LMFDB - Embedded newform 1849.2.a.r.1.19

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			1.19
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	$-0.143687\pi$
$2$	2.54512	1.79967	0.899834	+	0.436232i	$0.143687\pi$
$3$	2.55827	1.47702	0.738509	−	0.674244i	$-0.235530\pi$
$3$	2.55827	1.47702	0.738509	+	0.674244i	$0.235530\pi$
$4$	4.47761	2.23881
$5$	−1.89319	−0.846661	−0.423331	−	0.905975i	$-0.639139\pi$
$5$	−1.89319	−0.846661	−0.423331	+	0.905975i	$0.639139\pi$
$6$	6.51109	2.65814
$7$	1.19543	0.451829	0.225914	−	0.974147i	$-0.427463\pi$
$7$	1.19543	0.451829	0.225914	+	0.974147i	$0.427463\pi$
$8$	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	$-0.753105\pi$
$19$	−6.22425	−1.42794	−0.713970	+	0.700176i	$0.753105\pi$
$20$	−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	$-0.702792\pi$
$29$	−6.40682	−1.18972	−0.594858	+	0.803831i	$0.702792\pi$
$30$	−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	$-0.531798\pi$
$37$	−1.21329	−0.199463	−0.0997315	+	0.995014i	$0.531798\pi$
$38$	−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	$-0.371582\pi$
$61$	6.13231	0.785162	0.392581	+	0.919717i	$0.371582\pi$
$62$	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	$-0.390231\pi$
$71$	5.69699	0.676109	0.338054	+	0.941127i	$0.390231\pi$
$72$	22.3525	2.63426
$73$	10.9903	1.28632	0.643159	−	0.765733i	$-0.277623\pi$
$73$	10.9903	1.28632	0.643159	+	0.765733i	$0.277623\pi$
$74$	−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	$-0.474965\pi$
$89$	1.48243	0.157137	0.0785685	+	0.996909i	$0.474965\pi$
$90$	−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	$-0.560949\pi$
$113$	−4.04603	−0.380619	−0.190309	+	0.981724i	$0.560949\pi$
$114$	−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	$-0.618140\pi$
$131$	−8.30225	−0.725371	−0.362685	+	0.931912i	$0.618140\pi$
$132$	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	$-0.741675\pi$
$137$	−16.1144	−1.37675	−0.688374	+	0.725356i	$0.741675\pi$
$138$	−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	$-0.732857\pi$
$149$	−16.3084	−1.33604	−0.668018	+	0.744145i	$0.732857\pi$
$150$	−9.21854	−0.752691
$151$	−0.616437	−0.0501649	−0.0250825	−	0.999685i	$-0.507985\pi$
$151$	−0.616437	−0.0501649	−0.0250825	+	0.999685i	$0.507985\pi$
$152$	−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	$-0.300307\pi$
$157$	14.7103	1.17401	0.587004	+	0.809584i	$0.300307\pi$
$158$	−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	$-0.554718\pi$
$163$	−4.36778	−0.342111	−0.171056	+	0.985261i	$0.554718\pi$
$164$	−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	$-0.831548\pi$
$179$	−23.0979	−1.72642	−0.863208	+	0.504848i	$0.831548\pi$
$180$	−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	$-0.390696\pi$
$191$	9.30601	0.673359	0.336680	+	0.941619i	$0.390696\pi$
$192$	−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	$-0.454998\pi$
$199$	3.97547	0.281814	0.140907	+	0.990023i	$0.454998\pi$
$200$	−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	$-0.833124\pi$
$211$	−25.1500	−1.73139	−0.865697	+	0.500568i	$0.833124\pi$
$212$	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	$-0.688291\pi$
$223$	−16.6545	−1.11527	−0.557634	+	0.830087i	$0.688291\pi$
$224$	6.50660	0.434741
$225$	−5.01872	−0.334581
$226$	−10.2976	−0.684988
$227$	−5.61979	−0.372998	−0.186499	−	0.982455i	$-0.559714\pi$
$227$	−5.61979	−0.372998	−0.186499	+	0.982455i	$0.559714\pi$
$228$	−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	$-0.227986\pi$
$233$	23.0272	1.50856	0.754281	+	0.656552i	$0.227986\pi$
$234$	−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	$-0.453467\pi$
$241$	4.52273	0.291334	0.145667	+	0.989334i	$0.453467\pi$
$242$	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	$-0.489151\pi$
$257$	1.09262	0.0681555	0.0340777	+	0.999419i	$0.489151\pi$
$258$	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	$-0.405615\pi$
$263$	9.47719	0.584389	0.292194	+	0.956359i	$0.405615\pi$
$264$	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	$-0.649961\pi$
$277$	−15.1082	−0.907763	−0.453881	+	0.891062i	$0.649961\pi$
$278$	−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	$-0.139098\pi$
$313$	32.0586	1.81206	0.906030	+	0.423213i	$0.139098\pi$
$314$	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	$-0.542038\pi$
$331$	−4.79148	−0.263364	−0.131682	+	0.991292i	$0.542038\pi$
$332$	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	$-0.393432\pi$
$347$	12.2413	0.657145	0.328573	+	0.944479i	$0.393432\pi$
$348$	−73.3897	−3.93410
$349$	9.11712	0.488028	0.244014	−	0.969772i	$-0.421536\pi$
$349$	9.11712	0.488028	0.244014	+	0.969772i	$0.421536\pi$
$350$	−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	$-0.336441\pi$
$373$	18.9857	0.983041	0.491520	+	0.870866i	$0.336441\pi$
$374$	−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	$-0.619810\pi$
$383$	−14.3870	−0.735140	−0.367570	+	0.929996i	$0.619810\pi$
$384$	−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	$-0.387373\pi$
$389$	13.6677	0.692980	0.346490	+	0.938054i	$0.387373\pi$
$390$	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	$-0.120537\pi$
$409$	37.5820	1.85831	0.929154	+	0.369692i	$0.120537\pi$
$410$	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	$-0.564726\pi$
$419$	−8.26746	−0.403892	−0.201946	+	0.979397i	$0.564726\pi$
$420$	−25.9245	−1.26499
$421$	−14.6244	−0.712750	−0.356375	−	0.934343i	$-0.615988\pi$
$421$	−14.6244	−0.712750	−0.356375	+	0.934343i	$0.615988\pi$
$422$	−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	$-0.592701\pi$
$433$	−11.9495	−0.574257	−0.287129	+	0.957892i	$0.592701\pi$
$434$	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	$-0.392458\pi$
$449$	14.0471	0.662925	0.331463	+	0.943468i	$0.392458\pi$
$450$	−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	$-0.814770\pi$
$457$	−35.7181	−1.67082	−0.835410	+	0.549627i	$0.814770\pi$
$458$	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	$-0.491233\pi$
$463$	1.18519	0.0550806	0.0275403	+	0.999621i	$0.491233\pi$
$464$	−45.4487	−2.10990
$465$	−44.6404	−2.07015
$466$	58.6069	2.71491
$467$	26.2199	1.21331	0.606655	−	0.794965i	$-0.292511\pi$
$467$	26.2199	1.21331	0.606655	+	0.794965i	$0.292511\pi$
$468$	−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	$-0.388389\pi$
$491$	15.2226	0.686988	0.343494	+	0.939155i	$0.388389\pi$
$492$	−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	$-0.669096\pi$
$499$	−22.6329	−1.01319	−0.506594	+	0.862185i	$0.669096\pi$
$500$	54.3868	2.43225
$501$	37.2870	1.66586
$502$	−16.7287	−0.746637
$503$	10.2918	0.458888	0.229444	−	0.973322i	$-0.426309\pi$
$503$	10.2918	0.458888	0.229444	+	0.973322i	$0.426309\pi$
$504$	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	$-0.847033\pi$
$521$	−40.4803	−1.77347	−0.886736	+	0.462275i	$0.847033\pi$
$522$	−57.8008	−2.52987
$523$	−33.3575	−1.45862	−0.729311	−	0.684182i	$-0.760159\pi$
$523$	−33.3575	−1.45862	−0.729311	+	0.684182i	$0.760159\pi$
$524$	−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	$-0.655568\pi$
$571$	−22.4383	−0.939013	−0.469506	+	0.882929i	$0.655568\pi$
$572$	−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	$-0.637038\pi$
$577$	−20.0498	−0.834685	−0.417342	+	0.908749i	$0.637038\pi$
$578$	−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	$-0.611986\pi$
$587$	−16.6980	−0.689201	−0.344601	+	0.938749i	$0.611986\pi$
$588$	−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	$-0.681436\pi$
$593$	−26.2817	−1.07926	−0.539631	+	0.841902i	$0.681436\pi$
$594$	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	$-0.373132\pi$
$601$	19.0287	0.776195	0.388098	+	0.921618i	$0.373132\pi$
$602$	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	$-0.510526\pi$
$607$	−1.62907	−0.0661218	−0.0330609	+	0.999453i	$0.510526\pi$
$608$	−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	$-0.851438\pi$
$631$	−44.8663	−1.78610	−0.893049	+	0.449960i	$0.851438\pi$
$632$	−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	$-0.281148\pi$
$641$	32.1357	1.26928	0.634641	+	0.772807i	$0.281148\pi$
$642$	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	$-0.410191\pi$
$647$	14.1637	0.556831	0.278416	+	0.960461i	$0.410191\pi$
$648$	−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	$-0.429984\pi$
$653$	11.1514	0.436387	0.218194	+	0.975906i	$0.429984\pi$
$654$	−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	$-0.493095\pi$
$673$	1.12541	0.0433813	0.0216907	+	0.999765i	$0.493095\pi$
$674$	−8.94681	−0.344618
$675$	−1.97307	−0.0759435
$676$	−53.5953	−2.06136
$677$	3.40399	0.130826	0.0654130	−	0.997858i	$-0.479164\pi$
$677$	3.40399	0.130826	0.0654130	+	0.997858i	$0.479164\pi$
$678$	−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	$-0.653936\pi$
$691$	−24.4454	−0.929946	−0.464973	+	0.885325i	$0.653936\pi$
$692$	−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	$-0.348751\pi$
$727$	24.6702	0.914968	0.457484	+	0.889218i	$0.348751\pi$
$728$	−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	$-0.612955\pi$
$733$	−18.8142	−0.694917	−0.347459	+	0.937695i	$0.612955\pi$
$734$	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	$-0.441910\pi$
$739$	9.86714	0.362968	0.181484	+	0.983394i	$0.441910\pi$
$740$	10.2850	0.378084
$741$	16.1634	0.593777
$742$	21.4834	0.788679
$743$	−3.00273	−0.110159	−0.0550797	−	0.998482i	$-0.517541\pi$
$743$	−3.00273	−0.110159	−0.0550797	+	0.998482i	$0.517541\pi$
$744$	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	$-0.494949\pi$
$751$	0.869629	0.0317332	0.0158666	+	0.999874i	$0.494949\pi$
$752$	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	$-0.253196\pi$
$757$	38.5175	1.39994	0.699972	+	0.714170i	$0.253196\pi$
$758$	76.8497	2.79131
$759$	−44.5264	−1.61621
$760$	74.3058	2.69536
$761$	−10.5663	−0.383029	−0.191514	−	0.981490i	$-0.561340\pi$
$761$	−10.5663	−0.383029	−0.191514	+	0.981490i	$0.561340\pi$
$762$	−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	$-0.404866\pi$
$773$	16.3728	0.588887	0.294444	+	0.955669i	$0.404866\pi$
$774$	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	$-0.433411\pi$
$811$	11.8282	0.415344	0.207672	+	0.978199i	$0.433411\pi$
$812$	−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	$-0.686247\pi$
$829$	−31.8037	−1.10459	−0.552294	+	0.833649i	$0.686247\pi$
$830$	−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	$-0.374135\pi$
$839$	22.3146	0.770384	0.385192	+	0.922836i	$0.374135\pi$
$840$	−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	$-0.750889\pi$
$859$	−41.5643	−1.41816	−0.709079	+	0.705129i	$0.750889\pi$
$860$	0	0
$861$	−0.462252	−0.0157535
$862$	−24.6007	−0.837903
$863$	31.7216	1.07982	0.539908	−	0.841724i	$-0.318459\pi$
$863$	31.7216	1.07982	0.539908	+	0.841724i	$0.318459\pi$
$864$	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	$-0.811221\pi$
$887$	−49.3932	−1.65846	−0.829231	+	0.558906i	$0.811221\pi$
$888$	−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	$-0.387499\pi$
$911$	20.8938	0.692243	0.346121	+	0.938190i	$0.387499\pi$
$912$	−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	$-0.860054\pi$
$929$	−55.1618	−1.80980	−0.904900	+	0.425625i	$0.860054\pi$
$930$	−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	$-0.736616\pi$
$937$	−41.4317	−1.35352	−0.676758	+	0.736206i	$0.736616\pi$
$938$	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	$-0.481138\pi$
$953$	3.65648	0.118445	0.0592225	+	0.998245i	$0.481138\pi$
$954$	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	$-0.594676\pi$
$983$	−18.3770	−0.586136	−0.293068	+	0.956092i	$0.594676\pi$
$984$	−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	$-0.562567\pi$
$991$	−12.2959	−0.390593	−0.195296	+	0.980744i	$0.562567\pi$
$992$	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	$-0.888106\pi$
$997$	−59.2888	−1.87770	−0.938848	+	0.344331i	$0.888106\pi$
$998$	−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.r.1.19	yes	20
43.42	odd	2		1849.2.a.p.1.2	✓	20

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

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