LMFDB - Embedded newform 1045.4.a.g.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.96023 q^{2} -4.00535 q^{3} +0.762942 q^{4} -5.00000 q^{5} +11.8567 q^{6} +6.39895 q^{7} +21.4233 q^{8} -10.9572 q^{9} +O(q^{10})$	\(q-2.96023 q^{2} -4.00535 q^{3} +0.762942 q^{4} -5.00000 q^{5} +11.8567 q^{6} +6.39895 q^{7} +21.4233 q^{8} -10.9572 q^{9} +14.8011 q^{10} -11.0000 q^{11} -3.05585 q^{12} -47.4056 q^{13} -18.9423 q^{14} +20.0267 q^{15} -69.5215 q^{16} +80.5355 q^{17} +32.4357 q^{18} -19.0000 q^{19} -3.81471 q^{20} -25.6300 q^{21} +32.5625 q^{22} -55.8588 q^{23} -85.8079 q^{24} +25.0000 q^{25} +140.331 q^{26} +152.032 q^{27} +4.88203 q^{28} -100.001 q^{29} -59.2837 q^{30} +174.227 q^{31} +34.4126 q^{32} +44.0588 q^{33} -238.403 q^{34} -31.9948 q^{35} -8.35969 q^{36} -47.6032 q^{37} +56.2443 q^{38} +189.876 q^{39} -107.117 q^{40} -143.802 q^{41} +75.8707 q^{42} -47.0770 q^{43} -8.39236 q^{44} +54.7859 q^{45} +165.355 q^{46} -432.112 q^{47} +278.458 q^{48} -302.053 q^{49} -74.0057 q^{50} -322.573 q^{51} -36.1677 q^{52} -146.270 q^{53} -450.048 q^{54} +55.0000 q^{55} +137.087 q^{56} +76.1016 q^{57} +296.027 q^{58} -757.043 q^{59} +15.2792 q^{60} +57.8895 q^{61} -515.752 q^{62} -70.1145 q^{63} +454.303 q^{64} +237.028 q^{65} -130.424 q^{66} -343.284 q^{67} +61.4439 q^{68} +223.734 q^{69} +94.7117 q^{70} -722.293 q^{71} -234.739 q^{72} -700.016 q^{73} +140.916 q^{74} -100.134 q^{75} -14.4959 q^{76} -70.3885 q^{77} -562.076 q^{78} +84.7134 q^{79} +347.607 q^{80} -313.096 q^{81} +425.686 q^{82} +226.310 q^{83} -19.5542 q^{84} -402.677 q^{85} +139.359 q^{86} +400.541 q^{87} -235.657 q^{88} +43.0474 q^{89} -162.179 q^{90} -303.346 q^{91} -42.6170 q^{92} -697.840 q^{93} +1279.15 q^{94} +95.0000 q^{95} -137.834 q^{96} +737.305 q^{97} +894.147 q^{98} +120.529 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * 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 $

$2$	−2.96023	−1.04660	−0.523299	−	0.852149i	$-0.675299\pi$
$2$	−2.96023	−1.04660	−0.523299	+	0.852149i	$0.675299\pi$
$3$	−4.00535	−0.770830	−0.385415	−	0.922743i	$-0.625942\pi$
$3$	−4.00535	−0.770830	−0.385415	+	0.922743i	$0.625942\pi$
$4$	0.762942	0.0953677
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	11.8567	0.806749
$7$	6.39895	0.345511	0.172755	−	0.984965i	$-0.444733\pi$
$7$	6.39895	0.345511	0.172755	+	0.984965i	$0.444733\pi$
$8$	21.4233	0.946786
$9$	−10.9572	−0.405822
$10$	14.8011	0.468053
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−3.05585	−0.0735123
$13$	−47.4056	−1.01138	−0.505691	−	0.862715i	$-0.668762\pi$
$13$	−47.4056	−1.01138	−0.505691	+	0.862715i	$0.668762\pi$
$14$	−18.9423	−0.361611
$15$	20.0267	0.344726
$16$	−69.5215	−1.08627
$17$	80.5355	1.14898	0.574492	−	0.818510i	$-0.305200\pi$
$17$	80.5355	1.14898	0.574492	+	0.818510i	$0.305200\pi$
$18$	32.4357	0.424732
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−3.81471	−0.0426497
$21$	−25.6300	−0.266330
$22$	32.5625	0.315561
$23$	−55.8588	−0.506407	−0.253204	−	0.967413i	$-0.581484\pi$
$23$	−55.8588	−0.506407	−0.253204	+	0.967413i	$0.581484\pi$
$24$	−85.8079	−0.729811
$25$	25.0000	0.200000
$26$	140.331	1.05851
$27$	152.032	1.08365
$28$	4.88203	0.0329506
$29$	−100.001	−0.640338	−0.320169	−	0.947360i	$-0.603740\pi$
$29$	−100.001	−0.640338	−0.320169	+	0.947360i	$0.603740\pi$
$30$	−59.2837	−0.360789
$31$	174.227	1.00942	0.504711	−	0.863288i	$-0.331599\pi$
$31$	174.227	1.00942	0.504711	+	0.863288i	$0.331599\pi$
$32$	34.4126	0.190105
$33$	44.0588	0.232414
$34$	−238.403	−1.20252
$35$	−31.9948	−0.154517
$36$	−8.35969	−0.0387023
$37$	−47.6032	−0.211511	−0.105756	−	0.994392i	$-0.533726\pi$
$37$	−47.6032	−0.211511	−0.105756	+	0.994392i	$0.533726\pi$
$38$	56.2443	0.240106
$39$	189.876	0.779603
$40$	−107.117	−0.423416
$41$	−143.802	−0.547758	−0.273879	−	0.961764i	$-0.588307\pi$
$41$	−143.802	−0.547758	−0.273879	+	0.961764i	$0.588307\pi$
$42$	75.8707	0.278741
$43$	−47.0770	−0.166958	−0.0834788	−	0.996510i	$-0.526603\pi$
$43$	−47.0770	−0.166958	−0.0834788	+	0.996510i	$0.526603\pi$
$44$	−8.39236	−0.0287544
$45$	54.7859	0.181489
$46$	165.355	0.530005
$47$	−432.112	−1.34106	−0.670532	−	0.741881i	$-0.733934\pi$
$47$	−432.112	−1.34106	−0.670532	+	0.741881i	$0.733934\pi$
$48$	278.458	0.837331
$49$	−302.053	−0.880622
$50$	−74.0057	−0.209320
$51$	−322.573	−0.885671
$52$	−36.1677	−0.0964531
$53$	−146.270	−0.379089	−0.189545	−	0.981872i	$-0.560701\pi$
$53$	−146.270	−0.379089	−0.189545	+	0.981872i	$0.560701\pi$
$54$	−450.048	−1.13415
$55$	55.0000	0.134840
$56$	137.087	0.327125
$57$	76.1016	0.176840
$58$	296.027	0.670177
$59$	−757.043	−1.67048	−0.835242	−	0.549882i	$-0.814673\pi$
$59$	−757.043	−1.67048	−0.835242	+	0.549882i	$0.814673\pi$
$60$	15.2792	0.0328757
$61$	57.8895	0.121508	0.0607540	−	0.998153i	$-0.480649\pi$
$61$	57.8895	0.121508	0.0607540	+	0.998153i	$0.480649\pi$
$62$	−515.752	−1.05646
$63$	−70.1145	−0.140216
$64$	454.303	0.887310
$65$	237.028	0.452303
$66$	−130.424	−0.243244
$67$	−343.284	−0.625952	−0.312976	−	0.949761i	$-0.601326\pi$
$67$	−343.284	−0.625952	−0.312976	+	0.949761i	$0.601326\pi$
$68$	61.4439	0.109576
$69$	223.734	0.390354
$70$	94.7117	0.161717
$71$	−722.293	−1.20733	−0.603665	−	0.797238i	$-0.706293\pi$
$71$	−722.293	−1.20733	−0.603665	+	0.797238i	$0.706293\pi$
$72$	−234.739	−0.384226
$73$	−700.016	−1.12234	−0.561169	−	0.827701i	$-0.689648\pi$
$73$	−700.016	−1.12234	−0.561169	+	0.827701i	$0.689648\pi$
$74$	140.916	0.221367
$75$	−100.134	−0.154166
$76$	−14.4959	−0.0218789
$77$	−70.3885	−0.104175
$78$	−562.076	−0.815931
$79$	84.7134	0.120646	0.0603228	−	0.998179i	$-0.480787\pi$
$79$	84.7134	0.120646	0.0603228	+	0.998179i	$0.480787\pi$
$80$	347.607	0.485796
$81$	−313.096	−0.429487
$82$	425.686	0.573283
$83$	226.310	0.299286	0.149643	−	0.988740i	$-0.452188\pi$
$83$	226.310	0.299286	0.149643	+	0.988740i	$0.452188\pi$
$84$	−19.5542	−0.0253993
$85$	−402.677	−0.513841
$86$	139.359	0.174738
$87$	400.541	0.493592
$88$	−235.657	−0.285467
$89$	43.0474	0.0512698	0.0256349	−	0.999671i	$-0.491839\pi$
$89$	43.0474	0.0512698	0.0256349	+	0.999671i	$0.491839\pi$
$90$	−162.179	−0.189946
$91$	−303.346	−0.349443
$92$	−42.6170	−0.0482949
$93$	−697.840	−0.778093
$94$	1279.15	1.40356
$95$	95.0000	0.102598
$96$	−137.834	−0.146538
$97$	737.305	0.771773	0.385886	−	0.922546i	$-0.373896\pi$
$97$	737.305	0.771773	0.385886	+	0.922546i	$0.373896\pi$
$98$	894.147	0.921658
$99$	120.529	0.122360
$100$	19.0735	0.0190735
$101$	82.5893	0.0813658	0.0406829	−	0.999172i	$-0.487047\pi$
$101$	82.5893	0.0813658	0.0406829	+	0.999172i	$0.487047\pi$
$102$	954.888	0.926942
$103$	−1250.06	−1.19585	−0.597924	−	0.801553i	$-0.704007\pi$
$103$	−1250.06	−1.19585	−0.597924	+	0.801553i	$0.704007\pi$
$104$	−1015.59	−0.957562
$105$	128.150	0.119106
$106$	432.993	0.396754
$107$	−72.8801	−0.0658466	−0.0329233	−	0.999458i	$-0.510482\pi$
$107$	−72.8801	−0.0658466	−0.0329233	+	0.999458i	$0.510482\pi$
$108$	115.991	0.103345
$109$	1942.34	1.70681	0.853404	−	0.521250i	$-0.174534\pi$
$109$	1942.34	1.70681	0.853404	+	0.521250i	$0.174534\pi$
$110$	−162.812	−0.141123
$111$	190.667	0.163039
$112$	−444.864	−0.375319
$113$	−78.7530	−0.0655616	−0.0327808	−	0.999463i	$-0.510436\pi$
$113$	−78.7530	−0.0655616	−0.0327808	+	0.999463i	$0.510436\pi$
$114$	−225.278	−0.185081
$115$	279.294	0.226472
$116$	−76.2953	−0.0610676
$117$	519.432	0.410440
$118$	2241.02	1.74833
$119$	515.343	0.396986
$120$	429.040	0.326381
$121$	121.000	0.0909091
$122$	−171.366	−0.127170
$123$	575.977	0.422228
$124$	132.925	0.0962663
$125$	−125.000	−0.0894427
$126$	207.555	0.146750
$127$	−1026.00	−0.716869	−0.358435	−	0.933555i	$-0.616689\pi$
$127$	−1026.00	−0.716869	−0.358435	+	0.933555i	$0.616689\pi$
$128$	−1620.14	−1.11876
$129$	188.560	0.128696
$130$	−701.657	−0.473380
$131$	2215.34	1.47752	0.738760	−	0.673968i	$-0.235412\pi$
$131$	2215.34	1.47752	0.738760	+	0.673968i	$0.235412\pi$
$132$	33.6143	0.0221648
$133$	−121.580	−0.0792656
$134$	1016.20	0.655120
$135$	−760.159	−0.484623
$136$	1725.34	1.08784
$137$	853.481	0.532247	0.266123	−	0.963939i	$-0.414257\pi$
$137$	853.481	0.532247	0.266123	+	0.963939i	$0.414257\pi$
$138$	−662.304	−0.408544
$139$	−63.5254	−0.0387637	−0.0193819	−	0.999812i	$-0.506170\pi$
$139$	−63.5254	−0.0387637	−0.0193819	+	0.999812i	$0.506170\pi$
$140$	−24.4101	−0.0147359
$141$	1730.76	1.03373
$142$	2138.15	1.26359
$143$	521.462	0.304943
$144$	761.759	0.440833
$145$	500.007	0.286368
$146$	2072.21	1.17464
$147$	1209.83	0.678810
$148$	−36.3185	−0.0201713
$149$	2631.81	1.44702	0.723511	−	0.690313i	$-0.242527\pi$
$149$	2631.81	1.44702	0.723511	+	0.690313i	$0.242527\pi$
$150$	296.418	0.161350
$151$	456.674	0.246116	0.123058	−	0.992399i	$-0.460730\pi$
$151$	456.674	0.246116	0.123058	+	0.992399i	$0.460730\pi$
$152$	−407.043	−0.217208
$153$	−882.442	−0.466282
$154$	208.366	0.109030
$155$	−871.135	−0.451428
$156$	144.864	0.0743489
$157$	−1389.66	−0.706414	−0.353207	−	0.935545i	$-0.614909\pi$
$157$	−1389.66	−0.706414	−0.353207	+	0.935545i	$0.614909\pi$
$158$	−250.771	−0.126268
$159$	585.863	0.292213
$160$	−172.063	−0.0850173
$161$	−357.438	−0.174969
$162$	926.836	0.449501
$163$	−3831.52	−1.84115	−0.920577	−	0.390561i	$-0.872281\pi$
$163$	−3831.52	−1.84115	−0.920577	+	0.390561i	$0.872281\pi$
$164$	−109.713	−0.0522385
$165$	−220.294	−0.103939
$166$	−669.928	−0.313232
$167$	958.898	0.444322	0.222161	−	0.975010i	$-0.428689\pi$
$167$	958.898	0.444322	0.222161	+	0.975010i	$0.428689\pi$
$168$	−549.081	−0.252158
$169$	50.2933	0.0228918
$170$	1192.02	0.537785
$171$	208.186	0.0931019
$172$	−35.9170	−0.0159224
$173$	459.269	0.201835	0.100918	−	0.994895i	$-0.467822\pi$
$173$	459.269	0.201835	0.100918	+	0.994895i	$0.467822\pi$
$174$	−1185.69	−0.516592
$175$	159.974	0.0691022
$176$	764.736	0.327524
$177$	3032.22	1.28766
$178$	−127.430	−0.0536589
$179$	1609.13	0.671909	0.335954	−	0.941878i	$-0.390941\pi$
$179$	1609.13	0.671909	0.335954	+	0.941878i	$0.390941\pi$
$180$	41.7985	0.0173082
$181$	−2278.88	−0.935844	−0.467922	−	0.883770i	$-0.654997\pi$
$181$	−2278.88	−0.935844	−0.467922	+	0.883770i	$0.654997\pi$
$182$	897.974	0.365727
$183$	−231.867	−0.0936619
$184$	−1196.68	−0.479460
$185$	238.016	0.0945907
$186$	2065.76	0.814351
$187$	−885.890	−0.346432
$188$	−329.676	−0.127894
$189$	972.844	0.374413
$190$	−281.222	−0.107379
$191$	−555.631	−0.210492	−0.105246	−	0.994446i	$-0.533563\pi$
$191$	−555.631	−0.210492	−0.105246	+	0.994446i	$0.533563\pi$
$192$	−1819.64	−0.683965
$193$	−1842.37	−0.687133	−0.343567	−	0.939128i	$-0.611635\pi$
$193$	−1842.37	−0.687133	−0.343567	+	0.939128i	$0.611635\pi$
$194$	−2182.59	−0.807736
$195$	−949.380	−0.348649
$196$	−230.449	−0.0839829
$197$	1170.83	0.423443	0.211721	−	0.977330i	$-0.432093\pi$
$197$	1170.83	0.423443	0.211721	+	0.977330i	$0.432093\pi$
$198$	−356.793	−0.128062
$199$	777.641	0.277013	0.138506	−	0.990362i	$-0.455770\pi$
$199$	777.641	0.277013	0.138506	+	0.990362i	$0.455770\pi$
$200$	535.583	0.189357
$201$	1374.97	0.482502
$202$	−244.483	−0.0851573
$203$	−639.905	−0.221244
$204$	−246.104	−0.0844644
$205$	719.010	0.244965
$206$	3700.47	1.25157
$207$	612.055	0.205511
$208$	3295.71	1.09864
$209$	209.000	0.0691714
$210$	−379.353	−0.124657
$211$	4687.18	1.52928	0.764641	−	0.644456i	$-0.222916\pi$
$211$	4687.18	1.52928	0.764641	+	0.644456i	$0.222916\pi$
$212$	−111.596	−0.0361529
$213$	2893.03	0.930645
$214$	215.742	0.0689149
$215$	235.385	0.0746657
$216$	3257.03	1.02598
$217$	1114.87	0.348767
$218$	−5749.76	−1.78634
$219$	2803.81	0.865132
$220$	41.9618	0.0128594
$221$	−3817.84	−1.16206
$222$	−564.419	−0.170637
$223$	4631.80	1.39089	0.695445	−	0.718579i	$-0.255207\pi$
$223$	4631.80	1.39089	0.695445	+	0.718579i	$0.255207\pi$
$224$	220.205	0.0656832
$225$	−273.930	−0.0811643
$226$	233.127	0.0686166
$227$	4232.36	1.23750	0.618748	−	0.785589i	$-0.287640\pi$
$227$	4232.36	1.23750	0.618748	+	0.785589i	$0.287640\pi$
$228$	58.0611	0.0168649
$229$	1868.43	0.539167	0.269583	−	0.962977i	$-0.413114\pi$
$229$	1868.43	0.539167	0.269583	+	0.962977i	$0.413114\pi$
$230$	−826.774	−0.237025
$231$	281.930	0.0803015
$232$	−2142.36	−0.606264
$233$	−521.200	−0.146545	−0.0732724	−	0.997312i	$-0.523344\pi$
$233$	−521.200	−0.146545	−0.0732724	+	0.997312i	$0.523344\pi$
$234$	−1537.64	−0.429566
$235$	2160.56	0.599742
$236$	−577.580	−0.159310
$237$	−339.307	−0.0929973
$238$	−1525.53	−0.415485
$239$	3983.83	1.07821	0.539105	−	0.842238i	$-0.318763\pi$
$239$	3983.83	1.07821	0.539105	+	0.842238i	$0.318763\pi$
$240$	−1392.29	−0.374466
$241$	130.872	0.0349800	0.0174900	−	0.999847i	$-0.494432\pi$
$241$	130.872	0.0349800	0.0174900	+	0.999847i	$0.494432\pi$
$242$	−358.187	−0.0951453
$243$	−2850.80	−0.752587
$244$	44.1663	0.0115879
$245$	1510.27	0.393826
$246$	−1705.02	−0.441904
$247$	900.707	0.232027
$248$	3732.52	0.955708
$249$	−906.449	−0.230698
$250$	370.028	0.0936106
$251$	5402.80	1.35865	0.679327	−	0.733836i	$-0.262272\pi$
$251$	5402.80	1.35865	0.679327	+	0.733836i	$0.262272\pi$
$252$	−53.4933	−0.0133721
$253$	614.447	0.152688
$254$	3037.18	0.750274
$255$	1612.86	0.396084
$256$	1161.56	0.283584
$257$	−3536.01	−0.858251	−0.429125	−	0.903245i	$-0.641178\pi$
$257$	−3536.01	−0.858251	−0.429125	+	0.903245i	$0.641178\pi$
$258$	−558.180	−0.134693
$259$	−304.611	−0.0730795
$260$	180.839	0.0431351
$261$	1095.73	0.259863
$262$	−6557.91	−1.54637
$263$	2670.37	0.626092	0.313046	−	0.949738i	$-0.398651\pi$
$263$	2670.37	0.626092	0.313046	+	0.949738i	$0.398651\pi$
$264$	943.887	0.220046
$265$	731.350	0.169534
$266$	359.905	0.0829593
$267$	−172.420	−0.0395203
$268$	−261.905	−0.0596956
$269$	7966.67	1.80571	0.902856	−	0.429943i	$-0.141466\pi$
$269$	7966.67	1.80571	0.902856	+	0.429943i	$0.141466\pi$
$270$	2250.24	0.507205
$271$	−7807.89	−1.75017	−0.875085	−	0.483970i	$-0.839194\pi$
$271$	−7807.89	−1.75017	−0.875085	+	0.483970i	$0.839194\pi$
$272$	−5598.94	−1.24811
$273$	1215.01	0.269361
$274$	−2526.50	−0.557048
$275$	−275.000	−0.0603023
$276$	170.696	0.0372272
$277$	−1398.63	−0.303378	−0.151689	−	0.988428i	$-0.548471\pi$
$277$	−1398.63	−0.303378	−0.151689	+	0.988428i	$0.548471\pi$
$278$	188.050	0.0405700
$279$	−1909.04	−0.409646
$280$	−685.434	−0.146295
$281$	2993.33	0.635470	0.317735	−	0.948180i	$-0.397078\pi$
$281$	2993.33	0.635470	0.317735	+	0.948180i	$0.397078\pi$
$282$	−5123.44	−1.08190
$283$	3702.77	0.777763	0.388881	−	0.921288i	$-0.372862\pi$
$283$	3702.77	0.777763	0.388881	+	0.921288i	$0.372862\pi$
$284$	−551.067	−0.115140
$285$	−380.508	−0.0790855
$286$	−1543.65	−0.319153
$287$	−920.182	−0.189256
$288$	−377.065	−0.0771485
$289$	1572.97	0.320164
$290$	−1480.14	−0.299712
$291$	−2953.16	−0.594905
$292$	−534.072	−0.107035
$293$	2431.51	0.484813	0.242407	−	0.970175i	$-0.422063\pi$
$293$	2431.51	0.484813	0.242407	+	0.970175i	$0.422063\pi$
$294$	−3581.37	−0.710441
$295$	3785.22	0.747064
$296$	−1019.82	−0.200256
$297$	−1672.35	−0.326732
$298$	−7790.76	−1.51445
$299$	2648.02	0.512171
$300$	−76.3962	−0.0147025
$301$	−301.244	−0.0576857
$302$	−1351.86	−0.257585
$303$	−330.799	−0.0627192
$304$	1320.91	0.249208
$305$	−289.447	−0.0543400
$306$	2612.23	0.488010
$307$	−1112.29	−0.206781	−0.103391	−	0.994641i	$-0.532969\pi$
$307$	−1112.29	−0.206781	−0.103391	+	0.994641i	$0.532969\pi$
$308$	−53.7023	−0.00993497
$309$	5006.94	0.921795
$310$	2578.76	0.472463
$311$	−2641.10	−0.481553	−0.240777	−	0.970581i	$-0.577402\pi$
$311$	−2641.10	−0.481553	−0.240777	+	0.970581i	$0.577402\pi$
$312$	4067.78	0.738117
$313$	−7475.77	−1.35002	−0.675009	−	0.737810i	$-0.735860\pi$
$313$	−7475.77	−1.35002	−0.675009	+	0.737810i	$0.735860\pi$
$314$	4113.71	0.739331
$315$	350.572	0.0627064
$316$	64.6314	0.0115057
$317$	3511.92	0.622236	0.311118	−	0.950371i	$-0.399297\pi$
$317$	3511.92	0.622236	0.311118	+	0.950371i	$0.399297\pi$
$318$	−1734.29	−0.305830
$319$	1100.02	0.193069
$320$	−2271.51	−0.396817
$321$	291.910	0.0507565
$322$	1058.10	0.183123
$323$	−1530.17	−0.263595
$324$	−238.874	−0.0409592
$325$	−1185.14	−0.202276
$326$	11342.2	1.92695
$327$	−7779.74	−1.31566
$328$	−3080.72	−0.518610
$329$	−2765.06	−0.463352
$330$	652.121	0.108782
$331$	10453.5	1.73587	0.867937	−	0.496674i	$-0.165445\pi$
$331$	10453.5	1.73587	0.867937	+	0.496674i	$0.165445\pi$
$332$	172.661	0.0285422
$333$	521.597	0.0858358
$334$	−2838.56	−0.465027
$335$	1716.42	0.279934
$336$	1781.84	0.289307
$337$	−6622.09	−1.07041	−0.535205	−	0.844722i	$-0.679766\pi$
$337$	−6622.09	−1.07041	−0.535205	+	0.844722i	$0.679766\pi$
$338$	−148.880	−0.0239585
$339$	315.433	0.0505368
$340$	−307.219	−0.0490039
$341$	−1916.50	−0.304352
$342$	−616.279	−0.0974402
$343$	−4127.67	−0.649775
$344$	−1008.55	−0.158073
$345$	−1118.67	−0.174572
$346$	−1359.54	−0.211241
$347$	6670.99	1.03204	0.516020	−	0.856577i	$-0.327413\pi$
$347$	6670.99	1.03204	0.516020	+	0.856577i	$0.327413\pi$
$348$	305.589	0.0470727
$349$	−3242.13	−0.497271	−0.248635	−	0.968597i	$-0.579982\pi$
$349$	−3242.13	−0.497271	−0.248635	+	0.968597i	$0.579982\pi$
$350$	−473.559	−0.0723222
$351$	−7207.16	−1.09598
$352$	−378.539	−0.0573187
$353$	5514.16	0.831414	0.415707	−	0.909499i	$-0.363534\pi$
$353$	5514.16	0.831414	0.415707	+	0.909499i	$0.363534\pi$
$354$	−8976.06	−1.34766
$355$	3611.46	0.539934
$356$	32.8427	0.00488949
$357$	−2064.13	−0.306009
$358$	−4763.37	−0.703218
$359$	−2282.01	−0.335487	−0.167744	−	0.985831i	$-0.553648\pi$
$359$	−2282.01	−0.335487	−0.167744	+	0.985831i	$0.553648\pi$
$360$	1173.70	0.171831
$361$	361.000	0.0526316
$362$	6746.00	0.979453
$363$	−484.647	−0.0700754
$364$	−231.436	−0.0333256
$365$	3500.08	0.501925
$366$	686.380	0.0980264
$367$	6860.89	0.975846	0.487923	−	0.872887i	$-0.337755\pi$
$367$	6860.89	0.975846	0.487923	+	0.872887i	$0.337755\pi$
$368$	3883.39	0.550097
$369$	1575.66	0.222292
$370$	−704.581	−0.0989985
$371$	−935.975	−0.130980
$372$	−532.411	−0.0742049
$373$	−7510.14	−1.04252	−0.521261	−	0.853398i	$-0.674538\pi$
$373$	−7510.14	−1.04252	−0.521261	+	0.853398i	$0.674538\pi$
$374$	2622.44	0.362575
$375$	500.669	0.0689451
$376$	−9257.28	−1.26970
$377$	4740.63	0.647626
$378$	−2879.84	−0.391859
$379$	−3875.03	−0.525189	−0.262595	−	0.964906i	$-0.584578\pi$
$379$	−3875.03	−0.525189	−0.262595	+	0.964906i	$0.584578\pi$
$380$	72.4795	0.00978452
$381$	4109.47	0.552584
$382$	1644.79	0.220301
$383$	−13817.8	−1.84349	−0.921745	−	0.387796i	$-0.873236\pi$
$383$	−13817.8	−1.84349	−0.921745	+	0.387796i	$0.873236\pi$
$384$	6489.22	0.862374
$385$	351.942	0.0465887
$386$	5453.83	0.719152
$387$	515.832	0.0677550
$388$	562.520	0.0736022
$389$	6722.21	0.876169	0.438085	−	0.898934i	$-0.355657\pi$
$389$	6722.21	0.876169	0.438085	+	0.898934i	$0.355657\pi$
$390$	2810.38	0.364895
$391$	−4498.62	−0.581854
$392$	−6470.99	−0.833761
$393$	−8873.21	−1.13892
$394$	−3465.92	−0.443174
$395$	−423.567	−0.0539544
$396$	91.9566	0.0116692
$397$	−6654.55	−0.841265	−0.420633	−	0.907231i	$-0.638192\pi$
$397$	−6654.55	−0.841265	−0.420633	+	0.907231i	$0.638192\pi$
$398$	−2301.99	−0.289921
$399$	486.971	0.0611003
$400$	−1738.04	−0.217255
$401$	7199.16	0.896531	0.448265	−	0.893901i	$-0.352042\pi$
$401$	7199.16	0.896531	0.448265	+	0.893901i	$0.352042\pi$
$402$	−4070.22	−0.504986
$403$	−8259.34	−1.02091
$404$	63.0109	0.00775967
$405$	1565.48	0.192073
$406$	1894.26	0.231553
$407$	523.635	0.0637731
$408$	−6910.58	−0.838541
$409$	5543.10	0.670143	0.335072	−	0.942193i	$-0.391239\pi$
$409$	5543.10	0.670143	0.335072	+	0.942193i	$0.391239\pi$
$410$	−2128.43	−0.256380
$411$	−3418.49	−0.410271
$412$	−953.725	−0.114045
$413$	−4844.28	−0.577171
$414$	−1811.82	−0.215088
$415$	−1131.55	−0.133845
$416$	−1631.35	−0.192268
$417$	254.442	0.0298802
$418$	−618.687	−0.0723947
$419$	−14070.3	−1.64052	−0.820262	−	0.571989i	$-0.806172\pi$
$419$	−14070.3	−1.64052	−0.820262	+	0.571989i	$0.806172\pi$
$420$	97.7711	0.0113589
$421$	8852.04	1.02476	0.512378	−	0.858760i	$-0.328765\pi$
$421$	8852.04	1.02476	0.512378	+	0.858760i	$0.328765\pi$
$422$	−13875.1	−1.60054
$423$	4734.73	0.544233
$424$	−3133.59	−0.358917
$425$	2013.39	0.229797
$426$	−8564.04	−0.974012
$427$	370.432	0.0419823
$428$	−55.6033	−0.00627964
$429$	−2088.64	−0.235059
$430$	−696.793	−0.0781450
$431$	−6582.28	−0.735631	−0.367816	−	0.929899i	$-0.619894\pi$
$431$	−6582.28	−0.735631	−0.367816	+	0.929899i	$0.619894\pi$
$432$	−10569.5	−1.17714
$433$	−5416.95	−0.601205	−0.300603	−	0.953749i	$-0.597188\pi$
$433$	−5416.95	−0.601205	−0.300603	+	0.953749i	$0.597188\pi$
$434$	−3300.27	−0.365018
$435$	−2002.70	−0.220741
$436$	1481.89	0.162774
$437$	1061.32	0.116178
$438$	−8299.91	−0.905445
$439$	6867.93	0.746671	0.373335	−	0.927696i	$-0.378214\pi$
$439$	6867.93	0.746671	0.373335	+	0.927696i	$0.378214\pi$
$440$	1178.28	0.127665
$441$	3309.65	0.357376
$442$	11301.7	1.21621
$443$	4336.21	0.465055	0.232527	−	0.972590i	$-0.425300\pi$
$443$	4336.21	0.465055	0.232527	+	0.972590i	$0.425300\pi$
$444$	145.468	0.0155487
$445$	−215.237	−0.0229286
$446$	−13711.2	−1.45570
$447$	−10541.3	−1.11541
$448$	2907.06	0.306575
$449$	13884.3	1.45933	0.729666	−	0.683804i	$-0.239675\pi$
$449$	13884.3	1.45933	0.729666	+	0.683804i	$0.239675\pi$
$450$	810.894	0.0849464
$451$	1581.82	0.165155
$452$	−60.0839	−0.00625246
$453$	−1829.14	−0.189714
$454$	−12528.7	−1.29516
$455$	1516.73	0.156276
$456$	1630.35	0.167430
$457$	10193.5	1.04339	0.521697	−	0.853131i	$-0.325299\pi$
$457$	10193.5	1.04339	0.521697	+	0.853131i	$0.325299\pi$
$458$	−5530.97	−0.564291
$459$	12244.0	1.24510
$460$	213.085	0.0215981
$461$	3806.60	0.384579	0.192289	−	0.981338i	$-0.438409\pi$
$461$	3806.60	0.384579	0.192289	+	0.981338i	$0.438409\pi$
$462$	−834.578	−0.0840434
$463$	−13212.2	−1.32619	−0.663093	−	0.748537i	$-0.730757\pi$
$463$	−13212.2	−1.32619	−0.663093	+	0.748537i	$0.730757\pi$
$464$	6952.25	0.695582
$465$	3489.20	0.347974
$466$	1542.87	0.153373
$467$	−9980.45	−0.988951	−0.494475	−	0.869192i	$-0.664640\pi$
$467$	−9980.45	−0.988951	−0.494475	+	0.869192i	$0.664640\pi$
$468$	396.296	0.0391428
$469$	−2196.66	−0.216273
$470$	−6395.75	−0.627689
$471$	5566.07	0.544525
$472$	−16218.4	−1.58159
$473$	517.847	0.0503396
$474$	1004.43	0.0973308
$475$	−475.000	−0.0458831
$476$	393.176	0.0378597
$477$	1602.71	0.153843
$478$	−11793.0	−1.12845
$479$	−11563.2	−1.10300	−0.551499	−	0.834175i	$-0.685944\pi$
$479$	−11563.2	−1.10300	−0.551499	+	0.834175i	$0.685944\pi$
$480$	689.172	0.0655339
$481$	2256.66	0.213919
$482$	−387.410	−0.0366100
$483$	1431.66	0.134872
$484$	92.3159	0.00866979
$485$	−3686.52	−0.345147
$486$	8439.01	0.787657
$487$	12099.1	1.12580	0.562899	−	0.826526i	$-0.309686\pi$
$487$	12099.1	1.12580	0.562899	+	0.826526i	$0.309686\pi$
$488$	1240.19	0.115042
$489$	15346.6	1.41922
$490$	−4470.73	−0.412178
$491$	−13152.0	−1.20884	−0.604422	−	0.796664i	$-0.706596\pi$
$491$	−13152.0	−1.20884	−0.604422	+	0.796664i	$0.706596\pi$
$492$	439.437	0.0402670
$493$	−8053.67	−0.735738
$494$	−2666.30	−0.242839
$495$	−602.645	−0.0547210
$496$	−12112.5	−1.09651
$497$	−4621.92	−0.417145
$498$	2683.30	0.241449
$499$	18622.0	1.67062	0.835308	−	0.549782i	$-0.185289\pi$
$499$	18622.0	1.67062	0.835308	+	0.549782i	$0.185289\pi$
$500$	−95.3677	−0.00852995
$501$	−3840.72	−0.342497
$502$	−15993.5	−1.42196
$503$	1780.07	0.157792	0.0788960	−	0.996883i	$-0.474861\pi$
$503$	1780.07	0.157792	0.0788960	+	0.996883i	$0.474861\pi$
$504$	−1502.09	−0.132754
$505$	−412.947	−0.0363879
$506$	−1818.90	−0.159803
$507$	−201.442	−0.0176457
$508$	−782.775	−0.0683662
$509$	−9302.53	−0.810074	−0.405037	−	0.914300i	$-0.632741\pi$
$509$	−9302.53	−0.810074	−0.405037	+	0.914300i	$0.632741\pi$
$510$	−4774.44	−0.414541
$511$	−4479.37	−0.387780
$512$	9522.64	0.821963
$513$	−2888.60	−0.248606
$514$	10467.4	0.898244
$515$	6250.31	0.534799
$516$	143.860	0.0122734
$517$	4753.23	0.404346
$518$	901.716	0.0764848
$519$	−1839.53	−0.155581
$520$	5077.93	0.428235
$521$	−8052.10	−0.677099	−0.338550	−	0.940949i	$-0.609936\pi$
$521$	−8052.10	−0.677099	−0.338550	+	0.940949i	$0.609936\pi$
$522$	−3243.62	−0.271972
$523$	−8847.50	−0.739721	−0.369861	−	0.929087i	$-0.620595\pi$
$523$	−8847.50	−0.739721	−0.369861	+	0.929087i	$0.620595\pi$
$524$	1690.18	0.140908
$525$	−640.751	−0.0532660
$526$	−7904.91	−0.655267
$527$	14031.5	1.15981
$528$	−3063.03	−0.252465
$529$	−9046.79	−0.743551
$530$	−2164.96	−0.177434
$531$	8295.06	0.677919
$532$	−92.7585	−0.00755938
$533$	6817.02	0.553993
$534$	510.402	0.0413619
$535$	364.401	0.0294475
$536$	−7354.28	−0.592643
$537$	−6445.11	−0.517927
$538$	−23583.2	−1.88985
$539$	3322.59	0.265518
$540$	−579.957	−0.0462173
$541$	−1259.37	−0.100083	−0.0500414	−	0.998747i	$-0.515935\pi$
$541$	−1259.37	−0.100083	−0.0500414	+	0.998747i	$0.515935\pi$
$542$	23113.1	1.83172
$543$	9127.71	0.721376
$544$	2771.44	0.218427
$545$	−9711.68	−0.763308
$546$	−3596.70	−0.281913
$547$	14159.7	1.10681	0.553405	−	0.832913i	$-0.313328\pi$
$547$	14159.7	1.10681	0.553405	+	0.832913i	$0.313328\pi$
$548$	651.156	0.0507591
$549$	−634.305	−0.0493106
$550$	814.062	0.0631122
$551$	1900.03	0.146904
$552$	4793.13	0.369582
$553$	542.077	0.0416844
$554$	4140.27	0.317515
$555$	−953.337	−0.0729133
$556$	−48.4662	−0.00369681
$557$	5348.10	0.406834	0.203417	−	0.979092i	$-0.434795\pi$
$557$	5348.10	0.406834	0.203417	+	0.979092i	$0.434795\pi$
$558$	5651.18	0.428734
$559$	2231.72	0.168858
$560$	2224.32	0.167848
$561$	3548.30	0.267040
$562$	−8860.93	−0.665081
$563$	10197.8	0.763382	0.381691	−	0.924290i	$-0.375342\pi$
$563$	10197.8	0.763382	0.381691	+	0.924290i	$0.375342\pi$
$564$	1320.47	0.0985847
$565$	393.765	0.0293200
$566$	−10961.0	−0.814005
$567$	−2003.49	−0.148393
$568$	−15473.9	−1.14308
$569$	−4973.75	−0.366451	−0.183225	−	0.983071i	$-0.558654\pi$
$569$	−4973.75	−0.366451	−0.183225	+	0.983071i	$0.558654\pi$
$570$	1126.39	0.0827707
$571$	6501.42	0.476490	0.238245	−	0.971205i	$-0.423428\pi$
$571$	6501.42	0.476490	0.238245	+	0.971205i	$0.423428\pi$
$572$	397.845	0.0290817
$573$	2225.49	0.162254
$574$	2723.95	0.198076
$575$	−1396.47	−0.101281
$576$	−4977.88	−0.360089
$577$	3537.35	0.255220	0.127610	−	0.991824i	$-0.459269\pi$
$577$	3537.35	0.255220	0.127610	+	0.991824i	$0.459269\pi$
$578$	−4656.34	−0.335083
$579$	7379.33	0.529663
$580$	381.476	0.0273103
$581$	1448.14	0.103407
$582$	8742.03	0.622627
$583$	1608.97	0.114300
$584$	−14996.7	−1.06262
$585$	−2597.16	−0.183554
$586$	−7197.82	−0.507405
$587$	9851.41	0.692694	0.346347	−	0.938107i	$-0.387422\pi$
$587$	9851.41	0.692694	0.346347	+	0.938107i	$0.387422\pi$
$588$	923.029	0.0647365
$589$	−3310.31	−0.231577
$590$	−11205.1	−0.781875
$591$	−4689.58	−0.326402
$592$	3309.44	0.229759
$593$	4766.46	0.330076	0.165038	−	0.986287i	$-0.447225\pi$
$593$	4766.46	0.330076	0.165038	+	0.986287i	$0.447225\pi$
$594$	4950.53	0.341958
$595$	−2576.71	−0.177538
$596$	2007.92	0.137999
$597$	−3114.72	−0.213530
$598$	−7838.75	−0.536037
$599$	−22768.7	−1.55310	−0.776548	−	0.630058i	$-0.783031\pi$
$599$	−22768.7	−1.55310	−0.776548	+	0.630058i	$0.783031\pi$
$600$	−2145.20	−0.145962
$601$	4421.46	0.300091	0.150046	−	0.988679i	$-0.452058\pi$
$601$	4421.46	0.300091	0.150046	+	0.988679i	$0.452058\pi$
$602$	891.749	0.0603737
$603$	3761.42	0.254025
$604$	348.415	0.0234716
$605$	−605.000	−0.0406558
$606$	979.240	0.0656418
$607$	28610.9	1.91315	0.956575	−	0.291488i	$-0.0941503\pi$
$607$	28610.9	1.91315	0.956575	+	0.291488i	$0.0941503\pi$
$608$	−653.839	−0.0436130
$609$	2563.04	0.170541
$610$	856.830	0.0568722
$611$	20484.5	1.35633
$612$	−673.252	−0.0444683
$613$	777.932	0.0512567	0.0256284	−	0.999672i	$-0.491841\pi$
$613$	777.932	0.0512567	0.0256284	+	0.999672i	$0.491841\pi$
$614$	3292.64	0.216417
$615$	−2879.89	−0.188826
$616$	−1507.96	−0.0986319
$617$	2081.79	0.135834	0.0679172	−	0.997691i	$-0.478365\pi$
$617$	2081.79	0.135834	0.0679172	+	0.997691i	$0.478365\pi$
$618$	−14821.7	−0.964749
$619$	7488.28	0.486235	0.243117	−	0.969997i	$-0.421830\pi$
$619$	7488.28	0.486235	0.243117	+	0.969997i	$0.421830\pi$
$620$	−664.625	−0.0430516
$621$	−8492.32	−0.548768
$622$	7818.25	0.503993
$623$	275.458	0.0177143
$624$	−13200.5	−0.846861
$625$	625.000	0.0400000
$626$	22130.0	1.41293
$627$	−837.118	−0.0533194
$628$	−1060.23	−0.0673691
$629$	−3833.75	−0.243023
$630$	−1037.77	−0.0656284
$631$	−22699.8	−1.43212	−0.716058	−	0.698041i	$-0.754055\pi$
$631$	−22699.8	−1.43212	−0.716058	+	0.698041i	$0.754055\pi$
$632$	1814.84	0.114226
$633$	−18773.8	−1.17882
$634$	−10396.1	−0.651231
$635$	5129.98	0.320594
$636$	446.979	0.0278677
$637$	14319.0	0.890645
$638$	−3256.30	−0.202066
$639$	7914.29	0.489960
$640$	8100.70	0.500325
$641$	19445.2	1.19819	0.599095	−	0.800678i	$-0.295527\pi$
$641$	19445.2	1.19819	0.599095	+	0.800678i	$0.295527\pi$
$642$	−864.121	−0.0531217
$643$	3490.51	0.214078	0.107039	−	0.994255i	$-0.465863\pi$
$643$	3490.51	0.214078	0.107039	+	0.994255i	$0.465863\pi$
$644$	−272.704	−0.0166864
$645$	−942.799	−0.0575546
$646$	4529.66	0.275878
$647$	2957.99	0.179738	0.0898691	−	0.995954i	$-0.471355\pi$
$647$	2957.99	0.179738	0.0898691	+	0.995954i	$0.471355\pi$
$648$	−6707.56	−0.406633
$649$	8327.47	0.503670
$650$	3508.28	0.211702
$651$	−4465.44	−0.268840
$652$	−2923.23	−0.175587
$653$	5397.23	0.323445	0.161723	−	0.986836i	$-0.448295\pi$
$653$	5397.23	0.323445	0.161723	+	0.986836i	$0.448295\pi$
$654$	23029.8	1.37697
$655$	−11076.7	−0.660767
$656$	9997.32	0.595015
$657$	7670.21	0.455469
$658$	8185.22	0.484944
$659$	27128.7	1.60362	0.801809	−	0.597580i	$-0.203871\pi$
$659$	27128.7	1.60362	0.801809	+	0.597580i	$0.203871\pi$
$660$	−168.072	−0.00991239
$661$	−27935.5	−1.64382	−0.821910	−	0.569617i	$-0.807092\pi$
$661$	−27935.5	−1.64382	−0.821910	+	0.569617i	$0.807092\pi$
$662$	−30944.6	−1.81676
$663$	15291.8	0.895751
$664$	4848.31	0.283360
$665$	607.900	0.0354487
$666$	−1544.05	−0.0898356
$667$	5585.97	0.324272
$668$	731.583	0.0423740
$669$	−18552.0	−1.07214
$670$	−5080.99	−0.292979
$671$	−636.784	−0.0366360
$672$	−881.996	−0.0506305
$673$	−18817.0	−1.07777	−0.538886	−	0.842378i	$-0.681155\pi$
$673$	−18817.0	−1.07777	−0.538886	+	0.842378i	$0.681155\pi$
$674$	19602.9	1.12029
$675$	3800.79	0.216730
$676$	38.3709	0.00218314
$677$	32195.3	1.82772	0.913858	−	0.406033i	$-0.133088\pi$
$677$	32195.3	1.82772	0.913858	+	0.406033i	$0.133088\pi$
$678$	−933.754	−0.0528917
$679$	4717.98	0.266656
$680$	−8626.69	−0.486498
$681$	−16952.1	−0.953899
$682$	5673.27	0.318535
$683$	−2926.91	−0.163975	−0.0819876	−	0.996633i	$-0.526127\pi$
$683$	−2926.91	−0.163975	−0.0819876	+	0.996633i	$0.526127\pi$
$684$	158.834	0.00887891
$685$	−4267.40	−0.238028
$686$	12218.8	0.680054
$687$	−7483.70	−0.415606
$688$	3272.86	0.181362
$689$	6934.02	0.383404
$690$	3311.52	0.182706
$691$	14860.0	0.818093	0.409047	−	0.912513i	$-0.365861\pi$
$691$	14860.0	0.818093	0.409047	+	0.912513i	$0.365861\pi$
$692$	350.395	0.0192486
$693$	771.259	0.0422766
$694$	−19747.7	−1.08013
$695$	317.627	0.0173357
$696$	8580.92	0.467326
$697$	−11581.2	−0.629366
$698$	9597.45	0.520443
$699$	2087.59	0.112961
$700$	122.051	0.00659012
$701$	2134.43	0.115002	0.0575010	−	0.998345i	$-0.481687\pi$
$701$	2134.43	0.115002	0.0575010	+	0.998345i	$0.481687\pi$
$702$	21334.8	1.14705
$703$	904.461	0.0485240
$704$	−4997.33	−0.267534
$705$	−8653.80	−0.462299
$706$	−16323.2	−0.870157
$707$	528.485	0.0281128
$708$	2313.41	0.122801
$709$	6595.19	0.349348	0.174674	−	0.984626i	$-0.444113\pi$
$709$	6595.19	0.349348	0.174674	+	0.984626i	$0.444113\pi$
$710$	−10690.8	−0.565094
$711$	−928.221	−0.0489606
$712$	922.219	0.0485416
$713$	−9732.12	−0.511179
$714$	6110.28	0.320268
$715$	−2607.31	−0.136375
$716$	1227.67	0.0640784
$717$	−15956.6	−0.831117
$718$	6755.27	0.351120
$719$	10445.3	0.541784	0.270892	−	0.962610i	$-0.412681\pi$
$719$	10445.3	0.541784	0.270892	+	0.962610i	$0.412681\pi$
$720$	−3808.80	−0.197146
$721$	−7999.09	−0.413178
$722$	−1068.64	−0.0550841
$723$	−524.187	−0.0269637
$724$	−1738.65	−0.0892493
$725$	−2500.04	−0.128068
$726$	1434.67	0.0733408
$727$	−2202.60	−0.112366	−0.0561830	−	0.998420i	$-0.517893\pi$
$727$	−2202.60	−0.112366	−0.0561830	+	0.998420i	$0.517893\pi$
$728$	−6498.69	−0.330848
$729$	19872.0	1.00960
$730$	−10361.0	−0.525314
$731$	−3791.37	−0.191832
$732$	−176.901	−0.00893232
$733$	−18935.3	−0.954149	−0.477075	−	0.878863i	$-0.658303\pi$
$733$	−18935.3	−0.954149	−0.477075	+	0.878863i	$0.658303\pi$
$734$	−20309.8	−1.02132
$735$	−6049.15	−0.303573
$736$	−1922.25	−0.0962704
$737$	3776.12	0.188732
$738$	−4664.32	−0.232651
$739$	−3303.33	−0.164432	−0.0822158	−	0.996615i	$-0.526200\pi$
$739$	−3303.33	−0.164432	−0.0822158	+	0.996615i	$0.526200\pi$
$740$	181.592	0.00902090
$741$	−3607.64	−0.178853
$742$	2770.70	0.137083
$743$	−29787.2	−1.47078	−0.735388	−	0.677646i	$-0.763000\pi$
$743$	−29787.2	−1.47078	−0.735388	+	0.677646i	$0.763000\pi$
$744$	−14950.1	−0.736688
$745$	−13159.1	−0.647128
$746$	22231.7	1.09110
$747$	−2479.72	−0.121457
$748$	−675.883	−0.0330384
$749$	−466.356	−0.0227507
$750$	−1482.09	−0.0721578
$751$	32488.5	1.57859	0.789295	−	0.614014i	$-0.210446\pi$
$751$	32488.5	1.57859	0.789295	+	0.614014i	$0.210446\pi$
$752$	30041.1	1.45676
$753$	−21640.1	−1.04729
$754$	−14033.3	−0.677804
$755$	−2283.37	−0.110067
$756$	742.223	0.0357069
$757$	−4948.06	−0.237570	−0.118785	−	0.992920i	$-0.537900\pi$
$757$	−4948.06	−0.237570	−0.118785	+	0.992920i	$0.537900\pi$
$758$	11471.0	0.549662
$759$	−2461.07	−0.117696
$760$	2035.22	0.0971382
$761$	−1292.29	−0.0615577	−0.0307789	−	0.999526i	$-0.509799\pi$
$761$	−1292.29	−0.0615577	−0.0307789	+	0.999526i	$0.509799\pi$
$762$	−12165.0	−0.578334
$763$	12428.9	0.589721
$764$	−423.914	−0.0200742
$765$	4412.21	0.208528
$766$	40903.8	1.92939
$767$	35888.1	1.68950
$768$	−4652.45	−0.218595
$769$	895.454	0.0419907	0.0209954	−	0.999780i	$-0.493316\pi$
$769$	895.454	0.0419907	0.0209954	+	0.999780i	$0.493316\pi$
$770$	−1041.83	−0.0487596
$771$	14163.0	0.661565
$772$	−1405.62	−0.0655303
$773$	33241.3	1.54671	0.773356	−	0.633972i	$-0.218577\pi$
$773$	33241.3	1.54671	0.773356	+	0.633972i	$0.218577\pi$
$774$	−1526.98	−0.0709123
$775$	4355.68	0.201885
$776$	15795.5	0.730704
$777$	1220.07	0.0563318
$778$	−19899.3	−0.916997
$779$	2732.24	0.125664
$780$	−724.322	−0.0332498
$781$	7945.22	0.364023
$782$	13316.9	0.608967
$783$	−15203.4	−0.693902
$784$	20999.2	0.956596
$785$	6948.30	0.315918
$786$	26266.7	1.19199
$787$	17278.5	0.782608	0.391304	−	0.920262i	$-0.372024\pi$
$787$	17278.5	0.782608	0.391304	+	0.920262i	$0.372024\pi$
$788$	893.275	0.0403828
$789$	−10695.8	−0.482611
$790$	1253.85	0.0564686
$791$	−503.937	−0.0226522
$792$	2582.13	0.115849
$793$	−2744.29	−0.122891
$794$	19699.0	0.880467
$795$	−2929.31	−0.130682
$796$	593.295	0.0264181
$797$	11016.9	0.489637	0.244818	−	0.969569i	$-0.421272\pi$
$797$	11016.9	0.489637	0.244818	+	0.969569i	$0.421272\pi$
$798$	−1441.54	−0.0639475
$799$	−34800.4	−1.54086
$800$	860.315	0.0380209
$801$	−471.678	−0.0208064
$802$	−21311.1	−0.938307
$803$	7700.18	0.338398
$804$	1049.02	0.0460151
$805$	1787.19	0.0782486
$806$	24449.5	1.06848
$807$	−31909.3	−1.39190
$808$	1769.34	0.0770361
$809$	16144.2	0.701607	0.350803	−	0.936449i	$-0.385909\pi$
$809$	16144.2	0.701607	0.350803	+	0.936449i	$0.385909\pi$
$810$	−4634.18	−0.201023
$811$	−6193.96	−0.268187	−0.134093	−	0.990969i	$-0.542812\pi$
$811$	−6193.96	−0.268187	−0.134093	+	0.990969i	$0.542812\pi$
$812$	−488.210	−0.0210995
$813$	31273.3	1.34908
$814$	−1550.08	−0.0667448
$815$	19157.6	0.823389
$816$	22425.7	0.962080
$817$	894.463	0.0383027
$818$	−16408.8	−0.701371
$819$	3323.82	0.141812
$820$	548.563	0.0233618
$821$	−5571.35	−0.236835	−0.118417	−	0.992964i	$-0.537782\pi$
$821$	−5571.35	−0.236835	−0.118417	+	0.992964i	$0.537782\pi$
$822$	10119.5	0.429389
$823$	−13904.7	−0.588926	−0.294463	−	0.955663i	$-0.595141\pi$
$823$	−13904.7	−0.588926	−0.294463	+	0.955663i	$0.595141\pi$
$824$	−26780.5	−1.13221
$825$	1101.47	0.0464828
$826$	14340.2	0.604066
$827$	−11994.7	−0.504348	−0.252174	−	0.967682i	$-0.581146\pi$
$827$	−11994.7	−0.504348	−0.252174	+	0.967682i	$0.581146\pi$
$828$	466.963	0.0195991
$829$	−16647.9	−0.697473	−0.348737	−	0.937221i	$-0.613389\pi$
$829$	−16647.9	−0.697473	−0.348737	+	0.937221i	$0.613389\pi$
$830$	3349.64	0.140082
$831$	5602.01	0.233853
$832$	−21536.5	−0.897408
$833$	−24326.0	−1.01182
$834$	−753.205	−0.0312726
$835$	−4794.49	−0.198707
$836$	159.455	0.00659672
$837$	26488.0	1.09386
$838$	41651.3	1.71697
$839$	32993.8	1.35765	0.678827	−	0.734298i	$-0.262489\pi$
$839$	32993.8	1.35765	0.678827	+	0.734298i	$0.262489\pi$
$840$	2745.40	0.112768
$841$	−14388.7	−0.589967
$842$	−26204.1	−1.07251
$843$	−11989.3	−0.489839
$844$	3576.04	0.145844
$845$	−251.467	−0.0102375
$846$	−14015.9	−0.569593
$847$	774.273	0.0314101
$848$	10168.9	0.411794
$849$	−14830.9	−0.599523
$850$	−5960.08	−0.240505
$851$	2659.06	0.107111
$852$	2207.22	0.0887535
$853$	−16109.2	−0.646621	−0.323310	−	0.946293i	$-0.604796\pi$
$853$	−16109.2	−0.646621	−0.323310	+	0.946293i	$0.604796\pi$
$854$	−1096.56	−0.0439386
$855$	−1040.93	−0.0416364
$856$	−1561.34	−0.0623427
$857$	−12388.0	−0.493776	−0.246888	−	0.969044i	$-0.579408\pi$
$857$	−12388.0	−0.493776	−0.246888	+	0.969044i	$0.579408\pi$
$858$	6182.84	0.246012
$859$	43349.4	1.72184	0.860921	−	0.508739i	$-0.169888\pi$
$859$	43349.4	1.72184	0.860921	+	0.508739i	$0.169888\pi$
$860$	179.585	0.00712070
$861$	3685.65	0.145885
$862$	19485.0	0.769911
$863$	28411.1	1.12065	0.560327	−	0.828271i	$-0.310675\pi$
$863$	28411.1	1.12065	0.560327	+	0.828271i	$0.310675\pi$
$864$	5231.81	0.206007
$865$	−2296.34	−0.0902636
$866$	16035.4	0.629220
$867$	−6300.28	−0.246792
$868$	850.581	0.0332611
$869$	−931.848	−0.0363760
$870$	5928.46	0.231027
$871$	16273.6	0.633076
$872$	41611.3	1.61598
$873$	−8078.78	−0.313202
$874$	−3141.74	−0.121592
$875$	−799.869	−0.0309034
$876$	2139.14	0.0825057
$877$	−8548.95	−0.329165	−0.164582	−	0.986363i	$-0.552628\pi$
$877$	−8548.95	−0.329165	−0.164582	+	0.986363i	$0.552628\pi$
$878$	−20330.6	−0.781464
$879$	−9739.05	−0.373709
$880$	−3823.68	−0.146473
$881$	24601.5	0.940802	0.470401	−	0.882453i	$-0.344109\pi$
$881$	24601.5	0.940802	0.470401	+	0.882453i	$0.344109\pi$
$882$	−9797.33	−0.374029
$883$	−18007.8	−0.686308	−0.343154	−	0.939279i	$-0.611495\pi$
$883$	−18007.8	−0.686308	−0.343154	+	0.939279i	$0.611495\pi$
$884$	−2912.79	−0.110823
$885$	−15161.1	−0.575859
$886$	−12836.2	−0.486726
$887$	13726.3	0.519597	0.259799	−	0.965663i	$-0.416344\pi$
$887$	13726.3	0.519597	0.259799	+	0.965663i	$0.416344\pi$
$888$	4084.73	0.154363
$889$	−6565.30	−0.247686
$890$	637.150	0.0239970
$891$	3444.06	0.129495
$892$	3533.79	0.132646
$893$	8210.13	0.307661
$894$	31204.7	1.16738
$895$	−8045.63	−0.300487
$896$	−10367.2	−0.386544
$897$	−10606.3	−0.394797
$898$	−41100.6	−1.52733
$899$	−17423.0	−0.646372
$900$	−208.992	−0.00774046
$901$	−11779.9	−0.435568
$902$	−4682.55	−0.172851
$903$	1206.59	0.0444658
$904$	−1687.15	−0.0620728
$905$	11394.4	0.418522
$906$	5414.66	0.198554
$907$	29409.6	1.07666	0.538330	−	0.842734i	$-0.319055\pi$
$907$	29409.6	1.07666	0.538330	+	0.842734i	$0.319055\pi$
$908$	3229.04	0.118017
$909$	−904.947	−0.0330200
$910$	−4489.87	−0.163558
$911$	−19169.0	−0.697142	−0.348571	−	0.937282i	$-0.613333\pi$
$911$	−19169.0	−0.697142	−0.348571	+	0.937282i	$0.613333\pi$
$912$	−5290.70	−0.192097
$913$	−2489.41	−0.0902381
$914$	−30175.0	−1.09201
$915$	1159.34	0.0418869
$916$	1425.50	0.0514191
$917$	14175.9	0.510499
$918$	−36244.9	−1.30311
$919$	−11781.8	−0.422902	−0.211451	−	0.977389i	$-0.567819\pi$
$919$	−11781.8	−0.422902	−0.211451	+	0.977389i	$0.567819\pi$
$920$	5983.41	0.214421
$921$	4455.12	0.159393
$922$	−11268.4	−0.402499
$923$	34240.7	1.22107
$924$	215.096	0.00765817
$925$	−1190.08	−0.0423023
$926$	39111.2	1.38798
$927$	13697.2	0.485301
$928$	−3441.31	−0.121731
$929$	−18019.5	−0.636383	−0.318191	−	0.948026i	$-0.603075\pi$
$929$	−18019.5	−0.636383	−0.318191	+	0.948026i	$0.603075\pi$
$930$	−10328.8	−0.364189
$931$	5739.02	0.202029
$932$	−397.645	−0.0139756
$933$	10578.5	0.371195
$934$	29544.4	1.03503
$935$	4429.45	0.154929
$936$	11128.0	0.388599
$937$	−43929.6	−1.53161	−0.765804	−	0.643074i	$-0.777659\pi$
$937$	−43929.6	−1.53161	−0.765804	+	0.643074i	$0.777659\pi$
$938$	6502.60	0.226351
$939$	29943.1	1.04063
$940$	1648.38	0.0571960
$941$	16720.1	0.579233	0.289616	−	0.957143i	$-0.406472\pi$
$941$	16720.1	0.579233	0.289616	+	0.957143i	$0.406472\pi$
$942$	−16476.8	−0.569899
$943$	8032.61	0.277389
$944$	52630.7	1.81460
$945$	−4864.22	−0.167442
$946$	−1532.95	−0.0526854
$947$	3507.74	0.120366	0.0601829	−	0.998187i	$-0.480832\pi$
$947$	3507.74	0.120366	0.0601829	+	0.998187i	$0.480832\pi$
$948$	−258.871	−0.00886894
$949$	33184.7	1.13511
$950$	1406.11	0.0480212
$951$	−14066.5	−0.479638
$952$	11040.4	0.375861
$953$	−5262.17	−0.178865	−0.0894325	−	0.995993i	$-0.528505\pi$
$953$	−5262.17	−0.178865	−0.0894325	+	0.995993i	$0.528505\pi$
$954$	−4744.38	−0.161011
$955$	2778.15	0.0941350
$956$	3039.43	0.102826
$957$	−4405.95	−0.148824
$958$	34229.7	1.15440
$959$	5461.38	0.183897
$960$	9098.20	0.305878
$961$	564.066	0.0189341
$962$	−6680.22	−0.223887
$963$	798.561	0.0267220
$964$	99.8475	0.00333597
$965$	9211.85	0.307295
$966$	−4238.05	−0.141156
$967$	13394.0	0.445422	0.222711	−	0.974884i	$-0.428509\pi$
$967$	13394.0	0.445422	0.222711	+	0.974884i	$0.428509\pi$
$968$	2592.22	0.0860715
$969$	6128.88	0.203187
$970$	10912.9	0.361230
$971$	49902.1	1.64926	0.824632	−	0.565669i	$-0.191382\pi$
$971$	49902.1	1.64926	0.824632	+	0.565669i	$0.191382\pi$
$972$	−2174.99	−0.0717725
$973$	−406.496	−0.0133933
$974$	−35816.1	−1.17826
$975$	4746.90	0.155921
$976$	−4024.56	−0.131991
$977$	25750.0	0.843209	0.421605	−	0.906780i	$-0.361467\pi$
$977$	25750.0	0.843209	0.421605	+	0.906780i	$0.361467\pi$
$978$	−45429.4	−1.48535
$979$	−473.521	−0.0154584
$980$	1152.25	0.0375583
$981$	−21282.5	−0.692660
$982$	38933.0	1.26517
$983$	37270.4	1.20930	0.604650	−	0.796491i	$-0.293313\pi$
$983$	37270.4	1.20930	0.604650	+	0.796491i	$0.293313\pi$
$984$	12339.3	0.399760
$985$	−5854.15	−0.189369
$986$	23840.7	0.770022
$987$	11075.0	0.357166
$988$	687.187	0.0221279
$989$	2629.67	0.0845486
$990$	1783.97	0.0572709
$991$	46457.3	1.48917	0.744583	−	0.667530i	$-0.232648\pi$
$991$	46457.3	1.48917	0.744583	+	0.667530i	$0.232648\pi$
$992$	5995.61	0.191896
$993$	−41869.8	−1.33806
$994$	13681.9	0.436584
$995$	−3888.21	−0.123884
$996$	−691.568	−0.0220012
$997$	−12162.7	−0.386356	−0.193178	−	0.981164i	$-0.561880\pi$
$997$	−12162.7	−0.386356	−0.193178	+	0.981164i	$0.561880\pi$
$998$	−55125.5	−1.74846
$999$	−7237.20	−0.229204

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.g.1.6	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.g.1.6	✓	23	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:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q-2.96023 q^{2} -4.00535 q^{3} +0.762942 q^{4} -5.00000 q^{5} +11.8567 q^{6} +6.39895 q^{7} +21.4233 q^{8} -10.9572 q^{9} +O(q^{10})\)	\(q-2.96023 q^{2} -4.00535 q^{3} +0.762942 q^{4} -5.00000 q^{5} +11.8567 q^{6} +6.39895 q^{7} +21.4233 q^{8} -10.9572 q^{9} +14.8011 q^{10} -11.0000 q^{11} -3.05585 q^{12} -47.4056 q^{13} -18.9423 q^{14} +20.0267 q^{15} -69.5215 q^{16} +80.5355 q^{17} +32.4357 q^{18} -19.0000 q^{19} -3.81471 q^{20} -25.6300 q^{21} +32.5625 q^{22} -55.8588 q^{23} -85.8079 q^{24} +25.0000 q^{25} +140.331 q^{26} +152.032 q^{27} +4.88203 q^{28} -100.001 q^{29} -59.2837 q^{30} +174.227 q^{31} +34.4126 q^{32} +44.0588 q^{33} -238.403 q^{34} -31.9948 q^{35} -8.35969 q^{36} -47.6032 q^{37} +56.2443 q^{38} +189.876 q^{39} -107.117 q^{40} -143.802 q^{41} +75.8707 q^{42} -47.0770 q^{43} -8.39236 q^{44} +54.7859 q^{45} +165.355 q^{46} -432.112 q^{47} +278.458 q^{48} -302.053 q^{49} -74.0057 q^{50} -322.573 q^{51} -36.1677 q^{52} -146.270 q^{53} -450.048 q^{54} +55.0000 q^{55} +137.087 q^{56} +76.1016 q^{57} +296.027 q^{58} -757.043 q^{59} +15.2792 q^{60} +57.8895 q^{61} -515.752 q^{62} -70.1145 q^{63} +454.303 q^{64} +237.028 q^{65} -130.424 q^{66} -343.284 q^{67} +61.4439 q^{68} +223.734 q^{69} +94.7117 q^{70} -722.293 q^{71} -234.739 q^{72} -700.016 q^{73} +140.916 q^{74} -100.134 q^{75} -14.4959 q^{76} -70.3885 q^{77} -562.076 q^{78} +84.7134 q^{79} +347.607 q^{80} -313.096 q^{81} +425.686 q^{82} +226.310 q^{83} -19.5542 q^{84} -402.677 q^{85} +139.359 q^{86} +400.541 q^{87} -235.657 q^{88} +43.0474 q^{89} -162.179 q^{90} -303.346 q^{91} -42.6170 q^{92} -697.840 q^{93} +1279.15 q^{94} +95.0000 q^{95} -137.834 q^{96} +737.305 q^{97} +894.147 q^{98} +120.529 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9	\( 23 q + 6 q^{2} + 9 q^{3} + 92 q^{4} - 115 q^{5} - 25 q^{6} - 37 q^{7} + 42 q^{8} + 170 q^{9} - 30 q^{10} - 253 q^{11} + 44 q^{12} - 37 q^{13} + 61 q^{14} - 45 q^{15} + 588 q^{16} - 73 q^{17} + 391 q^{18} - 437 q^{19} - 460 q^{20} - 127 q^{21} - 66 q^{22} - 175 q^{23} + 16 q^{24} + 575 q^{25} + 719 q^{26} + 21 q^{27} + 253 q^{28} + 71 q^{29} + 125 q^{30} + 302 q^{31} + 1107 q^{32} - 99 q^{33} + 1267 q^{34} + 185 q^{35} + 703 q^{36} - 500 q^{37} - 114 q^{38} + 457 q^{39} - 210 q^{40} + 770 q^{41} + 2596 q^{42} - 902 q^{43} - 1012 q^{44} - 850 q^{45} - 1101 q^{46} + 356 q^{47} + 1221 q^{48} + 908 q^{49} + 150 q^{50} - 451 q^{51} - 358 q^{52} + 1327 q^{53} + 2534 q^{54} + 1265 q^{55} + 3135 q^{56} - 171 q^{57} + 1014 q^{58} + 3619 q^{59} - 220 q^{60} - 1432 q^{61} + 1826 q^{62} + 1658 q^{63} + 4006 q^{64} + 185 q^{65} + 275 q^{66} - 605 q^{67} + 5128 q^{68} + 3099 q^{69} - 305 q^{70} + 3230 q^{71} + 2152 q^{72} - 637 q^{73} + 5063 q^{74} + 225 q^{75} - 1748 q^{76} + 407 q^{77} + 7230 q^{78} + 2074 q^{79} - 2940 q^{80} + 2291 q^{81} + 530 q^{82} + 3882 q^{83} + 5096 q^{84} + 365 q^{85} + 2262 q^{86} - 27 q^{87} - 462 q^{88} - 210 q^{89} - 1955 q^{90} + 4133 q^{91} - 6064 q^{92} + 824 q^{93} - 392 q^{94} + 2185 q^{95} + 2462 q^{96} + 2032 q^{97} + 7896 q^{98} - 1870 q^{99}+O(q^{100}) \) 23 * q + 6 * q^2 + 9 * q^3 + 92 * q^4 - 115 * q^5 - 25 * q^6 - 37 * q^7 + 42 * q^8 + 170 * q^9 - 30 * q^10 - 253 * q^11 + 44 * q^12 - 37 * q^13 + 61 * q^14 - 45 * q^15 + 588 * q^16 - 73 * q^17 + 391 * q^18 - 437 * q^19 - 460 * q^20 - 127 * q^21 - 66 * q^22 - 175 * q^23 + 16 * q^24 + 575 * q^25 + 719 * q^26 + 21 * q^27 + 253 * q^28 + 71 * q^29 + 125 * q^30 + 302 * q^31 + 1107 * q^32 - 99 * q^33 + 1267 * q^34 + 185 * q^35 + 703 * q^36 - 500 * q^37 - 114 * q^38 + 457 * q^39 - 210 * q^40 + 770 * q^41 + 2596 * q^42 - 902 * q^43 - 1012 * q^44 - 850 * q^45 - 1101 * q^46 + 356 * q^47 + 1221 * q^48 + 908 * q^49 + 150 * q^50 - 451 * q^51 - 358 * q^52 + 1327 * q^53 + 2534 * q^54 + 1265 * q^55 + 3135 * q^56 - 171 * q^57 + 1014 * q^58 + 3619 * q^59 - 220 * q^60 - 1432 * q^61 + 1826 * q^62 + 1658 * q^63 + 4006 * q^64 + 185 * q^65 + 275 * q^66 - 605 * q^67 + 5128 * q^68 + 3099 * q^69 - 305 * q^70 + 3230 * q^71 + 2152 * q^72 - 637 * q^73 + 5063 * q^74 + 225 * q^75 - 1748 * q^76 + 407 * q^77 + 7230 * q^78 + 2074 * q^79 - 2940 * q^80 + 2291 * q^81 + 530 * q^82 + 3882 * q^83 + 5096 * q^84 + 365 * q^85 + 2262 * q^86 - 27 * q^87 - 462 * q^88 - 210 * q^89 - 1955 * q^90 + 4133 * q^91 - 6064 * q^92 + 824 * q^93 - 392 * q^94 + 2185 * q^95 + 2462 * q^96 + 2032 * q^97 + 7896 * q^98 - 1870 * q^99

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