LMFDB - Embedded newform 1849.2.a.r.1.11

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			1.11
Root			$0.415785$ 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)

$f(q)$	$=$	$q+0.415785 q^{2} -2.94504 q^{3} -1.82712 q^{4} +2.53117 q^{5} -1.22450 q^{6} +3.71316 q^{7} -1.59126 q^{8} +5.67325 q^{9} +O(q^{10})$	\(q+0.415785 q^{2} -2.94504 q^{3} -1.82712 q^{4} +2.53117 q^{5} -1.22450 q^{6} +3.71316 q^{7} -1.59126 q^{8} +5.67325 q^{9} +1.05242 q^{10} -0.395832 q^{11} +5.38095 q^{12} +2.52890 q^{13} +1.54388 q^{14} -7.45440 q^{15} +2.99263 q^{16} -0.637607 q^{17} +2.35885 q^{18} +2.15999 q^{19} -4.62476 q^{20} -10.9354 q^{21} -0.164581 q^{22} +4.49305 q^{23} +4.68632 q^{24} +1.40683 q^{25} +1.05148 q^{26} -7.87282 q^{27} -6.78441 q^{28} -5.74734 q^{29} -3.09942 q^{30} +3.56167 q^{31} +4.42681 q^{32} +1.16574 q^{33} -0.265107 q^{34} +9.39865 q^{35} -10.3657 q^{36} -8.24564 q^{37} +0.898090 q^{38} -7.44771 q^{39} -4.02775 q^{40} -2.58851 q^{41} -4.54677 q^{42} +0.723234 q^{44} +14.3600 q^{45} +1.86814 q^{46} -6.39735 q^{47} -8.81339 q^{48} +6.78758 q^{49} +0.584937 q^{50} +1.87778 q^{51} -4.62062 q^{52} +12.8401 q^{53} -3.27340 q^{54} -1.00192 q^{55} -5.90860 q^{56} -6.36125 q^{57} -2.38966 q^{58} +4.94389 q^{59} +13.6201 q^{60} -7.43998 q^{61} +1.48089 q^{62} +21.0657 q^{63} -4.14465 q^{64} +6.40108 q^{65} +0.484697 q^{66} +1.67835 q^{67} +1.16499 q^{68} -13.2322 q^{69} +3.90781 q^{70} +3.23679 q^{71} -9.02761 q^{72} -3.56964 q^{73} -3.42841 q^{74} -4.14316 q^{75} -3.94657 q^{76} -1.46979 q^{77} -3.09664 q^{78} +15.3283 q^{79} +7.57485 q^{80} +6.16600 q^{81} -1.07626 q^{82} +13.0711 q^{83} +19.9803 q^{84} -1.61389 q^{85} +16.9261 q^{87} +0.629871 q^{88} +15.8675 q^{89} +5.97065 q^{90} +9.39022 q^{91} -8.20936 q^{92} -10.4892 q^{93} -2.65992 q^{94} +5.46730 q^{95} -13.0371 q^{96} -4.92219 q^{97} +2.82217 q^{98} -2.24565 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.415785	0.294004	0.147002	−	0.989136i	$-0.453038\pi$
$2$	0.415785	0.294004	0.147002	+	0.989136i	$0.453038\pi$
$3$	−2.94504	−1.70032	−0.850159	−	0.526526i	$-0.823494\pi$
$3$	−2.94504	−1.70032	−0.850159	+	0.526526i	$0.823494\pi$
$4$	−1.82712	−0.913562
$5$	2.53117	1.13197	0.565987	−	0.824414i	$-0.308495\pi$
$5$	2.53117	1.13197	0.565987	+	0.824414i	$0.308495\pi$
$6$	−1.22450	−0.499901
$7$	3.71316	1.40344	0.701722	−	0.712451i	$-0.252415\pi$
$7$	3.71316	1.40344	0.701722	+	0.712451i	$0.252415\pi$
$8$	−1.59126	−0.562595
$9$	5.67325	1.89108
$10$	1.05242	0.332805
$11$	−0.395832	−0.119348	−0.0596739	−	0.998218i	$-0.519006\pi$
$11$	−0.395832	−0.119348	−0.0596739	+	0.998218i	$0.519006\pi$
$12$	5.38095	1.55335
$13$	2.52890	0.701391	0.350696	−	0.936490i	$-0.385945\pi$
$13$	2.52890	0.701391	0.350696	+	0.936490i	$0.385945\pi$
$14$	1.54388	0.412618
$15$	−7.45440	−1.92472
$16$	2.99263	0.748156
$17$	−0.637607	−0.154642	−0.0773212	−	0.997006i	$-0.524637\pi$
$17$	−0.637607	−0.154642	−0.0773212	+	0.997006i	$0.524637\pi$
$18$	2.35885	0.555986
$19$	2.15999	0.495536	0.247768	−	0.968819i	$-0.420303\pi$
$19$	2.15999	0.495536	0.247768	+	0.968819i	$0.420303\pi$
$20$	−4.62476	−1.03413
$21$	−10.9354	−2.38630
$22$	−0.164581	−0.0350888
$23$	4.49305	0.936866	0.468433	−	0.883499i	$-0.344819\pi$
$23$	4.49305	0.936866	0.468433	+	0.883499i	$0.344819\pi$
$24$	4.68632	0.956591
$25$	1.40683	0.281366
$26$	1.05148	0.206212
$27$	−7.87282	−1.51512
$28$	−6.78441	−1.28213
$29$	−5.74734	−1.06725	−0.533627	−	0.845720i	$-0.679171\pi$
$29$	−5.74734	−1.06725	−0.533627	+	0.845720i	$0.679171\pi$
$30$	−3.09942	−0.565875
$31$	3.56167	0.639694	0.319847	−	0.947469i	$-0.396368\pi$
$31$	3.56167	0.639694	0.319847	+	0.947469i	$0.396368\pi$
$32$	4.42681	0.782556
$33$	1.16574	0.202929
$34$	−0.265107	−0.0454655
$35$	9.39865	1.58866
$36$	−10.3657	−1.72762
$37$	−8.24564	−1.35557	−0.677787	−	0.735258i	$-0.737061\pi$
$37$	−8.24564	−1.35557	−0.677787	+	0.735258i	$0.737061\pi$
$38$	0.898090	0.145689
$39$	−7.44771	−1.19259
$40$	−4.02775	−0.636843
$41$	−2.58851	−0.404257	−0.202128	−	0.979359i	$-0.564786\pi$
$41$	−2.58851	−0.404257	−0.202128	+	0.979359i	$0.564786\pi$
$42$	−4.54677	−0.701582
$43$	0	0
$43$	0	0
$44$	0.723234	0.109032
$45$	14.3600	2.14066
$46$	1.86814	0.275443
$47$	−6.39735	−0.933149	−0.466574	−	0.884482i	$-0.654512\pi$
$47$	−6.39735	−0.933149	−0.466574	+	0.884482i	$0.654512\pi$
$48$	−8.81339	−1.27210
$49$	6.78758	0.969654
$50$	0.584937	0.0827226
$51$	1.87778	0.262941
$52$	−4.62062	−0.640764
$53$	12.8401	1.76372	0.881861	−	0.471509i	$-0.156291\pi$
$53$	12.8401	1.76372	0.881861	+	0.471509i	$0.156291\pi$
$54$	−3.27340	−0.445453
$55$	−1.00192	−0.135099
$56$	−5.90860	−0.789570
$57$	−6.36125	−0.842568
$58$	−2.38966	−0.313777
$59$	4.94389	0.643640	0.321820	−	0.946801i	$-0.395705\pi$
$59$	4.94389	0.643640	0.321820	+	0.946801i	$0.395705\pi$
$60$	13.6201	1.75835
$61$	−7.43998	−0.952592	−0.476296	−	0.879285i	$-0.658021\pi$
$61$	−7.43998	−0.952592	−0.476296	+	0.879285i	$0.658021\pi$
$62$	1.48089	0.188073
$63$	21.0657	2.65403
$64$	−4.14465	−0.518082
$65$	6.40108	0.793957
$66$	0.484697	0.0596621
$67$	1.67835	0.205043	0.102521	−	0.994731i	$-0.467309\pi$
$67$	1.67835	0.205043	0.102521	+	0.994731i	$0.467309\pi$
$68$	1.16499	0.141275
$69$	−13.2322	−1.59297
$70$	3.90781	0.467073
$71$	3.23679	0.384136	0.192068	−	0.981382i	$-0.438481\pi$
$71$	3.23679	0.384136	0.192068	+	0.981382i	$0.438481\pi$
$72$	−9.02761	−1.06391
$73$	−3.56964	−0.417795	−0.208897	−	0.977938i	$-0.566987\pi$
$73$	−3.56964	−0.417795	−0.208897	+	0.977938i	$0.566987\pi$
$74$	−3.42841	−0.398545
$75$	−4.14316	−0.478411
$76$	−3.94657	−0.452702
$77$	−1.46979	−0.167498
$78$	−3.09664	−0.350626
$79$	15.3283	1.72457	0.862284	−	0.506425i	$-0.169033\pi$
$79$	15.3283	1.72457	0.862284	+	0.506425i	$0.169033\pi$
$80$	7.57485	0.846894
$81$	6.16600	0.685111
$82$	−1.07626	−0.118853
$83$	13.0711	1.43474	0.717370	−	0.696692i	$-0.245346\pi$
$83$	13.0711	1.43474	0.717370	+	0.696692i	$0.245346\pi$
$84$	19.9803	2.18003
$85$	−1.61389	−0.175051
$86$	0	0
$87$	16.9261	1.81467
$88$	0.629871	0.0671445
$89$	15.8675	1.68195	0.840977	−	0.541071i	$-0.181981\pi$
$89$	15.8675	1.68195	0.840977	+	0.541071i	$0.181981\pi$
$90$	5.97065	0.629362
$91$	9.39022	0.984363
$92$	−8.20936	−0.855885
$93$	−10.4892	−1.08768
$94$	−2.65992	−0.274350
$95$	5.46730	0.560933
$96$	−13.0371	−1.33059
$97$	−4.92219	−0.499773	−0.249886	−	0.968275i	$-0.580393\pi$
$97$	−4.92219	−0.499773	−0.249886	+	0.968275i	$0.580393\pi$
$98$	2.82217	0.285082
$99$	−2.24565	−0.225697
$100$	−2.57045	−0.257045
$101$	4.47515	0.445294	0.222647	−	0.974899i	$-0.428530\pi$
$101$	4.47515	0.445294	0.222647	+	0.974899i	$0.428530\pi$
$102$	0.780750	0.0773058
$103$	15.7955	1.55638	0.778189	−	0.628030i	$-0.216139\pi$
$103$	15.7955	1.55638	0.778189	+	0.628030i	$0.216139\pi$
$104$	−4.02414	−0.394599
$105$	−27.6794	−2.70123
$106$	5.33871	0.518542
$107$	18.4053	1.77931	0.889654	−	0.456636i	$-0.150946\pi$
$107$	18.4053	1.77931	0.889654	+	0.456636i	$0.150946\pi$
$108$	14.3846	1.38416
$109$	9.28179	0.889034	0.444517	−	0.895770i	$-0.353375\pi$
$109$	9.28179	0.889034	0.444517	+	0.895770i	$0.353375\pi$
$110$	−0.416582	−0.0397196
$111$	24.2837	2.30491
$112$	11.1121	1.05000
$113$	−6.24895	−0.587852	−0.293926	−	0.955828i	$-0.594962\pi$
$113$	−6.24895	−0.587852	−0.293926	+	0.955828i	$0.594962\pi$
$114$	−2.64491	−0.247719
$115$	11.3727	1.06051
$116$	10.5011	0.975002
$117$	14.3471	1.32639
$118$	2.05559	0.189233
$119$	−2.36754	−0.217032
$120$	11.8619	1.08284
$121$	−10.8433	−0.985756
$122$	−3.09343	−0.280066
$123$	7.62325	0.687365
$124$	−6.50761	−0.584400
$125$	−9.09493	−0.813476
$126$	8.75879	0.780295
$127$	−12.7044	−1.12734	−0.563669	−	0.826001i	$-0.690611\pi$
$127$	−12.7044	−1.12734	−0.563669	+	0.826001i	$0.690611\pi$
$128$	−10.5769	−0.934874
$129$	0	0
$130$	2.66147	0.233427
$131$	−12.0338	−1.05139	−0.525697	−	0.850672i	$-0.676196\pi$
$131$	−12.0338	−1.05139	−0.525697	+	0.850672i	$0.676196\pi$
$132$	−2.12995	−0.185389
$133$	8.02039	0.695456
$134$	0.697831	0.0602835
$135$	−19.9274	−1.71508
$136$	1.01460	0.0870010
$137$	22.9407	1.95996	0.979980	−	0.199096i	$-0.0638005\pi$
$137$	22.9407	1.95996	0.979980	+	0.199096i	$0.0638005\pi$
$138$	−5.50175	−0.468340
$139$	−15.6743	−1.32947	−0.664737	−	0.747077i	$-0.731457\pi$
$139$	−15.6743	−1.32947	−0.664737	+	0.747077i	$0.731457\pi$
$140$	−17.1725	−1.45134
$141$	18.8404	1.58665
$142$	1.34581	0.112938
$143$	−1.00102	−0.0837096
$144$	16.9779	1.41483
$145$	−14.5475	−1.20810
$146$	−1.48420	−0.122833
$147$	−19.9897	−1.64872
$148$	15.0658	1.23840
$149$	1.43160	0.117282	0.0586408	−	0.998279i	$-0.481323\pi$
$149$	1.43160	0.117282	0.0586408	+	0.998279i	$0.481323\pi$
$150$	−1.72266	−0.140655
$151$	13.7399	1.11814	0.559068	−	0.829122i	$-0.311159\pi$
$151$	13.7399	1.11814	0.559068	+	0.829122i	$0.311159\pi$
$152$	−3.43710	−0.278786
$153$	−3.61730	−0.292441
$154$	−0.611116	−0.0492451
$155$	9.01519	0.724118
$156$	13.6079	1.08950
$157$	4.28393	0.341895	0.170947	−	0.985280i	$-0.445317\pi$
$157$	4.28393	0.341895	0.170947	+	0.985280i	$0.445317\pi$
$158$	6.37327	0.507030
$159$	−37.8146	−2.99889
$160$	11.2050	0.885833
$161$	16.6834	1.31484
$162$	2.56373	0.201425
$163$	1.57789	0.123590	0.0617951	−	0.998089i	$-0.480317\pi$
$163$	1.57789	0.123590	0.0617951	+	0.998089i	$0.480317\pi$
$164$	4.72952	0.369314
$165$	2.95069	0.229711
$166$	5.43477	0.421820
$167$	−0.311746	−0.0241236	−0.0120618	−	0.999927i	$-0.503839\pi$
$167$	−0.311746	−0.0241236	−0.0120618	+	0.999927i	$0.503839\pi$
$168$	17.4011	1.34252
$169$	−6.60466	−0.508050
$170$	−0.671031	−0.0514657
$171$	12.2542	0.937099
$172$	0	0
$173$	−11.1224	−0.845624	−0.422812	−	0.906218i	$-0.638957\pi$
$173$	−11.1224	−0.845624	−0.422812	+	0.906218i	$0.638957\pi$
$174$	7.03763	0.533521
$175$	5.22378	0.394881
$176$	−1.18458	−0.0892909
$177$	−14.5600	−1.09439
$178$	6.59747	0.494501
$179$	−11.8212	−0.883561	−0.441780	−	0.897123i	$-0.645653\pi$
$179$	−11.8212	−0.883561	−0.441780	+	0.897123i	$0.645653\pi$
$180$	−26.2374	−1.95562
$181$	14.3122	1.06382	0.531910	−	0.846801i	$-0.321474\pi$
$181$	14.3122	1.06382	0.531910	+	0.846801i	$0.321474\pi$
$182$	3.90431	0.289407
$183$	21.9110	1.61971
$184$	−7.14961	−0.527076
$185$	−20.8711	−1.53448
$186$	−4.36127	−0.319784
$187$	0.252385	0.0184562
$188$	11.6887	0.852489
$189$	−29.2331	−2.12639
$190$	2.27322	0.164917
$191$	10.5236	0.761461	0.380730	−	0.924686i	$-0.375673\pi$
$191$	10.5236	0.761461	0.380730	+	0.924686i	$0.375673\pi$
$192$	12.2062	0.880904
$193$	−11.1172	−0.800235	−0.400118	−	0.916464i	$-0.631031\pi$
$193$	−11.1172	−0.800235	−0.400118	+	0.916464i	$0.631031\pi$
$194$	−2.04657	−0.146935
$195$	−18.8514	−1.34998
$196$	−12.4017	−0.885839
$197$	−8.33464	−0.593819	−0.296909	−	0.954906i	$-0.595956\pi$
$197$	−8.33464	−0.593819	−0.296909	+	0.954906i	$0.595956\pi$
$198$	−0.933708	−0.0663558
$199$	−6.84962	−0.485557	−0.242778	−	0.970082i	$-0.578059\pi$
$199$	−6.84962	−0.485557	−0.242778	+	0.970082i	$0.578059\pi$
$200$	−2.23863	−0.158295
$201$	−4.94280	−0.348638
$202$	1.86070	0.130918
$203$	−21.3408	−1.49783
$204$	−3.43093	−0.240213
$205$	−6.55195	−0.457608
$206$	6.56753	0.457582
$207$	25.4902	1.77169
$208$	7.56806	0.524750
$209$	−0.854993	−0.0591411
$210$	−11.5087	−0.794173
$211$	1.01933	0.0701739	0.0350870	−	0.999384i	$-0.488829\pi$
$211$	1.01933	0.0701739	0.0350870	+	0.999384i	$0.488829\pi$
$212$	−23.4604	−1.61127
$213$	−9.53246	−0.653153
$214$	7.65264	0.523124
$215$	0	0
$216$	12.5277	0.852401
$217$	13.2251	0.897775
$218$	3.85922	0.261380
$219$	10.5127	0.710384
$220$	1.83063	0.123421
$221$	−1.61244	−0.108465
$222$	10.0968	0.677653
$223$	14.5698	0.975669	0.487835	−	0.872936i	$-0.337787\pi$
$223$	14.5698	0.975669	0.487835	+	0.872936i	$0.337787\pi$
$224$	16.4374	1.09827
$225$	7.98128	0.532086
$226$	−2.59822	−0.172831
$227$	−14.8346	−0.984607	−0.492303	−	0.870424i	$-0.663845\pi$
$227$	−14.8346	−0.984607	−0.492303	+	0.870424i	$0.663845\pi$
$228$	11.6228	0.769738
$229$	−4.94448	−0.326740	−0.163370	−	0.986565i	$-0.552236\pi$
$229$	−4.94448	−0.326740	−0.163370	+	0.986565i	$0.552236\pi$
$230$	4.72859	0.311794
$231$	4.32858	0.284800
$232$	9.14551	0.600432
$233$	19.3398	1.26699	0.633497	−	0.773745i	$-0.281619\pi$
$233$	19.3398	1.26699	0.633497	+	0.773745i	$0.281619\pi$
$234$	5.96530	0.389964
$235$	−16.1928	−1.05630
$236$	−9.03310	−0.588005
$237$	−45.1424	−2.93231
$238$	−0.984385	−0.0638082
$239$	14.5406	0.940556	0.470278	−	0.882518i	$-0.344154\pi$
$239$	14.5406	0.940556	0.470278	+	0.882518i	$0.344154\pi$
$240$	−22.3082	−1.43999
$241$	2.63236	0.169565	0.0847827	−	0.996399i	$-0.472980\pi$
$241$	2.63236	0.169565	0.0847827	+	0.996399i	$0.472980\pi$
$242$	−4.50848	−0.289816
$243$	5.45935	0.350217
$244$	13.5938	0.870252
$245$	17.1805	1.09762
$246$	3.16963	0.202088
$247$	5.46240	0.347564
$248$	−5.66754	−0.359889
$249$	−38.4949	−2.43952
$250$	−3.78153	−0.239165
$251$	−6.10858	−0.385570	−0.192785	−	0.981241i	$-0.561752\pi$
$251$	−6.10858	−0.385570	−0.192785	+	0.981241i	$0.561752\pi$
$252$	−38.4896	−2.42462
$253$	−1.77850	−0.111813
$254$	−5.28231	−0.331442
$255$	4.75297	0.297643
$256$	3.89160	0.243225
$257$	−6.63380	−0.413805	−0.206903	−	0.978362i	$-0.566338\pi$
$257$	−6.63380	−0.413805	−0.206903	+	0.978362i	$0.566338\pi$
$258$	0	0
$259$	−30.6174	−1.90247
$260$	−11.6956	−0.725328
$261$	−32.6061	−2.01827
$262$	−5.00345	−0.309114
$263$	8.22323	0.507066	0.253533	−	0.967327i	$-0.418407\pi$
$263$	8.22323	0.507066	0.253533	+	0.967327i	$0.418407\pi$
$264$	−1.85500	−0.114167
$265$	32.5005	1.99649
$266$	3.33476	0.204467
$267$	−46.7304	−2.85986
$268$	−3.06655	−0.187319
$269$	−8.46677	−0.516228	−0.258114	−	0.966114i	$-0.583101\pi$
$269$	−8.46677	−0.516228	−0.258114	+	0.966114i	$0.583101\pi$
$270$	−8.28553	−0.504241
$271$	26.0765	1.58403	0.792016	−	0.610500i	$-0.209031\pi$
$271$	26.0765	1.58403	0.792016	+	0.610500i	$0.209031\pi$
$272$	−1.90812	−0.115697
$273$	−27.6546	−1.67373
$274$	9.53841	0.576236
$275$	−0.556868	−0.0335804
$276$	24.1769	1.45528
$277$	−7.83631	−0.470838	−0.235419	−	0.971894i	$-0.575646\pi$
$277$	−7.83631	−0.470838	−0.235419	+	0.971894i	$0.575646\pi$
$278$	−6.51712	−0.390871
$279$	20.2062	1.20972
$280$	−14.9557	−0.893773
$281$	25.1919	1.50282	0.751412	−	0.659833i	$-0.229373\pi$
$281$	25.1919	1.50282	0.751412	+	0.659833i	$0.229373\pi$
$282$	7.83356	0.466482
$283$	−1.02789	−0.0611016	−0.0305508	−	0.999533i	$-0.509726\pi$
$283$	−1.02789	−0.0611016	−0.0305508	+	0.999533i	$0.509726\pi$
$284$	−5.91401	−0.350932
$285$	−16.1014	−0.953765
$286$	−0.416209	−0.0246110
$287$	−9.61155	−0.567352
$288$	25.1144	1.47988
$289$	−16.5935	−0.976086
$290$	−6.04863	−0.355188
$291$	14.4960	0.849773
$292$	6.52217	0.381681
$293$	−1.26059	−0.0736443	−0.0368222	−	0.999322i	$-0.511724\pi$
$293$	−1.26059	−0.0736443	−0.0368222	+	0.999322i	$0.511724\pi$
$294$	−8.31140	−0.484731
$295$	12.5138	0.728584
$296$	13.1209	0.762640
$297$	3.11631	0.180827
$298$	0.595239	0.0344813
$299$	11.3625	0.657110
$300$	7.57007	0.437058
$301$	0	0
$302$	5.71284	0.328737
$303$	−13.1795	−0.757142
$304$	6.46404	0.370738
$305$	−18.8319	−1.07831
$306$	−1.50402	−0.0859790
$307$	31.3312	1.78817	0.894084	−	0.447900i	$-0.147828\pi$
$307$	31.3312	1.78817	0.894084	+	0.447900i	$0.147828\pi$
$308$	2.68549	0.153020
$309$	−46.5184	−2.64634
$310$	3.74838	0.212894
$311$	17.3132	0.981739	0.490870	−	0.871233i	$-0.336679\pi$
$311$	17.3132	0.981739	0.490870	+	0.871233i	$0.336679\pi$
$312$	11.8512	0.670944
$313$	16.7164	0.944867	0.472433	−	0.881366i	$-0.343376\pi$
$313$	16.7164	0.944867	0.472433	+	0.881366i	$0.343376\pi$
$314$	1.78119	0.100519
$315$	53.3209	3.00429
$316$	−28.0067	−1.57550
$317$	−21.5162	−1.20847	−0.604235	−	0.796806i	$-0.706521\pi$
$317$	−21.5162	−1.20847	−0.604235	+	0.796806i	$0.706521\pi$
$318$	−15.7227	−0.881686
$319$	2.27498	0.127375
$320$	−10.4908	−0.586455
$321$	−54.2043	−3.02539
$322$	6.93672	0.386568
$323$	−1.37722	−0.0766308
$324$	−11.2660	−0.625891
$325$	3.55773	0.197347
$326$	0.656064	0.0363360
$327$	−27.3352	−1.51164
$328$	4.11898	0.227433
$329$	−23.7544	−1.30962
$330$	1.22685	0.0675359
$331$	1.12793	0.0619966	0.0309983	−	0.999519i	$-0.490131\pi$
$331$	1.12793	0.0619966	0.0309983	+	0.999519i	$0.490131\pi$
$332$	−23.8825	−1.31072
$333$	−46.7796	−2.56350
$334$	−0.129619	−0.00709245
$335$	4.24819	0.232103
$336$	−32.7256	−1.78533
$337$	2.90779	0.158398	0.0791988	−	0.996859i	$-0.474764\pi$
$337$	2.90779	0.158398	0.0791988	+	0.996859i	$0.474764\pi$
$338$	−2.74611	−0.149369
$339$	18.4034	0.999535
$340$	2.94878	0.159920
$341$	−1.40982	−0.0763462
$342$	5.09509	0.275511
$343$	−0.788757	−0.0425889
$344$	0	0
$345$	−33.4930	−1.80320
$346$	−4.62454	−0.248617
$347$	−26.8951	−1.44381	−0.721903	−	0.691994i	$-0.756732\pi$
$347$	−26.8951	−1.44381	−0.721903	+	0.691994i	$0.756732\pi$
$348$	−30.9261	−1.65781
$349$	0.912753	0.0488585	0.0244293	−	0.999702i	$-0.492223\pi$
$349$	0.912753	0.0488585	0.0244293	+	0.999702i	$0.492223\pi$
$350$	2.17197	0.116097
$351$	−19.9096	−1.06269
$352$	−1.75227	−0.0933964
$353$	14.2133	0.756500	0.378250	−	0.925704i	$-0.376526\pi$
$353$	14.2133	0.756500	0.378250	+	0.925704i	$0.376526\pi$
$354$	−6.05380	−0.321756
$355$	8.19286	0.434832
$356$	−28.9919	−1.53657
$357$	6.97249	0.369023
$358$	−4.91509	−0.259770
$359$	14.9251	0.787717	0.393859	−	0.919171i	$-0.371140\pi$
$359$	14.9251	0.787717	0.393859	+	0.919171i	$0.371140\pi$
$360$	−22.8504	−1.20432
$361$	−14.3344	−0.754445
$362$	5.95081	0.312768
$363$	31.9340	1.67610
$364$	−17.1571	−0.899276
$365$	−9.03537	−0.472933
$366$	9.11027	0.476201
$367$	−1.19586	−0.0624235	−0.0312117	−	0.999513i	$-0.509937\pi$
$367$	−1.19586	−0.0624235	−0.0312117	+	0.999513i	$0.509937\pi$
$368$	13.4460	0.700923
$369$	−14.6852	−0.764483
$370$	−8.67789	−0.451142
$371$	47.6773	2.47528
$372$	19.1651	0.993666
$373$	−24.1630	−1.25111	−0.625556	−	0.780179i	$-0.715128\pi$
$373$	−24.1630	−1.25111	−0.625556	+	0.780179i	$0.715128\pi$
$374$	0.104938	0.00542621
$375$	26.7849	1.38317
$376$	10.1798	0.524985
$377$	−14.5345	−0.748563
$378$	−12.1547	−0.625168
$379$	−10.4902	−0.538845	−0.269423	−	0.963022i	$-0.586833\pi$
$379$	−10.4902	−0.538845	−0.269423	+	0.963022i	$0.586833\pi$
$380$	−9.98944	−0.512447
$381$	37.4151	1.91683
$382$	4.37555	0.223873
$383$	−6.72135	−0.343445	−0.171723	−	0.985145i	$-0.554933\pi$
$383$	−6.72135	−0.343445	−0.171723	+	0.985145i	$0.554933\pi$
$384$	31.1494	1.58958
$385$	−3.72029	−0.189603
$386$	−4.62237	−0.235272
$387$	0	0
$388$	8.99345	0.456573
$389$	−8.81075	−0.446723	−0.223361	−	0.974736i	$-0.571703\pi$
$389$	−8.81075	−0.446723	−0.223361	+	0.974736i	$0.571703\pi$
$390$	−7.83814	−0.396899
$391$	−2.86480	−0.144879
$392$	−10.8008	−0.545523
$393$	35.4399	1.78771
$394$	−3.46542	−0.174585
$395$	38.7985	1.95217
$396$	4.10309	0.206188
$397$	20.6191	1.03484	0.517421	−	0.855731i	$-0.326892\pi$
$397$	20.6191	1.03484	0.517421	+	0.855731i	$0.326892\pi$
$398$	−2.84797	−0.142756
$399$	−23.6204	−1.18250
$400$	4.21011	0.210505
$401$	34.4329	1.71950	0.859748	−	0.510719i	$-0.170621\pi$
$401$	34.4329	1.71950	0.859748	+	0.510719i	$0.170621\pi$
$402$	−2.05514	−0.102501
$403$	9.00711	0.448676
$404$	−8.17665	−0.406804
$405$	15.6072	0.775528
$406$	−8.87318	−0.440368
$407$	3.26389	0.161785
$408$	−2.98803	−0.147929
$409$	−13.2312	−0.654241	−0.327120	−	0.944983i	$-0.606078\pi$
$409$	−13.2312	−0.654241	−0.327120	+	0.944983i	$0.606078\pi$
$410$	−2.72420	−0.134539
$411$	−67.5614	−3.33256
$412$	−28.8603	−1.42185
$413$	18.3575	0.903313
$414$	10.5984	0.520885
$415$	33.0852	1.62409
$416$	11.1950	0.548878
$417$	46.1613	2.26053
$418$	−0.355493	−0.0173877
$419$	−2.06677	−0.100968	−0.0504841	−	0.998725i	$-0.516076\pi$
$419$	−2.06677	−0.100968	−0.0504841	+	0.998725i	$0.516076\pi$
$420$	50.5736	2.46774
$421$	22.5468	1.09886	0.549432	−	0.835539i	$-0.314844\pi$
$421$	22.5468	1.09886	0.549432	+	0.835539i	$0.314844\pi$
$422$	0.423824	0.0206314
$423$	−36.2937	−1.76466
$424$	−20.4319	−0.992261
$425$	−0.897003	−0.0435110
$426$	−3.96345	−0.192030
$427$	−27.6259	−1.33691
$428$	−33.6287	−1.62551
$429$	2.94804	0.142333
$430$	0	0
$431$	−1.26613	−0.0609873	−0.0304937	−	0.999535i	$-0.509708\pi$
$431$	−1.26613	−0.0609873	−0.0304937	+	0.999535i	$0.509708\pi$
$432$	−23.5604	−1.13355
$433$	13.8466	0.665426	0.332713	−	0.943028i	$-0.392036\pi$
$433$	13.8466	0.665426	0.332713	+	0.943028i	$0.392036\pi$
$434$	5.49877	0.263950
$435$	42.8429	2.05416
$436$	−16.9590	−0.812187
$437$	9.70495	0.464251
$438$	4.37103	0.208856
$439$	−24.7184	−1.17974	−0.589872	−	0.807497i	$-0.700822\pi$
$439$	−24.7184	−1.17974	−0.589872	+	0.807497i	$0.700822\pi$
$440$	1.59431	0.0760059
$441$	38.5076	1.83370
$442$	−0.670430	−0.0318891
$443$	−18.3580	−0.872214	−0.436107	−	0.899895i	$-0.643643\pi$
$443$	−18.3580	−0.872214	−0.436107	+	0.899895i	$0.643643\pi$
$444$	−44.3694	−2.10568
$445$	40.1634	1.90393
$446$	6.05792	0.286851
$447$	−4.21613	−0.199416
$448$	−15.3898	−0.727098
$449$	−4.03512	−0.190429	−0.0952146	−	0.995457i	$-0.530354\pi$
$449$	−4.03512	−0.190429	−0.0952146	+	0.995457i	$0.530354\pi$
$450$	3.31849	0.156435
$451$	1.02461	0.0482472
$452$	11.4176	0.537039
$453$	−40.4645	−1.90119
$454$	−6.16800	−0.289478
$455$	23.7683	1.11427
$456$	10.1224	0.474025
$457$	0.550267	0.0257404	0.0128702	−	0.999917i	$-0.495903\pi$
$457$	0.550267	0.0257404	0.0128702	+	0.999917i	$0.495903\pi$
$458$	−2.05584	−0.0960630
$459$	5.01976	0.234302
$460$	−20.7793	−0.968840
$461$	24.5744	1.14454	0.572271	−	0.820065i	$-0.306063\pi$
$461$	24.5744	1.14454	0.572271	+	0.820065i	$0.306063\pi$
$462$	1.79976	0.0837324
$463$	16.9551	0.787970	0.393985	−	0.919117i	$-0.371096\pi$
$463$	16.9551	0.787970	0.393985	+	0.919117i	$0.371096\pi$
$464$	−17.1996	−0.798473
$465$	−26.5501	−1.23123
$466$	8.04120	0.372501
$467$	−25.5118	−1.18054	−0.590272	−	0.807204i	$-0.700980\pi$
$467$	−25.5118	−1.18054	−0.590272	+	0.807204i	$0.700980\pi$
$468$	−26.2139	−1.21174
$469$	6.23198	0.287766
$470$	−6.73271	−0.310557
$471$	−12.6163	−0.581330
$472$	−7.86701	−0.362109
$473$	0	0
$474$	−18.7695	−0.862112
$475$	3.03873	0.139427
$476$	4.32578	0.198272
$477$	72.8450	3.33534
$478$	6.04578	0.276527
$479$	7.32944	0.334891	0.167445	−	0.985881i	$-0.446448\pi$
$479$	7.32944	0.334891	0.167445	+	0.985881i	$0.446448\pi$
$480$	−32.9992	−1.50620
$481$	−20.8524	−0.950788
$482$	1.09450	0.0498529
$483$	−49.1334	−2.23565
$484$	19.8121	0.900549
$485$	−12.4589	−0.565730
$486$	2.26991	0.102965
$487$	−26.3391	−1.19354	−0.596769	−	0.802413i	$-0.703549\pi$
$487$	−26.3391	−1.19354	−0.596769	+	0.802413i	$0.703549\pi$
$488$	11.8389	0.535924
$489$	−4.64695	−0.210143
$490$	7.14340	0.322706
$491$	22.6596	1.02261	0.511307	−	0.859398i	$-0.329161\pi$
$491$	22.6596	1.02261	0.511307	+	0.859398i	$0.329161\pi$
$492$	−13.9286	−0.627951
$493$	3.66454	0.165043
$494$	2.27118	0.102185
$495$	−5.68413	−0.255483
$496$	10.6587	0.478591
$497$	12.0187	0.539113
$498$	−16.0056	−0.717228
$499$	8.01993	0.359021	0.179511	−	0.983756i	$-0.442549\pi$
$499$	8.01993	0.359021	0.179511	+	0.983756i	$0.442549\pi$
$500$	16.6176	0.743160
$501$	0.918104	0.0410179
$502$	−2.53985	−0.113359
$503$	5.77174	0.257349	0.128675	−	0.991687i	$-0.458928\pi$
$503$	5.77174	0.257349	0.128675	+	0.991687i	$0.458928\pi$
$504$	−33.5210	−1.49314
$505$	11.3274	0.504061
$506$	−0.739471	−0.0328735
$507$	19.4510	0.863847
$508$	23.2126	1.02989
$509$	38.4709	1.70519	0.852597	−	0.522569i	$-0.175026\pi$
$509$	38.4709	1.70519	0.852597	+	0.522569i	$0.175026\pi$
$510$	1.97621	0.0875082
$511$	−13.2546	−0.586351
$512$	22.7719	1.00638
$513$	−17.0052	−0.750798
$514$	−2.75823	−0.121660
$515$	39.9811	1.76178
$516$	0	0
$517$	2.53227	0.111369
$518$	−12.7302	−0.559335
$519$	32.7560	1.43783
$520$	−10.1858	−0.446676
$521$	1.31273	0.0575118	0.0287559	−	0.999586i	$-0.490845\pi$
$521$	1.31273	0.0575118	0.0287559	+	0.999586i	$0.490845\pi$
$522$	−13.5571	−0.593378
$523$	−22.3085	−0.975481	−0.487740	−	0.872989i	$-0.662179\pi$
$523$	−22.3085	−0.975481	−0.487740	+	0.872989i	$0.662179\pi$
$524$	21.9872	0.960514
$525$	−15.3842	−0.671423
$526$	3.41909	0.149080
$527$	−2.27094	−0.0989238
$528$	3.48862	0.151823
$529$	−2.81247	−0.122281
$530$	13.5132	0.586976
$531$	28.0479	1.21718
$532$	−14.6542	−0.635342
$533$	−6.54608	−0.283542
$534$	−19.4298	−0.840810
$535$	46.5870	2.01413
$536$	−2.67069	−0.115356
$537$	34.8140	1.50233
$538$	−3.52035	−0.151773
$539$	−2.68674	−0.115726
$540$	36.4099	1.56683
$541$	−2.37609	−0.102156	−0.0510781	−	0.998695i	$-0.516266\pi$
$541$	−2.37609	−0.102156	−0.0510781	+	0.998695i	$0.516266\pi$
$542$	10.8422	0.465712
$543$	−42.1501	−1.80883
$544$	−2.82256	−0.121016
$545$	23.4938	1.00636
$546$	−11.4983	−0.492084
$547$	−36.2821	−1.55131	−0.775656	−	0.631156i	$-0.782581\pi$
$547$	−36.2821	−1.55131	−0.775656	+	0.631156i	$0.782581\pi$
$548$	−41.9156	−1.79054
$549$	−42.2089	−1.80143
$550$	−0.231537	−0.00987277
$551$	−12.4142	−0.528862
$552$	21.0559	0.896198
$553$	56.9164	2.42033
$554$	−3.25822	−0.138428
$555$	61.4663	2.60910
$556$	28.6388	1.21456
$557$	18.7262	0.793454	0.396727	−	0.917937i	$-0.370146\pi$
$557$	18.7262	0.793454	0.396727	+	0.917937i	$0.370146\pi$
$558$	8.40144	0.355661
$559$	0	0
$560$	28.1266	1.18857
$561$	−0.743284	−0.0313815
$562$	10.4744	0.441836
$563$	29.3267	1.23597	0.617986	−	0.786189i	$-0.287949\pi$
$563$	29.3267	1.23597	0.617986	+	0.786189i	$0.287949\pi$
$564$	−34.4238	−1.44950
$565$	−15.8172	−0.665433
$566$	−0.427380	−0.0179641
$567$	22.8954	0.961515
$568$	−5.15057	−0.216113
$569$	16.9257	0.709563	0.354781	−	0.934949i	$-0.384555\pi$
$569$	16.9257	0.709563	0.354781	+	0.934949i	$0.384555\pi$
$570$	−6.69472	−0.280411
$571$	45.7613	1.91505	0.957525	−	0.288351i	$-0.0931071\pi$
$571$	45.7613	1.91505	0.957525	+	0.288351i	$0.0931071\pi$
$572$	1.82899	0.0764738
$573$	−30.9924	−1.29473
$574$	−3.99633	−0.166804
$575$	6.32095	0.263602
$576$	−23.5136	−0.979735
$577$	−45.1965	−1.88155	−0.940777	−	0.339025i	$-0.889903\pi$
$577$	−45.1965	−1.88155	−0.940777	+	0.339025i	$0.889903\pi$
$578$	−6.89930	−0.286973
$579$	32.7406	1.36065
$580$	26.5801	1.10368
$581$	48.5352	2.01358
$582$	6.02723	0.249837
$583$	−5.08252	−0.210497
$584$	5.68022	0.235049
$585$	36.3149	1.50144
$586$	−0.524133	−0.0216517
$587$	−34.7261	−1.43330	−0.716650	−	0.697433i	$-0.754325\pi$
$587$	−34.7261	−1.43330	−0.716650	+	0.697433i	$0.754325\pi$
$588$	36.5236	1.50621
$589$	7.69316	0.316991
$590$	5.20306	0.214207
$591$	24.5458	1.00968
$592$	−24.6761	−1.01418
$593$	−23.4746	−0.963985	−0.481993	−	0.876175i	$-0.660087\pi$
$593$	−23.4746	−0.963985	−0.481993	+	0.876175i	$0.660087\pi$
$594$	1.29572	0.0531639
$595$	−5.99264	−0.245674
$596$	−2.61572	−0.107144
$597$	20.1724	0.825601
$598$	4.72435	0.193193
$599$	−27.8123	−1.13638	−0.568191	−	0.822897i	$-0.692356\pi$
$599$	−27.8123	−1.13638	−0.568191	+	0.822897i	$0.692356\pi$
$600$	6.59284	0.269152
$601$	8.95134	0.365133	0.182566	−	0.983194i	$-0.441560\pi$
$601$	8.95134	0.365133	0.182566	+	0.983194i	$0.441560\pi$
$602$	0	0
$603$	9.52169	0.387753
$604$	−25.1045	−1.02149
$605$	−27.4463	−1.11585
$606$	−5.47983	−0.222603
$607$	−20.1316	−0.817117	−0.408558	−	0.912732i	$-0.633968\pi$
$607$	−20.1316	−0.817117	−0.408558	+	0.912732i	$0.633968\pi$
$608$	9.56185	0.387784
$609$	62.8495	2.54679
$610$	−7.83000	−0.317028
$611$	−16.1783	−0.654502
$612$	6.60925	0.267163
$613$	−11.5891	−0.468079	−0.234040	−	0.972227i	$-0.575195\pi$
$613$	−11.5891	−0.468079	−0.234040	+	0.972227i	$0.575195\pi$
$614$	13.0270	0.525729
$615$	19.2958	0.778080
$616$	2.33882	0.0942336
$617$	3.11550	0.125425	0.0627126	−	0.998032i	$-0.480025\pi$
$617$	3.11550	0.125425	0.0627126	+	0.998032i	$0.480025\pi$
$618$	−19.3416	−0.778034
$619$	2.46418	0.0990437	0.0495219	−	0.998773i	$-0.484230\pi$
$619$	2.46418	0.0990437	0.0495219	+	0.998773i	$0.484230\pi$
$620$	−16.4719	−0.661526
$621$	−35.3730	−1.41947
$622$	7.19854	0.288635
$623$	58.9187	2.36053
$624$	−22.2882	−0.892243
$625$	−30.0550	−1.20220
$626$	6.95042	0.277795
$627$	2.51799	0.100559
$628$	−7.82727	−0.312342
$629$	5.25747	0.209629
$630$	22.1700	0.883274
$631$	23.8769	0.950523	0.475261	−	0.879845i	$-0.342354\pi$
$631$	23.8769	0.950523	0.475261	+	0.879845i	$0.342354\pi$
$632$	−24.3913	−0.970233
$633$	−3.00198	−0.119318
$634$	−8.94610	−0.355295
$635$	−32.1571	−1.27612
$636$	69.0918	2.73967
$637$	17.1651	0.680107
$638$	0.945902	0.0374486
$639$	18.3631	0.726433
$640$	−26.7719	−1.05825
$641$	−30.0675	−1.18760	−0.593798	−	0.804614i	$-0.702372\pi$
$641$	−30.0675	−1.18760	−0.593798	+	0.804614i	$0.702372\pi$
$642$	−22.5373	−0.889477
$643$	9.06936	0.357661	0.178830	−	0.983880i	$-0.442769\pi$
$643$	9.06936	0.357661	0.178830	+	0.983880i	$0.442769\pi$
$644$	−30.4827	−1.20119
$645$	0	0
$646$	−0.572628	−0.0225298
$647$	34.7745	1.36713	0.683564	−	0.729890i	$-0.260429\pi$
$647$	34.7745	1.36713	0.683564	+	0.729890i	$0.260429\pi$
$648$	−9.81170	−0.385440
$649$	−1.95695	−0.0768171
$650$	1.47925	0.0580209
$651$	−38.9483	−1.52650
$652$	−2.88300	−0.112907
$653$	−4.45165	−0.174207	−0.0871033	−	0.996199i	$-0.527761\pi$
$653$	−4.45165	−0.174207	−0.0871033	+	0.996199i	$0.527761\pi$
$654$	−11.3656	−0.444429
$655$	−30.4595	−1.19015
$656$	−7.74643	−0.302447
$657$	−20.2514	−0.790084
$658$	−9.87671	−0.385034
$659$	21.7501	0.847264	0.423632	−	0.905834i	$-0.360755\pi$
$659$	21.7501	0.847264	0.423632	+	0.905834i	$0.360755\pi$
$660$	−5.39127	−0.209855
$661$	2.32739	0.0905250	0.0452625	−	0.998975i	$-0.485588\pi$
$661$	2.32739	0.0905250	0.0452625	+	0.998975i	$0.485588\pi$
$662$	0.468975	0.0182272
$663$	4.74871	0.184425
$664$	−20.7995	−0.807178
$665$	20.3010	0.787238
$666$	−19.4502	−0.753681
$667$	−25.8231	−0.999875
$668$	0.569598	0.0220384
$669$	−42.9088	−1.65895
$670$	1.76633	0.0682393
$671$	2.94498	0.113690
$672$	−48.4089	−1.86741
$673$	−12.3959	−0.477826	−0.238913	−	0.971041i	$-0.576791\pi$
$673$	−12.3959	−0.477826	−0.238913	+	0.971041i	$0.576791\pi$
$674$	1.20902	0.0465696
$675$	−11.0757	−0.426304
$676$	12.0675	0.464135
$677$	−5.52343	−0.212283	−0.106141	−	0.994351i	$-0.533850\pi$
$677$	−5.52343	−0.212283	−0.106141	+	0.994351i	$0.533850\pi$
$678$	7.65185	0.293868
$679$	−18.2769	−0.701403
$680$	2.56812	0.0984829
$681$	43.6885	1.67415
$682$	−0.586183	−0.0224461
$683$	−9.73275	−0.372413	−0.186207	−	0.982511i	$-0.559619\pi$
$683$	−9.73275	−0.372413	−0.186207	+	0.982511i	$0.559619\pi$
$684$	−22.3898	−0.856097
$685$	58.0670	2.21862
$686$	−0.327953	−0.0125213
$687$	14.5617	0.555563
$688$	0	0
$689$	32.4713	1.23706
$690$	−13.9259	−0.530149
$691$	−36.5741	−1.39134	−0.695672	−	0.718360i	$-0.744893\pi$
$691$	−36.5741	−1.39134	−0.695672	+	0.718360i	$0.744893\pi$
$692$	20.3221	0.772529
$693$	−8.33848	−0.316753
$694$	−11.1826	−0.424485
$695$	−39.6743	−1.50493
$696$	−26.9339	−1.02093
$697$	1.65045	0.0625152
$698$	0.379509	0.0143646
$699$	−56.9565	−2.15429
$700$	−9.54449	−0.360748
$701$	−49.4519	−1.86777	−0.933886	−	0.357570i	$-0.883605\pi$
$701$	−49.4519	−1.86777	−0.933886	+	0.357570i	$0.883605\pi$
$702$	−8.27810	−0.312437
$703$	−17.8105	−0.671736
$704$	1.64059	0.0618319
$705$	47.6883	1.79605
$706$	5.90969	0.222414
$707$	16.6170	0.624945
$708$	26.6028	0.999795
$709$	−34.4953	−1.29550	−0.647749	−	0.761854i	$-0.724290\pi$
$709$	−34.4953	−1.29550	−0.647749	+	0.761854i	$0.724290\pi$
$710$	3.40647	0.127842
$711$	86.9612	3.26130
$712$	−25.2493	−0.946259
$713$	16.0028	0.599308
$714$	2.89905	0.108494
$715$	−2.53375	−0.0947571
$716$	21.5989	0.807187
$717$	−42.8228	−1.59924
$718$	6.20563	0.231592
$719$	6.32998	0.236068	0.118034	−	0.993010i	$-0.462341\pi$
$719$	6.32998	0.236068	0.118034	+	0.993010i	$0.462341\pi$
$720$	42.9740	1.60155
$721$	58.6513	2.18429
$722$	−5.96004	−0.221810
$723$	−7.75241	−0.288315
$724$	−26.1502	−0.971866
$725$	−8.08552	−0.300289
$726$	13.2777	0.492780
$727$	28.5688	1.05956	0.529779	−	0.848136i	$-0.322275\pi$
$727$	28.5688	1.05956	0.529779	+	0.848136i	$0.322275\pi$
$728$	−14.9423	−0.553798
$729$	−34.5760	−1.28059
$730$	−3.75677	−0.139044
$731$	0	0
$732$	−40.0341	−1.47970
$733$	3.19408	0.117976	0.0589880	−	0.998259i	$-0.481213\pi$
$733$	3.19408	0.117976	0.0589880	+	0.998259i	$0.481213\pi$
$734$	−0.497221	−0.0183528
$735$	−50.5973	−1.86631
$736$	19.8899	0.733150
$737$	−0.664344	−0.0244714
$738$	−6.10590	−0.224761
$739$	13.5181	0.497273	0.248637	−	0.968597i	$-0.420018\pi$
$739$	13.5181	0.497273	0.248637	+	0.968597i	$0.420018\pi$
$740$	38.1341	1.40184
$741$	−16.0870	−0.590970
$742$	19.8235	0.727744
$743$	−24.5060	−0.899036	−0.449518	−	0.893271i	$-0.648404\pi$
$743$	−24.5060	−0.899036	−0.449518	+	0.893271i	$0.648404\pi$
$744$	16.6911	0.611926
$745$	3.62364	0.132760
$746$	−10.0466	−0.367832
$747$	74.1557	2.71321
$748$	−0.461139	−0.0168609
$749$	68.3419	2.49716
$750$	11.1368	0.406657
$751$	−23.0717	−0.841898	−0.420949	−	0.907084i	$-0.638303\pi$
$751$	−23.0717	−0.841898	−0.420949	+	0.907084i	$0.638303\pi$
$752$	−19.1449	−0.698141
$753$	17.9900	0.655592
$754$	−6.04320	−0.220081
$755$	34.7780	1.26570
$756$	53.4124	1.94259
$757$	−32.1570	−1.16877	−0.584383	−	0.811478i	$-0.698663\pi$
$757$	−32.1570	−1.16877	−0.584383	+	0.811478i	$0.698663\pi$
$758$	−4.36166	−0.158423
$759$	5.23774	0.190118
$760$	−8.69989	−0.315578
$761$	6.94959	0.251923	0.125961	−	0.992035i	$-0.459798\pi$
$761$	6.94959	0.251923	0.125961	+	0.992035i	$0.459798\pi$
$762$	15.5566	0.563557
$763$	34.4648	1.24771
$764$	−19.2279	−0.695641
$765$	−9.15601	−0.331036
$766$	−2.79464	−0.100974
$767$	12.5026	0.451443
$768$	−11.4609	−0.413559
$769$	−30.9389	−1.11568	−0.557842	−	0.829947i	$-0.688371\pi$
$769$	−30.9389	−1.11568	−0.557842	+	0.829947i	$0.688371\pi$
$770$	−1.54684	−0.0557442
$771$	19.5368	0.703601
$772$	20.3125	0.731064
$773$	−18.3168	−0.658810	−0.329405	−	0.944189i	$-0.606848\pi$
$773$	−18.3168	−0.658810	−0.329405	+	0.944189i	$0.606848\pi$
$774$	0	0
$775$	5.01065	0.179988
$776$	7.83248	0.281170
$777$	90.1694	3.23481
$778$	−3.66337	−0.131338
$779$	−5.59115	−0.200324
$780$	34.4439	1.23329
$781$	−1.28122	−0.0458458
$782$	−1.19114	−0.0425951
$783$	45.2477	1.61702
$784$	20.3127	0.725453
$785$	10.8434	0.387016
$786$	14.7354	0.525593
$787$	−25.1524	−0.896588	−0.448294	−	0.893886i	$-0.647968\pi$
$787$	−25.1524	−0.896588	−0.448294	+	0.893886i	$0.647968\pi$
$788$	15.2284	0.542490
$789$	−24.2177	−0.862174
$790$	16.1318	0.573945
$791$	−23.2034	−0.825017
$792$	3.57342	0.126976
$793$	−18.8150	−0.668140
$794$	8.57310	0.304248
$795$	−95.7151	−3.39467
$796$	12.5151	0.443586
$797$	−52.3347	−1.85379	−0.926895	−	0.375320i	$-0.877533\pi$
$797$	−52.3347	−1.85379	−0.926895	+	0.375320i	$0.877533\pi$
$798$	−9.82098	−0.347659
$799$	4.07899	0.144304
$800$	6.22775	0.220184
$801$	90.0204	3.18071
$802$	14.3167	0.505539
$803$	1.41298	0.0498629
$804$	9.03110	0.318502
$805$	42.2286	1.48836
$806$	3.74502	0.131913
$807$	24.9350	0.877752
$808$	−7.12112	−0.250520
$809$	40.5257	1.42481	0.712404	−	0.701769i	$-0.247606\pi$
$809$	40.5257	1.42481	0.712404	+	0.701769i	$0.247606\pi$
$810$	6.48923	0.228008
$811$	−22.1499	−0.777788	−0.388894	−	0.921282i	$-0.627143\pi$
$811$	−22.1499	−0.777788	−0.388894	+	0.921282i	$0.627143\pi$
$812$	38.9923	1.36836
$813$	−76.7962	−2.69336
$814$	1.35708	0.0475655
$815$	3.99392	0.139901
$816$	5.61948	0.196721
$817$	0	0
$818$	−5.50133	−0.192349
$819$	53.2731	1.86151
$820$	11.9712	0.418053
$821$	−30.4772	−1.06366	−0.531830	−	0.846851i	$-0.678495\pi$
$821$	−30.4772	−1.06366	−0.531830	+	0.846851i	$0.678495\pi$
$822$	−28.0910	−0.979785
$823$	−15.8210	−0.551485	−0.275743	−	0.961231i	$-0.588924\pi$
$823$	−15.8210	−0.551485	−0.275743	+	0.961231i	$0.588924\pi$
$824$	−25.1348	−0.875611
$825$	1.64000	0.0570973
$826$	7.63276	0.265578
$827$	12.7590	0.443675	0.221837	−	0.975084i	$-0.428795\pi$
$827$	12.7590	0.443675	0.221837	+	0.975084i	$0.428795\pi$
$828$	−46.5737	−1.61855
$829$	19.8531	0.689525	0.344763	−	0.938690i	$-0.387959\pi$
$829$	19.8531	0.689525	0.344763	+	0.938690i	$0.387959\pi$
$830$	13.7563	0.477489
$831$	23.0782	0.800575
$832$	−10.4814	−0.363378
$833$	−4.32780	−0.149950
$834$	19.1932	0.664605
$835$	−0.789083	−0.0273073
$836$	1.56218	0.0540291
$837$	−28.0404	−0.969217
$838$	−0.859330	−0.0296851
$839$	−10.9168	−0.376888	−0.188444	−	0.982084i	$-0.560344\pi$
$839$	−10.9168	−0.376888	−0.188444	+	0.982084i	$0.560344\pi$
$840$	44.0451	1.51970
$841$	4.03191	0.139031
$842$	9.37461	0.323070
$843$	−74.1912	−2.55528
$844$	−1.86245	−0.0641082
$845$	−16.7175	−0.575100
$846$	−15.0904	−0.518818
$847$	−40.2630	−1.38345
$848$	38.4256	1.31954
$849$	3.02717	0.103892
$850$	−0.372960	−0.0127924
$851$	−37.0481	−1.26999
$852$	17.4170	0.596696
$853$	−55.9637	−1.91616	−0.958081	−	0.286499i	$-0.907509\pi$
$853$	−55.9637	−1.91616	−0.958081	+	0.286499i	$0.907509\pi$
$854$	−11.4864	−0.393057
$855$	31.0174	1.06077
$856$	−29.2876	−1.00103
$857$	10.2257	0.349305	0.174652	−	0.984630i	$-0.444120\pi$
$857$	10.2257	0.349305	0.174652	+	0.984630i	$0.444120\pi$
$858$	1.22575	0.0418465
$859$	−16.2390	−0.554067	−0.277033	−	0.960860i	$-0.589351\pi$
$859$	−16.2390	−0.554067	−0.277033	+	0.960860i	$0.589351\pi$
$860$	0	0
$861$	28.3064	0.964679
$862$	−0.526437	−0.0179305
$863$	−50.7616	−1.72795	−0.863973	−	0.503539i	$-0.832031\pi$
$863$	−50.7616	−1.72795	−0.863973	+	0.503539i	$0.832031\pi$
$864$	−34.8514	−1.18567
$865$	−28.1528	−0.957224
$866$	5.75721	0.195638
$867$	48.8684	1.65966
$868$	−24.1638	−0.820173
$869$	−6.06743	−0.205823
$870$	17.8134	0.603932
$871$	4.24438	0.143815
$872$	−14.7697	−0.500166
$873$	−27.9248	−0.945111
$874$	4.03517	0.136492
$875$	−33.7710	−1.14167
$876$	−19.2080	−0.648979
$877$	22.8346	0.771070	0.385535	−	0.922693i	$-0.374017\pi$
$877$	22.8346	0.771070	0.385535	+	0.922693i	$0.374017\pi$
$878$	−10.2775	−0.346850
$879$	3.71248	0.125219
$880$	−2.99837	−0.101075
$881$	−5.05280	−0.170233	−0.0851166	−	0.996371i	$-0.527126\pi$
$881$	−5.05280	−0.170233	−0.0851166	+	0.996371i	$0.527126\pi$
$882$	16.0109	0.539114
$883$	6.44819	0.216999	0.108499	−	0.994097i	$-0.465395\pi$
$883$	6.44819	0.216999	0.108499	+	0.994097i	$0.465395\pi$
$884$	2.94613	0.0990892
$885$	−36.8537	−1.23882
$886$	−7.63297	−0.256435
$887$	−21.1033	−0.708580	−0.354290	−	0.935136i	$-0.615277\pi$
$887$	−21.1033	−0.708580	−0.354290	+	0.935136i	$0.615277\pi$
$888$	−38.6417	−1.29673
$889$	−47.1737	−1.58215
$890$	16.6993	0.559763
$891$	−2.44070	−0.0817666
$892$	−26.6209	−0.891334
$893$	−13.8182	−0.462408
$894$	−1.75300	−0.0586292
$895$	−29.9216	−1.00017
$896$	−39.2737	−1.31204
$897$	−33.4630	−1.11730
$898$	−1.67774	−0.0559870
$899$	−20.4701	−0.682716
$900$	−14.5828	−0.486093
$901$	−8.18693	−0.272746
$902$	0.426019	0.0141849
$903$	0	0
$904$	9.94370	0.330723
$905$	36.2267	1.20422
$906$	−16.8245	−0.558957
$907$	−33.6351	−1.11683	−0.558417	−	0.829561i	$-0.688591\pi$
$907$	−33.6351	−1.11683	−0.558417	+	0.829561i	$0.688591\pi$
$908$	27.1046	0.899499
$909$	25.3886	0.842088
$910$	9.88248	0.327601
$911$	−18.7318	−0.620613	−0.310307	−	0.950636i	$-0.600432\pi$
$911$	−18.7318	−0.620613	−0.310307	+	0.950636i	$0.600432\pi$
$912$	−19.0368	−0.630373
$913$	−5.17397	−0.171233
$914$	0.228793	0.00756778
$915$	55.4606	1.83347
$916$	9.03417	0.298497
$917$	−44.6833	−1.47557
$918$	2.08714	0.0688858
$919$	−37.4391	−1.23500	−0.617502	−	0.786570i	$-0.711855\pi$
$919$	−37.4391	−1.23500	−0.617502	+	0.786570i	$0.711855\pi$
$920$	−18.0969	−0.596637
$921$	−92.2716	−3.04045
$922$	10.2176	0.336500
$923$	8.18552	0.269430
$924$	−7.90886	−0.260182
$925$	−11.6002	−0.381412
$926$	7.04967	0.231667
$927$	89.6119	2.94324
$928$	−25.4424	−0.835186
$929$	52.4298	1.72017	0.860083	−	0.510155i	$-0.170412\pi$
$929$	52.4298	1.72017	0.860083	+	0.510155i	$0.170412\pi$
$930$	−11.0391	−0.361987
$931$	14.6611	0.480498
$932$	−35.3362	−1.15748
$933$	−50.9879	−1.66927
$934$	−10.6074	−0.347085
$935$	0.638830	0.0208920
$936$	−22.8299	−0.746220
$937$	22.6095	0.738622	0.369311	−	0.929306i	$-0.379594\pi$
$937$	22.6095	0.738622	0.369311	+	0.929306i	$0.379594\pi$
$938$	2.59116	0.0846044
$939$	−49.2304	−1.60657
$940$	29.5862	0.964995
$941$	48.6505	1.58596	0.792980	−	0.609248i	$-0.208529\pi$
$941$	48.6505	1.58596	0.792980	+	0.609248i	$0.208529\pi$
$942$	−5.24568	−0.170913
$943$	−11.6303	−0.378735
$944$	14.7952	0.481543
$945$	−73.9939	−2.40702
$946$	0	0
$947$	2.09904	0.0682096	0.0341048	−	0.999418i	$-0.489142\pi$
$947$	2.09904	0.0682096	0.0341048	+	0.999418i	$0.489142\pi$
$948$	82.4807	2.67885
$949$	−9.02726	−0.293037
$950$	1.26346	0.0409920
$951$	63.3660	2.05478
$952$	3.76736	0.122101
$953$	−11.9068	−0.385699	−0.192849	−	0.981228i	$-0.561773\pi$
$953$	−11.9068	−0.385699	−0.192849	+	0.981228i	$0.561773\pi$
$954$	30.2878	0.980605
$955$	26.6370	0.861954
$956$	−26.5675	−0.859256
$957$	−6.69991	−0.216577
$958$	3.04747	0.0984592
$959$	85.1827	2.75069
$960$	30.8959	0.997160
$961$	−18.3145	−0.590791
$962$	−8.67011	−0.279536
$963$	104.418	3.36482
$964$	−4.80965	−0.154908
$965$	−28.1396	−0.905845
$966$	−20.4289	−0.657289
$967$	−9.46299	−0.304309	−0.152155	−	0.988357i	$-0.548621\pi$
$967$	−9.46299	−0.304309	−0.152155	+	0.988357i	$0.548621\pi$
$968$	17.2545	0.554581
$969$	4.05597	0.130297
$970$	−5.18022	−0.166327
$971$	21.2911	0.683264	0.341632	−	0.939834i	$-0.389020\pi$
$971$	21.2911	0.683264	0.341632	+	0.939834i	$0.389020\pi$
$972$	−9.97490	−0.319945
$973$	−58.2011	−1.86584
$974$	−10.9514	−0.350905
$975$	−10.4776	−0.335553
$976$	−22.2651	−0.712688
$977$	−39.6851	−1.26964	−0.634819	−	0.772661i	$-0.718925\pi$
$977$	−39.6851	−1.26964	−0.634819	+	0.772661i	$0.718925\pi$
$978$	−1.93213	−0.0617828
$979$	−6.28087	−0.200738
$980$	−31.3909	−1.00275
$981$	52.6579	1.68124
$982$	9.42152	0.300653
$983$	−37.3151	−1.19017	−0.595084	−	0.803663i	$-0.702881\pi$
$983$	−37.3151	−1.19017	−0.595084	+	0.803663i	$0.702881\pi$
$984$	−12.1306	−0.386708
$985$	−21.0964	−0.672187
$986$	1.52366	0.0485232
$987$	69.9576	2.22677
$988$	−9.98048	−0.317521
$989$	0	0
$990$	−2.36338	−0.0751130
$991$	−51.5273	−1.63682	−0.818410	−	0.574635i	$-0.805144\pi$
$991$	−51.5273	−1.63682	−0.818410	+	0.574635i	$0.805144\pi$
$992$	15.7668	0.500597
$993$	−3.32179	−0.105414
$994$	4.99720	0.158501
$995$	−17.3376	−0.549638
$996$	70.3350	2.22865
$997$	−50.8136	−1.60928	−0.804642	−	0.593761i	$-0.797643\pi$
$997$	−50.8136	−1.60928	−0.804642	+	0.593761i	$0.797643\pi$
$998$	3.33456	0.105554
$999$	64.9164	2.05386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.r.1.11	yes	20
43.42	odd	2		1849.2.a.p.1.10	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.10	✓	20	43.42	odd	2
1849.2.a.r.1.11	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)

\(f(q)\)	\(=\)	\(q+0.415785 q^{2} -2.94504 q^{3} -1.82712 q^{4} +2.53117 q^{5} -1.22450 q^{6} +3.71316 q^{7} -1.59126 q^{8} +5.67325 q^{9} +O(q^{10})\)	\(q+0.415785 q^{2} -2.94504 q^{3} -1.82712 q^{4} +2.53117 q^{5} -1.22450 q^{6} +3.71316 q^{7} -1.59126 q^{8} +5.67325 q^{9} +1.05242 q^{10} -0.395832 q^{11} +5.38095 q^{12} +2.52890 q^{13} +1.54388 q^{14} -7.45440 q^{15} +2.99263 q^{16} -0.637607 q^{17} +2.35885 q^{18} +2.15999 q^{19} -4.62476 q^{20} -10.9354 q^{21} -0.164581 q^{22} +4.49305 q^{23} +4.68632 q^{24} +1.40683 q^{25} +1.05148 q^{26} -7.87282 q^{27} -6.78441 q^{28} -5.74734 q^{29} -3.09942 q^{30} +3.56167 q^{31} +4.42681 q^{32} +1.16574 q^{33} -0.265107 q^{34} +9.39865 q^{35} -10.3657 q^{36} -8.24564 q^{37} +0.898090 q^{38} -7.44771 q^{39} -4.02775 q^{40} -2.58851 q^{41} -4.54677 q^{42} +0.723234 q^{44} +14.3600 q^{45} +1.86814 q^{46} -6.39735 q^{47} -8.81339 q^{48} +6.78758 q^{49} +0.584937 q^{50} +1.87778 q^{51} -4.62062 q^{52} +12.8401 q^{53} -3.27340 q^{54} -1.00192 q^{55} -5.90860 q^{56} -6.36125 q^{57} -2.38966 q^{58} +4.94389 q^{59} +13.6201 q^{60} -7.43998 q^{61} +1.48089 q^{62} +21.0657 q^{63} -4.14465 q^{64} +6.40108 q^{65} +0.484697 q^{66} +1.67835 q^{67} +1.16499 q^{68} -13.2322 q^{69} +3.90781 q^{70} +3.23679 q^{71} -9.02761 q^{72} -3.56964 q^{73} -3.42841 q^{74} -4.14316 q^{75} -3.94657 q^{76} -1.46979 q^{77} -3.09664 q^{78} +15.3283 q^{79} +7.57485 q^{80} +6.16600 q^{81} -1.07626 q^{82} +13.0711 q^{83} +19.9803 q^{84} -1.61389 q^{85} +16.9261 q^{87} +0.629871 q^{88} +15.8675 q^{89} +5.97065 q^{90} +9.39022 q^{91} -8.20936 q^{92} -10.4892 q^{93} -2.65992 q^{94} +5.46730 q^{95} -13.0371 q^{96} -4.92219 q^{97} +2.82217 q^{98} -2.24565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 5 q^{2} - q^{3} + 23 q^{4} - 3 q^{5} + 5 q^{6} - 2 q^{7} + 9 q^{8} + 27 q^{9}+O(q^{10}) \) 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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