LMFDB - Embedded newform 114.4.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [114,4,Mod(1,114)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(114, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("114.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 114 = 2 \cdot 3 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	114.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:	$6.72621774065$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	114.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.3693 q^{5} -6.00000 q^{6} -22.7386 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.3693 q^{5} -6.00000 q^{6} -22.7386 q^{7} +8.00000 q^{8} +9.00000 q^{9} +42.7386 q^{10} +54.7386 q^{11} -12.0000 q^{12} -8.10795 q^{13} -45.4773 q^{14} -64.1080 q^{15} +16.0000 q^{16} +72.7386 q^{17} +18.0000 q^{18} +19.0000 q^{19} +85.4773 q^{20} +68.2159 q^{21} +109.477 q^{22} -27.3693 q^{23} -24.0000 q^{24} +331.648 q^{25} -16.2159 q^{26} -27.0000 q^{27} -90.9545 q^{28} -224.955 q^{29} -128.216 q^{30} -305.062 q^{31} +32.0000 q^{32} -164.216 q^{33} +145.477 q^{34} -485.909 q^{35} +36.0000 q^{36} -165.585 q^{37} +38.0000 q^{38} +24.3239 q^{39} +170.955 q^{40} +371.080 q^{41} +136.432 q^{42} -222.955 q^{43} +218.955 q^{44} +192.324 q^{45} -54.7386 q^{46} -541.062 q^{47} -48.0000 q^{48} +174.045 q^{49} +663.295 q^{50} -218.216 q^{51} -32.4318 q^{52} -452.773 q^{53} -54.0000 q^{54} +1169.73 q^{55} -181.909 q^{56} -57.0000 q^{57} -449.909 q^{58} +341.727 q^{59} -256.432 q^{60} -254.250 q^{61} -610.125 q^{62} -204.648 q^{63} +64.0000 q^{64} -173.261 q^{65} -328.432 q^{66} +251.602 q^{67} +290.955 q^{68} +82.1080 q^{69} -971.818 q^{70} +353.727 q^{71} +72.0000 q^{72} +481.170 q^{73} -331.170 q^{74} -994.943 q^{75} +76.0000 q^{76} -1244.68 q^{77} +48.6477 q^{78} +583.403 q^{79} +341.909 q^{80} +81.0000 q^{81} +742.159 q^{82} -523.386 q^{83} +272.864 q^{84} +1554.37 q^{85} -445.909 q^{86} +674.864 q^{87} +437.909 q^{88} -412.523 q^{89} +384.648 q^{90} +184.364 q^{91} -109.477 q^{92} +915.187 q^{93} -1082.12 q^{94} +406.017 q^{95} -96.0000 q^{96} -1390.10 q^{97} +348.091 q^{98} +492.648 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) $ 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} + 16 q^{8} + 18 q^{9} + 36 q^{10} + 60 q^{11} - 24 q^{12} + 58 q^{13} + 8 q^{14} - 54 q^{15} + 32 q^{16} + 96 q^{17} + 36 q^{18} + 38 q^{19} + 72 q^{20} - 12 q^{21} + 120 q^{22} - 30 q^{23} - 48 q^{24} + 218 q^{25} + 116 q^{26} - 54 q^{27} + 16 q^{28} - 252 q^{29} - 108 q^{30} - 338 q^{31} + 64 q^{32} - 180 q^{33} + 192 q^{34} - 576 q^{35} + 72 q^{36} - 158 q^{37} + 76 q^{38} - 174 q^{39} + 144 q^{40} - 24 q^{42} - 248 q^{43} + 240 q^{44} + 162 q^{45} - 60 q^{46} - 810 q^{47} - 96 q^{48} + 546 q^{49} + 436 q^{50} - 288 q^{51} + 232 q^{52} + 84 q^{53} - 108 q^{54} + 1152 q^{55} + 32 q^{56} - 114 q^{57} - 504 q^{58} - 504 q^{59} - 216 q^{60} + 580 q^{61} - 676 q^{62} + 36 q^{63} + 128 q^{64} - 396 q^{65} - 360 q^{66} - 140 q^{67} + 384 q^{68} + 90 q^{69} - 1152 q^{70} - 480 q^{71} + 144 q^{72} + 616 q^{73} - 316 q^{74} - 654 q^{75} + 152 q^{76} - 1104 q^{77} - 348 q^{78} + 202 q^{79} + 288 q^{80} + 162 q^{81} - 552 q^{83} - 48 q^{84} + 1476 q^{85} - 496 q^{86} + 756 q^{87} + 480 q^{88} - 924 q^{89} + 324 q^{90} + 1952 q^{91} - 120 q^{92} + 1014 q^{93} - 1620 q^{94} + 342 q^{95} - 192 q^{96} + 40 q^{97} + 1092 q^{98} + 540 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 + 16 * q^8 + 18 * q^9 + 36 * q^10 + 60 * q^11 - 24 * q^12 + 58 * q^13 + 8 * q^14 - 54 * q^15 + 32 * q^16 + 96 * q^17 + 36 * q^18 + 38 * q^19 + 72 * q^20 - 12 * q^21 + 120 * q^22 - 30 * q^23 - 48 * q^24 + 218 * q^25 + 116 * q^26 - 54 * q^27 + 16 * q^28 - 252 * q^29 - 108 * q^30 - 338 * q^31 + 64 * q^32 - 180 * q^33 + 192 * q^34 - 576 * q^35 + 72 * q^36 - 158 * q^37 + 76 * q^38 - 174 * q^39 + 144 * q^40 - 24 * q^42 - 248 * q^43 + 240 * q^44 + 162 * q^45 - 60 * q^46 - 810 * q^47 - 96 * q^48 + 546 * q^49 + 436 * q^50 - 288 * q^51 + 232 * q^52 + 84 * q^53 - 108 * q^54 + 1152 * q^55 + 32 * q^56 - 114 * q^57 - 504 * q^58 - 504 * q^59 - 216 * q^60 + 580 * q^61 - 676 * q^62 + 36 * q^63 + 128 * q^64 - 396 * q^65 - 360 * q^66 - 140 * q^67 + 384 * q^68 + 90 * q^69 - 1152 * q^70 - 480 * q^71 + 144 * q^72 + 616 * q^73 - 316 * q^74 - 654 * q^75 + 152 * q^76 - 1104 * q^77 - 348 * q^78 + 202 * q^79 + 288 * q^80 + 162 * q^81 - 552 * q^83 - 48 * q^84 + 1476 * q^85 - 496 * q^86 + 756 * q^87 + 480 * q^88 - 924 * q^89 + 324 * q^90 + 1952 * q^91 - 120 * q^92 + 1014 * q^93 - 1620 * q^94 + 342 * q^95 - 192 * q^96 + 40 * q^97 + 1092 * q^98 + 540 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	4.00000	0.500000
$5$	21.3693	1.91133	0.955665	−	0.294456i	$-0.0951386\pi$
$5$	21.3693	1.91133	0.955665	+	0.294456i	$0.0951386\pi$
$6$	−6.00000	−0.408248
$7$	−22.7386	−1.22777	−0.613885	−	0.789395i	$-0.710394\pi$
$7$	−22.7386	−1.22777	−0.613885	+	0.789395i	$0.710394\pi$
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	42.7386	1.35151
$11$	54.7386	1.50039	0.750196	−	0.661215i	$-0.229959\pi$
$11$	54.7386	1.50039	0.750196	+	0.661215i	$0.229959\pi$
$12$	−12.0000	−0.288675
$13$	−8.10795	−0.172980	−0.0864900	−	0.996253i	$-0.527565\pi$
$13$	−8.10795	−0.172980	−0.0864900	+	0.996253i	$0.527565\pi$
$14$	−45.4773	−0.868165
$15$	−64.1080	−1.10351
$16$	16.0000	0.250000
$17$	72.7386	1.03775	0.518874	−	0.854851i	$-0.326351\pi$
$17$	72.7386	1.03775	0.518874	+	0.854851i	$0.326351\pi$
$18$	18.0000	0.235702
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	85.4773	0.955665
$21$	68.2159	0.708854
$22$	109.477	1.06094
$23$	−27.3693	−0.248126	−0.124063	−	0.992274i	$-0.539592\pi$
$23$	−27.3693	−0.248126	−0.124063	+	0.992274i	$0.539592\pi$
$24$	−24.0000	−0.204124
$25$	331.648	2.65318
$26$	−16.2159	−0.122315
$27$	−27.0000	−0.192450
$28$	−90.9545	−0.613885
$29$	−224.955	−1.44045	−0.720224	−	0.693741i	$-0.755961\pi$
$29$	−224.955	−1.44045	−0.720224	+	0.693741i	$0.755961\pi$
$30$	−128.216	−0.780297
$31$	−305.062	−1.76745	−0.883723	−	0.468010i	$-0.844971\pi$
$31$	−305.062	−1.76745	−0.883723	+	0.468010i	$0.844971\pi$
$32$	32.0000	0.176777
$33$	−164.216	−0.866252
$34$	145.477	0.733798
$35$	−485.909	−2.34667
$36$	36.0000	0.166667
$37$	−165.585	−0.735731	−0.367865	−	0.929879i	$-0.619911\pi$
$37$	−165.585	−0.735731	−0.367865	+	0.929879i	$0.619911\pi$
$38$	38.0000	0.162221
$39$	24.3239	0.0998701
$40$	170.955	0.675757
$41$	371.080	1.41348	0.706742	−	0.707471i	$-0.250164\pi$
$41$	371.080	1.41348	0.706742	+	0.707471i	$0.250164\pi$
$42$	136.432	0.501235
$43$	−222.955	−0.790703	−0.395352	−	0.918530i	$-0.629377\pi$
$43$	−222.955	−0.790703	−0.395352	+	0.918530i	$0.629377\pi$
$44$	218.955	0.750196
$45$	192.324	0.637110
$46$	−54.7386	−0.175452
$47$	−541.062	−1.67919	−0.839597	−	0.543211i	$-0.817209\pi$
$47$	−541.062	−1.67919	−0.839597	+	0.543211i	$0.817209\pi$
$48$	−48.0000	−0.144338
$49$	174.045	0.507421
$50$	663.295	1.87608
$51$	−218.216	−0.599144
$52$	−32.4318	−0.0864900
$53$	−452.773	−1.17345	−0.586727	−	0.809784i	$-0.699584\pi$
$53$	−452.773	−1.17345	−0.586727	+	0.809784i	$0.699584\pi$
$54$	−54.0000	−0.136083
$55$	1169.73	2.86775
$56$	−181.909	−0.434083
$57$	−57.0000	−0.132453
$58$	−449.909	−1.01855
$59$	341.727	0.754052	0.377026	−	0.926203i	$-0.376947\pi$
$59$	341.727	0.754052	0.377026	+	0.926203i	$0.376947\pi$
$60$	−256.432	−0.551753
$61$	−254.250	−0.533662	−0.266831	−	0.963743i	$-0.585977\pi$
$61$	−254.250	−0.533662	−0.266831	+	0.963743i	$0.585977\pi$
$62$	−610.125	−1.24977
$63$	−204.648	−0.409257
$64$	64.0000	0.125000
$65$	−173.261	−0.330622
$66$	−328.432	−0.612533
$67$	251.602	0.458778	0.229389	−	0.973335i	$-0.426327\pi$
$67$	251.602	0.458778	0.229389	+	0.973335i	$0.426327\pi$
$68$	290.955	0.518874
$69$	82.1080	0.143256
$70$	−971.818	−1.65935
$71$	353.727	0.591263	0.295632	−	0.955302i	$-0.404470\pi$
$71$	353.727	0.591263	0.295632	+	0.955302i	$0.404470\pi$
$72$	72.0000	0.117851
$73$	481.170	0.771462	0.385731	−	0.922611i	$-0.373949\pi$
$73$	481.170	0.771462	0.385731	+	0.922611i	$0.373949\pi$
$74$	−331.170	−0.520240
$75$	−994.943	−1.53182
$76$	76.0000	0.114708
$77$	−1244.68	−1.84214
$78$	48.6477	0.0706188
$79$	583.403	0.830861	0.415430	−	0.909625i	$-0.363631\pi$
$79$	583.403	0.830861	0.415430	+	0.909625i	$0.363631\pi$
$80$	341.909	0.477832
$81$	81.0000	0.111111
$82$	742.159	0.999485
$83$	−523.386	−0.692158	−0.346079	−	0.938205i	$-0.612487\pi$
$83$	−523.386	−0.692158	−0.346079	+	0.938205i	$0.612487\pi$
$84$	272.864	0.354427
$85$	1554.37	1.98348
$86$	−445.909	−0.559112
$87$	674.864	0.831643
$88$	437.909	0.530469
$89$	−412.523	−0.491318	−0.245659	−	0.969356i	$-0.579004\pi$
$89$	−412.523	−0.491318	−0.245659	+	0.969356i	$0.579004\pi$
$90$	384.648	0.450505
$91$	184.364	0.212380
$92$	−109.477	−0.124063
$93$	915.187	1.02044
$94$	−1082.12	−1.18737
$95$	406.017	0.438489
$96$	−96.0000	−0.102062
$97$	−1390.10	−1.45509	−0.727544	−	0.686061i	$-0.759338\pi$
$97$	−1390.10	−1.45509	−0.727544	+	0.686061i	$0.759338\pi$
$98$	348.091	0.358801
$99$	492.648	0.500131
$100$	1326.59	1.32659
$101$	336.688	0.331700	0.165850	−	0.986151i	$-0.446963\pi$
$101$	336.688	0.331700	0.165850	+	0.986151i	$0.446963\pi$
$102$	−436.432	−0.423659
$103$	−257.528	−0.246359	−0.123180	−	0.992384i	$-0.539309\pi$
$103$	−257.528	−0.246359	−0.123180	+	0.992384i	$0.539309\pi$
$104$	−64.8636	−0.0611577
$105$	1457.73	1.35485
$106$	−905.545	−0.829758
$107$	−382.523	−0.345606	−0.172803	−	0.984956i	$-0.555282\pi$
$107$	−382.523	−0.345606	−0.172803	+	0.984956i	$0.555282\pi$
$108$	−108.000	−0.0962250
$109$	−284.756	−0.250226	−0.125113	−	0.992142i	$-0.539929\pi$
$109$	−284.756	−0.250226	−0.125113	+	0.992142i	$0.539929\pi$
$110$	2339.45	2.02780
$111$	496.756	0.424774
$112$	−363.818	−0.306943
$113$	1249.85	1.04050	0.520249	−	0.854015i	$-0.325839\pi$
$113$	1249.85	1.04050	0.520249	+	0.854015i	$0.325839\pi$
$114$	−114.000	−0.0936586
$115$	−584.864	−0.474251
$116$	−899.818	−0.720224
$117$	−72.9716	−0.0576600
$118$	683.454	0.533196
$119$	−1653.98	−1.27412
$120$	−512.864	−0.390149
$121$	1665.32	1.25118
$122$	−508.500	−0.377356
$123$	−1113.24	−0.816076
$124$	−1220.25	−0.883723
$125$	4415.92	3.15978
$126$	−409.295	−0.289388
$127$	414.210	0.289411	0.144706	−	0.989475i	$-0.453777\pi$
$127$	414.210	0.289411	0.144706	+	0.989475i	$0.453777\pi$
$128$	128.000	0.0883883
$129$	668.864	0.456513
$130$	−346.523	−0.233785
$131$	2042.95	1.36255	0.681274	−	0.732029i	$-0.261426\pi$
$131$	2042.95	1.36255	0.681274	+	0.732029i	$0.261426\pi$
$132$	−656.864	−0.433126
$133$	−432.034	−0.281670
$134$	503.204	0.324405
$135$	−576.972	−0.367836
$136$	581.909	0.366899
$137$	564.193	0.351841	0.175921	−	0.984404i	$-0.443710\pi$
$137$	564.193	0.351841	0.175921	+	0.984404i	$0.443710\pi$
$138$	164.216	0.101297
$139$	−749.659	−0.457448	−0.228724	−	0.973491i	$-0.573455\pi$
$139$	−749.659	−0.457448	−0.228724	+	0.973491i	$0.573455\pi$
$140$	−1943.64	−1.17334
$141$	1623.19	0.969483
$142$	707.454	0.418086
$143$	−443.818	−0.259538
$144$	144.000	0.0833333
$145$	−4807.12	−2.75317
$146$	962.341	0.545506
$147$	−522.136	−0.292960
$148$	−662.341	−0.367865
$149$	−659.960	−0.362859	−0.181430	−	0.983404i	$-0.558072\pi$
$149$	−659.960	−0.362859	−0.181430	+	0.983404i	$0.558072\pi$
$150$	−1989.89	−1.08316
$151$	−870.438	−0.469107	−0.234554	−	0.972103i	$-0.575363\pi$
$151$	−870.438	−0.469107	−0.234554	+	0.972103i	$0.575363\pi$
$152$	152.000	0.0811107
$153$	654.648	0.345916
$154$	−2489.36	−1.30259
$155$	−6518.98	−3.37817
$156$	97.2954	0.0499350
$157$	340.875	0.173279	0.0866394	−	0.996240i	$-0.472387\pi$
$157$	340.875	0.173279	0.0866394	+	0.996240i	$0.472387\pi$
$158$	1166.81	0.587507
$159$	1358.32	0.677495
$160$	683.818	0.337879
$161$	622.341	0.304642
$162$	162.000	0.0785674
$163$	447.386	0.214982	0.107491	−	0.994206i	$-0.465718\pi$
$163$	447.386	0.214982	0.107491	+	0.994206i	$0.465718\pi$
$164$	1484.32	0.706742
$165$	−3509.18	−1.65569
$166$	−1046.77	−0.489430
$167$	2165.08	1.00323	0.501613	−	0.865092i	$-0.332740\pi$
$167$	2165.08	1.00323	0.501613	+	0.865092i	$0.332740\pi$
$168$	545.727	0.250618
$169$	−2131.26	−0.970078
$170$	3108.75	1.40253
$171$	171.000	0.0764719
$172$	−891.818	−0.395352
$173$	614.784	0.270180	0.135090	−	0.990833i	$-0.456868\pi$
$173$	614.784	0.270180	0.135090	+	0.990833i	$0.456868\pi$
$174$	1349.73	0.588061
$175$	−7541.22	−3.25750
$176$	875.818	0.375098
$177$	−1025.18	−0.435352
$178$	−825.045	−0.347414
$179$	−2425.11	−1.01263	−0.506317	−	0.862347i	$-0.668993\pi$
$179$	−2425.11	−1.01263	−0.506317	+	0.862347i	$0.668993\pi$
$180$	769.295	0.318555
$181$	3139.68	1.28934	0.644669	−	0.764462i	$-0.276995\pi$
$181$	3139.68	1.28934	0.644669	+	0.764462i	$0.276995\pi$
$182$	368.727	0.150175
$183$	762.750	0.308110
$184$	−218.955	−0.0877258
$185$	−3538.44	−1.40622
$186$	1830.37	0.721557
$187$	3981.61	1.55703
$188$	−2164.25	−0.839597
$189$	613.943	0.236285
$190$	812.034	0.310059
$191$	4714.68	1.78608	0.893042	−	0.449974i	$-0.148567\pi$
$191$	4714.68	1.78608	0.893042	+	0.449974i	$0.148567\pi$
$192$	−192.000	−0.0721688
$193$	−1606.83	−0.599286	−0.299643	−	0.954051i	$-0.596868\pi$
$193$	−1606.83	−0.599286	−0.299643	+	0.954051i	$0.596868\pi$
$194$	−2780.20	−1.02890
$195$	519.784	0.190885
$196$	696.182	0.253711
$197$	−4161.01	−1.50487	−0.752435	−	0.658667i	$-0.771121\pi$
$197$	−4161.01	−1.50487	−0.752435	+	0.658667i	$0.771121\pi$
$198$	985.295	0.353646
$199$	−2200.10	−0.783724	−0.391862	−	0.920024i	$-0.628169\pi$
$199$	−2200.10	−0.783724	−0.391862	+	0.920024i	$0.628169\pi$
$200$	2653.18	0.938041
$201$	−754.807	−0.264875
$202$	673.375	0.234547
$203$	5115.16	1.76854
$204$	−872.864	−0.299572
$205$	7929.72	2.70164
$206$	−515.057	−0.174202
$207$	−246.324	−0.0827087
$208$	−129.727	−0.0432450
$209$	1040.03	0.344214
$210$	2915.45	0.958026
$211$	−1739.57	−0.567568	−0.283784	−	0.958888i	$-0.591590\pi$
$211$	−1739.57	−0.567568	−0.283784	+	0.958888i	$0.591590\pi$
$212$	−1811.09	−0.586727
$213$	−1061.18	−0.341366
$214$	−765.045	−0.244381
$215$	−4764.39	−1.51130
$216$	−216.000	−0.0680414
$217$	6936.70	2.17002
$218$	−569.511	−0.176937
$219$	−1443.51	−0.445404
$220$	4678.91	1.43387
$221$	−589.761	−0.179510
$222$	993.511	0.300361
$223$	−94.0399	−0.0282394	−0.0141197	−	0.999900i	$-0.504495\pi$
$223$	−94.0399	−0.0282394	−0.0141197	+	0.999900i	$0.504495\pi$
$224$	−727.636	−0.217041
$225$	2984.83	0.884394
$226$	2499.70	0.735743
$227$	1410.48	0.412408	0.206204	−	0.978509i	$-0.433889\pi$
$227$	1410.48	0.412408	0.206204	+	0.978509i	$0.433889\pi$
$228$	−228.000	−0.0662266
$229$	3651.11	1.05359	0.526796	−	0.849992i	$-0.323393\pi$
$229$	3651.11	1.05359	0.526796	+	0.849992i	$0.323393\pi$
$230$	−1169.73	−0.335346
$231$	3734.05	1.06356
$232$	−1799.64	−0.509275
$233$	−7015.86	−1.97264	−0.986319	−	0.164850i	$-0.947286\pi$
$233$	−7015.86	−1.97264	−0.986319	+	0.164850i	$0.947286\pi$
$234$	−145.943	−0.0407718
$235$	−11562.1	−3.20949
$236$	1366.91	0.377026
$237$	−1750.21	−0.479698
$238$	−3307.95	−0.900936
$239$	316.460	0.0856491	0.0428245	−	0.999083i	$-0.486364\pi$
$239$	316.460	0.0856491	0.0428245	+	0.999083i	$0.486364\pi$
$240$	−1025.73	−0.275877
$241$	3681.10	0.983903	0.491952	−	0.870623i	$-0.336284\pi$
$241$	3681.10	0.983903	0.491952	+	0.870623i	$0.336284\pi$
$242$	3330.64	0.884717
$243$	−243.000	−0.0641500
$244$	−1017.00	−0.266831
$245$	3719.23	0.969849
$246$	−2226.48	−0.577053
$247$	−154.051	−0.0396843
$248$	−2440.50	−0.624887
$249$	1570.16	0.399618
$250$	8831.84	2.23430
$251$	7650.66	1.92393	0.961963	−	0.273181i	$-0.0880757\pi$
$251$	7650.66	1.92393	0.961963	+	0.273181i	$0.0880757\pi$
$252$	−818.591	−0.204628
$253$	−1498.16	−0.372286
$254$	828.420	0.204645
$255$	−4663.12	−1.14516
$256$	256.000	0.0625000
$257$	1979.70	0.480508	0.240254	−	0.970710i	$-0.422769\pi$
$257$	1979.70	0.480508	0.240254	+	0.970710i	$0.422769\pi$
$258$	1337.73	0.322803
$259$	3765.18	0.903309
$260$	−693.045	−0.165311
$261$	−2024.59	−0.480150
$262$	4085.91	0.963467
$263$	−1982.93	−0.464914	−0.232457	−	0.972607i	$-0.574677\pi$
$263$	−1982.93	−0.464914	−0.232457	+	0.972607i	$0.574677\pi$
$264$	−1313.73	−0.306266
$265$	−9675.44	−2.24286
$266$	−864.068	−0.199171
$267$	1237.57	0.283663
$268$	1006.41	0.229389
$269$	−1255.56	−0.284582	−0.142291	−	0.989825i	$-0.545447\pi$
$269$	−1255.56	−0.284582	−0.142291	+	0.989825i	$0.545447\pi$
$270$	−1153.94	−0.260099
$271$	−18.4887	−0.00414431	−0.00207215	−	0.999998i	$-0.500660\pi$
$271$	−18.4887	−0.00414431	−0.00207215	+	0.999998i	$0.500660\pi$
$272$	1163.82	0.259437
$273$	−553.091	−0.122618
$274$	1128.39	0.248789
$275$	18153.9	3.98081
$276$	328.432	0.0716278
$277$	1266.98	0.274821	0.137410	−	0.990514i	$-0.456122\pi$
$277$	1266.98	0.274821	0.137410	+	0.990514i	$0.456122\pi$
$278$	−1499.32	−0.323465
$279$	−2745.56	−0.589149
$280$	−3887.27	−0.829675
$281$	2698.15	0.572804	0.286402	−	0.958109i	$-0.407541\pi$
$281$	2698.15	0.572804	0.286402	+	0.958109i	$0.407541\pi$
$282$	3246.37	0.685528
$283$	3548.00	0.745253	0.372627	−	0.927981i	$-0.378457\pi$
$283$	3548.00	0.745253	0.372627	+	0.927981i	$0.378457\pi$
$284$	1414.91	0.295632
$285$	−1218.05	−0.253162
$286$	−887.636	−0.183521
$287$	−8437.84	−1.73544
$288$	288.000	0.0589256
$289$	377.909	0.0769202
$290$	−9614.25	−1.94679
$291$	4170.31	0.840095
$292$	1924.68	0.385731
$293$	−4325.53	−0.862459	−0.431229	−	0.902242i	$-0.641920\pi$
$293$	−4325.53	−0.862459	−0.431229	+	0.902242i	$0.641920\pi$
$294$	−1044.27	−0.207154
$295$	7302.48	1.44124
$296$	−1324.68	−0.260120
$297$	−1477.94	−0.288751
$298$	−1319.92	−0.256580
$299$	221.909	0.0429208
$300$	−3979.77	−0.765908
$301$	5069.68	0.970803
$302$	−1740.88	−0.331709
$303$	−1010.06	−0.191507
$304$	304.000	0.0573539
$305$	−5433.15	−1.02000
$306$	1309.30	0.244599
$307$	4108.76	0.763842	0.381921	−	0.924195i	$-0.375263\pi$
$307$	4108.76	0.763842	0.381921	+	0.924195i	$0.375263\pi$
$308$	−4978.73	−0.921069
$309$	772.585	0.142236
$310$	−13038.0	−2.38873
$311$	−10394.8	−1.89529	−0.947644	−	0.319330i	$-0.896542\pi$
$311$	−10394.8	−1.89529	−0.947644	+	0.319330i	$0.896542\pi$
$312$	194.591	0.0353094
$313$	2629.15	0.474787	0.237393	−	0.971414i	$-0.423707\pi$
$313$	2629.15	0.474787	0.237393	+	0.971414i	$0.423707\pi$
$314$	681.750	0.122527
$315$	−4373.18	−0.782225
$316$	2333.61	0.415430
$317$	−8890.03	−1.57512	−0.787562	−	0.616236i	$-0.788657\pi$
$317$	−8890.03	−1.57512	−0.787562	+	0.616236i	$0.788657\pi$
$318$	2716.64	0.479061
$319$	−12313.7	−2.16124
$320$	1367.64	0.238916
$321$	1147.57	0.199536
$322$	1244.68	0.215414
$323$	1382.03	0.238076
$324$	324.000	0.0555556
$325$	−2688.98	−0.458948
$326$	894.773	0.152015
$327$	854.267	0.144468
$328$	2968.64	0.499742
$329$	12303.0	2.06166
$330$	−7018.36	−1.17075
$331$	302.716	0.0502682	0.0251341	−	0.999684i	$-0.491999\pi$
$331$	302.716	0.0502682	0.0251341	+	0.999684i	$0.491999\pi$
$332$	−2093.55	−0.346079
$333$	−1490.27	−0.245244
$334$	4330.16	0.709389
$335$	5376.57	0.876875
$336$	1091.45	0.177213
$337$	−5927.86	−0.958194	−0.479097	−	0.877762i	$-0.659036\pi$
$337$	−5927.86	−0.958194	−0.479097	+	0.877762i	$0.659036\pi$
$338$	−4262.52	−0.685949
$339$	−3749.56	−0.600731
$340$	6217.50	0.991739
$341$	−16698.7	−2.65186
$342$	342.000	0.0540738
$343$	3841.80	0.604774
$344$	−1783.64	−0.279556
$345$	1754.59	0.273809
$346$	1229.57	0.191046
$347$	12248.1	1.89485	0.947423	−	0.319985i	$-0.103678\pi$
$347$	12248.1	1.89485	0.947423	+	0.319985i	$0.103678\pi$
$348$	2699.45	0.415822
$349$	11493.4	1.76282	0.881411	−	0.472350i	$-0.156594\pi$
$349$	11493.4	1.76282	0.881411	+	0.472350i	$0.156594\pi$
$350$	−15082.4	−2.30340
$351$	218.915	0.0332900
$352$	1751.64	0.265234
$353$	1653.89	0.249370	0.124685	−	0.992196i	$-0.460208\pi$
$353$	1653.89	0.249370	0.124685	+	0.992196i	$0.460208\pi$
$354$	−2050.36	−0.307841
$355$	7558.91	1.13010
$356$	−1650.09	−0.245659
$357$	4961.93	0.735611
$358$	−4850.23	−0.716040
$359$	998.722	0.146826	0.0734130	−	0.997302i	$-0.476611\pi$
$359$	998.722	0.146826	0.0734130	+	0.997302i	$0.476611\pi$
$360$	1538.59	0.225252
$361$	361.000	0.0526316
$362$	6279.35	0.911700
$363$	−4995.95	−0.722368
$364$	737.455	0.106190
$365$	10282.3	1.47452
$366$	1525.50	0.217867
$367$	4653.99	0.661951	0.330976	−	0.943639i	$-0.392622\pi$
$367$	4653.99	0.661951	0.330976	+	0.943639i	$0.392622\pi$
$368$	−437.909	−0.0620315
$369$	3339.72	0.471162
$370$	−7076.89	−0.994351
$371$	10295.4	1.44073
$372$	3660.75	0.510218
$373$	11333.1	1.57320	0.786599	−	0.617464i	$-0.211840\pi$
$373$	11333.1	1.57320	0.786599	+	0.617464i	$0.211840\pi$
$374$	7963.23	1.10099
$375$	−13247.8	−1.82430
$376$	−4328.50	−0.593684
$377$	1823.92	0.249169
$378$	1227.89	0.167078
$379$	−11569.9	−1.56809	−0.784043	−	0.620707i	$-0.786846\pi$
$379$	−11569.9	−1.56809	−0.784043	+	0.620707i	$0.786846\pi$
$380$	1624.07	0.219245
$381$	−1242.63	−0.167092
$382$	9429.35	1.26295
$383$	6334.91	0.845166	0.422583	−	0.906324i	$-0.361123\pi$
$383$	6334.91	0.845166	0.422583	+	0.906324i	$0.361123\pi$
$384$	−384.000	−0.0510310
$385$	−26598.0	−3.52093
$386$	−3213.66	−0.423759
$387$	−2006.59	−0.263568
$388$	−5560.41	−0.727544
$389$	−9212.97	−1.20081	−0.600406	−	0.799695i	$-0.704994\pi$
$389$	−9212.97	−1.20081	−0.600406	+	0.799695i	$0.704994\pi$
$390$	1039.57	0.134976
$391$	−1990.81	−0.257492
$392$	1392.36	0.179400
$393$	−6128.86	−0.786667
$394$	−8322.01	−1.06410
$395$	12466.9	1.58805
$396$	1970.59	0.250065
$397$	−2805.03	−0.354611	−0.177306	−	0.984156i	$-0.556738\pi$
$397$	−2805.03	−0.354611	−0.177306	+	0.984156i	$0.556738\pi$
$398$	−4400.20	−0.554177
$399$	1296.10	0.162622
$400$	5306.36	0.663295
$401$	−13872.4	−1.72757	−0.863785	−	0.503861i	$-0.831913\pi$
$401$	−13872.4	−1.72757	−0.863785	+	0.503861i	$0.831913\pi$
$402$	−1509.61	−0.187295
$403$	2473.43	0.305733
$404$	1346.75	0.165850
$405$	1730.91	0.212370
$406$	10230.3	1.25055
$407$	−9063.91	−1.10389
$408$	−1745.73	−0.211829
$409$	8128.35	0.982692	0.491346	−	0.870964i	$-0.336505\pi$
$409$	8128.35	0.982692	0.491346	+	0.870964i	$0.336505\pi$
$410$	15859.4	1.91035
$411$	−1692.58	−0.203136
$412$	−1030.11	−0.123180
$413$	−7770.41	−0.925804
$414$	−492.648	−0.0584838
$415$	−11184.4	−1.32294
$416$	−259.454	−0.0305788
$417$	2248.98	0.264108
$418$	2080.07	0.243396
$419$	−1031.78	−0.120300	−0.0601502	−	0.998189i	$-0.519158\pi$
$419$	−1031.78	−0.120300	−0.0601502	+	0.998189i	$0.519158\pi$
$420$	5830.91	0.677427
$421$	−843.335	−0.0976286	−0.0488143	−	0.998808i	$-0.515544\pi$
$421$	−843.335	−0.0976286	−0.0488143	+	0.998808i	$0.515544\pi$
$422$	−3479.14	−0.401331
$423$	−4869.56	−0.559731
$424$	−3622.18	−0.414879
$425$	24123.6	2.75333
$426$	−2122.36	−0.241382
$427$	5781.30	0.655214
$428$	−1530.09	−0.172803
$429$	1331.45	0.149844
$430$	−9528.77	−1.06865
$431$	10461.4	1.16916	0.584580	−	0.811336i	$-0.301259\pi$
$431$	10461.4	1.16916	0.584580	+	0.811336i	$0.301259\pi$
$432$	−432.000	−0.0481125
$433$	−7440.65	−0.825808	−0.412904	−	0.910775i	$-0.635485\pi$
$433$	−7440.65	−0.825808	−0.412904	+	0.910775i	$0.635485\pi$
$434$	13873.4	1.53444
$435$	14421.4	1.58954
$436$	−1139.02	−0.125113
$437$	−520.017	−0.0569240
$438$	−2887.02	−0.314948
$439$	755.392	0.0821250	0.0410625	−	0.999157i	$-0.486926\pi$
$439$	755.392	0.0821250	0.0410625	+	0.999157i	$0.486926\pi$
$440$	9357.82	1.01390
$441$	1566.41	0.169140
$442$	−1179.52	−0.126932
$443$	−15130.1	−1.62270	−0.811349	−	0.584562i	$-0.801266\pi$
$443$	−15130.1	−1.62270	−0.811349	+	0.584562i	$0.801266\pi$
$444$	1987.02	0.212387
$445$	−8815.33	−0.939071
$446$	−188.080	−0.0199682
$447$	1979.88	0.209497
$448$	−1455.27	−0.153471
$449$	−1225.61	−0.128820	−0.0644101	−	0.997924i	$-0.520517\pi$
$449$	−1225.61	−0.128820	−0.0644101	+	0.997924i	$0.520517\pi$
$450$	5969.66	0.625361
$451$	20312.4	2.12078
$452$	4999.41	0.520249
$453$	2611.31	0.270839
$454$	2820.95	0.291617
$455$	3939.73	0.405928
$456$	−456.000	−0.0468293
$457$	6333.39	0.648279	0.324139	−	0.946009i	$-0.394925\pi$
$457$	6333.39	0.648279	0.324139	+	0.946009i	$0.394925\pi$
$458$	7302.23	0.745001
$459$	−1963.94	−0.199715
$460$	−2339.45	−0.237125
$461$	14291.6	1.44388	0.721938	−	0.691958i	$-0.243252\pi$
$461$	14291.6	1.44388	0.721938	+	0.691958i	$0.243252\pi$
$462$	7468.09	0.752050
$463$	−7529.19	−0.755748	−0.377874	−	0.925857i	$-0.623345\pi$
$463$	−7529.19	−0.755748	−0.377874	+	0.925857i	$0.623345\pi$
$464$	−3599.27	−0.360112
$465$	19556.9	1.95039
$466$	−14031.7	−1.39487
$467$	−592.636	−0.0587236	−0.0293618	−	0.999569i	$-0.509347\pi$
$467$	−592.636	−0.0587236	−0.0293618	+	0.999569i	$0.509347\pi$
$468$	−291.886	−0.0288300
$469$	−5721.09	−0.563274
$470$	−23124.3	−2.26945
$471$	−1022.62	−0.100043
$472$	2733.82	0.266598
$473$	−12204.2	−1.18637
$474$	−3500.42	−0.339198
$475$	6301.31	0.608682
$476$	−6615.91	−0.637058
$477$	−4074.95	−0.391152
$478$	632.921	0.0605630
$479$	13979.2	1.33346	0.666729	−	0.745301i	$-0.267694\pi$
$479$	13979.2	1.33346	0.666729	+	0.745301i	$0.267694\pi$
$480$	−2051.45	−0.195074
$481$	1342.56	0.127267
$482$	7362.20	0.695725
$483$	−1867.02	−0.175885
$484$	6661.27	0.625589
$485$	−29705.5	−2.78115
$486$	−486.000	−0.0453609
$487$	10744.9	0.999788	0.499894	−	0.866086i	$-0.333372\pi$
$487$	10744.9	0.999788	0.499894	+	0.866086i	$0.333372\pi$
$488$	−2034.00	−0.188678
$489$	−1342.16	−0.124120
$490$	7438.47	0.685787
$491$	5322.05	0.489166	0.244583	−	0.969628i	$-0.421349\pi$
$491$	5322.05	0.489166	0.244583	+	0.969628i	$0.421349\pi$
$492$	−4452.95	−0.408038
$493$	−16362.9	−1.49482
$494$	−308.102	−0.0280611
$495$	10527.5	0.955915
$496$	−4881.00	−0.441862
$497$	−8043.27	−0.725936
$498$	3140.32	0.282572
$499$	43.7958	0.00392899	0.00196450	−	0.999998i	$-0.499375\pi$
$499$	43.7958	0.00392899	0.00196450	+	0.999998i	$0.499375\pi$
$500$	17663.7	1.57989
$501$	−6495.24	−0.579213
$502$	15301.3	1.36042
$503$	−3382.49	−0.299837	−0.149918	−	0.988698i	$-0.547901\pi$
$503$	−3382.49	−0.299837	−0.149918	+	0.988698i	$0.547901\pi$
$504$	−1637.18	−0.144694
$505$	7194.78	0.633987
$506$	−2996.32	−0.263246
$507$	6393.78	0.560075
$508$	1656.84	0.144706
$509$	−13475.5	−1.17346	−0.586731	−	0.809782i	$-0.699586\pi$
$509$	−13475.5	−1.17346	−0.586731	+	0.809782i	$0.699586\pi$
$510$	−9326.25	−0.809752
$511$	−10941.2	−0.947179
$512$	512.000	0.0441942
$513$	−513.000	−0.0441511
$514$	3959.41	0.339771
$515$	−5503.20	−0.470874
$516$	2675.45	0.228256
$517$	−29617.0	−2.51945
$518$	7530.36	0.638736
$519$	−1844.35	−0.155989
$520$	−1386.09	−0.116893
$521$	16318.7	1.37224	0.686119	−	0.727489i	$-0.259313\pi$
$521$	16318.7	1.37224	0.686119	+	0.727489i	$0.259313\pi$
$522$	−4049.18	−0.339517
$523$	13301.2	1.11209	0.556044	−	0.831153i	$-0.312319\pi$
$523$	13301.2	1.11209	0.556044	+	0.831153i	$0.312319\pi$
$524$	8171.82	0.681274
$525$	22623.6	1.88072
$526$	−3965.85	−0.328744
$527$	−22189.8	−1.83416
$528$	−2627.45	−0.216563
$529$	−11417.9	−0.938434
$530$	−19350.9	−1.58594
$531$	3075.54	0.251351
$532$	−1728.14	−0.140835
$533$	−3008.69	−0.244505
$534$	2475.14	0.200580
$535$	−8174.25	−0.660568
$536$	2012.82	0.162202
$537$	7275.34	0.584645
$538$	−2511.11	−0.201230
$539$	9527.01	0.761331
$540$	−2307.89	−0.183918
$541$	23091.4	1.83508	0.917539	−	0.397645i	$-0.130172\pi$
$541$	23091.4	1.83508	0.917539	+	0.397645i	$0.130172\pi$
$542$	−36.9774	−0.00293047
$543$	−9419.03	−0.744400
$544$	2327.64	0.183450
$545$	−6085.03	−0.478265
$546$	−1106.18	−0.0867037
$547$	−8357.56	−0.653278	−0.326639	−	0.945149i	$-0.605916\pi$
$547$	−8357.56	−0.653278	−0.326639	+	0.945149i	$0.605916\pi$
$548$	2256.77	0.175921
$549$	−2288.25	−0.177887
$550$	36307.9	2.81486
$551$	−4274.14	−0.330462
$552$	656.864	0.0506485
$553$	−13265.8	−1.02011
$554$	2533.95	0.194327
$555$	10615.3	0.811884
$556$	−2998.64	−0.228724
$557$	−17353.4	−1.32009	−0.660044	−	0.751227i	$-0.729462\pi$
$557$	−17353.4	−1.32009	−0.660044	+	0.751227i	$0.729462\pi$
$558$	−5491.12	−0.416591
$559$	1807.70	0.136776
$560$	−7774.55	−0.586669
$561$	−11944.8	−0.898951
$562$	5396.30	0.405034
$563$	5675.64	0.424866	0.212433	−	0.977176i	$-0.431861\pi$
$563$	5675.64	0.424866	0.212433	+	0.977176i	$0.431861\pi$
$564$	6492.75	0.484741
$565$	26708.5	1.98873
$566$	7096.00	0.526974
$567$	−1841.83	−0.136419
$568$	2829.82	0.209043
$569$	−1674.57	−0.123377	−0.0616885	−	0.998095i	$-0.519649\pi$
$569$	−1674.57	−0.123377	−0.0616885	+	0.998095i	$0.519649\pi$
$570$	−2436.10	−0.179012
$571$	1490.84	0.109264	0.0546320	−	0.998507i	$-0.482601\pi$
$571$	1490.84	0.109264	0.0546320	+	0.998507i	$0.482601\pi$
$572$	−1775.27	−0.129769
$573$	−14144.0	−1.03120
$574$	−16875.7	−1.22714
$575$	−9076.97	−0.658323
$576$	576.000	0.0416667
$577$	−7929.18	−0.572090	−0.286045	−	0.958216i	$-0.592341\pi$
$577$	−7929.18	−0.572090	−0.286045	+	0.958216i	$0.592341\pi$
$578$	755.818	0.0543908
$579$	4820.49	0.345998
$580$	−19228.5	−1.37659
$581$	11901.1	0.849811
$582$	8340.61	0.594037
$583$	−24784.2	−1.76064
$584$	3849.36	0.272753
$585$	−1559.35	−0.110207
$586$	−8651.07	−0.609850
$587$	−13353.5	−0.938939	−0.469470	−	0.882949i	$-0.655555\pi$
$587$	−13353.5	−0.938939	−0.469470	+	0.882949i	$0.655555\pi$
$588$	−2088.55	−0.146480
$589$	−5796.19	−0.405480
$590$	14605.0	1.01911
$591$	12483.0	0.868837
$592$	−2649.36	−0.183933
$593$	−9165.02	−0.634675	−0.317338	−	0.948313i	$-0.602789\pi$
$593$	−9165.02	−0.634675	−0.317338	+	0.948313i	$0.602789\pi$
$594$	−2955.89	−0.204178
$595$	−35344.4	−2.43526
$596$	−2639.84	−0.181430
$597$	6600.31	0.452483
$598$	443.818	0.0303496
$599$	13524.6	0.922536	0.461268	−	0.887261i	$-0.347395\pi$
$599$	13524.6	0.922536	0.461268	+	0.887261i	$0.347395\pi$
$600$	−7959.54	−0.541578
$601$	−26518.9	−1.79988	−0.899941	−	0.436012i	$-0.856390\pi$
$601$	−26518.9	−1.79988	−0.899941	+	0.436012i	$0.856390\pi$
$602$	10139.4	0.686461
$603$	2264.42	0.152926
$604$	−3481.75	−0.234554
$605$	35586.7	2.39141
$606$	−2020.13	−0.135416
$607$	13559.2	0.906674	0.453337	−	0.891339i	$-0.350233\pi$
$607$	13559.2	0.906674	0.453337	+	0.891339i	$0.350233\pi$
$608$	608.000	0.0405554
$609$	−15345.5	−1.02107
$610$	−10866.3	−0.721252
$611$	4386.91	0.290467
$612$	2618.59	0.172958
$613$	−25283.6	−1.66590	−0.832949	−	0.553350i	$-0.813349\pi$
$613$	−25283.6	−1.66590	−0.832949	+	0.553350i	$0.813349\pi$
$614$	8217.52	0.540118
$615$	−23789.1	−1.55979
$616$	−9957.45	−0.651294
$617$	7900.16	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$617$	7900.16	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$618$	1545.17	0.100576
$619$	−8291.45	−0.538387	−0.269194	−	0.963086i	$-0.586757\pi$
$619$	−8291.45	−0.538387	−0.269194	+	0.963086i	$0.586757\pi$
$620$	−26075.9	−1.68909
$621$	738.972	0.0477519
$622$	−20789.6	−1.34017
$623$	9380.20	0.603226
$624$	389.182	0.0249675
$625$	52909.2	3.38619
$626$	5258.30	0.335725
$627$	−3120.10	−0.198732
$628$	1363.50	0.0866394
$629$	−12044.4	−0.763503
$630$	−8746.36	−0.553117
$631$	20749.3	1.30906	0.654529	−	0.756037i	$-0.272867\pi$
$631$	20749.3	1.30906	0.654529	+	0.756037i	$0.272867\pi$
$632$	4667.23	0.293754
$633$	5218.70	0.327685
$634$	−17780.1	−1.11378
$635$	8851.39	0.553160
$636$	5433.27	0.338747
$637$	−1411.15	−0.0877738
$638$	−24627.4	−1.52823
$639$	3183.54	0.197088
$640$	2735.27	0.168939
$641$	10132.5	0.624352	0.312176	−	0.950024i	$-0.398942\pi$
$641$	10132.5	0.624352	0.312176	+	0.950024i	$0.398942\pi$
$642$	2295.14	0.141093
$643$	−1371.80	−0.0841343	−0.0420671	−	0.999115i	$-0.513394\pi$
$643$	−1371.80	−0.0841343	−0.0420671	+	0.999115i	$0.513394\pi$
$644$	2489.36	0.152321
$645$	14293.2	0.872547
$646$	2764.07	0.168345
$647$	16558.2	1.00614	0.503069	−	0.864246i	$-0.332204\pi$
$647$	16558.2	1.00614	0.503069	+	0.864246i	$0.332204\pi$
$648$	648.000	0.0392837
$649$	18705.7	1.13137
$650$	−5377.97	−0.324525
$651$	−20810.1	−1.25286
$652$	1789.55	0.107491
$653$	14690.4	0.880370	0.440185	−	0.897907i	$-0.354913\pi$
$653$	14690.4	0.880370	0.440185	+	0.897907i	$0.354913\pi$
$654$	1708.53	0.102154
$655$	43656.5	2.60428
$656$	5937.27	0.353371
$657$	4330.53	0.257154
$658$	24606.0	1.45782
$659$	−8334.23	−0.492648	−0.246324	−	0.969187i	$-0.579223\pi$
$659$	−8334.23	−0.492648	−0.246324	+	0.969187i	$0.579223\pi$
$660$	−14036.7	−0.827847
$661$	20324.2	1.19594	0.597972	−	0.801517i	$-0.295973\pi$
$661$	20324.2	1.19594	0.597972	+	0.801517i	$0.295973\pi$
$662$	605.432	0.0355450
$663$	1769.28	0.103640
$664$	−4187.09	−0.244715
$665$	−9232.27	−0.538364
$666$	−2980.53	−0.173413
$667$	6156.85	0.357413
$668$	8660.32	0.501613
$669$	282.120	0.0163040
$670$	10753.1	0.620045
$671$	−13917.3	−0.800702
$672$	2182.91	0.125309
$673$	31410.9	1.79911	0.899556	−	0.436805i	$-0.143890\pi$
$673$	31410.9	1.79911	0.899556	+	0.436805i	$0.143890\pi$
$674$	−11855.7	−0.677545
$675$	−8954.49	−0.510605
$676$	−8525.04	−0.485039
$677$	−91.4773	−0.00519314	−0.00259657	−	0.999997i	$-0.500827\pi$
$677$	−91.4773	−0.00519314	−0.00259657	+	0.999997i	$0.500827\pi$
$678$	−7499.11	−0.424781
$679$	31609.0	1.78651
$680$	12435.0	0.701265
$681$	−4231.43	−0.238104
$682$	−33397.4	−1.87515
$683$	−25918.6	−1.45205	−0.726024	−	0.687670i	$-0.758634\pi$
$683$	−25918.6	−1.45205	−0.726024	+	0.687670i	$0.758634\pi$
$684$	684.000	0.0382360
$685$	12056.4	0.672485
$686$	7683.59	0.427640
$687$	−10953.3	−0.608291
$688$	−3567.27	−0.197676
$689$	3671.06	0.202984
$690$	3509.18	0.193612
$691$	17339.8	0.954613	0.477307	−	0.878737i	$-0.341613\pi$
$691$	17339.8	0.954613	0.477307	+	0.878737i	$0.341613\pi$
$692$	2459.14	0.135090
$693$	−11202.1	−0.614046
$694$	24496.2	1.33986
$695$	−16019.7	−0.874334
$696$	5398.91	0.294030
$697$	26991.8	1.46684
$698$	22986.7	1.24650
$699$	21047.6	1.13890
$700$	−30164.9	−1.62875
$701$	21860.1	1.17781	0.588905	−	0.808202i	$-0.299559\pi$
$701$	21860.1	1.17781	0.588905	+	0.808202i	$0.299559\pi$
$702$	437.829	0.0235396
$703$	−3146.12	−0.168788
$704$	3503.27	0.187549
$705$	34686.4	1.85300
$706$	3307.77	0.176331
$707$	−7655.82	−0.407251
$708$	−4100.73	−0.217676
$709$	27875.5	1.47657	0.738285	−	0.674489i	$-0.235636\pi$
$709$	27875.5	1.47657	0.738285	+	0.674489i	$0.235636\pi$
$710$	15117.8	0.799101
$711$	5250.63	0.276954
$712$	−3300.18	−0.173707
$713$	8349.35	0.438549
$714$	9923.86	0.520156
$715$	−9484.09	−0.496063
$716$	−9700.45	−0.506317
$717$	−949.381	−0.0494495
$718$	1997.44	0.103822
$719$	3333.87	0.172924	0.0864620	−	0.996255i	$-0.472444\pi$
$719$	3333.87	0.172924	0.0864620	+	0.996255i	$0.472444\pi$
$720$	3077.18	0.159277
$721$	5855.84	0.302473
$722$	722.000	0.0372161
$723$	−11043.3	−0.568057
$724$	12558.7	0.644669
$725$	−74605.7	−3.82177
$726$	−9991.91	−0.510791
$727$	−2521.55	−0.128637	−0.0643184	−	0.997929i	$-0.520487\pi$
$727$	−2521.55	−0.128637	−0.0643184	+	0.997929i	$0.520487\pi$
$728$	1474.91	0.0750876
$729$	729.000	0.0370370
$730$	20564.6	1.04264
$731$	−16217.4	−0.820551
$732$	3051.00	0.154055
$733$	16822.1	0.847664	0.423832	−	0.905741i	$-0.360685\pi$
$733$	16822.1	0.847664	0.423832	+	0.905741i	$0.360685\pi$
$734$	9307.98	0.468070
$735$	−11157.7	−0.559943
$736$	−875.818	−0.0438629
$737$	13772.4	0.688347
$738$	6679.43	0.333162
$739$	−23858.7	−1.18763	−0.593815	−	0.804602i	$-0.702379\pi$
$739$	−23858.7	−1.18763	−0.593815	+	0.804602i	$0.702379\pi$
$740$	−14153.8	−0.703112
$741$	462.153	0.0229118
$742$	20590.9	1.01875
$743$	22343.1	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$743$	22343.1	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$744$	7321.50	0.360778
$745$	−14102.9	−0.693544
$746$	22666.1	1.11242
$747$	−4710.48	−0.230719
$748$	15926.5	0.778514
$749$	8698.04	0.424325
$750$	−26495.5	−1.28997
$751$	−12535.9	−0.609110	−0.304555	−	0.952495i	$-0.598508\pi$
$751$	−12535.9	−0.609110	−0.304555	+	0.952495i	$0.598508\pi$
$752$	−8657.00	−0.419798
$753$	−22952.0	−1.11078
$754$	3647.84	0.176189
$755$	−18600.7	−0.896619
$756$	2455.77	0.118142
$757$	−18795.1	−0.902404	−0.451202	−	0.892422i	$-0.649005\pi$
$757$	−18795.1	−0.902404	−0.451202	+	0.892422i	$0.649005\pi$
$758$	−23139.8	−1.10880
$759$	4494.48	0.214940
$760$	3248.14	0.155029
$761$	−16941.9	−0.807023	−0.403511	−	0.914975i	$-0.632210\pi$
$761$	−16941.9	−0.807023	−0.403511	+	0.914975i	$0.632210\pi$
$762$	−2485.26	−0.118152
$763$	6474.95	0.307220
$764$	18858.7	0.893042
$765$	13989.4	0.661159
$766$	12669.8	0.597623
$767$	−2770.71	−0.130436
$768$	−768.000	−0.0360844
$769$	−9643.99	−0.452238	−0.226119	−	0.974100i	$-0.572604\pi$
$769$	−9643.99	−0.452238	−0.226119	+	0.974100i	$0.572604\pi$
$770$	−53196.0	−2.48968
$771$	−5939.11	−0.277422
$772$	−6427.32	−0.299643
$773$	−2745.06	−0.127727	−0.0638634	−	0.997959i	$-0.520342\pi$
$773$	−2745.06	−0.127727	−0.0638634	+	0.997959i	$0.520342\pi$
$774$	−4013.18	−0.186371
$775$	−101173.	−4.68936
$776$	−11120.8	−0.514451
$777$	−11295.5	−0.521526
$778$	−18425.9	−0.849103
$779$	7050.51	0.324276
$780$	2079.14	0.0954423
$781$	19362.5	0.887127
$782$	−3981.61	−0.182074
$783$	6073.77	0.277214
$784$	2784.73	0.126855
$785$	7284.26	0.331193
$786$	−12257.7	−0.556258
$787$	1312.23	0.0594357	0.0297178	−	0.999558i	$-0.490539\pi$
$787$	1312.23	0.0594357	0.0297178	+	0.999558i	$0.490539\pi$
$788$	−16644.0	−0.752435
$789$	5948.78	0.268418
$790$	24933.9	1.12292
$791$	−28419.9	−1.27749
$792$	3941.18	0.176823
$793$	2061.45	0.0923129
$794$	−5610.07	−0.250748
$795$	29026.3	1.29492
$796$	−8800.41	−0.391862
$797$	26003.1	1.15568	0.577841	−	0.816150i	$-0.303896\pi$
$797$	26003.1	1.15568	0.577841	+	0.816150i	$0.303896\pi$
$798$	2592.20	0.114991
$799$	−39356.1	−1.74258
$800$	10612.7	0.469021
$801$	−3712.70	−0.163773
$802$	−27744.8	−1.22158
$803$	26338.6	1.15750
$804$	−3019.23	−0.132438
$805$	13299.0	0.582271
$806$	4946.86	0.216186
$807$	3766.67	0.164304
$808$	2693.50	0.117274
$809$	−20785.7	−0.903320	−0.451660	−	0.892190i	$-0.649168\pi$
$809$	−20785.7	−0.903320	−0.451660	+	0.892190i	$0.649168\pi$
$810$	3461.83	0.150168
$811$	41004.6	1.77542	0.887710	−	0.460402i	$-0.152295\pi$
$811$	41004.6	1.77542	0.887710	+	0.460402i	$0.152295\pi$
$812$	20460.6	0.884270
$813$	55.4661	0.00239272
$814$	−18127.8	−0.780565
$815$	9560.34	0.410901
$816$	−3491.45	−0.149786
$817$	−4236.14	−0.181400
$818$	16256.7	0.694868
$819$	1659.27	0.0707933
$820$	31718.9	1.35082
$821$	32751.4	1.39224	0.696121	−	0.717924i	$-0.254908\pi$
$821$	32751.4	1.39224	0.696121	+	0.717924i	$0.254908\pi$
$822$	−3385.16	−0.143639
$823$	−1560.05	−0.0660751	−0.0330375	−	0.999454i	$-0.510518\pi$
$823$	−1560.05	−0.0660751	−0.0330375	+	0.999454i	$0.510518\pi$
$824$	−2060.23	−0.0871012
$825$	−54461.8	−2.29832
$826$	−15540.8	−0.654642
$827$	32096.7	1.34959	0.674796	−	0.738004i	$-0.264231\pi$
$827$	32096.7	1.34959	0.674796	+	0.738004i	$0.264231\pi$
$828$	−985.295	−0.0413543
$829$	−39115.0	−1.63874	−0.819372	−	0.573262i	$-0.805678\pi$
$829$	−39115.0	−1.63874	−0.819372	+	0.573262i	$0.805678\pi$
$830$	−22368.8	−0.935461
$831$	−3800.93	−0.158668
$832$	−518.909	−0.0216225
$833$	12659.8	0.526575
$834$	4497.95	0.186752
$835$	46266.3	1.91750
$836$	4160.14	0.172107
$837$	8236.69	0.340145
$838$	−2063.57	−0.0850653
$839$	25854.6	1.06389	0.531943	−	0.846780i	$-0.321462\pi$
$839$	25854.6	1.06389	0.531943	+	0.846780i	$0.321462\pi$
$840$	11661.8	0.479013
$841$	26215.5	1.07489
$842$	−1686.67	−0.0690338
$843$	−8094.44	−0.330709
$844$	−6958.27	−0.283784
$845$	−45543.6	−1.85414
$846$	−9739.12	−0.395790
$847$	−37867.1	−1.53616
$848$	−7244.36	−0.293364
$849$	−10644.0	−0.430272
$850$	48247.2	1.94690
$851$	4531.95	0.182554
$852$	−4244.73	−0.170683
$853$	21196.4	0.850822	0.425411	−	0.905000i	$-0.360130\pi$
$853$	21196.4	0.850822	0.425411	+	0.905000i	$0.360130\pi$
$854$	11562.6	0.463307
$855$	3654.15	0.146163
$856$	−3060.18	−0.122190
$857$	−35760.2	−1.42537	−0.712686	−	0.701483i	$-0.752522\pi$
$857$	−35760.2	−1.42537	−0.712686	+	0.701483i	$0.752522\pi$
$858$	2662.91	0.105956
$859$	−21145.9	−0.839918	−0.419959	−	0.907543i	$-0.637956\pi$
$859$	−21145.9	−0.839918	−0.419959	+	0.907543i	$0.637956\pi$
$860$	−19057.5	−0.755648
$861$	25313.5	1.00195
$862$	20922.8	0.826721
$863$	−36704.3	−1.44777	−0.723887	−	0.689919i	$-0.757646\pi$
$863$	−36704.3	−1.44777	−0.723887	+	0.689919i	$0.757646\pi$
$864$	−864.000	−0.0340207
$865$	13137.5	0.516403
$866$	−14881.3	−0.583934
$867$	−1133.73	−0.0444099
$868$	27746.8	1.08501
$869$	31934.7	1.24662
$870$	28842.7	1.12398
$871$	−2039.98	−0.0793594
$872$	−2278.05	−0.0884683
$873$	−12510.9	−0.485029
$874$	−1040.03	−0.0402513
$875$	−100412.	−3.87948
$876$	−5774.05	−0.222702
$877$	24118.9	0.928664	0.464332	−	0.885661i	$-0.346294\pi$
$877$	24118.9	0.928664	0.464332	+	0.885661i	$0.346294\pi$
$878$	1510.78	0.0580712
$879$	12976.6	0.497941
$880$	18715.6	0.716936
$881$	−14331.9	−0.548073	−0.274037	−	0.961719i	$-0.588359\pi$
$881$	−14331.9	−0.548073	−0.274037	+	0.961719i	$0.588359\pi$
$882$	3132.82	0.119600
$883$	−51819.2	−1.97492	−0.987460	−	0.157870i	$-0.949537\pi$
$883$	−51819.2	−1.97492	−0.987460	+	0.157870i	$0.949537\pi$
$884$	−2359.05	−0.0897548
$885$	−21907.4	−0.832102
$886$	−30260.3	−1.14742
$887$	8915.24	0.337480	0.168740	−	0.985661i	$-0.446030\pi$
$887$	8915.24	0.337480	0.168740	+	0.985661i	$0.446030\pi$
$888$	3974.05	0.150180
$889$	−9418.57	−0.355330
$890$	−17630.7	−0.664024
$891$	4433.83	0.166710
$892$	−376.159	−0.0141197
$893$	−10280.2	−0.385233
$894$	3959.76	0.148137
$895$	−51823.0	−1.93548
$896$	−2910.55	−0.108521
$897$	−665.727	−0.0247804
$898$	−2451.23	−0.0910896
$899$	68625.2	2.54592
$900$	11939.3	0.442197
$901$	−32934.1	−1.21775
$902$	40624.8	1.49962
$903$	−15209.0	−0.560493
$904$	9998.82	0.367871
$905$	67092.7	2.46435
$906$	5222.63	0.191512
$907$	31147.5	1.14028	0.570141	−	0.821547i	$-0.306889\pi$
$907$	31147.5	1.14028	0.570141	+	0.821547i	$0.306889\pi$
$908$	5641.91	0.206204
$909$	3030.19	0.110567
$910$	7879.45	0.287034
$911$	−49926.6	−1.81574	−0.907872	−	0.419247i	$-0.862294\pi$
$911$	−49926.6	−1.81574	−0.907872	+	0.419247i	$0.862294\pi$
$912$	−912.000	−0.0331133
$913$	−28649.5	−1.03851
$914$	12666.8	0.458402
$915$	16299.4	0.588899
$916$	14604.5	0.526796
$917$	−46454.0	−1.67290
$918$	−3927.89	−0.141220
$919$	134.681	0.00483430	0.00241715	−	0.999997i	$-0.499231\pi$
$919$	134.681	0.00483430	0.00241715	+	0.999997i	$0.499231\pi$
$920$	−4678.91	−0.167673
$921$	−12326.3	−0.441004
$922$	28583.2	1.02097
$923$	−2868.00	−0.102277
$924$	14936.2	0.531780
$925$	−54916.0	−1.95203
$926$	−15058.4	−0.534394
$927$	−2317.75	−0.0821198
$928$	−7198.55	−0.254638
$929$	23279.6	0.822152	0.411076	−	0.911601i	$-0.365153\pi$
$929$	23279.6	0.822152	0.411076	+	0.911601i	$0.365153\pi$
$930$	39113.9	1.37913
$931$	3306.86	0.116410
$932$	−28063.5	−0.986319
$933$	31184.4	1.09424
$934$	−1185.27	−0.0415239
$935$	85084.4	2.97600
$936$	−583.772	−0.0203859
$937$	−46692.7	−1.62794	−0.813972	−	0.580904i	$-0.802699\pi$
$937$	−46692.7	−1.62794	−0.813972	+	0.580904i	$0.802699\pi$
$938$	−11442.2	−0.398295
$939$	−7887.44	−0.274118
$940$	−46248.5	−1.60475
$941$	32111.4	1.11244	0.556218	−	0.831037i	$-0.312252\pi$
$941$	32111.4	1.11244	0.556218	+	0.831037i	$0.312252\pi$
$942$	−2045.25	−0.0707408
$943$	−10156.2	−0.350722
$944$	5467.64	0.188513
$945$	13119.5	0.451618
$946$	−24408.5	−0.838887
$947$	23641.5	0.811240	0.405620	−	0.914042i	$-0.367056\pi$
$947$	23641.5	0.811240	0.405620	+	0.914042i	$0.367056\pi$
$948$	−7000.84	−0.239849
$949$	−3901.31	−0.133448
$950$	12602.6	0.430403
$951$	26670.1	0.909398
$952$	−13231.8	−0.450468
$953$	−7361.42	−0.250220	−0.125110	−	0.992143i	$-0.539928\pi$
$953$	−7361.42	−0.250220	−0.125110	+	0.992143i	$0.539928\pi$
$954$	−8149.91	−0.276586
$955$	100749.	3.41379
$956$	1265.84	0.0428245
$957$	36941.1	1.24779
$958$	27958.4	0.942896
$959$	−12829.0	−0.431981
$960$	−4102.91	−0.137938
$961$	63272.1	2.12387
$962$	2685.11	0.0899912
$963$	−3442.70	−0.115202
$964$	14724.4	0.491952
$965$	−34336.9	−1.14543
$966$	−3734.05	−0.124369
$967$	24365.6	0.810284	0.405142	−	0.914254i	$-0.367222\pi$
$967$	24365.6	0.810284	0.405142	+	0.914254i	$0.367222\pi$
$968$	13322.5	0.442358
$969$	−4146.10	−0.137453
$970$	−59411.1	−1.96657
$971$	32522.6	1.07487	0.537436	−	0.843305i	$-0.319393\pi$
$971$	32522.6	1.07487	0.537436	+	0.843305i	$0.319393\pi$
$972$	−972.000	−0.0320750
$973$	17046.2	0.561641
$974$	21489.8	0.706957
$975$	8066.95	0.264973
$976$	−4068.00	−0.133415
$977$	−30882.3	−1.01127	−0.505636	−	0.862747i	$-0.668742\pi$
$977$	−30882.3	−1.01127	−0.505636	+	0.862747i	$0.668742\pi$
$978$	−2684.32	−0.0877659
$979$	−22580.9	−0.737170
$980$	14876.9	0.484925
$981$	−2562.80	−0.0834087
$982$	10644.1	0.345893
$983$	33962.7	1.10198	0.550988	−	0.834513i	$-0.314251\pi$
$983$	33962.7	1.10198	0.550988	+	0.834513i	$0.314251\pi$
$984$	−8905.91	−0.288526
$985$	−88917.8	−2.87630
$986$	−32725.8	−1.05700
$987$	−36909.1	−1.19030
$988$	−616.204	−0.0198422
$989$	6102.11	0.196194
$990$	21055.1	0.675934
$991$	−26955.6	−0.864050	−0.432025	−	0.901862i	$-0.642201\pi$
$991$	−26955.6	−0.864050	−0.432025	+	0.901862i	$0.642201\pi$
$992$	−9762.00	−0.312443
$993$	−908.149	−0.0290224
$994$	−16086.5	−0.513314
$995$	−47014.7	−1.49796
$996$	6280.64	0.199809
$997$	7081.56	0.224950	0.112475	−	0.993655i	$-0.464122\pi$
$997$	7081.56	0.224950	0.112475	+	0.993655i	$0.464122\pi$
$998$	87.5915	0.00277822
$999$	4470.80	0.141591

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	114.4.a.e.1.2	✓	2
3.2	odd	2		342.4.a.f.1.1		2
4.3	odd	2		912.4.a.m.1.2		2
19.18	odd	2		2166.4.a.n.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.4.a.e.1.2	✓	2	1.1	even	1	trivial
342.4.a.f.1.1		2	3.2	odd	2
912.4.a.m.1.2		2	4.3	odd	2
2166.4.a.n.1.2		2	19.18	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:	\( N \)	\(=\)	\( 114 = 2 \cdot 3 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	114.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.3693 q^{5} -6.00000 q^{6} -22.7386 q^{7} +8.00000 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+2.00000 q^{2} -3.00000 q^{3} +4.00000 q^{4} +21.3693 q^{5} -6.00000 q^{6} -22.7386 q^{7} +8.00000 q^{8} +9.00000 q^{9} +42.7386 q^{10} +54.7386 q^{11} -12.0000 q^{12} -8.10795 q^{13} -45.4773 q^{14} -64.1080 q^{15} +16.0000 q^{16} +72.7386 q^{17} +18.0000 q^{18} +19.0000 q^{19} +85.4773 q^{20} +68.2159 q^{21} +109.477 q^{22} -27.3693 q^{23} -24.0000 q^{24} +331.648 q^{25} -16.2159 q^{26} -27.0000 q^{27} -90.9545 q^{28} -224.955 q^{29} -128.216 q^{30} -305.062 q^{31} +32.0000 q^{32} -164.216 q^{33} +145.477 q^{34} -485.909 q^{35} +36.0000 q^{36} -165.585 q^{37} +38.0000 q^{38} +24.3239 q^{39} +170.955 q^{40} +371.080 q^{41} +136.432 q^{42} -222.955 q^{43} +218.955 q^{44} +192.324 q^{45} -54.7386 q^{46} -541.062 q^{47} -48.0000 q^{48} +174.045 q^{49} +663.295 q^{50} -218.216 q^{51} -32.4318 q^{52} -452.773 q^{53} -54.0000 q^{54} +1169.73 q^{55} -181.909 q^{56} -57.0000 q^{57} -449.909 q^{58} +341.727 q^{59} -256.432 q^{60} -254.250 q^{61} -610.125 q^{62} -204.648 q^{63} +64.0000 q^{64} -173.261 q^{65} -328.432 q^{66} +251.602 q^{67} +290.955 q^{68} +82.1080 q^{69} -971.818 q^{70} +353.727 q^{71} +72.0000 q^{72} +481.170 q^{73} -331.170 q^{74} -994.943 q^{75} +76.0000 q^{76} -1244.68 q^{77} +48.6477 q^{78} +583.403 q^{79} +341.909 q^{80} +81.0000 q^{81} +742.159 q^{82} -523.386 q^{83} +272.864 q^{84} +1554.37 q^{85} -445.909 q^{86} +674.864 q^{87} +437.909 q^{88} -412.523 q^{89} +384.648 q^{90} +184.364 q^{91} -109.477 q^{92} +915.187 q^{93} -1082.12 q^{94} +406.017 q^{95} -96.0000 q^{96} -1390.10 q^{97} +348.091 q^{98} +492.648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} + 16 q^{8} + 18 q^{9}+O(q^{10}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 + 16 * q^8 + 18 * q^9	\( 2 q + 4 q^{2} - 6 q^{3} + 8 q^{4} + 18 q^{5} - 12 q^{6} + 4 q^{7} + 16 q^{8} + 18 q^{9} + 36 q^{10} + 60 q^{11} - 24 q^{12} + 58 q^{13} + 8 q^{14} - 54 q^{15} + 32 q^{16} + 96 q^{17} + 36 q^{18} + 38 q^{19} + 72 q^{20} - 12 q^{21} + 120 q^{22} - 30 q^{23} - 48 q^{24} + 218 q^{25} + 116 q^{26} - 54 q^{27} + 16 q^{28} - 252 q^{29} - 108 q^{30} - 338 q^{31} + 64 q^{32} - 180 q^{33} + 192 q^{34} - 576 q^{35} + 72 q^{36} - 158 q^{37} + 76 q^{38} - 174 q^{39} + 144 q^{40} - 24 q^{42} - 248 q^{43} + 240 q^{44} + 162 q^{45} - 60 q^{46} - 810 q^{47} - 96 q^{48} + 546 q^{49} + 436 q^{50} - 288 q^{51} + 232 q^{52} + 84 q^{53} - 108 q^{54} + 1152 q^{55} + 32 q^{56} - 114 q^{57} - 504 q^{58} - 504 q^{59} - 216 q^{60} + 580 q^{61} - 676 q^{62} + 36 q^{63} + 128 q^{64} - 396 q^{65} - 360 q^{66} - 140 q^{67} + 384 q^{68} + 90 q^{69} - 1152 q^{70} - 480 q^{71} + 144 q^{72} + 616 q^{73} - 316 q^{74} - 654 q^{75} + 152 q^{76} - 1104 q^{77} - 348 q^{78} + 202 q^{79} + 288 q^{80} + 162 q^{81} - 552 q^{83} - 48 q^{84} + 1476 q^{85} - 496 q^{86} + 756 q^{87} + 480 q^{88} - 924 q^{89} + 324 q^{90} + 1952 q^{91} - 120 q^{92} + 1014 q^{93} - 1620 q^{94} + 342 q^{95} - 192 q^{96} + 40 q^{97} + 1092 q^{98} + 540 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 - 6 * q^3 + 8 * q^4 + 18 * q^5 - 12 * q^6 + 4 * q^7 + 16 * q^8 + 18 * q^9 + 36 * q^10 + 60 * q^11 - 24 * q^12 + 58 * q^13 + 8 * q^14 - 54 * q^15 + 32 * q^16 + 96 * q^17 + 36 * q^18 + 38 * q^19 + 72 * q^20 - 12 * q^21 + 120 * q^22 - 30 * q^23 - 48 * q^24 + 218 * q^25 + 116 * q^26 - 54 * q^27 + 16 * q^28 - 252 * q^29 - 108 * q^30 - 338 * q^31 + 64 * q^32 - 180 * q^33 + 192 * q^34 - 576 * q^35 + 72 * q^36 - 158 * q^37 + 76 * q^38 - 174 * q^39 + 144 * q^40 - 24 * q^42 - 248 * q^43 + 240 * q^44 + 162 * q^45 - 60 * q^46 - 810 * q^47 - 96 * q^48 + 546 * q^49 + 436 * q^50 - 288 * q^51 + 232 * q^52 + 84 * q^53 - 108 * q^54 + 1152 * q^55 + 32 * q^56 - 114 * q^57 - 504 * q^58 - 504 * q^59 - 216 * q^60 + 580 * q^61 - 676 * q^62 + 36 * q^63 + 128 * q^64 - 396 * q^65 - 360 * q^66 - 140 * q^67 + 384 * q^68 + 90 * q^69 - 1152 * q^70 - 480 * q^71 + 144 * q^72 + 616 * q^73 - 316 * q^74 - 654 * q^75 + 152 * q^76 - 1104 * q^77 - 348 * q^78 + 202 * q^79 + 288 * q^80 + 162 * q^81 - 552 * q^83 - 48 * q^84 + 1476 * q^85 - 496 * q^86 + 756 * q^87 + 480 * q^88 - 924 * q^89 + 324 * q^90 + 1952 * q^91 - 120 * q^92 + 1014 * q^93 - 1620 * q^94 + 342 * q^95 - 192 * q^96 + 40 * q^97 + 1092 * q^98 + 540 * q^99

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