LMFDB - Embedded newform 169.4.a.k.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [169,4,Mod(1,169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("169.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	169.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:	$9.97132279097$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 46x^{7} + 145x^{6} + 680x^{5} - 1501x^{4} - 3203x^{3} + 4784x^{2} + 3584x - 4096 $ x^9 - 4x^8 - 46x^7 + 145x^6 + 680x^5 - 1501x^4 - 3203x^3 + 4784x^2 + 3584x - 4096
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$2.72763$ of defining polynomial
Character	$\chi$	$=$	169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.72763 q^{2} +6.89591 q^{3} -5.01528 q^{4} -20.8281 q^{5} +11.9136 q^{6} -7.56566 q^{7} -22.4856 q^{8} +20.5536 q^{9} +O(q^{10})$	\(q+1.72763 q^{2} +6.89591 q^{3} -5.01528 q^{4} -20.8281 q^{5} +11.9136 q^{6} -7.56566 q^{7} -22.4856 q^{8} +20.5536 q^{9} -35.9833 q^{10} +4.40295 q^{11} -34.5850 q^{12} -13.0707 q^{14} -143.629 q^{15} +1.27533 q^{16} -73.0087 q^{17} +35.5091 q^{18} -55.9424 q^{19} +104.459 q^{20} -52.1722 q^{21} +7.60668 q^{22} -33.6244 q^{23} -155.059 q^{24} +308.809 q^{25} -44.4536 q^{27} +37.9439 q^{28} +121.429 q^{29} -248.138 q^{30} +84.1320 q^{31} +182.088 q^{32} +30.3624 q^{33} -126.132 q^{34} +157.578 q^{35} -103.082 q^{36} -171.716 q^{37} -96.6479 q^{38} +468.332 q^{40} -93.5714 q^{41} -90.1344 q^{42} +441.776 q^{43} -22.0820 q^{44} -428.093 q^{45} -58.0907 q^{46} -272.528 q^{47} +8.79455 q^{48} -285.761 q^{49} +533.508 q^{50} -503.461 q^{51} -480.202 q^{53} -76.7995 q^{54} -91.7049 q^{55} +170.119 q^{56} -385.774 q^{57} +209.785 q^{58} -350.534 q^{59} +720.338 q^{60} -484.467 q^{61} +145.349 q^{62} -155.502 q^{63} +304.379 q^{64} +52.4550 q^{66} +967.552 q^{67} +366.159 q^{68} -231.871 q^{69} +272.237 q^{70} +402.749 q^{71} -462.162 q^{72} +351.621 q^{73} -296.662 q^{74} +2129.52 q^{75} +280.567 q^{76} -33.3112 q^{77} -820.078 q^{79} -26.5626 q^{80} -861.496 q^{81} -161.657 q^{82} -192.314 q^{83} +261.658 q^{84} +1520.63 q^{85} +763.226 q^{86} +837.366 q^{87} -99.0031 q^{88} +813.571 q^{89} -739.587 q^{90} +168.636 q^{92} +580.167 q^{93} -470.829 q^{94} +1165.17 q^{95} +1255.67 q^{96} -788.293 q^{97} -493.690 q^{98} +90.4966 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) $ 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.72763	0.610811	0.305405	−	0.952222i	$-0.401208\pi$
$2$	1.72763	0.610811	0.305405	+	0.952222i	$0.401208\pi$
$3$	6.89591	1.32712	0.663560	−	0.748123i	$-0.269045\pi$
$3$	6.89591	1.32712	0.663560	+	0.748123i	$0.269045\pi$
$4$	−5.01528	−0.626910
$5$	−20.8281	−1.86292	−0.931460	−	0.363844i	$-0.881464\pi$
$5$	−20.8281	−1.86292	−0.931460	+	0.363844i	$0.881464\pi$
$6$	11.9136	0.810619
$7$	−7.56566	−0.408507	−0.204254	−	0.978918i	$-0.565477\pi$
$7$	−7.56566	−0.408507	−0.204254	+	0.978918i	$0.565477\pi$
$8$	−22.4856	−0.993734
$9$	20.5536	0.761246
$10$	−35.9833	−1.13789
$11$	4.40295	0.120685	0.0603427	−	0.998178i	$-0.480781\pi$
$11$	4.40295	0.120685	0.0603427	+	0.998178i	$0.480781\pi$
$12$	−34.5850	−0.831985
$13$	0	0
$13$	0	0
$14$	−13.0707	−0.249521
$15$	−143.629	−2.47232
$16$	1.27533	0.0199270
$17$	−73.0087	−1.04160	−0.520800	−	0.853679i	$-0.674366\pi$
$17$	−73.0087	−1.04160	−0.520800	+	0.853679i	$0.674366\pi$
$18$	35.5091	0.464977
$19$	−55.9424	−0.675477	−0.337738	−	0.941240i	$-0.609662\pi$
$19$	−55.9424	−0.675477	−0.337738	+	0.941240i	$0.609662\pi$
$20$	104.459	1.16788
$21$	−52.1722	−0.542138
$22$	7.60668	0.0737159
$23$	−33.6244	−0.304834	−0.152417	−	0.988316i	$-0.548706\pi$
$23$	−33.6244	−0.304834	−0.152417	+	0.988316i	$0.548706\pi$
$24$	−155.059	−1.31880
$25$	308.809	2.47047
$26$	0	0
$27$	−44.4536	−0.316855
$28$	37.9439	0.256098
$29$	121.429	0.777546	0.388773	−	0.921334i	$-0.372899\pi$
$29$	121.429	0.777546	0.388773	+	0.921334i	$0.372899\pi$
$30$	−248.138	−1.51012
$31$	84.1320	0.487437	0.243719	−	0.969846i	$-0.421633\pi$
$31$	84.1320	0.487437	0.243719	+	0.969846i	$0.421633\pi$
$32$	182.088	1.00591
$33$	30.3624	0.160164
$34$	−126.132	−0.636220
$35$	157.578	0.761016
$36$	−103.082	−0.477233
$37$	−171.716	−0.762969	−0.381485	−	0.924375i	$-0.624587\pi$
$37$	−171.716	−0.762969	−0.381485	+	0.924375i	$0.624587\pi$
$38$	−96.6479	−0.412588
$39$	0	0
$40$	468.332	1.85125
$41$	−93.5714	−0.356424	−0.178212	−	0.983992i	$-0.557031\pi$
$41$	−93.5714	−0.356424	−0.178212	+	0.983992i	$0.557031\pi$
$42$	−90.1344	−0.331144
$43$	441.776	1.56675	0.783374	−	0.621551i	$-0.213497\pi$
$43$	441.776	1.56675	0.783374	+	0.621551i	$0.213497\pi$
$44$	−22.0820	−0.0756589
$45$	−428.093	−1.41814
$46$	−58.0907	−0.186196
$47$	−272.528	−0.845794	−0.422897	−	0.906178i	$-0.638987\pi$
$47$	−272.528	−0.845794	−0.422897	+	0.906178i	$0.638987\pi$
$48$	8.79455	0.0264455
$49$	−285.761	−0.833122
$50$	533.508	1.50899
$51$	−503.461	−1.38233
$52$	0	0
$53$	−480.202	−1.24454	−0.622272	−	0.782801i	$-0.713790\pi$
$53$	−480.202	−1.24454	−0.622272	+	0.782801i	$0.713790\pi$
$54$	−76.7995	−0.193539
$55$	−91.7049	−0.224827
$56$	170.119	0.405948
$57$	−385.774	−0.896438
$58$	209.785	0.474934
$59$	−350.534	−0.773486	−0.386743	−	0.922187i	$-0.626400\pi$
$59$	−350.534	−0.773486	−0.386743	+	0.922187i	$0.626400\pi$
$60$	720.338	1.54992
$61$	−484.467	−1.01688	−0.508439	−	0.861098i	$-0.669777\pi$
$61$	−484.467	−1.01688	−0.508439	+	0.861098i	$0.669777\pi$
$62$	145.349	0.297732
$63$	−155.502	−0.310974
$64$	304.379	0.594491
$65$	0	0
$66$	52.4550	0.0978298
$67$	967.552	1.76426	0.882129	−	0.471008i	$-0.156110\pi$
$67$	967.552	1.76426	0.882129	+	0.471008i	$0.156110\pi$
$68$	366.159	0.652990
$69$	−231.871	−0.404551
$70$	272.237	0.464837
$71$	402.749	0.673205	0.336603	−	0.941647i	$-0.390722\pi$
$71$	402.749	0.673205	0.336603	+	0.941647i	$0.390722\pi$
$72$	−462.162	−0.756476
$73$	351.621	0.563754	0.281877	−	0.959450i	$-0.409043\pi$
$73$	351.621	0.563754	0.281877	+	0.959450i	$0.409043\pi$
$74$	−296.662	−0.466030
$75$	2129.52	3.27861
$76$	280.567	0.423463
$77$	−33.3112	−0.0493009
$78$	0	0
$79$	−820.078	−1.16792	−0.583962	−	0.811781i	$-0.698498\pi$
$79$	−820.078	−1.16792	−0.583962	+	0.811781i	$0.698498\pi$
$80$	−26.5626	−0.0371224
$81$	−861.496	−1.18175
$82$	−161.657	−0.217708
$83$	−192.314	−0.254328	−0.127164	−	0.991882i	$-0.540587\pi$
$83$	−192.314	−0.254328	−0.127164	+	0.991882i	$0.540587\pi$
$84$	261.658	0.339872
$85$	1520.63	1.94042
$86$	763.226	0.956986
$87$	837.366	1.03190
$88$	−99.0031	−0.119929
$89$	813.571	0.968970	0.484485	−	0.874799i	$-0.339007\pi$
$89$	813.571	0.968970	0.484485	+	0.874799i	$0.339007\pi$
$90$	−739.587	−0.866215
$91$	0	0
$92$	168.636	0.191103
$93$	580.167	0.646887
$94$	−470.829	−0.516620
$95$	1165.17	1.25836
$96$	1255.67	1.33496
$97$	−788.293	−0.825144	−0.412572	−	0.910925i	$-0.635370\pi$
$97$	−788.293	−0.825144	−0.412572	+	0.910925i	$0.635370\pi$
$98$	−493.690	−0.508880
$99$	90.4966	0.0918712
$100$	−1548.76	−1.54876
$101$	−1593.06	−1.56946	−0.784730	−	0.619838i	$-0.787198\pi$
$101$	−1593.06	−1.56946	−0.784730	+	0.619838i	$0.787198\pi$
$102$	−869.797	−0.844340
$103$	−134.659	−0.128819	−0.0644094	−	0.997924i	$-0.520516\pi$
$103$	−134.659	−0.128819	−0.0644094	+	0.997924i	$0.520516\pi$
$104$	0	0
$105$	1086.65	1.00996
$106$	−829.613	−0.760180
$107$	−779.219	−0.704018	−0.352009	−	0.935997i	$-0.614501\pi$
$107$	−779.219	−0.704018	−0.352009	+	0.935997i	$0.614501\pi$
$108$	222.947	0.198640
$109$	−1341.63	−1.17894	−0.589472	−	0.807789i	$-0.700664\pi$
$109$	−1341.63	−1.17894	−0.589472	+	0.807789i	$0.700664\pi$
$110$	−158.433	−0.137327
$111$	−1184.14	−1.01255
$112$	−9.64870	−0.00814032
$113$	1222.14	1.01742	0.508712	−	0.860937i	$-0.330122\pi$
$113$	1222.14	1.01742	0.508712	+	0.860937i	$0.330122\pi$
$114$	−666.476	−0.547554
$115$	700.332	0.567881
$116$	−609.002	−0.487452
$117$	0	0
$118$	−605.595	−0.472454
$119$	552.359	0.425501
$120$	3229.58	2.45683
$121$	−1311.61	−0.985435
$122$	−836.981	−0.621120
$123$	−645.260	−0.473018
$124$	−421.946	−0.305579
$125$	−3828.38	−2.73937
$126$	−268.650	−0.189947
$127$	448.886	0.313640	0.156820	−	0.987627i	$-0.449876\pi$
$127$	448.886	0.313640	0.156820	+	0.987627i	$0.449876\pi$
$128$	−930.851	−0.642784
$129$	3046.45	2.07926
$130$	0	0
$131$	1787.67	1.19229	0.596144	−	0.802877i	$-0.296699\pi$
$131$	1787.67	1.19229	0.596144	+	0.802877i	$0.296699\pi$
$132$	−152.276	−0.100408
$133$	423.241	0.275937
$134$	1671.58	1.07763
$135$	925.883	0.590276
$136$	1641.65	1.03507
$137$	−830.034	−0.517625	−0.258812	−	0.965928i	$-0.583331\pi$
$137$	−830.034	−0.517625	−0.258812	+	0.965928i	$0.583331\pi$
$138$	−400.588	−0.247104
$139$	989.347	0.603707	0.301854	−	0.953354i	$-0.402395\pi$
$139$	989.347	0.603707	0.301854	+	0.953354i	$0.402395\pi$
$140$	−790.299	−0.477089
$141$	−1879.33	−1.12247
$142$	695.803	0.411201
$143$	0	0
$144$	26.2126	0.0151693
$145$	−2529.14	−1.44851
$146$	607.471	0.344347
$147$	−1970.58	−1.10565
$148$	861.202	0.478313
$149$	469.575	0.258182	0.129091	−	0.991633i	$-0.458794\pi$
$149$	469.575	0.258182	0.129091	+	0.991633i	$0.458794\pi$
$150$	3679.03	2.00261
$151$	1936.82	1.04382	0.521908	−	0.853002i	$-0.325220\pi$
$151$	1936.82	1.04382	0.521908	+	0.853002i	$0.325220\pi$
$152$	1257.90	0.671244
$153$	−1500.59	−0.792914
$154$	−57.5496	−0.0301135
$155$	−1752.31	−0.908056
$156$	0	0
$157$	−2891.58	−1.46989	−0.734946	−	0.678125i	$-0.762793\pi$
$157$	−2891.58	−1.46989	−0.734946	+	0.678125i	$0.762793\pi$
$158$	−1416.79	−0.713380
$159$	−3311.43	−1.65166
$160$	−3792.55	−1.87392
$161$	254.391	0.124527
$162$	−1488.35	−0.721826
$163$	2293.98	1.10232	0.551160	−	0.834399i	$-0.314185\pi$
$163$	2293.98	1.10232	0.551160	+	0.834399i	$0.314185\pi$
$164$	469.287	0.223446
$165$	−632.389	−0.298372
$166$	−332.249	−0.155346
$167$	−1126.64	−0.522050	−0.261025	−	0.965332i	$-0.584061\pi$
$167$	−1126.64	−0.522050	−0.261025	+	0.965332i	$0.584061\pi$
$168$	1173.12	0.538741
$169$	0	0
$170$	2627.09	1.18523
$171$	−1149.82	−0.514204
$172$	−2215.63	−0.982210
$173$	−511.900	−0.224966	−0.112483	−	0.993654i	$-0.535880\pi$
$173$	−511.900	−0.224966	−0.112483	+	0.993654i	$0.535880\pi$
$174$	1446.66	0.630293
$175$	−2336.34	−1.00920
$176$	5.61520	0.00240490
$177$	−2417.26	−1.02651
$178$	1405.55	0.591857
$179$	363.391	0.151738	0.0758690	−	0.997118i	$-0.475827\pi$
$179$	363.391	0.151738	0.0758690	+	0.997118i	$0.475827\pi$
$180$	2147.01	0.889046
$181$	780.933	0.320698	0.160349	−	0.987060i	$-0.448738\pi$
$181$	780.933	0.320698	0.160349	+	0.987060i	$0.448738\pi$
$182$	0	0
$183$	−3340.84	−1.34952
$184$	756.066	0.302924
$185$	3576.50	1.42135
$186$	1002.32	0.395126
$187$	−321.453	−0.125706
$188$	1366.81	0.530237
$189$	336.321	0.129438
$190$	2012.99	0.768619
$191$	−4009.74	−1.51903	−0.759515	−	0.650490i	$-0.774563\pi$
$191$	−4009.74	−1.51903	−0.759515	+	0.650490i	$0.774563\pi$
$192$	2098.97	0.788960
$193$	165.316	0.0616564	0.0308282	−	0.999525i	$-0.490186\pi$
$193$	165.316	0.0616564	0.0308282	+	0.999525i	$0.490186\pi$
$194$	−1361.88	−0.504007
$195$	0	0
$196$	1433.17	0.522293
$197$	−821.276	−0.297023	−0.148511	−	0.988911i	$-0.547448\pi$
$197$	−821.276	−0.297023	−0.148511	+	0.988911i	$0.547448\pi$
$198$	156.345	0.0561159
$199$	−38.5703	−0.0137396	−0.00686978	−	0.999976i	$-0.502187\pi$
$199$	−38.5703	−0.0137396	−0.00686978	+	0.999976i	$0.502187\pi$
$200$	−6943.76	−2.45499
$201$	6672.16	2.34138
$202$	−2752.23	−0.958643
$203$	−918.693	−0.317633
$204$	2525.00	0.866596
$205$	1948.91	0.663990
$206$	−232.641	−0.0786839
$207$	−691.104	−0.232053
$208$	0	0
$209$	−246.311	−0.0815202
$210$	1877.33	0.616894
$211$	5044.01	1.64571	0.822853	−	0.568255i	$-0.192381\pi$
$211$	5044.01	1.64571	0.822853	+	0.568255i	$0.192381\pi$
$212$	2408.35	0.780217
$213$	2777.33	0.893423
$214$	−1346.20	−0.430022
$215$	−9201.33	−2.91872
$216$	999.567	0.314870
$217$	−636.514	−0.199122
$218$	−2317.85	−0.720112
$219$	2424.75	0.748169
$220$	459.926	0.140946
$221$	0	0
$222$	−2045.75	−0.618477
$223$	6127.78	1.84012	0.920059	−	0.391780i	$-0.128140\pi$
$223$	6127.78	1.84012	0.920059	+	0.391780i	$0.128140\pi$
$224$	−1377.62	−0.410920
$225$	6347.14	1.88063
$226$	2111.41	0.621454
$227$	−6342.92	−1.85460	−0.927300	−	0.374318i	$-0.877877\pi$
$227$	−6342.92	−1.85460	−0.927300	+	0.374318i	$0.877877\pi$
$228$	1934.76	0.561987
$229$	1334.67	0.385141	0.192571	−	0.981283i	$-0.438318\pi$
$229$	1334.67	0.385141	0.192571	+	0.981283i	$0.438318\pi$
$230$	1209.92	0.346868
$231$	−229.711	−0.0654281
$232$	−2730.41	−0.772674
$233$	−5392.39	−1.51617	−0.758084	−	0.652157i	$-0.773864\pi$
$233$	−5392.39	−1.51617	−0.758084	+	0.652157i	$0.773864\pi$
$234$	0	0
$235$	5676.24	1.57565
$236$	1758.03	0.484907
$237$	−5655.19	−1.54997
$238$	954.274	0.259901
$239$	−3748.70	−1.01458	−0.507288	−	0.861777i	$-0.669352\pi$
$239$	−3748.70	−1.01458	−0.507288	+	0.861777i	$0.669352\pi$
$240$	−183.173	−0.0492658
$241$	3829.47	1.02356	0.511780	−	0.859116i	$-0.328986\pi$
$241$	3829.47	1.02356	0.511780	+	0.859116i	$0.328986\pi$
$242$	−2265.99	−0.601914
$243$	−4740.56	−1.25147
$244$	2429.74	0.637492
$245$	5951.85	1.55204
$246$	−1114.77	−0.288924
$247$	0	0
$248$	−1891.76	−0.484383
$249$	−1326.18	−0.337524
$250$	−6614.04	−1.67323
$251$	−476.577	−0.119846	−0.0599229	−	0.998203i	$-0.519085\pi$
$251$	−476.577	−0.119846	−0.0599229	+	0.998203i	$0.519085\pi$
$252$	779.886	0.194953
$253$	−148.047	−0.0367890
$254$	775.511	0.191574
$255$	10486.1	2.57517
$256$	−4043.20	−0.987110
$257$	254.676	0.0618142	0.0309071	−	0.999522i	$-0.490160\pi$
$257$	254.676	0.0618142	0.0309071	+	0.999522i	$0.490160\pi$
$258$	5263.14	1.27003
$259$	1299.14	0.311679
$260$	0	0
$261$	2495.81	0.591904
$262$	3088.44	0.728262
$263$	2244.97	0.526352	0.263176	−	0.964748i	$-0.415230\pi$
$263$	2244.97	0.526352	0.263176	+	0.964748i	$0.415230\pi$
$264$	−682.717	−0.159160
$265$	10001.7	2.31848
$266$	731.205	0.168545
$267$	5610.32	1.28594
$268$	−4852.55	−1.10603
$269$	4585.32	1.03930	0.519650	−	0.854379i	$-0.326062\pi$
$269$	4585.32	1.03930	0.519650	+	0.854379i	$0.326062\pi$
$270$	1599.59	0.360547
$271$	−7972.80	−1.78713	−0.893566	−	0.448932i	$-0.851805\pi$
$271$	−7972.80	−1.78713	−0.893566	+	0.448932i	$0.851805\pi$
$272$	−93.1099	−0.0207560
$273$	0	0
$274$	−1433.99	−0.316171
$275$	1359.67	0.298149
$276$	1162.90	0.253617
$277$	5308.79	1.15153	0.575766	−	0.817615i	$-0.304704\pi$
$277$	5308.79	1.15153	0.575766	+	0.817615i	$0.304704\pi$
$278$	1709.23	0.368751
$279$	1729.22	0.371059
$280$	−3543.25	−0.756248
$281$	6534.86	1.38732	0.693661	−	0.720302i	$-0.255997\pi$
$281$	6534.86	1.38732	0.693661	+	0.720302i	$0.255997\pi$
$282$	−3246.80	−0.685617
$283$	4192.60	0.880652	0.440326	−	0.897838i	$-0.354863\pi$
$283$	4192.60	0.880652	0.440326	+	0.897838i	$0.354863\pi$
$284$	−2019.90	−0.422039
$285$	8034.93	1.66999
$286$	0	0
$287$	707.930	0.145602
$288$	3742.58	0.765741
$289$	417.265	0.0849308
$290$	−4369.42	−0.884763
$291$	−5436.00	−1.09506
$292$	−1763.48	−0.353424
$293$	−2393.89	−0.477313	−0.238656	−	0.971104i	$-0.576707\pi$
$293$	−2393.89	−0.477313	−0.238656	+	0.971104i	$0.576707\pi$
$294$	−3404.44	−0.675344
$295$	7300.96	1.44094
$296$	3861.13	0.758189
$297$	−195.727	−0.0382398
$298$	811.253	0.157700
$299$	0	0
$300$	−10680.1	−2.05539
$301$	−3342.32	−0.640028
$302$	3346.12	0.637574
$303$	−10985.6	−2.08286
$304$	−71.3448	−0.0134602
$305$	10090.5	1.89436
$306$	−2592.48	−0.484320
$307$	−821.783	−0.152774	−0.0763870	−	0.997078i	$-0.524338\pi$
$307$	−821.783	−0.152774	−0.0763870	+	0.997078i	$0.524338\pi$
$308$	167.065	0.0309072
$309$	−928.596	−0.170958
$310$	−3027.34	−0.554650
$311$	−5490.98	−1.00117	−0.500587	−	0.865686i	$-0.666882\pi$
$311$	−5490.98	−1.00117	−0.500587	+	0.865686i	$0.666882\pi$
$312$	0	0
$313$	−315.481	−0.0569714	−0.0284857	−	0.999594i	$-0.509069\pi$
$313$	−315.481	−0.0569714	−0.0284857	+	0.999594i	$0.509069\pi$
$314$	−4995.59	−0.897826
$315$	3238.80	0.579320
$316$	4112.92	0.732184
$317$	−8295.72	−1.46982	−0.734912	−	0.678163i	$-0.762777\pi$
$317$	−8295.72	−1.46982	−0.734912	+	0.678163i	$0.762777\pi$
$318$	−5720.94	−1.00885
$319$	534.647	0.0938385
$320$	−6339.64	−1.10749
$321$	−5373.42	−0.934316
$322$	439.494	0.0760623
$323$	4084.28	0.703577
$324$	4320.65	0.740852
$325$	0	0
$326$	3963.15	0.673309
$327$	−9251.77	−1.56460
$328$	2104.01	0.354191
$329$	2061.86	0.345513
$330$	−1092.54	−0.182249
$331$	855.068	0.141990	0.0709951	−	0.997477i	$-0.477383\pi$
$331$	855.068	0.141990	0.0709951	+	0.997477i	$0.477383\pi$
$332$	964.511	0.159441
$333$	−3529.38	−0.580807
$334$	−1946.43	−0.318874
$335$	−20152.2	−3.28667
$336$	−66.5366	−0.0108032
$337$	−3400.09	−0.549598	−0.274799	−	0.961502i	$-0.588611\pi$
$337$	−3400.09	−0.549598	−0.274799	+	0.961502i	$0.588611\pi$
$338$	0	0
$339$	8427.75	1.35024
$340$	−7626.39	−1.21647
$341$	370.429	0.0588265
$342$	−1986.47	−0.314081
$343$	4756.99	0.748844
$344$	−9933.60	−1.55693
$345$	4829.43	0.753645
$346$	−884.375	−0.137411
$347$	3544.33	0.548327	0.274164	−	0.961683i	$-0.411599\pi$
$347$	3544.33	0.548327	0.274164	+	0.961683i	$0.411599\pi$
$348$	−4199.63	−0.646907
$349$	8339.20	1.27905	0.639523	−	0.768772i	$-0.279132\pi$
$349$	8339.20	1.27905	0.639523	+	0.768772i	$0.279132\pi$
$350$	−4036.34	−0.616433
$351$	0	0
$352$	801.726	0.121398
$353$	4321.95	0.651655	0.325827	−	0.945429i	$-0.394357\pi$
$353$	4321.95	0.651655	0.325827	+	0.945429i	$0.394357\pi$
$354$	−4176.13	−0.627002
$355$	−8388.49	−1.25413
$356$	−4080.29	−0.607458
$357$	3809.02	0.564691
$358$	627.806	0.0926832
$359$	−4535.87	−0.666836	−0.333418	−	0.942779i	$-0.608202\pi$
$359$	−4535.87	−0.666836	−0.333418	+	0.942779i	$0.608202\pi$
$360$	9625.93	1.40925
$361$	−3729.45	−0.543731
$362$	1349.17	0.195886
$363$	−9044.78	−1.30779
$364$	0	0
$365$	−7323.58	−1.05023
$366$	−5771.75	−0.824301
$367$	225.362	0.0320540	0.0160270	−	0.999872i	$-0.494898\pi$
$367$	225.362	0.0320540	0.0160270	+	0.999872i	$0.494898\pi$
$368$	−42.8821	−0.00607442
$369$	−1923.23	−0.271326
$370$	6178.89	0.868176
$371$	3633.04	0.508405
$372$	−2909.70	−0.405540
$373$	2929.26	0.406625	0.203313	−	0.979114i	$-0.434829\pi$
$373$	2929.26	0.406625	0.203313	+	0.979114i	$0.434829\pi$
$374$	−555.354	−0.0767825
$375$	−26400.2	−3.63546
$376$	6127.97	0.840495
$377$	0	0
$378$	581.039	0.0790620
$379$	−7810.76	−1.05861	−0.529303	−	0.848433i	$-0.677547\pi$
$379$	−7810.76	−1.05861	−0.529303	+	0.848433i	$0.677547\pi$
$380$	−5843.67	−0.788878
$381$	3095.48	0.416237
$382$	−6927.36	−0.927839
$383$	−5576.15	−0.743937	−0.371968	−	0.928245i	$-0.621317\pi$
$383$	−5576.15	−0.743937	−0.371968	+	0.928245i	$0.621317\pi$
$384$	−6419.07	−0.853052
$385$	693.809	0.0918435
$386$	285.605	0.0376604
$387$	9080.09	1.19268
$388$	3953.51	0.517292
$389$	12425.7	1.61956	0.809778	−	0.586737i	$-0.199588\pi$
$389$	12425.7	1.61956	0.809778	+	0.586737i	$0.199588\pi$
$390$	0	0
$391$	2454.87	0.317515
$392$	6425.51	0.827902
$393$	12327.6	1.58231
$394$	−1418.86	−0.181425
$395$	17080.6	2.17575
$396$	−453.866	−0.0575950
$397$	6418.82	0.811464	0.405732	−	0.913992i	$-0.367017\pi$
$397$	6418.82	0.811464	0.405732	+	0.913992i	$0.367017\pi$
$398$	−66.6353	−0.00839227
$399$	2918.63	0.366202
$400$	393.832	0.0492290
$401$	−1553.37	−0.193446	−0.0967230	−	0.995311i	$-0.530836\pi$
$401$	−1553.37	−0.193446	−0.0967230	+	0.995311i	$0.530836\pi$
$402$	11527.0	1.43014
$403$	0	0
$404$	7989.65	0.983911
$405$	17943.3	2.20151
$406$	−1587.16	−0.194014
$407$	−756.055	−0.0920792
$408$	11320.7	1.37367
$409$	8761.54	1.05924	0.529622	−	0.848234i	$-0.322334\pi$
$409$	8761.54	1.05924	0.529622	+	0.848234i	$0.322334\pi$
$410$	3367.01	0.405572
$411$	−5723.84	−0.686949
$412$	675.353	0.0807578
$413$	2652.03	0.315975
$414$	−1193.97	−0.141741
$415$	4005.54	0.473793
$416$	0	0
$417$	6822.45	0.801191
$418$	−425.536	−0.0497934
$419$	8261.71	0.963272	0.481636	−	0.876371i	$-0.340043\pi$
$419$	8261.71	0.963272	0.481636	+	0.876371i	$0.340043\pi$
$420$	−5449.83	−0.633154
$421$	−4431.95	−0.513064	−0.256532	−	0.966536i	$-0.582580\pi$
$421$	−4431.95	−0.513064	−0.256532	+	0.966536i	$0.582580\pi$
$422$	8714.20	1.00521
$423$	−5601.45	−0.643857
$424$	10797.6	1.23674
$425$	−22545.7	−2.57324
$426$	4798.20	0.545713
$427$	3665.31	0.415402
$428$	3908.00	0.441356
$429$	0	0
$430$	−15896.5	−1.78279
$431$	−10076.0	−1.12609	−0.563045	−	0.826426i	$-0.690370\pi$
$431$	−10076.0	−1.12609	−0.563045	+	0.826426i	$0.690370\pi$
$432$	−56.6929	−0.00631398
$433$	99.6760	0.0110626	0.00553132	−	0.999985i	$-0.498239\pi$
$433$	99.6760	0.0110626	0.00553132	+	0.999985i	$0.498239\pi$
$434$	−1099.66	−0.121626
$435$	−17440.7	−1.92234
$436$	6728.66	0.739092
$437$	1881.03	0.205908
$438$	4189.07	0.456990
$439$	11428.8	1.24252	0.621258	−	0.783606i	$-0.286622\pi$
$439$	11428.8	1.24252	0.621258	+	0.783606i	$0.286622\pi$
$440$	2062.04	0.223418
$441$	−5873.42	−0.634210
$442$	0	0
$443$	−4786.94	−0.513396	−0.256698	−	0.966492i	$-0.582635\pi$
$443$	−4786.94	−0.513396	−0.256698	+	0.966492i	$0.582635\pi$
$444$	5938.78	0.634779
$445$	−16945.1	−1.80511
$446$	10586.6	1.12396
$447$	3238.15	0.342638
$448$	−2302.83	−0.242854
$449$	−6476.36	−0.680709	−0.340355	−	0.940297i	$-0.610547\pi$
$449$	−6476.36	−0.680709	−0.340355	+	0.940297i	$0.610547\pi$
$450$	10965.5	1.14871
$451$	−411.990	−0.0430152
$452$	−6129.36	−0.637834
$453$	13356.2	1.38527
$454$	−10958.2	−1.13281
$455$	0	0
$456$	8674.37	0.890822
$457$	−928.362	−0.0950261	−0.0475131	−	0.998871i	$-0.515130\pi$
$457$	−928.362	−0.0950261	−0.0475131	+	0.998871i	$0.515130\pi$
$458$	2305.82	0.235248
$459$	3245.50	0.330037
$460$	−3512.36	−0.356010
$461$	−10539.2	−1.06477	−0.532386	−	0.846502i	$-0.678704\pi$
$461$	−10539.2	−1.06477	−0.532386	+	0.846502i	$0.678704\pi$
$462$	−396.857	−0.0399642
$463$	−4928.72	−0.494724	−0.247362	−	0.968923i	$-0.579564\pi$
$463$	−4928.72	−0.494724	−0.247362	+	0.968923i	$0.579564\pi$
$464$	154.862	0.0154942
$465$	−12083.8	−1.20510
$466$	−9316.07	−0.926092
$467$	326.459	0.0323484	0.0161742	−	0.999869i	$-0.494851\pi$
$467$	326.459	0.0323484	0.0161742	+	0.999869i	$0.494851\pi$
$468$	0	0
$469$	−7320.17	−0.720712
$470$	9806.46	0.962422
$471$	−19940.1	−1.95072
$472$	7881.99	0.768640
$473$	1945.12	0.189083
$474$	−9770.09	−0.946741
$475$	−17275.5	−1.66874
$476$	−2770.24	−0.266751
$477$	−9869.89	−0.947403
$478$	−6476.38	−0.619713
$479$	1863.51	0.177757	0.0888787	−	0.996042i	$-0.471672\pi$
$479$	1863.51	0.177757	0.0888787	+	0.996042i	$0.471672\pi$
$480$	−26153.1	−2.48692
$481$	0	0
$482$	6615.92	0.625201
$483$	1754.26	0.165262
$484$	6578.12	0.617779
$485$	16418.6	1.53718
$486$	−8189.94	−0.764410
$487$	17379.2	1.61710	0.808548	−	0.588431i	$-0.200254\pi$
$487$	17379.2	1.61710	0.808548	+	0.588431i	$0.200254\pi$
$488$	10893.5	1.01051
$489$	15819.1	1.46291
$490$	10282.6	0.948002
$491$	19258.2	1.77008	0.885041	−	0.465513i	$-0.154130\pi$
$491$	19258.2	1.77008	0.885041	+	0.465513i	$0.154130\pi$
$492$	3236.16	0.296540
$493$	−8865.39	−0.809892
$494$	0	0
$495$	−1884.87	−0.171149
$496$	107.296	0.00971315
$497$	−3047.07	−0.275009
$498$	−2291.16	−0.206163
$499$	−13088.2	−1.17416	−0.587082	−	0.809528i	$-0.699723\pi$
$499$	−13088.2	−1.17416	−0.587082	+	0.809528i	$0.699723\pi$
$500$	19200.4	1.71734
$501$	−7769.24	−0.692823
$502$	−823.351	−0.0732031
$503$	18837.0	1.66978	0.834890	−	0.550416i	$-0.185531\pi$
$503$	18837.0	1.66978	0.834890	+	0.550416i	$0.185531\pi$
$504$	3496.56	0.309026
$505$	33180.4	2.92378
$506$	−255.770	−0.0224711
$507$	0	0
$508$	−2251.29	−0.196624
$509$	153.195	0.0133403	0.00667017	−	0.999978i	$-0.497877\pi$
$509$	153.195	0.0133403	0.00667017	+	0.999978i	$0.497877\pi$
$510$	18116.2	1.57294
$511$	−2660.24	−0.230298
$512$	461.635	0.0398468
$513$	2486.84	0.214029
$514$	439.987	0.0377568
$515$	2804.69	0.239979
$516$	−15278.8	−1.30351
$517$	−1199.93	−0.102075
$518$	2244.44	0.190377
$519$	−3530.02	−0.298556
$520$	0	0
$521$	−10847.8	−0.912192	−0.456096	−	0.889931i	$-0.650753\pi$
$521$	−10847.8	−0.912192	−0.456096	+	0.889931i	$0.650753\pi$
$522$	4311.85	0.361541
$523$	19849.9	1.65961	0.829804	−	0.558055i	$-0.188452\pi$
$523$	19849.9	1.65961	0.829804	+	0.558055i	$0.188452\pi$
$524$	−8965.69	−0.747458
$525$	−16111.2	−1.33934
$526$	3878.48	0.321501
$527$	−6142.36	−0.507714
$528$	38.7219	0.00319158
$529$	−11036.4	−0.907076
$530$	17279.2	1.41615
$531$	−7204.76	−0.588813
$532$	−2122.67	−0.172988
$533$	0	0
$534$	9692.57	0.785465
$535$	16229.6	1.31153
$536$	−21756.0	−1.75320
$537$	2505.91	0.201374
$538$	7921.75	0.634815
$539$	−1258.19	−0.100546
$540$	−4643.56	−0.370050
$541$	−15828.2	−1.25787	−0.628936	−	0.777457i	$-0.716509\pi$
$541$	−15828.2	−1.25787	−0.628936	+	0.777457i	$0.716509\pi$
$542$	−13774.1	−1.09160
$543$	5385.25	0.425604
$544$	−13294.0	−1.04775
$545$	27943.6	2.19628
$546$	0	0
$547$	6963.82	0.544335	0.272168	−	0.962250i	$-0.412259\pi$
$547$	6963.82	0.544335	0.272168	+	0.962250i	$0.412259\pi$
$548$	4162.85	0.324504
$549$	−9957.55	−0.774094
$550$	2349.01	0.182113
$551$	−6793.04	−0.525215
$552$	5213.77	0.402016
$553$	6204.43	0.477106
$554$	9171.64	0.703368
$555$	24663.3	1.88630
$556$	−4961.85	−0.378470
$557$	−18832.8	−1.43262	−0.716311	−	0.697781i	$-0.754171\pi$
$557$	−18832.8	−1.43262	−0.716311	+	0.697781i	$0.754171\pi$
$558$	2987.45	0.226647
$559$	0	0
$560$	200.964	0.0151648
$561$	−2216.72	−0.166827
$562$	11289.8	0.847391
$563$	−19661.5	−1.47182	−0.735911	−	0.677079i	$-0.763246\pi$
$563$	−19661.5	−1.47182	−0.735911	+	0.677079i	$0.763246\pi$
$564$	9425.38	0.703688
$565$	−25454.8	−1.89538
$566$	7243.28	0.537912
$567$	6517.79	0.482754
$568$	−9056.08	−0.668987
$569$	2153.41	0.158656	0.0793282	−	0.996849i	$-0.474722\pi$
$569$	2153.41	0.158656	0.0793282	+	0.996849i	$0.474722\pi$
$570$	13881.4	1.02005
$571$	4437.31	0.325212	0.162606	−	0.986691i	$-0.448010\pi$
$571$	4437.31	0.325212	0.162606	+	0.986691i	$0.448010\pi$
$572$	0	0
$573$	−27650.8	−2.01593
$574$	1223.04	0.0889352
$575$	−10383.5	−0.753082
$576$	6256.10	0.452554
$577$	14826.7	1.06975	0.534873	−	0.844933i	$-0.320360\pi$
$577$	14826.7	1.06975	0.534873	+	0.844933i	$0.320360\pi$
$578$	720.881	0.0518767
$579$	1140.00	0.0818254
$580$	12684.3	0.908084
$581$	1454.99	0.103895
$582$	−9391.41	−0.668877
$583$	−2114.30	−0.150198
$584$	−7906.41	−0.560222
$585$	0	0
$586$	−4135.77	−0.291548
$587$	−17009.7	−1.19602	−0.598011	−	0.801488i	$-0.704042\pi$
$587$	−17009.7	−1.19602	−0.598011	+	0.801488i	$0.704042\pi$
$588$	9883.02	0.693145
$589$	−4706.54	−0.329252
$590$	12613.4	0.880143
$591$	−5663.45	−0.394184
$592$	−218.994	−0.0152037
$593$	9173.23	0.635244	0.317622	−	0.948217i	$-0.397116\pi$
$593$	9173.23	0.635244	0.317622	+	0.948217i	$0.397116\pi$
$594$	−338.144	−0.0233573
$595$	−11504.6	−0.792675
$596$	−2355.05	−0.161857
$597$	−265.977	−0.0182340
$598$	0	0
$599$	2983.22	0.203491	0.101745	−	0.994810i	$-0.467557\pi$
$599$	2983.22	0.203491	0.101745	+	0.994810i	$0.467557\pi$
$600$	−47883.6	−3.25806
$601$	−18196.0	−1.23499	−0.617496	−	0.786574i	$-0.711853\pi$
$601$	−18196.0	−1.23499	−0.617496	+	0.786574i	$0.711853\pi$
$602$	−5774.31	−0.390936
$603$	19886.7	1.34303
$604$	−9713.71	−0.654380
$605$	27318.4	1.83579
$606$	−18979.1	−1.27223
$607$	−16672.6	−1.11486	−0.557428	−	0.830225i	$-0.688212\pi$
$607$	−16672.6	−1.11486	−0.557428	+	0.830225i	$0.688212\pi$
$608$	−10186.5	−0.679466
$609$	−6335.23	−0.421537
$610$	17432.7	1.15710
$611$	0	0
$612$	7525.90	0.497086
$613$	16756.9	1.10409	0.552044	−	0.833815i	$-0.313848\pi$
$613$	16756.9	1.10409	0.552044	+	0.833815i	$0.313848\pi$
$614$	−1419.74	−0.0933160
$615$	13439.5	0.881194
$616$	749.024	0.0489920
$617$	11985.4	0.782035	0.391017	−	0.920383i	$-0.372123\pi$
$617$	11985.4	0.782035	0.391017	+	0.920383i	$0.372123\pi$
$618$	−1604.27	−0.104423
$619$	−22471.2	−1.45912	−0.729560	−	0.683917i	$-0.760275\pi$
$619$	−22471.2	−1.45912	−0.729560	+	0.683917i	$0.760275\pi$
$620$	8788.31	0.569270
$621$	1494.73	0.0965882
$622$	−9486.40	−0.611527
$623$	−6155.20	−0.395832
$624$	0	0
$625$	41136.7	2.63275
$626$	−545.035	−0.0347987
$627$	−1698.54	−0.108187
$628$	14502.1	0.921491
$629$	12536.7	0.794709
$630$	5595.47	0.353855
$631$	−991.566	−0.0625572	−0.0312786	−	0.999511i	$-0.509958\pi$
$631$	−991.566	−0.0625572	−0.0312786	+	0.999511i	$0.509958\pi$
$632$	18440.0	1.16061
$633$	34783.0	2.18405
$634$	−14332.0	−0.897784
$635$	−9349.44	−0.584285
$636$	16607.8	1.03544
$637$	0	0
$638$	923.674	0.0573175
$639$	8277.96	0.512474
$640$	19387.8	1.19746
$641$	−8483.76	−0.522759	−0.261379	−	0.965236i	$-0.584177\pi$
$641$	−8483.76	−0.522759	−0.261379	+	0.965236i	$0.584177\pi$
$642$	−9283.31	−0.570690
$643$	−22527.5	−1.38165	−0.690823	−	0.723024i	$-0.742752\pi$
$643$	−22527.5	−1.38165	−0.690823	+	0.723024i	$0.742752\pi$
$644$	−1275.84	−0.0780671
$645$	−63451.6	−3.87350
$646$	7056.14	0.429752
$647$	5486.17	0.333359	0.166680	−	0.986011i	$-0.446695\pi$
$647$	5486.17	0.333359	0.166680	+	0.986011i	$0.446695\pi$
$648$	19371.3	1.17435
$649$	−1543.39	−0.0933485
$650$	0	0
$651$	−4389.35	−0.264258
$652$	−11505.0	−0.691056
$653$	20436.1	1.22470	0.612348	−	0.790588i	$-0.290225\pi$
$653$	20436.1	1.22470	0.612348	+	0.790588i	$0.290225\pi$
$654$	−15983.7	−0.955674
$655$	−37233.8	−2.22114
$656$	−119.334	−0.00710246
$657$	7227.08	0.429156
$658$	3562.13	0.211043
$659$	−8613.12	−0.509134	−0.254567	−	0.967055i	$-0.581933\pi$
$659$	−8613.12	−0.509134	−0.254567	+	0.967055i	$0.581933\pi$
$660$	3171.61	0.187053
$661$	−29268.8	−1.72228	−0.861138	−	0.508372i	$-0.830248\pi$
$661$	−29268.8	−1.72228	−0.861138	+	0.508372i	$0.830248\pi$
$662$	1477.24	0.0867291
$663$	0	0
$664$	4324.31	0.252735
$665$	−8815.30	−0.514049
$666$	−6097.47	−0.354763
$667$	−4082.99	−0.237022
$668$	5650.44	0.327279
$669$	42256.6	2.44206
$670$	−34815.7	−2.00753
$671$	−2133.08	−0.122722
$672$	−9499.94	−0.545340
$673$	−22540.5	−1.29104	−0.645522	−	0.763741i	$-0.723360\pi$
$673$	−22540.5	−1.29104	−0.645522	+	0.763741i	$0.723360\pi$
$674$	−5874.10	−0.335700
$675$	−13727.7	−0.782782
$676$	0	0
$677$	−18727.0	−1.06313	−0.531563	−	0.847019i	$-0.678395\pi$
$677$	−18727.0	−1.06313	−0.531563	+	0.847019i	$0.678395\pi$
$678$	14560.1	0.824743
$679$	5963.96	0.337078
$680$	−34192.3	−1.92826
$681$	−43740.2	−2.46128
$682$	639.965	0.0359319
$683$	5047.76	0.282793	0.141396	−	0.989953i	$-0.454841\pi$
$683$	5047.76	0.282793	0.141396	+	0.989953i	$0.454841\pi$
$684$	5766.67	0.322360
$685$	17288.0	0.964293
$686$	8218.34	0.457402
$687$	9203.76	0.511128
$688$	563.408	0.0312206
$689$	0	0
$690$	8343.48	0.460335
$691$	16446.9	0.905456	0.452728	−	0.891649i	$-0.350451\pi$
$691$	16446.9	0.905456	0.452728	+	0.891649i	$0.350451\pi$
$692$	2567.32	0.141033
$693$	−684.667	−0.0375301
$694$	6123.30	0.334924
$695$	−20606.2	−1.12466
$696$	−18828.7	−1.02543
$697$	6831.52	0.371252
$698$	14407.1	0.781255
$699$	−37185.5	−2.01214
$700$	11717.4	0.632681
$701$	14841.3	0.799639	0.399820	−	0.916594i	$-0.369073\pi$
$701$	14841.3	0.799639	0.399820	+	0.916594i	$0.369073\pi$
$702$	0	0
$703$	9606.18	0.515368
$704$	1340.17	0.0717464
$705$	39142.9	2.09107
$706$	7466.74	0.398038
$707$	12052.6	0.641136
$708$	12123.2	0.643529
$709$	19329.7	1.02390	0.511949	−	0.859016i	$-0.328924\pi$
$709$	19329.7	1.02390	0.511949	+	0.859016i	$0.328924\pi$
$710$	−14492.2	−0.766034
$711$	−16855.6	−0.889077
$712$	−18293.7	−0.962899
$713$	−2828.89	−0.148587
$714$	6580.59	0.344919
$715$	0	0
$716$	−1822.51	−0.0951261
$717$	−25850.7	−1.34646
$718$	−7836.32	−0.407310
$719$	21340.1	1.10689	0.553443	−	0.832887i	$-0.313314\pi$
$719$	21340.1	1.10689	0.553443	+	0.832887i	$0.313314\pi$
$720$	−545.958	−0.0282592
$721$	1018.78	0.0526234
$722$	−6443.12	−0.332117
$723$	26407.7	1.35839
$724$	−3916.60	−0.201049
$725$	37498.4	1.92090
$726$	−15626.1	−0.798812
$727$	−15092.3	−0.769934	−0.384967	−	0.922930i	$-0.625787\pi$
$727$	−15092.3	−0.769934	−0.384967	+	0.922930i	$0.625787\pi$
$728$	0	0
$729$	−9430.08	−0.479098
$730$	−12652.5	−0.641491
$731$	−32253.4	−1.63192
$732$	16755.3	0.846027
$733$	9108.56	0.458980	0.229490	−	0.973311i	$-0.426294\pi$
$733$	9108.56	0.458980	0.229490	+	0.973311i	$0.426294\pi$
$734$	389.343	0.0195789
$735$	41043.4	2.05974
$736$	−6122.62	−0.306634
$737$	4260.08	0.212920
$738$	−3322.64	−0.165729
$739$	19072.7	0.949394	0.474697	−	0.880149i	$-0.342558\pi$
$739$	19072.7	0.949394	0.474697	+	0.880149i	$0.342558\pi$
$740$	−17937.2	−0.891059
$741$	0	0
$742$	6276.57	0.310539
$743$	−12276.5	−0.606168	−0.303084	−	0.952964i	$-0.598016\pi$
$743$	−12276.5	−0.606168	−0.303084	+	0.952964i	$0.598016\pi$
$744$	−13045.4	−0.642834
$745$	−9780.33	−0.480971
$746$	5060.68	0.248371
$747$	−3952.76	−0.193606
$748$	1612.18	0.0788063
$749$	5895.30	0.287596
$750$	−45609.8	−2.22058
$751$	28665.5	1.39283	0.696417	−	0.717637i	$-0.254776\pi$
$751$	28665.5	1.39283	0.696417	+	0.717637i	$0.254776\pi$
$752$	−347.563	−0.0168541
$753$	−3286.44	−0.159050
$754$	0	0
$755$	−40340.3	−1.94455
$756$	−1686.74	−0.0811459
$757$	−28617.8	−1.37402	−0.687008	−	0.726650i	$-0.741076\pi$
$757$	−28617.8	−1.37402	−0.687008	+	0.726650i	$0.741076\pi$
$758$	−13494.1	−0.646608
$759$	−1020.92	−0.0488233
$760$	−26199.6	−1.25047
$761$	−26417.3	−1.25838	−0.629189	−	0.777252i	$-0.716613\pi$
$761$	−26417.3	−1.25838	−0.629189	+	0.777252i	$0.716613\pi$
$762$	5347.86	0.254242
$763$	10150.3	0.481607
$764$	20110.0	0.952295
$765$	31254.5	1.47713
$766$	−9633.54	−0.454404
$767$	0	0
$768$	−27881.6	−1.31001
$769$	−31524.2	−1.47827	−0.739136	−	0.673556i	$-0.764766\pi$
$769$	−31524.2	−1.47827	−0.739136	+	0.673556i	$0.764766\pi$
$770$	1198.65	0.0560990
$771$	1756.22	0.0820348
$772$	−829.105	−0.0386530
$773$	−21666.6	−1.00814	−0.504071	−	0.863662i	$-0.668165\pi$
$773$	−21666.6	−1.00814	−0.504071	+	0.863662i	$0.668165\pi$
$774$	15687.1	0.728501
$775$	25980.7	1.20420
$776$	17725.3	0.819974
$777$	8958.77	0.413635
$778$	21467.0	0.989242
$779$	5234.61	0.240756
$780$	0	0
$781$	1773.28	0.0812460
$782$	4241.12	0.193941
$783$	−5397.97	−0.246370
$784$	−364.438	−0.0166016
$785$	60226.0	2.73829
$786$	21297.7	0.966491
$787$	−26271.2	−1.18992	−0.594959	−	0.803756i	$-0.702832\pi$
$787$	−26271.2	−1.18992	−0.594959	+	0.803756i	$0.702832\pi$
$788$	4118.93	0.186207
$789$	15481.1	0.698532
$790$	29509.1	1.32897
$791$	−9246.28	−0.415626
$792$	−2034.87	−0.0912956
$793$	0	0
$794$	11089.4	0.495651
$795$	68970.7	3.07690
$796$	193.441	0.00861348
$797$	−336.983	−0.0149769	−0.00748843	−	0.999972i	$-0.502384\pi$
$797$	−336.983	−0.0149769	−0.00748843	+	0.999972i	$0.502384\pi$
$798$	5042.33	0.223680
$799$	19896.9	0.880980
$800$	56230.5	2.48506
$801$	16721.8	0.737625
$802$	−2683.66	−0.118159
$803$	1548.17	0.0680369
$804$	−33462.7	−1.46784
$805$	−5298.47	−0.231983
$806$	0	0
$807$	31619.9	1.37927
$808$	35821.0	1.55963
$809$	6478.70	0.281556	0.140778	−	0.990041i	$-0.455040\pi$
$809$	6478.70	0.281556	0.140778	+	0.990041i	$0.455040\pi$
$810$	30999.5	1.34470
$811$	−36823.2	−1.59437	−0.797186	−	0.603734i	$-0.793679\pi$
$811$	−36823.2	−1.59437	−0.797186	+	0.603734i	$0.793679\pi$
$812$	4607.50	0.199128
$813$	−54979.7	−2.37174
$814$	−1306.19	−0.0562430
$815$	−47779.2	−2.05354
$816$	−642.078	−0.0275456
$817$	−24714.0	−1.05830
$818$	15136.7	0.646997
$819$	0	0
$820$	−9774.34	−0.416262
$821$	8524.53	0.362373	0.181187	−	0.983449i	$-0.442006\pi$
$821$	8524.53	0.362373	0.181187	+	0.983449i	$0.442006\pi$
$822$	−9888.70	−0.419596
$823$	−13313.1	−0.563869	−0.281935	−	0.959434i	$-0.590976\pi$
$823$	−13313.1	−0.563869	−0.281935	+	0.959434i	$0.590976\pi$
$824$	3027.89	0.128012
$825$	9376.16	0.395680
$826$	4581.73	0.193001
$827$	41849.9	1.75969	0.879845	−	0.475261i	$-0.157646\pi$
$827$	41849.9	1.75969	0.879845	+	0.475261i	$0.157646\pi$
$828$	3466.08	0.145477
$829$	−13017.2	−0.545362	−0.272681	−	0.962104i	$-0.587910\pi$
$829$	−13017.2	−0.545362	−0.272681	+	0.962104i	$0.587910\pi$
$830$	6920.10	0.289398
$831$	36609.0	1.52822
$832$	0	0
$833$	20863.0	0.867780
$834$	11786.7	0.489376
$835$	23465.8	0.972537
$836$	1235.32	0.0511058
$837$	−3739.97	−0.154447
$838$	14273.2	0.588377
$839$	−20209.4	−0.831593	−0.415796	−	0.909458i	$-0.636497\pi$
$839$	−20209.4	−0.831593	−0.415796	+	0.909458i	$0.636497\pi$
$840$	−24433.9	−1.00363
$841$	−9643.94	−0.395422
$842$	−7656.78	−0.313385
$843$	45063.9	1.84114
$844$	−25297.1	−1.03171
$845$	0	0
$846$	−9677.25	−0.393275
$847$	9923.23	0.402557
$848$	−612.414	−0.0248000
$849$	28911.8	1.16873
$850$	−38950.7	−1.57176
$851$	5773.84	0.232579
$852$	−13929.1	−0.560096
$853$	7958.24	0.319443	0.159721	−	0.987162i	$-0.448940\pi$
$853$	7958.24	0.319443	0.159721	+	0.987162i	$0.448940\pi$
$854$	6332.31	0.253732
$855$	23948.5	0.957920
$856$	17521.2	0.699607
$857$	−2144.65	−0.0854840	−0.0427420	−	0.999086i	$-0.513609\pi$
$857$	−2144.65	−0.0854840	−0.0427420	+	0.999086i	$0.513609\pi$
$858$	0	0
$859$	−41723.5	−1.65726	−0.828632	−	0.559794i	$-0.810880\pi$
$859$	−41723.5	−1.65726	−0.828632	+	0.559794i	$0.810880\pi$
$860$	46147.3	1.82978
$861$	4881.82	0.193231
$862$	−17407.7	−0.687828
$863$	−10393.8	−0.409977	−0.204989	−	0.978764i	$-0.565716\pi$
$863$	−10393.8	−0.409977	−0.204989	+	0.978764i	$0.565716\pi$
$864$	−8094.48	−0.318727
$865$	10661.9	0.419093
$866$	172.204	0.00675718
$867$	2877.43	0.112713
$868$	3192.30	0.124831
$869$	−3610.76	−0.140951
$870$	−30131.2	−1.17419
$871$	0	0
$872$	30167.4	1.17156
$873$	−16202.3	−0.628138
$874$	3249.73	0.125771
$875$	28964.2	1.11905
$876$	−12160.8	−0.469035
$877$	−5005.59	−0.192733	−0.0963664	−	0.995346i	$-0.530722\pi$
$877$	−5005.59	−0.192733	−0.0963664	+	0.995346i	$0.530722\pi$
$878$	19744.7	0.758942
$879$	−16508.1	−0.633451
$880$	−116.954	−0.00448013
$881$	−14597.1	−0.558218	−0.279109	−	0.960259i	$-0.590039\pi$
$881$	−14597.1	−0.558218	−0.279109	+	0.960259i	$0.590039\pi$
$882$	−10147.1	−0.387382
$883$	−5629.11	−0.214535	−0.107268	−	0.994230i	$-0.534210\pi$
$883$	−5629.11	−0.214535	−0.107268	+	0.994230i	$0.534210\pi$
$884$	0	0
$885$	50346.8	1.91230
$886$	−8270.07	−0.313588
$887$	17014.9	0.644085	0.322043	−	0.946725i	$-0.395631\pi$
$887$	17014.9	0.644085	0.322043	+	0.946725i	$0.395631\pi$
$888$	26626.0	1.00621
$889$	−3396.12	−0.128124
$890$	−29275.0	−1.10258
$891$	−3793.12	−0.142620
$892$	−30732.5	−1.15359
$893$	15245.9	0.571315
$894$	5594.33	0.209287
$895$	−7568.73	−0.282676
$896$	7042.51	0.262582
$897$	0	0
$898$	−11188.8	−0.415784
$899$	10216.1	0.379005
$900$	−31832.7	−1.17899
$901$	35058.9	1.29632
$902$	−711.768	−0.0262741
$903$	−23048.4	−0.849393
$904$	−27480.5	−1.01105
$905$	−16265.3	−0.597434
$906$	23074.5	0.846137
$907$	−19185.3	−0.702358	−0.351179	−	0.936308i	$-0.614219\pi$
$907$	−19185.3	−0.702358	−0.351179	+	0.936308i	$0.614219\pi$
$908$	31811.5	1.16267
$909$	−32743.2	−1.19474
$910$	0	0
$911$	30427.5	1.10659	0.553297	−	0.832984i	$-0.313369\pi$
$911$	30427.5	1.10659	0.553297	+	0.832984i	$0.313369\pi$
$912$	−491.988	−0.0178633
$913$	−846.750	−0.0306937
$914$	−1603.87	−0.0580430
$915$	69583.3	2.51405
$916$	−6693.74	−0.241449
$917$	−13524.9	−0.487059
$918$	5607.03	0.201590
$919$	−39750.8	−1.42683	−0.713415	−	0.700742i	$-0.752853\pi$
$919$	−39750.8	−1.42683	−0.713415	+	0.700742i	$0.752853\pi$
$920$	−15747.4	−0.564322
$921$	−5666.95	−0.202749
$922$	−18207.9	−0.650374
$923$	0	0
$924$	1152.07	0.0410176
$925$	−53027.2	−1.88489
$926$	−8515.03	−0.302183
$927$	−2767.73	−0.0980628
$928$	22110.9	0.782138
$929$	−6260.88	−0.221112	−0.110556	−	0.993870i	$-0.535263\pi$
$929$	−6260.88	−0.221112	−0.110556	+	0.993870i	$0.535263\pi$
$930$	−20876.3	−0.736087
$931$	15986.1	0.562755
$932$	27044.4	0.950502
$933$	−37865.3	−1.32868
$934$	564.002	0.0197588
$935$	6695.26	0.234180
$936$	0	0
$937$	−24497.3	−0.854101	−0.427050	−	0.904228i	$-0.640447\pi$
$937$	−24497.3	−0.854101	−0.427050	+	0.904228i	$0.640447\pi$
$938$	−12646.6	−0.440219
$939$	−2175.53	−0.0756078
$940$	−28467.9	−0.987789
$941$	−10199.1	−0.353326	−0.176663	−	0.984271i	$-0.556530\pi$
$941$	−10199.1	−0.353326	−0.176663	+	0.984271i	$0.556530\pi$
$942$	−34449.1	−1.19152
$943$	3146.28	0.108650
$944$	−447.046	−0.0154133
$945$	−7004.92	−0.241132
$946$	3360.45	0.115494
$947$	−101.128	−0.00347015	−0.00173507	−	0.999998i	$-0.500552\pi$
$947$	−101.128	−0.00347015	−0.00173507	+	0.999998i	$0.500552\pi$
$948$	28362.4	0.971695
$949$	0	0
$950$	−29845.7	−1.01929
$951$	−57206.6	−1.95063
$952$	−12420.1	−0.422835
$953$	−42197.1	−1.43431	−0.717155	−	0.696914i	$-0.754556\pi$
$953$	−42197.1	−1.43431	−0.717155	+	0.696914i	$0.754556\pi$
$954$	−17051.6	−0.578684
$955$	83515.2	2.82983
$956$	18800.8	0.636048
$957$	3686.88	0.124535
$958$	3219.46	0.108576
$959$	6279.75	0.211453
$960$	−43717.6	−1.46977
$961$	−22712.8	−0.762405
$962$	0	0
$963$	−16015.8	−0.535931
$964$	−19205.9	−0.641681
$965$	−3443.21	−0.114861
$966$	3030.72	0.100944
$967$	1221.07	0.0406069	0.0203035	−	0.999794i	$-0.493537\pi$
$967$	1221.07	0.0406069	0.0203035	+	0.999794i	$0.493537\pi$
$968$	29492.5	0.979260
$969$	28164.8	0.933730
$970$	28365.4	0.938924
$971$	−42915.7	−1.41836	−0.709181	−	0.705026i	$-0.750935\pi$
$971$	−42915.7	−1.41836	−0.709181	+	0.705026i	$0.750935\pi$
$972$	23775.2	0.784559
$973$	−7485.06	−0.246619
$974$	30024.8	0.987739
$975$	0	0
$976$	−617.853	−0.0202633
$977$	31253.6	1.02343	0.511714	−	0.859156i	$-0.329011\pi$
$977$	31253.6	1.02343	0.511714	+	0.859156i	$0.329011\pi$
$978$	27329.6	0.893562
$979$	3582.11	0.116941
$980$	−29850.2	−0.972989
$981$	−27575.4	−0.897466
$982$	33271.1	1.08119
$983$	−41688.2	−1.35264	−0.676320	−	0.736608i	$-0.736426\pi$
$983$	−41688.2	−1.35264	−0.676320	+	0.736608i	$0.736426\pi$
$984$	14509.1	0.470054
$985$	17105.6	0.553329
$986$	−15316.1	−0.494691
$987$	14218.4	0.458537
$988$	0	0
$989$	−14854.4	−0.477597
$990$	−3256.36	−0.104539
$991$	20371.9	0.653013	0.326506	−	0.945195i	$-0.394129\pi$
$991$	20371.9	0.653013	0.326506	+	0.945195i	$0.394129\pi$
$992$	15319.5	0.490316
$993$	5896.47	0.188438
$994$	−5264.21	−0.167979
$995$	803.344	0.0255957
$996$	6651.18	0.211597
$997$	18696.5	0.593907	0.296954	−	0.954892i	$-0.404029\pi$
$997$	18696.5	0.593907	0.296954	+	0.954892i	$0.404029\pi$
$998$	−22611.6	−0.717191
$999$	7633.37	0.241751

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.k.1.7	✓	9
3.2	odd	2		1521.4.a.bh.1.3		9
13.2	odd	12		169.4.e.h.147.12		36
13.3	even	3		169.4.c.l.22.3		18
13.4	even	6		169.4.c.k.146.7		18
13.5	odd	4		169.4.b.g.168.7		18
13.6	odd	12		169.4.e.h.23.7		36
13.7	odd	12		169.4.e.h.23.12		36
13.8	odd	4		169.4.b.g.168.12		18
13.9	even	3		169.4.c.l.146.3		18
13.10	even	6		169.4.c.k.22.7		18
13.11	odd	12		169.4.e.h.147.7		36
13.12	even	2		169.4.a.l.1.3	yes	9
39.38	odd	2		1521.4.a.bg.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.7	✓	9	1.1	even	1	trivial
169.4.a.l.1.3	yes	9	13.12	even	2
169.4.b.g.168.7		18	13.5	odd	4
169.4.b.g.168.12		18	13.8	odd	4
169.4.c.k.22.7		18	13.10	even	6
169.4.c.k.146.7		18	13.4	even	6
169.4.c.l.22.3		18	13.3	even	3
169.4.c.l.146.3		18	13.9	even	3
169.4.e.h.23.7		36	13.6	odd	12
169.4.e.h.23.12		36	13.7	odd	12
169.4.e.h.147.7		36	13.11	odd	12
169.4.e.h.147.12		36	13.2	odd	12
1521.4.a.bg.1.7		9	39.38	odd	2
1521.4.a.bh.1.3		9	3.2	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:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)

\(f(q)\)	\(=\)	\(q+1.72763 q^{2} +6.89591 q^{3} -5.01528 q^{4} -20.8281 q^{5} +11.9136 q^{6} -7.56566 q^{7} -22.4856 q^{8} +20.5536 q^{9} +O(q^{10})\)	\(q+1.72763 q^{2} +6.89591 q^{3} -5.01528 q^{4} -20.8281 q^{5} +11.9136 q^{6} -7.56566 q^{7} -22.4856 q^{8} +20.5536 q^{9} -35.9833 q^{10} +4.40295 q^{11} -34.5850 q^{12} -13.0707 q^{14} -143.629 q^{15} +1.27533 q^{16} -73.0087 q^{17} +35.5091 q^{18} -55.9424 q^{19} +104.459 q^{20} -52.1722 q^{21} +7.60668 q^{22} -33.6244 q^{23} -155.059 q^{24} +308.809 q^{25} -44.4536 q^{27} +37.9439 q^{28} +121.429 q^{29} -248.138 q^{30} +84.1320 q^{31} +182.088 q^{32} +30.3624 q^{33} -126.132 q^{34} +157.578 q^{35} -103.082 q^{36} -171.716 q^{37} -96.6479 q^{38} +468.332 q^{40} -93.5714 q^{41} -90.1344 q^{42} +441.776 q^{43} -22.0820 q^{44} -428.093 q^{45} -58.0907 q^{46} -272.528 q^{47} +8.79455 q^{48} -285.761 q^{49} +533.508 q^{50} -503.461 q^{51} -480.202 q^{53} -76.7995 q^{54} -91.7049 q^{55} +170.119 q^{56} -385.774 q^{57} +209.785 q^{58} -350.534 q^{59} +720.338 q^{60} -484.467 q^{61} +145.349 q^{62} -155.502 q^{63} +304.379 q^{64} +52.4550 q^{66} +967.552 q^{67} +366.159 q^{68} -231.871 q^{69} +272.237 q^{70} +402.749 q^{71} -462.162 q^{72} +351.621 q^{73} -296.662 q^{74} +2129.52 q^{75} +280.567 q^{76} -33.3112 q^{77} -820.078 q^{79} -26.5626 q^{80} -861.496 q^{81} -161.657 q^{82} -192.314 q^{83} +261.658 q^{84} +1520.63 q^{85} +763.226 q^{86} +837.366 q^{87} -99.0031 q^{88} +813.571 q^{89} -739.587 q^{90} +168.636 q^{92} +580.167 q^{93} -470.829 q^{94} +1165.17 q^{95} +1255.67 q^{96} -788.293 q^{97} -493.690 q^{98} +90.4966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9}+O(q^{10}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9	\( 9 q - 5 q^{2} + q^{3} + 37 q^{4} - 30 q^{5} - 48 q^{6} - 38 q^{7} - 60 q^{8} + 66 q^{9} - 147 q^{10} - 181 q^{11} + 39 q^{12} - 147 q^{14} - 218 q^{15} + 269 q^{16} - 55 q^{17} - 79 q^{18} - 161 q^{19} - 370 q^{20} - 188 q^{21} + 340 q^{22} - 204 q^{23} - 798 q^{24} + 307 q^{25} - 668 q^{27} - 344 q^{28} + 280 q^{29} + 521 q^{30} - 706 q^{31} - 680 q^{32} - 500 q^{33} - 216 q^{34} + 20 q^{35} - 909 q^{36} - 298 q^{37} - 739 q^{38} + 13 q^{40} - 1201 q^{41} - 4 q^{42} - 533 q^{43} - 355 q^{44} + 90 q^{45} + 840 q^{46} - 956 q^{47} - 132 q^{48} + 403 q^{49} + 1156 q^{50} + 470 q^{51} - 278 q^{53} + 2555 q^{54} - 250 q^{55} + 250 q^{56} + 810 q^{57} + 2877 q^{58} - 1377 q^{59} + 3157 q^{60} - 136 q^{61} + 2035 q^{62} + 944 q^{63} + 284 q^{64} + 3279 q^{66} + 931 q^{67} - 1536 q^{68} - 2050 q^{69} + 4854 q^{70} - 2046 q^{71} + 4342 q^{72} + 45 q^{73} - 1990 q^{74} + 2393 q^{75} + 3608 q^{76} - 718 q^{77} + 412 q^{79} + 787 q^{80} - 835 q^{81} + 2757 q^{82} - 3709 q^{83} + 1539 q^{84} + 2106 q^{85} - 125 q^{86} - 786 q^{87} - 636 q^{88} - 1663 q^{89} - 1280 q^{90} + 4010 q^{92} + 1186 q^{93} - 2531 q^{94} - 1614 q^{95} + 3084 q^{96} + 1087 q^{97} + 282 q^{98} - 1357 q^{99}+O(q^{100}) \) 9 * q - 5 * q^2 + q^3 + 37 * q^4 - 30 * q^5 - 48 * q^6 - 38 * q^7 - 60 * q^8 + 66 * q^9 - 147 * q^10 - 181 * q^11 + 39 * q^12 - 147 * q^14 - 218 * q^15 + 269 * q^16 - 55 * q^17 - 79 * q^18 - 161 * q^19 - 370 * q^20 - 188 * q^21 + 340 * q^22 - 204 * q^23 - 798 * q^24 + 307 * q^25 - 668 * q^27 - 344 * q^28 + 280 * q^29 + 521 * q^30 - 706 * q^31 - 680 * q^32 - 500 * q^33 - 216 * q^34 + 20 * q^35 - 909 * q^36 - 298 * q^37 - 739 * q^38 + 13 * q^40 - 1201 * q^41 - 4 * q^42 - 533 * q^43 - 355 * q^44 + 90 * q^45 + 840 * q^46 - 956 * q^47 - 132 * q^48 + 403 * q^49 + 1156 * q^50 + 470 * q^51 - 278 * q^53 + 2555 * q^54 - 250 * q^55 + 250 * q^56 + 810 * q^57 + 2877 * q^58 - 1377 * q^59 + 3157 * q^60 - 136 * q^61 + 2035 * q^62 + 944 * q^63 + 284 * q^64 + 3279 * q^66 + 931 * q^67 - 1536 * q^68 - 2050 * q^69 + 4854 * q^70 - 2046 * q^71 + 4342 * q^72 + 45 * q^73 - 1990 * q^74 + 2393 * q^75 + 3608 * q^76 - 718 * q^77 + 412 * q^79 + 787 * q^80 - 835 * q^81 + 2757 * q^82 - 3709 * q^83 + 1539 * q^84 + 2106 * q^85 - 125 * q^86 - 786 * q^87 - 636 * q^88 - 1663 * q^89 - 1280 * q^90 + 4010 * q^92 + 1186 * q^93 - 2531 * q^94 - 1614 * q^95 + 3084 * q^96 + 1087 * q^97 + 282 * q^98 - 1357 * q^99

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