LMFDB - Embedded newform 1682.2.a.r.1.1

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:	$0$
Dimension:	$6$
Coefficient field:	$\Q(\zeta_{28})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - 7x^{4} + 14x^{2} - 7$ x^6 - 7x^4 + 14x^2 - 7
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			1.1
Root			$1.56366$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} -0.246980 q^{3} +1.00000 q^{4} -3.43143 q^{5} +0.246980 q^{6} +1.57064 q^{7} -1.00000 q^{8} -2.93900 q^{9} +3.43143 q^{10} -4.37431 q^{11} -0.246980 q^{12} -4.54503 q^{13} -1.57064 q^{14} +0.847493 q^{15} +1.00000 q^{16} +2.21432 q^{17} +2.93900 q^{18} -0.423299 q^{19} -3.43143 q^{20} -0.387917 q^{21} +4.37431 q^{22} -6.15540 q^{23} +0.246980 q^{24} +6.77471 q^{25} +4.54503 q^{26} +1.46681 q^{27} +1.57064 q^{28} -0.847493 q^{30} +7.46854 q^{31} -1.00000 q^{32} +1.08036 q^{33} -2.21432 q^{34} -5.38955 q^{35} -2.93900 q^{36} -1.58863 q^{37} +0.423299 q^{38} +1.12253 q^{39} +3.43143 q^{40} +4.56642 q^{41} +0.387917 q^{42} +9.26439 q^{43} -4.37431 q^{44} +10.0850 q^{45} +6.15540 q^{46} -4.31029 q^{47} -0.246980 q^{48} -4.53308 q^{49} -6.77471 q^{50} -0.546892 q^{51} -4.54503 q^{52} +6.53728 q^{53} -1.46681 q^{54} +15.0101 q^{55} -1.57064 q^{56} +0.104546 q^{57} -14.5556 q^{59} +0.847493 q^{60} -6.16373 q^{61} -7.46854 q^{62} -4.61612 q^{63} +1.00000 q^{64} +15.5959 q^{65} -1.08036 q^{66} -8.23637 q^{67} +2.21432 q^{68} +1.52026 q^{69} +5.38955 q^{70} +12.6713 q^{71} +2.93900 q^{72} +8.64084 q^{73} +1.58863 q^{74} -1.67322 q^{75} -0.423299 q^{76} -6.87048 q^{77} -1.12253 q^{78} +6.44491 q^{79} -3.43143 q^{80} +8.45473 q^{81} -4.56642 q^{82} +0.615687 q^{83} -0.387917 q^{84} -7.59829 q^{85} -9.26439 q^{86} +4.37431 q^{88} -5.06665 q^{89} -10.0850 q^{90} -7.13862 q^{91} -6.15540 q^{92} -1.84458 q^{93} +4.31029 q^{94} +1.45252 q^{95} +0.246980 q^{96} +8.13902 q^{97} +4.53308 q^{98} +12.8561 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 6 q^{2} + 8 q^{3} + 6 q^{4} - 6 q^{5} - 8 q^{6} + 2 q^{7} - 6 q^{8} + 2 q^{9} + 6 q^{10} + 2 q^{11} + 8 q^{12} - 8 q^{13} - 2 q^{14} - 8 q^{15} + 6 q^{16} + 12 q^{17} - 2 q^{18} + 4 q^{19} - 6 q^{20}+ \cdots + 24 q^{99}+O(q^{100})$ 6 * q - 6 * q^2 + 8 * q^3 + 6 * q^4 - 6 * q^5 - 8 * q^6 + 2 * q^7 - 6 * q^8 + 2 * q^9 + 6 * q^10 + 2 * q^11 + 8 * q^12 - 8 * q^13 - 2 * q^14 - 8 * q^15 + 6 * q^16 + 12 * q^17 - 2 * q^18 + 4 * q^19 - 6 * q^20 + 12 * q^21 - 2 * q^22 + 6 * q^23 - 8 * q^24 + 4 * q^25 + 8 * q^26 + 2 * q^27 + 2 * q^28 + 8 * q^30 + 26 * q^31 - 6 * q^32 + 12 * q^33 - 12 * q^34 - 16 * q^35 + 2 * q^36 + 20 * q^37 - 4 * q^38 - 6 * q^39 + 6 * q^40 + 24 * q^41 - 12 * q^42 + 26 * q^43 + 2 * q^44 - 2 * q^45 - 6 * q^46 + 18 * q^47 + 8 * q^48 - 4 * q^49 - 4 * q^50 + 16 * q^51 - 8 * q^52 + 8 * q^53 - 2 * q^54 + 26 * q^55 - 2 * q^56 + 10 * q^57 - 24 * q^59 - 8 * q^60 + 18 * q^61 - 26 * q^62 + 24 * q^63 + 6 * q^64 - 6 * q^65 - 12 * q^66 + 8 * q^67 + 12 * q^68 + 22 * q^69 + 16 * q^70 + 20 * q^71 - 2 * q^72 + 12 * q^73 - 20 * q^74 + 10 * q^75 + 4 * q^76 + 10 * q^77 + 6 * q^78 + 20 * q^79 - 6 * q^80 + 6 * q^81 - 24 * q^82 - 28 * q^83 + 12 * q^84 + 16 * q^85 - 26 * q^86 - 2 * q^88 + 2 * q^90 - 12 * q^91 + 6 * q^92 + 30 * q^93 - 18 * q^94 - 32 * q^95 - 8 * q^96 + 44 * q^97 + 4 * q^98 + 24 * 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
$3$	−0.246980	−0.142594	−0.0712969	−	0.997455i	$-0.522714\pi$
$3$	−0.246980	−0.142594	−0.0712969	+	0.997455i	$0.522714\pi$
$4$	1.00000	0.500000
$5$	−3.43143	−1.53458	−0.767291	−	0.641299i	$-0.778396\pi$
$5$	−3.43143	−1.53458	−0.767291	+	0.641299i	$0.778396\pi$
$6$	0.246980	0.100829
$7$	1.57064	0.593648	0.296824	−	0.954932i	$-0.404073\pi$
$7$	1.57064	0.593648	0.296824	+	0.954932i	$0.404073\pi$
$8$	−1.00000	−0.353553
$9$	−2.93900	−0.979667
$10$	3.43143	1.08511
$11$	−4.37431	−1.31890	−0.659451	−	0.751747i	$-0.729211\pi$
$11$	−4.37431	−1.31890	−0.659451	+	0.751747i	$0.729211\pi$
$12$	−0.246980	−0.0712969
$13$	−4.54503	−1.26056	−0.630282	−	0.776366i	$-0.717061\pi$
$13$	−4.54503	−1.26056	−0.630282	+	0.776366i	$0.717061\pi$
$14$	−1.57064	−0.419772
$15$	0.847493	0.218822
$16$	1.00000	0.250000
$17$	2.21432	0.537052	0.268526	−	0.963272i	$-0.413464\pi$
$17$	2.21432	0.537052	0.268526	+	0.963272i	$0.413464\pi$
$18$	2.93900	0.692729
$19$	−0.423299	−0.0971114	−0.0485557	−	0.998820i	$-0.515462\pi$
$19$	−0.423299	−0.0971114	−0.0485557	+	0.998820i	$0.515462\pi$
$20$	−3.43143	−0.767291
$21$	−0.387917	−0.0846504
$22$	4.37431	0.932605
$23$	−6.15540	−1.28349	−0.641744	−	0.766919i	$-0.721789\pi$
$23$	−6.15540	−1.28349	−0.641744	+	0.766919i	$0.721789\pi$
$24$	0.246980	0.0504145
$25$	6.77471	1.35494
$26$	4.54503	0.891353
$27$	1.46681	0.282288
$28$	1.57064	0.296824
$29$	0	0
$29$	0	0
$30$	−0.847493	−0.154730
$31$	7.46854	1.34139	0.670694	−	0.741734i	$-0.265996\pi$
$31$	7.46854	1.34139	0.670694	+	0.741734i	$0.265996\pi$
$32$	−1.00000	−0.176777
$33$	1.08036	0.188067
$34$	−2.21432	−0.379753
$35$	−5.38955	−0.911001
$36$	−2.93900	−0.489834
$37$	−1.58863	−0.261169	−0.130584	−	0.991437i	$-0.541685\pi$
$37$	−1.58863	−0.261169	−0.130584	+	0.991437i	$0.541685\pi$
$38$	0.423299	0.0686681
$39$	1.12253	0.179748
$40$	3.43143	0.542557
$41$	4.56642	0.713155	0.356577	−	0.934266i	$-0.383944\pi$
$41$	4.56642	0.713155	0.356577	+	0.934266i	$0.383944\pi$
$42$	0.387917	0.0598569
$43$	9.26439	1.41281	0.706403	−	0.707810i	$-0.250317\pi$
$43$	9.26439	1.41281	0.706403	+	0.707810i	$0.250317\pi$
$44$	−4.37431	−0.659451
$45$	10.0850	1.50338
$46$	6.15540	0.907564
$47$	−4.31029	−0.628721	−0.314361	−	0.949304i	$-0.601790\pi$
$47$	−4.31029	−0.628721	−0.314361	+	0.949304i	$0.601790\pi$
$48$	−0.246980	−0.0356484
$49$	−4.53308	−0.647583
$50$	−6.77471	−0.958089
$51$	−0.546892	−0.0765802
$52$	−4.54503	−0.630282
$53$	6.53728	0.897964	0.448982	−	0.893541i	$-0.351787\pi$
$53$	6.53728	0.897964	0.448982	+	0.893541i	$0.351787\pi$
$54$	−1.46681	−0.199608
$55$	15.0101	2.02396
$56$	−1.57064	−0.209886
$57$	0.104546	0.0138475
$58$	0	0
$59$	−14.5556	−1.89498	−0.947490	−	0.319787i	$-0.896389\pi$
$59$	−14.5556	−1.89498	−0.947490	+	0.319787i	$0.896389\pi$
$60$	0.847493	0.109411
$61$	−6.16373	−0.789184	−0.394592	−	0.918856i	$-0.629114\pi$
$61$	−6.16373	−0.789184	−0.394592	+	0.918856i	$0.629114\pi$
$62$	−7.46854	−0.948505
$63$	−4.61612	−0.581577
$64$	1.00000	0.125000
$65$	15.5959	1.93444
$66$	−1.08036	−0.132984
$67$	−8.23637	−1.00623	−0.503116	−	0.864219i	$-0.667813\pi$
$67$	−8.23637	−1.00623	−0.503116	+	0.864219i	$0.667813\pi$
$68$	2.21432	0.268526
$69$	1.52026	0.183017
$70$	5.38955	0.644175
$71$	12.6713	1.50381	0.751904	−	0.659272i	$-0.229135\pi$
$71$	12.6713	1.50381	0.751904	+	0.659272i	$0.229135\pi$
$72$	2.93900	0.346365
$73$	8.64084	1.01133	0.505667	−	0.862729i	$-0.331246\pi$
$73$	8.64084	1.01133	0.505667	+	0.862729i	$0.331246\pi$
$74$	1.58863	0.184674
$75$	−1.67322	−0.193206
$76$	−0.423299	−0.0485557
$77$	−6.87048	−0.782963
$78$	−1.12253	−0.127101
$79$	6.44491	0.725109	0.362554	−	0.931963i	$-0.381905\pi$
$79$	6.44491	0.725109	0.362554	+	0.931963i	$0.381905\pi$
$80$	−3.43143	−0.383646
$81$	8.45473	0.939415
$82$	−4.56642	−0.504277
$83$	0.615687	0.0675804	0.0337902	−	0.999429i	$-0.489242\pi$
$83$	0.615687	0.0675804	0.0337902	+	0.999429i	$0.489242\pi$
$84$	−0.387917	−0.0423252
$85$	−7.59829	−0.824150
$86$	−9.26439	−0.999005
$87$	0	0
$88$	4.37431	0.466303
$89$	−5.06665	−0.537063	−0.268532	−	0.963271i	$-0.586538\pi$
$89$	−5.06665	−0.537063	−0.268532	+	0.963271i	$0.586538\pi$
$90$	−10.0850	−1.06305
$91$	−7.13862	−0.748330
$92$	−6.15540	−0.641744
$93$	−1.84458	−0.191274
$94$	4.31029	0.444573
$95$	1.45252	0.149025
$96$	0.246980	0.0252073
$97$	8.13902	0.826392	0.413196	−	0.910642i	$-0.364412\pi$
$97$	8.13902	0.826392	0.413196	+	0.910642i	$0.364412\pi$
$98$	4.53308	0.457910
$99$	12.8561	1.29209
$100$	6.77471	0.677471
$101$	16.1534	1.60732	0.803662	−	0.595086i	$-0.202882\pi$
$101$	16.1534	1.60732	0.803662	+	0.595086i	$0.202882\pi$
$102$	0.546892	0.0541504
$103$	−0.486918	−0.0479774	−0.0239887	−	0.999712i	$-0.507637\pi$
$103$	−0.486918	−0.0479774	−0.0239887	+	0.999712i	$0.507637\pi$
$104$	4.54503	0.445677
$105$	1.33111	0.129903
$106$	−6.53728	−0.634957
$107$	−5.41656	−0.523638	−0.261819	−	0.965117i	$-0.584322\pi$
$107$	−5.41656	−0.523638	−0.261819	+	0.965117i	$0.584322\pi$
$108$	1.46681	0.141144
$109$	6.88139	0.659117	0.329559	−	0.944135i	$-0.393100\pi$
$109$	6.88139	0.659117	0.329559	+	0.944135i	$0.393100\pi$
$110$	−15.0101	−1.43116
$111$	0.392358	0.0372410
$112$	1.57064	0.148412
$113$	5.15909	0.485326	0.242663	−	0.970111i	$-0.421979\pi$
$113$	5.15909	0.485326	0.242663	+	0.970111i	$0.421979\pi$
$114$	−0.104546	−0.00979164
$115$	21.1218	1.96962
$116$	0	0
$117$	13.3578	1.23493
$118$	14.5556	1.33995
$119$	3.47791	0.318819
$120$	−0.847493	−0.0773652
$121$	8.13455	0.739504
$122$	6.16373	0.558037
$123$	−1.12781	−0.101691
$124$	7.46854	0.670694
$125$	−6.08981	−0.544689
$126$	4.61612	0.411237
$127$	−14.6339	−1.29855	−0.649276	−	0.760553i	$-0.724928\pi$
$127$	−14.6339	−1.29855	−0.649276	+	0.760553i	$0.724928\pi$
$128$	−1.00000	−0.0883883
$129$	−2.28812	−0.201457
$130$	−15.5959	−1.36785
$131$	−12.7288	−1.11212	−0.556059	−	0.831143i	$-0.687687\pi$
$131$	−12.7288	−1.11212	−0.556059	+	0.831143i	$0.687687\pi$
$132$	1.08036	0.0940336
$133$	−0.664851	−0.0576499
$134$	8.23637	0.711514
$135$	−5.03326	−0.433194
$136$	−2.21432	−0.189876
$137$	−7.19607	−0.614802	−0.307401	−	0.951580i	$-0.599459\pi$
$137$	−7.19607	−0.614802	−0.307401	+	0.951580i	$0.599459\pi$
$138$	−1.52026	−0.129413
$139$	5.60448	0.475366	0.237683	−	0.971343i	$-0.423612\pi$
$139$	5.60448	0.475366	0.237683	+	0.971343i	$0.423612\pi$
$140$	−5.38955	−0.455501
$141$	1.06455	0.0896517
$142$	−12.6713	−1.06335
$143$	19.8813	1.66256
$144$	−2.93900	−0.244917
$145$	0	0
$146$	−8.64084	−0.715121
$147$	1.11958	0.0923412
$148$	−1.58863	−0.130584
$149$	15.4995	1.26977	0.634884	−	0.772607i	$-0.281048\pi$
$149$	15.4995	1.26977	0.634884	+	0.772607i	$0.281048\pi$
$150$	1.67322	0.136618
$151$	2.02651	0.164915	0.0824575	−	0.996595i	$-0.473723\pi$
$151$	2.02651	0.164915	0.0824575	+	0.996595i	$0.473723\pi$
$152$	0.423299	0.0343341
$153$	−6.50789	−0.526132
$154$	6.87048	0.553639
$155$	−25.6278	−2.05847
$156$	1.12253	0.0898742
$157$	−0.631946	−0.0504348	−0.0252174	−	0.999682i	$-0.508028\pi$
$157$	−0.631946	−0.0504348	−0.0252174	+	0.999682i	$0.508028\pi$
$158$	−6.44491	−0.512729
$159$	−1.61457	−0.128044
$160$	3.43143	0.271278
$161$	−9.66793	−0.761940
$162$	−8.45473	−0.664266
$163$	24.3103	1.90413	0.952065	−	0.305897i	$-0.0989563\pi$
$163$	24.3103	1.90413	0.952065	+	0.305897i	$0.0989563\pi$
$164$	4.56642	0.356577
$165$	−3.70719	−0.288605
$166$	−0.615687	−0.0477866
$167$	14.8620	1.15006	0.575029	−	0.818133i	$-0.304991\pi$
$167$	14.8620	1.15006	0.575029	+	0.818133i	$0.304991\pi$
$168$	0.387917	0.0299284
$169$	7.65727	0.589020
$170$	7.59829	0.582762
$171$	1.24408	0.0951368
$172$	9.26439	0.706403
$173$	−0.183131	−0.0139232	−0.00696160	−	0.999976i	$-0.502216\pi$
$173$	−0.183131	−0.0139232	−0.00696160	+	0.999976i	$0.502216\pi$
$174$	0	0
$175$	10.6407	0.804359
$176$	−4.37431	−0.329726
$177$	3.59494	0.270212
$178$	5.06665	0.379761
$179$	−7.94570	−0.593890	−0.296945	−	0.954895i	$-0.595968\pi$
$179$	−7.94570	−0.593890	−0.296945	+	0.954895i	$0.595968\pi$
$180$	10.0850	0.751690
$181$	1.79743	0.133602	0.0668011	−	0.997766i	$-0.478721\pi$
$181$	1.79743	0.133602	0.0668011	+	0.997766i	$0.478721\pi$
$182$	7.13862	0.529150
$183$	1.52231	0.112533
$184$	6.15540	0.453782
$185$	5.45126	0.400785
$186$	1.84458	0.135251
$187$	−9.68612	−0.708319
$188$	−4.31029	−0.314361
$189$	2.30384	0.167580
$190$	−1.45252	−0.105377
$191$	−20.0665	−1.45196	−0.725981	−	0.687715i	$-0.758614\pi$
$191$	−20.0665	−1.45196	−0.725981	+	0.687715i	$0.758614\pi$
$192$	−0.246980	−0.0178242
$193$	7.08658	0.510103	0.255051	−	0.966927i	$-0.417908\pi$
$193$	7.08658	0.510103	0.255051	+	0.966927i	$0.417908\pi$
$194$	−8.13902	−0.584348
$195$	−3.85188	−0.275839
$196$	−4.53308	−0.323791
$197$	4.62674	0.329641	0.164821	−	0.986324i	$-0.447295\pi$
$197$	4.62674	0.329641	0.164821	+	0.986324i	$0.447295\pi$
$198$	−12.8561	−0.913642
$199$	3.58103	0.253852	0.126926	−	0.991912i	$-0.459489\pi$
$199$	3.58103	0.253852	0.126926	+	0.991912i	$0.459489\pi$
$200$	−6.77471	−0.479045
$201$	2.03421	0.143482
$202$	−16.1534	−1.13655
$203$	0	0
$204$	−0.546892	−0.0382901
$205$	−15.6694	−1.09439
$206$	0.486918	0.0339252
$207$	18.0907	1.25739
$208$	−4.54503	−0.315141
$209$	1.85164	0.128080
$210$	−1.33111	−0.0918553
$211$	1.81599	0.125018	0.0625091	−	0.998044i	$-0.480090\pi$
$211$	1.81599	0.125018	0.0625091	+	0.998044i	$0.480090\pi$
$212$	6.53728	0.448982
$213$	−3.12956	−0.214434
$214$	5.41656	0.370268
$215$	−31.7901	−2.16807
$216$	−1.46681	−0.0998039
$217$	11.7304	0.796312
$218$	−6.88139	−0.466066
$219$	−2.13411	−0.144210
$220$	15.0101	1.01198
$221$	−10.0641	−0.676988
$222$	−0.392358	−0.0263334
$223$	20.9351	1.40192	0.700958	−	0.713203i	$-0.252756\pi$
$223$	20.9351	1.40192	0.700958	+	0.713203i	$0.252756\pi$
$224$	−1.57064	−0.104943
$225$	−19.9109	−1.32739
$226$	−5.15909	−0.343177
$227$	2.77986	0.184506	0.0922530	−	0.995736i	$-0.470593\pi$
$227$	2.77986	0.184506	0.0922530	+	0.995736i	$0.470593\pi$
$228$	0.104546	0.00692374
$229$	−24.0260	−1.58768	−0.793842	−	0.608124i	$-0.791922\pi$
$229$	−24.0260	−1.58768	−0.793842	+	0.608124i	$0.791922\pi$
$230$	−21.1218	−1.39273
$231$	1.69687	0.111646
$232$	0	0
$233$	−3.18472	−0.208638	−0.104319	−	0.994544i	$-0.533266\pi$
$233$	−3.18472	−0.208638	−0.104319	+	0.994544i	$0.533266\pi$
$234$	−13.3578	−0.873229
$235$	14.7905	0.964824
$236$	−14.5556	−0.947490
$237$	−1.59176	−0.103396
$238$	−3.47791	−0.225439
$239$	12.8020	0.828094	0.414047	−	0.910255i	$-0.364115\pi$
$239$	12.8020	0.828094	0.414047	+	0.910255i	$0.364115\pi$
$240$	0.847493	0.0547055
$241$	0.472390	0.0304293	0.0152146	−	0.999884i	$-0.495157\pi$
$241$	0.472390	0.0304293	0.0152146	+	0.999884i	$0.495157\pi$
$242$	−8.13455	−0.522909
$243$	−6.48858	−0.416243
$244$	−6.16373	−0.394592
$245$	15.5549	0.993769
$246$	1.12781	0.0719067
$247$	1.92390	0.122415
$248$	−7.46854	−0.474253
$249$	−0.152062	−0.00963655
$250$	6.08981	0.385153
$251$	0.992245	0.0626300	0.0313150	−	0.999510i	$-0.490031\pi$
$251$	0.992245	0.0626300	0.0313150	+	0.999510i	$0.490031\pi$
$252$	−4.61612	−0.290788
$253$	26.9256	1.69280
$254$	14.6339	0.918215
$255$	1.87662	0.117519
$256$	1.00000	0.0625000
$257$	−21.2077	−1.32290	−0.661450	−	0.749989i	$-0.730059\pi$
$257$	−21.2077	−1.32290	−0.661450	+	0.749989i	$0.730059\pi$
$258$	2.28812	0.142452
$259$	−2.49517	−0.155042
$260$	15.5959	0.967219
$261$	0	0
$262$	12.7288	0.786386
$263$	4.59716	0.283473	0.141737	−	0.989904i	$-0.454731\pi$
$263$	4.59716	0.283473	0.141737	+	0.989904i	$0.454731\pi$
$264$	−1.08036	−0.0664918
$265$	−22.4322	−1.37800
$266$	0.664851	0.0407647
$267$	1.25136	0.0765819
$268$	−8.23637	−0.503116
$269$	2.23745	0.136420	0.0682098	−	0.997671i	$-0.478271\pi$
$269$	2.23745	0.136420	0.0682098	+	0.997671i	$0.478271\pi$
$270$	5.03326	0.306315
$271$	5.66146	0.343909	0.171955	−	0.985105i	$-0.444992\pi$
$271$	5.66146	0.343909	0.171955	+	0.985105i	$0.444992\pi$
$272$	2.21432	0.134263
$273$	1.76309	0.106707
$274$	7.19607	0.434730
$275$	−29.6347	−1.78704
$276$	1.52026	0.0915087
$277$	−20.7393	−1.24611	−0.623053	−	0.782180i	$-0.714108\pi$
$277$	−20.7393	−1.24611	−0.623053	+	0.782180i	$0.714108\pi$
$278$	−5.60448	−0.336135
$279$	−21.9500	−1.31411
$280$	5.38955	0.322087
$281$	18.6958	1.11530	0.557648	−	0.830077i	$-0.311704\pi$
$281$	18.6958	1.11530	0.557648	+	0.830077i	$0.311704\pi$
$282$	−1.06455	−0.0633933
$283$	−8.59049	−0.510651	−0.255326	−	0.966855i	$-0.582183\pi$
$283$	−8.59049	−0.510651	−0.255326	+	0.966855i	$0.582183\pi$
$284$	12.6713	0.751904
$285$	−0.358743	−0.0212501
$286$	−19.8813	−1.17561
$287$	7.17222	0.423363
$288$	2.93900	0.173182
$289$	−12.0968	−0.711575
$290$	0	0
$291$	−2.01017	−0.117838
$292$	8.64084	0.505667
$293$	7.69030	0.449272	0.224636	−	0.974443i	$-0.427881\pi$
$293$	7.69030	0.449272	0.224636	+	0.974443i	$0.427881\pi$
$294$	−1.11958	−0.0652951
$295$	49.9466	2.90800
$296$	1.58863	0.0923370
$297$	−6.41628	−0.372311
$298$	−15.4995	−0.897862
$299$	27.9764	1.61792
$300$	−1.67322	−0.0966032
$301$	14.5511	0.838709
$302$	−2.02651	−0.116612
$303$	−3.98956	−0.229194
$304$	−0.423299	−0.0242778
$305$	21.1504	1.21107
$306$	6.50789	0.372031
$307$	14.7061	0.839322	0.419661	−	0.907681i	$-0.362149\pi$
$307$	14.7061	0.839322	0.419661	+	0.907681i	$0.362149\pi$
$308$	−6.87048	−0.391482
$309$	0.120259	0.00684128
$310$	25.6278	1.45556
$311$	16.9331	0.960186	0.480093	−	0.877218i	$-0.340603\pi$
$311$	16.9331	0.960186	0.480093	+	0.877218i	$0.340603\pi$
$312$	−1.12253	−0.0635507
$313$	16.1943	0.915356	0.457678	−	0.889118i	$-0.348681\pi$
$313$	16.1943	0.915356	0.457678	+	0.889118i	$0.348681\pi$
$314$	0.631946	0.0356628
$315$	15.8399	0.892478
$316$	6.44491	0.362554
$317$	−10.7887	−0.605956	−0.302978	−	0.952997i	$-0.597981\pi$
$317$	−10.7887	−0.605956	−0.302978	+	0.952997i	$0.597981\pi$
$318$	1.61457	0.0905408
$319$	0	0
$320$	−3.43143	−0.191823
$321$	1.33778	0.0746676
$322$	9.66793	0.538773
$323$	−0.937319	−0.0521538
$324$	8.45473	0.469707
$325$	−30.7913	−1.70799
$326$	−24.3103	−1.34642
$327$	−1.69956	−0.0939860
$328$	−4.56642	−0.252138
$329$	−6.76994	−0.373239
$330$	3.70719	0.204074
$331$	26.4094	1.45159	0.725796	−	0.687910i	$-0.241471\pi$
$331$	26.4094	1.45159	0.725796	+	0.687910i	$0.241471\pi$
$332$	0.615687	0.0337902
$333$	4.66897	0.255858
$334$	−14.8620	−0.813213
$335$	28.2625	1.54415
$336$	−0.387917	−0.0211626
$337$	−20.6226	−1.12338	−0.561691	−	0.827347i	$-0.689849\pi$
$337$	−20.6226	−1.12338	−0.561691	+	0.827347i	$0.689849\pi$
$338$	−7.65727	−0.416500
$339$	−1.27419	−0.0692045
$340$	−7.59829	−0.412075
$341$	−32.6697	−1.76916
$342$	−1.24408	−0.0672719
$343$	−18.1144	−0.978083
$344$	−9.26439	−0.499502
$345$	−5.21666	−0.280855
$346$	0.183131	0.00984519
$347$	5.27973	0.283431	0.141715	−	0.989907i	$-0.454738\pi$
$347$	5.27973	0.283431	0.141715	+	0.989907i	$0.454738\pi$
$348$	0	0
$349$	−20.1993	−1.08124	−0.540622	−	0.841266i	$-0.681811\pi$
$349$	−20.1993	−1.08124	−0.540622	+	0.841266i	$0.681811\pi$
$350$	−10.6407	−0.568767
$351$	−6.66670	−0.355842
$352$	4.37431	0.233151
$353$	−12.7715	−0.679757	−0.339878	−	0.940469i	$-0.610386\pi$
$353$	−12.7715	−0.679757	−0.339878	+	0.940469i	$0.610386\pi$
$354$	−3.59494	−0.191069
$355$	−43.4808	−2.30772
$356$	−5.06665	−0.268532
$357$	−0.858973	−0.0454617
$358$	7.94570	0.419943
$359$	−2.41249	−0.127327	−0.0636633	−	0.997971i	$-0.520278\pi$
$359$	−2.41249	−0.127327	−0.0636633	+	0.997971i	$0.520278\pi$
$360$	−10.0850	−0.531525
$361$	−18.8208	−0.990569
$362$	−1.79743	−0.0944710
$363$	−2.00907	−0.105449
$364$	−7.13862	−0.374165
$365$	−29.6505	−1.55198
$366$	−1.52231	−0.0795726
$367$	26.3208	1.37393	0.686967	−	0.726688i	$-0.258942\pi$
$367$	26.3208	1.37393	0.686967	+	0.726688i	$0.258942\pi$
$368$	−6.15540	−0.320872
$369$	−13.4207	−0.698654
$370$	−5.45126	−0.283398
$371$	10.2677	0.533074
$372$	−1.84458	−0.0956368
$373$	−23.8819	−1.23656	−0.618278	−	0.785960i	$-0.712169\pi$
$373$	−23.8819	−1.23656	−0.618278	+	0.785960i	$0.712169\pi$
$374$	9.68612	0.500857
$375$	1.50406	0.0776693
$376$	4.31029	0.222286
$377$	0	0
$378$	−2.30384	−0.118497
$379$	5.55826	0.285509	0.142754	−	0.989758i	$-0.454404\pi$
$379$	5.55826	0.285509	0.142754	+	0.989758i	$0.454404\pi$
$380$	1.45252	0.0745127
$381$	3.61429	0.185165
$382$	20.0665	1.02669
$383$	23.2565	1.18835	0.594176	−	0.804335i	$-0.297478\pi$
$383$	23.2565	1.18835	0.594176	+	0.804335i	$0.297478\pi$
$384$	0.246980	0.0126036
$385$	23.5756	1.20152
$386$	−7.08658	−0.360697
$387$	−27.2281	−1.38408
$388$	8.13902	0.413196
$389$	−23.8652	−1.21001	−0.605006	−	0.796221i	$-0.706829\pi$
$389$	−23.8652	−1.21001	−0.605006	+	0.796221i	$0.706829\pi$
$390$	3.85188	0.195048
$391$	−13.6300	−0.689300
$392$	4.53308	0.228955
$393$	3.14374	0.158581
$394$	−4.62674	−0.233092
$395$	−22.1152	−1.11274
$396$	12.8561	0.646043
$397$	−6.97686	−0.350159	−0.175079	−	0.984554i	$-0.556018\pi$
$397$	−6.97686	−0.350159	−0.175079	+	0.984554i	$0.556018\pi$
$398$	−3.58103	−0.179501
$399$	0.164205	0.00822052
$400$	6.77471	0.338736
$401$	8.70652	0.434783	0.217391	−	0.976085i	$-0.430245\pi$
$401$	8.70652	0.434783	0.217391	+	0.976085i	$0.430245\pi$
$402$	−2.03421	−0.101457
$403$	−33.9447	−1.69091
$404$	16.1534	0.803662
$405$	−29.0118	−1.44161
$406$	0	0
$407$	6.94914	0.344456
$408$	0.546892	0.0270752
$409$	6.65711	0.329173	0.164587	−	0.986363i	$-0.447371\pi$
$409$	6.65711	0.329173	0.164587	+	0.986363i	$0.447371\pi$
$410$	15.6694	0.773854
$411$	1.77728	0.0876669
$412$	−0.486918	−0.0239887
$413$	−22.8617	−1.12495
$414$	−18.0907	−0.889110
$415$	−2.11269	−0.103708
$416$	4.54503	0.222838
$417$	−1.38419	−0.0677842
$418$	−1.85164	−0.0905666
$419$	−23.1279	−1.12987	−0.564935	−	0.825135i	$-0.691099\pi$
$419$	−23.1279	−1.12987	−0.564935	+	0.825135i	$0.691099\pi$
$420$	1.33111	0.0649515
$421$	−1.62673	−0.0792820	−0.0396410	−	0.999214i	$-0.512621\pi$
$421$	−1.62673	−0.0792820	−0.0396410	+	0.999214i	$0.512621\pi$
$422$	−1.81599	−0.0884012
$423$	12.6680	0.615937
$424$	−6.53728	−0.317478
$425$	15.0014	0.727674
$426$	3.12956	0.151628
$427$	−9.68102	−0.468497
$428$	−5.41656	−0.261819
$429$	−4.91028	−0.237071
$430$	31.7901	1.53306
$431$	−0.392815	−0.0189212	−0.00946061	−	0.999955i	$-0.503011\pi$
$431$	−0.392815	−0.0189212	−0.00946061	+	0.999955i	$0.503011\pi$
$432$	1.46681	0.0705720
$433$	−36.9800	−1.77714	−0.888572	−	0.458738i	$-0.848302\pi$
$433$	−36.9800	−1.77714	−0.888572	+	0.458738i	$0.848302\pi$
$434$	−11.7304	−0.563078
$435$	0	0
$436$	6.88139	0.329559
$437$	2.60557	0.124641
$438$	2.13411	0.101972
$439$	−13.6888	−0.653332	−0.326666	−	0.945140i	$-0.605925\pi$
$439$	−13.6888	−0.653332	−0.326666	+	0.945140i	$0.605925\pi$
$440$	−15.0101	−0.715580
$441$	13.3227	0.634415
$442$	10.0641	0.478703
$443$	−4.37363	−0.207797	−0.103899	−	0.994588i	$-0.533132\pi$
$443$	−4.37363	−0.207797	−0.103899	+	0.994588i	$0.533132\pi$
$444$	0.392358	0.0186205
$445$	17.3858	0.824168
$446$	−20.9351	−0.991304
$447$	−3.82806	−0.181061
$448$	1.57064	0.0742059
$449$	−19.7322	−0.931218	−0.465609	−	0.884990i	$-0.654165\pi$
$449$	−19.7322	−0.931218	−0.465609	+	0.884990i	$0.654165\pi$
$450$	19.9109	0.938609
$451$	−19.9749	−0.940582
$452$	5.15909	0.242663
$453$	−0.500506	−0.0235158
$454$	−2.77986	−0.130466
$455$	24.4957	1.14837
$456$	−0.104546	−0.00489582
$457$	36.3594	1.70082	0.850412	−	0.526118i	$-0.176353\pi$
$457$	36.3594	1.70082	0.850412	+	0.526118i	$0.176353\pi$
$458$	24.0260	1.12266
$459$	3.24799	0.151603
$460$	21.1218	0.984810
$461$	−8.77942	−0.408898	−0.204449	−	0.978877i	$-0.565540\pi$
$461$	−8.77942	−0.408898	−0.204449	+	0.978877i	$0.565540\pi$
$462$	−1.69687	−0.0789454
$463$	−9.83531	−0.457086	−0.228543	−	0.973534i	$-0.573396\pi$
$463$	−9.83531	−0.457086	−0.228543	+	0.973534i	$0.573396\pi$
$464$	0	0
$465$	6.32953	0.293525
$466$	3.18472	0.147529
$467$	4.43077	0.205032	0.102516	−	0.994731i	$-0.467311\pi$
$467$	4.43077	0.205032	0.102516	+	0.994731i	$0.467311\pi$
$468$	13.3578	0.617466
$469$	−12.9364	−0.597347
$470$	−14.7905	−0.682234
$471$	0.156078	0.00719168
$472$	14.5556	0.669976
$473$	−40.5253	−1.86335
$474$	1.59176	0.0731120
$475$	−2.86773	−0.131580
$476$	3.47791	0.159410
$477$	−19.2131	−0.879706
$478$	−12.8020	−0.585551
$479$	−21.1257	−0.965259	−0.482629	−	0.875825i	$-0.660318\pi$
$479$	−21.1257	−0.965259	−0.482629	+	0.875825i	$0.660318\pi$
$480$	−0.847493	−0.0386826
$481$	7.22035	0.329220
$482$	−0.472390	−0.0215168
$483$	2.38778	0.108648
$484$	8.13455	0.369752
$485$	−27.9285	−1.26817
$486$	6.48858	0.294328
$487$	−20.8892	−0.946578	−0.473289	−	0.880907i	$-0.656933\pi$
$487$	−20.8892	−0.946578	−0.473289	+	0.880907i	$0.656933\pi$
$488$	6.16373	0.279019
$489$	−6.00415	−0.271517
$490$	−15.5549	−0.702701
$491$	9.03901	0.407925	0.203962	−	0.978979i	$-0.434618\pi$
$491$	9.03901	0.407925	0.203962	+	0.978979i	$0.434618\pi$
$492$	−1.12781	−0.0508457
$493$	0	0
$494$	−1.92390	−0.0865605
$495$	−44.1148	−1.98281
$496$	7.46854	0.335347
$497$	19.9021	0.892732
$498$	0.152062	0.00681407
$499$	−41.3877	−1.85277	−0.926385	−	0.376579i	$-0.877100\pi$
$499$	−41.3877	−1.85277	−0.926385	+	0.376579i	$0.877100\pi$
$500$	−6.08981	−0.272345
$501$	−3.67061	−0.163991
$502$	−0.992245	−0.0442861
$503$	40.0349	1.78507	0.892535	−	0.450979i	$-0.148925\pi$
$503$	40.0349	1.78507	0.892535	+	0.450979i	$0.148925\pi$
$504$	4.61612	0.205618
$505$	−55.4293	−2.46657
$506$	−26.9256	−1.19699
$507$	−1.89119	−0.0839906
$508$	−14.6339	−0.649276
$509$	−21.0580	−0.933377	−0.466689	−	0.884422i	$-0.654553\pi$
$509$	−21.0580	−0.933377	−0.466689	+	0.884422i	$0.654553\pi$
$510$	−1.87662	−0.0830982
$511$	13.5717	0.600376
$512$	−1.00000	−0.0441942
$513$	−0.620900	−0.0274134
$514$	21.2077	0.935432
$515$	1.67082	0.0736253
$516$	−2.28812	−0.100729
$517$	18.8545	0.829222
$518$	2.49517	0.109631
$519$	0.0452297	0.00198536
$520$	−15.5959	−0.683927
$521$	12.1752	0.533406	0.266703	−	0.963779i	$-0.414066\pi$
$521$	12.1752	0.533406	0.266703	+	0.963779i	$0.414066\pi$
$522$	0	0
$523$	24.2771	1.06156	0.530781	−	0.847509i	$-0.321899\pi$
$523$	24.2771	1.06156	0.530781	+	0.847509i	$0.321899\pi$
$524$	−12.7288	−0.556059
$525$	−2.62803	−0.114696
$526$	−4.59716	−0.200446
$527$	16.5377	0.720395
$528$	1.08036	0.0470168
$529$	14.8889	0.647344
$530$	22.4322	0.974393
$531$	42.7790	1.85645
$532$	−0.664851	−0.0288250
$533$	−20.7545	−0.898977
$534$	−1.25136	−0.0541516
$535$	18.5865	0.803566
$536$	8.23637	0.355757
$537$	1.96243	0.0846850
$538$	−2.23745	−0.0964632
$539$	19.8291	0.854098
$540$	−5.03326	−0.216597
$541$	−12.3846	−0.532458	−0.266229	−	0.963910i	$-0.585778\pi$
$541$	−12.3846	−0.532458	−0.266229	+	0.963910i	$0.585778\pi$
$542$	−5.66146	−0.243180
$543$	−0.443929	−0.0190508
$544$	−2.21432	−0.0949382
$545$	−23.6130	−1.01147
$546$	−1.76309	−0.0754534
$547$	32.2704	1.37978	0.689891	−	0.723914i	$-0.257659\pi$
$547$	32.2704	1.37978	0.689891	+	0.723914i	$0.257659\pi$
$548$	−7.19607	−0.307401
$549$	18.1152	0.773138
$550$	29.6347	1.26363
$551$	0	0
$552$	−1.52026	−0.0647064
$553$	10.1226	0.430459
$554$	20.7393	0.881130
$555$	−1.34635	−0.0571494
$556$	5.60448	0.237683
$557$	39.2120	1.66147	0.830733	−	0.556671i	$-0.187922\pi$
$557$	39.2120	1.66147	0.830733	+	0.556671i	$0.187922\pi$
$558$	21.9500	0.929219
$559$	−42.1069	−1.78093
$560$	−5.38955	−0.227750
$561$	2.39227	0.101002
$562$	−18.6958	−0.788634
$563$	42.8169	1.80452	0.902259	−	0.431195i	$-0.141908\pi$
$563$	42.8169	1.80452	0.902259	+	0.431195i	$0.141908\pi$
$564$	1.06455	0.0448258
$565$	−17.7031	−0.744773
$566$	8.59049	0.361085
$567$	13.2794	0.557681
$568$	−12.6713	−0.531677
$569$	36.8744	1.54586	0.772929	−	0.634493i	$-0.218791\pi$
$569$	36.8744	1.54586	0.772929	+	0.634493i	$0.218791\pi$
$570$	0.358743	0.0150261
$571$	−18.8019	−0.786834	−0.393417	−	0.919360i	$-0.628707\pi$
$571$	−18.8019	−0.786834	−0.393417	+	0.919360i	$0.628707\pi$
$572$	19.8813	0.831280
$573$	4.95602	0.207041
$574$	−7.17222	−0.299363
$575$	−41.7011	−1.73905
$576$	−2.93900	−0.122458
$577$	−6.13904	−0.255572	−0.127786	−	0.991802i	$-0.540787\pi$
$577$	−6.13904	−0.255572	−0.127786	+	0.991802i	$0.540787\pi$
$578$	12.0968	0.503160
$579$	−1.75024	−0.0727375
$580$	0	0
$581$	0.967025	0.0401190
$582$	2.01017	0.0833243
$583$	−28.5961	−1.18433
$584$	−8.64084	−0.357561
$585$	−45.8365	−1.89511
$586$	−7.69030	−0.317683
$587$	−29.5333	−1.21897	−0.609484	−	0.792798i	$-0.708623\pi$
$587$	−29.5333	−1.21897	−0.609484	+	0.792798i	$0.708623\pi$
$588$	1.11958	0.0461706
$589$	−3.16142	−0.130264
$590$	−49.9466	−2.05627
$591$	−1.14271	−0.0470048
$592$	−1.58863	−0.0652921
$593$	18.0707	0.742074	0.371037	−	0.928618i	$-0.379002\pi$
$593$	18.0707	0.742074	0.371037	+	0.928618i	$0.379002\pi$
$594$	6.41628	0.263263
$595$	−11.9342	−0.489255
$596$	15.4995	0.634884
$597$	−0.884441	−0.0361978
$598$	−27.9764	−1.14404
$599$	10.6956	0.437010	0.218505	−	0.975836i	$-0.429882\pi$
$599$	10.6956	0.437010	0.218505	+	0.975836i	$0.429882\pi$
$600$	1.67322	0.0683088
$601$	4.26071	0.173798	0.0868989	−	0.996217i	$-0.472304\pi$
$601$	4.26071	0.173798	0.0868989	+	0.996217i	$0.472304\pi$
$602$	−14.5511	−0.593057
$603$	24.2067	0.985772
$604$	2.02651	0.0824575
$605$	−27.9131	−1.13483
$606$	3.98956	0.162065
$607$	10.6644	0.432856	0.216428	−	0.976299i	$-0.430559\pi$
$607$	10.6644	0.432856	0.216428	+	0.976299i	$0.430559\pi$
$608$	0.423299	0.0171670
$609$	0	0
$610$	−21.1504	−0.856354
$611$	19.5904	0.792543
$612$	−6.50789	−0.263066
$613$	−12.9410	−0.522681	−0.261340	−	0.965247i	$-0.584165\pi$
$613$	−12.9410	−0.522681	−0.261340	+	0.965247i	$0.584165\pi$
$614$	−14.7061	−0.593490
$615$	3.87001	0.156054
$616$	6.87048	0.276819
$617$	−13.1331	−0.528719	−0.264360	−	0.964424i	$-0.585161\pi$
$617$	−13.1331	−0.528719	−0.264360	+	0.964424i	$0.585161\pi$
$618$	−0.120259	−0.00483752
$619$	−22.1454	−0.890101	−0.445050	−	0.895506i	$-0.646814\pi$
$619$	−22.1454	−0.890101	−0.445050	+	0.895506i	$0.646814\pi$
$620$	−25.6278	−1.02924
$621$	−9.02881	−0.362314
$622$	−16.9331	−0.678954
$623$	−7.95789	−0.318826
$624$	1.12253	0.0449371
$625$	−12.9768	−0.519073
$626$	−16.1943	−0.647254
$627$	−0.457317	−0.0182635
$628$	−0.631946	−0.0252174
$629$	−3.51773	−0.140261
$630$	−15.8399	−0.631077
$631$	9.77633	0.389190	0.194595	−	0.980884i	$-0.437661\pi$
$631$	9.77633	0.389190	0.194595	+	0.980884i	$0.437661\pi$
$632$	−6.44491	−0.256365
$633$	−0.448513	−0.0178268
$634$	10.7887	0.428476
$635$	50.2154	1.99274
$636$	−1.61457	−0.0640220
$637$	20.6030	0.816319
$638$	0	0
$639$	−37.2410	−1.47323
$640$	3.43143	0.135639
$641$	8.50080	0.335762	0.167881	−	0.985807i	$-0.446308\pi$
$641$	8.50080	0.335762	0.167881	+	0.985807i	$0.446308\pi$
$642$	−1.33778	−0.0527979
$643$	40.2359	1.58675	0.793374	−	0.608734i	$-0.208322\pi$
$643$	40.2359	1.58675	0.793374	+	0.608734i	$0.208322\pi$
$644$	−9.66793	−0.380970
$645$	7.85151	0.309153
$646$	0.937319	0.0368783
$647$	21.2127	0.833956	0.416978	−	0.908916i	$-0.363089\pi$
$647$	21.2127	0.833956	0.416978	+	0.908916i	$0.363089\pi$
$648$	−8.45473	−0.332133
$649$	63.6707	2.49929
$650$	30.7913	1.20773
$651$	−2.89717	−0.113549
$652$	24.3103	0.952065
$653$	27.2722	1.06724	0.533622	−	0.845723i	$-0.320830\pi$
$653$	27.2722	1.06724	0.533622	+	0.845723i	$0.320830\pi$
$654$	1.69956	0.0664582
$655$	43.6779	1.70664
$656$	4.56642	0.178289
$657$	−25.3955	−0.990771
$658$	6.76994	0.263920
$659$	−18.3346	−0.714216	−0.357108	−	0.934063i	$-0.616237\pi$
$659$	−18.3346	−0.714216	−0.357108	+	0.934063i	$0.616237\pi$
$660$	−3.70719	−0.144302
$661$	−17.0116	−0.661673	−0.330837	−	0.943688i	$-0.607331\pi$
$661$	−17.0116	−0.661673	−0.330837	+	0.943688i	$0.607331\pi$
$662$	−26.4094	−1.02643
$663$	2.48564	0.0965342
$664$	−0.615687	−0.0238933
$665$	2.28139	0.0884686
$666$	−4.66897	−0.180919
$667$	0	0
$668$	14.8620	0.575029
$669$	−5.17053	−0.199904
$670$	−28.2625	−1.09188
$671$	26.9620	1.04086
$672$	0.387917	0.0149642
$673$	4.90116	0.188926	0.0944629	−	0.995528i	$-0.469887\pi$
$673$	4.90116	0.188926	0.0944629	+	0.995528i	$0.469887\pi$
$674$	20.6226	0.794351
$675$	9.93723	0.382484
$676$	7.65727	0.294510
$677$	37.4400	1.43894	0.719468	−	0.694525i	$-0.244386\pi$
$677$	37.4400	1.43894	0.719468	+	0.694525i	$0.244386\pi$
$678$	1.27419	0.0489350
$679$	12.7835	0.490586
$680$	7.59829	0.291381
$681$	−0.686570	−0.0263094
$682$	32.6697	1.25099
$683$	41.7713	1.59833	0.799167	−	0.601109i	$-0.205274\pi$
$683$	41.7713	1.59833	0.799167	+	0.601109i	$0.205274\pi$
$684$	1.24408	0.0475684
$685$	24.6928	0.943464
$686$	18.1144	0.691609
$687$	5.93393	0.226394
$688$	9.26439	0.353202
$689$	−29.7121	−1.13194
$690$	5.21666	0.198595
$691$	−1.25355	−0.0476872	−0.0238436	−	0.999716i	$-0.507590\pi$
$691$	−1.25355	−0.0476872	−0.0238436	+	0.999716i	$0.507590\pi$
$692$	−0.183131	−0.00696160
$693$	20.1923	0.767043
$694$	−5.27973	−0.200416
$695$	−19.2314	−0.729488
$696$	0	0
$697$	10.1115	0.383001
$698$	20.1993	0.764555
$699$	0.786561	0.0297505
$700$	10.6407	0.402179
$701$	−36.1405	−1.36501	−0.682503	−	0.730882i	$-0.739109\pi$
$701$	−36.1405	−1.36501	−0.682503	+	0.730882i	$0.739109\pi$
$702$	6.66670	0.251618
$703$	0.672463	0.0253624
$704$	−4.37431	−0.164863
$705$	−3.65295	−0.137578
$706$	12.7715	0.480661
$707$	25.3712	0.954184
$708$	3.59494	0.135106
$709$	27.5004	1.03280	0.516400	−	0.856347i	$-0.327272\pi$
$709$	27.5004	1.03280	0.516400	+	0.856347i	$0.327272\pi$
$710$	43.4808	1.63180
$711$	−18.9416	−0.710365
$712$	5.06665	0.189881
$713$	−45.9718	−1.72166
$714$	0.858973	0.0321462
$715$	−68.2214	−2.55134
$716$	−7.94570	−0.296945
$717$	−3.16184	−0.118081
$718$	2.41249	0.0900335
$719$	−3.76398	−0.140373	−0.0701865	−	0.997534i	$-0.522359\pi$
$719$	−3.76398	−0.140373	−0.0701865	+	0.997534i	$0.522359\pi$
$720$	10.0850	0.375845
$721$	−0.764774	−0.0284817
$722$	18.8208	0.700438
$723$	−0.116671	−0.00433903
$724$	1.79743	0.0668011
$725$	0	0
$726$	2.00907	0.0745635
$727$	39.7913	1.47578	0.737890	−	0.674921i	$-0.235823\pi$
$727$	39.7913	1.47578	0.737890	+	0.674921i	$0.235823\pi$
$728$	7.13862	0.264575
$729$	−23.7616	−0.880061
$730$	29.6505	1.09741
$731$	20.5143	0.758750
$732$	1.52231	0.0562664
$733$	45.6514	1.68617	0.843087	−	0.537777i	$-0.180736\pi$
$733$	45.6514	1.68617	0.843087	+	0.537777i	$0.180736\pi$
$734$	−26.3208	−0.971518
$735$	−3.84175	−0.141705
$736$	6.15540	0.226891
$737$	36.0284	1.32712
$738$	13.4207	0.494023
$739$	47.5516	1.74922	0.874608	−	0.484831i	$-0.161119\pi$
$739$	47.5516	1.74922	0.874608	+	0.484831i	$0.161119\pi$
$740$	5.45126	0.200392
$741$	−0.475165	−0.0174556
$742$	−10.2677	−0.376940
$743$	19.1947	0.704185	0.352092	−	0.935965i	$-0.385470\pi$
$743$	19.1947	0.704185	0.352092	+	0.935965i	$0.385470\pi$
$744$	1.84458	0.0676254
$745$	−53.1854	−1.94856
$746$	23.8819	0.874377
$747$	−1.80950	−0.0662063
$748$	−9.68612	−0.354159
$749$	−8.50748	−0.310857
$750$	−1.50406	−0.0549205
$751$	3.63323	0.132578	0.0662892	−	0.997800i	$-0.478884\pi$
$751$	3.63323	0.132578	0.0662892	+	0.997800i	$0.478884\pi$
$752$	−4.31029	−0.157180
$753$	−0.245064	−0.00893064
$754$	0	0
$755$	−6.95382	−0.253076
$756$	2.30384	0.0837898
$757$	40.3662	1.46714	0.733568	−	0.679616i	$-0.237854\pi$
$757$	40.3662	1.46714	0.733568	+	0.679616i	$0.237854\pi$
$758$	−5.55826	−0.201885
$759$	−6.65007	−0.241382
$760$	−1.45252	−0.0526884
$761$	28.6289	1.03780	0.518899	−	0.854836i	$-0.326342\pi$
$761$	28.6289	1.03780	0.518899	+	0.854836i	$0.326342\pi$
$762$	−3.61429	−0.130932
$763$	10.8082	0.391283
$764$	−20.0665	−0.725981
$765$	22.3314	0.807393
$766$	−23.2565	−0.840291
$767$	66.1556	2.38874
$768$	−0.246980	−0.00891211
$769$	−42.7668	−1.54221	−0.771106	−	0.636707i	$-0.780296\pi$
$769$	−42.7668	−1.54221	−0.771106	+	0.636707i	$0.780296\pi$
$770$	−23.5756	−0.849604
$771$	5.23787	0.188637
$772$	7.08658	0.255051
$773$	45.5294	1.63758	0.818788	−	0.574095i	$-0.194646\pi$
$773$	45.5294	1.63758	0.818788	+	0.574095i	$0.194646\pi$
$774$	27.2281	0.978692
$775$	50.5972	1.81751
$776$	−8.13902	−0.292174
$777$	0.616255	0.0221080
$778$	23.8652	0.855608
$779$	−1.93296	−0.0692555
$780$	−3.85188	−0.137919
$781$	−55.4282	−1.98338
$782$	13.6300	0.487409
$783$	0	0
$784$	−4.53308	−0.161896
$785$	2.16848	0.0773963
$786$	−3.14374	−0.112134
$787$	−32.6352	−1.16332	−0.581659	−	0.813433i	$-0.697596\pi$
$787$	−32.6352	−1.16332	−0.581659	+	0.813433i	$0.697596\pi$
$788$	4.62674	0.164821
$789$	−1.13541	−0.0404215
$790$	22.1152	0.786825
$791$	8.10309	0.288113
$792$	−12.8561	−0.456821
$793$	28.0143	0.994817
$794$	6.97686	0.247600
$795$	5.54030	0.196494
$796$	3.58103	0.126926
$797$	10.1161	0.358330	0.179165	−	0.983819i	$-0.442660\pi$
$797$	10.1161	0.358330	0.179165	+	0.983819i	$0.442660\pi$
$798$	−0.164205	−0.00581278
$799$	−9.54438	−0.337656
$800$	−6.77471	−0.239522
$801$	14.8909	0.526143
$802$	−8.70652	−0.307438
$803$	−37.7977	−1.33385
$804$	2.03421	0.0717412
$805$	33.1748	1.16926
$806$	33.9447	1.19565
$807$	−0.552604	−0.0194526
$808$	−16.1534	−0.568275
$809$	−8.02875	−0.282276	−0.141138	−	0.989990i	$-0.545076\pi$
$809$	−8.02875	−0.282276	−0.141138	+	0.989990i	$0.545076\pi$
$810$	29.0118	1.01937
$811$	6.02885	0.211702	0.105851	−	0.994382i	$-0.466243\pi$
$811$	6.02885	0.211702	0.105851	+	0.994382i	$0.466243\pi$
$812$	0	0
$813$	−1.39826	−0.0490393
$814$	−6.94914	−0.243567
$815$	−83.4191	−2.92204
$816$	−0.546892	−0.0191451
$817$	−3.92160	−0.137200
$818$	−6.65711	−0.232760
$819$	20.9804	0.733115
$820$	−15.6694	−0.547197
$821$	−18.0676	−0.630563	−0.315281	−	0.948998i	$-0.602099\pi$
$821$	−18.0676	−0.630563	−0.315281	+	0.948998i	$0.602099\pi$
$822$	−1.77728	−0.0619898
$823$	−11.9594	−0.416879	−0.208440	−	0.978035i	$-0.566838\pi$
$823$	−11.9594	−0.416879	−0.208440	+	0.978035i	$0.566838\pi$
$824$	0.486918	0.0169626
$825$	7.31916	0.254820
$826$	22.8617	0.795460
$827$	24.3798	0.847767	0.423884	−	0.905717i	$-0.360667\pi$
$827$	24.3798	0.847767	0.423884	+	0.905717i	$0.360667\pi$
$828$	18.0907	0.628696
$829$	38.6656	1.34291	0.671456	−	0.741044i	$-0.265669\pi$
$829$	38.6656	1.34291	0.671456	+	0.741044i	$0.265669\pi$
$830$	2.11269	0.0733324
$831$	5.12219	0.177687
$832$	−4.54503	−0.157570
$833$	−10.0377	−0.347785
$834$	1.38419	0.0479307
$835$	−50.9980	−1.76486
$836$	1.85164	0.0640402
$837$	10.9549	0.378658
$838$	23.1279	0.798939
$839$	29.7333	1.02651	0.513254	−	0.858237i	$-0.328440\pi$
$839$	29.7333	1.02651	0.513254	+	0.858237i	$0.328440\pi$
$840$	−1.33111	−0.0459277
$841$	0	0
$842$	1.62673	0.0560609
$843$	−4.61748	−0.159034
$844$	1.81599	0.0625091
$845$	−26.2754	−0.903900
$846$	−12.6680	−0.435533
$847$	12.7765	0.439005
$848$	6.53728	0.224491
$849$	2.12167	0.0728157
$850$	−15.0014	−0.514544
$851$	9.77862	0.335207
$852$	−3.12956	−0.107217
$853$	40.4719	1.38573	0.692865	−	0.721067i	$-0.256348\pi$
$853$	40.4719	1.38573	0.692865	+	0.721067i	$0.256348\pi$
$854$	9.68102	0.331278
$855$	−4.26896	−0.145995
$856$	5.41656	0.185134
$857$	21.9116	0.748487	0.374243	−	0.927331i	$-0.377902\pi$
$857$	21.9116	0.748487	0.374243	+	0.927331i	$0.377902\pi$
$858$	4.91028	0.167634
$859$	7.88078	0.268889	0.134444	−	0.990921i	$-0.457075\pi$
$859$	7.88078	0.268889	0.134444	+	0.990921i	$0.457075\pi$
$860$	−31.7901	−1.08403
$861$	−1.77139	−0.0603689
$862$	0.392815	0.0133793
$863$	25.5643	0.870218	0.435109	−	0.900378i	$-0.356710\pi$
$863$	25.5643	0.870218	0.435109	+	0.900378i	$0.356710\pi$
$864$	−1.46681	−0.0499020
$865$	0.628402	0.0213663
$866$	36.9800	1.25663
$867$	2.98766	0.101466
$868$	11.7304	0.398156
$869$	−28.1920	−0.956348
$870$	0	0
$871$	37.4345	1.26842
$872$	−6.88139	−0.233033
$873$	−23.9206	−0.809589
$874$	−2.60557	−0.0881347
$875$	−9.56492	−0.323353
$876$	−2.13411	−0.0721050
$877$	50.5828	1.70806	0.854029	−	0.520225i	$-0.174152\pi$
$877$	50.5828	1.70806	0.854029	+	0.520225i	$0.174152\pi$
$878$	13.6888	0.461976
$879$	−1.89935	−0.0640634
$880$	15.0101	0.505991
$881$	−38.2432	−1.28845	−0.644223	−	0.764838i	$-0.722819\pi$
$881$	−38.2432	−1.28845	−0.644223	+	0.764838i	$0.722819\pi$
$882$	−13.3227	−0.448599
$883$	−17.4595	−0.587559	−0.293779	−	0.955873i	$-0.594913\pi$
$883$	−17.4595	−0.587559	−0.293779	+	0.955873i	$0.594913\pi$
$884$	−10.0641	−0.338494
$885$	−12.3358	−0.414663
$886$	4.37363	0.146935
$887$	4.18979	0.140679	0.0703397	−	0.997523i	$-0.477592\pi$
$887$	4.18979	0.140679	0.0703397	+	0.997523i	$0.477592\pi$
$888$	−0.392358	−0.0131667
$889$	−22.9847	−0.770882
$890$	−17.3858	−0.582775
$891$	−36.9836	−1.23900
$892$	20.9351	0.700958
$893$	1.82454	0.0610560
$894$	3.82806	0.128029
$895$	27.2651	0.911373
$896$	−1.57064	−0.0524715
$897$	−6.90961	−0.230705
$898$	19.7322	0.658471
$899$	0	0
$900$	−19.9109	−0.663696
$901$	14.4756	0.482253
$902$	19.9749	0.665092
$903$	−3.59381	−0.119595
$904$	−5.15909	−0.171589
$905$	−6.16777	−0.205023
$906$	0.500506	0.0166282
$907$	18.1221	0.601735	0.300867	−	0.953666i	$-0.402724\pi$
$907$	18.1221	0.601735	0.300867	+	0.953666i	$0.402724\pi$
$908$	2.77986	0.0922530
$909$	−47.4749	−1.57464
$910$	−24.4957	−0.812023
$911$	3.83336	0.127005	0.0635025	−	0.997982i	$-0.479773\pi$
$911$	3.83336	0.127005	0.0635025	+	0.997982i	$0.479773\pi$
$912$	0.104546	0.00346187
$913$	−2.69320	−0.0891320
$914$	−36.3594	−1.20266
$915$	−5.22372	−0.172691
$916$	−24.0260	−0.793842
$917$	−19.9924	−0.660206
$918$	−3.24799	−0.107200
$919$	39.2248	1.29391	0.646954	−	0.762529i	$-0.276043\pi$
$919$	39.2248	1.29391	0.646954	+	0.762529i	$0.276043\pi$
$920$	−21.1218	−0.696366
$921$	−3.63211	−0.119682
$922$	8.77942	0.289135
$923$	−57.5915	−1.89565
$924$	1.69687	0.0558228
$925$	−10.7625	−0.353869
$926$	9.83531	0.323208
$927$	1.43105	0.0470019
$928$	0	0
$929$	−16.8372	−0.552409	−0.276204	−	0.961099i	$-0.589077\pi$
$929$	−16.8372	−0.552409	−0.276204	+	0.961099i	$0.589077\pi$
$930$	−6.32953	−0.207554
$931$	1.91885	0.0628876
$932$	−3.18472	−0.104319
$933$	−4.18212	−0.136916
$934$	−4.43077	−0.144979
$935$	33.2372	1.08697
$936$	−13.3578	−0.436615
$937$	25.4711	0.832105	0.416053	−	0.909341i	$-0.363413\pi$
$937$	25.4711	0.832105	0.416053	+	0.909341i	$0.363413\pi$
$938$	12.9364	0.422388
$939$	−3.99966	−0.130524
$940$	14.7905	0.482412
$941$	−29.5773	−0.964194	−0.482097	−	0.876118i	$-0.660125\pi$
$941$	−29.5773	−0.964194	−0.482097	+	0.876118i	$0.660125\pi$
$942$	−0.156078	−0.00508529
$943$	−28.1081	−0.915326
$944$	−14.5556	−0.473745
$945$	−7.90546	−0.257165
$946$	40.5253	1.31759
$947$	−46.5543	−1.51281	−0.756406	−	0.654103i	$-0.773046\pi$
$947$	−46.5543	−1.51281	−0.756406	+	0.654103i	$0.773046\pi$
$948$	−1.59176	−0.0516980
$949$	−39.2729	−1.27485
$950$	2.86773	0.0930414
$951$	2.66460	0.0864056
$952$	−3.47791	−0.112720
$953$	4.54252	0.147147	0.0735734	−	0.997290i	$-0.476560\pi$
$953$	4.54252	0.147147	0.0735734	+	0.997290i	$0.476560\pi$
$954$	19.2131	0.622046
$955$	68.8568	2.22816
$956$	12.8020	0.414047
$957$	0	0
$958$	21.1257	0.682541
$959$	−11.3025	−0.364975
$960$	0.847493	0.0273527
$961$	24.7790	0.799324
$962$	−7.22035	−0.232793
$963$	15.9193	0.512991
$964$	0.472390	0.0152146
$965$	−24.3171	−0.782795
$966$	−2.38778	−0.0768256
$967$	0.591500	0.0190213	0.00951067	−	0.999955i	$-0.496973\pi$
$967$	0.591500	0.0190213	0.00951067	+	0.999955i	$0.496973\pi$
$968$	−8.13455	−0.261454
$969$	0.231499	0.00743681
$970$	27.9285	0.896729
$971$	−32.3652	−1.03865	−0.519324	−	0.854577i	$-0.673816\pi$
$971$	−32.3652	−1.03865	−0.519324	+	0.854577i	$0.673816\pi$
$972$	−6.48858	−0.208121
$973$	8.80264	0.282200
$974$	20.8892	0.669332
$975$	7.60481	0.243549
$976$	−6.16373	−0.197296
$977$	−31.9020	−1.02063	−0.510317	−	0.859986i	$-0.670472\pi$
$977$	−31.9020	−1.02063	−0.510317	+	0.859986i	$0.670472\pi$
$978$	6.00415	0.191991
$979$	22.1631	0.708334
$980$	15.5549	0.496884
$981$	−20.2244	−0.645716
$982$	−9.03901	−0.288446
$983$	−15.1053	−0.481786	−0.240893	−	0.970552i	$-0.577440\pi$
$983$	−15.1053	−0.481786	−0.240893	+	0.970552i	$0.577440\pi$
$984$	1.12781	0.0359533
$985$	−15.8763	−0.505862
$986$	0	0
$987$	1.67204	0.0532215
$988$	1.92390	0.0612075
$989$	−57.0260	−1.81332
$990$	44.1148	1.40206
$991$	−49.9779	−1.58760	−0.793801	−	0.608178i	$-0.791901\pi$
$991$	−49.9779	−1.58760	−0.793801	+	0.608178i	$0.791901\pi$
$992$	−7.46854	−0.237126
$993$	−6.52259	−0.206988
$994$	−19.9021	−0.631257
$995$	−12.2881	−0.389558
$996$	−0.152062	−0.00481827
$997$	−0.167080	−0.00529147	−0.00264574	−	0.999997i	$-0.500842\pi$
$997$	−0.167080	−0.00529147	−0.00264574	+	0.999997i	$0.500842\pi$
$998$	41.3877	1.31011
$999$	−2.33022	−0.0737248

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.r.1.1		6
29.8	odd	28		58.2.e.a.35.1	yes	12
29.11	odd	28		58.2.e.a.5.1	✓	12
29.12	odd	4		1682.2.b.j.1681.2		12
29.17	odd	4		1682.2.b.j.1681.12		12
29.28	even	2		1682.2.a.s.1.5		6
87.8	even	28		522.2.n.a.325.2		12
87.11	even	28		522.2.n.a.469.2		12
116.11	even	28		464.2.y.c.353.1		12
116.95	even	28		464.2.y.c.209.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
58.2.e.a.5.1	✓	12	29.11	odd	28
58.2.e.a.35.1	yes	12	29.8	odd	28
464.2.y.c.209.1		12	116.95	even	28
464.2.y.c.353.1		12	116.11	even	28
522.2.n.a.325.2		12	87.8	even	28
522.2.n.a.469.2		12	87.11	even	28
1682.2.a.r.1.1		6	1.1	even	1	trivial
1682.2.a.s.1.5		6	29.28	even	2
1682.2.b.j.1681.2		12	29.12	odd	4
1682.2.b.j.1681.12		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.r.1.1

Level	$1682$
Weight	$2$
Character	1682.1
Self dual	yes
Analytic conductor	$13.431$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$