LMFDB - Embedded newform 255.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [255,2,Mod(1,255)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("255.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(255, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$255 = 3 \cdot 5 \cdot 17$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	255.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.03618525154$
Analytic rank:	$0$
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			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	255.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966 q^{2} -1.00000 q^{3} -1.85410 q^{4} +1.00000 q^{5} -0.381966 q^{6} +2.23607 q^{7} -1.47214 q^{8} +1.00000 q^{9} +0.381966 q^{10} +5.47214 q^{11} +1.85410 q^{12} +5.23607 q^{13} +0.854102 q^{14} -1.00000 q^{15} +3.14590 q^{16} -1.00000 q^{17} +0.381966 q^{18} -8.23607 q^{19} -1.85410 q^{20} -2.23607 q^{21} +2.09017 q^{22} +2.76393 q^{23} +1.47214 q^{24} +1.00000 q^{25} +2.00000 q^{26} -1.00000 q^{27} -4.14590 q^{28} +4.70820 q^{29} -0.381966 q^{30} -2.76393 q^{31} +4.14590 q^{32} -5.47214 q^{33} -0.381966 q^{34} +2.23607 q^{35} -1.85410 q^{36} +0.527864 q^{37} -3.14590 q^{38} -5.23607 q^{39} -1.47214 q^{40} -6.70820 q^{41} -0.854102 q^{42} -10.4721 q^{43} -10.1459 q^{44} +1.00000 q^{45} +1.05573 q^{46} +11.9443 q^{47} -3.14590 q^{48} -2.00000 q^{49} +0.381966 q^{50} +1.00000 q^{51} -9.70820 q^{52} +7.76393 q^{53} -0.381966 q^{54} +5.47214 q^{55} -3.29180 q^{56} +8.23607 q^{57} +1.79837 q^{58} +3.23607 q^{59} +1.85410 q^{60} -9.23607 q^{61} -1.05573 q^{62} +2.23607 q^{63} -4.70820 q^{64} +5.23607 q^{65} -2.09017 q^{66} -0.763932 q^{67} +1.85410 q^{68} -2.76393 q^{69} +0.854102 q^{70} -11.4164 q^{71} -1.47214 q^{72} +13.9443 q^{73} +0.201626 q^{74} -1.00000 q^{75} +15.2705 q^{76} +12.2361 q^{77} -2.00000 q^{78} -13.4164 q^{79} +3.14590 q^{80} +1.00000 q^{81} -2.56231 q^{82} -11.4164 q^{83} +4.14590 q^{84} -1.00000 q^{85} -4.00000 q^{86} -4.70820 q^{87} -8.05573 q^{88} +2.29180 q^{89} +0.381966 q^{90} +11.7082 q^{91} -5.12461 q^{92} +2.76393 q^{93} +4.56231 q^{94} -8.23607 q^{95} -4.14590 q^{96} -3.52786 q^{97} -0.763932 q^{98} +5.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 3 q^{2} - 2 q^{3} + 3 q^{4} + 2 q^{5} - 3 q^{6} + 6 q^{8} + 2 q^{9} + 3 q^{10} + 2 q^{11} - 3 q^{12} + 6 q^{13} - 5 q^{14} - 2 q^{15} + 13 q^{16} - 2 q^{17} + 3 q^{18} - 12 q^{19} + 3 q^{20} - 7 q^{22}+ \cdots + 2 q^{99}+O(q^{100})$ 2 * q + 3 * q^2 - 2 * q^3 + 3 * q^4 + 2 * q^5 - 3 * q^6 + 6 * q^8 + 2 * q^9 + 3 * q^10 + 2 * q^11 - 3 * q^12 + 6 * q^13 - 5 * q^14 - 2 * q^15 + 13 * q^16 - 2 * q^17 + 3 * q^18 - 12 * q^19 + 3 * q^20 - 7 * q^22 + 10 * q^23 - 6 * q^24 + 2 * q^25 + 4 * q^26 - 2 * q^27 - 15 * q^28 - 4 * q^29 - 3 * q^30 - 10 * q^31 + 15 * q^32 - 2 * q^33 - 3 * q^34 + 3 * q^36 + 10 * q^37 - 13 * q^38 - 6 * q^39 + 6 * q^40 + 5 * q^42 - 12 * q^43 - 27 * q^44 + 2 * q^45 + 20 * q^46 + 6 * q^47 - 13 * q^48 - 4 * q^49 + 3 * q^50 + 2 * q^51 - 6 * q^52 + 20 * q^53 - 3 * q^54 + 2 * q^55 - 20 * q^56 + 12 * q^57 - 21 * q^58 + 2 * q^59 - 3 * q^60 - 14 * q^61 - 20 * q^62 + 4 * q^64 + 6 * q^65 + 7 * q^66 - 6 * q^67 - 3 * q^68 - 10 * q^69 - 5 * q^70 + 4 * q^71 + 6 * q^72 + 10 * q^73 + 25 * q^74 - 2 * q^75 - 3 * q^76 + 20 * q^77 - 4 * q^78 + 13 * q^80 + 2 * q^81 + 15 * q^82 + 4 * q^83 + 15 * q^84 - 2 * q^85 - 8 * q^86 + 4 * q^87 - 34 * q^88 + 18 * q^89 + 3 * q^90 + 10 * q^91 + 30 * q^92 + 10 * q^93 - 11 * q^94 - 12 * q^95 - 15 * q^96 - 16 * q^97 - 6 * q^98 + 2 * 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

)

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

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.85410	−0.927051
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.381966	−0.155937
$7$	2.23607	0.845154	0.422577	−	0.906327i	$-0.361126\pi$
$7$	2.23607	0.845154	0.422577	+	0.906327i	$0.361126\pi$
$8$	−1.47214	−0.520479
$9$	1.00000	0.333333
$10$	0.381966	0.120788
$11$	5.47214	1.64991	0.824956	−	0.565198i	$-0.191200\pi$
$11$	5.47214	1.64991	0.824956	+	0.565198i	$0.191200\pi$
$12$	1.85410	0.535233
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	0.854102	0.228268
$15$	−1.00000	−0.258199
$16$	3.14590	0.786475
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0.381966	0.0900303
$19$	−8.23607	−1.88948	−0.944742	−	0.327815i	$-0.893688\pi$
$19$	−8.23607	−1.88948	−0.944742	+	0.327815i	$0.893688\pi$
$20$	−1.85410	−0.414590
$21$	−2.23607	−0.487950
$22$	2.09017	0.445626
$23$	2.76393	0.576320	0.288160	−	0.957582i	$-0.406957\pi$
$23$	2.76393	0.576320	0.288160	+	0.957582i	$0.406957\pi$
$24$	1.47214	0.300498
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	−1.00000	−0.192450
$28$	−4.14590	−0.783501
$29$	4.70820	0.874292	0.437146	−	0.899391i	$-0.355989\pi$
$29$	4.70820	0.874292	0.437146	+	0.899391i	$0.355989\pi$
$30$	−0.381966	−0.0697371
$31$	−2.76393	−0.496417	−0.248208	−	0.968707i	$-0.579842\pi$
$31$	−2.76393	−0.496417	−0.248208	+	0.968707i	$0.579842\pi$
$32$	4.14590	0.732898
$33$	−5.47214	−0.952577
$34$	−0.381966	−0.0655066
$35$	2.23607	0.377964
$36$	−1.85410	−0.309017
$37$	0.527864	0.0867803	0.0433902	−	0.999058i	$-0.486184\pi$
$37$	0.527864	0.0867803	0.0433902	+	0.999058i	$0.486184\pi$
$38$	−3.14590	−0.510332
$39$	−5.23607	−0.838442
$40$	−1.47214	−0.232765
$41$	−6.70820	−1.04765	−0.523823	−	0.851827i	$-0.675495\pi$
$41$	−6.70820	−1.04765	−0.523823	+	0.851827i	$0.675495\pi$
$42$	−0.854102	−0.131791
$43$	−10.4721	−1.59699	−0.798493	−	0.602004i	$-0.794369\pi$
$43$	−10.4721	−1.59699	−0.798493	+	0.602004i	$0.794369\pi$
$44$	−10.1459	−1.52955
$45$	1.00000	0.149071
$46$	1.05573	0.155659
$47$	11.9443	1.74225	0.871126	−	0.491060i	$-0.163391\pi$
$47$	11.9443	1.74225	0.871126	+	0.491060i	$0.163391\pi$
$48$	−3.14590	−0.454071
$49$	−2.00000	−0.285714
$50$	0.381966	0.0540182
$51$	1.00000	0.140028
$52$	−9.70820	−1.34629
$53$	7.76393	1.06646	0.533229	−	0.845971i	$-0.320978\pi$
$53$	7.76393	1.06646	0.533229	+	0.845971i	$0.320978\pi$
$54$	−0.381966	−0.0519790
$55$	5.47214	0.737863
$56$	−3.29180	−0.439885
$57$	8.23607	1.09089
$58$	1.79837	0.236138
$59$	3.23607	0.421300	0.210650	−	0.977562i	$-0.432442\pi$
$59$	3.23607	0.421300	0.210650	+	0.977562i	$0.432442\pi$
$60$	1.85410	0.239364
$61$	−9.23607	−1.18256	−0.591279	−	0.806467i	$-0.701377\pi$
$61$	−9.23607	−1.18256	−0.591279	+	0.806467i	$0.701377\pi$
$62$	−1.05573	−0.134078
$63$	2.23607	0.281718
$64$	−4.70820	−0.588525
$65$	5.23607	0.649454
$66$	−2.09017	−0.257282
$67$	−0.763932	−0.0933292	−0.0466646	−	0.998911i	$-0.514859\pi$
$67$	−0.763932	−0.0933292	−0.0466646	+	0.998911i	$0.514859\pi$
$68$	1.85410	0.224843
$69$	−2.76393	−0.332738
$70$	0.854102	0.102085
$71$	−11.4164	−1.35488	−0.677439	−	0.735579i	$-0.736910\pi$
$71$	−11.4164	−1.35488	−0.677439	+	0.735579i	$0.736910\pi$
$72$	−1.47214	−0.173493
$73$	13.9443	1.63205	0.816027	−	0.578014i	$-0.196172\pi$
$73$	13.9443	1.63205	0.816027	+	0.578014i	$0.196172\pi$
$74$	0.201626	0.0234386
$75$	−1.00000	−0.115470
$76$	15.2705	1.75165
$77$	12.2361	1.39443
$78$	−2.00000	−0.226455
$79$	−13.4164	−1.50946	−0.754732	−	0.656033i	$-0.772233\pi$
$79$	−13.4164	−1.50946	−0.754732	+	0.656033i	$0.772233\pi$
$80$	3.14590	0.351722
$81$	1.00000	0.111111
$82$	−2.56231	−0.282959
$83$	−11.4164	−1.25311	−0.626557	−	0.779376i	$-0.715536\pi$
$83$	−11.4164	−1.25311	−0.626557	+	0.779376i	$0.715536\pi$
$84$	4.14590	0.452355
$85$	−1.00000	−0.108465
$86$	−4.00000	−0.431331
$87$	−4.70820	−0.504772
$88$	−8.05573	−0.858743
$89$	2.29180	0.242930	0.121465	−	0.992596i	$-0.461241\pi$
$89$	2.29180	0.242930	0.121465	+	0.992596i	$0.461241\pi$
$90$	0.381966	0.0402628
$91$	11.7082	1.22735
$92$	−5.12461	−0.534278
$93$	2.76393	0.286606
$94$	4.56231	0.470566
$95$	−8.23607	−0.845003
$96$	−4.14590	−0.423139
$97$	−3.52786	−0.358200	−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786	−0.358200	−0.179100	+	0.983831i	$0.557319\pi$
$98$	−0.763932	−0.0771688
$99$	5.47214	0.549970
$100$	−1.85410	−0.185410
$101$	0.472136	0.0469793	0.0234896	−	0.999724i	$-0.492522\pi$
$101$	0.472136	0.0469793	0.0234896	+	0.999724i	$0.492522\pi$
$102$	0.381966	0.0378203
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	−7.70820	−0.755852
$105$	−2.23607	−0.218218
$106$	2.96556	0.288040
$107$	2.47214	0.238990	0.119495	−	0.992835i	$-0.461872\pi$
$107$	2.47214	0.238990	0.119495	+	0.992835i	$0.461872\pi$
$108$	1.85410	0.178411
$109$	8.47214	0.811483	0.405742	−	0.913988i	$-0.367013\pi$
$109$	8.47214	0.811483	0.405742	+	0.913988i	$0.367013\pi$
$110$	2.09017	0.199290
$111$	−0.527864	−0.0501026
$112$	7.03444	0.664692
$113$	0.291796	0.0274499	0.0137249	−	0.999906i	$-0.495631\pi$
$113$	0.291796	0.0274499	0.0137249	+	0.999906i	$0.495631\pi$
$114$	3.14590	0.294640
$115$	2.76393	0.257738
$116$	−8.72949	−0.810513
$117$	5.23607	0.484075
$118$	1.23607	0.113789
$119$	−2.23607	−0.204980
$120$	1.47214	0.134387
$121$	18.9443	1.72221
$122$	−3.52786	−0.319398
$123$	6.70820	0.604858
$124$	5.12461	0.460204
$125$	1.00000	0.0894427
$126$	0.854102	0.0760895
$127$	−5.23607	−0.464626	−0.232313	−	0.972641i	$-0.574629\pi$
$127$	−5.23607	−0.464626	−0.232313	+	0.972641i	$0.574629\pi$
$128$	−10.0902	−0.891853
$129$	10.4721	0.922020
$130$	2.00000	0.175412
$131$	−15.4164	−1.34694	−0.673469	−	0.739216i	$-0.735196\pi$
$131$	−15.4164	−1.34694	−0.673469	+	0.739216i	$0.735196\pi$
$132$	10.1459	0.883087
$133$	−18.4164	−1.59691
$134$	−0.291796	−0.0252073
$135$	−1.00000	−0.0860663
$136$	1.47214	0.126235
$137$	1.76393	0.150703	0.0753514	−	0.997157i	$-0.475992\pi$
$137$	1.76393	0.150703	0.0753514	+	0.997157i	$0.475992\pi$
$138$	−1.05573	−0.0898695
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	−4.14590	−0.350392
$141$	−11.9443	−1.00589
$142$	−4.36068	−0.365940
$143$	28.6525	2.39604
$144$	3.14590	0.262158
$145$	4.70820	0.390995
$146$	5.32624	0.440803
$147$	2.00000	0.164957
$148$	−0.978714	−0.0804498
$149$	18.1803	1.48939	0.744696	−	0.667404i	$-0.232595\pi$
$149$	18.1803	1.48939	0.744696	+	0.667404i	$0.232595\pi$
$150$	−0.381966	−0.0311874
$151$	3.18034	0.258812	0.129406	−	0.991592i	$-0.458693\pi$
$151$	3.18034	0.258812	0.129406	+	0.991592i	$0.458693\pi$
$152$	12.1246	0.983436
$153$	−1.00000	−0.0808452
$154$	4.67376	0.376622
$155$	−2.76393	−0.222004
$156$	9.70820	0.777278
$157$	4.00000	0.319235	0.159617	−	0.987179i	$-0.448974\pi$
$157$	4.00000	0.319235	0.159617	+	0.987179i	$0.448974\pi$
$158$	−5.12461	−0.407692
$159$	−7.76393	−0.615720
$160$	4.14590	0.327762
$161$	6.18034	0.487079
$162$	0.381966	0.0300101
$163$	−15.1803	−1.18902	−0.594508	−	0.804090i	$-0.702653\pi$
$163$	−15.1803	−1.18902	−0.594508	+	0.804090i	$0.702653\pi$
$164$	12.4377	0.971221
$165$	−5.47214	−0.426005
$166$	−4.36068	−0.338454
$167$	−18.9443	−1.46595	−0.732976	−	0.680255i	$-0.761869\pi$
$167$	−18.9443	−1.46595	−0.732976	+	0.680255i	$0.761869\pi$
$168$	3.29180	0.253968
$169$	14.4164	1.10895
$170$	−0.381966	−0.0292955
$171$	−8.23607	−0.629828
$172$	19.4164	1.48049
$173$	−0.763932	−0.0580807	−0.0290403	−	0.999578i	$-0.509245\pi$
$173$	−0.763932	−0.0580807	−0.0290403	+	0.999578i	$0.509245\pi$
$174$	−1.79837	−0.136334
$175$	2.23607	0.169031
$176$	17.2148	1.29761
$177$	−3.23607	−0.243238
$178$	0.875388	0.0656131
$179$	−10.0000	−0.747435	−0.373718	−	0.927543i	$-0.621917\pi$
$179$	−10.0000	−0.747435	−0.373718	+	0.927543i	$0.621917\pi$
$180$	−1.85410	−0.138197
$181$	−11.7082	−0.870264	−0.435132	−	0.900367i	$-0.643298\pi$
$181$	−11.7082	−0.870264	−0.435132	+	0.900367i	$0.643298\pi$
$182$	4.47214	0.331497
$183$	9.23607	0.682750
$184$	−4.06888	−0.299962
$185$	0.527864	0.0388093
$186$	1.05573	0.0774097
$187$	−5.47214	−0.400162
$188$	−22.1459	−1.61516
$189$	−2.23607	−0.162650
$190$	−3.14590	−0.228227
$191$	−2.29180	−0.165829	−0.0829143	−	0.996557i	$-0.526423\pi$
$191$	−2.29180	−0.165829	−0.0829143	+	0.996557i	$0.526423\pi$
$192$	4.70820	0.339785
$193$	−6.94427	−0.499860	−0.249930	−	0.968264i	$-0.580408\pi$
$193$	−6.94427	−0.499860	−0.249930	+	0.968264i	$0.580408\pi$
$194$	−1.34752	−0.0967466
$195$	−5.23607	−0.374963
$196$	3.70820	0.264872
$197$	−9.52786	−0.678832	−0.339416	−	0.940636i	$-0.610229\pi$
$197$	−9.52786	−0.678832	−0.339416	+	0.940636i	$0.610229\pi$
$198$	2.09017	0.148542
$199$	19.8885	1.40986	0.704931	−	0.709276i	$-0.250978\pi$
$199$	19.8885	1.40986	0.704931	+	0.709276i	$0.250978\pi$
$200$	−1.47214	−0.104096
$201$	0.763932	0.0538836
$202$	0.180340	0.0126887
$203$	10.5279	0.738911
$204$	−1.85410	−0.129813
$205$	−6.70820	−0.468521
$206$	1.52786	0.106451
$207$	2.76393	0.192107
$208$	16.4721	1.14214
$209$	−45.0689	−3.11748
$210$	−0.854102	−0.0589386
$211$	23.4164	1.61205	0.806026	−	0.591880i	$-0.201614\pi$
$211$	23.4164	1.61205	0.806026	+	0.591880i	$0.201614\pi$
$212$	−14.3951	−0.988661
$213$	11.4164	0.782239
$214$	0.944272	0.0645491
$215$	−10.4721	−0.714194
$216$	1.47214	0.100166
$217$	−6.18034	−0.419549
$218$	3.23607	0.219174
$219$	−13.9443	−0.942267
$220$	−10.1459	−0.684036
$221$	−5.23607	−0.352216
$222$	−0.201626	−0.0135323
$223$	6.94427	0.465023	0.232511	−	0.972594i	$-0.425306\pi$
$223$	6.94427	0.465023	0.232511	+	0.972594i	$0.425306\pi$
$224$	9.27051	0.619412
$225$	1.00000	0.0666667
$226$	0.111456	0.00741395
$227$	−11.7082	−0.777101	−0.388550	−	0.921427i	$-0.627024\pi$
$227$	−11.7082	−0.777101	−0.388550	+	0.921427i	$0.627024\pi$
$228$	−15.2705	−1.01131
$229$	−12.4164	−0.820499	−0.410250	−	0.911973i	$-0.634558\pi$
$229$	−12.4164	−0.820499	−0.410250	+	0.911973i	$0.634558\pi$
$230$	1.05573	0.0696126
$231$	−12.2361	−0.805074
$232$	−6.93112	−0.455050
$233$	4.65248	0.304794	0.152397	−	0.988319i	$-0.451301\pi$
$233$	4.65248	0.304794	0.152397	+	0.988319i	$0.451301\pi$
$234$	2.00000	0.130744
$235$	11.9443	0.779158
$236$	−6.00000	−0.390567
$237$	13.4164	0.871489
$238$	−0.854102	−0.0553632
$239$	6.76393	0.437522	0.218761	−	0.975778i	$-0.429798\pi$
$239$	6.76393	0.437522	0.218761	+	0.975778i	$0.429798\pi$
$240$	−3.14590	−0.203067
$241$	−19.1246	−1.23192	−0.615962	−	0.787776i	$-0.711233\pi$
$241$	−19.1246	−1.23192	−0.615962	+	0.787776i	$0.711233\pi$
$242$	7.23607	0.465152
$243$	−1.00000	−0.0641500
$244$	17.1246	1.09629
$245$	−2.00000	−0.127775
$246$	2.56231	0.163367
$247$	−43.1246	−2.74395
$248$	4.06888	0.258374
$249$	11.4164	0.723485
$250$	0.381966	0.0241577
$251$	−18.6525	−1.17733	−0.588667	−	0.808376i	$-0.700347\pi$
$251$	−18.6525	−1.17733	−0.588667	+	0.808376i	$0.700347\pi$
$252$	−4.14590	−0.261167
$253$	15.1246	0.950876
$254$	−2.00000	−0.125491
$255$	1.00000	0.0626224
$256$	5.56231	0.347644
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	4.00000	0.249029
$259$	1.18034	0.0733428
$260$	−9.70820	−0.602077
$261$	4.70820	0.291431
$262$	−5.88854	−0.363795
$263$	24.4164	1.50558	0.752790	−	0.658261i	$-0.228708\pi$
$263$	24.4164	1.50558	0.752790	+	0.658261i	$0.228708\pi$
$264$	8.05573	0.495796
$265$	7.76393	0.476935
$266$	−7.03444	−0.431309
$267$	−2.29180	−0.140256
$268$	1.41641	0.0865209
$269$	−9.29180	−0.566531	−0.283265	−	0.959042i	$-0.591418\pi$
$269$	−9.29180	−0.566531	−0.283265	+	0.959042i	$0.591418\pi$
$270$	−0.381966	−0.0232457
$271$	1.52786	0.0928111	0.0464056	−	0.998923i	$-0.485223\pi$
$271$	1.52786	0.0928111	0.0464056	+	0.998923i	$0.485223\pi$
$272$	−3.14590	−0.190748
$273$	−11.7082	−0.708613
$274$	0.673762	0.0407035
$275$	5.47214	0.329982
$276$	5.12461	0.308465
$277$	−17.4164	−1.04645	−0.523225	−	0.852194i	$-0.675271\pi$
$277$	−17.4164	−1.04645	−0.523225	+	0.852194i	$0.675271\pi$
$278$	−5.34752	−0.320723
$279$	−2.76393	−0.165472
$280$	−3.29180	−0.196722
$281$	21.8885	1.30576	0.652881	−	0.757461i	$-0.273560\pi$
$281$	21.8885	1.30576	0.652881	+	0.757461i	$0.273560\pi$
$282$	−4.56231	−0.271681
$283$	4.70820	0.279874	0.139937	−	0.990160i	$-0.455310\pi$
$283$	4.70820	0.279874	0.139937	+	0.990160i	$0.455310\pi$
$284$	21.1672	1.25604
$285$	8.23607	0.487863
$286$	10.9443	0.647148
$287$	−15.0000	−0.885422
$288$	4.14590	0.244299
$289$	1.00000	0.0588235
$290$	1.79837	0.105604
$291$	3.52786	0.206807
$292$	−25.8541	−1.51300
$293$	−16.2361	−0.948521	−0.474261	−	0.880385i	$-0.657285\pi$
$293$	−16.2361	−0.948521	−0.474261	+	0.880385i	$0.657285\pi$
$294$	0.763932	0.0445534
$295$	3.23607	0.188411
$296$	−0.777088	−0.0451673
$297$	−5.47214	−0.317526
$298$	6.94427	0.402271
$299$	14.4721	0.836945
$300$	1.85410	0.107047
$301$	−23.4164	−1.34970
$302$	1.21478	0.0699028
$303$	−0.472136	−0.0271235
$304$	−25.9098	−1.48603
$305$	−9.23607	−0.528856
$306$	−0.381966	−0.0218355
$307$	−32.0000	−1.82634	−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000	−1.82634	−0.913168	+	0.407583i	$0.866372\pi$
$308$	−22.6869	−1.29271
$309$	−4.00000	−0.227552
$310$	−1.05573	−0.0599613
$311$	14.4164	0.817479	0.408740	−	0.912651i	$-0.365968\pi$
$311$	14.4164	0.817479	0.408740	+	0.912651i	$0.365968\pi$
$312$	7.70820	0.436391
$313$	6.52786	0.368977	0.184488	−	0.982835i	$-0.440937\pi$
$313$	6.52786	0.368977	0.184488	+	0.982835i	$0.440937\pi$
$314$	1.52786	0.0862224
$315$	2.23607	0.125988
$316$	24.8754	1.39935
$317$	−3.70820	−0.208273	−0.104137	−	0.994563i	$-0.533208\pi$
$317$	−3.70820	−0.208273	−0.104137	+	0.994563i	$0.533208\pi$
$318$	−2.96556	−0.166300
$319$	25.7639	1.44250
$320$	−4.70820	−0.263197
$321$	−2.47214	−0.137981
$322$	2.36068	0.131556
$323$	8.23607	0.458267
$324$	−1.85410	−0.103006
$325$	5.23607	0.290445
$326$	−5.79837	−0.321142
$327$	−8.47214	−0.468510
$328$	9.87539	0.545277
$329$	26.7082	1.47247
$330$	−2.09017	−0.115060
$331$	−14.7082	−0.808436	−0.404218	−	0.914663i	$-0.632456\pi$
$331$	−14.7082	−0.808436	−0.404218	+	0.914663i	$0.632456\pi$
$332$	21.1672	1.16170
$333$	0.527864	0.0289268
$334$	−7.23607	−0.395940
$335$	−0.763932	−0.0417381
$336$	−7.03444	−0.383760
$337$	15.4721	0.842821	0.421410	−	0.906870i	$-0.361535\pi$
$337$	15.4721	0.842821	0.421410	+	0.906870i	$0.361535\pi$
$338$	5.50658	0.299518
$339$	−0.291796	−0.0158482
$340$	1.85410	0.100553
$341$	−15.1246	−0.819044
$342$	−3.14590	−0.170111
$343$	−20.1246	−1.08663
$344$	15.4164	0.831197
$345$	−2.76393	−0.148805
$346$	−0.291796	−0.0156871
$347$	22.6525	1.21605	0.608024	−	0.793918i	$-0.291962\pi$
$347$	22.6525	1.21605	0.608024	+	0.793918i	$0.291962\pi$
$348$	8.72949	0.467950
$349$	−11.4721	−0.614089	−0.307045	−	0.951695i	$-0.599340\pi$
$349$	−11.4721	−0.614089	−0.307045	+	0.951695i	$0.599340\pi$
$350$	0.854102	0.0456537
$351$	−5.23607	−0.279481
$352$	22.6869	1.20922
$353$	2.34752	0.124946	0.0624731	−	0.998047i	$-0.480101\pi$
$353$	2.34752	0.124946	0.0624731	+	0.998047i	$0.480101\pi$
$354$	−1.23607	−0.0656963
$355$	−11.4164	−0.605920
$356$	−4.24922	−0.225208
$357$	2.23607	0.118345
$358$	−3.81966	−0.201875
$359$	23.2361	1.22635	0.613176	−	0.789946i	$-0.289891\pi$
$359$	23.2361	1.22635	0.613176	+	0.789946i	$0.289891\pi$
$360$	−1.47214	−0.0775884
$361$	48.8328	2.57015
$362$	−4.47214	−0.235050
$363$	−18.9443	−0.994316
$364$	−21.7082	−1.13782
$365$	13.9443	0.729877
$366$	3.52786	0.184404
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	8.69505	0.453261
$369$	−6.70820	−0.349215
$370$	0.201626	0.0104820
$371$	17.3607	0.901322
$372$	−5.12461	−0.265699
$373$	12.9443	0.670229	0.335114	−	0.942177i	$-0.391225\pi$
$373$	12.9443	0.670229	0.335114	+	0.942177i	$0.391225\pi$
$374$	−2.09017	−0.108080
$375$	−1.00000	−0.0516398
$376$	−17.5836	−0.906805
$377$	24.6525	1.26967
$378$	−0.854102	−0.0439303
$379$	8.94427	0.459436	0.229718	−	0.973257i	$-0.426220\pi$
$379$	8.94427	0.459436	0.229718	+	0.973257i	$0.426220\pi$
$380$	15.2705	0.783361
$381$	5.23607	0.268252
$382$	−0.875388	−0.0447888
$383$	24.4164	1.24762	0.623810	−	0.781576i	$-0.285584\pi$
$383$	24.4164	1.24762	0.623810	+	0.781576i	$0.285584\pi$
$384$	10.0902	0.514912
$385$	12.2361	0.623608
$386$	−2.65248	−0.135007
$387$	−10.4721	−0.532329
$388$	6.54102	0.332070
$389$	−15.0557	−0.763356	−0.381678	−	0.924295i	$-0.624654\pi$
$389$	−15.0557	−0.763356	−0.381678	+	0.924295i	$0.624654\pi$
$390$	−2.00000	−0.101274
$391$	−2.76393	−0.139778
$392$	2.94427	0.148708
$393$	15.4164	0.777655
$394$	−3.63932	−0.183346
$395$	−13.4164	−0.675053
$396$	−10.1459	−0.509851
$397$	3.00000	0.150566	0.0752828	−	0.997162i	$-0.476014\pi$
$397$	3.00000	0.150566	0.0752828	+	0.997162i	$0.476014\pi$
$398$	7.59675	0.380791
$399$	18.4164	0.921974
$400$	3.14590	0.157295
$401$	−8.70820	−0.434867	−0.217433	−	0.976075i	$-0.569769\pi$
$401$	−8.70820	−0.434867	−0.217433	+	0.976075i	$0.569769\pi$
$402$	0.291796	0.0145535
$403$	−14.4721	−0.720908
$404$	−0.875388	−0.0435522
$405$	1.00000	0.0496904
$406$	4.02129	0.199573
$407$	2.88854	0.143180
$408$	−1.47214	−0.0728816
$409$	29.9443	1.48065	0.740324	−	0.672250i	$-0.234672\pi$
$409$	29.9443	1.48065	0.740324	+	0.672250i	$0.234672\pi$
$410$	−2.56231	−0.126543
$411$	−1.76393	−0.0870084
$412$	−7.41641	−0.365380
$413$	7.23607	0.356064
$414$	1.05573	0.0518862
$415$	−11.4164	−0.560409
$416$	21.7082	1.06433
$417$	14.0000	0.685583
$418$	−17.2148	−0.842002
$419$	29.0000	1.41674	0.708371	−	0.705840i	$-0.249430\pi$
$419$	29.0000	1.41674	0.708371	+	0.705840i	$0.249430\pi$
$420$	4.14590	0.202299
$421$	−7.58359	−0.369602	−0.184801	−	0.982776i	$-0.559164\pi$
$421$	−7.58359	−0.369602	−0.184801	+	0.982776i	$0.559164\pi$
$422$	8.94427	0.435400
$423$	11.9443	0.580750
$424$	−11.4296	−0.555069
$425$	−1.00000	−0.0485071
$426$	4.36068	0.211276
$427$	−20.6525	−0.999443
$428$	−4.58359	−0.221556
$429$	−28.6525	−1.38335
$430$	−4.00000	−0.192897
$431$	−7.36068	−0.354551	−0.177276	−	0.984161i	$-0.556728\pi$
$431$	−7.36068	−0.354551	−0.177276	+	0.984161i	$0.556728\pi$
$432$	−3.14590	−0.151357
$433$	18.3607	0.882358	0.441179	−	0.897419i	$-0.354560\pi$
$433$	18.3607	0.882358	0.441179	+	0.897419i	$0.354560\pi$
$434$	−2.36068	−0.113316
$435$	−4.70820	−0.225741
$436$	−15.7082	−0.752287
$437$	−22.7639	−1.08895
$438$	−5.32624	−0.254497
$439$	22.7639	1.08646	0.543232	−	0.839583i	$-0.317201\pi$
$439$	22.7639	1.08646	0.543232	+	0.839583i	$0.317201\pi$
$440$	−8.05573	−0.384042
$441$	−2.00000	−0.0952381
$442$	−2.00000	−0.0951303
$443$	−0.944272	−0.0448637	−0.0224319	−	0.999748i	$-0.507141\pi$
$443$	−0.944272	−0.0448637	−0.0224319	+	0.999748i	$0.507141\pi$
$444$	0.978714	0.0464477
$445$	2.29180	0.108642
$446$	2.65248	0.125598
$447$	−18.1803	−0.859901
$448$	−10.5279	−0.497395
$449$	−12.4721	−0.588596	−0.294298	−	0.955714i	$-0.595086\pi$
$449$	−12.4721	−0.588596	−0.294298	+	0.955714i	$0.595086\pi$
$450$	0.381966	0.0180061
$451$	−36.7082	−1.72852
$452$	−0.541020	−0.0254474
$453$	−3.18034	−0.149425
$454$	−4.47214	−0.209888
$455$	11.7082	0.548889
$456$	−12.1246	−0.567787
$457$	12.7639	0.597072	0.298536	−	0.954398i	$-0.403502\pi$
$457$	12.7639	0.597072	0.298536	+	0.954398i	$0.403502\pi$
$458$	−4.74265	−0.221609
$459$	1.00000	0.0466760
$460$	−5.12461	−0.238936
$461$	22.6525	1.05503	0.527515	−	0.849545i	$-0.323124\pi$
$461$	22.6525	1.05503	0.527515	+	0.849545i	$0.323124\pi$
$462$	−4.67376	−0.217443
$463$	−18.3607	−0.853293	−0.426647	−	0.904418i	$-0.640305\pi$
$463$	−18.3607	−0.853293	−0.426647	+	0.904418i	$0.640305\pi$
$464$	14.8115	0.687608
$465$	2.76393	0.128174
$466$	1.77709	0.0823220
$467$	1.94427	0.0899702	0.0449851	−	0.998988i	$-0.485676\pi$
$467$	1.94427	0.0899702	0.0449851	+	0.998988i	$0.485676\pi$
$468$	−9.70820	−0.448762
$469$	−1.70820	−0.0788775
$470$	4.56231	0.210443
$471$	−4.00000	−0.184310
$472$	−4.76393	−0.219278
$473$	−57.3050	−2.63488
$474$	5.12461	0.235381
$475$	−8.23607	−0.377897
$476$	4.14590	0.190027
$477$	7.76393	0.355486
$478$	2.58359	0.118171
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−4.14590	−0.189233
$481$	2.76393	0.126024
$482$	−7.30495	−0.332731
$483$	−6.18034	−0.281215
$484$	−35.1246	−1.59657
$485$	−3.52786	−0.160192
$486$	−0.381966	−0.0173263
$487$	12.0000	0.543772	0.271886	−	0.962329i	$-0.412353\pi$
$487$	12.0000	0.543772	0.271886	+	0.962329i	$0.412353\pi$
$488$	13.5967	0.615496
$489$	15.1803	0.686479
$490$	−0.763932	−0.0345109
$491$	35.8885	1.61963	0.809814	−	0.586687i	$-0.199568\pi$
$491$	35.8885	1.61963	0.809814	+	0.586687i	$0.199568\pi$
$492$	−12.4377	−0.560735
$493$	−4.70820	−0.212047
$494$	−16.4721	−0.741116
$495$	5.47214	0.245954
$496$	−8.69505	−0.390419
$497$	−25.5279	−1.14508
$498$	4.36068	0.195407
$499$	20.1803	0.903396	0.451698	−	0.892171i	$-0.350818\pi$
$499$	20.1803	0.903396	0.451698	+	0.892171i	$0.350818\pi$
$500$	−1.85410	−0.0829180
$501$	18.9443	0.846368
$502$	−7.12461	−0.317987
$503$	1.41641	0.0631545	0.0315773	−	0.999501i	$-0.489947\pi$
$503$	1.41641	0.0631545	0.0315773	+	0.999501i	$0.489947\pi$
$504$	−3.29180	−0.146628
$505$	0.472136	0.0210098
$506$	5.77709	0.256823
$507$	−14.4164	−0.640255
$508$	9.70820	0.430732
$509$	−37.0132	−1.64058	−0.820290	−	0.571948i	$-0.806188\pi$
$509$	−37.0132	−1.64058	−0.820290	+	0.571948i	$0.806188\pi$
$510$	0.381966	0.0169137
$511$	31.1803	1.37934
$512$	22.3050	0.985749
$513$	8.23607	0.363631
$514$	−0.763932	−0.0336956
$515$	4.00000	0.176261
$516$	−19.4164	−0.854760
$517$	65.3607	2.87456
$518$	0.450850	0.0198092
$519$	0.763932	0.0335329
$520$	−7.70820	−0.338027
$521$	12.2361	0.536072	0.268036	−	0.963409i	$-0.413625\pi$
$521$	12.2361	0.536072	0.268036	+	0.963409i	$0.413625\pi$
$522$	1.79837	0.0787127
$523$	−14.9443	−0.653467	−0.326734	−	0.945116i	$-0.605948\pi$
$523$	−14.9443	−0.653467	−0.326734	+	0.945116i	$0.605948\pi$
$524$	28.5836	1.24868
$525$	−2.23607	−0.0975900
$526$	9.32624	0.406643
$527$	2.76393	0.120399
$528$	−17.2148	−0.749177
$529$	−15.3607	−0.667856
$530$	2.96556	0.128816
$531$	3.23607	0.140433
$532$	34.1459	1.48041
$533$	−35.1246	−1.52142
$534$	−0.875388	−0.0378818
$535$	2.47214	0.106880
$536$	1.12461	0.0485758
$537$	10.0000	0.431532
$538$	−3.54915	−0.153015
$539$	−10.9443	−0.471403
$540$	1.85410	0.0797878
$541$	−13.8197	−0.594154	−0.297077	−	0.954854i	$-0.596012\pi$
$541$	−13.8197	−0.594154	−0.297077	+	0.954854i	$0.596012\pi$
$542$	0.583592	0.0250674
$543$	11.7082	0.502447
$544$	−4.14590	−0.177754
$545$	8.47214	0.362906
$546$	−4.47214	−0.191390
$547$	16.7082	0.714391	0.357196	−	0.934030i	$-0.383733\pi$
$547$	16.7082	0.714391	0.357196	+	0.934030i	$0.383733\pi$
$548$	−3.27051	−0.139709
$549$	−9.23607	−0.394186
$550$	2.09017	0.0891251
$551$	−38.7771	−1.65196
$552$	4.06888	0.173183
$553$	−30.0000	−1.27573
$554$	−6.65248	−0.282637
$555$	−0.527864	−0.0224066
$556$	25.9574	1.10084
$557$	16.4721	0.697947	0.348973	−	0.937133i	$-0.386530\pi$
$557$	16.4721	0.697947	0.348973	+	0.937133i	$0.386530\pi$
$558$	−1.05573	−0.0446925
$559$	−54.8328	−2.31918
$560$	7.03444	0.297259
$561$	5.47214	0.231034
$562$	8.36068	0.352674
$563$	−19.8328	−0.835854	−0.417927	−	0.908481i	$-0.637243\pi$
$563$	−19.8328	−0.835854	−0.417927	+	0.908481i	$0.637243\pi$
$564$	22.1459	0.932511
$565$	0.291796	0.0122760
$566$	1.79837	0.0755913
$567$	2.23607	0.0939060
$568$	16.8065	0.705185
$569$	−39.4164	−1.65242	−0.826211	−	0.563361i	$-0.809508\pi$
$569$	−39.4164	−1.65242	−0.826211	+	0.563361i	$0.809508\pi$
$570$	3.14590	0.131767
$571$	−15.7082	−0.657368	−0.328684	−	0.944440i	$-0.606605\pi$
$571$	−15.7082	−0.657368	−0.328684	+	0.944440i	$0.606605\pi$
$572$	−53.1246	−2.22125
$573$	2.29180	0.0957412
$574$	−5.72949	−0.239144
$575$	2.76393	0.115264
$576$	−4.70820	−0.196175
$577$	−14.8328	−0.617498	−0.308749	−	0.951144i	$-0.599910\pi$
$577$	−14.8328	−0.617498	−0.308749	+	0.951144i	$0.599910\pi$
$578$	0.381966	0.0158877
$579$	6.94427	0.288594
$580$	−8.72949	−0.362472
$581$	−25.5279	−1.05907
$582$	1.34752	0.0558567
$583$	42.4853	1.75956
$584$	−20.5279	−0.849449
$585$	5.23607	0.216485
$586$	−6.20163	−0.256187
$587$	−1.47214	−0.0607616	−0.0303808	−	0.999538i	$-0.509672\pi$
$587$	−1.47214	−0.0607616	−0.0303808	+	0.999538i	$0.509672\pi$
$588$	−3.70820	−0.152924
$589$	22.7639	0.937971
$590$	1.23607	0.0508881
$591$	9.52786	0.391924
$592$	1.66061	0.0682505
$593$	45.1803	1.85533	0.927667	−	0.373408i	$-0.121811\pi$
$593$	45.1803	1.85533	0.927667	+	0.373408i	$0.121811\pi$
$594$	−2.09017	−0.0857607
$595$	−2.23607	−0.0916698
$596$	−33.7082	−1.38074
$597$	−19.8885	−0.813984
$598$	5.52786	0.226051
$599$	−21.4164	−0.875051	−0.437525	−	0.899206i	$-0.644145\pi$
$599$	−21.4164	−0.875051	−0.437525	+	0.899206i	$0.644145\pi$
$600$	1.47214	0.0600997
$601$	−33.5967	−1.37044	−0.685220	−	0.728336i	$-0.740294\pi$
$601$	−33.5967	−1.37044	−0.685220	+	0.728336i	$0.740294\pi$
$602$	−8.94427	−0.364541
$603$	−0.763932	−0.0311097
$604$	−5.89667	−0.239932
$605$	18.9443	0.770194
$606$	−0.180340	−0.00732581
$607$	3.18034	0.129086	0.0645430	−	0.997915i	$-0.479441\pi$
$607$	3.18034	0.129086	0.0645430	+	0.997915i	$0.479441\pi$
$608$	−34.1459	−1.38480
$609$	−10.5279	−0.426611
$610$	−3.52786	−0.142839
$611$	62.5410	2.53014
$612$	1.85410	0.0749476
$613$	−4.58359	−0.185130	−0.0925648	−	0.995707i	$-0.529507\pi$
$613$	−4.58359	−0.185130	−0.0925648	+	0.995707i	$0.529507\pi$
$614$	−12.2229	−0.493277
$615$	6.70820	0.270501
$616$	−18.0132	−0.725771
$617$	35.7082	1.43756	0.718779	−	0.695239i	$-0.244701\pi$
$617$	35.7082	1.43756	0.718779	+	0.695239i	$0.244701\pi$
$618$	−1.52786	−0.0614597
$619$	−26.3607	−1.05953	−0.529763	−	0.848146i	$-0.677719\pi$
$619$	−26.3607	−1.05953	−0.529763	+	0.848146i	$0.677719\pi$
$620$	5.12461	0.205809
$621$	−2.76393	−0.110913
$622$	5.50658	0.220794
$623$	5.12461	0.205313
$624$	−16.4721	−0.659413
$625$	1.00000	0.0400000
$626$	2.49342	0.0996572
$627$	45.0689	1.79988
$628$	−7.41641	−0.295947
$629$	−0.527864	−0.0210473
$630$	0.854102	0.0340282
$631$	1.87539	0.0746580	0.0373290	−	0.999303i	$-0.488115\pi$
$631$	1.87539	0.0746580	0.0373290	+	0.999303i	$0.488115\pi$
$632$	19.7508	0.785644
$633$	−23.4164	−0.930719
$634$	−1.41641	−0.0562527
$635$	−5.23607	−0.207787
$636$	14.3951	0.570804
$637$	−10.4721	−0.414921
$638$	9.84095	0.389607
$639$	−11.4164	−0.451626
$640$	−10.0902	−0.398849
$641$	3.29180	0.130018	0.0650091	−	0.997885i	$-0.479292\pi$
$641$	3.29180	0.130018	0.0650091	+	0.997885i	$0.479292\pi$
$642$	−0.944272	−0.0372674
$643$	49.6525	1.95810	0.979051	−	0.203614i	$-0.0652687\pi$
$643$	49.6525	1.95810	0.979051	+	0.203614i	$0.0652687\pi$
$644$	−11.4590	−0.451547
$645$	10.4721	0.412340
$646$	3.14590	0.123774
$647$	8.58359	0.337456	0.168728	−	0.985663i	$-0.446034\pi$
$647$	8.58359	0.337456	0.168728	+	0.985663i	$0.446034\pi$
$648$	−1.47214	−0.0578310
$649$	17.7082	0.695108
$650$	2.00000	0.0784465
$651$	6.18034	0.242227
$652$	28.1459	1.10228
$653$	−10.5836	−0.414168	−0.207084	−	0.978323i	$-0.566397\pi$
$653$	−10.5836	−0.414168	−0.207084	+	0.978323i	$0.566397\pi$
$654$	−3.23607	−0.126540
$655$	−15.4164	−0.602369
$656$	−21.1033	−0.823946
$657$	13.9443	0.544018
$658$	10.2016	0.397701
$659$	18.9443	0.737964	0.368982	−	0.929437i	$-0.379706\pi$
$659$	18.9443	0.737964	0.368982	+	0.929437i	$0.379706\pi$
$660$	10.1459	0.394929
$661$	−21.9443	−0.853533	−0.426767	−	0.904362i	$-0.640347\pi$
$661$	−21.9443	−0.853533	−0.426767	+	0.904362i	$0.640347\pi$
$662$	−5.61803	−0.218351
$663$	5.23607	0.203352
$664$	16.8065	0.652219
$665$	−18.4164	−0.714158
$666$	0.201626	0.00781285
$667$	13.0132	0.503871
$668$	35.1246	1.35901
$669$	−6.94427	−0.268481
$670$	−0.291796	−0.0112731
$671$	−50.5410	−1.95111
$672$	−9.27051	−0.357618
$673$	10.0000	0.385472	0.192736	−	0.981251i	$-0.438264\pi$
$673$	10.0000	0.385472	0.192736	+	0.981251i	$0.438264\pi$
$674$	5.90983	0.227638
$675$	−1.00000	−0.0384900
$676$	−26.7295	−1.02806
$677$	−34.0000	−1.30673	−0.653363	−	0.757045i	$-0.726642\pi$
$677$	−34.0000	−1.30673	−0.653363	+	0.757045i	$0.726642\pi$
$678$	−0.111456	−0.00428045
$679$	−7.88854	−0.302735
$680$	1.47214	0.0564538
$681$	11.7082	0.448659
$682$	−5.77709	−0.221216
$683$	−15.0557	−0.576091	−0.288046	−	0.957617i	$-0.593006\pi$
$683$	−15.0557	−0.576091	−0.288046	+	0.957617i	$0.593006\pi$
$684$	15.2705	0.583883
$685$	1.76393	0.0673964
$686$	−7.68692	−0.293488
$687$	12.4164	0.473716
$688$	−32.9443	−1.25599
$689$	40.6525	1.54874
$690$	−1.05573	−0.0401909
$691$	21.5279	0.818959	0.409479	−	0.912319i	$-0.365710\pi$
$691$	21.5279	0.818959	0.409479	+	0.912319i	$0.365710\pi$
$692$	1.41641	0.0538437
$693$	12.2361	0.464810
$694$	8.65248	0.328443
$695$	−14.0000	−0.531050
$696$	6.93112	0.262723
$697$	6.70820	0.254091
$698$	−4.38197	−0.165860
$699$	−4.65248	−0.175973
$700$	−4.14590	−0.156700
$701$	−24.7639	−0.935321	−0.467660	−	0.883908i	$-0.654903\pi$
$701$	−24.7639	−0.935321	−0.467660	+	0.883908i	$0.654903\pi$
$702$	−2.00000	−0.0754851
$703$	−4.34752	−0.163970
$704$	−25.7639	−0.971015
$705$	−11.9443	−0.449847
$706$	0.896674	0.0337468
$707$	1.05573	0.0397047
$708$	6.00000	0.225494
$709$	15.3050	0.574790	0.287395	−	0.957812i	$-0.407211\pi$
$709$	15.3050	0.574790	0.287395	+	0.957812i	$0.407211\pi$
$710$	−4.36068	−0.163653
$711$	−13.4164	−0.503155
$712$	−3.37384	−0.126440
$713$	−7.63932	−0.286095
$714$	0.854102	0.0319640
$715$	28.6525	1.07154
$716$	18.5410	0.692910
$717$	−6.76393	−0.252604
$718$	8.87539	0.331227
$719$	−40.8885	−1.52489	−0.762443	−	0.647056i	$-0.776000\pi$
$719$	−40.8885	−1.52489	−0.762443	+	0.647056i	$0.776000\pi$
$720$	3.14590	0.117241
$721$	8.94427	0.333102
$722$	18.6525	0.694173
$723$	19.1246	0.711252
$724$	21.7082	0.806779
$725$	4.70820	0.174858
$726$	−7.23607	−0.268556
$727$	−22.5836	−0.837579	−0.418790	−	0.908083i	$-0.637545\pi$
$727$	−22.5836	−0.837579	−0.418790	+	0.908083i	$0.637545\pi$
$728$	−17.2361	−0.638811
$729$	1.00000	0.0370370
$730$	5.32624	0.197133
$731$	10.4721	0.387326
$732$	−17.1246	−0.632944
$733$	19.1246	0.706384	0.353192	−	0.935551i	$-0.385096\pi$
$733$	19.1246	0.706384	0.353192	+	0.935551i	$0.385096\pi$
$734$	6.11146	0.225578
$735$	2.00000	0.0737711
$736$	11.4590	0.422384
$737$	−4.18034	−0.153985
$738$	−2.56231	−0.0943198
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	−0.978714	−0.0359782
$741$	43.1246	1.58422
$742$	6.63119	0.243439
$743$	−23.8885	−0.876386	−0.438193	−	0.898881i	$-0.644381\pi$
$743$	−23.8885	−0.876386	−0.438193	+	0.898881i	$0.644381\pi$
$744$	−4.06888	−0.149173
$745$	18.1803	0.666076
$746$	4.94427	0.181023
$747$	−11.4164	−0.417705
$748$	10.1459	0.370971
$749$	5.52786	0.201984
$750$	−0.381966	−0.0139474
$751$	20.0689	0.732324	0.366162	−	0.930551i	$-0.380672\pi$
$751$	20.0689	0.732324	0.366162	+	0.930551i	$0.380672\pi$
$752$	37.5755	1.37024
$753$	18.6525	0.679734
$754$	9.41641	0.342925
$755$	3.18034	0.115744
$756$	4.14590	0.150785
$757$	50.8328	1.84755	0.923775	−	0.382936i	$-0.125087\pi$
$757$	50.8328	1.84755	0.923775	+	0.382936i	$0.125087\pi$
$758$	3.41641	0.124090
$759$	−15.1246	−0.548989
$760$	12.1246	0.439806
$761$	−4.29180	−0.155577	−0.0777887	−	0.996970i	$-0.524786\pi$
$761$	−4.29180	−0.155577	−0.0777887	+	0.996970i	$0.524786\pi$
$762$	2.00000	0.0724524
$763$	18.9443	0.685829
$764$	4.24922	0.153732
$765$	−1.00000	−0.0361551
$766$	9.32624	0.336971
$767$	16.9443	0.611822
$768$	−5.56231	−0.200712
$769$	7.00000	0.252426	0.126213	−	0.992003i	$-0.459718\pi$
$769$	7.00000	0.252426	0.126213	+	0.992003i	$0.459718\pi$
$770$	4.67376	0.168431
$771$	2.00000	0.0720282
$772$	12.8754	0.463395
$773$	6.12461	0.220287	0.110143	−	0.993916i	$-0.464869\pi$
$773$	6.12461	0.220287	0.110143	+	0.993916i	$0.464869\pi$
$774$	−4.00000	−0.143777
$775$	−2.76393	−0.0992834
$776$	5.19350	0.186436
$777$	−1.18034	−0.0423445
$778$	−5.75078	−0.206175
$779$	55.2492	1.97951
$780$	9.70820	0.347609
$781$	−62.4721	−2.23543
$782$	−1.05573	−0.0377528
$783$	−4.70820	−0.168257
$784$	−6.29180	−0.224707
$785$	4.00000	0.142766
$786$	5.88854	0.210037
$787$	−27.6525	−0.985704	−0.492852	−	0.870113i	$-0.664046\pi$
$787$	−27.6525	−0.985704	−0.492852	+	0.870113i	$0.664046\pi$
$788$	17.6656	0.629312
$789$	−24.4164	−0.869247
$790$	−5.12461	−0.182326
$791$	0.652476	0.0231994
$792$	−8.05573	−0.286248
$793$	−48.3607	−1.71734
$794$	1.14590	0.0406664
$795$	−7.76393	−0.275358
$796$	−36.8754	−1.30701
$797$	0.236068	0.00836196	0.00418098	−	0.999991i	$-0.498669\pi$
$797$	0.236068	0.00836196	0.00418098	+	0.999991i	$0.498669\pi$
$798$	7.03444	0.249017
$799$	−11.9443	−0.422558
$800$	4.14590	0.146580
$801$	2.29180	0.0809766
$802$	−3.32624	−0.117454
$803$	76.3050	2.69274
$804$	−1.41641	−0.0499529
$805$	6.18034	0.217828
$806$	−5.52786	−0.194711
$807$	9.29180	0.327087
$808$	−0.695048	−0.0244517
$809$	−38.9443	−1.36921	−0.684604	−	0.728915i	$-0.740025\pi$
$809$	−38.9443	−1.36921	−0.684604	+	0.728915i	$0.740025\pi$
$810$	0.381966	0.0134209
$811$	−24.1803	−0.849087	−0.424543	−	0.905408i	$-0.639565\pi$
$811$	−24.1803	−0.849087	−0.424543	+	0.905408i	$0.639565\pi$
$812$	−19.5197	−0.685008
$813$	−1.52786	−0.0535845
$814$	1.10333	0.0386715
$815$	−15.1803	−0.531744
$816$	3.14590	0.110128
$817$	86.2492	3.01748
$818$	11.4377	0.399910
$819$	11.7082	0.409118
$820$	12.4377	0.434343
$821$	−9.41641	−0.328635	−0.164317	−	0.986408i	$-0.552542\pi$
$821$	−9.41641	−0.328635	−0.164317	+	0.986408i	$0.552542\pi$
$822$	−0.673762	−0.0235002
$823$	−26.8197	−0.934875	−0.467437	−	0.884026i	$-0.654823\pi$
$823$	−26.8197	−0.934875	−0.467437	+	0.884026i	$0.654823\pi$
$824$	−5.88854	−0.205137
$825$	−5.47214	−0.190515
$826$	2.76393	0.0961695
$827$	13.7082	0.476681	0.238340	−	0.971182i	$-0.423397\pi$
$827$	13.7082	0.476681	0.238340	+	0.971182i	$0.423397\pi$
$828$	−5.12461	−0.178093
$829$	−9.47214	−0.328981	−0.164490	−	0.986379i	$-0.552598\pi$
$829$	−9.47214	−0.328981	−0.164490	+	0.986379i	$0.552598\pi$
$830$	−4.36068	−0.151361
$831$	17.4164	0.604168
$832$	−24.6525	−0.854671
$833$	2.00000	0.0692959
$834$	5.34752	0.185170
$835$	−18.9443	−0.655594
$836$	83.5623	2.89006
$837$	2.76393	0.0955355
$838$	11.0770	0.382649
$839$	−18.4164	−0.635805	−0.317903	−	0.948123i	$-0.602978\pi$
$839$	−18.4164	−0.635805	−0.317903	+	0.948123i	$0.602978\pi$
$840$	3.29180	0.113578
$841$	−6.83282	−0.235614
$842$	−2.89667	−0.0998260
$843$	−21.8885	−0.753881
$844$	−43.4164	−1.49445
$845$	14.4164	0.495940
$846$	4.56231	0.156855
$847$	42.3607	1.45553
$848$	24.4245	0.838742
$849$	−4.70820	−0.161585
$850$	−0.381966	−0.0131013
$851$	1.45898	0.0500132
$852$	−21.1672	−0.725176
$853$	−34.0000	−1.16414	−0.582069	−	0.813139i	$-0.697757\pi$
$853$	−34.0000	−1.16414	−0.582069	+	0.813139i	$0.697757\pi$
$854$	−7.88854	−0.269940
$855$	−8.23607	−0.281668
$856$	−3.63932	−0.124389
$857$	−51.1935	−1.74874	−0.874368	−	0.485263i	$-0.838724\pi$
$857$	−51.1935	−1.74874	−0.874368	+	0.485263i	$0.838724\pi$
$858$	−10.9443	−0.373631
$859$	1.18034	0.0402727	0.0201363	−	0.999797i	$-0.493590\pi$
$859$	1.18034	0.0402727	0.0201363	+	0.999797i	$0.493590\pi$
$860$	19.4164	0.662094
$861$	15.0000	0.511199
$862$	−2.81153	−0.0957611
$863$	−35.7214	−1.21597	−0.607985	−	0.793949i	$-0.708022\pi$
$863$	−35.7214	−1.21597	−0.607985	+	0.793949i	$0.708022\pi$
$864$	−4.14590	−0.141046
$865$	−0.763932	−0.0259745
$866$	7.01316	0.238317
$867$	−1.00000	−0.0339618
$868$	11.4590	0.388943
$869$	−73.4164	−2.49048
$870$	−1.79837	−0.0609706
$871$	−4.00000	−0.135535
$872$	−12.4721	−0.422360
$873$	−3.52786	−0.119400
$874$	−8.69505	−0.294114
$875$	2.23607	0.0755929
$876$	25.8541	0.873529
$877$	−5.00000	−0.168838	−0.0844190	−	0.996430i	$-0.526903\pi$
$877$	−5.00000	−0.168838	−0.0844190	+	0.996430i	$0.526903\pi$
$878$	8.69505	0.293444
$879$	16.2361	0.547629
$880$	17.2148	0.580310
$881$	20.7082	0.697677	0.348839	−	0.937183i	$-0.386576\pi$
$881$	20.7082	0.697677	0.348839	+	0.937183i	$0.386576\pi$
$882$	−0.763932	−0.0257229
$883$	13.3475	0.449180	0.224590	−	0.974453i	$-0.427896\pi$
$883$	13.3475	0.449180	0.224590	+	0.974453i	$0.427896\pi$
$884$	9.70820	0.326522
$885$	−3.23607	−0.108779
$886$	−0.360680	−0.0121173
$887$	36.9443	1.24047	0.620234	−	0.784417i	$-0.287038\pi$
$887$	36.9443	1.24047	0.620234	+	0.784417i	$0.287038\pi$
$888$	0.777088	0.0260774
$889$	−11.7082	−0.392681
$890$	0.875388	0.0293431
$891$	5.47214	0.183323
$892$	−12.8754	−0.431100
$893$	−98.3738	−3.29195
$894$	−6.94427	−0.232251
$895$	−10.0000	−0.334263
$896$	−22.5623	−0.753754
$897$	−14.4721	−0.483211
$898$	−4.76393	−0.158974
$899$	−13.0132	−0.434013
$900$	−1.85410	−0.0618034
$901$	−7.76393	−0.258654
$902$	−14.0213	−0.466858
$903$	23.4164	0.779249
$904$	−0.429563	−0.0142871
$905$	−11.7082	−0.389194
$906$	−1.21478	−0.0403584
$907$	8.12461	0.269773	0.134887	−	0.990861i	$-0.456933\pi$
$907$	8.12461	0.269773	0.134887	+	0.990861i	$0.456933\pi$
$908$	21.7082	0.720412
$909$	0.472136	0.0156598
$910$	4.47214	0.148250
$911$	−6.16718	−0.204328	−0.102164	−	0.994768i	$-0.532577\pi$
$911$	−6.16718	−0.204328	−0.102164	+	0.994768i	$0.532577\pi$
$912$	25.9098	0.857960
$913$	−62.4721	−2.06753
$914$	4.87539	0.161264
$915$	9.23607	0.305335
$916$	23.0213	0.760645
$917$	−34.4721	−1.13837
$918$	0.381966	0.0126068
$919$	43.5410	1.43629	0.718143	−	0.695896i	$-0.244992\pi$
$919$	43.5410	1.43629	0.718143	+	0.695896i	$0.244992\pi$
$920$	−4.06888	−0.134147
$921$	32.0000	1.05444
$922$	8.65248	0.284954
$923$	−59.7771	−1.96759
$924$	22.6869	0.746345
$925$	0.527864	0.0173561
$926$	−7.01316	−0.230467
$927$	4.00000	0.131377
$928$	19.5197	0.640767
$929$	27.5410	0.903592	0.451796	−	0.892121i	$-0.350784\pi$
$929$	27.5410	0.903592	0.451796	+	0.892121i	$0.350784\pi$
$930$	1.05573	0.0346187
$931$	16.4721	0.539852
$932$	−8.62616	−0.282559
$933$	−14.4164	−0.471972
$934$	0.742646	0.0243001
$935$	−5.47214	−0.178958
$936$	−7.70820	−0.251951
$937$	−10.1803	−0.332577	−0.166289	−	0.986077i	$-0.553178\pi$
$937$	−10.1803	−0.332577	−0.166289	+	0.986077i	$0.553178\pi$
$938$	−0.652476	−0.0213041
$939$	−6.52786	−0.213029
$940$	−22.1459	−0.722320
$941$	2.36068	0.0769560	0.0384780	−	0.999259i	$-0.487749\pi$
$941$	2.36068	0.0769560	0.0384780	+	0.999259i	$0.487749\pi$
$942$	−1.52786	−0.0497805
$943$	−18.5410	−0.603779
$944$	10.1803	0.331342
$945$	−2.23607	−0.0727393
$946$	−21.8885	−0.711658
$947$	−30.4721	−0.990211	−0.495106	−	0.868833i	$-0.664871\pi$
$947$	−30.4721	−0.990211	−0.495106	+	0.868833i	$0.664871\pi$
$948$	−24.8754	−0.807915
$949$	73.0132	2.37011
$950$	−3.14590	−0.102066
$951$	3.70820	0.120247
$952$	3.29180	0.106688
$953$	52.5967	1.70378	0.851888	−	0.523724i	$-0.175458\pi$
$953$	52.5967	1.70378	0.851888	+	0.523724i	$0.175458\pi$
$954$	2.96556	0.0960135
$955$	−2.29180	−0.0741608
$956$	−12.5410	−0.405606
$957$	−25.7639	−0.832830
$958$	0	0
$959$	3.94427	0.127367
$960$	4.70820	0.151957
$961$	−23.3607	−0.753570
$962$	1.05573	0.0340380
$963$	2.47214	0.0796635
$964$	35.4590	1.14206
$965$	−6.94427	−0.223544
$966$	−2.36068	−0.0759536
$967$	8.18034	0.263062	0.131531	−	0.991312i	$-0.458011\pi$
$967$	8.18034	0.263062	0.131531	+	0.991312i	$0.458011\pi$
$968$	−27.8885	−0.896372
$969$	−8.23607	−0.264581
$970$	−1.34752	−0.0432664
$971$	−1.70820	−0.0548189	−0.0274094	−	0.999624i	$-0.508726\pi$
$971$	−1.70820	−0.0548189	−0.0274094	+	0.999624i	$0.508726\pi$
$972$	1.85410	0.0594703
$973$	−31.3050	−1.00359
$974$	4.58359	0.146868
$975$	−5.23607	−0.167688
$976$	−29.0557	−0.930051
$977$	34.3607	1.09930	0.549648	−	0.835397i	$-0.314762\pi$
$977$	34.3607	1.09930	0.549648	+	0.835397i	$0.314762\pi$
$978$	5.79837	0.185412
$979$	12.5410	0.400813
$980$	3.70820	0.118454
$981$	8.47214	0.270494
$982$	13.7082	0.437446
$983$	−24.0689	−0.767678	−0.383839	−	0.923400i	$-0.625398\pi$
$983$	−24.0689	−0.767678	−0.383839	+	0.923400i	$0.625398\pi$
$984$	−9.87539	−0.314816
$985$	−9.52786	−0.303583
$986$	−1.79837	−0.0572719
$987$	−26.7082	−0.850131
$988$	79.9574	2.54378
$989$	−28.9443	−0.920374
$990$	2.09017	0.0664300
$991$	11.3475	0.360466	0.180233	−	0.983624i	$-0.442315\pi$
$991$	11.3475	0.360466	0.180233	+	0.983624i	$0.442315\pi$
$992$	−11.4590	−0.363823
$993$	14.7082	0.466751
$994$	−9.75078	−0.309276
$995$	19.8885	0.630509
$996$	−21.1672	−0.670708
$997$	−17.8328	−0.564771	−0.282385	−	0.959301i	$-0.591126\pi$
$997$	−17.8328	−0.564771	−0.282385	+	0.959301i	$0.591126\pi$
$998$	7.70820	0.243999
$999$	−0.527864	−0.0167009

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	255.2.a.b.1.1	✓	2
3.2	odd	2		765.2.a.d.1.2		2
4.3	odd	2		4080.2.a.bp.1.1		2
5.2	odd	4		1275.2.b.e.1174.3		4
5.3	odd	4		1275.2.b.e.1174.2		4
5.4	even	2		1275.2.a.h.1.2		2
15.14	odd	2		3825.2.a.z.1.1		2
17.16	even	2		4335.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.a.b.1.1	✓	2	1.1	even	1	trivial
765.2.a.d.1.2		2	3.2	odd	2
1275.2.a.h.1.2		2	5.4	even	2
1275.2.b.e.1174.2		4	5.3	odd	4
1275.2.b.e.1174.3		4	5.2	odd	4
3825.2.a.z.1.1		2	15.14	odd	2
4080.2.a.bp.1.1		2	4.3	odd	2
4335.2.a.m.1.1		2	17.16	even	2

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

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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	255.2.a.b.1.1

Level	$255$
Weight	$2$
Character	255.1
Self dual	yes
Analytic conductor	$2.036$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$