LMFDB - Embedded newform 639.2.a.c.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 213)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.30278$ 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)

$f(q)$	$=$	$q+1.30278 q^{2} -0.302776 q^{4} -1.30278 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})$	\(q+1.30278 q^{2} -0.302776 q^{4} -1.30278 q^{5} -1.00000 q^{7} -3.00000 q^{8} -1.69722 q^{10} -3.00000 q^{11} +0.302776 q^{13} -1.30278 q^{14} -3.30278 q^{16} -3.00000 q^{17} +1.60555 q^{19} +0.394449 q^{20} -3.90833 q^{22} -6.90833 q^{23} -3.30278 q^{25} +0.394449 q^{26} +0.302776 q^{28} -1.69722 q^{29} +2.00000 q^{31} +1.69722 q^{32} -3.90833 q^{34} +1.30278 q^{35} -2.30278 q^{37} +2.09167 q^{38} +3.90833 q^{40} +3.90833 q^{41} +1.09167 q^{43} +0.908327 q^{44} -9.00000 q^{46} +9.90833 q^{47} -6.00000 q^{49} -4.30278 q^{50} -0.0916731 q^{52} +6.51388 q^{53} +3.90833 q^{55} +3.00000 q^{56} -2.21110 q^{58} +3.00000 q^{59} -15.6056 q^{61} +2.60555 q^{62} +8.81665 q^{64} -0.394449 q^{65} +1.09167 q^{67} +0.908327 q^{68} +1.69722 q^{70} +1.00000 q^{71} +9.81665 q^{73} -3.00000 q^{74} -0.486122 q^{76} +3.00000 q^{77} -2.69722 q^{79} +4.30278 q^{80} +5.09167 q^{82} -6.39445 q^{83} +3.90833 q^{85} +1.42221 q^{86} +9.00000 q^{88} -3.00000 q^{89} -0.302776 q^{91} +2.09167 q^{92} +12.9083 q^{94} -2.09167 q^{95} -4.51388 q^{97} -7.81665 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} + q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - q^2 + 3 * q^4 + q^5 - 2 * q^7 - 6 * q^8	$ 2 q - q^{2} + 3 q^{4} + q^{5} - 2 q^{7} - 6 q^{8} - 7 q^{10} - 6 q^{11} - 3 q^{13} + q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} + 8 q^{20} + 3 q^{22} - 3 q^{23} - 3 q^{25} + 8 q^{26} - 3 q^{28} - 7 q^{29} + 4 q^{31} + 7 q^{32} + 3 q^{34} - q^{35} - q^{37} + 15 q^{38} - 3 q^{40} - 3 q^{41} + 13 q^{43} - 9 q^{44} - 18 q^{46} + 9 q^{47} - 12 q^{49} - 5 q^{50} - 11 q^{52} - 5 q^{53} - 3 q^{55} + 6 q^{56} + 10 q^{58} + 6 q^{59} - 24 q^{61} - 2 q^{62} - 4 q^{64} - 8 q^{65} + 13 q^{67} - 9 q^{68} + 7 q^{70} + 2 q^{71} - 2 q^{73} - 6 q^{74} - 19 q^{76} + 6 q^{77} - 9 q^{79} + 5 q^{80} + 21 q^{82} - 20 q^{83} - 3 q^{85} - 26 q^{86} + 18 q^{88} - 6 q^{89} + 3 q^{91} + 15 q^{92} + 15 q^{94} - 15 q^{95} + 9 q^{97} + 6 q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 + q^5 - 2 * q^7 - 6 * q^8 - 7 * q^10 - 6 * q^11 - 3 * q^13 + q^14 - 3 * q^16 - 6 * q^17 - 4 * q^19 + 8 * q^20 + 3 * q^22 - 3 * q^23 - 3 * q^25 + 8 * q^26 - 3 * q^28 - 7 * q^29 + 4 * q^31 + 7 * q^32 + 3 * q^34 - q^35 - q^37 + 15 * q^38 - 3 * q^40 - 3 * q^41 + 13 * q^43 - 9 * q^44 - 18 * q^46 + 9 * q^47 - 12 * q^49 - 5 * q^50 - 11 * q^52 - 5 * q^53 - 3 * q^55 + 6 * q^56 + 10 * q^58 + 6 * q^59 - 24 * q^61 - 2 * q^62 - 4 * q^64 - 8 * q^65 + 13 * q^67 - 9 * q^68 + 7 * q^70 + 2 * q^71 - 2 * q^73 - 6 * q^74 - 19 * q^76 + 6 * q^77 - 9 * q^79 + 5 * q^80 + 21 * q^82 - 20 * q^83 - 3 * q^85 - 26 * q^86 + 18 * q^88 - 6 * q^89 + 3 * q^91 + 15 * q^92 + 15 * q^94 - 15 * q^95 + 9 * q^97 + 6 * 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 $

$2$	1.30278	0.921201	0.460601	−	0.887607i	$-0.347634\pi$
$2$	1.30278	0.921201	0.460601	+	0.887607i	$0.347634\pi$
$3$	0	0
$3$	0	0
$4$	−0.302776	−0.151388
$5$	−1.30278	−0.582619	−0.291309	−	0.956629i	$-0.594091\pi$
$5$	−1.30278	−0.582619	−0.291309	+	0.956629i	$0.594091\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−1.69722	−0.536709
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	0.302776	0.0839749	0.0419874	−	0.999118i	$-0.486631\pi$
$13$	0.302776	0.0839749	0.0419874	+	0.999118i	$0.486631\pi$
$14$	−1.30278	−0.348181
$15$	0	0
$16$	−3.30278	−0.825694
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	1.60555	0.368339	0.184169	−	0.982895i	$-0.441041\pi$
$19$	1.60555	0.368339	0.184169	+	0.982895i	$0.441041\pi$
$20$	0.394449	0.0882014
$21$	0	0
$22$	−3.90833	−0.833258
$23$	−6.90833	−1.44049	−0.720243	−	0.693722i	$-0.755970\pi$
$23$	−6.90833	−1.44049	−0.720243	+	0.693722i	$0.755970\pi$
$24$	0	0
$25$	−3.30278	−0.660555
$26$	0.394449	0.0773578
$27$	0	0
$28$	0.302776	0.0572192
$29$	−1.69722	−0.315167	−0.157583	−	0.987506i	$-0.550370\pi$
$29$	−1.69722	−0.315167	−0.157583	+	0.987506i	$0.550370\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.69722	0.300030
$33$	0	0
$34$	−3.90833	−0.670273
$35$	1.30278	0.220209
$36$	0	0
$37$	−2.30278	−0.378574	−0.189287	−	0.981922i	$-0.560618\pi$
$37$	−2.30278	−0.378574	−0.189287	+	0.981922i	$0.560618\pi$
$38$	2.09167	0.339314
$39$	0	0
$40$	3.90833	0.617961
$41$	3.90833	0.610378	0.305189	−	0.952292i	$-0.401280\pi$
$41$	3.90833	0.610378	0.305189	+	0.952292i	$0.401280\pi$
$42$	0	0
$43$	1.09167	0.166479	0.0832393	−	0.996530i	$-0.473473\pi$
$43$	1.09167	0.166479	0.0832393	+	0.996530i	$0.473473\pi$
$44$	0.908327	0.136935
$45$	0	0
$46$	−9.00000	−1.32698
$47$	9.90833	1.44528	0.722639	−	0.691226i	$-0.242929\pi$
$47$	9.90833	1.44528	0.722639	+	0.691226i	$0.242929\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	−4.30278	−0.608504
$51$	0	0
$52$	−0.0916731	−0.0127128
$53$	6.51388	0.894750	0.447375	−	0.894346i	$-0.352359\pi$
$53$	6.51388	0.894750	0.447375	+	0.894346i	$0.352359\pi$
$54$	0	0
$55$	3.90833	0.526999
$56$	3.00000	0.400892
$57$	0	0
$58$	−2.21110	−0.290332
$59$	3.00000	0.390567	0.195283	−	0.980747i	$-0.437437\pi$
$59$	3.00000	0.390567	0.195283	+	0.980747i	$0.437437\pi$
$60$	0	0
$61$	−15.6056	−1.99809	−0.999043	−	0.0437377i	$-0.986073\pi$
$61$	−15.6056	−1.99809	−0.999043	+	0.0437377i	$0.986073\pi$
$62$	2.60555	0.330905
$63$	0	0
$64$	8.81665	1.10208
$65$	−0.394449	−0.0489253
$66$	0	0
$67$	1.09167	0.133369	0.0666845	−	0.997774i	$-0.478758\pi$
$67$	1.09167	0.133369	0.0666845	+	0.997774i	$0.478758\pi$
$68$	0.908327	0.110151
$69$	0	0
$70$	1.69722	0.202857
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	0	0
$73$	9.81665	1.14895	0.574476	−	0.818521i	$-0.305206\pi$
$73$	9.81665	1.14895	0.574476	+	0.818521i	$0.305206\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	−0.486122	−0.0557620
$77$	3.00000	0.341882
$78$	0	0
$79$	−2.69722	−0.303461	−0.151731	−	0.988422i	$-0.548485\pi$
$79$	−2.69722	−0.303461	−0.151731	+	0.988422i	$0.548485\pi$
$80$	4.30278	0.481065
$81$	0	0
$82$	5.09167	0.562281
$83$	−6.39445	−0.701882	−0.350941	−	0.936398i	$-0.614138\pi$
$83$	−6.39445	−0.701882	−0.350941	+	0.936398i	$0.614138\pi$
$84$	0	0
$85$	3.90833	0.423918
$86$	1.42221	0.153360
$87$	0	0
$88$	9.00000	0.959403
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	−0.302776	−0.0317395
$92$	2.09167	0.218072
$93$	0	0
$94$	12.9083	1.33139
$95$	−2.09167	−0.214601
$96$	0	0
$97$	−4.51388	−0.458315	−0.229157	−	0.973389i	$-0.573597\pi$
$97$	−4.51388	−0.458315	−0.229157	+	0.973389i	$0.573597\pi$
$98$	−7.81665	−0.789601
$99$	0	0
$100$	1.00000	0.100000
$101$	6.39445	0.636271	0.318136	−	0.948045i	$-0.396943\pi$
$101$	6.39445	0.636271	0.318136	+	0.948045i	$0.396943\pi$
$102$	0	0
$103$	−1.78890	−0.176265	−0.0881327	−	0.996109i	$-0.528090\pi$
$103$	−1.78890	−0.176265	−0.0881327	+	0.996109i	$0.528090\pi$
$104$	−0.908327	−0.0890688
$105$	0	0
$106$	8.48612	0.824245
$107$	−0.908327	−0.0878113	−0.0439056	−	0.999036i	$-0.513980\pi$
$107$	−0.908327	−0.0878113	−0.0439056	+	0.999036i	$0.513980\pi$
$108$	0	0
$109$	4.21110	0.403350	0.201675	−	0.979452i	$-0.435361\pi$
$109$	4.21110	0.403350	0.201675	+	0.979452i	$0.435361\pi$
$110$	5.09167	0.485472
$111$	0	0
$112$	3.30278	0.312083
$113$	−19.0278	−1.78998	−0.894990	−	0.446085i	$-0.852818\pi$
$113$	−19.0278	−1.78998	−0.894990	+	0.446085i	$0.852818\pi$
$114$	0	0
$115$	9.00000	0.839254
$116$	0.513878	0.0477124
$117$	0	0
$118$	3.90833	0.359791
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−20.3305	−1.84064
$123$	0	0
$124$	−0.605551	−0.0543801
$125$	10.8167	0.967471
$126$	0	0
$127$	19.6056	1.73971	0.869856	−	0.493306i	$-0.164212\pi$
$127$	19.6056	1.73971	0.869856	+	0.493306i	$0.164212\pi$
$128$	8.09167	0.715210
$129$	0	0
$130$	−0.513878	−0.0450701
$131$	−17.2111	−1.50374	−0.751871	−	0.659311i	$-0.770848\pi$
$131$	−17.2111	−1.50374	−0.751871	+	0.659311i	$0.770848\pi$
$132$	0	0
$133$	−1.60555	−0.139219
$134$	1.42221	0.122860
$135$	0	0
$136$	9.00000	0.771744
$137$	7.81665	0.667822	0.333911	−	0.942605i	$-0.391632\pi$
$137$	7.81665	0.667822	0.333911	+	0.942605i	$0.391632\pi$
$138$	0	0
$139$	−1.90833	−0.161862	−0.0809311	−	0.996720i	$-0.525789\pi$
$139$	−1.90833	−0.161862	−0.0809311	+	0.996720i	$0.525789\pi$
$140$	−0.394449	−0.0333370
$141$	0	0
$142$	1.30278	0.109327
$143$	−0.908327	−0.0759581
$144$	0	0
$145$	2.21110	0.183622
$146$	12.7889	1.05842
$147$	0	0
$148$	0.697224	0.0573115
$149$	6.90833	0.565952	0.282976	−	0.959127i	$-0.408678\pi$
$149$	6.90833	0.565952	0.282976	+	0.959127i	$0.408678\pi$
$150$	0	0
$151$	−21.7250	−1.76795	−0.883977	−	0.467531i	$-0.845144\pi$
$151$	−21.7250	−1.76795	−0.883977	+	0.467531i	$0.845144\pi$
$152$	−4.81665	−0.390682
$153$	0	0
$154$	3.90833	0.314942
$155$	−2.60555	−0.209283
$156$	0	0
$157$	−23.8167	−1.90078	−0.950388	−	0.311067i	$-0.899314\pi$
$157$	−23.8167	−1.90078	−0.950388	+	0.311067i	$0.899314\pi$
$158$	−3.51388	−0.279549
$159$	0	0
$160$	−2.21110	−0.174803
$161$	6.90833	0.544452
$162$	0	0
$163$	−6.21110	−0.486491	−0.243246	−	0.969965i	$-0.578212\pi$
$163$	−6.21110	−0.486491	−0.243246	+	0.969965i	$0.578212\pi$
$164$	−1.18335	−0.0924038
$165$	0	0
$166$	−8.33053	−0.646575
$167$	−6.39445	−0.494817	−0.247409	−	0.968911i	$-0.579579\pi$
$167$	−6.39445	−0.494817	−0.247409	+	0.968911i	$0.579579\pi$
$168$	0	0
$169$	−12.9083	−0.992948
$170$	5.09167	0.390513
$171$	0	0
$172$	−0.330532	−0.0252028
$173$	−4.81665	−0.366203	−0.183102	−	0.983094i	$-0.558614\pi$
$173$	−4.81665	−0.366203	−0.183102	+	0.983094i	$0.558614\pi$
$174$	0	0
$175$	3.30278	0.249666
$176$	9.90833	0.746868
$177$	0	0
$178$	−3.90833	−0.292941
$179$	24.9083	1.86174	0.930868	−	0.365356i	$-0.119053\pi$
$179$	24.9083	1.86174	0.930868	+	0.365356i	$0.119053\pi$
$180$	0	0
$181$	−8.69722	−0.646460	−0.323230	−	0.946321i	$-0.604769\pi$
$181$	−8.69722	−0.646460	−0.323230	+	0.946321i	$0.604769\pi$
$182$	−0.394449	−0.0292385
$183$	0	0
$184$	20.7250	1.52787
$185$	3.00000	0.220564
$186$	0	0
$187$	9.00000	0.658145
$188$	−3.00000	−0.218797
$189$	0	0
$190$	−2.72498	−0.197691
$191$	3.00000	0.217072	0.108536	−	0.994092i	$-0.465384\pi$
$191$	3.00000	0.217072	0.108536	+	0.994092i	$0.465384\pi$
$192$	0	0
$193$	9.42221	0.678225	0.339113	−	0.940746i	$-0.389873\pi$
$193$	9.42221	0.678225	0.339113	+	0.940746i	$0.389873\pi$
$194$	−5.88057	−0.422200
$195$	0	0
$196$	1.81665	0.129761
$197$	−20.7250	−1.47659	−0.738297	−	0.674476i	$-0.764370\pi$
$197$	−20.7250	−1.47659	−0.738297	+	0.674476i	$0.764370\pi$
$198$	0	0
$199$	15.0278	1.06529	0.532645	−	0.846339i	$-0.321198\pi$
$199$	15.0278	1.06529	0.532645	+	0.846339i	$0.321198\pi$
$200$	9.90833	0.700625
$201$	0	0
$202$	8.33053	0.586134
$203$	1.69722	0.119122
$204$	0	0
$205$	−5.09167	−0.355618
$206$	−2.33053	−0.162376
$207$	0	0
$208$	−1.00000	−0.0693375
$209$	−4.81665	−0.333175
$210$	0	0
$211$	−0.605551	−0.0416879	−0.0208439	−	0.999783i	$-0.506635\pi$
$211$	−0.605551	−0.0416879	−0.0208439	+	0.999783i	$0.506635\pi$
$212$	−1.97224	−0.135454
$213$	0	0
$214$	−1.18335	−0.0808919
$215$	−1.42221	−0.0969936
$216$	0	0
$217$	−2.00000	−0.135769
$218$	5.48612	0.371567
$219$	0	0
$220$	−1.18335	−0.0797812
$221$	−0.908327	−0.0611007
$222$	0	0
$223$	15.4222	1.03275	0.516374	−	0.856363i	$-0.327282\pi$
$223$	15.4222	1.03275	0.516374	+	0.856363i	$0.327282\pi$
$224$	−1.69722	−0.113401
$225$	0	0
$226$	−24.7889	−1.64893
$227$	−9.78890	−0.649712	−0.324856	−	0.945763i	$-0.605316\pi$
$227$	−9.78890	−0.649712	−0.324856	+	0.945763i	$0.605316\pi$
$228$	0	0
$229$	−11.4222	−0.754801	−0.377400	−	0.926050i	$-0.623182\pi$
$229$	−11.4222	−0.754801	−0.377400	+	0.926050i	$0.623182\pi$
$230$	11.7250	0.773122
$231$	0	0
$232$	5.09167	0.334285
$233$	17.0917	1.11971	0.559856	−	0.828590i	$-0.310856\pi$
$233$	17.0917	1.11971	0.559856	+	0.828590i	$0.310856\pi$
$234$	0	0
$235$	−12.9083	−0.842046
$236$	−0.908327	−0.0591270
$237$	0	0
$238$	3.90833	0.253339
$239$	−9.00000	−0.582162	−0.291081	−	0.956698i	$-0.594015\pi$
$239$	−9.00000	−0.582162	−0.291081	+	0.956698i	$0.594015\pi$
$240$	0	0
$241$	−16.9083	−1.08916	−0.544581	−	0.838709i	$-0.683311\pi$
$241$	−16.9083	−1.08916	−0.544581	+	0.838709i	$0.683311\pi$
$242$	−2.60555	−0.167491
$243$	0	0
$244$	4.72498	0.302486
$245$	7.81665	0.499388
$246$	0	0
$247$	0.486122	0.0309312
$248$	−6.00000	−0.381000
$249$	0	0
$250$	14.0917	0.891236
$251$	−24.5139	−1.54730	−0.773651	−	0.633612i	$-0.781572\pi$
$251$	−24.5139	−1.54730	−0.773651	+	0.633612i	$0.781572\pi$
$252$	0	0
$253$	20.7250	1.30297
$254$	25.5416	1.60262
$255$	0	0
$256$	−7.09167	−0.443230
$257$	−16.4222	−1.02439	−0.512195	−	0.858869i	$-0.671167\pi$
$257$	−16.4222	−1.02439	−0.512195	+	0.858869i	$0.671167\pi$
$258$	0	0
$259$	2.30278	0.143088
$260$	0.119429	0.00740670
$261$	0	0
$262$	−22.4222	−1.38525
$263$	−8.09167	−0.498954	−0.249477	−	0.968381i	$-0.580259\pi$
$263$	−8.09167	−0.498954	−0.249477	+	0.968381i	$0.580259\pi$
$264$	0	0
$265$	−8.48612	−0.521298
$266$	−2.09167	−0.128249
$267$	0	0
$268$	−0.330532	−0.0201905
$269$	3.39445	0.206963	0.103482	−	0.994631i	$-0.467002\pi$
$269$	3.39445	0.206963	0.103482	+	0.994631i	$0.467002\pi$
$270$	0	0
$271$	−28.2389	−1.71539	−0.857694	−	0.514160i	$-0.828104\pi$
$271$	−28.2389	−1.71539	−0.857694	+	0.514160i	$0.828104\pi$
$272$	9.90833	0.600781
$273$	0	0
$274$	10.1833	0.615198
$275$	9.90833	0.597495
$276$	0	0
$277$	17.9083	1.07601	0.538004	−	0.842943i	$-0.319179\pi$
$277$	17.9083	1.07601	0.538004	+	0.842943i	$0.319179\pi$
$278$	−2.48612	−0.149108
$279$	0	0
$280$	−3.90833	−0.233567
$281$	4.30278	0.256682	0.128341	−	0.991730i	$-0.459035\pi$
$281$	4.30278	0.256682	0.128341	+	0.991730i	$0.459035\pi$
$282$	0	0
$283$	33.0278	1.96330	0.981648	−	0.190701i	$-0.0610761\pi$
$283$	33.0278	1.96330	0.981648	+	0.190701i	$0.0610761\pi$
$284$	−0.302776	−0.0179664
$285$	0	0
$286$	−1.18335	−0.0699727
$287$	−3.90833	−0.230701
$288$	0	0
$289$	−8.00000	−0.470588
$290$	2.88057	0.169153
$291$	0	0
$292$	−2.97224	−0.173937
$293$	15.1194	0.883287	0.441643	−	0.897191i	$-0.354396\pi$
$293$	15.1194	0.883287	0.441643	+	0.897191i	$0.354396\pi$
$294$	0	0
$295$	−3.90833	−0.227552
$296$	6.90833	0.401538
$297$	0	0
$298$	9.00000	0.521356
$299$	−2.09167	−0.120965
$300$	0	0
$301$	−1.09167	−0.0629230
$302$	−28.3028	−1.62864
$303$	0	0
$304$	−5.30278	−0.304135
$305$	20.3305	1.16412
$306$	0	0
$307$	13.7250	0.783326	0.391663	−	0.920109i	$-0.371900\pi$
$307$	13.7250	0.783326	0.391663	+	0.920109i	$0.371900\pi$
$308$	−0.908327	−0.0517567
$309$	0	0
$310$	−3.39445	−0.192792
$311$	15.5139	0.879711	0.439856	−	0.898068i	$-0.355030\pi$
$311$	15.5139	0.879711	0.439856	+	0.898068i	$0.355030\pi$
$312$	0	0
$313$	−32.4222	−1.83261	−0.916306	−	0.400480i	$-0.868844\pi$
$313$	−32.4222	−1.83261	−0.916306	+	0.400480i	$0.868844\pi$
$314$	−31.0278	−1.75100
$315$	0	0
$316$	0.816654	0.0459404
$317$	13.3028	0.747158	0.373579	−	0.927598i	$-0.378130\pi$
$317$	13.3028	0.747158	0.373579	+	0.927598i	$0.378130\pi$
$318$	0	0
$319$	5.09167	0.285079
$320$	−11.4861	−0.642094
$321$	0	0
$322$	9.00000	0.501550
$323$	−4.81665	−0.268006
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−8.09167	−0.448156
$327$	0	0
$328$	−11.7250	−0.647404
$329$	−9.90833	−0.546264
$330$	0	0
$331$	13.4861	0.741264	0.370632	−	0.928780i	$-0.379141\pi$
$331$	13.4861	0.741264	0.370632	+	0.928780i	$0.379141\pi$
$332$	1.93608	0.106256
$333$	0	0
$334$	−8.33053	−0.455826
$335$	−1.42221	−0.0777034
$336$	0	0
$337$	−27.8444	−1.51678	−0.758391	−	0.651800i	$-0.774014\pi$
$337$	−27.8444	−1.51678	−0.758391	+	0.651800i	$0.774014\pi$
$338$	−16.8167	−0.914705
$339$	0	0
$340$	−1.18335	−0.0641760
$341$	−6.00000	−0.324918
$342$	0	0
$343$	13.0000	0.701934
$344$	−3.27502	−0.176577
$345$	0	0
$346$	−6.27502	−0.337347
$347$	−4.18335	−0.224574	−0.112287	−	0.993676i	$-0.535818\pi$
$347$	−4.18335	−0.224574	−0.112287	+	0.993676i	$0.535818\pi$
$348$	0	0
$349$	16.2111	0.867760	0.433880	−	0.900971i	$-0.357144\pi$
$349$	16.2111	0.867760	0.433880	+	0.900971i	$0.357144\pi$
$350$	4.30278	0.229993
$351$	0	0
$352$	−5.09167	−0.271387
$353$	−32.7250	−1.74177	−0.870887	−	0.491482i	$-0.836455\pi$
$353$	−32.7250	−1.74177	−0.870887	+	0.491482i	$0.836455\pi$
$354$	0	0
$355$	−1.30278	−0.0691442
$356$	0.908327	0.0481412
$357$	0	0
$358$	32.4500	1.71503
$359$	16.0278	0.845913	0.422956	−	0.906150i	$-0.360992\pi$
$359$	16.0278	0.845913	0.422956	+	0.906150i	$0.360992\pi$
$360$	0	0
$361$	−16.4222	−0.864327
$362$	−11.3305	−0.595520
$363$	0	0
$364$	0.0916731	0.00480498
$365$	−12.7889	−0.669401
$366$	0	0
$367$	−21.7250	−1.13403	−0.567017	−	0.823706i	$-0.691903\pi$
$367$	−21.7250	−1.13403	−0.567017	+	0.823706i	$0.691903\pi$
$368$	22.8167	1.18940
$369$	0	0
$370$	3.90833	0.203184
$371$	−6.51388	−0.338184
$372$	0	0
$373$	−17.5416	−0.908271	−0.454136	−	0.890933i	$-0.650052\pi$
$373$	−17.5416	−0.908271	−0.454136	+	0.890933i	$0.650052\pi$
$374$	11.7250	0.606284
$375$	0	0
$376$	−29.7250	−1.53295
$377$	−0.513878	−0.0264661
$378$	0	0
$379$	6.02776	0.309625	0.154813	−	0.987944i	$-0.450523\pi$
$379$	6.02776	0.309625	0.154813	+	0.987944i	$0.450523\pi$
$380$	0.633308	0.0324880
$381$	0	0
$382$	3.90833	0.199967
$383$	1.97224	0.100777	0.0503885	−	0.998730i	$-0.483954\pi$
$383$	1.97224	0.100777	0.0503885	+	0.998730i	$0.483954\pi$
$384$	0	0
$385$	−3.90833	−0.199187
$386$	12.2750	0.624782
$387$	0	0
$388$	1.36669	0.0693833
$389$	20.8444	1.05685	0.528427	−	0.848979i	$-0.322782\pi$
$389$	20.8444	1.05685	0.528427	+	0.848979i	$0.322782\pi$
$390$	0	0
$391$	20.7250	1.04811
$392$	18.0000	0.909137
$393$	0	0
$394$	−27.0000	−1.36024
$395$	3.51388	0.176802
$396$	0	0
$397$	10.4861	0.526283	0.263142	−	0.964757i	$-0.415241\pi$
$397$	10.4861	0.526283	0.263142	+	0.964757i	$0.415241\pi$
$398$	19.5778	0.981346
$399$	0	0
$400$	10.9083	0.545416
$401$	−32.7250	−1.63421	−0.817104	−	0.576491i	$-0.804422\pi$
$401$	−32.7250	−1.63421	−0.817104	+	0.576491i	$0.804422\pi$
$402$	0	0
$403$	0.605551	0.0301647
$404$	−1.93608	−0.0963237
$405$	0	0
$406$	2.21110	0.109735
$407$	6.90833	0.342433
$408$	0	0
$409$	26.1194	1.29152	0.645761	−	0.763539i	$-0.276540\pi$
$409$	26.1194	1.29152	0.645761	+	0.763539i	$0.276540\pi$
$410$	−6.63331	−0.327596
$411$	0	0
$412$	0.541635	0.0266844
$413$	−3.00000	−0.147620
$414$	0	0
$415$	8.33053	0.408930
$416$	0.513878	0.0251950
$417$	0	0
$418$	−6.27502	−0.306921
$419$	0.908327	0.0443747	0.0221873	−	0.999754i	$-0.492937\pi$
$419$	0.908327	0.0443747	0.0221873	+	0.999754i	$0.492937\pi$
$420$	0	0
$421$	15.4222	0.751632	0.375816	−	0.926694i	$-0.377362\pi$
$421$	15.4222	0.751632	0.375816	+	0.926694i	$0.377362\pi$
$422$	−0.788897	−0.0384029
$423$	0	0
$424$	−19.5416	−0.949026
$425$	9.90833	0.480624
$426$	0	0
$427$	15.6056	0.755206
$428$	0.275019	0.0132936
$429$	0	0
$430$	−1.85281	−0.0893506
$431$	−7.42221	−0.357515	−0.178758	−	0.983893i	$-0.557208\pi$
$431$	−7.42221	−0.357515	−0.178758	+	0.983893i	$0.557208\pi$
$432$	0	0
$433$	−14.5416	−0.698826	−0.349413	−	0.936969i	$-0.613619\pi$
$433$	−14.5416	−0.698826	−0.349413	+	0.936969i	$0.613619\pi$
$434$	−2.60555	−0.125070
$435$	0	0
$436$	−1.27502	−0.0610623
$437$	−11.0917	−0.530587
$438$	0	0
$439$	−28.5139	−1.36089	−0.680447	−	0.732798i	$-0.738214\pi$
$439$	−28.5139	−1.36089	−0.680447	+	0.732798i	$0.738214\pi$
$440$	−11.7250	−0.558967
$441$	0	0
$442$	−1.18335	−0.0562860
$443$	18.5139	0.879621	0.439810	−	0.898091i	$-0.355046\pi$
$443$	18.5139	0.879621	0.439810	+	0.898091i	$0.355046\pi$
$444$	0	0
$445$	3.90833	0.185272
$446$	20.0917	0.951368
$447$	0	0
$448$	−8.81665	−0.416548
$449$	−37.0278	−1.74745	−0.873724	−	0.486422i	$-0.838302\pi$
$449$	−37.0278	−1.74745	−0.873724	+	0.486422i	$0.838302\pi$
$450$	0	0
$451$	−11.7250	−0.552108
$452$	5.76114	0.270981
$453$	0	0
$454$	−12.7527	−0.598516
$455$	0.394449	0.0184920
$456$	0	0
$457$	23.2389	1.08707	0.543534	−	0.839387i	$-0.317086\pi$
$457$	23.2389	1.08707	0.543534	+	0.839387i	$0.317086\pi$
$458$	−14.8806	−0.695323
$459$	0	0
$460$	−2.72498	−0.127053
$461$	−26.4500	−1.23190	−0.615949	−	0.787786i	$-0.711227\pi$
$461$	−26.4500	−1.23190	−0.615949	+	0.787786i	$0.711227\pi$
$462$	0	0
$463$	−29.1472	−1.35458	−0.677292	−	0.735714i	$-0.736847\pi$
$463$	−29.1472	−1.35458	−0.677292	+	0.735714i	$0.736847\pi$
$464$	5.60555	0.260231
$465$	0	0
$466$	22.2666	1.03148
$467$	28.0278	1.29697	0.648485	−	0.761227i	$-0.275403\pi$
$467$	28.0278	1.29697	0.648485	+	0.761227i	$0.275403\pi$
$468$	0	0
$469$	−1.09167	−0.0504088
$470$	−16.8167	−0.775694
$471$	0	0
$472$	−9.00000	−0.414259
$473$	−3.27502	−0.150586
$474$	0	0
$475$	−5.30278	−0.243308
$476$	−0.908327	−0.0416331
$477$	0	0
$478$	−11.7250	−0.536288
$479$	19.6972	0.899989	0.449995	−	0.893031i	$-0.351426\pi$
$479$	19.6972	0.899989	0.449995	+	0.893031i	$0.351426\pi$
$480$	0	0
$481$	−0.697224	−0.0317907
$482$	−22.0278	−1.00334
$483$	0	0
$484$	0.605551	0.0275251
$485$	5.88057	0.267023
$486$	0	0
$487$	22.3305	1.01189	0.505946	−	0.862565i	$-0.331143\pi$
$487$	22.3305	1.01189	0.505946	+	0.862565i	$0.331143\pi$
$488$	46.8167	2.11929
$489$	0	0
$490$	10.1833	0.460037
$491$	−32.0555	−1.44665	−0.723323	−	0.690510i	$-0.757386\pi$
$491$	−32.0555	−1.44665	−0.723323	+	0.690510i	$0.757386\pi$
$492$	0	0
$493$	5.09167	0.229317
$494$	0.633308	0.0284939
$495$	0	0
$496$	−6.60555	−0.296598
$497$	−1.00000	−0.0448561
$498$	0	0
$499$	33.0278	1.47853	0.739263	−	0.673417i	$-0.235174\pi$
$499$	33.0278	1.47853	0.739263	+	0.673417i	$0.235174\pi$
$500$	−3.27502	−0.146463
$501$	0	0
$502$	−31.9361	−1.42538
$503$	12.5139	0.557966	0.278983	−	0.960296i	$-0.410003\pi$
$503$	12.5139	0.557966	0.278983	+	0.960296i	$0.410003\pi$
$504$	0	0
$505$	−8.33053	−0.370704
$506$	27.0000	1.20030
$507$	0	0
$508$	−5.93608	−0.263371
$509$	7.81665	0.346467	0.173234	−	0.984881i	$-0.444578\pi$
$509$	7.81665	0.346467	0.173234	+	0.984881i	$0.444578\pi$
$510$	0	0
$511$	−9.81665	−0.434263
$512$	−25.4222	−1.12351
$513$	0	0
$514$	−21.3944	−0.943669
$515$	2.33053	0.102696
$516$	0	0
$517$	−29.7250	−1.30730
$518$	3.00000	0.131812
$519$	0	0
$520$	1.18335	0.0518932
$521$	1.81665	0.0795890	0.0397945	−	0.999208i	$-0.487330\pi$
$521$	1.81665	0.0795890	0.0397945	+	0.999208i	$0.487330\pi$
$522$	0	0
$523$	7.09167	0.310097	0.155049	−	0.987907i	$-0.450447\pi$
$523$	7.09167	0.310097	0.155049	+	0.987907i	$0.450447\pi$
$524$	5.21110	0.227648
$525$	0	0
$526$	−10.5416	−0.459637
$527$	−6.00000	−0.261364
$528$	0	0
$529$	24.7250	1.07500
$530$	−11.0555	−0.480221
$531$	0	0
$532$	0.486122	0.0210761
$533$	1.18335	0.0512564
$534$	0	0
$535$	1.18335	0.0511605
$536$	−3.27502	−0.141459
$537$	0	0
$538$	4.42221	0.190655
$539$	18.0000	0.775315
$540$	0	0
$541$	4.21110	0.181049	0.0905247	−	0.995894i	$-0.471146\pi$
$541$	4.21110	0.181049	0.0905247	+	0.995894i	$0.471146\pi$
$542$	−36.7889	−1.58022
$543$	0	0
$544$	−5.09167	−0.218304
$545$	−5.48612	−0.235000
$546$	0	0
$547$	30.9361	1.32273	0.661366	−	0.750064i	$-0.269977\pi$
$547$	30.9361	1.32273	0.661366	+	0.750064i	$0.269977\pi$
$548$	−2.36669	−0.101100
$549$	0	0
$550$	12.9083	0.550413
$551$	−2.72498	−0.116088
$552$	0	0
$553$	2.69722	0.114698
$554$	23.3305	0.991219
$555$	0	0
$556$	0.577795	0.0245040
$557$	−23.6056	−1.00020	−0.500100	−	0.865968i	$-0.666703\pi$
$557$	−23.6056	−1.00020	−0.500100	+	0.865968i	$0.666703\pi$
$558$	0	0
$559$	0.330532	0.0139800
$560$	−4.30278	−0.181825
$561$	0	0
$562$	5.60555	0.236456
$563$	3.90833	0.164716	0.0823582	−	0.996603i	$-0.473755\pi$
$563$	3.90833	0.164716	0.0823582	+	0.996603i	$0.473755\pi$
$564$	0	0
$565$	24.7889	1.04288
$566$	43.0278	1.80859
$567$	0	0
$568$	−3.00000	−0.125877
$569$	−7.42221	−0.311155	−0.155578	−	0.987824i	$-0.549724\pi$
$569$	−7.42221	−0.311155	−0.155578	+	0.987824i	$0.549724\pi$
$570$	0	0
$571$	29.9083	1.25162	0.625812	−	0.779974i	$-0.284768\pi$
$571$	29.9083	1.25162	0.625812	+	0.779974i	$0.284768\pi$
$572$	0.275019	0.0114991
$573$	0	0
$574$	−5.09167	−0.212522
$575$	22.8167	0.951520
$576$	0	0
$577$	−23.3028	−0.970107	−0.485054	−	0.874484i	$-0.661200\pi$
$577$	−23.3028	−0.970107	−0.485054	+	0.874484i	$0.661200\pi$
$578$	−10.4222	−0.433507
$579$	0	0
$580$	−0.669468	−0.0277981
$581$	6.39445	0.265286
$582$	0	0
$583$	−19.5416	−0.809332
$584$	−29.4500	−1.21865
$585$	0	0
$586$	19.6972	0.813685
$587$	17.4500	0.720237	0.360118	−	0.932907i	$-0.382736\pi$
$587$	17.4500	0.720237	0.360118	+	0.932907i	$0.382736\pi$
$588$	0	0
$589$	3.21110	0.132311
$590$	−5.09167	−0.209621
$591$	0	0
$592$	7.60555	0.312586
$593$	−9.78890	−0.401982	−0.200991	−	0.979593i	$-0.564416\pi$
$593$	−9.78890	−0.401982	−0.200991	+	0.979593i	$0.564416\pi$
$594$	0	0
$595$	−3.90833	−0.160226
$596$	−2.09167	−0.0856783
$597$	0	0
$598$	−2.72498	−0.111433
$599$	36.1194	1.47580	0.737900	−	0.674910i	$-0.235818\pi$
$599$	36.1194	1.47580	0.737900	+	0.674910i	$0.235818\pi$
$600$	0	0
$601$	25.7250	1.04934	0.524672	−	0.851305i	$-0.324188\pi$
$601$	25.7250	1.04934	0.524672	+	0.851305i	$0.324188\pi$
$602$	−1.42221	−0.0579648
$603$	0	0
$604$	6.57779	0.267647
$605$	2.60555	0.105931
$606$	0	0
$607$	−19.2389	−0.780881	−0.390441	−	0.920628i	$-0.627677\pi$
$607$	−19.2389	−0.780881	−0.390441	+	0.920628i	$0.627677\pi$
$608$	2.72498	0.110513
$609$	0	0
$610$	26.4861	1.07239
$611$	3.00000	0.121367
$612$	0	0
$613$	43.4500	1.75493	0.877464	−	0.479643i	$-0.159234\pi$
$613$	43.4500	1.75493	0.877464	+	0.479643i	$0.159234\pi$
$614$	17.8806	0.721601
$615$	0	0
$616$	−9.00000	−0.362620
$617$	4.97224	0.200175	0.100087	−	0.994979i	$-0.468088\pi$
$617$	4.97224	0.200175	0.100087	+	0.994979i	$0.468088\pi$
$618$	0	0
$619$	24.3028	0.976811	0.488406	−	0.872617i	$-0.337579\pi$
$619$	24.3028	0.976811	0.488406	+	0.872617i	$0.337579\pi$
$620$	0.788897	0.0316829
$621$	0	0
$622$	20.2111	0.810391
$623$	3.00000	0.120192
$624$	0	0
$625$	2.42221	0.0968882
$626$	−42.2389	−1.68820
$627$	0	0
$628$	7.21110	0.287754
$629$	6.90833	0.275453
$630$	0	0
$631$	38.6333	1.53797	0.768984	−	0.639268i	$-0.220763\pi$
$631$	38.6333	1.53797	0.768984	+	0.639268i	$0.220763\pi$
$632$	8.09167	0.321869
$633$	0	0
$634$	17.3305	0.688283
$635$	−25.5416	−1.01359
$636$	0	0
$637$	−1.81665	−0.0719784
$638$	6.63331	0.262615
$639$	0	0
$640$	−10.5416	−0.416695
$641$	−15.3944	−0.608044	−0.304022	−	0.952665i	$-0.598330\pi$
$641$	−15.3944	−0.608044	−0.304022	+	0.952665i	$0.598330\pi$
$642$	0	0
$643$	5.51388	0.217446	0.108723	−	0.994072i	$-0.465324\pi$
$643$	5.51388	0.217446	0.108723	+	0.994072i	$0.465324\pi$
$644$	−2.09167	−0.0824235
$645$	0	0
$646$	−6.27502	−0.246887
$647$	46.5416	1.82974	0.914870	−	0.403748i	$-0.132293\pi$
$647$	46.5416	1.82974	0.914870	+	0.403748i	$0.132293\pi$
$648$	0	0
$649$	−9.00000	−0.353281
$650$	−1.30278	−0.0510991
$651$	0	0
$652$	1.88057	0.0736488
$653$	39.5139	1.54630	0.773149	−	0.634225i	$-0.218681\pi$
$653$	39.5139	1.54630	0.773149	+	0.634225i	$0.218681\pi$
$654$	0	0
$655$	22.4222	0.876108
$656$	−12.9083	−0.503985
$657$	0	0
$658$	−12.9083	−0.503219
$659$	14.2111	0.553586	0.276793	−	0.960930i	$-0.410728\pi$
$659$	14.2111	0.553586	0.276793	+	0.960930i	$0.410728\pi$
$660$	0	0
$661$	2.78890	0.108476	0.0542378	−	0.998528i	$-0.482727\pi$
$661$	2.78890	0.108476	0.0542378	+	0.998528i	$0.482727\pi$
$662$	17.5694	0.682854
$663$	0	0
$664$	19.1833	0.744458
$665$	2.09167	0.0811116
$666$	0	0
$667$	11.7250	0.453993
$668$	1.93608	0.0749093
$669$	0	0
$670$	−1.85281	−0.0715805
$671$	46.8167	1.80734
$672$	0	0
$673$	27.5416	1.06165	0.530826	−	0.847481i	$-0.321882\pi$
$673$	27.5416	1.06165	0.530826	+	0.847481i	$0.321882\pi$
$674$	−36.2750	−1.39726
$675$	0	0
$676$	3.90833	0.150320
$677$	−42.9083	−1.64910	−0.824550	−	0.565788i	$-0.808572\pi$
$677$	−42.9083	−1.64910	−0.824550	+	0.565788i	$0.808572\pi$
$678$	0	0
$679$	4.51388	0.173227
$680$	−11.7250	−0.449632
$681$	0	0
$682$	−7.81665	−0.299315
$683$	−40.0278	−1.53162	−0.765810	−	0.643067i	$-0.777662\pi$
$683$	−40.0278	−1.53162	−0.765810	+	0.643067i	$0.777662\pi$
$684$	0	0
$685$	−10.1833	−0.389086
$686$	16.9361	0.646623
$687$	0	0
$688$	−3.60555	−0.137460
$689$	1.97224	0.0751365
$690$	0	0
$691$	−11.6972	−0.444983	−0.222492	−	0.974935i	$-0.571419\pi$
$691$	−11.6972	−0.444983	−0.222492	+	0.974935i	$0.571419\pi$
$692$	1.45837	0.0554387
$693$	0	0
$694$	−5.44996	−0.206878
$695$	2.48612	0.0943040
$696$	0	0
$697$	−11.7250	−0.444115
$698$	21.1194	0.799382
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	21.6333	0.817079	0.408539	−	0.912741i	$-0.366038\pi$
$701$	21.6333	0.817079	0.408539	+	0.912741i	$0.366038\pi$
$702$	0	0
$703$	−3.69722	−0.139443
$704$	−26.4500	−0.996870
$705$	0	0
$706$	−42.6333	−1.60453
$707$	−6.39445	−0.240488
$708$	0	0
$709$	45.6611	1.71484	0.857419	−	0.514620i	$-0.172067\pi$
$709$	45.6611	1.71484	0.857419	+	0.514620i	$0.172067\pi$
$710$	−1.69722	−0.0636957
$711$	0	0
$712$	9.00000	0.337289
$713$	−13.8167	−0.517438
$714$	0	0
$715$	1.18335	0.0442546
$716$	−7.54163	−0.281844
$717$	0	0
$718$	20.8806	0.779256
$719$	29.7250	1.10855	0.554277	−	0.832332i	$-0.312995\pi$
$719$	29.7250	1.10855	0.554277	+	0.832332i	$0.312995\pi$
$720$	0	0
$721$	1.78890	0.0666220
$722$	−21.3944	−0.796219
$723$	0	0
$724$	2.63331	0.0978661
$725$	5.60555	0.208185
$726$	0	0
$727$	47.7527	1.77105	0.885525	−	0.464591i	$-0.153799\pi$
$727$	47.7527	1.77105	0.885525	+	0.464591i	$0.153799\pi$
$728$	0.908327	0.0336648
$729$	0	0
$730$	−16.6611	−0.616654
$731$	−3.27502	−0.121131
$732$	0	0
$733$	5.78890	0.213818	0.106909	−	0.994269i	$-0.465905\pi$
$733$	5.78890	0.213818	0.106909	+	0.994269i	$0.465905\pi$
$734$	−28.3028	−1.04467
$735$	0	0
$736$	−11.7250	−0.432189
$737$	−3.27502	−0.120637
$738$	0	0
$739$	−12.4861	−0.459309	−0.229655	−	0.973272i	$-0.573760\pi$
$739$	−12.4861	−0.459309	−0.229655	+	0.973272i	$0.573760\pi$
$740$	−0.908327	−0.0333908
$741$	0	0
$742$	−8.48612	−0.311535
$743$	34.6611	1.27159	0.635796	−	0.771858i	$-0.280672\pi$
$743$	34.6611	1.27159	0.635796	+	0.771858i	$0.280672\pi$
$744$	0	0
$745$	−9.00000	−0.329734
$746$	−22.8528	−0.836701
$747$	0	0
$748$	−2.72498	−0.0996352
$749$	0.908327	0.0331895
$750$	0	0
$751$	−10.7889	−0.393692	−0.196846	−	0.980434i	$-0.563070\pi$
$751$	−10.7889	−0.393692	−0.196846	+	0.980434i	$0.563070\pi$
$752$	−32.7250	−1.19336
$753$	0	0
$754$	−0.669468	−0.0243806
$755$	28.3028	1.03004
$756$	0	0
$757$	5.90833	0.214742	0.107371	−	0.994219i	$-0.465757\pi$
$757$	5.90833	0.214742	0.107371	+	0.994219i	$0.465757\pi$
$758$	7.85281	0.285227
$759$	0	0
$760$	6.27502	0.227619
$761$	−20.8806	−0.756920	−0.378460	−	0.925618i	$-0.623546\pi$
$761$	−20.8806	−0.756920	−0.378460	+	0.925618i	$0.623546\pi$
$762$	0	0
$763$	−4.21110	−0.152452
$764$	−0.908327	−0.0328621
$765$	0	0
$766$	2.56939	0.0928359
$767$	0.908327	0.0327978
$768$	0	0
$769$	−48.2111	−1.73854	−0.869268	−	0.494340i	$-0.835410\pi$
$769$	−48.2111	−1.73854	−0.869268	+	0.494340i	$0.835410\pi$
$770$	−5.09167	−0.183491
$771$	0	0
$772$	−2.85281	−0.102675
$773$	−5.36669	−0.193027	−0.0965133	−	0.995332i	$-0.530769\pi$
$773$	−5.36669	−0.193027	−0.0965133	+	0.995332i	$0.530769\pi$
$774$	0	0
$775$	−6.60555	−0.237278
$776$	13.5416	0.486116
$777$	0	0
$778$	27.1556	0.973575
$779$	6.27502	0.224826
$780$	0	0
$781$	−3.00000	−0.107348
$782$	27.0000	0.965518
$783$	0	0
$784$	19.8167	0.707738
$785$	31.0278	1.10743
$786$	0	0
$787$	−46.1194	−1.64398	−0.821990	−	0.569502i	$-0.807136\pi$
$787$	−46.1194	−1.64398	−0.821990	+	0.569502i	$0.807136\pi$
$788$	6.27502	0.223538
$789$	0	0
$790$	4.57779	0.162871
$791$	19.0278	0.676549
$792$	0	0
$793$	−4.72498	−0.167789
$794$	13.6611	0.484813
$795$	0	0
$796$	−4.55004	−0.161272
$797$	45.9083	1.62616	0.813078	−	0.582155i	$-0.197790\pi$
$797$	45.9083	1.62616	0.813078	+	0.582155i	$0.197790\pi$
$798$	0	0
$799$	−29.7250	−1.05159
$800$	−5.60555	−0.198186
$801$	0	0
$802$	−42.6333	−1.50543
$803$	−29.4500	−1.03927
$804$	0	0
$805$	−9.00000	−0.317208
$806$	0.788897	0.0277877
$807$	0	0
$808$	−19.1833	−0.674868
$809$	6.63331	0.233215	0.116607	−	0.993178i	$-0.462798\pi$
$809$	6.63331	0.233215	0.116607	+	0.993178i	$0.462798\pi$
$810$	0	0
$811$	−37.3583	−1.31183	−0.655913	−	0.754836i	$-0.727716\pi$
$811$	−37.3583	−1.31183	−0.655913	+	0.754836i	$0.727716\pi$
$812$	−0.513878	−0.0180336
$813$	0	0
$814$	9.00000	0.315450
$815$	8.09167	0.283439
$816$	0	0
$817$	1.75274	0.0613205
$818$	34.0278	1.18975
$819$	0	0
$820$	1.54163	0.0538362
$821$	−24.3944	−0.851372	−0.425686	−	0.904871i	$-0.639967\pi$
$821$	−24.3944	−0.851372	−0.425686	+	0.904871i	$0.639967\pi$
$822$	0	0
$823$	−34.9083	−1.21683	−0.608414	−	0.793620i	$-0.708194\pi$
$823$	−34.9083	−1.21683	−0.608414	+	0.793620i	$0.708194\pi$
$824$	5.36669	0.186958
$825$	0	0
$826$	−3.90833	−0.135988
$827$	17.0917	0.594336	0.297168	−	0.954825i	$-0.403958\pi$
$827$	17.0917	0.594336	0.297168	+	0.954825i	$0.403958\pi$
$828$	0	0
$829$	−19.0000	−0.659897	−0.329949	−	0.943999i	$-0.607031\pi$
$829$	−19.0000	−0.659897	−0.329949	+	0.943999i	$0.607031\pi$
$830$	10.8528	0.376707
$831$	0	0
$832$	2.66947	0.0925472
$833$	18.0000	0.623663
$834$	0	0
$835$	8.33053	0.288290
$836$	1.45837	0.0504386
$837$	0	0
$838$	1.18335	0.0408780
$839$	−37.8167	−1.30558	−0.652788	−	0.757541i	$-0.726401\pi$
$839$	−37.8167	−1.30558	−0.652788	+	0.757541i	$0.726401\pi$
$840$	0	0
$841$	−26.1194	−0.900670
$842$	20.0917	0.692405
$843$	0	0
$844$	0.183346	0.00631104
$845$	16.8167	0.578510
$846$	0	0
$847$	2.00000	0.0687208
$848$	−21.5139	−0.738790
$849$	0	0
$850$	12.9083	0.442752
$851$	15.9083	0.545330
$852$	0	0
$853$	24.3028	0.832111	0.416056	−	0.909339i	$-0.363412\pi$
$853$	24.3028	0.832111	0.416056	+	0.909339i	$0.363412\pi$
$854$	20.3305	0.695696
$855$	0	0
$856$	2.72498	0.0931379
$857$	11.7250	0.400518	0.200259	−	0.979743i	$-0.435822\pi$
$857$	11.7250	0.400518	0.200259	+	0.979743i	$0.435822\pi$
$858$	0	0
$859$	−7.23886	−0.246987	−0.123493	−	0.992345i	$-0.539410\pi$
$859$	−7.23886	−0.246987	−0.123493	+	0.992345i	$0.539410\pi$
$860$	0.430609	0.0146836
$861$	0	0
$862$	−9.66947	−0.329343
$863$	−20.0917	−0.683929	−0.341964	−	0.939713i	$-0.611092\pi$
$863$	−20.0917	−0.683929	−0.341964	+	0.939713i	$0.611092\pi$
$864$	0	0
$865$	6.27502	0.213357
$866$	−18.9445	−0.643760
$867$	0	0
$868$	0.605551	0.0205537
$869$	8.09167	0.274491
$870$	0	0
$871$	0.330532	0.0111997
$872$	−12.6333	−0.427818
$873$	0	0
$874$	−14.4500	−0.488777
$875$	−10.8167	−0.365670
$876$	0	0
$877$	28.8444	0.974007	0.487003	−	0.873400i	$-0.338090\pi$
$877$	28.8444	0.974007	0.487003	+	0.873400i	$0.338090\pi$
$878$	−37.1472	−1.25366
$879$	0	0
$880$	−12.9083	−0.435140
$881$	−48.7527	−1.64252	−0.821261	−	0.570553i	$-0.806729\pi$
$881$	−48.7527	−1.64252	−0.821261	+	0.570553i	$0.806729\pi$
$882$	0	0
$883$	26.6333	0.896282	0.448141	−	0.893963i	$-0.352086\pi$
$883$	26.6333	0.896282	0.448141	+	0.893963i	$0.352086\pi$
$884$	0.275019	0.00924990
$885$	0	0
$886$	24.1194	0.810308
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	0	0
$889$	−19.6056	−0.657549
$890$	5.09167	0.170673
$891$	0	0
$892$	−4.66947	−0.156345
$893$	15.9083	0.532352
$894$	0	0
$895$	−32.4500	−1.08468
$896$	−8.09167	−0.270324
$897$	0	0
$898$	−48.2389	−1.60975
$899$	−3.39445	−0.113211
$900$	0	0
$901$	−19.5416	−0.651026
$902$	−15.2750	−0.508603
$903$	0	0
$904$	57.0833	1.89856
$905$	11.3305	0.376640
$906$	0	0
$907$	−5.54163	−0.184007	−0.0920035	−	0.995759i	$-0.529327\pi$
$907$	−5.54163	−0.184007	−0.0920035	+	0.995759i	$0.529327\pi$
$908$	2.96384	0.0983585
$909$	0	0
$910$	0.513878	0.0170349
$911$	−28.9361	−0.958695	−0.479348	−	0.877625i	$-0.659127\pi$
$911$	−28.9361	−0.958695	−0.479348	+	0.877625i	$0.659127\pi$
$912$	0	0
$913$	19.1833	0.634876
$914$	30.2750	1.00141
$915$	0	0
$916$	3.45837	0.114268
$917$	17.2111	0.568361
$918$	0	0
$919$	−41.6611	−1.37427	−0.687136	−	0.726529i	$-0.741132\pi$
$919$	−41.6611	−1.37427	−0.687136	+	0.726529i	$0.741132\pi$
$920$	−27.0000	−0.890164
$921$	0	0
$922$	−34.4584	−1.13483
$923$	0.302776	0.00996598
$924$	0	0
$925$	7.60555	0.250069
$926$	−37.9722	−1.24785
$927$	0	0
$928$	−2.88057	−0.0945594
$929$	−28.4222	−0.932502	−0.466251	−	0.884652i	$-0.654396\pi$
$929$	−28.4222	−0.932502	−0.466251	+	0.884652i	$0.654396\pi$
$930$	0	0
$931$	−9.63331	−0.315719
$932$	−5.17494	−0.169511
$933$	0	0
$934$	36.5139	1.19477
$935$	−11.7250	−0.383448
$936$	0	0
$937$	−31.9083	−1.04240	−0.521200	−	0.853435i	$-0.674515\pi$
$937$	−31.9083	−1.04240	−0.521200	+	0.853435i	$0.674515\pi$
$938$	−1.42221	−0.0464366
$939$	0	0
$940$	3.90833	0.127476
$941$	−35.4500	−1.15564	−0.577818	−	0.816166i	$-0.696096\pi$
$941$	−35.4500	−1.15564	−0.577818	+	0.816166i	$0.696096\pi$
$942$	0	0
$943$	−27.0000	−0.879241
$944$	−9.90833	−0.322489
$945$	0	0
$946$	−4.26662	−0.138720
$947$	−50.8444	−1.65222	−0.826111	−	0.563508i	$-0.809451\pi$
$947$	−50.8444	−1.65222	−0.826111	+	0.563508i	$0.809451\pi$
$948$	0	0
$949$	2.97224	0.0964831
$950$	−6.90833	−0.224136
$951$	0	0
$952$	−9.00000	−0.291692
$953$	−30.6333	−0.992310	−0.496155	−	0.868234i	$-0.665255\pi$
$953$	−30.6333	−0.992310	−0.496155	+	0.868234i	$0.665255\pi$
$954$	0	0
$955$	−3.90833	−0.126470
$956$	2.72498	0.0881322
$957$	0	0
$958$	25.6611	0.829071
$959$	−7.81665	−0.252413
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−0.908327	−0.0292856
$963$	0	0
$964$	5.11943	0.164886
$965$	−12.2750	−0.395147
$966$	0	0
$967$	6.69722	0.215368	0.107684	−	0.994185i	$-0.465656\pi$
$967$	6.69722	0.215368	0.107684	+	0.994185i	$0.465656\pi$
$968$	6.00000	0.192847
$969$	0	0
$970$	7.66106	0.245982
$971$	−28.2666	−0.907119	−0.453559	−	0.891226i	$-0.649846\pi$
$971$	−28.2666	−0.907119	−0.453559	+	0.891226i	$0.649846\pi$
$972$	0	0
$973$	1.90833	0.0611782
$974$	29.0917	0.932157
$975$	0	0
$976$	51.5416	1.64981
$977$	45.2389	1.44732	0.723660	−	0.690157i	$-0.242459\pi$
$977$	45.2389	1.44732	0.723660	+	0.690157i	$0.242459\pi$
$978$	0	0
$979$	9.00000	0.287641
$980$	−2.36669	−0.0756012
$981$	0	0
$982$	−41.7611	−1.33265
$983$	5.33053	0.170018	0.0850088	−	0.996380i	$-0.472908\pi$
$983$	5.33053	0.170018	0.0850088	+	0.996380i	$0.472908\pi$
$984$	0	0
$985$	27.0000	0.860292
$986$	6.63331	0.211248
$987$	0	0
$988$	−0.147186	−0.00468261
$989$	−7.54163	−0.239810
$990$	0	0
$991$	51.7805	1.64486	0.822431	−	0.568865i	$-0.192617\pi$
$991$	51.7805	1.64486	0.822431	+	0.568865i	$0.192617\pi$
$992$	3.39445	0.107774
$993$	0	0
$994$	−1.30278	−0.0413215
$995$	−19.5778	−0.620658
$996$	0	0
$997$	−35.1472	−1.11312	−0.556561	−	0.830807i	$-0.687880\pi$
$997$	−35.1472	−1.11312	−0.556561	+	0.830807i	$0.687880\pi$
$998$	43.0278	1.36202
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.c.1.2		2
3.2	odd	2		213.2.a.d.1.1	✓	2
12.11	even	2		3408.2.a.m.1.2		2
15.14	odd	2		5325.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
213.2.a.d.1.1	✓	2	3.2	odd	2
639.2.a.c.1.2		2	1.1	even	1	trivial
3408.2.a.m.1.2		2	12.11	even	2
5325.2.a.q.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.30278 q^{2} -0.302776 q^{4} -1.30278 q^{5} -1.00000 q^{7} -3.00000 q^{8} +O(q^{10})\)	\(q+1.30278 q^{2} -0.302776 q^{4} -1.30278 q^{5} -1.00000 q^{7} -3.00000 q^{8} -1.69722 q^{10} -3.00000 q^{11} +0.302776 q^{13} -1.30278 q^{14} -3.30278 q^{16} -3.00000 q^{17} +1.60555 q^{19} +0.394449 q^{20} -3.90833 q^{22} -6.90833 q^{23} -3.30278 q^{25} +0.394449 q^{26} +0.302776 q^{28} -1.69722 q^{29} +2.00000 q^{31} +1.69722 q^{32} -3.90833 q^{34} +1.30278 q^{35} -2.30278 q^{37} +2.09167 q^{38} +3.90833 q^{40} +3.90833 q^{41} +1.09167 q^{43} +0.908327 q^{44} -9.00000 q^{46} +9.90833 q^{47} -6.00000 q^{49} -4.30278 q^{50} -0.0916731 q^{52} +6.51388 q^{53} +3.90833 q^{55} +3.00000 q^{56} -2.21110 q^{58} +3.00000 q^{59} -15.6056 q^{61} +2.60555 q^{62} +8.81665 q^{64} -0.394449 q^{65} +1.09167 q^{67} +0.908327 q^{68} +1.69722 q^{70} +1.00000 q^{71} +9.81665 q^{73} -3.00000 q^{74} -0.486122 q^{76} +3.00000 q^{77} -2.69722 q^{79} +4.30278 q^{80} +5.09167 q^{82} -6.39445 q^{83} +3.90833 q^{85} +1.42221 q^{86} +9.00000 q^{88} -3.00000 q^{89} -0.302776 q^{91} +2.09167 q^{92} +12.9083 q^{94} -2.09167 q^{95} -4.51388 q^{97} -7.81665 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} + q^{5} - 2 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - q^2 + 3 * q^4 + q^5 - 2 * q^7 - 6 * q^8	\( 2 q - q^{2} + 3 q^{4} + q^{5} - 2 q^{7} - 6 q^{8} - 7 q^{10} - 6 q^{11} - 3 q^{13} + q^{14} - 3 q^{16} - 6 q^{17} - 4 q^{19} + 8 q^{20} + 3 q^{22} - 3 q^{23} - 3 q^{25} + 8 q^{26} - 3 q^{28} - 7 q^{29} + 4 q^{31} + 7 q^{32} + 3 q^{34} - q^{35} - q^{37} + 15 q^{38} - 3 q^{40} - 3 q^{41} + 13 q^{43} - 9 q^{44} - 18 q^{46} + 9 q^{47} - 12 q^{49} - 5 q^{50} - 11 q^{52} - 5 q^{53} - 3 q^{55} + 6 q^{56} + 10 q^{58} + 6 q^{59} - 24 q^{61} - 2 q^{62} - 4 q^{64} - 8 q^{65} + 13 q^{67} - 9 q^{68} + 7 q^{70} + 2 q^{71} - 2 q^{73} - 6 q^{74} - 19 q^{76} + 6 q^{77} - 9 q^{79} + 5 q^{80} + 21 q^{82} - 20 q^{83} - 3 q^{85} - 26 q^{86} + 18 q^{88} - 6 q^{89} + 3 q^{91} + 15 q^{92} + 15 q^{94} - 15 q^{95} + 9 q^{97} + 6 q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 + q^5 - 2 * q^7 - 6 * q^8 - 7 * q^10 - 6 * q^11 - 3 * q^13 + q^14 - 3 * q^16 - 6 * q^17 - 4 * q^19 + 8 * q^20 + 3 * q^22 - 3 * q^23 - 3 * q^25 + 8 * q^26 - 3 * q^28 - 7 * q^29 + 4 * q^31 + 7 * q^32 + 3 * q^34 - q^35 - q^37 + 15 * q^38 - 3 * q^40 - 3 * q^41 + 13 * q^43 - 9 * q^44 - 18 * q^46 + 9 * q^47 - 12 * q^49 - 5 * q^50 - 11 * q^52 - 5 * q^53 - 3 * q^55 + 6 * q^56 + 10 * q^58 + 6 * q^59 - 24 * q^61 - 2 * q^62 - 4 * q^64 - 8 * q^65 + 13 * q^67 - 9 * q^68 + 7 * q^70 + 2 * q^71 - 2 * q^73 - 6 * q^74 - 19 * q^76 + 6 * q^77 - 9 * q^79 + 5 * q^80 + 21 * q^82 - 20 * q^83 - 3 * q^85 - 26 * q^86 + 18 * q^88 - 6 * q^89 + 3 * q^91 + 15 * q^92 + 15 * q^94 - 15 * q^95 + 9 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

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