LMFDB - Embedded newform 1045.4.a.b.1.8

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			1.8
Root			$-0.822685$ 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)

$f(q)$	$=$	$q-1.82269 q^{2} +5.45780 q^{3} -4.67782 q^{4} +5.00000 q^{5} -9.94786 q^{6} +19.7082 q^{7} +23.1077 q^{8} +2.78762 q^{9} +O(q^{10})$	\(q-1.82269 q^{2} +5.45780 q^{3} -4.67782 q^{4} +5.00000 q^{5} -9.94786 q^{6} +19.7082 q^{7} +23.1077 q^{8} +2.78762 q^{9} -9.11343 q^{10} +11.0000 q^{11} -25.5306 q^{12} -35.9280 q^{13} -35.9218 q^{14} +27.2890 q^{15} -4.69546 q^{16} -72.8033 q^{17} -5.08095 q^{18} +19.0000 q^{19} -23.3891 q^{20} +107.563 q^{21} -20.0495 q^{22} -204.250 q^{23} +126.117 q^{24} +25.0000 q^{25} +65.4854 q^{26} -132.146 q^{27} -92.1914 q^{28} +206.218 q^{29} -49.7393 q^{30} -292.446 q^{31} -176.303 q^{32} +60.0358 q^{33} +132.697 q^{34} +98.5410 q^{35} -13.0400 q^{36} -79.3846 q^{37} -34.6310 q^{38} -196.088 q^{39} +115.538 q^{40} -85.3761 q^{41} -196.054 q^{42} -276.991 q^{43} -51.4560 q^{44} +13.9381 q^{45} +372.284 q^{46} +20.1798 q^{47} -25.6269 q^{48} +45.4131 q^{49} -45.5671 q^{50} -397.346 q^{51} +168.065 q^{52} +514.889 q^{53} +240.861 q^{54} +55.0000 q^{55} +455.411 q^{56} +103.698 q^{57} -375.870 q^{58} -613.457 q^{59} -127.653 q^{60} +487.292 q^{61} +533.037 q^{62} +54.9390 q^{63} +358.909 q^{64} -179.640 q^{65} -109.426 q^{66} +578.772 q^{67} +340.560 q^{68} -1114.76 q^{69} -179.609 q^{70} -637.951 q^{71} +64.4154 q^{72} +582.918 q^{73} +144.693 q^{74} +136.445 q^{75} -88.8786 q^{76} +216.790 q^{77} +357.407 q^{78} -129.282 q^{79} -23.4773 q^{80} -796.495 q^{81} +155.614 q^{82} -860.012 q^{83} -503.162 q^{84} -364.016 q^{85} +504.867 q^{86} +1125.50 q^{87} +254.184 q^{88} +845.789 q^{89} -25.4048 q^{90} -708.076 q^{91} +955.445 q^{92} -1596.11 q^{93} -36.7814 q^{94} +95.0000 q^{95} -962.227 q^{96} +18.0305 q^{97} -82.7739 q^{98} +30.6638 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.82269	−0.644417	−0.322208	−	0.946669i	$-0.604425\pi$
$2$	−1.82269	−0.644417	−0.322208	+	0.946669i	$0.604425\pi$
$3$	5.45780	1.05035	0.525177	−	0.850993i	$-0.323999\pi$
$3$	5.45780	1.05035	0.525177	+	0.850993i	$0.323999\pi$
$4$	−4.67782	−0.584727
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	−9.94786	−0.676866
$7$	19.7082	1.06414	0.532071	−	0.846700i	$-0.321414\pi$
$7$	19.7082	1.06414	0.532071	+	0.846700i	$0.321414\pi$
$8$	23.1077	1.02122
$9$	2.78762	0.103245
$10$	−9.11343	−0.288192
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	−25.5306	−0.614171
$13$	−35.9280	−0.766510	−0.383255	−	0.923643i	$-0.625197\pi$
$13$	−35.9280	−0.766510	−0.383255	+	0.923643i	$0.625197\pi$
$14$	−35.9218	−0.685751
$15$	27.2890	0.469733
$16$	−4.69546	−0.0733666
$17$	−72.8033	−1.03867	−0.519335	−	0.854571i	$-0.673820\pi$
$17$	−72.8033	−1.03867	−0.519335	+	0.854571i	$0.673820\pi$
$18$	−5.08095	−0.0665329
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	−23.3891	−0.261498
$21$	107.563	1.11773
$22$	−20.0495	−0.194299
$23$	−204.250	−1.85170	−0.925850	−	0.377891i	$-0.876650\pi$
$23$	−204.250	−1.85170	−0.925850	+	0.377891i	$0.876650\pi$
$24$	126.117	1.07265
$25$	25.0000	0.200000
$26$	65.4854	0.493952
$27$	−132.146	−0.941911
$28$	−92.1914	−0.622233
$29$	206.218	1.32047	0.660236	−	0.751058i	$-0.270456\pi$
$29$	206.218	1.32047	0.660236	+	0.751058i	$0.270456\pi$
$30$	−49.7393	−0.302704
$31$	−292.446	−1.69435	−0.847175	−	0.531314i	$-0.821698\pi$
$31$	−292.446	−1.69435	−0.847175	+	0.531314i	$0.821698\pi$
$32$	−176.303	−0.973946
$33$	60.0358	0.316694
$34$	132.697	0.669336
$35$	98.5410	0.475899
$36$	−13.0400	−0.0603703
$37$	−79.3846	−0.352723	−0.176361	−	0.984325i	$-0.556433\pi$
$37$	−79.3846	−0.352723	−0.176361	+	0.984325i	$0.556433\pi$
$38$	−34.6310	−0.147839
$39$	−196.088	−0.805108
$40$	115.538	0.456705
$41$	−85.3761	−0.325207	−0.162604	−	0.986691i	$-0.551989\pi$
$41$	−85.3761	−0.325207	−0.162604	+	0.986691i	$0.551989\pi$
$42$	−196.054	−0.720282
$43$	−276.991	−0.982341	−0.491171	−	0.871063i	$-0.663431\pi$
$43$	−276.991	−0.982341	−0.491171	+	0.871063i	$0.663431\pi$
$44$	−51.4560	−0.176302
$45$	13.9381	0.0461726
$46$	372.284	1.19327
$47$	20.1798	0.0626282	0.0313141	−	0.999510i	$-0.490031\pi$
$47$	20.1798	0.0626282	0.0313141	+	0.999510i	$0.490031\pi$
$48$	−25.6269	−0.0770610
$49$	45.4131	0.132400
$50$	−45.5671	−0.128883
$51$	−397.346	−1.09097
$52$	168.065	0.448199
$53$	514.889	1.33444	0.667221	−	0.744860i	$-0.267484\pi$
$53$	514.889	1.33444	0.667221	+	0.744860i	$0.267484\pi$
$54$	240.861	0.606983
$55$	55.0000	0.134840
$56$	455.411	1.08673
$57$	103.698	0.240968
$58$	−375.870	−0.850934
$59$	−613.457	−1.35365	−0.676825	−	0.736144i	$-0.736645\pi$
$59$	−613.457	−1.35365	−0.676825	+	0.736144i	$0.736645\pi$
$60$	−127.653	−0.274666
$61$	487.292	1.02281	0.511405	−	0.859340i	$-0.329125\pi$
$61$	487.292	1.02281	0.511405	+	0.859340i	$0.329125\pi$
$62$	533.037	1.09187
$63$	54.9390	0.109868
$64$	358.909	0.700993
$65$	−179.640	−0.342794
$66$	−109.426	−0.204083
$67$	578.772	1.05535	0.527674	−	0.849447i	$-0.323064\pi$
$67$	578.772	1.05535	0.527674	+	0.849447i	$0.323064\pi$
$68$	340.560	0.607339
$69$	−1114.76	−1.94494
$70$	−179.609	−0.306677
$71$	−637.951	−1.06635	−0.533175	−	0.846005i	$-0.679001\pi$
$71$	−637.951	−1.06635	−0.533175	+	0.846005i	$0.679001\pi$
$72$	64.4154	0.105436
$73$	582.918	0.934594	0.467297	−	0.884100i	$-0.345228\pi$
$73$	582.918	0.934594	0.467297	+	0.884100i	$0.345228\pi$
$74$	144.693	0.227300
$75$	136.445	0.210071
$76$	−88.8786	−0.134146
$77$	216.790	0.320851
$78$	357.407	0.518825
$79$	−129.282	−0.184119	−0.0920594	−	0.995754i	$-0.529345\pi$
$79$	−129.282	−0.184119	−0.0920594	+	0.995754i	$0.529345\pi$
$80$	−23.4773	−0.0328106
$81$	−796.495	−1.09259
$82$	155.614	0.209569
$83$	−860.012	−1.13733	−0.568666	−	0.822568i	$-0.692540\pi$
$83$	−860.012	−1.13733	−0.568666	+	0.822568i	$0.692540\pi$
$84$	−503.162	−0.653566
$85$	−364.016	−0.464507
$86$	504.867	0.633037
$87$	1125.50	1.38696
$88$	254.184	0.307911
$89$	845.789	1.00734	0.503671	−	0.863896i	$-0.331982\pi$
$89$	845.789	1.00734	0.503671	+	0.863896i	$0.331982\pi$
$90$	−25.4048	−0.0297544
$91$	−708.076	−0.815676
$92$	955.445	1.08274
$93$	−1596.11	−1.77967
$94$	−36.7814	−0.0403586
$95$	95.0000	0.102598
$96$	−962.227	−1.02299
$97$	18.0305	0.0188734	0.00943668	−	0.999955i	$-0.496996\pi$
$97$	18.0305	0.0188734	0.00943668	+	0.999955i	$0.496996\pi$
$98$	−82.7739	−0.0853206
$99$	30.6638	0.0311296
$100$	−116.945	−0.116945
$101$	−276.262	−0.272169	−0.136084	−	0.990697i	$-0.543452\pi$
$101$	−276.262	−0.272169	−0.136084	+	0.990697i	$0.543452\pi$
$102$	724.237	0.703040
$103$	−1075.32	−1.02868	−0.514340	−	0.857586i	$-0.671963\pi$
$103$	−1075.32	−1.02868	−0.514340	+	0.857586i	$0.671963\pi$
$104$	−830.212	−0.782779
$105$	537.817	0.499863
$106$	−938.480	−0.859936
$107$	−1142.68	−1.03240	−0.516201	−	0.856467i	$-0.672654\pi$
$107$	−1142.68	−1.03240	−0.516201	+	0.856467i	$0.672654\pi$
$108$	618.157	0.550761
$109$	−1421.68	−1.24929	−0.624643	−	0.780911i	$-0.714755\pi$
$109$	−1421.68	−1.24929	−0.624643	+	0.780911i	$0.714755\pi$
$110$	−100.248	−0.0868931
$111$	−433.265	−0.370484
$112$	−92.5392	−0.0780726
$113$	1578.80	1.31435	0.657174	−	0.753739i	$-0.271752\pi$
$113$	1578.80	1.31435	0.657174	+	0.753739i	$0.271752\pi$
$114$	−189.009	−0.155284
$115$	−1021.25	−0.828106
$116$	−964.650	−0.772116
$117$	−100.154	−0.0791385
$118$	1118.14	0.872314
$119$	−1434.82	−1.10529
$120$	630.586	0.479703
$121$	121.000	0.0909091
$122$	−888.180	−0.659115
$123$	−465.966	−0.341583
$124$	1368.01	0.990732
$125$	125.000	0.0894427
$126$	−100.136	−0.0708005
$127$	−488.529	−0.341338	−0.170669	−	0.985328i	$-0.554593\pi$
$127$	−488.529	−0.341338	−0.170669	+	0.985328i	$0.554593\pi$
$128$	756.247	0.522214
$129$	−1511.76	−1.03181
$130$	327.427	0.220902
$131$	−433.057	−0.288827	−0.144413	−	0.989517i	$-0.546130\pi$
$131$	−433.057	−0.288827	−0.144413	+	0.989517i	$0.546130\pi$
$132$	−280.837	−0.185180
$133$	374.456	0.244131
$134$	−1054.92	−0.680083
$135$	−660.732	−0.421235
$136$	−1682.31	−1.06072
$137$	−1787.53	−1.11474	−0.557369	−	0.830265i	$-0.688189\pi$
$137$	−1787.53	−1.11474	−0.557369	+	0.830265i	$0.688189\pi$
$138$	2031.85	1.25335
$139$	2516.59	1.53564	0.767822	−	0.640663i	$-0.221341\pi$
$139$	2516.59	1.53564	0.767822	+	0.640663i	$0.221341\pi$
$140$	−460.957	−0.278271
$141$	110.137	0.0657818
$142$	1162.78	0.687174
$143$	−395.208	−0.231112
$144$	−13.0892	−0.00757475
$145$	1031.09	0.590533
$146$	−1062.48	−0.602268
$147$	247.856	0.139067
$148$	371.347	0.206247
$149$	1652.59	0.908628	0.454314	−	0.890842i	$-0.349884\pi$
$149$	1652.59	0.908628	0.454314	+	0.890842i	$0.349884\pi$
$150$	−248.696	−0.135373
$151$	−1629.82	−0.878364	−0.439182	−	0.898398i	$-0.644732\pi$
$151$	−1629.82	−0.878364	−0.439182	+	0.898398i	$0.644732\pi$
$152$	439.046	0.234285
$153$	−202.948	−0.107238
$154$	−395.140	−0.206762
$155$	−1462.23	−0.757736
$156$	917.264	0.470768
$157$	964.135	0.490104	0.245052	−	0.969510i	$-0.421195\pi$
$157$	964.135	0.490104	0.245052	+	0.969510i	$0.421195\pi$
$158$	235.641	0.118649
$159$	2810.16	1.40164
$160$	−881.515	−0.435562
$161$	−4025.40	−1.97047
$162$	1451.76	0.704080
$163$	−2701.32	−1.29806	−0.649031	−	0.760762i	$-0.724825\pi$
$163$	−2701.32	−1.29806	−0.649031	+	0.760762i	$0.724825\pi$
$164$	399.374	0.190158
$165$	300.179	0.141630
$166$	1567.53	0.732916
$167$	2816.13	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$167$	2816.13	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$168$	2485.54	1.14145
$169$	−906.179	−0.412462
$170$	663.487	0.299336
$171$	52.9648	0.0236861
$172$	1295.71	0.574402
$173$	−2037.73	−0.895525	−0.447763	−	0.894152i	$-0.647779\pi$
$173$	−2037.73	−0.895525	−0.447763	+	0.894152i	$0.647779\pi$
$174$	−2051.43	−0.893783
$175$	492.705	0.212829
$176$	−51.6501	−0.0221209
$177$	−3348.13	−1.42181
$178$	−1541.61	−0.649148
$179$	3530.41	1.47416	0.737081	−	0.675804i	$-0.236204\pi$
$179$	3530.41	1.47416	0.737081	+	0.675804i	$0.236204\pi$
$180$	−65.1999	−0.0269984
$181$	−3378.99	−1.38762	−0.693808	−	0.720160i	$-0.744068\pi$
$181$	−3378.99	−1.38762	−0.693808	+	0.720160i	$0.744068\pi$
$182$	1290.60	0.525635
$183$	2659.54	1.07431
$184$	−4719.75	−1.89100
$185$	−396.923	−0.157742
$186$	2909.21	1.14685
$187$	−800.836	−0.313171
$188$	−94.3974	−0.0366204
$189$	−2604.37	−1.00233
$190$	−173.155	−0.0661157
$191$	2608.93	0.988352	0.494176	−	0.869362i	$-0.335470\pi$
$191$	2608.93	0.988352	0.494176	+	0.869362i	$0.335470\pi$
$192$	1958.85	0.736292
$193$	−2015.11	−0.751558	−0.375779	−	0.926709i	$-0.622625\pi$
$193$	−2015.11	−0.751558	−0.375779	+	0.926709i	$0.622625\pi$
$194$	−32.8638	−0.0121623
$195$	−980.440	−0.360055
$196$	−212.434	−0.0774178
$197$	−2266.37	−0.819655	−0.409828	−	0.912163i	$-0.634411\pi$
$197$	−2266.37	−0.819655	−0.409828	+	0.912163i	$0.634411\pi$
$198$	−55.8905	−0.0200604
$199$	1419.18	0.505543	0.252771	−	0.967526i	$-0.418658\pi$
$199$	1419.18	0.505543	0.252771	+	0.967526i	$0.418658\pi$
$200$	577.692	0.204245
$201$	3158.82	1.10849
$202$	503.538	0.175390
$203$	4064.18	1.40517
$204$	1858.71	0.637921
$205$	−426.880	−0.145437
$206$	1959.96	0.662898
$207$	−569.372	−0.191179
$208$	168.699	0.0562363
$209$	209.000	0.0691714
$210$	−980.272	−0.322120
$211$	1247.45	0.407006	0.203503	−	0.979074i	$-0.434767\pi$
$211$	1247.45	0.407006	0.203503	+	0.979074i	$0.434767\pi$
$212$	−2408.56	−0.780284
$213$	−3481.81	−1.12005
$214$	2082.75	0.665297
$215$	−1384.95	−0.439316
$216$	−3053.60	−0.961902
$217$	−5763.58	−1.80303
$218$	2591.27	0.805060
$219$	3181.45	0.981655
$220$	−257.280	−0.0788446
$221$	2615.68	0.796151
$222$	789.706	0.238746
$223$	−6276.31	−1.88472	−0.942360	−	0.334599i	$-0.891399\pi$
$223$	−6276.31	−1.88472	−0.942360	+	0.334599i	$0.891399\pi$
$224$	−3474.62	−1.03642
$225$	69.6905	0.0206490
$226$	−2877.66	−0.846988
$227$	1848.80	0.540568	0.270284	−	0.962781i	$-0.412882\pi$
$227$	1848.80	0.540568	0.270284	+	0.962781i	$0.412882\pi$
$228$	−485.082	−0.140901
$229$	2222.16	0.641243	0.320621	−	0.947207i	$-0.396108\pi$
$229$	2222.16	0.641243	0.320621	+	0.947207i	$0.396108\pi$
$230$	1861.42	0.533645
$231$	1183.20	0.337007
$232$	4765.21	1.34850
$233$	−5616.29	−1.57912	−0.789560	−	0.613673i	$-0.789691\pi$
$233$	−5616.29	−1.57912	−0.789560	+	0.613673i	$0.789691\pi$
$234$	182.548	0.0509981
$235$	100.899	0.0280082
$236$	2869.64	0.791516
$237$	−705.597	−0.193390
$238$	2615.23	0.712269
$239$	168.262	0.0455397	0.0227699	−	0.999741i	$-0.492752\pi$
$239$	168.262	0.0455397	0.0227699	+	0.999741i	$0.492752\pi$
$240$	−128.135	−0.0344627
$241$	−1845.40	−0.493249	−0.246624	−	0.969111i	$-0.579321\pi$
$241$	−1845.40	−0.493249	−0.246624	+	0.969111i	$0.579321\pi$
$242$	−220.545	−0.0585833
$243$	−779.159	−0.205692
$244$	−2279.46	−0.598064
$245$	227.066	0.0592110
$246$	849.309	0.220122
$247$	−682.632	−0.175849
$248$	−6757.74	−1.73031
$249$	−4693.78	−1.19460
$250$	−227.836	−0.0576384
$251$	−4937.73	−1.24170	−0.620850	−	0.783930i	$-0.713212\pi$
$251$	−4937.73	−1.24170	−0.620850	+	0.783930i	$0.713212\pi$
$252$	−256.994	−0.0642426
$253$	−2246.75	−0.558309
$254$	890.435	0.219964
$255$	−1986.73	−0.487897
$256$	−4249.67	−1.03752
$257$	3318.44	0.805441	0.402721	−	0.915323i	$-0.368065\pi$
$257$	3318.44	0.805441	0.402721	+	0.915323i	$0.368065\pi$
$258$	2755.46	0.664914
$259$	−1564.53	−0.375347
$260$	840.323	0.200441
$261$	574.857	0.136332
$262$	789.326	0.186125
$263$	−5816.96	−1.36384	−0.681918	−	0.731428i	$-0.738854\pi$
$263$	−5816.96	−1.36384	−0.681918	+	0.731428i	$0.738854\pi$
$264$	1387.29	0.323416
$265$	2574.44	0.596780
$266$	−682.515	−0.157322
$267$	4616.15	1.05807
$268$	−2707.39	−0.617090
$269$	−4464.44	−1.01190	−0.505951	−	0.862562i	$-0.668858\pi$
$269$	−4464.44	−1.01190	−0.505951	+	0.862562i	$0.668858\pi$
$270$	1204.31	0.271451
$271$	−7784.89	−1.74501	−0.872507	−	0.488602i	$-0.837507\pi$
$271$	−7784.89	−1.74501	−0.872507	+	0.488602i	$0.837507\pi$
$272$	341.845	0.0762037
$273$	−3864.54	−0.856749
$274$	3258.11	0.718355
$275$	275.000	0.0603023
$276$	5214.63	1.13726
$277$	4208.34	0.912833	0.456416	−	0.889766i	$-0.349133\pi$
$277$	4208.34	0.912833	0.456416	+	0.889766i	$0.349133\pi$
$278$	−4586.95	−0.989594
$279$	−815.228	−0.174933
$280$	2277.05	0.486000
$281$	−7087.01	−1.50454	−0.752270	−	0.658855i	$-0.771041\pi$
$281$	−7087.01	−1.50454	−0.752270	+	0.658855i	$0.771041\pi$
$282$	−200.746	−0.0423909
$283$	−1155.91	−0.242798	−0.121399	−	0.992604i	$-0.538738\pi$
$283$	−1155.91	−0.242798	−0.121399	+	0.992604i	$0.538738\pi$
$284$	2984.22	0.623524
$285$	518.491	0.107764
$286$	720.340	0.148932
$287$	−1682.61	−0.346067
$288$	−491.466	−0.100555
$289$	387.316	0.0788350
$290$	−1879.35	−0.380549
$291$	98.4067	0.0198237
$292$	−2726.78	−0.546483
$293$	7537.67	1.50292	0.751460	−	0.659779i	$-0.229350\pi$
$293$	7537.67	1.50292	0.751460	+	0.659779i	$0.229350\pi$
$294$	−451.763	−0.0896169
$295$	−3067.29	−0.605371
$296$	−1834.39	−0.360209
$297$	−1453.61	−0.283997
$298$	−3012.15	−0.585535
$299$	7338.30	1.41935
$300$	−638.265	−0.122834
$301$	−5458.99	−1.04535
$302$	2970.65	0.566032
$303$	−1507.78	−0.285874
$304$	−89.2138	−0.0168315
$305$	2436.46	0.457414
$306$	369.910	0.0691057
$307$	10479.7	1.94824	0.974120	−	0.226030i	$-0.0725748\pi$
$307$	10479.7	1.94824	0.974120	+	0.226030i	$0.0725748\pi$
$308$	−1014.11	−0.187610
$309$	−5868.86	−1.08048
$310$	2665.18	0.488298
$311$	5615.48	1.02387	0.511937	−	0.859023i	$-0.328928\pi$
$311$	5615.48	1.02387	0.511937	+	0.859023i	$0.328928\pi$
$312$	−4531.14	−0.822196
$313$	478.309	0.0863759	0.0431879	−	0.999067i	$-0.486249\pi$
$313$	478.309	0.0863759	0.0431879	+	0.999067i	$0.486249\pi$
$314$	−1757.31	−0.315831
$315$	274.695	0.0491343
$316$	604.759	0.107659
$317$	−6206.55	−1.09967	−0.549833	−	0.835274i	$-0.685309\pi$
$317$	−6206.55	−1.09967	−0.549833	+	0.835274i	$0.685309\pi$
$318$	−5122.04	−0.903238
$319$	2268.40	0.398137
$320$	1794.54	0.313494
$321$	−6236.52	−1.08439
$322$	7337.04	1.26981
$323$	−1383.26	−0.238287
$324$	3725.86	0.638865
$325$	−898.200	−0.153302
$326$	4923.66	0.836492
$327$	−7759.24	−1.31219
$328$	−1972.84	−0.332110
$329$	397.707	0.0666453
$330$	−547.132	−0.0912686
$331$	−1957.22	−0.325011	−0.162506	−	0.986708i	$-0.551958\pi$
$331$	−1957.22	−0.325011	−0.162506	+	0.986708i	$0.551958\pi$
$332$	4022.98	0.665029
$333$	−221.294	−0.0364169
$334$	−5132.92	−0.840901
$335$	2893.86	0.471966
$336$	−505.061	−0.0820039
$337$	7968.85	1.28810	0.644052	−	0.764982i	$-0.277252\pi$
$337$	7968.85	1.28810	0.644052	+	0.764982i	$0.277252\pi$
$338$	1651.68	0.265797
$339$	8616.80	1.38053
$340$	1702.80	0.271610
$341$	−3216.90	−0.510866
$342$	−96.5381	−0.0152637
$343$	−5864.90	−0.923250
$344$	−6400.61	−1.00319
$345$	−5573.79	−0.869805
$346$	3714.14	0.577091
$347$	−5679.28	−0.878616	−0.439308	−	0.898336i	$-0.644776\pi$
$347$	−5679.28	−0.878616	−0.439308	+	0.898336i	$0.644776\pi$
$348$	−5264.87	−0.810996
$349$	3320.01	0.509216	0.254608	−	0.967044i	$-0.418054\pi$
$349$	3320.01	0.509216	0.254608	+	0.967044i	$0.418054\pi$
$350$	−898.046	−0.137150
$351$	4747.76	0.721984
$352$	−1939.33	−0.293656
$353$	1314.91	0.198259	0.0991296	−	0.995075i	$-0.468394\pi$
$353$	1314.91	0.198259	0.0991296	+	0.995075i	$0.468394\pi$
$354$	6102.59	0.916240
$355$	−3189.76	−0.476887
$356$	−3956.45	−0.589020
$357$	−7830.97	−1.16095
$358$	−6434.82	−0.949975
$359$	3378.45	0.496679	0.248339	−	0.968673i	$-0.420115\pi$
$359$	3378.45	0.496679	0.248339	+	0.968673i	$0.420115\pi$
$360$	322.077	0.0471526
$361$	361.000	0.0526316
$362$	6158.84	0.894202
$363$	660.394	0.0954868
$364$	3312.25	0.476948
$365$	2914.59	0.417963
$366$	−4847.51	−0.692305
$367$	−13.5188	−0.00192282	−0.000961409	−	1.00000i	$-0.500306\pi$
$367$	−13.5188	−0.00192282	−0.000961409		1.00000i	$0.500306\pi$
$368$	959.050	0.135853
$369$	−237.996	−0.0335761
$370$	723.465	0.101652
$371$	10147.5	1.42004
$372$	7466.32	1.04062
$373$	9695.73	1.34591	0.672957	−	0.739682i	$-0.265024\pi$
$373$	9695.73	1.34591	0.672957	+	0.739682i	$0.265024\pi$
$374$	1459.67	0.201812
$375$	682.225	0.0939466
$376$	466.308	0.0639575
$377$	−7408.99	−1.01216
$378$	4746.94	0.645916
$379$	4337.90	0.587923	0.293962	−	0.955817i	$-0.405026\pi$
$379$	4337.90	0.587923	0.293962	+	0.955817i	$0.405026\pi$
$380$	−444.393	−0.0599918
$381$	−2666.30	−0.358526
$382$	−4755.25	−0.636910
$383$	−4230.62	−0.564424	−0.282212	−	0.959352i	$-0.591068\pi$
$383$	−4230.62	−0.564424	−0.282212	+	0.959352i	$0.591068\pi$
$384$	4127.45	0.548510
$385$	1083.95	0.143489
$386$	3672.91	0.484317
$387$	−772.144	−0.101422
$388$	−84.3432	−0.0110358
$389$	12601.9	1.64253	0.821264	−	0.570548i	$-0.193269\pi$
$389$	12601.9	1.64253	0.821264	+	0.570548i	$0.193269\pi$
$390$	1787.03	0.232025
$391$	14870.1	1.92331
$392$	1049.39	0.135210
$393$	−2363.54	−0.303371
$394$	4130.88	0.528199
$395$	−646.411	−0.0823404
$396$	−143.440	−0.0182023
$397$	10090.9	1.27569	0.637844	−	0.770165i	$-0.279826\pi$
$397$	10090.9	1.27569	0.637844	+	0.770165i	$0.279826\pi$
$398$	−2586.72	−0.325780
$399$	2043.71	0.256424
$400$	−117.387	−0.0146733
$401$	−1377.05	−0.171488	−0.0857439	−	0.996317i	$-0.527327\pi$
$401$	−1377.05	−0.171488	−0.0857439	+	0.996317i	$0.527327\pi$
$402$	−5757.54	−0.714329
$403$	10507.0	1.29874
$404$	1292.30	0.159145
$405$	−3982.47	−0.488619
$406$	−7407.73	−0.905515
$407$	−873.230	−0.106350
$408$	−9181.74	−1.11413
$409$	−4576.33	−0.553264	−0.276632	−	0.960976i	$-0.589218\pi$
$409$	−4576.33	−0.553264	−0.276632	+	0.960976i	$0.589218\pi$
$410$	778.069	0.0937221
$411$	−9755.99	−1.17087
$412$	5030.13	0.601497
$413$	−12090.1	−1.44048
$414$	1037.79	0.123199
$415$	−4300.06	−0.508630
$416$	6334.21	0.746539
$417$	13735.1	1.61297
$418$	−380.941	−0.0445752
$419$	13076.4	1.52464	0.762320	−	0.647200i	$-0.224060\pi$
$419$	13076.4	1.52464	0.762320	+	0.647200i	$0.224060\pi$
$420$	−2515.81	−0.292283
$421$	10417.8	1.20601	0.603007	−	0.797736i	$-0.293969\pi$
$421$	10417.8	1.20601	0.603007	+	0.797736i	$0.293969\pi$
$422$	−2273.72	−0.262281
$423$	56.2536	0.00646606
$424$	11897.9	1.36276
$425$	−1820.08	−0.207734
$426$	6346.25	0.721777
$427$	9603.65	1.08841
$428$	5345.25	0.603674
$429$	−2156.97	−0.242749
$430$	2524.33	0.283103
$431$	−14686.1	−1.64131	−0.820657	−	0.571422i	$-0.806392\pi$
$431$	−14686.1	−1.64131	−0.820657	+	0.571422i	$0.806392\pi$
$432$	620.489	0.0691048
$433$	6024.29	0.668612	0.334306	−	0.942465i	$-0.391498\pi$
$433$	6024.29	0.668612	0.334306	+	0.942465i	$0.391498\pi$
$434$	10505.2	1.16190
$435$	5627.48	0.620269
$436$	6650.35	0.730491
$437$	−3880.75	−0.424809
$438$	−5798.78	−0.632595
$439$	−11568.1	−1.25767	−0.628834	−	0.777540i	$-0.716468\pi$
$439$	−11568.1	−1.25767	−0.628834	+	0.777540i	$0.716468\pi$
$440$	1270.92	0.137702
$441$	126.595	0.0136696
$442$	−4767.55	−0.513053
$443$	−7686.34	−0.824355	−0.412177	−	0.911104i	$-0.635232\pi$
$443$	−7686.34	−0.824355	−0.412177	+	0.911104i	$0.635232\pi$
$444$	2026.74	0.216632
$445$	4228.94	0.450497
$446$	11439.7	1.21455
$447$	9019.52	0.954382
$448$	7073.44	0.745957
$449$	−5888.44	−0.618915	−0.309458	−	0.950913i	$-0.600147\pi$
$449$	−5888.44	−0.618915	−0.309458	+	0.950913i	$0.600147\pi$
$450$	−127.024	−0.0133066
$451$	−939.137	−0.0980537
$452$	−7385.36	−0.768535
$453$	−8895.24	−0.922594
$454$	−3369.77	−0.348351
$455$	−3540.38	−0.364782
$456$	2396.23	0.246082
$457$	11569.3	1.18422	0.592111	−	0.805857i	$-0.298295\pi$
$457$	11569.3	1.18422	0.592111	+	0.805857i	$0.298295\pi$
$458$	−4050.30	−0.413228
$459$	9620.69	0.978334
$460$	4777.23	0.484216
$461$	10378.6	1.04854	0.524271	−	0.851552i	$-0.324338\pi$
$461$	10378.6	1.04854	0.524271	+	0.851552i	$0.324338\pi$
$462$	−2156.60	−0.217173
$463$	3571.96	0.358538	0.179269	−	0.983800i	$-0.442627\pi$
$463$	3571.96	0.358538	0.179269	+	0.983800i	$0.442627\pi$
$464$	−968.289	−0.0968786
$465$	−7980.56	−0.795892
$466$	10236.7	1.01761
$467$	−8086.19	−0.801251	−0.400626	−	0.916242i	$-0.631207\pi$
$467$	−8086.19	−0.801251	−0.400626	+	0.916242i	$0.631207\pi$
$468$	468.500	0.0462744
$469$	11406.6	1.12304
$470$	−183.907	−0.0180489
$471$	5262.06	0.514783
$472$	−14175.6	−1.38238
$473$	−3046.90	−0.296187
$474$	1286.08	0.124624
$475$	475.000	0.0458831
$476$	6711.83	0.646295
$477$	1435.31	0.137775
$478$	−306.690	−0.0293466
$479$	−13286.0	−1.26733	−0.633667	−	0.773606i	$-0.718451\pi$
$479$	−13286.0	−1.26733	−0.633667	+	0.773606i	$0.718451\pi$
$480$	−4811.14	−0.457494
$481$	2852.13	0.270366
$482$	3363.59	0.317858
$483$	−21969.9	−2.06970
$484$	−566.016	−0.0531570
$485$	90.1523	0.00844042
$486$	1420.16	0.132551
$487$	14124.5	1.31425	0.657127	−	0.753780i	$-0.271771\pi$
$487$	14124.5	1.31425	0.657127	+	0.753780i	$0.271771\pi$
$488$	11260.2	1.04452
$489$	−14743.3	−1.36343
$490$	−413.869	−0.0381565
$491$	506.534	0.0465572	0.0232786	−	0.999729i	$-0.492590\pi$
$491$	506.534	0.0465572	0.0232786	+	0.999729i	$0.492590\pi$
$492$	2179.70	0.199733
$493$	−15013.3	−1.37153
$494$	1244.22	0.113320
$495$	153.319	0.0139216
$496$	1373.17	0.124309
$497$	−12572.9	−1.13475
$498$	8555.28	0.769822
$499$	11053.1	0.991592	0.495796	−	0.868439i	$-0.334876\pi$
$499$	11053.1	0.991592	0.495796	+	0.868439i	$0.334876\pi$
$500$	−584.727	−0.0522996
$501$	15369.9	1.37061
$502$	8999.92	0.800171
$503$	−5946.16	−0.527090	−0.263545	−	0.964647i	$-0.584892\pi$
$503$	−5946.16	−0.527090	−0.263545	+	0.964647i	$0.584892\pi$
$504$	1269.51	0.112199
$505$	−1381.31	−0.121718
$506$	4095.12	0.359783
$507$	−4945.75	−0.433232
$508$	2285.25	0.199590
$509$	14421.1	1.25580	0.627902	−	0.778293i	$-0.283914\pi$
$509$	14421.1	1.25580	0.627902	+	0.778293i	$0.283914\pi$
$510$	3621.18	0.314409
$511$	11488.3	0.994542
$512$	1695.83	0.146379
$513$	−2510.78	−0.216089
$514$	−6048.47	−0.519040
$515$	−5376.58	−0.460040
$516$	7071.74	0.603326
$517$	221.978	0.0188831
$518$	2851.64	0.241880
$519$	−11121.5	−0.940619
$520$	−4151.06	−0.350069
$521$	−1713.43	−0.144082	−0.0720410	−	0.997402i	$-0.522951\pi$
$521$	−1713.43	−0.144082	−0.0720410	+	0.997402i	$0.522951\pi$
$522$	−1047.78	−0.0878548
$523$	17611.4	1.47245	0.736225	−	0.676736i	$-0.236606\pi$
$523$	17611.4	1.47245	0.736225	+	0.676736i	$0.236606\pi$
$524$	2025.76	0.168885
$525$	2689.09	0.223545
$526$	10602.5	0.878879
$527$	21291.0	1.75987
$528$	−281.896	−0.0232348
$529$	29551.1	2.42880
$530$	−4692.40	−0.384575
$531$	−1710.09	−0.139758
$532$	−1751.64	−0.142750
$533$	3067.39	0.249275
$534$	−8413.79	−0.681836
$535$	−5713.40	−0.461704
$536$	13374.1	1.07775
$537$	19268.3	1.54839
$538$	8137.26	0.652086
$539$	499.545	0.0399200
$540$	3090.78	0.246308
$541$	14642.9	1.16368	0.581838	−	0.813305i	$-0.302334\pi$
$541$	14642.9	1.16368	0.581838	+	0.813305i	$0.302334\pi$
$542$	14189.4	1.12452
$543$	−18441.9	−1.45749
$544$	12835.4	1.01161
$545$	−7108.39	−0.558697
$546$	7043.84	0.552104
$547$	8298.08	0.648629	0.324315	−	0.945949i	$-0.394866\pi$
$547$	8298.08	0.648629	0.324315	+	0.945949i	$0.394866\pi$
$548$	8361.74	0.651817
$549$	1358.38	0.105600
$550$	−501.238	−0.0388598
$551$	3918.14	0.302937
$552$	−25759.5	−1.98622
$553$	−2547.92	−0.195929
$554$	−7670.48	−0.588245
$555$	−2166.33	−0.165686
$556$	−11772.2	−0.897933
$557$	8624.45	0.656068	0.328034	−	0.944666i	$-0.393614\pi$
$557$	8624.45	0.656068	0.328034	+	0.944666i	$0.393614\pi$
$558$	1485.90	0.112730
$559$	9951.72	0.752975
$560$	−462.696	−0.0349151
$561$	−4370.81	−0.328940
$562$	12917.4	0.969551
$563$	−16886.7	−1.26410	−0.632052	−	0.774926i	$-0.717787\pi$
$563$	−16886.7	−1.26410	−0.632052	+	0.774926i	$0.717787\pi$
$564$	−515.203	−0.0384644
$565$	7894.02	0.587794
$566$	2106.87	0.156463
$567$	−15697.5	−1.16267
$568$	−14741.6	−1.08898
$569$	−17071.5	−1.25778	−0.628888	−	0.777496i	$-0.716490\pi$
$569$	−17071.5	−1.25778	−0.628888	+	0.777496i	$0.716490\pi$
$570$	−945.046	−0.0694450
$571$	−21838.4	−1.60054	−0.800270	−	0.599640i	$-0.795310\pi$
$571$	−21838.4	−1.60054	−0.800270	+	0.599640i	$0.795310\pi$
$572$	1848.71	0.135137
$573$	14239.0	1.03812
$574$	3066.87	0.223011
$575$	−5106.26	−0.370340
$576$	1000.50	0.0723742
$577$	−7287.98	−0.525828	−0.262914	−	0.964819i	$-0.584684\pi$
$577$	−7287.98	−0.525828	−0.262914	+	0.964819i	$0.584684\pi$
$578$	−705.956	−0.0508026
$579$	−10998.1	−0.789403
$580$	−4823.25	−0.345301
$581$	−16949.3	−1.21028
$582$	−179.364	−0.0127747
$583$	5663.78	0.402349
$584$	13469.9	0.954430
$585$	−500.768	−0.0353918
$586$	−13738.8	−0.968506
$587$	−6857.93	−0.482209	−0.241105	−	0.970499i	$-0.577510\pi$
$587$	−6857.93	−0.482209	−0.241105	+	0.970499i	$0.577510\pi$
$588$	−1159.43	−0.0813161
$589$	−5556.47	−0.388710
$590$	5590.70	0.390111
$591$	−12369.4	−0.860929
$592$	372.747	0.0258781
$593$	6947.11	0.481085	0.240543	−	0.970639i	$-0.422675\pi$
$593$	6947.11	0.481085	0.240543	+	0.970639i	$0.422675\pi$
$594$	2649.47	0.183012
$595$	−7174.11	−0.494302
$596$	−7730.52	−0.531300
$597$	7745.61	0.530999
$598$	−13375.4	−0.914651
$599$	5517.87	0.376384	0.188192	−	0.982132i	$-0.439737\pi$
$599$	5517.87	0.376384	0.188192	+	0.982132i	$0.439737\pi$
$600$	3152.93	0.214530
$601$	−15498.1	−1.05188	−0.525939	−	0.850522i	$-0.676286\pi$
$601$	−15498.1	−1.05188	−0.525939	+	0.850522i	$0.676286\pi$
$602$	9950.02	0.673642
$603$	1613.40	0.108959
$604$	7624.01	0.513603
$605$	605.000	0.0406558
$606$	2748.21	0.184222
$607$	274.817	0.0183764	0.00918820	−	0.999958i	$-0.497075\pi$
$607$	274.817	0.0183764	0.00918820	+	0.999958i	$0.497075\pi$
$608$	−3349.76	−0.223438
$609$	22181.5	1.47593
$610$	−4440.90	−0.294765
$611$	−725.019	−0.0480052
$612$	949.353	0.0627048
$613$	10074.9	0.663817	0.331908	−	0.943312i	$-0.392308\pi$
$613$	10074.9	0.663817	0.331908	+	0.943312i	$0.392308\pi$
$614$	−19101.3	−1.25548
$615$	−2329.83	−0.152761
$616$	5009.52	0.327661
$617$	1203.77	0.0785442	0.0392721	−	0.999229i	$-0.487496\pi$
$617$	1203.77	0.0785442	0.0392721	+	0.999229i	$0.487496\pi$
$618$	10697.1	0.696278
$619$	4900.43	0.318199	0.159099	−	0.987263i	$-0.449141\pi$
$619$	4900.43	0.318199	0.159099	+	0.987263i	$0.449141\pi$
$620$	6840.04	0.443069
$621$	26990.9	1.74414
$622$	−10235.3	−0.659801
$623$	16669.0	1.07196
$624$	920.724	0.0590680
$625$	625.000	0.0400000
$626$	−871.807	−0.0556620
$627$	1140.68	0.0726546
$628$	−4510.05	−0.286577
$629$	5779.46	0.366363
$630$	−500.682	−0.0316629
$631$	−15988.8	−1.00872	−0.504361	−	0.863493i	$-0.668272\pi$
$631$	−15988.8	−1.00872	−0.504361	+	0.863493i	$0.668272\pi$
$632$	−2987.41	−0.188027
$633$	6808.36	0.427501
$634$	11312.6	0.708644
$635$	−2442.65	−0.152651
$636$	−13145.4	−0.819575
$637$	−1631.60	−0.101486
$638$	−4134.57	−0.256566
$639$	−1778.37	−0.110096
$640$	3781.23	0.233541
$641$	16743.5	1.03172	0.515858	−	0.856674i	$-0.327473\pi$
$641$	16743.5	1.03172	0.515858	+	0.856674i	$0.327473\pi$
$642$	11367.2	0.698798
$643$	20583.7	1.26243	0.631216	−	0.775607i	$-0.282556\pi$
$643$	20583.7	1.26243	0.631216	+	0.775607i	$0.282556\pi$
$644$	18830.1	1.15219
$645$	−7558.80	−0.461438
$646$	2521.25	0.153556
$647$	4585.84	0.278652	0.139326	−	0.990247i	$-0.455506\pi$
$647$	4585.84	0.278652	0.139326	+	0.990247i	$0.455506\pi$
$648$	−18405.1	−1.11578
$649$	−6748.03	−0.408141
$650$	1637.14	0.0987904
$651$	−31456.5	−1.89382
$652$	12636.3	0.759012
$653$	19442.3	1.16514	0.582569	−	0.812781i	$-0.302048\pi$
$653$	19442.3	1.16514	0.582569	+	0.812781i	$0.302048\pi$
$654$	14142.7	0.845599
$655$	−2165.28	−0.129167
$656$	400.880	0.0238594
$657$	1624.95	0.0964923
$658$	−724.895	−0.0429474
$659$	11625.0	0.687170	0.343585	−	0.939122i	$-0.388359\pi$
$659$	11625.0	0.687170	0.343585	+	0.939122i	$0.388359\pi$
$660$	−1404.18	−0.0828148
$661$	−13339.5	−0.784940	−0.392470	−	0.919765i	$-0.628379\pi$
$661$	−13339.5	−0.784940	−0.392470	+	0.919765i	$0.628379\pi$
$662$	3567.40	0.209443
$663$	14275.8	0.836241
$664$	−19872.9	−1.16147
$665$	1872.28	0.109179
$666$	403.349	0.0234677
$667$	−42120.0	−2.44512
$668$	−13173.4	−0.763012
$669$	−34254.9	−1.97963
$670$	−5274.60	−0.304142
$671$	5360.21	0.308389
$672$	−18963.8	−1.08861
$673$	−2786.29	−0.159589	−0.0797946	−	0.996811i	$-0.525426\pi$
$673$	−2786.29	−0.159589	−0.0797946	+	0.996811i	$0.525426\pi$
$674$	−14524.7	−0.830076
$675$	−3303.66	−0.188382
$676$	4238.94	0.241178
$677$	−23976.6	−1.36114	−0.680572	−	0.732681i	$-0.738269\pi$
$677$	−23976.6	−1.36114	−0.680572	+	0.732681i	$0.738269\pi$
$678$	−15705.7	−0.889637
$679$	355.348	0.0200839
$680$	−8411.57	−0.474366
$681$	10090.4	0.567788
$682$	5863.40	0.329210
$683$	−8020.08	−0.449312	−0.224656	−	0.974438i	$-0.572126\pi$
$683$	−8020.08	−0.449312	−0.224656	+	0.974438i	$0.572126\pi$
$684$	−247.760	−0.0138499
$685$	−8937.65	−0.498526
$686$	10689.9	0.594958
$687$	12128.1	0.673532
$688$	1300.60	0.0720711
$689$	−18498.9	−1.02286
$690$	10159.3	0.560517
$691$	28546.0	1.57155	0.785774	−	0.618513i	$-0.212265\pi$
$691$	28546.0	1.57155	0.785774	+	0.618513i	$0.212265\pi$
$692$	9532.14	0.523638
$693$	604.328	0.0331263
$694$	10351.5	0.566195
$695$	12583.0	0.686761
$696$	26007.6	1.41640
$697$	6215.66	0.337783
$698$	−6051.34	−0.328147
$699$	−30652.6	−1.65864
$700$	−2304.78	−0.124447
$701$	786.871	0.0423962	0.0211981	−	0.999775i	$-0.493252\pi$
$701$	786.871	0.0423962	0.0211981	+	0.999775i	$0.493252\pi$
$702$	−8653.66	−0.465259
$703$	−1508.31	−0.0809202
$704$	3948.00	0.211357
$705$	550.687	0.0294185
$706$	−2396.66	−0.127761
$707$	−5444.62	−0.289627
$708$	15661.9	0.831373
$709$	23963.3	1.26934	0.634669	−	0.772784i	$-0.281136\pi$
$709$	23963.3	1.26934	0.634669	+	0.772784i	$0.281136\pi$
$710$	5813.92	0.307314
$711$	−360.390	−0.0190094
$712$	19544.2	1.02872
$713$	59732.1	3.13743
$714$	14273.4	0.748135
$715$	−1976.04	−0.103356
$716$	−16514.6	−0.861983
$717$	918.344	0.0478329
$718$	−6157.85	−0.320068
$719$	−19152.2	−0.993403	−0.496702	−	0.867921i	$-0.665456\pi$
$719$	−19152.2	−0.993403	−0.496702	+	0.867921i	$0.665456\pi$
$720$	−65.4458	−0.00338753
$721$	−21192.5	−1.09466
$722$	−657.989	−0.0339167
$723$	−10071.9	−0.518086
$724$	15806.3	0.811377
$725$	5155.45	0.264094
$726$	−1203.69	−0.0615333
$727$	−15462.0	−0.788794	−0.394397	−	0.918940i	$-0.629047\pi$
$727$	−15462.0	−0.788794	−0.394397	+	0.918940i	$0.629047\pi$
$728$	−16362.0	−0.832989
$729$	17252.9	0.876536
$730$	−5312.38	−0.269342
$731$	20165.8	1.02033
$732$	−12440.9	−0.628180
$733$	−29264.9	−1.47466	−0.737328	−	0.675535i	$-0.763913\pi$
$733$	−29264.9	−1.47466	−0.737328	+	0.675535i	$0.763913\pi$
$734$	24.6405	0.00123910
$735$	1239.28	0.0621926
$736$	36009.9	1.80346
$737$	6366.49	0.318199
$738$	433.792	0.0216370
$739$	3863.44	0.192312	0.0961562	−	0.995366i	$-0.469345\pi$
$739$	3863.44	0.192312	0.0961562	+	0.995366i	$0.469345\pi$
$740$	1856.73	0.0922363
$741$	−3725.67	−0.184704
$742$	−18495.8	−0.915095
$743$	−18488.9	−0.912909	−0.456454	−	0.889747i	$-0.650881\pi$
$743$	−18488.9	−0.912909	−0.456454	+	0.889747i	$0.650881\pi$
$744$	−36882.4	−1.81744
$745$	8262.96	0.406351
$746$	−17672.3	−0.867329
$747$	−2397.39	−0.117424
$748$	3746.17	0.183119
$749$	−22520.2	−1.09862
$750$	−1243.48	−0.0605407
$751$	22892.3	1.11232	0.556160	−	0.831075i	$-0.312274\pi$
$751$	22892.3	1.11232	0.556160	+	0.831075i	$0.312274\pi$
$752$	−94.7535	−0.00459482
$753$	−26949.1	−1.30422
$754$	13504.3	0.652250
$755$	−8149.11	−0.392816
$756$	12182.8	0.586088
$757$	−10149.6	−0.487312	−0.243656	−	0.969862i	$-0.578347\pi$
$757$	−10149.6	−0.487312	−0.243656	+	0.969862i	$0.578347\pi$
$758$	−7906.63	−0.378868
$759$	−12262.3	−0.586422
$760$	2195.23	0.104775
$761$	−22008.9	−1.04839	−0.524193	−	0.851599i	$-0.675633\pi$
$761$	−22008.9	−1.04839	−0.524193	+	0.851599i	$0.675633\pi$
$762$	4859.82	0.231040
$763$	−28018.7	−1.32942
$764$	−12204.1	−0.577916
$765$	−1014.74	−0.0479581
$766$	7711.09	0.363724
$767$	22040.3	1.03759
$768$	−23193.9	−1.08976
$769$	−8290.50	−0.388769	−0.194384	−	0.980925i	$-0.562271\pi$
$769$	−8290.50	−0.388769	−0.194384	+	0.980925i	$0.562271\pi$
$770$	−1975.70	−0.0924667
$771$	18111.4	0.845999
$772$	9426.32	0.439457
$773$	24285.8	1.13001	0.565006	−	0.825087i	$-0.308874\pi$
$773$	24285.8	1.13001	0.565006	+	0.825087i	$0.308874\pi$
$774$	1407.38	0.0653580
$775$	−7311.15	−0.338870
$776$	416.642	0.0192739
$777$	−8538.88	−0.394248
$778$	−22969.4	−1.05847
$779$	−1622.15	−0.0746077
$780$	4586.32	0.210534
$781$	−7017.47	−0.321517
$782$	−27103.5	−1.23941
$783$	−27250.9	−1.24377
$784$	−213.236	−0.00971373
$785$	4820.67	0.219181
$786$	4307.99	0.195497
$787$	−14490.8	−0.656343	−0.328172	−	0.944618i	$-0.606432\pi$
$787$	−14490.8	−0.656343	−0.328172	+	0.944618i	$0.606432\pi$
$788$	10601.7	0.479275
$789$	−31747.8	−1.43251
$790$	1178.20	0.0530615
$791$	31115.4	1.39865
$792$	708.569	0.0317903
$793$	−17507.4	−0.783994
$794$	−18392.6	−0.822075
$795$	14050.8	0.626831
$796$	−6638.67	−0.295605
$797$	−29286.0	−1.30159	−0.650793	−	0.759256i	$-0.725563\pi$
$797$	−29286.0	−1.30159	−0.650793	+	0.759256i	$0.725563\pi$
$798$	−3725.03	−0.165244
$799$	−1469.15	−0.0650500
$800$	−4407.58	−0.194789
$801$	2357.74	0.104003
$802$	2509.93	0.110510
$803$	6412.10	0.281791
$804$	−14776.4	−0.648164
$805$	−20127.0	−0.881223
$806$	−19150.9	−0.836927
$807$	−24366.0	−1.06286
$808$	−6383.76	−0.277946
$809$	−36361.4	−1.58022	−0.790111	−	0.612964i	$-0.789977\pi$
$809$	−36361.4	−1.58022	−0.790111	+	0.612964i	$0.789977\pi$
$810$	7258.80	0.314874
$811$	10849.5	0.469761	0.234881	−	0.972024i	$-0.424530\pi$
$811$	10849.5	0.469761	0.234881	+	0.972024i	$0.424530\pi$
$812$	−19011.5	−0.821642
$813$	−42488.4	−1.83288
$814$	1591.62	0.0685336
$815$	−13506.6	−0.580511
$816$	1865.72	0.0800409
$817$	−5262.82	−0.225365
$818$	8341.21	0.356533
$819$	−1973.85	−0.0842146
$820$	1996.87	0.0850411
$821$	−7846.49	−0.333550	−0.166775	−	0.985995i	$-0.553335\pi$
$821$	−7846.49	−0.333550	−0.166775	+	0.985995i	$0.553335\pi$
$822$	17782.1	0.754528
$823$	2867.35	0.121445	0.0607227	−	0.998155i	$-0.480659\pi$
$823$	2867.35	0.121445	0.0607227	+	0.998155i	$0.480659\pi$
$824$	−24848.0	−1.05051
$825$	1500.90	0.0633388
$826$	22036.5	0.928267
$827$	−36370.5	−1.52929	−0.764647	−	0.644450i	$-0.777086\pi$
$827$	−36370.5	−1.52929	−0.764647	+	0.644450i	$0.777086\pi$
$828$	2663.42	0.111788
$829$	5329.67	0.223290	0.111645	−	0.993748i	$-0.464388\pi$
$829$	5329.67	0.223290	0.111645	+	0.993748i	$0.464388\pi$
$830$	7837.66	0.327770
$831$	22968.3	0.958798
$832$	−12894.9	−0.537319
$833$	−3306.23	−0.137520
$834$	−25034.7	−1.03943
$835$	14080.7	0.583570
$836$	−977.664	−0.0404464
$837$	38645.7	1.59593
$838$	−23834.2	−0.982504
$839$	−29484.0	−1.21323	−0.606616	−	0.794995i	$-0.707473\pi$
$839$	−29484.0	−1.21323	−0.606616	+	0.794995i	$0.707473\pi$
$840$	12427.7	0.510472
$841$	18136.8	0.743647
$842$	−18988.3	−0.777175
$843$	−38679.5	−1.58030
$844$	−5835.36	−0.237988
$845$	−4530.90	−0.184459
$846$	−102.533	−0.00416683
$847$	2384.69	0.0967403
$848$	−2417.64	−0.0979035
$849$	−6308.75	−0.255024
$850$	3317.44	0.133867
$851$	16214.3	0.653137
$852$	16287.3	0.654922
$853$	−35139.5	−1.41049	−0.705247	−	0.708961i	$-0.749164\pi$
$853$	−35139.5	−1.41049	−0.705247	+	0.708961i	$0.749164\pi$
$854$	−17504.4	−0.701393
$855$	264.824	0.0105927
$856$	−26404.7	−1.05431
$857$	−44215.5	−1.76240	−0.881198	−	0.472748i	$-0.843262\pi$
$857$	−44215.5	−1.76240	−0.881198	+	0.472748i	$0.843262\pi$
$858$	3931.47	0.156432
$859$	45882.4	1.82245	0.911227	−	0.411905i	$-0.135136\pi$
$859$	45882.4	1.82245	0.911227	+	0.411905i	$0.135136\pi$
$860$	6478.56	0.256880
$861$	−9183.35	−0.363493
$862$	26768.2	1.05769
$863$	18039.8	0.711567	0.355783	−	0.934568i	$-0.384214\pi$
$863$	18039.8	0.711567	0.355783	+	0.934568i	$0.384214\pi$
$864$	23297.8	0.917370
$865$	−10188.7	−0.400491
$866$	−10980.4	−0.430864
$867$	2113.90	0.0828047
$868$	26961.0	1.05428
$869$	−1422.10	−0.0555139
$870$	−10257.1	−0.399712
$871$	−20794.1	−0.808934
$872$	−32851.7	−1.27580
$873$	50.2620	0.00194858
$874$	7073.39	0.273754
$875$	2463.52	0.0951798
$876$	−14882.2	−0.574001
$877$	−44998.3	−1.73259	−0.866296	−	0.499530i	$-0.833506\pi$
$877$	−44998.3	−1.73259	−0.866296	+	0.499530i	$0.833506\pi$
$878$	21085.0	0.810462
$879$	41139.1	1.57860
$880$	−258.251	−0.00989275
$881$	17688.9	0.676452	0.338226	−	0.941065i	$-0.390173\pi$
$881$	17688.9	0.676452	0.338226	+	0.941065i	$0.390173\pi$
$882$	−230.742	−0.00880894
$883$	28505.2	1.08638	0.543191	−	0.839609i	$-0.317216\pi$
$883$	28505.2	1.08638	0.543191	+	0.839609i	$0.317216\pi$
$884$	−12235.7	−0.465531
$885$	−16740.7	−0.635854
$886$	14009.8	0.531228
$887$	−5169.21	−0.195677	−0.0978383	−	0.995202i	$-0.531193\pi$
$887$	−5169.21	−0.195677	−0.0978383	+	0.995202i	$0.531193\pi$
$888$	−10011.8	−0.378347
$889$	−9628.04	−0.363233
$890$	−7708.03	−0.290308
$891$	−8761.44	−0.329427
$892$	29359.4	1.10205
$893$	383.416	0.0143679
$894$	−16439.7	−0.615019
$895$	17652.0	0.659265
$896$	14904.3	0.555710
$897$	40051.0	1.49082
$898$	10732.8	0.398839
$899$	−60307.6	−2.23734
$900$	−325.999	−0.0120741
$901$	−37485.6	−1.38604
$902$	1711.75	0.0631874
$903$	−29794.1	−1.09799
$904$	36482.5	1.34224
$905$	−16895.0	−0.620561
$906$	16213.2	0.594535
$907$	5924.33	0.216884	0.108442	−	0.994103i	$-0.465414\pi$
$907$	5924.33	0.216884	0.108442	+	0.994103i	$0.465414\pi$
$908$	−8648.33	−0.316085
$909$	−770.112	−0.0281001
$910$	6453.00	0.235071
$911$	21230.0	0.772098	0.386049	−	0.922478i	$-0.373840\pi$
$911$	21230.0	0.772098	0.386049	+	0.922478i	$0.373840\pi$
$912$	−486.912	−0.0176790
$913$	−9460.13	−0.342919
$914$	−21087.2	−0.763132
$915$	13297.7	0.480447
$916$	−10394.9	−0.374952
$917$	−8534.77	−0.307353
$918$	−17535.5	−0.630455
$919$	19920.5	0.715034	0.357517	−	0.933907i	$-0.383623\pi$
$919$	19920.5	0.715034	0.357517	+	0.933907i	$0.383623\pi$
$920$	−23598.7	−0.845682
$921$	57196.3	2.04634
$922$	−18916.8	−0.675697
$923$	22920.3	0.817369
$924$	−5534.79	−0.197057
$925$	−1984.61	−0.0705446
$926$	−6510.56	−0.231048
$927$	−2997.57	−0.106206
$928$	−36356.8	−1.28607
$929$	2873.92	0.101497	0.0507483	−	0.998711i	$-0.483839\pi$
$929$	2873.92	0.101497	0.0507483	+	0.998711i	$0.483839\pi$
$930$	14546.0	0.512886
$931$	862.850	0.0303746
$932$	26272.0	0.923355
$933$	30648.2	1.07543
$934$	14738.6	0.516340
$935$	−4004.18	−0.140054
$936$	−2314.32	−0.0808181
$937$	8982.08	0.313161	0.156580	−	0.987665i	$-0.449953\pi$
$937$	8982.08	0.313161	0.156580	+	0.987665i	$0.449953\pi$
$938$	−20790.6	−0.723706
$939$	2610.52	0.0907253
$940$	−471.987	−0.0163771
$941$	31708.0	1.09846	0.549231	−	0.835671i	$-0.314921\pi$
$941$	31708.0	1.09846	0.549231	+	0.835671i	$0.314921\pi$
$942$	−9591.08	−0.331735
$943$	17438.1	0.602187
$944$	2880.47	0.0993127
$945$	−13021.8	−0.448254
$946$	5553.53	0.190868
$947$	−2027.14	−0.0695600	−0.0347800	−	0.999395i	$-0.511073\pi$
$947$	−2027.14	−0.0695600	−0.0347800	+	0.999395i	$0.511073\pi$
$948$	3300.65	0.113080
$949$	−20943.1	−0.716376
$950$	−865.775	−0.0295679
$951$	−33874.1	−1.15504
$952$	−33155.4	−1.12875
$953$	−19599.1	−0.666188	−0.333094	−	0.942894i	$-0.608093\pi$
$953$	−19599.1	−0.666188	−0.333094	+	0.942894i	$0.608093\pi$
$954$	−2616.12	−0.0887842
$955$	13044.6	0.442004
$956$	−787.101	−0.0266283
$957$	12380.5	0.418185
$958$	24216.2	0.816691
$959$	−35229.0	−1.18624
$960$	9794.26	0.329280
$961$	55733.6	1.87082
$962$	−5198.53	−0.174228
$963$	−3185.36	−0.106591
$964$	8632.47	0.288416
$965$	−10075.5	−0.336107
$966$	40044.1	1.33375
$967$	35843.1	1.19197	0.595985	−	0.802996i	$-0.296762\pi$
$967$	35843.1	1.19197	0.595985	+	0.802996i	$0.296762\pi$
$968$	2796.03	0.0928386
$969$	−7549.57	−0.250286
$970$	−164.319	−0.00543915
$971$	45040.8	1.48860	0.744299	−	0.667847i	$-0.232784\pi$
$971$	45040.8	1.48860	0.744299	+	0.667847i	$0.232784\pi$
$972$	3644.77	0.120274
$973$	49597.5	1.63414
$974$	−25744.5	−0.846927
$975$	−4902.20	−0.161022
$976$	−2288.06	−0.0750401
$977$	24854.0	0.813870	0.406935	−	0.913457i	$-0.366598\pi$
$977$	24854.0	0.813870	0.406935	+	0.913457i	$0.366598\pi$
$978$	26872.4	0.878614
$979$	9303.68	0.303725
$980$	−1062.17	−0.0346223
$981$	−3963.10	−0.128983
$982$	−923.253	−0.0300022
$983$	−19643.9	−0.637380	−0.318690	−	0.947859i	$-0.603243\pi$
$983$	−19643.9	−0.637380	−0.318690	+	0.947859i	$0.603243\pi$
$984$	−10767.4	−0.348833
$985$	−11331.8	−0.366561
$986$	27364.6	0.883840
$987$	2170.61	0.0700013
$988$	3193.23	0.102824
$989$	56575.4	1.81900
$990$	−279.452	−0.00897129
$991$	−46967.4	−1.50552	−0.752759	−	0.658296i	$-0.771277\pi$
$991$	−46967.4	−1.50552	−0.752759	+	0.658296i	$0.771277\pi$
$992$	51559.1	1.65020
$993$	−10682.1	−0.341377
$994$	22916.4	0.731251
$995$	7095.90	0.226086
$996$	21956.6	0.698517
$997$	31784.1	1.00964	0.504821	−	0.863224i	$-0.331558\pi$
$997$	31784.1	1.00964	0.504821	+	0.863224i	$0.331558\pi$
$998$	−20146.3	−0.638998
$999$	10490.4	0.332233

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.b.1.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.b.1.8	✓	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)

\(f(q)\)	\(=\)	\(q-1.82269 q^{2} +5.45780 q^{3} -4.67782 q^{4} +5.00000 q^{5} -9.94786 q^{6} +19.7082 q^{7} +23.1077 q^{8} +2.78762 q^{9} +O(q^{10})\)	\(q-1.82269 q^{2} +5.45780 q^{3} -4.67782 q^{4} +5.00000 q^{5} -9.94786 q^{6} +19.7082 q^{7} +23.1077 q^{8} +2.78762 q^{9} -9.11343 q^{10} +11.0000 q^{11} -25.5306 q^{12} -35.9280 q^{13} -35.9218 q^{14} +27.2890 q^{15} -4.69546 q^{16} -72.8033 q^{17} -5.08095 q^{18} +19.0000 q^{19} -23.3891 q^{20} +107.563 q^{21} -20.0495 q^{22} -204.250 q^{23} +126.117 q^{24} +25.0000 q^{25} +65.4854 q^{26} -132.146 q^{27} -92.1914 q^{28} +206.218 q^{29} -49.7393 q^{30} -292.446 q^{31} -176.303 q^{32} +60.0358 q^{33} +132.697 q^{34} +98.5410 q^{35} -13.0400 q^{36} -79.3846 q^{37} -34.6310 q^{38} -196.088 q^{39} +115.538 q^{40} -85.3761 q^{41} -196.054 q^{42} -276.991 q^{43} -51.4560 q^{44} +13.9381 q^{45} +372.284 q^{46} +20.1798 q^{47} -25.6269 q^{48} +45.4131 q^{49} -45.5671 q^{50} -397.346 q^{51} +168.065 q^{52} +514.889 q^{53} +240.861 q^{54} +55.0000 q^{55} +455.411 q^{56} +103.698 q^{57} -375.870 q^{58} -613.457 q^{59} -127.653 q^{60} +487.292 q^{61} +533.037 q^{62} +54.9390 q^{63} +358.909 q^{64} -179.640 q^{65} -109.426 q^{66} +578.772 q^{67} +340.560 q^{68} -1114.76 q^{69} -179.609 q^{70} -637.951 q^{71} +64.4154 q^{72} +582.918 q^{73} +144.693 q^{74} +136.445 q^{75} -88.8786 q^{76} +216.790 q^{77} +357.407 q^{78} -129.282 q^{79} -23.4773 q^{80} -796.495 q^{81} +155.614 q^{82} -860.012 q^{83} -503.162 q^{84} -364.016 q^{85} +504.867 q^{86} +1125.50 q^{87} +254.184 q^{88} +845.789 q^{89} -25.4048 q^{90} -708.076 q^{91} +955.445 q^{92} -1596.11 q^{93} -36.7814 q^{94} +95.0000 q^{95} -962.227 q^{96} +18.0305 q^{97} -82.7739 q^{98} +30.6638 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 12 q^{2} - 21 q^{3} + 72 q^{4} + 100 q^{5} - 45 q^{6} - 131 q^{7} - 108 q^{8} + 159 q^{9}+O(q^{10}) \) 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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