0.691023i	$0.242840\pi$
$882$	−3.46015	−0.116509
$883$	22.7450	0.765429	0.382715	−	0.923867i	$-0.374989\pi$
$883$	22.7450	0.765429	0.382715	+	0.923867i	$0.374989\pi$
$884$	−1.74182	−0.0585839
$885$	−43.1326	−1.44989
$886$	−16.1827	−0.543669
$887$	−45.3202	−1.52170	−0.760852	−	0.648926i	$-0.775219\pi$
$887$	−45.3202	−1.52170	−0.760852	+	0.648926i	$0.775219\pi$
$888$	9.93620	0.333437
$889$	−28.8730	−0.968370
$890$	−14.8108	−0.496459
$891$	1.36789	0.0458261
$892$	−23.0827	−0.772866
$893$	5.29861	0.177311
$894$	−43.4634	−1.45363
$895$	−12.4126	−0.414906
$896$	2.53637	0.0847342
$897$	−13.6627	−0.456184
$898$	27.3050	0.911178
$899$	−6.36551	−0.212302
$900$	−21.4226	−0.714085
$901$	−4.17353	−0.139040
$902$	−1.05144	−0.0350091
$903$	−39.8363	−1.32567
$904$	2.36333	0.0786030
$905$	0.646320	0.0214844
$906$	40.6144	1.34932
$907$	16.7116	0.554900	0.277450	−	0.960740i	$-0.410511\pi$
$907$	16.7116	0.554900	0.277450	+	0.960740i	$0.410511\pi$
$908$	−2.95811	−0.0981684
$909$	28.3365	0.939863
$910$	2.49894	0.0828391
$911$	−9.91377	−0.328458	−0.164229	−	0.986422i	$-0.552514\pi$
$911$	−9.91377	−0.328458	−0.164229	+	0.986422i	$0.552514\pi$
$912$	−3.24682	−0.107513
$913$	0.612331	0.0202652
$914$	−10.5724	−0.349703
$915$	−2.46503	−0.0814913
$916$	−28.5272	−0.942566
$917$	40.9543	1.35243
$918$	−20.2199	−0.667356
$919$	41.8962	1.38203	0.691014	−	0.722842i	$-0.257164\pi$
$919$	41.8962	1.38203	0.691014	+	0.722842i	$0.257164\pi$
$920$	6.85103	0.225872
$921$	42.8774	1.41286
$922$	34.4944	1.13601
$923$	6.40815	0.210927
$924$	−1.05197	−0.0346072
$925$	11.5561	0.379963
$926$	−7.64417	−0.251203
$927$	88.0424	2.89169
$928$	3.70442	0.121604
$929$	6.89624	0.226258	0.113129	−	0.993580i	$-0.463913\pi$
$929$	6.89624	0.226258	0.113129	+	0.993580i	$0.463913\pi$
$930$	−6.33048	−0.207585
$931$	−0.609921	−0.0199893
$932$	−3.91475	−0.128232
$933$	65.1207	2.13196
$934$	−2.34336	−0.0766770
$935$	−0.362254	−0.0118470
$936$	−4.92605	−0.161013
$937$	53.5823	1.75046	0.875229	−	0.483709i	$-0.160711\pi$
$937$	53.5823	1.75046	0.875229	+	0.483709i	$0.160711\pi$
$938$	12.9106	0.421545
$939$	16.9597	0.553460
$940$	−6.01213	−0.196094
$941$	−26.8872	−0.876497	−0.438248	−	0.898854i	$-0.644401\pi$
$941$	−26.8872	−0.876497	−0.438248	+	0.898854i	$0.644401\pi$
$942$	52.3854	1.70681
$943$	42.9223	1.39774
$944$	−11.7080	−0.381062
$945$	29.0089	0.943658
$946$	0.715481	0.0232623
$947$	−5.49723	−0.178636	−0.0893181	−	0.996003i	$-0.528469\pi$
$947$	−5.49723	−0.178636	−0.0893181	+	0.996003i	$0.528469\pi$
$948$	39.5264	1.28376
$949$	−2.00772	−0.0651734
$950$	−3.77615	−0.122515
$951$	89.0051	2.88619
$952$	5.47482	0.177440
$953$	53.2061	1.72351	0.861757	−	0.507322i	$-0.169365\pi$
$953$	53.2061	1.72351	0.861757	+	0.507322i	$0.169365\pi$
$954$	−11.8031	−0.382141
$955$	21.0318	0.680574
$956$	27.0971	0.876384
$957$	−1.53642	−0.0496654
$958$	−0.134962	−0.00436043
$959$	−54.6178	−1.76370
$960$	3.68404	0.118902
$961$	−28.0473	−0.904750
$962$	2.65729	0.0856746
$963$	79.0856	2.54850
$964$	26.8742	0.865561
$965$	6.60242	0.212539
$966$	42.9439	1.38170
$967$	−28.0150	−0.900903	−0.450451	−	0.892801i	$-0.648737\pi$
$967$	−28.0150	−0.900903	−0.450451	+	0.892801i	$0.648737\pi$
$968$	−10.9811	−0.352946
$969$	−7.00833	−0.225140
$970$	−11.6508	−0.374085
$971$	15.9083	0.510521	0.255261	−	0.966872i	$-0.417839\pi$
$971$	15.9083	0.510521	0.255261	+	0.966872i	$0.417839\pi$
$972$	−1.92512	−0.0617483
$973$	1.11998	0.0359048
$974$	−27.9801	−0.896540
$975$	−8.54468	−0.273649
$976$	−0.669111	−0.0214177
$977$	3.73835	0.119600	0.0598001	−	0.998210i	$-0.480954\pi$
$977$	3.73835	0.119600	0.0598001	+	0.998210i	$0.480954\pi$
$978$	−7.82923	−0.250351
$979$	1.66741	0.0532908
$980$	0.692053	0.0221068
$981$	−97.8213	−3.12320
$982$	35.1814	1.12268
$983$	−22.5899	−0.720507	−0.360253	−	0.932855i	$-0.617310\pi$
$983$	−22.5899	−0.720507	−0.360253	+	0.932855i	$0.617310\pi$
$984$	23.0808	0.735790
$985$	−0.975678	−0.0310877
$986$	7.99608	0.254647
$987$	−37.6855	−1.19954
$988$	−0.868314	−0.0276247
$989$	−29.2077	−0.928752
$990$	−1.02449	−0.0325604
$991$	−9.21541	−0.292737	−0.146369	−	0.989230i	$-0.546759\pi$
$991$	−9.21541	−0.292737	−0.146369	+	0.989230i	$0.546759\pi$
$992$	−1.71836	−0.0545578
$993$	35.6802	1.13228
$994$	−20.1418	−0.638859
$995$	−28.1983	−0.893948
$996$	−13.4417	−0.425916
$997$	−37.0961	−1.17484	−0.587422	−	0.809281i	$-0.699857\pi$
$997$	−37.0961	−1.17484	−0.587422	+	0.809281i	$0.699857\pi$
$998$	12.4147	0.392981
$999$	30.8471	0.975959

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8042.2.a.a.1.5	✓	67

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8042.2.a.a.1.5	✓	67	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:	\( N \)	\(=\)	\( 8042 = 2 \cdot 4021 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -3.01737 q^{3} +1.00000 q^{4} -1.22094 q^{5} -3.01737 q^{6} +2.53637 q^{7} +1.00000 q^{8} +6.10451 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -3.01737 q^{3} +1.00000 q^{4} -1.22094 q^{5} -3.01737 q^{6} +2.53637 q^{7} +1.00000 q^{8} +6.10451 q^{9} -1.22094 q^{10} +0.137455 q^{11} -3.01737 q^{12} -0.806951 q^{13} +2.53637 q^{14} +3.68404 q^{15} +1.00000 q^{16} +2.15852 q^{17} +6.10451 q^{18} +1.07604 q^{19} -1.22094 q^{20} -7.65317 q^{21} +0.137455 q^{22} -5.61126 q^{23} -3.01737 q^{24} -3.50930 q^{25} -0.806951 q^{26} -9.36747 q^{27} +2.53637 q^{28} +3.70442 q^{29} +3.68404 q^{30} -1.71836 q^{31} +1.00000 q^{32} -0.414753 q^{33} +2.15852 q^{34} -3.09677 q^{35} +6.10451 q^{36} -3.29300 q^{37} +1.07604 q^{38} +2.43487 q^{39} -1.22094 q^{40} -7.64932 q^{41} -7.65317 q^{42} +5.20520 q^{43} +0.137455 q^{44} -7.45327 q^{45} -5.61126 q^{46} +4.92416 q^{47} -3.01737 q^{48} -0.566819 q^{49} -3.50930 q^{50} -6.51306 q^{51} -0.806951 q^{52} -1.93351 q^{53} -9.36747 q^{54} -0.167825 q^{55} +2.53637 q^{56} -3.24682 q^{57} +3.70442 q^{58} -11.7080 q^{59} +3.68404 q^{60} -0.669111 q^{61} -1.71836 q^{62} +15.4833 q^{63} +1.00000 q^{64} +0.985242 q^{65} -0.414753 q^{66} +5.09018 q^{67} +2.15852 q^{68} +16.9312 q^{69} -3.09677 q^{70} -7.94119 q^{71} +6.10451 q^{72} +2.48803 q^{73} -3.29300 q^{74} +10.5888 q^{75} +1.07604 q^{76} +0.348637 q^{77} +2.43487 q^{78} -13.0996 q^{79} -1.22094 q^{80} +9.95156 q^{81} -7.64932 q^{82} +4.45477 q^{83} -7.65317 q^{84} -2.63543 q^{85} +5.20520 q^{86} -11.1776 q^{87} +0.137455 q^{88} +12.1306 q^{89} -7.45327 q^{90} -2.04673 q^{91} -5.61126 q^{92} +5.18491 q^{93} +4.92416 q^{94} -1.31379 q^{95} -3.01737 q^{96} +9.54246 q^{97} -0.566819 q^{98} +0.839097 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9}+O(q^{10}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9	\( 67 q + 67 q^{2} - 11 q^{3} + 67 q^{4} - 20 q^{5} - 11 q^{6} - 40 q^{7} + 67 q^{8} + 24 q^{9} - 20 q^{10} - 13 q^{11} - 11 q^{12} - 51 q^{13} - 40 q^{14} - 31 q^{15} + 67 q^{16} - 34 q^{17} + 24 q^{18} - 33 q^{19} - 20 q^{20} - 39 q^{21} - 13 q^{22} - 43 q^{23} - 11 q^{24} - 9 q^{25} - 51 q^{26} - 29 q^{27} - 40 q^{28} - 63 q^{29} - 31 q^{30} - 43 q^{31} + 67 q^{32} - 49 q^{33} - 34 q^{34} - 20 q^{35} + 24 q^{36} - 77 q^{37} - 33 q^{38} - 40 q^{39} - 20 q^{40} - 50 q^{41} - 39 q^{42} - 56 q^{43} - 13 q^{44} - 48 q^{45} - 43 q^{46} - 48 q^{47} - 11 q^{48} + q^{49} - 9 q^{50} - 18 q^{51} - 51 q^{52} - 91 q^{53} - 29 q^{54} - 58 q^{55} - 40 q^{56} - 65 q^{57} - 63 q^{58} - 17 q^{59} - 31 q^{60} - 45 q^{61} - 43 q^{62} - 67 q^{63} + 67 q^{64} - 65 q^{65} - 49 q^{66} - 112 q^{67} - 34 q^{68} - 57 q^{69} - 20 q^{70} - 75 q^{71} + 24 q^{72} - 79 q^{73} - 77 q^{74} - 5 q^{75} - 33 q^{76} - 85 q^{77} - 40 q^{78} - 80 q^{79} - 20 q^{80} - 77 q^{81} - 50 q^{82} - 22 q^{83} - 39 q^{84} - 134 q^{85} - 56 q^{86} - 49 q^{87} - 13 q^{88} - 77 q^{89} - 48 q^{90} - 17 q^{91} - 43 q^{92} - 97 q^{93} - 48 q^{94} - 73 q^{95} - 11 q^{96} - 87 q^{97} + q^{98} - 44 q^{99}+O(q^{100}) \) 67 * q + 67 * q^2 - 11 * q^3 + 67 * q^4 - 20 * q^5 - 11 * q^6 - 40 * q^7 + 67 * q^8 + 24 * q^9 - 20 * q^10 - 13 * q^11 - 11 * q^12 - 51 * q^13 - 40 * q^14 - 31 * q^15 + 67 * q^16 - 34 * q^17 + 24 * q^18 - 33 * q^19 - 20 * q^20 - 39 * q^21 - 13 * q^22 - 43 * q^23 - 11 * q^24 - 9 * q^25 - 51 * q^26 - 29 * q^27 - 40 * q^28 - 63 * q^29 - 31 * q^30 - 43 * q^31 + 67 * q^32 - 49 * q^33 - 34 * q^34 - 20 * q^35 + 24 * q^36 - 77 * q^37 - 33 * q^38 - 40 * q^39 - 20 * q^40 - 50 * q^41 - 39 * q^42 - 56 * q^43 - 13 * q^44 - 48 * q^45 - 43 * q^46 - 48 * q^47 - 11 * q^48 + q^49 - 9 * q^50 - 18 * q^51 - 51 * q^52 - 91 * q^53 - 29 * q^54 - 58 * q^55 - 40 * q^56 - 65 * q^57 - 63 * q^58 - 17 * q^59 - 31 * q^60 - 45 * q^61 - 43 * q^62 - 67 * q^63 + 67 * q^64 - 65 * q^65 - 49 * q^66 - 112 * q^67 - 34 * q^68 - 57 * q^69 - 20 * q^70 - 75 * q^71 + 24 * q^72 - 79 * q^73 - 77 * q^74 - 5 * q^75 - 33 * q^76 - 85 * q^77 - 40 * q^78 - 80 * q^79 - 20 * q^80 - 77 * q^81 - 50 * q^82 - 22 * q^83 - 39 * q^84 - 134 * q^85 - 56 * q^86 - 49 * q^87 - 13 * q^88 - 77 * q^89 - 48 * q^90 - 17 * q^91 - 43 * q^92 - 97 * q^93 - 48 * q^94 - 73 * q^95 - 11 * q^96 - 87 * q^97 + q^98 - 44 * q^99

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