LMFDB - Embedded newform 83.4.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [83,4,Mod(1,83)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(83, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("83.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 83 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	83.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:	$4.89715853048$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 25x^{5} + 20x^{4} + 156x^{3} - 134x^{2} - 124x + 60 $ x^7 - x^6 - 25x^5 + 20x^4 + 156x^3 - 134x^2 - 124*x + 60
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.54565$ of defining polynomial
Character	$\chi$	$=$	83.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.80719 q^{2} +7.94657 q^{3} +6.49468 q^{4} -12.5913 q^{5} -30.2541 q^{6} -28.3547 q^{7} +5.73105 q^{8} +36.1480 q^{9} +O(q^{10})$	\(q-3.80719 q^{2} +7.94657 q^{3} +6.49468 q^{4} -12.5913 q^{5} -30.2541 q^{6} -28.3547 q^{7} +5.73105 q^{8} +36.1480 q^{9} +47.9375 q^{10} +0.0120009 q^{11} +51.6104 q^{12} -57.9258 q^{13} +107.951 q^{14} -100.058 q^{15} -73.7766 q^{16} -61.8085 q^{17} -137.622 q^{18} +32.2279 q^{19} -81.7765 q^{20} -225.322 q^{21} -0.0456895 q^{22} -17.8687 q^{23} +45.5422 q^{24} +33.5412 q^{25} +220.534 q^{26} +72.6949 q^{27} -184.154 q^{28} +198.682 q^{29} +380.939 q^{30} -322.743 q^{31} +235.033 q^{32} +0.0953656 q^{33} +235.317 q^{34} +357.022 q^{35} +234.769 q^{36} -244.164 q^{37} -122.698 q^{38} -460.311 q^{39} -72.1615 q^{40} +272.534 q^{41} +857.844 q^{42} +422.829 q^{43} +0.0779416 q^{44} -455.150 q^{45} +68.0296 q^{46} +284.858 q^{47} -586.271 q^{48} +460.987 q^{49} -127.698 q^{50} -491.166 q^{51} -376.209 q^{52} +142.581 q^{53} -276.763 q^{54} -0.151107 q^{55} -162.502 q^{56} +256.101 q^{57} -756.418 q^{58} -785.267 q^{59} -649.843 q^{60} +845.983 q^{61} +1228.74 q^{62} -1024.96 q^{63} -304.602 q^{64} +729.362 q^{65} -0.363075 q^{66} -304.834 q^{67} -401.426 q^{68} -141.995 q^{69} -1359.25 q^{70} -308.116 q^{71} +207.166 q^{72} -623.191 q^{73} +929.580 q^{74} +266.538 q^{75} +209.310 q^{76} -0.340280 q^{77} +1752.49 q^{78} -157.159 q^{79} +928.944 q^{80} -398.320 q^{81} -1037.59 q^{82} +83.0000 q^{83} -1463.39 q^{84} +778.251 q^{85} -1609.79 q^{86} +1578.84 q^{87} +0.0687775 q^{88} +1123.01 q^{89} +1732.84 q^{90} +1642.47 q^{91} -116.052 q^{92} -2564.70 q^{93} -1084.51 q^{94} -405.791 q^{95} +1867.71 q^{96} +800.163 q^{97} -1755.06 q^{98} +0.433806 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 5 q^{2} - 11 q^{3} + 13 q^{4} - 17 q^{5} - 27 q^{6} - 46 q^{7} - 57 q^{8} + 16 q^{9}+O(q^{10}) $ 7 * q - 5 * q^2 - 11 * q^3 + 13 * q^4 - 17 * q^5 - 27 * q^6 - 46 * q^7 - 57 * q^8 + 16 * q^9	\( 7 q - 5 q^{2} - 11 q^{3} + 13 q^{4} - 17 q^{5} - 27 q^{6} - 46 q^{7} - 57 q^{8} + 16 q^{9} - 65 q^{10} - 59 q^{11} - 89 q^{12} - 205 q^{13} + 30 q^{14} - 136 q^{15} - 163 q^{16} - 148 q^{17} - 188 q^{18} - 29 q^{19} + 3 q^{20} - 197 q^{21} - 127 q^{22} + 159 q^{23} + 465 q^{24} - 108 q^{25} + 603 q^{26} + 217 q^{27} + 24 q^{28} + 29 q^{29} + 1000 q^{30} - 420 q^{31} + 459 q^{32} - 47 q^{33} + 688 q^{34} + 300 q^{35} + 1000 q^{36} - 1091 q^{37} + 327 q^{38} + 286 q^{39} + 119 q^{40} + 127 q^{41} + 1115 q^{42} - 49 q^{43} + 795 q^{44} - 347 q^{45} + 11 q^{46} + 150 q^{47} + 119 q^{48} - 521 q^{49} + 448 q^{50} - 95 q^{51} - 1129 q^{52} - 785 q^{53} + 717 q^{54} - 66 q^{55} + 290 q^{56} - 434 q^{57} - 143 q^{58} - 81 q^{59} - 952 q^{60} - 195 q^{61} + 1096 q^{62} - 913 q^{63} - 1203 q^{64} - 534 q^{65} + 1517 q^{66} - 1825 q^{67} - 578 q^{68} - 1910 q^{69} - 2012 q^{70} - 666 q^{71} - 1284 q^{72} - 1594 q^{73} + 713 q^{74} + 819 q^{75} - 581 q^{76} - 1307 q^{77} - 610 q^{78} - 968 q^{79} + 2347 q^{80} - 1193 q^{81} - 3961 q^{82} + 581 q^{83} - 1655 q^{84} - 2407 q^{85} - 1197 q^{86} + 3079 q^{87} - 1315 q^{88} + 1616 q^{89} + 313 q^{90} + 1077 q^{91} + 1127 q^{92} - 827 q^{93} + 534 q^{94} + 2252 q^{95} + 2221 q^{96} + 50 q^{97} + 1071 q^{98} + 3184 q^{99}+O(q^{100}) \) 7 * q - 5 * q^2 - 11 * q^3 + 13 * q^4 - 17 * q^5 - 27 * q^6 - 46 * q^7 - 57 * q^8 + 16 * q^9 - 65 * q^10 - 59 * q^11 - 89 * q^12 - 205 * q^13 + 30 * q^14 - 136 * q^15 - 163 * q^16 - 148 * q^17 - 188 * q^18 - 29 * q^19 + 3 * q^20 - 197 * q^21 - 127 * q^22 + 159 * q^23 + 465 * q^24 - 108 * q^25 + 603 * q^26 + 217 * q^27 + 24 * q^28 + 29 * q^29 + 1000 * q^30 - 420 * q^31 + 459 * q^32 - 47 * q^33 + 688 * q^34 + 300 * q^35 + 1000 * q^36 - 1091 * q^37 + 327 * q^38 + 286 * q^39 + 119 * q^40 + 127 * q^41 + 1115 * q^42 - 49 * q^43 + 795 * q^44 - 347 * q^45 + 11 * q^46 + 150 * q^47 + 119 * q^48 - 521 * q^49 + 448 * q^50 - 95 * q^51 - 1129 * q^52 - 785 * q^53 + 717 * q^54 - 66 * q^55 + 290 * q^56 - 434 * q^57 - 143 * q^58 - 81 * q^59 - 952 * q^60 - 195 * q^61 + 1096 * q^62 - 913 * q^63 - 1203 * q^64 - 534 * q^65 + 1517 * q^66 - 1825 * q^67 - 578 * q^68 - 1910 * q^69 - 2012 * q^70 - 666 * q^71 - 1284 * q^72 - 1594 * q^73 + 713 * q^74 + 819 * q^75 - 581 * q^76 - 1307 * q^77 - 610 * q^78 - 968 * q^79 + 2347 * q^80 - 1193 * q^81 - 3961 * q^82 + 581 * q^83 - 1655 * q^84 - 2407 * q^85 - 1197 * q^86 + 3079 * q^87 - 1315 * q^88 + 1616 * q^89 + 313 * q^90 + 1077 * q^91 + 1127 * q^92 - 827 * q^93 + 534 * q^94 + 2252 * q^95 + 2221 * q^96 + 50 * q^97 + 1071 * q^98 + 3184 * 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 $

$2$	−3.80719	−1.34604	−0.673022	−	0.739622i	$-0.735004\pi$
$2$	−3.80719	−1.34604	−0.673022	+	0.739622i	$0.735004\pi$
$3$	7.94657	1.52932	0.764659	−	0.644435i	$-0.222907\pi$
$3$	7.94657	1.52932	0.764659	+	0.644435i	$0.222907\pi$
$4$	6.49468	0.811834
$5$	−12.5913	−1.12620	−0.563101	−	0.826388i	$-0.690392\pi$
$5$	−12.5913	−1.12620	−0.563101	+	0.826388i	$0.690392\pi$
$6$	−30.2541	−2.05853
$7$	−28.3547	−1.53101	−0.765504	−	0.643431i	$-0.777510\pi$
$7$	−28.3547	−1.53101	−0.765504	+	0.643431i	$0.777510\pi$
$8$	5.73105	0.253279
$9$	36.1480	1.33881
$10$	47.9375	1.51592
$11$	0.0120009	0.000328945 0	0.000164472	−	1.00000i	$-0.499948\pi$
$11$	0.0120009	0.000328945 0	0.000164472		1.00000i	$0.499948\pi$
$12$	51.6104	1.24155
$13$	−57.9258	−1.23583	−0.617913	−	0.786247i	$-0.712021\pi$
$13$	−57.9258	−1.23583	−0.617913	+	0.786247i	$0.712021\pi$
$14$	107.951	2.06080
$15$	−100.058	−1.72232
$16$	−73.7766	−1.15276
$17$	−61.8085	−0.881810	−0.440905	−	0.897554i	$-0.645342\pi$
$17$	−61.8085	−0.881810	−0.440905	+	0.897554i	$0.645342\pi$
$18$	−137.622	−1.80210
$19$	32.2279	0.389136	0.194568	−	0.980889i	$-0.437670\pi$
$19$	32.2279	0.389136	0.194568	+	0.980889i	$0.437670\pi$
$20$	−81.7765	−0.914289
$21$	−225.322	−2.34140
$22$	−0.0456895	−0.000442774 0
$23$	−17.8687	−0.161995	−0.0809975	−	0.996714i	$-0.525811\pi$
$23$	−17.8687	−0.161995	−0.0809975	+	0.996714i	$0.525811\pi$
$24$	45.5422	0.387344
$25$	33.5412	0.268330
$26$	220.534	1.66348
$27$	72.6949	0.518153
$28$	−184.154	−1.24292
$29$	198.682	1.27222	0.636108	−	0.771600i	$-0.280543\pi$
$29$	198.682	1.27222	0.636108	+	0.771600i	$0.280543\pi$
$30$	380.939	2.31832
$31$	−322.743	−1.86988	−0.934941	−	0.354803i	$-0.884548\pi$
$31$	−322.743	−1.86988	−0.934941	+	0.354803i	$0.884548\pi$
$32$	235.033	1.29839
$33$	0.0953656	0.000503061 0
$34$	235.317	1.18696
$35$	357.022	1.72422
$36$	234.769	1.08689
$37$	−244.164	−1.08488	−0.542438	−	0.840096i	$-0.682499\pi$
$37$	−244.164	−1.08488	−0.542438	+	0.840096i	$0.682499\pi$
$38$	−122.698	−0.523794
$39$	−460.311	−1.88997
$40$	−72.1615	−0.285243
$41$	272.534	1.03811	0.519057	−	0.854740i	$-0.326283\pi$
$41$	272.534	1.03811	0.519057	+	0.854740i	$0.326283\pi$
$42$	857.844	3.15162
$43$	422.829	1.49955	0.749777	−	0.661691i	$-0.230161\pi$
$43$	422.829	1.49955	0.749777	+	0.661691i	$0.230161\pi$
$44$	0.0779416	0.000267049 0
$45$	−455.150	−1.50777
$46$	68.0296	0.218052
$47$	284.858	0.884059	0.442029	−	0.897001i	$-0.354259\pi$
$47$	284.858	0.884059	0.442029	+	0.897001i	$0.354259\pi$
$48$	−586.271	−1.76294
$49$	460.987	1.34398
$50$	−127.698	−0.361184
$51$	−491.166	−1.34857
$52$	−376.209	−1.00329
$53$	142.581	0.369528	0.184764	−	0.982783i	$-0.440848\pi$
$53$	142.581	0.369528	0.184764	+	0.982783i	$0.440848\pi$
$54$	−276.763	−0.697457
$55$	−0.151107	−0.000370458 0
$56$	−162.502	−0.387772
$57$	256.101	0.595112
$58$	−756.418	−1.71246
$59$	−785.267	−1.73276	−0.866382	−	0.499382i	$-0.833560\pi$
$59$	−785.267	−1.73276	−0.866382	+	0.499382i	$0.833560\pi$
$60$	−649.843	−1.39824
$61$	845.983	1.77569	0.887845	−	0.460143i	$-0.152202\pi$
$61$	845.983	1.77569	0.887845	+	0.460143i	$0.152202\pi$
$62$	1228.74	2.51694
$63$	−1024.96	−2.04973
$64$	−304.602	−0.594925
$65$	729.362	1.39179
$66$	−0.363075	−0.000677142 0
$67$	−304.834	−0.555842	−0.277921	−	0.960604i	$-0.589645\pi$
$67$	−304.834	−0.555842	−0.277921	+	0.960604i	$0.589645\pi$
$68$	−401.426	−0.715884
$69$	−141.995	−0.247742
$70$	−1359.25	−2.32088
$71$	−308.116	−0.515022	−0.257511	−	0.966275i	$-0.582902\pi$
$71$	−308.116	−0.515022	−0.257511	+	0.966275i	$0.582902\pi$
$72$	207.166	0.339093
$73$	−623.191	−0.999164	−0.499582	−	0.866266i	$-0.666513\pi$
$73$	−623.191	−0.999164	−0.499582	+	0.866266i	$0.666513\pi$
$74$	929.580	1.46029
$75$	266.538	0.410362
$76$	209.310	0.315914
$77$	−0.340280	−0.000503617 0
$78$	1752.49	2.54398
$79$	−157.159	−0.223819	−0.111910	−	0.993718i	$-0.535697\pi$
$79$	−157.159	−0.223819	−0.111910	+	0.993718i	$0.535697\pi$
$80$	928.944	1.29824
$81$	−398.320	−0.546392
$82$	−1037.59	−1.39735
$83$	83.0000	0.109764
$83$	83.0000	0.109764
$84$	−1463.39	−1.90083
$85$	778.251	0.993096
$86$	−1609.79	−2.01847
$87$	1578.84	1.94562
$88$	0.0687775	8.33148e−5 0
$89$	1123.01	1.33752	0.668758	−	0.743480i	$-0.266826\pi$
$89$	1123.01	1.33752	0.668758	+	0.743480i	$0.266826\pi$
$90$	1732.84	2.02953
$91$	1642.47	1.89206
$92$	−116.052	−0.131513
$93$	−2564.70	−2.85964
$94$	−1084.51	−1.18998
$95$	−405.791	−0.438245
$96$	1867.71	1.98564
$97$	800.163	0.837570	0.418785	−	0.908086i	$-0.362456\pi$
$97$	800.163	0.837570	0.418785	+	0.908086i	$0.362456\pi$
$98$	−1755.06	−1.80906
$99$	0.433806	0.000440396 0
$100$	217.839	0.217839
$101$	−379.926	−0.374298	−0.187149	−	0.982332i	$-0.559925\pi$
$101$	−379.926	−0.374298	−0.187149	+	0.982332i	$0.559925\pi$
$102$	1869.96	1.81523
$103$	−544.100	−0.520503	−0.260251	−	0.965541i	$-0.583805\pi$
$103$	−544.100	−0.520503	−0.260251	+	0.965541i	$0.583805\pi$
$104$	−331.976	−0.313009
$105$	2837.10	2.63689
$106$	−542.832	−0.497401
$107$	−758.333	−0.685148	−0.342574	−	0.939491i	$-0.611299\pi$
$107$	−758.333	−0.685148	−0.342574	+	0.939491i	$0.611299\pi$
$108$	472.130	0.420655
$109$	−2115.61	−1.85907	−0.929533	−	0.368738i	$-0.879790\pi$
$109$	−2115.61	−1.85907	−0.929533	+	0.368738i	$0.879790\pi$
$110$	0.575291	0.000498653 0
$111$	−1940.27	−1.65912
$112$	2091.91	1.76488
$113$	−2154.91	−1.79396	−0.896979	−	0.442074i	$-0.854243\pi$
$113$	−2154.91	−1.79396	−0.896979	+	0.442074i	$0.854243\pi$
$114$	−975.025	−0.801047
$115$	224.991	0.182439
$116$	1290.37	1.03283
$117$	−2093.90	−1.65454
$118$	2989.66	2.33238
$119$	1752.56	1.35006
$120$	−573.436	−0.436228
$121$	−1331.00	−1.00000
$122$	−3220.82	−2.39016
$123$	2165.71	1.58761
$124$	−2096.11	−1.51803
$125$	1151.59	0.824008
$126$	3902.23	2.75903
$127$	−1899.16	−1.32695	−0.663477	−	0.748196i	$-0.730920\pi$
$127$	−1899.16	−1.32695	−0.663477	+	0.748196i	$0.730920\pi$
$128$	−720.588	−0.497591
$129$	3360.04	2.29329
$130$	−2776.82	−1.87341
$131$	1854.99	1.23719	0.618593	−	0.785712i	$-0.287703\pi$
$131$	1854.99	1.23719	0.618593	+	0.785712i	$0.287703\pi$
$132$	0.619369	0.000408402 0
$133$	−913.810	−0.595770
$134$	1160.56	0.748188
$135$	−915.324	−0.583545
$136$	−354.228	−0.223344
$137$	−1889.10	−1.17808	−0.589039	−	0.808105i	$-0.700493\pi$
$137$	−1889.10	−1.17808	−0.589039	+	0.808105i	$0.700493\pi$
$138$	540.602	0.333472
$139$	1185.22	0.723230	0.361615	−	0.932327i	$-0.382225\pi$
$139$	1185.22	0.723230	0.361615	+	0.932327i	$0.382225\pi$
$140$	2318.74	1.39978
$141$	2263.64	1.35201
$142$	1173.05	0.693243
$143$	−0.695159	−0.000406518 0
$144$	−2666.87	−1.54333
$145$	−2501.66	−1.43277
$146$	2372.61	1.34492
$147$	3663.26	2.05538
$148$	−1585.77	−0.880739
$149$	133.908	0.0736253	0.0368126	−	0.999322i	$-0.488280\pi$
$149$	133.908	0.0736253	0.0368126	+	0.999322i	$0.488280\pi$
$150$	−1014.76	−0.552365
$151$	1062.50	0.572616	0.286308	−	0.958138i	$-0.407572\pi$
$151$	1062.50	0.572616	0.286308	+	0.958138i	$0.407572\pi$
$152$	184.700	0.0985600
$153$	−2234.25	−1.18058
$154$	1.29551	0.000677891 0
$155$	4063.76	2.10586
$156$	−2989.57	−1.53434
$157$	−849.040	−0.431597	−0.215799	−	0.976438i	$-0.569235\pi$
$157$	−849.040	−0.431597	−0.215799	+	0.976438i	$0.569235\pi$
$158$	598.333	0.301271
$159$	1133.03	0.565126
$160$	−2959.37	−1.46224
$161$	506.661	0.248016
$162$	1516.48	0.735468
$163$	1239.50	0.595613	0.297807	−	0.954626i	$-0.403745\pi$
$163$	1239.50	0.595613	0.297807	+	0.954626i	$0.403745\pi$
$164$	1770.02	0.842776
$165$	−1.20078	−0.000566548 0
$166$	−315.997	−0.147748
$167$	−555.704	−0.257495	−0.128748	−	0.991677i	$-0.541096\pi$
$167$	−555.704	−0.257495	−0.128748	+	0.991677i	$0.541096\pi$
$168$	−1291.33	−0.593027
$169$	1158.40	0.527264
$170$	−2962.95	−1.33675
$171$	1164.97	0.520980
$172$	2746.14	1.21739
$173$	−1467.33	−0.644850	−0.322425	−	0.946595i	$-0.604498\pi$
$173$	−1467.33	−0.644850	−0.322425	+	0.946595i	$0.604498\pi$
$174$	−6010.93	−2.61889
$175$	−951.050	−0.410815
$176$	−0.885382	−0.000379194 0
$177$	−6240.18	−2.64995
$178$	−4275.51	−1.80035
$179$	−3443.83	−1.43801	−0.719006	−	0.695004i	$-0.755402\pi$
$179$	−3443.83	−1.43801	−0.719006	+	0.695004i	$0.755402\pi$
$180$	−2956.05	−1.22406
$181$	−1109.02	−0.455430	−0.227715	−	0.973728i	$-0.573125\pi$
$181$	−1109.02	−0.455430	−0.227715	+	0.973728i	$0.573125\pi$
$182$	−6253.18	−2.54679
$183$	6722.67	2.71559
$184$	−102.407	−0.0410300
$185$	3074.35	1.22179
$186$	9764.29	3.84921
$187$	−0.741755	−0.000290067 0
$188$	1850.06	0.717710
$189$	−2061.24	−0.793297
$190$	1544.92	0.589898
$191$	740.969	0.280705	0.140352	−	0.990102i	$-0.455176\pi$
$191$	740.969	0.280705	0.140352	+	0.990102i	$0.455176\pi$
$192$	−2420.54	−0.909829
$193$	−1702.99	−0.635150	−0.317575	−	0.948233i	$-0.602868\pi$
$193$	−1702.99	−0.635150	−0.317575	+	0.948233i	$0.602868\pi$
$194$	−3046.37	−1.12741
$195$	5795.93	2.12849
$196$	2993.96	1.09109
$197$	−1909.10	−0.690445	−0.345223	−	0.938521i	$-0.612197\pi$
$197$	−1909.10	−0.690445	−0.345223	+	0.938521i	$0.612197\pi$
$198$	−1.65158	−0.000592792 0
$199$	4129.79	1.47112	0.735561	−	0.677459i	$-0.236919\pi$
$199$	4129.79	1.47112	0.735561	+	0.677459i	$0.236919\pi$
$200$	192.227	0.0679624
$201$	−2422.39	−0.850060
$202$	1446.45	0.503821
$203$	−5633.55	−1.94777
$204$	−3189.96	−1.09481
$205$	−3431.56	−1.16912
$206$	2071.49	0.700620
$207$	−645.918	−0.216881
$208$	4273.57	1.42461
$209$	0.386762	0.000128004 0
$210$	−10801.4	−3.54936
$211$	5043.06	1.64540	0.822698	−	0.568478i	$-0.192468\pi$
$211$	5043.06	1.64540	0.822698	+	0.568478i	$0.192468\pi$
$212$	926.017	0.299996
$213$	−2448.46	−0.787633
$214$	2887.12	0.922240
$215$	−5323.97	−1.68880
$216$	416.618	0.131237
$217$	9151.26	2.86280
$218$	8054.51	2.50239
$219$	−4952.23	−1.52804
$220$	−0.981388	−0.000300751 0
$221$	3580.31	1.08976
$222$	7386.97	2.23325
$223$	−3214.36	−0.965245	−0.482622	−	0.875829i	$-0.660316\pi$
$223$	−3214.36	−0.965245	−0.482622	+	0.875829i	$0.660316\pi$
$224$	−6664.28	−1.98784
$225$	1212.45	0.359244
$226$	8204.16	2.41475
$227$	12.9954	0.00379970	0.00189985	−	0.999998i	$-0.499395\pi$
$227$	12.9954	0.00379970	0.00189985	+	0.999998i	$0.499395\pi$
$228$	1663.29	0.483133
$229$	2372.00	0.684481	0.342240	−	0.939612i	$-0.388814\pi$
$229$	2372.00	0.684481	0.342240	+	0.939612i	$0.388814\pi$
$230$	−856.582	−0.245571
$231$	−2.70406	−0.000770190 0
$232$	1138.65	0.322226
$233$	−1593.27	−0.447977	−0.223988	−	0.974592i	$-0.571908\pi$
$233$	−1593.27	−0.447977	−0.223988	+	0.974592i	$0.571908\pi$
$234$	7971.87	2.22708
$235$	−3586.73	−0.995629
$236$	−5100.05	−1.40672
$237$	−1248.87	−0.342291
$238$	−6672.32	−1.81724
$239$	−1320.09	−0.357277	−0.178639	−	0.983915i	$-0.557169\pi$
$239$	−1320.09	−0.357277	−0.178639	+	0.983915i	$0.557169\pi$
$240$	7381.92	1.98542
$241$	−690.697	−0.184613	−0.0923064	−	0.995731i	$-0.529424\pi$
$241$	−690.697	−0.184613	−0.0923064	+	0.995731i	$0.529424\pi$
$242$	5067.37	1.34604
$243$	−5128.04	−1.35376
$244$	5494.39	1.44157
$245$	−5804.43	−1.51360
$246$	−8245.26	−2.13699
$247$	−1866.83	−0.480904
$248$	−1849.66	−0.473602
$249$	659.565	0.167864
$250$	−4384.30	−1.10915
$251$	775.388	0.194988	0.0974942	−	0.995236i	$-0.468917\pi$
$251$	775.388	0.194988	0.0974942	+	0.995236i	$0.468917\pi$
$252$	−6656.80	−1.66404
$253$	−0.214440	−5.32874e−5 0
$254$	7230.46	1.78614
$255$	6184.42	1.51876
$256$	5180.23	1.26470
$257$	393.192	0.0954344	0.0477172	−	0.998861i	$-0.484805\pi$
$257$	393.192	0.0954344	0.0477172	+	0.998861i	$0.484805\pi$
$258$	−12792.3	−3.08688
$259$	6923.20	1.66095
$260$	4736.97	1.12990
$261$	7181.94	1.70326
$262$	−7062.30	−1.66531
$263$	4383.00	1.02763	0.513816	−	0.857900i	$-0.328231\pi$
$263$	4383.00	1.02763	0.513816	+	0.857900i	$0.328231\pi$
$264$	0.546545	0.000127415 0
$265$	−1795.28	−0.416163
$266$	3479.05	0.801932
$267$	8924.08	2.04549
$268$	−1979.80	−0.451252
$269$	7098.66	1.60897	0.804485	−	0.593973i	$-0.202441\pi$
$269$	7098.66	1.60897	0.804485	+	0.593973i	$0.202441\pi$
$270$	3484.81	0.785477
$271$	−1608.53	−0.360559	−0.180280	−	0.983615i	$-0.557700\pi$
$271$	−1608.53	−0.360559	−0.180280	+	0.983615i	$0.557700\pi$
$272$	4560.02	1.01651
$273$	13052.0	2.89356
$274$	7192.15	1.58574
$275$	0.402523	8.82657e−5 0
$276$	−922.212	−0.201125
$277$	−6087.31	−1.32040	−0.660200	−	0.751090i	$-0.729529\pi$
$277$	−6087.31	−1.32040	−0.660200	+	0.751090i	$0.729529\pi$
$278$	−4512.35	−0.973500
$279$	−11666.5	−2.50342
$280$	2046.11	0.436710
$281$	−3490.58	−0.741033	−0.370517	−	0.928826i	$-0.620819\pi$
$281$	−3490.58	−0.741033	−0.370517	+	0.928826i	$0.620819\pi$
$282$	−8618.11	−1.81986
$283$	317.132	0.0666133	0.0333066	−	0.999445i	$-0.489396\pi$
$283$	317.132	0.0666133	0.0333066	+	0.999445i	$0.489396\pi$
$284$	−2001.11	−0.418113
$285$	−3224.65	−0.670217
$286$	2.64660	0.000547192 0
$287$	−7727.61	−1.58936
$288$	8495.96	1.73830
$289$	−1092.71	−0.222411
$290$	9524.30	1.92857
$291$	6358.55	1.28091
$292$	−4047.42	−0.811156
$293$	5795.82	1.15562	0.577808	−	0.816173i	$-0.303908\pi$
$293$	5795.82	1.15562	0.577808	+	0.816173i	$0.303908\pi$
$294$	−13946.7	−2.76663
$295$	9887.54	1.95144
$296$	−1399.32	−0.274776
$297$	0.872401	0.000170444 0
$298$	−509.813	−0.0991029
$299$	1035.06	0.200198
$300$	1731.08	0.333146
$301$	−11989.2	−2.29583
$302$	−4045.14	−0.770767
$303$	−3019.11	−0.572420
$304$	−2377.66	−0.448580
$305$	−10652.0	−1.99978
$306$	8506.22	1.58911
$307$	−1917.58	−0.356488	−0.178244	−	0.983986i	$-0.557042\pi$
$307$	−1917.58	−0.356488	−0.178244	+	0.983986i	$0.557042\pi$
$308$	−2.21001	−0.000408854 0
$309$	−4323.73	−0.796014
$310$	−15471.5	−2.83459
$311$	−5502.29	−1.00324	−0.501618	−	0.865089i	$-0.667262\pi$
$311$	−5502.29	−1.00324	−0.501618	+	0.865089i	$0.667262\pi$
$312$	−2638.07	−0.478690
$313$	1361.44	0.245857	0.122928	−	0.992416i	$-0.460771\pi$
$313$	1361.44	0.245857	0.122928	+	0.992416i	$0.460771\pi$
$314$	3232.45	0.580949
$315$	12905.6	2.30841
$316$	−1020.69	−0.181704
$317$	3270.64	0.579488	0.289744	−	0.957104i	$-0.406430\pi$
$317$	3270.64	0.579488	0.289744	+	0.957104i	$0.406430\pi$
$318$	−4313.65	−0.760685
$319$	2.38435	0.000418489 0
$320$	3835.33	0.670005
$321$	−6026.15	−1.04781
$322$	−1928.96	−0.333840
$323$	−1991.96	−0.343144
$324$	−2586.96	−0.443580
$325$	−1942.90	−0.331609
$326$	−4719.00	−0.801722
$327$	−16811.8	−2.84310
$328$	1561.91	0.262932
$329$	−8077.04	−1.35350
$330$	4.57159	0.000762599 0
$331$	6625.13	1.10015	0.550076	−	0.835115i	$-0.314599\pi$
$331$	6625.13	1.10015	0.550076	+	0.835115i	$0.314599\pi$
$332$	539.058	0.0891104
$333$	−8826.05	−1.45245
$334$	2115.67	0.346600
$335$	3838.26	0.625990
$336$	16623.5	2.69907
$337$	−3981.65	−0.643604	−0.321802	−	0.946807i	$-0.604288\pi$
$337$	−3981.65	−0.643604	−0.321802	+	0.946807i	$0.604288\pi$
$338$	−4410.24	−0.709721
$339$	−17124.2	−2.74353
$340$	5054.49	0.806229
$341$	−3.87319	−0.000615088 0
$342$	−4435.27	−0.701262
$343$	−3345.47	−0.526642
$344$	2423.25	0.379806
$345$	1787.90	0.279007
$346$	5586.40	0.867996
$347$	−1462.11	−0.226197	−0.113098	−	0.993584i	$-0.536078\pi$
$347$	−1462.11	−0.226197	−0.113098	+	0.993584i	$0.536078\pi$
$348$	10254.0	1.57952
$349$	−2700.65	−0.414220	−0.207110	−	0.978318i	$-0.566406\pi$
$349$	−2700.65	−0.414220	−0.207110	+	0.978318i	$0.566406\pi$
$350$	3620.83	0.552975
$351$	−4210.91	−0.640347
$352$	2.82059	0.000427097 0
$353$	5438.22	0.819964	0.409982	−	0.912094i	$-0.365535\pi$
$353$	5438.22	0.819964	0.409982	+	0.912094i	$0.365535\pi$
$354$	23757.5	3.56694
$355$	3879.58	0.580019
$356$	7293.59	1.08584
$357$	13926.8	2.06467
$358$	13111.3	1.93563
$359$	7341.00	1.07923	0.539615	−	0.841912i	$-0.318570\pi$
$359$	7341.00	1.07923	0.539615	+	0.841912i	$0.318570\pi$
$360$	−2608.49	−0.381888
$361$	−5820.36	−0.848573
$362$	4222.25	0.613028
$363$	−10576.9	−1.52932
$364$	10667.3	1.53604
$365$	7846.80	1.12526
$366$	−25594.4	−3.65531
$367$	6132.29	0.872215	0.436108	−	0.899894i	$-0.356357\pi$
$367$	6132.29	0.872215	0.436108	+	0.899894i	$0.356357\pi$
$368$	1318.29	0.186741
$369$	9851.55	1.38984
$370$	−11704.6	−1.64458
$371$	−4042.83	−0.565751
$372$	−16656.9	−2.32156
$373$	9946.05	1.38066	0.690331	−	0.723493i	$-0.257465\pi$
$373$	9946.05	1.38066	0.690331	+	0.723493i	$0.257465\pi$
$374$	2.82400	0.000390443 0
$375$	9151.16	1.26017
$376$	1632.53	0.223914
$377$	−11508.8	−1.57224
$378$	7847.52	1.06781
$379$	−2295.97	−0.311177	−0.155589	−	0.987822i	$-0.549727\pi$
$379$	−2295.97	−0.311177	−0.155589	+	0.987822i	$0.549727\pi$
$380$	−2635.48	−0.355783
$381$	−15091.8	−2.02934
$382$	−2821.01	−0.377841
$383$	5692.84	0.759505	0.379753	−	0.925088i	$-0.376009\pi$
$383$	5692.84	0.759505	0.379753	+	0.925088i	$0.376009\pi$
$384$	−5726.20	−0.760974
$385$	4.28457	0.000567174 0
$386$	6483.60	0.854939
$387$	15284.4	2.00762
$388$	5196.80	0.679968
$389$	7446.02	0.970510	0.485255	−	0.874373i	$-0.338727\pi$
$389$	7446.02	0.970510	0.485255	+	0.874373i	$0.338727\pi$
$390$	−22066.2	−2.86504
$391$	1104.44	0.142849
$392$	2641.94	0.340403
$393$	14740.8	1.89205
$394$	7268.30	0.929369
$395$	1978.83	0.252066
$396$	2.81743	0.000357528 0
$397$	−5425.96	−0.685948	−0.342974	−	0.939345i	$-0.611434\pi$
$397$	−5425.96	−0.685948	−0.342974	+	0.939345i	$0.611434\pi$
$398$	−15722.9	−1.98019
$399$	−7261.66	−0.911122
$400$	−2474.56	−0.309320
$401$	−1679.58	−0.209163	−0.104581	−	0.994516i	$-0.533350\pi$
$401$	−1679.58	−0.209163	−0.104581	+	0.994516i	$0.533350\pi$
$402$	9222.48	1.14422
$403$	18695.1	2.31085
$404$	−2467.50	−0.303868
$405$	5015.37	0.615348
$406$	21448.0	2.62179
$407$	−2.93018	−0.000356864 0
$408$	−2814.90	−0.341564
$409$	−2691.48	−0.325391	−0.162696	−	0.986676i	$-0.552019\pi$
$409$	−2691.48	−0.325391	−0.162696	+	0.986676i	$0.552019\pi$
$410$	13064.6	1.57369
$411$	−15011.9	−1.80165
$412$	−3533.76	−0.422562
$413$	22266.0	2.65287
$414$	2459.13	0.291932
$415$	−1045.08	−0.123617
$416$	−13614.5	−1.60458
$417$	9418.43	1.10605
$418$	−1.47248	−0.000172299 0
$419$	1381.06	0.161025	0.0805124	−	0.996754i	$-0.474344\pi$
$419$	1381.06	0.161025	0.0805124	+	0.996754i	$0.474344\pi$
$420$	18426.1	2.14071
$421$	−11390.6	−1.31863	−0.659317	−	0.751865i	$-0.729155\pi$
$421$	−11390.6	−1.31863	−0.659317	+	0.751865i	$0.729155\pi$
$422$	−19199.9	−2.21478
$423$	10297.0	1.18359
$424$	817.139	0.0935938
$425$	−2073.13	−0.236616
$426$	9321.75	1.06019
$427$	−23987.6	−2.71859
$428$	−4925.13	−0.556227
$429$	−5.52413	−0.000621696 0
$430$	20269.4	2.27320
$431$	4521.68	0.505341	0.252670	−	0.967552i	$-0.418691\pi$
$431$	4521.68	0.505341	0.252670	+	0.967552i	$0.418691\pi$
$432$	−5363.18	−0.597306
$433$	−17310.3	−1.92120	−0.960602	−	0.277928i	$-0.910352\pi$
$433$	−17310.3	−1.92120	−0.960602	+	0.277928i	$0.910352\pi$
$434$	−34840.6	−3.85346
$435$	−19879.6	−2.19116
$436$	−13740.2	−1.50925
$437$	−575.871	−0.0630381
$438$	18854.1	2.05681
$439$	8411.60	0.914496	0.457248	−	0.889339i	$-0.348835\pi$
$439$	8411.60	0.914496	0.457248	+	0.889339i	$0.348835\pi$
$440$	−0.865999	−9.38293e−5 0
$441$	16663.7	1.79934
$442$	−13630.9	−1.46687
$443$	−11828.2	−1.26857	−0.634283	−	0.773101i	$-0.718705\pi$
$443$	−11828.2	−1.26857	−0.634283	+	0.773101i	$0.718705\pi$
$444$	−12601.4	−1.34693
$445$	−14140.2	−1.50631
$446$	12237.7	1.29926
$447$	1064.11	0.112596
$448$	8636.87	0.910834
$449$	15313.0	1.60950	0.804750	−	0.593614i	$-0.202299\pi$
$449$	15313.0	1.60950	0.804750	+	0.593614i	$0.202299\pi$
$450$	−4616.01	−0.483558
$451$	3.27064	0.000341482 0
$452$	−13995.5	−1.45640
$453$	8443.23	0.875712
$454$	−49.4758	−0.00511457
$455$	−20680.8	−2.13084
$456$	1467.73	0.150730
$457$	−1120.84	−0.114728	−0.0573638	−	0.998353i	$-0.518269\pi$
$457$	−1120.84	−0.114728	−0.0573638	+	0.998353i	$0.518269\pi$
$458$	−9030.64	−0.921341
$459$	−4493.16	−0.456913
$460$	1461.24	0.148110
$461$	−11420.3	−1.15379	−0.576896	−	0.816818i	$-0.695736\pi$
$461$	−11420.3	−1.15379	−0.576896	+	0.816818i	$0.695736\pi$
$462$	10.2949	0.00103671
$463$	−14926.7	−1.49828	−0.749138	−	0.662414i	$-0.769532\pi$
$463$	−14926.7	−1.49828	−0.749138	+	0.662414i	$0.769532\pi$
$464$	−14658.1	−1.46656
$465$	32292.9	3.22054
$466$	6065.88	0.602996
$467$	12705.7	1.25900	0.629498	−	0.777002i	$-0.283261\pi$
$467$	12705.7	1.25900	0.629498	+	0.777002i	$0.283261\pi$
$468$	−13599.2	−1.34321
$469$	8643.47	0.850999
$470$	13655.4	1.34016
$471$	−6746.95	−0.660049
$472$	−4500.41	−0.438873
$473$	5.07431	0.000493270 0
$474$	4754.69	0.460739
$475$	1080.96	0.104417
$476$	11382.3	1.09602
$477$	5154.01	0.494729
$478$	5025.82	0.480911
$479$	3713.69	0.354243	0.177122	−	0.984189i	$-0.443321\pi$
$479$	3713.69	0.354243	0.177122	+	0.984189i	$0.443321\pi$
$480$	−23516.9	−2.23624
$481$	14143.4	1.34072
$482$	2629.61	0.248497
$483$	4026.22	0.379295
$484$	−8644.41	−0.811834
$485$	−10075.1	−0.943272
$486$	19523.4	1.82222
$487$	−5738.58	−0.533963	−0.266981	−	0.963702i	$-0.586026\pi$
$487$	−5738.58	−0.533963	−0.266981	+	0.963702i	$0.586026\pi$
$488$	4848.38	0.449745
$489$	9849.76	0.910882
$490$	22098.5	2.03737
$491$	−4298.02	−0.395045	−0.197522	−	0.980298i	$-0.563290\pi$
$491$	−4298.02	−0.395045	−0.197522	+	0.980298i	$0.563290\pi$
$492$	14065.6	1.28887
$493$	−12280.2	−1.12185
$494$	7107.35	0.647318
$495$	−5.46219	−0.000495974 0
$496$	23810.9	2.15552
$497$	8736.51	0.788503
$498$	−2511.09	−0.225953
$499$	21386.2	1.91859	0.959296	−	0.282404i	$-0.0911317\pi$
$499$	21386.2	1.91859	0.959296	+	0.282404i	$0.0911317\pi$
$500$	7479.18	0.668958
$501$	−4415.94	−0.393792
$502$	−2952.05	−0.262463
$503$	13066.2	1.15824	0.579119	−	0.815243i	$-0.303397\pi$
$503$	13066.2	1.15824	0.579119	+	0.815243i	$0.303397\pi$
$504$	−5874.12	−0.519155
$505$	4783.77	0.421535
$506$	0.816413	7.17272e−5 0
$507$	9205.30	0.806354
$508$	−12334.4	−1.07727
$509$	−16716.4	−1.45568	−0.727841	−	0.685746i	$-0.759476\pi$
$509$	−16716.4	−1.45568	−0.727841	+	0.685746i	$0.759476\pi$
$510$	−23545.3	−2.04432
$511$	17670.4	1.52973
$512$	−13957.4	−1.20476
$513$	2342.80	0.201632
$514$	−1496.96	−0.128459
$515$	6850.94	0.586191
$516$	21822.4	1.86178
$517$	3.41854	0.000290807 0
$518$	−26357.9	−2.23571
$519$	−11660.2	−0.986181
$520$	4180.01	0.352511
$521$	1471.21	0.123714	0.0618568	−	0.998085i	$-0.480298\pi$
$521$	1471.21	0.123714	0.0618568	+	0.998085i	$0.480298\pi$
$522$	−27343.0	−2.29266
$523$	4187.22	0.350085	0.175042	−	0.984561i	$-0.443994\pi$
$523$	4187.22	0.350085	0.175042	+	0.984561i	$0.443994\pi$
$524$	12047.6	1.00439
$525$	−7557.59	−0.628267
$526$	−16686.9	−1.38324
$527$	19948.3	1.64888
$528$	−7.03575	−0.000579908 0
$529$	−11847.7	−0.973758
$530$	6834.97	0.560174
$531$	−28385.8	−2.31985
$532$	−5934.90	−0.483667
$533$	−15786.7	−1.28293
$534$	−33975.6	−2.75331
$535$	9548.41	0.771615
$536$	−1747.02	−0.140783
$537$	−27366.6	−2.19918
$538$	−27025.9	−2.16574
$539$	5.53223	0.000442097 0
$540$	−5944.73	−0.473742
$541$	−1020.26	−0.0810801	−0.0405401	−	0.999178i	$-0.512908\pi$
$541$	−1020.26	−0.0810801	−0.0405401	+	0.999178i	$0.512908\pi$
$542$	6123.99	0.485328
$543$	−8812.90	−0.696497
$544$	−14527.0	−1.14493
$545$	26638.3	2.09368
$546$	−49691.3	−3.89486
$547$	8385.19	0.655439	0.327719	−	0.944775i	$-0.393720\pi$
$547$	8385.19	0.655439	0.327719	+	0.944775i	$0.393720\pi$
$548$	−12269.1	−0.956404
$549$	30580.6	2.37732
$550$	−1.53248	−0.000118810 0
$551$	6403.09	0.495065
$552$	−813.781	−0.0627479
$553$	4456.18	0.342669
$554$	23175.5	1.77732
$555$	24430.5	1.86850
$556$	7697.62	0.587143
$557$	−4275.70	−0.325255	−0.162628	−	0.986688i	$-0.551997\pi$
$557$	−4275.70	−0.325255	−0.162628	+	0.986688i	$0.551997\pi$
$558$	44416.5	3.36972
$559$	−24492.7	−1.85319
$560$	−26339.9	−1.98761
$561$	−5.89441	−0.000443604 0
$562$	13289.3	0.997463
$563$	−22908.5	−1.71488	−0.857442	−	0.514581i	$-0.827948\pi$
$563$	−22908.5	−1.71488	−0.857442	+	0.514581i	$0.827948\pi$
$564$	14701.6	1.09761
$565$	27133.2	2.02036
$566$	−1207.38	−0.0896644
$567$	11294.2	0.836531
$568$	−1765.83	−0.130444
$569$	−6800.06	−0.501008	−0.250504	−	0.968116i	$-0.580596\pi$
$569$	−6800.06	−0.501008	−0.250504	+	0.968116i	$0.580596\pi$
$570$	12276.8	0.902141
$571$	−4495.10	−0.329447	−0.164723	−	0.986340i	$-0.552673\pi$
$571$	−4495.10	−0.329447	−0.164723	+	0.986340i	$0.552673\pi$
$572$	−4.51483	−0.000330026 0
$573$	5888.16	0.429287
$574$	29420.4	2.13935
$575$	−599.339	−0.0434681
$576$	−11010.7	−0.796493
$577$	8162.39	0.588917	0.294458	−	0.955664i	$-0.404861\pi$
$577$	8162.39	0.588917	0.294458	+	0.955664i	$0.404861\pi$
$578$	4160.13	0.299375
$579$	−13532.9	−0.971346
$580$	−16247.5	−1.16317
$581$	−2353.44	−0.168050
$582$	−24208.2	−1.72416
$583$	1.71109	0.000121554 0
$584$	−3571.54	−0.253067
$585$	26365.0	1.86334
$586$	−22065.8	−1.55551
$587$	15820.6	1.11241	0.556207	−	0.831044i	$-0.312256\pi$
$587$	15820.6	1.11241	0.556207	+	0.831044i	$0.312256\pi$
$588$	23791.7	1.66863
$589$	−10401.3	−0.727638
$590$	−37643.7	−2.62673
$591$	−15170.8	−1.05591
$592$	18013.6	1.25060
$593$	22738.1	1.57461	0.787304	−	0.616564i	$-0.211476\pi$
$593$	22738.1	1.57461	0.787304	+	0.616564i	$0.211476\pi$
$594$	−3.32139	−0.000229425 0
$595$	−22067.0	−1.52044
$596$	869.689	0.0597716
$597$	32817.7	2.24981
$598$	−3940.67	−0.269475
$599$	−3740.61	−0.255154	−0.127577	−	0.991829i	$-0.540720\pi$
$599$	−3740.61	−0.255154	−0.127577	+	0.991829i	$0.540720\pi$
$600$	1527.54	0.103936
$601$	−9343.19	−0.634138	−0.317069	−	0.948403i	$-0.602699\pi$
$601$	−9343.19	−0.634138	−0.317069	+	0.948403i	$0.602699\pi$
$602$	45645.0	3.09029
$603$	−11019.1	−0.744169
$604$	6900.59	0.464870
$605$	16759.0	1.12620
$606$	11494.3	0.770503
$607$	19404.0	1.29751	0.648753	−	0.760999i	$-0.275291\pi$
$607$	19404.0	1.29751	0.648753	+	0.760999i	$0.275291\pi$
$608$	7574.61	0.505248
$609$	−44767.4	−2.97876
$610$	40554.3	2.69180
$611$	−16500.6	−1.09254
$612$	−14510.7	−0.958435
$613$	−4485.36	−0.295533	−0.147767	−	0.989022i	$-0.547208\pi$
$613$	−4485.36	−0.295533	−0.147767	+	0.989022i	$0.547208\pi$
$614$	7300.57	0.479849
$615$	−27269.1	−1.78796
$616$	−1.95016	−0.000127556 0
$617$	13462.3	0.878401	0.439201	−	0.898389i	$-0.355262\pi$
$617$	13462.3	0.878401	0.439201	+	0.898389i	$0.355262\pi$
$618$	16461.3	1.07147
$619$	−17325.2	−1.12497	−0.562487	−	0.826806i	$-0.690155\pi$
$619$	−17325.2	−1.12497	−0.562487	+	0.826806i	$0.690155\pi$
$620$	26392.8	1.70961
$621$	−1298.96	−0.0839383
$622$	20948.3	1.35040
$623$	−31842.6	−2.04775
$624$	33960.2	2.17868
$625$	−18692.6	−1.19633
$626$	−5183.26	−0.330934
$627$	3.07343	0.000195759 0
$628$	−5514.24	−0.350385
$629$	15091.4	0.956654
$630$	−49134.2	−3.10723
$631$	18788.5	1.18536	0.592678	−	0.805439i	$-0.298071\pi$
$631$	18788.5	1.18536	0.592678	+	0.805439i	$0.298071\pi$
$632$	−900.685	−0.0566888
$633$	40075.0	2.51633
$634$	−12452.0	−0.780016
$635$	23912.9	1.49442
$636$	7358.66	0.458789
$637$	−26703.0	−1.66093
$638$	−9.07766	−0.000563304 0
$639$	−11137.7	−0.689519
$640$	9073.15	0.560387
$641$	−18559.4	−1.14361	−0.571804	−	0.820390i	$-0.693756\pi$
$641$	−18559.4	−1.14361	−0.571804	+	0.820390i	$0.693756\pi$
$642$	22942.7	1.41040
$643$	−26595.2	−1.63113	−0.815563	−	0.578669i	$-0.803572\pi$
$643$	−26595.2	−1.63113	−0.815563	+	0.578669i	$0.803572\pi$
$644$	3290.60	0.201348
$645$	−42307.3	−2.58271
$646$	7583.76	0.461887
$647$	24817.6	1.50800	0.754002	−	0.656872i	$-0.228121\pi$
$647$	24817.6	1.50800	0.754002	+	0.656872i	$0.228121\pi$
$648$	−2282.79	−0.138390
$649$	−9.42387	−0.000569984 0
$650$	7397.00	0.446360
$651$	72721.2	4.37814
$652$	8050.14	0.483539
$653$	−31474.2	−1.88619	−0.943095	−	0.332524i	$-0.892100\pi$
$653$	−31474.2	−1.88619	−0.943095	+	0.332524i	$0.892100\pi$
$654$	64005.7	3.82694
$655$	−23356.8	−1.39332
$656$	−20106.6	−1.19669
$657$	−22527.1	−1.33769
$658$	30750.8	1.82187
$659$	−18582.3	−1.09843	−0.549213	−	0.835683i	$-0.685072\pi$
$659$	−18582.3	−1.09843	−0.549213	+	0.835683i	$0.685072\pi$
$660$	−7.79867	−0.000459943 0
$661$	5434.90	0.319808	0.159904	−	0.987133i	$-0.448882\pi$
$661$	5434.90	0.319808	0.159904	+	0.987133i	$0.448882\pi$
$662$	−25223.1	−1.48085
$663$	28451.2	1.66659
$664$	475.677	0.0278010
$665$	11506.1	0.670957
$666$	33602.4	1.95506
$667$	−3550.19	−0.206093
$668$	−3609.12	−0.209043
$669$	−25543.1	−1.47617
$670$	−14613.0	−0.842611
$671$	10.1525	0.000584104 0
$672$	−52958.1	−3.04004
$673$	−7800.55	−0.446789	−0.223394	−	0.974728i	$-0.571714\pi$
$673$	−7800.55	−0.446789	−0.223394	+	0.974728i	$0.571714\pi$
$674$	15158.9	0.866319
$675$	2438.28	0.139036
$676$	7523.43	0.428051
$677$	−18258.1	−1.03651	−0.518253	−	0.855227i	$-0.673417\pi$
$677$	−18258.1	−1.03651	−0.518253	+	0.855227i	$0.673417\pi$
$678$	65194.9	3.69291
$679$	−22688.3	−1.28233
$680$	4460.20	0.251530
$681$	103.269	0.00581095
$682$	14.7460	0.000827935 0
$683$	24491.4	1.37209	0.686046	−	0.727559i	$-0.259345\pi$
$683$	24491.4	1.37209	0.686046	+	0.727559i	$0.259345\pi$
$684$	7566.11	0.422950
$685$	23786.2	1.32675
$686$	12736.8	0.708884
$687$	18849.2	1.04679
$688$	−31194.9	−1.72862
$689$	−8259.12	−0.456672
$690$	−6806.89	−0.375556
$691$	−31743.2	−1.74757	−0.873784	−	0.486314i	$-0.838341\pi$
$691$	−31743.2	−1.74757	−0.873784	+	0.486314i	$0.838341\pi$
$692$	−9529.83	−0.523511
$693$	−12.3004	−0.000674249 0
$694$	5566.53	0.304471
$695$	−14923.5	−0.814503
$696$	9048.40	0.492785
$697$	−16844.9	−0.915419
$698$	10281.9	0.557558
$699$	−12661.0	−0.685099
$700$	−6176.76	−0.333514
$701$	11102.1	0.598173	0.299086	−	0.954226i	$-0.403318\pi$
$701$	11102.1	0.598173	0.299086	+	0.954226i	$0.403318\pi$
$702$	16031.7	0.861935
$703$	−7868.90	−0.422164
$704$	−3.65548	−0.000195697 0
$705$	−28502.2	−1.52263
$706$	−20704.3	−1.10371
$707$	10772.7	0.573053
$708$	−40527.9	−2.15132
$709$	27736.6	1.46921	0.734604	−	0.678496i	$-0.237368\pi$
$709$	27736.6	1.46921	0.734604	+	0.678496i	$0.237368\pi$
$710$	−14770.3	−0.780731
$711$	−5680.97	−0.299652
$712$	6436.03	0.338765
$713$	5767.00	0.302912
$714$	−53022.1	−2.77913
$715$	8.75297	0.000457822 0
$716$	−22366.6	−1.16743
$717$	−10490.2	−0.546391
$718$	−27948.6	−1.45269
$719$	1870.01	0.0969952	0.0484976	−	0.998823i	$-0.484557\pi$
$719$	1870.01	0.0969952	0.0484976	+	0.998823i	$0.484557\pi$
$720$	33579.4	1.73810
$721$	15427.8	0.796894
$722$	22159.2	1.14222
$723$	−5488.67	−0.282332
$724$	−7202.72	−0.369734
$725$	6664.03	0.341374
$726$	40268.2	2.05853
$727$	87.1488	0.00444590	0.00222295	−	0.999998i	$-0.499292\pi$
$727$	87.1488	0.00444590	0.00222295	+	0.999998i	$0.499292\pi$
$728$	9413.06	0.479219
$729$	−29995.7	−1.52394
$730$	−29874.2	−1.51465
$731$	−26134.4	−1.32232
$732$	43661.5	2.20461
$733$	−27078.3	−1.36447	−0.682237	−	0.731131i	$-0.738993\pi$
$733$	−27078.3	−1.36447	−0.682237	+	0.731131i	$0.738993\pi$
$734$	−23346.8	−1.17404
$735$	−46125.3	−2.31477
$736$	−4199.74	−0.210332
$737$	−3.65827	−0.000182841 0
$738$	−37506.7	−1.87079
$739$	−25252.9	−1.25703	−0.628514	−	0.777799i	$-0.716336\pi$
$739$	−25252.9	−1.25703	−0.628514	+	0.777799i	$0.716336\pi$
$740$	19966.9	0.991890
$741$	−14834.9	−0.735455
$742$	15391.8	0.761525
$743$	18660.7	0.921391	0.460695	−	0.887558i	$-0.347600\pi$
$743$	18660.7	0.921391	0.460695	+	0.887558i	$0.347600\pi$
$744$	−14698.4	−0.724288
$745$	−1686.08	−0.0829169
$746$	−37866.5	−1.85843
$747$	3000.28	0.146954
$748$	−4.81746	−0.000235486 0
$749$	21502.3	1.04897
$750$	−34840.2	−1.69624
$751$	−15777.0	−0.766591	−0.383296	−	0.923626i	$-0.625211\pi$
$751$	−15777.0	−0.766591	−0.383296	+	0.923626i	$0.625211\pi$
$752$	−21015.8	−1.01911
$753$	6161.68	0.298199
$754$	43816.1	2.11630
$755$	−13378.3	−0.644881
$756$	−13387.1	−0.644025
$757$	−30368.3	−1.45806	−0.729031	−	0.684481i	$-0.760029\pi$
$757$	−30368.3	−1.45806	−0.729031	+	0.684481i	$0.760029\pi$
$758$	8741.20	0.418858
$759$	−1.70406	−8.14934e−5 0
$760$	−2325.61	−0.110998
$761$	1004.61	0.0478542	0.0239271	−	0.999714i	$-0.492383\pi$
$761$	1004.61	0.0478542	0.0239271	+	0.999714i	$0.492383\pi$
$762$	57457.4	2.73158
$763$	59987.3	2.84625
$764$	4812.35	0.227886
$765$	28132.2	1.32957
$766$	−21673.7	−1.02233
$767$	45487.2	2.14139
$768$	41165.0	1.93413
$769$	7589.94	0.355917	0.177958	−	0.984038i	$-0.443051\pi$
$769$	7589.94	0.355917	0.177958	+	0.984038i	$0.443051\pi$
$770$	−16.3122	−0.000763441 0
$771$	3124.53	0.145950
$772$	−11060.4	−0.515636
$773$	11264.2	0.524121	0.262060	−	0.965052i	$-0.415598\pi$
$773$	11264.2	0.524121	0.262060	+	0.965052i	$0.415598\pi$
$774$	−58190.6	−2.70235
$775$	−10825.2	−0.501745
$776$	4585.78	0.212139
$777$	55015.7	2.54012
$778$	−28348.4	−1.30635
$779$	8783.19	0.403967
$780$	37642.7	1.72798
$781$	−3.69765	−0.000169414 0
$782$	−4204.81	−0.192281
$783$	14443.1	0.659203
$784$	−34010.0	−1.54929
$785$	10690.5	0.486065
$786$	−56121.0	−2.54678
$787$	14838.7	0.672099	0.336050	−	0.941844i	$-0.390909\pi$
$787$	14838.7	0.672099	0.336050	+	0.941844i	$0.390909\pi$
$788$	−12399.0	−0.560527
$789$	34829.8	1.57158
$790$	−7533.79	−0.339292
$791$	61101.8	2.74656
$792$	2.48617	0.000111543 0
$793$	−49004.3	−2.19444
$794$	20657.7	0.923316
$795$	−14266.3	−0.636446
$796$	26821.7	1.19431
$797$	−13669.1	−0.607510	−0.303755	−	0.952750i	$-0.598240\pi$
$797$	−13669.1	−0.607510	−0.303755	+	0.952750i	$0.598240\pi$
$798$	27646.5	1.22641
$799$	−17606.6	−0.779572
$800$	7883.30	0.348396
$801$	40594.5	1.79068
$802$	6394.48	0.281542
$803$	−7.47882	−0.000328670 0
$804$	−15732.6	−0.690108
$805$	−6379.53	−0.279316
$806$	−71175.9	−3.11050
$807$	56410.0	2.46063
$808$	−2177.38	−0.0948018
$809$	23146.5	1.00592	0.502959	−	0.864310i	$-0.332245\pi$
$809$	23146.5	1.00592	0.502959	+	0.864310i	$0.332245\pi$
$810$	−19094.5	−0.828285
$811$	18130.5	0.785017	0.392509	−	0.919748i	$-0.371607\pi$
$811$	18130.5	0.785017	0.392509	+	0.919748i	$0.371607\pi$
$812$	−36588.1	−1.58127
$813$	−12782.3	−0.551409
$814$	11.1557	0.000480355 0
$815$	−15606.9	−0.670781
$816$	36236.5	1.55457
$817$	13626.9	0.583530
$818$	10247.0	0.437991
$819$	59371.8	2.53311
$820$	−22286.9	−0.949136
$821$	−15374.7	−0.653570	−0.326785	−	0.945099i	$-0.605965\pi$
$821$	−15374.7	−0.653570	−0.326785	+	0.945099i	$0.605965\pi$
$822$	57152.9	2.42511
$823$	−4249.01	−0.179965	−0.0899825	−	0.995943i	$-0.528681\pi$
$823$	−4249.01	−0.179965	−0.0899825	+	0.995943i	$0.528681\pi$
$824$	−3118.27	−0.131833
$825$	3.19868	0.000134986 0
$826$	−84770.7	−3.57089
$827$	−8260.64	−0.347340	−0.173670	−	0.984804i	$-0.555563\pi$
$827$	−8260.64	−0.347340	−0.173670	+	0.984804i	$0.555563\pi$
$828$	−4195.03	−0.176072
$829$	692.249	0.0290022	0.0145011	−	0.999895i	$-0.495384\pi$
$829$	692.249	0.0290022	0.0145011	+	0.999895i	$0.495384\pi$
$830$	3978.81	0.166393
$831$	−48373.2	−2.01931
$832$	17644.3	0.735223
$833$	−28492.9	−1.18514
$834$	−35857.7	−1.48879
$835$	6997.05	0.289991
$836$	2.51189	0.000103918 0
$837$	−23461.8	−0.968885
$838$	−5257.97	−0.216746
$839$	5494.12	0.226076	0.113038	−	0.993591i	$-0.463942\pi$
$839$	5494.12	0.226076	0.113038	+	0.993591i	$0.463942\pi$
$840$	16259.6	0.667868
$841$	15085.4	0.618533
$842$	43366.3	1.77494
$843$	−27738.1	−1.13328
$844$	32753.1	1.33579
$845$	−14585.8	−0.593806
$846$	−39202.7	−1.59316
$847$	37740.0	1.53101
$848$	−10519.1	−0.425977
$849$	2520.11	0.101873
$850$	7892.81	0.318496
$851$	4362.91	0.175744
$852$	−15902.0	−0.639428
$853$	23413.5	0.939814	0.469907	−	0.882716i	$-0.344287\pi$
$853$	23413.5	0.939814	0.469907	+	0.882716i	$0.344287\pi$
$854$	91325.2	3.65935
$855$	−14668.5	−0.586729
$856$	−4346.05	−0.173534
$857$	11206.0	0.446663	0.223331	−	0.974743i	$-0.428307\pi$
$857$	11206.0	0.446663	0.223331	+	0.974743i	$0.428307\pi$
$858$	21.0314	0.000836830 0
$859$	15439.9	0.613275	0.306637	−	0.951826i	$-0.400796\pi$
$859$	15439.9	0.613275	0.306637	+	0.951826i	$0.400796\pi$
$860$	−34577.5	−1.37103
$861$	−61408.0	−2.43064
$862$	−17214.9	−0.680211
$863$	18198.8	0.717839	0.358919	−	0.933369i	$-0.383145\pi$
$863$	18198.8	0.717839	0.358919	+	0.933369i	$0.383145\pi$
$864$	17085.7	0.672763
$865$	18475.6	0.726231
$866$	65903.7	2.58603
$867$	−8683.26	−0.340137
$868$	59434.5	2.32412
$869$	−1.88604	−7.36242e−5 0
$870$	75685.5	2.94940
$871$	17657.8	0.686924
$872$	−12124.6	−0.470863
$873$	28924.3	1.12135
$874$	2192.45	0.0848520
$875$	−32652.8	−1.26156
$876$	−32163.1	−1.24052
$877$	−45701.9	−1.75969	−0.879843	−	0.475265i	$-0.842352\pi$
$877$	−45701.9	−1.75969	−0.879843	+	0.475265i	$0.842352\pi$
$878$	−32024.5	−1.23095
$879$	46056.9	1.76730
$880$	11.1481	0.000427049 0
$881$	11224.3	0.429236	0.214618	−	0.976698i	$-0.431149\pi$
$881$	11224.3	0.429236	0.214618	+	0.976698i	$0.431149\pi$
$882$	−63441.9	−2.42200
$883$	−39097.1	−1.49006	−0.745030	−	0.667031i	$-0.767565\pi$
$883$	−39097.1	−1.49006	−0.745030	+	0.667031i	$0.767565\pi$
$884$	23252.9	0.884707
$885$	78572.1	2.98437
$886$	45032.2	1.70755
$887$	−21256.0	−0.804630	−0.402315	−	0.915501i	$-0.631794\pi$
$887$	−21256.0	−0.804630	−0.402315	+	0.915501i	$0.631794\pi$
$888$	−11119.8	−0.420220
$889$	53850.1	2.03158
$890$	53834.3	2.02756
$891$	−4.78018	−0.000179733 0
$892$	−20876.2	−0.783619
$893$	9180.36	0.344019
$894$	−4051.26	−0.151560
$895$	43362.4	1.61949
$896$	20432.0	0.761815
$897$	8225.18	0.306166
$898$	−58299.5	−2.16646
$899$	−64123.1	−2.37889
$900$	7874.45	0.291646
$901$	−8812.72	−0.325854
$902$	−12.4519	−0.000459650 0
$903$	−95272.8	−3.51105
$904$	−12349.9	−0.454372
$905$	13964.0	0.512906
$906$	−32145.0	−1.17875
$907$	−15625.4	−0.572030	−0.286015	−	0.958225i	$-0.592331\pi$
$907$	−15625.4	−0.572030	−0.286015	+	0.958225i	$0.592331\pi$
$908$	84.4006	0.00308473
$909$	−13733.6	−0.501115
$910$	78735.7	2.86820
$911$	9438.69	0.343269	0.171634	−	0.985161i	$-0.445095\pi$
$911$	9438.69	0.343269	0.171634	+	0.985161i	$0.445095\pi$
$912$	−18894.3	−0.686021
$913$	0.996071	3.61064e−5 0
$914$	4267.23	0.154428
$915$	−84647.2	−3.05831
$916$	15405.4	0.555685
$917$	−52597.6	−1.89414
$918$	17106.3	0.615025
$919$	−16365.1	−0.587416	−0.293708	−	0.955895i	$-0.594889\pi$
$919$	−16365.1	−0.587416	−0.293708	+	0.955895i	$0.594889\pi$
$920$	1289.43	0.0462080
$921$	−15238.1	−0.545184
$922$	43479.3	1.55305
$923$	17847.8	0.636478
$924$	−17.5620	−0.000625267 0
$925$	−8189.58	−0.291104
$926$	56828.7	2.01675
$927$	−19668.1	−0.696856
$928$	46696.7	1.65183
$929$	−16566.3	−0.585062	−0.292531	−	0.956256i	$-0.594497\pi$
$929$	−16566.3	−0.585062	−0.292531	+	0.956256i	$0.594497\pi$
$930$	−122945.	−4.33498
$931$	14856.6	0.522992
$932$	−10347.8	−0.363683
$933$	−43724.4	−1.53427
$934$	−48373.1	−1.69466
$935$	9.33967	0.000326674 0
$936$	−12000.2	−0.419060
$937$	41515.3	1.44744	0.723718	−	0.690096i	$-0.242432\pi$
$937$	41515.3	1.44744	0.723718	+	0.690096i	$0.242432\pi$
$938$	−32907.3	−1.14548
$939$	10818.8	0.375993
$940$	−23294.7	−0.808286
$941$	35646.7	1.23491	0.617454	−	0.786607i	$-0.288164\pi$
$941$	35646.7	1.23491	0.617454	+	0.786607i	$0.288164\pi$
$942$	25686.9	0.888455
$943$	−4869.83	−0.168169
$944$	57934.3	1.99746
$945$	25953.7	0.893412
$946$	−19.3188	−0.000663964 0
$947$	12963.0	0.444817	0.222408	−	0.974954i	$-0.428608\pi$
$947$	12963.0	0.444817	0.222408	+	0.974954i	$0.428608\pi$
$948$	−8111.02	−0.277884
$949$	36098.9	1.23479
$950$	−4115.43	−0.140550
$951$	25990.4	0.886221
$952$	10044.0	0.341941
$953$	34776.3	1.18207	0.591036	−	0.806645i	$-0.298719\pi$
$953$	34776.3	1.18207	0.591036	+	0.806645i	$0.298719\pi$
$954$	−19622.3	−0.665927
$955$	−9329.78	−0.316130
$956$	−8573.53	−0.290050
$957$	18.9474	0.000640002 0
$958$	−14138.7	−0.476827
$959$	53564.7	1.80365
$960$	30477.7	1.02465
$961$	74372.0	2.49646
$962$	−53846.7	−1.80466
$963$	−27412.2	−0.917285
$964$	−4485.85	−0.149875
$965$	21442.9	0.715306
$966$	−15328.6	−0.510547
$967$	−16661.6	−0.554087	−0.277043	−	0.960857i	$-0.589355\pi$
$967$	−16661.6	−0.554087	−0.277043	+	0.960857i	$0.589355\pi$
$968$	−7628.03	−0.253279
$969$	−15829.2	−0.524776
$970$	38357.8	1.26969
$971$	−23206.6	−0.766978	−0.383489	−	0.923545i	$-0.625278\pi$
$971$	−23206.6	−0.766978	−0.383489	+	0.923545i	$0.625278\pi$
$972$	−33304.9	−1.09903
$973$	−33606.5	−1.10727
$974$	21847.9	0.718738
$975$	−15439.4	−0.507135
$976$	−62413.8	−2.04694
$977$	−9650.59	−0.316018	−0.158009	−	0.987438i	$-0.550508\pi$
$977$	−9650.59	−0.316018	−0.158009	+	0.987438i	$0.550508\pi$
$978$	−37499.9	−1.22609
$979$	13.4771	0.000439969 0
$980$	−37697.9	−1.22879
$981$	−76474.8	−2.48894
$982$	16363.4	0.531748
$983$	7770.03	0.252111	0.126056	−	0.992023i	$-0.459768\pi$
$983$	7770.03	0.252111	0.126056	+	0.992023i	$0.459768\pi$
$984$	12411.8	0.402107
$985$	24038.1	0.777580
$986$	46753.1	1.51006
$987$	−64184.8	−2.06993
$988$	−12124.4	−0.390414
$989$	−7555.41	−0.242920
$990$	20.7956	0.000667603 0
$991$	5538.92	0.177548	0.0887738	−	0.996052i	$-0.471705\pi$
$991$	5538.92	0.177548	0.0887738	+	0.996052i	$0.471705\pi$
$992$	−75855.2	−2.42783
$993$	52647.0	1.68248
$994$	−33261.5	−1.06136
$995$	−51999.5	−1.65678
$996$	4283.66	0.136278
$997$	11375.2	0.361339	0.180670	−	0.983544i	$-0.442174\pi$
$997$	11375.2	0.361339	0.180670	+	0.983544i	$0.442174\pi$
$998$	−81421.2	−2.58251
$999$	−17749.5	−0.562132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	83.4.a.a.1.2	✓	7
3.2	odd	2		747.4.a.a.1.6		7
4.3	odd	2		1328.4.a.f.1.1		7
5.4	even	2		2075.4.a.c.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
83.4.a.a.1.2	✓	7	1.1	even	1	trivial
747.4.a.a.1.6		7	3.2	odd	2
1328.4.a.f.1.1		7	4.3	odd	2
2075.4.a.c.1.6		7	5.4	even	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:	\( N \)	\(=\)	\( 83 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	83.a (trivial)

\(f(q)\)	\(=\)	\(q-3.80719 q^{2} +7.94657 q^{3} +6.49468 q^{4} -12.5913 q^{5} -30.2541 q^{6} -28.3547 q^{7} +5.73105 q^{8} +36.1480 q^{9} +O(q^{10})\)	\(q-3.80719 q^{2} +7.94657 q^{3} +6.49468 q^{4} -12.5913 q^{5} -30.2541 q^{6} -28.3547 q^{7} +5.73105 q^{8} +36.1480 q^{9} +47.9375 q^{10} +0.0120009 q^{11} +51.6104 q^{12} -57.9258 q^{13} +107.951 q^{14} -100.058 q^{15} -73.7766 q^{16} -61.8085 q^{17} -137.622 q^{18} +32.2279 q^{19} -81.7765 q^{20} -225.322 q^{21} -0.0456895 q^{22} -17.8687 q^{23} +45.5422 q^{24} +33.5412 q^{25} +220.534 q^{26} +72.6949 q^{27} -184.154 q^{28} +198.682 q^{29} +380.939 q^{30} -322.743 q^{31} +235.033 q^{32} +0.0953656 q^{33} +235.317 q^{34} +357.022 q^{35} +234.769 q^{36} -244.164 q^{37} -122.698 q^{38} -460.311 q^{39} -72.1615 q^{40} +272.534 q^{41} +857.844 q^{42} +422.829 q^{43} +0.0779416 q^{44} -455.150 q^{45} +68.0296 q^{46} +284.858 q^{47} -586.271 q^{48} +460.987 q^{49} -127.698 q^{50} -491.166 q^{51} -376.209 q^{52} +142.581 q^{53} -276.763 q^{54} -0.151107 q^{55} -162.502 q^{56} +256.101 q^{57} -756.418 q^{58} -785.267 q^{59} -649.843 q^{60} +845.983 q^{61} +1228.74 q^{62} -1024.96 q^{63} -304.602 q^{64} +729.362 q^{65} -0.363075 q^{66} -304.834 q^{67} -401.426 q^{68} -141.995 q^{69} -1359.25 q^{70} -308.116 q^{71} +207.166 q^{72} -623.191 q^{73} +929.580 q^{74} +266.538 q^{75} +209.310 q^{76} -0.340280 q^{77} +1752.49 q^{78} -157.159 q^{79} +928.944 q^{80} -398.320 q^{81} -1037.59 q^{82} +83.0000 q^{83} -1463.39 q^{84} +778.251 q^{85} -1609.79 q^{86} +1578.84 q^{87} +0.0687775 q^{88} +1123.01 q^{89} +1732.84 q^{90} +1642.47 q^{91} -116.052 q^{92} -2564.70 q^{93} -1084.51 q^{94} -405.791 q^{95} +1867.71 q^{96} +800.163 q^{97} -1755.06 q^{98} +0.433806 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 5 q^{2} - 11 q^{3} + 13 q^{4} - 17 q^{5} - 27 q^{6} - 46 q^{7} - 57 q^{8} + 16 q^{9}+O(q^{10}) \) 7 * q - 5 * q^2 - 11 * q^3 + 13 * q^4 - 17 * q^5 - 27 * q^6 - 46 * q^7 - 57 * q^8 + 16 * q^9	\( 7 q - 5 q^{2} - 11 q^{3} + 13 q^{4} - 17 q^{5} - 27 q^{6} - 46 q^{7} - 57 q^{8} + 16 q^{9} - 65 q^{10} - 59 q^{11} - 89 q^{12} - 205 q^{13} + 30 q^{14} - 136 q^{15} - 163 q^{16} - 148 q^{17} - 188 q^{18} - 29 q^{19} + 3 q^{20} - 197 q^{21} - 127 q^{22} + 159 q^{23} + 465 q^{24} - 108 q^{25} + 603 q^{26} + 217 q^{27} + 24 q^{28} + 29 q^{29} + 1000 q^{30} - 420 q^{31} + 459 q^{32} - 47 q^{33} + 688 q^{34} + 300 q^{35} + 1000 q^{36} - 1091 q^{37} + 327 q^{38} + 286 q^{39} + 119 q^{40} + 127 q^{41} + 1115 q^{42} - 49 q^{43} + 795 q^{44} - 347 q^{45} + 11 q^{46} + 150 q^{47} + 119 q^{48} - 521 q^{49} + 448 q^{50} - 95 q^{51} - 1129 q^{52} - 785 q^{53} + 717 q^{54} - 66 q^{55} + 290 q^{56} - 434 q^{57} - 143 q^{58} - 81 q^{59} - 952 q^{60} - 195 q^{61} + 1096 q^{62} - 913 q^{63} - 1203 q^{64} - 534 q^{65} + 1517 q^{66} - 1825 q^{67} - 578 q^{68} - 1910 q^{69} - 2012 q^{70} - 666 q^{71} - 1284 q^{72} - 1594 q^{73} + 713 q^{74} + 819 q^{75} - 581 q^{76} - 1307 q^{77} - 610 q^{78} - 968 q^{79} + 2347 q^{80} - 1193 q^{81} - 3961 q^{82} + 581 q^{83} - 1655 q^{84} - 2407 q^{85} - 1197 q^{86} + 3079 q^{87} - 1315 q^{88} + 1616 q^{89} + 313 q^{90} + 1077 q^{91} + 1127 q^{92} - 827 q^{93} + 534 q^{94} + 2252 q^{95} + 2221 q^{96} + 50 q^{97} + 1071 q^{98} + 3184 q^{99}+O(q^{100}) \) 7 * q - 5 * q^2 - 11 * q^3 + 13 * q^4 - 17 * q^5 - 27 * q^6 - 46 * q^7 - 57 * q^8 + 16 * q^9 - 65 * q^10 - 59 * q^11 - 89 * q^12 - 205 * q^13 + 30 * q^14 - 136 * q^15 - 163 * q^16 - 148 * q^17 - 188 * q^18 - 29 * q^19 + 3 * q^20 - 197 * q^21 - 127 * q^22 + 159 * q^23 + 465 * q^24 - 108 * q^25 + 603 * q^26 + 217 * q^27 + 24 * q^28 + 29 * q^29 + 1000 * q^30 - 420 * q^31 + 459 * q^32 - 47 * q^33 + 688 * q^34 + 300 * q^35 + 1000 * q^36 - 1091 * q^37 + 327 * q^38 + 286 * q^39 + 119 * q^40 + 127 * q^41 + 1115 * q^42 - 49 * q^43 + 795 * q^44 - 347 * q^45 + 11 * q^46 + 150 * q^47 + 119 * q^48 - 521 * q^49 + 448 * q^50 - 95 * q^51 - 1129 * q^52 - 785 * q^53 + 717 * q^54 - 66 * q^55 + 290 * q^56 - 434 * q^57 - 143 * q^58 - 81 * q^59 - 952 * q^60 - 195 * q^61 + 1096 * q^62 - 913 * q^63 - 1203 * q^64 - 534 * q^65 + 1517 * q^66 - 1825 * q^67 - 578 * q^68 - 1910 * q^69 - 2012 * q^70 - 666 * q^71 - 1284 * q^72 - 1594 * q^73 + 713 * q^74 + 819 * q^75 - 581 * q^76 - 1307 * q^77 - 610 * q^78 - 968 * q^79 + 2347 * q^80 - 1193 * q^81 - 3961 * q^82 + 581 * q^83 - 1655 * q^84 - 2407 * q^85 - 1197 * q^86 + 3079 * q^87 - 1315 * q^88 + 1616 * q^89 + 313 * q^90 + 1077 * q^91 + 1127 * q^92 - 827 * q^93 + 534 * q^94 + 2252 * q^95 + 2221 * q^96 + 50 * q^97 + 1071 * q^98 + 3184 * q^99

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