LMFDB - Embedded newform 847.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [847,2,Mod(1,847)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(847, 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("847.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 847 = 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	847.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:	$6.76332905120$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.46673$ of defining polynomial
Character	$\chi$	$=$	847.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46673 q^{2} -1.61803 q^{3} +4.08477 q^{4} -3.46673 q^{5} +3.99126 q^{6} +1.00000 q^{7} -5.14256 q^{8} -0.381966 q^{9} +O(q^{10})$	\(q-2.46673 q^{2} -1.61803 q^{3} +4.08477 q^{4} -3.46673 q^{5} +3.99126 q^{6} +1.00000 q^{7} -5.14256 q^{8} -0.381966 q^{9} +8.55150 q^{10} -6.60929 q^{12} +0.653752 q^{13} -2.46673 q^{14} +5.60929 q^{15} +4.51578 q^{16} +1.13715 q^{17} +0.942208 q^{18} +6.07602 q^{19} -14.1608 q^{20} -1.61803 q^{21} -6.66708 q^{23} +8.32083 q^{24} +7.01823 q^{25} -1.61263 q^{26} +5.47214 q^{27} +4.08477 q^{28} +4.57357 q^{29} -13.8366 q^{30} +2.79631 q^{31} -0.854102 q^{32} -2.80505 q^{34} -3.46673 q^{35} -1.56024 q^{36} -0.439758 q^{37} -14.9879 q^{38} -1.05779 q^{39} +17.8279 q^{40} +5.90315 q^{41} +3.99126 q^{42} -8.70820 q^{43} +1.32417 q^{45} +16.4459 q^{46} -0.604703 q^{47} -7.30669 q^{48} +1.00000 q^{49} -17.3121 q^{50} -1.83995 q^{51} +2.67042 q^{52} +9.82247 q^{53} -13.4983 q^{54} -5.14256 q^{56} -9.83121 q^{57} -11.2818 q^{58} +1.69406 q^{59} +22.9126 q^{60} -6.85818 q^{61} -6.89775 q^{62} -0.381966 q^{63} -6.92472 q^{64} -2.26638 q^{65} -6.17828 q^{67} +4.64501 q^{68} +10.7876 q^{69} +8.55150 q^{70} -5.41687 q^{71} +1.96428 q^{72} -6.70198 q^{73} +1.08477 q^{74} -11.3557 q^{75} +24.8191 q^{76} +2.60929 q^{78} -2.65375 q^{79} -15.6550 q^{80} -7.70820 q^{81} -14.5615 q^{82} -6.69658 q^{83} -6.60929 q^{84} -3.94221 q^{85} +21.4808 q^{86} -7.40020 q^{87} -0.698213 q^{89} -3.26638 q^{90} +0.653752 q^{91} -27.2335 q^{92} -4.52452 q^{93} +1.49164 q^{94} -21.0639 q^{95} +1.38197 q^{96} -14.8587 q^{97} -2.46673 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9	$ 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9} + 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} + 3 q^{17} + 3 q^{18} - 3 q^{19} - 17 q^{20} - 2 q^{21} - 8 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} - 12 q^{30} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} - 5 q^{39} + 13 q^{40} - 4 q^{41} + q^{42} - 8 q^{43} + 9 q^{45} + 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} - 33 q^{50} + 11 q^{51} + 17 q^{52} - 9 q^{53} - 2 q^{54} - 9 q^{56} - 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} + 19 q^{61} - 10 q^{62} - 6 q^{63} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} + 6 q^{72} + 11 q^{73} - 8 q^{74} - 5 q^{75} + 26 q^{76} - 9 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} + q^{83} - 7 q^{84} - 15 q^{85} + 4 q^{86} - 6 q^{87} - 17 q^{89} - 16 q^{90} - 17 q^{92} - 11 q^{93} + 20 q^{94} - 17 q^{95} + 10 q^{96} - 15 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9 + 14 * q^10 - 7 * q^12 - 2 * q^14 + 3 * q^15 - 4 * q^16 + 3 * q^17 + 3 * q^18 - 3 * q^19 - 17 * q^20 - 2 * q^21 - 8 * q^23 + 12 * q^24 - 12 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 - 12 * q^30 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - q^36 - 7 * q^37 - 20 * q^38 - 5 * q^39 + 13 * q^40 - 4 * q^41 + q^42 - 8 * q^43 + 9 * q^45 + 3 * q^46 - 14 * q^47 - 3 * q^48 + 4 * q^49 - 33 * q^50 + 11 * q^51 + 17 * q^52 - 9 * q^53 - 2 * q^54 - 9 * q^56 - 6 * q^57 + 3 * q^58 - 25 * q^59 + 21 * q^60 + 19 * q^61 - 10 * q^62 - 6 * q^63 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^69 + 14 * q^70 - 7 * q^71 + 6 * q^72 + 11 * q^73 - 8 * q^74 - 5 * q^75 + 26 * q^76 - 9 * q^78 - 8 * q^79 - 4 * q^80 - 4 * q^81 + 3 * q^82 + q^83 - 7 * q^84 - 15 * q^85 + 4 * q^86 - 6 * q^87 - 17 * q^89 - 16 * q^90 - 17 * q^92 - 11 * q^93 + 20 * q^94 - 17 * q^95 + 10 * q^96 - 15 * q^97 - 2 * q^98

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 $

$2$	−2.46673	−1.74424	−0.872121	−	0.489290i	$-0.837256\pi$
$2$	−2.46673	−1.74424	−0.872121	+	0.489290i	$0.837256\pi$
$3$	−1.61803	−0.934172	−0.467086	−	0.884212i	$-0.654696\pi$
$3$	−1.61803	−0.934172	−0.467086	+	0.884212i	$0.654696\pi$
$4$	4.08477	2.04238
$5$	−3.46673	−1.55037	−0.775185	−	0.631735i	$-0.782343\pi$
$5$	−3.46673	−1.55037	−0.775185	+	0.631735i	$0.782343\pi$
$6$	3.99126	1.62942
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−5.14256	−1.81817
$9$	−0.381966	−0.127322
$10$	8.55150	2.70422
$11$	0	0
$11$	0	0
$12$	−6.60929	−1.90794
$13$	0.653752	0.181318	0.0906590	−	0.995882i	$-0.471103\pi$
$13$	0.653752	0.181318	0.0906590	+	0.995882i	$0.471103\pi$
$14$	−2.46673	−0.659262
$15$	5.60929	1.44831
$16$	4.51578	1.12894
$17$	1.13715	0.275800	0.137900	−	0.990446i	$-0.455965\pi$
$17$	1.13715	0.275800	0.137900	+	0.990446i	$0.455965\pi$
$18$	0.942208	0.222080
$19$	6.07602	1.39393	0.696967	−	0.717103i	$-0.254532\pi$
$19$	6.07602	1.39393	0.696967	+	0.717103i	$0.254532\pi$
$20$	−14.1608	−3.16645
$21$	−1.61803	−0.353084
$22$	0	0
$23$	−6.66708	−1.39018	−0.695091	−	0.718921i	$-0.744636\pi$
$23$	−6.66708	−1.39018	−0.695091	+	0.718921i	$0.744636\pi$
$24$	8.32083	1.69848
$25$	7.01823	1.40365
$26$	−1.61263	−0.316263
$27$	5.47214	1.05311
$28$	4.08477	0.771948
$29$	4.57357	0.849291	0.424646	−	0.905360i	$-0.360399\pi$
$29$	4.57357	0.849291	0.424646	+	0.905360i	$0.360399\pi$
$30$	−13.8366	−2.52621
$31$	2.79631	0.502232	0.251116	−	0.967957i	$-0.419202\pi$
$31$	2.79631	0.502232	0.251116	+	0.967957i	$0.419202\pi$
$32$	−0.854102	−0.150985
$33$	0	0
$34$	−2.80505	−0.481063
$35$	−3.46673	−0.585985
$36$	−1.56024	−0.260040
$37$	−0.439758	−0.0722958	−0.0361479	−	0.999346i	$-0.511509\pi$
$37$	−0.439758	−0.0722958	−0.0361479	+	0.999346i	$0.511509\pi$
$38$	−14.9879	−2.43136
$39$	−1.05779	−0.169382
$40$	17.8279	2.81883
$41$	5.90315	0.921917	0.460959	−	0.887422i	$-0.347506\pi$
$41$	5.90315	0.921917	0.460959	+	0.887422i	$0.347506\pi$
$42$	3.99126	0.615864
$43$	−8.70820	−1.32799	−0.663994	−	0.747738i	$-0.731140\pi$
$43$	−8.70820	−1.32799	−0.663994	+	0.747738i	$0.731140\pi$
$44$	0	0
$45$	1.32417	0.197396
$46$	16.4459	2.42482
$47$	−0.604703	−0.0882051	−0.0441025	−	0.999027i	$-0.514043\pi$
$47$	−0.604703	−0.0882051	−0.0441025	+	0.999027i	$0.514043\pi$
$48$	−7.30669	−1.05463
$49$	1.00000	0.142857
$50$	−17.3121	−2.44830
$51$	−1.83995	−0.257645
$52$	2.67042	0.370321
$53$	9.82247	1.34922	0.674610	−	0.738175i	$-0.264312\pi$
$53$	9.82247	1.34922	0.674610	+	0.738175i	$0.264312\pi$
$54$	−13.4983	−1.83688
$55$	0	0
$56$	−5.14256	−0.687203
$57$	−9.83121	−1.30218
$58$	−11.2818	−1.48137
$59$	1.69406	0.220547	0.110274	−	0.993901i	$-0.464827\pi$
$59$	1.69406	0.220547	0.110274	+	0.993901i	$0.464827\pi$
$60$	22.9126	2.95801
$61$	−6.85818	−0.878100	−0.439050	−	0.898463i	$-0.644685\pi$
$61$	−6.85818	−0.878100	−0.439050	+	0.898463i	$0.644685\pi$
$62$	−6.89775	−0.876015
$63$	−0.381966	−0.0481232
$64$	−6.92472	−0.865590
$65$	−2.26638	−0.281110
$66$	0	0
$67$	−6.17828	−0.754797	−0.377398	−	0.926051i	$-0.623181\pi$
$67$	−6.17828	−0.754797	−0.377398	+	0.926051i	$0.623181\pi$
$68$	4.64501	0.563290
$69$	10.7876	1.29867
$70$	8.55150	1.02210
$71$	−5.41687	−0.642864	−0.321432	−	0.946933i	$-0.604164\pi$
$71$	−5.41687	−0.642864	−0.321432	+	0.946933i	$0.604164\pi$
$72$	1.96428	0.231493
$73$	−6.70198	−0.784408	−0.392204	−	0.919878i	$-0.628287\pi$
$73$	−6.70198	−0.784408	−0.392204	+	0.919878i	$0.628287\pi$
$74$	1.08477	0.126101
$75$	−11.3557	−1.31125
$76$	24.8191	2.84695
$77$	0	0
$78$	2.60929	0.295444
$79$	−2.65375	−0.298570	−0.149285	−	0.988794i	$-0.547697\pi$
$79$	−2.65375	−0.298570	−0.149285	+	0.988794i	$0.547697\pi$
$80$	−15.6550	−1.75028
$81$	−7.70820	−0.856467
$82$	−14.5615	−1.60805
$83$	−6.69658	−0.735045	−0.367522	−	0.930015i	$-0.619794\pi$
$83$	−6.69658	−0.735045	−0.367522	+	0.930015i	$0.619794\pi$
$84$	−6.60929	−0.721133
$85$	−3.94221	−0.427592
$86$	21.4808	2.31633
$87$	−7.40020	−0.793384
$88$	0	0
$89$	−0.698213	−0.0740105	−0.0370052	−	0.999315i	$-0.511782\pi$
$89$	−0.698213	−0.0740105	−0.0370052	+	0.999315i	$0.511782\pi$
$90$	−3.26638	−0.344307
$91$	0.653752	0.0685318
$92$	−27.2335	−2.83929
$93$	−4.52452	−0.469171
$94$	1.49164	0.153851
$95$	−21.0639	−2.16111
$96$	1.38197	0.141046
$97$	−14.8587	−1.50867	−0.754336	−	0.656489i	$-0.772041\pi$
$97$	−14.8587	−1.50867	−0.754336	+	0.656489i	$0.772041\pi$
$98$	−2.46673	−0.249178
$99$	0	0
$100$	28.6678	2.86678
$101$	−8.61959	−0.857682	−0.428841	−	0.903380i	$-0.641078\pi$
$101$	−8.61959	−0.857682	−0.428841	+	0.903380i	$0.641078\pi$
$102$	4.53867	0.449396
$103$	−0.932958	−0.0919271	−0.0459636	−	0.998943i	$-0.514636\pi$
$103$	−0.932958	−0.0919271	−0.0459636	+	0.998943i	$0.514636\pi$
$104$	−3.36196	−0.329667
$105$	5.60929	0.547411
$106$	−24.2294	−2.35337
$107$	−6.62212	−0.640184	−0.320092	−	0.947386i	$-0.603714\pi$
$107$	−6.62212	−0.640184	−0.320092	+	0.947386i	$0.603714\pi$
$108$	22.3524	2.15086
$109$	4.12507	0.395110	0.197555	−	0.980292i	$-0.436700\pi$
$109$	4.12507	0.395110	0.197555	+	0.980292i	$0.436700\pi$
$110$	0	0
$111$	0.711544	0.0675368
$112$	4.51578	0.426701
$113$	−18.8258	−1.77098	−0.885491	−	0.464656i	$-0.846178\pi$
$113$	−18.8258	−1.77098	−0.885491	+	0.464656i	$0.846178\pi$
$114$	24.2510	2.27131
$115$	23.1130	2.15530
$116$	18.6820	1.73458
$117$	−0.249711	−0.0230858
$118$	−4.17878	−0.384688
$119$	1.13715	0.104243
$120$	−28.8461	−2.63328
$121$	0	0
$122$	16.9173	1.53162
$123$	−9.55150	−0.861230
$124$	11.4223	1.02575
$125$	−6.99666	−0.625800
$126$	0.942208	0.0839385
$127$	7.96635	0.706899	0.353449	−	0.935454i	$-0.385009\pi$
$127$	7.96635	0.706899	0.353449	+	0.935454i	$0.385009\pi$
$128$	18.7896	1.66078
$129$	14.0902	1.24057
$130$	5.59056	0.490324
$131$	4.80505	0.419819	0.209910	−	0.977721i	$-0.432683\pi$
$131$	4.80505	0.419819	0.209910	+	0.977721i	$0.432683\pi$
$132$	0	0
$133$	6.07602	0.526858
$134$	15.2401	1.31655
$135$	−18.9704	−1.63271
$136$	−5.84788	−0.501452
$137$	−21.8777	−1.86914	−0.934571	−	0.355778i	$-0.884216\pi$
$137$	−21.8777	−1.86914	−0.934571	+	0.355778i	$0.884216\pi$
$138$	−26.6100	−2.26520
$139$	19.8137	1.68058	0.840289	−	0.542139i	$-0.182385\pi$
$139$	19.8137	1.68058	0.840289	+	0.542139i	$0.182385\pi$
$140$	−14.1608	−1.19680
$141$	0.978431	0.0823987
$142$	13.3620	1.12131
$143$	0	0
$144$	−1.72487	−0.143740
$145$	−15.8553	−1.31672
$146$	16.5320	1.36820
$147$	−1.61803	−0.133453
$148$	−1.79631	−0.147656
$149$	−3.16211	−0.259050	−0.129525	−	0.991576i	$-0.541345\pi$
$149$	−3.16211	−0.259050	−0.129525	+	0.991576i	$0.541345\pi$
$150$	28.0115	2.28713
$151$	−8.92806	−0.726555	−0.363278	−	0.931681i	$-0.618342\pi$
$151$	−8.92806	−0.726555	−0.363278	+	0.931681i	$0.618342\pi$
$152$	−31.2463	−2.53441
$153$	−0.434354	−0.0351155
$154$	0	0
$155$	−9.69406	−0.778645
$156$	−4.32083	−0.345944
$157$	1.13968	0.0909561	0.0454780	−	0.998965i	$-0.485519\pi$
$157$	1.13968	0.0909561	0.0454780	+	0.998965i	$0.485519\pi$
$158$	6.54609	0.520779
$159$	−15.8931	−1.26040
$160$	2.96094	0.234083
$161$	−6.66708	−0.525440
$162$	19.0141	1.49389
$163$	−5.06728	−0.396900	−0.198450	−	0.980111i	$-0.563591\pi$
$163$	−5.06728	−0.396900	−0.198450	+	0.980111i	$0.563591\pi$
$164$	24.1130	1.88291
$165$	0	0
$166$	16.5187	1.28210
$167$	19.7069	1.52496	0.762482	−	0.647009i	$-0.223981\pi$
$167$	19.7069	1.52496	0.762482	+	0.647009i	$0.223981\pi$
$168$	8.32083	0.641966
$169$	−12.5726	−0.967124
$170$	9.72437	0.745825
$171$	−2.32083	−0.177479
$172$	−35.5710	−2.71226
$173$	14.3948	1.09442	0.547208	−	0.836997i	$-0.315691\pi$
$173$	14.3948	1.09442	0.547208	+	0.836997i	$0.315691\pi$
$174$	18.2543	1.38385
$175$	7.01823	0.530528
$176$	0	0
$177$	−2.74104	−0.206029
$178$	1.72230	0.129092
$179$	−4.66420	−0.348619	−0.174309	−	0.984691i	$-0.555769\pi$
$179$	−4.66420	−0.348619	−0.174309	+	0.984691i	$0.555769\pi$
$180$	5.40894	0.403159
$181$	9.90805	0.736459	0.368230	−	0.929735i	$-0.379964\pi$
$181$	9.90805	0.736459	0.368230	+	0.929735i	$0.379964\pi$
$182$	−1.61263	−0.119536
$183$	11.0968	0.820297
$184$	34.2859	2.52759
$185$	1.52452	0.112085
$186$	11.1608	0.818349
$187$	0	0
$188$	−2.47007	−0.180148
$189$	5.47214	0.398039
$190$	51.9591	3.76951
$191$	9.94295	0.719447	0.359723	−	0.933059i	$-0.382871\pi$
$191$	9.94295	0.719447	0.359723	+	0.933059i	$0.382871\pi$
$192$	11.2044	0.808610
$193$	−3.70665	−0.266810	−0.133405	−	0.991062i	$-0.542591\pi$
$193$	−3.70665	−0.266810	−0.133405	+	0.991062i	$0.542591\pi$
$194$	36.6524	2.63149
$195$	3.66708	0.262605
$196$	4.08477	0.291769
$197$	−5.91982	−0.421770	−0.210885	−	0.977511i	$-0.567635\pi$
$197$	−5.91982	−0.421770	−0.210885	+	0.977511i	$0.567635\pi$
$198$	0	0
$199$	11.4842	0.814095	0.407047	−	0.913407i	$-0.366558\pi$
$199$	11.4842	0.814095	0.407047	+	0.913407i	$0.366558\pi$
$200$	−36.0917	−2.55207
$201$	9.99666	0.705110
$202$	21.2622	1.49600
$203$	4.57357	0.321002
$204$	−7.51578	−0.526210
$205$	−20.4646	−1.42931
$206$	2.30136	0.160343
$207$	2.54660	0.177001
$208$	2.95220	0.204698
$209$	0	0
$210$	−13.8366	−0.954817
$211$	−8.72487	−0.600645	−0.300323	−	0.953838i	$-0.597094\pi$
$211$	−8.72487	−0.600645	−0.300323	+	0.953838i	$0.597094\pi$
$212$	40.1225	2.75562
$213$	8.76467	0.600546
$214$	16.3350	1.11664
$215$	30.1890	2.05887
$216$	−28.1408	−1.91474
$217$	2.79631	0.189826
$218$	−10.1754	−0.689168
$219$	10.8440	0.732772
$220$	0	0
$221$	0.743416	0.0500076
$222$	−1.75519	−0.117801
$223$	10.3328	0.691938	0.345969	−	0.938246i	$-0.387550\pi$
$223$	10.3328	0.691938	0.345969	+	0.938246i	$0.387550\pi$
$224$	−0.854102	−0.0570671
$225$	−2.68073	−0.178715
$226$	46.4382	3.08902
$227$	13.2690	0.880691	0.440346	−	0.897828i	$-0.354856\pi$
$227$	13.2690	0.880691	0.440346	+	0.897828i	$0.354856\pi$
$228$	−40.1582	−2.65954
$229$	−2.49623	−0.164955	−0.0824777	−	0.996593i	$-0.526283\pi$
$229$	−2.49623	−0.164955	−0.0824777	+	0.996593i	$0.526283\pi$
$230$	−57.0135	−3.75936
$231$	0	0
$232$	−23.5199	−1.54415
$233$	9.60389	0.629171	0.314586	−	0.949229i	$-0.398134\pi$
$233$	9.60389	0.629171	0.314586	+	0.949229i	$0.398134\pi$
$234$	0.615970	0.0402672
$235$	2.09634	0.136750
$236$	6.91982	0.450442
$237$	4.29386	0.278916
$238$	−2.80505	−0.181825
$239$	−5.56327	−0.359858	−0.179929	−	0.983680i	$-0.557587\pi$
$239$	−5.56327	−0.359858	−0.179929	+	0.983680i	$0.557587\pi$
$240$	25.3303	1.63507
$241$	−15.0208	−0.967572	−0.483786	−	0.875186i	$-0.660739\pi$
$241$	−15.0208	−0.967572	−0.483786	+	0.875186i	$0.660739\pi$
$242$	0	0
$243$	−3.94427	−0.253025
$244$	−28.0141	−1.79342
$245$	−3.46673	−0.221481
$246$	23.5610	1.50219
$247$	3.97221	0.252746
$248$	−14.3802	−0.913143
$249$	10.8353	0.686659
$250$	17.2589	1.09155
$251$	27.6131	1.74292	0.871460	−	0.490466i	$-0.163173\pi$
$251$	27.6131	1.74292	0.871460	+	0.490466i	$0.163173\pi$
$252$	−1.56024	−0.0982860
$253$	0	0
$254$	−19.6508	−1.23300
$255$	6.37863	0.399445
$256$	−32.4995	−2.03122
$257$	−28.4680	−1.77578	−0.887892	−	0.460052i	$-0.847831\pi$
$257$	−28.4680	−1.77578	−0.887892	+	0.460052i	$0.847831\pi$
$258$	−34.7567	−2.16386
$259$	−0.439758	−0.0273253
$260$	−9.25764	−0.574134
$261$	−1.74695	−0.108133
$262$	−11.8528	−0.732267
$263$	−14.1803	−0.874397	−0.437199	−	0.899365i	$-0.644029\pi$
$263$	−14.1803	−0.874397	−0.437199	+	0.899365i	$0.644029\pi$
$264$	0	0
$265$	−34.0519	−2.09179
$266$	−14.9879	−0.918968
$267$	1.12973	0.0691385
$268$	−25.2368	−1.54158
$269$	−24.1937	−1.47511	−0.737557	−	0.675285i	$-0.764021\pi$
$269$	−24.1937	−1.47511	−0.737557	+	0.675285i	$0.764021\pi$
$270$	46.7950	2.84785
$271$	−7.44975	−0.452540	−0.226270	−	0.974065i	$-0.572653\pi$
$271$	−7.44975	−0.452540	−0.226270	+	0.974065i	$0.572653\pi$
$272$	5.13514	0.311363
$273$	−1.05779	−0.0640205
$274$	53.9665	3.26024
$275$	0	0
$276$	44.0647	2.65238
$277$	19.1890	1.15296	0.576478	−	0.817113i	$-0.304427\pi$
$277$	19.1890	1.15296	0.576478	+	0.817113i	$0.304427\pi$
$278$	−48.8751	−2.93134
$279$	−1.06810	−0.0639452
$280$	17.8279	1.06542
$281$	1.90063	0.113382	0.0566910	−	0.998392i	$-0.481945\pi$
$281$	1.90063	0.113382	0.0566910	+	0.998392i	$0.481945\pi$
$282$	−2.41353	−0.143723
$283$	−7.25178	−0.431073	−0.215537	−	0.976496i	$-0.569150\pi$
$283$	−7.25178	−0.431073	−0.215537	+	0.976496i	$0.569150\pi$
$284$	−22.1266	−1.31297
$285$	34.0822	2.01885
$286$	0	0
$287$	5.90315	0.348452
$288$	0.326238	0.0192238
$289$	−15.7069	−0.923934
$290$	39.1109	2.29667
$291$	24.0419	1.40936
$292$	−27.3760	−1.60206
$293$	3.27097	0.191092	0.0955460	−	0.995425i	$-0.469540\pi$
$293$	3.27097	0.191092	0.0955460	+	0.995425i	$0.469540\pi$
$294$	3.99126	0.232775
$295$	−5.87284	−0.341930
$296$	2.26148	0.131446
$297$	0	0
$298$	7.80008	0.451846
$299$	−4.35862	−0.252065
$300$	−46.3855	−2.67807
$301$	−8.70820	−0.501933
$302$	22.0231	1.26729
$303$	13.9468	0.801222
$304$	27.4380	1.57368
$305$	23.7755	1.36138
$306$	1.07144	0.0612499
$307$	−31.6121	−1.80420	−0.902099	−	0.431530i	$-0.857974\pi$
$307$	−31.6121	−1.80420	−0.902099	+	0.431530i	$0.857974\pi$
$308$	0	0
$309$	1.50956	0.0858758
$310$	23.9126	1.35815
$311$	−9.03829	−0.512514	−0.256257	−	0.966609i	$-0.582489\pi$
$311$	−9.03829	−0.512514	−0.256257	+	0.966609i	$0.582489\pi$
$312$	5.43976	0.307966
$313$	14.4990	0.819534	0.409767	−	0.912190i	$-0.365610\pi$
$313$	14.4990	0.819534	0.409767	+	0.912190i	$0.365610\pi$
$314$	−2.81128	−0.158649
$315$	1.32417	0.0746087
$316$	−10.8400	−0.609795
$317$	−18.4174	−1.03442	−0.517211	−	0.855858i	$-0.673030\pi$
$317$	−18.4174	−1.03442	−0.517211	+	0.855858i	$0.673030\pi$
$318$	39.2040	2.19845
$319$	0	0
$320$	24.0061	1.34198
$321$	10.7148	0.598042
$322$	16.4459	0.916494
$323$	6.90937	0.384448
$324$	−31.4862	−1.74923
$325$	4.58818	0.254506
$326$	12.4996	0.692290
$327$	−6.67450	−0.369101
$328$	−30.3573	−1.67620
$329$	−0.604703	−0.0333384
$330$	0	0
$331$	−6.47653	−0.355982	−0.177991	−	0.984032i	$-0.556960\pi$
$331$	−6.47653	−0.355982	−0.177991	+	0.984032i	$0.556960\pi$
$332$	−27.3540	−1.50124
$333$	0.167973	0.00920485
$334$	−48.6116	−2.65991
$335$	21.4184	1.17021
$336$	−7.30669	−0.398612
$337$	−6.25682	−0.340831	−0.170415	−	0.985372i	$-0.554511\pi$
$337$	−6.25682	−0.340831	−0.170415	+	0.985372i	$0.554511\pi$
$338$	31.0133	1.68690
$339$	30.4608	1.65440
$340$	−16.1030	−0.873308
$341$	0	0
$342$	5.72487	0.309566
$343$	1.00000	0.0539949
$344$	44.7824	2.41451
$345$	−37.3976	−2.01342
$346$	−35.5081	−1.90893
$347$	−22.7156	−1.21944	−0.609719	−	0.792617i	$-0.708718\pi$
$347$	−22.7156	−1.21944	−0.609719	+	0.792617i	$0.708718\pi$
$348$	−30.2281	−1.62039
$349$	−29.6941	−1.58949	−0.794743	−	0.606946i	$-0.792394\pi$
$349$	−29.6941	−1.58949	−0.794743	+	0.606946i	$0.792394\pi$
$350$	−17.3121	−0.925370
$351$	3.57742	0.190948
$352$	0	0
$353$	−6.82506	−0.363262	−0.181631	−	0.983367i	$-0.558138\pi$
$353$	−6.82506	−0.363262	−0.181631	+	0.983367i	$0.558138\pi$
$354$	6.76141	0.359365
$355$	18.7788	0.996676
$356$	−2.85204	−0.151158
$357$	−1.83995	−0.0973807
$358$	11.5053	0.608076
$359$	7.25970	0.383152	0.191576	−	0.981478i	$-0.438640\pi$
$359$	7.25970	0.383152	0.191576	+	0.981478i	$0.438640\pi$
$360$	−6.80964	−0.358900
$361$	17.9180	0.943055
$362$	−24.4405	−1.28456
$363$	0	0
$364$	2.67042	0.139968
$365$	23.2340	1.21612
$366$	−27.3728	−1.43080
$367$	−36.3366	−1.89676	−0.948378	−	0.317143i	$-0.897277\pi$
$367$	−36.3366	−1.89676	−0.948378	+	0.317143i	$0.897277\pi$
$368$	−30.1071	−1.56944
$369$	−2.25480	−0.117380
$370$	−3.76059	−0.195504
$371$	9.82247	0.509957
$372$	−18.4816	−0.958227
$373$	14.2913	0.739977	0.369989	−	0.929036i	$-0.379362\pi$
$373$	14.2913	0.739977	0.369989	+	0.929036i	$0.379362\pi$
$374$	0	0
$375$	11.3208	0.584605
$376$	3.10972	0.160372
$377$	2.98998	0.153992
$378$	−13.4983	−0.694277
$379$	−2.54528	−0.130742	−0.0653710	−	0.997861i	$-0.520823\pi$
$379$	−2.54528	−0.130742	−0.0653710	+	0.997861i	$0.520823\pi$
$380$	−86.0413	−4.41382
$381$	−12.8898	−0.660365
$382$	−24.5266	−1.25489
$383$	−23.1367	−1.18223	−0.591115	−	0.806587i	$-0.701312\pi$
$383$	−23.1367	−1.18223	−0.591115	+	0.806587i	$0.701312\pi$
$384$	−30.4023	−1.55146
$385$	0	0
$386$	9.14330	0.465382
$387$	3.32624	0.169082
$388$	−60.6943	−3.08128
$389$	30.2615	1.53432	0.767158	−	0.641458i	$-0.221670\pi$
$389$	30.2615	1.53432	0.767158	+	0.641458i	$0.221670\pi$
$390$	−9.04571	−0.458047
$391$	−7.58150	−0.383413
$392$	−5.14256	−0.259738
$393$	−7.77474	−0.392184
$394$	14.6026	0.735669
$395$	9.19985	0.462894
$396$	0	0
$397$	−22.6740	−1.13798	−0.568989	−	0.822345i	$-0.692665\pi$
$397$	−22.6740	−1.13798	−0.568989	+	0.822345i	$0.692665\pi$
$398$	−28.3285	−1.41998
$399$	−9.83121	−0.492176
$400$	31.6928	1.58464
$401$	−16.2186	−0.809917	−0.404959	−	0.914335i	$-0.632714\pi$
$401$	−16.2186	−0.809917	−0.404959	+	0.914335i	$0.632714\pi$
$402$	−24.6591	−1.22988
$403$	1.82809	0.0910637
$404$	−35.2090	−1.75171
$405$	26.7223	1.32784
$406$	−11.2818	−0.559905
$407$	0	0
$408$	9.46207	0.468442
$409$	35.0614	1.73368	0.866838	−	0.498590i	$-0.166149\pi$
$409$	35.0614	1.73368	0.866838	+	0.498590i	$0.166149\pi$
$410$	50.4808	2.49307
$411$	35.3989	1.74610
$412$	−3.81092	−0.187750
$413$	1.69406	0.0833590
$414$	−6.28178	−0.308732
$415$	23.2152	1.13959
$416$	−0.558371	−0.0273764
$417$	−32.0593	−1.56995
$418$	0	0
$419$	28.2633	1.38075	0.690376	−	0.723451i	$-0.257445\pi$
$419$	28.2633	1.38075	0.690376	+	0.723451i	$0.257445\pi$
$420$	22.9126	1.11802
$421$	−13.7947	−0.672311	−0.336156	−	0.941806i	$-0.609127\pi$
$421$	−13.7947	−0.672311	−0.336156	+	0.941806i	$0.609127\pi$
$422$	21.5219	1.04767
$423$	0.230976	0.0112304
$424$	−50.5126	−2.45311
$425$	7.98081	0.387126
$426$	−21.6201	−1.04750
$427$	−6.85818	−0.331891
$428$	−27.0498	−1.30750
$429$	0	0
$430$	−74.4682	−3.59117
$431$	−28.1700	−1.35690	−0.678451	−	0.734645i	$-0.737349\pi$
$431$	−28.1700	−1.35690	−0.678451	+	0.734645i	$0.737349\pi$
$432$	24.7110	1.18891
$433$	14.1793	0.681415	0.340708	−	0.940169i	$-0.389333\pi$
$433$	14.1793	0.681415	0.340708	+	0.940169i	$0.389333\pi$
$434$	−6.89775	−0.331102
$435$	25.6545	1.23004
$436$	16.8499	0.806966
$437$	−40.5093	−1.93782
$438$	−26.7493	−1.27813
$439$	−28.0185	−1.33725	−0.668625	−	0.743599i	$-0.733117\pi$
$439$	−28.0185	−1.33725	−0.668625	+	0.743599i	$0.733117\pi$
$440$	0	0
$441$	−0.381966	−0.0181889
$442$	−1.83381	−0.0872254
$443$	17.3370	0.823706	0.411853	−	0.911250i	$-0.364882\pi$
$443$	17.3370	0.823706	0.411853	+	0.911250i	$0.364882\pi$
$444$	2.90649	0.137936
$445$	2.42052	0.114744
$446$	−25.4883	−1.20691
$447$	5.11640	0.241998
$448$	−6.92472	−0.327162
$449$	−29.5215	−1.39321	−0.696603	−	0.717457i	$-0.745306\pi$
$449$	−29.5215	−1.39321	−0.696603	+	0.717457i	$0.745306\pi$
$450$	6.61263	0.311722
$451$	0	0
$452$	−76.8990	−3.61702
$453$	14.4459	0.678728
$454$	−32.7309	−1.53614
$455$	−2.26638	−0.106250
$456$	50.5576	2.36757
$457$	−9.69725	−0.453618	−0.226809	−	0.973939i	$-0.572829\pi$
$457$	−9.69725	−0.453618	−0.226809	+	0.973939i	$0.572829\pi$
$458$	6.15752	0.287722
$459$	6.22266	0.290449
$460$	94.4111	4.40194
$461$	19.2216	0.895240	0.447620	−	0.894224i	$-0.352272\pi$
$461$	19.2216	0.895240	0.447620	+	0.894224i	$0.352272\pi$
$462$	0	0
$463$	20.5327	0.954235	0.477117	−	0.878840i	$-0.341682\pi$
$463$	20.5327	0.954235	0.477117	+	0.878840i	$0.341682\pi$
$464$	20.6532	0.958803
$465$	15.6853	0.727389
$466$	−23.6902	−1.09743
$467$	33.6714	1.55813	0.779064	−	0.626944i	$-0.215695\pi$
$467$	33.6714	1.55813	0.779064	+	0.626944i	$0.215695\pi$
$468$	−1.02001	−0.0471500
$469$	−6.17828	−0.285286
$470$	−5.17112	−0.238526
$471$	−1.84403	−0.0849686
$472$	−8.71178	−0.400992
$473$	0	0
$474$	−10.5918	−0.486498
$475$	42.6429	1.95659
$476$	4.64501	0.212904
$477$	−3.75185	−0.171785
$478$	13.7231	0.627680
$479$	−14.1708	−0.647479	−0.323740	−	0.946146i	$-0.604940\pi$
$479$	−14.1708	−0.647479	−0.323740	+	0.946146i	$0.604940\pi$
$480$	−4.79091	−0.218674
$481$	−0.287493	−0.0131085
$482$	37.0522	1.68768
$483$	10.7876	0.490851
$484$	0	0
$485$	51.5111	2.33900
$486$	9.72946	0.441337
$487$	27.7415	1.25709	0.628543	−	0.777775i	$-0.283652\pi$
$487$	27.7415	1.25709	0.628543	+	0.777775i	$0.283652\pi$
$488$	35.2686	1.59653
$489$	8.19903	0.370773
$490$	8.55150	0.386317
$491$	3.44295	0.155378	0.0776891	−	0.996978i	$-0.475246\pi$
$491$	3.44295	0.155378	0.0776891	+	0.996978i	$0.475246\pi$
$492$	−39.0156	−1.75896
$493$	5.20086	0.234235
$494$	−9.79837	−0.440850
$495$	0	0
$496$	12.6275	0.566992
$497$	−5.41687	−0.242980
$498$	−26.7278	−1.19770
$499$	−2.58923	−0.115910	−0.0579550	−	0.998319i	$-0.518458\pi$
$499$	−2.58923	−0.115910	−0.0579550	+	0.998319i	$0.518458\pi$
$500$	−28.5797	−1.27812
$501$	−31.8864	−1.42458
$502$	−68.1140	−3.04008
$503$	22.9026	1.02118	0.510589	−	0.859825i	$-0.329427\pi$
$503$	22.9026	1.02118	0.510589	+	0.859825i	$0.329427\pi$
$504$	1.96428	0.0874961
$505$	29.8818	1.32972
$506$	0	0
$507$	20.3429	0.903460
$508$	32.5407	1.44376
$509$	−24.0985	−1.06815	−0.534074	−	0.845438i	$-0.679340\pi$
$509$	−24.0985	−1.06815	−0.534074	+	0.845438i	$0.679340\pi$
$510$	−15.7344	−0.696729
$511$	−6.70198	−0.296478
$512$	42.5884	1.88216
$513$	33.2488	1.46797
$514$	70.2229	3.09740
$515$	3.23432	0.142521
$516$	57.5550	2.53372
$517$	0	0
$518$	1.08477	0.0476619
$519$	−23.2913	−1.02237
$520$	11.6550	0.511105
$521$	−33.5057	−1.46791	−0.733956	−	0.679197i	$-0.762328\pi$
$521$	−33.5057	−1.46791	−0.733956	+	0.679197i	$0.762328\pi$
$522$	4.30926	0.188611
$523$	31.1574	1.36242	0.681208	−	0.732090i	$-0.261455\pi$
$523$	31.1574	1.36242	0.681208	+	0.732090i	$0.261455\pi$
$524$	19.6275	0.857432
$525$	−11.3557	−0.495605
$526$	34.9791	1.52516
$527$	3.17983	0.138516
$528$	0	0
$529$	21.4500	0.932608
$530$	83.9968	3.64859
$531$	−0.647072	−0.0280805
$532$	24.8191	1.07605
$533$	3.85919	0.167160
$534$	−2.78675	−0.120594
$535$	22.9571	0.992522
$536$	31.7721	1.37235
$537$	7.54683	0.325670
$538$	59.6793	2.57296
$539$	0	0
$540$	−77.4898	−3.33463
$541$	−22.6071	−0.971957	−0.485979	−	0.873971i	$-0.661537\pi$
$541$	−22.6071	−0.971957	−0.485979	+	0.873971i	$0.661537\pi$
$542$	18.3765	0.789340
$543$	−16.0316	−0.687980
$544$	−0.971245	−0.0416418
$545$	−14.3005	−0.612567
$546$	2.60929	0.111667
$547$	27.4442	1.17343	0.586715	−	0.809794i	$-0.300421\pi$
$547$	27.4442	1.17343	0.586715	+	0.809794i	$0.300421\pi$
$548$	−89.3654	−3.81750
$549$	2.61959	0.111801
$550$	0	0
$551$	27.7891	1.18386
$552$	−55.4757	−2.36120
$553$	−2.65375	−0.112849
$554$	−47.3341	−2.01103
$555$	−2.46673	−0.104707
$556$	80.9344	3.43238
$557$	30.2504	1.28175	0.640875	−	0.767645i	$-0.278572\pi$
$557$	30.2504	1.28175	0.640875	+	0.767645i	$0.278572\pi$
$558$	2.63470	0.111536
$559$	−5.69300	−0.240788
$560$	−15.6550	−0.661544
$561$	0	0
$562$	−4.68834	−0.197766
$563$	−7.46234	−0.314500	−0.157250	−	0.987559i	$-0.550263\pi$
$563$	−7.46234	−0.314500	−0.157250	+	0.987559i	$0.550263\pi$
$564$	3.99666	0.168290
$565$	65.2640	2.74568
$566$	17.8882	0.751896
$567$	−7.70820	−0.323714
$568$	27.8565	1.16883
$569$	−35.6483	−1.49446	−0.747228	−	0.664568i	$-0.768616\pi$
$569$	−35.6483	−1.49446	−0.747228	+	0.664568i	$0.768616\pi$
$570$	−84.0716	−3.52137
$571$	−25.8902	−1.08347	−0.541737	−	0.840548i	$-0.682233\pi$
$571$	−25.8902	−1.08347	−0.541737	+	0.840548i	$0.682233\pi$
$572$	0	0
$573$	−16.0880	−0.672087
$574$	−14.5615	−0.607785
$575$	−46.7911	−1.95132
$576$	2.64501	0.110209
$577$	8.16382	0.339864	0.169932	−	0.985456i	$-0.445645\pi$
$577$	8.16382	0.339864	0.169932	+	0.985456i	$0.445645\pi$
$578$	38.7447	1.61157
$579$	5.99748	0.249247
$580$	−64.7654	−2.68924
$581$	−6.69658	−0.277821
$582$	−59.3048	−2.45826
$583$	0	0
$584$	34.4653	1.42619
$585$	0.865681	0.0357915
$586$	−8.06860	−0.333311
$587$	−12.4634	−0.514419	−0.257210	−	0.966356i	$-0.582803\pi$
$587$	−12.4634	−0.514419	−0.257210	+	0.966356i	$0.582803\pi$
$588$	−6.60929	−0.272563
$589$	16.9904	0.700079
$590$	14.4867	0.596409
$591$	9.57847	0.394006
$592$	−1.98585	−0.0816180
$593$	23.6707	0.972037	0.486019	−	0.873948i	$-0.338449\pi$
$593$	23.6707	0.972037	0.486019	+	0.873948i	$0.338449\pi$
$594$	0	0
$595$	−3.94221	−0.161615
$596$	−12.9165	−0.529080
$597$	−18.5819	−0.760505
$598$	10.7515	0.439663
$599$	38.9809	1.59272	0.796358	−	0.604826i	$-0.206757\pi$
$599$	38.9809	1.59272	0.796358	+	0.604826i	$0.206757\pi$
$600$	58.3975	2.38407
$601$	−30.5510	−1.24620	−0.623100	−	0.782142i	$-0.714127\pi$
$601$	−30.5510	−1.24620	−0.623100	+	0.782142i	$0.714127\pi$
$602$	21.4808	0.875492
$603$	2.35989	0.0961022
$604$	−36.4690	−1.48390
$605$	0	0
$606$	−34.4030	−1.39753
$607$	−37.6173	−1.52684	−0.763420	−	0.645903i	$-0.776481\pi$
$607$	−37.6173	−1.52684	−0.763420	+	0.645903i	$0.776481\pi$
$608$	−5.18954	−0.210464
$609$	−7.40020	−0.299871
$610$	−58.6477	−2.37458
$611$	−0.395326	−0.0159932
$612$	−1.77423	−0.0717192
$613$	−17.6272	−0.711956	−0.355978	−	0.934494i	$-0.615852\pi$
$613$	−17.6272	−0.711956	−0.355978	+	0.934494i	$0.615852\pi$
$614$	77.9786	3.14696
$615$	33.1125	1.33522
$616$	0	0
$617$	−44.4849	−1.79089	−0.895447	−	0.445168i	$-0.853144\pi$
$617$	−44.4849	−1.79089	−0.895447	+	0.445168i	$0.853144\pi$
$618$	−3.72368	−0.149788
$619$	−6.20424	−0.249369	−0.124685	−	0.992196i	$-0.539792\pi$
$619$	−6.20424	−0.249369	−0.124685	+	0.992196i	$0.539792\pi$
$620$	−39.5979	−1.59029
$621$	−36.4832	−1.46402
$622$	22.2950	0.893949
$623$	−0.698213	−0.0279733
$624$	−4.77676	−0.191223
$625$	−10.8356	−0.433424
$626$	−35.7652	−1.42947
$627$	0	0
$628$	4.65531	0.185767
$629$	−0.500073	−0.0199392
$630$	−3.26638	−0.130136
$631$	44.8057	1.78369	0.891844	−	0.452344i	$-0.149412\pi$
$631$	44.8057	1.78369	0.891844	+	0.452344i	$0.149412\pi$
$632$	13.6471	0.542851
$633$	14.1171	0.561106
$634$	45.4307	1.80428
$635$	−27.6172	−1.09595
$636$	−64.9195	−2.57423
$637$	0.653752	0.0259026
$638$	0	0
$639$	2.06906	0.0818507
$640$	−65.1386	−2.57483
$641$	−10.2756	−0.405863	−0.202932	−	0.979193i	$-0.565047\pi$
$641$	−10.2756	−0.405863	−0.202932	+	0.979193i	$0.565047\pi$
$642$	−26.4306	−1.04313
$643$	−16.4446	−0.648511	−0.324255	−	0.945970i	$-0.605114\pi$
$643$	−16.4446	−0.648511	−0.324255	+	0.945970i	$0.605114\pi$
$644$	−27.2335	−1.07315
$645$	−48.8468	−1.92334
$646$	−17.0436	−0.670570
$647$	−26.9686	−1.06025	−0.530123	−	0.847920i	$-0.677854\pi$
$647$	−26.9686	−1.06025	−0.530123	+	0.847920i	$0.677854\pi$
$648$	39.6399	1.55720
$649$	0	0
$650$	−11.3178	−0.443921
$651$	−4.52452	−0.177330
$652$	−20.6986	−0.810621
$653$	7.10602	0.278080	0.139040	−	0.990287i	$-0.455598\pi$
$653$	7.10602	0.278080	0.139040	+	0.990287i	$0.455598\pi$
$654$	16.4642	0.643802
$655$	−16.6578	−0.650875
$656$	26.6573	1.04079
$657$	2.55993	0.0998724
$658$	1.49164	0.0581502
$659$	32.6279	1.27100	0.635502	−	0.772099i	$-0.280793\pi$
$659$	32.6279	1.27100	0.635502	+	0.772099i	$0.280793\pi$
$660$	0	0
$661$	−33.8165	−1.31531	−0.657654	−	0.753320i	$-0.728451\pi$
$661$	−33.8165	−1.31531	−0.657654	+	0.753320i	$0.728451\pi$
$662$	15.9759	0.620919
$663$	−1.20287	−0.0467157
$664$	34.4375	1.33644
$665$	−21.0639	−0.816824
$666$	−0.414344	−0.0160555
$667$	−30.4924	−1.18067
$668$	80.4980	3.11456
$669$	−16.7189	−0.646390
$670$	−52.8335	−2.04114
$671$	0	0
$672$	1.38197	0.0533105
$673$	−23.7496	−0.915478	−0.457739	−	0.889087i	$-0.651341\pi$
$673$	−23.7496	−0.915478	−0.457739	+	0.889087i	$0.651341\pi$
$674$	15.4339	0.594491
$675$	38.4047	1.47820
$676$	−51.3562	−1.97524
$677$	45.6311	1.75375	0.876874	−	0.480721i	$-0.159625\pi$
$677$	45.6311	1.75375	0.876874	+	0.480721i	$0.159625\pi$
$678$	−75.1386	−2.88568
$679$	−14.8587	−0.570224
$680$	20.2730	0.777435
$681$	−21.4696	−0.822717
$682$	0	0
$683$	−28.8727	−1.10478	−0.552392	−	0.833585i	$-0.686285\pi$
$683$	−28.8727	−1.10478	−0.552392	+	0.833585i	$0.686285\pi$
$684$	−9.48006	−0.362479
$685$	75.8442	2.89786
$686$	−2.46673	−0.0941803
$687$	4.03898	0.154097
$688$	−39.3243	−1.49923
$689$	6.42145	0.244638
$690$	92.2498	3.51189
$691$	−27.0635	−1.02954	−0.514772	−	0.857327i	$-0.672123\pi$
$691$	−27.0635	−1.02954	−0.514772	+	0.857327i	$0.672123\pi$
$692$	58.7994	2.23522
$693$	0	0
$694$	56.0334	2.12700
$695$	−68.6889	−2.60552
$696$	38.0559	1.44251
$697$	6.71279	0.254265
$698$	73.2473	2.77245
$699$	−15.5394	−0.587755
$700$	28.6678	1.08354
$701$	−38.5156	−1.45471	−0.727357	−	0.686259i	$-0.759252\pi$
$701$	−38.5156	−1.45471	−0.727357	+	0.686259i	$0.759252\pi$
$702$	−8.82453	−0.333060
$703$	−2.67198	−0.100776
$704$	0	0
$705$	−3.39196	−0.127748
$706$	16.8356	0.633616
$707$	−8.61959	−0.324173
$708$	−11.1965	−0.420790
$709$	−1.89464	−0.0711548	−0.0355774	−	0.999367i	$-0.511327\pi$
$709$	−1.89464	−0.0711548	−0.0355774	+	0.999367i	$0.511327\pi$
$710$	−46.3223	−1.73845
$711$	1.01364	0.0380146
$712$	3.59060	0.134564
$713$	−18.6432	−0.698194
$714$	4.53867	0.169856
$715$	0	0
$716$	−19.0522	−0.712013
$717$	9.00156	0.336169
$718$	−17.9077	−0.668311
$719$	14.5772	0.543639	0.271819	−	0.962348i	$-0.412375\pi$
$719$	14.5772	0.543639	0.271819	+	0.962348i	$0.412375\pi$
$720$	5.97968	0.222849
$721$	−0.932958	−0.0347452
$722$	−44.1990	−1.64492
$723$	24.3041	0.903879
$724$	40.4721	1.50413
$725$	32.0984	1.19210
$726$	0	0
$727$	4.04780	0.150125	0.0750623	−	0.997179i	$-0.476084\pi$
$727$	4.04780	0.150125	0.0750623	+	0.997179i	$0.476084\pi$
$728$	−3.36196	−0.124602
$729$	29.5066	1.09284
$730$	−57.3120	−2.12121
$731$	−9.90257	−0.366260
$732$	45.3277	1.67536
$733$	23.5012	0.868036	0.434018	−	0.900904i	$-0.357095\pi$
$733$	23.5012	0.868036	0.434018	+	0.900904i	$0.357095\pi$
$734$	89.6327	3.30840
$735$	5.60929	0.206902
$736$	5.69437	0.209897
$737$	0	0
$738$	5.56199	0.204740
$739$	32.6786	1.20210	0.601051	−	0.799210i	$-0.294749\pi$
$739$	32.6786	1.20210	0.601051	+	0.799210i	$0.294749\pi$
$740$	6.22732	0.228921
$741$	−6.42717	−0.236108
$742$	−24.2294	−0.889489
$743$	18.1058	0.664237	0.332119	−	0.943238i	$-0.392237\pi$
$743$	18.1058	0.664237	0.332119	+	0.943238i	$0.392237\pi$
$744$	23.2676	0.853033
$745$	10.9622	0.401624
$746$	−35.2529	−1.29070
$747$	2.55787	0.0935874
$748$	0	0
$749$	−6.62212	−0.241967
$750$	−27.9255	−1.01969
$751$	9.14357	0.333654	0.166827	−	0.985986i	$-0.446648\pi$
$751$	9.14357	0.333654	0.166827	+	0.985986i	$0.446648\pi$
$752$	−2.73071	−0.0995787
$753$	−44.6789	−1.62819
$754$	−7.37548	−0.268599
$755$	30.9512	1.12643
$756$	22.3524	0.812949
$757$	47.3509	1.72100	0.860499	−	0.509452i	$-0.170152\pi$
$757$	47.3509	1.72100	0.860499	+	0.509452i	$0.170152\pi$
$758$	6.27851	0.228046
$759$	0	0
$760$	108.323	3.92927
$761$	−3.44828	−0.125000	−0.0625000	−	0.998045i	$-0.519907\pi$
$761$	−3.44828	−0.125000	−0.0625000	+	0.998045i	$0.519907\pi$
$762$	31.7957	1.15184
$763$	4.12507	0.149338
$764$	40.6146	1.46939
$765$	1.50579	0.0544419
$766$	57.0720	2.06210
$767$	1.10749	0.0399892
$768$	52.5854	1.89751
$769$	−34.9787	−1.26137	−0.630683	−	0.776041i	$-0.717225\pi$
$769$	−34.9787	−1.26137	−0.630683	+	0.776041i	$0.717225\pi$
$770$	0	0
$771$	46.0622	1.65889
$772$	−15.1408	−0.544928
$773$	25.3794	0.912832	0.456416	−	0.889766i	$-0.349133\pi$
$773$	25.3794	0.912832	0.456416	+	0.889766i	$0.349133\pi$
$774$	−8.20494	−0.294920
$775$	19.6251	0.704956
$776$	76.4117	2.74302
$777$	0.711544	0.0255265
$778$	−74.6469	−2.67622
$779$	35.8677	1.28509
$780$	14.9792	0.536340
$781$	0	0
$782$	18.7015	0.668765
$783$	25.0272	0.894399
$784$	4.51578	0.161278
$785$	−3.95095	−0.141016
$786$	19.1782	0.684064
$787$	−29.7552	−1.06066	−0.530329	−	0.847792i	$-0.677932\pi$
$787$	−29.7552	−1.06066	−0.530329	+	0.847792i	$0.677932\pi$
$788$	−24.1811	−0.861415
$789$	22.9443	0.816838
$790$	−22.6936	−0.807400
$791$	−18.8258	−0.669369
$792$	0	0
$793$	−4.48355	−0.159215
$794$	55.9308	1.98491
$795$	55.0971	1.95409
$796$	46.9103	1.66269
$797$	−6.24330	−0.221149	−0.110575	−	0.993868i	$-0.535269\pi$
$797$	−6.24330	−0.221149	−0.110575	+	0.993868i	$0.535269\pi$
$798$	24.2510	0.858475
$799$	−0.687641	−0.0243270
$800$	−5.99428	−0.211930
$801$	0.266694	0.00942316
$802$	40.0069	1.41269
$803$	0	0
$804$	40.8340	1.44010
$805$	23.1130	0.814626
$806$	−4.50941	−0.158837
$807$	39.1462	1.37801
$808$	44.3268	1.55941
$809$	−39.1860	−1.37771	−0.688854	−	0.724900i	$-0.741886\pi$
$809$	−39.1860	−1.37771	−0.688854	+	0.724900i	$0.741886\pi$
$810$	−65.9167	−2.31608
$811$	12.3809	0.434753	0.217376	−	0.976088i	$-0.430250\pi$
$811$	12.3809	0.434753	0.217376	+	0.976088i	$0.430250\pi$
$812$	18.6820	0.655609
$813$	12.0539	0.422750
$814$	0	0
$815$	17.5669	0.615341
$816$	−8.30883	−0.290867
$817$	−52.9112	−1.85113
$818$	−86.4871	−3.02395
$819$	−0.249711	−0.00872560
$820$	−83.5933	−2.91920
$821$	−47.6987	−1.66470	−0.832349	−	0.554252i	$-0.813004\pi$
$821$	−47.6987	−1.66470	−0.832349	+	0.554252i	$0.813004\pi$
$822$	−87.3196	−3.04562
$823$	−14.5207	−0.506161	−0.253080	−	0.967445i	$-0.581444\pi$
$823$	−14.5207	−0.506161	−0.253080	+	0.967445i	$0.581444\pi$
$824$	4.79779	0.167139
$825$	0	0
$826$	−4.17878	−0.145398
$827$	30.9372	1.07579	0.537896	−	0.843011i	$-0.319219\pi$
$827$	30.9372	1.07579	0.537896	+	0.843011i	$0.319219\pi$
$828$	10.4023	0.361504
$829$	0.0858057	0.00298015	0.00149008	−	0.999999i	$-0.499526\pi$
$829$	0.0858057	0.00298015	0.00149008	+	0.999999i	$0.499526\pi$
$830$	−57.2658	−1.98772
$831$	−31.0485	−1.07706
$832$	−4.52705	−0.156947
$833$	1.13715	0.0394000
$834$	79.0816	2.73837
$835$	−68.3185	−2.36426
$836$	0	0
$837$	15.3018	0.528907
$838$	−69.7179	−2.40837
$839$	10.6905	0.369078	0.184539	−	0.982825i	$-0.440921\pi$
$839$	10.6905	0.369078	0.184539	+	0.982825i	$0.440921\pi$
$840$	−28.8461	−0.995285
$841$	−8.08244	−0.278705
$842$	34.0278	1.17267
$843$	−3.07528	−0.105918
$844$	−35.6391	−1.22675
$845$	43.5859	1.49940
$846$	−0.569756	−0.0195886
$847$	0	0
$848$	44.3561	1.52319
$849$	11.7336	0.402697
$850$	−19.6865	−0.675242
$851$	2.93190	0.100504
$852$	35.8016	1.22654
$853$	−21.2003	−0.725884	−0.362942	−	0.931812i	$-0.618228\pi$
$853$	−21.2003	−0.725884	−0.362942	+	0.931812i	$0.618228\pi$
$854$	16.9173	0.578898
$855$	8.04571	0.275157
$856$	34.0546	1.16396
$857$	9.45359	0.322929	0.161464	−	0.986879i	$-0.448378\pi$
$857$	9.45359	0.322929	0.161464	+	0.986879i	$0.448378\pi$
$858$	0	0
$859$	−38.8261	−1.32473	−0.662365	−	0.749181i	$-0.730447\pi$
$859$	−38.8261	−1.32473	−0.662365	+	0.749181i	$0.730447\pi$
$860$	123.315	4.20501
$861$	−9.55150	−0.325514
$862$	69.4879	2.36677
$863$	−37.6046	−1.28008	−0.640038	−	0.768343i	$-0.721081\pi$
$863$	−37.6046	−1.28008	−0.640038	+	0.768343i	$0.721081\pi$
$864$	−4.67376	−0.159005
$865$	−49.9029	−1.69675
$866$	−34.9766	−1.18855
$867$	25.4143	0.863114
$868$	11.4223	0.387697
$869$	0	0
$870$	−63.2828	−2.14549
$871$	−4.03906	−0.136858
$872$	−21.2134	−0.718377
$873$	5.67551	0.192087
$874$	99.9257	3.38004
$875$	−6.99666	−0.236530
$876$	44.2953	1.49660
$877$	−22.0086	−0.743178	−0.371589	−	0.928397i	$-0.621187\pi$
$877$	−22.0086	−0.743178	−0.371589	+	0.928397i	$0.621187\pi$
$878$	69.1142	2.33249
$879$	−5.29254	−0.178513
$880$	0	0
$881$	−6.92969	−0.233467	−0.116734	−	0.993163i	$-0.537242\pi$
$881$	−6.92969	−0.233467	−0.116734	+	0.993163i	$0.537242\pi$
$882$	0.942208	0.0317258
$883$	42.3388	1.42481	0.712407	−	0.701767i	$-0.247605\pi$
$883$	42.3388	1.42481	0.712407	+	0.701767i	$0.247605\pi$
$884$	3.03668	0.102135
$885$	9.50245	0.319421
$886$	−42.7657	−1.43674
$887$	−8.05647	−0.270510	−0.135255	−	0.990811i	$-0.543185\pi$
$887$	−8.05647	−0.270510	−0.135255	+	0.990811i	$0.543185\pi$
$888$	−3.65916	−0.122793
$889$	7.96635	0.267183
$890$	−5.97077	−0.200141
$891$	0	0
$892$	42.2072	1.41320
$893$	−3.67419	−0.122952
$894$	−12.6208	−0.422102
$895$	16.1695	0.540488
$896$	18.7896	0.627717
$897$	7.05239	0.235472
$898$	72.8216	2.43009
$899$	12.7891	0.426541
$900$	−10.9501	−0.365004
$901$	11.1697	0.372115
$902$	0	0
$903$	14.0902	0.468891
$904$	96.8128	3.21995
$905$	−34.3485	−1.14178
$906$	−35.6342	−1.18387
$907$	2.89429	0.0961033	0.0480517	−	0.998845i	$-0.484699\pi$
$907$	2.89429	0.0961033	0.0480517	+	0.998845i	$0.484699\pi$
$908$	54.2006	1.79871
$909$	3.29239	0.109202
$910$	5.59056	0.185325
$911$	−20.2500	−0.670912	−0.335456	−	0.942056i	$-0.608890\pi$
$911$	−20.2500	−0.670912	−0.335456	+	0.942056i	$0.608890\pi$
$912$	−44.3956	−1.47008
$913$	0	0
$914$	23.9205	0.791220
$915$	−38.4695	−1.27176
$916$	−10.1965	−0.336902
$917$	4.80505	0.158677
$918$	−15.3496	−0.506613
$919$	−18.7261	−0.617718	−0.308859	−	0.951108i	$-0.599947\pi$
$919$	−18.7261	−0.617718	−0.308859	+	0.951108i	$0.599947\pi$
$920$	−118.860	−3.91869
$921$	51.1494	1.68543
$922$	−47.4146	−1.56152
$923$	−3.54128	−0.116563
$924$	0	0
$925$	−3.08632	−0.101478
$926$	−50.6486	−1.66442
$927$	0.356358	0.0117043
$928$	−3.90630	−0.128230
$929$	−45.2701	−1.48526	−0.742631	−	0.669700i	$-0.766423\pi$
$929$	−45.2701	−1.48526	−0.742631	+	0.669700i	$0.766423\pi$
$930$	−38.6915	−1.26874
$931$	6.07602	0.199134
$932$	39.2296	1.28501
$933$	14.6243	0.478777
$934$	−83.0584	−2.71775
$935$	0	0
$936$	1.28415	0.0419738
$937$	−29.4599	−0.962413	−0.481207	−	0.876607i	$-0.659801\pi$
$937$	−29.4599	−0.962413	−0.481207	+	0.876607i	$0.659801\pi$
$938$	15.2401	0.497609
$939$	−23.4599	−0.765586
$940$	8.56308	0.279297
$941$	−58.2346	−1.89839	−0.949197	−	0.314683i	$-0.898102\pi$
$941$	−58.2346	−1.89839	−0.949197	+	0.314683i	$0.898102\pi$
$942$	4.54874	0.148206
$943$	−39.3568	−1.28163
$944$	7.64998	0.248986
$945$	−18.9704	−0.617108
$946$	0	0
$947$	45.3642	1.47414	0.737069	−	0.675818i	$-0.236209\pi$
$947$	45.3642	1.47414	0.737069	+	0.675818i	$0.236209\pi$
$948$	17.5394	0.569654
$949$	−4.38143	−0.142227
$950$	−105.189	−3.41277
$951$	29.7999	0.966329
$952$	−5.84788	−0.189531
$953$	41.3375	1.33905	0.669527	−	0.742788i	$-0.266497\pi$
$953$	41.3375	1.33905	0.669527	+	0.742788i	$0.266497\pi$
$954$	9.25480	0.299635
$955$	−34.4695	−1.11541
$956$	−22.7247	−0.734968
$957$	0	0
$958$	34.9555	1.12936
$959$	−21.8777	−0.706469
$960$	−38.8428	−1.25364
$961$	−23.1807	−0.747763
$962$	0.709167	0.0228645
$963$	2.52942	0.0815095
$964$	−61.3563	−1.97615
$965$	12.8499	0.413654
$966$	−26.6100	−0.856164
$967$	6.52818	0.209932	0.104966	−	0.994476i	$-0.466527\pi$
$967$	6.52818	0.209932	0.104966	+	0.994476i	$0.466527\pi$
$968$	0	0
$969$	−11.1796	−0.359140
$970$	−127.064	−4.07978
$971$	28.8587	0.926119	0.463060	−	0.886327i	$-0.346752\pi$
$971$	28.8587	0.926119	0.463060	+	0.886327i	$0.346752\pi$
$972$	−16.1114	−0.516774
$973$	19.8137	0.635199
$974$	−68.4308	−2.19266
$975$	−7.42383	−0.237753
$976$	−30.9700	−0.991327
$977$	−8.98453	−0.287441	−0.143720	−	0.989618i	$-0.545907\pi$
$977$	−8.98453	−0.287441	−0.143720	+	0.989618i	$0.545907\pi$
$978$	−20.2248	−0.646718
$979$	0	0
$980$	−14.1608	−0.452350
$981$	−1.57564	−0.0503062
$982$	−8.49284	−0.271017
$983$	36.8819	1.17635	0.588175	−	0.808734i	$-0.299847\pi$
$983$	36.8819	1.17635	0.588175	+	0.808734i	$0.299847\pi$
$984$	49.1191	1.56586
$985$	20.5224	0.653899
$986$	−12.8291	−0.408562
$987$	0.978431	0.0311438
$988$	16.2255	0.516203
$989$	58.0583	1.84615
$990$	0	0
$991$	−2.98352	−0.0947746	−0.0473873	−	0.998877i	$-0.515089\pi$
$991$	−2.98352	−0.0947746	−0.0473873	+	0.998877i	$0.515089\pi$
$992$	−2.38833	−0.0758297
$993$	10.4792	0.332549
$994$	13.3620	0.423815
$995$	−39.8127	−1.26215
$996$	44.2596	1.40242
$997$	−62.8553	−1.99065	−0.995324	−	0.0965930i	$-0.969205\pi$
$997$	−62.8553	−1.99065	−0.995324	+	0.0965930i	$0.969205\pi$
$998$	6.38694	0.202175
$999$	−2.40642	−0.0761357

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	847.2.a.k.1.1		4
3.2	odd	2		7623.2.a.co.1.4		4
7.6	odd	2		5929.2.a.bb.1.1		4
11.2	odd	10		77.2.f.a.15.1	✓	8
11.3	even	5		847.2.f.s.372.1		8
11.4	even	5		847.2.f.s.148.1		8
11.5	even	5		847.2.f.q.729.2		8
11.6	odd	10		77.2.f.a.36.1	yes	8
11.7	odd	10		847.2.f.p.148.2		8
11.8	odd	10		847.2.f.p.372.2		8
11.9	even	5		847.2.f.q.323.2		8
11.10	odd	2		847.2.a.l.1.4		4
33.2	even	10		693.2.m.g.631.2		8
33.17	even	10		693.2.m.g.190.2		8
33.32	even	2		7623.2.a.ch.1.1		4
77.2	odd	30		539.2.q.c.312.1		16
77.6	even	10		539.2.f.d.344.1		8
77.13	even	10		539.2.f.d.246.1		8
77.17	even	30		539.2.q.b.520.1		16
77.24	even	30		539.2.q.b.422.2		16
77.39	odd	30		539.2.q.c.520.1		16
77.46	odd	30		539.2.q.c.422.2		16
77.61	even	30		539.2.q.b.410.2		16
77.68	even	30		539.2.q.b.312.1		16
77.72	odd	30		539.2.q.c.410.2		16
77.76	even	2		5929.2.a.bi.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.15.1	✓	8	11.2	odd	10
77.2.f.a.36.1	yes	8	11.6	odd	10
539.2.f.d.246.1		8	77.13	even	10
539.2.f.d.344.1		8	77.6	even	10
539.2.q.b.312.1		16	77.68	even	30
539.2.q.b.410.2		16	77.61	even	30
539.2.q.b.422.2		16	77.24	even	30
539.2.q.b.520.1		16	77.17	even	30
539.2.q.c.312.1		16	77.2	odd	30
539.2.q.c.410.2		16	77.72	odd	30
539.2.q.c.422.2		16	77.46	odd	30
539.2.q.c.520.1		16	77.39	odd	30
693.2.m.g.190.2		8	33.17	even	10
693.2.m.g.631.2		8	33.2	even	10
847.2.a.k.1.1		4	1.1	even	1	trivial
847.2.a.l.1.4		4	11.10	odd	2
847.2.f.p.148.2		8	11.7	odd	10
847.2.f.p.372.2		8	11.8	odd	10
847.2.f.q.323.2		8	11.9	even	5
847.2.f.q.729.2		8	11.5	even	5
847.2.f.s.148.1		8	11.4	even	5
847.2.f.s.372.1		8	11.3	even	5
5929.2.a.bb.1.1		4	7.6	odd	2
5929.2.a.bi.1.4		4	77.76	even	2
7623.2.a.ch.1.1		4	33.32	even	2
7623.2.a.co.1.4		4	3.2	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:	\( N \)	\(=\)	\( 847 = 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	847.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46673 q^{2} -1.61803 q^{3} +4.08477 q^{4} -3.46673 q^{5} +3.99126 q^{6} +1.00000 q^{7} -5.14256 q^{8} -0.381966 q^{9} +O(q^{10})\)	\(q-2.46673 q^{2} -1.61803 q^{3} +4.08477 q^{4} -3.46673 q^{5} +3.99126 q^{6} +1.00000 q^{7} -5.14256 q^{8} -0.381966 q^{9} +8.55150 q^{10} -6.60929 q^{12} +0.653752 q^{13} -2.46673 q^{14} +5.60929 q^{15} +4.51578 q^{16} +1.13715 q^{17} +0.942208 q^{18} +6.07602 q^{19} -14.1608 q^{20} -1.61803 q^{21} -6.66708 q^{23} +8.32083 q^{24} +7.01823 q^{25} -1.61263 q^{26} +5.47214 q^{27} +4.08477 q^{28} +4.57357 q^{29} -13.8366 q^{30} +2.79631 q^{31} -0.854102 q^{32} -2.80505 q^{34} -3.46673 q^{35} -1.56024 q^{36} -0.439758 q^{37} -14.9879 q^{38} -1.05779 q^{39} +17.8279 q^{40} +5.90315 q^{41} +3.99126 q^{42} -8.70820 q^{43} +1.32417 q^{45} +16.4459 q^{46} -0.604703 q^{47} -7.30669 q^{48} +1.00000 q^{49} -17.3121 q^{50} -1.83995 q^{51} +2.67042 q^{52} +9.82247 q^{53} -13.4983 q^{54} -5.14256 q^{56} -9.83121 q^{57} -11.2818 q^{58} +1.69406 q^{59} +22.9126 q^{60} -6.85818 q^{61} -6.89775 q^{62} -0.381966 q^{63} -6.92472 q^{64} -2.26638 q^{65} -6.17828 q^{67} +4.64501 q^{68} +10.7876 q^{69} +8.55150 q^{70} -5.41687 q^{71} +1.96428 q^{72} -6.70198 q^{73} +1.08477 q^{74} -11.3557 q^{75} +24.8191 q^{76} +2.60929 q^{78} -2.65375 q^{79} -15.6550 q^{80} -7.70820 q^{81} -14.5615 q^{82} -6.69658 q^{83} -6.60929 q^{84} -3.94221 q^{85} +21.4808 q^{86} -7.40020 q^{87} -0.698213 q^{89} -3.26638 q^{90} +0.653752 q^{91} -27.2335 q^{92} -4.52452 q^{93} +1.49164 q^{94} -21.0639 q^{95} +1.38197 q^{96} -14.8587 q^{97} -2.46673 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9	\( 4 q - 2 q^{2} - 2 q^{3} + 4 q^{4} - 6 q^{5} + q^{6} + 4 q^{7} - 9 q^{8} - 6 q^{9} + 14 q^{10} - 7 q^{12} - 2 q^{14} + 3 q^{15} - 4 q^{16} + 3 q^{17} + 3 q^{18} - 3 q^{19} - 17 q^{20} - 2 q^{21} - 8 q^{23} + 12 q^{24} - 12 q^{26} + 4 q^{27} + 4 q^{28} - 3 q^{29} - 12 q^{30} - 3 q^{31} + 10 q^{32} - 12 q^{34} - 6 q^{35} - q^{36} - 7 q^{37} - 20 q^{38} - 5 q^{39} + 13 q^{40} - 4 q^{41} + q^{42} - 8 q^{43} + 9 q^{45} + 3 q^{46} - 14 q^{47} - 3 q^{48} + 4 q^{49} - 33 q^{50} + 11 q^{51} + 17 q^{52} - 9 q^{53} - 2 q^{54} - 9 q^{56} - 6 q^{57} + 3 q^{58} - 25 q^{59} + 21 q^{60} + 19 q^{61} - 10 q^{62} - 6 q^{63} + 3 q^{64} - 12 q^{65} - 15 q^{67} + q^{68} + 14 q^{69} + 14 q^{70} - 7 q^{71} + 6 q^{72} + 11 q^{73} - 8 q^{74} - 5 q^{75} + 26 q^{76} - 9 q^{78} - 8 q^{79} - 4 q^{80} - 4 q^{81} + 3 q^{82} + q^{83} - 7 q^{84} - 15 q^{85} + 4 q^{86} - 6 q^{87} - 17 q^{89} - 16 q^{90} - 17 q^{92} - 11 q^{93} + 20 q^{94} - 17 q^{95} + 10 q^{96} - 15 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 2 * q^3 + 4 * q^4 - 6 * q^5 + q^6 + 4 * q^7 - 9 * q^8 - 6 * q^9 + 14 * q^10 - 7 * q^12 - 2 * q^14 + 3 * q^15 - 4 * q^16 + 3 * q^17 + 3 * q^18 - 3 * q^19 - 17 * q^20 - 2 * q^21 - 8 * q^23 + 12 * q^24 - 12 * q^26 + 4 * q^27 + 4 * q^28 - 3 * q^29 - 12 * q^30 - 3 * q^31 + 10 * q^32 - 12 * q^34 - 6 * q^35 - q^36 - 7 * q^37 - 20 * q^38 - 5 * q^39 + 13 * q^40 - 4 * q^41 + q^42 - 8 * q^43 + 9 * q^45 + 3 * q^46 - 14 * q^47 - 3 * q^48 + 4 * q^49 - 33 * q^50 + 11 * q^51 + 17 * q^52 - 9 * q^53 - 2 * q^54 - 9 * q^56 - 6 * q^57 + 3 * q^58 - 25 * q^59 + 21 * q^60 + 19 * q^61 - 10 * q^62 - 6 * q^63 + 3 * q^64 - 12 * q^65 - 15 * q^67 + q^68 + 14 * q^69 + 14 * q^70 - 7 * q^71 + 6 * q^72 + 11 * q^73 - 8 * q^74 - 5 * q^75 + 26 * q^76 - 9 * q^78 - 8 * q^79 - 4 * q^80 - 4 * q^81 + 3 * q^82 + q^83 - 7 * q^84 - 15 * q^85 + 4 * q^86 - 6 * q^87 - 17 * q^89 - 16 * q^90 - 17 * q^92 - 11 * q^93 + 20 * q^94 - 17 * q^95 + 10 * q^96 - 15 * q^97 - 2 * q^98

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