LMFDB - Embedded newform 1045.4.a.c.1.5

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} - x^{19} - 105 x^{18} + 103 x^{17} + 4500 x^{16} - 4345 x^{15} - 101844 x^{14} + 95592 x^{13} + \cdots + 150528 $ x^20 - x^19 - 105x^18 + 103x^17 + 4500x^16 - 4345x^15 - 101844x^14 + 95592x^13 + 1317797x^12 - 1160501x^11 - 9914845x^10 + 7570653x^9 + 42786958x^8 - 23777633x^7 - 102801526x^6 + 28436356x^5 + 122325928x^4 + 411232x^3 - 47350496x^2 - 4782848x + 150528
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.81607$ 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)

$f(q)$	$=$	$q-2.81607 q^{2} +3.33662 q^{3} -0.0697562 q^{4} -5.00000 q^{5} -9.39614 q^{6} +35.3821 q^{7} +22.7250 q^{8} -15.8670 q^{9} +O(q^{10})$	\(q-2.81607 q^{2} +3.33662 q^{3} -0.0697562 q^{4} -5.00000 q^{5} -9.39614 q^{6} +35.3821 q^{7} +22.7250 q^{8} -15.8670 q^{9} +14.0803 q^{10} -11.0000 q^{11} -0.232750 q^{12} +16.5610 q^{13} -99.6385 q^{14} -16.6831 q^{15} -63.4371 q^{16} -35.7214 q^{17} +44.6825 q^{18} +19.0000 q^{19} +0.348781 q^{20} +118.057 q^{21} +30.9768 q^{22} -96.3766 q^{23} +75.8246 q^{24} +25.0000 q^{25} -46.6369 q^{26} -143.031 q^{27} -2.46812 q^{28} -163.372 q^{29} +46.9807 q^{30} +130.338 q^{31} -3.15672 q^{32} -36.7028 q^{33} +100.594 q^{34} -176.911 q^{35} +1.10682 q^{36} +7.26988 q^{37} -53.5053 q^{38} +55.2577 q^{39} -113.625 q^{40} -315.838 q^{41} -332.456 q^{42} -250.799 q^{43} +0.767318 q^{44} +79.3350 q^{45} +271.403 q^{46} -280.397 q^{47} -211.665 q^{48} +908.895 q^{49} -70.4017 q^{50} -119.189 q^{51} -1.15523 q^{52} +504.085 q^{53} +402.784 q^{54} +55.0000 q^{55} +804.059 q^{56} +63.3957 q^{57} +460.066 q^{58} +49.5715 q^{59} +1.16375 q^{60} +360.508 q^{61} -367.039 q^{62} -561.408 q^{63} +516.386 q^{64} -82.8050 q^{65} +103.358 q^{66} -370.276 q^{67} +2.49179 q^{68} -321.572 q^{69} +498.193 q^{70} +309.651 q^{71} -360.577 q^{72} +42.6753 q^{73} -20.4725 q^{74} +83.4154 q^{75} -1.32537 q^{76} -389.203 q^{77} -155.610 q^{78} -771.511 q^{79} +317.185 q^{80} -48.8298 q^{81} +889.423 q^{82} -363.907 q^{83} -8.23518 q^{84} +178.607 q^{85} +706.268 q^{86} -545.109 q^{87} -249.975 q^{88} +0.664243 q^{89} -223.413 q^{90} +585.964 q^{91} +6.72286 q^{92} +434.886 q^{93} +789.617 q^{94} -95.0000 q^{95} -10.5327 q^{96} -573.501 q^{97} -2559.51 q^{98} +174.537 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) $ 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.81607	−0.995631	−0.497815	−	0.867283i	$-0.665864\pi$
$2$	−2.81607	−0.995631	−0.497815	+	0.867283i	$0.665864\pi$
$3$	3.33662	0.642132	0.321066	−	0.947057i	$-0.395959\pi$
$3$	3.33662	0.642132	0.321066	+	0.947057i	$0.395959\pi$
$4$	−0.0697562	−0.00871953
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	−9.39614	−0.639326
$7$	35.3821	1.91046	0.955228	−	0.295871i	$-0.0956099\pi$
$7$	35.3821	1.91046	0.955228	+	0.295871i	$0.0956099\pi$
$8$	22.7250	1.00431
$9$	−15.8670	−0.587666
$10$	14.0803	0.445260
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	−0.232750	−0.00559909
$13$	16.5610	0.353323	0.176661	−	0.984272i	$-0.443470\pi$
$13$	16.5610	0.353323	0.176661	+	0.984272i	$0.443470\pi$
$14$	−99.6385	−1.90211
$15$	−16.6831	−0.287170
$16$	−63.4371	−0.991204
$17$	−35.7214	−0.509631	−0.254815	−	0.966990i	$-0.582015\pi$
$17$	−35.7214	−0.509631	−0.254815	+	0.966990i	$0.582015\pi$
$18$	44.6825	0.585099
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0.348781	0.00389949
$21$	118.057	1.22676
$22$	30.9768	0.300194
$23$	−96.3766	−0.873735	−0.436868	−	0.899526i	$-0.643912\pi$
$23$	−96.3766	−0.873735	−0.436868	+	0.899526i	$0.643912\pi$
$24$	75.8246	0.644901
$25$	25.0000	0.200000
$26$	−46.6369	−0.351779
$27$	−143.031	−1.01949
$28$	−2.46812	−0.0166583
$29$	−163.372	−1.04612	−0.523058	−	0.852297i	$-0.675209\pi$
$29$	−163.372	−1.04612	−0.523058	+	0.852297i	$0.675209\pi$
$30$	46.9807	0.285915
$31$	130.338	0.755139	0.377570	−	0.925981i	$-0.376760\pi$
$31$	130.338	0.755139	0.377570	+	0.925981i	$0.376760\pi$
$32$	−3.15672	−0.0174386
$33$	−36.7028	−0.193610
$34$	100.594	0.507404
$35$	−176.911	−0.854382
$36$	1.10682	0.00512417
$37$	7.26988	0.0323016	0.0161508	−	0.999870i	$-0.494859\pi$
$37$	7.26988	0.0323016	0.0161508	+	0.999870i	$0.494859\pi$
$38$	−53.5053	−0.228413
$39$	55.2577	0.226880
$40$	−113.625	−0.449142
$41$	−315.838	−1.20307	−0.601533	−	0.798848i	$-0.705443\pi$
$41$	−315.838	−1.20307	−0.601533	+	0.798848i	$0.705443\pi$
$42$	−332.456	−1.22140
$43$	−250.799	−0.889454	−0.444727	−	0.895666i	$-0.646699\pi$
$43$	−250.799	−0.889454	−0.444727	+	0.895666i	$0.646699\pi$
$44$	0.767318	0.00262904
$45$	79.3350	0.262812
$46$	271.403	0.869917
$47$	−280.397	−0.870214	−0.435107	−	0.900379i	$-0.643290\pi$
$47$	−280.397	−0.870214	−0.435107	+	0.900379i	$0.643290\pi$
$48$	−211.665	−0.636484
$49$	908.895	2.64984
$50$	−70.4017	−0.199126
$51$	−119.189	−0.327250
$52$	−1.15523	−0.00308081
$53$	504.085	1.30644	0.653221	−	0.757167i	$-0.273417\pi$
$53$	504.085	1.30644	0.653221	+	0.757167i	$0.273417\pi$
$54$	402.784	1.01504
$55$	55.0000	0.134840
$56$	804.059	1.91869
$57$	63.3957	0.147315
$58$	460.066	1.04154
$59$	49.5715	0.109384	0.0546921	−	0.998503i	$-0.482582\pi$
$59$	49.5715	0.109384	0.0546921	+	0.998503i	$0.482582\pi$
$60$	1.16375	0.00250399
$61$	360.508	0.756695	0.378347	−	0.925664i	$-0.376493\pi$
$61$	360.508	0.756695	0.378347	+	0.925664i	$0.376493\pi$
$62$	−367.039	−0.751840
$63$	−561.408	−1.12271
$64$	516.386	1.00857
$65$	−82.8050	−0.158011
$66$	103.358	0.192764
$67$	−370.276	−0.675170	−0.337585	−	0.941295i	$-0.609610\pi$
$67$	−370.276	−0.675170	−0.337585	+	0.941295i	$0.609610\pi$
$68$	2.49179	0.00444374
$69$	−321.572	−0.561053
$70$	498.193	0.850649
$71$	309.651	0.517590	0.258795	−	0.965932i	$-0.416675\pi$
$71$	309.651	0.517590	0.258795	+	0.965932i	$0.416675\pi$
$72$	−360.577	−0.590200
$73$	42.6753	0.0684214	0.0342107	−	0.999415i	$-0.489108\pi$
$73$	42.6753	0.0684214	0.0342107	+	0.999415i	$0.489108\pi$
$74$	−20.4725	−0.0321605
$75$	83.4154	0.128426
$76$	−1.32537	−0.00200040
$77$	−389.203	−0.576024
$78$	−155.610	−0.225889
$79$	−771.511	−1.09876	−0.549378	−	0.835574i	$-0.685135\pi$
$79$	−771.511	−1.09876	−0.549378	+	0.835574i	$0.685135\pi$
$80$	317.185	0.443280
$81$	−48.8298	−0.0669819
$82$	889.423	1.19781
$83$	−363.907	−0.481253	−0.240626	−	0.970618i	$-0.577353\pi$
$83$	−363.907	−0.481253	−0.240626	+	0.970618i	$0.577353\pi$
$84$	−8.23518	−0.0106968
$85$	178.607	0.227914
$86$	706.268	0.885568
$87$	−545.109	−0.671744
$88$	−249.975	−0.302811
$89$	0.664243	0.000791119 0	0.000395559	−	1.00000i	$-0.499874\pi$
$89$	0.664243	0.000791119 0	0.000395559		1.00000i	$0.499874\pi$
$90$	−223.413	−0.261664
$91$	585.964	0.675008
$92$	6.72286	0.00761855
$93$	434.886	0.484899
$94$	789.617	0.866412
$95$	−95.0000	−0.102598
$96$	−10.5327	−0.0111979
$97$	−573.501	−0.600311	−0.300156	−	0.953890i	$-0.597039\pi$
$97$	−573.501	−0.600311	−0.300156	+	0.953890i	$0.597039\pi$
$98$	−2559.51	−2.63826
$99$	174.537	0.177188
$100$	−1.74391	−0.00174391
$101$	−970.755	−0.956374	−0.478187	−	0.878258i	$-0.658706\pi$
$101$	−970.755	−0.956374	−0.478187	+	0.878258i	$0.658706\pi$
$102$	335.644	0.325820
$103$	425.291	0.406846	0.203423	−	0.979091i	$-0.434793\pi$
$103$	425.291	0.406846	0.203423	+	0.979091i	$0.434793\pi$
$104$	376.349	0.354846
$105$	−590.283	−0.548626
$106$	−1419.54	−1.30073
$107$	905.406	0.818027	0.409014	−	0.912528i	$-0.365873\pi$
$107$	905.406	0.818027	0.409014	+	0.912528i	$0.365873\pi$
$108$	9.97728	0.00888948
$109$	−827.160	−0.726858	−0.363429	−	0.931622i	$-0.618394\pi$
$109$	−827.160	−0.726858	−0.363429	+	0.931622i	$0.618394\pi$
$110$	−154.884	−0.134251
$111$	24.2568	0.0207419
$112$	−2244.54	−1.89365
$113$	−2239.00	−1.86396	−0.931980	−	0.362510i	$-0.881920\pi$
$113$	−2239.00	−1.86396	−0.931980	+	0.362510i	$0.881920\pi$
$114$	−178.527	−0.146672
$115$	481.883	0.390746
$116$	11.3962	0.00912163
$117$	−262.773	−0.207636
$118$	−139.597	−0.108906
$119$	−1263.90	−0.973627
$120$	−379.123	−0.288409
$121$	121.000	0.0909091
$122$	−1015.22	−0.753388
$123$	−1053.83	−0.772527
$124$	−9.09185	−0.00658445
$125$	−125.000	−0.0894427
$126$	1580.96	1.11781
$127$	−26.8256	−0.0187432	−0.00937161	−	0.999956i	$-0.502983\pi$
$127$	−26.8256	−0.0187432	−0.00937161	+	0.999956i	$0.502983\pi$
$128$	−1428.93	−0.986722
$129$	−836.821	−0.571147
$130$	233.185	0.157320
$131$	1073.61	0.716047	0.358023	−	0.933713i	$-0.383451\pi$
$131$	1073.61	0.716047	0.358023	+	0.933713i	$0.383451\pi$
$132$	2.56025	0.00168819
$133$	672.261	0.438289
$134$	1042.72	0.672220
$135$	715.154	0.455930
$136$	−811.769	−0.511828
$137$	720.676	0.449427	0.224713	−	0.974425i	$-0.427855\pi$
$137$	720.676	0.449427	0.224713	+	0.974425i	$0.427855\pi$
$138$	905.568	0.558602
$139$	−107.451	−0.0655674	−0.0327837	−	0.999462i	$-0.510437\pi$
$139$	−107.451	−0.0655674	−0.0327837	+	0.999462i	$0.510437\pi$
$140$	12.3406	0.00744980
$141$	−935.577	−0.558793
$142$	−872.000	−0.515328
$143$	−182.171	−0.106531
$144$	1006.56	0.582497
$145$	816.858	0.467837
$146$	−120.177	−0.0681225
$147$	3032.64	1.70155
$148$	−0.507119	−0.000281655 0
$149$	1961.09	1.07825	0.539123	−	0.842227i	$-0.318756\pi$
$149$	1961.09	1.07825	0.539123	+	0.842227i	$0.318756\pi$
$150$	−234.904	−0.127865
$151$	902.642	0.486463	0.243232	−	0.969968i	$-0.421792\pi$
$151$	902.642	0.486463	0.243232	+	0.969968i	$0.421792\pi$
$152$	431.775	0.230405
$153$	566.792	0.299493
$154$	1096.02	0.573507
$155$	−651.688	−0.337708
$156$	−3.85457	−0.00197829
$157$	−2620.69	−1.33219	−0.666096	−	0.745866i	$-0.732036\pi$
$157$	−2620.69	−1.33219	−0.666096	+	0.745866i	$0.732036\pi$
$158$	2172.63	1.09396
$159$	1681.94	0.838908
$160$	15.7836	0.00779876
$161$	−3410.01	−1.66923
$162$	137.508	0.0666892
$163$	887.061	0.426258	0.213129	−	0.977024i	$-0.431635\pi$
$163$	887.061	0.426258	0.213129	+	0.977024i	$0.431635\pi$
$164$	22.0317	0.0104902
$165$	183.514	0.0865851
$166$	1024.79	0.479150
$167$	−2935.44	−1.36019	−0.680093	−	0.733125i	$-0.738061\pi$
$167$	−2935.44	−1.36019	−0.680093	+	0.733125i	$0.738061\pi$
$168$	2682.84	1.23205
$169$	−1922.73	−0.875163
$170$	−502.970	−0.226918
$171$	−301.473	−0.134820
$172$	17.4948	0.00775562
$173$	−2867.13	−1.26002	−0.630010	−	0.776587i	$-0.716949\pi$
$173$	−2867.13	−1.26002	−0.630010	+	0.776587i	$0.716949\pi$
$174$	1535.06	0.668809
$175$	884.553	0.382091
$176$	697.808	0.298859
$177$	165.401	0.0702390
$178$	−1.87055	−0.000787662 0
$179$	−4338.27	−1.81150	−0.905748	−	0.423816i	$-0.860690\pi$
$179$	−4338.27	−1.81150	−0.905748	+	0.423816i	$0.860690\pi$
$180$	−5.53411	−0.00229160
$181$	275.587	0.113172	0.0565862	−	0.998398i	$-0.481978\pi$
$181$	275.587	0.113172	0.0565862	+	0.998398i	$0.481978\pi$
$182$	−1650.11	−0.672058
$183$	1202.88	0.485898
$184$	−2190.16	−0.877503
$185$	−36.3494	−0.0144457
$186$	−1224.67	−0.482780
$187$	392.936	0.153659
$188$	19.5594	0.00758786
$189$	−5060.73	−1.94769
$190$	267.527	0.102150
$191$	731.067	0.276954	0.138477	−	0.990366i	$-0.455779\pi$
$191$	731.067	0.276954	0.138477	+	0.990366i	$0.455779\pi$
$192$	1722.98	0.647633
$193$	1773.81	0.661564	0.330782	−	0.943707i	$-0.392688\pi$
$193$	1773.81	0.661564	0.330782	+	0.943707i	$0.392688\pi$
$194$	1615.02	0.597688
$195$	−276.289	−0.101464
$196$	−63.4011	−0.0231054
$197$	1060.34	0.383484	0.191742	−	0.981445i	$-0.438586\pi$
$197$	1060.34	0.383484	0.191742	+	0.981445i	$0.438586\pi$
$198$	−491.508	−0.176414
$199$	−4735.68	−1.68695	−0.843476	−	0.537167i	$-0.819494\pi$
$199$	−4735.68	−1.68695	−0.843476	+	0.537167i	$0.819494\pi$
$200$	568.125	0.200862
$201$	−1235.47	−0.433549
$202$	2733.71	0.952195
$203$	−5780.44	−1.99856
$204$	8.31415	0.00285347
$205$	1579.19	0.538027
$206$	−1197.65	−0.405068
$207$	1529.21	0.513465
$208$	−1050.58	−0.350215
$209$	−209.000	−0.0691714
$210$	1662.28	0.546229
$211$	−131.852	−0.0430192	−0.0215096	−	0.999769i	$-0.506847\pi$
$211$	−131.852	−0.0430192	−0.0215096	+	0.999769i	$0.506847\pi$
$212$	−35.1631	−0.0113916
$213$	1033.19	0.332361
$214$	−2549.69	−0.814453
$215$	1254.00	0.397776
$216$	−3250.37	−1.02389
$217$	4611.62	1.44266
$218$	2329.34	0.723682
$219$	142.391	0.0439356
$220$	−3.83659	−0.00117574
$221$	−591.583	−0.180064
$222$	−68.3088	−0.0206513
$223$	15.1854	0.00456006	0.00228003	−	0.999997i	$-0.499274\pi$
$223$	15.1854	0.00456006	0.00228003	+	0.999997i	$0.499274\pi$
$224$	−111.691	−0.0333156
$225$	−396.675	−0.117533
$226$	6305.18	1.85582
$227$	−1126.42	−0.329352	−0.164676	−	0.986348i	$-0.552658\pi$
$227$	−1126.42	−0.329352	−0.164676	+	0.986348i	$0.552658\pi$
$228$	−4.42224	−0.00128452
$229$	−3232.93	−0.932918	−0.466459	−	0.884543i	$-0.654470\pi$
$229$	−3232.93	−0.932918	−0.466459	+	0.884543i	$0.654470\pi$
$230$	−1357.02	−0.389039
$231$	−1298.62	−0.369884
$232$	−3712.62	−1.05063
$233$	−3006.01	−0.845195	−0.422598	−	0.906317i	$-0.638882\pi$
$233$	−3006.01	−0.845195	−0.422598	+	0.906317i	$0.638882\pi$
$234$	739.988	0.206729
$235$	1401.98	0.389172
$236$	−3.45792	−0.000953777 0
$237$	−2574.24	−0.705547
$238$	3559.23	0.969373
$239$	−923.389	−0.249912	−0.124956	−	0.992162i	$-0.539879\pi$
$239$	−923.389	−0.249912	−0.124956	+	0.992162i	$0.539879\pi$
$240$	1058.33	0.284644
$241$	−5944.82	−1.58896	−0.794480	−	0.607290i	$-0.792257\pi$
$241$	−5944.82	−1.58896	−0.794480	+	0.607290i	$0.792257\pi$
$242$	−340.744	−0.0905119
$243$	3698.90	0.976480
$244$	−25.1477	−0.00659802
$245$	−4544.48	−1.18504
$246$	2967.66	0.769152
$247$	314.659	0.0810578
$248$	2961.92	0.758395
$249$	−1214.22	−0.309028
$250$	352.009	0.0890519
$251$	−3644.44	−0.916474	−0.458237	−	0.888830i	$-0.651519\pi$
$251$	−3644.44	−0.916474	−0.458237	+	0.888830i	$0.651519\pi$
$252$	39.1617	0.00978950
$253$	1060.14	0.263441
$254$	75.5428	0.0186613
$255$	595.944	0.146351
$256$	−107.137	−0.0261566
$257$	3492.93	0.847794	0.423897	−	0.905710i	$-0.360662\pi$
$257$	3492.93	0.847794	0.423897	+	0.905710i	$0.360662\pi$
$258$	2356.55	0.568652
$259$	257.224	0.0617108
$260$	5.77617	0.00137778
$261$	2592.22	0.614767
$262$	−3023.37	−0.712918
$263$	−4129.23	−0.968134	−0.484067	−	0.875031i	$-0.660841\pi$
$263$	−4129.23	−0.968134	−0.484067	+	0.875031i	$0.660841\pi$
$264$	−834.070	−0.194445
$265$	−2520.43	−0.584259
$266$	−1893.13	−0.436374
$267$	2.21632	0.000508003 0
$268$	25.8291	0.00588717
$269$	−6512.16	−1.47603	−0.738017	−	0.674782i	$-0.764238\pi$
$269$	−6512.16	−1.47603	−0.738017	+	0.674782i	$0.764238\pi$
$270$	−2013.92	−0.453938
$271$	2674.98	0.599607	0.299803	−	0.954001i	$-0.403079\pi$
$271$	2674.98	0.599607	0.299803	+	0.954001i	$0.403079\pi$
$272$	2266.06	0.505148
$273$	1955.14	0.433444
$274$	−2029.47	−0.447463
$275$	−275.000	−0.0603023
$276$	22.4316	0.00489212
$277$	−5144.41	−1.11588	−0.557938	−	0.829883i	$-0.688407\pi$
$277$	−5144.41	−1.11588	−0.557938	+	0.829883i	$0.688407\pi$
$278$	302.589	0.0652809
$279$	−2068.06	−0.443770
$280$	−4020.29	−0.858066
$281$	6593.16	1.39970	0.699849	−	0.714291i	$-0.253251\pi$
$281$	6593.16	1.39970	0.699849	+	0.714291i	$0.253251\pi$
$282$	2634.65	0.556351
$283$	6742.17	1.41619	0.708093	−	0.706119i	$-0.249556\pi$
$283$	6742.17	1.41619	0.708093	+	0.706119i	$0.249556\pi$
$284$	−21.6001	−0.00451314
$285$	−316.979	−0.0658814
$286$	513.006	0.106065
$287$	−11175.0	−2.29840
$288$	50.0876	0.0102480
$289$	−3636.98	−0.740277
$290$	−2300.33	−0.465793
$291$	−1913.55	−0.385479
$292$	−2.97687	−0.000596602 0
$293$	514.102	0.102506	0.0512529	−	0.998686i	$-0.483679\pi$
$293$	514.102	0.102506	0.0512529	+	0.998686i	$0.483679\pi$
$294$	−8540.11	−1.69411
$295$	−247.858	−0.0489181
$296$	165.208	0.0324409
$297$	1573.34	0.307388
$298$	−5522.56	−1.07353
$299$	−1596.09	−0.308711
$300$	−5.81874	−0.00111982
$301$	−8873.81	−1.69926
$302$	−2541.90	−0.484338
$303$	−3239.04	−0.614118
$304$	−1205.30	−0.227398
$305$	−1802.54	−0.338404
$306$	−1596.12	−0.298184
$307$	−2887.23	−0.536751	−0.268376	−	0.963314i	$-0.586487\pi$
$307$	−2887.23	−0.536751	−0.268376	+	0.963314i	$0.586487\pi$
$308$	27.1494	0.00502266
$309$	1419.03	0.261249
$310$	1835.20	0.336233
$311$	6666.03	1.21542	0.607711	−	0.794158i	$-0.292088\pi$
$311$	6666.03	1.21542	0.607711	+	0.794158i	$0.292088\pi$
$312$	1255.73	0.227858
$313$	6935.87	1.25252	0.626260	−	0.779614i	$-0.284585\pi$
$313$	6935.87	1.25252	0.626260	+	0.779614i	$0.284585\pi$
$314$	7380.06	1.32637
$315$	2807.04	0.502091
$316$	53.8177	0.00958064
$317$	−2548.21	−0.451488	−0.225744	−	0.974187i	$-0.572481\pi$
$317$	−2548.21	−0.451488	−0.225744	+	0.974187i	$0.572481\pi$
$318$	−4736.46	−0.835243
$319$	1797.09	0.315416
$320$	−2581.93	−0.451045
$321$	3020.99	0.525282
$322$	9602.82	1.66194
$323$	−678.707	−0.116917
$324$	3.40618	0.000584050 0
$325$	414.025	0.0706646
$326$	−2498.02	−0.424395
$327$	−2759.91	−0.466739
$328$	−7177.43	−1.20825
$329$	−9921.04	−1.66251
$330$	−516.788	−0.0862068
$331$	2780.92	0.461792	0.230896	−	0.972978i	$-0.425834\pi$
$331$	2780.92	0.461792	0.230896	+	0.972978i	$0.425834\pi$
$332$	25.3848	0.00419630
$333$	−115.351	−0.0189826
$334$	8266.40	1.35424
$335$	1851.38	0.301945
$336$	−7489.17	−1.21597
$337$	−10910.3	−1.76357	−0.881783	−	0.471655i	$-0.843657\pi$
$337$	−10910.3	−1.76357	−0.881783	+	0.471655i	$0.843657\pi$
$338$	5414.55	0.871339
$339$	−7470.69	−1.19691
$340$	−12.4590	−0.00198730
$341$	−1433.71	−0.227683
$342$	848.968	0.134231
$343$	20022.6	3.15195
$344$	−5699.41	−0.893290
$345$	1607.86	0.250911
$346$	8074.02	1.25452
$347$	−6518.77	−1.00849	−0.504245	−	0.863561i	$-0.668229\pi$
$347$	−6518.77	−1.00849	−0.504245	+	0.863561i	$0.668229\pi$
$348$	38.0247	0.00585729
$349$	−4855.11	−0.744665	−0.372333	−	0.928099i	$-0.621442\pi$
$349$	−4855.11	−0.744665	−0.372333	+	0.928099i	$0.621442\pi$
$350$	−2490.96	−0.380422
$351$	−2368.73	−0.360210
$352$	34.7239	0.00525792
$353$	4649.09	0.700981	0.350491	−	0.936566i	$-0.386015\pi$
$353$	4649.09	0.700981	0.350491	+	0.936566i	$0.386015\pi$
$354$	−465.781	−0.0699321
$355$	−1548.26	−0.231473
$356$	−0.0463350	−6.89818e−6 0
$357$	−4217.15	−0.625197
$358$	12216.9	1.80358
$359$	−2432.15	−0.357559	−0.178780	−	0.983889i	$-0.557215\pi$
$359$	−2432.15	−0.357559	−0.178780	+	0.983889i	$0.557215\pi$
$360$	1802.89	0.263946
$361$	361.000	0.0526316
$362$	−776.071	−0.112678
$363$	403.731	0.0583756
$364$	−40.8746	−0.00588575
$365$	−213.376	−0.0305990
$366$	−3387.39	−0.483775
$367$	−7051.07	−1.00290	−0.501448	−	0.865188i	$-0.667199\pi$
$367$	−7051.07	−1.00290	−0.501448	+	0.865188i	$0.667199\pi$
$368$	6113.85	0.866050
$369$	5011.41	0.707001
$370$	102.362	0.0143826
$371$	17835.6	2.49590
$372$	−30.3360	−0.00422809
$373$	−6658.95	−0.924363	−0.462182	−	0.886785i	$-0.652933\pi$
$373$	−6658.95	−0.924363	−0.462182	+	0.886785i	$0.652933\pi$
$374$	−1106.53	−0.152988
$375$	−417.077	−0.0574340
$376$	−6372.01	−0.873967
$377$	−2705.60	−0.369617
$378$	14251.4	1.93918
$379$	9402.70	1.27437	0.637183	−	0.770713i	$-0.280100\pi$
$379$	9402.70	1.27437	0.637183	+	0.770713i	$0.280100\pi$
$380$	6.62684	0.000894604 0
$381$	−89.5068	−0.0120356
$382$	−2058.73	−0.275743
$383$	8799.50	1.17398	0.586989	−	0.809595i	$-0.300313\pi$
$383$	8799.50	1.17398	0.586989	+	0.809595i	$0.300313\pi$
$384$	−4767.78	−0.633606
$385$	1946.02	0.257606
$386$	−4995.18	−0.658673
$387$	3979.43	0.522702
$388$	40.0052	0.00523443
$389$	−7090.90	−0.924223	−0.462112	−	0.886822i	$-0.652908\pi$
$389$	−7090.90	−0.924223	−0.462112	+	0.886822i	$0.652908\pi$
$390$	778.048	0.101020
$391$	3442.71	0.445282
$392$	20654.6	2.66127
$393$	3582.24	0.459796
$394$	−2986.00	−0.381809
$395$	3857.56	0.491379
$396$	−12.1750	−0.00154500
$397$	−1343.40	−0.169832	−0.0849159	−	0.996388i	$-0.527062\pi$
$397$	−1343.40	−0.169832	−0.0849159	+	0.996388i	$0.527062\pi$
$398$	13336.0	1.67958
$399$	2243.08	0.281439
$400$	−1585.93	−0.198241
$401$	−1765.83	−0.219904	−0.109952	−	0.993937i	$-0.535070\pi$
$401$	−1765.83	−0.219904	−0.109952	+	0.993937i	$0.535070\pi$
$402$	3479.17	0.431654
$403$	2158.52	0.266808
$404$	67.7162	0.00833913
$405$	244.149	0.0299552
$406$	16278.1	1.98983
$407$	−79.9687	−0.00973931
$408$	−2708.56	−0.328661
$409$	11987.7	1.44927	0.724636	−	0.689131i	$-0.242008\pi$
$409$	11987.7	1.44927	0.724636	+	0.689131i	$0.242008\pi$
$410$	−4447.11	−0.535676
$411$	2404.62	0.288591
$412$	−29.6667	−0.00354750
$413$	1753.95	0.208973
$414$	−4306.35	−0.511221
$415$	1819.53	0.215223
$416$	−52.2784	−0.00616144
$417$	−358.523	−0.0421029
$418$	588.558	0.0688692
$419$	14960.5	1.74432	0.872160	−	0.489220i	$-0.162719\pi$
$419$	14960.5	1.74432	0.872160	+	0.489220i	$0.162719\pi$
$420$	41.1759	0.00478376
$421$	9875.94	1.14329	0.571643	−	0.820502i	$-0.306306\pi$
$421$	9875.94	1.14329	0.571643	+	0.820502i	$0.306306\pi$
$422$	371.304	0.0428312
$423$	4449.05	0.511396
$424$	11455.3	1.31208
$425$	−893.036	−0.101926
$426$	−2909.53	−0.330909
$427$	12755.6	1.44563
$428$	−63.1577	−0.00713281
$429$	−607.835	−0.0684069
$430$	−3531.34	−0.396038
$431$	11892.2	1.32907	0.664535	−	0.747257i	$-0.268630\pi$
$431$	11892.2	1.32907	0.664535	+	0.747257i	$0.268630\pi$
$432$	9073.45	1.01052
$433$	−2273.84	−0.252364	−0.126182	−	0.992007i	$-0.540272\pi$
$433$	−2273.84	−0.252364	−0.126182	+	0.992007i	$0.540272\pi$
$434$	−12986.6	−1.43636
$435$	2725.54	0.300413
$436$	57.6995	0.00633786
$437$	−1831.15	−0.200449
$438$	−400.983	−0.0437436
$439$	−10428.2	−1.13374	−0.566870	−	0.823808i	$-0.691846\pi$
$439$	−10428.2	−1.13374	−0.566870	+	0.823808i	$0.691846\pi$
$440$	1249.87	0.135421
$441$	−14421.4	−1.55722
$442$	1665.94	0.179277
$443$	3826.67	0.410407	0.205204	−	0.978719i	$-0.434214\pi$
$443$	3826.67	0.410407	0.205204	+	0.978719i	$0.434214\pi$
$444$	−1.69206	−0.000180860 0
$445$	−3.32121	−0.000353799 0
$446$	−42.7633	−0.00454013
$447$	6543.40	0.692376
$448$	18270.8	1.92682
$449$	−6096.21	−0.640753	−0.320376	−	0.947290i	$-0.603809\pi$
$449$	−6096.21	−0.640753	−0.320376	+	0.947290i	$0.603809\pi$
$450$	1117.06	0.117020
$451$	3474.22	0.362738
$452$	156.184	0.0162528
$453$	3011.77	0.312374
$454$	3172.07	0.327913
$455$	−2929.82	−0.301873
$456$	1440.67	0.147950
$457$	−199.059	−0.0203755	−0.0101877	−	0.999948i	$-0.503243\pi$
$457$	−199.059	−0.0203755	−0.0101877	+	0.999948i	$0.503243\pi$
$458$	9104.16	0.928842
$459$	5109.26	0.519564
$460$	−33.6143	−0.00340712
$461$	5245.95	0.529996	0.264998	−	0.964249i	$-0.414629\pi$
$461$	5245.95	0.529996	0.264998	+	0.964249i	$0.414629\pi$
$462$	3657.01	0.368267
$463$	14613.6	1.46685	0.733427	−	0.679769i	$-0.237920\pi$
$463$	14613.6	1.46685	0.733427	+	0.679769i	$0.237920\pi$
$464$	10363.8	1.03691
$465$	−2174.43	−0.216853
$466$	8465.14	0.841503
$467$	−1392.53	−0.137984	−0.0689920	−	0.997617i	$-0.521978\pi$
$467$	−1392.53	−0.137984	−0.0689920	+	0.997617i	$0.521978\pi$
$468$	18.3301	0.00181049
$469$	−13101.2	−1.28988
$470$	−3948.08	−0.387471
$471$	−8744.25	−0.855443
$472$	1126.51	0.109856
$473$	2758.79	0.268181
$474$	7249.23	0.702464
$475$	475.000	0.0458831
$476$	88.1649	0.00848956
$477$	−7998.31	−0.767752
$478$	2600.33	0.248820
$479$	−6679.96	−0.637192	−0.318596	−	0.947891i	$-0.603211\pi$
$479$	−6679.96	−0.637192	−0.318596	+	0.947891i	$0.603211\pi$
$480$	52.6637	0.00500783
$481$	120.396	0.0114129
$482$	16741.0	1.58202
$483$	−11377.9	−1.07187
$484$	−8.44050	−0.000792684 0
$485$	2867.50	0.268467
$486$	−10416.4	−0.972214
$487$	−2320.34	−0.215903	−0.107951	−	0.994156i	$-0.534429\pi$
$487$	−2320.34	−0.215903	−0.107951	+	0.994156i	$0.534429\pi$
$488$	8192.55	0.759957
$489$	2959.78	0.273714
$490$	12797.6	1.17987
$491$	−8455.23	−0.777147	−0.388573	−	0.921418i	$-0.627032\pi$
$491$	−8455.23	−0.777147	−0.388573	+	0.921418i	$0.627032\pi$
$492$	73.5113	0.00673607
$493$	5835.87	0.533133
$494$	−886.102	−0.0807037
$495$	−872.685	−0.0792409
$496$	−8268.23	−0.748497
$497$	10956.1	0.988832
$498$	3419.32	0.307678
$499$	13473.2	1.20870	0.604352	−	0.796718i	$-0.293432\pi$
$499$	13473.2	1.20870	0.604352	+	0.796718i	$0.293432\pi$
$500$	8.71953	0.000779898 0
$501$	−9794.44	−0.873420
$502$	10263.0	0.912470
$503$	4812.36	0.426585	0.213293	−	0.976988i	$-0.431581\pi$
$503$	4812.36	0.426585	0.213293	+	0.976988i	$0.431581\pi$
$504$	−12758.0	−1.12755
$505$	4853.78	0.427703
$506$	−2985.43	−0.262290
$507$	−6415.42	−0.561970
$508$	1.87125	0.000163432 0
$509$	8693.16	0.757009	0.378505	−	0.925599i	$-0.376438\pi$
$509$	8693.16	0.757009	0.378505	+	0.925599i	$0.376438\pi$
$510$	−1678.22	−0.145711
$511$	1509.94	0.130716
$512$	11733.1	1.01276
$513$	−2717.58	−0.233887
$514$	−9836.33	−0.844090
$515$	−2126.45	−0.181947
$516$	58.3735	0.00498013
$517$	3084.37	0.262380
$518$	−724.360	−0.0614412
$519$	−9566.50	−0.809100
$520$	−1881.74	−0.158692
$521$	−4186.69	−0.352058	−0.176029	−	0.984385i	$-0.556325\pi$
$521$	−4186.69	−0.352058	−0.176029	+	0.984385i	$0.556325\pi$
$522$	−7299.86	−0.612081
$523$	3761.19	0.314465	0.157233	−	0.987562i	$-0.449743\pi$
$523$	3761.19	0.314465	0.157233	+	0.987562i	$0.449743\pi$
$524$	−74.8912	−0.00624359
$525$	2951.42	0.245353
$526$	11628.2	0.963904
$527$	−4655.84	−0.384842
$528$	2328.32	0.191907
$529$	−2878.56	−0.236587
$530$	7097.69	0.581706
$531$	−786.551	−0.0642814
$532$	−46.8943	−0.00382167
$533$	−5230.60	−0.425071
$534$	−6.24132	−0.000505783 0
$535$	−4527.03	−0.365833
$536$	−8414.52	−0.678082
$537$	−14475.2	−1.16322
$538$	18338.7	1.46959
$539$	−9997.85	−0.798957
$540$	−49.8864	−0.00397550
$541$	−1941.09	−0.154258	−0.0771292	−	0.997021i	$-0.524575\pi$
$541$	−1941.09	−0.154258	−0.0771292	+	0.997021i	$0.524575\pi$
$542$	−7532.92	−0.596987
$543$	919.527	0.0726716
$544$	112.762	0.00888722
$545$	4135.80	0.325061
$546$	−5505.80	−0.431550
$547$	−11815.2	−0.923551	−0.461776	−	0.886997i	$-0.652787\pi$
$547$	−11815.2	−0.923551	−0.461776	+	0.886997i	$0.652787\pi$
$548$	−50.2716	−0.00391879
$549$	−5720.18	−0.444684
$550$	774.419	0.0600388
$551$	−3104.06	−0.239995
$552$	−7307.71	−0.563473
$553$	−27297.7	−2.09913
$554$	14487.0	1.11100
$555$	−121.284	−0.00927607
$556$	7.49537	0.000571717 0
$557$	−6769.46	−0.514957	−0.257479	−	0.966284i	$-0.582892\pi$
$557$	−6769.46	−0.514957	−0.257479	+	0.966284i	$0.582892\pi$
$558$	5823.81	0.441831
$559$	−4153.49	−0.314265
$560$	11222.7	0.846867
$561$	1311.08	0.0986697
$562$	−18566.8	−1.39358
$563$	−6761.89	−0.506180	−0.253090	−	0.967443i	$-0.581447\pi$
$563$	−6761.89	−0.506180	−0.253090	+	0.967443i	$0.581447\pi$
$564$	65.2623	0.00487241
$565$	11195.0	0.833588
$566$	−18986.4	−1.41000
$567$	−1727.70	−0.127966
$568$	7036.83	0.519822
$569$	−11385.7	−0.838860	−0.419430	−	0.907788i	$-0.637770\pi$
$569$	−11385.7	−0.838860	−0.419430	+	0.907788i	$0.637770\pi$
$570$	892.633	0.0655935
$571$	17478.8	1.28102	0.640512	−	0.767948i	$-0.278722\pi$
$571$	17478.8	1.28102	0.640512	+	0.767948i	$0.278722\pi$
$572$	12.7076	0.000928898 0
$573$	2439.29	0.177841
$574$	31469.7	2.28836
$575$	−2409.41	−0.174747
$576$	−8193.50	−0.592701
$577$	−5534.31	−0.399300	−0.199650	−	0.979867i	$-0.563981\pi$
$577$	−5534.31	−0.399300	−0.199650	+	0.979867i	$0.563981\pi$
$578$	10242.0	0.737042
$579$	5918.53	0.424811
$580$	−56.9809	−0.00407932
$581$	−12875.8	−0.919412
$582$	5388.69	0.383795
$583$	−5544.94	−0.393907
$584$	969.796	0.0687165
$585$	1313.87	0.0928576
$586$	−1447.75	−0.102058
$587$	25907.1	1.82163	0.910817	−	0.412810i	$-0.135453\pi$
$587$	25907.1	1.82163	0.910817	+	0.412810i	$0.135453\pi$
$588$	−211.545	−0.0148367
$589$	2476.41	0.173241
$590$	697.984	0.0487043
$591$	3537.96	0.246248
$592$	−461.180	−0.0320175
$593$	−25291.4	−1.75142	−0.875712	−	0.482835i	$-0.839607\pi$
$593$	−25291.4	−1.75142	−0.875712	+	0.482835i	$0.839607\pi$
$594$	−4430.63	−0.306045
$595$	6319.50	0.435419
$596$	−136.798	−0.00940179
$597$	−15801.1	−1.08325
$598$	4494.71	0.307362
$599$	21055.0	1.43620	0.718099	−	0.695941i	$-0.245012\pi$
$599$	21055.0	1.43620	0.718099	+	0.695941i	$0.245012\pi$
$600$	1895.61	0.128980
$601$	28962.6	1.96574	0.982870	−	0.184300i	$-0.0590017\pi$
$601$	28962.6	1.96574	0.982870	+	0.184300i	$0.0590017\pi$
$602$	24989.3	1.69184
$603$	5875.17	0.396775
$604$	−62.9649	−0.00424173
$605$	−605.000	−0.0406558
$606$	9121.35	0.611435
$607$	−7890.64	−0.527629	−0.263815	−	0.964573i	$-0.584981\pi$
$607$	−7890.64	−0.527629	−0.263815	+	0.964573i	$0.584981\pi$
$608$	−59.9776	−0.00400068
$609$	−19287.1	−1.28334
$610$	5076.08	0.336925
$611$	−4643.65	−0.307467
$612$	−39.5372	−0.00261144
$613$	−5050.70	−0.332783	−0.166391	−	0.986060i	$-0.553212\pi$
$613$	−5050.70	−0.332783	−0.166391	+	0.986060i	$0.553212\pi$
$614$	8130.63	0.534406
$615$	5269.16	0.345485
$616$	−8844.64	−0.578508
$617$	23340.9	1.52297	0.761484	−	0.648184i	$-0.224471\pi$
$617$	23340.9	1.52297	0.761484	+	0.648184i	$0.224471\pi$
$618$	−3996.09	−0.260107
$619$	−5647.72	−0.366722	−0.183361	−	0.983046i	$-0.558698\pi$
$619$	−5647.72	−0.366722	−0.183361	+	0.983046i	$0.558698\pi$
$620$	45.4593	0.00294466
$621$	13784.8	0.890765
$622$	−18772.0	−1.21011
$623$	23.5023	0.00151140
$624$	−3505.39	−0.224884
$625$	625.000	0.0400000
$626$	−19531.9	−1.24705
$627$	−697.353	−0.0444172
$628$	182.810	0.0116161
$629$	−259.690	−0.0164619
$630$	−7904.82	−0.499898
$631$	−9535.77	−0.601605	−0.300803	−	0.953686i	$-0.597255\pi$
$631$	−9535.77	−0.601605	−0.300803	+	0.953686i	$0.597255\pi$
$632$	−17532.6	−1.10349
$633$	−439.939	−0.0276240
$634$	7175.94	0.449516
$635$	134.128	0.00838223
$636$	−117.326	−0.00731488
$637$	15052.2	0.936249
$638$	−5060.72	−0.314038
$639$	−4913.24	−0.304170
$640$	7144.63	0.441275
$641$	−8928.98	−0.550193	−0.275096	−	0.961417i	$-0.588710\pi$
$641$	−8928.98	−0.550193	−0.275096	+	0.961417i	$0.588710\pi$
$642$	−8507.32	−0.522986
$643$	14311.6	0.877753	0.438877	−	0.898547i	$-0.355377\pi$
$643$	14311.6	0.877753	0.438877	+	0.898547i	$0.355377\pi$
$644$	237.869	0.0145549
$645$	4184.11	0.255425
$646$	1911.29	0.116406
$647$	−25100.4	−1.52519	−0.762595	−	0.646876i	$-0.776075\pi$
$647$	−25100.4	−1.52519	−0.762595	+	0.646876i	$0.776075\pi$
$648$	−1109.66	−0.0672707
$649$	−545.287	−0.0329805
$650$	−1165.92	−0.0703558
$651$	15387.2	0.926378
$652$	−61.8780	−0.00371676
$653$	−25852.1	−1.54927	−0.774635	−	0.632409i	$-0.782066\pi$
$653$	−25852.1	−1.54927	−0.774635	+	0.632409i	$0.782066\pi$
$654$	7772.11	0.464699
$655$	−5368.07	−0.320226
$656$	20035.9	1.19248
$657$	−677.129	−0.0402090
$658$	27938.3	1.65524
$659$	−31004.7	−1.83274	−0.916368	−	0.400336i	$-0.868893\pi$
$659$	−31004.7	−1.83274	−0.916368	+	0.400336i	$0.868893\pi$
$660$	−12.8012	−0.000754981 0
$661$	6075.32	0.357493	0.178746	−	0.983895i	$-0.442796\pi$
$661$	6075.32	0.357493	0.178746	+	0.983895i	$0.442796\pi$
$662$	−7831.26	−0.459774
$663$	−1973.89	−0.115625
$664$	−8269.78	−0.483328
$665$	−3361.30	−0.196009
$666$	324.837	0.0188996
$667$	15745.2	0.914028
$668$	204.765	0.0118602
$669$	50.6680	0.00292816
$670$	−5213.61	−0.300626
$671$	−3965.59	−0.228152
$672$	−372.671	−0.0213930
$673$	3403.29	0.194929	0.0974644	−	0.995239i	$-0.468927\pi$
$673$	3403.29	0.194929	0.0974644	+	0.995239i	$0.468927\pi$
$674$	30724.1	1.75586
$675$	−3575.77	−0.203898
$676$	134.123	0.00763101
$677$	21049.5	1.19497	0.597486	−	0.801879i	$-0.296166\pi$
$677$	21049.5	1.19497	0.597486	+	0.801879i	$0.296166\pi$
$678$	21038.0	1.19168
$679$	−20291.7	−1.14687
$680$	4058.85	0.228897
$681$	−3758.42	−0.211487
$682$	4037.43	0.226688
$683$	−18806.7	−1.05361	−0.526806	−	0.849985i	$-0.676611\pi$
$683$	−18806.7	−1.05361	−0.526806	+	0.849985i	$0.676611\pi$
$684$	21.0296	0.00117557
$685$	−3603.38	−0.200990
$686$	−56385.0	−3.13818
$687$	−10787.1	−0.599057
$688$	15910.0	0.881631
$689$	8348.16	0.461596
$690$	−4527.84	−0.249814
$691$	−24254.6	−1.33529	−0.667647	−	0.744478i	$-0.732698\pi$
$691$	−24254.6	−1.33529	−0.667647	+	0.744478i	$0.732698\pi$
$692$	200.000	0.0109868
$693$	6175.49	0.338510
$694$	18357.3	1.00408
$695$	537.255	0.0293226
$696$	−12387.6	−0.674641
$697$	11282.2	0.613119
$698$	13672.3	0.741412
$699$	−10029.9	−0.542727
$700$	−61.7031	−0.00333165
$701$	2422.32	0.130513	0.0652565	−	0.997869i	$-0.479213\pi$
$701$	2422.32	0.130513	0.0652565	+	0.997869i	$0.479213\pi$
$702$	6670.51	0.358636
$703$	138.128	0.00741050
$704$	−5680.25	−0.304094
$705$	4677.88	0.249900
$706$	−13092.2	−0.697918
$707$	−34347.4	−1.82711
$708$	−11.5378	−0.000612451 0
$709$	−25061.8	−1.32753	−0.663764	−	0.747942i	$-0.731042\pi$
$709$	−25061.8	−1.32753	−0.663764	+	0.747942i	$0.731042\pi$
$710$	4360.00	0.230462
$711$	12241.6	0.645702
$712$	15.0949	0.000794530 0
$713$	−12561.5	−0.659792
$714$	11875.8	0.622465
$715$	910.855	0.0476420
$716$	302.622	0.0157954
$717$	−3080.99	−0.160477
$718$	6849.10	0.355997
$719$	−12118.3	−0.628562	−0.314281	−	0.949330i	$-0.601763\pi$
$719$	−12118.3	−0.628562	−0.314281	+	0.949330i	$0.601763\pi$
$720$	−5032.78	−0.260501
$721$	15047.7	0.777261
$722$	−1016.60	−0.0524016
$723$	−19835.6	−1.02032
$724$	−19.2239	−0.000986809 0
$725$	−4084.29	−0.209223
$726$	−1136.93	−0.0581206
$727$	3705.97	0.189060	0.0945300	−	0.995522i	$-0.469865\pi$
$727$	3705.97	0.189060	0.0945300	+	0.995522i	$0.469865\pi$
$728$	13316.0	0.677918
$729$	13660.2	0.694011
$730$	600.883	0.0304653
$731$	8958.91	0.453293
$732$	−83.9082	−0.00423680
$733$	24264.3	1.22268	0.611338	−	0.791370i	$-0.290632\pi$
$733$	24264.3	1.22268	0.611338	+	0.791370i	$0.290632\pi$
$734$	19856.3	0.998515
$735$	−15163.2	−0.760955
$736$	304.233	0.0152367
$737$	4073.04	0.203572
$738$	−14112.5	−0.703912
$739$	26472.4	1.31773	0.658864	−	0.752262i	$-0.271037\pi$
$739$	26472.4	1.31773	0.658864	+	0.752262i	$0.271037\pi$
$740$	2.53560	0.000125960 0
$741$	1049.90	0.0520498
$742$	−50226.3	−2.48499
$743$	9403.97	0.464331	0.232166	−	0.972676i	$-0.425419\pi$
$743$	9403.97	0.464331	0.232166	+	0.972676i	$0.425419\pi$
$744$	9882.79	0.486990
$745$	−9805.44	−0.482206
$746$	18752.1	0.920325
$747$	5774.11	0.282816
$748$	−27.4097	−0.00133984
$749$	32035.2	1.56280
$750$	1174.52	0.0571831
$751$	−6747.81	−0.327871	−0.163935	−	0.986471i	$-0.552419\pi$
$751$	−6747.81	−0.327871	−0.163935	+	0.986471i	$0.552419\pi$
$752$	17787.6	0.862560
$753$	−12160.1	−0.588498
$754$	7619.15	0.368002
$755$	−4513.21	−0.217553
$756$	353.017	0.0169830
$757$	22692.4	1.08952	0.544762	−	0.838591i	$-0.316620\pi$
$757$	22692.4	1.08952	0.544762	+	0.838591i	$0.316620\pi$
$758$	−26478.7	−1.26880
$759$	3537.29	0.169164
$760$	−2158.87	−0.103040
$761$	36689.5	1.74769	0.873846	−	0.486203i	$-0.161618\pi$
$761$	36689.5	1.74769	0.873846	+	0.486203i	$0.161618\pi$
$762$	252.057	0.0119830
$763$	−29266.7	−1.38863
$764$	−50.9964	−0.00241490
$765$	−2833.96	−0.133937
$766$	−24780.0	−1.16885
$767$	820.954	0.0386479
$768$	−357.476	−0.0167960
$769$	30205.0	1.41641	0.708206	−	0.706006i	$-0.249505\pi$
$769$	30205.0	1.41641	0.708206	+	0.706006i	$0.249505\pi$
$770$	−5480.12	−0.256480
$771$	11654.6	0.544396
$772$	−123.734	−0.00576852
$773$	−29371.6	−1.36665	−0.683327	−	0.730113i	$-0.739468\pi$
$773$	−29371.6	−1.36665	−0.683327	+	0.730113i	$0.739468\pi$
$774$	−11206.3	−0.520418
$775$	3258.44	0.151028
$776$	−13032.8	−0.602900
$777$	858.257	0.0396265
$778$	19968.5	0.920185
$779$	−6000.93	−0.276002
$780$	19.2728	0.000884716 0
$781$	−3406.17	−0.156059
$782$	−9694.91	−0.443337
$783$	23367.2	1.06651
$784$	−57657.7	−2.62653
$785$	13103.5	0.595774
$786$	−10087.8	−0.457787
$787$	−33921.9	−1.53645	−0.768224	−	0.640181i	$-0.778859\pi$
$787$	−33921.9	−1.53645	−0.768224	+	0.640181i	$0.778859\pi$
$788$	−73.9656	−0.00334380
$789$	−13777.7	−0.621670
$790$	−10863.1	−0.489232
$791$	−79220.6	−3.56101
$792$	3966.35	0.177952
$793$	5970.38	0.267357
$794$	3783.10	0.169090
$795$	−8409.69	−0.375171
$796$	330.343	0.0147094
$797$	−24073.9	−1.06994	−0.534969	−	0.844872i	$-0.679677\pi$
$797$	−24073.9	−1.06994	−0.534969	+	0.844872i	$0.679677\pi$
$798$	−6316.66	−0.280209
$799$	10016.2	0.443488
$800$	−78.9179	−0.00348771
$801$	−10.5395	−0.000464914 0
$802$	4972.71	0.218943
$803$	−469.428	−0.0206298
$804$	86.1816	0.00378034
$805$	17050.0	0.746503
$806$	−6078.54	−0.265642
$807$	−21728.6	−0.947809
$808$	−22060.4	−0.960498
$809$	−670.425	−0.0291358	−0.0145679	−	0.999894i	$-0.504637\pi$
$809$	−670.425	−0.0291358	−0.0145679	+	0.999894i	$0.504637\pi$
$810$	−687.541	−0.0298243
$811$	10775.0	0.466536	0.233268	−	0.972412i	$-0.425058\pi$
$811$	10775.0	0.466536	0.233268	+	0.972412i	$0.425058\pi$
$812$	403.221	0.0174265
$813$	8925.38	0.385027
$814$	225.197	0.00969676
$815$	−4435.31	−0.190628
$816$	7560.99	0.324372
$817$	−4765.19	−0.204055
$818$	−33758.1	−1.44294
$819$	−9297.48	−0.396679
$820$	−110.158	−0.00469134
$821$	−28550.6	−1.21367	−0.606835	−	0.794828i	$-0.707561\pi$
$821$	−28550.6	−1.21367	−0.606835	+	0.794828i	$0.707561\pi$
$822$	−6771.57	−0.287330
$823$	31286.3	1.32512	0.662558	−	0.749011i	$-0.269471\pi$
$823$	31286.3	1.32512	0.662558	+	0.749011i	$0.269471\pi$
$824$	9664.72	0.408600
$825$	−917.569	−0.0387220
$826$	−4939.23	−0.208060
$827$	24268.3	1.02043	0.510213	−	0.860048i	$-0.329566\pi$
$827$	24268.3	1.02043	0.510213	+	0.860048i	$0.329566\pi$
$828$	−106.672	−0.00447717
$829$	−2245.33	−0.0940696	−0.0470348	−	0.998893i	$-0.514977\pi$
$829$	−2245.33	−0.0940696	−0.0470348	+	0.998893i	$0.514977\pi$
$830$	−5123.94	−0.214282
$831$	−17164.9	−0.716540
$832$	8551.88	0.356350
$833$	−32467.1	−1.35044
$834$	1009.62	0.0419190
$835$	14677.2	0.608294
$836$	14.5790	0.000603142 0
$837$	−18642.3	−0.769858
$838$	−42129.9	−1.73670
$839$	−1461.78	−0.0601506	−0.0300753	−	0.999548i	$-0.509575\pi$
$839$	−1461.78	−0.0601506	−0.0300753	+	0.999548i	$0.509575\pi$
$840$	−13414.2	−0.550992
$841$	2301.29	0.0943579
$842$	−27811.3	−1.13829
$843$	21998.8	0.898791
$844$	9.19748	0.000375107 0
$845$	9613.67	0.391385
$846$	−12528.8	−0.509161
$847$	4281.24	0.173678
$848$	−31977.7	−1.29495
$849$	22496.0	0.909378
$850$	2514.85	0.101481
$851$	−700.646	−0.0282231
$852$	−72.0713	−0.00289803
$853$	−6468.16	−0.259631	−0.129816	−	0.991538i	$-0.541439\pi$
$853$	−6468.16	−0.259631	−0.129816	+	0.991538i	$0.541439\pi$
$854$	−35920.5	−1.43932
$855$	1507.36	0.0602933
$856$	20575.3	0.821555
$857$	22875.3	0.911790	0.455895	−	0.890034i	$-0.349319\pi$
$857$	22875.3	0.911790	0.455895	+	0.890034i	$0.349319\pi$
$858$	1711.71	0.0681080
$859$	29145.0	1.15764	0.578821	−	0.815455i	$-0.303513\pi$
$859$	29145.0	1.15764	0.578821	+	0.815455i	$0.303513\pi$
$860$	−87.4740	−0.00346842
$861$	−37286.8	−1.47588
$862$	−33489.4	−1.32326
$863$	43299.0	1.70790	0.853949	−	0.520357i	$-0.174201\pi$
$863$	43299.0	1.70790	0.853949	+	0.520357i	$0.174201\pi$
$864$	451.507	0.0177785
$865$	14335.6	0.563498
$866$	6403.28	0.251261
$867$	−12135.2	−0.475355
$868$	−321.689	−0.0125793
$869$	8486.62	0.331288
$870$	−7675.32	−0.299101
$871$	−6132.15	−0.238553
$872$	−18797.2	−0.729992
$873$	9099.73	0.352783
$874$	5156.66	0.199573
$875$	−4422.77	−0.170876
$876$	−9.93266	−0.000383098 0
$877$	28197.3	1.08569	0.542847	−	0.839832i	$-0.317346\pi$
$877$	28197.3	1.08569	0.542847	+	0.839832i	$0.317346\pi$
$878$	29366.6	1.12879
$879$	1715.36	0.0658222
$880$	−3489.04	−0.133654
$881$	−36316.5	−1.38880	−0.694402	−	0.719588i	$-0.744331\pi$
$881$	−36316.5	−1.38880	−0.694402	+	0.719588i	$0.744331\pi$
$882$	40611.8	1.55042
$883$	−16385.4	−0.624478	−0.312239	−	0.950004i	$-0.601079\pi$
$883$	−16385.4	−0.624478	−0.312239	+	0.950004i	$0.601079\pi$
$884$	41.2666	0.00157007
$885$	−827.006	−0.0314119
$886$	−10776.2	−0.408614
$887$	19539.7	0.739661	0.369830	−	0.929099i	$-0.379416\pi$
$887$	19539.7	0.739661	0.369830	+	0.929099i	$0.379416\pi$
$888$	551.235	0.0208314
$889$	−949.148	−0.0358081
$890$	9.35277	0.000352253 0
$891$	537.128	0.0201958
$892$	−1.05928	−3.97615e−5 0
$893$	−5327.54	−0.199641
$894$	−18426.7	−0.689351
$895$	21691.4	0.810126
$896$	−50558.4	−1.88509
$897$	−5325.55	−0.198233
$898$	17167.4	0.637953
$899$	−21293.5	−0.789963
$900$	27.6705	0.00102483
$901$	−18006.6	−0.665803
$902$	−9783.65	−0.361153
$903$	−29608.5	−1.09115
$904$	−50881.3	−1.87200
$905$	−1377.93	−0.0506122
$906$	−8481.35	−0.311009
$907$	44433.5	1.62667	0.813336	−	0.581794i	$-0.197649\pi$
$907$	44433.5	1.62667	0.813336	+	0.581794i	$0.197649\pi$
$908$	78.5745	0.00287179
$909$	15403.0	0.562029
$910$	8250.57	0.300554
$911$	−7675.35	−0.279139	−0.139570	−	0.990212i	$-0.544572\pi$
$911$	−7675.35	−0.279139	−0.139570	+	0.990212i	$0.544572\pi$
$912$	−4021.64	−0.146019
$913$	4002.98	0.145103
$914$	560.564	0.0202865
$915$	−6014.39	−0.217300
$916$	225.517	0.00813460
$917$	37986.8	1.36798
$918$	−14388.0	−0.517294
$919$	−19102.2	−0.685661	−0.342831	−	0.939397i	$-0.611386\pi$
$919$	−19102.2	−0.685661	−0.342831	+	0.939397i	$0.611386\pi$
$920$	10950.8	0.392431
$921$	−9633.56	−0.344665
$922$	−14772.9	−0.527680
$923$	5128.14	0.182876
$924$	90.5870	0.00322521
$925$	181.747	0.00646033
$926$	−41153.0	−1.46044
$927$	−6748.08	−0.239090
$928$	515.718	0.0182427
$929$	−37268.3	−1.31618	−0.658091	−	0.752939i	$-0.728636\pi$
$929$	−37268.3	−1.31618	−0.658091	+	0.752939i	$0.728636\pi$
$930$	6123.35	0.215906
$931$	17269.0	0.607915
$932$	209.688	0.00736970
$933$	22242.0	0.780461
$934$	3921.45	0.137381
$935$	−1964.68	−0.0687186
$936$	−5971.52	−0.208531
$937$	−21507.6	−0.749863	−0.374931	−	0.927053i	$-0.622334\pi$
$937$	−21507.6	−0.749863	−0.374931	+	0.927053i	$0.622334\pi$
$938$	36893.8	1.28425
$939$	23142.3	0.804283
$940$	−97.7971	−0.00339339
$941$	11451.4	0.396710	0.198355	−	0.980130i	$-0.436440\pi$
$941$	11451.4	0.396710	0.198355	+	0.980130i	$0.436440\pi$
$942$	24624.4	0.851706
$943$	30439.4	1.05116
$944$	−3144.67	−0.108422
$945$	25303.7	0.871035
$946$	−7768.95	−0.267009
$947$	16012.3	0.549450	0.274725	−	0.961523i	$-0.411413\pi$
$947$	16012.3	0.549450	0.274725	+	0.961523i	$0.411413\pi$
$948$	179.569	0.00615204
$949$	706.746	0.0241749
$950$	−1337.63	−0.0456827
$951$	−8502.40	−0.289915
$952$	−28722.1	−0.977825
$953$	−24428.9	−0.830357	−0.415179	−	0.909740i	$-0.636281\pi$
$953$	−24428.9	−0.830357	−0.415179	+	0.909740i	$0.636281\pi$
$954$	22523.8	0.764397
$955$	−3655.33	−0.123857
$956$	64.4121	0.00217912
$957$	5996.19	0.202539
$958$	18811.2	0.634408
$959$	25499.0	0.858610
$960$	−8614.91	−0.289630
$961$	−12803.1	−0.429765
$962$	−339.045	−0.0113630
$963$	−14366.1	−0.480727
$964$	414.688	0.0138550
$965$	−8869.06	−0.295860
$966$	32040.9	1.06718
$967$	22318.7	0.742213	0.371107	−	0.928590i	$-0.378978\pi$
$967$	22318.7	0.742213	0.371107	+	0.928590i	$0.378978\pi$
$968$	2749.72	0.0913011
$969$	−2264.59	−0.0750764
$970$	−8075.09	−0.267294
$971$	−19197.0	−0.634459	−0.317230	−	0.948349i	$-0.602753\pi$
$971$	−19197.0	−0.634459	−0.317230	+	0.948349i	$0.602753\pi$
$972$	−258.021	−0.00851444
$973$	−3801.84	−0.125264
$974$	6534.24	0.214960
$975$	1381.44	0.0453760
$976$	−22869.6	−0.750039
$977$	−32172.6	−1.05353	−0.526763	−	0.850012i	$-0.676594\pi$
$977$	−32172.6	−1.05353	−0.526763	+	0.850012i	$0.676594\pi$
$978$	−8334.95	−0.272518
$979$	−7.30667	−0.000238531 0
$980$	317.005	0.0103330
$981$	13124.5	0.427150
$982$	23810.5	0.773751
$983$	44722.4	1.45109	0.725546	−	0.688173i	$-0.241587\pi$
$983$	44722.4	1.45109	0.725546	+	0.688173i	$0.241587\pi$
$984$	−23948.3	−0.775858
$985$	−5301.72	−0.171499
$986$	−16434.2	−0.530803
$987$	−33102.7	−1.06755
$988$	−21.9494	−0.000706786 0
$989$	24171.2	0.777147
$990$	2457.54	0.0788947
$991$	−39887.1	−1.27856	−0.639282	−	0.768973i	$-0.720768\pi$
$991$	−39887.1	−1.27856	−0.639282	+	0.768973i	$0.720768\pi$
$992$	−411.438	−0.0131685
$993$	9278.86	0.296531
$994$	−30853.2	−0.984512
$995$	23678.4	0.754428
$996$	84.6992	0.00269458
$997$	−268.045	−0.00851462	−0.00425731	−	0.999991i	$-0.501355\pi$
$997$	−268.045	−0.00851462	−0.00425731	+	0.999991i	$0.501355\pi$
$998$	−37941.4	−1.20342
$999$	−1039.82	−0.0329312

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1045.4.a.c.1.5	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.4.a.c.1.5	✓	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)

\(f(q)\)	\(=\)	\(q-2.81607 q^{2} +3.33662 q^{3} -0.0697562 q^{4} -5.00000 q^{5} -9.39614 q^{6} +35.3821 q^{7} +22.7250 q^{8} -15.8670 q^{9} +O(q^{10})\)	\(q-2.81607 q^{2} +3.33662 q^{3} -0.0697562 q^{4} -5.00000 q^{5} -9.39614 q^{6} +35.3821 q^{7} +22.7250 q^{8} -15.8670 q^{9} +14.0803 q^{10} -11.0000 q^{11} -0.232750 q^{12} +16.5610 q^{13} -99.6385 q^{14} -16.6831 q^{15} -63.4371 q^{16} -35.7214 q^{17} +44.6825 q^{18} +19.0000 q^{19} +0.348781 q^{20} +118.057 q^{21} +30.9768 q^{22} -96.3766 q^{23} +75.8246 q^{24} +25.0000 q^{25} -46.6369 q^{26} -143.031 q^{27} -2.46812 q^{28} -163.372 q^{29} +46.9807 q^{30} +130.338 q^{31} -3.15672 q^{32} -36.7028 q^{33} +100.594 q^{34} -176.911 q^{35} +1.10682 q^{36} +7.26988 q^{37} -53.5053 q^{38} +55.2577 q^{39} -113.625 q^{40} -315.838 q^{41} -332.456 q^{42} -250.799 q^{43} +0.767318 q^{44} +79.3350 q^{45} +271.403 q^{46} -280.397 q^{47} -211.665 q^{48} +908.895 q^{49} -70.4017 q^{50} -119.189 q^{51} -1.15523 q^{52} +504.085 q^{53} +402.784 q^{54} +55.0000 q^{55} +804.059 q^{56} +63.3957 q^{57} +460.066 q^{58} +49.5715 q^{59} +1.16375 q^{60} +360.508 q^{61} -367.039 q^{62} -561.408 q^{63} +516.386 q^{64} -82.8050 q^{65} +103.358 q^{66} -370.276 q^{67} +2.49179 q^{68} -321.572 q^{69} +498.193 q^{70} +309.651 q^{71} -360.577 q^{72} +42.6753 q^{73} -20.4725 q^{74} +83.4154 q^{75} -1.32537 q^{76} -389.203 q^{77} -155.610 q^{78} -771.511 q^{79} +317.185 q^{80} -48.8298 q^{81} +889.423 q^{82} -363.907 q^{83} -8.23518 q^{84} +178.607 q^{85} +706.268 q^{86} -545.109 q^{87} -249.975 q^{88} +0.664243 q^{89} -223.413 q^{90} +585.964 q^{91} +6.72286 q^{92} +434.886 q^{93} +789.617 q^{94} -95.0000 q^{95} -10.5327 q^{96} -573.501 q^{97} -2559.51 q^{98} +174.537 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9}+O(q^{10}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9	\( 20 q - q^{2} - 8 q^{3} + 51 q^{4} - 100 q^{5} - 54 q^{6} + 49 q^{7} + 9 q^{8} + 146 q^{9} + 5 q^{10} - 220 q^{11} - 59 q^{12} + 60 q^{13} - 89 q^{14} + 40 q^{15} + 275 q^{16} - 155 q^{17} + 45 q^{18} + 380 q^{19} - 255 q^{20} + 105 q^{21} + 11 q^{22} - 154 q^{23} - 397 q^{24} + 500 q^{25} + 176 q^{26} - 206 q^{27} + 155 q^{28} - 305 q^{29} + 270 q^{30} - 759 q^{31} - 254 q^{32} + 88 q^{33} - 565 q^{34} - 245 q^{35} + 705 q^{36} + 698 q^{37} - 19 q^{38} - 758 q^{39} - 45 q^{40} + 547 q^{41} + 106 q^{42} - 925 q^{43} - 561 q^{44} - 730 q^{45} - 254 q^{46} - 681 q^{47} - 540 q^{48} + 213 q^{49} - 25 q^{50} - 899 q^{51} + 889 q^{52} - 419 q^{53} - 2241 q^{54} + 1100 q^{55} - 2473 q^{56} - 152 q^{57} - 1440 q^{58} - 2829 q^{59} + 295 q^{60} - 959 q^{61} + 1575 q^{62} - 426 q^{63} + 93 q^{64} - 300 q^{65} + 594 q^{66} - 1020 q^{67} - 4218 q^{68} - 572 q^{69} + 445 q^{70} + 106 q^{71} + 210 q^{72} + 558 q^{73} - 3439 q^{74} - 200 q^{75} + 969 q^{76} - 539 q^{77} - 3599 q^{78} + 536 q^{79} - 1375 q^{80} - 2128 q^{81} - 1255 q^{82} - 4179 q^{83} - 2024 q^{84} + 775 q^{85} - 1119 q^{86} - 557 q^{87} - 99 q^{88} - 4120 q^{89} - 225 q^{90} - 111 q^{91} - 2831 q^{92} + 801 q^{93} + 1213 q^{94} - 1900 q^{95} - 6147 q^{96} + 1414 q^{97} - 7869 q^{98} - 1606 q^{99}+O(q^{100}) \) 20 * q - q^2 - 8 * q^3 + 51 * q^4 - 100 * q^5 - 54 * q^6 + 49 * q^7 + 9 * q^8 + 146 * q^9 + 5 * q^10 - 220 * q^11 - 59 * q^12 + 60 * q^13 - 89 * q^14 + 40 * q^15 + 275 * q^16 - 155 * q^17 + 45 * q^18 + 380 * q^19 - 255 * q^20 + 105 * q^21 + 11 * q^22 - 154 * q^23 - 397 * q^24 + 500 * q^25 + 176 * q^26 - 206 * q^27 + 155 * q^28 - 305 * q^29 + 270 * q^30 - 759 * q^31 - 254 * q^32 + 88 * q^33 - 565 * q^34 - 245 * q^35 + 705 * q^36 + 698 * q^37 - 19 * q^38 - 758 * q^39 - 45 * q^40 + 547 * q^41 + 106 * q^42 - 925 * q^43 - 561 * q^44 - 730 * q^45 - 254 * q^46 - 681 * q^47 - 540 * q^48 + 213 * q^49 - 25 * q^50 - 899 * q^51 + 889 * q^52 - 419 * q^53 - 2241 * q^54 + 1100 * q^55 - 2473 * q^56 - 152 * q^57 - 1440 * q^58 - 2829 * q^59 + 295 * q^60 - 959 * q^61 + 1575 * q^62 - 426 * q^63 + 93 * q^64 - 300 * q^65 + 594 * q^66 - 1020 * q^67 - 4218 * q^68 - 572 * q^69 + 445 * q^70 + 106 * q^71 + 210 * q^72 + 558 * q^73 - 3439 * q^74 - 200 * q^75 + 969 * q^76 - 539 * q^77 - 3599 * q^78 + 536 * q^79 - 1375 * q^80 - 2128 * q^81 - 1255 * q^82 - 4179 * q^83 - 2024 * q^84 + 775 * q^85 - 1119 * q^86 - 557 * q^87 - 99 * q^88 - 4120 * q^89 - 225 * q^90 - 111 * q^91 - 2831 * q^92 + 801 * q^93 + 1213 * q^94 - 1900 * q^95 - 6147 * q^96 + 1414 * q^97 - 7869 * q^98 - 1606 * q^99

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