LMFDB - Embedded newform 507.4.a.o.1.4

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			1.4
Root			$-0.100291$ 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)

$f(q)$	$=$	$q-2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} -15.3991 q^{5} -7.04181 q^{6} +10.1317 q^{7} +24.6236 q^{8} +9.00000 q^{9} +O(q^{10})$	\(q-2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} -15.3991 q^{5} -7.04181 q^{6} +10.1317 q^{7} +24.6236 q^{8} +9.00000 q^{9} +36.1458 q^{10} -15.0669 q^{11} -7.47096 q^{12} -23.7819 q^{14} -46.1972 q^{15} -37.8757 q^{16} +90.8352 q^{17} -21.1254 q^{18} -114.640 q^{19} +38.3486 q^{20} +30.3952 q^{21} +35.3661 q^{22} +75.7635 q^{23} +73.8709 q^{24} +112.132 q^{25} +27.0000 q^{27} -25.2313 q^{28} +214.817 q^{29} +108.437 q^{30} +284.476 q^{31} -108.084 q^{32} -45.2007 q^{33} -213.215 q^{34} -156.019 q^{35} -22.4129 q^{36} -358.878 q^{37} +269.091 q^{38} -379.181 q^{40} -313.154 q^{41} -71.3458 q^{42} -296.702 q^{43} +37.5214 q^{44} -138.592 q^{45} -177.837 q^{46} +316.691 q^{47} -113.627 q^{48} -240.348 q^{49} -263.203 q^{50} +272.506 q^{51} +163.911 q^{53} -63.3763 q^{54} +232.016 q^{55} +249.480 q^{56} -343.920 q^{57} -504.233 q^{58} -254.149 q^{59} +115.046 q^{60} -935.247 q^{61} -667.742 q^{62} +91.1856 q^{63} +556.709 q^{64} +106.098 q^{66} +240.494 q^{67} -226.209 q^{68} +227.291 q^{69} +366.220 q^{70} -947.455 q^{71} +221.613 q^{72} +430.712 q^{73} +842.384 q^{74} +336.395 q^{75} +285.490 q^{76} -152.654 q^{77} -496.620 q^{79} +583.251 q^{80} +81.0000 q^{81} +735.058 q^{82} -392.527 q^{83} -75.6938 q^{84} -1398.78 q^{85} +696.439 q^{86} +644.451 q^{87} -371.002 q^{88} -979.895 q^{89} +325.312 q^{90} -188.675 q^{92} +853.428 q^{93} -743.360 q^{94} +1765.35 q^{95} -324.253 q^{96} +553.356 q^{97} +564.162 q^{98} -135.602 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.34727	−0.829886	−0.414943	−	0.909848i	$-0.636198\pi$
$2$	−2.34727	−0.829886	−0.414943	+	0.909848i	$0.636198\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	−2.49032	−0.311290
$5$	−15.3991	−1.37734	−0.688668	−	0.725077i	$-0.741804\pi$
$5$	−15.3991	−1.37734	−0.688668	+	0.725077i	$0.741804\pi$
$6$	−7.04181	−0.479135
$7$	10.1317	0.547062	0.273531	−	0.961863i	$-0.411808\pi$
$7$	10.1317	0.547062	0.273531	+	0.961863i	$0.411808\pi$
$8$	24.6236	1.08822
$9$	9.00000	0.333333
$10$	36.1458	1.14303
$11$	−15.0669	−0.412985	−0.206493	−	0.978448i	$-0.566205\pi$
$11$	−15.0669	−0.412985	−0.206493	+	0.978448i	$0.566205\pi$
$12$	−7.47096	−0.179723
$13$	0	0
$13$	0	0
$14$	−23.7819	−0.453999
$15$	−46.1972	−0.795205
$16$	−37.8757	−0.591809
$17$	90.8352	1.29593	0.647964	−	0.761671i	$-0.275621\pi$
$17$	90.8352	1.29593	0.647964	+	0.761671i	$0.275621\pi$
$18$	−21.1254	−0.276629
$19$	−114.640	−1.38422	−0.692110	−	0.721792i	$-0.743319\pi$
$19$	−114.640	−1.38422	−0.692110	+	0.721792i	$0.743319\pi$
$20$	38.3486	0.428751
$21$	30.3952	0.315847
$22$	35.3661	0.342731
$23$	75.7635	0.686860	0.343430	−	0.939178i	$-0.388411\pi$
$23$	75.7635	0.686860	0.343430	+	0.939178i	$0.388411\pi$
$24$	73.8709	0.628284
$25$	112.132	0.897052
$26$	0	0
$27$	27.0000	0.192450
$28$	−25.2313	−0.170295
$29$	214.817	1.37553	0.687767	−	0.725931i	$-0.258591\pi$
$29$	214.817	1.37553	0.687767	+	0.725931i	$0.258591\pi$
$30$	108.437	0.659929
$31$	284.476	1.64817	0.824087	−	0.566463i	$-0.191689\pi$
$31$	284.476	1.64817	0.824087	+	0.566463i	$0.191689\pi$
$32$	−108.084	−0.597087
$33$	−45.2007	−0.238437
$34$	−213.215	−1.07547
$35$	−156.019	−0.753488
$36$	−22.4129	−0.103763
$37$	−358.878	−1.59457	−0.797286	−	0.603602i	$-0.793732\pi$
$37$	−358.878	−1.59457	−0.797286	+	0.603602i	$0.793732\pi$
$38$	269.091	1.14874
$39$	0	0
$40$	−379.181	−1.49884
$41$	−313.154	−1.19284	−0.596421	−	0.802672i	$-0.703411\pi$
$41$	−313.154	−1.19284	−0.596421	+	0.802672i	$0.703411\pi$
$42$	−71.3458	−0.262116
$43$	−296.702	−1.05225	−0.526123	−	0.850409i	$-0.676355\pi$
$43$	−296.702	−1.05225	−0.526123	+	0.850409i	$0.676355\pi$
$44$	37.5214	0.128558
$45$	−138.592	−0.459112
$46$	−177.837	−0.570015
$47$	316.691	0.982854	0.491427	−	0.870919i	$-0.336475\pi$
$47$	316.691	0.982854	0.491427	+	0.870919i	$0.336475\pi$
$48$	−113.627	−0.341681
$49$	−240.348	−0.700723
$50$	−263.203	−0.744451
$51$	272.506	0.748204
$52$	0	0
$53$	163.911	0.424810	0.212405	−	0.977182i	$-0.431870\pi$
$53$	163.911	0.424810	0.212405	+	0.977182i	$0.431870\pi$
$54$	−63.3763	−0.159712
$55$	232.016	0.568819
$56$	249.480	0.595324
$57$	−343.920	−0.799180
$58$	−504.233	−1.14154
$59$	−254.149	−0.560803	−0.280401	−	0.959883i	$-0.590468\pi$
$59$	−254.149	−0.560803	−0.280401	+	0.959883i	$0.590468\pi$
$60$	115.046	0.247539
$61$	−935.247	−1.96305	−0.981526	−	0.191330i	$-0.938720\pi$
$61$	−935.247	−1.96305	−0.981526	+	0.191330i	$0.938720\pi$
$62$	−667.742	−1.36780
$63$	91.1856	0.182354
$64$	556.709	1.08732
$65$	0	0
$66$	106.098	0.197876
$67$	240.494	0.438522	0.219261	−	0.975666i	$-0.429635\pi$
$67$	240.494	0.438522	0.219261	+	0.975666i	$0.429635\pi$
$68$	−226.209	−0.403409
$69$	227.291	0.396559
$70$	366.220	0.625309
$71$	−947.455	−1.58369	−0.791847	−	0.610720i	$-0.790880\pi$
$71$	−947.455	−1.58369	−0.791847	+	0.610720i	$0.790880\pi$
$72$	221.613	0.362740
$73$	430.712	0.690562	0.345281	−	0.938499i	$-0.387784\pi$
$73$	430.712	0.690562	0.345281	+	0.938499i	$0.387784\pi$
$74$	842.384	1.32331
$75$	336.395	0.517913
$76$	285.490	0.430894
$77$	−152.654	−0.225929
$78$	0	0
$79$	−496.620	−0.707268	−0.353634	−	0.935384i	$-0.615054\pi$
$79$	−496.620	−0.707268	−0.353634	+	0.935384i	$0.615054\pi$
$80$	583.251	0.815119
$81$	81.0000	0.111111
$82$	735.058	0.989922
$83$	−392.527	−0.519102	−0.259551	−	0.965729i	$-0.583575\pi$
$83$	−392.527	−0.519102	−0.259551	+	0.965729i	$0.583575\pi$
$84$	−75.6938	−0.0983199
$85$	−1398.78	−1.78493
$86$	696.439	0.873244
$87$	644.451	0.794165
$88$	−371.002	−0.449419
$89$	−979.895	−1.16706	−0.583532	−	0.812090i	$-0.698330\pi$
$89$	−979.895	−1.16706	−0.583532	+	0.812090i	$0.698330\pi$
$90$	325.312	0.381010
$91$	0	0
$92$	−188.675	−0.213813
$93$	853.428	0.951574
$94$	−743.360	−0.815656
$95$	1765.35	1.90654
$96$	−324.253	−0.344729
$97$	553.356	0.579225	0.289613	−	0.957144i	$-0.406474\pi$
$97$	553.356	0.579225	0.289613	+	0.957144i	$0.406474\pi$
$98$	564.162	0.581520
$99$	−135.602	−0.137662
$100$	−279.243	−0.279243
$101$	−763.951	−0.752633	−0.376317	−	0.926491i	$-0.622810\pi$
$101$	−763.951	−0.752633	−0.376317	+	0.926491i	$0.622810\pi$
$102$	−639.645	−0.620924
$103$	182.518	0.174602	0.0873010	−	0.996182i	$-0.472176\pi$
$103$	182.518	0.174602	0.0873010	+	0.996182i	$0.472176\pi$
$104$	0	0
$105$	−468.058	−0.435027
$106$	−384.744	−0.352544
$107$	183.699	0.165971	0.0829855	−	0.996551i	$-0.473554\pi$
$107$	183.699	0.165971	0.0829855	+	0.996551i	$0.473554\pi$
$108$	−67.2386	−0.0599078
$109$	−1774.42	−1.55925	−0.779627	−	0.626244i	$-0.784592\pi$
$109$	−1774.42	−1.55925	−0.779627	+	0.626244i	$0.784592\pi$
$110$	−544.605	−0.472055
$111$	−1076.63	−0.920627
$112$	−383.747	−0.323756
$113$	417.288	0.347391	0.173695	−	0.984799i	$-0.444429\pi$
$113$	417.288	0.347391	0.173695	+	0.984799i	$0.444429\pi$
$114$	807.273	0.663228
$115$	−1166.69	−0.946037
$116$	−534.963	−0.428190
$117$	0	0
$118$	596.556	0.465402
$119$	920.318	0.708953
$120$	−1137.54	−0.865358
$121$	−1103.99	−0.829443
$122$	2195.28	1.62911
$123$	−939.463	−0.688687
$124$	−708.436	−0.513060
$125$	198.162	0.141793
$126$	−214.037	−0.151333
$127$	−1951.69	−1.36366	−0.681828	−	0.731513i	$-0.738815\pi$
$127$	−1951.69	−1.36366	−0.681828	+	0.731513i	$0.738815\pi$
$128$	−442.072	−0.305266
$129$	−890.105	−0.607514
$130$	0	0
$131$	−1475.61	−0.984159	−0.492080	−	0.870550i	$-0.663763\pi$
$131$	−1475.61	−0.984159	−0.492080	+	0.870550i	$0.663763\pi$
$132$	112.564	0.0742231
$133$	−1161.50	−0.757255
$134$	−564.503	−0.363923
$135$	−415.775	−0.265068
$136$	2236.69	1.41026
$137$	1900.71	1.18532	0.592660	−	0.805453i	$-0.298078\pi$
$137$	1900.71	1.18532	0.592660	+	0.805453i	$0.298078\pi$
$138$	−533.512	−0.329099
$139$	−2326.76	−1.41981	−0.709903	−	0.704299i	$-0.751261\pi$
$139$	−2326.76	−1.41981	−0.709903	+	0.704299i	$0.751261\pi$
$140$	388.538	0.234553
$141$	950.073	0.567451
$142$	2223.93	1.31428
$143$	0	0
$144$	−340.882	−0.197270
$145$	−3307.98	−1.89457
$146$	−1011.00	−0.573088
$147$	−721.044	−0.404563
$148$	893.721	0.496374
$149$	1370.92	0.753761	0.376881	−	0.926262i	$-0.376997\pi$
$149$	1370.92	0.753761	0.376881	+	0.926262i	$0.376997\pi$
$150$	−789.609	−0.429809
$151$	−1177.57	−0.634628	−0.317314	−	0.948320i	$-0.602781\pi$
$151$	−1177.57	−0.634628	−0.317314	+	0.948320i	$0.602781\pi$
$152$	−2822.85	−1.50634
$153$	817.517	0.431976
$154$	358.320	0.187495
$155$	−4380.67	−2.27009
$156$	0	0
$157$	1621.57	0.824302	0.412151	−	0.911116i	$-0.364778\pi$
$157$	1621.57	0.824302	0.412151	+	0.911116i	$0.364778\pi$
$158$	1165.70	0.586951
$159$	491.734	0.245264
$160$	1664.40	0.822389
$161$	767.616	0.375755
$162$	−190.129	−0.0922095
$163$	133.130	0.0639727	0.0319864	−	0.999488i	$-0.489817\pi$
$163$	133.130	0.0639727	0.0319864	+	0.999488i	$0.489817\pi$
$164$	779.854	0.371320
$165$	696.049	0.328408
$166$	921.368	0.430795
$167$	490.724	0.227385	0.113693	−	0.993516i	$-0.463732\pi$
$167$	490.724	0.227385	0.113693	+	0.993516i	$0.463732\pi$
$168$	748.440	0.343711
$169$	0	0
$170$	3283.31	1.48129
$171$	−1031.76	−0.461407
$172$	738.882	0.327554
$173$	2008.13	0.882518	0.441259	−	0.897380i	$-0.354532\pi$
$173$	2008.13	0.882518	0.441259	+	0.897380i	$0.354532\pi$
$174$	−1512.70	−0.659066
$175$	1136.09	0.490743
$176$	570.670	0.244408
$177$	−762.446	−0.323779
$178$	2300.08	0.968529
$179$	−2152.60	−0.898843	−0.449421	−	0.893320i	$-0.648370\pi$
$179$	−2152.60	−0.898843	−0.449421	+	0.893320i	$0.648370\pi$
$180$	345.138	0.142917
$181$	834.690	0.342774	0.171387	−	0.985204i	$-0.445175\pi$
$181$	834.690	0.342774	0.171387	+	0.985204i	$0.445175\pi$
$182$	0	0
$183$	−2805.74	−1.13337
$184$	1865.57	0.747455
$185$	5526.39	2.19626
$186$	−2003.23	−0.789697
$187$	−1368.60	−0.535200
$188$	−788.662	−0.305953
$189$	273.557	0.105282
$190$	−4143.75	−1.58221
$191$	4464.71	1.69139	0.845695	−	0.533667i	$-0.179186\pi$
$191$	4464.71	1.69139	0.845695	+	0.533667i	$0.179186\pi$
$192$	1670.13	0.627766
$193$	3299.84	1.23071	0.615357	−	0.788248i	$-0.289012\pi$
$193$	3299.84	1.23071	0.615357	+	0.788248i	$0.289012\pi$
$194$	−1298.88	−0.480691
$195$	0	0
$196$	598.543	0.218128
$197$	−2973.59	−1.07543	−0.537714	−	0.843127i	$-0.680712\pi$
$197$	−2973.59	−1.07543	−0.537714	+	0.843127i	$0.680712\pi$
$198$	318.295	0.114244
$199$	−5430.82	−1.93457	−0.967287	−	0.253684i	$-0.918358\pi$
$199$	−5430.82	−1.93457	−0.967287	+	0.253684i	$0.918358\pi$
$200$	2761.08	0.976191
$201$	721.481	0.253181
$202$	1793.20	0.624600
$203$	2176.47	0.752503
$204$	−678.626	−0.232909
$205$	4822.29	1.64294
$206$	−428.419	−0.144900
$207$	681.872	0.228953
$208$	0	0
$209$	1727.27	0.571663
$210$	1098.66	0.361022
$211$	1228.25	0.400742	0.200371	−	0.979720i	$-0.435785\pi$
$211$	1228.25	0.400742	0.200371	+	0.979720i	$0.435785\pi$
$212$	−408.192	−0.132239
$213$	−2842.37	−0.914346
$214$	−431.192	−0.137737
$215$	4568.93	1.44930
$216$	664.838	0.209428
$217$	2882.24	0.901654
$218$	4165.05	1.29400
$219$	1292.14	0.398696
$220$	−577.795	−0.177068
$221$	0	0
$222$	2527.15	0.764015
$223$	685.256	0.205776	0.102888	−	0.994693i	$-0.467192\pi$
$223$	685.256	0.205776	0.102888	+	0.994693i	$0.467192\pi$
$224$	−1095.08	−0.326644
$225$	1009.18	0.299017
$226$	−979.487	−0.288294
$227$	−287.067	−0.0839354	−0.0419677	−	0.999119i	$-0.513363\pi$
$227$	−287.067	−0.0839354	−0.0419677	+	0.999119i	$0.513363\pi$
$228$	856.470	0.248777
$229$	2302.23	0.664347	0.332174	−	0.943218i	$-0.392218\pi$
$229$	2302.23	0.664347	0.332174	+	0.943218i	$0.392218\pi$
$230$	2738.53	0.785102
$231$	−457.961	−0.130440
$232$	5289.57	1.49688
$233$	−970.620	−0.272908	−0.136454	−	0.990646i	$-0.543571\pi$
$233$	−970.620	−0.272908	−0.136454	+	0.990646i	$0.543571\pi$
$234$	0	0
$235$	−4876.75	−1.35372
$236$	632.912	0.174572
$237$	−1489.86	−0.408341
$238$	−2160.24	−0.588350
$239$	5007.40	1.35524	0.677619	−	0.735413i	$-0.263012\pi$
$239$	5007.40	1.35524	0.677619	+	0.735413i	$0.263012\pi$
$240$	1749.75	0.470609
$241$	540.092	0.144358	0.0721792	−	0.997392i	$-0.477005\pi$
$241$	540.092	0.144358	0.0721792	+	0.997392i	$0.477005\pi$
$242$	2591.36	0.688343
$243$	243.000	0.0641500
$244$	2329.07	0.611078
$245$	3701.14	0.965130
$246$	2205.17	0.571531
$247$	0	0
$248$	7004.83	1.79358
$249$	−1177.58	−0.299704
$250$	−465.141	−0.117672
$251$	−6087.80	−1.53091	−0.765455	−	0.643489i	$-0.777486\pi$
$251$	−6087.80	−1.53091	−0.765455	+	0.643489i	$0.777486\pi$
$252$	−227.081	−0.0567650
$253$	−1141.52	−0.283663
$254$	4581.14	1.13168
$255$	−4196.34	−1.03053
$256$	−3416.01	−0.833987
$257$	−5096.34	−1.23697	−0.618484	−	0.785797i	$-0.712253\pi$
$257$	−5096.34	−1.23697	−0.618484	+	0.785797i	$0.712253\pi$
$258$	2089.32	0.504167
$259$	−3636.06	−0.872330
$260$	0	0
$261$	1933.35	0.458511
$262$	3463.66	0.816739
$263$	3405.50	0.798450	0.399225	−	0.916853i	$-0.369279\pi$
$263$	3405.50	0.798450	0.399225	+	0.916853i	$0.369279\pi$
$264$	−1113.00	−0.259472
$265$	−2524.08	−0.585106
$266$	2726.36	0.628435
$267$	−2939.68	−0.673804
$268$	−598.906	−0.136507
$269$	−2720.44	−0.616611	−0.308306	−	0.951287i	$-0.599762\pi$
$269$	−2720.44	−0.616611	−0.308306	+	0.951287i	$0.599762\pi$
$270$	975.937	0.219976
$271$	−6954.27	−1.55883	−0.779413	−	0.626511i	$-0.784482\pi$
$271$	−6954.27	−1.55883	−0.779413	+	0.626511i	$0.784482\pi$
$272$	−3440.45	−0.766941
$273$	0	0
$274$	−4461.49	−0.983680
$275$	−1689.47	−0.370470
$276$	−566.026	−0.123445
$277$	−6563.96	−1.42379	−0.711895	−	0.702286i	$-0.752163\pi$
$277$	−6563.96	−1.42379	−0.711895	+	0.702286i	$0.752163\pi$
$278$	5461.53	1.17828
$279$	2560.28	0.549392
$280$	−3841.76	−0.819961
$281$	652.800	0.138586	0.0692932	−	0.997596i	$-0.477926\pi$
$281$	652.800	0.138586	0.0692932	+	0.997596i	$0.477926\pi$
$282$	−2230.08	−0.470919
$283$	6010.62	1.26252	0.631262	−	0.775570i	$-0.282537\pi$
$283$	6010.62	1.26252	0.631262	+	0.775570i	$0.282537\pi$
$284$	2359.47	0.492988
$285$	5296.05	1.10074
$286$	0	0
$287$	−3172.80	−0.652558
$288$	−972.759	−0.199029
$289$	3338.04	0.679430
$290$	7764.73	1.57228
$291$	1660.07	0.334416
$292$	−1072.61	−0.214965
$293$	1912.72	0.381372	0.190686	−	0.981651i	$-0.438929\pi$
$293$	1912.72	0.381372	0.190686	+	0.981651i	$0.438929\pi$
$294$	1692.49	0.335741
$295$	3913.66	0.772413
$296$	−8836.87	−1.73525
$297$	−406.806	−0.0794791
$298$	−3217.93	−0.625535
$299$	0	0
$300$	−837.730	−0.161221
$301$	−3006.10	−0.575644
$302$	2764.06	0.526669
$303$	−2291.85	−0.434533
$304$	4342.07	0.819194
$305$	14401.9	2.70378
$306$	−1918.93	−0.358491
$307$	1983.07	0.368665	0.184332	−	0.982864i	$-0.440988\pi$
$307$	1983.07	0.368665	0.184332	+	0.982864i	$0.440988\pi$
$308$	380.157	0.0703294
$309$	547.553	0.100807
$310$	10282.6	1.88391
$311$	1893.76	0.345291	0.172645	−	0.984984i	$-0.444769\pi$
$311$	1893.76	0.345291	0.172645	+	0.984984i	$0.444769\pi$
$312$	0	0
$313$	4574.19	0.826034	0.413017	−	0.910723i	$-0.364475\pi$
$313$	4574.19	0.826034	0.413017	+	0.910723i	$0.364475\pi$
$314$	−3806.26	−0.684076
$315$	−1404.17	−0.251163
$316$	1236.74	0.220165
$317$	−7594.53	−1.34559	−0.672794	−	0.739830i	$-0.734906\pi$
$317$	−7594.53	−1.34559	−0.672794	+	0.739830i	$0.734906\pi$
$318$	−1154.23	−0.203541
$319$	−3236.62	−0.568076
$320$	−8572.81	−1.49761
$321$	551.098	0.0958234
$322$	−1801.80	−0.311834
$323$	−10413.3	−1.79385
$324$	−201.716	−0.0345878
$325$	0	0
$326$	−312.492	−0.0530901
$327$	−5323.27	−0.900236
$328$	−7710.99	−1.29807
$329$	3208.63	0.537682
$330$	−1633.82	−0.272541
$331$	1738.58	0.288705	0.144352	−	0.989526i	$-0.453890\pi$
$331$	1738.58	0.288705	0.144352	+	0.989526i	$0.453890\pi$
$332$	977.519	0.161591
$333$	−3229.90	−0.531524
$334$	−1151.86	−0.188704
$335$	−3703.38	−0.603992
$336$	−1151.24	−0.186921
$337$	−2710.61	−0.438149	−0.219074	−	0.975708i	$-0.570304\pi$
$337$	−2710.61	−0.438149	−0.219074	+	0.975708i	$0.570304\pi$
$338$	0	0
$339$	1251.86	0.200566
$340$	3483.41	0.555630
$341$	−4286.17	−0.680672
$342$	2421.82	0.382915
$343$	−5910.33	−0.930401
$344$	−7305.87	−1.14508
$345$	−3500.06	−0.546195
$346$	−4713.63	−0.732389
$347$	−2506.81	−0.387817	−0.193908	−	0.981020i	$-0.562116\pi$
$347$	−2506.81	−0.387817	−0.193908	+	0.981020i	$0.562116\pi$
$348$	−1604.89	−0.247216
$349$	−7536.48	−1.15593	−0.577963	−	0.816063i	$-0.696152\pi$
$349$	−7536.48	−1.15593	−0.577963	+	0.816063i	$0.696152\pi$
$350$	−2666.70	−0.407261
$351$	0	0
$352$	1628.50	0.246588
$353$	8992.88	1.35593	0.677964	−	0.735095i	$-0.262862\pi$
$353$	8992.88	1.35593	0.677964	+	0.735095i	$0.262862\pi$
$354$	1789.67	0.268700
$355$	14589.9	2.18128
$356$	2440.25	0.363295
$357$	2760.96	0.409314
$358$	5052.74	0.745937
$359$	4566.64	0.671359	0.335679	−	0.941976i	$-0.391034\pi$
$359$	4566.64	0.671359	0.335679	+	0.941976i	$0.391034\pi$
$360$	−3412.63	−0.499615
$361$	6283.31	0.916068
$362$	−1959.24	−0.284463
$363$	−3311.97	−0.478879
$364$	0	0
$365$	−6632.57	−0.951136
$366$	6585.84	0.940566
$367$	−6449.50	−0.917332	−0.458666	−	0.888609i	$-0.651673\pi$
$367$	−6449.50	−0.917332	−0.458666	+	0.888609i	$0.651673\pi$
$368$	−2869.60	−0.406490
$369$	−2818.39	−0.397614
$370$	−12971.9	−1.82264
$371$	1660.71	0.232398
$372$	−2125.31	−0.296215
$373$	7648.89	1.06178	0.530891	−	0.847440i	$-0.321857\pi$
$373$	7648.89	1.06178	0.530891	+	0.847440i	$0.321857\pi$
$374$	3212.49	0.444154
$375$	594.487	0.0818645
$376$	7798.08	1.06956
$377$	0	0
$378$	−642.112	−0.0873722
$379$	10297.8	1.39567	0.697837	−	0.716256i	$-0.254146\pi$
$379$	10297.8	1.39567	0.697837	+	0.716256i	$0.254146\pi$
$380$	−4396.28	−0.593486
$381$	−5855.06	−0.787307
$382$	−10479.9	−1.40366
$383$	−10258.0	−1.36856	−0.684282	−	0.729217i	$-0.739884\pi$
$383$	−10258.0	−1.36856	−0.684282	+	0.729217i	$0.739884\pi$
$384$	−1326.22	−0.176245
$385$	2350.73	0.311180
$386$	−7745.63	−1.02135
$387$	−2670.31	−0.350749
$388$	−1378.03	−0.180307
$389$	−4771.57	−0.621924	−0.310962	−	0.950422i	$-0.600651\pi$
$389$	−4771.57	−0.621924	−0.310962	+	0.950422i	$0.600651\pi$
$390$	0	0
$391$	6882.00	0.890122
$392$	−5918.24	−0.762541
$393$	−4426.84	−0.568205
$394$	6979.82	0.892483
$395$	7647.50	0.974145
$396$	337.693	0.0428528
$397$	−2291.22	−0.289655	−0.144827	−	0.989457i	$-0.546263\pi$
$397$	−2291.22	−0.289655	−0.144827	+	0.989457i	$0.546263\pi$
$398$	12747.6	1.60548
$399$	−3484.50	−0.437201
$400$	−4247.07	−0.530883
$401$	−7534.63	−0.938308	−0.469154	−	0.883116i	$-0.655441\pi$
$401$	−7534.63	−0.938308	−0.469154	+	0.883116i	$0.655441\pi$
$402$	−1693.51	−0.210111
$403$	0	0
$404$	1902.48	0.234287
$405$	−1247.33	−0.153037
$406$	−5108.76	−0.624491
$407$	5407.18	0.658535
$408$	6710.08	0.814211
$409$	1517.97	0.183517	0.0917587	−	0.995781i	$-0.470751\pi$
$409$	1517.97	0.183517	0.0917587	+	0.995781i	$0.470751\pi$
$410$	−11319.2	−1.36345
$411$	5702.14	0.684345
$412$	−454.528	−0.0543519
$413$	−2574.97	−0.306794
$414$	−1600.54	−0.190005
$415$	6044.56	0.714978
$416$	0	0
$417$	−6980.28	−0.819726
$418$	−4054.36	−0.474415
$419$	8080.97	0.942199	0.471100	−	0.882080i	$-0.343857\pi$
$419$	8080.97	0.942199	0.471100	+	0.882080i	$0.343857\pi$
$420$	1165.61	0.135419
$421$	−8073.13	−0.934585	−0.467293	−	0.884103i	$-0.654771\pi$
$421$	−8073.13	−0.934585	−0.467293	+	0.884103i	$0.654771\pi$
$422$	−2883.05	−0.332570
$423$	2850.22	0.327618
$424$	4036.09	0.462287
$425$	10185.5	1.16252
$426$	6671.80	0.758803
$427$	−9475.68	−1.07391
$428$	−457.470	−0.0516651
$429$	0	0
$430$	−10724.5	−1.20275
$431$	−2241.39	−0.250496	−0.125248	−	0.992125i	$-0.539973\pi$
$431$	−2241.39	−0.250496	−0.125248	+	0.992125i	$0.539973\pi$
$432$	−1022.65	−0.113894
$433$	10237.9	1.13626	0.568130	−	0.822939i	$-0.307667\pi$
$433$	10237.9	1.13626	0.568130	+	0.822939i	$0.307667\pi$
$434$	−6765.39	−0.748270
$435$	−9923.94	−1.09383
$436$	4418.88	0.485380
$437$	−8685.52	−0.950766
$438$	−3033.00	−0.330872
$439$	−5416.69	−0.588894	−0.294447	−	0.955668i	$-0.595135\pi$
$439$	−5416.69	−0.588894	−0.294447	+	0.955668i	$0.595135\pi$
$440$	5713.08	0.619001
$441$	−2163.13	−0.233574
$442$	0	0
$443$	−2537.44	−0.272138	−0.136069	−	0.990699i	$-0.543447\pi$
$443$	−2537.44	−0.272138	−0.136069	+	0.990699i	$0.543447\pi$
$444$	2681.16	0.286582
$445$	15089.5	1.60744
$446$	−1608.48	−0.170771
$447$	4112.77	0.435184
$448$	5640.43	0.594833
$449$	−7790.61	−0.818846	−0.409423	−	0.912345i	$-0.634270\pi$
$449$	−7790.61	−0.818846	−0.409423	+	0.912345i	$0.634270\pi$
$450$	−2368.83	−0.248150
$451$	4718.26	0.492626
$452$	−1039.18	−0.108139
$453$	−3532.70	−0.366403
$454$	673.825	0.0696568
$455$	0	0
$456$	−8468.55	−0.869685
$457$	−6599.71	−0.675539	−0.337770	−	0.941229i	$-0.609672\pi$
$457$	−6599.71	−0.675539	−0.337770	+	0.941229i	$0.609672\pi$
$458$	−5403.95	−0.551332
$459$	2452.55	0.249401
$460$	2905.43	0.294492
$461$	−4482.57	−0.452872	−0.226436	−	0.974026i	$-0.572707\pi$
$461$	−4482.57	−0.452872	−0.226436	+	0.974026i	$0.572707\pi$
$462$	1074.96	0.108250
$463$	3805.86	0.382016	0.191008	−	0.981589i	$-0.438824\pi$
$463$	3805.86	0.382016	0.191008	+	0.981589i	$0.438824\pi$
$464$	−8136.35	−0.814053
$465$	−13142.0	−1.31064
$466$	2278.31	0.226482
$467$	14778.8	1.46441	0.732207	−	0.681082i	$-0.238490\pi$
$467$	14778.8	1.46441	0.732207	+	0.681082i	$0.238490\pi$
$468$	0	0
$469$	2436.62	0.239899
$470$	11447.0	1.12343
$471$	4864.71	0.475911
$472$	−6258.06	−0.610277
$473$	4470.37	0.434562
$474$	3497.11	0.338877
$475$	−12854.8	−1.24172
$476$	−2291.89	−0.220690
$477$	1475.20	0.141603
$478$	−11753.7	−1.12469
$479$	−11171.2	−1.06560	−0.532801	−	0.846240i	$-0.678861\pi$
$479$	−11171.2	−1.06560	−0.532801	+	0.846240i	$0.678861\pi$
$480$	4993.20	0.474807
$481$	0	0
$482$	−1267.74	−0.119801
$483$	2302.85	0.216942
$484$	2749.29	0.258197
$485$	−8521.18	−0.797787
$486$	−570.387	−0.0532372
$487$	−6046.57	−0.562621	−0.281310	−	0.959617i	$-0.590769\pi$
$487$	−6046.57	−0.562621	−0.281310	+	0.959617i	$0.590769\pi$
$488$	−23029.2	−2.13623
$489$	399.390	0.0369347
$490$	−8687.57	−0.800948
$491$	−1035.04	−0.0951338	−0.0475669	−	0.998868i	$-0.515147\pi$
$491$	−1035.04	−0.0951338	−0.0475669	+	0.998868i	$0.515147\pi$
$492$	2339.56	0.214381
$493$	19512.9	1.78259
$494$	0	0
$495$	2088.15	0.189606
$496$	−10774.7	−0.975404
$497$	−9599.37	−0.866379
$498$	2764.10	0.248720
$499$	11698.3	1.04947	0.524736	−	0.851265i	$-0.324164\pi$
$499$	11698.3	1.04947	0.524736	+	0.851265i	$0.324164\pi$
$500$	−493.488	−0.0441389
$501$	1472.17	0.131281
$502$	14289.7	1.27048
$503$	−13552.0	−1.20130	−0.600651	−	0.799511i	$-0.705092\pi$
$503$	−13552.0	−1.20130	−0.600651	+	0.799511i	$0.705092\pi$
$504$	2245.32	0.198441
$505$	11764.1	1.03663
$506$	2679.46	0.235408
$507$	0	0
$508$	4860.33	0.424492
$509$	−5076.46	−0.442064	−0.221032	−	0.975267i	$-0.570942\pi$
$509$	−5076.46	−0.442064	−0.221032	+	0.975267i	$0.570942\pi$
$510$	9849.94	0.855221
$511$	4363.86	0.377781
$512$	11554.9	0.997380
$513$	−3095.28	−0.266393
$514$	11962.5	1.02654
$515$	−2810.60	−0.240486
$516$	2216.65	0.189113
$517$	−4771.55	−0.405904
$518$	8534.81	0.723934
$519$	6024.40	0.509522
$520$	0	0
$521$	−8493.73	−0.714236	−0.357118	−	0.934059i	$-0.616241\pi$
$521$	−8493.73	−0.714236	−0.357118	+	0.934059i	$0.616241\pi$
$522$	−4538.10	−0.380512
$523$	−1384.53	−0.115758	−0.0578789	−	0.998324i	$-0.518434\pi$
$523$	−1384.53	−0.115758	−0.0578789	+	0.998324i	$0.518434\pi$
$524$	3674.75	0.306359
$525$	3408.26	0.283331
$526$	−7993.64	−0.662622
$527$	25840.4	2.13592
$528$	1712.01	0.141109
$529$	−6426.89	−0.528223
$530$	5924.71	0.485571
$531$	−2287.34	−0.186934
$532$	2892.51	0.235726
$533$	0	0
$534$	6900.24	0.559181
$535$	−2828.80	−0.228598
$536$	5921.82	0.477208
$537$	−6457.80	−0.518947
$538$	6385.62	0.511717
$539$	3621.30	0.289388
$540$	1035.41	0.0825131
$541$	−5026.83	−0.399483	−0.199742	−	0.979849i	$-0.564010\pi$
$541$	−5026.83	−0.399483	−0.199742	+	0.979849i	$0.564010\pi$
$542$	16323.6	1.29365
$543$	2504.07	0.197900
$544$	−9817.87	−0.773782
$545$	27324.5	2.14762
$546$	0	0
$547$	1540.36	0.120404	0.0602021	−	0.998186i	$-0.480825\pi$
$547$	1540.36	0.120404	0.0602021	+	0.998186i	$0.480825\pi$
$548$	−4733.38	−0.368978
$549$	−8417.23	−0.654351
$550$	3965.65	0.307447
$551$	−24626.6	−1.90404
$552$	5596.72	0.431544
$553$	−5031.63	−0.386920
$554$	15407.4	1.18158
$555$	16579.2	1.26801
$556$	5794.38	0.441972
$557$	8550.51	0.650443	0.325221	−	0.945638i	$-0.394561\pi$
$557$	8550.51	0.650443	0.325221	+	0.945638i	$0.394561\pi$
$558$	−6009.68	−0.455932
$559$	0	0
$560$	5909.35	0.445921
$561$	−4105.81	−0.308998
$562$	−1532.30	−0.115011
$563$	−6569.35	−0.491768	−0.245884	−	0.969299i	$-0.579078\pi$
$563$	−6569.35	−0.491768	−0.245884	+	0.969299i	$0.579078\pi$
$564$	−2365.99	−0.176642
$565$	−6425.85	−0.478473
$566$	−14108.5	−1.04775
$567$	820.671	0.0607847
$568$	−23329.8	−1.72341
$569$	23766.1	1.75102	0.875508	−	0.483204i	$-0.160527\pi$
$569$	23766.1	1.75102	0.875508	+	0.483204i	$0.160527\pi$
$570$	−12431.3	−0.913488
$571$	−24971.7	−1.83018	−0.915091	−	0.403248i	$-0.867881\pi$
$571$	−24971.7	−1.83018	−0.915091	+	0.403248i	$0.867881\pi$
$572$	0	0
$573$	13394.1	0.976524
$574$	7447.41	0.541549
$575$	8495.48	0.616150
$576$	5010.38	0.362441
$577$	16643.4	1.20082	0.600409	−	0.799693i	$-0.295004\pi$
$577$	16643.4	1.20082	0.600409	+	0.799693i	$0.295004\pi$
$578$	−7835.28	−0.563849
$579$	9899.53	0.710553
$580$	8237.93	0.589761
$581$	−3976.98	−0.283981
$582$	−3896.63	−0.277527
$583$	−2469.64	−0.175441
$584$	10605.7	0.751484
$585$	0	0
$586$	−4489.66	−0.316495
$587$	6720.67	0.472558	0.236279	−	0.971685i	$-0.424072\pi$
$587$	6720.67	0.472558	0.236279	+	0.971685i	$0.424072\pi$
$588$	1795.63	0.125936
$589$	−32612.3	−2.28144
$590$	−9186.41	−0.641014
$591$	−8920.77	−0.620899
$592$	13592.8	0.943681
$593$	−26349.8	−1.82471	−0.912357	−	0.409395i	$-0.865740\pi$
$593$	−26349.8	−1.82471	−0.912357	+	0.409395i	$0.865740\pi$
$594$	954.884	0.0659586
$595$	−14172.1	−0.976466
$596$	−3414.04	−0.234638
$597$	−16292.4	−1.11693
$598$	0	0
$599$	−6136.81	−0.418603	−0.209302	−	0.977851i	$-0.567119\pi$
$599$	−6136.81	−0.418603	−0.209302	+	0.977851i	$0.567119\pi$
$600$	8283.25	0.563604
$601$	12493.7	0.847966	0.423983	−	0.905670i	$-0.360632\pi$
$601$	12493.7	0.847966	0.423983	+	0.905670i	$0.360632\pi$
$602$	7056.14	0.477719
$603$	2164.44	0.146174
$604$	2932.51	0.197553
$605$	17000.4	1.14242
$606$	5379.60	0.360613
$607$	18696.6	1.25020	0.625099	−	0.780545i	$-0.285059\pi$
$607$	18696.6	1.25020	0.625099	+	0.780545i	$0.285059\pi$
$608$	12390.8	0.826501
$609$	6529.40	0.434458
$610$	−33805.3	−2.24383
$611$	0	0
$612$	−2035.88	−0.134470
$613$	−1238.64	−0.0816123	−0.0408062	−	0.999167i	$-0.512993\pi$
$613$	−1238.64	−0.0816123	−0.0408062	+	0.999167i	$0.512993\pi$
$614$	−4654.81	−0.305949
$615$	14466.9	0.948553
$616$	−3758.89	−0.245860
$617$	−16549.5	−1.07983	−0.539917	−	0.841718i	$-0.681544\pi$
$617$	−16549.5	−1.07983	−0.539917	+	0.841718i	$0.681544\pi$
$618$	−1285.26	−0.0836579
$619$	−13945.4	−0.905513	−0.452756	−	0.891634i	$-0.649559\pi$
$619$	−13945.4	−0.905513	−0.452756	+	0.891634i	$0.649559\pi$
$620$	10909.3	0.706656
$621$	2045.62	0.132186
$622$	−4445.17	−0.286552
$623$	−9928.03	−0.638456
$624$	0	0
$625$	−17068.0	−1.09235
$626$	−10736.9	−0.685513
$627$	5181.80	0.330050
$628$	−4038.23	−0.256597
$629$	−32598.8	−2.06645
$630$	3295.98	0.208436
$631$	−15343.1	−0.967987	−0.483994	−	0.875072i	$-0.660814\pi$
$631$	−15343.1	−0.967987	−0.483994	+	0.875072i	$0.660814\pi$
$632$	−12228.6	−0.769663
$633$	3684.76	0.231368
$634$	17826.4	1.11668
$635$	30054.2	1.87821
$636$	−1224.58	−0.0763484
$637$	0	0
$638$	7597.23	0.471438
$639$	−8527.10	−0.527898
$640$	6807.51	0.420454
$641$	11067.7	0.681979	0.340989	−	0.940067i	$-0.389238\pi$
$641$	11067.7	0.681979	0.340989	+	0.940067i	$0.389238\pi$
$642$	−1293.58	−0.0795224
$643$	25118.7	1.54057	0.770284	−	0.637701i	$-0.220115\pi$
$643$	25118.7	1.54057	0.770284	+	0.637701i	$0.220115\pi$
$644$	−1911.61	−0.116969
$645$	13706.8	0.836751
$646$	24442.9	1.48869
$647$	−4447.03	−0.270217	−0.135109	−	0.990831i	$-0.543138\pi$
$647$	−4447.03	−0.270217	−0.135109	+	0.990831i	$0.543138\pi$
$648$	1994.51	0.120913
$649$	3829.23	0.231603
$650$	0	0
$651$	8646.71	0.520570
$652$	−331.537	−0.0199141
$653$	−19426.2	−1.16418	−0.582088	−	0.813126i	$-0.697764\pi$
$653$	−19426.2	−1.16418	−0.582088	+	0.813126i	$0.697764\pi$
$654$	12495.1	0.747093
$655$	22723.1	1.35552
$656$	11861.0	0.705934
$657$	3876.41	0.230187
$658$	−7531.52	−0.446215
$659$	−14099.4	−0.833438	−0.416719	−	0.909035i	$-0.636820\pi$
$659$	−14099.4	−0.833438	−0.416719	+	0.909035i	$0.636820\pi$
$660$	−1733.38	−0.102230
$661$	−2754.25	−0.162070	−0.0810348	−	0.996711i	$-0.525823\pi$
$661$	−2754.25	−0.162070	−0.0810348	+	0.996711i	$0.525823\pi$
$662$	−4080.93	−0.239592
$663$	0	0
$664$	−9665.44	−0.564898
$665$	17886.0	1.04299
$666$	7581.45	0.441104
$667$	16275.3	0.944800
$668$	−1222.06	−0.0707828
$669$	2055.77	0.118805
$670$	8692.83	0.501244
$671$	14091.3	0.810712
$672$	−3285.25	−0.188588
$673$	11936.8	0.683700	0.341850	−	0.939754i	$-0.388946\pi$
$673$	11936.8	0.683700	0.341850	+	0.939754i	$0.388946\pi$
$674$	6362.53	0.363613
$675$	3027.55	0.172638
$676$	0	0
$677$	−10255.2	−0.582186	−0.291093	−	0.956695i	$-0.594019\pi$
$677$	−10255.2	−0.582186	−0.291093	+	0.956695i	$0.594019\pi$
$678$	−2938.46	−0.166447
$679$	5606.46	0.316872
$680$	−34443.0	−1.94239
$681$	−861.202	−0.0484601
$682$	10060.8	0.564880
$683$	−10605.7	−0.594169	−0.297085	−	0.954851i	$-0.596014\pi$
$683$	−10605.7	−0.594169	−0.297085	+	0.954851i	$0.596014\pi$
$684$	2569.41	0.143631
$685$	−29269.2	−1.63258
$686$	13873.1	0.772127
$687$	6906.69	0.383561
$688$	11237.8	0.622728
$689$	0	0
$690$	8215.60	0.453279
$691$	−4660.14	−0.256556	−0.128278	−	0.991738i	$-0.540945\pi$
$691$	−4660.14	−0.256556	−0.128278	+	0.991738i	$0.540945\pi$
$692$	−5000.90	−0.274719
$693$	−1373.88	−0.0753096
$694$	5884.15	0.321844
$695$	35829.9	1.95555
$696$	15868.7	0.864227
$697$	−28445.4	−1.54584
$698$	17690.2	0.959287
$699$	−2911.86	−0.157563
$700$	−2829.22	−0.152764
$701$	−24016.9	−1.29402	−0.647008	−	0.762483i	$-0.723980\pi$
$701$	−24016.9	−1.29402	−0.647008	+	0.762483i	$0.723980\pi$
$702$	0	0
$703$	41141.7	2.20724
$704$	−8387.88	−0.449048
$705$	−14630.2	−0.781570
$706$	−21108.7	−1.12527
$707$	−7740.15	−0.411737
$708$	1898.74	0.100789
$709$	7252.56	0.384169	0.192084	−	0.981378i	$-0.438475\pi$
$709$	7252.56	0.384169	0.192084	+	0.981378i	$0.438475\pi$
$710$	−34246.5	−1.81021
$711$	−4469.58	−0.235756
$712$	−24128.6	−1.27002
$713$	21552.9	1.13207
$714$	−6480.71	−0.339684
$715$	0	0
$716$	5360.66	0.279801
$717$	15022.2	0.782447
$718$	−10719.1	−0.557151
$719$	10951.7	0.568050	0.284025	−	0.958817i	$-0.408330\pi$
$719$	10951.7	0.568050	0.284025	+	0.958817i	$0.408330\pi$
$720$	5249.26	0.271706
$721$	1849.22	0.0955182
$722$	−14748.6	−0.760231
$723$	1620.28	0.0833454
$724$	−2078.65	−0.106702
$725$	24087.7	1.23393
$726$	7774.08	0.397415
$727$	−27856.0	−1.42108	−0.710538	−	0.703658i	$-0.751549\pi$
$727$	−27856.0	−1.42108	−0.710538	+	0.703658i	$0.751549\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	15568.4	0.789334
$731$	−26951.0	−1.36364
$732$	6987.20	0.352806
$733$	31213.8	1.57286	0.786431	−	0.617678i	$-0.211927\pi$
$733$	31213.8	1.57286	0.786431	+	0.617678i	$0.211927\pi$
$734$	15138.7	0.761281
$735$	11103.4	0.557218
$736$	−8188.85	−0.410116
$737$	−3623.49	−0.181103
$738$	6615.52	0.329974
$739$	−14423.1	−0.717946	−0.358973	−	0.933348i	$-0.616873\pi$
$739$	−14423.1	−0.717946	−0.358973	+	0.933348i	$0.616873\pi$
$740$	−13762.5	−0.683674
$741$	0	0
$742$	−3898.13	−0.192864
$743$	−13469.7	−0.665079	−0.332539	−	0.943089i	$-0.607905\pi$
$743$	−13469.7	−0.665079	−0.332539	+	0.943089i	$0.607905\pi$
$744$	21014.5	1.03552
$745$	−21111.0	−1.03818
$746$	−17954.0	−0.881158
$747$	−3532.75	−0.173034
$748$	3408.26	0.166602
$749$	1861.19	0.0907965
$750$	−1395.42	−0.0679381
$751$	−32033.6	−1.55649	−0.778245	−	0.627961i	$-0.783890\pi$
$751$	−32033.6	−1.55649	−0.778245	+	0.627961i	$0.783890\pi$
$752$	−11994.9	−0.581661
$753$	−18263.4	−0.883871
$754$	0	0
$755$	18133.4	0.874096
$756$	−681.244	−0.0327733
$757$	−26097.2	−1.25300	−0.626498	−	0.779423i	$-0.715512\pi$
$757$	−26097.2	−1.25300	−0.626498	+	0.779423i	$0.715512\pi$
$758$	−24171.6	−1.15825
$759$	−3424.56	−0.163773
$760$	43469.3	2.07473
$761$	−18238.2	−0.868772	−0.434386	−	0.900727i	$-0.643035\pi$
$761$	−18238.2	−0.868772	−0.434386	+	0.900727i	$0.643035\pi$
$762$	13743.4	0.653375
$763$	−17978.0	−0.853009
$764$	−11118.6	−0.526513
$765$	−12589.0	−0.594976
$766$	24078.4	1.13575
$767$	0	0
$768$	−10248.0	−0.481502
$769$	18817.0	0.882390	0.441195	−	0.897411i	$-0.354555\pi$
$769$	18817.0	0.882390	0.441195	+	0.897411i	$0.354555\pi$
$770$	−5517.79	−0.258243
$771$	−15289.0	−0.714164
$772$	−8217.67	−0.383109
$773$	13795.6	0.641904	0.320952	−	0.947095i	$-0.395997\pi$
$773$	13795.6	0.641904	0.320952	+	0.947095i	$0.395997\pi$
$774$	6267.95	0.291081
$775$	31898.7	1.47850
$776$	13625.6	0.630325
$777$	−10908.2	−0.503640
$778$	11200.2	0.516126
$779$	35900.0	1.65116
$780$	0	0
$781$	14275.2	0.654043
$782$	−16153.9	−0.738699
$783$	5800.05	0.264722
$784$	9103.36	0.414694
$785$	−24970.7	−1.13534
$786$	10391.0	0.471545
$787$	40545.4	1.83645	0.918227	−	0.396055i	$-0.129621\pi$
$787$	40545.4	1.83645	0.918227	+	0.396055i	$0.129621\pi$
$788$	7405.19	0.334770
$789$	10216.5	0.460985
$790$	−17950.7	−0.808429
$791$	4227.85	0.190044
$792$	−3339.01	−0.149806
$793$	0	0
$794$	5378.11	0.240380
$795$	−7572.25	−0.337811
$796$	13524.5	0.602214
$797$	−31576.4	−1.40338	−0.701690	−	0.712482i	$-0.747571\pi$
$797$	−31576.4	−1.40338	−0.701690	+	0.712482i	$0.747571\pi$
$798$	8179.07	0.362827
$799$	28766.7	1.27371
$800$	−12119.7	−0.535619
$801$	−8819.05	−0.389021
$802$	17685.8	0.778688
$803$	−6489.50	−0.285192
$804$	−1796.72	−0.0788126
$805$	−11820.6	−0.517541
$806$	0	0
$807$	−8161.33	−0.356001
$808$	−18811.2	−0.819031
$809$	31130.1	1.35287	0.676437	−	0.736501i	$-0.263523\pi$
$809$	31130.1	1.35287	0.676437	+	0.736501i	$0.263523\pi$
$810$	2927.81	0.127003
$811$	8733.71	0.378153	0.189076	−	0.981962i	$-0.439451\pi$
$811$	8733.71	0.378153	0.189076	+	0.981962i	$0.439451\pi$
$812$	−5420.10	−0.234247
$813$	−20862.8	−0.899988
$814$	−12692.1	−0.546509
$815$	−2050.08	−0.0881119
$816$	−10321.4	−0.442794
$817$	34013.8	1.45654
$818$	−3563.08	−0.152298
$819$	0	0
$820$	−12009.0	−0.511431
$821$	36960.1	1.57115	0.785576	−	0.618765i	$-0.212367\pi$
$821$	36960.1	1.57115	0.785576	+	0.618765i	$0.212367\pi$
$822$	−13384.5	−0.567928
$823$	20509.7	0.868679	0.434340	−	0.900749i	$-0.356982\pi$
$823$	20509.7	0.868679	0.434340	+	0.900749i	$0.356982\pi$
$824$	4494.25	0.190006
$825$	−5068.42	−0.213891
$826$	6044.15	0.254604
$827$	11533.4	0.484952	0.242476	−	0.970157i	$-0.422040\pi$
$827$	11533.4	0.484952	0.242476	+	0.970157i	$0.422040\pi$
$828$	−1698.08	−0.0712709
$829$	34096.4	1.42849	0.714244	−	0.699897i	$-0.246771\pi$
$829$	34096.4	1.42849	0.714244	+	0.699897i	$0.246771\pi$
$830$	−14188.2	−0.593350
$831$	−19691.9	−0.822026
$832$	0	0
$833$	−21832.1	−0.908087
$834$	16384.6	0.680279
$835$	−7556.69	−0.313186
$836$	−4301.45	−0.177953
$837$	7680.85	0.317191
$838$	−18968.2	−0.781917
$839$	1855.42	0.0763484	0.0381742	−	0.999271i	$-0.487846\pi$
$839$	1855.42	0.0763484	0.0381742	+	0.999271i	$0.487846\pi$
$840$	−11525.3	−0.473405
$841$	21757.3	0.892094
$842$	18949.8	0.775599
$843$	1958.40	0.0800129
$844$	−3058.75	−0.124747
$845$	0	0
$846$	−6690.24	−0.271885
$847$	−11185.3	−0.453757
$848$	−6208.26	−0.251406
$849$	18031.9	0.728918
$850$	−23908.1	−0.964755
$851$	−27189.9	−1.09525
$852$	7078.40	0.284627
$853$	28668.7	1.15076	0.575380	−	0.817886i	$-0.304854\pi$
$853$	28668.7	1.15076	0.575380	+	0.817886i	$0.304854\pi$
$854$	22242.0	0.891224
$855$	15888.1	0.635512
$856$	4523.35	0.180613
$857$	449.310	0.0179091	0.00895457	−	0.999960i	$-0.497150\pi$
$857$	449.310	0.0179091	0.00895457	+	0.999960i	$0.497150\pi$
$858$	0	0
$859$	33466.9	1.32931	0.664654	−	0.747151i	$-0.268579\pi$
$859$	33466.9	1.32931	0.664654	+	0.747151i	$0.268579\pi$
$860$	−11378.1	−0.451151
$861$	−9518.39	−0.376755
$862$	5261.14	0.207883
$863$	25097.5	0.989953	0.494976	−	0.868906i	$-0.335177\pi$
$863$	25097.5	0.989953	0.494976	+	0.868906i	$0.335177\pi$
$864$	−2918.28	−0.114910
$865$	−30923.4	−1.21552
$866$	−24031.0	−0.942965
$867$	10014.1	0.392269
$868$	−7177.69	−0.280676
$869$	7482.53	0.292091
$870$	23294.2	0.907755
$871$	0	0
$872$	−43692.7	−1.69681
$873$	4980.21	0.193075
$874$	20387.3	0.789027
$875$	2007.73	0.0775698
$876$	−3217.83	−0.124110
$877$	−27015.2	−1.04018	−0.520090	−	0.854112i	$-0.674101\pi$
$877$	−27015.2	−1.04018	−0.520090	+	0.854112i	$0.674101\pi$
$878$	12714.4	0.488715
$879$	5738.15	0.220185
$880$	−8787.79	−0.336632
$881$	48638.7	1.86002	0.930010	−	0.367534i	$-0.119798\pi$
$881$	48638.7	1.86002	0.930010	+	0.367534i	$0.119798\pi$
$882$	5077.46	0.193840
$883$	−25479.0	−0.971048	−0.485524	−	0.874223i	$-0.661371\pi$
$883$	−25479.0	−0.971048	−0.485524	+	0.874223i	$0.661371\pi$
$884$	0	0
$885$	11741.0	0.445953
$886$	5956.05	0.225843
$887$	42359.7	1.60350	0.801748	−	0.597663i	$-0.203904\pi$
$887$	42359.7	1.60350	0.801748	+	0.597663i	$0.203904\pi$
$888$	−26510.6	−1.00184
$889$	−19774.0	−0.746005
$890$	−35419.1	−1.33399
$891$	−1220.42	−0.0458873
$892$	−1706.51	−0.0640561
$893$	−36305.4	−1.36049
$894$	−9653.78	−0.361153
$895$	33148.1	1.23801
$896$	−4478.96	−0.167000
$897$	0	0
$898$	18286.7	0.679548
$899$	61110.3	2.26712
$900$	−2513.19	−0.0930811
$901$	14888.9	0.550524
$902$	−11075.0	−0.408823
$903$	−9018.31	−0.332348
$904$	10275.1	0.378038
$905$	−12853.5	−0.472114
$906$	8292.19	0.304072
$907$	−5314.91	−0.194574	−0.0972871	−	0.995256i	$-0.531016\pi$
$907$	−5314.91	−0.194574	−0.0972871	+	0.995256i	$0.531016\pi$
$908$	714.890	0.0261283
$909$	−6875.56	−0.250878
$910$	0	0
$911$	2471.71	0.0898919	0.0449459	−	0.998989i	$-0.485688\pi$
$911$	2471.71	0.0898919	0.0449459	+	0.998989i	$0.485688\pi$
$912$	13026.2	0.472962
$913$	5914.17	0.214382
$914$	15491.3	0.560620
$915$	43205.8	1.56103
$916$	−5733.29	−0.206805
$917$	−14950.5	−0.538396
$918$	−5756.80	−0.206975
$919$	−9636.13	−0.345883	−0.172942	−	0.984932i	$-0.555327\pi$
$919$	−9636.13	−0.345883	−0.172942	+	0.984932i	$0.555327\pi$
$920$	−28728.1	−1.02950
$921$	5949.22	0.212849
$922$	10521.8	0.375832
$923$	0	0
$924$	1140.47	0.0406047
$925$	−40241.5	−1.43041
$926$	−8933.38	−0.317029
$927$	1642.66	0.0582007
$928$	−23218.3	−0.821314
$929$	53249.1	1.88057	0.940283	−	0.340394i	$-0.110561\pi$
$929$	53249.1	1.88057	0.940283	+	0.340394i	$0.110561\pi$
$930$	30847.8	1.08768
$931$	27553.5	0.969955
$932$	2417.16	0.0849534
$933$	5681.28	0.199354
$934$	−34689.8	−1.21530
$935$	21075.3	0.737149
$936$	0	0
$937$	−41122.6	−1.43374	−0.716871	−	0.697205i	$-0.754427\pi$
$937$	−41122.6	−1.43374	−0.716871	+	0.697205i	$0.754427\pi$
$938$	−5719.40	−0.199088
$939$	13722.6	0.476911
$940$	12144.7	0.421399
$941$	8005.71	0.277342	0.138671	−	0.990339i	$-0.455717\pi$
$941$	8005.71	0.277342	0.138671	+	0.990339i	$0.455717\pi$
$942$	−11418.8	−0.394952
$943$	−23725.7	−0.819315
$944$	9626.07	0.331888
$945$	−4212.52	−0.145009
$946$	−10493.2	−0.360637
$947$	17468.6	0.599424	0.299712	−	0.954030i	$-0.403109\pi$
$947$	17468.6	0.599424	0.299712	+	0.954030i	$0.403109\pi$
$948$	3710.23	0.127113
$949$	0	0
$950$	30173.6	1.03048
$951$	−22783.6	−0.776875
$952$	22661.6	0.771498
$953$	30681.4	1.04288	0.521442	−	0.853287i	$-0.325394\pi$
$953$	30681.4	1.04288	0.521442	+	0.853287i	$0.325394\pi$
$954$	−3462.70	−0.117515
$955$	−68752.5	−2.32961
$956$	−12470.0	−0.421872
$957$	−9709.87	−0.327979
$958$	26221.8	0.884328
$959$	19257.5	0.648444
$960$	−25718.4	−0.864644
$961$	51135.6	1.71648
$962$	0	0
$963$	1653.29	0.0553237
$964$	−1345.00	−0.0449373
$965$	−50814.5	−1.69511
$966$	−5405.41	−0.180037
$967$	−15616.0	−0.519313	−0.259656	−	0.965701i	$-0.583609\pi$
$967$	−15616.0	−0.519313	−0.259656	+	0.965701i	$0.583609\pi$
$968$	−27184.2	−0.902617
$969$	−31240.0	−1.03568
$970$	20001.5	0.662072
$971$	7185.98	0.237496	0.118748	−	0.992924i	$-0.462112\pi$
$971$	7185.98	0.237496	0.118748	+	0.992924i	$0.462112\pi$
$972$	−605.148	−0.0199693
$973$	−23574.1	−0.776723
$974$	14192.9	0.466911
$975$	0	0
$976$	35423.2	1.16175
$977$	−33165.3	−1.08603	−0.543015	−	0.839723i	$-0.682717\pi$
$977$	−33165.3	−1.08603	−0.543015	+	0.839723i	$0.682717\pi$
$978$	−937.477	−0.0306516
$979$	14764.0	0.481980
$980$	−9217.01	−0.300435
$981$	−15969.8	−0.519752
$982$	2429.52	0.0789502
$983$	11658.7	0.378287	0.189143	−	0.981949i	$-0.439429\pi$
$983$	11658.7	0.378287	0.189143	+	0.981949i	$0.439429\pi$
$984$	−23133.0	−0.749444
$985$	45790.5	1.48123
$986$	−45802.1	−1.47935
$987$	9625.89	0.310431
$988$	0	0
$989$	−22479.2	−0.722746
$990$	−4901.45	−0.157352
$991$	−1957.32	−0.0627409	−0.0313704	−	0.999508i	$-0.509987\pi$
$991$	−1957.32	−0.0627409	−0.0313704	+	0.999508i	$0.509987\pi$
$992$	−30747.4	−0.984104
$993$	5215.75	0.166684
$994$	22532.3	0.718996
$995$	83629.5	2.66456
$996$	2932.56	0.0932948
$997$	−11434.8	−0.363233	−0.181617	−	0.983369i	$-0.558133\pi$
$997$	−11434.8	−0.363233	−0.181617	+	0.983369i	$0.558133\pi$
$998$	−27459.0	−0.870942
$999$	−9689.70	−0.306876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.a.o.1.4	✓	9
3.2	odd	2		1521.4.a.bi.1.6		9
13.5	odd	4		507.4.b.k.337.12		18
13.8	odd	4		507.4.b.k.337.7		18
13.12	even	2		507.4.a.p.1.6	yes	9
39.38	odd	2		1521.4.a.bf.1.4		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.4.a.o.1.4	✓	9	1.1	even	1	trivial
507.4.a.p.1.6	yes	9	13.12	even	2
507.4.b.k.337.7		18	13.8	odd	4
507.4.b.k.337.12		18	13.5	odd	4
1521.4.a.bf.1.4		9	39.38	odd	2
1521.4.a.bi.1.6		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)

\(f(q)\)	\(=\)	\(q-2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} -15.3991 q^{5} -7.04181 q^{6} +10.1317 q^{7} +24.6236 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q-2.34727 q^{2} +3.00000 q^{3} -2.49032 q^{4} -15.3991 q^{5} -7.04181 q^{6} +10.1317 q^{7} +24.6236 q^{8} +9.00000 q^{9} +36.1458 q^{10} -15.0669 q^{11} -7.47096 q^{12} -23.7819 q^{14} -46.1972 q^{15} -37.8757 q^{16} +90.8352 q^{17} -21.1254 q^{18} -114.640 q^{19} +38.3486 q^{20} +30.3952 q^{21} +35.3661 q^{22} +75.7635 q^{23} +73.8709 q^{24} +112.132 q^{25} +27.0000 q^{27} -25.2313 q^{28} +214.817 q^{29} +108.437 q^{30} +284.476 q^{31} -108.084 q^{32} -45.2007 q^{33} -213.215 q^{34} -156.019 q^{35} -22.4129 q^{36} -358.878 q^{37} +269.091 q^{38} -379.181 q^{40} -313.154 q^{41} -71.3458 q^{42} -296.702 q^{43} +37.5214 q^{44} -138.592 q^{45} -177.837 q^{46} +316.691 q^{47} -113.627 q^{48} -240.348 q^{49} -263.203 q^{50} +272.506 q^{51} +163.911 q^{53} -63.3763 q^{54} +232.016 q^{55} +249.480 q^{56} -343.920 q^{57} -504.233 q^{58} -254.149 q^{59} +115.046 q^{60} -935.247 q^{61} -667.742 q^{62} +91.1856 q^{63} +556.709 q^{64} +106.098 q^{66} +240.494 q^{67} -226.209 q^{68} +227.291 q^{69} +366.220 q^{70} -947.455 q^{71} +221.613 q^{72} +430.712 q^{73} +842.384 q^{74} +336.395 q^{75} +285.490 q^{76} -152.654 q^{77} -496.620 q^{79} +583.251 q^{80} +81.0000 q^{81} +735.058 q^{82} -392.527 q^{83} -75.6938 q^{84} -1398.78 q^{85} +696.439 q^{86} +644.451 q^{87} -371.002 q^{88} -979.895 q^{89} +325.312 q^{90} -188.675 q^{92} +853.428 q^{93} -743.360 q^{94} +1765.35 q^{95} -324.253 q^{96} +553.356 q^{97} +564.162 q^{98} -135.602 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 6 q^{2} + 27 q^{3} + 44 q^{4} - 33 q^{5} - 18 q^{6} - 83 q^{7} - 87 q^{8} + 81 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

Fields

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