LMFDB - Embedded newform 507.4.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(1,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("507.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.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:	$29.9139683729$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 56x^{7} - 27x^{6} + 945x^{5} + 763x^{4} - 4139x^{3} - 2478x^{2} + 63x + 27 $ x^9 - 56x^7 - 27x^6 + 945x^5 + 763x^4 - 4139x^3 - 2478x^2 + 63*x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 13^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-4.83218$ of defining polynomial
Character	$\chi$	$=$	507.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.03025 q^{2} +3.00000 q^{3} +8.24289 q^{4} -8.08864 q^{5} -12.0907 q^{6} -5.95078 q^{7} -0.978887 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-4.03025 q^{2} +3.00000 q^{3} +8.24289 q^{4} -8.08864 q^{5} -12.0907 q^{6} -5.95078 q^{7} -0.978887 q^{8} +9.00000 q^{9} +32.5992 q^{10} -17.2359 q^{11} +24.7287 q^{12} +23.9831 q^{14} -24.2659 q^{15} -61.9979 q^{16} +92.9299 q^{17} -36.2722 q^{18} -13.3832 q^{19} -66.6738 q^{20} -17.8523 q^{21} +69.4648 q^{22} +219.710 q^{23} -2.93666 q^{24} -59.5738 q^{25} +27.0000 q^{27} -49.0516 q^{28} -199.485 q^{29} +97.7977 q^{30} -307.777 q^{31} +257.698 q^{32} -51.7076 q^{33} -374.531 q^{34} +48.1337 q^{35} +74.1860 q^{36} +333.777 q^{37} +53.9376 q^{38} +7.91787 q^{40} +200.689 q^{41} +71.9493 q^{42} +116.806 q^{43} -142.073 q^{44} -72.7978 q^{45} -885.487 q^{46} -338.610 q^{47} -185.994 q^{48} -307.588 q^{49} +240.097 q^{50} +278.790 q^{51} -26.6215 q^{53} -108.817 q^{54} +139.415 q^{55} +5.82514 q^{56} -40.1496 q^{57} +803.976 q^{58} +280.058 q^{59} -200.021 q^{60} -207.084 q^{61} +1240.42 q^{62} -53.5570 q^{63} -542.603 q^{64} +208.394 q^{66} +285.981 q^{67} +766.011 q^{68} +659.131 q^{69} -193.991 q^{70} +317.673 q^{71} -8.80999 q^{72} +63.0668 q^{73} -1345.20 q^{74} -178.721 q^{75} -110.316 q^{76} +102.567 q^{77} -623.835 q^{79} +501.479 q^{80} +81.0000 q^{81} -808.824 q^{82} -659.874 q^{83} -147.155 q^{84} -751.677 q^{85} -470.755 q^{86} -598.456 q^{87} +16.8720 q^{88} -1273.19 q^{89} +293.393 q^{90} +1811.05 q^{92} -923.331 q^{93} +1364.68 q^{94} +108.252 q^{95} +773.094 q^{96} -603.746 q^{97} +1239.66 q^{98} -155.123 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) $ 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9} - 54 q^{10} - 85 q^{11} + 132 q^{12} + 158 q^{14} - 99 q^{15} + 216 q^{16} + 178 q^{17} - 54 q^{18} - 352 q^{19} - 402 q^{20} - 249 q^{21} - 630 q^{22} + 150 q^{23} - 261 q^{24} - 20 q^{25} + 243 q^{27} - 940 q^{28} - 97 q^{29} - 162 q^{30} - 717 q^{31} - 707 q^{32} - 255 q^{33} - 632 q^{34} - 418 q^{35} + 396 q^{36} - 1108 q^{37} - 660 q^{38} - 1506 q^{40} - 334 q^{41} + 474 q^{42} + 242 q^{43} + 307 q^{44} - 297 q^{45} - 979 q^{46} + 184 q^{47} + 648 q^{48} - 38 q^{49} + 2031 q^{50} + 534 q^{51} - 151 q^{53} - 162 q^{54} + 2064 q^{55} + 2276 q^{56} - 1056 q^{57} - 1161 q^{58} - 537 q^{59} - 1206 q^{60} - 1340 q^{61} + 347 q^{62} - 747 q^{63} + 893 q^{64} - 1890 q^{66} - 2308 q^{67} + 2785 q^{68} + 450 q^{69} + 1420 q^{70} - 96 q^{71} - 783 q^{72} - 2505 q^{73} - 1191 q^{74} - 60 q^{75} - 2409 q^{76} - 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 729 q^{81} + 1517 q^{82} - 1539 q^{83} - 2820 q^{84} - 4296 q^{85} + 3763 q^{86} - 291 q^{87} - 3716 q^{88} + 592 q^{89} - 486 q^{90} + 515 q^{92} - 2151 q^{93} - 692 q^{94} + 4158 q^{95} - 2121 q^{96} - 1445 q^{97} - 1457 q^{98} - 765 q^{99}+O(q^{100}) \) 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9 - 54 * q^10 - 85 * q^11 + 132 * q^12 + 158 * q^14 - 99 * q^15 + 216 * q^16 + 178 * q^17 - 54 * q^18 - 352 * q^19 - 402 * q^20 - 249 * q^21 - 630 * q^22 + 150 * q^23 - 261 * q^24 - 20 * q^25 + 243 * q^27 - 940 * q^28 - 97 * q^29 - 162 * q^30 - 717 * q^31 - 707 * q^32 - 255 * q^33 - 632 * q^34 - 418 * q^35 + 396 * q^36 - 1108 * q^37 - 660 * q^38 - 1506 * q^40 - 334 * q^41 + 474 * q^42 + 242 * q^43 + 307 * q^44 - 297 * q^45 - 979 * q^46 + 184 * q^47 + 648 * q^48 - 38 * q^49 + 2031 * q^50 + 534 * q^51 - 151 * q^53 - 162 * q^54 + 2064 * q^55 + 2276 * q^56 - 1056 * q^57 - 1161 * q^58 - 537 * q^59 - 1206 * q^60 - 1340 * q^61 + 347 * q^62 - 747 * q^63 + 893 * q^64 - 1890 * q^66 - 2308 * q^67 + 2785 * q^68 + 450 * q^69 + 1420 * q^70 - 96 * q^71 - 783 * q^72 - 2505 * q^73 - 1191 * q^74 - 60 * q^75 - 2409 * q^76 - 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 729 * q^81 + 1517 * q^82 - 1539 * q^83 - 2820 * q^84 - 4296 * q^85 + 3763 * q^86 - 291 * q^87 - 3716 * q^88 + 592 * q^89 - 486 * q^90 + 515 * q^92 - 2151 * q^93 - 692 * q^94 + 4158 * q^95 - 2121 * q^96 - 1445 * q^97 - 1457 * q^98 - 765 * 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 $

$2$	−4.03025	−1.42491	−0.712454	−	0.701719i	$-0.752416\pi$
$2$	−4.03025	−1.42491	−0.712454	+	0.701719i	$0.752416\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	8.24289	1.03036
$5$	−8.08864	−0.723470	−0.361735	−	0.932281i	$-0.617816\pi$
$5$	−8.08864	−0.723470	−0.361735	+	0.932281i	$0.617816\pi$
$6$	−12.0907	−0.822671
$7$	−5.95078	−0.321312	−0.160656	−	0.987010i	$-0.551361\pi$
$7$	−5.95078	−0.321312	−0.160656	+	0.987010i	$0.551361\pi$
$8$	−0.978887	−0.0432611
$9$	9.00000	0.333333
$10$	32.5992	1.03088
$11$	−17.2359	−0.472437	−0.236219	−	0.971700i	$-0.575908\pi$
$11$	−17.2359	−0.472437	−0.236219	+	0.971700i	$0.575908\pi$
$12$	24.7287	0.594879
$13$	0	0
$13$	0	0
$14$	23.9831	0.457840
$15$	−24.2659	−0.417696
$16$	−61.9979	−0.968718
$17$	92.9299	1.32581	0.662907	−	0.748702i	$-0.269323\pi$
$17$	92.9299	1.32581	0.662907	+	0.748702i	$0.269323\pi$
$18$	−36.2722	−0.474969
$19$	−13.3832	−0.161596	−0.0807979	−	0.996731i	$-0.525747\pi$
$19$	−13.3832	−0.161596	−0.0807979	+	0.996731i	$0.525747\pi$
$20$	−66.6738	−0.745435
$21$	−17.8523	−0.185510
$22$	69.4648	0.673179
$23$	219.710	1.99186	0.995930	−	0.0901293i	$-0.0287280\pi$
$23$	219.710	1.99186	0.995930	+	0.0901293i	$0.0287280\pi$
$24$	−2.93666	−0.0249768
$25$	−59.5738	−0.476591
$26$	0	0
$27$	27.0000	0.192450
$28$	−49.0516	−0.331067
$29$	−199.485	−1.27736	−0.638681	−	0.769471i	$-0.720520\pi$
$29$	−199.485	−1.27736	−0.638681	+	0.769471i	$0.720520\pi$
$30$	97.7977	0.595178
$31$	−307.777	−1.78317	−0.891586	−	0.452851i	$-0.850407\pi$
$31$	−307.777	−1.78317	−0.891586	+	0.452851i	$0.850407\pi$
$32$	257.698	1.42359
$33$	−51.7076	−0.272762
$34$	−374.531	−1.88916
$35$	48.1337	0.232460
$36$	74.1860	0.343454
$37$	333.777	1.48304	0.741521	−	0.670930i	$-0.234105\pi$
$37$	333.777	1.48304	0.741521	+	0.670930i	$0.234105\pi$
$38$	53.9376	0.230259
$39$	0	0
$40$	7.91787	0.0312981
$41$	200.689	0.764446	0.382223	−	0.924070i	$-0.375159\pi$
$41$	200.689	0.764446	0.382223	+	0.924070i	$0.375159\pi$
$42$	71.9493	0.264334
$43$	116.806	0.414248	0.207124	−	0.978315i	$-0.433590\pi$
$43$	116.806	0.414248	0.207124	+	0.978315i	$0.433590\pi$
$44$	−142.073	−0.486781
$45$	−72.7978	−0.241157
$46$	−885.487	−2.83822
$47$	−338.610	−1.05088	−0.525440	−	0.850831i	$-0.676099\pi$
$47$	−338.610	−1.05088	−0.525440	+	0.850831i	$0.676099\pi$
$48$	−185.994	−0.559289
$49$	−307.588	−0.896759
$50$	240.097	0.679097
$51$	278.790	0.765459
$52$	0	0
$53$	−26.6215	−0.0689951	−0.0344975	−	0.999405i	$-0.510983\pi$
$53$	−26.6215	−0.0689951	−0.0344975	+	0.999405i	$0.510983\pi$
$54$	−108.817	−0.274224
$55$	139.415	0.341794
$56$	5.82514	0.0139003
$57$	−40.1496	−0.0932973
$58$	803.976	1.82012
$59$	280.058	0.617973	0.308987	−	0.951066i	$-0.400010\pi$
$59$	280.058	0.617973	0.308987	+	0.951066i	$0.400010\pi$
$60$	−200.021	−0.430377
$61$	−207.084	−0.434663	−0.217332	−	0.976098i	$-0.569735\pi$
$61$	−207.084	−0.434663	−0.217332	+	0.976098i	$0.569735\pi$
$62$	1240.42	2.54086
$63$	−53.5570	−0.107104
$64$	−542.603	−1.05977
$65$	0	0
$66$	208.394	0.388660
$67$	285.981	0.521465	0.260732	−	0.965411i	$-0.416036\pi$
$67$	285.981	0.521465	0.260732	+	0.965411i	$0.416036\pi$
$68$	766.011	1.36607
$69$	659.131	1.15000
$70$	−193.991	−0.331233
$71$	317.673	0.530998	0.265499	−	0.964111i	$-0.414463\pi$
$71$	317.673	0.530998	0.265499	+	0.964111i	$0.414463\pi$
$72$	−8.80999	−0.0144204
$73$	63.0668	0.101115	0.0505576	−	0.998721i	$-0.483900\pi$
$73$	63.0668	0.101115	0.0505576	+	0.998721i	$0.483900\pi$
$74$	−1345.20	−2.11320
$75$	−178.721	−0.275160
$76$	−110.316	−0.166502
$77$	102.567	0.151800
$78$	0	0
$79$	−623.835	−0.888443	−0.444221	−	0.895917i	$-0.646520\pi$
$79$	−623.835	−0.888443	−0.444221	+	0.895917i	$0.646520\pi$
$80$	501.479	0.700838
$81$	81.0000	0.111111
$82$	−808.824	−1.08926
$83$	−659.874	−0.872658	−0.436329	−	0.899787i	$-0.643722\pi$
$83$	−659.874	−0.872658	−0.436329	+	0.899787i	$0.643722\pi$
$84$	−147.155	−0.191142
$85$	−751.677	−0.959186
$86$	−470.755	−0.590265
$87$	−598.456	−0.737486
$88$	16.8720	0.0204382
$89$	−1273.19	−1.51638	−0.758191	−	0.652032i	$-0.773917\pi$
$89$	−1273.19	−1.51638	−0.758191	+	0.652032i	$0.773917\pi$
$90$	293.393	0.343626
$91$	0	0
$92$	1811.05	2.05233
$93$	−923.331	−1.02952
$94$	1364.68	1.49741
$95$	108.252	0.116910
$96$	773.094	0.821912
$97$	−603.746	−0.631970	−0.315985	−	0.948764i	$-0.602335\pi$
$97$	−603.746	−0.631970	−0.315985	+	0.948764i	$0.602335\pi$
$98$	1239.66	1.27780
$99$	−155.123	−0.157479
$100$	−491.060	−0.491060
$101$	−740.588	−0.729616	−0.364808	−	0.931083i	$-0.618865\pi$
$101$	−740.588	−0.729616	−0.364808	+	0.931083i	$0.618865\pi$
$102$	−1123.59	−1.09071
$103$	−1888.57	−1.80666	−0.903332	−	0.428942i	$-0.858887\pi$
$103$	−1888.57	−1.80666	−0.903332	+	0.428942i	$0.858887\pi$
$104$	0	0
$105$	144.401	0.134211
$106$	107.291	0.0983116
$107$	−919.463	−0.830728	−0.415364	−	0.909655i	$-0.636346\pi$
$107$	−919.463	−0.830728	−0.415364	+	0.909655i	$0.636346\pi$
$108$	222.558	0.198293
$109$	−1570.40	−1.37998	−0.689988	−	0.723821i	$-0.742384\pi$
$109$	−1570.40	−1.37998	−0.689988	+	0.723821i	$0.742384\pi$
$110$	−561.876	−0.487025
$111$	1001.33	0.856234
$112$	368.936	0.311260
$113$	−1324.66	−1.10277	−0.551386	−	0.834250i	$-0.685901\pi$
$113$	−1324.66	−1.10277	−0.551386	+	0.834250i	$0.685901\pi$
$114$	161.813	0.132940
$115$	−1777.16	−1.44105
$116$	−1644.34	−1.31614
$117$	0	0
$118$	−1128.70	−0.880555
$119$	−553.006	−0.426000
$120$	23.7536	0.0180700
$121$	−1033.92	−0.776803
$122$	834.601	0.619354
$123$	602.066	0.441353
$124$	−2536.97	−1.83731
$125$	1492.95	1.06827
$126$	215.848	0.152613
$127$	−2350.39	−1.64223	−0.821115	−	0.570763i	$-0.806648\pi$
$127$	−2350.39	−1.64223	−0.821115	+	0.570763i	$0.806648\pi$
$128$	125.240	0.0864824
$129$	350.417	0.239166
$130$	0	0
$131$	2308.69	1.53978	0.769890	−	0.638177i	$-0.220311\pi$
$131$	2308.69	1.53978	0.769890	+	0.638177i	$0.220311\pi$
$132$	−426.220	−0.281043
$133$	79.6405	0.0519226
$134$	−1152.57	−0.743039
$135$	−218.393	−0.139232
$136$	−90.9680	−0.0573562
$137$	374.912	0.233802	0.116901	−	0.993144i	$-0.462704\pi$
$137$	374.912	0.233802	0.116901	+	0.993144i	$0.462704\pi$
$138$	−2656.46	−1.63864
$139$	487.711	0.297605	0.148802	−	0.988867i	$-0.452458\pi$
$139$	487.711	0.297605	0.148802	+	0.988867i	$0.452458\pi$
$140$	396.761	0.239517
$141$	−1015.83	−0.606726
$142$	−1280.30	−0.756623
$143$	0	0
$144$	−557.981	−0.322906
$145$	1613.57	0.924134
$146$	−254.175	−0.144080
$147$	−922.765	−0.517744
$148$	2751.28	1.52807
$149$	−1055.91	−0.580558	−0.290279	−	0.956942i	$-0.593748\pi$
$149$	−1055.91	−0.580558	−0.290279	+	0.956942i	$0.593748\pi$
$150$	720.292	0.392077
$151$	888.560	0.478874	0.239437	−	0.970912i	$-0.423037\pi$
$151$	888.560	0.478874	0.239437	+	0.970912i	$0.423037\pi$
$152$	13.1007	0.00699081
$153$	836.369	0.441938
$154$	−413.370	−0.216301
$155$	2489.50	1.29007
$156$	0	0
$157$	−3648.56	−1.85469	−0.927346	−	0.374206i	$-0.877915\pi$
$157$	−3648.56	−1.85469	−0.927346	+	0.374206i	$0.877915\pi$
$158$	2514.21	1.26595
$159$	−79.8644	−0.0398343
$160$	−2084.43	−1.02993
$161$	−1307.45	−0.640008
$162$	−326.450	−0.158323
$163$	−1386.49	−0.666249	−0.333125	−	0.942883i	$-0.608103\pi$
$163$	−1386.49	−0.666249	−0.333125	+	0.942883i	$0.608103\pi$
$164$	1654.25	0.787655
$165$	418.244	0.197335
$166$	2659.46	1.24346
$167$	3376.19	1.56442	0.782209	−	0.623017i	$-0.214093\pi$
$167$	3376.19	1.56442	0.782209	+	0.623017i	$0.214093\pi$
$168$	17.4754	0.00802535
$169$	0	0
$170$	3029.44	1.36675
$171$	−120.449	−0.0538652
$172$	962.815	0.426825
$173$	341.817	0.150219	0.0751095	−	0.997175i	$-0.476069\pi$
$173$	341.817	0.150219	0.0751095	+	0.997175i	$0.476069\pi$
$174$	2411.93	1.05085
$175$	354.511	0.153134
$176$	1068.59	0.457658
$177$	840.174	0.356787
$178$	5131.28	2.16070
$179$	−2885.55	−1.20489	−0.602446	−	0.798159i	$-0.705807\pi$
$179$	−2885.55	−1.20489	−0.602446	+	0.798159i	$0.705807\pi$
$180$	−600.064	−0.248478
$181$	3795.62	1.55871	0.779354	−	0.626584i	$-0.215547\pi$
$181$	3795.62	1.55871	0.779354	+	0.626584i	$0.215547\pi$
$182$	0	0
$183$	−621.253	−0.250953
$184$	−215.072	−0.0861701
$185$	−2699.80	−1.07294
$186$	3721.25	1.46696
$187$	−1601.73	−0.626363
$188$	−2791.12	−1.08278
$189$	−160.671	−0.0618365
$190$	−436.282	−0.166586
$191$	−2805.90	−1.06297	−0.531486	−	0.847067i	$-0.678366\pi$
$191$	−2805.90	−1.06297	−0.531486	+	0.847067i	$0.678366\pi$
$192$	−1627.81	−0.611859
$193$	2485.64	0.927048	0.463524	−	0.886084i	$-0.346585\pi$
$193$	2485.64	0.927048	0.463524	+	0.886084i	$0.346585\pi$
$194$	2433.24	0.900499
$195$	0	0
$196$	−2535.41	−0.923985
$197$	−2750.70	−0.994818	−0.497409	−	0.867516i	$-0.665715\pi$
$197$	−2750.70	−0.994818	−0.497409	+	0.867516i	$0.665715\pi$
$198$	625.183	0.224393
$199$	3998.53	1.42436	0.712182	−	0.701994i	$-0.247707\pi$
$199$	3998.53	1.42436	0.712182	+	0.701994i	$0.247707\pi$
$200$	58.3161	0.0206178
$201$	857.943	0.301068
$202$	2984.75	1.03964
$203$	1187.09	0.410432
$204$	2298.03	0.788698
$205$	−1623.30	−0.553054
$206$	7611.41	2.57433
$207$	1977.39	0.663953
$208$	0	0
$209$	230.671	0.0763438
$210$	−581.973	−0.191238
$211$	−1375.80	−0.448882	−0.224441	−	0.974488i	$-0.572056\pi$
$211$	−1375.80	−0.448882	−0.224441	+	0.974488i	$0.572056\pi$
$212$	−219.438	−0.0710898
$213$	953.020	0.306572
$214$	3705.66	1.18371
$215$	−944.799	−0.299696
$216$	−26.4300	−0.00832561
$217$	1831.51	0.572955
$218$	6329.12	1.96634
$219$	189.200	0.0583789
$220$	1149.18	0.352171
$221$	0	0
$222$	−4035.61	−1.22005
$223$	−1694.37	−0.508805	−0.254403	−	0.967098i	$-0.581879\pi$
$223$	−1694.37	−0.508805	−0.254403	+	0.967098i	$0.581879\pi$
$224$	−1533.50	−0.457418
$225$	−536.164	−0.158864
$226$	5338.70	1.57135
$227$	405.400	0.118534	0.0592672	−	0.998242i	$-0.481124\pi$
$227$	405.400	0.118534	0.0592672	+	0.998242i	$0.481124\pi$
$228$	−330.949	−0.0961299
$229$	−2359.54	−0.680884	−0.340442	−	0.940265i	$-0.610577\pi$
$229$	−2359.54	−0.680884	−0.340442	+	0.940265i	$0.610577\pi$
$230$	7162.39	2.05337
$231$	307.701	0.0876416
$232$	195.274	0.0552601
$233$	938.507	0.263878	0.131939	−	0.991258i	$-0.457880\pi$
$233$	938.507	0.263878	0.131939	+	0.991258i	$0.457880\pi$
$234$	0	0
$235$	2738.90	0.760280
$236$	2308.49	0.636736
$237$	−1871.51	−0.512943
$238$	2228.75	0.607010
$239$	−4664.25	−1.26237	−0.631183	−	0.775634i	$-0.717430\pi$
$239$	−4664.25	−1.26237	−0.631183	+	0.775634i	$0.717430\pi$
$240$	1504.44	0.404629
$241$	3774.94	1.00898	0.504492	−	0.863416i	$-0.331680\pi$
$241$	3774.94	1.00898	0.504492	+	0.863416i	$0.331680\pi$
$242$	4166.97	1.10687
$243$	243.000	0.0641500
$244$	−1706.97	−0.447860
$245$	2487.97	0.648778
$246$	−2426.47	−0.628887
$247$	0	0
$248$	301.279	0.0771421
$249$	−1979.62	−0.503829
$250$	−6016.96	−1.52219
$251$	−960.695	−0.241588	−0.120794	−	0.992678i	$-0.538544\pi$
$251$	−960.695	−0.241588	−0.120794	+	0.992678i	$0.538544\pi$
$252$	−441.464	−0.110356
$253$	−3786.90	−0.941029
$254$	9472.64	2.34003
$255$	−2255.03	−0.553787
$256$	3836.08	0.936542
$257$	−16.7302	−0.00406070	−0.00203035	−	0.999998i	$-0.500646\pi$
$257$	−16.7302	−0.00406070	−0.00203035	+	0.999998i	$0.500646\pi$
$258$	−1412.27	−0.340790
$259$	−1986.23	−0.476519
$260$	0	0
$261$	−1795.37	−0.425788
$262$	−9304.58	−2.19404
$263$	−400.736	−0.0939560	−0.0469780	−	0.998896i	$-0.514959\pi$
$263$	−400.736	−0.0939560	−0.0469780	+	0.998896i	$0.514959\pi$
$264$	50.6159	0.0118000
$265$	215.332	0.0499159
$266$	−320.971	−0.0739849
$267$	−3819.57	−0.875484
$268$	2357.31	0.537297
$269$	2236.97	0.507027	0.253514	−	0.967332i	$-0.418414\pi$
$269$	2236.97	0.507027	0.253514	+	0.967332i	$0.418414\pi$
$270$	880.179	0.198393
$271$	4018.89	0.900850	0.450425	−	0.892814i	$-0.351273\pi$
$271$	4018.89	0.900850	0.450425	+	0.892814i	$0.351273\pi$
$272$	−5761.46	−1.28434
$273$	0	0
$274$	−1510.99	−0.333147
$275$	1026.81	0.225159
$276$	5433.14	1.18492
$277$	6792.95	1.47346	0.736731	−	0.676186i	$-0.236368\pi$
$277$	6792.95	1.47346	0.736731	+	0.676186i	$0.236368\pi$
$278$	−1965.59	−0.424059
$279$	−2769.99	−0.594391
$280$	−47.1175	−0.0100565
$281$	7286.80	1.54695	0.773477	−	0.633824i	$-0.218516\pi$
$281$	7286.80	1.54695	0.773477	+	0.633824i	$0.218516\pi$
$282$	4094.04	0.864528
$283$	−2429.77	−0.510369	−0.255185	−	0.966892i	$-0.582136\pi$
$283$	−2429.77	−0.510369	−0.255185	+	0.966892i	$0.582136\pi$
$284$	2618.54	0.547120
$285$	324.756	0.0674979
$286$	0	0
$287$	−1194.25	−0.245626
$288$	2319.28	0.474531
$289$	3722.97	0.757780
$290$	−6503.07	−1.31681
$291$	−1811.24	−0.364868
$292$	519.853	0.104185
$293$	−4211.25	−0.839672	−0.419836	−	0.907600i	$-0.637912\pi$
$293$	−4211.25	−0.839672	−0.419836	+	0.907600i	$0.637912\pi$
$294$	3718.97	0.737737
$295$	−2265.29	−0.447085
$296$	−326.730	−0.0641580
$297$	−465.368	−0.0909206
$298$	4255.56	0.827242
$299$	0	0
$300$	−1473.18	−0.283514
$301$	−695.084	−0.133103
$302$	−3581.12	−0.682351
$303$	−2221.76	−0.421244
$304$	829.731	0.156541
$305$	1675.03	0.314466
$306$	−3370.78	−0.629720
$307$	−8212.27	−1.52671	−0.763353	−	0.645982i	$-0.776448\pi$
$307$	−8212.27	−1.52671	−0.763353	+	0.645982i	$0.776448\pi$
$308$	845.447	0.156408
$309$	−5665.71	−1.04308
$310$	−10033.3	−1.83823
$311$	−4238.55	−0.772818	−0.386409	−	0.922328i	$-0.626285\pi$
$311$	−4238.55	−0.772818	−0.386409	+	0.922328i	$0.626285\pi$
$312$	0	0
$313$	3807.82	0.687638	0.343819	−	0.939036i	$-0.388279\pi$
$313$	3807.82	0.687638	0.343819	+	0.939036i	$0.388279\pi$
$314$	14704.6	2.64276
$315$	433.204	0.0774865
$316$	−5142.20	−0.915416
$317$	10497.7	1.85996	0.929981	−	0.367608i	$-0.119823\pi$
$317$	10497.7	1.85996	0.929981	+	0.367608i	$0.119823\pi$
$318$	321.873	0.0567602
$319$	3438.31	0.603474
$320$	4388.92	0.766713
$321$	−2758.39	−0.479621
$322$	5269.34	0.911953
$323$	−1243.70	−0.214246
$324$	667.674	0.114485
$325$	0	0
$326$	5587.91	0.949343
$327$	−4711.21	−0.796730
$328$	−196.452	−0.0330708
$329$	2014.99	0.337660
$330$	−1685.63	−0.281184
$331$	−11905.2	−1.97695	−0.988474	−	0.151389i	$-0.951625\pi$
$331$	−11905.2	−1.97695	−0.988474	+	0.151389i	$0.951625\pi$
$332$	−5439.27	−0.899152
$333$	3003.99	0.494347
$334$	−13606.9	−2.22915
$335$	−2313.20	−0.377264
$336$	1106.81	0.179706
$337$	8981.18	1.45174	0.725869	−	0.687833i	$-0.241438\pi$
$337$	8981.18	1.45174	0.725869	+	0.687833i	$0.241438\pi$
$338$	0	0
$339$	−3973.97	−0.636686
$340$	−6195.99	−0.988308
$341$	5304.80	0.842437
$342$	485.439	0.0767530
$343$	3871.51	0.609451
$344$	−114.339	−0.0179208
$345$	−5331.48	−0.831992
$346$	−1377.61	−0.214048
$347$	−5712.87	−0.883813	−0.441906	−	0.897061i	$-0.645698\pi$
$347$	−5712.87	−0.883813	−0.441906	+	0.897061i	$0.645698\pi$
$348$	−4933.01	−0.759876
$349$	−9507.28	−1.45820	−0.729102	−	0.684406i	$-0.760062\pi$
$349$	−9507.28	−1.45820	−0.729102	+	0.684406i	$0.760062\pi$
$350$	−1428.77	−0.218202
$351$	0	0
$352$	−4441.65	−0.672559
$353$	3972.96	0.599035	0.299518	−	0.954091i	$-0.403174\pi$
$353$	3972.96	0.599035	0.299518	+	0.954091i	$0.403174\pi$
$354$	−3386.11	−0.508389
$355$	−2569.55	−0.384162
$356$	−10494.8	−1.56242
$357$	−1659.02	−0.245951
$358$	11629.5	1.71686
$359$	109.866	0.0161519	0.00807594	−	0.999967i	$-0.497429\pi$
$359$	109.866	0.0161519	0.00807594	+	0.999967i	$0.497429\pi$
$360$	71.2609	0.0104327
$361$	−6679.89	−0.973887
$362$	−15297.3	−2.22101
$363$	−3101.77	−0.448487
$364$	0	0
$365$	−510.125	−0.0731539
$366$	2503.80	0.357584
$367$	7020.02	0.998480	0.499240	−	0.866464i	$-0.333613\pi$
$367$	7020.02	0.998480	0.499240	+	0.866464i	$0.333613\pi$
$368$	−13621.6	−1.92955
$369$	1806.20	0.254815
$370$	10880.9	1.52884
$371$	158.418	0.0221689
$372$	−7610.91	−1.06077
$373$	2227.18	0.309166	0.154583	−	0.987980i	$-0.450597\pi$
$373$	2227.18	0.309166	0.154583	+	0.987980i	$0.450597\pi$
$374$	6455.36	0.892510
$375$	4478.86	0.616766
$376$	331.461	0.0454622
$377$	0	0
$378$	647.544	0.0881113
$379$	11004.6	1.49147	0.745734	−	0.666244i	$-0.232099\pi$
$379$	11004.6	1.49147	0.745734	+	0.666244i	$0.232099\pi$
$380$	892.309	0.120459
$381$	−7051.16	−0.948142
$382$	11308.5	1.51464
$383$	4436.85	0.591938	0.295969	−	0.955198i	$-0.404357\pi$
$383$	4436.85	0.591938	0.295969	+	0.955198i	$0.404357\pi$
$384$	375.720	0.0499306
$385$	−829.627	−0.109823
$386$	−10017.7	−1.32096
$387$	1051.25	0.138083
$388$	−4976.61	−0.651157
$389$	−2561.58	−0.333874	−0.166937	−	0.985968i	$-0.553388\pi$
$389$	−2561.58	−0.333874	−0.166937	+	0.985968i	$0.553388\pi$
$390$	0	0
$391$	20417.7	2.64083
$392$	301.094	0.0387948
$393$	6926.07	0.888992
$394$	11086.0	1.41752
$395$	5045.98	0.642762
$396$	−1278.66	−0.162260
$397$	−2206.90	−0.278996	−0.139498	−	0.990222i	$-0.544549\pi$
$397$	−2206.90	−0.278996	−0.139498	+	0.990222i	$0.544549\pi$
$398$	−16115.1	−2.02959
$399$	238.922	0.0299775
$400$	3693.45	0.461682
$401$	8071.02	1.00511	0.502553	−	0.864546i	$-0.332394\pi$
$401$	8071.02	1.00511	0.502553	+	0.864546i	$0.332394\pi$
$402$	−3457.72	−0.428994
$403$	0	0
$404$	−6104.58	−0.751768
$405$	−655.180	−0.0803856
$406$	−4784.28	−0.584827
$407$	−5752.93	−0.700644
$408$	−272.904	−0.0331146
$409$	−6298.36	−0.761452	−0.380726	−	0.924688i	$-0.624326\pi$
$409$	−6298.36	−0.761452	−0.380726	+	0.924688i	$0.624326\pi$
$410$	6542.29	0.788051
$411$	1124.74	0.134986
$412$	−15567.3	−1.86152
$413$	−1666.56	−0.198562
$414$	−7969.38	−0.946072
$415$	5337.49	0.631342
$416$	0	0
$417$	1463.13	0.171822
$418$	−929.662	−0.108783
$419$	−1432.01	−0.166965	−0.0834824	−	0.996509i	$-0.526604\pi$
$419$	−1432.01	−0.166965	−0.0834824	+	0.996509i	$0.526604\pi$
$420$	1190.28	0.138285
$421$	−9641.43	−1.11614	−0.558069	−	0.829794i	$-0.688458\pi$
$421$	−9641.43	−1.11614	−0.558069	+	0.829794i	$0.688458\pi$
$422$	5544.82	0.639615
$423$	−3047.49	−0.350293
$424$	26.0594	0.00298480
$425$	−5536.19	−0.631870
$426$	−3840.90	−0.436837
$427$	1232.31	0.139662
$428$	−7579.03	−0.855949
$429$	0	0
$430$	3807.77	0.427040
$431$	−3370.61	−0.376698	−0.188349	−	0.982102i	$-0.560314\pi$
$431$	−3370.61	−0.376698	−0.188349	+	0.982102i	$0.560314\pi$
$432$	−1673.94	−0.186430
$433$	6248.91	0.693541	0.346770	−	0.937950i	$-0.387278\pi$
$433$	6248.91	0.693541	0.346770	+	0.937950i	$0.387278\pi$
$434$	−7381.45	−0.816407
$435$	4840.70	0.533549
$436$	−12944.7	−1.42187
$437$	−2940.43	−0.321876
$438$	−762.524	−0.0831845
$439$	2866.16	0.311604	0.155802	−	0.987788i	$-0.450204\pi$
$439$	2866.16	0.311604	0.155802	+	0.987788i	$0.450204\pi$
$440$	−136.471	−0.0147864
$441$	−2768.29	−0.298920
$442$	0	0
$443$	10776.3	1.15575	0.577877	−	0.816124i	$-0.303881\pi$
$443$	10776.3	1.15575	0.577877	+	0.816124i	$0.303881\pi$
$444$	8253.85	0.882230
$445$	10298.4	1.09706
$446$	6828.74	0.725000
$447$	−3167.72	−0.335185
$448$	3228.91	0.340517
$449$	−13834.9	−1.45414	−0.727069	−	0.686564i	$-0.759118\pi$
$449$	−13834.9	−1.45414	−0.727069	+	0.686564i	$0.759118\pi$
$450$	2160.87	0.226366
$451$	−3459.04	−0.361153
$452$	−10919.0	−1.13625
$453$	2665.68	0.276478
$454$	−1633.86	−0.168901
$455$	0	0
$456$	39.3020	0.00403615
$457$	−17233.0	−1.76395	−0.881976	−	0.471295i	$-0.843787\pi$
$457$	−17233.0	−1.76395	−0.881976	+	0.471295i	$0.843787\pi$
$458$	9509.51	0.970197
$459$	2509.11	0.255153
$460$	−14648.9	−1.48480
$461$	−11485.0	−1.16033	−0.580164	−	0.814500i	$-0.697012\pi$
$461$	−11485.0	−1.16033	−0.580164	+	0.814500i	$0.697012\pi$
$462$	−1240.11	−0.124881
$463$	−1613.61	−0.161967	−0.0809837	−	0.996715i	$-0.525806\pi$
$463$	−1613.61	−0.161967	−0.0809837	+	0.996715i	$0.525806\pi$
$464$	12367.7	1.23740
$465$	7468.49	0.744824
$466$	−3782.41	−0.376002
$467$	−8149.20	−0.807495	−0.403747	−	0.914871i	$-0.632293\pi$
$467$	−8149.20	−0.807495	−0.403747	+	0.914871i	$0.632293\pi$
$468$	0	0
$469$	−1701.81	−0.167553
$470$	−11038.4	−1.08333
$471$	−10945.7	−1.07081
$472$	−274.145	−0.0267342
$473$	−2013.24	−0.195706
$474$	7542.63	0.730896
$475$	797.289	0.0770150
$476$	−4558.36	−0.438933
$477$	−239.593	−0.0229984
$478$	18798.1	1.79875
$479$	−1827.62	−0.174334	−0.0871670	−	0.996194i	$-0.527781\pi$
$479$	−1827.62	−0.174334	−0.0871670	+	0.996194i	$0.527781\pi$
$480$	−6253.28	−0.594629
$481$	0	0
$482$	−15213.9	−1.43771
$483$	−3922.34	−0.369509
$484$	−8522.52	−0.800387
$485$	4883.49	0.457212
$486$	−979.350	−0.0914078
$487$	7838.95	0.729398	0.364699	−	0.931125i	$-0.381172\pi$
$487$	7838.95	0.729398	0.364699	+	0.931125i	$0.381172\pi$
$488$	202.712	0.0188040
$489$	−4159.48	−0.384659
$490$	−10027.1	−0.924449
$491$	17196.1	1.58055	0.790273	−	0.612755i	$-0.209939\pi$
$491$	17196.1	1.58055	0.790273	+	0.612755i	$0.209939\pi$
$492$	4962.76	0.454753
$493$	−18538.2	−1.69354
$494$	0	0
$495$	1254.73	0.113931
$496$	19081.5	1.72739
$497$	−1890.40	−0.170616
$498$	7978.37	0.717910
$499$	−5355.28	−0.480431	−0.240216	−	0.970720i	$-0.577218\pi$
$499$	−5355.28	−0.480431	−0.240216	+	0.970720i	$0.577218\pi$
$500$	12306.2	1.10070
$501$	10128.6	0.903217
$502$	3871.84	0.344240
$503$	2979.80	0.264141	0.132071	−	0.991240i	$-0.457837\pi$
$503$	2979.80	0.264141	0.132071	+	0.991240i	$0.457837\pi$
$504$	52.4263	0.00463344
$505$	5990.35	0.527856
$506$	15262.1	1.34088
$507$	0	0
$508$	−19374.0	−1.69209
$509$	−15353.7	−1.33701	−0.668507	−	0.743706i	$-0.733066\pi$
$509$	−15353.7	−1.33701	−0.668507	+	0.743706i	$0.733066\pi$
$510$	9088.33	0.789094
$511$	−375.297	−0.0324895
$512$	−16462.3	−1.42097
$513$	−361.347	−0.0310991
$514$	67.4267	0.00578611
$515$	15276.0	1.30707
$516$	2888.44	0.246428
$517$	5836.24	0.496475
$518$	8005.00	0.678995
$519$	1025.45	0.0867289
$520$	0	0
$521$	433.801	0.0364783	0.0182391	−	0.999834i	$-0.494194\pi$
$521$	433.801	0.0364783	0.0182391	+	0.999834i	$0.494194\pi$
$522$	7235.78	0.606708
$523$	18900.3	1.58021	0.790106	−	0.612970i	$-0.210026\pi$
$523$	18900.3	1.58021	0.790106	+	0.612970i	$0.210026\pi$
$524$	19030.3	1.58653
$525$	1063.53	0.0884121
$526$	1615.06	0.133879
$527$	−28601.7	−2.36415
$528$	3205.76	0.264229
$529$	36105.7	2.96751
$530$	−867.839	−0.0711255
$531$	2520.52	0.205991
$532$	656.468	0.0534990
$533$	0	0
$534$	15393.8	1.24748
$535$	7437.21	0.601007
$536$	−279.943	−0.0225592
$537$	−8656.64	−0.695645
$538$	−9015.53	−0.722467
$539$	5301.55	0.423662
$540$	−1800.19	−0.143459
$541$	−13247.5	−1.05278	−0.526391	−	0.850243i	$-0.676455\pi$
$541$	−13247.5	−1.05278	−0.526391	+	0.850243i	$0.676455\pi$
$542$	−16197.1	−1.28363
$543$	11386.9	0.899920
$544$	23947.9	1.88742
$545$	12702.4	0.998372
$546$	0	0
$547$	−5543.52	−0.433316	−0.216658	−	0.976248i	$-0.569516\pi$
$547$	−5543.52	−0.433316	−0.216658	+	0.976248i	$0.569516\pi$
$548$	3090.36	0.240901
$549$	−1863.76	−0.144888
$550$	−4138.28	−0.320831
$551$	2669.76	0.206416
$552$	−645.215	−0.0497503
$553$	3712.31	0.285467
$554$	−27377.3	−2.09955
$555$	−8099.40	−0.619460
$556$	4020.14	0.306640
$557$	−6733.19	−0.512198	−0.256099	−	0.966651i	$-0.582437\pi$
$557$	−6733.19	−0.512198	−0.256099	+	0.966651i	$0.582437\pi$
$558$	11163.7	0.846952
$559$	0	0
$560$	−2984.19	−0.225188
$561$	−4805.18	−0.361631
$562$	−29367.6	−2.20427
$563$	24282.4	1.81773	0.908863	−	0.417096i	$-0.136952\pi$
$563$	24282.4	1.81773	0.908863	+	0.417096i	$0.136952\pi$
$564$	−8373.37	−0.625146
$565$	10714.7	0.797823
$566$	9792.55	0.727229
$567$	−482.013	−0.0357013
$568$	−310.966	−0.0229716
$569$	−19876.2	−1.46441	−0.732207	−	0.681082i	$-0.761510\pi$
$569$	−19876.2	−1.46441	−0.732207	+	0.681082i	$0.761510\pi$
$570$	−1308.85	−0.0961782
$571$	225.014	0.0164913	0.00824567	−	0.999966i	$-0.497375\pi$
$571$	225.014	0.0164913	0.00824567	+	0.999966i	$0.497375\pi$
$572$	0	0
$573$	−8417.70	−0.613707
$574$	4813.14	0.349994
$575$	−13089.0	−0.949302
$576$	−4883.43	−0.353257
$577$	12258.1	0.884424	0.442212	−	0.896910i	$-0.354194\pi$
$577$	12258.1	0.884424	0.442212	+	0.896910i	$0.354194\pi$
$578$	−15004.5	−1.07977
$579$	7456.92	0.535231
$580$	13300.4	0.952191
$581$	3926.77	0.280395
$582$	7299.73	0.519903
$583$	458.844	0.0325958
$584$	−61.7353	−0.00437436
$585$	0	0
$586$	16972.4	1.19645
$587$	−10282.3	−0.722992	−0.361496	−	0.932374i	$-0.617734\pi$
$587$	−10282.3	−0.722992	−0.361496	+	0.932374i	$0.617734\pi$
$588$	−7606.24	−0.533463
$589$	4119.04	0.288153
$590$	9129.67	0.637055
$591$	−8252.10	−0.574359
$592$	−20693.5	−1.43665
$593$	−12355.1	−0.855586	−0.427793	−	0.903877i	$-0.640709\pi$
$593$	−12355.1	−0.855586	−0.427793	+	0.903877i	$0.640709\pi$
$594$	1875.55	0.129553
$595$	4473.07	0.308198
$596$	−8703.71	−0.598184
$597$	11995.6	0.822357
$598$	0	0
$599$	13035.2	0.889157	0.444578	−	0.895740i	$-0.353354\pi$
$599$	13035.2	0.889157	0.444578	+	0.895740i	$0.353354\pi$
$600$	174.948	0.0119037
$601$	−5153.36	−0.349767	−0.174884	−	0.984589i	$-0.555955\pi$
$601$	−5153.36	−0.349767	−0.174884	+	0.984589i	$0.555955\pi$
$602$	2801.36	0.189659
$603$	2573.83	0.173822
$604$	7324.30	0.493413
$605$	8363.05	0.561994
$606$	8954.25	0.600234
$607$	−1406.55	−0.0940530	−0.0470265	−	0.998894i	$-0.514975\pi$
$607$	−1406.55	−0.0940530	−0.0470265	+	0.998894i	$0.514975\pi$
$608$	−3448.83	−0.230047
$609$	3561.28	0.236963
$610$	−6750.79	−0.448085
$611$	0	0
$612$	6894.10	0.455355
$613$	15984.3	1.05318	0.526590	−	0.850120i	$-0.323470\pi$
$613$	15984.3	1.05318	0.526590	+	0.850120i	$0.323470\pi$
$614$	33097.5	2.17541
$615$	−4869.90	−0.319306
$616$	−100.401	−0.00656703
$617$	10324.4	0.673652	0.336826	−	0.941567i	$-0.390647\pi$
$617$	10324.4	0.673652	0.336826	+	0.941567i	$0.390647\pi$
$618$	22834.2	1.48629
$619$	7423.00	0.481996	0.240998	−	0.970526i	$-0.422525\pi$
$619$	7423.00	0.481996	0.240998	+	0.970526i	$0.422525\pi$
$620$	20520.6	1.32924
$621$	5932.18	0.383334
$622$	17082.4	1.10119
$623$	7576.48	0.487232
$624$	0	0
$625$	−4629.23	−0.296271
$626$	−15346.4	−0.979820
$627$	692.014	0.0440771
$628$	−30074.6	−1.91100
$629$	31017.8	1.96624
$630$	−1745.92	−0.110411
$631$	−1190.30	−0.0750954	−0.0375477	−	0.999295i	$-0.511955\pi$
$631$	−1190.30	−0.0750954	−0.0375477	+	0.999295i	$0.511955\pi$
$632$	610.665	0.0384350
$633$	−4127.41	−0.259162
$634$	−42308.2	−2.65027
$635$	19011.5	1.18810
$636$	−658.313	−0.0410437
$637$	0	0
$638$	−13857.2	−0.859894
$639$	2859.06	0.176999
$640$	−1013.02	−0.0625674
$641$	705.064	0.0434451	0.0217226	−	0.999764i	$-0.493085\pi$
$641$	705.064	0.0434451	0.0217226	+	0.999764i	$0.493085\pi$
$642$	11117.0	0.683415
$643$	−14641.4	−0.897976	−0.448988	−	0.893538i	$-0.648215\pi$
$643$	−14641.4	−0.897976	−0.448988	+	0.893538i	$0.648215\pi$
$644$	−10777.1	−0.659439
$645$	−2834.40	−0.173030
$646$	5012.42	0.305280
$647$	1520.83	0.0924111	0.0462056	−	0.998932i	$-0.485287\pi$
$647$	1520.83	0.0924111	0.0462056	+	0.998932i	$0.485287\pi$
$648$	−79.2899	−0.00480679
$649$	−4827.04	−0.291954
$650$	0	0
$651$	5494.54	0.330796
$652$	−11428.7	−0.686477
$653$	−22054.4	−1.32168	−0.660838	−	0.750529i	$-0.729799\pi$
$653$	−22054.4	−1.32168	−0.660838	+	0.750529i	$0.729799\pi$
$654$	18987.3	1.13527
$655$	−18674.2	−1.11398
$656$	−12442.3	−0.740532
$657$	567.601	0.0337051
$658$	−8120.92	−0.481134
$659$	−12652.3	−0.747895	−0.373947	−	0.927450i	$-0.621996\pi$
$659$	−12652.3	−0.747895	−0.373947	+	0.927450i	$0.621996\pi$
$660$	3447.54	0.203326
$661$	10893.2	0.640994	0.320497	−	0.947250i	$-0.396150\pi$
$661$	10893.2	0.640994	0.320497	+	0.947250i	$0.396150\pi$
$662$	47981.0	2.81697
$663$	0	0
$664$	645.942	0.0377522
$665$	−644.184	−0.0375645
$666$	−12106.8	−0.704399
$667$	−43829.0	−2.54433
$668$	27829.6	1.61191
$669$	−5083.12	−0.293759
$670$	9322.76	0.537567
$671$	3569.28	0.205351
$672$	−4600.51	−0.264090
$673$	−9019.75	−0.516621	−0.258310	−	0.966062i	$-0.583166\pi$
$673$	−9019.75	−0.516621	−0.258310	+	0.966062i	$0.583166\pi$
$674$	−36196.4	−2.06859
$675$	−1608.49	−0.0917199
$676$	0	0
$677$	−24923.6	−1.41491	−0.707454	−	0.706760i	$-0.750156\pi$
$677$	−24923.6	−1.41491	−0.707454	+	0.706760i	$0.750156\pi$
$678$	16016.1	0.907218
$679$	3592.76	0.203060
$680$	735.807	0.0414955
$681$	1216.20	0.0684359
$682$	−21379.7	−1.20039
$683$	24634.2	1.38009	0.690044	−	0.723767i	$-0.257591\pi$
$683$	24634.2	1.38009	0.690044	+	0.723767i	$0.257591\pi$
$684$	−992.846	−0.0555006
$685$	−3032.53	−0.169149
$686$	−15603.1	−0.868411
$687$	−7078.61	−0.393109
$688$	−7241.70	−0.401290
$689$	0	0
$690$	21487.2	1.18551
$691$	10340.2	0.569264	0.284632	−	0.958637i	$-0.408129\pi$
$691$	10340.2	0.569264	0.284632	+	0.958637i	$0.408129\pi$
$692$	2817.56	0.154780
$693$	923.102	0.0505999
$694$	23024.3	1.25935
$695$	−3944.92	−0.215308
$696$	585.821	0.0319045
$697$	18650.0	1.01351
$698$	38316.7	2.07780
$699$	2815.52	0.152350
$700$	2922.19	0.157784
$701$	−20833.9	−1.12252	−0.561258	−	0.827641i	$-0.689683\pi$
$701$	−20833.9	−1.12252	−0.561258	+	0.827641i	$0.689683\pi$
$702$	0	0
$703$	−4467.00	−0.239653
$704$	9352.23	0.500676
$705$	8216.69	0.438948
$706$	−16012.0	−0.853569
$707$	4407.07	0.234434
$708$	6925.46	0.367619
$709$	−9189.56	−0.486772	−0.243386	−	0.969930i	$-0.578258\pi$
$709$	−9189.56	−0.486772	−0.243386	+	0.969930i	$0.578258\pi$
$710$	10355.9	0.547395
$711$	−5614.52	−0.296148
$712$	1246.31	0.0656004
$713$	−67621.8	−3.55183
$714$	6686.25	0.350457
$715$	0	0
$716$	−23785.2	−1.24147
$717$	−13992.8	−0.728827
$718$	−442.788	−0.0230149
$719$	7503.22	0.389183	0.194592	−	0.980884i	$-0.437662\pi$
$719$	7503.22	0.389183	0.194592	+	0.980884i	$0.437662\pi$
$720$	4513.31	0.233613
$721$	11238.5	0.580503
$722$	26921.6	1.38770
$723$	11324.8	0.582537
$724$	31286.8	1.60603
$725$	11884.1	0.608779
$726$	12500.9	0.639053
$727$	−20727.2	−1.05740	−0.528700	−	0.848809i	$-0.677320\pi$
$727$	−20727.2	−1.05740	−0.528700	+	0.848809i	$0.677320\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	2055.93	0.104237
$731$	10854.7	0.549216
$732$	−5120.92	−0.258572
$733$	−17361.3	−0.874836	−0.437418	−	0.899258i	$-0.644107\pi$
$733$	−17361.3	−0.874836	−0.437418	+	0.899258i	$0.644107\pi$
$734$	−28292.4	−1.42274
$735$	7463.92	0.374572
$736$	56618.9	2.83560
$737$	−4929.13	−0.246359
$738$	−7279.42	−0.363088
$739$	−18093.8	−0.900663	−0.450331	−	0.892861i	$-0.648694\pi$
$739$	−18093.8	−0.900663	−0.450331	+	0.892861i	$0.648694\pi$
$740$	−22254.1	−1.10551
$741$	0	0
$742$	−638.466	−0.0315887
$743$	14875.3	0.734482	0.367241	−	0.930126i	$-0.380302\pi$
$743$	14875.3	0.734482	0.367241	+	0.930126i	$0.380302\pi$
$744$	903.837	0.0445380
$745$	8540.85	0.420017
$746$	−8976.07	−0.440533
$747$	−5938.87	−0.290886
$748$	−13202.9	−0.645380
$749$	5471.52	0.266923
$750$	−18050.9	−0.878834
$751$	−17750.9	−0.862502	−0.431251	−	0.902232i	$-0.641928\pi$
$751$	−17750.9	−0.862502	−0.431251	+	0.902232i	$0.641928\pi$
$752$	20993.1	1.01801
$753$	−2882.09	−0.139481
$754$	0	0
$755$	−7187.25	−0.346451
$756$	−1324.39	−0.0637139
$757$	−10220.3	−0.490702	−0.245351	−	0.969434i	$-0.578903\pi$
$757$	−10220.3	−0.490702	−0.245351	+	0.969434i	$0.578903\pi$
$758$	−44351.1	−2.12520
$759$	−11360.7	−0.543303
$760$	−105.967	−0.00505765
$761$	14112.0	0.672221	0.336110	−	0.941823i	$-0.390888\pi$
$761$	14112.0	0.672221	0.336110	+	0.941823i	$0.390888\pi$
$762$	28417.9	1.35101
$763$	9345.13	0.443403
$764$	−23128.7	−1.09525
$765$	−6765.10	−0.319729
$766$	−17881.6	−0.843457
$767$	0	0
$768$	11508.2	0.540713
$769$	22736.7	1.06620	0.533100	−	0.846052i	$-0.321027\pi$
$769$	22736.7	1.06620	0.533100	+	0.846052i	$0.321027\pi$
$770$	3343.60	0.156487
$771$	−50.1905	−0.00234444
$772$	20488.8	0.955194
$773$	343.173	0.0159678	0.00798388	−	0.999968i	$-0.497459\pi$
$773$	343.173	0.0159678	0.00798388	+	0.999968i	$0.497459\pi$
$774$	−4236.80	−0.196755
$775$	18335.4	0.849844
$776$	590.999	0.0273397
$777$	−5958.69	−0.275118
$778$	10323.8	0.475740
$779$	−2685.86	−0.123531
$780$	0	0
$781$	−5475.37	−0.250863
$782$	−82288.3	−3.76294
$783$	−5386.11	−0.245829
$784$	19069.8	0.868706
$785$	29511.9	1.34181
$786$	−27913.8	−1.26673
$787$	19086.4	0.864496	0.432248	−	0.901755i	$-0.357721\pi$
$787$	19086.4	0.864496	0.432248	+	0.901755i	$0.357721\pi$
$788$	−22673.7	−1.02502
$789$	−1202.21	−0.0542455
$790$	−20336.6	−0.915876
$791$	7882.74	0.354334
$792$	151.848	0.00681272
$793$	0	0
$794$	8894.36	0.397543
$795$	645.995	0.0288190
$796$	32959.5	1.46761
$797$	12031.1	0.534709	0.267354	−	0.963598i	$-0.413851\pi$
$797$	12031.1	0.534709	0.267354	+	0.963598i	$0.413851\pi$
$798$	−962.913	−0.0427152
$799$	−31467.0	−1.39327
$800$	−15352.1	−0.678471
$801$	−11458.7	−0.505461
$802$	−32528.2	−1.43218
$803$	−1087.01	−0.0477706
$804$	7071.93	0.310208
$805$	10575.5	0.463027
$806$	0	0
$807$	6710.91	0.292732
$808$	724.952	0.0315640
$809$	10175.4	0.442210	0.221105	−	0.975250i	$-0.429034\pi$
$809$	10175.4	0.442210	0.221105	+	0.975250i	$0.429034\pi$
$810$	2640.54	0.114542
$811$	−26754.1	−1.15840	−0.579201	−	0.815185i	$-0.696635\pi$
$811$	−26754.1	−1.15840	−0.579201	+	0.815185i	$0.696635\pi$
$812$	9785.08	0.422893
$813$	12056.7	0.520106
$814$	23185.7	0.998353
$815$	11214.9	0.482011
$816$	−17284.4	−0.741513
$817$	−1563.23	−0.0669408
$818$	25383.9	1.08500
$819$	0	0
$820$	−13380.7	−0.569845
$821$	15777.5	0.670693	0.335347	−	0.942095i	$-0.391147\pi$
$821$	15777.5	0.670693	0.335347	+	0.942095i	$0.391147\pi$
$822$	−4532.97	−0.192342
$823$	−13863.9	−0.587197	−0.293599	−	0.955929i	$-0.594853\pi$
$823$	−13863.9	−0.587197	−0.293599	+	0.955929i	$0.594853\pi$
$824$	1848.70	0.0781583
$825$	3080.42	0.129996
$826$	6716.66	0.282933
$827$	26835.0	1.12835	0.564175	−	0.825655i	$-0.309194\pi$
$827$	26835.0	1.12835	0.564175	+	0.825655i	$0.309194\pi$
$828$	16299.4	0.684111
$829$	625.251	0.0261953	0.0130976	−	0.999914i	$-0.495831\pi$
$829$	625.251	0.0261953	0.0130976	+	0.999914i	$0.495831\pi$
$830$	−21511.4	−0.899604
$831$	20378.9	0.850704
$832$	0	0
$833$	−28584.2	−1.18893
$834$	−5896.78	−0.244831
$835$	−27308.8	−1.13181
$836$	1901.40	0.0786617
$837$	−8309.98	−0.343172
$838$	5771.35	0.237909
$839$	27307.4	1.12367	0.561833	−	0.827251i	$-0.310096\pi$
$839$	27307.4	1.12367	0.561833	+	0.827251i	$0.310096\pi$
$840$	−141.353	−0.00580610
$841$	15405.4	0.631656
$842$	38857.3	1.59039
$843$	21860.4	0.893135
$844$	−11340.6	−0.462510
$845$	0	0
$846$	12282.1	0.499135
$847$	6152.66	0.249596
$848$	1650.48	0.0668368
$849$	−7289.30	−0.294662
$850$	22312.2	0.900356
$851$	73334.2	2.95401
$852$	7855.63	0.315880
$853$	−44801.9	−1.79834	−0.899172	−	0.437595i	$-0.855830\pi$
$853$	−44801.9	−1.79834	−0.899172	+	0.437595i	$0.855830\pi$
$854$	−4966.53	−0.199006
$855$	974.268	0.0389699
$856$	900.051	0.0359382
$857$	−25167.1	−1.00314	−0.501571	−	0.865116i	$-0.667244\pi$
$857$	−25167.1	−1.00314	−0.501571	+	0.865116i	$0.667244\pi$
$858$	0	0
$859$	4059.10	0.161228	0.0806138	−	0.996745i	$-0.474312\pi$
$859$	4059.10	0.161228	0.0806138	+	0.996745i	$0.474312\pi$
$860$	−7787.87	−0.308795
$861$	−3582.76	−0.141812
$862$	13584.4	0.536759
$863$	−818.924	−0.0323019	−0.0161509	−	0.999870i	$-0.505141\pi$
$863$	−818.924	−0.0323019	−0.0161509	+	0.999870i	$0.505141\pi$
$864$	6957.85	0.273971
$865$	−2764.84	−0.108679
$866$	−25184.6	−0.988231
$867$	11168.9	0.437505
$868$	15096.9	0.590350
$869$	10752.3	0.419733
$870$	−19509.2	−0.760258
$871$	0	0
$872$	1537.25	0.0596993
$873$	−5433.71	−0.210657
$874$	11850.7	0.458644
$875$	−8884.23	−0.343248
$876$	1559.56	0.0601513
$877$	−190.251	−0.00732533	−0.00366267	−	0.999993i	$-0.501166\pi$
$877$	−190.251	−0.00732533	−0.00366267	+	0.999993i	$0.501166\pi$
$878$	−11551.3	−0.444007
$879$	−12633.8	−0.484785
$880$	−8643.43	−0.331102
$881$	−19803.4	−0.757315	−0.378657	−	0.925537i	$-0.623614\pi$
$881$	−19803.4	−0.757315	−0.378657	+	0.925537i	$0.623614\pi$
$882$	11156.9	0.425933
$883$	19652.1	0.748974	0.374487	−	0.927232i	$-0.377819\pi$
$883$	19652.1	0.748974	0.374487	+	0.927232i	$0.377819\pi$
$884$	0	0
$885$	−6795.87	−0.258125
$886$	−43431.2	−1.64684
$887$	−26295.0	−0.995379	−0.497689	−	0.867355i	$-0.665818\pi$
$887$	−26295.0	−0.995379	−0.497689	+	0.867355i	$0.665818\pi$
$888$	−980.189	−0.0370417
$889$	13986.6	0.527668
$890$	−41505.1	−1.56321
$891$	−1396.11	−0.0524930
$892$	−13966.5	−0.524253
$893$	4531.69	0.169818
$894$	12766.7	0.477608
$895$	23340.2	0.871704
$896$	−745.275	−0.0277878
$897$	0	0
$898$	55758.0	2.07201
$899$	61397.0	2.27776
$900$	−4419.54	−0.163687
$901$	−2473.93	−0.0914746
$902$	13940.8	0.514609
$903$	−2085.25	−0.0768470
$904$	1296.69	0.0477072
$905$	−30701.4	−1.12768
$906$	−10743.3	−0.393956
$907$	42417.3	1.55286	0.776429	−	0.630204i	$-0.217029\pi$
$907$	42417.3	1.55286	0.776429	+	0.630204i	$0.217029\pi$
$908$	3341.66	0.122133
$909$	−6665.29	−0.243205
$910$	0	0
$911$	−872.245	−0.0317220	−0.0158610	−	0.999874i	$-0.505049\pi$
$911$	−872.245	−0.0317220	−0.0158610	+	0.999874i	$0.505049\pi$
$912$	2489.19	0.0903788
$913$	11373.5	0.412276
$914$	69453.2	2.51347
$915$	5025.10	0.181557
$916$	−19449.4	−0.701557
$917$	−13738.5	−0.494749
$918$	−10112.3	−0.363569
$919$	−17181.4	−0.616716	−0.308358	−	0.951270i	$-0.599779\pi$
$919$	−17181.4	−0.616716	−0.308358	+	0.951270i	$0.599779\pi$
$920$	1739.64	0.0623415
$921$	−24636.8	−0.881444
$922$	46287.5	1.65336
$923$	0	0
$924$	2536.34	0.0903025
$925$	−19884.4	−0.706804
$926$	6503.25	0.230788
$927$	−16997.1	−0.602221
$928$	−51407.0	−1.81845
$929$	56042.9	1.97923	0.989617	−	0.143728i	$-0.0459091\pi$
$929$	56042.9	1.97923	0.989617	+	0.143728i	$0.0459091\pi$
$930$	−30099.9	−1.06130
$931$	4116.52	0.144912
$932$	7736.00	0.271890
$933$	−12715.7	−0.446187
$934$	32843.3	1.15060
$935$	12955.8	0.453155
$936$	0	0
$937$	−36672.5	−1.27859	−0.639295	−	0.768961i	$-0.720774\pi$
$937$	−36672.5	−1.27859	−0.639295	+	0.768961i	$0.720774\pi$
$938$	6858.71	0.238747
$939$	11423.5	0.397008
$940$	22576.4	0.783363
$941$	−21069.0	−0.729895	−0.364947	−	0.931028i	$-0.618913\pi$
$941$	−21069.0	−0.729895	−0.364947	+	0.931028i	$0.618913\pi$
$942$	44113.7	1.52580
$943$	44093.4	1.52267
$944$	−17363.0	−0.598642
$945$	1299.61	0.0447369
$946$	8113.87	0.278863
$947$	−1838.70	−0.0630938	−0.0315469	−	0.999502i	$-0.510043\pi$
$947$	−1838.70	−0.0630938	−0.0315469	+	0.999502i	$0.510043\pi$
$948$	−15426.6	−0.528516
$949$	0	0
$950$	−3213.27	−0.109739
$951$	31493.0	1.07385
$952$	541.330	0.0184292
$953$	13599.8	0.462266	0.231133	−	0.972922i	$-0.425757\pi$
$953$	13599.8	0.462266	0.231133	+	0.972922i	$0.425757\pi$
$954$	965.620	0.0327705
$955$	22695.9	0.769029
$956$	−38446.9	−1.30069
$957$	10314.9	0.348416
$958$	7365.75	0.248410
$959$	−2231.02	−0.0751235
$960$	13166.8	0.442662
$961$	64935.6	2.17971
$962$	0	0
$963$	−8275.17	−0.276909
$964$	31116.4	1.03962
$965$	−20105.5	−0.670692
$966$	15808.0	0.526516
$967$	2081.30	0.0692141	0.0346070	−	0.999401i	$-0.488982\pi$
$967$	2081.30	0.0692141	0.0346070	+	0.999401i	$0.488982\pi$
$968$	1012.10	0.0336054
$969$	−3731.10	−0.123695
$970$	−19681.6	−0.651484
$971$	1636.62	0.0540904	0.0270452	−	0.999634i	$-0.491390\pi$
$971$	1636.62	0.0540904	0.0270452	+	0.999634i	$0.491390\pi$
$972$	2003.02	0.0660977
$973$	−2902.26	−0.0956240
$974$	−31592.9	−1.03932
$975$	0	0
$976$	12838.8	0.421066
$977$	−29387.1	−0.962311	−0.481156	−	0.876635i	$-0.659783\pi$
$977$	−29387.1	−0.962311	−0.481156	+	0.876635i	$0.659783\pi$
$978$	16763.7	0.548104
$979$	21944.6	0.716396
$980$	20508.1	0.668476
$981$	−14133.6	−0.459992
$982$	−69304.4	−2.25213
$983$	24084.4	0.781457	0.390728	−	0.920506i	$-0.372223\pi$
$983$	24084.4	0.781457	0.390728	+	0.920506i	$0.372223\pi$
$984$	−589.355	−0.0190934
$985$	22249.4	0.719722
$986$	74713.4	2.41314
$987$	6044.98	0.194948
$988$	0	0
$989$	25663.4	0.825125
$990$	−5056.88	−0.162342
$991$	1413.43	0.0453068	0.0226534	−	0.999743i	$-0.492789\pi$
$991$	1413.43	0.0453068	0.0226534	+	0.999743i	$0.492789\pi$
$992$	−79313.5	−2.53851
$993$	−35715.7	−1.14139
$994$	7618.79	0.243112
$995$	−32342.7	−1.03049
$996$	−16317.8	−0.519126
$997$	−33357.4	−1.05962	−0.529809	−	0.848117i	$-0.677737\pi$
$997$	−33357.4	−1.05962	−0.529809	+	0.848117i	$0.677737\pi$
$998$	21583.1	0.684570
$999$	9011.97	0.285411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.o.1.3	✓	9
3.2	odd	2		1521.4.a.bi.1.7		9
13.5	odd	4		507.4.b.k.337.15		18
13.8	odd	4		507.4.b.k.337.4		18
13.12	even	2		507.4.a.p.1.7	yes	9
39.38	odd	2		1521.4.a.bf.1.3		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.3	✓	9	1.1	even	1	trivial
507.4.a.p.1.7	yes	9	13.12	even	2
507.4.b.k.337.4		18	13.8	odd	4
507.4.b.k.337.15		18	13.5	odd	4
1521.4.a.bf.1.3		9	39.38	odd	2
1521.4.a.bi.1.7		9	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.a (trivial)

\(f(q)\)	\(=\)	\(q-4.03025 q^{2} +3.00000 q^{3} +8.24289 q^{4} -8.08864 q^{5} -12.0907 q^{6} -5.95078 q^{7} -0.978887 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-4.03025 q^{2} +3.00000 q^{3} +8.24289 q^{4} -8.08864 q^{5} -12.0907 q^{6} -5.95078 q^{7} -0.978887 q^{8} +9.00000 q^{9} +32.5992 q^{10} -17.2359 q^{11} +24.7287 q^{12} +23.9831 q^{14} -24.2659 q^{15} -61.9979 q^{16} +92.9299 q^{17} -36.2722 q^{18} -13.3832 q^{19} -66.6738 q^{20} -17.8523 q^{21} +69.4648 q^{22} +219.710 q^{23} -2.93666 q^{24} -59.5738 q^{25} +27.0000 q^{27} -49.0516 q^{28} -199.485 q^{29} +97.7977 q^{30} -307.777 q^{31} +257.698 q^{32} -51.7076 q^{33} -374.531 q^{34} +48.1337 q^{35} +74.1860 q^{36} +333.777 q^{37} +53.9376 q^{38} +7.91787 q^{40} +200.689 q^{41} +71.9493 q^{42} +116.806 q^{43} -142.073 q^{44} -72.7978 q^{45} -885.487 q^{46} -338.610 q^{47} -185.994 q^{48} -307.588 q^{49} +240.097 q^{50} +278.790 q^{51} -26.6215 q^{53} -108.817 q^{54} +139.415 q^{55} +5.82514 q^{56} -40.1496 q^{57} +803.976 q^{58} +280.058 q^{59} -200.021 q^{60} -207.084 q^{61} +1240.42 q^{62} -53.5570 q^{63} -542.603 q^{64} +208.394 q^{66} +285.981 q^{67} +766.011 q^{68} +659.131 q^{69} -193.991 q^{70} +317.673 q^{71} -8.80999 q^{72} +63.0668 q^{73} -1345.20 q^{74} -178.721 q^{75} -110.316 q^{76} +102.567 q^{77} -623.835 q^{79} +501.479 q^{80} +81.0000 q^{81} -808.824 q^{82} -659.874 q^{83} -147.155 q^{84} -751.677 q^{85} -470.755 q^{86} -598.456 q^{87} +16.8720 q^{88} -1273.19 q^{89} +293.393 q^{90} +1811.05 q^{92} -923.331 q^{93} +1364.68 q^{94} +108.252 q^{95} +773.094 q^{96} -603.746 q^{97} +1239.66 q^{98} -155.123 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9} - 54 q^{10} - 85 q^{11} + 132 q^{12} + 158 q^{14} - 99 q^{15} + 216 q^{16} + 178 q^{17} - 54 q^{18} - 352 q^{19} - 402 q^{20} - 249 q^{21} - 630 q^{22} + 150 q^{23} - 261 q^{24} - 20 q^{25} + 243 q^{27} - 940 q^{28} - 97 q^{29} - 162 q^{30} - 717 q^{31} - 707 q^{32} - 255 q^{33} - 632 q^{34} - 418 q^{35} + 396 q^{36} - 1108 q^{37} - 660 q^{38} - 1506 q^{40} - 334 q^{41} + 474 q^{42} + 242 q^{43} + 307 q^{44} - 297 q^{45} - 979 q^{46} + 184 q^{47} + 648 q^{48} - 38 q^{49} + 2031 q^{50} + 534 q^{51} - 151 q^{53} - 162 q^{54} + 2064 q^{55} + 2276 q^{56} - 1056 q^{57} - 1161 q^{58} - 537 q^{59} - 1206 q^{60} - 1340 q^{61} + 347 q^{62} - 747 q^{63} + 893 q^{64} - 1890 q^{66} - 2308 q^{67} + 2785 q^{68} + 450 q^{69} + 1420 q^{70} - 96 q^{71} - 783 q^{72} - 2505 q^{73} - 1191 q^{74} - 60 q^{75} - 2409 q^{76} - 2142 q^{77} - 1591 q^{79} + 2671 q^{80} + 729 q^{81} + 1517 q^{82} - 1539 q^{83} - 2820 q^{84} - 4296 q^{85} + 3763 q^{86} - 291 q^{87} - 3716 q^{88} + 592 q^{89} - 486 q^{90} + 515 q^{92} - 2151 q^{93} - 692 q^{94} + 4158 q^{95} - 2121 q^{96} - 1445 q^{97} - 1457 q^{98} - 765 q^{99}+O(q^{100}) \) 9 * q - 6 * q^2 + 27 * q^3 + 44 * q^4 - 33 * q^5 - 18 * q^6 - 83 * q^7 - 87 * q^8 + 81 * q^9 - 54 * q^10 - 85 * q^11 + 132 * q^12 + 158 * q^14 - 99 * q^15 + 216 * q^16 + 178 * q^17 - 54 * q^18 - 352 * q^19 - 402 * q^20 - 249 * q^21 - 630 * q^22 + 150 * q^23 - 261 * q^24 - 20 * q^25 + 243 * q^27 - 940 * q^28 - 97 * q^29 - 162 * q^30 - 717 * q^31 - 707 * q^32 - 255 * q^33 - 632 * q^34 - 418 * q^35 + 396 * q^36 - 1108 * q^37 - 660 * q^38 - 1506 * q^40 - 334 * q^41 + 474 * q^42 + 242 * q^43 + 307 * q^44 - 297 * q^45 - 979 * q^46 + 184 * q^47 + 648 * q^48 - 38 * q^49 + 2031 * q^50 + 534 * q^51 - 151 * q^53 - 162 * q^54 + 2064 * q^55 + 2276 * q^56 - 1056 * q^57 - 1161 * q^58 - 537 * q^59 - 1206 * q^60 - 1340 * q^61 + 347 * q^62 - 747 * q^63 + 893 * q^64 - 1890 * q^66 - 2308 * q^67 + 2785 * q^68 + 450 * q^69 + 1420 * q^70 - 96 * q^71 - 783 * q^72 - 2505 * q^73 - 1191 * q^74 - 60 * q^75 - 2409 * q^76 - 2142 * q^77 - 1591 * q^79 + 2671 * q^80 + 729 * q^81 + 1517 * q^82 - 1539 * q^83 - 2820 * q^84 - 4296 * q^85 + 3763 * q^86 - 291 * q^87 - 3716 * q^88 + 592 * q^89 - 486 * q^90 + 515 * q^92 - 2151 * q^93 - 692 * q^94 + 4158 * q^95 - 2121 * q^96 - 1445 * q^97 - 1457 * q^98 - 765 * q^99

Introduction

L-functions

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