LMFDB - Embedded newform 639.2.a.g.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:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 71)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.713538$ 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)

$f(q)$	$=$	$q+2.49086 q^{2} +4.20440 q^{4} -1.49086 q^{5} +1.42708 q^{7} +5.49086 q^{8} +O(q^{10})$	\(q+2.49086 q^{2} +4.20440 q^{4} -1.49086 q^{5} +1.42708 q^{7} +5.49086 q^{8} -3.71354 q^{10} +1.42708 q^{11} +4.00000 q^{13} +3.55465 q^{14} +5.26819 q^{16} -1.42708 q^{17} +3.28646 q^{19} -6.26819 q^{20} +3.55465 q^{22} -8.40880 q^{23} -2.77733 q^{25} +9.96345 q^{26} +6.00000 q^{28} +3.77733 q^{29} +2.98173 q^{31} +2.14061 q^{32} -3.55465 q^{34} -2.12758 q^{35} -3.12234 q^{37} +8.18613 q^{38} -8.18613 q^{40} -5.42708 q^{41} -3.93621 q^{43} +6.00000 q^{44} -20.9452 q^{46} -2.40880 q^{47} -4.96345 q^{49} -6.91794 q^{50} +16.8176 q^{52} -4.98173 q^{53} -2.12758 q^{55} +7.83588 q^{56} +9.40880 q^{58} +6.57292 q^{59} +6.12758 q^{61} +7.42708 q^{62} -5.20440 q^{64} -5.96345 q^{65} -13.9635 q^{67} -6.00000 q^{68} -5.29950 q^{70} -1.00000 q^{71} -1.49086 q^{73} -7.77733 q^{74} +13.8176 q^{76} +2.03655 q^{77} +14.0272 q^{79} -7.85415 q^{80} -13.5181 q^{82} -7.28646 q^{83} +2.12758 q^{85} -9.80457 q^{86} +7.83588 q^{88} +5.04028 q^{89} +5.70830 q^{91} -35.3540 q^{92} -6.00000 q^{94} -4.89967 q^{95} +5.55465 q^{97} -12.3633 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 4 q^{4} + 3 q^{5} + 2 q^{7} + 9 q^{8}+O(q^{10}) $ 3 * q + 4 * q^4 + 3 * q^5 + 2 * q^7 + 9 * q^8	$ 3 q + 4 q^{4} + 3 q^{5} + 2 q^{7} + 9 q^{8} - 10 q^{10} + 2 q^{11} + 12 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 11 q^{19} - 5 q^{20} - 2 q^{22} - 8 q^{23} - 2 q^{25} + 18 q^{28} + 5 q^{29} - 6 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 9 q^{37} + q^{38} - q^{40} - 14 q^{41} - 17 q^{43} + 18 q^{44} - 18 q^{46} + 10 q^{47} + 15 q^{49} - 11 q^{50} + 16 q^{52} + 4 q^{55} + 4 q^{56} + 11 q^{58} + 22 q^{59} + 8 q^{61} + 20 q^{62} - 7 q^{64} + 12 q^{65} - 12 q^{67} - 18 q^{68} - 24 q^{70} - 3 q^{71} + 3 q^{73} - 17 q^{74} + 7 q^{76} + 36 q^{77} + 7 q^{79} - 19 q^{80} + 2 q^{82} - 23 q^{83} - 4 q^{85} + 12 q^{86} + 4 q^{88} - 13 q^{89} + 8 q^{91} - 44 q^{92} - 18 q^{94} + 10 q^{95} + 4 q^{97} - 40 q^{98}+O(q^{100}) $ 3 * q + 4 * q^4 + 3 * q^5 + 2 * q^7 + 9 * q^8 - 10 * q^10 + 2 * q^11 + 12 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 11 * q^19 - 5 * q^20 - 2 * q^22 - 8 * q^23 - 2 * q^25 + 18 * q^28 + 5 * q^29 - 6 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 9 * q^37 + q^38 - q^40 - 14 * q^41 - 17 * q^43 + 18 * q^44 - 18 * q^46 + 10 * q^47 + 15 * q^49 - 11 * q^50 + 16 * q^52 + 4 * q^55 + 4 * q^56 + 11 * q^58 + 22 * q^59 + 8 * q^61 + 20 * q^62 - 7 * q^64 + 12 * q^65 - 12 * q^67 - 18 * q^68 - 24 * q^70 - 3 * q^71 + 3 * q^73 - 17 * q^74 + 7 * q^76 + 36 * q^77 + 7 * q^79 - 19 * q^80 + 2 * q^82 - 23 * q^83 - 4 * q^85 + 12 * q^86 + 4 * q^88 - 13 * q^89 + 8 * q^91 - 44 * q^92 - 18 * q^94 + 10 * q^95 + 4 * q^97 - 40 * 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 $

$2$	2.49086	1.76131	0.880653	−	0.473761i	$-0.157104\pi$
$2$	2.49086	1.76131	0.880653	+	0.473761i	$0.157104\pi$
$3$	0	0
$3$	0	0
$4$	4.20440	2.10220
$5$	−1.49086	−0.666734	−0.333367	−	0.942797i	$-0.608185\pi$
$5$	−1.49086	−0.666734	−0.333367	+	0.942797i	$0.608185\pi$
$6$	0	0
$7$	1.42708	0.539384	0.269692	−	0.962947i	$-0.413078\pi$
$7$	1.42708	0.539384	0.269692	+	0.962947i	$0.413078\pi$
$8$	5.49086	1.94131
$9$	0	0
$10$	−3.71354	−1.17432
$11$	1.42708	0.430280	0.215140	−	0.976583i	$-0.430979\pi$
$11$	1.42708	0.430280	0.215140	+	0.976583i	$0.430979\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	3.55465	0.950021
$15$	0	0
$16$	5.26819	1.31705
$17$	−1.42708	−0.346117	−0.173058	−	0.984912i	$-0.555365\pi$
$17$	−1.42708	−0.346117	−0.173058	+	0.984912i	$0.555365\pi$
$18$	0	0
$19$	3.28646	0.753966	0.376983	−	0.926220i	$-0.376961\pi$
$19$	3.28646	0.753966	0.376983	+	0.926220i	$0.376961\pi$
$20$	−6.26819	−1.40161
$21$	0	0
$22$	3.55465	0.757854
$23$	−8.40880	−1.75336	−0.876678	−	0.481077i	$-0.840246\pi$
$23$	−8.40880	−1.75336	−0.876678	+	0.481077i	$0.840246\pi$
$24$	0	0
$25$	−2.77733	−0.555465
$26$	9.96345	1.95399
$27$	0	0
$28$	6.00000	1.13389
$29$	3.77733	0.701432	0.350716	−	0.936482i	$-0.385938\pi$
$29$	3.77733	0.701432	0.350716	+	0.936482i	$0.385938\pi$
$30$	0	0
$31$	2.98173	0.535534	0.267767	−	0.963484i	$-0.413714\pi$
$31$	2.98173	0.535534	0.267767	+	0.963484i	$0.413714\pi$
$32$	2.14061	0.378411
$33$	0	0
$34$	−3.55465	−0.609618
$35$	−2.12758	−0.359626
$36$	0	0
$37$	−3.12234	−0.513310	−0.256655	−	0.966503i	$-0.582620\pi$
$37$	−3.12234	−0.513310	−0.256655	+	0.966503i	$0.582620\pi$
$38$	8.18613	1.32797
$39$	0	0
$40$	−8.18613	−1.29434
$41$	−5.42708	−0.847567	−0.423783	−	0.905764i	$-0.639298\pi$
$41$	−5.42708	−0.847567	−0.423783	+	0.905764i	$0.639298\pi$
$42$	0	0
$43$	−3.93621	−0.600267	−0.300133	−	0.953897i	$-0.597031\pi$
$43$	−3.93621	−0.600267	−0.300133	+	0.953897i	$0.597031\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	−20.9452	−3.08820
$47$	−2.40880	−0.351360	−0.175680	−	0.984447i	$-0.556212\pi$
$47$	−2.40880	−0.351360	−0.175680	+	0.984447i	$0.556212\pi$
$48$	0	0
$49$	−4.96345	−0.709065
$50$	−6.91794	−0.978344
$51$	0	0
$52$	16.8176	2.33218
$53$	−4.98173	−0.684293	−0.342146	−	0.939647i	$-0.611154\pi$
$53$	−4.98173	−0.684293	−0.342146	+	0.939647i	$0.611154\pi$
$54$	0	0
$55$	−2.12758	−0.286882
$56$	7.83588	1.04711
$57$	0	0
$58$	9.40880	1.23544
$59$	6.57292	0.855722	0.427861	−	0.903845i	$-0.359267\pi$
$59$	6.57292	0.855722	0.427861	+	0.903845i	$0.359267\pi$
$60$	0	0
$61$	6.12758	0.784556	0.392278	−	0.919847i	$-0.371687\pi$
$61$	6.12758	0.784556	0.392278	+	0.919847i	$0.371687\pi$
$62$	7.42708	0.943240
$63$	0	0
$64$	−5.20440	−0.650550
$65$	−5.96345	−0.739675
$66$	0	0
$67$	−13.9635	−1.70591	−0.852954	−	0.521987i	$-0.825191\pi$
$67$	−13.9635	−1.70591	−0.852954	+	0.521987i	$0.825191\pi$
$68$	−6.00000	−0.727607
$69$	0	0
$70$	−5.29950	−0.633411
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	0	0
$73$	−1.49086	−0.174492	−0.0872462	−	0.996187i	$-0.527807\pi$
$73$	−1.49086	−0.174492	−0.0872462	+	0.996187i	$0.527807\pi$
$74$	−7.77733	−0.904096
$75$	0	0
$76$	13.8176	1.58499
$77$	2.03655	0.232086
$78$	0	0
$79$	14.0272	1.57819	0.789094	−	0.614272i	$-0.210550\pi$
$79$	14.0272	1.57819	0.789094	+	0.614272i	$0.210550\pi$
$80$	−7.85415	−0.878121
$81$	0	0
$82$	−13.5181	−1.49283
$83$	−7.28646	−0.799793	−0.399897	−	0.916560i	$-0.630954\pi$
$83$	−7.28646	−0.799793	−0.399897	+	0.916560i	$0.630954\pi$
$84$	0	0
$85$	2.12758	0.230768
$86$	−9.80457	−1.05725
$87$	0	0
$88$	7.83588	0.835308
$89$	5.04028	0.534269	0.267134	−	0.963659i	$-0.413923\pi$
$89$	5.04028	0.534269	0.267134	+	0.963659i	$0.413923\pi$
$90$	0	0
$91$	5.70830	0.598393
$92$	−35.3540	−3.68591
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−4.89967	−0.502695
$96$	0	0
$97$	5.55465	0.563989	0.281995	−	0.959416i	$-0.409004\pi$
$97$	5.55465	0.563989	0.281995	+	0.959416i	$0.409004\pi$
$98$	−12.3633	−1.24888
$99$	0	0
$100$	−11.6770	−1.16770
$101$	−2.84111	−0.282701	−0.141351	−	0.989960i	$-0.545145\pi$
$101$	−2.84111	−0.282701	−0.141351	+	0.989960i	$0.545145\pi$
$102$	0	0
$103$	5.61320	0.553085	0.276543	−	0.961002i	$-0.410811\pi$
$103$	5.61320	0.553085	0.276543	+	0.961002i	$0.410811\pi$
$104$	21.9635	2.15369
$105$	0	0
$106$	−12.4088	−1.20525
$107$	13.9635	1.34990	0.674949	−	0.737864i	$-0.264166\pi$
$107$	13.9635	1.34990	0.674949	+	0.737864i	$0.264166\pi$
$108$	0	0
$109$	−5.74078	−0.549867	−0.274934	−	0.961463i	$-0.588656\pi$
$109$	−5.74078	−0.549867	−0.274934	+	0.961463i	$0.588656\pi$
$110$	−5.29950	−0.505288
$111$	0	0
$112$	7.51811	0.710394
$113$	7.68223	0.722683	0.361342	−	0.932434i	$-0.382319\pi$
$113$	7.68223	0.722683	0.361342	+	0.932434i	$0.382319\pi$
$114$	0	0
$115$	12.5364	1.16902
$116$	15.8814	1.47455
$117$	0	0
$118$	16.3723	1.50719
$119$	−2.03655	−0.186690
$120$	0	0
$121$	−8.96345	−0.814859
$122$	15.2630	1.38184
$123$	0	0
$124$	12.5364	1.12580
$125$	11.5949	1.03708
$126$	0	0
$127$	−17.1093	−1.51821	−0.759103	−	0.650971i	$-0.774362\pi$
$127$	−17.1093	−1.51821	−0.759103	+	0.650971i	$0.774362\pi$
$128$	−17.2447	−1.52423
$129$	0	0
$130$	−14.8542	−1.30280
$131$	11.0455	0.965051	0.482526	−	0.875882i	$-0.339720\pi$
$131$	11.0455	0.965051	0.482526	+	0.875882i	$0.339720\pi$
$132$	0	0
$133$	4.69003	0.406677
$134$	−34.7811	−3.00463
$135$	0	0
$136$	−7.83588	−0.671921
$137$	21.5181	1.83842	0.919208	−	0.393773i	$-0.128830\pi$
$137$	21.5181	1.83842	0.919208	+	0.393773i	$0.128830\pi$
$138$	0	0
$139$	14.4088	1.22214	0.611069	−	0.791577i	$-0.290740\pi$
$139$	14.4088	1.22214	0.611069	+	0.791577i	$0.290740\pi$
$140$	−8.94518	−0.756006
$141$	0	0
$142$	−2.49086	−0.209029
$143$	5.70830	0.477352
$144$	0	0
$145$	−5.63148	−0.467669
$146$	−3.71354	−0.307335
$147$	0	0
$148$	−13.1276	−1.07908
$149$	−5.10930	−0.418570	−0.209285	−	0.977855i	$-0.567114\pi$
$149$	−5.10930	−0.418570	−0.209285	+	0.977855i	$0.567114\pi$
$150$	0	0
$151$	−7.64975	−0.622528	−0.311264	−	0.950324i	$-0.600752\pi$
$151$	−7.64975	−0.622528	−0.311264	+	0.950324i	$0.600752\pi$
$152$	18.0455	1.46368
$153$	0	0
$154$	5.07276	0.408774
$155$	−4.44535	−0.357059
$156$	0	0
$157$	−10.1861	−0.812942	−0.406471	−	0.913664i	$-0.633241\pi$
$157$	−10.1861	−0.812942	−0.406471	+	0.913664i	$0.633241\pi$
$158$	34.9399	2.77967
$159$	0	0
$160$	−3.19136	−0.252299
$161$	−12.0000	−0.945732
$162$	0	0
$163$	12.5364	0.981925	0.490962	−	0.871181i	$-0.336645\pi$
$163$	12.5364	0.981925	0.490962	+	0.871181i	$0.336645\pi$
$164$	−22.8176	−1.78176
$165$	0	0
$166$	−18.1496	−1.40868
$167$	−20.0638	−1.55258	−0.776291	−	0.630374i	$-0.782901\pi$
$167$	−20.0638	−1.55258	−0.776291	+	0.630374i	$0.782901\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	5.29950	0.406453
$171$	0	0
$172$	−16.5494	−1.26188
$173$	19.2264	1.46176	0.730878	−	0.682508i	$-0.239111\pi$
$173$	19.2264	1.46176	0.730878	+	0.682508i	$0.239111\pi$
$174$	0	0
$175$	−3.96345	−0.299609
$176$	7.51811	0.566699
$177$	0	0
$178$	12.5547	0.941011
$179$	−18.7173	−1.39899	−0.699497	−	0.714635i	$-0.746593\pi$
$179$	−18.7173	−1.39899	−0.699497	+	0.714635i	$0.746593\pi$
$180$	0	0
$181$	3.46362	0.257449	0.128724	−	0.991680i	$-0.458912\pi$
$181$	3.46362	0.257449	0.128724	+	0.991680i	$0.458912\pi$
$182$	14.2186	1.05395
$183$	0	0
$184$	−46.1716	−3.40381
$185$	4.65498	0.342241
$186$	0	0
$187$	−2.03655	−0.148927
$188$	−10.1276	−0.738629
$189$	0	0
$190$	−12.2044	−0.885400
$191$	24.7225	1.78886	0.894429	−	0.447210i	$-0.147582\pi$
$191$	24.7225	1.78886	0.894429	+	0.447210i	$0.147582\pi$
$192$	0	0
$193$	25.7628	1.85445	0.927223	−	0.374510i	$-0.122189\pi$
$193$	25.7628	1.85445	0.927223	+	0.374510i	$0.122189\pi$
$194$	13.8359	0.993358
$195$	0	0
$196$	−20.8684	−1.49060
$197$	−9.96345	−0.709867	−0.354933	−	0.934892i	$-0.615496\pi$
$197$	−9.96345	−0.709867	−0.354933	+	0.934892i	$0.615496\pi$
$198$	0	0
$199$	−18.6770	−1.32398	−0.661988	−	0.749514i	$-0.730287\pi$
$199$	−18.6770	−1.32398	−0.661988	+	0.749514i	$0.730287\pi$
$200$	−15.2499	−1.07833
$201$	0	0
$202$	−7.07683	−0.497924
$203$	5.39053	0.378341
$204$	0	0
$205$	8.09103	0.565102
$206$	13.9817	0.974153
$207$	0	0
$208$	21.0728	1.46113
$209$	4.69003	0.324416
$210$	0	0
$211$	−6.81761	−0.469343	−0.234672	−	0.972075i	$-0.575402\pi$
$211$	−6.81761	−0.469343	−0.234672	+	0.972075i	$0.575402\pi$
$212$	−20.9452	−1.43852
$213$	0	0
$214$	34.7811	2.37758
$215$	5.86836	0.400219
$216$	0	0
$217$	4.25515	0.288858
$218$	−14.2995	−0.968484
$219$	0	0
$220$	−8.94518	−0.603084
$221$	−5.70830	−0.383982
$222$	0	0
$223$	8.96869	0.600588	0.300294	−	0.953847i	$-0.402915\pi$
$223$	8.96869	0.600588	0.300294	+	0.953847i	$0.402915\pi$
$224$	3.05482	0.204109
$225$	0	0
$226$	19.1354	1.27287
$227$	−5.13538	−0.340847	−0.170424	−	0.985371i	$-0.554514\pi$
$227$	−5.13538	−0.340847	−0.170424	+	0.985371i	$0.554514\pi$
$228$	0	0
$229$	8.21744	0.543024	0.271512	−	0.962435i	$-0.412476\pi$
$229$	8.21744	0.543024	0.271512	+	0.962435i	$0.412476\pi$
$230$	31.2264	2.05901
$231$	0	0
$232$	20.7408	1.36170
$233$	5.44011	0.356394	0.178197	−	0.983995i	$-0.442974\pi$
$233$	5.44011	0.356394	0.178197	+	0.983995i	$0.442974\pi$
$234$	0	0
$235$	3.59120	0.234264
$236$	27.6352	1.79890
$237$	0	0
$238$	−5.07276	−0.328818
$239$	4.60947	0.298162	0.149081	−	0.988825i	$-0.452369\pi$
$239$	4.60947	0.298162	0.149081	+	0.988825i	$0.452369\pi$
$240$	0	0
$241$	26.4088	1.70114	0.850570	−	0.525861i	$-0.176257\pi$
$241$	26.4088	1.70114	0.850570	+	0.525861i	$0.176257\pi$
$242$	−22.3267	−1.43522
$243$	0	0
$244$	25.7628	1.64929
$245$	7.39983	0.472758
$246$	0	0
$247$	13.1458	0.836450
$248$	16.3723	1.03964
$249$	0	0
$250$	28.8814	1.82662
$251$	−10.3958	−0.656175	−0.328087	−	0.944647i	$-0.606404\pi$
$251$	−10.3958	−0.656175	−0.328087	+	0.944647i	$0.606404\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	−42.6169	−2.67402
$255$	0	0
$256$	−32.5453	−2.03408
$257$	26.2081	1.63482	0.817409	−	0.576057i	$-0.195409\pi$
$257$	26.2081	1.63482	0.817409	+	0.576057i	$0.195409\pi$
$258$	0	0
$259$	−4.45582	−0.276871
$260$	−25.0728	−1.55495
$261$	0	0
$262$	27.5129	1.69975
$263$	−0.191363	−0.0118000	−0.00589998	−	0.999983i	$-0.501878\pi$
$263$	−0.191363	−0.0118000	−0.00589998	+	0.999983i	$0.501878\pi$
$264$	0	0
$265$	7.42708	0.456242
$266$	11.6822	0.716283
$267$	0	0
$268$	−58.7080	−3.58616
$269$	10.2812	0.626858	0.313429	−	0.949612i	$-0.398522\pi$
$269$	10.2812	0.626858	0.313429	+	0.949612i	$0.398522\pi$
$270$	0	0
$271$	4.79560	0.291312	0.145656	−	0.989335i	$-0.453471\pi$
$271$	4.79560	0.291312	0.145656	+	0.989335i	$0.453471\pi$
$272$	−7.51811	−0.455852
$273$	0	0
$274$	53.5987	3.23801
$275$	−3.96345	−0.239005
$276$	0	0
$277$	−2.55989	−0.153809	−0.0769043	−	0.997038i	$-0.524504\pi$
$277$	−2.55989	−0.153809	−0.0769043	+	0.997038i	$0.524504\pi$
$278$	35.8904	2.15256
$279$	0	0
$280$	−11.6822	−0.698147
$281$	−5.14585	−0.306976	−0.153488	−	0.988151i	$-0.549051\pi$
$281$	−5.14585	−0.306976	−0.153488	+	0.988151i	$0.549051\pi$
$282$	0	0
$283$	−3.05482	−0.181590	−0.0907950	−	0.995870i	$-0.528941\pi$
$283$	−3.05482	−0.181590	−0.0907950	+	0.995870i	$0.528941\pi$
$284$	−4.20440	−0.249485
$285$	0	0
$286$	14.2186	0.840764
$287$	−7.74485	−0.457164
$288$	0	0
$289$	−14.9635	−0.880203
$290$	−14.0272	−0.823708
$291$	0	0
$292$	−6.26819	−0.366818
$293$	−9.67176	−0.565030	−0.282515	−	0.959263i	$-0.591169\pi$
$293$	−9.67176	−0.565030	−0.282515	+	0.959263i	$0.591169\pi$
$294$	0	0
$295$	−9.79933	−0.570539
$296$	−17.1443	−0.996495
$297$	0	0
$298$	−12.7266	−0.737231
$299$	−33.6352	−1.94517
$300$	0	0
$301$	−5.61727	−0.323774
$302$	−19.0545	−1.09646
$303$	0	0
$304$	17.3137	0.993009
$305$	−9.13538	−0.523090
$306$	0	0
$307$	−29.6352	−1.69137	−0.845685	−	0.533682i	$-0.820808\pi$
$307$	−29.6352	−1.69137	−0.845685	+	0.533682i	$0.820808\pi$
$308$	8.56246	0.487891
$309$	0	0
$310$	−11.0728	−0.628890
$311$	11.8489	0.671890	0.335945	−	0.941882i	$-0.390944\pi$
$311$	11.8489	0.671890	0.335945	+	0.941882i	$0.390944\pi$
$312$	0	0
$313$	−8.52334	−0.481768	−0.240884	−	0.970554i	$-0.577437\pi$
$313$	−8.52334	−0.481768	−0.240884	+	0.970554i	$0.577437\pi$
$314$	−25.3723	−1.43184
$315$	0	0
$316$	58.9762	3.31767
$317$	1.01827	0.0571919	0.0285959	−	0.999591i	$-0.490896\pi$
$317$	1.01827	0.0571919	0.0285959	+	0.999591i	$0.490896\pi$
$318$	0	0
$319$	5.39053	0.301812
$320$	7.75905	0.433744
$321$	0	0
$322$	−29.8904	−1.66572
$323$	−4.69003	−0.260960
$324$	0	0
$325$	−11.1093	−0.616233
$326$	31.2264	1.72947
$327$	0	0
$328$	−29.7993	−1.64539
$329$	−3.43754	−0.189518
$330$	0	0
$331$	1.43754	0.0790146	0.0395073	−	0.999219i	$-0.487421\pi$
$331$	1.43754	0.0790146	0.0395073	+	0.999219i	$0.487421\pi$
$332$	−30.6352	−1.68133
$333$	0	0
$334$	−49.9762	−2.73457
$335$	20.8176	1.13739
$336$	0	0
$337$	22.1276	1.20537	0.602683	−	0.797981i	$-0.294098\pi$
$337$	22.1276	1.20537	0.602683	+	0.797981i	$0.294098\pi$
$338$	7.47259	0.406455
$339$	0	0
$340$	8.94518	0.485121
$341$	4.25515	0.230429
$342$	0	0
$343$	−17.0728	−0.921842
$344$	−21.6132	−1.16531
$345$	0	0
$346$	47.8904	2.57460
$347$	−12.6900	−0.681237	−0.340618	−	0.940202i	$-0.610636\pi$
$347$	−12.6900	−0.681237	−0.340618	+	0.940202i	$0.610636\pi$
$348$	0	0
$349$	−17.3540	−0.928938	−0.464469	−	0.885590i	$-0.653755\pi$
$349$	−17.3540	−0.928938	−0.464469	+	0.885590i	$0.653755\pi$
$350$	−9.87242	−0.527703
$351$	0	0
$352$	3.05482	0.162822
$353$	9.14585	0.486784	0.243392	−	0.969928i	$-0.421740\pi$
$353$	9.14585	0.486784	0.243392	+	0.969928i	$0.421740\pi$
$354$	0	0
$355$	1.49086	0.0791268
$356$	21.1914	1.12314
$357$	0	0
$358$	−46.6222	−2.46406
$359$	−7.73555	−0.408266	−0.204133	−	0.978943i	$-0.565438\pi$
$359$	−7.73555	−0.408266	−0.204133	+	0.978943i	$0.565438\pi$
$360$	0	0
$361$	−8.19917	−0.431535
$362$	8.62741	0.453447
$363$	0	0
$364$	24.0000	1.25794
$365$	2.22267	0.116340
$366$	0	0
$367$	−23.3007	−1.21628	−0.608142	−	0.793828i	$-0.708085\pi$
$367$	−23.3007	−1.21628	−0.608142	+	0.793828i	$0.708085\pi$
$368$	−44.2992	−2.30925
$369$	0	0
$370$	11.5949	0.602792
$371$	−7.10930	−0.369097
$372$	0	0
$373$	6.76429	0.350242	0.175121	−	0.984547i	$-0.443968\pi$
$373$	6.76429	0.350242	0.175121	+	0.984547i	$0.443968\pi$
$374$	−5.07276	−0.262306
$375$	0	0
$376$	−13.2264	−0.682100
$377$	15.1093	0.778169
$378$	0	0
$379$	7.20440	0.370065	0.185033	−	0.982732i	$-0.440761\pi$
$379$	7.20440	0.370065	0.185033	+	0.982732i	$0.440761\pi$
$380$	−20.6002	−1.05677
$381$	0	0
$382$	61.5804	3.15073
$383$	27.4816	1.40424	0.702121	−	0.712058i	$-0.252237\pi$
$383$	27.4816	1.40424	0.702121	+	0.712058i	$0.252237\pi$
$384$	0	0
$385$	−3.03621	−0.154740
$386$	64.1716	3.26625
$387$	0	0
$388$	23.3540	1.18562
$389$	17.5547	0.890056	0.445028	−	0.895517i	$-0.353194\pi$
$389$	17.5547	0.890056	0.445028	+	0.895517i	$0.353194\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	−27.2537	−1.37652
$393$	0	0
$394$	−24.8176	−1.25029
$395$	−20.9127	−1.05223
$396$	0	0
$397$	10.6640	0.535209	0.267604	−	0.963529i	$-0.413768\pi$
$397$	10.6640	0.535209	0.267604	+	0.963529i	$0.413768\pi$
$398$	−46.5218	−2.33193
$399$	0	0
$400$	−14.6315	−0.731574
$401$	22.3462	1.11592	0.557958	−	0.829870i	$-0.311585\pi$
$401$	22.3462	1.11592	0.557958	+	0.829870i	$0.311585\pi$
$402$	0	0
$403$	11.9269	0.594122
$404$	−11.9452	−0.594295
$405$	0	0
$406$	13.4271	0.666375
$407$	−4.45582	−0.220867
$408$	0	0
$409$	31.3320	1.54927	0.774633	−	0.632411i	$-0.217934\pi$
$409$	31.3320	1.54927	0.774633	+	0.632411i	$0.217934\pi$
$410$	20.1537	0.995318
$411$	0	0
$412$	23.6002	1.16270
$413$	9.38006	0.461563
$414$	0	0
$415$	10.8631	0.533250
$416$	8.56246	0.419809
$417$	0	0
$418$	11.6822	0.571396
$419$	−0.350250	−0.0171108	−0.00855541	−	0.999963i	$-0.502723\pi$
$419$	−0.350250	−0.0171108	−0.00855541	+	0.999963i	$0.502723\pi$
$420$	0	0
$421$	32.2992	1.57417	0.787083	−	0.616848i	$-0.211591\pi$
$421$	32.2992	1.57417	0.787083	+	0.616848i	$0.211591\pi$
$422$	−16.9817	−0.826658
$423$	0	0
$424$	−27.3540	−1.32843
$425$	3.96345	0.192256
$426$	0	0
$427$	8.74452	0.423177
$428$	58.7080	2.83776
$429$	0	0
$430$	14.6173	0.704908
$431$	6.95822	0.335166	0.167583	−	0.985858i	$-0.446404\pi$
$431$	6.95822	0.335166	0.167583	+	0.985858i	$0.446404\pi$
$432$	0	0
$433$	38.7080	1.86019	0.930093	−	0.367324i	$-0.119726\pi$
$433$	38.7080	1.86019	0.930093	+	0.367324i	$0.119726\pi$
$434$	10.5990	0.508768
$435$	0	0
$436$	−24.1365	−1.15593
$437$	−27.6352	−1.32197
$438$	0	0
$439$	−24.6900	−1.17839	−0.589195	−	0.807991i	$-0.700555\pi$
$439$	−24.6900	−1.17839	−0.589195	+	0.807991i	$0.700555\pi$
$440$	−11.6822	−0.556928
$441$	0	0
$442$	−14.2186	−0.676310
$443$	9.64568	0.458280	0.229140	−	0.973393i	$-0.426409\pi$
$443$	9.64568	0.458280	0.229140	+	0.973393i	$0.426409\pi$
$444$	0	0
$445$	−7.51437	−0.356215
$446$	22.3398	1.05782
$447$	0	0
$448$	−7.42708	−0.350896
$449$	−5.39053	−0.254395	−0.127197	−	0.991877i	$-0.540598\pi$
$449$	−5.39053	−0.254395	−0.127197	+	0.991877i	$0.540598\pi$
$450$	0	0
$451$	−7.74485	−0.364691
$452$	32.2992	1.51922
$453$	0	0
$454$	−12.7915	−0.600336
$455$	−8.51030	−0.398969
$456$	0	0
$457$	−28.7915	−1.34681	−0.673405	−	0.739273i	$-0.735169\pi$
$457$	−28.7915	−1.34681	−0.673405	+	0.739273i	$0.735169\pi$
$458$	20.4685	0.956431
$459$	0	0
$460$	52.7080	2.45752
$461$	14.3723	0.669383	0.334691	−	0.942328i	$-0.391368\pi$
$461$	14.3723	0.669383	0.334691	+	0.942328i	$0.391368\pi$
$462$	0	0
$463$	−10.4689	−0.486529	−0.243264	−	0.969960i	$-0.578218\pi$
$463$	−10.4689	−0.486529	−0.243264	+	0.969960i	$0.578218\pi$
$464$	19.8997	0.923819
$465$	0	0
$466$	13.5506	0.627719
$467$	−6.73705	−0.311753	−0.155877	−	0.987777i	$-0.549820\pi$
$467$	−6.73705	−0.311753	−0.155877	+	0.987777i	$0.549820\pi$
$468$	0	0
$469$	−19.9269	−0.920139
$470$	8.94518	0.412610
$471$	0	0
$472$	36.0910	1.66122
$473$	−5.61727	−0.258283
$474$	0	0
$475$	−9.12758	−0.418802
$476$	−8.56246	−0.392459
$477$	0	0
$478$	11.4816	0.525154
$479$	−22.6640	−1.03554	−0.517771	−	0.855519i	$-0.673238\pi$
$479$	−22.6640	−1.03554	−0.517771	+	0.855519i	$0.673238\pi$
$480$	0	0
$481$	−12.4894	−0.569466
$482$	65.7807	2.99623
$483$	0	0
$484$	−37.6860	−1.71300
$485$	−8.28123	−0.376031
$486$	0	0
$487$	−20.0545	−0.908755	−0.454378	−	0.890809i	$-0.650138\pi$
$487$	−20.0545	−0.908755	−0.454378	+	0.890809i	$0.650138\pi$
$488$	33.6457	1.52307
$489$	0	0
$490$	18.4320	0.832672
$491$	20.9452	0.945243	0.472621	−	0.881266i	$-0.343308\pi$
$491$	20.9452	0.945243	0.472621	+	0.881266i	$0.343308\pi$
$492$	0	0
$493$	−5.39053	−0.242777
$494$	32.7445	1.47325
$495$	0	0
$496$	15.7083	0.705324
$497$	−1.42708	−0.0640131
$498$	0	0
$499$	−15.4596	−0.692065	−0.346032	−	0.938223i	$-0.612471\pi$
$499$	−15.4596	−0.692065	−0.346032	+	0.938223i	$0.612471\pi$
$500$	48.7497	2.18016
$501$	0	0
$502$	−25.8944	−1.15573
$503$	4.60540	0.205345	0.102672	−	0.994715i	$-0.467261\pi$
$503$	4.60540	0.205345	0.102672	+	0.994715i	$0.467261\pi$
$504$	0	0
$505$	4.23571	0.188487
$506$	−29.8904	−1.32879
$507$	0	0
$508$	−71.9344	−3.19157
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	−2.12758	−0.0941184
$512$	−46.5767	−2.05842
$513$	0	0
$514$	65.2809	2.87942
$515$	−8.36852	−0.368761
$516$	0	0
$517$	−3.43754	−0.151183
$518$	−11.0988	−0.487655
$519$	0	0
$520$	−32.7445	−1.43594
$521$	1.74601	0.0764943	0.0382471	−	0.999268i	$-0.487823\pi$
$521$	1.74601	0.0764943	0.0382471	+	0.999268i	$0.487823\pi$
$522$	0	0
$523$	−4.53638	−0.198362	−0.0991810	−	0.995069i	$-0.531622\pi$
$523$	−4.53638	−0.198362	−0.0991810	+	0.995069i	$0.531622\pi$
$524$	46.4398	2.02873
$525$	0	0
$526$	−0.476660	−0.0207834
$527$	−4.25515	−0.185357
$528$	0	0
$529$	47.7080	2.07426
$530$	18.4998	0.803581
$531$	0	0
$532$	19.7188	0.854917
$533$	−21.7083	−0.940291
$534$	0	0
$535$	−20.8176	−0.900024
$536$	−76.6714	−3.31170
$537$	0	0
$538$	25.6091	1.10409
$539$	−7.08323	−0.305096
$540$	0	0
$541$	−2.12758	−0.0914716	−0.0457358	−	0.998954i	$-0.514563\pi$
$541$	−2.12758	−0.0914716	−0.0457358	+	0.998954i	$0.514563\pi$
$542$	11.9452	0.513089
$543$	0	0
$544$	−3.05482	−0.130974
$545$	8.55872	0.366615
$546$	0	0
$547$	−17.5311	−0.749578	−0.374789	−	0.927110i	$-0.622285\pi$
$547$	−17.5311	−0.749578	−0.374789	+	0.927110i	$0.622285\pi$
$548$	90.4708	3.86472
$549$	0	0
$550$	−9.87242	−0.420962
$551$	12.4140	0.528856
$552$	0	0
$553$	20.0179	0.851249
$554$	−6.37633	−0.270904
$555$	0	0
$556$	60.5804	2.56918
$557$	−29.5506	−1.25210	−0.626049	−	0.779784i	$-0.715329\pi$
$557$	−29.5506	−1.25210	−0.626049	+	0.779784i	$0.715329\pi$
$558$	0	0
$559$	−15.7448	−0.665936
$560$	−11.2085	−0.473644
$561$	0	0
$562$	−12.8176	−0.540678
$563$	−37.1638	−1.56627	−0.783133	−	0.621854i	$-0.786380\pi$
$563$	−37.1638	−1.56627	−0.783133	+	0.621854i	$0.786380\pi$
$564$	0	0
$565$	−11.4532	−0.481838
$566$	−7.60914	−0.319836
$567$	0	0
$568$	−5.49086	−0.230392
$569$	32.1443	1.34756	0.673781	−	0.738931i	$-0.264669\pi$
$569$	32.1443	1.34756	0.673781	+	0.738931i	$0.264669\pi$
$570$	0	0
$571$	13.6042	0.569320	0.284660	−	0.958629i	$-0.408119\pi$
$571$	13.6042	0.569320	0.284660	+	0.958629i	$0.408119\pi$
$572$	24.0000	1.00349
$573$	0	0
$574$	−19.2914	−0.805206
$575$	23.3540	0.973928
$576$	0	0
$577$	25.0350	1.04222	0.521111	−	0.853489i	$-0.325518\pi$
$577$	25.0350	1.04222	0.521111	+	0.853489i	$0.325518\pi$
$578$	−37.2719	−1.55031
$579$	0	0
$580$	−23.6770	−0.983134
$581$	−10.3983	−0.431396
$582$	0	0
$583$	−7.10930	−0.294437
$584$	−8.18613	−0.338744
$585$	0	0
$586$	−24.0910	−0.995191
$587$	−30.8762	−1.27440	−0.637198	−	0.770700i	$-0.719907\pi$
$587$	−30.8762	−1.27440	−0.637198	+	0.770700i	$0.719907\pi$
$588$	0	0
$589$	9.79933	0.403774
$590$	−24.4088	−1.00489
$591$	0	0
$592$	−16.4491	−0.676053
$593$	−7.77733	−0.319376	−0.159688	−	0.987167i	$-0.551049\pi$
$593$	−7.77733	−0.319376	−0.159688	+	0.987167i	$0.551049\pi$
$594$	0	0
$595$	3.03621	0.124473
$596$	−21.4816	−0.879919
$597$	0	0
$598$	−83.7807	−3.42605
$599$	−3.42708	−0.140027	−0.0700133	−	0.997546i	$-0.522304\pi$
$599$	−3.42708	−0.140027	−0.0700133	+	0.997546i	$0.522304\pi$
$600$	0	0
$601$	2.48189	0.101239	0.0506193	−	0.998718i	$-0.483880\pi$
$601$	2.48189	0.101239	0.0506193	+	0.998718i	$0.483880\pi$
$602$	−13.9919	−0.570266
$603$	0	0
$604$	−32.1626	−1.30868
$605$	13.3633	0.543295
$606$	0	0
$607$	−0.0365455	−0.00148334	−0.000741669	−	1.00000i	$-0.500236\pi$
$607$	−0.0365455	−0.00148334	−0.000741669		1.00000i	$0.500236\pi$
$608$	7.03505	0.285309
$609$	0	0
$610$	−22.7550	−0.921322
$611$	−9.63521	−0.389799
$612$	0	0
$613$	24.7120	0.998110	0.499055	−	0.866570i	$-0.333681\pi$
$613$	24.7120	0.998110	0.499055	+	0.866570i	$0.333681\pi$
$614$	−73.8173	−2.97902
$615$	0	0
$616$	11.1824	0.450551
$617$	−26.4360	−1.06428	−0.532138	−	0.846658i	$-0.678611\pi$
$617$	−26.4360	−1.06428	−0.532138	+	0.846658i	$0.678611\pi$
$618$	0	0
$619$	−23.5076	−0.944852	−0.472426	−	0.881370i	$-0.656622\pi$
$619$	−23.5076	−0.944852	−0.472426	+	0.881370i	$0.656622\pi$
$620$	−18.6900	−0.750610
$621$	0	0
$622$	29.5140	1.18341
$623$	7.19286	0.288176
$624$	0	0
$625$	−3.39983	−0.135993
$626$	−21.2305	−0.848541
$627$	0	0
$628$	−42.8266	−1.70897
$629$	4.45582	0.177665
$630$	0	0
$631$	−12.1276	−0.482791	−0.241396	−	0.970427i	$-0.577605\pi$
$631$	−12.1276	−0.482791	−0.241396	+	0.970427i	$0.577605\pi$
$632$	77.0217	3.06376
$633$	0	0
$634$	2.53638	0.100732
$635$	25.5076	1.01224
$636$	0	0
$637$	−19.8538	−0.786637
$638$	13.4271	0.531583
$639$	0	0
$640$	25.7095	1.01626
$641$	−2.55989	−0.101109	−0.0505547	−	0.998721i	$-0.516099\pi$
$641$	−2.55989	−0.101109	−0.0505547	+	0.998721i	$0.516099\pi$
$642$	0	0
$643$	1.63521	0.0644865	0.0322432	−	0.999480i	$-0.489735\pi$
$643$	1.63521	0.0644865	0.0322432	+	0.999480i	$0.489735\pi$
$644$	−50.4528	−1.98812
$645$	0	0
$646$	−11.6822	−0.459631
$647$	−3.10930	−0.122239	−0.0611197	−	0.998130i	$-0.519467\pi$
$647$	−3.10930	−0.122239	−0.0611197	+	0.998130i	$0.519467\pi$
$648$	0	0
$649$	9.38006	0.368200
$650$	−27.6718	−1.08538
$651$	0	0
$652$	52.7080	2.06420
$653$	−31.0728	−1.21597	−0.607985	−	0.793948i	$-0.708022\pi$
$653$	−31.0728	−1.21597	−0.607985	+	0.793948i	$0.708022\pi$
$654$	0	0
$655$	−16.4674	−0.643433
$656$	−28.5909	−1.11629
$657$	0	0
$658$	−8.56246	−0.333799
$659$	7.34245	0.286021	0.143011	−	0.989721i	$-0.454322\pi$
$659$	7.34245	0.286021	0.143011	+	0.989721i	$0.454322\pi$
$660$	0	0
$661$	−24.0910	−0.937032	−0.468516	−	0.883455i	$-0.655211\pi$
$661$	−24.0910	−0.937032	−0.468516	+	0.883455i	$0.655211\pi$
$662$	3.58073	0.139169
$663$	0	0
$664$	−40.0090	−1.55265
$665$	−6.99220	−0.271146
$666$	0	0
$667$	−31.7628	−1.22986
$668$	−84.3562	−3.26384
$669$	0	0
$670$	51.8538	2.00329
$671$	8.74452	0.337578
$672$	0	0
$673$	−12.4349	−0.479329	−0.239665	−	0.970856i	$-0.577038\pi$
$673$	−12.4349	−0.479329	−0.239665	+	0.970856i	$0.577038\pi$
$674$	55.1168	2.12302
$675$	0	0
$676$	12.6132	0.485123
$677$	8.22267	0.316023	0.158012	−	0.987437i	$-0.449492\pi$
$677$	8.22267	0.316023	0.158012	+	0.987437i	$0.449492\pi$
$678$	0	0
$679$	7.92691	0.304207
$680$	11.6822	0.447993
$681$	0	0
$682$	10.5990	0.405857
$683$	−18.3282	−0.701311	−0.350655	−	0.936505i	$-0.614041\pi$
$683$	−18.3282	−0.701311	−0.350655	+	0.936505i	$0.614041\pi$
$684$	0	0
$685$	−32.0806	−1.22574
$686$	−42.5259	−1.62365
$687$	0	0
$688$	−20.7367	−0.790580
$689$	−19.9269	−0.759155
$690$	0	0
$691$	23.2809	0.885647	0.442823	−	0.896609i	$-0.353977\pi$
$691$	23.2809	0.885647	0.442823	+	0.896609i	$0.353977\pi$
$692$	80.8355	3.07291
$693$	0	0
$694$	−31.6091	−1.19987
$695$	−21.4816	−0.814842
$696$	0	0
$697$	7.74485	0.293357
$698$	−43.2264	−1.63614
$699$	0	0
$700$	−16.6640	−0.629838
$701$	−39.1533	−1.47880	−0.739400	−	0.673266i	$-0.764891\pi$
$701$	−39.1533	−1.47880	−0.739400	+	0.673266i	$0.764891\pi$
$702$	0	0
$703$	−10.2615	−0.387018
$704$	−7.42708	−0.279918
$705$	0	0
$706$	22.7811	0.857377
$707$	−4.05448	−0.152485
$708$	0	0
$709$	25.5912	0.961098	0.480549	−	0.876968i	$-0.340437\pi$
$709$	25.5912	0.961098	0.480549	+	0.876968i	$0.340437\pi$
$710$	3.71354	0.139367
$711$	0	0
$712$	27.6755	1.03718
$713$	−25.0728	−0.938982
$714$	0	0
$715$	−8.51030	−0.318267
$716$	−78.6949	−2.94097
$717$	0	0
$718$	−19.2682	−0.719082
$719$	−22.8851	−0.853471	−0.426736	−	0.904376i	$-0.640337\pi$
$719$	−22.8851	−0.853471	−0.426736	+	0.904376i	$0.640337\pi$
$720$	0	0
$721$	8.01047	0.298325
$722$	−20.4230	−0.760066
$723$	0	0
$724$	14.5625	0.541209
$725$	−10.4909	−0.389621
$726$	0	0
$727$	42.7080	1.58395	0.791975	−	0.610553i	$-0.209053\pi$
$727$	42.7080	1.58395	0.791975	+	0.610553i	$0.209053\pi$
$728$	31.3435	1.16167
$729$	0	0
$730$	5.53638	0.204911
$731$	5.61727	0.207762
$732$	0	0
$733$	−28.9452	−1.06911	−0.534557	−	0.845132i	$-0.679522\pi$
$733$	−28.9452	−1.06911	−0.534557	+	0.845132i	$0.679522\pi$
$734$	−58.0388	−2.14225
$735$	0	0
$736$	−18.0000	−0.663489
$737$	−19.9269	−0.734017
$738$	0	0
$739$	−45.4890	−1.67334	−0.836671	−	0.547707i	$-0.815501\pi$
$739$	−45.4890	−1.67334	−0.836671	+	0.547707i	$0.815501\pi$
$740$	19.5714	0.719460
$741$	0	0
$742$	−17.7083	−0.650092
$743$	29.9895	1.10021	0.550105	−	0.835096i	$-0.314588\pi$
$743$	29.9895	1.10021	0.550105	+	0.835096i	$0.314588\pi$
$744$	0	0
$745$	7.61727	0.279075
$746$	16.8489	0.616883
$747$	0	0
$748$	−8.56246	−0.313074
$749$	19.9269	0.728113
$750$	0	0
$751$	9.55465	0.348654	0.174327	−	0.984688i	$-0.444225\pi$
$751$	9.55465	0.348654	0.174327	+	0.984688i	$0.444225\pi$
$752$	−12.6900	−0.462758
$753$	0	0
$754$	37.6352	1.37059
$755$	11.4047	0.415061
$756$	0	0
$757$	−12.2812	−0.446369	−0.223184	−	0.974776i	$-0.571645\pi$
$757$	−12.2812	−0.446369	−0.223184	+	0.974776i	$0.571645\pi$
$758$	17.9452	0.651798
$759$	0	0
$760$	−26.9034	−0.975889
$761$	14.8542	0.538463	0.269231	−	0.963076i	$-0.413230\pi$
$761$	14.8542	0.538463	0.269231	+	0.963076i	$0.413230\pi$
$762$	0	0
$763$	−8.19253	−0.296589
$764$	103.943	3.76054
$765$	0	0
$766$	68.4528	2.47330
$767$	26.2917	0.949338
$768$	0	0
$769$	12.6640	0.456674	0.228337	−	0.973582i	$-0.426671\pi$
$769$	12.6640	0.456674	0.228337	+	0.973582i	$0.426671\pi$
$770$	−7.56279	−0.272544
$771$	0	0
$772$	108.317	3.89842
$773$	35.3174	1.27028	0.635140	−	0.772397i	$-0.280942\pi$
$773$	35.3174	1.27028	0.635140	+	0.772397i	$0.280942\pi$
$774$	0	0
$775$	−8.28123	−0.297470
$776$	30.4998	1.09488
$777$	0	0
$778$	43.7262	1.56766
$779$	−17.8359	−0.639037
$780$	0	0
$781$	−1.42708	−0.0510648
$782$	29.8904	1.06888
$783$	0	0
$784$	−26.1484	−0.933872
$785$	15.1861	0.542016
$786$	0	0
$787$	−37.5401	−1.33816	−0.669080	−	0.743190i	$-0.733312\pi$
$787$	−37.5401	−1.33816	−0.669080	+	0.743190i	$0.733312\pi$
$788$	−41.8904	−1.49228
$789$	0	0
$790$	−52.0907	−1.85330
$791$	10.9631	0.389804
$792$	0	0
$793$	24.5103	0.870386
$794$	26.5625	0.942666
$795$	0	0
$796$	−78.5256	−2.78327
$797$	−6.37783	−0.225914	−0.112957	−	0.993600i	$-0.536032\pi$
$797$	−6.37783	−0.225914	−0.112957	+	0.993600i	$0.536032\pi$
$798$	0	0
$799$	3.43754	0.121612
$800$	−5.94518	−0.210194
$801$	0	0
$802$	55.6613	1.96547
$803$	−2.12758	−0.0750805
$804$	0	0
$805$	17.8904	0.630552
$806$	29.7083	1.04643
$807$	0	0
$808$	−15.6002	−0.548812
$809$	10.2812	0.361469	0.180734	−	0.983532i	$-0.442153\pi$
$809$	10.2812	0.361469	0.180734	+	0.983532i	$0.442153\pi$
$810$	0	0
$811$	5.38936	0.189246	0.0946231	−	0.995513i	$-0.469835\pi$
$811$	5.38936	0.189246	0.0946231	+	0.995513i	$0.469835\pi$
$812$	22.6640	0.795349
$813$	0	0
$814$	−11.0988	−0.389014
$815$	−18.6900	−0.654683
$816$	0	0
$817$	−12.9362	−0.452581
$818$	78.0437	2.72873
$819$	0	0
$820$	34.0179	1.18796
$821$	−36.2682	−1.26577	−0.632884	−	0.774246i	$-0.718129\pi$
$821$	−36.2682	−1.26577	−0.632884	+	0.774246i	$0.718129\pi$
$822$	0	0
$823$	0.153652	0.00535598	0.00267799	−	0.999996i	$-0.499148\pi$
$823$	0.153652	0.00535598	0.00267799	+	0.999996i	$0.499148\pi$
$824$	30.8213	1.07371
$825$	0	0
$826$	23.3645	0.812953
$827$	11.2995	0.392922	0.196461	−	0.980512i	$-0.437055\pi$
$827$	11.2995	0.392922	0.196461	+	0.980512i	$0.437055\pi$
$828$	0	0
$829$	−24.5494	−0.852636	−0.426318	−	0.904573i	$-0.640190\pi$
$829$	−24.5494	−0.852636	−0.426318	+	0.904573i	$0.640190\pi$
$830$	27.0586	0.939216
$831$	0	0
$832$	−20.8176	−0.721721
$833$	7.08323	0.245419
$834$	0	0
$835$	29.9124	1.03516
$836$	19.7188	0.681988
$837$	0	0
$838$	−0.872425	−0.0301374
$839$	−22.8851	−0.790082	−0.395041	−	0.918663i	$-0.629270\pi$
$839$	−22.8851	−0.790082	−0.395041	+	0.918663i	$0.629270\pi$
$840$	0	0
$841$	−14.7318	−0.507993
$842$	80.4528	2.77259
$843$	0	0
$844$	−28.6640	−0.986654
$845$	−4.47259	−0.153862
$846$	0	0
$847$	−12.7915	−0.439522
$848$	−26.2447	−0.901246
$849$	0	0
$850$	9.87242	0.338621
$851$	26.2552	0.900015
$852$	0	0
$853$	−48.1936	−1.65012	−0.825059	−	0.565047i	$-0.808858\pi$
$853$	−48.1936	−1.65012	−0.825059	+	0.565047i	$0.808858\pi$
$854$	21.7814	0.745344
$855$	0	0
$856$	76.6714	2.62058
$857$	51.4491	1.75747	0.878734	−	0.477313i	$-0.158389\pi$
$857$	51.4491	1.75747	0.878734	+	0.477313i	$0.158389\pi$
$858$	0	0
$859$	24.8281	0.847123	0.423561	−	0.905867i	$-0.360780\pi$
$859$	24.8281	0.847123	0.423561	+	0.905867i	$0.360780\pi$
$860$	24.6729	0.841340
$861$	0	0
$862$	17.3320	0.590329
$863$	−48.3462	−1.64572	−0.822862	−	0.568242i	$-0.807624\pi$
$863$	−48.3462	−1.64572	−0.822862	+	0.568242i	$0.807624\pi$
$864$	0	0
$865$	−28.6640	−0.974604
$866$	96.4163	3.27636
$867$	0	0
$868$	17.8904	0.607239
$869$	20.0179	0.679062
$870$	0	0
$871$	−55.8538	−1.89253
$872$	−31.5218	−1.06746
$873$	0	0
$874$	−68.8355	−2.32840
$875$	16.5468	0.559386
$876$	0	0
$877$	39.7826	1.34336	0.671681	−	0.740841i	$-0.265573\pi$
$877$	39.7826	1.34336	0.671681	+	0.740841i	$0.265573\pi$
$878$	−61.4995	−2.07551
$879$	0	0
$880$	−11.2085	−0.377837
$881$	−2.38156	−0.0802368	−0.0401184	−	0.999195i	$-0.512774\pi$
$881$	−2.38156	−0.0802368	−0.0401184	+	0.999195i	$0.512774\pi$
$882$	0	0
$883$	50.4528	1.69787	0.848936	−	0.528495i	$-0.177244\pi$
$883$	50.4528	1.69787	0.848936	+	0.528495i	$0.177244\pi$
$884$	−24.0000	−0.807207
$885$	0	0
$886$	24.0261	0.807172
$887$	−0.583393	−0.0195884	−0.00979421	−	0.999952i	$-0.503118\pi$
$887$	−0.583393	−0.0195884	−0.00979421	+	0.999952i	$0.503118\pi$
$888$	0	0
$889$	−24.4163	−0.818896
$890$	−18.7173	−0.627404
$891$	0	0
$892$	37.7080	1.26256
$893$	−7.91644	−0.264914
$894$	0	0
$895$	27.9049	0.932758
$896$	−24.6095	−0.822145
$897$	0	0
$898$	−13.4271	−0.448067
$899$	11.2630	0.375641
$900$	0	0
$901$	7.10930	0.236845
$902$	−19.2914	−0.642332
$903$	0	0
$904$	42.1821	1.40295
$905$	−5.16379	−0.171650
$906$	0	0
$907$	31.8463	1.05744	0.528720	−	0.848796i	$-0.322672\pi$
$907$	31.8463	1.05744	0.528720	+	0.848796i	$0.322672\pi$
$908$	−21.5912	−0.716529
$909$	0	0
$910$	−21.1980	−0.702707
$911$	55.2339	1.82998	0.914990	−	0.403476i	$-0.132198\pi$
$911$	55.2339	1.82998	0.914990	+	0.403476i	$0.132198\pi$
$912$	0	0
$913$	−10.3983	−0.344135
$914$	−71.7158	−2.37215
$915$	0	0
$916$	34.5494	1.14154
$917$	15.7628	0.520533
$918$	0	0
$919$	−22.6640	−0.747615	−0.373807	−	0.927506i	$-0.621948\pi$
$919$	−22.6640	−0.747615	−0.373807	+	0.927506i	$0.621948\pi$
$920$	68.8355	2.26944
$921$	0	0
$922$	35.7993	1.17899
$923$	−4.00000	−0.131662
$924$	0	0
$925$	8.67176	0.285126
$926$	−26.0765	−0.856927
$927$	0	0
$928$	8.08580	0.265429
$929$	−54.9291	−1.80217	−0.901083	−	0.433646i	$-0.857227\pi$
$929$	−54.9291	−1.80217	−0.901083	+	0.433646i	$0.857227\pi$
$930$	0	0
$931$	−16.3122	−0.534611
$932$	22.8724	0.749211
$933$	0	0
$934$	−16.7811	−0.549093
$935$	3.03621	0.0992947
$936$	0	0
$937$	15.1354	0.494451	0.247226	−	0.968958i	$-0.420481\pi$
$937$	15.1354	0.494451	0.247226	+	0.968958i	$0.420481\pi$
$938$	−49.6352	−1.62065
$939$	0	0
$940$	15.0988	0.492470
$941$	−50.2577	−1.63835	−0.819177	−	0.573540i	$-0.805570\pi$
$941$	−50.2577	−1.63835	−0.819177	+	0.573540i	$0.805570\pi$
$942$	0	0
$943$	45.6352	1.48609
$944$	34.6274	1.12703
$945$	0	0
$946$	−13.9919	−0.454915
$947$	−13.2096	−0.429255	−0.214628	−	0.976696i	$-0.568854\pi$
$947$	−13.2096	−0.429255	−0.214628	+	0.976696i	$0.568854\pi$
$948$	0	0
$949$	−5.96345	−0.193582
$950$	−22.7355	−0.737639
$951$	0	0
$952$	−11.1824	−0.362423
$953$	27.7826	0.899965	0.449983	−	0.893037i	$-0.351430\pi$
$953$	27.7826	0.899965	0.449983	+	0.893037i	$0.351430\pi$
$954$	0	0
$955$	−36.8579	−1.19269
$956$	19.3801	0.626796
$957$	0	0
$958$	−56.4528	−1.82391
$959$	30.7080	0.991612
$960$	0	0
$961$	−22.1093	−0.713203
$962$	−31.1093	−1.00300
$963$	0	0
$964$	111.033	3.57614
$965$	−38.4088	−1.23642
$966$	0	0
$967$	−0.518440	−0.0166719	−0.00833595	−	0.999965i	$-0.502653\pi$
$967$	−0.518440	−0.0166719	−0.00833595	+	0.999965i	$0.502653\pi$
$968$	−49.2171	−1.58190
$969$	0	0
$970$	−20.6274	−0.662306
$971$	7.59867	0.243853	0.121926	−	0.992539i	$-0.461093\pi$
$971$	7.59867	0.243853	0.121926	+	0.992539i	$0.461093\pi$
$972$	0	0
$973$	20.5625	0.659202
$974$	−49.9530	−1.60060
$975$	0	0
$976$	32.2812	1.03330
$977$	−4.52184	−0.144667	−0.0723333	−	0.997381i	$-0.523045\pi$
$977$	−4.52184	−0.144667	−0.0723333	+	0.997381i	$0.523045\pi$
$978$	0	0
$979$	7.19286	0.229885
$980$	31.1119	0.993832
$981$	0	0
$982$	52.1716	1.66486
$983$	47.7027	1.52148	0.760740	−	0.649056i	$-0.224836\pi$
$983$	47.7027	1.52148	0.760740	+	0.649056i	$0.224836\pi$
$984$	0	0
$985$	14.8542	0.473293
$986$	−13.4271	−0.427605
$987$	0	0
$988$	55.2704	1.75839
$989$	33.0988	1.05248
$990$	0	0
$991$	−36.5364	−1.16062	−0.580308	−	0.814397i	$-0.697068\pi$
$991$	−36.5364	−1.16062	−0.580308	+	0.814397i	$0.697068\pi$
$992$	6.38273	0.202652
$993$	0	0
$994$	−3.55465	−0.112747
$995$	27.8448	0.882741
$996$	0	0
$997$	−33.7251	−1.06808	−0.534042	−	0.845458i	$-0.679328\pi$
$997$	−33.7251	−1.06808	−0.534042	+	0.845458i	$0.679328\pi$
$998$	−38.5076	−1.21894
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.g.1.3		3
3.2	odd	2		71.2.a.b.1.1	✓	3
12.11	even	2		1136.2.a.i.1.3		3
15.14	odd	2		1775.2.a.e.1.3		3
21.20	even	2		3479.2.a.l.1.1		3
24.5	odd	2		4544.2.a.w.1.3		3
24.11	even	2		4544.2.a.t.1.1		3
33.32	even	2		8591.2.a.f.1.3		3
213.212	even	2		5041.2.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.a.b.1.1	✓	3	3.2	odd	2
639.2.a.g.1.3		3	1.1	even	1	trivial
1136.2.a.i.1.3		3	12.11	even	2
1775.2.a.e.1.3		3	15.14	odd	2
3479.2.a.l.1.1		3	21.20	even	2
4544.2.a.t.1.1		3	24.11	even	2
4544.2.a.w.1.3		3	24.5	odd	2
5041.2.a.b.1.1		3	213.212	even	2
8591.2.a.f.1.3		3	33.32	even	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)

\(f(q)\)	\(=\)	\(q+2.49086 q^{2} +4.20440 q^{4} -1.49086 q^{5} +1.42708 q^{7} +5.49086 q^{8} +O(q^{10})\)	\(q+2.49086 q^{2} +4.20440 q^{4} -1.49086 q^{5} +1.42708 q^{7} +5.49086 q^{8} -3.71354 q^{10} +1.42708 q^{11} +4.00000 q^{13} +3.55465 q^{14} +5.26819 q^{16} -1.42708 q^{17} +3.28646 q^{19} -6.26819 q^{20} +3.55465 q^{22} -8.40880 q^{23} -2.77733 q^{25} +9.96345 q^{26} +6.00000 q^{28} +3.77733 q^{29} +2.98173 q^{31} +2.14061 q^{32} -3.55465 q^{34} -2.12758 q^{35} -3.12234 q^{37} +8.18613 q^{38} -8.18613 q^{40} -5.42708 q^{41} -3.93621 q^{43} +6.00000 q^{44} -20.9452 q^{46} -2.40880 q^{47} -4.96345 q^{49} -6.91794 q^{50} +16.8176 q^{52} -4.98173 q^{53} -2.12758 q^{55} +7.83588 q^{56} +9.40880 q^{58} +6.57292 q^{59} +6.12758 q^{61} +7.42708 q^{62} -5.20440 q^{64} -5.96345 q^{65} -13.9635 q^{67} -6.00000 q^{68} -5.29950 q^{70} -1.00000 q^{71} -1.49086 q^{73} -7.77733 q^{74} +13.8176 q^{76} +2.03655 q^{77} +14.0272 q^{79} -7.85415 q^{80} -13.5181 q^{82} -7.28646 q^{83} +2.12758 q^{85} -9.80457 q^{86} +7.83588 q^{88} +5.04028 q^{89} +5.70830 q^{91} -35.3540 q^{92} -6.00000 q^{94} -4.89967 q^{95} +5.55465 q^{97} -12.3633 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 4 q^{4} + 3 q^{5} + 2 q^{7} + 9 q^{8}+O(q^{10}) \) 3 * q + 4 * q^4 + 3 * q^5 + 2 * q^7 + 9 * q^8	\( 3 q + 4 q^{4} + 3 q^{5} + 2 q^{7} + 9 q^{8} - 10 q^{10} + 2 q^{11} + 12 q^{13} - 2 q^{14} + 2 q^{16} - 2 q^{17} + 11 q^{19} - 5 q^{20} - 2 q^{22} - 8 q^{23} - 2 q^{25} + 18 q^{28} + 5 q^{29} - 6 q^{31} + 3 q^{32} + 2 q^{34} + 4 q^{35} + 9 q^{37} + q^{38} - q^{40} - 14 q^{41} - 17 q^{43} + 18 q^{44} - 18 q^{46} + 10 q^{47} + 15 q^{49} - 11 q^{50} + 16 q^{52} + 4 q^{55} + 4 q^{56} + 11 q^{58} + 22 q^{59} + 8 q^{61} + 20 q^{62} - 7 q^{64} + 12 q^{65} - 12 q^{67} - 18 q^{68} - 24 q^{70} - 3 q^{71} + 3 q^{73} - 17 q^{74} + 7 q^{76} + 36 q^{77} + 7 q^{79} - 19 q^{80} + 2 q^{82} - 23 q^{83} - 4 q^{85} + 12 q^{86} + 4 q^{88} - 13 q^{89} + 8 q^{91} - 44 q^{92} - 18 q^{94} + 10 q^{95} + 4 q^{97} - 40 q^{98}+O(q^{100}) \) 3 * q + 4 * q^4 + 3 * q^5 + 2 * q^7 + 9 * q^8 - 10 * q^10 + 2 * q^11 + 12 * q^13 - 2 * q^14 + 2 * q^16 - 2 * q^17 + 11 * q^19 - 5 * q^20 - 2 * q^22 - 8 * q^23 - 2 * q^25 + 18 * q^28 + 5 * q^29 - 6 * q^31 + 3 * q^32 + 2 * q^34 + 4 * q^35 + 9 * q^37 + q^38 - q^40 - 14 * q^41 - 17 * q^43 + 18 * q^44 - 18 * q^46 + 10 * q^47 + 15 * q^49 - 11 * q^50 + 16 * q^52 + 4 * q^55 + 4 * q^56 + 11 * q^58 + 22 * q^59 + 8 * q^61 + 20 * q^62 - 7 * q^64 + 12 * q^65 - 12 * q^67 - 18 * q^68 - 24 * q^70 - 3 * q^71 + 3 * q^73 - 17 * q^74 + 7 * q^76 + 36 * q^77 + 7 * q^79 - 19 * q^80 + 2 * q^82 - 23 * q^83 - 4 * q^85 + 12 * q^86 + 4 * q^88 - 13 * q^89 + 8 * q^91 - 44 * q^92 - 18 * q^94 + 10 * q^95 + 4 * q^97 - 40 * q^98

Introduction

L-functions

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