LMFDB - Embedded newform 1045.4.a.f.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:	$1$
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)

$f(q)$	$=$	$q-3.46924 q^{2} -9.89028 q^{3} +4.03563 q^{4} -5.00000 q^{5} +34.3118 q^{6} +31.6510 q^{7} +13.7534 q^{8} +70.8177 q^{9} +O(q^{10})$	\(q-3.46924 q^{2} -9.89028 q^{3} +4.03563 q^{4} -5.00000 q^{5} +34.3118 q^{6} +31.6510 q^{7} +13.7534 q^{8} +70.8177 q^{9} +17.3462 q^{10} +11.0000 q^{11} -39.9135 q^{12} -57.8297 q^{13} -109.805 q^{14} +49.4514 q^{15} -79.9987 q^{16} +19.6654 q^{17} -245.684 q^{18} -19.0000 q^{19} -20.1781 q^{20} -313.038 q^{21} -38.1616 q^{22} +88.1872 q^{23} -136.025 q^{24} +25.0000 q^{25} +200.625 q^{26} -433.370 q^{27} +127.732 q^{28} -213.921 q^{29} -171.559 q^{30} +107.715 q^{31} +167.508 q^{32} -108.793 q^{33} -68.2238 q^{34} -158.255 q^{35} +285.794 q^{36} +185.074 q^{37} +65.9156 q^{38} +571.952 q^{39} -68.7668 q^{40} -464.309 q^{41} +1086.00 q^{42} -169.206 q^{43} +44.3919 q^{44} -354.089 q^{45} -305.943 q^{46} +368.413 q^{47} +791.210 q^{48} +658.788 q^{49} -86.7310 q^{50} -194.496 q^{51} -233.379 q^{52} +194.202 q^{53} +1503.46 q^{54} -55.0000 q^{55} +435.308 q^{56} +187.915 q^{57} +742.143 q^{58} +277.803 q^{59} +199.567 q^{60} -533.456 q^{61} -373.688 q^{62} +2241.45 q^{63} +58.8648 q^{64} +289.148 q^{65} +377.429 q^{66} -589.251 q^{67} +79.3620 q^{68} -872.197 q^{69} +549.025 q^{70} -14.8649 q^{71} +973.982 q^{72} -21.9206 q^{73} -642.065 q^{74} -247.257 q^{75} -76.6769 q^{76} +348.161 q^{77} -1984.24 q^{78} +354.943 q^{79} +399.994 q^{80} +2374.07 q^{81} +1610.80 q^{82} -511.756 q^{83} -1263.30 q^{84} -98.3268 q^{85} +587.015 q^{86} +2115.74 q^{87} +151.287 q^{88} -1108.67 q^{89} +1228.42 q^{90} -1830.37 q^{91} +355.891 q^{92} -1065.33 q^{93} -1278.11 q^{94} +95.0000 q^{95} -1656.70 q^{96} +400.249 q^{97} -2285.49 q^{98} +778.995 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) $ 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * 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 $

$2$	−3.46924	−1.22656	−0.613281	−	0.789865i	$-0.710151\pi$
$2$	−3.46924	−1.22656	−0.613281	+	0.789865i	$0.710151\pi$
$3$	−9.89028	−1.90339	−0.951693	−	0.307051i	$-0.900658\pi$
$3$	−9.89028	−1.90339	−0.951693	+	0.307051i	$0.900658\pi$
$4$	4.03563	0.504453
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	34.3118	2.33462
$7$	31.6510	1.70900	0.854498	−	0.519455i	$-0.173865\pi$
$7$	31.6510	1.70900	0.854498	+	0.519455i	$0.173865\pi$
$8$	13.7534	0.607819
$9$	70.8177	2.62288
$10$	17.3462	0.548535
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−39.9135	−0.960169
$13$	−57.8297	−1.23377	−0.616887	−	0.787051i	$-0.711607\pi$
$13$	−57.8297	−1.23377	−0.616887	+	0.787051i	$0.711607\pi$
$14$	−109.805	−2.09619
$15$	49.4514	0.851220
$16$	−79.9987	−1.24998
$17$	19.6654	0.280562	0.140281	−	0.990112i	$-0.455199\pi$
$17$	19.6654	0.280562	0.140281	+	0.990112i	$0.455199\pi$
$18$	−245.684	−3.21712
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−20.1781	−0.225598
$21$	−313.038	−3.25288
$22$	−38.1616	−0.369822
$23$	88.1872	0.799492	0.399746	−	0.916626i	$-0.369098\pi$
$23$	88.1872	0.799492	0.399746	+	0.916626i	$0.369098\pi$
$24$	−136.025	−1.15691
$25$	25.0000	0.200000
$26$	200.625	1.51330
$27$	−433.370	−3.08897
$28$	127.732	0.862108
$29$	−213.921	−1.36980	−0.684899	−	0.728638i	$-0.740154\pi$
$29$	−213.921	−1.36980	−0.684899	+	0.728638i	$0.740154\pi$
$30$	−171.559	−1.04407
$31$	107.715	0.624068	0.312034	−	0.950071i	$-0.398990\pi$
$31$	107.715	0.624068	0.312034	+	0.950071i	$0.398990\pi$
$32$	167.508	0.925359
$33$	−108.793	−0.573893
$34$	−68.2238	−0.344126
$35$	−158.255	−0.764286
$36$	285.794	1.32312
$37$	185.074	0.822322	0.411161	−	0.911563i	$-0.365123\pi$
$37$	185.074	0.822322	0.411161	+	0.911563i	$0.365123\pi$
$38$	65.9156	0.281393
$39$	571.952	2.34835
$40$	−68.7668	−0.271825
$41$	−464.309	−1.76861	−0.884304	−	0.466912i	$-0.845367\pi$
$41$	−464.309	−1.76861	−0.884304	+	0.466912i	$0.845367\pi$
$42$	1086.00	3.98985
$43$	−169.206	−0.600084	−0.300042	−	0.953926i	$-0.597001\pi$
$43$	−169.206	−0.600084	−0.300042	+	0.953926i	$0.597001\pi$
$44$	44.3919	0.152098
$45$	−354.089	−1.17299
$46$	−305.943	−0.980626
$47$	368.413	1.14337	0.571687	−	0.820472i	$-0.306289\pi$
$47$	368.413	1.14337	0.571687	+	0.820472i	$0.306289\pi$
$48$	791.210	2.37919
$49$	658.788	1.92066
$50$	−86.7310	−0.245312
$51$	−194.496	−0.534017
$52$	−233.379	−0.622382
$53$	194.202	0.503314	0.251657	−	0.967816i	$-0.419024\pi$
$53$	194.202	0.503314	0.251657	+	0.967816i	$0.419024\pi$
$54$	1503.46	3.78881
$55$	−55.0000	−0.134840
$56$	435.308	1.03876
$57$	187.915	0.436667
$58$	742.143	1.68014
$59$	277.803	0.612997	0.306499	−	0.951871i	$-0.400843\pi$
$59$	277.803	0.612997	0.306499	+	0.951871i	$0.400843\pi$
$60$	199.567	0.429401
$61$	−533.456	−1.11971	−0.559853	−	0.828592i	$-0.689142\pi$
$61$	−533.456	−1.11971	−0.559853	+	0.828592i	$0.689142\pi$
$62$	−373.688	−0.765458
$63$	2241.45	4.48249
$64$	58.8648	0.114970
$65$	289.148	0.551761
$66$	377.429	0.703915
$67$	−589.251	−1.07445	−0.537227	−	0.843437i	$-0.680528\pi$
$67$	−589.251	−1.07445	−0.537227	+	0.843437i	$0.680528\pi$
$68$	79.3620	0.141530
$69$	−872.197	−1.52174
$70$	549.025	0.937444
$71$	−14.8649	−0.0248470	−0.0124235	−	0.999923i	$-0.503955\pi$
$71$	−14.8649	−0.0248470	−0.0124235	+	0.999923i	$0.503955\pi$
$72$	973.982	1.59423
$73$	−21.9206	−0.0351453	−0.0175727	−	0.999846i	$-0.505594\pi$
$73$	−21.9206	−0.0351453	−0.0175727	+	0.999846i	$0.505594\pi$
$74$	−642.065	−1.00863
$75$	−247.257	−0.380677
$76$	−76.6769	−0.115730
$77$	348.161	0.515281
$78$	−1984.24	−2.88040
$79$	354.943	0.505497	0.252748	−	0.967532i	$-0.418666\pi$
$79$	354.943	0.505497	0.252748	+	0.967532i	$0.418666\pi$
$80$	399.994	0.559008
$81$	2374.07	3.25661
$82$	1610.80	2.16931
$83$	−511.756	−0.676778	−0.338389	−	0.941006i	$-0.609882\pi$
$83$	−511.756	−0.676778	−0.338389	+	0.941006i	$0.609882\pi$
$84$	−1263.30	−1.64092
$85$	−98.3268	−0.125471
$86$	587.015	0.736041
$87$	2115.74	2.60725
$88$	151.287	0.183264
$89$	−1108.67	−1.32044	−0.660220	−	0.751072i	$-0.729537\pi$
$89$	−1108.67	−1.32044	−0.660220	+	0.751072i	$0.729537\pi$
$90$	1228.42	1.43874
$91$	−1830.37	−2.10851
$92$	355.891	0.403306
$93$	−1065.33	−1.18784
$94$	−1278.11	−1.40242
$95$	95.0000	0.102598
$96$	−1656.70	−1.76132
$97$	400.249	0.418960	0.209480	−	0.977813i	$-0.432823\pi$
$97$	400.249	0.418960	0.209480	+	0.977813i	$0.432823\pi$
$98$	−2285.49	−2.35581
$99$	778.995	0.790828
$100$	100.891	0.100891
$101$	−755.420	−0.744229	−0.372115	−	0.928187i	$-0.621367\pi$
$101$	−755.420	−0.744229	−0.372115	+	0.928187i	$0.621367\pi$
$102$	674.753	0.655005
$103$	834.267	0.798085	0.399043	−	0.916932i	$-0.369343\pi$
$103$	834.267	0.798085	0.399043	+	0.916932i	$0.369343\pi$
$104$	−795.353	−0.749911
$105$	1565.19	1.45473
$106$	−673.732	−0.617346
$107$	−753.619	−0.680889	−0.340445	−	0.940265i	$-0.610578\pi$
$107$	−753.619	−0.680889	−0.340445	+	0.940265i	$0.610578\pi$
$108$	−1748.92	−1.55824
$109$	310.403	0.272763	0.136382	−	0.990656i	$-0.456453\pi$
$109$	310.403	0.272763	0.136382	+	0.990656i	$0.456453\pi$
$110$	190.808	0.165390
$111$	−1830.43	−1.56520
$112$	−2532.04	−2.13621
$113$	606.396	0.504823	0.252411	−	0.967620i	$-0.418776\pi$
$113$	606.396	0.504823	0.252411	+	0.967620i	$0.418776\pi$
$114$	−651.924	−0.535599
$115$	−440.936	−0.357544
$116$	−863.305	−0.690999
$117$	−4095.37	−3.23604
$118$	−963.764	−0.751879
$119$	622.429	0.479479
$120$	680.123	0.517387
$121$	121.000	0.0909091
$122$	1850.69	1.37339
$123$	4592.15	3.36634
$124$	434.696	0.314813
$125$	−125.000	−0.0894427
$126$	−7776.14	−5.49805
$127$	1992.59	1.39223	0.696117	−	0.717928i	$-0.254909\pi$
$127$	1992.59	1.39223	0.696117	+	0.717928i	$0.254909\pi$
$128$	−1544.28	−1.06638
$129$	1673.49	1.14219
$130$	−1003.13	−0.676769
$131$	−2779.74	−1.85394	−0.926972	−	0.375131i	$-0.877598\pi$
$131$	−2779.74	−1.85394	−0.926972	+	0.375131i	$0.877598\pi$
$132$	−439.048	−0.289502
$133$	−601.370	−0.392070
$134$	2044.25	1.31788
$135$	2166.85	1.38143
$136$	270.465	0.170531
$137$	2352.99	1.46737	0.733685	−	0.679489i	$-0.237799\pi$
$137$	2352.99	1.46737	0.733685	+	0.679489i	$0.237799\pi$
$138$	3025.86	1.86651
$139$	2155.27	1.31516	0.657582	−	0.753383i	$-0.271579\pi$
$139$	2155.27	1.31516	0.657582	+	0.753383i	$0.271579\pi$
$140$	−638.659	−0.385546
$141$	−3643.71	−2.17628
$142$	51.5699	0.0304764
$143$	−636.127	−0.371997
$144$	−5665.33	−3.27855
$145$	1069.60	0.612592
$146$	76.0478	0.0431079
$147$	−6515.60	−3.65577
$148$	746.888	0.414823
$149$	−1891.07	−1.03975	−0.519874	−	0.854243i	$-0.674021\pi$
$149$	−1891.07	−1.03975	−0.519874	+	0.854243i	$0.674021\pi$
$150$	857.794	0.466924
$151$	−1742.81	−0.939260	−0.469630	−	0.882863i	$-0.655613\pi$
$151$	−1742.81	−0.939260	−0.469630	+	0.882863i	$0.655613\pi$
$152$	−261.314	−0.139443
$153$	1392.66	0.735880
$154$	−1207.86	−0.632024
$155$	−538.573	−0.279092
$156$	2308.18	1.18463
$157$	364.953	0.185518	0.0927592	−	0.995689i	$-0.470431\pi$
$157$	364.953	0.185518	0.0927592	+	0.995689i	$0.470431\pi$
$158$	−1231.38	−0.620023
$159$	−1920.71	−0.958001
$160$	−837.539	−0.413833
$161$	2791.22	1.36633
$162$	−8236.23	−3.99444
$163$	3526.38	1.69452	0.847261	−	0.531176i	$-0.178250\pi$
$163$	3526.38	1.69452	0.847261	+	0.531176i	$0.178250\pi$
$164$	−1873.78	−0.892180
$165$	543.966	0.256653
$166$	1775.40	0.830109
$167$	609.763	0.282544	0.141272	−	0.989971i	$-0.454881\pi$
$167$	609.763	0.282544	0.141272	+	0.989971i	$0.454881\pi$
$168$	−4305.32	−1.97716
$169$	1147.27	0.522200
$170$	341.119	0.153898
$171$	−1345.54	−0.601730
$172$	−682.851	−0.302715
$173$	4154.31	1.82570	0.912852	−	0.408291i	$-0.133875\pi$
$173$	4154.31	1.82570	0.912852	+	0.408291i	$0.133875\pi$
$174$	−7340.01	−3.19796
$175$	791.276	0.341799
$176$	−879.986	−0.376883
$177$	−2747.55	−1.16677
$178$	3846.26	1.61960
$179$	3786.04	1.58091	0.790453	−	0.612523i	$-0.209845\pi$
$179$	3786.04	1.58091	0.790453	+	0.612523i	$0.209845\pi$
$180$	−1428.97	−0.591717
$181$	−433.687	−0.178098	−0.0890490	−	0.996027i	$-0.528383\pi$
$181$	−433.687	−0.178098	−0.0890490	+	0.996027i	$0.528383\pi$
$182$	6349.99	2.58622
$183$	5276.03	2.13123
$184$	1212.87	0.485946
$185$	−925.368	−0.367754
$186$	3695.88	1.45696
$187$	216.319	0.0845926
$188$	1486.78	0.576778
$189$	−13716.6	−5.27903
$190$	−329.578	−0.125843
$191$	−2473.80	−0.937161	−0.468581	−	0.883421i	$-0.655234\pi$
$191$	−2473.80	−0.937161	−0.468581	+	0.883421i	$0.655234\pi$
$192$	−582.190	−0.218833
$193$	3296.03	1.22929	0.614645	−	0.788804i	$-0.289299\pi$
$193$	3296.03	1.22929	0.614645	+	0.788804i	$0.289299\pi$
$194$	−1388.56	−0.513881
$195$	−2859.76	−1.05021
$196$	2658.62	0.968885
$197$	−3026.41	−1.09453	−0.547267	−	0.836958i	$-0.684332\pi$
$197$	−3026.41	−1.09453	−0.547267	+	0.836958i	$0.684332\pi$
$198$	−2702.52	−0.969999
$199$	−1999.79	−0.712369	−0.356184	−	0.934416i	$-0.615922\pi$
$199$	−1999.79	−0.712369	−0.356184	+	0.934416i	$0.615922\pi$
$200$	343.834	0.121564
$201$	5827.86	2.04510
$202$	2620.73	0.912843
$203$	−6770.82	−2.34098
$204$	−784.913	−0.269387
$205$	2321.55	0.790945
$206$	−2894.27	−0.978901
$207$	6245.22	2.09697
$208$	4626.30	1.54219
$209$	−209.000	−0.0691714
$210$	−5430.01	−1.78432
$211$	1253.09	0.408844	0.204422	−	0.978883i	$-0.434468\pi$
$211$	1253.09	0.408844	0.204422	+	0.978883i	$0.434468\pi$
$212$	783.725	0.253898
$213$	147.018	0.0472935
$214$	2614.49	0.835152
$215$	846.029	0.268366
$216$	−5960.29	−1.87753
$217$	3409.28	1.06653
$218$	−1076.86	−0.334561
$219$	216.801	0.0668952
$220$	−221.959	−0.0680205
$221$	−1137.24	−0.346150
$222$	6350.20	1.91981
$223$	−5279.12	−1.58527	−0.792637	−	0.609694i	$-0.791292\pi$
$223$	−5279.12	−1.58527	−0.792637	+	0.609694i	$0.791292\pi$
$224$	5301.80	1.58143
$225$	1770.44	0.524576
$226$	−2103.73	−0.619196
$227$	−5597.22	−1.63657	−0.818283	−	0.574816i	$-0.805074\pi$
$227$	−5597.22	−1.63657	−0.818283	+	0.574816i	$0.805074\pi$
$228$	758.356	0.220278
$229$	4522.58	1.30507	0.652533	−	0.757760i	$-0.273706\pi$
$229$	4522.58	1.30507	0.652533	+	0.757760i	$0.273706\pi$
$230$	1529.71	0.438549
$231$	−3443.41	−0.980780
$232$	−2942.13	−0.832588
$233$	−6897.95	−1.93948	−0.969742	−	0.244133i	$-0.921497\pi$
$233$	−6897.95	−1.93948	−0.969742	+	0.244133i	$0.921497\pi$
$234$	14207.8	3.96920
$235$	−1842.06	−0.511332
$236$	1121.11	0.309228
$237$	−3510.49	−0.962155
$238$	−2159.36	−0.588110
$239$	−1231.88	−0.333406	−0.166703	−	0.986007i	$-0.553312\pi$
$239$	−1231.88	−0.333406	−0.166703	+	0.986007i	$0.553312\pi$
$240$	−3956.05	−1.06401
$241$	−5170.89	−1.38210	−0.691051	−	0.722806i	$-0.742852\pi$
$241$	−5170.89	−1.38210	−0.691051	+	0.722806i	$0.742852\pi$
$242$	−419.778	−0.111506
$243$	−11779.3	−3.10963
$244$	−2152.83	−0.564839
$245$	−3293.94	−0.858947
$246$	−15931.3	−4.12903
$247$	1098.76	0.283047
$248$	1481.44	0.379320
$249$	5061.41	1.28817
$250$	433.655	0.109707
$251$	5522.98	1.38887	0.694437	−	0.719554i	$-0.255654\pi$
$251$	5522.98	1.38887	0.694437	+	0.719554i	$0.255654\pi$
$252$	9045.67	2.26121
$253$	970.060	0.241056
$254$	−6912.77	−1.70766
$255$	972.480	0.238820
$256$	4886.56	1.19301
$257$	−7524.36	−1.82629	−0.913145	−	0.407636i	$-0.866353\pi$
$257$	−7524.36	−1.82629	−0.913145	+	0.407636i	$0.866353\pi$
$258$	−5805.75	−1.40097
$259$	5857.77	1.40534
$260$	1166.90	0.278338
$261$	−15149.4	−3.59281
$262$	9643.57	2.27398
$263$	−5860.37	−1.37401	−0.687007	−	0.726650i	$-0.741076\pi$
$263$	−5860.37	−1.37401	−0.687007	+	0.726650i	$0.741076\pi$
$264$	−1496.27	−0.348823
$265$	−971.008	−0.225089
$266$	2086.30	0.480898
$267$	10965.1	2.51331
$268$	−2378.00	−0.542012
$269$	1297.53	0.294095	0.147048	−	0.989129i	$-0.453023\pi$
$269$	1297.53	0.294095	0.147048	+	0.989129i	$0.453023\pi$
$270$	−7517.32	−1.69441
$271$	4232.08	0.948636	0.474318	−	0.880353i	$-0.342695\pi$
$271$	4232.08	0.948636	0.474318	+	0.880353i	$0.342695\pi$
$272$	−1573.20	−0.350697
$273$	18102.9	4.01332
$274$	−8163.10	−1.79982
$275$	275.000	0.0603023
$276$	−3519.86	−0.767647
$277$	−231.982	−0.0503193	−0.0251596	−	0.999683i	$-0.508009\pi$
$277$	−231.982	−0.0503193	−0.0251596	+	0.999683i	$0.508009\pi$
$278$	−7477.15	−1.61313
$279$	7628.10	1.63685
$280$	−2176.54	−0.464547
$281$	2968.28	0.630153	0.315077	−	0.949066i	$-0.397970\pi$
$281$	2968.28	0.630153	0.315077	+	0.949066i	$0.397970\pi$
$282$	12640.9	2.66934
$283$	3306.15	0.694453	0.347226	−	0.937781i	$-0.387124\pi$
$283$	3306.15	0.694453	0.347226	+	0.937781i	$0.387124\pi$
$284$	−59.9891	−0.0125342
$285$	−939.577	−0.195283
$286$	2206.88	0.456277
$287$	−14695.9	−3.02254
$288$	11862.5	2.42710
$289$	−4526.27	−0.921285
$290$	−3710.72	−0.751382
$291$	−3958.58	−0.797443
$292$	−88.4633	−0.0177292
$293$	1741.03	0.347140	0.173570	−	0.984822i	$-0.444470\pi$
$293$	1741.03	0.347140	0.173570	+	0.984822i	$0.444470\pi$
$294$	22604.2	4.48402
$295$	−1389.01	−0.274141
$296$	2545.39	0.499823
$297$	−4767.07	−0.931358
$298$	6560.57	1.27531
$299$	−5099.84	−0.986393
$300$	−997.837	−0.192034
$301$	−5355.54	−1.02554
$302$	6046.24	1.15206
$303$	7471.32	1.41656
$304$	1519.98	0.286765
$305$	2667.28	0.500748
$306$	−4831.46	−0.902602
$307$	−2080.56	−0.386789	−0.193394	−	0.981121i	$-0.561950\pi$
$307$	−2080.56	−0.386789	−0.193394	+	0.981121i	$0.561950\pi$
$308$	1405.05	0.259935
$309$	−8251.14	−1.51906
$310$	1868.44	0.342323
$311$	−1433.41	−0.261354	−0.130677	−	0.991425i	$-0.541715\pi$
$311$	−1433.41	−0.261354	−0.130677	+	0.991425i	$0.541715\pi$
$312$	7866.27	1.42737
$313$	3901.69	0.704590	0.352295	−	0.935889i	$-0.385401\pi$
$313$	3901.69	0.704590	0.352295	+	0.935889i	$0.385401\pi$
$314$	−1266.11	−0.227550
$315$	−11207.3	−2.00463
$316$	1432.42	0.254999
$317$	6038.74	1.06993	0.534967	−	0.844873i	$-0.320324\pi$
$317$	6038.74	1.06993	0.534967	+	0.844873i	$0.320324\pi$
$318$	6663.40	1.17505
$319$	−2353.13	−0.413010
$320$	−294.324	−0.0514163
$321$	7453.51	1.29599
$322$	−9683.40	−1.67588
$323$	−373.642	−0.0643653
$324$	9580.87	1.64281
$325$	−1445.74	−0.246755
$326$	−12233.9	−2.07844
$327$	−3069.97	−0.519173
$328$	−6385.81	−1.07499
$329$	11660.6	1.95402
$330$	−1887.15	−0.314800
$331$	5582.13	0.926954	0.463477	−	0.886109i	$-0.346602\pi$
$331$	5582.13	0.926954	0.463477	+	0.886109i	$0.346602\pi$
$332$	−2065.26	−0.341403
$333$	13106.5	2.15685
$334$	−2115.41	−0.346558
$335$	2946.26	0.480511
$336$	25042.6	4.06603
$337$	3436.74	0.555523	0.277761	−	0.960650i	$-0.410408\pi$
$337$	3436.74	0.555523	0.277761	+	0.960650i	$0.410408\pi$
$338$	−3980.17	−0.640510
$339$	−5997.43	−0.960872
$340$	−396.810	−0.0632943
$341$	1184.86	0.188164
$342$	4667.99	0.738058
$343$	9995.01	1.57341
$344$	−2327.15	−0.364742
$345$	4360.98	0.680544
$346$	−14412.3	−2.23934
$347$	10210.5	1.57963	0.789814	−	0.613347i	$-0.210177\pi$
$347$	10210.5	1.57963	0.789814	+	0.613347i	$0.210177\pi$
$348$	8538.33	1.31524
$349$	−483.597	−0.0741729	−0.0370865	−	0.999312i	$-0.511808\pi$
$349$	−483.597	−0.0741729	−0.0370865	+	0.999312i	$0.511808\pi$
$350$	−2745.13	−0.419238
$351$	25061.6	3.81109
$352$	1842.59	0.279006
$353$	−1566.86	−0.236248	−0.118124	−	0.992999i	$-0.537688\pi$
$353$	−1566.86	−0.236248	−0.118124	+	0.992999i	$0.537688\pi$
$354$	9531.90	1.43112
$355$	74.3245	0.0111119
$356$	−4474.20	−0.666101
$357$	−6156.00	−0.912633
$358$	−13134.7	−1.93908
$359$	−12225.7	−1.79735	−0.898675	−	0.438616i	$-0.855469\pi$
$359$	−12225.7	−1.79735	−0.898675	+	0.438616i	$0.855469\pi$
$360$	−4869.91	−0.712963
$361$	361.000	0.0526316
$362$	1504.57	0.218448
$363$	−1196.72	−0.173035
$364$	−7386.69	−1.06365
$365$	109.603	0.0157175
$366$	−18303.8	−2.61409
$367$	−8685.70	−1.23539	−0.617697	−	0.786416i	$-0.711934\pi$
$367$	−8685.70	−1.23539	−0.617697	+	0.786416i	$0.711934\pi$
$368$	−7054.87	−0.999349
$369$	−32881.3	−4.63884
$370$	3210.32	0.451073
$371$	6146.68	0.860161
$372$	−4299.26	−0.599211
$373$	10263.7	1.42476	0.712379	−	0.701795i	$-0.247618\pi$
$373$	10263.7	1.42476	0.712379	+	0.701795i	$0.247618\pi$
$374$	−750.462	−0.103758
$375$	1236.29	0.170244
$376$	5066.92	0.694964
$377$	12371.0	1.69002
$378$	47586.2	6.47505
$379$	−12368.3	−1.67629	−0.838147	−	0.545444i	$-0.816361\pi$
$379$	−12368.3	−1.67629	−0.838147	+	0.545444i	$0.816361\pi$
$380$	383.384	0.0517558
$381$	−19707.3	−2.64996
$382$	8582.20	1.14949
$383$	−14233.0	−1.89888	−0.949441	−	0.313945i	$-0.898349\pi$
$383$	−14233.0	−1.89888	−0.949441	+	0.313945i	$0.898349\pi$
$384$	15273.4	2.02973
$385$	−1740.81	−0.230441
$386$	−11434.7	−1.50780
$387$	−11982.8	−1.57395
$388$	1615.26	0.211346
$389$	−10642.3	−1.38711	−0.693555	−	0.720404i	$-0.743956\pi$
$389$	−10642.3	−1.38711	−0.693555	+	0.720404i	$0.743956\pi$
$390$	9921.20	1.28815
$391$	1734.23	0.224307
$392$	9060.55	1.16742
$393$	27492.4	3.52877
$394$	10499.4	1.34251
$395$	−1774.72	−0.226065
$396$	3143.73	0.398936
$397$	5774.96	0.730068	0.365034	−	0.930994i	$-0.381057\pi$
$397$	5774.96	0.730068	0.365034	+	0.930994i	$0.381057\pi$
$398$	6937.75	0.873764
$399$	5947.72	0.746261
$400$	−1999.97	−0.249996
$401$	11332.2	1.41123	0.705616	−	0.708595i	$-0.250671\pi$
$401$	11332.2	1.41123	0.705616	+	0.708595i	$0.250671\pi$
$402$	−20218.2	−2.50844
$403$	−6229.10	−0.769959
$404$	−3048.59	−0.375429
$405$	−11870.4	−1.45640
$406$	23489.6	2.87135
$407$	2035.81	0.247939
$408$	−2674.97	−0.324586
$409$	−8565.37	−1.03553	−0.517763	−	0.855524i	$-0.673235\pi$
$409$	−8565.37	−1.03553	−0.517763	+	0.855524i	$0.673235\pi$
$410$	−8054.00	−0.970143
$411$	−23271.8	−2.79297
$412$	3366.79	0.402597
$413$	8792.74	1.04761
$414$	−21666.2	−2.57206
$415$	2558.78	0.302664
$416$	−9686.93	−1.14168
$417$	−21316.2	−2.50326
$418$	725.071	0.0848430
$419$	689.115	0.0803472	0.0401736	−	0.999193i	$-0.487209\pi$
$419$	689.115	0.0803472	0.0401736	+	0.999193i	$0.487209\pi$
$420$	6316.52	0.733844
$421$	1158.36	0.134097	0.0670485	−	0.997750i	$-0.478642\pi$
$421$	1158.36	0.134097	0.0670485	+	0.997750i	$0.478642\pi$
$422$	−4347.26	−0.501473
$423$	26090.2	2.99893
$424$	2670.93	0.305924
$425$	491.634	0.0561124
$426$	−510.041	−0.0580084
$427$	−16884.4	−1.91357
$428$	−3041.33	−0.343477
$429$	6291.47	0.708054
$430$	−2935.08	−0.329167
$431$	−13508.3	−1.50968	−0.754840	−	0.655909i	$-0.772286\pi$
$431$	−13508.3	−1.50968	−0.754840	+	0.655909i	$0.772286\pi$
$432$	34669.0	3.86115
$433$	1525.95	0.169359	0.0846796	−	0.996408i	$-0.473013\pi$
$433$	1525.95	0.169359	0.0846796	+	0.996408i	$0.473013\pi$
$434$	−11827.6	−1.30816
$435$	−10578.7	−1.16600
$436$	1252.67	0.137596
$437$	−1675.56	−0.183416
$438$	−752.134	−0.0820510
$439$	−11012.2	−1.19723	−0.598615	−	0.801037i	$-0.704282\pi$
$439$	−11012.2	−1.19723	−0.598615	+	0.801037i	$0.704282\pi$
$440$	−756.435	−0.0819582
$441$	46653.9	5.03767
$442$	3945.36	0.424574
$443$	−9349.14	−1.00269	−0.501344	−	0.865248i	$-0.667161\pi$
$443$	−9349.14	−1.00269	−0.501344	+	0.865248i	$0.667161\pi$
$444$	−7386.94	−0.789569
$445$	5543.37	0.590519
$446$	18314.5	1.94444
$447$	18703.2	1.97904
$448$	1863.13	0.196484
$449$	2378.72	0.250019	0.125010	−	0.992156i	$-0.460104\pi$
$449$	2378.72	0.250019	0.125010	+	0.992156i	$0.460104\pi$
$450$	−6142.09	−0.643424
$451$	−5107.40	−0.533255
$452$	2447.19	0.254659
$453$	17236.9	1.78777
$454$	19418.1	2.00735
$455$	9151.85	0.942957
$456$	2584.47	0.265414
$457$	−8831.38	−0.903971	−0.451985	−	0.892025i	$-0.649284\pi$
$457$	−8831.38	−0.903971	−0.451985	+	0.892025i	$0.649284\pi$
$458$	−15689.9	−1.60074
$459$	−8522.37	−0.866646
$460$	−1779.45	−0.180364
$461$	3717.60	0.375588	0.187794	−	0.982208i	$-0.439866\pi$
$461$	3717.60	0.375588	0.187794	+	0.982208i	$0.439866\pi$
$462$	11946.0	1.20299
$463$	−4762.13	−0.478002	−0.239001	−	0.971019i	$-0.576820\pi$
$463$	−4762.13	−0.478002	−0.239001	+	0.971019i	$0.576820\pi$
$464$	17113.4	1.71222
$465$	5326.64	0.531219
$466$	23930.6	2.37890
$467$	−4039.17	−0.400237	−0.200118	−	0.979772i	$-0.564133\pi$
$467$	−4039.17	−0.400237	−0.200118	+	0.979772i	$0.564133\pi$
$468$	−16527.4	−1.63243
$469$	−18650.4	−1.83624
$470$	6390.56	0.627180
$471$	−3609.49	−0.353113
$472$	3820.72	0.372591
$473$	−1861.26	−0.180932
$474$	12178.7	1.18014
$475$	−475.000	−0.0458831
$476$	2511.89	0.241875
$477$	13752.9	1.32013
$478$	4273.70	0.408943
$479$	6268.36	0.597930	0.298965	−	0.954264i	$-0.403359\pi$
$479$	6268.36	0.597930	0.298965	+	0.954264i	$0.403359\pi$
$480$	8283.50	0.787684
$481$	−10702.8	−1.01456
$482$	17939.1	1.69523
$483$	−27605.9	−2.60065
$484$	488.311	0.0458594
$485$	−2001.25	−0.187365
$486$	40865.1	3.81415
$487$	−16940.9	−1.57632	−0.788159	−	0.615472i	$-0.788965\pi$
$487$	−16940.9	−1.57632	−0.788159	+	0.615472i	$0.788965\pi$
$488$	−7336.82	−0.680578
$489$	−34876.9	−3.22533
$490$	11427.5	1.05355
$491$	13672.8	1.25671	0.628353	−	0.777928i	$-0.283729\pi$
$491$	13672.8	1.25671	0.628353	+	0.777928i	$0.283729\pi$
$492$	18532.2	1.69816
$493$	−4206.83	−0.384313
$494$	−3811.88	−0.347175
$495$	−3894.98	−0.353669
$496$	−8617.03	−0.780072
$497$	−470.489	−0.0424634
$498$	−17559.3	−1.58002
$499$	16612.1	1.49030	0.745149	−	0.666898i	$-0.232378\pi$
$499$	16612.1	1.49030	0.745149	+	0.666898i	$0.232378\pi$
$500$	−504.453	−0.0451197
$501$	−6030.73	−0.537790
$502$	−19160.5	−1.70354
$503$	−6323.13	−0.560506	−0.280253	−	0.959926i	$-0.590418\pi$
$503$	−6323.13	−0.560506	−0.280253	+	0.959926i	$0.590418\pi$
$504$	30827.5	2.72454
$505$	3777.10	0.332829
$506$	−3365.37	−0.295670
$507$	−11346.9	−0.993948
$508$	8041.35	0.702317
$509$	21624.3	1.88307	0.941533	−	0.336920i	$-0.109385\pi$
$509$	21624.3	1.88307	0.941533	+	0.336920i	$0.109385\pi$
$510$	−3373.77	−0.292927
$511$	−693.809	−0.0600632
$512$	−4598.40	−0.396919
$513$	8234.03	0.708657
$514$	26103.8	2.24006
$515$	−4171.34	−0.356915
$516$	6753.59	0.576183
$517$	4052.54	0.344740
$518$	−20322.0	−1.72374
$519$	−41087.4	−3.47502
$520$	3976.76	0.335370
$521$	−8536.09	−0.717799	−0.358899	−	0.933376i	$-0.616848\pi$
$521$	−8536.09	−0.717799	−0.358899	+	0.933376i	$0.616848\pi$
$522$	52556.9	4.40681
$523$	13374.6	1.11822	0.559110	−	0.829093i	$-0.311143\pi$
$523$	13374.6	1.11822	0.559110	+	0.829093i	$0.311143\pi$
$524$	−11218.0	−0.935228
$525$	−7825.94	−0.650576
$526$	20331.0	1.68531
$527$	2118.24	0.175090
$528$	8703.31	0.717354
$529$	−4390.01	−0.360813
$530$	3368.66	0.276085
$531$	19673.4	1.60782
$532$	−2426.90	−0.197781
$533$	26850.9	2.18206
$534$	−38040.6	−3.08273
$535$	3768.10	0.304503
$536$	−8104.18	−0.653074
$537$	−37445.0	−3.00907
$538$	−4501.43	−0.360726
$539$	7246.67	0.579102
$540$	8744.59	0.696865
$541$	−6608.12	−0.525148	−0.262574	−	0.964912i	$-0.584571\pi$
$541$	−6608.12	−0.525148	−0.262574	+	0.964912i	$0.584571\pi$
$542$	−14682.1	−1.16356
$543$	4289.29	0.338989
$544$	3294.10	0.259620
$545$	−1552.01	−0.121983
$546$	−62803.2	−4.92258
$547$	19595.8	1.53173	0.765866	−	0.643001i	$-0.222311\pi$
$547$	19595.8	1.53173	0.765866	+	0.643001i	$0.222311\pi$
$548$	9495.80	0.740220
$549$	−37778.2	−2.93685
$550$	−954.041	−0.0739644
$551$	4064.50	0.314253
$552$	−11995.6	−0.924943
$553$	11234.3	0.863891
$554$	804.800	0.0617197
$555$	9152.16	0.699977
$556$	8697.87	0.663438
$557$	−8092.08	−0.615570	−0.307785	−	0.951456i	$-0.599588\pi$
$557$	−8092.08	−0.615570	−0.307785	+	0.951456i	$0.599588\pi$
$558$	−26463.7	−2.00770
$559$	9785.12	0.740369
$560$	12660.2	0.955342
$561$	−2139.46	−0.161012
$562$	−10297.7	−0.772921
$563$	−15391.6	−1.15219	−0.576093	−	0.817384i	$-0.695423\pi$
$563$	−15391.6	−1.15219	−0.576093	+	0.817384i	$0.695423\pi$
$564$	−14704.6	−1.09783
$565$	−3031.98	−0.225763
$566$	−11469.8	−0.851789
$567$	75141.8	5.56554
$568$	−204.442	−0.0151025
$569$	−14564.8	−1.07309	−0.536546	−	0.843871i	$-0.680271\pi$
$569$	−14564.8	−1.07309	−0.536546	+	0.843871i	$0.680271\pi$
$570$	3259.62	0.239527
$571$	−5214.69	−0.382186	−0.191093	−	0.981572i	$-0.561203\pi$
$571$	−5214.69	−0.382186	−0.191093	+	0.981572i	$0.561203\pi$
$572$	−2567.17	−0.187655
$573$	24466.6	1.78378
$574$	50983.5	3.70733
$575$	2204.68	0.159898
$576$	4168.67	0.301553
$577$	−6482.46	−0.467710	−0.233855	−	0.972272i	$-0.575134\pi$
$577$	−6482.46	−0.467710	−0.233855	+	0.972272i	$0.575134\pi$
$578$	15702.7	1.13001
$579$	−32598.6	−2.33981
$580$	4316.53	0.309024
$581$	−16197.6	−1.15661
$582$	13733.3	0.978113
$583$	2136.22	0.151755
$584$	−301.482	−0.0213620
$585$	20476.8	1.44720
$586$	−6040.05	−0.425789
$587$	−25467.3	−1.79071	−0.895355	−	0.445352i	$-0.853078\pi$
$587$	−25467.3	−1.79071	−0.895355	+	0.445352i	$0.853078\pi$
$588$	−26294.5	−1.84416
$589$	−2046.58	−0.143171
$590$	4818.82	0.336250
$591$	29932.1	2.08332
$592$	−14805.7	−1.02789
$593$	−7875.74	−0.545393	−0.272697	−	0.962100i	$-0.587915\pi$
$593$	−7875.74	−0.545393	−0.272697	+	0.962100i	$0.587915\pi$
$594$	16538.1	1.14237
$595$	−3112.14	−0.214429
$596$	−7631.65	−0.524504
$597$	19778.5	1.35591
$598$	17692.6	1.20987
$599$	11100.3	0.757169	0.378584	−	0.925567i	$-0.376411\pi$
$599$	11100.3	0.757169	0.378584	+	0.925567i	$0.376411\pi$
$600$	−3400.62	−0.231383
$601$	−6391.32	−0.433789	−0.216895	−	0.976195i	$-0.569593\pi$
$601$	−6391.32	−0.433789	−0.216895	+	0.976195i	$0.569593\pi$
$602$	18579.6	1.25789
$603$	−41729.4	−2.81816
$604$	−7033.35	−0.473813
$605$	−605.000	−0.0406558
$606$	−25919.8	−1.73749
$607$	−7508.99	−0.502110	−0.251055	−	0.967973i	$-0.580777\pi$
$607$	−7508.99	−0.502110	−0.251055	+	0.967973i	$0.580777\pi$
$608$	−3182.65	−0.212292
$609$	66965.3	4.45578
$610$	−9253.44	−0.614198
$611$	−21305.2	−1.41067
$612$	5620.24	0.371217
$613$	−9149.61	−0.602854	−0.301427	−	0.953489i	$-0.597463\pi$
$613$	−9149.61	−0.602854	−0.301427	+	0.953489i	$0.597463\pi$
$614$	7217.98	0.474420
$615$	−22960.8	−1.50547
$616$	4788.39	0.313198
$617$	13581.3	0.886161	0.443081	−	0.896482i	$-0.353886\pi$
$617$	13581.3	0.886161	0.443081	+	0.896482i	$0.353886\pi$
$618$	28625.2	1.86323
$619$	−15206.7	−0.987414	−0.493707	−	0.869628i	$-0.664358\pi$
$619$	−15206.7	−0.987414	−0.493707	+	0.869628i	$0.664358\pi$
$620$	−2173.48	−0.140789
$621$	−38217.7	−2.46960
$622$	4972.83	0.320566
$623$	−35090.7	−2.25663
$624$	−45755.4	−2.93539
$625$	625.000	0.0400000
$626$	−13535.9	−0.864223
$627$	2067.07	0.131660
$628$	1472.81	0.0935854
$629$	3639.54	0.230712
$630$	38880.7	2.45880
$631$	−7772.85	−0.490384	−0.245192	−	0.969475i	$-0.578851\pi$
$631$	−7772.85	−0.490384	−0.245192	+	0.969475i	$0.578851\pi$
$632$	4881.66	0.307250
$633$	−12393.4	−0.778188
$634$	−20949.8	−1.31234
$635$	−9962.95	−0.622626
$636$	−7751.27	−0.483267
$637$	−38097.5	−2.36967
$638$	8163.58	0.506582
$639$	−1052.70	−0.0651707
$640$	7721.40	0.476898
$641$	18173.4	1.11982	0.559910	−	0.828553i	$-0.310836\pi$
$641$	18173.4	1.11982	0.559910	+	0.828553i	$0.310836\pi$
$642$	−25858.0	−1.58962
$643$	27208.7	1.66875	0.834375	−	0.551198i	$-0.185829\pi$
$643$	27208.7	1.66875	0.834375	+	0.551198i	$0.185829\pi$
$644$	11264.3	0.689248
$645$	−8367.46	−0.510804
$646$	1296.25	0.0789480
$647$	28852.2	1.75316	0.876582	−	0.481253i	$-0.159818\pi$
$647$	28852.2	1.75316	0.876582	+	0.481253i	$0.159818\pi$
$648$	32651.5	1.97943
$649$	3055.83	0.184826
$650$	5015.63	0.302660
$651$	−33718.7	−2.03002
$652$	14231.1	0.854808
$653$	−5134.90	−0.307724	−0.153862	−	0.988092i	$-0.549171\pi$
$653$	−5134.90	−0.307724	−0.153862	+	0.988092i	$0.549171\pi$
$654$	10650.5	0.636798
$655$	13898.7	0.829109
$656$	37144.2	2.21072
$657$	−1552.37	−0.0921820
$658$	−40453.6	−2.39673
$659$	−30609.8	−1.80939	−0.904695	−	0.426059i	$-0.859902\pi$
$659$	−30609.8	−1.80939	−0.904695	+	0.426059i	$0.859902\pi$
$660$	2195.24	0.129469
$661$	20311.7	1.19521	0.597604	−	0.801791i	$-0.296119\pi$
$661$	20311.7	1.19521	0.597604	+	0.801791i	$0.296119\pi$
$662$	−19365.8	−1.13697
$663$	11247.6	0.658857
$664$	−7038.37	−0.411358
$665$	3006.85	0.175339
$666$	−45469.6	−2.64551
$667$	−18865.1	−1.09514
$668$	2460.77	0.142530
$669$	52212.0	3.01739
$670$	−10221.3	−0.589376
$671$	−5868.02	−0.337604
$672$	−52436.3	−3.01008
$673$	18830.5	1.07855	0.539274	−	0.842130i	$-0.318699\pi$
$673$	18830.5	1.07855	0.539274	+	0.842130i	$0.318699\pi$
$674$	−11922.9	−0.681383
$675$	−10834.2	−0.617793
$676$	4629.97	0.263425
$677$	−30018.2	−1.70413	−0.852063	−	0.523440i	$-0.824648\pi$
$677$	−30018.2	−1.70413	−0.852063	+	0.523440i	$0.824648\pi$
$678$	20806.5	1.17857
$679$	12668.3	0.716001
$680$	−1352.32	−0.0762636
$681$	55358.1	3.11502
$682$	−4110.56	−0.230794
$683$	−33401.3	−1.87125	−0.935625	−	0.352996i	$-0.885163\pi$
$683$	−33401.3	−1.87125	−0.935625	+	0.352996i	$0.885163\pi$
$684$	−5430.08	−0.303544
$685$	−11765.0	−0.656228
$686$	−34675.1	−1.92989
$687$	−44729.6	−2.48405
$688$	13536.2	0.750094
$689$	−11230.6	−0.620976
$690$	−15129.3	−0.834729
$691$	−9458.35	−0.520713	−0.260357	−	0.965513i	$-0.583840\pi$
$691$	−9458.35	−0.520713	−0.260357	+	0.965513i	$0.583840\pi$
$692$	16765.3	0.920982
$693$	24656.0	1.35152
$694$	−35422.8	−1.93751
$695$	−10776.4	−0.588159
$696$	29098.5	1.58474
$697$	−9130.81	−0.496204
$698$	1677.71	0.0909777
$699$	68222.7	3.69159
$700$	3193.29	0.172422
$701$	10159.5	0.547385	0.273693	−	0.961817i	$-0.411755\pi$
$701$	10159.5	0.547385	0.273693	+	0.961817i	$0.411755\pi$
$702$	−86944.9	−4.67453
$703$	−3516.40	−0.188654
$704$	647.513	0.0346649
$705$	18218.5	0.973262
$706$	5435.81	0.289773
$707$	−23909.8	−1.27188
$708$	−11088.1	−0.588581
$709$	−9428.61	−0.499434	−0.249717	−	0.968319i	$-0.580338\pi$
$709$	−9428.61	−0.499434	−0.249717	+	0.968319i	$0.580338\pi$
$710$	−257.849	−0.0136295
$711$	25136.3	1.32586
$712$	−15248.0	−0.802589
$713$	9499.05	0.498937
$714$	21356.6	1.11940
$715$	3180.63	0.166362
$716$	15279.1	0.797493
$717$	12183.7	0.634600
$718$	42413.9	2.20456
$719$	16092.0	0.834673	0.417337	−	0.908752i	$-0.362964\pi$
$719$	16092.0	0.834673	0.417337	+	0.908752i	$0.362964\pi$
$720$	28326.6	1.46621
$721$	26405.4	1.36392
$722$	−1252.40	−0.0645559
$723$	51141.6	2.63067
$724$	−1750.20	−0.0898421
$725$	−5348.02	−0.273960
$726$	4151.72	0.212238
$727$	−8555.12	−0.436440	−0.218220	−	0.975900i	$-0.570025\pi$
$727$	−8555.12	−0.436440	−0.218220	+	0.975900i	$0.570025\pi$
$728$	−25173.7	−1.28159
$729$	52400.3	2.66221
$730$	−380.239	−0.0192784
$731$	−3327.49	−0.168361
$732$	21292.1	1.07511
$733$	−12903.8	−0.650223	−0.325112	−	0.945676i	$-0.605402\pi$
$733$	−12903.8	−0.650223	−0.325112	+	0.945676i	$0.605402\pi$
$734$	30132.8	1.51529
$735$	32578.0	1.63491
$736$	14772.1	0.739817
$737$	−6481.76	−0.323960
$738$	114073.	5.68983
$739$	6278.59	0.312533	0.156266	−	0.987715i	$-0.450054\pi$
$739$	6278.59	0.312533	0.156266	+	0.987715i	$0.450054\pi$
$740$	−3734.44	−0.185515
$741$	−10867.1	−0.538748
$742$	−21324.3	−1.05504
$743$	−19236.0	−0.949799	−0.474899	−	0.880040i	$-0.657516\pi$
$743$	−19236.0	−0.949799	−0.474899	+	0.880040i	$0.657516\pi$
$744$	−14651.8	−0.721992
$745$	9455.35	0.464989
$746$	−35607.3	−1.74755
$747$	−36241.4	−1.77511
$748$	872.982	0.0426730
$749$	−23852.8	−1.16364
$750$	−4288.97	−0.208815
$751$	−1434.88	−0.0697195	−0.0348598	−	0.999392i	$-0.511098\pi$
$751$	−1434.88	−0.0697195	−0.0348598	+	0.999392i	$0.511098\pi$
$752$	−29472.6	−1.42919
$753$	−54623.8	−2.64356
$754$	−42917.9	−2.07292
$755$	8714.07	0.420050
$756$	−55355.1	−2.66302
$757$	−23165.1	−1.11222	−0.556108	−	0.831110i	$-0.687706\pi$
$757$	−23165.1	−1.11222	−0.556108	+	0.831110i	$0.687706\pi$
$758$	42908.5	2.05608
$759$	−9594.17	−0.458822
$760$	1306.57	0.0623609
$761$	−33583.5	−1.59974	−0.799869	−	0.600175i	$-0.795098\pi$
$761$	−33583.5	−1.59974	−0.799869	+	0.600175i	$0.795098\pi$
$762$	68369.3	3.25034
$763$	9824.56	0.466151
$764$	−9983.33	−0.472754
$765$	−6963.28	−0.329095
$766$	49377.7	2.32910
$767$	−16065.2	−0.756300
$768$	−48329.4	−2.27075
$769$	11042.9	0.517839	0.258920	−	0.965899i	$-0.416634\pi$
$769$	11042.9	0.517839	0.258920	+	0.965899i	$0.416634\pi$
$770$	6039.28	0.282650
$771$	74418.0	3.47613
$772$	13301.5	0.620120
$773$	12955.2	0.602803	0.301401	−	0.953497i	$-0.402546\pi$
$773$	12955.2	0.602803	0.301401	+	0.953497i	$0.402546\pi$
$774$	41571.1	1.93055
$775$	2692.86	0.124814
$776$	5504.77	0.254652
$777$	−57935.0	−2.67491
$778$	36920.7	1.70138
$779$	8821.88	0.405746
$780$	−11540.9	−0.529784
$781$	−163.514	−0.00749166
$782$	−6016.47	−0.275126
$783$	92706.9	4.23126
$784$	−52702.2	−2.40079
$785$	−1824.76	−0.0829663
$786$	−95377.6	−4.32825
$787$	9128.72	0.413473	0.206737	−	0.978397i	$-0.433716\pi$
$787$	9128.72	0.413473	0.206737	+	0.978397i	$0.433716\pi$
$788$	−12213.5	−0.552141
$789$	57960.7	2.61528
$790$	6156.92	0.277283
$791$	19193.1	0.862739
$792$	10713.8	0.480680
$793$	30849.6	1.38146
$794$	−20034.7	−0.895474
$795$	9603.55	0.428431
$796$	−8070.40	−0.359357
$797$	21173.0	0.941010	0.470505	−	0.882397i	$-0.344072\pi$
$797$	21173.0	0.941010	0.470505	+	0.882397i	$0.344072\pi$
$798$	−20634.1	−0.915335
$799$	7244.97	0.320787
$800$	4187.70	0.185072
$801$	−78513.8	−3.46336
$802$	−39314.2	−1.73096
$803$	−241.126	−0.0105967
$804$	23519.1	1.03166
$805$	−13956.1	−0.611040
$806$	21610.2	0.944402
$807$	−12832.9	−0.559777
$808$	−10389.6	−0.452356
$809$	10416.8	0.452701	0.226351	−	0.974046i	$-0.427320\pi$
$809$	10416.8	0.452701	0.226351	+	0.974046i	$0.427320\pi$
$810$	41181.1	1.78637
$811$	−44865.7	−1.94260	−0.971299	−	0.237863i	$-0.923553\pi$
$811$	−44865.7	−1.94260	−0.971299	+	0.237863i	$0.923553\pi$
$812$	−27324.5	−1.18091
$813$	−41856.5	−1.80562
$814$	−7062.71	−0.304113
$815$	−17631.9	−0.757814
$816$	15559.4	0.667511
$817$	3214.91	0.137669
$818$	29715.3	1.27014
$819$	−129623.	−5.53038
$820$	9368.89	0.398995
$821$	26899.0	1.14346	0.571730	−	0.820442i	$-0.306272\pi$
$821$	26899.0	1.14346	0.571730	+	0.820442i	$0.306272\pi$
$822$	80735.4	3.42575
$823$	8784.15	0.372049	0.186024	−	0.982545i	$-0.440440\pi$
$823$	8784.15	0.372049	0.186024	+	0.982545i	$0.440440\pi$
$824$	11474.0	0.485091
$825$	−2719.83	−0.114779
$826$	−30504.1	−1.28496
$827$	−42934.7	−1.80530	−0.902651	−	0.430374i	$-0.858382\pi$
$827$	−42934.7	−1.80530	−0.902651	+	0.430374i	$0.858382\pi$
$828$	25203.4	1.05782
$829$	9968.45	0.417634	0.208817	−	0.977955i	$-0.433039\pi$
$829$	9968.45	0.417634	0.208817	+	0.977955i	$0.433039\pi$
$830$	−8877.02	−0.371236
$831$	2294.37	0.0957770
$832$	−3404.13	−0.141848
$833$	12955.3	0.538865
$834$	73951.2	3.07041
$835$	−3048.81	−0.126358
$836$	−843.446	−0.0348938
$837$	−46680.2	−1.92772
$838$	−2390.70	−0.0985508
$839$	4276.78	0.175984	0.0879921	−	0.996121i	$-0.471955\pi$
$839$	4276.78	0.175984	0.0879921	+	0.996121i	$0.471955\pi$
$840$	21526.6	0.884213
$841$	21373.2	0.876345
$842$	−4018.61	−0.164478
$843$	−29357.2	−1.19942
$844$	5056.99	0.206243
$845$	−5736.37	−0.233535
$846$	−90513.0	−3.67837
$847$	3829.77	0.155363
$848$	−15535.9	−0.629133
$849$	−32698.8	−1.32181
$850$	−1705.60	−0.0688253
$851$	16321.1	0.657440
$852$	593.310	0.0238573
$853$	−18513.4	−0.743127	−0.371564	−	0.928408i	$-0.621178\pi$
$853$	−18513.4	−0.743127	−0.371564	+	0.928408i	$0.621178\pi$
$854$	58576.2	2.34711
$855$	6727.68	0.269102
$856$	−10364.8	−0.413857
$857$	−28924.9	−1.15292	−0.576462	−	0.817124i	$-0.695567\pi$
$857$	−28924.9	−1.15292	−0.576462	+	0.817124i	$0.695567\pi$
$858$	−21826.6	−0.868472
$859$	16763.7	0.665856	0.332928	−	0.942952i	$-0.391963\pi$
$859$	16763.7	0.665856	0.332928	+	0.942952i	$0.391963\pi$
$860$	3414.26	0.135378
$861$	145346.	5.75307
$862$	46863.6	1.85172
$863$	6982.88	0.275434	0.137717	−	0.990472i	$-0.456023\pi$
$863$	6982.88	0.275434	0.137717	+	0.990472i	$0.456023\pi$
$864$	−72592.9	−2.85840
$865$	−20771.6	−0.816479
$866$	−5293.89	−0.207730
$867$	44766.1	1.75356
$868$	13758.6	0.538014
$869$	3904.38	0.152413
$870$	36700.0	1.43017
$871$	34076.2	1.32564
$872$	4269.08	0.165790
$873$	28344.7	1.09888
$874$	5812.91	0.224971
$875$	−3956.38	−0.152857
$876$	874.927	0.0337455
$877$	−11354.5	−0.437187	−0.218594	−	0.975816i	$-0.570147\pi$
$877$	−11354.5	−0.437187	−0.218594	+	0.975816i	$0.570147\pi$
$878$	38204.0	1.46848
$879$	−17219.3	−0.660741
$880$	4399.93	0.168547
$881$	−9261.20	−0.354163	−0.177082	−	0.984196i	$-0.556666\pi$
$881$	−9261.20	−0.354163	−0.177082	+	0.984196i	$0.556666\pi$
$882$	−161853.	−6.17901
$883$	30432.5	1.15984	0.579918	−	0.814675i	$-0.303085\pi$
$883$	30432.5	1.15984	0.579918	+	0.814675i	$0.303085\pi$
$884$	−4589.48	−0.174616
$885$	13737.7	0.521795
$886$	32434.4	1.22986
$887$	8821.76	0.333941	0.166971	−	0.985962i	$-0.446602\pi$
$887$	8821.76	0.333941	0.166971	+	0.985962i	$0.446602\pi$
$888$	−25174.6	−0.951356
$889$	63067.5	2.37932
$890$	−19231.3	−0.724308
$891$	26114.8	0.981906
$892$	−21304.6	−0.799697
$893$	−6999.85	−0.262308
$894$	−64885.9	−2.42742
$895$	−18930.2	−0.707002
$896$	−48878.0	−1.82243
$897$	50438.9	1.87749
$898$	−8252.34	−0.306664
$899$	−23042.4	−0.854847
$900$	7144.85	0.264624
$901$	3819.05	0.141211
$902$	17718.8	0.654070
$903$	52967.8	1.95200
$904$	8339.99	0.306841
$905$	2168.44	0.0796478
$906$	−59799.1	−2.19282
$907$	−17218.2	−0.630342	−0.315171	−	0.949035i	$-0.602062\pi$
$907$	−17218.2	−0.630342	−0.315171	+	0.949035i	$0.602062\pi$
$908$	−22588.3	−0.825571
$909$	−53497.2	−1.95202
$910$	−31750.0	−1.15659
$911$	20160.3	0.733196	0.366598	−	0.930379i	$-0.380522\pi$
$911$	20160.3	0.733196	0.366598	+	0.930379i	$0.380522\pi$
$912$	−15033.0	−0.545825
$913$	−5629.32	−0.204056
$914$	30638.2	1.10878
$915$	−26380.2	−0.953116
$916$	18251.4	0.658345
$917$	−87981.5	−3.16838
$918$	29566.2	1.06299
$919$	−4141.21	−0.148646	−0.0743232	−	0.997234i	$-0.523680\pi$
$919$	−4141.21	−0.148646	−0.0743232	+	0.997234i	$0.523680\pi$
$920$	−6064.36	−0.217322
$921$	20577.4	0.736208
$922$	−12897.2	−0.460681
$923$	859.632	0.0306556
$924$	−13896.3	−0.494757
$925$	4626.84	0.164464
$926$	16521.0	0.586299
$927$	59080.9	2.09328
$928$	−35833.5	−1.26755
$929$	−7305.59	−0.258007	−0.129003	−	0.991644i	$-0.541178\pi$
$929$	−7305.59	−0.258007	−0.129003	+	0.991644i	$0.541178\pi$
$930$	−18479.4	−0.651573
$931$	−12517.0	−0.440631
$932$	−27837.5	−0.978379
$933$	14176.8	0.497457
$934$	14012.9	0.490915
$935$	−1081.59	−0.0378309
$936$	−56325.1	−1.96693
$937$	−9967.40	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$937$	−9967.40	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$938$	64702.7	2.25226
$939$	−38588.9	−1.34111
$940$	−7433.88	−0.257943
$941$	−47477.1	−1.64475	−0.822376	−	0.568945i	$-0.807352\pi$
$941$	−47477.1	−1.64475	−0.822376	+	0.568945i	$0.807352\pi$
$942$	12522.2	0.433115
$943$	−40946.2	−1.41399
$944$	−22223.9	−0.766234
$945$	68583.0	2.36085
$946$	6457.17	0.221925
$947$	−16724.7	−0.573895	−0.286948	−	0.957946i	$-0.592641\pi$
$947$	−16724.7	−0.573895	−0.286948	+	0.957946i	$0.592641\pi$
$948$	−14167.0	−0.485362
$949$	1267.66	0.0433614
$950$	1647.89	0.0562785
$951$	−59724.8	−2.03650
$952$	8560.49	0.291436
$953$	18384.1	0.624890	0.312445	−	0.949936i	$-0.398852\pi$
$953$	18384.1	0.624890	0.312445	+	0.949936i	$0.398852\pi$
$954$	−47712.2	−1.61922
$955$	12369.0	0.419111
$956$	−4971.43	−0.168188
$957$	23273.1	0.786117
$958$	−21746.4	−0.733398
$959$	74474.7	2.50773
$960$	2910.95	0.0978651
$961$	−18188.6	−0.610539
$962$	37130.4	1.24442
$963$	−53369.6	−1.78589
$964$	−20867.8	−0.697206
$965$	−16480.1	−0.549755
$966$	95771.6	3.18986
$967$	40306.3	1.34040	0.670198	−	0.742183i	$-0.266209\pi$
$967$	40306.3	1.34040	0.670198	+	0.742183i	$0.266209\pi$
$968$	1664.16	0.0552562
$969$	3695.42	0.122512
$970$	6942.80	0.229814
$971$	−8989.51	−0.297103	−0.148552	−	0.988905i	$-0.547461\pi$
$971$	−8989.51	−0.297103	−0.148552	+	0.988905i	$0.547461\pi$
$972$	−47536.7	−1.56866
$973$	68216.5	2.24761
$974$	58772.1	1.93345
$975$	14298.8	0.469670
$976$	42675.8	1.39961
$977$	−23010.4	−0.753498	−0.376749	−	0.926315i	$-0.622958\pi$
$977$	−23010.4	−0.753498	−0.376749	+	0.926315i	$0.622958\pi$
$978$	120996.	3.95607
$979$	−12195.4	−0.398128
$980$	−13293.1	−0.433299
$981$	21982.0	0.715425
$982$	−47434.1	−1.54143
$983$	−17743.4	−0.575714	−0.287857	−	0.957673i	$-0.592943\pi$
$983$	−17743.4	−0.575714	−0.287857	+	0.957673i	$0.592943\pi$
$984$	63157.5	2.04613
$985$	15132.1	0.489490
$986$	14594.5	0.471383
$987$	−115327.	−3.71925
$988$	4434.20	0.142784
$989$	−14921.8	−0.479763
$990$	13512.6	0.433797
$991$	−38107.0	−1.22150	−0.610750	−	0.791823i	$-0.709132\pi$
$991$	−38107.0	−1.22150	−0.610750	+	0.791823i	$0.709132\pi$
$992$	18043.0	0.577487
$993$	−55208.9	−1.76435
$994$	1632.24	0.0520840
$995$	9998.95	0.318581
$996$	20426.0	0.649821
$997$	6654.13	0.211373	0.105686	−	0.994400i	$-0.466296\pi$
$997$	6654.13	0.211373	0.105686	+	0.994400i	$0.466296\pi$
$998$	−57631.3	−1.82794
$999$	−80205.3	−2.54012

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.f.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)

\(f(q)\)	\(=\)	\(q-3.46924 q^{2} -9.89028 q^{3} +4.03563 q^{4} -5.00000 q^{5} +34.3118 q^{6} +31.6510 q^{7} +13.7534 q^{8} +70.8177 q^{9} +O(q^{10})\)	\(q-3.46924 q^{2} -9.89028 q^{3} +4.03563 q^{4} -5.00000 q^{5} +34.3118 q^{6} +31.6510 q^{7} +13.7534 q^{8} +70.8177 q^{9} +17.3462 q^{10} +11.0000 q^{11} -39.9135 q^{12} -57.8297 q^{13} -109.805 q^{14} +49.4514 q^{15} -79.9987 q^{16} +19.6654 q^{17} -245.684 q^{18} -19.0000 q^{19} -20.1781 q^{20} -313.038 q^{21} -38.1616 q^{22} +88.1872 q^{23} -136.025 q^{24} +25.0000 q^{25} +200.625 q^{26} -433.370 q^{27} +127.732 q^{28} -213.921 q^{29} -171.559 q^{30} +107.715 q^{31} +167.508 q^{32} -108.793 q^{33} -68.2238 q^{34} -158.255 q^{35} +285.794 q^{36} +185.074 q^{37} +65.9156 q^{38} +571.952 q^{39} -68.7668 q^{40} -464.309 q^{41} +1086.00 q^{42} -169.206 q^{43} +44.3919 q^{44} -354.089 q^{45} -305.943 q^{46} +368.413 q^{47} +791.210 q^{48} +658.788 q^{49} -86.7310 q^{50} -194.496 q^{51} -233.379 q^{52} +194.202 q^{53} +1503.46 q^{54} -55.0000 q^{55} +435.308 q^{56} +187.915 q^{57} +742.143 q^{58} +277.803 q^{59} +199.567 q^{60} -533.456 q^{61} -373.688 q^{62} +2241.45 q^{63} +58.8648 q^{64} +289.148 q^{65} +377.429 q^{66} -589.251 q^{67} +79.3620 q^{68} -872.197 q^{69} +549.025 q^{70} -14.8649 q^{71} +973.982 q^{72} -21.9206 q^{73} -642.065 q^{74} -247.257 q^{75} -76.6769 q^{76} +348.161 q^{77} -1984.24 q^{78} +354.943 q^{79} +399.994 q^{80} +2374.07 q^{81} +1610.80 q^{82} -511.756 q^{83} -1263.30 q^{84} -98.3268 q^{85} +587.015 q^{86} +2115.74 q^{87} +151.287 q^{88} -1108.67 q^{89} +1228.42 q^{90} -1830.37 q^{91} +355.891 q^{92} -1065.33 q^{93} -1278.11 q^{94} +95.0000 q^{95} -1656.70 q^{96} +400.249 q^{97} -2285.49 q^{98} +778.995 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9}+O(q^{10}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9	\( 23 q - 2 q^{2} - 9 q^{3} + 98 q^{4} - 115 q^{5} - 61 q^{6} + 13 q^{7} - 54 q^{8} + 170 q^{9} + 10 q^{10} + 253 q^{11} - 76 q^{12} - 37 q^{13} - 191 q^{14} + 45 q^{15} + 214 q^{16} - 51 q^{17} - 63 q^{18} - 437 q^{19} - 490 q^{20} - 479 q^{21} - 22 q^{22} + 101 q^{23} - 598 q^{24} + 575 q^{25} - 197 q^{26} - 627 q^{27} + 279 q^{28} - 357 q^{29} + 305 q^{30} - 90 q^{31} - 19 q^{32} - 99 q^{33} + 71 q^{34} - 65 q^{35} + 573 q^{36} - 378 q^{37} + 38 q^{38} + 193 q^{39} + 270 q^{40} - 830 q^{41} + 1480 q^{42} + 260 q^{43} + 1078 q^{44} - 850 q^{45} - 919 q^{46} - 1468 q^{47} + 837 q^{48} + 1200 q^{49} - 50 q^{50} - 1147 q^{51} - 1222 q^{52} + 185 q^{53} - 1406 q^{54} - 1265 q^{55} - 2299 q^{56} + 171 q^{57} - 958 q^{58} - 3665 q^{59} + 380 q^{60} - 2528 q^{61} - 1722 q^{62} + 172 q^{63} - 120 q^{64} + 185 q^{65} - 671 q^{66} + 329 q^{67} - 2240 q^{68} - 1337 q^{69} + 955 q^{70} - 3190 q^{71} - 2488 q^{72} - 2183 q^{73} - 1613 q^{74} - 225 q^{75} - 1862 q^{76} + 143 q^{77} - 2748 q^{78} - 3546 q^{79} - 1070 q^{80} - 2077 q^{81} + 2202 q^{82} - 4324 q^{83} - 8608 q^{84} + 255 q^{85} - 3626 q^{86} + 2921 q^{87} - 594 q^{88} - 4630 q^{89} + 315 q^{90} - 5043 q^{91} + 108 q^{92} - 5644 q^{93} - 8328 q^{94} + 2185 q^{95} - 2016 q^{96} - 774 q^{97} - 6388 q^{98} + 1870 q^{99}+O(q^{100}) \) 23 * q - 2 * q^2 - 9 * q^3 + 98 * q^4 - 115 * q^5 - 61 * q^6 + 13 * q^7 - 54 * q^8 + 170 * q^9 + 10 * q^10 + 253 * q^11 - 76 * q^12 - 37 * q^13 - 191 * q^14 + 45 * q^15 + 214 * q^16 - 51 * q^17 - 63 * q^18 - 437 * q^19 - 490 * q^20 - 479 * q^21 - 22 * q^22 + 101 * q^23 - 598 * q^24 + 575 * q^25 - 197 * q^26 - 627 * q^27 + 279 * q^28 - 357 * q^29 + 305 * q^30 - 90 * q^31 - 19 * q^32 - 99 * q^33 + 71 * q^34 - 65 * q^35 + 573 * q^36 - 378 * q^37 + 38 * q^38 + 193 * q^39 + 270 * q^40 - 830 * q^41 + 1480 * q^42 + 260 * q^43 + 1078 * q^44 - 850 * q^45 - 919 * q^46 - 1468 * q^47 + 837 * q^48 + 1200 * q^49 - 50 * q^50 - 1147 * q^51 - 1222 * q^52 + 185 * q^53 - 1406 * q^54 - 1265 * q^55 - 2299 * q^56 + 171 * q^57 - 958 * q^58 - 3665 * q^59 + 380 * q^60 - 2528 * q^61 - 1722 * q^62 + 172 * q^63 - 120 * q^64 + 185 * q^65 - 671 * q^66 + 329 * q^67 - 2240 * q^68 - 1337 * q^69 + 955 * q^70 - 3190 * q^71 - 2488 * q^72 - 2183 * q^73 - 1613 * q^74 - 225 * q^75 - 1862 * q^76 + 143 * q^77 - 2748 * q^78 - 3546 * q^79 - 1070 * q^80 - 2077 * q^81 + 2202 * q^82 - 4324 * q^83 - 8608 * q^84 + 255 * q^85 - 3626 * q^86 + 2921 * q^87 - 594 * q^88 - 4630 * q^89 + 315 * q^90 - 5043 * q^91 + 108 * q^92 - 5644 * q^93 - 8328 * q^94 + 2185 * q^95 - 2016 * q^96 - 774 * q^97 - 6388 * q^98 + 1870 * q^99

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