LMFDB - Embedded newform 1849.2.a.p.1.2

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			1.2
Root			$2.54512$ of defining polynomial
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54512 q^{2} -2.55827 q^{3} +4.47761 q^{4} +1.89319 q^{5} +6.51109 q^{6} -1.19543 q^{7} -6.30581 q^{8} +3.54474 q^{9} +O(q^{10})$	\(q-2.54512 q^{2} -2.55827 q^{3} +4.47761 q^{4} +1.89319 q^{5} +6.51109 q^{6} -1.19543 q^{7} -6.30581 q^{8} +3.54474 q^{9} -4.81839 q^{10} +3.87615 q^{11} -11.4549 q^{12} -1.01508 q^{13} +3.04250 q^{14} -4.84329 q^{15} +7.09380 q^{16} -0.458716 q^{17} -9.02177 q^{18} +6.22425 q^{19} +8.47699 q^{20} +3.05822 q^{21} -9.86526 q^{22} -4.49025 q^{23} +16.1320 q^{24} -1.41582 q^{25} +2.58349 q^{26} -1.39359 q^{27} -5.35266 q^{28} +6.40682 q^{29} +12.3267 q^{30} +9.21695 q^{31} -5.44291 q^{32} -9.91624 q^{33} +1.16748 q^{34} -2.26317 q^{35} +15.8720 q^{36} +1.21329 q^{37} -15.8414 q^{38} +2.59684 q^{39} -11.9381 q^{40} -0.151151 q^{41} -7.78353 q^{42} +17.3559 q^{44} +6.71087 q^{45} +11.4282 q^{46} +8.19536 q^{47} -18.1478 q^{48} -5.57096 q^{49} +3.60343 q^{50} +1.17352 q^{51} -4.54513 q^{52} +7.06109 q^{53} +3.54684 q^{54} +7.33830 q^{55} +7.53814 q^{56} -15.9233 q^{57} -16.3061 q^{58} -10.2036 q^{59} -21.6864 q^{60} -6.13231 q^{61} -23.4582 q^{62} -4.23747 q^{63} -0.334759 q^{64} -1.92174 q^{65} +25.2380 q^{66} +7.17972 q^{67} -2.05395 q^{68} +11.4873 q^{69} +5.76004 q^{70} -5.69699 q^{71} -22.3525 q^{72} -10.9903 q^{73} -3.08795 q^{74} +3.62205 q^{75} +27.8698 q^{76} -4.63366 q^{77} -6.60926 q^{78} -15.3276 q^{79} +13.4299 q^{80} -7.06905 q^{81} +0.384696 q^{82} +8.73939 q^{83} +13.6935 q^{84} -0.868437 q^{85} -16.3904 q^{87} -24.4423 q^{88} -1.48243 q^{89} -17.0799 q^{90} +1.21345 q^{91} -20.1056 q^{92} -23.5794 q^{93} -20.8581 q^{94} +11.7837 q^{95} +13.9244 q^{96} -4.51129 q^{97} +14.1787 q^{98} +13.7399 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.54512	−1.79967	−0.899834	−	0.436232i	$-0.856313\pi$
$2$	−2.54512	−1.79967	−0.899834	+	0.436232i	$0.856313\pi$
$3$	−2.55827	−1.47702	−0.738509	−	0.674244i	$-0.764470\pi$
$3$	−2.55827	−1.47702	−0.738509	+	0.674244i	$0.764470\pi$
$4$	4.47761	2.23881
$5$	1.89319	0.846661	0.423331	−	0.905975i	$-0.360861\pi$
$5$	1.89319	0.846661	0.423331	+	0.905975i	$0.360861\pi$
$6$	6.51109	2.65814
$7$	−1.19543	−0.451829	−0.225914	−	0.974147i	$-0.572537\pi$
$7$	−1.19543	−0.451829	−0.225914	+	0.974147i	$0.572537\pi$
$8$	−6.30581	−2.22944
$9$	3.54474	1.18158
$10$	−4.81839	−1.52371
$11$	3.87615	1.16870	0.584352	−	0.811500i	$-0.301349\pi$
$11$	3.87615	1.16870	0.584352	+	0.811500i	$0.301349\pi$
$12$	−11.4549	−3.30676
$13$	−1.01508	−0.281532	−0.140766	−	0.990043i	$-0.544957\pi$
$13$	−1.01508	−0.281532	−0.140766	+	0.990043i	$0.544957\pi$
$14$	3.04250	0.813142
$15$	−4.84329	−1.25053
$16$	7.09380	1.77345
$17$	−0.458716	−0.111255	−0.0556275	−	0.998452i	$-0.517716\pi$
$17$	−0.458716	−0.111255	−0.0556275	+	0.998452i	$0.517716\pi$
$18$	−9.02177	−2.12645
$19$	6.22425	1.42794	0.713970	−	0.700176i	$-0.246895\pi$
$19$	6.22425	1.42794	0.713970	+	0.700176i	$0.246895\pi$
$20$	8.47699	1.89551
$21$	3.05822	0.667359
$22$	−9.86526	−2.10328
$23$	−4.49025	−0.936283	−0.468141	−	0.883654i	$-0.655076\pi$
$23$	−4.49025	−0.936283	−0.468141	+	0.883654i	$0.655076\pi$
$24$	16.1320	3.29292
$25$	−1.41582	−0.283164
$26$	2.58349	0.506664
$27$	−1.39359	−0.268196
$28$	−5.35266	−1.01156
$29$	6.40682	1.18972	0.594858	−	0.803831i	$-0.297208\pi$
$29$	6.40682	1.18972	0.594858	+	0.803831i	$0.297208\pi$
$30$	12.3267	2.25055
$31$	9.21695	1.65541	0.827707	−	0.561161i	$-0.189645\pi$
$31$	9.21695	1.65541	0.827707	+	0.561161i	$0.189645\pi$
$32$	−5.44291	−0.962180
$33$	−9.91624	−1.72620
$34$	1.16748	0.200222
$35$	−2.26317	−0.382546
$36$	15.8720	2.64533
$37$	1.21329	0.199463	0.0997315	−	0.995014i	$-0.468202\pi$
$37$	1.21329	0.199463	0.0997315	+	0.995014i	$0.468202\pi$
$38$	−15.8414	−2.56982
$39$	2.59684	0.415828
$40$	−11.9381	−1.88758
$41$	−0.151151	−0.0236058	−0.0118029	−	0.999930i	$-0.503757\pi$
$41$	−0.151151	−0.0236058	−0.0118029	+	0.999930i	$0.503757\pi$
$42$	−7.78353	−1.20102
$43$	0	0
$43$	0	0
$44$	17.3559	2.61650
$45$	6.71087	1.00040
$46$	11.4282	1.68500
$47$	8.19536	1.19542	0.597708	−	0.801714i	$-0.296078\pi$
$47$	8.19536	1.19542	0.597708	+	0.801714i	$0.296078\pi$
$48$	−18.1478	−2.61942
$49$	−5.57096	−0.795851
$50$	3.60343	0.509602
$51$	1.17352	0.164325
$52$	−4.54513	−0.630296
$53$	7.06109	0.969915	0.484958	−	0.874538i	$-0.338835\pi$
$53$	7.06109	0.969915	0.484958	+	0.874538i	$0.338835\pi$
$54$	3.54684	0.482664
$55$	7.33830	0.989497
$56$	7.53814	1.00733
$57$	−15.9233	−2.10909
$58$	−16.3061	−2.14109
$59$	−10.2036	−1.32840	−0.664199	−	0.747556i	$-0.731227\pi$
$59$	−10.2036	−1.32840	−0.664199	+	0.747556i	$0.731227\pi$
$60$	−21.6864	−2.79970
$61$	−6.13231	−0.785162	−0.392581	−	0.919717i	$-0.628418\pi$
$61$	−6.13231	−0.785162	−0.392581	+	0.919717i	$0.628418\pi$
$62$	−23.4582	−2.97920
$63$	−4.23747	−0.533872
$64$	−0.334759	−0.0418448
$65$	−1.92174	−0.238362
$66$	25.2380	3.10658
$67$	7.17972	0.877142	0.438571	−	0.898697i	$-0.355485\pi$
$67$	7.17972	0.877142	0.438571	+	0.898697i	$0.355485\pi$
$68$	−2.05395	−0.249078
$69$	11.4873	1.38291
$70$	5.76004	0.688456
$71$	−5.69699	−0.676109	−0.338054	−	0.941127i	$-0.609769\pi$
$71$	−5.69699	−0.676109	−0.338054	+	0.941127i	$0.609769\pi$
$72$	−22.3525	−2.63426
$73$	−10.9903	−1.28632	−0.643159	−	0.765733i	$-0.722377\pi$
$73$	−10.9903	−1.28632	−0.643159	+	0.765733i	$0.722377\pi$
$74$	−3.08795	−0.358967
$75$	3.62205	0.418239
$76$	27.8698	3.19688
$77$	−4.63366	−0.528054
$78$	−6.60926	−0.748352
$79$	−15.3276	−1.72449	−0.862246	−	0.506489i	$-0.830943\pi$
$79$	−15.3276	−1.72449	−0.862246	+	0.506489i	$0.830943\pi$
$80$	13.4299	1.50151
$81$	−7.06905	−0.785450
$82$	0.384696	0.0424826
$83$	8.73939	0.959273	0.479636	−	0.877467i	$-0.340769\pi$
$83$	8.73939	0.959273	0.479636	+	0.877467i	$0.340769\pi$
$84$	13.6935	1.49409
$85$	−0.868437	−0.0941953
$86$	0	0
$87$	−16.3904	−1.75723
$88$	−24.4423	−2.60556
$89$	−1.48243	−0.157137	−0.0785685	−	0.996909i	$-0.525035\pi$
$89$	−1.48243	−0.157137	−0.0785685	+	0.996909i	$0.525035\pi$
$90$	−17.0799	−1.80038
$91$	1.21345	0.127204
$92$	−20.1056	−2.09616
$93$	−23.5794	−2.44507
$94$	−20.8581	−2.15135
$95$	11.7837	1.20898
$96$	13.9244	1.42116
$97$	−4.51129	−0.458053	−0.229026	−	0.973420i	$-0.573554\pi$
$97$	−4.51129	−0.458053	−0.229026	+	0.973420i	$0.573554\pi$
$98$	14.1787	1.43227
$99$	13.7399	1.38092
$100$	−6.33950	−0.633950
$101$	2.72636	0.271283	0.135641	−	0.990758i	$-0.456691\pi$
$101$	2.72636	0.271283	0.135641	+	0.990758i	$0.456691\pi$
$102$	−2.98674	−0.295731
$103$	6.81242	0.671248	0.335624	−	0.941996i	$-0.391053\pi$
$103$	6.81242	0.671248	0.335624	+	0.941996i	$0.391053\pi$
$104$	6.40089	0.627659
$105$	5.78980	0.565027
$106$	−17.9713	−1.74553
$107$	12.2697	1.18616	0.593079	−	0.805145i	$-0.297912\pi$
$107$	12.2697	1.18616	0.593079	+	0.805145i	$0.297912\pi$
$108$	−6.23994	−0.600439
$109$	−2.87348	−0.275230	−0.137615	−	0.990486i	$-0.543944\pi$
$109$	−2.87348	−0.275230	−0.137615	+	0.990486i	$0.543944\pi$
$110$	−18.6768	−1.78077
$111$	−3.10391	−0.294610
$112$	−8.48012	−0.801296
$113$	4.04603	0.380619	0.190309	−	0.981724i	$-0.439051\pi$
$113$	4.04603	0.380619	0.190309	+	0.981724i	$0.439051\pi$
$114$	40.5266	3.79567
$115$	−8.50091	−0.792714
$116$	28.6873	2.66354
$117$	−3.59819	−0.332652
$118$	25.9694	2.39067
$119$	0.548361	0.0502682
$120$	30.5409	2.78799
$121$	4.02456	0.365870
$122$	15.6074	1.41303
$123$	0.386684	0.0348661
$124$	41.2700	3.70615
$125$	−12.1464	−1.08641
$126$	10.7849	0.960792
$127$	−6.83166	−0.606212	−0.303106	−	0.952957i	$-0.598024\pi$
$127$	−6.83166	−0.606212	−0.303106	+	0.952957i	$0.598024\pi$
$128$	11.7378	1.03749
$129$	0	0
$130$	4.89105	0.428973
$131$	8.30225	0.725371	0.362685	−	0.931912i	$-0.381860\pi$
$131$	8.30225	0.725371	0.362685	+	0.931912i	$0.381860\pi$
$132$	−44.4011	−3.86462
$133$	−7.44063	−0.645185
$134$	−18.2732	−1.57856
$135$	−2.63833	−0.227071
$136$	2.89258	0.248036
$137$	16.1144	1.37675	0.688374	−	0.725356i	$-0.258325\pi$
$137$	16.1144	1.37675	0.688374	+	0.725356i	$0.258325\pi$
$138$	−29.2364	−2.48877
$139$	−0.387060	−0.0328300	−0.0164150	−	0.999865i	$-0.505225\pi$
$139$	−0.387060	−0.0328300	−0.0164150	+	0.999865i	$0.505225\pi$
$140$	−10.1336	−0.856447
$141$	−20.9659	−1.76565
$142$	14.4995	1.21677
$143$	−3.93460	−0.329028
$144$	25.1457	2.09547
$145$	12.1293	1.00729
$146$	27.9716	2.31495
$147$	14.2520	1.17549
$148$	5.43263	0.446559
$149$	16.3084	1.33604	0.668018	−	0.744145i	$-0.267143\pi$
$149$	16.3084	1.33604	0.668018	+	0.744145i	$0.267143\pi$
$150$	−9.21854	−0.752691
$151$	0.616437	0.0501649	0.0250825	−	0.999685i	$-0.492015\pi$
$151$	0.616437	0.0501649	0.0250825	+	0.999685i	$0.492015\pi$
$152$	−39.2490	−3.18351
$153$	−1.62603	−0.131457
$154$	11.7932	0.950323
$155$	17.4495	1.40157
$156$	11.6277	0.930958
$157$	−14.7103	−1.17401	−0.587004	−	0.809584i	$-0.699693\pi$
$157$	−14.7103	−1.17401	−0.587004	+	0.809584i	$0.699693\pi$
$158$	39.0106	3.10352
$159$	−18.0642	−1.43258
$160$	−10.3045	−0.814641
$161$	5.36777	0.423039
$162$	17.9915	1.41355
$163$	4.36778	0.342111	0.171056	−	0.985261i	$-0.445282\pi$
$163$	4.36778	0.342111	0.171056	+	0.985261i	$0.445282\pi$
$164$	−0.676794	−0.0528488
$165$	−18.7734	−1.46150
$166$	−22.2428	−1.72637
$167$	14.5751	1.12785	0.563927	−	0.825825i	$-0.309290\pi$
$167$	14.5751	1.12785	0.563927	+	0.825825i	$0.309290\pi$
$168$	−19.2846	−1.48784
$169$	−11.9696	−0.920740
$170$	2.21027	0.169520
$171$	22.0633	1.68723
$172$	0	0
$173$	−4.82163	−0.366581	−0.183291	−	0.983059i	$-0.558675\pi$
$173$	−4.82163	−0.366581	−0.183291	+	0.983059i	$0.558675\pi$
$174$	41.7154	3.16243
$175$	1.69251	0.127942
$176$	27.4967	2.07264
$177$	26.1036	1.96207
$178$	3.77295	0.282794
$179$	23.0979	1.72642	0.863208	−	0.504848i	$-0.168452\pi$
$179$	23.0979	1.72642	0.863208	+	0.504848i	$0.168452\pi$
$180$	30.0487	2.23970
$181$	−5.81928	−0.432544	−0.216272	−	0.976333i	$-0.569390\pi$
$181$	−5.81928	−0.432544	−0.216272	+	0.976333i	$0.569390\pi$
$182$	−3.08837	−0.228926
$183$	15.6881	1.15970
$184$	28.3147	2.08739
$185$	2.29698	0.168878
$186$	60.0124	4.40032
$187$	−1.77805	−0.130024
$188$	36.6957	2.67631
$189$	1.66593	0.121179
$190$	−29.9909	−2.17577
$191$	−9.30601	−0.673359	−0.336680	−	0.941619i	$-0.609304\pi$
$191$	−9.30601	−0.673359	−0.336680	+	0.941619i	$0.609304\pi$
$192$	0.856402	0.0618055
$193$	21.8426	1.57226	0.786131	−	0.618059i	$-0.212081\pi$
$193$	21.8426	1.57226	0.786131	+	0.618059i	$0.212081\pi$
$194$	11.4818	0.824343
$195$	4.91632	0.352065
$196$	−24.9446	−1.78176
$197$	7.76057	0.552918	0.276459	−	0.961026i	$-0.410839\pi$
$197$	7.76057	0.552918	0.276459	+	0.961026i	$0.410839\pi$
$198$	−34.9698	−2.48519
$199$	−3.97547	−0.281814	−0.140907	−	0.990023i	$-0.545002\pi$
$199$	−3.97547	−0.281814	−0.140907	+	0.990023i	$0.545002\pi$
$200$	8.92791	0.631299
$201$	−18.3676	−1.29555
$202$	−6.93890	−0.488219
$203$	−7.65888	−0.537548
$204$	5.25456	0.367893
$205$	−0.286157	−0.0199861
$206$	−17.3384	−1.20802
$207$	−15.9168	−1.10629
$208$	−7.20076	−0.499283
$209$	24.1261	1.66884
$210$	−14.7357	−1.01686
$211$	25.1500	1.73139	0.865697	−	0.500568i	$-0.166876\pi$
$211$	25.1500	1.73139	0.865697	+	0.500568i	$0.166876\pi$
$212$	31.6168	2.17145
$213$	14.5744	0.998624
$214$	−31.2278	−2.13469
$215$	0	0
$216$	8.78770	0.597927
$217$	−11.0182	−0.747964
$218$	7.31334	0.495322
$219$	28.1161	1.89991
$220$	32.8581	2.21529
$221$	0.465632	0.0313218
$222$	7.89981	0.530201
$223$	16.6545	1.11527	0.557634	−	0.830087i	$-0.311709\pi$
$223$	16.6545	1.11527	0.557634	+	0.830087i	$0.311709\pi$
$224$	6.50660	0.434741
$225$	−5.01872	−0.334581
$226$	−10.2976	−0.684988
$227$	5.61979	0.372998	0.186499	−	0.982455i	$-0.440286\pi$
$227$	5.61979	0.372998	0.186499	+	0.982455i	$0.440286\pi$
$228$	−71.2984	−4.72185
$229$	12.3018	0.812923	0.406462	−	0.913668i	$-0.366762\pi$
$229$	12.3018	0.812923	0.406462	+	0.913668i	$0.366762\pi$
$230$	21.6358	1.42662
$231$	11.8541	0.779945
$232$	−40.4002	−2.65240
$233$	−23.0272	−1.50856	−0.754281	−	0.656552i	$-0.772014\pi$
$233$	−23.0272	−1.50856	−0.754281	+	0.656552i	$0.772014\pi$
$234$	9.15780	0.598664
$235$	15.5154	1.01211
$236$	−45.6878	−2.97402
$237$	39.2122	2.54711
$238$	−1.39564	−0.0904661
$239$	−1.09901	−0.0710889	−0.0355444	−	0.999368i	$-0.511317\pi$
$239$	−1.09901	−0.0710889	−0.0355444	+	0.999368i	$0.511317\pi$
$240$	−34.3574	−2.21776
$241$	−4.52273	−0.291334	−0.145667	−	0.989334i	$-0.546533\pi$
$241$	−4.52273	−0.291334	−0.145667	+	0.989334i	$0.546533\pi$
$242$	−10.2430	−0.658444
$243$	22.2653	1.42832
$244$	−27.4581	−1.75783
$245$	−10.5469	−0.673816
$246$	−0.984155	−0.0627475
$247$	−6.31810	−0.402011
$248$	−58.1204	−3.69065
$249$	−22.3577	−1.41686
$250$	30.9140	1.95517
$251$	−6.57285	−0.414875	−0.207437	−	0.978248i	$-0.566512\pi$
$251$	−6.57285	−0.414875	−0.207437	+	0.978248i	$0.566512\pi$
$252$	−18.9738	−1.19524
$253$	−17.4049	−1.09424
$254$	17.3874	1.09098
$255$	2.22170	0.139128
$256$	−29.2046	−1.82529
$257$	−1.09262	−0.0681555	−0.0340777	−	0.999419i	$-0.510849\pi$
$257$	−1.09262	−0.0681555	−0.0340777	+	0.999419i	$0.510849\pi$
$258$	0	0
$259$	−1.45039	−0.0901231
$260$	−8.60480	−0.533647
$261$	22.7105	1.40574
$262$	−21.1302	−1.30543
$263$	−9.47719	−0.584389	−0.292194	−	0.956359i	$-0.594385\pi$
$263$	−9.47719	−0.584389	−0.292194	+	0.956359i	$0.594385\pi$
$264$	62.5300	3.84845
$265$	13.3680	0.821190
$266$	18.9373	1.16112
$267$	3.79245	0.232094
$268$	32.1480	1.96375
$269$	4.04265	0.246484	0.123242	−	0.992377i	$-0.460671\pi$
$269$	4.04265	0.246484	0.123242	+	0.992377i	$0.460671\pi$
$270$	6.71485	0.408653
$271$	14.8056	0.899378	0.449689	−	0.893185i	$-0.351535\pi$
$271$	14.8056	0.899378	0.449689	+	0.893185i	$0.351535\pi$
$272$	−3.25404	−0.197305
$273$	−3.10433	−0.187883
$274$	−41.0131	−2.47769
$275$	−5.48794	−0.330935
$276$	51.4356	3.09606
$277$	15.1082	0.907763	0.453881	−	0.891062i	$-0.350039\pi$
$277$	15.1082	0.907763	0.453881	+	0.891062i	$0.350039\pi$
$278$	0.985112	0.0590831
$279$	32.6717	1.95600
$280$	14.2711	0.852864
$281$	17.6185	1.05103	0.525516	−	0.850784i	$-0.323872\pi$
$281$	17.6185	1.05103	0.525516	+	0.850784i	$0.323872\pi$
$282$	53.3607	3.17758
$283$	2.68332	0.159507	0.0797534	−	0.996815i	$-0.474587\pi$
$283$	2.68332	0.159507	0.0797534	+	0.996815i	$0.474587\pi$
$284$	−25.5089	−1.51368
$285$	−30.1459	−1.78569
$286$	10.0140	0.592141
$287$	0.180690	0.0106658
$288$	−19.2937	−1.13689
$289$	−16.7896	−0.987622
$290$	−30.8706	−1.81278
$291$	11.5411	0.676551
$292$	−49.2103	−2.87982
$293$	10.9704	0.640896	0.320448	−	0.947266i	$-0.396167\pi$
$293$	10.9704	0.640896	0.320448	+	0.947266i	$0.396167\pi$
$294$	−36.2730	−2.11548
$295$	−19.3174	−1.12470
$296$	−7.65076	−0.444691
$297$	−5.40175	−0.313442
$298$	−41.5068	−2.40442
$299$	4.55796	0.263594
$300$	16.2182	0.936356
$301$	0	0
$302$	−1.56890	−0.0902802
$303$	−6.97475	−0.400689
$304$	44.1536	2.53238
$305$	−11.6096	−0.664766
$306$	4.13843	0.236578
$307$	−12.8523	−0.733522	−0.366761	−	0.930315i	$-0.619533\pi$
$307$	−12.8523	−0.733522	−0.366761	+	0.930315i	$0.619533\pi$
$308$	−20.7477	−1.18221
$309$	−17.4280	−0.991445
$310$	−44.4109	−2.52237
$311$	−15.7397	−0.892516	−0.446258	−	0.894904i	$-0.647244\pi$
$311$	−15.7397	−0.892516	−0.446258	+	0.894904i	$0.647244\pi$
$312$	−16.3752	−0.927064
$313$	−32.0586	−1.81206	−0.906030	−	0.423213i	$-0.860902\pi$
$313$	−32.0586	−1.81206	−0.906030	+	0.423213i	$0.860902\pi$
$314$	37.4394	2.11283
$315$	−8.02235	−0.452008
$316$	−68.6312	−3.86081
$317$	21.2160	1.19161	0.595803	−	0.803130i	$-0.296834\pi$
$317$	21.2160	1.19161	0.595803	+	0.803130i	$0.296834\pi$
$318$	45.9754	2.57817
$319$	24.8338	1.39043
$320$	−0.633762	−0.0354284
$321$	−31.3892	−1.75197
$322$	−13.6616	−0.761331
$323$	−2.85516	−0.158865
$324$	−31.6525	−1.75847
$325$	1.43717	0.0797198
$326$	−11.1165	−0.615687
$327$	7.35114	0.406519
$328$	0.953128	0.0526277
$329$	−9.79695	−0.540123
$330$	47.7804	2.63022
$331$	4.79148	0.263364	0.131682	−	0.991292i	$-0.457962\pi$
$331$	4.79148	0.263364	0.131682	+	0.991292i	$0.457962\pi$
$332$	39.1316	2.14763
$333$	4.30078	0.235681
$334$	−37.0953	−2.02976
$335$	13.5926	0.742642
$336$	21.6944	1.18353
$337$	−3.51529	−0.191490	−0.0957449	−	0.995406i	$-0.530523\pi$
$337$	−3.51529	−0.191490	−0.0957449	+	0.995406i	$0.530523\pi$
$338$	30.4641	1.65703
$339$	−10.3508	−0.562180
$340$	−3.88853	−0.210885
$341$	35.7263	1.93469
$342$	−56.1537	−3.03645
$343$	15.0277	0.811417
$344$	0	0
$345$	21.7476	1.17085
$346$	12.2716	0.659725
$347$	−12.2413	−0.657145	−0.328573	−	0.944479i	$-0.606568\pi$
$347$	−12.2413	−0.657145	−0.328573	+	0.944479i	$0.606568\pi$
$348$	−73.3897	−3.93410
$349$	−9.11712	−0.488028	−0.244014	−	0.969772i	$-0.578464\pi$
$349$	−9.11712	−0.488028	−0.244014	+	0.969772i	$0.578464\pi$
$350$	−4.30764	−0.230253
$351$	1.41460	0.0755057
$352$	−21.0976	−1.12450
$353$	29.8078	1.58651	0.793254	−	0.608891i	$-0.208385\pi$
$353$	29.8078	1.58651	0.793254	+	0.608891i	$0.208385\pi$
$354$	−66.4366	−3.53107
$355$	−10.7855	−0.572435
$356$	−6.63774	−0.351799
$357$	−1.40286	−0.0742470
$358$	−58.7868	−3.10698
$359$	30.3878	1.60381	0.801903	−	0.597454i	$-0.203821\pi$
$359$	30.3878	1.60381	0.801903	+	0.597454i	$0.203821\pi$
$360$	−42.3175	−2.23033
$361$	19.7413	1.03901
$362$	14.8108	0.778436
$363$	−10.2959	−0.540396
$364$	5.43337	0.284786
$365$	−20.8068	−1.08908
$366$	−39.9280	−2.08707
$367$	28.4908	1.48721	0.743603	−	0.668621i	$-0.233115\pi$
$367$	28.4908	1.48721	0.743603	+	0.668621i	$0.233115\pi$
$368$	−31.8530	−1.66045
$369$	−0.535789	−0.0278921
$370$	−5.84609	−0.303924
$371$	−8.44101	−0.438236
$372$	−105.580	−5.47405
$373$	−18.9857	−0.983041	−0.491520	−	0.870866i	$-0.663559\pi$
$373$	−18.9857	−0.983041	−0.491520	+	0.870866i	$0.663559\pi$
$374$	4.52535	0.234000
$375$	31.0737	1.60464
$376$	−51.6784	−2.66511
$377$	−6.50342	−0.334943
$378$	−4.23998	−0.218081
$379$	30.1950	1.55101	0.775506	−	0.631340i	$-0.217495\pi$
$379$	30.1950	1.55101	0.775506	+	0.631340i	$0.217495\pi$
$380$	52.7629	2.70668
$381$	17.4772	0.895385
$382$	23.6849	1.21182
$383$	14.3870	0.735140	0.367570	−	0.929996i	$-0.380190\pi$
$383$	14.3870	0.735140	0.367570	+	0.929996i	$0.380190\pi$
$384$	−30.0285	−1.53239
$385$	−8.77241	−0.447083
$386$	−55.5919	−2.82955
$387$	0	0
$388$	−20.1998	−1.02549
$389$	−13.6677	−0.692980	−0.346490	−	0.938054i	$-0.612627\pi$
$389$	−13.6677	−0.692980	−0.346490	+	0.938054i	$0.612627\pi$
$390$	−12.5126	−0.633601
$391$	2.05975	0.104166
$392$	35.1294	1.77430
$393$	−21.2394	−1.07139
$394$	−19.7516	−0.995069
$395$	−29.0181	−1.46006
$396$	61.5222	3.09161
$397$	23.9951	1.20428	0.602140	−	0.798390i	$-0.294315\pi$
$397$	23.9951	1.20428	0.602140	+	0.798390i	$0.294315\pi$
$398$	10.1180	0.507171
$399$	19.0351	0.952949
$400$	−10.0436	−0.502178
$401$	14.5709	0.727637	0.363819	−	0.931470i	$-0.381473\pi$
$401$	14.5709	0.727637	0.363819	+	0.931470i	$0.381473\pi$
$402$	46.7478	2.33157
$403$	−9.35593	−0.466052
$404$	12.2076	0.607350
$405$	−13.3831	−0.665010
$406$	19.4927	0.967408
$407$	4.70288	0.233113
$408$	−7.39999	−0.366354
$409$	−37.5820	−1.85831	−0.929154	−	0.369692i	$-0.879463\pi$
$409$	−37.5820	−1.85831	−0.929154	+	0.369692i	$0.879463\pi$
$410$	0.728303	0.0359683
$411$	−41.2250	−2.03348
$412$	30.5034	1.50279
$413$	12.1977	0.600208
$414$	40.5100	1.99096
$415$	16.5453	0.812179
$416$	5.52498	0.270884
$417$	0.990202	0.0484904
$418$	−61.4038	−3.00336
$419$	8.26746	0.403892	0.201946	−	0.979397i	$-0.435274\pi$
$419$	8.26746	0.403892	0.201946	+	0.979397i	$0.435274\pi$
$420$	25.9245	1.26499
$421$	14.6244	0.712750	0.356375	−	0.934343i	$-0.384012\pi$
$421$	14.6244	0.712750	0.356375	+	0.934343i	$0.384012\pi$
$422$	−64.0095	−3.11594
$423$	29.0504	1.41248
$424$	−44.5259	−2.16237
$425$	0.649460	0.0315034
$426$	−37.0936	−1.79719
$427$	7.33073	0.354759
$428$	54.9390	2.65558
$429$	10.0658	0.485979
$430$	0	0
$431$	−9.66584	−0.465587	−0.232794	−	0.972526i	$-0.574787\pi$
$431$	−9.66584	−0.465587	−0.232794	+	0.972526i	$0.574787\pi$
$432$	−9.88582	−0.475632
$433$	11.9495	0.574257	0.287129	−	0.957892i	$-0.407299\pi$
$433$	11.9495	0.574257	0.287129	+	0.957892i	$0.407299\pi$
$434$	28.0426	1.34609
$435$	−31.0301	−1.48778
$436$	−12.8663	−0.616186
$437$	−27.9485	−1.33696
$438$	−71.5588	−3.41921
$439$	9.90146	0.472571	0.236286	−	0.971684i	$-0.424070\pi$
$439$	9.90146	0.472571	0.236286	+	0.971684i	$0.424070\pi$
$440$	−46.2740	−2.20603
$441$	−19.7476	−0.940361
$442$	−1.18509	−0.0563689
$443$	−19.9352	−0.947148	−0.473574	−	0.880754i	$-0.657036\pi$
$443$	−19.9352	−0.947148	−0.473574	+	0.880754i	$0.657036\pi$
$444$	−13.8981	−0.659576
$445$	−2.80652	−0.133042
$446$	−42.3877	−2.00711
$447$	−41.7213	−1.97335
$448$	0.400179	0.0189067
$449$	−14.0471	−0.662925	−0.331463	−	0.943468i	$-0.607542\pi$
$449$	−14.0471	−0.662925	−0.331463	+	0.943468i	$0.607542\pi$
$450$	12.7732	0.602135
$451$	−0.585883	−0.0275882
$452$	18.1166	0.852132
$453$	−1.57701	−0.0740944
$454$	−14.3030	−0.671273
$455$	2.29730	0.107699
$456$	100.409	4.70210
$457$	35.7181	1.67082	0.835410	−	0.549627i	$-0.185230\pi$
$457$	35.7181	1.67082	0.835410	+	0.549627i	$0.185230\pi$
$458$	−31.3094	−1.46299
$459$	0.639260	0.0298381
$460$	−38.0638	−1.77473
$461$	12.2957	0.572666	0.286333	−	0.958130i	$-0.407564\pi$
$461$	12.2957	0.572666	0.286333	+	0.958130i	$0.407564\pi$
$462$	−30.1702	−1.40364
$463$	−1.18519	−0.0550806	−0.0275403	−	0.999621i	$-0.508767\pi$
$463$	−1.18519	−0.0550806	−0.0275403	+	0.999621i	$0.508767\pi$
$464$	45.4487	2.10990
$465$	−44.6404	−2.07015
$466$	58.6069	2.71491
$467$	−26.2199	−1.21331	−0.606655	−	0.794965i	$-0.707489\pi$
$467$	−26.2199	−1.21331	−0.606655	+	0.794965i	$0.707489\pi$
$468$	−16.1113	−0.744745
$469$	−8.58282	−0.396318
$470$	−39.4885	−1.82147
$471$	37.6328	1.73403
$472$	64.3421	2.96158
$473$	0	0
$474$	−99.7996	−4.58395
$475$	−8.81243	−0.404342
$476$	2.45535	0.112541
$477$	25.0297	1.14603
$478$	2.79710	0.127936
$479$	41.6739	1.90413	0.952065	−	0.305897i	$-0.0989562\pi$
$479$	41.6739	1.90413	0.952065	+	0.305897i	$0.0989562\pi$
$480$	26.3616	1.20324
$481$	−1.23158	−0.0561552
$482$	11.5109	0.524305
$483$	−13.7322	−0.624836
$484$	18.0204	0.819111
$485$	−8.54075	−0.387815
$486$	−56.6677	−2.57050
$487$	−35.8772	−1.62575	−0.812876	−	0.582436i	$-0.802100\pi$
$487$	−35.8772	−1.62575	−0.812876	+	0.582436i	$0.802100\pi$
$488$	38.6692	1.75047
$489$	−11.1740	−0.505304
$490$	26.8431	1.21265
$491$	−15.2226	−0.686988	−0.343494	−	0.939155i	$-0.611611\pi$
$491$	−15.2226	−0.686988	−0.343494	+	0.939155i	$0.611611\pi$
$492$	1.73142	0.0780585
$493$	−2.93891	−0.132362
$494$	16.0803	0.723487
$495$	26.0124	1.16917
$496$	65.3832	2.93579
$497$	6.81034	0.305485
$498$	56.9030	2.54988
$499$	22.6329	1.01319	0.506594	−	0.862185i	$-0.330904\pi$
$499$	22.6329	1.01319	0.506594	+	0.862185i	$0.330904\pi$
$500$	−54.3868	−2.43225
$501$	−37.2870	−1.66586
$502$	16.7287	0.746637
$503$	−10.2918	−0.458888	−0.229444	−	0.973322i	$-0.573691\pi$
$503$	−10.2918	−0.458888	−0.229444	+	0.973322i	$0.573691\pi$
$504$	26.7207	1.19024
$505$	5.16152	0.229685
$506$	44.2975	1.96926
$507$	30.6215	1.35995
$508$	−30.5895	−1.35719
$509$	13.0008	0.576251	0.288125	−	0.957593i	$-0.406968\pi$
$509$	13.0008	0.576251	0.288125	+	0.957593i	$0.406968\pi$
$510$	−5.65447	−0.250384
$511$	13.1381	0.581195
$512$	50.8534	2.24743
$513$	−8.67403	−0.382968
$514$	2.78083	0.122657
$515$	12.8972	0.568320
$516$	0	0
$517$	31.7665	1.39709
$518$	3.69142	0.162192
$519$	12.3350	0.541447
$520$	12.1181	0.531415
$521$	40.4803	1.77347	0.886736	−	0.462275i	$-0.152967\pi$
$521$	40.4803	1.77347	0.886736	+	0.462275i	$0.152967\pi$
$522$	−57.8008	−2.52987
$523$	33.3575	1.45862	0.729311	−	0.684182i	$-0.239841\pi$
$523$	33.3575	1.45862	0.729311	+	0.684182i	$0.239841\pi$
$524$	37.1743	1.62397
$525$	−4.32990	−0.188972
$526$	24.1206	1.05171
$527$	−4.22796	−0.184173
$528$	−70.3438	−3.06132
$529$	−2.83763	−0.123375
$530$	−34.0231	−1.47787
$531$	−36.1691	−1.56961
$532$	−33.3163	−1.44444
$533$	0.153430	0.00664578
$534$	−9.65222	−0.417692
$535$	23.2289	1.00427
$536$	−45.2740	−1.95554
$537$	−59.0906	−2.54995
$538$	−10.2890	−0.443590
$539$	−21.5939	−0.930114
$540$	−11.8134	−0.508368
$541$	−5.65148	−0.242976	−0.121488	−	0.992593i	$-0.538767\pi$
$541$	−5.65148	−0.242976	−0.121488	+	0.992593i	$0.538767\pi$
$542$	−37.6820	−1.61858
$543$	14.8873	0.638875
$544$	2.49675	0.107047
$545$	−5.44005	−0.233026
$546$	7.90089	0.338127
$547$	37.4343	1.60057	0.800287	−	0.599617i	$-0.204680\pi$
$547$	37.4343	1.60057	0.800287	+	0.599617i	$0.204680\pi$
$548$	72.1542	3.08227
$549$	−21.7374	−0.927731
$550$	13.9675	0.595574
$551$	39.8776	1.69884
$552$	−72.4366	−3.08311
$553$	18.3231	0.779176
$554$	−38.4521	−1.63367
$555$	−5.87630	−0.249435
$556$	−1.73310	−0.0735000
$557$	−0.420756	−0.0178280	−0.00891400	−	0.999960i	$-0.502837\pi$
$557$	−0.420756	−0.0178280	−0.00891400	+	0.999960i	$0.502837\pi$
$558$	−83.1532	−3.52016
$559$	0	0
$560$	−16.0545	−0.678426
$561$	4.54874	0.192048
$562$	−44.8411	−1.89151
$563$	−15.7137	−0.662255	−0.331128	−	0.943586i	$-0.607429\pi$
$563$	−15.7137	−0.662255	−0.331128	+	0.943586i	$0.607429\pi$
$564$	−93.8774	−3.95295
$565$	7.65992	0.322255
$566$	−6.82936	−0.287059
$567$	8.45053	0.354889
$568$	35.9242	1.50735
$569$	38.1299	1.59849	0.799245	−	0.601006i	$-0.205233\pi$
$569$	38.1299	1.59849	0.799245	+	0.601006i	$0.205233\pi$
$570$	76.7247	3.21365
$571$	22.4383	0.939013	0.469506	−	0.882929i	$-0.344432\pi$
$571$	22.4383	0.939013	0.469506	+	0.882929i	$0.344432\pi$
$572$	−17.6176	−0.736629
$573$	23.8073	0.994563
$574$	−0.459876	−0.0191948
$575$	6.35740	0.265122
$576$	−1.18663	−0.0494430
$577$	20.0498	0.834685	0.417342	−	0.908749i	$-0.362962\pi$
$577$	20.0498	0.834685	0.417342	+	0.908749i	$0.362962\pi$
$578$	42.7314	1.77739
$579$	−55.8792	−2.32226
$580$	54.3105	2.25512
$581$	−10.4473	−0.433427
$582$	−29.3734	−1.21757
$583$	27.3699	1.13354
$584$	69.3028	2.86777
$585$	−6.81206	−0.281644
$586$	−27.9209	−1.15340
$587$	16.6980	0.689201	0.344601	−	0.938749i	$-0.388014\pi$
$587$	16.6980	0.689201	0.344601	+	0.938749i	$0.388014\pi$
$588$	63.8150	2.63168
$589$	57.3686	2.36383
$590$	49.1650	2.02409
$591$	−19.8536	−0.816669
$592$	8.60681	0.353738
$593$	26.2817	1.07926	0.539631	−	0.841902i	$-0.318564\pi$
$593$	26.2817	1.07926	0.539631	+	0.841902i	$0.318564\pi$
$594$	13.7481	0.564091
$595$	1.03815	0.0425601
$596$	73.0227	2.99113
$597$	10.1703	0.416244
$598$	−11.6005	−0.474381
$599$	−12.3178	−0.503290	−0.251645	−	0.967820i	$-0.580972\pi$
$599$	−12.3178	−0.503290	−0.251645	+	0.967820i	$0.580972\pi$
$600$	−22.8400	−0.932439
$601$	−19.0287	−0.776195	−0.388098	−	0.921618i	$-0.626868\pi$
$601$	−19.0287	−0.776195	−0.388098	+	0.921618i	$0.626868\pi$
$602$	0	0
$603$	25.4502	1.03641
$604$	2.76017	0.112310
$605$	7.61928	0.309768
$606$	17.7516	0.721108
$607$	1.62907	0.0661218	0.0330609	−	0.999453i	$-0.489474\pi$
$607$	1.62907	0.0661218	0.0330609	+	0.999453i	$0.489474\pi$
$608$	−33.8780	−1.37394
$609$	19.5935	0.793968
$610$	29.5479	1.19636
$611$	−8.31893	−0.336548
$612$	−7.28072	−0.294306
$613$	−31.5764	−1.27536	−0.637680	−	0.770301i	$-0.720106\pi$
$613$	−31.5764	−1.27536	−0.637680	+	0.770301i	$0.720106\pi$
$614$	32.7107	1.32010
$615$	0.732067	0.0295198
$616$	29.2190	1.17727
$617$	−29.7042	−1.19585	−0.597923	−	0.801553i	$-0.704007\pi$
$617$	−29.7042	−1.19585	−0.597923	+	0.801553i	$0.704007\pi$
$618$	44.3563	1.78427
$619$	−13.4001	−0.538596	−0.269298	−	0.963057i	$-0.586792\pi$
$619$	−13.4001	−0.538596	−0.269298	+	0.963057i	$0.586792\pi$
$620$	78.1320	3.13786
$621$	6.25755	0.251107
$622$	40.0593	1.60623
$623$	1.77213	0.0709990
$624$	18.4215	0.737449
$625$	−15.9163	−0.636654
$626$	81.5929	3.26111
$627$	−61.7212	−2.46491
$628$	−65.8670	−2.62838
$629$	−0.556554	−0.0221912
$630$	20.4178	0.813465
$631$	44.8663	1.78610	0.893049	−	0.449960i	$-0.148562\pi$
$631$	44.8663	1.78610	0.893049	+	0.449960i	$0.148562\pi$
$632$	96.6532	3.84466
$633$	−64.3403	−2.55730
$634$	−53.9971	−2.14450
$635$	−12.9337	−0.513256
$636$	−80.8843	−3.20727
$637$	5.65495	0.224057
$638$	−63.2049	−2.50231
$639$	−20.1944	−0.798876
$640$	22.2220	0.878400
$641$	−32.1357	−1.26928	−0.634641	−	0.772807i	$-0.718852\pi$
$641$	−32.1357	−1.26928	−0.634641	+	0.772807i	$0.718852\pi$
$642$	79.8891	3.15297
$643$	1.72355	0.0679703	0.0339851	−	0.999422i	$-0.489180\pi$
$643$	1.72355	0.0679703	0.0339851	+	0.999422i	$0.489180\pi$
$644$	24.0348	0.947104
$645$	0	0
$646$	7.26672	0.285905
$647$	−14.1637	−0.556831	−0.278416	−	0.960461i	$-0.589809\pi$
$647$	−14.1637	−0.556831	−0.278416	+	0.960461i	$0.589809\pi$
$648$	44.5761	1.75111
$649$	−39.5508	−1.55250
$650$	−3.65776	−0.143469
$651$	28.1875	1.10475
$652$	19.5573	0.765921
$653$	−11.1514	−0.436387	−0.218194	−	0.975906i	$-0.570016\pi$
$653$	−11.1514	−0.436387	−0.218194	+	0.975906i	$0.570016\pi$
$654$	−18.7095	−0.731599
$655$	15.7178	0.614144
$656$	−1.07223	−0.0418636
$657$	−38.9577	−1.51989
$658$	24.9344	0.972043
$659$	27.4144	1.06791	0.533957	−	0.845511i	$-0.320704\pi$
$659$	27.4144	1.06791	0.533957	+	0.845511i	$0.320704\pi$
$660$	−84.0598	−3.27202
$661$	−48.9800	−1.90510	−0.952550	−	0.304383i	$-0.901550\pi$
$661$	−48.9800	−1.90510	−0.952550	+	0.304383i	$0.901550\pi$
$662$	−12.1949	−0.473968
$663$	−1.19121	−0.0462629
$664$	−55.1090	−2.13864
$665$	−14.0866	−0.546253
$666$	−10.9460	−0.424148
$667$	−28.7682	−1.11391
$668$	65.2616	2.52505
$669$	−42.6067	−1.64727
$670$	−34.5947	−1.33651
$671$	−23.7698	−0.917622
$672$	−16.6456	−0.642119
$673$	−1.12541	−0.0433813	−0.0216907	−	0.999765i	$-0.506905\pi$
$673$	−1.12541	−0.0433813	−0.0216907	+	0.999765i	$0.506905\pi$
$674$	8.94681	0.344618
$675$	1.97307	0.0759435
$676$	−53.5953	−2.06136
$677$	−3.40399	−0.130826	−0.0654130	−	0.997858i	$-0.520836\pi$
$677$	−3.40399	−0.130826	−0.0654130	+	0.997858i	$0.520836\pi$
$678$	26.3441	1.01174
$679$	5.39292	0.206961
$680$	5.47621	0.210003
$681$	−14.3769	−0.550925
$682$	−90.9276	−3.48180
$683$	−14.5192	−0.555563	−0.277782	−	0.960644i	$-0.589599\pi$
$683$	−14.5192	−0.555563	−0.277782	+	0.960644i	$0.589599\pi$
$684$	98.7911	3.77737
$685$	30.5077	1.16564
$686$	−38.2471	−1.46028
$687$	−31.4712	−1.20070
$688$	0	0
$689$	−7.16756	−0.273062
$690$	−55.3502	−2.10715
$691$	24.4454	0.929946	0.464973	−	0.885325i	$-0.346064\pi$
$691$	24.4454	0.929946	0.464973	+	0.885325i	$0.346064\pi$
$692$	−21.5894	−0.820705
$693$	−16.4251	−0.623938
$694$	31.1554	1.18264
$695$	−0.732778	−0.0277959
$696$	103.355	3.91764
$697$	0.0693352	0.00262626
$698$	23.2041	0.878289
$699$	58.9097	2.22817
$700$	7.57841	0.286437
$701$	−26.3852	−0.996555	−0.498277	−	0.867018i	$-0.666034\pi$
$701$	−26.3852	−0.996555	−0.498277	+	0.867018i	$0.666034\pi$
$702$	−3.60032	−0.135885
$703$	7.55180	0.284821
$704$	−1.29758	−0.0489042
$705$	−39.6925	−1.49491
$706$	−75.8642	−2.85519
$707$	−3.25916	−0.122573
$708$	116.882	4.39269
$709$	26.2339	0.985233	0.492617	−	0.870246i	$-0.336040\pi$
$709$	26.2339	0.985233	0.492617	+	0.870246i	$0.336040\pi$
$710$	27.4504	1.03019
$711$	−54.3324	−2.03762
$712$	9.34791	0.350328
$713$	−41.3864	−1.54993
$714$	3.57043	0.133620
$715$	−7.44895	−0.278575
$716$	103.423	3.86511
$717$	2.81156	0.104999
$718$	−77.3405	−2.88632
$719$	−15.4928	−0.577783	−0.288892	−	0.957362i	$-0.593287\pi$
$719$	−15.4928	−0.577783	−0.288892	+	0.957362i	$0.593287\pi$
$720$	47.6056	1.77416
$721$	−8.14375	−0.303289
$722$	−50.2438	−1.86988
$723$	11.5703	0.430306
$724$	−26.0565	−0.968383
$725$	−9.07091	−0.336885
$726$	26.2043	0.972533
$727$	−24.6702	−0.914968	−0.457484	−	0.889218i	$-0.651249\pi$
$727$	−24.6702	−0.914968	−0.457484	+	0.889218i	$0.651249\pi$
$728$	−7.65180	−0.283595
$729$	−35.7534	−1.32420
$730$	52.9556	1.95998
$731$	0	0
$732$	70.2452	2.59634
$733$	18.8142	0.694917	0.347459	−	0.937695i	$-0.387045\pi$
$733$	18.8142	0.694917	0.347459	+	0.937695i	$0.387045\pi$
$734$	−72.5123	−2.67648
$735$	26.9818	0.995238
$736$	24.4401	0.900872
$737$	27.8297	1.02512
$738$	1.36365	0.0501965
$739$	−9.86714	−0.362968	−0.181484	−	0.983394i	$-0.558090\pi$
$739$	−9.86714	−0.362968	−0.181484	+	0.983394i	$0.558090\pi$
$740$	10.2850	0.378084
$741$	16.1634	0.593777
$742$	21.4834	0.788679
$743$	3.00273	0.110159	0.0550797	−	0.998482i	$-0.482459\pi$
$743$	3.00273	0.110159	0.0550797	+	0.998482i	$0.482459\pi$
$744$	148.688	5.45115
$745$	30.8749	1.13117
$746$	48.3207	1.76915
$747$	30.9788	1.13346
$748$	−7.96143	−0.291099
$749$	−14.6675	−0.535940
$750$	−79.0862	−2.88782
$751$	−0.869629	−0.0317332	−0.0158666	−	0.999874i	$-0.505051\pi$
$751$	−0.869629	−0.0317332	−0.0158666	+	0.999874i	$0.505051\pi$
$752$	58.1362	2.12001
$753$	16.8151	0.612777
$754$	16.5520	0.602787
$755$	1.16703	0.0424727
$756$	7.45939	0.271295
$757$	−38.5175	−1.39994	−0.699972	−	0.714170i	$-0.746804\pi$
$757$	−38.5175	−1.39994	−0.699972	+	0.714170i	$0.746804\pi$
$758$	−76.8497	−2.79131
$759$	44.5264	1.61621
$760$	−74.3058	−2.69536
$761$	10.5663	0.383029	0.191514	−	0.981490i	$-0.438660\pi$
$761$	10.5663	0.383029	0.191514	+	0.981490i	$0.438660\pi$
$762$	−44.4816	−1.61140
$763$	3.43504	0.124357
$764$	−41.6687	−1.50752
$765$	−3.07838	−0.111299
$766$	−36.6165	−1.32301
$767$	10.3575	0.373986
$768$	74.7132	2.69598
$769$	2.75025	0.0991767	0.0495883	−	0.998770i	$-0.484209\pi$
$769$	2.75025	0.0991767	0.0495883	+	0.998770i	$0.484209\pi$
$770$	22.3268	0.804602
$771$	2.79520	0.100667
$772$	97.8026	3.51999
$773$	−16.3728	−0.588887	−0.294444	−	0.955669i	$-0.595134\pi$
$773$	−16.3728	−0.588887	−0.294444	+	0.955669i	$0.595134\pi$
$774$	0	0
$775$	−13.0496	−0.468754
$776$	28.4474	1.02120
$777$	3.71050	0.133113
$778$	34.7859	1.24713
$779$	−0.940799	−0.0337076
$780$	22.0134	0.788206
$781$	−22.0824	−0.790171
$782$	−5.24230	−0.187464
$783$	−8.92845	−0.319077
$784$	−39.5192	−1.41140
$785$	−27.8494	−0.993987
$786$	54.0567	1.92814
$787$	−23.4769	−0.836859	−0.418430	−	0.908249i	$-0.637419\pi$
$787$	−23.4769	−0.836859	−0.418430	+	0.908249i	$0.637419\pi$
$788$	34.7489	1.23788
$789$	24.2452	0.863152
$790$	73.8545	2.62763
$791$	−4.83674	−0.171975
$792$	−86.6416	−3.07867
$793$	6.22477	0.221048
$794$	−61.0704	−2.16731
$795$	−34.1989	−1.21291
$796$	−17.8006	−0.630927
$797$	10.8875	0.385655	0.192827	−	0.981233i	$-0.438234\pi$
$797$	10.8875	0.385655	0.192827	+	0.981233i	$0.438234\pi$
$798$	−48.4466	−1.71499
$799$	−3.75934	−0.132996
$800$	7.70619	0.272455
$801$	−5.25482	−0.185670
$802$	−37.0847	−1.30951
$803$	−42.6001	−1.50332
$804$	−82.2432	−2.90049
$805$	10.1622	0.358171
$806$	23.8119	0.838739
$807$	−10.3422	−0.364062
$808$	−17.1919	−0.604809
$809$	−16.7151	−0.587672	−0.293836	−	0.955856i	$-0.594932\pi$
$809$	−16.7151	−0.587672	−0.293836	+	0.955856i	$0.594932\pi$
$810$	34.0615	1.19680
$811$	−11.8282	−0.415344	−0.207672	−	0.978199i	$-0.566589\pi$
$811$	−11.8282	−0.415344	−0.207672	+	0.978199i	$0.566589\pi$
$812$	−34.2935	−1.20347
$813$	−37.8768	−1.32840
$814$	−11.9694	−0.419527
$815$	8.26906	0.289652
$816$	8.32470	0.291423
$817$	0	0
$818$	95.6504	3.34434
$819$	4.30137	0.150302
$820$	−1.28130	−0.0447450
$821$	40.8863	1.42694	0.713471	−	0.700684i	$-0.247122\pi$
$821$	40.8863	1.42694	0.713471	+	0.700684i	$0.247122\pi$
$822$	104.922	3.65959
$823$	−35.0225	−1.22081	−0.610404	−	0.792090i	$-0.708993\pi$
$823$	−35.0225	−1.22081	−0.610404	+	0.792090i	$0.708993\pi$
$824$	−42.9579	−1.49651
$825$	14.0396	0.488797
$826$	−31.0445	−1.08018
$827$	−37.1876	−1.29314	−0.646570	−	0.762855i	$-0.723797\pi$
$827$	−37.1876	−1.29314	−0.646570	+	0.762855i	$0.723797\pi$
$828$	−71.2692	−2.47677
$829$	31.8037	1.10459	0.552294	−	0.833649i	$-0.313753\pi$
$829$	31.8037	1.10459	0.552294	+	0.833649i	$0.313753\pi$
$830$	−42.1098	−1.46165
$831$	−38.6508	−1.34078
$832$	0.339806	0.0117807
$833$	2.55549	0.0885423
$834$	−2.52018	−0.0872667
$835$	27.5934	0.954910
$836$	108.028	3.73621
$837$	−12.8446	−0.443975
$838$	−21.0416	−0.726871
$839$	−22.3146	−0.770384	−0.385192	−	0.922836i	$-0.625865\pi$
$839$	−22.3146	−0.770384	−0.385192	+	0.922836i	$0.625865\pi$
$840$	−36.5094	−1.25970
$841$	12.0473	0.415424
$842$	−37.2208	−1.28271
$843$	−45.0729	−1.55239
$844$	112.612	3.87626
$845$	−22.6608	−0.779555
$846$	−73.9366	−2.54199
$847$	−4.81107	−0.165310
$848$	50.0900	1.72010
$849$	−6.86466	−0.235594
$850$	−1.65295	−0.0566957
$851$	−5.44796	−0.186754
$852$	65.2587	2.23573
$853$	−9.03270	−0.309274	−0.154637	−	0.987971i	$-0.549421\pi$
$853$	−9.03270	−0.309274	−0.154637	+	0.987971i	$0.549421\pi$
$854$	−18.6575	−0.638448
$855$	41.7701	1.42851
$856$	−77.3705	−2.64447
$857$	−36.2027	−1.23666	−0.618330	−	0.785919i	$-0.712190\pi$
$857$	−36.2027	−1.23666	−0.618330	+	0.785919i	$0.712190\pi$
$858$	−25.6185	−0.874602
$859$	41.5643	1.41816	0.709079	−	0.705129i	$-0.249111\pi$
$859$	41.5643	1.41816	0.709079	+	0.705129i	$0.249111\pi$
$860$	0	0
$861$	−0.462252	−0.0157535
$862$	24.6007	0.837903
$863$	−31.7216	−1.07982	−0.539908	−	0.841724i	$-0.681541\pi$
$863$	−31.7216	−1.07982	−0.539908	+	0.841724i	$0.681541\pi$
$864$	7.58517	0.258053
$865$	−9.12826	−0.310370
$866$	−30.4129	−1.03347
$867$	42.9523	1.45874
$868$	−49.3352	−1.67455
$869$	−59.4122	−2.01542
$870$	78.9752	2.67751
$871$	−7.28797	−0.246944
$872$	18.1196	0.613609
$873$	−15.9914	−0.541225
$874$	71.1321	2.40608
$875$	14.5201	0.490869
$876$	125.893	4.25354
$877$	15.5081	0.523671	0.261836	−	0.965113i	$-0.415672\pi$
$877$	15.5081	0.523671	0.261836	+	0.965113i	$0.415672\pi$
$878$	−25.2004	−0.850471
$879$	−28.0651	−0.946614
$880$	52.0565	1.75482
$881$	−3.69055	−0.124338	−0.0621689	−	0.998066i	$-0.519802\pi$
$881$	−3.69055	−0.124338	−0.0621689	+	0.998066i	$0.519802\pi$
$882$	50.2599	1.69234
$883$	−55.5238	−1.86853	−0.934263	−	0.356585i	$-0.883941\pi$
$883$	−55.5238	−1.86853	−0.934263	+	0.356585i	$0.883941\pi$
$884$	2.08492	0.0701235
$885$	49.4191	1.66120
$886$	50.7373	1.70455
$887$	49.3932	1.65846	0.829231	−	0.558906i	$-0.188779\pi$
$887$	49.3932	1.65846	0.829231	+	0.558906i	$0.188779\pi$
$888$	19.5727	0.656817
$889$	8.16675	0.273904
$890$	7.14292	0.239431
$891$	−27.4007	−0.917958
$892$	74.5725	2.49687
$893$	51.0100	1.70698
$894$	106.185	3.55137
$895$	43.7287	1.46169
$896$	−14.0317	−0.468766
$897$	−11.6605	−0.389332
$898$	35.7516	1.19305
$899$	59.0513	1.96947
$900$	−22.4719	−0.749063
$901$	−3.23903	−0.107908
$902$	1.49114	0.0496495
$903$	0	0
$904$	−25.5135	−0.848567
$905$	−11.0170	−0.366218
$906$	4.01368	0.133345
$907$	−37.6244	−1.24930	−0.624648	−	0.780906i	$-0.714758\pi$
$907$	−37.6244	−1.24930	−0.624648	+	0.780906i	$0.714758\pi$
$908$	25.1632	0.835071
$909$	9.66422	0.320542
$910$	−5.84689	−0.193822
$911$	−20.8938	−0.692243	−0.346121	−	0.938190i	$-0.612501\pi$
$911$	−20.8938	−0.692243	−0.346121	+	0.938190i	$0.612501\pi$
$912$	−112.957	−3.74037
$913$	33.8752	1.12111
$914$	−90.9066	−3.00692
$915$	29.7006	0.981871
$916$	55.0826	1.81998
$917$	−9.92473	−0.327743
$918$	−1.62699	−0.0536987
$919$	50.0456	1.65085	0.825426	−	0.564511i	$-0.190935\pi$
$919$	50.0456	1.65085	0.825426	+	0.564511i	$0.190935\pi$
$920$	53.6052	1.76731
$921$	32.8798	1.08342
$922$	−31.2939	−1.03061
$923$	5.78290	0.190346
$924$	53.0783	1.74615
$925$	−1.71780	−0.0564808
$926$	3.01645	0.0991269
$927$	24.1483	0.793133
$928$	−34.8717	−1.14472
$929$	55.1618	1.80980	0.904900	−	0.425625i	$-0.139946\pi$
$929$	55.1618	1.80980	0.904900	+	0.425625i	$0.139946\pi$
$930$	113.615	3.72558
$931$	−34.6750	−1.13643
$932$	−103.107	−3.37738
$933$	40.2664	1.31826
$934$	66.7326	2.18356
$935$	−3.36620	−0.110086
$936$	22.6895	0.741629
$937$	41.4317	1.35352	0.676758	−	0.736206i	$-0.263384\pi$
$937$	41.4317	1.35352	0.676758	+	0.736206i	$0.263384\pi$
$938$	21.8443	0.713241
$939$	82.0146	2.67644
$940$	69.4720	2.26593
$941$	−3.62142	−0.118055	−0.0590274	−	0.998256i	$-0.518800\pi$
$941$	−3.62142	−0.118055	−0.0590274	+	0.998256i	$0.518800\pi$
$942$	−95.7799	−3.12068
$943$	0.678705	0.0221017
$944$	−72.3824	−2.35585
$945$	3.15393	0.102597
$946$	0	0
$947$	−32.6704	−1.06165	−0.530823	−	0.847483i	$-0.678117\pi$
$947$	−32.6704	−1.06165	−0.530823	+	0.847483i	$0.678117\pi$
$948$	175.577	5.70248
$949$	11.1560	0.362140
$950$	22.4287	0.727682
$951$	−54.2761	−1.76002
$952$	−3.45786	−0.112070
$953$	−3.65648	−0.118445	−0.0592225	−	0.998245i	$-0.518862\pi$
$953$	−3.65648	−0.118445	−0.0592225	+	0.998245i	$0.518862\pi$
$954$	−63.7035	−2.06248
$955$	−17.6181	−0.570107
$956$	−4.92093	−0.159154
$957$	−63.5315	−2.05368
$958$	−106.065	−3.42680
$959$	−19.2636	−0.622055
$960$	1.62133	0.0523283
$961$	53.9522	1.74039
$962$	3.13451	0.101061
$963$	43.4929	1.40154
$964$	−20.2510	−0.652241
$965$	41.3522	1.33117
$966$	34.9500	1.12450
$967$	−21.8428	−0.702417	−0.351209	−	0.936297i	$-0.614229\pi$
$967$	−21.8428	−0.702417	−0.351209	+	0.936297i	$0.614229\pi$
$968$	−25.3782	−0.815685
$969$	7.30427	0.234647
$970$	21.7372	0.697939
$971$	47.5721	1.52666	0.763330	−	0.646008i	$-0.223563\pi$
$971$	47.5721	1.52666	0.763330	+	0.646008i	$0.223563\pi$
$972$	99.6953	3.19773
$973$	0.462701	0.0148335
$974$	91.3117	2.92582
$975$	−3.67667	−0.117748
$976$	−43.5014	−1.39245
$977$	−10.1705	−0.325383	−0.162691	−	0.986677i	$-0.552017\pi$
$977$	−10.1705	−0.325383	−0.162691	+	0.986677i	$0.552017\pi$
$978$	28.4390	0.909380
$979$	−5.74611	−0.183647
$980$	−47.2249	−1.50854
$981$	−10.1857	−0.325206
$982$	38.7434	1.23635
$983$	18.3770	0.586136	0.293068	−	0.956092i	$-0.405324\pi$
$983$	18.3770	0.586136	0.293068	+	0.956092i	$0.405324\pi$
$984$	−2.43836	−0.0777320
$985$	14.6923	0.468134
$986$	7.47986	0.238207
$987$	25.0632	0.797772
$988$	−28.2900	−0.900025
$989$	0	0
$990$	−66.2045	−2.10412
$991$	12.2959	0.390593	0.195296	−	0.980744i	$-0.437433\pi$
$991$	12.2959	0.390593	0.195296	+	0.980744i	$0.437433\pi$
$992$	−50.1671	−1.59281
$993$	−12.2579	−0.388993
$994$	−17.3331	−0.549773
$995$	−7.52634	−0.238601
$996$	−100.109	−3.17208
$997$	59.2888	1.87770	0.938848	−	0.344331i	$-0.111894\pi$
$997$	59.2888	1.87770	0.938848	+	0.344331i	$0.111894\pi$
$998$	−57.6033	−1.82340
$999$	−1.69082	−0.0534951

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.2.a.p.1.2	✓	20
43.42	odd	2		1849.2.a.r.1.19	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.2.a.p.1.2	✓	20	1.1	even	1	trivial
1849.2.a.r.1.19	yes	20	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)

\(f(q)\)	\(=\)	\(q-2.54512 q^{2} -2.55827 q^{3} +4.47761 q^{4} +1.89319 q^{5} +6.51109 q^{6} -1.19543 q^{7} -6.30581 q^{8} +3.54474 q^{9} +O(q^{10})\)	\(q-2.54512 q^{2} -2.55827 q^{3} +4.47761 q^{4} +1.89319 q^{5} +6.51109 q^{6} -1.19543 q^{7} -6.30581 q^{8} +3.54474 q^{9} -4.81839 q^{10} +3.87615 q^{11} -11.4549 q^{12} -1.01508 q^{13} +3.04250 q^{14} -4.84329 q^{15} +7.09380 q^{16} -0.458716 q^{17} -9.02177 q^{18} +6.22425 q^{19} +8.47699 q^{20} +3.05822 q^{21} -9.86526 q^{22} -4.49025 q^{23} +16.1320 q^{24} -1.41582 q^{25} +2.58349 q^{26} -1.39359 q^{27} -5.35266 q^{28} +6.40682 q^{29} +12.3267 q^{30} +9.21695 q^{31} -5.44291 q^{32} -9.91624 q^{33} +1.16748 q^{34} -2.26317 q^{35} +15.8720 q^{36} +1.21329 q^{37} -15.8414 q^{38} +2.59684 q^{39} -11.9381 q^{40} -0.151151 q^{41} -7.78353 q^{42} +17.3559 q^{44} +6.71087 q^{45} +11.4282 q^{46} +8.19536 q^{47} -18.1478 q^{48} -5.57096 q^{49} +3.60343 q^{50} +1.17352 q^{51} -4.54513 q^{52} +7.06109 q^{53} +3.54684 q^{54} +7.33830 q^{55} +7.53814 q^{56} -15.9233 q^{57} -16.3061 q^{58} -10.2036 q^{59} -21.6864 q^{60} -6.13231 q^{61} -23.4582 q^{62} -4.23747 q^{63} -0.334759 q^{64} -1.92174 q^{65} +25.2380 q^{66} +7.17972 q^{67} -2.05395 q^{68} +11.4873 q^{69} +5.76004 q^{70} -5.69699 q^{71} -22.3525 q^{72} -10.9903 q^{73} -3.08795 q^{74} +3.62205 q^{75} +27.8698 q^{76} -4.63366 q^{77} -6.60926 q^{78} -15.3276 q^{79} +13.4299 q^{80} -7.06905 q^{81} +0.384696 q^{82} +8.73939 q^{83} +13.6935 q^{84} -0.868437 q^{85} -16.3904 q^{87} -24.4423 q^{88} -1.48243 q^{89} -17.0799 q^{90} +1.21345 q^{91} -20.1056 q^{92} -23.5794 q^{93} -20.8581 q^{94} +11.7837 q^{95} +13.9244 q^{96} -4.51129 q^{97} +14.1787 q^{98} +13.7399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 5 q^{2} + q^{3} + 23 q^{4} + 3 q^{5} + 5 q^{6} + 2 q^{7} - 9 q^{8} + 27 q^{9}+O(q^{10}) \) 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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