LMFDB - Embedded newform 1849.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,2,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1849.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:	$14.7643393337$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 5 x^{17} - 15 x^{16} + 106 x^{15} + 47 x^{14} - 897 x^{13} + 364 x^{12} + 3855 x^{11} + \cdots - 27 $ x^18 - 5x^17 - 15x^16 + 106x^15 + 47x^14 - 897x^13 + 364x^12 + 3855x^11 - 3223x^10 - 8851x^9 + 9909x^8 + 10400x^7 - 14483x^6 - 5118x^5 + 9945x^4 - 27x^3 - 2493x^2 + 513*x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.31808$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.31808 q^{2} -1.34851 q^{3} +3.37350 q^{4} +4.07240 q^{5} +3.12595 q^{6} +2.77841 q^{7} -3.18389 q^{8} -1.18153 q^{9} +O(q^{10})$	\(q-2.31808 q^{2} -1.34851 q^{3} +3.37350 q^{4} +4.07240 q^{5} +3.12595 q^{6} +2.77841 q^{7} -3.18389 q^{8} -1.18153 q^{9} -9.44015 q^{10} +0.752881 q^{11} -4.54919 q^{12} -1.72089 q^{13} -6.44058 q^{14} -5.49166 q^{15} +0.633520 q^{16} -3.25039 q^{17} +2.73888 q^{18} +0.0478129 q^{19} +13.7383 q^{20} -3.74670 q^{21} -1.74524 q^{22} +2.03896 q^{23} +4.29350 q^{24} +11.5844 q^{25} +3.98917 q^{26} +5.63882 q^{27} +9.37297 q^{28} -0.747535 q^{29} +12.7301 q^{30} -3.44106 q^{31} +4.89924 q^{32} -1.01527 q^{33} +7.53468 q^{34} +11.3148 q^{35} -3.98589 q^{36} +7.83004 q^{37} -0.110834 q^{38} +2.32064 q^{39} -12.9661 q^{40} +8.40047 q^{41} +8.68517 q^{42} +2.53985 q^{44} -4.81165 q^{45} -4.72647 q^{46} -7.83357 q^{47} -0.854306 q^{48} +0.719553 q^{49} -26.8537 q^{50} +4.38318 q^{51} -5.80544 q^{52} +1.78846 q^{53} -13.0712 q^{54} +3.06603 q^{55} -8.84616 q^{56} -0.0644761 q^{57} +1.73285 q^{58} +5.78675 q^{59} -18.5261 q^{60} +2.94434 q^{61} +7.97665 q^{62} -3.28277 q^{63} -12.6239 q^{64} -7.00816 q^{65} +2.35347 q^{66} +5.60017 q^{67} -10.9652 q^{68} -2.74955 q^{69} -26.2286 q^{70} +7.27515 q^{71} +3.76186 q^{72} -10.1884 q^{73} -18.1507 q^{74} -15.6217 q^{75} +0.161297 q^{76} +2.09181 q^{77} -5.37943 q^{78} +13.4104 q^{79} +2.57995 q^{80} -4.05941 q^{81} -19.4730 q^{82} +5.67937 q^{83} -12.6395 q^{84} -13.2369 q^{85} +1.00806 q^{87} -2.39709 q^{88} +12.3263 q^{89} +11.1538 q^{90} -4.78134 q^{91} +6.87843 q^{92} +4.64029 q^{93} +18.1589 q^{94} +0.194713 q^{95} -6.60666 q^{96} -10.4454 q^{97} -1.66798 q^{98} -0.889550 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) $ 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} + 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} + 25 q^{18} + 31 q^{19} + 25 q^{20} + 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} + 27 q^{26} + 23 q^{27} + 20 q^{28} + 37 q^{29} - 17 q^{30} - 12 q^{31} + 39 q^{32} + 38 q^{33} + 14 q^{34} + 16 q^{35} + 47 q^{36} - 19 q^{37} + 56 q^{38} + 46 q^{39} + 6 q^{40} - 7 q^{41} - q^{42} + 7 q^{44} + 23 q^{45} - 47 q^{46} - q^{47} + 15 q^{48} - 6 q^{49} - 3 q^{50} + 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} + 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} + 28 q^{61} + 33 q^{62} + 26 q^{63} + 10 q^{64} + 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} + 7 q^{69} - 34 q^{70} + 86 q^{71} - 2 q^{72} - 27 q^{73} - 79 q^{74} + 31 q^{75} + 59 q^{76} + 43 q^{77} + 91 q^{78} + 17 q^{79} + 8 q^{80} - 10 q^{81} + 13 q^{82} - 12 q^{83} - 32 q^{84} + 28 q^{85} - 43 q^{87} - 23 q^{88} + 51 q^{89} + 10 q^{90} - 20 q^{91} + 18 q^{92} - 30 q^{93} - 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} - 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 + 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 + 25 * q^18 + 31 * q^19 + 25 * q^20 + 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 + 27 * q^26 + 23 * q^27 + 20 * q^28 + 37 * q^29 - 17 * q^30 - 12 * q^31 + 39 * q^32 + 38 * q^33 + 14 * q^34 + 16 * q^35 + 47 * q^36 - 19 * q^37 + 56 * q^38 + 46 * q^39 + 6 * q^40 - 7 * q^41 - q^42 + 7 * q^44 + 23 * q^45 - 47 * q^46 - q^47 + 15 * q^48 - 6 * q^49 - 3 * q^50 + 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 + 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 + 28 * q^61 + 33 * q^62 + 26 * q^63 + 10 * q^64 + 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 + 7 * q^69 - 34 * q^70 + 86 * q^71 - 2 * q^72 - 27 * q^73 - 79 * q^74 + 31 * q^75 + 59 * q^76 + 43 * q^77 + 91 * q^78 + 17 * q^79 + 8 * q^80 - 10 * q^81 + 13 * q^82 - 12 * q^83 - 32 * q^84 + 28 * q^85 - 43 * q^87 - 23 * q^88 + 51 * q^89 + 10 * q^90 - 20 * q^91 + 18 * q^92 - 30 * q^93 - 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 - 5 * q^98 + 38 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.31808	−1.63913	−0.819566	−	0.572985i	$-0.805785\pi$
$2$	−2.31808	−1.63913	−0.819566	+	0.572985i	$0.805785\pi$
$3$	−1.34851	−0.778561	−0.389281	−	0.921119i	$-0.627276\pi$
$3$	−1.34851	−0.778561	−0.389281	+	0.921119i	$0.627276\pi$
$4$	3.37350	1.68675
$5$	4.07240	1.82123	0.910616	−	0.413253i	$-0.135608\pi$
$5$	4.07240	1.82123	0.910616	+	0.413253i	$0.135608\pi$
$6$	3.12595	1.27616
$7$	2.77841	1.05014	0.525070	−	0.851059i	$-0.324039\pi$
$7$	2.77841	1.05014	0.525070	+	0.851059i	$0.324039\pi$
$8$	−3.18389	−1.12568
$9$	−1.18153	−0.393843
$10$	−9.44015	−2.98524
$11$	0.752881	0.227002	0.113501	−	0.993538i	$-0.463793\pi$
$11$	0.752881	0.227002	0.113501	+	0.993538i	$0.463793\pi$
$12$	−4.54919	−1.31324
$13$	−1.72089	−0.477290	−0.238645	−	0.971107i	$-0.576703\pi$
$13$	−1.72089	−0.477290	−0.238645	+	0.971107i	$0.576703\pi$
$14$	−6.44058	−1.72132
$15$	−5.49166	−1.41794
$16$	0.633520	0.158380
$17$	−3.25039	−0.788336	−0.394168	−	0.919038i	$-0.628967\pi$
$17$	−3.25039	−0.788336	−0.394168	+	0.919038i	$0.628967\pi$
$18$	2.73888	0.645560
$19$	0.0478129	0.0109690	0.00548452	−	0.999985i	$-0.498254\pi$
$19$	0.0478129	0.0109690	0.00548452	+	0.999985i	$0.498254\pi$
$20$	13.7383	3.07197
$21$	−3.74670	−0.817598
$22$	−1.74524	−0.372087
$23$	2.03896	0.425152	0.212576	−	0.977145i	$-0.431815\pi$
$23$	2.03896	0.425152	0.212576	+	0.977145i	$0.431815\pi$
$24$	4.29350	0.876408
$25$	11.5844	2.31689
$26$	3.98917	0.782341
$27$	5.63882	1.08519
$28$	9.37297	1.77133
$29$	−0.747535	−0.138814	−0.0694069	−	0.997588i	$-0.522111\pi$
$29$	−0.747535	−0.138814	−0.0694069	+	0.997588i	$0.522111\pi$
$30$	12.7301	2.32419
$31$	−3.44106	−0.618032	−0.309016	−	0.951057i	$-0.600000\pi$
$31$	−3.44106	−0.618032	−0.309016	+	0.951057i	$0.600000\pi$
$32$	4.89924	0.866071
$33$	−1.01527	−0.176735
$34$	7.53468	1.29219
$35$	11.3148	1.91255
$36$	−3.98589	−0.664315
$37$	7.83004	1.28725	0.643625	−	0.765341i	$-0.277430\pi$
$37$	7.83004	1.28725	0.643625	+	0.765341i	$0.277430\pi$
$38$	−0.110834	−0.0179797
$39$	2.32064	0.371599
$40$	−12.9661	−2.05012
$41$	8.40047	1.31193	0.655966	−	0.754790i	$-0.272261\pi$
$41$	8.40047	1.31193	0.655966	+	0.754790i	$0.272261\pi$
$42$	8.68517	1.34015
$43$	0	0
$43$	0	0
$44$	2.53985	0.382897
$45$	−4.81165	−0.717279
$46$	−4.72647	−0.696880
$47$	−7.83357	−1.14264	−0.571322	−	0.820726i	$-0.693569\pi$
$47$	−7.83357	−1.14264	−0.571322	+	0.820726i	$0.693569\pi$
$48$	−0.854306	−0.123308
$49$	0.719553	0.102793
$50$	−26.8537	−3.79768
$51$	4.38318	0.613768
$52$	−5.80544	−0.805069
$53$	1.78846	0.245664	0.122832	−	0.992427i	$-0.460802\pi$
$53$	1.78846	0.245664	0.122832	+	0.992427i	$0.460802\pi$
$54$	−13.0712	−1.77877
$55$	3.06603	0.413424
$56$	−8.84616	−1.18212
$57$	−0.0644761	−0.00854006
$58$	1.73285	0.227534
$59$	5.78675	0.753371	0.376686	−	0.926341i	$-0.377064\pi$
$59$	5.78675	0.753371	0.376686	+	0.926341i	$0.377064\pi$
$60$	−18.5261	−2.39171
$61$	2.94434	0.376985	0.188492	−	0.982075i	$-0.439640\pi$
$61$	2.94434	0.376985	0.188492	+	0.982075i	$0.439640\pi$
$62$	7.97665	1.01304
$63$	−3.28277	−0.413590
$64$	−12.6239	−1.57798
$65$	−7.00816	−0.869256
$66$	2.35347	0.289692
$67$	5.60017	0.684170	0.342085	−	0.939669i	$-0.388867\pi$
$67$	5.60017	0.684170	0.342085	+	0.939669i	$0.388867\pi$
$68$	−10.9652	−1.32973
$69$	−2.74955	−0.331007
$70$	−26.2286	−3.13492
$71$	7.27515	0.863401	0.431700	−	0.902017i	$-0.357914\pi$
$71$	7.27515	0.863401	0.431700	+	0.902017i	$0.357914\pi$
$72$	3.76186	0.443339
$73$	−10.1884	−1.19246	−0.596230	−	0.802813i	$-0.703335\pi$
$73$	−10.1884	−1.19246	−0.596230	+	0.802813i	$0.703335\pi$
$74$	−18.1507	−2.10997
$75$	−15.6217	−1.80384
$76$	0.161297	0.0185020
$77$	2.09181	0.238384
$78$	−5.37943	−0.609100
$79$	13.4104	1.50879	0.754393	−	0.656423i	$-0.227931\pi$
$79$	13.4104	1.50879	0.754393	+	0.656423i	$0.227931\pi$
$80$	2.57995	0.288447
$81$	−4.05941	−0.451045
$82$	−19.4730	−2.15043
$83$	5.67937	0.623392	0.311696	−	0.950182i	$-0.399103\pi$
$83$	5.67937	0.623392	0.311696	+	0.950182i	$0.399103\pi$
$84$	−12.6395	−1.37908
$85$	−13.2369	−1.43574
$86$	0	0
$87$	1.00806	0.108075
$88$	−2.39709	−0.255531
$89$	12.3263	1.30658	0.653290	−	0.757108i	$-0.273388\pi$
$89$	12.3263	1.30658	0.653290	+	0.757108i	$0.273388\pi$
$90$	11.1538	1.17571
$91$	−4.78134	−0.501221
$92$	6.87843	0.717126
$93$	4.64029	0.481176
$94$	18.1589	1.87294
$95$	0.194713	0.0199772
$96$	−6.60666	−0.674289
$97$	−10.4454	−1.06057	−0.530287	−	0.847818i	$-0.677916\pi$
$97$	−10.4454	−1.06057	−0.530287	+	0.847818i	$0.677916\pi$
$98$	−1.66798	−0.168492
$99$	−0.889550	−0.0894032
$100$	39.0801	3.90801
$101$	2.29223	0.228085	0.114043	−	0.993476i	$-0.463620\pi$
$101$	2.29223	0.228085	0.114043	+	0.993476i	$0.463620\pi$
$102$	−10.1606	−1.00605
$103$	−5.31015	−0.523224	−0.261612	−	0.965173i	$-0.584254\pi$
$103$	−5.31015	−0.523224	−0.261612	+	0.965173i	$0.584254\pi$
$104$	5.47914	0.537274
$105$	−15.2581	−1.48904
$106$	−4.14580	−0.402676
$107$	−3.59557	−0.347597	−0.173798	−	0.984781i	$-0.555604\pi$
$107$	−3.59557	−0.347597	−0.173798	+	0.984781i	$0.555604\pi$
$108$	19.0226	1.83045
$109$	5.56466	0.532998	0.266499	−	0.963835i	$-0.414133\pi$
$109$	5.56466	0.532998	0.266499	+	0.963835i	$0.414133\pi$
$110$	−7.10732	−0.677656
$111$	−10.5589	−1.00220
$112$	1.76018	0.166321
$113$	−0.140198	−0.0131887	−0.00659435	−	0.999978i	$-0.502099\pi$
$113$	−0.140198	−0.0131887	−0.00659435	+	0.999978i	$0.502099\pi$
$114$	0.149461	0.0139983
$115$	8.30345	0.774301
$116$	−2.52181	−0.234144
$117$	2.03328	0.187977
$118$	−13.4142	−1.23487
$119$	−9.03092	−0.827863
$120$	17.4849	1.59614
$121$	−10.4332	−0.948470
$122$	−6.82523	−0.617927
$123$	−11.3281	−1.02142
$124$	−11.6084	−1.04247
$125$	26.8144	2.39836
$126$	7.60972	0.677928
$127$	1.72954	0.153472	0.0767358	−	0.997051i	$-0.475550\pi$
$127$	1.72954	0.153472	0.0767358	+	0.997051i	$0.475550\pi$
$128$	19.4647	1.72045
$129$	0	0
$130$	16.2455	1.42482
$131$	21.8959	1.91305	0.956527	−	0.291645i	$-0.0942026\pi$
$131$	21.8959	1.91305	0.956527	+	0.291645i	$0.0942026\pi$
$132$	−3.42500	−0.298108
$133$	0.132844	0.0115190
$134$	−12.9817	−1.12144
$135$	22.9635	1.97639
$136$	10.3489	0.887412
$137$	−21.3631	−1.82518	−0.912588	−	0.408880i	$-0.865919\pi$
$137$	−21.3631	−1.82518	−0.912588	+	0.408880i	$0.865919\pi$
$138$	6.37368	0.542564
$139$	12.1855	1.03356	0.516780	−	0.856118i	$-0.327130\pi$
$139$	12.1855	1.03356	0.516780	+	0.856118i	$0.327130\pi$
$140$	38.1705	3.22599
$141$	10.5636	0.889618
$142$	−16.8644	−1.41523
$143$	−1.29563	−0.108346
$144$	−0.748521	−0.0623768
$145$	−3.04426	−0.252812
$146$	23.6175	1.95460
$147$	−0.970323	−0.0800309
$148$	26.4147	2.17127
$149$	8.26736	0.677288	0.338644	−	0.940915i	$-0.390032\pi$
$149$	8.26736	0.677288	0.338644	+	0.940915i	$0.390032\pi$
$150$	36.2124	2.95673
$151$	4.57068	0.371957	0.185978	−	0.982554i	$-0.440455\pi$
$151$	4.57068	0.371957	0.185978	+	0.982554i	$0.440455\pi$
$152$	−0.152231	−0.0123476
$153$	3.84043	0.310480
$154$	−4.84899	−0.390743
$155$	−14.0134	−1.12558
$156$	7.82868	0.626796
$157$	10.0177	0.799499	0.399749	−	0.916625i	$-0.369097\pi$
$157$	10.0177	0.799499	0.399749	+	0.916625i	$0.369097\pi$
$158$	−31.0864	−2.47310
$159$	−2.41176	−0.191265
$160$	19.9516	1.57732
$161$	5.66506	0.446469
$162$	9.41004	0.739323
$163$	−21.3751	−1.67423	−0.837115	−	0.547026i	$-0.815760\pi$
$163$	−21.3751	−1.67423	−0.837115	+	0.547026i	$0.815760\pi$
$164$	28.3390	2.21290
$165$	−4.13457	−0.321876
$166$	−13.1653	−1.02182
$167$	21.9623	1.69949	0.849746	−	0.527192i	$-0.176755\pi$
$167$	21.9623	1.69949	0.849746	+	0.527192i	$0.176755\pi$
$168$	11.9291	0.920351
$169$	−10.0385	−0.772194
$170$	30.6842	2.35337
$171$	−0.0564923	−0.00432007
$172$	0	0
$173$	−15.5770	−1.18430	−0.592151	−	0.805827i	$-0.701721\pi$
$173$	−15.5770	−1.18430	−0.592151	+	0.805827i	$0.701721\pi$
$174$	−2.33676	−0.177149
$175$	32.1863	2.43305
$176$	0.476965	0.0359526
$177$	−7.80348	−0.586545
$178$	−28.5733	−2.14166
$179$	6.98022	0.521726	0.260863	−	0.965376i	$-0.415993\pi$
$179$	6.98022	0.521726	0.260863	+	0.965376i	$0.415993\pi$
$180$	−16.2321	−1.20987
$181$	−5.50450	−0.409146	−0.204573	−	0.978851i	$-0.565581\pi$
$181$	−5.50450	−0.409146	−0.204573	+	0.978851i	$0.565581\pi$
$182$	11.0835	0.821567
$183$	−3.97047	−0.293506
$184$	−6.49183	−0.478584
$185$	31.8870	2.34438
$186$	−10.7566	−0.788710
$187$	−2.44716	−0.178954
$188$	−26.4266	−1.92736
$189$	15.6669	1.13960
$190$	−0.451361	−0.0327452
$191$	19.0059	1.37522	0.687610	−	0.726080i	$-0.258660\pi$
$191$	19.0059	1.37522	0.687610	+	0.726080i	$0.258660\pi$
$192$	17.0234	1.22856
$193$	−3.07951	−0.221668	−0.110834	−	0.993839i	$-0.535352\pi$
$193$	−3.07951	−0.221668	−0.110834	+	0.993839i	$0.535352\pi$
$194$	24.2134	1.73842
$195$	9.45056	0.676769
$196$	2.42741	0.173387
$197$	11.7435	0.836692	0.418346	−	0.908288i	$-0.362610\pi$
$197$	11.7435	0.836692	0.418346	+	0.908288i	$0.362610\pi$
$198$	2.06205	0.146544
$199$	−0.267688	−0.0189759	−0.00948796	−	0.999955i	$-0.503020\pi$
$199$	−0.267688	−0.0189759	−0.00948796	+	0.999955i	$0.503020\pi$
$200$	−36.8836	−2.60807
$201$	−7.55187	−0.532668
$202$	−5.31357	−0.373861
$203$	−2.07696	−0.145774
$204$	14.7867	1.03527
$205$	34.2101	2.38933
$206$	12.3094	0.857633
$207$	−2.40909	−0.167443
$208$	−1.09022	−0.0755931
$209$	0.0359974	0.00249000
$210$	35.3695	2.44073
$211$	−16.3897	−1.12832	−0.564158	−	0.825667i	$-0.690799\pi$
$211$	−16.3897	−1.12832	−0.564158	+	0.825667i	$0.690799\pi$
$212$	6.03339	0.414375
$213$	−9.81059	−0.672210
$214$	8.33482	0.569757
$215$	0	0
$216$	−17.9534	−1.22157
$217$	−9.56066	−0.649020
$218$	−12.8993	−0.873653
$219$	13.7391	0.928403
$220$	10.3433	0.697343
$221$	5.59358	0.376265
$222$	24.4763	1.64274
$223$	−4.86950	−0.326086	−0.163043	−	0.986619i	$-0.552131\pi$
$223$	−4.86950	−0.326086	−0.163043	+	0.986619i	$0.552131\pi$
$224$	13.6121	0.909495
$225$	−13.6873	−0.912489
$226$	0.324990	0.0216180
$227$	19.2778	1.27951	0.639755	−	0.768579i	$-0.279036\pi$
$227$	19.2778	1.27951	0.639755	+	0.768579i	$0.279036\pi$
$228$	−0.217510	−0.0144050
$229$	−7.22209	−0.477249	−0.238625	−	0.971112i	$-0.576697\pi$
$229$	−7.22209	−0.477249	−0.238625	+	0.971112i	$0.576697\pi$
$230$	−19.2481	−1.26918
$231$	−2.82082	−0.185597
$232$	2.38007	0.156259
$233$	−8.80509	−0.576841	−0.288420	−	0.957504i	$-0.593130\pi$
$233$	−8.80509	−0.576841	−0.288420	+	0.957504i	$0.593130\pi$
$234$	−4.71332	−0.308119
$235$	−31.9014	−2.08102
$236$	19.5216	1.27075
$237$	−18.0840	−1.17468
$238$	20.9344	1.35698
$239$	6.02041	0.389428	0.194714	−	0.980860i	$-0.437622\pi$
$239$	6.02041	0.389428	0.194714	+	0.980860i	$0.437622\pi$
$240$	−3.47908	−0.224573
$241$	17.5995	1.13369	0.566843	−	0.823826i	$-0.308165\pi$
$241$	17.5995	1.13369	0.566843	+	0.823826i	$0.308165\pi$
$242$	24.1849	1.55467
$243$	−11.4423	−0.734025
$244$	9.93276	0.635880
$245$	2.93031	0.187210
$246$	26.2594	1.67424
$247$	−0.0822809	−0.00523541
$248$	10.9560	0.695704
$249$	−7.65868	−0.485349
$250$	−62.1581	−3.93122
$251$	−17.3837	−1.09725	−0.548624	−	0.836069i	$-0.684848\pi$
$251$	−17.3837	−1.09725	−0.548624	+	0.836069i	$0.684848\pi$
$252$	−11.0744	−0.697623
$253$	1.53509	0.0965105
$254$	−4.00921	−0.251560
$255$	17.8501	1.11781
$256$	−19.8730	−1.24206
$257$	24.3517	1.51902	0.759508	−	0.650498i	$-0.225440\pi$
$257$	24.3517	1.51902	0.759508	+	0.650498i	$0.225440\pi$
$258$	0	0
$259$	21.7550	1.35179
$260$	−23.6421	−1.46622
$261$	0.883233	0.0546708
$262$	−50.7565	−3.13575
$263$	20.5178	1.26518	0.632591	−	0.774486i	$-0.281992\pi$
$263$	20.5178	1.26518	0.632591	+	0.774486i	$0.281992\pi$
$264$	3.23250	0.198947
$265$	7.28333	0.447412
$266$	−0.307943	−0.0188812
$267$	−16.6220	−1.01725
$268$	18.8922	1.15403
$269$	−17.0217	−1.03783	−0.518917	−	0.854825i	$-0.673664\pi$
$269$	−17.0217	−1.03783	−0.518917	+	0.854825i	$0.673664\pi$
$270$	−53.2313	−3.23956
$271$	3.45584	0.209928	0.104964	−	0.994476i	$-0.466527\pi$
$271$	3.45584	0.209928	0.104964	+	0.994476i	$0.466527\pi$
$272$	−2.05919	−0.124857
$273$	6.44768	0.390231
$274$	49.5215	2.99170
$275$	8.72171	0.525939
$276$	−9.27562	−0.558326
$277$	−24.0955	−1.44776	−0.723880	−	0.689926i	$-0.757643\pi$
$277$	−24.0955	−1.44776	−0.723880	+	0.689926i	$0.757643\pi$
$278$	−28.2470	−1.69414
$279$	4.06571	0.243407
$280$	−36.0251	−2.15291
$281$	5.75323	0.343209	0.171604	−	0.985166i	$-0.445105\pi$
$281$	5.75323	0.343209	0.171604	+	0.985166i	$0.445105\pi$
$282$	−24.4874	−1.45820
$283$	−15.9566	−0.948524	−0.474262	−	0.880384i	$-0.657285\pi$
$283$	−15.9566	−0.948524	−0.474262	+	0.880384i	$0.657285\pi$
$284$	24.5427	1.45634
$285$	−0.262572	−0.0155534
$286$	3.00337	0.177593
$287$	23.3399	1.37771
$288$	−5.78858	−0.341096
$289$	−6.43494	−0.378526
$290$	7.05685	0.414392
$291$	14.0857	0.825721
$292$	−34.3706	−2.01138
$293$	−6.05218	−0.353572	−0.176786	−	0.984249i	$-0.556570\pi$
$293$	−6.05218	−0.353572	−0.176786	+	0.984249i	$0.556570\pi$
$294$	2.24929	0.131181
$295$	23.5660	1.37206
$296$	−24.9300	−1.44903
$297$	4.24536	0.246341
$298$	−19.1644	−1.11016
$299$	−3.50883	−0.202921
$300$	−52.6998	−3.04263
$301$	0	0
$302$	−10.5952	−0.609686
$303$	−3.09109	−0.177578
$304$	0.0302904	0.00173728
$305$	11.9905	0.686577
$306$	−8.90243	−0.508918
$307$	23.4952	1.34094	0.670472	−	0.741935i	$-0.266092\pi$
$307$	23.4952	1.34094	0.670472	+	0.741935i	$0.266092\pi$
$308$	7.05674	0.402095
$309$	7.16077	0.407362
$310$	32.4841	1.84497
$311$	−1.17805	−0.0668011	−0.0334006	−	0.999442i	$-0.510634\pi$
$311$	−1.17805	−0.0668011	−0.0334006	+	0.999442i	$0.510634\pi$
$312$	−7.38866	−0.418301
$313$	29.7576	1.68200	0.840998	−	0.541038i	$-0.181968\pi$
$313$	29.7576	1.68200	0.840998	+	0.541038i	$0.181968\pi$
$314$	−23.2218	−1.31048
$315$	−13.3687	−0.753243
$316$	45.2400	2.54495
$317$	−26.8302	−1.50694	−0.753468	−	0.657484i	$-0.771621\pi$
$317$	−26.8302	−1.50694	−0.753468	+	0.657484i	$0.771621\pi$
$318$	5.59065	0.313508
$319$	−0.562805	−0.0315110
$320$	−51.4094	−2.87388
$321$	4.84865	0.270625
$322$	−13.1321	−0.731821
$323$	−0.155411	−0.00864729
$324$	−13.6944	−0.760802
$325$	−19.9356	−1.10583
$326$	49.5493	2.74428
$327$	−7.50398	−0.414971
$328$	−26.7462	−1.47681
$329$	−21.7649	−1.19994
$330$	9.58427	0.527597
$331$	2.78711	0.153193	0.0765967	−	0.997062i	$-0.475595\pi$
$331$	2.78711	0.153193	0.0765967	+	0.997062i	$0.475595\pi$
$332$	19.1594	1.05151
$333$	−9.25141	−0.506974
$334$	−50.9104	−2.78569
$335$	22.8061	1.24603
$336$	−2.37361	−0.129491
$337$	9.04893	0.492927	0.246463	−	0.969152i	$-0.420731\pi$
$337$	9.04893	0.492927	0.246463	+	0.969152i	$0.420731\pi$
$338$	23.2701	1.26573
$339$	0.189058	0.0102682
$340$	−44.6547	−2.42174
$341$	−2.59071	−0.140295
$342$	0.130954	0.00708117
$343$	−17.4496	−0.942192
$344$	0	0
$345$	−11.1973	−0.602840
$346$	36.1089	1.94123
$347$	−10.5387	−0.565748	−0.282874	−	0.959157i	$-0.591288\pi$
$347$	−10.5387	−0.565748	−0.282874	+	0.959157i	$0.591288\pi$
$348$	3.40068	0.182296
$349$	12.2864	0.657677	0.328839	−	0.944386i	$-0.393343\pi$
$349$	12.2864	0.657677	0.328839	+	0.944386i	$0.393343\pi$
$350$	−74.6105	−3.98810
$351$	−9.70381	−0.517951
$352$	3.68854	0.196600
$353$	4.28314	0.227969	0.113984	−	0.993483i	$-0.463639\pi$
$353$	4.28314	0.227969	0.113984	+	0.993483i	$0.463639\pi$
$354$	18.0891	0.961425
$355$	29.6273	1.57245
$356$	41.5827	2.20388
$357$	12.1783	0.644542
$358$	−16.1807	−0.855178
$359$	−34.2349	−1.80685	−0.903424	−	0.428748i	$-0.858955\pi$
$359$	−34.2349	−1.80685	−0.903424	+	0.428748i	$0.858955\pi$
$360$	15.3198	0.807424
$361$	−18.9977	−0.999880
$362$	12.7599	0.670645
$363$	14.0692	0.738442
$364$	−16.1299	−0.845435
$365$	−41.4912	−2.17175
$366$	9.20388	0.481094
$367$	−30.9297	−1.61452	−0.807260	−	0.590196i	$-0.799050\pi$
$367$	−30.9297	−1.61452	−0.807260	+	0.590196i	$0.799050\pi$
$368$	1.29172	0.0673356
$369$	−9.92538	−0.516695
$370$	−73.9168	−3.84275
$371$	4.96908	0.257982
$372$	15.6540	0.811624
$373$	−7.94094	−0.411166	−0.205583	−	0.978640i	$-0.565909\pi$
$373$	−7.94094	−0.411166	−0.205583	+	0.978640i	$0.565909\pi$
$374$	5.67272	0.293329
$375$	−36.1595	−1.86727
$376$	24.9413	1.28625
$377$	1.28643	0.0662544
$378$	−36.3173	−1.86796
$379$	22.3291	1.14697	0.573484	−	0.819217i	$-0.305592\pi$
$379$	22.3291	1.14697	0.573484	+	0.819217i	$0.305592\pi$
$380$	0.656866	0.0336965
$381$	−2.33229	−0.119487
$382$	−44.0573	−2.25417
$383$	25.1218	1.28366	0.641832	−	0.766845i	$-0.278175\pi$
$383$	25.1218	1.28366	0.641832	+	0.766845i	$0.278175\pi$
$384$	−26.2483	−1.33948
$385$	8.51869	0.434153
$386$	7.13855	0.363343
$387$	0	0
$388$	−35.2377	−1.78892
$389$	17.6085	0.892787	0.446393	−	0.894837i	$-0.352708\pi$
$389$	17.6085	0.892787	0.446393	+	0.894837i	$0.352708\pi$
$390$	−21.9072	−1.10931
$391$	−6.62742	−0.335163
$392$	−2.29098	−0.115712
$393$	−29.5268	−1.48943
$394$	−27.2225	−1.37145
$395$	54.6124	2.74785
$396$	−3.00090	−0.150801
$397$	1.44764	0.0726549	0.0363275	−	0.999340i	$-0.488434\pi$
$397$	1.44764	0.0726549	0.0363275	+	0.999340i	$0.488434\pi$
$398$	0.620523	0.0311040
$399$	−0.179141	−0.00896826
$400$	7.33897	0.366948
$401$	7.72115	0.385576	0.192788	−	0.981240i	$-0.438247\pi$
$401$	7.72115	0.385576	0.192788	+	0.981240i	$0.438247\pi$
$402$	17.5059	0.873113
$403$	5.92169	0.294980
$404$	7.73284	0.384723
$405$	−16.5315	−0.821458
$406$	4.81456	0.238942
$407$	5.89509	0.292209
$408$	−13.9556	−0.690904
$409$	−32.0321	−1.58388	−0.791942	−	0.610596i	$-0.790930\pi$
$409$	−32.0321	−1.58388	−0.791942	+	0.610596i	$0.790930\pi$
$410$	−79.3017	−3.91643
$411$	28.8083	1.42101
$412$	−17.9138	−0.882549
$413$	16.0780	0.791145
$414$	5.58446	0.274461
$415$	23.1287	1.13534
$416$	−8.43106	−0.413367
$417$	−16.4322	−0.804690
$418$	−0.0834450	−0.00408143
$419$	22.0904	1.07919	0.539594	−	0.841926i	$-0.318578\pi$
$419$	22.0904	1.07919	0.539594	+	0.841926i	$0.318578\pi$
$420$	−51.4732	−2.51163
$421$	15.2193	0.741744	0.370872	−	0.928684i	$-0.379059\pi$
$421$	15.2193	0.741744	0.370872	+	0.928684i	$0.379059\pi$
$422$	37.9927	1.84946
$423$	9.25558	0.450022
$424$	−5.69428	−0.276538
$425$	−37.6540	−1.82649
$426$	22.7417	1.10184
$427$	8.18059	0.395887
$428$	−12.1297	−0.586309
$429$	1.74716	0.0843539
$430$	0	0
$431$	16.5109	0.795301	0.397651	−	0.917537i	$-0.369826\pi$
$431$	16.5109	0.795301	0.397651	+	0.917537i	$0.369826\pi$
$432$	3.57231	0.171873
$433$	−25.2365	−1.21279	−0.606395	−	0.795164i	$-0.707385\pi$
$433$	−25.2365	−1.21279	−0.606395	+	0.795164i	$0.707385\pi$
$434$	22.1624	1.06383
$435$	4.10521	0.196830
$436$	18.7724	0.899035
$437$	0.0974885	0.00466351
$438$	−31.8484	−1.52178
$439$	18.9663	0.905212	0.452606	−	0.891711i	$-0.350494\pi$
$439$	18.9663	0.905212	0.452606	+	0.891711i	$0.350494\pi$
$440$	−9.76193	−0.465382
$441$	−0.850172	−0.0404844
$442$	−12.9664	−0.616748
$443$	37.4944	1.78141	0.890707	−	0.454578i	$-0.150210\pi$
$443$	37.4944	1.78141	0.890707	+	0.454578i	$0.150210\pi$
$444$	−35.6204	−1.69047
$445$	50.1974	2.37959
$446$	11.2879	0.534497
$447$	−11.1486	−0.527310
$448$	−35.0743	−1.65710
$449$	18.0895	0.853696	0.426848	−	0.904323i	$-0.359624\pi$
$449$	18.0895	0.853696	0.426848	+	0.904323i	$0.359624\pi$
$450$	31.7284	1.49569
$451$	6.32455	0.297812
$452$	−0.472958	−0.0222461
$453$	−6.16360	−0.289591
$454$	−44.6874	−2.09728
$455$	−19.4715	−0.912840
$456$	0.205285	0.00961335
$457$	−7.09968	−0.332109	−0.166055	−	0.986117i	$-0.553103\pi$
$457$	−7.09968	−0.332109	−0.166055	+	0.986117i	$0.553103\pi$
$458$	16.7414	0.782274
$459$	−18.3284	−0.855496
$460$	28.0117	1.30605
$461$	−2.02137	−0.0941447	−0.0470724	−	0.998891i	$-0.514989\pi$
$461$	−2.02137	−0.0941447	−0.0470724	+	0.998891i	$0.514989\pi$
$462$	6.53890	0.304217
$463$	−23.8272	−1.10734	−0.553671	−	0.832736i	$-0.686773\pi$
$463$	−23.8272	−1.10734	−0.553671	+	0.832736i	$0.686773\pi$
$464$	−0.473578	−0.0219853
$465$	18.8971	0.876333
$466$	20.4109	0.945517
$467$	−17.3717	−0.803864	−0.401932	−	0.915669i	$-0.631661\pi$
$467$	−17.3717	−0.803864	−0.401932	+	0.915669i	$0.631661\pi$
$468$	6.85929	0.317071
$469$	15.5596	0.718474
$470$	73.9501	3.41107
$471$	−13.5089	−0.622458
$472$	−18.4244	−0.848052
$473$	0	0
$474$	41.9202	1.92546
$475$	0.553885	0.0254140
$476$	−30.4658	−1.39640
$477$	−2.11312	−0.0967530
$478$	−13.9558	−0.638324
$479$	7.72741	0.353074	0.176537	−	0.984294i	$-0.443510\pi$
$479$	7.72741	0.353074	0.176537	+	0.984294i	$0.443510\pi$
$480$	−26.9049	−1.22804
$481$	−13.4747	−0.614391
$482$	−40.7972	−1.85826
$483$	−7.63937	−0.347603
$484$	−35.1963	−1.59983
$485$	−42.5380	−1.93155
$486$	26.5242	1.20316
$487$	17.4818	0.792176	0.396088	−	0.918213i	$-0.370367\pi$
$487$	17.4818	0.792176	0.396088	+	0.918213i	$0.370367\pi$
$488$	−9.37448	−0.424363
$489$	28.8245	1.30349
$490$	−6.79269	−0.306863
$491$	21.7140	0.979939	0.489970	−	0.871740i	$-0.337008\pi$
$491$	21.7140	0.979939	0.489970	+	0.871740i	$0.337008\pi$
$492$	−38.2154	−1.72288
$493$	2.42978	0.109432
$494$	0.190734	0.00858152
$495$	−3.62260	−0.162824
$496$	−2.17998	−0.0978839
$497$	20.2133	0.906692
$498$	17.7534	0.795551
$499$	35.8911	1.60670	0.803352	−	0.595504i	$-0.203048\pi$
$499$	35.8911	1.60670	0.803352	+	0.595504i	$0.203048\pi$
$500$	90.4586	4.04543
$501$	−29.6163	−1.32316
$502$	40.2968	1.79853
$503$	0.0909739	0.00405633	0.00202816	−	0.999998i	$-0.499354\pi$
$503$	0.0909739	0.00405633	0.00202816	+	0.999998i	$0.499354\pi$
$504$	10.4520	0.465568
$505$	9.33486	0.415396
$506$	−3.55847	−0.158193
$507$	13.5370	0.601201
$508$	5.83460	0.258869
$509$	22.6851	1.00550	0.502749	−	0.864432i	$-0.332322\pi$
$509$	22.6851	1.00550	0.502749	+	0.864432i	$0.332322\pi$
$510$	−41.3779	−1.83224
$511$	−28.3075	−1.25225
$512$	7.13789	0.315453
$513$	0.269608	0.0119035
$514$	−56.4492	−2.48987
$515$	−21.6250	−0.952913
$516$	0	0
$517$	−5.89775	−0.259383
$518$	−50.4300	−2.21577
$519$	21.0058	0.922051
$520$	22.3132	0.978501
$521$	−4.71739	−0.206673	−0.103336	−	0.994646i	$-0.532952\pi$
$521$	−4.71739	−0.206673	−0.103336	+	0.994646i	$0.532952\pi$
$522$	−2.04741	−0.0896126
$523$	−1.70620	−0.0746067	−0.0373034	−	0.999304i	$-0.511877\pi$
$523$	−1.70620	−0.0746067	−0.0373034	+	0.999304i	$0.511877\pi$
$524$	73.8659	3.22685
$525$	−43.4034	−1.89428
$526$	−47.5619	−2.07380
$527$	11.1848	0.487217
$528$	−0.643191	−0.0279913
$529$	−18.8427	−0.819246
$530$	−16.8834	−0.733366
$531$	−6.83721	−0.296710
$532$	0.448149	0.0194297
$533$	−14.4563	−0.626172
$534$	38.5313	1.66741
$535$	−14.6426	−0.633054
$536$	−17.8304	−0.770154
$537$	−9.41288	−0.406196
$538$	39.4578	1.70114
$539$	0.541738	0.0233343
$540$	77.4676	3.33367
$541$	11.3669	0.488701	0.244351	−	0.969687i	$-0.421425\pi$
$541$	11.3669	0.488701	0.244351	+	0.969687i	$0.421425\pi$
$542$	−8.01093	−0.344099
$543$	7.42286	0.318545
$544$	−15.9244	−0.682755
$545$	22.6615	0.970712
$546$	−14.9462	−0.639640
$547$	23.3375	0.997839	0.498919	−	0.866648i	$-0.333730\pi$
$547$	23.3375	0.997839	0.498919	+	0.866648i	$0.333730\pi$
$548$	−72.0686	−3.07862
$549$	−3.47883	−0.148473
$550$	−20.2176	−0.862083
$551$	−0.0357418	−0.00152265
$552$	8.75427	0.372607
$553$	37.2595	1.58444
$554$	55.8554	2.37307
$555$	−42.9999	−1.82524
$556$	41.1078	1.74336
$557$	5.77628	0.244749	0.122374	−	0.992484i	$-0.460949\pi$
$557$	5.77628	0.244749	0.122374	+	0.992484i	$0.460949\pi$
$558$	−9.42464	−0.398977
$559$	0	0
$560$	7.16814	0.302909
$561$	3.30001	0.139327
$562$	−13.3365	−0.562565
$563$	3.80118	0.160201	0.0801003	−	0.996787i	$-0.474476\pi$
$563$	3.80118	0.160201	0.0801003	+	0.996787i	$0.474476\pi$
$564$	35.6364	1.50056
$565$	−0.570942	−0.0240197
$566$	36.9888	1.55476
$567$	−11.2787	−0.473661
$568$	−23.1633	−0.971910
$569$	−36.2770	−1.52081	−0.760405	−	0.649449i	$-0.774999\pi$
$569$	−36.2770	−1.52081	−0.760405	+	0.649449i	$0.774999\pi$
$570$	0.608664	0.0254941
$571$	−26.5511	−1.11113	−0.555563	−	0.831474i	$-0.687497\pi$
$571$	−26.5511	−1.11113	−0.555563	+	0.831474i	$0.687497\pi$
$572$	−4.37081	−0.182753
$573$	−25.6296	−1.07069
$574$	−54.1039	−2.25825
$575$	23.6202	0.985029
$576$	14.9155	0.621477
$577$	13.0252	0.542247	0.271123	−	0.962545i	$-0.412605\pi$
$577$	13.0252	0.542247	0.271123	+	0.962545i	$0.412605\pi$
$578$	14.9167	0.620454
$579$	4.15274	0.172582
$580$	−10.2698	−0.426431
$581$	15.7796	0.654649
$582$	−32.6519	−1.35347
$583$	1.34650	0.0557663
$584$	32.4387	1.34232
$585$	8.28034	0.342350
$586$	14.0294	0.579551
$587$	5.76434	0.237920	0.118960	−	0.992899i	$-0.462044\pi$
$587$	5.76434	0.237920	0.118960	+	0.992899i	$0.462044\pi$
$588$	−3.27339	−0.134992
$589$	−0.164527	−0.00677922
$590$	−54.6278	−2.24899
$591$	−15.8362	−0.651416
$592$	4.96048	0.203875
$593$	−8.10141	−0.332685	−0.166342	−	0.986068i	$-0.553196\pi$
$593$	−8.10141	−0.332685	−0.166342	+	0.986068i	$0.553196\pi$
$594$	−9.84110	−0.403785
$595$	−36.7775	−1.50773
$596$	27.8900	1.14242
$597$	0.360980	0.0147739
$598$	8.13375	0.332614
$599$	−8.49887	−0.347254	−0.173627	−	0.984811i	$-0.555549\pi$
$599$	−8.49887	−0.347254	−0.173627	+	0.984811i	$0.555549\pi$
$600$	49.7378	2.03054
$601$	−35.6791	−1.45538	−0.727691	−	0.685905i	$-0.759406\pi$
$601$	−35.6791	−1.45538	−0.727691	+	0.685905i	$0.759406\pi$
$602$	0	0
$603$	−6.61676	−0.269455
$604$	15.4192	0.627399
$605$	−42.4880	−1.72738
$606$	7.16539	0.291074
$607$	−32.8574	−1.33364	−0.666820	−	0.745219i	$-0.732345\pi$
$607$	−32.8574	−1.33364	−0.666820	+	0.745219i	$0.732345\pi$
$608$	0.234247	0.00949996
$609$	2.80079	0.113494
$610$	−27.7951	−1.12539
$611$	13.4807	0.545372
$612$	12.9557	0.523703
$613$	−2.23216	−0.0901562	−0.0450781	−	0.998983i	$-0.514354\pi$
$613$	−2.23216	−0.0901562	−0.0450781	+	0.998983i	$0.514354\pi$
$614$	−54.4639	−2.19798
$615$	−46.1325	−1.86024
$616$	−6.66011	−0.268343
$617$	−13.1580	−0.529722	−0.264861	−	0.964287i	$-0.585326\pi$
$617$	−13.1580	−0.529722	−0.264861	+	0.964287i	$0.585326\pi$
$618$	−16.5993	−0.667720
$619$	−13.8696	−0.557466	−0.278733	−	0.960369i	$-0.589914\pi$
$619$	−13.8696	−0.557466	−0.278733	+	0.960369i	$0.589914\pi$
$620$	−47.2741	−1.89857
$621$	11.4973	0.461371
$622$	2.73082	0.109496
$623$	34.2474	1.37209
$624$	1.47017	0.0588539
$625$	51.2769	2.05108
$626$	−68.9805	−2.75701
$627$	−0.0485428	−0.00193861
$628$	33.7947	1.34856
$629$	−25.4507	−1.01479
$630$	30.9898	1.23466
$631$	5.35589	0.213214	0.106607	−	0.994301i	$-0.466001\pi$
$631$	5.35589	0.213214	0.106607	+	0.994301i	$0.466001\pi$
$632$	−42.6972	−1.69841
$633$	22.1017	0.878462
$634$	62.1947	2.47007
$635$	7.04337	0.279508
$636$	−8.13606	−0.322616
$637$	−1.23827	−0.0490622
$638$	1.30463	0.0516507
$639$	−8.59579	−0.340044
$640$	79.2680	3.13334
$641$	−5.06425	−0.200026	−0.100013	−	0.994986i	$-0.531888\pi$
$641$	−5.06425	−0.200026	−0.100013	+	0.994986i	$0.531888\pi$
$642$	−11.2396	−0.443590
$643$	20.6566	0.814615	0.407308	−	0.913291i	$-0.366468\pi$
$643$	20.6566	0.814615	0.407308	+	0.913291i	$0.366468\pi$
$644$	19.1111	0.753082
$645$	0	0
$646$	0.360255	0.0141740
$647$	−30.4923	−1.19878	−0.599389	−	0.800458i	$-0.704590\pi$
$647$	−30.4923	−1.19878	−0.599389	+	0.800458i	$0.704590\pi$
$648$	12.9247	0.507731
$649$	4.35674	0.171017
$650$	46.2123	1.81259
$651$	12.8926	0.505302
$652$	−72.1091	−2.82401
$653$	−41.6324	−1.62920	−0.814602	−	0.580021i	$-0.803044\pi$
$653$	−41.6324	−1.62920	−0.814602	+	0.580021i	$0.803044\pi$
$654$	17.3948	0.680192
$655$	89.1688	3.48411
$656$	5.32186	0.207784
$657$	12.0379	0.469642
$658$	50.4527	1.96685
$659$	−21.8882	−0.852643	−0.426321	−	0.904572i	$-0.640191\pi$
$659$	−21.8882	−0.852643	−0.426321	+	0.904572i	$0.640191\pi$
$660$	−13.9480	−0.542925
$661$	−5.46508	−0.212567	−0.106283	−	0.994336i	$-0.533895\pi$
$661$	−5.46508	−0.212567	−0.106283	+	0.994336i	$0.533895\pi$
$662$	−6.46075	−0.251104
$663$	−7.54298	−0.292945
$664$	−18.0825	−0.701738
$665$	0.540993	0.0209788
$666$	21.4455	0.830997
$667$	−1.52419	−0.0590170
$668$	74.0898	2.86662
$669$	6.56655	0.253878
$670$	−52.8665	−2.04241
$671$	2.21674	0.0855764
$672$	−18.3560	−0.708098
$673$	−3.55403	−0.136998	−0.0684990	−	0.997651i	$-0.521821\pi$
$673$	−3.55403	−0.136998	−0.0684990	+	0.997651i	$0.521821\pi$
$674$	−20.9762	−0.807972
$675$	65.3226	2.51427
$676$	−33.8650	−1.30250
$677$	−9.75140	−0.374777	−0.187388	−	0.982286i	$-0.560002\pi$
$677$	−9.75140	−0.374777	−0.187388	+	0.982286i	$0.560002\pi$
$678$	−0.438252	−0.0168310
$679$	−29.0217	−1.11375
$680$	42.1449	1.61618
$681$	−25.9962	−0.996176
$682$	6.00547	0.229961
$683$	−20.9787	−0.802730	−0.401365	−	0.915918i	$-0.631464\pi$
$683$	−20.9787	−0.802730	−0.401365	+	0.915918i	$0.631464\pi$
$684$	−0.190577	−0.00728689
$685$	−86.9992	−3.32407
$686$	40.4497	1.54438
$687$	9.73904	0.371568
$688$	0	0
$689$	−3.07775	−0.117253
$690$	25.9562	0.988135
$691$	−8.88049	−0.337830	−0.168915	−	0.985631i	$-0.554026\pi$
$691$	−8.88049	−0.337830	−0.168915	+	0.985631i	$0.554026\pi$
$692$	−52.5492	−1.99762
$693$	−2.47153	−0.0938858
$694$	24.4296	0.927335
$695$	49.6242	1.88235
$696$	−3.20955	−0.121658
$697$	−27.3048	−1.03424
$698$	−28.4809	−1.07802
$699$	11.8737	0.449106
$700$	108.581	4.10396
$701$	−25.0985	−0.947958	−0.473979	−	0.880536i	$-0.657183\pi$
$701$	−25.0985	−0.947958	−0.473979	+	0.880536i	$0.657183\pi$
$702$	22.4942	0.848990
$703$	0.374377	0.0141199
$704$	−9.50428	−0.358206
$705$	43.0193	1.62020
$706$	−9.92867	−0.373670
$707$	6.36874	0.239521
$708$	−26.3251	−0.989357
$709$	49.9971	1.87768	0.938841	−	0.344351i	$-0.111901\pi$
$709$	49.9971	1.87768	0.938841	+	0.344351i	$0.111901\pi$
$710$	−68.6785	−2.57746
$711$	−15.8447	−0.594224
$712$	−39.2455	−1.47079
$713$	−7.01617	−0.262758
$714$	−28.2302	−1.05649
$715$	−5.27632	−0.197323
$716$	23.5478	0.880022
$717$	−8.11857	−0.303194
$718$	79.3593	2.96166
$719$	10.9732	0.409231	0.204616	−	0.978842i	$-0.434406\pi$
$719$	10.9732	0.409231	0.204616	+	0.978842i	$0.434406\pi$
$720$	−3.04828	−0.113603
$721$	−14.7538	−0.549458
$722$	44.0383	1.63893
$723$	−23.7331	−0.882644
$724$	−18.5695	−0.690128
$725$	−8.65977	−0.321616
$726$	−32.6136	−1.21040
$727$	−23.5916	−0.874965	−0.437482	−	0.899227i	$-0.644130\pi$
$727$	−23.5916	−0.874965	−0.437482	+	0.899227i	$0.644130\pi$
$728$	15.2233	0.564213
$729$	27.6083	1.02253
$730$	96.1799	3.55978
$731$	0	0
$732$	−13.3944	−0.495071
$733$	35.2905	1.30348	0.651742	−	0.758441i	$-0.274039\pi$
$733$	35.2905	1.30348	0.651742	+	0.758441i	$0.274039\pi$
$734$	71.6977	2.64641
$735$	−3.95154	−0.145755
$736$	9.98934	0.368212
$737$	4.21627	0.155308
$738$	23.0079	0.846931
$739$	21.9799	0.808544	0.404272	−	0.914639i	$-0.367525\pi$
$739$	21.9799	0.808544	0.404272	+	0.914639i	$0.367525\pi$
$740$	107.571	3.95439
$741$	0.110956	0.00407608
$742$	−11.5187	−0.422866
$743$	−1.84090	−0.0675362	−0.0337681	−	0.999430i	$-0.510751\pi$
$743$	−1.84090	−0.0675362	−0.0337681	+	0.999430i	$0.510751\pi$
$744$	−14.7742	−0.541648
$745$	33.6680	1.23350
$746$	18.4078	0.673956
$747$	−6.71034	−0.245518
$748$	−8.25551	−0.301851
$749$	−9.98996	−0.365025
$750$	83.8206	3.06070
$751$	32.0186	1.16838	0.584188	−	0.811618i	$-0.301413\pi$
$751$	32.0186	1.16838	0.584188	+	0.811618i	$0.301413\pi$
$752$	−4.96272	−0.180972
$753$	23.4420	0.854275
$754$	−2.98204	−0.108600
$755$	18.6136	0.677419
$756$	52.8525	1.92223
$757$	−17.0455	−0.619530	−0.309765	−	0.950813i	$-0.600250\pi$
$757$	−17.0455	−0.619530	−0.309765	+	0.950813i	$0.600250\pi$
$758$	−51.7606	−1.88003
$759$	−2.07008	−0.0751393
$760$	−0.619946	−0.0224878
$761$	−2.85155	−0.103368	−0.0516842	−	0.998663i	$-0.516459\pi$
$761$	−2.85155	−0.103368	−0.0516842	+	0.998663i	$0.516459\pi$
$762$	5.40645	0.195855
$763$	15.4609	0.559722
$764$	64.1165	2.31966
$765$	15.6398	0.565457
$766$	−58.2344	−2.10409
$767$	−9.95838	−0.359576
$768$	26.7989	0.967022
$769$	47.9163	1.72790	0.863952	−	0.503573i	$-0.167982\pi$
$769$	47.9163	1.72790	0.863952	+	0.503573i	$0.167982\pi$
$770$	−19.7470	−0.711634
$771$	−32.8384	−1.18265
$772$	−10.3887	−0.373899
$773$	−12.7995	−0.460366	−0.230183	−	0.973147i	$-0.573933\pi$
$773$	−12.7995	−0.460366	−0.230183	+	0.973147i	$0.573933\pi$
$774$	0	0
$775$	−39.8627	−1.43191
$776$	33.2572	1.19386
$777$	−29.3368	−1.05245
$778$	−40.8180	−1.46339
$779$	0.401651	0.0143906
$780$	31.8815	1.14154
$781$	5.47732	0.195994
$782$	15.3629	0.549376
$783$	−4.21522	−0.150640
$784$	0.455851	0.0162804
$785$	40.7960	1.45607
$786$	68.4455	2.44137
$787$	13.2393	0.471931	0.235966	−	0.971761i	$-0.424175\pi$
$787$	13.2393	0.471931	0.235966	+	0.971761i	$0.424175\pi$
$788$	39.6168	1.41129
$789$	−27.6684	−0.985021
$790$	−126.596	−4.50409
$791$	−0.389527	−0.0138500
$792$	2.83223	0.100639
$793$	−5.06690	−0.179931
$794$	−3.35575	−0.119091
$795$	−9.82163	−0.348337
$796$	−0.903047	−0.0320077
$797$	11.5949	0.410714	0.205357	−	0.978687i	$-0.434164\pi$
$797$	11.5949	0.410714	0.205357	+	0.978687i	$0.434164\pi$
$798$	0.415263	0.0147002
$799$	25.4622	0.900788
$800$	56.7549	2.00659
$801$	−14.5638	−0.514587
$802$	−17.8982	−0.632009
$803$	−7.67065	−0.270691
$804$	−25.4763	−0.898479
$805$	23.0704	0.813124
$806$	−13.7270	−0.483512
$807$	22.9539	0.808017
$808$	−7.29821	−0.256750
$809$	−24.0859	−0.846814	−0.423407	−	0.905940i	$-0.639166\pi$
$809$	−24.0859	−0.846814	−0.423407	+	0.905940i	$0.639166\pi$
$810$	38.3214	1.34648
$811$	19.3461	0.679335	0.339667	−	0.940546i	$-0.389686\pi$
$811$	19.3461	0.679335	0.339667	+	0.940546i	$0.389686\pi$
$812$	−7.00662	−0.245884
$813$	−4.66023	−0.163441
$814$	−13.6653	−0.478968
$815$	−87.0481	−3.04916
$816$	2.77683	0.0972086
$817$	0	0
$818$	74.2530	2.59620
$819$	5.64929	0.197402
$820$	115.408	4.03021
$821$	−5.19452	−0.181290	−0.0906449	−	0.995883i	$-0.528893\pi$
$821$	−5.19452	−0.181290	−0.0906449	+	0.995883i	$0.528893\pi$
$822$	−66.7801	−2.32922
$823$	−54.2369	−1.89058	−0.945290	−	0.326230i	$-0.894222\pi$
$823$	−54.2369	−1.89058	−0.945290	+	0.326230i	$0.894222\pi$
$824$	16.9069	0.588981
$825$	−11.7613	−0.409475
$826$	−37.2700	−1.29679
$827$	3.36419	0.116984	0.0584922	−	0.998288i	$-0.481371\pi$
$827$	3.36419	0.116984	0.0584922	+	0.998288i	$0.481371\pi$
$828$	−8.12706	−0.282435
$829$	−40.4505	−1.40490	−0.702452	−	0.711731i	$-0.747912\pi$
$829$	−40.4505	−1.40490	−0.702452	+	0.711731i	$0.747912\pi$
$830$	−53.6142	−1.86097
$831$	32.4930	1.12717
$832$	21.7243	0.753156
$833$	−2.33883	−0.0810357
$834$	38.0913	1.31899
$835$	89.4392	3.09517
$836$	0.121438	0.00420000
$837$	−19.4035	−0.670683
$838$	−51.2074	−1.76893
$839$	0.0819671	0.00282982	0.00141491	−	0.999999i	$-0.499550\pi$
$839$	0.0819671	0.00282982	0.00141491	+	0.999999i	$0.499550\pi$
$840$	48.5801	1.67617
$841$	−28.4412	−0.980731
$842$	−35.2796	−1.21582
$843$	−7.75827	−0.267209
$844$	−55.2908	−1.90319
$845$	−40.8809	−1.40635
$846$	−21.4552	−0.737645
$847$	−28.9876	−0.996026
$848$	1.13303	0.0389083
$849$	21.5177	0.738484
$850$	87.2850	2.99385
$851$	15.9651	0.547277
$852$	−33.0961	−1.13385
$853$	−24.1500	−0.826882	−0.413441	−	0.910531i	$-0.635673\pi$
$853$	−24.1500	−0.826882	−0.413441	+	0.910531i	$0.635673\pi$
$854$	−18.9633	−0.648910
$855$	−0.230059	−0.00786785
$856$	11.4479	0.391281
$857$	42.0860	1.43763	0.718815	−	0.695201i	$-0.244685\pi$
$857$	42.0860	1.43763	0.718815	+	0.695201i	$0.244685\pi$
$858$	−4.05007	−0.138267
$859$	2.35350	0.0803004	0.0401502	−	0.999194i	$-0.487216\pi$
$859$	2.35350	0.0803004	0.0401502	+	0.999194i	$0.487216\pi$
$860$	0	0
$861$	−31.4741	−1.07263
$862$	−38.2736	−1.30360
$863$	30.6614	1.04373	0.521863	−	0.853029i	$-0.325237\pi$
$863$	30.6614	1.04373	0.521863	+	0.853029i	$0.325237\pi$
$864$	27.6259	0.939853
$865$	−63.4360	−2.15689
$866$	58.5003	1.98792
$867$	8.67756	0.294706
$868$	−32.2529	−1.09474
$869$	10.0964	0.342498
$870$	−9.51621	−0.322630
$871$	−9.63730	−0.326547
$872$	−17.7173	−0.599983
$873$	12.3416	0.417699
$874$	−0.225986	−0.00764410
$875$	74.5015	2.51861
$876$	46.3490	1.56599
$877$	−46.1697	−1.55904	−0.779521	−	0.626377i	$-0.784537\pi$
$877$	−46.1697	−1.55904	−0.779521	+	0.626377i	$0.784537\pi$
$878$	−43.9654	−1.48376
$879$	8.16141	0.275277
$880$	1.94239	0.0654781
$881$	10.3059	0.347216	0.173608	−	0.984815i	$-0.444457\pi$
$881$	10.3059	0.347216	0.173608	+	0.984815i	$0.444457\pi$
$882$	1.97077	0.0663592
$883$	11.8119	0.397501	0.198750	−	0.980050i	$-0.436312\pi$
$883$	11.8119	0.397501	0.198750	+	0.980050i	$0.436312\pi$
$884$	18.8700	0.634665
$885$	−31.7789	−1.06824
$886$	−86.9152	−2.91997
$887$	22.2425	0.746829	0.373415	−	0.927665i	$-0.378187\pi$
$887$	22.2425	0.746829	0.373415	+	0.927665i	$0.378187\pi$
$888$	33.6183	1.12816
$889$	4.80536	0.161167
$890$	−116.362	−3.90045
$891$	−3.05625	−0.102388
$892$	−16.4273	−0.550025
$893$	−0.374546	−0.0125337
$894$	25.8434	0.864331
$895$	28.4262	0.950184
$896$	54.0809	1.80672
$897$	4.73168	0.157986
$898$	−41.9329	−1.39932
$899$	2.57231	0.0857914
$900$	−46.1743	−1.53914
$901$	−5.81321	−0.193666
$902$	−14.6608	−0.488152
$903$	0	0
$904$	0.446375	0.0148462
$905$	−22.4165	−0.745151
$906$	14.2877	0.474678
$907$	35.0559	1.16401	0.582006	−	0.813185i	$-0.302268\pi$
$907$	35.0559	1.16401	0.582006	+	0.813185i	$0.302268\pi$
$908$	65.0336	2.15822
$909$	−2.70833	−0.0898296
$910$	45.1366	1.49626
$911$	−55.7864	−1.84829	−0.924143	−	0.382046i	$-0.875219\pi$
$911$	−55.7864	−1.84829	−0.924143	+	0.382046i	$0.875219\pi$
$912$	−0.0408469	−0.00135257
$913$	4.27589	0.141511
$914$	16.4576	0.544370
$915$	−16.1693	−0.534542
$916$	−24.3637	−0.805001
$917$	60.8357	2.00897
$918$	42.4867	1.40227
$919$	43.9635	1.45022	0.725111	−	0.688632i	$-0.241789\pi$
$919$	43.9635	1.45022	0.725111	+	0.688632i	$0.241789\pi$
$920$	−26.4373	−0.871612
$921$	−31.6835	−1.04401
$922$	4.68571	0.154316
$923$	−12.5197	−0.412092
$924$	−9.51606	−0.313055
$925$	90.7065	2.98241
$926$	55.2333	1.81508
$927$	6.27408	0.206068
$928$	−3.66235	−0.120223
$929$	5.76233	0.189056	0.0945280	−	0.995522i	$-0.469866\pi$
$929$	5.76233	0.189056	0.0945280	+	0.995522i	$0.469866\pi$
$930$	−43.8051	−1.43642
$931$	0.0344039	0.00112754
$932$	−29.7040	−0.972987
$933$	1.58861	0.0520087
$934$	40.2689	1.31764
$935$	−9.96582	−0.325917
$936$	−6.47376	−0.211601
$937$	21.9384	0.716695	0.358347	−	0.933588i	$-0.383340\pi$
$937$	21.9384	0.716695	0.358347	+	0.933588i	$0.383340\pi$
$938$	−36.0684	−1.17767
$939$	−40.1283	−1.30954
$940$	−107.620	−3.51016
$941$	−18.9321	−0.617169	−0.308585	−	0.951197i	$-0.599855\pi$
$941$	−18.9321	−0.617169	−0.308585	+	0.951197i	$0.599855\pi$
$942$	31.3148	1.02029
$943$	17.1282	0.557771
$944$	3.66602	0.119319
$945$	63.8021	2.07548
$946$	0	0
$947$	40.2434	1.30774	0.653868	−	0.756609i	$-0.273145\pi$
$947$	40.2434	1.30774	0.653868	+	0.756609i	$0.273145\pi$
$948$	−61.0064	−1.98140
$949$	17.5331	0.569149
$950$	−1.28395	−0.0416569
$951$	36.1808	1.17324
$952$	28.7535	0.931906
$953$	−36.6510	−1.18724	−0.593621	−	0.804745i	$-0.702302\pi$
$953$	−36.6510	−1.18724	−0.593621	+	0.804745i	$0.702302\pi$
$954$	4.89838	0.158591
$955$	77.3997	2.50460
$956$	20.3099	0.656869
$957$	0.758947	0.0245333
$958$	−17.9128	−0.578735
$959$	−59.3555	−1.91669
$960$	69.3260	2.23749
$961$	−19.1591	−0.618036
$962$	31.2354	1.00707
$963$	4.24826	0.136898
$964$	59.3721	1.91225
$965$	−12.5410	−0.403709
$966$	17.7087	0.569768
$967$	−8.56099	−0.275303	−0.137651	−	0.990481i	$-0.543955\pi$
$967$	−8.56099	−0.275303	−0.137651	+	0.990481i	$0.543955\pi$
$968$	33.2181	1.06767
$969$	0.209573	0.00673244
$970$	98.6065	3.16606
$971$	−49.4838	−1.58801	−0.794005	−	0.607911i	$-0.792008\pi$
$971$	−49.4838	−1.58801	−0.794005	+	0.607911i	$0.792008\pi$
$972$	−38.6007	−1.23812
$973$	33.8563	1.08538
$974$	−40.5243	−1.29848
$975$	26.8833	0.860953
$976$	1.86530	0.0597068
$977$	18.3542	0.587203	0.293601	−	0.955928i	$-0.405146\pi$
$977$	18.3542	0.587203	0.293601	+	0.955928i	$0.405146\pi$
$978$	−66.8177	−2.13659
$979$	9.28021	0.296597
$980$	9.88540	0.315778
$981$	−6.57480	−0.209917
$982$	−50.3348	−1.60625
$983$	4.71634	0.150428	0.0752140	−	0.997167i	$-0.476036\pi$
$983$	4.71634	0.150428	0.0752140	+	0.997167i	$0.476036\pi$
$984$	36.0674	1.14979
$985$	47.8243	1.52381
$986$	−5.63244	−0.179373
$987$	29.3501	0.934223
$988$	−0.277575	−0.00883083
$989$	0	0
$990$	8.39749	0.266890
$991$	−45.2891	−1.43866	−0.719328	−	0.694671i	$-0.755550\pi$
$991$	−45.2891	−1.43866	−0.719328	+	0.694671i	$0.755550\pi$
$992$	−16.8586	−0.535260
$993$	−3.75844	−0.119270
$994$	−46.8561	−1.48619
$995$	−1.09013	−0.0345596
$996$	−25.8366	−0.818663
$997$	29.2819	0.927368	0.463684	−	0.886001i	$-0.346527\pi$
$997$	29.2819	0.927368	0.463684	+	0.886001i	$0.346527\pi$
$998$	−83.1984	−2.63360
$999$	44.1522	1.39691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.o.1.1		18
43.19	odd	42		43.2.g.a.17.1	✓	36
43.34	odd	42		43.2.g.a.38.1	yes	36
43.42	odd	2		1849.2.a.n.1.18		18
129.62	even	42		387.2.y.c.361.3		36
129.77	even	42		387.2.y.c.253.3		36
172.19	even	42		688.2.bg.c.17.1		36
172.163	even	42		688.2.bg.c.81.1		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.17.1	✓	36	43.19	odd	42
43.2.g.a.38.1	yes	36	43.34	odd	42
387.2.y.c.253.3		36	129.77	even	42
387.2.y.c.361.3		36	129.62	even	42
688.2.bg.c.17.1		36	172.19	even	42
688.2.bg.c.81.1		36	172.163	even	42
1849.2.a.n.1.18		18	43.42	odd	2
1849.2.a.o.1.1		18	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q-2.31808 q^{2} -1.34851 q^{3} +3.37350 q^{4} +4.07240 q^{5} +3.12595 q^{6} +2.77841 q^{7} -3.18389 q^{8} -1.18153 q^{9} +O(q^{10})\)	\(q-2.31808 q^{2} -1.34851 q^{3} +3.37350 q^{4} +4.07240 q^{5} +3.12595 q^{6} +2.77841 q^{7} -3.18389 q^{8} -1.18153 q^{9} -9.44015 q^{10} +0.752881 q^{11} -4.54919 q^{12} -1.72089 q^{13} -6.44058 q^{14} -5.49166 q^{15} +0.633520 q^{16} -3.25039 q^{17} +2.73888 q^{18} +0.0478129 q^{19} +13.7383 q^{20} -3.74670 q^{21} -1.74524 q^{22} +2.03896 q^{23} +4.29350 q^{24} +11.5844 q^{25} +3.98917 q^{26} +5.63882 q^{27} +9.37297 q^{28} -0.747535 q^{29} +12.7301 q^{30} -3.44106 q^{31} +4.89924 q^{32} -1.01527 q^{33} +7.53468 q^{34} +11.3148 q^{35} -3.98589 q^{36} +7.83004 q^{37} -0.110834 q^{38} +2.32064 q^{39} -12.9661 q^{40} +8.40047 q^{41} +8.68517 q^{42} +2.53985 q^{44} -4.81165 q^{45} -4.72647 q^{46} -7.83357 q^{47} -0.854306 q^{48} +0.719553 q^{49} -26.8537 q^{50} +4.38318 q^{51} -5.80544 q^{52} +1.78846 q^{53} -13.0712 q^{54} +3.06603 q^{55} -8.84616 q^{56} -0.0644761 q^{57} +1.73285 q^{58} +5.78675 q^{59} -18.5261 q^{60} +2.94434 q^{61} +7.97665 q^{62} -3.28277 q^{63} -12.6239 q^{64} -7.00816 q^{65} +2.35347 q^{66} +5.60017 q^{67} -10.9652 q^{68} -2.74955 q^{69} -26.2286 q^{70} +7.27515 q^{71} +3.76186 q^{72} -10.1884 q^{73} -18.1507 q^{74} -15.6217 q^{75} +0.161297 q^{76} +2.09181 q^{77} -5.37943 q^{78} +13.4104 q^{79} +2.57995 q^{80} -4.05941 q^{81} -19.4730 q^{82} +5.67937 q^{83} -12.6395 q^{84} -13.2369 q^{85} +1.00806 q^{87} -2.39709 q^{88} +12.3263 q^{89} +11.1538 q^{90} -4.78134 q^{91} +6.87843 q^{92} +4.64029 q^{93} +18.1589 q^{94} +0.194713 q^{95} -6.60666 q^{96} -10.4454 q^{97} -1.66798 q^{98} -0.889550 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9	\( 18 q + 5 q^{2} + 5 q^{3} + 19 q^{4} + 11 q^{5} + 4 q^{6} + 6 q^{7} + 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} + 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} + 25 q^{18} + 31 q^{19} + 25 q^{20} + 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} + 27 q^{26} + 23 q^{27} + 20 q^{28} + 37 q^{29} - 17 q^{30} - 12 q^{31} + 39 q^{32} + 38 q^{33} + 14 q^{34} + 16 q^{35} + 47 q^{36} - 19 q^{37} + 56 q^{38} + 46 q^{39} + 6 q^{40} - 7 q^{41} - q^{42} + 7 q^{44} + 23 q^{45} - 47 q^{46} - q^{47} + 15 q^{48} - 6 q^{49} - 3 q^{50} + 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} + 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} + 28 q^{61} + 33 q^{62} + 26 q^{63} + 10 q^{64} + 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} + 7 q^{69} - 34 q^{70} + 86 q^{71} - 2 q^{72} - 27 q^{73} - 79 q^{74} + 31 q^{75} + 59 q^{76} + 43 q^{77} + 91 q^{78} + 17 q^{79} + 8 q^{80} - 10 q^{81} + 13 q^{82} - 12 q^{83} - 32 q^{84} + 28 q^{85} - 43 q^{87} - 23 q^{88} + 51 q^{89} + 10 q^{90} - 20 q^{91} + 18 q^{92} - 30 q^{93} - 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} - 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q + 5 * q^2 + 5 * q^3 + 19 * q^4 + 11 * q^5 + 4 * q^6 + 6 * q^7 + 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 + 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 + 25 * q^18 + 31 * q^19 + 25 * q^20 + 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 + 27 * q^26 + 23 * q^27 + 20 * q^28 + 37 * q^29 - 17 * q^30 - 12 * q^31 + 39 * q^32 + 38 * q^33 + 14 * q^34 + 16 * q^35 + 47 * q^36 - 19 * q^37 + 56 * q^38 + 46 * q^39 + 6 * q^40 - 7 * q^41 - q^42 + 7 * q^44 + 23 * q^45 - 47 * q^46 - q^47 + 15 * q^48 - 6 * q^49 - 3 * q^50 + 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 + 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 + 28 * q^61 + 33 * q^62 + 26 * q^63 + 10 * q^64 + 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 + 7 * q^69 - 34 * q^70 + 86 * q^71 - 2 * q^72 - 27 * q^73 - 79 * q^74 + 31 * q^75 + 59 * q^76 + 43 * q^77 + 91 * q^78 + 17 * q^79 + 8 * q^80 - 10 * q^81 + 13 * q^82 - 12 * q^83 - 32 * q^84 + 28 * q^85 - 43 * q^87 - 23 * q^88 + 51 * q^89 + 10 * q^90 - 20 * q^91 + 18 * q^92 - 30 * q^93 - 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 - 5 * q^98 + 38 * q^99

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