LMFDB - Embedded newform 35.4.b.a.29.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [35,4,Mod(29,35)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(35, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("35.29");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 35 = 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	35.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$2.06506685020$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 55x^{8} + 983x^{6} + 6409x^{4} + 13560x^{2} + 3600 $ x^10 + 55x^8 + 983x^6 + 6409x^4 + 13560x^2 + 3600
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			29.8
Root			$1.85474i$ of defining polynomial
Character	$\chi$	$=$	35.29
Dual form			35.4.b.a.29.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.85474i q^{2} -8.98858i q^{3} -0.149548 q^{4} +(3.91321 - 10.4731i) q^{5} +25.6601 q^{6} +7.00000i q^{7} +22.4110i q^{8} -53.7945 q^{9} +(29.8981 + 11.1712i) q^{10} +37.4408 q^{11} +1.34423i q^{12} -3.96370i q^{13} -19.9832 q^{14} +(-94.1387 - 35.1742i) q^{15} -65.1740 q^{16} +51.6780i q^{17} -153.569i q^{18} -25.9323 q^{19} +(-0.585214 + 1.56624i) q^{20} +62.9200 q^{21} +106.884i q^{22} +173.454i q^{23} +201.443 q^{24} +(-94.3736 - 81.9673i) q^{25} +11.3154 q^{26} +240.845i q^{27} -1.04684i q^{28} +245.676 q^{29} +(100.413 - 268.742i) q^{30} -172.074 q^{31} -6.76690i q^{32} -336.539i q^{33} -147.527 q^{34} +(73.3120 + 27.3925i) q^{35} +8.04488 q^{36} -250.699i q^{37} -74.0300i q^{38} -35.6281 q^{39} +(234.714 + 87.6990i) q^{40} -48.8649 q^{41} +179.620i q^{42} +143.612i q^{43} -5.59920 q^{44} +(-210.509 + 563.398i) q^{45} -495.167 q^{46} -36.6415i q^{47} +585.822i q^{48} -49.0000 q^{49} +(233.995 - 269.412i) q^{50} +464.511 q^{51} +0.592765i q^{52} -645.286i q^{53} -687.549 q^{54} +(146.514 - 392.123i) q^{55} -156.877 q^{56} +233.094i q^{57} +701.343i q^{58} -395.495 q^{59} +(14.0783 + 5.26024i) q^{60} +47.5130 q^{61} -491.228i q^{62} -376.562i q^{63} -502.074 q^{64} +(-41.5125 - 15.5108i) q^{65} +960.733 q^{66} +263.189i q^{67} -7.72835i q^{68} +1559.11 q^{69} +(-78.1984 + 209.287i) q^{70} -268.177 q^{71} -1205.59i q^{72} -199.757i q^{73} +715.680 q^{74} +(-736.769 + 848.284i) q^{75} +3.87813 q^{76} +262.085i q^{77} -101.709i q^{78} -473.640 q^{79} +(-255.040 + 682.577i) q^{80} +712.399 q^{81} -139.497i q^{82} +72.7028i q^{83} -9.40958 q^{84} +(541.231 + 202.227i) q^{85} -409.975 q^{86} -2208.28i q^{87} +839.086i q^{88} +1552.25 q^{89} +(-1608.36 - 600.950i) q^{90} +27.7459 q^{91} -25.9398i q^{92} +1546.70i q^{93} +104.602 q^{94} +(-101.478 + 271.593i) q^{95} -60.8248 q^{96} +243.338i q^{97} -139.882i q^{98} -2014.11 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9} - 16 q^{10} + 84 q^{11} - 56 q^{14} + 8 q^{15} + 148 q^{16} + 72 q^{19} - 68 q^{20} + 140 q^{21} + 72 q^{24} - 362 q^{25} - 620 q^{26} + 88 q^{29} + 52 q^{30}+ \cdots - 5304 q^{99}+O(q^{100}) $ 10 * q - 36 * q^4 + 6 * q^5 + 12 * q^6 - 46 * q^9 - 16 * q^10 + 84 * q^11 - 56 * q^14 + 8 * q^15 + 148 * q^16 + 72 * q^19 - 68 * q^20 + 140 * q^21 + 72 * q^24 - 362 * q^25 - 620 * q^26 + 88 * q^29 + 52 * q^30 + 120 * q^31 + 964 * q^34 - 28 * q^35 - 420 * q^36 + 212 * q^39 + 1396 * q^40 - 852 * q^41 - 1424 * q^44 - 510 * q^45 + 176 * q^46 - 490 * q^49 + 1644 * q^50 + 1276 * q^51 - 996 * q^54 - 1136 * q^55 + 588 * q^56 + 864 * q^59 - 1692 * q^60 - 884 * q^61 - 2724 * q^64 + 520 * q^65 + 1148 * q^66 + 4808 * q^69 + 756 * q^70 + 880 * q^71 + 3024 * q^74 + 720 * q^75 - 2672 * q^76 - 3580 * q^79 - 2316 * q^80 - 302 * q^81 - 1904 * q^84 + 1572 * q^85 + 2432 * q^86 - 1492 * q^89 - 7748 * q^90 + 476 * q^91 + 1652 * q^94 - 1628 * q^95 + 4080 * q^96 - 5304 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/35\mathbb{Z}\right)^\times$.

$n$	$22$	$31$
$\chi(n)$	$-1$	$1$

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 $

$2$			2.85474i			1.00930i	0.863323	+	0.504652i	$0.168379\pi$
$2$			2.85474i			1.00930i	−0.863323	+	0.504652i	$0.831621\pi$
$3$		−	8.98858i		−	1.72985i	−0.501899	−	0.864926i	$-0.667365\pi$
$3$		−	8.98858i		−	1.72985i	0.501899	−	0.864926i	$-0.332635\pi$
$4$	−0.149548			−0.0186935
$5$	3.91321	−	10.4731i	0.350008	−	0.936747i
$5$	3.91321	−	10.4731i	0.350008	−	0.936747i
$6$	25.6601			1.74595
$7$			7.00000i			0.377964i
$7$			7.00000i			0.377964i
$8$			22.4110i			0.990436i
$9$	−53.7945			−1.99239
$10$	29.8981	+	11.1712i	0.945462	+	0.353265i
$11$	37.4408			1.02626			0.513128	−	0.858312i	$-0.328487\pi$
$11$	37.4408			1.02626			0.513128	+	0.858312i	$0.328487\pi$
$12$			1.34423i			0.0323371i
$13$		−	3.96370i		−	0.0845641i	−0.999106	−	0.0422821i	$-0.986537\pi$
$13$		−	3.96370i		−	0.0845641i	0.999106	−	0.0422821i	$-0.0134628\pi$
$14$	−19.9832			−0.381481
$15$	−94.1387	−	35.1742i	−1.62043	−	0.605463i
$16$	−65.1740			−1.01834
$17$			51.6780i			0.737279i	0.929572	+	0.368640i	$0.120176\pi$
$17$			51.6780i			0.737279i	−0.929572	+	0.368640i	$0.879824\pi$
$18$		−	153.569i		−	2.01093i
$19$	−25.9323			−0.313120			−0.156560	−	0.987668i	$-0.550040\pi$
$19$	−25.9323			−0.313120			−0.156560	+	0.987668i	$0.550040\pi$
$20$	−0.585214	+	1.56624i	−0.00654289	+	0.0175111i
$21$	62.9200			0.653823
$22$			106.884i			1.03580i
$23$			173.454i			1.57251i	0.617902	+	0.786255i	$0.287983\pi$
$23$			173.454i			1.57251i	−0.617902	+	0.786255i	$0.712017\pi$
$24$	201.443			1.71331
$25$	−94.3736	−	81.9673i	−0.754988	−	0.655738i
$26$	11.3154			0.0853509
$27$			240.845i			1.71669i
$28$		−	1.04684i		−	0.00706549i
$29$	245.676			1.57314			0.786568	−	0.617503i	$-0.211856\pi$
$29$	245.676			1.57314			0.786568	+	0.617503i	$0.211856\pi$
$30$	100.413	−	268.742i	0.611096	−	1.63551i
$31$	−172.074			−0.996951			−0.498475	−	0.866904i	$-0.666107\pi$
$31$	−172.074			−0.996951			−0.498475	+	0.866904i	$0.666107\pi$
$32$		−	6.76690i		−	0.0373822i
$33$		−	336.539i		−	1.77527i
$34$	−147.527			−0.744139
$35$	73.3120	+	27.3925i	0.354057	+	0.132291i
$36$	8.04488			0.0372448
$37$		−	250.699i		−	1.11391i	−0.830543	−	0.556954i	$-0.811970\pi$
$37$		−	250.699i		−	1.11391i	0.830543	−	0.556954i	$-0.188030\pi$
$38$		−	74.0300i		−	0.316033i
$39$	−35.6281			−0.146283
$40$	234.714	+	87.6990i	0.927788	+	0.346661i
$41$	−48.8649			−0.186132			−0.0930661	−	0.995660i	$-0.529667\pi$
$41$	−48.8649			−0.186132			−0.0930661	+	0.995660i	$0.529667\pi$
$42$			179.620i			0.659906i
$43$			143.612i			0.509317i	0.967031	+	0.254658i	$0.0819630\pi$
$43$			143.612i			0.509317i	−0.967031	+	0.254658i	$0.918037\pi$
$44$	−5.59920			−0.0191844
$45$	−210.509	+	563.398i	−0.697353	+	1.86636i
$46$	−495.167			−1.58714
$47$		−	36.6415i		−	0.113717i	−0.998382	−	0.0568587i	$-0.981892\pi$
$47$		−	36.6415i		−	0.113717i	0.998382	−	0.0568587i	$-0.0181085\pi$
$48$			585.822i			1.76159i
$49$	−49.0000			−0.142857
$50$	233.995	−	269.412i	0.661839	−	0.762012i
$51$	464.511			1.27538
$52$			0.592765i			0.00158080i
$53$		−	645.286i		−	1.67239i	−0.548430	−	0.836196i	$-0.684774\pi$
$53$		−	645.286i		−	1.67239i	0.548430	−	0.836196i	$-0.315226\pi$
$54$	−687.549			−1.73266
$55$	146.514	−	392.123i	0.359198	−	0.961342i
$56$	−156.877			−0.374350
$57$			233.094i			0.541651i
$58$			701.343i			1.58777i
$59$	−395.495			−0.872696			−0.436348	−	0.899778i	$-0.643728\pi$
$59$	−395.495			−0.872696			−0.436348	+	0.899778i	$0.643728\pi$
$60$	14.0783	+	5.26024i	0.0302916	+	0.0113182i
$61$	47.5130			0.0997282			0.0498641	−	0.998756i	$-0.484121\pi$
$61$	47.5130			0.0997282			0.0498641	+	0.998756i	$0.484121\pi$
$62$		−	491.228i		−	1.00623i
$63$		−	376.562i		−	0.753053i
$64$	−502.074			−0.980614
$65$	−41.5125	−	15.5108i	−0.0792152	−	0.0295981i
$66$	960.733			1.79179
$67$			263.189i			0.479906i	0.970785	+	0.239953i	$0.0771320\pi$
$67$			263.189i			0.479906i	−0.970785	+	0.239953i	$0.922868\pi$
$68$		−	7.72835i		−	0.0137824i
$69$	1559.11			2.72021
$70$	−78.1984	+	209.287i	−0.133521	+	0.357351i
$71$	−268.177			−0.448264			−0.224132	−	0.974559i	$-0.571955\pi$
$71$	−268.177			−0.448264			−0.224132	+	0.974559i	$0.571955\pi$
$72$		−	1205.59i		−	1.97333i
$73$		−	199.757i		−	0.320271i	−0.987095	−	0.160136i	$-0.948807\pi$
$73$		−	199.757i		−	0.320271i	0.987095	−	0.160136i	$-0.0511931\pi$
$74$	715.680			1.12427
$75$	−736.769	+	848.284i	−1.13433	+	1.30602i
$76$	3.87813			0.00585331
$77$			262.085i			0.387888i
$78$		−	101.709i		−	0.147644i
$79$	−473.640			−0.674540			−0.337270	−	0.941408i	$-0.609503\pi$
$79$	−473.640			−0.674540			−0.337270	+	0.941408i	$0.609503\pi$
$80$	−255.040	+	682.577i	−0.356429	+	0.953930i
$81$	712.399			0.977227
$82$		−	139.497i		−	0.187864i
$83$			72.7028i			0.0961466i	0.998844	+	0.0480733i	$0.0153081\pi$
$83$			72.7028i			0.0961466i	−0.998844	+	0.0480733i	$0.984692\pi$
$84$	−9.40958			−0.0122223
$85$	541.231	+	202.227i	0.690644	+	0.258054i
$86$	−409.975			−0.514055
$87$		−	2208.28i		−	2.72129i
$88$			839.086i			1.01644i
$89$	1552.25			1.84874			0.924369	−	0.381500i	$-0.124592\pi$
$89$	1552.25			1.84874			0.924369	+	0.381500i	$0.124592\pi$
$90$	−1608.36	−	600.950i	−1.88373	−	0.703841i
$91$	27.7459			0.0319622
$92$		−	25.9398i		−	0.0293958i
$93$			1546.70i			1.72458i
$94$	104.602			0.114775
$95$	−101.478	+	271.593i	−0.109594	+	0.293314i
$96$	−60.8248			−0.0646657
$97$			243.338i			0.254714i	0.991857	+	0.127357i	$0.0406494\pi$
$97$			243.338i			0.254714i	−0.991857	+	0.127357i	$0.959351\pi$
$98$		−	139.882i		−	0.144186i
$99$	−2014.11			−2.04470
$100$	14.1134	+	12.2581i	0.0141134	+	0.0122581i
$101$	−1539.34			−1.51653			−0.758265	−	0.651946i	$-0.773953\pi$
$101$	−1539.34			−1.51653			−0.758265	+	0.651946i	$0.773953\pi$
$102$			1326.06i			1.28725i
$103$		−	948.628i		−	0.907486i	−0.891133	−	0.453743i	$-0.850088\pi$
$103$		−	948.628i		−	0.907486i	0.891133	−	0.453743i	$-0.149912\pi$
$104$	88.8306			0.0837554
$105$	246.219	−	658.971i	0.228843	−	0.612466i
$106$	1842.12			1.68795
$107$			863.983i			0.780602i	0.920687	+	0.390301i	$0.127629\pi$
$107$			863.983i			0.780602i	−0.920687	+	0.390301i	$0.872371\pi$
$108$		−	36.0179i		−	0.0320910i
$109$	−886.319			−0.778844			−0.389422	−	0.921060i	$-0.627325\pi$
$109$	−886.319			−0.778844			−0.389422	+	0.921060i	$0.627325\pi$
$110$	1119.41	+	418.259i	0.970286	+	0.362540i
$111$	−2253.42			−1.92690
$112$		−	456.218i		−	0.384898i
$113$		−	765.957i		−	0.637657i	−0.947812	−	0.318828i	$-0.896711\pi$
$113$		−	765.957i		−	0.637657i	0.947812	−	0.318828i	$-0.103289\pi$
$114$	−665.424			−0.546690
$115$	1816.61	+	678.763i	1.47304	+	0.550391i
$116$	−36.7405			−0.0294075
$117$			213.226i			0.168485i
$118$		−	1129.04i		−	0.880816i
$119$	−361.746			−0.278665
$120$	788.289	−	2109.74i	0.599672	−	1.60494i
$121$	70.8116			0.0532018
$122$			135.637i			0.100656i
$123$			439.226i			0.321981i
$124$	25.7334			0.0186365
$125$	−1227.76	+	667.633i	−0.878513	+	0.477719i
$126$	1074.99			0.760059
$127$			505.042i			0.352876i	0.984312	+	0.176438i	$0.0564575\pi$
$127$			505.042i			0.352876i	−0.984312	+	0.176438i	$0.943543\pi$
$128$		−	1487.43i		−	1.02712i
$129$	1290.87			0.881043
$130$	44.2794	−	118.507i	0.0298735	−	0.0799521i
$131$	672.930			0.448811			0.224405	−	0.974496i	$-0.427956\pi$
$131$	672.930			0.448811			0.224405	+	0.974496i	$0.427956\pi$
$132$			50.3289i			0.0331861i
$133$		−	181.526i		−	0.118348i
$134$	−751.337			−0.484371
$135$	2522.40	+	942.476i	1.60810	+	0.600855i
$136$	−1158.16			−0.730228
$137$			1552.28i			0.968032i	0.875059	+	0.484016i	$0.160822\pi$
$137$			1552.28i			0.968032i	−0.875059	+	0.484016i	$0.839178\pi$
$138$			4450.85i			2.74552i
$139$	1072.02			0.654154			0.327077	−	0.944998i	$-0.393936\pi$
$139$	1072.02			0.654154			0.327077	+	0.944998i	$0.393936\pi$
$140$	−10.9637	−	4.09650i	−0.00661857	−	0.00247298i
$141$	−329.355			−0.196714
$142$		−	765.575i		−	0.452434i
$143$		−	148.404i		−	0.0867845i
$144$	3506.01			2.02894
$145$	961.384	−	2573.00i	0.550611	−	1.47363i
$146$	570.255			0.323251
$147$			440.440i			0.247122i
$148$			37.4915i			0.0208229i
$149$	−645.936			−0.355149			−0.177574	−	0.984107i	$-0.556825\pi$
$149$	−645.936			−0.355149			−0.177574	+	0.984107i	$0.556825\pi$
$150$	−2421.63	−	2103.29i	−1.31817	−	1.14488i
$151$	243.194			0.131065			0.0655326	−	0.997850i	$-0.479125\pi$
$151$	243.194			0.131065			0.0655326	+	0.997850i	$0.479125\pi$
$152$		−	581.169i		−	0.310125i
$153$		−	2779.99i		−	1.46895i
$154$	−748.186			−0.391497
$155$	−673.363	+	1802.16i	−0.348941	+	0.933890i
$156$	5.32811			0.00273455
$157$		−	1552.56i		−	0.789223i	−0.918848	−	0.394611i	$-0.870879\pi$
$157$		−	1552.56i		−	0.789223i	0.918848	−	0.394611i	$-0.129121\pi$
$158$		−	1352.12i		−	0.680815i
$159$	−5800.20			−2.89299
$160$	−70.8707	−	26.4803i	−0.0350176	−	0.0130841i
$161$	−1214.18			−0.594353
$162$			2033.71i			0.986319i
$163$		−	2553.65i		−	1.22710i	−0.789656	−	0.613550i	$-0.789741\pi$
$163$		−	2553.65i		−	1.22710i	0.789656	−	0.613550i	$-0.210259\pi$
$164$	7.30766			0.00347947
$165$	−3524.63	−	1316.95i	−1.66298	−	0.621360i
$166$	−207.548			−0.0970411
$167$			3573.14i			1.65568i	0.560966	+	0.827839i	$0.310430\pi$
$167$			3573.14i			1.65568i	−0.560966	+	0.827839i	$0.689570\pi$
$168$			1410.10i			0.647570i
$169$	2181.29			0.992849
$170$	−577.305	+	1545.07i	−0.260455	+	0.697069i
$171$	1395.01			0.623856
$172$		−	21.4769i		−	0.00952093i
$173$		−	2234.71i		−	0.982090i	−0.871134	−	0.491045i	$-0.836615\pi$
$173$		−	2234.71i		−	0.982090i	0.871134	−	0.491045i	$-0.163385\pi$
$174$	6304.07			2.74661
$175$	573.771	−	660.615i	0.247846	−	0.285359i
$176$	−2440.17			−1.04508
$177$			3554.94i			1.50964i
$178$			4431.26i			1.86594i
$179$	1830.53			0.764361			0.382180	−	0.924088i	$-0.375173\pi$
$179$	1830.53			0.764361			0.382180	+	0.924088i	$0.375173\pi$
$180$	31.4813	−	84.2552i	0.0130360	−	0.0348889i
$181$	2437.22			1.00087			0.500433	−	0.865775i	$-0.333174\pi$
$181$	2437.22			1.00087			0.500433	+	0.865775i	$0.333174\pi$
$182$			79.2075i			0.0322596i
$183$		−	427.075i		−	0.172515i
$184$	−3887.29			−1.55747
$185$	−2625.60	−	981.037i	−1.04345	−	0.389877i
$186$	−4415.44			−1.74062
$187$			1934.86i			0.756638i
$188$			5.47968i			0.00212578i
$189$	−1685.91			−0.648847
$190$	−775.327	−	289.695i	−0.296043	−	0.110614i
$191$	5079.50			1.92429			0.962145	−	0.272538i	$-0.0878632\pi$
$191$	5079.50			1.92429			0.962145	+	0.272538i	$0.0878632\pi$
$192$			4512.93i			1.69632i
$193$			2805.09i			1.04619i	0.852274	+	0.523095i	$0.175223\pi$
$193$			2805.09i			1.04619i	−0.852274	+	0.523095i	$0.824777\pi$
$194$	−694.667			−0.257084
$195$	−139.420	+	373.138i	−0.0512004	+	0.137031i
$196$	7.32786			0.00267050
$197$		−	3107.79i		−	1.12396i	−0.827149	−	0.561982i	$-0.810039\pi$
$197$		−	3107.79i		−	1.12396i	0.827149	−	0.561982i	$-0.189961\pi$
$198$		−	5749.76i		−	2.06373i
$199$	2145.63			0.764321			0.382161	−	0.924096i	$-0.375180\pi$
$199$	2145.63			0.764321			0.382161	+	0.924096i	$0.375180\pi$
$200$	1836.97	−	2115.01i	0.649467	−	0.747768i
$201$	2365.70			0.830166
$202$		−	4394.41i		−	1.53064i
$203$			1719.73i			0.594590i
$204$	−69.4669			−0.0238414
$205$	−191.219	+	511.769i	−0.0651478	+	0.174359i
$206$	2708.09			0.915929
$207$		−	9330.89i		−	3.13305i
$208$			258.331i			0.0861154i
$209$	−970.925			−0.321341
$210$	1881.19	+	702.893i	0.618164	+	0.230972i
$211$	2837.45			0.925772			0.462886	−	0.886418i	$-0.346814\pi$
$211$	2837.45			0.925772			0.462886	+	0.886418i	$0.346814\pi$
$212$			96.5013i			0.0312629i
$213$			2410.53i			0.775430i
$214$	−2466.45			−0.787864
$215$	1504.07	+	561.984i	0.477101	+	0.178265i
$216$	−5397.57			−1.70027
$217$		−	1204.52i		−	0.376812i
$218$		−	2530.21i		−	0.786089i
$219$	−1795.53			−0.554022
$220$	−21.9109	+	58.6413i	−0.00671468	+	0.0179709i
$221$	204.836			0.0623474
$222$		−	6432.94i		−	1.94482i
$223$			4741.40i			1.42380i	0.702280	+	0.711901i	$0.252166\pi$
$223$			4741.40i			1.42380i	−0.702280	+	0.711901i	$0.747834\pi$
$224$	47.3683			0.0141291
$225$	5076.78	+	4409.39i	1.50423	+	1.30649i
$226$	2186.61			0.643589
$227$		−	960.790i		−	0.280925i	−0.990086	−	0.140462i	$-0.955141\pi$
$227$		−	960.790i		−	0.280925i	0.990086	−	0.140462i	$-0.0448589\pi$
$228$		−	34.8589i		−	0.0101254i
$229$	−744.006			−0.214696			−0.107348	−	0.994222i	$-0.534236\pi$
$229$	−744.006			−0.214696			−0.107348	+	0.994222i	$0.534236\pi$
$230$	−1937.69	+	5185.96i	−0.555512	+	1.48675i
$231$	2355.78			0.670990
$232$			5505.86i			1.55809i
$233$		−	1550.56i		−	0.435968i	−0.975952	−	0.217984i	$-0.930052\pi$
$233$		−	1550.56i		−	0.435968i	0.975952	−	0.217984i	$-0.0699480\pi$
$234$	−608.704			−0.170052
$235$	−383.752	−	143.386i	−0.106524	−	0.0398020i
$236$	59.1456			0.0163138
$237$			4257.35i			1.16685i
$238$		−	1032.69i		−	0.281258i
$239$	−2775.00			−0.751045			−0.375523	−	0.926813i	$-0.622537\pi$
$239$	−2775.00			−0.751045			−0.375523	+	0.926813i	$0.622537\pi$
$240$	6135.40	+	2292.44i	1.65016	+	0.616569i
$241$	−2550.20			−0.681630			−0.340815	−	0.940130i	$-0.610703\pi$
$241$	−2550.20			−0.681630			−0.340815	+	0.940130i	$0.610703\pi$
$242$			202.149i			0.0536968i
$243$			99.3558i			0.0262291i
$244$	−7.10549			−0.00186427
$245$	−191.747	+	513.184i	−0.0500012	+	0.133821i
$246$	−1253.88			−0.324977
$247$			102.788i			0.0264787i
$248$		−	3856.36i		−	0.987416i
$249$	653.494			0.166319
$250$	−1905.92	−	3504.93i	−0.482164	−	0.886686i
$251$	−2933.00			−0.737568			−0.368784	−	0.929515i	$-0.620226\pi$
$251$	−2933.00			−0.737568			−0.368784	+	0.929515i	$0.620226\pi$
$252$			56.3141i			0.0140772i
$253$			6494.26i			1.61380i
$254$	−1441.76			−0.356159
$255$	1817.73	−	4864.90i	0.446395	−	1.19471i
$256$	229.626			0.0560611
$257$			2725.22i			0.661459i	0.943726	+	0.330729i	$0.107295\pi$
$257$			2725.22i			0.661459i	−0.943726	+	0.330729i	$0.892705\pi$
$258$			3685.09i			0.889240i
$259$	1754.89			0.421018
$260$	6.20811	+	2.31961i	0.00148081	+	0.000553294i
$261$	−13216.0			−3.13430
$262$			1921.04i			0.452986i
$263$		−	3027.26i		−	0.709767i	−0.934910	−	0.354884i	$-0.884520\pi$
$263$		−	3027.26i		−	0.709767i	0.934910	−	0.354884i	$-0.115480\pi$
$264$	7542.19			1.75829
$265$	−6758.17	−	2525.14i	−1.56661	−	0.585351i
$266$	518.210			0.119449
$267$		−	13952.5i		−	3.19804i
$268$		−	39.3595i		−	0.00897114i
$269$	1442.46			0.326946			0.163473	−	0.986548i	$-0.447730\pi$
$269$	1442.46			0.326946			0.163473	+	0.986548i	$0.447730\pi$
$270$	−2690.52	+	7200.80i	−0.606445	+	1.62306i
$271$	−6464.45			−1.44903			−0.724516	−	0.689258i	$-0.757937\pi$
$271$	−6464.45			−1.44903			−0.724516	+	0.689258i	$0.757937\pi$
$272$		−	3368.06i		−	0.750804i
$273$		−	249.396i		−	0.0552900i
$274$	−4431.36			−0.977038
$275$	−3533.42	−	3068.92i	−0.774812	−	0.672955i
$276$	−233.162			−0.0508503
$277$		−	876.614i		−	0.190147i	−0.995470	−	0.0950733i	$-0.969691\pi$
$277$		−	876.614i		−	0.190147i	0.995470	−	0.0950733i	$-0.0303086\pi$
$278$			3060.33i			0.660240i
$279$	9256.66			1.98631
$280$	−613.893	+	1643.00i	−0.131025	+	0.350671i
$281$	6252.19			1.32731			0.663655	−	0.748038i	$-0.269004\pi$
$281$	6252.19			1.32731			0.663655	+	0.748038i	$0.269004\pi$
$282$		−	940.224i		−	0.198544i
$283$			2250.07i			0.472625i	0.971677	+	0.236312i	$0.0759389\pi$
$283$			2250.07i			0.472625i	−0.971677	+	0.236312i	$0.924061\pi$
$284$	40.1054			0.00837963
$285$	2441.23	+	912.147i	0.507390	+	0.189582i
$286$	423.655			0.0875919
$287$		−	342.054i		−	0.0703513i
$288$			364.022i			0.0744799i
$289$	2242.39			0.456419
$290$	7345.26	+	2744.50i	1.48734	+	0.555733i
$291$	2187.26			0.440617
$292$			29.8733i			0.00598700i
$293$		−	5917.86i		−	1.17995i	−0.807422	−	0.589975i	$-0.799138\pi$
$293$		−	5917.86i		−	1.17995i	0.807422	−	0.589975i	$-0.200862\pi$
$294$	−1257.34			−0.249421
$295$	−1547.66	+	4142.08i	−0.305451	+	0.817495i
$296$	5618.41			1.10325
$297$			9017.41i			1.76176i
$298$		−	1843.98i		−	0.358453i
$299$	687.522			0.132978
$300$	110.183	−	126.859i	0.0212046	−	0.0244141i
$301$	−1005.28			−0.192504
$302$			694.256i			0.132284i
$303$			13836.4i			2.62337i
$304$	1690.11			0.318864
$305$	185.929	−	497.611i	0.0349057	−	0.0934200i
$306$	7936.16			1.48261
$307$		−	9458.47i		−	1.75838i	−0.476469	−	0.879191i	$-0.658084\pi$
$307$		−	9458.47i		−	1.75838i	0.476469	−	0.879191i	$-0.341916\pi$
$308$		−	39.1944i		−	0.00725100i
$309$	−8526.81			−1.56982
$310$	−5144.70	−	1922.28i	−0.942579	−	0.352187i
$311$	−7576.78			−1.38148			−0.690739	−	0.723104i	$-0.742715\pi$
$311$	−7576.78			−1.38148			−0.690739	+	0.723104i	$0.742715\pi$
$312$		−	798.461i		−	0.144884i
$313$		−	9172.41i		−	1.65641i	−0.560427	−	0.828204i	$-0.689363\pi$
$313$		−	9172.41i		−	1.65641i	0.560427	−	0.828204i	$-0.310637\pi$
$314$	4432.16			0.796565
$315$	−3943.79	−	1473.57i	−0.705419	−	0.263575i
$316$	70.8320			0.0126095
$317$			3077.94i			0.545345i	0.962107	+	0.272672i	$0.0879075\pi$
$317$			3077.94i			0.545345i	−0.962107	+	0.272672i	$0.912092\pi$
$318$		−	16558.1i		−	2.91991i
$319$	9198.31			1.61444
$320$	−1964.72	+	5258.30i	−0.343223	+	0.918587i
$321$	7765.98			1.35033
$322$		−	3466.17i		−	0.599882i
$323$		−	1340.13i		−	0.230857i
$324$	−106.538			−0.0182678
$325$	−324.894	+	374.069i	−0.0554519	+	0.0638449i
$326$	7290.01			1.23852
$327$			7966.75i			1.34728i
$328$		−	1095.11i		−	0.184352i
$329$	256.491			0.0429811
$330$	3759.55	−	10061.9i	0.627141	−	1.67845i
$331$	3234.50			0.537113			0.268557	−	0.963264i	$-0.413453\pi$
$331$	3234.50			0.537113			0.268557	+	0.963264i	$0.413453\pi$
$332$		−	10.8726i		−	0.00179732i
$333$			13486.2i			2.21934i
$334$	−10200.4			−1.67108
$335$	2756.42	+	1029.92i	0.449550	+	0.167971i
$336$	−4100.75			−0.665817
$337$			3777.84i			0.610658i	0.952247	+	0.305329i	$0.0987665\pi$
$337$			3777.84i			0.610658i	−0.952247	+	0.305329i	$0.901234\pi$
$338$			6227.02i			1.00209i
$339$	−6884.87			−1.10305
$340$	−80.9401	−	30.2427i	−0.0129106	−	0.00482394i
$341$	−6442.60			−1.02313
$342$			3982.41i			0.629660i
$343$		−	343.000i		−	0.0539949i
$344$	−3218.49			−0.504446
$345$	6101.12	−	16328.8i	0.952096	−	2.54815i
$346$	6379.51			0.991227
$347$			8244.08i			1.27540i	0.770283	+	0.637702i	$0.220115\pi$
$347$			8244.08i			1.27540i	−0.770283	+	0.637702i	$0.779885\pi$
$348$			330.245i			0.0508706i
$349$	−7173.78			−1.10030			−0.550148	−	0.835067i	$-0.685429\pi$
$349$	−7173.78			−1.10030			−0.550148	+	0.835067i	$0.685429\pi$
$350$	1885.88	+	1637.97i	0.288014	+	0.250152i
$351$	954.637			0.145170
$352$		−	253.358i		−	0.0383637i
$353$		−	4191.51i		−	0.631987i	−0.948761	−	0.315994i	$-0.897662\pi$
$353$		−	4191.51i		−	0.631987i	0.948761	−	0.315994i	$-0.102338\pi$
$354$	−10148.4			−1.52368
$355$	−1049.43	+	2808.65i	−0.156896	+	0.419910i
$356$	−232.136			−0.0345594
$357$			3251.58i			0.482050i
$358$			5225.70i			0.771472i
$359$	3136.29			0.461078			0.230539	−	0.973063i	$-0.425951\pi$
$359$	3136.29			0.461078			0.230539	+	0.973063i	$0.425951\pi$
$360$	−12626.3	−	4717.73i	−1.84851	−	0.690683i
$361$	−6186.52			−0.901956
$362$			6957.62i			1.01018i
$363$		−	636.496i		−	0.0920313i
$364$	−4.14936			−0.000597487
$365$	−2092.08	−	781.691i	−0.300013	−	0.112098i
$366$	1219.19			0.174120
$367$		−	1723.30i		−	0.245110i	−0.992462	−	0.122555i	$-0.960891\pi$
$367$		−	1723.30i		−	0.245110i	0.992462	−	0.122555i	$-0.0391088\pi$
$368$		−	11304.7i		−	1.60136i
$369$	2628.67			0.370848
$370$	2800.61	−	7495.42i	0.393504	−	1.05316i
$371$	4517.00			0.632105
$372$		−	231.307i		−	0.0322384i
$373$		−	2818.55i		−	0.391258i	−0.980678	−	0.195629i	$-0.937325\pi$
$373$		−	2818.55i		−	0.391258i	0.980678	−	0.195629i	$-0.0626748\pi$
$374$	−5523.53			−0.763677
$375$	6001.07	+	11035.8i	0.826384	+	1.51970i
$376$	821.174			0.112630
$377$		−	973.788i		−	0.133031i
$378$		−	4812.84i		−	0.654884i
$379$	10466.1			1.41849			0.709246	−	0.704961i	$-0.249036\pi$
$379$	10466.1			1.41849			0.709246	+	0.704961i	$0.249036\pi$
$380$	15.1759	−	40.6162i	0.00204871	−	0.00548307i
$381$	4539.61			0.610423
$382$			14500.6i			1.94219i
$383$			258.055i			0.0344282i	0.999852	+	0.0172141i	$0.00547969\pi$
$383$			258.055i			0.0344282i	−0.999852	+	0.0172141i	$0.994520\pi$
$384$	−13369.9			−1.77677
$385$	2744.86	+	1025.60i	0.363353	+	0.135764i
$386$	−8007.81			−1.05592
$387$		−	7725.54i		−	1.01476i
$388$		−	36.3908i		−	0.00476150i
$389$	−4573.87			−0.596156			−0.298078	−	0.954542i	$-0.596346\pi$
$389$	−4573.87			−0.596156			−0.298078	+	0.954542i	$0.596346\pi$
$390$	−1065.21	−	398.008i	−0.138305	−	0.0516768i
$391$	−8963.77			−1.15938
$392$		−	1098.14i		−	0.141491i
$393$		−	6048.69i		−	0.776376i
$394$	8871.94			1.13442
$395$	−1853.45	+	4960.50i	−0.236094	+	0.631873i
$396$	301.206			0.0382227
$397$			3624.55i			0.458215i	0.973401	+	0.229107i	$0.0735807\pi$
$397$			3624.55i			0.458215i	−0.973401	+	0.229107i	$0.926419\pi$
$398$			6125.23i			0.771432i
$399$	−1631.66			−0.204725
$400$	6150.70	+	5342.14i	0.768838	+	0.667767i
$401$	6358.32			0.791819			0.395910	−	0.918289i	$-0.370429\pi$
$401$	6358.32			0.791819			0.395910	+	0.918289i	$0.370429\pi$
$402$			6753.45i			0.837890i
$403$			682.052i			0.0843063i
$404$	230.205			0.0283493
$405$	2787.77	−	7461.05i	0.342038	−	0.915414i
$406$	−4909.40			−0.600121
$407$		−	9386.35i		−	1.14316i
$408$			10410.2i			1.26319i
$409$	6536.39			0.790228			0.395114	−	0.918632i	$-0.370705\pi$
$409$	6536.39			0.790228			0.395114	+	0.918632i	$0.370705\pi$
$410$	−1460.97	−	545.880i	−0.175981	−	0.0657539i
$411$	13952.8			1.67455
$412$			141.866i			0.0169641i
$413$		−	2768.47i		−	0.329848i
$414$	26637.3			3.16220
$415$	761.427	+	284.501i	0.0900650	+	0.0336521i
$416$	−26.8220			−0.00316119
$417$		−	9635.91i		−	1.13159i
$418$		−	2771.74i		−	0.324331i
$419$	−6333.56			−0.738460			−0.369230	−	0.929338i	$-0.620379\pi$
$419$	−6333.56			−0.738460			−0.369230	+	0.929338i	$0.620379\pi$
$420$	−36.8217	+	98.5479i	−0.00427789	+	0.0114492i
$421$	−8139.62			−0.942282			−0.471141	−	0.882058i	$-0.656158\pi$
$421$	−8139.62			−0.942282			−0.471141	+	0.882058i	$0.656158\pi$
$422$			8100.18i			0.934385i
$423$			1971.11i			0.226569i
$424$	14461.5			1.65640
$425$	4235.90	−	4877.03i	0.483462	−	0.556637i
$426$	−6881.43			−0.782645
$427$			332.591i			0.0376937i
$428$		−	129.207i		−	0.0145922i
$429$	−1333.94			−0.150124
$430$	−1604.32	+	4293.73i	−0.179924	+	0.481539i
$431$	−14367.6			−1.60571			−0.802856	−	0.596173i	$-0.796687\pi$
$431$	−14367.6			−1.60571			−0.802856	+	0.596173i	$0.796687\pi$
$432$		−	15696.8i		−	1.74818i
$433$			8399.05i			0.932176i	0.884738	+	0.466088i	$0.154337\pi$
$433$			8399.05i			0.932176i	−0.884738	+	0.466088i	$0.845663\pi$
$434$	3438.59			0.380318
$435$	−23127.6	−	8641.47i	−2.54916	−	0.952475i
$436$	132.547			0.0145593
$437$		−	4498.07i		−	0.492384i
$438$		−	5125.78i		−	0.559176i
$439$	17860.8			1.94180			0.970901	−	0.239482i	$-0.0769777\pi$
$439$	17860.8			1.94180			0.970901	+	0.239482i	$0.0769777\pi$
$440$	8787.87	+	3283.52i	0.952148	+	0.355763i
$441$	2635.93			0.284627
$442$			584.754i			0.0629274i
$443$		−	1901.57i		−	0.203942i	−0.994787	−	0.101971i	$-0.967485\pi$
$443$		−	1901.57i		−	0.203942i	0.994787	−	0.101971i	$-0.0325148\pi$
$444$	336.996			0.0360205
$445$	6074.26	−	16256.9i	0.647074	−	1.73180i
$446$	−13535.5			−1.43705
$447$			5806.05i			0.614355i
$448$		−	3514.52i		−	0.370637i
$449$	5185.68			0.545050			0.272525	−	0.962149i	$-0.412141\pi$
$449$	5185.68			0.545050			0.272525	+	0.962149i	$0.412141\pi$
$450$	−12587.7	+	14492.9i	−1.31864	+	1.51823i
$451$	−1829.54			−0.191019
$452$			114.548i			0.0119201i
$453$		−	2185.97i		−	0.226723i
$454$	2742.81			0.283538
$455$	108.576	−	290.587i	0.0111870	−	0.0299405i
$456$	−5223.88			−0.536471
$457$		−	11198.8i		−	1.14630i	−0.819451	−	0.573149i	$-0.805722\pi$
$457$		−	11198.8i		−	1.14630i	0.819451	−	0.573149i	$-0.194278\pi$
$458$		−	2123.94i		−	0.216693i
$459$	−12446.4			−1.26568
$460$	−271.671	−	101.508i	−0.0275364	−	0.0102888i
$461$	17270.7			1.74485			0.872427	−	0.488744i	$-0.162545\pi$
$461$	17270.7			1.74485			0.872427	+	0.488744i	$0.162545\pi$
$462$			6725.13i			0.677232i
$463$		−	385.660i		−	0.0387109i	−0.999813	−	0.0193554i	$-0.993839\pi$
$463$		−	385.660i		−	0.0387109i	0.999813	−	0.0193554i	$-0.00616141\pi$
$464$	−16011.7			−1.60199
$465$	16198.9	+	6052.58i	1.61549	+	0.603616i
$466$	4426.44			0.440024
$467$			5035.36i			0.498947i	0.968382	+	0.249474i	$0.0802576\pi$
$467$			5035.36i			0.498947i	−0.968382	+	0.249474i	$0.919742\pi$
$468$		−	31.8875i		−	0.00314957i
$469$	−1842.33			−0.181387
$470$	409.330	−	1095.51i	0.0401723	−	0.107515i
$471$	−13955.3			−1.36524
$472$		−	8863.45i		−	0.864350i
$473$			5376.94i			0.522689i
$474$	−12153.6			−1.17771
$475$	2447.32	+	2125.60i	0.236402	+	0.205324i
$476$	54.0985			0.00520924
$477$			34712.8i			3.33206i
$478$		−	7921.91i		−	0.758033i
$479$	8681.99			0.828163			0.414082	−	0.910240i	$-0.364103\pi$
$479$	8681.99			0.828163			0.414082	+	0.910240i	$0.364103\pi$
$480$	−238.020	+	637.027i	−0.0226335	+	0.0605754i
$481$	−993.695			−0.0941967
$482$		−	7280.16i		−	0.687972i
$483$			10913.8i			1.02814i
$484$	−10.5898			−0.000994530
$485$	2548.51	+	952.233i	0.238602	+	0.0891519i
$486$	−283.635			−0.0264731
$487$			890.476i			0.0828569i	0.999141	+	0.0414284i	$0.0131909\pi$
$487$			890.476i			0.0828569i	−0.999141	+	0.0414284i	$0.986809\pi$
$488$			1064.81i			0.0987744i
$489$	−22953.7			−2.12270
$490$	−1465.01	−	547.389i	−0.135066	−	0.0504664i
$491$	1562.48			0.143613			0.0718063	−	0.997419i	$-0.477124\pi$
$491$	1562.48			0.143613			0.0718063	+	0.997419i	$0.477124\pi$
$492$		−	65.6855i		−	0.00601897i
$493$			12696.1i			1.15984i
$494$	−293.433			−0.0267250
$495$	−7881.63	+	21094.1i	−0.715663	+	1.91537i
$496$	11214.8			1.01524
$497$		−	1877.24i		−	0.169428i
$498$			1865.56i			0.167867i
$499$	−8234.33			−0.738716			−0.369358	−	0.929287i	$-0.620422\pi$
$499$	−8234.33			−0.738716			−0.369358	+	0.929287i	$0.620422\pi$
$500$	183.609	−	99.8433i	0.0164225	−	0.00893026i
$501$	32117.5			2.86408
$502$		−	8372.97i		−	0.744430i
$503$		−	72.5340i		−	0.00642969i	−0.999995	−	0.00321484i	$-0.998977\pi$
$503$		−	72.5340i		−	0.00642969i	0.999995	−	0.00321484i	$-0.00102332\pi$
$504$	8439.13			0.745850
$505$	−6023.75	+	16121.7i	−0.530798	+	1.42061i
$506$	−18539.4			−1.62881
$507$		−	19606.7i		−	1.71748i
$508$		−	75.5281i		−	0.00659649i
$509$	−7793.44			−0.678660			−0.339330	−	0.940667i	$-0.610200\pi$
$509$	−7793.44			−0.678660			−0.339330	+	0.940667i	$0.610200\pi$
$510$	13888.0	+	5189.15i	1.20583	+	0.450548i
$511$	1398.30			0.121051
$512$		−	11243.9i		−	0.970537i
$513$		−	6245.65i		−	0.537529i
$514$	−7779.81			−0.667612
$515$	−9935.12	−	3712.18i	−0.850084	−	0.317628i
$516$	−193.047			−0.0164698
$517$		−	1371.89i		−	0.116703i
$518$			5009.76i			0.424935i
$519$	−20086.8			−1.69887
$520$	347.613	−	930.336i	0.0293151	−	0.0784576i
$521$	4645.42			0.390633			0.195316	−	0.980740i	$-0.437427\pi$
$521$	4645.42			0.390633			0.195316	+	0.980740i	$0.437427\pi$
$522$		−	37728.4i		−	3.16346i
$523$		−	8783.88i		−	0.734402i	−0.930142	−	0.367201i	$-0.880316\pi$
$523$		−	8783.88i		−	0.734402i	0.930142	−	0.367201i	$-0.119684\pi$
$524$	−100.636			−0.00838986
$525$	−5937.99	−	5157.38i	−0.493629	−	0.428737i
$526$	8642.04			0.716371
$527$		−	8892.46i		−	0.735031i
$528$			21933.6i			1.80784i
$529$	−17919.4			−1.47279
$530$	7208.62	−	19292.8i	0.590797	−	1.58118i
$531$	21275.5			1.73875
$532$			27.1469i			0.00221234i
$533$			193.686i			0.0157401i
$534$	39830.7			3.22780
$535$	9048.62	+	3380.95i	0.731226	+	0.273217i
$536$	−5898.34			−0.475316
$537$		−	16453.9i		−	1.32223i
$538$			4117.86i			0.329988i
$539$	−1834.60			−0.146608
$540$	−377.221	−	140.946i	−0.0300611	−	0.0112321i
$541$	−7054.13			−0.560593			−0.280296	−	0.959913i	$-0.590433\pi$
$541$	−7054.13			−0.560593			−0.280296	+	0.959913i	$0.590433\pi$
$542$		−	18454.3i		−	1.46251i
$543$		−	21907.1i		−	1.73135i
$544$	349.700			0.0275611
$545$	−3468.35	+	9282.55i	−0.272602	+	0.729579i
$546$	711.962			0.0558043
$547$		−	5776.83i		−	0.451553i	−0.974179	−	0.225776i	$-0.927508\pi$
$547$		−	5776.83i		−	0.451553i	0.974179	−	0.225776i	$-0.0724919\pi$
$548$		−	232.141i		−	0.0180959i
$549$	−2555.94			−0.198697
$550$	8760.97	−	10087.0i	0.679216	−	0.782020i
$551$	−6370.95			−0.492580
$552$			34941.2i			2.69419i
$553$		−	3315.48i		−	0.254952i
$554$	2502.51			0.191916
$555$	−8818.12	+	23600.4i	−0.674430	+	1.80501i
$556$	−160.318			−0.0122284
$557$			20562.6i			1.56421i	0.623145	+	0.782106i	$0.285854\pi$
$557$			20562.6i			1.56421i	−0.623145	+	0.782106i	$0.714146\pi$
$558$			26425.4i			2.00479i
$559$	569.235			0.0430699
$560$	−4778.04	−	1785.28i	−0.360552	−	0.134717i
$561$	17391.7			1.30887
$562$			17848.4i			1.33966i
$563$			24009.5i			1.79730i	0.438666	+	0.898650i	$0.355451\pi$
$563$			24009.5i			1.79730i	−0.438666	+	0.898650i	$0.644549\pi$
$564$	49.2545			0.00367729
$565$	−8021.98	−	2997.35i	−0.597323	−	0.223185i
$566$	−6423.37			−0.477022
$567$			4986.79i			0.369357i
$568$		−	6010.11i		−	0.443977i
$569$	24157.5			1.77985			0.889925	−	0.456107i	$-0.150757\pi$
$569$	24157.5			1.77985			0.889925	+	0.456107i	$0.150757\pi$
$570$	−2603.94	+	6969.08i	−0.191346	+	0.512110i
$571$	−706.993			−0.0518157			−0.0259078	−	0.999664i	$-0.508248\pi$
$571$	−706.993			−0.0518157			−0.0259078	+	0.999664i	$0.508248\pi$
$572$			22.1936i			0.00162231i
$573$		−	45657.4i		−	3.32874i
$574$	976.477			0.0710059
$575$	14217.6	−	16369.5i	1.03115	−	1.18723i
$576$	27008.9			1.95377
$577$			16057.2i			1.15853i	0.815139	+	0.579265i	$0.196660\pi$
$577$			16057.2i			1.15853i	−0.815139	+	0.579265i	$0.803340\pi$
$578$			6401.44i			0.460665i
$579$	25213.8			1.80976
$580$	−143.773	+	384.788i	−0.0102929	+	0.0275474i
$581$	−508.919			−0.0363400
$582$			6244.07i			0.444717i
$583$		−	24160.0i		−	1.71630i
$584$	4476.76			0.317208
$585$	2233.14	+	834.397i	0.157827	+	0.0589710i
$586$	16894.0			1.19093
$587$			8605.63i			0.605098i	0.953134	+	0.302549i	$0.0978376\pi$
$587$			8605.63i			0.605098i	−0.953134	+	0.302549i	$0.902162\pi$
$588$		−	65.8671i		−	0.00461958i
$589$	4462.28			0.312165
$590$	−11824.6	−	4418.16i	−0.825101	−	0.308293i
$591$	−27934.6			−1.94429
$592$			16339.0i			1.13434i
$593$			20355.6i			1.40962i	0.709397	+	0.704809i	$0.248967\pi$
$593$			20355.6i			1.40962i	−0.709397	+	0.704809i	$0.751033\pi$
$594$	−25742.4			−1.77815
$595$	−1415.59	+	3788.62i	−0.0975352	+	0.261039i
$596$	96.5986			0.00663898
$597$		−	19286.2i		−	1.32216i
$598$			1962.70i			0.134215i
$599$	−22635.7			−1.54402			−0.772010	−	0.635610i	$-0.780749\pi$
$599$	−22635.7			−1.54402			−0.772010	+	0.635610i	$0.780749\pi$
$600$	−19010.9	−	16511.7i	−1.29353	−	1.12348i
$601$	−22553.8			−1.53077			−0.765383	−	0.643575i	$-0.777450\pi$
$601$	−22553.8			−1.53077			−0.765383	+	0.643575i	$0.777450\pi$
$602$		−	2869.83i		−	0.194295i
$603$		−	14158.1i		−	0.956159i
$604$	−36.3692			−0.00245007
$605$	277.101	−	741.620i	0.0186211	−	0.0498366i
$606$	−39499.4			−2.64778
$607$		−	17534.2i		−	1.17248i	−0.810139	−	0.586238i	$-0.800608\pi$
$607$		−	17534.2i		−	1.17248i	0.810139	−	0.586238i	$-0.199392\pi$
$608$			175.481i			0.0117051i
$609$	15458.0			1.02855
$610$	1420.55	+	530.778i	0.0942892	+	0.0352304i
$611$	−145.236			−0.00961641
$612$			415.743i			0.0274598i
$613$			10445.5i			0.688238i	0.938926	+	0.344119i	$0.111822\pi$
$613$			10445.5i			0.688238i	−0.938926	+	0.344119i	$0.888178\pi$
$614$	27001.5			1.77474
$615$	4600.08	+	1718.78i	0.301615	+	0.112696i
$616$	−5873.60			−0.384179
$617$			13218.9i			0.862516i	0.902229	+	0.431258i	$0.141930\pi$
$617$			13218.9i			0.862516i	−0.902229	+	0.431258i	$0.858070\pi$
$618$		−	24341.8i		−	1.58442i
$619$	−23438.9			−1.52195			−0.760976	−	0.648780i	$-0.775280\pi$
$619$	−23438.9			−1.52195			−0.760976	+	0.648780i	$0.775280\pi$
$620$	100.700	−	269.510i	0.00652294	−	0.0174577i
$621$	−41775.5			−2.69951
$622$		−	21629.8i		−	1.39433i
$623$			10865.7i			0.698757i
$624$	2322.02			0.148967
$625$	2187.74	+	15471.1i	0.140015	+	0.990149i
$626$	26184.9			1.67182
$627$			8727.23i			0.555873i
$628$			232.183i			0.0147534i
$629$	12955.6			0.821262
$630$	4206.65	−	11258.5i	0.266027	−	0.711982i
$631$	−874.004			−0.0551403			−0.0275702	−	0.999620i	$-0.508777\pi$
$631$	−874.004			−0.0551403			−0.0275702	+	0.999620i	$0.508777\pi$
$632$		−	10614.7i		−	0.668089i
$633$		−	25504.6i		−	1.60145i
$634$	−8786.72			−0.550419
$635$	5289.38	+	1976.34i	0.330555	+	0.123509i
$636$	867.410			0.0540802
$637$			194.222i			0.0120806i
$638$			26258.8i			1.62946i
$639$	14426.4			0.893116
$640$	−15578.0	−	5820.62i	−0.962151	−	0.359500i
$641$	23977.0			1.47743			0.738715	−	0.674017i	$-0.235433\pi$
$641$	23977.0			1.47743			0.738715	+	0.674017i	$0.235433\pi$
$642$			22169.9i			1.36289i
$643$			27698.0i			1.69876i	0.527782	+	0.849380i	$0.323024\pi$
$643$			27698.0i			1.69876i	−0.527782	+	0.849380i	$0.676976\pi$
$644$	181.579			0.0111106
$645$	5051.44	−	13519.4i	0.308372	−	0.825314i
$646$	3825.72			0.233004
$647$		−	10965.1i		−	0.666282i	−0.942877	−	0.333141i	$-0.891891\pi$
$647$		−	10965.1i		−	0.666282i	0.942877	−	0.333141i	$-0.108109\pi$
$648$			15965.6i			0.967881i
$649$	−14807.6			−0.895610
$650$	−1067.87	−	927.488i	−0.0644389	−	0.0559678i
$651$	−10826.9			−0.651829
$652$			381.894i			0.0229388i
$653$			12336.2i			0.739285i	0.929174	+	0.369642i	$0.120520\pi$
$653$			12336.2i			0.739285i	−0.929174	+	0.369642i	$0.879480\pi$
$654$	−22743.0			−1.35982
$655$	2633.32	−	7047.70i	0.157087	−	0.420422i
$656$	3184.72			0.189547
$657$			10745.8i			0.638105i
$658$			732.215i			0.0433810i
$659$	25275.6			1.49408			0.747040	−	0.664779i	$-0.231474\pi$
$659$	25275.6			1.49408			0.747040	+	0.664779i	$0.231474\pi$
$660$	527.102	+	196.947i	0.0310870	+	0.0116154i
$661$	−4447.92			−0.261731			−0.130865	−	0.991400i	$-0.541776\pi$
$661$	−4447.92			−0.261731			−0.130865	+	0.991400i	$0.541776\pi$
$662$			9233.67i			0.542110i
$663$		−	1841.19i		−	0.107852i
$664$	−1629.34			−0.0952270
$665$	−1901.15	−	710.349i	−0.110862	−	0.0414228i
$666$	−38499.7			−2.23999
$667$			42613.6i			2.47377i
$668$		−	534.357i		−	0.0309505i
$669$	42618.5			2.46297
$670$	−2940.14	+	7868.87i	−0.169534	+	0.453733i
$671$	1778.92			0.102347
$672$		−	425.774i		−	0.0244413i
$673$		−	30358.9i		−	1.73885i	−0.494061	−	0.869427i	$-0.664488\pi$
$673$		−	30358.9i		−	1.73885i	0.494061	−	0.869427i	$-0.335512\pi$
$674$	−10784.7			−0.616340
$675$	19741.4	−	22729.4i	1.12570	−	1.29608i
$676$	−326.208			−0.0185599
$677$			6916.48i			0.392647i	0.980539	+	0.196324i	$0.0629003\pi$
$677$			6916.48i			0.392647i	−0.980539	+	0.196324i	$0.937100\pi$
$678$		−	19654.5i		−	1.11331i
$679$	−1703.37			−0.0962728
$680$	−4532.11	+	12129.5i	−0.255586	+	0.684039i
$681$	−8636.14			−0.485958
$682$		−	18392.0i		−	1.03265i
$683$		−	4532.72i		−	0.253938i	−0.991907	−	0.126969i	$-0.959475\pi$
$683$		−	4532.72i		−	0.253938i	0.991907	−	0.126969i	$-0.0405249\pi$
$684$	−208.622			−0.0116621
$685$	16257.3	+	6074.40i	0.906800	+	0.338819i
$686$	979.176			0.0544973
$687$			6687.56i			0.371392i
$688$		−	9359.77i		−	0.518660i
$689$	−2557.72			−0.141424
$690$	46614.4	+	17417.1i	2.57185	+	0.960954i
$691$	27235.2			1.49939			0.749694	−	0.661785i	$-0.230201\pi$
$691$	27235.2			1.49939			0.749694	+	0.661785i	$0.230201\pi$
$692$			334.197i			0.0183587i
$693$		−	14098.8i		−	0.772825i
$694$	−23534.7			−1.28727
$695$	4195.03	−	11227.4i	0.228959	−	0.612776i
$696$	49489.8			2.69527
$697$		−	2525.24i		−	0.137231i
$698$		−	20479.3i		−	1.11053i
$699$	−13937.3			−0.754160
$700$	−85.8064	+	98.7938i	−0.00463311	+	0.00533436i
$701$	−17144.3			−0.923726			−0.461863	−	0.886951i	$-0.652819\pi$
$701$	−17144.3			−0.923726			−0.461863	+	0.886951i	$0.652819\pi$
$702$			2725.24i			0.146521i
$703$			6501.19i			0.348787i
$704$	−18798.1			−1.00636
$705$	−1288.84	+	3449.39i	−0.0688516	+	0.184271i
$706$	11965.7			0.637867
$707$		−	10775.3i		−	0.573195i
$708$		−	531.635i		−	0.0282204i
$709$	−16724.1			−0.885877			−0.442939	−	0.896552i	$-0.646064\pi$
$709$	−16724.1			−0.885877			−0.442939	+	0.896552i	$0.646064\pi$
$710$	−8017.98	−	2995.86i	−0.423816	−	0.158356i
$711$	25479.2			1.34395
$712$			34787.4i			1.83106i
$713$		−	29847.0i		−	1.56771i
$714$	−9282.42			−0.486535
$715$	−1554.26	−	580.737i	−0.0812951	−	0.0303753i
$716$	−273.753			−0.0142886
$717$			24943.3i			1.29920i
$718$			8953.30i			0.465368i
$719$	−4308.66			−0.223485			−0.111743	−	0.993737i	$-0.535643\pi$
$719$	−4308.66			−0.223485			−0.111743	+	0.993737i	$0.535643\pi$
$720$	13719.7	−	36718.9i	0.710145	−	1.90060i
$721$	6640.39			0.342997
$722$		−	17660.9i		−	0.910347i
$723$			22922.7i			1.17912i
$724$	−364.481			−0.0187097
$725$	−23185.4	−	20137.4i	−1.18770	−	1.03157i
$726$	1817.03			0.0928875
$727$		−	29435.6i		−	1.50166i	−0.660496	−	0.750830i	$-0.729654\pi$
$727$		−	29435.6i		−	1.50166i	0.660496	−	0.750830i	$-0.270346\pi$
$728$			621.814i			0.0316566i
$729$	20127.8			1.02260
$730$	2231.53	−	5972.36i	0.113140	−	0.302804i
$731$	−7421.58			−0.375509
$732$			63.8683i			0.00322492i
$733$			6587.69i			0.331954i	0.986130	+	0.165977i	$0.0530777\pi$
$733$			6587.69i			0.331954i	−0.986130	+	0.165977i	$0.946922\pi$
$734$	4919.57			0.247390
$735$	4612.80	+	1723.54i	0.231490	+	0.0864947i
$736$	1173.75			0.0587839
$737$			9854.01i			0.492506i
$738$			7504.16i			0.374298i
$739$	−3684.46			−0.183404			−0.0917018	−	0.995787i	$-0.529231\pi$
$739$	−3684.46			−0.183404			−0.0917018	+	0.995787i	$0.529231\pi$
$740$	392.654	+	146.712i	0.0195058	+	0.00728818i
$741$	923.917			0.0458042
$742$			12894.9i			0.637986i
$743$		−	12271.9i		−	0.605940i	−0.953000	−	0.302970i	$-0.902022\pi$
$743$		−	12271.9i		−	0.605940i	0.953000	−	0.302970i	$-0.0979783\pi$
$744$	−34663.2			−1.70808
$745$	−2527.68	+	6764.98i	−0.124305	+	0.332684i
$746$	8046.24			0.394898
$747$		−	3911.01i		−	0.191561i
$748$		−	289.355i		−	0.0141442i
$749$	−6047.88			−0.295040
$750$	−31504.4	+	17131.5i	−1.53384	+	0.834072i
$751$	−30871.0			−1.50000			−0.749999	−	0.661439i	$-0.769946\pi$
$751$	−30871.0			−1.50000			−0.749999	+	0.661439i	$0.769946\pi$
$752$			2388.08i			0.115803i
$753$			26363.5i			1.27588i
$754$	2779.91			0.134269
$755$	951.669	−	2547.00i	0.0458739	−	0.122775i
$756$	252.125			0.0121292
$757$			11442.1i			0.549368i	0.961535	+	0.274684i	$0.0885732\pi$
$757$			11442.1i			0.549368i	−0.961535	+	0.274684i	$0.911427\pi$
$758$			29878.1i			1.43169i
$759$	58374.2			2.79163
$760$	−6086.66	−	2274.24i	−0.290509	−	0.108546i
$761$	14423.3			0.687048			0.343524	−	0.939144i	$-0.388379\pi$
$761$	14423.3			0.687048			0.343524	+	0.939144i	$0.388379\pi$
$762$			12959.4i			0.616102i
$763$		−	6204.23i		−	0.294375i
$764$	−759.630			−0.0359718
$765$	−29115.3	−	10878.7i	−1.37603	−	0.514144i
$766$	−736.681			−0.0347485
$767$			1567.63i			0.0737988i
$768$		−	2064.01i		−	0.0969775i
$769$	−26772.8			−1.25546			−0.627731	−	0.778430i	$-0.716016\pi$
$769$	−26772.8			−1.25546			−0.627731	+	0.778430i	$0.716016\pi$
$770$	−2927.81	+	7835.86i	−0.137027	+	0.366734i
$771$	24495.9			1.14423
$772$		−	419.496i		−	0.0195570i
$773$		−	27669.3i		−	1.28745i	−0.765258	−	0.643724i	$-0.777389\pi$
$773$		−	27669.3i		−	1.28745i	0.765258	−	0.643724i	$-0.222611\pi$
$774$	22054.4			1.02420
$775$	16239.3	+	14104.5i	0.752686	+	0.653739i
$776$	−5453.45			−0.252278
$777$		−	15774.0i		−	0.728299i
$778$		−	13057.2i		−	0.601702i
$779$	1267.18

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:	\( N \)	\(=\)	\( 35 = 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	35.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.85474i q^{2} -8.98858i q^{3} -0.149548 q^{4} +(3.91321 - 10.4731i) q^{5} +25.6601 q^{6} +7.00000i q^{7} +22.4110i q^{8} -53.7945 q^{9} +(29.8981 + 11.1712i) q^{10} +37.4408 q^{11} +1.34423i q^{12} -3.96370i q^{13} -19.9832 q^{14} +(-94.1387 - 35.1742i) q^{15} -65.1740 q^{16} +51.6780i q^{17} -153.569i q^{18} -25.9323 q^{19} +(-0.585214 + 1.56624i) q^{20} +62.9200 q^{21} +106.884i q^{22} +173.454i q^{23} +201.443 q^{24} +(-94.3736 - 81.9673i) q^{25} +11.3154 q^{26} +240.845i q^{27} -1.04684i q^{28} +245.676 q^{29} +(100.413 - 268.742i) q^{30} -172.074 q^{31} -6.76690i q^{32} -336.539i q^{33} -147.527 q^{34} +(73.3120 + 27.3925i) q^{35} +8.04488 q^{36} -250.699i q^{37} -74.0300i q^{38} -35.6281 q^{39} +(234.714 + 87.6990i) q^{40} -48.8649 q^{41} +179.620i q^{42} +143.612i q^{43} -5.59920 q^{44} +(-210.509 + 563.398i) q^{45} -495.167 q^{46} -36.6415i q^{47} +585.822i q^{48} -49.0000 q^{49} +(233.995 - 269.412i) q^{50} +464.511 q^{51} +0.592765i q^{52} -645.286i q^{53} -687.549 q^{54} +(146.514 - 392.123i) q^{55} -156.877 q^{56} +233.094i q^{57} +701.343i q^{58} -395.495 q^{59} +(14.0783 + 5.26024i) q^{60} +47.5130 q^{61} -491.228i q^{62} -376.562i q^{63} -502.074 q^{64} +(-41.5125 - 15.5108i) q^{65} +960.733 q^{66} +263.189i q^{67} -7.72835i q^{68} +1559.11 q^{69} +(-78.1984 + 209.287i) q^{70} -268.177 q^{71} -1205.59i q^{72} -199.757i q^{73} +715.680 q^{74} +(-736.769 + 848.284i) q^{75} +3.87813 q^{76} +262.085i q^{77} -101.709i q^{78} -473.640 q^{79} +(-255.040 + 682.577i) q^{80} +712.399 q^{81} -139.497i q^{82} +72.7028i q^{83} -9.40958 q^{84} +(541.231 + 202.227i) q^{85} -409.975 q^{86} -2208.28i q^{87} +839.086i q^{88} +1552.25 q^{89} +(-1608.36 - 600.950i) q^{90} +27.7459 q^{91} -25.9398i q^{92} +1546.70i q^{93} +104.602 q^{94} +(-101.478 + 271.593i) q^{95} -60.8248 q^{96} +243.338i q^{97} -139.882i q^{98} -2014.11 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 36 q^{4} + 6 q^{5} + 12 q^{6} - 46 q^{9} - 16 q^{10} + 84 q^{11} - 56 q^{14} + 8 q^{15} + 148 q^{16} + 72 q^{19} - 68 q^{20} + 140 q^{21} + 72 q^{24} - 362 q^{25} - 620 q^{26} + 88 q^{29} + 52 q^{30}+ \cdots - 5304 q^{99}+O(q^{100}) \) 10 * q - 36 * q^4 + 6 * q^5 + 12 * q^6 - 46 * q^9 - 16 * q^10 + 84 * q^11 - 56 * q^14 + 8 * q^15 + 148 * q^16 + 72 * q^19 - 68 * q^20 + 140 * q^21 + 72 * q^24 - 362 * q^25 - 620 * q^26 + 88 * q^29 + 52 * q^30 + 120 * q^31 + 964 * q^34 - 28 * q^35 - 420 * q^36 + 212 * q^39 + 1396 * q^40 - 852 * q^41 - 1424 * q^44 - 510 * q^45 + 176 * q^46 - 490 * q^49 + 1644 * q^50 + 1276 * q^51 - 996 * q^54 - 1136 * q^55 + 588 * q^56 + 864 * q^59 - 1692 * q^60 - 884 * q^61 - 2724 * q^64 + 520 * q^65 + 1148 * q^66 + 4808 * q^69 + 756 * q^70 + 880 * q^71 + 3024 * q^74 + 720 * q^75 - 2672 * q^76 - 3580 * q^79 - 2316 * q^80 - 302 * q^81 - 1904 * q^84 + 1572 * q^85 + 2432 * q^86 - 1492 * q^89 - 7748 * q^90 + 476 * q^91 + 1652 * q^94 - 1628 * q^95 + 4080 * q^96 - 5304 * q^99

Introduction

L-functions

Modular forms

Varieties

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