LMFDB - Embedded newform 2015.4.a.a.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2015,4,Mod(1,2015)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2015, 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("2015.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2015 = 5 \cdot 13 \cdot 31 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2015.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:	$118.888848662$
Analytic rank:	$1$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	2015.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.612379 q^{2} -1.60947 q^{3} -7.62499 q^{4} +5.00000 q^{5} +0.985605 q^{6} -21.6794 q^{7} +9.56842 q^{8} -24.4096 q^{9} +O(q^{10})$	\(q-0.612379 q^{2} -1.60947 q^{3} -7.62499 q^{4} +5.00000 q^{5} +0.985605 q^{6} -21.6794 q^{7} +9.56842 q^{8} -24.4096 q^{9} -3.06190 q^{10} -35.9413 q^{11} +12.2722 q^{12} +13.0000 q^{13} +13.2760 q^{14} -8.04734 q^{15} +55.1404 q^{16} -120.185 q^{17} +14.9479 q^{18} +88.6496 q^{19} -38.1250 q^{20} +34.8923 q^{21} +22.0097 q^{22} +98.9123 q^{23} -15.4001 q^{24} +25.0000 q^{25} -7.96093 q^{26} +82.7422 q^{27} +165.305 q^{28} +127.648 q^{29} +4.92802 q^{30} +31.0000 q^{31} -110.314 q^{32} +57.8464 q^{33} +73.5990 q^{34} -108.397 q^{35} +186.123 q^{36} +435.427 q^{37} -54.2872 q^{38} -20.9231 q^{39} +47.8421 q^{40} +127.374 q^{41} -21.3673 q^{42} -128.140 q^{43} +274.052 q^{44} -122.048 q^{45} -60.5718 q^{46} -60.3060 q^{47} -88.7468 q^{48} +126.997 q^{49} -15.3095 q^{50} +193.435 q^{51} -99.1249 q^{52} +97.6443 q^{53} -50.6696 q^{54} -179.707 q^{55} -207.438 q^{56} -142.679 q^{57} -78.1690 q^{58} -69.9762 q^{59} +61.3609 q^{60} -615.837 q^{61} -18.9838 q^{62} +529.186 q^{63} -373.569 q^{64} +65.0000 q^{65} -35.4239 q^{66} +311.970 q^{67} +916.413 q^{68} -159.196 q^{69} +66.3801 q^{70} -124.835 q^{71} -233.561 q^{72} +207.568 q^{73} -266.646 q^{74} -40.2367 q^{75} -675.953 q^{76} +779.186 q^{77} +12.8129 q^{78} -309.227 q^{79} +275.702 q^{80} +525.889 q^{81} -78.0014 q^{82} +1209.46 q^{83} -266.054 q^{84} -600.927 q^{85} +78.4705 q^{86} -205.446 q^{87} -343.901 q^{88} +646.094 q^{89} +74.7397 q^{90} -281.832 q^{91} -754.206 q^{92} -49.8935 q^{93} +36.9301 q^{94} +443.248 q^{95} +177.547 q^{96} -1230.03 q^{97} -77.7701 q^{98} +877.313 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q - 9 q^{2} - 17 q^{3} + 117 q^{4} + 185 q^{5} - 21 q^{6} - 64 q^{7} - 87 q^{8} + 196 q^{9}+O(q^{10}) $ 37 * q - 9 * q^2 - 17 * q^3 + 117 * q^4 + 185 * q^5 - 21 * q^6 - 64 * q^7 - 87 * q^8 + 196 * q^9	\( 37 q - 9 q^{2} - 17 q^{3} + 117 q^{4} + 185 q^{5} - 21 q^{6} - 64 q^{7} - 87 q^{8} + 196 q^{9} - 45 q^{10} - 191 q^{11} - 76 q^{12} + 481 q^{13} - 242 q^{14} - 85 q^{15} + 125 q^{16} - 272 q^{17} - 306 q^{18} - 359 q^{19} + 585 q^{20} - 680 q^{21} - 191 q^{22} - 218 q^{23} + 171 q^{24} + 925 q^{25} - 117 q^{26} - 476 q^{27} - 997 q^{28} - 921 q^{29} - 105 q^{30} + 1147 q^{31} - 160 q^{32} - 938 q^{33} - 1250 q^{34} - 320 q^{35} + 182 q^{36} - 1651 q^{37} - 596 q^{38} - 221 q^{39} - 435 q^{40} - 1750 q^{41} + 72 q^{42} - 487 q^{43} - 364 q^{44} + 980 q^{45} - 1625 q^{46} - 784 q^{47} + 13 q^{48} - 11 q^{49} - 225 q^{50} - 1652 q^{51} + 1521 q^{52} - 719 q^{53} - 960 q^{54} - 955 q^{55} - 2610 q^{56} - 612 q^{57} + 137 q^{58} - 2017 q^{59} - 380 q^{60} - 1493 q^{61} - 279 q^{62} - 244 q^{63} - 1963 q^{64} + 2405 q^{65} - 1641 q^{66} - 2645 q^{67} + 163 q^{68} - 3656 q^{69} - 1210 q^{70} - 1546 q^{71} - 5448 q^{72} - 4272 q^{73} + 2937 q^{74} - 425 q^{75} - 5612 q^{76} - 656 q^{77} - 273 q^{78} - 3974 q^{79} + 625 q^{80} - 767 q^{81} - 1300 q^{82} + 937 q^{83} - 3067 q^{84} - 1360 q^{85} - 2617 q^{86} - 384 q^{87} - 2069 q^{88} - 2298 q^{89} - 1530 q^{90} - 832 q^{91} - 769 q^{92} - 527 q^{93} - 162 q^{94} - 1795 q^{95} + 3378 q^{96} - 7254 q^{97} + 453 q^{98} - 6339 q^{99}+O(q^{100}) \) 37 * q - 9 * q^2 - 17 * q^3 + 117 * q^4 + 185 * q^5 - 21 * q^6 - 64 * q^7 - 87 * q^8 + 196 * q^9 - 45 * q^10 - 191 * q^11 - 76 * q^12 + 481 * q^13 - 242 * q^14 - 85 * q^15 + 125 * q^16 - 272 * q^17 - 306 * q^18 - 359 * q^19 + 585 * q^20 - 680 * q^21 - 191 * q^22 - 218 * q^23 + 171 * q^24 + 925 * q^25 - 117 * q^26 - 476 * q^27 - 997 * q^28 - 921 * q^29 - 105 * q^30 + 1147 * q^31 - 160 * q^32 - 938 * q^33 - 1250 * q^34 - 320 * q^35 + 182 * q^36 - 1651 * q^37 - 596 * q^38 - 221 * q^39 - 435 * q^40 - 1750 * q^41 + 72 * q^42 - 487 * q^43 - 364 * q^44 + 980 * q^45 - 1625 * q^46 - 784 * q^47 + 13 * q^48 - 11 * q^49 - 225 * q^50 - 1652 * q^51 + 1521 * q^52 - 719 * q^53 - 960 * q^54 - 955 * q^55 - 2610 * q^56 - 612 * q^57 + 137 * q^58 - 2017 * q^59 - 380 * q^60 - 1493 * q^61 - 279 * q^62 - 244 * q^63 - 1963 * q^64 + 2405 * q^65 - 1641 * q^66 - 2645 * q^67 + 163 * q^68 - 3656 * q^69 - 1210 * q^70 - 1546 * q^71 - 5448 * q^72 - 4272 * q^73 + 2937 * q^74 - 425 * q^75 - 5612 * q^76 - 656 * q^77 - 273 * q^78 - 3974 * q^79 + 625 * q^80 - 767 * q^81 - 1300 * q^82 + 937 * q^83 - 3067 * q^84 - 1360 * q^85 - 2617 * q^86 - 384 * q^87 - 2069 * q^88 - 2298 * q^89 - 1530 * q^90 - 832 * q^91 - 769 * q^92 - 527 * q^93 - 162 * q^94 - 1795 * q^95 + 3378 * q^96 - 7254 * q^97 + 453 * q^98 - 6339 * 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 $

$2$	−0.612379	−0.216509	−0.108254	−	0.994123i	$-0.534526\pi$
$2$	−0.612379	−0.216509	−0.108254	+	0.994123i	$0.534526\pi$
$3$	−1.60947	−0.309742	−0.154871	−	0.987935i	$-0.549496\pi$
$3$	−1.60947	−0.309742	−0.154871	+	0.987935i	$0.549496\pi$
$4$	−7.62499	−0.953124
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0.985605	0.0670619
$7$	−21.6794	−1.17058	−0.585289	−	0.810825i	$-0.699019\pi$
$7$	−21.6794	−1.17058	−0.585289	+	0.810825i	$0.699019\pi$
$8$	9.56842	0.422868
$9$	−24.4096	−0.904060
$10$	−3.06190	−0.0968256
$11$	−35.9413	−0.985156	−0.492578	−	0.870268i	$-0.663945\pi$
$11$	−35.9413	−0.985156	−0.492578	+	0.870268i	$0.663945\pi$
$12$	12.2722	0.295223
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	13.2760	0.253440
$15$	−8.04734	−0.138521
$16$	55.1404	0.861569
$17$	−120.185	−1.71466	−0.857331	−	0.514766i	$-0.827879\pi$
$17$	−120.185	−1.71466	−0.857331	+	0.514766i	$0.827879\pi$
$18$	14.9479	0.195737
$19$	88.6496	1.07040	0.535200	−	0.844725i	$-0.320236\pi$
$19$	88.6496	1.07040	0.535200	+	0.844725i	$0.320236\pi$
$20$	−38.1250	−0.426250
$21$	34.8923	0.362578
$22$	22.0097	0.213295
$23$	98.9123	0.896724	0.448362	−	0.893852i	$-0.352008\pi$
$23$	98.9123	0.896724	0.448362	+	0.893852i	$0.352008\pi$
$24$	−15.4001	−0.130980
$25$	25.0000	0.200000
$26$	−7.96093	−0.0600487
$27$	82.7422	0.589768
$28$	165.305	1.11571
$29$	127.648	0.817367	0.408684	−	0.912676i	$-0.365988\pi$
$29$	127.648	0.817367	0.408684	+	0.912676i	$0.365988\pi$
$30$	4.92802	0.0299910
$31$	31.0000	0.179605
$31$	31.0000	0.179605
$32$	−110.314	−0.609406
$33$	57.8464	0.305144
$34$	73.5990	0.371239
$35$	−108.397	−0.523498
$36$	186.123	0.861681
$37$	435.427	1.93470	0.967348	−	0.253451i	$-0.0815657\pi$
$37$	435.427	1.93470	0.967348	+	0.253451i	$0.0815657\pi$
$38$	−54.2872	−0.231751
$39$	−20.9231	−0.0859071
$40$	47.8421	0.189112
$41$	127.374	0.485184	0.242592	−	0.970128i	$-0.422002\pi$
$41$	127.374	0.485184	0.242592	+	0.970128i	$0.422002\pi$
$42$	−21.3673	−0.0785012
$43$	−128.140	−0.454447	−0.227224	−	0.973843i	$-0.572965\pi$
$43$	−128.140	−0.454447	−0.227224	+	0.973843i	$0.572965\pi$
$44$	274.052	0.938975
$45$	−122.048	−0.404308
$46$	−60.5718	−0.194148
$47$	−60.3060	−0.187160	−0.0935801	−	0.995612i	$-0.529831\pi$
$47$	−60.3060	−0.187160	−0.0935801	+	0.995612i	$0.529831\pi$
$48$	−88.7468	−0.266865
$49$	126.997	0.370253
$50$	−15.3095	−0.0433017
$51$	193.435	0.531103
$52$	−99.1249	−0.264349
$53$	97.6443	0.253065	0.126533	−	0.991962i	$-0.459615\pi$
$53$	97.6443	0.253065	0.126533	+	0.991962i	$0.459615\pi$
$54$	−50.6696	−0.127690
$55$	−179.707	−0.440575
$56$	−207.438	−0.495000
$57$	−142.679	−0.331548
$58$	−78.1690	−0.176967
$59$	−69.9762	−0.154409	−0.0772044	−	0.997015i	$-0.524599\pi$
$59$	−69.9762	−0.154409	−0.0772044	+	0.997015i	$0.524599\pi$
$60$	61.3609	0.132028
$61$	−615.837	−1.29262	−0.646310	−	0.763075i	$-0.723689\pi$
$61$	−615.837	−1.29262	−0.646310	+	0.763075i	$0.723689\pi$
$62$	−18.9838	−0.0388861
$63$	529.186	1.05827
$64$	−373.569	−0.729628
$65$	65.0000	0.124035
$66$	−35.4239	−0.0660664
$67$	311.970	0.568854	0.284427	−	0.958698i	$-0.408197\pi$
$67$	311.970	0.568854	0.284427	+	0.958698i	$0.408197\pi$
$68$	916.413	1.63429
$69$	−159.196	−0.277753
$70$	66.3801	0.113342
$71$	−124.835	−0.208664	−0.104332	−	0.994543i	$-0.533270\pi$
$71$	−124.835	−0.208664	−0.104332	+	0.994543i	$0.533270\pi$
$72$	−233.561	−0.382298
$73$	207.568	0.332794	0.166397	−	0.986059i	$-0.446787\pi$
$73$	207.568	0.332794	0.166397	+	0.986059i	$0.446787\pi$
$74$	−266.646	−0.418879
$75$	−40.2367	−0.0619485
$76$	−675.953	−1.02022
$77$	779.186	1.15320
$78$	12.8129	0.0185996
$79$	−309.227	−0.440389	−0.220194	−	0.975456i	$-0.570669\pi$
$79$	−309.227	−0.440389	−0.220194	+	0.975456i	$0.570669\pi$
$80$	275.702	0.385306
$81$	525.889	0.721384
$82$	−78.0014	−0.105046
$83$	1209.46	1.59946	0.799732	−	0.600358i	$-0.204975\pi$
$83$	1209.46	1.59946	0.799732	+	0.600358i	$0.204975\pi$
$84$	−266.054	−0.345581
$85$	−600.927	−0.766820
$86$	78.4705	0.0983918
$87$	−205.446	−0.253173
$88$	−343.901	−0.416591
$89$	646.094	0.769504	0.384752	−	0.923020i	$-0.374287\pi$
$89$	646.094	0.769504	0.384752	+	0.923020i	$0.374287\pi$
$90$	74.7397	0.0875361
$91$	−281.832	−0.324660
$92$	−754.206	−0.854689
$93$	−49.8935	−0.0556314
$94$	36.9301	0.0405218
$95$	443.248	0.478698
$96$	177.547	0.188759
$97$	−1230.03	−1.28754	−0.643768	−	0.765221i	$-0.722630\pi$
$97$	−1230.03	−1.28754	−0.643768	+	0.765221i	$0.722630\pi$
$98$	−77.7701	−0.0801630
$99$	877.313	0.890639
$100$	−190.625	−0.190625
$101$	18.5531	0.0182782	0.00913911	−	0.999958i	$-0.497091\pi$
$101$	18.5531	0.0182782	0.00913911	+	0.999958i	$0.497091\pi$
$102$	−118.455	−0.114988
$103$	−887.543	−0.849050	−0.424525	−	0.905416i	$-0.639559\pi$
$103$	−887.543	−0.849050	−0.424525	+	0.905416i	$0.639559\pi$
$104$	124.389	0.117283
$105$	174.462	0.162150
$106$	−59.7953	−0.0547909
$107$	−831.259	−0.751036	−0.375518	−	0.926815i	$-0.622535\pi$
$107$	−831.259	−0.751036	−0.375518	+	0.926815i	$0.622535\pi$
$108$	−630.908	−0.562122
$109$	154.948	0.136159	0.0680796	−	0.997680i	$-0.478313\pi$
$109$	154.948	0.136159	0.0680796	+	0.997680i	$0.478313\pi$
$110$	110.049	0.0953883
$111$	−700.806	−0.599257
$112$	−1195.41	−1.00853
$113$	1306.09	1.08731	0.543656	−	0.839308i	$-0.317040\pi$
$113$	1306.09	1.08731	0.543656	+	0.839308i	$0.317040\pi$
$114$	87.3735	0.0717831
$115$	494.562	0.401027
$116$	−973.315	−0.779052
$117$	−317.325	−0.250741
$118$	42.8519	0.0334309
$119$	2605.55	2.00715
$120$	−77.0003	−0.0585761
$121$	−39.2226	−0.0294685
$122$	377.125	0.279863
$123$	−205.005	−0.150282
$124$	−236.375	−0.171186
$125$	125.000	0.0894427
$126$	−324.062	−0.229125
$127$	566.403	0.395749	0.197875	−	0.980227i	$-0.436596\pi$
$127$	566.403	0.395749	0.197875	+	0.980227i	$0.436596\pi$
$128$	1111.28	0.767376
$129$	206.238	0.140762
$130$	−39.8046	−0.0268546
$131$	262.182	0.174862	0.0874310	−	0.996171i	$-0.472134\pi$
$131$	262.182	0.174862	0.0874310	+	0.996171i	$0.472134\pi$
$132$	−441.078	−0.290840
$133$	−1921.87	−1.25299
$134$	−191.044	−0.123162
$135$	413.711	0.263752
$136$	−1149.98	−0.725076
$137$	−2315.99	−1.44429	−0.722146	−	0.691740i	$-0.756844\pi$
$137$	−2315.99	−1.44429	−0.722146	+	0.691740i	$0.756844\pi$
$138$	97.4884	0.0601360
$139$	876.470	0.534829	0.267415	−	0.963582i	$-0.413831\pi$
$139$	876.470	0.534829	0.267415	+	0.963582i	$0.413831\pi$
$140$	826.527	0.498959
$141$	97.0606	0.0579715
$142$	76.4462	0.0451776
$143$	−467.237	−0.273233
$144$	−1345.96	−0.778910
$145$	638.240	0.365538
$146$	−127.110	−0.0720528
$147$	−204.397	−0.114683
$148$	−3320.13	−1.84401
$149$	1449.79	0.797122	0.398561	−	0.917142i	$-0.369510\pi$
$149$	1449.79	0.797122	0.398561	+	0.917142i	$0.369510\pi$
$150$	24.6401	0.0134124
$151$	3079.70	1.65975	0.829876	−	0.557948i	$-0.188411\pi$
$151$	3079.70	1.65975	0.829876	+	0.557948i	$0.188411\pi$
$152$	848.236	0.452639
$153$	2933.68	1.55016
$154$	−477.157	−0.249678
$155$	155.000	0.0803219
$156$	159.538	0.0818801
$157$	60.8144	0.0309141	0.0154571	−	0.999881i	$-0.495080\pi$
$157$	60.8144	0.0309141	0.0154571	+	0.999881i	$0.495080\pi$
$158$	189.364	0.0953480
$159$	−157.155	−0.0783851
$160$	−551.571	−0.272534
$161$	−2144.36	−1.04968
$162$	−322.043	−0.156186
$163$	197.719	0.0950095	0.0475047	−	0.998871i	$-0.484873\pi$
$163$	197.719	0.0950095	0.0475047	+	0.998871i	$0.484873\pi$
$164$	−971.228	−0.462440
$165$	289.232	0.136465
$166$	−740.648	−0.346298
$167$	−4044.52	−1.87410	−0.937050	−	0.349196i	$-0.886455\pi$
$167$	−4044.52	−1.87410	−0.937050	+	0.349196i	$0.886455\pi$
$168$	333.864	0.153323
$169$	169.000	0.0769231
$170$	367.995	0.166023
$171$	−2163.90	−0.967706
$172$	977.070	0.433145
$173$	−643.286	−0.282706	−0.141353	−	0.989959i	$-0.545145\pi$
$173$	−643.286	−0.282706	−0.141353	+	0.989959i	$0.545145\pi$
$174$	125.811	0.0548142
$175$	−541.985	−0.234116
$176$	−1981.82	−0.848780
$177$	112.624	0.0478270
$178$	−395.655	−0.166604
$179$	−1624.63	−0.678384	−0.339192	−	0.940717i	$-0.610154\pi$
$179$	−1624.63	−0.678384	−0.339192	+	0.940717i	$0.610154\pi$
$180$	930.615	0.385355
$181$	1859.65	0.763681	0.381841	−	0.924228i	$-0.375290\pi$
$181$	1859.65	0.763681	0.381841	+	0.924228i	$0.375290\pi$
$182$	172.588	0.0702917
$183$	991.170	0.400379
$184$	946.434	0.379196
$185$	2177.14	0.865223
$186$	30.5537	0.0120447
$187$	4319.62	1.68921
$188$	459.833	0.178387
$189$	−1793.80	−0.690369
$190$	−271.436	−0.103642
$191$	1605.41	0.608185	0.304092	−	0.952643i	$-0.401647\pi$
$191$	1605.41	0.608185	0.304092	+	0.952643i	$0.401647\pi$
$192$	601.248	0.225997
$193$	−2751.25	−1.02611	−0.513056	−	0.858355i	$-0.671487\pi$
$193$	−2751.25	−1.02611	−0.513056	+	0.858355i	$0.671487\pi$
$194$	753.247	0.278763
$195$	−104.615	−0.0384188
$196$	−968.349	−0.352897
$197$	1542.50	0.557862	0.278931	−	0.960311i	$-0.410020\pi$
$197$	1542.50	0.557862	0.278931	+	0.960311i	$0.410020\pi$
$198$	−537.248	−0.192831
$199$	−1648.23	−0.587135	−0.293568	−	0.955938i	$-0.594843\pi$
$199$	−1648.23	−0.587135	−0.293568	+	0.955938i	$0.594843\pi$
$200$	239.210	0.0845737
$201$	−502.106	−0.176198
$202$	−11.3615	−0.00395739
$203$	−2767.33	−0.956792
$204$	−1474.94	−0.506207
$205$	636.872	0.216981
$206$	543.513	0.183827
$207$	−2414.41	−0.810692
$208$	716.826	0.238956
$209$	−3186.18	−1.05451
$210$	−106.837	−0.0351068
$211$	−4596.99	−1.49986	−0.749928	−	0.661520i	$-0.769912\pi$
$211$	−4596.99	−1.49986	−0.749928	+	0.661520i	$0.769912\pi$
$212$	−744.537	−0.241203
$213$	200.918	0.0646322
$214$	509.046	0.162606
$215$	−640.702	−0.203235
$216$	791.711	0.249394
$217$	−672.062	−0.210242
$218$	−94.8871	−0.0294796
$219$	−334.074	−0.103080
$220$	1370.26	0.419923
$221$	−1562.41	−0.475562
$222$	429.159	0.129744
$223$	4681.06	1.40568	0.702841	−	0.711347i	$-0.251915\pi$
$223$	4681.06	1.40568	0.702841	+	0.711347i	$0.251915\pi$
$224$	2391.55	0.713357
$225$	−610.240	−0.180812
$226$	−799.820	−0.235412
$227$	3937.43	1.15126	0.575630	−	0.817710i	$-0.304757\pi$
$227$	3937.43	1.15126	0.575630	+	0.817710i	$0.304757\pi$
$228$	1087.92	0.316007
$229$	1446.29	0.417351	0.208675	−	0.977985i	$-0.433085\pi$
$229$	1446.29	0.417351	0.208675	+	0.977985i	$0.433085\pi$
$230$	−302.859	−0.0868258
$231$	−1254.08	−0.357195
$232$	1221.39	0.345639
$233$	−3369.98	−0.947532	−0.473766	−	0.880651i	$-0.657106\pi$
$233$	−3369.98	−0.947532	−0.473766	+	0.880651i	$0.657106\pi$
$234$	194.323	0.0542876
$235$	−301.530	−0.0837006
$236$	533.568	0.147171
$237$	497.690	0.136407
$238$	−1595.58	−0.434564
$239$	5182.67	1.40267	0.701336	−	0.712830i	$-0.252587\pi$
$239$	5182.67	1.40267	0.701336	+	0.712830i	$0.252587\pi$
$240$	−443.734	−0.119345
$241$	−1915.57	−0.512002	−0.256001	−	0.966677i	$-0.582405\pi$
$241$	−1915.57	−0.512002	−0.256001	+	0.966677i	$0.582405\pi$
$242$	24.0191	0.00638019
$243$	−3080.44	−0.813211
$244$	4695.75	1.23203
$245$	634.984	0.165582
$246$	125.541	0.0325373
$247$	1152.44	0.296876
$248$	296.621	0.0759494
$249$	−1946.59	−0.495421
$250$	−76.5474	−0.0193651
$251$	−338.305	−0.0850743	−0.0425371	−	0.999095i	$-0.513544\pi$
$251$	−338.305	−0.0850743	−0.0425371	+	0.999095i	$0.513544\pi$
$252$	−4035.04	−1.00866
$253$	−3555.04	−0.883412
$254$	−346.853	−0.0856831
$255$	967.173	0.237517
$256$	2308.03	0.563484
$257$	1883.84	0.457241	0.228620	−	0.973516i	$-0.426579\pi$
$257$	1883.84	0.457241	0.228620	+	0.973516i	$0.426579\pi$
$258$	−126.296	−0.0304761
$259$	−9439.80	−2.26471
$260$	−495.624	−0.118220
$261$	−3115.84	−0.738949
$262$	−160.555	−0.0378592
$263$	−7509.05	−1.76056	−0.880281	−	0.474452i	$-0.842646\pi$
$263$	−7509.05	−1.76056	−0.880281	+	0.474452i	$0.842646\pi$
$264$	553.499	0.129036
$265$	488.221	0.113174
$266$	1176.91	0.271283
$267$	−1039.87	−0.238348
$268$	−2378.77	−0.542188
$269$	1020.53	0.231311	0.115656	−	0.993289i	$-0.463103\pi$
$269$	1020.53	0.231311	0.115656	+	0.993289i	$0.463103\pi$
$270$	−253.348	−0.0571047
$271$	−3133.05	−0.702285	−0.351142	−	0.936322i	$-0.614207\pi$
$271$	−3133.05	−0.702285	−0.351142	+	0.936322i	$0.614207\pi$
$272$	−6627.08	−1.47730
$273$	453.600	0.100561
$274$	1418.26	0.312702
$275$	−898.533	−0.197031
$276$	1213.87	0.264733
$277$	5179.88	1.12357	0.561784	−	0.827284i	$-0.310115\pi$
$277$	5179.88	1.12357	0.561784	+	0.827284i	$0.310115\pi$
$278$	−536.732	−0.115795
$279$	−756.698	−0.162374
$280$	−1037.19	−0.221371
$281$	−6927.31	−1.47064	−0.735318	−	0.677722i	$-0.762967\pi$
$281$	−6927.31	−1.47064	−0.735318	+	0.677722i	$0.762967\pi$
$282$	−59.4379	−0.0125513
$283$	−692.954	−0.145554	−0.0727771	−	0.997348i	$-0.523186\pi$
$283$	−692.954	−0.145554	−0.0727771	+	0.997348i	$0.523186\pi$
$284$	951.864	0.198883
$285$	−713.394	−0.148273
$286$	286.126	0.0591573
$287$	−2761.40	−0.567945
$288$	2692.73	0.550939
$289$	9531.54	1.94006
$290$	−390.845	−0.0791421
$291$	1979.70	0.398804
$292$	−1582.70	−0.317194
$293$	−1760.61	−0.351045	−0.175522	−	0.984475i	$-0.556161\pi$
$293$	−1760.61	−0.351045	−0.175522	+	0.984475i	$0.556161\pi$
$294$	125.169	0.0248299
$295$	−349.881	−0.0690537
$296$	4166.35	0.818122
$297$	−2973.86	−0.581013
$298$	−887.819	−0.172584
$299$	1285.86	0.248706
$300$	306.805	0.0590446
$301$	2778.01	0.531966
$302$	−1885.94	−0.359351
$303$	−29.8606	−0.00566154
$304$	4888.18	0.922224
$305$	−3079.18	−0.578077
$306$	−1796.52	−0.335622
$307$	−4628.76	−0.860512	−0.430256	−	0.902707i	$-0.641577\pi$
$307$	−4628.76	−0.860512	−0.430256	+	0.902707i	$0.641577\pi$
$308$	−5941.29	−1.09914
$309$	1428.47	0.262987
$310$	−94.9188	−0.0173904
$311$	−6649.78	−1.21246	−0.606229	−	0.795290i	$-0.707319\pi$
$311$	−6649.78	−1.21246	−0.606229	+	0.795290i	$0.707319\pi$
$312$	−200.201	−0.0363274
$313$	4.87640	0.000880609 0	0.000440304	−	1.00000i	$-0.499860\pi$
$313$	4.87640	0.000880609 0	0.000440304		1.00000i	$0.499860\pi$
$314$	−37.2414	−0.00669317
$315$	2645.93	0.473274
$316$	2357.85	0.419745
$317$	−4104.79	−0.727281	−0.363640	−	0.931539i	$-0.618466\pi$
$317$	−4104.79	−0.727281	−0.363640	+	0.931539i	$0.618466\pi$
$318$	96.2386	0.0169711
$319$	−4587.84	−0.805234
$320$	−1867.85	−0.326299
$321$	1337.89	0.232628
$322$	1313.16	0.227266
$323$	−10654.4	−1.83537
$324$	−4009.90	−0.687568
$325$	325.000	0.0554700
$326$	−121.079	−0.0205704
$327$	−249.384	−0.0421743
$328$	1218.77	0.205169
$329$	1307.40	0.219086
$330$	−177.120	−0.0295458
$331$	−7832.34	−1.30062	−0.650308	−	0.759670i	$-0.725360\pi$
$331$	−7832.34	−1.30062	−0.650308	+	0.759670i	$0.725360\pi$
$332$	−9222.12	−1.52449
$333$	−10628.6	−1.74908
$334$	2476.78	0.405759
$335$	1559.85	0.254399
$336$	1923.98	0.312386
$337$	−7656.27	−1.23758	−0.618789	−	0.785557i	$-0.712376\pi$
$337$	−7656.27	−1.23758	−0.618789	+	0.785557i	$0.712376\pi$
$338$	−103.492	−0.0166545
$339$	−2102.10	−0.336787
$340$	4582.06	0.730874
$341$	−1114.18	−0.176939
$342$	1325.13	0.209517
$343$	4682.82	0.737168
$344$	−1226.10	−0.192171
$345$	−795.981	−0.124215
$346$	393.935	0.0612083
$347$	7152.54	1.10654	0.553268	−	0.833003i	$-0.313380\pi$
$347$	7152.54	1.10654	0.553268	+	0.833003i	$0.313380\pi$
$348$	1566.52	0.241305
$349$	−5230.53	−0.802246	−0.401123	−	0.916024i	$-0.631380\pi$
$349$	−5230.53	−0.802246	−0.401123	+	0.916024i	$0.631380\pi$
$350$	331.900	0.0506881
$351$	1075.65	0.163572
$352$	3964.84	0.600359
$353$	5657.62	0.853044	0.426522	−	0.904477i	$-0.359739\pi$
$353$	5657.62	0.853044	0.426522	+	0.904477i	$0.359739\pi$
$354$	−68.9689	−0.0103550
$355$	−624.174	−0.0933175
$356$	−4926.46	−0.733433
$357$	−4193.55	−0.621698
$358$	994.891	0.146876
$359$	11449.8	1.68327	0.841637	−	0.540044i	$-0.181592\pi$
$359$	11449.8	1.68327	0.841637	+	0.540044i	$0.181592\pi$
$360$	−1167.81	−0.170969
$361$	999.752	0.145758
$362$	−1138.81	−0.165344
$363$	63.1275	0.00912764
$364$	2148.97	0.309441
$365$	1037.84	0.148830
$366$	−606.972	−0.0866856
$367$	−10224.2	−1.45423	−0.727113	−	0.686518i	$-0.759138\pi$
$367$	−10224.2	−1.45423	−0.727113	+	0.686518i	$0.759138\pi$
$368$	5454.07	0.772590
$369$	−3109.16	−0.438635
$370$	−1333.23	−0.187328
$371$	−2116.87	−0.296233
$372$	380.438	0.0530236
$373$	−2875.94	−0.399224	−0.199612	−	0.979875i	$-0.563968\pi$
$373$	−2875.94	−0.399224	−0.199612	+	0.979875i	$0.563968\pi$
$374$	−2645.25	−0.365728
$375$	−201.184	−0.0277042
$376$	−577.033	−0.0791441
$377$	1659.42	0.226697
$378$	1098.49	0.149471
$379$	−7888.83	−1.06919	−0.534594	−	0.845109i	$-0.679535\pi$
$379$	−7888.83	−1.06919	−0.534594	+	0.845109i	$0.679535\pi$
$380$	−3379.76	−0.456258
$381$	−911.608	−0.122580
$382$	−983.118	−0.131677
$383$	3369.46	0.449533	0.224767	−	0.974413i	$-0.427838\pi$
$383$	3369.46	0.449533	0.224767	+	0.974413i	$0.427838\pi$
$384$	−1788.57	−0.237689
$385$	3895.93	0.515727
$386$	1684.81	0.222162
$387$	3127.86	0.410848
$388$	9378.99	1.22718
$389$	−6096.10	−0.794561	−0.397281	−	0.917697i	$-0.630046\pi$
$389$	−6096.10	−0.794561	−0.397281	+	0.917697i	$0.630046\pi$
$390$	64.0643	0.00831801
$391$	−11887.8	−1.53758
$392$	1215.16	0.156568
$393$	−421.973	−0.0541622
$394$	−944.597	−0.120782
$395$	−1546.13	−0.196948
$396$	−6689.51	−0.848890
$397$	15014.1	1.89808	0.949041	−	0.315153i	$-0.102056\pi$
$397$	15014.1	1.89808	0.949041	+	0.315153i	$0.102056\pi$
$398$	1009.34	0.127120
$399$	3093.19	0.388103
$400$	1378.51	0.172314
$401$	−6941.58	−0.864454	−0.432227	−	0.901765i	$-0.642272\pi$
$401$	−6941.58	−0.864454	−0.432227	+	0.901765i	$0.642272\pi$
$402$	307.479	0.0381484
$403$	403.000	0.0498135
$404$	−141.467	−0.0174214
$405$	2629.44	0.322613
$406$	1694.66	0.207154
$407$	−15649.8	−1.90598
$408$	1850.86	0.224587
$409$	1688.94	0.204188	0.102094	−	0.994775i	$-0.467446\pi$
$409$	1688.94	0.204188	0.102094	+	0.994775i	$0.467446\pi$
$410$	−390.007	−0.0469782
$411$	3727.51	0.447359
$412$	6767.51	0.809250
$413$	1517.04	0.180748
$414$	1478.53	0.175522
$415$	6047.30	0.715302
$416$	−1434.08	−0.169019
$417$	−1410.65	−0.165659
$418$	1951.15	0.228311
$419$	−266.428	−0.0310641	−0.0155320	−	0.999879i	$-0.504944\pi$
$419$	−266.428	−0.0310641	−0.0155320	+	0.999879i	$0.504944\pi$
$420$	−1330.27	−0.154549
$421$	−8978.93	−1.03945	−0.519723	−	0.854335i	$-0.673965\pi$
$421$	−8978.93	−1.03945	−0.519723	+	0.854335i	$0.673965\pi$
$422$	2815.10	0.324732
$423$	1472.05	0.169204
$424$	934.301	0.107013
$425$	−3004.64	−0.342932
$426$	−123.038	−0.0139934
$427$	13351.0	1.51311
$428$	6338.35	0.715831
$429$	752.003	0.0846318
$430$	392.353	0.0440022
$431$	6649.43	0.743137	0.371568	−	0.928406i	$-0.378820\pi$
$431$	6649.43	0.743137	0.371568	+	0.928406i	$0.378820\pi$
$432$	4562.44	0.508126
$433$	2593.67	0.287862	0.143931	−	0.989588i	$-0.454026\pi$
$433$	2593.67	0.287862	0.143931	+	0.989588i	$0.454026\pi$
$434$	411.556	0.0455192
$435$	−1027.23	−0.113223
$436$	−1181.48	−0.129777
$437$	8768.54	0.959854
$438$	204.580	0.0223178
$439$	−494.266	−0.0537359	−0.0268679	−	0.999639i	$-0.508553\pi$
$439$	−494.266	−0.0537359	−0.0268679	+	0.999639i	$0.508553\pi$
$440$	−1719.51	−0.186305
$441$	−3099.94	−0.334731
$442$	956.787	0.102963
$443$	−6488.86	−0.695926	−0.347963	−	0.937508i	$-0.613126\pi$
$443$	−6488.86	−0.695926	−0.347963	+	0.937508i	$0.613126\pi$
$444$	5343.64	0.571167
$445$	3230.47	0.344133
$446$	−2866.58	−0.304342
$447$	−2333.39	−0.246902
$448$	8098.76	0.854086
$449$	−4744.79	−0.498709	−0.249355	−	0.968412i	$-0.580218\pi$
$449$	−4744.79	−0.498709	−0.249355	+	0.968412i	$0.580218\pi$
$450$	373.698	0.0391474
$451$	−4578.00	−0.477981
$452$	−9958.89	−1.03634
$453$	−4956.68	−0.514095
$454$	−2411.20	−0.249258
$455$	−1409.16	−0.145192
$456$	−1365.21	−0.140201
$457$	16405.6	1.67926	0.839629	−	0.543161i	$-0.182773\pi$
$457$	16405.6	1.67926	0.839629	+	0.543161i	$0.182773\pi$
$458$	−885.676	−0.0903601
$459$	−9944.40	−1.01125
$460$	−3771.03	−0.382228
$461$	−9893.64	−0.999551	−0.499775	−	0.866155i	$-0.666584\pi$
$461$	−9893.64	−0.999551	−0.499775	+	0.866155i	$0.666584\pi$
$462$	767.970	0.0773359
$463$	−13974.5	−1.40270	−0.701348	−	0.712819i	$-0.747418\pi$
$463$	−13974.5	−1.40270	−0.701348	+	0.712819i	$0.747418\pi$
$464$	7038.57	0.704219
$465$	−249.468	−0.0248791
$466$	2063.71	0.205149
$467$	13573.9	1.34502	0.672511	−	0.740087i	$-0.265216\pi$
$467$	13573.9	1.34502	0.672511	+	0.740087i	$0.265216\pi$
$468$	2419.60	0.238987
$469$	−6763.32	−0.665888
$470$	184.651	0.0181219
$471$	−97.8788	−0.00957541
$472$	−669.561	−0.0652946
$473$	4605.54	0.447701
$474$	−304.775	−0.0295333
$475$	2216.24	0.214080
$476$	−19867.3	−1.91306
$477$	−2383.46	−0.228786
$478$	−3173.76	−0.303691
$479$	808.908	0.0771606	0.0385803	−	0.999256i	$-0.487716\pi$
$479$	808.908	0.0771606	0.0385803	+	0.999256i	$0.487716\pi$
$480$	887.736	0.0844155
$481$	5660.55	0.536588
$482$	1173.05	0.110853
$483$	3451.28	0.325132
$484$	299.072	0.0280871
$485$	−6150.17	−0.575803
$486$	1886.40	0.176067
$487$	15786.3	1.46888	0.734439	−	0.678675i	$-0.237445\pi$
$487$	15786.3	1.46888	0.734439	+	0.678675i	$0.237445\pi$
$488$	−5892.58	−0.546608
$489$	−318.222	−0.0294285
$490$	−388.851	−0.0358500
$491$	−6625.51	−0.608972	−0.304486	−	0.952517i	$-0.598485\pi$
$491$	−6625.51	−0.608972	−0.304486	+	0.952517i	$0.598485\pi$
$492$	1563.16	0.143237
$493$	−15341.4	−1.40151
$494$	−705.733	−0.0642762
$495$	4386.57	0.398306
$496$	1709.35	0.154742
$497$	2706.34	0.244258
$498$	1192.05	0.107263
$499$	13394.4	1.20163	0.600815	−	0.799388i	$-0.294843\pi$
$499$	13394.4	1.20163	0.600815	+	0.799388i	$0.294843\pi$
$500$	−953.124	−0.0852500
$501$	6509.53	0.580488
$502$	207.171	0.0184193
$503$	−1212.27	−0.107460	−0.0537302	−	0.998555i	$-0.517111\pi$
$503$	−1212.27	−0.107460	−0.0537302	+	0.998555i	$0.517111\pi$
$504$	5063.47	0.447510
$505$	92.7654	0.00817427
$506$	2177.03	0.191266
$507$	−272.000	−0.0238263
$508$	−4318.82	−0.377198
$509$	−22515.2	−1.96065	−0.980325	−	0.197392i	$-0.936753\pi$
$509$	−22515.2	−1.96065	−0.980325	+	0.197392i	$0.936753\pi$
$510$	−592.277	−0.0514244
$511$	−4499.95	−0.389562
$512$	−10303.6	−0.889375
$513$	7335.06	0.631288
$514$	−1153.63	−0.0989966
$515$	−4437.71	−0.379707
$516$	−1572.56	−0.134163
$517$	2167.48	0.184382
$518$	5780.74	0.490330
$519$	1035.35	0.0875660
$520$	621.947	0.0524504
$521$	17030.9	1.43213	0.716063	−	0.698035i	$-0.245942\pi$
$521$	17030.9	1.43213	0.716063	+	0.698035i	$0.245942\pi$
$522$	1908.07	0.159989
$523$	13252.8	1.10804	0.554018	−	0.832504i	$-0.313094\pi$
$523$	13252.8	1.10804	0.554018	+	0.832504i	$0.313094\pi$
$524$	−1999.13	−0.166665
$525$	872.308	0.0725155
$526$	4598.39	0.381177
$527$	−3725.75	−0.307962
$528$	3189.68	0.262903
$529$	−2383.35	−0.195887
$530$	−298.976	−0.0245032
$531$	1708.09	0.139595
$532$	14654.3	1.19425
$533$	1655.87	0.134566
$534$	636.794	0.0516044
$535$	−4156.30	−0.335874
$536$	2985.06	0.240550
$537$	2614.80	0.210124
$538$	−624.950	−0.0500809
$539$	−4564.43	−0.364757
$540$	−3154.54	−0.251389
$541$	6438.01	0.511630	0.255815	−	0.966726i	$-0.417656\pi$
$541$	6438.01	0.511630	0.255815	+	0.966726i	$0.417656\pi$
$542$	1918.61	0.152051
$543$	−2993.04	−0.236544
$544$	13258.2	1.04492
$545$	774.741	0.0608922
$546$	−277.775	−0.0217723
$547$	−8147.42	−0.636853	−0.318426	−	0.947948i	$-0.603154\pi$
$547$	−8147.42	−0.636853	−0.318426	+	0.947948i	$0.603154\pi$
$548$	17659.4	1.37659
$549$	15032.3	1.16861
$550$	550.243	0.0426589
$551$	11315.9	0.874910
$552$	−1523.26	−0.117453
$553$	6703.85	0.515509
$554$	−3172.05	−0.243262
$555$	−3504.03	−0.267996
$556$	−6683.08	−0.509758
$557$	4842.49	0.368372	0.184186	−	0.982891i	$-0.441035\pi$
$557$	4842.49	0.368372	0.184186	+	0.982891i	$0.441035\pi$
$558$	463.386	0.0351554
$559$	−1665.83	−0.126041
$560$	−5977.06	−0.451030
$561$	−6952.29	−0.523219
$562$	4242.14	0.318405
$563$	−5676.32	−0.424917	−0.212459	−	0.977170i	$-0.568147\pi$
$563$	−5676.32	−0.424917	−0.212459	+	0.977170i	$0.568147\pi$
$564$	−740.086	−0.0552540
$565$	6530.43	0.486261
$566$	424.350	0.0315137
$567$	−11401.0	−0.844436
$568$	−1194.47	−0.0882375
$569$	10822.3	0.797352	0.398676	−	0.917092i	$-0.369470\pi$
$569$	10822.3	0.797352	0.398676	+	0.917092i	$0.369470\pi$
$570$	436.867	0.0321024
$571$	−14065.0	−1.03083	−0.515413	−	0.856942i	$-0.672361\pi$
$571$	−14065.0	−1.03083	−0.515413	+	0.856942i	$0.672361\pi$
$572$	3562.68	0.260425
$573$	−2583.85	−0.188381
$574$	1691.02	0.122965
$575$	2472.81	0.179345
$576$	9118.68	0.659627
$577$	−3855.09	−0.278145	−0.139072	−	0.990282i	$-0.544412\pi$
$577$	−3855.09	−0.278145	−0.139072	+	0.990282i	$0.544412\pi$
$578$	−5836.91	−0.420041
$579$	4428.05	0.317830
$580$	−4866.58	−0.348403
$581$	−26220.4	−1.87230
$582$	−1212.33	−0.0863446
$583$	−3509.46	−0.249309
$584$	1986.10	0.140728
$585$	−1586.62	−0.112135
$586$	1078.16	0.0760042
$587$	1886.20	0.132626	0.0663132	−	0.997799i	$-0.478876\pi$
$587$	1886.20	0.132626	0.0663132	+	0.997799i	$0.478876\pi$
$588$	1558.53	0.109307
$589$	2748.14	0.192250
$590$	214.260	0.0149507
$591$	−2482.61	−0.172793
$592$	24009.6	1.66688
$593$	−3395.68	−0.235150	−0.117575	−	0.993064i	$-0.537512\pi$
$593$	−3395.68	−0.235150	−0.117575	+	0.993064i	$0.537512\pi$
$594$	1821.13	0.125794
$595$	13027.7	0.897623
$596$	−11054.6	−0.759756
$597$	2652.78	0.181861
$598$	−787.434	−0.0538471
$599$	−7629.34	−0.520411	−0.260206	−	0.965553i	$-0.583790\pi$
$599$	−7629.34	−0.520411	−0.260206	+	0.965553i	$0.583790\pi$
$600$	−385.002	−0.0261960
$601$	−11473.4	−0.778721	−0.389361	−	0.921085i	$-0.627304\pi$
$601$	−11473.4	−0.778721	−0.389361	+	0.921085i	$0.627304\pi$
$602$	−1701.19	−0.115175
$603$	−7615.07	−0.514278
$604$	−23482.7	−1.58195
$605$	−196.113	−0.0131787
$606$	18.2860	0.00122577
$607$	20716.6	1.38528	0.692638	−	0.721285i	$-0.256448\pi$
$607$	20716.6	1.38528	0.692638	+	0.721285i	$0.256448\pi$
$608$	−9779.31	−0.652308
$609$	4453.94	0.296359
$610$	1885.63	0.125159
$611$	−783.978	−0.0519089
$612$	−22369.3	−1.47749
$613$	−16833.0	−1.10910	−0.554549	−	0.832151i	$-0.687109\pi$
$613$	−16833.0	−1.10910	−0.554549	+	0.832151i	$0.687109\pi$
$614$	2834.56	0.186308
$615$	−1025.03	−0.0672081
$616$	7455.58	0.487652
$617$	9932.89	0.648108	0.324054	−	0.946039i	$-0.394954\pi$
$617$	9932.89	0.648108	0.324054	+	0.946039i	$0.394954\pi$
$618$	−874.766	−0.0569389
$619$	10409.9	0.675946	0.337973	−	0.941156i	$-0.390259\pi$
$619$	10409.9	0.675946	0.337973	+	0.941156i	$0.390259\pi$
$620$	−1181.87	−0.0765568
$621$	8184.22	0.528859
$622$	4072.19	0.262508
$623$	−14006.9	−0.900764
$624$	−1153.71	−0.0740149
$625$	625.000	0.0400000
$626$	−2.98621	−0.000190659 0
$627$	5128.06	0.326627
$628$	−463.709	−0.0294650
$629$	−52332.0	−3.31735
$630$	−1620.31	−0.102468
$631$	−12187.6	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$631$	−12187.6	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$632$	−2958.81	−0.186226
$633$	7398.70	0.464569
$634$	2513.69	0.157463
$635$	2832.01	0.176984
$636$	1198.31	0.0747107
$637$	1650.96	0.102690
$638$	2809.50	0.174340
$639$	3047.17	0.188645
$640$	5556.40	0.343181
$641$	−16384.4	−1.00959	−0.504794	−	0.863240i	$-0.668431\pi$
$641$	−16384.4	−1.00959	−0.504794	+	0.863240i	$0.668431\pi$
$642$	−819.293	−0.0503659
$643$	3346.41	0.205240	0.102620	−	0.994721i	$-0.467277\pi$
$643$	3346.41	0.205240	0.102620	+	0.994721i	$0.467277\pi$
$644$	16350.7	1.00048
$645$	1031.19	0.0629505
$646$	6524.52	0.397375
$647$	25152.4	1.52835	0.764174	−	0.645010i	$-0.223147\pi$
$647$	25152.4	1.52835	0.764174	+	0.645010i	$0.223147\pi$
$648$	5031.92	0.305050
$649$	2515.04	0.152117
$650$	−199.023	−0.0120097
$651$	1081.66	0.0651209
$652$	−1507.61	−0.0905558
$653$	−10583.6	−0.634254	−0.317127	−	0.948383i	$-0.602718\pi$
$653$	−10583.6	−0.634254	−0.317127	+	0.948383i	$0.602718\pi$
$654$	152.718	0.00913109
$655$	1310.91	0.0782007
$656$	7023.48	0.418019
$657$	−5066.65	−0.300866
$658$	−800.623	−0.0474340
$659$	−15843.7	−0.936545	−0.468272	−	0.883584i	$-0.655123\pi$
$659$	−15843.7	−0.936545	−0.468272	+	0.883584i	$0.655123\pi$
$660$	−2205.39	−0.130068
$661$	−13367.8	−0.786608	−0.393304	−	0.919408i	$-0.628668\pi$
$661$	−13367.8	−0.786608	−0.393304	+	0.919408i	$0.628668\pi$
$662$	4796.36	0.281595
$663$	2514.65	0.147302
$664$	11572.6	0.676362
$665$	−9609.35	−0.560353
$666$	6508.74	0.378691
$667$	12626.0	0.732952
$668$	30839.5	1.78625
$669$	−7534.02	−0.435399
$670$	−955.219	−0.0550796
$671$	22134.0	1.27343
$672$	−3849.12	−0.220957
$673$	−17442.1	−0.999026	−0.499513	−	0.866306i	$-0.666488\pi$
$673$	−17442.1	−0.999026	−0.499513	+	0.866306i	$0.666488\pi$
$674$	4688.54	0.267946
$675$	2068.55	0.117954
$676$	−1288.62	−0.0733172
$677$	−1237.43	−0.0702484	−0.0351242	−	0.999383i	$-0.511183\pi$
$677$	−1237.43	−0.0702484	−0.0351242	+	0.999383i	$0.511183\pi$
$678$	1287.28	0.0729172
$679$	26666.4	1.50716
$680$	−5749.92	−0.324264
$681$	−6337.16	−0.356594
$682$	682.301	0.0383089
$683$	30192.6	1.69149	0.845744	−	0.533588i	$-0.179157\pi$
$683$	30192.6	1.69149	0.845744	+	0.533588i	$0.179157\pi$
$684$	16499.7	0.922344
$685$	−11579.9	−0.645907
$686$	−2867.66	−0.159603
$687$	−2327.75	−0.129271
$688$	−7065.72	−0.391538
$689$	1269.38	0.0701877
$690$	487.442	0.0268936
$691$	22534.1	1.24058	0.620289	−	0.784374i	$-0.287015\pi$
$691$	22534.1	1.24058	0.620289	+	0.784374i	$0.287015\pi$
$692$	4905.05	0.269454
$693$	−19019.6	−1.04256
$694$	−4380.06	−0.239575
$695$	4382.35	0.239183
$696$	−1965.79	−0.107059
$697$	−15308.5	−0.831926
$698$	3203.07	0.173693
$699$	5423.88	0.293491
$700$	4132.63	0.223141
$701$	8021.64	0.432202	0.216101	−	0.976371i	$-0.430666\pi$
$701$	8021.64	0.432202	0.216101	+	0.976371i	$0.430666\pi$
$702$	−658.704	−0.0354148
$703$	38600.4	2.07090
$704$	13426.6	0.718797
$705$	485.303	0.0259256
$706$	−3464.60	−0.184691
$707$	−402.220	−0.0213961
$708$	−858.761	−0.0455850
$709$	−26544.5	−1.40606	−0.703031	−	0.711159i	$-0.748171\pi$
$709$	−26544.5	−1.40606	−0.703031	+	0.711159i	$0.748171\pi$
$710$	382.231	0.0202040
$711$	7548.10	0.398138
$712$	6182.10	0.325399
$713$	3066.28	0.161056
$714$	2568.04	0.134603
$715$	−2336.18	−0.122194
$716$	12387.8	0.646584
$717$	−8341.34	−0.434467
$718$	−7011.59	−0.364443
$719$	−8584.57	−0.445272	−0.222636	−	0.974902i	$-0.571466\pi$
$719$	−8584.57	−0.445272	−0.222636	+	0.974902i	$0.571466\pi$
$720$	−6729.78	−0.348339
$721$	19241.4	0.993880
$722$	−612.227	−0.0315578
$723$	3083.04	0.158589
$724$	−14179.8	−0.727883
$725$	3191.20	0.163473
$726$	−38.6580	−0.00197621
$727$	11004.4	0.561390	0.280695	−	0.959797i	$-0.409435\pi$
$727$	11004.4	0.561390	0.280695	+	0.959797i	$0.409435\pi$
$728$	−2696.69	−0.137288
$729$	−9241.12	−0.469498
$730$	−635.551	−0.0322230
$731$	15400.6	0.779224
$732$	−7557.66	−0.381611
$733$	11383.0	0.573591	0.286795	−	0.957992i	$-0.407410\pi$
$733$	11383.0	0.573591	0.286795	+	0.957992i	$0.407410\pi$
$734$	6261.11	0.314852
$735$	−1021.99	−0.0512878
$736$	−10911.4	−0.546468
$737$	−11212.6	−0.560409
$738$	1903.98	0.0949683
$739$	30554.1	1.52091	0.760453	−	0.649393i	$-0.224977\pi$
$739$	30554.1	1.52091	0.760453	+	0.649393i	$0.224977\pi$
$740$	−16600.6	−0.824664
$741$	−1854.82	−0.0919550
$742$	1296.33	0.0641370
$743$	12906.7	0.637281	0.318641	−	0.947876i	$-0.396774\pi$
$743$	12906.7	0.637281	0.318641	+	0.947876i	$0.396774\pi$
$744$	−477.402	−0.0235247
$745$	7248.93	0.356484
$746$	1761.17	0.0864355
$747$	−29522.4	−1.44601
$748$	−32937.1	−1.61002
$749$	18021.2	0.879146
$750$	123.201	0.00599820
$751$	−1184.83	−0.0575699	−0.0287849	−	0.999586i	$-0.509164\pi$
$751$	−1184.83	−0.0575699	−0.0287849	+	0.999586i	$0.509164\pi$
$752$	−3325.30	−0.161252
$753$	544.492	0.0263511
$754$	−1016.20	−0.0490818
$755$	15398.5	0.742264
$756$	13677.7	0.658008
$757$	−12741.3	−0.611744	−0.305872	−	0.952073i	$-0.598948\pi$
$757$	−12741.3	−0.611744	−0.305872	+	0.952073i	$0.598948\pi$
$758$	4830.95	0.231488
$759$	5721.72	0.273630
$760$	4241.18	0.202426
$761$	16380.5	0.780280	0.390140	−	0.920755i	$-0.372427\pi$
$761$	16380.5	0.780280	0.390140	+	0.920755i	$0.372427\pi$
$762$	558.249	0.0265397
$763$	−3359.19	−0.159385
$764$	−12241.2	−0.579675
$765$	14668.4	0.693251
$766$	−2063.38	−0.0973278
$767$	−909.690	−0.0428253
$768$	−3714.70	−0.174535
$769$	18132.7	0.850300	0.425150	−	0.905123i	$-0.360221\pi$
$769$	18132.7	0.850300	0.425150	+	0.905123i	$0.360221\pi$
$770$	−2385.79	−0.111659
$771$	−3031.98	−0.141627
$772$	20978.3	0.978011
$773$	−30948.6	−1.44003	−0.720016	−	0.693957i	$-0.755866\pi$
$773$	−30948.6	−1.44003	−0.720016	+	0.693957i	$0.755866\pi$
$774$	−1915.44	−0.0889521
$775$	775.000	0.0359211
$776$	−11769.5	−0.544458
$777$	15193.1	0.701478
$778$	3733.12	0.172029
$779$	11291.7	0.519341
$780$	797.692	0.0366179
$781$	4486.72	0.205567
$782$	7279.85	0.332899
$783$	10561.9	0.482057
$784$	7002.66	0.318999
$785$	304.072	0.0138252
$786$	258.408	0.0117266
$787$	36146.8	1.63722	0.818610	−	0.574349i	$-0.194745\pi$
$787$	36146.8	1.63722	0.818610	+	0.574349i	$0.194745\pi$
$788$	−11761.6	−0.531712
$789$	12085.6	0.545321
$790$	946.819	0.0426409
$791$	−28315.2	−1.27278
$792$	8394.50	0.376623
$793$	−8005.88	−0.358508
$794$	−9194.35	−0.410951
$795$	−785.777	−0.0350549
$796$	12567.7	0.559613
$797$	−26215.1	−1.16510	−0.582552	−	0.812794i	$-0.697946\pi$
$797$	−26215.1	−1.16510	−0.582552	+	0.812794i	$0.697946\pi$
$798$	−1894.21	−0.0840277
$799$	7247.90	0.320917
$800$	−2757.85	−0.121881
$801$	−15770.9	−0.695677
$802$	4250.88	0.187162
$803$	−7460.26	−0.327854
$804$	3828.55	0.167939
$805$	−10721.8	−0.469433
$806$	−246.789	−0.0107851
$807$	−1642.51	−0.0716469
$808$	177.524	0.00772928
$809$	−376.244	−0.0163511	−0.00817555	−	0.999967i	$-0.502602\pi$
$809$	−376.244	−0.0163511	−0.00817555	+	0.999967i	$0.502602\pi$
$810$	−1610.22	−0.0698484
$811$	19429.4	0.841255	0.420627	−	0.907233i	$-0.361810\pi$
$811$	19429.4	0.841255	0.420627	+	0.907233i	$0.361810\pi$
$812$	21100.9	0.911941
$813$	5042.54	0.217527
$814$	9583.62	0.412661
$815$	988.595	0.0424895
$816$	10666.1	0.457582
$817$	−11359.6	−0.486441
$818$	−1034.27	−0.0442084
$819$	6879.42	0.293512
$820$	−4856.14	−0.206810
$821$	−36837.1	−1.56592	−0.782962	−	0.622070i	$-0.786292\pi$
$821$	−36837.1	−1.56592	−0.782962	+	0.622070i	$0.786292\pi$
$822$	−2282.65	−0.0968570
$823$	−16390.6	−0.694219	−0.347109	−	0.937825i	$-0.612837\pi$
$823$	−16390.6	−0.694219	−0.347109	+	0.937825i	$0.612837\pi$
$824$	−8492.38	−0.359036
$825$	1446.16	0.0610289
$826$	−929.005	−0.0391334
$827$	8965.18	0.376965	0.188482	−	0.982077i	$-0.439643\pi$
$827$	8965.18	0.376965	0.188482	+	0.982077i	$0.439643\pi$
$828$	18409.9	0.772690
$829$	−27290.8	−1.14336	−0.571682	−	0.820475i	$-0.693709\pi$
$829$	−27290.8	−1.14336	−0.571682	+	0.820475i	$0.693709\pi$
$830$	−3703.24	−0.154869
$831$	−8336.85	−0.348017
$832$	−4856.40	−0.202362
$833$	−15263.2	−0.634858
$834$	863.853	0.0358667
$835$	−20222.6	−0.838123
$836$	24294.6	1.00508
$837$	2565.01	0.105925
$838$	163.155	0.00672564
$839$	−39986.3	−1.64539	−0.822693	−	0.568486i	$-0.807529\pi$
$839$	−39986.3	−1.64539	−0.822693	+	0.568486i	$0.807529\pi$
$840$	1669.32	0.0685679
$841$	−8094.97	−0.331911
$842$	5498.51	0.225049
$843$	11149.3	0.455518
$844$	35052.0	1.42955
$845$	845.000	0.0344010
$846$	−901.450	−0.0366341
$847$	850.322	0.0344952
$848$	5384.15	0.218033
$849$	1115.29	0.0450843
$850$	1839.98	0.0742478
$851$	43069.1	1.73489
$852$	−1532.00	−0.0616025
$853$	−25379.0	−1.01871	−0.509356	−	0.860556i	$-0.670116\pi$
$853$	−25379.0	−1.01871	−0.509356	+	0.860556i	$0.670116\pi$
$854$	−8175.86	−0.327602
$855$	−10819.5	−0.432771
$856$	−7953.84	−0.317589
$857$	27203.1	1.08430	0.542148	−	0.840283i	$-0.317611\pi$
$857$	27203.1	1.08430	0.542148	+	0.840283i	$0.317611\pi$
$858$	−460.511	−0.0183235
$859$	−39793.9	−1.58062	−0.790309	−	0.612708i	$-0.790080\pi$
$859$	−39793.9	−1.58062	−0.790309	+	0.612708i	$0.790080\pi$
$860$	4885.35	0.193708
$861$	4444.39	0.175917
$862$	−4071.97	−0.160896
$863$	−43054.9	−1.69827	−0.849135	−	0.528176i	$-0.822876\pi$
$863$	−43054.9	−1.69827	−0.849135	+	0.528176i	$0.822876\pi$
$864$	−9127.63	−0.359408
$865$	−3216.43	−0.126430
$866$	−1588.31	−0.0623245
$867$	−15340.7	−0.600920
$868$	5124.46	0.200387
$869$	11114.0	0.433851
$870$	629.053	0.0245137
$871$	4055.61	0.157772
$872$	1482.61	0.0575774
$873$	30024.6	1.16401
$874$	−5369.67	−0.207817
$875$	−2709.93	−0.104700
$876$	2547.31	0.0982484
$877$	−6612.86	−0.254619	−0.127309	−	0.991863i	$-0.540634\pi$
$877$	−6612.86	−0.254619	−0.127309	+	0.991863i	$0.540634\pi$
$878$	302.678	0.0116343
$879$	2833.65	0.108733
$880$	−9909.10	−0.379586
$881$	−11150.2	−0.426400	−0.213200	−	0.977009i	$-0.568389\pi$
$881$	−11150.2	−0.426400	−0.213200	+	0.977009i	$0.568389\pi$
$882$	1898.34	0.0724721
$883$	28841.2	1.09919	0.549594	−	0.835432i	$-0.314783\pi$
$883$	28841.2	1.09919	0.549594	+	0.835432i	$0.314783\pi$
$884$	11913.4	0.453269
$885$	563.122	0.0213889
$886$	3973.64	0.150674
$887$	8152.03	0.308589	0.154295	−	0.988025i	$-0.450690\pi$
$887$	8152.03	0.308589	0.154295	+	0.988025i	$0.450690\pi$
$888$	−6705.61	−0.253407
$889$	−12279.3	−0.463255
$890$	−1978.27	−0.0745077
$891$	−18901.1	−0.710675
$892$	−35693.0	−1.33979
$893$	−5346.10	−0.200336
$894$	1428.92	0.0534565
$895$	−8123.16	−0.303383
$896$	−24091.9	−0.898274
$897$	−2069.55	−0.0770349
$898$	2905.61	0.107975
$899$	3957.09	0.146803
$900$	4653.08	0.172336
$901$	−11735.4	−0.433922
$902$	2803.47	0.103487
$903$	−4471.12	−0.164772
$904$	12497.2	0.459790
$905$	9298.23	0.341529
$906$	3035.37	0.111306
$907$	−19105.4	−0.699431	−0.349716	−	0.936856i	$-0.613722\pi$
$907$	−19105.4	−0.699431	−0.349716	+	0.936856i	$0.613722\pi$
$908$	−30022.8	−1.09729
$909$	−452.873	−0.0165246
$910$	862.941	0.0314354
$911$	48230.2	1.75405	0.877025	−	0.480445i	$-0.159525\pi$
$911$	48230.2	1.75405	0.877025	+	0.480445i	$0.159525\pi$
$912$	−7867.37	−0.285652
$913$	−43469.6	−1.57572
$914$	−10046.4	−0.363574
$915$	4955.85	0.179055
$916$	−11027.9	−0.397787
$917$	−5683.95	−0.204690
$918$	6089.74	0.218945
$919$	21076.0	0.756512	0.378256	−	0.925701i	$-0.376524\pi$
$919$	21076.0	0.756512	0.378256	+	0.925701i	$0.376524\pi$
$920$	4732.17	0.169582
$921$	7449.84	0.266537
$922$	6058.66	0.216411
$923$	−1622.85	−0.0578731
$924$	9562.32	0.340451
$925$	10885.7	0.386939
$926$	8557.67	0.303696
$927$	21664.6	0.767592
$928$	−14081.4	−0.498108
$929$	22309.0	0.787873	0.393937	−	0.919138i	$-0.371113\pi$
$929$	22309.0	0.787873	0.393937	+	0.919138i	$0.371113\pi$
$930$	152.769	0.00538654
$931$	11258.2	0.396319
$932$	25696.1	0.903115
$933$	10702.6	0.375550
$934$	−8312.37	−0.291209
$935$	21598.1	0.755437
$936$	−3036.30	−0.106030
$937$	−51620.9	−1.79977	−0.899883	−	0.436131i	$-0.856348\pi$
$937$	−51620.9	−1.79977	−0.899883	+	0.436131i	$0.856348\pi$
$938$	4141.72	0.144170
$939$	−7.84842	−0.000272762 0
$940$	2299.16	0.0797771
$941$	29965.7	1.03810	0.519050	−	0.854744i	$-0.326286\pi$
$941$	29965.7	1.03810	0.519050	+	0.854744i	$0.326286\pi$
$942$	59.9389	0.00207316
$943$	12598.9	0.435076
$944$	−3858.52	−0.133034
$945$	−8969.00	−0.308743
$946$	−2820.33	−0.0969312
$947$	−5146.71	−0.176606	−0.0883029	−	0.996094i	$-0.528144\pi$
$947$	−5146.71	−0.176606	−0.0883029	+	0.996094i	$0.528144\pi$
$948$	−3794.89	−0.130013
$949$	2698.38	0.0923005
$950$	−1357.18	−0.0463502
$951$	6606.53	0.225270
$952$	24931.0	0.848758
$953$	41815.1	1.42133	0.710664	−	0.703532i	$-0.248395\pi$
$953$	41815.1	1.42133	0.710664	+	0.703532i	$0.248395\pi$
$954$	1459.58	0.0495342
$955$	8027.04	0.271988
$956$	−39517.8	−1.33692
$957$	7383.98	0.249415
$958$	−495.358	−0.0167059
$959$	50209.2	1.69066
$960$	3006.24	0.101069
$961$	961.000	0.0322581
$962$	−3466.40	−0.116176
$963$	20290.7	0.678982
$964$	14606.2	0.488001
$965$	−13756.3	−0.458891
$966$	−2113.49	−0.0703939
$967$	−12139.1	−0.403687	−0.201844	−	0.979418i	$-0.564693\pi$
$967$	−12139.1	−0.403687	−0.201844	+	0.979418i	$0.564693\pi$
$968$	−375.298	−0.0124613
$969$	17147.9	0.568493
$970$	3766.23	0.124666
$971$	−588.290	−0.0194430	−0.00972148	−	0.999953i	$-0.503094\pi$
$971$	−588.290	−0.0194430	−0.00972148	+	0.999953i	$0.503094\pi$
$972$	23488.3	0.775091
$973$	−19001.4	−0.626059
$974$	−9667.17	−0.318025
$975$	−523.077	−0.0171814
$976$	−33957.5	−1.11368
$977$	28950.3	0.948005	0.474002	−	0.880524i	$-0.342809\pi$
$977$	28950.3	0.948005	0.474002	+	0.880524i	$0.342809\pi$
$978$	194.873	0.00637152
$979$	−23221.5	−0.758081
$980$	−4841.75	−0.157820
$981$	−3782.23	−0.123096
$982$	4057.32	0.131848
$983$	34576.1	1.12188	0.560938	−	0.827858i	$-0.310440\pi$
$983$	34576.1	1.12188	0.560938	+	0.827858i	$0.310440\pi$
$984$	−1961.57	−0.0635495
$985$	7712.52	0.249483
$986$	9394.77	0.303439
$987$	−2104.22	−0.0678601
$988$	−8787.38	−0.282959
$989$	−12674.7	−0.407514
$990$	−2686.24	−0.0862367
$991$	−12201.2	−0.391103	−0.195552	−	0.980693i	$-0.562650\pi$
$991$	−12201.2	−0.391103	−0.195552	+	0.980693i	$0.562650\pi$
$992$	−3419.74	−0.109452
$993$	12605.9	0.402856
$994$	−1657.31	−0.0528839
$995$	−8241.15	−0.262575
$996$	14842.7	0.472198
$997$	28342.4	0.900314	0.450157	−	0.892950i	$-0.351368\pi$
$997$	28342.4	0.900314	0.450157	+	0.892950i	$0.351368\pi$
$998$	−8202.42	−0.260163
$999$	36028.2	1.14102

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2015.4.a.a.1.18	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2015.4.a.a.1.18	✓	37	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:	\( N \)	\(=\)	\( 2015 = 5 \cdot 13 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2015.a (trivial)

\(f(q)\)	\(=\)	\(q-0.612379 q^{2} -1.60947 q^{3} -7.62499 q^{4} +5.00000 q^{5} +0.985605 q^{6} -21.6794 q^{7} +9.56842 q^{8} -24.4096 q^{9} +O(q^{10})\)	\(q-0.612379 q^{2} -1.60947 q^{3} -7.62499 q^{4} +5.00000 q^{5} +0.985605 q^{6} -21.6794 q^{7} +9.56842 q^{8} -24.4096 q^{9} -3.06190 q^{10} -35.9413 q^{11} +12.2722 q^{12} +13.0000 q^{13} +13.2760 q^{14} -8.04734 q^{15} +55.1404 q^{16} -120.185 q^{17} +14.9479 q^{18} +88.6496 q^{19} -38.1250 q^{20} +34.8923 q^{21} +22.0097 q^{22} +98.9123 q^{23} -15.4001 q^{24} +25.0000 q^{25} -7.96093 q^{26} +82.7422 q^{27} +165.305 q^{28} +127.648 q^{29} +4.92802 q^{30} +31.0000 q^{31} -110.314 q^{32} +57.8464 q^{33} +73.5990 q^{34} -108.397 q^{35} +186.123 q^{36} +435.427 q^{37} -54.2872 q^{38} -20.9231 q^{39} +47.8421 q^{40} +127.374 q^{41} -21.3673 q^{42} -128.140 q^{43} +274.052 q^{44} -122.048 q^{45} -60.5718 q^{46} -60.3060 q^{47} -88.7468 q^{48} +126.997 q^{49} -15.3095 q^{50} +193.435 q^{51} -99.1249 q^{52} +97.6443 q^{53} -50.6696 q^{54} -179.707 q^{55} -207.438 q^{56} -142.679 q^{57} -78.1690 q^{58} -69.9762 q^{59} +61.3609 q^{60} -615.837 q^{61} -18.9838 q^{62} +529.186 q^{63} -373.569 q^{64} +65.0000 q^{65} -35.4239 q^{66} +311.970 q^{67} +916.413 q^{68} -159.196 q^{69} +66.3801 q^{70} -124.835 q^{71} -233.561 q^{72} +207.568 q^{73} -266.646 q^{74} -40.2367 q^{75} -675.953 q^{76} +779.186 q^{77} +12.8129 q^{78} -309.227 q^{79} +275.702 q^{80} +525.889 q^{81} -78.0014 q^{82} +1209.46 q^{83} -266.054 q^{84} -600.927 q^{85} +78.4705 q^{86} -205.446 q^{87} -343.901 q^{88} +646.094 q^{89} +74.7397 q^{90} -281.832 q^{91} -754.206 q^{92} -49.8935 q^{93} +36.9301 q^{94} +443.248 q^{95} +177.547 q^{96} -1230.03 q^{97} -77.7701 q^{98} +877.313 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q - 9 q^{2} - 17 q^{3} + 117 q^{4} + 185 q^{5} - 21 q^{6} - 64 q^{7} - 87 q^{8} + 196 q^{9}+O(q^{10}) \) 37 * q - 9 * q^2 - 17 * q^3 + 117 * q^4 + 185 * q^5 - 21 * q^6 - 64 * q^7 - 87 * q^8 + 196 * q^9	\( 37 q - 9 q^{2} - 17 q^{3} + 117 q^{4} + 185 q^{5} - 21 q^{6} - 64 q^{7} - 87 q^{8} + 196 q^{9} - 45 q^{10} - 191 q^{11} - 76 q^{12} + 481 q^{13} - 242 q^{14} - 85 q^{15} + 125 q^{16} - 272 q^{17} - 306 q^{18} - 359 q^{19} + 585 q^{20} - 680 q^{21} - 191 q^{22} - 218 q^{23} + 171 q^{24} + 925 q^{25} - 117 q^{26} - 476 q^{27} - 997 q^{28} - 921 q^{29} - 105 q^{30} + 1147 q^{31} - 160 q^{32} - 938 q^{33} - 1250 q^{34} - 320 q^{35} + 182 q^{36} - 1651 q^{37} - 596 q^{38} - 221 q^{39} - 435 q^{40} - 1750 q^{41} + 72 q^{42} - 487 q^{43} - 364 q^{44} + 980 q^{45} - 1625 q^{46} - 784 q^{47} + 13 q^{48} - 11 q^{49} - 225 q^{50} - 1652 q^{51} + 1521 q^{52} - 719 q^{53} - 960 q^{54} - 955 q^{55} - 2610 q^{56} - 612 q^{57} + 137 q^{58} - 2017 q^{59} - 380 q^{60} - 1493 q^{61} - 279 q^{62} - 244 q^{63} - 1963 q^{64} + 2405 q^{65} - 1641 q^{66} - 2645 q^{67} + 163 q^{68} - 3656 q^{69} - 1210 q^{70} - 1546 q^{71} - 5448 q^{72} - 4272 q^{73} + 2937 q^{74} - 425 q^{75} - 5612 q^{76} - 656 q^{77} - 273 q^{78} - 3974 q^{79} + 625 q^{80} - 767 q^{81} - 1300 q^{82} + 937 q^{83} - 3067 q^{84} - 1360 q^{85} - 2617 q^{86} - 384 q^{87} - 2069 q^{88} - 2298 q^{89} - 1530 q^{90} - 832 q^{91} - 769 q^{92} - 527 q^{93} - 162 q^{94} - 1795 q^{95} + 3378 q^{96} - 7254 q^{97} + 453 q^{98} - 6339 q^{99}+O(q^{100}) \) 37 * q - 9 * q^2 - 17 * q^3 + 117 * q^4 + 185 * q^5 - 21 * q^6 - 64 * q^7 - 87 * q^8 + 196 * q^9 - 45 * q^10 - 191 * q^11 - 76 * q^12 + 481 * q^13 - 242 * q^14 - 85 * q^15 + 125 * q^16 - 272 * q^17 - 306 * q^18 - 359 * q^19 + 585 * q^20 - 680 * q^21 - 191 * q^22 - 218 * q^23 + 171 * q^24 + 925 * q^25 - 117 * q^26 - 476 * q^27 - 997 * q^28 - 921 * q^29 - 105 * q^30 + 1147 * q^31 - 160 * q^32 - 938 * q^33 - 1250 * q^34 - 320 * q^35 + 182 * q^36 - 1651 * q^37 - 596 * q^38 - 221 * q^39 - 435 * q^40 - 1750 * q^41 + 72 * q^42 - 487 * q^43 - 364 * q^44 + 980 * q^45 - 1625 * q^46 - 784 * q^47 + 13 * q^48 - 11 * q^49 - 225 * q^50 - 1652 * q^51 + 1521 * q^52 - 719 * q^53 - 960 * q^54 - 955 * q^55 - 2610 * q^56 - 612 * q^57 + 137 * q^58 - 2017 * q^59 - 380 * q^60 - 1493 * q^61 - 279 * q^62 - 244 * q^63 - 1963 * q^64 + 2405 * q^65 - 1641 * q^66 - 2645 * q^67 + 163 * q^68 - 3656 * q^69 - 1210 * q^70 - 1546 * q^71 - 5448 * q^72 - 4272 * q^73 + 2937 * q^74 - 425 * q^75 - 5612 * q^76 - 656 * q^77 - 273 * q^78 - 3974 * q^79 + 625 * q^80 - 767 * q^81 - 1300 * q^82 + 937 * q^83 - 3067 * q^84 - 1360 * q^85 - 2617 * q^86 - 384 * q^87 - 2069 * q^88 - 2298 * q^89 - 1530 * q^90 - 832 * q^91 - 769 * q^92 - 527 * q^93 - 162 * q^94 - 1795 * q^95 + 3378 * q^96 - 7254 * q^97 + 453 * q^98 - 6339 * q^99

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