LMFDB - Embedded newform 16.50.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,50,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 50, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 50);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 50 $
Character orbit:	$[\chi]$	$=$	16.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$243.305928158$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 104434803447206332x + 4289992005756109702361620 $ x^3 - x^2 - 104434803447206332*x + 4289992005756109702361620
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{5}\cdot 5^{2}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.00244e8$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.77152e11 q^{3} -2.33026e17 q^{5} +5.15430e20 q^{7} +2.19235e23 q^{9} +O(q^{10})$	$q+6.77152e11 q^{3} -2.33026e17 q^{5} +5.15430e20 q^{7} +2.19235e23 q^{9} +5.21317e25 q^{11} +1.37568e27 q^{13} -1.57794e29 q^{15} -9.93263e29 q^{17} -1.85532e31 q^{19} +3.49024e32 q^{21} +2.69345e33 q^{23} +3.65374e34 q^{25} -1.35867e34 q^{27} -3.88028e34 q^{29} +8.58347e35 q^{31} +3.53011e37 q^{33} -1.20108e38 q^{35} +2.50854e38 q^{37} +9.31545e38 q^{39} +2.12382e38 q^{41} +3.48149e39 q^{43} -5.10874e40 q^{45} -2.11562e40 q^{47} +8.74461e39 q^{49} -6.72589e41 q^{51} -2.52544e42 q^{53} -1.21480e43 q^{55} -1.25633e43 q^{57} +4.29058e43 q^{59} -1.42312e43 q^{61} +1.13000e44 q^{63} -3.20569e44 q^{65} -2.95546e44 q^{67} +1.82387e45 q^{69} +2.56483e45 q^{71} -5.17938e45 q^{73} +2.47414e46 q^{75} +2.68703e46 q^{77} -1.28546e46 q^{79} -6.16630e46 q^{81} -8.58374e46 q^{83} +2.31456e47 q^{85} -2.62754e46 q^{87} +6.95931e47 q^{89} +7.09067e47 q^{91} +5.81231e47 q^{93} +4.32337e48 q^{95} +7.53593e48 q^{97} +1.14291e49 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots + 38\!\cdots\!99 q^{9}+O(q^{10}) $ 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9	$ 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots - 17\!\cdots\!68 q^{99}+O(q^{100}) $ 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9 + 25860036791834065092767004 * q^11 + 2945123797958353151076283242 * q^13 - 87846213583565151681178171080 * q^15 + 3910755572889939740390354668854 * q^17 - 14590272981610371514592320211580 * q^19 - 109864176920553355284360667668384 * q^21 + 4953158032634305739950894912759608 * q^23 + 39451396326519132102111422427450525 * q^25 - 196120894647932767255303897131684600 * q^27 + 937305925294643190481854503019954330 * q^29 - 2414809991776326638635666091994140256 * q^31 + 84513992666209345011409783180743651984 * q^33 - 300439012903055674136683294221406953360 * q^35 + 309377978155169809207437071391240173874 * q^37 - 360871814296103457146322338824042742568 * q^39 - 1143109642820632592659114174552039005954 * q^41 + 17579019372148890319048762066144112105148 * q^43 - 114813225075613761568802306330842527416790 * q^45 + 218573936001854642959028674713638582342256 * q^47 + 373628008074429359506251920014708170372651 * q^49 - 2241123510744223933356304325509677976541016 * q^51 + 740200780025011930130058449331923084936802 * q^53 - 3200488473448119911679443277764576117207640 * q^55 - 20594915296543361145254552753753824314692880 * q^57 + 53014363316602767635595514099864508327238540 * q^59 - 124730102606725385116407342714381742110559494 * q^61 + 468292624427612329618287149072325587596605768 * q^63 - 421321721827999315000625667485604880336092420 * q^65 - 1111804215897361218882144911380577470099878924 * q^67 + 770909887444909922528222240570561038770401568 * q^69 + 4013080456871179368491803943114613092268032424 * q^71 - 10015638764428635672946257750737669286684666018 * q^73 + 38646070268253498119014909069282536501398301900 * q^75 - 23360330575813840356220747501745396153372866272 * q^77 - 36077368003505469428776991593224667583970712560 * q^79 - 14427585993163108762153736521422002728666768917 * q^81 - 150749458884727999220762915504103393007529125692 * q^83 + 566911972337878875844467446456481028710158526660 * q^85 - 1072056267743755243448622937355132996639037778920 * q^87 + 542251936064510107856207107418847041962289942670 * q^89 + 1459783720923935217191336434024489699257545191344 * q^91 - 2368291579513879043404530519265491460552020418176 * q^93 + 2813579016011586836853932078889401271258883877400 * q^95 + 4343844254337968328695747517284119056283595359014 * q^97 - 17595360353038055002493296165642581030249046200468 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	6.77152e11	1.38425	0.692126	−	0.721777i	$-0.256674\pi$
$3$	6.77152e11	1.38425	0.692126	+	0.721777i	$0.256674\pi$
$4$	0	0
$5$	−2.33026e17	−1.74839	−0.874196	−	0.485573i	$-0.838611\pi$
$5$	−2.33026e17	−1.74839	−0.874196	+	0.485573i	$0.838611\pi$
$6$	0	0
$7$	5.15430e20	1.01688	0.508438	−	0.861099i	$-0.330223\pi$
$7$	5.15430e20	1.01688	0.508438	+	0.861099i	$0.330223\pi$
$8$	0	0
$9$	2.19235e23	0.916153
$10$	0	0
$11$	5.21317e25	1.59581	0.797905	−	0.602783i	$-0.205942\pi$
$11$	5.21317e25	1.59581	0.797905	+	0.602783i	$0.205942\pi$
$12$	0	0
$13$	1.37568e27	0.702920	0.351460	−	0.936203i	$-0.385685\pi$
$13$	1.37568e27	0.702920	0.351460	+	0.936203i	$0.385685\pi$
$14$	0	0
$15$	−1.57794e29	−2.42021
$16$	0	0
$17$	−9.93263e29	−0.709685	−0.354843	−	0.934926i	$-0.615466\pi$
$17$	−9.93263e29	−0.709685	−0.354843	+	0.934926i	$0.615466\pi$
$18$	0	0
$19$	−1.85532e31	−0.868871	−0.434436	−	0.900703i	$-0.643052\pi$
$19$	−1.85532e31	−0.868871	−0.434436	+	0.900703i	$0.643052\pi$
$20$	0	0
$21$	3.49024e32	1.40761
$22$	0	0
$23$	2.69345e33	1.16944	0.584718	−	0.811236i	$-0.301205\pi$
$23$	2.69345e33	1.16944	0.584718	+	0.811236i	$0.301205\pi$
$24$	0	0
$25$	3.65374e34	2.05687
$26$	0	0
$27$	−1.35867e34	−0.116065
$28$	0	0
$29$	−3.88028e34	−0.0575588	−0.0287794	−	0.999586i	$-0.509162\pi$
$29$	−3.88028e34	−0.0575588	−0.0287794	+	0.999586i	$0.509162\pi$
$30$	0	0
$31$	8.58347e35	0.248486	0.124243	−	0.992252i	$-0.460350\pi$
$31$	8.58347e35	0.248486	0.124243	+	0.992252i	$0.460350\pi$
$32$	0	0
$33$	3.53011e37	2.20900
$34$	0	0
$35$	−1.20108e38	−1.77790
$36$	0	0
$37$	2.50854e38	0.951652	0.475826	−	0.879539i	$-0.342149\pi$
$37$	2.50854e38	0.951652	0.475826	+	0.879539i	$0.342149\pi$
$38$	0	0
$39$	9.31545e38	0.973018
$40$	0	0
$41$	2.12382e38	0.0651498	0.0325749	−	0.999469i	$-0.489629\pi$
$41$	2.12382e38	0.0651498	0.0325749	+	0.999469i	$0.489629\pi$
$42$	0	0
$43$	3.48149e39	0.332497	0.166249	−	0.986084i	$-0.446835\pi$
$43$	3.48149e39	0.332497	0.166249	+	0.986084i	$0.446835\pi$
$44$	0	0
$45$	−5.10874e40	−1.60179
$46$	0	0
$47$	−2.11562e40	−0.228581	−0.114290	−	0.993447i	$-0.536459\pi$
$47$	−2.11562e40	−0.228581	−0.114290	+	0.993447i	$0.536459\pi$
$48$	0	0
$49$	8.74461e39	0.0340358
$50$	0	0
$51$	−6.72589e41	−0.982383
$52$	0	0
$53$	−2.52544e42	−1.43740	−0.718701	−	0.695319i	$-0.755263\pi$
$53$	−2.52544e42	−1.43740	−0.718701	+	0.695319i	$0.755263\pi$
$54$	0	0
$55$	−1.21480e43	−2.79010
$56$	0	0
$57$	−1.25633e43	−1.20274
$58$	0	0
$59$	4.29058e43	1.76458	0.882290	−	0.470707i	$-0.156001\pi$
$59$	4.29058e43	1.76458	0.882290	+	0.470707i	$0.156001\pi$
$60$	0	0
$61$	−1.42312e43	−0.258619	−0.129309	−	0.991604i	$-0.541276\pi$
$61$	−1.42312e43	−0.258619	−0.129309	+	0.991604i	$0.541276\pi$
$62$	0	0
$63$	1.13000e44	0.931614
$64$	0	0
$65$	−3.20569e44	−1.22898
$66$	0	0
$67$	−2.95546e44	−0.539254	−0.269627	−	0.962965i	$-0.586900\pi$
$67$	−2.95546e44	−0.539254	−0.269627	+	0.962965i	$0.586900\pi$
$68$	0	0
$69$	1.82387e45	1.61879
$70$	0	0
$71$	2.56483e45	1.13039	0.565197	−	0.824956i	$-0.308800\pi$
$71$	2.56483e45	1.13039	0.565197	+	0.824956i	$0.308800\pi$
$72$	0	0
$73$	−5.17938e45	−1.15576	−0.577880	−	0.816122i	$-0.696120\pi$
$73$	−5.17938e45	−1.15576	−0.577880	+	0.816122i	$0.696120\pi$
$74$	0	0
$75$	2.47414e46	2.84723
$76$	0	0
$77$	2.68703e46	1.62274
$78$	0	0
$79$	−1.28546e46	−0.414185	−0.207092	−	0.978321i	$-0.566400\pi$
$79$	−1.28546e46	−0.414185	−0.207092	+	0.978321i	$0.566400\pi$
$80$	0	0
$81$	−6.16630e46	−1.07682
$82$	0	0
$83$	−8.58374e46	−0.824638	−0.412319	−	0.911040i	$-0.635281\pi$
$83$	−8.58374e46	−0.824638	−0.412319	+	0.911040i	$0.635281\pi$
$84$	0	0
$85$	2.31456e47	1.24081
$86$	0	0
$87$	−2.62754e46	−0.0796759
$88$	0	0
$89$	6.95931e47	1.20924	0.604618	−	0.796516i	$-0.293326\pi$
$89$	6.95931e47	1.20924	0.604618	+	0.796516i	$0.293326\pi$
$90$	0	0
$91$	7.09067e47	0.714782
$92$	0	0
$93$	5.81231e47	0.343967
$94$	0	0
$95$	4.32337e48	1.51913
$96$	0	0
$97$	7.53593e48	1.58939	0.794693	−	0.607011i	$-0.207632\pi$
$97$	7.53593e48	1.58939	0.794693	+	0.607011i	$0.207632\pi$
$98$	0	0
$99$	1.14291e49	1.46201
$100$	0	0
$101$	2.43523e49	1.90839	0.954195	−	0.299185i	$-0.0967147\pi$
$101$	2.43523e49	1.90839	0.954195	+	0.299185i	$0.0967147\pi$
$102$	0	0
$103$	−4.35062e48	−0.210882	−0.105441	−	0.994426i	$-0.533625\pi$
$103$	−4.35062e48	−0.210882	−0.105441	+	0.994426i	$0.533625\pi$
$104$	0	0
$105$	−8.13316e49	−2.46106
$106$	0	0
$107$	−4.63551e49	−0.883477	−0.441739	−	0.897144i	$-0.645638\pi$
$107$	−4.63551e49	−0.883477	−0.441739	+	0.897144i	$0.645638\pi$
$108$	0	0
$109$	−3.16577e49	−0.383293	−0.191646	−	0.981464i	$-0.561383\pi$
$109$	−3.16577e49	−0.383293	−0.191646	+	0.981464i	$0.561383\pi$
$110$	0	0
$111$	1.69866e50	1.31733
$112$	0	0
$113$	−2.77559e50	−1.38974	−0.694869	−	0.719136i	$-0.744538\pi$
$113$	−2.77559e50	−1.38974	−0.694869	+	0.719136i	$0.744538\pi$
$114$	0	0
$115$	−6.27643e50	−2.04463
$116$	0	0
$117$	3.01597e50	0.643982
$118$	0	0
$119$	−5.11958e50	−0.721661
$120$	0	0
$121$	1.65053e51	1.54661
$122$	0	0
$123$	1.43815e50	0.0901837
$124$	0	0
$125$	−4.37479e51	−1.84783
$126$	0	0
$127$	1.11605e51	0.319516	0.159758	−	0.987156i	$-0.448929\pi$
$127$	1.11605e51	0.319516	0.159758	+	0.987156i	$0.448929\pi$
$128$	0	0
$129$	2.35750e51	0.460260
$130$	0	0
$131$	−3.32081e51	−0.444730	−0.222365	−	0.974963i	$-0.571378\pi$
$131$	−3.32081e51	−0.444730	−0.222365	+	0.974963i	$0.571378\pi$
$132$	0	0
$133$	−9.56287e51	−0.883534
$134$	0	0
$135$	3.16605e51	0.202927
$136$	0	0
$137$	2.87128e52	1.28358	0.641788	−	0.766882i	$-0.278193\pi$
$137$	2.87128e52	1.28358	0.641788	+	0.766882i	$0.278193\pi$
$138$	0	0
$139$	2.07847e52	0.651450	0.325725	−	0.945465i	$-0.394392\pi$
$139$	2.07847e52	0.651450	0.325725	+	0.945465i	$0.394392\pi$
$140$	0	0
$141$	−1.43260e52	−0.316414
$142$	0	0
$143$	7.17166e52	1.12173
$144$	0	0
$145$	9.04205e51	0.100635
$146$	0	0
$147$	5.92142e51	0.0471142
$148$	0	0
$149$	2.65487e53	1.51698	0.758489	−	0.651686i	$-0.225938\pi$
$149$	2.65487e53	1.51698	0.758489	+	0.651686i	$0.225938\pi$
$150$	0	0
$151$	2.97824e52	0.122751	0.0613754	−	0.998115i	$-0.480451\pi$
$151$	2.97824e52	0.122751	0.0613754	+	0.998115i	$0.480451\pi$
$152$	0	0
$153$	−2.17758e53	−0.650180
$154$	0	0
$155$	−2.00017e53	−0.434450
$156$	0	0
$157$	−7.16451e52	−0.113670	−0.0568349	−	0.998384i	$-0.518101\pi$
$157$	−7.16451e52	−0.113670	−0.0568349	+	0.998384i	$0.518101\pi$
$158$	0	0
$159$	−1.71011e54	−1.98973
$160$	0	0
$161$	1.38828e54	1.18917
$162$	0	0
$163$	8.62792e53	0.546147	0.273074	−	0.961993i	$-0.411960\pi$
$163$	8.62792e53	0.546147	0.273074	+	0.961993i	$0.411960\pi$
$164$	0	0
$165$	−8.22606e54	−3.86220
$166$	0	0
$167$	−1.80874e53	−0.0632154	−0.0316077	−	0.999500i	$-0.510063\pi$
$167$	−1.80874e53	−0.0632154	−0.0316077	+	0.999500i	$0.510063\pi$
$168$	0	0
$169$	−1.93773e54	−0.505904
$170$	0	0
$171$	−4.06751e54	−0.796019
$172$	0	0
$173$	8.09453e54	1.19141	0.595706	−	0.803202i	$-0.296872\pi$
$173$	8.09453e54	1.19141	0.595706	+	0.803202i	$0.296872\pi$
$174$	0	0
$175$	1.88325e55	2.09158
$176$	0	0
$177$	2.90537e55	2.44262
$178$	0	0
$179$	9.52961e54	0.608381	0.304191	−	0.952611i	$-0.401614\pi$
$179$	9.52961e54	0.608381	0.304191	+	0.952611i	$0.401614\pi$
$180$	0	0
$181$	1.57599e55	0.766354	0.383177	−	0.923675i	$-0.374830\pi$
$181$	1.57599e55	0.766354	0.383177	+	0.923675i	$0.374830\pi$
$182$	0	0
$183$	−9.63667e54	−0.357993
$184$	0	0
$185$	−5.84553e55	−1.66386
$186$	0	0
$187$	−5.17805e55	−1.13252
$188$	0	0
$189$	−7.00298e54	−0.118024
$190$	0	0
$191$	1.13259e56	1.47487	0.737435	−	0.675419i	$-0.236037\pi$
$191$	1.13259e56	1.47487	0.737435	+	0.675419i	$0.236037\pi$
$192$	0	0
$193$	5.85668e55	0.590878	0.295439	−	0.955362i	$-0.404534\pi$
$193$	5.85668e55	0.590878	0.295439	+	0.955362i	$0.404534\pi$
$194$	0	0
$195$	−2.17074e56	−1.70122
$196$	0	0
$197$	6.36238e55	0.388327	0.194163	−	0.980969i	$-0.437801\pi$
$197$	6.36238e55	0.388327	0.194163	+	0.980969i	$0.437801\pi$
$198$	0	0
$199$	1.38808e56	0.661475	0.330738	−	0.943723i	$-0.392702\pi$
$199$	1.38808e56	0.661475	0.330738	+	0.943723i	$0.392702\pi$
$200$	0	0
$201$	−2.00130e56	−0.746463
$202$	0	0
$203$	−2.00001e55	−0.0585302
$204$	0	0
$205$	−4.94905e55	−0.113907
$206$	0	0
$207$	5.90498e56	1.07138
$208$	0	0
$209$	−9.67210e56	−1.38655
$210$	0	0
$211$	−5.64284e55	−0.0640586	−0.0320293	−	0.999487i	$-0.510197\pi$
$211$	−5.64284e55	−0.0640586	−0.0320293	+	0.999487i	$0.510197\pi$
$212$	0	0
$213$	1.73678e57	1.56475
$214$	0	0
$215$	−8.11277e56	−0.581336
$216$	0	0
$217$	4.42418e56	0.252679
$218$	0	0
$219$	−3.50722e57	−1.59986
$220$	0	0
$221$	−1.36641e57	−0.498852
$222$	0	0
$223$	6.70689e57	1.96360	0.981798	−	0.189928i	$-0.0608256\pi$
$223$	6.70689e57	1.96360	0.981798	+	0.189928i	$0.0608256\pi$
$224$	0	0
$225$	8.01028e57	1.88441
$226$	0	0
$227$	−1.77430e57	−0.336042	−0.168021	−	0.985783i	$-0.553738\pi$
$227$	−1.77430e57	−0.336042	−0.168021	+	0.985783i	$0.553738\pi$
$228$	0	0
$229$	1.42321e57	0.217420	0.108710	−	0.994074i	$-0.465328\pi$
$229$	1.42321e57	0.217420	0.108710	+	0.994074i	$0.465328\pi$
$230$	0	0
$231$	1.81952e58	2.24628
$232$	0	0
$233$	6.14657e57	0.614346	0.307173	−	0.951654i	$-0.400617\pi$
$233$	6.14657e57	0.614346	0.307173	+	0.951654i	$0.400617\pi$
$234$	0	0
$235$	4.92994e57	0.399649
$236$	0	0
$237$	−8.70449e57	−0.573336
$238$	0	0
$239$	1.87389e58	1.00460	0.502301	−	0.864693i	$-0.332487\pi$
$239$	1.87389e58	1.00460	0.502301	+	0.864693i	$0.332487\pi$
$240$	0	0
$241$	1.41041e58	0.616491	0.308245	−	0.951307i	$-0.400258\pi$
$241$	1.41041e58	0.616491	0.308245	+	0.951307i	$0.400258\pi$
$242$	0	0
$243$	−3.85039e58	−1.37452
$244$	0	0
$245$	−2.03772e57	−0.0595080
$246$	0	0
$247$	−2.55233e58	−0.610747
$248$	0	0
$249$	−5.81249e58	−1.14151
$250$	0	0
$251$	−2.19883e58	−0.354965	−0.177482	−	0.984124i	$-0.556795\pi$
$251$	−2.19883e58	−0.354965	−0.177482	+	0.984124i	$0.556795\pi$
$252$	0	0
$253$	1.40414e59	1.86620
$254$	0	0
$255$	1.56731e59	1.71759
$256$	0	0
$257$	−1.39270e58	−0.126036	−0.0630180	−	0.998012i	$-0.520073\pi$
$257$	−1.39270e58	−0.126036	−0.0630180	+	0.998012i	$0.520073\pi$
$258$	0	0
$259$	1.29298e59	0.967712
$260$	0	0
$261$	−8.50692e57	−0.0527327
$262$	0	0
$263$	−1.50291e59	−0.772710	−0.386355	−	0.922350i	$-0.626266\pi$
$263$	−1.50291e59	−0.772710	−0.386355	+	0.922350i	$0.626266\pi$
$264$	0	0
$265$	5.88493e59	2.51314
$266$	0	0
$267$	4.71250e59	1.67389
$268$	0	0
$269$	4.46165e59	1.31997	0.659986	−	0.751278i	$-0.270562\pi$
$269$	4.46165e59	1.31997	0.659986	+	0.751278i	$0.270562\pi$
$270$	0	0
$271$	1.87553e59	0.462782	0.231391	−	0.972861i	$-0.425672\pi$
$271$	1.87553e59	0.462782	0.231391	+	0.972861i	$0.425672\pi$
$272$	0	0
$273$	4.80146e59	0.989438
$274$	0	0
$275$	1.90476e60	3.28238
$276$	0	0
$277$	−5.17979e59	−0.747408	−0.373704	−	0.927548i	$-0.621912\pi$
$277$	−5.17979e59	−0.747408	−0.373704	+	0.927548i	$0.621912\pi$
$278$	0	0
$279$	1.88180e59	0.227651
$280$	0	0
$281$	−1.19166e60	−1.21017	−0.605085	−	0.796161i	$-0.706861\pi$
$281$	−1.19166e60	−1.21017	−0.605085	+	0.796161i	$0.706861\pi$
$282$	0	0
$283$	−2.07435e60	−1.77058	−0.885288	−	0.465044i	$-0.846039\pi$
$283$	−2.07435e60	−1.77058	−0.885288	+	0.465044i	$0.846039\pi$
$284$	0	0
$285$	2.92758e60	2.10285
$286$	0	0
$287$	1.09468e59	0.0662492
$288$	0	0
$289$	−9.72261e59	−0.496347
$290$	0	0
$291$	5.10297e60	2.20011
$292$	0	0
$293$	4.37111e60	1.59344	0.796719	−	0.604350i	$-0.206567\pi$
$293$	4.37111e60	1.59344	0.796719	+	0.604350i	$0.206567\pi$
$294$	0	0
$295$	−9.99814e60	−3.08518
$296$	0	0
$297$	−7.08297e59	−0.185218
$298$	0	0
$299$	3.70533e60	0.822020
$300$	0	0
$301$	1.79447e60	0.338108
$302$	0	0
$303$	1.64902e61	2.64169
$304$	0	0
$305$	3.31623e60	0.452167
$306$	0	0
$307$	4.69486e60	0.545421	0.272711	−	0.962096i	$-0.412080\pi$
$307$	4.69486e60	0.545421	0.272711	+	0.962096i	$0.412080\pi$
$308$	0	0
$309$	−2.94603e60	−0.291914
$310$	0	0
$311$	1.13490e61	0.960127	0.480063	−	0.877234i	$-0.340614\pi$
$311$	1.13490e61	0.960127	0.480063	+	0.877234i	$0.340614\pi$
$312$	0	0
$313$	2.24105e61	1.62037	0.810186	−	0.586173i	$-0.199366\pi$
$313$	2.24105e61	1.62037	0.810186	+	0.586173i	$0.199366\pi$
$314$	0	0
$315$	−2.63320e61	−1.62883
$316$	0	0
$317$	−1.97282e61	−1.04505	−0.522523	−	0.852625i	$-0.675009\pi$
$317$	−1.97282e61	−1.04505	−0.522523	+	0.852625i	$0.675009\pi$
$318$	0	0
$319$	−2.02286e60	−0.0918530
$320$	0	0
$321$	−3.13895e61	−1.22296
$322$	0	0
$323$	1.84282e61	0.616625
$324$	0	0
$325$	5.02638e61	1.44582
$326$	0	0
$327$	−2.14371e61	−0.530574
$328$	0	0
$329$	−1.09045e61	−0.232438
$330$	0	0
$331$	8.48788e60	0.155960	0.0779801	−	0.996955i	$-0.475153\pi$
$331$	8.48788e60	0.155960	0.0779801	+	0.996955i	$0.475153\pi$
$332$	0	0
$333$	5.49959e61	0.871859
$334$	0	0
$335$	6.88699e61	0.942826
$336$	0	0
$337$	−1.13716e62	−1.34552	−0.672760	−	0.739861i	$-0.734891\pi$
$337$	−1.13716e62	−1.34552	−0.672760	+	0.739861i	$0.734891\pi$
$338$	0	0
$339$	−1.87949e62	−1.92375
$340$	0	0
$341$	4.47471e61	0.396536
$342$	0	0
$343$	−1.27919e62	−0.982265
$344$	0	0
$345$	−4.25009e62	−2.83029
$346$	0	0
$347$	−1.85460e62	−1.07196	−0.535979	−	0.844231i	$-0.680057\pi$
$347$	−1.85460e62	−1.07196	−0.535979	+	0.844231i	$0.680057\pi$
$348$	0	0
$349$	1.74990e62	0.878598	0.439299	−	0.898341i	$-0.355227\pi$
$349$	1.74990e62	0.878598	0.439299	+	0.898341i	$0.355227\pi$
$350$	0	0
$351$	−1.86909e61	−0.0815843
$352$	0	0
$353$	−4.46780e62	−1.69673	−0.848367	−	0.529409i	$-0.822414\pi$
$353$	−4.46780e62	−1.69673	−0.848367	+	0.529409i	$0.822414\pi$
$354$	0	0
$355$	−5.97672e62	−1.97637
$356$	0	0
$357$	−3.46673e62	−0.998961
$358$	0	0
$359$	6.15371e62	1.54640	0.773202	−	0.634160i	$-0.218654\pi$
$359$	6.15371e62	1.54640	0.773202	+	0.634160i	$0.218654\pi$
$360$	0	0
$361$	−1.11739e62	−0.245063
$362$	0	0
$363$	1.11766e63	2.14090
$364$	0	0
$365$	1.20693e63	2.02072
$366$	0	0
$367$	−1.78399e62	−0.261261	−0.130631	−	0.991431i	$-0.541700\pi$
$367$	−1.78399e62	−0.261261	−0.130631	+	0.991431i	$0.541700\pi$
$368$	0	0
$369$	4.65615e61	0.0596872
$370$	0	0
$371$	−1.30169e63	−1.46166
$372$	0	0
$373$	−6.15312e61	−0.0605661	−0.0302830	−	0.999541i	$-0.509641\pi$
$373$	−6.15312e61	−0.0605661	−0.0302830	+	0.999541i	$0.509641\pi$
$374$	0	0
$375$	−2.96240e63	−2.55786
$376$	0	0
$377$	−5.33802e61	−0.0404592
$378$	0	0
$379$	1.93924e63	1.29113	0.645567	−	0.763704i	$-0.276621\pi$
$379$	1.93924e63	1.29113	0.645567	+	0.763704i	$0.276621\pi$
$380$	0	0
$381$	7.55736e62	0.442291
$382$	0	0
$383$	−4.92299e61	−0.0253430	−0.0126715	−	0.999920i	$-0.504034\pi$
$383$	−4.92299e61	−0.0253430	−0.0126715	+	0.999920i	$0.504034\pi$
$384$	0	0
$385$	−6.26146e63	−2.83719
$386$	0	0
$387$	7.63264e62	0.304619
$388$	0	0
$389$	1.52402e63	0.536075	0.268038	−	0.963408i	$-0.413625\pi$
$389$	1.52402e63	0.536075	0.268038	+	0.963408i	$0.413625\pi$
$390$	0	0
$391$	−2.67530e63	−0.829931
$392$	0	0
$393$	−2.24869e63	−0.615619
$394$	0	0
$395$	2.99545e63	0.724157
$396$	0	0
$397$	−4.22139e63	−0.901753	−0.450877	−	0.892586i	$-0.648889\pi$
$397$	−4.22139e63	−0.901753	−0.450877	+	0.892586i	$0.648889\pi$
$398$	0	0
$399$	−6.47551e63	−1.22303
$400$	0	0
$401$	7.10002e63	1.18638	0.593188	−	0.805064i	$-0.297869\pi$
$401$	7.10002e63	1.18638	0.593188	+	0.805064i	$0.297869\pi$
$402$	0	0
$403$	1.18081e63	0.174665
$404$	0	0
$405$	1.43691e64	1.88270
$406$	0	0
$407$	1.30774e64	1.51866
$408$	0	0
$409$	−4.19928e63	−0.432468	−0.216234	−	0.976342i	$-0.569377\pi$
$409$	−4.19928e63	−0.432468	−0.216234	+	0.976342i	$0.569377\pi$
$410$	0	0
$411$	1.94429e64	1.77679
$412$	0	0
$413$	2.21149e64	1.79436
$414$	0	0
$415$	2.00023e64	1.44179
$416$	0	0
$417$	1.40744e64	0.901771
$418$	0	0
$419$	−2.50397e64	−1.42687	−0.713436	−	0.700720i	$-0.752862\pi$
$419$	−2.50397e64	−1.42687	−0.713436	+	0.700720i	$0.752862\pi$
$420$	0	0
$421$	−2.02007e63	−0.102437	−0.0512184	−	0.998687i	$-0.516310\pi$
$421$	−2.02007e63	−0.102437	−0.0512184	+	0.998687i	$0.516310\pi$
$422$	0	0
$423$	−4.63818e63	−0.209415
$424$	0	0
$425$	−3.62913e64	−1.45973
$426$	0	0
$427$	−7.33518e63	−0.262983
$428$	0	0
$429$	4.85630e64	1.55275
$430$	0	0
$431$	6.45676e64	1.84213	0.921067	−	0.389403i	$-0.127319\pi$
$431$	6.45676e64	1.84213	0.921067	+	0.389403i	$0.127319\pi$
$432$	0	0
$433$	5.64458e64	1.43773	0.718867	−	0.695148i	$-0.244661\pi$
$433$	5.64458e64	1.43773	0.718867	+	0.695148i	$0.244661\pi$
$434$	0	0
$435$	6.12283e63	0.139305
$436$	0	0
$437$	−4.99721e64	−1.01609
$438$	0	0
$439$	−4.39799e64	−0.799602	−0.399801	−	0.916602i	$-0.630921\pi$
$439$	−4.39799e64	−0.799602	−0.399801	+	0.916602i	$0.630921\pi$
$440$	0	0
$441$	1.91712e63	0.0311820
$442$	0	0
$443$	−5.44392e64	−0.792538	−0.396269	−	0.918134i	$-0.629695\pi$
$443$	−5.44392e64	−0.792538	−0.396269	+	0.918134i	$0.629695\pi$
$444$	0	0
$445$	−1.62170e65	−2.11422
$446$	0	0
$447$	1.79775e65	2.09988
$448$	0	0
$449$	6.51981e64	0.682652	0.341326	−	0.939945i	$-0.389124\pi$
$449$	6.51981e64	0.682652	0.341326	+	0.939945i	$0.389124\pi$
$450$	0	0
$451$	1.10718e64	0.103967
$452$	0	0
$453$	2.01672e64	0.169918
$454$	0	0
$455$	−1.65231e65	−1.24972
$456$	0	0
$457$	1.64423e65	1.11691	0.558453	−	0.829536i	$-0.311395\pi$
$457$	1.64423e65	1.11691	0.558453	+	0.829536i	$0.311395\pi$
$458$	0	0
$459$	1.34951e64	0.0823696
$460$	0	0
$461$	7.21325e64	0.395784	0.197892	−	0.980224i	$-0.436590\pi$
$461$	7.21325e64	0.395784	0.197892	+	0.980224i	$0.436590\pi$
$462$	0	0
$463$	−1.94522e65	−0.959916	−0.479958	−	0.877291i	$-0.659348\pi$
$463$	−1.94522e65	−0.959916	−0.479958	+	0.877291i	$0.659348\pi$
$464$	0	0
$465$	−1.35442e65	−0.601388
$466$	0	0
$467$	7.84448e64	0.313545	0.156773	−	0.987635i	$-0.449891\pi$
$467$	7.84448e64	0.313545	0.156773	+	0.987635i	$0.449891\pi$
$468$	0	0
$469$	−1.52334e65	−0.548354
$470$	0	0
$471$	−4.85146e64	−0.157348
$472$	0	0
$473$	1.81496e65	0.530603
$474$	0	0
$475$	−6.77886e65	−1.78716
$476$	0	0
$477$	−5.53665e65	−1.31688
$478$	0	0
$479$	3.87609e65	0.832097	0.416048	−	0.909342i	$-0.363415\pi$
$479$	3.87609e65	0.832097	0.416048	+	0.909342i	$0.363415\pi$
$480$	0	0
$481$	3.45095e65	0.668935
$482$	0	0
$483$	9.40079e65	1.64611
$484$	0	0
$485$	−1.75607e66	−2.77887
$486$	0	0
$487$	−8.09553e65	−1.15821	−0.579103	−	0.815255i	$-0.696597\pi$
$487$	−8.09553e65	−1.15821	−0.579103	+	0.815255i	$0.696597\pi$
$488$	0	0
$489$	5.84241e65	0.756005
$490$	0	0
$491$	1.15921e66	1.35727	0.678634	−	0.734477i	$-0.262572\pi$
$491$	1.15921e66	1.35727	0.678634	+	0.734477i	$0.262572\pi$
$492$	0	0
$493$	3.85414e64	0.0408486
$494$	0	0
$495$	−2.66327e66	−2.55616
$496$	0	0
$497$	1.32199e66	1.14947
$498$	0	0
$499$	−1.98775e66	−1.56638	−0.783191	−	0.621781i	$-0.786409\pi$
$499$	−1.98775e66	−1.56638	−0.783191	+	0.621781i	$0.786409\pi$
$500$	0	0
$501$	−1.22479e65	−0.0875061
$502$	0	0
$503$	1.31100e66	0.849548	0.424774	−	0.905300i	$-0.360354\pi$
$503$	1.31100e66	0.849548	0.424774	+	0.905300i	$0.360354\pi$
$504$	0	0
$505$	−5.67472e66	−3.33661
$506$	0	0
$507$	−1.31213e66	−0.700299
$508$	0	0
$509$	7.52616e65	0.364744	0.182372	−	0.983230i	$-0.441623\pi$
$509$	7.52616e65	0.364744	0.182372	+	0.983230i	$0.441623\pi$
$510$	0	0
$511$	−2.66961e66	−1.17526
$512$	0	0
$513$	2.52076e65	0.100846
$514$	0	0
$515$	1.01381e66	0.368704
$516$	0	0
$517$	−1.10291e66	−0.364772
$518$	0	0
$519$	5.48123e66	1.64922
$520$	0	0
$521$	2.28801e66	0.626519	0.313260	−	0.949667i	$-0.398579\pi$
$521$	2.28801e66	0.626519	0.313260	+	0.949667i	$0.398579\pi$
$522$	0	0
$523$	−3.63925e65	−0.0907237	−0.0453618	−	0.998971i	$-0.514444\pi$
$523$	−3.63925e65	−0.0907237	−0.0453618	+	0.998971i	$0.514444\pi$
$524$	0	0
$525$	1.27524e67	2.89528
$526$	0	0
$527$	−8.52564e65	−0.176347
$528$	0	0
$529$	1.94992e66	0.367581
$530$	0	0
$531$	9.40644e66	1.61663
$532$	0	0
$533$	2.92170e65	0.0457951
$534$	0	0
$535$	1.08019e67	1.54466
$536$	0	0
$537$	6.45299e66	0.842153
$538$	0	0
$539$	4.55871e65	0.0543147
$540$	0	0
$541$	−1.63939e67	−1.78382	−0.891908	−	0.452218i	$-0.850633\pi$
$541$	−1.63939e67	−1.78382	−0.891908	+	0.452218i	$0.850633\pi$
$542$	0	0
$543$	1.06719e67	1.06083
$544$	0	0
$545$	7.37707e66	0.670146
$546$	0	0
$547$	−1.43390e67	−1.19078	−0.595388	−	0.803439i	$-0.703001\pi$
$547$	−1.43390e67	−1.19078	−0.595388	+	0.803439i	$0.703001\pi$
$548$	0	0
$549$	−3.11997e66	−0.236934
$550$	0	0
$551$	7.19915e65	0.0500112
$552$	0	0
$553$	−6.62563e66	−0.421174
$554$	0	0
$555$	−3.95831e67	−2.30320
$556$	0	0
$557$	1.82851e67	0.974193	0.487097	−	0.873348i	$-0.338056\pi$
$557$	1.82851e67	0.974193	0.487097	+	0.873348i	$0.338056\pi$
$558$	0	0
$559$	4.78942e66	0.233719
$560$	0	0
$561$	−3.50632e67	−1.56770
$562$	0	0
$563$	2.90085e67	1.18869	0.594346	−	0.804209i	$-0.297411\pi$
$563$	2.90085e67	1.18869	0.594346	+	0.804209i	$0.297411\pi$
$564$	0	0
$565$	6.46783e67	2.42981
$566$	0	0
$567$	−3.17830e67	−1.09499
$568$	0	0
$569$	−4.00130e66	−0.126459	−0.0632296	−	0.997999i	$-0.520140\pi$
$569$	−4.00130e66	−0.126459	−0.0632296	+	0.997999i	$0.520140\pi$
$570$	0	0
$571$	4.01640e67	1.16480	0.582401	−	0.812902i	$-0.302113\pi$
$571$	4.01640e67	1.16480	0.582401	+	0.812902i	$0.302113\pi$
$572$	0	0
$573$	7.66932e67	2.04159
$574$	0	0
$575$	9.84116e67	2.40538
$576$	0	0
$577$	−4.66796e67	−1.04790	−0.523950	−	0.851749i	$-0.675542\pi$
$577$	−4.66796e67	−1.04790	−0.523950	+	0.851749i	$0.675542\pi$
$578$	0	0
$579$	3.96586e67	0.817924
$580$	0	0
$581$	−4.42432e67	−0.838554
$582$	0	0
$583$	−1.31656e68	−2.29382
$584$	0	0
$585$	−7.02799e67	−1.12593
$586$	0	0
$587$	−2.48745e67	−0.366539	−0.183270	−	0.983063i	$-0.558668\pi$
$587$	−2.48745e67	−0.366539	−0.183270	+	0.983063i	$0.558668\pi$
$588$	0	0
$589$	−1.59251e67	−0.215902
$590$	0	0
$591$	4.30830e67	0.537542
$592$	0	0
$593$	4.14071e66	0.0475592	0.0237796	−	0.999717i	$-0.492430\pi$
$593$	4.14071e66	0.0475592	0.0237796	+	0.999717i	$0.492430\pi$
$594$	0	0
$595$	1.19299e68	1.26175
$596$	0	0
$597$	9.39940e67	0.915649
$598$	0	0
$599$	−2.10487e68	−1.88915	−0.944577	−	0.328291i	$-0.893527\pi$
$599$	−2.10487e68	−1.88915	−0.944577	+	0.328291i	$0.893527\pi$
$600$	0	0
$601$	1.59794e68	1.32171	0.660854	−	0.750514i	$-0.270194\pi$
$601$	1.59794e68	1.32171	0.660854	+	0.750514i	$0.270194\pi$
$602$	0	0
$603$	−6.47941e67	−0.494039
$604$	0	0
$605$	−3.84615e68	−2.70408
$606$	0	0
$607$	−2.59164e68	−1.68055	−0.840275	−	0.542160i	$-0.817607\pi$
$607$	−2.59164e68	−1.68055	−0.840275	+	0.542160i	$0.817607\pi$
$608$	0	0
$609$	−1.35431e67	−0.0810205
$610$	0	0
$611$	−2.91042e67	−0.160674
$612$	0	0
$613$	−5.81382e67	−0.296265	−0.148132	−	0.988968i	$-0.547326\pi$
$613$	−5.81382e67	−0.296265	−0.148132	+	0.988968i	$0.547326\pi$
$614$	0	0
$615$	−3.35125e67	−0.157676
$616$	0	0
$617$	−3.39743e68	−1.47626	−0.738131	−	0.674657i	$-0.764291\pi$
$617$	−3.39743e68	−1.47626	−0.738131	+	0.674657i	$0.764291\pi$
$618$	0	0
$619$	−3.60992e67	−0.144903	−0.0724514	−	0.997372i	$-0.523082\pi$
$619$	−3.60992e67	−0.144903	−0.0724514	+	0.997372i	$0.523082\pi$
$620$	0	0
$621$	−3.65950e67	−0.135731
$622$	0	0
$623$	3.58704e68	1.22964
$624$	0	0
$625$	3.70404e68	1.17386
$626$	0	0
$627$	−6.54948e68	−1.91934
$628$	0	0
$629$	−2.49164e68	−0.675373
$630$	0	0
$631$	1.97407e68	0.495044	0.247522	−	0.968882i	$-0.420384\pi$
$631$	1.97407e68	0.495044	0.247522	+	0.968882i	$0.420384\pi$
$632$	0	0
$633$	−3.82106e67	−0.0886733
$634$	0	0
$635$	−2.60069e68	−0.558640
$636$	0	0
$637$	1.20298e67	0.0239245
$638$	0	0
$639$	5.62301e68	1.03561
$640$	0	0
$641$	−8.01714e67	−0.136772	−0.0683861	−	0.997659i	$-0.521785\pi$
$641$	−8.01714e67	−0.136772	−0.0683861	+	0.997659i	$0.521785\pi$
$642$	0	0
$643$	−1.38094e67	−0.0218276	−0.0109138	−	0.999940i	$-0.503474\pi$
$643$	−1.38094e67	−0.0218276	−0.0109138	+	0.999940i	$0.503474\pi$
$644$	0	0
$645$	−5.49357e68	−0.804715
$646$	0	0
$647$	3.29500e67	0.0447405	0.0223702	−	0.999750i	$-0.492879\pi$
$647$	3.29500e67	0.0447405	0.0223702	+	0.999750i	$0.492879\pi$
$648$	0	0
$649$	2.23675e69	2.81593
$650$	0	0
$651$	2.99584e68	0.349771
$652$	0	0
$653$	−1.29443e69	−1.40186	−0.700929	−	0.713231i	$-0.747231\pi$
$653$	−1.29443e69	−1.40186	−0.700929	+	0.713231i	$0.747231\pi$
$654$	0	0
$655$	7.73833e68	0.777563
$656$	0	0
$657$	−1.13550e69	−1.05885
$658$	0	0
$659$	1.29161e69	1.11799	0.558995	−	0.829171i	$-0.311187\pi$
$659$	1.29161e69	1.11799	0.558995	+	0.829171i	$0.311187\pi$
$660$	0	0
$661$	9.63417e68	0.774245	0.387123	−	0.922028i	$-0.373469\pi$
$661$	9.63417e68	0.774245	0.387123	+	0.922028i	$0.373469\pi$
$662$	0	0
$663$	−9.25269e68	−0.690536
$664$	0	0
$665$	2.22840e69	1.54476
$666$	0	0
$667$	−1.04513e68	−0.0673114
$668$	0	0
$669$	4.54158e69	2.71811
$670$	0	0
$671$	−7.41896e68	−0.412706
$672$	0	0
$673$	8.62520e68	0.446067	0.223034	−	0.974811i	$-0.428404\pi$
$673$	8.62520e68	0.446067	0.223034	+	0.974811i	$0.428404\pi$
$674$	0	0
$675$	−4.96422e68	−0.238731
$676$	0	0
$677$	−2.63131e69	−1.17693	−0.588464	−	0.808524i	$-0.700267\pi$
$677$	−2.63131e69	−1.17693	−0.588464	+	0.808524i	$0.700267\pi$
$678$	0	0
$679$	3.88425e69	1.61621
$680$	0	0
$681$	−1.20147e69	−0.465167
$682$	0	0
$683$	−1.91092e69	−0.688551	−0.344275	−	0.938869i	$-0.611875\pi$
$683$	−1.91092e69	−0.688551	−0.344275	+	0.938869i	$0.611875\pi$
$684$	0	0
$685$	−6.69082e69	−2.24420
$686$	0	0
$687$	9.63727e68	0.300964
$688$	0	0
$689$	−3.47420e69	−1.01038
$690$	0	0
$691$	3.63593e69	0.984926	0.492463	−	0.870333i	$-0.336097\pi$
$691$	3.63593e69	0.984926	0.492463	+	0.870333i	$0.336097\pi$
$692$	0	0
$693$	5.89090e69	1.48668
$694$	0	0
$695$	−4.84337e69	−1.13899
$696$	0	0
$697$	−2.10951e68	−0.0462358
$698$	0	0
$699$	4.16216e69	0.850409
$700$	0	0
$701$	8.99753e69	1.71408	0.857041	−	0.515248i	$-0.172300\pi$
$701$	8.99753e69	1.71408	0.857041	+	0.515248i	$0.172300\pi$
$702$	0	0
$703$	−4.65414e69	−0.826863
$704$	0	0
$705$	3.33832e69	0.553215
$706$	0	0
$707$	1.25519e70	1.94060
$708$	0	0
$709$	−7.36602e69	−1.06267	−0.531337	−	0.847160i	$-0.678310\pi$
$709$	−7.36602e69	−1.06267	−0.531337	+	0.847160i	$0.678310\pi$
$710$	0	0
$711$	−2.81817e69	−0.379457
$712$	0	0
$713$	2.31191e69	0.290588
$714$	0	0
$715$	−1.67118e70	−1.96122
$716$	0	0
$717$	1.26890e70	1.39062
$718$	0	0
$719$	7.15587e69	0.732495	0.366247	−	0.930518i	$-0.380642\pi$
$719$	7.15587e69	0.732495	0.366247	+	0.930518i	$0.380642\pi$
$720$	0	0
$721$	−2.24244e69	−0.214441
$722$	0	0
$723$	9.55058e69	0.853378
$724$	0	0
$725$	−1.41775e69	−0.118391
$726$	0	0
$727$	1.60013e69	0.124900	0.0624501	−	0.998048i	$-0.480109\pi$
$727$	1.60013e69	0.124900	0.0624501	+	0.998048i	$0.480109\pi$
$728$	0	0
$729$	−1.13171e70	−0.825866
$730$	0	0
$731$	−3.45804e69	−0.235968
$732$	0	0
$733$	2.09305e69	0.133577	0.0667887	−	0.997767i	$-0.478725\pi$
$733$	2.09305e69	0.133577	0.0667887	+	0.997767i	$0.478725\pi$
$734$	0	0
$735$	−1.37984e69	−0.0823740
$736$	0	0
$737$	−1.54073e70	−0.860546
$738$	0	0
$739$	−2.82972e70	−1.47895	−0.739477	−	0.673182i	$-0.764927\pi$
$739$	−2.82972e70	−1.47895	−0.739477	+	0.673182i	$0.764927\pi$
$740$	0	0
$741$	−1.72831e70	−0.845427
$742$	0	0
$743$	−3.69750e70	−1.69310	−0.846550	−	0.532310i	$-0.821324\pi$
$743$	−3.69750e70	−1.69310	−0.846550	+	0.532310i	$0.821324\pi$
$744$	0	0
$745$	−6.18653e70	−2.65227
$746$	0	0
$747$	−1.88186e70	−0.755495
$748$	0	0
$749$	−2.38928e70	−0.898386
$750$	0	0
$751$	−3.61337e70	−1.27272	−0.636361	−	0.771391i	$-0.719561\pi$
$751$	−3.61337e70	−1.27272	−0.636361	+	0.771391i	$0.719561\pi$
$752$	0	0
$753$	−1.48894e70	−0.491361
$754$	0	0
$755$	−6.94006e69	−0.214616
$756$	0	0
$757$	3.99376e70	1.15753	0.578766	−	0.815493i	$-0.303534\pi$
$757$	3.99376e70	1.15753	0.578766	+	0.815493i	$0.303534\pi$
$758$	0	0
$759$	9.50816e70	2.58329
$760$	0	0
$761$	6.65810e70	1.69600	0.848000	−	0.529996i	$-0.177807\pi$
$761$	6.65810e70	1.69600	0.848000	+	0.529996i	$0.177807\pi$
$762$	0	0
$763$	−1.63173e70	−0.389761
$764$	0	0
$765$	5.07432e70	1.13677
$766$	0	0
$767$	5.90246e70	1.24036
$768$	0	0
$769$	−5.73078e70	−1.12984	−0.564922	−	0.825144i	$-0.691094\pi$
$769$	−5.73078e70	−1.12984	−0.564922	+	0.825144i	$0.691094\pi$
$770$	0	0
$771$	−9.43066e69	−0.174466
$772$	0	0
$773$	−6.58392e70	−1.14311	−0.571553	−	0.820565i	$-0.693659\pi$
$773$	−6.58392e70	−1.14311	−0.571553	+	0.820565i	$0.693659\pi$
$774$	0	0
$775$	3.13618e70	0.511104
$776$	0	0
$777$	8.75540e70	1.33956
$778$	0	0
$779$	−3.94036e69	−0.0566068
$780$	0	0
$781$	1.33709e71	1.80389
$782$	0	0
$783$	5.27201e68	0.00668056
$784$	0	0
$785$	1.66952e70	0.198739
$786$	0	0
$787$	−1.47428e71	−1.64892	−0.824460	−	0.565920i	$-0.808521\pi$
$787$	−1.47428e71	−1.64892	−0.824460	+	0.565920i	$0.808521\pi$
$788$	0	0
$789$	−1.01770e71	−1.06963
$790$	0	0
$791$	−1.43062e71	−1.41319
$792$	0	0
$793$	−1.95776e70	−0.181788
$794$	0	0
$795$	3.98499e71	3.47882
$796$	0	0
$797$	3.58327e70	0.294138	0.147069	−	0.989126i	$-0.453016\pi$
$797$	3.58327e70	0.294138	0.147069	+	0.989126i	$0.453016\pi$
$798$	0	0
$799$	2.10137e70	0.162220
$800$	0	0
$801$	1.52572e71	1.10784
$802$	0	0
$803$	−2.70010e71	−1.84437
$804$	0	0
$805$	−3.23506e71	−2.07914
$806$	0	0
$807$	3.02122e71	1.82717
$808$	0	0
$809$	4.95238e70	0.281887	0.140943	−	0.990018i	$-0.454986\pi$
$809$	4.95238e70	0.281887	0.140943	+	0.990018i	$0.454986\pi$
$810$	0	0
$811$	3.54799e71	1.90095	0.950477	−	0.310796i	$-0.100596\pi$
$811$	3.54799e71	1.90095	0.950477	+	0.310796i	$0.100596\pi$
$812$	0	0
$813$	1.27002e71	0.640607
$814$	0	0
$815$	−2.01053e71	−0.954880
$816$	0	0
$817$	−6.45928e70	−0.288897
$818$	0	0
$819$	1.55452e71	0.654850
$820$	0	0
$821$	−1.95327e71	−0.775095	−0.387547	−	0.921850i	$-0.626678\pi$
$821$	−1.95327e71	−0.775095	−0.387547	+	0.921850i	$0.626678\pi$
$822$	0	0
$823$	−2.05807e71	−0.769421	−0.384710	−	0.923037i	$-0.625699\pi$
$823$	−2.05807e71	−0.769421	−0.384710	+	0.923037i	$0.625699\pi$
$824$	0	0
$825$	1.28981e72	4.54364
$826$	0	0
$827$	8.99522e70	0.298625	0.149313	−	0.988790i	$-0.452294\pi$
$827$	8.99522e70	0.298625	0.149313	+	0.988790i	$0.452294\pi$
$828$	0	0
$829$	−4.10043e71	−1.28305	−0.641526	−	0.767102i	$-0.721698\pi$
$829$	−4.10043e71	−1.28305	−0.641526	+	0.767102i	$0.721698\pi$
$830$	0	0
$831$	−3.50750e71	−1.03460
$832$	0	0
$833$	−8.68569e69	−0.0241547
$834$	0	0
$835$	4.21483e70	0.110525
$836$	0	0
$837$	−1.16621e70	−0.0288405
$838$	0	0
$839$	6.23847e69	0.0145516	0.00727579	−	0.999974i	$-0.497684\pi$
$839$	6.23847e69	0.0145516	0.00727579	+	0.999974i	$0.497684\pi$
$840$	0	0
$841$	−4.52961e71	−0.996687
$842$	0	0
$843$	−8.06932e71	−1.67518
$844$	0	0
$845$	4.51540e71	0.884518
$846$	0	0
$847$	8.50731e71	1.57271
$848$	0	0
$849$	−1.40465e72	−2.45092
$850$	0	0
$851$	6.75661e71	1.11290
$852$	0	0
$853$	1.20122e72	1.86798	0.933989	−	0.357301i	$-0.116303\pi$
$853$	1.20122e72	1.86798	0.933989	+	0.357301i	$0.116303\pi$
$854$	0	0
$855$	9.47834e71	1.39175
$856$	0	0
$857$	3.84535e71	0.533219	0.266610	−	0.963805i	$-0.414097\pi$
$857$	3.84535e71	0.533219	0.266610	+	0.963805i	$0.414097\pi$
$858$	0	0
$859$	−4.56460e71	−0.597820	−0.298910	−	0.954281i	$-0.596623\pi$
$859$	−4.56460e71	−0.597820	−0.298910	+	0.954281i	$0.596623\pi$
$860$	0	0
$861$	7.41264e70	0.0917056
$862$	0	0
$863$	−9.22876e71	−1.07865	−0.539323	−	0.842099i	$-0.681320\pi$
$863$	−9.22876e71	−1.07865	−0.539323	+	0.842099i	$0.681320\pi$
$864$	0	0
$865$	−1.88623e72	−2.08306
$866$	0	0
$867$	−6.58368e71	−0.687070
$868$	0	0
$869$	−6.70131e71	−0.660960
$870$	0	0
$871$	−4.06578e71	−0.379052
$872$	0	0
$873$	1.65214e72	1.45612
$874$	0	0
$875$	−2.25490e72	−1.87901
$876$	0	0
$877$	1.86935e72	1.47299	0.736496	−	0.676442i	$-0.236479\pi$
$877$	1.86935e72	1.47299	0.736496	+	0.676442i	$0.236479\pi$
$878$	0	0
$879$	2.95991e72	2.20572
$880$	0	0
$881$	−2.06246e72	−1.45370	−0.726850	−	0.686796i	$-0.759017\pi$
$881$	−2.06246e72	−1.45370	−0.726850	+	0.686796i	$0.759017\pi$
$882$	0	0
$883$	−1.01095e72	−0.674051	−0.337026	−	0.941495i	$-0.609421\pi$
$883$	−1.01095e72	−0.674051	−0.337026	+	0.941495i	$0.609421\pi$
$884$	0	0
$885$	−6.77026e72	−4.27066
$886$	0	0
$887$	−3.84733e71	−0.229631	−0.114815	−	0.993387i	$-0.536628\pi$
$887$	−3.84733e71	−0.229631	−0.114815	+	0.993387i	$0.536628\pi$
$888$	0	0
$889$	5.75247e71	0.324908
$890$	0	0
$891$	−3.21460e72	−1.71839
$892$	0	0
$893$	3.92515e71	0.198607
$894$	0	0
$895$	−2.22064e72	−1.06369
$896$	0	0
$897$	2.50907e72	1.13788
$898$	0	0
$899$	−3.33062e70	−0.0143025
$900$	0	0
$901$	2.50843e72	1.02010
$902$	0	0
$903$	1.21513e72	0.468027
$904$	0	0
$905$	−3.67247e72	−1.33989
$906$	0	0
$907$	1.65047e72	0.570465	0.285232	−	0.958458i	$-0.407929\pi$
$907$	1.65047e72	0.570465	0.285232	+	0.958458i	$0.407929\pi$
$908$	0	0
$909$	5.33888e72	1.74838
$910$	0	0
$911$	−1.74723e72	−0.542188	−0.271094	−	0.962553i	$-0.587386\pi$
$911$	−1.74723e72	−0.542188	−0.271094	+	0.962553i	$0.587386\pi$
$912$	0	0
$913$	−4.47485e72	−1.31597
$914$	0	0
$915$	2.24559e72	0.625913
$916$	0	0
$917$	−1.71164e72	−0.452235
$918$	0	0
$919$	2.26784e72	0.568044	0.284022	−	0.958818i	$-0.408331\pi$
$919$	2.26784e72	0.568044	0.284022	+	0.958818i	$0.408331\pi$
$920$	0	0
$921$	3.17913e72	0.755000
$922$	0	0
$923$	3.52839e72	0.794576
$924$	0	0
$925$	9.16554e72	1.95743
$926$	0	0
$927$	−9.53809e71	−0.193200
$928$	0	0
$929$	−5.22436e72	−1.00380	−0.501901	−	0.864925i	$-0.667366\pi$
$929$	−5.22436e72	−1.00380	−0.501901	+	0.864925i	$0.667366\pi$
$930$	0	0
$931$	−1.62240e71	−0.0295728
$932$	0	0
$933$	7.68500e72	1.32906
$934$	0	0
$935$	1.20662e73	1.98009
$936$	0	0
$937$	2.17978e72	0.339464	0.169732	−	0.985490i	$-0.445710\pi$
$937$	2.17978e72	0.339464	0.169732	+	0.985490i	$0.445710\pi$
$938$	0	0
$939$	1.51753e73	2.24300
$940$	0	0
$941$	−6.39953e72	−0.897847	−0.448924	−	0.893570i	$-0.648192\pi$
$941$	−6.39953e72	−0.897847	−0.448924	+	0.893570i	$0.648192\pi$
$942$	0	0
$943$	5.72040e71	0.0761885
$944$	0	0
$945$	1.63188e72	0.206352
$946$	0	0
$947$	−8.32607e72	−0.999691	−0.499845	−	0.866115i	$-0.666610\pi$
$947$	−8.32607e72	−0.999691	−0.499845	+	0.866115i	$0.666610\pi$
$948$	0	0
$949$	−7.12517e72	−0.812406
$950$	0	0
$951$	−1.33590e73	−1.44661
$952$	0	0
$953$	1.86907e73	1.92243	0.961214	−	0.275804i	$-0.0889441\pi$
$953$	1.86907e73	1.92243	0.961214	+	0.275804i	$0.0889441\pi$
$954$	0	0
$955$	−2.63922e73	−2.57865
$956$	0	0
$957$	−1.36978e72	−0.127148
$958$	0	0
$959$	1.47994e73	1.30524
$960$	0	0
$961$	−1.11955e73	−0.938255
$962$	0	0
$963$	−1.01627e73	−0.809401
$964$	0	0
$965$	−1.36476e73	−1.03309
$966$	0	0
$967$	−7.70074e72	−0.554095	−0.277047	−	0.960856i	$-0.589356\pi$
$967$	−7.70074e72	−0.554095	−0.277047	+	0.960856i	$0.589356\pi$
$968$	0	0
$969$	1.24787e73	0.853564
$970$	0	0
$971$	−2.89962e72	−0.188569	−0.0942845	−	0.995545i	$-0.530056\pi$
$971$	−2.89962e72	−0.188569	−0.0942845	+	0.995545i	$0.530056\pi$
$972$	0	0
$973$	1.07131e73	0.662444
$974$	0	0
$975$	3.40362e73	2.00137
$976$	0	0
$977$	1.83564e73	1.02653	0.513263	−	0.858231i	$-0.328437\pi$
$977$	1.83564e73	1.02653	0.513263	+	0.858231i	$0.328437\pi$
$978$	0	0
$979$	3.62801e73	1.92971
$980$	0	0
$981$	−6.94048e72	−0.351155
$982$	0	0
$983$	1.42481e73	0.685799	0.342900	−	0.939372i	$-0.388591\pi$
$983$	1.42481e73	0.685799	0.342900	+	0.939372i	$0.388591\pi$
$984$	0	0
$985$	−1.48260e73	−0.678947
$986$	0	0
$987$	−7.38403e72	−0.321753
$988$	0	0
$989$	9.37722e72	0.388834
$990$	0	0
$991$	1.87344e73	0.739323	0.369662	−	0.929166i	$-0.379474\pi$
$991$	1.87344e73	0.739323	0.369662	+	0.929166i	$0.379474\pi$
$992$	0	0
$993$	5.74758e72	0.215888
$994$	0	0
$995$	−3.23458e73	−1.15652
$996$	0	0
$997$	−4.87977e73	−1.66099	−0.830497	−	0.557023i	$-0.811943\pi$
$997$	−4.87977e73	−1.66099	−0.830497	+	0.557023i	$0.811943\pi$
$998$	0	0
$999$	−3.40827e72	−0.110453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.50.a.b.1.3		3
4.3	odd	2		2.50.a.b.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.50.a.b.1.1	✓	3	4.3	odd	2
16.50.a.b.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 50 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+6.77152e11 q^{3} -2.33026e17 q^{5} +5.15430e20 q^{7} +2.19235e23 q^{9} +O(q^{10})\)	\(q+6.77152e11 q^{3} -2.33026e17 q^{5} +5.15430e20 q^{7} +2.19235e23 q^{9} +5.21317e25 q^{11} +1.37568e27 q^{13} -1.57794e29 q^{15} -9.93263e29 q^{17} -1.85532e31 q^{19} +3.49024e32 q^{21} +2.69345e33 q^{23} +3.65374e34 q^{25} -1.35867e34 q^{27} -3.88028e34 q^{29} +8.58347e35 q^{31} +3.53011e37 q^{33} -1.20108e38 q^{35} +2.50854e38 q^{37} +9.31545e38 q^{39} +2.12382e38 q^{41} +3.48149e39 q^{43} -5.10874e40 q^{45} -2.11562e40 q^{47} +8.74461e39 q^{49} -6.72589e41 q^{51} -2.52544e42 q^{53} -1.21480e43 q^{55} -1.25633e43 q^{57} +4.29058e43 q^{59} -1.42312e43 q^{61} +1.13000e44 q^{63} -3.20569e44 q^{65} -2.95546e44 q^{67} +1.82387e45 q^{69} +2.56483e45 q^{71} -5.17938e45 q^{73} +2.47414e46 q^{75} +2.68703e46 q^{77} -1.28546e46 q^{79} -6.16630e46 q^{81} -8.58374e46 q^{83} +2.31456e47 q^{85} -2.62754e46 q^{87} +6.95931e47 q^{89} +7.09067e47 q^{91} +5.81231e47 q^{93} +4.32337e48 q^{95} +7.53593e48 q^{97} +1.14291e49 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots + 38\!\cdots\!99 q^{9}+O(q^{10}) \) 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9	\( 3 q + 16203614388 q^{3} - 10\!\cdots\!30 q^{5}+ \cdots - 17\!\cdots\!68 q^{99}+O(q^{100}) \) 3 * q + 16203614388 * q^3 - 101813016401840430 * q^5 + 125479179230203797096 * q^7 + 389432931519696922052199 * q^9 + 25860036791834065092767004 * q^11 + 2945123797958353151076283242 * q^13 - 87846213583565151681178171080 * q^15 + 3910755572889939740390354668854 * q^17 - 14590272981610371514592320211580 * q^19 - 109864176920553355284360667668384 * q^21 + 4953158032634305739950894912759608 * q^23 + 39451396326519132102111422427450525 * q^25 - 196120894647932767255303897131684600 * q^27 + 937305925294643190481854503019954330 * q^29 - 2414809991776326638635666091994140256 * q^31 + 84513992666209345011409783180743651984 * q^33 - 300439012903055674136683294221406953360 * q^35 + 309377978155169809207437071391240173874 * q^37 - 360871814296103457146322338824042742568 * q^39 - 1143109642820632592659114174552039005954 * q^41 + 17579019372148890319048762066144112105148 * q^43 - 114813225075613761568802306330842527416790 * q^45 + 218573936001854642959028674713638582342256 * q^47 + 373628008074429359506251920014708170372651 * q^49 - 2241123510744223933356304325509677976541016 * q^51 + 740200780025011930130058449331923084936802 * q^53 - 3200488473448119911679443277764576117207640 * q^55 - 20594915296543361145254552753753824314692880 * q^57 + 53014363316602767635595514099864508327238540 * q^59 - 124730102606725385116407342714381742110559494 * q^61 + 468292624427612329618287149072325587596605768 * q^63 - 421321721827999315000625667485604880336092420 * q^65 - 1111804215897361218882144911380577470099878924 * q^67 + 770909887444909922528222240570561038770401568 * q^69 + 4013080456871179368491803943114613092268032424 * q^71 - 10015638764428635672946257750737669286684666018 * q^73 + 38646070268253498119014909069282536501398301900 * q^75 - 23360330575813840356220747501745396153372866272 * q^77 - 36077368003505469428776991593224667583970712560 * q^79 - 14427585993163108762153736521422002728666768917 * q^81 - 150749458884727999220762915504103393007529125692 * q^83 + 566911972337878875844467446456481028710158526660 * q^85 - 1072056267743755243448622937355132996639037778920 * q^87 + 542251936064510107856207107418847041962289942670 * q^89 + 1459783720923935217191336434024489699257545191344 * q^91 - 2368291579513879043404530519265491460552020418176 * q^93 + 2813579016011586836853932078889401271258883877400 * q^95 + 4343844254337968328695747517284119056283595359014 * q^97 - 17595360353038055002493296165642581030249046200468 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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