LMFDB - Embedded newform 8027.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8027,2,Mod(1,8027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8027, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8027 = 23 \cdot 349 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8027.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:	$64.0959177025$
Analytic rank:	$0$
Dimension:	$176$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	8027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.77938 q^{2} +1.07296 q^{3} +5.72495 q^{4} +0.733926 q^{5} -2.98217 q^{6} -2.15808 q^{7} -10.3531 q^{8} -1.84875 q^{9} +O(q^{10})$	\(q-2.77938 q^{2} +1.07296 q^{3} +5.72495 q^{4} +0.733926 q^{5} -2.98217 q^{6} -2.15808 q^{7} -10.3531 q^{8} -1.84875 q^{9} -2.03986 q^{10} -2.49200 q^{11} +6.14265 q^{12} -5.66398 q^{13} +5.99813 q^{14} +0.787474 q^{15} +17.3252 q^{16} -6.34198 q^{17} +5.13839 q^{18} -6.25095 q^{19} +4.20170 q^{20} -2.31554 q^{21} +6.92623 q^{22} +1.00000 q^{23} -11.1084 q^{24} -4.46135 q^{25} +15.7424 q^{26} -5.20252 q^{27} -12.3549 q^{28} -4.49732 q^{29} -2.18869 q^{30} -0.531708 q^{31} -27.4472 q^{32} -2.67382 q^{33} +17.6268 q^{34} -1.58387 q^{35} -10.5840 q^{36} +5.87088 q^{37} +17.3738 q^{38} -6.07723 q^{39} -7.59839 q^{40} -10.2857 q^{41} +6.43576 q^{42} +10.2016 q^{43} -14.2666 q^{44} -1.35685 q^{45} -2.77938 q^{46} +0.810322 q^{47} +18.5893 q^{48} -2.34269 q^{49} +12.3998 q^{50} -6.80469 q^{51} -32.4260 q^{52} -1.04820 q^{53} +14.4598 q^{54} -1.82895 q^{55} +22.3428 q^{56} -6.70702 q^{57} +12.4998 q^{58} +8.71970 q^{59} +4.50826 q^{60} -1.25973 q^{61} +1.47782 q^{62} +3.98976 q^{63} +41.6358 q^{64} -4.15695 q^{65} +7.43157 q^{66} +6.08227 q^{67} -36.3075 q^{68} +1.07296 q^{69} +4.40218 q^{70} -9.04273 q^{71} +19.1403 q^{72} -2.30362 q^{73} -16.3174 q^{74} -4.78686 q^{75} -35.7864 q^{76} +5.37795 q^{77} +16.8909 q^{78} -2.45578 q^{79} +12.7154 q^{80} -0.0358428 q^{81} +28.5878 q^{82} +0.414136 q^{83} -13.2563 q^{84} -4.65454 q^{85} -28.3542 q^{86} -4.82545 q^{87} +25.7999 q^{88} -7.75727 q^{89} +3.77120 q^{90} +12.2233 q^{91} +5.72495 q^{92} -0.570502 q^{93} -2.25219 q^{94} -4.58774 q^{95} -29.4498 q^{96} -14.0088 q^{97} +6.51121 q^{98} +4.60711 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) $ 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.77938	−1.96532	−0.982659	−	0.185420i	$-0.940635\pi$
$2$	−2.77938	−1.96532	−0.982659	+	0.185420i	$0.940635\pi$
$3$	1.07296	0.619474	0.309737	−	0.950822i	$-0.399759\pi$
$3$	1.07296	0.619474	0.309737	+	0.950822i	$0.399759\pi$
$4$	5.72495	2.86248
$5$	0.733926	0.328222	0.164111	−	0.986442i	$-0.447525\pi$
$5$	0.733926	0.328222	0.164111	+	0.986442i	$0.447525\pi$
$6$	−2.98217	−1.21746
$7$	−2.15808	−0.815678	−0.407839	−	0.913054i	$-0.633718\pi$
$7$	−2.15808	−0.815678	−0.407839	+	0.913054i	$0.633718\pi$
$8$	−10.3531	−3.66036
$9$	−1.84875	−0.616252
$10$	−2.03986	−0.645060
$11$	−2.49200	−0.751368	−0.375684	−	0.926748i	$-0.622592\pi$
$11$	−2.49200	−0.751368	−0.375684	+	0.926748i	$0.622592\pi$
$12$	6.14265	1.77323
$13$	−5.66398	−1.57091	−0.785453	−	0.618921i	$-0.787570\pi$
$13$	−5.66398	−1.57091	−0.785453	+	0.618921i	$0.787570\pi$
$14$	5.99813	1.60307
$15$	0.787474	0.203325
$16$	17.3252	4.33130
$17$	−6.34198	−1.53816	−0.769078	−	0.639155i	$-0.779284\pi$
$17$	−6.34198	−1.53816	−0.769078	+	0.639155i	$0.779284\pi$
$18$	5.13839	1.21113
$19$	−6.25095	−1.43407	−0.717033	−	0.697039i	$-0.754500\pi$
$19$	−6.25095	−1.43407	−0.717033	+	0.697039i	$0.754500\pi$
$20$	4.20170	0.939528
$21$	−2.31554	−0.505292
$22$	6.92623	1.47668
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−11.1084	−2.26750
$25$	−4.46135	−0.892270
$26$	15.7424	3.08733
$27$	−5.20252	−1.00123
$28$	−12.3549	−2.33486
$29$	−4.49732	−0.835131	−0.417565	−	0.908647i	$-0.637117\pi$
$29$	−4.49732	−0.835131	−0.417565	+	0.908647i	$0.637117\pi$
$30$	−2.18869	−0.399598
$31$	−0.531708	−0.0954977	−0.0477488	−	0.998859i	$-0.515205\pi$
$31$	−0.531708	−0.0954977	−0.0477488	+	0.998859i	$0.515205\pi$
$32$	−27.4472	−4.85202
$33$	−2.67382	−0.465453
$34$	17.6268	3.02297
$35$	−1.58387	−0.267723
$36$	−10.5840	−1.76401
$37$	5.87088	0.965166	0.482583	−	0.875850i	$-0.339699\pi$
$37$	5.87088	0.965166	0.482583	+	0.875850i	$0.339699\pi$
$38$	17.3738	2.81840
$39$	−6.07723	−0.973136
$40$	−7.59839	−1.20141
$41$	−10.2857	−1.60635	−0.803176	−	0.595742i	$-0.796858\pi$
$41$	−10.2857	−1.60635	−0.803176	+	0.595742i	$0.796858\pi$
$42$	6.43576	0.993059
$43$	10.2016	1.55574	0.777868	−	0.628428i	$-0.216301\pi$
$43$	10.2016	1.55574	0.777868	+	0.628428i	$0.216301\pi$
$44$	−14.2666	−2.15077
$45$	−1.35685	−0.202267
$46$	−2.77938	−0.409797
$47$	0.810322	0.118198	0.0590988	−	0.998252i	$-0.481177\pi$
$47$	0.810322	0.118198	0.0590988	+	0.998252i	$0.481177\pi$
$48$	18.5893	2.68313
$49$	−2.34269	−0.334669
$50$	12.3998	1.75360
$51$	−6.80469	−0.952848
$52$	−32.4260	−4.49668
$53$	−1.04820	−0.143981	−0.0719903	−	0.997405i	$-0.522935\pi$
$53$	−1.04820	−0.143981	−0.0719903	+	0.997405i	$0.522935\pi$
$54$	14.4598	1.96773
$55$	−1.82895	−0.246615
$56$	22.3428	2.98568
$57$	−6.70702	−0.888367
$58$	12.4998	1.64130
$59$	8.71970	1.13521	0.567604	−	0.823301i	$-0.307870\pi$
$59$	8.71970	1.13521	0.567604	+	0.823301i	$0.307870\pi$
$60$	4.50826	0.582013
$61$	−1.25973	−0.161292	−0.0806459	−	0.996743i	$-0.525698\pi$
$61$	−1.25973	−0.161292	−0.0806459	+	0.996743i	$0.525698\pi$
$62$	1.47782	0.187683
$63$	3.98976	0.502663
$64$	41.6358	5.20447
$65$	−4.15695	−0.515606
$66$	7.43157	0.914764
$67$	6.08227	0.743068	0.371534	−	0.928419i	$-0.378832\pi$
$67$	6.08227	0.743068	0.371534	+	0.928419i	$0.378832\pi$
$68$	−36.3075	−4.40293
$69$	1.07296	0.129169
$70$	4.40218	0.526162
$71$	−9.04273	−1.07318	−0.536588	−	0.843845i	$-0.680287\pi$
$71$	−9.04273	−1.07318	−0.536588	+	0.843845i	$0.680287\pi$
$72$	19.1403	2.25570
$73$	−2.30362	−0.269619	−0.134809	−	0.990872i	$-0.543042\pi$
$73$	−2.30362	−0.269619	−0.134809	+	0.990872i	$0.543042\pi$
$74$	−16.3174	−1.89686
$75$	−4.78686	−0.552739
$76$	−35.7864	−4.10498
$77$	5.37795	0.612874
$78$	16.8909	1.91252
$79$	−2.45578	−0.276297	−0.138148	−	0.990412i	$-0.544115\pi$
$79$	−2.45578	−0.276297	−0.138148	+	0.990412i	$0.544115\pi$
$80$	12.7154	1.42163
$81$	−0.0358428	−0.00398253
$82$	28.5878	3.15699
$83$	0.414136	0.0454573	0.0227286	−	0.999742i	$-0.492765\pi$
$83$	0.414136	0.0454573	0.0227286	+	0.999742i	$0.492765\pi$
$84$	−13.2563	−1.44639
$85$	−4.65454	−0.504856
$86$	−28.3542	−3.05752
$87$	−4.82545	−0.517342
$88$	25.7999	2.75028
$89$	−7.75727	−0.822269	−0.411135	−	0.911575i	$-0.634867\pi$
$89$	−7.75727	−0.822269	−0.411135	+	0.911575i	$0.634867\pi$
$90$	3.77120	0.397520
$91$	12.2233	1.28135
$92$	5.72495	0.596868
$93$	−0.570502	−0.0591584
$94$	−2.25219	−0.232296
$95$	−4.58774	−0.470692
$96$	−29.4498	−3.00570
$97$	−14.0088	−1.42238	−0.711189	−	0.703001i	$-0.751843\pi$
$97$	−14.0088	−1.42238	−0.711189	+	0.703001i	$0.751843\pi$
$98$	6.51121	0.657732
$99$	4.60711	0.463032
$100$	−25.5410	−2.55410
$101$	−16.1286	−1.60485	−0.802427	−	0.596750i	$-0.796458\pi$
$101$	−16.1286	−1.60485	−0.802427	+	0.596750i	$0.796458\pi$
$102$	18.9128	1.87265
$103$	−3.61298	−0.355997	−0.177999	−	0.984031i	$-0.556962\pi$
$103$	−3.61298	−0.355997	−0.177999	+	0.984031i	$0.556962\pi$
$104$	58.6396	5.75008
$105$	−1.69943	−0.165848
$106$	2.91333	0.282968
$107$	8.62515	0.833824	0.416912	−	0.908947i	$-0.363112\pi$
$107$	8.62515	0.833824	0.416912	+	0.908947i	$0.363112\pi$
$108$	−29.7842	−2.86599
$109$	−2.38299	−0.228249	−0.114125	−	0.993466i	$-0.536406\pi$
$109$	−2.38299	−0.228249	−0.114125	+	0.993466i	$0.536406\pi$
$110$	5.08334	0.484678
$111$	6.29922	0.597896
$112$	−37.3892	−3.53295
$113$	−15.7369	−1.48040	−0.740201	−	0.672385i	$-0.765270\pi$
$113$	−15.7369	−1.48040	−0.740201	+	0.672385i	$0.765270\pi$
$114$	18.6414	1.74592
$115$	0.733926	0.0684390
$116$	−25.7469	−2.39054
$117$	10.4713	0.968073
$118$	−24.2354	−2.23105
$119$	13.6865	1.25464
$120$	−8.15277	−0.744243
$121$	−4.78991	−0.435447
$122$	3.50127	0.316990
$123$	−11.0361	−0.995094
$124$	−3.04401	−0.273360
$125$	−6.94394	−0.621084
$126$	−11.0891	−0.987893
$127$	−8.74485	−0.775980	−0.387990	−	0.921664i	$-0.626831\pi$
$127$	−8.74485	−0.775980	−0.387990	+	0.921664i	$0.626831\pi$
$128$	−60.8273	−5.37642
$129$	10.9460	0.963738
$130$	11.5537	1.01333
$131$	−4.68062	−0.408948	−0.204474	−	0.978872i	$-0.565548\pi$
$131$	−4.68062	−0.408948	−0.204474	+	0.978872i	$0.565548\pi$
$132$	−15.3075	−1.33235
$133$	13.4901	1.16974
$134$	−16.9050	−1.46037
$135$	−3.81827	−0.328624
$136$	65.6589	5.63020
$137$	3.36342	0.287356	0.143678	−	0.989624i	$-0.454107\pi$
$137$	3.36342	0.287356	0.143678	+	0.989624i	$0.454107\pi$
$138$	−2.98217	−0.253859
$139$	−18.1258	−1.53741	−0.768704	−	0.639604i	$-0.779098\pi$
$139$	−18.1258	−1.53741	−0.768704	+	0.639604i	$0.779098\pi$
$140$	−9.06760	−0.766352
$141$	0.869444	0.0732204
$142$	25.1332	2.10913
$143$	14.1147	1.18033
$144$	−32.0300	−2.66917
$145$	−3.30070	−0.274108
$146$	6.40265	0.529887
$147$	−2.51361	−0.207319
$148$	33.6105	2.76277
$149$	13.5979	1.11398	0.556990	−	0.830519i	$-0.311956\pi$
$149$	13.5979	1.11398	0.556990	+	0.830519i	$0.311956\pi$
$150$	13.3045	1.08631
$151$	16.5747	1.34883	0.674415	−	0.738353i	$-0.264396\pi$
$151$	16.5747	1.34883	0.674415	+	0.738353i	$0.264396\pi$
$152$	64.7165	5.24920
$153$	11.7248	0.947891
$154$	−14.9474	−1.20449
$155$	−0.390235	−0.0313444
$156$	−34.7919	−2.78558
$157$	−4.11891	−0.328725	−0.164362	−	0.986400i	$-0.552557\pi$
$157$	−4.11891	−0.328725	−0.164362	+	0.986400i	$0.552557\pi$
$158$	6.82555	0.543011
$159$	−1.12467	−0.0891923
$160$	−20.1442	−1.59254
$161$	−2.15808	−0.170081
$162$	0.0996207	0.00782694
$163$	13.5000	1.05740	0.528701	−	0.848808i	$-0.322679\pi$
$163$	13.5000	1.05740	0.528701	+	0.848808i	$0.322679\pi$
$164$	−58.8850	−4.59815
$165$	−1.96239	−0.152772
$166$	−1.15104	−0.0893380
$167$	−10.8543	−0.839932	−0.419966	−	0.907540i	$-0.637958\pi$
$167$	−10.8543	−0.839932	−0.419966	+	0.907540i	$0.637958\pi$
$168$	23.9729	1.84955
$169$	19.0807	1.46775
$170$	12.9367	0.992203
$171$	11.5565	0.883745
$172$	58.4039	4.45326
$173$	1.92183	0.146114	0.0730572	−	0.997328i	$-0.476724\pi$
$173$	1.92183	0.146114	0.0730572	+	0.997328i	$0.476724\pi$
$174$	13.4117	1.01674
$175$	9.62796	0.727805
$176$	−43.1745	−3.25440
$177$	9.35590	0.703233
$178$	21.5604	1.61602
$179$	15.2725	1.14152	0.570761	−	0.821116i	$-0.306648\pi$
$179$	15.2725	1.14152	0.570761	+	0.821116i	$0.306648\pi$
$180$	−7.76790	−0.578985
$181$	−20.7905	−1.54535	−0.772674	−	0.634803i	$-0.781081\pi$
$181$	−20.7905	−1.54535	−0.772674	+	0.634803i	$0.781081\pi$
$182$	−33.9733	−2.51827
$183$	−1.35164	−0.0999162
$184$	−10.3531	−0.763238
$185$	4.30879	0.316789
$186$	1.58564	0.116265
$187$	15.8042	1.15572
$188$	4.63906	0.338338
$189$	11.2275	0.816678
$190$	12.7511	0.925059
$191$	−5.27262	−0.381514	−0.190757	−	0.981637i	$-0.561094\pi$
$191$	−5.27262	−0.381514	−0.190757	+	0.981637i	$0.561094\pi$
$192$	44.6736	3.22404
$193$	8.48768	0.610957	0.305478	−	0.952199i	$-0.401184\pi$
$193$	8.48768	0.610957	0.305478	+	0.952199i	$0.401184\pi$
$194$	38.9358	2.79542
$195$	−4.46024	−0.319405
$196$	−13.4118	−0.957984
$197$	−25.1915	−1.79482	−0.897410	−	0.441199i	$-0.854553\pi$
$197$	−25.1915	−1.79482	−0.897410	+	0.441199i	$0.854553\pi$
$198$	−12.8049	−0.910004
$199$	−3.70561	−0.262684	−0.131342	−	0.991337i	$-0.541929\pi$
$199$	−3.70561	−0.262684	−0.131342	+	0.991337i	$0.541929\pi$
$200$	46.1887	3.26603
$201$	6.52604	0.460312
$202$	44.8275	3.15405
$203$	9.70557	0.681198
$204$	−38.9566	−2.72751
$205$	−7.54893	−0.527240
$206$	10.0418	0.699648
$207$	−1.84875	−0.128497
$208$	−98.1296	−6.80407
$209$	15.5774	1.07751
$210$	4.72337	0.325944
$211$	−21.3792	−1.47181	−0.735903	−	0.677087i	$-0.763242\pi$
$211$	−21.3792	−1.47181	−0.735903	+	0.677087i	$0.763242\pi$
$212$	−6.00087	−0.412141
$213$	−9.70250	−0.664804
$214$	−23.9726	−1.63873
$215$	7.48725	0.510626
$216$	53.8621	3.66485
$217$	1.14747	0.0778953
$218$	6.62324	0.448583
$219$	−2.47170	−0.167022
$220$	−10.4706	−0.705931
$221$	35.9208	2.41630
$222$	−17.5079	−1.17506
$223$	7.10724	0.475936	0.237968	−	0.971273i	$-0.423519\pi$
$223$	7.10724	0.475936	0.237968	+	0.971273i	$0.423519\pi$
$224$	59.2332	3.95769
$225$	8.24795	0.549863
$226$	43.7388	2.90946
$227$	9.81408	0.651383	0.325692	−	0.945476i	$-0.394403\pi$
$227$	9.81408	0.651383	0.325692	+	0.945476i	$0.394403\pi$
$228$	−38.3974	−2.54293
$229$	−10.8468	−0.716775	−0.358388	−	0.933573i	$-0.616673\pi$
$229$	−10.8468	−0.716775	−0.358388	+	0.933573i	$0.616673\pi$
$230$	−2.03986	−0.134504
$231$	5.77033	0.379660
$232$	46.5610	3.05688
$233$	2.92032	0.191317	0.0956584	−	0.995414i	$-0.469504\pi$
$233$	2.92032	0.191317	0.0956584	+	0.995414i	$0.469504\pi$
$234$	−29.1038	−1.90257
$235$	0.594717	0.0387950
$236$	49.9199	3.24951
$237$	−2.63496	−0.171159
$238$	−38.0400	−2.46577
$239$	7.40108	0.478736	0.239368	−	0.970929i	$-0.423060\pi$
$239$	7.40108	0.478736	0.239368	+	0.970929i	$0.423060\pi$
$240$	13.6431	0.880662
$241$	26.5264	1.70872	0.854358	−	0.519684i	$-0.173950\pi$
$241$	26.5264	1.70872	0.854358	+	0.519684i	$0.173950\pi$
$242$	13.3130	0.855791
$243$	15.5691	0.998759
$244$	−7.21190	−0.461694
$245$	−1.71936	−0.109846
$246$	30.6736	1.95568
$247$	35.4053	2.25278
$248$	5.50481	0.349556
$249$	0.444351	0.0281596
$250$	19.2998	1.22063
$251$	−0.250558	−0.0158151	−0.00790756	−	0.999969i	$-0.502517\pi$
$251$	−0.250558	−0.0158151	−0.00790756	+	0.999969i	$0.502517\pi$
$252$	22.8412	1.43886
$253$	−2.49200	−0.156671
$254$	24.3053	1.52505
$255$	−4.99414	−0.312745
$256$	85.7905	5.36191
$257$	10.7268	0.669118	0.334559	−	0.942375i	$-0.391413\pi$
$257$	10.7268	0.669118	0.334559	+	0.942375i	$0.391413\pi$
$258$	−30.4230	−1.89405
$259$	−12.6698	−0.787265
$260$	−23.7983	−1.47591
$261$	8.31443	0.514651
$262$	13.0092	0.803713
$263$	8.48462	0.523184	0.261592	−	0.965179i	$-0.415752\pi$
$263$	8.48462	0.523184	0.261592	+	0.965179i	$0.415752\pi$
$264$	27.6823	1.70373
$265$	−0.769298	−0.0472576
$266$	−37.4940	−2.29890
$267$	−8.32325	−0.509375
$268$	34.8207	2.12702
$269$	−16.4049	−1.00022	−0.500111	−	0.865961i	$-0.666708\pi$
$269$	−16.4049	−1.00022	−0.500111	+	0.865961i	$0.666708\pi$
$270$	10.6124	0.645852
$271$	24.8720	1.51086	0.755432	−	0.655227i	$-0.227427\pi$
$271$	24.8720	1.51086	0.755432	+	0.655227i	$0.227427\pi$
$272$	−109.876	−6.66221
$273$	13.1152	0.793766
$274$	−9.34821	−0.564746
$275$	11.1177	0.670423
$276$	6.14265	0.369744
$277$	17.1419	1.02996	0.514979	−	0.857203i	$-0.327800\pi$
$277$	17.1419	1.02996	0.514979	+	0.857203i	$0.327800\pi$
$278$	50.3784	3.02150
$279$	0.982998	0.0588506
$280$	16.3979	0.979964
$281$	−23.0529	−1.37522	−0.687610	−	0.726080i	$-0.741340\pi$
$281$	−23.0529	−1.37522	−0.687610	+	0.726080i	$0.741340\pi$
$282$	−2.41652	−0.143901
$283$	−19.4824	−1.15811	−0.579054	−	0.815289i	$-0.696578\pi$
$283$	−19.4824	−1.15811	−0.579054	+	0.815289i	$0.696578\pi$
$284$	−51.7692	−3.07194
$285$	−4.92246	−0.291581
$286$	−39.2300	−2.31972
$287$	22.1973	1.31027
$288$	50.7431	2.99007
$289$	23.2207	1.36592
$290$	9.17390	0.538710
$291$	−15.0309	−0.881126
$292$	−13.1881	−0.771778
$293$	7.29960	0.426447	0.213224	−	0.977003i	$-0.431604\pi$
$293$	7.29960	0.426447	0.213224	+	0.977003i	$0.431604\pi$
$294$	6.98628	0.407448
$295$	6.39962	0.372600
$296$	−60.7816	−3.53286
$297$	12.9647	0.752289
$298$	−37.7936	−2.18933
$299$	−5.66398	−0.327557
$300$	−27.4045	−1.58220
$301$	−22.0160	−1.26898
$302$	−46.0674	−2.65088
$303$	−17.3053	−0.994166
$304$	−108.299	−6.21137
$305$	−0.924549	−0.0529395
$306$	−32.5876	−1.86291
$307$	32.9419	1.88009	0.940047	−	0.341044i	$-0.110780\pi$
$307$	32.9419	1.88009	0.940047	+	0.341044i	$0.110780\pi$
$308$	30.7885	1.75434
$309$	−3.87658	−0.220531
$310$	1.08461	0.0616018
$311$	−14.0869	−0.798793	−0.399397	−	0.916778i	$-0.630780\pi$
$311$	−14.0869	−0.798793	−0.399397	+	0.916778i	$0.630780\pi$
$312$	62.9180	3.56203
$313$	23.2143	1.31215	0.656075	−	0.754695i	$-0.272215\pi$
$313$	23.2143	1.31215	0.656075	+	0.754695i	$0.272215\pi$
$314$	11.4480	0.646049
$315$	2.92819	0.164985
$316$	−14.0592	−0.790893
$317$	−3.85515	−0.216527	−0.108263	−	0.994122i	$-0.534529\pi$
$317$	−3.85515	−0.216527	−0.108263	+	0.994122i	$0.534529\pi$
$318$	3.12589	0.175291
$319$	11.2073	0.627490
$320$	30.5576	1.70822
$321$	9.25445	0.516533
$322$	5.99813	0.334263
$323$	39.6434	2.20582
$324$	−0.205198	−0.0113999
$325$	25.2690	1.40167
$326$	−37.5216	−2.07813
$327$	−2.55686	−0.141395
$328$	106.488	5.87983
$329$	−1.74874	−0.0964112
$330$	5.45423	0.300245
$331$	9.56895	0.525957	0.262979	−	0.964802i	$-0.415295\pi$
$331$	9.56895	0.525957	0.262979	+	0.964802i	$0.415295\pi$
$332$	2.37091	0.130120
$333$	−10.8538	−0.594785
$334$	30.1682	1.65073
$335$	4.46394	0.243891
$336$	−40.1171	−2.18857
$337$	19.9366	1.08602	0.543008	−	0.839727i	$-0.317285\pi$
$337$	19.9366	1.08602	0.543008	+	0.839727i	$0.317285\pi$
$338$	−53.0325	−2.88459
$339$	−16.8851	−0.917072
$340$	−26.6471	−1.44514
$341$	1.32502	0.0717539
$342$	−32.1198	−1.73684
$343$	20.1623	1.08866
$344$	−105.618	−5.69455
$345$	0.787474	0.0423962
$346$	−5.34151	−0.287161
$347$	−30.1310	−1.61751	−0.808757	−	0.588142i	$-0.799859\pi$
$347$	−30.1310	−1.61751	−0.808757	+	0.588142i	$0.799859\pi$
$348$	−27.6255	−1.48088
$349$	−1.00000	−0.0535288
$349$	−1.00000	−0.0535288
$350$	−26.7598	−1.43037
$351$	29.4670	1.57283
$352$	68.3985	3.64565
$353$	−20.7845	−1.10625	−0.553124	−	0.833099i	$-0.686565\pi$
$353$	−20.7845	−1.10625	−0.553124	+	0.833099i	$0.686565\pi$
$354$	−26.0036	−1.38208
$355$	−6.63670	−0.352240
$356$	−44.4100	−2.35373
$357$	14.6851	0.777217
$358$	−42.4481	−2.24345
$359$	−13.8233	−0.729567	−0.364783	−	0.931092i	$-0.618857\pi$
$359$	−13.8233	−0.729567	−0.364783	+	0.931092i	$0.618857\pi$
$360$	14.0476	0.740371
$361$	20.0743	1.05654
$362$	57.7848	3.03710
$363$	−5.13939	−0.269748
$364$	69.9780	3.66785
$365$	−1.69069	−0.0884948
$366$	3.75672	0.196367
$367$	15.7662	0.822990	0.411495	−	0.911412i	$-0.365007\pi$
$367$	15.7662	0.822990	0.411495	+	0.911412i	$0.365007\pi$
$368$	17.3252	0.903138
$369$	19.0157	0.989917
$370$	−11.9758	−0.622591
$371$	2.26209	0.117442
$372$	−3.26610	−0.169339
$373$	26.3868	1.36626	0.683129	−	0.730297i	$-0.260619\pi$
$373$	26.3868	1.36626	0.683129	+	0.730297i	$0.260619\pi$
$374$	−43.9260	−2.27136
$375$	−7.45057	−0.384746
$376$	−8.38932	−0.432646
$377$	25.4727	1.31191
$378$	−31.2054	−1.60503
$379$	−7.81830	−0.401599	−0.200800	−	0.979632i	$-0.564354\pi$
$379$	−7.81830	−0.401599	−0.200800	+	0.979632i	$0.564354\pi$
$380$	−26.2646	−1.34734
$381$	−9.38289	−0.480700
$382$	14.6546	0.749796
$383$	2.84229	0.145234	0.0726172	−	0.997360i	$-0.476865\pi$
$383$	2.84229	0.145234	0.0726172	+	0.997360i	$0.476865\pi$
$384$	−65.2653	−3.33055
$385$	3.94702	0.201159
$386$	−23.5905	−1.20072
$387$	−18.8603	−0.958724
$388$	−80.1997	−4.07152
$389$	−29.4065	−1.49097	−0.745484	−	0.666524i	$-0.767781\pi$
$389$	−29.4065	−1.49097	−0.745484	+	0.666524i	$0.767781\pi$
$390$	12.3967	0.627732
$391$	−6.34198	−0.320728
$392$	24.2540	1.22501
$393$	−5.02212	−0.253333
$394$	70.0167	3.52739
$395$	−1.80236	−0.0906867
$396$	26.3755	1.32542
$397$	−20.5940	−1.03358	−0.516791	−	0.856112i	$-0.672873\pi$
$397$	−20.5940	−1.03358	−0.516791	+	0.856112i	$0.672873\pi$
$398$	10.2993	0.516257
$399$	14.4743	0.724621
$400$	−77.2938	−3.86469
$401$	2.43088	0.121392	0.0606962	−	0.998156i	$-0.480668\pi$
$401$	2.43088	0.121392	0.0606962	+	0.998156i	$0.480668\pi$
$402$	−18.1384	−0.904659
$403$	3.01159	0.150018
$404$	−92.3354	−4.59386
$405$	−0.0263060	−0.00130715
$406$	−26.9755	−1.33877
$407$	−14.6303	−0.725195
$408$	70.4494	3.48777
$409$	−22.3476	−1.10502	−0.552510	−	0.833506i	$-0.686330\pi$
$409$	−22.3476	−1.10502	−0.552510	+	0.833506i	$0.686330\pi$
$410$	20.9813	1.03619
$411$	3.60882	0.178010
$412$	−20.6841	−1.01903
$413$	−18.8178	−0.925965
$414$	5.13839	0.252538
$415$	0.303945	0.0149201
$416$	155.460	7.62207
$417$	−19.4483	−0.952385
$418$	−43.2955	−2.11765
$419$	−34.7494	−1.69762	−0.848810	−	0.528698i	$-0.822680\pi$
$419$	−34.7494	−1.69762	−0.848810	+	0.528698i	$0.822680\pi$
$420$	−9.72918	−0.474735
$421$	2.64290	0.128807	0.0644036	−	0.997924i	$-0.479486\pi$
$421$	2.64290	0.128807	0.0644036	+	0.997924i	$0.479486\pi$
$422$	59.4210	2.89257
$423$	−1.49809	−0.0728395
$424$	10.8520	0.527021
$425$	28.2938	1.37245
$426$	26.9669	1.30655
$427$	2.71860	0.131562
$428$	49.3786	2.38680
$429$	15.1445	0.731183
$430$	−20.8099	−1.00354
$431$	−0.616863	−0.0297132	−0.0148566	−	0.999890i	$-0.504729\pi$
$431$	−0.616863	−0.0297132	−0.0148566	+	0.999890i	$0.504729\pi$
$432$	−90.1348	−4.33661
$433$	22.7536	1.09347	0.546733	−	0.837307i	$-0.315871\pi$
$433$	22.7536	1.09347	0.546733	+	0.837307i	$0.315871\pi$
$434$	−3.18926	−0.153089
$435$	−3.54152	−0.169803
$436$	−13.6425	−0.653358
$437$	−6.25095	−0.299023
$438$	6.86979	0.328251
$439$	−19.2503	−0.918765	−0.459383	−	0.888239i	$-0.651929\pi$
$439$	−19.2503	−0.918765	−0.459383	+	0.888239i	$0.651929\pi$
$440$	18.9352	0.902701
$441$	4.33105	0.206241
$442$	−99.8377	−4.74879
$443$	−21.1932	−1.00692	−0.503461	−	0.864018i	$-0.667940\pi$
$443$	−21.1932	−1.00692	−0.503461	+	0.864018i	$0.667940\pi$
$444$	36.0628	1.71146
$445$	−5.69327	−0.269887
$446$	−19.7537	−0.935365
$447$	14.5900	0.690082
$448$	−89.8534	−4.24517
$449$	30.8234	1.45465	0.727323	−	0.686296i	$-0.240764\pi$
$449$	30.8234	1.45465	0.727323	+	0.686296i	$0.240764\pi$
$450$	−22.9242	−1.08066
$451$	25.6319	1.20696
$452$	−90.0930	−4.23762
$453$	17.7840	0.835565
$454$	−27.2771	−1.28018
$455$	8.97103	0.420568
$456$	69.4383	3.25174
$457$	25.1681	1.17731	0.588656	−	0.808383i	$-0.299657\pi$
$457$	25.1681	1.17731	0.588656	+	0.808383i	$0.299657\pi$
$458$	30.1473	1.40869
$459$	32.9943	1.54004
$460$	4.20170	0.195905
$461$	−26.5726	−1.23761	−0.618805	−	0.785545i	$-0.712383\pi$
$461$	−26.5726	−1.23761	−0.618805	+	0.785545i	$0.712383\pi$
$462$	−16.0379	−0.746153
$463$	21.7853	1.01245	0.506224	−	0.862402i	$-0.331041\pi$
$463$	21.7853	1.01245	0.506224	+	0.862402i	$0.331041\pi$
$464$	−77.9169	−3.61720
$465$	−0.418707	−0.0194171
$466$	−8.11669	−0.375998
$467$	−23.7522	−1.09912	−0.549561	−	0.835454i	$-0.685205\pi$
$467$	−23.7522	−1.09912	−0.549561	+	0.835454i	$0.685205\pi$
$468$	59.9478	2.77109
$469$	−13.1260	−0.606104
$470$	−1.65294	−0.0762446
$471$	−4.41943	−0.203636
$472$	−90.2757	−4.15528
$473$	−25.4225	−1.16893
$474$	7.32355	0.336382
$475$	27.8877	1.27957
$476$	78.3546	3.59138
$477$	1.93786	0.0887283
$478$	−20.5704	−0.940868
$479$	26.0076	1.18832	0.594160	−	0.804347i	$-0.297485\pi$
$479$	26.0076	1.18832	0.594160	+	0.804347i	$0.297485\pi$
$480$	−21.6140	−0.986537
$481$	−33.2526	−1.51619
$482$	−73.7270	−3.35817
$483$	−2.31554	−0.105361
$484$	−27.4220	−1.24646
$485$	−10.2814	−0.466855
$486$	−43.2725	−1.96288
$487$	−34.8165	−1.57769	−0.788844	−	0.614594i	$-0.789320\pi$
$487$	−34.8165	−1.57769	−0.788844	+	0.614594i	$0.789320\pi$
$488$	13.0421	0.590387
$489$	14.4850	0.655033
$490$	4.77875	0.215882
$491$	−11.9085	−0.537421	−0.268711	−	0.963221i	$-0.586598\pi$
$491$	−11.9085	−0.537421	−0.268711	+	0.963221i	$0.586598\pi$
$492$	−63.1813	−2.84843
$493$	28.5219	1.28456
$494$	−98.4047	−4.42744
$495$	3.38128	0.151977
$496$	−9.21195	−0.413629
$497$	19.5150	0.875365
$498$	−1.23502	−0.0553426
$499$	−4.57926	−0.204996	−0.102498	−	0.994733i	$-0.532684\pi$
$499$	−4.57926	−0.204996	−0.102498	+	0.994733i	$0.532684\pi$
$500$	−39.7537	−1.77784
$501$	−11.6462	−0.520316
$502$	0.696397	0.0310817
$503$	−39.4638	−1.75960	−0.879802	−	0.475339i	$-0.842325\pi$
$503$	−39.4638	−1.75960	−0.879802	+	0.475339i	$0.842325\pi$
$504$	−41.3063	−1.83993
$505$	−11.8372	−0.526748
$506$	6.92623	0.307908
$507$	20.4728	0.909231
$508$	−50.0639	−2.22123
$509$	−34.6796	−1.53715	−0.768573	−	0.639762i	$-0.779033\pi$
$509$	−34.6796	−1.53715	−0.768573	+	0.639762i	$0.779033\pi$
$510$	13.8806	0.614644
$511$	4.97141	0.219922
$512$	−116.790	−5.16144
$513$	32.5207	1.43582
$514$	−29.8138	−1.31503
$515$	−2.65166	−0.116846
$516$	62.6651	2.75868
$517$	−2.01933	−0.0888099
$518$	35.2143	1.54723
$519$	2.06205	0.0905141
$520$	43.0371	1.88730
$521$	−8.22397	−0.360299	−0.180149	−	0.983639i	$-0.557658\pi$
$521$	−8.22397	−0.360299	−0.180149	+	0.983639i	$0.557658\pi$
$522$	−23.1090	−1.01145
$523$	9.77095	0.427254	0.213627	−	0.976915i	$-0.431472\pi$
$523$	9.77095	0.427254	0.213627	+	0.976915i	$0.431472\pi$
$524$	−26.7963	−1.17060
$525$	10.3304	0.450857
$526$	−23.5820	−1.02822
$527$	3.37208	0.146890
$528$	−46.3245	−2.01602
$529$	1.00000	0.0434783
$530$	2.13817	0.0928763
$531$	−16.1206	−0.699574
$532$	77.2299	3.34834
$533$	58.2579	2.52343
$534$	23.1335	1.00108
$535$	6.33022	0.273679
$536$	−62.9702	−2.71990
$537$	16.3868	0.707144
$538$	45.5954	1.96576
$539$	5.83798	0.251460
$540$	−21.8594	−0.940680
$541$	−2.26469	−0.0973667	−0.0486833	−	0.998814i	$-0.515503\pi$
$541$	−2.26469	−0.0973667	−0.0486833	+	0.998814i	$0.515503\pi$
$542$	−69.1286	−2.96933
$543$	−22.3074	−0.957304
$544$	174.069	7.46316
$545$	−1.74894	−0.0749164
$546$	−36.4520	−1.56000
$547$	−16.6070	−0.710063	−0.355032	−	0.934854i	$-0.615530\pi$
$547$	−16.6070	−0.710063	−0.355032	+	0.934854i	$0.615530\pi$
$548$	19.2554	0.822550
$549$	2.32893	0.0993964
$550$	−30.9003	−1.31760
$551$	28.1125	1.19763
$552$	−11.1084	−0.472806
$553$	5.29977	0.225369
$554$	−47.6439	−2.02419
$555$	4.62317	0.196242
$556$	−103.769	−4.40080
$557$	0.177868	0.00753652	0.00376826	−	0.999993i	$-0.498801\pi$
$557$	0.177868	0.00753652	0.00376826	+	0.999993i	$0.498801\pi$
$558$	−2.73213	−0.115660
$559$	−57.7819	−2.44391
$560$	−27.4409	−1.15959
$561$	16.9573	0.715939
$562$	64.0727	2.70275
$563$	23.6017	0.994692	0.497346	−	0.867552i	$-0.334308\pi$
$563$	23.6017	0.994692	0.497346	+	0.867552i	$0.334308\pi$
$564$	4.97753	0.209592
$565$	−11.5497	−0.485901
$566$	54.1489	2.27605
$567$	0.0773516	0.00324846
$568$	93.6200	3.92821
$569$	−23.9893	−1.00568	−0.502842	−	0.864378i	$-0.667712\pi$
$569$	−23.9893	−1.00568	−0.502842	+	0.864378i	$0.667712\pi$
$570$	13.6814	0.573050
$571$	7.76209	0.324833	0.162417	−	0.986722i	$-0.448071\pi$
$571$	7.76209	0.324833	0.162417	+	0.986722i	$0.448071\pi$
$572$	80.8059	3.37866
$573$	−5.65732	−0.236338
$574$	−61.6948	−2.57509
$575$	−4.46135	−0.186051
$576$	−76.9743	−3.20726
$577$	−42.3041	−1.76114	−0.880570	−	0.473915i	$-0.842840\pi$
$577$	−42.3041	−1.76114	−0.880570	+	0.473915i	$0.842840\pi$
$578$	−64.5391	−2.68447
$579$	9.10695	0.378472
$580$	−18.8964	−0.784628
$581$	−0.893738	−0.0370785
$582$	41.7766	1.73169
$583$	2.61211	0.108182
$584$	23.8496	0.986903
$585$	7.68517	0.317743
$586$	−20.2884	−0.838105
$587$	−12.7211	−0.525055	−0.262528	−	0.964925i	$-0.584556\pi$
$587$	−12.7211	−0.525055	−0.262528	+	0.964925i	$0.584556\pi$
$588$	−14.3903	−0.593446
$589$	3.32368	0.136950
$590$	−17.7870	−0.732278
$591$	−27.0295	−1.11184
$592$	101.714	4.18043
$593$	35.3328	1.45094	0.725472	−	0.688252i	$-0.241622\pi$
$593$	35.3328	1.45094	0.725472	+	0.688252i	$0.241622\pi$
$594$	−36.0339	−1.47849
$595$	10.0449	0.411800
$596$	77.8471	3.18874
$597$	−3.97597	−0.162726
$598$	15.7424	0.643753
$599$	15.6172	0.638101	0.319050	−	0.947738i	$-0.396636\pi$
$599$	15.6172	0.638101	0.319050	+	0.947738i	$0.396636\pi$
$600$	49.5586	2.02322
$601$	−23.7703	−0.969611	−0.484806	−	0.874622i	$-0.661110\pi$
$601$	−23.7703	−0.969611	−0.484806	+	0.874622i	$0.661110\pi$
$602$	61.1907	2.49395
$603$	−11.2446	−0.457917
$604$	94.8894	3.86099
$605$	−3.51544	−0.142923
$606$	48.0981	1.95385
$607$	11.6679	0.473584	0.236792	−	0.971560i	$-0.423904\pi$
$607$	11.6679	0.473584	0.236792	+	0.971560i	$0.423904\pi$
$608$	171.571	6.95812
$609$	10.4137	0.421985
$610$	2.56967	0.104043
$611$	−4.58965	−0.185677
$612$	67.1237	2.71332
$613$	−4.56271	−0.184286	−0.0921431	−	0.995746i	$-0.529372\pi$
$613$	−4.56271	−0.184286	−0.0921431	+	0.995746i	$0.529372\pi$
$614$	−91.5581	−3.69499
$615$	−8.09970	−0.326612
$616$	−55.6783	−2.24334
$617$	4.45037	0.179165	0.0895825	−	0.995979i	$-0.471447\pi$
$617$	4.45037	0.179165	0.0895825	+	0.995979i	$0.471447\pi$
$618$	10.7745	0.433414
$619$	17.8374	0.716945	0.358473	−	0.933540i	$-0.383298\pi$
$619$	17.8374	0.716945	0.358473	+	0.933540i	$0.383298\pi$
$620$	−2.23408	−0.0897227
$621$	−5.20252	−0.208770
$622$	39.1528	1.56988
$623$	16.7408	0.670707
$624$	−105.289	−4.21494
$625$	17.2104	0.688417
$626$	−64.5214	−2.57879
$627$	16.7139	0.667490
$628$	−23.5806	−0.940967
$629$	−37.2330	−1.48458
$630$	−8.13856	−0.324248
$631$	36.4207	1.44989	0.724943	−	0.688809i	$-0.241866\pi$
$631$	36.4207	1.44989	0.724943	+	0.688809i	$0.241866\pi$
$632$	25.4249	1.01135
$633$	−22.9391	−0.911746
$634$	10.7149	0.425544
$635$	−6.41808	−0.254694
$636$	−6.43870	−0.255311
$637$	13.2689	0.525734
$638$	−31.1494	−1.23322
$639$	16.7178	0.661346
$640$	−44.6427	−1.76466
$641$	25.8592	1.02138	0.510688	−	0.859766i	$-0.329391\pi$
$641$	25.8592	1.02138	0.510688	+	0.859766i	$0.329391\pi$
$642$	−25.7216	−1.01515
$643$	−4.03188	−0.159002	−0.0795010	−	0.996835i	$-0.525333\pi$
$643$	−4.03188	−0.159002	−0.0795010	+	0.996835i	$0.525333\pi$
$644$	−12.3549	−0.486852
$645$	8.03353	0.316320
$646$	−110.184	−4.33513
$647$	45.2789	1.78010	0.890048	−	0.455867i	$-0.150671\pi$
$647$	45.2789	1.78010	0.890048	+	0.455867i	$0.150671\pi$
$648$	0.371083	0.0145775
$649$	−21.7295	−0.852959
$650$	−70.2322	−2.75473
$651$	1.23119	0.0482542
$652$	77.2869	3.02679
$653$	13.4356	0.525774	0.262887	−	0.964827i	$-0.415325\pi$
$653$	13.4356	0.525774	0.262887	+	0.964827i	$0.415325\pi$
$654$	7.10648	0.277885
$655$	−3.43523	−0.134226
$656$	−178.201	−6.95759
$657$	4.25884	0.166153
$658$	4.86042	0.189479
$659$	−4.85141	−0.188984	−0.0944921	−	0.995526i	$-0.530123\pi$
$659$	−4.85141	−0.188984	−0.0944921	+	0.995526i	$0.530123\pi$
$660$	−11.2346	−0.437306
$661$	25.3089	0.984402	0.492201	−	0.870482i	$-0.336192\pi$
$661$	25.3089	0.984402	0.492201	+	0.870482i	$0.336192\pi$
$662$	−26.5958	−1.03367
$663$	38.5417	1.49683
$664$	−4.28757	−0.166390
$665$	9.90070	0.383933
$666$	30.1669	1.16894
$667$	−4.49732	−0.174137
$668$	−62.1404	−2.40428
$669$	7.62579	0.294830
$670$	−12.4070	−0.479324
$671$	3.13925	0.121189
$672$	63.5550	2.45169
$673$	−12.7372	−0.490983	−0.245492	−	0.969399i	$-0.578949\pi$
$673$	−12.7372	−0.490983	−0.245492	+	0.969399i	$0.578949\pi$
$674$	−55.4114	−2.13437
$675$	23.2103	0.893365
$676$	109.236	4.20139
$677$	−37.1815	−1.42900	−0.714500	−	0.699636i	$-0.753346\pi$
$677$	−37.1815	−1.42900	−0.714500	+	0.699636i	$0.753346\pi$
$678$	46.9301	1.80234
$679$	30.2321	1.16020
$680$	48.1888	1.84796
$681$	10.5301	0.403515
$682$	−3.68273	−0.141019
$683$	−8.61310	−0.329571	−0.164786	−	0.986329i	$-0.552693\pi$
$683$	−8.61310	−0.329571	−0.164786	+	0.986329i	$0.552693\pi$
$684$	66.1603	2.52970
$685$	2.46850	0.0943165
$686$	−56.0386	−2.13956
$687$	−11.6382	−0.444024
$688$	176.745	6.73836
$689$	5.93696	0.226180
$690$	−2.18869	−0.0833220
$691$	−24.0088	−0.913338	−0.456669	−	0.889637i	$-0.650958\pi$
$691$	−24.0088	−0.913338	−0.456669	+	0.889637i	$0.650958\pi$
$692$	11.0024	0.418249
$693$	−9.94251	−0.377685
$694$	83.7454	3.17893
$695$	−13.3030	−0.504611
$696$	49.9582	1.89366
$697$	65.2315	2.47082
$698$	2.77938	0.105201
$699$	3.13339	0.118516
$700$	55.1196	2.08333
$701$	0.139269	0.00526012	0.00263006	−	0.999997i	$-0.499163\pi$
$701$	0.139269	0.00526012	0.00263006	+	0.999997i	$0.499163\pi$
$702$	−81.9000	−3.09112
$703$	−36.6986	−1.38411
$704$	−103.757	−3.91047
$705$	0.638108	0.0240325
$706$	57.7681	2.17413
$707$	34.8068	1.30904
$708$	53.5621	2.01299
$709$	−15.5777	−0.585033	−0.292517	−	0.956260i	$-0.594493\pi$
$709$	−15.5777	−0.585033	−0.292517	+	0.956260i	$0.594493\pi$
$710$	18.4459	0.692263
$711$	4.54013	0.170268
$712$	80.3115	3.00980
$713$	−0.531708	−0.0199126
$714$	−40.8154	−1.52748
$715$	10.3591	0.387409
$716$	87.4345	3.26758
$717$	7.94107	0.296565
$718$	38.4202	1.43383
$719$	−2.52288	−0.0940877	−0.0470438	−	0.998893i	$-0.514980\pi$
$719$	−2.52288	−0.0940877	−0.0470438	+	0.998893i	$0.514980\pi$
$720$	−23.5077	−0.876080
$721$	7.79710	0.290379
$722$	−55.7942	−2.07645
$723$	28.4618	1.05851
$724$	−119.025	−4.42352
$725$	20.0641	0.745162
$726$	14.2843	0.530141
$727$	−27.1085	−1.00540	−0.502700	−	0.864461i	$-0.667660\pi$
$727$	−27.1085	−1.00540	−0.502700	+	0.864461i	$0.667660\pi$
$728$	−126.549	−4.69022
$729$	16.8126	0.622688
$730$	4.69907	0.173920
$731$	−64.6986	−2.39296
$732$	−7.73808	−0.286008
$733$	−8.67264	−0.320331	−0.160166	−	0.987090i	$-0.551203\pi$
$733$	−8.67264	−0.320331	−0.160166	+	0.987090i	$0.551203\pi$
$734$	−43.8203	−1.61744
$735$	−1.84480	−0.0680467
$736$	−27.4472	−1.01172
$737$	−15.1571	−0.558317
$738$	−52.8518	−1.94550
$739$	7.03904	0.258935	0.129468	−	0.991584i	$-0.458673\pi$
$739$	7.03904	0.258935	0.129468	+	0.991584i	$0.458673\pi$
$740$	24.6676	0.906800
$741$	37.9885	1.39554
$742$	−6.28721	−0.230811
$743$	43.2112	1.58527	0.792633	−	0.609700i	$-0.208710\pi$
$743$	43.2112	1.58527	0.792633	+	0.609700i	$0.208710\pi$
$744$	5.90645	0.216541
$745$	9.97983	0.365633
$746$	−73.3391	−2.68513
$747$	−0.765635	−0.0280131
$748$	90.4785	3.30822
$749$	−18.6138	−0.680132
$750$	20.7080	0.756148
$751$	−7.78872	−0.284214	−0.142107	−	0.989851i	$-0.545388\pi$
$751$	−7.78872	−0.284214	−0.142107	+	0.989851i	$0.545388\pi$
$752$	14.0390	0.511949
$753$	−0.268839	−0.00979706
$754$	−70.7984	−2.57833
$755$	12.1646	0.442715
$756$	64.2768	2.33772
$757$	35.3499	1.28481	0.642406	−	0.766364i	$-0.277936\pi$
$757$	35.3499	1.28481	0.642406	+	0.766364i	$0.277936\pi$
$758$	21.7300	0.789271
$759$	−2.67382	−0.0970537
$760$	47.4971	1.72290
$761$	−45.1325	−1.63605	−0.818026	−	0.575181i	$-0.804931\pi$
$761$	−45.1325	−1.63605	−0.818026	+	0.575181i	$0.804931\pi$
$762$	26.0786	0.944728
$763$	5.14269	0.186178
$764$	−30.1855	−1.09207
$765$	8.60511	0.311118
$766$	−7.89982	−0.285432
$767$	−49.3883	−1.78331
$768$	92.0499	3.32157
$769$	35.5296	1.28123	0.640615	−	0.767862i	$-0.278679\pi$
$769$	35.5296	1.28123	0.640615	+	0.767862i	$0.278679\pi$
$770$	−10.9703	−0.395341
$771$	11.5094	0.414502
$772$	48.5916	1.74885
$773$	−33.7122	−1.21255	−0.606273	−	0.795257i	$-0.707336\pi$
$773$	−33.7122	−1.21255	−0.606273	+	0.795257i	$0.707336\pi$
$774$	52.4200	1.88420
$775$	2.37214	0.0852097
$776$	145.034	5.20642
$777$	−13.5942	−0.487691
$778$	81.7318	2.93023
$779$	64.2952	2.30362
$780$	−25.5347	−0.914288
$781$	22.5345	0.806349
$782$	17.6268	0.630332
$783$	23.3974	0.836155
$784$	−40.5875	−1.44955
$785$	−3.02297	−0.107895
$786$	13.9584	0.497879
$787$	−10.1125	−0.360471	−0.180236	−	0.983623i	$-0.557686\pi$
$787$	−10.1125	−0.360471	−0.180236	+	0.983623i	$0.557686\pi$
$788$	−144.220	−5.13763
$789$	9.10367	0.324099
$790$	5.00945	0.178228
$791$	33.9615	1.20753
$792$	−47.6977	−1.69486
$793$	7.13509	0.253374
$794$	57.2384	2.03132
$795$	−0.825427	−0.0292749
$796$	−21.2144	−0.751926
$797$	−46.4061	−1.64379	−0.821893	−	0.569641i	$-0.807082\pi$
$797$	−46.4061	−1.64379	−0.821893	+	0.569641i	$0.807082\pi$
$798$	−40.2296	−1.42411
$799$	−5.13904	−0.181806
$800$	122.452	4.32932
$801$	14.3413	0.506725
$802$	−6.75635	−0.238575
$803$	5.74064	0.202583
$804$	37.3613	1.31763
$805$	−1.58387	−0.0558242
$806$	−8.37035	−0.294833
$807$	−17.6018	−0.619612
$808$	166.980	5.87435
$809$	−20.1895	−0.709826	−0.354913	−	0.934899i	$-0.615490\pi$
$809$	−20.1895	−0.709826	−0.354913	+	0.934899i	$0.615490\pi$
$810$	0.0731143	0.00256897
$811$	−3.88458	−0.136406	−0.0682030	−	0.997671i	$-0.521727\pi$
$811$	−3.88458	−0.136406	−0.0682030	+	0.997671i	$0.521727\pi$
$812$	55.5640	1.94991
$813$	26.6866	0.935941
$814$	40.6630	1.42524
$815$	9.90800	0.347062
$816$	−117.893	−4.12707
$817$	−63.7699	−2.23103
$818$	62.1126	2.17172
$819$	−22.5979	−0.789636
$820$	−43.2173	−1.50921
$821$	−28.6055	−0.998340	−0.499170	−	0.866504i	$-0.666362\pi$
$821$	−28.6055	−0.998340	−0.499170	+	0.866504i	$0.666362\pi$
$822$	−10.0303	−0.349846
$823$	−34.1518	−1.19046	−0.595228	−	0.803557i	$-0.702938\pi$
$823$	−34.1518	−1.19046	−0.595228	+	0.803557i	$0.702938\pi$
$824$	37.4054	1.30308
$825$	11.9289	0.415310
$826$	52.3019	1.81982
$827$	11.1702	0.388426	0.194213	−	0.980959i	$-0.437785\pi$
$827$	11.1702	0.388426	0.194213	+	0.980959i	$0.437785\pi$
$828$	−10.5840	−0.367821
$829$	−0.377626	−0.0131155	−0.00655774	−	0.999978i	$-0.502087\pi$
$829$	−0.377626	−0.0131155	−0.00655774	+	0.999978i	$0.502087\pi$
$830$	−0.844779	−0.0293227
$831$	18.3926	0.638032
$832$	−235.824	−8.17573
$833$	14.8573	0.514773
$834$	54.0541	1.87174
$835$	−7.96626	−0.275684
$836$	89.1799	3.08435
$837$	2.76623	0.0956148
$838$	96.5818	3.33636
$839$	−47.2668	−1.63183	−0.815915	−	0.578172i	$-0.803766\pi$
$839$	−47.2668	−1.63183	−0.815915	+	0.578172i	$0.803766\pi$
$840$	17.5943	0.607063
$841$	−8.77415	−0.302557
$842$	−7.34563	−0.253147
$843$	−24.7348	−0.851913
$844$	−122.395	−4.21301
$845$	14.0038	0.481746
$846$	4.16375	0.143153
$847$	10.3370	0.355184
$848$	−18.1602	−0.623623
$849$	−20.9038	−0.717418
$850$	−78.6392	−2.69730
$851$	5.87088	0.201251
$852$	−55.5464	−1.90299
$853$	−49.9716	−1.71099	−0.855497	−	0.517807i	$-0.826749\pi$
$853$	−49.9716	−1.71099	−0.855497	+	0.517807i	$0.826749\pi$
$854$	−7.55602	−0.258562
$855$	8.48160	0.290064
$856$	−89.2967	−3.05210
$857$	8.23835	0.281417	0.140708	−	0.990051i	$-0.455062\pi$
$857$	8.23835	0.281417	0.140708	+	0.990051i	$0.455062\pi$
$858$	−42.0923	−1.43701
$859$	48.0245	1.63858	0.819288	−	0.573382i	$-0.194369\pi$
$859$	48.0245	1.63858	0.819288	+	0.573382i	$0.194369\pi$
$860$	42.8642	1.46166
$861$	23.8169	0.811676
$862$	1.71450	0.0583960
$863$	−10.9120	−0.371450	−0.185725	−	0.982602i	$-0.559463\pi$
$863$	−10.9120	−0.371450	−0.185725	+	0.982602i	$0.559463\pi$
$864$	142.795	4.85797
$865$	1.41048	0.0479579
$866$	−63.2408	−2.14901
$867$	24.9149	0.846153
$868$	6.56921	0.222974
$869$	6.11982	0.207601
$870$	9.84323	0.333717
$871$	−34.4499	−1.16729
$872$	24.6713	0.835475
$873$	25.8988	0.876542
$874$	17.3738	0.587676
$875$	14.9856	0.506605
$876$	−14.1504	−0.478097
$877$	−27.7805	−0.938079	−0.469040	−	0.883177i	$-0.655400\pi$
$877$	−27.7805	−0.938079	−0.469040	+	0.883177i	$0.655400\pi$
$878$	53.5038	1.80567
$879$	7.83219	0.264173
$880$	−31.6869	−1.06816
$881$	−8.55089	−0.288087	−0.144043	−	0.989571i	$-0.546011\pi$
$881$	−8.55089	−0.288087	−0.144043	+	0.989571i	$0.546011\pi$
$882$	−12.0376	−0.405328
$883$	1.42088	0.0478165	0.0239082	−	0.999714i	$-0.492389\pi$
$883$	1.42088	0.0478165	0.0239082	+	0.999714i	$0.492389\pi$
$884$	205.645	6.91660
$885$	6.86654	0.230816
$886$	58.9041	1.97892
$887$	2.42617	0.0814628	0.0407314	−	0.999170i	$-0.487031\pi$
$887$	2.42617	0.0814628	0.0407314	+	0.999170i	$0.487031\pi$
$888$	−65.2163	−2.18852
$889$	18.8721	0.632950
$890$	15.8238	0.530413
$891$	0.0893204	0.00299234
$892$	40.6886	1.36236
$893$	−5.06528	−0.169503
$894$	−40.5511	−1.35623
$895$	11.2089	0.374672
$896$	131.270	4.38543
$897$	−6.07723	−0.202913
$898$	−85.6699	−2.85884
$899$	2.39126	0.0797530
$900$	47.2191	1.57397
$901$	6.64763	0.221465
$902$	−71.2409	−2.37206
$903$	−23.6223	−0.786100
$904$	162.925	5.41881
$905$	−15.2587	−0.507217
$906$	−49.4285	−1.64215
$907$	−40.2815	−1.33753	−0.668763	−	0.743475i	$-0.733176\pi$
$907$	−40.2815	−1.33753	−0.668763	+	0.743475i	$0.733176\pi$
$908$	56.1852	1.86457
$909$	29.8178	0.988994
$910$	−24.9339	−0.826551
$911$	−39.4654	−1.30755	−0.653774	−	0.756690i	$-0.726815\pi$
$911$	−39.4654	−1.30755	−0.653774	+	0.756690i	$0.726815\pi$
$912$	−116.201	−3.84778
$913$	−1.03203	−0.0341551
$914$	−69.9516	−2.31379
$915$	−0.992005	−0.0327947
$916$	−62.0973	−2.05175
$917$	10.1012	0.333570
$918$	−91.7037	−3.02667
$919$	−30.4427	−1.00421	−0.502105	−	0.864807i	$-0.667441\pi$
$919$	−30.4427	−1.00421	−0.502105	+	0.864807i	$0.667441\pi$
$920$	−7.59839	−0.250511
$921$	35.3454	1.16467
$922$	73.8554	2.43230
$923$	51.2179	1.68586
$924$	33.0349	1.08677
$925$	−26.1921	−0.861189
$926$	−60.5496	−1.98978
$927$	6.67951	0.219384
$928$	123.439	4.05207
$929$	44.3241	1.45423	0.727113	−	0.686517i	$-0.240861\pi$
$929$	44.3241	1.45423	0.727113	+	0.686517i	$0.240861\pi$
$930$	1.16375	0.0381607
$931$	14.6440	0.479938
$932$	16.7187	0.547640
$933$	−15.1147	−0.494832
$934$	66.0165	2.16013
$935$	11.5991	0.379333
$936$	−108.410	−3.54350
$937$	−11.1132	−0.363054	−0.181527	−	0.983386i	$-0.558104\pi$
$937$	−11.1132	−0.363054	−0.181527	+	0.983386i	$0.558104\pi$
$938$	36.4823	1.19119
$939$	24.9081	0.812844
$940$	3.40473	0.111050
$941$	34.5270	1.12555	0.562774	−	0.826610i	$-0.309734\pi$
$941$	34.5270	1.12555	0.562774	+	0.826610i	$0.309734\pi$
$942$	12.2833	0.400211
$943$	−10.2857	−0.334948
$944$	151.071	4.91693
$945$	8.24014	0.268052
$946$	70.6589	2.29732
$947$	52.1941	1.69608	0.848040	−	0.529932i	$-0.177782\pi$
$947$	52.1941	1.69608	0.848040	+	0.529932i	$0.177782\pi$
$948$	−15.0850	−0.489938
$949$	13.0477	0.423546
$950$	−77.5105	−2.51477
$951$	−4.13642	−0.134133
$952$	−141.697	−4.59243
$953$	−37.4211	−1.21219	−0.606094	−	0.795393i	$-0.707265\pi$
$953$	−37.4211	−1.21219	−0.606094	+	0.795393i	$0.707265\pi$
$954$	−5.38604	−0.174379
$955$	−3.86972	−0.125221
$956$	42.3708	1.37037
$957$	12.0250	0.388714
$958$	−72.2851	−2.33543
$959$	−7.25853	−0.234390
$960$	32.7871	1.05820
$961$	−30.7173	−0.990880
$962$	92.4215	2.97979
$963$	−15.9458	−0.513845
$964$	151.863	4.89116
$965$	6.22933	0.200529
$966$	6.43576	0.207067
$967$	−22.7597	−0.731902	−0.365951	−	0.930634i	$-0.619256\pi$
$967$	−22.7597	−0.731902	−0.365951	+	0.930634i	$0.619256\pi$
$968$	49.5903	1.59389
$969$	42.5358	1.36645
$970$	28.5760	0.917519
$971$	7.31002	0.234590	0.117295	−	0.993097i	$-0.462578\pi$
$971$	7.31002	0.234590	0.117295	+	0.993097i	$0.462578\pi$
$972$	89.1325	2.85893
$973$	39.1169	1.25403
$974$	96.7684	3.10066
$975$	27.1127	0.868301
$976$	−21.8251	−0.698603
$977$	−1.09574	−0.0350558	−0.0175279	−	0.999846i	$-0.505580\pi$
$977$	−1.09574	−0.0350558	−0.0175279	+	0.999846i	$0.505580\pi$
$978$	−40.2592	−1.28735
$979$	19.3312	0.617826
$980$	−9.84325	−0.314431
$981$	4.40557	0.140659
$982$	33.0981	1.05620
$983$	31.9922	1.02039	0.510196	−	0.860058i	$-0.329573\pi$
$983$	31.9922	1.02039	0.510196	+	0.860058i	$0.329573\pi$
$984$	114.258	3.64240
$985$	−18.4887	−0.589099
$986$	−79.2731	−2.52457
$987$	−1.87633	−0.0597243
$988$	202.694	6.44854
$989$	10.2016	0.324393
$990$	−9.39785	−0.298683
$991$	52.4198	1.66517	0.832585	−	0.553897i	$-0.186860\pi$
$991$	52.4198	1.66517	0.832585	+	0.553897i	$0.186860\pi$
$992$	14.5939	0.463357
$993$	10.2671	0.325817
$994$	−54.2395	−1.72037
$995$	−2.71964	−0.0862185
$996$	2.54389	0.0806063
$997$	45.2745	1.43386	0.716929	−	0.697146i	$-0.245547\pi$
$997$	45.2745	1.43386	0.716929	+	0.697146i	$0.245547\pi$
$998$	12.7275	0.402883
$999$	−30.5434	−0.966350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8027.2.a.f.1.2	✓	176

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8027.2.a.f.1.2	✓	176	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:	\( N \)	\(=\)	\( 8027 = 23 \cdot 349 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8027.a (trivial)

\(f(q)\)	\(=\)	\(q-2.77938 q^{2} +1.07296 q^{3} +5.72495 q^{4} +0.733926 q^{5} -2.98217 q^{6} -2.15808 q^{7} -10.3531 q^{8} -1.84875 q^{9} +O(q^{10})\)	\(q-2.77938 q^{2} +1.07296 q^{3} +5.72495 q^{4} +0.733926 q^{5} -2.98217 q^{6} -2.15808 q^{7} -10.3531 q^{8} -1.84875 q^{9} -2.03986 q^{10} -2.49200 q^{11} +6.14265 q^{12} -5.66398 q^{13} +5.99813 q^{14} +0.787474 q^{15} +17.3252 q^{16} -6.34198 q^{17} +5.13839 q^{18} -6.25095 q^{19} +4.20170 q^{20} -2.31554 q^{21} +6.92623 q^{22} +1.00000 q^{23} -11.1084 q^{24} -4.46135 q^{25} +15.7424 q^{26} -5.20252 q^{27} -12.3549 q^{28} -4.49732 q^{29} -2.18869 q^{30} -0.531708 q^{31} -27.4472 q^{32} -2.67382 q^{33} +17.6268 q^{34} -1.58387 q^{35} -10.5840 q^{36} +5.87088 q^{37} +17.3738 q^{38} -6.07723 q^{39} -7.59839 q^{40} -10.2857 q^{41} +6.43576 q^{42} +10.2016 q^{43} -14.2666 q^{44} -1.35685 q^{45} -2.77938 q^{46} +0.810322 q^{47} +18.5893 q^{48} -2.34269 q^{49} +12.3998 q^{50} -6.80469 q^{51} -32.4260 q^{52} -1.04820 q^{53} +14.4598 q^{54} -1.82895 q^{55} +22.3428 q^{56} -6.70702 q^{57} +12.4998 q^{58} +8.71970 q^{59} +4.50826 q^{60} -1.25973 q^{61} +1.47782 q^{62} +3.98976 q^{63} +41.6358 q^{64} -4.15695 q^{65} +7.43157 q^{66} +6.08227 q^{67} -36.3075 q^{68} +1.07296 q^{69} +4.40218 q^{70} -9.04273 q^{71} +19.1403 q^{72} -2.30362 q^{73} -16.3174 q^{74} -4.78686 q^{75} -35.7864 q^{76} +5.37795 q^{77} +16.8909 q^{78} -2.45578 q^{79} +12.7154 q^{80} -0.0358428 q^{81} +28.5878 q^{82} +0.414136 q^{83} -13.2563 q^{84} -4.65454 q^{85} -28.3542 q^{86} -4.82545 q^{87} +25.7999 q^{88} -7.75727 q^{89} +3.77120 q^{90} +12.2233 q^{91} +5.72495 q^{92} -0.570502 q^{93} -2.25219 q^{94} -4.58774 q^{95} -29.4498 q^{96} -14.0088 q^{97} +6.51121 q^{98} +4.60711 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9}+O(q^{10}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9	\( 176 q + 19 q^{2} + 22 q^{3} + 203 q^{4} + 28 q^{5} + 9 q^{6} + 30 q^{7} + 51 q^{8} + 204 q^{9} + 18 q^{10} + 11 q^{11} + 46 q^{12} + 87 q^{13} - 6 q^{14} + 29 q^{15} + 257 q^{16} + 14 q^{17} + 72 q^{18} + 34 q^{19} + 45 q^{20} + 23 q^{21} + 62 q^{22} + 176 q^{23} + 33 q^{24} + 272 q^{25} + 31 q^{26} + 82 q^{27} + 80 q^{28} + 75 q^{29} - 30 q^{30} + 73 q^{31} + 71 q^{32} + 30 q^{33} + 23 q^{34} + 44 q^{35} + 264 q^{36} + 236 q^{37} - 21 q^{38} + 17 q^{39} + 43 q^{40} + 51 q^{41} + 38 q^{42} + 51 q^{43} + 12 q^{44} + 127 q^{45} + 19 q^{46} + 45 q^{47} + 61 q^{48} + 268 q^{49} + 55 q^{50} - 3 q^{51} + 166 q^{52} + 63 q^{53} - 32 q^{54} + 11 q^{55} - 9 q^{56} + 72 q^{57} + 98 q^{58} + 95 q^{59} - 7 q^{60} + 73 q^{61} + 12 q^{62} + 19 q^{63} + 365 q^{64} + 19 q^{65} - 28 q^{66} + 138 q^{67} + 16 q^{68} + 22 q^{69} + 100 q^{70} + 85 q^{71} + 129 q^{72} + 118 q^{73} - 21 q^{74} - 12 q^{75} + 52 q^{76} + 75 q^{77} + 97 q^{78} + 74 q^{79} + 8 q^{80} + 280 q^{81} + 67 q^{82} + 10 q^{83} - 51 q^{84} + 169 q^{85} - 39 q^{86} - 6 q^{87} + 159 q^{88} + 38 q^{89} + 22 q^{90} + 90 q^{91} + 203 q^{92} + 230 q^{93} + 63 q^{94} + 30 q^{95} + 107 q^{96} + 161 q^{97} + 58 q^{98} + 7 q^{99}+O(q^{100}) \) 176 * q + 19 * q^2 + 22 * q^3 + 203 * q^4 + 28 * q^5 + 9 * q^6 + 30 * q^7 + 51 * q^8 + 204 * q^9 + 18 * q^10 + 11 * q^11 + 46 * q^12 + 87 * q^13 - 6 * q^14 + 29 * q^15 + 257 * q^16 + 14 * q^17 + 72 * q^18 + 34 * q^19 + 45 * q^20 + 23 * q^21 + 62 * q^22 + 176 * q^23 + 33 * q^24 + 272 * q^25 + 31 * q^26 + 82 * q^27 + 80 * q^28 + 75 * q^29 - 30 * q^30 + 73 * q^31 + 71 * q^32 + 30 * q^33 + 23 * q^34 + 44 * q^35 + 264 * q^36 + 236 * q^37 - 21 * q^38 + 17 * q^39 + 43 * q^40 + 51 * q^41 + 38 * q^42 + 51 * q^43 + 12 * q^44 + 127 * q^45 + 19 * q^46 + 45 * q^47 + 61 * q^48 + 268 * q^49 + 55 * q^50 - 3 * q^51 + 166 * q^52 + 63 * q^53 - 32 * q^54 + 11 * q^55 - 9 * q^56 + 72 * q^57 + 98 * q^58 + 95 * q^59 - 7 * q^60 + 73 * q^61 + 12 * q^62 + 19 * q^63 + 365 * q^64 + 19 * q^65 - 28 * q^66 + 138 * q^67 + 16 * q^68 + 22 * q^69 + 100 * q^70 + 85 * q^71 + 129 * q^72 + 118 * q^73 - 21 * q^74 - 12 * q^75 + 52 * q^76 + 75 * q^77 + 97 * q^78 + 74 * q^79 + 8 * q^80 + 280 * q^81 + 67 * q^82 + 10 * q^83 - 51 * q^84 + 169 * q^85 - 39 * q^86 - 6 * q^87 + 159 * q^88 + 38 * q^89 + 22 * q^90 + 90 * q^91 + 203 * q^92 + 230 * q^93 + 63 * q^94 + 30 * q^95 + 107 * q^96 + 161 * q^97 + 58 * q^98 + 7 * q^99

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