LMFDB - Embedded newform 1045.4.a.b.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,4,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1045.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1045 = 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1045.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:	$61.6569959560$
Analytic rank:	$1$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} - 82 x^{18} + 700 x^{17} + 2826 x^{16} - 25467 x^{15} - 53768 x^{14} + 498499 x^{13} + \cdots + 17756160 $ x^20 - 8x^19 - 82x^18 + 700x^17 + 2826x^16 - 25467x^15 - 53768x^14 + 498499x^13 + 623613x^12 - 5673747x^11 - 4539454x^10 + 37893109x^9 + 19879768x^8 - 143049638x^7 - 45064360x^6 + 280461480x^5 + 43097920x^4 - 240447168x^3 - 36068096x^2 + 74703872*x + 17756160
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3}\cdot 7 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Root			$4.23917$ of defining polynomial
Character	$\chi$	$=$	1045.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23917 q^{2} +2.30947 q^{3} +2.49220 q^{4} +5.00000 q^{5} +7.48077 q^{6} -5.43134 q^{7} -17.8407 q^{8} -21.6663 q^{9} +O(q^{10})$	\(q+3.23917 q^{2} +2.30947 q^{3} +2.49220 q^{4} +5.00000 q^{5} +7.48077 q^{6} -5.43134 q^{7} -17.8407 q^{8} -21.6663 q^{9} +16.1958 q^{10} +11.0000 q^{11} +5.75566 q^{12} +31.6680 q^{13} -17.5930 q^{14} +11.5474 q^{15} -77.7265 q^{16} -13.7285 q^{17} -70.1809 q^{18} +19.0000 q^{19} +12.4610 q^{20} -12.5435 q^{21} +35.6308 q^{22} -11.5367 q^{23} -41.2026 q^{24} +25.0000 q^{25} +102.578 q^{26} -112.394 q^{27} -13.5360 q^{28} -211.239 q^{29} +37.4038 q^{30} -267.088 q^{31} -109.044 q^{32} +25.4042 q^{33} -44.4688 q^{34} -27.1567 q^{35} -53.9968 q^{36} +271.257 q^{37} +61.5442 q^{38} +73.1365 q^{39} -89.2034 q^{40} -102.529 q^{41} -40.6306 q^{42} +20.1265 q^{43} +27.4142 q^{44} -108.332 q^{45} -37.3692 q^{46} -296.277 q^{47} -179.507 q^{48} -313.501 q^{49} +80.9792 q^{50} -31.7055 q^{51} +78.9230 q^{52} -311.224 q^{53} -364.061 q^{54} +55.0000 q^{55} +96.8989 q^{56} +43.8800 q^{57} -684.237 q^{58} -399.608 q^{59} +28.7783 q^{60} -139.691 q^{61} -865.143 q^{62} +117.677 q^{63} +268.602 q^{64} +158.340 q^{65} +82.2884 q^{66} -66.9022 q^{67} -34.2141 q^{68} -26.6436 q^{69} -87.9651 q^{70} -219.623 q^{71} +386.542 q^{72} +8.47416 q^{73} +878.646 q^{74} +57.7368 q^{75} +47.3518 q^{76} -59.7448 q^{77} +236.901 q^{78} -1042.99 q^{79} -388.633 q^{80} +325.421 q^{81} -332.108 q^{82} -1008.39 q^{83} -31.2610 q^{84} -68.6424 q^{85} +65.1931 q^{86} -487.850 q^{87} -196.248 q^{88} -206.246 q^{89} -350.904 q^{90} -172.000 q^{91} -28.7517 q^{92} -616.833 q^{93} -959.691 q^{94} +95.0000 q^{95} -251.833 q^{96} +26.1949 q^{97} -1015.48 q^{98} -238.330 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) $ 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * 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 $

$2$	3.23917	1.14522	0.572609	−	0.819829i	$-0.305931\pi$
$2$	3.23917	1.14522	0.572609	+	0.819829i	$0.305931\pi$
$3$	2.30947	0.444458	0.222229	−	0.974994i	$-0.428667\pi$
$3$	2.30947	0.444458	0.222229	+	0.974994i	$0.428667\pi$
$4$	2.49220	0.311525
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	7.48077	0.509002
$7$	−5.43134	−0.293265	−0.146632	−	0.989191i	$-0.546843\pi$
$7$	−5.43134	−0.293265	−0.146632	+	0.989191i	$0.546843\pi$
$8$	−17.8407	−0.788454
$9$	−21.6663	−0.802457
$10$	16.1958	0.512157
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	5.75566	0.138460
$13$	31.6680	0.675626	0.337813	−	0.941213i	$-0.390313\pi$
$13$	31.6680	0.675626	0.337813	+	0.941213i	$0.390313\pi$
$14$	−17.5930	−0.335852
$15$	11.5474	0.198768
$16$	−77.7265	−1.21448
$17$	−13.7285	−0.195861	−0.0979307	−	0.995193i	$-0.531222\pi$
$17$	−13.7285	−0.195861	−0.0979307	+	0.995193i	$0.531222\pi$
$18$	−70.1809	−0.918988
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	12.4610	0.139318
$21$	−12.5435	−0.130344
$22$	35.6308	0.345296
$23$	−11.5367	−0.104590	−0.0522949	−	0.998632i	$-0.516654\pi$
$23$	−11.5367	−0.104590	−0.0522949	+	0.998632i	$0.516654\pi$
$24$	−41.2026	−0.350435
$25$	25.0000	0.200000
$26$	102.578	0.773739
$27$	−112.394	−0.801117
$28$	−13.5360	−0.0913593
$29$	−211.239	−1.35262	−0.676311	−	0.736617i	$-0.736422\pi$
$29$	−211.239	−1.35262	−0.676311	+	0.736617i	$0.736422\pi$
$30$	37.4038	0.227632
$31$	−267.088	−1.54743	−0.773717	−	0.633531i	$-0.781605\pi$
$31$	−267.088	−1.54743	−0.773717	+	0.633531i	$0.781605\pi$
$32$	−109.044	−0.602387
$33$	25.4042	0.134009
$34$	−44.4688	−0.224304
$35$	−27.1567	−0.131152
$36$	−53.9968	−0.249985
$37$	271.257	1.20525	0.602626	−	0.798023i	$-0.294121\pi$
$37$	271.257	1.20525	0.602626	+	0.798023i	$0.294121\pi$
$38$	61.5442	0.262731
$39$	73.1365	0.300287
$40$	−89.2034	−0.352608
$41$	−102.529	−0.390545	−0.195272	−	0.980749i	$-0.562559\pi$
$41$	−102.529	−0.390545	−0.195272	+	0.980749i	$0.562559\pi$
$42$	−40.6306	−0.149272
$43$	20.1265	0.0713783	0.0356891	−	0.999363i	$-0.488637\pi$
$43$	20.1265	0.0713783	0.0356891	+	0.999363i	$0.488637\pi$
$44$	27.4142	0.0939282
$45$	−108.332	−0.358870
$46$	−37.3692	−0.119778
$47$	−296.277	−0.919499	−0.459749	−	0.888049i	$-0.652061\pi$
$47$	−296.277	−0.919499	−0.459749	+	0.888049i	$0.652061\pi$
$48$	−179.507	−0.539784
$49$	−313.501	−0.913996
$50$	80.9792	0.229044
$51$	−31.7055	−0.0870522
$52$	78.9230	0.210474
$53$	−311.224	−0.806601	−0.403301	−	0.915068i	$-0.632137\pi$
$53$	−311.224	−0.806601	−0.403301	+	0.915068i	$0.632137\pi$
$54$	−364.061	−0.917453
$55$	55.0000	0.134840
$56$	96.8989	0.231226
$57$	43.8800	0.101966
$58$	−684.237	−1.54905
$59$	−399.608	−0.881771	−0.440885	−	0.897563i	$-0.645335\pi$
$59$	−399.608	−0.881771	−0.440885	+	0.897563i	$0.645335\pi$
$60$	28.7783	0.0619211
$61$	−139.691	−0.293206	−0.146603	−	0.989195i	$-0.546834\pi$
$61$	−139.691	−0.293206	−0.146603	+	0.989195i	$0.546834\pi$
$62$	−865.143	−1.77215
$63$	117.677	0.235333
$64$	268.602	0.524613
$65$	158.340	0.302149
$66$	82.2884	0.153470
$67$	−66.9022	−0.121991	−0.0609955	−	0.998138i	$-0.519428\pi$
$67$	−66.9022	−0.121991	−0.0609955	+	0.998138i	$0.519428\pi$
$68$	−34.2141	−0.0610157
$69$	−26.6436	−0.0464858
$70$	−87.9651	−0.150198
$71$	−219.623	−0.367106	−0.183553	−	0.983010i	$-0.558760\pi$
$71$	−219.623	−0.367106	−0.183553	+	0.983010i	$0.558760\pi$
$72$	386.542	0.632701
$73$	8.47416	0.0135867	0.00679333	−	0.999977i	$-0.497838\pi$
$73$	8.47416	0.0135867	0.00679333	+	0.999977i	$0.497838\pi$
$74$	878.646	1.38028
$75$	57.7368	0.0888916
$76$	47.3518	0.0714687
$77$	−59.7448	−0.0884227
$78$	236.901	0.343895
$79$	−1042.99	−1.48539	−0.742693	−	0.669632i	$-0.766452\pi$
$79$	−1042.99	−1.48539	−0.742693	+	0.669632i	$0.766452\pi$
$80$	−388.633	−0.543131
$81$	325.421	0.446394
$82$	−332.108	−0.447259
$83$	−1008.39	−1.33356	−0.666778	−	0.745256i	$-0.732327\pi$
$83$	−1008.39	−1.33356	−0.666778	+	0.745256i	$0.732327\pi$
$84$	−31.2610	−0.0406054
$85$	−68.6424	−0.0875919
$86$	65.1931	0.0817437
$87$	−487.850	−0.601184
$88$	−196.248	−0.237728
$89$	−206.246	−0.245641	−0.122821	−	0.992429i	$-0.539194\pi$
$89$	−206.246	−0.245641	−0.122821	+	0.992429i	$0.539194\pi$
$90$	−350.904	−0.410984
$91$	−172.000	−0.198137
$92$	−28.7517	−0.0325823
$93$	−616.833	−0.687770
$94$	−959.691	−1.05303
$95$	95.0000	0.102598
$96$	−251.833	−0.267736
$97$	26.1949	0.0274195	0.0137098	−	0.999906i	$-0.495636\pi$
$97$	26.1949	0.0274195	0.0137098	+	0.999906i	$0.495636\pi$
$98$	−1015.48	−1.04672
$99$	−238.330	−0.241950
$100$	62.3049	0.0623049
$101$	371.747	0.366240	0.183120	−	0.983091i	$-0.441380\pi$
$101$	371.747	0.366240	0.183120	+	0.983091i	$0.441380\pi$
$102$	−102.700	−0.0996938
$103$	1961.28	1.87622	0.938108	−	0.346344i	$-0.112577\pi$
$103$	1961.28	1.87622	0.938108	+	0.346344i	$0.112577\pi$
$104$	−564.980	−0.532700
$105$	−62.7177	−0.0582916
$106$	−1008.11	−0.923735
$107$	1052.43	0.950865	0.475433	−	0.879752i	$-0.342292\pi$
$107$	1052.43	0.950865	0.475433	+	0.879752i	$0.342292\pi$
$108$	−280.107	−0.249568
$109$	434.678	0.381969	0.190985	−	0.981593i	$-0.438832\pi$
$109$	434.678	0.381969	0.190985	+	0.981593i	$0.438832\pi$
$110$	178.154	0.154421
$111$	626.460	0.535685
$112$	422.159	0.356164
$113$	325.220	0.270744	0.135372	−	0.990795i	$-0.456777\pi$
$113$	325.220	0.270744	0.135372	+	0.990795i	$0.456777\pi$
$114$	142.135	0.116773
$115$	−57.6834	−0.0467740
$116$	−526.448	−0.421375
$117$	−686.130	−0.542160
$118$	−1294.40	−1.00982
$119$	74.5641	0.0574393
$120$	−206.013	−0.156719
$121$	121.000	0.0909091
$122$	−452.481	−0.335784
$123$	−236.788	−0.173581
$124$	−665.637	−0.482064
$125$	125.000	0.0894427
$126$	381.176	0.269507
$127$	2388.91	1.66914	0.834571	−	0.550900i	$-0.185715\pi$
$127$	2388.91	1.66914	0.834571	+	0.550900i	$0.185715\pi$
$128$	1742.39	1.20318
$129$	46.4816	0.0317247
$130$	512.890	0.346026
$131$	1295.71	0.864176	0.432088	−	0.901831i	$-0.357777\pi$
$131$	1295.71	0.864176	0.432088	+	0.901831i	$0.357777\pi$
$132$	63.3123	0.0417472
$133$	−103.196	−0.0672796
$134$	−216.707	−0.139706
$135$	−561.968	−0.358270
$136$	244.925	0.154428
$137$	−114.447	−0.0713714	−0.0356857	−	0.999363i	$-0.511362\pi$
$137$	−114.447	−0.0713714	−0.0356857	+	0.999363i	$0.511362\pi$
$138$	−86.3032	−0.0532363
$139$	1368.75	0.835223	0.417612	−	0.908626i	$-0.362867\pi$
$139$	1368.75	0.835223	0.417612	+	0.908626i	$0.362867\pi$
$140$	−67.6799	−0.0408571
$141$	−684.244	−0.408679
$142$	−711.397	−0.420416
$143$	348.348	0.203709
$144$	1684.05	0.974565
$145$	−1056.19	−0.604911
$146$	27.4492	0.0155597
$147$	−724.021	−0.406233
$148$	676.026	0.375466
$149$	−791.245	−0.435042	−0.217521	−	0.976056i	$-0.569797\pi$
$149$	−791.245	−0.435042	−0.217521	+	0.976056i	$0.569797\pi$
$150$	187.019	0.101800
$151$	1020.40	0.549925	0.274963	−	0.961455i	$-0.411335\pi$
$151$	1020.40	0.549925	0.274963	+	0.961455i	$0.411335\pi$
$152$	−338.973	−0.180884
$153$	297.446	0.157170
$154$	−193.523	−0.101263
$155$	−1335.44	−0.692034
$156$	182.271	0.0935469
$157$	1409.87	0.716686	0.358343	−	0.933590i	$-0.383342\pi$
$157$	1409.87	0.716686	0.358343	+	0.933590i	$0.383342\pi$
$158$	−3378.42	−1.70109
$159$	−718.763	−0.358501
$160$	−545.218	−0.269396
$161$	62.6597	0.0306725
$162$	1054.09	0.511218
$163$	−3618.54	−1.73881	−0.869404	−	0.494101i	$-0.835497\pi$
$163$	−3618.54	−1.73881	−0.869404	+	0.494101i	$0.835497\pi$
$164$	−255.522	−0.121664
$165$	127.021	0.0599307
$166$	−3266.34	−1.52721
$167$	−2746.63	−1.27270	−0.636348	−	0.771402i	$-0.719556\pi$
$167$	−2746.63	−1.27270	−0.636348	+	0.771402i	$0.719556\pi$
$168$	223.785	0.102770
$169$	−1194.14	−0.543530
$170$	−222.344	−0.100312
$171$	−411.660	−0.184096
$172$	50.1593	0.0222361
$173$	284.105	0.124856	0.0624280	−	0.998049i	$-0.480116\pi$
$173$	284.105	0.124856	0.0624280	+	0.998049i	$0.480116\pi$
$174$	−1580.23	−0.688486
$175$	−135.784	−0.0586530
$176$	−854.992	−0.366179
$177$	−922.883	−0.391910
$178$	−668.066	−0.281313
$179$	−1109.50	−0.463286	−0.231643	−	0.972801i	$-0.574410\pi$
$179$	−1109.50	−0.463286	−0.231643	+	0.972801i	$0.574410\pi$
$180$	−269.984	−0.111797
$181$	759.095	0.311730	0.155865	−	0.987778i	$-0.450184\pi$
$181$	759.095	0.311730	0.155865	+	0.987778i	$0.450184\pi$
$182$	−557.137	−0.226910
$183$	−322.612	−0.130318
$184$	205.822	0.0824642
$185$	1356.28	0.539005
$186$	−1998.02	−0.787647
$187$	−151.013	−0.0590545
$188$	−738.381	−0.286447
$189$	610.448	0.234939
$190$	307.721	0.117497
$191$	1225.59	0.464294	0.232147	−	0.972681i	$-0.425425\pi$
$191$	1225.59	0.464294	0.232147	+	0.972681i	$0.425425\pi$
$192$	620.328	0.233168
$193$	2083.98	0.777244	0.388622	−	0.921397i	$-0.372951\pi$
$193$	2083.98	0.777244	0.388622	+	0.921397i	$0.372951\pi$
$194$	84.8497	0.0314013
$195$	365.682	0.134293
$196$	−781.305	−0.284732
$197$	−3176.53	−1.14883	−0.574413	−	0.818566i	$-0.694770\pi$
$197$	−3176.53	−1.14883	−0.574413	+	0.818566i	$0.694770\pi$
$198$	−771.989	−0.277085
$199$	−4948.10	−1.76262	−0.881311	−	0.472538i	$-0.843338\pi$
$199$	−4948.10	−1.76262	−0.881311	+	0.472538i	$0.843338\pi$
$200$	−446.017	−0.157691
$201$	−154.509	−0.0542199
$202$	1204.15	0.419425
$203$	1147.31	0.396676
$204$	−79.0165	−0.0271189
$205$	−512.645	−0.174657
$206$	6352.90	2.14868
$207$	249.958	0.0839288
$208$	−2461.45	−0.820532
$209$	209.000	0.0691714
$210$	−203.153	−0.0667566
$211$	3950.96	1.28908	0.644538	−	0.764572i	$-0.277050\pi$
$211$	3950.96	1.28908	0.644538	+	0.764572i	$0.277050\pi$
$212$	−775.631	−0.251276
$213$	−507.214	−0.163163
$214$	3409.01	1.08895
$215$	100.633	0.0319213
$216$	2005.18	0.631644
$217$	1450.65	0.453808
$218$	1408.00	0.437438
$219$	19.5709	0.00603870
$220$	137.071	0.0420060
$221$	−434.754	−0.132329
$222$	2029.21	0.613476
$223$	−2528.31	−0.759228	−0.379614	−	0.925145i	$-0.623943\pi$
$223$	−2528.31	−0.759228	−0.379614	+	0.925145i	$0.623943\pi$
$224$	592.254	0.176659
$225$	−541.658	−0.160491
$226$	1053.44	0.310061
$227$	−2005.39	−0.586353	−0.293176	−	0.956058i	$-0.594712\pi$
$227$	−2005.39	−0.586353	−0.293176	+	0.956058i	$0.594712\pi$
$228$	109.358	0.0317648
$229$	4430.93	1.27862	0.639310	−	0.768949i	$-0.279220\pi$
$229$	4430.93	1.27862	0.639310	+	0.768949i	$0.279220\pi$
$230$	−186.846	−0.0535664
$231$	−137.979	−0.0393002
$232$	3768.64	1.06648
$233$	393.186	0.110551	0.0552757	−	0.998471i	$-0.482396\pi$
$233$	393.186	0.110551	0.0552757	+	0.998471i	$0.482396\pi$
$234$	−2222.49	−0.620892
$235$	−1481.39	−0.411212
$236$	−995.901	−0.274693
$237$	−2408.76	−0.660192
$238$	241.525	0.0657805
$239$	−4097.15	−1.10888	−0.554440	−	0.832224i	$-0.687068\pi$
$239$	−4097.15	−1.10888	−0.554440	+	0.832224i	$0.687068\pi$
$240$	−897.536	−0.241399
$241$	−5189.76	−1.38714	−0.693572	−	0.720387i	$-0.743964\pi$
$241$	−5189.76	−1.38714	−0.693572	+	0.720387i	$0.743964\pi$
$242$	391.939	0.104111
$243$	3786.18	0.999520
$244$	−348.136	−0.0913408
$245$	−1567.50	−0.408751
$246$	−766.995	−0.198788
$247$	601.693	0.154999
$248$	4765.04	1.22008
$249$	−2328.85	−0.592710
$250$	404.896	0.102431
$251$	2135.02	0.536897	0.268449	−	0.963294i	$-0.413489\pi$
$251$	2135.02	0.536897	0.268449	+	0.963294i	$0.413489\pi$
$252$	293.275	0.0733119
$253$	−126.903	−0.0315350
$254$	7738.06	1.91153
$255$	−158.528	−0.0389309
$256$	3495.09	0.853294
$257$	−5230.94	−1.26964	−0.634820	−	0.772660i	$-0.718926\pi$
$257$	−5230.94	−1.26964	−0.634820	+	0.772660i	$0.718926\pi$
$258$	150.562	0.0363317
$259$	−1473.29	−0.353458
$260$	394.615	0.0941269
$261$	4576.77	1.08542
$262$	4197.04	0.989671
$263$	7421.74	1.74009	0.870046	−	0.492970i	$-0.164089\pi$
$263$	7421.74	1.74009	0.870046	+	0.492970i	$0.164089\pi$
$264$	−453.228	−0.105660
$265$	−1556.12	−0.360723
$266$	−334.267	−0.0770498
$267$	−476.320	−0.109177
$268$	−166.733	−0.0380032
$269$	1803.49	0.408775	0.204388	−	0.978890i	$-0.434480\pi$
$269$	1803.49	0.408775	0.204388	+	0.978890i	$0.434480\pi$
$270$	−1820.31	−0.410298
$271$	−578.625	−0.129701	−0.0648504	−	0.997895i	$-0.520657\pi$
$271$	−578.625	−0.129701	−0.0648504	+	0.997895i	$0.520657\pi$
$272$	1067.07	0.237869
$273$	−397.229	−0.0880638
$274$	−370.713	−0.0817358
$275$	275.000	0.0603023
$276$	−66.4012	−0.0144815
$277$	−410.493	−0.0890402	−0.0445201	−	0.999008i	$-0.514176\pi$
$277$	−410.493	−0.0890402	−0.0445201	+	0.999008i	$0.514176\pi$
$278$	4433.62	0.956513
$279$	5786.82	1.24175
$280$	484.495	0.103407
$281$	−6129.22	−1.30121	−0.650603	−	0.759418i	$-0.725484\pi$
$281$	−6129.22	−1.30121	−0.650603	+	0.759418i	$0.725484\pi$
$282$	−2216.38	−0.468026
$283$	7954.56	1.67085	0.835423	−	0.549607i	$-0.185223\pi$
$283$	7954.56	1.67085	0.835423	+	0.549607i	$0.185223\pi$
$284$	−547.345	−0.114363
$285$	219.400	0.0456005
$286$	1128.36	0.233291
$287$	556.870	0.114533
$288$	2362.58	0.483389
$289$	−4724.53	−0.961638
$290$	−3421.18	−0.692755
$291$	60.4965	0.0121868
$292$	21.1193	0.00423258
$293$	5204.46	1.03771	0.518853	−	0.854864i	$-0.326359\pi$
$293$	5204.46	1.03771	0.518853	+	0.854864i	$0.326359\pi$
$294$	−2345.22	−0.465225
$295$	−1998.04	−0.394340
$296$	−4839.41	−0.950287
$297$	−1236.33	−0.241546
$298$	−2562.97	−0.498218
$299$	−365.344	−0.0706635
$300$	143.892	0.0276919
$301$	−109.314	−0.0209327
$302$	3305.23	0.629784
$303$	858.540	0.162778
$304$	−1476.80	−0.278620
$305$	−698.453	−0.131126
$306$	963.476	0.179994
$307$	904.067	0.168071	0.0840355	−	0.996463i	$-0.473219\pi$
$307$	904.067	0.168071	0.0840355	+	0.996463i	$0.473219\pi$
$308$	−148.896	−0.0275459
$309$	4529.51	0.833899
$310$	−4325.72	−0.792530
$311$	5952.79	1.08538	0.542688	−	0.839934i	$-0.317407\pi$
$311$	5952.79	1.08538	0.542688	+	0.839934i	$0.317407\pi$
$312$	−1304.80	−0.236763
$313$	5040.03	0.910158	0.455079	−	0.890451i	$-0.349611\pi$
$313$	5040.03	0.910158	0.455079	+	0.890451i	$0.349611\pi$
$314$	4566.79	0.820761
$315$	588.387	0.105244
$316$	−2599.34	−0.462735
$317$	2870.32	0.508558	0.254279	−	0.967131i	$-0.418162\pi$
$317$	2870.32	0.508558	0.254279	+	0.967131i	$0.418162\pi$
$318$	−2328.19	−0.410561
$319$	−2323.62	−0.407831
$320$	1343.01	0.234614
$321$	2430.57	0.422620
$322$	202.965	0.0351267
$323$	−260.841	−0.0449337
$324$	811.014	0.139063
$325$	791.701	0.135125
$326$	−11721.0	−1.99132
$327$	1003.88	0.169769
$328$	1829.19	0.307927
$329$	1609.18	0.269657
$330$	411.442	0.0686338
$331$	−1698.83	−0.282103	−0.141052	−	0.990002i	$-0.545048\pi$
$331$	−1698.83	−0.282103	−0.141052	+	0.990002i	$0.545048\pi$
$332$	−2513.11	−0.415436
$333$	−5877.14	−0.967163
$334$	−8896.78	−1.45752
$335$	−334.511	−0.0545561
$336$	974.966	0.158300
$337$	3598.08	0.581602	0.290801	−	0.956784i	$-0.406078\pi$
$337$	3598.08	0.581602	0.290801	+	0.956784i	$0.406078\pi$
$338$	−3868.00	−0.622461
$339$	751.086	0.120334
$340$	−171.070	−0.0272870
$341$	−2937.97	−0.466569
$342$	−1333.44	−0.210830
$343$	3565.68	0.561308
$344$	−359.071	−0.0562785
$345$	−133.218	−0.0207891
$346$	920.262	0.142987
$347$	7078.89	1.09514	0.547572	−	0.836759i	$-0.315552\pi$
$347$	7078.89	1.09514	0.547572	+	0.836759i	$0.315552\pi$
$348$	−1215.82	−0.187284
$349$	−2035.16	−0.312148	−0.156074	−	0.987745i	$-0.549884\pi$
$349$	−2035.16	−0.312148	−0.156074	+	0.987745i	$0.549884\pi$
$350$	−439.826	−0.0671705
$351$	−3559.28	−0.541255
$352$	−1199.48	−0.181626
$353$	5464.38	0.823908	0.411954	−	0.911205i	$-0.364846\pi$
$353$	5464.38	0.823908	0.411954	+	0.911205i	$0.364846\pi$
$354$	−2989.37	−0.448823
$355$	−1098.12	−0.164175
$356$	−514.007	−0.0765233
$357$	172.204	0.0255294
$358$	−3593.86	−0.530563
$359$	−13423.2	−1.97340	−0.986701	−	0.162546i	$-0.948029\pi$
$359$	−13423.2	−1.97340	−0.986701	+	0.162546i	$0.948029\pi$
$360$	1932.71	0.282952
$361$	361.000	0.0526316
$362$	2458.83	0.356999
$363$	279.446	0.0404053
$364$	−428.658	−0.0617247
$365$	42.3708	0.00607614
$366$	−1044.99	−0.149242
$367$	11613.9	1.65188	0.825939	−	0.563760i	$-0.190646\pi$
$367$	11613.9	1.65188	0.825939	+	0.563760i	$0.190646\pi$
$368$	896.706	0.127022
$369$	2221.43	0.313395
$370$	4393.23	0.617279
$371$	1690.36	0.236548
$372$	−1537.27	−0.214257
$373$	−8052.47	−1.11780	−0.558902	−	0.829234i	$-0.688777\pi$
$373$	−8052.47	−1.11780	−0.558902	+	0.829234i	$0.688777\pi$
$374$	−489.157	−0.0676302
$375$	288.684	0.0397536
$376$	5285.79	0.724983
$377$	−6689.51	−0.913865
$378$	1977.34	0.269057
$379$	−8313.59	−1.12676	−0.563378	−	0.826200i	$-0.690498\pi$
$379$	−8313.59	−1.12676	−0.563378	+	0.826200i	$0.690498\pi$
$380$	236.759	0.0319618
$381$	5517.11	0.741864
$382$	3969.87	0.531718
$383$	−95.0078	−0.0126754	−0.00633769	−	0.999980i	$-0.502017\pi$
$383$	−95.0078	−0.0126754	−0.00633769	+	0.999980i	$0.502017\pi$
$384$	4024.01	0.534765
$385$	−298.724	−0.0395438
$386$	6750.35	0.890114
$387$	−436.068	−0.0572780
$388$	65.2829	0.00854185
$389$	4366.57	0.569136	0.284568	−	0.958656i	$-0.408150\pi$
$389$	4366.57	0.569136	0.284568	+	0.958656i	$0.408150\pi$
$390$	1184.51	0.153794
$391$	158.381	0.0204851
$392$	5593.06	0.720644
$393$	2992.42	0.384090
$394$	−10289.3	−1.31566
$395$	−5214.95	−0.664285
$396$	−593.965	−0.0753734
$397$	−2970.21	−0.375493	−0.187746	−	0.982218i	$-0.560118\pi$
$397$	−2970.21	−0.375493	−0.187746	+	0.982218i	$0.560118\pi$
$398$	−16027.7	−2.01859
$399$	−238.327	−0.0299030
$400$	−1943.16	−0.242895
$401$	10452.7	1.30171	0.650853	−	0.759204i	$-0.274411\pi$
$401$	10452.7	1.30171	0.650853	+	0.759204i	$0.274411\pi$
$402$	−500.479	−0.0620936
$403$	−8458.16	−1.04549
$404$	926.468	0.114093
$405$	1627.11	0.199633
$406$	3716.32	0.454281
$407$	2983.83	0.363397
$408$	565.649	0.0686367
$409$	615.228	0.0743791	0.0371895	−	0.999308i	$-0.488159\pi$
$409$	615.228	0.0743791	0.0371895	+	0.999308i	$0.488159\pi$
$410$	−1660.54	−0.200020
$411$	−264.312	−0.0317216
$412$	4887.89	0.584487
$413$	2170.41	0.258593
$414$	809.654	0.0961167
$415$	−5041.95	−0.596385
$416$	−3453.20	−0.406988
$417$	3161.10	0.371222
$418$	676.986	0.0792164
$419$	−2654.93	−0.309551	−0.154775	−	0.987950i	$-0.549465\pi$
$419$	−2654.93	−0.309551	−0.154775	+	0.987950i	$0.549465\pi$
$420$	−156.305	−0.0181593
$421$	−1584.72	−0.183455	−0.0917276	−	0.995784i	$-0.529239\pi$
$421$	−1584.72	−0.183455	−0.0917276	+	0.995784i	$0.529239\pi$
$422$	12797.8	1.47627
$423$	6419.24	0.737858
$424$	5552.45	0.635968
$425$	−343.212	−0.0391723
$426$	−1642.95	−0.186857
$427$	758.707	0.0859869
$428$	2622.87	0.296218
$429$	804.501	0.0905400
$430$	325.966	0.0365569
$431$	−6350.35	−0.709711	−0.354856	−	0.934921i	$-0.615470\pi$
$431$	−6350.35	−0.709711	−0.354856	+	0.934921i	$0.615470\pi$
$432$	8735.96	0.972938
$433$	−638.655	−0.0708817	−0.0354409	−	0.999372i	$-0.511284\pi$
$433$	−638.655	−0.0708817	−0.0354409	+	0.999372i	$0.511284\pi$
$434$	4698.89	0.519710
$435$	−2439.25	−0.268857
$436$	1083.30	0.118993
$437$	−219.197	−0.0239945
$438$	63.3932	0.00691563
$439$	−11640.4	−1.26552	−0.632761	−	0.774347i	$-0.718078\pi$
$439$	−11640.4	−1.26552	−0.632761	+	0.774347i	$0.718078\pi$
$440$	−981.238	−0.106315
$441$	6792.41	0.733442
$442$	−1408.24	−0.151546
$443$	−3795.16	−0.407029	−0.203514	−	0.979072i	$-0.565236\pi$
$443$	−3795.16	−0.407029	−0.203514	+	0.979072i	$0.565236\pi$
$444$	1561.26	0.166879
$445$	−1031.23	−0.109854
$446$	−8189.60	−0.869482
$447$	−1827.36	−0.193358
$448$	−1458.87	−0.153851
$449$	724.564	0.0761566	0.0380783	−	0.999275i	$-0.487876\pi$
$449$	724.564	0.0761566	0.0380783	+	0.999275i	$0.487876\pi$
$450$	−1754.52	−0.183798
$451$	−1127.82	−0.117754
$452$	810.511	0.0843435
$453$	2356.58	0.244419
$454$	−6495.78	−0.671502
$455$	−860.000	−0.0886097
$456$	−782.849	−0.0803953
$457$	−7918.86	−0.810566	−0.405283	−	0.914191i	$-0.632827\pi$
$457$	−7918.86	−0.810566	−0.405283	+	0.914191i	$0.632827\pi$
$458$	14352.5	1.46430
$459$	1542.99	0.156908
$460$	−143.758	−0.0145712
$461$	1270.29	0.128337	0.0641686	−	0.997939i	$-0.479560\pi$
$461$	1270.29	0.128337	0.0641686	+	0.997939i	$0.479560\pi$
$462$	−446.937	−0.0450073
$463$	−4768.88	−0.478679	−0.239340	−	0.970936i	$-0.576931\pi$
$463$	−4768.88	−0.478679	−0.239340	+	0.970936i	$0.576931\pi$
$464$	16418.8	1.64273
$465$	−3084.17	−0.307580
$466$	1273.59	0.126605
$467$	7966.66	0.789407	0.394704	−	0.918809i	$-0.370847\pi$
$467$	7966.66	0.789407	0.394704	+	0.918809i	$0.370847\pi$
$468$	−1709.97	−0.168896
$469$	363.369	0.0357757
$470$	−4798.45	−0.470928
$471$	3256.05	0.318537
$472$	7129.27	0.695236
$473$	221.392	0.0215214
$474$	−7802.36	−0.756064
$475$	475.000	0.0458831
$476$	185.828	0.0178938
$477$	6743.08	0.647263
$478$	−13271.3	−1.26991
$479$	−3117.02	−0.297328	−0.148664	−	0.988888i	$-0.547497\pi$
$479$	−3117.02	−0.297328	−0.148664	+	0.988888i	$0.547497\pi$
$480$	−1259.17	−0.119735
$481$	8590.17	0.814300
$482$	−16810.5	−1.58858
$483$	144.711	0.0136326
$484$	301.556	0.0283204
$485$	130.975	0.0122624
$486$	12264.1	1.14467
$487$	−5219.57	−0.485670	−0.242835	−	0.970068i	$-0.578077\pi$
$487$	−5219.57	−0.485670	−0.242835	+	0.970068i	$0.578077\pi$
$488$	2492.18	0.231179
$489$	−8356.92	−0.772828
$490$	−5077.40	−0.468109
$491$	8266.08	0.759762	0.379881	−	0.925035i	$-0.375965\pi$
$491$	8266.08	0.759762	0.379881	+	0.925035i	$0.375965\pi$
$492$	−590.122	−0.0540747
$493$	2899.98	0.264926
$494$	1948.98	0.177508
$495$	−1191.65	−0.108203
$496$	20759.8	1.87932
$497$	1192.85	0.107659
$498$	−7543.53	−0.678783
$499$	−5424.50	−0.486641	−0.243320	−	0.969946i	$-0.578237\pi$
$499$	−5424.50	−0.486641	−0.243320	+	0.969946i	$0.578237\pi$
$500$	311.525	0.0278636
$501$	−6343.26	−0.565660
$502$	6915.68	0.614864
$503$	−17615.7	−1.56152	−0.780760	−	0.624832i	$-0.785168\pi$
$503$	−17615.7	−1.56152	−0.780760	+	0.624832i	$0.785168\pi$
$504$	−2099.44	−0.185549
$505$	1858.74	0.163787
$506$	−411.061	−0.0361144
$507$	−2757.82	−0.241576
$508$	5953.62	0.519979
$509$	19429.7	1.69196	0.845979	−	0.533216i	$-0.179017\pi$
$509$	19429.7	1.69196	0.845979	+	0.533216i	$0.179017\pi$
$510$	−513.498	−0.0445844
$511$	−46.0261	−0.00398449
$512$	−2617.97	−0.225975
$513$	−2135.48	−0.183789
$514$	−16943.9	−1.45401
$515$	9806.38	0.839069
$516$	115.841	0.00988301
$517$	−3259.05	−0.277239
$518$	−4772.23	−0.404787
$519$	656.132	0.0554933
$520$	−2824.90	−0.238231
$521$	17477.8	1.46971	0.734854	−	0.678225i	$-0.237250\pi$
$521$	17477.8	1.46971	0.734854	+	0.678225i	$0.237250\pi$
$522$	14824.9	1.24304
$523$	17343.9	1.45009	0.725045	−	0.688701i	$-0.241819\pi$
$523$	17343.9	1.45009	0.725045	+	0.688701i	$0.241819\pi$
$524$	3229.18	0.269212
$525$	−313.588	−0.0260688
$526$	24040.3	1.99279
$527$	3666.72	0.303083
$528$	−1974.58	−0.162751
$529$	−12033.9	−0.989061
$530$	−5040.53	−0.413107
$531$	8658.03	0.707583
$532$	−257.184	−0.0209593
$533$	−3246.89	−0.263862
$534$	−1542.88	−0.125032
$535$	5262.17	0.425240
$536$	1193.58	0.0961844
$537$	−2562.37	−0.205911
$538$	5841.79	0.468137
$539$	−3448.51	−0.275580
$540$	−1400.53	−0.111610
$541$	−4907.88	−0.390030	−0.195015	−	0.980800i	$-0.562476\pi$
$541$	−4907.88	−0.390030	−0.195015	+	0.980800i	$0.562476\pi$
$542$	−1874.26	−0.148536
$543$	1753.11	0.138551
$544$	1497.00	0.117984
$545$	2173.39	0.170822
$546$	−1286.69	−0.100852
$547$	19746.4	1.54350	0.771751	−	0.635925i	$-0.219381\pi$
$547$	19746.4	1.54350	0.771751	+	0.635925i	$0.219381\pi$
$548$	−285.225	−0.0222339
$549$	3026.58	0.235285
$550$	890.771	0.0690593
$551$	−4013.53	−0.310313
$552$	475.341	0.0366519
$553$	5664.84	0.435612
$554$	−1329.66	−0.101970
$555$	3132.30	0.239565
$556$	3411.20	0.260193
$557$	−2637.62	−0.200645	−0.100323	−	0.994955i	$-0.531988\pi$
$557$	−2637.62	−0.200645	−0.100323	+	0.994955i	$0.531988\pi$
$558$	18744.5	1.42207
$559$	637.367	0.0482250
$560$	2110.80	0.159281
$561$	−348.761	−0.0262472
$562$	−19853.6	−1.49016
$563$	−13343.5	−0.998864	−0.499432	−	0.866353i	$-0.666458\pi$
$563$	−13343.5	−0.998864	−0.499432	+	0.866353i	$0.666458\pi$
$564$	−1705.27	−0.127314
$565$	1626.10	0.121080
$566$	25766.1	1.91348
$567$	−1767.47	−0.130912
$568$	3918.23	0.289446
$569$	−3687.90	−0.271713	−0.135857	−	0.990729i	$-0.543379\pi$
$569$	−3687.90	−0.271713	−0.135857	+	0.990729i	$0.543379\pi$
$570$	710.673	0.0522225
$571$	−11373.2	−0.833544	−0.416772	−	0.909011i	$-0.636839\pi$
$571$	−11373.2	−0.833544	−0.416772	+	0.909011i	$0.636839\pi$
$572$	868.153	0.0634603
$573$	2830.46	0.206359
$574$	1803.79	0.131165
$575$	−288.417	−0.0209179
$576$	−5819.62	−0.420979
$577$	−17425.5	−1.25725	−0.628625	−	0.777709i	$-0.716382\pi$
$577$	−17425.5	−1.25725	−0.628625	+	0.777709i	$0.716382\pi$
$578$	−15303.5	−1.10129
$579$	4812.89	0.345452
$580$	−2632.24	−0.188445
$581$	5476.91	0.391086
$582$	195.958	0.0139566
$583$	−3423.46	−0.243199
$584$	−151.185	−0.0107125
$585$	−3430.65	−0.242462
$586$	16858.1	1.18840
$587$	14603.7	1.02685	0.513423	−	0.858136i	$-0.328377\pi$
$587$	14603.7	1.02685	0.513423	+	0.858136i	$0.328377\pi$
$588$	−1804.40	−0.126552
$589$	−5074.68	−0.355006
$590$	−6471.98	−0.451605
$591$	−7336.12	−0.510605
$592$	−21083.9	−1.46375
$593$	−1262.80	−0.0874488	−0.0437244	−	0.999044i	$-0.513922\pi$
$593$	−1262.80	−0.0874488	−0.0437244	+	0.999044i	$0.513922\pi$
$594$	−4004.68	−0.276623
$595$	372.820	0.0256876
$596$	−1971.94	−0.135526
$597$	−11427.5	−0.783411
$598$	−1183.41	−0.0809251
$599$	−17481.0	−1.19241	−0.596204	−	0.802833i	$-0.703325\pi$
$599$	−17481.0	−1.19241	−0.596204	+	0.802833i	$0.703325\pi$
$600$	−1030.06	−0.0700870
$601$	9273.35	0.629398	0.314699	−	0.949192i	$-0.398096\pi$
$601$	9273.35	0.629398	0.314699	+	0.949192i	$0.398096\pi$
$602$	−354.086	−0.0239726
$603$	1449.52	0.0978926
$604$	2543.03	0.171315
$605$	605.000	0.0406558
$606$	2780.95	0.186417
$607$	9419.30	0.629848	0.314924	−	0.949117i	$-0.398021\pi$
$607$	9419.30	0.629848	0.314924	+	0.949117i	$0.398021\pi$
$608$	−2071.83	−0.138197
$609$	2649.68	0.176306
$610$	−2262.40	−0.150167
$611$	−9382.51	−0.621237
$612$	741.294	0.0489625
$613$	−3577.77	−0.235734	−0.117867	−	0.993029i	$-0.537606\pi$
$613$	−3577.77	−0.235734	−0.117867	+	0.993029i	$0.537606\pi$
$614$	2928.42	0.192478
$615$	−1183.94	−0.0776277
$616$	1065.89	0.0697173
$617$	19601.7	1.27899	0.639493	−	0.768797i	$-0.279144\pi$
$617$	19601.7	1.27899	0.639493	+	0.768797i	$0.279144\pi$
$618$	14671.8	0.954997
$619$	−21115.9	−1.37111	−0.685557	−	0.728019i	$-0.740441\pi$
$619$	−21115.9	−1.37111	−0.685557	+	0.728019i	$0.740441\pi$
$620$	−3328.18	−0.215586
$621$	1296.65	0.0837886
$622$	19282.1	1.24299
$623$	1120.19	0.0720380
$624$	−5684.64	−0.364692
$625$	625.000	0.0400000
$626$	16325.5	1.04233
$627$	482.680	0.0307438
$628$	3513.67	0.223265
$629$	−3723.94	−0.236063
$630$	1905.88	0.120527
$631$	21001.1	1.32495	0.662473	−	0.749086i	$-0.269507\pi$
$631$	21001.1	1.32495	0.662473	+	0.749086i	$0.269507\pi$
$632$	18607.7	1.17116
$633$	9124.63	0.572941
$634$	9297.43	0.582410
$635$	11944.5	0.746463
$636$	−1791.30	−0.111682
$637$	−9927.95	−0.617519
$638$	−7526.60	−0.467055
$639$	4758.44	0.294587
$640$	8711.97	0.538080
$641$	−22840.6	−1.40741	−0.703705	−	0.710492i	$-0.748472\pi$
$641$	−22840.6	−1.40741	−0.703705	+	0.710492i	$0.748472\pi$
$642$	7873.01	0.483992
$643$	−26784.5	−1.64273	−0.821367	−	0.570400i	$-0.806788\pi$
$643$	−26784.5	−1.64273	−0.821367	+	0.570400i	$0.806788\pi$
$644$	156.160	0.00955524
$645$	232.408	0.0141877
$646$	−844.908	−0.0514589
$647$	−31238.9	−1.89819	−0.949094	−	0.314994i	$-0.897997\pi$
$647$	−31238.9	−1.89819	−0.949094	+	0.314994i	$0.897997\pi$
$648$	−5805.74	−0.351961
$649$	−4395.68	−0.265864
$650$	2564.45	0.154748
$651$	3350.23	0.201699
$652$	−9018.11	−0.541682
$653$	5244.16	0.314272	0.157136	−	0.987577i	$-0.449774\pi$
$653$	5244.16	0.314272	0.157136	+	0.987577i	$0.449774\pi$
$654$	3251.73	0.194423
$655$	6478.57	0.386471
$656$	7969.22	0.474307
$657$	−183.604	−0.0109027
$658$	5212.41	0.308816
$659$	−25318.5	−1.49661	−0.748307	−	0.663352i	$-0.769133\pi$
$659$	−25318.5	−1.49661	−0.748307	+	0.663352i	$0.769133\pi$
$660$	316.561	0.0186699
$661$	−3761.25	−0.221324	−0.110662	−	0.993858i	$-0.535297\pi$
$661$	−3761.25	−0.221324	−0.110662	+	0.993858i	$0.535297\pi$
$662$	−5502.80	−0.323070
$663$	−1004.05	−0.0588147
$664$	17990.4	1.05145
$665$	−515.978	−0.0300884
$666$	−19037.0	−1.10761
$667$	2436.99	0.141470
$668$	−6845.14	−0.396476
$669$	−5839.05	−0.337445
$670$	−1083.54	−0.0624786
$671$	−1536.60	−0.0884048
$672$	1367.79	0.0785175
$673$	−19620.2	−1.12378	−0.561890	−	0.827212i	$-0.689926\pi$
$673$	−19620.2	−1.12378	−0.561890	+	0.827212i	$0.689926\pi$
$674$	11654.8	0.666062
$675$	−2809.84	−0.160223
$676$	−2976.02	−0.169323
$677$	2796.15	0.158737	0.0793683	−	0.996845i	$-0.474710\pi$
$677$	2796.15	0.158737	0.0793683	+	0.996845i	$0.474710\pi$
$678$	2432.89	0.137809
$679$	−142.274	−0.00804118
$680$	1224.63	0.0690622
$681$	−4631.38	−0.260609
$682$	−9516.58	−0.534323
$683$	7915.89	0.443475	0.221737	−	0.975106i	$-0.428827\pi$
$683$	7915.89	0.443475	0.221737	+	0.975106i	$0.428827\pi$
$684$	−1025.94	−0.0573505
$685$	−572.235	−0.0319182
$686$	11549.8	0.642820
$687$	10233.1	0.568293
$688$	−1564.36	−0.0866873
$689$	−9855.85	−0.544961
$690$	−431.516	−0.0238080
$691$	−12417.2	−0.683607	−0.341804	−	0.939771i	$-0.611038\pi$
$691$	−12417.2	−0.683607	−0.341804	+	0.939771i	$0.611038\pi$
$692$	708.045	0.0388957
$693$	1294.45	0.0709554
$694$	22929.7	1.25418
$695$	6843.76	0.373523
$696$	8703.57	0.474006
$697$	1407.57	0.0764926
$698$	−6592.21	−0.357477
$699$	908.052	0.0491354
$700$	−338.400	−0.0182719
$701$	16511.8	0.889646	0.444823	−	0.895619i	$-0.353267\pi$
$701$	16511.8	0.889646	0.444823	+	0.895619i	$0.353267\pi$
$702$	−11529.1	−0.619855
$703$	5153.88	0.276504
$704$	2954.62	0.158177
$705$	−3421.22	−0.182767
$706$	17700.0	0.943555
$707$	−2019.09	−0.107405
$708$	−2300.01	−0.122090
$709$	19448.5	1.03019	0.515094	−	0.857134i	$-0.327757\pi$
$709$	19448.5	1.03019	0.515094	+	0.857134i	$0.327757\pi$
$710$	−3556.98	−0.188016
$711$	22597.8	1.19196
$712$	3679.58	0.193677
$713$	3081.31	0.161846
$714$	557.796	0.0292367
$715$	1741.74	0.0911013
$716$	−2765.10	−0.144325
$717$	−9462.25	−0.492851
$718$	−43480.1	−2.25998
$719$	24846.3	1.28875	0.644375	−	0.764710i	$-0.277118\pi$
$719$	24846.3	1.28875	0.644375	+	0.764710i	$0.277118\pi$
$720$	8420.25	0.435839
$721$	−10652.4	−0.550228
$722$	1169.34	0.0602746
$723$	−11985.6	−0.616527
$724$	1891.81	0.0971115
$725$	−5280.96	−0.270524
$726$	905.173	0.0462729
$727$	−15093.4	−0.769990	−0.384995	−	0.922919i	$-0.625797\pi$
$727$	−15093.4	−0.769990	−0.384995	+	0.922919i	$0.625797\pi$
$728$	3068.60	0.156222
$729$	−42.2984	−0.00214898
$730$	137.246	0.00695850
$731$	−276.306	−0.0139803
$732$	−804.012	−0.0405972
$733$	−15363.6	−0.774170	−0.387085	−	0.922044i	$-0.626518\pi$
$733$	−15363.6	−0.774170	−0.387085	+	0.922044i	$0.626518\pi$
$734$	37619.3	1.89176
$735$	−3620.10	−0.181673
$736$	1258.00	0.0630035
$737$	−735.924	−0.0367817
$738$	7195.57	0.358906
$739$	26801.3	1.33410	0.667052	−	0.745011i	$-0.267556\pi$
$739$	26801.3	1.33410	0.667052	+	0.745011i	$0.267556\pi$
$740$	3380.13	0.167914
$741$	1389.59	0.0688906
$742$	5475.37	0.270899
$743$	−16289.7	−0.804323	−0.402161	−	0.915569i	$-0.631741\pi$
$743$	−16289.7	−0.804323	−0.402161	+	0.915569i	$0.631741\pi$
$744$	11004.7	0.542275
$745$	−3956.23	−0.194557
$746$	−26083.3	−1.28013
$747$	21848.1	1.07012
$748$	−376.355	−0.0183969
$749$	−5716.13	−0.278855
$750$	935.096	0.0455265
$751$	15853.4	0.770307	0.385153	−	0.922853i	$-0.374149\pi$
$751$	15853.4	0.770307	0.385153	+	0.922853i	$0.374149\pi$
$752$	23028.6	1.11671
$753$	4930.77	0.238628
$754$	−21668.4	−1.04658
$755$	5101.98	0.245934
$756$	1521.36	0.0731895
$757$	18414.9	0.884148	0.442074	−	0.896979i	$-0.354243\pi$
$757$	18414.9	0.884148	0.442074	+	0.896979i	$0.354243\pi$
$758$	−26929.1	−1.29038
$759$	−293.080	−0.0140160
$760$	−1694.87	−0.0808937
$761$	20336.0	0.968700	0.484350	−	0.874874i	$-0.339056\pi$
$761$	20336.0	0.968700	0.484350	+	0.874874i	$0.339056\pi$
$762$	17870.8	0.849596
$763$	−2360.89	−0.112018
$764$	3054.40	0.144639
$765$	1487.23	0.0702887
$766$	−307.746	−0.0145161
$767$	−12654.8	−0.595747
$768$	8071.82	0.379254
$769$	−36179.9	−1.69660	−0.848298	−	0.529519i	$-0.822372\pi$
$769$	−36179.9	−1.69660	−0.848298	+	0.529519i	$0.822372\pi$
$770$	−967.616	−0.0452863
$771$	−12080.7	−0.564302
$772$	5193.69	0.242131
$773$	−19989.3	−0.930098	−0.465049	−	0.885285i	$-0.653963\pi$
$773$	−19989.3	−0.930098	−0.465049	+	0.885285i	$0.653963\pi$
$774$	−1412.50	−0.0655958
$775$	−6677.21	−0.309487
$776$	−467.335	−0.0216190
$777$	−3402.52	−0.157098
$778$	14144.0	0.651785
$779$	−1948.05	−0.0895971
$780$	911.353	0.0418355
$781$	−2415.86	−0.110687
$782$	513.023	0.0234599
$783$	23741.9	1.08361
$784$	24367.3	1.11003
$785$	7049.34	0.320512
$786$	9692.94	0.439867
$787$	−13266.3	−0.600882	−0.300441	−	0.953800i	$-0.597134\pi$
$787$	−13266.3	−0.600882	−0.300441	+	0.953800i	$0.597134\pi$
$788$	−7916.55	−0.357887
$789$	17140.3	0.773398
$790$	−16892.1	−0.760751
$791$	−1766.38	−0.0793998
$792$	4251.97	0.190766
$793$	−4423.73	−0.198097
$794$	−9621.01	−0.430021
$795$	−3593.81	−0.160326
$796$	−12331.6	−0.549100
$797$	−10780.6	−0.479131	−0.239566	−	0.970880i	$-0.577005\pi$
$797$	−10780.6	−0.479131	−0.239566	+	0.970880i	$0.577005\pi$
$798$	−771.982	−0.0342454
$799$	4067.43	0.180094
$800$	−2726.09	−0.120477
$801$	4468.60	0.197116
$802$	33858.1	1.49074
$803$	93.2158	0.00409653
$804$	−385.066	−0.0168908
$805$	313.298	0.0137172
$806$	−27397.4	−1.19731
$807$	4165.10	0.181683
$808$	−6632.23	−0.288764
$809$	−31894.8	−1.38611	−0.693054	−	0.720886i	$-0.743735\pi$
$809$	−31894.8	−1.38611	−0.693054	+	0.720886i	$0.743735\pi$
$810$	5270.47	0.228624
$811$	27055.7	1.17146	0.585731	−	0.810506i	$-0.300808\pi$
$811$	27055.7	1.17146	0.585731	+	0.810506i	$0.300808\pi$
$812$	2859.32	0.123574
$813$	−1336.32	−0.0576466
$814$	9665.11	0.416169
$815$	−18092.7	−0.777619
$816$	2464.36	0.105723
$817$	382.404	0.0163753
$818$	1992.82	0.0851803
$819$	3726.61	0.158997
$820$	−1277.61	−0.0544099
$821$	45128.6	1.91839	0.959197	−	0.282740i	$-0.0912434\pi$
$821$	45128.6	1.91839	0.959197	+	0.282740i	$0.0912434\pi$
$822$	−856.152	−0.0363281
$823$	−41382.3	−1.75273	−0.876365	−	0.481648i	$-0.840038\pi$
$823$	−41382.3	−1.75273	−0.876365	+	0.481648i	$0.840038\pi$
$824$	−34990.5	−1.47931
$825$	635.105	0.0268018
$826$	7030.31	0.296145
$827$	−28718.6	−1.20755	−0.603775	−	0.797155i	$-0.706337\pi$
$827$	−28718.6	−1.20755	−0.603775	+	0.797155i	$0.706337\pi$
$828$	622.944	0.0261459
$829$	−18228.5	−0.763693	−0.381846	−	0.924226i	$-0.624712\pi$
$829$	−18228.5	−0.763693	−0.381846	+	0.924226i	$0.624712\pi$
$830$	−16331.7	−0.682991
$831$	−948.023	−0.0395747
$832$	8506.09	0.354442
$833$	4303.88	0.179017
$834$	10239.3	0.425130
$835$	−13733.1	−0.569167
$836$	520.869	0.0215486
$837$	30019.0	1.23968
$838$	−8599.76	−0.354503
$839$	−32471.7	−1.33617	−0.668087	−	0.744084i	$-0.732886\pi$
$839$	−32471.7	−1.33617	−0.668087	+	0.744084i	$0.732886\pi$
$840$	1118.93	0.0459603
$841$	20232.7	0.829584
$842$	−5133.18	−0.210096
$843$	−14155.3	−0.578332
$844$	9846.57	0.401579
$845$	−5970.68	−0.243074
$846$	20793.0	0.845009
$847$	−657.193	−0.0266605
$848$	24190.3	0.979599
$849$	18370.8	0.742621
$850$	−1111.72	−0.0448608
$851$	−3129.40	−0.126057
$852$	−1264.08	−0.0508294
$853$	7683.56	0.308417	0.154209	−	0.988038i	$-0.450717\pi$
$853$	7683.56	0.308417	0.154209	+	0.988038i	$0.450717\pi$
$854$	2457.58	0.0984738
$855$	−2058.30	−0.0823303
$856$	−18776.1	−0.749714
$857$	26419.6	1.05306	0.526532	−	0.850156i	$-0.323492\pi$
$857$	26419.6	1.05306	0.526532	+	0.850156i	$0.323492\pi$
$858$	2605.91	0.103688
$859$	7781.66	0.309088	0.154544	−	0.987986i	$-0.450609\pi$
$859$	7781.66	0.309088	0.154544	+	0.987986i	$0.450609\pi$
$860$	250.796	0.00994428
$861$	1286.08	0.0509051
$862$	−20569.8	−0.812774
$863$	−28649.8	−1.13007	−0.565035	−	0.825067i	$-0.691138\pi$
$863$	−28649.8	−1.13007	−0.565035	+	0.825067i	$0.691138\pi$
$864$	12255.8	0.482582
$865$	1420.52	0.0558373
$866$	−2068.71	−0.0811750
$867$	−10911.2	−0.427408
$868$	3615.30	0.141373
$869$	−11472.9	−0.447861
$870$	−7901.13	−0.307900
$871$	−2118.66	−0.0824203
$872$	−7754.96	−0.301165
$873$	−567.548	−0.0220030
$874$	−710.015	−0.0274790
$875$	−678.918	−0.0262304
$876$	48.7744	0.00188120
$877$	−25076.1	−0.965518	−0.482759	−	0.875753i	$-0.660365\pi$
$877$	−25076.1	−0.965518	−0.482759	+	0.875753i	$0.660365\pi$
$878$	−37705.0	−1.44930
$879$	12019.6	0.461217
$880$	−4274.96	−0.163760
$881$	1168.40	0.0446813	0.0223407	−	0.999750i	$-0.492888\pi$
$881$	1168.40	0.0446813	0.0223407	+	0.999750i	$0.492888\pi$
$882$	22001.7	0.839951
$883$	6260.91	0.238614	0.119307	−	0.992857i	$-0.461933\pi$
$883$	6260.91	0.238614	0.119307	+	0.992857i	$0.461933\pi$
$884$	−1083.49	−0.0412238
$885$	−4614.41	−0.175268
$886$	−12293.2	−0.466137
$887$	15019.9	0.568568	0.284284	−	0.958740i	$-0.408244\pi$
$887$	15019.9	0.568568	0.284284	+	0.958740i	$0.408244\pi$
$888$	−11176.5	−0.422363
$889$	−12975.0	−0.489501
$890$	−3340.33	−0.125807
$891$	3579.63	0.134593
$892$	−6301.04	−0.236518
$893$	−5629.26	−0.210948
$894$	−5919.12	−0.221437
$895$	−5547.51	−0.207188
$896$	−9463.55	−0.352851
$897$	−843.752	−0.0314070
$898$	2346.98	0.0872159
$899$	56419.3	2.09309
$900$	−1349.92	−0.0499970
$901$	4272.63	0.157982
$902$	−3653.19	−0.134854
$903$	−252.458	−0.00930373
$904$	−5802.14	−0.213469
$905$	3795.47	0.139410
$906$	7633.35	0.279913
$907$	12443.0	0.455528	0.227764	−	0.973716i	$-0.426858\pi$
$907$	12443.0	0.455528	0.227764	+	0.973716i	$0.426858\pi$
$908$	−4997.82	−0.182663
$909$	−8054.40	−0.293892
$910$	−2785.68	−0.101477
$911$	−23474.5	−0.853727	−0.426863	−	0.904316i	$-0.640381\pi$
$911$	−23474.5	−0.853727	−0.426863	+	0.904316i	$0.640381\pi$
$912$	−3410.64	−0.123835
$913$	−11092.3	−0.402083
$914$	−25650.5	−0.928275
$915$	−1613.06	−0.0582798
$916$	11042.7	0.398322
$917$	−7037.47	−0.253433
$918$	4998.01	0.179694
$919$	−28645.0	−1.02819	−0.514097	−	0.857732i	$-0.671873\pi$
$919$	−28645.0	−1.02819	−0.514097	+	0.857732i	$0.671873\pi$
$920$	1029.11	0.0368791
$921$	2087.92	0.0747005
$922$	4114.69	0.146974
$923$	−6955.04	−0.248026
$924$	−343.871	−0.0122430
$925$	6781.42	0.241051
$926$	−15447.2	−0.548192
$927$	−42493.6	−1.50558
$928$	23034.2	0.814801
$929$	20547.0	0.725648	0.362824	−	0.931858i	$-0.381813\pi$
$929$	20547.0	0.725648	0.362824	+	0.931858i	$0.381813\pi$
$930$	−9990.12	−0.352246
$931$	−5956.51	−0.209685
$932$	979.897	0.0344395
$933$	13747.8	0.482404
$934$	25805.3	0.904043
$935$	−755.066	−0.0264100
$936$	12241.0	0.427469
$937$	−15313.2	−0.533895	−0.266947	−	0.963711i	$-0.586015\pi$
$937$	−15313.2	−0.533895	−0.266947	+	0.963711i	$0.586015\pi$
$938$	1177.01	0.0409710
$939$	11639.8	0.404527
$940$	−3691.90	−0.128103
$941$	−43396.2	−1.50337	−0.751687	−	0.659520i	$-0.770760\pi$
$941$	−43396.2	−1.50337	−0.751687	+	0.659520i	$0.770760\pi$
$942$	10546.9	0.364794
$943$	1182.84	0.0408470
$944$	31060.1	1.07089
$945$	3052.24	0.105068
$946$	717.125	0.0246466
$947$	3610.45	0.123890	0.0619450	−	0.998080i	$-0.480270\pi$
$947$	3610.45	0.123890	0.0619450	+	0.998080i	$0.480270\pi$
$948$	−6003.10	−0.205666
$949$	268.360	0.00917949
$950$	1538.60	0.0525462
$951$	6628.92	0.226033
$952$	−1330.27	−0.0452883
$953$	−27494.4	−0.934554	−0.467277	−	0.884111i	$-0.654765\pi$
$953$	−27494.4	−0.934554	−0.467277	+	0.884111i	$0.654765\pi$
$954$	21842.0	0.741257
$955$	6127.93	0.207639
$956$	−10210.9	−0.345444
$957$	−5366.35	−0.181264
$958$	−10096.5	−0.340505
$959$	621.601	0.0209307
$960$	3101.64	0.104276
$961$	41545.1	1.39455
$962$	27825.0	0.932551
$963$	−22802.4	−0.763028
$964$	−12933.9	−0.432130
$965$	10419.9	0.347594
$966$	468.742	0.0156124
$967$	−26457.0	−0.879833	−0.439916	−	0.898039i	$-0.644992\pi$
$967$	−26457.0	−0.879833	−0.439916	+	0.898039i	$0.644992\pi$
$968$	−2158.72	−0.0716777
$969$	−602.405	−0.0199712
$970$	424.249	0.0140431
$971$	−6966.09	−0.230229	−0.115114	−	0.993352i	$-0.536723\pi$
$971$	−6966.09	−0.230229	−0.115114	+	0.993352i	$0.536723\pi$
$972$	9435.90	0.311375
$973$	−7434.16	−0.244942
$974$	−16907.1	−0.556198
$975$	1828.41	0.0600575
$976$	10857.7	0.356092
$977$	−33212.0	−1.08756	−0.543780	−	0.839228i	$-0.683007\pi$
$977$	−33212.0	−1.08756	−0.543780	+	0.839228i	$0.683007\pi$
$978$	−27069.4	−0.885057
$979$	−2268.71	−0.0740636
$980$	−3906.53	−0.127336
$981$	−9417.89	−0.306514
$982$	26775.2	0.870093
$983$	−28008.4	−0.908780	−0.454390	−	0.890803i	$-0.650143\pi$
$983$	−28008.4	−0.908780	−0.454390	+	0.890803i	$0.650143\pi$
$984$	4224.46	0.136860
$985$	−15882.7	−0.513770
$986$	9393.53	0.303398
$987$	3716.36	0.119851
$988$	1499.54	0.0482861
$989$	−232.193	−0.00746543
$990$	−3859.95	−0.123916
$991$	19141.3	0.613567	0.306784	−	0.951779i	$-0.400747\pi$
$991$	19141.3	0.613567	0.306784	+	0.951779i	$0.400747\pi$
$992$	29124.3	0.932154
$993$	−3923.40	−0.125383
$994$	3863.84	0.123293
$995$	−24740.5	−0.788268
$996$	−5803.95	−0.184644
$997$	−36206.9	−1.15014	−0.575068	−	0.818106i	$-0.695024\pi$
$997$	−36206.9	−1.15014	−0.575068	+	0.818106i	$0.695024\pi$
$998$	−17570.8	−0.557310
$999$	−30487.5	−0.965548

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.17	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.17	✓	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1045 = 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1045.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23917 q^{2} +2.30947 q^{3} +2.49220 q^{4} +5.00000 q^{5} +7.48077 q^{6} -5.43134 q^{7} -17.8407 q^{8} -21.6663 q^{9} +O(q^{10})\)	\(q+3.23917 q^{2} +2.30947 q^{3} +2.49220 q^{4} +5.00000 q^{5} +7.48077 q^{6} -5.43134 q^{7} -17.8407 q^{8} -21.6663 q^{9} +16.1958 q^{10} +11.0000 q^{11} +5.75566 q^{12} +31.6680 q^{13} -17.5930 q^{14} +11.5474 q^{15} -77.7265 q^{16} -13.7285 q^{17} -70.1809 q^{18} +19.0000 q^{19} +12.4610 q^{20} -12.5435 q^{21} +35.6308 q^{22} -11.5367 q^{23} -41.2026 q^{24} +25.0000 q^{25} +102.578 q^{26} -112.394 q^{27} -13.5360 q^{28} -211.239 q^{29} +37.4038 q^{30} -267.088 q^{31} -109.044 q^{32} +25.4042 q^{33} -44.4688 q^{34} -27.1567 q^{35} -53.9968 q^{36} +271.257 q^{37} +61.5442 q^{38} +73.1365 q^{39} -89.2034 q^{40} -102.529 q^{41} -40.6306 q^{42} +20.1265 q^{43} +27.4142 q^{44} -108.332 q^{45} -37.3692 q^{46} -296.277 q^{47} -179.507 q^{48} -313.501 q^{49} +80.9792 q^{50} -31.7055 q^{51} +78.9230 q^{52} -311.224 q^{53} -364.061 q^{54} +55.0000 q^{55} +96.8989 q^{56} +43.8800 q^{57} -684.237 q^{58} -399.608 q^{59} +28.7783 q^{60} -139.691 q^{61} -865.143 q^{62} +117.677 q^{63} +268.602 q^{64} +158.340 q^{65} +82.2884 q^{66} -66.9022 q^{67} -34.2141 q^{68} -26.6436 q^{69} -87.9651 q^{70} -219.623 q^{71} +386.542 q^{72} +8.47416 q^{73} +878.646 q^{74} +57.7368 q^{75} +47.3518 q^{76} -59.7448 q^{77} +236.901 q^{78} -1042.99 q^{79} -388.633 q^{80} +325.421 q^{81} -332.108 q^{82} -1008.39 q^{83} -31.2610 q^{84} -68.6424 q^{85} +65.1931 q^{86} -487.850 q^{87} -196.248 q^{88} -206.246 q^{89} -350.904 q^{90} -172.000 q^{91} -28.7517 q^{92} -616.833 q^{93} -959.691 q^{94} +95.0000 q^{95} -251.833 q^{96} +26.1949 q^{97} -1015.48 q^{98} -238.330 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9} - 60 q^{10} + 220 q^{11} - 196 q^{12} - 223 q^{13} - 11 q^{14} - 105 q^{15} + 380 q^{16} - 471 q^{17} + 113 q^{18} + 380 q^{19} + 360 q^{20} - 57 q^{21} - 132 q^{22} - 653 q^{23} - 486 q^{24} + 500 q^{25} - 145 q^{26} - 177 q^{27} - 747 q^{28} + 51 q^{29} - 225 q^{30} - 90 q^{31} - 381 q^{32} - 231 q^{33} + 517 q^{34} - 655 q^{35} + 875 q^{36} - 96 q^{37} - 228 q^{38} + 97 q^{39} - 540 q^{40} - 1284 q^{41} - 638 q^{42} - 1592 q^{43} + 792 q^{44} + 795 q^{45} - 35 q^{46} - 2030 q^{47} - 1471 q^{48} + 765 q^{49} - 300 q^{50} - 185 q^{51} - 3242 q^{52} - 943 q^{53} - 730 q^{54} + 1100 q^{55} + 887 q^{56} - 399 q^{57} - 492 q^{58} - 515 q^{59} - 980 q^{60} + 446 q^{61} - 770 q^{62} - 3980 q^{63} + 2526 q^{64} - 1115 q^{65} - 495 q^{66} - 1719 q^{67} - 1808 q^{68} + 317 q^{69} - 55 q^{70} - 90 q^{71} + 1058 q^{72} - 3763 q^{73} - 1313 q^{74} - 525 q^{75} + 1368 q^{76} - 1441 q^{77} + 4286 q^{78} + 2 q^{79} + 1900 q^{80} + 308 q^{81} - 3830 q^{82} - 3436 q^{83} + 5734 q^{84} - 2355 q^{85} + 1922 q^{86} - 2719 q^{87} - 1188 q^{88} + 1700 q^{89} + 565 q^{90} + 2007 q^{91} - 3980 q^{92} - 2276 q^{93} + 2700 q^{94} + 1900 q^{95} - 6252 q^{96} - 1956 q^{97} - 2356 q^{98} + 1749 q^{99}+O(q^{100}) \) 20 * q - 12 * q^2 - 21 * q^3 + 72 * q^4 + 100 * q^5 - 45 * q^6 - 131 * q^7 - 108 * q^8 + 159 * q^9 - 60 * q^10 + 220 * q^11 - 196 * q^12 - 223 * q^13 - 11 * q^14 - 105 * q^15 + 380 * q^16 - 471 * q^17 + 113 * q^18 + 380 * q^19 + 360 * q^20 - 57 * q^21 - 132 * q^22 - 653 * q^23 - 486 * q^24 + 500 * q^25 - 145 * q^26 - 177 * q^27 - 747 * q^28 + 51 * q^29 - 225 * q^30 - 90 * q^31 - 381 * q^32 - 231 * q^33 + 517 * q^34 - 655 * q^35 + 875 * q^36 - 96 * q^37 - 228 * q^38 + 97 * q^39 - 540 * q^40 - 1284 * q^41 - 638 * q^42 - 1592 * q^43 + 792 * q^44 + 795 * q^45 - 35 * q^46 - 2030 * q^47 - 1471 * q^48 + 765 * q^49 - 300 * q^50 - 185 * q^51 - 3242 * q^52 - 943 * q^53 - 730 * q^54 + 1100 * q^55 + 887 * q^56 - 399 * q^57 - 492 * q^58 - 515 * q^59 - 980 * q^60 + 446 * q^61 - 770 * q^62 - 3980 * q^63 + 2526 * q^64 - 1115 * q^65 - 495 * q^66 - 1719 * q^67 - 1808 * q^68 + 317 * q^69 - 55 * q^70 - 90 * q^71 + 1058 * q^72 - 3763 * q^73 - 1313 * q^74 - 525 * q^75 + 1368 * q^76 - 1441 * q^77 + 4286 * q^78 + 2 * q^79 + 1900 * q^80 + 308 * q^81 - 3830 * q^82 - 3436 * q^83 + 5734 * q^84 - 2355 * q^85 + 1922 * q^86 - 2719 * q^87 - 1188 * q^88 + 1700 * q^89 + 565 * q^90 + 2007 * q^91 - 3980 * q^92 - 2276 * q^93 + 2700 * q^94 + 1900 * q^95 - 6252 * q^96 - 1956 * q^97 - 2356 * q^98 + 1749 * q^99

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