LMFDB - Embedded newform 45.8.b.c.19.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [45,8,Mod(19,45)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("45.19"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	45.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,-528] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$14.0573261468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-65}, \sqrt{85})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 143x^{2} + 1521 $ x^4 + 143*x^2 + 1521
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.1
Root			$-11.4642i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.8.b.c.19.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-16.1245i q^{2} -132.000 q^{4} +(-276.586 + 40.3113i) q^{5} +891.964i q^{7} +64.4981i q^{8} +(650.000 + 4459.82i) q^{10} +3872.21 q^{11} +9811.61i q^{13} +14382.5 q^{14} -15856.0 q^{16} +10916.3i q^{17} +14924.0 q^{19} +(36509.4 - 5321.09i) q^{20} -62437.5i q^{22} +225.743i q^{23} +(74875.0 - 22299.1i) q^{25} +158207. q^{26} -117739. i q^{28} +136634. q^{29} -244192. q^{31} +263926. i q^{32} +176020. q^{34} +(-35956.2 - 246705. i) q^{35} +511095. i q^{37} -240642. i q^{38} +(-2600.00 - 17839.3i) q^{40} -616234. q^{41} +656486. i q^{43} -511132. q^{44} +3640.00 q^{46} +1.32102e6i q^{47} +27943.0 q^{49} +(-359562. - 1.20732e6i) q^{50} -1.29513e6i q^{52} -1.10593e6i q^{53} +(-1.07100e6 + 156094. i) q^{55} -57530.0 q^{56} -2.20315e6i q^{58} -1.96210e6 q^{59} +1.76244e6 q^{61} +3.93748e6i q^{62} +2.22611e6 q^{64} +(-395518. - 2.71376e6i) q^{65} -993648. i q^{67} -1.44095e6i q^{68} +(-3.97800e6 + 579777. i) q^{70} -454708. q^{71} +4.05665e6i q^{73} +8.24117e6 q^{74} -1.96997e6 q^{76} +3.45387e6i q^{77} +2.16822e6 q^{79} +(4.38555e6 - 639176. i) q^{80} +9.93648e6i q^{82} +748371. i q^{83} +(-440050. - 3.01930e6i) q^{85} +1.05855e7 q^{86} +249750. i q^{88} -5.31046e6 q^{89} -8.75160e6 q^{91} -29798.1i q^{92} +2.13008e7 q^{94} +(-4.12777e6 + 601606. i) q^{95} -1.13048e7i q^{97} -450567. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 528 q^{4} + 2600 q^{10} - 63424 q^{16} + 59696 q^{19} + 299500 q^{25} - 976768 q^{31} + 704080 q^{34} - 10400 q^{40} + 14560 q^{46} + 111772 q^{49} - 4284000 q^{55} + 7049768 q^{61} + 8904448 q^{64}+ \cdots + 85203040 q^{94}+O(q^{100}) $ 4 * q - 528 * q^4 + 2600 * q^10 - 63424 * q^16 + 59696 * q^19 + 299500 * q^25 - 976768 * q^31 + 704080 * q^34 - 10400 * q^40 + 14560 * q^46 + 111772 * q^49 - 4284000 * q^55 + 7049768 * q^61 + 8904448 * q^64 - 15912000 * q^70 - 7879872 * q^76 + 8672864 * q^79 - 1760200 * q^85 - 35006400 * q^91 + 85203040 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	16.1245i		−	1.42522i	−0.701561	−	0.712610i	$-0.747513\pi$
$2$		−	16.1245i		−	1.42522i	0.701561	−	0.712610i	$-0.252487\pi$
$3$		0			0
$3$		0			0
$4$	−132.000			−1.03125
$5$	−276.586	+	40.3113i	−0.989545	+	0.144222i
$5$	−276.586	+	40.3113i	−0.989545	+	0.144222i
$6$		0			0
$7$			891.964i			0.982888i	0.870909	+	0.491444i	$0.163531\pi$
$7$			891.964i			0.982888i	−0.870909	+	0.491444i	$0.836469\pi$
$8$			64.4981i			0.0445381i
$9$		0			0
$10$	650.000	+	4459.82i	0.205548	+	1.41032i
$11$	3872.21			0.877171			0.438586	−	0.898689i	$-0.355480\pi$
$11$	3872.21			0.877171			0.438586	+	0.898689i	$0.355480\pi$
$12$		0			0
$13$			9811.61i			1.23862i	0.785146	+	0.619310i	$0.212588\pi$
$13$			9811.61i			1.23862i	−0.785146	+	0.619310i	$0.787412\pi$
$14$	14382.5			1.40083
$15$		0			0
$16$	−15856.0			−0.967773
$17$			10916.3i			0.538895i	0.963015	+	0.269447i	$0.0868411\pi$
$17$			10916.3i			0.538895i	−0.963015	+	0.269447i	$0.913159\pi$
$18$		0			0
$19$	14924.0			0.499169			0.249585	−	0.968353i	$-0.419706\pi$
$19$	14924.0			0.499169			0.249585	+	0.968353i	$0.419706\pi$
$20$	36509.4	−	5321.09i	1.02047	−	0.148729i
$21$		0			0
$22$		−	62437.5i		−	1.25016i
$23$			225.743i			0.00386872i	0.999998	+	0.00193436i	$0.000615726\pi$
$23$			225.743i			0.00386872i	−0.999998	+	0.00193436i	$0.999384\pi$
$24$		0			0
$25$	74875.0	−	22299.1i	0.958400	−	0.285429i
$26$	158207.			1.76531
$27$		0			0
$28$		−	117739.i		−	1.01360i
$29$	136634.			1.04031			0.520157	−	0.854070i	$-0.325873\pi$
$29$	136634.			1.04031			0.520157	+	0.854070i	$0.325873\pi$
$30$		0			0
$31$	−244192.			−1.47220			−0.736098	−	0.676875i	$-0.763334\pi$
$31$	−244192.			−1.47220			−0.736098	+	0.676875i	$0.763334\pi$
$32$			263926.i			1.42383i
$33$		0			0
$34$	176020.			0.768043
$35$	−35956.2	−	246705.i	−0.141754	−	0.972613i
$36$		0			0
$37$			511095.i			1.65881i	0.558650	+	0.829404i	$0.311320\pi$
$37$			511095.i			1.65881i	−0.558650	+	0.829404i	$0.688680\pi$
$38$		−	240642.i		−	0.711425i
$39$		0			0
$40$	−2600.00	−	17839.3i	−0.00642338	−	0.0440725i
$41$	−616234.			−1.39638			−0.698188	−	0.715914i	$-0.746010\pi$
$41$	−616234.			−1.39638			−0.698188	+	0.715914i	$0.746010\pi$
$42$		0			0
$43$			656486.i			1.25917i	0.776930	+	0.629587i	$0.216776\pi$
$43$			656486.i			1.25917i	−0.776930	+	0.629587i	$0.783224\pi$
$44$	−511132.			−0.904583
$45$		0			0
$46$	3640.00			0.00551377
$47$			1.32102e6i			1.85595i	0.372644	+	0.927974i	$0.378451\pi$
$47$			1.32102e6i			1.85595i	−0.372644	+	0.927974i	$0.621549\pi$
$48$		0			0
$49$	27943.0			0.0339302
$50$	−359562.	−	1.20732e6i	−0.406798	−	1.36593i
$51$		0			0
$52$		−	1.29513e6i		−	1.27733i
$53$		−	1.10593e6i		−	1.02038i	−0.860061	−	0.510191i	$-0.829575\pi$
$53$		−	1.10593e6i		−	1.02038i	0.860061	−	0.510191i	$-0.170425\pi$
$54$		0			0
$55$	−1.07100e6	+	156094.i	−0.868001	+	0.126507i
$56$	−57530.0			−0.0437760
$57$		0			0
$58$		−	2.20315e6i		−	1.48268i
$59$	−1.96210e6			−1.24377			−0.621885	−	0.783109i	$-0.713633\pi$
$59$	−1.96210e6			−1.24377			−0.621885	+	0.783109i	$0.713633\pi$
$60$		0			0
$61$	1.76244e6			0.994169			0.497085	−	0.867702i	$-0.334404\pi$
$61$	1.76244e6			0.994169			0.497085	+	0.867702i	$0.334404\pi$
$62$			3.93748e6i			2.09820i
$63$		0			0
$64$	2.22611e6			1.06149
$65$	−395518.	−	2.71376e6i	−0.178636	−	1.22567i
$66$		0			0
$67$		−	993648.i		−	0.403618i	−0.979425	−	0.201809i	$-0.935318\pi$
$67$		−	993648.i		−	0.403618i	0.979425	−	0.201809i	$-0.0646821\pi$
$68$		−	1.44095e6i		−	0.555735i
$69$		0			0
$70$	−3.97800e6	+	579777.i	−1.38619	+	0.202031i
$71$	−454708.			−0.150775			−0.0753873	−	0.997154i	$-0.524019\pi$
$71$	−454708.			−0.150775			−0.0753873	+	0.997154i	$0.524019\pi$
$72$		0			0
$73$			4.05665e6i			1.22050i	0.792209	+	0.610250i	$0.208931\pi$
$73$			4.05665e6i			1.22050i	−0.792209	+	0.610250i	$0.791069\pi$
$74$	8.24117e6			2.36416
$75$		0			0
$76$	−1.96997e6			−0.514768
$77$			3.45387e6i			0.862161i
$78$		0			0
$79$	2.16822e6			0.494775			0.247387	−	0.968917i	$-0.420428\pi$
$79$	2.16822e6			0.494775			0.247387	+	0.968917i	$0.420428\pi$
$80$	4.38555e6	−	639176.i	0.957656	−	0.139574i
$81$		0			0
$82$			9.93648e6i			1.99014i
$83$			748371.i			0.143663i	0.997417	+	0.0718313i	$0.0228843\pi$
$83$			748371.i			0.143663i	−0.997417	+	0.0718313i	$0.977116\pi$
$84$		0			0
$85$	−440050.	−	3.01930e6i	−0.0777205	−	0.533261i
$86$	1.05855e7			1.79460
$87$		0			0
$88$			249750.i			0.0390675i
$89$	−5.31046e6			−0.798485			−0.399243	−	0.916845i	$-0.630727\pi$
$89$	−5.31046e6			−0.798485			−0.399243	+	0.916845i	$0.630727\pi$
$90$		0			0
$91$	−8.75160e6			−1.21743
$92$		−	29798.1i		−	0.00398962i
$93$		0			0
$94$	2.13008e7			2.64513
$95$	−4.12777e6	+	601606.i	−0.493950	+	0.0719912i
$96$		0			0
$97$		−	1.13048e7i		−	1.25765i	−0.777547	−	0.628825i	$-0.783536\pi$
$97$		−	1.13048e7i		−	1.25765i	0.777547	−	0.628825i	$-0.216464\pi$
$98$		−	450567.i		−	0.0483580i
$99$		0			0
$100$	−9.88350e6	+	2.94348e6i	−0.988350	+	0.294348i
$101$	−9.78728e6			−0.945230			−0.472615	−	0.881269i	$-0.656690\pi$
$101$	−9.78728e6			−0.945230			−0.472615	+	0.881269i	$0.656690\pi$
$102$		0			0
$103$		−	1.92673e7i		−	1.73736i	−0.495370	−	0.868682i	$-0.664967\pi$
$103$		−	1.92673e7i		−	1.73736i	0.495370	−	0.868682i	$-0.335033\pi$
$104$	−632830.			−0.0551658
$105$		0			0
$106$	−1.78326e7			−1.45427
$107$		−	2.24713e7i		−	1.77331i	−0.462429	−	0.886656i	$-0.653022\pi$
$107$		−	2.24713e7i		−	1.77331i	0.462429	−	0.886656i	$-0.346978\pi$
$108$		0			0
$109$	1.70298e7			1.25955			0.629776	−	0.776777i	$-0.283147\pi$
$109$	1.70298e7			1.25955			0.629776	+	0.776777i	$0.283147\pi$
$110$	2.51694e6	+	1.72694e7i	0.180301	+	1.23709i
$111$		0			0
$112$		−	1.41430e7i		−	0.951213i
$113$			1.37331e7i			0.895355i	0.894195	+	0.447677i	$0.147749\pi$
$113$			1.37331e7i			0.895355i	−0.894195	+	0.447677i	$0.852251\pi$
$114$		0			0
$115$	−9100.00	−	62437.5i	−0.000557955	−	0.00382827i
$116$	−1.80356e7			−1.07282
$117$		0			0
$118$			3.16380e7i			1.77264i
$119$	−9.73695e6			−0.529674
$120$		0			0
$121$	−4.49317e6			−0.230571
$122$		−	2.84185e7i		−	1.41691i
$123$		0			0
$124$	3.22333e7			1.51820
$125$	−1.98105e7	+	9.18593e6i	−0.907215	+	0.420667i
$126$		0			0
$127$			2.85330e7i			1.23605i	0.786160	+	0.618024i	$0.212066\pi$
$127$			2.85330e7i			1.23605i	−0.786160	+	0.618024i	$0.787934\pi$
$128$		−	2.11244e6i		−	0.0890327i
$129$		0			0
$130$	−4.37580e7	+	6.37754e6i	−1.74685	+	0.254596i
$131$	−9.19760e6			−0.357458			−0.178729	−	0.983898i	$-0.557199\pi$
$131$	−9.19760e6			−0.357458			−0.178729	+	0.983898i	$0.557199\pi$
$132$		0			0
$133$			1.33117e7i			0.490627i
$134$	−1.60221e7			−0.575244
$135$		0			0
$136$	−704080.			−0.0240014
$137$		−	3.64799e7i		−	1.21208i	−0.795434	−	0.606040i	$-0.792757\pi$
$137$		−	3.64799e7i		−	1.21208i	0.795434	−	0.606040i	$-0.207243\pi$
$138$		0			0
$139$	1.05728e7			0.333916			0.166958	−	0.985964i	$-0.446606\pi$
$139$	1.05728e7			0.333916			0.166958	+	0.985964i	$0.446606\pi$
$140$	4.74622e6	+	3.25651e7i	0.146184	+	1.00301i
$141$		0			0
$142$			7.33195e6i			0.214887i
$143$			3.79926e7i			1.08648i
$144$		0			0
$145$	−3.77910e7	+	5.50788e6i	−1.02944	+	0.150036i
$146$	6.54116e7			1.73948
$147$		0			0
$148$		−	6.74646e7i		−	1.71065i
$149$	2.63327e7			0.652143			0.326072	−	0.945345i	$-0.394275\pi$
$149$	2.63327e7			0.652143			0.326072	+	0.945345i	$0.394275\pi$
$150$		0			0
$151$	2.87125e7			0.678660			0.339330	−	0.940667i	$-0.389800\pi$
$151$	2.87125e7			0.678660			0.339330	+	0.940667i	$0.389800\pi$
$152$			962569.i			0.0222320i
$153$		0			0
$154$	5.56920e7			1.22877
$155$	6.75402e7	−	9.84369e6i	1.45680	−	0.212323i
$156$		0			0
$157$			994540.i			0.0205104i	0.999947	+	0.0102552i	$0.00326438\pi$
$157$			994540.i			0.0205104i	−0.999947	+	0.0102552i	$0.996736\pi$
$158$		−	3.49614e7i		−	0.705163i
$159$		0			0
$160$	−1.06392e7	−	7.29983e7i	−0.205347	−	1.40894i
$161$	−201355.			−0.00380252
$162$		0			0
$163$		−	8.91072e6i		−	0.161160i	−0.996748	−	0.0805798i	$-0.974323\pi$
$163$		−	8.91072e6i		−	0.161160i	0.996748	−	0.0805798i	$-0.0256772\pi$
$164$	8.13429e7			1.44001
$165$		0			0
$166$	1.20671e7			0.204751
$167$			1.85304e7i			0.307877i	0.988080	+	0.153938i	$0.0491957\pi$
$167$			1.85304e7i			0.307877i	−0.988080	+	0.153938i	$0.950804\pi$
$168$		0			0
$169$	−3.35191e7			−0.534181
$170$	−4.86847e7	+	7.09559e6i	−0.760014	+	0.110769i
$171$		0			0
$172$		−	8.66561e7i		−	1.29852i
$173$			4.68600e7i			0.688083i	0.938954	+	0.344042i	$0.111796\pi$
$173$			4.68600e7i			0.688083i	−0.938954	+	0.344042i	$0.888204\pi$
$174$		0			0
$175$	1.98900e7	+	6.67858e7i	0.280544	+	0.942000i
$176$	−6.13977e7			−0.848903
$177$		0			0
$178$			8.56286e7i			1.13802i
$179$	−3.64596e6			−0.0475145			−0.0237573	−	0.999718i	$-0.507563\pi$
$179$	−3.64596e6			−0.0475145			−0.0237573	+	0.999718i	$0.507563\pi$
$180$		0			0
$181$	−8.98324e7			−1.12605			−0.563025	−	0.826440i	$-0.690363\pi$
$181$	−8.98324e7			−1.12605			−0.563025	+	0.826440i	$0.690363\pi$
$182$			1.41115e8i			1.73510i
$183$		0			0
$184$	−14560.0			−0.000172305
$185$	−2.06029e7	−	1.41362e8i	−0.239237	−	1.64147i
$186$		0			0
$187$			4.22702e7i			0.472703i
$188$		−	1.74374e8i		−	1.91395i
$189$		0			0
$190$	9.70060e6	+	6.65584e7i	0.102603	+	0.703988i
$191$	1.69641e8			1.76163			0.880816	−	0.473459i	$-0.156995\pi$
$191$	1.69641e8			1.76163			0.880816	+	0.473459i	$0.156995\pi$
$192$		0			0
$193$			8.59407e7i			0.860495i	0.902711	+	0.430248i	$0.141574\pi$
$193$			8.59407e7i			0.860495i	−0.902711	+	0.430248i	$0.858426\pi$
$194$	−1.82284e8			−1.79243
$195$		0			0
$196$	−3.68848e6			−0.0349905
$197$			7.48293e7i			0.697333i	0.937247	+	0.348666i	$0.113365\pi$
$197$			7.48293e7i			0.697333i	−0.937247	+	0.348666i	$0.886635\pi$
$198$		0			0
$199$	8.39132e7			0.754822			0.377411	−	0.926046i	$-0.376814\pi$
$199$	8.39132e7			0.754822			0.377411	+	0.926046i	$0.376814\pi$
$200$	1.43825e6	+	4.82929e6i	0.0127124	+	0.0426853i
$201$		0			0
$202$			1.57815e8i			1.34716i
$203$			1.21872e8i			1.02251i
$204$		0			0
$205$	1.70442e8	−	2.48412e7i	1.38178	−	0.201388i
$206$	−3.10676e8			−2.47613
$207$		0			0
$208$		−	1.55573e8i		−	1.19870i
$209$	5.77888e7			0.437857
$210$		0			0
$211$	−9.68059e7			−0.709436			−0.354718	−	0.934973i	$-0.615423\pi$
$211$	−9.68059e7			−0.709436			−0.354718	+	0.934973i	$0.615423\pi$
$212$			1.45983e8i			1.05227i
$213$		0			0
$214$	−3.62339e8			−2.52736
$215$	−2.64638e7	−	1.81575e8i	−0.181601	−	1.24601i
$216$		0			0
$217$		−	2.17811e8i		−	1.44700i
$218$		−	2.74597e8i		−	1.79514i
$219$		0			0
$220$	1.41372e8	−	2.06044e7i	0.895126	−	0.130461i
$221$	−1.07106e8			−0.667486
$222$		0			0
$223$		−	1.32857e8i		−	0.802265i	−0.916020	−	0.401133i	$-0.868617\pi$
$223$		−	1.32857e8i		−	0.802265i	0.916020	−	0.401133i	$-0.131383\pi$
$224$	−2.35413e8			−1.39946
$225$		0			0
$226$	2.21440e8			1.27608
$227$			9.72571e7i			0.551862i	0.961177	+	0.275931i	$0.0889862\pi$
$227$			9.72571e7i			0.551862i	−0.961177	+	0.275931i	$0.911014\pi$
$228$		0			0
$229$	−2.84016e7			−0.156286			−0.0781429	−	0.996942i	$-0.524899\pi$
$229$	−2.84016e7			−0.156286			−0.0781429	+	0.996942i	$0.524899\pi$
$230$	−1.00677e6	+	146733.i	−0.00545613	+	0.000795208i
$231$		0			0
$232$			8.81261e6i			0.0463336i
$233$			1.36738e8i			0.708180i	0.935211	+	0.354090i	$0.115209\pi$
$233$			1.36738e8i			0.708180i	−0.935211	+	0.354090i	$0.884791\pi$
$234$		0			0
$235$	−5.32519e7	−	3.65375e8i	−0.267669	−	1.83655i
$236$	2.58998e8			1.28264
$237$		0			0
$238$			1.57004e8i			0.754901i
$239$	2.93583e8			1.39104			0.695518	−	0.718509i	$-0.255175\pi$
$239$	2.93583e8			1.39104			0.695518	+	0.718509i	$0.255175\pi$
$240$		0			0
$241$	2.01709e8			0.928251			0.464126	−	0.885769i	$-0.346369\pi$
$241$	2.01709e8			0.928251			0.464126	+	0.885769i	$0.346369\pi$
$242$			7.24502e7i			0.328614i
$243$		0			0
$244$	−2.32642e8			−1.02524
$245$	−7.72865e6	+	1.12642e6i	−0.0335755	+	0.00489349i
$246$		0			0
$247$			1.46428e8i			0.618281i
$248$		−	1.57499e7i		−	0.0655688i
$249$		0			0
$250$	1.48119e8	+	3.19435e8i	0.599543	+	1.29298i
$251$	2.23310e8			0.891353			0.445676	−	0.895194i	$-0.352963\pi$
$251$	2.23310e8			0.891353			0.445676	+	0.895194i	$0.352963\pi$
$252$		0			0
$253$			874125.i			0.00339353i
$254$	4.60081e8			1.76164
$255$		0			0
$256$	2.50880e8			0.934602
$257$		−	3.81922e8i		−	1.40349i	−0.712430	−	0.701744i	$-0.752405\pi$
$257$		−	3.81922e8i		−	1.40349i	0.712430	−	0.701744i	$-0.247595\pi$
$258$		0			0
$259$	−4.55879e8			−1.63042
$260$	5.22084e7	+	3.58216e8i	0.184219	+	1.26397i
$261$		0			0
$262$			1.48307e8i			0.509456i
$263$			2.76045e8i			0.935695i	0.883809	+	0.467848i	$0.154970\pi$
$263$			2.76045e8i			0.935695i	−0.883809	+	0.467848i	$0.845030\pi$
$264$		0			0
$265$	4.45816e7	+	3.05886e8i	0.147162	+	1.00971i
$266$	2.14644e8			0.699252
$267$		0			0
$268$			1.31162e8i			0.416231i
$269$	−4.22435e8			−1.32321			−0.661603	−	0.749855i	$-0.730123\pi$
$269$	−4.22435e8			−1.32321			−0.661603	+	0.749855i	$0.730123\pi$
$270$		0			0
$271$	−1.71153e6			−0.00522386			−0.00261193	−	0.999997i	$-0.500831\pi$
$271$	−1.71153e6			−0.00522386			−0.00261193	+	0.999997i	$0.500831\pi$
$272$		−	1.73089e8i		−	0.521528i
$273$		0			0
$274$	−5.88220e8			−1.72748
$275$	2.89932e8	−	8.63468e7i	0.840681	−	0.250370i
$276$		0			0
$277$		−	1.61917e8i		−	0.457735i	−0.973458	−	0.228868i	$-0.926498\pi$
$277$		−	1.61917e8i		−	0.457735i	0.973458	−	0.228868i	$-0.0735023\pi$
$278$		−	1.70481e8i		−	0.475903i
$279$		0			0
$280$	1.59120e7	−	2.31911e6i	0.0433183	−	0.00631346i
$281$	−3.59164e8			−0.965652			−0.482826	−	0.875716i	$-0.660390\pi$
$281$	−3.59164e8			−0.965652			−0.482826	+	0.875716i	$0.660390\pi$
$282$		0			0
$283$			2.05767e8i			0.539664i	0.962907	+	0.269832i	$0.0869682\pi$
$283$			2.05767e8i			0.539664i	−0.962907	+	0.269832i	$0.913032\pi$
$284$	6.00214e7			0.155486
$285$		0			0
$286$	6.12612e8			1.54848
$287$		−	5.49659e8i		−	1.37248i
$288$		0			0
$289$	2.91173e8			0.709592
$290$	8.88119e7	+	6.09362e8i	0.213835	+	1.46718i
$291$		0			0
$292$		−	5.35478e8i		−	1.25864i
$293$			7.21304e8i			1.67526i	0.546240	+	0.837629i	$0.316059\pi$
$293$			7.21304e8i			1.67526i	−0.546240	+	0.837629i	$0.683941\pi$
$294$		0			0
$295$	5.42691e8	−	7.90949e7i	1.23077	−	0.179379i
$296$	−3.29647e7			−0.0738801
$297$		0			0
$298$		−	4.24602e8i		−	0.929447i
$299$	−2.21490e6			−0.00479188
$300$		0			0
$301$	−5.85562e8			−1.23763
$302$		−	4.62976e8i		−	0.967239i
$303$		0			0
$304$	−2.36635e8			−0.483083
$305$	−4.87467e8	+	7.10463e7i	−0.983776	+	0.143381i
$306$		0			0
$307$			2.99711e8i			0.591177i	0.955315	+	0.295589i	$0.0955158\pi$
$307$			2.99711e8i			0.591177i	−0.955315	+	0.295589i	$0.904484\pi$
$308$		−	4.55911e8i		−	0.889104i
$309$		0			0
$310$	−1.58725e8	−	1.08905e9i	−0.302607	−	2.07627i
$311$	−1.72058e8			−0.324349			−0.162175	−	0.986762i	$-0.551851\pi$
$311$	−1.72058e8			−0.324349			−0.162175	+	0.986762i	$0.551851\pi$
$312$		0			0
$313$		−	3.05567e8i		−	0.563251i	−0.959524	−	0.281625i	$-0.909126\pi$
$313$		−	3.05567e8i		−	0.563251i	0.959524	−	0.281625i	$-0.0908735\pi$
$314$	1.60365e7			0.0292318
$315$		0			0
$316$	−2.86205e8			−0.510237
$317$		−	4.39480e8i		−	0.774875i	−0.921896	−	0.387437i	$-0.873360\pi$
$317$		−	4.39480e8i		−	0.774875i	0.921896	−	0.387437i	$-0.126640\pi$
$318$		0			0
$319$	5.29074e8			0.912534
$320$	−6.15712e8	+	8.97374e7i	−1.05040	+	0.153091i
$321$		0			0
$322$			3.24675e6i			0.00541943i
$323$			1.62915e8i			0.269000i
$324$		0			0
$325$	2.18790e8	+	7.34644e8i	0.353538	+	1.18709i
$326$	−1.43681e8			−0.229688
$327$		0			0
$328$		−	3.97459e7i		−	0.0621919i
$329$	−1.17830e9			−1.82419
$330$		0			0
$331$	−9.61785e8			−1.45774			−0.728870	−	0.684652i	$-0.759954\pi$
$331$	−9.61785e8			−1.45774			−0.728870	+	0.684652i	$0.759954\pi$
$332$		−	9.87850e7i		−	0.148152i
$333$		0			0
$334$	2.98794e8			0.438792
$335$	4.00552e7	+	2.74829e8i	0.0582106	+	0.399399i
$336$		0			0
$337$			5.23454e8i			0.745031i	0.928026	+	0.372515i	$0.121505\pi$
$337$			5.23454e8i			0.745031i	−0.928026	+	0.372515i	$0.878495\pi$
$338$			5.40479e8i			0.761325i
$339$		0			0
$340$	5.80866e7	+	3.98547e8i	0.0801493	+	0.549925i
$341$	−9.45562e8			−1.29137
$342$		0			0
$343$			7.59495e8i			1.01624i
$344$	−4.23420e7			−0.0560812
$345$		0			0
$346$	7.55595e8			0.980670
$347$			1.36908e8i			0.175905i	0.996125	+	0.0879523i	$0.0280323\pi$
$347$			1.36908e8i			0.175905i	−0.996125	+	0.0879523i	$0.971968\pi$
$348$		0			0
$349$	−1.82702e8			−0.230067			−0.115034	−	0.993362i	$-0.536698\pi$
$349$	−1.82702e8			−0.230067			−0.115034	+	0.993362i	$0.536698\pi$
$350$	1.07689e9	−	3.20717e8i	1.34256	−	0.399837i
$351$		0			0
$352$			1.02198e9i			1.24894i
$353$		−	1.14182e9i		−	1.38161i	−0.723042	−	0.690804i	$-0.757257\pi$
$353$		−	1.14182e9i		−	1.38161i	0.723042	−	0.690804i	$-0.242743\pi$
$354$		0			0
$355$	1.25766e8	−	1.83299e7i	0.149198	−	0.0217450i
$356$	7.00980e8			0.823438
$357$		0			0
$358$			5.87894e7i			0.0677186i
$359$	1.46808e9			1.67463			0.837317	−	0.546718i	$-0.184123\pi$
$359$	1.46808e9			1.67463			0.837317	+	0.546718i	$0.184123\pi$
$360$		0			0
$361$	−6.71146e8			−0.750830
$362$			1.44850e9i			1.60487i
$363$		0			0
$364$	1.15521e9			1.25547
$365$	−1.63529e8	−	1.12201e9i	−0.176023	−	1.20774i
$366$		0			0
$367$		−	3.71113e8i		−	0.391900i	−0.980614	−	0.195950i	$-0.937221\pi$
$367$		−	3.71113e8i		−	0.391900i	0.980614	−	0.195950i	$-0.0627790\pi$
$368$		−	3.57938e6i		−	0.00374404i
$369$		0			0
$370$	−2.27939e9	+	3.32212e8i	−2.33945	+	0.340965i
$371$	9.86452e8			1.00292
$372$		0			0
$373$		−	4.24232e8i		−	0.423274i	−0.977348	−	0.211637i	$-0.932120\pi$
$373$		−	4.24232e8i		−	0.423274i	0.977348	−	0.211637i	$-0.0678795\pi$
$374$	6.81586e8			0.673706
$375$		0			0
$376$	−8.52030e7			−0.0826604
$377$			1.34060e9i			1.28856i
$378$		0			0
$379$	1.03619e9			0.977688			0.488844	−	0.872371i	$-0.337419\pi$
$379$	1.03619e9			0.977688			0.488844	+	0.872371i	$0.337419\pi$
$380$	5.44866e8	−	7.94119e7i	0.509386	−	0.0742409i
$381$		0			0
$382$		−	2.73539e9i		−	2.51071i
$383$			7.40719e8i			0.673687i	0.941561	+	0.336843i	$0.109359\pi$
$383$			7.40719e8i			0.673687i	−0.941561	+	0.336843i	$0.890641\pi$
$384$		0			0
$385$	−1.39230e8	−	9.55294e8i	−0.124343	−	0.853148i
$386$	1.38575e9			1.22639
$387$		0			0
$388$			1.49223e9i			1.29695i
$389$	−1.22689e9			−1.05677			−0.528387	−	0.849003i	$-0.677203\pi$
$389$	−1.22689e9			−1.05677			−0.528387	+	0.849003i	$0.677203\pi$
$390$		0			0
$391$	−2.46428e6			−0.00208483
$392$			1.80227e6i			0.00151119i
$393$		0			0
$394$	1.20659e9			0.993852
$395$	−5.99699e8	+	8.74036e7i	−0.489602	+	0.0713575i
$396$		0			0
$397$		−	1.16332e9i		−	0.933109i	−0.884492	−	0.466555i	$-0.845495\pi$
$397$		−	1.16332e9i		−	0.933109i	0.884492	−	0.466555i	$-0.154505\pi$
$398$		−	1.35306e9i		−	1.07579i
$399$		0			0
$400$	−1.18722e9	+	3.53575e8i	−0.927514	+	0.276230i
$401$	7.05638e7			0.0546483			0.0273242	−	0.999627i	$-0.491301\pi$
$401$	7.05638e7			0.0546483			0.0273242	+	0.999627i	$0.491301\pi$
$402$		0			0
$403$		−	2.39592e9i		−	1.82349i
$404$	1.29192e9			0.974768
$405$		0			0
$406$	1.96513e9			1.45731
$407$			1.97907e9i			1.45506i
$408$		0			0
$409$	4.84419e8			0.350098			0.175049	−	0.984560i	$-0.443992\pi$
$409$	4.84419e8			0.350098			0.175049	+	0.984560i	$0.443992\pi$
$410$	−4.00552e8	−	2.74829e9i	−0.287022	−	1.96934i
$411$		0			0
$412$			2.54329e9i			1.79166i
$413$		−	1.75013e9i		−	1.22249i
$414$		0			0
$415$	−3.01678e7	−	2.06989e8i	−0.0207193	−	0.142161i
$416$	−2.58954e9			−1.76358
$417$		0			0
$418$		−	9.31817e8i		−	0.624042i
$419$	2.05312e9			1.36353			0.681767	−	0.731570i	$-0.261212\pi$
$419$	2.05312e9			1.36353			0.681767	+	0.731570i	$0.261212\pi$
$420$		0			0
$421$	−4.64439e8			−0.303348			−0.151674	−	0.988431i	$-0.548466\pi$
$421$	−4.64439e8			−0.303348			−0.151674	+	0.988431i	$0.548466\pi$
$422$			1.56095e9i			1.01110i
$423$		0			0
$424$	7.13305e7			0.0454459
$425$	2.43424e8	+	8.17358e8i	0.153816	+	0.516477i
$426$		0			0
$427$			1.57204e9i			0.977158i
$428$			2.96621e9i			1.82873i
$429$		0			0
$430$	−2.92781e9	+	4.26716e8i	−1.77584	+	0.258821i
$431$	1.66845e9			1.00379			0.501896	−	0.864928i	$-0.332636\pi$
$431$	1.66845e9			1.00379			0.501896	+	0.864928i	$0.332636\pi$
$432$		0			0
$433$			1.08260e9i			0.640854i	0.947273	+	0.320427i	$0.103826\pi$
$433$			1.08260e9i			0.640854i	−0.947273	+	0.320427i	$0.896174\pi$
$434$	−3.51209e9			−2.06230
$435$		0			0
$436$	−2.24793e9			−1.29891
$437$			3.36899e6i			0.00193115i
$438$		0			0
$439$	1.94290e9			1.09603			0.548016	−	0.836468i	$-0.315383\pi$
$439$	1.94290e9			1.09603			0.548016	+	0.836468i	$0.315383\pi$
$440$	−1.00677e7	−	6.90774e7i	−0.00563440	−	0.0386591i
$441$		0			0
$442$			1.72704e9i			0.951315i
$443$		−	3.25401e9i		−	1.77830i	−0.457615	−	0.889150i	$-0.651296\pi$
$443$		−	3.25401e9i		−	1.77830i	0.457615	−	0.889150i	$-0.348704\pi$
$444$		0			0
$445$	1.46880e9	−	2.14071e8i	0.790137	−	0.115159i
$446$	−2.14226e9			−1.14340
$447$		0			0
$448$			1.98561e9i			1.04333i
$449$	−1.59381e9			−0.830950			−0.415475	−	0.909605i	$-0.636385\pi$
$449$	−1.59381e9			−0.830950			−0.415475	+	0.909605i	$0.636385\pi$
$450$		0			0
$451$	−2.38619e9			−1.22486
$452$		−	1.81277e9i		−	0.923335i
$453$		0			0
$454$	1.56822e9			0.786525
$455$	2.42057e9	−	3.52788e8i	1.20470	−	0.175580i
$456$		0			0
$457$		−	1.03584e9i		−	0.507675i	−0.967247	−	0.253838i	$-0.918307\pi$
$457$		−	1.03584e9i		−	0.507675i	0.967247	−	0.253838i	$-0.0816929\pi$
$458$			4.57963e8i			0.222741i
$459$		0			0
$460$	1.20120e6	+	8.24175e6i	0.000575391	+	0.00394791i
$461$	−2.37166e9			−1.12745			−0.563727	−	0.825961i	$-0.690633\pi$
$461$	−2.37166e9			−1.12745			−0.563727	+	0.825961i	$0.690633\pi$
$462$		0			0
$463$			2.47025e9i			1.15666i	0.815801	+	0.578332i	$0.196296\pi$
$463$			2.47025e9i			1.15666i	−0.815801	+	0.578332i	$0.803704\pi$
$464$	−2.16646e9			−1.00679
$465$		0			0
$466$	2.20484e9			1.00931
$467$		−	6.83727e8i		−	0.310652i	−0.987863	−	0.155326i	$-0.950357\pi$
$467$		−	6.83727e8i		−	0.310652i	0.987863	−	0.155326i	$-0.0496428\pi$
$468$		0			0
$469$	8.86298e8			0.396712
$470$	−5.89150e9	+	8.58661e8i	−2.61748	+	0.381487i
$471$		0			0
$472$		−	1.26552e8i		−	0.0553951i
$473$			2.54205e9i			1.10451i
$474$		0			0
$475$	1.11743e9	−	3.32792e8i	0.478404	−	0.142477i
$476$	1.28528e9			0.546226
$477$		0			0
$478$		−	4.73389e9i		−	1.98253i
$479$	7.36812e8			0.306325			0.153162	−	0.988201i	$-0.451054\pi$
$479$	7.36812e8			0.306325			0.153162	+	0.988201i	$0.451054\pi$
$480$		0			0
$481$	−5.01467e9			−2.05463
$482$		−	3.25246e9i		−	1.32296i
$483$		0			0
$484$	5.93099e8			0.237776
$485$	4.55709e8	+	3.12674e9i	0.181381	+	1.24450i
$486$		0			0
$487$		−	4.17299e9i		−	1.63718i	−0.574380	−	0.818589i	$-0.694757\pi$
$487$		−	4.17299e9i		−	1.63718i	0.574380	−	0.818589i	$-0.305243\pi$
$488$			1.13674e8i			0.0442784i
$489$		0			0
$490$	1.81630e7	+	1.24621e8i	0.00697429	+	0.0478524i
$491$	2.52822e9			0.963894			0.481947	−	0.876200i	$-0.339930\pi$
$491$	2.52822e9			0.963894			0.481947	+	0.876200i	$0.339930\pi$
$492$		0			0
$493$			1.49153e9i			0.560620i
$494$	2.36109e9			0.881186
$495$		0			0
$496$	3.87191e9			1.42475
$497$		−	4.05583e8i		−	0.148195i
$498$		0			0
$499$	−1.08872e9			−0.392252			−0.196126	−	0.980579i	$-0.562836\pi$
$499$	−1.08872e9			−0.392252			−0.196126	+	0.980579i	$0.562836\pi$
$500$	2.61499e9	−	1.21254e9i	0.935566	−	0.433813i
$501$		0			0
$502$		−	3.60076e9i		−	1.27037i
$503$		−	1.07775e9i		−	0.377600i	−0.982016	−	0.188800i	$-0.939540\pi$
$503$		−	1.07775e9i		−	0.377600i	0.982016	−	0.188800i	$-0.0604598\pi$
$504$		0			0
$505$	2.70703e9	−	3.94538e8i	0.935348	−	0.136323i
$506$	1.40948e7			0.00483652
$507$		0			0
$508$		−	3.76636e9i		−	1.27467i
$509$	3.85873e9			1.29698			0.648488	−	0.761225i	$-0.275401\pi$
$509$	3.85873e9			1.29698			0.648488	+	0.761225i	$0.275401\pi$
$510$		0			0
$511$	−3.61839e9			−1.19962
$512$		−	4.31571e9i		−	1.42105i
$513$		0			0
$514$	−6.15830e9			−2.00028
$515$	7.76690e8	+	5.32908e9i	0.250566	+	1.71920i
$516$		0			0
$517$			5.11525e9i			1.62798i
$518$			7.35082e9i			2.32371i
$519$		0			0
$520$	1.75032e8	−	2.55102e7i	0.0545891	−	0.00795613i
$521$	6.39661e8			0.198161			0.0990805	−	0.995079i	$-0.468410\pi$
$521$	6.39661e8			0.198161			0.0990805	+	0.995079i	$0.468410\pi$
$522$		0			0
$523$		−	9.53972e8i		−	0.291595i	−0.989314	−	0.145797i	$-0.953425\pi$
$523$		−	9.53972e8i		−	0.291595i	0.989314	−	0.145797i	$-0.0465747\pi$
$524$	1.21408e9			0.368629
$525$		0			0
$526$	4.45109e9			1.33357
$527$		−	2.66567e9i		−	0.793359i
$528$		0			0
$529$	3.40477e9			0.999985
$530$	4.93226e9	−	7.18856e8i	1.43906	−	0.209738i
$531$		0			0
$532$		−	1.75714e9i		−	0.505960i
$533$		−	6.04625e9i		−	1.72958i
$534$		0			0
$535$	9.05848e8	+	6.21526e9i	0.255751	+	1.75477i
$536$	6.40884e7			0.0179764
$537$		0			0
$538$			6.81156e9i			1.88586i
$539$	1.08201e8			0.0297626
$540$		0			0
$541$	4.12373e9			1.11969			0.559847	−	0.828596i	$-0.310860\pi$
$541$	4.12373e9			1.11969			0.559847	+	0.828596i	$0.310860\pi$
$542$			2.75976e7i			0.00744515i
$543$		0			0
$544$	−2.88110e9			−0.767293
$545$	−4.71020e9	+	6.86492e8i	−1.24638	+	0.181655i
$546$		0			0
$547$			3.87126e9i			1.01134i	0.862727	+	0.505669i	$0.168754\pi$
$547$			3.87126e9i			1.01134i	−0.862727	+	0.505669i	$0.831246\pi$
$548$			4.81534e9i			1.24996i
$549$		0			0
$550$	−1.39230e9	−	4.67501e9i	−0.356832	−	1.19815i
$551$	2.03912e9			0.519293
$552$		0			0
$553$			1.93397e9i			0.486309i
$554$	−2.61084e9			−0.652373
$555$		0			0
$556$	−1.39560e9			−0.344350
$557$		−	1.38658e9i		−	0.339978i	−0.985446	−	0.169989i	$-0.945627\pi$
$557$		−	1.38658e9i		−	0.339978i	0.985446	−	0.169989i	$-0.0543732\pi$
$558$		0			0
$559$	−6.44118e9			−1.55964
$560$	5.70122e8	+	3.91176e9i	0.137186	+	0.941269i
$561$		0			0
$562$			5.79134e9i			1.37627i
$563$			2.02834e9i			0.479028i	0.970893	+	0.239514i	$0.0769881\pi$
$563$			2.02834e9i			0.479028i	−0.970893	+	0.239514i	$0.923012\pi$
$564$		0			0
$565$	−5.53600e8	−	3.79840e9i	−0.129130	−	0.885994i
$566$	3.31790e9			0.769140
$567$		0			0
$568$		−	2.93278e7i		−	0.00671522i
$569$	−7.16780e8			−0.163115			−0.0815573	−	0.996669i	$-0.525989\pi$
$569$	−7.16780e8			−0.163115			−0.0815573	+	0.996669i	$0.525989\pi$
$570$		0			0
$571$	3.19832e9			0.718944			0.359472	−	0.933156i	$-0.382957\pi$
$571$	3.19832e9			0.718944			0.359472	+	0.933156i	$0.382957\pi$
$572$		−	5.01502e9i		−	1.12043i
$573$		0			0
$574$	−8.86298e9			−1.95609
$575$	5.03387e6	+	1.69025e7i	0.00110424	+	0.00370778i
$576$		0			0
$577$			2.14993e9i			0.465917i	0.972487	+	0.232959i	$0.0748407\pi$
$577$			2.14993e9i			0.465917i	−0.972487	+	0.232959i	$0.925159\pi$
$578$		−	4.69503e9i		−	1.01132i
$579$		0			0
$580$	4.98841e9	−	7.27040e8i	1.06161	−	0.154725i
$581$	−6.67520e8			−0.141204
$582$		0			0
$583$		−	4.28240e9i		−	0.895050i
$584$	−2.61646e8			−0.0543588
$585$		0			0
$586$	1.16307e10			2.38761
$587$			6.42578e9i			1.31127i	0.755078	+	0.655635i	$0.227599\pi$
$587$			6.42578e9i			1.31127i	−0.755078	+	0.655635i	$0.772401\pi$
$588$		0			0
$589$	−3.64432e9			−0.734874
$590$	−1.27537e9	−	8.75063e9i	−0.255654	−	1.75411i
$591$		0			0
$592$		−	8.10393e9i		−	1.60535i
$593$		−	3.04761e9i		−	0.600162i	−0.953914	−	0.300081i	$-0.902986\pi$
$593$		−	3.04761e9i		−	0.600162i	0.953914	−	0.300081i	$-0.0970137\pi$
$594$		0			0
$595$	2.69311e9	−	3.92509e8i	0.524136	−	0.0763906i
$596$	−3.47591e9			−0.672523
$597$		0			0
$598$			3.57142e7i			0.00682947i
$599$	−4.90779e9			−0.933023			−0.466512	−	0.884515i	$-0.654489\pi$
$599$	−4.90779e9			−0.933023			−0.466512	+	0.884515i	$0.654489\pi$
$600$		0			0
$601$	−3.13473e9			−0.589033			−0.294516	−	0.955646i	$-0.595159\pi$
$601$	−3.13473e9			−0.589033			−0.294516	+	0.955646i	$0.595159\pi$
$602$			9.44190e9i			1.76389i
$603$		0			0
$604$	−3.79005e9			−0.699868
$605$	1.24275e9	−	1.81126e8i	0.228160	−	0.0332534i
$606$		0			0
$607$		−	8.96199e9i		−	1.62646i	−0.581940	−	0.813231i	$-0.697706\pi$
$607$		−	8.96199e9i		−	1.62646i	0.581940	−	0.813231i	$-0.302294\pi$
$608$			3.93883e9i			0.710731i
$609$		0			0
$610$	1.14559e9	+	7.86018e9i	0.204350	+	1.40210i
$611$	−1.29613e10			−2.29882
$612$		0			0
$613$		−	9.33008e9i		−	1.63596i	−0.575243	−	0.817982i	$-0.695093\pi$
$613$		−	9.33008e9i		−	1.63596i	0.575243	−	0.817982i	$-0.304907\pi$
$614$	4.83269e9			0.842558
$615$		0			0
$616$	−2.22768e8			−0.0383990
$617$		−	6.31024e7i		−	0.0108155i	−0.999985	−	0.00540777i	$-0.998279\pi$
$617$		−	6.31024e7i		−	0.0108155i	0.999985	−	0.00540777i	$-0.00172135\pi$
$618$		0			0
$619$	9.58488e9			1.62431			0.812156	−	0.583440i	$-0.198294\pi$
$619$	9.58488e9			1.62431			0.812156	+	0.583440i	$0.198294\pi$
$620$	−8.91530e9	+	1.29937e9i	−1.50233	+	0.218958i
$621$		0			0
$622$			2.77435e9i			0.462269i
$623$		−	4.73674e9i		−	0.784822i
$624$		0			0
$625$	5.10902e9	−	3.33929e9i	0.837061	−	0.547109i
$626$	−4.92712e9			−0.802756
$627$		0			0
$628$		−	1.31279e8i		−	0.0211513i
$629$	−5.57927e9			−0.893923
$630$		0			0
$631$	−6.88427e9			−1.09083			−0.545413	−	0.838168i	$-0.683627\pi$
$631$	−6.88427e9			−1.09083			−0.545413	+	0.838168i	$0.683627\pi$
$632$			1.39846e8i			0.0220363i
$633$		0			0
$634$	−7.08640e9			−1.10437
$635$	−1.15020e9	−	7.89185e9i	−0.178265	−	1.22312i
$636$		0			0
$637$			2.74166e8i			0.0420267i
$638$		−	8.53106e9i		−	1.30056i
$639$		0			0
$640$	8.51552e7	+	5.84272e8i	0.0128405	+	0.0881019i
$641$	4.62211e9			0.693167			0.346583	−	0.938019i	$-0.387342\pi$
$641$	4.62211e9			0.693167			0.346583	+	0.938019i	$0.387342\pi$
$642$		0			0
$643$			4.44867e9i			0.659921i	0.943995	+	0.329961i	$0.107035\pi$
$643$			4.44867e9i			0.659921i	−0.943995	+	0.329961i	$0.892965\pi$
$644$	2.65788e7			0.00392135
$645$		0			0
$646$	2.62692e9			0.383384
$647$			5.26642e9i			0.764453i	0.924069	+	0.382226i	$0.124843\pi$
$647$			5.26642e9i			0.764453i	−0.924069	+	0.382226i	$0.875157\pi$
$648$		0			0
$649$	−7.59767e9			−1.09100
$650$	1.18458e10	−	3.52788e9i	1.69187	−	0.503869i
$651$		0			0
$652$			1.17622e9i			0.166196i
$653$			6.40447e9i			0.900092i	0.893005	+	0.450046i	$0.148592\pi$
$653$			6.40447e9i			0.900092i	−0.893005	+	0.450046i	$0.851408\pi$
$654$		0			0
$655$	2.54393e9	−	3.70767e8i	0.353721	−	0.0515533i
$656$	9.77101e9			1.35138
$657$		0			0
$658$			1.89995e10i			2.59987i
$659$	−7.30551e9			−0.994378			−0.497189	−	0.867642i	$-0.665634\pi$
$659$	−7.30551e9			−0.994378			−0.497189	+	0.867642i	$0.665634\pi$
$660$		0			0
$661$	−1.00223e10			−1.34978			−0.674889	−	0.737919i	$-0.735809\pi$
$661$	−1.00223e10			−1.34978			−0.674889	+	0.737919i	$0.735809\pi$
$662$			1.55083e10i			2.07760i
$663$		0			0
$664$	−4.82685e7			−0.00639846
$665$	−5.36611e8	−	3.68183e9i	−0.0707593	−	0.485498i
$666$		0			0
$667$			3.08441e7i			0.00402469i
$668$		−	2.44601e9i		−	0.317498i
$669$		0			0
$670$	4.43149e9	−	6.45871e8i	0.569230	−	0.0829629i
$671$	6.82454e9			0.872057
$672$		0			0
$673$		−	3.65625e8i		−	0.0462363i	−0.999733	−	0.0231182i	$-0.992641\pi$
$673$		−	3.65625e8i		−	0.0462363i	0.999733	−	0.0231182i	$-0.00735939\pi$
$674$	8.44045e9			1.06183
$675$		0			0
$676$	4.42452e9			0.550874
$677$			9.06259e9i			1.12251i	0.827641	+	0.561257i	$0.189682\pi$
$677$			9.06259e9i			1.12251i	−0.827641	+	0.561257i	$0.810318\pi$
$678$		0			0
$679$	1.00834e10			1.23613
$680$	1.94739e8	−	2.83824e7i	0.0237504	−	0.00346153i
$681$		0			0
$682$			1.52467e10i			1.84048i
$683$		−	5.51289e9i		−	0.662075i	−0.943618	−	0.331038i	$-0.892601\pi$
$683$		−	5.51289e9i		−	0.662075i	0.943618	−	0.331038i	$-0.107399\pi$
$684$		0			0
$685$	1.47055e9	+	1.00898e10i	0.174809	+	1.19941i
$686$	1.22465e10			1.44836
$687$		0			0
$688$		−	1.04092e10i		−	1.21859i
$689$	1.08510e10			1.26387
$690$		0			0
$691$	1.71515e9			0.197755			0.0988777	−	0.995100i	$-0.468475\pi$
$691$	1.71515e9			0.197755			0.0988777	+	0.995100i	$0.468475\pi$
$692$		−	6.18552e9i		−	0.709586i
$693$		0			0
$694$	2.20758e9			0.250703
$695$	−2.92428e9	+	4.26202e8i	−0.330425	+	0.0481580i
$696$		0			0
$697$		−	6.72700e9i		−	0.752500i
$698$			2.94598e9i			0.327896i
$699$		0			0
$700$	−2.62548e9	−	8.81573e9i	−0.289311	−	0.971438i
$701$	−1.10662e10			−1.21335			−0.606675	−	0.794950i	$-0.707497\pi$
$701$	−1.10662e10			−1.21335			−0.606675	+	0.794950i	$0.707497\pi$
$702$		0			0
$703$			7.62759e9i			0.828025i
$704$	8.61997e9			0.931111
$705$		0			0
$706$	−1.84112e10			−1.96909
$707$		−	8.72991e9i		−	0.929055i
$708$		0			0
$709$	−3.69647e9			−0.389516			−0.194758	−	0.980851i	$-0.562392\pi$
$709$	−3.69647e9			−0.389516			−0.194758	+	0.980851i	$0.562392\pi$
$710$	−2.95560e8	−	2.02792e9i	−0.0309914	−	0.212640i
$711$		0			0
$712$		−	3.42514e8i		−	0.0355630i
$713$		−	5.51247e7i		−	0.00569551i
$714$		0			0
$715$	−1.53153e9	−	1.05082e10i	−0.156695	−	1.07512i
$716$	4.81267e8			0.0489994
$717$		0			0
$718$		−	2.36721e10i		−	2.38672i
$719$	−4.99915e9			−0.501586			−0.250793	−	0.968041i	$-0.580691\pi$
$719$	−4.99915e9			−0.501586			−0.250793	+	0.968041i	$0.580691\pi$
$720$		0			0
$721$	1.71858e10			1.70764
$722$			1.08219e10i			1.07010i
$723$		0			0
$724$	1.18579e10			1.16124
$725$	1.02304e10	−	3.04681e9i	0.997037	−	0.296935i
$726$		0			0
$727$			2.91969e9i			0.281817i	0.990023	+	0.140908i	$0.0450023\pi$
$727$			2.91969e9i			0.281817i	−0.990023	+	0.140908i	$0.954998\pi$
$728$		−	5.64461e8i		−	0.0542218i
$729$		0			0
$730$	−1.80919e10	+	2.63682e9i	−1.72129	+	0.250871i
$731$	−7.16639e9			−0.678562
$732$		0			0
$733$		−	5.21237e9i		−	0.488845i	−0.969669	−	0.244422i	$-0.921402\pi$
$733$		−	5.21237e9i		−	0.488845i	0.969669	−	0.244422i	$-0.0785983\pi$
$734$	−5.98402e9			−0.558543
$735$		0			0
$736$	−5.95795e7			−0.00550839
$737$		−	3.84761e9i		−	0.354042i
$738$		0			0
$739$	−5.26869e8			−0.0480227			−0.0240114	−	0.999712i	$-0.507644\pi$
$739$	−5.26869e8			−0.0480227			−0.0240114	+	0.999712i	$0.507644\pi$
$740$	2.71958e9	+	1.86598e10i	0.246713	+	1.69276i
$741$		0			0
$742$		−	1.59061e10i		−	1.42938i
$743$			1.67008e10i			1.49375i	0.664965	+	0.746875i	$0.268447\pi$
$743$			1.67008e10i			1.49375i	−0.664965	+	0.746875i	$0.731553\pi$
$744$		0			0
$745$	−7.28326e9	+	1.06150e9i	−0.645325	+	0.0940535i
$746$	−6.84053e9			−0.603259
$747$		0			0
$748$		−	5.57966e9i		−	0.487475i
$749$	2.00436e10			1.74297
$750$		0			0
$751$	−9.03609e8			−0.0778468			−0.0389234	−	0.999242i	$-0.512393\pi$
$751$	−9.03609e8			−0.0778468			−0.0389234	+	0.999242i	$0.512393\pi$
$752$		−	2.09460e10i		−	1.79614i
$753$		0			0
$754$	2.16165e10			1.83647
$755$	−7.94149e9	+	1.15744e9i	−0.671565	+	0.0978777i
$756$		0			0
$757$			1.02084e10i			0.855307i	0.903943	+	0.427654i	$0.140660\pi$
$757$			1.02084e10i			0.855307i	−0.903943	+	0.427654i	$0.859340\pi$
$758$		−	1.67080e10i		−	1.39342i
$759$		0			0
$760$	−3.88024e7	−	2.66233e8i	−0.00320635	−	0.0219996i
$761$	1.81052e10			1.48922			0.744608	−	0.667502i	$-0.232637\pi$
$761$	1.81052e10			1.48922			0.744608	+	0.667502i	$0.232637\pi$
$762$		0			0
$763$			1.51899e10i			1.23800i
$764$	−2.23927e10			−1.81668
$765$		0			0
$766$	1.19437e10			0.960151
$767$		−	1.92514e10i		−	1.54056i
$768$		0			0
$769$	4.21234e9			0.334026			0.167013	−	0.985955i	$-0.446588\pi$
$769$	4.21234e9			0.334026			0.167013	+	0.985955i	$0.446588\pi$
$770$	−1.54036e10	+	2.24502e9i	−1.21592	+	0.177216i
$771$		0			0
$772$		−	1.13442e10i		−	0.887386i
$773$			1.99796e10i			1.55582i	0.628379	+	0.777908i	$0.283719\pi$
$773$			1.99796e10i			1.55582i	−0.628379	+	0.777908i	$0.716281\pi$
$774$		0			0
$775$	−1.82839e10	+	5.44526e9i	−1.41095	+	0.420207i
$776$	7.29135e8			0.0560134
$777$		0			0
$778$			1.97830e10i			1.50614i
$779$	−9.19668e9			−0.697028
$780$		0			0
$781$	−1.76072e9			−0.132255
$782$			3.97353e7i			0.00297135i
$783$		0			0
$784$	−4.43064e8			−0.0328368
$785$	−4.00912e7	−	2.75076e8i	−0.00295805	−	0.0202959i
$786$		0			0
$787$			1.58792e9i			0.116123i	0.998313	+	0.0580613i	$0.0184919\pi$
$787$			1.58792e9i			0.116123i	−0.998313	+	0.0580613i	$0.981508\pi$
$788$		−	9.87747e9i		−	0.719124i
$789$		0			0
$790$	1.40934e9	+	9.66985e9i	0.101700	+	0.697791i
$791$	−1.22495e10			−0.880034
$792$		0			0
$793$			1.72924e10i			1.23140i
$794$	−1.87580e10			−1.32989
$795$		0			0
$796$	−1.10765e10			−0.778410
$797$		−	6.84247e9i		−	0.478750i	−0.970927	−	0.239375i	$-0.923058\pi$
$797$		−	6.84247e9i		−	0.478750i	0.970927	−	0.239375i	$-0.0769425\pi$
$798$		0			0
$799$	−1.44206e10			−1.00016
$800$	5.88531e9	+	1.97615e10i	0.406401	+	1.36460i
$801$		0			0
$802$		−	1.13781e9i		−	0.0778858i
$803$			1.57082e10i			1.07059i
$804$		0			0
$805$	5.56920e7	−	8.11687e6i	0.00376277	−	0.000548407i
$806$	−3.86330e10			−2.59888
$807$		0			0
$808$		−	6.31261e8i		−	0.0420987i
$809$	−9.65265e9			−0.640954			−0.320477	−	0.947256i	$-0.603843\pi$
$809$	−9.65265e9			−0.640954			−0.320477	+	0.947256i	$0.603843\pi$
$810$		0			0
$811$	−1.52539e9			−0.100417			−0.0502086	−	0.998739i	$-0.515989\pi$
$811$	−1.52539e9			−0.100417			−0.0502086	+	0.998739i	$0.515989\pi$
$812$		−	1.60871e10i		−	1.05447i
$813$		0			0
$814$	3.19115e10			2.07378
$815$	3.59203e8	+	2.46458e9i	0.0232428	+	0.159475i
$816$		0			0
$817$			9.79739e9i			0.628540i
$818$		−	7.81102e9i		−	0.498966i
$819$		0			0
$820$	−2.24983e10	+	3.27904e9i	−1.42496	+	0.207682i
$821$	2.60390e10			1.64219			0.821096	−	0.570790i	$-0.193363\pi$
$821$	2.60390e10			1.64219			0.821096	+	0.570790i	$0.193363\pi$
$822$		0			0
$823$		−	1.13082e10i		−	0.707120i	−0.935412	−	0.353560i	$-0.884971\pi$
$823$		−	1.13082e10i		−	0.707120i	0.935412	−	0.353560i	$-0.115029\pi$
$824$	1.24270e9			0.0773789
$825$		0			0
$826$	−2.82199e10			−1.74231
$827$		−	1.80755e10i		−	1.11127i	−0.831425	−	0.555637i	$-0.812474\pi$
$827$		−	1.80755e10i		−	1.11127i	0.831425	−	0.555637i	$-0.187526\pi$
$828$		0			0
$829$	2.51806e10			1.53506			0.767531	−	0.641012i	$-0.221485\pi$
$829$	2.51806e10			1.53506			0.767531	+	0.641012i	$0.221485\pi$
$830$	−3.33760e9	+	4.86441e8i	−0.202610	+	0.0295296i
$831$		0			0
$832$			2.18417e10i			1.31479i
$833$			3.05034e8i			0.0182848i
$834$		0			0
$835$	−7.46984e8	−	5.12525e9i	−0.0444026	−	0.304658i
$836$	−7.62813e9			−0.451540
$837$		0			0
$838$		−	3.31056e10i		−	1.94333i
$839$	2.72950e10			1.59557			0.797787	−	0.602939i	$-0.206004\pi$
$839$	2.72950e10			1.59557			0.797787	+	0.602939i	$0.206004\pi$
$840$		0			0
$841$	1.41888e9			0.0822544
$842$			7.48885e9i			0.432338i
$843$		0			0
$844$	1.27784e10			0.731606
$845$	9.27092e9	−	1.35120e9i	0.528597	−	0.0770407i
$846$		0			0
$847$		−	4.00775e9i		−	0.226625i
$848$			1.75357e10i			0.987499i
$849$		0			0
$850$	1.31795e10	−	3.92509e9i	0.736093	−	0.219222i
$851$	−1.15376e8			−0.00641746
$852$		0			0
$853$		−	1.29870e10i		−	0.716453i	−0.933635	−	0.358226i	$-0.883382\pi$
$853$		−	1.29870e10i		−	0.716453i	0.933635	−	0.358226i	$-0.116618\pi$
$854$	2.53483e10			1.39266
$855$		0			0
$856$	1.44936e9			0.0789800
$857$		−	8.31784e9i		−	0.451417i	−0.974195	−	0.225708i	$-0.927530\pi$
$857$		−	8.31784e9i		−	0.451417i	0.974195	−	0.225708i	$-0.0724697\pi$
$858$		0			0
$859$	8.94316e9			0.481410			0.240705	−	0.970598i	$-0.422621\pi$
$859$	8.94316e9			0.481410			0.240705	+	0.970598i	$0.422621\pi$
$860$	3.49322e9	+	2.39679e10i	0.187276	+	1.28495i
$861$		0			0
$862$		−	2.69030e10i		−	1.43062i
$863$			4.59600e9i			0.243412i	0.992566	+	0.121706i	$0.0388365\pi$
$863$			4.59600e9i			0.243412i	−0.992566	+	0.121706i	$0.961163\pi$
$864$		0			0
$865$	−1.88899e9	−	1.29608e10i	−0.0992368	−	0.680890i
$866$	1.74563e10			0.913357
$867$		0			0
$868$			2.87510e10i			1.49222i
$869$	8.39578e9			0.434002
$870$		0			0
$871$	9.74928e9			0.499930
$872$			1.09839e9i			0.0560980i
$873$		0			0
$874$	5.43234e7			0.00275231
$875$	−8.19352e9	−	1.76703e10i	−0.413469	−	0.891691i
$876$		0			0
$877$		−	2.17651e10i		−	1.08959i	−0.838569	−	0.544795i	$-0.816608\pi$
$877$		−	2.17651e10i		−	1.08959i	0.838569	−	0.544795i	$-0.183392\pi$
$878$		−	3.13282e10i		−	1.56209i
$879$		0			0
$880$	1.69818e10	−	2.47502e9i	0.840028	−	0.122431i
$881$	3.32060e10			1.63607			0.818034	−	0.575170i	$-0.195064\pi$
$881$	3.32060e10			1.63607			0.818034	+	0.575170i	$0.195064\pi$
$882$		0			0
$883$		−	3.47892e10i		−	1.70052i	−0.526361	−	0.850261i	$-0.676444\pi$
$883$		−	3.47892e10i		−	1.70052i	0.526361	−	0.850261i	$-0.323556\pi$
$884$	1.41380e10			0.688345
$885$		0			0
$886$	−5.24693e10			−2.53447
$887$			3.48616e10i			1.67731i	0.544660	+	0.838657i	$0.316659\pi$
$887$			3.48616e10i			1.67731i	−0.544660	+	0.838657i	$0.683341\pi$
$888$		0			0
$889$	−2.54504e10			−1.21490
$890$	−3.45180e9	−	2.36837e10i	−0.164127	−	1.12612i
$891$		0			0
$892$			1.75371e10i			0.827336i
$893$			1.97149e10i			0.926432i
$894$		0			0
$895$	1.00842e9	−	1.46973e8i	0.0470178	−	0.00685264i
$896$	1.88422e9			0.0875092
$897$		0			0
$898$			2.56994e10i			1.18429i
$899$	−3.33648e10			−1.53155
$900$		0			0
$901$	1.20727e10			0.549879
$902$			3.84761e10i			1.74570i
$903$		0			0
$904$	−8.85761e8			−0.0398774
$905$	2.48464e10	−	3.62126e9i	1.11428	−	0.162401i
$906$		0			0
$907$			1.82643e10i			0.812788i	0.913698	+	0.406394i	$0.133214\pi$
$907$			1.82643e10i			0.812788i	−0.913698	+	0.406394i	$0.866786\pi$
$908$		−	1.28379e10i		−	0.569108i
$909$		0			0
$910$	−5.68854e9	−	3.90306e10i	−0.250240	−	1.71696i
$911$	−1.62009e10			−0.709945			−0.354973	−	0.934877i	$-0.615510\pi$
$911$	−1.62009e10			−0.709945			−0.354973	+	0.934877i	$0.615510\pi$
$912$		0			0
$913$			2.89785e9i			0.126017i
$914$	−1.67024e10			−0.723548
$915$		0			0
$916$	3.74902e9			0.161170
$917$		−	8.20393e9i		−	0.351341i
$918$		0			0
$919$	2.47808e10			1.05320			0.526599	−	0.850114i	$-0.323467\pi$
$919$	2.47808e10			1.05320			0.526599	+	0.850114i	$0.323467\pi$
$920$	4.02710e6	−	586932.i	0.000170504	−	2.48502e-5i
$921$		0			0
$922$			3.82418e10i			1.60687i
$923$		−	4.46141e9i		−	0.186753i
$924$		0			0
$925$	1.13970e10	+	3.82683e10i	0.473471	+	1.58980i
$926$	3.98316e10			1.64850
$927$		0			0
$928$			3.60612e10i			1.48123i
$929$	1.75952e9			0.0720011			0.0360005	−	0.999352i	$-0.488538\pi$
$929$	1.75952e9			0.0720011			0.0360005	+	0.999352i	$0.488538\pi$
$930$		0			0
$931$	4.17021e8			0.0169369
$932$		−	1.80494e10i		−	0.730311i
$933$		0			0
$934$	−1.10248e10			−0.442747
$935$	−1.70397e9	−	1.16914e10i	−0.0681742	−	0.467761i
$936$		0			0
$937$		−	4.17587e10i		−	1.65828i	−0.559040	−	0.829141i	$-0.688830\pi$
$937$		−	4.17587e10i		−	1.65828i	0.559040	−	0.829141i	$-0.311170\pi$
$938$		−	1.42911e10i		−	0.565401i
$939$		0			0
$940$	7.02925e9	+	4.82295e10i	0.276033	+	1.89394i
$941$	−2.01970e10			−0.790173			−0.395087	−	0.918644i	$-0.629285\pi$
$941$	−2.01970e10			−0.790173			−0.395087	+	0.918644i	$0.629285\pi$
$942$		0			0
$943$		−	1.39111e8i		−	0.00540219i
$944$	3.11111e10			1.20369
$945$		0			0
$946$	4.09893e10			1.57417
$947$			2.65395e10i			1.01547i	0.861513	+	0.507736i	$0.169517\pi$
$947$			2.65395e10i			1.01547i	−0.861513	+	0.507736i	$0.830483\pi$
$948$		0			0
$949$	−3.98023e10			−1.51174
$950$	−5.36611e9	−	1.80181e10i	−0.203061	−	0.681830i
$951$		0			0
$952$		−	6.28014e8i		−	0.0235907i
$953$		−	3.15558e9i		−	0.118101i	−0.998255	−	0.0590505i	$-0.981193\pi$
$953$		−	3.15558e9i		−	0.118101i	0.998255	−	0.0590505i	$-0.0188073\pi$
$954$		0			0
$955$	−4.69205e10	+	6.83847e9i	−1.74321	+	0.254066i
$956$	−3.87530e10			−1.43451
$957$		0			0
$958$		−	1.18807e10i		−	0.436580i
$959$	3.25387e10			1.19134
$960$		0			0
$961$	3.21171e10			1.16736
$962$			8.08591e10i			2.92830i
$963$		0			0
$964$	−2.66256e10			−0.957259
$965$	−3.46438e9	−	2.37700e10i	−0.124102	−	0.851499i
$966$		0			0
$967$			1.84578e10i			0.656428i	0.944603	+	0.328214i	$0.106447\pi$
$967$			1.84578e10i			0.656428i	−0.944603	+	0.328214i	$0.893553\pi$
$968$		−	2.89801e8i		−	0.0102692i
$969$		0			0
$970$	5.04172e10	−	7.34809e9i	1.77369	−	0.258508i
$971$	−3.98868e10			−1.39818			−0.699089	−	0.715035i	$-0.746411\pi$
$971$	−3.98868e10			−1.39818			−0.699089	+	0.715035i	$0.746411\pi$
$972$		0			0
$973$			9.43053e9i			0.328202i
$974$	−6.72874e10			−2.33334
$975$		0			0
$976$	−2.79453e10			−0.962131
$977$			3.05548e10i			1.04821i	0.851653	+	0.524106i	$0.175600\pi$
$977$			3.05548e10i			1.04821i	−0.851653	+	0.524106i	$0.824400\pi$
$978$		0			0
$979$	−2.05632e10			−0.700408
$980$	1.02018e9	−	1.48687e8i	0.0346247	−	0.00504641i
$981$		0			0
$982$		−	4.07663e10i		−	1.37376i
$983$		−	3.06865e10i		−	1.03041i	−0.857067	−	0.515206i	$-0.827716\pi$
$983$		−	3.06865e10i		−	1.03041i	0.857067	−	0.515206i	$-0.172284\pi$
$984$		0			0
$985$	−3.01647e9	−	2.06968e10i	−0.100571	−	0.690042i
$986$	2.40503e10			0.799007
$987$		0			0
$988$		−	1.93285e10i		−	0.637602i
$989$	−1.48197e8			−0.00487139
$990$		0			0
$991$	5.44706e10			1.77789			0.888944	−	0.458016i	$-0.151440\pi$
$991$	5.44706e10			1.77789			0.888944	+	0.458016i	$0.151440\pi$
$992$		−	6.44486e10i		−	2.09615i
$993$		0			0
$994$	−6.53983e9			−0.211210
$995$	−2.32092e10	+	3.38265e9i	−0.746930	+	0.108862i
$996$		0			0
$997$		−	1.17569e10i		−	0.375717i	−0.982196	−	0.187858i	$-0.939845\pi$
$997$		−	1.17569e10i		−	0.375717i	0.982196	−	0.187858i	$-0.0601546\pi$
$998$			1.75551e10i			0.559045i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.8.b.c.19.1	✓	4
3.2	odd	2	inner	45.8.b.c.19.4	yes	4
5.2	odd	4		225.8.a.ba.1.3		4
5.3	odd	4		225.8.a.ba.1.2		4
5.4	even	2	inner	45.8.b.c.19.3	yes	4
15.2	even	4		225.8.a.ba.1.1		4
15.8	even	4		225.8.a.ba.1.4		4
15.14	odd	2	inner	45.8.b.c.19.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.8.b.c.19.1	✓	4	1.1	even	1	trivial
45.8.b.c.19.2	yes	4	15.14	odd	2	inner
45.8.b.c.19.3	yes	4	5.4	even	2	inner
45.8.b.c.19.4	yes	4	3.2	odd	2	inner
225.8.a.ba.1.1		4	15.2	even	4
225.8.a.ba.1.2		4	5.3	odd	4
225.8.a.ba.1.3		4	5.2	odd	4
225.8.a.ba.1.4		4	15.8	even	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-16.1245i q^{2} -132.000 q^{4} +(-276.586 + 40.3113i) q^{5} +891.964i q^{7} +64.4981i q^{8} +(650.000 + 4459.82i) q^{10} +3872.21 q^{11} +9811.61i q^{13} +14382.5 q^{14} -15856.0 q^{16} +10916.3i q^{17} +14924.0 q^{19} +(36509.4 - 5321.09i) q^{20} -62437.5i q^{22} +225.743i q^{23} +(74875.0 - 22299.1i) q^{25} +158207. q^{26} -117739. i q^{28} +136634. q^{29} -244192. q^{31} +263926. i q^{32} +176020. q^{34} +(-35956.2 - 246705. i) q^{35} +511095. i q^{37} -240642. i q^{38} +(-2600.00 - 17839.3i) q^{40} -616234. q^{41} +656486. i q^{43} -511132. q^{44} +3640.00 q^{46} +1.32102e6i q^{47} +27943.0 q^{49} +(-359562. - 1.20732e6i) q^{50} -1.29513e6i q^{52} -1.10593e6i q^{53} +(-1.07100e6 + 156094. i) q^{55} -57530.0 q^{56} -2.20315e6i q^{58} -1.96210e6 q^{59} +1.76244e6 q^{61} +3.93748e6i q^{62} +2.22611e6 q^{64} +(-395518. - 2.71376e6i) q^{65} -993648. i q^{67} -1.44095e6i q^{68} +(-3.97800e6 + 579777. i) q^{70} -454708. q^{71} +4.05665e6i q^{73} +8.24117e6 q^{74} -1.96997e6 q^{76} +3.45387e6i q^{77} +2.16822e6 q^{79} +(4.38555e6 - 639176. i) q^{80} +9.93648e6i q^{82} +748371. i q^{83} +(-440050. - 3.01930e6i) q^{85} +1.05855e7 q^{86} +249750. i q^{88} -5.31046e6 q^{89} -8.75160e6 q^{91} -29798.1i q^{92} +2.13008e7 q^{94} +(-4.12777e6 + 601606. i) q^{95} -1.13048e7i q^{97} -450567. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 528 q^{4} + 2600 q^{10} - 63424 q^{16} + 59696 q^{19} + 299500 q^{25} - 976768 q^{31} + 704080 q^{34} - 10400 q^{40} + 14560 q^{46} + 111772 q^{49} - 4284000 q^{55} + 7049768 q^{61} + 8904448 q^{64}+ \cdots + 85203040 q^{94}+O(q^{100}) \) 4 * q - 528 * q^4 + 2600 * q^10 - 63424 * q^16 + 59696 * q^19 + 299500 * q^25 - 976768 * q^31 + 704080 * q^34 - 10400 * q^40 + 14560 * q^46 + 111772 * q^49 - 4284000 * q^55 + 7049768 * q^61 + 8904448 * q^64 - 15912000 * q^70 - 7879872 * q^76 + 8672864 * q^79 - 1760200 * q^85 - 35006400 * q^91 + 85203040 * q^94

Introduction

L-functions

Modular forms

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