LMFDB - Embedded newform 507.4.a.o.1.9

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			1.9
Root			$5.39246$ 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)

$f(q)$	$=$	$q+4.83750 q^{2} +3.00000 q^{3} +15.4014 q^{4} -21.1983 q^{5} +14.5125 q^{6} -16.2806 q^{7} +35.8043 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q+4.83750 q^{2} +3.00000 q^{3} +15.4014 q^{4} -21.1983 q^{5} +14.5125 q^{6} -16.2806 q^{7} +35.8043 q^{8} +9.00000 q^{9} -102.547 q^{10} -30.7532 q^{11} +46.2042 q^{12} -78.7575 q^{14} -63.5948 q^{15} +49.9922 q^{16} +46.2371 q^{17} +43.5375 q^{18} -144.865 q^{19} -326.483 q^{20} -48.8418 q^{21} -148.769 q^{22} +8.38045 q^{23} +107.413 q^{24} +324.366 q^{25} +27.0000 q^{27} -250.744 q^{28} -242.958 q^{29} -307.640 q^{30} -87.9353 q^{31} -44.5973 q^{32} -92.2597 q^{33} +223.672 q^{34} +345.121 q^{35} +138.613 q^{36} +49.6950 q^{37} -700.783 q^{38} -758.990 q^{40} +107.947 q^{41} -236.272 q^{42} -35.4166 q^{43} -473.643 q^{44} -190.784 q^{45} +40.5404 q^{46} +374.815 q^{47} +149.977 q^{48} -77.9418 q^{49} +1569.12 q^{50} +138.711 q^{51} -348.583 q^{53} +130.613 q^{54} +651.915 q^{55} -582.916 q^{56} -434.594 q^{57} -1175.31 q^{58} +679.430 q^{59} -979.450 q^{60} -230.403 q^{61} -425.387 q^{62} -146.525 q^{63} -615.677 q^{64} -446.306 q^{66} -295.642 q^{67} +712.117 q^{68} +25.1413 q^{69} +1669.52 q^{70} +329.215 q^{71} +322.239 q^{72} +48.9973 q^{73} +240.399 q^{74} +973.099 q^{75} -2231.12 q^{76} +500.681 q^{77} -107.942 q^{79} -1059.75 q^{80} +81.0000 q^{81} +522.192 q^{82} +515.654 q^{83} -752.233 q^{84} -980.147 q^{85} -171.328 q^{86} -728.874 q^{87} -1101.10 q^{88} +984.453 q^{89} -922.919 q^{90} +129.071 q^{92} -263.806 q^{93} +1813.17 q^{94} +3070.88 q^{95} -133.792 q^{96} +487.072 q^{97} -377.043 q^{98} -276.779 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	4.83750	1.71031	0.855157	−	0.518368i	$-0.173460\pi$
$2$	4.83750	1.71031	0.855157	+	0.518368i	$0.173460\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	15.4014	1.92518
$5$	−21.1983	−1.89603	−0.948015	−	0.318225i	$-0.896913\pi$
$5$	−21.1983	−1.89603	−0.948015	+	0.318225i	$0.896913\pi$
$6$	14.5125	0.987451
$7$	−16.2806	−0.879070	−0.439535	−	0.898225i	$-0.644857\pi$
$7$	−16.2806	−0.879070	−0.439535	+	0.898225i	$0.644857\pi$
$8$	35.8043	1.58234
$9$	9.00000	0.333333
$10$	−102.547	−3.24281
$11$	−30.7532	−0.842950	−0.421475	−	0.906840i	$-0.638487\pi$
$11$	−30.7532	−0.842950	−0.421475	+	0.906840i	$0.638487\pi$
$12$	46.2042	1.11150
$13$	0	0
$13$	0	0
$14$	−78.7575	−1.50349
$15$	−63.5948	−1.09467
$16$	49.9922	0.781128
$17$	46.2371	0.659656	0.329828	−	0.944041i	$-0.393009\pi$
$17$	46.2371	0.659656	0.329828	+	0.944041i	$0.393009\pi$
$18$	43.5375	0.570105
$19$	−144.865	−1.74917	−0.874585	−	0.484873i	$-0.838866\pi$
$19$	−144.865	−1.74917	−0.874585	+	0.484873i	$0.838866\pi$
$20$	−326.483	−3.65019
$21$	−48.8418	−0.507531
$22$	−148.769	−1.44171
$23$	8.38045	0.0759758	0.0379879	−	0.999278i	$-0.487905\pi$
$23$	8.38045	0.0759758	0.0379879	+	0.999278i	$0.487905\pi$
$24$	107.413	0.913566
$25$	324.366	2.59493
$26$	0	0
$27$	27.0000	0.192450
$28$	−250.744	−1.69237
$29$	−242.958	−1.55573	−0.777865	−	0.628431i	$-0.783697\pi$
$29$	−242.958	−1.55573	−0.777865	+	0.628431i	$0.783697\pi$
$30$	−307.640	−1.87224
$31$	−87.9353	−0.509472	−0.254736	−	0.967011i	$-0.581989\pi$
$31$	−87.9353	−0.509472	−0.254736	+	0.967011i	$0.581989\pi$
$32$	−44.5973	−0.246368
$33$	−92.2597	−0.486677
$34$	223.672	1.12822
$35$	345.121	1.66674
$36$	138.613	0.641726
$37$	49.6950	0.220806	0.110403	−	0.993887i	$-0.464786\pi$
$37$	49.6950	0.220806	0.110403	+	0.993887i	$0.464786\pi$
$38$	−700.783	−2.99163
$39$	0	0
$40$	−758.990	−3.00017
$41$	107.947	0.411181	0.205591	−	0.978638i	$-0.434088\pi$
$41$	107.947	0.411181	0.205591	+	0.978638i	$0.434088\pi$
$42$	−236.272	−0.868039
$43$	−35.4166	−0.125604	−0.0628021	−	0.998026i	$-0.520004\pi$
$43$	−35.4166	−0.125604	−0.0628021	+	0.998026i	$0.520004\pi$
$44$	−473.643	−1.62283
$45$	−190.784	−0.632010
$46$	40.5404	0.129943
$47$	374.815	1.16324	0.581621	−	0.813460i	$-0.302419\pi$
$47$	374.815	1.16324	0.581621	+	0.813460i	$0.302419\pi$
$48$	149.977	0.450985
$49$	−77.9418	−0.227236
$50$	1569.12	4.43815
$51$	138.711	0.380853
$52$	0	0
$53$	−348.583	−0.903426	−0.451713	−	0.892163i	$-0.649187\pi$
$53$	−348.583	−0.903426	−0.451713	+	0.892163i	$0.649187\pi$
$54$	130.613	0.329150
$55$	651.915	1.59826
$56$	−582.916	−1.39099
$57$	−434.594	−1.00988
$58$	−1175.31	−2.66079
$59$	679.430	1.49923	0.749613	−	0.661877i	$-0.230240\pi$
$59$	679.430	1.49923	0.749613	+	0.661877i	$0.230240\pi$
$60$	−979.450	−2.10744
$61$	−230.403	−0.483608	−0.241804	−	0.970325i	$-0.577739\pi$
$61$	−230.403	−0.483608	−0.241804	+	0.970325i	$0.577739\pi$
$62$	−425.387	−0.871358
$63$	−146.525	−0.293023
$64$	−615.677	−1.20249
$65$	0	0
$66$	−446.306	−0.832372
$67$	−295.642	−0.539082	−0.269541	−	0.962989i	$-0.586872\pi$
$67$	−295.642	−0.539082	−0.269541	+	0.962989i	$0.586872\pi$
$68$	712.117	1.26995
$69$	25.1413	0.0438647
$70$	1669.52	2.85066
$71$	329.215	0.550290	0.275145	−	0.961403i	$-0.411274\pi$
$71$	329.215	0.550290	0.275145	+	0.961403i	$0.411274\pi$
$72$	322.239	0.527448
$73$	48.9973	0.0785575	0.0392787	−	0.999228i	$-0.487494\pi$
$73$	48.9973	0.0785575	0.0392787	+	0.999228i	$0.487494\pi$
$74$	240.399	0.377647
$75$	973.099	1.49818
$76$	−2231.12	−3.36746
$77$	500.681	0.741012
$78$	0	0
$79$	−107.942	−0.153727	−0.0768636	−	0.997042i	$-0.524491\pi$
$79$	−107.942	−0.153727	−0.0768636	+	0.997042i	$0.524491\pi$
$80$	−1059.75	−1.48104
$81$	81.0000	0.111111
$82$	522.192	0.703250
$83$	515.654	0.681932	0.340966	−	0.940076i	$-0.389246\pi$
$83$	515.654	0.681932	0.340966	+	0.940076i	$0.389246\pi$
$84$	−752.233	−0.977088
$85$	−980.147	−1.25073
$86$	−171.328	−0.214823
$87$	−728.874	−0.898201
$88$	−1101.10	−1.33384
$89$	984.453	1.17249	0.586246	−	0.810133i	$-0.300605\pi$
$89$	984.453	1.17249	0.586246	+	0.810133i	$0.300605\pi$
$90$	−922.919	−1.08094
$91$	0	0
$92$	129.071	0.146267
$93$	−263.806	−0.294144
$94$	1813.17	1.98951
$95$	3070.88	3.31648
$96$	−133.792	−0.142241
$97$	487.072	0.509842	0.254921	−	0.966962i	$-0.417951\pi$
$97$	487.072	0.509842	0.254921	+	0.966962i	$0.417951\pi$
$98$	−377.043	−0.388644
$99$	−276.779	−0.280983
$100$	4995.70	4.99570
$101$	766.375	0.755021	0.377511	−	0.926005i	$-0.376780\pi$
$101$	766.375	0.755021	0.377511	+	0.926005i	$0.376780\pi$
$102$	671.016	0.651378
$103$	1229.87	1.17653	0.588266	−	0.808668i	$-0.299811\pi$
$103$	1229.87	1.17653	0.588266	+	0.808668i	$0.299811\pi$
$104$	0	0
$105$	1035.36	0.962295
$106$	−1686.27	−1.54514
$107$	76.8261	0.0694117	0.0347059	−	0.999398i	$-0.488951\pi$
$107$	76.8261	0.0694117	0.0347059	+	0.999398i	$0.488951\pi$
$108$	415.838	0.370500
$109$	−626.001	−0.550092	−0.275046	−	0.961431i	$-0.588693\pi$
$109$	−626.001	−0.550092	−0.275046	+	0.961431i	$0.588693\pi$
$110$	3153.64	2.73353
$111$	149.085	0.127482
$112$	−813.904	−0.686667
$113$	−1343.21	−1.11822	−0.559108	−	0.829095i	$-0.688856\pi$
$113$	−1343.21	−1.11822	−0.559108	+	0.829095i	$0.688856\pi$
$114$	−2102.35	−1.72722
$115$	−177.651	−0.144052
$116$	−3741.90	−2.99506
$117$	0	0
$118$	3286.74	2.56415
$119$	−752.769	−0.579884
$120$	−2276.97	−1.73215
$121$	−385.238	−0.289435
$122$	−1114.58	−0.827122
$123$	323.840	0.237396
$124$	−1354.33	−0.980824
$125$	−4226.22	−3.02404
$126$	−708.817	−0.501162
$127$	−2146.69	−1.49990	−0.749952	−	0.661492i	$-0.769923\pi$
$127$	−2146.69	−1.49990	−0.749952	+	0.661492i	$0.769923\pi$
$128$	−2621.56	−1.81028
$129$	−106.250	−0.0725177
$130$	0	0
$131$	−798.626	−0.532644	−0.266322	−	0.963884i	$-0.585808\pi$
$131$	−798.626	−0.532644	−0.266322	+	0.963884i	$0.585808\pi$
$132$	−1420.93	−0.936940
$133$	2358.48	1.53764
$134$	−1430.17	−0.921999
$135$	−572.353	−0.364891
$136$	1655.49	1.04380
$137$	601.153	0.374891	0.187445	−	0.982275i	$-0.439979\pi$
$137$	601.153	0.374891	0.187445	+	0.982275i	$0.439979\pi$
$138$	121.621	0.0750224
$139$	−2134.18	−1.30229	−0.651146	−	0.758953i	$-0.725711\pi$
$139$	−2134.18	−1.30229	−0.651146	+	0.758953i	$0.725711\pi$
$140$	5315.35	3.20878
$141$	1124.44	0.671598
$142$	1592.58	0.941169
$143$	0	0
$144$	449.930	0.260376
$145$	5150.29	2.94971
$146$	237.024	0.134358
$147$	−233.825	−0.131194
$148$	765.373	0.425090
$149$	−3439.93	−1.89134	−0.945670	−	0.325127i	$-0.894593\pi$
$149$	−3439.93	−1.89134	−0.945670	+	0.325127i	$0.894593\pi$
$150$	4707.37	2.56237
$151$	−2224.04	−1.19861	−0.599305	−	0.800521i	$-0.704556\pi$
$151$	−2224.04	−1.19861	−0.599305	+	0.800521i	$0.704556\pi$
$152$	−5186.78	−2.76779
$153$	416.134	0.219885
$154$	2422.05	1.26736
$155$	1864.07	0.965975
$156$	0	0
$157$	2465.32	1.25321	0.626604	−	0.779338i	$-0.284444\pi$
$157$	2465.32	1.25321	0.626604	+	0.779338i	$0.284444\pi$
$158$	−522.171	−0.262922
$159$	−1045.75	−0.521593
$160$	945.386	0.467121
$161$	−136.439	−0.0667881
$162$	391.838	0.190035
$163$	243.565	0.117040	0.0585199	−	0.998286i	$-0.481362\pi$
$163$	243.565	0.117040	0.0585199	+	0.998286i	$0.481362\pi$
$164$	1662.53	0.791597
$165$	1955.75	0.922755
$166$	2494.47	1.16632
$167$	−409.099	−0.189563	−0.0947814	−	0.995498i	$-0.530215\pi$
$167$	−409.099	−0.189563	−0.0947814	+	0.995498i	$0.530215\pi$
$168$	−1748.75	−0.803089
$169$	0	0
$170$	−4741.46	−2.13914
$171$	−1303.78	−0.583056
$172$	−545.466	−0.241810
$173$	−2618.97	−1.15096	−0.575481	−	0.817815i	$-0.695185\pi$
$173$	−2618.97	−1.15096	−0.575481	+	0.817815i	$0.695185\pi$
$174$	−3525.93	−1.53621
$175$	−5280.88	−2.28113
$176$	−1537.42	−0.658452
$177$	2038.29	0.865578
$178$	4762.29	2.00533
$179$	163.311	0.0681925	0.0340963	−	0.999419i	$-0.489145\pi$
$179$	163.311	0.0681925	0.0340963	+	0.999419i	$0.489145\pi$
$180$	−2938.35	−1.21673
$181$	−3313.07	−1.36054	−0.680272	−	0.732960i	$-0.738138\pi$
$181$	−3313.07	−1.36054	−0.680272	+	0.732960i	$0.738138\pi$
$182$	0	0
$183$	−691.209	−0.279211
$184$	300.056	0.120220
$185$	−1053.45	−0.418654
$186$	−1276.16	−0.503079
$187$	−1421.94	−0.556057
$188$	5772.68	2.23945
$189$	−439.576	−0.169177
$190$	14855.4	5.67222
$191$	−4281.90	−1.62213	−0.811066	−	0.584954i	$-0.801112\pi$
$191$	−4281.90	−1.62213	−0.811066	+	0.584954i	$0.801112\pi$
$192$	−1847.03	−0.694261
$193$	−1877.33	−0.700171	−0.350086	−	0.936718i	$-0.613848\pi$
$193$	−1877.33	−0.700171	−0.350086	+	0.936718i	$0.613848\pi$
$194$	2356.21	0.871990
$195$	0	0
$196$	−1200.41	−0.437468
$197$	1991.26	0.720158	0.360079	−	0.932922i	$-0.382750\pi$
$197$	1991.26	0.720158	0.360079	+	0.932922i	$0.382750\pi$
$198$	−1338.92	−0.480570
$199$	1345.05	0.479137	0.239568	−	0.970879i	$-0.422994\pi$
$199$	1345.05	0.479137	0.239568	+	0.970879i	$0.422994\pi$
$200$	11613.7	4.10607
$201$	−886.927	−0.311239
$202$	3707.34	1.29132
$203$	3955.51	1.36760
$204$	2136.35	0.733208
$205$	−2288.28	−0.779612
$206$	5949.50	2.01224
$207$	75.4240	0.0253253
$208$	0	0
$209$	4455.05	1.47446
$210$	5008.56	1.64583
$211$	−288.763	−0.0942147	−0.0471073	−	0.998890i	$-0.515000\pi$
$211$	−288.763	−0.0942147	−0.0471073	+	0.998890i	$0.515000\pi$
$212$	−5368.67	−1.73925
$213$	987.644	0.317710
$214$	371.646	0.118716
$215$	750.771	0.238150
$216$	966.717	0.304522
$217$	1431.64	0.447862
$218$	−3028.28	−0.940830
$219$	146.992	0.0453552
$220$	10040.4	3.07693
$221$	0	0
$222$	721.198	0.218035
$223$	1798.41	0.540046	0.270023	−	0.962854i	$-0.412969\pi$
$223$	1798.41	0.540046	0.270023	+	0.962854i	$0.412969\pi$
$224$	726.072	0.216575
$225$	2919.30	0.864977
$226$	−6497.77	−1.91250
$227$	−2486.10	−0.726909	−0.363454	−	0.931612i	$-0.618403\pi$
$227$	−2486.10	−0.726909	−0.363454	+	0.931612i	$0.618403\pi$
$228$	−6693.36	−1.94420
$229$	−6074.71	−1.75296	−0.876480	−	0.481438i	$-0.840115\pi$
$229$	−6074.71	−1.75296	−0.876480	+	0.481438i	$0.840115\pi$
$230$	−859.386	−0.246375
$231$	1502.04	0.427824
$232$	−8698.95	−2.46170
$233$	6367.16	1.79024	0.895121	−	0.445822i	$-0.147089\pi$
$233$	6367.16	1.79024	0.895121	+	0.445822i	$0.147089\pi$
$234$	0	0
$235$	−7945.42	−2.20554
$236$	10464.2	2.88627
$237$	−323.827	−0.0887544
$238$	−3641.52	−0.991784
$239$	1886.43	0.510556	0.255278	−	0.966868i	$-0.417833\pi$
$239$	1886.43	0.510556	0.255278	+	0.966868i	$0.417833\pi$
$240$	−3179.24	−0.855081
$241$	−5847.91	−1.56306	−0.781529	−	0.623869i	$-0.785560\pi$
$241$	−5847.91	−1.56306	−0.781529	+	0.623869i	$0.785560\pi$
$242$	−1863.59	−0.495025
$243$	243.000	0.0641500
$244$	−3548.53	−0.931031
$245$	1652.23	0.430845
$246$	1566.58	0.406021
$247$	0	0
$248$	−3148.46	−0.806160
$249$	1546.96	0.393713
$250$	−20444.3	−5.17205
$251$	−2388.65	−0.600678	−0.300339	−	0.953832i	$-0.597100\pi$
$251$	−2388.65	−0.600678	−0.300339	+	0.953832i	$0.597100\pi$
$252$	−2256.70	−0.564122
$253$	−257.726	−0.0640438
$254$	−10384.6	−2.56531
$255$	−2940.44	−0.722108
$256$	−7756.38	−1.89365
$257$	4447.21	1.07941	0.539707	−	0.841853i	$-0.318535\pi$
$257$	4447.21	1.07941	0.539707	+	0.841853i	$0.318535\pi$
$258$	−513.984	−0.124028
$259$	−809.064	−0.194104
$260$	0	0
$261$	−2186.62	−0.518577
$262$	−3863.36	−0.910988
$263$	7590.87	1.77975	0.889873	−	0.456208i	$-0.150793\pi$
$263$	7590.87	1.77975	0.889873	+	0.456208i	$0.150793\pi$
$264$	−3303.30	−0.770091
$265$	7389.36	1.71292
$266$	11409.2	2.62985
$267$	2953.36	0.676939
$268$	−4553.31	−1.03783
$269$	−557.911	−0.126455	−0.0632275	−	0.997999i	$-0.520139\pi$
$269$	−557.911	−0.126455	−0.0632275	+	0.997999i	$0.520139\pi$
$270$	−2768.76	−0.624079
$271$	2707.41	0.606877	0.303439	−	0.952851i	$-0.401865\pi$
$271$	2707.41	0.606877	0.303439	+	0.952851i	$0.401865\pi$
$272$	2311.50	0.515276
$273$	0	0
$274$	2908.08	0.641181
$275$	−9975.31	−2.18740
$276$	387.212	0.0844472
$277$	−1231.35	−0.267092	−0.133546	−	0.991043i	$-0.542636\pi$
$277$	−1231.35	−0.267092	−0.133546	+	0.991043i	$0.542636\pi$
$278$	−10324.1	−2.22733
$279$	−791.417	−0.169824
$280$	12356.8	2.63736
$281$	4601.44	0.976864	0.488432	−	0.872602i	$-0.337569\pi$
$281$	4601.44	0.976864	0.488432	+	0.872602i	$0.337569\pi$
$282$	5439.50	1.14864
$283$	−54.5677	−0.0114619	−0.00573094	−	0.999984i	$-0.501824\pi$
$283$	−54.5677	−0.0114619	−0.00573094	+	0.999984i	$0.501824\pi$
$284$	5070.37	1.05941
$285$	9212.63	1.91477
$286$	0	0
$287$	−1757.44	−0.361457
$288$	−401.376	−0.0821226
$289$	−2775.13	−0.564854
$290$	24914.5	5.04494
$291$	1461.22	0.294357
$292$	754.627	0.151237
$293$	−4745.24	−0.946144	−0.473072	−	0.881024i	$-0.656855\pi$
$293$	−4745.24	−0.946144	−0.473072	+	0.881024i	$0.656855\pi$
$294$	−1131.13	−0.224384
$295$	−14402.7	−2.84258
$296$	1779.30	0.349390
$297$	−830.337	−0.162226
$298$	−16640.6	−3.23479
$299$	0	0
$300$	14987.1	2.88427
$301$	576.604	0.110415
$302$	−10758.8	−2.05000
$303$	2299.13	0.435912
$304$	−7242.10	−1.36633
$305$	4884.15	0.916936
$306$	2013.05	0.376073
$307$	−1028.82	−0.191264	−0.0956320	−	0.995417i	$-0.530487\pi$
$307$	−1028.82	−0.191264	−0.0956320	+	0.995417i	$0.530487\pi$
$308$	7711.20	1.42658
$309$	3689.61	0.679271
$310$	9017.46	1.65212
$311$	−2437.84	−0.444492	−0.222246	−	0.974991i	$-0.571339\pi$
$311$	−2437.84	−0.444492	−0.222246	+	0.974991i	$0.571339\pi$
$312$	0	0
$313$	7934.20	1.43280	0.716402	−	0.697688i	$-0.245788\pi$
$313$	7934.20	1.43280	0.716402	+	0.697688i	$0.245788\pi$
$314$	11926.0	2.14338
$315$	3106.09	0.555581
$316$	−1662.46	−0.295952
$317$	−6942.72	−1.23010	−0.615050	−	0.788488i	$-0.710864\pi$
$317$	−6942.72	−1.23010	−0.615050	+	0.788488i	$0.710864\pi$
$318$	−5058.81	−0.892089
$319$	7471.75	1.31140
$320$	13051.3	2.27997
$321$	230.478	0.0400749
$322$	−660.023	−0.114229
$323$	−6698.12	−1.15385
$324$	1247.51	0.213909
$325$	0	0
$326$	1178.25	0.200175
$327$	−1878.00	−0.317596
$328$	3864.96	0.650630
$329$	−6102.21	−1.02257
$330$	9460.92	1.57820
$331$	5439.27	0.903230	0.451615	−	0.892213i	$-0.350848\pi$
$331$	5439.27	0.903230	0.451615	+	0.892213i	$0.350848\pi$
$332$	7941.79	1.31284
$333$	447.255	0.0736018
$334$	−1979.01	−0.324212
$335$	6267.10	1.02211
$336$	−2441.71	−0.396447
$337$	−1663.87	−0.268952	−0.134476	−	0.990917i	$-0.542935\pi$
$337$	−1663.87	−0.268952	−0.134476	+	0.990917i	$0.542935\pi$
$338$	0	0
$339$	−4029.63	−0.645602
$340$	−15095.6	−2.40787
$341$	2704.29	0.429460
$342$	−6307.04	−0.997210
$343$	6853.19	1.07883
$344$	−1268.07	−0.198749
$345$	−532.953	−0.0831687
$346$	−12669.3	−1.96851
$347$	9268.10	1.43383	0.716913	−	0.697163i	$-0.245554\pi$
$347$	9268.10	1.43383	0.716913	+	0.697163i	$0.245554\pi$
$348$	−11225.7	−1.72920
$349$	−5239.45	−0.803614	−0.401807	−	0.915724i	$-0.631618\pi$
$349$	−5239.45	−0.803614	−0.401807	+	0.915724i	$0.631618\pi$
$350$	−25546.3	−3.90144
$351$	0	0
$352$	1371.51	0.207676
$353$	−1218.01	−0.183649	−0.0918244	−	0.995775i	$-0.529270\pi$
$353$	−1218.01	−0.183649	−0.0918244	+	0.995775i	$0.529270\pi$
$354$	9860.23	1.48041
$355$	−6978.78	−1.04337
$356$	15162.0	2.25726
$357$	−2258.31	−0.334796
$358$	790.018	0.116631
$359$	−5316.06	−0.781534	−0.390767	−	0.920490i	$-0.627790\pi$
$359$	−5316.06	−0.781534	−0.390767	+	0.920490i	$0.627790\pi$
$360$	−6830.91	−1.00006
$361$	14126.7	2.05959
$362$	−16027.0	−2.32696
$363$	−1155.72	−0.167106
$364$	0	0
$365$	−1038.66	−0.148947
$366$	−3343.73	−0.477539
$367$	−289.101	−0.0411197	−0.0205599	−	0.999789i	$-0.506545\pi$
$367$	−289.101	−0.0411197	−0.0205599	+	0.999789i	$0.506545\pi$
$368$	418.957	0.0593469
$369$	971.520	0.137060
$370$	−5096.05	−0.716030
$371$	5675.15	0.794175
$372$	−4062.98	−0.566279
$373$	6163.98	0.855654	0.427827	−	0.903861i	$-0.359279\pi$
$373$	6163.98	0.855654	0.427827	+	0.903861i	$0.359279\pi$
$374$	−6878.64	−0.951032
$375$	−12678.7	−1.74593
$376$	13420.0	1.84065
$377$	0	0
$378$	−2126.45	−0.289346
$379$	2440.03	0.330702	0.165351	−	0.986235i	$-0.447124\pi$
$379$	2440.03	0.330702	0.165351	+	0.986235i	$0.447124\pi$
$380$	47295.9	6.38481
$381$	−6440.06	−0.865970
$382$	−20713.7	−2.77436
$383$	6577.28	0.877502	0.438751	−	0.898609i	$-0.355421\pi$
$383$	6577.28	0.877502	0.438751	+	0.898609i	$0.355421\pi$
$384$	−7864.68	−1.04516
$385$	−10613.6	−1.40498
$386$	−9081.58	−1.19751
$387$	−318.750	−0.0418681
$388$	7501.59	0.981535
$389$	4964.94	0.647127	0.323563	−	0.946206i	$-0.395119\pi$
$389$	4964.94	0.647127	0.323563	+	0.946206i	$0.395119\pi$
$390$	0	0
$391$	387.488	0.0501179
$392$	−2790.65	−0.359565
$393$	−2395.88	−0.307522
$394$	9632.70	1.23170
$395$	2288.19	0.291471
$396$	−4262.79	−0.540943
$397$	12073.9	1.52637	0.763187	−	0.646178i	$-0.223634\pi$
$397$	12073.9	1.52637	0.763187	+	0.646178i	$0.223634\pi$
$398$	6506.69	0.819475
$399$	7075.45	0.887758
$400$	16215.8	2.02697
$401$	4916.90	0.612315	0.306158	−	0.951981i	$-0.400957\pi$
$401$	4916.90	0.612315	0.306158	+	0.951981i	$0.400957\pi$
$402$	−4290.51	−0.532316
$403$	0	0
$404$	11803.3	1.45355
$405$	−1717.06	−0.210670
$406$	19134.8	2.33902
$407$	−1528.28	−0.186128
$408$	4966.47	0.602639
$409$	−15350.9	−1.85588	−0.927939	−	0.372733i	$-0.878421\pi$
$409$	−15350.9	−1.85588	−0.927939	+	0.372733i	$0.878421\pi$
$410$	−11069.6	−1.33338
$411$	1803.46	0.216443
$412$	18941.7	2.26503
$413$	−11061.5	−1.31792
$414$	364.864	0.0433142
$415$	−10931.0	−1.29296
$416$	0	0
$417$	−6402.53	−0.751878
$418$	21551.3	2.52179
$419$	−5488.58	−0.639939	−0.319970	−	0.947428i	$-0.603673\pi$
$419$	−5488.58	−0.639939	−0.319970	+	0.947428i	$0.603673\pi$
$420$	15946.0	1.85259
$421$	927.681	0.107393	0.0536964	−	0.998557i	$-0.482900\pi$
$421$	927.681	0.107393	0.0536964	+	0.998557i	$0.482900\pi$
$422$	−1396.89	−0.161137
$423$	3373.33	0.387747
$424$	−12480.8	−1.42953
$425$	14997.8	1.71176
$426$	4777.73	0.543384
$427$	3751.10	0.425126
$428$	1183.23	0.133630
$429$	0	0
$430$	3631.85	0.407311
$431$	−11002.3	−1.22961	−0.614806	−	0.788678i	$-0.710766\pi$
$431$	−11002.3	−1.22961	−0.614806	+	0.788678i	$0.710766\pi$
$432$	1349.79	0.150328
$433$	−9596.76	−1.06511	−0.532553	−	0.846397i	$-0.678767\pi$
$433$	−9596.76	−1.06511	−0.532553	+	0.846397i	$0.678767\pi$
$434$	6925.56	0.765985
$435$	15450.9	1.70302
$436$	−9641.30	−1.05902
$437$	−1214.03	−0.132895
$438$	711.073	0.0775716
$439$	−10346.3	−1.12483	−0.562416	−	0.826854i	$-0.690128\pi$
$439$	−10346.3	−1.12483	−0.562416	+	0.826854i	$0.690128\pi$
$440$	23341.4	2.52899
$441$	−701.476	−0.0757452
$442$	0	0
$443$	5650.38	0.606000	0.303000	−	0.952991i	$-0.402012\pi$
$443$	5650.38	0.606000	0.303000	+	0.952991i	$0.402012\pi$
$444$	2296.12	0.245426
$445$	−20868.7	−2.22308
$446$	8699.80	0.923649
$447$	−10319.8	−1.09197
$448$	10023.6	1.05708
$449$	11987.8	1.25999	0.629997	−	0.776597i	$-0.283056\pi$
$449$	11987.8	1.25999	0.629997	+	0.776597i	$0.283056\pi$
$450$	14122.1	1.47938
$451$	−3319.71	−0.346605
$452$	−20687.3	−2.15276
$453$	−6672.13	−0.692018
$454$	−12026.5	−1.24324
$455$	0	0
$456$	−15560.3	−1.59798
$457$	−16437.6	−1.68253	−0.841265	−	0.540623i	$-0.818189\pi$
$457$	−16437.6	−1.68253	−0.841265	+	0.540623i	$0.818189\pi$
$458$	−29386.4	−2.99811
$459$	1248.40	0.126951
$460$	−2736.08	−0.277326
$461$	8847.70	0.893880	0.446940	−	0.894564i	$-0.352514\pi$
$461$	8847.70	0.893880	0.446940	+	0.894564i	$0.352514\pi$
$462$	7266.14	0.731713
$463$	10269.6	1.03081	0.515407	−	0.856945i	$-0.327641\pi$
$463$	10269.6	1.03081	0.515407	+	0.856945i	$0.327641\pi$
$464$	−12146.0	−1.21523
$465$	5592.22	0.557706
$466$	30801.1	3.06188
$467$	14730.4	1.45962	0.729811	−	0.683649i	$-0.239608\pi$
$467$	14730.4	1.45962	0.729811	+	0.683649i	$0.239608\pi$
$468$	0	0
$469$	4813.24	0.473891
$470$	−38436.0	−3.77217
$471$	7395.95	0.723540
$472$	24326.6	2.37229
$473$	1089.18	0.105878
$474$	−1566.51	−0.151798
$475$	−46989.2	−4.53897
$476$	−11593.7	−1.11638
$477$	−3137.25	−0.301142
$478$	9125.59	0.873211
$479$	635.077	0.0605792	0.0302896	−	0.999541i	$-0.490357\pi$
$479$	635.077	0.0605792	0.0302896	+	0.999541i	$0.490357\pi$
$480$	2836.16	0.269692
$481$	0	0
$482$	−28289.3	−2.67332
$483$	−409.316	−0.0385601
$484$	−5933.22	−0.557214
$485$	−10325.1	−0.966675
$486$	1175.51	0.109717
$487$	−16611.1	−1.54563	−0.772815	−	0.634632i	$-0.781152\pi$
$487$	−16611.1	−1.54563	−0.772815	+	0.634632i	$0.781152\pi$
$488$	−8249.43	−0.765234
$489$	730.696	0.0675730
$490$	7992.66	0.736881
$491$	−2584.14	−0.237516	−0.118758	−	0.992923i	$-0.537891\pi$
$491$	−2584.14	−0.237516	−0.118758	+	0.992923i	$0.537891\pi$
$492$	4987.59	0.457029
$493$	−11233.7	−1.02625
$494$	0	0
$495$	5867.24	0.532753
$496$	−4396.08	−0.397963
$497$	−5359.82	−0.483744
$498$	7483.42	0.673374
$499$	2183.88	0.195919	0.0979597	−	0.995190i	$-0.468768\pi$
$499$	2183.88	0.195919	0.0979597	+	0.995190i	$0.468768\pi$
$500$	−65089.8	−5.82180
$501$	−1227.30	−0.109444
$502$	−11555.1	−1.02735
$503$	−17214.9	−1.52600	−0.762998	−	0.646401i	$-0.776273\pi$
$503$	−17214.9	−1.52600	−0.762998	+	0.646401i	$0.776273\pi$
$504$	−5246.25	−0.463664
$505$	−16245.8	−1.43154
$506$	−1246.75	−0.109535
$507$	0	0
$508$	−33062.0	−2.88758
$509$	20260.6	1.76431	0.882157	−	0.470955i	$-0.156091\pi$
$509$	20260.6	1.76431	0.882157	+	0.470955i	$0.156091\pi$
$510$	−14224.4	−1.23503
$511$	−797.705	−0.0690575
$512$	−16549.0	−1.42846
$513$	−3911.34	−0.336628
$514$	21513.4	1.84614
$515$	−26071.1	−2.23074
$516$	−1636.40	−0.139609
$517$	−11526.8	−0.980554
$518$	−3913.85	−0.331978
$519$	−7856.90	−0.664508
$520$	0	0
$521$	−20731.5	−1.74331	−0.871653	−	0.490124i	$-0.836952\pi$
$521$	−20731.5	−1.74331	−0.871653	+	0.490124i	$0.836952\pi$
$522$	−10577.8	−0.886930
$523$	−944.353	−0.0789554	−0.0394777	−	0.999220i	$-0.512569\pi$
$523$	−944.353	−0.0789554	−0.0394777	+	0.999220i	$0.512569\pi$
$524$	−12300.0	−1.02543
$525$	−15842.6	−1.31701
$526$	36720.9	3.04393
$527$	−4065.87	−0.336076
$528$	−4612.27	−0.380158
$529$	−12096.8	−0.994228
$530$	35746.0	2.92964
$531$	6114.87	0.499742
$532$	36324.0	2.96023
$533$	0	0
$534$	14286.9	1.15778
$535$	−1628.58	−0.131607
$536$	−10585.3	−0.853012
$537$	489.934	0.0393710
$538$	−2698.89	−0.216278
$539$	2396.96	0.191548
$540$	−8815.05	−0.702480
$541$	4883.06	0.388058	0.194029	−	0.980996i	$-0.437844\pi$
$541$	4883.06	0.388058	0.194029	+	0.980996i	$0.437844\pi$
$542$	13097.1	1.03795
$543$	−9939.21	−0.785511
$544$	−2062.05	−0.162518
$545$	13270.1	1.04299
$546$	0	0
$547$	−16269.1	−1.27169	−0.635847	−	0.771815i	$-0.719349\pi$
$547$	−16269.1	−1.27169	−0.635847	+	0.771815i	$0.719349\pi$
$548$	9258.61	0.721730
$549$	−2073.63	−0.161203
$550$	−48255.6	−3.74114
$551$	35196.0	2.72124
$552$	900.169	0.0694090
$553$	1757.37	0.135137
$554$	−5956.63	−0.456811
$555$	−3160.34	−0.241710
$556$	−32869.3	−2.50714
$557$	−15661.0	−1.19134	−0.595670	−	0.803229i	$-0.703113\pi$
$557$	−15661.0	−1.19134	−0.595670	+	0.803229i	$0.703113\pi$
$558$	−3828.48	−0.290453
$559$	0	0
$560$	17253.3	1.30194
$561$	−4265.82	−0.321040
$562$	22259.5	1.67074
$563$	−9915.18	−0.742230	−0.371115	−	0.928587i	$-0.621024\pi$
$563$	−9915.18	−0.742230	−0.371115	+	0.928587i	$0.621024\pi$
$564$	17318.0	1.29294
$565$	28473.7	2.12017
$566$	−263.971	−0.0196034
$567$	−1318.73	−0.0976745
$568$	11787.3	0.870748
$569$	−11299.1	−0.832482	−0.416241	−	0.909254i	$-0.636653\pi$
$569$	−11299.1	−0.832482	−0.416241	+	0.909254i	$0.636653\pi$
$570$	44566.1	3.27486
$571$	−17619.6	−1.29134	−0.645672	−	0.763615i	$-0.723423\pi$
$571$	−17619.6	−1.29134	−0.645672	+	0.763615i	$0.723423\pi$
$572$	0	0
$573$	−12845.7	−0.936539
$574$	−8501.61	−0.618206
$575$	2718.33	0.197152
$576$	−5541.10	−0.400832
$577$	22153.5	1.59837	0.799186	−	0.601083i	$-0.205264\pi$
$577$	22153.5	1.59837	0.799186	+	0.601083i	$0.205264\pi$
$578$	−13424.7	−0.966078
$579$	−5631.99	−0.404244
$580$	79321.7	5.67872
$581$	−8395.15	−0.599466
$582$	7068.63	0.503444
$583$	10720.1	0.761543
$584$	1754.31	0.124305
$585$	0	0
$586$	−22955.1	−1.61820
$587$	9891.28	0.695497	0.347748	−	0.937588i	$-0.386946\pi$
$587$	9891.28	0.695497	0.347748	+	0.937588i	$0.386946\pi$
$588$	−3601.24	−0.252573
$589$	12738.7	0.891153
$590$	−69673.3	−4.86170
$591$	5973.77	0.415783
$592$	2484.36	0.172477
$593$	9746.23	0.674924	0.337462	−	0.941339i	$-0.390432\pi$
$593$	9746.23	0.674924	0.337462	+	0.941339i	$0.390432\pi$
$594$	−4016.76	−0.277457
$595$	15957.4	1.09948
$596$	−52979.7	−3.64116
$597$	4035.16	0.276630
$598$	0	0
$599$	−8460.53	−0.577109	−0.288554	−	0.957464i	$-0.593175\pi$
$599$	−8460.53	−0.577109	−0.288554	+	0.957464i	$0.593175\pi$
$600$	34841.2	2.37064
$601$	6792.99	0.461052	0.230526	−	0.973066i	$-0.425955\pi$
$601$	6792.99	0.461052	0.230526	+	0.973066i	$0.425955\pi$
$602$	2789.32	0.188844
$603$	−2660.78	−0.179694
$604$	−34253.4	−2.30754
$605$	8166.38	0.548778
$606$	11122.0	0.745546
$607$	19073.4	1.27539	0.637697	−	0.770287i	$-0.279887\pi$
$607$	19073.4	1.27539	0.637697	+	0.770287i	$0.279887\pi$
$608$	6460.58	0.430939
$609$	11866.5	0.789582
$610$	23627.1	1.56825
$611$	0	0
$612$	6409.05	0.423318
$613$	14465.2	0.953091	0.476545	−	0.879150i	$-0.341889\pi$
$613$	14465.2	0.953091	0.476545	+	0.879150i	$0.341889\pi$
$614$	−4976.93	−0.327122
$615$	−6864.85	−0.450109
$616$	17926.6	1.17254
$617$	−12897.3	−0.841536	−0.420768	−	0.907168i	$-0.638239\pi$
$617$	−12897.3	−0.841536	−0.420768	+	0.907168i	$0.638239\pi$
$618$	17848.5	1.16177
$619$	27735.8	1.80096	0.900481	−	0.434895i	$-0.143214\pi$
$619$	27735.8	1.80096	0.900481	+	0.434895i	$0.143214\pi$
$620$	28709.4	1.85967
$621$	226.272	0.0146216
$622$	−11793.0	−0.760222
$623$	−16027.5	−1.03070
$624$	0	0
$625$	49042.7	3.13873
$626$	38381.7	2.45055
$627$	13365.2	0.851281
$628$	37969.3	2.41265
$629$	2297.75	0.145656
$630$	15025.7	0.950219
$631$	−28560.1	−1.80184	−0.900918	−	0.433990i	$-0.857105\pi$
$631$	−28560.1	−1.80184	−0.900918	+	0.433990i	$0.857105\pi$
$632$	−3864.80	−0.243249
$633$	−866.290	−0.0543949
$634$	−33585.4	−2.10386
$635$	45506.1	2.84386
$636$	−16106.0	−1.00416
$637$	0	0
$638$	36144.6	2.24291
$639$	2962.93	0.183430
$640$	55572.5	3.43234
$641$	1862.53	0.114767	0.0573833	−	0.998352i	$-0.481724\pi$
$641$	1862.53	0.114767	0.0573833	+	0.998352i	$0.481724\pi$
$642$	1114.94	0.0685407
$643$	−29495.2	−1.80899	−0.904494	−	0.426487i	$-0.859751\pi$
$643$	−29495.2	−1.80899	−0.904494	+	0.426487i	$0.859751\pi$
$644$	−2101.35	−0.128579
$645$	2252.31	0.137496
$646$	−32402.2	−1.97345
$647$	27780.7	1.68805	0.844027	−	0.536301i	$-0.180179\pi$
$647$	27780.7	1.68805	0.844027	+	0.536301i	$0.180179\pi$
$648$	2900.15	0.175816
$649$	−20894.7	−1.26377
$650$	0	0
$651$	4294.92	0.258573
$652$	3751.25	0.225322
$653$	276.678	0.0165808	0.00829039	−	0.999966i	$-0.497361\pi$
$653$	276.678	0.0165808	0.00829039	+	0.999966i	$0.497361\pi$
$654$	−9084.84	−0.543188
$655$	16929.5	1.00991
$656$	5396.49	0.321185
$657$	440.975	0.0261858
$658$	−29519.5	−1.74892
$659$	18114.9	1.07080	0.535398	−	0.844600i	$-0.320162\pi$
$659$	18114.9	1.07080	0.535398	+	0.844600i	$0.320162\pi$
$660$	30121.2	1.77647
$661$	−27094.4	−1.59433	−0.797163	−	0.603764i	$-0.793667\pi$
$661$	−27094.4	−1.59433	−0.797163	+	0.603764i	$0.793667\pi$
$662$	26312.5	1.54481
$663$	0	0
$664$	18462.6	1.07905
$665$	−49995.8	−2.91542
$666$	2163.60	0.125882
$667$	−2036.10	−0.118198
$668$	−6300.70	−0.364942
$669$	5395.23	0.311796
$670$	30317.1	1.74814
$671$	7085.64	0.407657
$672$	2178.22	0.125039
$673$	2410.31	0.138054	0.0690272	−	0.997615i	$-0.478010\pi$
$673$	2410.31	0.138054	0.0690272	+	0.997615i	$0.478010\pi$
$674$	−8048.99	−0.459993
$675$	8757.89	0.499395
$676$	0	0
$677$	12508.3	0.710091	0.355046	−	0.934849i	$-0.384465\pi$
$677$	12508.3	0.710091	0.355046	+	0.934849i	$0.384465\pi$
$678$	−19493.3	−1.10418
$679$	−7929.83	−0.448187
$680$	−35093.5	−1.97908
$681$	−7458.30	−0.419681
$682$	13082.0	0.734511
$683$	16793.8	0.940844	0.470422	−	0.882442i	$-0.344102\pi$
$683$	16793.8	0.940844	0.470422	+	0.882442i	$0.344102\pi$
$684$	−20080.1	−1.12249
$685$	−12743.4	−0.710804
$686$	33152.3	1.84513
$687$	−18224.1	−1.01207
$688$	−1770.56	−0.0981131
$689$	0	0
$690$	−2578.16	−0.142245
$691$	26026.9	1.43286	0.716432	−	0.697657i	$-0.245774\pi$
$691$	26026.9	1.43286	0.716432	+	0.697657i	$0.245774\pi$
$692$	−40335.8	−2.21580
$693$	4506.13	0.247004
$694$	44834.4	2.45229
$695$	45240.8	2.46918
$696$	−26096.9	−1.42126
$697$	4991.14	0.271238
$698$	−25345.8	−1.37443
$699$	19101.5	1.03360
$700$	−81333.0	−4.39157
$701$	5079.08	0.273658	0.136829	−	0.990595i	$-0.456309\pi$
$701$	5079.08	0.273658	0.136829	+	0.990595i	$0.456309\pi$
$702$	0	0
$703$	−7199.04	−0.386226
$704$	18934.1	1.01364
$705$	−23836.3	−1.27337
$706$	−5892.11	−0.314097
$707$	−12477.1	−0.663717
$708$	31392.6	1.66639
$709$	2530.72	0.134052	0.0670262	−	0.997751i	$-0.478649\pi$
$709$	2530.72	0.134052	0.0670262	+	0.997751i	$0.478649\pi$
$710$	−33759.9	−1.78449
$711$	−971.480	−0.0512424
$712$	35247.7	1.85529
$713$	−736.937	−0.0387076
$714$	−10924.6	−0.572607
$715$	0	0
$716$	2515.22	0.131283
$717$	5659.28	0.294770
$718$	−25716.4	−1.33667
$719$	9503.67	0.492944	0.246472	−	0.969150i	$-0.420729\pi$
$719$	9503.67	0.492944	0.246472	+	0.969150i	$0.420729\pi$
$720$	−9537.73	−0.493681
$721$	−20023.0	−1.03425
$722$	68338.2	3.52255
$723$	−17543.7	−0.902432
$724$	−51026.0	−2.61929
$725$	−78807.4	−4.03701
$726$	−5590.77	−0.285803
$727$	33343.1	1.70100	0.850500	−	0.525975i	$-0.176300\pi$
$727$	33343.1	1.70100	0.850500	+	0.525975i	$0.176300\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−5024.50	−0.254747
$731$	−1637.56	−0.0828556
$732$	−10645.6	−0.537531
$733$	−32288.1	−1.62699	−0.813497	−	0.581569i	$-0.802439\pi$
$733$	−32288.1	−1.62699	−0.813497	+	0.581569i	$0.802439\pi$
$734$	−1398.53	−0.0703277
$735$	4956.69	0.248749
$736$	−373.746	−0.0187180
$737$	9091.96	0.454419
$738$	4699.73	0.234417
$739$	14168.8	0.705288	0.352644	−	0.935758i	$-0.385283\pi$
$739$	14168.8	0.705288	0.352644	+	0.935758i	$0.385283\pi$
$740$	−16224.6	−0.805983
$741$	0	0
$742$	27453.5	1.35829
$743$	−13248.6	−0.654165	−0.327083	−	0.944996i	$-0.606066\pi$
$743$	−13248.6	−0.654165	−0.327083	+	0.944996i	$0.606066\pi$
$744$	−9445.39	−0.465437
$745$	72920.5	3.58604
$746$	29818.3	1.46344
$747$	4640.88	0.227311
$748$	−21899.9	−1.07051
$749$	−1250.78	−0.0610178
$750$	−61333.0	−2.98609
$751$	5113.98	0.248484	0.124242	−	0.992252i	$-0.460350\pi$
$751$	5113.98	0.248484	0.124242	+	0.992252i	$0.460350\pi$
$752$	18737.8	0.908641
$753$	−7165.95	−0.346802
$754$	0	0
$755$	47145.9	2.27260
$756$	−6770.10	−0.325696
$757$	27380.4	1.31461	0.657303	−	0.753626i	$-0.271697\pi$
$757$	27380.4	1.31461	0.657303	+	0.753626i	$0.271697\pi$
$758$	11803.6	0.565604
$759$	−773.178	−0.0369757
$760$	109951.	5.24781
$761$	10613.8	0.505586	0.252793	−	0.967520i	$-0.418651\pi$
$761$	10613.8	0.505586	0.252793	+	0.967520i	$0.418651\pi$
$762$	−31153.8	−1.48108
$763$	10191.7	0.483569
$764$	−65947.3	−3.12289
$765$	−8821.32	−0.416909
$766$	31817.6	1.50080
$767$	0	0
$768$	−23269.2	−1.09330
$769$	−9172.22	−0.430115	−0.215058	−	0.976601i	$-0.568994\pi$
$769$	−9172.22	−0.430115	−0.215058	+	0.976601i	$0.568994\pi$
$770$	−51343.2	−2.40296
$771$	13341.6	0.623200
$772$	−28913.5	−1.34795
$773$	−14952.1	−0.695718	−0.347859	−	0.937547i	$-0.613091\pi$
$773$	−14952.1	−0.695718	−0.347859	+	0.937547i	$0.613091\pi$
$774$	−1541.95	−0.0716076
$775$	−28523.2	−1.32205
$776$	17439.3	0.806745
$777$	−2427.19	−0.112066
$778$	24017.9	1.10679
$779$	−15637.7	−0.719226
$780$	0	0
$781$	−10124.4	−0.463867
$782$	1874.47	0.0857174
$783$	−6559.87	−0.299400
$784$	−3896.48	−0.177500
$785$	−52260.4	−2.37612
$786$	−11590.1	−0.525959
$787$	2671.52	0.121003	0.0605015	−	0.998168i	$-0.480730\pi$
$787$	2671.52	0.121003	0.0605015	+	0.998168i	$0.480730\pi$
$788$	30668.1	1.38643
$789$	22772.6	1.02754
$790$	11069.1	0.498508
$791$	21868.3	0.982991
$792$	−9909.89	−0.444612
$793$	0	0
$794$	58407.4	2.61058
$795$	22168.1	0.988957
$796$	20715.7	0.922423
$797$	−12949.7	−0.575536	−0.287768	−	0.957700i	$-0.592913\pi$
$797$	−12949.7	−0.575536	−0.287768	+	0.957700i	$0.592913\pi$
$798$	34227.5	1.51835
$799$	17330.4	0.767339
$800$	−14465.9	−0.639307
$801$	8860.08	0.390831
$802$	23785.5	1.04725
$803$	−1506.82	−0.0662200
$804$	−13659.9	−0.599190
$805$	2892.27	0.126632
$806$	0	0
$807$	−1673.73	−0.0730089
$808$	27439.6	1.19470
$809$	−11640.4	−0.505876	−0.252938	−	0.967482i	$-0.581397\pi$
$809$	−11640.4	−0.505876	−0.252938	+	0.967482i	$0.581397\pi$
$810$	−8306.28	−0.360312
$811$	−18135.3	−0.785225	−0.392613	−	0.919704i	$-0.628429\pi$
$811$	−18135.3	−0.785225	−0.392613	+	0.919704i	$0.628429\pi$
$812$	60920.4	2.63286
$813$	8122.24	0.350381
$814$	−7393.06	−0.318337
$815$	−5163.16	−0.221911
$816$	6934.49	0.297495
$817$	5130.61	0.219703
$818$	−74260.0	−3.17413
$819$	0	0
$820$	−35242.8	−1.50089
$821$	10789.1	0.458640	0.229320	−	0.973351i	$-0.426350\pi$
$821$	10789.1	0.458640	0.229320	+	0.973351i	$0.426350\pi$
$822$	8724.24	0.370186
$823$	−12284.7	−0.520312	−0.260156	−	0.965567i	$-0.583774\pi$
$823$	−12284.7	−0.520312	−0.260156	+	0.965567i	$0.583774\pi$
$824$	44034.7	1.86168
$825$	−29925.9	−1.26289
$826$	−53510.2	−2.25407
$827$	36077.2	1.51696	0.758480	−	0.651696i	$-0.225942\pi$
$827$	36077.2	1.51696	0.758480	+	0.651696i	$0.225942\pi$
$828$	1161.64	0.0487556
$829$	−8861.83	−0.371271	−0.185636	−	0.982619i	$-0.559434\pi$
$829$	−8861.83	−0.371271	−0.185636	+	0.982619i	$0.559434\pi$
$830$	−52878.5	−2.21137
$831$	−3694.04	−0.154205
$832$	0	0
$833$	−3603.80	−0.149897
$834$	−30972.2	−1.28595
$835$	8672.18	0.359417
$836$	68614.1	2.83860
$837$	−2374.25	−0.0980480
$838$	−26551.0	−1.09450
$839$	5833.37	0.240036	0.120018	−	0.992772i	$-0.461705\pi$
$839$	5833.37	0.240036	0.120018	+	0.992772i	$0.461705\pi$
$840$	37070.4	1.52268
$841$	34639.6	1.42030
$842$	4487.66	0.183676
$843$	13804.3	0.563993
$844$	−4447.37	−0.181380
$845$	0	0
$846$	16318.5	0.663170
$847$	6271.92	0.254434
$848$	−17426.4	−0.705692
$849$	−163.703	−0.00661752
$850$	72551.7	2.92765
$851$	416.466	0.0167759
$852$	15211.1	0.611648
$853$	−28649.5	−1.14999	−0.574995	−	0.818157i	$-0.694996\pi$
$853$	−28649.5	−1.14999	−0.574995	+	0.818157i	$0.694996\pi$
$854$	18146.0	0.727098
$855$	27637.9	1.10549
$856$	2750.71	0.109833
$857$	32336.1	1.28889	0.644445	−	0.764651i	$-0.277089\pi$
$857$	32336.1	1.28889	0.644445	+	0.764651i	$0.277089\pi$
$858$	0	0
$859$	−13878.5	−0.551254	−0.275627	−	0.961265i	$-0.588885\pi$
$859$	−13878.5	−0.551254	−0.275627	+	0.961265i	$0.588885\pi$
$860$	11562.9	0.458480
$861$	−5272.31	−0.208688
$862$	−53223.7	−2.10302
$863$	−32934.7	−1.29908	−0.649542	−	0.760326i	$-0.725039\pi$
$863$	−32934.7	−1.29908	−0.649542	+	0.760326i	$0.725039\pi$
$864$	−1204.13	−0.0474135
$865$	55517.5	2.18226
$866$	−46424.3	−1.82167
$867$	−8325.38	−0.326119
$868$	22049.3	0.862213
$869$	3319.57	0.129584
$870$	74743.6	2.91270
$871$	0	0
$872$	−22413.5	−0.870434
$873$	4383.65	0.169947
$874$	−5872.87	−0.227292
$875$	68805.4	2.65834
$876$	2263.88	0.0873167
$877$	−18375.7	−0.707531	−0.353765	−	0.935334i	$-0.615099\pi$
$877$	−18375.7	−0.707531	−0.353765	+	0.935334i	$0.615099\pi$
$878$	−50050.2	−1.92382
$879$	−14235.7	−0.546256
$880$	32590.7	1.24845
$881$	46883.4	1.79290	0.896448	−	0.443150i	$-0.146139\pi$
$881$	46883.4	1.79290	0.896448	+	0.443150i	$0.146139\pi$
$882$	−3393.39	−0.129548
$883$	1050.05	0.0400194	0.0200097	−	0.999800i	$-0.493630\pi$
$883$	1050.05	0.0400194	0.0200097	+	0.999800i	$0.493630\pi$
$884$	0	0
$885$	−43208.2	−1.64116
$886$	27333.7	1.03645
$887$	27916.8	1.05677	0.528385	−	0.849005i	$-0.322798\pi$
$887$	27916.8	1.05677	0.528385	+	0.849005i	$0.322798\pi$
$888$	5337.89	0.201720
$889$	34949.4	1.31852
$890$	−100952.	−3.80217
$891$	−2491.01	−0.0936611
$892$	27698.0	1.03968
$893$	−54297.4	−2.03471
$894$	−49921.9	−1.86761
$895$	−3461.92	−0.129295
$896$	42680.6	1.59136
$897$	0	0
$898$	57990.8	2.15499
$899$	21364.6	0.792601
$900$	44961.3	1.66523
$901$	−16117.5	−0.595950
$902$	−16059.1	−0.592804
$903$	1729.81	0.0637481
$904$	−48092.7	−1.76940
$905$	70231.3	2.57963
$906$	−32276.4	−1.18357
$907$	23220.1	0.850066	0.425033	−	0.905178i	$-0.360263\pi$
$907$	23220.1	0.850066	0.425033	+	0.905178i	$0.360263\pi$
$908$	−38289.5	−1.39943
$909$	6897.38	0.251674
$910$	0	0
$911$	16344.9	0.594435	0.297217	−	0.954810i	$-0.403941\pi$
$911$	16344.9	0.594435	0.297217	+	0.954810i	$0.403941\pi$
$912$	−21726.3	−0.788848
$913$	−15858.0	−0.574834
$914$	−79516.7	−2.87766
$915$	14652.4	0.529393
$916$	−93559.0	−3.37476
$917$	13002.1	0.468231
$918$	6039.15	0.217126
$919$	−32931.7	−1.18206	−0.591032	−	0.806648i	$-0.701279\pi$
$919$	−32931.7	−1.18206	−0.591032	+	0.806648i	$0.701279\pi$
$920$	−6360.67	−0.227940
$921$	−3086.47	−0.110426
$922$	42800.8	1.52882
$923$	0	0
$924$	23133.6	0.823636
$925$	16119.4	0.572975
$926$	49679.0	1.76302
$927$	11068.8	0.392177
$928$	10835.3	0.383282
$929$	−56305.5	−1.98851	−0.994253	−	0.107056i	$-0.965858\pi$
$929$	−56305.5	−1.98851	−0.994253	+	0.107056i	$0.965858\pi$
$930$	27052.4	0.953852
$931$	11291.0	0.397473
$932$	98063.3	3.44653
$933$	−7313.51	−0.256628
$934$	71258.5	2.49641
$935$	30142.7	1.05430
$936$	0	0
$937$	7095.61	0.247389	0.123695	−	0.992320i	$-0.460526\pi$
$937$	7095.61	0.247389	0.123695	+	0.992320i	$0.460526\pi$
$938$	23284.0	0.810502
$939$	23802.6	0.827230
$940$	−122371.	−4.24606
$941$	−5185.17	−0.179630	−0.0898150	−	0.995958i	$-0.528628\pi$
$941$	−5185.17	−0.179630	−0.0898150	+	0.995958i	$0.528628\pi$
$942$	35777.9	1.23748
$943$	904.641	0.0312398
$944$	33966.2	1.17109
$945$	9318.26	0.320765
$946$	5268.89	0.181085
$947$	38451.7	1.31944	0.659722	−	0.751510i	$-0.270674\pi$
$947$	38451.7	1.31944	0.659722	+	0.751510i	$0.270674\pi$
$948$	−4987.39	−0.170868
$949$	0	0
$950$	−227310.	−7.76307
$951$	−20828.2	−0.710199
$952$	−26952.4	−0.917575
$953$	−20930.5	−0.711445	−0.355722	−	0.934592i	$-0.615765\pi$
$953$	−20930.5	−0.711445	−0.355722	+	0.934592i	$0.615765\pi$
$954$	−15176.4	−0.515048
$955$	90768.8	3.07561
$956$	29053.7	0.982910
$957$	22415.2	0.757139
$958$	3072.19	0.103609
$959$	−9787.14	−0.329555
$960$	39153.9	1.31634
$961$	−22058.4	−0.740438
$962$	0	0
$963$	691.435	0.0231372
$964$	−90066.1	−3.00916
$965$	39796.1	1.32755
$966$	−1980.07	−0.0659499
$967$	−50788.1	−1.68897	−0.844486	−	0.535578i	$-0.820094\pi$
$967$	−50788.1	−1.68897	−0.844486	+	0.535578i	$0.820094\pi$
$968$	−13793.2	−0.457986
$969$	−20094.4	−0.666175
$970$	−49947.6	−1.65332
$971$	15277.5	0.504922	0.252461	−	0.967607i	$-0.418760\pi$
$971$	15277.5	0.504922	0.252461	+	0.967607i	$0.418760\pi$
$972$	3742.54	0.123500
$973$	34745.7	1.14481
$974$	−80356.3	−2.64351
$975$	0	0
$976$	−11518.4	−0.377760
$977$	1293.19	0.0423469	0.0211734	−	0.999776i	$-0.493260\pi$
$977$	1293.19	0.0423469	0.0211734	+	0.999776i	$0.493260\pi$
$978$	3534.74	0.115571
$979$	−30275.1	−0.988353
$980$	25446.7	0.829453
$981$	−5634.01	−0.183364
$982$	−12500.8	−0.406227
$983$	8474.63	0.274973	0.137487	−	0.990504i	$-0.456098\pi$
$983$	8474.63	0.274973	0.137487	+	0.990504i	$0.456098\pi$
$984$	11594.9	0.375641
$985$	−42211.1	−1.36544
$986$	−54343.0	−1.75521
$987$	−18306.6	−0.590382
$988$	0	0
$989$	−296.807	−0.00954289
$990$	28382.8	0.911175
$991$	−7080.71	−0.226969	−0.113484	−	0.993540i	$-0.536201\pi$
$991$	−7080.71	−0.226969	−0.113484	+	0.993540i	$0.536201\pi$
$992$	3921.68	0.125518
$993$	16317.8	0.521480
$994$	−25928.1	−0.827354
$995$	−28512.8	−0.908458
$996$	23825.4	0.757968
$997$	19423.1	0.616985	0.308493	−	0.951227i	$-0.400175\pi$
$997$	19423.1	0.616985	0.308493	+	0.951227i	$0.400175\pi$
$998$	10564.5	0.335084
$999$	1341.76	0.0424940

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.o.1.9	✓	9
3.2	odd	2		1521.4.a.bi.1.1		9
13.5	odd	4		507.4.b.k.337.2		18
13.8	odd	4		507.4.b.k.337.17		18
13.12	even	2		507.4.a.p.1.1	yes	9
39.38	odd	2		1521.4.a.bf.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.9	✓	9	1.1	even	1	trivial
507.4.a.p.1.1	yes	9	13.12	even	2
507.4.b.k.337.2		18	13.5	odd	4
507.4.b.k.337.17		18	13.8	odd	4
1521.4.a.bf.1.9		9	39.38	odd	2
1521.4.a.bi.1.1		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)

\(f(q)\)	\(=\)	\(q+4.83750 q^{2} +3.00000 q^{3} +15.4014 q^{4} -21.1983 q^{5} +14.5125 q^{6} -16.2806 q^{7} +35.8043 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+4.83750 q^{2} +3.00000 q^{3} +15.4014 q^{4} -21.1983 q^{5} +14.5125 q^{6} -16.2806 q^{7} +35.8043 q^{8} +9.00000 q^{9} -102.547 q^{10} -30.7532 q^{11} +46.2042 q^{12} -78.7575 q^{14} -63.5948 q^{15} +49.9922 q^{16} +46.2371 q^{17} +43.5375 q^{18} -144.865 q^{19} -326.483 q^{20} -48.8418 q^{21} -148.769 q^{22} +8.38045 q^{23} +107.413 q^{24} +324.366 q^{25} +27.0000 q^{27} -250.744 q^{28} -242.958 q^{29} -307.640 q^{30} -87.9353 q^{31} -44.5973 q^{32} -92.2597 q^{33} +223.672 q^{34} +345.121 q^{35} +138.613 q^{36} +49.6950 q^{37} -700.783 q^{38} -758.990 q^{40} +107.947 q^{41} -236.272 q^{42} -35.4166 q^{43} -473.643 q^{44} -190.784 q^{45} +40.5404 q^{46} +374.815 q^{47} +149.977 q^{48} -77.9418 q^{49} +1569.12 q^{50} +138.711 q^{51} -348.583 q^{53} +130.613 q^{54} +651.915 q^{55} -582.916 q^{56} -434.594 q^{57} -1175.31 q^{58} +679.430 q^{59} -979.450 q^{60} -230.403 q^{61} -425.387 q^{62} -146.525 q^{63} -615.677 q^{64} -446.306 q^{66} -295.642 q^{67} +712.117 q^{68} +25.1413 q^{69} +1669.52 q^{70} +329.215 q^{71} +322.239 q^{72} +48.9973 q^{73} +240.399 q^{74} +973.099 q^{75} -2231.12 q^{76} +500.681 q^{77} -107.942 q^{79} -1059.75 q^{80} +81.0000 q^{81} +522.192 q^{82} +515.654 q^{83} -752.233 q^{84} -980.147 q^{85} -171.328 q^{86} -728.874 q^{87} -1101.10 q^{88} +984.453 q^{89} -922.919 q^{90} +129.071 q^{92} -263.806 q^{93} +1813.17 q^{94} +3070.88 q^{95} -133.792 q^{96} +487.072 q^{97} -377.043 q^{98} -276.779 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) 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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