LMFDB - Embedded newform 1849.2.a.r.1.9

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:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 5 x^{19} - 19 x^{18} + 127 x^{17} + 95 x^{16} - 1293 x^{15} + 329 x^{14} + 6765 x^{13} + \cdots - 43 $ x^20 - 5x^19 - 19x^18 + 127x^17 + 95x^16 - 1293x^15 + 329x^14 + 6765x^13 - 5068x^12 - 19360x^11 + 20357x^10 + 29618x^9 - 37889x^8 - 21700x^7 + 32885x^6 + 6093x^5 - 11330x^4 - 1004x^3 + 1386x^2 + 87*x - 43
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.277120$ 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)

$f(q)$	$=$	$q-0.277120 q^{2} -3.15485 q^{3} -1.92320 q^{4} +0.330136 q^{5} +0.874272 q^{6} -1.68241 q^{7} +1.08720 q^{8} +6.95305 q^{9} +O(q^{10})$	\(q-0.277120 q^{2} -3.15485 q^{3} -1.92320 q^{4} +0.330136 q^{5} +0.874272 q^{6} -1.68241 q^{7} +1.08720 q^{8} +6.95305 q^{9} -0.0914873 q^{10} +4.70976 q^{11} +6.06741 q^{12} -2.45403 q^{13} +0.466229 q^{14} -1.04153 q^{15} +3.54512 q^{16} -2.98833 q^{17} -1.92683 q^{18} -3.21031 q^{19} -0.634918 q^{20} +5.30773 q^{21} -1.30517 q^{22} -4.77558 q^{23} -3.42995 q^{24} -4.89101 q^{25} +0.680063 q^{26} -12.4713 q^{27} +3.23561 q^{28} -6.73807 q^{29} +0.288628 q^{30} +7.27801 q^{31} -3.15683 q^{32} -14.8585 q^{33} +0.828128 q^{34} -0.555422 q^{35} -13.3721 q^{36} -4.80685 q^{37} +0.889642 q^{38} +7.74210 q^{39} +0.358924 q^{40} -0.0870863 q^{41} -1.47088 q^{42} -9.05782 q^{44} +2.29545 q^{45} +1.32341 q^{46} +6.96882 q^{47} -11.1843 q^{48} -4.16951 q^{49} +1.35540 q^{50} +9.42772 q^{51} +4.71961 q^{52} +1.51425 q^{53} +3.45604 q^{54} +1.55486 q^{55} -1.82911 q^{56} +10.1280 q^{57} +1.86726 q^{58} -1.29165 q^{59} +2.00307 q^{60} +6.16047 q^{61} -2.01689 q^{62} -11.6979 q^{63} -6.21543 q^{64} -0.810164 q^{65} +4.11761 q^{66} -2.64081 q^{67} +5.74717 q^{68} +15.0662 q^{69} +0.153919 q^{70} +0.474031 q^{71} +7.55935 q^{72} +6.86360 q^{73} +1.33207 q^{74} +15.4304 q^{75} +6.17408 q^{76} -7.92372 q^{77} -2.14549 q^{78} +6.15013 q^{79} +1.17037 q^{80} +18.4857 q^{81} +0.0241334 q^{82} -4.71008 q^{83} -10.2079 q^{84} -0.986555 q^{85} +21.2576 q^{87} +5.12045 q^{88} -6.17384 q^{89} -0.636116 q^{90} +4.12868 q^{91} +9.18441 q^{92} -22.9610 q^{93} -1.93120 q^{94} -1.05984 q^{95} +9.95930 q^{96} +1.51108 q^{97} +1.15546 q^{98} +32.7472 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) $ 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9} + 21 q^{11} - 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} + 9 q^{18} - 7 q^{19} - 17 q^{20} + 8 q^{21} + 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} + 58 q^{26} - 22 q^{27} - 28 q^{28} - 12 q^{29} - 24 q^{30} + 26 q^{31} + 3 q^{32} - 17 q^{33} + 54 q^{34} + 26 q^{35} + 14 q^{36} - 19 q^{37} + 5 q^{38} - 7 q^{39} - 16 q^{40} + 27 q^{41} - 52 q^{42} + 32 q^{44} + 75 q^{45} + 59 q^{46} + 45 q^{47} - 66 q^{48} + 20 q^{49} + 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} - 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} + 30 q^{61} + 72 q^{62} - 21 q^{63} - 7 q^{64} - 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} - 35 q^{69} - 80 q^{70} - 31 q^{71} - 15 q^{72} - 26 q^{73} + 87 q^{74} + 73 q^{75} - 68 q^{76} - 21 q^{77} - 50 q^{78} + 39 q^{79} - 60 q^{80} + 16 q^{81} + 45 q^{82} + 13 q^{83} + 31 q^{84} - 22 q^{85} + 61 q^{87} + 50 q^{88} - 4 q^{89} - 13 q^{90} - 25 q^{91} + 5 q^{92} + 67 q^{93} + 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} - 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9 + 21 * q^11 - 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 + 9 * q^18 - 7 * q^19 - 17 * q^20 + 8 * q^21 + 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 + 58 * q^26 - 22 * q^27 - 28 * q^28 - 12 * q^29 - 24 * q^30 + 26 * q^31 + 3 * q^32 - 17 * q^33 + 54 * q^34 + 26 * q^35 + 14 * q^36 - 19 * q^37 + 5 * q^38 - 7 * q^39 - 16 * q^40 + 27 * q^41 - 52 * q^42 + 32 * q^44 + 75 * q^45 + 59 * q^46 + 45 * q^47 - 66 * q^48 + 20 * q^49 + 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 - 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 + 30 * q^61 + 72 * q^62 - 21 * q^63 - 7 * q^64 - 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 - 35 * q^69 - 80 * q^70 - 31 * q^71 - 15 * q^72 - 26 * q^73 + 87 * q^74 + 73 * q^75 - 68 * q^76 - 21 * q^77 - 50 * q^78 + 39 * q^79 - 60 * q^80 + 16 * q^81 + 45 * q^82 + 13 * q^83 + 31 * q^84 - 22 * q^85 + 61 * q^87 + 50 * q^88 - 4 * q^89 - 13 * q^90 - 25 * q^91 + 5 * q^92 + 67 * q^93 + 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 - 35 * q^98 + 13 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.277120	−0.195954	−0.0979769	−	0.995189i	$-0.531237\pi$
$2$	−0.277120	−0.195954	−0.0979769	+	0.995189i	$0.531237\pi$
$3$	−3.15485	−1.82145	−0.910725	−	0.413012i	$-0.864477\pi$
$3$	−3.15485	−1.82145	−0.910725	+	0.413012i	$0.864477\pi$
$4$	−1.92320	−0.961602
$5$	0.330136	0.147641	0.0738206	−	0.997272i	$-0.476481\pi$
$5$	0.330136	0.147641	0.0738206	+	0.997272i	$0.476481\pi$
$6$	0.874272	0.356920
$7$	−1.68241	−0.635890	−0.317945	−	0.948109i	$-0.602993\pi$
$7$	−1.68241	−0.635890	−0.317945	+	0.948109i	$0.602993\pi$
$8$	1.08720	0.384383
$9$	6.95305	2.31768
$10$	−0.0914873	−0.0289308
$11$	4.70976	1.42004	0.710022	−	0.704179i	$-0.248685\pi$
$11$	4.70976	1.42004	0.710022	+	0.704179i	$0.248685\pi$
$12$	6.06741	1.75151
$13$	−2.45403	−0.680626	−0.340313	−	0.940312i	$-0.610533\pi$
$13$	−2.45403	−0.680626	−0.340313	+	0.940312i	$0.610533\pi$
$14$	0.466229	0.124605
$15$	−1.04153	−0.268921
$16$	3.54512	0.886281
$17$	−2.98833	−0.724777	−0.362388	−	0.932027i	$-0.618039\pi$
$17$	−2.98833	−0.724777	−0.362388	+	0.932027i	$0.618039\pi$
$18$	−1.92683	−0.454159
$19$	−3.21031	−0.736495	−0.368248	−	0.929728i	$-0.620042\pi$
$19$	−3.21031	−0.736495	−0.368248	+	0.929728i	$0.620042\pi$
$20$	−0.634918	−0.141972
$21$	5.30773	1.15824
$22$	−1.30517	−0.278263
$23$	−4.77558	−0.995776	−0.497888	−	0.867241i	$-0.665891\pi$
$23$	−4.77558	−0.995776	−0.497888	+	0.867241i	$0.665891\pi$
$24$	−3.42995	−0.700135
$25$	−4.89101	−0.978202
$26$	0.680063	0.133371
$27$	−12.4713	−2.40009
$28$	3.23561	0.611473
$29$	−6.73807	−1.25123	−0.625614	−	0.780133i	$-0.715152\pi$
$29$	−6.73807	−1.25123	−0.625614	+	0.780133i	$0.715152\pi$
$30$	0.288628	0.0526961
$31$	7.27801	1.30717	0.653585	−	0.756853i	$-0.273264\pi$
$31$	7.27801	1.30717	0.653585	+	0.756853i	$0.273264\pi$
$32$	−3.15683	−0.558053
$33$	−14.8585	−2.58654
$34$	0.828128	0.142023
$35$	−0.555422	−0.0938835
$36$	−13.3721	−2.22869
$37$	−4.80685	−0.790240	−0.395120	−	0.918629i	$-0.629297\pi$
$37$	−4.80685	−0.790240	−0.395120	+	0.918629i	$0.629297\pi$
$38$	0.889642	0.144319
$39$	7.74210	1.23973
$40$	0.358924	0.0567508
$41$	−0.0870863	−0.0136006	−0.00680029	−	0.999977i	$-0.502165\pi$
$41$	−0.0870863	−0.0136006	−0.00680029	+	0.999977i	$0.502165\pi$
$42$	−1.47088	−0.226962
$43$	0	0
$43$	0	0
$44$	−9.05782	−1.36552
$45$	2.29545	0.342185
$46$	1.32341	0.195126
$47$	6.96882	1.01651	0.508254	−	0.861207i	$-0.330291\pi$
$47$	6.96882	1.01651	0.508254	+	0.861207i	$0.330291\pi$
$48$	−11.1843	−1.61432
$49$	−4.16951	−0.595644
$50$	1.35540	0.191682
$51$	9.42772	1.32015
$52$	4.71961	0.654492
$53$	1.51425	0.207998	0.103999	−	0.994577i	$-0.466836\pi$
$53$	1.51425	0.207998	0.103999	+	0.994577i	$0.466836\pi$
$54$	3.45604	0.470307
$55$	1.55486	0.209657
$56$	−1.82911	−0.244425
$57$	10.1280	1.34149
$58$	1.86726	0.245183
$59$	−1.29165	−0.168159	−0.0840796	−	0.996459i	$-0.526795\pi$
$59$	−1.29165	−0.168159	−0.0840796	+	0.996459i	$0.526795\pi$
$60$	2.00307	0.258595
$61$	6.16047	0.788768	0.394384	−	0.918946i	$-0.370958\pi$
$61$	6.16047	0.788768	0.394384	+	0.918946i	$0.370958\pi$
$62$	−2.01689	−0.256145
$63$	−11.6979	−1.47379
$64$	−6.21543	−0.776928
$65$	−0.810164	−0.100488
$66$	4.11761	0.506842
$67$	−2.64081	−0.322626	−0.161313	−	0.986903i	$-0.551573\pi$
$67$	−2.64081	−0.322626	−0.161313	+	0.986903i	$0.551573\pi$
$68$	5.74717	0.696947
$69$	15.0662	1.81376
$70$	0.153919	0.0183968
$71$	0.474031	0.0562572	0.0281286	−	0.999604i	$-0.491045\pi$
$71$	0.474031	0.0562572	0.0281286	+	0.999604i	$0.491045\pi$
$72$	7.55935	0.890878
$73$	6.86360	0.803324	0.401662	−	0.915788i	$-0.368433\pi$
$73$	6.86360	0.803324	0.401662	+	0.915788i	$0.368433\pi$
$74$	1.33207	0.154851
$75$	15.4304	1.78175
$76$	6.17408	0.708215
$77$	−7.92372	−0.902992
$78$	−2.14549	−0.242929
$79$	6.15013	0.691943	0.345972	−	0.938245i	$-0.387549\pi$
$79$	6.15013	0.691943	0.345972	+	0.938245i	$0.387549\pi$
$80$	1.17037	0.130852
$81$	18.4857	2.05397
$82$	0.0241334	0.00266509
$83$	−4.71008	−0.516999	−0.258499	−	0.966011i	$-0.583228\pi$
$83$	−4.71008	−0.516999	−0.258499	+	0.966011i	$0.583228\pi$
$84$	−10.2079	−1.11377
$85$	−0.986555	−0.107007
$86$	0	0
$87$	21.2576	2.27905
$88$	5.12045	0.545841
$89$	−6.17384	−0.654426	−0.327213	−	0.944951i	$-0.606109\pi$
$89$	−6.17384	−0.654426	−0.327213	+	0.944951i	$0.606109\pi$
$90$	−0.636116	−0.0670525
$91$	4.12868	0.432803
$92$	9.18441	0.957541
$93$	−22.9610	−2.38095
$94$	−1.93120	−0.199188
$95$	−1.05984	−0.108737
$96$	9.95930	1.01647
$97$	1.51108	0.153427	0.0767136	−	0.997053i	$-0.475557\pi$
$97$	1.51108	0.153427	0.0767136	+	0.997053i	$0.475557\pi$
$98$	1.15546	0.116719
$99$	32.7472	3.29121
$100$	9.40641	0.940641
$101$	1.89488	0.188547	0.0942737	−	0.995546i	$-0.469947\pi$
$101$	1.89488	0.188547	0.0942737	+	0.995546i	$0.469947\pi$
$102$	−2.61261	−0.258687
$103$	−2.41685	−0.238139	−0.119070	−	0.992886i	$-0.537991\pi$
$103$	−2.41685	−0.238139	−0.119070	+	0.992886i	$0.537991\pi$
$104$	−2.66802	−0.261621
$105$	1.75227	0.171004
$106$	−0.419628	−0.0407579
$107$	−18.5194	−1.79033	−0.895167	−	0.445731i	$-0.852944\pi$
$107$	−18.5194	−1.79033	−0.895167	+	0.445731i	$0.852944\pi$
$108$	23.9848	2.30794
$109$	16.3101	1.56222	0.781112	−	0.624392i	$-0.214653\pi$
$109$	16.3101	1.56222	0.781112	+	0.624392i	$0.214653\pi$
$110$	−0.430883	−0.0410831
$111$	15.1649	1.43938
$112$	−5.96434	−0.563577
$113$	11.9502	1.12418	0.562092	−	0.827075i	$-0.309997\pi$
$113$	11.9502	1.12418	0.562092	+	0.827075i	$0.309997\pi$
$114$	−2.80668	−0.262870
$115$	−1.57659	−0.147018
$116$	12.9587	1.20318
$117$	−17.0630	−1.57748
$118$	0.357944	0.0329514
$119$	5.02759	0.460878
$120$	−1.13235	−0.103369
$121$	11.1818	1.01653
$122$	−1.70719	−0.154562
$123$	0.274744	0.0247728
$124$	−13.9971	−1.25698
$125$	−3.26538	−0.292064
$126$	3.24171	0.288795
$127$	−0.334512	−0.0296832	−0.0148416	−	0.999890i	$-0.504724\pi$
$127$	−0.334512	−0.0296832	−0.0148416	+	0.999890i	$0.504724\pi$
$128$	8.03607	0.710295
$129$	0	0
$130$	0.224513	0.0196911
$131$	12.5235	1.09419	0.547093	−	0.837072i	$-0.315734\pi$
$131$	12.5235	1.09419	0.547093	+	0.837072i	$0.315734\pi$
$132$	28.5760	2.48722
$133$	5.40104	0.468330
$134$	0.731822	0.0632198
$135$	−4.11721	−0.354353
$136$	−3.24891	−0.278592
$137$	−17.0229	−1.45436	−0.727181	−	0.686445i	$-0.759170\pi$
$137$	−17.0229	−1.45436	−0.727181	+	0.686445i	$0.759170\pi$
$138$	−4.17515	−0.355413
$139$	16.9845	1.44061	0.720305	−	0.693658i	$-0.244002\pi$
$139$	16.9845	1.44061	0.720305	+	0.693658i	$0.244002\pi$
$140$	1.06819	0.0902786
$141$	−21.9856	−1.85152
$142$	−0.131364	−0.0110238
$143$	−11.5579	−0.966520
$144$	24.6494	2.05412
$145$	−2.22448	−0.184733
$146$	−1.90204	−0.157414
$147$	13.1542	1.08494
$148$	9.24455	0.759897
$149$	−20.1214	−1.64841	−0.824204	−	0.566293i	$-0.808377\pi$
$149$	−20.1214	−1.64841	−0.824204	+	0.566293i	$0.808377\pi$
$150$	−4.27607	−0.349140
$151$	−11.7694	−0.957782	−0.478891	−	0.877874i	$-0.658961\pi$
$151$	−11.7694	−0.957782	−0.478891	+	0.877874i	$0.658961\pi$
$152$	−3.49025	−0.283096
$153$	−20.7780	−1.67980
$154$	2.19583	0.176945
$155$	2.40273	0.192992
$156$	−14.8896	−1.19212
$157$	13.5051	1.07783	0.538913	−	0.842361i	$-0.318835\pi$
$157$	13.5051	1.07783	0.538913	+	0.842361i	$0.318835\pi$
$158$	−1.70433	−0.135589
$159$	−4.77721	−0.378857
$160$	−1.04218	−0.0823916
$161$	8.03446	0.633204
$162$	−5.12277	−0.402483
$163$	7.31784	0.573178	0.286589	−	0.958054i	$-0.407479\pi$
$163$	7.31784	0.573178	0.286589	+	0.958054i	$0.407479\pi$
$164$	0.167485	0.0130784
$165$	−4.90534	−0.381880
$166$	1.30526	0.101308
$167$	18.2930	1.41556	0.707778	−	0.706435i	$-0.249698\pi$
$167$	18.2930	1.41556	0.707778	+	0.706435i	$0.249698\pi$
$168$	5.77057	0.445209
$169$	−6.97772	−0.536748
$170$	0.273394	0.0209684
$171$	−22.3214	−1.70696
$172$	0	0
$173$	−9.47235	−0.720169	−0.360085	−	0.932920i	$-0.617252\pi$
$173$	−9.47235	−0.720169	−0.360085	+	0.932920i	$0.617252\pi$
$174$	−5.89091	−0.446589
$175$	8.22867	0.622029
$176$	16.6967	1.25856
$177$	4.07497	0.306294
$178$	1.71090	0.128237
$179$	17.2757	1.29124	0.645622	−	0.763657i	$-0.276598\pi$
$179$	17.2757	1.29124	0.645622	+	0.763657i	$0.276598\pi$
$180$	−4.41462	−0.329046
$181$	−8.88341	−0.660299	−0.330149	−	0.943929i	$-0.607099\pi$
$181$	−8.88341	−0.660299	−0.330149	+	0.943929i	$0.607099\pi$
$182$	−1.14414	−0.0848094
$183$	−19.4353	−1.43670
$184$	−5.19201	−0.382760
$185$	−1.58691	−0.116672
$186$	6.36296	0.466555
$187$	−14.0743	−1.02922
$188$	−13.4025	−0.977476
$189$	20.9817	1.52620
$190$	0.293702	0.0213074
$191$	11.8494	0.857393	0.428697	−	0.903448i	$-0.358973\pi$
$191$	11.8494	0.857393	0.428697	+	0.903448i	$0.358973\pi$
$192$	19.6087	1.41514
$193$	−0.606593	−0.0436635	−0.0218318	−	0.999762i	$-0.506950\pi$
$193$	−0.606593	−0.0436635	−0.0218318	+	0.999762i	$0.506950\pi$
$194$	−0.418752	−0.0300646
$195$	2.55594	0.183035
$196$	8.01882	0.572773
$197$	−22.3390	−1.59159	−0.795795	−	0.605566i	$-0.792947\pi$
$197$	−22.3390	−1.59159	−0.795795	+	0.605566i	$0.792947\pi$
$198$	−9.07490	−0.644925
$199$	12.2596	0.869059	0.434530	−	0.900658i	$-0.356915\pi$
$199$	12.2596	0.869059	0.434530	+	0.900658i	$0.356915\pi$
$200$	−5.31751	−0.376004
$201$	8.33135	0.587648
$202$	−0.525110	−0.0369466
$203$	11.3362	0.795644
$204$	−18.1314	−1.26945
$205$	−0.0287503	−0.00200801
$206$	0.669758	0.0466642
$207$	−33.2048	−2.30789
$208$	−8.69985	−0.603226
$209$	−15.1198	−1.04586
$210$	−0.485590	−0.0335089
$211$	22.6896	1.56202	0.781008	−	0.624521i	$-0.214706\pi$
$211$	22.6896	1.56202	0.781008	+	0.624521i	$0.214706\pi$
$212$	−2.91220	−0.200011
$213$	−1.49550	−0.102470
$214$	5.13209	0.350823
$215$	0	0
$216$	−13.5587	−0.922556
$217$	−12.2446	−0.831216
$218$	−4.51986	−0.306123
$219$	−21.6536	−1.46321
$220$	−2.99031	−0.201607
$221$	7.33346	0.493302
$222$	−4.20249	−0.282053
$223$	20.2297	1.35468	0.677339	−	0.735671i	$-0.263133\pi$
$223$	20.2297	1.35468	0.677339	+	0.735671i	$0.263133\pi$
$224$	5.31106	0.354860
$225$	−34.0074	−2.26716
$226$	−3.31165	−0.220288
$227$	23.3251	1.54814	0.774071	−	0.633099i	$-0.218217\pi$
$227$	23.3251	1.54814	0.774071	+	0.633099i	$0.218217\pi$
$228$	−19.4783	−1.28998
$229$	−7.10045	−0.469211	−0.234605	−	0.972091i	$-0.575380\pi$
$229$	−7.10045	−0.469211	−0.234605	+	0.972091i	$0.575380\pi$
$230$	0.436905	0.0288086
$231$	24.9981	1.64476
$232$	−7.32563	−0.480951
$233$	8.21943	0.538473	0.269236	−	0.963074i	$-0.413229\pi$
$233$	8.21943	0.538473	0.269236	+	0.963074i	$0.413229\pi$
$234$	4.72851	0.309112
$235$	2.30066	0.150078
$236$	2.48412	0.161702
$237$	−19.4027	−1.26034
$238$	−1.39325	−0.0903108
$239$	12.2552	0.792723	0.396361	−	0.918095i	$-0.370273\pi$
$239$	12.2552	0.792723	0.396361	+	0.918095i	$0.370273\pi$
$240$	−3.69234	−0.238340
$241$	21.7581	1.40156	0.700782	−	0.713376i	$-0.252835\pi$
$241$	21.7581	1.40156	0.700782	+	0.713376i	$0.252835\pi$
$242$	−3.09870	−0.199192
$243$	−20.9059	−1.34111
$244$	−11.8479	−0.758481
$245$	−1.37650	−0.0879416
$246$	−0.0761371	−0.00485432
$247$	7.87820	0.501278
$248$	7.91266	0.502454
$249$	14.8596	0.941687
$250$	0.904902	0.0572310
$251$	23.7231	1.49739	0.748694	−	0.662916i	$-0.230681\pi$
$251$	23.7231	1.49739	0.748694	+	0.662916i	$0.230681\pi$
$252$	22.4974	1.41720
$253$	−22.4918	−1.41405
$254$	0.0927002	0.00581653
$255$	3.11243	0.194908
$256$	10.2039	0.637743
$257$	22.1466	1.38147	0.690735	−	0.723108i	$-0.257287\pi$
$257$	22.1466	1.38147	0.690735	+	0.723108i	$0.257287\pi$
$258$	0	0
$259$	8.08707	0.502506
$260$	1.55811	0.0966299
$261$	−46.8501	−2.89995
$262$	−3.47053	−0.214410
$263$	16.7892	1.03527	0.517633	−	0.855603i	$-0.326813\pi$
$263$	16.7892	1.03527	0.517633	+	0.855603i	$0.326813\pi$
$264$	−16.1542	−0.994223
$265$	0.499906	0.0307090
$266$	−1.49674	−0.0917709
$267$	19.4775	1.19200
$268$	5.07882	0.310238
$269$	22.5689	1.37605	0.688024	−	0.725688i	$-0.258478\pi$
$269$	22.5689	1.37605	0.688024	+	0.725688i	$0.258478\pi$
$270$	1.14096	0.0694367
$271$	−18.3993	−1.11768	−0.558840	−	0.829276i	$-0.688753\pi$
$271$	−18.3993	−1.11768	−0.558840	+	0.829276i	$0.688753\pi$
$272$	−10.5940	−0.642356
$273$	−13.0253	−0.788330
$274$	4.71739	0.284988
$275$	−23.0355	−1.38909
$276$	−28.9754	−1.74411
$277$	−22.8424	−1.37247	−0.686234	−	0.727381i	$-0.740737\pi$
$277$	−22.8424	−1.37247	−0.686234	+	0.727381i	$0.740737\pi$
$278$	−4.70676	−0.282293
$279$	50.6044	3.02961
$280$	−0.603855	−0.0360873
$281$	−8.28280	−0.494111	−0.247055	−	0.969001i	$-0.579463\pi$
$281$	−8.28280	−0.494111	−0.247055	+	0.969001i	$0.579463\pi$
$282$	6.09265	0.362812
$283$	−4.03859	−0.240069	−0.120035	−	0.992770i	$-0.538301\pi$
$283$	−4.03859	−0.240069	−0.120035	+	0.992770i	$0.538301\pi$
$284$	−0.911659	−0.0540970
$285$	3.34362	0.198059
$286$	3.20293	0.189393
$287$	0.146514	0.00864848
$288$	−21.9496	−1.29339
$289$	−8.06988	−0.474699
$290$	0.616448	0.0361991
$291$	−4.76723	−0.279460
$292$	−13.2001	−0.772478
$293$	27.8582	1.62749	0.813746	−	0.581221i	$-0.197425\pi$
$293$	27.8582	1.62749	0.813746	+	0.581221i	$0.197425\pi$
$294$	−3.64528	−0.212597
$295$	−0.426421	−0.0248272
$296$	−5.22600	−0.303755
$297$	−58.7366	−3.40824
$298$	5.57604	0.323012
$299$	11.7194	0.677752
$300$	−29.6758	−1.71333
$301$	0	0
$302$	3.26155	0.187681
$303$	−5.97805	−0.343430
$304$	−11.3809	−0.652741
$305$	2.03379	0.116455
$306$	5.75801	0.329164
$307$	25.2714	1.44231	0.721157	−	0.692772i	$-0.243611\pi$
$307$	25.2714	1.44231	0.721157	+	0.692772i	$0.243611\pi$
$308$	15.2389	0.868319
$309$	7.62478	0.433759
$310$	−0.665846	−0.0378175
$311$	−15.0489	−0.853346	−0.426673	−	0.904406i	$-0.640315\pi$
$311$	−15.0489	−0.853346	−0.426673	+	0.904406i	$0.640315\pi$
$312$	8.41720	0.476530
$313$	8.60352	0.486300	0.243150	−	0.969989i	$-0.421819\pi$
$313$	8.60352	0.486300	0.243150	+	0.969989i	$0.421819\pi$
$314$	−3.74255	−0.211204
$315$	−3.86188	−0.217592
$316$	−11.8279	−0.665374
$317$	1.86247	0.104607	0.0523034	−	0.998631i	$-0.483344\pi$
$317$	1.86247	0.104607	0.0523034	+	0.998631i	$0.483344\pi$
$318$	1.32386	0.0742385
$319$	−31.7347	−1.77680
$320$	−2.05193	−0.114707
$321$	58.4257	3.26100
$322$	−2.22651	−0.124079
$323$	9.59346	0.533795
$324$	−35.5518	−1.97510
$325$	12.0027	0.665790
$326$	−2.02792	−0.112316
$327$	−51.4558	−2.84551
$328$	−0.0946802	−0.00522784
$329$	−11.7244	−0.646387
$330$	1.35937	0.0748308
$331$	−3.22312	−0.177159	−0.0885793	−	0.996069i	$-0.528233\pi$
$331$	−3.22312	−0.177159	−0.0885793	+	0.996069i	$0.528233\pi$
$332$	9.05845	0.497147
$333$	−33.4222	−1.83153
$334$	−5.06937	−0.277384
$335$	−0.871826	−0.0476329
$336$	18.8166	1.02653
$337$	−4.68934	−0.255445	−0.127722	−	0.991810i	$-0.540767\pi$
$337$	−4.68934	−0.255445	−0.127722	+	0.991810i	$0.540767\pi$
$338$	1.93367	0.105178
$339$	−37.7011	−2.04764
$340$	1.89735	0.102898
$341$	34.2777	1.85624
$342$	6.18572	0.334485
$343$	18.7917	1.01465
$344$	0	0
$345$	4.97389	0.267785
$346$	2.62498	0.141120
$347$	8.17679	0.438953	0.219477	−	0.975618i	$-0.429565\pi$
$347$	8.17679	0.438953	0.219477	+	0.975618i	$0.429565\pi$
$348$	−40.8827	−2.19154
$349$	−21.9077	−1.17269	−0.586345	−	0.810061i	$-0.699434\pi$
$349$	−21.9077	−1.17269	−0.586345	+	0.810061i	$0.699434\pi$
$350$	−2.28033	−0.121889
$351$	30.6049	1.63357
$352$	−14.8679	−0.792461
$353$	11.5744	0.616044	0.308022	−	0.951379i	$-0.400333\pi$
$353$	11.5744	0.616044	0.308022	+	0.951379i	$0.400333\pi$
$354$	−1.12926	−0.0600194
$355$	0.156495	0.00830588
$356$	11.8736	0.629297
$357$	−15.8613	−0.839467
$358$	−4.78744	−0.253024
$359$	−13.7479	−0.725587	−0.362793	−	0.931870i	$-0.618177\pi$
$359$	−13.7479	−0.725587	−0.362793	+	0.931870i	$0.618177\pi$
$360$	2.49561	0.131530
$361$	−8.69393	−0.457575
$362$	2.46177	0.129388
$363$	−35.2768	−1.85155
$364$	−7.94030	−0.416185
$365$	2.26592	0.118604
$366$	5.38593	0.281527
$367$	−14.0031	−0.730954	−0.365477	−	0.930820i	$-0.619094\pi$
$367$	−14.0031	−0.730954	−0.365477	+	0.930820i	$0.619094\pi$
$368$	−16.9300	−0.882538
$369$	−0.605515	−0.0315218
$370$	0.439765	0.0228623
$371$	−2.54758	−0.132264
$372$	44.1587	2.28952
$373$	−9.14022	−0.473262	−0.236631	−	0.971600i	$-0.576043\pi$
$373$	−9.14022	−0.473262	−0.236631	+	0.971600i	$0.576043\pi$
$374$	3.90028	0.201679
$375$	10.3018	0.531980
$376$	7.57650	0.390728
$377$	16.5355	0.851619
$378$	−5.81446	−0.299064
$379$	−16.2291	−0.833634	−0.416817	−	0.908990i	$-0.636854\pi$
$379$	−16.2291	−0.833634	−0.416817	+	0.908990i	$0.636854\pi$
$380$	2.03828	0.104562
$381$	1.05534	0.0540665
$382$	−3.28371	−0.168009
$383$	−4.62623	−0.236389	−0.118195	−	0.992990i	$-0.537711\pi$
$383$	−4.62623	−0.236389	−0.118195	+	0.992990i	$0.537711\pi$
$384$	−25.3526	−1.29377
$385$	−2.61590	−0.133319
$386$	0.168099	0.00855603
$387$	0	0
$388$	−2.90612	−0.147536
$389$	−8.25092	−0.418338	−0.209169	−	0.977879i	$-0.567076\pi$
$389$	−8.25092	−0.418338	−0.209169	+	0.977879i	$0.567076\pi$
$390$	−0.708304	−0.0358664
$391$	14.2710	0.721716
$392$	−4.53309	−0.228956
$393$	−39.5098	−1.99301
$394$	6.19060	0.311878
$395$	2.03038	0.102159
$396$	−62.9795	−3.16484
$397$	−10.1545	−0.509638	−0.254819	−	0.966989i	$-0.582016\pi$
$397$	−10.1545	−0.509638	−0.254819	+	0.966989i	$0.582016\pi$
$398$	−3.39738	−0.170295
$399$	−17.0395	−0.853039
$400$	−17.3392	−0.866962
$401$	4.22369	0.210921	0.105460	−	0.994423i	$-0.466368\pi$
$401$	4.22369	0.210921	0.105460	+	0.994423i	$0.466368\pi$
$402$	−2.30879	−0.115152
$403$	−17.8605	−0.889694
$404$	−3.64424	−0.181308
$405$	6.10280	0.303251
$406$	−3.14149	−0.155909
$407$	−22.6391	−1.12218
$408$	10.2498	0.507442
$409$	−13.6200	−0.673467	−0.336733	−	0.941600i	$-0.609322\pi$
$409$	−13.6200	−0.673467	−0.336733	+	0.941600i	$0.609322\pi$
$410$	0.00796729	0.000393476 0
$411$	53.7046	2.64905
$412$	4.64809	0.228995
$413$	2.17309	0.106931
$414$	9.20173	0.452240
$415$	−1.55497	−0.0763303
$416$	7.74696	0.379826
$417$	−53.5836	−2.62400
$418$	4.18999	0.204939
$419$	−35.2336	−1.72128	−0.860638	−	0.509218i	$-0.829935\pi$
$419$	−35.2336	−1.72128	−0.860638	+	0.509218i	$0.829935\pi$
$420$	−3.36998	−0.164438
$421$	11.2735	0.549438	0.274719	−	0.961524i	$-0.411415\pi$
$421$	11.2735	0.549438	0.274719	+	0.961524i	$0.411415\pi$
$422$	−6.28775	−0.306083
$423$	48.4546	2.35594
$424$	1.64629	0.0799508
$425$	14.6160	0.708978
$426$	0.414432	0.0200793
$427$	−10.3644	−0.501570
$428$	35.6165	1.72159
$429$	36.4634	1.76047
$430$	0	0
$431$	−34.5576	−1.66458	−0.832291	−	0.554339i	$-0.812971\pi$
$431$	−34.5576	−1.66458	−0.832291	+	0.554339i	$0.812971\pi$
$432$	−44.2121	−2.12716
$433$	19.9503	0.958750	0.479375	−	0.877610i	$-0.340863\pi$
$433$	19.9503	0.958750	0.479375	+	0.877610i	$0.340863\pi$
$434$	3.39322	0.162880
$435$	7.01789	0.336482
$436$	−31.3676	−1.50224
$437$	15.3311	0.733384
$438$	6.00065	0.286722
$439$	−4.30315	−0.205378	−0.102689	−	0.994714i	$-0.532745\pi$
$439$	−4.30315	−0.205378	−0.102689	+	0.994714i	$0.532745\pi$
$440$	1.69044	0.0805887
$441$	−28.9908	−1.38051
$442$	−2.03225	−0.0966644
$443$	14.1621	0.672864	0.336432	−	0.941708i	$-0.390780\pi$
$443$	14.1621	0.672864	0.336432	+	0.941708i	$0.390780\pi$
$444$	−29.1651	−1.38411
$445$	−2.03821	−0.0966202
$446$	−5.60605	−0.265454
$447$	63.4798	3.00249
$448$	10.4569	0.494041
$449$	37.4853	1.76904	0.884521	−	0.466499i	$-0.154485\pi$
$449$	37.4853	1.76904	0.884521	+	0.466499i	$0.154485\pi$
$450$	9.42415	0.444259
$451$	−0.410155	−0.0193134
$452$	−22.9827	−1.08102
$453$	37.1307	1.74455
$454$	−6.46386	−0.303364
$455$	1.36303	0.0638996
$456$	11.0112	0.515646
$457$	24.2546	1.13458	0.567291	−	0.823517i	$-0.307991\pi$
$457$	24.2546	1.13458	0.567291	+	0.823517i	$0.307991\pi$
$458$	1.96768	0.0919436
$459$	37.2682	1.73953
$460$	3.03210	0.141372
$461$	12.7583	0.594214	0.297107	−	0.954844i	$-0.403978\pi$
$461$	12.7583	0.594214	0.297107	+	0.954844i	$0.403978\pi$
$462$	−6.92749	−0.322296
$463$	21.3861	0.993898	0.496949	−	0.867780i	$-0.334454\pi$
$463$	21.3861	0.993898	0.496949	+	0.867780i	$0.334454\pi$
$464$	−23.8873	−1.10894
$465$	−7.58025	−0.351526
$466$	−2.27777	−0.105516
$467$	−7.31804	−0.338638	−0.169319	−	0.985561i	$-0.554157\pi$
$467$	−7.31804	−0.338638	−0.169319	+	0.985561i	$0.554157\pi$
$468$	32.8157	1.51690
$469$	4.44291	0.205155
$470$	−0.637559	−0.0294084
$471$	−42.6066	−1.96321
$472$	−1.40429	−0.0646376
$473$	0	0
$474$	5.37688	0.246968
$475$	15.7016	0.720441
$476$	−9.66908	−0.443181
$477$	10.5286	0.482072
$478$	−3.39617	−0.155337
$479$	23.9089	1.09243	0.546214	−	0.837646i	$-0.316069\pi$
$479$	23.9089	1.09243	0.546214	+	0.837646i	$0.316069\pi$
$480$	3.28792	0.150072
$481$	11.7962	0.537859
$482$	−6.02962	−0.274642
$483$	−25.3475	−1.15335
$484$	−21.5049	−0.977495
$485$	0.498862	0.0226522
$486$	5.79344	0.262796
$487$	21.0344	0.953161	0.476581	−	0.879131i	$-0.341876\pi$
$487$	21.0344	0.953161	0.476581	+	0.879131i	$0.341876\pi$
$488$	6.69767	0.303189
$489$	−23.0867	−1.04402
$490$	0.381457	0.0172325
$491$	31.2479	1.41020	0.705099	−	0.709109i	$-0.250903\pi$
$491$	31.2479	1.41020	0.705099	+	0.709109i	$0.250903\pi$
$492$	−0.528388	−0.0238216
$493$	20.1356	0.906862
$494$	−2.18321	−0.0982273
$495$	10.8110	0.485919
$496$	25.8015	1.15852
$497$	−0.797514	−0.0357734
$498$	−4.11789	−0.184527
$499$	−24.2579	−1.08593	−0.542967	−	0.839754i	$-0.682699\pi$
$499$	−24.2579	−1.08593	−0.542967	+	0.839754i	$0.682699\pi$
$500$	6.27999	0.280849
$501$	−57.7117	−2.57837
$502$	−6.57415	−0.293419
$503$	43.6157	1.94473	0.972364	−	0.233470i	$-0.0750082\pi$
$503$	43.6157	1.94473	0.972364	+	0.233470i	$0.0750082\pi$
$504$	−12.7179	−0.566500
$505$	0.625567	0.0278374
$506$	6.23293	0.277088
$507$	22.0136	0.977660
$508$	0.643336	0.0285434
$509$	6.58138	0.291714	0.145857	−	0.989306i	$-0.453406\pi$
$509$	6.58138	0.291714	0.145857	+	0.989306i	$0.453406\pi$
$510$	−0.862517	−0.0381929
$511$	−11.5474	−0.510825
$512$	−18.8999	−0.835263
$513$	40.0366	1.76766
$514$	−6.13728	−0.270704
$515$	−0.797888	−0.0351591
$516$	0	0
$517$	32.8215	1.44349
$518$	−2.24109	−0.0984679
$519$	29.8838	1.31175
$520$	−0.880810	−0.0386261
$521$	−12.9451	−0.567137	−0.283569	−	0.958952i	$-0.591518\pi$
$521$	−12.9451	−0.567137	−0.283569	+	0.958952i	$0.591518\pi$
$522$	12.9831	0.568256
$523$	−20.0741	−0.877779	−0.438890	−	0.898541i	$-0.644628\pi$
$523$	−20.0741	−0.877779	−0.438890	+	0.898541i	$0.644628\pi$
$524$	−24.0853	−1.05217
$525$	−25.9602	−1.13299
$526$	−4.65263	−0.202864
$527$	−21.7491	−0.947406
$528$	−52.6754	−2.29240
$529$	−0.193876	−0.00842940
$530$	−0.138534	−0.00601754
$531$	−8.98094	−0.389740
$532$	−10.3873	−0.450347
$533$	0.213713	0.00925692
$534$	−5.39762	−0.233578
$535$	−6.11390	−0.264327
$536$	−2.87109	−0.124012
$537$	−54.5021	−2.35194
$538$	−6.25429	−0.269642
$539$	−19.6374	−0.845841
$540$	7.91823	0.340746
$541$	14.0745	0.605109	0.302554	−	0.953132i	$-0.402161\pi$
$541$	14.0745	0.605109	0.302554	+	0.953132i	$0.402161\pi$
$542$	5.09883	0.219013
$543$	28.0258	1.20270
$544$	9.43364	0.404464
$545$	5.38454	0.230648
$546$	3.60959	0.154476
$547$	−32.4873	−1.38906	−0.694528	−	0.719465i	$-0.744387\pi$
$547$	−32.4873	−1.38906	−0.694528	+	0.719465i	$0.744387\pi$
$548$	32.7385	1.39852
$549$	42.8341	1.82811
$550$	6.38360	0.272197
$551$	21.6313	0.921524
$552$	16.3800	0.697178
$553$	−10.3470	−0.440000
$554$	6.33010	0.268940
$555$	5.00646	0.212512
$556$	−32.6647	−1.38529
$557$	4.65754	0.197346	0.0986731	−	0.995120i	$-0.468540\pi$
$557$	4.65754	0.197346	0.0986731	+	0.995120i	$0.468540\pi$
$558$	−14.0235	−0.593662
$559$	0	0
$560$	−1.96904	−0.0832072
$561$	44.4023	1.87467
$562$	2.29533	0.0968228
$563$	−11.1458	−0.469738	−0.234869	−	0.972027i	$-0.575466\pi$
$563$	−11.1458	−0.469738	−0.234869	+	0.972027i	$0.575466\pi$
$564$	42.2827	1.78042
$565$	3.94520	0.165976
$566$	1.11918	0.0470425
$567$	−31.1005	−1.30610
$568$	0.515367	0.0216243
$569$	38.9746	1.63390	0.816950	−	0.576708i	$-0.195663\pi$
$569$	38.9746	1.63390	0.816950	+	0.576708i	$0.195663\pi$
$570$	−0.926586	−0.0388104
$571$	−17.3762	−0.727171	−0.363585	−	0.931561i	$-0.618447\pi$
$571$	−17.3762	−0.727171	−0.363585	+	0.931561i	$0.618447\pi$
$572$	22.2282	0.929408
$573$	−37.3831	−1.56170
$574$	−0.0406021	−0.00169470
$575$	23.3574	0.974071
$576$	−43.2162	−1.80067
$577$	5.68461	0.236654	0.118327	−	0.992975i	$-0.462247\pi$
$577$	5.68461	0.236654	0.118327	+	0.992975i	$0.462247\pi$
$578$	2.23633	0.0930189
$579$	1.91371	0.0795310
$580$	4.27813	0.177640
$581$	7.92427	0.328754
$582$	1.32110	0.0547612
$583$	7.13172	0.295366
$584$	7.46211	0.308784
$585$	−5.63311	−0.232900
$586$	−7.72007	−0.318913
$587$	−14.1858	−0.585510	−0.292755	−	0.956188i	$-0.594572\pi$
$587$	−14.1858	−0.585510	−0.292755	+	0.956188i	$0.594572\pi$
$588$	−25.2981	−1.04328
$589$	−23.3647	−0.962724
$590$	0.118170	0.00486498
$591$	70.4762	2.89900
$592$	−17.0409	−0.700375
$593$	−2.99513	−0.122995	−0.0614977	−	0.998107i	$-0.519588\pi$
$593$	−2.99513	−0.122995	−0.0614977	+	0.998107i	$0.519588\pi$
$594$	16.2771	0.667857
$595$	1.65979	0.0680446
$596$	38.6975	1.58511
$597$	−38.6771	−1.58295
$598$	−3.24769	−0.132808
$599$	25.6516	1.04809	0.524047	−	0.851689i	$-0.324421\pi$
$599$	25.6516	1.04809	0.524047	+	0.851689i	$0.324421\pi$
$600$	16.7759	0.684874
$601$	−32.6995	−1.33384	−0.666921	−	0.745129i	$-0.732388\pi$
$601$	−32.6995	−1.33384	−0.666921	+	0.745129i	$0.732388\pi$
$602$	0	0
$603$	−18.3617	−0.747745
$604$	22.6350	0.921006
$605$	3.69151	0.150081
$606$	1.65664	0.0672964
$607$	46.4051	1.88353	0.941763	−	0.336278i	$-0.109168\pi$
$607$	46.4051	1.88353	0.941763	+	0.336278i	$0.109168\pi$
$608$	10.1344	0.411003
$609$	−35.7639	−1.44923
$610$	−0.563605	−0.0228197
$611$	−17.1017	−0.691862
$612$	39.9604	1.61530
$613$	22.8751	0.923916	0.461958	−	0.886902i	$-0.347147\pi$
$613$	22.8751	0.923916	0.461958	+	0.886902i	$0.347147\pi$
$614$	−7.00321	−0.282627
$615$	0.0907027	0.00365749
$616$	−8.61467	−0.347095
$617$	36.5536	1.47159	0.735795	−	0.677204i	$-0.236808\pi$
$617$	36.5536	1.47159	0.735795	+	0.677204i	$0.236808\pi$
$618$	−2.11298	−0.0849966
$619$	3.89880	0.156706	0.0783531	−	0.996926i	$-0.475034\pi$
$619$	3.89880	0.156706	0.0783531	+	0.996926i	$0.475034\pi$
$620$	−4.62095	−0.185582
$621$	59.5574	2.38996
$622$	4.17036	0.167216
$623$	10.3869	0.416143
$624$	27.4467	1.09875
$625$	23.3770	0.935081
$626$	−2.38421	−0.0952922
$627$	47.7005	1.90497
$628$	−25.9731	−1.03644
$629$	14.3644	0.572748
$630$	1.07021	0.0426380
$631$	13.8145	0.549945	0.274973	−	0.961452i	$-0.411331\pi$
$631$	13.8145	0.549945	0.274973	+	0.961452i	$0.411331\pi$
$632$	6.68642	0.265971
$633$	−71.5822	−2.84514
$634$	−0.516128	−0.0204981
$635$	−0.110435	−0.00438246
$636$	9.18755	0.364310
$637$	10.2321	0.405411
$638$	8.79433	0.348171
$639$	3.29596	0.130386
$640$	2.65299	0.104869
$641$	−10.5318	−0.415982	−0.207991	−	0.978131i	$-0.566692\pi$
$641$	−10.5318	−0.415982	−0.207991	+	0.978131i	$0.566692\pi$
$642$	−16.1910	−0.639006
$643$	−26.8974	−1.06073	−0.530366	−	0.847769i	$-0.677945\pi$
$643$	−26.8974	−1.06073	−0.530366	+	0.847769i	$0.677945\pi$
$644$	−15.4519	−0.608890
$645$	0	0
$646$	−2.65854	−0.104599
$647$	−43.9827	−1.72914	−0.864570	−	0.502513i	$-0.832409\pi$
$647$	−43.9827	−1.72914	−0.864570	+	0.502513i	$0.832409\pi$
$648$	20.0977	0.789512
$649$	−6.08338	−0.238793
$650$	−3.32619	−0.130464
$651$	38.6297	1.51402
$652$	−14.0737	−0.551169
$653$	−39.7369	−1.55502	−0.777512	−	0.628868i	$-0.783519\pi$
$653$	−39.7369	−1.55502	−0.777512	+	0.628868i	$0.783519\pi$
$654$	14.2595	0.557589
$655$	4.13447	0.161547
$656$	−0.308732	−0.0120539
$657$	47.7229	1.86185
$658$	3.24907	0.126662
$659$	14.4512	0.562938	0.281469	−	0.959570i	$-0.409178\pi$
$659$	14.4512	0.562938	0.281469	+	0.959570i	$0.409178\pi$
$660$	9.43397	0.367217
$661$	41.2901	1.60600	0.803000	−	0.595979i	$-0.203236\pi$
$661$	41.2901	1.60600	0.803000	+	0.595979i	$0.203236\pi$
$662$	0.893191	0.0347149
$663$	−23.1359	−0.898526
$664$	−5.12080	−0.198726
$665$	1.78308	0.0691448
$666$	9.26198	0.358894
$667$	32.1782	1.24594
$668$	−35.1812	−1.36120
$669$	−63.8214	−2.46748
$670$	0.241601	0.00933385
$671$	29.0143	1.12009
$672$	−16.7556	−0.646361
$673$	−1.37373	−0.0529533	−0.0264767	−	0.999649i	$-0.508429\pi$
$673$	−1.37373	−0.0529533	−0.0264767	+	0.999649i	$0.508429\pi$
$674$	1.29951	0.0500553
$675$	60.9970	2.34778
$676$	13.4196	0.516138
$677$	−36.0293	−1.38472	−0.692359	−	0.721554i	$-0.743428\pi$
$677$	−36.0293	−1.38472	−0.692359	+	0.721554i	$0.743428\pi$
$678$	10.4478	0.401244
$679$	−2.54225	−0.0975628
$680$	−1.07258	−0.0411317
$681$	−73.5871	−2.81986
$682$	−9.49904	−0.363737
$683$	17.4967	0.669493	0.334746	−	0.942308i	$-0.391349\pi$
$683$	17.4967	0.669493	0.334746	+	0.942308i	$0.391349\pi$
$684$	42.9287	1.64142
$685$	−5.61986	−0.214724
$686$	−5.20755	−0.198825
$687$	22.4008	0.854644
$688$	0	0
$689$	−3.71601	−0.141569
$690$	−1.37837	−0.0524735
$691$	−20.1302	−0.765790	−0.382895	−	0.923792i	$-0.625073\pi$
$691$	−20.1302	−0.765790	−0.382895	+	0.923792i	$0.625073\pi$
$692$	18.2173	0.692517
$693$	−55.0940	−2.09285
$694$	−2.26596	−0.0860145
$695$	5.60720	0.212693
$696$	23.1112	0.876029
$697$	0.260243	0.00985739
$698$	6.07106	0.229793
$699$	−25.9310	−0.980802
$700$	−15.8254	−0.598144
$701$	6.32349	0.238835	0.119417	−	0.992844i	$-0.461897\pi$
$701$	6.32349	0.238835	0.119417	+	0.992844i	$0.461897\pi$
$702$	−8.48123	−0.320104
$703$	15.4315	0.582008
$704$	−29.2731	−1.10327
$705$	−7.25822	−0.273360
$706$	−3.20751	−0.120716
$707$	−3.18796	−0.119895
$708$	−7.83700	−0.294533
$709$	−42.3779	−1.59154	−0.795768	−	0.605601i	$-0.792933\pi$
$709$	−42.3779	−1.59154	−0.795768	+	0.605601i	$0.792933\pi$
$710$	−0.0433679	−0.00162757
$711$	42.7621	1.60370
$712$	−6.71220	−0.251550
$713$	−34.7567	−1.30165
$714$	4.39548	0.164497
$715$	−3.81567	−0.142698
$716$	−33.2247	−1.24166
$717$	−38.6633	−1.44391
$718$	3.80983	0.142181
$719$	34.3385	1.28061	0.640306	−	0.768120i	$-0.278808\pi$
$719$	34.3385	1.28061	0.640306	+	0.768120i	$0.278808\pi$
$720$	8.13765	0.303272
$721$	4.06612	0.151430
$722$	2.40926	0.0896635
$723$	−68.6435	−2.55288
$724$	17.0846	0.634945
$725$	32.9560	1.22395
$726$	9.77593	0.362819
$727$	20.7180	0.768389	0.384195	−	0.923252i	$-0.374479\pi$
$727$	20.7180	0.768389	0.384195	+	0.923252i	$0.374479\pi$
$728$	4.48870	0.166362
$729$	10.4976	0.388799
$730$	−0.627933	−0.0232408
$731$	0	0
$732$	37.3781	1.38154
$733$	−20.2283	−0.747148	−0.373574	−	0.927600i	$-0.621868\pi$
$733$	−20.2283	−0.747148	−0.373574	+	0.927600i	$0.621868\pi$
$734$	3.88054	0.143233
$735$	4.34266	0.160181
$736$	15.0757	0.555696
$737$	−12.4376	−0.458144
$738$	0.167801	0.00617682
$739$	5.38694	0.198162	0.0990810	−	0.995079i	$-0.468410\pi$
$739$	5.38694	0.198162	0.0990810	+	0.995079i	$0.468410\pi$
$740$	3.05195	0.112192
$741$	−24.8545	−0.913053
$742$	0.705985	0.0259175
$743$	19.8517	0.728290	0.364145	−	0.931342i	$-0.381361\pi$
$743$	19.8517	0.728290	0.364145	+	0.931342i	$0.381361\pi$
$744$	−24.9632	−0.915196
$745$	−6.64279	−0.243373
$746$	2.53294	0.0927375
$747$	−32.7494	−1.19824
$748$	27.0678	0.989696
$749$	31.1571	1.13845
$750$	−2.85483	−0.104244
$751$	−11.1535	−0.406998	−0.203499	−	0.979075i	$-0.565231\pi$
$751$	−11.1535	−0.406998	−0.203499	+	0.979075i	$0.565231\pi$
$752$	24.7053	0.900911
$753$	−74.8426	−2.72742
$754$	−4.58231	−0.166878
$755$	−3.88551	−0.141408
$756$	−40.3521	−1.46759
$757$	13.8616	0.503808	0.251904	−	0.967752i	$-0.418943\pi$
$757$	13.8616	0.503808	0.251904	+	0.967752i	$0.418943\pi$
$758$	4.49742	0.163354
$759$	70.9581	2.57562
$760$	−1.15225	−0.0417967
$761$	−32.5418	−1.17964	−0.589819	−	0.807536i	$-0.700801\pi$
$761$	−32.5418	−1.17964	−0.589819	+	0.807536i	$0.700801\pi$
$762$	−0.292455	−0.0105945
$763$	−27.4402	−0.993402
$764$	−22.7888	−0.824471
$765$	−6.85956	−0.248008
$766$	1.28202	0.0463214
$767$	3.16976	0.114454
$768$	−32.1917	−1.16162
$769$	42.7581	1.54190	0.770949	−	0.636897i	$-0.219783\pi$
$769$	42.7581	1.54190	0.770949	+	0.636897i	$0.219783\pi$
$770$	0.724920	0.0261243
$771$	−69.8692	−2.51628
$772$	1.16660	0.0419869
$773$	−25.1380	−0.904152	−0.452076	−	0.891979i	$-0.649316\pi$
$773$	−25.1380	−0.904152	−0.452076	+	0.891979i	$0.649316\pi$
$774$	0	0
$775$	−35.5968	−1.27868
$776$	1.64285	0.0589748
$777$	−25.5134	−0.915290
$778$	2.28650	0.0819749
$779$	0.279574	0.0100168
$780$	−4.91560	−0.176007
$781$	2.23257	0.0798877
$782$	−3.95479	−0.141423
$783$	84.0322	3.00307
$784$	−14.7814	−0.527908
$785$	4.45853	0.159132
$786$	10.9490	0.390537
$787$	1.54450	0.0550553	0.0275277	−	0.999621i	$-0.491237\pi$
$787$	1.54450	0.0550553	0.0275277	+	0.999621i	$0.491237\pi$
$788$	42.9625	1.53048
$789$	−52.9673	−1.88569
$790$	−0.562659	−0.0200185
$791$	−20.1051	−0.714857
$792$	35.6027	1.26509
$793$	−15.1180	−0.536856
$794$	2.81401	0.0998654
$795$	−1.57713	−0.0559349
$796$	−23.5777	−0.835689
$797$	−20.1477	−0.713667	−0.356833	−	0.934168i	$-0.616144\pi$
$797$	−20.1477	−0.713667	−0.356833	+	0.934168i	$0.616144\pi$
$798$	4.72198	0.167156
$799$	−20.8252	−0.736741
$800$	15.4401	0.545889
$801$	−42.9270	−1.51675
$802$	−1.17047	−0.0413307
$803$	32.3259	1.14076
$804$	−16.0229	−0.565083
$805$	2.65246	0.0934870
$806$	4.94951	0.174339
$807$	−71.2013	−2.50640
$808$	2.06011	0.0724745
$809$	−26.5658	−0.934004	−0.467002	−	0.884256i	$-0.654666\pi$
$809$	−26.5658	−0.934004	−0.467002	+	0.884256i	$0.654666\pi$
$810$	−1.69121	−0.0594231
$811$	−40.3943	−1.41843	−0.709217	−	0.704990i	$-0.750951\pi$
$811$	−40.3943	−1.41843	−0.709217	+	0.704990i	$0.750951\pi$
$812$	−21.8018	−0.765093
$813$	58.0470	2.03580
$814$	6.27375	0.219895
$815$	2.41588	0.0846247
$816$	33.4224	1.17002
$817$	0	0
$818$	3.77438	0.131968
$819$	28.7069	1.00310
$820$	0.0552927	0.00193090
$821$	20.2695	0.707409	0.353705	−	0.935357i	$-0.384922\pi$
$821$	20.2695	0.707409	0.353705	+	0.935357i	$0.384922\pi$
$822$	−14.8826	−0.519091
$823$	−27.9276	−0.973495	−0.486748	−	0.873543i	$-0.661817\pi$
$823$	−27.9276	−0.973495	−0.486748	+	0.873543i	$0.661817\pi$
$824$	−2.62760	−0.0915366
$825$	72.6733	2.53016
$826$	−0.602207	−0.0209535
$827$	13.5783	0.472164	0.236082	−	0.971733i	$-0.424137\pi$
$827$	13.5783	0.472164	0.236082	+	0.971733i	$0.424137\pi$
$828$	63.8596	2.21928
$829$	16.8903	0.586625	0.293313	−	0.956017i	$-0.405242\pi$
$829$	16.8903	0.586625	0.293313	+	0.956017i	$0.405242\pi$
$830$	0.430913	0.0149572
$831$	72.0643	2.49988
$832$	15.2529	0.528798
$833$	12.4599	0.431709
$834$	14.8491	0.514182
$835$	6.03918	0.208994
$836$	29.0784	1.00570
$837$	−90.7660	−3.13733
$838$	9.76396	0.337290
$839$	32.4216	1.11932	0.559659	−	0.828723i	$-0.310932\pi$
$839$	32.4216	1.11932	0.559659	+	0.828723i	$0.310932\pi$
$840$	1.90507	0.0657312
$841$	16.4016	0.565573
$842$	−3.12413	−0.107664
$843$	26.1310	0.899998
$844$	−43.6367	−1.50204
$845$	−2.30359	−0.0792461
$846$	−13.4277	−0.461655
$847$	−18.8123	−0.646399
$848$	5.36819	0.184344
$849$	12.7411	0.437275
$850$	−4.05038	−0.138927
$851$	22.9555	0.786903
$852$	2.87614	0.0985351
$853$	−40.5438	−1.38819	−0.694097	−	0.719881i	$-0.744196\pi$
$853$	−40.5438	−1.38819	−0.694097	+	0.719881i	$0.744196\pi$
$854$	2.87219	0.0982844
$855$	−7.36910	−0.252018
$856$	−20.1342	−0.688174
$857$	28.2862	0.966237	0.483118	−	0.875555i	$-0.339504\pi$
$857$	28.2862	0.966237	0.483118	+	0.875555i	$0.339504\pi$
$858$	−10.1047	−0.344970
$859$	7.11076	0.242616	0.121308	−	0.992615i	$-0.461291\pi$
$859$	7.11076	0.242616	0.121308	+	0.992615i	$0.461291\pi$
$860$	0	0
$861$	−0.462230	−0.0157528
$862$	9.57662	0.326181
$863$	10.2232	0.348002	0.174001	−	0.984745i	$-0.444330\pi$
$863$	10.2232	0.348002	0.174001	+	0.984745i	$0.444330\pi$
$864$	39.3696	1.33938
$865$	−3.12716	−0.106327
$866$	−5.52864	−0.187871
$867$	25.4592	0.864640
$868$	23.5488	0.799299
$869$	28.9656	0.982590
$870$	−1.94480	−0.0659349
$871$	6.48063	0.219588
$872$	17.7323	0.600492
$873$	10.5066	0.355596
$874$	−4.24855	−0.143709
$875$	5.49369	0.185721
$876$	41.6443	1.40703
$877$	47.7597	1.61273	0.806365	−	0.591418i	$-0.201432\pi$
$877$	47.7597	1.61273	0.806365	+	0.591418i	$0.201432\pi$
$878$	1.19249	0.0402446
$879$	−87.8882	−2.96440
$880$	5.51217	0.185815
$881$	43.5236	1.46635	0.733173	−	0.680042i	$-0.238038\pi$
$881$	43.5236	1.46635	0.733173	+	0.680042i	$0.238038\pi$
$882$	8.03394	0.270517
$883$	−23.1874	−0.780319	−0.390159	−	0.920747i	$-0.627580\pi$
$883$	−23.1874	−0.780319	−0.390159	+	0.920747i	$0.627580\pi$
$884$	−14.1038	−0.474360
$885$	1.34529	0.0452215
$886$	−3.92462	−0.131850
$887$	−18.2513	−0.612818	−0.306409	−	0.951900i	$-0.599128\pi$
$887$	−18.2513	−0.612818	−0.306409	+	0.951900i	$0.599128\pi$
$888$	16.4872	0.553275
$889$	0.562786	0.0188752
$890$	0.564828	0.0189331
$891$	87.0633	2.91673
$892$	−38.9058	−1.30266
$893$	−22.3721	−0.748653
$894$	−17.5916	−0.588350
$895$	5.70332	0.190641
$896$	−13.5199	−0.451669
$897$	−36.9730	−1.23449
$898$	−10.3880	−0.346651
$899$	−49.0398	−1.63557
$900$	65.4032	2.18011
$901$	−4.52507	−0.150752
$902$	0.113662	0.00378454
$903$	0	0
$904$	12.9923	0.432117
$905$	−2.93273	−0.0974873
$906$	−10.2897	−0.341852
$907$	−55.1763	−1.83210	−0.916049	−	0.401066i	$-0.868640\pi$
$907$	−55.1763	−1.83210	−0.916049	+	0.401066i	$0.868640\pi$
$908$	−44.8590	−1.48870
$909$	13.1752	0.436993
$910$	−0.377722	−0.0125214
$911$	3.32587	0.110191	0.0550955	−	0.998481i	$-0.482454\pi$
$911$	3.32587	0.110191	0.0550955	+	0.998481i	$0.482454\pi$
$912$	35.9051	1.18894
$913$	−22.1833	−0.734161
$914$	−6.72145	−0.222326
$915$	−6.41630	−0.212116
$916$	13.6556	0.451194
$917$	−21.0697	−0.695782
$918$	−10.3278	−0.340868
$919$	32.8210	1.08267	0.541333	−	0.840808i	$-0.317920\pi$
$919$	32.8210	1.08267	0.541333	+	0.840808i	$0.317920\pi$
$920$	−1.71407	−0.0565111
$921$	−79.7273	−2.62710
$922$	−3.53559	−0.116438
$923$	−1.16329	−0.0382901
$924$	−48.0765	−1.58160
$925$	23.5103	0.773015
$926$	−5.92653	−0.194758
$927$	−16.8045	−0.551931
$928$	21.2709	0.698252
$929$	−3.23330	−0.106081	−0.0530406	−	0.998592i	$-0.516891\pi$
$929$	−3.23330	−0.106081	−0.0530406	+	0.998592i	$0.516891\pi$
$930$	2.10064	0.0688828
$931$	13.3854	0.438689
$932$	−15.8076	−0.517797
$933$	47.4770	1.55433
$934$	2.02798	0.0663574
$935$	−4.64643	−0.151955
$936$	−18.5509	−0.606355
$937$	−4.99621	−0.163219	−0.0816095	−	0.996664i	$-0.526006\pi$
$937$	−4.99621	−0.163219	−0.0816095	+	0.996664i	$0.526006\pi$
$938$	−1.23122	−0.0402008
$939$	−27.1428	−0.885771
$940$	−4.42463	−0.144316
$941$	31.0314	1.01159	0.505797	−	0.862652i	$-0.331198\pi$
$941$	31.0314	1.01159	0.505797	+	0.862652i	$0.331198\pi$
$942$	11.8072	0.384698
$943$	0.415887	0.0135431
$944$	−4.57908	−0.149036
$945$	6.92682	0.225329
$946$	0	0
$947$	24.8621	0.807911	0.403956	−	0.914779i	$-0.367635\pi$
$947$	24.8621	0.807911	0.403956	+	0.914779i	$0.367635\pi$
$948$	37.3153	1.21195
$949$	−16.8435	−0.546763
$950$	−4.35125	−0.141173
$951$	−5.87580	−0.190536
$952$	5.46599	0.177154
$953$	−24.7810	−0.802734	−0.401367	−	0.915917i	$-0.631465\pi$
$953$	−24.7810	−0.802734	−0.401367	+	0.915917i	$0.631465\pi$
$954$	−2.91769	−0.0944639
$955$	3.91191	0.126587
$956$	−23.5693	−0.762284
$957$	100.118	3.23635
$958$	−6.62565	−0.214065
$959$	28.6394	0.924815
$960$	6.47353	0.208932
$961$	21.9695	0.708693
$962$	−3.26896	−0.105395
$963$	−128.766	−4.14943
$964$	−41.8453	−1.34775
$965$	−0.200258	−0.00644654
$966$	7.02430	0.226003
$967$	44.8178	1.44124	0.720621	−	0.693329i	$-0.243857\pi$
$967$	44.8178	1.44124	0.720621	+	0.693329i	$0.243857\pi$
$968$	12.1568	0.390736
$969$	−30.2659	−0.972280
$970$	−0.138245	−0.00443878
$971$	2.65395	0.0851694	0.0425847	−	0.999093i	$-0.486441\pi$
$971$	2.65395	0.0851694	0.0425847	+	0.999093i	$0.486441\pi$
$972$	40.2062	1.28962
$973$	−28.5749	−0.916069
$974$	−5.82907	−0.186775
$975$	−37.8667	−1.21270
$976$	21.8396	0.699070
$977$	40.4223	1.29322	0.646612	−	0.762819i	$-0.276185\pi$
$977$	40.4223	1.29322	0.646612	+	0.762819i	$0.276185\pi$
$978$	6.39779	0.204579
$979$	−29.0773	−0.929314
$980$	2.64730	0.0845648
$981$	113.405	3.62074
$982$	−8.65942	−0.276333
$983$	10.9910	0.350559	0.175279	−	0.984519i	$-0.443917\pi$
$983$	10.9910	0.350559	0.175279	+	0.984519i	$0.443917\pi$
$984$	0.298701	0.00952225
$985$	−7.37491	−0.234984
$986$	−5.57998	−0.177703
$987$	36.9886	1.17736
$988$	−15.1514	−0.482030
$989$	0	0
$990$	−2.99595	−0.0952176
$991$	8.74235	0.277710	0.138855	−	0.990313i	$-0.455658\pi$
$991$	8.74235	0.277710	0.138855	+	0.990313i	$0.455658\pi$
$992$	−22.9754	−0.729470
$993$	10.1684	0.322686
$994$	0.221007	0.00700992
$995$	4.04733	0.128309
$996$	−28.5780	−0.905529
$997$	14.7508	0.467164	0.233582	−	0.972337i	$-0.424955\pi$
$997$	14.7508	0.467164	0.233582	+	0.972337i	$0.424955\pi$
$998$	6.72236	0.212793
$999$	59.9474	1.89665

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.r.1.9	yes	20
43.42	odd	2		1849.2.a.p.1.12	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.12	✓	20	43.42	odd	2
1849.2.a.r.1.9	yes	20	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)

\(f(q)\)	\(=\)	\(q-0.277120 q^{2} -3.15485 q^{3} -1.92320 q^{4} +0.330136 q^{5} +0.874272 q^{6} -1.68241 q^{7} +1.08720 q^{8} +6.95305 q^{9} +O(q^{10})\)	\(q-0.277120 q^{2} -3.15485 q^{3} -1.92320 q^{4} +0.330136 q^{5} +0.874272 q^{6} -1.68241 q^{7} +1.08720 q^{8} +6.95305 q^{9} -0.0914873 q^{10} +4.70976 q^{11} +6.06741 q^{12} -2.45403 q^{13} +0.466229 q^{14} -1.04153 q^{15} +3.54512 q^{16} -2.98833 q^{17} -1.92683 q^{18} -3.21031 q^{19} -0.634918 q^{20} +5.30773 q^{21} -1.30517 q^{22} -4.77558 q^{23} -3.42995 q^{24} -4.89101 q^{25} +0.680063 q^{26} -12.4713 q^{27} +3.23561 q^{28} -6.73807 q^{29} +0.288628 q^{30} +7.27801 q^{31} -3.15683 q^{32} -14.8585 q^{33} +0.828128 q^{34} -0.555422 q^{35} -13.3721 q^{36} -4.80685 q^{37} +0.889642 q^{38} +7.74210 q^{39} +0.358924 q^{40} -0.0870863 q^{41} -1.47088 q^{42} -9.05782 q^{44} +2.29545 q^{45} +1.32341 q^{46} +6.96882 q^{47} -11.1843 q^{48} -4.16951 q^{49} +1.35540 q^{50} +9.42772 q^{51} +4.71961 q^{52} +1.51425 q^{53} +3.45604 q^{54} +1.55486 q^{55} -1.82911 q^{56} +10.1280 q^{57} +1.86726 q^{58} -1.29165 q^{59} +2.00307 q^{60} +6.16047 q^{61} -2.01689 q^{62} -11.6979 q^{63} -6.21543 q^{64} -0.810164 q^{65} +4.11761 q^{66} -2.64081 q^{67} +5.74717 q^{68} +15.0662 q^{69} +0.153919 q^{70} +0.474031 q^{71} +7.55935 q^{72} +6.86360 q^{73} +1.33207 q^{74} +15.4304 q^{75} +6.17408 q^{76} -7.92372 q^{77} -2.14549 q^{78} +6.15013 q^{79} +1.17037 q^{80} +18.4857 q^{81} +0.0241334 q^{82} -4.71008 q^{83} -10.2079 q^{84} -0.986555 q^{85} +21.2576 q^{87} +5.12045 q^{88} -6.17384 q^{89} -0.636116 q^{90} +4.12868 q^{91} +9.18441 q^{92} -22.9610 q^{93} -1.93120 q^{94} -1.05984 q^{95} +9.95930 q^{96} +1.51108 q^{97} +1.15546 q^{98} +32.7472 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9} + 21 q^{11} - 12 q^{12} + 11 q^{13} + 8 q^{14} + 4 q^{15} + 29 q^{16} + 15 q^{17} + 9 q^{18} - 7 q^{19} - 17 q^{20} + 8 q^{21} + 47 q^{22} + 26 q^{23} - q^{24} + 11 q^{25} + 58 q^{26} - 22 q^{27} - 28 q^{28} - 12 q^{29} - 24 q^{30} + 26 q^{31} + 3 q^{32} - 17 q^{33} + 54 q^{34} + 26 q^{35} + 14 q^{36} - 19 q^{37} + 5 q^{38} - 7 q^{39} - 16 q^{40} + 27 q^{41} - 52 q^{42} + 32 q^{44} + 75 q^{45} + 59 q^{46} + 45 q^{47} - 66 q^{48} + 20 q^{49} + 75 q^{51} + 11 q^{52} + 3 q^{53} + 57 q^{54} - 2 q^{55} + 87 q^{56} - 24 q^{57} - 46 q^{58} + 66 q^{59} + 29 q^{60} + 30 q^{61} + 72 q^{62} - 21 q^{63} - 7 q^{64} - 6 q^{65} + 41 q^{66} - 6 q^{67} + 28 q^{68} - 35 q^{69} - 80 q^{70} - 31 q^{71} - 15 q^{72} - 26 q^{73} + 87 q^{74} + 73 q^{75} - 68 q^{76} - 21 q^{77} - 50 q^{78} + 39 q^{79} - 60 q^{80} + 16 q^{81} + 45 q^{82} + 13 q^{83} + 31 q^{84} - 22 q^{85} + 61 q^{87} + 50 q^{88} - 4 q^{89} - 13 q^{90} - 25 q^{91} + 5 q^{92} + 67 q^{93} + 42 q^{94} + 79 q^{95} + 36 q^{96} - 2 q^{97} - 35 q^{98} + 13 q^{99}+O(q^{100}) \) 20 * q + 5 * q^2 - q^3 + 23 * q^4 - 3 * q^5 + 5 * q^6 - 2 * q^7 + 9 * q^8 + 27 * q^9 + 21 * q^11 - 12 * q^12 + 11 * q^13 + 8 * q^14 + 4 * q^15 + 29 * q^16 + 15 * q^17 + 9 * q^18 - 7 * q^19 - 17 * q^20 + 8 * q^21 + 47 * q^22 + 26 * q^23 - q^24 + 11 * q^25 + 58 * q^26 - 22 * q^27 - 28 * q^28 - 12 * q^29 - 24 * q^30 + 26 * q^31 + 3 * q^32 - 17 * q^33 + 54 * q^34 + 26 * q^35 + 14 * q^36 - 19 * q^37 + 5 * q^38 - 7 * q^39 - 16 * q^40 + 27 * q^41 - 52 * q^42 + 32 * q^44 + 75 * q^45 + 59 * q^46 + 45 * q^47 - 66 * q^48 + 20 * q^49 + 75 * q^51 + 11 * q^52 + 3 * q^53 + 57 * q^54 - 2 * q^55 + 87 * q^56 - 24 * q^57 - 46 * q^58 + 66 * q^59 + 29 * q^60 + 30 * q^61 + 72 * q^62 - 21 * q^63 - 7 * q^64 - 6 * q^65 + 41 * q^66 - 6 * q^67 + 28 * q^68 - 35 * q^69 - 80 * q^70 - 31 * q^71 - 15 * q^72 - 26 * q^73 + 87 * q^74 + 73 * q^75 - 68 * q^76 - 21 * q^77 - 50 * q^78 + 39 * q^79 - 60 * q^80 + 16 * q^81 + 45 * q^82 + 13 * q^83 + 31 * q^84 - 22 * q^85 + 61 * q^87 + 50 * q^88 - 4 * q^89 - 13 * q^90 - 25 * q^91 + 5 * q^92 + 67 * q^93 + 42 * q^94 + 79 * q^95 + 36 * q^96 - 2 * q^97 - 35 * q^98 + 13 * q^99

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