LMFDB - Embedded newform 8007.2.a.f.1.31

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8007,2,Mod(1,8007)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8007, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8007.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8007 = 3 \cdot 17 \cdot 157 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8007.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:	$63.9362168984$
Analytic rank:	$1$
Dimension:	$48$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.31
Character	$\chi$	$=$	8007.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.901623 q^{2} -1.00000 q^{3} -1.18708 q^{4} -2.82523 q^{5} -0.901623 q^{6} -0.878690 q^{7} -2.87354 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.901623 q^{2} -1.00000 q^{3} -1.18708 q^{4} -2.82523 q^{5} -0.901623 q^{6} -0.878690 q^{7} -2.87354 q^{8} +1.00000 q^{9} -2.54729 q^{10} +4.81398 q^{11} +1.18708 q^{12} -5.14267 q^{13} -0.792247 q^{14} +2.82523 q^{15} -0.216697 q^{16} -1.00000 q^{17} +0.901623 q^{18} -0.823997 q^{19} +3.35376 q^{20} +0.878690 q^{21} +4.34039 q^{22} +2.70529 q^{23} +2.87354 q^{24} +2.98191 q^{25} -4.63675 q^{26} -1.00000 q^{27} +1.04307 q^{28} -7.58704 q^{29} +2.54729 q^{30} +6.60216 q^{31} +5.55170 q^{32} -4.81398 q^{33} -0.901623 q^{34} +2.48250 q^{35} -1.18708 q^{36} +7.06009 q^{37} -0.742935 q^{38} +5.14267 q^{39} +8.11840 q^{40} +2.65942 q^{41} +0.792247 q^{42} -2.47244 q^{43} -5.71456 q^{44} -2.82523 q^{45} +2.43915 q^{46} +6.04421 q^{47} +0.216697 q^{48} -6.22790 q^{49} +2.68855 q^{50} +1.00000 q^{51} +6.10474 q^{52} -2.88461 q^{53} -0.901623 q^{54} -13.6006 q^{55} +2.52495 q^{56} +0.823997 q^{57} -6.84065 q^{58} +4.30346 q^{59} -3.35376 q^{60} +12.7081 q^{61} +5.95266 q^{62} -0.878690 q^{63} +5.43894 q^{64} +14.5292 q^{65} -4.34039 q^{66} +5.27025 q^{67} +1.18708 q^{68} -2.70529 q^{69} +2.23828 q^{70} +0.559293 q^{71} -2.87354 q^{72} +8.67777 q^{73} +6.36554 q^{74} -2.98191 q^{75} +0.978147 q^{76} -4.22999 q^{77} +4.63675 q^{78} -11.9113 q^{79} +0.612219 q^{80} +1.00000 q^{81} +2.39780 q^{82} +6.37939 q^{83} -1.04307 q^{84} +2.82523 q^{85} -2.22921 q^{86} +7.58704 q^{87} -13.8332 q^{88} -0.915887 q^{89} -2.54729 q^{90} +4.51881 q^{91} -3.21139 q^{92} -6.60216 q^{93} +5.44960 q^{94} +2.32798 q^{95} -5.55170 q^{96} -0.734920 q^{97} -5.61522 q^{98} +4.81398 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) $ 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * 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 $

$2$	0.901623	0.637544	0.318772	−	0.947831i	$-0.396730\pi$
$2$	0.901623	0.637544	0.318772	+	0.947831i	$0.396730\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.18708	−0.593538
$5$	−2.82523	−1.26348	−0.631740	−	0.775180i	$-0.717659\pi$
$5$	−2.82523	−1.26348	−0.631740	+	0.775180i	$0.717659\pi$
$6$	−0.901623	−0.368086
$7$	−0.878690	−0.332113	−0.166057	−	0.986116i	$-0.553103\pi$
$7$	−0.878690	−0.332113	−0.166057	+	0.986116i	$0.553103\pi$
$8$	−2.87354	−1.01595
$9$	1.00000	0.333333
$10$	−2.54729	−0.805523
$11$	4.81398	1.45147	0.725734	−	0.687975i	$-0.241500\pi$
$11$	4.81398	1.45147	0.725734	+	0.687975i	$0.241500\pi$
$12$	1.18708	0.342679
$13$	−5.14267	−1.42632	−0.713160	−	0.701001i	$-0.752737\pi$
$13$	−5.14267	−1.42632	−0.713160	+	0.701001i	$0.752737\pi$
$14$	−0.792247	−0.211737
$15$	2.82523	0.729470
$16$	−0.216697	−0.0541743
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.901623	0.212515
$19$	−0.823997	−0.189038	−0.0945190	−	0.995523i	$-0.530131\pi$
$19$	−0.823997	−0.189038	−0.0945190	+	0.995523i	$0.530131\pi$
$20$	3.35376	0.749923
$21$	0.878690	0.191746
$22$	4.34039	0.925374
$23$	2.70529	0.564092	0.282046	−	0.959401i	$-0.408987\pi$
$23$	2.70529	0.564092	0.282046	+	0.959401i	$0.408987\pi$
$24$	2.87354	0.586559
$25$	2.98191	0.596381
$26$	−4.63675	−0.909342
$27$	−1.00000	−0.192450
$28$	1.04307	0.197122
$29$	−7.58704	−1.40888	−0.704439	−	0.709765i	$-0.748801\pi$
$29$	−7.58704	−1.40888	−0.704439	+	0.709765i	$0.748801\pi$
$30$	2.54729	0.465069
$31$	6.60216	1.18578	0.592892	−	0.805282i	$-0.297986\pi$
$31$	6.60216	1.18578	0.592892	+	0.805282i	$0.297986\pi$
$32$	5.55170	0.981412
$33$	−4.81398	−0.838006
$34$	−0.901623	−0.154627
$35$	2.48250	0.419619
$36$	−1.18708	−0.197846
$37$	7.06009	1.16067	0.580336	−	0.814377i	$-0.302921\pi$
$37$	7.06009	1.16067	0.580336	+	0.814377i	$0.302921\pi$
$38$	−0.742935	−0.120520
$39$	5.14267	0.823487
$40$	8.11840	1.28363
$41$	2.65942	0.415332	0.207666	−	0.978200i	$-0.433413\pi$
$41$	2.65942	0.415332	0.207666	+	0.978200i	$0.433413\pi$
$42$	0.792247	0.122246
$43$	−2.47244	−0.377044	−0.188522	−	0.982069i	$-0.560370\pi$
$43$	−2.47244	−0.377044	−0.188522	+	0.982069i	$0.560370\pi$
$44$	−5.71456	−0.861502
$45$	−2.82523	−0.421160
$46$	2.43915	0.359633
$47$	6.04421	0.881639	0.440819	−	0.897596i	$-0.354688\pi$
$47$	6.04421	0.881639	0.440819	+	0.897596i	$0.354688\pi$
$48$	0.216697	0.0312776
$49$	−6.22790	−0.889701
$50$	2.68855	0.380219
$51$	1.00000	0.140028
$52$	6.10474	0.846576
$53$	−2.88461	−0.396231	−0.198116	−	0.980179i	$-0.563482\pi$
$53$	−2.88461	−0.396231	−0.198116	+	0.980179i	$0.563482\pi$
$54$	−0.901623	−0.122695
$55$	−13.6006	−1.83390
$56$	2.52495	0.337411
$57$	0.823997	0.109141
$58$	−6.84065	−0.898221
$59$	4.30346	0.560263	0.280132	−	0.959962i	$-0.409622\pi$
$59$	4.30346	0.560263	0.280132	+	0.959962i	$0.409622\pi$
$60$	−3.35376	−0.432968
$61$	12.7081	1.62711	0.813555	−	0.581489i	$-0.197529\pi$
$61$	12.7081	1.62711	0.813555	+	0.581489i	$0.197529\pi$
$62$	5.95266	0.755989
$63$	−0.878690	−0.110704
$64$	5.43894	0.679867
$65$	14.5292	1.80213
$66$	−4.34039	−0.534265
$67$	5.27025	0.643864	0.321932	−	0.946763i	$-0.395668\pi$
$67$	5.27025	0.643864	0.321932	+	0.946763i	$0.395668\pi$
$68$	1.18708	0.143954
$69$	−2.70529	−0.325679
$70$	2.23828	0.267525
$71$	0.559293	0.0663759	0.0331879	−	0.999449i	$-0.489434\pi$
$71$	0.559293	0.0663759	0.0331879	+	0.999449i	$0.489434\pi$
$72$	−2.87354	−0.338650
$73$	8.67777	1.01566	0.507828	−	0.861458i	$-0.330449\pi$
$73$	8.67777	1.01566	0.507828	+	0.861458i	$0.330449\pi$
$74$	6.36554	0.739979
$75$	−2.98191	−0.344321
$76$	0.978147	0.112201
$77$	−4.22999	−0.482052
$78$	4.63675	0.525009
$79$	−11.9113	−1.34012	−0.670062	−	0.742305i	$-0.733732\pi$
$79$	−11.9113	−1.34012	−0.670062	+	0.742305i	$0.733732\pi$
$80$	0.612219	0.0684482
$81$	1.00000	0.111111
$82$	2.39780	0.264792
$83$	6.37939	0.700229	0.350114	−	0.936707i	$-0.386143\pi$
$83$	6.37939	0.700229	0.350114	+	0.936707i	$0.386143\pi$
$84$	−1.04307	−0.113808
$85$	2.82523	0.306439
$86$	−2.22921	−0.240382
$87$	7.58704	0.813416
$88$	−13.8332	−1.47462
$89$	−0.915887	−0.0970838	−0.0485419	−	0.998821i	$-0.515457\pi$
$89$	−0.915887	−0.0970838	−0.0485419	+	0.998821i	$0.515457\pi$
$90$	−2.54729	−0.268508
$91$	4.51881	0.473700
$92$	−3.21139	−0.334810
$93$	−6.60216	−0.684612
$94$	5.44960	0.562083
$95$	2.32798	0.238846
$96$	−5.55170	−0.566618
$97$	−0.734920	−0.0746199	−0.0373099	−	0.999304i	$-0.511879\pi$
$97$	−0.734920	−0.0746199	−0.0373099	+	0.999304i	$0.511879\pi$
$98$	−5.61522	−0.567223
$99$	4.81398	0.483823
$100$	−3.53975	−0.353975
$101$	−6.61199	−0.657917	−0.328959	−	0.944344i	$-0.606698\pi$
$101$	−6.61199	−0.657917	−0.328959	+	0.944344i	$0.606698\pi$
$102$	0.901623	0.0892740
$103$	−0.797810	−0.0786106	−0.0393053	−	0.999227i	$-0.512514\pi$
$103$	−0.797810	−0.0786106	−0.0393053	+	0.999227i	$0.512514\pi$
$104$	14.7777	1.44907
$105$	−2.48250	−0.242267
$106$	−2.60083	−0.252615
$107$	−9.88807	−0.955916	−0.477958	−	0.878383i	$-0.658623\pi$
$107$	−9.88807	−0.955916	−0.477958	+	0.878383i	$0.658623\pi$
$108$	1.18708	0.114226
$109$	7.86949	0.753761	0.376880	−	0.926262i	$-0.376997\pi$
$109$	7.86949	0.753761	0.376880	+	0.926262i	$0.376997\pi$
$110$	−12.2626	−1.16919
$111$	−7.06009	−0.670114
$112$	0.190410	0.0179920
$113$	7.60312	0.715241	0.357621	−	0.933867i	$-0.383588\pi$
$113$	7.60312	0.715241	0.357621	+	0.933867i	$0.383588\pi$
$114$	0.742935	0.0695822
$115$	−7.64306	−0.712719
$116$	9.00639	0.836223
$117$	−5.14267	−0.475440
$118$	3.88010	0.357192
$119$	0.878690	0.0805493
$120$	−8.11840	−0.741105
$121$	12.1744	1.10676
$122$	11.4579	1.03735
$123$	−2.65942	−0.239792
$124$	−7.83727	−0.703808
$125$	5.70157	0.509964
$126$	−0.792247	−0.0705789
$127$	13.5975	1.20659	0.603293	−	0.797519i	$-0.293855\pi$
$127$	13.5975	1.20659	0.603293	+	0.797519i	$0.293855\pi$
$128$	−6.19954	−0.547967
$129$	2.47244	0.217687
$130$	13.0999	1.14893
$131$	−6.85124	−0.598596	−0.299298	−	0.954160i	$-0.596752\pi$
$131$	−6.85124	−0.598596	−0.299298	+	0.954160i	$0.596752\pi$
$132$	5.71456	0.497388
$133$	0.724038	0.0627820
$134$	4.75178	0.410491
$135$	2.82523	0.243157
$136$	2.87354	0.246404
$137$	−13.6270	−1.16424	−0.582119	−	0.813104i	$-0.697776\pi$
$137$	−13.6270	−1.16424	−0.582119	+	0.813104i	$0.697776\pi$
$138$	−2.43915	−0.207634
$139$	2.30778	0.195744	0.0978718	−	0.995199i	$-0.468796\pi$
$139$	2.30778	0.195744	0.0978718	+	0.995199i	$0.468796\pi$
$140$	−2.94691	−0.249060
$141$	−6.04421	−0.509014
$142$	0.504271	0.0423175
$143$	−24.7567	−2.07026
$144$	−0.216697	−0.0180581
$145$	21.4351	1.78009
$146$	7.82408	0.647525
$147$	6.22790	0.513669
$148$	−8.38087	−0.688903
$149$	5.05450	0.414081	0.207040	−	0.978332i	$-0.433617\pi$
$149$	5.05450	0.414081	0.207040	+	0.978332i	$0.433617\pi$
$150$	−2.68855	−0.219520
$151$	−14.5908	−1.18739	−0.593693	−	0.804692i	$-0.702330\pi$
$151$	−14.5908	−1.18739	−0.593693	+	0.804692i	$0.702330\pi$
$152$	2.36779	0.192053
$153$	−1.00000	−0.0808452
$154$	−3.81386	−0.307329
$155$	−18.6526	−1.49821
$156$	−6.10474	−0.488771
$157$	−1.00000	−0.0798087
$157$	−1.00000	−0.0798087
$158$	−10.7395	−0.854387
$159$	2.88461	0.228764
$160$	−15.6848	−1.23999
$161$	−2.37711	−0.187343
$162$	0.901623	0.0708382
$163$	−20.9990	−1.64477	−0.822386	−	0.568931i	$-0.807357\pi$
$163$	−20.9990	−1.64477	−0.822386	+	0.568931i	$0.807357\pi$
$164$	−3.15694	−0.246515
$165$	13.6006	1.05880
$166$	5.75180	0.446426
$167$	−15.0163	−1.16200	−0.580998	−	0.813905i	$-0.697337\pi$
$167$	−15.0163	−1.16200	−0.580998	+	0.813905i	$0.697337\pi$
$168$	−2.52495	−0.194804
$169$	13.4471	1.03439
$170$	2.54729	0.195368
$171$	−0.823997	−0.0630126
$172$	2.93498	0.223790
$173$	−3.09532	−0.235333	−0.117666	−	0.993053i	$-0.537541\pi$
$173$	−3.09532	−0.235333	−0.117666	+	0.993053i	$0.537541\pi$
$174$	6.84065	0.518588
$175$	−2.62017	−0.198066
$176$	−1.04318	−0.0786324
$177$	−4.30346	−0.323468
$178$	−0.825784	−0.0618952
$179$	−18.7948	−1.40479	−0.702394	−	0.711789i	$-0.747885\pi$
$179$	−18.7948	−1.40479	−0.702394	+	0.711789i	$0.747885\pi$
$180$	3.35376	0.249974
$181$	21.5460	1.60150	0.800751	−	0.598998i	$-0.204434\pi$
$181$	21.5460	1.60150	0.800751	+	0.598998i	$0.204434\pi$
$182$	4.07426	0.302005
$183$	−12.7081	−0.939412
$184$	−7.77376	−0.573089
$185$	−19.9464	−1.46649
$186$	−5.95266	−0.436470
$187$	−4.81398	−0.352033
$188$	−7.17494	−0.523286
$189$	0.878690	0.0639153
$190$	2.09896	0.152274
$191$	−14.7815	−1.06955	−0.534775	−	0.844995i	$-0.679604\pi$
$191$	−14.7815	−1.06955	−0.534775	+	0.844995i	$0.679604\pi$
$192$	−5.43894	−0.392521
$193$	−8.44007	−0.607529	−0.303765	−	0.952747i	$-0.598244\pi$
$193$	−8.44007	−0.607529	−0.303765	+	0.952747i	$0.598244\pi$
$194$	−0.662621	−0.0475734
$195$	−14.5292	−1.04046
$196$	7.39300	0.528071
$197$	24.0682	1.71479	0.857396	−	0.514658i	$-0.172081\pi$
$197$	24.0682	1.71479	0.857396	+	0.514658i	$0.172081\pi$
$198$	4.34039	0.308458
$199$	−3.80884	−0.270002	−0.135001	−	0.990845i	$-0.543104\pi$
$199$	−3.80884	−0.270002	−0.135001	+	0.990845i	$0.543104\pi$
$200$	−8.56863	−0.605893
$201$	−5.27025	−0.371735
$202$	−5.96152	−0.419451
$203$	6.66665	0.467907
$204$	−1.18708	−0.0831120
$205$	−7.51347	−0.524764
$206$	−0.719324	−0.0501177
$207$	2.70529	0.188031
$208$	1.11440	0.0772700
$209$	−3.96670	−0.274383
$210$	−2.23828	−0.154456
$211$	−12.9833	−0.893809	−0.446905	−	0.894582i	$-0.647474\pi$
$211$	−12.9833	−0.893809	−0.446905	+	0.894582i	$0.647474\pi$
$212$	3.42425	0.235178
$213$	−0.559293	−0.0383221
$214$	−8.91531	−0.609438
$215$	6.98522	0.476388
$216$	2.87354	0.195520
$217$	−5.80125	−0.393815
$218$	7.09532	0.480555
$219$	−8.67777	−0.586389
$220$	16.1449	1.08849
$221$	5.14267	0.345934
$222$	−6.36554	−0.427227
$223$	−12.2485	−0.820219	−0.410110	−	0.912036i	$-0.634510\pi$
$223$	−12.2485	−0.820219	−0.410110	+	0.912036i	$0.634510\pi$
$224$	−4.87822	−0.325940
$225$	2.98191	0.198794
$226$	6.85514	0.455997
$227$	7.45130	0.494560	0.247280	−	0.968944i	$-0.420463\pi$
$227$	7.45130	0.494560	0.247280	+	0.968944i	$0.420463\pi$
$228$	−0.978147	−0.0647794
$229$	−4.65306	−0.307483	−0.153741	−	0.988111i	$-0.549132\pi$
$229$	−4.65306	−0.307483	−0.153741	+	0.988111i	$0.549132\pi$
$230$	−6.89116	−0.454389
$231$	4.22999	0.278313
$232$	21.8017	1.43135
$233$	−2.11476	−0.138542	−0.0692712	−	0.997598i	$-0.522067\pi$
$233$	−2.11476	−0.138542	−0.0692712	+	0.997598i	$0.522067\pi$
$234$	−4.63675	−0.303114
$235$	−17.0763	−1.11393
$236$	−5.10854	−0.332538
$237$	11.9113	0.773721
$238$	0.792247	0.0513537
$239$	−5.97051	−0.386200	−0.193100	−	0.981179i	$-0.561854\pi$
$239$	−5.97051	−0.386200	−0.193100	+	0.981179i	$0.561854\pi$
$240$	−0.612219	−0.0395186
$241$	0.110401	0.00711155	0.00355578	−	0.999994i	$-0.498868\pi$
$241$	0.110401	0.00711155	0.00355578	+	0.999994i	$0.498868\pi$
$242$	10.9767	0.705608
$243$	−1.00000	−0.0641500
$244$	−15.0855	−0.965751
$245$	17.5952	1.12412
$246$	−2.39780	−0.152878
$247$	4.23755	0.269629
$248$	−18.9716	−1.20470
$249$	−6.37939	−0.404277
$250$	5.14067	0.325125
$251$	−1.45020	−0.0915355	−0.0457678	−	0.998952i	$-0.514573\pi$
$251$	−1.45020	−0.0915355	−0.0457678	+	0.998952i	$0.514573\pi$
$252$	1.04307	0.0657073
$253$	13.0232	0.818762
$254$	12.2599	0.769252
$255$	−2.82523	−0.176923
$256$	−16.4675	−1.02922
$257$	−10.1713	−0.634467	−0.317233	−	0.948347i	$-0.602754\pi$
$257$	−10.1713	−0.634467	−0.317233	+	0.948347i	$0.602754\pi$
$258$	2.22921	0.138785
$259$	−6.20363	−0.385475
$260$	−17.2473	−1.06963
$261$	−7.58704	−0.469626
$262$	−6.17724	−0.381631
$263$	2.42569	0.149575	0.0747873	−	0.997200i	$-0.476172\pi$
$263$	2.42569	0.149575	0.0747873	+	0.997200i	$0.476172\pi$
$264$	13.8332	0.851372
$265$	8.14967	0.500630
$266$	0.652809	0.0400263
$267$	0.915887	0.0560514
$268$	−6.25619	−0.382158
$269$	17.9118	1.09210	0.546049	−	0.837753i	$-0.316131\pi$
$269$	17.9118	1.09210	0.546049	+	0.837753i	$0.316131\pi$
$270$	2.54729	0.155023
$271$	−14.4200	−0.875952	−0.437976	−	0.898987i	$-0.644304\pi$
$271$	−14.4200	−0.875952	−0.437976	+	0.898987i	$0.644304\pi$
$272$	0.216697	0.0131392
$273$	−4.51881	−0.273491
$274$	−12.2865	−0.742252
$275$	14.3548	0.865628
$276$	3.21139	0.193303
$277$	−7.40497	−0.444921	−0.222461	−	0.974942i	$-0.571409\pi$
$277$	−7.40497	−0.444921	−0.222461	+	0.974942i	$0.571409\pi$
$278$	2.08075	0.124795
$279$	6.60216	0.395261
$280$	−7.13356	−0.426312
$281$	−0.470128	−0.0280455	−0.0140227	−	0.999902i	$-0.504464\pi$
$281$	−0.470128	−0.0280455	−0.0140227	+	0.999902i	$0.504464\pi$
$282$	−5.44960	−0.324519
$283$	2.75781	0.163935	0.0819673	−	0.996635i	$-0.473880\pi$
$283$	2.75781	0.163935	0.0819673	+	0.996635i	$0.473880\pi$
$284$	−0.663924	−0.0393966
$285$	−2.32798	−0.137898
$286$	−22.3212	−1.31988
$287$	−2.33681	−0.137937
$288$	5.55170	0.327137
$289$	1.00000	0.0588235
$290$	19.3264	1.13488
$291$	0.734920	0.0430818
$292$	−10.3012	−0.602831
$293$	6.64041	0.387937	0.193968	−	0.981008i	$-0.437864\pi$
$293$	6.64041	0.387937	0.193968	+	0.981008i	$0.437864\pi$
$294$	5.61522	0.327486
$295$	−12.1583	−0.707881
$296$	−20.2875	−1.17918
$297$	−4.81398	−0.279335
$298$	4.55725	0.263994
$299$	−13.9124	−0.804576
$300$	3.53975	0.204368
$301$	2.17251	0.125221
$302$	−13.1554	−0.757010
$303$	6.61199	0.379849
$304$	0.178558	0.0102410
$305$	−35.9033	−2.05582
$306$	−0.901623	−0.0515423
$307$	28.8938	1.64906	0.824528	−	0.565822i	$-0.191441\pi$
$307$	28.8938	1.64906	0.824528	+	0.565822i	$0.191441\pi$
$308$	5.02132	0.286116
$309$	0.797810	0.0453858
$310$	−16.8176	−0.955176
$311$	−4.99039	−0.282979	−0.141490	−	0.989940i	$-0.545189\pi$
$311$	−4.99039	−0.282979	−0.141490	+	0.989940i	$0.545189\pi$
$312$	−14.7777	−0.836621
$313$	−19.2970	−1.09073	−0.545365	−	0.838199i	$-0.683609\pi$
$313$	−19.2970	−1.09073	−0.545365	+	0.838199i	$0.683609\pi$
$314$	−0.901623	−0.0508815
$315$	2.48250	0.139873
$316$	14.1396	0.795415
$317$	−29.5183	−1.65791	−0.828957	−	0.559312i	$-0.811065\pi$
$317$	−29.5183	−1.65791	−0.828957	+	0.559312i	$0.811065\pi$
$318$	2.60083	0.145847
$319$	−36.5238	−2.04494
$320$	−15.3662	−0.858998
$321$	9.88807	0.551898
$322$	−2.14326	−0.119439
$323$	0.823997	0.0458484
$324$	−1.18708	−0.0659487
$325$	−15.3350	−0.850631
$326$	−18.9332	−1.04861
$327$	−7.86949	−0.435184
$328$	−7.64196	−0.421957
$329$	−5.31098	−0.292804
$330$	12.2626	0.675033
$331$	−26.2865	−1.44484	−0.722419	−	0.691456i	$-0.756970\pi$
$331$	−26.2865	−1.44484	−0.722419	+	0.691456i	$0.756970\pi$
$332$	−7.57282	−0.415613
$333$	7.06009	0.386891
$334$	−13.5390	−0.740823
$335$	−14.8897	−0.813509
$336$	−0.190410	−0.0103877
$337$	−4.32529	−0.235614	−0.117807	−	0.993037i	$-0.537586\pi$
$337$	−4.32529	−0.235614	−0.117807	+	0.993037i	$0.537586\pi$
$338$	12.1242	0.659469
$339$	−7.60312	−0.412945
$340$	−3.35376	−0.181883
$341$	31.7827	1.72113
$342$	−0.742935	−0.0401733
$343$	11.6232	0.627595
$344$	7.10467	0.383058
$345$	7.64306	0.411488
$346$	−2.79081	−0.150035
$347$	22.9267	1.23077	0.615384	−	0.788227i	$-0.289001\pi$
$347$	22.9267	1.23077	0.615384	+	0.788227i	$0.289001\pi$
$348$	−9.00639	−0.482793
$349$	0.973027	0.0520850	0.0260425	−	0.999661i	$-0.491709\pi$
$349$	0.973027	0.0520850	0.0260425	+	0.999661i	$0.491709\pi$
$350$	−2.36240	−0.126276
$351$	5.14267	0.274496
$352$	26.7258	1.42449
$353$	−25.9428	−1.38080	−0.690398	−	0.723430i	$-0.742565\pi$
$353$	−25.9428	−1.38080	−0.690398	+	0.723430i	$0.742565\pi$
$354$	−3.88010	−0.206225
$355$	−1.58013	−0.0838646
$356$	1.08723	0.0576229
$357$	−0.878690	−0.0465052
$358$	−16.9458	−0.895613
$359$	33.1794	1.75114	0.875570	−	0.483091i	$-0.160486\pi$
$359$	33.1794	1.75114	0.875570	+	0.483091i	$0.160486\pi$
$360$	8.11840	0.427877
$361$	−18.3210	−0.964265
$362$	19.4264	1.02103
$363$	−12.1744	−0.638989
$364$	−5.36418	−0.281159
$365$	−24.5167	−1.28326
$366$	−11.4579	−0.598916
$367$	17.3166	0.903917	0.451958	−	0.892039i	$-0.350726\pi$
$367$	17.3166	0.903917	0.451958	+	0.892039i	$0.350726\pi$
$368$	−0.586229	−0.0305593
$369$	2.65942	0.138444
$370$	−17.9841	−0.934948
$371$	2.53467	0.131594
$372$	7.83727	0.406344
$373$	−24.7665	−1.28236	−0.641181	−	0.767390i	$-0.721555\pi$
$373$	−24.7665	−1.28236	−0.641181	+	0.767390i	$0.721555\pi$
$374$	−4.34039	−0.224436
$375$	−5.70157	−0.294428
$376$	−17.3683	−0.895701
$377$	39.0177	2.00951
$378$	0.792247	0.0407488
$379$	−21.3604	−1.09721	−0.548605	−	0.836082i	$-0.684841\pi$
$379$	−21.3604	−1.09721	−0.548605	+	0.836082i	$0.684841\pi$
$380$	−2.76349	−0.141764
$381$	−13.5975	−0.696623
$382$	−13.3273	−0.681885
$383$	−7.39582	−0.377909	−0.188954	−	0.981986i	$-0.560510\pi$
$383$	−7.39582	−0.377909	−0.188954	+	0.981986i	$0.560510\pi$
$384$	6.19954	0.316369
$385$	11.9507	0.609063
$386$	−7.60976	−0.387326
$387$	−2.47244	−0.125681
$388$	0.872406	0.0442897
$389$	20.9675	1.06309	0.531547	−	0.847029i	$-0.321611\pi$
$389$	20.9675	1.06309	0.531547	+	0.847029i	$0.321611\pi$
$390$	−13.0999	−0.663338
$391$	−2.70529	−0.136812
$392$	17.8961	0.903891
$393$	6.85124	0.345600
$394$	21.7005	1.09325
$395$	33.6521	1.69322
$396$	−5.71456	−0.287167
$397$	−9.72098	−0.487882	−0.243941	−	0.969790i	$-0.578440\pi$
$397$	−9.72098	−0.487882	−0.243941	+	0.969790i	$0.578440\pi$
$398$	−3.43414	−0.172138
$399$	−0.724038	−0.0362472
$400$	−0.646171	−0.0323086
$401$	−27.8355	−1.39004	−0.695019	−	0.718992i	$-0.744604\pi$
$401$	−27.8355	−1.39004	−0.695019	+	0.718992i	$0.744604\pi$
$402$	−4.75178	−0.236997
$403$	−33.9528	−1.69131
$404$	7.84893	0.390499
$405$	−2.82523	−0.140387
$406$	6.01080	0.298311
$407$	33.9871	1.68468
$408$	−2.87354	−0.142261
$409$	−10.3677	−0.512651	−0.256325	−	0.966591i	$-0.582512\pi$
$409$	−10.3677	−0.512651	−0.256325	+	0.966591i	$0.582512\pi$
$410$	−6.77432	−0.334560
$411$	13.6270	0.672173
$412$	0.947061	0.0466584
$413$	−3.78141	−0.186071
$414$	2.43915	0.119878
$415$	−18.0232	−0.884725
$416$	−28.5506	−1.39981
$417$	−2.30778	−0.113013
$418$	−3.57647	−0.174931
$419$	−7.30960	−0.357098	−0.178549	−	0.983931i	$-0.557140\pi$
$419$	−7.30960	−0.357098	−0.178549	+	0.983931i	$0.557140\pi$
$420$	2.94691	0.143795
$421$	10.1206	0.493249	0.246624	−	0.969111i	$-0.420679\pi$
$421$	10.1206	0.493249	0.246624	+	0.969111i	$0.420679\pi$
$422$	−11.7061	−0.569843
$423$	6.04421	0.293880
$424$	8.28903	0.402551
$425$	−2.98191	−0.144644
$426$	−0.504271	−0.0244320
$427$	−11.1665	−0.540385
$428$	11.7379	0.567373
$429$	24.7567	1.19526
$430$	6.29803	0.303718
$431$	19.7501	0.951331	0.475665	−	0.879626i	$-0.342207\pi$
$431$	19.7501	0.951331	0.475665	+	0.879626i	$0.342207\pi$
$432$	0.216697	0.0104259
$433$	24.0596	1.15623	0.578115	−	0.815955i	$-0.303788\pi$
$433$	24.0596	1.15623	0.578115	+	0.815955i	$0.303788\pi$
$434$	−5.23054	−0.251074
$435$	−21.4351	−1.02773
$436$	−9.34169	−0.447386
$437$	−2.22915	−0.106635
$438$	−7.82408	−0.373849
$439$	6.74427	0.321886	0.160943	−	0.986964i	$-0.448546\pi$
$439$	6.74427	0.321886	0.160943	+	0.986964i	$0.448546\pi$
$440$	39.0818	1.86315
$441$	−6.22790	−0.296567
$442$	4.63675	0.220548
$443$	38.7949	1.84320	0.921601	−	0.388139i	$-0.126882\pi$
$443$	38.7949	1.84320	0.921601	+	0.388139i	$0.126882\pi$
$444$	8.38087	0.397738
$445$	2.58759	0.122663
$446$	−11.0435	−0.522926
$447$	−5.05450	−0.239070
$448$	−4.77914	−0.225793
$449$	−23.2150	−1.09558	−0.547791	−	0.836615i	$-0.684531\pi$
$449$	−23.2150	−1.09558	−0.547791	+	0.836615i	$0.684531\pi$
$450$	2.68855	0.126740
$451$	12.8024	0.602842
$452$	−9.02548	−0.424523
$453$	14.5908	0.685537
$454$	6.71826	0.315304
$455$	−12.7667	−0.598511
$456$	−2.36779	−0.110882
$457$	−24.2862	−1.13606	−0.568030	−	0.823008i	$-0.692294\pi$
$457$	−24.2862	−1.13606	−0.568030	+	0.823008i	$0.692294\pi$
$458$	−4.19530	−0.196034
$459$	1.00000	0.0466760
$460$	9.07289	0.423026
$461$	13.7110	0.638586	0.319293	−	0.947656i	$-0.396555\pi$
$461$	13.7110	0.638586	0.319293	+	0.947656i	$0.396555\pi$
$462$	3.81386	0.177437
$463$	−19.4030	−0.901732	−0.450866	−	0.892592i	$-0.648885\pi$
$463$	−19.4030	−0.901732	−0.450866	+	0.892592i	$0.648885\pi$
$464$	1.64409	0.0763250
$465$	18.6526	0.864994
$466$	−1.90671	−0.0883268
$467$	28.3586	1.31228	0.656139	−	0.754640i	$-0.272188\pi$
$467$	28.3586	1.31228	0.656139	+	0.754640i	$0.272188\pi$
$468$	6.10474	0.282192
$469$	−4.63092	−0.213836
$470$	−15.3963	−0.710181
$471$	1.00000	0.0460776
$472$	−12.3662	−0.569200
$473$	−11.9023	−0.547268
$474$	10.7395	0.493281
$475$	−2.45708	−0.112739
$476$	−1.04307	−0.0478091
$477$	−2.88461	−0.132077
$478$	−5.38315	−0.246219
$479$	−26.1238	−1.19363	−0.596813	−	0.802380i	$-0.703567\pi$
$479$	−26.1238	−1.19363	−0.596813	+	0.802380i	$0.703567\pi$
$480$	15.6848	0.715911
$481$	−36.3077	−1.65549
$482$	0.0995401	0.00453393
$483$	2.37711	0.108162
$484$	−14.4519	−0.656905
$485$	2.07632	0.0942807
$486$	−0.901623	−0.0408984
$487$	−11.7214	−0.531147	−0.265574	−	0.964091i	$-0.585561\pi$
$487$	−11.7214	−0.531147	−0.265574	+	0.964091i	$0.585561\pi$
$488$	−36.5173	−1.65306
$489$	20.9990	0.949609
$490$	15.8643	0.716675
$491$	25.1444	1.13475	0.567375	−	0.823459i	$-0.307959\pi$
$491$	25.1444	1.13475	0.567375	+	0.823459i	$0.307959\pi$
$492$	3.15694	0.142326
$493$	7.58704	0.341703
$494$	3.82067	0.171900
$495$	−13.6006	−0.611300
$496$	−1.43067	−0.0642390
$497$	−0.491445	−0.0220443
$498$	−5.75180	−0.257744
$499$	−20.6991	−0.926620	−0.463310	−	0.886196i	$-0.653338\pi$
$499$	−20.6991	−0.926620	−0.463310	+	0.886196i	$0.653338\pi$
$500$	−6.76820	−0.302683
$501$	15.0163	0.670878
$502$	−1.30753	−0.0583579
$503$	−26.8753	−1.19831	−0.599156	−	0.800632i	$-0.704497\pi$
$503$	−26.8753	−1.19831	−0.599156	+	0.800632i	$0.704497\pi$
$504$	2.52495	0.112470
$505$	18.6804	0.831265
$506$	11.7420	0.521996
$507$	−13.4471	−0.597206
$508$	−16.1413	−0.716155
$509$	43.0113	1.90644	0.953221	−	0.302273i	$-0.0977454\pi$
$509$	43.0113	1.90644	0.953221	+	0.302273i	$0.0977454\pi$
$510$	−2.54729	−0.112796
$511$	−7.62507	−0.337313
$512$	−2.44842	−0.108206
$513$	0.823997	0.0363804
$514$	−9.17066	−0.404500
$515$	2.25399	0.0993228
$516$	−2.93498	−0.129205
$517$	29.0967	1.27967
$518$	−5.59333	−0.245757
$519$	3.09532	0.135869
$520$	−41.7503	−1.83087
$521$	9.88187	0.432933	0.216466	−	0.976290i	$-0.430547\pi$
$521$	9.88187	0.432933	0.216466	+	0.976290i	$0.430547\pi$
$522$	−6.84065	−0.299407
$523$	−35.0848	−1.53415	−0.767075	−	0.641557i	$-0.778289\pi$
$523$	−35.0848	−1.53415	−0.767075	+	0.641557i	$0.778289\pi$
$524$	8.13295	0.355290
$525$	2.62017	0.114354
$526$	2.18706	0.0953603
$527$	−6.60216	−0.287595
$528$	1.04318	0.0453984
$529$	−15.6814	−0.681800
$530$	7.34792	0.319173
$531$	4.30346	0.186754
$532$	−0.859488	−0.0372635
$533$	−13.6765	−0.592397
$534$	0.825784	0.0357352
$535$	27.9360	1.20778
$536$	−15.1443	−0.654134
$537$	18.7948	0.811055
$538$	16.1496	0.696261
$539$	−29.9810	−1.29137
$540$	−3.35376	−0.144323
$541$	26.8771	1.15554	0.577768	−	0.816201i	$-0.303924\pi$
$541$	26.8771	1.15554	0.577768	+	0.816201i	$0.303924\pi$
$542$	−13.0014	−0.558458
$543$	−21.5460	−0.924627
$544$	−5.55170	−0.238027
$545$	−22.2331	−0.952361
$546$	−4.07426	−0.174362
$547$	13.7078	0.586103	0.293052	−	0.956097i	$-0.405329\pi$
$547$	13.7078	0.586103	0.293052	+	0.956097i	$0.405329\pi$
$548$	16.1763	0.691019
$549$	12.7081	0.542370
$550$	12.9426	0.551876
$551$	6.25170	0.266331
$552$	7.77376	0.330873
$553$	10.4663	0.445073
$554$	−6.67649	−0.283657
$555$	19.9464	0.846676
$556$	−2.73952	−0.116181
$557$	−15.1966	−0.643901	−0.321950	−	0.946757i	$-0.604338\pi$
$557$	−15.1966	−0.643901	−0.321950	+	0.946757i	$0.604338\pi$
$558$	5.95266	0.251996
$559$	12.7150	0.537786
$560$	−0.537951	−0.0227326
$561$	4.81398	0.203246
$562$	−0.423878	−0.0178802
$563$	8.62789	0.363622	0.181811	−	0.983333i	$-0.441804\pi$
$563$	8.62789	0.363622	0.181811	+	0.983333i	$0.441804\pi$
$564$	7.17494	0.302119
$565$	−21.4805	−0.903693
$566$	2.48650	0.104515
$567$	−0.878690	−0.0369015
$568$	−1.60715	−0.0674346
$569$	15.7209	0.659054	0.329527	−	0.944146i	$-0.393111\pi$
$569$	15.7209	0.659054	0.329527	+	0.944146i	$0.393111\pi$
$570$	−2.09896	−0.0879157
$571$	−6.26602	−0.262225	−0.131112	−	0.991368i	$-0.541855\pi$
$571$	−6.26602	−0.262225	−0.131112	+	0.991368i	$0.541855\pi$
$572$	29.3881	1.22878
$573$	14.7815	0.617505
$574$	−2.10692	−0.0879411
$575$	8.06692	0.336414
$576$	5.43894	0.226622
$577$	−29.7294	−1.23765	−0.618825	−	0.785529i	$-0.712391\pi$
$577$	−29.7294	−1.23765	−0.618825	+	0.785529i	$0.712391\pi$
$578$	0.901623	0.0375026
$579$	8.44007	0.350757
$580$	−25.4451	−1.05655
$581$	−5.60550	−0.232555
$582$	0.662621	0.0274665
$583$	−13.8864	−0.575117
$584$	−24.9359	−1.03186
$585$	14.5292	0.600709
$586$	5.98714	0.247327
$587$	13.0419	0.538297	0.269148	−	0.963099i	$-0.413258\pi$
$587$	13.0419	0.538297	0.269148	+	0.963099i	$0.413258\pi$
$588$	−7.39300	−0.304882
$589$	−5.44016	−0.224158
$590$	−10.9622	−0.451305
$591$	−24.0682	−0.990035
$592$	−1.52990	−0.0628786
$593$	−33.7402	−1.38554	−0.692772	−	0.721156i	$-0.743611\pi$
$593$	−33.7402	−1.38554	−0.692772	+	0.721156i	$0.743611\pi$
$594$	−4.34039	−0.178088
$595$	−2.48250	−0.101772
$596$	−6.00007	−0.245773
$597$	3.80884	0.155885
$598$	−12.5438	−0.512952
$599$	37.4511	1.53021	0.765104	−	0.643906i	$-0.222687\pi$
$599$	37.4511	1.53021	0.765104	+	0.643906i	$0.222687\pi$
$600$	8.56863	0.349813
$601$	−0.0910357	−0.00371342	−0.00185671	−	0.999998i	$-0.500591\pi$
$601$	−0.0910357	−0.00371342	−0.00185671	+	0.999998i	$0.500591\pi$
$602$	1.95879	0.0798341
$603$	5.27025	0.214621
$604$	17.3204	0.704758
$605$	−34.3954	−1.39837
$606$	5.96152	0.242170
$607$	−0.823536	−0.0334263	−0.0167132	−	0.999860i	$-0.505320\pi$
$607$	−0.823536	−0.0334263	−0.0167132	+	0.999860i	$0.505320\pi$
$608$	−4.57459	−0.185524
$609$	−6.66665	−0.270146
$610$	−32.3713	−1.31067
$611$	−31.0834	−1.25750
$612$	1.18708	0.0479847
$613$	−5.87403	−0.237250	−0.118625	−	0.992939i	$-0.537849\pi$
$613$	−5.87403	−0.237250	−0.118625	+	0.992939i	$0.537849\pi$
$614$	26.0513	1.05134
$615$	7.51347	0.302973
$616$	12.1551	0.489741
$617$	−33.7166	−1.35738	−0.678690	−	0.734425i	$-0.737452\pi$
$617$	−33.7166	−1.35738	−0.678690	+	0.734425i	$0.737452\pi$
$618$	0.719324	0.0289354
$619$	−27.6436	−1.11109	−0.555546	−	0.831486i	$-0.687491\pi$
$619$	−27.6436	−1.11109	−0.555546	+	0.831486i	$0.687491\pi$
$620$	22.1421	0.889247
$621$	−2.70529	−0.108560
$622$	−4.49945	−0.180412
$623$	0.804780	0.0322428
$624$	−1.11440	−0.0446118
$625$	−31.0178	−1.24071
$626$	−17.3986	−0.695388
$627$	3.96670	0.158415
$628$	1.18708	0.0473695
$629$	−7.06009	−0.281504
$630$	2.23828	0.0891751
$631$	19.8445	0.789998	0.394999	−	0.918682i	$-0.370745\pi$
$631$	19.8445	0.789998	0.394999	+	0.918682i	$0.370745\pi$
$632$	34.2275	1.36150
$633$	12.9833	0.516041
$634$	−26.6144	−1.05699
$635$	−38.4161	−1.52450
$636$	−3.42425	−0.135780
$637$	32.0281	1.26900
$638$	−32.9307	−1.30374
$639$	0.559293	0.0221253
$640$	17.5151	0.692345
$641$	23.1925	0.916047	0.458023	−	0.888940i	$-0.348558\pi$
$641$	23.1925	0.916047	0.458023	+	0.888940i	$0.348558\pi$
$642$	8.91531	0.351859
$643$	−30.5209	−1.20363	−0.601814	−	0.798636i	$-0.705555\pi$
$643$	−30.5209	−1.20363	−0.601814	+	0.798636i	$0.705555\pi$
$644$	2.82181	0.111195
$645$	−6.98522	−0.275043
$646$	0.742935	0.0292304
$647$	−17.1517	−0.674305	−0.337152	−	0.941450i	$-0.609464\pi$
$647$	−17.1517	−0.674305	−0.337152	+	0.941450i	$0.609464\pi$
$648$	−2.87354	−0.112883
$649$	20.7168	0.813205
$650$	−13.8264	−0.542314
$651$	5.80125	0.227369
$652$	24.9275	0.976234
$653$	−27.9655	−1.09437	−0.547187	−	0.837011i	$-0.684301\pi$
$653$	−27.9655	−1.09437	−0.547187	+	0.837011i	$0.684301\pi$
$654$	−7.09532	−0.277449
$655$	19.3563	0.756314
$656$	−0.576290	−0.0225003
$657$	8.67777	0.338552
$658$	−4.78850	−0.186675
$659$	−50.6218	−1.97195	−0.985973	−	0.166903i	$-0.946623\pi$
$659$	−50.6218	−1.97195	−0.985973	+	0.166903i	$0.946623\pi$
$660$	−16.1449	−0.628440
$661$	45.8747	1.78432	0.892160	−	0.451719i	$-0.149189\pi$
$661$	45.8747	1.78432	0.892160	+	0.451719i	$0.149189\pi$
$662$	−23.7005	−0.921147
$663$	−5.14267	−0.199725
$664$	−18.3314	−0.711398
$665$	−2.04557	−0.0793238
$666$	6.36554	0.246660
$667$	−20.5251	−0.794737
$668$	17.8255	0.689689
$669$	12.2485	0.473554
$670$	−13.4249	−0.518647
$671$	61.1766	2.36170
$672$	4.87822	0.188182
$673$	16.8397	0.649124	0.324562	−	0.945864i	$-0.394783\pi$
$673$	16.8397	0.649124	0.324562	+	0.945864i	$0.394783\pi$
$674$	−3.89978	−0.150214
$675$	−2.98191	−0.114774
$676$	−15.9627	−0.613950
$677$	−26.7675	−1.02876	−0.514379	−	0.857563i	$-0.671977\pi$
$677$	−26.7675	−1.02876	−0.514379	+	0.857563i	$0.671977\pi$
$678$	−6.85514	−0.263270
$679$	0.645767	0.0247823
$680$	−8.11840	−0.311327
$681$	−7.45130	−0.285534
$682$	28.6560	1.09729
$683$	−3.37662	−0.129203	−0.0646014	−	0.997911i	$-0.520578\pi$
$683$	−3.37662	−0.129203	−0.0646014	+	0.997911i	$0.520578\pi$
$684$	0.978147	0.0374004
$685$	38.4995	1.47099
$686$	10.4798	0.400119
$687$	4.65306	0.177525
$688$	0.535772	0.0204261
$689$	14.8346	0.565153
$690$	6.89116	0.262342
$691$	23.2760	0.885459	0.442729	−	0.896655i	$-0.354010\pi$
$691$	23.2760	0.885459	0.442729	+	0.896655i	$0.354010\pi$
$692$	3.67438	0.139679
$693$	−4.22999	−0.160684
$694$	20.6712	0.784668
$695$	−6.52001	−0.247318
$696$	−21.8017	−0.826390
$697$	−2.65942	−0.100733
$698$	0.877304	0.0332064
$699$	2.11476	0.0799875
$700$	3.11034	0.117560
$701$	29.3464	1.10840	0.554200	−	0.832384i	$-0.313024\pi$
$701$	29.3464	1.10840	0.554200	+	0.832384i	$0.313024\pi$
$702$	4.63675	0.175003
$703$	−5.81750	−0.219411
$704$	26.1829	0.986806
$705$	17.0763	0.643129
$706$	−23.3906	−0.880318
$707$	5.80988	0.218503
$708$	5.10854	0.191991
$709$	33.8224	1.27023	0.635113	−	0.772419i	$-0.280953\pi$
$709$	33.8224	1.27023	0.635113	+	0.772419i	$0.280953\pi$
$710$	−1.42468	−0.0534673
$711$	−11.9113	−0.446708
$712$	2.63184	0.0986323
$713$	17.8608	0.668891
$714$	−0.792247	−0.0296491
$715$	69.9433	2.61573
$716$	22.3108	0.833795
$717$	5.97051	0.222973
$718$	29.9153	1.11643
$719$	3.63404	0.135527	0.0677634	−	0.997701i	$-0.478414\pi$
$719$	3.63404	0.135527	0.0677634	+	0.997701i	$0.478414\pi$
$720$	0.612219	0.0228161
$721$	0.701027	0.0261076
$722$	−16.5187	−0.614761
$723$	−0.110401	−0.00410586
$724$	−25.5767	−0.950552
$725$	−22.6238	−0.840228
$726$	−10.9767	−0.407383
$727$	−20.9761	−0.777960	−0.388980	−	0.921246i	$-0.627172\pi$
$727$	−20.9761	−0.777960	−0.388980	+	0.921246i	$0.627172\pi$
$728$	−12.9850	−0.481256
$729$	1.00000	0.0370370
$730$	−22.1048	−0.818135
$731$	2.47244	0.0914467
$732$	15.0855	0.557577
$733$	13.4772	0.497791	0.248895	−	0.968530i	$-0.419932\pi$
$733$	13.4772	0.497791	0.248895	+	0.968530i	$0.419932\pi$
$734$	15.6130	0.576286
$735$	−17.5952	−0.649010
$736$	15.0190	0.553606
$737$	25.3709	0.934548
$738$	2.39780	0.0882641
$739$	27.2751	1.00333	0.501665	−	0.865062i	$-0.332721\pi$
$739$	27.2751	1.00333	0.501665	+	0.865062i	$0.332721\pi$
$740$	23.6778	0.870415
$741$	−4.23755	−0.155670
$742$	2.28532	0.0838967
$743$	−31.9774	−1.17314	−0.586568	−	0.809900i	$-0.699521\pi$
$743$	−31.9774	−1.17314	−0.586568	+	0.809900i	$0.699521\pi$
$744$	18.9716	0.695532
$745$	−14.2801	−0.523182
$746$	−22.3300	−0.817561
$747$	6.37939	0.233410
$748$	5.71456	0.208945
$749$	8.68855	0.317473
$750$	−5.14067	−0.187711
$751$	27.8372	1.01579	0.507897	−	0.861418i	$-0.330423\pi$
$751$	27.8372	1.01579	0.507897	+	0.861418i	$0.330423\pi$
$752$	−1.30976	−0.0477622
$753$	1.45020	0.0528481
$754$	35.1792	1.28115
$755$	41.2224	1.50024
$756$	−1.04307	−0.0379361
$757$	−1.36797	−0.0497198	−0.0248599	−	0.999691i	$-0.507914\pi$
$757$	−1.36797	−0.0497198	−0.0248599	+	0.999691i	$0.507914\pi$
$758$	−19.2590	−0.699519
$759$	−13.0232	−0.472712
$760$	−6.68954	−0.242655
$761$	46.1362	1.67244	0.836218	−	0.548397i	$-0.184762\pi$
$761$	46.1362	1.67244	0.836218	+	0.548397i	$0.184762\pi$
$762$	−12.2599	−0.444128
$763$	−6.91484	−0.250334
$764$	17.5467	0.634819
$765$	2.82523	0.102146
$766$	−6.66824	−0.240933
$767$	−22.1313	−0.799115
$768$	16.4675	0.594220
$769$	−23.2948	−0.840031	−0.420016	−	0.907517i	$-0.637975\pi$
$769$	−23.2948	−0.840031	−0.420016	+	0.907517i	$0.637975\pi$
$770$	10.7750	0.388304
$771$	10.1713	0.366310
$772$	10.0190	0.360592
$773$	26.7362	0.961635	0.480818	−	0.876821i	$-0.340340\pi$
$773$	26.7362	0.961635	0.480818	+	0.876821i	$0.340340\pi$
$774$	−2.22921	−0.0801274
$775$	19.6870	0.707179
$776$	2.11182	0.0758100
$777$	6.20363	0.222554
$778$	18.9048	0.677768
$779$	−2.19136	−0.0785135
$780$	17.2473	0.617552
$781$	2.69242	0.0963425
$782$	−2.43915	−0.0872239
$783$	7.58704	0.271139
$784$	1.34957	0.0481989
$785$	2.82523	0.100837
$786$	6.17724	0.220335
$787$	45.5050	1.62208	0.811040	−	0.584991i	$-0.198902\pi$
$787$	45.5050	1.62208	0.811040	+	0.584991i	$0.198902\pi$
$788$	−28.5708	−1.01779
$789$	−2.42569	−0.0863569
$790$	30.3415	1.07950
$791$	−6.68078	−0.237541
$792$	−13.8332	−0.491540
$793$	−65.3537	−2.32078
$794$	−8.76466	−0.311046
$795$	−8.14967	−0.289039
$796$	4.52138	0.160256
$797$	5.23937	0.185588	0.0927941	−	0.995685i	$-0.470420\pi$
$797$	5.23937	0.185588	0.0927941	+	0.995685i	$0.470420\pi$
$798$	−0.652809	−0.0231092
$799$	−6.04421	−0.213829
$800$	16.5547	0.585295
$801$	−0.915887	−0.0323613
$802$	−25.0971	−0.886209
$803$	41.7746	1.47419
$804$	6.25619	0.220639
$805$	6.71588	0.236704
$806$	−30.6126	−1.07828
$807$	−17.9118	−0.630524
$808$	18.9998	0.668411
$809$	40.1169	1.41044	0.705218	−	0.708990i	$-0.250849\pi$
$809$	40.1169	1.41044	0.705218	+	0.708990i	$0.250849\pi$
$810$	−2.54729	−0.0895026
$811$	2.53994	0.0891894	0.0445947	−	0.999005i	$-0.485800\pi$
$811$	2.53994	0.0891894	0.0445947	+	0.999005i	$0.485800\pi$
$812$	−7.91382	−0.277721
$813$	14.4200	0.505731
$814$	30.6436	1.07406
$815$	59.3270	2.07813
$816$	−0.216697	−0.00758593
$817$	2.03729	0.0712757
$818$	−9.34777	−0.326837
$819$	4.51881	0.157900
$820$	8.91907	0.311467
$821$	24.8007	0.865551	0.432776	−	0.901502i	$-0.357534\pi$
$821$	24.8007	0.865551	0.432776	+	0.901502i	$0.357534\pi$
$822$	12.2865	0.428539
$823$	−25.7022	−0.895921	−0.447961	−	0.894053i	$-0.647850\pi$
$823$	−25.7022	−0.895921	−0.447961	+	0.894053i	$0.647850\pi$
$824$	2.29254	0.0798644
$825$	−14.3548	−0.499771
$826$	−3.40941	−0.118628
$827$	−30.5023	−1.06067	−0.530335	−	0.847788i	$-0.677934\pi$
$827$	−30.5023	−1.06067	−0.530335	+	0.847788i	$0.677934\pi$
$828$	−3.21139	−0.111603
$829$	2.29307	0.0796417	0.0398209	−	0.999207i	$-0.487321\pi$
$829$	2.29307	0.0796417	0.0398209	+	0.999207i	$0.487321\pi$
$830$	−16.2501	−0.564051
$831$	7.40497	0.256875
$832$	−27.9707	−0.969708
$833$	6.22790	0.215784
$834$	−2.08075	−0.0720505
$835$	42.4244	1.46816
$836$	4.70878	0.162857
$837$	−6.60216	−0.228204
$838$	−6.59051	−0.227665
$839$	31.9689	1.10369	0.551844	−	0.833947i	$-0.313924\pi$
$839$	31.9689	1.10369	0.551844	+	0.833947i	$0.313924\pi$
$840$	7.13356	0.246131
$841$	28.5631	0.984936
$842$	9.12498	0.314468
$843$	0.470128	0.0161921
$844$	15.4122	0.530510
$845$	−37.9910	−1.30693
$846$	5.44960	0.187361
$847$	−10.6975	−0.367570
$848$	0.625087	0.0214656
$849$	−2.75781	−0.0946477
$850$	−2.68855	−0.0922166
$851$	19.0996	0.654726
$852$	0.663924	0.0227456
$853$	22.4462	0.768544	0.384272	−	0.923220i	$-0.374453\pi$
$853$	22.4462	0.768544	0.384272	+	0.923220i	$0.374453\pi$
$854$	−10.0680	−0.344519
$855$	2.32798	0.0796152
$856$	28.4138	0.971163
$857$	−56.9347	−1.94485	−0.972425	−	0.233214i	$-0.925076\pi$
$857$	−56.9347	−1.94485	−0.972425	+	0.233214i	$0.925076\pi$
$858$	22.3212	0.762034
$859$	−18.8210	−0.642166	−0.321083	−	0.947051i	$-0.604047\pi$
$859$	−18.8210	−0.642166	−0.321083	+	0.947051i	$0.604047\pi$
$860$	−8.29198	−0.282754
$861$	2.33681	0.0796382
$862$	17.8072	0.606515
$863$	8.07558	0.274896	0.137448	−	0.990509i	$-0.456110\pi$
$863$	8.07558	0.274896	0.137448	+	0.990509i	$0.456110\pi$
$864$	−5.55170	−0.188873
$865$	8.74498	0.297338
$866$	21.6927	0.737147
$867$	−1.00000	−0.0339618
$868$	6.88653	0.233744
$869$	−57.3406	−1.94515
$870$	−19.3264	−0.655226
$871$	−27.1032	−0.918356
$872$	−22.6133	−0.765783
$873$	−0.734920	−0.0248733
$874$	−2.00985	−0.0679843
$875$	−5.00991	−0.169366
$876$	10.3012	0.348044
$877$	−32.7860	−1.10711	−0.553553	−	0.832814i	$-0.686728\pi$
$877$	−32.7860	−1.10711	−0.553553	+	0.832814i	$0.686728\pi$
$878$	6.08079	0.205217
$879$	−6.64041	−0.223975
$880$	2.94721	0.0993504
$881$	24.3734	0.821162	0.410581	−	0.911824i	$-0.365326\pi$
$881$	24.3734	0.821162	0.410581	+	0.911824i	$0.365326\pi$
$882$	−5.61522	−0.189074
$883$	20.9762	0.705904	0.352952	−	0.935641i	$-0.385178\pi$
$883$	20.9762	0.705904	0.352952	+	0.935641i	$0.385178\pi$
$884$	−6.10474	−0.205325
$885$	12.1583	0.408696
$886$	34.9784	1.17512
$887$	37.1600	1.24771	0.623855	−	0.781540i	$-0.285566\pi$
$887$	37.1600	1.24771	0.623855	+	0.781540i	$0.285566\pi$
$888$	20.2875	0.680803
$889$	−11.9480	−0.400724
$890$	2.33303	0.0782033
$891$	4.81398	0.161274
$892$	14.5399	0.486831
$893$	−4.98041	−0.166663
$894$	−4.55725	−0.152417
$895$	53.0995	1.77492
$896$	5.44747	0.181987
$897$	13.9124	0.464522
$898$	−20.9312	−0.698482
$899$	−50.0909	−1.67062
$900$	−3.53975	−0.117992
$901$	2.88461	0.0961002
$902$	11.5429	0.384338
$903$	−2.17251	−0.0722966
$904$	−21.8479	−0.726649
$905$	−60.8723	−2.02346
$906$	13.1554	0.437060
$907$	−49.6774	−1.64951	−0.824755	−	0.565491i	$-0.808687\pi$
$907$	−49.6774	−1.64951	−0.824755	+	0.565491i	$0.808687\pi$
$908$	−8.84526	−0.293540
$909$	−6.61199	−0.219306
$910$	−11.5107	−0.381577
$911$	23.1227	0.766088	0.383044	−	0.923730i	$-0.374876\pi$
$911$	23.1227	0.766088	0.383044	+	0.923730i	$0.374876\pi$
$912$	−0.178558	−0.00591265
$913$	30.7102	1.01636
$914$	−21.8970	−0.724287
$915$	35.9033	1.18693
$916$	5.52354	0.182503
$917$	6.02012	0.198802
$918$	0.901623	0.0297580
$919$	8.04319	0.265320	0.132660	−	0.991162i	$-0.457648\pi$
$919$	8.04319	0.265320	0.132660	+	0.991162i	$0.457648\pi$
$920$	21.9626	0.724087
$921$	−28.8938	−0.952083
$922$	12.3622	0.407126
$923$	−2.87626	−0.0946733
$924$	−5.02132	−0.165189
$925$	21.0525	0.692203
$926$	−17.4942	−0.574894
$927$	−0.797810	−0.0262035
$928$	−42.1210	−1.38269
$929$	−0.00389609	−0.000127827 0	−6.39133e−5	−	1.00000i	$-0.500020\pi$
$929$	−0.00389609	−0.000127827 0	−6.39133e−5		1.00000i	$0.500020\pi$
$930$	16.8176	0.551471
$931$	5.13178	0.168187
$932$	2.51038	0.0822302
$933$	4.99039	0.163378
$934$	25.5687	0.836635
$935$	13.6006	0.444786
$936$	14.7777	0.483024
$937$	−21.2904	−0.695525	−0.347763	−	0.937583i	$-0.613059\pi$
$937$	−21.2904	−0.695525	−0.347763	+	0.937583i	$0.613059\pi$
$938$	−4.17534	−0.136330
$939$	19.2970	0.629733
$940$	20.2708	0.661162
$941$	2.02523	0.0660206	0.0330103	−	0.999455i	$-0.489491\pi$
$941$	2.02523	0.0660206	0.0330103	+	0.999455i	$0.489491\pi$
$942$	0.901623	0.0293765
$943$	7.19451	0.234286
$944$	−0.932550	−0.0303519
$945$	−2.48250	−0.0807556
$946$	−10.7314	−0.348907
$947$	−28.6393	−0.930653	−0.465326	−	0.885139i	$-0.654063\pi$
$947$	−28.6393	−0.930653	−0.465326	+	0.885139i	$0.654063\pi$
$948$	−14.1396	−0.459233
$949$	−44.6269	−1.44865
$950$	−2.21536	−0.0718758
$951$	29.5183	0.957197
$952$	−2.52495	−0.0818341
$953$	3.55625	0.115198	0.0575991	−	0.998340i	$-0.481655\pi$
$953$	3.55625	0.115198	0.0575991	+	0.998340i	$0.481655\pi$
$954$	−2.60083	−0.0842049
$955$	41.7610	1.35135
$956$	7.08745	0.229225
$957$	36.5238	1.18065
$958$	−23.5538	−0.760988
$959$	11.9739	0.386659
$960$	15.3662	0.495943
$961$	12.5885	0.406082
$962$	−32.7359	−1.05545
$963$	−9.88807	−0.318639
$964$	−0.131054	−0.00422098
$965$	23.8451	0.767601
$966$	2.14326	0.0689582
$967$	−43.3529	−1.39413	−0.697067	−	0.717006i	$-0.745512\pi$
$967$	−43.3529	−1.39413	−0.697067	+	0.717006i	$0.745512\pi$
$968$	−34.9835	−1.12441
$969$	−0.823997	−0.0264706
$970$	1.87205	0.0601080
$971$	−26.9132	−0.863685	−0.431842	−	0.901949i	$-0.642136\pi$
$971$	−26.9132	−0.863685	−0.431842	+	0.901949i	$0.642136\pi$
$972$	1.18708	0.0380755
$973$	−2.02783	−0.0650091
$974$	−10.5683	−0.338629
$975$	15.3350	0.491112
$976$	−2.75382	−0.0881476
$977$	56.1951	1.79784	0.898920	−	0.438112i	$-0.144353\pi$
$977$	56.1951	1.79784	0.898920	+	0.438112i	$0.144353\pi$
$978$	18.9332	0.605417
$979$	−4.40906	−0.140914
$980$	−20.8869	−0.667207
$981$	7.86949	0.251254
$982$	22.6708	0.723453
$983$	29.6813	0.946686	0.473343	−	0.880878i	$-0.343047\pi$
$983$	29.6813	0.946686	0.473343	+	0.880878i	$0.343047\pi$
$984$	7.64196	0.243617
$985$	−67.9982	−2.16660
$986$	6.84065	0.217851
$987$	5.31098	0.169051
$988$	−5.03029	−0.160035
$989$	−6.68868	−0.212688
$990$	−12.2626	−0.389731
$991$	−19.2055	−0.610083	−0.305042	−	0.952339i	$-0.598670\pi$
$991$	−19.2055	−0.610083	−0.305042	+	0.952339i	$0.598670\pi$
$992$	36.6532	1.16374
$993$	26.2865	0.834177
$994$	−0.443098	−0.0140542
$995$	10.7608	0.341141
$996$	7.57282	0.239954
$997$	−29.7547	−0.942340	−0.471170	−	0.882042i	$-0.656168\pi$
$997$	−29.7547	−0.942340	−0.471170	+	0.882042i	$0.656168\pi$
$998$	−18.6628	−0.590761
$999$	−7.06009	−0.223371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8007.2.a.f.1.31	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8007.2.a.f.1.31	✓	48	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:	\( N \)	\(=\)	\( 8007 = 3 \cdot 17 \cdot 157 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8007.a (trivial)

\(f(q)\)	\(=\)	\(q+0.901623 q^{2} -1.00000 q^{3} -1.18708 q^{4} -2.82523 q^{5} -0.901623 q^{6} -0.878690 q^{7} -2.87354 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.901623 q^{2} -1.00000 q^{3} -1.18708 q^{4} -2.82523 q^{5} -0.901623 q^{6} -0.878690 q^{7} -2.87354 q^{8} +1.00000 q^{9} -2.54729 q^{10} +4.81398 q^{11} +1.18708 q^{12} -5.14267 q^{13} -0.792247 q^{14} +2.82523 q^{15} -0.216697 q^{16} -1.00000 q^{17} +0.901623 q^{18} -0.823997 q^{19} +3.35376 q^{20} +0.878690 q^{21} +4.34039 q^{22} +2.70529 q^{23} +2.87354 q^{24} +2.98191 q^{25} -4.63675 q^{26} -1.00000 q^{27} +1.04307 q^{28} -7.58704 q^{29} +2.54729 q^{30} +6.60216 q^{31} +5.55170 q^{32} -4.81398 q^{33} -0.901623 q^{34} +2.48250 q^{35} -1.18708 q^{36} +7.06009 q^{37} -0.742935 q^{38} +5.14267 q^{39} +8.11840 q^{40} +2.65942 q^{41} +0.792247 q^{42} -2.47244 q^{43} -5.71456 q^{44} -2.82523 q^{45} +2.43915 q^{46} +6.04421 q^{47} +0.216697 q^{48} -6.22790 q^{49} +2.68855 q^{50} +1.00000 q^{51} +6.10474 q^{52} -2.88461 q^{53} -0.901623 q^{54} -13.6006 q^{55} +2.52495 q^{56} +0.823997 q^{57} -6.84065 q^{58} +4.30346 q^{59} -3.35376 q^{60} +12.7081 q^{61} +5.95266 q^{62} -0.878690 q^{63} +5.43894 q^{64} +14.5292 q^{65} -4.34039 q^{66} +5.27025 q^{67} +1.18708 q^{68} -2.70529 q^{69} +2.23828 q^{70} +0.559293 q^{71} -2.87354 q^{72} +8.67777 q^{73} +6.36554 q^{74} -2.98191 q^{75} +0.978147 q^{76} -4.22999 q^{77} +4.63675 q^{78} -11.9113 q^{79} +0.612219 q^{80} +1.00000 q^{81} +2.39780 q^{82} +6.37939 q^{83} -1.04307 q^{84} +2.82523 q^{85} -2.22921 q^{86} +7.58704 q^{87} -13.8332 q^{88} -0.915887 q^{89} -2.54729 q^{90} +4.51881 q^{91} -3.21139 q^{92} -6.60216 q^{93} +5.44960 q^{94} +2.32798 q^{95} -5.55170 q^{96} -0.734920 q^{97} -5.61522 q^{98} +4.81398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9}+O(q^{10}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9	\( 48 q - q^{2} - 48 q^{3} + 45 q^{4} + q^{5} + q^{6} - 13 q^{7} - 6 q^{8} + 48 q^{9} - 20 q^{10} + 5 q^{11} - 45 q^{12} - 8 q^{13} + 4 q^{14} - q^{15} + 39 q^{16} - 48 q^{17} - q^{18} - 6 q^{19} + 6 q^{20} + 13 q^{21} - 35 q^{22} - 8 q^{23} + 6 q^{24} + 13 q^{25} + 17 q^{26} - 48 q^{27} - 38 q^{28} + q^{29} + 20 q^{30} - 21 q^{31} - 3 q^{32} - 5 q^{33} + q^{34} + 19 q^{35} + 45 q^{36} - 58 q^{37} - 14 q^{38} + 8 q^{39} - 54 q^{40} - 3 q^{41} - 4 q^{42} - 33 q^{43} + 2 q^{44} + q^{45} - 26 q^{46} + 9 q^{47} - 39 q^{48} + 11 q^{49} + 4 q^{50} + 48 q^{51} - 31 q^{52} - 33 q^{53} + q^{54} - 21 q^{55} + 6 q^{57} - 55 q^{58} + 77 q^{59} - 6 q^{60} - 29 q^{61} - 46 q^{62} - 13 q^{63} + 24 q^{64} - 49 q^{65} + 35 q^{66} - 44 q^{67} - 45 q^{68} + 8 q^{69} + 4 q^{70} + 22 q^{71} - 6 q^{72} - 63 q^{73} - 16 q^{74} - 13 q^{75} - 46 q^{76} - 30 q^{77} - 17 q^{78} - 46 q^{79} - 14 q^{80} + 48 q^{81} - 75 q^{82} + 11 q^{83} + 38 q^{84} - q^{85} + 8 q^{86} - q^{87} - 116 q^{88} + 10 q^{89} - 20 q^{90} - 67 q^{91} - 64 q^{92} + 21 q^{93} - 16 q^{94} - 8 q^{95} + 3 q^{96} - 96 q^{97} - 46 q^{98} + 5 q^{99}+O(q^{100}) \) 48 * q - q^2 - 48 * q^3 + 45 * q^4 + q^5 + q^6 - 13 * q^7 - 6 * q^8 + 48 * q^9 - 20 * q^10 + 5 * q^11 - 45 * q^12 - 8 * q^13 + 4 * q^14 - q^15 + 39 * q^16 - 48 * q^17 - q^18 - 6 * q^19 + 6 * q^20 + 13 * q^21 - 35 * q^22 - 8 * q^23 + 6 * q^24 + 13 * q^25 + 17 * q^26 - 48 * q^27 - 38 * q^28 + q^29 + 20 * q^30 - 21 * q^31 - 3 * q^32 - 5 * q^33 + q^34 + 19 * q^35 + 45 * q^36 - 58 * q^37 - 14 * q^38 + 8 * q^39 - 54 * q^40 - 3 * q^41 - 4 * q^42 - 33 * q^43 + 2 * q^44 + q^45 - 26 * q^46 + 9 * q^47 - 39 * q^48 + 11 * q^49 + 4 * q^50 + 48 * q^51 - 31 * q^52 - 33 * q^53 + q^54 - 21 * q^55 + 6 * q^57 - 55 * q^58 + 77 * q^59 - 6 * q^60 - 29 * q^61 - 46 * q^62 - 13 * q^63 + 24 * q^64 - 49 * q^65 + 35 * q^66 - 44 * q^67 - 45 * q^68 + 8 * q^69 + 4 * q^70 + 22 * q^71 - 6 * q^72 - 63 * q^73 - 16 * q^74 - 13 * q^75 - 46 * q^76 - 30 * q^77 - 17 * q^78 - 46 * q^79 - 14 * q^80 + 48 * q^81 - 75 * q^82 + 11 * q^83 + 38 * q^84 - q^85 + 8 * q^86 - q^87 - 116 * q^88 + 10 * q^89 - 20 * q^90 - 67 * q^91 - 64 * q^92 + 21 * q^93 - 16 * q^94 - 8 * q^95 + 3 * q^96 - 96 * q^97 - 46 * q^98 + 5 * q^99

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