LMFDB - Embedded newform 1045.4.a.d.1.13

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:	$22$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
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)

$f(q)$	$=$	$q+1.04826 q^{2} +7.96396 q^{3} -6.90116 q^{4} +5.00000 q^{5} +8.34827 q^{6} -11.0379 q^{7} -15.6202 q^{8} +36.4247 q^{9} +O(q^{10})$	\(q+1.04826 q^{2} +7.96396 q^{3} -6.90116 q^{4} +5.00000 q^{5} +8.34827 q^{6} -11.0379 q^{7} -15.6202 q^{8} +36.4247 q^{9} +5.24128 q^{10} -11.0000 q^{11} -54.9605 q^{12} -37.4361 q^{13} -11.5706 q^{14} +39.8198 q^{15} +38.8352 q^{16} +33.1237 q^{17} +38.1824 q^{18} -19.0000 q^{19} -34.5058 q^{20} -87.9055 q^{21} -11.5308 q^{22} -24.0790 q^{23} -124.399 q^{24} +25.0000 q^{25} -39.2426 q^{26} +75.0576 q^{27} +76.1744 q^{28} +18.0847 q^{29} +41.7414 q^{30} -62.5041 q^{31} +165.671 q^{32} -87.6036 q^{33} +34.7221 q^{34} -55.1896 q^{35} -251.372 q^{36} -266.833 q^{37} -19.9169 q^{38} -298.139 q^{39} -78.1012 q^{40} -29.4174 q^{41} -92.1475 q^{42} -384.108 q^{43} +75.9127 q^{44} +182.123 q^{45} -25.2410 q^{46} -443.356 q^{47} +309.282 q^{48} -221.164 q^{49} +26.2064 q^{50} +263.795 q^{51} +258.352 q^{52} -264.474 q^{53} +78.6796 q^{54} -55.0000 q^{55} +172.415 q^{56} -151.315 q^{57} +18.9574 q^{58} +654.606 q^{59} -274.803 q^{60} +91.9399 q^{61} -65.5203 q^{62} -402.052 q^{63} -137.016 q^{64} -187.180 q^{65} -91.8310 q^{66} -475.332 q^{67} -228.592 q^{68} -191.764 q^{69} -57.8528 q^{70} -879.594 q^{71} -568.962 q^{72} -94.5002 q^{73} -279.709 q^{74} +199.099 q^{75} +131.122 q^{76} +121.417 q^{77} -312.527 q^{78} +349.276 q^{79} +194.176 q^{80} -385.710 q^{81} -30.8370 q^{82} -1461.89 q^{83} +606.650 q^{84} +165.618 q^{85} -402.644 q^{86} +144.026 q^{87} +171.823 q^{88} +1291.14 q^{89} +190.912 q^{90} +413.216 q^{91} +166.173 q^{92} -497.780 q^{93} -464.751 q^{94} -95.0000 q^{95} +1319.40 q^{96} -1170.73 q^{97} -231.837 q^{98} -400.671 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) $ 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * 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 $

$2$	1.04826	0.370615	0.185307	−	0.982681i	$-0.440672\pi$
$2$	1.04826	0.370615	0.185307	+	0.982681i	$0.440672\pi$
$3$	7.96396	1.53266	0.766332	−	0.642444i	$-0.222080\pi$
$3$	7.96396	1.53266	0.766332	+	0.642444i	$0.222080\pi$
$4$	−6.90116	−0.862645
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	8.34827	0.568028
$7$	−11.0379	−0.595991	−0.297996	−	0.954567i	$-0.596318\pi$
$7$	−11.0379	−0.595991	−0.297996	+	0.954567i	$0.596318\pi$
$8$	−15.6202	−0.690323
$9$	36.4247	1.34906
$10$	5.24128	0.165744
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−54.9605	−1.32215
$13$	−37.4361	−0.798685	−0.399342	−	0.916802i	$-0.630762\pi$
$13$	−37.4361	−0.798685	−0.399342	+	0.916802i	$0.630762\pi$
$14$	−11.5706	−0.220883
$15$	39.8198	0.685429
$16$	38.8352	0.606801
$17$	33.1237	0.472569	0.236284	−	0.971684i	$-0.424070\pi$
$17$	33.1237	0.472569	0.236284	+	0.971684i	$0.424070\pi$
$18$	38.1824	0.499982
$19$	−19.0000	−0.229416
$19$	−19.0000	−0.229416
$20$	−34.5058	−0.385786
$21$	−87.9055	−0.913455
$22$	−11.5308	−0.111745
$23$	−24.0790	−0.218296	−0.109148	−	0.994025i	$-0.534812\pi$
$23$	−24.0790	−0.218296	−0.109148	+	0.994025i	$0.534812\pi$
$24$	−124.399	−1.05803
$25$	25.0000	0.200000
$26$	−39.2426	−0.296004
$27$	75.0576	0.534994
$28$	76.1744	0.514129
$29$	18.0847	0.115801	0.0579007	−	0.998322i	$-0.481559\pi$
$29$	18.0847	0.115801	0.0579007	+	0.998322i	$0.481559\pi$
$30$	41.7414	0.254030
$31$	−62.5041	−0.362131	−0.181066	−	0.983471i	$-0.557955\pi$
$31$	−62.5041	−0.362131	−0.181066	+	0.983471i	$0.557955\pi$
$32$	165.671	0.915213
$33$	−87.6036	−0.462116
$34$	34.7221	0.175141
$35$	−55.1896	−0.266535
$36$	−251.372	−1.16376
$37$	−266.833	−1.18560	−0.592798	−	0.805351i	$-0.701977\pi$
$37$	−266.833	−1.18560	−0.592798	+	0.805351i	$0.701977\pi$
$38$	−19.9169	−0.0850248
$39$	−298.139	−1.22412
$40$	−78.1012	−0.308722
$41$	−29.4174	−0.112054	−0.0560271	−	0.998429i	$-0.517843\pi$
$41$	−29.4174	−0.112054	−0.0560271	+	0.998429i	$0.517843\pi$
$42$	−92.1475	−0.338540
$43$	−384.108	−1.36223	−0.681115	−	0.732176i	$-0.738505\pi$
$43$	−384.108	−1.36223	−0.681115	+	0.732176i	$0.738505\pi$
$44$	75.9127	0.260097
$45$	182.123	0.603319
$46$	−25.2410	−0.0809039
$47$	−443.356	−1.37596	−0.687980	−	0.725729i	$-0.741503\pi$
$47$	−443.356	−1.37596	−0.687980	+	0.725729i	$0.741503\pi$
$48$	309.282	0.930022
$49$	−221.164	−0.644794
$50$	26.2064	0.0741229
$51$	263.795	0.724289
$52$	258.352	0.688981
$53$	−264.474	−0.685441	−0.342720	−	0.939437i	$-0.611348\pi$
$53$	−264.474	−0.685441	−0.342720	+	0.939437i	$0.611348\pi$
$54$	78.6796	0.198277
$55$	−55.0000	−0.134840
$56$	172.415	0.411427
$57$	−151.315	−0.351617
$58$	18.9574	0.0429177
$59$	654.606	1.44445	0.722224	−	0.691659i	$-0.243120\pi$
$59$	654.606	1.44445	0.722224	+	0.691659i	$0.243120\pi$
$60$	−274.803	−0.591281
$61$	91.9399	0.192979	0.0964894	−	0.995334i	$-0.469239\pi$
$61$	91.9399	0.192979	0.0964894	+	0.995334i	$0.469239\pi$
$62$	−65.5203	−0.134211
$63$	−402.052	−0.804029
$64$	−137.016	−0.267609
$65$	−187.180	−0.357183
$66$	−91.8310	−0.171267
$67$	−475.332	−0.866732	−0.433366	−	0.901218i	$-0.642674\pi$
$67$	−475.332	−0.866732	−0.433366	+	0.901218i	$0.642674\pi$
$68$	−228.592	−0.407659
$69$	−191.764	−0.334575
$70$	−57.8528	−0.0987819
$71$	−879.594	−1.47026	−0.735131	−	0.677925i	$-0.762880\pi$
$71$	−879.594	−1.47026	−0.735131	+	0.677925i	$0.762880\pi$
$72$	−568.962	−0.931289
$73$	−94.5002	−0.151513	−0.0757563	−	0.997126i	$-0.524137\pi$
$73$	−94.5002	−0.151513	−0.0757563	+	0.997126i	$0.524137\pi$
$74$	−279.709	−0.439399
$75$	199.099	0.306533
$76$	131.122	0.197904
$77$	121.417	0.179698
$78$	−312.527	−0.453675
$79$	349.276	0.497426	0.248713	−	0.968577i	$-0.419992\pi$
$79$	349.276	0.497426	0.248713	+	0.968577i	$0.419992\pi$
$80$	194.176	0.271370
$81$	−385.710	−0.529095
$82$	−30.8370	−0.0415289
$83$	−1461.89	−1.93329	−0.966645	−	0.256120i	$-0.917556\pi$
$83$	−1461.89	−1.93329	−0.966645	+	0.256120i	$0.917556\pi$
$84$	606.650	0.787987
$85$	165.618	0.211339
$86$	−402.644	−0.504863
$87$	144.026	0.177485
$88$	171.823	0.208140
$89$	1291.14	1.53775	0.768877	−	0.639396i	$-0.220816\pi$
$89$	1291.14	1.53775	0.768877	+	0.639396i	$0.220816\pi$
$90$	190.912	0.223599
$91$	413.216	0.476009
$92$	166.173	0.188312
$93$	−497.780	−0.555026
$94$	−464.751	−0.509951
$95$	−95.0000	−0.102598
$96$	1319.40	1.40271
$97$	−1170.73	−1.22546	−0.612732	−	0.790290i	$-0.709930\pi$
$97$	−1170.73	−1.22546	−0.612732	+	0.790290i	$0.709930\pi$
$98$	−231.837	−0.238970
$99$	−400.671	−0.406757
$100$	−172.529	−0.172529
$101$	1313.45	1.29399	0.646995	−	0.762494i	$-0.276025\pi$
$101$	1313.45	1.29399	0.646995	+	0.762494i	$0.276025\pi$
$102$	276.525	0.268432
$103$	687.581	0.657760	0.328880	−	0.944372i	$-0.393329\pi$
$103$	687.581	0.657760	0.328880	+	0.944372i	$0.393329\pi$
$104$	584.761	0.551351
$105$	−439.528	−0.408509
$106$	−277.237	−0.254034
$107$	1521.15	1.37435	0.687175	−	0.726492i	$-0.258850\pi$
$107$	1521.15	1.37435	0.687175	+	0.726492i	$0.258850\pi$
$108$	−517.984	−0.461510
$109$	334.693	0.294108	0.147054	−	0.989128i	$-0.453021\pi$
$109$	334.693	0.294108	0.147054	+	0.989128i	$0.453021\pi$
$110$	−57.6541	−0.0499737
$111$	−2125.05	−1.81712
$112$	−428.660	−0.361648
$113$	−1982.52	−1.65044	−0.825220	−	0.564811i	$-0.808949\pi$
$113$	−1982.52	−1.65044	−0.825220	+	0.564811i	$0.808949\pi$
$114$	−158.617	−0.130315
$115$	−120.395	−0.0976251
$116$	−124.805	−0.0998954
$117$	−1363.60	−1.07747
$118$	686.195	0.535334
$119$	−365.616	−0.281647
$120$	−621.995	−0.473167
$121$	121.000	0.0909091
$122$	96.3766	0.0715207
$123$	−234.279	−0.171741
$124$	431.351	0.312391
$125$	125.000	0.0894427
$126$	−421.454	−0.297985
$127$	287.279	0.200724	0.100362	−	0.994951i	$-0.468000\pi$
$127$	287.279	0.200724	0.100362	+	0.994951i	$0.468000\pi$
$128$	−1469.00	−1.01439
$129$	−3059.02	−2.08784
$130$	−196.213	−0.132377
$131$	30.3310	0.0202293	0.0101146	−	0.999949i	$-0.496780\pi$
$131$	30.3310	0.0202293	0.0101146	+	0.999949i	$0.496780\pi$
$132$	604.566	0.398642
$133$	209.720	0.136730
$134$	−498.270	−0.321224
$135$	375.288	0.239257
$136$	−517.399	−0.326225
$137$	−1059.27	−0.660578	−0.330289	−	0.943880i	$-0.607146\pi$
$137$	−1059.27	−0.660578	−0.330289	+	0.943880i	$0.607146\pi$
$138$	−201.018	−0.123998
$139$	889.429	0.542737	0.271368	−	0.962476i	$-0.412524\pi$
$139$	889.429	0.542737	0.271368	+	0.962476i	$0.412524\pi$
$140$	380.872	0.229925
$141$	−3530.87	−2.10889
$142$	−922.041	−0.544901
$143$	411.797	0.240813
$144$	1414.56	0.818611
$145$	90.4234	0.0517879
$146$	−99.0605	−0.0561528
$147$	−1761.34	−0.988254
$148$	1841.46	1.02275
$149$	127.560	0.0701349	0.0350674	−	0.999385i	$-0.488835\pi$
$149$	127.560	0.0701349	0.0350674	+	0.999385i	$0.488835\pi$
$150$	208.707	0.113606
$151$	−348.061	−0.187581	−0.0937907	−	0.995592i	$-0.529898\pi$
$151$	−348.061	−0.187581	−0.0937907	+	0.995592i	$0.529898\pi$
$152$	296.785	0.158371
$153$	1206.52	0.637524
$154$	127.276	0.0665988
$155$	−312.521	−0.161950
$156$	2057.51	1.05598
$157$	2607.66	1.32557	0.662783	−	0.748812i	$-0.269375\pi$
$157$	2607.66	1.32557	0.662783	+	0.748812i	$0.269375\pi$
$158$	366.131	0.184353
$159$	−2106.26	−1.05055
$160$	828.356	0.409296
$161$	265.782	0.130103
$162$	−404.323	−0.196090
$163$	−3834.16	−1.84242	−0.921211	−	0.389064i	$-0.872799\pi$
$163$	−3834.16	−1.84242	−0.921211	+	0.389064i	$0.872799\pi$
$164$	203.014	0.0966629
$165$	−438.018	−0.206664
$166$	−1532.43	−0.716506
$167$	3810.29	1.76556	0.882782	−	0.469784i	$-0.155668\pi$
$167$	3810.29	1.76556	0.882782	+	0.469784i	$0.155668\pi$
$168$	1373.10	0.630579
$169$	−795.539	−0.362103
$170$	173.610	0.0783254
$171$	−692.068	−0.309496
$172$	2650.79	1.17512
$173$	3729.48	1.63900	0.819499	−	0.573080i	$-0.194252\pi$
$173$	3729.48	1.63900	0.819499	+	0.573080i	$0.194252\pi$
$174$	150.976	0.0657784
$175$	−275.948	−0.119198
$176$	−427.188	−0.182957
$177$	5213.26	2.21386
$178$	1353.44	0.569914
$179$	444.692	0.185686	0.0928431	−	0.995681i	$-0.470404\pi$
$179$	444.692	0.185686	0.0928431	+	0.995681i	$0.470404\pi$
$180$	−1256.86	−0.520450
$181$	951.390	0.390697	0.195349	−	0.980734i	$-0.437416\pi$
$181$	951.390	0.390697	0.195349	+	0.980734i	$0.437416\pi$
$182$	433.157	0.176416
$183$	732.206	0.295772
$184$	376.120	0.150695
$185$	−1334.16	−0.530215
$186$	−521.801	−0.205701
$187$	−364.360	−0.142485
$188$	3059.67	1.18697
$189$	−828.479	−0.318852
$190$	−99.5844	−0.0380243
$191$	3751.95	1.42137	0.710684	−	0.703511i	$-0.248385\pi$
$191$	3751.95	1.42137	0.710684	+	0.703511i	$0.248385\pi$
$192$	−1091.19	−0.410156
$193$	−2874.79	−1.07219	−0.536093	−	0.844159i	$-0.680100\pi$
$193$	−2874.79	−1.07219	−0.536093	+	0.844159i	$0.680100\pi$
$194$	−1227.23	−0.454175
$195$	−1490.70	−0.547441
$196$	1526.29	0.556228
$197$	5240.28	1.89520	0.947600	−	0.319458i	$-0.103501\pi$
$197$	5240.28	1.89520	0.947600	+	0.319458i	$0.103501\pi$
$198$	−420.006	−0.150750
$199$	−802.103	−0.285727	−0.142863	−	0.989742i	$-0.545631\pi$
$199$	−802.103	−0.285727	−0.142863	+	0.989742i	$0.545631\pi$
$200$	−390.506	−0.138065
$201$	−3785.52	−1.32841
$202$	1376.83	0.479572
$203$	−199.617	−0.0690166
$204$	−1820.49	−0.624804
$205$	−147.087	−0.0501122
$206$	720.761	0.243776
$207$	−877.069	−0.294495
$208$	−1453.84	−0.484643
$209$	209.000	0.0691714
$210$	−460.738	−0.151400
$211$	−2663.80	−0.869115	−0.434558	−	0.900644i	$-0.643095\pi$
$211$	−2663.80	−0.869115	−0.434558	+	0.900644i	$0.643095\pi$
$212$	1825.18	0.591292
$213$	−7005.05	−2.25342
$214$	1594.56	0.509355
$215$	−1920.54	−0.609208
$216$	−1172.42	−0.369319
$217$	689.915	0.215827
$218$	350.845	0.109001
$219$	−752.596	−0.232218
$220$	379.564	0.116319
$221$	−1240.02	−0.377433
$222$	−2227.59	−0.673452
$223$	−3308.15	−0.993409	−0.496705	−	0.867920i	$-0.665457\pi$
$223$	−3308.15	−0.993409	−0.496705	+	0.867920i	$0.665457\pi$
$224$	−1828.66	−0.545459
$225$	910.616	0.269812
$226$	−2078.19	−0.611677
$227$	4226.06	1.23565	0.617827	−	0.786314i	$-0.288013\pi$
$227$	4226.06	1.23565	0.617827	+	0.786314i	$0.288013\pi$
$228$	1044.25	0.303321
$229$	3509.30	1.01267	0.506334	−	0.862337i	$-0.331000\pi$
$229$	3509.30	1.01267	0.506334	+	0.862337i	$0.331000\pi$
$230$	−126.205	−0.0361813
$231$	966.961	0.275417
$232$	−282.487	−0.0799404
$233$	4911.19	1.38087	0.690435	−	0.723394i	$-0.257419\pi$
$233$	4911.19	1.38087	0.690435	+	0.723394i	$0.257419\pi$
$234$	−1429.40	−0.399328
$235$	−2216.78	−0.615348
$236$	−4517.54	−1.24605
$237$	2781.62	0.762388
$238$	−383.259	−0.104382
$239$	−2783.08	−0.753231	−0.376616	−	0.926370i	$-0.622912\pi$
$239$	−2783.08	−0.753231	−0.376616	+	0.926370i	$0.622912\pi$
$240$	1546.41	0.415919
$241$	206.579	0.0552156	0.0276078	−	0.999619i	$-0.491211\pi$
$241$	206.579	0.0552156	0.0276078	+	0.999619i	$0.491211\pi$
$242$	126.839	0.0336922
$243$	−5098.34	−1.34592
$244$	−634.492	−0.166472
$245$	−1105.82	−0.288361
$246$	−245.584	−0.0636499
$247$	711.286	0.183231
$248$	976.329	0.249988
$249$	−11642.4	−2.96309
$250$	131.032	0.0331488
$251$	162.703	0.0409153	0.0204577	−	0.999791i	$-0.493488\pi$
$251$	162.703	0.0409153	0.0204577	+	0.999791i	$0.493488\pi$
$252$	2774.63	0.693591
$253$	264.869	0.0658188
$254$	301.142	0.0743911
$255$	1318.98	0.323912
$256$	−443.758	−0.108339
$257$	−1691.34	−0.410517	−0.205259	−	0.978708i	$-0.565804\pi$
$257$	−1691.34	−0.410517	−0.205259	+	0.978708i	$0.565804\pi$
$258$	−3206.64	−0.773785
$259$	2945.28	0.706605
$260$	1291.76	0.308122
$261$	658.728	0.156223
$262$	31.7947	0.00749727
$263$	−5063.18	−1.18711	−0.593553	−	0.804795i	$-0.702275\pi$
$263$	−5063.18	−1.18711	−0.593553	+	0.804795i	$0.702275\pi$
$264$	1368.39	0.319009
$265$	−1322.37	−0.306538
$266$	219.841	0.0506741
$267$	10282.6	2.35686
$268$	3280.34	0.747682
$269$	−1248.85	−0.283062	−0.141531	−	0.989934i	$-0.545203\pi$
$269$	−1248.85	−0.283062	−0.141531	+	0.989934i	$0.545203\pi$
$270$	393.398	0.0886720
$271$	−4321.40	−0.968659	−0.484329	−	0.874886i	$-0.660936\pi$
$271$	−4321.40	−0.968659	−0.484329	+	0.874886i	$0.660936\pi$
$272$	1286.37	0.286755
$273$	3290.84	0.729563
$274$	−1110.38	−0.244820
$275$	−275.000	−0.0603023
$276$	1323.39	0.288620
$277$	4149.72	0.900118	0.450059	−	0.892999i	$-0.351403\pi$
$277$	4149.72	0.900118	0.450059	+	0.892999i	$0.351403\pi$
$278$	932.350	0.201146
$279$	−2276.69	−0.488537
$280$	862.074	0.183996
$281$	−6315.65	−1.34078	−0.670391	−	0.742008i	$-0.733874\pi$
$281$	−6315.65	−1.34078	−0.670391	+	0.742008i	$0.733874\pi$
$282$	−3701.26	−0.781584
$283$	−1813.27	−0.380874	−0.190437	−	0.981699i	$-0.560991\pi$
$283$	−1813.27	−0.380874	−0.190437	+	0.981699i	$0.560991\pi$
$284$	6070.22	1.26831
$285$	−756.576	−0.157248
$286$	431.669	0.0892487
$287$	324.706	0.0667833
$288$	6034.52	1.23468
$289$	−3815.82	−0.776679
$290$	94.7869	0.0191934
$291$	−9323.68	−1.87823
$292$	652.161	0.130701
$293$	−4938.55	−0.984686	−0.492343	−	0.870401i	$-0.663859\pi$
$293$	−4938.55	−0.984686	−0.492343	+	0.870401i	$0.663859\pi$
$294$	−1846.34	−0.366261
$295$	3273.03	0.645977
$296$	4167.99	0.818445
$297$	−825.633	−0.161307
$298$	133.715	0.0259930
$299$	901.423	0.174350
$300$	−1374.01	−0.264429
$301$	4239.75	0.811878
$302$	−364.857	−0.0695204
$303$	10460.3	1.98325
$304$	−737.870	−0.139210
$305$	459.700	0.0863027
$306$	1264.74	0.236276
$307$	929.663	0.172830	0.0864148	−	0.996259i	$-0.472459\pi$
$307$	929.663	0.172830	0.0864148	+	0.996259i	$0.472459\pi$
$308$	−837.918	−0.155016
$309$	5475.86	1.00813
$310$	−327.602	−0.0600211
$311$	3928.13	0.716219	0.358109	−	0.933680i	$-0.383421\pi$
$311$	3928.13	0.716219	0.358109	+	0.933680i	$0.383421\pi$
$312$	4657.01	0.845036
$313$	−6147.26	−1.11011	−0.555054	−	0.831814i	$-0.687302\pi$
$313$	−6147.26	−1.11011	−0.555054	+	0.831814i	$0.687302\pi$
$314$	2733.50	0.491274
$315$	−2010.26	−0.359573
$316$	−2410.41	−0.429102
$317$	1558.00	0.276044	0.138022	−	0.990429i	$-0.455926\pi$
$317$	1558.00	0.276044	0.138022	+	0.990429i	$0.455926\pi$
$318$	−2207.91	−0.389350
$319$	−198.931	−0.0349154
$320$	−685.080	−0.119679
$321$	12114.4	2.10642
$322$	278.608	0.0482180
$323$	−629.349	−0.108415
$324$	2661.85	0.456421
$325$	−935.902	−0.159737
$326$	−4019.19	−0.682828
$327$	2665.48	0.450769
$328$	459.506	0.0773536
$329$	4893.73	0.820061
$330$	−459.155	−0.0765929
$331$	−1182.77	−0.196408	−0.0982038	−	0.995166i	$-0.531310\pi$
$331$	−1182.77	−0.196408	−0.0982038	+	0.995166i	$0.531310\pi$
$332$	10088.7	1.66774
$333$	−9719.30	−1.59944
$334$	3994.16	0.654344
$335$	−2376.66	−0.387614
$336$	−3413.83	−0.554285
$337$	2966.74	0.479550	0.239775	−	0.970828i	$-0.422926\pi$
$337$	2966.74	0.479550	0.239775	+	0.970828i	$0.422926\pi$
$338$	−833.929	−0.134201
$339$	−15788.7	−2.52957
$340$	−1142.96	−0.182311
$341$	687.545	0.109187
$342$	−725.465	−0.114704
$343$	6227.20	0.980283
$344$	5999.86	0.940380
$345$	−958.821	−0.149627
$346$	3909.45	0.607437
$347$	−4718.76	−0.730018	−0.365009	−	0.931004i	$-0.618934\pi$
$347$	−4718.76	−0.730018	−0.365009	+	0.931004i	$0.618934\pi$
$348$	−993.943	−0.153106
$349$	−1934.27	−0.296674	−0.148337	−	0.988937i	$-0.547392\pi$
$349$	−1934.27	−0.296674	−0.148337	+	0.988937i	$0.547392\pi$
$350$	−289.264	−0.0441766
$351$	−2809.86	−0.427291
$352$	−1822.38	−0.275947
$353$	6422.77	0.968412	0.484206	−	0.874954i	$-0.339108\pi$
$353$	6422.77	0.968412	0.484206	+	0.874954i	$0.339108\pi$
$354$	5464.83	0.820487
$355$	−4397.97	−0.657521
$356$	−8910.33	−1.32654
$357$	−2911.75	−0.431670
$358$	466.151	0.0688180
$359$	2550.80	0.375002	0.187501	−	0.982264i	$-0.439961\pi$
$359$	2550.80	0.375002	0.187501	+	0.982264i	$0.439961\pi$
$360$	−2844.81	−0.416485
$361$	361.000	0.0526316
$362$	997.301	0.144798
$363$	963.639	0.139333
$364$	−2851.67	−0.410627
$365$	−472.501	−0.0677585
$366$	767.540	0.109617
$367$	−6855.57	−0.975089	−0.487544	−	0.873098i	$-0.662107\pi$
$367$	−6855.57	−0.975089	−0.487544	+	0.873098i	$0.662107\pi$
$368$	−935.114	−0.132462
$369$	−1071.52	−0.151168
$370$	−1398.55	−0.196505
$371$	2919.25	0.408517
$372$	3435.26	0.478790
$373$	−6756.43	−0.937894	−0.468947	−	0.883226i	$-0.655367\pi$
$373$	−6756.43	−0.937894	−0.468947	+	0.883226i	$0.655367\pi$
$374$	−381.943	−0.0528070
$375$	995.495	0.137086
$376$	6925.33	0.949858
$377$	−677.019	−0.0924888
$378$	−868.458	−0.118171
$379$	−13568.5	−1.83896	−0.919481	−	0.393136i	$-0.871390\pi$
$379$	−13568.5	−1.83896	−0.919481	+	0.393136i	$0.871390\pi$
$380$	655.610	0.0885055
$381$	2287.88	0.307642
$382$	3933.00	0.526780
$383$	3751.89	0.500556	0.250278	−	0.968174i	$-0.419478\pi$
$383$	3751.89	0.500556	0.250278	+	0.968174i	$0.419478\pi$
$384$	−11699.0	−1.55472
$385$	607.085	0.0803635
$386$	−3013.52	−0.397368
$387$	−13991.0	−1.83773
$388$	8079.42	1.05714
$389$	10050.8	1.31001	0.655004	−	0.755625i	$-0.272667\pi$
$389$	10050.8	1.31001	0.655004	+	0.755625i	$0.272667\pi$
$390$	−1562.63	−0.202890
$391$	−797.584	−0.103160
$392$	3454.64	0.445117
$393$	241.555	0.0310047
$394$	5493.16	0.702389
$395$	1746.38	0.222456
$396$	2765.10	0.350887
$397$	−9742.71	−1.23167	−0.615835	−	0.787875i	$-0.711181\pi$
$397$	−9742.71	−1.23167	−0.615835	+	0.787875i	$0.711181\pi$
$398$	−840.810	−0.105894
$399$	1670.20	0.209561
$400$	970.881	0.121360
$401$	1232.38	0.153472	0.0767358	−	0.997051i	$-0.475550\pi$
$401$	1232.38	0.153472	0.0767358	+	0.997051i	$0.475550\pi$
$402$	−3968.20	−0.492328
$403$	2339.91	0.289229
$404$	−9064.32	−1.11625
$405$	−1928.55	−0.236618
$406$	−209.250	−0.0255786
$407$	2935.16	0.357471
$408$	−4120.55	−0.499994
$409$	1746.93	0.211198	0.105599	−	0.994409i	$-0.466324\pi$
$409$	1746.93	0.211198	0.105599	+	0.994409i	$0.466324\pi$
$410$	−154.185	−0.0185723
$411$	−8435.94	−1.01244
$412$	−4745.10	−0.567414
$413$	−7225.49	−0.860879
$414$	−919.393	−0.109144
$415$	−7309.44	−0.864593
$416$	−6202.08	−0.730967
$417$	7083.38	0.831833
$418$	219.086	0.0256360
$419$	−4130.83	−0.481633	−0.240817	−	0.970571i	$-0.577415\pi$
$419$	−4130.83	−0.481633	−0.240817	+	0.970571i	$0.577415\pi$
$420$	3033.25	0.352399
$421$	13994.9	1.62012	0.810058	−	0.586350i	$-0.199436\pi$
$421$	13994.9	1.62012	0.810058	+	0.586350i	$0.199436\pi$
$422$	−2792.34	−0.322107
$423$	−16149.1	−1.85626
$424$	4131.15	0.473176
$425$	828.091	0.0945137
$426$	−7343.09	−0.835150
$427$	−1014.82	−0.115014
$428$	−10497.7	−1.18558
$429$	3279.53	0.369085
$430$	−2013.22	−0.225781
$431$	−138.145	−0.0154390	−0.00771951	−	0.999970i	$-0.502457\pi$
$431$	−138.145	−0.0154390	−0.00771951	+	0.999970i	$0.502457\pi$
$432$	2914.88	0.324635
$433$	8200.97	0.910193	0.455096	−	0.890442i	$-0.349605\pi$
$433$	8200.97	0.910193	0.455096	+	0.890442i	$0.349605\pi$
$434$	723.208	0.0799887
$435$	720.128	0.0793736
$436$	−2309.77	−0.253711
$437$	457.501	0.0500806
$438$	−788.914	−0.0860634
$439$	1012.27	0.110053	0.0550264	−	0.998485i	$-0.482476\pi$
$439$	1012.27	0.110053	0.0550264	+	0.998485i	$0.482476\pi$
$440$	859.113	0.0930832
$441$	−8055.84	−0.869867
$442$	−1299.86	−0.139882
$443$	15574.1	1.67031	0.835157	−	0.550011i	$-0.185377\pi$
$443$	15574.1	1.67031	0.835157	+	0.550011i	$0.185377\pi$
$444$	14665.3	1.56753
$445$	6455.68	0.687705
$446$	−3467.79	−0.368172
$447$	1015.88	0.107493
$448$	1512.37	0.159493
$449$	7488.49	0.787090	0.393545	−	0.919305i	$-0.371249\pi$
$449$	7488.49	0.787090	0.393545	+	0.919305i	$0.371249\pi$
$450$	954.560	0.0999964
$451$	323.591	0.0337856
$452$	13681.7	1.42374
$453$	−2771.94	−0.287499
$454$	4429.99	0.457951
$455$	2066.08	0.212878
$456$	2363.58	0.242730
$457$	16782.0	1.71779	0.858895	−	0.512151i	$-0.171151\pi$
$457$	16782.0	1.71779	0.858895	+	0.512151i	$0.171151\pi$
$458$	3678.65	0.375310
$459$	2486.18	0.252821
$460$	830.865	0.0842158
$461$	−15154.1	−1.53101	−0.765504	−	0.643431i	$-0.777510\pi$
$461$	−15154.1	−1.53101	−0.765504	+	0.643431i	$0.777510\pi$
$462$	1013.62	0.102074
$463$	−4679.58	−0.469716	−0.234858	−	0.972030i	$-0.575462\pi$
$463$	−4679.58	−0.469716	−0.234858	+	0.972030i	$0.575462\pi$
$464$	702.323	0.0702683
$465$	−2488.90	−0.248215
$466$	5148.19	0.511771
$467$	158.295	0.0156853	0.00784264	−	0.999969i	$-0.497504\pi$
$467$	158.295	0.0156853	0.00784264	+	0.999969i	$0.497504\pi$
$468$	9410.40	0.929478
$469$	5246.67	0.516565
$470$	−2323.76	−0.228057
$471$	20767.3	2.03165
$472$	−10225.1	−0.997137
$473$	4225.19	0.410728
$474$	2915.86	0.282552
$475$	−475.000	−0.0458831
$476$	2523.17	0.242961
$477$	−9633.39	−0.924702
$478$	−2917.38	−0.279159
$479$	6768.21	0.645610	0.322805	−	0.946465i	$-0.395374\pi$
$479$	6768.21	0.645610	0.322805	+	0.946465i	$0.395374\pi$
$480$	6596.99	0.627313
$481$	9989.18	0.946918
$482$	216.548	0.0204637
$483$	2116.68	0.199404
$484$	−835.040	−0.0784223
$485$	−5853.67	−0.548045
$486$	−5344.36	−0.498817
$487$	−9326.01	−0.867766	−0.433883	−	0.900969i	$-0.642857\pi$
$487$	−9326.01	−0.867766	−0.433883	+	0.900969i	$0.642857\pi$
$488$	−1436.12	−0.133218
$489$	−30535.1	−2.82381
$490$	−1159.19	−0.106871
$491$	7527.05	0.691835	0.345918	−	0.938265i	$-0.387568\pi$
$491$	7527.05	0.691835	0.345918	+	0.938265i	$0.387568\pi$
$492$	1616.79	0.148152
$493$	599.030	0.0547241
$494$	745.610	0.0679080
$495$	−2003.36	−0.181907
$496$	−2427.36	−0.219741
$497$	9708.89	0.876264
$498$	−12204.2	−1.09816
$499$	13375.8	1.19996	0.599981	−	0.800014i	$-0.295175\pi$
$499$	13375.8	1.19996	0.599981	+	0.800014i	$0.295175\pi$
$500$	−862.645	−0.0771573
$501$	30345.0	2.70602
$502$	170.555	0.0151638
$503$	7937.23	0.703586	0.351793	−	0.936078i	$-0.385572\pi$
$503$	7937.23	0.703586	0.351793	+	0.936078i	$0.385572\pi$
$504$	6280.15	0.555040
$505$	6567.24	0.578690
$506$	277.651	0.0243934
$507$	−6335.64	−0.554982
$508$	−1982.56	−0.173153
$509$	10465.8	0.911368	0.455684	−	0.890142i	$-0.349395\pi$
$509$	10465.8	0.911368	0.455684	+	0.890142i	$0.349395\pi$
$510$	1382.63	0.120047
$511$	1043.09	0.0903001
$512$	11286.8	0.974241
$513$	−1426.09	−0.122736
$514$	−1772.96	−0.152144
$515$	3437.90	0.294159
$516$	21110.8	1.80107
$517$	4876.92	0.414868
$518$	3087.41	0.261878
$519$	29701.4	2.51204
$520$	2923.80	0.246572
$521$	5125.90	0.431036	0.215518	−	0.976500i	$-0.430856\pi$
$521$	5125.90	0.431036	0.215518	+	0.976500i	$0.430856\pi$
$522$	690.516	0.0578986
$523$	22097.2	1.84750	0.923751	−	0.382993i	$-0.125107\pi$
$523$	22097.2	1.84750	0.923751	+	0.382993i	$0.125107\pi$
$524$	−209.319	−0.0174507
$525$	−2197.64	−0.182691
$526$	−5307.51	−0.439959
$527$	−2070.36	−0.171132
$528$	−3402.11	−0.280412
$529$	−11587.2	−0.952347
$530$	−1386.19	−0.113608
$531$	23843.8	1.94865
$532$	−1447.31	−0.117949
$533$	1101.27	0.0894960
$534$	10778.8	0.873488
$535$	7605.77	0.614628
$536$	7424.80	0.598325
$537$	3541.51	0.284595
$538$	−1309.12	−0.104907
$539$	2432.81	0.194413
$540$	−2589.92	−0.206393
$541$	−10895.6	−0.865875	−0.432937	−	0.901424i	$-0.642523\pi$
$541$	−10895.6	−0.865875	−0.432937	+	0.901424i	$0.642523\pi$
$542$	−4529.94	−0.358999
$543$	7576.83	0.598808
$544$	5487.64	0.432501
$545$	1673.47	0.131529
$546$	3449.64	0.270387
$547$	−22760.8	−1.77913	−0.889564	−	0.456811i	$-0.848992\pi$
$547$	−22760.8	−1.77913	−0.889564	+	0.456811i	$0.848992\pi$
$548$	7310.16	0.569844
$549$	3348.88	0.260340
$550$	−288.271	−0.0223489
$551$	−343.609	−0.0265667
$552$	2995.40	0.230965
$553$	−3855.28	−0.296462
$554$	4349.98	0.333597
$555$	−10625.2	−0.812641
$556$	−6138.09	−0.468189
$557$	2987.17	0.227236	0.113618	−	0.993525i	$-0.463756\pi$
$557$	2987.17	0.227236	0.113618	+	0.993525i	$0.463756\pi$
$558$	−2386.56	−0.181059
$559$	14379.5	1.08799
$560$	−2143.30	−0.161734
$561$	−2901.75	−0.218381
$562$	−6620.42	−0.496914
$563$	4745.68	0.355251	0.177626	−	0.984098i	$-0.443158\pi$
$563$	4745.68	0.355251	0.177626	+	0.984098i	$0.443158\pi$
$564$	24367.1	1.81922
$565$	−9912.60	−0.738099
$566$	−1900.77	−0.141158
$567$	4257.44	0.315336
$568$	13739.5	1.01496
$569$	−4143.95	−0.305314	−0.152657	−	0.988279i	$-0.548783\pi$
$569$	−4143.95	−0.305314	−0.152657	+	0.988279i	$0.548783\pi$
$570$	−793.086	−0.0582784
$571$	8334.49	0.610836	0.305418	−	0.952218i	$-0.401204\pi$
$571$	8334.49	0.610836	0.305418	+	0.952218i	$0.401204\pi$
$572$	−2841.88	−0.207736
$573$	29880.4	2.17848
$574$	340.376	0.0247509
$575$	−601.975	−0.0436593
$576$	−4990.76	−0.361022
$577$	−1220.01	−0.0880240	−0.0440120	−	0.999031i	$-0.514014\pi$
$577$	−1220.01	−0.0880240	−0.0440120	+	0.999031i	$0.514014\pi$
$578$	−3999.96	−0.287849
$579$	−22894.7	−1.64330
$580$	−624.026	−0.0446746
$581$	16136.2	1.15222
$582$	−9773.61	−0.696098
$583$	2909.22	0.206668
$584$	1476.12	0.104593
$585$	−6817.98	−0.481861
$586$	−5176.86	−0.364939
$587$	4883.50	0.343379	0.171690	−	0.985151i	$-0.445077\pi$
$587$	4883.50	0.343379	0.171690	+	0.985151i	$0.445077\pi$
$588$	12155.3	0.852512
$589$	1187.58	0.0830786
$590$	3430.98	0.239409
$591$	41733.4	2.90471
$592$	−10362.5	−0.719421
$593$	−4671.28	−0.323485	−0.161742	−	0.986833i	$-0.551711\pi$
$593$	−4671.28	−0.323485	−0.161742	+	0.986833i	$0.551711\pi$
$594$	−865.475	−0.0597826
$595$	−1828.08	−0.125956
$596$	−880.310	−0.0605015
$597$	−6387.92	−0.437923
$598$	944.923	0.0646167
$599$	−7427.52	−0.506645	−0.253322	−	0.967382i	$-0.581523\pi$
$599$	−7427.52	−0.506645	−0.253322	+	0.967382i	$0.581523\pi$
$600$	−3109.97	−0.211607
$601$	−14580.4	−0.989597	−0.494799	−	0.869008i	$-0.664758\pi$
$601$	−14580.4	−0.989597	−0.494799	+	0.869008i	$0.664758\pi$
$602$	4444.35	0.300894
$603$	−17313.8	−1.16927
$604$	2402.02	0.161816
$605$	605.000	0.0406558
$606$	10965.0	0.735023
$607$	−1039.72	−0.0695235	−0.0347617	−	0.999396i	$-0.511067\pi$
$607$	−1039.72	−0.0695235	−0.0347617	+	0.999396i	$0.511067\pi$
$608$	−3147.75	−0.209964
$609$	−1589.74	−0.105779
$610$	481.883	0.0319850
$611$	16597.5	1.09896
$612$	−8326.37	−0.549957
$613$	5567.16	0.366811	0.183406	−	0.983037i	$-0.441288\pi$
$613$	5567.16	0.366811	0.183406	+	0.983037i	$0.441288\pi$
$614$	974.525	0.0640532
$615$	−1171.39	−0.0768051
$616$	−1896.56	−0.124050
$617$	7260.89	0.473764	0.236882	−	0.971538i	$-0.423875\pi$
$617$	7260.89	0.473764	0.236882	+	0.971538i	$0.423875\pi$
$618$	5740.11	0.373626
$619$	7721.80	0.501398	0.250699	−	0.968065i	$-0.419340\pi$
$619$	7721.80	0.501398	0.250699	+	0.968065i	$0.419340\pi$
$620$	2156.75	0.139705
$621$	−1807.31	−0.116787
$622$	4117.69	0.265441
$623$	−14251.4	−0.916488
$624$	−11578.3	−0.742795
$625$	625.000	0.0400000
$626$	−6443.91	−0.411422
$627$	1664.47	0.106017
$628$	−17995.9	−1.14349
$629$	−8838.48	−0.560276
$630$	−2107.27	−0.133263
$631$	−13258.3	−0.836458	−0.418229	−	0.908342i	$-0.637349\pi$
$631$	−13258.3	−0.836458	−0.418229	+	0.908342i	$0.637349\pi$
$632$	−5455.78	−0.343385
$633$	−21214.4	−1.33206
$634$	1633.18	0.102306
$635$	1436.40	0.0897663
$636$	14535.7	0.906252
$637$	8279.53	0.514987
$638$	−208.531	−0.0129402
$639$	−32038.9	−1.98347
$640$	−7344.99	−0.453650
$641$	13365.3	0.823551	0.411775	−	0.911285i	$-0.364909\pi$
$641$	13365.3	0.823551	0.411775	+	0.911285i	$0.364909\pi$
$642$	12699.0	0.780670
$643$	−24397.8	−1.49635	−0.748177	−	0.663499i	$-0.769070\pi$
$643$	−24397.8	−1.49635	−0.748177	+	0.663499i	$0.769070\pi$
$644$	−1834.20	−0.112232
$645$	−15295.1	−0.933712
$646$	−659.720	−0.0401801
$647$	−919.667	−0.0558823	−0.0279412	−	0.999610i	$-0.508895\pi$
$647$	−919.667	−0.0558823	−0.0279412	+	0.999610i	$0.508895\pi$
$648$	6024.89	0.365247
$649$	−7200.67	−0.435518
$650$	−981.066	−0.0592009
$651$	5494.45	0.330791
$652$	26460.2	1.58936
$653$	−2309.51	−0.138404	−0.0692022	−	0.997603i	$-0.522045\pi$
$653$	−2309.51	−0.138404	−0.0692022	+	0.997603i	$0.522045\pi$
$654$	2794.11	0.167062
$655$	151.655	0.00904681
$656$	−1142.43	−0.0679946
$657$	−3442.14	−0.204400
$658$	5129.88	0.303927
$659$	−27468.2	−1.62369	−0.811844	−	0.583875i	$-0.801536\pi$
$659$	−27468.2	−1.62369	−0.811844	+	0.583875i	$0.801536\pi$
$660$	3022.83	0.178278
$661$	−937.888	−0.0551885	−0.0275942	−	0.999619i	$-0.508785\pi$
$661$	−937.888	−0.0551885	−0.0275942	+	0.999619i	$0.508785\pi$
$662$	−1239.85	−0.0727915
$663$	−9875.47	−0.578479
$664$	22835.0	1.33460
$665$	1048.60	0.0611474
$666$	−10188.3	−0.592777
$667$	−435.461	−0.0252790
$668$	−26295.4	−1.52305
$669$	−26346.0	−1.52256
$670$	−2491.35	−0.143656
$671$	−1011.34	−0.0581853
$672$	−14563.4	−0.836006
$673$	5516.01	0.315939	0.157969	−	0.987444i	$-0.449505\pi$
$673$	5516.01	0.315939	0.157969	+	0.987444i	$0.449505\pi$
$674$	3109.90	0.177728
$675$	1876.44	0.106999
$676$	5490.14	0.312366
$677$	−15528.2	−0.881529	−0.440765	−	0.897623i	$-0.645293\pi$
$677$	−15528.2	−0.881529	−0.440765	+	0.897623i	$0.645293\pi$
$678$	−16550.6	−0.937496
$679$	12922.5	0.730366
$680$	−2587.00	−0.145892
$681$	33656.2	1.89384
$682$	720.724	0.0404662
$683$	−23734.0	−1.32966	−0.664828	−	0.746997i	$-0.731495\pi$
$683$	−23734.0	−1.32966	−0.664828	+	0.746997i	$0.731495\pi$
$684$	4776.07	0.266985
$685$	−5296.33	−0.295419
$686$	6527.70	0.363307
$687$	27947.9	1.55208
$688$	−14916.9	−0.826603
$689$	9900.89	0.547451
$690$	−1005.09	−0.0554538
$691$	28121.8	1.54820	0.774098	−	0.633066i	$-0.218204\pi$
$691$	28121.8	1.54820	0.774098	+	0.633066i	$0.218204\pi$
$692$	−25737.7	−1.41387
$693$	4422.57	0.242424
$694$	−4946.47	−0.270556
$695$	4447.14	0.242719
$696$	−2249.71	−0.122522
$697$	−974.411	−0.0529533
$698$	−2027.61	−0.109952
$699$	39112.5	2.11641
$700$	1904.36	0.102826
$701$	−27488.2	−1.48105	−0.740525	−	0.672029i	$-0.765423\pi$
$701$	−27488.2	−1.48105	−0.740525	+	0.672029i	$0.765423\pi$
$702$	−2945.46	−0.158360
$703$	5069.83	0.271994
$704$	1507.18	0.0806873
$705$	−17654.4	−0.943123
$706$	6732.71	0.358908
$707$	−14497.7	−0.771207
$708$	−35977.5	−1.90977
$709$	−10774.0	−0.570697	−0.285349	−	0.958424i	$-0.592109\pi$
$709$	−10774.0	−0.570697	−0.285349	+	0.958424i	$0.592109\pi$
$710$	−4610.20	−0.243687
$711$	12722.3	0.671058
$712$	−20167.9	−1.06155
$713$	1505.04	0.0790519
$714$	−3052.26	−0.159983
$715$	2058.98	0.107695
$716$	−3068.89	−0.160181
$717$	−22164.3	−1.15445
$718$	2673.89	0.138981
$719$	14116.2	0.732191	0.366096	−	0.930577i	$-0.380694\pi$
$719$	14116.2	0.732191	0.366096	+	0.930577i	$0.380694\pi$
$720$	7072.80	0.366094
$721$	−7589.46	−0.392020
$722$	378.421	0.0195060
$723$	1645.19	0.0846269
$724$	−6565.69	−0.337033
$725$	452.117	0.0231603
$726$	1010.14	0.0516389
$727$	16455.2	0.839461	0.419731	−	0.907649i	$-0.362125\pi$
$727$	16455.2	0.839461	0.419731	+	0.907649i	$0.362125\pi$
$728$	−6454.54	−0.328600
$729$	−30188.8	−1.53375
$730$	−495.302	−0.0251123
$731$	−12723.1	−0.643748
$732$	−5053.07	−0.255146
$733$	26.0178	0.00131103	0.000655517	−	1.00000i	$-0.499791\pi$
$733$	26.0178	0.00131103	0.000655517		1.00000i	$0.499791\pi$
$734$	−7186.39	−0.361382
$735$	−8806.72	−0.441960
$736$	−3989.20	−0.199788
$737$	5228.65	0.261329
$738$	−1123.23	−0.0560251
$739$	2269.93	0.112992	0.0564958	−	0.998403i	$-0.482007\pi$
$739$	2269.93	0.112992	0.0564958	+	0.998403i	$0.482007\pi$
$740$	9207.28	0.457387
$741$	5664.65	0.280831
$742$	3060.12	0.151402
$743$	7462.91	0.368490	0.184245	−	0.982880i	$-0.441016\pi$
$743$	7462.91	0.368490	0.184245	+	0.982880i	$0.441016\pi$
$744$	7775.44	0.383147
$745$	637.798	0.0313653
$746$	−7082.47	−0.347597
$747$	−53248.8	−2.60813
$748$	2514.51	0.122914
$749$	−16790.4	−0.819101
$750$	1043.53	0.0508060
$751$	16795.4	0.816075	0.408038	−	0.912965i	$-0.366213\pi$
$751$	16795.4	0.816075	0.408038	+	0.912965i	$0.366213\pi$
$752$	−17217.8	−0.834934
$753$	1295.76	0.0627095
$754$	−709.690	−0.0342777
$755$	−1740.30	−0.0838889
$756$	5717.46	0.275056
$757$	−27937.5	−1.34135	−0.670677	−	0.741750i	$-0.733996\pi$
$757$	−27937.5	−1.34135	−0.670677	+	0.741750i	$0.733996\pi$
$758$	−14223.3	−0.681546
$759$	2109.41	0.100878
$760$	1483.92	0.0708257
$761$	14073.0	0.670365	0.335182	−	0.942153i	$-0.391202\pi$
$761$	14073.0	0.670365	0.335182	+	0.942153i	$0.391202\pi$
$762$	2398.28	0.114017
$763$	−3694.32	−0.175286
$764$	−25892.8	−1.22614
$765$	6032.59	0.285109
$766$	3932.95	0.185513
$767$	−24505.9	−1.15366
$768$	−3534.07	−0.166048
$769$	−2610.36	−0.122408	−0.0612042	−	0.998125i	$-0.519494\pi$
$769$	−2610.36	−0.122408	−0.0612042	+	0.998125i	$0.519494\pi$
$770$	636.381	0.0297839
$771$	−13469.8	−0.629186
$772$	19839.4	0.924916
$773$	−30962.1	−1.44066	−0.720330	−	0.693631i	$-0.756010\pi$
$773$	−30962.1	−1.44066	−0.720330	+	0.693631i	$0.756010\pi$
$774$	−14666.2	−0.681091
$775$	−1562.60	−0.0724262
$776$	18287.2	0.845967
$777$	23456.1	1.08299
$778$	10535.8	0.485509
$779$	558.930	0.0257070
$780$	10287.5	0.472247
$781$	9675.54	0.443301
$782$	−836.073	−0.0382326
$783$	1357.39	0.0619530
$784$	−8588.98	−0.391262
$785$	13038.3	0.592811
$786$	253.212	0.0114908
$787$	13229.1	0.599196	0.299598	−	0.954066i	$-0.403148\pi$
$787$	13229.1	0.599196	0.299598	+	0.954066i	$0.403148\pi$
$788$	−36164.0	−1.63489
$789$	−40322.9	−1.81944
$790$	1830.66	0.0824454
$791$	21882.9	0.983648
$792$	6258.58	0.280794
$793$	−3441.87	−0.154129
$794$	−10212.9	−0.456475
$795$	−10531.3	−0.469821
$796$	5535.44	0.246481
$797$	36863.8	1.63837	0.819187	−	0.573527i	$-0.194425\pi$
$797$	36863.8	1.63837	0.819187	+	0.573527i	$0.194425\pi$
$798$	1750.80	0.0776664
$799$	−14685.6	−0.650236
$800$	4141.78	0.183043
$801$	47029.2	2.07452
$802$	1291.85	0.0568788
$803$	1039.50	0.0456827
$804$	26124.5	1.14595
$805$	1328.91	0.0581837
$806$	2452.83	0.107192
$807$	−9945.80	−0.433840
$808$	−20516.4	−0.893272
$809$	−21740.6	−0.944821	−0.472411	−	0.881379i	$-0.656616\pi$
$809$	−21740.6	−0.944821	−0.472411	+	0.881379i	$0.656616\pi$
$810$	−2021.62	−0.0876943
$811$	15339.3	0.664164	0.332082	−	0.943251i	$-0.392249\pi$
$811$	15339.3	0.664164	0.332082	+	0.943251i	$0.392249\pi$
$812$	1377.59	0.0595368
$813$	−34415.5	−1.48463
$814$	3076.80	0.132484
$815$	−19170.8	−0.823956
$816$	10244.6	0.439499
$817$	7298.05	0.312517
$818$	1831.23	0.0782733
$819$	15051.3	0.642166
$820$	1015.07	0.0432290
$821$	−9498.02	−0.403755	−0.201878	−	0.979411i	$-0.564704\pi$
$821$	−9498.02	−0.403755	−0.201878	+	0.979411i	$0.564704\pi$
$822$	−8843.03	−0.375227
$823$	1535.86	0.0650508	0.0325254	−	0.999471i	$-0.489645\pi$
$823$	1535.86	0.0650508	0.0325254	+	0.999471i	$0.489645\pi$
$824$	−10740.2	−0.454067
$825$	−2190.09	−0.0924232
$826$	−7574.16	−0.319054
$827$	−6547.21	−0.275295	−0.137647	−	0.990481i	$-0.543954\pi$
$827$	−6547.21	−0.275295	−0.137647	+	0.990481i	$0.543954\pi$
$828$	6052.79	0.254045
$829$	28305.0	1.18585	0.592927	−	0.805256i	$-0.297972\pi$
$829$	28305.0	1.18585	0.592927	+	0.805256i	$0.297972\pi$
$830$	−7662.17	−0.320431
$831$	33048.2	1.37958
$832$	5129.34	0.213736
$833$	−7325.78	−0.304710
$834$	7425.20	0.308290
$835$	19051.4	0.789584
$836$	−1442.34	−0.0596704
$837$	−4691.41	−0.193738
$838$	−4330.17	−0.178500
$839$	26278.6	1.08133	0.540666	−	0.841238i	$-0.318172\pi$
$839$	26278.6	1.08133	0.540666	+	0.841238i	$0.318172\pi$
$840$	6865.52	0.282004
$841$	−24061.9	−0.986590
$842$	14670.2	0.600438
$843$	−50297.6	−2.05497
$844$	18383.3	0.749738
$845$	−3977.70	−0.161937
$846$	−16928.4	−0.687955
$847$	−1335.59	−0.0541810
$848$	−10270.9	−0.415926
$849$	−14440.8	−0.583753
$850$	868.052	0.0350282
$851$	6425.07	0.258811
$852$	48343.0	1.94390
$853$	−45415.7	−1.82298	−0.911492	−	0.411319i	$-0.865068\pi$
$853$	−45415.7	−1.82298	−0.911492	+	0.411319i	$0.865068\pi$
$854$	−1063.80	−0.0426257
$855$	−3460.34	−0.138411
$856$	−23760.8	−0.948747
$857$	33414.4	1.33187	0.665936	−	0.746009i	$-0.268033\pi$
$857$	33414.4	1.33187	0.665936	+	0.746009i	$0.268033\pi$
$858$	3437.79	0.136788
$859$	−16878.6	−0.670419	−0.335210	−	0.942144i	$-0.608807\pi$
$859$	−16878.6	−0.670419	−0.335210	+	0.942144i	$0.608807\pi$
$860$	13254.0	0.525530
$861$	2585.95	0.102356
$862$	−144.812	−0.00572193
$863$	−3399.18	−0.134078	−0.0670390	−	0.997750i	$-0.521355\pi$
$863$	−3399.18	−0.134078	−0.0670390	+	0.997750i	$0.521355\pi$
$864$	12434.9	0.489633
$865$	18647.4	0.732983
$866$	8596.72	0.337331
$867$	−30389.1	−1.19039
$868$	−4761.21	−0.186182
$869$	−3842.04	−0.149980
$870$	754.879	0.0294170
$871$	17794.6	0.692246
$872$	−5227.99	−0.203030
$873$	−42643.6	−1.65323
$874$	479.578	0.0185606
$875$	−1379.74	−0.0533071
$876$	5193.78	0.200322
$877$	26774.6	1.03092	0.515458	−	0.856915i	$-0.327622\pi$
$877$	26774.6	1.03092	0.515458	+	0.856915i	$0.327622\pi$
$878$	1061.12	0.0407872
$879$	−39330.4	−1.50919
$880$	−2135.94	−0.0818210
$881$	−38922.1	−1.48844	−0.744222	−	0.667933i	$-0.767179\pi$
$881$	−38922.1	−1.48844	−0.744222	+	0.667933i	$0.767179\pi$
$882$	−8444.59	−0.322385
$883$	−23186.3	−0.883672	−0.441836	−	0.897096i	$-0.645673\pi$
$883$	−23186.3	−0.883672	−0.441836	+	0.897096i	$0.645673\pi$
$884$	8557.57	0.325591
$885$	26066.3	0.990066
$886$	16325.7	0.619043
$887$	−18041.1	−0.682931	−0.341466	−	0.939894i	$-0.610923\pi$
$887$	−18041.1	−0.682931	−0.341466	+	0.939894i	$0.610923\pi$
$888$	33193.7	1.25440
$889$	−3170.96	−0.119629
$890$	6767.21	0.254873
$891$	4242.81	0.159528
$892$	22830.1	0.856959
$893$	8423.77	0.315667
$894$	1064.90	0.0398386
$895$	2223.46	0.0830414
$896$	16214.7	0.604569
$897$	7178.90	0.267220
$898$	7849.86	0.291707
$899$	−1130.37	−0.0419353
$900$	−6284.31	−0.232752
$901$	−8760.36	−0.323918
$902$	339.206	0.0125214
$903$	33765.2	1.24434
$904$	30967.4	1.13934
$905$	4756.95	0.174725
$906$	−2905.71	−0.106551
$907$	−13454.0	−0.492537	−0.246269	−	0.969202i	$-0.579205\pi$
$907$	−13454.0	−0.492537	−0.246269	+	0.969202i	$0.579205\pi$
$908$	−29164.7	−1.06593
$909$	47841.9	1.74567
$910$	2165.78	0.0788956
$911$	−21983.0	−0.799485	−0.399742	−	0.916628i	$-0.630900\pi$
$911$	−21983.0	−0.799485	−0.399742	+	0.916628i	$0.630900\pi$
$912$	−5876.36	−0.213362
$913$	16080.8	0.582909
$914$	17591.9	0.636638
$915$	3661.03	0.132273
$916$	−24218.2	−0.873574
$917$	−334.791	−0.0120565
$918$	2606.16	0.0936993
$919$	−35442.9	−1.27220	−0.636101	−	0.771606i	$-0.719454\pi$
$919$	−35442.9	−1.27220	−0.636101	+	0.771606i	$0.719454\pi$
$920$	1880.60	0.0673929
$921$	7403.80	0.264890
$922$	−15885.3	−0.567414
$923$	32928.6	1.17428
$924$	−6673.15	−0.237587
$925$	−6670.82	−0.237119
$926$	−4905.40	−0.174084
$927$	25044.9	0.887359
$928$	2996.11	0.105983
$929$	−8224.86	−0.290473	−0.145236	−	0.989397i	$-0.546394\pi$
$929$	−8224.86	−0.290473	−0.145236	+	0.989397i	$0.546394\pi$
$930$	−2609.01	−0.0919921
$931$	4202.12	0.147926
$932$	−33892.9	−1.19120
$933$	31283.5	1.09772
$934$	165.934	0.00581320
$935$	−1821.80	−0.0637211
$936$	21299.7	0.743806
$937$	−34837.7	−1.21462	−0.607309	−	0.794466i	$-0.707751\pi$
$937$	−34837.7	−1.21462	−0.607309	+	0.794466i	$0.707751\pi$
$938$	5499.86	0.191446
$939$	−48956.5	−1.70142
$940$	15298.4	0.530827
$941$	−44537.3	−1.54290	−0.771452	−	0.636287i	$-0.780469\pi$
$941$	−44537.3	−1.54290	−0.771452	+	0.636287i	$0.780469\pi$
$942$	21769.4	0.752958
$943$	708.341	0.0244610
$944$	25421.8	0.876492
$945$	−4142.39	−0.142595
$946$	4429.08	0.152222
$947$	−37070.5	−1.27205	−0.636024	−	0.771669i	$-0.719422\pi$
$947$	−37070.5	−1.27205	−0.636024	+	0.771669i	$0.719422\pi$
$948$	−19196.4	−0.657670
$949$	3537.72	0.121011
$950$	−497.922	−0.0170050
$951$	12407.8	0.423083
$952$	5711.01	0.194427
$953$	−46438.7	−1.57849	−0.789243	−	0.614080i	$-0.789527\pi$
$953$	−46438.7	−1.57849	−0.789243	+	0.614080i	$0.789527\pi$
$954$	−10098.3	−0.342708
$955$	18759.7	0.635655
$956$	19206.5	0.649771
$957$	−1584.28	−0.0535136
$958$	7094.82	0.239273
$959$	11692.1	0.393699
$960$	−5455.95	−0.183427
$961$	−25884.2	−0.868861
$962$	10471.2	0.350942
$963$	55407.5	1.85408
$964$	−1425.64	−0.0476314
$965$	−14374.0	−0.479496
$966$	2218.82	0.0739020
$967$	−14014.4	−0.466051	−0.233026	−	0.972471i	$-0.574863\pi$
$967$	−14014.4	−0.466051	−0.233026	+	0.972471i	$0.574863\pi$
$968$	−1890.05	−0.0627567
$969$	−5012.11	−0.166163
$970$	−6136.15	−0.203113
$971$	−31532.3	−1.04214	−0.521070	−	0.853514i	$-0.674467\pi$
$971$	−31532.3	−1.04214	−0.521070	+	0.853514i	$0.674467\pi$
$972$	35184.4	1.16105
$973$	−9817.44	−0.323466
$974$	−9776.05	−0.321607
$975$	−7453.49	−0.244823
$976$	3570.51	0.117100
$977$	−34613.1	−1.13344	−0.566719	−	0.823911i	$-0.691788\pi$
$977$	−34613.1	−1.13344	−0.566719	+	0.823911i	$0.691788\pi$
$978$	−32008.6	−1.04655
$979$	−14202.5	−0.463650
$980$	7631.45	0.248753
$981$	12191.1	0.396770
$982$	7890.28	0.256404
$983$	−49032.7	−1.59095	−0.795473	−	0.605989i	$-0.792777\pi$
$983$	−49032.7	−1.59095	−0.795473	+	0.605989i	$0.792777\pi$
$984$	3659.49	0.118557
$985$	26201.4	0.847560
$986$	627.938	0.0202816
$987$	38973.5	1.25688
$988$	−4908.69	−0.158063
$989$	9248.93	0.297370
$990$	−2100.03	−0.0674175
$991$	28633.5	0.917834	0.458917	−	0.888479i	$-0.348238\pi$
$991$	28633.5	0.917834	0.458917	+	0.888479i	$0.348238\pi$
$992$	−10355.1	−0.331427
$993$	−9419.53	−0.301027
$994$	10177.4	0.324756
$995$	−4010.52	−0.127781
$996$	80346.2	2.55609
$997$	−13315.3	−0.422968	−0.211484	−	0.977381i	$-0.567830\pi$
$997$	−13315.3	−0.422968	−0.211484	+	0.977381i	$0.567830\pi$
$998$	14021.2	0.444724
$999$	−20027.8	−0.634287

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.d.1.13	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.d.1.13	✓	22	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)

\(f(q)\)	\(=\)	\(q+1.04826 q^{2} +7.96396 q^{3} -6.90116 q^{4} +5.00000 q^{5} +8.34827 q^{6} -11.0379 q^{7} -15.6202 q^{8} +36.4247 q^{9} +O(q^{10})\)	\(q+1.04826 q^{2} +7.96396 q^{3} -6.90116 q^{4} +5.00000 q^{5} +8.34827 q^{6} -11.0379 q^{7} -15.6202 q^{8} +36.4247 q^{9} +5.24128 q^{10} -11.0000 q^{11} -54.9605 q^{12} -37.4361 q^{13} -11.5706 q^{14} +39.8198 q^{15} +38.8352 q^{16} +33.1237 q^{17} +38.1824 q^{18} -19.0000 q^{19} -34.5058 q^{20} -87.9055 q^{21} -11.5308 q^{22} -24.0790 q^{23} -124.399 q^{24} +25.0000 q^{25} -39.2426 q^{26} +75.0576 q^{27} +76.1744 q^{28} +18.0847 q^{29} +41.7414 q^{30} -62.5041 q^{31} +165.671 q^{32} -87.6036 q^{33} +34.7221 q^{34} -55.1896 q^{35} -251.372 q^{36} -266.833 q^{37} -19.9169 q^{38} -298.139 q^{39} -78.1012 q^{40} -29.4174 q^{41} -92.1475 q^{42} -384.108 q^{43} +75.9127 q^{44} +182.123 q^{45} -25.2410 q^{46} -443.356 q^{47} +309.282 q^{48} -221.164 q^{49} +26.2064 q^{50} +263.795 q^{51} +258.352 q^{52} -264.474 q^{53} +78.6796 q^{54} -55.0000 q^{55} +172.415 q^{56} -151.315 q^{57} +18.9574 q^{58} +654.606 q^{59} -274.803 q^{60} +91.9399 q^{61} -65.5203 q^{62} -402.052 q^{63} -137.016 q^{64} -187.180 q^{65} -91.8310 q^{66} -475.332 q^{67} -228.592 q^{68} -191.764 q^{69} -57.8528 q^{70} -879.594 q^{71} -568.962 q^{72} -94.5002 q^{73} -279.709 q^{74} +199.099 q^{75} +131.122 q^{76} +121.417 q^{77} -312.527 q^{78} +349.276 q^{79} +194.176 q^{80} -385.710 q^{81} -30.8370 q^{82} -1461.89 q^{83} +606.650 q^{84} +165.618 q^{85} -402.644 q^{86} +144.026 q^{87} +171.823 q^{88} +1291.14 q^{89} +190.912 q^{90} +413.216 q^{91} +166.173 q^{92} -497.780 q^{93} -464.751 q^{94} -95.0000 q^{95} +1319.40 q^{96} -1170.73 q^{97} -231.837 q^{98} -400.671 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9}+O(q^{10}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9	\( 22 q - 4 q^{2} - 21 q^{3} + 74 q^{4} + 110 q^{5} - 9 q^{6} - 41 q^{7} - 78 q^{8} + 209 q^{9} - 20 q^{10} - 242 q^{11} - 196 q^{12} - q^{13} - 63 q^{14} - 105 q^{15} + 6 q^{16} + 187 q^{17} - 361 q^{18} - 418 q^{19} + 370 q^{20} - 107 q^{21} + 44 q^{22} - 361 q^{23} + 208 q^{24} + 550 q^{25} - 365 q^{26} - 1467 q^{27} - 773 q^{28} - 319 q^{29} - 45 q^{30} - 402 q^{31} - 873 q^{32} + 231 q^{33} - 717 q^{34} - 205 q^{35} + 725 q^{36} - 838 q^{37} + 76 q^{38} - 607 q^{39} - 390 q^{40} - 392 q^{41} - 1350 q^{42} - 610 q^{43} - 814 q^{44} + 1045 q^{45} - 605 q^{46} - 1866 q^{47} - 1637 q^{48} + 379 q^{49} - 100 q^{50} - 2659 q^{51} - 638 q^{52} - 1303 q^{53} + 2338 q^{54} - 1210 q^{55} + 727 q^{56} + 399 q^{57} + 44 q^{58} - 2417 q^{59} - 980 q^{60} + 918 q^{61} - 1634 q^{62} - 374 q^{63} - 1716 q^{64} - 5 q^{65} + 99 q^{66} - 2339 q^{67} + 4940 q^{68} + 127 q^{69} - 315 q^{70} - 2370 q^{71} - 3306 q^{72} + 2207 q^{73} + 2051 q^{74} - 525 q^{75} - 1406 q^{76} + 451 q^{77} + 1380 q^{78} + 586 q^{79} + 30 q^{80} + 1950 q^{81} - 1566 q^{82} - 2870 q^{83} + 3076 q^{84} + 935 q^{85} - 1246 q^{86} - 1811 q^{87} + 858 q^{88} - 1768 q^{89} - 1805 q^{90} - 2195 q^{91} - 6728 q^{92} - 2916 q^{93} + 672 q^{94} - 2090 q^{95} + 6022 q^{96} - 4022 q^{97} + 1162 q^{98} - 2299 q^{99}+O(q^{100}) \) 22 * q - 4 * q^2 - 21 * q^3 + 74 * q^4 + 110 * q^5 - 9 * q^6 - 41 * q^7 - 78 * q^8 + 209 * q^9 - 20 * q^10 - 242 * q^11 - 196 * q^12 - q^13 - 63 * q^14 - 105 * q^15 + 6 * q^16 + 187 * q^17 - 361 * q^18 - 418 * q^19 + 370 * q^20 - 107 * q^21 + 44 * q^22 - 361 * q^23 + 208 * q^24 + 550 * q^25 - 365 * q^26 - 1467 * q^27 - 773 * q^28 - 319 * q^29 - 45 * q^30 - 402 * q^31 - 873 * q^32 + 231 * q^33 - 717 * q^34 - 205 * q^35 + 725 * q^36 - 838 * q^37 + 76 * q^38 - 607 * q^39 - 390 * q^40 - 392 * q^41 - 1350 * q^42 - 610 * q^43 - 814 * q^44 + 1045 * q^45 - 605 * q^46 - 1866 * q^47 - 1637 * q^48 + 379 * q^49 - 100 * q^50 - 2659 * q^51 - 638 * q^52 - 1303 * q^53 + 2338 * q^54 - 1210 * q^55 + 727 * q^56 + 399 * q^57 + 44 * q^58 - 2417 * q^59 - 980 * q^60 + 918 * q^61 - 1634 * q^62 - 374 * q^63 - 1716 * q^64 - 5 * q^65 + 99 * q^66 - 2339 * q^67 + 4940 * q^68 + 127 * q^69 - 315 * q^70 - 2370 * q^71 - 3306 * q^72 + 2207 * q^73 + 2051 * q^74 - 525 * q^75 - 1406 * q^76 + 451 * q^77 + 1380 * q^78 + 586 * q^79 + 30 * q^80 + 1950 * q^81 - 1566 * q^82 - 2870 * q^83 + 3076 * q^84 + 935 * q^85 - 1246 * q^86 - 1811 * q^87 + 858 * q^88 - 1768 * q^89 - 1805 * q^90 - 2195 * q^91 - 6728 * q^92 - 2916 * q^93 + 672 * q^94 - 2090 * q^95 + 6022 * q^96 - 4022 * q^97 + 1162 * q^98 - 2299 * q^99

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