LMFDB - Embedded newform 315.4.d.c.64.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,4,Mod(64,315)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(315, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("315.64");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	315.d (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:	$18.5856016518$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 7^{4} $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.8
Root			$1.85474i$ of defining polynomial
Character	$\chi$	$=$	315.64
Dual form			315.4.d.c.64.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.85474i q^{2} -0.149548 q^{4} +(-3.91321 - 10.4731i) q^{5} -7.00000i q^{7} +22.4110i q^{8} +(29.8981 - 11.1712i) q^{10} -37.4408 q^{11} +3.96370i q^{13} +19.9832 q^{14} -65.1740 q^{16} +51.6780i q^{17} -25.9323 q^{19} +(0.585214 + 1.56624i) q^{20} -106.884i q^{22} +173.454i q^{23} +(-94.3736 + 81.9673i) q^{25} -11.3154 q^{26} +1.04684i q^{28} -245.676 q^{29} -172.074 q^{31} -6.76690i q^{32} -147.527 q^{34} +(-73.3120 + 27.3925i) q^{35} +250.699i q^{37} -74.0300i q^{38} +(234.714 - 87.6990i) q^{40} +48.8649 q^{41} -143.612i q^{43} +5.59920 q^{44} -495.167 q^{46} -36.6415i q^{47} -49.0000 q^{49} +(-233.995 - 269.412i) q^{50} -0.592765i q^{52} -645.286i q^{53} +(146.514 + 392.123i) q^{55} +156.877 q^{56} -701.343i q^{58} +395.495 q^{59} +47.5130 q^{61} -491.228i q^{62} -502.074 q^{64} +(41.5125 - 15.5108i) q^{65} -263.189i q^{67} -7.72835i q^{68} +(-78.1984 - 209.287i) q^{70} +268.177 q^{71} +199.757i q^{73} -715.680 q^{74} +3.87813 q^{76} +262.085i q^{77} -473.640 q^{79} +(255.040 + 682.577i) q^{80} +139.497i q^{82} +72.7028i q^{83} +(541.231 - 202.227i) q^{85} +409.975 q^{86} -839.086i q^{88} -1552.25 q^{89} +27.7459 q^{91} -25.9398i q^{92} +104.602 q^{94} +(101.478 + 271.593i) q^{95} -243.338i q^{97} -139.882i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 36 q^{4} - 6 q^{5} - 16 q^{10} - 84 q^{11} + 56 q^{14} + 148 q^{16} + 72 q^{19} + 68 q^{20} - 362 q^{25} + 620 q^{26} - 88 q^{29} + 120 q^{31} + 964 q^{34} + 28 q^{35} + 1396 q^{40} + 852 q^{41}+ \cdots + 1628 q^{95}+O(q^{100}) $ 10 * q - 36 * q^4 - 6 * q^5 - 16 * q^10 - 84 * q^11 + 56 * q^14 + 148 * q^16 + 72 * q^19 + 68 * q^20 - 362 * q^25 + 620 * q^26 - 88 * q^29 + 120 * q^31 + 964 * q^34 + 28 * q^35 + 1396 * q^40 + 852 * q^41 + 1424 * q^44 + 176 * q^46 - 490 * q^49 - 1644 * q^50 - 1136 * q^55 - 588 * q^56 - 864 * q^59 - 884 * q^61 - 2724 * q^64 - 520 * q^65 + 756 * q^70 - 880 * q^71 - 3024 * q^74 - 2672 * q^76 - 3580 * q^79 + 2316 * q^80 + 1572 * q^85 - 2432 * q^86 + 1492 * q^89 + 476 * q^91 + 1652 * q^94 + 1628 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/315\mathbb{Z}\right)^\times$.

$n$	$127$	$136$	$281$
$\chi(n)$	$-1$	$1$	$1$

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	$0.831621\pi$
$3$		0			0
$3$		0			0
$4$	−0.149548			−0.0186935
$5$	−3.91321	−	10.4731i	−0.350008	−	0.936747i
$5$	−3.91321	−	10.4731i	−0.350008	−	0.936747i
$6$		0			0
$7$		−	7.00000i		−	0.377964i
$7$		−	7.00000i		−	0.377964i
$8$			22.4110i			0.990436i
$9$		0			0
$10$	29.8981	−	11.1712i	0.945462	−	0.353265i
$11$	−37.4408			−1.02626			−0.513128	−	0.858312i	$-0.671513\pi$
$11$	−37.4408			−1.02626			−0.513128	+	0.858312i	$0.671513\pi$
$12$		0			0
$13$			3.96370i			0.0845641i	0.999106	+	0.0422821i	$0.0134628\pi$
$13$			3.96370i			0.0845641i	−0.999106	+	0.0422821i	$0.986537\pi$
$14$	19.9832			0.381481
$15$		0			0
$16$	−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	$0.879824\pi$
$18$		0			0
$19$	−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
$21$		0			0
$22$		−	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	$0.712017\pi$
$24$		0			0
$25$	−94.3736	+	81.9673i	−0.754988	+	0.655738i
$26$	−11.3154			−0.0853509
$27$		0			0
$28$			1.04684i			0.00706549i
$29$	−245.676			−1.57314			−0.786568	−	0.617503i	$-0.788144\pi$
$29$	−245.676			−1.57314			−0.786568	+	0.617503i	$0.788144\pi$
$30$		0			0
$31$	−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
$33$		0			0
$34$	−147.527			−0.744139
$35$	−73.3120	+	27.3925i	−0.354057	+	0.132291i
$36$		0			0
$37$			250.699i			1.11391i	0.830543	+	0.556954i	$0.188030\pi$
$37$			250.699i			1.11391i	−0.830543	+	0.556954i	$0.811970\pi$
$38$		−	74.0300i		−	0.316033i
$39$		0			0
$40$	234.714	−	87.6990i	0.927788	−	0.346661i
$41$	48.8649			0.186132			0.0930661	−	0.995660i	$-0.470333\pi$
$41$	48.8649			0.186132			0.0930661	+	0.995660i	$0.470333\pi$
$42$		0			0
$43$		−	143.612i		−	0.509317i	−0.967031	−	0.254658i	$-0.918037\pi$
$43$		−	143.612i		−	0.509317i	0.967031	−	0.254658i	$-0.0819630\pi$
$44$	5.59920			0.0191844
$45$		0			0
$46$	−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	$-0.0181085\pi$
$48$		0			0
$49$	−49.0000			−0.142857
$50$	−233.995	−	269.412i	−0.661839	−	0.762012i
$51$		0			0
$52$		−	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	$-0.315226\pi$
$54$		0			0
$55$	146.514	+	392.123i	0.359198	+	0.961342i
$56$	156.877			0.374350
$57$		0			0
$58$		−	701.343i		−	1.58777i
$59$	395.495			0.872696			0.436348	−	0.899778i	$-0.356272\pi$
$59$	395.495			0.872696			0.436348	+	0.899778i	$0.356272\pi$
$60$		0			0
$61$	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
$63$		0			0
$64$	−502.074			−0.980614
$65$	41.5125	−	15.5108i	0.0792152	−	0.0295981i
$66$		0			0
$67$		−	263.189i		−	0.479906i	−0.970785	−	0.239953i	$-0.922868\pi$
$67$		−	263.189i		−	0.479906i	0.970785	−	0.239953i	$-0.0771320\pi$
$68$		−	7.72835i		−	0.0137824i
$69$		0			0
$70$	−78.1984	−	209.287i	−0.133521	−	0.357351i
$71$	268.177			0.448264			0.224132	−	0.974559i	$-0.428045\pi$
$71$	268.177			0.448264			0.224132	+	0.974559i	$0.428045\pi$
$72$		0			0
$73$			199.757i			0.320271i	0.987095	+	0.160136i	$0.0511931\pi$
$73$			199.757i			0.320271i	−0.987095	+	0.160136i	$0.948807\pi$
$74$	−715.680			−1.12427
$75$		0			0
$76$	3.87813			0.00585331
$77$			262.085i			0.387888i
$78$		0			0
$79$	−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
$81$		0			0
$82$			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	$0.984692\pi$
$84$		0			0
$85$	541.231	−	202.227i	0.690644	−	0.258054i
$86$	409.975			0.514055
$87$		0			0
$88$		−	839.086i		−	1.01644i
$89$	−1552.25			−1.84874			−0.924369	−	0.381500i	$-0.875408\pi$
$89$	−1552.25			−1.84874			−0.924369	+	0.381500i	$0.875408\pi$
$90$		0			0
$91$	27.7459			0.0319622
$92$		−	25.9398i		−	0.0293958i
$93$		0			0
$94$	104.602			0.114775
$95$	101.478	+	271.593i	0.109594	+	0.293314i
$96$		0			0
$97$		−	243.338i		−	0.254714i	−0.991857	−	0.127357i	$-0.959351\pi$
$97$		−	243.338i		−	0.254714i	0.991857	−	0.127357i	$-0.0406494\pi$
$98$		−	139.882i		−	0.144186i
$99$		0			0
$100$	14.1134	−	12.2581i	0.0141134	−	0.0122581i
$101$	1539.34			1.51653			0.758265	−	0.651946i	$-0.226047\pi$
$101$	1539.34			1.51653			0.758265	+	0.651946i	$0.226047\pi$
$102$		0			0
$103$			948.628i			0.907486i	0.891133	+	0.453743i	$0.149912\pi$
$103$			948.628i			0.907486i	−0.891133	+	0.453743i	$0.850088\pi$
$104$	−88.8306			−0.0837554
$105$		0			0
$106$	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	$0.872371\pi$
$108$		0			0
$109$	−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
$111$		0			0
$112$			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	$-0.103289\pi$
$114$		0			0
$115$	1816.61	−	678.763i	1.47304	−	0.550391i
$116$	36.7405			0.0294075
$117$		0			0
$118$			1129.04i			0.880816i
$119$	361.746			0.278665
$120$		0			0
$121$	70.8116			0.0532018
$122$			135.637i			0.100656i
$123$		0			0
$124$	25.7334			0.0186365
$125$	1227.76	+	667.633i	0.878513	+	0.477719i
$126$		0			0
$127$		−	505.042i		−	0.352876i	−0.984312	−	0.176438i	$-0.943543\pi$
$127$		−	505.042i		−	0.352876i	0.984312	−	0.176438i	$-0.0564575\pi$
$128$		−	1487.43i		−	1.02712i
$129$		0			0
$130$	44.2794	+	118.507i	0.0298735	+	0.0799521i
$131$	−672.930			−0.448811			−0.224405	−	0.974496i	$-0.572044\pi$
$131$	−672.930			−0.448811			−0.224405	+	0.974496i	$0.572044\pi$
$132$		0			0
$133$			181.526i			0.118348i
$134$	751.337			0.484371
$135$		0			0
$136$	−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	$0.839178\pi$
$138$		0			0
$139$	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
$141$		0			0
$142$			765.575i			0.452434i
$143$		−	148.404i		−	0.0867845i
$144$		0			0
$145$	961.384	+	2573.00i	0.550611	+	1.47363i
$146$	−570.255			−0.323251
$147$		0			0
$148$		−	37.4915i		−	0.0208229i
$149$	645.936			0.355149			0.177574	−	0.984107i	$-0.443175\pi$
$149$	645.936			0.355149			0.177574	+	0.984107i	$0.443175\pi$
$150$		0			0
$151$	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
$153$		0			0
$154$	−748.186			−0.391497
$155$	673.363	+	1802.16i	0.348941	+	0.933890i
$156$		0			0
$157$			1552.56i			0.789223i	0.918848	+	0.394611i	$0.129121\pi$
$157$			1552.56i			0.789223i	−0.918848	+	0.394611i	$0.870879\pi$
$158$		−	1352.12i		−	0.680815i
$159$		0			0
$160$	−70.8707	+	26.4803i	−0.0350176	+	0.0130841i
$161$	1214.18			0.594353
$162$		0			0
$163$			2553.65i			1.22710i	0.789656	+	0.613550i	$0.210259\pi$
$163$			2553.65i			1.22710i	−0.789656	+	0.613550i	$0.789741\pi$
$164$	−7.30766			−0.00347947
$165$		0			0
$166$	−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	$0.689570\pi$
$168$		0			0
$169$	2181.29			0.992849
$170$	577.305	+	1545.07i	0.260455	+	0.697069i
$171$		0			0
$172$			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	$-0.163385\pi$
$174$		0			0
$175$	573.771	+	660.615i	0.247846	+	0.285359i
$176$	2440.17			1.04508
$177$		0			0
$178$		−	4431.26i		−	1.86594i
$179$	−1830.53			−0.764361			−0.382180	−	0.924088i	$-0.624827\pi$
$179$	−1830.53			−0.764361			−0.382180	+	0.924088i	$0.624827\pi$
$180$		0			0
$181$	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
$183$		0			0
$184$	−3887.29			−1.55747
$185$	2625.60	−	981.037i	1.04345	−	0.389877i
$186$		0			0
$187$		−	1934.86i		−	0.756638i
$188$			5.47968i			0.00212578i
$189$		0			0
$190$	−775.327	+	289.695i	−0.296043	+	0.110614i
$191$	−5079.50			−1.92429			−0.962145	−	0.272538i	$-0.912137\pi$
$191$	−5079.50			−1.92429			−0.962145	+	0.272538i	$0.912137\pi$
$192$		0			0
$193$		−	2805.09i		−	1.04619i	−0.852274	−	0.523095i	$-0.824777\pi$
$193$		−	2805.09i		−	1.04619i	0.852274	−	0.523095i	$-0.175223\pi$
$194$	694.667			0.257084
$195$		0			0
$196$	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	$-0.189961\pi$
$198$		0			0
$199$	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
$201$		0			0
$202$			4394.41i			1.53064i
$203$			1719.73i			0.594590i
$204$		0			0
$205$	−191.219	−	511.769i	−0.0651478	−	0.174359i
$206$	−2708.09			−0.915929
$207$		0			0
$208$		−	258.331i		−	0.0861154i
$209$	970.925			0.321341
$210$		0			0
$211$	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
$213$		0			0
$214$	−2466.45			−0.787864
$215$	−1504.07	+	561.984i	−0.477101	+	0.178265i
$216$		0			0
$217$			1204.52i			0.376812i
$218$		−	2530.21i		−	0.786089i
$219$		0			0
$220$	−21.9109	−	58.6413i	−0.00671468	−	0.0179709i
$221$	−204.836			−0.0623474
$222$		0			0
$223$		−	4741.40i		−	1.42380i	−0.702280	−	0.711901i	$-0.747834\pi$
$223$		−	4741.40i		−	1.42380i	0.702280	−	0.711901i	$-0.252166\pi$
$224$	−47.3683			−0.0141291
$225$		0			0
$226$	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	$-0.0448589\pi$
$228$		0			0
$229$	−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
$231$		0			0
$232$		−	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	$-0.0699480\pi$
$234$		0			0
$235$	−383.752	+	143.386i	−0.106524	+	0.0398020i
$236$	−59.1456			−0.0163138
$237$		0			0
$238$			1032.69i			0.281258i
$239$	2775.00			0.751045			0.375523	−	0.926813i	$-0.377463\pi$
$239$	2775.00			0.751045			0.375523	+	0.926813i	$0.377463\pi$
$240$		0			0
$241$	−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
$243$		0			0
$244$	−7.10549			−0.00186427
$245$	191.747	+	513.184i	0.0500012	+	0.133821i
$246$		0			0
$247$		−	102.788i		−	0.0264787i
$248$		−	3856.36i		−	0.987416i
$249$		0			0
$250$	−1905.92	+	3504.93i	−0.482164	+	0.886686i
$251$	2933.00			0.737568			0.368784	−	0.929515i	$-0.379774\pi$
$251$	2933.00			0.737568			0.368784	+	0.929515i	$0.379774\pi$
$252$		0			0
$253$		−	6494.26i		−	1.61380i
$254$	1441.76			0.356159
$255$		0			0
$256$	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	$0.892705\pi$
$258$		0			0
$259$	1754.89			0.421018
$260$	−6.20811	+	2.31961i	−0.00148081	+	0.000553294i
$261$		0			0
$262$		−	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	$-0.115480\pi$
$264$		0			0
$265$	−6758.17	+	2525.14i	−1.56661	+	0.585351i
$266$	−518.210			−0.119449
$267$		0			0
$268$			39.3595i			0.00897114i
$269$	−1442.46			−0.326946			−0.163473	−	0.986548i	$-0.552270\pi$
$269$	−1442.46			−0.326946			−0.163473	+	0.986548i	$0.552270\pi$
$270$		0			0
$271$	−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
$273$		0			0
$274$	−4431.36			−0.977038
$275$	3533.42	−	3068.92i	0.774812	−	0.672955i
$276$		0			0
$277$			876.614i			0.190147i	0.995470	+	0.0950733i	$0.0303086\pi$
$277$			876.614i			0.190147i	−0.995470	+	0.0950733i	$0.969691\pi$
$278$			3060.33i			0.660240i
$279$		0			0
$280$	−613.893	−	1643.00i	−0.131025	−	0.350671i
$281$	−6252.19			−1.32731			−0.663655	−	0.748038i	$-0.730996\pi$
$281$	−6252.19			−1.32731			−0.663655	+	0.748038i	$0.730996\pi$
$282$		0			0
$283$		−	2250.07i		−	0.472625i	−0.971677	−	0.236312i	$-0.924061\pi$
$283$		−	2250.07i		−	0.472625i	0.971677	−	0.236312i	$-0.0759389\pi$
$284$	−40.1054			−0.00837963
$285$		0			0
$286$	423.655			0.0875919
$287$		−	342.054i		−	0.0703513i
$288$		0			0
$289$	2242.39			0.456419
$290$	−7345.26	+	2744.50i	−1.48734	+	0.555733i
$291$		0			0
$292$		−	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	$-0.200862\pi$
$294$		0			0
$295$	−1547.66	−	4142.08i	−0.305451	−	0.817495i
$296$	−5618.41			−1.10325
$297$		0			0
$298$			1843.98i			0.358453i
$299$	−687.522			−0.132978
$300$		0			0
$301$	−1005.28			−0.192504
$302$			694.256i			0.132284i
$303$		0			0
$304$	1690.11			0.318864
$305$	−185.929	−	497.611i	−0.0349057	−	0.0934200i
$306$		0			0
$307$			9458.47i			1.75838i	0.476469	+	0.879191i	$0.341916\pi$
$307$			9458.47i			1.75838i	−0.476469	+	0.879191i	$0.658084\pi$
$308$		−	39.1944i		−	0.00725100i
$309$		0			0
$310$	−5144.70	+	1922.28i	−0.942579	+	0.352187i
$311$	7576.78			1.38148			0.690739	−	0.723104i	$-0.257285\pi$
$311$	7576.78			1.38148			0.690739	+	0.723104i	$0.257285\pi$
$312$		0			0
$313$			9172.41i			1.65641i	0.560427	+	0.828204i	$0.310637\pi$
$313$			9172.41i			1.65641i	−0.560427	+	0.828204i	$0.689363\pi$
$314$	−4432.16			−0.796565
$315$		0			0
$316$	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	$0.912092\pi$
$318$		0			0
$319$	9198.31			1.61444
$320$	1964.72	+	5258.30i	0.343223	+	0.918587i
$321$		0			0
$322$			3466.17i			0.599882i
$323$		−	1340.13i		−	0.230857i
$324$		0			0
$325$	−324.894	−	374.069i	−0.0554519	−	0.0638449i
$326$	−7290.01			−1.23852
$327$		0			0
$328$			1095.11i			0.184352i
$329$	−256.491			−0.0429811
$330$		0			0
$331$	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
$333$		0			0
$334$	−10200.4			−1.67108
$335$	−2756.42	+	1029.92i	−0.449550	+	0.167971i
$336$		0			0
$337$		−	3777.84i		−	0.610658i	−0.952247	−	0.305329i	$-0.901234\pi$
$337$		−	3777.84i		−	0.610658i	0.952247	−	0.305329i	$-0.0987665\pi$
$338$			6227.02i			1.00209i
$339$		0			0
$340$	−80.9401	+	30.2427i	−0.0129106	+	0.00482394i
$341$	6442.60			1.02313
$342$		0			0
$343$			343.000i			0.0539949i
$344$	3218.49			0.504446
$345$		0			0
$346$	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	$0.779885\pi$
$348$		0			0
$349$	−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
$351$		0			0
$352$			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	$-0.102338\pi$
$354$		0			0
$355$	−1049.43	−	2808.65i	−0.156896	−	0.419910i
$356$	232.136			0.0345594
$357$		0			0
$358$		−	5225.70i		−	0.771472i
$359$	−3136.29			−0.461078			−0.230539	−	0.973063i	$-0.574049\pi$
$359$	−3136.29			−0.461078			−0.230539	+	0.973063i	$0.574049\pi$
$360$		0			0
$361$	−6186.52			−0.901956
$362$			6957.62i			1.01018i
$363$		0			0
$364$	−4.14936			−0.000597487
$365$	2092.08	−	781.691i	0.300013	−	0.112098i
$366$		0			0
$367$			1723.30i			0.245110i	0.992462	+	0.122555i	$0.0391088\pi$
$367$			1723.30i			0.245110i	−0.992462	+	0.122555i	$0.960891\pi$
$368$		−	11304.7i		−	1.60136i
$369$		0			0
$370$	2800.61	+	7495.42i	0.393504	+	1.05316i
$371$	−4517.00			−0.632105
$372$		0			0
$373$			2818.55i			0.391258i	0.980678	+	0.195629i	$0.0626748\pi$
$373$			2818.55i			0.391258i	−0.980678	+	0.195629i	$0.937325\pi$
$374$	5523.53			0.763677
$375$		0			0
$376$	821.174			0.112630
$377$		−	973.788i		−	0.133031i
$378$		0			0
$379$	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
$381$		0			0
$382$		−	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	$0.994520\pi$
$384$		0			0
$385$	2744.86	−	1025.60i	0.363353	−	0.135764i
$386$	8007.81			1.05592
$387$		0			0
$388$			36.3908i			0.00476150i
$389$	4573.87			0.596156			0.298078	−	0.954542i	$-0.403654\pi$
$389$	4573.87			0.596156			0.298078	+	0.954542i	$0.403654\pi$
$390$		0			0
$391$	−8963.77			−1.15938
$392$		−	1098.14i		−	0.141491i
$393$		0			0
$394$	8871.94			1.13442
$395$	1853.45	+	4960.50i	0.236094	+	0.631873i
$396$		0			0
$397$		−	3624.55i		−	0.458215i	−0.973401	−	0.229107i	$-0.926419\pi$
$397$		−	3624.55i		−	0.458215i	0.973401	−	0.229107i	$-0.0735807\pi$
$398$			6125.23i			0.771432i
$399$		0			0
$400$	6150.70	−	5342.14i	0.768838	−	0.667767i
$401$	−6358.32			−0.791819			−0.395910	−	0.918289i	$-0.629571\pi$
$401$	−6358.32			−0.791819			−0.395910	+	0.918289i	$0.629571\pi$
$402$		0			0
$403$		−	682.052i		−	0.0843063i
$404$	−230.205			−0.0283493
$405$		0			0
$406$	−4909.40			−0.600121
$407$		−	9386.35i		−	1.14316i
$408$		0			0
$409$	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
$411$		0			0
$412$		−	141.866i		−	0.0169641i
$413$		−	2768.47i		−	0.329848i
$414$		0			0
$415$	761.427	−	284.501i	0.0900650	−	0.0336521i
$416$	26.8220			0.00316119
$417$		0			0
$418$			2771.74i			0.324331i
$419$	6333.56			0.738460			0.369230	−	0.929338i	$-0.379621\pi$
$419$	6333.56			0.738460			0.369230	+	0.929338i	$0.379621\pi$
$420$		0			0
$421$	−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
$423$		0			0
$424$	14461.5			1.65640
$425$	−4235.90	−	4877.03i	−0.483462	−	0.556637i
$426$		0			0
$427$		−	332.591i		−	0.0376937i
$428$		−	129.207i		−	0.0145922i
$429$		0			0
$430$	−1604.32	−	4293.73i	−0.179924	−	0.481539i
$431$	14367.6			1.60571			0.802856	−	0.596173i	$-0.203313\pi$
$431$	14367.6			1.60571			0.802856	+	0.596173i	$0.203313\pi$
$432$		0			0
$433$		−	8399.05i		−	0.932176i	−0.884738	−	0.466088i	$-0.845663\pi$
$433$		−	8399.05i		−	0.932176i	0.884738	−	0.466088i	$-0.154337\pi$
$434$	−3438.59			−0.380318
$435$		0			0
$436$	132.547			0.0145593
$437$		−	4498.07i		−	0.492384i
$438$		0			0
$439$	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
$441$		0			0
$442$		−	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	$-0.0325148\pi$
$444$		0			0
$445$	6074.26	+	16256.9i	0.647074	+	1.73180i
$446$	13535.5			1.43705
$447$		0			0
$448$			3514.52i			0.370637i
$449$	−5185.68			−0.545050			−0.272525	−	0.962149i	$-0.587859\pi$
$449$	−5185.68			−0.545050			−0.272525	+	0.962149i	$0.587859\pi$
$450$		0			0
$451$	−1829.54			−0.191019
$452$			114.548i			0.0119201i
$453$		0			0
$454$	2742.81			0.283538
$455$	−108.576	−	290.587i	−0.0111870	−	0.0299405i
$456$		0			0
$457$			11198.8i			1.14630i	0.819451	+	0.573149i	$0.194278\pi$
$457$			11198.8i			1.14630i	−0.819451	+	0.573149i	$0.805722\pi$
$458$		−	2123.94i		−	0.216693i
$459$		0			0
$460$	−271.671	+	101.508i	−0.0275364	+	0.0102888i
$461$	−17270.7			−1.74485			−0.872427	−	0.488744i	$-0.837455\pi$
$461$	−17270.7			−1.74485			−0.872427	+	0.488744i	$0.837455\pi$
$462$		0			0
$463$			385.660i			0.0387109i	0.999813	+	0.0193554i	$0.00616141\pi$
$463$			385.660i			0.0387109i	−0.999813	+	0.0193554i	$0.993839\pi$
$464$	16011.7			1.60199
$465$		0			0
$466$	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	$0.919742\pi$
$468$		0			0
$469$	−1842.33			−0.181387
$470$	−409.330	−	1095.51i	−0.0401723	−	0.107515i
$471$		0			0
$472$			8863.45i			0.864350i
$473$			5376.94i			0.522689i
$474$		0			0
$475$	2447.32	−	2125.60i	0.236402	−	0.205324i
$476$	−54.0985			−0.00520924
$477$		0			0
$478$			7921.91i			0.758033i
$479$	−8681.99			−0.828163			−0.414082	−	0.910240i	$-0.635897\pi$
$479$	−8681.99			−0.828163			−0.414082	+	0.910240i	$0.635897\pi$
$480$		0			0
$481$	−993.695			−0.0941967
$482$		−	7280.16i		−	0.687972i
$483$		0			0
$484$	−10.5898			−0.000994530
$485$	−2548.51	+	952.233i	−0.238602	+	0.0891519i
$486$		0			0
$487$		−	890.476i		−	0.0828569i	−0.999141	−	0.0414284i	$-0.986809\pi$
$487$		−	890.476i		−	0.0828569i	0.999141	−	0.0414284i	$-0.0131909\pi$
$488$			1064.81i			0.0987744i
$489$		0			0
$490$	−1465.01	+	547.389i	−0.135066	+	0.0504664i
$491$	−1562.48			−0.143613			−0.0718063	−	0.997419i	$-0.522876\pi$
$491$	−1562.48			−0.143613			−0.0718063	+	0.997419i	$0.522876\pi$
$492$		0			0
$493$		−	12696.1i		−	1.15984i
$494$	293.433			0.0267250
$495$		0			0
$496$	11214.8			1.01524
$497$		−	1877.24i		−	0.169428i
$498$		0			0
$499$	−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
$501$		0			0
$502$			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	$-0.00102332\pi$
$504$		0			0
$505$	−6023.75	−	16121.7i	−0.530798	−	1.42061i
$506$	18539.4			1.62881
$507$		0			0
$508$			75.5281i			0.00659649i
$509$	7793.44			0.678660			0.339330	−	0.940667i	$-0.389800\pi$
$509$	7793.44			0.678660			0.339330	+	0.940667i	$0.389800\pi$
$510$		0			0
$511$	1398.30			0.121051
$512$		−	11243.9i		−	0.970537i
$513$		0			0
$514$	−7779.81			−0.667612
$515$	9935.12	−	3712.18i	0.850084	−	0.317628i
$516$		0			0
$517$			1371.89i			0.116703i
$518$			5009.76i			0.424935i
$519$		0			0
$520$	347.613	+	930.336i	0.0293151	+	0.0784576i
$521$	−4645.42			−0.390633			−0.195316	−	0.980740i	$-0.562573\pi$
$521$	−4645.42			−0.390633			−0.195316	+	0.980740i	$0.562573\pi$
$522$		0			0
$523$			8783.88i			0.734402i	0.930142	+	0.367201i	$0.119684\pi$
$523$			8783.88i			0.734402i	−0.930142	+	0.367201i	$0.880316\pi$
$524$	100.636			0.00838986
$525$		0			0
$526$	8642.04			0.716371
$527$		−	8892.46i		−	0.735031i
$528$		0			0
$529$	−17919.4			−1.47279
$530$	−7208.62	−	19292.8i	−0.590797	−	1.58118i
$531$		0			0
$532$		−	27.1469i		−	0.00221234i
$533$			193.686i			0.0157401i
$534$		0			0
$535$	9048.62	−	3380.95i	0.731226	−	0.273217i
$536$	5898.34			0.475316
$537$		0			0
$538$		−	4117.86i		−	0.329988i
$539$	1834.60			0.146608
$540$		0			0
$541$	−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
$543$		0			0
$544$	349.700			0.0275611
$545$	3468.35	+	9282.55i	0.272602	+	0.729579i
$546$		0			0
$547$			5776.83i			0.451553i	0.974179	+	0.225776i	$0.0724919\pi$
$547$			5776.83i			0.451553i	−0.974179	+	0.225776i	$0.927508\pi$
$548$		−	232.141i		−	0.0180959i
$549$		0			0
$550$	8760.97	+	10087.0i	0.679216	+	0.782020i
$551$	6370.95			0.492580
$552$		0			0
$553$			3315.48i			0.254952i
$554$	−2502.51			−0.191916
$555$		0			0
$556$	−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	$0.714146\pi$
$558$		0			0
$559$	569.235			0.0430699
$560$	4778.04	−	1785.28i	0.360552	−	0.134717i
$561$		0			0
$562$		−	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	$0.644549\pi$
$564$		0			0
$565$	−8021.98	+	2997.35i	−0.597323	+	0.223185i
$566$	6423.37			0.477022
$567$		0			0
$568$			6010.11i			0.443977i
$569$	−24157.5			−1.77985			−0.889925	−	0.456107i	$-0.849243\pi$
$569$	−24157.5			−1.77985			−0.889925	+	0.456107i	$0.849243\pi$
$570$		0			0
$571$	−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
$573$		0			0
$574$	976.477			0.0710059
$575$	−14217.6	−	16369.5i	−1.03115	−	1.18723i
$576$		0			0
$577$		−	16057.2i		−	1.15853i	−0.815139	−	0.579265i	$-0.803340\pi$
$577$		−	16057.2i		−	1.15853i	0.815139	−	0.579265i	$-0.196660\pi$
$578$			6401.44i			0.460665i
$579$		0			0
$580$	−143.773	−	384.788i	−0.0102929	−	0.0275474i
$581$	508.919			0.0363400
$582$		0			0
$583$			24160.0i			1.71630i
$584$	−4476.76			−0.317208
$585$		0			0
$586$	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	$0.902162\pi$
$588$		0			0
$589$	4462.28			0.312165
$590$	11824.6	−	4418.16i	0.825101	−	0.308293i
$591$		0			0
$592$		−	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	$0.751033\pi$
$594$		0			0
$595$	−1415.59	−	3788.62i	−0.0975352	−	0.261039i
$596$	−96.5986			−0.00663898
$597$		0			0
$598$		−	1962.70i		−	0.134215i
$599$	22635.7			1.54402			0.772010	−	0.635610i	$-0.219251\pi$
$599$	22635.7			1.54402			0.772010	+	0.635610i	$0.219251\pi$
$600$		0			0
$601$	−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
$603$		0			0
$604$	−36.3692			−0.00245007
$605$	−277.101	−	741.620i	−0.0186211	−	0.0498366i
$606$		0			0
$607$			17534.2i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$607$			17534.2i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$608$			175.481i			0.0117051i
$609$		0			0
$610$	1420.55	−	530.778i	0.0942892	−	0.0352304i
$611$	145.236			0.00961641
$612$		0			0
$613$		−	10445.5i		−	0.688238i	−0.938926	−	0.344119i	$-0.888178\pi$
$613$		−	10445.5i		−	0.688238i	0.938926	−	0.344119i	$-0.111822\pi$
$614$	−27001.5			−1.77474
$615$		0			0
$616$	−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	$0.858070\pi$
$618$		0			0
$619$	−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
$621$		0			0
$622$			21629.8i			1.39433i
$623$			10865.7i			0.698757i
$624$		0			0
$625$	2187.74	−	15471.1i	0.140015	−	0.990149i
$626$	−26184.9			−1.67182
$627$		0			0
$628$		−	232.183i		−	0.0147534i
$629$	−12955.6			−0.821262
$630$		0			0
$631$	−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
$633$		0			0
$634$	−8786.72			−0.550419
$635$	−5289.38	+	1976.34i	−0.330555	+	0.123509i
$636$		0			0
$637$		−	194.222i		−	0.0120806i
$638$			26258.8i			1.62946i
$639$		0			0
$640$	−15578.0	+	5820.62i	−0.962151	+	0.359500i
$641$	−23977.0			−1.47743			−0.738715	−	0.674017i	$-0.764567\pi$
$641$	−23977.0			−1.47743			−0.738715	+	0.674017i	$0.764567\pi$
$642$		0			0
$643$		−	27698.0i		−	1.69876i	−0.527782	−	0.849380i	$-0.676976\pi$
$643$		−	27698.0i		−	1.69876i	0.527782	−	0.849380i	$-0.323024\pi$
$644$	−181.579			−0.0111106
$645$		0			0
$646$	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	$-0.108109\pi$
$648$		0			0
$649$	−14807.6			−0.895610
$650$	1067.87	−	927.488i	0.0644389	−	0.0559678i
$651$		0			0
$652$		−	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	$0.879480\pi$
$654$		0			0
$655$	2633.32	+	7047.70i	0.157087	+	0.420422i
$656$	−3184.72			−0.189547
$657$		0			0
$658$		−	732.215i		−	0.0433810i
$659$	−25275.6			−1.49408			−0.747040	−	0.664779i	$-0.768526\pi$
$659$	−25275.6			−1.49408			−0.747040	+	0.664779i	$0.768526\pi$
$660$		0			0
$661$	−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
$663$		0			0
$664$	−1629.34			−0.0952270
$665$	1901.15	−	710.349i	0.110862	−	0.0414228i
$666$		0			0
$667$		−	42613.6i		−	2.47377i
$668$		−	534.357i		−	0.0309505i
$669$		0			0
$670$	−2940.14	−	7868.87i	−0.169534	−	0.453733i
$671$	−1778.92			−0.102347
$672$		0			0
$673$			30358.9i			1.73885i	0.494061	+	0.869427i	$0.335512\pi$
$673$			30358.9i			1.73885i	−0.494061	+	0.869427i	$0.664488\pi$
$674$	10784.7			0.616340
$675$		0			0
$676$	−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	$0.937100\pi$
$678$		0			0
$679$	−1703.37			−0.0962728
$680$	4532.11	+	12129.5i	0.255586	+	0.684039i
$681$		0			0
$682$			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	$-0.0405249\pi$
$684$		0			0
$685$	16257.3	−	6074.40i	0.906800	−	0.338819i
$686$	−979.176			−0.0544973
$687$		0			0
$688$			9359.77i			0.518660i
$689$	2557.72			0.141424
$690$		0			0
$691$	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
$693$		0			0
$694$	−23534.7			−1.28727
$695$	−4195.03	−	11227.4i	−0.228959	−	0.612776i
$696$		0			0
$697$			2525.24i			0.137231i
$698$		−	20479.3i		−	1.11053i
$699$		0			0
$700$	−85.8064	−	98.7938i	−0.00463311	−	0.00533436i
$701$	17144.3			0.923726			0.461863	−	0.886951i	$-0.347181\pi$
$701$	17144.3			0.923726			0.461863	+	0.886951i	$0.347181\pi$
$702$		0			0
$703$		−	6501.19i		−	0.348787i
$704$	18798.1			1.00636
$705$		0			0
$706$	11965.7			0.637867
$707$		−	10775.3i		−	0.573195i
$708$		0			0
$709$	−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
$711$		0			0
$712$		−	34787.4i		−	1.83106i
$713$		−	29847.0i		−	1.56771i
$714$		0			0
$715$	−1554.26	+	580.737i	−0.0812951	+	0.0303753i
$716$	273.753			0.0142886
$717$		0			0
$718$		−	8953.30i		−	0.465368i
$719$	4308.66			0.223485			0.111743	−	0.993737i	$-0.464357\pi$
$719$	4308.66			0.223485			0.111743	+	0.993737i	$0.464357\pi$
$720$		0			0
$721$	6640.39			0.342997
$722$		−	17660.9i		−	0.910347i
$723$		0			0
$724$	−364.481			−0.0187097
$725$	23185.4	−	20137.4i	1.18770	−	1.03157i
$726$		0			0
$727$			29435.6i			1.50166i	0.660496	+	0.750830i	$0.270346\pi$
$727$			29435.6i			1.50166i	−0.660496	+	0.750830i	$0.729654\pi$
$728$			621.814i			0.0316566i
$729$		0			0
$730$	2231.53	+	5972.36i	0.113140	+	0.302804i
$731$	7421.58			0.375509
$732$		0			0
$733$		−	6587.69i		−	0.331954i	−0.986130	−	0.165977i	$-0.946922\pi$
$733$		−	6587.69i		−	0.331954i	0.986130	−	0.165977i	$-0.0530777\pi$
$734$	−4919.57			−0.247390
$735$		0			0
$736$	1173.75			0.0587839
$737$			9854.01i			0.492506i
$738$		0			0
$739$	−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
$741$		0			0
$742$		−	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	$-0.0979783\pi$
$744$		0			0
$745$	−2527.68	−	6764.98i	−0.124305	−	0.332684i
$746$	−8046.24			−0.394898
$747$		0			0
$748$			289.355i			0.0141442i
$749$	6047.88			0.295040
$750$		0			0
$751$	−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
$753$		0			0
$754$	2779.91			0.134269
$755$	−951.669	−	2547.00i	−0.0458739	−	0.122775i
$756$		0			0
$757$		−	11442.1i		−	0.549368i	−0.961535	−	0.274684i	$-0.911427\pi$
$757$		−	11442.1i		−	0.549368i	0.961535	−	0.274684i	$-0.0885732\pi$
$758$			29878.1i			1.43169i
$759$		0			0
$760$	−6086.66	+	2274.24i	−0.290509	+	0.108546i
$761$	−14423.3			−0.687048			−0.343524	−	0.939144i	$-0.611621\pi$
$761$	−14423.3			−0.687048			−0.343524	+	0.939144i	$0.611621\pi$
$762$		0			0
$763$			6204.23i			0.294375i
$764$	759.630			0.0359718
$765$		0			0
$766$	−736.681			−0.0347485
$767$			1567.63i			0.0737988i
$768$		0			0
$769$	−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
$771$		0			0
$772$			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	$-0.222611\pi$
$774$		0			0
$775$	16239.3	−	14104.5i	0.752686	−	0.653739i
$776$	5453.45			0.252278
$777$		0			0
$778$			13057.2i			0.601702i
$779$	−1267.18			−0.0582816
$780$		0			0
$781$	−10040.7			−0.460034
$782$		−	25589.2i		−	1.17017i
$783$		0			0
$784$	3193.53			0.145478
$785$	16260.2	−	6075.51i	0.739302	−	0.276235i
$786$		0			0
$787$			10934.5i			0.495263i	0.968854	+	0.247632i	$0.0796523\pi$
$787$			10934.5i			0.495263i	−0.968854	+	0.247632i	$0.920348\pi$
$788$

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:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	315.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.85474i q^{2} -0.149548 q^{4} +(-3.91321 - 10.4731i) q^{5} -7.00000i q^{7} +22.4110i q^{8} +(29.8981 - 11.1712i) q^{10} -37.4408 q^{11} +3.96370i q^{13} +19.9832 q^{14} -65.1740 q^{16} +51.6780i q^{17} -25.9323 q^{19} +(0.585214 + 1.56624i) q^{20} -106.884i q^{22} +173.454i q^{23} +(-94.3736 + 81.9673i) q^{25} -11.3154 q^{26} +1.04684i q^{28} -245.676 q^{29} -172.074 q^{31} -6.76690i q^{32} -147.527 q^{34} +(-73.3120 + 27.3925i) q^{35} +250.699i q^{37} -74.0300i q^{38} +(234.714 - 87.6990i) q^{40} +48.8649 q^{41} -143.612i q^{43} +5.59920 q^{44} -495.167 q^{46} -36.6415i q^{47} -49.0000 q^{49} +(-233.995 - 269.412i) q^{50} -0.592765i q^{52} -645.286i q^{53} +(146.514 + 392.123i) q^{55} +156.877 q^{56} -701.343i q^{58} +395.495 q^{59} +47.5130 q^{61} -491.228i q^{62} -502.074 q^{64} +(41.5125 - 15.5108i) q^{65} -263.189i q^{67} -7.72835i q^{68} +(-78.1984 - 209.287i) q^{70} +268.177 q^{71} +199.757i q^{73} -715.680 q^{74} +3.87813 q^{76} +262.085i q^{77} -473.640 q^{79} +(255.040 + 682.577i) q^{80} +139.497i q^{82} +72.7028i q^{83} +(541.231 - 202.227i) q^{85} +409.975 q^{86} -839.086i q^{88} -1552.25 q^{89} +27.7459 q^{91} -25.9398i q^{92} +104.602 q^{94} +(101.478 + 271.593i) q^{95} -243.338i q^{97} -139.882i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 36 q^{4} - 6 q^{5} - 16 q^{10} - 84 q^{11} + 56 q^{14} + 148 q^{16} + 72 q^{19} + 68 q^{20} - 362 q^{25} + 620 q^{26} - 88 q^{29} + 120 q^{31} + 964 q^{34} + 28 q^{35} + 1396 q^{40} + 852 q^{41}+ \cdots + 1628 q^{95}+O(q^{100}) \) 10 * q - 36 * q^4 - 6 * q^5 - 16 * q^10 - 84 * q^11 + 56 * q^14 + 148 * q^16 + 72 * q^19 + 68 * q^20 - 362 * q^25 + 620 * q^26 - 88 * q^29 + 120 * q^31 + 964 * q^34 + 28 * q^35 + 1396 * q^40 + 852 * q^41 + 1424 * q^44 + 176 * q^46 - 490 * q^49 - 1644 * q^50 - 1136 * q^55 - 588 * q^56 - 864 * q^59 - 884 * q^61 - 2724 * q^64 - 520 * q^65 + 756 * q^70 - 880 * q^71 - 3024 * q^74 - 2672 * q^76 - 3580 * q^79 + 2316 * q^80 + 1572 * q^85 - 2432 * q^86 + 1492 * q^89 + 476 * q^91 + 1652 * q^94 + 1628 * q^95

Introduction

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