LMFDB - Embedded newform 1682.2.a.q.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1682,2,Mod(1,1682)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1682, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1682.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$1682 = 2 \cdot 29^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1682.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:	$13.4308376200$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.13716913.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 2x^{5} - 13x^{4} + 17x^{3} + 52x^{2} - 32x - 64$ x^6 - 2x^5 - 13x^4 + 17x^3 + 52x^2 - 32*x - 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 58)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.63883$ of defining polynomial
Character	$\chi$	$=$	1682.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +2.44077 q^{3} +1.00000 q^{4} -2.88581 q^{5} -2.44077 q^{6} -3.04359 q^{7} -1.00000 q^{8} +2.95734 q^{9} +2.88581 q^{10} +5.48435 q^{11} +2.44077 q^{12} -0.107544 q^{13} +3.04359 q^{14} -7.04359 q^{15} +1.00000 q^{16} -0.816005 q^{17} -2.95734 q^{18} -2.04266 q^{19} -2.88581 q^{20} -7.42869 q^{21} -5.48435 q^{22} -9.16509 q^{23} -2.44077 q^{24} +3.32789 q^{25} +0.107544 q^{26} -0.104116 q^{27} -3.04359 q^{28} +7.04359 q^{30} +3.44170 q^{31} -1.00000 q^{32} +13.3860 q^{33} +0.816005 q^{34} +8.78321 q^{35} +2.95734 q^{36} -5.90758 q^{37} +2.04266 q^{38} -0.262489 q^{39} +2.88581 q^{40} +2.43376 q^{41} +7.42869 q^{42} -5.48435 q^{43} +5.48435 q^{44} -8.53433 q^{45} +9.16509 q^{46} -5.91000 q^{47} +2.44077 q^{48} +2.26342 q^{49} -3.32789 q^{50} -1.99168 q^{51} -0.107544 q^{52} +3.95070 q^{53} +0.104116 q^{54} -15.8268 q^{55} +3.04359 q^{56} -4.98565 q^{57} +1.13359 q^{59} -7.04359 q^{60} -8.16584 q^{61} -3.44170 q^{62} -9.00093 q^{63} +1.00000 q^{64} +0.310351 q^{65} -13.3860 q^{66} +0.956413 q^{67} -0.816005 q^{68} -22.3699 q^{69} -8.78321 q^{70} +2.38979 q^{71} -2.95734 q^{72} -8.89410 q^{73} +5.90758 q^{74} +8.12261 q^{75} -2.04266 q^{76} -16.6921 q^{77} +0.262489 q^{78} +4.20062 q^{79} -2.88581 q^{80} -9.12615 q^{81} -2.43376 q^{82} -15.4732 q^{83} -7.42869 q^{84} +2.35483 q^{85} +5.48435 q^{86} -5.48435 q^{88} -5.54154 q^{89} +8.53433 q^{90} +0.327319 q^{91} -9.16509 q^{92} +8.40038 q^{93} +5.91000 q^{94} +5.89472 q^{95} -2.44077 q^{96} -11.9217 q^{97} -2.26342 q^{98} +16.2191 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{2} - 2 q^{3} + 6 q^{4} + 2 q^{6} - 3 q^{7} - 6 q^{8} + 12 q^{9} + q^{11} - 2 q^{12} - 3 q^{13} + 3 q^{14} - 27 q^{15} + 6 q^{16} + 6 q^{17} - 12 q^{18} - 18 q^{19} + 10 q^{21} - q^{22} - 14 q^{23}+ \cdots + 4 q^{99}+O(q^{100})$ 6 * q - 6 * q^2 - 2 * q^3 + 6 * q^4 + 2 * q^6 - 3 * q^7 - 6 * q^8 + 12 * q^9 + q^11 - 2 * q^12 - 3 * q^13 + 3 * q^14 - 27 * q^15 + 6 * q^16 + 6 * q^17 - 12 * q^18 - 18 * q^19 + 10 * q^21 - q^22 - 14 * q^23 + 2 * q^24 + 4 * q^25 + 3 * q^26 - 23 * q^27 - 3 * q^28 + 27 * q^30 - 17 * q^31 - 6 * q^32 + 20 * q^33 - 6 * q^34 - 9 * q^35 + 12 * q^36 - 12 * q^37 + 18 * q^38 - 10 * q^39 + 15 * q^41 - 10 * q^42 - q^43 + q^44 + 18 * q^45 + 14 * q^46 - 8 * q^47 - 2 * q^48 + q^49 - 4 * q^50 - 11 * q^51 - 3 * q^52 + 3 * q^53 + 23 * q^54 - 18 * q^55 + 3 * q^56 - 19 * q^57 + 19 * q^59 - 27 * q^60 - 15 * q^61 + 17 * q^62 - 33 * q^63 + 6 * q^64 + 11 * q^65 - 20 * q^66 + 21 * q^67 + 6 * q^68 + 11 * q^69 + 9 * q^70 - 9 * q^71 - 12 * q^72 - 31 * q^73 + 12 * q^74 + q^75 - 18 * q^76 - 33 * q^77 + 10 * q^78 + 2 * q^79 + 2 * q^81 - 15 * q^82 - 2 * q^83 + 10 * q^84 - 19 * q^85 + q^86 - q^88 - 2 * q^89 - 18 * q^90 - 22 * q^91 - 14 * q^92 + q^93 + 8 * q^94 + 18 * q^95 + 2 * q^96 - 43 * q^97 - q^98 + 4 * q^99

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

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	2.44077	1.40918	0.704589	−	0.709616i	$-0.251132\pi$
$3$	2.44077	1.40918	0.704589	+	0.709616i	$0.251132\pi$
$4$	1.00000	0.500000
$5$	−2.88581	−1.29057	−0.645286	−	0.763941i	$-0.723262\pi$
$5$	−2.88581	−1.29057	−0.645286	+	0.763941i	$0.723262\pi$
$6$	−2.44077	−0.996439
$7$	−3.04359	−1.15037	−0.575184	−	0.818024i	$-0.695069\pi$
$7$	−3.04359	−1.15037	−0.575184	+	0.818024i	$0.695069\pi$
$8$	−1.00000	−0.353553
$9$	2.95734	0.985781
$10$	2.88581	0.912573
$11$	5.48435	1.65359	0.826797	−	0.562500i	$-0.190160\pi$
$11$	5.48435	1.65359	0.826797	+	0.562500i	$0.190160\pi$
$12$	2.44077	0.704589
$13$	−0.107544	−0.0298273	−0.0149136	−	0.999889i	$-0.504747\pi$
$13$	−0.107544	−0.0298273	−0.0149136	+	0.999889i	$0.504747\pi$
$14$	3.04359	0.813433
$15$	−7.04359	−1.81865
$16$	1.00000	0.250000
$17$	−0.816005	−0.197910	−0.0989551	−	0.995092i	$-0.531550\pi$
$17$	−0.816005	−0.197910	−0.0989551	+	0.995092i	$0.531550\pi$
$18$	−2.95734	−0.697052
$19$	−2.04266	−0.468618	−0.234309	−	0.972162i	$-0.575283\pi$
$19$	−2.04266	−0.468618	−0.234309	+	0.972162i	$0.575283\pi$
$20$	−2.88581	−0.645286
$21$	−7.42869	−1.62107
$22$	−5.48435	−1.16927
$23$	−9.16509	−1.91105	−0.955527	−	0.294903i	$-0.904713\pi$
$23$	−9.16509	−1.91105	−0.955527	+	0.294903i	$0.904713\pi$
$24$	−2.44077	−0.498219
$25$	3.32789	0.665578
$26$	0.107544	0.0210911
$27$	−0.104116	−0.0200371
$28$	−3.04359	−0.575184
$29$	0	0
$29$	0	0
$30$	7.04359	1.28598
$31$	3.44170	0.618147	0.309073	−	0.951038i	$-0.399981\pi$
$31$	3.44170	0.618147	0.309073	+	0.951038i	$0.399981\pi$
$32$	−1.00000	−0.176777
$33$	13.3860	2.33021
$34$	0.816005	0.139944
$35$	8.78321	1.48463
$36$	2.95734	0.492891
$37$	−5.90758	−0.971200	−0.485600	−	0.874181i	$-0.661399\pi$
$37$	−5.90758	−0.971200	−0.485600	+	0.874181i	$0.661399\pi$
$38$	2.04266	0.331363
$39$	−0.262489	−0.0420319
$40$	2.88581	0.456286
$41$	2.43376	0.380090	0.190045	−	0.981775i	$-0.439137\pi$
$41$	2.43376	0.380090	0.190045	+	0.981775i	$0.439137\pi$
$42$	7.42869	1.14627
$43$	−5.48435	−0.836356	−0.418178	−	0.908365i	$-0.637331\pi$
$43$	−5.48435	−0.836356	−0.418178	+	0.908365i	$0.637331\pi$
$44$	5.48435	0.826797
$45$	−8.53433	−1.27222
$46$	9.16509	1.35132
$47$	−5.91000	−0.862062	−0.431031	−	0.902337i	$-0.641850\pi$
$47$	−5.91000	−0.862062	−0.431031	+	0.902337i	$0.641850\pi$
$48$	2.44077	0.352294
$49$	2.26342	0.323346
$50$	−3.32789	−0.470635
$51$	−1.99168	−0.278891
$52$	−0.107544	−0.0149136
$53$	3.95070	0.542670	0.271335	−	0.962485i	$-0.412535\pi$
$53$	3.95070	0.542670	0.271335	+	0.962485i	$0.412535\pi$
$54$	0.104116	0.0141683
$55$	−15.8268	−2.13408
$56$	3.04359	0.406716
$57$	−4.98565	−0.660365
$58$	0	0
$59$	1.13359	0.147581	0.0737904	−	0.997274i	$-0.476490\pi$
$59$	1.13359	0.147581	0.0737904	+	0.997274i	$0.476490\pi$
$60$	−7.04359	−0.909323
$61$	−8.16584	−1.04553	−0.522764	−	0.852477i	$-0.675099\pi$
$61$	−8.16584	−1.04553	−0.522764	+	0.852477i	$0.675099\pi$
$62$	−3.44170	−0.437096
$63$	−9.00093	−1.13401
$64$	1.00000	0.125000
$65$	0.310351	0.0384943
$66$	−13.3860	−1.64771
$67$	0.956413	0.116844	0.0584222	−	0.998292i	$-0.481393\pi$
$67$	0.956413	0.116844	0.0584222	+	0.998292i	$0.481393\pi$
$68$	−0.816005	−0.0989551
$69$	−22.3699	−2.69301
$70$	−8.78321	−1.04979
$71$	2.38979	0.283616	0.141808	−	0.989894i	$-0.454709\pi$
$71$	2.38979	0.283616	0.141808	+	0.989894i	$0.454709\pi$
$72$	−2.95734	−0.348526
$73$	−8.89410	−1.04098	−0.520488	−	0.853869i	$-0.674250\pi$
$73$	−8.89410	−1.04098	−0.520488	+	0.853869i	$0.674250\pi$
$74$	5.90758	0.686742
$75$	8.12261	0.937918
$76$	−2.04266	−0.234309
$77$	−16.6921	−1.90224
$78$	0.262489	0.0297211
$79$	4.20062	0.472607	0.236303	−	0.971679i	$-0.424064\pi$
$79$	4.20062	0.472607	0.236303	+	0.971679i	$0.424064\pi$
$80$	−2.88581	−0.322643
$81$	−9.12615	−1.01402
$82$	−2.43376	−0.268764
$83$	−15.4732	−1.69840	−0.849202	−	0.528068i	$-0.822917\pi$
$83$	−15.4732	−1.69840	−0.849202	+	0.528068i	$0.822917\pi$
$84$	−7.42869	−0.810536
$85$	2.35483	0.255418
$86$	5.48435	0.591393
$87$	0	0
$88$	−5.48435	−0.584634
$89$	−5.54154	−0.587402	−0.293701	−	0.955897i	$-0.594887\pi$
$89$	−5.54154	−0.587402	−0.293701	+	0.955897i	$0.594887\pi$
$90$	8.53433	0.899597
$91$	0.327319	0.0343124
$92$	−9.16509	−0.955527
$93$	8.40038	0.871079
$94$	5.91000	0.609570
$95$	5.89472	0.604785
$96$	−2.44077	−0.249110
$97$	−11.9217	−1.21046	−0.605232	−	0.796049i	$-0.706920\pi$
$97$	−11.9217	−1.21046	−0.605232	+	0.796049i	$0.706920\pi$
$98$	−2.26342	−0.228640
$99$	16.2191	1.63008
$100$	3.32789	0.332789
$101$	9.40543	0.935875	0.467937	−	0.883762i	$-0.344997\pi$
$101$	9.40543	0.935875	0.467937	+	0.883762i	$0.344997\pi$
$102$	1.99168	0.197205
$103$	−11.3271	−1.11610	−0.558048	−	0.829808i	$-0.688450\pi$
$103$	−11.3271	−1.11610	−0.558048	+	0.829808i	$0.688450\pi$
$104$	0.107544	0.0105455
$105$	21.4378	2.09211
$106$	−3.95070	−0.383725
$107$	10.4570	1.01092	0.505458	−	0.862851i	$-0.331324\pi$
$107$	10.4570	1.01092	0.505458	+	0.862851i	$0.331324\pi$
$108$	−0.104116	−0.0100185
$109$	−13.9287	−1.33413	−0.667064	−	0.745000i	$-0.732449\pi$
$109$	−13.9287	−1.33413	−0.667064	+	0.745000i	$0.732449\pi$
$110$	15.8268	1.50903
$111$	−14.4190	−1.36859
$112$	−3.04359	−0.287592
$113$	13.1190	1.23414	0.617068	−	0.786910i	$-0.288321\pi$
$113$	13.1190	1.23414	0.617068	+	0.786910i	$0.288321\pi$
$114$	4.98565	0.466949
$115$	26.4487	2.46635
$116$	0	0
$117$	−0.318044	−0.0294032
$118$	−1.13359	−0.104355
$119$	2.48358	0.227670
$120$	7.04359	0.642989
$121$	19.0781	1.73438
$122$	8.16584	0.739300
$123$	5.94024	0.535614
$124$	3.44170	0.309073
$125$	4.82538	0.431595
$126$	9.00093	0.801867
$127$	1.34000	0.118906	0.0594529	−	0.998231i	$-0.481064\pi$
$127$	1.34000	0.118906	0.0594529	+	0.998231i	$0.481064\pi$
$128$	−1.00000	−0.0883883
$129$	−13.3860	−1.17857
$130$	−0.310351	−0.0272196
$131$	−5.10731	−0.446228	−0.223114	−	0.974792i	$-0.571622\pi$
$131$	−5.10731	−0.446228	−0.223114	+	0.974792i	$0.571622\pi$
$132$	13.3860	1.16510
$133$	6.21700	0.539083
$134$	−0.956413	−0.0826215
$135$	0.300458	0.0258593
$136$	0.816005	0.0699718
$137$	−1.19304	−0.101928	−0.0509642	−	0.998700i	$-0.516229\pi$
$137$	−1.19304	−0.101928	−0.0509642	+	0.998700i	$0.516229\pi$
$138$	22.3699	1.90425
$139$	10.0823	0.855172	0.427586	−	0.903975i	$-0.359364\pi$
$139$	10.0823	0.855172	0.427586	+	0.903975i	$0.359364\pi$
$140$	8.78321	0.742317
$141$	−14.4249	−1.21480
$142$	−2.38979	−0.200547
$143$	−0.589808	−0.0493223
$144$	2.95734	0.246445
$145$	0	0
$146$	8.89410	0.736081
$147$	5.52448	0.455651
$148$	−5.90758	−0.485600
$149$	−1.75096	−0.143444	−0.0717220	−	0.997425i	$-0.522849\pi$
$149$	−1.75096	−0.143444	−0.0717220	+	0.997425i	$0.522849\pi$
$150$	−8.12261	−0.663208
$151$	−0.564094	−0.0459053	−0.0229527	−	0.999737i	$-0.507307\pi$
$151$	−0.564094	−0.0459053	−0.0229527	+	0.999737i	$0.507307\pi$
$152$	2.04266	0.165681
$153$	−2.41321	−0.195096
$154$	16.6921	1.34509
$155$	−9.93208	−0.797764
$156$	−0.262489	−0.0210160
$157$	−17.3463	−1.38439	−0.692193	−	0.721712i	$-0.743355\pi$
$157$	−17.3463	−1.38439	−0.692193	+	0.721712i	$0.743355\pi$
$158$	−4.20062	−0.334183
$159$	9.64273	0.764718
$160$	2.88581	0.228143
$161$	27.8948	2.19841
$162$	9.12615	0.717018
$163$	−11.6189	−0.910060	−0.455030	−	0.890476i	$-0.650372\pi$
$163$	−11.6189	−0.910060	−0.455030	+	0.890476i	$0.650372\pi$
$164$	2.43376	0.190045
$165$	−38.6295	−3.00730
$166$	15.4732	1.20095
$167$	−8.50687	−0.658281	−0.329141	−	0.944281i	$-0.606759\pi$
$167$	−8.50687	−0.658281	−0.329141	+	0.944281i	$0.606759\pi$
$168$	7.42869	0.573136
$169$	−12.9884	−0.999110
$170$	−2.35483	−0.180608
$171$	−6.04084	−0.461954
$172$	−5.48435	−0.418178
$173$	10.1868	0.774484	0.387242	−	0.921978i	$-0.373428\pi$
$173$	10.1868	0.774484	0.387242	+	0.921978i	$0.373428\pi$
$174$	0	0
$175$	−10.1287	−0.765660
$176$	5.48435	0.413399
$177$	2.76683	0.207967
$178$	5.54154	0.415356
$179$	4.78942	0.357978	0.178989	−	0.983851i	$-0.442717\pi$
$179$	4.78942	0.357978	0.178989	+	0.983851i	$0.442717\pi$
$180$	−8.53433	−0.636111
$181$	0.737152	0.0547921	0.0273960	−	0.999625i	$-0.491278\pi$
$181$	0.737152	0.0547921	0.0273960	+	0.999625i	$0.491278\pi$
$182$	−0.327319	−0.0242625
$183$	−19.9309	−1.47334
$184$	9.16509	0.675660
$185$	17.0481	1.25340
$186$	−8.40038	−0.615946
$187$	−4.47526	−0.327263
$188$	−5.91000	−0.431031
$189$	0.316885	0.0230500
$190$	−5.89472	−0.427648
$191$	0.120340	0.00870752	0.00435376	−	0.999991i	$-0.498614\pi$
$191$	0.120340	0.00870752	0.00435376	+	0.999991i	$0.498614\pi$
$192$	2.44077	0.176147
$193$	21.1388	1.52161	0.760804	−	0.648982i	$-0.224805\pi$
$193$	21.1388	1.52161	0.760804	+	0.648982i	$0.224805\pi$
$194$	11.9217	0.855928
$195$	0.757494	0.0542453
$196$	2.26342	0.161673
$197$	−16.0211	−1.14146	−0.570728	−	0.821139i	$-0.693339\pi$
$197$	−16.0211	−1.14146	−0.570728	+	0.821139i	$0.693339\pi$
$198$	−16.2191	−1.15264
$199$	13.8572	0.982311	0.491155	−	0.871072i	$-0.336575\pi$
$199$	13.8572	0.982311	0.491155	+	0.871072i	$0.336575\pi$
$200$	−3.32789	−0.235318
$201$	2.33438	0.164655
$202$	−9.40543	−0.661763
$203$	0	0
$204$	−1.99168	−0.139445
$205$	−7.02337	−0.490533
$206$	11.3271	0.789200
$207$	−27.1043	−1.88388
$208$	−0.107544	−0.00745682
$209$	−11.2027	−0.774904
$210$	−21.4378	−1.47935
$211$	24.3125	1.67374	0.836870	−	0.547401i	$-0.184383\pi$
$211$	24.3125	1.67374	0.836870	+	0.547401i	$0.184383\pi$
$212$	3.95070	0.271335
$213$	5.83291	0.399665
$214$	−10.4570	−0.714825
$215$	15.8268	1.07938
$216$	0.104116	0.00708417
$217$	−10.4751	−0.711096
$218$	13.9287	0.943371
$219$	−21.7084	−1.46692
$220$	−15.8268	−1.06704
$221$	0.0877563	0.00590313
$222$	14.4190	0.967741
$223$	10.4249	0.698105	0.349052	−	0.937103i	$-0.386504\pi$
$223$	10.4249	0.698105	0.349052	+	0.937103i	$0.386504\pi$
$224$	3.04359	0.203358
$225$	9.84172	0.656115
$226$	−13.1190	−0.872665
$227$	20.9815	1.39259	0.696294	−	0.717757i	$-0.254831\pi$
$227$	20.9815	1.39259	0.696294	+	0.717757i	$0.254831\pi$
$228$	−4.98565	−0.330183
$229$	−0.984111	−0.0650319	−0.0325159	−	0.999471i	$-0.510352\pi$
$229$	−0.984111	−0.0650319	−0.0325159	+	0.999471i	$0.510352\pi$
$230$	−26.4487	−1.74398
$231$	−40.7415	−2.68060
$232$	0	0
$233$	17.3778	1.13846	0.569229	−	0.822179i	$-0.307242\pi$
$233$	17.3778	1.13846	0.569229	+	0.822179i	$0.307242\pi$
$234$	0.318044	0.0207912
$235$	17.0551	1.11255
$236$	1.13359	0.0737904
$237$	10.2527	0.665987
$238$	−2.48358	−0.160987
$239$	−1.15163	−0.0744930	−0.0372465	−	0.999306i	$-0.511859\pi$
$239$	−1.15163	−0.0744930	−0.0372465	+	0.999306i	$0.511859\pi$
$240$	−7.04359	−0.454662
$241$	−19.7078	−1.26949	−0.634747	−	0.772720i	$-0.718896\pi$
$241$	−19.7078	−1.26949	−0.634747	+	0.772720i	$0.718896\pi$
$242$	−19.0781	−1.22639
$243$	−21.9625	−1.40889
$244$	−8.16584	−0.522764
$245$	−6.53179	−0.417301
$246$	−5.94024	−0.378736
$247$	0.219675	0.0139776
$248$	−3.44170	−0.218548
$249$	−37.7665	−2.39335
$250$	−4.82538	−0.305184
$251$	8.50449	0.536799	0.268399	−	0.963308i	$-0.413505\pi$
$251$	8.50449	0.536799	0.268399	+	0.963308i	$0.413505\pi$
$252$	−9.00093	−0.567005
$253$	−50.2646	−3.16011
$254$	−1.34000	−0.0840791
$255$	5.74760	0.359929
$256$	1.00000	0.0625000
$257$	22.7220	1.41736	0.708681	−	0.705529i	$-0.249291\pi$
$257$	22.7220	1.41736	0.708681	+	0.705529i	$0.249291\pi$
$258$	13.3860	0.833378
$259$	17.9802	1.11724
$260$	0.310351	0.0192471
$261$	0	0
$262$	5.10731	0.315531
$263$	26.0425	1.60585	0.802926	−	0.596079i	$-0.203275\pi$
$263$	26.0425	1.60585	0.802926	+	0.596079i	$0.203275\pi$
$264$	−13.3860	−0.823853
$265$	−11.4010	−0.700355
$266$	−6.21700	−0.381189
$267$	−13.5256	−0.827753
$268$	0.956413	0.0584222
$269$	−1.45648	−0.0888029	−0.0444014	−	0.999014i	$-0.514138\pi$
$269$	−1.45648	−0.0888029	−0.0444014	+	0.999014i	$0.514138\pi$
$270$	−0.300458	−0.0182853
$271$	10.9282	0.663839	0.331919	−	0.943308i	$-0.392304\pi$
$271$	10.9282	0.663839	0.331919	+	0.943308i	$0.392304\pi$
$272$	−0.816005	−0.0494776
$273$	0.798909	0.0483522
$274$	1.19304	0.0720743
$275$	18.2513	1.10060
$276$	−22.3699	−1.34651
$277$	−2.47242	−0.148553	−0.0742766	−	0.997238i	$-0.523665\pi$
$277$	−2.47242	−0.148553	−0.0742766	+	0.997238i	$0.523665\pi$
$278$	−10.0823	−0.604698
$279$	10.1783	0.609358
$280$	−8.78321	−0.524897
$281$	−0.186647	−0.0111344	−0.00556721	−	0.999985i	$-0.501772\pi$
$281$	−0.186647	−0.0111344	−0.00556721	+	0.999985i	$0.501772\pi$
$282$	14.4249	0.858992
$283$	−18.4487	−1.09666	−0.548331	−	0.836261i	$-0.684737\pi$
$283$	−18.4487	−1.09666	−0.548331	+	0.836261i	$0.684737\pi$
$284$	2.38979	0.141808
$285$	14.3876	0.852250
$286$	0.589808	0.0348761
$287$	−7.40736	−0.437243
$288$	−2.95734	−0.174263
$289$	−16.3341	−0.960832
$290$	0	0
$291$	−29.0981	−1.70576
$292$	−8.89410	−0.520488
$293$	1.55630	0.0909202	0.0454601	−	0.998966i	$-0.485525\pi$
$293$	1.55630	0.0909202	0.0454601	+	0.998966i	$0.485525\pi$
$294$	−5.52448	−0.322194
$295$	−3.27132	−0.190464
$296$	5.90758	0.343371
$297$	−0.571007	−0.0331332
$298$	1.75096	0.101430
$299$	0.985649	0.0570016
$300$	8.12261	0.468959
$301$	16.6921	0.962117
$302$	0.564094	0.0324600
$303$	22.9565	1.31881
$304$	−2.04266	−0.117154
$305$	23.5651	1.34933
$306$	2.41321	0.137954
$307$	−19.6000	−1.11863	−0.559316	−	0.828955i	$-0.688936\pi$
$307$	−19.6000	−1.11863	−0.559316	+	0.828955i	$0.688936\pi$
$308$	−16.6921	−0.951121
$309$	−27.6469	−1.57278
$310$	9.93208	0.564104
$311$	26.4681	1.50087	0.750433	−	0.660946i	$-0.229845\pi$
$311$	26.4681	1.50087	0.750433	+	0.660946i	$0.229845\pi$
$312$	0.262489	0.0148605
$313$	21.8049	1.23249	0.616244	−	0.787555i	$-0.288654\pi$
$313$	21.8049	1.23249	0.616244	+	0.787555i	$0.288654\pi$
$314$	17.3463	0.978909
$315$	25.9750	1.46352
$316$	4.20062	0.236303
$317$	−18.9285	−1.06313	−0.531564	−	0.847018i	$-0.678396\pi$
$317$	−18.9285	−1.06313	−0.531564	+	0.847018i	$0.678396\pi$
$318$	−9.64273	−0.540737
$319$	0	0
$320$	−2.88581	−0.161322
$321$	25.5231	1.42456
$322$	−27.8948	−1.55451
$323$	1.66682	0.0927442
$324$	−9.12615	−0.507008
$325$	−0.357894	−0.0198524
$326$	11.6189	0.643510
$327$	−33.9967	−1.88002
$328$	−2.43376	−0.134382
$329$	17.9876	0.991688
$330$	38.6295	2.12648
$331$	−17.7806	−0.977311	−0.488656	−	0.872477i	$-0.662513\pi$
$331$	−17.7806	−0.977311	−0.488656	+	0.872477i	$0.662513\pi$
$332$	−15.4732	−0.849202
$333$	−17.4707	−0.957391
$334$	8.50687	0.465475
$335$	−2.76003	−0.150796
$336$	−7.42869	−0.405268
$337$	−2.83904	−0.154652	−0.0773261	−	0.997006i	$-0.524638\pi$
$337$	−2.83904	−0.154652	−0.0773261	+	0.997006i	$0.524638\pi$
$338$	12.9884	0.706478
$339$	32.0205	1.73912
$340$	2.35483	0.127709
$341$	18.8755	1.02216
$342$	6.04084	0.326651
$343$	14.4162	0.778401
$344$	5.48435	0.295697
$345$	64.5551	3.47553
$346$	−10.1868	−0.547643
$347$	−7.14376	−0.383497	−0.191749	−	0.981444i	$-0.561416\pi$
$347$	−7.14376	−0.383497	−0.191749	+	0.981444i	$0.561416\pi$
$348$	0	0
$349$	8.26091	0.442197	0.221098	−	0.975252i	$-0.429036\pi$
$349$	8.26091	0.442197	0.221098	+	0.975252i	$0.429036\pi$
$350$	10.1287	0.541403
$351$	0.0111970	0.000597651 0
$352$	−5.48435	−0.292317
$353$	18.1818	0.967718	0.483859	−	0.875146i	$-0.339235\pi$
$353$	18.1818	0.967718	0.483859	+	0.875146i	$0.339235\pi$
$354$	−2.76683	−0.147055
$355$	−6.89647	−0.366027
$356$	−5.54154	−0.293701
$357$	6.06184	0.320827
$358$	−4.78942	−0.253129
$359$	19.8073	1.04539	0.522695	−	0.852520i	$-0.324927\pi$
$359$	19.8073	1.04539	0.522695	+	0.852520i	$0.324927\pi$
$360$	8.53433	0.449799
$361$	−14.8276	−0.780398
$362$	−0.737152	−0.0387438
$363$	46.5653	2.44404
$364$	0.327319	0.0171562
$365$	25.6667	1.34345
$366$	19.9309	1.04181
$367$	−3.83357	−0.200111	−0.100055	−	0.994982i	$-0.531902\pi$
$367$	−3.83357	−0.200111	−0.100055	+	0.994982i	$0.531902\pi$
$368$	−9.16509	−0.477764
$369$	7.19747	0.374685
$370$	−17.0481	−0.886291
$371$	−12.0243	−0.624270
$372$	8.40038	0.435539
$373$	18.1776	0.941202	0.470601	−	0.882346i	$-0.344037\pi$
$373$	18.1776	0.941202	0.470601	+	0.882346i	$0.344037\pi$
$374$	4.47526	0.231410
$375$	11.7776	0.608194
$376$	5.91000	0.304785
$377$	0	0
$378$	−0.316885	−0.0162988
$379$	−37.3670	−1.91941	−0.959707	−	0.281003i	$-0.909333\pi$
$379$	−37.3670	−1.91941	−0.959707	+	0.281003i	$0.909333\pi$
$380$	5.89472	0.302393
$381$	3.27063	0.167559
$382$	−0.120340	−0.00615715
$383$	−11.8246	−0.604208	−0.302104	−	0.953275i	$-0.597689\pi$
$383$	−11.8246	−0.604208	−0.302104	+	0.953275i	$0.597689\pi$
$384$	−2.44077	−0.124555
$385$	48.1702	2.45498
$386$	−21.1388	−1.07594
$387$	−16.2191	−0.824464
$388$	−11.9217	−0.605232
$389$	29.8660	1.51426	0.757132	−	0.653262i	$-0.226600\pi$
$389$	29.8660	1.51426	0.757132	+	0.653262i	$0.226600\pi$
$390$	−0.757494	−0.0383572
$391$	7.47876	0.378217
$392$	−2.26342	−0.114320
$393$	−12.4658	−0.628815
$394$	16.0211	0.807131
$395$	−12.1222	−0.609934
$396$	16.2191	0.815041
$397$	−8.10324	−0.406690	−0.203345	−	0.979107i	$-0.565181\pi$
$397$	−8.10324	−0.406690	−0.203345	+	0.979107i	$0.565181\pi$
$398$	−13.8572	−0.694599
$399$	15.1743	0.759663
$400$	3.32789	0.166395
$401$	5.94737	0.296997	0.148499	−	0.988913i	$-0.452556\pi$
$401$	5.94737	0.296997	0.148499	+	0.988913i	$0.452556\pi$
$402$	−2.33438	−0.116428
$403$	−0.370133	−0.0184377
$404$	9.40543	0.467937
$405$	26.3363	1.30866
$406$	0	0
$407$	−32.3993	−1.60597
$408$	1.99168	0.0986027
$409$	−13.1395	−0.649706	−0.324853	−	0.945765i	$-0.605315\pi$
$409$	−13.1395	−0.649706	−0.324853	+	0.945765i	$0.605315\pi$
$410$	7.02337	0.346859
$411$	−2.91194	−0.143635
$412$	−11.3271	−0.558048
$413$	−3.45018	−0.169772
$414$	27.1043	1.33211
$415$	44.6527	2.19191
$416$	0.107544	0.00527277
$417$	24.6086	1.20509
$418$	11.2027	0.547940
$419$	22.1612	1.08265	0.541324	−	0.840814i	$-0.317923\pi$
$419$	22.1612	1.08265	0.541324	+	0.840814i	$0.317923\pi$
$420$	21.4378	1.04606
$421$	−23.8521	−1.16248	−0.581240	−	0.813732i	$-0.697432\pi$
$421$	−23.8521	−1.16248	−0.581240	+	0.813732i	$0.697432\pi$
$422$	−24.3125	−1.18351
$423$	−17.4779	−0.849804
$424$	−3.95070	−0.191863
$425$	−2.71558	−0.131725
$426$	−5.83291	−0.282606
$427$	24.8534	1.20274
$428$	10.4570	0.505458
$429$	−1.43958	−0.0695038
$430$	−15.8268	−0.763236
$431$	−0.776969	−0.0374253	−0.0187126	−	0.999825i	$-0.505957\pi$
$431$	−0.776969	−0.0374253	−0.0187126	+	0.999825i	$0.505957\pi$
$432$	−0.104116	−0.00500926
$433$	−0.612460	−0.0294329	−0.0147165	−	0.999892i	$-0.504685\pi$
$433$	−0.612460	−0.0294329	−0.0147165	+	0.999892i	$0.504685\pi$
$434$	10.4751	0.502821
$435$	0	0
$436$	−13.9287	−0.667064
$437$	18.7211	0.895554
$438$	21.7084	1.03727
$439$	−10.8953	−0.520004	−0.260002	−	0.965608i	$-0.583723\pi$
$439$	−10.8953	−0.520004	−0.260002	+	0.965608i	$0.583723\pi$
$440$	15.8268	0.754513
$441$	6.69371	0.318748
$442$	−0.0877563	−0.00417414
$443$	34.8279	1.65472	0.827362	−	0.561669i	$-0.189841\pi$
$443$	34.8279	1.65472	0.827362	+	0.561669i	$0.189841\pi$
$444$	−14.4190	−0.684297
$445$	15.9918	0.758085
$446$	−10.4249	−0.493635
$447$	−4.27368	−0.202138
$448$	−3.04359	−0.143796
$449$	−5.02242	−0.237022	−0.118511	−	0.992953i	$-0.537812\pi$
$449$	−5.02242	−0.237022	−0.118511	+	0.992953i	$0.537812\pi$
$450$	−9.84172	−0.463943
$451$	13.3476	0.628514
$452$	13.1190	0.617068
$453$	−1.37682	−0.0646887
$454$	−20.9815	−0.984708
$455$	−0.944580	−0.0442826
$456$	4.98565	0.233474
$457$	38.2426	1.78891	0.894456	−	0.447156i	$-0.147563\pi$
$457$	38.2426	1.78891	0.894456	+	0.447156i	$0.147563\pi$
$458$	0.984111	0.0459845
$459$	0.0849588	0.00396554
$460$	26.4487	1.23318
$461$	−25.9962	−1.21076	−0.605381	−	0.795935i	$-0.706979\pi$
$461$	−25.9962	−1.21076	−0.605381	+	0.795935i	$0.706979\pi$
$462$	40.7415	1.89547
$463$	17.3502	0.806331	0.403166	−	0.915127i	$-0.367910\pi$
$463$	17.3502	0.806331	0.403166	+	0.915127i	$0.367910\pi$
$464$	0	0
$465$	−24.2419	−1.12419
$466$	−17.3778	−0.805012
$467$	−29.2399	−1.35306	−0.676530	−	0.736415i	$-0.736517\pi$
$467$	−29.2399	−1.35306	−0.676530	+	0.736415i	$0.736517\pi$
$468$	−0.318044	−0.0147016
$469$	−2.91093	−0.134414
$470$	−17.0551	−0.786694
$471$	−42.3383	−1.95085
$472$	−1.13359	−0.0521777
$473$	−30.0781	−1.38299
$474$	−10.2527	−0.470924
$475$	−6.79774	−0.311902
$476$	2.48358	0.113835
$477$	11.6836	0.534954
$478$	1.15163	0.0526745
$479$	−42.9871	−1.96413	−0.982065	−	0.188543i	$-0.939624\pi$
$479$	−42.9871	−1.96413	−0.982065	+	0.188543i	$0.939624\pi$
$480$	7.04359	0.321494
$481$	0.635324	0.0289683
$482$	19.7078	0.897668
$483$	68.0846	3.09796
$484$	19.0781	0.867188
$485$	34.4037	1.56219
$486$	21.9625	0.996237
$487$	−9.99343	−0.452845	−0.226423	−	0.974029i	$-0.572703\pi$
$487$	−9.99343	−0.452845	−0.226423	+	0.974029i	$0.572703\pi$
$488$	8.16584	0.369650
$489$	−28.3590	−1.28244
$490$	6.53179	0.295076
$491$	−3.80604	−0.171764	−0.0858821	−	0.996305i	$-0.527371\pi$
$491$	−3.80604	−0.171764	−0.0858821	+	0.996305i	$0.527371\pi$
$492$	5.94024	0.267807
$493$	0	0
$494$	−0.219675	−0.00988365
$495$	−46.8053	−2.10374
$496$	3.44170	0.154537
$497$	−7.27352	−0.326262
$498$	37.7665	1.69236
$499$	−28.3943	−1.27110	−0.635552	−	0.772058i	$-0.719227\pi$
$499$	−28.3943	−1.27110	−0.635552	+	0.772058i	$0.719227\pi$
$500$	4.82538	0.215798
$501$	−20.7633	−0.927635
$502$	−8.50449	−0.379574
$503$	−1.63425	−0.0728675	−0.0364337	−	0.999336i	$-0.511600\pi$
$503$	−1.63425	−0.0728675	−0.0364337	+	0.999336i	$0.511600\pi$
$504$	9.00093	0.400933
$505$	−27.1423	−1.20781
$506$	50.2646	2.23453
$507$	−31.7017	−1.40792
$508$	1.34000	0.0594529
$509$	36.4706	1.61653	0.808267	−	0.588817i	$-0.200406\pi$
$509$	36.4706	1.61653	0.808267	+	0.588817i	$0.200406\pi$
$510$	−5.74760	−0.254508
$511$	27.0700	1.19750
$512$	−1.00000	−0.0441942
$513$	0.212672	0.00938972
$514$	−22.7220	−1.00223
$515$	32.6880	1.44040
$516$	−13.3860	−0.589287
$517$	−32.4125	−1.42550
$518$	−17.9802	−0.790006
$519$	24.8635	1.09139
$520$	−0.310351	−0.0136098
$521$	−37.4549	−1.64093	−0.820465	−	0.571697i	$-0.806285\pi$
$521$	−37.4549	−1.64093	−0.820465	+	0.571697i	$0.806285\pi$
$522$	0	0
$523$	37.9515	1.65950	0.829751	−	0.558133i	$-0.188482\pi$
$523$	37.9515	1.65950	0.829751	+	0.558133i	$0.188482\pi$
$524$	−5.10731	−0.223114
$525$	−24.7219	−1.07895
$526$	−26.0425	−1.13551
$527$	−2.80844	−0.122338
$528$	13.3860	0.582552
$529$	60.9989	2.65213
$530$	11.4010	0.495226
$531$	3.35241	0.145482
$532$	6.21700	0.269541
$533$	−0.261736	−0.0113370
$534$	13.5256	0.585310
$535$	−30.1769	−1.30466
$536$	−0.956413	−0.0413108
$537$	11.6899	0.504455
$538$	1.45648	0.0627931
$539$	12.4134	0.534683
$540$	0.300458	0.0129296
$541$	−25.2604	−1.08603	−0.543014	−	0.839724i	$-0.682717\pi$
$541$	−25.2604	−1.08603	−0.543014	+	0.839724i	$0.682717\pi$
$542$	−10.9282	−0.469405
$543$	1.79922	0.0772118
$544$	0.816005	0.0349859
$545$	40.1956	1.72179
$546$	−0.798909	−0.0341902
$547$	−14.9623	−0.639744	−0.319872	−	0.947461i	$-0.603640\pi$
$547$	−14.9623	−0.639744	−0.319872	+	0.947461i	$0.603640\pi$
$548$	−1.19304	−0.0509642
$549$	−24.1492	−1.03066
$550$	−18.2513	−0.778240
$551$	0	0
$552$	22.3699	0.952124
$553$	−12.7850	−0.543672
$554$	2.47242	0.105043
$555$	41.6105	1.76627
$556$	10.0823	0.427586
$557$	23.7019	1.00428	0.502141	−	0.864786i	$-0.332546\pi$
$557$	23.7019	1.00428	0.502141	+	0.864786i	$0.332546\pi$
$558$	−10.1783	−0.430881
$559$	0.589808	0.0249462
$560$	8.78321	0.371158
$561$	−10.9231	−0.461172
$562$	0.186647	0.00787322
$563$	−38.7490	−1.63308	−0.816538	−	0.577292i	$-0.804109\pi$
$563$	−38.7490	−1.63308	−0.816538	+	0.577292i	$0.804109\pi$
$564$	−14.4249	−0.607399
$565$	−37.8590	−1.59274
$566$	18.4487	0.775457
$567$	27.7762	1.16649
$568$	−2.38979	−0.100273
$569$	−1.15229	−0.0483064	−0.0241532	−	0.999708i	$-0.507689\pi$
$569$	−1.15229	−0.0483064	−0.0241532	+	0.999708i	$0.507689\pi$
$570$	−14.3876	−0.602632
$571$	16.7044	0.699057	0.349528	−	0.936926i	$-0.386342\pi$
$571$	16.7044	0.699057	0.349528	+	0.936926i	$0.386342\pi$
$572$	−0.589808	−0.0246611
$573$	0.293723	0.0122704
$574$	7.40736	0.309177
$575$	−30.5004	−1.27196
$576$	2.95734	0.123223
$577$	13.9742	0.581752	0.290876	−	0.956761i	$-0.406053\pi$
$577$	13.9742	0.581752	0.290876	+	0.956761i	$0.406053\pi$
$578$	16.3341	0.679410
$579$	51.5950	2.14422
$580$	0	0
$581$	47.0940	1.95379
$582$	29.0981	1.20615
$583$	21.6670	0.897356
$584$	8.89410	0.368040
$585$	0.917814	0.0379469
$586$	−1.55630	−0.0642903
$587$	−36.7791	−1.51804	−0.759018	−	0.651069i	$-0.774321\pi$
$587$	−36.7791	−1.51804	−0.759018	+	0.651069i	$0.774321\pi$
$588$	5.52448	0.227826
$589$	−7.03021	−0.289675
$590$	3.27132	0.134678
$591$	−39.1038	−1.60851
$592$	−5.90758	−0.242800
$593$	28.8630	1.18526	0.592630	−	0.805475i	$-0.298090\pi$
$593$	28.8630	1.18526	0.592630	+	0.805475i	$0.298090\pi$
$594$	0.571007	0.0234287
$595$	−7.16714	−0.293824
$596$	−1.75096	−0.0717220
$597$	33.8222	1.38425
$598$	−0.985649	−0.0403062
$599$	7.65942	0.312955	0.156478	−	0.987681i	$-0.449986\pi$
$599$	7.65942	0.312955	0.156478	+	0.987681i	$0.449986\pi$
$600$	−8.12261	−0.331604
$601$	30.5315	1.24540	0.622702	−	0.782459i	$-0.286035\pi$
$601$	30.5315	1.24540	0.622702	+	0.782459i	$0.286035\pi$
$602$	−16.6921	−0.680319
$603$	2.82844	0.115183
$604$	−0.564094	−0.0229527
$605$	−55.0558	−2.23834
$606$	−22.9565	−0.932542
$607$	17.3469	0.704089	0.352045	−	0.935983i	$-0.385486\pi$
$607$	17.3469	0.704089	0.352045	+	0.935983i	$0.385486\pi$
$608$	2.04266	0.0828407
$609$	0	0
$610$	−23.5651	−0.954121
$611$	0.635584	0.0257130
$612$	−2.41321	−0.0975481
$613$	42.0660	1.69903	0.849515	−	0.527564i	$-0.176895\pi$
$613$	42.0660	1.69903	0.849515	+	0.527564i	$0.176895\pi$
$614$	19.6000	0.790992
$615$	−17.1424	−0.691249
$616$	16.6921	0.672544
$617$	−38.1341	−1.53522	−0.767610	−	0.640917i	$-0.778554\pi$
$617$	−38.1341	−1.53522	−0.767610	+	0.640917i	$0.778554\pi$
$618$	27.6469	1.11212
$619$	−11.0550	−0.444339	−0.222170	−	0.975008i	$-0.571314\pi$
$619$	−11.0550	−0.444339	−0.222170	+	0.975008i	$0.571314\pi$
$620$	−9.93208	−0.398882
$621$	0.954229	0.0382919
$622$	−26.4681	−1.06127
$623$	16.8661	0.675728
$624$	−0.262489	−0.0105080
$625$	−30.5646	−1.22258
$626$	−21.8049	−0.871500
$627$	−27.3431	−1.09198
$628$	−17.3463	−0.692193
$629$	4.82061	0.192210
$630$	−25.9750	−1.03487
$631$	−5.25268	−0.209106	−0.104553	−	0.994519i	$-0.533341\pi$
$631$	−5.25268	−0.209106	−0.104553	+	0.994519i	$0.533341\pi$
$632$	−4.20062	−0.167092
$633$	59.3411	2.35860
$634$	18.9285	0.751746
$635$	−3.86699	−0.153457
$636$	9.64273	0.382359
$637$	−0.243417	−0.00964452
$638$	0	0
$639$	7.06742	0.279583
$640$	2.88581	0.114072
$641$	−42.8000	−1.69050	−0.845249	−	0.534373i	$-0.820548\pi$
$641$	−42.8000	−1.69050	−0.845249	+	0.534373i	$0.820548\pi$
$642$	−25.5231	−1.00732
$643$	5.70529	0.224995	0.112497	−	0.993652i	$-0.464115\pi$
$643$	5.70529	0.224995	0.112497	+	0.993652i	$0.464115\pi$
$644$	27.8948	1.09921
$645$	38.6295	1.52104
$646$	−1.66682	−0.0655801
$647$	−43.8776	−1.72501	−0.862503	−	0.506052i	$-0.831104\pi$
$647$	−43.8776	−1.72501	−0.862503	+	0.506052i	$0.831104\pi$
$648$	9.12615	0.358509
$649$	6.21700	0.244039
$650$	0.357894	0.0140378
$651$	−25.5673	−1.00206
$652$	−11.6189	−0.455030
$653$	8.56897	0.335330	0.167665	−	0.985844i	$-0.446377\pi$
$653$	8.56897	0.335330	0.167665	+	0.985844i	$0.446377\pi$
$654$	33.9967	1.32938
$655$	14.7387	0.575890
$656$	2.43376	0.0950224
$657$	−26.3029	−1.02617
$658$	−17.9876	−0.701229
$659$	−41.1046	−1.60121	−0.800603	−	0.599195i	$-0.795487\pi$
$659$	−41.1046	−1.60121	−0.800603	+	0.599195i	$0.795487\pi$
$660$	−38.6295	−1.50365
$661$	4.19096	0.163009	0.0815047	−	0.996673i	$-0.474027\pi$
$661$	4.19096	0.163009	0.0815047	+	0.996673i	$0.474027\pi$
$662$	17.7806	0.691063
$663$	0.214193	0.00831855
$664$	15.4732	0.600477
$665$	−17.9411	−0.695725
$666$	17.4707	0.676977
$667$	0	0
$668$	−8.50687	−0.329141
$669$	25.4448	0.983753
$670$	2.76003	0.106629
$671$	−44.7844	−1.72888
$672$	7.42869	0.286568
$673$	26.9200	1.03769	0.518845	−	0.854869i	$-0.326362\pi$
$673$	26.9200	1.03769	0.518845	+	0.854869i	$0.326362\pi$
$674$	2.83904	0.109356
$675$	−0.346485	−0.0133362
$676$	−12.9884	−0.499555
$677$	−7.71912	−0.296670	−0.148335	−	0.988937i	$-0.547391\pi$
$677$	−7.71912	−0.296670	−0.148335	+	0.988937i	$0.547391\pi$
$678$	−32.0205	−1.22974
$679$	36.2847	1.39248
$680$	−2.35483	−0.0903038
$681$	51.2108	1.96240
$682$	−18.8755	−0.722779
$683$	−20.9011	−0.799758	−0.399879	−	0.916568i	$-0.630948\pi$
$683$	−20.9011	−0.799758	−0.399879	+	0.916568i	$0.630948\pi$
$684$	−6.04084	−0.230977
$685$	3.44289	0.131546
$686$	−14.4162	−0.550413
$687$	−2.40199	−0.0916415
$688$	−5.48435	−0.209089
$689$	−0.424873	−0.0161864
$690$	−64.5551	−2.45757
$691$	11.7693	0.447726	0.223863	−	0.974621i	$-0.428133\pi$
$691$	11.7693	0.447726	0.223863	+	0.974621i	$0.428133\pi$
$692$	10.1868	0.387242
$693$	−49.3643	−1.87519
$694$	7.14376	0.271173
$695$	−29.0957	−1.10366
$696$	0	0
$697$	−1.98596	−0.0752236
$698$	−8.26091	−0.312680
$699$	42.4152	1.60429
$700$	−10.1287	−0.382830
$701$	−40.9645	−1.54721	−0.773605	−	0.633669i	$-0.781548\pi$
$701$	−40.9645	−1.54721	−0.773605	+	0.633669i	$0.781548\pi$
$702$	−0.0111970	−0.000422603 0
$703$	12.0672	0.455121
$704$	5.48435	0.206699
$705$	41.6276	1.56779
$706$	−18.1818	−0.684280
$707$	−28.6262	−1.07660
$708$	2.76683	0.103984
$709$	−27.2842	−1.02468	−0.512339	−	0.858783i	$-0.671221\pi$
$709$	−27.2842	−1.02468	−0.512339	+	0.858783i	$0.671221\pi$
$710$	6.89647	0.258820
$711$	12.4227	0.465887
$712$	5.54154	0.207678
$713$	−31.5435	−1.18131
$714$	−6.06184	−0.226859
$715$	1.70207	0.0636540
$716$	4.78942	0.178989
$717$	−2.81087	−0.104974
$718$	−19.8073	−0.739203
$719$	−23.1238	−0.862372	−0.431186	−	0.902263i	$-0.641905\pi$
$719$	−23.1238	−0.862372	−0.431186	+	0.902263i	$0.641905\pi$
$720$	−8.53433	−0.318056
$721$	34.4752	1.28392
$722$	14.8276	0.551824
$723$	−48.1023	−1.78894
$724$	0.737152	0.0273960
$725$	0	0
$726$	−46.5653	−1.72820
$727$	−47.1118	−1.74728	−0.873639	−	0.486574i	$-0.838246\pi$
$727$	−47.1118	−1.74728	−0.873639	+	0.486574i	$0.838246\pi$
$728$	−0.327319	−0.0121312
$729$	−26.2268	−0.971363
$730$	−25.6667	−0.949966
$731$	4.47526	0.165523
$732$	−19.9309	−0.736668
$733$	−24.4234	−0.902099	−0.451050	−	0.892499i	$-0.648950\pi$
$733$	−24.4234	−0.902099	−0.451050	+	0.892499i	$0.648950\pi$
$734$	3.83357	0.141500
$735$	−15.9426	−0.588051
$736$	9.16509	0.337830
$737$	5.24531	0.193213
$738$	−7.19747	−0.264942
$739$	12.8289	0.471920	0.235960	−	0.971763i	$-0.424177\pi$
$739$	12.8289	0.471920	0.235960	+	0.971763i	$0.424177\pi$
$740$	17.0481	0.626702
$741$	0.536176	0.0196969
$742$	12.0243	0.441425
$743$	−0.349393	−0.0128180	−0.00640899	−	0.999979i	$-0.502040\pi$
$743$	−0.349393	−0.0128180	−0.00640899	+	0.999979i	$0.502040\pi$
$744$	−8.40038	−0.307973
$745$	5.05293	0.185125
$746$	−18.1776	−0.665530
$747$	−45.7596	−1.67425
$748$	−4.47526	−0.163632
$749$	−31.8268	−1.16292
$750$	−11.7776	−0.430058
$751$	−38.7195	−1.41289	−0.706447	−	0.707765i	$-0.749703\pi$
$751$	−38.7195	−1.41289	−0.706447	+	0.707765i	$0.749703\pi$
$752$	−5.91000	−0.215515
$753$	20.7575	0.756445
$754$	0	0
$755$	1.62787	0.0592442
$756$	0.316885	0.0115250
$757$	42.6278	1.54933	0.774666	−	0.632370i	$-0.217918\pi$
$757$	42.6278	1.54933	0.774666	+	0.632370i	$0.217918\pi$
$758$	37.3670	1.35723
$759$	−122.684	−4.45315
$760$	−5.89472	−0.213824
$761$	−8.66810	−0.314218	−0.157109	−	0.987581i	$-0.550217\pi$
$761$	−8.66810	−0.314218	−0.157109	+	0.987581i	$0.550217\pi$
$762$	−3.27063	−0.118482
$763$	42.3932	1.53474
$764$	0.120340	0.00435376
$765$	6.96405	0.251786
$766$	11.8246	0.427240
$767$	−0.121911	−0.00440193
$768$	2.44077	0.0880736
$769$	21.8200	0.786850	0.393425	−	0.919357i	$-0.371290\pi$
$769$	21.8200	0.786850	0.393425	+	0.919357i	$0.371290\pi$
$770$	−48.1702	−1.73593
$771$	55.4592	1.99731
$772$	21.1388	0.760804
$773$	39.8500	1.43331	0.716653	−	0.697430i	$-0.245673\pi$
$773$	39.8500	1.43331	0.716653	+	0.697430i	$0.245673\pi$
$774$	16.2191	0.582984
$775$	11.4536	0.411425
$776$	11.9217	0.427964
$777$	43.8855	1.57439
$778$	−29.8660	−1.07075
$779$	−4.97134	−0.178117
$780$	0.757494	0.0271226
$781$	13.1064	0.468985
$782$	−7.47876	−0.267440
$783$	0	0
$784$	2.26342	0.0808364
$785$	50.0582	1.78665
$786$	12.4658	0.444639
$787$	32.5707	1.16102	0.580510	−	0.814253i	$-0.302853\pi$
$787$	32.5707	1.16102	0.580510	+	0.814253i	$0.302853\pi$
$788$	−16.0211	−0.570728
$789$	63.5638	2.26293
$790$	12.1222	0.431288
$791$	−39.9289	−1.41971
$792$	−16.2191	−0.576321
$793$	0.878186	0.0311853
$794$	8.10324	0.287573
$795$	−27.8271	−0.986924
$796$	13.8572	0.491155
$797$	13.5804	0.481044	0.240522	−	0.970644i	$-0.422681\pi$
$797$	13.5804	0.481044	0.240522	+	0.970644i	$0.422681\pi$
$798$	−15.1743	−0.537163
$799$	4.82259	0.170611
$800$	−3.32789	−0.117659
$801$	−16.3882	−0.579050
$802$	−5.94737	−0.210009
$803$	−48.7784	−1.72135
$804$	2.33438	0.0823273
$805$	−80.4989	−2.83721
$806$	0.370133	0.0130374
$807$	−3.55492	−0.125139
$808$	−9.40543	−0.330882
$809$	−2.36666	−0.0832073	−0.0416037	−	0.999134i	$-0.513247\pi$
$809$	−2.36666	−0.0832073	−0.0416037	+	0.999134i	$0.513247\pi$
$810$	−26.3363	−0.925364
$811$	−45.5629	−1.59993	−0.799965	−	0.600047i	$-0.795148\pi$
$811$	−45.5629	−1.59993	−0.799965	+	0.600047i	$0.795148\pi$
$812$	0	0
$813$	26.6731	0.935466
$814$	32.3993	1.13559
$815$	33.5298	1.17450
$816$	−1.99168	−0.0697227
$817$	11.2027	0.391931
$818$	13.1395	0.459411
$819$	0.967995	0.0338245
$820$	−7.02337	−0.245267
$821$	38.8974	1.35753	0.678765	−	0.734356i	$-0.262516\pi$
$821$	38.8974	1.35753	0.678765	+	0.734356i	$0.262516\pi$
$822$	2.91194	0.101565
$823$	−9.56146	−0.333291	−0.166646	−	0.986017i	$-0.553294\pi$
$823$	−9.56146	−0.333291	−0.166646	+	0.986017i	$0.553294\pi$
$824$	11.3271	0.394600
$825$	44.5473	1.55094
$826$	3.45018	0.120047
$827$	−1.22644	−0.0426476	−0.0213238	−	0.999773i	$-0.506788\pi$
$827$	−1.22644	−0.0426476	−0.0213238	+	0.999773i	$0.506788\pi$
$828$	−27.1043	−0.941940
$829$	−45.1704	−1.56883	−0.784417	−	0.620234i	$-0.787038\pi$
$829$	−45.1704	−1.56883	−0.784417	+	0.620234i	$0.787038\pi$
$830$	−44.6527	−1.54992
$831$	−6.03459	−0.209338
$832$	−0.107544	−0.00372841
$833$	−1.84696	−0.0639934
$834$	−24.6086	−0.852126
$835$	24.5492	0.849560
$836$	−11.2027	−0.387452
$837$	−0.358334	−0.0123858
$838$	−22.1612	−0.765547
$839$	4.50926	0.155677	0.0778384	−	0.996966i	$-0.475198\pi$
$839$	4.50926	0.155677	0.0778384	+	0.996966i	$0.475198\pi$
$840$	−21.4378	−0.739673
$841$	0	0
$842$	23.8521	0.821998
$843$	−0.455561	−0.0156904
$844$	24.3125	0.836870
$845$	37.4821	1.28942
$846$	17.4779	0.600902
$847$	−58.0659	−1.99517
$848$	3.95070	0.135667
$849$	−45.0290	−1.54539
$850$	2.71558	0.0931435
$851$	54.1435	1.85602
$852$	5.83291	0.199832
$853$	11.4706	0.392747	0.196373	−	0.980529i	$-0.437084\pi$
$853$	11.4706	0.392747	0.196373	+	0.980529i	$0.437084\pi$
$854$	−24.8534	−0.850467
$855$	17.4327	0.596186
$856$	−10.4570	−0.357413
$857$	−5.18116	−0.176985	−0.0884926	−	0.996077i	$-0.528205\pi$
$857$	−5.18116	−0.176985	−0.0884926	+	0.996077i	$0.528205\pi$
$858$	1.43958	0.0491466
$859$	−24.6785	−0.842020	−0.421010	−	0.907056i	$-0.638324\pi$
$859$	−24.6785	−0.842020	−0.421010	+	0.907056i	$0.638324\pi$
$860$	15.8268	0.539689
$861$	−18.0796	−0.616153
$862$	0.776969	0.0264637
$863$	36.2370	1.23352	0.616762	−	0.787150i	$-0.288444\pi$
$863$	36.2370	1.23352	0.616762	+	0.787150i	$0.288444\pi$
$864$	0.104116	0.00354208
$865$	−29.3970	−0.999529
$866$	0.612460	0.0208122
$867$	−39.8678	−1.35398
$868$	−10.4751	−0.355548
$869$	23.0377	0.781500
$870$	0	0
$871$	−0.102856	−0.00348516
$872$	13.9287	0.471686
$873$	−35.2565	−1.19325
$874$	−18.7211	−0.633252
$875$	−14.6865	−0.496493
$876$	−21.7084	−0.733460
$877$	−27.3664	−0.924099	−0.462049	−	0.886854i	$-0.652886\pi$
$877$	−27.3664	−0.924099	−0.462049	+	0.886854i	$0.652886\pi$
$878$	10.8953	0.367698
$879$	3.79857	0.128123
$880$	−15.8268	−0.533521
$881$	3.17405	0.106936	0.0534682	−	0.998570i	$-0.482972\pi$
$881$	3.17405	0.106936	0.0534682	+	0.998570i	$0.482972\pi$
$882$	−6.69371	−0.225389
$883$	8.67680	0.291998	0.145999	−	0.989285i	$-0.453360\pi$
$883$	8.67680	0.291998	0.145999	+	0.989285i	$0.453360\pi$
$884$	0.0877563	0.00295156
$885$	−7.98453	−0.268397
$886$	−34.8279	−1.17007
$887$	−2.46214	−0.0826706	−0.0413353	−	0.999145i	$-0.513161\pi$
$887$	−2.46214	−0.0826706	−0.0413353	+	0.999145i	$0.513161\pi$
$888$	14.4190	0.483871
$889$	−4.07841	−0.136785
$890$	−15.9918	−0.536047
$891$	−50.0510	−1.67677
$892$	10.4249	0.349052
$893$	12.0721	0.403977
$894$	4.27368	0.142933
$895$	−13.8213	−0.461997
$896$	3.04359	0.101679
$897$	2.40574	0.0803253
$898$	5.02242	0.167600
$899$	0	0
$900$	9.84172	0.328057
$901$	−3.22379	−0.107400
$902$	−13.3476	−0.444427
$903$	40.7415	1.35579
$904$	−13.1190	−0.436333
$905$	−2.12728	−0.0707132
$906$	1.37682	0.0457418
$907$	−6.91867	−0.229731	−0.114865	−	0.993381i	$-0.536644\pi$
$907$	−6.91867	−0.229731	−0.114865	+	0.993381i	$0.536644\pi$
$908$	20.9815	0.696294
$909$	27.8151	0.922568
$910$	0.944580	0.0313125
$911$	−51.8794	−1.71884	−0.859421	−	0.511269i	$-0.829176\pi$
$911$	−51.8794	−1.71884	−0.859421	+	0.511269i	$0.829176\pi$
$912$	−4.98565	−0.165091
$913$	−84.8605	−2.80847
$914$	−38.2426	−1.26495
$915$	57.5168	1.90145
$916$	−0.984111	−0.0325159
$917$	15.5446	0.513326
$918$	−0.0849588	−0.00280406
$919$	−27.1032	−0.894052	−0.447026	−	0.894521i	$-0.647517\pi$
$919$	−27.1032	−0.894052	−0.447026	+	0.894521i	$0.647517\pi$
$920$	−26.4487	−0.871988
$921$	−47.8390	−1.57635
$922$	25.9962	0.856139
$923$	−0.257007	−0.00845949
$924$	−40.7415	−1.34030
$925$	−19.6598	−0.646410
$926$	−17.3502	−0.570162
$927$	−33.4983	−1.10023
$928$	0	0
$929$	−40.7022	−1.33539	−0.667697	−	0.744433i	$-0.732720\pi$
$929$	−40.7022	−1.33539	−0.667697	+	0.744433i	$0.732720\pi$
$930$	24.2419	0.794923
$931$	−4.62339	−0.151525
$932$	17.3778	0.569229
$933$	64.6024	2.11499
$934$	29.2399	0.956757
$935$	12.9147	0.422357
$936$	0.318044	0.0103956
$937$	2.92447	0.0955383	0.0477691	−	0.998858i	$-0.484789\pi$
$937$	2.92447	0.0955383	0.0477691	+	0.998858i	$0.484789\pi$
$938$	2.91093	0.0950451
$939$	53.2208	1.73679
$940$	17.0551	0.556277
$941$	30.1902	0.984171	0.492085	−	0.870547i	$-0.336235\pi$
$941$	30.1902	0.984171	0.492085	+	0.870547i	$0.336235\pi$
$942$	42.3383	1.37946
$943$	−22.3056	−0.726372
$944$	1.13359	0.0368952
$945$	−0.914469	−0.0297477
$946$	30.0781	0.977924
$947$	−25.3391	−0.823410	−0.411705	−	0.911317i	$-0.635067\pi$
$947$	−25.3391	−0.823410	−0.411705	+	0.911317i	$0.635067\pi$
$948$	10.2527	0.332993
$949$	0.956505	0.0310495
$950$	6.79774	0.220548
$951$	−46.2000	−1.49814
$952$	−2.48358	−0.0804933
$953$	33.2844	1.07819	0.539094	−	0.842245i	$-0.318767\pi$
$953$	33.2844	1.07819	0.539094	+	0.842245i	$0.318767\pi$
$954$	−11.6836	−0.378269
$955$	−0.347279	−0.0112377
$956$	−1.15163	−0.0372465
$957$	0	0
$958$	42.9871	1.38885
$959$	3.63113	0.117255
$960$	−7.04359	−0.227331
$961$	−19.1547	−0.617894
$962$	−0.635324	−0.0204837
$963$	30.9249	0.996541
$964$	−19.7078	−0.634747
$965$	−61.0027	−1.96375
$966$	−68.0846	−2.19059
$967$	37.6906	1.21205	0.606024	−	0.795447i	$-0.292764\pi$
$967$	37.6906	1.21205	0.606024	+	0.795447i	$0.292764\pi$
$968$	−19.0781	−0.613194
$969$	4.06831	0.130693
$970$	−34.4037	−1.10464
$971$	23.2873	0.747324	0.373662	−	0.927565i	$-0.378102\pi$
$971$	23.2873	0.747324	0.373662	+	0.927565i	$0.378102\pi$
$972$	−21.9625	−0.704446
$973$	−30.6864	−0.983762
$974$	9.99343	0.320210
$975$	−0.873537	−0.0279756
$976$	−8.16584	−0.261382
$977$	−14.6622	−0.469085	−0.234542	−	0.972106i	$-0.575359\pi$
$977$	−14.6622	−0.469085	−0.234542	+	0.972106i	$0.575359\pi$
$978$	28.3590	0.906820
$979$	−30.3917	−0.971324
$980$	−6.53179	−0.208651
$981$	−41.1920	−1.31516
$982$	3.80604	0.121456
$983$	−17.2640	−0.550636	−0.275318	−	0.961353i	$-0.588783\pi$
$983$	−17.2640	−0.550636	−0.275318	+	0.961353i	$0.588783\pi$
$984$	−5.94024	−0.189368
$985$	46.2338	1.47313
$986$	0	0
$987$	43.9035	1.39746
$988$	0.219675	0.00698880
$989$	50.2646	1.59832
$990$	46.8053	1.48757
$991$	9.40478	0.298753	0.149376	−	0.988780i	$-0.452273\pi$
$991$	9.40478	0.298753	0.149376	+	0.988780i	$0.452273\pi$
$992$	−3.44170	−0.109274
$993$	−43.3983	−1.37720
$994$	7.27352	0.230702
$995$	−39.9892	−1.26774
$996$	−37.7665	−1.19668
$997$	46.5435	1.47405	0.737023	−	0.675867i	$-0.236231\pi$
$997$	46.5435	1.47405	0.737023	+	0.675867i	$0.236231\pi$
$998$	28.3943	0.898806
$999$	0.615071	0.0194600

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1682.2.a.q.1.6		6
29.4	even	14		58.2.d.b.45.1	✓	12
29.12	odd	4		1682.2.b.i.1681.6		12
29.17	odd	4		1682.2.b.i.1681.7		12
29.22	even	14		58.2.d.b.49.1	yes	12
29.28	even	2		1682.2.a.t.1.1		6
87.62	odd	14		522.2.k.h.451.1		12
87.80	odd	14		522.2.k.h.397.1		12
116.51	odd	14		464.2.u.h.49.2		12
116.91	odd	14		464.2.u.h.161.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.d.b.45.1	✓	12	29.4	even	14
58.2.d.b.49.1	yes	12	29.22	even	14
464.2.u.h.49.2		12	116.51	odd	14
464.2.u.h.161.2		12	116.91	odd	14
522.2.k.h.397.1		12	87.80	odd	14
522.2.k.h.451.1		12	87.62	odd	14
1682.2.a.q.1.6		6	1.1	even	1	trivial
1682.2.a.t.1.1		6	29.28	even	2
1682.2.b.i.1681.6		12	29.12	odd	4
1682.2.b.i.1681.7		12	29.17	odd	4

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1682.2.a.q.1.6

Level	$1682$
Weight	$2$
Character	1682.1
Self dual	yes
Analytic conductor	$13.431$
Analytic rank	$1$
Dimension	$6$
CM	no
Inner twists	$1$