LMFDB - Embedded newform 1849.4.a.i.1.31

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1849,4,Mod(1,1849)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1849, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1849.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1849 = 43^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1849.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$109.094531601$
Analytic rank:	$1$
Dimension:	$50$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.31
Character	$\chi$	$=$	1849.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.29424 q^{2} -3.36009 q^{3} -6.32495 q^{4} +1.78174 q^{5} -4.34876 q^{6} -27.5745 q^{7} -18.5399 q^{8} -15.7098 q^{9} +O(q^{10})$	\(q+1.29424 q^{2} -3.36009 q^{3} -6.32495 q^{4} +1.78174 q^{5} -4.34876 q^{6} -27.5745 q^{7} -18.5399 q^{8} -15.7098 q^{9} +2.30599 q^{10} +9.45755 q^{11} +21.2524 q^{12} +38.4667 q^{13} -35.6879 q^{14} -5.98680 q^{15} +26.6046 q^{16} +54.0990 q^{17} -20.3322 q^{18} +37.3489 q^{19} -11.2694 q^{20} +92.6528 q^{21} +12.2403 q^{22} -100.967 q^{23} +62.2958 q^{24} -121.825 q^{25} +49.7850 q^{26} +143.509 q^{27} +174.407 q^{28} +175.001 q^{29} -7.74835 q^{30} -247.586 q^{31} +182.752 q^{32} -31.7783 q^{33} +70.0170 q^{34} -49.1304 q^{35} +99.3634 q^{36} +234.362 q^{37} +48.3384 q^{38} -129.252 q^{39} -33.0332 q^{40} +325.321 q^{41} +119.915 q^{42} -59.8185 q^{44} -27.9907 q^{45} -130.675 q^{46} +148.643 q^{47} -89.3938 q^{48} +417.351 q^{49} -157.671 q^{50} -181.778 q^{51} -243.300 q^{52} +735.610 q^{53} +185.735 q^{54} +16.8509 q^{55} +511.228 q^{56} -125.496 q^{57} +226.493 q^{58} -275.860 q^{59} +37.8662 q^{60} +30.9134 q^{61} -320.436 q^{62} +433.188 q^{63} +23.6879 q^{64} +68.5375 q^{65} -41.1286 q^{66} -27.3051 q^{67} -342.173 q^{68} +339.259 q^{69} -63.5864 q^{70} -66.0822 q^{71} +291.257 q^{72} -511.898 q^{73} +303.320 q^{74} +409.345 q^{75} -236.230 q^{76} -260.787 q^{77} -167.282 q^{78} -1190.03 q^{79} +47.4023 q^{80} -58.0396 q^{81} +421.043 q^{82} -144.868 q^{83} -586.024 q^{84} +96.3902 q^{85} -588.021 q^{87} -175.342 q^{88} +769.313 q^{89} -36.2266 q^{90} -1060.70 q^{91} +638.611 q^{92} +831.914 q^{93} +192.379 q^{94} +66.5460 q^{95} -614.063 q^{96} +1881.67 q^{97} +540.151 q^{98} -148.576 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) $ 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * 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 $

$2$	1.29424	0.457582	0.228791	−	0.973476i	$-0.426523\pi$
$2$	1.29424	0.457582	0.228791	+	0.973476i	$0.426523\pi$
$3$	−3.36009	−0.646650	−0.323325	−	0.946288i	$-0.604801\pi$
$3$	−3.36009	−0.646650	−0.323325	+	0.946288i	$0.604801\pi$
$4$	−6.32495	−0.790618
$5$	1.78174	0.159363	0.0796817	−	0.996820i	$-0.474610\pi$
$5$	1.78174	0.159363	0.0796817	+	0.996820i	$0.474610\pi$
$6$	−4.34876	−0.295896
$7$	−27.5745	−1.48888	−0.744440	−	0.667689i	$-0.767284\pi$
$7$	−27.5745	−1.48888	−0.744440	+	0.667689i	$0.767284\pi$
$8$	−18.5399	−0.819355
$9$	−15.7098	−0.581843
$10$	2.30599	0.0729218
$11$	9.45755	0.259233	0.129616	−	0.991564i	$-0.458625\pi$
$11$	9.45755	0.259233	0.129616	+	0.991564i	$0.458625\pi$
$12$	21.2524	0.511254
$13$	38.4667	0.820672	0.410336	−	0.911934i	$-0.365411\pi$
$13$	38.4667	0.820672	0.410336	+	0.911934i	$0.365411\pi$
$14$	−35.6879	−0.681285
$15$	−5.98680	−0.103052
$16$	26.6046	0.415696
$17$	54.0990	0.771820	0.385910	−	0.922536i	$-0.373888\pi$
$17$	54.0990	0.771820	0.385910	+	0.922536i	$0.373888\pi$
$18$	−20.3322	−0.266241
$19$	37.3489	0.450970	0.225485	−	0.974247i	$-0.427603\pi$
$19$	37.3489	0.450970	0.225485	+	0.974247i	$0.427603\pi$
$20$	−11.2694	−0.125996
$21$	92.6528	0.962786
$22$	12.2403	0.118620
$23$	−100.967	−0.915351	−0.457675	−	0.889119i	$-0.651318\pi$
$23$	−100.967	−0.915351	−0.457675	+	0.889119i	$0.651318\pi$
$24$	62.2958	0.529836
$25$	−121.825	−0.974603
$26$	49.7850	0.375525
$27$	143.509	1.02290
$28$	174.407	1.17714
$29$	175.001	1.12058	0.560292	−	0.828295i	$-0.310689\pi$
$29$	175.001	1.12058	0.560292	+	0.828295i	$0.310689\pi$
$30$	−7.74835	−0.0471549
$31$	−247.586	−1.43445	−0.717223	−	0.696844i	$-0.754587\pi$
$31$	−247.586	−1.43445	−0.717223	+	0.696844i	$0.754587\pi$
$32$	182.752	1.00957
$33$	−31.7783	−0.167633
$34$	70.0170	0.353171
$35$	−49.1304	−0.237273
$36$	99.3634	0.460016
$37$	234.362	1.04132	0.520660	−	0.853764i	$-0.325686\pi$
$37$	234.362	1.04132	0.520660	+	0.853764i	$0.325686\pi$
$38$	48.3384	0.206356
$39$	−129.252	−0.530688
$40$	−33.0332	−0.130575
$41$	325.321	1.23919	0.619593	−	0.784923i	$-0.287298\pi$
$41$	325.321	1.23919	0.619593	+	0.784923i	$0.287298\pi$
$42$	119.915	0.440554
$43$	0	0
$43$	0	0
$44$	−59.8185	−0.204954
$45$	−27.9907	−0.0927245
$46$	−130.675	−0.418848
$47$	148.643	0.461315	0.230657	−	0.973035i	$-0.425912\pi$
$47$	148.643	0.461315	0.230657	+	0.973035i	$0.425912\pi$
$48$	−89.3938	−0.268810
$49$	417.351	1.21677
$50$	−157.671	−0.445961
$51$	−181.778	−0.499098
$52$	−243.300	−0.648838
$53$	735.610	1.90649	0.953244	−	0.302202i	$-0.0977216\pi$
$53$	735.610	1.90649	0.953244	+	0.302202i	$0.0977216\pi$
$54$	185.735	0.468061
$55$	16.8509	0.0413122
$56$	511.228	1.21992
$57$	−125.496	−0.291620
$58$	226.493	0.512760
$59$	−275.860	−0.608710	−0.304355	−	0.952559i	$-0.598441\pi$
$59$	−275.860	−0.608710	−0.304355	+	0.952559i	$0.598441\pi$
$60$	37.8662	0.0814751
$61$	30.9134	0.0648861	0.0324431	−	0.999474i	$-0.489671\pi$
$61$	30.9134	0.0648861	0.0324431	+	0.999474i	$0.489671\pi$
$62$	−320.436	−0.656377
$63$	433.188	0.866295
$64$	23.6879	0.0462654
$65$	68.5375	0.130785
$66$	−41.1286	−0.0767058
$67$	−27.3051	−0.0497888	−0.0248944	−	0.999690i	$-0.507925\pi$
$67$	−27.3051	−0.0497888	−0.0248944	+	0.999690i	$0.507925\pi$
$68$	−342.173	−0.610215
$69$	339.259	0.591912
$70$	−63.5864	−0.108572
$71$	−66.0822	−0.110458	−0.0552290	−	0.998474i	$-0.517589\pi$
$71$	−66.0822	−0.110458	−0.0552290	+	0.998474i	$0.517589\pi$
$72$	291.257	0.476736
$73$	−511.898	−0.820728	−0.410364	−	0.911922i	$-0.634598\pi$
$73$	−511.898	−0.820728	−0.410364	+	0.911922i	$0.634598\pi$
$74$	303.320	0.476490
$75$	409.345	0.630228
$76$	−236.230	−0.356545
$77$	−260.787	−0.385967
$78$	−167.282	−0.242833
$79$	−1190.03	−1.69479	−0.847397	−	0.530959i	$-0.821832\pi$
$79$	−1190.03	−1.69479	−0.847397	+	0.530959i	$0.821832\pi$
$80$	47.4023	0.0662467
$81$	−58.0396	−0.0796154
$82$	421.043	0.567030
$83$	−144.868	−0.191582	−0.0957909	−	0.995401i	$-0.530538\pi$
$83$	−144.868	−0.191582	−0.0957909	+	0.995401i	$0.530538\pi$
$84$	−586.024	−0.761196
$85$	96.3902	0.123000
$86$	0	0
$87$	−588.021	−0.724626
$88$	−175.342	−0.212404
$89$	769.313	0.916258	0.458129	−	0.888886i	$-0.348520\pi$
$89$	769.313	0.916258	0.458129	+	0.888886i	$0.348520\pi$
$90$	−36.2266	−0.0424291
$91$	−1060.70	−1.22188
$92$	638.611	0.723693
$93$	831.914	0.927585
$94$	192.379	0.211090
$95$	66.5460	0.0718681
$96$	−614.063	−0.652839
$97$	1881.67	1.96964	0.984820	−	0.173577i	$-0.0555325\pi$
$97$	1881.67	1.96964	0.984820	+	0.173577i	$0.0555325\pi$
$98$	540.151	0.556771
$99$	−148.576	−0.150833
$100$	770.539	0.770539
$101$	−664.679	−0.654832	−0.327416	−	0.944880i	$-0.606178\pi$
$101$	−664.679	−0.654832	−0.327416	+	0.944880i	$0.606178\pi$
$102$	−235.264	−0.228378
$103$	−757.340	−0.724495	−0.362247	−	0.932082i	$-0.617990\pi$
$103$	−757.340	−0.724495	−0.362247	+	0.932082i	$0.617990\pi$
$104$	−713.168	−0.672422
$105$	165.083	0.153433
$106$	952.055	0.872375
$107$	−1312.68	−1.18599	−0.592996	−	0.805205i	$-0.702055\pi$
$107$	−1312.68	−1.18599	−0.592996	+	0.805205i	$0.702055\pi$
$108$	−907.686	−0.808723
$109$	314.072	0.275988	0.137994	−	0.990433i	$-0.455935\pi$
$109$	314.072	0.275988	0.137994	+	0.990433i	$0.455935\pi$
$110$	21.8090	0.0189037
$111$	−787.478	−0.673370
$112$	−733.606	−0.618922
$113$	−2100.85	−1.74895	−0.874477	−	0.485068i	$-0.838795\pi$
$113$	−2100.85	−1.74895	−0.874477	+	0.485068i	$0.838795\pi$
$114$	−162.422	−0.133440
$115$	−179.896	−0.145873
$116$	−1106.87	−0.885955
$117$	−604.302	−0.477502
$118$	−357.028	−0.278535
$119$	−1491.75	−1.14915
$120$	110.995	0.0844365
$121$	−1241.55	−0.932798
$122$	40.0093	0.0296907
$123$	−1093.11	−0.801321
$124$	1565.97	1.13410
$125$	−439.778	−0.314679
$126$	560.649	0.396401
$127$	−1771.00	−1.23741	−0.618703	−	0.785625i	$-0.712342\pi$
$127$	−1771.00	−1.23741	−0.618703	+	0.785625i	$0.712342\pi$
$128$	−1431.36	−0.988400
$129$	0	0
$130$	88.7038	0.0598449
$131$	2833.28	1.88966	0.944828	−	0.327567i	$-0.106229\pi$
$131$	2833.28	1.88966	0.944828	+	0.327567i	$0.106229\pi$
$132$	200.996	0.132534
$133$	−1029.88	−0.671441
$134$	−35.3393	−0.0227825
$135$	255.695	0.163013
$136$	−1002.99	−0.632395
$137$	384.344	0.239684	0.119842	−	0.992793i	$-0.461761\pi$
$137$	384.344	0.239684	0.119842	+	0.992793i	$0.461761\pi$
$138$	439.081	0.270848
$139$	1099.42	0.670872	0.335436	−	0.942063i	$-0.391116\pi$
$139$	1099.42	0.670872	0.335436	+	0.942063i	$0.391116\pi$
$140$	310.747	0.187592
$141$	−499.454	−0.298310
$142$	−85.5261	−0.0505436
$143$	363.800	0.212745
$144$	−417.951	−0.241870
$145$	311.806	0.178580
$146$	−662.518	−0.375550
$147$	−1402.34	−0.786823
$148$	−1482.33	−0.823287
$149$	2307.28	1.26859	0.634294	−	0.773092i	$-0.281291\pi$
$149$	2307.28	1.26859	0.634294	+	0.773092i	$0.281291\pi$
$150$	529.790	0.288381
$151$	2158.29	1.16317	0.581586	−	0.813485i	$-0.302432\pi$
$151$	2158.29	1.16317	0.581586	+	0.813485i	$0.302432\pi$
$152$	−692.445	−0.369505
$153$	−849.883	−0.449078
$154$	−337.520	−0.176611
$155$	−441.134	−0.228598
$156$	817.510	0.419572
$157$	−2032.88	−1.03338	−0.516692	−	0.856171i	$-0.672837\pi$
$157$	−2032.88	−1.03338	−0.516692	+	0.856171i	$0.672837\pi$
$158$	−1540.18	−0.775508
$159$	−2471.72	−1.23283
$160$	325.615	0.160888
$161$	2784.11	1.36285
$162$	−75.1171	−0.0364306
$163$	41.5668	0.0199740	0.00998701	−	0.999950i	$-0.496821\pi$
$163$	41.5668	0.0199740	0.00998701	+	0.999950i	$0.496821\pi$
$164$	−2057.64	−0.979724
$165$	−56.6205	−0.0267145
$166$	−187.493	−0.0876644
$167$	−3405.82	−1.57814	−0.789072	−	0.614301i	$-0.789438\pi$
$167$	−3405.82	−1.57814	−0.789072	+	0.614301i	$0.789438\pi$
$168$	−1717.77	−0.788863
$169$	−717.315	−0.326498
$170$	124.752	0.0562825
$171$	−586.743	−0.262394
$172$	0	0
$173$	841.476	0.369805	0.184902	−	0.982757i	$-0.440803\pi$
$173$	841.476	0.369805	0.184902	+	0.982757i	$0.440803\pi$
$174$	−761.040	−0.331576
$175$	3359.27	1.45107
$176$	251.614	0.107762
$177$	926.915	0.393622
$178$	995.674	0.419263
$179$	757.562	0.316329	0.158164	−	0.987413i	$-0.449442\pi$
$179$	757.562	0.316329	0.158164	+	0.987413i	$0.449442\pi$
$180$	177.039	0.0733097
$181$	−2451.87	−1.00688	−0.503442	−	0.864029i	$-0.667933\pi$
$181$	−2451.87	−1.00688	−0.503442	+	0.864029i	$0.667933\pi$
$182$	−1372.80	−0.559112
$183$	−103.872	−0.0419586
$184$	1871.92	0.749997
$185$	417.571	0.165948
$186$	1076.69	0.424446
$187$	511.644	0.200081
$188$	−940.159	−0.364724
$189$	−3957.18	−1.52298
$190$	86.1263	0.0328856
$191$	957.463	0.362720	0.181360	−	0.983417i	$-0.441950\pi$
$191$	957.463	0.362720	0.181360	+	0.983417i	$0.441950\pi$
$192$	−79.5935	−0.0299175
$193$	4925.30	1.83695	0.918474	−	0.395481i	$-0.129422\pi$
$193$	4925.30	1.83695	0.918474	+	0.395481i	$0.129422\pi$
$194$	2435.34	0.901273
$195$	−230.292	−0.0845722
$196$	−2639.72	−0.961998
$197$	−1146.48	−0.414636	−0.207318	−	0.978274i	$-0.566474\pi$
$197$	−1146.48	−0.414636	−0.207318	+	0.978274i	$0.566474\pi$
$198$	−192.293	−0.0690184
$199$	3188.35	1.13576	0.567880	−	0.823111i	$-0.307764\pi$
$199$	3188.35	1.13576	0.567880	+	0.823111i	$0.307764\pi$
$200$	2258.63	0.798546
$201$	91.7478	0.0321960
$202$	−860.253	−0.299640
$203$	−4825.57	−1.66842
$204$	1149.74	0.394596
$205$	579.637	0.197481
$206$	−980.179	−0.331516
$207$	1586.17	0.532591
$208$	1023.39	0.341150
$209$	353.230	0.116906
$210$	213.656	0.0702081
$211$	−5892.40	−1.92251	−0.961256	−	0.275659i	$-0.911104\pi$
$211$	−5892.40	−1.92251	−0.961256	+	0.275659i	$0.911104\pi$
$212$	−4652.70	−1.50730
$213$	222.042	0.0714277
$214$	−1698.92	−0.542689
$215$	0	0
$216$	−2660.64	−0.838118
$217$	6827.06	2.13572
$218$	406.484	0.126287
$219$	1720.03	0.530724
$220$	−106.581	−0.0326622
$221$	2081.01	0.633411
$222$	−1019.18	−0.308122
$223$	−1185.81	−0.356088	−0.178044	−	0.984023i	$-0.556977\pi$
$223$	−1185.81	−0.356088	−0.178044	+	0.984023i	$0.556977\pi$
$224$	−5039.28	−1.50313
$225$	1913.85	0.567066
$226$	−2719.01	−0.800290
$227$	4647.73	1.35895	0.679473	−	0.733700i	$-0.262208\pi$
$227$	4647.73	1.35895	0.679473	+	0.733700i	$0.262208\pi$
$228$	793.756	0.230560
$229$	−1956.95	−0.564711	−0.282355	−	0.959310i	$-0.591116\pi$
$229$	−1956.95	−0.564711	−0.282355	+	0.959310i	$0.591116\pi$
$230$	−232.829	−0.0667490
$231$	876.269	0.249585
$232$	−3244.51	−0.918157
$233$	−2211.01	−0.621665	−0.310832	−	0.950465i	$-0.600608\pi$
$233$	−2211.01	−0.621665	−0.310832	+	0.950465i	$0.600608\pi$
$234$	−782.111	−0.218497
$235$	264.842	0.0735167
$236$	1744.80	0.481257
$237$	3998.61	1.09594
$238$	−1930.68	−0.525830
$239$	808.463	0.218808	0.109404	−	0.993997i	$-0.465106\pi$
$239$	808.463	0.218808	0.109404	+	0.993997i	$0.465106\pi$
$240$	−159.276	−0.0428385
$241$	−3816.46	−1.02008	−0.510041	−	0.860150i	$-0.670370\pi$
$241$	−3816.46	−1.02008	−0.510041	+	0.860150i	$0.670370\pi$
$242$	−1606.87	−0.426832
$243$	−3679.72	−0.971416
$244$	−195.526	−0.0513002
$245$	743.609	0.193908
$246$	−1414.74	−0.366670
$247$	1436.69	0.370099
$248$	4590.22	1.17532
$249$	486.769	0.123886
$250$	−569.177	−0.143992
$251$	4999.26	1.25717	0.628586	−	0.777740i	$-0.283634\pi$
$251$	4999.26	1.25717	0.628586	+	0.777740i	$0.283634\pi$
$252$	−2739.89	−0.684909
$253$	−954.900	−0.237289
$254$	−2292.09	−0.566215
$255$	−323.880	−0.0795379
$256$	−2042.02	−0.498540
$257$	−2878.94	−0.698767	−0.349383	−	0.936980i	$-0.613609\pi$
$257$	−2878.94	−0.698767	−0.349383	+	0.936980i	$0.613609\pi$
$258$	0	0
$259$	−6462.40	−1.55040
$260$	−433.496	−0.103401
$261$	−2749.23	−0.652004
$262$	3666.94	0.864673
$263$	7196.10	1.68719	0.843594	−	0.536982i	$-0.180436\pi$
$263$	7196.10	1.68719	0.843594	+	0.536982i	$0.180436\pi$
$264$	589.166	0.137351
$265$	1310.66	0.303824
$266$	−1332.91	−0.307239
$267$	−2584.96	−0.592499
$268$	172.703	0.0393640
$269$	8400.53	1.90405	0.952025	−	0.306019i	$-0.0989972\pi$
$269$	8400.53	1.90405	0.952025	+	0.306019i	$0.0989972\pi$
$270$	330.930	0.0745917
$271$	−692.143	−0.155147	−0.0775733	−	0.996987i	$-0.524717\pi$
$271$	−692.143	−0.155147	−0.0775733	+	0.996987i	$0.524717\pi$
$272$	1439.28	0.320843
$273$	3564.04	0.790131
$274$	497.432	0.109675
$275$	−1152.17	−0.252649
$276$	−2145.79	−0.467977
$277$	−2581.54	−0.559963	−0.279981	−	0.960005i	$-0.590328\pi$
$277$	−2581.54	−0.559963	−0.279981	+	0.960005i	$0.590328\pi$
$278$	1422.91	0.306979
$279$	3889.52	0.834623
$280$	910.873	0.194411
$281$	−5866.29	−1.24539	−0.622693	−	0.782466i	$-0.713961\pi$
$281$	−5866.29	−1.24539	−0.622693	+	0.782466i	$0.713961\pi$
$282$	−646.413	−0.136501
$283$	−7501.69	−1.57572	−0.787861	−	0.615854i	$-0.788811\pi$
$283$	−7501.69	−1.57572	−0.787861	+	0.615854i	$0.788811\pi$
$284$	417.966	0.0873301
$285$	−223.601	−0.0464736
$286$	470.844	0.0973483
$287$	−8970.56	−1.84500
$288$	−2870.99	−0.587412
$289$	−1986.30	−0.404294
$290$	403.552	0.0817151
$291$	−6322.61	−1.27367
$292$	3237.73	0.648883
$293$	−4809.72	−0.958999	−0.479500	−	0.877542i	$-0.659182\pi$
$293$	−4809.72	−0.958999	−0.479500	+	0.877542i	$0.659182\pi$
$294$	−1814.96	−0.360036
$295$	−491.509	−0.0970060
$296$	−4345.04	−0.853211
$297$	1357.24	0.265169
$298$	2986.17	0.580483
$299$	−3883.86	−0.751203
$300$	−2589.09	−0.498270
$301$	0	0
$302$	2793.34	0.532247
$303$	2233.39	0.423448
$304$	993.652	0.187467
$305$	55.0795	0.0103405
$306$	−1099.95	−0.205490
$307$	−2532.95	−0.470890	−0.235445	−	0.971888i	$-0.575655\pi$
$307$	−2532.95	−0.470890	−0.235445	+	0.971888i	$0.575655\pi$
$308$	1649.46	0.305152
$309$	2544.73	0.468495
$310$	−570.932	−0.104602
$311$	−9641.95	−1.75802	−0.879011	−	0.476802i	$-0.841796\pi$
$311$	−9641.95	−1.75802	−0.879011	+	0.476802i	$0.841796\pi$
$312$	2396.31	0.434822
$313$	7799.67	1.40851	0.704254	−	0.709948i	$-0.251281\pi$
$313$	7799.67	1.40851	0.704254	+	0.709948i	$0.251281\pi$
$314$	−2631.03	−0.472858
$315$	771.827	0.138056
$316$	7526.87	1.33994
$317$	−8558.52	−1.51639	−0.758193	−	0.652031i	$-0.773917\pi$
$317$	−8558.52	−1.51639	−0.758193	+	0.652031i	$0.773917\pi$
$318$	−3198.99	−0.564122
$319$	1655.08	0.290492
$320$	42.2055	0.00737300
$321$	4410.72	0.766923
$322$	3603.30	0.623615
$323$	2020.54	0.348068
$324$	367.098	0.0629454
$325$	−4686.22	−0.799830
$326$	53.7974	0.00913976
$327$	−1055.31	−0.178468
$328$	−6031.42	−1.01533
$329$	−4098.75	−0.686843
$330$	−73.2804	−0.0122241
$331$	−3134.43	−0.520495	−0.260247	−	0.965542i	$-0.583804\pi$
$331$	−3134.43	−0.520495	−0.260247	+	0.965542i	$0.583804\pi$
$332$	916.280	0.151468
$333$	−3681.77	−0.605885
$334$	−4407.94	−0.722131
$335$	−48.6505	−0.00793451
$336$	2464.99	0.400226
$337$	−192.133	−0.0310569	−0.0155284	−	0.999879i	$-0.504943\pi$
$337$	−192.133	−0.0310569	−0.0155284	+	0.999879i	$0.504943\pi$
$338$	−928.377	−0.149400
$339$	7059.07	1.13096
$340$	−609.663	−0.0972459
$341$	−2341.56	−0.371855
$342$	−759.385	−0.120067
$343$	−2050.19	−0.322740
$344$	0	0
$345$	604.469	0.0943291
$346$	1089.07	0.169216
$347$	4066.72	0.629144	0.314572	−	0.949234i	$-0.398139\pi$
$347$	4066.72	0.629144	0.314572	+	0.949234i	$0.398139\pi$
$348$	3719.20	0.572903
$349$	−9154.26	−1.40406	−0.702029	−	0.712148i	$-0.747723\pi$
$349$	−9154.26	−1.40406	−0.702029	+	0.712148i	$0.747723\pi$
$350$	4347.70	0.663983
$351$	5520.31	0.839465
$352$	1728.38	0.261714
$353$	854.088	0.128778	0.0643888	−	0.997925i	$-0.479490\pi$
$353$	854.088	0.128778	0.0643888	+	0.997925i	$0.479490\pi$
$354$	1199.65	0.180115
$355$	−117.741	−0.0176029
$356$	−4865.86	−0.724411
$357$	5012.42	0.743097
$358$	980.465	0.144746
$359$	3079.71	0.452761	0.226380	−	0.974039i	$-0.427311\pi$
$359$	3079.71	0.452761	0.226380	+	0.974039i	$0.427311\pi$
$360$	518.944	0.0759743
$361$	−5464.06	−0.796626
$362$	−3173.30	−0.460732
$363$	4171.74	0.603195
$364$	6708.86	0.966043
$365$	−912.067	−0.130794
$366$	−134.435	−0.0191995
$367$	1288.57	0.183277	0.0916384	−	0.995792i	$-0.470790\pi$
$367$	1288.57	0.183277	0.0916384	+	0.995792i	$0.470790\pi$
$368$	−2686.18	−0.380508
$369$	−5110.72	−0.721012
$370$	540.436	0.0759350
$371$	−20284.1	−2.83853
$372$	−5261.81	−0.733366
$373$	−9257.47	−1.28508	−0.642538	−	0.766254i	$-0.722119\pi$
$373$	−9257.47	−1.28508	−0.642538	+	0.766254i	$0.722119\pi$
$374$	662.189	0.0915535
$375$	1477.69	0.203488
$376$	−2755.82	−0.377981
$377$	6731.72	0.919632
$378$	−5121.53	−0.696887
$379$	−7746.53	−1.04990	−0.524951	−	0.851133i	$-0.675916\pi$
$379$	−7746.53	−1.04990	−0.524951	+	0.851133i	$0.675916\pi$
$380$	−420.900	−0.0568203
$381$	5950.72	0.800169
$382$	1239.19	0.165974
$383$	247.436	0.0330114	0.0165057	−	0.999864i	$-0.494746\pi$
$383$	247.436	0.0330114	0.0165057	+	0.999864i	$0.494746\pi$
$384$	4809.49	0.639149
$385$	−464.653	−0.0615089
$386$	6374.51	0.840555
$387$	0	0
$388$	−11901.5	−1.55723
$389$	10078.8	1.31366	0.656831	−	0.754038i	$-0.271896\pi$
$389$	10078.8	1.31366	0.656831	+	0.754038i	$0.271896\pi$
$390$	−298.053	−0.0386987
$391$	−5462.21	−0.706486
$392$	−7737.64	−0.996964
$393$	−9520.09	−1.22195
$394$	−1483.82	−0.189730
$395$	−2120.32	−0.270088
$396$	939.735	0.119251
$397$	−58.9880	−0.00745723	−0.00372862	−	0.999993i	$-0.501187\pi$
$397$	−58.9880	−0.00745723	−0.00372862	+	0.999993i	$0.501187\pi$
$398$	4126.49	0.519704
$399$	3460.48	0.434188
$400$	−3241.11	−0.405139
$401$	7494.39	0.933297	0.466648	−	0.884443i	$-0.345461\pi$
$401$	7494.39	0.933297	0.466648	+	0.884443i	$0.345461\pi$
$402$	118.743	0.0147323
$403$	−9523.82	−1.17721
$404$	4204.06	0.517723
$405$	−103.411	−0.0126878
$406$	−6245.44	−0.763438
$407$	2216.49	0.269944
$408$	3370.14	0.408938
$409$	−9345.06	−1.12979	−0.564894	−	0.825163i	$-0.691083\pi$
$409$	−9345.06	−1.12979	−0.564894	+	0.825163i	$0.691083\pi$
$410$	750.188	0.0903637
$411$	−1291.43	−0.154992
$412$	4790.14	0.572799
$413$	7606.68	0.906296
$414$	2052.88	0.243704
$415$	−258.116	−0.0305311
$416$	7029.85	0.828526
$417$	−3694.14	−0.433820
$418$	457.163	0.0534942
$419$	11194.4	1.30521	0.652607	−	0.757697i	$-0.273675\pi$
$419$	11194.4	1.30521	0.652607	+	0.757697i	$0.273675\pi$
$420$	−1044.14	−0.121307
$421$	4066.28	0.470732	0.235366	−	0.971907i	$-0.424371\pi$
$421$	4066.28	0.470732	0.235366	+	0.971907i	$0.424371\pi$
$422$	−7626.17	−0.879707
$423$	−2335.15	−0.268413
$424$	−13638.1	−1.56209
$425$	−6590.63	−0.752218
$426$	287.376	0.0326840
$427$	−852.420	−0.0966077
$428$	8302.61	0.937668
$429$	−1222.40	−0.137572
$430$	0	0
$431$	−8136.58	−0.909340	−0.454670	−	0.890660i	$-0.650243\pi$
$431$	−8136.58	−0.909340	−0.454670	+	0.890660i	$0.650243\pi$
$432$	3817.99	0.425215
$433$	−7039.42	−0.781277	−0.390638	−	0.920544i	$-0.627746\pi$
$433$	−7039.42	−0.781277	−0.390638	+	0.920544i	$0.627746\pi$
$434$	8835.84	0.977267
$435$	−1047.70	−0.115479
$436$	−1986.49	−0.218201
$437$	−3771.01	−0.412796
$438$	2226.12	0.242850
$439$	12596.3	1.36945	0.684727	−	0.728800i	$-0.259922\pi$
$439$	12596.3	1.36945	0.684727	+	0.728800i	$0.259922\pi$
$440$	−312.413	−0.0338493
$441$	−6556.49	−0.707967
$442$	2693.32	0.289838
$443$	17582.4	1.88570	0.942851	−	0.333216i	$-0.108134\pi$
$443$	17582.4	1.88570	0.942851	+	0.333216i	$0.108134\pi$
$444$	4980.76	0.532379
$445$	1370.71	0.146018
$446$	−1534.72	−0.162939
$447$	−7752.67	−0.820333
$448$	−653.180	−0.0688836
$449$	1356.00	0.142524	0.0712621	−	0.997458i	$-0.477297\pi$
$449$	1356.00	0.142524	0.0712621	+	0.997458i	$0.477297\pi$
$450$	2476.98	0.259479
$451$	3076.74	0.321238
$452$	13287.8	1.38275
$453$	−7252.05	−0.752166
$454$	6015.27	0.621830
$455$	−1889.88	−0.194723
$456$	2326.68	0.238940
$457$	9380.12	0.960139	0.480069	−	0.877230i	$-0.340611\pi$
$457$	9380.12	0.960139	0.480069	+	0.877230i	$0.340611\pi$
$458$	−2532.76	−0.258402
$459$	7763.69	0.789494
$460$	1137.84	0.115330
$461$	8627.66	0.871649	0.435825	−	0.900032i	$-0.356457\pi$
$461$	8627.66	0.871649	0.435825	+	0.900032i	$0.356457\pi$
$462$	1134.10	0.114206
$463$	−2501.34	−0.251073	−0.125537	−	0.992089i	$-0.540065\pi$
$463$	−2501.34	−0.251073	−0.125537	+	0.992089i	$0.540065\pi$
$464$	4655.83	0.465823
$465$	1482.25	0.147823
$466$	−2861.57	−0.284463
$467$	10147.1	1.00546	0.502731	−	0.864443i	$-0.332329\pi$
$467$	10147.1	1.00546	0.502731	+	0.864443i	$0.332329\pi$
$468$	3822.18	0.377522
$469$	752.924	0.0741296
$470$	342.769	0.0336399
$471$	6830.66	0.668238
$472$	5114.41	0.498749
$473$	0	0
$474$	5175.15	0.501483
$475$	−4550.05	−0.439517
$476$	9435.25	0.908538
$477$	−11556.3	−1.10928
$478$	1046.34	0.100123
$479$	20100.3	1.91735	0.958673	−	0.284511i	$-0.0918313\pi$
$479$	20100.3	1.91735	0.958673	+	0.284511i	$0.0918313\pi$
$480$	−1094.10	−0.104039
$481$	9015.12	0.854582
$482$	−4939.41	−0.466772
$483$	−9354.87	−0.881287
$484$	7852.77	0.737488
$485$	3352.65	0.313888
$486$	−4762.43	−0.444503
$487$	−8302.85	−0.772563	−0.386281	−	0.922381i	$-0.626241\pi$
$487$	−8302.85	−0.772563	−0.386281	+	0.922381i	$0.626241\pi$
$488$	−573.131	−0.0531648
$489$	−139.669	−0.0129162
$490$	962.407	0.0887288
$491$	988.707	0.0908752	0.0454376	−	0.998967i	$-0.485532\pi$
$491$	988.707	0.0908752	0.0454376	+	0.998967i	$0.485532\pi$
$492$	6913.87	0.633539
$493$	9467.40	0.864889
$494$	1859.42	0.169351
$495$	−264.723	−0.0240372
$496$	−6586.92	−0.596294
$497$	1822.18	0.164459
$498$	629.995	0.0566882
$499$	−1586.04	−0.142286	−0.0711430	−	0.997466i	$-0.522665\pi$
$499$	−1586.04	−0.142286	−0.0711430	+	0.997466i	$0.522665\pi$
$500$	2781.57	0.248791
$501$	11443.9	1.02051
$502$	6470.23	0.575260
$503$	9056.14	0.802770	0.401385	−	0.915909i	$-0.368529\pi$
$503$	9056.14	0.802770	0.401385	+	0.915909i	$0.368529\pi$
$504$	−8031.26	−0.709803
$505$	−1184.28	−0.104356
$506$	−1235.87	−0.108579
$507$	2410.25	0.211130
$508$	11201.5	0.978316
$509$	10268.3	0.894170	0.447085	−	0.894491i	$-0.352462\pi$
$509$	10268.3	0.894170	0.447085	+	0.894491i	$0.352462\pi$
$510$	−419.178	−0.0363951
$511$	14115.3	1.22197
$512$	8807.99	0.760277
$513$	5359.90	0.461297
$514$	−3726.03	−0.319743
$515$	−1349.38	−0.115458
$516$	0	0
$517$	1405.80	0.119588
$518$	−8363.88	−0.709436
$519$	−2827.44	−0.239134
$520$	−1270.68	−0.107159
$521$	8690.28	0.730764	0.365382	−	0.930858i	$-0.380938\pi$
$521$	8690.28	0.730764	0.365382	+	0.930858i	$0.380938\pi$
$522$	−3558.16	−0.298346
$523$	−11938.1	−0.998124	−0.499062	−	0.866566i	$-0.666322\pi$
$523$	−11938.1	−0.998124	−0.499062	+	0.866566i	$0.666322\pi$
$524$	−17920.4	−1.49400
$525$	−11287.5	−0.938334
$526$	9313.46	0.772027
$527$	−13394.2	−1.10713
$528$	−845.446	−0.0696843
$529$	−1972.67	−0.162133
$530$	1696.31	0.139025
$531$	4333.69	0.354174
$532$	6513.92	0.530854
$533$	12514.0	1.01697
$534$	−3345.56	−0.271117
$535$	−2338.84	−0.189004
$536$	506.234	0.0407947
$537$	−2545.48	−0.204554
$538$	10872.3	0.871260
$539$	3947.12	0.315426
$540$	−1617.26	−0.128881
$541$	−11644.8	−0.925413	−0.462707	−	0.886511i	$-0.653122\pi$
$541$	−11644.8	−0.925413	−0.462707	+	0.886511i	$0.653122\pi$
$542$	−895.798	−0.0709923
$543$	8238.51	0.651102
$544$	9886.69	0.779206
$545$	559.594	0.0439823
$546$	4612.72	0.361550
$547$	−16635.8	−1.30035	−0.650177	−	0.759783i	$-0.725305\pi$
$547$	−16635.8	−1.30035	−0.650177	+	0.759783i	$0.725305\pi$
$548$	−2430.95	−0.189498
$549$	−485.642	−0.0377536
$550$	−1491.18	−0.115608
$551$	6536.12	0.505350
$552$	−6289.82	−0.484986
$553$	32814.4	2.52335
$554$	−3341.13	−0.256229
$555$	−1403.08	−0.107310
$556$	−6953.75	−0.530404
$557$	310.333	0.0236073	0.0118036	−	0.999930i	$-0.496243\pi$
$557$	310.333	0.0236073	0.0118036	+	0.999930i	$0.496243\pi$
$558$	5033.97	0.381908
$559$	0	0
$560$	−1307.09	−0.0986335
$561$	−1719.17	−0.129382
$562$	−7592.37	−0.569866
$563$	−5029.04	−0.376463	−0.188232	−	0.982125i	$-0.560276\pi$
$563$	−5029.04	−0.376463	−0.188232	+	0.982125i	$0.560276\pi$
$564$	3159.02	0.235849
$565$	−3743.17	−0.278719
$566$	−9708.97	−0.721022
$567$	1600.41	0.118538
$568$	1225.16	0.0905043
$569$	11414.7	0.841001	0.420500	−	0.907292i	$-0.361855\pi$
$569$	11414.7	0.841001	0.420500	+	0.907292i	$0.361855\pi$
$570$	−289.393	−0.0212655
$571$	−5046.78	−0.369879	−0.184940	−	0.982750i	$-0.559209\pi$
$571$	−5046.78	−0.369879	−0.184940	+	0.982750i	$0.559209\pi$
$572$	−2301.02	−0.168200
$573$	−3217.17	−0.234553
$574$	−11610.0	−0.844240
$575$	12300.3	0.892104
$576$	−372.131	−0.0269192
$577$	−1338.61	−0.0965807	−0.0482904	−	0.998833i	$-0.515377\pi$
$577$	−1338.61	−0.0965807	−0.0482904	+	0.998833i	$0.515377\pi$
$578$	−2570.74	−0.184998
$579$	−16549.5	−1.18786
$580$	−1972.16	−0.141189
$581$	3994.65	0.285243
$582$	−8182.96	−0.582808
$583$	6957.07	0.494224
$584$	9490.53	0.672468
$585$	−1076.71	−0.0760964
$586$	−6224.92	−0.438821
$587$	−8253.72	−0.580354	−0.290177	−	0.956973i	$-0.593714\pi$
$587$	−8253.72	−0.580354	−0.290177	+	0.956973i	$0.593714\pi$
$588$	8869.72	0.622077
$589$	−9247.09	−0.646892
$590$	−636.130	−0.0443882
$591$	3852.28	0.268125
$592$	6235.09	0.432873
$593$	17646.3	1.22200	0.610999	−	0.791631i	$-0.290768\pi$
$593$	17646.3	1.22200	0.610999	+	0.791631i	$0.290768\pi$
$594$	1756.59	0.121337
$595$	−2657.91	−0.183132
$596$	−14593.4	−1.00297
$597$	−10713.2	−0.734440
$598$	−5026.64	−0.343737
$599$	−19407.0	−1.32379	−0.661894	−	0.749598i	$-0.730247\pi$
$599$	−19407.0	−1.32379	−0.661894	+	0.749598i	$0.730247\pi$
$600$	−7589.21	−0.516380
$601$	2556.29	0.173500	0.0867499	−	0.996230i	$-0.472352\pi$
$601$	2556.29	0.173500	0.0867499	+	0.996230i	$0.472352\pi$
$602$	0	0
$603$	428.957	0.0289693
$604$	−13651.1	−0.919625
$605$	−2212.12	−0.148654
$606$	2890.53	0.193762
$607$	14431.3	0.964988	0.482494	−	0.875899i	$-0.339731\pi$
$607$	14431.3	0.964988	0.482494	+	0.875899i	$0.339731\pi$
$608$	6825.58	0.455286
$609$	16214.4	1.07888
$610$	71.2860	0.00473162
$611$	5717.80	0.378588
$612$	5375.46	0.355049
$613$	−10358.6	−0.682512	−0.341256	−	0.939970i	$-0.610852\pi$
$613$	−10358.6	−0.682512	−0.341256	+	0.939970i	$0.610852\pi$
$614$	−3278.25	−0.215471
$615$	−1947.63	−0.127701
$616$	4834.96	0.316244
$617$	−25620.7	−1.67172	−0.835860	−	0.548942i	$-0.815031\pi$
$617$	−25620.7	−1.67172	−0.835860	+	0.548942i	$0.815031\pi$
$618$	3293.49	0.214375
$619$	7346.23	0.477011	0.238505	−	0.971141i	$-0.423343\pi$
$619$	7346.23	0.477011	0.238505	+	0.971141i	$0.423343\pi$
$620$	2790.15	0.180734
$621$	−14489.7	−0.936312
$622$	−12479.0	−0.804440
$623$	−21213.4	−1.36420
$624$	−3438.68	−0.220605
$625$	14444.6	0.924455
$626$	10094.6	0.644509
$627$	−1186.88	−0.0755975
$628$	12857.8	0.817012
$629$	12678.7	0.803711
$630$	998.928	0.0631718
$631$	−10677.3	−0.673626	−0.336813	−	0.941572i	$-0.609349\pi$
$631$	−10677.3	−0.673626	−0.336813	+	0.941572i	$0.609349\pi$
$632$	22063.0	1.38864
$633$	19799.0	1.24319
$634$	−11076.8	−0.693871
$635$	−3155.45	−0.197197
$636$	15633.5	0.974699
$637$	16054.1	0.998566
$638$	2142.07	0.132924
$639$	1038.14	0.0642692
$640$	−2550.30	−0.157515
$641$	1114.95	0.0687018	0.0343509	−	0.999410i	$-0.489064\pi$
$641$	1114.95	0.0687018	0.0343509	+	0.999410i	$0.489064\pi$
$642$	5708.52	0.350930
$643$	9253.91	0.567556	0.283778	−	0.958890i	$-0.408412\pi$
$643$	9253.91	0.567556	0.283778	+	0.958890i	$0.408412\pi$
$644$	−17609.3	−1.07749
$645$	0	0
$646$	2615.06	0.159270
$647$	−18799.8	−1.14234	−0.571172	−	0.820831i	$-0.693511\pi$
$647$	−18799.8	−1.14234	−0.571172	+	0.820831i	$0.693511\pi$
$648$	1076.05	0.0652333
$649$	−2608.96	−0.157797
$650$	−6065.08	−0.365988
$651$	−22939.6	−1.38106
$652$	−262.908	−0.0157918
$653$	24082.8	1.44324	0.721618	−	0.692291i	$-0.243399\pi$
$653$	24082.8	1.44324	0.721618	+	0.692291i	$0.243399\pi$
$654$	−1365.82	−0.0816636
$655$	5048.16	0.301142
$656$	8655.03	0.515125
$657$	8041.80	0.477535
$658$	−5304.76	−0.314287
$659$	−23306.6	−1.37769	−0.688843	−	0.724911i	$-0.741881\pi$
$659$	−23306.6	−1.37769	−0.688843	+	0.724911i	$0.741881\pi$
$660$	358.122	0.0211210
$661$	8490.11	0.499587	0.249793	−	0.968299i	$-0.419637\pi$
$661$	8490.11	0.499587	0.249793	+	0.968299i	$0.419637\pi$
$662$	−4056.69	−0.238169
$663$	−6992.39	−0.409595
$664$	2685.83	0.156974
$665$	−1834.97	−0.107003
$666$	−4765.09	−0.277242
$667$	−17669.4	−1.02573
$668$	21541.6	1.24771
$669$	3984.43	0.230264
$670$	−62.9654	−0.00363069
$671$	292.365	0.0168206
$672$	16932.5	0.972000
$673$	6648.43	0.380800	0.190400	−	0.981707i	$-0.439022\pi$
$673$	6648.43	0.380800	0.190400	+	0.981707i	$0.439022\pi$
$674$	−248.666	−0.0142111
$675$	−17483.0	−0.996921
$676$	4536.98	0.258135
$677$	−18154.8	−1.03064	−0.515320	−	0.856998i	$-0.672327\pi$
$677$	−18154.8	−1.03064	−0.515320	+	0.856998i	$0.672327\pi$
$678$	9136.11	0.517508
$679$	−51886.2	−2.93256
$680$	−1787.06	−0.100781
$681$	−15616.8	−0.878763
$682$	−3030.54	−0.170154
$683$	1949.71	0.109229	0.0546147	−	0.998508i	$-0.482607\pi$
$683$	1949.71	0.109229	0.0546147	+	0.998508i	$0.482607\pi$
$684$	3711.12	0.207454
$685$	684.799	0.0381968
$686$	−2653.43	−0.147680
$687$	6575.53	0.365170
$688$	0	0
$689$	28296.5	1.56460
$690$	782.327	0.0431633
$691$	6988.90	0.384762	0.192381	−	0.981320i	$-0.438379\pi$
$691$	6988.90	0.384762	0.192381	+	0.981320i	$0.438379\pi$
$692$	−5322.29	−0.292374
$693$	4096.90	0.224572
$694$	5263.31	0.287885
$695$	1958.87	0.106912
$696$	10901.9	0.593726
$697$	17599.6	0.956429
$698$	−11847.8	−0.642472
$699$	7429.19	0.402000
$700$	−21247.2	−1.14724
$701$	−6698.64	−0.360919	−0.180459	−	0.983582i	$-0.557758\pi$
$701$	−6698.64	−0.360919	−0.180459	+	0.983582i	$0.557758\pi$
$702$	7144.59	0.384124
$703$	8753.17	0.469604
$704$	224.029	0.0119935
$705$	−889.896	−0.0475396
$706$	1105.39	0.0589263
$707$	18328.2	0.974968
$708$	−5862.69	−0.311205
$709$	−33993.1	−1.80062	−0.900309	−	0.435251i	$-0.856659\pi$
$709$	−33993.1	−1.80062	−0.900309	+	0.435251i	$0.856659\pi$
$710$	−152.385	−0.00805479
$711$	18695.1	0.986105
$712$	−14263.0	−0.750741
$713$	24998.0	1.31302
$714$	6487.27	0.340028
$715$	648.196	0.0339037
$716$	−4791.54	−0.250095
$717$	−2716.51	−0.141492
$718$	3985.88	0.207175
$719$	6671.11	0.346023	0.173011	−	0.984920i	$-0.444650\pi$
$719$	6671.11	0.346023	0.173011	+	0.984920i	$0.444650\pi$
$720$	−744.679	−0.0385452
$721$	20883.3	1.07869
$722$	−7071.79	−0.364522
$723$	12823.7	0.659637
$724$	15507.9	0.796061
$725$	−21319.6	−1.09213
$726$	5399.23	0.276011
$727$	20828.5	1.06257	0.531285	−	0.847193i	$-0.321710\pi$
$727$	20828.5	1.06257	0.531285	+	0.847193i	$0.321710\pi$
$728$	19665.2	1.00116
$729$	13931.3	0.707782
$730$	−1180.43	−0.0598490
$731$	0	0
$732$	656.984	0.0331733
$733$	19082.3	0.961556	0.480778	−	0.876842i	$-0.340354\pi$
$733$	19082.3	0.961556	0.480778	+	0.876842i	$0.340354\pi$
$734$	1667.71	0.0838642
$735$	−2498.60	−0.125391
$736$	−18451.9	−0.924111
$737$	−258.240	−0.0129069
$738$	−6614.49	−0.329922
$739$	−21641.2	−1.07725	−0.538623	−	0.842547i	$-0.681055\pi$
$739$	−21641.2	−1.07725	−0.538623	+	0.842547i	$0.681055\pi$
$740$	−2641.11	−0.131202
$741$	−4827.41	−0.239324
$742$	−26252.4	−1.29886
$743$	−22127.2	−1.09255	−0.546277	−	0.837605i	$-0.683955\pi$
$743$	−22127.2	−1.09255	−0.546277	+	0.837605i	$0.683955\pi$
$744$	−15423.6	−0.760022
$745$	4110.96	0.202166
$746$	−11981.4	−0.588028
$747$	2275.84	0.111471
$748$	−3236.12	−0.158188
$749$	36196.3	1.76580
$750$	1912.49	0.0931123
$751$	−18774.4	−0.912234	−0.456117	−	0.889920i	$-0.650760\pi$
$751$	−18774.4	−0.912234	−0.456117	+	0.889920i	$0.650760\pi$
$752$	3954.58	0.191767
$753$	−16798.0	−0.812952
$754$	8712.45	0.420807
$755$	3845.50	0.185367
$756$	25029.0	1.20409
$757$	−39416.8	−1.89251	−0.946253	−	0.323429i	$-0.895164\pi$
$757$	−39416.8	−1.89251	−0.946253	+	0.323429i	$0.895164\pi$
$758$	−10025.9	−0.480416
$759$	3208.55	0.153443
$760$	−1233.75	−0.0588855
$761$	−562.546	−0.0267967	−0.0133983	−	0.999910i	$-0.504265\pi$
$761$	−562.546	−0.0267967	−0.0133983	+	0.999910i	$0.504265\pi$
$762$	7701.64	0.366143
$763$	−8660.37	−0.410913
$764$	−6055.91	−0.286774
$765$	−1514.27	−0.0715666
$766$	320.241	0.0151054
$767$	−10611.4	−0.499551
$768$	6861.38	0.322381
$769$	−25134.3	−1.17863	−0.589315	−	0.807903i	$-0.700602\pi$
$769$	−25134.3	−1.17863	−0.589315	+	0.807903i	$0.700602\pi$
$770$	−601.372	−0.0281454
$771$	9673.49	0.451858
$772$	−31152.3	−1.45233
$773$	22440.0	1.04413	0.522063	−	0.852907i	$-0.325163\pi$
$773$	22440.0	1.04413	0.522063	+	0.852907i	$0.325163\pi$
$774$	0	0
$775$	30162.3	1.39802
$776$	−34886.1	−1.61384
$777$	21714.3	1.00257
$778$	13044.3	0.601108
$779$	12150.4	0.558836
$780$	1456.59	0.0668643
$781$	−624.976	−0.0286343
$782$	−7069.40	−0.323275
$783$	25114.3	1.14625
$784$	11103.4	0.505805
$785$	−3622.05	−0.164683
$786$	−12321.3	−0.559141
$787$	12520.6	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$787$	12520.6	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$788$	7251.43	0.327819
$789$	−24179.6	−1.09102
$790$	−2744.20	−0.123588
$791$	57929.9	2.60398
$792$	2754.58	0.123586
$793$	1189.14	0.0532502
$794$	−76.3445	−0.00341230
$795$	−4403.95	−0.196468
$796$	−20166.2	−0.897954
$797$	−682.158	−0.0303178	−0.0151589	−	0.999885i	$-0.504825\pi$
$797$	−682.158	−0.0303178	−0.0151589	+	0.999885i	$0.504825\pi$
$798$	4478.69	0.198677
$799$	8041.44	0.356052
$800$	−22263.8	−0.983931
$801$	−12085.7	−0.533118
$802$	9699.52	0.427060
$803$	−4841.30	−0.212759
$804$	−580.300	−0.0254547
$805$	4960.55	0.217188
$806$	−12326.1	−0.538670
$807$	−28226.6	−1.23126
$808$	12323.1	0.536540
$809$	−20230.9	−0.879211	−0.439605	−	0.898191i	$-0.644882\pi$
$809$	−20230.9	−0.879211	−0.439605	+	0.898191i	$0.644882\pi$
$810$	−133.839	−0.00580570
$811$	−30575.2	−1.32385	−0.661924	−	0.749571i	$-0.730260\pi$
$811$	−30575.2	−1.32385	−0.661924	+	0.749571i	$0.730260\pi$
$812$	30521.5	1.31908
$813$	2325.67	0.100326
$814$	2868.66	0.123522
$815$	74.0611	0.00318313
$816$	−4836.12	−0.207473
$817$	0	0
$818$	−12094.7	−0.516971
$819$	16663.3	0.710944
$820$	−3666.17	−0.156132
$821$	−24080.7	−1.02365	−0.511827	−	0.859088i	$-0.671031\pi$
$821$	−24080.7	−1.02365	−0.511827	+	0.859088i	$0.671031\pi$
$822$	−1671.42	−0.0709214
$823$	−37732.6	−1.59815	−0.799073	−	0.601234i	$-0.794676\pi$
$823$	−37732.6	−1.59815	−0.799073	+	0.601234i	$0.794676\pi$
$824$	14041.0	0.593619
$825$	3871.40	0.163376
$826$	9844.86	0.414705
$827$	−4683.21	−0.196918	−0.0984590	−	0.995141i	$-0.531391\pi$
$827$	−4683.21	−0.196918	−0.0984590	+	0.995141i	$0.531391\pi$
$828$	−10032.4	−0.421076
$829$	−32363.7	−1.35590	−0.677948	−	0.735110i	$-0.737130\pi$
$829$	−32363.7	−1.35590	−0.677948	+	0.735110i	$0.737130\pi$
$830$	−334.063	−0.0139705
$831$	8674.22	0.362100
$832$	911.193	0.0379687
$833$	22578.3	0.939125
$834$	−4781.10	−0.198508
$835$	−6068.27	−0.251498
$836$	−2234.16	−0.0924282
$837$	−35530.8	−1.46729
$838$	14488.3	0.597243
$839$	−41429.4	−1.70477	−0.852385	−	0.522915i	$-0.824845\pi$
$839$	−41429.4	−1.70477	−0.852385	+	0.522915i	$0.824845\pi$
$840$	−3060.62	−0.125716
$841$	6236.50	0.255710
$842$	5262.73	0.215399
$843$	19711.3	0.805329
$844$	37269.2	1.51997
$845$	−1278.07	−0.0520317
$846$	−3022.23	−0.122821
$847$	34235.2	1.38883
$848$	19570.6	0.792520
$849$	25206.4	1.01894
$850$	−8529.85	−0.344202
$851$	−23662.8	−0.953173
$852$	−1404.41	−0.0564720
$853$	3138.27	0.125970	0.0629849	−	0.998014i	$-0.479938\pi$
$853$	3138.27	0.125970	0.0629849	+	0.998014i	$0.479938\pi$
$854$	−1103.23	−0.0442060
$855$	−1045.42	−0.0418160
$856$	24336.9	0.971749
$857$	28519.2	1.13675	0.568376	−	0.822769i	$-0.307572\pi$
$857$	28519.2	1.13675	0.568376	+	0.822769i	$0.307572\pi$
$858$	−1582.08	−0.0629503
$859$	11047.9	0.438823	0.219411	−	0.975632i	$-0.429586\pi$
$859$	11047.9	0.438823	0.219411	+	0.975632i	$0.429586\pi$
$860$	0	0
$861$	30141.9	1.19307
$862$	−10530.7	−0.416098
$863$	−38205.6	−1.50699	−0.753496	−	0.657452i	$-0.771634\pi$
$863$	−38205.6	−1.50699	−0.753496	+	0.657452i	$0.771634\pi$
$864$	26226.5	1.03269
$865$	1499.29	0.0589333
$866$	−9110.68	−0.357498
$867$	6674.15	0.261437
$868$	−43180.8	−1.68854
$869$	−11254.8	−0.439346
$870$	−1355.97	−0.0528411
$871$	−1050.34	−0.0408603
$872$	−5822.86	−0.226132
$873$	−29560.7	−1.14602
$874$	−4880.58	−0.188888
$875$	12126.6	0.468520
$876$	−10879.1	−0.419600
$877$	17849.2	0.687258	0.343629	−	0.939106i	$-0.388344\pi$
$877$	17849.2	0.687258	0.343629	+	0.939106i	$0.388344\pi$
$878$	16302.6	0.626637
$879$	16161.1	0.620137
$880$	448.310	0.0171733
$881$	−3455.04	−0.132126	−0.0660632	−	0.997815i	$-0.521044\pi$
$881$	−3455.04	−0.132126	−0.0660632	+	0.997815i	$0.521044\pi$
$882$	−8485.65	−0.323953
$883$	−2949.64	−0.112416	−0.0562081	−	0.998419i	$-0.517901\pi$
$883$	−2949.64	−0.112416	−0.0562081	+	0.998419i	$0.517901\pi$
$884$	−13162.3	−0.500786
$885$	1651.52	0.0627290
$886$	22755.8	0.862863
$887$	16846.9	0.637728	0.318864	−	0.947800i	$-0.396699\pi$
$887$	16846.9	0.637728	0.318864	+	0.947800i	$0.396699\pi$
$888$	14599.8	0.551729
$889$	48834.3	1.84235
$890$	1774.03	0.0668152
$891$	−548.913	−0.0206389
$892$	7500.17	0.281530
$893$	5551.66	0.208039
$894$	−10033.8	−0.375370
$895$	1349.78	0.0504112
$896$	39468.9	1.47161
$897$	13050.1	0.485766
$898$	1754.98	0.0652165
$899$	−43328.0	−1.60742
$900$	−12105.0	−0.448333
$901$	39795.8	1.47147
$902$	3982.04	0.146993
$903$	0	0
$904$	38949.6	1.43301
$905$	−4368.58	−0.160460
$906$	−9385.88	−0.344178
$907$	−12483.3	−0.457004	−0.228502	−	0.973543i	$-0.573383\pi$
$907$	−12483.3	−0.457004	−0.228502	+	0.973543i	$0.573383\pi$
$908$	−29396.7	−1.07441
$909$	10442.0	0.381010
$910$	−2445.96	−0.0891019
$911$	30326.7	1.10293	0.551465	−	0.834198i	$-0.314069\pi$
$911$	30326.7	1.10293	0.551465	+	0.834198i	$0.314069\pi$
$912$	−3338.76	−0.121225
$913$	−1370.09	−0.0496643
$914$	12140.1	0.439343
$915$	−185.072	−0.00668667
$916$	12377.6	0.446471
$917$	−78126.2	−2.81347
$918$	10048.1	0.361259
$919$	23434.7	0.841175	0.420587	−	0.907252i	$-0.361824\pi$
$919$	23434.7	0.841175	0.420587	+	0.907252i	$0.361824\pi$
$920$	3335.26	0.119522
$921$	8510.96	0.304501
$922$	11166.2	0.398851
$923$	−2541.96	−0.0906497
$924$	−5542.35	−0.197327
$925$	−28551.2	−1.01487
$926$	−3237.32	−0.114887
$927$	11897.6	0.421542
$928$	31981.8	1.13131
$929$	−8192.13	−0.289317	−0.144658	−	0.989482i	$-0.546208\pi$
$929$	−8192.13	−0.289317	−0.144658	+	0.989482i	$0.546208\pi$
$930$	1918.38	0.0676412
$931$	15587.6	0.548726
$932$	13984.5	0.491500
$933$	32397.9	1.13683
$934$	13132.7	0.460082
$935$	911.615	0.0318856
$936$	11203.7	0.391244
$937$	12436.4	0.433596	0.216798	−	0.976216i	$-0.430439\pi$
$937$	12436.4	0.433596	0.216798	+	0.976216i	$0.430439\pi$
$938$	974.463	0.0339204
$939$	−26207.6	−0.910813
$940$	−1675.11	−0.0581236
$941$	−15625.9	−0.541327	−0.270664	−	0.962674i	$-0.587243\pi$
$941$	−15625.9	−0.541327	−0.270664	+	0.962674i	$0.587243\pi$
$942$	8840.50	0.305774
$943$	−32846.7	−1.13429
$944$	−7339.12	−0.253038
$945$	−7050.65	−0.242706
$946$	0	0
$947$	−9888.65	−0.339322	−0.169661	−	0.985502i	$-0.554267\pi$
$947$	−9888.65	−0.339322	−0.169661	+	0.985502i	$0.554267\pi$
$948$	−25291.0	−0.866470
$949$	−19691.0	−0.673548
$950$	−5888.85	−0.201115
$951$	28757.4	0.980571
$952$	27656.9	0.941560
$953$	7170.55	0.243732	0.121866	−	0.992547i	$-0.461112\pi$
$953$	7170.55	0.243732	0.121866	+	0.992547i	$0.461112\pi$
$954$	−14956.6	−0.507585
$955$	1705.95	0.0578043
$956$	−5113.48	−0.172994
$957$	−5561.24	−0.187847
$958$	26014.6	0.877343
$959$	−10598.1	−0.356861
$960$	−141.815	−0.00476775
$961$	31508.0	1.05764
$962$	11667.7	0.391042
$963$	20621.8	0.690062
$964$	24138.9	0.806496
$965$	8775.59	0.292742
$966$	−12107.4	−0.403261
$967$	314.417	0.0104560	0.00522801	−	0.999986i	$-0.498336\pi$
$967$	314.417	0.0104560	0.00522801	+	0.999986i	$0.498336\pi$
$968$	23018.3	0.764293
$969$	−6789.21	−0.225078
$970$	4339.12	0.143630
$971$	4278.97	0.141420	0.0707100	−	0.997497i	$-0.477474\pi$
$971$	4278.97	0.141420	0.0707100	+	0.997497i	$0.477474\pi$
$972$	23274.0	0.768020
$973$	−30315.8	−0.998849
$974$	−10745.9	−0.353511
$975$	15746.1	0.517210
$976$	822.437	0.0269729
$977$	−23808.5	−0.779634	−0.389817	−	0.920892i	$-0.627462\pi$
$977$	−23808.5	−0.779634	−0.389817	+	0.920892i	$0.627462\pi$
$978$	−180.764	−0.00591023
$979$	7275.81	0.237524
$980$	−4703.29	−0.153307
$981$	−4934.00	−0.160582
$982$	1279.62	0.0415829
$983$	26887.8	0.872418	0.436209	−	0.899845i	$-0.356321\pi$
$983$	26887.8	0.872418	0.436209	+	0.899845i	$0.356321\pi$
$984$	20266.1	0.656566
$985$	−2042.73	−0.0660778
$986$	12253.1	0.395758
$987$	13772.2	0.444147
$988$	−9086.99	−0.292607
$989$	0	0
$990$	−342.615	−0.0109990
$991$	−17203.2	−0.551441	−0.275720	−	0.961238i	$-0.588916\pi$
$991$	−17203.2	−0.551441	−0.275720	+	0.961238i	$0.588916\pi$
$992$	−45246.8	−1.44817
$993$	10532.0	0.336578
$994$	2358.34	0.0752534
$995$	5680.81	0.180999
$996$	−3078.79	−0.0979469
$997$	−41515.7	−1.31877	−0.659385	−	0.751805i	$-0.729183\pi$
$997$	−41515.7	−1.31877	−0.659385	+	0.751805i	$0.729183\pi$
$998$	−2052.71	−0.0651076
$999$	33633.0	1.06517

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1849.4.a.i.1.31	✓	50
43.42	odd	2		1849.4.a.j.1.20	yes	50

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1849.4.a.i.1.31	✓	50	1.1	even	1	trivial
1849.4.a.j.1.20	yes	50	43.42	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1849 = 43^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1849.a (trivial)

\(f(q)\)	\(=\)	\(q+1.29424 q^{2} -3.36009 q^{3} -6.32495 q^{4} +1.78174 q^{5} -4.34876 q^{6} -27.5745 q^{7} -18.5399 q^{8} -15.7098 q^{9} +O(q^{10})\)	\(q+1.29424 q^{2} -3.36009 q^{3} -6.32495 q^{4} +1.78174 q^{5} -4.34876 q^{6} -27.5745 q^{7} -18.5399 q^{8} -15.7098 q^{9} +2.30599 q^{10} +9.45755 q^{11} +21.2524 q^{12} +38.4667 q^{13} -35.6879 q^{14} -5.98680 q^{15} +26.6046 q^{16} +54.0990 q^{17} -20.3322 q^{18} +37.3489 q^{19} -11.2694 q^{20} +92.6528 q^{21} +12.2403 q^{22} -100.967 q^{23} +62.2958 q^{24} -121.825 q^{25} +49.7850 q^{26} +143.509 q^{27} +174.407 q^{28} +175.001 q^{29} -7.74835 q^{30} -247.586 q^{31} +182.752 q^{32} -31.7783 q^{33} +70.0170 q^{34} -49.1304 q^{35} +99.3634 q^{36} +234.362 q^{37} +48.3384 q^{38} -129.252 q^{39} -33.0332 q^{40} +325.321 q^{41} +119.915 q^{42} -59.8185 q^{44} -27.9907 q^{45} -130.675 q^{46} +148.643 q^{47} -89.3938 q^{48} +417.351 q^{49} -157.671 q^{50} -181.778 q^{51} -243.300 q^{52} +735.610 q^{53} +185.735 q^{54} +16.8509 q^{55} +511.228 q^{56} -125.496 q^{57} +226.493 q^{58} -275.860 q^{59} +37.8662 q^{60} +30.9134 q^{61} -320.436 q^{62} +433.188 q^{63} +23.6879 q^{64} +68.5375 q^{65} -41.1286 q^{66} -27.3051 q^{67} -342.173 q^{68} +339.259 q^{69} -63.5864 q^{70} -66.0822 q^{71} +291.257 q^{72} -511.898 q^{73} +303.320 q^{74} +409.345 q^{75} -236.230 q^{76} -260.787 q^{77} -167.282 q^{78} -1190.03 q^{79} +47.4023 q^{80} -58.0396 q^{81} +421.043 q^{82} -144.868 q^{83} -586.024 q^{84} +96.3902 q^{85} -588.021 q^{87} -175.342 q^{88} +769.313 q^{89} -36.2266 q^{90} -1060.70 q^{91} +638.611 q^{92} +831.914 q^{93} +192.379 q^{94} +66.5460 q^{95} -614.063 q^{96} +1881.67 q^{97} +540.151 q^{98} -148.576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9}+O(q^{10}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9	\( 50 q - 10 q^{2} - 2 q^{3} + 186 q^{4} + 8 q^{5} - 51 q^{6} - 6 q^{7} - 138 q^{8} + 360 q^{9} - 137 q^{10} - 252 q^{11} - 48 q^{12} - 192 q^{13} - 272 q^{14} - 314 q^{15} + 542 q^{16} - 236 q^{17} - 386 q^{18} + 12 q^{19} + 108 q^{20} - 408 q^{21} + 1235 q^{22} - 630 q^{23} - 613 q^{24} + 1098 q^{25} + 1493 q^{26} + 10 q^{27} - 242 q^{28} + 208 q^{29} - 48 q^{30} - 932 q^{31} - 1124 q^{32} + 254 q^{33} + 765 q^{34} - 1452 q^{35} + 747 q^{36} - 90 q^{37} - 1213 q^{38} - 1610 q^{39} - 1693 q^{40} - 1354 q^{41} - 16 q^{42} - 2704 q^{44} + 4508 q^{45} + 233 q^{46} - 3484 q^{47} - 376 q^{48} + 1324 q^{49} - 408 q^{50} + 4054 q^{51} - 2176 q^{52} - 726 q^{53} - 6497 q^{54} - 3288 q^{55} - 7097 q^{56} - 870 q^{57} + 275 q^{58} - 4370 q^{59} - 3891 q^{60} + 1172 q^{61} - 1546 q^{62} - 3686 q^{63} + 606 q^{64} + 2610 q^{65} - 4697 q^{66} - 344 q^{67} - 3221 q^{68} + 136 q^{69} - 1310 q^{70} + 162 q^{71} - 5814 q^{72} + 746 q^{73} - 4332 q^{74} + 236 q^{75} + 1338 q^{76} + 2024 q^{77} - 2782 q^{78} - 2656 q^{79} - 5713 q^{80} - 86 q^{81} - 4168 q^{82} - 3514 q^{83} - 4269 q^{84} - 7558 q^{85} - 10278 q^{87} + 11692 q^{88} + 2640 q^{89} - 8286 q^{90} - 5946 q^{91} - 4271 q^{92} - 2 q^{93} + 9062 q^{94} - 12140 q^{95} - 700 q^{96} - 3864 q^{97} - 2826 q^{98} - 8174 q^{99}+O(q^{100}) \) 50 * q - 10 * q^2 - 2 * q^3 + 186 * q^4 + 8 * q^5 - 51 * q^6 - 6 * q^7 - 138 * q^8 + 360 * q^9 - 137 * q^10 - 252 * q^11 - 48 * q^12 - 192 * q^13 - 272 * q^14 - 314 * q^15 + 542 * q^16 - 236 * q^17 - 386 * q^18 + 12 * q^19 + 108 * q^20 - 408 * q^21 + 1235 * q^22 - 630 * q^23 - 613 * q^24 + 1098 * q^25 + 1493 * q^26 + 10 * q^27 - 242 * q^28 + 208 * q^29 - 48 * q^30 - 932 * q^31 - 1124 * q^32 + 254 * q^33 + 765 * q^34 - 1452 * q^35 + 747 * q^36 - 90 * q^37 - 1213 * q^38 - 1610 * q^39 - 1693 * q^40 - 1354 * q^41 - 16 * q^42 - 2704 * q^44 + 4508 * q^45 + 233 * q^46 - 3484 * q^47 - 376 * q^48 + 1324 * q^49 - 408 * q^50 + 4054 * q^51 - 2176 * q^52 - 726 * q^53 - 6497 * q^54 - 3288 * q^55 - 7097 * q^56 - 870 * q^57 + 275 * q^58 - 4370 * q^59 - 3891 * q^60 + 1172 * q^61 - 1546 * q^62 - 3686 * q^63 + 606 * q^64 + 2610 * q^65 - 4697 * q^66 - 344 * q^67 - 3221 * q^68 + 136 * q^69 - 1310 * q^70 + 162 * q^71 - 5814 * q^72 + 746 * q^73 - 4332 * q^74 + 236 * q^75 + 1338 * q^76 + 2024 * q^77 - 2782 * q^78 - 2656 * q^79 - 5713 * q^80 - 86 * q^81 - 4168 * q^82 - 3514 * q^83 - 4269 * q^84 - 7558 * q^85 - 10278 * q^87 + 11692 * q^88 + 2640 * q^89 - 8286 * q^90 - 5946 * q^91 - 4271 * q^92 - 2 * q^93 + 9062 * q^94 - 12140 * q^95 - 700 * q^96 - 3864 * q^97 - 2826 * q^98 - 8174 * q^99

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