LMFDB - Embedded newform 483.2.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [483,2,Mod(1,483)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(483, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("483.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$483 = 3 \cdot 7 \cdot 23$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	483.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:	$3.85677441763$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x - 1$ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	483.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} -1.00000 q^{3} +0.618034 q^{4} -0.618034 q^{5} +1.61803 q^{6} -1.00000 q^{7} +2.23607 q^{8} +1.00000 q^{9} +1.00000 q^{10} +2.23607 q^{11} -0.618034 q^{12} -2.38197 q^{13} +1.61803 q^{14} +0.618034 q^{15} -4.85410 q^{16} +6.70820 q^{17} -1.61803 q^{18} -3.47214 q^{19} -0.381966 q^{20} +1.00000 q^{21} -3.61803 q^{22} -1.00000 q^{23} -2.23607 q^{24} -4.61803 q^{25} +3.85410 q^{26} -1.00000 q^{27} -0.618034 q^{28} -8.23607 q^{29} -1.00000 q^{30} +6.70820 q^{31} +3.38197 q^{32} -2.23607 q^{33} -10.8541 q^{34} +0.618034 q^{35} +0.618034 q^{36} -11.0000 q^{37} +5.61803 q^{38} +2.38197 q^{39} -1.38197 q^{40} -1.47214 q^{41} -1.61803 q^{42} -1.61803 q^{43} +1.38197 q^{44} -0.618034 q^{45} +1.61803 q^{46} -7.23607 q^{47} +4.85410 q^{48} +1.00000 q^{49} +7.47214 q^{50} -6.70820 q^{51} -1.47214 q^{52} -13.0902 q^{53} +1.61803 q^{54} -1.38197 q^{55} -2.23607 q^{56} +3.47214 q^{57} +13.3262 q^{58} +9.38197 q^{59} +0.381966 q^{60} -4.85410 q^{61} -10.8541 q^{62} -1.00000 q^{63} +4.23607 q^{64} +1.47214 q^{65} +3.61803 q^{66} -5.09017 q^{67} +4.14590 q^{68} +1.00000 q^{69} -1.00000 q^{70} +4.38197 q^{71} +2.23607 q^{72} -12.7082 q^{73} +17.7984 q^{74} +4.61803 q^{75} -2.14590 q^{76} -2.23607 q^{77} -3.85410 q^{78} -9.47214 q^{79} +3.00000 q^{80} +1.00000 q^{81} +2.38197 q^{82} -9.18034 q^{83} +0.618034 q^{84} -4.14590 q^{85} +2.61803 q^{86} +8.23607 q^{87} +5.00000 q^{88} +11.6180 q^{89} +1.00000 q^{90} +2.38197 q^{91} -0.618034 q^{92} -6.70820 q^{93} +11.7082 q^{94} +2.14590 q^{95} -3.38197 q^{96} +10.4164 q^{97} -1.61803 q^{98} +2.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - q^{2} - 2 q^{3} - q^{4} + q^{5} + q^{6} - 2 q^{7} + 2 q^{9} + 2 q^{10} + q^{12} - 7 q^{13} + q^{14} - q^{15} - 3 q^{16} - q^{18} + 2 q^{19} - 3 q^{20} + 2 q^{21} - 5 q^{22} - 2 q^{23} - 7 q^{25}+ \cdots - q^{98}+O(q^{100})$ 2 * q - q^2 - 2 * q^3 - q^4 + q^5 + q^6 - 2 * q^7 + 2 * q^9 + 2 * q^10 + q^12 - 7 * q^13 + q^14 - q^15 - 3 * q^16 - q^18 + 2 * q^19 - 3 * q^20 + 2 * q^21 - 5 * q^22 - 2 * q^23 - 7 * q^25 + q^26 - 2 * q^27 + q^28 - 12 * q^29 - 2 * q^30 + 9 * q^32 - 15 * q^34 - q^35 - q^36 - 22 * q^37 + 9 * q^38 + 7 * q^39 - 5 * q^40 + 6 * q^41 - q^42 - q^43 + 5 * q^44 + q^45 + q^46 - 10 * q^47 + 3 * q^48 + 2 * q^49 + 6 * q^50 + 6 * q^52 - 15 * q^53 + q^54 - 5 * q^55 - 2 * q^57 + 11 * q^58 + 21 * q^59 + 3 * q^60 - 3 * q^61 - 15 * q^62 - 2 * q^63 + 4 * q^64 - 6 * q^65 + 5 * q^66 + q^67 + 15 * q^68 + 2 * q^69 - 2 * q^70 + 11 * q^71 - 12 * q^73 + 11 * q^74 + 7 * q^75 - 11 * q^76 - q^78 - 10 * q^79 + 6 * q^80 + 2 * q^81 + 7 * q^82 + 4 * q^83 - q^84 - 15 * q^85 + 3 * q^86 + 12 * q^87 + 10 * q^88 + 21 * q^89 + 2 * q^90 + 7 * q^91 + q^92 + 10 * q^94 + 11 * q^95 - 9 * q^96 - 6 * q^97 - q^98

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0.618034	0.309017
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	1.61803	0.660560
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	−0.618034	−0.178411
$13$	−2.38197	−0.660639	−0.330319	−	0.943869i	$-0.607156\pi$
$13$	−2.38197	−0.660639	−0.330319	+	0.943869i	$0.607156\pi$
$14$	1.61803	0.432438
$15$	0.618034	0.159576
$16$	−4.85410	−1.21353
$17$	6.70820	1.62698	0.813489	−	0.581580i	$-0.197565\pi$
$17$	6.70820	1.62698	0.813489	+	0.581580i	$0.197565\pi$
$18$	−1.61803	−0.381374
$19$	−3.47214	−0.796563	−0.398281	−	0.917263i	$-0.630393\pi$
$19$	−3.47214	−0.796563	−0.398281	+	0.917263i	$0.630393\pi$
$20$	−0.381966	−0.0854102
$21$	1.00000	0.218218
$22$	−3.61803	−0.771367
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−2.23607	−0.456435
$25$	−4.61803	−0.923607
$26$	3.85410	0.755852
$27$	−1.00000	−0.192450
$28$	−0.618034	−0.116797
$29$	−8.23607	−1.52940	−0.764700	−	0.644387i	$-0.777113\pi$
$29$	−8.23607	−1.52940	−0.764700	+	0.644387i	$0.777113\pi$
$30$	−1.00000	−0.182574
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	3.38197	0.597853
$33$	−2.23607	−0.389249
$34$	−10.8541	−1.86146
$35$	0.618034	0.104467
$36$	0.618034	0.103006
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	5.61803	0.911365
$39$	2.38197	0.381420
$40$	−1.38197	−0.218508
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	−1.61803	−0.249668
$43$	−1.61803	−0.246748	−0.123374	−	0.992360i	$-0.539371\pi$
$43$	−1.61803	−0.246748	−0.123374	+	0.992360i	$0.539371\pi$
$44$	1.38197	0.208339
$45$	−0.618034	−0.0921311
$46$	1.61803	0.238566
$47$	−7.23607	−1.05549	−0.527744	−	0.849403i	$-0.676962\pi$
$47$	−7.23607	−1.05549	−0.527744	+	0.849403i	$0.676962\pi$
$48$	4.85410	0.700629
$49$	1.00000	0.142857
$50$	7.47214	1.05672
$51$	−6.70820	−0.939336
$52$	−1.47214	−0.204149
$53$	−13.0902	−1.79807	−0.899037	−	0.437874i	$-0.855732\pi$
$53$	−13.0902	−1.79807	−0.899037	+	0.437874i	$0.855732\pi$
$54$	1.61803	0.220187
$55$	−1.38197	−0.186344
$56$	−2.23607	−0.298807
$57$	3.47214	0.459896
$58$	13.3262	1.74982
$59$	9.38197	1.22143	0.610714	−	0.791851i	$-0.290883\pi$
$59$	9.38197	1.22143	0.610714	+	0.791851i	$0.290883\pi$
$60$	0.381966	0.0493116
$61$	−4.85410	−0.621504	−0.310752	−	0.950491i	$-0.600581\pi$
$61$	−4.85410	−0.621504	−0.310752	+	0.950491i	$0.600581\pi$
$62$	−10.8541	−1.37847
$63$	−1.00000	−0.125988
$64$	4.23607	0.529508
$65$	1.47214	0.182596
$66$	3.61803	0.445349
$67$	−5.09017	−0.621863	−0.310932	−	0.950432i	$-0.600641\pi$
$67$	−5.09017	−0.621863	−0.310932	+	0.950432i	$0.600641\pi$
$68$	4.14590	0.502764
$69$	1.00000	0.120386
$70$	−1.00000	−0.119523
$71$	4.38197	0.520044	0.260022	−	0.965603i	$-0.416270\pi$
$71$	4.38197	0.520044	0.260022	+	0.965603i	$0.416270\pi$
$72$	2.23607	0.263523
$73$	−12.7082	−1.48738	−0.743691	−	0.668523i	$-0.766927\pi$
$73$	−12.7082	−1.48738	−0.743691	+	0.668523i	$0.766927\pi$
$74$	17.7984	2.06902
$75$	4.61803	0.533245
$76$	−2.14590	−0.246151
$77$	−2.23607	−0.254824
$78$	−3.85410	−0.436391
$79$	−9.47214	−1.06570	−0.532849	−	0.846210i	$-0.678879\pi$
$79$	−9.47214	−1.06570	−0.532849	+	0.846210i	$0.678879\pi$
$80$	3.00000	0.335410
$81$	1.00000	0.111111
$82$	2.38197	0.263044
$83$	−9.18034	−1.00767	−0.503837	−	0.863799i	$-0.668079\pi$
$83$	−9.18034	−1.00767	−0.503837	+	0.863799i	$0.668079\pi$
$84$	0.618034	0.0674330
$85$	−4.14590	−0.449686
$86$	2.61803	0.282310
$87$	8.23607	0.882999
$88$	5.00000	0.533002
$89$	11.6180	1.23151	0.615755	−	0.787938i	$-0.288851\pi$
$89$	11.6180	1.23151	0.615755	+	0.787938i	$0.288851\pi$
$90$	1.00000	0.105409
$91$	2.38197	0.249698
$92$	−0.618034	−0.0644345
$93$	−6.70820	−0.695608
$94$	11.7082	1.20761
$95$	2.14590	0.220164
$96$	−3.38197	−0.345170
$97$	10.4164	1.05763	0.528813	−	0.848738i	$-0.322637\pi$
$97$	10.4164	1.05763	0.528813	+	0.848738i	$0.322637\pi$
$98$	−1.61803	−0.163446
$99$	2.23607	0.224733
$100$	−2.85410	−0.285410
$101$	4.14590	0.412532	0.206266	−	0.978496i	$-0.433869\pi$
$101$	4.14590	0.412532	0.206266	+	0.978496i	$0.433869\pi$
$102$	10.8541	1.07472
$103$	−7.41641	−0.730760	−0.365380	−	0.930858i	$-0.619061\pi$
$103$	−7.41641	−0.730760	−0.365380	+	0.930858i	$0.619061\pi$
$104$	−5.32624	−0.522281
$105$	−0.618034	−0.0603139
$106$	21.1803	2.05722
$107$	3.32624	0.321560	0.160780	−	0.986990i	$-0.448599\pi$
$107$	3.32624	0.321560	0.160780	+	0.986990i	$0.448599\pi$
$108$	−0.618034	−0.0594703
$109$	19.2705	1.84578	0.922890	−	0.385064i	$-0.125820\pi$
$109$	19.2705	1.84578	0.922890	+	0.385064i	$0.125820\pi$
$110$	2.23607	0.213201
$111$	11.0000	1.04407
$112$	4.85410	0.458670
$113$	−0.909830	−0.0855896	−0.0427948	−	0.999084i	$-0.513626\pi$
$113$	−0.909830	−0.0855896	−0.0427948	+	0.999084i	$0.513626\pi$
$114$	−5.61803	−0.526177
$115$	0.618034	0.0576320
$116$	−5.09017	−0.472610
$117$	−2.38197	−0.220213
$118$	−15.1803	−1.39746
$119$	−6.70820	−0.614940
$120$	1.38197	0.126156
$121$	−6.00000	−0.545455
$122$	7.85410	0.711077
$123$	1.47214	0.132738
$124$	4.14590	0.372313
$125$	5.94427	0.531672
$126$	1.61803	0.144146
$127$	22.2705	1.97619	0.988094	−	0.153851i	$-0.0491675\pi$
$127$	22.2705	1.97619	0.988094	+	0.153851i	$0.0491675\pi$
$128$	−13.6180	−1.20368
$129$	1.61803	0.142460
$130$	−2.38197	−0.208912
$131$	15.1803	1.32631	0.663156	−	0.748481i	$-0.269216\pi$
$131$	15.1803	1.32631	0.663156	+	0.748481i	$0.269216\pi$
$132$	−1.38197	−0.120285
$133$	3.47214	0.301072
$134$	8.23607	0.711488
$135$	0.618034	0.0531919
$136$	15.0000	1.28624
$137$	−16.4164	−1.40255	−0.701274	−	0.712892i	$-0.747385\pi$
$137$	−16.4164	−1.40255	−0.701274	+	0.712892i	$0.747385\pi$
$138$	−1.61803	−0.137736
$139$	−11.3820	−0.965406	−0.482703	−	0.875784i	$-0.660345\pi$
$139$	−11.3820	−0.965406	−0.482703	+	0.875784i	$0.660345\pi$
$140$	0.381966	0.0322820
$141$	7.23607	0.609387
$142$	−7.09017	−0.594994
$143$	−5.32624	−0.445402
$144$	−4.85410	−0.404508
$145$	5.09017	0.422716
$146$	20.5623	1.70175
$147$	−1.00000	−0.0824786
$148$	−6.79837	−0.558823
$149$	−19.2361	−1.57588	−0.787940	−	0.615752i	$-0.788852\pi$
$149$	−19.2361	−1.57588	−0.787940	+	0.615752i	$0.788852\pi$
$150$	−7.47214	−0.610097
$151$	−15.2361	−1.23989	−0.619947	−	0.784644i	$-0.712846\pi$
$151$	−15.2361	−1.23989	−0.619947	+	0.784644i	$0.712846\pi$
$152$	−7.76393	−0.629738
$153$	6.70820	0.542326
$154$	3.61803	0.291549
$155$	−4.14590	−0.333007
$156$	1.47214	0.117865
$157$	−0.291796	−0.0232879	−0.0116439	−	0.999932i	$-0.503706\pi$
$157$	−0.291796	−0.0232879	−0.0116439	+	0.999932i	$0.503706\pi$
$158$	15.3262	1.21929
$159$	13.0902	1.03812
$160$	−2.09017	−0.165242
$161$	1.00000	0.0788110
$162$	−1.61803	−0.127125
$163$	−0.618034	−0.0484082	−0.0242041	−	0.999707i	$-0.507705\pi$
$163$	−0.618034	−0.0484082	−0.0242041	+	0.999707i	$0.507705\pi$
$164$	−0.909830	−0.0710458
$165$	1.38197	0.107586
$166$	14.8541	1.15290
$167$	−7.18034	−0.555631	−0.277816	−	0.960634i	$-0.589610\pi$
$167$	−7.18034	−0.555631	−0.277816	+	0.960634i	$0.589610\pi$
$168$	2.23607	0.172516
$169$	−7.32624	−0.563557
$170$	6.70820	0.514496
$171$	−3.47214	−0.265521
$172$	−1.00000	−0.0762493
$173$	−3.47214	−0.263982	−0.131991	−	0.991251i	$-0.542137\pi$
$173$	−3.47214	−0.263982	−0.131991	+	0.991251i	$0.542137\pi$
$174$	−13.3262	−1.01026
$175$	4.61803	0.349091
$176$	−10.8541	−0.818159
$177$	−9.38197	−0.705192
$178$	−18.7984	−1.40900
$179$	1.85410	0.138582	0.0692910	−	0.997596i	$-0.477926\pi$
$179$	1.85410	0.138582	0.0692910	+	0.997596i	$0.477926\pi$
$180$	−0.381966	−0.0284701
$181$	−1.94427	−0.144517	−0.0722583	−	0.997386i	$-0.523021\pi$
$181$	−1.94427	−0.144517	−0.0722583	+	0.997386i	$0.523021\pi$
$182$	−3.85410	−0.285685
$183$	4.85410	0.358826
$184$	−2.23607	−0.164845
$185$	6.79837	0.499826
$186$	10.8541	0.795861
$187$	15.0000	1.09691
$188$	−4.47214	−0.326164
$189$	1.00000	0.0727393
$190$	−3.47214	−0.251895
$191$	16.1803	1.17077	0.585384	−	0.810756i	$-0.300944\pi$
$191$	16.1803	1.17077	0.585384	+	0.810756i	$0.300944\pi$
$192$	−4.23607	−0.305712
$193$	8.29180	0.596857	0.298428	−	0.954432i	$-0.403538\pi$
$193$	8.29180	0.596857	0.298428	+	0.954432i	$0.403538\pi$
$194$	−16.8541	−1.21005
$195$	−1.47214	−0.105422
$196$	0.618034	0.0441453
$197$	−25.5066	−1.81727	−0.908634	−	0.417593i	$-0.862874\pi$
$197$	−25.5066	−1.81727	−0.908634	+	0.417593i	$0.862874\pi$
$198$	−3.61803	−0.257122
$199$	−6.90983	−0.489825	−0.244912	−	0.969545i	$-0.578759\pi$
$199$	−6.90983	−0.489825	−0.244912	+	0.969545i	$0.578759\pi$
$200$	−10.3262	−0.730175
$201$	5.09017	0.359033
$202$	−6.70820	−0.471988
$203$	8.23607	0.578059
$204$	−4.14590	−0.290271
$205$	0.909830	0.0635453
$206$	12.0000	0.836080
$207$	−1.00000	−0.0695048
$208$	11.5623	0.801702
$209$	−7.76393	−0.537042
$210$	1.00000	0.0690066
$211$	10.4164	0.717095	0.358548	−	0.933511i	$-0.383272\pi$
$211$	10.4164	0.717095	0.358548	+	0.933511i	$0.383272\pi$
$212$	−8.09017	−0.555635
$213$	−4.38197	−0.300247
$214$	−5.38197	−0.367904
$215$	1.00000	0.0681994
$216$	−2.23607	−0.152145
$217$	−6.70820	−0.455383
$218$	−31.1803	−2.11180
$219$	12.7082	0.858741
$220$	−0.854102	−0.0575835
$221$	−15.9787	−1.07484
$222$	−17.7984	−1.19455
$223$	−17.8541	−1.19560	−0.597800	−	0.801646i	$-0.703958\pi$
$223$	−17.8541	−1.19560	−0.597800	+	0.801646i	$0.703958\pi$
$224$	−3.38197	−0.225967
$225$	−4.61803	−0.307869
$226$	1.47214	0.0979250
$227$	24.3262	1.61459	0.807295	−	0.590149i	$-0.200931\pi$
$227$	24.3262	1.61459	0.807295	+	0.590149i	$0.200931\pi$
$228$	2.14590	0.142116
$229$	−16.3262	−1.07887	−0.539434	−	0.842028i	$-0.681362\pi$
$229$	−16.3262	−1.07887	−0.539434	+	0.842028i	$0.681362\pi$
$230$	−1.00000	−0.0659380
$231$	2.23607	0.147122
$232$	−18.4164	−1.20910
$233$	−11.0902	−0.726541	−0.363271	−	0.931684i	$-0.618340\pi$
$233$	−11.0902	−0.726541	−0.363271	+	0.931684i	$0.618340\pi$
$234$	3.85410	0.251951
$235$	4.47214	0.291730
$236$	5.79837	0.377442
$237$	9.47214	0.615281
$238$	10.8541	0.703567
$239$	−4.79837	−0.310381	−0.155191	−	0.987885i	$-0.549599\pi$
$239$	−4.79837	−0.310381	−0.155191	+	0.987885i	$0.549599\pi$
$240$	−3.00000	−0.193649
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	9.70820	0.624067
$243$	−1.00000	−0.0641500
$244$	−3.00000	−0.192055
$245$	−0.618034	−0.0394847
$246$	−2.38197	−0.151869
$247$	8.27051	0.526240
$248$	15.0000	0.952501
$249$	9.18034	0.581780
$250$	−9.61803	−0.608298
$251$	23.1246	1.45961	0.729806	−	0.683654i	$-0.239610\pi$
$251$	23.1246	1.45961	0.729806	+	0.683654i	$0.239610\pi$
$252$	−0.618034	−0.0389325
$253$	−2.23607	−0.140580
$254$	−36.0344	−2.26100
$255$	4.14590	0.259626
$256$	13.5623	0.847644
$257$	−3.23607	−0.201860	−0.100930	−	0.994894i	$-0.532182\pi$
$257$	−3.23607	−0.201860	−0.100930	+	0.994894i	$0.532182\pi$
$258$	−2.61803	−0.162992
$259$	11.0000	0.683507
$260$	0.909830	0.0564253
$261$	−8.23607	−0.509800
$262$	−24.5623	−1.51746
$263$	−0.0557281	−0.00343634	−0.00171817	−	0.999999i	$-0.500547\pi$
$263$	−0.0557281	−0.00343634	−0.00171817	+	0.999999i	$0.500547\pi$
$264$	−5.00000	−0.307729
$265$	8.09017	0.496975
$266$	−5.61803	−0.344464
$267$	−11.6180	−0.711012
$268$	−3.14590	−0.192166
$269$	−32.0902	−1.95657	−0.978286	−	0.207259i	$-0.933546\pi$
$269$	−32.0902	−1.95657	−0.978286	+	0.207259i	$0.933546\pi$
$270$	−1.00000	−0.0608581
$271$	21.9443	1.33302	0.666510	−	0.745496i	$-0.267787\pi$
$271$	21.9443	1.33302	0.666510	+	0.745496i	$0.267787\pi$
$272$	−32.5623	−1.97438
$273$	−2.38197	−0.144163
$274$	26.5623	1.60469
$275$	−10.3262	−0.622696
$276$	0.618034	0.0372013
$277$	4.27051	0.256590	0.128295	−	0.991736i	$-0.459050\pi$
$277$	4.27051	0.256590	0.128295	+	0.991736i	$0.459050\pi$
$278$	18.4164	1.10454
$279$	6.70820	0.401610
$280$	1.38197	0.0825883
$281$	−3.70820	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$281$	−3.70820	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$282$	−11.7082	−0.697213
$283$	−26.2148	−1.55831	−0.779154	−	0.626833i	$-0.784351\pi$
$283$	−26.2148	−1.55831	−0.779154	+	0.626833i	$0.784351\pi$
$284$	2.70820	0.160702
$285$	−2.14590	−0.127112
$286$	8.61803	0.509595
$287$	1.47214	0.0868974
$288$	3.38197	0.199284
$289$	28.0000	1.64706
$290$	−8.23607	−0.483639
$291$	−10.4164	−0.610621
$292$	−7.85410	−0.459627
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	1.61803	0.0943657
$295$	−5.79837	−0.337594
$296$	−24.5967	−1.42966
$297$	−2.23607	−0.129750
$298$	31.1246	1.80300
$299$	2.38197	0.137753
$300$	2.85410	0.164782
$301$	1.61803	0.0932619
$302$	24.6525	1.41859
$303$	−4.14590	−0.238176
$304$	16.8541	0.966649
$305$	3.00000	0.171780
$306$	−10.8541	−0.620488
$307$	16.1246	0.920280	0.460140	−	0.887846i	$-0.347799\pi$
$307$	16.1246	0.920280	0.460140	+	0.887846i	$0.347799\pi$
$308$	−1.38197	−0.0787448
$309$	7.41641	0.421905
$310$	6.70820	0.381000
$311$	15.3262	0.869071	0.434536	−	0.900655i	$-0.356913\pi$
$311$	15.3262	0.869071	0.434536	+	0.900655i	$0.356913\pi$
$312$	5.32624	0.301539
$313$	−2.47214	−0.139733	−0.0698667	−	0.997556i	$-0.522257\pi$
$313$	−2.47214	−0.139733	−0.0698667	+	0.997556i	$0.522257\pi$
$314$	0.472136	0.0266442
$315$	0.618034	0.0348223
$316$	−5.85410	−0.329319
$317$	14.5066	0.814771	0.407385	−	0.913256i	$-0.366441\pi$
$317$	14.5066	0.814771	0.407385	+	0.913256i	$0.366441\pi$
$318$	−21.1803	−1.18773
$319$	−18.4164	−1.03112
$320$	−2.61803	−0.146353
$321$	−3.32624	−0.185652
$322$	−1.61803	−0.0901695
$323$	−23.2918	−1.29599
$324$	0.618034	0.0343352
$325$	11.0000	0.610170
$326$	1.00000	0.0553849
$327$	−19.2705	−1.06566
$328$	−3.29180	−0.181759
$329$	7.23607	0.398937
$330$	−2.23607	−0.123091
$331$	−13.4164	−0.737432	−0.368716	−	0.929542i	$-0.620203\pi$
$331$	−13.4164	−0.737432	−0.368716	+	0.929542i	$0.620203\pi$
$332$	−5.67376	−0.311388
$333$	−11.0000	−0.602796
$334$	11.6180	0.635711
$335$	3.14590	0.171879
$336$	−4.85410	−0.264813
$337$	−7.50658	−0.408909	−0.204455	−	0.978876i	$-0.565542\pi$
$337$	−7.50658	−0.408909	−0.204455	+	0.978876i	$0.565542\pi$
$338$	11.8541	0.644778
$339$	0.909830	0.0494152
$340$	−2.56231	−0.138961
$341$	15.0000	0.812296
$342$	5.61803	0.303788
$343$	−1.00000	−0.0539949
$344$	−3.61803	−0.195071
$345$	−0.618034	−0.0332738
$346$	5.61803	0.302027
$347$	5.18034	0.278095	0.139048	−	0.990286i	$-0.455596\pi$
$347$	5.18034	0.278095	0.139048	+	0.990286i	$0.455596\pi$
$348$	5.09017	0.272862
$349$	−11.8541	−0.634536	−0.317268	−	0.948336i	$-0.602765\pi$
$349$	−11.8541	−0.634536	−0.317268	+	0.948336i	$0.602765\pi$
$350$	−7.47214	−0.399402
$351$	2.38197	0.127140
$352$	7.56231	0.403072
$353$	30.3050	1.61297	0.806485	−	0.591255i	$-0.201367\pi$
$353$	30.3050	1.61297	0.806485	+	0.591255i	$0.201367\pi$
$354$	15.1803	0.806826
$355$	−2.70820	−0.143737
$356$	7.18034	0.380557
$357$	6.70820	0.355036
$358$	−3.00000	−0.158555
$359$	23.0902	1.21865	0.609326	−	0.792920i	$-0.291440\pi$
$359$	23.0902	1.21865	0.609326	+	0.792920i	$0.291440\pi$
$360$	−1.38197	−0.0728360
$361$	−6.94427	−0.365488
$362$	3.14590	0.165345
$363$	6.00000	0.314918
$364$	1.47214	0.0771609
$365$	7.85410	0.411102
$366$	−7.85410	−0.410540
$367$	10.8541	0.566580	0.283290	−	0.959034i	$-0.408574\pi$
$367$	10.8541	0.566580	0.283290	+	0.959034i	$0.408574\pi$
$368$	4.85410	0.253038
$369$	−1.47214	−0.0766363
$370$	−11.0000	−0.571863
$371$	13.0902	0.679608
$372$	−4.14590	−0.214955
$373$	24.4164	1.26423	0.632117	−	0.774873i	$-0.282186\pi$
$373$	24.4164	1.26423	0.632117	+	0.774873i	$0.282186\pi$
$374$	−24.2705	−1.25500
$375$	−5.94427	−0.306961
$376$	−16.1803	−0.834437
$377$	19.6180	1.01038
$378$	−1.61803	−0.0832227
$379$	7.41641	0.380955	0.190478	−	0.981692i	$-0.438996\pi$
$379$	7.41641	0.380955	0.190478	+	0.981692i	$0.438996\pi$
$380$	1.32624	0.0680346
$381$	−22.2705	−1.14095
$382$	−26.1803	−1.33950
$383$	−0.708204	−0.0361875	−0.0180938	−	0.999836i	$-0.505760\pi$
$383$	−0.708204	−0.0361875	−0.0180938	+	0.999836i	$0.505760\pi$
$384$	13.6180	0.694942
$385$	1.38197	0.0704315
$386$	−13.4164	−0.682877
$387$	−1.61803	−0.0822493
$388$	6.43769	0.326824
$389$	5.29180	0.268305	0.134152	−	0.990961i	$-0.457169\pi$
$389$	5.29180	0.268305	0.134152	+	0.990961i	$0.457169\pi$
$390$	2.38197	0.120616
$391$	−6.70820	−0.339248
$392$	2.23607	0.112938
$393$	−15.1803	−0.765747
$394$	41.2705	2.07918
$395$	5.85410	0.294552
$396$	1.38197	0.0694464
$397$	22.0689	1.10761	0.553803	−	0.832648i	$-0.313176\pi$
$397$	22.0689	1.10761	0.553803	+	0.832648i	$0.313176\pi$
$398$	11.1803	0.560420
$399$	−3.47214	−0.173824
$400$	22.4164	1.12082
$401$	33.1803	1.65695	0.828474	−	0.560028i	$-0.189210\pi$
$401$	33.1803	1.65695	0.828474	+	0.560028i	$0.189210\pi$
$402$	−8.23607	−0.410778
$403$	−15.9787	−0.795956
$404$	2.56231	0.127479
$405$	−0.618034	−0.0307104
$406$	−13.3262	−0.661370
$407$	−24.5967	−1.21922
$408$	−15.0000	−0.742611
$409$	6.41641	0.317271	0.158635	−	0.987337i	$-0.449291\pi$
$409$	6.41641	0.317271	0.158635	+	0.987337i	$0.449291\pi$
$410$	−1.47214	−0.0727036
$411$	16.4164	0.809762
$412$	−4.58359	−0.225817
$413$	−9.38197	−0.461656
$414$	1.61803	0.0795220
$415$	5.67376	0.278514
$416$	−8.05573	−0.394965
$417$	11.3820	0.557377
$418$	12.5623	0.614442
$419$	15.6738	0.765713	0.382857	−	0.923808i	$-0.374940\pi$
$419$	15.6738	0.765713	0.382857	+	0.923808i	$0.374940\pi$
$420$	−0.381966	−0.0186380
$421$	−10.2705	−0.500554	−0.250277	−	0.968174i	$-0.580522\pi$
$421$	−10.2705	−0.500554	−0.250277	+	0.968174i	$0.580522\pi$
$422$	−16.8541	−0.820445
$423$	−7.23607	−0.351830
$424$	−29.2705	−1.42150
$425$	−30.9787	−1.50269
$426$	7.09017	0.343520
$427$	4.85410	0.234906
$428$	2.05573	0.0993674
$429$	5.32624	0.257153
$430$	−1.61803	−0.0780285
$431$	0.673762	0.0324540	0.0162270	−	0.999868i	$-0.494835\pi$
$431$	0.673762	0.0324540	0.0162270	+	0.999868i	$0.494835\pi$
$432$	4.85410	0.233543
$433$	19.1246	0.919070	0.459535	−	0.888160i	$-0.348016\pi$
$433$	19.1246	0.919070	0.459535	+	0.888160i	$0.348016\pi$
$434$	10.8541	0.521014
$435$	−5.09017	−0.244055
$436$	11.9098	0.570377
$437$	3.47214	0.166095
$438$	−20.5623	−0.982505
$439$	23.6525	1.12887	0.564436	−	0.825477i	$-0.309094\pi$
$439$	23.6525	1.12887	0.564436	+	0.825477i	$0.309094\pi$
$440$	−3.09017	−0.147318
$441$	1.00000	0.0476190
$442$	25.8541	1.22975
$443$	−9.52786	−0.452682	−0.226341	−	0.974048i	$-0.572676\pi$
$443$	−9.52786	−0.452682	−0.226341	+	0.974048i	$0.572676\pi$
$444$	6.79837	0.322637
$445$	−7.18034	−0.340381
$446$	28.8885	1.36791
$447$	19.2361	0.909835
$448$	−4.23607	−0.200135
$449$	−18.4377	−0.870129	−0.435064	−	0.900399i	$-0.643274\pi$
$449$	−18.4377	−0.870129	−0.435064	+	0.900399i	$0.643274\pi$
$450$	7.47214	0.352240
$451$	−3.29180	−0.155005
$452$	−0.562306	−0.0264486
$453$	15.2361	0.715853
$454$	−39.3607	−1.84729
$455$	−1.47214	−0.0690148
$456$	7.76393	0.363579
$457$	−14.3262	−0.670153	−0.335077	−	0.942191i	$-0.608762\pi$
$457$	−14.3262	−0.670153	−0.335077	+	0.942191i	$0.608762\pi$
$458$	26.4164	1.23436
$459$	−6.70820	−0.313112
$460$	0.381966	0.0178093
$461$	25.4508	1.18536	0.592682	−	0.805436i	$-0.298069\pi$
$461$	25.4508	1.18536	0.592682	+	0.805436i	$0.298069\pi$
$462$	−3.61803	−0.168326
$463$	17.2918	0.803618	0.401809	−	0.915724i	$-0.368382\pi$
$463$	17.2918	0.803618	0.401809	+	0.915724i	$0.368382\pi$
$464$	39.9787	1.85597
$465$	4.14590	0.192261
$466$	17.9443	0.831252
$467$	−15.1803	−0.702462	−0.351231	−	0.936289i	$-0.614237\pi$
$467$	−15.1803	−0.702462	−0.351231	+	0.936289i	$0.614237\pi$
$468$	−1.47214	−0.0680495
$469$	5.09017	0.235042
$470$	−7.23607	−0.333775
$471$	0.291796	0.0134453
$472$	20.9787	0.965624
$473$	−3.61803	−0.166357
$474$	−15.3262	−0.703957
$475$	16.0344	0.735711
$476$	−4.14590	−0.190027
$477$	−13.0902	−0.599358
$478$	7.76393	0.355114
$479$	21.0000	0.959514	0.479757	−	0.877401i	$-0.340725\pi$
$479$	21.0000	0.959514	0.479757	+	0.877401i	$0.340725\pi$
$480$	2.09017	0.0954028
$481$	26.2016	1.19469
$482$	17.7984	0.810694
$483$	−1.00000	−0.0455016
$484$	−3.70820	−0.168555
$485$	−6.43769	−0.292321
$486$	1.61803	0.0733955
$487$	−29.2918	−1.32734	−0.663669	−	0.748026i	$-0.731002\pi$
$487$	−29.2918	−1.32734	−0.663669	+	0.748026i	$0.731002\pi$
$488$	−10.8541	−0.491342
$489$	0.618034	0.0279485
$490$	1.00000	0.0451754
$491$	9.79837	0.442194	0.221097	−	0.975252i	$-0.429036\pi$
$491$	9.79837	0.442194	0.221097	+	0.975252i	$0.429036\pi$
$492$	0.909830	0.0410183
$493$	−55.2492	−2.48830
$494$	−13.3820	−0.602083
$495$	−1.38197	−0.0621148
$496$	−32.5623	−1.46209
$497$	−4.38197	−0.196558
$498$	−14.8541	−0.665628
$499$	−38.3951	−1.71880	−0.859401	−	0.511302i	$-0.829163\pi$
$499$	−38.3951	−1.71880	−0.859401	+	0.511302i	$0.829163\pi$
$500$	3.67376	0.164296
$501$	7.18034	0.320794
$502$	−37.4164	−1.66998
$503$	36.3262	1.61971	0.809853	−	0.586632i	$-0.199547\pi$
$503$	36.3262	1.61971	0.809853	+	0.586632i	$0.199547\pi$
$504$	−2.23607	−0.0996024
$505$	−2.56231	−0.114021
$506$	3.61803	0.160841
$507$	7.32624	0.325370
$508$	13.7639	0.610676
$509$	−15.5967	−0.691314	−0.345657	−	0.938361i	$-0.612344\pi$
$509$	−15.5967	−0.691314	−0.345657	+	0.938361i	$0.612344\pi$
$510$	−6.70820	−0.297044
$511$	12.7082	0.562178
$512$	5.29180	0.233867
$513$	3.47214	0.153299
$514$	5.23607	0.230953
$515$	4.58359	0.201977
$516$	1.00000	0.0440225
$517$	−16.1803	−0.711611
$518$	−17.7984	−0.782016
$519$	3.47214	0.152410
$520$	3.29180	0.144355
$521$	3.52786	0.154559	0.0772793	−	0.997009i	$-0.475377\pi$
$521$	3.52786	0.154559	0.0772793	+	0.997009i	$0.475377\pi$
$522$	13.3262	0.583274
$523$	21.4164	0.936474	0.468237	−	0.883603i	$-0.344889\pi$
$523$	21.4164	0.936474	0.468237	+	0.883603i	$0.344889\pi$
$524$	9.38197	0.409853
$525$	−4.61803	−0.201548
$526$	0.0901699	0.00393160
$527$	45.0000	1.96023
$528$	10.8541	0.472364
$529$	1.00000	0.0434783
$530$	−13.0902	−0.568601
$531$	9.38197	0.407143
$532$	2.14590	0.0930365
$533$	3.50658	0.151887
$534$	18.7984	0.813485
$535$	−2.05573	−0.0888769
$536$	−11.3820	−0.491626
$537$	−1.85410	−0.0800104
$538$	51.9230	2.23856
$539$	2.23607	0.0963143
$540$	0.381966	0.0164372
$541$	−5.12461	−0.220324	−0.110162	−	0.993914i	$-0.535137\pi$
$541$	−5.12461	−0.220324	−0.110162	+	0.993914i	$0.535137\pi$
$542$	−35.5066	−1.52514
$543$	1.94427	0.0834367
$544$	22.6869	0.972694
$545$	−11.9098	−0.510161
$546$	3.85410	0.164940
$547$	12.7984	0.547219	0.273609	−	0.961841i	$-0.411782\pi$
$547$	12.7984	0.547219	0.273609	+	0.961841i	$0.411782\pi$
$548$	−10.1459	−0.433411
$549$	−4.85410	−0.207168
$550$	16.7082	0.712440
$551$	28.5967	1.21826
$552$	2.23607	0.0951734
$553$	9.47214	0.402796
$554$	−6.90983	−0.293571
$555$	−6.79837	−0.288575
$556$	−7.03444	−0.298327
$557$	−5.81966	−0.246587	−0.123293	−	0.992370i	$-0.539346\pi$
$557$	−5.81966	−0.246587	−0.123293	+	0.992370i	$0.539346\pi$
$558$	−10.8541	−0.459491
$559$	3.85410	0.163011
$560$	−3.00000	−0.126773
$561$	−15.0000	−0.633300
$562$	6.00000	0.253095
$563$	−23.5066	−0.990684	−0.495342	−	0.868698i	$-0.664957\pi$
$563$	−23.5066	−0.990684	−0.495342	+	0.868698i	$0.664957\pi$
$564$	4.47214	0.188311
$565$	0.562306	0.0236564
$566$	42.4164	1.78289
$567$	−1.00000	−0.0419961
$568$	9.79837	0.411131
$569$	−16.3607	−0.685875	−0.342938	−	0.939358i	$-0.611422\pi$
$569$	−16.3607	−0.685875	−0.342938	+	0.939358i	$0.611422\pi$
$570$	3.47214	0.145432
$571$	−44.4164	−1.85877	−0.929384	−	0.369113i	$-0.879661\pi$
$571$	−44.4164	−1.85877	−0.929384	+	0.369113i	$0.879661\pi$
$572$	−3.29180	−0.137637
$573$	−16.1803	−0.675943
$574$	−2.38197	−0.0994213
$575$	4.61803	0.192585
$576$	4.23607	0.176503
$577$	−36.8328	−1.53337	−0.766685	−	0.642023i	$-0.778095\pi$
$577$	−36.8328	−1.53337	−0.766685	+	0.642023i	$0.778095\pi$
$578$	−45.3050	−1.88444
$579$	−8.29180	−0.344595
$580$	3.14590	0.130626
$581$	9.18034	0.380865
$582$	16.8541	0.698625
$583$	−29.2705	−1.21226
$584$	−28.4164	−1.17588
$585$	1.47214	0.0608653
$586$	−19.4164	−0.802084
$587$	−31.0344	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$587$	−31.0344	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$588$	−0.618034	−0.0254873
$589$	−23.2918	−0.959722
$590$	9.38197	0.386249
$591$	25.5066	1.04920
$592$	53.3951	2.19453
$593$	−32.0689	−1.31691	−0.658456	−	0.752620i	$-0.728790\pi$
$593$	−32.0689	−1.31691	−0.658456	+	0.752620i	$0.728790\pi$
$594$	3.61803	0.148450
$595$	4.14590	0.169965
$596$	−11.8885	−0.486974
$597$	6.90983	0.282801
$598$	−3.85410	−0.157606
$599$	−15.5066	−0.633582	−0.316791	−	0.948495i	$-0.602605\pi$
$599$	−15.5066	−0.633582	−0.316791	+	0.948495i	$0.602605\pi$
$600$	10.3262	0.421567
$601$	0.909830	0.0371127	0.0185564	−	0.999828i	$-0.494093\pi$
$601$	0.909830	0.0371127	0.0185564	+	0.999828i	$0.494093\pi$
$602$	−2.61803	−0.106703
$603$	−5.09017	−0.207288
$604$	−9.41641	−0.383148
$605$	3.70820	0.150760
$606$	6.70820	0.272502
$607$	−33.7984	−1.37183	−0.685917	−	0.727680i	$-0.740599\pi$
$607$	−33.7984	−1.37183	−0.685917	+	0.727680i	$0.740599\pi$
$608$	−11.7426	−0.476227
$609$	−8.23607	−0.333742
$610$	−4.85410	−0.196537
$611$	17.2361	0.697297
$612$	4.14590	0.167588
$613$	−11.6525	−0.470639	−0.235320	−	0.971918i	$-0.575614\pi$
$613$	−11.6525	−0.470639	−0.235320	+	0.971918i	$0.575614\pi$
$614$	−26.0902	−1.05291
$615$	−0.909830	−0.0366879
$616$	−5.00000	−0.201456
$617$	−2.67376	−0.107642	−0.0538208	−	0.998551i	$-0.517140\pi$
$617$	−2.67376	−0.107642	−0.0538208	+	0.998551i	$0.517140\pi$
$618$	−12.0000	−0.482711
$619$	−19.6180	−0.788515	−0.394258	−	0.919000i	$-0.628998\pi$
$619$	−19.6180	−0.788515	−0.394258	+	0.919000i	$0.628998\pi$
$620$	−2.56231	−0.102905
$621$	1.00000	0.0401286
$622$	−24.7984	−0.994324
$623$	−11.6180	−0.465467
$624$	−11.5623	−0.462863
$625$	19.4164	0.776656
$626$	4.00000	0.159872
$627$	7.76393	0.310062
$628$	−0.180340	−0.00719634
$629$	−73.7902	−2.94221
$630$	−1.00000	−0.0398410
$631$	16.1246	0.641911	0.320955	−	0.947094i	$-0.395996\pi$
$631$	16.1246	0.641911	0.320955	+	0.947094i	$0.395996\pi$
$632$	−21.1803	−0.842509
$633$	−10.4164	−0.414015
$634$	−23.4721	−0.932198
$635$	−13.7639	−0.546205
$636$	8.09017	0.320796
$637$	−2.38197	−0.0943769
$638$	29.7984	1.17973
$639$	4.38197	0.173348
$640$	8.41641	0.332688
$641$	17.7984	0.702994	0.351497	−	0.936189i	$-0.385673\pi$
$641$	17.7984	0.702994	0.351497	+	0.936189i	$0.385673\pi$
$642$	5.38197	0.212409
$643$	−11.9787	−0.472394	−0.236197	−	0.971705i	$-0.575901\pi$
$643$	−11.9787	−0.472394	−0.236197	+	0.971705i	$0.575901\pi$
$644$	0.618034	0.0243540
$645$	−1.00000	−0.0393750
$646$	37.6869	1.48277
$647$	−24.0344	−0.944891	−0.472446	−	0.881360i	$-0.656629\pi$
$647$	−24.0344	−0.944891	−0.472446	+	0.881360i	$0.656629\pi$
$648$	2.23607	0.0878410
$649$	20.9787	0.823487
$650$	−17.7984	−0.698110
$651$	6.70820	0.262915
$652$	−0.381966	−0.0149589
$653$	27.7426	1.08565	0.542827	−	0.839845i	$-0.317354\pi$
$653$	27.7426	1.08565	0.542827	+	0.839845i	$0.317354\pi$
$654$	31.1803	1.21925
$655$	−9.38197	−0.366584
$656$	7.14590	0.279000
$657$	−12.7082	−0.495794
$658$	−11.7082	−0.456433
$659$	9.65248	0.376007	0.188004	−	0.982168i	$-0.439798\pi$
$659$	9.65248	0.376007	0.188004	+	0.982168i	$0.439798\pi$
$660$	0.854102	0.0332459
$661$	14.4164	0.560733	0.280367	−	0.959893i	$-0.409544\pi$
$661$	14.4164	0.560733	0.280367	+	0.959893i	$0.409544\pi$
$662$	21.7082	0.843713
$663$	15.9787	0.620562
$664$	−20.5279	−0.796636
$665$	−2.14590	−0.0832144
$666$	17.7984	0.689673
$667$	8.23607	0.318902
$668$	−4.43769	−0.171700
$669$	17.8541	0.690279
$670$	−5.09017	−0.196650
$671$	−10.8541	−0.419018
$672$	3.38197	0.130462
$673$	12.4164	0.478617	0.239309	−	0.970944i	$-0.423079\pi$
$673$	12.4164	0.478617	0.239309	+	0.970944i	$0.423079\pi$
$674$	12.1459	0.467843
$675$	4.61803	0.177748
$676$	−4.52786	−0.174149
$677$	−1.25735	−0.0483240	−0.0241620	−	0.999708i	$-0.507692\pi$
$677$	−1.25735	−0.0483240	−0.0241620	+	0.999708i	$0.507692\pi$
$678$	−1.47214	−0.0565370
$679$	−10.4164	−0.399745
$680$	−9.27051	−0.355508
$681$	−24.3262	−0.932183
$682$	−24.2705	−0.929366
$683$	26.0132	0.995366	0.497683	−	0.867359i	$-0.334184\pi$
$683$	26.0132	0.995366	0.497683	+	0.867359i	$0.334184\pi$
$684$	−2.14590	−0.0820505
$685$	10.1459	0.387655
$686$	1.61803	0.0617768
$687$	16.3262	0.622885
$688$	7.85410	0.299435
$689$	31.1803	1.18788
$690$	1.00000	0.0380693
$691$	−7.72949	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$691$	−7.72949	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$692$	−2.14590	−0.0815748
$693$	−2.23607	−0.0849412
$694$	−8.38197	−0.318175
$695$	7.03444	0.266832
$696$	18.4164	0.698072
$697$	−9.87539	−0.374057
$698$	19.1803	0.725987
$699$	11.0902	0.419469
$700$	2.85410	0.107875
$701$	37.4508	1.41450	0.707250	−	0.706964i	$-0.249936\pi$
$701$	37.4508	1.41450	0.707250	+	0.706964i	$0.249936\pi$
$702$	−3.85410	−0.145464
$703$	38.1935	1.44049
$704$	9.47214	0.356995
$705$	−4.47214	−0.168430
$706$	−49.0344	−1.84544
$707$	−4.14590	−0.155923
$708$	−5.79837	−0.217916
$709$	−2.56231	−0.0962294	−0.0481147	−	0.998842i	$-0.515321\pi$
$709$	−2.56231	−0.0962294	−0.0481147	+	0.998842i	$0.515321\pi$
$710$	4.38197	0.164452
$711$	−9.47214	−0.355233
$712$	25.9787	0.973593
$713$	−6.70820	−0.251224
$714$	−10.8541	−0.406205
$715$	3.29180	0.123106
$716$	1.14590	0.0428242
$717$	4.79837	0.179199
$718$	−37.3607	−1.39429
$719$	−3.00000	−0.111881	−0.0559406	−	0.998434i	$-0.517816\pi$
$719$	−3.00000	−0.111881	−0.0559406	+	0.998434i	$0.517816\pi$
$720$	3.00000	0.111803
$721$	7.41641	0.276201
$722$	11.2361	0.418163
$723$	11.0000	0.409094
$724$	−1.20163	−0.0446581
$725$	38.0344	1.41256
$726$	−9.70820	−0.360305
$727$	18.8885	0.700537	0.350269	−	0.936649i	$-0.386090\pi$
$727$	18.8885	0.700537	0.350269	+	0.936649i	$0.386090\pi$
$728$	5.32624	0.197404
$729$	1.00000	0.0370370
$730$	−12.7082	−0.470352
$731$	−10.8541	−0.401453
$732$	3.00000	0.110883
$733$	25.5967	0.945437	0.472719	−	0.881213i	$-0.343273\pi$
$733$	25.5967	0.945437	0.472719	+	0.881213i	$0.343273\pi$
$734$	−17.5623	−0.648237
$735$	0.618034	0.0227965
$736$	−3.38197	−0.124661
$737$	−11.3820	−0.419260
$738$	2.38197	0.0876814
$739$	45.2492	1.66452	0.832260	−	0.554386i	$-0.187047\pi$
$739$	45.2492	1.66452	0.832260	+	0.554386i	$0.187047\pi$
$740$	4.20163	0.154455
$741$	−8.27051	−0.303825
$742$	−21.1803	−0.777555
$743$	11.6738	0.428269	0.214134	−	0.976804i	$-0.431307\pi$
$743$	11.6738	0.428269	0.214134	+	0.976804i	$0.431307\pi$
$744$	−15.0000	−0.549927
$745$	11.8885	0.435563
$746$	−39.5066	−1.44644
$747$	−9.18034	−0.335891
$748$	9.27051	0.338963
$749$	−3.32624	−0.121538
$750$	9.61803	0.351201
$751$	32.9787	1.20341	0.601705	−	0.798718i	$-0.294488\pi$
$751$	32.9787	1.20341	0.601705	+	0.798718i	$0.294488\pi$
$752$	35.1246	1.28086
$753$	−23.1246	−0.842708
$754$	−31.7426	−1.15600
$755$	9.41641	0.342698
$756$	0.618034	0.0224777
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	−12.0000	−0.435860
$759$	2.23607	0.0811641
$760$	4.79837	0.174055
$761$	−14.4721	−0.524615	−0.262307	−	0.964984i	$-0.584483\pi$
$761$	−14.4721	−0.524615	−0.262307	+	0.964984i	$0.584483\pi$
$762$	36.0344	1.30539
$763$	−19.2705	−0.697639
$764$	10.0000	0.361787
$765$	−4.14590	−0.149895
$766$	1.14590	0.0414030
$767$	−22.3475	−0.806922
$768$	−13.5623	−0.489388
$769$	33.2361	1.19852	0.599262	−	0.800553i	$-0.295461\pi$
$769$	33.2361	1.19852	0.599262	+	0.800553i	$0.295461\pi$
$770$	−2.23607	−0.0805823
$771$	3.23607	0.116544
$772$	5.12461	0.184439
$773$	−19.9443	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$773$	−19.9443	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$774$	2.61803	0.0941033
$775$	−30.9787	−1.11279
$776$	23.2918	0.836127
$777$	−11.0000	−0.394623
$778$	−8.56231	−0.306974
$779$	5.11146	0.183137
$780$	−0.909830	−0.0325771
$781$	9.79837	0.350613
$782$	10.8541	0.388142
$783$	8.23607	0.294333
$784$	−4.85410	−0.173361
$785$	0.180340	0.00643661
$786$	24.5623	0.876108
$787$	−48.5755	−1.73153	−0.865764	−	0.500452i	$-0.833167\pi$
$787$	−48.5755	−1.73153	−0.865764	+	0.500452i	$0.833167\pi$
$788$	−15.7639	−0.561567
$789$	0.0557281	0.00198397
$790$	−9.47214	−0.337003
$791$	0.909830	0.0323498
$792$	5.00000	0.177667
$793$	11.5623	0.410590
$794$	−35.7082	−1.26724
$795$	−8.09017	−0.286929
$796$	−4.27051	−0.151364
$797$	44.1803	1.56495	0.782474	−	0.622683i	$-0.213957\pi$
$797$	44.1803	1.56495	0.782474	+	0.622683i	$0.213957\pi$
$798$	5.61803	0.198876
$799$	−48.5410	−1.71726
$800$	−15.6180	−0.552181
$801$	11.6180	0.410503
$802$	−53.6869	−1.89575
$803$	−28.4164	−1.00279
$804$	3.14590	0.110947
$805$	−0.618034	−0.0217828
$806$	25.8541	0.910672
$807$	32.0902	1.12963
$808$	9.27051	0.326135
$809$	−9.43769	−0.331812	−0.165906	−	0.986142i	$-0.553055\pi$
$809$	−9.43769	−0.331812	−0.165906	+	0.986142i	$0.553055\pi$
$810$	1.00000	0.0351364
$811$	8.52786	0.299454	0.149727	−	0.988727i	$-0.452161\pi$
$811$	8.52786	0.299454	0.149727	+	0.988727i	$0.452161\pi$
$812$	5.09017	0.178630
$813$	−21.9443	−0.769619
$814$	39.7984	1.39493
$815$	0.381966	0.0133797
$816$	32.5623	1.13991
$817$	5.61803	0.196550
$818$	−10.3820	−0.362997
$819$	2.38197	0.0832326
$820$	0.562306	0.0196366
$821$	−29.1246	−1.01646	−0.508228	−	0.861223i	$-0.669699\pi$
$821$	−29.1246	−1.01646	−0.508228	+	0.861223i	$0.669699\pi$
$822$	−26.5623	−0.926467
$823$	−39.8541	−1.38923	−0.694613	−	0.719383i	$-0.744425\pi$
$823$	−39.8541	−1.38923	−0.694613	+	0.719383i	$0.744425\pi$
$824$	−16.5836	−0.577717
$825$	10.3262	0.359513
$826$	15.1803	0.528192
$827$	−28.2148	−0.981124	−0.490562	−	0.871406i	$-0.663208\pi$
$827$	−28.2148	−0.981124	−0.490562	+	0.871406i	$0.663208\pi$
$828$	−0.618034	−0.0214782
$829$	−30.4164	−1.05641	−0.528203	−	0.849118i	$-0.677134\pi$
$829$	−30.4164	−1.05641	−0.528203	+	0.849118i	$0.677134\pi$
$830$	−9.18034	−0.318654
$831$	−4.27051	−0.148142
$832$	−10.0902	−0.349814
$833$	6.70820	0.232425
$834$	−18.4164	−0.637708
$835$	4.43769	0.153573
$836$	−4.79837	−0.165955
$837$	−6.70820	−0.231869
$838$	−25.3607	−0.876070
$839$	16.7426	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$839$	16.7426	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$840$	−1.38197	−0.0476824
$841$	38.8328	1.33906
$842$	16.6180	0.572695
$843$	3.70820	0.127717
$844$	6.43769	0.221595
$845$	4.52786	0.155763
$846$	11.7082	0.402536
$847$	6.00000	0.206162
$848$	63.5410	2.18201
$849$	26.2148	0.899689
$850$	50.1246	1.71926
$851$	11.0000	0.377075
$852$	−2.70820	−0.0927815
$853$	−47.1246	−1.61352	−0.806758	−	0.590882i	$-0.798780\pi$
$853$	−47.1246	−1.61352	−0.806758	+	0.590882i	$0.798780\pi$
$854$	−7.85410	−0.268762
$855$	2.14590	0.0733882
$856$	7.43769	0.254215
$857$	39.7082	1.35641	0.678203	−	0.734874i	$-0.262759\pi$
$857$	39.7082	1.35641	0.678203	+	0.734874i	$0.262759\pi$
$858$	−8.61803	−0.294215
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0.618034	0.0210748
$861$	−1.47214	−0.0501703
$862$	−1.09017	−0.0371313
$863$	−8.06888	−0.274668	−0.137334	−	0.990525i	$-0.543853\pi$
$863$	−8.06888	−0.274668	−0.137334	+	0.990525i	$0.543853\pi$
$864$	−3.38197	−0.115057
$865$	2.14590	0.0729627
$866$	−30.9443	−1.05153
$867$	−28.0000	−0.950930
$868$	−4.14590	−0.140721
$869$	−21.1803	−0.718494
$870$	8.23607	0.279229
$871$	12.1246	0.410827
$872$	43.0902	1.45922
$873$	10.4164	0.352542
$874$	−5.61803	−0.190033
$875$	−5.94427	−0.200953
$876$	7.85410	0.265366
$877$	16.5836	0.559988	0.279994	−	0.960002i	$-0.409667\pi$
$877$	16.5836	0.559988	0.279994	+	0.960002i	$0.409667\pi$
$878$	−38.2705	−1.29157
$879$	−12.0000	−0.404750
$880$	6.70820	0.226134
$881$	1.81966	0.0613059	0.0306530	−	0.999530i	$-0.490241\pi$
$881$	1.81966	0.0613059	0.0306530	+	0.999530i	$0.490241\pi$
$882$	−1.61803	−0.0544820
$883$	−3.90983	−0.131576	−0.0657881	−	0.997834i	$-0.520956\pi$
$883$	−3.90983	−0.131576	−0.0657881	+	0.997834i	$0.520956\pi$
$884$	−9.87539	−0.332145
$885$	5.79837	0.194910
$886$	15.4164	0.517924
$887$	9.79837	0.328997	0.164499	−	0.986377i	$-0.447399\pi$
$887$	9.79837	0.328997	0.164499	+	0.986377i	$0.447399\pi$
$888$	24.5967	0.825413
$889$	−22.2705	−0.746929
$890$	11.6180	0.389437
$891$	2.23607	0.0749111
$892$	−11.0344	−0.369460
$893$	25.1246	0.840763
$894$	−31.1246	−1.04096
$895$	−1.14590	−0.0383031
$896$	13.6180	0.454947
$897$	−2.38197	−0.0795315
$898$	29.8328	0.995534
$899$	−55.2492	−1.84266
$900$	−2.85410	−0.0951367
$901$	−87.8115	−2.92543
$902$	5.32624	0.177344
$903$	−1.61803	−0.0538448
$904$	−2.03444	−0.0676645
$905$	1.20163	0.0399434
$906$	−24.6525	−0.819024
$907$	48.9787	1.62631	0.813156	−	0.582046i	$-0.197748\pi$
$907$	48.9787	1.62631	0.813156	+	0.582046i	$0.197748\pi$
$908$	15.0344	0.498935
$909$	4.14590	0.137511
$910$	2.38197	0.0789614
$911$	−7.59675	−0.251691	−0.125846	−	0.992050i	$-0.540164\pi$
$911$	−7.59675	−0.251691	−0.125846	+	0.992050i	$0.540164\pi$
$912$	−16.8541	−0.558095
$913$	−20.5279	−0.679373
$914$	23.1803	0.766737
$915$	−3.00000	−0.0991769
$916$	−10.0902	−0.333389
$917$	−15.1803	−0.501299
$918$	10.8541	0.358239
$919$	−50.1246	−1.65346	−0.826729	−	0.562600i	$-0.809801\pi$
$919$	−50.1246	−1.65346	−0.826729	+	0.562600i	$0.809801\pi$
$920$	1.38197	0.0455621
$921$	−16.1246	−0.531324
$922$	−41.1803	−1.35620
$923$	−10.4377	−0.343561
$924$	1.38197	0.0454633
$925$	50.7984	1.67024
$926$	−27.9787	−0.919438
$927$	−7.41641	−0.243587
$928$	−27.8541	−0.914356
$929$	28.9098	0.948501	0.474250	−	0.880390i	$-0.342719\pi$
$929$	28.9098	0.948501	0.474250	+	0.880390i	$0.342719\pi$
$930$	−6.70820	−0.219971
$931$	−3.47214	−0.113795
$932$	−6.85410	−0.224514
$933$	−15.3262	−0.501759
$934$	24.5623	0.803703
$935$	−9.27051	−0.303178
$936$	−5.32624	−0.174094
$937$	2.70820	0.0884732	0.0442366	−	0.999021i	$-0.485914\pi$
$937$	2.70820	0.0884732	0.0442366	+	0.999021i	$0.485914\pi$
$938$	−8.23607	−0.268917
$939$	2.47214	0.0806751
$940$	2.76393	0.0901495
$941$	−15.0557	−0.490803	−0.245401	−	0.969422i	$-0.578920\pi$
$941$	−15.0557	−0.490803	−0.245401	+	0.969422i	$0.578920\pi$
$942$	−0.472136	−0.0153830
$943$	1.47214	0.0479393
$944$	−45.5410	−1.48223
$945$	−0.618034	−0.0201046
$946$	5.85410	0.190333
$947$	−18.9443	−0.615606	−0.307803	−	0.951450i	$-0.599594\pi$
$947$	−18.9443	−0.615606	−0.307803	+	0.951450i	$0.599594\pi$
$948$	5.85410	0.190132
$949$	30.2705	0.982622
$950$	−25.9443	−0.841743
$951$	−14.5066	−0.470408
$952$	−15.0000	−0.486153
$953$	12.1591	0.393870	0.196935	−	0.980417i	$-0.436901\pi$
$953$	12.1591	0.393870	0.196935	+	0.980417i	$0.436901\pi$
$954$	21.1803	0.685739
$955$	−10.0000	−0.323592
$956$	−2.96556	−0.0959130
$957$	18.4164	0.595318
$958$	−33.9787	−1.09780
$959$	16.4164	0.530113
$960$	2.61803	0.0844967
$961$	14.0000	0.451613
$962$	−42.3951	−1.36687
$963$	3.32624	0.107187
$964$	−6.79837	−0.218961
$965$	−5.12461	−0.164967
$966$	1.61803	0.0520594
$967$	50.3607	1.61949	0.809745	−	0.586782i	$-0.199605\pi$
$967$	50.3607	1.61949	0.809745	+	0.586782i	$0.199605\pi$
$968$	−13.4164	−0.431220
$969$	23.2918	0.748240
$970$	10.4164	0.334451
$971$	−19.6869	−0.631783	−0.315892	−	0.948795i	$-0.602304\pi$
$971$	−19.6869	−0.631783	−0.315892	+	0.948795i	$0.602304\pi$
$972$	−0.618034	−0.0198234
$973$	11.3820	0.364889
$974$	47.3951	1.51864
$975$	−11.0000	−0.352282
$976$	23.5623	0.754211
$977$	49.7984	1.59319	0.796596	−	0.604513i	$-0.206632\pi$
$977$	49.7984	1.59319	0.796596	+	0.604513i	$0.206632\pi$
$978$	−1.00000	−0.0319765
$979$	25.9787	0.830283
$980$	−0.381966	−0.0122015
$981$	19.2705	0.615260
$982$	−15.8541	−0.505925
$983$	−7.40325	−0.236127	−0.118064	−	0.993006i	$-0.537669\pi$
$983$	−7.40325	−0.236127	−0.118064	+	0.993006i	$0.537669\pi$
$984$	3.29180	0.104939
$985$	15.7639	0.502281
$986$	89.3951	2.84692
$987$	−7.23607	−0.230327
$988$	5.11146	0.162617
$989$	1.61803	0.0514505
$990$	2.23607	0.0710669
$991$	−35.5755	−1.13009	−0.565046	−	0.825059i	$-0.691142\pi$
$991$	−35.5755	−1.13009	−0.565046	+	0.825059i	$0.691142\pi$
$992$	22.6869	0.720310
$993$	13.4164	0.425757
$994$	7.09017	0.224887
$995$	4.27051	0.135384
$996$	5.67376	0.179780
$997$	57.7214	1.82805	0.914027	−	0.405654i	$-0.132956\pi$
$997$	57.7214	1.82805	0.914027	+	0.405654i	$0.132956\pi$
$998$	62.1246	1.96652
$999$	11.0000	0.348025

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	483.2.a.d.1.1	✓	2
3.2	odd	2		1449.2.a.h.1.2		2
4.3	odd	2		7728.2.a.bn.1.1		2
7.6	odd	2		3381.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.d.1.1	✓	2	1.1	even	1	trivial
1449.2.a.h.1.2		2	3.2	odd	2
3381.2.a.r.1.1		2	7.6	odd	2
7728.2.a.bn.1.1		2	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	483.2.a.d.1.1

Level	$483$
Weight	$2$
Character	483.1
Self dual	yes
Analytic conductor	$3.857$
Analytic rank	$1$
Dimension	$2$
CM	no
Inner twists	$1$