LMFDB - Embedded newform 403.2.a.d.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [403,2,Mod(1,403)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(403, 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("403.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 403 = 13 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	403.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:	$3.21797120146$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 7x^{6} + 19x^{5} + 21x^{4} - 31x^{3} - 29x^{2} - x + 1 $ x^8 - 3x^7 - 7x^6 + 19x^5 + 21x^4 - 31x^3 - 29x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.247616$ of defining polynomial
Character	$\chi$	$=$	403.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.24762 q^{2} -2.42435 q^{3} -0.443454 q^{4} +0.848757 q^{5} +3.02465 q^{6} -1.14028 q^{7} +3.04849 q^{8} +2.87746 q^{9} +O(q^{10})$	\(q-1.24762 q^{2} -2.42435 q^{3} -0.443454 q^{4} +0.848757 q^{5} +3.02465 q^{6} -1.14028 q^{7} +3.04849 q^{8} +2.87746 q^{9} -1.05892 q^{10} +6.08227 q^{11} +1.07509 q^{12} -1.00000 q^{13} +1.42263 q^{14} -2.05768 q^{15} -2.91644 q^{16} -7.37689 q^{17} -3.58996 q^{18} +0.506539 q^{19} -0.376385 q^{20} +2.76444 q^{21} -7.58834 q^{22} -1.14165 q^{23} -7.39060 q^{24} -4.27961 q^{25} +1.24762 q^{26} +0.297090 q^{27} +0.505663 q^{28} -2.56988 q^{29} +2.56720 q^{30} -1.00000 q^{31} -2.45839 q^{32} -14.7455 q^{33} +9.20352 q^{34} -0.967822 q^{35} -1.27602 q^{36} +7.75719 q^{37} -0.631966 q^{38} +2.42435 q^{39} +2.58743 q^{40} -1.65078 q^{41} -3.44896 q^{42} -9.58780 q^{43} -2.69721 q^{44} +2.44226 q^{45} +1.42434 q^{46} -5.58932 q^{47} +7.07046 q^{48} -5.69976 q^{49} +5.33931 q^{50} +17.8841 q^{51} +0.443454 q^{52} -11.1260 q^{53} -0.370654 q^{54} +5.16237 q^{55} -3.47614 q^{56} -1.22803 q^{57} +3.20622 q^{58} -10.5509 q^{59} +0.912487 q^{60} -3.26259 q^{61} +1.24762 q^{62} -3.28111 q^{63} +8.90000 q^{64} -0.848757 q^{65} +18.3968 q^{66} +7.13535 q^{67} +3.27131 q^{68} +2.76775 q^{69} +1.20747 q^{70} +10.2568 q^{71} +8.77190 q^{72} +9.30178 q^{73} -9.67800 q^{74} +10.3753 q^{75} -0.224627 q^{76} -6.93550 q^{77} -3.02465 q^{78} +12.8603 q^{79} -2.47535 q^{80} -9.35262 q^{81} +2.05954 q^{82} +3.80522 q^{83} -1.22590 q^{84} -6.26119 q^{85} +11.9619 q^{86} +6.23028 q^{87} +18.5418 q^{88} -12.4692 q^{89} -3.04700 q^{90} +1.14028 q^{91} +0.506268 q^{92} +2.42435 q^{93} +6.97333 q^{94} +0.429928 q^{95} +5.95998 q^{96} -11.2208 q^{97} +7.11111 q^{98} +17.5015 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) $ 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9} + 9 q^{10} - 5 q^{11} - 9 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 3 q^{16} - 11 q^{17} - 30 q^{18} - 9 q^{19} - 31 q^{20} - 16 q^{21} - 2 q^{22} + 13 q^{24} + 19 q^{25} + 5 q^{26} - 9 q^{27} + 16 q^{28} - 12 q^{29} - 7 q^{30} - 8 q^{31} - 25 q^{32} - 14 q^{33} + 22 q^{34} + 7 q^{35} + 37 q^{36} - 9 q^{37} - 9 q^{38} + 3 q^{39} + 55 q^{40} - 25 q^{41} - 3 q^{42} + 7 q^{43} - 26 q^{44} - 45 q^{45} + 5 q^{46} - 17 q^{47} - 9 q^{48} - 11 q^{50} - 10 q^{51} - 9 q^{52} - 15 q^{53} + 54 q^{54} + 7 q^{55} - 14 q^{56} - 7 q^{57} - 5 q^{58} - 15 q^{59} + 61 q^{60} + 11 q^{61} + 5 q^{62} - 21 q^{63} + 47 q^{64} + 15 q^{65} + 83 q^{66} + 18 q^{67} - 16 q^{68} - 15 q^{69} - 24 q^{70} - 7 q^{71} - 21 q^{72} + 24 q^{73} + 48 q^{74} - 17 q^{75} - 3 q^{76} - 49 q^{77} + 33 q^{79} - 16 q^{80} + 20 q^{81} - q^{82} - 13 q^{83} - 6 q^{84} + q^{85} + 19 q^{86} + 18 q^{87} + 37 q^{88} - 23 q^{89} + 117 q^{90} + 4 q^{91} + 22 q^{92} + 3 q^{93} + 10 q^{94} + 43 q^{95} + 46 q^{96} - 17 q^{97} + 52 q^{98} + 51 q^{99}+O(q^{100}) \) 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9 + 9 * q^10 - 5 * q^11 - 9 * q^12 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 3 * q^16 - 11 * q^17 - 30 * q^18 - 9 * q^19 - 31 * q^20 - 16 * q^21 - 2 * q^22 + 13 * q^24 + 19 * q^25 + 5 * q^26 - 9 * q^27 + 16 * q^28 - 12 * q^29 - 7 * q^30 - 8 * q^31 - 25 * q^32 - 14 * q^33 + 22 * q^34 + 7 * q^35 + 37 * q^36 - 9 * q^37 - 9 * q^38 + 3 * q^39 + 55 * q^40 - 25 * q^41 - 3 * q^42 + 7 * q^43 - 26 * q^44 - 45 * q^45 + 5 * q^46 - 17 * q^47 - 9 * q^48 - 11 * q^50 - 10 * q^51 - 9 * q^52 - 15 * q^53 + 54 * q^54 + 7 * q^55 - 14 * q^56 - 7 * q^57 - 5 * q^58 - 15 * q^59 + 61 * q^60 + 11 * q^61 + 5 * q^62 - 21 * q^63 + 47 * q^64 + 15 * q^65 + 83 * q^66 + 18 * q^67 - 16 * q^68 - 15 * q^69 - 24 * q^70 - 7 * q^71 - 21 * q^72 + 24 * q^73 + 48 * q^74 - 17 * q^75 - 3 * q^76 - 49 * q^77 + 33 * q^79 - 16 * q^80 + 20 * q^81 - q^82 - 13 * q^83 - 6 * q^84 + q^85 + 19 * q^86 + 18 * q^87 + 37 * q^88 - 23 * q^89 + 117 * q^90 + 4 * q^91 + 22 * q^92 + 3 * q^93 + 10 * q^94 + 43 * q^95 + 46 * q^96 - 17 * q^97 + 52 * q^98 + 51 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.24762	−0.882198	−0.441099	−	0.897458i	$-0.645411\pi$
$2$	−1.24762	−0.882198	−0.441099	+	0.897458i	$0.645411\pi$
$3$	−2.42435	−1.39970	−0.699849	−	0.714291i	$-0.746749\pi$
$3$	−2.42435	−1.39970	−0.699849	+	0.714291i	$0.746749\pi$
$4$	−0.443454	−0.221727
$5$	0.848757	0.379576	0.189788	−	0.981825i	$-0.439220\pi$
$5$	0.848757	0.379576	0.189788	+	0.981825i	$0.439220\pi$
$6$	3.02465	1.23481
$7$	−1.14028	−0.430986	−0.215493	−	0.976505i	$-0.569136\pi$
$7$	−1.14028	−0.430986	−0.215493	+	0.976505i	$0.569136\pi$
$8$	3.04849	1.07780
$9$	2.87746	0.959152
$10$	−1.05892	−0.334861
$11$	6.08227	1.83387	0.916937	−	0.399032i	$-0.130654\pi$
$11$	6.08227	1.83387	0.916937	+	0.399032i	$0.130654\pi$
$12$	1.07509	0.310351
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	1.42263	0.380215
$15$	−2.05768	−0.531291
$16$	−2.91644	−0.729110
$17$	−7.37689	−1.78916	−0.894579	−	0.446910i	$-0.852525\pi$
$17$	−7.37689	−1.78916	−0.894579	+	0.446910i	$0.852525\pi$
$18$	−3.58996	−0.846162
$19$	0.506539	0.116208	0.0581040	−	0.998311i	$-0.481494\pi$
$19$	0.506539	0.116208	0.0581040	+	0.998311i	$0.481494\pi$
$20$	−0.376385	−0.0841622
$21$	2.76444	0.603250
$22$	−7.58834	−1.61784
$23$	−1.14165	−0.238050	−0.119025	−	0.992891i	$-0.537977\pi$
$23$	−1.14165	−0.238050	−0.119025	+	0.992891i	$0.537977\pi$
$24$	−7.39060	−1.50860
$25$	−4.27961	−0.855922
$26$	1.24762	0.244678
$27$	0.297090	0.0571749
$28$	0.505663	0.0955612
$29$	−2.56988	−0.477215	−0.238607	−	0.971116i	$-0.576691\pi$
$29$	−2.56988	−0.477215	−0.238607	+	0.971116i	$0.576691\pi$
$30$	2.56720	0.468704
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	−2.45839	−0.434586
$33$	−14.7455	−2.56687
$34$	9.20352	1.57839
$35$	−0.967822	−0.163592
$36$	−1.27602	−0.212670
$37$	7.75719	1.27527	0.637637	−	0.770337i	$-0.279912\pi$
$37$	7.75719	1.27527	0.637637	+	0.770337i	$0.279912\pi$
$38$	−0.631966	−0.102518
$39$	2.42435	0.388206
$40$	2.58743	0.409109
$41$	−1.65078	−0.257809	−0.128905	−	0.991657i	$-0.541146\pi$
$41$	−1.65078	−0.257809	−0.128905	+	0.991657i	$0.541146\pi$
$42$	−3.44896	−0.532186
$43$	−9.58780	−1.46213	−0.731063	−	0.682310i	$-0.760975\pi$
$43$	−9.58780	−1.46213	−0.731063	+	0.682310i	$0.760975\pi$
$44$	−2.69721	−0.406619
$45$	2.44226	0.364071
$46$	1.42434	0.210007
$47$	−5.58932	−0.815286	−0.407643	−	0.913141i	$-0.633649\pi$
$47$	−5.58932	−0.815286	−0.407643	+	0.913141i	$0.633649\pi$
$48$	7.07046	1.02053
$49$	−5.69976	−0.814251
$50$	5.33931	0.755093
$51$	17.8841	2.50428
$52$	0.443454	0.0614960
$53$	−11.1260	−1.52828	−0.764140	−	0.645050i	$-0.776836\pi$
$53$	−11.1260	−1.52828	−0.764140	+	0.645050i	$0.776836\pi$
$54$	−0.370654	−0.0504396
$55$	5.16237	0.696094
$56$	−3.47614	−0.464519
$57$	−1.22803	−0.162656
$58$	3.20622	0.420998
$59$	−10.5509	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$59$	−10.5509	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$60$	0.912487	0.117802
$61$	−3.26259	−0.417732	−0.208866	−	0.977944i	$-0.566977\pi$
$61$	−3.26259	−0.417732	−0.208866	+	0.977944i	$0.566977\pi$
$62$	1.24762	0.158447
$63$	−3.28111	−0.413381
$64$	8.90000	1.11250
$65$	−0.848757	−0.105275
$66$	18.3968	2.26448
$67$	7.13535	0.871722	0.435861	−	0.900014i	$-0.356444\pi$
$67$	7.13535	0.871722	0.435861	+	0.900014i	$0.356444\pi$
$68$	3.27131	0.396705
$69$	2.76775	0.333197
$70$	1.20747	0.144320
$71$	10.2568	1.21726	0.608631	−	0.793454i	$-0.291719\pi$
$71$	10.2568	1.21726	0.608631	+	0.793454i	$0.291719\pi$
$72$	8.77190	1.03378
$73$	9.30178	1.08869	0.544346	−	0.838861i	$-0.316778\pi$
$73$	9.30178	1.08869	0.544346	+	0.838861i	$0.316778\pi$
$74$	−9.67800	−1.12504
$75$	10.3753	1.19803
$76$	−0.224627	−0.0257665
$77$	−6.93550	−0.790374
$78$	−3.02465	−0.342475
$79$	12.8603	1.44689	0.723447	−	0.690380i	$-0.242556\pi$
$79$	12.8603	1.44689	0.723447	+	0.690380i	$0.242556\pi$
$80$	−2.47535	−0.276752
$81$	−9.35262	−1.03918
$82$	2.05954	0.227439
$83$	3.80522	0.417677	0.208838	−	0.977950i	$-0.433032\pi$
$83$	3.80522	0.417677	0.208838	+	0.977950i	$0.433032\pi$
$84$	−1.22590	−0.133757
$85$	−6.26119	−0.679121
$86$	11.9619	1.28988
$87$	6.23028	0.667956
$88$	18.5418	1.97656
$89$	−12.4692	−1.32174	−0.660868	−	0.750502i	$-0.729812\pi$
$89$	−12.4692	−1.32174	−0.660868	+	0.750502i	$0.729812\pi$
$90$	−3.04700	−0.321182
$91$	1.14028	0.119534
$92$	0.506268	0.0527821
$93$	2.42435	0.251393
$94$	6.97333	0.719244
$95$	0.429928	0.0441097
$96$	5.95998	0.608288
$97$	−11.2208	−1.13930	−0.569650	−	0.821887i	$-0.692921\pi$
$97$	−11.2208	−1.13930	−0.569650	+	0.821887i	$0.692921\pi$
$98$	7.11111	0.718331
$99$	17.5015	1.75896
$100$	1.89781	0.189781
$101$	−8.30857	−0.826733	−0.413367	−	0.910565i	$-0.635647\pi$
$101$	−8.30857	−0.826733	−0.413367	+	0.910565i	$0.635647\pi$
$102$	−22.3125	−2.20927
$103$	−11.8093	−1.16361	−0.581804	−	0.813329i	$-0.697653\pi$
$103$	−11.8093	−1.16361	−0.581804	+	0.813329i	$0.697653\pi$
$104$	−3.04849	−0.298929
$105$	2.34634	0.228979
$106$	13.8810	1.34825
$107$	−10.6791	−1.03239	−0.516196	−	0.856471i	$-0.672652\pi$
$107$	−10.6791	−1.03239	−0.516196	+	0.856471i	$0.672652\pi$
$108$	−0.131746	−0.0126772
$109$	−6.63365	−0.635389	−0.317694	−	0.948193i	$-0.602909\pi$
$109$	−6.63365	−0.635389	−0.317694	+	0.948193i	$0.602909\pi$
$110$	−6.44066	−0.614092
$111$	−18.8061	−1.78500
$112$	3.32556	0.314236
$113$	−0.846821	−0.0796623	−0.0398311	−	0.999206i	$-0.512682\pi$
$113$	−0.846821	−0.0796623	−0.0398311	+	0.999206i	$0.512682\pi$
$114$	1.53210	0.143495
$115$	−0.968980	−0.0903579
$116$	1.13962	0.105811
$117$	−2.87746	−0.266021
$118$	13.1634	1.21179
$119$	8.41173	0.771102
$120$	−6.27283	−0.572628
$121$	25.9940	2.36309
$122$	4.07046	0.368522
$123$	4.00207	0.360855
$124$	0.443454	0.0398233
$125$	−7.87614	−0.704463
$126$	4.09357	0.364684
$127$	−11.6920	−1.03750	−0.518750	−	0.854926i	$-0.673603\pi$
$127$	−11.6920	−1.03750	−0.518750	+	0.854926i	$0.673603\pi$
$128$	−6.18701	−0.546860
$129$	23.2442	2.04653
$130$	1.05892	0.0928737
$131$	8.82499	0.771043	0.385521	−	0.922699i	$-0.374022\pi$
$131$	8.82499	0.771043	0.385521	+	0.922699i	$0.374022\pi$
$132$	6.53897	0.569144
$133$	−0.577597	−0.0500840
$134$	−8.90218	−0.769032
$135$	0.252157	0.0217022
$136$	−22.4884	−1.92836
$137$	−6.00990	−0.513461	−0.256730	−	0.966483i	$-0.582645\pi$
$137$	−6.00990	−0.513461	−0.256730	+	0.966483i	$0.582645\pi$
$138$	−3.45308	−0.293946
$139$	5.61773	0.476490	0.238245	−	0.971205i	$-0.423428\pi$
$139$	5.61773	0.476490	0.238245	+	0.971205i	$0.423428\pi$
$140$	0.429185	0.0362727
$141$	13.5505	1.14115
$142$	−12.7966	−1.07387
$143$	−6.08227	−0.508625
$144$	−8.39193	−0.699327
$145$	−2.18120	−0.181139
$146$	−11.6051	−0.960441
$147$	13.8182	1.13970
$148$	−3.43996	−0.282763
$149$	−21.7637	−1.78295	−0.891477	−	0.453065i	$-0.850330\pi$
$149$	−21.7637	−1.78295	−0.891477	+	0.453065i	$0.850330\pi$
$150$	−12.9443	−1.05690
$151$	8.56841	0.697287	0.348644	−	0.937255i	$-0.386642\pi$
$151$	8.56841	0.697287	0.348644	+	0.937255i	$0.386642\pi$
$152$	1.54418	0.125250
$153$	−21.2267	−1.71607
$154$	8.65284	0.697266
$155$	−0.848757	−0.0681738
$156$	−1.07509	−0.0860758
$157$	−11.5984	−0.925656	−0.462828	−	0.886448i	$-0.653165\pi$
$157$	−11.5984	−0.925656	−0.462828	+	0.886448i	$0.653165\pi$
$158$	−16.0447	−1.27645
$159$	26.9734	2.13913
$160$	−2.08657	−0.164958
$161$	1.30180	0.102596
$162$	11.6685	0.916762
$163$	17.4320	1.36538	0.682689	−	0.730710i	$-0.260811\pi$
$163$	17.4320	1.36538	0.682689	+	0.730710i	$0.260811\pi$
$164$	0.732047	0.0571633
$165$	−12.5154	−0.974320
$166$	−4.74745	−0.368473
$167$	−0.855902	−0.0662317	−0.0331158	−	0.999452i	$-0.510543\pi$
$167$	−0.855902	−0.0662317	−0.0331158	+	0.999452i	$0.510543\pi$
$168$	8.42737	0.650186
$169$	1.00000	0.0769231
$170$	7.81156	0.599119
$171$	1.45754	0.111461
$172$	4.25175	0.324193
$173$	8.98510	0.683125	0.341562	−	0.939859i	$-0.389044\pi$
$173$	8.98510	0.683125	0.341562	+	0.939859i	$0.389044\pi$
$174$	−7.77300	−0.589269
$175$	4.87996	0.368891
$176$	−17.7386	−1.33710
$177$	25.5790	1.92263
$178$	15.5568	1.16603
$179$	−12.1286	−0.906538	−0.453269	−	0.891374i	$-0.649742\pi$
$179$	−12.1286	−0.906538	−0.453269	+	0.891374i	$0.649742\pi$
$180$	−1.08303	−0.0807243
$181$	−4.05407	−0.301337	−0.150668	−	0.988584i	$-0.548143\pi$
$181$	−4.05407	−0.301337	−0.150668	+	0.988584i	$0.548143\pi$
$182$	−1.42263	−0.105453
$183$	7.90965	0.584698
$184$	−3.48030	−0.256571
$185$	6.58397	0.484063
$186$	−3.02465	−0.221778
$187$	−44.8682	−3.28109
$188$	2.47861	0.180771
$189$	−0.338766	−0.0246416
$190$	−0.536386	−0.0389135
$191$	20.1432	1.45751	0.728755	−	0.684774i	$-0.240099\pi$
$191$	20.1432	1.45751	0.728755	+	0.684774i	$0.240099\pi$
$192$	−21.5767	−1.55716
$193$	4.42629	0.318611	0.159306	−	0.987229i	$-0.449074\pi$
$193$	4.42629	0.318611	0.159306	+	0.987229i	$0.449074\pi$
$194$	13.9993	1.00509
$195$	2.05768	0.147354
$196$	2.52758	0.180541
$197$	−0.469825	−0.0334737	−0.0167368	−	0.999860i	$-0.505328\pi$
$197$	−0.469825	−0.0334737	−0.0167368	+	0.999860i	$0.505328\pi$
$198$	−21.8351	−1.55175
$199$	−17.1855	−1.21825	−0.609123	−	0.793076i	$-0.708478\pi$
$199$	−17.1855	−1.21825	−0.609123	+	0.793076i	$0.708478\pi$
$200$	−13.0464	−0.922517
$201$	−17.2986	−1.22015
$202$	10.3659	0.729342
$203$	2.93039	0.205673
$204$	−7.93079	−0.555266
$205$	−1.40111	−0.0978581
$206$	14.7335	1.02653
$207$	−3.28504	−0.228326
$208$	2.91644	0.202219
$209$	3.08091	0.213111
$210$	−2.92733	−0.202005
$211$	−13.4075	−0.923011	−0.461505	−	0.887137i	$-0.652690\pi$
$211$	−13.4075	−0.923011	−0.461505	+	0.887137i	$0.652690\pi$
$212$	4.93389	0.338861
$213$	−24.8661	−1.70380
$214$	13.3235	0.910774
$215$	−8.13771	−0.554987
$216$	0.905676	0.0616234
$217$	1.14028	0.0774074
$218$	8.27625	0.560539
$219$	−22.5507	−1.52384
$220$	−2.28927	−0.154343
$221$	7.37689	0.496223
$222$	23.4628	1.57472
$223$	−11.5695	−0.774749	−0.387375	−	0.921922i	$-0.626618\pi$
$223$	−11.5695	−0.774749	−0.387375	+	0.921922i	$0.626618\pi$
$224$	2.80325	0.187300
$225$	−12.3144	−0.820960
$226$	1.05651	0.0702779
$227$	−13.9317	−0.924682	−0.462341	−	0.886702i	$-0.652990\pi$
$227$	−13.9317	−0.924682	−0.462341	+	0.886702i	$0.652990\pi$
$228$	0.544573	0.0360652
$229$	9.24104	0.610665	0.305333	−	0.952246i	$-0.401232\pi$
$229$	9.24104	0.610665	0.305333	+	0.952246i	$0.401232\pi$
$230$	1.20892	0.0797135
$231$	16.8141	1.10628
$232$	−7.83426	−0.514344
$233$	22.0969	1.44761	0.723807	−	0.690002i	$-0.242391\pi$
$233$	22.0969	1.44761	0.723807	+	0.690002i	$0.242391\pi$
$234$	3.58996	0.234683
$235$	−4.74398	−0.309463
$236$	4.67882	0.304566
$237$	−31.1778	−2.02521
$238$	−10.4946	−0.680265
$239$	−7.88700	−0.510168	−0.255084	−	0.966919i	$-0.582103\pi$
$239$	−7.88700	−0.510168	−0.255084	+	0.966919i	$0.582103\pi$
$240$	6.00110	0.387370
$241$	22.4020	1.44304	0.721520	−	0.692393i	$-0.243444\pi$
$241$	22.4020	1.44304	0.721520	+	0.692393i	$0.243444\pi$
$242$	−32.4306	−2.08471
$243$	21.7827	1.39736
$244$	1.44681	0.0926224
$245$	−4.83771	−0.309070
$246$	−4.99305	−0.318345
$247$	−0.506539	−0.0322303
$248$	−3.04849	−0.193579
$249$	−9.22516	−0.584621
$250$	9.82639	0.621476
$251$	22.1030	1.39513	0.697564	−	0.716523i	$-0.254268\pi$
$251$	22.1030	1.39513	0.697564	+	0.716523i	$0.254268\pi$
$252$	1.45502	0.0916577
$253$	−6.94380	−0.436553
$254$	14.5872	0.915281
$255$	15.1793	0.950564
$256$	−10.0810	−0.630062
$257$	18.4679	1.15200	0.575999	−	0.817450i	$-0.304613\pi$
$257$	18.4679	1.15200	0.575999	+	0.817450i	$0.304613\pi$
$258$	−28.9998	−1.80545
$259$	−8.84539	−0.549626
$260$	0.376385	0.0233424
$261$	−7.39472	−0.457721
$262$	−11.0102	−0.680212
$263$	14.3945	0.887601	0.443800	−	0.896126i	$-0.353630\pi$
$263$	14.3945	0.887601	0.443800	+	0.896126i	$0.353630\pi$
$264$	−44.9516	−2.76658
$265$	−9.44331	−0.580098
$266$	0.720619	0.0441840
$267$	30.2298	1.85003
$268$	−3.16420	−0.193284
$269$	9.74643	0.594250	0.297125	−	0.954839i	$-0.403972\pi$
$269$	9.74643	0.594250	0.297125	+	0.954839i	$0.403972\pi$
$270$	−0.314595	−0.0191457
$271$	22.9643	1.39498	0.697491	−	0.716593i	$-0.254300\pi$
$271$	22.9643	1.39498	0.697491	+	0.716593i	$0.254300\pi$
$272$	21.5143	1.30449
$273$	−2.76444	−0.167311
$274$	7.49805	0.452974
$275$	−26.0298	−1.56965
$276$	−1.22737	−0.0738789
$277$	18.4300	1.10735	0.553676	−	0.832732i	$-0.313225\pi$
$277$	18.4300	1.10735	0.553676	+	0.832732i	$0.313225\pi$
$278$	−7.00877	−0.420358
$279$	−2.87746	−0.172269
$280$	−2.95040	−0.176320
$281$	−22.4432	−1.33885	−0.669426	−	0.742879i	$-0.733460\pi$
$281$	−22.4432	−1.33885	−0.669426	+	0.742879i	$0.733460\pi$
$282$	−16.9058	−1.00672
$283$	−17.0038	−1.01077	−0.505385	−	0.862894i	$-0.668649\pi$
$283$	−17.0038	−1.01077	−0.505385	+	0.862894i	$0.668649\pi$
$284$	−4.54843	−0.269900
$285$	−1.04230	−0.0617403
$286$	7.58834	0.448708
$287$	1.88236	0.111112
$288$	−7.07390	−0.416834
$289$	37.4185	2.20109
$290$	2.72130	0.159801
$291$	27.2031	1.59467
$292$	−4.12491	−0.241392
$293$	−23.4585	−1.37046	−0.685230	−	0.728326i	$-0.740298\pi$
$293$	−23.4585	−1.37046	−0.685230	+	0.728326i	$0.740298\pi$
$294$	−17.2398	−1.00545
$295$	−8.95512	−0.521387
$296$	23.6477	1.37450
$297$	1.80698	0.104852
$298$	27.1528	1.57292
$299$	1.14165	0.0660231
$300$	−4.60095	−0.265636
$301$	10.9328	0.630156
$302$	−10.6901	−0.615145
$303$	20.1428	1.15718
$304$	−1.47729	−0.0847284
$305$	−2.76915	−0.158561
$306$	26.4827	1.51392
$307$	0.860307	0.0491003	0.0245502	−	0.999699i	$-0.492185\pi$
$307$	0.860307	0.0491003	0.0245502	+	0.999699i	$0.492185\pi$
$308$	3.07558	0.175247
$309$	28.6299	1.62870
$310$	1.05892	0.0601428
$311$	−29.7892	−1.68919	−0.844595	−	0.535406i	$-0.820159\pi$
$311$	−29.7892	−1.68919	−0.844595	+	0.535406i	$0.820159\pi$
$312$	7.39060	0.418410
$313$	7.53334	0.425810	0.212905	−	0.977073i	$-0.431708\pi$
$313$	7.53334	0.425810	0.212905	+	0.977073i	$0.431708\pi$
$314$	14.4704	0.816612
$315$	−2.78487	−0.156909
$316$	−5.70294	−0.320816
$317$	−7.93443	−0.445642	−0.222821	−	0.974859i	$-0.571527\pi$
$317$	−7.93443	−0.445642	−0.222821	+	0.974859i	$0.571527\pi$
$318$	−33.6524	−1.88714
$319$	−15.6307	−0.875152
$320$	7.55394	0.422278
$321$	25.8899	1.44504
$322$	−1.62414	−0.0905100
$323$	−3.73668	−0.207915
$324$	4.14746	0.230414
$325$	4.27961	0.237390
$326$	−21.7484	−1.20453
$327$	16.0823	0.889352
$328$	−5.03240	−0.277868
$329$	6.37340	0.351377
$330$	15.6144	0.859543
$331$	22.2788	1.22455	0.612276	−	0.790644i	$-0.290254\pi$
$331$	22.2788	1.22455	0.612276	+	0.790644i	$0.290254\pi$
$332$	−1.68744	−0.0926102
$333$	22.3210	1.22318
$334$	1.06784	0.0584294
$335$	6.05618	0.330885
$336$	−8.06232	−0.439836
$337$	27.3078	1.48755	0.743774	−	0.668431i	$-0.233034\pi$
$337$	27.3078	1.48755	0.743774	+	0.668431i	$0.233034\pi$
$338$	−1.24762	−0.0678614
$339$	2.05299	0.111503
$340$	2.77655	0.150579
$341$	−6.08227	−0.329373
$342$	−1.81845	−0.0983308
$343$	14.4813	0.781917
$344$	−29.2283	−1.57589
$345$	2.34914	0.126474
$346$	−11.2100	−0.602651
$347$	−15.9671	−0.857160	−0.428580	−	0.903504i	$-0.640986\pi$
$347$	−15.9671	−0.857160	−0.428580	+	0.903504i	$0.640986\pi$
$348$	−2.76284	−0.148104
$349$	17.8869	0.957464	0.478732	−	0.877961i	$-0.341097\pi$
$349$	17.8869	0.957464	0.478732	+	0.877961i	$0.341097\pi$
$350$	−6.08832	−0.325434
$351$	−0.297090	−0.0158575
$352$	−14.9526	−0.796975
$353$	11.4688	0.610422	0.305211	−	0.952285i	$-0.401273\pi$
$353$	11.4688	0.610422	0.305211	+	0.952285i	$0.401273\pi$
$354$	−31.9127	−1.69614
$355$	8.70555	0.462043
$356$	5.52954	0.293065
$357$	−20.3930	−1.07931
$358$	15.1319	0.799746
$359$	22.6622	1.19607	0.598034	−	0.801471i	$-0.295949\pi$
$359$	22.6622	1.19607	0.598034	+	0.801471i	$0.295949\pi$
$360$	7.44521	0.392397
$361$	−18.7434	−0.986496
$362$	5.05792	0.265839
$363$	−63.0185	−3.30761
$364$	−0.505663	−0.0265039
$365$	7.89495	0.413241
$366$	−9.86820	−0.515819
$367$	−1.41627	−0.0739287	−0.0369644	−	0.999317i	$-0.511769\pi$
$367$	−1.41627	−0.0739287	−0.0369644	+	0.999317i	$0.511769\pi$
$368$	3.32954	0.173564
$369$	−4.75006	−0.247278
$370$	−8.21427	−0.427040
$371$	12.6868	0.658667
$372$	−1.07509	−0.0557406
$373$	−14.9384	−0.773483	−0.386742	−	0.922188i	$-0.626399\pi$
$373$	−14.9384	−0.773483	−0.386742	+	0.922188i	$0.626399\pi$
$374$	55.9783	2.89457
$375$	19.0945	0.986035
$376$	−17.0390	−0.878720
$377$	2.56988	0.132356
$378$	0.422650	0.0217388
$379$	11.1186	0.571126	0.285563	−	0.958360i	$-0.407819\pi$
$379$	11.1186	0.571126	0.285563	+	0.958360i	$0.407819\pi$
$380$	−0.190654	−0.00978032
$381$	28.3455	1.45219
$382$	−25.1310	−1.28581
$383$	12.0925	0.617900	0.308950	−	0.951078i	$-0.400022\pi$
$383$	12.0925	0.617900	0.308950	+	0.951078i	$0.400022\pi$
$384$	14.9995	0.765438
$385$	−5.88656	−0.300007
$386$	−5.52231	−0.281078
$387$	−27.5885	−1.40240
$388$	4.97591	0.252614
$389$	16.2456	0.823684	0.411842	−	0.911255i	$-0.364886\pi$
$389$	16.2456	0.823684	0.411842	+	0.911255i	$0.364886\pi$
$390$	−2.56720	−0.129995
$391$	8.42180	0.425909
$392$	−17.3757	−0.877604
$393$	−21.3948	−1.07923
$394$	0.586162	0.0295304
$395$	10.9153	0.549206
$396$	−7.76110	−0.390010
$397$	8.43695	0.423439	0.211719	−	0.977331i	$-0.432094\pi$
$397$	8.43695	0.423439	0.211719	+	0.977331i	$0.432094\pi$
$398$	21.4409	1.07473
$399$	1.40030	0.0701025
$400$	12.4812	0.624062
$401$	−23.1069	−1.15390	−0.576951	−	0.816779i	$-0.695757\pi$
$401$	−23.1069	−1.15390	−0.576951	+	0.816779i	$0.695757\pi$
$402$	21.5820	1.07641
$403$	1.00000	0.0498135
$404$	3.68447	0.183309
$405$	−7.93810	−0.394447
$406$	−3.65600	−0.181444
$407$	47.1814	2.33869
$408$	54.5196	2.69912
$409$	−2.52043	−0.124627	−0.0623136	−	0.998057i	$-0.519848\pi$
$409$	−2.52043	−0.124627	−0.0623136	+	0.998057i	$0.519848\pi$
$410$	1.74805	0.0863302
$411$	14.5701	0.718689
$412$	5.23690	0.258004
$413$	12.0310	0.592005
$414$	4.09846	0.201429
$415$	3.22970	0.158540
$416$	2.45839	0.120532
$417$	−13.6193	−0.666941
$418$	−3.84379	−0.188006
$419$	−30.7749	−1.50345	−0.751726	−	0.659476i	$-0.770778\pi$
$419$	−30.7749	−1.50345	−0.751726	+	0.659476i	$0.770778\pi$
$420$	−1.04049	−0.0507708
$421$	−10.9747	−0.534873	−0.267437	−	0.963575i	$-0.586177\pi$
$421$	−10.9747	−0.534873	−0.267437	+	0.963575i	$0.586177\pi$
$422$	16.7274	0.814278
$423$	−16.0830	−0.781983
$424$	−33.9177	−1.64719
$425$	31.5702	1.53138
$426$	31.0233	1.50309
$427$	3.72027	0.180037
$428$	4.73571	0.228909
$429$	14.7455	0.711921
$430$	10.1527	0.489609
$431$	−30.4203	−1.46530	−0.732648	−	0.680608i	$-0.761716\pi$
$431$	−30.4203	−1.46530	−0.732648	+	0.680608i	$0.761716\pi$
$432$	−0.866445	−0.0416868
$433$	−1.06120	−0.0509982	−0.0254991	−	0.999675i	$-0.508117\pi$
$433$	−1.06120	−0.0509982	−0.0254991	+	0.999675i	$0.508117\pi$
$434$	−1.42263	−0.0682886
$435$	5.28799	0.253540
$436$	2.94172	0.140883
$437$	−0.578288	−0.0276633
$438$	28.1347	1.34433
$439$	−3.28884	−0.156968	−0.0784839	−	0.996915i	$-0.525008\pi$
$439$	−3.28884	−0.156968	−0.0784839	+	0.996915i	$0.525008\pi$
$440$	15.7374	0.750253
$441$	−16.4008	−0.780990
$442$	−9.20352	−0.437767
$443$	−8.86739	−0.421302	−0.210651	−	0.977561i	$-0.567558\pi$
$443$	−8.86739	−0.421302	−0.210651	+	0.977561i	$0.567558\pi$
$444$	8.33965	0.395782
$445$	−10.5834	−0.501699
$446$	14.4343	0.683482
$447$	52.7628	2.49560
$448$	−10.1485	−0.479472
$449$	8.65406	0.408410	0.204205	−	0.978928i	$-0.434539\pi$
$449$	8.65406	0.408410	0.204205	+	0.978928i	$0.434539\pi$
$450$	15.3636	0.724249
$451$	−10.0405	−0.472789
$452$	0.375526	0.0176633
$453$	−20.7728	−0.975991
$454$	17.3815	0.815752
$455$	0.967822	0.0453722
$456$	−3.74363	−0.175311
$457$	−16.7376	−0.782950	−0.391475	−	0.920189i	$-0.628035\pi$
$457$	−16.7376	−0.782950	−0.391475	+	0.920189i	$0.628035\pi$
$458$	−11.5293	−0.538727
$459$	−2.19160	−0.102295
$460$	0.429698	0.0200348
$461$	20.0259	0.932699	0.466350	−	0.884600i	$-0.345569\pi$
$461$	20.0259	0.932699	0.466350	+	0.884600i	$0.345569\pi$
$462$	−20.9775	−0.975961
$463$	−18.2333	−0.847375	−0.423688	−	0.905808i	$-0.639265\pi$
$463$	−18.2333	−0.847375	−0.423688	+	0.905808i	$0.639265\pi$
$464$	7.49490	0.347942
$465$	2.05768	0.0954227
$466$	−27.5684	−1.27708
$467$	−28.1665	−1.30339	−0.651694	−	0.758482i	$-0.725941\pi$
$467$	−28.1665	−1.30339	−0.651694	+	0.758482i	$0.725941\pi$
$468$	1.27602	0.0589840
$469$	−8.13631	−0.375700
$470$	5.91866	0.273007
$471$	28.1186	1.29564
$472$	−32.1642	−1.48048
$473$	−58.3156	−2.68135
$474$	38.8979	1.78664
$475$	−2.16779	−0.0994650
$476$	−3.73022	−0.170974
$477$	−32.0147	−1.46585
$478$	9.83995	0.450069
$479$	32.8660	1.50169	0.750843	−	0.660481i	$-0.229648\pi$
$479$	32.8660	1.50169	0.750843	+	0.660481i	$0.229648\pi$
$480$	5.05858	0.230891
$481$	−7.75719	−0.353698
$482$	−27.9491	−1.27305
$483$	−3.15601	−0.143603
$484$	−11.5272	−0.523961
$485$	−9.52373	−0.432450
$486$	−27.1765	−1.23275
$487$	3.21935	0.145882	0.0729412	−	0.997336i	$-0.476761\pi$
$487$	3.21935	0.145882	0.0729412	+	0.997336i	$0.476761\pi$
$488$	−9.94598	−0.450233
$489$	−42.2611	−1.91111
$490$	6.03560	0.272661
$491$	−6.13513	−0.276875	−0.138437	−	0.990371i	$-0.544208\pi$
$491$	−6.13513	−0.276875	−0.138437	+	0.990371i	$0.544208\pi$
$492$	−1.77474	−0.0800112
$493$	18.9577	0.853813
$494$	0.631966	0.0284335
$495$	14.8545	0.667660
$496$	2.91644	0.130952
$497$	−11.6957	−0.524623
$498$	11.5095	0.515751
$499$	22.1282	0.990596	0.495298	−	0.868723i	$-0.335059\pi$
$499$	22.1282	0.990596	0.495298	+	0.868723i	$0.335059\pi$
$500$	3.49270	0.156198
$501$	2.07500	0.0927043
$502$	−27.5760	−1.23078
$503$	−0.0236299	−0.00105361	−0.000526803	−	1.00000i	$-0.500168\pi$
$503$	−0.0236299	−0.00105361	−0.000526803		1.00000i	$0.500168\pi$
$504$	−10.0024	−0.445544
$505$	−7.05196	−0.313808
$506$	8.66320	0.385126
$507$	−2.42435	−0.107669
$508$	5.18488	0.230042
$509$	−28.0899	−1.24506	−0.622532	−	0.782595i	$-0.713896\pi$
$509$	−28.0899	−1.24506	−0.622532	+	0.782595i	$0.713896\pi$
$510$	−18.9379	−0.838585
$511$	−10.6067	−0.469211
$512$	24.9512	1.10270
$513$	0.150488	0.00664419
$514$	−23.0409	−1.01629
$515$	−10.0233	−0.441678
$516$	−10.3077	−0.453772
$517$	−33.9958	−1.49513
$518$	11.0356	0.484878
$519$	−21.7830	−0.956168
$520$	−2.58743	−0.113466
$521$	0.513055	0.0224774	0.0112387	−	0.999937i	$-0.496423\pi$
$521$	0.513055	0.0224774	0.0112387	+	0.999937i	$0.496423\pi$
$522$	9.22577	0.403801
$523$	−13.8949	−0.607580	−0.303790	−	0.952739i	$-0.598252\pi$
$523$	−13.8949	−0.607580	−0.303790	+	0.952739i	$0.598252\pi$
$524$	−3.91348	−0.170961
$525$	−11.8307	−0.516335
$526$	−17.9588	−0.783039
$527$	7.37689	0.321342
$528$	43.0045	1.87153
$529$	−21.6966	−0.943332
$530$	11.7816	0.511761
$531$	−30.3597	−1.31750
$532$	0.256138	0.0111050
$533$	1.65078	0.0715034
$534$	−37.7151	−1.63209
$535$	−9.06400	−0.391871
$536$	21.7521	0.939547
$537$	29.4040	1.26888
$538$	−12.1598	−0.524246
$539$	−34.6675	−1.49323
$540$	−0.111820	−0.00481197
$541$	7.39583	0.317971	0.158986	−	0.987281i	$-0.449178\pi$
$541$	7.39583	0.317971	0.158986	+	0.987281i	$0.449178\pi$
$542$	−28.6506	−1.23065
$543$	9.82847	0.421780
$544$	18.1352	0.777542
$545$	−5.63036	−0.241178
$546$	3.44896	0.147602
$547$	11.9786	0.512167	0.256083	−	0.966655i	$-0.417568\pi$
$547$	11.9786	0.512167	0.256083	+	0.966655i	$0.417568\pi$
$548$	2.66512	0.113848
$549$	−9.38796	−0.400668
$550$	32.4751	1.38474
$551$	−1.30174	−0.0554562
$552$	8.43745	0.359122
$553$	−14.6643	−0.623591
$554$	−22.9936	−0.976904
$555$	−15.9618	−0.677542
$556$	−2.49120	−0.105651
$557$	44.2214	1.87372	0.936860	−	0.349704i	$-0.113718\pi$
$557$	44.2214	1.87372	0.936860	+	0.349704i	$0.113718\pi$
$558$	3.58996	0.151975
$559$	9.58780	0.405521
$560$	2.82260	0.119276
$561$	108.776	4.59253
$562$	28.0005	1.18113
$563$	5.59177	0.235665	0.117833	−	0.993033i	$-0.462405\pi$
$563$	5.59177	0.235665	0.117833	+	0.993033i	$0.462405\pi$
$564$	−6.00900	−0.253025
$565$	−0.718746	−0.0302379
$566$	21.2142	0.891698
$567$	10.6646	0.447872
$568$	31.2679	1.31197
$569$	−17.6209	−0.738706	−0.369353	−	0.929289i	$-0.620421\pi$
$569$	−17.6209	−0.738706	−0.369353	+	0.929289i	$0.620421\pi$
$570$	1.30038	0.0544671
$571$	−19.1101	−0.799732	−0.399866	−	0.916574i	$-0.630943\pi$
$571$	−19.1101	−0.799732	−0.399866	+	0.916574i	$0.630943\pi$
$572$	2.69721	0.112776
$573$	−48.8341	−2.04007
$574$	−2.34846	−0.0980229
$575$	4.88580	0.203752
$576$	25.6094	1.06706
$577$	23.1100	0.962081	0.481040	−	0.876698i	$-0.340259\pi$
$577$	23.1100	0.962081	0.481040	+	0.876698i	$0.340259\pi$
$578$	−46.6839	−1.94179
$579$	−10.7309	−0.445960
$580$	0.967264	0.0401634
$581$	−4.33902	−0.180013
$582$	−33.9390	−1.40682
$583$	−67.6716	−2.80267
$584$	28.3564	1.17340
$585$	−2.44226	−0.100975
$586$	29.2672	1.20902
$587$	14.3558	0.592527	0.296264	−	0.955106i	$-0.404259\pi$
$587$	14.3558	0.592527	0.296264	+	0.955106i	$0.404259\pi$
$588$	−6.12773	−0.252703
$589$	−0.506539	−0.0208716
$590$	11.1726	0.459967
$591$	1.13902	0.0468530
$592$	−22.6234	−0.929816
$593$	24.0583	0.987956	0.493978	−	0.869474i	$-0.335542\pi$
$593$	24.0583	0.987956	0.493978	+	0.869474i	$0.335542\pi$
$594$	−2.25442	−0.0924999
$595$	7.13952	0.292692
$596$	9.65122	0.395329
$597$	41.6635	1.70518
$598$	−1.42434	−0.0582454
$599$	−40.6507	−1.66094	−0.830471	−	0.557062i	$-0.811929\pi$
$599$	−40.6507	−1.66094	−0.830471	+	0.557062i	$0.811929\pi$
$600$	31.6289	1.29124
$601$	11.8772	0.484480	0.242240	−	0.970216i	$-0.422118\pi$
$601$	11.8772	0.484480	0.242240	+	0.970216i	$0.422118\pi$
$602$	−13.6399	−0.555922
$603$	20.5317	0.836114
$604$	−3.79970	−0.154607
$605$	22.0626	0.896972
$606$	−25.1305	−1.02086
$607$	−42.2976	−1.71681	−0.858404	−	0.512975i	$-0.828543\pi$
$607$	−42.2976	−1.71681	−0.858404	+	0.512975i	$0.828543\pi$
$608$	−1.24527	−0.0505023
$609$	−7.10427	−0.287880
$610$	3.45483	0.139882
$611$	5.58932	0.226120
$612$	9.41305	0.380500
$613$	−5.53687	−0.223632	−0.111816	−	0.993729i	$-0.535667\pi$
$613$	−5.53687	−0.223632	−0.111816	+	0.993729i	$0.535667\pi$
$614$	−1.07333	−0.0433162
$615$	3.39679	0.136972
$616$	−21.1428	−0.851869
$617$	8.73793	0.351776	0.175888	−	0.984410i	$-0.443720\pi$
$617$	8.73793	0.351776	0.175888	+	0.984410i	$0.443720\pi$
$618$	−35.7192	−1.43684
$619$	−38.9759	−1.56657	−0.783287	−	0.621660i	$-0.786459\pi$
$619$	−38.9759	−1.56657	−0.783287	+	0.621660i	$0.786459\pi$
$620$	0.376385	0.0151160
$621$	−0.339171	−0.0136105
$622$	37.1655	1.49020
$623$	14.2184	0.569650
$624$	−7.07046	−0.283045
$625$	14.7131	0.588525
$626$	−9.39872	−0.375648
$627$	−7.46919	−0.298291
$628$	5.14338	0.205243
$629$	−57.2240	−2.28167
$630$	3.47444	0.138425
$631$	32.5885	1.29733	0.648663	−	0.761076i	$-0.275328\pi$
$631$	32.5885	1.29733	0.648663	+	0.761076i	$0.275328\pi$
$632$	39.2045	1.55947
$633$	32.5044	1.29194
$634$	9.89912	0.393144
$635$	−9.92370	−0.393810
$636$	−11.9615	−0.474303
$637$	5.69976	0.225833
$638$	19.5011	0.772057
$639$	29.5136	1.16754
$640$	−5.25127	−0.207575
$641$	14.0765	0.555990	0.277995	−	0.960583i	$-0.410330\pi$
$641$	14.0765	0.555990	0.277995	+	0.960583i	$0.410330\pi$
$642$	−32.3007	−1.27481
$643$	49.6636	1.95854	0.979270	−	0.202558i	$-0.0649256\pi$
$643$	49.6636	1.95854	0.979270	+	0.202558i	$0.0649256\pi$
$644$	−0.577288	−0.0227483
$645$	19.7286	0.776814
$646$	4.66194	0.183422
$647$	27.2588	1.07165	0.535826	−	0.844328i	$-0.320000\pi$
$647$	27.2588	1.07165	0.535826	+	0.844328i	$0.320000\pi$
$648$	−28.5114	−1.12003
$649$	−64.1732	−2.51902
$650$	−5.33931	−0.209425
$651$	−2.76444	−0.108347
$652$	−7.73028	−0.302741
$653$	11.0004	0.430480	0.215240	−	0.976561i	$-0.430947\pi$
$653$	11.0004	0.430480	0.215240	+	0.976561i	$0.430947\pi$
$654$	−20.0645	−0.784584
$655$	7.49027	0.292669
$656$	4.81441	0.187971
$657$	26.7655	1.04422
$658$	−7.95156	−0.309984
$659$	−11.1552	−0.434546	−0.217273	−	0.976111i	$-0.569716\pi$
$659$	−11.1552	−0.434546	−0.217273	+	0.976111i	$0.569716\pi$
$660$	5.54999	0.216033
$661$	1.33816	0.0520484	0.0260242	−	0.999661i	$-0.491715\pi$
$661$	1.33816	0.0520484	0.0260242	+	0.999661i	$0.491715\pi$
$662$	−27.7953	−1.08030
$663$	−17.8841	−0.694562
$664$	11.6002	0.450174
$665$	−0.490240	−0.0190107
$666$	−27.8480	−1.07909
$667$	2.93389	0.113601
$668$	0.379553	0.0146853
$669$	28.0484	1.08441
$670$	−7.55579	−0.291906
$671$	−19.8440	−0.766067
$672$	−6.79606	−0.262164
$673$	−4.11650	−0.158680	−0.0793398	−	0.996848i	$-0.525281\pi$
$673$	−4.11650	−0.158680	−0.0793398	+	0.996848i	$0.525281\pi$
$674$	−34.0696	−1.31231
$675$	−1.27143	−0.0489373
$676$	−0.443454	−0.0170559
$677$	−38.2410	−1.46972	−0.734861	−	0.678217i	$-0.762753\pi$
$677$	−38.2410	−1.46972	−0.734861	+	0.678217i	$0.762753\pi$
$678$	−2.56134	−0.0983677
$679$	12.7949	0.491022
$680$	−19.0872	−0.731960
$681$	33.7753	1.29427
$682$	7.58834	0.290573
$683$	21.3265	0.816037	0.408019	−	0.912974i	$-0.366220\pi$
$683$	21.3265	0.816037	0.408019	+	0.912974i	$0.366220\pi$
$684$	−0.646353	−0.0247139
$685$	−5.10095	−0.194897
$686$	−18.0671	−0.689805
$687$	−22.4035	−0.854746
$688$	27.9623	1.06605
$689$	11.1260	0.423869
$690$	−2.93083	−0.111575
$691$	−14.0257	−0.533563	−0.266782	−	0.963757i	$-0.585960\pi$
$691$	−14.0257	−0.533563	−0.266782	+	0.963757i	$0.585960\pi$
$692$	−3.98448	−0.151467
$693$	−19.9566	−0.758089
$694$	19.9208	0.756185
$695$	4.76809	0.180864
$696$	18.9930	0.719926
$697$	12.1776	0.461261
$698$	−22.3160	−0.844673
$699$	−53.5705	−2.02622
$700$	−2.16404	−0.0817930
$701$	−28.8864	−1.09103	−0.545513	−	0.838102i	$-0.683665\pi$
$701$	−28.8864	−1.09103	−0.545513	+	0.838102i	$0.683665\pi$
$702$	0.370654	0.0139894
$703$	3.92932	0.148197
$704$	54.1322	2.04019
$705$	11.5010	0.433154
$706$	−14.3086	−0.538513
$707$	9.47411	0.356311
$708$	−11.3431	−0.426299
$709$	7.33077	0.275313	0.137656	−	0.990480i	$-0.456043\pi$
$709$	7.33077	0.275313	0.137656	+	0.990480i	$0.456043\pi$
$710$	−10.8612	−0.407613
$711$	37.0049	1.38779
$712$	−38.0124	−1.42457
$713$	1.14165	0.0427550
$714$	25.4426	0.952164
$715$	−5.16237	−0.193062
$716$	5.37850	0.201004
$717$	19.1208	0.714080
$718$	−28.2738	−1.05517
$719$	−11.8276	−0.441094	−0.220547	−	0.975376i	$-0.570784\pi$
$719$	−11.8276	−0.441094	−0.220547	+	0.975376i	$0.570784\pi$
$720$	−7.12271	−0.265448
$721$	13.4660	0.501499
$722$	23.3846	0.870284
$723$	−54.3102	−2.01982
$724$	1.79779	0.0668145
$725$	10.9981	0.408459
$726$	78.6229	2.91797
$727$	32.7421	1.21434	0.607168	−	0.794573i	$-0.292305\pi$
$727$	32.7421	1.21434	0.607168	+	0.794573i	$0.292305\pi$
$728$	3.47614	0.128834
$729$	−24.7510	−0.916703
$730$	−9.84987	−0.364560
$731$	70.7281	2.61597
$732$	−3.50756	−0.129643
$733$	9.73587	0.359603	0.179801	−	0.983703i	$-0.442455\pi$
$733$	9.73587	0.359603	0.179801	+	0.983703i	$0.442455\pi$
$734$	1.76696	0.0652198
$735$	11.7283	0.432604
$736$	2.80661	0.103453
$737$	43.3992	1.59863
$738$	5.92625	0.218148
$739$	7.88995	0.290236	0.145118	−	0.989414i	$-0.453644\pi$
$739$	7.88995	0.290236	0.145118	+	0.989414i	$0.453644\pi$
$740$	−2.91969	−0.107330
$741$	1.22803	0.0451127
$742$	−15.8283	−0.581075
$743$	−19.0952	−0.700536	−0.350268	−	0.936650i	$-0.613909\pi$
$743$	−19.0952	−0.700536	−0.350268	+	0.936650i	$0.613909\pi$
$744$	7.39060	0.270953
$745$	−18.4721	−0.676766
$746$	18.6374	0.682365
$747$	10.9493	0.400615
$748$	19.8970	0.727506
$749$	12.1772	0.444946
$750$	−23.8226	−0.869878
$751$	−44.9757	−1.64119	−0.820594	−	0.571512i	$-0.806357\pi$
$751$	−44.9757	−1.64119	−0.820594	+	0.571512i	$0.806357\pi$
$752$	16.3009	0.594433
$753$	−53.5853	−1.95276
$754$	−3.20622	−0.116764
$755$	7.27250	0.264673
$756$	0.150227	0.00546371
$757$	0.358681	0.0130365	0.00651825	−	0.999979i	$-0.497925\pi$
$757$	0.358681	0.0130365	0.00651825	+	0.999979i	$0.497925\pi$
$758$	−13.8718	−0.503846
$759$	16.8342	0.611042
$760$	1.31063	0.0475417
$761$	−15.0001	−0.543751	−0.271876	−	0.962332i	$-0.587644\pi$
$761$	−15.0001	−0.543751	−0.271876	+	0.962332i	$0.587644\pi$
$762$	−35.3644	−1.28112
$763$	7.56423	0.273844
$764$	−8.93258	−0.323169
$765$	−18.0163	−0.651380
$766$	−15.0869	−0.545110
$767$	10.5509	0.380970
$768$	24.4398	0.881896
$769$	−48.1534	−1.73645	−0.868227	−	0.496167i	$-0.834741\pi$
$769$	−48.1534	−1.73645	−0.868227	+	0.496167i	$0.834741\pi$
$770$	7.34416	0.264665
$771$	−44.7727	−1.61245
$772$	−1.96286	−0.0706448
$773$	−34.2344	−1.23132	−0.615662	−	0.788010i	$-0.711111\pi$
$773$	−34.2344	−1.23132	−0.615662	+	0.788010i	$0.711111\pi$
$774$	34.4198	1.23720
$775$	4.27961	0.153728
$776$	−34.2065	−1.22794
$777$	21.4443	0.769309
$778$	−20.2683	−0.726652
$779$	−0.836186	−0.0299595
$780$	−0.912487	−0.0326723
$781$	62.3848	2.23230
$782$	−10.5072	−0.375736
$783$	−0.763485	−0.0272847
$784$	16.6230	0.593679
$785$	−9.84426	−0.351357
$786$	26.6925	0.952091
$787$	16.5666	0.590535	0.295267	−	0.955415i	$-0.404591\pi$
$787$	16.5666	0.590535	0.295267	+	0.955415i	$0.404591\pi$
$788$	0.208346	0.00742202
$789$	−34.8972	−1.24237
$790$	−13.6180	−0.484508
$791$	0.965615	0.0343333
$792$	53.3531	1.89582
$793$	3.26259	0.115858
$794$	−10.5261	−0.373557
$795$	22.8939	0.811961
$796$	7.62097	0.270118
$797$	−35.1847	−1.24631	−0.623153	−	0.782100i	$-0.714149\pi$
$797$	−35.1847	−1.24631	−0.623153	+	0.782100i	$0.714149\pi$
$798$	−1.74703	−0.0618442
$799$	41.2318	1.45868
$800$	10.5209	0.371971
$801$	−35.8797	−1.26775
$802$	28.8285	1.01797
$803$	56.5760	1.99652
$804$	7.67112	0.270540
$805$	1.10491	0.0389430
$806$	−1.24762	−0.0439454
$807$	−23.6287	−0.831770
$808$	−25.3286	−0.891057
$809$	12.0448	0.423473	0.211737	−	0.977327i	$-0.432088\pi$
$809$	12.0448	0.423473	0.211737	+	0.977327i	$0.432088\pi$
$810$	9.90370	0.347981
$811$	−15.7184	−0.551949	−0.275975	−	0.961165i	$-0.589001\pi$
$811$	−15.7184	−0.551949	−0.275975	+	0.961165i	$0.589001\pi$
$812$	−1.29949	−0.0456032
$813$	−55.6735	−1.95255
$814$	−58.8642	−2.06319
$815$	14.7955	0.518264
$816$	−52.1580	−1.82590
$817$	−4.85659	−0.169911
$818$	3.14453	0.109946
$819$	3.28111	0.114651
$820$	0.621330	0.0216978
$821$	−6.54313	−0.228357	−0.114178	−	0.993460i	$-0.536424\pi$
$821$	−6.54313	−0.228357	−0.114178	+	0.993460i	$0.536424\pi$
$822$	−18.1779	−0.634026
$823$	−16.0972	−0.561112	−0.280556	−	0.959838i	$-0.590519\pi$
$823$	−16.0972	−0.561112	−0.280556	+	0.959838i	$0.590519\pi$
$824$	−36.0007	−1.25414
$825$	63.1051	2.19704
$826$	−15.0100	−0.522265
$827$	−2.83054	−0.0984275	−0.0492137	−	0.998788i	$-0.515672\pi$
$827$	−2.83054	−0.0984275	−0.0492137	+	0.998788i	$0.515672\pi$
$828$	1.45676	0.0506260
$829$	−11.6064	−0.403108	−0.201554	−	0.979477i	$-0.564599\pi$
$829$	−11.6064	−0.403108	−0.201554	+	0.979477i	$0.564599\pi$
$830$	−4.02943	−0.139864
$831$	−44.6807	−1.54996
$832$	−8.90000	−0.308552
$833$	42.0465	1.45682
$834$	16.9917	0.588374
$835$	−0.726453	−0.0251399
$836$	−1.36624	−0.0472524
$837$	−0.297090	−0.0102689
$838$	38.3952	1.32634
$839$	−17.7885	−0.614126	−0.307063	−	0.951689i	$-0.599346\pi$
$839$	−17.7885	−0.614126	−0.307063	+	0.951689i	$0.599346\pi$
$840$	7.15279	0.246795
$841$	−22.3957	−0.772266
$842$	13.6922	0.471864
$843$	54.4102	1.87399
$844$	5.94561	0.204656
$845$	0.848757	0.0291981
$846$	20.0654	0.689864
$847$	−29.6405	−1.01846
$848$	32.4485	1.11428
$849$	41.2230	1.41477
$850$	−39.3875	−1.35098
$851$	−8.85597	−0.303579
$852$	11.0270	0.377778
$853$	17.9052	0.613063	0.306531	−	0.951861i	$-0.400832\pi$
$853$	17.9052	0.613063	0.306531	+	0.951861i	$0.400832\pi$
$854$	−4.64147	−0.158828
$855$	1.23710	0.0423079
$856$	−32.5553	−1.11272
$857$	8.51005	0.290698	0.145349	−	0.989380i	$-0.453570\pi$
$857$	8.51005	0.290698	0.145349	+	0.989380i	$0.453570\pi$
$858$	−18.3968	−0.628055
$859$	47.1152	1.60755	0.803775	−	0.594934i	$-0.202822\pi$
$859$	47.1152	1.60755	0.803775	+	0.594934i	$0.202822\pi$
$860$	3.60870	0.123056
$861$	−4.56349	−0.155523
$862$	37.9529	1.29268
$863$	−3.92657	−0.133662	−0.0668309	−	0.997764i	$-0.521289\pi$
$863$	−3.92657	−0.133662	−0.0668309	+	0.997764i	$0.521289\pi$
$864$	−0.730362	−0.0248474
$865$	7.62617	0.259298
$866$	1.32397	0.0449905
$867$	−90.7154	−3.08086
$868$	−0.505663	−0.0171633
$869$	78.2197	2.65342
$870$	−6.59739	−0.223672
$871$	−7.13535	−0.241772
$872$	−20.2226	−0.684825
$873$	−32.2874	−1.09276
$874$	0.721482	0.0244045
$875$	8.98101	0.303614
$876$	10.0002	0.337876
$877$	27.6540	0.933808	0.466904	−	0.884308i	$-0.345369\pi$
$877$	27.6540	0.933808	0.466904	+	0.884308i	$0.345369\pi$
$878$	4.10321	0.138477
$879$	56.8716	1.91823
$880$	−15.0557	−0.507529
$881$	−47.2655	−1.59241	−0.796207	−	0.605024i	$-0.793163\pi$
$881$	−47.2655	−1.59241	−0.796207	+	0.605024i	$0.793163\pi$
$882$	20.4619	0.688988
$883$	34.7649	1.16993	0.584966	−	0.811058i	$-0.301108\pi$
$883$	34.7649	1.16993	0.584966	+	0.811058i	$0.301108\pi$
$884$	−3.27131	−0.110026
$885$	21.7103	0.729784
$886$	11.0631	0.371672
$887$	9.66079	0.324378	0.162189	−	0.986760i	$-0.448145\pi$
$887$	9.66079	0.324378	0.162189	+	0.986760i	$0.448145\pi$
$888$	−57.3303	−1.92388
$889$	13.3322	0.447148
$890$	13.2040	0.442598
$891$	−56.8851	−1.90572
$892$	5.13053	0.171783
$893$	−2.83121	−0.0947428
$894$	−65.8278	−2.20161
$895$	−10.2943	−0.344100
$896$	7.05494	0.235689
$897$	−2.76775	−0.0924124
$898$	−10.7969	−0.360299
$899$	2.56988	0.0857103
$900$	5.46087	0.182029
$901$	82.0756	2.73434
$902$	12.5267	0.417094
$903$	−26.5049	−0.882027
$904$	−2.58153	−0.0858604
$905$	−3.44092	−0.114380
$906$	25.9165	0.861017
$907$	22.2740	0.739597	0.369798	−	0.929112i	$-0.379427\pi$
$907$	22.2740	0.739597	0.369798	+	0.929112i	$0.379427\pi$
$908$	6.17808	0.205027
$909$	−23.9075	−0.792963
$910$	−1.20747	−0.0400273
$911$	19.9339	0.660441	0.330220	−	0.943904i	$-0.392877\pi$
$911$	19.9339	0.660441	0.330220	+	0.943904i	$0.392877\pi$
$912$	3.58146	0.118594
$913$	23.1444	0.765966
$914$	20.8821	0.690717
$915$	6.71337	0.221937
$916$	−4.09798	−0.135401
$917$	−10.0630	−0.332309
$918$	2.73427	0.0902445
$919$	−14.4899	−0.477977	−0.238989	−	0.971022i	$-0.576816\pi$
$919$	−14.4899	−0.477977	−0.238989	+	0.971022i	$0.576816\pi$
$920$	−2.95393	−0.0973882
$921$	−2.08568	−0.0687256
$922$	−24.9846	−0.822825
$923$	−10.2568	−0.337608
$924$	−7.45626	−0.245293
$925$	−33.1978	−1.09154
$926$	22.7482	0.747553
$927$	−33.9809	−1.11608
$928$	6.31776	0.207391
$929$	−19.6926	−0.646094	−0.323047	−	0.946383i	$-0.604707\pi$
$929$	−19.6926	−0.646094	−0.323047	+	0.946383i	$0.604707\pi$
$930$	−2.56720	−0.0841817
$931$	−2.88715	−0.0946225
$932$	−9.79895	−0.320975
$933$	72.2193	2.36435
$934$	35.1409	1.14985
$935$	−38.0822	−1.24542
$936$	−8.77190	−0.286719
$937$	21.4176	0.699681	0.349841	−	0.936809i	$-0.386236\pi$
$937$	21.4176	0.699681	0.349841	+	0.936809i	$0.386236\pi$
$938$	10.1510	0.331442
$939$	−18.2634	−0.596005
$940$	2.10374	0.0686163
$941$	7.47165	0.243569	0.121784	−	0.992557i	$-0.461138\pi$
$941$	7.47165	0.243569	0.121784	+	0.992557i	$0.461138\pi$
$942$	−35.0813	−1.14301
$943$	1.88461	0.0613714
$944$	30.7710	1.00151
$945$	−0.287530	−0.00935335
$946$	72.7555	2.36548
$947$	31.4944	1.02343	0.511715	−	0.859155i	$-0.329010\pi$
$947$	31.4944	1.02343	0.511715	+	0.859155i	$0.329010\pi$
$948$	13.8259	0.449045
$949$	−9.30178	−0.301949
$950$	2.70457	0.0877478
$951$	19.2358	0.623764
$952$	25.6431	0.831098
$953$	−23.6635	−0.766534	−0.383267	−	0.923638i	$-0.625201\pi$
$953$	−23.6635	−0.766534	−0.383267	+	0.923638i	$0.625201\pi$
$954$	39.9421	1.29317
$955$	17.0967	0.553236
$956$	3.49752	0.113118
$957$	37.8942	1.22495
$958$	−41.0041	−1.32478
$959$	6.85298	0.221294
$960$	−18.3134	−0.591061
$961$	1.00000	0.0322581
$962$	9.67800	0.312031
$963$	−30.7288	−0.990220
$964$	−9.93426	−0.319961
$965$	3.75685	0.120937
$966$	3.93749	0.126687
$967$	16.1708	0.520017	0.260009	−	0.965606i	$-0.416275\pi$
$967$	16.1708	0.520017	0.260009	+	0.965606i	$0.416275\pi$
$968$	79.2426	2.54695
$969$	9.05901	0.291017
$970$	11.8820	0.381507
$971$	−13.1072	−0.420631	−0.210315	−	0.977634i	$-0.567449\pi$
$971$	−13.1072	−0.420631	−0.210315	+	0.977634i	$0.567449\pi$
$972$	−9.65963	−0.309833
$973$	−6.40579	−0.205360
$974$	−4.01651	−0.128697
$975$	−10.3753	−0.332274
$976$	9.51515	0.304572
$977$	53.2887	1.70486	0.852428	−	0.522845i	$-0.175129\pi$
$977$	53.2887	1.70486	0.852428	+	0.522845i	$0.175129\pi$
$978$	52.7257	1.68598
$979$	−75.8413	−2.42390
$980$	2.14530	0.0685291
$981$	−19.0880	−0.609434
$982$	7.65429	0.244258
$983$	−11.4544	−0.365340	−0.182670	−	0.983174i	$-0.558474\pi$
$983$	−11.4544	−0.365340	−0.182670	+	0.983174i	$0.558474\pi$
$984$	12.2003	0.388931
$985$	−0.398768	−0.0127058
$986$	−23.6520	−0.753232
$987$	−15.4513	−0.491821
$988$	0.224627	0.00714633
$989$	10.9459	0.348059
$990$	−18.5327	−0.589008
$991$	56.1997	1.78524	0.892621	−	0.450807i	$-0.148864\pi$
$991$	56.1997	1.78524	0.892621	+	0.450807i	$0.148864\pi$
$992$	2.45839	0.0780539
$993$	−54.0114	−1.71400
$994$	14.5917	0.462821
$995$	−14.5863	−0.462416
$996$	4.09094	0.129626
$997$	−28.9682	−0.917431	−0.458715	−	0.888583i	$-0.651690\pi$
$997$	−28.9682	−0.917431	−0.458715	+	0.888583i	$0.651690\pi$
$998$	−27.6075	−0.873901
$999$	2.30458	0.0729138

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	403.2.a.d.1.4	✓	8
3.2	odd	2		3627.2.a.q.1.5		8
4.3	odd	2		6448.2.a.bf.1.7		8
13.12	even	2		5239.2.a.j.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
403.2.a.d.1.4	✓	8	1.1	even	1	trivial
3627.2.a.q.1.5		8	3.2	odd	2
5239.2.a.j.1.5		8	13.12	even	2
6448.2.a.bf.1.7		8	4.3	odd	2

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:	\( N \)	\(=\)	\( 403 = 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	403.a (trivial)

\(f(q)\)	\(=\)	\(q-1.24762 q^{2} -2.42435 q^{3} -0.443454 q^{4} +0.848757 q^{5} +3.02465 q^{6} -1.14028 q^{7} +3.04849 q^{8} +2.87746 q^{9} +O(q^{10})\)	\(q-1.24762 q^{2} -2.42435 q^{3} -0.443454 q^{4} +0.848757 q^{5} +3.02465 q^{6} -1.14028 q^{7} +3.04849 q^{8} +2.87746 q^{9} -1.05892 q^{10} +6.08227 q^{11} +1.07509 q^{12} -1.00000 q^{13} +1.42263 q^{14} -2.05768 q^{15} -2.91644 q^{16} -7.37689 q^{17} -3.58996 q^{18} +0.506539 q^{19} -0.376385 q^{20} +2.76444 q^{21} -7.58834 q^{22} -1.14165 q^{23} -7.39060 q^{24} -4.27961 q^{25} +1.24762 q^{26} +0.297090 q^{27} +0.505663 q^{28} -2.56988 q^{29} +2.56720 q^{30} -1.00000 q^{31} -2.45839 q^{32} -14.7455 q^{33} +9.20352 q^{34} -0.967822 q^{35} -1.27602 q^{36} +7.75719 q^{37} -0.631966 q^{38} +2.42435 q^{39} +2.58743 q^{40} -1.65078 q^{41} -3.44896 q^{42} -9.58780 q^{43} -2.69721 q^{44} +2.44226 q^{45} +1.42434 q^{46} -5.58932 q^{47} +7.07046 q^{48} -5.69976 q^{49} +5.33931 q^{50} +17.8841 q^{51} +0.443454 q^{52} -11.1260 q^{53} -0.370654 q^{54} +5.16237 q^{55} -3.47614 q^{56} -1.22803 q^{57} +3.20622 q^{58} -10.5509 q^{59} +0.912487 q^{60} -3.26259 q^{61} +1.24762 q^{62} -3.28111 q^{63} +8.90000 q^{64} -0.848757 q^{65} +18.3968 q^{66} +7.13535 q^{67} +3.27131 q^{68} +2.76775 q^{69} +1.20747 q^{70} +10.2568 q^{71} +8.77190 q^{72} +9.30178 q^{73} -9.67800 q^{74} +10.3753 q^{75} -0.224627 q^{76} -6.93550 q^{77} -3.02465 q^{78} +12.8603 q^{79} -2.47535 q^{80} -9.35262 q^{81} +2.05954 q^{82} +3.80522 q^{83} -1.22590 q^{84} -6.26119 q^{85} +11.9619 q^{86} +6.23028 q^{87} +18.5418 q^{88} -12.4692 q^{89} -3.04700 q^{90} +1.14028 q^{91} +0.506268 q^{92} +2.42435 q^{93} +6.97333 q^{94} +0.429928 q^{95} +5.95998 q^{96} -11.2208 q^{97} +7.11111 q^{98} +17.5015 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9}+O(q^{10}) \) 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9	\( 8 q - 5 q^{2} - 3 q^{3} + 9 q^{4} - 15 q^{5} - 4 q^{7} - 15 q^{8} + 9 q^{9} + 9 q^{10} - 5 q^{11} - 9 q^{12} - 8 q^{13} + 2 q^{14} + 2 q^{15} + 3 q^{16} - 11 q^{17} - 30 q^{18} - 9 q^{19} - 31 q^{20} - 16 q^{21} - 2 q^{22} + 13 q^{24} + 19 q^{25} + 5 q^{26} - 9 q^{27} + 16 q^{28} - 12 q^{29} - 7 q^{30} - 8 q^{31} - 25 q^{32} - 14 q^{33} + 22 q^{34} + 7 q^{35} + 37 q^{36} - 9 q^{37} - 9 q^{38} + 3 q^{39} + 55 q^{40} - 25 q^{41} - 3 q^{42} + 7 q^{43} - 26 q^{44} - 45 q^{45} + 5 q^{46} - 17 q^{47} - 9 q^{48} - 11 q^{50} - 10 q^{51} - 9 q^{52} - 15 q^{53} + 54 q^{54} + 7 q^{55} - 14 q^{56} - 7 q^{57} - 5 q^{58} - 15 q^{59} + 61 q^{60} + 11 q^{61} + 5 q^{62} - 21 q^{63} + 47 q^{64} + 15 q^{65} + 83 q^{66} + 18 q^{67} - 16 q^{68} - 15 q^{69} - 24 q^{70} - 7 q^{71} - 21 q^{72} + 24 q^{73} + 48 q^{74} - 17 q^{75} - 3 q^{76} - 49 q^{77} + 33 q^{79} - 16 q^{80} + 20 q^{81} - q^{82} - 13 q^{83} - 6 q^{84} + q^{85} + 19 q^{86} + 18 q^{87} + 37 q^{88} - 23 q^{89} + 117 q^{90} + 4 q^{91} + 22 q^{92} + 3 q^{93} + 10 q^{94} + 43 q^{95} + 46 q^{96} - 17 q^{97} + 52 q^{98} + 51 q^{99}+O(q^{100}) \) 8 * q - 5 * q^2 - 3 * q^3 + 9 * q^4 - 15 * q^5 - 4 * q^7 - 15 * q^8 + 9 * q^9 + 9 * q^10 - 5 * q^11 - 9 * q^12 - 8 * q^13 + 2 * q^14 + 2 * q^15 + 3 * q^16 - 11 * q^17 - 30 * q^18 - 9 * q^19 - 31 * q^20 - 16 * q^21 - 2 * q^22 + 13 * q^24 + 19 * q^25 + 5 * q^26 - 9 * q^27 + 16 * q^28 - 12 * q^29 - 7 * q^30 - 8 * q^31 - 25 * q^32 - 14 * q^33 + 22 * q^34 + 7 * q^35 + 37 * q^36 - 9 * q^37 - 9 * q^38 + 3 * q^39 + 55 * q^40 - 25 * q^41 - 3 * q^42 + 7 * q^43 - 26 * q^44 - 45 * q^45 + 5 * q^46 - 17 * q^47 - 9 * q^48 - 11 * q^50 - 10 * q^51 - 9 * q^52 - 15 * q^53 + 54 * q^54 + 7 * q^55 - 14 * q^56 - 7 * q^57 - 5 * q^58 - 15 * q^59 + 61 * q^60 + 11 * q^61 + 5 * q^62 - 21 * q^63 + 47 * q^64 + 15 * q^65 + 83 * q^66 + 18 * q^67 - 16 * q^68 - 15 * q^69 - 24 * q^70 - 7 * q^71 - 21 * q^72 + 24 * q^73 + 48 * q^74 - 17 * q^75 - 3 * q^76 - 49 * q^77 + 33 * q^79 - 16 * q^80 + 20 * q^81 - q^82 - 13 * q^83 - 6 * q^84 + q^85 + 19 * q^86 + 18 * q^87 + 37 * q^88 - 23 * q^89 + 117 * q^90 + 4 * q^91 + 22 * q^92 + 3 * q^93 + 10 * q^94 + 43 * q^95 + 46 * q^96 - 17 * q^97 + 52 * q^98 + 51 * q^99

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