LMFDB - Embedded newform 80.20.c.d.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [80,20,Mod(49,80)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(80, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("80.49");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 80 = 2^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	80.c (of order $2$, degree $1$, not 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:	$183.053357245$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.4
Character	$\chi$	$=$	80.49
Dual form			80.20.c.d.49.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-51907.0i q^{3} +(3.18724e6 - 2.98580e6i) q^{5} +9.49555e7i q^{7} -1.53207e9 q^{9} +O(q^{10})$	\(q-51907.0i q^{3} +(3.18724e6 - 2.98580e6i) q^{5} +9.49555e7i q^{7} -1.53207e9 q^{9} +2.75339e9 q^{11} +6.26869e10i q^{13} +(-1.54984e11 - 1.65440e11i) q^{15} -6.18891e11i q^{17} -5.01733e11 q^{19} +4.92885e12 q^{21} -4.34747e12i q^{23} +(1.24354e12 - 1.90329e13i) q^{25} +1.91957e13i q^{27} -1.23271e14 q^{29} -2.11787e13 q^{31} -1.42920e14i q^{33} +(2.83518e14 + 3.02646e14i) q^{35} -5.81760e14i q^{37} +3.25389e15 q^{39} -4.73607e14 q^{41} +5.02442e15i q^{43} +(-4.88308e15 + 4.57445e15i) q^{45} +1.08473e16i q^{47} +2.38236e15 q^{49} -3.21248e16 q^{51} +1.62044e16i q^{53} +(8.77572e15 - 8.22106e15i) q^{55} +2.60434e16i q^{57} -5.36470e16 q^{59} -6.07683e16 q^{61} -1.45478e17i q^{63} +(1.87170e17 + 1.99798e17i) q^{65} +4.31722e17i q^{67} -2.25664e17 q^{69} -2.10030e17 q^{71} -3.63041e17i q^{73} +(-9.87940e17 - 6.45482e16i) q^{75} +2.61449e17i q^{77} +6.38309e17 q^{79} -7.84278e17 q^{81} +1.00531e18i q^{83} +(-1.84788e18 - 1.97256e18i) q^{85} +6.39863e18i q^{87} -2.24662e18 q^{89} -5.95247e18 q^{91} +1.09932e18i q^{93} +(-1.59915e18 + 1.49807e18i) q^{95} +1.27007e19i q^{97} -4.21839e18 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 3935164 q^{5} - 10352560732 q^{9}+O(q^{10}) $ 28 * q - 3935164 * q^5 - 10352560732 * q^9	$ 28 q - 3935164 q^{5} - 10352560732 q^{9} - 18570995888 q^{11} - 79041415120 q^{15} + 1685869130672 q^{19} + 140419722832 q^{21} - 29558822439924 q^{25} + 160448197635496 q^{29} - 64251929934720 q^{31} + 88899634227664 q^{35} - 45\!\cdots\!04 q^{39}+ \cdots + 24\!\cdots\!92 q^{99}+O(q^{100}) $ 28 * q - 3935164 * q^5 - 10352560732 * q^9 - 18570995888 * q^11 - 79041415120 * q^15 + 1685869130672 * q^19 + 140419722832 * q^21 - 29558822439924 * q^25 + 160448197635496 * q^29 - 64251929934720 * q^31 + 88899634227664 * q^35 - 4563253596731104 * q^39 + 87868042694936 * q^41 + 3557698140217852 * q^45 - 28923781405647564 * q^49 - 14937379593456000 * q^51 - 28335039172010704 * q^55 + 83879503376835216 * q^59 + 2336306250785000 * q^61 + 23503427909569568 * q^65 - 46927873937137712 * q^69 - 104881848512783264 * q^71 - 1094024007292275040 * q^75 + 795721739900308672 * q^79 + 3593102866196377132 * q^81 + 834124526841485952 * q^85 + 1582238786752844568 * q^89 - 11017818924975052064 * q^91 - 30914057837649668144 * q^95 + 24571127463084047792 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/80\mathbb{Z}\right)^\times$.

$n$	$17$	$21$	$31$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$		−	51907.0i		−	1.52256i	−0.648425	−	0.761279i	$-0.724572\pi$
$3$		−	51907.0i		−	1.52256i	0.648425	−	0.761279i	$-0.275428\pi$
$4$		0			0
$5$	3.18724e6	−	2.98580e6i	0.729793	−	0.683668i
$5$	3.18724e6	−	2.98580e6i	0.729793	−	0.683668i
$6$		0			0
$7$			9.49555e7i			0.889382i	0.895684	+	0.444691i	$0.146687\pi$
$7$			9.49555e7i			0.889382i	−0.895684	+	0.444691i	$0.853313\pi$
$8$		0			0
$9$	−1.53207e9			−1.31818
$10$		0			0
$11$	2.75339e9			0.352077			0.176038	−	0.984383i	$-0.443672\pi$
$11$	2.75339e9			0.352077			0.176038	+	0.984383i	$0.443672\pi$
$12$		0			0
$13$			6.26869e10i			1.63951i	0.572712	+	0.819757i	$0.305892\pi$
$13$			6.26869e10i			1.63951i	−0.572712	+	0.819757i	$0.694108\pi$
$14$		0			0
$15$	−1.54984e11	−	1.65440e11i	−1.04092	−	1.11115i
$16$		0			0
$17$		−	6.18891e11i		−	1.26575i	−0.774252	−	0.632877i	$-0.781874\pi$
$17$		−	6.18891e11i		−	1.26575i	0.774252	−	0.632877i	$-0.218126\pi$
$18$		0			0
$19$	−5.01733e11			−0.356709			−0.178354	−	0.983966i	$-0.557077\pi$
$19$	−5.01733e11			−0.356709			−0.178354	+	0.983966i	$0.557077\pi$
$20$		0			0
$21$	4.92885e12			1.35414
$22$		0			0
$23$		−	4.34747e12i		−	0.503295i	−0.967819	−	0.251647i	$-0.919028\pi$
$23$		−	4.34747e12i		−	0.503295i	0.967819	−	0.251647i	$-0.0809723\pi$
$24$		0			0
$25$	1.24354e12	−	1.90329e13i	0.0651971	−	0.997872i
$26$		0			0
$27$			1.91957e13i			0.484449i
$28$		0			0
$29$	−1.23271e14			−1.57790			−0.788952	−	0.614455i	$-0.789376\pi$
$29$	−1.23271e14			−1.57790			−0.788952	+	0.614455i	$0.789376\pi$
$30$		0			0
$31$	−2.11787e13			−0.143868			−0.0719340	−	0.997409i	$-0.522917\pi$
$31$	−2.11787e13			−0.143868			−0.0719340	+	0.997409i	$0.522917\pi$
$32$		0			0
$33$		−	1.42920e14i		−	0.536057i
$34$		0			0
$35$	2.83518e14	+	3.02646e14i	0.608042	+	0.649065i
$36$		0			0
$37$		−	5.81760e14i		−	0.735915i	−0.929843	−	0.367958i	$-0.880057\pi$
$37$		−	5.81760e14i		−	0.735915i	0.929843	−	0.367958i	$-0.119943\pi$
$38$		0			0
$39$	3.25389e15			2.49625
$40$		0			0
$41$	−4.73607e14			−0.225929			−0.112964	−	0.993599i	$-0.536035\pi$
$41$	−4.73607e14			−0.225929			−0.112964	+	0.993599i	$0.536035\pi$
$42$		0			0
$43$			5.02442e15i			1.52453i	0.647266	+	0.762264i	$0.275912\pi$
$43$			5.02442e15i			1.52453i	−0.647266	+	0.762264i	$0.724088\pi$
$44$		0			0
$45$	−4.88308e15	+	4.57445e15i	−0.962000	+	0.901198i
$46$		0			0
$47$			1.08473e16i			1.41381i	0.707309	+	0.706905i	$0.249909\pi$
$47$			1.08473e16i			1.41381i	−0.707309	+	0.706905i	$0.750091\pi$
$48$		0			0
$49$	2.38236e15			0.208999
$50$		0			0
$51$	−3.21248e16			−1.92718
$52$		0			0
$53$			1.62044e16i			0.674549i	0.941406	+	0.337274i	$0.109505\pi$
$53$			1.62044e16i			0.674549i	−0.941406	+	0.337274i	$0.890495\pi$
$54$		0			0
$55$	8.77572e15	−	8.22106e15i	0.256943	−	0.240703i
$56$		0			0
$57$			2.60434e16i			0.543109i
$58$		0			0
$59$	−5.36470e16			−0.806216			−0.403108	−	0.915152i	$-0.632070\pi$
$59$	−5.36470e16			−0.806216			−0.403108	+	0.915152i	$0.632070\pi$
$60$		0			0
$61$	−6.07683e16			−0.665340			−0.332670	−	0.943043i	$-0.607949\pi$
$61$	−6.07683e16			−0.665340			−0.332670	+	0.943043i	$0.607949\pi$
$62$		0			0
$63$		−	1.45478e17i		−	1.17237i
$64$		0			0
$65$	1.87170e17	+	1.99798e17i	1.12088	+	1.19651i
$66$		0			0
$67$			4.31722e17i			1.93863i	0.245830	+	0.969313i	$0.420939\pi$
$67$			4.31722e17i			1.93863i	−0.245830	+	0.969313i	$0.579061\pi$
$68$		0			0
$69$	−2.25664e17			−0.766295
$70$		0			0
$71$	−2.10030e17			−0.543660			−0.271830	−	0.962345i	$-0.587629\pi$
$71$	−2.10030e17			−0.543660			−0.271830	+	0.962345i	$0.587629\pi$
$72$		0			0
$73$		−	3.63041e17i		−	0.721752i	−0.932614	−	0.360876i	$-0.882478\pi$
$73$		−	3.63041e17i		−	0.721752i	0.932614	−	0.360876i	$-0.117522\pi$
$74$		0			0
$75$	−9.87940e17	−	6.45482e16i	−1.51932	−	0.0992663i
$76$		0			0
$77$			2.61449e17i			0.313131i
$78$		0			0
$79$	6.38309e17			0.599202			0.299601	−	0.954065i	$-0.403146\pi$
$79$	6.38309e17			0.599202			0.299601	+	0.954065i	$0.403146\pi$
$80$		0			0
$81$	−7.84278e17			−0.580580
$82$		0			0
$83$			1.00531e18i			0.590280i	0.955454	+	0.295140i	$0.0953663\pi$
$83$			1.00531e18i			0.590280i	−0.955454	+	0.295140i	$0.904634\pi$
$84$		0			0
$85$	−1.84788e18	−	1.97256e18i	−0.865355	−	0.923739i
$86$		0			0
$87$			6.39863e18i			2.40245i
$88$		0			0
$89$	−2.24662e18			−0.679713			−0.339856	−	0.940477i	$-0.610378\pi$
$89$	−2.24662e18			−0.679713			−0.339856	+	0.940477i	$0.610378\pi$
$90$		0			0
$91$	−5.95247e18			−1.45815
$92$		0			0
$93$			1.09932e18i			0.219047i
$94$		0			0
$95$	−1.59915e18	+	1.49807e18i	−0.260324	+	0.243870i
$96$		0			0
$97$			1.27007e19i			1.69627i	0.529781	+	0.848135i	$0.322274\pi$
$97$			1.27007e19i			1.69627i	−0.529781	+	0.848135i	$0.677726\pi$
$98$		0			0
$99$	−4.21839e18			−0.464101
$100$		0			0
$101$	1.86126e19			1.69338			0.846690	−	0.532087i	$-0.178592\pi$
$101$	1.86126e19			1.69338			0.846690	+	0.532087i	$0.178592\pi$
$102$		0			0
$103$			1.72849e19i			1.30531i	0.757656	+	0.652654i	$0.226344\pi$
$103$			1.72849e19i			1.30531i	−0.757656	+	0.652654i	$0.773656\pi$
$104$		0			0
$105$	1.57094e19	−	1.47165e19i	0.988239	−	0.925779i
$106$		0			0
$107$		−	1.53090e19i		−	0.805008i	−0.915418	−	0.402504i	$-0.868140\pi$
$107$		−	1.53090e19i		−	0.805008i	0.915418	−	0.402504i	$-0.131860\pi$
$108$		0			0
$109$	4.19011e19			1.84788			0.923940	−	0.382537i	$-0.124950\pi$
$109$	4.19011e19			1.84788			0.923940	+	0.382537i	$0.124950\pi$
$110$		0			0
$111$	−3.01974e19			−1.12047
$112$		0			0
$113$		−	3.54178e19i		−	1.10911i	−0.832146	−	0.554557i	$-0.812888\pi$
$113$		−	3.54178e19i		−	1.10911i	0.832146	−	0.554557i	$-0.187112\pi$
$114$		0			0
$115$	−1.29807e19	−	1.38564e19i	−0.344086	−	0.367301i
$116$		0			0
$117$		−	9.60408e19i		−	2.16118i
$118$		0			0
$119$	5.87671e19			1.12574
$120$		0			0
$121$	−5.35779e19			−0.876042
$122$		0			0
$123$			2.45835e19i			0.343989i
$124$		0			0
$125$	−5.28649e19	−	6.43754e19i	−0.634633	−	0.772814i
$126$		0			0
$127$			7.80973e19i			0.806308i	0.915132	+	0.403154i	$0.132086\pi$
$127$			7.80973e19i			0.806308i	−0.915132	+	0.403154i	$0.867914\pi$
$128$		0			0
$129$	2.60802e20			2.32118
$130$		0			0
$131$	−1.40863e19			−0.108323			−0.0541615	−	0.998532i	$-0.517249\pi$
$131$	−1.40863e19			−0.108323			−0.0541615	+	0.998532i	$0.517249\pi$
$132$		0			0
$133$		−	4.76423e19i		−	0.317250i
$134$		0			0
$135$	5.73144e19	+	6.11813e19i	0.331202	+	0.353547i
$136$		0			0
$137$		−	2.84488e20i		−	1.42961i	−0.699323	−	0.714806i	$-0.746515\pi$
$137$		−	2.84488e20i		−	1.42961i	0.699323	−	0.714806i	$-0.253485\pi$
$138$		0			0
$139$	−2.56887e19			−0.112487			−0.0562434	−	0.998417i	$-0.517912\pi$
$139$	−2.56887e19			−0.112487			−0.0562434	+	0.998417i	$0.517912\pi$
$140$		0			0
$141$	5.63049e20			2.15261
$142$		0			0
$143$			1.72602e20i			0.577235i
$144$		0			0
$145$	−3.92895e20	+	3.68063e20i	−1.15154	+	1.07876i
$146$		0			0
$147$		−	1.23661e20i		−	0.318213i
$148$		0			0
$149$	−6.04560e19			−0.136826			−0.0684132	−	0.997657i	$-0.521794\pi$
$149$	−6.04560e19			−0.136826			−0.0684132	+	0.997657i	$0.521794\pi$
$150$		0			0
$151$	−9.35785e20			−1.86593			−0.932964	−	0.359969i	$-0.882787\pi$
$151$	−9.35785e20			−1.86593			−0.932964	+	0.359969i	$0.882787\pi$
$152$		0			0
$153$			9.48185e20i			1.66849i
$154$		0			0
$155$	−6.75018e19	+	6.32354e19i	−0.104994	+	0.0983578i
$156$		0			0
$157$			6.63878e20i			0.914201i	0.889415	+	0.457101i	$0.151112\pi$
$157$			6.63878e20i			0.914201i	−0.889415	+	0.457101i	$0.848888\pi$
$158$		0			0
$159$	8.41124e20			1.02704
$160$		0			0
$161$	4.12816e20			0.447622
$162$		0			0
$163$			9.73165e20i			0.938435i	0.883083	+	0.469218i	$0.155464\pi$
$163$			9.73165e20i			0.938435i	−0.883083	+	0.469218i	$0.844536\pi$
$164$		0			0
$165$	−4.26730e20	−	4.55521e20i	−0.366485	−	0.391211i
$166$		0			0
$167$		−	1.02768e20i		−	0.0787137i	−0.999225	−	0.0393569i	$-0.987469\pi$
$167$		−	1.02768e20i		−	0.0787137i	0.999225	−	0.0393569i	$-0.0125309\pi$
$168$		0			0
$169$	−2.46773e21			−1.68801
$170$		0			0
$171$	7.68691e20			0.470207
$172$		0			0
$173$			1.57020e21i			0.860039i	0.902820	+	0.430020i	$0.141493\pi$
$173$			1.57020e21i			0.860039i	−0.902820	+	0.430020i	$0.858507\pi$
$174$		0			0
$175$	1.80728e21	+	1.18081e20i	0.887490	+	0.0579851i
$176$		0			0
$177$			2.78465e21i			1.22751i
$178$		0			0
$179$	1.88879e21			0.748308			0.374154	−	0.927367i	$-0.377933\pi$
$179$	1.88879e21			0.748308			0.374154	+	0.927367i	$0.377933\pi$
$180$		0			0
$181$	5.78786e20			0.206334			0.103167	−	0.994664i	$-0.467102\pi$
$181$	5.78786e20			0.206334			0.103167	+	0.994664i	$0.467102\pi$
$182$		0			0
$183$			3.15430e21i			1.01302i
$184$		0			0
$185$	−1.73702e21	−	1.85421e21i	−0.503121	−	0.537066i
$186$		0			0
$187$		−	1.70405e21i		−	0.445643i
$188$		0			0
$189$	−1.82273e21			−0.430860
$190$		0			0
$191$	−4.82963e21			−1.03299			−0.516495	−	0.856290i	$-0.672764\pi$
$191$	−4.82963e21			−1.03299			−0.516495	+	0.856290i	$0.672764\pi$
$192$		0			0
$193$			4.40372e21i			0.853149i	0.904452	+	0.426575i	$0.140280\pi$
$193$			4.40372e21i			0.853149i	−0.904452	+	0.426575i	$0.859720\pi$
$194$		0			0
$195$	1.03709e22	−	9.71544e21i	1.82175	−	1.70661i
$196$		0			0
$197$			2.23795e21i			0.356797i	0.983958	+	0.178399i	$0.0570916\pi$
$197$			2.23795e21i			0.356797i	−0.983958	+	0.178399i	$0.942908\pi$
$198$		0			0
$199$	1.32253e22			1.91558			0.957789	−	0.287471i	$-0.0928145\pi$
$199$	1.32253e22			1.91558			0.957789	+	0.287471i	$0.0928145\pi$
$200$		0			0
$201$	2.24094e22			2.95167
$202$		0			0
$203$		−	1.17053e22i		−	1.40336i
$204$		0			0
$205$	−1.50950e21	+	1.41409e21i	−0.164881	+	0.154460i
$206$		0			0
$207$			6.66064e21i			0.663434i
$208$		0			0
$209$	−1.38147e21			−0.125589
$210$		0			0
$211$	1.14422e21			0.0950221			0.0475110	−	0.998871i	$-0.484871\pi$
$211$	1.14422e21			0.0950221			0.0475110	+	0.998871i	$0.484871\pi$
$212$		0			0
$213$			1.09020e22i			0.827754i
$214$		0			0
$215$	1.50019e22	+	1.60140e22i	1.04227	+	1.11259i
$216$		0			0
$217$		−	2.01104e21i		−	0.127954i
$218$		0			0
$219$	−1.88443e22			−1.09891
$220$		0			0
$221$	3.87964e22			2.07522
$222$		0			0
$223$			2.44066e22i			1.19842i	0.800591	+	0.599212i	$0.204519\pi$
$223$			2.44066e22i			1.19842i	−0.800591	+	0.599212i	$0.795481\pi$
$224$		0			0
$225$	−1.90518e21	+	2.91598e22i	−0.0859416	+	1.31538i
$226$		0			0
$227$			1.92778e22i			0.799487i	0.916627	+	0.399743i	$0.130901\pi$
$227$			1.92778e22i			0.799487i	−0.916627	+	0.399743i	$0.869099\pi$
$228$		0			0
$229$	1.86172e22			0.710359			0.355180	−	0.934798i	$-0.384420\pi$
$229$	1.86172e22			0.710359			0.355180	+	0.934798i	$0.384420\pi$
$230$		0			0
$231$	1.35710e22			0.476760
$232$		0			0
$233$		−	2.27033e22i		−	0.734867i	−0.930050	−	0.367433i	$-0.880237\pi$
$233$		−	2.27033e22i		−	0.734867i	0.930050	−	0.367433i	$-0.119763\pi$
$234$		0			0
$235$	3.23877e22	+	3.45729e22i	0.966576	+	1.03179i
$236$		0			0
$237$		−	3.31327e22i		−	0.912320i
$238$		0			0
$239$	2.52661e22			0.642329			0.321164	−	0.947023i	$-0.395926\pi$
$239$	2.52661e22			0.642329			0.321164	+	0.947023i	$0.395926\pi$
$240$		0			0
$241$	−5.86168e22			−1.37677			−0.688383	−	0.725347i	$-0.741679\pi$
$241$	−5.86168e22			−1.37677			−0.688383	+	0.725347i	$0.741679\pi$
$242$		0			0
$243$			6.30199e22i			1.36842i
$244$		0			0
$245$	7.59315e21	−	7.11323e21i	0.152526	−	0.142886i
$246$		0			0
$247$		−	3.14521e22i		−	0.584829i
$248$		0			0
$249$	5.21826e22			0.898735
$250$		0			0
$251$	−8.86574e22			−1.41519			−0.707595	−	0.706619i	$-0.750220\pi$
$251$	−8.86574e22			−1.41519			−0.707595	+	0.706619i	$0.750220\pi$
$252$		0			0
$253$		−	1.19703e22i		−	0.177198i
$254$		0			0
$255$	−1.02389e23	+	9.59180e22i	−1.40645	+	1.31755i
$256$		0			0
$257$			1.04451e23i			1.33213i	0.745893	+	0.666065i	$0.232023\pi$
$257$			1.04451e23i			1.33213i	−0.745893	+	0.666065i	$0.767977\pi$
$258$		0			0
$259$	5.52413e22			0.654510
$260$		0			0
$261$	1.88860e23			2.07996
$262$		0			0
$263$		−	1.57350e21i		−	0.0161171i	−0.999968	−	0.00805855i	$-0.997435\pi$
$263$		−	1.57350e21i		−	0.0161171i	0.999968	−	0.00805855i	$-0.00256515\pi$
$264$		0			0
$265$	4.83832e22	+	5.16475e22i	0.461167	+	0.492281i
$266$		0			0
$267$			1.16615e23i			1.03490i
$268$		0			0
$269$	9.71828e22			0.803420			0.401710	−	0.915767i	$-0.368416\pi$
$269$	9.71828e22			0.803420			0.401710	+	0.915767i	$0.368416\pi$
$270$		0			0
$271$	1.57587e22			0.121427			0.0607133	−	0.998155i	$-0.480662\pi$
$271$	1.57587e22			0.121427			0.0607133	+	0.998155i	$0.480662\pi$
$272$		0			0
$273$			3.08974e23i			2.22012i
$274$		0			0
$275$	3.42394e21	−	5.24050e22i	0.0229544	−	0.351328i
$276$		0			0
$277$		−	9.05740e22i		−	0.566821i	−0.958999	−	0.283411i	$-0.908534\pi$
$277$		−	9.05740e22i		−	0.566821i	0.958999	−	0.283411i	$-0.0914659\pi$
$278$		0			0
$279$	3.24473e22			0.189644
$280$		0			0
$281$	−1.95756e23			−1.06907			−0.534534	−	0.845147i	$-0.679513\pi$
$281$	−1.95756e23			−1.06907			−0.534534	+	0.845147i	$0.679513\pi$
$282$		0			0
$283$			1.80289e23i			0.920446i	0.887803	+	0.460223i	$0.152231\pi$
$283$			1.80289e23i			0.920446i	−0.887803	+	0.460223i	$0.847769\pi$
$284$		0			0
$285$	7.77604e22	+	8.30068e22i	0.371306	+	0.396358i
$286$		0			0
$287$		−	4.49716e22i		−	0.200937i
$288$		0			0
$289$	−1.43954e23			−0.602135
$290$		0			0
$291$	6.59253e23			2.58267
$292$		0			0
$293$		−	4.98213e23i		−	1.82883i	−0.404780	−	0.914414i	$-0.632652\pi$
$293$		−	4.98213e23i		−	1.82883i	0.404780	−	0.914414i	$-0.367348\pi$
$294$		0			0
$295$	−1.70986e23	+	1.60179e23i	−0.588371	+	0.551184i
$296$		0			0
$297$			5.28532e22i			0.170563i
$298$		0			0
$299$	2.72530e23			0.825159
$300$		0			0
$301$	−4.77096e23			−1.35589
$302$		0			0
$303$		−	9.66124e23i		−	2.57827i
$304$		0			0
$305$	−1.93683e23	+	1.81442e23i	−0.485561	+	0.454871i
$306$		0			0
$307$		−	7.55902e22i		−	0.178095i	−0.996027	−	0.0890473i	$-0.971618\pi$
$307$		−	7.55902e22i		−	0.178095i	0.996027	−	0.0890473i	$-0.0283822\pi$
$308$		0			0
$309$	8.97206e23			1.98741
$310$		0			0
$311$	9.02693e23			1.88069			0.940343	−	0.340228i	$-0.110504\pi$
$311$	9.02693e23			1.88069			0.940343	+	0.340228i	$0.110504\pi$
$312$		0			0
$313$			4.36458e23i			0.855601i	0.903873	+	0.427801i	$0.140711\pi$
$313$			4.36458e23i			0.855601i	−0.903873	+	0.427801i	$0.859289\pi$
$314$		0			0
$315$	−4.34369e23	−	4.63675e23i	−0.801509	−	0.855586i
$316$		0			0
$317$		−	9.56760e23i		−	1.66242i	−0.555961	−	0.831208i	$-0.687650\pi$
$317$		−	9.56760e23i		−	1.66242i	0.555961	−	0.831208i	$-0.312350\pi$
$318$		0			0
$319$	−3.39414e23			−0.555543
$320$		0			0
$321$	−7.94642e23			−1.22567
$322$		0			0
$323$			3.10518e23i			0.451506i
$324$		0			0
$325$	1.19311e24	+	7.79534e22i	1.63603	+	0.106892i
$326$		0			0
$327$		−	2.17496e24i		−	2.81350i
$328$		0			0
$329$	−1.03001e24			−1.25742
$330$		0			0
$331$	−7.98381e23			−0.920120			−0.460060	−	0.887888i	$-0.652172\pi$
$331$	−7.98381e23			−0.920120			−0.460060	+	0.887888i	$0.652172\pi$
$332$		0			0
$333$			8.91298e23i			0.970069i
$334$		0			0
$335$	1.28903e24	+	1.37600e24i	1.32538	+	1.41480i
$336$		0			0
$337$			1.05297e23i			0.102313i	0.998691	+	0.0511567i	$0.0162908\pi$
$337$			1.05297e23i			0.102313i	−0.998691	+	0.0511567i	$0.983709\pi$
$338$		0			0
$339$	−1.83843e24			−1.68869
$340$		0			0
$341$	−5.83133e22			−0.0506525
$342$		0			0
$343$			1.30861e24i			1.07526i
$344$		0			0
$345$	−7.19246e23	+	6.73787e23i	−0.559237	+	0.523891i
$346$		0			0
$347$		−	2.22010e24i		−	1.63397i	−0.576661	−	0.816983i	$-0.695645\pi$
$347$		−	2.22010e24i		−	1.63397i	0.576661	−	0.816983i	$-0.304355\pi$
$348$		0			0
$349$	−2.24343e24			−1.56340			−0.781700	−	0.623654i	$-0.785647\pi$
$349$	−2.24343e24			−1.56340			−0.781700	+	0.623654i	$0.785647\pi$
$350$		0			0
$351$	−1.20332e24			−0.794260
$352$		0			0
$353$			1.44469e24i			0.903472i	0.892152	+	0.451736i	$0.149195\pi$
$353$			1.44469e24i			0.903472i	−0.892152	+	0.451736i	$0.850805\pi$
$354$		0			0
$355$	−6.69417e23	+	6.27108e23i	−0.396760	+	0.371683i
$356$		0			0
$357$		−	3.05042e24i		−	1.71400i
$358$		0			0
$359$	−2.47412e24			−1.31833			−0.659163	−	0.752000i	$-0.729089\pi$
$359$	−2.47412e24			−1.31833			−0.659163	+	0.752000i	$0.729089\pi$
$360$		0			0
$361$	−1.72668e24			−0.872759
$362$		0			0
$363$			2.78107e24i			1.33382i
$364$		0			0
$365$	−1.08397e24	−	1.15710e24i	−0.493439	−	0.526730i
$366$		0			0
$367$			2.79620e24i			1.20848i	0.796801	+	0.604241i	$0.206524\pi$
$367$			2.79620e24i			1.20848i	−0.796801	+	0.604241i	$0.793476\pi$
$368$		0			0
$369$	7.25599e23			0.297815
$370$		0			0
$371$	−1.53870e24			−0.599932
$372$		0			0
$373$		−	4.66160e22i		−	0.0172704i	−0.999963	−	0.00863518i	$-0.997251\pi$
$373$		−	4.66160e22i		−	0.0172704i	0.999963	−	0.00863518i	$-0.00274870\pi$
$374$		0			0
$375$	−3.34153e24	+	2.74406e24i	−1.17665	+	0.966265i
$376$		0			0
$377$		−	7.72749e24i		−	2.58700i
$378$		0			0
$379$	4.68682e24			1.49213			0.746063	−	0.665875i	$-0.231942\pi$
$379$	4.68682e24			1.49213			0.746063	+	0.665875i	$0.231942\pi$
$380$		0			0
$381$	4.05380e24			1.22765
$382$		0			0
$383$		−	3.47940e24i		−	1.00257i	−0.865282	−	0.501286i	$-0.832860\pi$
$383$		−	3.47940e24i		−	1.00257i	0.865282	−	0.501286i	$-0.167140\pi$
$384$		0			0
$385$	7.80634e23	+	8.33303e23i	0.214077	+	0.228521i
$386$		0			0
$387$		−	7.69776e24i		−	2.00960i
$388$		0			0
$389$	−7.55787e23			−0.187879			−0.0939395	−	0.995578i	$-0.529946\pi$
$389$	−7.55787e23			−0.187879			−0.0939395	+	0.995578i	$0.529946\pi$
$390$		0			0
$391$	−2.69061e24			−0.637048
$392$		0			0
$393$			7.31179e23i			0.164928i
$394$		0			0
$395$	2.03445e24	−	1.90586e24i	0.437294	−	0.409655i
$396$		0			0
$397$		−	7.54400e24i		−	1.54558i	−0.634661	−	0.772791i	$-0.718860\pi$
$397$		−	7.54400e24i		−	1.54558i	0.634661	−	0.772791i	$-0.281140\pi$
$398$		0			0
$399$	−2.47297e24			−0.483032
$400$		0			0
$401$	6.32118e24			1.17741			0.588704	−	0.808349i	$-0.299638\pi$
$401$	6.32118e24			1.17741			0.588704	+	0.808349i	$0.299638\pi$
$402$		0			0
$403$		−	1.32763e24i		−	0.235873i
$404$		0			0
$405$	−2.49968e24	+	2.34169e24i	−0.423704	+	0.396924i
$406$		0			0
$407$		−	1.60181e24i		−	0.259099i
$408$		0			0
$409$	−3.00935e24			−0.464624			−0.232312	−	0.972641i	$-0.574629\pi$
$409$	−3.00935e24			−0.464624			−0.232312	+	0.972641i	$0.574629\pi$
$410$		0			0
$411$	−1.47669e25			−2.17667
$412$		0			0
$413$		−	5.09408e24i		−	0.717034i
$414$		0			0
$415$	3.00165e24	+	3.20417e24i	0.403555	+	0.430783i
$416$		0			0
$417$			1.33342e24i			0.171268i
$418$		0			0
$419$	2.13195e23			0.0261663			0.0130832	−	0.999914i	$-0.495835\pi$
$419$	2.13195e23			0.0261663			0.0130832	+	0.999914i	$0.495835\pi$
$420$		0			0
$421$	5.43076e24			0.637061			0.318530	−	0.947913i	$-0.396811\pi$
$421$	5.43076e24			0.637061			0.318530	+	0.947913i	$0.396811\pi$
$422$		0			0
$423$		−	1.66188e25i		−	1.86366i
$424$		0			0
$425$	−1.17793e25	−	7.69613e23i	−1.26306	−	0.0825235i
$426$		0			0
$427$		−	5.77028e24i		−	0.591742i
$428$		0			0
$429$	8.95922e24			0.878873
$430$		0			0
$431$	8.98233e24			0.843053			0.421527	−	0.906816i	$-0.361494\pi$
$431$	8.98233e24			0.843053			0.421527	+	0.906816i	$0.361494\pi$
$432$		0			0
$433$			3.26267e24i			0.293048i	0.989207	+	0.146524i	$0.0468085\pi$
$433$			3.26267e24i			0.293048i	−0.989207	+	0.146524i	$0.953192\pi$
$434$		0			0
$435$	1.91050e25	+	2.03940e25i	1.64248	+	1.75329i
$436$		0			0
$437$			2.18127e24i			0.179530i
$438$		0			0
$439$	2.11317e25			1.66541			0.832707	−	0.553714i	$-0.186790\pi$
$439$	2.11317e25			1.66541			0.832707	+	0.553714i	$0.186790\pi$
$440$		0			0
$441$	−3.64994e24			−0.275498
$442$		0			0
$443$			7.96861e24i			0.576166i	0.957606	+	0.288083i	$0.0930178\pi$
$443$			7.96861e24i			0.576166i	−0.957606	+	0.288083i	$0.906982\pi$
$444$		0			0
$445$	−7.16054e24	+	6.70796e24i	−0.496050	+	0.464698i
$446$		0			0
$447$			3.13809e24i			0.208326i
$448$		0			0
$449$	6.91113e24			0.439753			0.219877	−	0.975528i	$-0.429435\pi$
$449$	6.91113e24			0.439753			0.219877	+	0.975528i	$0.429435\pi$
$450$		0			0
$451$	−1.30402e24			−0.0795442
$452$		0			0
$453$			4.85737e25i			2.84098i
$454$		0			0
$455$	−1.89719e25	+	1.77728e25i	−1.06415	+	0.996893i
$456$		0			0
$457$		−	3.29466e24i		−	0.177258i	−0.996065	−	0.0886291i	$-0.971751\pi$
$457$		−	3.29466e24i		−	0.177258i	0.996065	−	0.0886291i	$-0.0282486\pi$
$458$		0			0
$459$	1.18800e25			0.613193
$460$		0			0
$461$	−3.23204e25			−1.60073			−0.800365	−	0.599513i	$-0.795361\pi$
$461$	−3.23204e25			−1.60073			−0.800365	+	0.599513i	$0.795361\pi$
$462$		0			0
$463$			4.83832e24i			0.229972i	0.993367	+	0.114986i	$0.0366823\pi$
$463$			4.83832e24i			0.229972i	−0.993367	+	0.114986i	$0.963318\pi$
$464$		0			0
$465$	3.28236e24	+	3.50381e24i	0.149755	+	0.159859i
$466$		0			0
$467$		−	2.95292e25i		−	1.29343i	−0.762734	−	0.646713i	$-0.776143\pi$
$467$		−	2.95292e25i		−	1.29343i	0.762734	−	0.646713i	$-0.223857\pi$
$468$		0			0
$469$	−4.09944e25			−1.72418
$470$		0			0
$471$	3.44599e25			1.39192
$472$		0			0
$473$			1.38342e25i			0.536751i
$474$		0			0
$475$	−6.23923e23	+	9.54944e24i	−0.0232564	+	0.355950i
$476$		0			0
$477$		−	2.48264e25i		−	0.889178i
$478$		0			0
$479$	−4.00260e25			−1.37770			−0.688850	−	0.724904i	$-0.741884\pi$
$479$	−4.00260e25			−1.37770			−0.688850	+	0.724904i	$0.741884\pi$
$480$		0			0
$481$	3.64688e25			1.20654
$482$		0			0
$483$		−	2.14280e25i		−	0.681530i
$484$		0			0
$485$	3.79216e25	+	4.04801e25i	1.15968	+	1.23793i
$486$		0			0
$487$		−	1.08371e24i		−	0.0318704i	−0.999873	−	0.0159352i	$-0.994927\pi$
$487$		−	1.08371e24i		−	0.0318704i	0.999873	−	0.0159352i	$-0.00507254\pi$
$488$		0			0
$489$	5.05140e25			1.42882
$490$		0			0
$491$	2.29078e24			0.0623318			0.0311659	−	0.999514i	$-0.490078\pi$
$491$	2.29078e24			0.0623318			0.0311659	+	0.999514i	$0.490078\pi$
$492$		0			0
$493$			7.62914e25i			1.99724i
$494$		0			0
$495$	−1.34450e25	+	1.25952e25i	−0.338698	+	0.317291i
$496$		0			0
$497$		−	1.99435e25i		−	0.483522i
$498$		0			0
$499$	−2.31509e25			−0.540272			−0.270136	−	0.962822i	$-0.587069\pi$
$499$	−2.31509e25			−0.540272			−0.270136	+	0.962822i	$0.587069\pi$
$500$		0			0
$501$	−5.33436e24			−0.119846
$502$		0			0
$503$		−	2.08813e25i		−	0.451712i	−0.974161	−	0.225856i	$-0.927482\pi$
$503$		−	2.08813e25i		−	0.451712i	0.974161	−	0.225856i	$-0.0725179\pi$
$504$		0			0
$505$	5.93229e25	−	5.55735e25i	1.23582	−	1.15771i
$506$		0			0
$507$			1.28092e26i			2.57009i
$508$		0			0
$509$	−1.56854e25			−0.303163			−0.151581	−	0.988445i	$-0.548437\pi$
$509$	−1.56854e25			−0.303163			−0.151581	+	0.988445i	$0.548437\pi$
$510$		0			0
$511$	3.44727e25			0.641914
$512$		0			0
$513$		−	9.63111e24i		−	0.172807i
$514$		0			0
$515$	5.16091e25	+	5.50911e25i	0.892397	+	0.952605i
$516$		0			0
$517$			2.98668e25i			0.497769i
$518$		0			0
$519$	8.15046e25			1.30946
$520$		0			0
$521$	−2.02646e25			−0.313892			−0.156946	−	0.987607i	$-0.550165\pi$
$521$	−2.02646e25			−0.313892			−0.156946	+	0.987607i	$0.550165\pi$
$522$		0			0
$523$		−	2.19165e25i		−	0.327344i	−0.986515	−	0.163672i	$-0.947666\pi$
$523$		−	2.19165e25i		−	0.327344i	0.986515	−	0.163672i	$-0.0523338\pi$
$524$		0			0
$525$	6.12920e24	−	9.38103e25i	0.0882857	−	1.35125i
$526$		0			0
$527$			1.31073e25i			0.182101i
$528$		0			0
$529$	5.57149e25			0.746694
$530$		0			0
$531$	8.21910e25			1.06274
$532$		0			0
$533$		−	2.96890e25i		−	0.370413i
$534$		0			0
$535$	−4.57094e25	−	4.87934e25i	−0.550358	−	0.587489i
$536$		0			0
$537$		−	9.80415e25i		−	1.13934i
$538$		0			0
$539$	6.55956e24			0.0735836
$540$		0			0
$541$	3.03853e25			0.329071			0.164536	−	0.986371i	$-0.447387\pi$
$541$	3.03853e25			0.329071			0.164536	+	0.986371i	$0.447387\pi$
$542$		0			0
$543$		−	3.00430e25i		−	0.314156i
$544$		0			0
$545$	1.33549e26	−	1.25108e26i	1.34857	−	1.26334i
$546$		0			0
$547$			1.47601e26i			1.43950i	0.694235	+	0.719748i	$0.255743\pi$
$547$			1.47601e26i			1.43950i	−0.694235	+	0.719748i	$0.744257\pi$
$548$		0			0
$549$	9.31013e25			0.877038
$550$		0			0
$551$	6.18493e25			0.562852
$552$		0			0
$553$			6.06109e25i			0.532920i
$554$		0			0
$555$	−9.62464e25	+	9.01633e25i	−0.817714	+	0.766031i
$556$		0			0
$557$			1.48048e26i			1.21557i	0.794103	+	0.607783i	$0.207941\pi$
$557$			1.48048e26i			1.21557i	−0.794103	+	0.607783i	$0.792059\pi$
$558$		0			0
$559$	−3.14965e26			−2.49948
$560$		0			0
$561$	−8.84520e25			−0.678516
$562$		0			0
$563$		−	2.39040e26i		−	1.77272i	−0.462994	−	0.886362i	$-0.653225\pi$
$563$		−	2.39040e26i		−	1.77272i	0.462994	−	0.886362i	$-0.346775\pi$
$564$		0			0
$565$	−1.05750e26	−	1.12885e26i	−0.758265	−	0.809424i
$566$		0			0
$567$		−	7.44714e25i		−	0.516358i
$568$		0			0
$569$	5.52796e25			0.370679			0.185340	−	0.982675i	$-0.440662\pi$
$569$	5.52796e25			0.370679			0.185340	+	0.982675i	$0.440662\pi$
$570$		0			0
$571$	−1.90520e26			−1.23566			−0.617830	−	0.786312i	$-0.711988\pi$
$571$	−1.90520e26			−1.23566			−0.617830	+	0.786312i	$0.711988\pi$
$572$		0			0
$573$			2.50691e26i			1.57279i
$574$		0			0
$575$	−8.27450e25	−	5.40624e24i	−0.502224	−	0.0328134i
$576$		0			0
$577$			2.81877e26i			1.65535i	0.561208	+	0.827675i	$0.310337\pi$
$577$			2.81877e26i			1.65535i	−0.561208	+	0.827675i	$0.689663\pi$
$578$		0			0
$579$	2.28584e26			1.29897
$580$		0			0
$581$	−9.54597e25			−0.524985
$582$		0			0
$583$			4.46172e25i			0.237493i
$584$		0			0
$585$	−2.86758e26	−	3.06105e26i	−1.47753	−	1.57721i
$586$		0			0
$587$			1.47526e26i			0.735880i	0.929850	+	0.367940i	$0.119937\pi$
$587$			1.47526e26i			0.735880i	−0.929850	+	0.367940i	$0.880063\pi$
$588$		0			0
$589$	1.06261e25			0.0513189
$590$		0			0
$591$	1.16165e26			0.543244
$592$		0			0
$593$			3.42954e26i			1.55316i	0.630018	+	0.776580i	$0.283047\pi$
$593$			3.42954e26i			1.55316i	−0.630018	+	0.776580i	$0.716953\pi$
$594$		0			0
$595$	1.87305e26	−	1.75467e26i	0.821558	−	0.769632i
$596$		0			0
$597$		−	6.86483e26i		−	2.91658i
$598$		0			0
$599$	−1.08138e25			−0.0445064			−0.0222532	−	0.999752i	$-0.507084\pi$
$599$	−1.08138e25			−0.0445064			−0.0222532	+	0.999752i	$0.507084\pi$
$600$		0			0
$601$	3.33608e24			0.0133024			0.00665119	−	0.999978i	$-0.497883\pi$
$601$	3.33608e24			0.0133024			0.00665119	+	0.999978i	$0.497883\pi$
$602$		0			0
$603$		−	6.61429e26i		−	2.55546i
$604$		0			0
$605$	−1.70766e26	+	1.59973e26i	−0.639330	+	0.598922i
$606$		0			0
$607$			2.55206e26i			0.925972i	0.886366	+	0.462986i	$0.153222\pi$
$607$			2.55206e26i			0.925972i	−0.886366	+	0.462986i	$0.846778\pi$
$608$		0			0
$609$	−6.07585e26			−2.13670
$610$		0			0
$611$	−6.79982e26			−2.31796
$612$		0			0
$613$		−	1.21928e26i		−	0.402929i	−0.979496	−	0.201464i	$-0.935430\pi$
$613$		−	1.21928e26i		−	0.402929i	0.979496	−	0.201464i	$-0.0645701\pi$
$614$		0			0
$615$	7.34013e25	+	7.83535e25i	0.235174	+	0.251041i
$616$		0			0
$617$			4.20744e26i			1.30710i	0.756883	+	0.653550i	$0.226721\pi$
$617$			4.20744e26i			1.30710i	−0.756883	+	0.653550i	$0.773279\pi$
$618$		0			0
$619$	3.15171e26			0.949478			0.474739	−	0.880127i	$-0.342542\pi$
$619$	3.15171e26			0.949478			0.474739	+	0.880127i	$0.342542\pi$
$620$		0			0
$621$	8.34527e25			0.243820
$622$		0			0
$623$		−	2.13329e26i		−	0.604525i
$624$		0			0
$625$	−3.60705e26	−	4.73362e25i	−0.991499	−	0.130117i
$626$		0			0
$627$			7.17078e25i			0.191216i
$628$		0			0
$629$	−3.60046e26			−0.931488
$630$		0			0
$631$	−3.34376e26			−0.839374			−0.419687	−	0.907669i	$-0.637860\pi$
$631$	−3.34376e26			−0.839374			−0.419687	+	0.907669i	$0.637860\pi$
$632$		0			0
$633$		−	5.93928e25i		−	0.144677i
$634$		0			0
$635$	2.33183e26	+	2.48915e26i	0.551247	+	0.588438i
$636$		0			0
$637$			1.49343e26i			0.342657i
$638$		0			0
$639$	3.21781e26			0.716643
$640$		0			0
$641$	−2.13375e26			−0.461309			−0.230655	−	0.973036i	$-0.574087\pi$
$641$	−2.13375e26			−0.461309			−0.230655	+	0.973036i	$0.574087\pi$
$642$		0			0
$643$		−	7.78778e26i		−	1.63459i	−0.576218	−	0.817296i	$-0.695472\pi$
$643$		−	7.78778e26i		−	1.63459i	0.576218	−	0.817296i	$-0.304528\pi$
$644$		0			0
$645$	8.31240e26	−	7.78702e26i	1.69398	−	1.58692i
$646$		0			0
$647$			3.74677e26i			0.741423i	0.928748	+	0.370711i	$0.120886\pi$
$647$			3.74677e26i			0.741423i	−0.928748	+	0.370711i	$0.879114\pi$
$648$		0			0
$649$	−1.47711e26			−0.283850
$650$		0			0
$651$	−1.04387e26			−0.194817
$652$		0			0
$653$			8.45287e25i			0.153225i	0.997061	+	0.0766124i	$0.0244104\pi$
$653$			8.45287e25i			0.153225i	−0.997061	+	0.0766124i	$0.975590\pi$
$654$		0			0
$655$	−4.48966e25	+	4.20589e25i	−0.0790535	+	0.0740570i
$656$		0			0
$657$			5.56204e26i			0.951400i
$658$		0			0
$659$	1.18088e25			0.0196243			0.00981216	−	0.999952i	$-0.496877\pi$
$659$	1.18088e25			0.0196243			0.00981216	+	0.999952i	$0.496877\pi$
$660$		0			0
$661$	2.62597e26			0.424009			0.212005	−	0.977269i	$-0.432001\pi$
$661$	2.62597e26			0.424009			0.212005	+	0.977269i	$0.432001\pi$
$662$		0			0
$663$		−	2.01380e27i		−	3.15964i
$664$		0			0
$665$	−1.42250e26	−	1.51848e26i	−0.216894	−	0.231527i
$666$		0			0
$667$			5.35918e26i			0.794151i
$668$		0			0
$669$	1.26687e27			1.82467
$670$		0			0
$671$	−1.67319e26			−0.234251
$672$		0			0
$673$		−	2.36213e26i		−	0.321485i	−0.986996	−	0.160742i	$-0.948611\pi$
$673$		−	2.36213e26i		−	0.321485i	0.986996	−	0.160742i	$-0.0513888\pi$
$674$		0			0
$675$	3.65350e26	+	2.38705e25i	0.483418	+	0.0315846i
$676$		0			0
$677$		−	6.49049e26i		−	0.834997i	−0.908678	−	0.417498i	$-0.862907\pi$
$677$		−	6.49049e26i		−	0.834997i	0.908678	−	0.417498i	$-0.137093\pi$
$678$		0			0
$679$	−1.20600e27			−1.50863
$680$		0			0
$681$	1.00065e27			1.21726
$682$		0			0
$683$			8.23990e26i			0.974822i	0.873173	+	0.487411i	$0.162059\pi$
$683$			8.23990e26i			0.974822i	−0.873173	+	0.487411i	$0.837941\pi$
$684$		0			0
$685$	−8.49422e26	−	9.06731e26i	−0.977379	−	1.04332i
$686$		0			0
$687$		−	9.66364e26i		−	1.08156i
$688$		0			0
$689$	−1.01581e27			−1.10593
$690$		0			0
$691$	6.97671e26			0.738940			0.369470	−	0.929243i	$-0.379539\pi$
$691$	6.97671e26			0.738940			0.369470	+	0.929243i	$0.379539\pi$
$692$		0			0
$693$		−	4.00559e26i		−	0.412763i
$694$		0			0
$695$	−8.18761e25	+	7.67012e25i	−0.0820921	+	0.0769036i
$696$		0			0
$697$			2.93111e26i			0.285970i
$698$		0			0
$699$	−1.17846e27			−1.11888
$700$		0			0
$701$	6.27862e26			0.580154			0.290077	−	0.957003i	$-0.406319\pi$
$701$	6.27862e26			0.580154			0.290077	+	0.957003i	$0.406319\pi$
$702$		0			0
$703$			2.91888e26i			0.262507i
$704$		0			0
$705$	1.79457e27	−	1.68115e27i	1.57096	−	1.47167i
$706$		0			0
$707$			1.76737e27i			1.50606i
$708$		0			0
$709$	−5.38698e26			−0.446896			−0.223448	−	0.974716i	$-0.571731\pi$
$709$	−5.38698e26			−0.446896			−0.223448	+	0.974716i	$0.571731\pi$
$710$		0			0
$711$	−9.77935e26			−0.789857
$712$		0			0
$713$			9.20740e25i			0.0724080i
$714$		0			0
$715$	5.15353e26	+	5.50123e26i	0.394637	+	0.421262i
$716$		0			0
$717$		−	1.31148e27i		−	0.977982i
$718$		0			0
$719$	8.72365e26			0.633539			0.316770	−	0.948502i	$-0.397402\pi$
$719$	8.72365e26			0.633539			0.316770	+	0.948502i	$0.397402\pi$
$720$		0			0
$721$	−1.64129e27			−1.16092
$722$		0			0
$723$			3.04262e27i			2.09621i
$724$		0			0
$725$	−1.53292e26	+	2.34621e27i	−0.102875	+	1.57455i
$726$		0			0
$727$			3.01357e27i			1.97017i	0.172065	+	0.985086i	$0.444956\pi$
$727$			3.01357e27i			1.97017i	−0.172065	+	0.985086i	$0.555044\pi$
$728$		0			0
$729$	2.35963e27			1.50291
$730$		0			0
$731$	3.10957e27			1.92968
$732$		0			0
$733$			3.19909e27i			1.93437i	0.254078	+	0.967184i	$0.418228\pi$
$733$			3.19909e27i			1.93437i	−0.254078	+	0.967184i	$0.581772\pi$
$734$		0			0
$735$	−3.69226e26	−	3.94137e26i	−0.217552	−	0.232230i
$736$		0			0
$737$			1.18870e27i			0.682545i
$738$		0			0
$739$	1.17417e27			0.657068			0.328534	−	0.944492i	$-0.393445\pi$
$739$	1.17417e27			0.657068			0.328534	+	0.944492i	$0.393445\pi$
$740$		0			0
$741$	−1.63258e27			−0.890435
$742$		0			0
$743$		−	1.77571e27i		−	0.944014i	−0.881595	−	0.472007i	$-0.843530\pi$
$743$		−	1.77571e27i		−	0.944014i	0.881595	−	0.472007i	$-0.156470\pi$
$744$		0			0
$745$	−1.92688e26	+	1.80509e26i	−0.0998550	+	0.0935438i
$746$		0			0
$747$		−	1.54021e27i		−	0.778096i
$748$		0			0
$749$	1.45367e27			0.715960
$750$		0			0
$751$	−3.75649e27			−1.80386			−0.901930	−	0.431882i	$-0.857850\pi$
$751$	−3.75649e27			−1.80386			−0.901930	+	0.431882i	$0.857850\pi$
$752$		0			0
$753$			4.60194e27i			2.15471i
$754$		0			0
$755$	−2.98257e27	+	2.79406e27i	−1.36174	+	1.27568i
$756$		0			0
$757$		−	9.52038e26i		−	0.423880i	−0.977283	−	0.211940i	$-0.932022\pi$
$757$		−	9.52038e26i		−	0.423880i	0.977283	−	0.211940i	$-0.0679782\pi$
$758$		0			0
$759$	−6.21341e26			−0.269795
$760$		0			0
$761$	−3.69849e27			−1.56628			−0.783141	−	0.621845i	$-0.786384\pi$
$761$	−3.69849e27			−1.56628			−0.783141	+	0.621845i	$0.786384\pi$
$762$		0			0
$763$			3.97874e27i			1.64347i
$764$		0			0
$765$	2.83109e27	+	3.02209e27i	1.14069	+	1.21766i
$766$		0			0
$767$		−	3.36297e27i		−	1.32180i
$768$		0			0
$769$	2.80215e27			1.07446			0.537232	−	0.843435i	$-0.319470\pi$
$769$	2.80215e27			1.07446			0.537232	+	0.843435i	$0.319470\pi$
$770$		0			0
$771$	5.42172e27			2.02825
$772$		0			0
$773$		−	2.21422e26i		−	0.0808194i	−0.999183	−	0.0404097i	$-0.987134\pi$
$773$		−	2.21422e26i		−	0.0808194i	0.999183	−	0.0404097i	$-0.0128663\pi$
$774$		0			0
$775$	−2.63365e25	+	4.03093e26i	−0.00937977	+	0.143562i
$776$		0			0
$777$		−	2.86741e27i		−	0.996529i
$778$		0			0
$779$	2.37624e26			0.0805907
$780$		0			0
$781$	−5.78295e26			−0.191410
$782$		0			0
$783$		−	2.36627e27i		−	0.764413i
$784$		0			0
$785$	1.98221e27	+	2.11594e27i	0.625010	+	0.667178i
$786$		0			0
$787$		−	5.93703e27i		−	1.82730i	−0.406504	−	0.913649i	$-0.633252\pi$
$787$		−	5.93703e27i		−	1.82730i	0.406504	−	0.913649i	$-0.366748\pi$
$788$		0			0
$789$	−8.16756e25			−0.0245392
$790$		0			0
$791$	3.36311e27			0.986426
$792$		0			0
$793$		−	3.80938e27i		−	1.09083i
$794$		0			0
$795$	2.68086e27	−	2.51142e27i	0.749527	−	0.702154i
$796$		0			0
$797$			1.92749e27i			0.526185i	0.964771	+	0.263093i	$0.0847425\pi$
$797$			1.92749e27i			0.526185i	−0.964771	+	0.263093i	$0.915258\pi$
$798$		0			0
$799$	6.71328e27			1.78954
$800$		0			0
$801$	3.44199e27			0.895984
$802$		0			0
$803$		−	9.99593e26i		−	0.254112i
$804$		0			0
$805$	1.31575e27	−	1.23259e27i	0.326671	−	0.306024i
$806$		0			0
$807$		−	5.04446e27i		−	1.22325i
$808$		0			0
$809$	−3.43062e27			−0.812572			−0.406286	−	0.913746i	$-0.633176\pi$
$809$	−3.43062e27			−0.812572			−0.406286	+	0.913746i	$0.633176\pi$
$810$		0			0
$811$	−1.59172e27			−0.368272			−0.184136	−	0.982901i	$-0.558949\pi$
$811$	−1.59172e27			−0.368272			−0.184136	+	0.982901i	$0.558949\pi$
$812$		0			0
$813$		−	8.17989e26i		−	0.184879i
$814$		0			0
$815$	2.90567e27	+	3.10171e27i	0.641578	+	0.684864i
$816$		0			0
$817$		−	2.52092e27i		−	0.543812i
$818$		0			0
$819$	9.11960e27			1.92211
$820$		0			0
$821$	−7.09546e27			−1.46124			−0.730618	−	0.682786i	$-0.760768\pi$
$821$	−7.09546e27			−1.46124			−0.730618	+	0.682786i	$0.760768\pi$
$822$		0			0
$823$			5.62780e27i			1.13251i	0.824232	+	0.566253i	$0.191607\pi$
$823$			5.62780e27i			1.13251i	−0.824232	+	0.566253i	$0.808393\pi$
$824$		0			0
$825$	−2.72018e27	−	1.77726e26i	−0.534916	−	0.0349493i
$826$		0			0
$827$			1.11492e27i			0.214260i	0.994245	+	0.107130i	$0.0341661\pi$
$827$			1.11492e27i			0.214260i	−0.994245	+	0.107130i	$0.965834\pi$
$828$		0			0
$829$	−2.95026e27			−0.554105			−0.277052	−	0.960855i	$-0.589358\pi$
$829$	−2.95026e27			−0.554105			−0.277052	+	0.960855i	$0.589358\pi$
$830$		0			0
$831$	−4.70142e27			−0.863018
$832$		0			0
$833$		−	1.47442e27i		−	0.264541i
$834$		0			0
$835$	−3.06844e26	−	3.27546e26i	−0.0538140	−	0.0574448i
$836$		0			0
$837$		−	4.06540e26i		−	0.0696966i
$838$		0			0
$839$	−6.78383e27			−1.13694			−0.568469	−	0.822705i	$-0.692464\pi$
$839$	−6.78383e27			−1.13694			−0.568469	+	0.822705i	$0.692464\pi$
$840$		0			0
$841$	9.09252e27			1.48978
$842$		0			0
$843$			1.01611e28i			1.62772i
$844$		0			0
$845$	−7.86525e27	+	7.36814e27i	−1.23190	+	1.15404i
$846$		0			0
$847$		−	5.08752e27i		−	0.779136i
$848$		0			0
$849$	9.35826e27			1.40143
$850$		0			0
$851$	−2.52919e27			−0.370382
$852$		0			0
$853$			2.09040e27i			0.299374i	0.988733	+	0.149687i	$0.0478265\pi$
$853$			2.09040e27i			0.299374i	−0.988733	+	0.149687i	$0.952173\pi$
$854$		0			0
$855$	2.45000e27	−	2.29515e27i	0.343154	−	0.321465i
$856$		0			0
$857$			1.53040e27i			0.209646i	0.994491	+	0.104823i	$0.0334276\pi$
$857$			1.53040e27i			0.209646i	−0.994491	+	0.104823i	$0.966572\pi$
$858$		0			0
$859$	−2.08735e27			−0.279679			−0.139839	−	0.990174i	$-0.544659\pi$
$859$	−2.08735e27			−0.279679			−0.139839	+	0.990174i	$0.544659\pi$
$860$		0			0
$861$	−2.33434e27			−0.305938
$862$		0			0
$863$		−	8.71784e27i		−	1.11765i	−0.829285	−	0.558826i	$-0.811252\pi$
$863$		−	8.71784e27i		−	1.11765i	0.829285	−	0.558826i	$-0.188748\pi$
$864$		0			0
$865$	4.68831e27	+	5.00462e27i	0.587981	+	0.627651i
$866$		0			0
$867$			7.47220e27i			0.916784i
$868$		0			0
$869$	1.75751e27			0.210965
$870$		0			0
$871$	−2.70634e28			−3.17840
$872$		0			0
$873$		−	1.94583e28i		−	2.23599i
$874$		0			0
$875$	6.11280e27	−	5.01981e27i	0.687327	−	0.564431i
$876$		0			0
$877$		−	3.82930e26i		−	0.0421331i	−0.999778	−	0.0210665i	$-0.993294\pi$
$877$		−	3.82930e26i		−	0.0421331i	0.999778	−	0.0210665i	$-0.00670618\pi$
$878$		0			0
$879$	−2.58607e28			−2.78450
$880$		0			0
$881$	−1.11821e28			−1.17829			−0.589143	−	0.808029i	$-0.700534\pi$
$881$	−1.11821e28			−1.17829			−0.589143	+	0.808029i	$0.700534\pi$
$882$		0			0
$883$			6.48578e27i			0.668861i	0.942420	+	0.334430i	$0.108544\pi$
$883$			6.48578e27i			0.668861i	−0.942420	+	0.334430i	$0.891456\pi$
$884$		0			0
$885$	8.31440e27	+	8.87536e27i	0.839209	+	0.895829i
$886$		0			0
$887$		−	9.49846e27i		−	0.938380i	−0.883097	−	0.469190i	$-0.844546\pi$
$887$		−	9.49846e27i		−	0.938380i	0.883097	−	0.469190i	$-0.155454\pi$
$888$		0			0
$889$	−7.41577e27			−0.717116
$890$		0			0
$891$	−2.15942e27			−0.204409
$892$		0			0
$893$		−	5.44244e27i		−	0.504318i
$894$		0			0
$895$	6.02004e27	−	5.63955e27i	0.546111	−	0.511594i
$896$		0			0
$897$		−	1.41462e28i		−	1.25635i
$898$		0			0
$899$	2.61073e27			0.227010
$900$		0			0
$901$	1.00288e28			0.853813
$902$		0			0
$903$			2.47646e28i			2.06442i
$904$		0			0
$905$	1.84473e27	−	1.72814e27i	0.150581	−	0.141064i
$906$		0			0
$907$			1.45654e28i			1.16427i	0.813093	+	0.582134i	$0.197782\pi$
$907$			1.45654e28i			1.16427i	−0.813093	+	0.582134i	$0.802218\pi$
$908$		0			0
$909$	−2.85158e28			−2.23218
$910$		0			0
$911$	−9.31168e27			−0.713844			−0.356922	−	0.934134i	$-0.616174\pi$
$911$	−9.31168e27			−0.713844			−0.356922	+	0.934134i	$0.616174\pi$
$912$		0			0
$913$			2.76801e27i			0.207824i
$914$		0			0
$915$	9.41808e27	+	1.00535e28i	0.692568	+	0.739294i
$916$		0			0
$917$		−	1.33758e27i		−	0.0963406i
$918$		0			0
$919$	1.77093e27			0.124941			0.0624705	−	0.998047i	$-0.480102\pi$
$919$	1.77093e27			0.124941			0.0624705	+	0.998047i	$0.480102\pi$
$920$		0			0
$921$	−3.92366e27			−0.271159
$922$		0			0
$923$		−	1.31662e28i		−	0.891339i
$924$		0			0
$925$	−1.10726e28	−	7.23440e26i	−0.734349	−	0.0479795i
$926$		0			0
$927$		−	2.64817e28i		−	1.72063i
$928$		0			0
$929$	−1.17089e27			−0.0745359			−0.0372680	−	0.999305i	$-0.511866\pi$
$929$	−1.17089e27			−0.0745359			−0.0372680	+	0.999305i	$0.511866\pi$
$930$		0			0
$931$	−1.19531e27			−0.0745517
$932$		0			0
$933$		−	4.68560e28i		−	2.86345i
$934$		0			0
$935$	−5.08794e27	−	5.43122e27i	−0.304671	−	0.325227i
$936$		0			0
$937$		−	7.69183e27i		−	0.451340i	−0.974204	−	0.225670i	$-0.927543\pi$
$937$		−	7.69183e27i		−	0.451340i	0.974204	−	0.225670i	$-0.0724571\pi$
$938$		0			0
$939$	2.26552e28			1.30270
$940$		0			0
$941$	1.18799e28			0.669440			0.334720	−	0.942318i	$-0.391358\pi$
$941$	1.18799e28			0.669440			0.334720	+	0.942318i	$0.391358\pi$
$942$		0			0
$943$			2.05899e27i			0.113709i
$944$		0			0
$945$	−5.80950e27	+	5.44231e27i	−0.314439	+	0.294565i
$946$		0			0
$947$		−	2.13232e28i		−	1.13117i	−0.824690	−	0.565586i	$-0.808650\pi$
$947$		−	2.13232e28i		−	1.13117i	0.824690	−	0.565586i	$-0.191350\pi$
$948$		0			0
$949$	2.27579e28			1.18332
$950$		0			0
$951$	−4.96625e28			−2.53112
$952$		0			0
$953$		−	1.15954e28i		−	0.579299i	−0.957133	−	0.289650i	$-0.906461\pi$
$953$		−	1.15954e28i		−	0.579299i	0.957133	−	0.289650i	$-0.0935387\pi$
$954$		0			0
$955$	−1.53932e28	+	1.44203e28i	−0.753870	+	0.706222i
$956$		0			0
$957$			1.76179e28i			0.845846i
$958$		0			0
$959$	2.70137e28			1.27147
$960$		0			0
$961$	−2.12221e28			−0.979302
$962$		0			0
$963$			2.34544e28i			1.06115i
$964$		0			0
$965$	1.31486e28	+	1.40357e28i	0.583270	+	0.622623i
$966$		0			0
$967$		−	2.71941e28i		−	1.18283i	−0.806366	−	0.591417i	$-0.798569\pi$
$967$		−	2.71941e28i		−	1.18283i	0.806366	−	0.591417i	$-0.201431\pi$
$968$		0			0
$969$	1.61181e28			0.687443
$970$		0			0
$971$	2.13235e28			0.891818			0.445909	−	0.895078i	$-0.352880\pi$
$971$	2.13235e28			0.891818			0.445909	+	0.895078i	$0.352880\pi$
$972$		0			0
$973$		−	2.43928e27i		−	0.100044i
$974$		0			0
$975$	4.04633e27	−	6.19309e28i	0.162748	−	2.49094i
$976$		0			0
$977$			4.37913e28i			1.72739i	0.504019	+	0.863693i	$0.331854\pi$
$977$			4.37913e28i			1.72739i	−0.504019	+	0.863693i	$0.668146\pi$
$978$		0			0
$979$	−6.18583e27			−0.239311
$980$		0			0
$981$	−6.41955e28			−2.43584
$982$		0			0
$983$		−	2.29758e28i		−	0.855091i	−0.903994	−	0.427546i	$-0.859378\pi$
$983$		−	2.29758e28i		−	0.855091i	0.903994	−	0.427546i	$-0.140622\pi$
$984$		0			0
$985$	6.68206e27	+	7.13289e27i	0.243931	+	0.260388i
$986$		0			0
$987$			5.34646e28i			1.91449i
$988$		0			0
$989$	2.18435e28			0.767287
$990$		0			0
$991$	4.81722e28			1.65996			0.829978	−	0.557796i	$-0.188353\pi$
$991$	4.81722e28			1.65996			0.829978	+	0.557796i	$0.188353\pi$
$992$		0			0
$993$			4.14415e28i			1.40094i
$994$		0			0
$995$	4.21521e28	−	3.94879e28i	1.39798	−	1.30962i
$996$		0			0
$997$			1.18301e28i			0.384933i	0.981304	+	0.192466i	$0.0616487\pi$
$997$			1.18301e28i			0.384933i	−0.981304	+	0.192466i	$0.938351\pi$
$998$		0			0
$999$	1.11673e28			0.356513

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	80.20.c.d.49.4		28
4.3	odd	2		40.20.c.a.9.25	yes	28
5.4	even	2	inner	80.20.c.d.49.25		28
20.19	odd	2		40.20.c.a.9.4	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.20.c.a.9.4	✓	28	20.19	odd	2
40.20.c.a.9.25	yes	28	4.3	odd	2
80.20.c.d.49.4		28	1.1	even	1	trivial
80.20.c.d.49.25		28	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 80 = 2^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	80.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-51907.0i q^{3} +(3.18724e6 - 2.98580e6i) q^{5} +9.49555e7i q^{7} -1.53207e9 q^{9} +O(q^{10})\)	\(q-51907.0i q^{3} +(3.18724e6 - 2.98580e6i) q^{5} +9.49555e7i q^{7} -1.53207e9 q^{9} +2.75339e9 q^{11} +6.26869e10i q^{13} +(-1.54984e11 - 1.65440e11i) q^{15} -6.18891e11i q^{17} -5.01733e11 q^{19} +4.92885e12 q^{21} -4.34747e12i q^{23} +(1.24354e12 - 1.90329e13i) q^{25} +1.91957e13i q^{27} -1.23271e14 q^{29} -2.11787e13 q^{31} -1.42920e14i q^{33} +(2.83518e14 + 3.02646e14i) q^{35} -5.81760e14i q^{37} +3.25389e15 q^{39} -4.73607e14 q^{41} +5.02442e15i q^{43} +(-4.88308e15 + 4.57445e15i) q^{45} +1.08473e16i q^{47} +2.38236e15 q^{49} -3.21248e16 q^{51} +1.62044e16i q^{53} +(8.77572e15 - 8.22106e15i) q^{55} +2.60434e16i q^{57} -5.36470e16 q^{59} -6.07683e16 q^{61} -1.45478e17i q^{63} +(1.87170e17 + 1.99798e17i) q^{65} +4.31722e17i q^{67} -2.25664e17 q^{69} -2.10030e17 q^{71} -3.63041e17i q^{73} +(-9.87940e17 - 6.45482e16i) q^{75} +2.61449e17i q^{77} +6.38309e17 q^{79} -7.84278e17 q^{81} +1.00531e18i q^{83} +(-1.84788e18 - 1.97256e18i) q^{85} +6.39863e18i q^{87} -2.24662e18 q^{89} -5.95247e18 q^{91} +1.09932e18i q^{93} +(-1.59915e18 + 1.49807e18i) q^{95} +1.27007e19i q^{97} -4.21839e18 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 3935164 q^{5} - 10352560732 q^{9}+O(q^{10}) \) 28 * q - 3935164 * q^5 - 10352560732 * q^9	\( 28 q - 3935164 q^{5} - 10352560732 q^{9} - 18570995888 q^{11} - 79041415120 q^{15} + 1685869130672 q^{19} + 140419722832 q^{21} - 29558822439924 q^{25} + 160448197635496 q^{29} - 64251929934720 q^{31} + 88899634227664 q^{35} - 45\!\cdots\!04 q^{39}+ \cdots + 24\!\cdots\!92 q^{99}+O(q^{100}) \) 28 * q - 3935164 * q^5 - 10352560732 * q^9 - 18570995888 * q^11 - 79041415120 * q^15 + 1685869130672 * q^19 + 140419722832 * q^21 - 29558822439924 * q^25 + 160448197635496 * q^29 - 64251929934720 * q^31 + 88899634227664 * q^35 - 4563253596731104 * q^39 + 87868042694936 * q^41 + 3557698140217852 * q^45 - 28923781405647564 * q^49 - 14937379593456000 * q^51 - 28335039172010704 * q^55 + 83879503376835216 * q^59 + 2336306250785000 * q^61 + 23503427909569568 * q^65 - 46927873937137712 * q^69 - 104881848512783264 * q^71 - 1094024007292275040 * q^75 + 795721739900308672 * q^79 + 3593102866196377132 * q^81 + 834124526841485952 * q^85 + 1582238786752844568 * q^89 - 11017818924975052064 * q^91 - 30914057837649668144 * q^95 + 24571127463084047792 * q^99

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