LMFDB - Embedded newform 507.4.a.l.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.1362828.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 23x^{2} + 48 $ x^4 - 23*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 39)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.52356$ 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)

$f(q)$	$=$	$q-1.52356 q^{2} +3.00000 q^{3} -5.67878 q^{4} +9.65841 q^{5} -4.57067 q^{6} +22.3639 q^{7} +20.8404 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-1.52356 q^{2} +3.00000 q^{3} -5.67878 q^{4} +9.65841 q^{5} -4.57067 q^{6} +22.3639 q^{7} +20.8404 q^{8} +9.00000 q^{9} -14.7151 q^{10} -50.3050 q^{11} -17.0363 q^{12} -34.0727 q^{14} +28.9752 q^{15} +13.6788 q^{16} +86.1454 q^{17} -13.7120 q^{18} +116.880 q^{19} -54.8480 q^{20} +67.0918 q^{21} +76.6424 q^{22} +72.0000 q^{23} +62.5211 q^{24} -31.7151 q^{25} +27.0000 q^{27} -127.000 q^{28} +14.1454 q^{29} -44.1454 q^{30} -196.215 q^{31} -187.563 q^{32} -150.915 q^{33} -131.247 q^{34} +216.000 q^{35} -51.1090 q^{36} -154.424 q^{37} -178.073 q^{38} +201.285 q^{40} -265.726 q^{41} -102.218 q^{42} +211.855 q^{43} +285.671 q^{44} +86.9257 q^{45} -109.696 q^{46} +67.5535 q^{47} +41.0363 q^{48} +157.145 q^{49} +48.3197 q^{50} +258.436 q^{51} +686.581 q^{53} -41.1360 q^{54} -485.866 q^{55} +466.073 q^{56} +350.639 q^{57} -21.5512 q^{58} -91.9304 q^{59} -164.544 q^{60} +329.006 q^{61} +298.945 q^{62} +201.275 q^{63} +176.333 q^{64} +229.927 q^{66} +768.370 q^{67} -489.201 q^{68} +216.000 q^{69} -329.088 q^{70} +264.969 q^{71} +187.563 q^{72} +771.306 q^{73} +235.273 q^{74} -95.1454 q^{75} -663.734 q^{76} -1125.02 q^{77} +1226.86 q^{79} +132.115 q^{80} +81.0000 q^{81} +404.849 q^{82} -514.019 q^{83} -381.000 q^{84} +832.027 q^{85} -322.772 q^{86} +42.4361 q^{87} -1048.38 q^{88} +527.889 q^{89} -132.436 q^{90} -408.872 q^{92} -588.646 q^{93} -102.921 q^{94} +1128.87 q^{95} -562.690 q^{96} -74.2755 q^{97} -239.420 q^{98} -452.745 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 14 * q^4 + 36 * q^9	$ 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9} + 88 q^{10} + 42 q^{12} + 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} + 20 q^{25} + 108 q^{27} - 384 q^{29} + 264 q^{30} + 864 q^{35} + 126 q^{36} - 492 q^{38} + 952 q^{40} + 252 q^{42} + 1288 q^{43} + 54 q^{48} + 188 q^{49} - 288 q^{51} + 984 q^{53} - 328 q^{55} + 1644 q^{56} + 288 q^{61} - 1668 q^{62} - 1314 q^{64} + 1140 q^{66} - 4380 q^{68} + 864 q^{69} + 3144 q^{74} + 60 q^{75} - 1416 q^{77} + 4320 q^{79} + 324 q^{81} + 3088 q^{82} - 1152 q^{87} - 1036 q^{88} + 792 q^{90} + 1008 q^{92} - 1660 q^{94} + 1872 q^{95}+O(q^{100}) $ 4 * q + 12 * q^3 + 14 * q^4 + 36 * q^9 + 88 * q^10 + 42 * q^12 + 84 * q^14 + 18 * q^16 - 96 * q^17 + 380 * q^22 + 288 * q^23 + 20 * q^25 + 108 * q^27 - 384 * q^29 + 264 * q^30 + 864 * q^35 + 126 * q^36 - 492 * q^38 + 952 * q^40 + 252 * q^42 + 1288 * q^43 + 54 * q^48 + 188 * q^49 - 288 * q^51 + 984 * q^53 - 328 * q^55 + 1644 * q^56 + 288 * q^61 - 1668 * q^62 - 1314 * q^64 + 1140 * q^66 - 4380 * q^68 + 864 * q^69 + 3144 * q^74 + 60 * q^75 - 1416 * q^77 + 4320 * q^79 + 324 * q^81 + 3088 * q^82 - 1152 * q^87 - 1036 * q^88 + 792 * q^90 + 1008 * q^92 - 1660 * q^94 + 1872 * q^95

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.52356	−0.538658	−0.269329	−	0.963048i	$-0.586802\pi$
$2$	−1.52356	−0.538658	−0.269329	+	0.963048i	$0.586802\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−5.67878	−0.709847
$5$	9.65841	0.863874	0.431937	−	0.901904i	$-0.357830\pi$
$5$	9.65841	0.863874	0.431937	+	0.901904i	$0.357830\pi$
$6$	−4.57067	−0.310994
$7$	22.3639	1.20754	0.603769	−	0.797159i	$-0.293665\pi$
$7$	22.3639	1.20754	0.603769	+	0.797159i	$0.293665\pi$
$8$	20.8404	0.921023
$9$	9.00000	0.333333
$10$	−14.7151	−0.465333
$11$	−50.3050	−1.37887	−0.689433	−	0.724349i	$-0.742140\pi$
$11$	−50.3050	−1.37887	−0.689433	+	0.724349i	$0.742140\pi$
$12$	−17.0363	−0.409831
$13$	0	0
$13$	0	0
$14$	−34.0727	−0.650450
$15$	28.9752	0.498758
$16$	13.6788	0.213731
$17$	86.1454	1.22902	0.614509	−	0.788910i	$-0.289354\pi$
$17$	86.1454	1.22902	0.614509	+	0.788910i	$0.289354\pi$
$18$	−13.7120	−0.179553
$19$	116.880	1.41127	0.705633	−	0.708578i	$-0.250663\pi$
$19$	116.880	1.41127	0.705633	+	0.708578i	$0.250663\pi$
$20$	−54.8480	−0.613219
$21$	67.0918	0.697173
$22$	76.6424	0.742737
$23$	72.0000	0.652741	0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000	0.652741	0.326370	+	0.945242i	$0.394174\pi$
$24$	62.5211	0.531753
$25$	−31.7151	−0.253721
$26$	0	0
$27$	27.0000	0.192450
$28$	−127.000	−0.857168
$29$	14.1454	0.0905768	0.0452884	−	0.998974i	$-0.485579\pi$
$29$	14.1454	0.0905768	0.0452884	+	0.998974i	$0.485579\pi$
$30$	−44.1454	−0.268660
$31$	−196.215	−1.13682	−0.568408	−	0.822747i	$-0.692441\pi$
$31$	−196.215	−1.13682	−0.568408	+	0.822747i	$0.692441\pi$
$32$	−187.563	−1.03615
$33$	−150.915	−0.796089
$34$	−131.247	−0.662021
$35$	216.000	1.04316
$36$	−51.1090	−0.236616
$37$	−154.424	−0.686138	−0.343069	−	0.939310i	$-0.611467\pi$
$37$	−154.424	−0.686138	−0.343069	+	0.939310i	$0.611467\pi$
$38$	−178.073	−0.760190
$39$	0	0
$40$	201.285	0.795648
$41$	−265.726	−1.01218	−0.506091	−	0.862480i	$-0.668910\pi$
$41$	−265.726	−1.01218	−0.506091	+	0.862480i	$0.668910\pi$
$42$	−102.218	−0.375538
$43$	211.855	0.751338	0.375669	−	0.926754i	$-0.377413\pi$
$43$	211.855	0.751338	0.375669	+	0.926754i	$0.377413\pi$
$44$	285.671	0.978785
$45$	86.9257	0.287958
$46$	−109.696	−0.351604
$47$	67.5535	0.209653	0.104827	−	0.994491i	$-0.466571\pi$
$47$	67.5535	0.209653	0.104827	+	0.994491i	$0.466571\pi$
$48$	41.0363	0.123398
$49$	157.145	0.458150
$50$	48.3197	0.136669
$51$	258.436	0.709574
$52$	0	0
$53$	686.581	1.77942	0.889710	−	0.456527i	$-0.150907\pi$
$53$	686.581	1.77942	0.889710	+	0.456527i	$0.150907\pi$
$54$	−41.1360	−0.103665
$55$	−485.866	−1.19117
$56$	466.073	1.11217
$57$	350.639	0.814795
$58$	−21.5512	−0.0487899
$59$	−91.9304	−0.202853	−0.101426	−	0.994843i	$-0.532341\pi$
$59$	−91.9304	−0.202853	−0.101426	+	0.994843i	$0.532341\pi$
$60$	−164.544	−0.354042
$61$	329.006	0.690572	0.345286	−	0.938498i	$-0.387782\pi$
$61$	329.006	0.690572	0.345286	+	0.938498i	$0.387782\pi$
$62$	298.945	0.612355
$63$	201.275	0.402513
$64$	176.333	0.344400
$65$	0	0
$66$	229.927	0.428820
$67$	768.370	1.40106	0.700532	−	0.713621i	$-0.252946\pi$
$67$	768.370	1.40106	0.700532	+	0.713621i	$0.252946\pi$
$68$	−489.201	−0.872416
$69$	216.000	0.376860
$70$	−329.088	−0.561908
$71$	264.969	0.442902	0.221451	−	0.975171i	$-0.428921\pi$
$71$	264.969	0.442902	0.221451	+	0.975171i	$0.428921\pi$
$72$	187.563	0.307008
$73$	771.306	1.23664	0.618319	−	0.785927i	$-0.287814\pi$
$73$	771.306	1.23664	0.618319	+	0.785927i	$0.287814\pi$
$74$	235.273	0.369594
$75$	−95.1454	−0.146486
$76$	−663.734	−1.00178
$77$	−1125.02	−1.66503
$78$	0	0
$79$	1226.86	1.74725	0.873624	−	0.486602i	$-0.161764\pi$
$79$	1226.86	1.74725	0.873624	+	0.486602i	$0.161764\pi$
$80$	132.115	0.184637
$81$	81.0000	0.111111
$82$	404.849	0.545220
$83$	−514.019	−0.679771	−0.339885	−	0.940467i	$-0.610388\pi$
$83$	−514.019	−0.679771	−0.339885	+	0.940467i	$0.610388\pi$
$84$	−381.000	−0.494886
$85$	832.027	1.06172
$86$	−322.772	−0.404714
$87$	42.4361	0.0522945
$88$	−1048.38	−1.26997
$89$	527.889	0.628720	0.314360	−	0.949304i	$-0.398210\pi$
$89$	527.889	0.628720	0.314360	+	0.949304i	$0.398210\pi$
$90$	−132.436	−0.155111
$91$	0	0
$92$	−408.872	−0.463346
$93$	−588.646	−0.656341
$94$	−102.921	−0.112931
$95$	1128.87	1.21916
$96$	−562.690	−0.598222
$97$	−74.2755	−0.0777478	−0.0388739	−	0.999244i	$-0.512377\pi$
$97$	−74.2755	−0.0777478	−0.0388739	+	0.999244i	$0.512377\pi$
$98$	−239.420	−0.246786
$99$	−452.745	−0.459622
$100$	180.103	0.180103
$101$	609.419	0.600390	0.300195	−	0.953878i	$-0.402948\pi$
$101$	609.419	0.600390	0.300195	+	0.953878i	$0.402948\pi$
$102$	−393.742	−0.382218
$103$	−32.4361	−0.0310293	−0.0155147	−	0.999880i	$-0.504939\pi$
$103$	−32.4361	−0.0310293	−0.0155147	+	0.999880i	$0.504939\pi$
$104$	0	0
$105$	648.000	0.602270
$106$	−1046.04	−0.958498
$107$	−1725.45	−1.55893	−0.779467	−	0.626444i	$-0.784510\pi$
$107$	−1725.45	−1.55893	−0.779467	+	0.626444i	$0.784510\pi$
$108$	−153.327	−0.136610
$109$	1273.43	1.11902	0.559508	−	0.828825i	$-0.310990\pi$
$109$	1273.43	1.11902	0.559508	+	0.828825i	$0.310990\pi$
$110$	740.244	0.641632
$111$	−463.271	−0.396142
$112$	305.911	0.258088
$113$	1114.73	0.928006	0.464003	−	0.885834i	$-0.346413\pi$
$113$	1114.73	0.928006	0.464003	+	0.885834i	$0.346413\pi$
$114$	−534.218	−0.438896
$115$	695.406	0.563886
$116$	−80.3284	−0.0642957
$117$	0	0
$118$	140.061	0.109268
$119$	1926.55	1.48409
$120$	603.855	0.459368
$121$	1199.59	0.901272
$122$	−501.259	−0.371982
$123$	−797.179	−0.584384
$124$	1114.26	0.806966
$125$	−1513.62	−1.08306
$126$	−306.654	−0.216817
$127$	1174.01	0.820289	0.410144	−	0.912021i	$-0.365478\pi$
$127$	1174.01	0.820289	0.410144	+	0.912021i	$0.365478\pi$
$128$	1231.85	0.850637
$129$	635.564	0.433785
$130$	0	0
$131$	−1445.16	−0.963851	−0.481925	−	0.876212i	$-0.660062\pi$
$131$	−1445.16	−0.963851	−0.481925	+	0.876212i	$0.660062\pi$
$132$	857.013	0.565102
$133$	2613.89	1.70416
$134$	−1170.65	−0.754695
$135$	260.777	0.166253
$136$	1795.30	1.13195
$137$	508.793	0.317293	0.158646	−	0.987335i	$-0.449287\pi$
$137$	508.793	0.317293	0.158646	+	0.987335i	$0.449287\pi$
$138$	−329.088	−0.202999
$139$	−757.018	−0.461938	−0.230969	−	0.972961i	$-0.574190\pi$
$139$	−757.018	−0.461938	−0.230969	+	0.972961i	$0.574190\pi$
$140$	−1226.62	−0.740486
$141$	202.661	0.121043
$142$	−403.695	−0.238573
$143$	0	0
$144$	123.109	0.0712436
$145$	136.622	0.0782470
$146$	−1175.13	−0.666125
$147$	471.436	0.264513
$148$	876.939	0.487054
$149$	−3247.79	−1.78570	−0.892851	−	0.450352i	$-0.851298\pi$
$149$	−3247.79	−1.78570	−0.892851	+	0.450352i	$0.851298\pi$
$150$	144.959	0.0789058
$151$	795.296	0.428611	0.214305	−	0.976767i	$-0.431251\pi$
$151$	795.296	0.428611	0.214305	+	0.976767i	$0.431251\pi$
$152$	2435.82	1.29981
$153$	775.308	0.409673
$154$	1714.03	0.896884
$155$	−1895.13	−0.982067
$156$	0	0
$157$	−65.2732	−0.0331807	−0.0165903	−	0.999862i	$-0.505281\pi$
$157$	−65.2732	−0.0331807	−0.0165903	+	0.999862i	$0.505281\pi$
$158$	−1869.19	−0.941169
$159$	2059.74	1.02735
$160$	−1811.56	−0.895104
$161$	1610.20	0.788210
$162$	−123.408	−0.0598509
$163$	−1855.39	−0.891568	−0.445784	−	0.895140i	$-0.647075\pi$
$163$	−1855.39	−0.891568	−0.445784	+	0.895140i	$0.647075\pi$
$164$	1509.00	0.718495
$165$	−1457.60	−0.687721
$166$	783.137	0.366164
$167$	−3532.54	−1.63686	−0.818432	−	0.574604i	$-0.805156\pi$
$167$	−3532.54	−1.63686	−0.818432	+	0.574604i	$0.805156\pi$
$168$	1398.22	0.642112
$169$	0	0
$170$	−1267.64	−0.571903
$171$	1051.92	0.470422
$172$	−1203.08	−0.533335
$173$	−3178.36	−1.39680	−0.698400	−	0.715708i	$-0.746104\pi$
$173$	−3178.36	−1.39680	−0.698400	+	0.715708i	$0.746104\pi$
$174$	−64.6537	−0.0281689
$175$	−709.275	−0.306378
$176$	−688.111	−0.294706
$177$	−275.791	−0.117117
$178$	−804.267	−0.338665
$179$	2741.16	1.14460	0.572302	−	0.820043i	$-0.306050\pi$
$179$	2741.16	1.14460	0.572302	+	0.820043i	$0.306050\pi$
$180$	−493.632	−0.204406
$181$	3871.09	1.58970	0.794850	−	0.606806i	$-0.207550\pi$
$181$	3871.09	1.58970	0.794850	+	0.606806i	$0.207550\pi$
$182$	0	0
$183$	987.018	0.398702
$184$	1500.51	0.601189
$185$	−1491.49	−0.592737
$186$	896.834	0.353544
$187$	−4333.54	−1.69465
$188$	−383.621	−0.148822
$189$	603.826	0.232391
$190$	−1719.90	−0.656708
$191$	−928.291	−0.351669	−0.175834	−	0.984420i	$-0.556262\pi$
$191$	−928.291	−0.351669	−0.175834	+	0.984420i	$0.556262\pi$
$192$	528.999	0.198840
$193$	−2261.51	−0.843456	−0.421728	−	0.906722i	$-0.638576\pi$
$193$	−2261.51	−0.843456	−0.421728	+	0.906722i	$0.638576\pi$
$194$	113.163	0.0418795
$195$	0	0
$196$	−892.394	−0.325216
$197$	−2265.59	−0.819374	−0.409687	−	0.912226i	$-0.634362\pi$
$197$	−2265.59	−0.819374	−0.409687	+	0.912226i	$0.634362\pi$
$198$	689.782	0.247579
$199$	−260.895	−0.0929366	−0.0464683	−	0.998920i	$-0.514797\pi$
$199$	−260.895	−0.0929366	−0.0464683	+	0.998920i	$0.514797\pi$
$200$	−660.955	−0.233683
$201$	2305.11	0.808905
$202$	−928.483	−0.323405
$203$	316.346	0.109375
$204$	−1467.60	−0.503690
$205$	−2566.49	−0.874399
$206$	49.4181	0.0167142
$207$	648.000	0.217580
$208$	0	0
$209$	−5879.63	−1.94595
$210$	−987.264	−0.324417
$211$	5851.22	1.90907	0.954537	−	0.298092i	$-0.0963502\pi$
$211$	5851.22	1.90907	0.954537	+	0.298092i	$0.0963502\pi$
$212$	−3898.94	−1.26312
$213$	794.907	0.255710
$214$	2628.82	0.839732
$215$	2046.18	0.649062
$216$	562.690	0.177251
$217$	−4388.15	−1.37275
$218$	−1940.15	−0.602767
$219$	2313.92	0.713973
$220$	2759.13	0.845547
$221$	0	0
$222$	705.820	0.213385
$223$	−3463.60	−1.04009	−0.520045	−	0.854139i	$-0.674085\pi$
$223$	−3463.60	−1.04009	−0.520045	+	0.854139i	$0.674085\pi$
$224$	−4194.65	−1.25119
$225$	−285.436	−0.0845737
$226$	−1698.35	−0.499878
$227$	5329.15	1.55819	0.779093	−	0.626908i	$-0.215680\pi$
$227$	5329.15	1.55819	0.779093	+	0.626908i	$0.215680\pi$
$228$	−1991.20	−0.578380
$229$	4773.95	1.37761	0.688803	−	0.724949i	$-0.258137\pi$
$229$	4773.95	1.37761	0.688803	+	0.724949i	$0.258137\pi$
$230$	−1059.49	−0.303742
$231$	−3375.05	−0.961308
$232$	294.795	0.0834233
$233$	−4813.78	−1.35348	−0.676741	−	0.736221i	$-0.736608\pi$
$233$	−4813.78	−1.35348	−0.676741	+	0.736221i	$0.736608\pi$
$234$	0	0
$235$	652.459	0.181114
$236$	522.052	0.143995
$237$	3680.58	1.00877
$238$	−2935.20	−0.799416
$239$	1683.19	0.455549	0.227775	−	0.973714i	$-0.426855\pi$
$239$	1683.19	0.455549	0.227775	+	0.973714i	$0.426855\pi$
$240$	396.346	0.106600
$241$	−664.861	−0.177707	−0.0888537	−	0.996045i	$-0.528320\pi$
$241$	−664.861	−0.177707	−0.0888537	+	0.996045i	$0.528320\pi$
$242$	−1827.65	−0.485477
$243$	243.000	0.0641500
$244$	−1868.35	−0.490201
$245$	1517.77	0.395784
$246$	1214.55	0.314783
$247$	0	0
$248$	−4089.20	−1.04703
$249$	−1542.06	−0.392466
$250$	2306.08	0.583398
$251$	2142.04	0.538662	0.269331	−	0.963048i	$-0.413198\pi$
$251$	2142.04	0.538662	0.269331	+	0.963048i	$0.413198\pi$
$252$	−1143.00	−0.285723
$253$	−3621.96	−0.900042
$254$	−1788.67	−0.441855
$255$	2496.08	0.612983
$256$	−3287.46	−0.802603
$257$	3152.62	0.765194	0.382597	−	0.923915i	$-0.375030\pi$
$257$	3152.62	0.765194	0.382597	+	0.923915i	$0.375030\pi$
$258$	−968.317	−0.233662
$259$	−3453.52	−0.828539
$260$	0	0
$261$	127.308	0.0301923
$262$	2201.79	0.519186
$263$	−2167.71	−0.508238	−0.254119	−	0.967173i	$-0.581786\pi$
$263$	−2167.71	−0.508238	−0.254119	+	0.967173i	$0.581786\pi$
$264$	−3145.13	−0.733216
$265$	6631.28	1.53719
$266$	−3982.40	−0.917958
$267$	1583.67	0.362992
$268$	−4363.40	−0.994542
$269$	−3248.98	−0.736409	−0.368204	−	0.929745i	$-0.620027\pi$
$269$	−3248.98	−0.736409	−0.368204	+	0.929745i	$0.620027\pi$
$270$	−397.308	−0.0895534
$271$	−4897.70	−1.09784	−0.548919	−	0.835876i	$-0.684960\pi$
$271$	−4897.70	−1.09784	−0.548919	+	0.835876i	$0.684960\pi$
$272$	1178.36	0.262679
$273$	0	0
$274$	−775.175	−0.170912
$275$	1595.43	0.349847
$276$	−1226.62	−0.267513
$277$	2900.33	0.629111	0.314555	−	0.949239i	$-0.398145\pi$
$277$	2900.33	0.629111	0.314555	+	0.949239i	$0.398145\pi$
$278$	1153.36	0.248827
$279$	−1765.94	−0.378939
$280$	4501.52	0.960776
$281$	4396.53	0.933364	0.466682	−	0.884425i	$-0.345449\pi$
$281$	4396.53	0.933364	0.466682	+	0.884425i	$0.345449\pi$
$282$	−308.764	−0.0652009
$283$	−559.151	−0.117449	−0.0587245	−	0.998274i	$-0.518703\pi$
$283$	−559.151	−0.117449	−0.0587245	+	0.998274i	$0.518703\pi$
$284$	−1504.70	−0.314393
$285$	3386.62	0.703880
$286$	0	0
$287$	−5942.69	−1.22225
$288$	−1688.07	−0.345384
$289$	2508.02	0.510487
$290$	−208.151	−0.0421484
$291$	−222.827	−0.0448877
$292$	−4380.08	−0.877825
$293$	−6918.65	−1.37950	−0.689748	−	0.724050i	$-0.742278\pi$
$293$	−6918.65	−1.37950	−0.689748	+	0.724050i	$0.742278\pi$
$294$	−718.259	−0.142482
$295$	−887.901	−0.175239
$296$	−3218.25	−0.631949
$297$	−1358.24	−0.265363
$298$	4948.19	0.961883
$299$	0	0
$300$	540.310	0.103983
$301$	4737.90	0.907270
$302$	−1211.68	−0.230875
$303$	1828.26	0.346635
$304$	1598.77	0.301631
$305$	3177.67	0.596567
$306$	−1181.22	−0.220674
$307$	−8980.94	−1.66961	−0.834803	−	0.550548i	$-0.814419\pi$
$307$	−8980.94	−1.66961	−0.834803	+	0.550548i	$0.814419\pi$
$308$	6388.73	1.18192
$309$	−97.3082	−0.0179148
$310$	2887.33	0.528998
$311$	−7943.13	−1.44827	−0.724137	−	0.689656i	$-0.757762\pi$
$311$	−7943.13	−1.44827	−0.724137	+	0.689656i	$0.757762\pi$
$312$	0	0
$313$	−5059.57	−0.913686	−0.456843	−	0.889547i	$-0.651020\pi$
$313$	−5059.57	−0.913686	−0.456843	+	0.889547i	$0.651020\pi$
$314$	99.4473	0.0178730
$315$	1944.00	0.347721
$316$	−6967.07	−1.24028
$317$	−8702.12	−1.54183	−0.770914	−	0.636939i	$-0.780200\pi$
$317$	−8702.12	−1.54183	−0.770914	+	0.636939i	$0.780200\pi$
$318$	−3138.13	−0.553389
$319$	−711.582	−0.124893
$320$	1703.10	0.297519
$321$	−5176.36	−0.900051
$322$	−2453.23	−0.424576
$323$	10068.6	1.73447
$324$	−459.981	−0.0788719
$325$	0	0
$326$	2826.79	0.480250
$327$	3820.30	0.646064
$328$	−5537.84	−0.932244
$329$	1510.76	0.253164
$330$	2220.73	0.370446
$331$	−1737.65	−0.288549	−0.144275	−	0.989538i	$-0.546085\pi$
$331$	−1737.65	−0.288549	−0.144275	+	0.989538i	$0.546085\pi$
$332$	2919.00	0.482533
$333$	−1389.81	−0.228713
$334$	5382.02	0.881710
$335$	7421.23	1.21034
$336$	917.734	0.149007
$337$	2917.47	0.471586	0.235793	−	0.971803i	$-0.424231\pi$
$337$	2917.47	0.471586	0.235793	+	0.971803i	$0.424231\pi$
$338$	0	0
$339$	3344.18	0.535785
$340$	−4724.90	−0.753658
$341$	9870.61	1.56752
$342$	−1602.65	−0.253397
$343$	−4156.44	−0.654305
$344$	4415.13	0.692000
$345$	2086.22	0.325560
$346$	4842.41	0.752397
$347$	5081.23	0.786095	0.393047	−	0.919518i	$-0.371421\pi$
$347$	5081.23	0.786095	0.393047	+	0.919518i	$0.371421\pi$
$348$	−240.985	−0.0371211
$349$	4266.14	0.654330	0.327165	−	0.944967i	$-0.393907\pi$
$349$	4266.14	0.654330	0.327165	+	0.944967i	$0.393907\pi$
$350$	1080.62	0.165033
$351$	0	0
$352$	9435.38	1.42871
$353$	3264.49	0.492213	0.246106	−	0.969243i	$-0.420849\pi$
$353$	3264.49	0.492213	0.246106	+	0.969243i	$0.420849\pi$
$354$	420.183	0.0630861
$355$	2559.18	0.382612
$356$	−2997.76	−0.446295
$357$	5779.65	0.856838
$358$	−4176.31	−0.616550
$359$	−4416.98	−0.649357	−0.324679	−	0.945824i	$-0.605256\pi$
$359$	−4416.98	−0.649357	−0.324679	+	0.945824i	$0.605256\pi$
$360$	1811.56	0.265216
$361$	6801.87	0.991670
$362$	−5897.82	−0.856305
$363$	3598.78	0.520350
$364$	0	0
$365$	7449.59	1.06830
$366$	−1503.78	−0.214764
$367$	−1740.22	−0.247516	−0.123758	−	0.992312i	$-0.539495\pi$
$367$	−1740.22	−0.247516	−0.123758	+	0.992312i	$0.539495\pi$
$368$	984.872	0.139511
$369$	−2391.54	−0.337394
$370$	2272.37	0.319283
$371$	15354.7	2.14872
$372$	3342.79	0.465902
$373$	1176.28	0.163285	0.0816427	−	0.996662i	$-0.473983\pi$
$373$	1176.28	0.163285	0.0816427	+	0.996662i	$0.473983\pi$
$374$	6602.39	0.912838
$375$	−4540.86	−0.625304
$376$	1407.84	0.193095
$377$	0	0
$378$	−919.962	−0.125179
$379$	−7135.54	−0.967092	−0.483546	−	0.875319i	$-0.660651\pi$
$379$	−7135.54	−0.967092	−0.483546	+	0.875319i	$0.660651\pi$
$380$	−6410.62	−0.865415
$381$	3522.04	0.473594
$382$	1414.30	0.189429
$383$	1942.87	0.259207	0.129603	−	0.991566i	$-0.458630\pi$
$383$	1942.87	0.259207	0.129603	+	0.991566i	$0.458630\pi$
$384$	3695.56	0.491116
$385$	−10865.9	−1.43838
$386$	3445.53	0.454334
$387$	1906.69	0.250446
$388$	421.794	0.0551891
$389$	−7545.92	−0.983531	−0.491766	−	0.870728i	$-0.663648\pi$
$389$	−7545.92	−0.983531	−0.491766	+	0.870728i	$0.663648\pi$
$390$	0	0
$391$	6202.47	0.802231
$392$	3274.97	0.421967
$393$	−4335.49	−0.556480
$394$	3451.75	0.441362
$395$	11849.5	1.50940
$396$	2571.04	0.326262
$397$	−415.922	−0.0525806	−0.0262903	−	0.999654i	$-0.508369\pi$
$397$	−415.922	−0.0525806	−0.0262903	+	0.999654i	$0.508369\pi$
$398$	397.489	0.0500611
$399$	7841.67	0.983896
$400$	−433.824	−0.0542280
$401$	958.178	0.119324	0.0596622	−	0.998219i	$-0.480998\pi$
$401$	958.178	0.119324	0.0596622	+	0.998219i	$0.480998\pi$
$402$	−3511.96	−0.435723
$403$	0	0
$404$	−3460.75	−0.426185
$405$	782.331	0.0959860
$406$	−481.970	−0.0589157
$407$	7768.29	0.946093
$408$	5385.90	0.653534
$409$	−4284.83	−0.518022	−0.259011	−	0.965874i	$-0.583397\pi$
$409$	−4284.83	−0.518022	−0.259011	+	0.965874i	$0.583397\pi$
$410$	3910.20	0.471002
$411$	1526.38	0.183189
$412$	184.197	0.0220261
$413$	−2055.92	−0.244953
$414$	−987.264	−0.117201
$415$	−4964.61	−0.587236
$416$	0	0
$417$	−2271.05	−0.266700
$418$	8957.95	1.04820
$419$	−9949.01	−1.16000	−0.580001	−	0.814616i	$-0.696948\pi$
$419$	−9949.01	−1.16000	−0.580001	+	0.814616i	$0.696948\pi$
$420$	−3679.85	−0.427520
$421$	377.250	0.0436724	0.0218362	−	0.999762i	$-0.493049\pi$
$421$	377.250	0.0436724	0.0218362	+	0.999762i	$0.493049\pi$
$422$	−8914.66	−1.02834
$423$	607.982	0.0698843
$424$	14308.6	1.63889
$425$	−2732.11	−0.311828
$426$	−1211.08	−0.137740
$427$	7357.86	0.833892
$428$	9798.47	1.10661
$429$	0	0
$430$	−3117.47	−0.349622
$431$	−2437.13	−0.272373	−0.136186	−	0.990683i	$-0.543485\pi$
$431$	−2437.13	−0.272373	−0.136186	+	0.990683i	$0.543485\pi$
$432$	369.327	0.0411325
$433$	11215.1	1.24471	0.622357	−	0.782733i	$-0.286175\pi$
$433$	11215.1	1.24471	0.622357	+	0.782733i	$0.286175\pi$
$434$	6685.58	0.739443
$435$	409.865	0.0451759
$436$	−7231.55	−0.794331
$437$	8415.34	0.921191
$438$	−3525.38	−0.384588
$439$	−1835.19	−0.199519	−0.0997596	−	0.995012i	$-0.531807\pi$
$439$	−1835.19	−0.199519	−0.0997596	+	0.995012i	$0.531807\pi$
$440$	−10125.6	−1.09709
$441$	1414.31	0.152717
$442$	0	0
$443$	11610.1	1.24518	0.622588	−	0.782550i	$-0.286081\pi$
$443$	11610.1	1.24518	0.622588	+	0.782550i	$0.286081\pi$
$444$	2630.82	0.281201
$445$	5098.56	0.543135
$446$	5276.99	0.560253
$447$	−9743.38	−1.03098
$448$	3943.50	0.415877
$449$	−14087.0	−1.48064	−0.740319	−	0.672255i	$-0.765326\pi$
$449$	−14087.0	−1.48064	−0.740319	+	0.672255i	$0.765326\pi$
$450$	434.878	0.0455563
$451$	13367.4	1.39566
$452$	−6330.29	−0.658743
$453$	2385.89	0.247459
$454$	−8119.26	−0.839330
$455$	0	0
$456$	7307.45	0.750445
$457$	2375.01	0.243103	0.121552	−	0.992585i	$-0.461213\pi$
$457$	2375.01	0.243103	0.121552	+	0.992585i	$0.461213\pi$
$458$	−7273.38	−0.742058
$459$	2325.92	0.236525
$460$	−3949.05	−0.400273
$461$	−6372.06	−0.643766	−0.321883	−	0.946779i	$-0.604316\pi$
$461$	−6372.06	−0.643766	−0.321883	+	0.946779i	$0.604316\pi$
$462$	5142.08	0.517816
$463$	−63.4732	−0.00637117	−0.00318558	−	0.999995i	$-0.501014\pi$
$463$	−63.4732	−0.00637117	−0.00318558	+	0.999995i	$0.501014\pi$
$464$	193.491	0.0193591
$465$	−5685.38	−0.566996
$466$	7334.06	0.729064
$467$	−7855.78	−0.778420	−0.389210	−	0.921149i	$-0.627252\pi$
$467$	−7855.78	−0.778420	−0.389210	+	0.921149i	$0.627252\pi$
$468$	0	0
$469$	17183.8	1.69184
$470$	−994.058	−0.0975584
$471$	−195.820	−0.0191569
$472$	−1915.86	−0.186832
$473$	−10657.3	−1.03599
$474$	−5607.57	−0.543384
$475$	−3706.85	−0.358068
$476$	−10940.4	−1.05348
$477$	6179.23	0.593140
$478$	−2564.43	−0.245385
$479$	−13033.3	−1.24323	−0.621613	−	0.783324i	$-0.713523\pi$
$479$	−13033.3	−1.24323	−0.621613	+	0.783324i	$0.713523\pi$
$480$	−5434.69	−0.516789
$481$	0	0
$482$	1012.95	0.0957235
$483$	4830.61	0.455073
$484$	−6812.23	−0.639766
$485$	−717.384	−0.0671643
$486$	−370.224	−0.0345549
$487$	−69.8976	−0.00650383	−0.00325191	−	0.999995i	$-0.501035\pi$
$487$	−69.8976	−0.00650383	−0.00325191	+	0.999995i	$0.501035\pi$
$488$	6856.60	0.636033
$489$	−5566.18	−0.514747
$490$	−2312.41	−0.213192
$491$	−2625.66	−0.241333	−0.120667	−	0.992693i	$-0.538503\pi$
$491$	−2625.66	−0.241333	−0.120667	+	0.992693i	$0.538503\pi$
$492$	4527.01	0.414823
$493$	1218.56	0.111321
$494$	0	0
$495$	−4372.80	−0.397056
$496$	−2683.99	−0.242973
$497$	5925.75	0.534821
$498$	2349.41	0.211405
$499$	7631.34	0.684621	0.342310	−	0.939587i	$-0.388791\pi$
$499$	7631.34	0.684621	0.342310	+	0.939587i	$0.388791\pi$
$500$	8595.51	0.768806
$501$	−10597.6	−0.945044
$502$	−3263.51	−0.290154
$503$	−4320.14	−0.382953	−0.191477	−	0.981497i	$-0.561328\pi$
$503$	−4320.14	−0.382953	−0.191477	+	0.981497i	$0.561328\pi$
$504$	4194.65	0.370724
$505$	5886.01	0.518662
$506$	5518.26	0.484815
$507$	0	0
$508$	−6666.95	−0.582280
$509$	−12450.7	−1.08422	−0.542109	−	0.840308i	$-0.682374\pi$
$509$	−12450.7	−1.08422	−0.542109	+	0.840308i	$0.682374\pi$
$510$	−3802.92	−0.330188
$511$	17249.4	1.49329
$512$	−4846.21	−0.418309
$513$	3155.75	0.271598
$514$	−4803.18	−0.412178
$515$	−313.281	−0.0268054
$516$	−3609.23	−0.307921
$517$	−3398.28	−0.289083
$518$	5261.63	0.446299
$519$	−9535.08	−0.806443
$520$	0	0
$521$	14373.1	1.20863	0.604314	−	0.796746i	$-0.293447\pi$
$521$	14373.1	1.20863	0.604314	+	0.796746i	$0.293447\pi$
$522$	−193.961	−0.0162633
$523$	−16946.7	−1.41688	−0.708439	−	0.705772i	$-0.750600\pi$
$523$	−16946.7	−1.41688	−0.708439	+	0.705772i	$0.750600\pi$
$524$	8206.76	0.684187
$525$	−2127.82	−0.176887
$526$	3302.62	0.273767
$527$	−16903.0	−1.39717
$528$	−2064.33	−0.170149
$529$	−6983.00	−0.573929
$530$	−10103.1	−0.828022
$531$	−827.373	−0.0676176
$532$	−14843.7	−1.20969
$533$	0	0
$534$	−2412.80	−0.195528
$535$	−16665.1	−1.34672
$536$	16013.1	1.29041
$537$	8223.49	0.660837
$538$	4950.00	0.396673
$539$	−7905.20	−0.631727
$540$	−1480.90	−0.118014
$541$	815.667	0.0648212	0.0324106	−	0.999475i	$-0.489682\pi$
$541$	815.667	0.0648212	0.0324106	+	0.999475i	$0.489682\pi$
$542$	7461.92	0.591359
$543$	11613.3	0.917814
$544$	−16157.7	−1.27345
$545$	12299.3	0.966689
$546$	0	0
$547$	−17971.4	−1.40476	−0.702378	−	0.711804i	$-0.747878\pi$
$547$	−17971.4	−1.40476	−0.702378	+	0.711804i	$0.747878\pi$
$548$	−2889.32	−0.225230
$549$	2961.05	0.230191
$550$	−2430.72	−0.188448
$551$	1653.31	0.127828
$552$	4501.52	0.347097
$553$	27437.4	2.10987
$554$	−4418.81	−0.338876
$555$	−4474.47	−0.342217
$556$	4298.94	0.327906
$557$	6760.83	0.514301	0.257151	−	0.966371i	$-0.417216\pi$
$557$	6760.83	0.514301	0.257151	+	0.966371i	$0.417216\pi$
$558$	2690.50	0.204118
$559$	0	0
$560$	2954.62	0.222956
$561$	−13000.6	−0.978408
$562$	−6698.36	−0.502764
$563$	−12962.7	−0.970359	−0.485179	−	0.874415i	$-0.661246\pi$
$563$	−12962.7	−0.970359	−0.485179	+	0.874415i	$0.661246\pi$
$564$	−1150.86	−0.0859222
$565$	10766.5	0.801681
$566$	851.898	0.0632649
$567$	1811.48	0.134171
$568$	5522.05	0.407923
$569$	−164.757	−0.0121388	−0.00606938	−	0.999982i	$-0.501932\pi$
$569$	−164.757	−0.0121388	−0.00606938	+	0.999982i	$0.501932\pi$
$570$	−5159.70	−0.379151
$571$	5216.55	0.382322	0.191161	−	0.981559i	$-0.438775\pi$
$571$	5216.55	0.382322	0.191161	+	0.981559i	$0.438775\pi$
$572$	0	0
$573$	−2784.87	−0.203036
$574$	9054.01	0.658375
$575$	−2283.49	−0.165614
$576$	1587.00	0.114800
$577$	12753.3	0.920151	0.460076	−	0.887880i	$-0.347822\pi$
$577$	12753.3	0.920151	0.460076	+	0.887880i	$0.347822\pi$
$578$	−3821.11	−0.274978
$579$	−6784.53	−0.486970
$580$	−775.844	−0.0555434
$581$	−11495.5	−0.820849
$582$	339.489	0.0241791
$583$	−34538.5	−2.45358
$584$	16074.3	1.13897
$585$	0	0
$586$	10540.9	0.743076
$587$	1575.38	0.110771	0.0553857	−	0.998465i	$-0.482361\pi$
$587$	1575.38	0.110771	0.0553857	+	0.998465i	$0.482361\pi$
$588$	−2677.18	−0.187764
$589$	−22933.6	−1.60435
$590$	1352.77	0.0943941
$591$	−6796.77	−0.473066
$592$	−2112.33	−0.146649
$593$	3845.95	0.266331	0.133165	−	0.991094i	$-0.457486\pi$
$593$	3845.95	0.266331	0.133165	+	0.991094i	$0.457486\pi$
$594$	2069.35	0.142940
$595$	18607.4	1.28207
$596$	18443.5	1.26758
$597$	−782.686	−0.0536570
$598$	0	0
$599$	6107.20	0.416583	0.208292	−	0.978067i	$-0.433210\pi$
$599$	6107.20	0.416583	0.208292	+	0.978067i	$0.433210\pi$
$600$	−1982.86	−0.134917
$601$	9638.90	0.654208	0.327104	−	0.944988i	$-0.393927\pi$
$601$	9638.90	0.654208	0.327104	+	0.944988i	$0.393927\pi$
$602$	−7218.46	−0.488708
$603$	6915.33	0.467022
$604$	−4516.31	−0.304248
$605$	11586.2	0.778586
$606$	−2785.45	−0.186718
$607$	11821.7	0.790489	0.395244	−	0.918576i	$-0.370660\pi$
$607$	11821.7	0.790489	0.395244	+	0.918576i	$0.370660\pi$
$608$	−21922.4	−1.46228
$609$	949.037	0.0631477
$610$	−4841.36	−0.321346
$611$	0	0
$612$	−4402.80	−0.290805
$613$	8107.86	0.534214	0.267107	−	0.963667i	$-0.413932\pi$
$613$	8107.86	0.534214	0.267107	+	0.963667i	$0.413932\pi$
$614$	13683.0	0.899347
$615$	−7699.48	−0.504834
$616$	−23445.8	−1.53354
$617$	27647.6	1.80397	0.901985	−	0.431768i	$-0.142110\pi$
$617$	27647.6	1.80397	0.901985	+	0.431768i	$0.142110\pi$
$618$	148.254	0.00964995
$619$	29181.9	1.89486	0.947432	−	0.319956i	$-0.103668\pi$
$619$	29181.9	1.89486	0.947432	+	0.319956i	$0.103668\pi$
$620$	10762.0	0.697118
$621$	1944.00	0.125620
$622$	12101.8	0.780125
$623$	11805.7	0.759204
$624$	0	0
$625$	−10654.8	−0.681905
$626$	7708.53	0.492165
$627$	−17638.9	−1.12349
$628$	370.672	0.0235532
$629$	−13302.9	−0.843277
$630$	−2961.79	−0.187303
$631$	−2209.34	−0.139386	−0.0696928	−	0.997569i	$-0.522202\pi$
$631$	−2209.34	−0.139386	−0.0696928	+	0.997569i	$0.522202\pi$
$632$	25568.2	1.60926
$633$	17553.7	1.10220
$634$	13258.2	0.830518
$635$	11339.1	0.708627
$636$	−11696.8	−0.729260
$637$	0	0
$638$	1084.13	0.0672748
$639$	2384.72	0.147634
$640$	11897.8	0.734844
$641$	−18256.0	−1.12491	−0.562455	−	0.826828i	$-0.690143\pi$
$641$	−18256.0	−1.12491	−0.562455	+	0.826828i	$0.690143\pi$
$642$	7886.47	0.484820
$643$	1281.61	0.0786033	0.0393016	−	0.999227i	$-0.487487\pi$
$643$	1281.61	0.0786033	0.0393016	+	0.999227i	$0.487487\pi$
$644$	−9143.99	−0.559509
$645$	6138.54	0.374736
$646$	−15340.1	−0.934287
$647$	16393.2	0.996107	0.498054	−	0.867146i	$-0.334048\pi$
$647$	16393.2	0.996107	0.498054	+	0.867146i	$0.334048\pi$
$648$	1688.07	0.102336
$649$	4624.56	0.279707
$650$	0	0
$651$	−13164.4	−0.792557
$652$	10536.4	0.632878
$653$	16759.2	1.00434	0.502172	−	0.864768i	$-0.332534\pi$
$653$	16759.2	1.00434	0.502172	+	0.864768i	$0.332534\pi$
$654$	−5820.44	−0.348008
$655$	−13958.0	−0.832646
$656$	−3634.81	−0.216335
$657$	6941.76	0.412213
$658$	−2301.73	−0.136369
$659$	−29659.3	−1.75320	−0.876601	−	0.481217i	$-0.840195\pi$
$659$	−29659.3	−1.75320	−0.876601	+	0.481217i	$0.840195\pi$
$660$	8277.38	0.488177
$661$	−10386.0	−0.611147	−0.305573	−	0.952169i	$-0.598848\pi$
$661$	−10386.0	−0.611147	−0.305573	+	0.952169i	$0.598848\pi$
$662$	2647.40	0.155429
$663$	0	0
$664$	−10712.4	−0.626084
$665$	25246.0	1.47218
$666$	2117.46	0.123198
$667$	1018.47	0.0591232
$668$	20060.5	1.16192
$669$	−10390.8	−0.600496
$670$	−11306.7	−0.651962
$671$	−16550.6	−0.952206
$672$	−12584.0	−0.722376
$673$	−19449.6	−1.11400	−0.557002	−	0.830511i	$-0.688049\pi$
$673$	−19449.6	−1.11400	−0.557002	+	0.830511i	$0.688049\pi$
$674$	−4444.92	−0.254024
$675$	−856.308	−0.0488286
$676$	0	0
$677$	−6629.48	−0.376354	−0.188177	−	0.982135i	$-0.560258\pi$
$677$	−6629.48	−0.376354	−0.188177	+	0.982135i	$0.560258\pi$
$678$	−5095.04	−0.288605
$679$	−1661.09	−0.0938835
$680$	17339.8	0.977867
$681$	15987.5	0.899619
$682$	−15038.4	−0.844356
$683$	−2526.40	−0.141537	−0.0707687	−	0.997493i	$-0.522545\pi$
$683$	−2526.40	−0.141537	−0.0707687	+	0.997493i	$0.522545\pi$
$684$	−5973.61	−0.333928
$685$	4914.13	0.274101
$686$	6332.57	0.352447
$687$	14321.9	0.795361
$688$	2897.91	0.160584
$689$	0	0
$690$	−3178.47	−0.175365
$691$	3808.76	0.209685	0.104842	−	0.994489i	$-0.466566\pi$
$691$	3808.76	0.209685	0.104842	+	0.994489i	$0.466566\pi$
$692$	18049.2	0.991515
$693$	−10125.2	−0.555011
$694$	−7741.54	−0.423436
$695$	−7311.59	−0.399056
$696$	884.384	0.0481645
$697$	−22891.1	−1.24399
$698$	−6499.69	−0.352460
$699$	−14441.3	−0.781433
$700$	4027.81	0.217482
$701$	33617.8	1.81131	0.905655	−	0.424015i	$-0.139380\pi$
$701$	33617.8	1.81131	0.905655	+	0.424015i	$0.139380\pi$
$702$	0	0
$703$	−18049.0	−0.968323
$704$	−8870.43	−0.474882
$705$	1957.38	0.104566
$706$	−4973.63	−0.265134
$707$	13629.0	0.724994
$708$	1566.16	0.0831353
$709$	26606.5	1.40935	0.704675	−	0.709530i	$-0.251093\pi$
$709$	26606.5	1.40935	0.704675	+	0.709530i	$0.251093\pi$
$710$	−3899.05	−0.206097
$711$	11041.7	0.582416
$712$	11001.4	0.579066
$713$	−14127.5	−0.742046
$714$	−8805.61	−0.461543
$715$	0	0
$716$	−15566.5	−0.812494
$717$	5049.56	0.263011
$718$	6729.51	0.349781
$719$	−16539.3	−0.857877	−0.428939	−	0.903334i	$-0.641112\pi$
$719$	−16539.3	−0.857877	−0.428939	+	0.903334i	$0.641112\pi$
$720$	1189.04	0.0615456
$721$	−725.398	−0.0374691
$722$	−10363.0	−0.534171
$723$	−1994.58	−0.102599
$724$	−21983.1	−1.12844
$725$	−448.622	−0.0229812
$726$	−5482.94	−0.280291
$727$	−12757.5	−0.650823	−0.325411	−	0.945573i	$-0.605503\pi$
$727$	−12757.5	−0.650823	−0.325411	+	0.945573i	$0.605503\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−11349.9	−0.575448
$731$	18250.3	0.923408
$732$	−5605.06	−0.283017
$733$	−20523.4	−1.03417	−0.517087	−	0.855933i	$-0.672984\pi$
$733$	−20523.4	−1.03417	−0.517087	+	0.855933i	$0.672984\pi$
$734$	2651.31	0.133327
$735$	4553.32	0.228506
$736$	−13504.6	−0.676338
$737$	−38652.9	−1.93188
$738$	3643.64	0.181740
$739$	7462.66	0.371473	0.185736	−	0.982600i	$-0.440533\pi$
$739$	7462.66	0.371473	0.185736	+	0.982600i	$0.440533\pi$
$740$	8469.84	0.420753
$741$	0	0
$742$	−23393.7	−1.15742
$743$	−18847.8	−0.930632	−0.465316	−	0.885145i	$-0.654059\pi$
$743$	−18847.8	−0.930632	−0.465316	+	0.885145i	$0.654059\pi$
$744$	−12267.6	−0.604506
$745$	−31368.5	−1.54262
$746$	−1792.13	−0.0879549
$747$	−4626.17	−0.226590
$748$	24609.2	1.20294
$749$	−38587.9	−1.88247
$750$	6918.25	0.336825
$751$	−20832.6	−1.01224	−0.506119	−	0.862464i	$-0.668920\pi$
$751$	−20832.6	−1.01224	−0.506119	+	0.862464i	$0.668920\pi$
$752$	924.050	0.0448093
$753$	6426.11	0.310996
$754$	0	0
$755$	7681.29	0.370266
$756$	−3429.00	−0.164962
$757$	−20860.9	−1.00159	−0.500795	−	0.865566i	$-0.666959\pi$
$757$	−20860.9	−1.00159	−0.500795	+	0.865566i	$0.666959\pi$
$758$	10871.4	0.520932
$759$	−10865.9	−0.519640
$760$	23526.1	1.12287
$761$	4464.86	0.212682	0.106341	−	0.994330i	$-0.466086\pi$
$761$	4464.86	0.212682	0.106341	+	0.994330i	$0.466086\pi$
$762$	−5366.01	−0.255105
$763$	28479.0	1.35126
$764$	5271.56	0.249631
$765$	7488.24	0.353906
$766$	−2960.08	−0.139624
$767$	0	0
$768$	−9862.38	−0.463383
$769$	−23797.7	−1.11595	−0.557977	−	0.829857i	$-0.688422\pi$
$769$	−23797.7	−1.11595	−0.557977	+	0.829857i	$0.688422\pi$
$770$	16554.8	0.774795
$771$	9457.85	0.441785
$772$	12842.6	0.598725
$773$	−29418.7	−1.36885	−0.684423	−	0.729085i	$-0.739946\pi$
$773$	−29418.7	−1.36885	−0.684423	+	0.729085i	$0.739946\pi$
$774$	−2904.95	−0.134905
$775$	6222.99	0.288434
$776$	−1547.93	−0.0716075
$777$	−10360.6	−0.478357
$778$	11496.6	0.529787
$779$	−31058.0	−1.42846
$780$	0	0
$781$	−13329.3	−0.610703
$782$	−9449.80	−0.432128
$783$	381.925	0.0174315
$784$	2149.56	0.0979208
$785$	−630.435	−0.0286640
$786$	6605.36	0.299752
$787$	−23896.0	−1.08234	−0.541170	−	0.840913i	$-0.682018\pi$
$787$	−23896.0	−1.08234	−0.541170	+	0.840913i	$0.682018\pi$
$788$	12865.8	0.581630
$789$	−6503.13	−0.293432
$790$	−18053.4	−0.813052
$791$	24929.7	1.12060
$792$	−9435.38	−0.423323
$793$	0	0
$794$	633.680	0.0283230
$795$	19893.9	0.887500
$796$	1481.57	0.0659708
$797$	−1034.67	−0.0459847	−0.0229923	−	0.999736i	$-0.507319\pi$
$797$	−1034.67	−0.0459847	−0.0229923	+	0.999736i	$0.507319\pi$
$798$	−11947.2	−0.529983
$799$	5819.42	0.257667
$800$	5948.59	0.262893
$801$	4751.00	0.209573
$802$	−1459.84	−0.0642751
$803$	−38800.6	−1.70516
$804$	−13090.2	−0.574199
$805$	15552.0	0.680914
$806$	0	0
$807$	−9746.95	−0.425166
$808$	12700.5	0.552973
$809$	29557.9	1.28455	0.642275	−	0.766474i	$-0.277991\pi$
$809$	29557.9	1.28455	0.642275	+	0.766474i	$0.277991\pi$
$810$	−1191.92	−0.0517037
$811$	30268.9	1.31058	0.655292	−	0.755376i	$-0.272546\pi$
$811$	30268.9	1.31058	0.655292	+	0.755376i	$0.272546\pi$
$812$	−1796.46	−0.0776396
$813$	−14693.1	−0.633837
$814$	−11835.4	−0.509621
$815$	−17920.2	−0.770203
$816$	3535.09	0.151658
$817$	24761.5	1.06034
$818$	6528.17	0.279037
$819$	0	0
$820$	14574.6	0.620690
$821$	39292.7	1.67031	0.835156	−	0.550013i	$-0.185377\pi$
$821$	39292.7	1.67031	0.835156	+	0.550013i	$0.185377\pi$
$822$	−2325.52	−0.0986763
$823$	−25988.8	−1.10074	−0.550372	−	0.834920i	$-0.685514\pi$
$823$	−25988.8	−1.10074	−0.550372	+	0.834920i	$0.685514\pi$
$824$	−675.980	−0.0285787
$825$	4786.29	0.201984
$826$	3132.31	0.131946
$827$	19585.1	0.823508	0.411754	−	0.911295i	$-0.364916\pi$
$827$	19585.1	0.823508	0.411754	+	0.911295i	$0.364916\pi$
$828$	−3679.85	−0.154449
$829$	666.472	0.0279222	0.0139611	−	0.999903i	$-0.495556\pi$
$829$	666.472	0.0279222	0.0139611	+	0.999903i	$0.495556\pi$
$830$	7563.86	0.316320
$831$	8700.98	0.363217
$832$	0	0
$833$	13537.3	0.563075
$834$	3460.07	0.143660
$835$	−34118.7	−1.41404
$836$	33389.1	1.38133
$837$	−5297.81	−0.218780
$838$	15157.9	0.624845
$839$	36379.7	1.49698	0.748490	−	0.663146i	$-0.230779\pi$
$839$	36379.7	1.49698	0.748490	+	0.663146i	$0.230779\pi$
$840$	13504.6	0.554704
$841$	−24188.9	−0.991796
$842$	−574.762	−0.0235245
$843$	13189.6	0.538878
$844$	−33227.8	−1.35515
$845$	0	0
$846$	−926.293	−0.0376438
$847$	26827.6	1.08832
$848$	9391.60	0.380317
$849$	−1677.45	−0.0678093
$850$	4162.52	0.167969
$851$	−11118.5	−0.447871
$852$	−4514.10	−0.181515
$853$	39951.4	1.60364	0.801822	−	0.597563i	$-0.203864\pi$
$853$	39951.4	1.60364	0.801822	+	0.597563i	$0.203864\pi$
$854$	−11210.1	−0.449183
$855$	10159.8	0.406385
$856$	−35959.1	−1.43581
$857$	17226.4	0.686629	0.343314	−	0.939221i	$-0.388450\pi$
$857$	17226.4	0.686629	0.343314	+	0.939221i	$0.388450\pi$
$858$	0	0
$859$	−33392.3	−1.32634	−0.663172	−	0.748467i	$-0.730790\pi$
$859$	−33392.3	−1.32634	−0.663172	+	0.748467i	$0.730790\pi$
$860$	−11619.8	−0.460735
$861$	−17828.1	−0.705666
$862$	3713.11	0.146716
$863$	−17381.9	−0.685614	−0.342807	−	0.939406i	$-0.611378\pi$
$863$	−17381.9	−0.685614	−0.342807	+	0.939406i	$0.611378\pi$
$864$	−5064.21	−0.199407
$865$	−30697.9	−1.20666
$866$	−17086.8	−0.670476
$867$	7524.07	0.294730
$868$	24919.3	0.974443
$869$	−61717.2	−2.40922
$870$	−624.452	−0.0243344
$871$	0	0
$872$	26538.8	1.03064
$873$	−668.480	−0.0259159
$874$	−12821.2	−0.496207
$875$	−33850.5	−1.30783
$876$	−13140.2	−0.506812
$877$	14335.6	0.551970	0.275985	−	0.961162i	$-0.410996\pi$
$877$	14335.6	0.551970	0.275985	+	0.961162i	$0.410996\pi$
$878$	2796.02	0.107473
$879$	−20756.0	−0.796452
$880$	−6646.06	−0.254589
$881$	5436.53	0.207901	0.103951	−	0.994582i	$-0.466852\pi$
$881$	5436.53	0.207901	0.103951	+	0.994582i	$0.466852\pi$
$882$	−2154.78	−0.0822620
$883$	21185.9	0.807430	0.403715	−	0.914885i	$-0.367719\pi$
$883$	21185.9	0.807430	0.403715	+	0.914885i	$0.367719\pi$
$884$	0	0
$885$	−2663.70	−0.101174
$886$	−17688.6	−0.670724
$887$	−12661.4	−0.479287	−0.239644	−	0.970861i	$-0.577031\pi$
$887$	−12661.4	−0.479287	−0.239644	+	0.970861i	$0.577031\pi$
$888$	−9654.75	−0.364856
$889$	26255.5	0.990531
$890$	−7767.94	−0.292564
$891$	−4074.71	−0.153207
$892$	19669.0	0.738305
$893$	7895.63	0.295876
$894$	14844.6	0.555343
$895$	26475.3	0.988794
$896$	27549.1	1.02718
$897$	0	0
$898$	21462.3	0.797558
$899$	−2775.54	−0.102969
$900$	1620.93	0.0600344
$901$	59145.8	2.18694
$902$	−20365.9	−0.751786
$903$	14213.7	0.523812
$904$	23231.3	0.854715
$905$	37388.6	1.37330
$906$	−3635.03	−0.133296
$907$	3545.27	0.129789	0.0648946	−	0.997892i	$-0.479329\pi$
$907$	3545.27	0.129789	0.0648946	+	0.997892i	$0.479329\pi$
$908$	−30263.1	−1.10607
$909$	5484.77	0.200130
$910$	0	0
$911$	−3913.88	−0.142341	−0.0711706	−	0.997464i	$-0.522673\pi$
$911$	−3913.88	−0.142341	−0.0711706	+	0.997464i	$0.522673\pi$
$912$	4796.32	0.174147
$913$	25857.7	0.937313
$914$	−3618.46	−0.130950
$915$	9533.02	0.344428
$916$	−27110.2	−0.977890
$917$	−32319.5	−1.16389
$918$	−3543.67	−0.127406
$919$	6917.79	0.248310	0.124155	−	0.992263i	$-0.460378\pi$
$919$	6917.79	0.248310	0.124155	+	0.992263i	$0.460378\pi$
$920$	14492.5	0.519352
$921$	−26942.8	−0.963948
$922$	9708.18	0.346770
$923$	0	0
$924$	19166.2	0.682382
$925$	4897.57	0.174088
$926$	96.7050	0.00343188
$927$	−291.925	−0.0103431
$928$	−2653.15	−0.0938512
$929$	−3753.03	−0.132543	−0.0662717	−	0.997802i	$-0.521110\pi$
$929$	−3753.03	−0.132543	−0.0662717	+	0.997802i	$0.521110\pi$
$930$	8662.00	0.305417
$931$	18367.1	0.646571
$932$	27336.4	0.960765
$933$	−23829.4	−0.836162
$934$	11968.7	0.419302
$935$	−41855.1	−1.46397
$936$	0	0
$937$	48189.0	1.68011	0.840056	−	0.542499i	$-0.182522\pi$
$937$	48189.0	1.68011	0.840056	+	0.542499i	$0.182522\pi$
$938$	−26180.4	−0.911323
$939$	−15178.7	−0.527517
$940$	−3705.17	−0.128563
$941$	26656.4	0.923456	0.461728	−	0.887022i	$-0.347230\pi$
$941$	26656.4	0.923456	0.461728	+	0.887022i	$0.347230\pi$
$942$	298.342	0.0103190
$943$	−19132.3	−0.660693
$944$	−1257.50	−0.0433559
$945$	5832.00	0.200757
$946$	16237.1	0.558047
$947$	31258.9	1.07263	0.536314	−	0.844018i	$-0.319816\pi$
$947$	31258.9	1.07263	0.536314	+	0.844018i	$0.319816\pi$
$948$	−20901.2	−0.716076
$949$	0	0
$950$	5647.60	0.192876
$951$	−26106.4	−0.890175
$952$	40150.0	1.36688
$953$	10602.4	0.360382	0.180191	−	0.983632i	$-0.442328\pi$
$953$	10602.4	0.360382	0.180191	+	0.983632i	$0.442328\pi$
$954$	−9414.40	−0.319499
$955$	−8965.81	−0.303798
$956$	−9558.44	−0.323370
$957$	−2134.75	−0.0721072
$958$	19856.9	0.669674
$959$	11378.6	0.383144
$960$	5109.29	0.171772
$961$	8709.45	0.292352
$962$	0	0
$963$	−15529.1	−0.519645
$964$	3775.60	0.126145
$965$	−21842.6	−0.728640
$966$	−7359.70	−0.245129
$967$	−4815.93	−0.160155	−0.0800774	−	0.996789i	$-0.525517\pi$
$967$	−4815.93	−0.160155	−0.0800774	+	0.996789i	$0.525517\pi$
$968$	25000.0	0.830092
$969$	30205.9	1.00140
$970$	1092.97	0.0361786
$971$	56729.5	1.87491	0.937454	−	0.348109i	$-0.113176\pi$
$971$	56729.5	1.87491	0.937454	+	0.348109i	$0.113176\pi$
$972$	−1379.94	−0.0455367
$973$	−16929.9	−0.557808
$974$	106.493	0.00350334
$975$	0	0
$976$	4500.40	0.147597
$977$	−55246.2	−1.80909	−0.904546	−	0.426375i	$-0.859790\pi$
$977$	−55246.2	−1.80909	−0.904546	+	0.426375i	$0.859790\pi$
$978$	8480.38	0.277273
$979$	−26555.4	−0.866921
$980$	−8619.11	−0.280946
$981$	11460.9	0.373005
$982$	4000.34	0.129996
$983$	−22820.6	−0.740451	−0.370226	−	0.928942i	$-0.620720\pi$
$983$	−22820.6	−0.740451	−0.370226	+	0.928942i	$0.620720\pi$
$984$	−16613.5	−0.538231
$985$	−21882.0	−0.707836
$986$	−1856.54	−0.0599637
$987$	4532.29	0.146164
$988$	0	0
$989$	15253.5	0.490429
$990$	6662.20	0.213877
$991$	−46490.4	−1.49023	−0.745115	−	0.666936i	$-0.767605\pi$
$991$	−46490.4	−1.49023	−0.745115	+	0.666936i	$0.767605\pi$
$992$	36802.8	1.17791
$993$	−5212.94	−0.166594
$994$	−9028.21	−0.288086
$995$	−2519.84	−0.0802856
$996$	8757.01	0.278591
$997$	24943.4	0.792344	0.396172	−	0.918176i	$-0.370338\pi$
$997$	24943.4	0.792344	0.396172	+	0.918176i	$0.370338\pi$
$998$	−11626.8	−0.368777
$999$	−4169.44	−0.132047

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.l.1.2		4
3.2	odd	2		1521.4.a.w.1.3		4
13.5	odd	4		39.4.b.b.25.3	yes	4
13.8	odd	4		39.4.b.b.25.2	✓	4
13.12	even	2	inner	507.4.a.l.1.3		4
39.5	even	4		117.4.b.e.64.2		4
39.8	even	4		117.4.b.e.64.3		4
39.38	odd	2		1521.4.a.w.1.2		4
52.31	even	4		624.4.c.c.337.2		4
52.47	even	4		624.4.c.c.337.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.b.25.2	✓	4	13.8	odd	4
39.4.b.b.25.3	yes	4	13.5	odd	4
117.4.b.e.64.2		4	39.5	even	4
117.4.b.e.64.3		4	39.8	even	4
507.4.a.l.1.2		4	1.1	even	1	trivial
507.4.a.l.1.3		4	13.12	even	2	inner
624.4.c.c.337.2		4	52.31	even	4
624.4.c.c.337.3		4	52.47	even	4
1521.4.a.w.1.2		4	39.38	odd	2
1521.4.a.w.1.3		4	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)

\(f(q)\)	\(=\)	\(q-1.52356 q^{2} +3.00000 q^{3} -5.67878 q^{4} +9.65841 q^{5} -4.57067 q^{6} +22.3639 q^{7} +20.8404 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-1.52356 q^{2} +3.00000 q^{3} -5.67878 q^{4} +9.65841 q^{5} -4.57067 q^{6} +22.3639 q^{7} +20.8404 q^{8} +9.00000 q^{9} -14.7151 q^{10} -50.3050 q^{11} -17.0363 q^{12} -34.0727 q^{14} +28.9752 q^{15} +13.6788 q^{16} +86.1454 q^{17} -13.7120 q^{18} +116.880 q^{19} -54.8480 q^{20} +67.0918 q^{21} +76.6424 q^{22} +72.0000 q^{23} +62.5211 q^{24} -31.7151 q^{25} +27.0000 q^{27} -127.000 q^{28} +14.1454 q^{29} -44.1454 q^{30} -196.215 q^{31} -187.563 q^{32} -150.915 q^{33} -131.247 q^{34} +216.000 q^{35} -51.1090 q^{36} -154.424 q^{37} -178.073 q^{38} +201.285 q^{40} -265.726 q^{41} -102.218 q^{42} +211.855 q^{43} +285.671 q^{44} +86.9257 q^{45} -109.696 q^{46} +67.5535 q^{47} +41.0363 q^{48} +157.145 q^{49} +48.3197 q^{50} +258.436 q^{51} +686.581 q^{53} -41.1360 q^{54} -485.866 q^{55} +466.073 q^{56} +350.639 q^{57} -21.5512 q^{58} -91.9304 q^{59} -164.544 q^{60} +329.006 q^{61} +298.945 q^{62} +201.275 q^{63} +176.333 q^{64} +229.927 q^{66} +768.370 q^{67} -489.201 q^{68} +216.000 q^{69} -329.088 q^{70} +264.969 q^{71} +187.563 q^{72} +771.306 q^{73} +235.273 q^{74} -95.1454 q^{75} -663.734 q^{76} -1125.02 q^{77} +1226.86 q^{79} +132.115 q^{80} +81.0000 q^{81} +404.849 q^{82} -514.019 q^{83} -381.000 q^{84} +832.027 q^{85} -322.772 q^{86} +42.4361 q^{87} -1048.38 q^{88} +527.889 q^{89} -132.436 q^{90} -408.872 q^{92} -588.646 q^{93} -102.921 q^{94} +1128.87 q^{95} -562.690 q^{96} -74.2755 q^{97} -239.420 q^{98} -452.745 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 14 * q^4 + 36 * q^9	\( 4 q + 12 q^{3} + 14 q^{4} + 36 q^{9} + 88 q^{10} + 42 q^{12} + 84 q^{14} + 18 q^{16} - 96 q^{17} + 380 q^{22} + 288 q^{23} + 20 q^{25} + 108 q^{27} - 384 q^{29} + 264 q^{30} + 864 q^{35} + 126 q^{36} - 492 q^{38} + 952 q^{40} + 252 q^{42} + 1288 q^{43} + 54 q^{48} + 188 q^{49} - 288 q^{51} + 984 q^{53} - 328 q^{55} + 1644 q^{56} + 288 q^{61} - 1668 q^{62} - 1314 q^{64} + 1140 q^{66} - 4380 q^{68} + 864 q^{69} + 3144 q^{74} + 60 q^{75} - 1416 q^{77} + 4320 q^{79} + 324 q^{81} + 3088 q^{82} - 1152 q^{87} - 1036 q^{88} + 792 q^{90} + 1008 q^{92} - 1660 q^{94} + 1872 q^{95}+O(q^{100}) \) 4 * q + 12 * q^3 + 14 * q^4 + 36 * q^9 + 88 * q^10 + 42 * q^12 + 84 * q^14 + 18 * q^16 - 96 * q^17 + 380 * q^22 + 288 * q^23 + 20 * q^25 + 108 * q^27 - 384 * q^29 + 264 * q^30 + 864 * q^35 + 126 * q^36 - 492 * q^38 + 952 * q^40 + 252 * q^42 + 1288 * q^43 + 54 * q^48 + 188 * q^49 - 288 * q^51 + 984 * q^53 - 328 * q^55 + 1644 * q^56 + 288 * q^61 - 1668 * q^62 - 1314 * q^64 + 1140 * q^66 - 4380 * q^68 + 864 * q^69 + 3144 * q^74 + 60 * q^75 - 1416 * q^77 + 4320 * q^79 + 324 * q^81 + 3088 * q^82 - 1152 * q^87 - 1036 * q^88 + 792 * q^90 + 1008 * q^92 - 1660 * q^94 + 1872 * q^95

Introduction

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