LMFDB - Embedded newform 1849.2.a.n.1.8

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

Embedding invariants

Embedding label			1.8
Root			$1.04477$ 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)

$f(q)$	$=$	$q-1.04477 q^{2} +2.52412 q^{3} -0.908449 q^{4} -0.499809 q^{5} -2.63714 q^{6} +3.30668 q^{7} +3.03867 q^{8} +3.37119 q^{9} +O(q^{10})$	\(q-1.04477 q^{2} +2.52412 q^{3} -0.908449 q^{4} -0.499809 q^{5} -2.63714 q^{6} +3.30668 q^{7} +3.03867 q^{8} +3.37119 q^{9} +0.522187 q^{10} -3.72276 q^{11} -2.29304 q^{12} -4.68695 q^{13} -3.45473 q^{14} -1.26158 q^{15} -1.35782 q^{16} -4.59028 q^{17} -3.52213 q^{18} -6.05844 q^{19} +0.454051 q^{20} +8.34647 q^{21} +3.88944 q^{22} -0.388946 q^{23} +7.66997 q^{24} -4.75019 q^{25} +4.89680 q^{26} +0.936938 q^{27} -3.00395 q^{28} -3.53498 q^{29} +1.31806 q^{30} -0.870839 q^{31} -4.65872 q^{32} -9.39670 q^{33} +4.79580 q^{34} -1.65271 q^{35} -3.06256 q^{36} +3.93313 q^{37} +6.32970 q^{38} -11.8304 q^{39} -1.51875 q^{40} -10.7837 q^{41} -8.72017 q^{42} +3.38193 q^{44} -1.68495 q^{45} +0.406360 q^{46} +3.49088 q^{47} -3.42731 q^{48} +3.93414 q^{49} +4.96287 q^{50} -11.5864 q^{51} +4.25785 q^{52} +4.77361 q^{53} -0.978888 q^{54} +1.86067 q^{55} +10.0479 q^{56} -15.2923 q^{57} +3.69325 q^{58} +6.72259 q^{59} +1.14608 q^{60} +5.24696 q^{61} +0.909829 q^{62} +11.1475 q^{63} +7.58295 q^{64} +2.34258 q^{65} +9.81742 q^{66} +10.5533 q^{67} +4.17003 q^{68} -0.981747 q^{69} +1.72670 q^{70} +0.899542 q^{71} +10.2439 q^{72} -0.966562 q^{73} -4.10922 q^{74} -11.9901 q^{75} +5.50379 q^{76} -12.3100 q^{77} +12.3601 q^{78} -1.00197 q^{79} +0.678652 q^{80} -7.74863 q^{81} +11.2665 q^{82} -3.41592 q^{83} -7.58234 q^{84} +2.29426 q^{85} -8.92272 q^{87} -11.3122 q^{88} -1.84065 q^{89} +1.76039 q^{90} -15.4982 q^{91} +0.353338 q^{92} -2.19810 q^{93} -3.64718 q^{94} +3.02806 q^{95} -11.7592 q^{96} +0.960596 q^{97} -4.11028 q^{98} -12.5501 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9}+O(q^{10}) $ 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9	\( 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} - 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} - 25 q^{18} - 31 q^{19} - 25 q^{20} - 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} - 27 q^{26} - 23 q^{27} - 20 q^{28} - 37 q^{29} + 17 q^{30} - 12 q^{31} - 39 q^{32} - 38 q^{33} - 14 q^{34} + 16 q^{35} + 47 q^{36} + 19 q^{37} + 56 q^{38} - 46 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} + 7 q^{44} - 23 q^{45} + 47 q^{46} - q^{47} - 15 q^{48} - 6 q^{49} + 3 q^{50} - 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} - 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} - 28 q^{61} - 33 q^{62} - 26 q^{63} + 10 q^{64} - 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} - 7 q^{69} + 34 q^{70} - 86 q^{71} + 2 q^{72} + 27 q^{73} - 79 q^{74} - 31 q^{75} - 59 q^{76} - 43 q^{77} + 91 q^{78} + 17 q^{79} - 8 q^{80} - 10 q^{81} - 13 q^{82} - 12 q^{83} - 32 q^{84} - 28 q^{85} - 43 q^{87} + 23 q^{88} - 51 q^{89} + 10 q^{90} + 20 q^{91} + 18 q^{92} + 30 q^{93} + 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} + 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 - 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 - 25 * q^18 - 31 * q^19 - 25 * q^20 - 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 - 27 * q^26 - 23 * q^27 - 20 * q^28 - 37 * q^29 + 17 * q^30 - 12 * q^31 - 39 * q^32 - 38 * q^33 - 14 * q^34 + 16 * q^35 + 47 * q^36 + 19 * q^37 + 56 * q^38 - 46 * q^39 + 6 * q^40 - 7 * q^41 + q^42 + 7 * q^44 - 23 * q^45 + 47 * q^46 - q^47 - 15 * q^48 - 6 * q^49 + 3 * q^50 - 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 - 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 - 28 * q^61 - 33 * q^62 - 26 * q^63 + 10 * q^64 - 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 - 7 * q^69 + 34 * q^70 - 86 * q^71 + 2 * q^72 + 27 * q^73 - 79 * q^74 - 31 * q^75 - 59 * q^76 - 43 * q^77 + 91 * q^78 + 17 * q^79 - 8 * q^80 - 10 * q^81 - 13 * q^82 - 12 * q^83 - 32 * q^84 - 28 * q^85 - 43 * q^87 + 23 * q^88 - 51 * q^89 + 10 * q^90 + 20 * q^91 + 18 * q^92 + 30 * q^93 + 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 + 5 * q^98 + 38 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.04477	−0.738766	−0.369383	−	0.929277i	$-0.620431\pi$
$2$	−1.04477	−0.738766	−0.369383	+	0.929277i	$0.620431\pi$
$3$	2.52412	1.45730	0.728651	−	0.684885i	$-0.240147\pi$
$3$	2.52412	1.45730	0.728651	+	0.684885i	$0.240147\pi$
$4$	−0.908449	−0.454224
$5$	−0.499809	−0.223521	−0.111761	−	0.993735i	$-0.535649\pi$
$5$	−0.499809	−0.223521	−0.111761	+	0.993735i	$0.535649\pi$
$6$	−2.63714	−1.07661
$7$	3.30668	1.24981	0.624904	−	0.780702i	$-0.285138\pi$
$7$	3.30668	1.24981	0.624904	+	0.780702i	$0.285138\pi$
$8$	3.03867	1.07433
$9$	3.37119	1.12373
$10$	0.522187	0.165130
$11$	−3.72276	−1.12245	−0.561227	−	0.827662i	$-0.689670\pi$
$11$	−3.72276	−1.12245	−0.561227	+	0.827662i	$0.689670\pi$
$12$	−2.29304	−0.661942
$13$	−4.68695	−1.29993	−0.649963	−	0.759966i	$-0.725216\pi$
$13$	−4.68695	−1.29993	−0.649963	+	0.759966i	$0.725216\pi$
$14$	−3.45473	−0.923316
$15$	−1.26158	−0.325738
$16$	−1.35782	−0.339456
$17$	−4.59028	−1.11331	−0.556653	−	0.830745i	$-0.687915\pi$
$17$	−4.59028	−1.11331	−0.556653	+	0.830745i	$0.687915\pi$
$18$	−3.52213	−0.830175
$19$	−6.05844	−1.38990	−0.694951	−	0.719057i	$-0.744574\pi$
$19$	−6.05844	−1.38990	−0.694951	+	0.719057i	$0.744574\pi$
$20$	0.454051	0.101529
$21$	8.34647	1.82135
$22$	3.88944	0.829231
$23$	−0.388946	−0.0811008	−0.0405504	−	0.999177i	$-0.512911\pi$
$23$	−0.388946	−0.0811008	−0.0405504	+	0.999177i	$0.512911\pi$
$24$	7.66997	1.56563
$25$	−4.75019	−0.950038
$26$	4.89680	0.960342
$27$	0.936938	0.180314
$28$	−3.00395	−0.567693
$29$	−3.53498	−0.656429	−0.328215	−	0.944603i	$-0.606447\pi$
$29$	−3.53498	−0.656429	−0.328215	+	0.944603i	$0.606447\pi$
$30$	1.31806	0.240644
$31$	−0.870839	−0.156407	−0.0782037	−	0.996937i	$-0.524918\pi$
$31$	−0.870839	−0.156407	−0.0782037	+	0.996937i	$0.524918\pi$
$32$	−4.65872	−0.823553
$33$	−9.39670	−1.63575
$34$	4.79580	0.822473
$35$	−1.65271	−0.279359
$36$	−3.06256	−0.510426
$37$	3.93313	0.646602	0.323301	−	0.946296i	$-0.395207\pi$
$37$	3.93313	0.646602	0.323301	+	0.946296i	$0.395207\pi$
$38$	6.32970	1.02681
$39$	−11.8304	−1.89439
$40$	−1.51875	−0.240136
$41$	−10.7837	−1.68412	−0.842062	−	0.539380i	$-0.818659\pi$
$41$	−10.7837	−1.68412	−0.842062	+	0.539380i	$0.818659\pi$
$42$	−8.72017	−1.34555
$43$	0	0
$43$	0	0
$44$	3.38193	0.509846
$45$	−1.68495	−0.251178
$46$	0.406360	0.0599146
$47$	3.49088	0.509197	0.254599	−	0.967047i	$-0.418057\pi$
$47$	3.49088	0.509197	0.254599	+	0.967047i	$0.418057\pi$
$48$	−3.42731	−0.494690
$49$	3.93414	0.562020
$50$	4.96287	0.701856
$51$	−11.5864	−1.62242
$52$	4.25785	0.590458
$53$	4.77361	0.655706	0.327853	−	0.944729i	$-0.393675\pi$
$53$	4.77361	0.655706	0.327853	+	0.944729i	$0.393675\pi$
$54$	−0.978888	−0.133210
$55$	1.86067	0.250892
$56$	10.0479	1.34271
$57$	−15.2923	−2.02551
$58$	3.69325	0.484948
$59$	6.72259	0.875207	0.437603	−	0.899168i	$-0.355827\pi$
$59$	6.72259	0.875207	0.437603	+	0.899168i	$0.355827\pi$
$60$	1.14608	0.147958
$61$	5.24696	0.671805	0.335902	−	0.941897i	$-0.390959\pi$
$61$	5.24696	0.671805	0.335902	+	0.941897i	$0.390959\pi$
$62$	0.909829	0.115548
$63$	11.1475	1.40445
$64$	7.58295	0.947869
$65$	2.34258	0.290561
$66$	9.81742	1.20844
$67$	10.5533	1.28929	0.644645	−	0.764482i	$-0.277005\pi$
$67$	10.5533	1.28929	0.644645	+	0.764482i	$0.277005\pi$
$68$	4.17003	0.505691
$69$	−0.981747	−0.118188
$70$	1.72670	0.206381
$71$	0.899542	0.106756	0.0533780	−	0.998574i	$-0.483001\pi$
$71$	0.899542	0.106756	0.0533780	+	0.998574i	$0.483001\pi$
$72$	10.2439	1.20726
$73$	−0.966562	−0.113128	−0.0565638	−	0.998399i	$-0.518014\pi$
$73$	−0.966562	−0.113128	−0.0565638	+	0.998399i	$0.518014\pi$
$74$	−4.10922	−0.477688
$75$	−11.9901	−1.38449
$76$	5.50379	0.631328
$77$	−12.3100	−1.40285
$78$	12.3601	1.39951
$79$	−1.00197	−0.112730	−0.0563650	−	0.998410i	$-0.517951\pi$
$79$	−1.00197	−0.112730	−0.0563650	+	0.998410i	$0.517951\pi$
$80$	0.678652	0.0758756
$81$	−7.74863	−0.860959
$82$	11.2665	1.24417
$83$	−3.41592	−0.374946	−0.187473	−	0.982270i	$-0.560030\pi$
$83$	−3.41592	−0.374946	−0.187473	+	0.982270i	$0.560030\pi$
$84$	−7.58234	−0.827301
$85$	2.29426	0.248847
$86$	0	0
$87$	−8.92272	−0.956616
$88$	−11.3122	−1.20589
$89$	−1.84065	−0.195108	−0.0975542	−	0.995230i	$-0.531102\pi$
$89$	−1.84065	−0.195108	−0.0975542	+	0.995230i	$0.531102\pi$
$90$	1.76039	0.185562
$91$	−15.4982	−1.62466
$92$	0.353338	0.0368380
$93$	−2.19810	−0.227933
$94$	−3.64718	−0.376178
$95$	3.02806	0.310673
$96$	−11.7592	−1.20017
$97$	0.960596	0.0975337	0.0487669	−	0.998810i	$-0.484471\pi$
$97$	0.960596	0.0975337	0.0487669	+	0.998810i	$0.484471\pi$
$98$	−4.11028	−0.415201
$99$	−12.5501	−1.26134
$100$	4.31531	0.431531
$101$	16.4388	1.63572	0.817862	−	0.575414i	$-0.195159\pi$
$101$	16.4388	1.63572	0.817862	+	0.575414i	$0.195159\pi$
$102$	12.1052	1.19859
$103$	−0.572128	−0.0563735	−0.0281867	−	0.999603i	$-0.508973\pi$
$103$	−0.572128	−0.0563735	−0.0281867	+	0.999603i	$0.508973\pi$
$104$	−14.2421	−1.39655
$105$	−4.17164	−0.407110
$106$	−4.98734	−0.484413
$107$	−16.1723	−1.56343	−0.781716	−	0.623634i	$-0.785656\pi$
$107$	−16.1723	−1.56343	−0.781716	+	0.623634i	$0.785656\pi$
$108$	−0.851160	−0.0819029
$109$	−6.52710	−0.625182	−0.312591	−	0.949888i	$-0.601197\pi$
$109$	−6.52710	−0.625182	−0.312591	+	0.949888i	$0.601197\pi$
$110$	−1.94397	−0.185351
$111$	9.92769	0.942295
$112$	−4.48989	−0.424255
$113$	−1.70054	−0.159974	−0.0799868	−	0.996796i	$-0.525488\pi$
$113$	−1.70054	−0.159974	−0.0799868	+	0.996796i	$0.525488\pi$
$114$	15.9769	1.49638
$115$	0.194399	0.0181278
$116$	3.21135	0.298166
$117$	−15.8006	−1.46077
$118$	−7.02358	−0.646573
$119$	−15.1786	−1.39142
$120$	−3.83352	−0.349951
$121$	2.85893	0.259902
$122$	−5.48189	−0.496307
$123$	−27.2193	−2.45428
$124$	0.791113	0.0710440
$125$	4.87323	0.435875
$126$	−11.6466	−1.03756
$127$	15.6958	1.39277	0.696387	−	0.717666i	$-0.254790\pi$
$127$	15.6958	1.39277	0.696387	+	0.717666i	$0.254790\pi$
$128$	1.39498	0.123300
$129$	0	0
$130$	−2.44746	−0.214657
$131$	−10.7262	−0.937151	−0.468576	−	0.883423i	$-0.655233\pi$
$131$	−10.7262	−0.937151	−0.468576	+	0.883423i	$0.655233\pi$
$132$	8.53642	0.743000
$133$	−20.0333	−1.73711
$134$	−11.0258	−0.952484
$135$	−0.468290	−0.0403040
$136$	−13.9483	−1.19606
$137$	−8.08569	−0.690807	−0.345404	−	0.938454i	$-0.612258\pi$
$137$	−8.08569	−0.690807	−0.345404	+	0.938454i	$0.612258\pi$
$138$	1.02570	0.0873137
$139$	12.8655	1.09124	0.545620	−	0.838032i	$-0.316294\pi$
$139$	12.8655	1.09124	0.545620	+	0.838032i	$0.316294\pi$
$140$	1.50140	0.126891
$141$	8.81141	0.742054
$142$	−0.939817	−0.0788677
$143$	17.4484	1.45911
$144$	−4.57748	−0.381457
$145$	1.76681	0.146726
$146$	1.00984	0.0835748
$147$	9.93025	0.819033
$148$	−3.57304	−0.293702
$149$	−7.44008	−0.609515	−0.304757	−	0.952430i	$-0.598575\pi$
$149$	−7.44008	−0.609515	−0.304757	+	0.952430i	$0.598575\pi$
$150$	12.5269	1.02282
$151$	−4.77560	−0.388633	−0.194316	−	0.980939i	$-0.562249\pi$
$151$	−4.77560	−0.388633	−0.194316	+	0.980939i	$0.562249\pi$
$152$	−18.4096	−1.49322
$153$	−15.4747	−1.25106
$154$	12.8611	1.03638
$155$	0.435253	0.0349604
$156$	10.7473	0.860476
$157$	13.4410	1.07271	0.536354	−	0.843993i	$-0.319801\pi$
$157$	13.4410	1.07271	0.536354	+	0.843993i	$0.319801\pi$
$158$	1.04683	0.0832811
$159$	12.0492	0.955562
$160$	2.32847	0.184082
$161$	−1.28612	−0.101360
$162$	8.09557	0.636048
$163$	18.7293	1.46699	0.733496	−	0.679694i	$-0.237887\pi$
$163$	18.7293	1.46699	0.733496	+	0.679694i	$0.237887\pi$
$164$	9.79640	0.764970
$165$	4.69655	0.365626
$166$	3.56886	0.276997
$167$	−13.8552	−1.07215	−0.536075	−	0.844170i	$-0.680094\pi$
$167$	−13.8552	−1.07215	−0.536075	+	0.844170i	$0.680094\pi$
$168$	25.3622	1.95673
$169$	8.96750	0.689808
$170$	−2.39698	−0.183840
$171$	−20.4242	−1.56188
$172$	0	0
$173$	−13.8705	−1.05456	−0.527278	−	0.849693i	$-0.676787\pi$
$173$	−13.8705	−1.05456	−0.527278	+	0.849693i	$0.676787\pi$
$174$	9.32222	0.706716
$175$	−15.7074	−1.18737
$176$	5.05485	0.381023
$177$	16.9686	1.27544
$178$	1.92306	0.144140
$179$	6.18438	0.462242	0.231121	−	0.972925i	$-0.425761\pi$
$179$	6.18438	0.462242	0.231121	+	0.972925i	$0.425761\pi$
$180$	1.53069	0.114091
$181$	3.70276	0.275224	0.137612	−	0.990486i	$-0.456057\pi$
$181$	3.70276	0.275224	0.137612	+	0.990486i	$0.456057\pi$
$182$	16.1922	1.20024
$183$	13.2440	0.979023
$184$	−1.18188	−0.0871292
$185$	−1.96581	−0.144529
$186$	2.29652	0.168389
$187$	17.0885	1.24963
$188$	−3.17128	−0.231290
$189$	3.09815	0.225358
$190$	−3.16364	−0.229515
$191$	11.4154	0.825991	0.412995	−	0.910733i	$-0.364483\pi$
$191$	11.4154	0.825991	0.412995	+	0.910733i	$0.364483\pi$
$192$	19.1403	1.38133
$193$	−8.44971	−0.608223	−0.304112	−	0.952636i	$-0.598360\pi$
$193$	−8.44971	−0.608223	−0.304112	+	0.952636i	$0.598360\pi$
$194$	−1.00360	−0.0720546
$195$	5.91295	0.423435
$196$	−3.57396	−0.255283
$197$	−21.6649	−1.54356	−0.771780	−	0.635890i	$-0.780633\pi$
$197$	−21.6649	−1.54356	−0.771780	+	0.635890i	$0.780633\pi$
$198$	13.1120	0.931833
$199$	−9.87849	−0.700267	−0.350134	−	0.936700i	$-0.613864\pi$
$199$	−9.87849	−0.700267	−0.350134	+	0.936700i	$0.613864\pi$
$200$	−14.4343	−1.02066
$201$	26.6378	1.87889
$202$	−17.1748	−1.20842
$203$	−11.6890	−0.820410
$204$	10.5257	0.736944
$205$	5.38977	0.376438
$206$	0.597744	0.0416468
$207$	−1.31121	−0.0911356
$208$	6.36405	0.441267
$209$	22.5541	1.56010
$210$	4.35841	0.300759
$211$	−15.2999	−1.05329	−0.526644	−	0.850086i	$-0.676550\pi$
$211$	−15.2999	−1.05329	−0.526644	+	0.850086i	$0.676550\pi$
$212$	−4.33658	−0.297838
$213$	2.27055	0.155576
$214$	16.8964	1.15501
$215$	0	0
$216$	2.84704	0.193717
$217$	−2.87959	−0.195479
$218$	6.81934	0.461864
$219$	−2.43972	−0.164861
$220$	−1.69032	−0.113961
$221$	21.5144	1.44722
$222$	−10.3722	−0.696135
$223$	11.7167	0.784608	0.392304	−	0.919836i	$-0.371678\pi$
$223$	11.7167	0.784608	0.392304	+	0.919836i	$0.371678\pi$
$224$	−15.4049	−1.02928
$225$	−16.0138	−1.06759
$226$	1.77668	0.118183
$227$	9.03975	0.599989	0.299995	−	0.953941i	$-0.403015\pi$
$227$	9.03975	0.599989	0.299995	+	0.953941i	$0.403015\pi$
$228$	13.8922	0.920035
$229$	−15.5978	−1.03073	−0.515366	−	0.856970i	$-0.672344\pi$
$229$	−15.5978	−1.03073	−0.515366	+	0.856970i	$0.672344\pi$
$230$	−0.203102	−0.0133922
$231$	−31.0719	−2.04438
$232$	−10.7416	−0.705223
$233$	−4.69718	−0.307722	−0.153861	−	0.988092i	$-0.549171\pi$
$233$	−4.69718	−0.307722	−0.153861	+	0.988092i	$0.549171\pi$
$234$	16.5081	1.07917
$235$	−1.74477	−0.113816
$236$	−6.10713	−0.397540
$237$	−2.52908	−0.164282
$238$	15.8582	1.02793
$239$	−4.11467	−0.266156	−0.133078	−	0.991106i	$-0.542486\pi$
$239$	−4.11467	−0.266156	−0.133078	+	0.991106i	$0.542486\pi$
$240$	1.71300	0.110574
$241$	3.26436	0.210276	0.105138	−	0.994458i	$-0.466472\pi$
$241$	3.26436	0.210276	0.105138	+	0.994458i	$0.466472\pi$
$242$	−2.98693	−0.192007
$243$	−22.3693	−1.43499
$244$	−4.76660	−0.305150
$245$	−1.96632	−0.125623
$246$	28.4380	1.81314
$247$	28.3956	1.80677
$248$	−2.64619	−0.168033
$249$	−8.62220	−0.546410
$250$	−5.09142	−0.322010
$251$	5.47251	0.345422	0.172711	−	0.984973i	$-0.444747\pi$
$251$	5.47251	0.345422	0.172711	+	0.984973i	$0.444747\pi$
$252$	−10.1269	−0.637935
$253$	1.44795	0.0910319
$254$	−16.3985	−1.02893
$255$	5.79099	0.362646
$256$	−16.6233	−1.03896
$257$	−8.41973	−0.525208	−0.262604	−	0.964904i	$-0.584581\pi$
$257$	−8.41973	−0.525208	−0.262604	+	0.964904i	$0.584581\pi$
$258$	0	0
$259$	13.0056	0.808128
$260$	−2.12811	−0.131980
$261$	−11.9171	−0.737650
$262$	11.2064	0.692336
$263$	6.11997	0.377374	0.188687	−	0.982037i	$-0.439577\pi$
$263$	6.11997	0.377374	0.188687	+	0.982037i	$0.439577\pi$
$264$	−28.5535	−1.75734
$265$	−2.38589	−0.146564
$266$	20.9303	1.28332
$267$	−4.64602	−0.284332
$268$	−9.58712	−0.585627
$269$	2.29453	0.139900	0.0699500	−	0.997550i	$-0.477716\pi$
$269$	2.29453	0.139900	0.0699500	+	0.997550i	$0.477716\pi$
$270$	0.489256	0.0297752
$271$	27.3969	1.66425	0.832123	−	0.554591i	$-0.187125\pi$
$271$	27.3969	1.66425	0.832123	+	0.554591i	$0.187125\pi$
$272$	6.23279	0.377918
$273$	−39.1195	−2.36762
$274$	8.44771	0.510345
$275$	17.6838	1.06637
$276$	0.891867	0.0536841
$277$	−26.5986	−1.59815	−0.799077	−	0.601229i	$-0.794678\pi$
$277$	−26.5986	−1.59815	−0.799077	+	0.601229i	$0.794678\pi$
$278$	−13.4416	−0.806172
$279$	−2.93577	−0.175760
$280$	−5.02203	−0.300124
$281$	16.6768	0.994852	0.497426	−	0.867506i	$-0.334279\pi$
$281$	16.6768	0.994852	0.497426	+	0.867506i	$0.334279\pi$
$282$	−9.20592	−0.548205
$283$	17.1076	1.01694	0.508471	−	0.861079i	$-0.330211\pi$
$283$	17.1076	1.01694	0.508471	+	0.861079i	$0.330211\pi$
$284$	−0.817188	−0.0484912
$285$	7.64320	0.452744
$286$	−18.2296	−1.07794
$287$	−35.6581	−2.10483
$288$	−15.7055	−0.925453
$289$	4.07065	0.239450
$290$	−1.84592	−0.108396
$291$	2.42466	0.142136
$292$	0.878072	0.0513853
$293$	−29.9122	−1.74749	−0.873744	−	0.486387i	$-0.838315\pi$
$293$	−29.9122	−1.74749	−0.873744	+	0.486387i	$0.838315\pi$
$294$	−10.3749	−0.605074
$295$	−3.36001	−0.195627
$296$	11.9515	0.694665
$297$	−3.48799	−0.202394
$298$	7.77319	0.450289
$299$	1.82297	0.105425
$300$	10.8924	0.628871
$301$	0	0
$302$	4.98942	0.287109
$303$	41.4936	2.38375
$304$	8.22630	0.471810
$305$	−2.62248	−0.150163
$306$	16.1676	0.924238
$307$	−13.4430	−0.767230	−0.383615	−	0.923493i	$-0.625321\pi$
$307$	−13.4430	−0.767230	−0.383615	+	0.923493i	$0.625321\pi$
$308$	11.1830	0.637209
$309$	−1.44412	−0.0821532
$310$	−0.454741	−0.0258275
$311$	15.1900	0.861347	0.430673	−	0.902508i	$-0.358276\pi$
$311$	15.1900	0.861347	0.430673	+	0.902508i	$0.358276\pi$
$312$	−35.9488	−2.03520
$313$	11.3920	0.643913	0.321956	−	0.946754i	$-0.395660\pi$
$313$	11.3920	0.643913	0.321956	+	0.946754i	$0.395660\pi$
$314$	−14.0428	−0.792480
$315$	−5.57160	−0.313924
$316$	0.910235	0.0512047
$317$	−22.8190	−1.28164	−0.640822	−	0.767689i	$-0.721406\pi$
$317$	−22.8190	−1.28164	−0.640822	+	0.767689i	$0.721406\pi$
$318$	−12.5887	−0.705937
$319$	13.1599	0.736811
$320$	−3.79003	−0.211869
$321$	−40.8208	−2.27840
$322$	1.34370	0.0748817
$323$	27.8099	1.54739
$324$	7.03924	0.391069
$325$	22.2639	1.23498
$326$	−19.5679	−1.08376
$327$	−16.4752	−0.911080
$328$	−32.7680	−1.80931
$329$	11.5432	0.636399
$330$	−4.90683	−0.270112
$331$	15.5508	0.854747	0.427373	−	0.904075i	$-0.359439\pi$
$331$	15.5508	0.854747	0.427373	+	0.904075i	$0.359439\pi$
$332$	3.10319	0.170310
$333$	13.2593	0.726607
$334$	14.4756	0.792068
$335$	−5.27463	−0.288184
$336$	−11.3330	−0.618267
$337$	14.9040	0.811871	0.405935	−	0.913902i	$-0.366946\pi$
$337$	14.9040	0.811871	0.405935	+	0.913902i	$0.366946\pi$
$338$	−9.36901	−0.509607
$339$	−4.29238	−0.233130
$340$	−2.08422	−0.113033
$341$	3.24192	0.175560
$342$	21.3386	1.15386
$343$	−10.1378	−0.547391
$344$	0	0
$345$	0.490686	0.0264176
$346$	14.4915	0.779071
$347$	27.1661	1.45835	0.729177	−	0.684325i	$-0.239903\pi$
$347$	27.1661	1.45835	0.729177	+	0.684325i	$0.239903\pi$
$348$	8.10583	0.434518
$349$	28.3556	1.51784	0.758920	−	0.651184i	$-0.225727\pi$
$349$	28.3556	1.51784	0.758920	+	0.651184i	$0.225727\pi$
$350$	16.4106	0.877185
$351$	−4.39138	−0.234395
$352$	17.3433	0.924401
$353$	−19.7233	−1.04977	−0.524883	−	0.851175i	$-0.675891\pi$
$353$	−19.7233	−1.04977	−0.524883	+	0.851175i	$0.675891\pi$
$354$	−17.7284	−0.942253
$355$	−0.449599	−0.0238622
$356$	1.67214	0.0886230
$357$	−38.3126	−2.02772
$358$	−6.46127	−0.341489
$359$	−28.1781	−1.48718	−0.743592	−	0.668633i	$-0.766880\pi$
$359$	−28.1781	−1.48718	−0.743592	+	0.668633i	$0.766880\pi$
$360$	−5.12001	−0.269848
$361$	17.7047	0.931829
$362$	−3.86854	−0.203326
$363$	7.21628	0.378756
$364$	14.0794	0.737959
$365$	0.483096	0.0252864
$366$	−13.8370	−0.723269
$367$	−29.5394	−1.54194	−0.770972	−	0.636869i	$-0.780229\pi$
$367$	−29.5394	−1.54194	−0.770972	+	0.636869i	$0.780229\pi$
$368$	0.528120	0.0275302
$369$	−36.3538	−1.89250
$370$	2.05383	0.106773
$371$	15.7848	0.819506
$372$	1.99687	0.103533
$373$	−32.1477	−1.66454	−0.832272	−	0.554367i	$-0.812960\pi$
$373$	−32.1477	−1.66454	−0.832272	+	0.554367i	$0.812960\pi$
$374$	−17.8536	−0.923188
$375$	12.3006	0.635202
$376$	10.6076	0.547047
$377$	16.5683	0.853309
$378$	−3.23687	−0.166487
$379$	−18.0279	−0.926033	−0.463017	−	0.886350i	$-0.653233\pi$
$379$	−18.0279	−0.926033	−0.463017	+	0.886350i	$0.653233\pi$
$380$	−2.75084	−0.141115
$381$	39.6180	2.02969
$382$	−11.9265	−0.610214
$383$	−22.9648	−1.17345	−0.586724	−	0.809787i	$-0.699583\pi$
$383$	−22.9648	−1.17345	−0.586724	+	0.809787i	$0.699583\pi$
$384$	3.52109	0.179685
$385$	6.15263	0.313567
$386$	8.82803	0.449335
$387$	0	0
$388$	−0.872652	−0.0443022
$389$	−22.8174	−1.15689	−0.578445	−	0.815721i	$-0.696340\pi$
$389$	−22.8174	−1.15689	−0.578445	+	0.815721i	$0.696340\pi$
$390$	−6.17770	−0.312820
$391$	1.78537	0.0902901
$392$	11.9545	0.603796
$393$	−27.0742	−1.36571
$394$	22.6349	1.14033
$395$	0.500791	0.0251975
$396$	11.4012	0.572930
$397$	−8.63189	−0.433222	−0.216611	−	0.976258i	$-0.569500\pi$
$397$	−8.63189	−0.433222	−0.216611	+	0.976258i	$0.569500\pi$
$398$	10.3208	0.517334
$399$	−50.5666	−2.53150
$400$	6.44992	0.322496
$401$	33.8253	1.68916	0.844578	−	0.535433i	$-0.179852\pi$
$401$	33.8253	1.68916	0.844578	+	0.535433i	$0.179852\pi$
$402$	−27.8305	−1.38806
$403$	4.08158	0.203318
$404$	−14.9338	−0.742986
$405$	3.87283	0.192443
$406$	12.2124	0.606091
$407$	−14.6421	−0.725781
$408$	−35.2073	−1.74302
$409$	−25.6569	−1.26865	−0.634326	−	0.773065i	$-0.718722\pi$
$409$	−25.6569	−1.26865	−0.634326	+	0.773065i	$0.718722\pi$
$410$	−5.63108	−0.278099
$411$	−20.4093	−1.00672
$412$	0.519749	0.0256062
$413$	22.2295	1.09384
$414$	1.36992	0.0673279
$415$	1.70731	0.0838083
$416$	21.8352	1.07056
$417$	32.4742	1.59027
$418$	−23.5639	−1.15255
$419$	−24.8323	−1.21314	−0.606570	−	0.795030i	$-0.707455\pi$
$419$	−24.8323	−1.21314	−0.606570	+	0.795030i	$0.707455\pi$
$420$	3.78972	0.184919
$421$	19.2945	0.940356	0.470178	−	0.882571i	$-0.344190\pi$
$421$	19.2945	0.940356	0.470178	+	0.882571i	$0.344190\pi$
$422$	15.9849	0.778134
$423$	11.7684	0.572201
$424$	14.5054	0.704446
$425$	21.8047	1.05768
$426$	−2.37221	−0.114934
$427$	17.3500	0.839627
$428$	14.6917	0.710149
$429$	44.0418	2.12636
$430$	0	0
$431$	32.4273	1.56197	0.780983	−	0.624552i	$-0.214718\pi$
$431$	32.4273	1.56197	0.780983	+	0.624552i	$0.214718\pi$
$432$	−1.27220	−0.0612085
$433$	−17.1636	−0.824828	−0.412414	−	0.910997i	$-0.635314\pi$
$433$	−17.1636	−0.824828	−0.412414	+	0.910997i	$0.635314\pi$
$434$	3.00852	0.144413
$435$	4.45965	0.213824
$436$	5.92953	0.283973
$437$	2.35641	0.112722
$438$	2.54896	0.121794
$439$	14.4303	0.688720	0.344360	−	0.938838i	$-0.388096\pi$
$439$	14.4303	0.688720	0.344360	+	0.938838i	$0.388096\pi$
$440$	5.65395	0.269542
$441$	13.2627	0.631559
$442$	−22.4777	−1.06915
$443$	19.0065	0.903024	0.451512	−	0.892265i	$-0.350885\pi$
$443$	19.0065	0.903024	0.451512	+	0.892265i	$0.350885\pi$
$444$	−9.01880	−0.428013
$445$	0.919972	0.0436109
$446$	−12.2413	−0.579642
$447$	−18.7797	−0.888248
$448$	25.0744	1.18465
$449$	5.33116	0.251593	0.125797	−	0.992056i	$-0.459851\pi$
$449$	5.33116	0.251593	0.125797	+	0.992056i	$0.459851\pi$
$450$	16.7308	0.788698
$451$	40.1449	1.89035
$452$	1.54486	0.0726639
$453$	−12.0542	−0.566355
$454$	−9.44449	−0.443252
$455$	7.74616	0.363146
$456$	−46.4681	−2.17607
$457$	7.95238	0.371997	0.185998	−	0.982550i	$-0.440448\pi$
$457$	7.95238	0.371997	0.185998	+	0.982550i	$0.440448\pi$
$458$	16.2962	0.761470
$459$	−4.30080	−0.200744
$460$	−0.176601	−0.00823407
$461$	42.5990	1.98403	0.992016	−	0.126115i	$-0.0402510\pi$
$461$	42.5990	1.98403	0.992016	+	0.126115i	$0.0402510\pi$
$462$	32.4631	1.51032
$463$	−5.69293	−0.264573	−0.132286	−	0.991212i	$-0.542232\pi$
$463$	−5.69293	−0.264573	−0.132286	+	0.991212i	$0.542232\pi$
$464$	4.79988	0.222829
$465$	1.09863	0.0509478
$466$	4.90749	0.227335
$467$	−7.38749	−0.341852	−0.170926	−	0.985284i	$-0.554676\pi$
$467$	−7.38749	−0.341852	−0.170926	+	0.985284i	$0.554676\pi$
$468$	14.3541	0.663516
$469$	34.8964	1.61136
$470$	1.82289	0.0840837
$471$	33.9267	1.56326
$472$	20.4277	0.940263
$473$	0	0
$474$	2.64232	0.121366
$475$	28.7788	1.32046
$476$	13.7890	0.632016
$477$	16.0928	0.736837
$478$	4.29889	0.196627
$479$	20.7081	0.946177	0.473089	−	0.881015i	$-0.343139\pi$
$479$	20.7081	0.946177	0.473089	+	0.881015i	$0.343139\pi$
$480$	5.87734	0.268263
$481$	−18.4344	−0.840535
$482$	−3.41051	−0.155345
$483$	−3.24633	−0.147713
$484$	−2.59719	−0.118054
$485$	−0.480114	−0.0218009
$486$	23.3709	1.06012
$487$	35.6007	1.61322	0.806610	−	0.591084i	$-0.201300\pi$
$487$	35.6007	1.61322	0.806610	+	0.591084i	$0.201300\pi$
$488$	15.9438	0.721741
$489$	47.2750	2.13785
$490$	2.05436	0.0928063
$491$	0.665932	0.0300531	0.0150265	−	0.999887i	$-0.495217\pi$
$491$	0.665932	0.0300531	0.0150265	+	0.999887i	$0.495217\pi$
$492$	24.7273	1.11479
$493$	16.2265	0.730806
$494$	−29.6670	−1.33478
$495$	6.27267	0.281935
$496$	1.18245	0.0530934
$497$	2.97450	0.133424
$498$	9.00824	0.403669
$499$	−7.19573	−0.322125	−0.161063	−	0.986944i	$-0.551492\pi$
$499$	−7.19573	−0.322125	−0.161063	+	0.986944i	$0.551492\pi$
$500$	−4.42708	−0.197985
$501$	−34.9723	−1.56245
$502$	−5.71753	−0.255186
$503$	−41.5468	−1.85248	−0.926240	−	0.376935i	$-0.876978\pi$
$503$	−41.5468	−1.85248	−0.926240	+	0.376935i	$0.876978\pi$
$504$	33.8735	1.50884
$505$	−8.21627	−0.365619
$506$	−1.51278	−0.0672513
$507$	22.6351	1.00526
$508$	−14.2588	−0.632632
$509$	−30.7587	−1.36335	−0.681677	−	0.731653i	$-0.738749\pi$
$509$	−30.7587	−1.36335	−0.681677	+	0.731653i	$0.738749\pi$
$510$	−6.05028	−0.267911
$511$	−3.19611	−0.141388
$512$	14.5777	0.644248
$513$	−5.67639	−0.250619
$514$	8.79671	0.388006
$515$	0.285955	0.0126007
$516$	0	0
$517$	−12.9957	−0.571550
$518$	−13.5879	−0.597018
$519$	−35.0109	−1.53681
$520$	7.11832	0.312159
$521$	−15.3022	−0.670402	−0.335201	−	0.942147i	$-0.608804\pi$
$521$	−15.3022	−0.670402	−0.335201	+	0.942147i	$0.608804\pi$
$522$	12.4507	0.544951
$523$	1.34406	0.0587716	0.0293858	−	0.999568i	$-0.490645\pi$
$523$	1.34406	0.0587716	0.0293858	+	0.999568i	$0.490645\pi$
$524$	9.74419	0.425677
$525$	−39.6473	−1.73035
$526$	−6.39399	−0.278791
$527$	3.99739	0.174129
$528$	12.7591	0.555266
$529$	−22.8487	−0.993423
$530$	2.49272	0.108277
$531$	22.6632	0.983497
$532$	18.1993	0.789038
$533$	50.5425	2.18924
$534$	4.85404	0.210055
$535$	8.08304	0.349460
$536$	32.0680	1.38512
$537$	15.6101	0.673627
$538$	−2.39727	−0.103353
$539$	−14.6458	−0.630841
$540$	0.425417	0.0183070
$541$	−0.198489	−0.00853373	−0.00426686	−	0.999991i	$-0.501358\pi$
$541$	−0.198489	−0.00853373	−0.00426686	+	0.999991i	$0.501358\pi$
$542$	−28.6236	−1.22949
$543$	9.34621	0.401084
$544$	21.3848	0.916867
$545$	3.26230	0.139742
$546$	40.8710	1.74912
$547$	7.93941	0.339465	0.169732	−	0.985490i	$-0.445710\pi$
$547$	7.93941	0.339465	0.169732	+	0.985490i	$0.445710\pi$
$548$	7.34544	0.313782
$549$	17.6885	0.754928
$550$	−18.4756	−0.787801
$551$	21.4165	0.912372
$552$	−2.98321	−0.126974
$553$	−3.31318	−0.140891
$554$	27.7895	1.18066
$555$	−4.96194	−0.210623
$556$	−11.6877	−0.495668
$557$	−17.5130	−0.742048	−0.371024	−	0.928623i	$-0.620993\pi$
$557$	−17.5130	−0.742048	−0.371024	+	0.928623i	$0.620993\pi$
$558$	3.06721	0.129845
$559$	0	0
$560$	2.24408	0.0948299
$561$	43.1334	1.82110
$562$	−17.4234	−0.734963
$563$	24.8082	1.04554	0.522771	−	0.852473i	$-0.324898\pi$
$563$	24.8082	1.04554	0.522771	+	0.852473i	$0.324898\pi$
$564$	−8.00471	−0.337059
$565$	0.849946	0.0357575
$566$	−17.8736	−0.751283
$567$	−25.6223	−1.07603
$568$	2.73341	0.114691
$569$	−28.8478	−1.20936	−0.604681	−	0.796468i	$-0.706699\pi$
$569$	−28.8478	−1.20936	−0.604681	+	0.796468i	$0.706699\pi$
$570$	−7.98541	−0.334472
$571$	32.6060	1.36452	0.682259	−	0.731111i	$-0.260998\pi$
$571$	32.6060	1.36452	0.682259	+	0.731111i	$0.260998\pi$
$572$	−15.8510	−0.662762
$573$	28.8139	1.20372
$574$	37.2547	1.55498
$575$	1.84757	0.0770489
$576$	25.5636	1.06515
$577$	−33.6229	−1.39974	−0.699870	−	0.714270i	$-0.746759\pi$
$577$	−33.6229	−1.39974	−0.699870	+	0.714270i	$0.746759\pi$
$578$	−4.25291	−0.176898
$579$	−21.3281	−0.886366
$580$	−1.60506	−0.0666465
$581$	−11.2954	−0.468610
$582$	−2.53322	−0.105005
$583$	−17.7710	−0.735999
$584$	−2.93706	−0.121537
$585$	7.89728	0.326513
$586$	31.2514	1.29098
$587$	16.2192	0.669439	0.334720	−	0.942318i	$-0.391358\pi$
$587$	16.2192	0.669439	0.334720	+	0.942318i	$0.391358\pi$
$588$	−9.02112	−0.372025
$589$	5.27593	0.217391
$590$	3.51045	0.144523
$591$	−54.6848	−2.24943
$592$	−5.34049	−0.219493
$593$	−44.2180	−1.81582	−0.907909	−	0.419168i	$-0.862322\pi$
$593$	−44.2180	−1.81582	−0.907909	+	0.419168i	$0.862322\pi$
$594$	3.64416	0.149522
$595$	7.58639	0.311012
$596$	6.75893	0.276857
$597$	−24.9345	−1.02050
$598$	−1.90459	−0.0778845
$599$	4.14316	0.169285	0.0846425	−	0.996411i	$-0.473025\pi$
$599$	4.14316	0.169285	0.0846425	+	0.996411i	$0.473025\pi$
$600$	−36.4338	−1.48741
$601$	19.9921	0.815496	0.407748	−	0.913094i	$-0.366314\pi$
$601$	19.9921	0.815496	0.407748	+	0.913094i	$0.366314\pi$
$602$	0	0
$603$	35.5772	1.44881
$604$	4.33839	0.176526
$605$	−1.42892	−0.0580937
$606$	−43.3514	−1.76103
$607$	−18.0883	−0.734183	−0.367091	−	0.930185i	$-0.619646\pi$
$607$	−18.0883	−0.734183	−0.367091	+	0.930185i	$0.619646\pi$
$608$	28.2246	1.14466
$609$	−29.5046	−1.19559
$610$	2.73989	0.110935
$611$	−16.3616	−0.661919
$612$	14.0580	0.568260
$613$	37.5137	1.51516	0.757582	−	0.652740i	$-0.226380\pi$
$613$	37.5137	1.51516	0.757582	+	0.652740i	$0.226380\pi$
$614$	14.0448	0.566804
$615$	13.6044	0.548583
$616$	−37.4059	−1.50713
$617$	−9.05821	−0.364670	−0.182335	−	0.983236i	$-0.558366\pi$
$617$	−9.05821	−0.364670	−0.182335	+	0.983236i	$0.558366\pi$
$618$	1.50878	0.0606920
$619$	4.96217	0.199446	0.0997232	−	0.995015i	$-0.468204\pi$
$619$	4.96217	0.199446	0.0997232	+	0.995015i	$0.468204\pi$
$620$	−0.395405	−0.0158798
$621$	−0.364418	−0.0146236
$622$	−15.8701	−0.636334
$623$	−6.08644	−0.243848
$624$	16.0636	0.643060
$625$	21.3153	0.852611
$626$	−11.9020	−0.475701
$627$	56.9294	2.27354
$628$	−12.2104	−0.487250
$629$	−18.0541	−0.719866
$630$	5.82106	0.231916
$631$	34.8666	1.38802	0.694009	−	0.719966i	$-0.255843\pi$
$631$	34.8666	1.38802	0.694009	+	0.719966i	$0.255843\pi$
$632$	−3.04464	−0.121109
$633$	−38.6188	−1.53496
$634$	23.8407	0.946836
$635$	−7.84488	−0.311315
$636$	−10.9461	−0.434039
$637$	−18.4391	−0.730584
$638$	−13.7491	−0.544331
$639$	3.03253	0.119965
$640$	−0.697221	−0.0275601
$641$	−43.9244	−1.73491	−0.867454	−	0.497518i	$-0.834245\pi$
$641$	−43.9244	−1.73491	−0.867454	+	0.497518i	$0.834245\pi$
$642$	42.6485	1.68320
$643$	23.0813	0.910236	0.455118	−	0.890431i	$-0.349597\pi$
$643$	23.0813	0.910236	0.455118	+	0.890431i	$0.349597\pi$
$644$	1.16837	0.0460404
$645$	0	0
$646$	−29.0551	−1.14316
$647$	−44.1509	−1.73575	−0.867876	−	0.496780i	$-0.834515\pi$
$647$	−44.1509	−1.73575	−0.867876	+	0.496780i	$0.834515\pi$
$648$	−23.5455	−0.924956
$649$	−25.0266	−0.982379
$650$	−23.2607	−0.912361
$651$	−7.26843	−0.284872
$652$	−17.0146	−0.666344
$653$	6.65908	0.260590	0.130295	−	0.991475i	$-0.458408\pi$
$653$	6.65908	0.260590	0.130295	+	0.991475i	$0.458408\pi$
$654$	17.2128	0.673075
$655$	5.36104	0.209473
$656$	14.6423	0.571686
$657$	−3.25847	−0.127125
$658$	−12.0601	−0.470150
$659$	−9.47092	−0.368935	−0.184467	−	0.982839i	$-0.559056\pi$
$659$	−9.47092	−0.368935	−0.184467	+	0.982839i	$0.559056\pi$
$660$	−4.26658	−0.166076
$661$	16.8401	0.655005	0.327503	−	0.944850i	$-0.393793\pi$
$661$	16.8401	0.655005	0.327503	+	0.944850i	$0.393793\pi$
$662$	−16.2470	−0.631458
$663$	54.3050	2.10903
$664$	−10.3798	−0.402816
$665$	10.0128	0.388281
$666$	−13.8530	−0.536792
$667$	1.37492	0.0532370
$668$	12.5868	0.486997
$669$	29.5744	1.14341
$670$	5.51079	0.212900
$671$	−19.5332	−0.754070
$672$	−38.8839	−1.49998
$673$	18.0156	0.694449	0.347225	−	0.937782i	$-0.387124\pi$
$673$	18.0156	0.694449	0.347225	+	0.937782i	$0.387124\pi$
$674$	−15.5713	−0.599783
$675$	−4.45063	−0.171305
$676$	−8.14652	−0.313328
$677$	5.40686	0.207802	0.103901	−	0.994588i	$-0.466867\pi$
$677$	5.40686	0.207802	0.103901	+	0.994588i	$0.466867\pi$
$678$	4.48456	0.172229
$679$	3.17638	0.121898
$680$	6.97150	0.267345
$681$	22.8174	0.874366
$682$	−3.38707	−0.129698
$683$	−36.8545	−1.41020	−0.705099	−	0.709109i	$-0.749098\pi$
$683$	−36.8545	−1.41020	−0.705099	+	0.709109i	$0.749098\pi$
$684$	18.5543	0.709443
$685$	4.04130	0.154410
$686$	10.5917	0.404394
$687$	−39.3708	−1.50209
$688$	0	0
$689$	−22.3737	−0.852369
$690$	−0.512655	−0.0195165
$691$	9.97585	0.379499	0.189750	−	0.981833i	$-0.439232\pi$
$691$	9.97585	0.379499	0.189750	+	0.981833i	$0.439232\pi$
$692$	12.6007	0.479005
$693$	−41.4993	−1.57643
$694$	−28.3824	−1.07738
$695$	−6.43031	−0.243915
$696$	−27.1132	−1.02772
$697$	49.5000	1.87495
$698$	−29.6252	−1.12133
$699$	−11.8562	−0.448445
$700$	14.2693	0.539330
$701$	11.6450	0.439827	0.219913	−	0.975519i	$-0.429422\pi$
$701$	11.6450	0.439827	0.219913	+	0.975519i	$0.429422\pi$
$702$	4.58800	0.173163
$703$	−23.8286	−0.898713
$704$	−28.2295	−1.06394
$705$	−4.40402	−0.165865
$706$	20.6064	0.775531
$707$	54.3580	2.04434
$708$	−15.4151	−0.579337
$709$	16.6947	0.626981	0.313491	−	0.949591i	$-0.398502\pi$
$709$	16.6947	0.626981	0.313491	+	0.949591i	$0.398502\pi$
$710$	0.469729	0.0176286
$711$	−3.37782	−0.126678
$712$	−5.59312	−0.209611
$713$	0.338709	0.0126848
$714$	40.0280	1.49801
$715$	−8.72085	−0.326141
$716$	−5.61819	−0.209962
$717$	−10.3859	−0.387869
$718$	29.4398	1.09868
$719$	−32.2362	−1.20221	−0.601104	−	0.799170i	$-0.705272\pi$
$719$	−32.2362	−1.20221	−0.601104	+	0.799170i	$0.705272\pi$
$720$	2.28787	0.0852637
$721$	−1.89184	−0.0704560
$722$	−18.4974	−0.688404
$723$	8.23964	0.306435
$724$	−3.36376	−0.125013
$725$	16.7918	0.623633
$726$	−7.53937	−0.279812
$727$	21.0335	0.780088	0.390044	−	0.920796i	$-0.372460\pi$
$727$	21.0335	0.780088	0.390044	+	0.920796i	$0.372460\pi$
$728$	−47.0941	−1.74542
$729$	−33.2170	−1.23026
$730$	−0.504726	−0.0186807
$731$	0	0
$732$	−12.0315	−0.444696
$733$	4.52071	0.166976	0.0834881	−	0.996509i	$-0.473394\pi$
$733$	4.52071	0.166976	0.0834881	+	0.996509i	$0.473394\pi$
$734$	30.8620	1.13914
$735$	−4.96322	−0.183071
$736$	1.81199	0.0667909
$737$	−39.2873	−1.44717
$738$	37.9815	1.39812
$739$	−6.99980	−0.257492	−0.128746	−	0.991678i	$-0.541095\pi$
$739$	−6.99980	−0.257492	−0.128746	+	0.991678i	$0.541095\pi$
$740$	1.78584	0.0656487
$741$	71.6740	2.63301
$742$	−16.4915	−0.605424
$743$	−23.0203	−0.844532	−0.422266	−	0.906472i	$-0.638765\pi$
$743$	−23.0203	−0.844532	−0.422266	+	0.906472i	$0.638765\pi$
$744$	−6.67931	−0.244876
$745$	3.71861	0.136240
$746$	33.5870	1.22971
$747$	−11.5157	−0.421338
$748$	−15.5240	−0.567614
$749$	−53.4766	−1.95399
$750$	−12.8514	−0.469266
$751$	14.3313	0.522957	0.261478	−	0.965209i	$-0.415790\pi$
$751$	14.3313	0.522957	0.261478	+	0.965209i	$0.415790\pi$
$752$	−4.74000	−0.172850
$753$	13.8133	0.503384
$754$	−17.3101	−0.630396
$755$	2.38689	0.0868676
$756$	−2.81451	−0.102363
$757$	36.4497	1.32479	0.662393	−	0.749156i	$-0.269541\pi$
$757$	36.4497	1.32479	0.662393	+	0.749156i	$0.269541\pi$
$758$	18.8351	0.684122
$759$	3.65481	0.132661
$760$	9.20128	0.333766
$761$	−44.4716	−1.61209	−0.806047	−	0.591852i	$-0.798397\pi$
$761$	−44.4716	−1.61209	−0.806047	+	0.591852i	$0.798397\pi$
$762$	−41.3919	−1.49947
$763$	−21.5830	−0.781358
$764$	−10.3703	−0.375185
$765$	7.73440	0.279638
$766$	23.9931	0.866904
$767$	−31.5084	−1.13770
$768$	−41.9593	−1.51408
$769$	2.51601	0.0907298	0.0453649	−	0.998970i	$-0.485555\pi$
$769$	2.51601	0.0907298	0.0453649	+	0.998970i	$0.485555\pi$
$770$	−6.42810	−0.231653
$771$	−21.2524	−0.765387
$772$	7.67613	0.276270
$773$	−44.8085	−1.61165	−0.805824	−	0.592155i	$-0.798277\pi$
$773$	−44.8085	−1.61165	−0.805824	+	0.592155i	$0.798277\pi$
$774$	0	0
$775$	4.13665	0.148593
$776$	2.91893	0.104784
$777$	32.8277	1.17769
$778$	23.8391	0.854672
$779$	65.3322	2.34077
$780$	−5.37162	−0.192335
$781$	−3.34878	−0.119829
$782$	−1.86531	−0.0667032
$783$	−3.31206	−0.118363
$784$	−5.34187	−0.190781
$785$	−6.71792	−0.239773
$786$	28.2864	1.00894
$787$	2.59659	0.0925583	0.0462792	−	0.998929i	$-0.485264\pi$
$787$	2.59659	0.0925583	0.0462792	+	0.998929i	$0.485264\pi$
$788$	19.6814	0.701122
$789$	15.4476	0.549948
$790$	−0.523213	−0.0186151
$791$	−5.62315	−0.199936
$792$	−38.1357	−1.35509
$793$	−24.5922	−0.873296
$794$	9.01836	0.320050
$795$	−6.02228	−0.213588
$796$	8.97410	0.318078
$797$	23.6408	0.837400	0.418700	−	0.908124i	$-0.362486\pi$
$797$	23.6408	0.837400	0.418700	+	0.908124i	$0.362486\pi$
$798$	52.8306	1.87018
$799$	−16.0241	−0.566892
$800$	22.1298	0.782407
$801$	−6.20518	−0.219249
$802$	−35.3398	−1.24789
$803$	3.59828	0.126980
$804$	−24.1991	−0.853435
$805$	0.642814	0.0226562
$806$	−4.26433	−0.150204
$807$	5.79168	0.203877
$808$	49.9522	1.75731
$809$	40.0359	1.40759	0.703793	−	0.710405i	$-0.251488\pi$
$809$	40.0359	1.40759	0.703793	+	0.710405i	$0.251488\pi$
$810$	−4.04623	−0.142170
$811$	−9.32389	−0.327406	−0.163703	−	0.986510i	$-0.552344\pi$
$811$	−9.32389	−0.327406	−0.163703	+	0.986510i	$0.552344\pi$
$812$	10.6189	0.372650
$813$	69.1533	2.42531
$814$	15.2976	0.536182
$815$	−9.36107	−0.327904
$816$	15.7323	0.550741
$817$	0	0
$818$	26.8057	0.937238
$819$	−52.2476	−1.82568
$820$	−4.89633	−0.170987
$821$	−1.87089	−0.0652944	−0.0326472	−	0.999467i	$-0.510394\pi$
$821$	−1.87089	−0.0652944	−0.0326472	+	0.999467i	$0.510394\pi$
$822$	21.3231	0.743727
$823$	7.72253	0.269191	0.134595	−	0.990901i	$-0.457027\pi$
$823$	7.72253	0.269191	0.134595	+	0.990901i	$0.457027\pi$
$824$	−1.73851	−0.0605638
$825$	44.6361	1.55403
$826$	−23.2247	−0.808092
$827$	10.8269	0.376490	0.188245	−	0.982122i	$-0.439720\pi$
$827$	10.8269	0.376490	0.188245	+	0.982122i	$0.439720\pi$
$828$	1.19117	0.0413960
$829$	−16.7060	−0.580223	−0.290111	−	0.956993i	$-0.593692\pi$
$829$	−16.7060	−0.580223	−0.290111	+	0.956993i	$0.593692\pi$
$830$	−1.78375	−0.0619148
$831$	−67.1380	−2.32899
$832$	−35.5409	−1.23216
$833$	−18.0588	−0.625700
$834$	−33.9282	−1.17484
$835$	6.92497	0.239648
$836$	−20.4893	−0.708636
$837$	−0.815922	−0.0282024
$838$	25.9442	0.896226
$839$	2.28441	0.0788666	0.0394333	−	0.999222i	$-0.487445\pi$
$839$	2.28441	0.0788666	0.0394333	+	0.999222i	$0.487445\pi$
$840$	−12.6762	−0.437371
$841$	−16.5039	−0.569101
$842$	−20.1584	−0.694704
$843$	42.0942	1.44980
$844$	13.8992	0.478429
$845$	−4.48203	−0.154187
$846$	−12.2953	−0.422723
$847$	9.45356	0.324828
$848$	−6.48172	−0.222583
$849$	43.1817	1.48199
$850$	−22.7810	−0.781381
$851$	−1.52977	−0.0524400
$852$	−2.06268	−0.0706663
$853$	14.0427	0.480811	0.240406	−	0.970673i	$-0.422720\pi$
$853$	14.0427	0.480811	0.240406	+	0.970673i	$0.422720\pi$
$854$	−18.1268	−0.620288
$855$	10.2082	0.349113
$856$	−49.1422	−1.67965
$857$	−40.2306	−1.37425	−0.687126	−	0.726538i	$-0.741128\pi$
$857$	−40.2306	−1.37425	−0.687126	+	0.726538i	$0.741128\pi$
$858$	−46.0137	−1.57088
$859$	−34.4229	−1.17449	−0.587247	−	0.809407i	$-0.699788\pi$
$859$	−34.4229	−1.17449	−0.587247	+	0.809407i	$0.699788\pi$
$860$	0	0
$861$	−90.0055	−3.06738
$862$	−33.8791	−1.15393
$863$	20.2575	0.689575	0.344787	−	0.938681i	$-0.387951\pi$
$863$	20.2575	0.689575	0.344787	+	0.938681i	$0.387951\pi$
$864$	−4.36493	−0.148498
$865$	6.93261	0.235716
$866$	17.9320	0.609355
$867$	10.2748	0.348951
$868$	2.61596	0.0887914
$869$	3.73008	0.126534
$870$	−4.65933	−0.157966
$871$	−49.4627	−1.67598
$872$	−19.8337	−0.671653
$873$	3.23835	0.109602
$874$	−2.46191	−0.0832754
$875$	16.1142	0.544760
$876$	2.21636	0.0748839
$877$	−43.7768	−1.47824	−0.739119	−	0.673575i	$-0.764758\pi$
$877$	−43.7768	−1.47824	−0.739119	+	0.673575i	$0.764758\pi$
$878$	−15.0764	−0.508803
$879$	−75.5020	−2.54662
$880$	−2.52646	−0.0851668
$881$	3.24793	0.109426	0.0547128	−	0.998502i	$-0.482576\pi$
$881$	3.24793	0.109426	0.0547128	+	0.998502i	$0.482576\pi$
$882$	−13.8566	−0.466575
$883$	−15.3073	−0.515131	−0.257565	−	0.966261i	$-0.582920\pi$
$883$	−15.3073	−0.515131	−0.257565	+	0.966261i	$0.582920\pi$
$884$	−19.5447	−0.657361
$885$	−8.48107	−0.285088
$886$	−19.8574	−0.667124
$887$	−10.3041	−0.345977	−0.172988	−	0.984924i	$-0.555342\pi$
$887$	−10.3041	−0.345977	−0.172988	+	0.984924i	$0.555342\pi$
$888$	30.1670	1.01234
$889$	51.9009	1.74070
$890$	−0.961162	−0.0322182
$891$	28.8463	0.966387
$892$	−10.6440	−0.356388
$893$	−21.1493	−0.707734
$894$	19.6205	0.656207
$895$	−3.09101	−0.103321
$896$	4.61274	0.154101
$897$	4.60140	0.153636
$898$	−5.56986	−0.185868
$899$	3.07840	0.102670
$900$	14.5477	0.484924
$901$	−21.9122	−0.730001
$902$	−41.9424	−1.39653
$903$	0	0
$904$	−5.16739	−0.171865
$905$	−1.85067	−0.0615183
$906$	12.5939	0.418404
$907$	−44.5746	−1.48008	−0.740038	−	0.672565i	$-0.765192\pi$
$907$	−44.5746	−1.48008	−0.740038	+	0.672565i	$0.765192\pi$
$908$	−8.21215	−0.272530
$909$	55.4185	1.83811
$910$	−8.09298	−0.268280
$911$	9.12024	0.302167	0.151084	−	0.988521i	$-0.451724\pi$
$911$	9.12024	0.302167	0.151084	+	0.988521i	$0.451724\pi$
$912$	20.7642	0.687571
$913$	12.7166	0.420859
$914$	−8.30843	−0.274819
$915$	−6.61945	−0.218832
$916$	14.1698	0.468184
$917$	−35.4681	−1.17126
$918$	4.49337	0.148303
$919$	−27.4060	−0.904039	−0.452020	−	0.892008i	$-0.649296\pi$
$919$	−27.4060	−0.904039	−0.452020	+	0.892008i	$0.649296\pi$
$920$	0.590713	0.0194752
$921$	−33.9317	−1.11809
$922$	−44.5062	−1.46574
$923$	−4.21611	−0.138775
$924$	28.2272	0.928607
$925$	−18.6831	−0.614296
$926$	5.94782	0.195458
$927$	−1.92875	−0.0633486
$928$	16.4685	0.540604
$929$	8.08472	0.265251	0.132625	−	0.991166i	$-0.457659\pi$
$929$	8.08472	0.265251	0.132625	+	0.991166i	$0.457659\pi$
$930$	−1.14782	−0.0376385
$931$	−23.8348	−0.781153
$932$	4.26714	0.139775
$933$	38.3415	1.25524
$934$	7.71825	0.252549
$935$	−8.54098	−0.279320
$936$	−48.0129	−1.56935
$937$	37.2600	1.21723	0.608615	−	0.793465i	$-0.291725\pi$
$937$	37.2600	1.21723	0.608615	+	0.793465i	$0.291725\pi$
$938$	−36.4588	−1.19042
$939$	28.7548	0.938376
$940$	1.58504	0.0516982
$941$	36.3058	1.18354	0.591768	−	0.806108i	$-0.298430\pi$
$941$	36.3058	1.18354	0.591768	+	0.806108i	$0.298430\pi$
$942$	−35.4457	−1.15488
$943$	4.19426	0.136584
$944$	−9.12809	−0.297094
$945$	−1.54848	−0.0503722
$946$	0	0
$947$	46.5566	1.51288	0.756442	−	0.654060i	$-0.226936\pi$
$947$	46.5566	1.51288	0.756442	+	0.654060i	$0.226936\pi$
$948$	2.29754	0.0746207
$949$	4.53023	0.147057
$950$	−30.0673	−0.975512
$951$	−57.5980	−1.86774
$952$	−46.1227	−1.49485
$953$	31.4936	1.02018	0.510089	−	0.860122i	$-0.329612\pi$
$953$	31.4936	1.02018	0.510089	+	0.860122i	$0.329612\pi$
$954$	−16.8133	−0.544350
$955$	−5.70552	−0.184626
$956$	3.73796	0.120894
$957$	33.2171	1.07376
$958$	−21.6353	−0.699004
$959$	−26.7368	−0.863376
$960$	−9.56649	−0.308757
$961$	−30.2416	−0.975537
$962$	19.2597	0.620959
$963$	−54.5199	−1.75688
$964$	−2.96550	−0.0955124
$965$	4.22324	0.135951
$966$	3.39167	0.109125
$967$	−1.18018	−0.0379519	−0.0189760	−	0.999820i	$-0.506041\pi$
$967$	−1.18018	−0.0379519	−0.0189760	+	0.999820i	$0.506041\pi$
$968$	8.68733	0.279221
$969$	70.1957	2.25501
$970$	0.501610	0.0161057
$971$	−39.8983	−1.28040	−0.640199	−	0.768209i	$-0.721148\pi$
$971$	−39.8983	−1.28040	−0.640199	+	0.768209i	$0.721148\pi$
$972$	20.3214	0.651809
$973$	42.5422	1.36384
$974$	−37.1946	−1.19179
$975$	56.1968	1.79974
$976$	−7.12445	−0.228048
$977$	−31.6088	−1.01125	−0.505627	−	0.862752i	$-0.668739\pi$
$977$	−31.6088	−1.01125	−0.505627	+	0.862752i	$0.668739\pi$
$978$	−49.3917	−1.57937
$979$	6.85229	0.219000
$980$	1.78630	0.0570612
$981$	−22.0041	−0.702537
$982$	−0.695748	−0.0222022
$983$	−24.3164	−0.775573	−0.387787	−	0.921749i	$-0.626760\pi$
$983$	−24.3164	−0.775573	−0.387787	+	0.921749i	$0.626760\pi$
$984$	−82.7104	−2.63671
$985$	10.8283	0.345018
$986$	−16.9531	−0.539895
$987$	29.1365	0.927425
$988$	−25.7960	−0.820679
$989$	0	0
$990$	−6.55351	−0.208284
$991$	−3.22998	−0.102604	−0.0513019	−	0.998683i	$-0.516337\pi$
$991$	−3.22998	−0.102604	−0.0513019	+	0.998683i	$0.516337\pi$
$992$	4.05700	0.128810
$993$	39.2520	1.24562
$994$	−3.10768	−0.0985695
$995$	4.93735	0.156525
$996$	7.83283	0.248193
$997$	−14.7265	−0.466394	−0.233197	−	0.972430i	$-0.574919\pi$
$997$	−14.7265	−0.466394	−0.233197	+	0.972430i	$0.574919\pi$
$998$	7.51791	0.237975
$999$	3.68509	0.116591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.n.1.8		18
43.25	even	21		43.2.g.a.23.2	yes	36
43.31	even	21		43.2.g.a.15.2	✓	36
43.42	odd	2		1849.2.a.o.1.11		18
129.68	odd	42		387.2.y.c.109.2		36
129.74	odd	42		387.2.y.c.316.2		36
172.31	odd	42		688.2.bg.c.273.1		36
172.111	odd	42		688.2.bg.c.625.1		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.g.a.15.2	✓	36	43.31	even	21
43.2.g.a.23.2	yes	36	43.25	even	21
387.2.y.c.109.2		36	129.68	odd	42
387.2.y.c.316.2		36	129.74	odd	42
688.2.bg.c.273.1		36	172.31	odd	42
688.2.bg.c.625.1		36	172.111	odd	42
1849.2.a.n.1.8		18	1.1	even	1	trivial
1849.2.a.o.1.11		18	43.42	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.04477 q^{2} +2.52412 q^{3} -0.908449 q^{4} -0.499809 q^{5} -2.63714 q^{6} +3.30668 q^{7} +3.03867 q^{8} +3.37119 q^{9} +O(q^{10})\)	\(q-1.04477 q^{2} +2.52412 q^{3} -0.908449 q^{4} -0.499809 q^{5} -2.63714 q^{6} +3.30668 q^{7} +3.03867 q^{8} +3.37119 q^{9} +0.522187 q^{10} -3.72276 q^{11} -2.29304 q^{12} -4.68695 q^{13} -3.45473 q^{14} -1.26158 q^{15} -1.35782 q^{16} -4.59028 q^{17} -3.52213 q^{18} -6.05844 q^{19} +0.454051 q^{20} +8.34647 q^{21} +3.88944 q^{22} -0.388946 q^{23} +7.66997 q^{24} -4.75019 q^{25} +4.89680 q^{26} +0.936938 q^{27} -3.00395 q^{28} -3.53498 q^{29} +1.31806 q^{30} -0.870839 q^{31} -4.65872 q^{32} -9.39670 q^{33} +4.79580 q^{34} -1.65271 q^{35} -3.06256 q^{36} +3.93313 q^{37} +6.32970 q^{38} -11.8304 q^{39} -1.51875 q^{40} -10.7837 q^{41} -8.72017 q^{42} +3.38193 q^{44} -1.68495 q^{45} +0.406360 q^{46} +3.49088 q^{47} -3.42731 q^{48} +3.93414 q^{49} +4.96287 q^{50} -11.5864 q^{51} +4.25785 q^{52} +4.77361 q^{53} -0.978888 q^{54} +1.86067 q^{55} +10.0479 q^{56} -15.2923 q^{57} +3.69325 q^{58} +6.72259 q^{59} +1.14608 q^{60} +5.24696 q^{61} +0.909829 q^{62} +11.1475 q^{63} +7.58295 q^{64} +2.34258 q^{65} +9.81742 q^{66} +10.5533 q^{67} +4.17003 q^{68} -0.981747 q^{69} +1.72670 q^{70} +0.899542 q^{71} +10.2439 q^{72} -0.966562 q^{73} -4.10922 q^{74} -11.9901 q^{75} +5.50379 q^{76} -12.3100 q^{77} +12.3601 q^{78} -1.00197 q^{79} +0.678652 q^{80} -7.74863 q^{81} +11.2665 q^{82} -3.41592 q^{83} -7.58234 q^{84} +2.29426 q^{85} -8.92272 q^{87} -11.3122 q^{88} -1.84065 q^{89} +1.76039 q^{90} -15.4982 q^{91} +0.353338 q^{92} -2.19810 q^{93} -3.64718 q^{94} +3.02806 q^{95} -11.7592 q^{96} +0.960596 q^{97} -4.11028 q^{98} -12.5501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9}+O(q^{10}) \) 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9	\( 18 q - 5 q^{2} - 5 q^{3} + 19 q^{4} - 11 q^{5} + 4 q^{6} - 6 q^{7} - 12 q^{8} + 15 q^{9} - 7 q^{10} - 2 q^{11} - 16 q^{12} - 7 q^{13} - 4 q^{14} + 3 q^{15} + 9 q^{16} - 11 q^{17} - 25 q^{18} - 31 q^{19} - 25 q^{20} - 19 q^{22} - 11 q^{23} + 18 q^{24} + 9 q^{25} - 27 q^{26} - 23 q^{27} - 20 q^{28} - 37 q^{29} + 17 q^{30} - 12 q^{31} - 39 q^{32} - 38 q^{33} - 14 q^{34} + 16 q^{35} + 47 q^{36} + 19 q^{37} + 56 q^{38} - 46 q^{39} + 6 q^{40} - 7 q^{41} + q^{42} + 7 q^{44} - 23 q^{45} + 47 q^{46} - q^{47} - 15 q^{48} - 6 q^{49} + 3 q^{50} - 38 q^{51} + 15 q^{52} + 3 q^{53} - 67 q^{54} - 28 q^{55} - 81 q^{56} + 46 q^{57} + 34 q^{58} + 17 q^{59} - 83 q^{60} - 28 q^{61} - 33 q^{62} - 26 q^{63} + 10 q^{64} - 16 q^{65} - 72 q^{66} + 18 q^{67} + 53 q^{68} - 7 q^{69} + 34 q^{70} - 86 q^{71} + 2 q^{72} + 27 q^{73} - 79 q^{74} - 31 q^{75} - 59 q^{76} - 43 q^{77} + 91 q^{78} + 17 q^{79} - 8 q^{80} - 10 q^{81} - 13 q^{82} - 12 q^{83} - 32 q^{84} - 28 q^{85} - 43 q^{87} + 23 q^{88} - 51 q^{89} + 10 q^{90} + 20 q^{91} + 18 q^{92} + 30 q^{93} + 15 q^{94} - q^{95} - 20 q^{96} - 19 q^{97} + 5 q^{98} + 38 q^{99}+O(q^{100}) \) 18 * q - 5 * q^2 - 5 * q^3 + 19 * q^4 - 11 * q^5 + 4 * q^6 - 6 * q^7 - 12 * q^8 + 15 * q^9 - 7 * q^10 - 2 * q^11 - 16 * q^12 - 7 * q^13 - 4 * q^14 + 3 * q^15 + 9 * q^16 - 11 * q^17 - 25 * q^18 - 31 * q^19 - 25 * q^20 - 19 * q^22 - 11 * q^23 + 18 * q^24 + 9 * q^25 - 27 * q^26 - 23 * q^27 - 20 * q^28 - 37 * q^29 + 17 * q^30 - 12 * q^31 - 39 * q^32 - 38 * q^33 - 14 * q^34 + 16 * q^35 + 47 * q^36 + 19 * q^37 + 56 * q^38 - 46 * q^39 + 6 * q^40 - 7 * q^41 + q^42 + 7 * q^44 - 23 * q^45 + 47 * q^46 - q^47 - 15 * q^48 - 6 * q^49 + 3 * q^50 - 38 * q^51 + 15 * q^52 + 3 * q^53 - 67 * q^54 - 28 * q^55 - 81 * q^56 + 46 * q^57 + 34 * q^58 + 17 * q^59 - 83 * q^60 - 28 * q^61 - 33 * q^62 - 26 * q^63 + 10 * q^64 - 16 * q^65 - 72 * q^66 + 18 * q^67 + 53 * q^68 - 7 * q^69 + 34 * q^70 - 86 * q^71 + 2 * q^72 + 27 * q^73 - 79 * q^74 - 31 * q^75 - 59 * q^76 - 43 * q^77 + 91 * q^78 + 17 * q^79 - 8 * q^80 - 10 * q^81 - 13 * q^82 - 12 * q^83 - 32 * q^84 - 28 * q^85 - 43 * q^87 + 23 * q^88 - 51 * q^89 + 10 * q^90 + 20 * q^91 + 18 * q^92 + 30 * q^93 + 15 * q^94 - q^95 - 20 * q^96 - 19 * q^97 + 5 * q^98 + 38 * q^99

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