LMFDB - Embedded newform 639.2.a.i.1.4

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			1.4
Root			$2.43828$ 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)

$f(q)$	$=$	$q+1.43828 q^{2} +0.0686587 q^{4} +3.94523 q^{5} +0.493058 q^{7} -2.77782 q^{8} +O(q^{10})$	\(q+1.43828 q^{2} +0.0686587 q^{4} +3.94523 q^{5} +0.493058 q^{7} -2.77782 q^{8} +5.67435 q^{10} -1.72913 q^{11} +5.67435 q^{13} +0.709157 q^{14} -4.13260 q^{16} +3.10251 q^{17} +0.675304 q^{19} +0.270874 q^{20} -2.48697 q^{22} -1.81167 q^{23} +10.5648 q^{25} +8.16132 q^{26} +0.0338527 q^{28} -9.56480 q^{29} -5.66827 q^{31} -0.388222 q^{32} +4.46229 q^{34} +1.94523 q^{35} +4.79778 q^{37} +0.971279 q^{38} -10.9591 q^{40} +3.69527 q^{41} +8.21610 q^{43} -0.118720 q^{44} -2.60569 q^{46} -11.4312 q^{47} -6.75689 q^{49} +15.1952 q^{50} +0.389593 q^{52} +6.77686 q^{53} -6.82179 q^{55} -1.36962 q^{56} -13.7569 q^{58} -5.55187 q^{59} -9.48226 q^{61} -8.15257 q^{62} +7.70683 q^{64} +22.3866 q^{65} +8.57560 q^{67} +0.213014 q^{68} +2.79778 q^{70} -1.00000 q^{71} +2.60945 q^{73} +6.90057 q^{74} +0.0463655 q^{76} -0.852560 q^{77} -0.891404 q^{79} -16.3041 q^{80} +5.31485 q^{82} -12.4684 q^{83} +12.2401 q^{85} +11.8171 q^{86} +4.80319 q^{88} -9.91446 q^{89} +2.79778 q^{91} -0.124387 q^{92} -16.4414 q^{94} +2.66423 q^{95} +0.486973 q^{97} -9.71833 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.43828	1.01702	0.508510	−	0.861056i	$-0.330197\pi$
$2$	1.43828	1.01702	0.508510	+	0.861056i	$0.330197\pi$
$3$	0	0
$3$	0	0
$4$	0.0686587	0.0343293
$5$	3.94523	1.76436	0.882179	−	0.470914i	$-0.156076\pi$
$5$	3.94523	1.76436	0.882179	+	0.470914i	$0.156076\pi$
$6$	0	0
$7$	0.493058	0.186358	0.0931792	−	0.995649i	$-0.470297\pi$
$7$	0.493058	0.186358	0.0931792	+	0.995649i	$0.470297\pi$
$8$	−2.77782	−0.982106
$9$	0	0
$10$	5.67435	1.79439
$11$	−1.72913	−0.521351	−0.260676	−	0.965426i	$-0.583945\pi$
$11$	−1.72913	−0.521351	−0.260676	+	0.965426i	$0.583945\pi$
$12$	0	0
$13$	5.67435	1.57378	0.786891	−	0.617092i	$-0.211689\pi$
$13$	5.67435	1.57378	0.786891	+	0.617092i	$0.211689\pi$
$14$	0.709157	0.189530
$15$	0	0
$16$	−4.13260	−1.03315
$17$	3.10251	0.752470	0.376235	−	0.926524i	$-0.377219\pi$
$17$	3.10251	0.752470	0.376235	+	0.926524i	$0.377219\pi$
$18$	0	0
$19$	0.675304	0.154925	0.0774627	−	0.996995i	$-0.475318\pi$
$19$	0.675304	0.154925	0.0774627	+	0.996995i	$0.475318\pi$
$20$	0.270874	0.0605693
$21$	0	0
$22$	−2.48697	−0.530224
$23$	−1.81167	−0.377759	−0.188880	−	0.982000i	$-0.560486\pi$
$23$	−1.81167	−0.377759	−0.188880	+	0.982000i	$0.560486\pi$
$24$	0	0
$25$	10.5648	2.11296
$26$	8.16132	1.60057
$27$	0	0
$28$	0.0338527	0.00639756
$29$	−9.56480	−1.77614	−0.888070	−	0.459709i	$-0.847954\pi$
$29$	−9.56480	−1.77614	−0.888070	+	0.459709i	$0.847954\pi$
$30$	0	0
$31$	−5.66827	−1.01805	−0.509025	−	0.860752i	$-0.669994\pi$
$31$	−5.66827	−1.01805	−0.509025	+	0.860752i	$0.669994\pi$
$32$	−0.388222	−0.0686287
$33$	0	0
$34$	4.46229	0.765276
$35$	1.94523	0.328803
$36$	0	0
$37$	4.79778	0.788751	0.394375	−	0.918949i	$-0.370961\pi$
$37$	4.79778	0.788751	0.394375	+	0.918949i	$0.370961\pi$
$38$	0.971279	0.157562
$39$	0	0
$40$	−10.9591	−1.73279
$41$	3.69527	0.577105	0.288552	−	0.957464i	$-0.406826\pi$
$41$	3.69527	0.577105	0.288552	+	0.957464i	$0.406826\pi$
$42$	0	0
$43$	8.21610	1.25294	0.626472	−	0.779444i	$-0.284498\pi$
$43$	8.21610	1.25294	0.626472	+	0.779444i	$0.284498\pi$
$44$	−0.118720	−0.0178976
$45$	0	0
$46$	−2.60569	−0.384188
$47$	−11.4312	−1.66742	−0.833709	−	0.552204i	$-0.813787\pi$
$47$	−11.4312	−1.66742	−0.833709	+	0.552204i	$0.813787\pi$
$48$	0	0
$49$	−6.75689	−0.965271
$50$	15.1952	2.14892
$51$	0	0
$52$	0.389593	0.0540269
$53$	6.77686	0.930874	0.465437	−	0.885081i	$-0.345897\pi$
$53$	6.77686	0.930874	0.465437	+	0.885081i	$0.345897\pi$
$54$	0	0
$55$	−6.82179	−0.919850
$56$	−1.36962	−0.183024
$57$	0	0
$58$	−13.7569	−1.80637
$59$	−5.55187	−0.722792	−0.361396	−	0.932412i	$-0.617700\pi$
$59$	−5.55187	−0.722792	−0.361396	+	0.932412i	$0.617700\pi$
$60$	0	0
$61$	−9.48226	−1.21408	−0.607039	−	0.794672i	$-0.707643\pi$
$61$	−9.48226	−1.21408	−0.607039	+	0.794672i	$0.707643\pi$
$62$	−8.15257	−1.03538
$63$	0	0
$64$	7.70683	0.963354
$65$	22.3866	2.77672
$66$	0	0
$67$	8.57560	1.04768	0.523838	−	0.851818i	$-0.324500\pi$
$67$	8.57560	1.04768	0.523838	+	0.851818i	$0.324500\pi$
$68$	0.213014	0.0258318
$69$	0	0
$70$	2.79778	0.334399
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	0	0
$73$	2.60945	0.305413	0.152707	−	0.988272i	$-0.451201\pi$
$73$	2.60945	0.305413	0.152707	+	0.988272i	$0.451201\pi$
$74$	6.90057	0.802175
$75$	0	0
$76$	0.0463655	0.00531849
$77$	−0.852560	−0.0971582
$78$	0	0
$79$	−0.891404	−0.100291	−0.0501454	−	0.998742i	$-0.515968\pi$
$79$	−0.891404	−0.100291	−0.0501454	+	0.998742i	$0.515968\pi$
$80$	−16.3041	−1.82285
$81$	0	0
$82$	5.31485	0.586927
$83$	−12.4684	−1.36858	−0.684291	−	0.729209i	$-0.739888\pi$
$83$	−12.4684	−1.36858	−0.684291	+	0.729209i	$0.739888\pi$
$84$	0	0
$85$	12.2401	1.32763
$86$	11.8171	1.27427
$87$	0	0
$88$	4.80319	0.512022
$89$	−9.91446	−1.05093	−0.525465	−	0.850815i	$-0.676109\pi$
$89$	−9.91446	−1.05093	−0.525465	+	0.850815i	$0.676109\pi$
$90$	0	0
$91$	2.79778	0.293288
$92$	−0.124387	−0.0129682
$93$	0	0
$94$	−16.4414	−1.69580
$95$	2.66423	0.273344
$96$	0	0
$97$	0.486973	0.0494446	0.0247223	−	0.999694i	$-0.492130\pi$
$97$	0.486973	0.0494446	0.0247223	+	0.999694i	$0.492130\pi$
$98$	−9.71833	−0.981699
$99$	0	0
$100$	0.725365	0.0725365
$101$	−5.21515	−0.518926	−0.259463	−	0.965753i	$-0.583546\pi$
$101$	−5.21515	−0.518926	−0.259463	+	0.965753i	$0.583546\pi$
$102$	0	0
$103$	4.74301	0.467343	0.233671	−	0.972316i	$-0.424926\pi$
$103$	4.74301	0.467343	0.233671	+	0.972316i	$0.424926\pi$
$104$	−15.7623	−1.54562
$105$	0	0
$106$	9.74705	0.946717
$107$	−19.5648	−1.89140	−0.945700	−	0.325040i	$-0.894622\pi$
$107$	−19.5648	−1.89140	−0.945700	+	0.325040i	$0.894622\pi$
$108$	0	0
$109$	−16.3189	−1.56307	−0.781533	−	0.623864i	$-0.785562\pi$
$109$	−16.3189	−1.56307	−0.781533	+	0.623864i	$0.785562\pi$
$110$	−9.81167	−0.935506
$111$	0	0
$112$	−2.03761	−0.192536
$113$	0.763932	0.0718647	0.0359323	−	0.999354i	$-0.488560\pi$
$113$	0.763932	0.0718647	0.0359323	+	0.999354i	$0.488560\pi$
$114$	0	0
$115$	−7.14744	−0.666502
$116$	−0.656707	−0.0609737
$117$	0	0
$118$	−7.98516	−0.735094
$119$	1.52972	0.140229
$120$	0	0
$121$	−8.01012	−0.728193
$122$	−13.6382	−1.23474
$123$	0	0
$124$	−0.389176	−0.0349490
$125$	21.9544	1.96366
$126$	0	0
$127$	−12.8834	−1.14322	−0.571609	−	0.820526i	$-0.693681\pi$
$127$	−12.8834	−1.14322	−0.571609	+	0.820526i	$0.693681\pi$
$128$	11.8611	1.04838
$129$	0	0
$130$	32.1983	2.82397
$131$	6.56199	0.573324	0.286662	−	0.958032i	$-0.407454\pi$
$131$	6.56199	0.573324	0.286662	+	0.958032i	$0.407454\pi$
$132$	0	0
$133$	0.332964	0.0288717
$134$	12.3341	1.06551
$135$	0	0
$136$	−8.61821	−0.739005
$137$	−12.4475	−1.06346	−0.531729	−	0.846915i	$-0.678457\pi$
$137$	−12.4475	−1.06346	−0.531729	+	0.846915i	$0.678457\pi$
$138$	0	0
$139$	−1.26308	−0.107133	−0.0535663	−	0.998564i	$-0.517059\pi$
$139$	−1.26308	−0.107133	−0.0535663	+	0.998564i	$0.517059\pi$
$140$	0.133557	0.0112876
$141$	0	0
$142$	−1.43828	−0.120698
$143$	−9.81167	−0.820493
$144$	0	0
$145$	−37.7353	−3.13375
$146$	3.75313	0.310611
$147$	0	0
$148$	0.329410	0.0270773
$149$	−1.49210	−0.122238	−0.0611190	−	0.998130i	$-0.519467\pi$
$149$	−1.49210	−0.122238	−0.0611190	+	0.998130i	$0.519467\pi$
$150$	0	0
$151$	0.794024	0.0646168	0.0323084	−	0.999478i	$-0.489714\pi$
$151$	0.794024	0.0646168	0.0323084	+	0.999478i	$0.489714\pi$
$152$	−1.87587	−0.152153
$153$	0	0
$154$	−1.22622	−0.0988118
$155$	−22.3626	−1.79621
$156$	0	0
$157$	10.6019	0.846126	0.423063	−	0.906100i	$-0.360955\pi$
$157$	10.6019	0.846126	0.423063	+	0.906100i	$0.360955\pi$
$158$	−1.28209	−0.101998
$159$	0	0
$160$	−1.53162	−0.121086
$161$	−0.893258	−0.0703986
$162$	0	0
$163$	25.2182	1.97524	0.987622	−	0.156852i	$-0.0501345\pi$
$163$	25.2182	1.97524	0.987622	+	0.156852i	$0.0501345\pi$
$164$	0.253713	0.0198116
$165$	0	0
$166$	−17.9331	−1.39188
$167$	23.9266	1.85150	0.925749	−	0.378139i	$-0.123436\pi$
$167$	23.9266	1.85150	0.925749	+	0.378139i	$0.123436\pi$
$168$	0	0
$169$	19.1983	1.47679
$170$	17.6047	1.35022
$171$	0	0
$172$	0.564106	0.0430127
$173$	22.5966	1.71799	0.858994	−	0.511985i	$-0.171090\pi$
$173$	22.5966	1.71799	0.858994	+	0.511985i	$0.171090\pi$
$174$	0	0
$175$	5.20906	0.393768
$176$	7.14579	0.538634
$177$	0	0
$178$	−14.2598	−1.06882
$179$	−2.18833	−0.163564	−0.0817818	−	0.996650i	$-0.526061\pi$
$179$	−2.18833	−0.163564	−0.0817818	+	0.996650i	$0.526061\pi$
$180$	0	0
$181$	−2.09471	−0.155699	−0.0778494	−	0.996965i	$-0.524805\pi$
$181$	−2.09471	−0.155699	−0.0778494	+	0.996965i	$0.524805\pi$
$182$	4.02401	0.298279
$183$	0	0
$184$	5.03248	0.370999
$185$	18.9283	1.39164
$186$	0	0
$187$	−5.36463	−0.392301
$188$	−0.784854	−0.0572414
$189$	0	0
$190$	3.83191	0.277996
$191$	16.0917	1.16436	0.582178	−	0.813062i	$-0.302201\pi$
$191$	16.0917	1.16436	0.582178	+	0.813062i	$0.302201\pi$
$192$	0	0
$193$	−8.87281	−0.638679	−0.319339	−	0.947640i	$-0.603461\pi$
$193$	−8.87281	−0.638679	−0.319339	+	0.947640i	$0.603461\pi$
$194$	0.700405	0.0502862
$195$	0	0
$196$	−0.463919	−0.0331371
$197$	26.5141	1.88905	0.944524	−	0.328441i	$-0.106523\pi$
$197$	26.5141	1.88905	0.944524	+	0.328441i	$0.106523\pi$
$198$	0	0
$199$	10.9043	0.772988	0.386494	−	0.922292i	$-0.373686\pi$
$199$	10.9043	0.772988	0.386494	+	0.922292i	$0.373686\pi$
$200$	−29.3471	−2.07515
$201$	0	0
$202$	−7.50086	−0.527758
$203$	−4.71600	−0.330999
$204$	0	0
$205$	14.5787	1.01822
$206$	6.82179	0.475297
$207$	0	0
$208$	−23.4498	−1.62595
$209$	−1.16769	−0.0807706
$210$	0	0
$211$	14.4475	0.994604	0.497302	−	0.867578i	$-0.334324\pi$
$211$	14.4475	0.994604	0.497302	+	0.867578i	$0.334324\pi$
$212$	0.465290	0.0319563
$213$	0	0
$214$	−28.1397	−1.92359
$215$	32.4144	2.21064
$216$	0	0
$217$	−2.79478	−0.189722
$218$	−23.4712	−1.58967
$219$	0	0
$220$	−0.468375	−0.0315778
$221$	17.6047	1.18422
$222$	0	0
$223$	−14.5935	−0.977255	−0.488627	−	0.872493i	$-0.662502\pi$
$223$	−14.5935	−0.977255	−0.488627	+	0.872493i	$0.662502\pi$
$224$	−0.191416	−0.0127895
$225$	0	0
$226$	1.09875	0.0730878
$227$	−7.25699	−0.481663	−0.240832	−	0.970567i	$-0.577420\pi$
$227$	−7.25699	−0.481663	−0.240832	+	0.970567i	$0.577420\pi$
$228$	0	0
$229$	−0.0936188	−0.00618651	−0.00309325	−	0.999995i	$-0.500985\pi$
$229$	−0.0936188	−0.00618651	−0.00309325	+	0.999995i	$0.500985\pi$
$230$	−10.2800	−0.677846
$231$	0	0
$232$	26.5693	1.74436
$233$	−26.1546	−1.71344	−0.856721	−	0.515780i	$-0.827502\pi$
$233$	−26.1546	−1.71344	−0.856721	+	0.515780i	$0.827502\pi$
$234$	0	0
$235$	−45.0988	−2.94192
$236$	−0.381184	−0.0248130
$237$	0	0
$238$	2.20017	0.142616
$239$	−7.74937	−0.501265	−0.250633	−	0.968082i	$-0.580639\pi$
$239$	−7.74937	−0.501265	−0.250633	+	0.968082i	$0.580639\pi$
$240$	0	0
$241$	24.7362	1.59340	0.796701	−	0.604374i	$-0.206577\pi$
$241$	24.7362	1.59340	0.796701	+	0.604374i	$0.206577\pi$
$242$	−11.5208	−0.740587
$243$	0	0
$244$	−0.651039	−0.0416785
$245$	−26.6575	−1.70308
$246$	0	0
$247$	3.83191	0.243819
$248$	15.7454	0.999834
$249$	0	0
$250$	31.5766	1.99708
$251$	5.17069	0.326371	0.163185	−	0.986595i	$-0.447823\pi$
$251$	5.17069	0.326371	0.163185	+	0.986595i	$0.447823\pi$
$252$	0	0
$253$	3.13260	0.196945
$254$	−18.5300	−1.16268
$255$	0	0
$256$	1.64589	0.102868
$257$	25.0151	1.56040	0.780198	−	0.625532i	$-0.215118\pi$
$257$	25.0151	1.56040	0.780198	+	0.625532i	$0.215118\pi$
$258$	0	0
$259$	2.36559	0.146990
$260$	1.53703	0.0953228
$261$	0	0
$262$	9.43801	0.583082
$263$	1.49334	0.0920830	0.0460415	−	0.998940i	$-0.485339\pi$
$263$	1.49334	0.0920830	0.0460415	+	0.998940i	$0.485339\pi$
$264$	0	0
$265$	26.7362	1.64239
$266$	0.478897	0.0293631
$267$	0	0
$268$	0.588789	0.0359660
$269$	−16.1803	−0.986533	−0.493266	−	0.869878i	$-0.664197\pi$
$269$	−16.1803	−0.986533	−0.493266	+	0.869878i	$0.664197\pi$
$270$	0	0
$271$	2.90433	0.176426	0.0882129	−	0.996102i	$-0.471884\pi$
$271$	2.90433	0.176426	0.0882129	+	0.996102i	$0.471884\pi$
$272$	−12.8214	−0.777415
$273$	0	0
$274$	−17.9030	−1.08156
$275$	−18.2679	−1.10159
$276$	0	0
$277$	3.69028	0.221728	0.110864	−	0.993836i	$-0.464638\pi$
$277$	3.69028	0.221728	0.110864	+	0.993836i	$0.464638\pi$
$278$	−1.81666	−0.108956
$279$	0	0
$280$	−5.40348	−0.322920
$281$	22.7035	1.35438	0.677188	−	0.735810i	$-0.263198\pi$
$281$	22.7035	1.35438	0.677188	+	0.735810i	$0.263198\pi$
$282$	0	0
$283$	31.5790	1.87717	0.938587	−	0.345042i	$-0.112135\pi$
$283$	31.5790	1.87717	0.938587	+	0.345042i	$0.112135\pi$
$284$	−0.0686587	−0.00407414
$285$	0	0
$286$	−14.1120	−0.834458
$287$	1.82198	0.107548
$288$	0	0
$289$	−7.37442	−0.433790
$290$	−54.2740	−3.18708
$291$	0	0
$292$	0.179162	0.0104846
$293$	−7.87752	−0.460210	−0.230105	−	0.973166i	$-0.573907\pi$
$293$	−7.87752	−0.460210	−0.230105	+	0.973166i	$0.573907\pi$
$294$	0	0
$295$	−21.9034	−1.27526
$296$	−13.3274	−0.774637
$297$	0	0
$298$	−2.14607	−0.124318
$299$	−10.2800	−0.594510
$300$	0	0
$301$	4.05101	0.233497
$302$	1.14203	0.0657165
$303$	0	0
$304$	−2.79077	−0.160061
$305$	−37.4096	−2.14207
$306$	0	0
$307$	−20.9147	−1.19367	−0.596833	−	0.802365i	$-0.703575\pi$
$307$	−20.9147	−1.19367	−0.596833	+	0.802365i	$0.703575\pi$
$308$	−0.0585356	−0.00333538
$309$	0	0
$310$	−32.1637	−1.82678
$311$	10.6464	0.603701	0.301851	−	0.953355i	$-0.402396\pi$
$311$	10.6464	0.603701	0.301851	+	0.953355i	$0.402396\pi$
$312$	0	0
$313$	−13.7809	−0.778943	−0.389471	−	0.921039i	$-0.627342\pi$
$313$	−13.7809	−0.778943	−0.389471	+	0.921039i	$0.627342\pi$
$314$	15.2486	0.860527
$315$	0	0
$316$	−0.0612026	−0.00344291
$317$	26.1134	1.46667	0.733337	−	0.679865i	$-0.237962\pi$
$317$	26.1134	1.46667	0.733337	+	0.679865i	$0.237962\pi$
$318$	0	0
$319$	16.5387	0.925992
$320$	30.4052	1.69970
$321$	0	0
$322$	−1.28476	−0.0715968
$323$	2.09514	0.116577
$324$	0	0
$325$	59.9484	3.32534
$326$	36.2710	2.00886
$327$	0	0
$328$	−10.2648	−0.566778
$329$	−5.63627	−0.310737
$330$	0	0
$331$	16.3730	0.899941	0.449970	−	0.893043i	$-0.351435\pi$
$331$	16.3730	0.899941	0.449970	+	0.893043i	$0.351435\pi$
$332$	−0.856062	−0.0469825
$333$	0	0
$334$	34.4133	1.88301
$335$	33.8327	1.84848
$336$	0	0
$337$	4.37290	0.238207	0.119104	−	0.992882i	$-0.461998\pi$
$337$	4.37290	0.238207	0.119104	+	0.992882i	$0.461998\pi$
$338$	27.6125	1.50192
$339$	0	0
$340$	0.840389	0.0455765
$341$	9.80115	0.530762
$342$	0	0
$343$	−6.78295	−0.366245
$344$	−22.8228	−1.23052
$345$	0	0
$346$	32.5003	1.74723
$347$	−0.828553	−0.0444790	−0.0222395	−	0.999753i	$-0.507080\pi$
$347$	−0.828553	−0.0444790	−0.0222395	+	0.999753i	$0.507080\pi$
$348$	0	0
$349$	−4.37290	−0.234076	−0.117038	−	0.993127i	$-0.537340\pi$
$349$	−4.37290	−0.234076	−0.117038	+	0.993127i	$0.537340\pi$
$350$	7.49210	0.400470
$351$	0	0
$352$	0.671285	0.0357796
$353$	−7.44000	−0.395991	−0.197996	−	0.980203i	$-0.563443\pi$
$353$	−7.44000	−0.395991	−0.197996	+	0.980203i	$0.563443\pi$
$354$	0	0
$355$	−3.94523	−0.209391
$356$	−0.680714	−0.0360777
$357$	0	0
$358$	−3.14744	−0.166347
$359$	−7.93415	−0.418748	−0.209374	−	0.977836i	$-0.567143\pi$
$359$	−7.93415	−0.418748	−0.209374	+	0.977836i	$0.567143\pi$
$360$	0	0
$361$	−18.5440	−0.975998
$362$	−3.01279	−0.158349
$363$	0	0
$364$	0.192092	0.0100684
$365$	10.2949	0.538859
$366$	0	0
$367$	−22.6224	−1.18088	−0.590439	−	0.807082i	$-0.701046\pi$
$367$	−22.6224	−1.18088	−0.590439	+	0.807082i	$0.701046\pi$
$368$	7.48691	0.390282
$369$	0	0
$370$	27.2243	1.41532
$371$	3.34139	0.173476
$372$	0	0
$373$	11.7261	0.607156	0.303578	−	0.952807i	$-0.401819\pi$
$373$	11.7261	0.607156	0.303578	+	0.952807i	$0.401819\pi$
$374$	−7.71586	−0.398978
$375$	0	0
$376$	31.7539	1.63758
$377$	−54.2740	−2.79526
$378$	0	0
$379$	27.6954	1.42261	0.711307	−	0.702881i	$-0.248103\pi$
$379$	27.6954	1.42261	0.711307	+	0.702881i	$0.248103\pi$
$380$	0.182922	0.00938372
$381$	0	0
$382$	23.1444	1.18417
$383$	−20.3245	−1.03853	−0.519267	−	0.854612i	$-0.673795\pi$
$383$	−20.3245	−1.03853	−0.519267	+	0.854612i	$0.673795\pi$
$384$	0	0
$385$	−3.36354	−0.171422
$386$	−12.7616	−0.649549
$387$	0	0
$388$	0.0334349	0.00169740
$389$	−8.97703	−0.455154	−0.227577	−	0.973760i	$-0.573080\pi$
$389$	−8.97703	−0.455154	−0.227577	+	0.973760i	$0.573080\pi$
$390$	0	0
$391$	−5.62072	−0.284252
$392$	18.7694	0.947998
$393$	0	0
$394$	38.1347	1.92120
$395$	−3.51679	−0.176949
$396$	0	0
$397$	2.01669	0.101215	0.0506074	−	0.998719i	$-0.483884\pi$
$397$	2.01669	0.101215	0.0506074	+	0.998719i	$0.483884\pi$
$398$	15.6835	0.786144
$399$	0	0
$400$	−43.6601	−2.18301
$401$	4.76093	0.237750	0.118875	−	0.992909i	$-0.462071\pi$
$401$	4.76093	0.237750	0.118875	+	0.992909i	$0.462071\pi$
$402$	0	0
$403$	−32.1637	−1.60219
$404$	−0.358065	−0.0178144
$405$	0	0
$406$	−6.78295	−0.336632
$407$	−8.29597	−0.411216
$408$	0	0
$409$	−16.3793	−0.809903	−0.404952	−	0.914338i	$-0.632712\pi$
$409$	−16.3793	−0.809903	−0.404952	+	0.914338i	$0.632712\pi$
$410$	20.9683	1.03555
$411$	0	0
$412$	0.325649	0.0160436
$413$	−2.73739	−0.134698
$414$	0	0
$415$	−49.1905	−2.41467
$416$	−2.20291	−0.108007
$417$	0	0
$418$	−1.67946	−0.0821453
$419$	15.8394	0.773807	0.386904	−	0.922120i	$-0.373545\pi$
$419$	15.8394	0.773807	0.386904	+	0.922120i	$0.373545\pi$
$420$	0	0
$421$	8.20317	0.399798	0.199899	−	0.979817i	$-0.435939\pi$
$421$	8.20317	0.399798	0.199899	+	0.979817i	$0.435939\pi$
$422$	20.7795	1.01153
$423$	0	0
$424$	−18.8249	−0.914217
$425$	32.7774	1.58994
$426$	0	0
$427$	−4.67530	−0.226254
$428$	−1.34329	−0.0649305
$429$	0	0
$430$	46.6210	2.24827
$431$	−22.5258	−1.08503	−0.542515	−	0.840046i	$-0.682528\pi$
$431$	−22.5258	−1.08503	−0.542515	+	0.840046i	$0.682528\pi$
$432$	0	0
$433$	−8.70041	−0.418115	−0.209057	−	0.977903i	$-0.567040\pi$
$433$	−8.70041	−0.418115	−0.209057	+	0.977903i	$0.567040\pi$
$434$	−4.01969	−0.192951
$435$	0	0
$436$	−1.12043	−0.0536590
$437$	−1.22343	−0.0585245
$438$	0	0
$439$	−7.68120	−0.366604	−0.183302	−	0.983057i	$-0.558679\pi$
$439$	−7.68120	−0.366604	−0.183302	+	0.983057i	$0.558679\pi$
$440$	18.9497	0.903391
$441$	0	0
$442$	25.3206	1.20438
$443$	15.1148	0.718124	0.359062	−	0.933314i	$-0.383097\pi$
$443$	15.1148	0.718124	0.359062	+	0.933314i	$0.383097\pi$
$444$	0	0
$445$	−39.1148	−1.85422
$446$	−20.9896	−0.993887
$447$	0	0
$448$	3.79992	0.179529
$449$	15.7285	0.742271	0.371136	−	0.928579i	$-0.378968\pi$
$449$	15.7285	0.742271	0.371136	+	0.928579i	$0.378968\pi$
$450$	0	0
$451$	−6.38959	−0.300874
$452$	0.0524506	0.00246707
$453$	0	0
$454$	−10.4376	−0.489861
$455$	11.0379	0.517464
$456$	0	0
$457$	39.2909	1.83795	0.918976	−	0.394313i	$-0.129018\pi$
$457$	39.2909	1.83795	0.918976	+	0.394313i	$0.129018\pi$
$458$	−0.134650	−0.00629180
$459$	0	0
$460$	−0.490734	−0.0228806
$461$	0.375432	0.0174856	0.00874281	−	0.999962i	$-0.497217\pi$
$461$	0.375432	0.0174856	0.00874281	+	0.999962i	$0.497217\pi$
$462$	0	0
$463$	40.3874	1.87696	0.938480	−	0.345334i	$-0.112234\pi$
$463$	40.3874	1.87696	0.938480	+	0.345334i	$0.112234\pi$
$464$	39.5275	1.83502
$465$	0	0
$466$	−37.6177	−1.74260
$467$	16.2271	0.750902	0.375451	−	0.926842i	$-0.377488\pi$
$467$	16.2271	0.750902	0.375451	+	0.926842i	$0.377488\pi$
$468$	0	0
$469$	4.22827	0.195243
$470$	−64.8649	−2.99199
$471$	0	0
$472$	15.4221	0.709859
$473$	−14.2067	−0.653223
$474$	0	0
$475$	7.13446	0.327351
$476$	0.105028	0.00481397
$477$	0	0
$478$	−11.1458	−0.509797
$479$	−31.9058	−1.45781	−0.728906	−	0.684613i	$-0.759971\pi$
$479$	−31.9058	−1.45781	−0.728906	+	0.684613i	$0.759971\pi$
$480$	0	0
$481$	27.2243	1.24132
$482$	35.5777	1.62052
$483$	0	0
$484$	−0.549964	−0.0249984
$485$	1.92122	0.0872380
$486$	0	0
$487$	24.0640	1.09045	0.545223	−	0.838291i	$-0.316445\pi$
$487$	24.0640	1.09045	0.545223	+	0.838291i	$0.316445\pi$
$488$	26.3400	1.19235
$489$	0	0
$490$	−38.3410	−1.73207
$491$	24.4924	1.10533	0.552663	−	0.833405i	$-0.313612\pi$
$491$	24.4924	1.10533	0.552663	+	0.833405i	$0.313612\pi$
$492$	0	0
$493$	−29.6749	−1.33649
$494$	5.51138	0.247969
$495$	0	0
$496$	23.4247	1.05180
$497$	−0.493058	−0.0221167
$498$	0	0
$499$	−24.7449	−1.10773	−0.553866	−	0.832606i	$-0.686848\pi$
$499$	−24.7449	−1.10773	−0.553866	+	0.832606i	$0.686848\pi$
$500$	1.50736	0.0674112
$501$	0	0
$502$	7.43691	0.331926
$503$	15.2000	0.677733	0.338867	−	0.940834i	$-0.389956\pi$
$503$	15.2000	0.677733	0.338867	+	0.940834i	$0.389956\pi$
$504$	0	0
$505$	−20.5749	−0.915572
$506$	4.50557	0.200297
$507$	0	0
$508$	−0.884558	−0.0392459
$509$	−22.0183	−0.975946	−0.487973	−	0.872859i	$-0.662264\pi$
$509$	−22.0183	−0.975946	−0.487973	+	0.872859i	$0.662264\pi$
$510$	0	0
$511$	1.28661	0.0569164
$512$	−21.3549	−0.943760
$513$	0	0
$514$	35.9787	1.58695
$515$	18.7122	0.824560
$516$	0	0
$517$	19.7661	0.869310
$518$	3.40238	0.149492
$519$	0	0
$520$	−62.1858	−2.72703
$521$	−12.2727	−0.537678	−0.268839	−	0.963185i	$-0.586640\pi$
$521$	−12.2727	−0.537678	−0.268839	+	0.963185i	$0.586640\pi$
$522$	0	0
$523$	14.6414	0.640224	0.320112	−	0.947380i	$-0.396280\pi$
$523$	14.6414	0.640224	0.320112	+	0.947380i	$0.396280\pi$
$524$	0.450538	0.0196818
$525$	0	0
$526$	2.14784	0.0936502
$527$	−17.5859	−0.766052
$528$	0	0
$529$	−19.7179	−0.857298
$530$	38.4543	1.67035
$531$	0	0
$532$	0.0228609	0.000991145 0
$533$	20.9683	0.908237
$534$	0	0
$535$	−77.1875	−3.33711
$536$	−23.8214	−1.02893
$537$	0	0
$538$	−23.2719	−1.00332
$539$	11.6835	0.503245
$540$	0	0
$541$	28.2230	1.21340	0.606701	−	0.794930i	$-0.292492\pi$
$541$	28.2230	1.21340	0.606701	+	0.794930i	$0.292492\pi$
$542$	4.17726	0.179428
$543$	0	0
$544$	−1.20446	−0.0516410
$545$	−64.3817	−2.75781
$546$	0	0
$547$	−42.1645	−1.80282	−0.901412	−	0.432963i	$-0.857468\pi$
$547$	−42.1645	−1.80282	−0.901412	+	0.432963i	$0.857468\pi$
$548$	−0.854626	−0.0365078
$549$	0	0
$550$	−26.2744	−1.12034
$551$	−6.45915	−0.275169
$552$	0	0
$553$	−0.439514	−0.0186900
$554$	5.30767	0.225501
$555$	0	0
$556$	−0.0867211	−0.00367779
$557$	3.46818	0.146952	0.0734758	−	0.997297i	$-0.476591\pi$
$557$	3.46818	0.146952	0.0734758	+	0.997297i	$0.476591\pi$
$558$	0	0
$559$	46.6210	1.97186
$560$	−8.03884	−0.339703
$561$	0	0
$562$	32.6540	1.37743
$563$	−4.19209	−0.176676	−0.0883378	−	0.996091i	$-0.528155\pi$
$563$	−4.19209	−0.176676	−0.0883378	+	0.996091i	$0.528155\pi$
$564$	0	0
$565$	3.01388	0.126795
$566$	45.4195	1.90912
$567$	0	0
$568$	2.77782	0.116555
$569$	3.29693	0.138214	0.0691072	−	0.997609i	$-0.477985\pi$
$569$	3.29693	0.138214	0.0691072	+	0.997609i	$0.477985\pi$
$570$	0	0
$571$	−31.1011	−1.30154	−0.650770	−	0.759275i	$-0.725554\pi$
$571$	−31.1011	−1.30154	−0.650770	+	0.759275i	$0.725554\pi$
$572$	−0.673656	−0.0281670
$573$	0	0
$574$	2.62053	0.109379
$575$	−19.1399	−0.798190
$576$	0	0
$577$	2.52074	0.104940	0.0524699	−	0.998623i	$-0.483291\pi$
$577$	2.52074	0.104940	0.0524699	+	0.998623i	$0.483291\pi$
$578$	−10.6065	−0.441173
$579$	0	0
$580$	−2.59086	−0.107579
$581$	−6.14763	−0.255047
$582$	0	0
$583$	−11.7180	−0.485312
$584$	−7.24858	−0.299948
$585$	0	0
$586$	−11.3301	−0.468042
$587$	29.9107	1.23455	0.617273	−	0.786749i	$-0.288237\pi$
$587$	29.9107	1.23455	0.617273	+	0.786749i	$0.288237\pi$
$588$	0	0
$589$	−3.82781	−0.157722
$590$	−31.5033	−1.29697
$591$	0	0
$592$	−19.8273	−0.814899
$593$	13.2928	0.545871	0.272936	−	0.962032i	$-0.412005\pi$
$593$	13.2928	0.545871	0.272936	+	0.962032i	$0.412005\pi$
$594$	0	0
$595$	6.03508	0.247414
$596$	−0.102446	−0.00419635
$597$	0	0
$598$	−14.7856	−0.604629
$599$	−5.38564	−0.220051	−0.110026	−	0.993929i	$-0.535093\pi$
$599$	−5.38564	−0.220051	−0.110026	+	0.993929i	$0.535093\pi$
$600$	0	0
$601$	8.27252	0.337443	0.168722	−	0.985664i	$-0.446036\pi$
$601$	8.27252	0.337443	0.168722	+	0.985664i	$0.446036\pi$
$602$	5.82651	0.237471
$603$	0	0
$604$	0.0545166	0.00221825
$605$	−31.6017	−1.28479
$606$	0	0
$607$	−9.95939	−0.404239	−0.202120	−	0.979361i	$-0.564783\pi$
$607$	−9.95939	−0.404239	−0.202120	+	0.979361i	$0.564783\pi$
$608$	−0.262168	−0.0106323
$609$	0	0
$610$	−53.8057	−2.17853
$611$	−64.8649	−2.62415
$612$	0	0
$613$	−1.56780	−0.0633229	−0.0316615	−	0.999499i	$-0.510080\pi$
$613$	−1.56780	−0.0633229	−0.0316615	+	0.999499i	$0.510080\pi$
$614$	−30.0813	−1.21398
$615$	0	0
$616$	2.36825	0.0954197
$617$	−30.2056	−1.21603	−0.608015	−	0.793925i	$-0.708034\pi$
$617$	−30.2056	−1.21603	−0.608015	+	0.793925i	$0.708034\pi$
$618$	0	0
$619$	−31.6178	−1.27083	−0.635413	−	0.772172i	$-0.719170\pi$
$619$	−31.6178	−1.27083	−0.635413	+	0.772172i	$0.719170\pi$
$620$	−1.53539	−0.0616626
$621$	0	0
$622$	15.3125	0.613976
$623$	−4.88840	−0.195850
$624$	0	0
$625$	33.7910	1.35164
$626$	−19.8208	−0.792200
$627$	0	0
$628$	0.727915	0.0290470
$629$	14.8852	0.593511
$630$	0	0
$631$	42.8421	1.70552	0.852759	−	0.522305i	$-0.174928\pi$
$631$	42.8421	1.70552	0.852759	+	0.522305i	$0.174928\pi$
$632$	2.47615	0.0984962
$633$	0	0
$634$	37.5585	1.49164
$635$	−50.8280	−2.01705
$636$	0	0
$637$	−38.3410	−1.51913
$638$	23.7874	0.941752
$639$	0	0
$640$	46.7945	1.84972
$641$	−21.8905	−0.864621	−0.432310	−	0.901725i	$-0.642302\pi$
$641$	−21.8905	−0.864621	−0.432310	+	0.901725i	$0.642302\pi$
$642$	0	0
$643$	28.3540	1.11817	0.559086	−	0.829110i	$-0.311152\pi$
$643$	28.3540	1.11817	0.559086	+	0.829110i	$0.311152\pi$
$644$	−0.0613299	−0.00241674
$645$	0	0
$646$	3.01340	0.118561
$647$	−30.6869	−1.20643	−0.603213	−	0.797580i	$-0.706113\pi$
$647$	−30.6869	−1.20643	−0.603213	+	0.797580i	$0.706113\pi$
$648$	0	0
$649$	9.59988	0.376828
$650$	86.2228	3.38194
$651$	0	0
$652$	1.73145	0.0678088
$653$	29.2487	1.14459	0.572294	−	0.820048i	$-0.306053\pi$
$653$	29.2487	1.14459	0.572294	+	0.820048i	$0.306053\pi$
$654$	0	0
$655$	25.8885	1.01155
$656$	−15.2711	−0.596236
$657$	0	0
$658$	−8.10655	−0.316026
$659$	−13.6539	−0.531880	−0.265940	−	0.963990i	$-0.585682\pi$
$659$	−13.6539	−0.531880	−0.265940	+	0.963990i	$0.585682\pi$
$660$	0	0
$661$	−25.2453	−0.981929	−0.490965	−	0.871180i	$-0.663356\pi$
$661$	−25.2453	−0.981929	−0.490965	+	0.871180i	$0.663356\pi$
$662$	23.5490	0.915258
$663$	0	0
$664$	34.6349	1.34409
$665$	1.31362	0.0509400
$666$	0	0
$667$	17.3283	0.670953
$668$	1.64277	0.0635607
$669$	0	0
$670$	48.6610	1.87994
$671$	16.3960	0.632961
$672$	0	0
$673$	42.3892	1.63398	0.816992	−	0.576649i	$-0.195640\pi$
$673$	42.3892	1.63398	0.816992	+	0.576649i	$0.195640\pi$
$674$	6.28947	0.242261
$675$	0	0
$676$	1.31813	0.0506972
$677$	4.63270	0.178049	0.0890246	−	0.996029i	$-0.471625\pi$
$677$	4.63270	0.178049	0.0890246	+	0.996029i	$0.471625\pi$
$678$	0	0
$679$	0.240106	0.00921442
$680$	−34.0008	−1.30387
$681$	0	0
$682$	14.0968	0.539795
$683$	4.09923	0.156853	0.0784264	−	0.996920i	$-0.475010\pi$
$683$	4.09923	0.156853	0.0784264	+	0.996920i	$0.475010\pi$
$684$	0	0
$685$	−49.1080	−1.87632
$686$	−9.75580	−0.372478
$687$	0	0
$688$	−33.9539	−1.29448
$689$	38.4543	1.46499
$690$	0	0
$691$	18.7906	0.714828	0.357414	−	0.933946i	$-0.383658\pi$
$691$	18.7906	0.714828	0.357414	+	0.933946i	$0.383658\pi$
$692$	1.55145	0.0589774
$693$	0	0
$694$	−1.19169	−0.0452361
$695$	−4.98312	−0.189020
$696$	0	0
$697$	11.4646	0.434254
$698$	−6.28947	−0.238060
$699$	0	0
$700$	0.357647	0.0135178
$701$	9.51878	0.359519	0.179760	−	0.983711i	$-0.442468\pi$
$701$	9.51878	0.359519	0.179760	+	0.983711i	$0.442468\pi$
$702$	0	0
$703$	3.23997	0.122198
$704$	−13.3261	−0.502246
$705$	0	0
$706$	−10.7008	−0.402731
$707$	−2.57137	−0.0967063
$708$	0	0
$709$	−23.2772	−0.874195	−0.437097	−	0.899414i	$-0.643994\pi$
$709$	−23.2772	−0.874195	−0.437097	+	0.899414i	$0.643994\pi$
$710$	−5.67435	−0.212955
$711$	0	0
$712$	27.5405	1.03213
$713$	10.2690	0.384578
$714$	0	0
$715$	−38.7092	−1.44764
$716$	−0.150248	−0.00561503
$717$	0	0
$718$	−11.4116	−0.425875
$719$	−1.83103	−0.0682858	−0.0341429	−	0.999417i	$-0.510870\pi$
$719$	−1.83103	−0.0682858	−0.0341429	+	0.999417i	$0.510870\pi$
$720$	0	0
$721$	2.33858	0.0870932
$722$	−26.6715	−0.992609
$723$	0	0
$724$	−0.143820	−0.00534504
$725$	−101.050	−3.75291
$726$	0	0
$727$	−30.9123	−1.14647	−0.573237	−	0.819389i	$-0.694313\pi$
$727$	−30.9123	−1.14647	−0.573237	+	0.819389i	$0.694313\pi$
$728$	−7.77173	−0.288040
$729$	0	0
$730$	14.8070	0.548030
$731$	25.4905	0.942802
$732$	0	0
$733$	−15.1752	−0.560509	−0.280255	−	0.959926i	$-0.590419\pi$
$733$	−15.1752	−0.560509	−0.280255	+	0.959926i	$0.590419\pi$
$734$	−32.5374	−1.20098
$735$	0	0
$736$	0.703330	0.0259251
$737$	−14.8283	−0.546207
$738$	0	0
$739$	−4.04349	−0.148742	−0.0743711	−	0.997231i	$-0.523695\pi$
$739$	−4.04349	−0.148742	−0.0743711	+	0.997231i	$0.523695\pi$
$740$	1.29959	0.0477741
$741$	0	0
$742$	4.80586	0.176429
$743$	25.2716	0.927124	0.463562	−	0.886064i	$-0.346571\pi$
$743$	25.2716	0.927124	0.463562	+	0.886064i	$0.346571\pi$
$744$	0	0
$745$	−5.88669	−0.215672
$746$	16.8655	0.617489
$747$	0	0
$748$	−0.368329	−0.0134674
$749$	−9.64658	−0.352478
$750$	0	0
$751$	−20.6478	−0.753450	−0.376725	−	0.926325i	$-0.622950\pi$
$751$	−20.6478	−0.753450	−0.376725	+	0.926325i	$0.622950\pi$
$752$	47.2408	1.72269
$753$	0	0
$754$	−78.0614	−2.84283
$755$	3.13260	0.114007
$756$	0	0
$757$	−30.9850	−1.12617	−0.563084	−	0.826400i	$-0.690385\pi$
$757$	−30.9850	−1.12617	−0.563084	+	0.826400i	$0.690385\pi$
$758$	39.8338	1.44683
$759$	0	0
$760$	−7.40074	−0.268453
$761$	31.9586	1.15850	0.579249	−	0.815151i	$-0.303346\pi$
$761$	31.9586	1.15850	0.579249	+	0.815151i	$0.303346\pi$
$762$	0	0
$763$	−8.04616	−0.291291
$764$	1.10484	0.0399716
$765$	0	0
$766$	−29.2324	−1.05621
$767$	−31.5033	−1.13752
$768$	0	0
$769$	−0.115771	−0.00417482	−0.00208741	−	0.999998i	$-0.500664\pi$
$769$	−0.115771	−0.00417482	−0.00208741	+	0.999998i	$0.500664\pi$
$770$	−4.83772	−0.174339
$771$	0	0
$772$	−0.609195	−0.0219254
$773$	−43.3339	−1.55861	−0.779307	−	0.626642i	$-0.784429\pi$
$773$	−43.3339	−1.55861	−0.779307	+	0.626642i	$0.784429\pi$
$774$	0	0
$775$	−59.8841	−2.15110
$776$	−1.35272	−0.0485599
$777$	0	0
$778$	−12.9115	−0.462900
$779$	2.49543	0.0894082
$780$	0	0
$781$	1.72913	0.0618730
$782$	−8.08419	−0.289090
$783$	0	0
$784$	27.9236	0.997270
$785$	41.8270	1.49287
$786$	0	0
$787$	24.8034	0.884146	0.442073	−	0.896979i	$-0.354243\pi$
$787$	24.8034	0.884146	0.442073	+	0.896979i	$0.354243\pi$
$788$	1.82042	0.0648498
$789$	0	0
$790$	−5.05814	−0.179960
$791$	0.376663	0.0133926
$792$	0	0
$793$	−53.8057	−1.91070
$794$	2.90057	0.102937
$795$	0	0
$796$	0.748677	0.0265362
$797$	16.9869	0.601706	0.300853	−	0.953671i	$-0.402729\pi$
$797$	16.9869	0.601706	0.300853	+	0.953671i	$0.402729\pi$
$798$	0	0
$799$	−35.4656	−1.25468
$800$	−4.10149	−0.145010
$801$	0	0
$802$	6.84757	0.241796
$803$	−4.51207	−0.159228
$804$	0	0
$805$	−3.52410	−0.124208
$806$	−46.2606	−1.62946
$807$	0	0
$808$	14.4867	0.509641
$809$	27.8352	0.978634	0.489317	−	0.872106i	$-0.337246\pi$
$809$	27.8352	0.978634	0.489317	+	0.872106i	$0.337246\pi$
$810$	0	0
$811$	4.92943	0.173096	0.0865479	−	0.996248i	$-0.472416\pi$
$811$	4.92943	0.173096	0.0865479	+	0.996248i	$0.472416\pi$
$812$	−0.323795	−0.0113630
$813$	0	0
$814$	−11.9320	−0.418215
$815$	99.4916	3.48504
$816$	0	0
$817$	5.54837	0.194113
$818$	−23.5580	−0.823688
$819$	0	0
$820$	1.00095	0.0349548
$821$	13.3924	0.467398	0.233699	−	0.972309i	$-0.424917\pi$
$821$	13.3924	0.467398	0.233699	+	0.972309i	$0.424917\pi$
$822$	0	0
$823$	−53.8128	−1.87580	−0.937899	−	0.346909i	$-0.887231\pi$
$823$	−53.8128	−1.87580	−0.937899	+	0.346909i	$0.887231\pi$
$824$	−13.1752	−0.458980
$825$	0	0
$826$	−3.93715	−0.136991
$827$	−31.9293	−1.11029	−0.555145	−	0.831753i	$-0.687337\pi$
$827$	−31.9293	−1.11029	−0.555145	+	0.831753i	$0.687337\pi$
$828$	0	0
$829$	−26.3391	−0.914796	−0.457398	−	0.889262i	$-0.651219\pi$
$829$	−26.3391	−0.914796	−0.457398	+	0.889262i	$0.651219\pi$
$830$	−70.7499	−2.45577
$831$	0	0
$832$	43.7313	1.51611
$833$	−20.9633	−0.726337
$834$	0	0
$835$	94.3959	3.26671
$836$	−0.0801718	−0.00277280
$837$	0	0
$838$	22.7816	0.786977
$839$	8.11707	0.280232	0.140116	−	0.990135i	$-0.455252\pi$
$839$	8.11707	0.280232	0.140116	+	0.990135i	$0.455252\pi$
$840$	0	0
$841$	62.4854	2.15467
$842$	11.7985	0.406602
$843$	0	0
$844$	0.991943	0.0341441
$845$	75.7415	2.60559
$846$	0	0
$847$	−3.94946	−0.135705
$848$	−28.0061	−0.961733
$849$	0	0
$850$	47.1432	1.61700
$851$	−8.69200	−0.297958
$852$	0	0
$853$	−33.5610	−1.14911	−0.574553	−	0.818467i	$-0.694824\pi$
$853$	−33.5610	−1.14911	−0.574553	+	0.818467i	$0.694824\pi$
$854$	−6.72441	−0.230105
$855$	0	0
$856$	54.3474	1.85756
$857$	43.8974	1.49951	0.749753	−	0.661718i	$-0.230172\pi$
$857$	43.8974	1.49951	0.749753	+	0.661718i	$0.230172\pi$
$858$	0	0
$859$	−26.5542	−0.906018	−0.453009	−	0.891506i	$-0.649649\pi$
$859$	−26.5542	−0.906018	−0.453009	+	0.891506i	$0.649649\pi$
$860$	2.22553	0.0758898
$861$	0	0
$862$	−32.3985	−1.10350
$863$	−2.80962	−0.0956406	−0.0478203	−	0.998856i	$-0.515227\pi$
$863$	−2.80962	−0.0956406	−0.0478203	+	0.998856i	$0.515227\pi$
$864$	0	0
$865$	89.1487	3.03115
$866$	−12.5136	−0.425231
$867$	0	0
$868$	−0.191886	−0.00651304
$869$	1.54135	0.0522867
$870$	0	0
$871$	48.6610	1.64881
$872$	45.3309	1.53510
$873$	0	0
$874$	−1.75964	−0.0595206
$875$	10.8248	0.365945
$876$	0	0
$877$	−24.6494	−0.832351	−0.416175	−	0.909284i	$-0.636630\pi$
$877$	−24.6494	−0.832351	−0.416175	+	0.909284i	$0.636630\pi$
$878$	−11.0477	−0.372843
$879$	0	0
$880$	28.1918	0.950344
$881$	−12.0806	−0.407007	−0.203503	−	0.979074i	$-0.565233\pi$
$881$	−12.0806	−0.407007	−0.203503	+	0.979074i	$0.565233\pi$
$882$	0	0
$883$	−25.9852	−0.874473	−0.437237	−	0.899347i	$-0.644043\pi$
$883$	−25.9852	−0.874473	−0.437237	+	0.899347i	$0.644043\pi$
$884$	1.20872	0.0406536
$885$	0	0
$886$	21.7393	0.730347
$887$	36.1516	1.21385	0.606925	−	0.794759i	$-0.292403\pi$
$887$	36.1516	1.21385	0.606925	+	0.794759i	$0.292403\pi$
$888$	0	0
$889$	−6.35227	−0.213048
$890$	−56.2581	−1.88578
$891$	0	0
$892$	−1.00197	−0.0335485
$893$	−7.71957	−0.258326
$894$	0	0
$895$	−8.63346	−0.288585
$896$	5.84819	0.195374
$897$	0	0
$898$	22.6220	0.754905
$899$	54.2158	1.80820
$900$	0	0
$901$	21.0253	0.700454
$902$	−9.19005	−0.305995
$903$	0	0
$904$	−2.12206	−0.0705787
$905$	−8.26411	−0.274708
$906$	0	0
$907$	−40.8249	−1.35557	−0.677784	−	0.735261i	$-0.737059\pi$
$907$	−40.8249	−1.35557	−0.677784	+	0.735261i	$0.737059\pi$
$908$	−0.498255	−0.0165352
$909$	0	0
$910$	15.8756	0.526271
$911$	52.7128	1.74645	0.873226	−	0.487315i	$-0.162024\pi$
$911$	52.7128	1.74645	0.873226	+	0.487315i	$0.162024\pi$
$912$	0	0
$913$	21.5594	0.713512
$914$	56.5115	1.86923
$915$	0	0
$916$	−0.00642774	−0.000212379 0
$917$	3.23544	0.106844
$918$	0	0
$919$	−28.0427	−0.925043	−0.462521	−	0.886608i	$-0.653055\pi$
$919$	−28.0427	−0.925043	−0.462521	+	0.886608i	$0.653055\pi$
$920$	19.8543	0.654576
$921$	0	0
$922$	0.539978	0.0177832
$923$	−5.67435	−0.186774
$924$	0	0
$925$	50.6876	1.66660
$926$	58.0885	1.90891
$927$	0	0
$928$	3.71327	0.121894
$929$	39.8392	1.30708	0.653541	−	0.756892i	$-0.273283\pi$
$929$	39.8392	1.30708	0.653541	+	0.756892i	$0.273283\pi$
$930$	0	0
$931$	−4.56296	−0.149545
$932$	−1.79574	−0.0588213
$933$	0	0
$934$	23.3392	0.763682
$935$	−21.1647	−0.692159
$936$	0	0
$937$	−18.7719	−0.613250	−0.306625	−	0.951830i	$-0.599200\pi$
$937$	−18.7719	−0.613250	−0.306625	+	0.951830i	$0.599200\pi$
$938$	6.08145	0.198566
$939$	0	0
$940$	−3.09643	−0.100994
$941$	−9.97279	−0.325104	−0.162552	−	0.986700i	$-0.551972\pi$
$941$	−9.97279	−0.325104	−0.162552	+	0.986700i	$0.551972\pi$
$942$	0	0
$943$	−6.69461	−0.218007
$944$	22.9437	0.746753
$945$	0	0
$946$	−20.4332	−0.664341
$947$	−4.00752	−0.130227	−0.0651135	−	0.997878i	$-0.520741\pi$
$947$	−4.00752	−0.130227	−0.0651135	+	0.997878i	$0.520741\pi$
$948$	0	0
$949$	14.8070	0.480654
$950$	10.2614	0.332923
$951$	0	0
$952$	−4.24928	−0.137720
$953$	−0.685568	−0.0222077	−0.0111039	−	0.999938i	$-0.503535\pi$
$953$	−0.685568	−0.0222077	−0.0111039	+	0.999938i	$0.503535\pi$
$954$	0	0
$955$	63.4854	2.05434
$956$	−0.532062	−0.0172081
$957$	0	0
$958$	−45.8896	−1.48262
$959$	−6.13732	−0.198184
$960$	0	0
$961$	1.12924	0.0364271
$962$	39.1563	1.26245
$963$	0	0
$964$	1.69836	0.0547004
$965$	−35.0052	−1.12686
$966$	0	0
$967$	5.64126	0.181411	0.0907053	−	0.995878i	$-0.471088\pi$
$967$	5.64126	0.181411	0.0907053	+	0.995878i	$0.471088\pi$
$968$	22.2506	0.715163
$969$	0	0
$970$	2.76326	0.0887228
$971$	−34.6057	−1.11055	−0.555275	−	0.831667i	$-0.687387\pi$
$971$	−34.6057	−1.11055	−0.555275	+	0.831667i	$0.687387\pi$
$972$	0	0
$973$	−0.622769	−0.0199651
$974$	34.6109	1.10900
$975$	0	0
$976$	39.1864	1.25433
$977$	15.5025	0.495969	0.247985	−	0.968764i	$-0.420232\pi$
$977$	15.5025	0.495969	0.247985	+	0.968764i	$0.420232\pi$
$978$	0	0
$979$	17.1433	0.547904
$980$	−1.83027	−0.0584657
$981$	0	0
$982$	35.2270	1.12414
$983$	−11.8098	−0.376673	−0.188336	−	0.982105i	$-0.560310\pi$
$983$	−11.8098	−0.376673	−0.188336	+	0.982105i	$0.560310\pi$
$984$	0	0
$985$	104.604	3.33296
$986$	−42.6809	−1.35924
$987$	0	0
$988$	0.263094	0.00837014
$989$	−14.8848	−0.473311
$990$	0	0
$991$	8.48389	0.269500	0.134750	−	0.990880i	$-0.456977\pi$
$991$	8.48389	0.269500	0.134750	+	0.990880i	$0.456977\pi$
$992$	2.20055	0.0698675
$993$	0	0
$994$	−0.709157	−0.0224931
$995$	43.0201	1.36383
$996$	0	0
$997$	46.2269	1.46402	0.732011	−	0.681293i	$-0.238582\pi$
$997$	46.2269	1.46402	0.732011	+	0.681293i	$0.238582\pi$
$998$	−35.5901	−1.12659
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.i.1.4		4
3.2	odd	2		213.2.a.e.1.1	✓	4
12.11	even	2		3408.2.a.w.1.1		4
15.14	odd	2		5325.2.a.w.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
213.2.a.e.1.1	✓	4	3.2	odd	2
639.2.a.i.1.4		4	1.1	even	1	trivial
3408.2.a.w.1.1		4	12.11	even	2
5325.2.a.w.1.4		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)

\(f(q)\)	\(=\)	\(q+1.43828 q^{2} +0.0686587 q^{4} +3.94523 q^{5} +0.493058 q^{7} -2.77782 q^{8} +O(q^{10})\)	\(q+1.43828 q^{2} +0.0686587 q^{4} +3.94523 q^{5} +0.493058 q^{7} -2.77782 q^{8} +5.67435 q^{10} -1.72913 q^{11} +5.67435 q^{13} +0.709157 q^{14} -4.13260 q^{16} +3.10251 q^{17} +0.675304 q^{19} +0.270874 q^{20} -2.48697 q^{22} -1.81167 q^{23} +10.5648 q^{25} +8.16132 q^{26} +0.0338527 q^{28} -9.56480 q^{29} -5.66827 q^{31} -0.388222 q^{32} +4.46229 q^{34} +1.94523 q^{35} +4.79778 q^{37} +0.971279 q^{38} -10.9591 q^{40} +3.69527 q^{41} +8.21610 q^{43} -0.118720 q^{44} -2.60569 q^{46} -11.4312 q^{47} -6.75689 q^{49} +15.1952 q^{50} +0.389593 q^{52} +6.77686 q^{53} -6.82179 q^{55} -1.36962 q^{56} -13.7569 q^{58} -5.55187 q^{59} -9.48226 q^{61} -8.15257 q^{62} +7.70683 q^{64} +22.3866 q^{65} +8.57560 q^{67} +0.213014 q^{68} +2.79778 q^{70} -1.00000 q^{71} +2.60945 q^{73} +6.90057 q^{74} +0.0463655 q^{76} -0.852560 q^{77} -0.891404 q^{79} -16.3041 q^{80} +5.31485 q^{82} -12.4684 q^{83} +12.2401 q^{85} +11.8171 q^{86} +4.80319 q^{88} -9.91446 q^{89} +2.79778 q^{91} -0.124387 q^{92} -16.4414 q^{94} +2.66423 q^{95} +0.486973 q^{97} -9.71833 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 5 q^{4} + 3 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 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

L-functions

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Varieties

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Database

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