LMFDB - Embedded newform 639.2.a.i.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,2,Mod(1,639)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("639.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	639.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:	$5.10244068916$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 213)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.13856$ of defining polynomial
Character	$\chi$	$=$	639.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.138564 q^{2} -1.98080 q^{4} -0.703671 q^{5} +3.84224 q^{7} -0.551597 q^{8} +O(q^{10})$	\(q+0.138564 q^{2} -1.98080 q^{4} -0.703671 q^{5} +3.84224 q^{7} -0.551597 q^{8} -0.0975037 q^{10} -0.606168 q^{11} -0.0975037 q^{13} +0.532397 q^{14} +3.88517 q^{16} -6.59150 q^{17} +8.14310 q^{19} +1.39383 q^{20} -0.0839932 q^{22} +8.05910 q^{23} -4.50485 q^{25} -0.0135105 q^{26} -7.61070 q^{28} +5.50485 q^{29} +5.85575 q^{31} +1.64154 q^{32} -0.913346 q^{34} -2.70367 q^{35} +1.62537 q^{37} +1.12834 q^{38} +0.388143 q^{40} +10.2169 q^{41} +4.69016 q^{43} +1.20070 q^{44} +1.11670 q^{46} +8.86028 q^{47} +7.76278 q^{49} -0.624211 q^{50} +0.193135 q^{52} -8.68900 q^{53} +0.426543 q^{55} -2.11936 q^{56} +0.762775 q^{58} -10.4202 q^{59} -3.16042 q^{61} +0.811397 q^{62} -7.54288 q^{64} +0.0686106 q^{65} +3.17696 q^{67} +13.0564 q^{68} -0.374632 q^{70} -1.00000 q^{71} -10.4337 q^{73} +0.225218 q^{74} -16.1298 q^{76} -2.32904 q^{77} -4.83326 q^{79} -2.73388 q^{80} +1.41569 q^{82} -12.8449 q^{83} +4.63825 q^{85} +0.649889 q^{86} +0.334360 q^{88} +3.45925 q^{89} -0.374632 q^{91} -15.9635 q^{92} +1.22772 q^{94} -5.73006 q^{95} -1.91601 q^{97} +1.07564 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8	$ 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 2 q^{11} + 5 q^{13} - q^{14} + 11 q^{16} + 8 q^{17} + 8 q^{19} + 6 q^{20} - 7 q^{22} + q^{23} - q^{25} + 12 q^{26} - 9 q^{28} + 5 q^{29} + 2 q^{31} - 17 q^{32} - 21 q^{34} - 5 q^{35} + 19 q^{37} - 3 q^{38} - 23 q^{40} + 19 q^{41} + 25 q^{43} + 19 q^{44} + 12 q^{46} - 7 q^{47} - 6 q^{49} + 31 q^{50} - 13 q^{52} + 5 q^{53} + 3 q^{55} + 8 q^{56} - 34 q^{58} - 10 q^{59} + 2 q^{61} - 4 q^{62} + 34 q^{64} + 16 q^{65} + 35 q^{67} + 45 q^{68} + 11 q^{70} - 4 q^{71} + 2 q^{73} - 20 q^{74} + 13 q^{76} - 16 q^{77} - q^{79} + 5 q^{80} - 5 q^{82} - 18 q^{83} + 11 q^{85} - 20 q^{86} - 40 q^{88} + 16 q^{89} + 11 q^{91} - 41 q^{92} - 5 q^{94} + 15 q^{95} - q^{97} + 10 q^{98}+O(q^{100}) $ 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 2 * q^11 + 5 * q^13 - q^14 + 11 * q^16 + 8 * q^17 + 8 * q^19 + 6 * q^20 - 7 * q^22 + q^23 - q^25 + 12 * q^26 - 9 * q^28 + 5 * q^29 + 2 * q^31 - 17 * q^32 - 21 * q^34 - 5 * q^35 + 19 * q^37 - 3 * q^38 - 23 * q^40 + 19 * q^41 + 25 * q^43 + 19 * q^44 + 12 * q^46 - 7 * q^47 - 6 * q^49 + 31 * q^50 - 13 * q^52 + 5 * q^53 + 3 * q^55 + 8 * q^56 - 34 * q^58 - 10 * q^59 + 2 * q^61 - 4 * q^62 + 34 * q^64 + 16 * q^65 + 35 * q^67 + 45 * q^68 + 11 * q^70 - 4 * q^71 + 2 * q^73 - 20 * q^74 + 13 * q^76 - 16 * q^77 - q^79 + 5 * q^80 - 5 * q^82 - 18 * q^83 + 11 * q^85 - 20 * q^86 - 40 * q^88 + 16 * q^89 + 11 * q^91 - 41 * q^92 - 5 * q^94 + 15 * q^95 - q^97 + 10 * q^98

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 $

$2$	0.138564	0.0979797	0.0489899	−	0.998799i	$-0.484400\pi$
$2$	0.138564	0.0979797	0.0489899	+	0.998799i	$0.484400\pi$
$3$	0	0
$3$	0	0
$4$	−1.98080	−0.990400
$5$	−0.703671	−0.314691	−0.157346	−	0.987544i	$-0.550294\pi$
$5$	−0.703671	−0.314691	−0.157346	+	0.987544i	$0.550294\pi$
$6$	0	0
$7$	3.84224	1.45223	0.726114	−	0.687574i	$-0.241324\pi$
$7$	3.84224	1.45223	0.726114	+	0.687574i	$0.241324\pi$
$8$	−0.551597	−0.195019
$9$	0	0
$10$	−0.0975037	−0.0308334
$11$	−0.606168	−0.182766	−0.0913832	−	0.995816i	$-0.529129\pi$
$11$	−0.606168	−0.182766	−0.0913832	+	0.995816i	$0.529129\pi$
$12$	0	0
$13$	−0.0975037	−0.0270427	−0.0135213	−	0.999909i	$-0.504304\pi$
$13$	−0.0975037	−0.0270427	−0.0135213	+	0.999909i	$0.504304\pi$
$14$	0.532397	0.142289
$15$	0	0
$16$	3.88517	0.971292
$17$	−6.59150	−1.59867	−0.799337	−	0.600883i	$-0.794816\pi$
$17$	−6.59150	−1.59867	−0.799337	+	0.600883i	$0.794816\pi$
$18$	0	0
$19$	8.14310	1.86815	0.934077	−	0.357071i	$-0.116225\pi$
$19$	8.14310	1.86815	0.934077	+	0.357071i	$0.116225\pi$
$20$	1.39383	0.311670
$21$	0	0
$22$	−0.0839932	−0.0179074
$23$	8.05910	1.68044	0.840220	−	0.542246i	$-0.182426\pi$
$23$	8.05910	1.68044	0.840220	+	0.542246i	$0.182426\pi$
$24$	0	0
$25$	−4.50485	−0.900969
$26$	−0.0135105	−0.00264963
$27$	0	0
$28$	−7.61070	−1.43829
$29$	5.50485	1.02222	0.511112	−	0.859514i	$-0.329234\pi$
$29$	5.50485	1.02222	0.511112	+	0.859514i	$0.329234\pi$
$30$	0	0
$31$	5.85575	1.05172	0.525862	−	0.850570i	$-0.323743\pi$
$31$	5.85575	1.05172	0.525862	+	0.850570i	$0.323743\pi$
$32$	1.64154	0.290186
$33$	0	0
$34$	−0.913346	−0.156638
$35$	−2.70367	−0.457004
$36$	0	0
$37$	1.62537	0.267209	0.133604	−	0.991035i	$-0.457345\pi$
$37$	1.62537	0.267209	0.133604	+	0.991035i	$0.457345\pi$
$38$	1.12834	0.183041
$39$	0	0
$40$	0.388143	0.0613708
$41$	10.2169	1.59561	0.797803	−	0.602918i	$-0.205995\pi$
$41$	10.2169	1.59561	0.797803	+	0.602918i	$0.205995\pi$
$42$	0	0
$43$	4.69016	0.715243	0.357621	−	0.933867i	$-0.383588\pi$
$43$	4.69016	0.715243	0.357621	+	0.933867i	$0.383588\pi$
$44$	1.20070	0.181012
$45$	0	0
$46$	1.11670	0.164649
$47$	8.86028	1.29240	0.646202	−	0.763166i	$-0.276356\pi$
$47$	8.86028	1.29240	0.646202	+	0.763166i	$0.276356\pi$
$48$	0	0
$49$	7.76278	1.10897
$50$	−0.624211	−0.0882767
$51$	0	0
$52$	0.193135	0.0267831
$53$	−8.68900	−1.19353	−0.596763	−	0.802417i	$-0.703547\pi$
$53$	−8.68900	−1.19353	−0.596763	+	0.802417i	$0.703547\pi$
$54$	0	0
$55$	0.426543	0.0575150
$56$	−2.11936	−0.283212
$57$	0	0
$58$	0.762775	0.100157
$59$	−10.4202	−1.35660	−0.678299	−	0.734786i	$-0.737283\pi$
$59$	−10.4202	−1.35660	−0.678299	+	0.734786i	$0.737283\pi$
$60$	0	0
$61$	−3.16042	−0.404651	−0.202325	−	0.979318i	$-0.564850\pi$
$61$	−3.16042	−0.404651	−0.202325	+	0.979318i	$0.564850\pi$
$62$	0.811397	0.103048
$63$	0	0
$64$	−7.54288	−0.942860
$65$	0.0686106	0.00851009
$66$	0	0
$67$	3.17696	0.388128	0.194064	−	0.980989i	$-0.437833\pi$
$67$	3.17696	0.388128	0.194064	+	0.980989i	$0.437833\pi$
$68$	13.0564	1.58333
$69$	0	0
$70$	−0.374632	−0.0447771
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	0	0
$73$	−10.4337	−1.22118	−0.610588	−	0.791948i	$-0.709067\pi$
$73$	−10.4337	−1.22118	−0.610588	+	0.791948i	$0.709067\pi$
$74$	0.225218	0.0261810
$75$	0	0
$76$	−16.1298	−1.85022
$77$	−2.32904	−0.265419
$78$	0	0
$79$	−4.83326	−0.543784	−0.271892	−	0.962328i	$-0.587649\pi$
$79$	−4.83326	−0.543784	−0.271892	+	0.962328i	$0.587649\pi$
$80$	−2.73388	−0.305657
$81$	0	0
$82$	1.41569	0.156337
$83$	−12.8449	−1.40991	−0.704955	−	0.709252i	$-0.749033\pi$
$83$	−12.8449	−1.40991	−0.704955	+	0.709252i	$0.749033\pi$
$84$	0	0
$85$	4.63825	0.503089
$86$	0.649889	0.0700793
$87$	0	0
$88$	0.334360	0.0356429
$89$	3.45925	0.366680	0.183340	−	0.983050i	$-0.441309\pi$
$89$	3.45925	0.366680	0.183340	+	0.983050i	$0.441309\pi$
$90$	0	0
$91$	−0.374632	−0.0392721
$92$	−15.9635	−1.66431
$93$	0	0
$94$	1.22772	0.126629
$95$	−5.73006	−0.587892
$96$	0	0
$97$	−1.91601	−0.194541	−0.0972705	−	0.995258i	$-0.531011\pi$
$97$	−1.91601	−0.194541	−0.0972705	+	0.995258i	$0.531011\pi$
$98$	1.07564	0.108656
$99$	0	0
$100$	8.92320	0.892320
$101$	11.5504	1.14931	0.574656	−	0.818395i	$-0.305136\pi$
$101$	11.5504	1.14931	0.574656	+	0.818395i	$0.305136\pi$
$102$	0	0
$103$	−3.07830	−0.303314	−0.151657	−	0.988433i	$-0.548461\pi$
$103$	−3.07830	−0.303314	−0.151657	+	0.988433i	$0.548461\pi$
$104$	0.0537827	0.00527383
$105$	0	0
$106$	−1.20399	−0.116941
$107$	−4.49515	−0.434563	−0.217281	−	0.976109i	$-0.569719\pi$
$107$	−4.49515	−0.434563	−0.217281	+	0.976109i	$0.569719\pi$
$108$	0	0
$109$	−9.29001	−0.889822	−0.444911	−	0.895575i	$-0.646765\pi$
$109$	−9.29001	−0.889822	−0.444911	+	0.895575i	$0.646765\pi$
$110$	0.0591036	0.00563531
$111$	0	0
$112$	14.9277	1.41054
$113$	5.23607	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$113$	5.23607	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$114$	0	0
$115$	−5.67096	−0.528820
$116$	−10.9040	−1.01241
$117$	0	0
$118$	−1.44387	−0.132919
$119$	−25.3261	−2.32164
$120$	0	0
$121$	−10.6326	−0.966596
$122$	−0.437922	−0.0396476
$123$	0	0
$124$	−11.5991	−1.04163
$125$	6.68829	0.598219
$126$	0	0
$127$	15.4062	1.36708	0.683539	−	0.729914i	$-0.260440\pi$
$127$	15.4062	1.36708	0.683539	+	0.729914i	$0.260440\pi$
$128$	−4.32825	−0.382567
$129$	0	0
$130$	0.00950697	0.000833817 0
$131$	14.0528	1.22780	0.613899	−	0.789385i	$-0.289600\pi$
$131$	14.0528	1.22780	0.613899	+	0.789385i	$0.289600\pi$
$132$	0	0
$133$	31.2877	2.71299
$134$	0.440214	0.0380287
$135$	0	0
$136$	3.63585	0.311771
$137$	−0.530524	−0.0453258	−0.0226629	−	0.999743i	$-0.507214\pi$
$137$	−0.530524	−0.0453258	−0.0226629	+	0.999743i	$0.507214\pi$
$138$	0	0
$139$	−14.8365	−1.25842	−0.629210	−	0.777236i	$-0.716621\pi$
$139$	−14.8365	−1.25842	−0.629210	+	0.777236i	$0.716621\pi$
$140$	5.35543	0.452617
$141$	0	0
$142$	−0.138564	−0.0116281
$143$	0.0591036	0.00494249
$144$	0	0
$145$	−3.87360	−0.321685
$146$	−1.44574	−0.119651
$147$	0	0
$148$	−3.21953	−0.264644
$149$	8.39836	0.688021	0.344010	−	0.938966i	$-0.388214\pi$
$149$	8.39836	0.688021	0.344010	+	0.938966i	$0.388214\pi$
$150$	0	0
$151$	6.94240	0.564964	0.282482	−	0.959273i	$-0.408842\pi$
$151$	6.94240	0.564964	0.282482	+	0.959273i	$0.408842\pi$
$152$	−4.49170	−0.364325
$153$	0	0
$154$	−0.322722	−0.0260056
$155$	−4.12052	−0.330968
$156$	0	0
$157$	16.2003	1.29293	0.646463	−	0.762945i	$-0.276247\pi$
$157$	16.2003	1.29293	0.646463	+	0.762945i	$0.276247\pi$
$158$	−0.669717	−0.0532798
$159$	0	0
$160$	−1.15510	−0.0913190
$161$	30.9650	2.44038
$162$	0	0
$163$	−7.91672	−0.620086	−0.310043	−	0.950723i	$-0.600343\pi$
$163$	−7.91672	−0.620086	−0.310043	+	0.950723i	$0.600343\pi$
$164$	−20.2376	−1.58029
$165$	0	0
$166$	−1.77984	−0.138143
$167$	22.0572	1.70684	0.853420	−	0.521224i	$-0.174524\pi$
$167$	22.0572	1.70684	0.853420	+	0.521224i	$0.174524\pi$
$168$	0	0
$169$	−12.9905	−0.999269
$170$	0.642696	0.0492925
$171$	0	0
$172$	−9.29027	−0.708376
$173$	−8.99947	−0.684217	−0.342109	−	0.939660i	$-0.611141\pi$
$173$	−8.99947	−0.684217	−0.342109	+	0.939660i	$0.611141\pi$
$174$	0	0
$175$	−17.3087	−1.30841
$176$	−2.35506	−0.177520
$177$	0	0
$178$	0.479329	0.0359272
$179$	−12.0591	−0.901340	−0.450670	−	0.892691i	$-0.648815\pi$
$179$	−12.0591	−0.901340	−0.450670	+	0.892691i	$0.648815\pi$
$180$	0	0
$181$	−4.85121	−0.360588	−0.180294	−	0.983613i	$-0.557705\pi$
$181$	−4.85121	−0.360588	−0.180294	+	0.983613i	$0.557705\pi$
$182$	−0.0519106	−0.00384787
$183$	0	0
$184$	−4.44537	−0.327717
$185$	−1.14372	−0.0840883
$186$	0	0
$187$	3.99555	0.292184
$188$	−17.5504	−1.28000
$189$	0	0
$190$	−0.793982	−0.0576015
$191$	−3.27331	−0.236848	−0.118424	−	0.992963i	$-0.537784\pi$
$191$	−3.27331	−0.236848	−0.118424	+	0.992963i	$0.537784\pi$
$192$	0	0
$193$	−15.5942	−1.12249	−0.561246	−	0.827649i	$-0.689678\pi$
$193$	−15.5942	−1.12249	−0.561246	+	0.827649i	$0.689678\pi$
$194$	−0.265490	−0.0190611
$195$	0	0
$196$	−15.3765	−1.09832
$197$	3.66580	0.261177	0.130589	−	0.991437i	$-0.458313\pi$
$197$	3.66580	0.261177	0.130589	+	0.991437i	$0.458313\pi$
$198$	0	0
$199$	−5.09181	−0.360949	−0.180475	−	0.983580i	$-0.557763\pi$
$199$	−5.09181	−0.360949	−0.180475	+	0.983580i	$0.557763\pi$
$200$	2.48486	0.175706
$201$	0	0
$202$	1.60048	0.112609
$203$	21.1509	1.48450
$204$	0	0
$205$	−7.18932	−0.502124
$206$	−0.426543	−0.0297187
$207$	0	0
$208$	−0.378818	−0.0262663
$209$	−4.93608	−0.341436
$210$	0	0
$211$	2.53052	0.174208	0.0871042	−	0.996199i	$-0.472239\pi$
$211$	2.53052	0.174208	0.0871042	+	0.996199i	$0.472239\pi$
$212$	17.2112	1.18207
$213$	0	0
$214$	−0.622868	−0.0425784
$215$	−3.30033	−0.225081
$216$	0	0
$217$	22.4992	1.52734
$218$	−1.28726	−0.0871845
$219$	0	0
$220$	−0.844896	−0.0569629
$221$	0.642696	0.0432324
$222$	0	0
$223$	0.633189	0.0424015	0.0212007	−	0.999775i	$-0.493251\pi$
$223$	0.633189	0.0424015	0.0212007	+	0.999775i	$0.493251\pi$
$224$	6.30718	0.421416
$225$	0	0
$226$	0.725532	0.0482617
$227$	−15.0783	−1.00078	−0.500391	−	0.865799i	$-0.666810\pi$
$227$	−15.0783	−1.00078	−0.500391	+	0.865799i	$0.666810\pi$
$228$	0	0
$229$	−7.20789	−0.476311	−0.238155	−	0.971227i	$-0.576543\pi$
$229$	−7.20789	−0.476311	−0.238155	+	0.971227i	$0.576543\pi$
$230$	−0.785793	−0.0518136
$231$	0	0
$232$	−3.03645	−0.199353
$233$	−5.17900	−0.339287	−0.169644	−	0.985505i	$-0.554262\pi$
$233$	−5.17900	−0.339287	−0.169644	+	0.985505i	$0.554262\pi$
$234$	0	0
$235$	−6.23472	−0.406709
$236$	20.6404	1.34357
$237$	0	0
$238$	−3.50929	−0.227474
$239$	−11.8713	−0.767890	−0.383945	−	0.923356i	$-0.625435\pi$
$239$	−11.8713	−0.767890	−0.383945	+	0.923356i	$0.625435\pi$
$240$	0	0
$241$	4.11420	0.265019	0.132510	−	0.991182i	$-0.457696\pi$
$241$	4.11420	0.265019	0.132510	+	0.991182i	$0.457696\pi$
$242$	−1.47329	−0.0947069
$243$	0	0
$244$	6.26017	0.400766
$245$	−5.46244	−0.348983
$246$	0	0
$247$	−0.793982	−0.0505199
$248$	−3.23001	−0.205106
$249$	0	0
$250$	0.926758	0.0586133
$251$	31.0606	1.96053	0.980264	−	0.197693i	$-0.0633448\pi$
$251$	31.0606	1.96053	0.980264	+	0.197693i	$0.0633448\pi$
$252$	0	0
$253$	−4.88517	−0.307128
$254$	2.13475	0.133946
$255$	0	0
$256$	14.4860	0.905376
$257$	−24.5320	−1.53026	−0.765131	−	0.643875i	$-0.777326\pi$
$257$	−24.5320	−1.53026	−0.765131	+	0.643875i	$0.777326\pi$
$258$	0	0
$259$	6.24505	0.388048
$260$	−0.135904	−0.00842840
$261$	0	0
$262$	1.94721	0.120299
$263$	11.0336	0.680360	0.340180	−	0.940360i	$-0.389512\pi$
$263$	11.0336	0.680360	0.340180	+	0.940360i	$0.389512\pi$
$264$	0	0
$265$	6.11420	0.375593
$266$	4.33536	0.265818
$267$	0	0
$268$	−6.29293	−0.384402
$269$	6.18034	0.376822	0.188411	−	0.982090i	$-0.439666\pi$
$269$	6.18034	0.376822	0.188411	+	0.982090i	$0.439666\pi$
$270$	0	0
$271$	−13.0918	−0.795271	−0.397636	−	0.917543i	$-0.630169\pi$
$271$	−13.0918	−0.795271	−0.397636	+	0.917543i	$0.630169\pi$
$272$	−25.6091	−1.55278
$273$	0	0
$274$	−0.0735117	−0.00444101
$275$	2.73069	0.164667
$276$	0	0
$277$	0.101949	0.00612553	0.00306277	−	0.999995i	$-0.499025\pi$
$277$	0.101949	0.00612553	0.00306277	+	0.999995i	$0.499025\pi$
$278$	−2.05582	−0.123300
$279$	0	0
$280$	1.49134	0.0891244
$281$	5.36823	0.320242	0.160121	−	0.987097i	$-0.448812\pi$
$281$	5.36823	0.320242	0.160121	+	0.987097i	$0.448812\pi$
$282$	0	0
$283$	16.0020	0.951222	0.475611	−	0.879656i	$-0.342227\pi$
$283$	16.0020	0.951222	0.475611	+	0.879656i	$0.342227\pi$
$284$	1.98080	0.117539
$285$	0	0
$286$	0.00818965	0.000484264 0
$287$	39.2556	2.31719
$288$	0	0
$289$	26.4479	1.55576
$290$	−0.536743	−0.0315186
$291$	0	0
$292$	20.6671	1.20945
$293$	−18.5177	−1.08182	−0.540909	−	0.841081i	$-0.681920\pi$
$293$	−18.5177	−1.08182	−0.540909	+	0.841081i	$0.681920\pi$
$294$	0	0
$295$	7.33241	0.426910
$296$	−0.896547	−0.0521108
$297$	0	0
$298$	1.16371	0.0674121
$299$	−0.785793	−0.0454436
$300$	0	0
$301$	18.0207	1.03870
$302$	0.961969	0.0553551
$303$	0	0
$304$	31.6373	1.81452
$305$	2.22390	0.127340
$306$	0	0
$307$	−13.7321	−0.783732	−0.391866	−	0.920022i	$-0.628170\pi$
$307$	−13.7321	−0.783732	−0.391866	+	0.920022i	$0.628170\pi$
$308$	4.61336	0.262871
$309$	0	0
$310$	−0.570957	−0.0324282
$311$	−26.4107	−1.49761	−0.748807	−	0.662788i	$-0.769373\pi$
$311$	−26.4107	−1.49761	−0.748807	+	0.662788i	$0.769373\pi$
$312$	0	0
$313$	4.81469	0.272142	0.136071	−	0.990699i	$-0.456552\pi$
$313$	4.81469	0.272142	0.136071	+	0.990699i	$0.456552\pi$
$314$	2.24479	0.126681
$315$	0	0
$316$	9.57372	0.538564
$317$	−12.4005	−0.696481	−0.348241	−	0.937405i	$-0.613221\pi$
$317$	−12.4005	−0.696481	−0.348241	+	0.937405i	$0.613221\pi$
$318$	0	0
$319$	−3.33686	−0.186828
$320$	5.30771	0.296710
$321$	0	0
$322$	4.29064	0.239108
$323$	−53.6752	−2.98657
$324$	0	0
$325$	0.439239	0.0243646
$326$	−1.09697	−0.0607558
$327$	0	0
$328$	−5.63559	−0.311173
$329$	34.0433	1.87687
$330$	0	0
$331$	6.94443	0.381701	0.190850	−	0.981619i	$-0.438876\pi$
$331$	6.94443	0.381701	0.190850	+	0.981619i	$0.438876\pi$
$332$	25.4432	1.39638
$333$	0	0
$334$	3.05634	0.167236
$335$	−2.23554	−0.122141
$336$	0	0
$337$	33.4352	1.82133	0.910667	−	0.413142i	$-0.135569\pi$
$337$	33.4352	1.82133	0.910667	+	0.413142i	$0.135569\pi$
$338$	−1.80002	−0.0979081
$339$	0	0
$340$	−9.18745	−0.498259
$341$	−3.54956	−0.192220
$342$	0	0
$343$	2.93076	0.158246
$344$	−2.58708	−0.139486
$345$	0	0
$346$	−1.24701	−0.0670394
$347$	−6.38095	−0.342547	−0.171274	−	0.985223i	$-0.554788\pi$
$347$	−6.38095	−0.342547	−0.171274	+	0.985223i	$0.554788\pi$
$348$	0	0
$349$	−33.4352	−1.78975	−0.894874	−	0.446320i	$-0.852734\pi$
$349$	−33.4352	−1.78975	−0.894874	+	0.446320i	$0.852734\pi$
$350$	−2.39836	−0.128198
$351$	0	0
$352$	−0.995048	−0.0530362
$353$	12.0624	0.642016	0.321008	−	0.947076i	$-0.395978\pi$
$353$	12.0624	0.642016	0.321008	+	0.947076i	$0.395978\pi$
$354$	0	0
$355$	0.703671	0.0373470
$356$	−6.85209	−0.363160
$357$	0	0
$358$	−1.67096	−0.0883130
$359$	12.5768	0.663780	0.331890	−	0.943318i	$-0.392314\pi$
$359$	12.5768	0.663780	0.331890	+	0.943318i	$0.392314\pi$
$360$	0	0
$361$	47.3100	2.49000
$362$	−0.672205	−0.0353303
$363$	0	0
$364$	0.742072	0.0388951
$365$	7.34192	0.384294
$366$	0	0
$367$	10.3588	0.540725	0.270363	−	0.962759i	$-0.412856\pi$
$367$	10.3588	0.540725	0.270363	+	0.962759i	$0.412856\pi$
$368$	31.3110	1.63220
$369$	0	0
$370$	−0.158479	−0.00823895
$371$	−33.3852	−1.73327
$372$	0	0
$373$	−11.5184	−0.596398	−0.298199	−	0.954504i	$-0.596386\pi$
$373$	−11.5184	−0.596398	−0.298199	+	0.954504i	$0.596386\pi$
$374$	0.553641	0.0286281
$375$	0	0
$376$	−4.88730	−0.252043
$377$	−0.536743	−0.0276437
$378$	0	0
$379$	−4.27394	−0.219538	−0.109769	−	0.993957i	$-0.535011\pi$
$379$	−4.27394	−0.219538	−0.109769	+	0.993957i	$0.535011\pi$
$380$	11.3501	0.582248
$381$	0	0
$382$	−0.453564	−0.0232064
$383$	31.8253	1.62619	0.813097	−	0.582128i	$-0.197780\pi$
$383$	31.8253	1.62619	0.813097	+	0.582128i	$0.197780\pi$
$384$	0	0
$385$	1.63888	0.0835250
$386$	−2.16079	−0.109981
$387$	0	0
$388$	3.79523	0.192673
$389$	12.1983	0.618478	0.309239	−	0.950984i	$-0.399926\pi$
$389$	12.1983	0.618478	0.309239	+	0.950984i	$0.399926\pi$
$390$	0	0
$391$	−53.1216	−2.68647
$392$	−4.28192	−0.216270
$393$	0	0
$394$	0.507949	0.0255901
$395$	3.40103	0.171124
$396$	0	0
$397$	−27.2421	−1.36724	−0.683621	−	0.729837i	$-0.739596\pi$
$397$	−27.2421	−1.36724	−0.683621	+	0.729837i	$0.739596\pi$
$398$	−0.705543	−0.0353657
$399$	0	0
$400$	−17.5021	−0.875104
$401$	−12.8885	−0.643619	−0.321809	−	0.946804i	$-0.604291\pi$
$401$	−12.8885	−0.643619	−0.321809	+	0.946804i	$0.604291\pi$
$402$	0	0
$403$	−0.570957	−0.0284414
$404$	−22.8791	−1.13828
$405$	0	0
$406$	2.93076	0.145451
$407$	−0.985245	−0.0488368
$408$	0	0
$409$	31.1216	1.53886	0.769432	−	0.638729i	$-0.220540\pi$
$409$	31.1216	1.53886	0.769432	+	0.638729i	$0.220540\pi$
$410$	−0.996183	−0.0491979
$411$	0	0
$412$	6.09750	0.300402
$413$	−40.0370	−1.97009
$414$	0	0
$415$	9.03859	0.443687
$416$	−0.160056	−0.00784740
$417$	0	0
$418$	−0.683965	−0.0334538
$419$	−7.42805	−0.362884	−0.181442	−	0.983402i	$-0.558076\pi$
$419$	−7.42805	−0.362884	−0.181442	+	0.983402i	$0.558076\pi$
$420$	0	0
$421$	24.6152	1.19967	0.599837	−	0.800123i	$-0.295232\pi$
$421$	24.6152	1.19967	0.599837	+	0.800123i	$0.295232\pi$
$422$	0.350640	0.0170689
$423$	0	0
$424$	4.79283	0.232760
$425$	29.6937	1.44036
$426$	0	0
$427$	−12.1431	−0.587646
$428$	8.90400	0.430391
$429$	0	0
$430$	−0.457308	−0.0220534
$431$	−22.5882	−1.08804	−0.544018	−	0.839074i	$-0.683098\pi$
$431$	−22.5882	−1.08804	−0.544018	+	0.839074i	$0.683098\pi$
$432$	0	0
$433$	−7.73451	−0.371697	−0.185848	−	0.982578i	$-0.559503\pi$
$433$	−7.73451	−0.371697	−0.185848	+	0.982578i	$0.559503\pi$
$434$	3.11758	0.149649
$435$	0	0
$436$	18.4017	0.881279
$437$	65.6261	3.13932
$438$	0	0
$439$	23.7808	1.13500	0.567498	−	0.823375i	$-0.307911\pi$
$439$	23.7808	1.13500	0.567498	+	0.823375i	$0.307911\pi$
$440$	−0.235280	−0.0112165
$441$	0	0
$442$	0.0890547	0.00423590
$443$	−21.5658	−1.02462	−0.512312	−	0.858800i	$-0.671211\pi$
$443$	−21.5658	−1.02462	−0.512312	+	0.858800i	$0.671211\pi$
$444$	0	0
$445$	−2.43418	−0.115391
$446$	0.0877373	0.00415448
$447$	0	0
$448$	−28.9815	−1.36925
$449$	7.55692	0.356633	0.178316	−	0.983973i	$-0.442935\pi$
$449$	7.55692	0.356633	0.178316	+	0.983973i	$0.442935\pi$
$450$	0	0
$451$	−6.19314	−0.291623
$452$	−10.3716	−0.487839
$453$	0	0
$454$	−2.08931	−0.0980564
$455$	0.263618	0.0123586
$456$	0	0
$457$	0.976796	0.0456926	0.0228463	−	0.999739i	$-0.492727\pi$
$457$	0.976796	0.0456926	0.0228463	+	0.999739i	$0.492727\pi$
$458$	−0.998756	−0.0466688
$459$	0	0
$460$	11.2330	0.523743
$461$	0.686256	0.0319621	0.0159811	−	0.999872i	$-0.494913\pi$
$461$	0.686256	0.0319621	0.0159811	+	0.999872i	$0.494913\pi$
$462$	0	0
$463$	−13.3729	−0.621494	−0.310747	−	0.950493i	$-0.600579\pi$
$463$	−13.3729	−0.621494	−0.310747	+	0.950493i	$0.600579\pi$
$464$	21.3873	0.992878
$465$	0	0
$466$	−0.717624	−0.0332433
$467$	−33.7161	−1.56020	−0.780098	−	0.625658i	$-0.784831\pi$
$467$	−33.7161	−1.56020	−0.780098	+	0.625658i	$0.784831\pi$
$468$	0	0
$469$	12.2066	0.563651
$470$	−0.863910	−0.0398492
$471$	0	0
$472$	5.74776	0.264562
$473$	−2.84302	−0.130722
$474$	0	0
$475$	−36.6834	−1.68315
$476$	50.1659	2.29935
$477$	0	0
$478$	−1.64494	−0.0752376
$479$	20.7479	0.947998	0.473999	−	0.880525i	$-0.342810\pi$
$479$	20.7479	0.947998	0.473999	+	0.880525i	$0.342810\pi$
$480$	0	0
$481$	−0.158479	−0.00722604
$482$	0.570082	0.0259665
$483$	0	0
$484$	21.0610	0.957317
$485$	1.34824	0.0612204
$486$	0	0
$487$	−20.3952	−0.924194	−0.462097	−	0.886829i	$-0.652903\pi$
$487$	−20.3952	−0.924194	−0.462097	+	0.886829i	$0.652903\pi$
$488$	1.74328	0.0789146
$489$	0	0
$490$	−0.756899	−0.0341932
$491$	20.7930	0.938374	0.469187	−	0.883099i	$-0.344547\pi$
$491$	20.7930	0.938374	0.469187	+	0.883099i	$0.344547\pi$
$492$	0	0
$493$	−36.2852	−1.63420
$494$	−0.110018	−0.00494992
$495$	0	0
$496$	22.7506	1.02153
$497$	−3.84224	−0.172348
$498$	0	0
$499$	18.8765	0.845030	0.422515	−	0.906356i	$-0.361147\pi$
$499$	18.8765	0.845030	0.422515	+	0.906356i	$0.361147\pi$
$500$	−13.2482	−0.592476
$501$	0	0
$502$	4.30389	0.192092
$503$	−35.1914	−1.56911	−0.784555	−	0.620060i	$-0.787108\pi$
$503$	−35.1914	−1.56911	−0.784555	+	0.620060i	$0.787108\pi$
$504$	0	0
$505$	−8.12771	−0.361679
$506$	−0.676910	−0.0300923
$507$	0	0
$508$	−30.5166	−1.35395
$509$	−0.783920	−0.0347467	−0.0173733	−	0.999849i	$-0.505530\pi$
$509$	−0.783920	−0.0347467	−0.0173733	+	0.999849i	$0.505530\pi$
$510$	0	0
$511$	−40.0889	−1.77343
$512$	10.6637	0.471275
$513$	0	0
$514$	−3.39925	−0.149935
$515$	2.16611	0.0954504
$516$	0	0
$517$	−5.37081	−0.236208
$518$	0.865340	0.0380209
$519$	0	0
$520$	−0.0378454	−0.00165963
$521$	22.4044	0.981555	0.490777	−	0.871285i	$-0.336713\pi$
$521$	22.4044	0.981555	0.490777	+	0.871285i	$0.336713\pi$
$522$	0	0
$523$	−32.5256	−1.42225	−0.711123	−	0.703067i	$-0.751813\pi$
$523$	−32.5256	−1.42225	−0.711123	+	0.703067i	$0.751813\pi$
$524$	−27.8358	−1.21601
$525$	0	0
$526$	1.52886	0.0666615
$527$	−38.5982	−1.68136
$528$	0	0
$529$	41.9492	1.82388
$530$	0.847210	0.0368005
$531$	0	0
$532$	−61.9747	−2.68694
$533$	−0.996183	−0.0431495
$534$	0	0
$535$	3.16311	0.136753
$536$	−1.75240	−0.0756923
$537$	0	0
$538$	0.856374	0.0369209
$539$	−4.70554	−0.202682
$540$	0	0
$541$	−39.4840	−1.69755	−0.848774	−	0.528756i	$-0.822659\pi$
$541$	−39.4840	−1.69755	−0.848774	+	0.528756i	$0.822659\pi$
$542$	−1.81406	−0.0779204
$543$	0	0
$544$	−10.8202	−0.463912
$545$	6.53712	0.280019
$546$	0	0
$547$	20.8706	0.892362	0.446181	−	0.894943i	$-0.352784\pi$
$547$	20.8706	0.892362	0.446181	+	0.894943i	$0.352784\pi$
$548$	1.05086	0.0448906
$549$	0	0
$550$	0.378376	0.0161340
$551$	44.8265	1.90967
$552$	0	0
$553$	−18.5705	−0.789699
$554$	0.0141265	0.000600178 0
$555$	0	0
$556$	29.3882	1.24634
$557$	−40.8373	−1.73033	−0.865165	−	0.501487i	$-0.832786\pi$
$557$	−40.8373	−1.73033	−0.865165	+	0.501487i	$0.832786\pi$
$558$	0	0
$559$	−0.457308	−0.0193421
$560$	−10.5042	−0.443884
$561$	0	0
$562$	0.743845	0.0313772
$563$	−4.74207	−0.199854	−0.0999272	−	0.994995i	$-0.531861\pi$
$563$	−4.74207	−0.199854	−0.0999272	+	0.994995i	$0.531861\pi$
$564$	0	0
$565$	−3.68447	−0.155007
$566$	2.21731	0.0932005
$567$	0	0
$568$	0.551597	0.0231445
$569$	9.22585	0.386768	0.193384	−	0.981123i	$-0.438054\pi$
$569$	9.22585	0.386768	0.193384	+	0.981123i	$0.438054\pi$
$570$	0	0
$571$	−45.8008	−1.91670	−0.958352	−	0.285589i	$-0.907811\pi$
$571$	−45.8008	−1.91670	−0.958352	+	0.285589i	$0.907811\pi$
$572$	−0.117072	−0.00489504
$573$	0	0
$574$	5.43943	0.227037
$575$	−36.3050	−1.51402
$576$	0	0
$577$	30.9641	1.28905	0.644526	−	0.764582i	$-0.277055\pi$
$577$	30.9641	1.28905	0.644526	+	0.764582i	$0.277055\pi$
$578$	3.66473	0.152433
$579$	0	0
$580$	7.67283	0.318597
$581$	−49.3531	−2.04751
$582$	0	0
$583$	5.26699	0.218137
$584$	5.75521	0.238152
$585$	0	0
$586$	−2.56590	−0.105996
$587$	25.8578	1.06726	0.533632	−	0.845717i	$-0.320827\pi$
$587$	25.8578	1.06726	0.533632	+	0.845717i	$0.320827\pi$
$588$	0	0
$589$	47.6839	1.96478
$590$	1.01601	0.0418285
$591$	0	0
$592$	6.31483	0.259538
$593$	1.45800	0.0598728	0.0299364	−	0.999552i	$-0.490470\pi$
$593$	1.45800	0.0598728	0.0299364	+	0.999552i	$0.490470\pi$
$594$	0	0
$595$	17.8213	0.730600
$596$	−16.6355	−0.681416
$597$	0	0
$598$	−0.108883	−0.00445255
$599$	30.1720	1.23279	0.616397	−	0.787436i	$-0.288592\pi$
$599$	30.1720	1.23279	0.616397	+	0.787436i	$0.288592\pi$
$600$	0	0
$601$	17.4199	0.710571	0.355285	−	0.934758i	$-0.384384\pi$
$601$	17.4199	0.710571	0.355285	+	0.934758i	$0.384384\pi$
$602$	2.49703	0.101771
$603$	0	0
$604$	−13.7515	−0.559541
$605$	7.48183	0.304180
$606$	0	0
$607$	−4.80321	−0.194956	−0.0974781	−	0.995238i	$-0.531078\pi$
$607$	−4.80321	−0.194956	−0.0974781	+	0.995238i	$0.531078\pi$
$608$	13.3672	0.542112
$609$	0	0
$610$	0.308153	0.0124768
$611$	−0.863910	−0.0349501
$612$	0	0
$613$	−8.61968	−0.348146	−0.174073	−	0.984733i	$-0.555693\pi$
$613$	−8.61968	−0.348146	−0.174073	+	0.984733i	$0.555693\pi$
$614$	−1.90278	−0.0767899
$615$	0	0
$616$	1.28469	0.0517616
$617$	−23.1997	−0.933985	−0.466992	−	0.884261i	$-0.654662\pi$
$617$	−23.1997	−0.933985	−0.466992	+	0.884261i	$0.654662\pi$
$618$	0	0
$619$	−18.5063	−0.743829	−0.371915	−	0.928267i	$-0.621299\pi$
$619$	−18.5063	−0.743829	−0.371915	+	0.928267i	$0.621299\pi$
$620$	8.16193	0.327791
$621$	0	0
$622$	−3.65958	−0.146736
$623$	13.2913	0.532503
$624$	0	0
$625$	17.8179	0.712715
$626$	0.667143	0.0266644
$627$	0	0
$628$	−32.0896	−1.28051
$629$	−10.7136	−0.427180
$630$	0	0
$631$	2.34777	0.0934633	0.0467317	−	0.998907i	$-0.485119\pi$
$631$	2.34777	0.0934633	0.0467317	+	0.998907i	$0.485119\pi$
$632$	2.66601	0.106048
$633$	0	0
$634$	−1.71827	−0.0682410
$635$	−10.8409	−0.430208
$636$	0	0
$637$	−0.756899	−0.0299894
$638$	−0.462370	−0.0183054
$639$	0	0
$640$	3.04567	0.120391
$641$	−12.5927	−0.497380	−0.248690	−	0.968583i	$-0.580000\pi$
$641$	−12.5927	−0.497380	−0.248690	+	0.968583i	$0.580000\pi$
$642$	0	0
$643$	33.1113	1.30578	0.652890	−	0.757452i	$-0.273556\pi$
$643$	33.1113	1.30578	0.652890	+	0.757452i	$0.273556\pi$
$644$	−61.3354	−2.41695
$645$	0	0
$646$	−7.43747	−0.292623
$647$	−4.11952	−0.161955	−0.0809776	−	0.996716i	$-0.525804\pi$
$647$	−4.11952	−0.161955	−0.0809776	+	0.996716i	$0.525804\pi$
$648$	0	0
$649$	6.31640	0.247941
$650$	0.0608629	0.00238724
$651$	0	0
$652$	15.6814	0.614133
$653$	−22.2460	−0.870554	−0.435277	−	0.900297i	$-0.643350\pi$
$653$	−22.2460	−0.870554	−0.435277	+	0.900297i	$0.643350\pi$
$654$	0	0
$655$	−9.88854	−0.386377
$656$	39.6943	1.54980
$657$	0	0
$658$	4.71718	0.183895
$659$	−46.4616	−1.80989	−0.904944	−	0.425530i	$-0.860088\pi$
$659$	−46.4616	−1.80989	−0.904944	+	0.425530i	$0.860088\pi$
$660$	0	0
$661$	28.3349	1.10210	0.551050	−	0.834472i	$-0.314227\pi$
$661$	28.3349	1.10210	0.551050	+	0.834472i	$0.314227\pi$
$662$	0.962250	0.0373989
$663$	0	0
$664$	7.08520	0.274959
$665$	−22.0163	−0.853754
$666$	0	0
$667$	44.3641	1.71779
$668$	−43.6910	−1.69045
$669$	0	0
$670$	−0.309766	−0.0119673
$671$	1.91575	0.0739566
$672$	0	0
$673$	−47.1712	−1.81832	−0.909158	−	0.416452i	$-0.863273\pi$
$673$	−47.1712	−1.81832	−0.909158	+	0.416452i	$0.863273\pi$
$674$	4.63293	0.178454
$675$	0	0
$676$	25.7316	0.989676
$677$	18.9559	0.728535	0.364267	−	0.931294i	$-0.381319\pi$
$677$	18.9559	0.728535	0.364267	+	0.931294i	$0.381319\pi$
$678$	0	0
$679$	−7.36175	−0.282518
$680$	−2.55844	−0.0981118
$681$	0	0
$682$	−0.491843	−0.0188336
$683$	−33.9074	−1.29743	−0.648715	−	0.761032i	$-0.724693\pi$
$683$	−33.9074	−1.29743	−0.648715	+	0.761032i	$0.724693\pi$
$684$	0	0
$685$	0.373315	0.0142636
$686$	0.406099	0.0155049
$687$	0	0
$688$	18.2221	0.694710
$689$	0.847210	0.0322761
$690$	0	0
$691$	14.2238	0.541099	0.270549	−	0.962706i	$-0.412795\pi$
$691$	14.2238	0.541099	0.270549	+	0.962706i	$0.412795\pi$
$692$	17.8262	0.677649
$693$	0	0
$694$	−0.884172	−0.0335627
$695$	10.4401	0.396014
$696$	0	0
$697$	−67.3445	−2.55085
$698$	−4.63293	−0.175359
$699$	0	0
$700$	34.2850	1.29585
$701$	−9.41063	−0.355435	−0.177717	−	0.984082i	$-0.556871\pi$
$701$	−9.41063	−0.355435	−0.177717	+	0.984082i	$0.556871\pi$
$702$	0	0
$703$	13.2355	0.499187
$704$	4.57225	0.172323
$705$	0	0
$706$	1.67142	0.0629046
$707$	44.3795	1.66906
$708$	0	0
$709$	−36.3434	−1.36491	−0.682453	−	0.730929i	$-0.739087\pi$
$709$	−36.3434	−1.36491	−0.682453	+	0.730929i	$0.739087\pi$
$710$	0.0975037	0.00365925
$711$	0	0
$712$	−1.90811	−0.0715095
$713$	47.1921	1.76736
$714$	0	0
$715$	−0.0415895	−0.00155536
$716$	23.8867	0.892687
$717$	0	0
$718$	1.74270	0.0650370
$719$	42.2616	1.57609	0.788045	−	0.615617i	$-0.211093\pi$
$719$	42.2616	1.57609	0.788045	+	0.615617i	$0.211093\pi$
$720$	0	0
$721$	−11.8276	−0.440482
$722$	6.55548	0.243970
$723$	0	0
$724$	9.60928	0.357126
$725$	−24.7985	−0.920993
$726$	0	0
$727$	−47.1476	−1.74861	−0.874304	−	0.485378i	$-0.838682\pi$
$727$	−47.1476	−1.74861	−0.874304	+	0.485378i	$0.838682\pi$
$728$	0.206646	0.00765881
$729$	0	0
$730$	1.01733	0.0376530
$731$	−30.9152	−1.14344
$732$	0	0
$733$	−0.302018	−0.0111553	−0.00557765	−	0.999984i	$-0.501775\pi$
$733$	−0.302018	−0.0111553	−0.00557765	+	0.999984i	$0.501775\pi$
$734$	1.43536	0.0529801
$735$	0	0
$736$	13.2293	0.487640
$737$	−1.92577	−0.0709368
$738$	0	0
$739$	−36.6548	−1.34837	−0.674184	−	0.738564i	$-0.735504\pi$
$739$	−36.6548	−1.34837	−0.674184	+	0.738564i	$0.735504\pi$
$740$	2.26549	0.0832811
$741$	0	0
$742$	−4.62600	−0.169826
$743$	21.1793	0.776992	0.388496	−	0.921451i	$-0.372995\pi$
$743$	21.1793	0.776992	0.388496	+	0.921451i	$0.372995\pi$
$744$	0	0
$745$	−5.90969	−0.216514
$746$	−1.59603	−0.0584349
$747$	0	0
$748$	−7.91439	−0.289379
$749$	−17.2714	−0.631085
$750$	0	0
$751$	40.8030	1.48892	0.744462	−	0.667665i	$-0.232706\pi$
$751$	40.8030	1.48892	0.744462	+	0.667665i	$0.232706\pi$
$752$	34.4237	1.25530
$753$	0	0
$754$	−0.0743734	−0.00270852
$755$	−4.88517	−0.177789
$756$	0	0
$757$	20.2383	0.735573	0.367787	−	0.929910i	$-0.380116\pi$
$757$	20.2383	0.735573	0.367787	+	0.929910i	$0.380116\pi$
$758$	−0.592215	−0.0215102
$759$	0	0
$760$	3.16068	0.114650
$761$	−4.03467	−0.146257	−0.0731283	−	0.997323i	$-0.523298\pi$
$761$	−4.03467	−0.146257	−0.0731283	+	0.997323i	$0.523298\pi$
$762$	0	0
$763$	−35.6944	−1.29222
$764$	6.48377	0.234575
$765$	0	0
$766$	4.40984	0.159334
$767$	1.01601	0.0366860
$768$	0	0
$769$	−38.9542	−1.40472	−0.702362	−	0.711820i	$-0.747871\pi$
$769$	−38.9542	−1.40472	−0.702362	+	0.711820i	$0.747871\pi$
$770$	0.227090	0.00818375
$771$	0	0
$772$	30.8889	1.11172
$773$	13.2419	0.476280	0.238140	−	0.971231i	$-0.423462\pi$
$773$	13.2419	0.476280	0.238140	+	0.971231i	$0.423462\pi$
$774$	0	0
$775$	−26.3792	−0.947570
$776$	1.05686	0.0379392
$777$	0	0
$778$	1.69025	0.0605983
$779$	83.1969	2.98084
$780$	0	0
$781$	0.606168	0.0216904
$782$	−7.36075	−0.263220
$783$	0	0
$784$	30.1597	1.07713
$785$	−11.3997	−0.406873
$786$	0	0
$787$	−23.4899	−0.837324	−0.418662	−	0.908142i	$-0.637501\pi$
$787$	−23.4899	−0.837324	−0.418662	+	0.908142i	$0.637501\pi$
$788$	−7.26122	−0.258670
$789$	0	0
$790$	0.471261	0.0167667
$791$	20.1182	0.715321
$792$	0	0
$793$	0.308153	0.0109428
$794$	−3.77478	−0.133962
$795$	0	0
$796$	10.0859	0.357484
$797$	−7.75709	−0.274770	−0.137385	−	0.990518i	$-0.543870\pi$
$797$	−7.75709	−0.274770	−0.137385	+	0.990518i	$0.543870\pi$
$798$	0	0
$799$	−58.4025	−2.06613
$800$	−7.39488	−0.261448
$801$	0	0
$802$	−1.78588	−0.0630616
$803$	6.32459	0.223190
$804$	0	0
$805$	−21.7892	−0.767967
$806$	−0.0791142	−0.00278668
$807$	0	0
$808$	−6.37118	−0.224137
$809$	−36.9845	−1.30031	−0.650153	−	0.759803i	$-0.725295\pi$
$809$	−36.9845	−1.30031	−0.650153	+	0.759803i	$0.725295\pi$
$810$	0	0
$811$	−19.5004	−0.684752	−0.342376	−	0.939563i	$-0.611232\pi$
$811$	−19.5004	−0.684752	−0.342376	+	0.939563i	$0.611232\pi$
$812$	−41.8957	−1.47025
$813$	0	0
$814$	−0.136520	−0.00478502
$815$	5.57077	0.195136
$816$	0	0
$817$	38.1924	1.33618
$818$	4.31234	0.150777
$819$	0	0
$820$	14.2406	0.497303
$821$	−9.36450	−0.326823	−0.163412	−	0.986558i	$-0.552250\pi$
$821$	−9.36450	−0.326823	−0.163412	+	0.986558i	$0.552250\pi$
$822$	0	0
$823$	−1.09341	−0.0381140	−0.0190570	−	0.999818i	$-0.506066\pi$
$823$	−1.09341	−0.0381140	−0.0190570	+	0.999818i	$0.506066\pi$
$824$	1.69798	0.0591520
$825$	0	0
$826$	−5.54769	−0.193029
$827$	−25.0969	−0.872704	−0.436352	−	0.899776i	$-0.643730\pi$
$827$	−25.0969	−0.872704	−0.436352	+	0.899776i	$0.643730\pi$
$828$	0	0
$829$	−24.5551	−0.852835	−0.426418	−	0.904526i	$-0.640225\pi$
$829$	−24.5551	−0.852835	−0.426418	+	0.904526i	$0.640225\pi$
$830$	1.25243	0.0434723
$831$	0	0
$832$	0.735459	0.0254974
$833$	−51.1683	−1.77288
$834$	0	0
$835$	−15.5210	−0.537128
$836$	9.77739	0.338158
$837$	0	0
$838$	−1.02926	−0.0355553
$839$	−1.22672	−0.0423511	−0.0211756	−	0.999776i	$-0.506741\pi$
$839$	−1.22672	−0.0423511	−0.0211756	+	0.999776i	$0.506741\pi$
$840$	0	0
$841$	1.30334	0.0449426
$842$	3.41079	0.117544
$843$	0	0
$844$	−5.01246	−0.172536
$845$	9.14104	0.314461
$846$	0	0
$847$	−40.8528	−1.40372
$848$	−33.7582	−1.15926
$849$	0	0
$850$	4.11449	0.141126
$851$	13.0990	0.449028
$852$	0	0
$853$	34.4672	1.18014	0.590068	−	0.807354i	$-0.299101\pi$
$853$	34.4672	1.18014	0.590068	+	0.807354i	$0.299101\pi$
$854$	−1.68260	−0.0575774
$855$	0	0
$856$	2.47951	0.0847480
$857$	−29.5815	−1.01048	−0.505242	−	0.862978i	$-0.668597\pi$
$857$	−29.5815	−1.01048	−0.505242	+	0.862978i	$0.668597\pi$
$858$	0	0
$859$	42.0109	1.43339	0.716697	−	0.697385i	$-0.245653\pi$
$859$	42.0109	1.43339	0.716697	+	0.697385i	$0.245653\pi$
$860$	6.53730	0.222920
$861$	0	0
$862$	−3.12992	−0.106605
$863$	15.9430	0.542707	0.271353	−	0.962480i	$-0.412529\pi$
$863$	15.9430	0.542707	0.271353	+	0.962480i	$0.412529\pi$
$864$	0	0
$865$	6.33267	0.215317
$866$	−1.07173	−0.0364188
$867$	0	0
$868$	−44.5663	−1.51268
$869$	2.92976	0.0993855
$870$	0	0
$871$	−0.309766	−0.0104960
$872$	5.12434	0.173532
$873$	0	0
$874$	9.09343	0.307590
$875$	25.6980	0.868750
$876$	0	0
$877$	−9.71381	−0.328012	−0.164006	−	0.986459i	$-0.552442\pi$
$877$	−9.71381	−0.328012	−0.164006	+	0.986459i	$0.552442\pi$
$878$	3.29517	0.111207
$879$	0	0
$880$	1.65719	0.0558639
$881$	23.1465	0.779824	0.389912	−	0.920852i	$-0.372505\pi$
$881$	23.1465	0.779824	0.389912	+	0.920852i	$0.372505\pi$
$882$	0	0
$883$	19.0469	0.640980	0.320490	−	0.947252i	$-0.396152\pi$
$883$	19.0469	0.640980	0.320490	+	0.947252i	$0.396152\pi$
$884$	−1.27305	−0.0428174
$885$	0	0
$886$	−2.98825	−0.100392
$887$	−6.94553	−0.233208	−0.116604	−	0.993178i	$-0.537201\pi$
$887$	−6.94553	−0.233208	−0.116604	+	0.993178i	$0.537201\pi$
$888$	0	0
$889$	59.1942	1.98531
$890$	−0.337290	−0.0113060
$891$	0	0
$892$	−1.25422	−0.0419944
$893$	72.1501	2.41441
$894$	0	0
$895$	8.48565	0.283644
$896$	−16.6302	−0.555575
$897$	0	0
$898$	1.04712	0.0349428
$899$	32.2350	1.07510
$900$	0	0
$901$	57.2736	1.90806
$902$	−0.858147	−0.0285732
$903$	0	0
$904$	−2.88820	−0.0960600
$905$	3.41366	0.113474
$906$	0	0
$907$	−17.2072	−0.571355	−0.285677	−	0.958326i	$-0.592219\pi$
$907$	−17.2072	−0.571355	−0.285677	+	0.958326i	$0.592219\pi$
$908$	29.8671	0.991175
$909$	0	0
$910$	0.0365280	0.00121089
$911$	48.5488	1.60849	0.804247	−	0.594295i	$-0.202569\pi$
$911$	48.5488	1.60849	0.804247	+	0.594295i	$0.202569\pi$
$912$	0	0
$913$	7.78616	0.257684
$914$	0.135349	0.00447695
$915$	0	0
$916$	14.2774	0.471738
$917$	53.9941	1.78304
$918$	0	0
$919$	17.3036	0.570794	0.285397	−	0.958409i	$-0.407875\pi$
$919$	17.3036	0.570794	0.285397	+	0.958409i	$0.407875\pi$
$920$	3.12808	0.103130
$921$	0	0
$922$	0.0950906	0.00313164
$923$	0.0975037	0.00320937
$924$	0	0
$925$	−7.32203	−0.240747
$926$	−1.85301	−0.0608938
$927$	0	0
$928$	9.03642	0.296635
$929$	−1.88322	−0.0617865	−0.0308933	−	0.999523i	$-0.509835\pi$
$929$	−1.88322	−0.0617865	−0.0308933	+	0.999523i	$0.509835\pi$
$930$	0	0
$931$	63.2130	2.07172
$932$	10.2586	0.336030
$933$	0	0
$934$	−4.67185	−0.152868
$935$	−2.81156	−0.0919478
$936$	0	0
$937$	6.80392	0.222274	0.111137	−	0.993805i	$-0.464551\pi$
$937$	6.80392	0.222274	0.111137	+	0.993805i	$0.464551\pi$
$938$	1.69140	0.0552263
$939$	0	0
$940$	12.3497	0.402804
$941$	−35.7516	−1.16547	−0.582735	−	0.812662i	$-0.698017\pi$
$941$	−35.7516	−1.16547	−0.582735	+	0.812662i	$0.698017\pi$
$942$	0	0
$943$	82.3388	2.68132
$944$	−40.4843	−1.31765
$945$	0	0
$946$	−0.393942	−0.0128081
$947$	14.6341	0.475543	0.237772	−	0.971321i	$-0.423583\pi$
$947$	14.6341	0.475543	0.237772	+	0.971321i	$0.423583\pi$
$948$	0	0
$949$	1.01733	0.0330239
$950$	−5.08301	−0.164915
$951$	0	0
$952$	13.9698	0.452763
$953$	6.82162	0.220974	0.110487	−	0.993878i	$-0.464759\pi$
$953$	6.82162	0.220974	0.110487	+	0.993878i	$0.464759\pi$
$954$	0	0
$955$	2.30334	0.0745342
$956$	23.5146	0.760518
$957$	0	0
$958$	2.87492	0.0928846
$959$	−2.03840	−0.0658234
$960$	0	0
$961$	3.28976	0.106121
$962$	−0.0219596	−0.000708005 0
$963$	0	0
$964$	−8.14941	−0.262475
$965$	10.9732	0.353239
$966$	0	0
$967$	−23.9284	−0.769484	−0.384742	−	0.923024i	$-0.625710\pi$
$967$	−23.9284	−0.769484	−0.384742	+	0.923024i	$0.625710\pi$
$968$	5.86488	0.188505
$969$	0	0
$970$	0.186818	0.00599836
$971$	−30.8833	−0.991092	−0.495546	−	0.868582i	$-0.665032\pi$
$971$	−30.8833	−0.991092	−0.495546	+	0.868582i	$0.665032\pi$
$972$	0	0
$973$	−57.0055	−1.82751
$974$	−2.82604	−0.0905522
$975$	0	0
$976$	−12.2788	−0.393034
$977$	14.4255	0.461514	0.230757	−	0.973011i	$-0.425880\pi$
$977$	14.4255	0.461514	0.230757	+	0.973011i	$0.425880\pi$
$978$	0	0
$979$	−2.09689	−0.0670168
$980$	10.8200	0.345632
$981$	0	0
$982$	2.88116	0.0919417
$983$	24.5403	0.782714	0.391357	−	0.920239i	$-0.372006\pi$
$983$	24.5403	0.782714	0.391357	+	0.920239i	$0.372006\pi$
$984$	0	0
$985$	−2.57952	−0.0821903
$986$	−5.02783	−0.160119
$987$	0	0
$988$	1.57272	0.0500349
$989$	37.7985	1.20192
$990$	0	0
$991$	22.4503	0.713157	0.356578	−	0.934265i	$-0.383943\pi$
$991$	22.4503	0.713157	0.356578	+	0.934265i	$0.383943\pi$
$992$	9.61243	0.305195
$993$	0	0
$994$	−0.532397	−0.0168866
$995$	3.58296	0.113588
$996$	0	0
$997$	40.1081	1.27024	0.635119	−	0.772414i	$-0.280951\pi$
$997$	40.1081	1.27024	0.635119	+	0.772414i	$0.280951\pi$
$998$	2.61561	0.0827958
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.i.1.3		4
3.2	odd	2		213.2.a.e.1.2	✓	4
12.11	even	2		3408.2.a.w.1.3		4
15.14	odd	2		5325.2.a.w.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
213.2.a.e.1.2	✓	4	3.2	odd	2
639.2.a.i.1.3		4	1.1	even	1	trivial
3408.2.a.w.1.3		4	12.11	even	2
5325.2.a.w.1.3		4	15.14	odd	2

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:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	639.a (trivial)

\(f(q)\)	\(=\)	\(q+0.138564 q^{2} -1.98080 q^{4} -0.703671 q^{5} +3.84224 q^{7} -0.551597 q^{8} +O(q^{10})\)	\(q+0.138564 q^{2} -1.98080 q^{4} -0.703671 q^{5} +3.84224 q^{7} -0.551597 q^{8} -0.0975037 q^{10} -0.606168 q^{11} -0.0975037 q^{13} +0.532397 q^{14} +3.88517 q^{16} -6.59150 q^{17} +8.14310 q^{19} +1.39383 q^{20} -0.0839932 q^{22} +8.05910 q^{23} -4.50485 q^{25} -0.0135105 q^{26} -7.61070 q^{28} +5.50485 q^{29} +5.85575 q^{31} +1.64154 q^{32} -0.913346 q^{34} -2.70367 q^{35} +1.62537 q^{37} +1.12834 q^{38} +0.388143 q^{40} +10.2169 q^{41} +4.69016 q^{43} +1.20070 q^{44} +1.11670 q^{46} +8.86028 q^{47} +7.76278 q^{49} -0.624211 q^{50} +0.193135 q^{52} -8.68900 q^{53} +0.426543 q^{55} -2.11936 q^{56} +0.762775 q^{58} -10.4202 q^{59} -3.16042 q^{61} +0.811397 q^{62} -7.54288 q^{64} +0.0686106 q^{65} +3.17696 q^{67} +13.0564 q^{68} -0.374632 q^{70} -1.00000 q^{71} -10.4337 q^{73} +0.225218 q^{74} -16.1298 q^{76} -2.32904 q^{77} -4.83326 q^{79} -2.73388 q^{80} +1.41569 q^{82} -12.8449 q^{83} +4.63825 q^{85} +0.649889 q^{86} +0.334360 q^{88} +3.45925 q^{89} -0.374632 q^{91} -15.9635 q^{92} +1.22772 q^{94} -5.73006 q^{95} -1.91601 q^{97} +1.07564 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8	\( 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8} + 5 q^{10} - 2 q^{11} + 5 q^{13} - q^{14} + 11 q^{16} + 8 q^{17} + 8 q^{19} + 6 q^{20} - 7 q^{22} + q^{23} - q^{25} + 12 q^{26} - 9 q^{28} + 5 q^{29} + 2 q^{31} - 17 q^{32} - 21 q^{34} - 5 q^{35} + 19 q^{37} - 3 q^{38} - 23 q^{40} + 19 q^{41} + 25 q^{43} + 19 q^{44} + 12 q^{46} - 7 q^{47} - 6 q^{49} + 31 q^{50} - 13 q^{52} + 5 q^{53} + 3 q^{55} + 8 q^{56} - 34 q^{58} - 10 q^{59} + 2 q^{61} - 4 q^{62} + 34 q^{64} + 16 q^{65} + 35 q^{67} + 45 q^{68} + 11 q^{70} - 4 q^{71} + 2 q^{73} - 20 q^{74} + 13 q^{76} - 16 q^{77} - q^{79} + 5 q^{80} - 5 q^{82} - 18 q^{83} + 11 q^{85} - 20 q^{86} - 40 q^{88} + 16 q^{89} + 11 q^{91} - 41 q^{92} - 5 q^{94} + 15 q^{95} - q^{97} + 10 q^{98}+O(q^{100}) \) 4 * q - 3 * q^2 + 5 * q^4 + 3 * q^5 + 6 * q^7 - 12 * q^8 + 5 * q^10 - 2 * q^11 + 5 * q^13 - q^14 + 11 * q^16 + 8 * q^17 + 8 * q^19 + 6 * q^20 - 7 * q^22 + q^23 - q^25 + 12 * q^26 - 9 * q^28 + 5 * q^29 + 2 * q^31 - 17 * q^32 - 21 * q^34 - 5 * q^35 + 19 * q^37 - 3 * q^38 - 23 * q^40 + 19 * q^41 + 25 * q^43 + 19 * q^44 + 12 * q^46 - 7 * q^47 - 6 * q^49 + 31 * q^50 - 13 * q^52 + 5 * q^53 + 3 * q^55 + 8 * q^56 - 34 * q^58 - 10 * q^59 + 2 * q^61 - 4 * q^62 + 34 * q^64 + 16 * q^65 + 35 * q^67 + 45 * q^68 + 11 * q^70 - 4 * q^71 + 2 * q^73 - 20 * q^74 + 13 * q^76 - 16 * q^77 - q^79 + 5 * q^80 - 5 * q^82 - 18 * q^83 + 11 * q^85 - 20 * q^86 - 40 * q^88 + 16 * q^89 + 11 * q^91 - 41 * q^92 - 5 * q^94 + 15 * q^95 - q^97 + 10 * q^98

Introduction

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Database

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