LMFDB - Embedded newform 169.4.a.l.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:	$0$
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			$-2.16135$ 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)

$f(q)$	$=$	$q+3.16135 q^{2} +7.08883 q^{3} +1.99415 q^{4} +13.6039 q^{5} +22.4103 q^{6} -14.3315 q^{7} -18.9866 q^{8} +23.2516 q^{9} +O(q^{10})$	\(q+3.16135 q^{2} +7.08883 q^{3} +1.99415 q^{4} +13.6039 q^{5} +22.4103 q^{6} -14.3315 q^{7} -18.9866 q^{8} +23.2516 q^{9} +43.0068 q^{10} +67.7223 q^{11} +14.1362 q^{12} -45.3068 q^{14} +96.4360 q^{15} -75.9766 q^{16} +0.337795 q^{17} +73.5064 q^{18} -40.5229 q^{19} +27.1282 q^{20} -101.593 q^{21} +214.094 q^{22} -155.632 q^{23} -134.593 q^{24} +60.0670 q^{25} -26.5720 q^{27} -28.5791 q^{28} -33.7925 q^{29} +304.868 q^{30} +157.397 q^{31} -88.2957 q^{32} +480.072 q^{33} +1.06789 q^{34} -194.964 q^{35} +46.3670 q^{36} +58.6204 q^{37} -128.107 q^{38} -258.293 q^{40} -59.3488 q^{41} -321.173 q^{42} -208.311 q^{43} +135.048 q^{44} +316.313 q^{45} -492.006 q^{46} +221.212 q^{47} -538.585 q^{48} -137.609 q^{49} +189.893 q^{50} +2.39457 q^{51} -409.639 q^{53} -84.0035 q^{54} +921.289 q^{55} +272.106 q^{56} -287.260 q^{57} -106.830 q^{58} -173.587 q^{59} +192.308 q^{60} +560.796 q^{61} +497.586 q^{62} -333.229 q^{63} +328.679 q^{64} +1517.68 q^{66} +269.074 q^{67} +0.673613 q^{68} -1103.25 q^{69} -616.351 q^{70} -60.9754 q^{71} -441.469 q^{72} -282.066 q^{73} +185.320 q^{74} +425.805 q^{75} -80.8086 q^{76} -970.560 q^{77} +984.026 q^{79} -1033.58 q^{80} -816.157 q^{81} -187.622 q^{82} -1201.86 q^{83} -202.592 q^{84} +4.59534 q^{85} -658.543 q^{86} -239.550 q^{87} -1285.82 q^{88} +539.899 q^{89} +999.976 q^{90} -310.352 q^{92} +1115.76 q^{93} +699.328 q^{94} -551.271 q^{95} -625.913 q^{96} -1587.25 q^{97} -435.030 q^{98} +1574.65 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	3.16135	1.11771	0.558853	−	0.829267i	$-0.311241\pi$
$2$	3.16135	1.11771	0.558853	+	0.829267i	$0.311241\pi$
$3$	7.08883	1.36425	0.682123	−	0.731237i	$-0.261057\pi$
$3$	7.08883	1.36425	0.682123	+	0.731237i	$0.261057\pi$
$4$	1.99415	0.249268
$5$	13.6039	1.21677	0.608386	−	0.793641i	$-0.291817\pi$
$5$	13.6039	1.21677	0.608386	+	0.793641i	$0.291817\pi$
$6$	22.4103	1.52483
$7$	−14.3315	−0.773827	−0.386913	−	0.922116i	$-0.626459\pi$
$7$	−14.3315	−0.773827	−0.386913	+	0.922116i	$0.626459\pi$
$8$	−18.9866	−0.839098
$9$	23.2516	0.861169
$10$	43.0068	1.36000
$11$	67.7223	1.85628	0.928138	−	0.372237i	$-0.121409\pi$
$11$	67.7223	1.85628	0.928138	+	0.372237i	$0.121409\pi$
$12$	14.1362	0.340063
$13$	0	0
$13$	0	0
$14$	−45.3068	−0.864911
$15$	96.4360	1.65998
$16$	−75.9766	−1.18713
$17$	0.337795	0.00481925	0.00240963	−	0.999997i	$-0.499233\pi$
$17$	0.337795	0.00481925	0.00240963	+	0.999997i	$0.499233\pi$
$18$	73.5064	0.962535
$19$	−40.5229	−0.489294	−0.244647	−	0.969612i	$-0.578672\pi$
$19$	−40.5229	−0.489294	−0.244647	+	0.969612i	$0.578672\pi$
$20$	27.1282	0.303303
$21$	−101.593	−1.05569
$22$	214.094	2.07477
$23$	−155.632	−1.41093	−0.705466	−	0.708744i	$-0.749262\pi$
$23$	−155.632	−1.41093	−0.705466	+	0.708744i	$0.749262\pi$
$24$	−134.593	−1.14474
$25$	60.0670	0.480536
$26$	0	0
$27$	−26.5720	−0.189399
$28$	−28.5791	−0.192890
$29$	−33.7925	−0.216383	−0.108192	−	0.994130i	$-0.534506\pi$
$29$	−33.7925	−0.216383	−0.108192	+	0.994130i	$0.534506\pi$
$30$	304.868	1.85537
$31$	157.397	0.911911	0.455956	−	0.890003i	$-0.349298\pi$
$31$	157.397	0.911911	0.455956	+	0.890003i	$0.349298\pi$
$32$	−88.2957	−0.487769
$33$	480.072	2.53242
$34$	1.06789	0.00538651
$35$	−194.964	−0.941571
$36$	46.3670	0.214662
$37$	58.6204	0.260463	0.130231	−	0.991484i	$-0.458428\pi$
$37$	58.6204	0.260463	0.130231	+	0.991484i	$0.458428\pi$
$38$	−128.107	−0.546888
$39$	0	0
$40$	−258.293	−1.02099
$41$	−59.3488	−0.226066	−0.113033	−	0.993591i	$-0.536057\pi$
$41$	−59.3488	−0.226066	−0.113033	+	0.993591i	$0.536057\pi$
$42$	−321.173	−1.17995
$43$	−208.311	−0.738769	−0.369385	−	0.929277i	$-0.620431\pi$
$43$	−208.311	−0.738769	−0.369385	+	0.929277i	$0.620431\pi$
$44$	135.048	0.462711
$45$	316.313	1.04785
$46$	−492.006	−1.57701
$47$	221.212	0.686533	0.343266	−	0.939238i	$-0.388467\pi$
$47$	221.212	0.686533	0.343266	+	0.939238i	$0.388467\pi$
$48$	−538.585	−1.61954
$49$	−137.609	−0.401192
$50$	189.893	0.537098
$51$	2.39457	0.00657465
$52$	0	0
$53$	−409.639	−1.06167	−0.530833	−	0.847477i	$-0.678121\pi$
$53$	−409.639	−1.06167	−0.530833	+	0.847477i	$0.678121\pi$
$54$	−84.0035	−0.211693
$55$	921.289	2.25867
$56$	272.106	0.649316
$57$	−287.260	−0.667518
$58$	−106.830	−0.241853
$59$	−173.587	−0.383036	−0.191518	−	0.981489i	$-0.561341\pi$
$59$	−173.587	−0.383036	−0.191518	+	0.981489i	$0.561341\pi$
$60$	192.308	0.413780
$61$	560.796	1.17709	0.588546	−	0.808464i	$-0.299701\pi$
$61$	560.796	1.17709	0.588546	+	0.808464i	$0.299701\pi$
$62$	497.586	1.01925
$63$	−333.229	−0.666396
$64$	328.679	0.641951
$65$	0	0
$66$	1517.68	2.83050
$67$	269.074	0.490635	0.245318	−	0.969443i	$-0.421108\pi$
$67$	269.074	0.490635	0.245318	+	0.969443i	$0.421108\pi$
$68$	0.673613	0.00120129
$69$	−1103.25	−1.92486
$70$	−616.351	−1.05240
$71$	−60.9754	−0.101922	−0.0509609	−	0.998701i	$-0.516228\pi$
$71$	−60.9754	−0.101922	−0.0509609	+	0.998701i	$0.516228\pi$
$72$	−441.469	−0.722605
$73$	−282.066	−0.452238	−0.226119	−	0.974100i	$-0.572604\pi$
$73$	−282.066	−0.452238	−0.226119	+	0.974100i	$0.572604\pi$
$74$	185.320	0.291121
$75$	425.805	0.655570
$76$	−80.8086	−0.121966
$77$	−970.560	−1.43644
$78$	0	0
$79$	984.026	1.40141	0.700706	−	0.713450i	$-0.252868\pi$
$79$	984.026	1.40141	0.700706	+	0.713450i	$0.252868\pi$
$80$	−1033.58	−1.44447
$81$	−816.157	−1.11956
$82$	−187.622	−0.252676
$83$	−1201.86	−1.58942	−0.794709	−	0.606991i	$-0.792376\pi$
$83$	−1201.86	−1.58942	−0.794709	+	0.606991i	$0.792376\pi$
$84$	−202.592	−0.263150
$85$	4.59534	0.00586394
$86$	−658.543	−0.825727
$87$	−239.550	−0.295200
$88$	−1285.82	−1.55760
$89$	539.899	0.643024	0.321512	−	0.946906i	$-0.395809\pi$
$89$	539.899	0.643024	0.321512	+	0.946906i	$0.395809\pi$
$90$	999.976	1.17119
$91$	0	0
$92$	−310.352	−0.351701
$93$	1115.76	1.24407
$94$	699.328	0.767342
$95$	−551.271	−0.595360
$96$	−625.913	−0.665438
$97$	−1587.25	−1.66146	−0.830728	−	0.556678i	$-0.812076\pi$
$97$	−1587.25	−1.66146	−0.830728	+	0.556678i	$0.812076\pi$
$98$	−435.030	−0.448415
$99$	1574.65	1.59857
$100$	119.782	0.119782
$101$	1160.73	1.14353	0.571765	−	0.820417i	$-0.306259\pi$
$101$	1160.73	1.14353	0.571765	+	0.820417i	$0.306259\pi$
$102$	7.57009	0.00734853
$103$	−82.2962	−0.0787270	−0.0393635	−	0.999225i	$-0.512533\pi$
$103$	−82.2962	−0.0787270	−0.0393635	+	0.999225i	$0.512533\pi$
$104$	0	0
$105$	−1382.07	−1.28454
$106$	−1295.01	−1.18663
$107$	1322.04	1.19446	0.597229	−	0.802071i	$-0.296269\pi$
$107$	1322.04	1.19446	0.597229	+	0.802071i	$0.296269\pi$
$108$	−52.9885	−0.0472113
$109$	2073.41	1.82199	0.910994	−	0.412420i	$-0.135316\pi$
$109$	2073.41	1.82199	0.910994	+	0.412420i	$0.135316\pi$
$110$	2912.52	2.52453
$111$	415.550	0.355336
$112$	1088.86	0.918636
$113$	−1602.82	−1.33434	−0.667172	−	0.744904i	$-0.732495\pi$
$113$	−1602.82	−1.33434	−0.667172	+	0.744904i	$0.732495\pi$
$114$	−908.131	−0.746090
$115$	−2117.20	−1.71678
$116$	−67.3872	−0.0539375
$117$	0	0
$118$	−548.771	−0.428122
$119$	−4.84110	−0.00372927
$120$	−1830.99	−1.39288
$121$	3255.30	2.44576
$122$	1772.87	1.31564
$123$	−420.714	−0.308410
$124$	313.872	0.227311
$125$	−883.344	−0.632070
$126$	−1053.45	−0.744835
$127$	614.495	0.429352	0.214676	−	0.976685i	$-0.431131\pi$
$127$	614.495	0.429352	0.214676	+	0.976685i	$0.431131\pi$
$128$	1745.43	1.20528
$129$	−1476.68	−1.00786
$130$	0	0
$131$	−330.171	−0.220208	−0.110104	−	0.993920i	$-0.535118\pi$
$131$	−330.171	−0.220208	−0.110104	+	0.993920i	$0.535118\pi$
$132$	957.333	0.631251
$133$	580.753	0.378629
$134$	850.636	0.548386
$135$	−361.484	−0.230456
$136$	−6.41358	−0.00404383
$137$	2302.54	1.43591	0.717953	−	0.696091i	$-0.245079\pi$
$137$	2302.54	1.43591	0.717953	+	0.696091i	$0.245079\pi$
$138$	−3487.75	−2.15143
$139$	1207.22	0.736656	0.368328	−	0.929696i	$-0.379930\pi$
$139$	1207.22	0.736656	0.368328	+	0.929696i	$0.379930\pi$
$140$	−388.788	−0.234704
$141$	1568.13	0.936600
$142$	−192.765	−0.113919
$143$	0	0
$144$	−1766.57	−1.02232
$145$	−459.711	−0.263289
$146$	−891.711	−0.505470
$147$	−975.486	−0.547325
$148$	116.898	0.0649251
$149$	338.938	0.186355	0.0931774	−	0.995650i	$-0.470298\pi$
$149$	338.938	0.186355	0.0931774	+	0.995650i	$0.470298\pi$
$150$	1346.12	0.732734
$151$	1694.81	0.913386	0.456693	−	0.889624i	$-0.349034\pi$
$151$	1694.81	0.913386	0.456693	+	0.889624i	$0.349034\pi$
$152$	769.393	0.410566
$153$	7.85426	0.00415019
$154$	−3068.28	−1.60551
$155$	2141.21	1.10959
$156$	0	0
$157$	9.59250	0.00487621	0.00243811	−	0.999997i	$-0.499224\pi$
$157$	9.59250	0.00487621	0.00243811	+	0.999997i	$0.499224\pi$
$158$	3110.85	1.56637
$159$	−2903.86	−1.44837
$160$	−1201.17	−0.593504
$161$	2230.43	1.09182
$162$	−2580.16	−1.25134
$163$	122.100	0.0586725	0.0293363	−	0.999570i	$-0.490661\pi$
$163$	122.100	0.0586725	0.0293363	+	0.999570i	$0.490661\pi$
$164$	−118.350	−0.0563512
$165$	6530.87	3.08138
$166$	−3799.51	−1.77650
$167$	2211.25	1.02462	0.512309	−	0.858801i	$-0.328790\pi$
$167$	2211.25	1.02462	0.512309	+	0.858801i	$0.328790\pi$
$168$	1928.92	0.885828
$169$	0	0
$170$	14.5275	0.00655416
$171$	−942.221	−0.421365
$172$	−415.402	−0.184152
$173$	−213.874	−0.0939914	−0.0469957	−	0.998895i	$-0.514965\pi$
$173$	−213.874	−0.0939914	−0.0469957	+	0.998895i	$0.514965\pi$
$174$	−757.301	−0.329947
$175$	−860.848	−0.371852
$176$	−5145.30	−2.20365
$177$	−1230.53	−0.522556
$178$	1706.81	0.718712
$179$	2967.41	1.23907	0.619537	−	0.784967i	$-0.287320\pi$
$179$	2967.41	1.23907	0.619537	+	0.784967i	$0.287320\pi$
$180$	630.774	0.261195
$181$	2329.44	0.956609	0.478304	−	0.878194i	$-0.341252\pi$
$181$	2329.44	0.956609	0.478304	+	0.878194i	$0.341252\pi$
$182$	0	0
$183$	3975.39	1.60584
$184$	2954.92	1.18391
$185$	797.467	0.316924
$186$	3527.30	1.39051
$187$	22.8762	0.00894586
$188$	441.128	0.171131
$189$	380.816	0.146562
$190$	−1742.76	−0.665438
$191$	−4479.76	−1.69709	−0.848544	−	0.529125i	$-0.822520\pi$
$191$	−4479.76	−1.69709	−0.848544	+	0.529125i	$0.822520\pi$
$192$	2329.95	0.875779
$193$	1302.63	0.485830	0.242915	−	0.970048i	$-0.421896\pi$
$193$	1302.63	0.485830	0.242915	+	0.970048i	$0.421896\pi$
$194$	−5017.87	−1.85702
$195$	0	0
$196$	−274.412	−0.100004
$197$	−740.954	−0.267974	−0.133987	−	0.990983i	$-0.542778\pi$
$197$	−740.954	−0.267974	−0.133987	+	0.990983i	$0.542778\pi$
$198$	4978.02	1.78673
$199$	−4688.74	−1.67023	−0.835115	−	0.550075i	$-0.814599\pi$
$199$	−4688.74	−1.67023	−0.835115	+	0.550075i	$0.814599\pi$
$200$	−1140.47	−0.403217
$201$	1907.42	0.669348
$202$	3669.46	1.27813
$203$	484.297	0.167443
$204$	4.77513	0.00163885
$205$	−807.377	−0.275071
$206$	−260.167	−0.0879937
$207$	−3618.68	−1.21505
$208$	0	0
$209$	−2744.30	−0.908265
$210$	−4369.21	−1.43573
$211$	−3205.11	−1.04573	−0.522865	−	0.852416i	$-0.675137\pi$
$211$	−3205.11	−1.04573	−0.522865	+	0.852416i	$0.675137\pi$
$212$	−816.880	−0.264640
$213$	−432.244	−0.139046
$214$	4179.45	1.33505
$215$	−2833.84	−0.898914
$216$	504.513	0.158925
$217$	−2255.72	−0.705661
$218$	6554.78	2.03645
$219$	−1999.52	−0.616964
$220$	1837.18	0.563014
$221$	0	0
$222$	1313.70	0.397161
$223$	1057.39	0.317525	0.158762	−	0.987317i	$-0.449250\pi$
$223$	1057.39	0.317525	0.158762	+	0.987317i	$0.449250\pi$
$224$	1265.41	0.377449
$225$	1396.65	0.413823
$226$	−5067.09	−1.49141
$227$	5159.68	1.50863	0.754317	−	0.656510i	$-0.227968\pi$
$227$	5159.68	1.50863	0.754317	+	0.656510i	$0.227968\pi$
$228$	−572.839	−0.166391
$229$	1698.25	0.490059	0.245030	−	0.969516i	$-0.421202\pi$
$229$	1698.25	0.490059	0.245030	+	0.969516i	$0.421202\pi$
$230$	−6693.22	−1.91886
$231$	−6880.14	−1.95965
$232$	641.606	0.181567
$233$	3162.23	0.889119	0.444559	−	0.895749i	$-0.353360\pi$
$233$	3162.23	0.889119	0.444559	+	0.895749i	$0.353360\pi$
$234$	0	0
$235$	3009.35	0.835354
$236$	−346.159	−0.0954788
$237$	6975.60	1.91187
$238$	−15.3044	−0.00416823
$239$	2350.04	0.636030	0.318015	−	0.948086i	$-0.396984\pi$
$239$	2350.04	0.636030	0.318015	+	0.948086i	$0.396984\pi$
$240$	−7326.88	−1.97062
$241$	−5167.18	−1.38111	−0.690555	−	0.723280i	$-0.742633\pi$
$241$	−5167.18	−1.38111	−0.690555	+	0.723280i	$0.742633\pi$
$242$	10291.2	2.73364
$243$	−5068.16	−1.33795
$244$	1118.31	0.293412
$245$	−1872.02	−0.488159
$246$	−1330.02	−0.344712
$247$	0	0
$248$	−2988.43	−0.765183
$249$	−8519.81	−2.16836
$250$	−2792.56	−0.706469
$251$	2894.01	0.727761	0.363880	−	0.931446i	$-0.381452\pi$
$251$	2894.01	0.727761	0.363880	+	0.931446i	$0.381452\pi$
$252$	−664.508	−0.166111
$253$	−10539.7	−2.61908
$254$	1942.64	0.479889
$255$	32.5756	0.00799986
$256$	2888.50	0.705201
$257$	−6162.63	−1.49577	−0.747887	−	0.663826i	$-0.768932\pi$
$257$	−6162.63	−1.49577	−0.747887	+	0.663826i	$0.768932\pi$
$258$	−4668.30	−1.12650
$259$	−840.116	−0.201553
$260$	0	0
$261$	−785.729	−0.186343
$262$	−1043.79	−0.246127
$263$	−199.427	−0.0467574	−0.0233787	−	0.999727i	$-0.507442\pi$
$263$	−199.427	−0.0467574	−0.0233787	+	0.999727i	$0.507442\pi$
$264$	−9114.94	−2.12495
$265$	−5572.70	−1.29181
$266$	1835.96	0.423196
$267$	3827.25	0.877243
$268$	536.572	0.122300
$269$	1108.00	0.251138	0.125569	−	0.992085i	$-0.459924\pi$
$269$	1108.00	0.251138	0.125569	+	0.992085i	$0.459924\pi$
$270$	−1142.78	−0.257582
$271$	406.054	0.0910186	0.0455093	−	0.998964i	$-0.485509\pi$
$271$	406.054	0.0910186	0.0455093	+	0.998964i	$0.485509\pi$
$272$	−25.6645	−0.00572110
$273$	0	0
$274$	7279.13	1.60492
$275$	4067.87	0.892007
$276$	−2200.04	−0.479806
$277$	288.810	0.0626460	0.0313230	−	0.999509i	$-0.490028\pi$
$277$	288.810	0.0626460	0.0313230	+	0.999509i	$0.490028\pi$
$278$	3816.45	0.823366
$279$	3659.72	0.785310
$280$	3701.71	0.790071
$281$	−5134.93	−1.09012	−0.545061	−	0.838396i	$-0.683494\pi$
$281$	−5134.93	−1.09012	−0.545061	+	0.838396i	$0.683494\pi$
$282$	4957.42	1.04684
$283$	5274.40	1.10788	0.553940	−	0.832556i	$-0.313124\pi$
$283$	5274.40	1.10788	0.553940	+	0.832556i	$0.313124\pi$
$284$	−121.594	−0.0254059
$285$	−3907.87	−0.812218
$286$	0	0
$287$	850.556	0.174936
$288$	−2053.01	−0.420052
$289$	−4912.89	−0.999977
$290$	−1453.31	−0.294280
$291$	−11251.8	−2.26664
$292$	−562.482	−0.112729
$293$	−1445.69	−0.288253	−0.144127	−	0.989559i	$-0.546037\pi$
$293$	−1445.69	−0.288253	−0.144127	+	0.989559i	$0.546037\pi$
$294$	−3083.86	−0.611749
$295$	−2361.47	−0.466068
$296$	−1113.00	−0.218554
$297$	−1799.52	−0.351578
$298$	1071.50	0.208290
$299$	0	0
$300$	849.117	0.163413
$301$	2985.40	0.571679
$302$	5357.88	1.02090
$303$	8228.19	1.56006
$304$	3078.79	0.580858
$305$	7629.04	1.43225
$306$	24.8301	0.00463870
$307$	−9421.42	−1.75149	−0.875747	−	0.482770i	$-0.839631\pi$
$307$	−9421.42	−1.75149	−0.875747	+	0.482770i	$0.839631\pi$
$308$	−1935.44	−0.358058
$309$	−583.384	−0.107403
$310$	6769.12	1.24019
$311$	7885.87	1.43784	0.718918	−	0.695095i	$-0.244638\pi$
$311$	7885.87	1.43784	0.718918	+	0.695095i	$0.244638\pi$
$312$	0	0
$313$	−550.423	−0.0993986	−0.0496993	−	0.998764i	$-0.515826\pi$
$313$	−550.423	−0.0993986	−0.0496993	+	0.998764i	$0.515826\pi$
$314$	30.3253	0.00545017
$315$	−4533.23	−0.810852
$316$	1962.29	0.349328
$317$	150.974	0.0267494	0.0133747	−	0.999911i	$-0.495743\pi$
$317$	150.974	0.0267494	0.0133747	+	0.999911i	$0.495743\pi$
$318$	−9180.14	−1.61886
$319$	−2288.51	−0.401667
$320$	4471.32	0.781108
$321$	9371.76	1.62953
$322$	7051.18	1.22033
$323$	−13.6884	−0.00235803
$324$	−1627.54	−0.279070
$325$	0	0
$326$	386.002	0.0655787
$327$	14698.1	2.48564
$328$	1126.83	0.189692
$329$	−3170.29	−0.531257
$330$	20646.4	3.44408
$331$	7079.29	1.17557	0.587784	−	0.809018i	$-0.300001\pi$
$331$	7079.29	1.17557	0.587784	+	0.809018i	$0.300001\pi$
$332$	−2396.69	−0.396191
$333$	1363.02	0.224303
$334$	6990.52	1.14522
$335$	3660.46	0.596992
$336$	7718.72	1.25325
$337$	−6633.34	−1.07223	−0.536114	−	0.844145i	$-0.680108\pi$
$337$	−6633.34	−1.07223	−0.536114	+	0.844145i	$0.680108\pi$
$338$	0	0
$339$	−11362.1	−1.82037
$340$	9.16378	0.00146169
$341$	10659.2	1.69276
$342$	−2978.69	−0.470963
$343$	6887.83	1.08428
$344$	3955.11	0.619900
$345$	−15008.5	−2.34212
$346$	−676.130	−0.105055
$347$	6404.17	0.990760	0.495380	−	0.868676i	$-0.335029\pi$
$347$	6404.17	0.990760	0.495380	+	0.868676i	$0.335029\pi$
$348$	−477.697	−0.0735840
$349$	168.098	0.0257825	0.0128913	−	0.999917i	$-0.495896\pi$
$349$	168.098	0.0257825	0.0128913	+	0.999917i	$0.495896\pi$
$350$	−2721.45	−0.415621
$351$	0	0
$352$	−5979.58	−0.905434
$353$	−8451.92	−1.27436	−0.637182	−	0.770713i	$-0.719900\pi$
$353$	−8451.92	−1.27436	−0.637182	+	0.770713i	$0.719900\pi$
$354$	−3890.14	−0.584065
$355$	−829.505	−0.124016
$356$	1076.64	0.160285
$357$	−34.3178	−0.00508764
$358$	9381.01	1.38492
$359$	2631.92	0.386929	0.193464	−	0.981107i	$-0.438028\pi$
$359$	2631.92	0.386929	0.193464	+	0.981107i	$0.438028\pi$
$360$	−6005.71	−0.879246
$361$	−5216.89	−0.760591
$362$	7364.19	1.06921
$363$	23076.3	3.33662
$364$	0	0
$365$	−3837.21	−0.550271
$366$	12567.6	1.79486
$367$	−13269.9	−1.88742	−0.943708	−	0.330780i	$-0.892688\pi$
$367$	−13269.9	−1.88742	−0.943708	+	0.330780i	$0.892688\pi$
$368$	11824.4	1.67496
$369$	−1379.95	−0.194681
$370$	2521.07	0.354228
$371$	5870.73	0.821545
$372$	2224.98	0.310108
$373$	−11680.1	−1.62137	−0.810684	−	0.585483i	$-0.800905\pi$
$373$	−11680.1	−1.62137	−0.810684	+	0.585483i	$0.800905\pi$
$374$	72.3198	0.00999885
$375$	−6261.88	−0.862299
$376$	−4200.06	−0.576068
$377$	0	0
$378$	1203.89	0.163814
$379$	−3378.23	−0.457857	−0.228928	−	0.973443i	$-0.573522\pi$
$379$	−3378.23	−0.457857	−0.228928	+	0.973443i	$0.573522\pi$
$380$	−1099.31	−0.148404
$381$	4356.06	0.585742
$382$	−14162.1	−1.89685
$383$	−4380.92	−0.584476	−0.292238	−	0.956346i	$-0.594400\pi$
$383$	−4380.92	−0.584476	−0.292238	+	0.956346i	$0.594400\pi$
$384$	12373.1	1.64430
$385$	−13203.4	−1.74782
$386$	4118.06	0.543015
$387$	−4843.55	−0.636205
$388$	−3165.22	−0.414148
$389$	−12484.0	−1.62716	−0.813578	−	0.581456i	$-0.802483\pi$
$389$	−12484.0	−1.62716	−0.813578	+	0.581456i	$0.802483\pi$
$390$	0	0
$391$	−52.5716	−0.00679964
$392$	2612.73	0.336639
$393$	−2340.53	−0.300417
$394$	−2342.42	−0.299516
$395$	13386.6	1.70520
$396$	3140.08	0.398472
$397$	9450.48	1.19473	0.597363	−	0.801971i	$-0.296215\pi$
$397$	9450.48	1.19473	0.597363	+	0.801971i	$0.296215\pi$
$398$	−14822.8	−1.86683
$399$	4116.86	0.516544
$400$	−4563.68	−0.570460
$401$	2906.03	0.361895	0.180948	−	0.983493i	$-0.442084\pi$
$401$	2906.03	0.361895	0.180948	+	0.983493i	$0.442084\pi$
$402$	6030.02	0.748134
$403$	0	0
$404$	2314.66	0.285046
$405$	−11102.9	−1.36225
$406$	1531.03	0.187152
$407$	3969.90	0.483491
$408$	−45.4648	−0.00551678
$409$	−5795.93	−0.700710	−0.350355	−	0.936617i	$-0.613939\pi$
$409$	−5795.93	−0.700710	−0.350355	+	0.936617i	$0.613939\pi$
$410$	−2552.40	−0.307449
$411$	16322.3	1.95893
$412$	−164.111	−0.0196242
$413$	2487.76	0.296404
$414$	−11439.9	−1.35807
$415$	−16350.1	−1.93396
$416$	0	0
$417$	8557.79	1.00498
$418$	−8675.71	−1.01517
$419$	−4142.47	−0.482990	−0.241495	−	0.970402i	$-0.577638\pi$
$419$	−4142.47	−0.482990	−0.241495	+	0.970402i	$0.577638\pi$
$420$	−2756.05	−0.320194
$421$	−1426.92	−0.165187	−0.0825935	−	0.996583i	$-0.526320\pi$
$421$	−1426.92	−0.165187	−0.0825935	+	0.996583i	$0.526320\pi$
$422$	−10132.5	−1.16882
$423$	5143.52	0.591221
$424$	7777.66	0.890841
$425$	20.2903	0.00231582
$426$	−1366.48	−0.155413
$427$	−8037.04	−0.910865
$428$	2636.35	0.297740
$429$	0	0
$430$	−8958.78	−1.00472
$431$	2543.08	0.284213	0.142106	−	0.989851i	$-0.454612\pi$
$431$	2543.08	0.284213	0.142106	+	0.989851i	$0.454612\pi$
$432$	2018.85	0.224842
$433$	−3448.44	−0.382728	−0.191364	−	0.981519i	$-0.561291\pi$
$433$	−3448.44	−0.382728	−0.191364	+	0.981519i	$0.561291\pi$
$434$	−7131.14	−0.788723
$435$	−3258.82	−0.359191
$436$	4134.68	0.454164
$437$	6306.65	0.690361
$438$	−6321.19	−0.689585
$439$	−12642.6	−1.37448	−0.687240	−	0.726431i	$-0.741178\pi$
$439$	−12642.6	−1.37448	−0.687240	+	0.726431i	$0.741178\pi$
$440$	−17492.2	−1.89524
$441$	−3199.62	−0.345494
$442$	0	0
$443$	8486.59	0.910181	0.455091	−	0.890445i	$-0.349607\pi$
$443$	8486.59	0.910181	0.455091	+	0.890445i	$0.349607\pi$
$444$	828.667	0.0885739
$445$	7344.74	0.782414
$446$	3342.78	0.354900
$447$	2402.67	0.254234
$448$	−4710.45	−0.496759
$449$	−11369.5	−1.19502	−0.597508	−	0.801863i	$-0.703842\pi$
$449$	−11369.5	−1.19502	−0.597508	+	0.801863i	$0.703842\pi$
$450$	4415.31	0.462532
$451$	−4019.23	−0.419642
$452$	−3196.26	−0.332610
$453$	12014.2	1.24608
$454$	16311.6	1.68621
$455$	0	0
$456$	5454.10	0.560113
$457$	−11289.0	−1.15553	−0.577766	−	0.816202i	$-0.696075\pi$
$457$	−11289.0	−1.15553	−0.577766	+	0.816202i	$0.696075\pi$
$458$	5368.77	0.547742
$459$	−8.97589	−0.000912764 0
$460$	−4222.01	−0.427940
$461$	−3834.12	−0.387359	−0.193680	−	0.981065i	$-0.562042\pi$
$461$	−3834.12	−0.387359	−0.193680	+	0.981065i	$0.562042\pi$
$462$	−21750.5	−2.19032
$463$	−1294.44	−0.129930	−0.0649651	−	0.997888i	$-0.520694\pi$
$463$	−1294.44	−0.129930	−0.0649651	+	0.997888i	$0.520694\pi$
$464$	2567.44	0.256876
$465$	15178.7	1.51375
$466$	9996.93	0.993774
$467$	13861.5	1.37352	0.686760	−	0.726884i	$-0.259032\pi$
$467$	13861.5	1.37352	0.686760	+	0.726884i	$0.259032\pi$
$468$	0	0
$469$	−3856.22	−0.379667
$470$	9513.61	0.933681
$471$	67.9997	0.00665235
$472$	3295.84	0.321405
$473$	−14107.3	−1.37136
$474$	22052.3	2.13691
$475$	−2434.09	−0.235124
$476$	−9.65386	−0.000929588 0
$477$	−9524.75	−0.914274
$478$	7429.29	0.710895
$479$	9627.66	0.918369	0.459185	−	0.888341i	$-0.348142\pi$
$479$	9627.66	0.918369	0.459185	+	0.888341i	$0.348142\pi$
$480$	−8514.88	−0.809686
$481$	0	0
$482$	−16335.3	−1.54368
$483$	15811.2	1.48951
$484$	6491.55	0.609650
$485$	−21592.9	−2.02162
$486$	−16022.2	−1.49544
$487$	11952.8	1.11219	0.556094	−	0.831120i	$-0.312300\pi$
$487$	11952.8	1.11219	0.556094	+	0.831120i	$0.312300\pi$
$488$	−10647.6	−0.987695
$489$	865.548	0.0800438
$490$	−5918.12	−0.545619
$491$	5410.12	0.497261	0.248631	−	0.968598i	$-0.420020\pi$
$491$	5410.12	0.497261	0.248631	+	0.968598i	$0.420020\pi$
$492$	−838.965	−0.0768769
$493$	−11.4149	−0.00104281
$494$	0	0
$495$	21421.4	1.94509
$496$	−11958.4	−1.08256
$497$	873.867	0.0788698
$498$	−26934.1	−2.42359
$499$	14472.9	1.29838	0.649192	−	0.760624i	$-0.275107\pi$
$499$	14472.9	1.29838	0.649192	+	0.760624i	$0.275107\pi$
$500$	−1761.52	−0.157555
$501$	15675.2	1.39783
$502$	9148.97	0.813423
$503$	−601.940	−0.0533582	−0.0266791	−	0.999644i	$-0.508493\pi$
$503$	−601.940	−0.0533582	−0.0266791	+	0.999644i	$0.508493\pi$
$504$	6326.90	0.559171
$505$	15790.4	1.39142
$506$	−33319.8	−2.92736
$507$	0	0
$508$	1225.39	0.107024
$509$	−17420.8	−1.51702	−0.758512	−	0.651659i	$-0.774073\pi$
$509$	−17420.8	−1.51702	−0.758512	+	0.651659i	$0.774073\pi$
$510$	102.983	0.00894149
$511$	4042.43	0.349954
$512$	−4831.90	−0.417074
$513$	1076.78	0.0926721
$514$	−19482.2	−1.67184
$515$	−1119.55	−0.0957929
$516$	−2944.71	−0.251228
$517$	14981.0	1.27439
$518$	−2655.90	−0.225277
$519$	−1516.11	−0.128227
$520$	0	0
$521$	12881.5	1.08320	0.541601	−	0.840636i	$-0.317818\pi$
$521$	12881.5	1.08320	0.541601	+	0.840636i	$0.317818\pi$
$522$	−2483.97	−0.208276
$523$	16534.0	1.38238	0.691188	−	0.722675i	$-0.257088\pi$
$523$	16534.0	1.38238	0.691188	+	0.722675i	$0.257088\pi$
$524$	−658.410	−0.0548907
$525$	−6102.41	−0.507297
$526$	−630.459	−0.0522611
$527$	53.1678	0.00439473
$528$	−36474.2	−3.00632
$529$	12054.2	0.990730
$530$	−17617.3	−1.44386
$531$	−4036.18	−0.329859
$532$	1158.11	0.0943802
$533$	0	0
$534$	12099.3	0.980501
$535$	17985.0	1.45338
$536$	−5108.80	−0.411691
$537$	21035.4	1.69040
$538$	3502.79	0.280699
$539$	−9319.18	−0.744723
$540$	−720.852	−0.0574454
$541$	19026.9	1.51207	0.756035	−	0.654531i	$-0.227134\pi$
$541$	19026.9	1.51207	0.756035	+	0.654531i	$0.227134\pi$
$542$	1283.68	0.101732
$543$	16513.0	1.30505
$544$	−29.8258	−0.00235068
$545$	28206.5	2.21694
$546$	0	0
$547$	8153.61	0.637336	0.318668	−	0.947866i	$-0.396764\pi$
$547$	8153.61	0.637336	0.318668	+	0.947866i	$0.396764\pi$
$548$	4591.60	0.357926
$549$	13039.4	1.01368
$550$	12860.0	0.997002
$551$	1369.37	0.105875
$552$	20946.9	1.61515
$553$	−14102.5	−1.08445
$554$	913.031	0.0700198
$555$	5653.11	0.432363
$556$	2407.38	0.183625
$557$	−1308.06	−0.0995051	−0.0497525	−	0.998762i	$-0.515843\pi$
$557$	−1308.06	−0.0995051	−0.0497525	+	0.998762i	$0.515843\pi$
$558$	11569.7	0.877746
$559$	0	0
$560$	14812.7	1.11777
$561$	162.166	0.0122044
$562$	−16233.3	−1.21844
$563$	−20736.1	−1.55226	−0.776129	−	0.630575i	$-0.782819\pi$
$563$	−20736.1	−1.55226	−0.776129	+	0.630575i	$0.782819\pi$
$564$	3127.09	0.233465
$565$	−21804.7	−1.62359
$566$	16674.2	1.23829
$567$	11696.7	0.866343
$568$	1157.72	0.0855224
$569$	−14982.0	−1.10383	−0.551913	−	0.833902i	$-0.686102\pi$
$569$	−14982.0	−1.10383	−0.551913	+	0.833902i	$0.686102\pi$
$570$	−12354.1	−0.907822
$571$	−5668.79	−0.415467	−0.207734	−	0.978185i	$-0.566609\pi$
$571$	−5668.79	−0.415467	−0.207734	+	0.978185i	$0.566609\pi$
$572$	0	0
$573$	−31756.2	−2.31525
$574$	2688.91	0.195527
$575$	−9348.32	−0.678004
$576$	7642.30	0.552828
$577$	−6872.94	−0.495882	−0.247941	−	0.968775i	$-0.579754\pi$
$577$	−6872.94	−0.495882	−0.247941	+	0.968775i	$0.579754\pi$
$578$	−15531.4	−1.11768
$579$	9234.10	0.662791
$580$	−916.731	−0.0656297
$581$	17224.5	1.22993
$582$	−35570.9	−2.53344
$583$	−27741.7	−1.97074
$584$	5355.49	0.379472
$585$	0	0
$586$	−4570.34	−0.322182
$587$	−18593.4	−1.30738	−0.653691	−	0.756762i	$-0.726780\pi$
$587$	−18593.4	−1.30738	−0.653691	+	0.756762i	$0.726780\pi$
$588$	−1945.26	−0.136431
$589$	−6378.17	−0.446193
$590$	−7465.44	−0.520928
$591$	−5252.50	−0.365582
$592$	−4453.77	−0.309204
$593$	−15612.6	−1.08117	−0.540583	−	0.841291i	$-0.681796\pi$
$593$	−15612.6	−1.08117	−0.540583	+	0.841291i	$0.681796\pi$
$594$	−5688.91	−0.392961
$595$	−65.8580	−0.00453767
$596$	675.892	0.0464523
$597$	−33237.7	−2.27861
$598$	0	0
$599$	−22979.8	−1.56749	−0.783747	−	0.621081i	$-0.786694\pi$
$599$	−22979.8	−1.56749	−0.783747	+	0.621081i	$0.786694\pi$
$600$	−8084.60	−0.550087
$601$	2781.77	0.188803	0.0944015	−	0.995534i	$-0.469906\pi$
$601$	2781.77	0.188803	0.0944015	+	0.995534i	$0.469906\pi$
$602$	9437.89	0.638970
$603$	6256.38	0.422520
$604$	3379.69	0.227678
$605$	44284.9	2.97593
$606$	26012.2	1.74369
$607$	−4078.57	−0.272725	−0.136363	−	0.990659i	$-0.543541\pi$
$607$	−4078.57	−0.272725	−0.136363	+	0.990659i	$0.543541\pi$
$608$	3578.00	0.238663
$609$	3433.10	0.228434
$610$	24118.1	1.60084
$611$	0	0
$612$	15.6625	0.00103451
$613$	26290.1	1.73221	0.866107	−	0.499858i	$-0.166615\pi$
$613$	26290.1	1.73221	0.866107	+	0.499858i	$0.166615\pi$
$614$	−29784.4	−1.95766
$615$	−5723.36	−0.375265
$616$	18427.6	1.20531
$617$	−4851.69	−0.316567	−0.158283	−	0.987394i	$-0.550596\pi$
$617$	−4851.69	−0.316567	−0.158283	+	0.987394i	$0.550596\pi$
$618$	−1844.28	−0.120045
$619$	−4957.05	−0.321875	−0.160937	−	0.986965i	$-0.551452\pi$
$619$	−4957.05	−0.321875	−0.160937	+	0.986965i	$0.551452\pi$
$620$	4269.89	0.276585
$621$	4135.45	0.267230
$622$	24930.0	1.60708
$623$	−7737.54	−0.497589
$624$	0	0
$625$	−19525.3	−1.24962
$626$	−1740.08	−0.111098
$627$	−19453.9	−1.23910
$628$	19.1289	0.00121548
$629$	19.8017	0.00125524
$630$	−14331.1	−0.906295
$631$	−2725.11	−0.171925	−0.0859627	−	0.996298i	$-0.527397\pi$
$631$	−2725.11	−0.171925	−0.0859627	+	0.996298i	$0.527397\pi$
$632$	−18683.3	−1.17592
$633$	−22720.5	−1.42663
$634$	477.283	0.0298980
$635$	8359.55	0.522423
$636$	−5790.73	−0.361034
$637$	0	0
$638$	−7234.77	−0.448946
$639$	−1417.77	−0.0877719
$640$	23744.8	1.46655
$641$	21396.3	1.31841	0.659207	−	0.751961i	$-0.270892\pi$
$641$	21396.3	1.31841	0.659207	+	0.751961i	$0.270892\pi$
$642$	29627.4	1.82134
$643$	21833.8	1.33910	0.669549	−	0.742768i	$-0.266487\pi$
$643$	21833.8	1.33910	0.669549	+	0.742768i	$0.266487\pi$
$644$	4447.80	0.272155
$645$	−20088.6	−1.22634
$646$	−43.2740	−0.00263559
$647$	−4869.06	−0.295861	−0.147931	−	0.988998i	$-0.547261\pi$
$647$	−4869.06	−0.295861	−0.147931	+	0.988998i	$0.547261\pi$
$648$	15496.1	0.939418
$649$	−11755.7	−0.711021
$650$	0	0
$651$	−15990.5	−0.962696
$652$	243.486	0.0146252
$653$	−16847.0	−1.00961	−0.504803	−	0.863235i	$-0.668435\pi$
$653$	−16847.0	−1.00961	−0.504803	+	0.863235i	$0.668435\pi$
$654$	46465.7	2.77822
$655$	−4491.63	−0.267943
$656$	4509.12	0.268371
$657$	−6558.49	−0.389454
$658$	−10022.4	−0.593790
$659$	−24741.0	−1.46248	−0.731240	−	0.682121i	$-0.761058\pi$
$659$	−24741.0	−1.46248	−0.731240	+	0.682121i	$0.761058\pi$
$660$	13023.5	0.768089
$661$	−16522.0	−0.972210	−0.486105	−	0.873900i	$-0.661583\pi$
$661$	−16522.0	−0.972210	−0.486105	+	0.873900i	$0.661583\pi$
$662$	22380.1	1.31394
$663$	0	0
$664$	22819.3	1.33368
$665$	7900.53	0.460706
$666$	4308.97	0.250705
$667$	5259.19	0.305302
$668$	4409.55	0.255405
$669$	7495.66	0.433182
$670$	11572.0	0.667262
$671$	37978.4	2.18501
$672$	8970.26	0.514934
$673$	−20721.5	−1.18686	−0.593429	−	0.804886i	$-0.702226\pi$
$673$	−20721.5	−1.18686	−0.593429	+	0.804886i	$0.702226\pi$
$674$	−20970.3	−1.19844
$675$	−1596.10	−0.0910133
$676$	0	0
$677$	32407.3	1.83975	0.919877	−	0.392207i	$-0.128288\pi$
$677$	32407.3	1.83975	0.919877	+	0.392207i	$0.128288\pi$
$678$	−35919.7	−2.03465
$679$	22747.7	1.28568
$680$	−87.2500	−0.00492042
$681$	36576.1	2.05815
$682$	33697.6	1.89201
$683$	10123.8	0.567168	0.283584	−	0.958947i	$-0.408477\pi$
$683$	10123.8	0.567168	0.283584	+	0.958947i	$0.408477\pi$
$684$	−1878.93	−0.105033
$685$	31323.6	1.74717
$686$	21774.9	1.21191
$687$	12038.6	0.668562
$688$	15826.7	0.877018
$689$	0	0
$690$	−47447.1	−2.61780
$691$	29704.5	1.63533	0.817665	−	0.575695i	$-0.195268\pi$
$691$	29704.5	1.63533	0.817665	+	0.575695i	$0.195268\pi$
$692$	−426.495	−0.0234291
$693$	−22567.0	−1.23701
$694$	20245.8	1.10738
$695$	16423.0	0.896343
$696$	4548.24	0.247702
$697$	−20.0477	−0.00108947
$698$	531.418	0.0288173
$699$	22416.5	1.21298
$700$	−1716.66	−0.0926908
$701$	−20585.2	−1.10912	−0.554558	−	0.832145i	$-0.687113\pi$
$701$	−20585.2	−1.10912	−0.554558	+	0.832145i	$0.687113\pi$
$702$	0	0
$703$	−2375.47	−0.127443
$704$	22258.9	1.19164
$705$	21332.8	1.13963
$706$	−26719.5	−1.42437
$707$	−16634.9	−0.884894
$708$	−2453.86	−0.130257
$709$	−29660.2	−1.57110	−0.785552	−	0.618796i	$-0.787621\pi$
$709$	−29660.2	−1.57110	−0.785552	+	0.618796i	$0.787621\pi$
$710$	−2622.36	−0.138613
$711$	22880.1	1.20685
$712$	−10250.8	−0.539560
$713$	−24495.9	−1.28664
$714$	−108.491	−0.00568649
$715$	0	0
$716$	5917.44	0.308862
$717$	16659.0	0.867702
$718$	8320.43	0.432473
$719$	1279.68	0.0663758	0.0331879	−	0.999449i	$-0.489434\pi$
$719$	1279.68	0.0663758	0.0331879	+	0.999449i	$0.489434\pi$
$720$	−24032.4	−1.24393
$721$	1179.43	0.0609211
$722$	−16492.4	−0.850118
$723$	−36629.3	−1.88417
$724$	4645.25	0.238452
$725$	−2029.82	−0.103980
$726$	72952.3	3.72936
$727$	6202.77	0.316435	0.158217	−	0.987404i	$-0.449425\pi$
$727$	6202.77	0.316435	0.158217	+	0.987404i	$0.449425\pi$
$728$	0	0
$729$	−13891.1	−0.705740
$730$	−12130.8	−0.615042
$731$	−70.3663	−0.00356032
$732$	7927.51	0.400286
$733$	35501.2	1.78891	0.894453	−	0.447162i	$-0.147565\pi$
$733$	35501.2	1.78891	0.894453	+	0.447162i	$0.147565\pi$
$734$	−41950.7	−2.10958
$735$	−13270.5	−0.665970
$736$	13741.6	0.688209
$737$	18222.3	0.910754
$738$	−4362.52	−0.217597
$739$	3893.52	0.193810	0.0969050	−	0.995294i	$-0.469106\pi$
$739$	3893.52	0.193810	0.0969050	+	0.995294i	$0.469106\pi$
$740$	1590.27	0.0789991
$741$	0	0
$742$	18559.5	0.918247
$743$	2214.99	0.109368	0.0546839	−	0.998504i	$-0.482585\pi$
$743$	2214.99	0.109368	0.0546839	+	0.998504i	$0.482585\pi$
$744$	−21184.5	−1.04390
$745$	4610.89	0.226751
$746$	−36924.8	−1.81221
$747$	−27945.2	−1.36876
$748$	45.6186	0.00222992
$749$	−18946.9	−0.924303
$750$	−19796.0	−0.963797
$751$	2604.34	0.126543	0.0632714	−	0.997996i	$-0.479847\pi$
$751$	2604.34	0.126543	0.0632714	+	0.997996i	$0.479847\pi$
$752$	−16806.9	−0.815006
$753$	20515.1	0.992845
$754$	0	0
$755$	23056.0	1.11138
$756$	759.403	0.0365334
$757$	−9685.72	−0.465037	−0.232519	−	0.972592i	$-0.574697\pi$
$757$	−9685.72	−0.465037	−0.232519	+	0.972592i	$0.574697\pi$
$758$	−10679.8	−0.511750
$759$	−74714.4	−3.57307
$760$	10466.8	0.499565
$761$	10105.6	0.481378	0.240689	−	0.970602i	$-0.422627\pi$
$761$	10105.6	0.481378	0.240689	+	0.970602i	$0.422627\pi$
$762$	13771.0	0.654687
$763$	−29715.0	−1.40990
$764$	−8933.29	−0.423030
$765$	106.849	0.00504984
$766$	−13849.6	−0.653273
$767$	0	0
$768$	20476.1	0.962068
$769$	4573.20	0.214452	0.107226	−	0.994235i	$-0.465803\pi$
$769$	4573.20	0.214452	0.107226	+	0.994235i	$0.465803\pi$
$770$	−41740.7	−1.95355
$771$	−43685.8	−2.04061
$772$	2597.63	0.121102
$773$	−7446.13	−0.346467	−0.173233	−	0.984881i	$-0.555421\pi$
$773$	−7446.13	−0.346467	−0.173233	+	0.984881i	$0.555421\pi$
$774$	−15312.2	−0.711091
$775$	9454.34	0.438206
$776$	30136.6	1.39412
$777$	−5955.44	−0.274968
$778$	−39466.3	−1.81868
$779$	2404.99	0.110613
$780$	0	0
$781$	−4129.39	−0.189195
$782$	−166.197	−0.00760000
$783$	897.935	0.0409829
$784$	10455.0	0.476269
$785$	130.496	0.00593324
$786$	−7399.23	−0.335779
$787$	−3267.11	−0.147980	−0.0739898	−	0.997259i	$-0.523573\pi$
$787$	−3267.11	−0.147980	−0.0739898	+	0.997259i	$0.523573\pi$
$788$	−1477.57	−0.0667973
$789$	−1413.71	−0.0637887
$790$	42319.8	1.90591
$791$	22970.8	1.03255
$792$	−29897.3	−1.34135
$793$	0	0
$794$	29876.3	1.33535
$795$	−39504.0	−1.76234
$796$	−9350.03	−0.416336
$797$	−3165.75	−0.140698	−0.0703492	−	0.997522i	$-0.522411\pi$
$797$	−3165.75	−0.140698	−0.0703492	+	0.997522i	$0.522411\pi$
$798$	13014.9	0.577344
$799$	74.7242	0.00330857
$800$	−5303.66	−0.234391
$801$	12553.5	0.553752
$802$	9186.97	0.404493
$803$	−19102.2	−0.839478
$804$	3803.67	0.166847
$805$	30342.6	1.32849
$806$	0	0
$807$	7854.45	0.342614
$808$	−22038.3	−0.959534
$809$	13589.7	0.590592	0.295296	−	0.955406i	$-0.404582\pi$
$809$	13589.7	0.590592	0.295296	+	0.955406i	$0.404582\pi$
$810$	−35100.3	−1.52259
$811$	−24977.7	−1.08149	−0.540744	−	0.841188i	$-0.681857\pi$
$811$	−24977.7	−1.08149	−0.540744	+	0.841188i	$0.681857\pi$
$812$	965.758	0.0417383
$813$	2878.45	0.124172
$814$	12550.3	0.540401
$815$	1661.04	0.0713911
$816$	−181.931	−0.00780499
$817$	8441.35	0.361476
$818$	−18323.0	−0.783188
$819$	0	0
$820$	−1610.03	−0.0685666
$821$	8317.38	0.353567	0.176784	−	0.984250i	$-0.443431\pi$
$821$	8317.38	0.353567	0.176784	+	0.984250i	$0.443431\pi$
$822$	51600.6	2.18951
$823$	14462.9	0.612571	0.306286	−	0.951940i	$-0.400914\pi$
$823$	14462.9	0.612571	0.306286	+	0.951940i	$0.400914\pi$
$824$	1562.53	0.0660597
$825$	28836.5	1.21692
$826$	7864.69	0.331293
$827$	−17881.0	−0.751854	−0.375927	−	0.926649i	$-0.622676\pi$
$827$	−17881.0	−0.751854	−0.375927	+	0.926649i	$0.622676\pi$
$828$	−7216.18	−0.302874
$829$	351.169	0.0147124	0.00735620	−	0.999973i	$-0.497658\pi$
$829$	351.169	0.0147124	0.00735620	+	0.999973i	$0.497658\pi$
$830$	−51688.3	−2.16160
$831$	2047.33	0.0854646
$832$	0	0
$833$	−46.4836	−0.00193345
$834$	27054.2	1.12327
$835$	30081.6	1.24673
$836$	−5472.54	−0.226402
$837$	−4182.34	−0.172716
$838$	−13095.8	−0.539841
$839$	33425.9	1.37543	0.687717	−	0.725978i	$-0.258613\pi$
$839$	33425.9	1.37543	0.687717	+	0.725978i	$0.258613\pi$
$840$	26240.8	1.07785
$841$	−23247.1	−0.953178
$842$	−4510.99	−0.184631
$843$	−36400.7	−1.48720
$844$	−6391.46	−0.260667
$845$	0	0
$846$	16260.5	0.660811
$847$	−46653.3	−1.89259
$848$	31123.0	1.26034
$849$	37389.3	1.51142
$850$	64.1449	0.00258841
$851$	−9123.18	−0.367495
$852$	−861.959	−0.0346599
$853$	35097.5	1.40881	0.704406	−	0.709797i	$-0.251213\pi$
$853$	35097.5	1.40881	0.704406	+	0.709797i	$0.251213\pi$
$854$	−25407.9	−1.01808
$855$	−12817.9	−0.512706
$856$	−25101.2	−1.00227
$857$	15015.8	0.598519	0.299259	−	0.954172i	$-0.403260\pi$
$857$	15015.8	0.598519	0.299259	+	0.954172i	$0.403260\pi$
$858$	0	0
$859$	19647.1	0.780383	0.390192	−	0.920734i	$-0.372409\pi$
$859$	19647.1	0.780383	0.390192	+	0.920734i	$0.372409\pi$
$860$	−5651.10	−0.224071
$861$	6029.45	0.238656
$862$	8039.57	0.317667
$863$	9035.14	0.356385	0.178192	−	0.983996i	$-0.442975\pi$
$863$	9035.14	0.356385	0.178192	+	0.983996i	$0.442975\pi$
$864$	2346.19	0.0923833
$865$	−2909.52	−0.114366
$866$	−10901.7	−0.427778
$867$	−34826.6	−1.36422
$868$	−4498.24	−0.175899
$869$	66640.5	2.60141
$870$	−10302.3	−0.401471
$871$	0	0
$872$	−39367.0	−1.52883
$873$	−36906.2	−1.43080
$874$	19937.5	0.771621
$875$	12659.6	0.489113
$876$	−3987.34	−0.153790
$877$	8254.16	0.317814	0.158907	−	0.987294i	$-0.449203\pi$
$877$	8254.16	0.317814	0.158907	+	0.987294i	$0.449203\pi$
$878$	−39967.6	−1.53627
$879$	−10248.3	−0.393248
$880$	−69996.4	−2.68134
$881$	−42107.4	−1.61026	−0.805128	−	0.593101i	$-0.797903\pi$
$881$	−42107.4	−1.61026	−0.805128	+	0.593101i	$0.797903\pi$
$882$	−10115.1	−0.386161
$883$	−17584.7	−0.670183	−0.335091	−	0.942186i	$-0.608767\pi$
$883$	−17584.7	−0.670183	−0.335091	+	0.942186i	$0.608767\pi$
$884$	0	0
$885$	−16740.1	−0.635832
$886$	26829.1	1.01732
$887$	−23845.8	−0.902666	−0.451333	−	0.892356i	$-0.649051\pi$
$887$	−23845.8	−0.902666	−0.451333	+	0.892356i	$0.649051\pi$
$888$	−7889.89	−0.298161
$889$	−8806.63	−0.332244
$890$	23219.3	0.874509
$891$	−55272.0	−2.07821
$892$	2108.59	0.0791488
$893$	−8964.14	−0.335917
$894$	7595.70	0.284159
$895$	40368.4	1.50767
$896$	−25014.6	−0.932679
$897$	0	0
$898$	−35943.1	−1.33568
$899$	−5318.83	−0.197322
$900$	2785.13	0.103153
$901$	−138.374	−0.00511644
$902$	−12706.2	−0.469036
$903$	21163.0	0.779912
$904$	30432.2	1.11965
$905$	31689.6	1.16398
$906$	37981.1	1.39276
$907$	30565.7	1.11898	0.559492	−	0.828836i	$-0.310996\pi$
$907$	30565.7	1.11898	0.559492	+	0.828836i	$0.310996\pi$
$908$	10289.2	0.376055
$909$	26988.7	0.984773
$910$	0	0
$911$	18556.9	0.674882	0.337441	−	0.941347i	$-0.390439\pi$
$911$	18556.9	0.674882	0.337441	+	0.941347i	$0.390439\pi$
$912$	21825.0	0.792433
$913$	−81392.9	−2.95040
$914$	−35688.6	−1.29155
$915$	54081.0	1.95395
$916$	3386.56	0.122156
$917$	4731.84	0.170402
$918$	−28.3760	−0.00102020
$919$	−9938.82	−0.356748	−0.178374	−	0.983963i	$-0.557084\pi$
$919$	−9938.82	−0.356748	−0.178374	+	0.983963i	$0.557084\pi$
$920$	40198.5	1.44055
$921$	−66786.9	−2.38947
$922$	−12121.0	−0.432954
$923$	0	0
$924$	−13720.0	−0.488479
$925$	3521.15	0.125162
$926$	−4092.18	−0.145224
$927$	−1913.52	−0.0677973
$928$	2983.73	0.105545
$929$	5953.61	0.210260	0.105130	−	0.994458i	$-0.466474\pi$
$929$	5953.61	0.210260	0.105130	+	0.994458i	$0.466474\pi$
$930$	47985.2	1.69193
$931$	5576.31	0.196301
$932$	6305.95	0.221629
$933$	55901.6	1.96156
$934$	43821.1	1.53519
$935$	311.207	0.0108851
$936$	0	0
$937$	−24568.0	−0.856564	−0.428282	−	0.903645i	$-0.640881\pi$
$937$	−24568.0	−0.856564	−0.428282	+	0.903645i	$0.640881\pi$
$938$	−12190.9	−0.424356
$939$	−3901.86	−0.135604
$940$	6001.08	0.208227
$941$	11024.8	0.381931	0.190965	−	0.981597i	$-0.438838\pi$
$941$	11024.8	0.381931	0.190965	+	0.981597i	$0.438838\pi$
$942$	214.971	0.00743538
$943$	9236.55	0.318964
$944$	13188.6	0.454715
$945$	5180.60	0.178333
$946$	−44598.0	−1.53278
$947$	−6171.46	−0.211769	−0.105885	−	0.994378i	$-0.533767\pi$
$947$	−6171.46	−0.211769	−0.105885	+	0.994378i	$0.533767\pi$
$948$	13910.4	0.476569
$949$	0	0
$950$	−7695.01	−0.262799
$951$	1070.23	0.0364928
$952$	91.9161	0.00312922
$953$	−18838.4	−0.640333	−0.320166	−	0.947361i	$-0.603739\pi$
$953$	−18838.4	−0.640333	−0.320166	+	0.947361i	$0.603739\pi$
$954$	−30111.1	−1.02189
$955$	−60942.3	−2.06497
$956$	4686.32	0.158542
$957$	−16222.8	−0.547973
$958$	30436.4	1.02647
$959$	−32998.8	−1.11114
$960$	31696.5	1.06562
$961$	−5017.33	−0.168418
$962$	0	0
$963$	30739.6	1.02863
$964$	−10304.1	−0.344267
$965$	17720.8	0.591144
$966$	49984.6	1.66483
$967$	−24934.5	−0.829203	−0.414602	−	0.910003i	$-0.636079\pi$
$967$	−24934.5	−0.829203	−0.414602	+	0.910003i	$0.636079\pi$
$968$	−61807.2	−2.05223
$969$	−97.0351	−0.00321694
$970$	−68262.8	−2.25957
$971$	32161.2	1.06293	0.531463	−	0.847082i	$-0.321643\pi$
$971$	32161.2	1.06293	0.531463	+	0.847082i	$0.321643\pi$
$972$	−10106.6	−0.333509
$973$	−17301.3	−0.570044
$974$	37787.2	1.24310
$975$	0	0
$976$	−42607.4	−1.39737
$977$	18864.0	0.617720	0.308860	−	0.951107i	$-0.400052\pi$
$977$	18864.0	0.617720	0.308860	+	0.951107i	$0.400052\pi$
$978$	2736.30	0.0894655
$979$	36563.1	1.19363
$980$	−3733.08	−0.121683
$981$	48210.0	1.56904
$982$	17103.3	0.555792
$983$	7883.83	0.255804	0.127902	−	0.991787i	$-0.459176\pi$
$983$	7883.83	0.255804	0.127902	+	0.991787i	$0.459176\pi$
$984$	7987.93	0.258786
$985$	−10079.9	−0.326063
$986$	−36.0867	−0.00116555
$987$	−22473.6	−0.724766
$988$	0	0
$989$	32419.7	1.04235
$990$	67720.6	2.17404
$991$	14172.3	0.454286	0.227143	−	0.973861i	$-0.427062\pi$
$991$	14172.3	0.454286	0.227143	+	0.973861i	$0.427062\pi$
$992$	−13897.4	−0.444802
$993$	50183.9	1.60376
$994$	2762.60	0.0881533
$995$	−63785.3	−2.03229
$996$	−16989.7	−0.540503
$997$	52462.7	1.66651	0.833256	−	0.552888i	$-0.186474\pi$
$997$	52462.7	1.66651	0.833256	+	0.552888i	$0.186474\pi$
$998$	45753.8	1.45121
$999$	−1557.66	−0.0493315

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.4.a.l.1.7	yes	9
3.2	odd	2		1521.4.a.bg.1.3		9
13.2	odd	12		169.4.e.h.147.14		36
13.3	even	3		169.4.c.k.22.3		18
13.4	even	6		169.4.c.l.146.7		18
13.5	odd	4		169.4.b.g.168.5		18
13.6	odd	12		169.4.e.h.23.5		36
13.7	odd	12		169.4.e.h.23.14		36
13.8	odd	4		169.4.b.g.168.14		18
13.9	even	3		169.4.c.k.146.3		18
13.10	even	6		169.4.c.l.22.7		18
13.11	odd	12		169.4.e.h.147.5		36
13.12	even	2		169.4.a.k.1.3	✓	9
39.38	odd	2		1521.4.a.bh.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.4.a.k.1.3	✓	9	13.12	even	2
169.4.a.l.1.7	yes	9	1.1	even	1	trivial
169.4.b.g.168.5		18	13.5	odd	4
169.4.b.g.168.14		18	13.8	odd	4
169.4.c.k.22.3		18	13.3	even	3
169.4.c.k.146.3		18	13.9	even	3
169.4.c.l.22.7		18	13.10	even	6
169.4.c.l.146.7		18	13.4	even	6
169.4.e.h.23.5		36	13.6	odd	12
169.4.e.h.23.14		36	13.7	odd	12
169.4.e.h.147.5		36	13.11	odd	12
169.4.e.h.147.14		36	13.2	odd	12
1521.4.a.bg.1.3		9	3.2	odd	2
1521.4.a.bh.1.7		9	39.38	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)

\(f(q)\)	\(=\)	\(q+3.16135 q^{2} +7.08883 q^{3} +1.99415 q^{4} +13.6039 q^{5} +22.4103 q^{6} -14.3315 q^{7} -18.9866 q^{8} +23.2516 q^{9} +O(q^{10})\)	\(q+3.16135 q^{2} +7.08883 q^{3} +1.99415 q^{4} +13.6039 q^{5} +22.4103 q^{6} -14.3315 q^{7} -18.9866 q^{8} +23.2516 q^{9} +43.0068 q^{10} +67.7223 q^{11} +14.1362 q^{12} -45.3068 q^{14} +96.4360 q^{15} -75.9766 q^{16} +0.337795 q^{17} +73.5064 q^{18} -40.5229 q^{19} +27.1282 q^{20} -101.593 q^{21} +214.094 q^{22} -155.632 q^{23} -134.593 q^{24} +60.0670 q^{25} -26.5720 q^{27} -28.5791 q^{28} -33.7925 q^{29} +304.868 q^{30} +157.397 q^{31} -88.2957 q^{32} +480.072 q^{33} +1.06789 q^{34} -194.964 q^{35} +46.3670 q^{36} +58.6204 q^{37} -128.107 q^{38} -258.293 q^{40} -59.3488 q^{41} -321.173 q^{42} -208.311 q^{43} +135.048 q^{44} +316.313 q^{45} -492.006 q^{46} +221.212 q^{47} -538.585 q^{48} -137.609 q^{49} +189.893 q^{50} +2.39457 q^{51} -409.639 q^{53} -84.0035 q^{54} +921.289 q^{55} +272.106 q^{56} -287.260 q^{57} -106.830 q^{58} -173.587 q^{59} +192.308 q^{60} +560.796 q^{61} +497.586 q^{62} -333.229 q^{63} +328.679 q^{64} +1517.68 q^{66} +269.074 q^{67} +0.673613 q^{68} -1103.25 q^{69} -616.351 q^{70} -60.9754 q^{71} -441.469 q^{72} -282.066 q^{73} +185.320 q^{74} +425.805 q^{75} -80.8086 q^{76} -970.560 q^{77} +984.026 q^{79} -1033.58 q^{80} -816.157 q^{81} -187.622 q^{82} -1201.86 q^{83} -202.592 q^{84} +4.59534 q^{85} -658.543 q^{86} -239.550 q^{87} -1285.82 q^{88} +539.899 q^{89} +999.976 q^{90} -310.352 q^{92} +1115.76 q^{93} +699.328 q^{94} -551.271 q^{95} -625.913 q^{96} -1587.25 q^{97} -435.030 q^{98} +1574.65 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + q^{3} + 37 q^{4} + 30 q^{5} + 48 q^{6} + 38 q^{7} + 60 q^{8} + 66 q^{9}+O(q^{10}) \) 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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