LMFDB - Embedded newform 639.2.a.d.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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
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			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{2} +0.618034 q^{4} -1.61803 q^{5} -3.00000 q^{7} -2.23607 q^{8} +O(q^{10})$	\(q+1.61803 q^{2} +0.618034 q^{4} -1.61803 q^{5} -3.00000 q^{7} -2.23607 q^{8} -2.61803 q^{10} -0.236068 q^{11} -5.85410 q^{13} -4.85410 q^{14} -4.85410 q^{16} +2.23607 q^{17} -1.76393 q^{19} -1.00000 q^{20} -0.381966 q^{22} +7.09017 q^{23} -2.38197 q^{25} -9.47214 q^{26} -1.85410 q^{28} +1.85410 q^{29} -2.00000 q^{31} -3.38197 q^{32} +3.61803 q^{34} +4.85410 q^{35} +11.5623 q^{37} -2.85410 q^{38} +3.61803 q^{40} -6.38197 q^{41} -11.5623 q^{43} -0.145898 q^{44} +11.4721 q^{46} -5.32624 q^{47} +2.00000 q^{49} -3.85410 q^{50} -3.61803 q^{52} -4.09017 q^{53} +0.381966 q^{55} +6.70820 q^{56} +3.00000 q^{58} +6.70820 q^{59} +6.70820 q^{61} -3.23607 q^{62} +4.23607 q^{64} +9.47214 q^{65} -2.90983 q^{67} +1.38197 q^{68} +7.85410 q^{70} +1.00000 q^{71} -2.76393 q^{73} +18.7082 q^{74} -1.09017 q^{76} +0.708204 q^{77} -6.09017 q^{79} +7.85410 q^{80} -10.3262 q^{82} +3.00000 q^{83} -3.61803 q^{85} -18.7082 q^{86} +0.527864 q^{88} -12.7082 q^{89} +17.5623 q^{91} +4.38197 q^{92} -8.61803 q^{94} +2.85410 q^{95} -14.5623 q^{97} +3.23607 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - q^{4} - q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + q^2 - q^4 - q^5 - 6 * q^7	$ 2 q + q^{2} - q^{4} - q^{5} - 6 q^{7} - 3 q^{10} + 4 q^{11} - 5 q^{13} - 3 q^{14} - 3 q^{16} - 8 q^{19} - 2 q^{20} - 3 q^{22} + 3 q^{23} - 7 q^{25} - 10 q^{26} + 3 q^{28} - 3 q^{29} - 4 q^{31} - 9 q^{32} + 5 q^{34} + 3 q^{35} + 3 q^{37} + q^{38} + 5 q^{40} - 15 q^{41} - 3 q^{43} - 7 q^{44} + 14 q^{46} + 5 q^{47} + 4 q^{49} - q^{50} - 5 q^{52} + 3 q^{53} + 3 q^{55} + 6 q^{58} - 2 q^{62} + 4 q^{64} + 10 q^{65} - 17 q^{67} + 5 q^{68} + 9 q^{70} + 2 q^{71} - 10 q^{73} + 24 q^{74} + 9 q^{76} - 12 q^{77} - q^{79} + 9 q^{80} - 5 q^{82} + 6 q^{83} - 5 q^{85} - 24 q^{86} + 10 q^{88} - 12 q^{89} + 15 q^{91} + 11 q^{92} - 15 q^{94} - q^{95} - 9 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + q^2 - q^4 - q^5 - 6 * q^7 - 3 * q^10 + 4 * q^11 - 5 * q^13 - 3 * q^14 - 3 * q^16 - 8 * q^19 - 2 * q^20 - 3 * q^22 + 3 * q^23 - 7 * q^25 - 10 * q^26 + 3 * q^28 - 3 * q^29 - 4 * q^31 - 9 * q^32 + 5 * q^34 + 3 * q^35 + 3 * q^37 + q^38 + 5 * q^40 - 15 * q^41 - 3 * q^43 - 7 * q^44 + 14 * q^46 + 5 * q^47 + 4 * q^49 - q^50 - 5 * q^52 + 3 * q^53 + 3 * q^55 + 6 * q^58 - 2 * q^62 + 4 * q^64 + 10 * q^65 - 17 * q^67 + 5 * q^68 + 9 * q^70 + 2 * q^71 - 10 * q^73 + 24 * q^74 + 9 * q^76 - 12 * q^77 - q^79 + 9 * q^80 - 5 * q^82 + 6 * q^83 - 5 * q^85 - 24 * q^86 + 10 * q^88 - 12 * q^89 + 15 * q^91 + 11 * q^92 - 15 * q^94 - q^95 - 9 * q^97 + 2 * 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 $

$2$	1.61803	1.14412	0.572061	−	0.820211i	$-0.306144\pi$
$2$	1.61803	1.14412	0.572061	+	0.820211i	$0.306144\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	−1.61803	−0.723607	−0.361803	−	0.932254i	$-0.617839\pi$
$5$	−1.61803	−0.723607	−0.361803	+	0.932254i	$0.617839\pi$
$6$	0	0
$7$	−3.00000	−1.13389	−0.566947	−	0.823754i	$-0.691875\pi$
$7$	−3.00000	−1.13389	−0.566947	+	0.823754i	$0.691875\pi$
$8$	−2.23607	−0.790569
$9$	0	0
$10$	−2.61803	−0.827895
$11$	−0.236068	−0.0711772	−0.0355886	−	0.999367i	$-0.511331\pi$
$11$	−0.236068	−0.0711772	−0.0355886	+	0.999367i	$0.511331\pi$
$12$	0	0
$13$	−5.85410	−1.62364	−0.811818	−	0.583911i	$-0.801522\pi$
$13$	−5.85410	−1.62364	−0.811818	+	0.583911i	$0.801522\pi$
$14$	−4.85410	−1.29731
$15$	0	0
$16$	−4.85410	−1.21353
$17$	2.23607	0.542326	0.271163	−	0.962533i	$-0.412592\pi$
$17$	2.23607	0.542326	0.271163	+	0.962533i	$0.412592\pi$
$18$	0	0
$19$	−1.76393	−0.404674	−0.202337	−	0.979316i	$-0.564854\pi$
$19$	−1.76393	−0.404674	−0.202337	+	0.979316i	$0.564854\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	−0.381966	−0.0814354
$23$	7.09017	1.47840	0.739201	−	0.673485i	$-0.235203\pi$
$23$	7.09017	1.47840	0.739201	+	0.673485i	$0.235203\pi$
$24$	0	0
$25$	−2.38197	−0.476393
$26$	−9.47214	−1.85764
$27$	0	0
$28$	−1.85410	−0.350392
$29$	1.85410	0.344298	0.172149	−	0.985071i	$-0.444929\pi$
$29$	1.85410	0.344298	0.172149	+	0.985071i	$0.444929\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−3.38197	−0.597853
$33$	0	0
$34$	3.61803	0.620488
$35$	4.85410	0.820493
$36$	0	0
$37$	11.5623	1.90083	0.950416	−	0.310982i	$-0.100658\pi$
$37$	11.5623	1.90083	0.950416	+	0.310982i	$0.100658\pi$
$38$	−2.85410	−0.462996
$39$	0	0
$40$	3.61803	0.572061
$41$	−6.38197	−0.996696	−0.498348	−	0.866977i	$-0.666060\pi$
$41$	−6.38197	−0.996696	−0.498348	+	0.866977i	$0.666060\pi$
$42$	0	0
$43$	−11.5623	−1.76324	−0.881618	−	0.471964i	$-0.843545\pi$
$43$	−11.5623	−1.76324	−0.881618	+	0.471964i	$0.843545\pi$
$44$	−0.145898	−0.0219950
$45$	0	0
$46$	11.4721	1.69147
$47$	−5.32624	−0.776912	−0.388456	−	0.921467i	$-0.626991\pi$
$47$	−5.32624	−0.776912	−0.388456	+	0.921467i	$0.626991\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	−3.85410	−0.545052
$51$	0	0
$52$	−3.61803	−0.501731
$53$	−4.09017	−0.561828	−0.280914	−	0.959733i	$-0.590638\pi$
$53$	−4.09017	−0.561828	−0.280914	+	0.959733i	$0.590638\pi$
$54$	0	0
$55$	0.381966	0.0515043
$56$	6.70820	0.896421
$57$	0	0
$58$	3.00000	0.393919
$59$	6.70820	0.873334	0.436667	−	0.899623i	$-0.356159\pi$
$59$	6.70820	0.873334	0.436667	+	0.899623i	$0.356159\pi$
$60$	0	0
$61$	6.70820	0.858898	0.429449	−	0.903091i	$-0.358708\pi$
$61$	6.70820	0.858898	0.429449	+	0.903091i	$0.358708\pi$
$62$	−3.23607	−0.410981
$63$	0	0
$64$	4.23607	0.529508
$65$	9.47214	1.17487
$66$	0	0
$67$	−2.90983	−0.355492	−0.177746	−	0.984076i	$-0.556881\pi$
$67$	−2.90983	−0.355492	−0.177746	+	0.984076i	$0.556881\pi$
$68$	1.38197	0.167588
$69$	0	0
$70$	7.85410	0.938745
$71$	1.00000	0.118678
$71$	1.00000	0.118678
$72$	0	0
$73$	−2.76393	−0.323494	−0.161747	−	0.986832i	$-0.551713\pi$
$73$	−2.76393	−0.323494	−0.161747	+	0.986832i	$0.551713\pi$
$74$	18.7082	2.17478
$75$	0	0
$76$	−1.09017	−0.125051
$77$	0.708204	0.0807073
$78$	0	0
$79$	−6.09017	−0.685198	−0.342599	−	0.939482i	$-0.611307\pi$
$79$	−6.09017	−0.685198	−0.342599	+	0.939482i	$0.611307\pi$
$80$	7.85410	0.878115
$81$	0	0
$82$	−10.3262	−1.14034
$83$	3.00000	0.329293	0.164646	−	0.986353i	$-0.447352\pi$
$83$	3.00000	0.329293	0.164646	+	0.986353i	$0.447352\pi$
$84$	0	0
$85$	−3.61803	−0.392431
$86$	−18.7082	−2.01736
$87$	0	0
$88$	0.527864	0.0562705
$89$	−12.7082	−1.34707	−0.673533	−	0.739157i	$-0.735224\pi$
$89$	−12.7082	−1.34707	−0.673533	+	0.739157i	$0.735224\pi$
$90$	0	0
$91$	17.5623	1.84103
$92$	4.38197	0.456852
$93$	0	0
$94$	−8.61803	−0.888882
$95$	2.85410	0.292825
$96$	0	0
$97$	−14.5623	−1.47858	−0.739289	−	0.673388i	$-0.764838\pi$
$97$	−14.5623	−1.47858	−0.739289	+	0.673388i	$0.764838\pi$
$98$	3.23607	0.326892
$99$	0	0
$100$	−1.47214	−0.147214
$101$	−9.47214	−0.942513	−0.471256	−	0.881996i	$-0.656199\pi$
$101$	−9.47214	−0.942513	−0.471256	+	0.881996i	$0.656199\pi$
$102$	0	0
$103$	15.4721	1.52451	0.762257	−	0.647274i	$-0.224091\pi$
$103$	15.4721	1.52451	0.762257	+	0.647274i	$0.224091\pi$
$104$	13.0902	1.28360
$105$	0	0
$106$	−6.61803	−0.642800
$107$	17.3820	1.68038	0.840189	−	0.542294i	$-0.182444\pi$
$107$	17.3820	1.68038	0.840189	+	0.542294i	$0.182444\pi$
$108$	0	0
$109$	−9.00000	−0.862044	−0.431022	−	0.902342i	$-0.641847\pi$
$109$	−9.00000	−0.862044	−0.431022	+	0.902342i	$0.641847\pi$
$110$	0.618034	0.0589272
$111$	0	0
$112$	14.5623	1.37601
$113$	5.23607	0.492568	0.246284	−	0.969198i	$-0.420790\pi$
$113$	5.23607	0.492568	0.246284	+	0.969198i	$0.420790\pi$
$114$	0	0
$115$	−11.4721	−1.06978
$116$	1.14590	0.106394
$117$	0	0
$118$	10.8541	0.999201
$119$	−6.70820	−0.614940
$120$	0	0
$121$	−10.9443	−0.994934
$122$	10.8541	0.982684
$123$	0	0
$124$	−1.23607	−0.111002
$125$	11.9443	1.06833
$126$	0	0
$127$	4.70820	0.417786	0.208893	−	0.977939i	$-0.433014\pi$
$127$	4.70820	0.417786	0.208893	+	0.977939i	$0.433014\pi$
$128$	13.6180	1.20368
$129$	0	0
$130$	15.3262	1.34420
$131$	−22.4721	−1.96340	−0.981700	−	0.190435i	$-0.939010\pi$
$131$	−22.4721	−1.96340	−0.981700	+	0.190435i	$0.939010\pi$
$132$	0	0
$133$	5.29180	0.458857
$134$	−4.70820	−0.406727
$135$	0	0
$136$	−5.00000	−0.428746
$137$	−9.70820	−0.829428	−0.414714	−	0.909952i	$-0.636118\pi$
$137$	−9.70820	−0.829428	−0.414714	+	0.909952i	$0.636118\pi$
$138$	0	0
$139$	0.0901699	0.00764811	0.00382406	−	0.999993i	$-0.498783\pi$
$139$	0.0901699	0.00764811	0.00382406	+	0.999993i	$0.498783\pi$
$140$	3.00000	0.253546
$141$	0	0
$142$	1.61803	0.135782
$143$	1.38197	0.115566
$144$	0	0
$145$	−3.00000	−0.249136
$146$	−4.47214	−0.370117
$147$	0	0
$148$	7.14590	0.587389
$149$	1.85410	0.151894	0.0759470	−	0.997112i	$-0.475802\pi$
$149$	1.85410	0.151894	0.0759470	+	0.997112i	$0.475802\pi$
$150$	0	0
$151$	4.67376	0.380345	0.190173	−	0.981751i	$-0.439095\pi$
$151$	4.67376	0.380345	0.190173	+	0.981751i	$0.439095\pi$
$152$	3.94427	0.319923
$153$	0	0
$154$	1.14590	0.0923391
$155$	3.23607	0.259927
$156$	0	0
$157$	4.76393	0.380203	0.190102	−	0.981764i	$-0.439118\pi$
$157$	4.76393	0.380203	0.190102	+	0.981764i	$0.439118\pi$
$158$	−9.85410	−0.783950
$159$	0	0
$160$	5.47214	0.432610
$161$	−21.2705	−1.67635
$162$	0	0
$163$	16.4164	1.28583	0.642916	−	0.765937i	$-0.277724\pi$
$163$	16.4164	1.28583	0.642916	+	0.765937i	$0.277724\pi$
$164$	−3.94427	−0.307996
$165$	0	0
$166$	4.85410	0.376751
$167$	−13.9443	−1.07904	−0.539520	−	0.841973i	$-0.681394\pi$
$167$	−13.9443	−1.07904	−0.539520	+	0.841973i	$0.681394\pi$
$168$	0	0
$169$	21.2705	1.63619
$170$	−5.85410	−0.448989
$171$	0	0
$172$	−7.14590	−0.544870
$173$	14.8885	1.13196	0.565978	−	0.824421i	$-0.308499\pi$
$173$	14.8885	1.13196	0.565978	+	0.824421i	$0.308499\pi$
$174$	0	0
$175$	7.14590	0.540179
$176$	1.14590	0.0863753
$177$	0	0
$178$	−20.5623	−1.54121
$179$	−6.32624	−0.472845	−0.236423	−	0.971650i	$-0.575975\pi$
$179$	−6.32624	−0.472845	−0.236423	+	0.971650i	$0.575975\pi$
$180$	0	0
$181$	−5.79837	−0.430990	−0.215495	−	0.976505i	$-0.569136\pi$
$181$	−5.79837	−0.430990	−0.215495	+	0.976505i	$0.569136\pi$
$182$	28.4164	2.10636
$183$	0	0
$184$	−15.8541	−1.16878
$185$	−18.7082	−1.37545
$186$	0	0
$187$	−0.527864	−0.0386012
$188$	−3.29180	−0.240079
$189$	0	0
$190$	4.61803	0.335027
$191$	−18.7082	−1.35368	−0.676839	−	0.736131i	$-0.736651\pi$
$191$	−18.7082	−1.35368	−0.676839	+	0.736131i	$0.736651\pi$
$192$	0	0
$193$	−14.4164	−1.03772	−0.518858	−	0.854861i	$-0.673643\pi$
$193$	−14.4164	−1.03772	−0.518858	+	0.854861i	$0.673643\pi$
$194$	−23.5623	−1.69167
$195$	0	0
$196$	1.23607	0.0882906
$197$	0.909830	0.0648227	0.0324114	−	0.999475i	$-0.489681\pi$
$197$	0.909830	0.0648227	0.0324114	+	0.999475i	$0.489681\pi$
$198$	0	0
$199$	4.29180	0.304237	0.152119	−	0.988362i	$-0.451390\pi$
$199$	4.29180	0.304237	0.152119	+	0.988362i	$0.451390\pi$
$200$	5.32624	0.376622
$201$	0	0
$202$	−15.3262	−1.07835
$203$	−5.56231	−0.390397
$204$	0	0
$205$	10.3262	0.721216
$206$	25.0344	1.74423
$207$	0	0
$208$	28.4164	1.97032
$209$	0.416408	0.0288035
$210$	0	0
$211$	−23.1246	−1.59196	−0.795982	−	0.605320i	$-0.793045\pi$
$211$	−23.1246	−1.59196	−0.795982	+	0.605320i	$0.793045\pi$
$212$	−2.52786	−0.173614
$213$	0	0
$214$	28.1246	1.92256
$215$	18.7082	1.27589
$216$	0	0
$217$	6.00000	0.407307
$218$	−14.5623	−0.986284
$219$	0	0
$220$	0.236068	0.0159157
$221$	−13.0902	−0.880540
$222$	0	0
$223$	−14.4164	−0.965394	−0.482697	−	0.875787i	$-0.660343\pi$
$223$	−14.4164	−0.965394	−0.482697	+	0.875787i	$0.660343\pi$
$224$	10.1459	0.677901
$225$	0	0
$226$	8.47214	0.563558
$227$	11.1803	0.742065	0.371033	−	0.928620i	$-0.379004\pi$
$227$	11.1803	0.742065	0.371033	+	0.928620i	$0.379004\pi$
$228$	0	0
$229$	0.527864	0.0348822	0.0174411	−	0.999848i	$-0.494448\pi$
$229$	0.527864	0.0348822	0.0174411	+	0.999848i	$0.494448\pi$
$230$	−18.5623	−1.22396
$231$	0	0
$232$	−4.14590	−0.272192
$233$	−11.6738	−0.764774	−0.382387	−	0.924002i	$-0.624898\pi$
$233$	−11.6738	−0.764774	−0.382387	+	0.924002i	$0.624898\pi$
$234$	0	0
$235$	8.61803	0.562179
$236$	4.14590	0.269875
$237$	0	0
$238$	−10.8541	−0.703567
$239$	25.1803	1.62878	0.814390	−	0.580317i	$-0.197072\pi$
$239$	25.1803	1.62878	0.814390	+	0.580317i	$0.197072\pi$
$240$	0	0
$241$	−25.8541	−1.66541	−0.832705	−	0.553718i	$-0.813209\pi$
$241$	−25.8541	−1.66541	−0.832705	+	0.553718i	$0.813209\pi$
$242$	−17.7082	−1.13833
$243$	0	0
$244$	4.14590	0.265414
$245$	−3.23607	−0.206745
$246$	0	0
$247$	10.3262	0.657043
$248$	4.47214	0.283981
$249$	0	0
$250$	19.3262	1.22230
$251$	27.7984	1.75462	0.877309	−	0.479926i	$-0.159337\pi$
$251$	27.7984	1.75462	0.877309	+	0.479926i	$0.159337\pi$
$252$	0	0
$253$	−1.67376	−0.105229
$254$	7.61803	0.477998
$255$	0	0
$256$	13.5623	0.847644
$257$	−8.47214	−0.528477	−0.264239	−	0.964457i	$-0.585121\pi$
$257$	−8.47214	−0.528477	−0.264239	+	0.964457i	$0.585121\pi$
$258$	0	0
$259$	−34.6869	−2.15534
$260$	5.85410	0.363056
$261$	0	0
$262$	−36.3607	−2.24637
$263$	9.43769	0.581953	0.290977	−	0.956730i	$-0.406020\pi$
$263$	9.43769	0.581953	0.290977	+	0.956730i	$0.406020\pi$
$264$	0	0
$265$	6.61803	0.406543
$266$	8.56231	0.524989
$267$	0	0
$268$	−1.79837	−0.109853
$269$	32.0689	1.95527	0.977637	−	0.210299i	$-0.0674437\pi$
$269$	32.0689	1.95527	0.977637	+	0.210299i	$0.0674437\pi$
$270$	0	0
$271$	−0.291796	−0.0177253	−0.00886267	−	0.999961i	$-0.502821\pi$
$271$	−0.291796	−0.0177253	−0.00886267	+	0.999961i	$0.502821\pi$
$272$	−10.8541	−0.658127
$273$	0	0
$274$	−15.7082	−0.948967
$275$	0.562306	0.0339083
$276$	0	0
$277$	17.0344	1.02350	0.511750	−	0.859134i	$-0.328997\pi$
$277$	17.0344	1.02350	0.511750	+	0.859134i	$0.328997\pi$
$278$	0.145898	0.00875038
$279$	0	0
$280$	−10.8541	−0.648657
$281$	−3.27051	−0.195102	−0.0975511	−	0.995231i	$-0.531101\pi$
$281$	−3.27051	−0.195102	−0.0975511	+	0.995231i	$0.531101\pi$
$282$	0	0
$283$	−1.34752	−0.0801020	−0.0400510	−	0.999198i	$-0.512752\pi$
$283$	−1.34752	−0.0801020	−0.0400510	+	0.999198i	$0.512752\pi$
$284$	0.618034	0.0366736
$285$	0	0
$286$	2.23607	0.132221
$287$	19.1459	1.13015
$288$	0	0
$289$	−12.0000	−0.705882
$290$	−4.85410	−0.285043
$291$	0	0
$292$	−1.70820	−0.0999651
$293$	−21.5066	−1.25643	−0.628214	−	0.778041i	$-0.716214\pi$
$293$	−21.5066	−1.25643	−0.628214	+	0.778041i	$0.716214\pi$
$294$	0	0
$295$	−10.8541	−0.631950
$296$	−25.8541	−1.50274
$297$	0	0
$298$	3.00000	0.173785
$299$	−41.5066	−2.40039
$300$	0	0
$301$	34.6869	1.99932
$302$	7.56231	0.435162
$303$	0	0
$304$	8.56231	0.491082
$305$	−10.8541	−0.621504
$306$	0	0
$307$	−12.8541	−0.733622	−0.366811	−	0.930295i	$-0.619550\pi$
$307$	−12.8541	−0.733622	−0.366811	+	0.930295i	$0.619550\pi$
$308$	0.437694	0.0249399
$309$	0	0
$310$	5.23607	0.297389
$311$	−22.1459	−1.25578	−0.627889	−	0.778303i	$-0.716081\pi$
$311$	−22.1459	−1.25578	−0.627889	+	0.778303i	$0.716081\pi$
$312$	0	0
$313$	−29.4164	−1.66271	−0.831357	−	0.555739i	$-0.812435\pi$
$313$	−29.4164	−1.66271	−0.831357	+	0.555739i	$0.812435\pi$
$314$	7.70820	0.434999
$315$	0	0
$316$	−3.76393	−0.211738
$317$	22.3820	1.25710	0.628548	−	0.777771i	$-0.283649\pi$
$317$	22.3820	1.25710	0.628548	+	0.777771i	$0.283649\pi$
$318$	0	0
$319$	−0.437694	−0.0245062
$320$	−6.85410	−0.383156
$321$	0	0
$322$	−34.4164	−1.91795
$323$	−3.94427	−0.219465
$324$	0	0
$325$	13.9443	0.773489
$326$	26.5623	1.47115
$327$	0	0
$328$	14.2705	0.787957
$329$	15.9787	0.880935
$330$	0	0
$331$	20.7426	1.14012	0.570059	−	0.821603i	$-0.306920\pi$
$331$	20.7426	1.14012	0.570059	+	0.821603i	$0.306920\pi$
$332$	1.85410	0.101757
$333$	0	0
$334$	−22.5623	−1.23455
$335$	4.70820	0.257237
$336$	0	0
$337$	14.4164	0.785312	0.392656	−	0.919685i	$-0.371556\pi$
$337$	14.4164	0.785312	0.392656	+	0.919685i	$0.371556\pi$
$338$	34.4164	1.87201
$339$	0	0
$340$	−2.23607	−0.121268
$341$	0.472136	0.0255676
$342$	0	0
$343$	15.0000	0.809924
$344$	25.8541	1.39396
$345$	0	0
$346$	24.0902	1.29510
$347$	−8.18034	−0.439144	−0.219572	−	0.975596i	$-0.570466\pi$
$347$	−8.18034	−0.439144	−0.219572	+	0.975596i	$0.570466\pi$
$348$	0	0
$349$	−26.4164	−1.41404	−0.707019	−	0.707195i	$-0.749960\pi$
$349$	−26.4164	−1.41404	−0.707019	+	0.707195i	$0.749960\pi$
$350$	11.5623	0.618031
$351$	0	0
$352$	0.798374	0.0425535
$353$	−8.03444	−0.427630	−0.213815	−	0.976874i	$-0.568589\pi$
$353$	−8.03444	−0.427630	−0.213815	+	0.976874i	$0.568589\pi$
$354$	0	0
$355$	−1.61803	−0.0858763
$356$	−7.85410	−0.416267
$357$	0	0
$358$	−10.2361	−0.540993
$359$	8.88854	0.469119	0.234560	−	0.972102i	$-0.424635\pi$
$359$	8.88854	0.469119	0.234560	+	0.972102i	$0.424635\pi$
$360$	0	0
$361$	−15.8885	−0.836239
$362$	−9.38197	−0.493105
$363$	0	0
$364$	10.8541	0.568910
$365$	4.47214	0.234082
$366$	0	0
$367$	33.9787	1.77367	0.886837	−	0.462082i	$-0.152897\pi$
$367$	33.9787	1.77367	0.886837	+	0.462082i	$0.152897\pi$
$368$	−34.4164	−1.79408
$369$	0	0
$370$	−30.2705	−1.57369
$371$	12.2705	0.637053
$372$	0	0
$373$	−23.1459	−1.19845	−0.599225	−	0.800581i	$-0.704524\pi$
$373$	−23.1459	−1.19845	−0.599225	+	0.800581i	$0.704524\pi$
$374$	−0.854102	−0.0441646
$375$	0	0
$376$	11.9098	0.614203
$377$	−10.8541	−0.559015
$378$	0	0
$379$	−15.1803	−0.779762	−0.389881	−	0.920865i	$-0.627484\pi$
$379$	−15.1803	−0.779762	−0.389881	+	0.920865i	$0.627484\pi$
$380$	1.76393	0.0904878
$381$	0	0
$382$	−30.2705	−1.54877
$383$	−9.47214	−0.484004	−0.242002	−	0.970276i	$-0.577804\pi$
$383$	−9.47214	−0.484004	−0.242002	+	0.970276i	$0.577804\pi$
$384$	0	0
$385$	−1.14590	−0.0584004
$386$	−23.3262	−1.18727
$387$	0	0
$388$	−9.00000	−0.456906
$389$	11.0557	0.560548	0.280274	−	0.959920i	$-0.409575\pi$
$389$	11.0557	0.560548	0.280274	+	0.959920i	$0.409575\pi$
$390$	0	0
$391$	15.8541	0.801776
$392$	−4.47214	−0.225877
$393$	0	0
$394$	1.47214	0.0741651
$395$	9.85410	0.495814
$396$	0	0
$397$	−30.5066	−1.53108	−0.765541	−	0.643388i	$-0.777528\pi$
$397$	−30.5066	−1.53108	−0.765541	+	0.643388i	$0.777528\pi$
$398$	6.94427	0.348085
$399$	0	0
$400$	11.5623	0.578115
$401$	−22.1459	−1.10591	−0.552957	−	0.833210i	$-0.686501\pi$
$401$	−22.1459	−1.10591	−0.552957	+	0.833210i	$0.686501\pi$
$402$	0	0
$403$	11.7082	0.583227
$404$	−5.85410	−0.291252
$405$	0	0
$406$	−9.00000	−0.446663
$407$	−2.72949	−0.135296
$408$	0	0
$409$	1.56231	0.0772511	0.0386255	−	0.999254i	$-0.487702\pi$
$409$	1.56231	0.0772511	0.0386255	+	0.999254i	$0.487702\pi$
$410$	16.7082	0.825159
$411$	0	0
$412$	9.56231	0.471101
$413$	−20.1246	−0.990267
$414$	0	0
$415$	−4.85410	−0.238278
$416$	19.7984	0.970695
$417$	0	0
$418$	0.673762	0.0329548
$419$	22.6180	1.10496	0.552482	−	0.833525i	$-0.313681\pi$
$419$	22.6180	1.10496	0.552482	+	0.833525i	$0.313681\pi$
$420$	0	0
$421$	−28.4164	−1.38493	−0.692465	−	0.721451i	$-0.743476\pi$
$421$	−28.4164	−1.38493	−0.692465	+	0.721451i	$0.743476\pi$
$422$	−37.4164	−1.82140
$423$	0	0
$424$	9.14590	0.444164
$425$	−5.32624	−0.258360
$426$	0	0
$427$	−20.1246	−0.973898
$428$	10.7426	0.519265
$429$	0	0
$430$	30.2705	1.45977
$431$	21.0689	1.01485	0.507426	−	0.861695i	$-0.330597\pi$
$431$	21.0689	1.01485	0.507426	+	0.861695i	$0.330597\pi$
$432$	0	0
$433$	16.1459	0.775922	0.387961	−	0.921676i	$-0.373179\pi$
$433$	16.1459	0.775922	0.387961	+	0.921676i	$0.373179\pi$
$434$	9.70820	0.466009
$435$	0	0
$436$	−5.56231	−0.266386
$437$	−12.5066	−0.598271
$438$	0	0
$439$	17.4377	0.832256	0.416128	−	0.909306i	$-0.363387\pi$
$439$	17.4377	0.832256	0.416128	+	0.909306i	$0.363387\pi$
$440$	−0.854102	−0.0407177
$441$	0	0
$442$	−21.1803	−1.00745
$443$	24.2705	1.15313	0.576563	−	0.817052i	$-0.304393\pi$
$443$	24.2705	1.15313	0.576563	+	0.817052i	$0.304393\pi$
$444$	0	0
$445$	20.5623	0.974747
$446$	−23.3262	−1.10453
$447$	0	0
$448$	−12.7082	−0.600406
$449$	14.2918	0.674472	0.337236	−	0.941420i	$-0.390508\pi$
$449$	14.2918	0.674472	0.337236	+	0.941420i	$0.390508\pi$
$450$	0	0
$451$	1.50658	0.0709420
$452$	3.23607	0.152212
$453$	0	0
$454$	18.0902	0.849014
$455$	−28.4164	−1.33218
$456$	0	0
$457$	36.1246	1.68984	0.844919	−	0.534894i	$-0.179649\pi$
$457$	36.1246	1.68984	0.844919	+	0.534894i	$0.179649\pi$
$458$	0.854102	0.0399096
$459$	0	0
$460$	−7.09017	−0.330581
$461$	−11.4721	−0.534311	−0.267155	−	0.963653i	$-0.586084\pi$
$461$	−11.4721	−0.534311	−0.267155	+	0.963653i	$0.586084\pi$
$462$	0	0
$463$	5.20163	0.241740	0.120870	−	0.992668i	$-0.461432\pi$
$463$	5.20163	0.241740	0.120870	+	0.992668i	$0.461432\pi$
$464$	−9.00000	−0.417815
$465$	0	0
$466$	−18.8885	−0.874995
$467$	−28.3050	−1.30980	−0.654898	−	0.755717i	$-0.727289\pi$
$467$	−28.3050	−1.30980	−0.654898	+	0.755717i	$0.727289\pi$
$468$	0	0
$469$	8.72949	0.403090
$470$	13.9443	0.643201
$471$	0	0
$472$	−15.0000	−0.690431
$473$	2.72949	0.125502
$474$	0	0
$475$	4.20163	0.192784
$476$	−4.14590	−0.190027
$477$	0	0
$478$	40.7426	1.86353
$479$	−18.9787	−0.867160	−0.433580	−	0.901115i	$-0.642750\pi$
$479$	−18.9787	−0.867160	−0.433580	+	0.901115i	$0.642750\pi$
$480$	0	0
$481$	−67.6869	−3.08626
$482$	−41.8328	−1.90543
$483$	0	0
$484$	−6.76393	−0.307451
$485$	23.5623	1.06991
$486$	0	0
$487$	5.90983	0.267800	0.133900	−	0.990995i	$-0.457250\pi$
$487$	5.90983	0.267800	0.133900	+	0.990995i	$0.457250\pi$
$488$	−15.0000	−0.679018
$489$	0	0
$490$	−5.23607	−0.236541
$491$	23.5279	1.06180	0.530899	−	0.847435i	$-0.321854\pi$
$491$	23.5279	1.06180	0.530899	+	0.847435i	$0.321854\pi$
$492$	0	0
$493$	4.14590	0.186722
$494$	16.7082	0.751738
$495$	0	0
$496$	9.70820	0.435911
$497$	−3.00000	−0.134568
$498$	0	0
$499$	1.70820	0.0764697	0.0382349	−	0.999269i	$-0.487827\pi$
$499$	1.70820	0.0764697	0.0382349	+	0.999269i	$0.487827\pi$
$500$	7.38197	0.330132
$501$	0	0
$502$	44.9787	2.00750
$503$	−28.7426	−1.28157	−0.640786	−	0.767720i	$-0.721391\pi$
$503$	−28.7426	−1.28157	−0.640786	+	0.767720i	$0.721391\pi$
$504$	0	0
$505$	15.3262	0.682009
$506$	−2.70820	−0.120394
$507$	0	0
$508$	2.90983	0.129103
$509$	−15.2361	−0.675327	−0.337663	−	0.941267i	$-0.609637\pi$
$509$	−15.2361	−0.675327	−0.337663	+	0.941267i	$0.609637\pi$
$510$	0	0
$511$	8.29180	0.366807
$512$	−5.29180	−0.233867
$513$	0	0
$514$	−13.7082	−0.604643
$515$	−25.0344	−1.10315
$516$	0	0
$517$	1.25735	0.0552984
$518$	−56.1246	−2.46597
$519$	0	0
$520$	−21.1803	−0.928819
$521$	−44.6525	−1.95626	−0.978130	−	0.207993i	$-0.933307\pi$
$521$	−44.6525	−1.95626	−0.978130	+	0.207993i	$0.933307\pi$
$522$	0	0
$523$	8.03444	0.351322	0.175661	−	0.984451i	$-0.443794\pi$
$523$	8.03444	0.351322	0.175661	+	0.984451i	$0.443794\pi$
$524$	−13.8885	−0.606724
$525$	0	0
$526$	15.2705	0.665826
$527$	−4.47214	−0.194809
$528$	0	0
$529$	27.2705	1.18567
$530$	10.7082	0.465135
$531$	0	0
$532$	3.27051	0.141795
$533$	37.3607	1.61827
$534$	0	0
$535$	−28.1246	−1.21593
$536$	6.50658	0.281041
$537$	0	0
$538$	51.8885	2.23707
$539$	−0.472136	−0.0203363
$540$	0	0
$541$	−19.0000	−0.816874	−0.408437	−	0.912787i	$-0.633926\pi$
$541$	−19.0000	−0.816874	−0.408437	+	0.912787i	$0.633926\pi$
$542$	−0.472136	−0.0202800
$543$	0	0
$544$	−7.56231	−0.324231
$545$	14.5623	0.623781
$546$	0	0
$547$	−13.6180	−0.582265	−0.291133	−	0.956683i	$-0.594032\pi$
$547$	−13.6180	−0.582265	−0.291133	+	0.956683i	$0.594032\pi$
$548$	−6.00000	−0.256307
$549$	0	0
$550$	0.909830	0.0387953
$551$	−3.27051	−0.139328
$552$	0	0
$553$	18.2705	0.776941
$554$	27.5623	1.17101
$555$	0	0
$556$	0.0557281	0.00236340
$557$	10.4164	0.441357	0.220679	−	0.975347i	$-0.429173\pi$
$557$	10.4164	0.441357	0.220679	+	0.975347i	$0.429173\pi$
$558$	0	0
$559$	67.6869	2.86285
$560$	−23.5623	−0.995689
$561$	0	0
$562$	−5.29180	−0.223221
$563$	41.0344	1.72940	0.864698	−	0.502293i	$-0.167510\pi$
$563$	41.0344	1.72940	0.864698	+	0.502293i	$0.167510\pi$
$564$	0	0
$565$	−8.47214	−0.356425
$566$	−2.18034	−0.0916465
$567$	0	0
$568$	−2.23607	−0.0938233
$569$	30.5967	1.28268	0.641341	−	0.767256i	$-0.278378\pi$
$569$	30.5967	1.28268	0.641341	+	0.767256i	$0.278378\pi$
$570$	0	0
$571$	−28.2705	−1.18308	−0.591542	−	0.806274i	$-0.701481\pi$
$571$	−28.2705	−1.18308	−0.591542	+	0.806274i	$0.701481\pi$
$572$	0.854102	0.0357118
$573$	0	0
$574$	30.9787	1.29303
$575$	−16.8885	−0.704301
$576$	0	0
$577$	11.1459	0.464010	0.232005	−	0.972715i	$-0.425471\pi$
$577$	11.1459	0.464010	0.232005	+	0.972715i	$0.425471\pi$
$578$	−19.4164	−0.807616
$579$	0	0
$580$	−1.85410	−0.0769874
$581$	−9.00000	−0.373383
$582$	0	0
$583$	0.965558	0.0399893
$584$	6.18034	0.255744
$585$	0	0
$586$	−34.7984	−1.43751
$587$	30.5410	1.26056	0.630281	−	0.776367i	$-0.282940\pi$
$587$	30.5410	1.26056	0.630281	+	0.776367i	$0.282940\pi$
$588$	0	0
$589$	3.52786	0.145363
$590$	−17.5623	−0.723029
$591$	0	0
$592$	−56.1246	−2.30671
$593$	−0.708204	−0.0290824	−0.0145412	−	0.999894i	$-0.504629\pi$
$593$	−0.708204	−0.0290824	−0.0145412	+	0.999894i	$0.504629\pi$
$594$	0	0
$595$	10.8541	0.444975
$596$	1.14590	0.0469378
$597$	0	0
$598$	−67.1591	−2.74634
$599$	41.6312	1.70100	0.850502	−	0.525972i	$-0.176298\pi$
$599$	41.6312	1.70100	0.850502	+	0.525972i	$0.176298\pi$
$600$	0	0
$601$	7.50658	0.306200	0.153100	−	0.988211i	$-0.451074\pi$
$601$	7.50658	0.306200	0.153100	+	0.988211i	$0.451074\pi$
$602$	56.1246	2.28747
$603$	0	0
$604$	2.88854	0.117533
$605$	17.7082	0.719941
$606$	0	0
$607$	11.2918	0.458320	0.229160	−	0.973389i	$-0.426402\pi$
$607$	11.2918	0.458320	0.229160	+	0.973389i	$0.426402\pi$
$608$	5.96556	0.241935
$609$	0	0
$610$	−17.5623	−0.711077
$611$	31.1803	1.26142
$612$	0	0
$613$	5.12461	0.206981	0.103491	−	0.994630i	$-0.466999\pi$
$613$	5.12461	0.206981	0.103491	+	0.994630i	$0.466999\pi$
$614$	−20.7984	−0.839354
$615$	0	0
$616$	−1.58359	−0.0638047
$617$	30.5410	1.22954	0.614768	−	0.788708i	$-0.289250\pi$
$617$	30.5410	1.22954	0.614768	+	0.788708i	$0.289250\pi$
$618$	0	0
$619$	28.3951	1.14130	0.570648	−	0.821195i	$-0.306692\pi$
$619$	28.3951	1.14130	0.570648	+	0.821195i	$0.306692\pi$
$620$	2.00000	0.0803219
$621$	0	0
$622$	−35.8328	−1.43677
$623$	38.1246	1.52743
$624$	0	0
$625$	−7.41641	−0.296656
$626$	−47.5967	−1.90235
$627$	0	0
$628$	2.94427	0.117489
$629$	25.8541	1.03087
$630$	0	0
$631$	21.9443	0.873588	0.436794	−	0.899562i	$-0.356114\pi$
$631$	21.9443	0.873588	0.436794	+	0.899562i	$0.356114\pi$
$632$	13.6180	0.541696
$633$	0	0
$634$	36.2148	1.43827
$635$	−7.61803	−0.302312
$636$	0	0
$637$	−11.7082	−0.463896
$638$	−0.708204	−0.0280381
$639$	0	0
$640$	−22.0344	−0.870988
$641$	48.6525	1.92166	0.960829	−	0.277143i	$-0.0893876\pi$
$641$	48.6525	1.92166	0.960829	+	0.277143i	$0.0893876\pi$
$642$	0	0
$643$	26.4377	1.04260	0.521300	−	0.853373i	$-0.325447\pi$
$643$	26.4377	1.04260	0.521300	+	0.853373i	$0.325447\pi$
$644$	−13.1459	−0.518021
$645$	0	0
$646$	−6.38197	−0.251095
$647$	19.0902	0.750512	0.375256	−	0.926921i	$-0.377555\pi$
$647$	19.0902	0.750512	0.375256	+	0.926921i	$0.377555\pi$
$648$	0	0
$649$	−1.58359	−0.0621614
$650$	22.5623	0.884966
$651$	0	0
$652$	10.1459	0.397344
$653$	−6.79837	−0.266041	−0.133020	−	0.991113i	$-0.542468\pi$
$653$	−6.79837	−0.266041	−0.133020	+	0.991113i	$0.542468\pi$
$654$	0	0
$655$	36.3607	1.42073
$656$	30.9787	1.20952
$657$	0	0
$658$	25.8541	1.00790
$659$	−14.2361	−0.554558	−0.277279	−	0.960789i	$-0.589433\pi$
$659$	−14.2361	−0.554558	−0.277279	+	0.960789i	$0.589433\pi$
$660$	0	0
$661$	−33.8885	−1.31811	−0.659056	−	0.752094i	$-0.729044\pi$
$661$	−33.8885	−1.31811	−0.659056	+	0.752094i	$0.729044\pi$
$662$	33.5623	1.30444
$663$	0	0
$664$	−6.70820	−0.260329
$665$	−8.56231	−0.332032
$666$	0	0
$667$	13.1459	0.509011
$668$	−8.61803	−0.333442
$669$	0	0
$670$	7.61803	0.294310
$671$	−1.58359	−0.0611339
$672$	0	0
$673$	10.2705	0.395899	0.197950	−	0.980212i	$-0.436572\pi$
$673$	10.2705	0.395899	0.197950	+	0.980212i	$0.436572\pi$
$674$	23.3262	0.898493
$675$	0	0
$676$	13.1459	0.505611
$677$	−12.9787	−0.498812	−0.249406	−	0.968399i	$-0.580235\pi$
$677$	−12.9787	−0.498812	−0.249406	+	0.968399i	$0.580235\pi$
$678$	0	0
$679$	43.6869	1.67655
$680$	8.09017	0.310244
$681$	0	0
$682$	0.763932	0.0292525
$683$	36.8885	1.41150	0.705750	−	0.708461i	$-0.250610\pi$
$683$	36.8885	1.41150	0.705750	+	0.708461i	$0.250610\pi$
$684$	0	0
$685$	15.7082	0.600180
$686$	24.2705	0.926652
$687$	0	0
$688$	56.1246	2.13973
$689$	23.9443	0.912204
$690$	0	0
$691$	−14.4377	−0.549236	−0.274618	−	0.961553i	$-0.588551\pi$
$691$	−14.4377	−0.549236	−0.274618	+	0.961553i	$0.588551\pi$
$692$	9.20163	0.349793
$693$	0	0
$694$	−13.2361	−0.502434
$695$	−0.145898	−0.00553423
$696$	0	0
$697$	−14.2705	−0.540534
$698$	−42.7426	−1.61783
$699$	0	0
$700$	4.41641	0.166925
$701$	−8.83282	−0.333611	−0.166805	−	0.985990i	$-0.553345\pi$
$701$	−8.83282	−0.333611	−0.166805	+	0.985990i	$0.553345\pi$
$702$	0	0
$703$	−20.3951	−0.769217
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−13.0000	−0.489261
$707$	28.4164	1.06871
$708$	0	0
$709$	42.2361	1.58621	0.793104	−	0.609086i	$-0.208463\pi$
$709$	42.2361	1.58621	0.793104	+	0.609086i	$0.208463\pi$
$710$	−2.61803	−0.0982531
$711$	0	0
$712$	28.4164	1.06495
$713$	−14.1803	−0.531058
$714$	0	0
$715$	−2.23607	−0.0836242
$716$	−3.90983	−0.146117
$717$	0	0
$718$	14.3820	0.536730
$719$	−21.9098	−0.817099	−0.408549	−	0.912736i	$-0.633965\pi$
$719$	−21.9098	−0.817099	−0.408549	+	0.912736i	$0.633965\pi$
$720$	0	0
$721$	−46.4164	−1.72864
$722$	−25.7082	−0.956760
$723$	0	0
$724$	−3.58359	−0.133183
$725$	−4.41641	−0.164021
$726$	0	0
$727$	43.2705	1.60481	0.802407	−	0.596777i	$-0.203552\pi$
$727$	43.2705	1.60481	0.802407	+	0.596777i	$0.203552\pi$
$728$	−39.2705	−1.45546
$729$	0	0
$730$	7.23607	0.267819
$731$	−25.8541	−0.956249
$732$	0	0
$733$	−41.8328	−1.54513	−0.772565	−	0.634935i	$-0.781027\pi$
$733$	−41.8328	−1.54513	−0.772565	+	0.634935i	$0.781027\pi$
$734$	54.9787	2.02930
$735$	0	0
$736$	−23.9787	−0.883867
$737$	0.686918	0.0253029
$738$	0	0
$739$	−39.3394	−1.44712	−0.723561	−	0.690260i	$-0.757496\pi$
$739$	−39.3394	−1.44712	−0.723561	+	0.690260i	$0.757496\pi$
$740$	−11.5623	−0.425039
$741$	0	0
$742$	19.8541	0.728867
$743$	−11.1246	−0.408122	−0.204061	−	0.978958i	$-0.565414\pi$
$743$	−11.1246	−0.408122	−0.204061	+	0.978958i	$0.565414\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	−37.4508	−1.37117
$747$	0	0
$748$	−0.326238	−0.0119284
$749$	−52.1459	−1.90537
$750$	0	0
$751$	3.41641	0.124666	0.0623332	−	0.998055i	$-0.480146\pi$
$751$	3.41641	0.124666	0.0623332	+	0.998055i	$0.480146\pi$
$752$	25.8541	0.942802
$753$	0	0
$754$	−17.5623	−0.639581
$755$	−7.56231	−0.275220
$756$	0	0
$757$	4.27051	0.155214	0.0776072	−	0.996984i	$-0.475272\pi$
$757$	4.27051	0.155214	0.0776072	+	0.996984i	$0.475272\pi$
$758$	−24.5623	−0.892143
$759$	0	0
$760$	−6.38197	−0.231498
$761$	34.4508	1.24884	0.624421	−	0.781088i	$-0.285335\pi$
$761$	34.4508	1.24884	0.624421	+	0.781088i	$0.285335\pi$
$762$	0	0
$763$	27.0000	0.977466
$764$	−11.5623	−0.418310
$765$	0	0
$766$	−15.3262	−0.553759
$767$	−39.2705	−1.41798
$768$	0	0
$769$	−29.0000	−1.04577	−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000	−1.04577	−0.522883	+	0.852404i	$0.675144\pi$
$770$	−1.85410	−0.0668172
$771$	0	0
$772$	−8.90983	−0.320672
$773$	40.0132	1.43917	0.719587	−	0.694403i	$-0.244331\pi$
$773$	40.0132	1.43917	0.719587	+	0.694403i	$0.244331\pi$
$774$	0	0
$775$	4.76393	0.171125
$776$	32.5623	1.16892
$777$	0	0
$778$	17.8885	0.641335
$779$	11.2574	0.403337
$780$	0	0
$781$	−0.236068	−0.00844718
$782$	25.6525	0.917331
$783$	0	0
$784$	−9.70820	−0.346722
$785$	−7.70820	−0.275118
$786$	0	0
$787$	−32.3951	−1.15476	−0.577381	−	0.816475i	$-0.695925\pi$
$787$	−32.3951	−1.15476	−0.577381	+	0.816475i	$0.695925\pi$
$788$	0.562306	0.0200313
$789$	0	0
$790$	15.9443	0.567272
$791$	−15.7082	−0.558519
$792$	0	0
$793$	−39.2705	−1.39454
$794$	−49.3607	−1.75174
$795$	0	0
$796$	2.65248	0.0940145
$797$	−43.0344	−1.52436	−0.762179	−	0.647366i	$-0.775870\pi$
$797$	−43.0344	−1.52436	−0.762179	+	0.647366i	$0.775870\pi$
$798$	0	0
$799$	−11.9098	−0.421339
$800$	8.05573	0.284813
$801$	0	0
$802$	−35.8328	−1.26530
$803$	0.652476	0.0230254
$804$	0	0
$805$	34.4164	1.21302
$806$	18.9443	0.667284
$807$	0	0
$808$	21.1803	0.745122
$809$	24.5967	0.864776	0.432388	−	0.901688i	$-0.357671\pi$
$809$	24.5967	0.864776	0.432388	+	0.901688i	$0.357671\pi$
$810$	0	0
$811$	11.1459	0.391385	0.195693	−	0.980665i	$-0.437304\pi$
$811$	11.1459	0.391385	0.195693	+	0.980665i	$0.437304\pi$
$812$	−3.43769	−0.120639
$813$	0	0
$814$	−4.41641	−0.154795
$815$	−26.5623	−0.930437
$816$	0	0
$817$	20.3951	0.713535
$818$	2.52786	0.0883847
$819$	0	0
$820$	6.38197	0.222868
$821$	−41.4721	−1.44739	−0.723694	−	0.690121i	$-0.757557\pi$
$821$	−41.4721	−1.44739	−0.723694	+	0.690121i	$0.757557\pi$
$822$	0	0
$823$	45.4508	1.58432	0.792159	−	0.610315i	$-0.208957\pi$
$823$	45.4508	1.58432	0.792159	+	0.610315i	$0.208957\pi$
$824$	−34.5967	−1.20523
$825$	0	0
$826$	−32.5623	−1.13299
$827$	−16.9098	−0.588012	−0.294006	−	0.955804i	$-0.594989\pi$
$827$	−16.9098	−0.588012	−0.294006	+	0.955804i	$0.594989\pi$
$828$	0	0
$829$	13.9443	0.484305	0.242152	−	0.970238i	$-0.422147\pi$
$829$	13.9443	0.484305	0.242152	+	0.970238i	$0.422147\pi$
$830$	−7.85410	−0.272620
$831$	0	0
$832$	−24.7984	−0.859729
$833$	4.47214	0.154950
$834$	0	0
$835$	22.5623	0.780801
$836$	0.257354	0.00890078
$837$	0	0
$838$	36.5967	1.26421
$839$	−17.8197	−0.615203	−0.307601	−	0.951515i	$-0.599526\pi$
$839$	−17.8197	−0.615203	−0.307601	+	0.951515i	$0.599526\pi$
$840$	0	0
$841$	−25.5623	−0.881459
$842$	−45.9787	−1.58453
$843$	0	0
$844$	−14.2918	−0.491944
$845$	−34.4164	−1.18396
$846$	0	0
$847$	32.8328	1.12815
$848$	19.8541	0.681793
$849$	0	0
$850$	−8.61803	−0.295596
$851$	81.9787	2.81019
$852$	0	0
$853$	9.27051	0.317416	0.158708	−	0.987326i	$-0.449267\pi$
$853$	9.27051	0.317416	0.158708	+	0.987326i	$0.449267\pi$
$854$	−32.5623	−1.11426
$855$	0	0
$856$	−38.8673	−1.32846
$857$	−46.3820	−1.58438	−0.792189	−	0.610276i	$-0.791059\pi$
$857$	−46.3820	−1.58438	−0.792189	+	0.610276i	$0.791059\pi$
$858$	0	0
$859$	11.2918	0.385271	0.192636	−	0.981270i	$-0.438296\pi$
$859$	11.2918	0.385271	0.192636	+	0.981270i	$0.438296\pi$
$860$	11.5623	0.394271
$861$	0	0
$862$	34.0902	1.16112
$863$	−2.74265	−0.0933607	−0.0466804	−	0.998910i	$-0.514864\pi$
$863$	−2.74265	−0.0933607	−0.0466804	+	0.998910i	$0.514864\pi$
$864$	0	0
$865$	−24.0902	−0.819090
$866$	26.1246	0.887750
$867$	0	0
$868$	3.70820	0.125865
$869$	1.43769	0.0487704
$870$	0	0
$871$	17.0344	0.577190
$872$	20.1246	0.681505
$873$	0	0
$874$	−20.2361	−0.684495
$875$	−35.8328	−1.21137
$876$	0	0
$877$	−37.8885	−1.27941	−0.639703	−	0.768623i	$-0.720942\pi$
$877$	−37.8885	−1.27941	−0.639703	+	0.768623i	$0.720942\pi$
$878$	28.2148	0.952203
$879$	0	0
$880$	−1.85410	−0.0625018
$881$	−21.2148	−0.714744	−0.357372	−	0.933962i	$-0.616327\pi$
$881$	−21.2148	−0.714744	−0.357372	+	0.933962i	$0.616327\pi$
$882$	0	0
$883$	35.8328	1.20587	0.602935	−	0.797790i	$-0.293998\pi$
$883$	35.8328	1.20587	0.602935	+	0.797790i	$0.293998\pi$
$884$	−8.09017	−0.272102
$885$	0	0
$886$	39.2705	1.31932
$887$	−20.9443	−0.703240	−0.351620	−	0.936143i	$-0.614369\pi$
$887$	−20.9443	−0.703240	−0.351620	+	0.936143i	$0.614369\pi$
$888$	0	0
$889$	−14.1246	−0.473724
$890$	33.2705	1.11523
$891$	0	0
$892$	−8.90983	−0.298323
$893$	9.39512	0.314396
$894$	0	0
$895$	10.2361	0.342154
$896$	−40.8541	−1.36484
$897$	0	0
$898$	23.1246	0.771678
$899$	−3.70820	−0.123676
$900$	0	0
$901$	−9.14590	−0.304694
$902$	2.43769	0.0811663
$903$	0	0
$904$	−11.7082	−0.389409
$905$	9.38197	0.311867
$906$	0	0
$907$	−8.27051	−0.274618	−0.137309	−	0.990528i	$-0.543845\pi$
$907$	−8.27051	−0.274618	−0.137309	+	0.990528i	$0.543845\pi$
$908$	6.90983	0.229311
$909$	0	0
$910$	−45.9787	−1.52418
$911$	−48.6312	−1.61122	−0.805612	−	0.592444i	$-0.798163\pi$
$911$	−48.6312	−1.61122	−0.805612	+	0.592444i	$0.798163\pi$
$912$	0	0
$913$	−0.708204	−0.0234381
$914$	58.4508	1.93338
$915$	0	0
$916$	0.326238	0.0107792
$917$	67.4164	2.22629
$918$	0	0
$919$	−44.1246	−1.45554	−0.727768	−	0.685823i	$-0.759442\pi$
$919$	−44.1246	−1.45554	−0.727768	+	0.685823i	$0.759442\pi$
$920$	25.6525	0.845737
$921$	0	0
$922$	−18.5623	−0.611317
$923$	−5.85410	−0.192690
$924$	0	0
$925$	−27.5410	−0.905543
$926$	8.41641	0.276580
$927$	0	0
$928$	−6.27051	−0.205840
$929$	−25.4164	−0.833885	−0.416943	−	0.908933i	$-0.636898\pi$
$929$	−25.4164	−0.833885	−0.416943	+	0.908933i	$0.636898\pi$
$930$	0	0
$931$	−3.52786	−0.115621
$932$	−7.21478	−0.236328
$933$	0	0
$934$	−45.7984	−1.49857
$935$	0.854102	0.0279321
$936$	0	0
$937$	20.5623	0.671741	0.335871	−	0.941908i	$-0.390970\pi$
$937$	20.5623	0.671741	0.335871	+	0.941908i	$0.390970\pi$
$938$	14.1246	0.461185
$939$	0	0
$940$	5.32624	0.173723
$941$	51.5967	1.68201	0.841003	−	0.541031i	$-0.181966\pi$
$941$	51.5967	1.68201	0.841003	+	0.541031i	$0.181966\pi$
$942$	0	0
$943$	−45.2492	−1.47352
$944$	−32.5623	−1.05981
$945$	0	0
$946$	4.41641	0.143590
$947$	−41.7771	−1.35757	−0.678786	−	0.734336i	$-0.737494\pi$
$947$	−41.7771	−1.35757	−0.678786	+	0.734336i	$0.737494\pi$
$948$	0	0
$949$	16.1803	0.525236
$950$	6.79837	0.220568
$951$	0	0
$952$	15.0000	0.486153
$953$	−33.0689	−1.07121	−0.535603	−	0.844470i	$-0.679916\pi$
$953$	−33.0689	−1.07121	−0.535603	+	0.844470i	$0.679916\pi$
$954$	0	0
$955$	30.2705	0.979531
$956$	15.5623	0.503321
$957$	0	0
$958$	−30.7082	−0.992137
$959$	29.1246	0.940483
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−109.520	−3.53106
$963$	0	0
$964$	−15.9787	−0.514640
$965$	23.3262	0.750898
$966$	0	0
$967$	−30.0902	−0.967635	−0.483817	−	0.875169i	$-0.660750\pi$
$967$	−30.0902	−0.967635	−0.483817	+	0.875169i	$0.660750\pi$
$968$	24.4721	0.786564
$969$	0	0
$970$	38.1246	1.22411
$971$	0.708204	0.0227273	0.0113637	−	0.999935i	$-0.496383\pi$
$971$	0.708204	0.0227273	0.0113637	+	0.999935i	$0.496383\pi$
$972$	0	0
$973$	−0.270510	−0.00867215
$974$	9.56231	0.306396
$975$	0	0
$976$	−32.5623	−1.04229
$977$	13.4721	0.431012	0.215506	−	0.976503i	$-0.430860\pi$
$977$	13.4721	0.431012	0.215506	+	0.976503i	$0.430860\pi$
$978$	0	0
$979$	3.00000	0.0958804
$980$	−2.00000	−0.0638877
$981$	0	0
$982$	38.0689	1.21483
$983$	−42.4377	−1.35355	−0.676776	−	0.736189i	$-0.736623\pi$
$983$	−42.4377	−1.35355	−0.676776	+	0.736189i	$0.736623\pi$
$984$	0	0
$985$	−1.47214	−0.0469062
$986$	6.70820	0.213633
$987$	0	0
$988$	6.38197	0.203037
$989$	−81.9787	−2.60677
$990$	0	0
$991$	−47.6869	−1.51482	−0.757412	−	0.652937i	$-0.773537\pi$
$991$	−47.6869	−1.51482	−0.757412	+	0.652937i	$0.773537\pi$
$992$	6.76393	0.214755
$993$	0	0
$994$	−4.85410	−0.153963
$995$	−6.94427	−0.220148
$996$	0	0
$997$	22.3951	0.709261	0.354630	−	0.935007i	$-0.384607\pi$
$997$	22.3951	0.709261	0.354630	+	0.935007i	$0.384607\pi$
$998$	2.76393	0.0874907
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.61803 q^{5} -3.00000 q^{7} -2.23607 q^{8} +O(q^{10})\)	\(q+1.61803 q^{2} +0.618034 q^{4} -1.61803 q^{5} -3.00000 q^{7} -2.23607 q^{8} -2.61803 q^{10} -0.236068 q^{11} -5.85410 q^{13} -4.85410 q^{14} -4.85410 q^{16} +2.23607 q^{17} -1.76393 q^{19} -1.00000 q^{20} -0.381966 q^{22} +7.09017 q^{23} -2.38197 q^{25} -9.47214 q^{26} -1.85410 q^{28} +1.85410 q^{29} -2.00000 q^{31} -3.38197 q^{32} +3.61803 q^{34} +4.85410 q^{35} +11.5623 q^{37} -2.85410 q^{38} +3.61803 q^{40} -6.38197 q^{41} -11.5623 q^{43} -0.145898 q^{44} +11.4721 q^{46} -5.32624 q^{47} +2.00000 q^{49} -3.85410 q^{50} -3.61803 q^{52} -4.09017 q^{53} +0.381966 q^{55} +6.70820 q^{56} +3.00000 q^{58} +6.70820 q^{59} +6.70820 q^{61} -3.23607 q^{62} +4.23607 q^{64} +9.47214 q^{65} -2.90983 q^{67} +1.38197 q^{68} +7.85410 q^{70} +1.00000 q^{71} -2.76393 q^{73} +18.7082 q^{74} -1.09017 q^{76} +0.708204 q^{77} -6.09017 q^{79} +7.85410 q^{80} -10.3262 q^{82} +3.00000 q^{83} -3.61803 q^{85} -18.7082 q^{86} +0.527864 q^{88} -12.7082 q^{89} +17.5623 q^{91} +4.38197 q^{92} -8.61803 q^{94} +2.85410 q^{95} -14.5623 q^{97} +3.23607 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - q^{4} - q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + q^2 - q^4 - q^5 - 6 * q^7	\( 2 q + q^{2} - q^{4} - q^{5} - 6 q^{7} - 3 q^{10} + 4 q^{11} - 5 q^{13} - 3 q^{14} - 3 q^{16} - 8 q^{19} - 2 q^{20} - 3 q^{22} + 3 q^{23} - 7 q^{25} - 10 q^{26} + 3 q^{28} - 3 q^{29} - 4 q^{31} - 9 q^{32} + 5 q^{34} + 3 q^{35} + 3 q^{37} + q^{38} + 5 q^{40} - 15 q^{41} - 3 q^{43} - 7 q^{44} + 14 q^{46} + 5 q^{47} + 4 q^{49} - q^{50} - 5 q^{52} + 3 q^{53} + 3 q^{55} + 6 q^{58} - 2 q^{62} + 4 q^{64} + 10 q^{65} - 17 q^{67} + 5 q^{68} + 9 q^{70} + 2 q^{71} - 10 q^{73} + 24 q^{74} + 9 q^{76} - 12 q^{77} - q^{79} + 9 q^{80} - 5 q^{82} + 6 q^{83} - 5 q^{85} - 24 q^{86} + 10 q^{88} - 12 q^{89} + 15 q^{91} + 11 q^{92} - 15 q^{94} - q^{95} - 9 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + q^2 - q^4 - q^5 - 6 * q^7 - 3 * q^10 + 4 * q^11 - 5 * q^13 - 3 * q^14 - 3 * q^16 - 8 * q^19 - 2 * q^20 - 3 * q^22 + 3 * q^23 - 7 * q^25 - 10 * q^26 + 3 * q^28 - 3 * q^29 - 4 * q^31 - 9 * q^32 + 5 * q^34 + 3 * q^35 + 3 * q^37 + q^38 + 5 * q^40 - 15 * q^41 - 3 * q^43 - 7 * q^44 + 14 * q^46 + 5 * q^47 + 4 * q^49 - q^50 - 5 * q^52 + 3 * q^53 + 3 * q^55 + 6 * q^58 - 2 * q^62 + 4 * q^64 + 10 * q^65 - 17 * q^67 + 5 * q^68 + 9 * q^70 + 2 * q^71 - 10 * q^73 + 24 * q^74 + 9 * q^76 - 12 * q^77 - q^79 + 9 * q^80 - 5 * q^82 + 6 * q^83 - 5 * q^85 - 24 * q^86 + 10 * q^88 - 12 * q^89 + 15 * q^91 + 11 * q^92 - 15 * q^94 - q^95 - 9 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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