LMFDB - Embedded newform 507.4.a.o.1.1

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			1.1
Root			$-3.27560$ 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)

$f(q)$	$=$	$q-5.52257 q^{2} +3.00000 q^{3} +22.4988 q^{4} +6.08065 q^{5} -16.5677 q^{6} -20.2718 q^{7} -80.0709 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-5.52257 q^{2} +3.00000 q^{3} +22.4988 q^{4} +6.08065 q^{5} -16.5677 q^{6} -20.2718 q^{7} -80.0709 q^{8} +9.00000 q^{9} -33.5808 q^{10} +48.8284 q^{11} +67.4965 q^{12} +111.953 q^{14} +18.2419 q^{15} +262.207 q^{16} -37.7513 q^{17} -49.7032 q^{18} -120.837 q^{19} +136.807 q^{20} -60.8155 q^{21} -269.658 q^{22} +74.8543 q^{23} -240.213 q^{24} -88.0257 q^{25} +27.0000 q^{27} -456.092 q^{28} -112.710 q^{29} -100.742 q^{30} -113.134 q^{31} -807.490 q^{32} +146.485 q^{33} +208.485 q^{34} -123.266 q^{35} +202.489 q^{36} +85.7704 q^{37} +667.331 q^{38} -486.883 q^{40} +133.993 q^{41} +335.858 q^{42} -319.135 q^{43} +1098.58 q^{44} +54.7258 q^{45} -413.388 q^{46} +401.982 q^{47} +786.620 q^{48} +67.9471 q^{49} +486.129 q^{50} -113.254 q^{51} -384.493 q^{53} -149.110 q^{54} +296.908 q^{55} +1623.18 q^{56} -362.511 q^{57} +622.450 q^{58} +121.629 q^{59} +410.422 q^{60} +220.043 q^{61} +624.790 q^{62} -182.446 q^{63} +2361.77 q^{64} -808.975 q^{66} -975.363 q^{67} -849.361 q^{68} +224.563 q^{69} +680.745 q^{70} -106.725 q^{71} -720.638 q^{72} -43.2566 q^{73} -473.673 q^{74} -264.077 q^{75} -2718.69 q^{76} -989.841 q^{77} +539.339 q^{79} +1594.39 q^{80} +81.0000 q^{81} -739.986 q^{82} -811.183 q^{83} -1368.28 q^{84} -229.553 q^{85} +1762.45 q^{86} -338.130 q^{87} -3909.73 q^{88} +1130.64 q^{89} -302.227 q^{90} +1684.13 q^{92} -339.401 q^{93} -2219.98 q^{94} -734.767 q^{95} -2422.47 q^{96} +229.088 q^{97} -375.243 q^{98} +439.456 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−5.52257	−1.95253	−0.976263	−	0.216591i	$-0.930506\pi$
$2$	−5.52257	−1.95253	−0.976263	+	0.216591i	$0.930506\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	22.4988	2.81235
$5$	6.08065	0.543870	0.271935	−	0.962316i	$-0.412336\pi$
$5$	6.08065	0.543870	0.271935	+	0.962316i	$0.412336\pi$
$6$	−16.5677	−1.12729
$7$	−20.2718	−1.09458	−0.547288	−	0.836944i	$-0.684340\pi$
$7$	−20.2718	−1.09458	−0.547288	+	0.836944i	$0.684340\pi$
$8$	−80.0709	−3.53867
$9$	9.00000	0.333333
$10$	−33.5808	−1.06192
$11$	48.8284	1.33839	0.669196	−	0.743086i	$-0.266639\pi$
$11$	48.8284	1.33839	0.669196	+	0.743086i	$0.266639\pi$
$12$	67.4965	1.62371
$13$	0	0
$13$	0	0
$14$	111.953	2.13719
$15$	18.2419	0.314003
$16$	262.207	4.09698
$17$	−37.7513	−0.538591	−0.269295	−	0.963058i	$-0.586791\pi$
$17$	−37.7513	−0.538591	−0.269295	+	0.963058i	$0.586791\pi$
$18$	−49.7032	−0.650842
$19$	−120.837	−1.45905	−0.729524	−	0.683955i	$-0.760258\pi$
$19$	−120.837	−1.45905	−0.729524	+	0.683955i	$0.760258\pi$
$20$	136.807	1.52955
$21$	−60.8155	−0.631954
$22$	−269.658	−2.61324
$23$	74.8543	0.678617	0.339309	−	0.940675i	$-0.389807\pi$
$23$	74.8543	0.678617	0.339309	+	0.940675i	$0.389807\pi$
$24$	−240.213	−2.04305
$25$	−88.0257	−0.704206
$26$	0	0
$27$	27.0000	0.192450
$28$	−456.092	−3.07834
$29$	−112.710	−0.721715	−0.360858	−	0.932621i	$-0.617516\pi$
$29$	−112.710	−0.721715	−0.360858	+	0.932621i	$0.617516\pi$
$30$	−100.742	−0.613099
$31$	−113.134	−0.655465	−0.327733	−	0.944770i	$-0.606285\pi$
$31$	−113.134	−0.655465	−0.327733	+	0.944770i	$0.606285\pi$
$32$	−807.490	−4.46079
$33$	146.485	0.772721
$34$	208.485	1.05161
$35$	−123.266	−0.595307
$36$	202.489	0.937451
$37$	85.7704	0.381096	0.190548	−	0.981678i	$-0.438973\pi$
$37$	85.7704	0.381096	0.190548	+	0.981678i	$0.438973\pi$
$38$	667.331	2.84883
$39$	0	0
$40$	−486.883	−1.92457
$41$	133.993	0.510395	0.255197	−	0.966889i	$-0.417860\pi$
$41$	133.993	0.510395	0.255197	+	0.966889i	$0.417860\pi$
$42$	335.858	1.23391
$43$	−319.135	−1.13181	−0.565903	−	0.824472i	$-0.691472\pi$
$43$	−319.135	−1.13181	−0.565903	+	0.824472i	$0.691472\pi$
$44$	1098.58	3.76403
$45$	54.7258	0.181290
$46$	−413.388	−1.32502
$47$	401.982	1.24755	0.623777	−	0.781602i	$-0.285597\pi$
$47$	401.982	1.24755	0.623777	+	0.781602i	$0.285597\pi$
$48$	786.620	2.36539
$49$	67.9471	0.198096
$50$	486.129	1.37498
$51$	−113.254	−0.310955
$52$	0	0
$53$	−384.493	−0.996494	−0.498247	−	0.867035i	$-0.666023\pi$
$53$	−384.493	−0.996494	−0.498247	+	0.867035i	$0.666023\pi$
$54$	−149.110	−0.375764
$55$	296.908	0.727911
$56$	1623.18	3.87334
$57$	−362.511	−0.842382
$58$	622.450	1.40917
$59$	121.629	0.268385	0.134192	−	0.990955i	$-0.457156\pi$
$59$	121.629	0.268385	0.134192	+	0.990955i	$0.457156\pi$
$60$	410.422	0.883088
$61$	220.043	0.461864	0.230932	−	0.972970i	$-0.425823\pi$
$61$	220.043	0.461864	0.230932	+	0.972970i	$0.425823\pi$
$62$	624.790	1.27981
$63$	−182.446	−0.364859
$64$	2361.77	4.61283
$65$	0	0
$66$	−808.975	−1.50876
$67$	−975.363	−1.77850	−0.889250	−	0.457421i	$-0.848773\pi$
$67$	−975.363	−1.77850	−0.889250	+	0.457421i	$0.848773\pi$
$68$	−849.361	−1.51471
$69$	224.563	0.391800
$70$	680.745	1.16235
$71$	−106.725	−0.178394	−0.0891969	−	0.996014i	$-0.528430\pi$
$71$	−106.725	−0.178394	−0.0891969	+	0.996014i	$0.528430\pi$
$72$	−720.638	−1.17956
$73$	−43.2566	−0.0693535	−0.0346767	−	0.999399i	$-0.511040\pi$
$73$	−43.2566	−0.0693535	−0.0346767	+	0.999399i	$0.511040\pi$
$74$	−473.673	−0.744100
$75$	−264.077	−0.406573
$76$	−2718.69	−4.10336
$77$	−989.841	−1.46497
$78$	0	0
$79$	539.339	0.768106	0.384053	−	0.923311i	$-0.374528\pi$
$79$	539.339	0.768106	0.384053	+	0.923311i	$0.374528\pi$
$80$	1594.39	2.22822
$81$	81.0000	0.111111
$82$	−739.986	−0.996558
$83$	−811.183	−1.07276	−0.536379	−	0.843977i	$-0.680208\pi$
$83$	−811.183	−1.07276	−0.536379	+	0.843977i	$0.680208\pi$
$84$	−1368.28	−1.77728
$85$	−229.553	−0.292923
$86$	1762.45	2.20988
$87$	−338.130	−0.416683
$88$	−3909.73	−4.73612
$89$	1130.64	1.34661	0.673304	−	0.739366i	$-0.264875\pi$
$89$	1130.64	1.34661	0.673304	+	0.739366i	$0.264875\pi$
$90$	−302.227	−0.353973
$91$	0	0
$92$	1684.13	1.90851
$93$	−339.401	−0.378433
$94$	−2219.98	−2.43588
$95$	−734.767	−0.793532
$96$	−2422.47	−2.57544
$97$	229.088	0.239797	0.119899	−	0.992786i	$-0.461743\pi$
$97$	229.088	0.239797	0.119899	+	0.992786i	$0.461743\pi$
$98$	−375.243	−0.386788
$99$	439.456	0.446131
$100$	−1980.48	−1.98048
$101$	−845.077	−0.832557	−0.416279	−	0.909237i	$-0.636666\pi$
$101$	−845.077	−0.832557	−0.416279	+	0.909237i	$0.636666\pi$
$102$	625.454	0.607148
$103$	−1095.09	−1.04760	−0.523800	−	0.851841i	$-0.675486\pi$
$103$	−1095.09	−1.04760	−0.523800	+	0.851841i	$0.675486\pi$
$104$	0	0
$105$	−369.798	−0.343700
$106$	2123.39	1.94568
$107$	1183.33	1.06913	0.534563	−	0.845129i	$-0.320476\pi$
$107$	1183.33	1.06913	0.534563	+	0.845129i	$0.320476\pi$
$108$	607.468	0.541238
$109$	239.968	0.210870	0.105435	−	0.994426i	$-0.466377\pi$
$109$	239.968	0.210870	0.105435	+	0.994426i	$0.466377\pi$
$110$	−1639.70	−1.42126
$111$	257.311	0.220026
$112$	−5315.41	−4.48446
$113$	−1031.22	−0.858490	−0.429245	−	0.903188i	$-0.641220\pi$
$113$	−1031.22	−0.858490	−0.429245	+	0.903188i	$0.641220\pi$
$114$	2001.99	1.64477
$115$	455.162	0.369079
$116$	−2535.85	−2.02972
$117$	0	0
$118$	−671.703	−0.524028
$119$	765.288	0.589528
$120$	−1460.65	−1.11115
$121$	1053.21	0.791294
$122$	−1215.21	−0.901800
$123$	401.979	0.294676
$124$	−2545.38	−1.84340
$125$	−1295.33	−0.926866
$126$	1007.57	0.712396
$127$	−18.0201	−0.0125908	−0.00629539	−	0.999980i	$-0.502004\pi$
$127$	−18.0201	−0.0125908	−0.00629539	+	0.999980i	$0.502004\pi$
$128$	−6583.12	−4.54587
$129$	−957.406	−0.653449
$130$	0	0
$131$	−1282.52	−0.855378	−0.427689	−	0.903926i	$-0.640672\pi$
$131$	−1282.52	−0.855378	−0.427689	+	0.903926i	$0.640672\pi$
$132$	3295.75	2.17317
$133$	2449.59	1.59704
$134$	5386.51	3.47257
$135$	164.177	0.104668
$136$	3022.78	1.90589
$137$	−1547.36	−0.964965	−0.482482	−	0.875906i	$-0.660265\pi$
$137$	−1547.36	−0.964965	−0.482482	+	0.875906i	$0.660265\pi$
$138$	−1240.16	−0.764999
$139$	−3096.87	−1.88973	−0.944867	−	0.327455i	$-0.893809\pi$
$139$	−3096.87	−1.88973	−0.944867	+	0.327455i	$0.893809\pi$
$140$	−2773.34	−1.67421
$141$	1205.95	0.720276
$142$	589.398	0.348318
$143$	0	0
$144$	2359.86	1.36566
$145$	−685.351	−0.392519
$146$	238.888	0.135414
$147$	203.841	0.114371
$148$	1929.73	1.07178
$149$	−420.879	−0.231408	−0.115704	−	0.993284i	$-0.536912\pi$
$149$	−420.879	−0.231408	−0.115704	+	0.993284i	$0.536912\pi$
$150$	1458.39	0.793845
$151$	2801.17	1.50964	0.754821	−	0.655931i	$-0.227724\pi$
$151$	2801.17	1.50964	0.754821	+	0.655931i	$0.227724\pi$
$152$	9675.52	5.16308
$153$	−339.762	−0.179530
$154$	5466.47	2.86039
$155$	−687.927	−0.356488
$156$	0	0
$157$	−2344.20	−1.19164	−0.595820	−	0.803118i	$-0.703173\pi$
$157$	−2344.20	−1.19164	−0.595820	+	0.803118i	$0.703173\pi$
$158$	−2978.54	−1.49975
$159$	−1153.48	−0.575326
$160$	−4910.06	−2.42609
$161$	−1517.43	−0.742798
$162$	−447.329	−0.216947
$163$	−3658.41	−1.75797	−0.878984	−	0.476851i	$-0.841778\pi$
$163$	−3658.41	−1.75797	−0.878984	+	0.476851i	$0.841778\pi$
$164$	3014.68	1.43541
$165$	890.725	0.420260
$166$	4479.82	2.09459
$167$	−1987.85	−0.921105	−0.460552	−	0.887632i	$-0.652349\pi$
$167$	−1987.85	−0.921105	−0.460552	+	0.887632i	$0.652349\pi$
$168$	4869.55	2.23627
$169$	0	0
$170$	1267.72	0.571940
$171$	−1087.53	−0.486349
$172$	−7180.17	−3.18304
$173$	−3577.11	−1.57204	−0.786019	−	0.618202i	$-0.787861\pi$
$173$	−3577.11	−1.57204	−0.786019	+	0.618202i	$0.787861\pi$
$174$	1867.35	0.813583
$175$	1784.44	0.770807
$176$	12803.1	5.48337
$177$	364.886	0.154952
$178$	−6244.07	−2.62928
$179$	770.144	0.321582	0.160791	−	0.986988i	$-0.448595\pi$
$179$	770.144	0.321582	0.160791	+	0.986988i	$0.448595\pi$
$180$	1231.27	0.509851
$181$	−1895.69	−0.778482	−0.389241	−	0.921136i	$-0.627263\pi$
$181$	−1895.69	−0.778482	−0.389241	+	0.921136i	$0.627263\pi$
$182$	0	0
$183$	660.130	0.266657
$184$	−5993.65	−2.40140
$185$	521.539	0.207267
$186$	1874.37	0.738900
$187$	−1843.34	−0.720846
$188$	9044.12	3.50857
$189$	−547.339	−0.210651
$190$	4057.81	1.54939
$191$	2203.83	0.834887	0.417443	−	0.908703i	$-0.362926\pi$
$191$	2203.83	0.834887	0.417443	+	0.908703i	$0.362926\pi$
$192$	7085.30	2.66322
$193$	−1016.38	−0.379069	−0.189534	−	0.981874i	$-0.560698\pi$
$193$	−1016.38	−0.379069	−0.189534	+	0.981874i	$0.560698\pi$
$194$	−1265.15	−0.468210
$195$	0	0
$196$	1528.73	0.557117
$197$	−4318.58	−1.56186	−0.780930	−	0.624619i	$-0.785254\pi$
$197$	−4318.58	−1.56186	−0.780930	+	0.624619i	$0.785254\pi$
$198$	−2426.93	−0.871082
$199$	1363.40	0.485672	0.242836	−	0.970067i	$-0.421922\pi$
$199$	1363.40	0.485672	0.242836	+	0.970067i	$0.421922\pi$
$200$	7048.30	2.49195
$201$	−2926.09	−1.02682
$202$	4667.00	1.62559
$203$	2284.84	0.789972
$204$	−2548.08	−0.874517
$205$	814.764	0.277588
$206$	6047.74	2.04546
$207$	673.688	0.226206
$208$	0	0
$209$	−5900.28	−1.95278
$210$	2042.23	0.671084
$211$	5288.34	1.72542	0.862711	−	0.505697i	$-0.168765\pi$
$211$	5288.34	1.72542	0.862711	+	0.505697i	$0.168765\pi$
$212$	−8650.64	−2.80249
$213$	−320.176	−0.102996
$214$	−6535.00	−2.08749
$215$	−1940.55	−0.615555
$216$	−2161.91	−0.681017
$217$	2293.43	0.717457
$218$	−1325.24	−0.411728
$219$	−129.770	−0.0400412
$220$	6680.09	2.04714
$221$	0	0
$222$	−1421.02	−0.429606
$223$	1258.01	0.377769	0.188884	−	0.981999i	$-0.439513\pi$
$223$	1258.01	0.377769	0.188884	+	0.981999i	$0.439513\pi$
$224$	16369.3	4.88268
$225$	−792.232	−0.234735
$226$	5695.01	1.67622
$227$	−1135.21	−0.331922	−0.165961	−	0.986132i	$-0.553073\pi$
$227$	−1135.21	−0.331922	−0.165961	+	0.986132i	$0.553073\pi$
$228$	−8156.07	−2.36908
$229$	218.002	0.0629083	0.0314541	−	0.999505i	$-0.489986\pi$
$229$	218.002	0.0629083	0.0314541	+	0.999505i	$0.489986\pi$
$230$	−2513.67	−0.720636
$231$	−2969.52	−0.845802
$232$	9024.80	2.55391
$233$	−2.56367	−0.000720823 0	−0.000360412	−	1.00000i	$-0.500115\pi$
$233$	−2.56367	−0.000720823 0	−0.000360412		1.00000i	$0.500115\pi$
$234$	0	0
$235$	2444.31	0.678507
$236$	2736.50	0.754793
$237$	1618.02	0.443466
$238$	−4226.36	−1.15107
$239$	−3971.13	−1.07477	−0.537387	−	0.843336i	$-0.680589\pi$
$239$	−3971.13	−1.07477	−0.537387	+	0.843336i	$0.680589\pi$
$240$	4783.16	1.28647
$241$	658.032	0.175882	0.0879411	−	0.996126i	$-0.471971\pi$
$241$	658.032	0.175882	0.0879411	+	0.996126i	$0.471971\pi$
$242$	−5816.44	−1.54502
$243$	243.000	0.0641500
$244$	4950.72	1.29892
$245$	413.162	0.107739
$246$	−2219.96	−0.575363
$247$	0	0
$248$	9058.72	2.31947
$249$	−2433.55	−0.619357
$250$	7153.58	1.80973
$251$	7091.01	1.78319	0.891595	−	0.452833i	$-0.149587\pi$
$251$	7091.01	1.78319	0.891595	+	0.452833i	$0.149587\pi$
$252$	−4104.83	−1.02611
$253$	3655.01	0.908256
$254$	99.5176	0.0245838
$255$	−688.658	−0.169119
$256$	17461.6	4.26309
$257$	792.530	0.192361	0.0961803	−	0.995364i	$-0.469337\pi$
$257$	792.530	0.192361	0.0961803	+	0.995364i	$0.469337\pi$
$258$	5287.35	1.27588
$259$	−1738.72	−0.417139
$260$	0	0
$261$	−1014.39	−0.240572
$262$	7082.83	1.67015
$263$	−337.467	−0.0791221	−0.0395610	−	0.999217i	$-0.512596\pi$
$263$	−337.467	−0.0791221	−0.0395610	+	0.999217i	$0.512596\pi$
$264$	−11729.2	−2.73440
$265$	−2337.97	−0.541963
$266$	−13528.0	−3.11826
$267$	3391.93	0.777464
$268$	−21944.5	−5.00177
$269$	−3523.93	−0.798728	−0.399364	−	0.916792i	$-0.630769\pi$
$269$	−3523.93	−0.798728	−0.399364	+	0.916792i	$0.630769\pi$
$270$	−906.682	−0.204366
$271$	−8313.62	−1.86353	−0.931765	−	0.363063i	$-0.881731\pi$
$271$	−8313.62	−1.86353	−0.931765	+	0.363063i	$0.881731\pi$
$272$	−9898.65	−2.20660
$273$	0	0
$274$	8545.43	1.88412
$275$	−4298.16	−0.942504
$276$	5052.40	1.10188
$277$	−2500.33	−0.542347	−0.271174	−	0.962530i	$-0.587412\pi$
$277$	−2500.33	−0.542347	−0.271174	+	0.962530i	$0.587412\pi$
$278$	17102.7	3.68975
$279$	−1018.20	−0.218488
$280$	9870.01	2.10659
$281$	3053.13	0.648165	0.324083	−	0.946029i	$-0.394944\pi$
$281$	3053.13	0.648165	0.324083	+	0.946029i	$0.394944\pi$
$282$	−6659.93	−1.40636
$283$	1158.80	0.243404	0.121702	−	0.992567i	$-0.461165\pi$
$283$	1158.80	0.243404	0.121702	+	0.992567i	$0.461165\pi$
$284$	−2401.19	−0.501706
$285$	−2204.30	−0.458146
$286$	0	0
$287$	−2716.28	−0.558666
$288$	−7267.41	−1.48693
$289$	−3487.84	−0.709920
$290$	3784.90	0.766403
$291$	687.263	0.138447
$292$	−973.223	−0.195046
$293$	8357.45	1.66637	0.833187	−	0.552991i	$-0.186514\pi$
$293$	8357.45	1.66637	0.833187	+	0.552991i	$0.186514\pi$
$294$	−1125.73	−0.223312
$295$	739.581	0.145966
$296$	−6867.71	−1.34857
$297$	1318.37	0.257574
$298$	2324.33	0.451829
$299$	0	0
$300$	−5941.43	−1.14343
$301$	6469.46	1.23885
$302$	−15469.7	−2.94761
$303$	−2535.23	−0.480677
$304$	−31684.3	−5.97769
$305$	1338.01	0.251194
$306$	1876.36	0.350537
$307$	8429.30	1.56705	0.783527	−	0.621357i	$-0.213418\pi$
$307$	8429.30	1.56705	0.783527	+	0.621357i	$0.213418\pi$
$308$	−22270.3	−4.12002
$309$	−3285.28	−0.604832
$310$	3799.13	0.696051
$311$	3602.87	0.656914	0.328457	−	0.944519i	$-0.393471\pi$
$311$	3602.87	0.656914	0.328457	+	0.944519i	$0.393471\pi$
$312$	0	0
$313$	6626.98	1.19674	0.598369	−	0.801220i	$-0.295816\pi$
$313$	6626.98	1.19674	0.598369	+	0.801220i	$0.295816\pi$
$314$	12946.0	2.32671
$315$	−1109.39	−0.198436
$316$	12134.5	2.16019
$317$	−3983.87	−0.705857	−0.352928	−	0.935650i	$-0.614814\pi$
$317$	−3983.87	−0.705857	−0.352928	+	0.935650i	$0.614814\pi$
$318$	6370.17	1.12334
$319$	−5503.46	−0.965938
$320$	14361.1	2.50878
$321$	3549.98	0.617260
$322$	8380.14	1.45033
$323$	4561.76	0.785829
$324$	1822.41	0.312484
$325$	0	0
$326$	20203.8	3.43248
$327$	719.905	0.121746
$328$	−10728.9	−1.80612
$329$	−8148.91	−1.36554
$330$	−4919.09	−0.820567
$331$	−4172.82	−0.692927	−0.346464	−	0.938063i	$-0.612618\pi$
$331$	−4172.82	−0.692927	−0.346464	+	0.938063i	$0.612618\pi$
$332$	−18250.7	−3.01697
$333$	771.933	0.127032
$334$	10978.1	1.79848
$335$	−5930.84	−0.967272
$336$	−15946.2	−2.58910
$337$	4157.51	0.672030	0.336015	−	0.941857i	$-0.390921\pi$
$337$	4157.51	0.672030	0.336015	+	0.941857i	$0.390921\pi$
$338$	0	0
$339$	−3093.67	−0.495649
$340$	−5164.66	−0.823804
$341$	−5524.14	−0.877270
$342$	6005.98	0.949609
$343$	5575.83	0.877744
$344$	25553.4	4.00509
$345$	1365.49	0.213088
$346$	19754.9	3.06944
$347$	4667.98	0.722162	0.361081	−	0.932535i	$-0.382408\pi$
$347$	4667.98	0.722162	0.361081	+	0.932535i	$0.382408\pi$
$348$	−7607.54	−1.17186
$349$	−11905.6	−1.82606	−0.913028	−	0.407896i	$-0.866263\pi$
$349$	−11905.6	−1.82606	−0.913028	+	0.407896i	$0.866263\pi$
$350$	−9854.72	−1.50502
$351$	0	0
$352$	−39428.4	−5.97029
$353$	1662.12	0.250610	0.125305	−	0.992118i	$-0.460009\pi$
$353$	1662.12	0.250610	0.125305	+	0.992118i	$0.460009\pi$
$354$	−2015.11	−0.302548
$355$	−648.959	−0.0970229
$356$	25438.2	3.78714
$357$	2295.87	0.340364
$358$	−4253.18	−0.627898
$359$	−12754.4	−1.87507	−0.937535	−	0.347890i	$-0.886898\pi$
$359$	−12754.4	−1.87507	−0.937535	+	0.347890i	$0.886898\pi$
$360$	−4381.95	−0.641524
$361$	7742.58	1.12882
$362$	10469.1	1.52001
$363$	3159.64	0.456854
$364$	0	0
$365$	−263.028	−0.0377192
$366$	−3645.62	−0.520655
$367$	4463.81	0.634901	0.317451	−	0.948275i	$-0.397173\pi$
$367$	4463.81	0.634901	0.317451	+	0.948275i	$0.397173\pi$
$368$	19627.3	2.78028
$369$	1205.94	0.170132
$370$	−2880.24	−0.404693
$371$	7794.38	1.09074
$372$	−7636.13	−1.06429
$373$	9454.46	1.31242	0.656211	−	0.754578i	$-0.272158\pi$
$373$	9454.46	1.31242	0.656211	+	0.754578i	$0.272158\pi$
$374$	10180.0	1.40747
$375$	−3886.00	−0.535126
$376$	−32187.0	−4.41468
$377$	0	0
$378$	3022.72	0.411302
$379$	−14085.9	−1.90908	−0.954542	−	0.298077i	$-0.903655\pi$
$379$	−14085.9	−1.90908	−0.954542	+	0.298077i	$0.903655\pi$
$380$	−16531.4	−2.23169
$381$	−54.0604	−0.00726929
$382$	−12170.8	−1.63014
$383$	8663.84	1.15588	0.577939	−	0.816080i	$-0.303857\pi$
$383$	8663.84	1.15588	0.577939	+	0.816080i	$0.303857\pi$
$384$	−19749.4	−2.62456
$385$	−6018.87	−0.796754
$386$	5613.01	0.740141
$387$	−2872.22	−0.377269
$388$	5154.20	0.674395
$389$	5085.58	0.662852	0.331426	−	0.943481i	$-0.392470\pi$
$389$	5085.58	0.662852	0.331426	+	0.943481i	$0.392470\pi$
$390$	0	0
$391$	−2825.85	−0.365497
$392$	−5440.58	−0.700997
$393$	−3847.57	−0.493853
$394$	23849.7	3.04957
$395$	3279.53	0.417750
$396$	9887.24	1.25468
$397$	−12983.7	−1.64140	−0.820700	−	0.571360i	$-0.806416\pi$
$397$	−12983.7	−1.64140	−0.820700	+	0.571360i	$0.806416\pi$
$398$	−7529.48	−0.948288
$399$	7348.76	0.922051
$400$	−23080.9	−2.88512
$401$	−6493.66	−0.808673	−0.404337	−	0.914610i	$-0.632498\pi$
$401$	−6493.66	−0.808673	−0.404337	+	0.914610i	$0.632498\pi$
$402$	16159.5	2.00489
$403$	0	0
$404$	−19013.2	−2.34145
$405$	492.532	0.0604300
$406$	−12618.2	−1.54244
$407$	4188.03	0.510056
$408$	9068.35	1.10037
$409$	12546.2	1.51680	0.758398	−	0.651792i	$-0.225983\pi$
$409$	12546.2	1.51680	0.758398	+	0.651792i	$0.225983\pi$
$410$	−4499.59	−0.541998
$411$	−4642.09	−0.557123
$412$	−24638.3	−2.94622
$413$	−2465.63	−0.293767
$414$	−3720.49	−0.441672
$415$	−4932.52	−0.583440
$416$	0	0
$417$	−9290.61	−1.09104
$418$	32584.7	3.81285
$419$	7945.58	0.926413	0.463206	−	0.886250i	$-0.346699\pi$
$419$	7945.58	0.926413	0.463206	+	0.886250i	$0.346699\pi$
$420$	−8320.01	−0.966607
$421$	284.758	0.0329650	0.0164825	−	0.999864i	$-0.494753\pi$
$421$	284.758	0.0329650	0.0164825	+	0.999864i	$0.494753\pi$
$422$	−29205.2	−3.36893
$423$	3617.84	0.415852
$424$	30786.7	3.52626
$425$	3323.09	0.379279
$426$	1768.19	0.201102
$427$	−4460.68	−0.505545
$428$	26623.4	3.00676
$429$	0	0
$430$	10716.8	1.20189
$431$	12512.1	1.39835	0.699173	−	0.714952i	$-0.253552\pi$
$431$	12512.1	1.39835	0.699173	+	0.714952i	$0.253552\pi$
$432$	7079.58	0.788464
$433$	−13217.2	−1.46692	−0.733460	−	0.679732i	$-0.762096\pi$
$433$	−13217.2	−1.46692	−0.733460	+	0.679732i	$0.762096\pi$
$434$	−12665.6	−1.40085
$435$	−2056.05	−0.226621
$436$	5399.01	0.593040
$437$	−9045.16	−0.990135
$438$	716.663	0.0781815
$439$	4635.40	0.503953	0.251977	−	0.967733i	$-0.418919\pi$
$439$	4635.40	0.503953	0.251977	+	0.967733i	$0.418919\pi$
$440$	−23773.7	−2.57583
$441$	611.524	0.0660321
$442$	0	0
$443$	−2945.92	−0.315947	−0.157974	−	0.987443i	$-0.550496\pi$
$443$	−2945.92	−0.315947	−0.157974	+	0.987443i	$0.550496\pi$
$444$	5789.20	0.618791
$445$	6875.05	0.732379
$446$	−6947.45	−0.737603
$447$	−1262.64	−0.133603
$448$	−47877.3	−5.04909
$449$	8668.29	0.911095	0.455548	−	0.890211i	$-0.349443\pi$
$449$	8668.29	0.911095	0.455548	+	0.890211i	$0.349443\pi$
$450$	4375.16	0.458326
$451$	6542.66	0.683108
$452$	−23201.3	−2.41438
$453$	8403.51	0.871592
$454$	6269.27	0.648087
$455$	0	0
$456$	29026.6	2.98091
$457$	5195.36	0.531791	0.265896	−	0.964002i	$-0.414332\pi$
$457$	5195.36	0.531791	0.265896	+	0.964002i	$0.414332\pi$
$458$	−1203.93	−0.122830
$459$	−1019.29	−0.103652
$460$	10240.6	1.03798
$461$	2931.19	0.296136	0.148068	−	0.988977i	$-0.452694\pi$
$461$	2931.19	0.296136	0.148068	+	0.988977i	$0.452694\pi$
$462$	16399.4	1.65145
$463$	396.589	0.0398079	0.0199039	−	0.999802i	$-0.493664\pi$
$463$	396.589	0.0398079	0.0199039	+	0.999802i	$0.493664\pi$
$464$	−29553.4	−2.95685
$465$	−2063.78	−0.205818
$466$	14.1581	0.00140743
$467$	16627.5	1.64760	0.823801	−	0.566880i	$-0.191850\pi$
$467$	16627.5	1.64760	0.823801	+	0.566880i	$0.191850\pi$
$468$	0	0
$469$	19772.4	1.94670
$470$	−13498.9	−1.32480
$471$	−7032.60	−0.687994
$472$	−9738.91	−0.949724
$473$	−15582.9	−1.51480
$474$	−8935.62	−0.865879
$475$	10636.8	1.02747
$476$	17218.1	1.65796
$477$	−3460.44	−0.332165
$478$	21930.9	2.09852
$479$	−8903.28	−0.849272	−0.424636	−	0.905364i	$-0.639598\pi$
$479$	−8903.28	−0.849272	−0.424636	+	0.905364i	$0.639598\pi$
$480$	−14730.2	−1.40070
$481$	0	0
$482$	−3634.03	−0.343414
$483$	−4552.30	−0.428855
$484$	23696.1	2.22540
$485$	1393.00	0.130418
$486$	−1341.99	−0.125255
$487$	469.526	0.0436884	0.0218442	−	0.999761i	$-0.493046\pi$
$487$	469.526	0.0436884	0.0218442	+	0.999761i	$0.493046\pi$
$488$	−17619.1	−1.63438
$489$	−10975.2	−1.01496
$490$	−2281.72	−0.210362
$491$	−18823.8	−1.73015	−0.865077	−	0.501639i	$-0.832731\pi$
$491$	−18823.8	−1.73015	−0.865077	+	0.501639i	$0.832731\pi$
$492$	9044.05	0.828734
$493$	4254.96	0.388709
$494$	0	0
$495$	2672.17	0.242637
$496$	−29664.4	−2.68543
$497$	2163.52	0.195265
$498$	13439.5	1.20931
$499$	5651.95	0.507046	0.253523	−	0.967329i	$-0.418411\pi$
$499$	5651.95	0.507046	0.253523	+	0.967329i	$0.418411\pi$
$500$	−29143.5	−2.60667
$501$	−5963.55	−0.531800
$502$	−39160.7	−3.48172
$503$	8022.35	0.711131	0.355565	−	0.934651i	$-0.384288\pi$
$503$	8022.35	0.711131	0.355565	+	0.934651i	$0.384288\pi$
$504$	14608.7	1.29111
$505$	−5138.62	−0.452803
$506$	−20185.1	−1.77339
$507$	0	0
$508$	−405.432	−0.0354097
$509$	−3940.56	−0.343148	−0.171574	−	0.985171i	$-0.554885\pi$
$509$	−3940.56	−0.343148	−0.171574	+	0.985171i	$0.554885\pi$
$510$	3803.16	0.330210
$511$	876.890	0.0759126
$512$	−43768.2	−3.77793
$513$	−3262.60	−0.280794
$514$	−4376.81	−0.375589
$515$	−6658.88	−0.569758
$516$	−21540.5	−1.83773
$517$	19628.1	1.66972
$518$	9602.22	0.814474
$519$	−10731.3	−0.907617
$520$	0	0
$521$	8644.84	0.726943	0.363472	−	0.931605i	$-0.381591\pi$
$521$	8644.84	0.726943	0.363472	+	0.931605i	$0.381591\pi$
$522$	5602.05	0.469722
$523$	1956.76	0.163600	0.0818001	−	0.996649i	$-0.473933\pi$
$523$	1956.76	0.163600	0.0818001	+	0.996649i	$0.473933\pi$
$524$	−28855.3	−2.40563
$525$	5353.33	0.445025
$526$	1863.69	0.154488
$527$	4270.95	0.353028
$528$	38409.4	3.16582
$529$	−6563.84	−0.539479
$530$	12911.6	1.05820
$531$	1094.66	0.0894615
$532$	55112.8	4.49144
$533$	0	0
$534$	−18732.2	−1.51802
$535$	7195.39	0.581465
$536$	78098.2	6.29352
$537$	2310.43	0.185666
$538$	19461.2	1.55954
$539$	3317.75	0.265131
$540$	3693.80	0.294363
$541$	−7005.37	−0.556718	−0.278359	−	0.960477i	$-0.589790\pi$
$541$	−7005.37	−0.556718	−0.278359	+	0.960477i	$0.589790\pi$
$542$	45912.6	3.63859
$543$	−5687.06	−0.449457
$544$	30483.8	2.40254
$545$	1459.16	0.114686
$546$	0	0
$547$	23216.7	1.81476	0.907382	−	0.420307i	$-0.138077\pi$
$547$	23216.7	1.81476	0.907382	+	0.420307i	$0.138077\pi$
$548$	−34813.9	−2.71382
$549$	1980.39	0.153955
$550$	23736.9	1.84026
$551$	13619.6	1.05302
$552$	−17980.9	−1.38645
$553$	−10933.4	−0.840751
$554$	13808.2	1.05895
$555$	1564.62	0.119665
$556$	−69675.9	−5.31460
$557$	14912.9	1.13443	0.567217	−	0.823568i	$-0.308020\pi$
$557$	14912.9	1.13443	0.567217	+	0.823568i	$0.308020\pi$
$558$	5623.11	0.426604
$559$	0	0
$560$	−32321.1	−2.43896
$561$	−5530.01	−0.416180
$562$	−16861.1	−1.26556
$563$	10854.3	0.812531	0.406266	−	0.913755i	$-0.366831\pi$
$563$	10854.3	0.812531	0.406266	+	0.913755i	$0.366831\pi$
$564$	27132.4	2.02567
$565$	−6270.50	−0.466906
$566$	−6399.55	−0.475253
$567$	−1642.02	−0.121620
$568$	8545.58	0.631276
$569$	17892.5	1.31827	0.659134	−	0.752026i	$-0.270923\pi$
$569$	17892.5	1.31827	0.659134	+	0.752026i	$0.270923\pi$
$570$	12173.4	0.894541
$571$	17319.8	1.26937	0.634686	−	0.772770i	$-0.281130\pi$
$571$	17319.8	1.26937	0.634686	+	0.772770i	$0.281130\pi$
$572$	0	0
$573$	6611.48	0.482022
$574$	15000.9	1.09081
$575$	−6589.10	−0.477886
$576$	21255.9	1.53761
$577$	9738.48	0.702632	0.351316	−	0.936257i	$-0.385734\pi$
$577$	9738.48	0.702632	0.351316	+	0.936257i	$0.385734\pi$
$578$	19261.8	1.38614
$579$	−3049.13	−0.218855
$580$	−15419.6	−1.10390
$581$	16444.2	1.17421
$582$	−3795.46	−0.270321
$583$	−18774.2	−1.33370
$584$	3463.59	0.245419
$585$	0	0
$586$	−46154.7	−3.25364
$587$	5960.31	0.419095	0.209547	−	0.977799i	$-0.432801\pi$
$587$	5960.31	0.419095	0.209547	+	0.977799i	$0.432801\pi$
$588$	4586.19	0.321652
$589$	13670.7	0.956355
$590$	−4084.39	−0.285003
$591$	−12955.7	−0.901740
$592$	22489.6	1.56134
$593$	17870.2	1.23751	0.618753	−	0.785586i	$-0.287638\pi$
$593$	17870.2	1.23751	0.618753	+	0.785586i	$0.287638\pi$
$594$	−7280.78	−0.502919
$595$	4653.45	0.320627
$596$	−9469.28	−0.650800
$597$	4090.20	0.280403
$598$	0	0
$599$	−17943.7	−1.22397	−0.611987	−	0.790868i	$-0.709629\pi$
$599$	−17943.7	−1.22397	−0.611987	+	0.790868i	$0.709629\pi$
$600$	21144.9	1.43873
$601$	6548.20	0.444437	0.222219	−	0.974997i	$-0.428670\pi$
$601$	6548.20	0.444437	0.222219	+	0.974997i	$0.428670\pi$
$602$	−35728.1	−2.41888
$603$	−8778.27	−0.592834
$604$	63023.0	4.24565
$605$	6404.21	0.430361
$606$	14001.0	0.938534
$607$	−14886.0	−0.995391	−0.497696	−	0.867352i	$-0.665820\pi$
$607$	−14886.0	−0.995391	−0.497696	+	0.867352i	$0.665820\pi$
$608$	97574.6	6.50851
$609$	6854.52	0.456091
$610$	−7389.24	−0.490462
$611$	0	0
$612$	−7644.25	−0.504903
$613$	8774.65	0.578148	0.289074	−	0.957307i	$-0.406653\pi$
$613$	8774.65	0.578148	0.289074	+	0.957307i	$0.406653\pi$
$614$	−46551.5	−3.05971
$615$	2444.29	0.160266
$616$	79257.4	5.18405
$617$	3810.41	0.248625	0.124312	−	0.992243i	$-0.460327\pi$
$617$	3810.41	0.248625	0.124312	+	0.992243i	$0.460327\pi$
$618$	18143.2	1.18095
$619$	16392.6	1.06441	0.532207	−	0.846614i	$-0.321363\pi$
$619$	16392.6	1.06441	0.532207	+	0.846614i	$0.321363\pi$
$620$	−15477.5	−1.00257
$621$	2021.07	0.130600
$622$	−19897.1	−1.28264
$623$	−22920.2	−1.47396
$624$	0	0
$625$	3126.74	0.200112
$626$	−36598.0	−2.33666
$627$	−17700.8	−1.12744
$628$	−52741.8	−3.35131
$629$	−3237.95	−0.205255
$630$	6126.70	0.387450
$631$	11077.3	0.698858	0.349429	−	0.936963i	$-0.386376\pi$
$631$	11077.3	0.698858	0.349429	+	0.936963i	$0.386376\pi$
$632$	−43185.4	−2.71807
$633$	15865.0	0.996173
$634$	22001.2	1.37820
$635$	−109.574	−0.00684775
$636$	−25951.9	−1.61802
$637$	0	0
$638$	30393.2	1.88602
$639$	−960.527	−0.0594646
$640$	−40029.6	−2.47236
$641$	−2143.45	−0.132077	−0.0660383	−	0.997817i	$-0.521036\pi$
$641$	−2143.45	−0.132077	−0.0660383	+	0.997817i	$0.521036\pi$
$642$	−19605.0	−1.20522
$643$	−9774.93	−0.599511	−0.299755	−	0.954016i	$-0.596905\pi$
$643$	−9774.93	−0.599511	−0.299755	+	0.954016i	$0.596905\pi$
$644$	−34140.5	−2.08901
$645$	−5821.65	−0.355391
$646$	−25192.6	−1.53435
$647$	−22447.6	−1.36400	−0.681999	−	0.731353i	$-0.738889\pi$
$647$	−22447.6	−1.36400	−0.681999	+	0.731353i	$0.738889\pi$
$648$	−6485.74	−0.393185
$649$	5938.93	0.359204
$650$	0	0
$651$	6880.29	0.414224
$652$	−82310.0	−4.94403
$653$	10582.0	0.634157	0.317079	−	0.948399i	$-0.397298\pi$
$653$	10582.0	0.634157	0.317079	+	0.948399i	$0.397298\pi$
$654$	−3975.73	−0.237711
$655$	−7798.57	−0.465214
$656$	35133.9	2.09108
$657$	−389.309	−0.0231178
$658$	45003.0	2.66626
$659$	−15114.6	−0.893445	−0.446722	−	0.894673i	$-0.647409\pi$
$659$	−15114.6	−0.893445	−0.446722	+	0.894673i	$0.647409\pi$
$660$	20040.3	1.18192
$661$	−2229.61	−0.131198	−0.0655990	−	0.997846i	$-0.520896\pi$
$661$	−2229.61	−0.131198	−0.0655990	+	0.997846i	$0.520896\pi$
$662$	23044.7	1.35296
$663$	0	0
$664$	64952.1	3.79613
$665$	14895.1	0.868581
$666$	−4263.06	−0.248033
$667$	−8436.83	−0.489768
$668$	−44724.3	−2.59047
$669$	3774.03	0.218105
$670$	32753.5	1.88862
$671$	10744.4	0.618155
$672$	49107.9	2.81901
$673$	−3588.22	−0.205521	−0.102761	−	0.994706i	$-0.532768\pi$
$673$	−3588.22	−0.205521	−0.102761	+	0.994706i	$0.532768\pi$
$674$	−22960.2	−1.31216
$675$	−2376.69	−0.135524
$676$	0	0
$677$	−26458.4	−1.50204	−0.751019	−	0.660280i	$-0.770438\pi$
$677$	−26458.4	−1.50204	−0.751019	+	0.660280i	$0.770438\pi$
$678$	17085.0	0.967767
$679$	−4644.03	−0.262476
$680$	18380.5	1.03656
$681$	−3405.62	−0.191635
$682$	30507.5	1.71289
$683$	23598.4	1.32206	0.661031	−	0.750358i	$-0.270119\pi$
$683$	23598.4	1.32206	0.661031	+	0.750358i	$0.270119\pi$
$684$	−24468.2	−1.36779
$685$	−9408.97	−0.524815
$686$	−30792.9	−1.71382
$687$	654.007	0.0363201
$688$	−83679.4	−4.63699
$689$	0	0
$690$	−7541.01	−0.416060
$691$	16433.6	0.904722	0.452361	−	0.891835i	$-0.350582\pi$
$691$	16433.6	0.904722	0.452361	+	0.891835i	$0.350582\pi$
$692$	−80480.8	−4.42113
$693$	−8908.57	−0.488324
$694$	−25779.2	−1.41004
$695$	−18831.0	−1.02777
$696$	27074.4	1.47450
$697$	−5058.41	−0.274894
$698$	65749.7	3.56542
$699$	−7.69102	−0.000416167 0
$700$	40147.9	2.16778
$701$	13899.4	0.748892	0.374446	−	0.927249i	$-0.377833\pi$
$701$	13899.4	0.748892	0.374446	+	0.927249i	$0.377833\pi$
$702$	0	0
$703$	−10364.2	−0.556038
$704$	115321.	6.17377
$705$	7332.93	0.391736
$706$	−9179.16	−0.489323
$707$	17131.3	0.911297
$708$	8209.50	0.435780
$709$	−32829.6	−1.73899	−0.869494	−	0.493944i	$-0.835555\pi$
$709$	−32829.6	−1.73899	−0.869494	+	0.493944i	$0.835555\pi$
$710$	3583.92	0.189440
$711$	4854.05	0.256035
$712$	−90531.7	−4.76519
$713$	−8468.55	−0.444810
$714$	−12679.1	−0.664570
$715$	0	0
$716$	17327.3	0.904403
$717$	−11913.4	−0.620521
$718$	70437.0	3.66112
$719$	−4683.94	−0.242951	−0.121475	−	0.992594i	$-0.538763\pi$
$719$	−4683.94	−0.242951	−0.121475	+	0.992594i	$0.538763\pi$
$720$	14349.5	0.742741
$721$	22199.5	1.14668
$722$	−42759.0	−2.20405
$723$	1974.10	0.101546
$724$	−42650.7	−2.18937
$725$	9921.39	0.508236
$726$	−17449.3	−0.892019
$727$	−35368.3	−1.80432	−0.902159	−	0.431405i	$-0.858018\pi$
$727$	−35368.3	−1.80432	−0.902159	+	0.431405i	$0.858018\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	1452.59	0.0736478
$731$	12047.8	0.609580
$732$	14852.2	0.749934
$733$	−33988.4	−1.71268	−0.856338	−	0.516416i	$-0.827266\pi$
$733$	−33988.4	−1.71268	−0.856338	+	0.516416i	$0.827266\pi$
$734$	−24651.7	−1.23966
$735$	1239.49	0.0622029
$736$	−60444.0	−3.02717
$737$	−47625.4	−2.38033
$738$	−6659.87	−0.332186
$739$	7777.31	0.387135	0.193568	−	0.981087i	$-0.437994\pi$
$739$	7777.31	0.387135	0.193568	+	0.981087i	$0.437994\pi$
$740$	11734.0	0.582907
$741$	0	0
$742$	−43045.0	−2.12969
$743$	7022.08	0.346723	0.173361	−	0.984858i	$-0.444537\pi$
$743$	7022.08	0.346723	0.173361	+	0.984858i	$0.444537\pi$
$744$	27176.2	1.33915
$745$	−2559.22	−0.125856
$746$	−52212.9	−2.56254
$747$	−7300.64	−0.357586
$748$	−41472.9	−2.02727
$749$	−23988.2	−1.17024
$750$	21460.7	1.04485
$751$	8214.69	0.399146	0.199573	−	0.979883i	$-0.436045\pi$
$751$	8214.69	0.399146	0.199573	+	0.979883i	$0.436045\pi$
$752$	105402.	5.11121
$753$	21273.0	1.02953
$754$	0	0
$755$	17032.9	0.821048
$756$	−12314.5	−0.592426
$757$	−20846.9	−1.00092	−0.500459	−	0.865760i	$-0.666835\pi$
$757$	−20846.9	−1.00092	−0.500459	+	0.865760i	$0.666835\pi$
$758$	77790.3	3.72753
$759$	10965.0	0.524382
$760$	58833.5	2.80804
$761$	3563.06	0.169725	0.0848627	−	0.996393i	$-0.472955\pi$
$761$	3563.06	0.169725	0.0848627	+	0.996393i	$0.472955\pi$
$762$	298.553	0.0141935
$763$	−4864.60	−0.230813
$764$	49583.6	2.34800
$765$	−2065.97	−0.0976410
$766$	−47846.7	−2.25688
$767$	0	0
$768$	52384.9	2.46130
$769$	−32557.9	−1.52675	−0.763373	−	0.645958i	$-0.776458\pi$
$769$	−32557.9	−1.52675	−0.763373	+	0.645958i	$0.776458\pi$
$770$	33239.7	1.55568
$771$	2377.59	0.111059
$772$	−22867.3	−1.06608
$773$	−20126.4	−0.936477	−0.468238	−	0.883602i	$-0.655111\pi$
$773$	−20126.4	−0.936477	−0.468238	+	0.883602i	$0.655111\pi$
$774$	15862.0	0.736627
$775$	9958.68	0.461583
$776$	−18343.2	−0.848562
$777$	−5216.17	−0.240835
$778$	−28085.5	−1.29423
$779$	−16191.3	−0.744690
$780$	0	0
$781$	−5211.22	−0.238761
$782$	15606.0	0.713642
$783$	−3043.17	−0.138894
$784$	17816.2	0.811597
$785$	−14254.3	−0.648097
$786$	21248.5	0.964260
$787$	−18828.4	−0.852806	−0.426403	−	0.904533i	$-0.640219\pi$
$787$	−18828.4	−0.852806	−0.426403	+	0.904533i	$0.640219\pi$
$788$	−97163.1	−4.39250
$789$	−1012.40	−0.0456812
$790$	−18111.5	−0.815667
$791$	20904.8	0.939682
$792$	−35187.6	−1.57871
$793$	0	0
$794$	71703.7	3.20487
$795$	−7013.90	−0.312902
$796$	30674.9	1.36588
$797$	7288.52	0.323930	0.161965	−	0.986796i	$-0.448217\pi$
$797$	7288.52	0.323930	0.161965	+	0.986796i	$0.448217\pi$
$798$	−40584.1	−1.80033
$799$	−15175.3	−0.671921
$800$	71079.9	3.14132
$801$	10175.8	0.448869
$802$	35861.7	1.57896
$803$	−2112.15	−0.0928221
$804$	−65833.6	−2.88778
$805$	−9226.97	−0.403985
$806$	0	0
$807$	−10571.8	−0.461146
$808$	67666.1	2.94614
$809$	−33799.0	−1.46886	−0.734430	−	0.678684i	$-0.762551\pi$
$809$	−33799.0	−1.46886	−0.734430	+	0.678684i	$0.762551\pi$
$810$	−2720.05	−0.117991
$811$	29055.3	1.25804	0.629019	−	0.777390i	$-0.283457\pi$
$811$	29055.3	1.25804	0.629019	+	0.777390i	$0.283457\pi$
$812$	51406.2	2.22168
$813$	−24940.9	−1.07591
$814$	−23128.7	−0.995898
$815$	−22245.5	−0.956106
$816$	−29696.0	−1.27398
$817$	38563.3	1.65136
$818$	−69287.3	−2.96158
$819$	0	0
$820$	18331.2	0.780676
$821$	8921.98	0.379268	0.189634	−	0.981855i	$-0.439270\pi$
$821$	8921.98	0.379268	0.189634	+	0.981855i	$0.439270\pi$
$822$	25636.3	1.08780
$823$	−9148.48	−0.387480	−0.193740	−	0.981053i	$-0.562062\pi$
$823$	−9148.48	−0.387480	−0.193740	+	0.981053i	$0.562062\pi$
$824$	87685.1	3.70711
$825$	−12894.5	−0.544155
$826$	13616.6	0.573588
$827$	20206.0	0.849615	0.424807	−	0.905284i	$-0.360342\pi$
$827$	20206.0	0.849615	0.424807	+	0.905284i	$0.360342\pi$
$828$	15157.2	0.636170
$829$	−10832.4	−0.453830	−0.226915	−	0.973915i	$-0.572864\pi$
$829$	−10832.4	−0.453830	−0.226915	+	0.973915i	$0.572864\pi$
$830$	27240.2	1.13918
$831$	−7500.98	−0.313124
$832$	0	0
$833$	−2565.09	−0.106693
$834$	51308.1	2.13028
$835$	−12087.4	−0.500961
$836$	−132749.	−5.49190
$837$	−3054.61	−0.126144
$838$	−43880.1	−1.80884
$839$	−7047.44	−0.289994	−0.144997	−	0.989432i	$-0.546317\pi$
$839$	−7047.44	−0.289994	−0.144997	+	0.989432i	$0.546317\pi$
$840$	29610.0	1.21624
$841$	−11685.4	−0.479127
$842$	−1572.60	−0.0643650
$843$	9159.39	0.374218
$844$	118981.	4.85250
$845$	0	0
$846$	−19979.8	−0.811961
$847$	−21350.5	−0.866132
$848$	−100817.	−4.08262
$849$	3476.40	0.140530
$850$	−18352.0	−0.740551
$851$	6420.28	0.258618
$852$	−7203.58	−0.289660
$853$	24085.5	0.966790	0.483395	−	0.875402i	$-0.339404\pi$
$853$	24085.5	0.966790	0.483395	+	0.875402i	$0.339404\pi$
$854$	24634.5	0.987089
$855$	−6612.90	−0.264511
$856$	−94749.9	−3.78328
$857$	−4161.31	−0.165866	−0.0829332	−	0.996555i	$-0.526429\pi$
$857$	−4161.31	−0.165866	−0.0829332	+	0.996555i	$0.526429\pi$
$858$	0	0
$859$	27387.5	1.08784	0.543918	−	0.839139i	$-0.316940\pi$
$859$	27387.5	1.08784	0.543918	+	0.839139i	$0.316940\pi$
$860$	−43660.1	−1.73116
$861$	−8148.84	−0.322546
$862$	−69099.1	−2.73031
$863$	19420.4	0.766022	0.383011	−	0.923744i	$-0.374887\pi$
$863$	19420.4	0.766022	0.383011	+	0.923744i	$0.374887\pi$
$864$	−21802.2	−0.858480
$865$	−21751.1	−0.854984
$866$	72992.8	2.86420
$867$	−10463.5	−0.409873
$868$	51599.5	2.01774
$869$	26335.1	1.02803
$870$	11354.7	0.442483
$871$	0	0
$872$	−19214.5	−0.746198
$873$	2061.79	0.0799324
$874$	49952.6	1.93326
$875$	26258.8	1.01453
$876$	−2919.67	−0.112610
$877$	−9908.85	−0.381526	−0.190763	−	0.981636i	$-0.561096\pi$
$877$	−9908.85	−0.381526	−0.190763	+	0.981636i	$0.561096\pi$
$878$	−25599.3	−0.983981
$879$	25072.4	0.962081
$880$	77851.4	2.98224
$881$	20323.3	0.777197	0.388598	−	0.921407i	$-0.372959\pi$
$881$	20323.3	0.777197	0.388598	+	0.921407i	$0.372959\pi$
$882$	−3377.19	−0.128929
$883$	12443.3	0.474238	0.237119	−	0.971481i	$-0.423797\pi$
$883$	12443.3	0.474238	0.237119	+	0.971481i	$0.423797\pi$
$884$	0	0
$885$	2218.74	0.0842737
$886$	16269.0	0.616895
$887$	42313.5	1.60174	0.800872	−	0.598835i	$-0.204369\pi$
$887$	42313.5	1.60174	0.800872	+	0.598835i	$0.204369\pi$
$888$	−20603.1	−0.778599
$889$	365.301	0.0137816
$890$	−37968.0	−1.42999
$891$	3955.10	0.148710
$892$	28303.7	1.06242
$893$	−48574.3	−1.82024
$894$	6973.00	0.260864
$895$	4682.97	0.174899
$896$	133452.	4.97580
$897$	0	0
$898$	−47871.3	−1.77894
$899$	12751.3	0.473059
$900$	−17824.3	−0.660159
$901$	14515.1	0.536702
$902$	−36132.3	−1.33379
$903$	19408.4	0.715249
$904$	82571.0	3.03791
$905$	−11527.0	−0.423393
$906$	−46409.0	−1.70181
$907$	−11002.3	−0.402786	−0.201393	−	0.979511i	$-0.564547\pi$
$907$	−11002.3	−0.402786	−0.201393	+	0.979511i	$0.564547\pi$
$908$	−25540.8	−0.933483
$909$	−7605.69	−0.277519
$910$	0	0
$911$	−40803.4	−1.48395	−0.741974	−	0.670428i	$-0.766110\pi$
$911$	−40803.4	−1.48395	−0.741974	+	0.670428i	$0.766110\pi$
$912$	−95052.8	−3.45122
$913$	−39608.8	−1.43577
$914$	−28691.8	−1.03834
$915$	4014.02	0.145027
$916$	4904.80	0.176920
$917$	25999.1	0.936276
$918$	5629.08	0.202383
$919$	2642.69	0.0948579	0.0474290	−	0.998875i	$-0.484897\pi$
$919$	2642.69	0.0948579	0.0474290	+	0.998875i	$0.484897\pi$
$920$	−36445.3	−1.30605
$921$	25287.9	0.904739
$922$	−16187.7	−0.578214
$923$	0	0
$924$	−66810.8	−2.37869
$925$	−7550.00	−0.268370
$926$	−2190.19	−0.0777258
$927$	−9855.84	−0.349200
$928$	91012.3	3.21942
$929$	12687.7	0.448083	0.224042	−	0.974580i	$-0.428075\pi$
$929$	12687.7	0.448083	0.224042	+	0.974580i	$0.428075\pi$
$930$	11397.4	0.401865
$931$	−8210.52	−0.289032
$932$	−57.6796	−0.00202721
$933$	10808.6	0.379269
$934$	−91826.7	−3.21698
$935$	−11208.7	−0.392046
$936$	0	0
$937$	−23066.2	−0.804205	−0.402103	−	0.915595i	$-0.631721\pi$
$937$	−23066.2	−0.804205	−0.402103	+	0.915595i	$0.631721\pi$
$938$	−109195.	−3.80099
$939$	19880.9	0.690937
$940$	54994.1	1.90820
$941$	12660.7	0.438605	0.219303	−	0.975657i	$-0.429622\pi$
$941$	12660.7	0.438605	0.219303	+	0.975657i	$0.429622\pi$
$942$	38838.1	1.34332
$943$	10029.9	0.346362
$944$	31891.8	1.09957
$945$	−3328.18	−0.114567
$946$	86057.5	2.95769
$947$	21059.1	0.722629	0.361315	−	0.932444i	$-0.382328\pi$
$947$	21059.1	0.722629	0.361315	+	0.932444i	$0.382328\pi$
$948$	36403.5	1.24718
$949$	0	0
$950$	−58742.3	−2.00616
$951$	−11951.6	−0.407526
$952$	−61277.3	−2.08614
$953$	35646.2	1.21164	0.605820	−	0.795602i	$-0.292845\pi$
$953$	35646.2	1.21164	0.605820	+	0.795602i	$0.292845\pi$
$954$	19110.5	0.648560
$955$	13400.7	0.454070
$956$	−89345.8	−3.02265
$957$	−16510.4	−0.557685
$958$	49169.0	1.65822
$959$	31367.9	1.05623
$960$	43083.2	1.44844
$961$	−16991.7	−0.570365
$962$	0	0
$963$	10649.9	0.356375
$964$	14805.0	0.494643
$965$	−6180.22	−0.206164
$966$	25140.4	0.837349
$967$	53452.1	1.77756	0.888782	−	0.458329i	$-0.151552\pi$
$967$	53452.1	1.77756	0.888782	+	0.458329i	$0.151552\pi$
$968$	−84331.7	−2.80013
$969$	13685.3	0.453699
$970$	−7692.95	−0.254645
$971$	6502.60	0.214911	0.107455	−	0.994210i	$-0.465730\pi$
$971$	6502.60	0.214911	0.107455	+	0.994210i	$0.465730\pi$
$972$	5467.22	0.180413
$973$	62779.2	2.06846
$974$	−2592.99	−0.0853027
$975$	0	0
$976$	57696.9	1.89225
$977$	−4351.84	−0.142505	−0.0712527	−	0.997458i	$-0.522700\pi$
$977$	−4351.84	−0.142505	−0.0712527	+	0.997458i	$0.522700\pi$
$978$	60611.5	1.98174
$979$	55207.5	1.80229
$980$	9295.67	0.302999
$981$	2159.71	0.0702899
$982$	103956.	3.37817
$983$	48698.2	1.58009	0.790047	−	0.613046i	$-0.210056\pi$
$983$	48698.2	1.58009	0.790047	+	0.613046i	$0.210056\pi$
$984$	−32186.8	−1.04276
$985$	−26259.8	−0.849448
$986$	−23498.3	−0.758964
$987$	−24446.7	−0.788397
$988$	0	0
$989$	−23888.6	−0.768063
$990$	−14757.3	−0.473755
$991$	46630.0	1.49470	0.747351	−	0.664429i	$-0.231325\pi$
$991$	46630.0	1.49470	0.747351	+	0.664429i	$0.231325\pi$
$992$	91354.3	2.92389
$993$	−12518.5	−0.400062
$994$	−11948.2	−0.381261
$995$	8290.35	0.264143
$996$	−54752.0	−1.74185
$997$	37175.5	1.18090	0.590452	−	0.807073i	$-0.298950\pi$
$997$	37175.5	1.18090	0.590452	+	0.807073i	$0.298950\pi$
$998$	−31213.3	−0.990021
$999$	2315.80	0.0733420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.o.1.1	✓	9
3.2	odd	2		1521.4.a.bi.1.9		9
13.5	odd	4		507.4.b.k.337.18		18
13.8	odd	4		507.4.b.k.337.1		18
13.12	even	2		507.4.a.p.1.9	yes	9
39.38	odd	2		1521.4.a.bf.1.1		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.1	✓	9	1.1	even	1	trivial
507.4.a.p.1.9	yes	9	13.12	even	2
507.4.b.k.337.1		18	13.8	odd	4
507.4.b.k.337.18		18	13.5	odd	4
1521.4.a.bf.1.1		9	39.38	odd	2
1521.4.a.bi.1.9		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)

\(f(q)\)	\(=\)	\(q-5.52257 q^{2} +3.00000 q^{3} +22.4988 q^{4} +6.08065 q^{5} -16.5677 q^{6} -20.2718 q^{7} -80.0709 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-5.52257 q^{2} +3.00000 q^{3} +22.4988 q^{4} +6.08065 q^{5} -16.5677 q^{6} -20.2718 q^{7} -80.0709 q^{8} +9.00000 q^{9} -33.5808 q^{10} +48.8284 q^{11} +67.4965 q^{12} +111.953 q^{14} +18.2419 q^{15} +262.207 q^{16} -37.7513 q^{17} -49.7032 q^{18} -120.837 q^{19} +136.807 q^{20} -60.8155 q^{21} -269.658 q^{22} +74.8543 q^{23} -240.213 q^{24} -88.0257 q^{25} +27.0000 q^{27} -456.092 q^{28} -112.710 q^{29} -100.742 q^{30} -113.134 q^{31} -807.490 q^{32} +146.485 q^{33} +208.485 q^{34} -123.266 q^{35} +202.489 q^{36} +85.7704 q^{37} +667.331 q^{38} -486.883 q^{40} +133.993 q^{41} +335.858 q^{42} -319.135 q^{43} +1098.58 q^{44} +54.7258 q^{45} -413.388 q^{46} +401.982 q^{47} +786.620 q^{48} +67.9471 q^{49} +486.129 q^{50} -113.254 q^{51} -384.493 q^{53} -149.110 q^{54} +296.908 q^{55} +1623.18 q^{56} -362.511 q^{57} +622.450 q^{58} +121.629 q^{59} +410.422 q^{60} +220.043 q^{61} +624.790 q^{62} -182.446 q^{63} +2361.77 q^{64} -808.975 q^{66} -975.363 q^{67} -849.361 q^{68} +224.563 q^{69} +680.745 q^{70} -106.725 q^{71} -720.638 q^{72} -43.2566 q^{73} -473.673 q^{74} -264.077 q^{75} -2718.69 q^{76} -989.841 q^{77} +539.339 q^{79} +1594.39 q^{80} +81.0000 q^{81} -739.986 q^{82} -811.183 q^{83} -1368.28 q^{84} -229.553 q^{85} +1762.45 q^{86} -338.130 q^{87} -3909.73 q^{88} +1130.64 q^{89} -302.227 q^{90} +1684.13 q^{92} -339.401 q^{93} -2219.98 q^{94} -734.767 q^{95} -2422.47 q^{96} +229.088 q^{97} -375.243 q^{98} +439.456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) 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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