LMFDB - Embedded newform 855.2.a.k.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [855,2,Mod(1,855)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(855, 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("855.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 855 = 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	855.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:	$6.82720937282$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*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			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	855.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +2.87936 q^{7} +1.53919 q^{8} +O(q^{10})$	\(q+2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +2.87936 q^{7} +1.53919 q^{8} -2.17009 q^{10} +4.24846 q^{11} -3.61757 q^{13} +6.24846 q^{14} -2.07838 q^{16} +6.63090 q^{17} +1.00000 q^{19} -2.70928 q^{20} +9.21953 q^{22} +4.63090 q^{23} +1.00000 q^{25} -7.85043 q^{26} +7.80098 q^{28} -0.986669 q^{29} +2.44748 q^{31} -7.58864 q^{32} +14.3896 q^{34} -2.87936 q^{35} -9.80098 q^{37} +2.17009 q^{38} -1.53919 q^{40} -6.92881 q^{41} -10.1412 q^{43} +11.5103 q^{44} +10.0494 q^{46} +10.2062 q^{47} +1.29072 q^{49} +2.17009 q^{50} -9.80098 q^{52} -10.7298 q^{53} -4.24846 q^{55} +4.43188 q^{56} -2.14116 q^{58} -0.921622 q^{59} +8.04945 q^{61} +5.31124 q^{62} -12.3112 q^{64} +3.61757 q^{65} +5.60197 q^{67} +17.9649 q^{68} -6.24846 q^{70} -0.340173 q^{71} -9.75872 q^{73} -21.2690 q^{74} +2.70928 q^{76} +12.2329 q^{77} -9.17727 q^{79} +2.07838 q^{80} -15.0361 q^{82} +5.89269 q^{83} -6.63090 q^{85} -22.0072 q^{86} +6.53919 q^{88} +14.0917 q^{89} -10.4163 q^{91} +12.5464 q^{92} +22.1483 q^{94} -1.00000 q^{95} -2.72261 q^{97} +2.80098 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8	$ 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8} - q^{10} + 4 q^{11} - 6 q^{13} + 10 q^{14} - 3 q^{16} + 16 q^{17} + 3 q^{19} - q^{20} + 4 q^{22} + 10 q^{23} + 3 q^{25} + 4 q^{26} + 14 q^{28} - 2 q^{29} + 8 q^{31} - 3 q^{32} + 14 q^{34} + 4 q^{35} - 20 q^{37} + q^{38} - 3 q^{40} + 10 q^{41} - 10 q^{43} + 18 q^{44} + 12 q^{46} + 6 q^{47} + 11 q^{49} + q^{50} - 20 q^{52} + 8 q^{53} - 4 q^{55} + 14 q^{58} - 6 q^{59} + 6 q^{61} - 10 q^{62} - 11 q^{64} + 6 q^{65} - 2 q^{67} + 4 q^{68} - 10 q^{70} + 10 q^{71} - 4 q^{73} - 22 q^{74} + q^{76} + 14 q^{77} + 12 q^{79} + 3 q^{80} - 26 q^{82} + 6 q^{83} - 16 q^{85} - 32 q^{86} + 18 q^{88} + 40 q^{89} - 4 q^{91} + 2 q^{92} + 12 q^{94} - 3 q^{95} - 2 q^{97} - q^{98}+O(q^{100}) $ 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8 - q^10 + 4 * q^11 - 6 * q^13 + 10 * q^14 - 3 * q^16 + 16 * q^17 + 3 * q^19 - q^20 + 4 * q^22 + 10 * q^23 + 3 * q^25 + 4 * q^26 + 14 * q^28 - 2 * q^29 + 8 * q^31 - 3 * q^32 + 14 * q^34 + 4 * q^35 - 20 * q^37 + q^38 - 3 * q^40 + 10 * q^41 - 10 * q^43 + 18 * q^44 + 12 * q^46 + 6 * q^47 + 11 * q^49 + q^50 - 20 * q^52 + 8 * q^53 - 4 * q^55 + 14 * q^58 - 6 * q^59 + 6 * q^61 - 10 * q^62 - 11 * q^64 + 6 * q^65 - 2 * q^67 + 4 * q^68 - 10 * q^70 + 10 * q^71 - 4 * q^73 - 22 * q^74 + q^76 + 14 * q^77 + 12 * q^79 + 3 * q^80 - 26 * q^82 + 6 * q^83 - 16 * q^85 - 32 * q^86 + 18 * q^88 + 40 * q^89 - 4 * q^91 + 2 * q^92 + 12 * q^94 - 3 * q^95 - 2 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	2.17009	1.53448	0.767241	−	0.641358i	$-0.221629\pi$
$2$	2.17009	1.53448	0.767241	+	0.641358i	$0.221629\pi$
$3$	0	0
$3$	0	0
$4$	2.70928	1.35464
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.87936	1.08830	0.544148	−	0.838989i	$-0.316853\pi$
$7$	2.87936	1.08830	0.544148	+	0.838989i	$0.316853\pi$
$8$	1.53919	0.544185
$9$	0	0
$10$	−2.17009	−0.686242
$11$	4.24846	1.28096	0.640480	−	0.767975i	$-0.278735\pi$
$11$	4.24846	1.28096	0.640480	+	0.767975i	$0.278735\pi$
$12$	0	0
$13$	−3.61757	−1.00333	−0.501666	−	0.865061i	$-0.667279\pi$
$13$	−3.61757	−1.00333	−0.501666	+	0.865061i	$0.667279\pi$
$14$	6.24846	1.66997
$15$	0	0
$16$	−2.07838	−0.519594
$17$	6.63090	1.60823	0.804114	−	0.594475i	$-0.202640\pi$
$17$	6.63090	1.60823	0.804114	+	0.594475i	$0.202640\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.70928	−0.605812
$21$	0	0
$22$	9.21953	1.96561
$23$	4.63090	0.965609	0.482804	−	0.875728i	$-0.339618\pi$
$23$	4.63090	0.965609	0.482804	+	0.875728i	$0.339618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−7.85043	−1.53960
$27$	0	0
$28$	7.80098	1.47425
$29$	−0.986669	−0.183220	−0.0916099	−	0.995795i	$-0.529201\pi$
$29$	−0.986669	−0.183220	−0.0916099	+	0.995795i	$0.529201\pi$
$30$	0	0
$31$	2.44748	0.439580	0.219790	−	0.975547i	$-0.429463\pi$
$31$	2.44748	0.439580	0.219790	+	0.975547i	$0.429463\pi$
$32$	−7.58864	−1.34149
$33$	0	0
$34$	14.3896	2.46780
$35$	−2.87936	−0.486701
$36$	0	0
$37$	−9.80098	−1.61127	−0.805636	−	0.592411i	$-0.798176\pi$
$37$	−9.80098	−1.61127	−0.805636	+	0.592411i	$0.798176\pi$
$38$	2.17009	0.352035
$39$	0	0
$40$	−1.53919	−0.243367
$41$	−6.92881	−1.08210	−0.541049	−	0.840991i	$-0.681973\pi$
$41$	−6.92881	−1.08210	−0.541049	+	0.840991i	$0.681973\pi$
$42$	0	0
$43$	−10.1412	−1.54651	−0.773256	−	0.634094i	$-0.781373\pi$
$43$	−10.1412	−1.54651	−0.773256	+	0.634094i	$0.781373\pi$
$44$	11.5103	1.73524
$45$	0	0
$46$	10.0494	1.48171
$47$	10.2062	1.48873	0.744364	−	0.667774i	$-0.232753\pi$
$47$	10.2062	1.48873	0.744364	+	0.667774i	$0.232753\pi$
$48$	0	0
$49$	1.29072	0.184389
$50$	2.17009	0.306897
$51$	0	0
$52$	−9.80098	−1.35915
$53$	−10.7298	−1.47385	−0.736925	−	0.675974i	$-0.763723\pi$
$53$	−10.7298	−1.47385	−0.736925	+	0.675974i	$0.763723\pi$
$54$	0	0
$55$	−4.24846	−0.572863
$56$	4.43188	0.592235
$57$	0	0
$58$	−2.14116	−0.281148
$59$	−0.921622	−0.119985	−0.0599925	−	0.998199i	$-0.519108\pi$
$59$	−0.921622	−0.119985	−0.0599925	+	0.998199i	$0.519108\pi$
$60$	0	0
$61$	8.04945	1.03063	0.515313	−	0.857002i	$-0.327676\pi$
$61$	8.04945	1.03063	0.515313	+	0.857002i	$0.327676\pi$
$62$	5.31124	0.674529
$63$	0	0
$64$	−12.3112	−1.53891
$65$	3.61757	0.448704
$66$	0	0
$67$	5.60197	0.684389	0.342195	−	0.939629i	$-0.388830\pi$
$67$	5.60197	0.684389	0.342195	+	0.939629i	$0.388830\pi$
$68$	17.9649	2.17857
$69$	0	0
$70$	−6.24846	−0.746834
$71$	−0.340173	−0.0403711	−0.0201856	−	0.999796i	$-0.506426\pi$
$71$	−0.340173	−0.0403711	−0.0201856	+	0.999796i	$0.506426\pi$
$72$	0	0
$73$	−9.75872	−1.14217	−0.571086	−	0.820890i	$-0.693478\pi$
$73$	−9.75872	−1.14217	−0.571086	+	0.820890i	$0.693478\pi$
$74$	−21.2690	−2.47247
$75$	0	0
$76$	2.70928	0.310775
$77$	12.2329	1.39406
$78$	0	0
$79$	−9.17727	−1.03252	−0.516262	−	0.856431i	$-0.672677\pi$
$79$	−9.17727	−1.03252	−0.516262	+	0.856431i	$0.672677\pi$
$80$	2.07838	0.232370
$81$	0	0
$82$	−15.0361	−1.66046
$83$	5.89269	0.646807	0.323404	−	0.946261i	$-0.395173\pi$
$83$	5.89269	0.646807	0.323404	+	0.946261i	$0.395173\pi$
$84$	0	0
$85$	−6.63090	−0.719222
$86$	−22.0072	−2.37310
$87$	0	0
$88$	6.53919	0.697080
$89$	14.0917	1.49372	0.746859	−	0.664982i	$-0.231561\pi$
$89$	14.0917	1.49372	0.746859	+	0.664982i	$0.231561\pi$
$90$	0	0
$91$	−10.4163	−1.09192
$92$	12.5464	1.30805
$93$	0	0
$94$	22.1483	2.28443
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−2.72261	−0.276439	−0.138219	−	0.990402i	$-0.544138\pi$
$97$	−2.72261	−0.276439	−0.138219	+	0.990402i	$0.544138\pi$
$98$	2.80098	0.282942
$99$	0	0
$100$	2.70928	0.270928
$101$	−10.8638	−1.08098	−0.540492	−	0.841349i	$-0.681762\pi$
$101$	−10.8638	−1.08098	−0.540492	+	0.841349i	$0.681762\pi$
$102$	0	0
$103$	12.6803	1.24943	0.624716	−	0.780852i	$-0.285215\pi$
$103$	12.6803	1.24943	0.624716	+	0.780852i	$0.285215\pi$
$104$	−5.56812	−0.545999
$105$	0	0
$106$	−23.2846	−2.26160
$107$	−8.49693	−0.821429	−0.410715	−	0.911764i	$-0.634721\pi$
$107$	−8.49693	−0.821429	−0.410715	+	0.911764i	$0.634721\pi$
$108$	0	0
$109$	−19.8576	−1.90202	−0.951008	−	0.309168i	$-0.899950\pi$
$109$	−19.8576	−1.90202	−0.951008	+	0.309168i	$0.899950\pi$
$110$	−9.21953	−0.879048
$111$	0	0
$112$	−5.98440	−0.565473
$113$	−8.88655	−0.835976	−0.417988	−	0.908452i	$-0.637265\pi$
$113$	−8.88655	−0.835976	−0.417988	+	0.908452i	$0.637265\pi$
$114$	0	0
$115$	−4.63090	−0.431833
$116$	−2.67316	−0.248196
$117$	0	0
$118$	−2.00000	−0.184115
$119$	19.0928	1.75023
$120$	0	0
$121$	7.04945	0.640859
$122$	17.4680	1.58148
$123$	0	0
$124$	6.63090	0.595472
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.0205	−1.51033	−0.755163	−	0.655537i	$-0.772443\pi$
$127$	−17.0205	−1.51033	−0.755163	+	0.655537i	$0.772443\pi$
$128$	−11.5392	−1.01993
$129$	0	0
$130$	7.85043	0.688528
$131$	−9.26898	−0.809835	−0.404917	−	0.914353i	$-0.632700\pi$
$131$	−9.26898	−0.809835	−0.404917	+	0.914353i	$0.632700\pi$
$132$	0	0
$133$	2.87936	0.249672
$134$	12.1568	1.05018
$135$	0	0
$136$	10.2062	0.875175
$137$	16.6537	1.42282	0.711410	−	0.702777i	$-0.248057\pi$
$137$	16.6537	1.42282	0.711410	+	0.702777i	$0.248057\pi$
$138$	0	0
$139$	12.1834	1.03338	0.516692	−	0.856171i	$-0.327163\pi$
$139$	12.1834	1.03338	0.516692	+	0.856171i	$0.327163\pi$
$140$	−7.80098	−0.659303
$141$	0	0
$142$	−0.738205	−0.0619488
$143$	−15.3691	−1.28523
$144$	0	0
$145$	0.986669	0.0819384
$146$	−21.1773	−1.75264
$147$	0	0
$148$	−26.5536	−2.18269
$149$	5.84324	0.478697	0.239349	−	0.970934i	$-0.423066\pi$
$149$	5.84324	0.478697	0.239349	+	0.970934i	$0.423066\pi$
$150$	0	0
$151$	6.99773	0.569467	0.284734	−	0.958607i	$-0.408095\pi$
$151$	6.99773	0.569467	0.284734	+	0.958607i	$0.408095\pi$
$152$	1.53919	0.124845
$153$	0	0
$154$	26.5464	2.13917
$155$	−2.44748	−0.196586
$156$	0	0
$157$	−18.8371	−1.50336	−0.751682	−	0.659526i	$-0.770757\pi$
$157$	−18.8371	−1.50336	−0.751682	+	0.659526i	$0.770757\pi$
$158$	−19.9155	−1.58439
$159$	0	0
$160$	7.58864	0.599934
$161$	13.3340	1.05087
$162$	0	0
$163$	11.1929	0.876693	0.438347	−	0.898806i	$-0.355564\pi$
$163$	11.1929	0.876693	0.438347	+	0.898806i	$0.355564\pi$
$164$	−18.7721	−1.46585
$165$	0	0
$166$	12.7877	0.992514
$167$	−10.6042	−0.820580	−0.410290	−	0.911955i	$-0.634573\pi$
$167$	−10.6042	−0.820580	−0.410290	+	0.911955i	$0.634573\pi$
$168$	0	0
$169$	0.0867882	0.00667602
$170$	−14.3896	−1.10363
$171$	0	0
$172$	−27.4752	−2.09496
$173$	10.1483	0.771564	0.385782	−	0.922590i	$-0.373932\pi$
$173$	10.1483	0.771564	0.385782	+	0.922590i	$0.373932\pi$
$174$	0	0
$175$	2.87936	0.217659
$176$	−8.82991	−0.665580
$177$	0	0
$178$	30.5802	2.29208
$179$	6.86376	0.513022	0.256511	−	0.966541i	$-0.417427\pi$
$179$	6.86376	0.513022	0.256511	+	0.966541i	$0.417427\pi$
$180$	0	0
$181$	−2.39803	−0.178244	−0.0891221	−	0.996021i	$-0.528406\pi$
$181$	−2.39803	−0.178244	−0.0891221	+	0.996021i	$0.528406\pi$
$182$	−22.6042	−1.67554
$183$	0	0
$184$	7.12783	0.525470
$185$	9.80098	0.720583
$186$	0	0
$187$	28.1711	2.06008
$188$	27.6514	2.01669
$189$	0	0
$190$	−2.17009	−0.157435
$191$	4.74539	0.343365	0.171682	−	0.985152i	$-0.445080\pi$
$191$	4.74539	0.343365	0.171682	+	0.985152i	$0.445080\pi$
$192$	0	0
$193$	8.66475	0.623702	0.311851	−	0.950131i	$-0.399051\pi$
$193$	8.66475	0.623702	0.311851	+	0.950131i	$0.399051\pi$
$194$	−5.90829	−0.424191
$195$	0	0
$196$	3.49693	0.249781
$197$	−15.6248	−1.11322	−0.556609	−	0.830775i	$-0.687898\pi$
$197$	−15.6248	−1.11322	−0.556609	+	0.830775i	$0.687898\pi$
$198$	0	0
$199$	11.0205	0.781224	0.390612	−	0.920555i	$-0.372263\pi$
$199$	11.0205	0.781224	0.390612	+	0.920555i	$0.372263\pi$
$200$	1.53919	0.108837
$201$	0	0
$202$	−23.5753	−1.65875
$203$	−2.84098	−0.199398
$204$	0	0
$205$	6.92881	0.483929
$206$	27.5174	1.91723
$207$	0	0
$208$	7.51867	0.521326
$209$	4.24846	0.293872
$210$	0	0
$211$	10.0267	0.690264	0.345132	−	0.938554i	$-0.387834\pi$
$211$	10.0267	0.690264	0.345132	+	0.938554i	$0.387834\pi$
$212$	−29.0700	−1.99653
$213$	0	0
$214$	−18.4391	−1.26047
$215$	10.1412	0.691621
$216$	0	0
$217$	7.04718	0.478394
$218$	−43.0928	−2.91861
$219$	0	0
$220$	−11.5103	−0.776022
$221$	−23.9877	−1.61359
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−21.8504	−1.45994
$225$	0	0
$226$	−19.2846	−1.28279
$227$	−13.1278	−0.871324	−0.435662	−	0.900110i	$-0.643486\pi$
$227$	−13.1278	−0.871324	−0.435662	+	0.900110i	$0.643486\pi$
$228$	0	0
$229$	9.44134	0.623901	0.311951	−	0.950098i	$-0.399018\pi$
$229$	9.44134	0.623901	0.311951	+	0.950098i	$0.399018\pi$
$230$	−10.0494	−0.662641
$231$	0	0
$232$	−1.51867	−0.0997056
$233$	13.7321	0.899617	0.449809	−	0.893125i	$-0.351492\pi$
$233$	13.7321	0.899617	0.449809	+	0.893125i	$0.351492\pi$
$234$	0	0
$235$	−10.2062	−0.665779
$236$	−2.49693	−0.162536
$237$	0	0
$238$	41.4329	2.68570
$239$	8.58864	0.555553	0.277776	−	0.960646i	$-0.410403\pi$
$239$	8.58864	0.555553	0.277776	+	0.960646i	$0.410403\pi$
$240$	0	0
$241$	8.15676	0.525423	0.262711	−	0.964874i	$-0.415383\pi$
$241$	8.15676	0.525423	0.262711	+	0.964874i	$0.415383\pi$
$242$	15.2979	0.983387
$243$	0	0
$244$	21.8082	1.39613
$245$	−1.29072	−0.0824614
$246$	0	0
$247$	−3.61757	−0.230180
$248$	3.76713	0.239213
$249$	0	0
$250$	−2.17009	−0.137248
$251$	−1.66701	−0.105221	−0.0526105	−	0.998615i	$-0.516754\pi$
$251$	−1.66701	−0.105221	−0.0526105	+	0.998615i	$0.516754\pi$
$252$	0	0
$253$	19.6742	1.23691
$254$	−36.9360	−2.31757
$255$	0	0
$256$	−0.418551	−0.0261594
$257$	14.1795	0.884496	0.442248	−	0.896893i	$-0.354181\pi$
$257$	14.1795	0.884496	0.442248	+	0.896893i	$0.354181\pi$
$258$	0	0
$259$	−28.2206	−1.75354
$260$	9.80098	0.607831
$261$	0	0
$262$	−20.1145	−1.24268
$263$	−15.9649	−0.984440	−0.492220	−	0.870471i	$-0.663814\pi$
$263$	−15.9649	−0.984440	−0.492220	+	0.870471i	$0.663814\pi$
$264$	0	0
$265$	10.7298	0.659126
$266$	6.24846	0.383118
$267$	0	0
$268$	15.1773	0.927100
$269$	21.2423	1.29517	0.647584	−	0.761994i	$-0.275780\pi$
$269$	21.2423	1.29517	0.647584	+	0.761994i	$0.275780\pi$
$270$	0	0
$271$	−25.9565	−1.57675	−0.788373	−	0.615197i	$-0.789076\pi$
$271$	−25.9565	−1.57675	−0.788373	+	0.615197i	$0.789076\pi$
$272$	−13.7815	−0.835627
$273$	0	0
$274$	36.1399	2.18329
$275$	4.24846	0.256192
$276$	0	0
$277$	20.5692	1.23588	0.617941	−	0.786225i	$-0.287967\pi$
$277$	20.5692	1.23588	0.617941	+	0.786225i	$0.287967\pi$
$278$	26.4391	1.58571
$279$	0	0
$280$	−4.43188	−0.264856
$281$	2.00719	0.119739	0.0598694	−	0.998206i	$-0.480932\pi$
$281$	2.00719	0.119739	0.0598694	+	0.998206i	$0.480932\pi$
$282$	0	0
$283$	6.29791	0.374372	0.187186	−	0.982324i	$-0.440063\pi$
$283$	6.29791	0.374372	0.187186	+	0.982324i	$0.440063\pi$
$284$	−0.921622	−0.0546882
$285$	0	0
$286$	−33.3523	−1.97216
$287$	−19.9506	−1.17764
$288$	0	0
$289$	26.9688	1.58640
$290$	2.14116	0.125733
$291$	0	0
$292$	−26.4391	−1.54723
$293$	16.6042	0.970030	0.485015	−	0.874506i	$-0.338814\pi$
$293$	16.6042	0.970030	0.485015	+	0.874506i	$0.338814\pi$
$294$	0	0
$295$	0.921622	0.0536589
$296$	−15.0856	−0.876831
$297$	0	0
$298$	12.6803	0.734553
$299$	−16.7526	−0.968827
$300$	0	0
$301$	−29.2001	−1.68306
$302$	15.1857	0.873838
$303$	0	0
$304$	−2.07838	−0.119203
$305$	−8.04945	−0.460910
$306$	0	0
$307$	20.0144	1.14228	0.571140	−	0.820852i	$-0.306501\pi$
$307$	20.0144	1.14228	0.571140	+	0.820852i	$0.306501\pi$
$308$	33.1422	1.88845
$309$	0	0
$310$	−5.31124	−0.301658
$311$	27.0277	1.53260	0.766300	−	0.642483i	$-0.222095\pi$
$311$	27.0277	1.53260	0.766300	+	0.642483i	$0.222095\pi$
$312$	0	0
$313$	−17.9421	−1.01415	−0.507075	−	0.861902i	$-0.669273\pi$
$313$	−17.9421	−1.01415	−0.507075	+	0.861902i	$0.669273\pi$
$314$	−40.8781	−2.30689
$315$	0	0
$316$	−24.8638	−1.39870
$317$	28.4885	1.60007	0.800037	−	0.599950i	$-0.204813\pi$
$317$	28.4885	1.60007	0.800037	+	0.599950i	$0.204813\pi$
$318$	0	0
$319$	−4.19183	−0.234697
$320$	12.3112	0.688219
$321$	0	0
$322$	28.9360	1.61254
$323$	6.63090	0.368953
$324$	0	0
$325$	−3.61757	−0.200666
$326$	24.2895	1.34527
$327$	0	0
$328$	−10.6647	−0.588862
$329$	29.3874	1.62018
$330$	0	0
$331$	20.7877	1.14259	0.571296	−	0.820744i	$-0.306441\pi$
$331$	20.7877	1.14259	0.571296	+	0.820744i	$0.306441\pi$
$332$	15.9649	0.876189
$333$	0	0
$334$	−23.0121	−1.25917
$335$	−5.60197	−0.306068
$336$	0	0
$337$	3.95774	0.215592	0.107796	−	0.994173i	$-0.465621\pi$
$337$	3.95774	0.215592	0.107796	+	0.994173i	$0.465621\pi$
$338$	0.188338	0.0102442
$339$	0	0
$340$	−17.9649	−0.974285
$341$	10.3980	0.563085
$342$	0	0
$343$	−16.4391	−0.887626
$344$	−15.6092	−0.841589
$345$	0	0
$346$	22.0228	1.18395
$347$	−19.2534	−1.03358	−0.516788	−	0.856113i	$-0.672872\pi$
$347$	−19.2534	−1.03358	−0.516788	+	0.856113i	$0.672872\pi$
$348$	0	0
$349$	21.6020	1.15633	0.578163	−	0.815921i	$-0.303770\pi$
$349$	21.6020	1.15633	0.578163	+	0.815921i	$0.303770\pi$
$350$	6.24846	0.333994
$351$	0	0
$352$	−32.2401	−1.71840
$353$	−12.8371	−0.683250	−0.341625	−	0.939836i	$-0.610977\pi$
$353$	−12.8371	−0.683250	−0.341625	+	0.939836i	$0.610977\pi$
$354$	0	0
$355$	0.340173	0.0180545
$356$	38.1783	2.02345
$357$	0	0
$358$	14.8950	0.787223
$359$	15.9493	0.841773	0.420887	−	0.907113i	$-0.361719\pi$
$359$	15.9493	0.841773	0.420887	+	0.907113i	$0.361719\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−5.20394	−0.273513
$363$	0	0
$364$	−28.2206	−1.47916
$365$	9.75872	0.510795
$366$	0	0
$367$	−18.4547	−0.963326	−0.481663	−	0.876357i	$-0.659967\pi$
$367$	−18.4547	−0.963326	−0.481663	+	0.876357i	$0.659967\pi$
$368$	−9.62475	−0.501725
$369$	0	0
$370$	21.2690	1.10572
$371$	−30.8950	−1.60399
$372$	0	0
$373$	−33.0772	−1.71267	−0.856335	−	0.516421i	$-0.827264\pi$
$373$	−33.0772	−1.71267	−0.856335	+	0.516421i	$0.827264\pi$
$374$	61.1338	3.16115
$375$	0	0
$376$	15.7093	0.810144
$377$	3.56934	0.183830
$378$	0	0
$379$	0.305100	0.0156720	0.00783598	−	0.999969i	$-0.497506\pi$
$379$	0.305100	0.0156720	0.00783598	+	0.999969i	$0.497506\pi$
$380$	−2.70928	−0.138983
$381$	0	0
$382$	10.2979	0.526887
$383$	−14.6225	−0.747174	−0.373587	−	0.927595i	$-0.621872\pi$
$383$	−14.6225	−0.747174	−0.373587	+	0.927595i	$0.621872\pi$
$384$	0	0
$385$	−12.2329	−0.623445
$386$	18.8033	0.957060
$387$	0	0
$388$	−7.37629	−0.374474
$389$	−0.639308	−0.0324142	−0.0162071	−	0.999869i	$-0.505159\pi$
$389$	−0.639308	−0.0324142	−0.0162071	+	0.999869i	$0.505159\pi$
$390$	0	0
$391$	30.7070	1.55292
$392$	1.98667	0.100342
$393$	0	0
$394$	−33.9071	−1.70821
$395$	9.17727	0.461759
$396$	0	0
$397$	−25.3028	−1.26991	−0.634956	−	0.772548i	$-0.718982\pi$
$397$	−25.3028	−1.26991	−0.634956	+	0.772548i	$0.718982\pi$
$398$	23.9155	1.19877
$399$	0	0
$400$	−2.07838	−0.103919
$401$	16.8710	0.842495	0.421248	−	0.906946i	$-0.361592\pi$
$401$	16.8710	0.842495	0.421248	+	0.906946i	$0.361592\pi$
$402$	0	0
$403$	−8.85392	−0.441045
$404$	−29.4329	−1.46434
$405$	0	0
$406$	−6.16517	−0.305972
$407$	−41.6391	−2.06398
$408$	0	0
$409$	38.5113	1.90426	0.952131	−	0.305691i	$-0.0988875\pi$
$409$	38.5113	1.90426	0.952131	+	0.305691i	$0.0988875\pi$
$410$	15.0361	0.742581
$411$	0	0
$412$	34.3545	1.69253
$413$	−2.65368	−0.130579
$414$	0	0
$415$	−5.89269	−0.289261
$416$	27.4524	1.34596
$417$	0	0
$418$	9.21953	0.450942
$419$	4.14957	0.202720	0.101360	−	0.994850i	$-0.467681\pi$
$419$	4.14957	0.202720	0.101360	+	0.994850i	$0.467681\pi$
$420$	0	0
$421$	27.8576	1.35770	0.678849	−	0.734278i	$-0.262479\pi$
$421$	27.8576	1.35770	0.678849	+	0.734278i	$0.262479\pi$
$422$	21.7587	1.05920
$423$	0	0
$424$	−16.5152	−0.802048
$425$	6.63090	0.321646
$426$	0	0
$427$	23.1773	1.12163
$428$	−23.0205	−1.11274
$429$	0	0
$430$	22.0072	1.06128
$431$	−5.39189	−0.259718	−0.129859	−	0.991532i	$-0.541452\pi$
$431$	−5.39189	−0.259718	−0.129859	+	0.991532i	$0.541452\pi$
$432$	0	0
$433$	−1.47519	−0.0708930	−0.0354465	−	0.999372i	$-0.511285\pi$
$433$	−1.47519	−0.0708930	−0.0354465	+	0.999372i	$0.511285\pi$
$434$	15.2930	0.734087
$435$	0	0
$436$	−53.7998	−2.57654
$437$	4.63090	0.221526
$438$	0	0
$439$	−27.5174	−1.31334	−0.656668	−	0.754180i	$-0.728035\pi$
$439$	−27.5174	−1.31334	−0.656668	+	0.754180i	$0.728035\pi$
$440$	−6.53919	−0.311744
$441$	0	0
$442$	−52.0554	−2.47602
$443$	−21.5936	−1.02594	−0.512970	−	0.858406i	$-0.671455\pi$
$443$	−21.5936	−1.02594	−0.512970	+	0.858406i	$0.671455\pi$
$444$	0	0
$445$	−14.0917	−0.668011
$446$	−8.68035	−0.411026
$447$	0	0
$448$	−35.4485	−1.67479
$449$	−16.6030	−0.783545	−0.391772	−	0.920062i	$-0.628138\pi$
$449$	−16.6030	−0.783545	−0.391772	+	0.920062i	$0.628138\pi$
$450$	0	0
$451$	−29.4368	−1.38612
$452$	−24.0761	−1.13244
$453$	0	0
$454$	−28.4885	−1.33703
$455$	10.4163	0.488323
$456$	0	0
$457$	−39.9299	−1.86784	−0.933920	−	0.357482i	$-0.883635\pi$
$457$	−39.9299	−1.86784	−0.933920	+	0.357482i	$0.883635\pi$
$458$	20.4885	0.957366
$459$	0	0
$460$	−12.5464	−0.584978
$461$	−8.99386	−0.418886	−0.209443	−	0.977821i	$-0.567165\pi$
$461$	−8.99386	−0.418886	−0.209443	+	0.977821i	$0.567165\pi$
$462$	0	0
$463$	−25.9265	−1.20491	−0.602454	−	0.798153i	$-0.705810\pi$
$463$	−25.9265	−1.20491	−0.602454	+	0.798153i	$0.705810\pi$
$464$	2.05067	0.0952000
$465$	0	0
$466$	29.7998	1.38045
$467$	−14.8020	−0.684956	−0.342478	−	0.939526i	$-0.611266\pi$
$467$	−14.8020	−0.684956	−0.342478	+	0.939526i	$0.611266\pi$
$468$	0	0
$469$	16.1301	0.744819
$470$	−22.1483	−1.02163
$471$	0	0
$472$	−1.41855	−0.0652941
$473$	−43.0843	−1.98102
$474$	0	0
$475$	1.00000	0.0458831
$476$	51.7275	2.37093
$477$	0	0
$478$	18.6381	0.852486
$479$	−37.7347	−1.72414	−0.862072	−	0.506787i	$-0.830833\pi$
$479$	−37.7347	−1.72414	−0.862072	+	0.506787i	$0.830833\pi$
$480$	0	0
$481$	35.4557	1.61664
$482$	17.7009	0.806252
$483$	0	0
$484$	19.0989	0.868132
$485$	2.72261	0.123627
$486$	0	0
$487$	39.1917	1.77594	0.887972	−	0.459898i	$-0.152114\pi$
$487$	39.1917	1.77594	0.887972	+	0.459898i	$0.152114\pi$
$488$	12.3896	0.560852
$489$	0	0
$490$	−2.80098	−0.126536
$491$	0.234088	0.0105643	0.00528213	−	0.999986i	$-0.498319\pi$
$491$	0.234088	0.0105643	0.00528213	+	0.999986i	$0.498319\pi$
$492$	0	0
$493$	−6.54250	−0.294659
$494$	−7.85043	−0.353208
$495$	0	0
$496$	−5.08679	−0.228404
$497$	−0.979481	−0.0439357
$498$	0	0
$499$	21.4908	0.962060	0.481030	−	0.876704i	$-0.340263\pi$
$499$	21.4908	0.962060	0.481030	+	0.876704i	$0.340263\pi$
$500$	−2.70928	−0.121162
$501$	0	0
$502$	−3.61757	−0.161460
$503$	−32.6453	−1.45558	−0.727790	−	0.685800i	$-0.759453\pi$
$503$	−32.6453	−1.45558	−0.727790	+	0.685800i	$0.759453\pi$
$504$	0	0
$505$	10.8638	0.483431
$506$	42.6947	1.89801
$507$	0	0
$508$	−46.1133	−2.04595
$509$	−7.59478	−0.336633	−0.168316	−	0.985733i	$-0.553833\pi$
$509$	−7.59478	−0.336633	−0.168316	+	0.985733i	$0.553833\pi$
$510$	0	0
$511$	−28.0989	−1.24302
$512$	22.1701	0.979789
$513$	0	0
$514$	30.7708	1.35724
$515$	−12.6803	−0.558763
$516$	0	0
$517$	43.3607	1.90700
$518$	−61.2411	−2.69078
$519$	0	0
$520$	5.56812	0.244178
$521$	31.2690	1.36992	0.684960	−	0.728581i	$-0.259820\pi$
$521$	31.2690	1.36992	0.684960	+	0.728581i	$0.259820\pi$
$522$	0	0
$523$	31.7275	1.38735	0.693674	−	0.720289i	$-0.255991\pi$
$523$	31.7275	1.38735	0.693674	+	0.720289i	$0.255991\pi$
$524$	−25.1122	−1.09703
$525$	0	0
$526$	−34.6453	−1.51061
$527$	16.2290	0.706946
$528$	0	0
$529$	−1.55479	−0.0675994
$530$	23.2846	1.01142
$531$	0	0
$532$	7.80098	0.338216
$533$	25.0654	1.08570
$534$	0	0
$535$	8.49693	0.367354
$536$	8.62249	0.372435
$537$	0	0
$538$	46.0977	1.98741
$539$	5.48360	0.236195
$540$	0	0
$541$	35.3607	1.52027	0.760137	−	0.649762i	$-0.225132\pi$
$541$	35.3607	1.52027	0.760137	+	0.649762i	$0.225132\pi$
$542$	−56.3279	−2.41949
$543$	0	0
$544$	−50.3195	−2.15743
$545$	19.8576	0.850607
$546$	0	0
$547$	−28.4801	−1.21772	−0.608861	−	0.793277i	$-0.708373\pi$
$547$	−28.4801	−1.21772	−0.608861	+	0.793277i	$0.708373\pi$
$548$	45.1194	1.92741
$549$	0	0
$550$	9.21953	0.393122
$551$	−0.986669	−0.0420335
$552$	0	0
$553$	−26.4247	−1.12369
$554$	44.6369	1.89644
$555$	0	0
$556$	33.0082	1.39986
$557$	−15.4452	−0.654435	−0.327217	−	0.944949i	$-0.606111\pi$
$557$	−15.4452	−0.654435	−0.327217	+	0.944949i	$0.606111\pi$
$558$	0	0
$559$	36.6863	1.55167
$560$	5.98440	0.252887
$561$	0	0
$562$	4.35577	0.183737
$563$	24.8599	1.04772	0.523860	−	0.851805i	$-0.324492\pi$
$563$	24.8599	1.04772	0.523860	+	0.851805i	$0.324492\pi$
$564$	0	0
$565$	8.88655	0.373860
$566$	13.6670	0.574467
$567$	0	0
$568$	−0.523590	−0.0219694
$569$	30.7454	1.28891	0.644457	−	0.764641i	$-0.277083\pi$
$569$	30.7454	1.28891	0.644457	+	0.764641i	$0.277083\pi$
$570$	0	0
$571$	29.5753	1.23769	0.618844	−	0.785514i	$-0.287601\pi$
$571$	29.5753	1.23769	0.618844	+	0.785514i	$0.287601\pi$
$572$	−41.6391	−1.74102
$573$	0	0
$574$	−43.2944	−1.80707
$575$	4.63090	0.193122
$576$	0	0
$577$	−6.47027	−0.269361	−0.134680	−	0.990889i	$-0.543001\pi$
$577$	−6.47027	−0.269361	−0.134680	+	0.990889i	$0.543001\pi$
$578$	58.5246	2.43430
$579$	0	0
$580$	2.67316	0.110997
$581$	16.9672	0.703918
$582$	0	0
$583$	−45.5851	−1.88794
$584$	−15.0205	−0.621553
$585$	0	0
$586$	36.0326	1.48849
$587$	28.0638	1.15832	0.579159	−	0.815215i	$-0.303381\pi$
$587$	28.0638	1.15832	0.579159	+	0.815215i	$0.303381\pi$
$588$	0	0
$589$	2.44748	0.100847
$590$	2.00000	0.0823387
$591$	0	0
$592$	20.3701	0.837208
$593$	8.92162	0.366367	0.183184	−	0.983079i	$-0.441360\pi$
$593$	8.92162	0.366367	0.183184	+	0.983079i	$0.441360\pi$
$594$	0	0
$595$	−19.0928	−0.782727
$596$	15.8310	0.648461
$597$	0	0
$598$	−36.3545	−1.48665
$599$	23.8720	0.975383	0.487692	−	0.873016i	$-0.337839\pi$
$599$	23.8720	0.975383	0.487692	+	0.873016i	$0.337839\pi$
$600$	0	0
$601$	5.91548	0.241297	0.120649	−	0.992695i	$-0.461503\pi$
$601$	5.91548	0.241297	0.120649	+	0.992695i	$0.461503\pi$
$602$	−63.3667	−2.58263
$603$	0	0
$604$	18.9588	0.771422
$605$	−7.04945	−0.286601
$606$	0	0
$607$	38.1978	1.55040	0.775200	−	0.631716i	$-0.217649\pi$
$607$	38.1978	1.55040	0.775200	+	0.631716i	$0.217649\pi$
$608$	−7.58864	−0.307760
$609$	0	0
$610$	−17.4680	−0.707259
$611$	−36.9216	−1.49369
$612$	0	0
$613$	−8.58145	−0.346601	−0.173301	−	0.984869i	$-0.555443\pi$
$613$	−8.58145	−0.346601	−0.173301	+	0.984869i	$0.555443\pi$
$614$	43.4329	1.75281
$615$	0	0
$616$	18.8287	0.758630
$617$	−42.2472	−1.70081	−0.850405	−	0.526129i	$-0.823643\pi$
$617$	−42.2472	−1.70081	−0.850405	+	0.526129i	$0.823643\pi$
$618$	0	0
$619$	−21.8888	−0.879786	−0.439893	−	0.898050i	$-0.644984\pi$
$619$	−21.8888	−0.879786	−0.439893	+	0.898050i	$0.644984\pi$
$620$	−6.63090	−0.266303
$621$	0	0
$622$	58.6525	2.35175
$623$	40.5751	1.62561
$624$	0	0
$625$	1.00000	0.0400000
$626$	−38.9360	−1.55620
$627$	0	0
$628$	−51.0349	−2.03651
$629$	−64.9893	−2.59129
$630$	0	0
$631$	34.5380	1.37493	0.687467	−	0.726215i	$-0.258722\pi$
$631$	34.5380	1.37493	0.687467	+	0.726215i	$0.258722\pi$
$632$	−14.1256	−0.561885
$633$	0	0
$634$	61.8225	2.45529
$635$	17.0205	0.675439
$636$	0	0
$637$	−4.66928	−0.185004
$638$	−9.09663	−0.360139
$639$	0	0
$640$	11.5392	0.456126
$641$	17.1967	0.679231	0.339615	−	0.940564i	$-0.389703\pi$
$641$	17.1967	0.679231	0.339615	+	0.940564i	$0.389703\pi$
$642$	0	0
$643$	−0.910559	−0.0359089	−0.0179545	−	0.999839i	$-0.505715\pi$
$643$	−0.910559	−0.0359089	−0.0179545	+	0.999839i	$0.505715\pi$
$644$	36.1256	1.42355
$645$	0	0
$646$	14.3896	0.566152
$647$	3.49920	0.137568	0.0687838	−	0.997632i	$-0.478088\pi$
$647$	3.49920	0.137568	0.0687838	+	0.997632i	$0.478088\pi$
$648$	0	0
$649$	−3.91548	−0.153696
$650$	−7.85043	−0.307919
$651$	0	0
$652$	30.3246	1.18760
$653$	−0.742080	−0.0290399	−0.0145199	−	0.999895i	$-0.504622\pi$
$653$	−0.742080	−0.0290399	−0.0145199	+	0.999895i	$0.504622\pi$
$654$	0	0
$655$	9.26898	0.362169
$656$	14.4007	0.562252
$657$	0	0
$658$	63.7731	2.48613
$659$	−25.1506	−0.979729	−0.489864	−	0.871799i	$-0.662954\pi$
$659$	−25.1506	−0.979729	−0.489864	+	0.871799i	$0.662954\pi$
$660$	0	0
$661$	17.7854	0.691771	0.345886	−	0.938277i	$-0.387579\pi$
$661$	17.7854	0.691771	0.345886	+	0.938277i	$0.387579\pi$
$662$	45.1110	1.75329
$663$	0	0
$664$	9.06997	0.351983
$665$	−2.87936	−0.111657
$666$	0	0
$667$	−4.56916	−0.176919
$668$	−28.7298	−1.11159
$669$	0	0
$670$	−12.1568	−0.469656
$671$	34.1978	1.32019
$672$	0	0
$673$	15.9577	0.615126	0.307563	−	0.951528i	$-0.400487\pi$
$673$	15.9577	0.615126	0.307563	+	0.951528i	$0.400487\pi$
$674$	8.58864	0.330822
$675$	0	0
$676$	0.235133	0.00904359
$677$	0.191828	0.00737255	0.00368627	−	0.999993i	$-0.498827\pi$
$677$	0.191828	0.00737255	0.00368627	+	0.999993i	$0.498827\pi$
$678$	0	0
$679$	−7.83937	−0.300847
$680$	−10.2062	−0.391390
$681$	0	0
$682$	22.5646	0.864044
$683$	−23.8166	−0.911316	−0.455658	−	0.890155i	$-0.650596\pi$
$683$	−23.8166	−0.911316	−0.455658	+	0.890155i	$0.650596\pi$
$684$	0	0
$685$	−16.6537	−0.636305
$686$	−35.6742	−1.36205
$687$	0	0
$688$	21.0772	0.803559
$689$	38.8157	1.47876
$690$	0	0
$691$	−22.3090	−0.848673	−0.424337	−	0.905504i	$-0.639493\pi$
$691$	−22.3090	−0.848673	−0.424337	+	0.905504i	$0.639493\pi$
$692$	27.4947	1.04519
$693$	0	0
$694$	−41.7815	−1.58600
$695$	−12.1834	−0.462143
$696$	0	0
$697$	−45.9442	−1.74026
$698$	46.8781	1.77436
$699$	0	0
$700$	7.80098	0.294849
$701$	12.7792	0.482665	0.241333	−	0.970442i	$-0.422416\pi$
$701$	12.7792	0.482665	0.241333	+	0.970442i	$0.422416\pi$
$702$	0	0
$703$	−9.80098	−0.369651
$704$	−52.3039	−1.97128
$705$	0	0
$706$	−27.8576	−1.04844
$707$	−31.2807	−1.17643
$708$	0	0
$709$	−37.1422	−1.39490	−0.697452	−	0.716631i	$-0.745683\pi$
$709$	−37.1422	−1.39490	−0.697452	+	0.716631i	$0.745683\pi$
$710$	0.738205	0.0277043
$711$	0	0
$712$	21.6898	0.812860
$713$	11.3340	0.424463
$714$	0	0
$715$	15.3691	0.574772
$716$	18.5958	0.694959
$717$	0	0
$718$	34.6114	1.29169
$719$	13.7515	0.512846	0.256423	−	0.966565i	$-0.417456\pi$
$719$	13.7515	0.512846	0.256423	+	0.966565i	$0.417456\pi$
$720$	0	0
$721$	36.5113	1.35975
$722$	2.17009	0.0807623
$723$	0	0
$724$	−6.49693	−0.241456
$725$	−0.986669	−0.0366440
$726$	0	0
$727$	42.1145	1.56194	0.780970	−	0.624568i	$-0.214725\pi$
$727$	42.1145	1.56194	0.780970	+	0.624568i	$0.214725\pi$
$728$	−16.0326	−0.594209
$729$	0	0
$730$	21.1773	0.783806
$731$	−67.2450	−2.48715
$732$	0	0
$733$	−14.2413	−0.526014	−0.263007	−	0.964794i	$-0.584714\pi$
$733$	−14.2413	−0.526014	−0.263007	+	0.964794i	$0.584714\pi$
$734$	−40.0482	−1.47821
$735$	0	0
$736$	−35.1422	−1.29536
$737$	23.7998	0.876675
$738$	0	0
$739$	−33.5441	−1.23394	−0.616970	−	0.786987i	$-0.711640\pi$
$739$	−33.5441	−1.23394	−0.616970	+	0.786987i	$0.711640\pi$
$740$	26.5536	0.976128
$741$	0	0
$742$	−67.0447	−2.46129
$743$	−50.5692	−1.85520	−0.927601	−	0.373572i	$-0.878133\pi$
$743$	−50.5692	−1.85520	−0.927601	+	0.373572i	$0.878133\pi$
$744$	0	0
$745$	−5.84324	−0.214080
$746$	−71.7803	−2.62806
$747$	0	0
$748$	76.3234	2.79066
$749$	−24.4657	−0.893958
$750$	0	0
$751$	−19.9916	−0.729503	−0.364752	−	0.931105i	$-0.618846\pi$
$751$	−19.9916	−0.729503	−0.364752	+	0.931105i	$0.618846\pi$
$752$	−21.2123	−0.773535
$753$	0	0
$754$	7.74578	0.282085
$755$	−6.99773	−0.254674
$756$	0	0
$757$	2.22899	0.0810140	0.0405070	−	0.999179i	$-0.487103\pi$
$757$	2.22899	0.0810140	0.0405070	+	0.999179i	$0.487103\pi$
$758$	0.662094	0.0240484
$759$	0	0
$760$	−1.53919	−0.0558322
$761$	−42.2967	−1.53325	−0.766627	−	0.642093i	$-0.778066\pi$
$761$	−42.2967	−1.53325	−0.766627	+	0.642093i	$0.778066\pi$
$762$	0	0
$763$	−57.1773	−2.06996
$764$	12.8566	0.465135
$765$	0	0
$766$	−31.7321	−1.14653
$767$	3.33403	0.120385
$768$	0	0
$769$	−38.6803	−1.39485	−0.697424	−	0.716658i	$-0.745671\pi$
$769$	−38.6803	−1.39485	−0.697424	+	0.716658i	$0.745671\pi$
$770$	−26.5464	−0.956665
$771$	0	0
$772$	23.4752	0.844890
$773$	−23.8225	−0.856837	−0.428419	−	0.903580i	$-0.640929\pi$
$773$	−23.8225	−0.856837	−0.428419	+	0.903580i	$0.640929\pi$
$774$	0	0
$775$	2.44748	0.0879161
$776$	−4.19061	−0.150434
$777$	0	0
$778$	−1.38735	−0.0497390
$779$	−6.92881	−0.248250
$780$	0	0
$781$	−1.44521	−0.0517138
$782$	66.6369	2.38293
$783$	0	0
$784$	−2.68261	−0.0958076
$785$	18.8371	0.672325
$786$	0	0
$787$	4.63931	0.165373	0.0826867	−	0.996576i	$-0.473650\pi$
$787$	4.63931	0.165373	0.0826867	+	0.996576i	$0.473650\pi$
$788$	−42.3318	−1.50801
$789$	0	0
$790$	19.9155	0.708561
$791$	−25.5876	−0.909790
$792$	0	0
$793$	−29.1194	−1.03406
$794$	−54.9093	−1.94866
$795$	0	0
$796$	29.8576	1.05828
$797$	−20.1750	−0.714635	−0.357318	−	0.933983i	$-0.616309\pi$
$797$	−20.1750	−0.714635	−0.357318	+	0.933983i	$0.616309\pi$
$798$	0	0
$799$	67.6763	2.39422
$800$	−7.58864	−0.268299
$801$	0	0
$802$	36.6114	1.29279
$803$	−41.4596	−1.46308
$804$	0	0
$805$	−13.3340	−0.469963
$806$	−19.2138	−0.676776
$807$	0	0
$808$	−16.7214	−0.588256
$809$	14.2101	0.499600	0.249800	−	0.968297i	$-0.419635\pi$
$809$	14.2101	0.499600	0.249800	+	0.968297i	$0.419635\pi$
$810$	0	0
$811$	32.2511	1.13249	0.566245	−	0.824237i	$-0.308396\pi$
$811$	32.2511	1.13249	0.566245	+	0.824237i	$0.308396\pi$
$812$	−7.69699	−0.270111
$813$	0	0
$814$	−90.3605	−3.16713
$815$	−11.1929	−0.392069
$816$	0	0
$817$	−10.1412	−0.354794
$818$	83.5729	2.92206
$819$	0	0
$820$	18.7721	0.655549
$821$	23.5441	0.821695	0.410848	−	0.911704i	$-0.365233\pi$
$821$	23.5441	0.821695	0.410848	+	0.911704i	$0.365233\pi$
$822$	0	0
$823$	8.21339	0.286301	0.143150	−	0.989701i	$-0.454277\pi$
$823$	8.21339	0.286301	0.143150	+	0.989701i	$0.454277\pi$
$824$	19.5174	0.679922
$825$	0	0
$826$	−5.75872	−0.200372
$827$	30.9998	1.07797	0.538985	−	0.842316i	$-0.318808\pi$
$827$	30.9998	1.07797	0.538985	+	0.842316i	$0.318808\pi$
$828$	0	0
$829$	−25.7854	−0.895563	−0.447782	−	0.894143i	$-0.647786\pi$
$829$	−25.7854	−0.895563	−0.447782	+	0.894143i	$0.647786\pi$
$830$	−12.7877	−0.443866
$831$	0	0
$832$	44.5367	1.54403
$833$	8.55866	0.296540
$834$	0	0
$835$	10.6042	0.366975
$836$	11.5103	0.398091
$837$	0	0
$838$	9.00492	0.311070
$839$	−35.1240	−1.21261	−0.606307	−	0.795231i	$-0.707350\pi$
$839$	−35.1240	−1.21261	−0.606307	+	0.795231i	$0.707350\pi$
$840$	0	0
$841$	−28.0265	−0.966430
$842$	60.4534	2.08336
$843$	0	0
$844$	27.1650	0.935057
$845$	−0.0867882	−0.00298561
$846$	0	0
$847$	20.2979	0.697445
$848$	22.3006	0.765805
$849$	0	0
$850$	14.3896	0.493560
$851$	−45.3874	−1.55586
$852$	0	0
$853$	6.22899	0.213277	0.106638	−	0.994298i	$-0.465991\pi$
$853$	6.22899	0.213277	0.106638	+	0.994298i	$0.465991\pi$
$854$	50.2967	1.72112
$855$	0	0
$856$	−13.0784	−0.447010
$857$	13.8804	0.474146	0.237073	−	0.971492i	$-0.423812\pi$
$857$	13.8804	0.474146	0.237073	+	0.971492i	$0.423812\pi$
$858$	0	0
$859$	−42.6369	−1.45475	−0.727375	−	0.686240i	$-0.759260\pi$
$859$	−42.6369	−1.45475	−0.727375	+	0.686240i	$0.759260\pi$
$860$	27.4752	0.936896
$861$	0	0
$862$	−11.7009	−0.398533
$863$	−44.4969	−1.51469	−0.757347	−	0.653013i	$-0.773505\pi$
$863$	−44.4969	−1.51469	−0.757347	+	0.653013i	$0.773505\pi$
$864$	0	0
$865$	−10.1483	−0.345054
$866$	−3.20128	−0.108784
$867$	0	0
$868$	19.0928	0.648050
$869$	−38.9893	−1.32262
$870$	0	0
$871$	−20.2655	−0.686670
$872$	−30.5646	−1.03505
$873$	0	0
$874$	10.0494	0.339928
$875$	−2.87936	−0.0973402
$876$	0	0
$877$	45.6319	1.54088	0.770441	−	0.637512i	$-0.220036\pi$
$877$	45.6319	1.54088	0.770441	+	0.637512i	$0.220036\pi$
$878$	−59.7152	−2.01529
$879$	0	0
$880$	8.82991	0.297656
$881$	28.3980	0.956754	0.478377	−	0.878155i	$-0.341225\pi$
$881$	28.3980	0.956754	0.478377	+	0.878155i	$0.341225\pi$
$882$	0	0
$883$	−18.6959	−0.629169	−0.314584	−	0.949230i	$-0.601865\pi$
$883$	−18.6959	−0.629169	−0.314584	+	0.949230i	$0.601865\pi$
$884$	−64.9893	−2.18583
$885$	0	0
$886$	−46.8599	−1.57429
$887$	39.8310	1.33739	0.668696	−	0.743536i	$-0.266853\pi$
$887$	39.8310	1.33739	0.668696	+	0.743536i	$0.266853\pi$
$888$	0	0
$889$	−49.0082	−1.64368
$890$	−30.5802	−1.02505
$891$	0	0
$892$	−10.8371	−0.362853
$893$	10.2062	0.341538
$894$	0	0
$895$	−6.86376	−0.229430
$896$	−33.2255	−1.10999
$897$	0	0
$898$	−36.0300	−1.20234
$899$	−2.41485	−0.0805398
$900$	0	0
$901$	−71.1482	−2.37029
$902$	−63.8804	−2.12698
$903$	0	0
$904$	−13.6781	−0.454926
$905$	2.39803	0.0797133
$906$	0	0
$907$	24.1568	0.802112	0.401056	−	0.916054i	$-0.368643\pi$
$907$	24.1568	0.802112	0.401056	+	0.916054i	$0.368643\pi$
$908$	−35.5669	−1.18033
$909$	0	0
$910$	22.6042	0.749323
$911$	48.5113	1.60725	0.803626	−	0.595135i	$-0.202901\pi$
$911$	48.5113	1.60725	0.803626	+	0.595135i	$0.202901\pi$
$912$	0	0
$913$	25.0349	0.828534
$914$	−86.6512	−2.86617
$915$	0	0
$916$	25.5792	0.845160
$917$	−26.6888	−0.881340
$918$	0	0
$919$	−13.5609	−0.447334	−0.223667	−	0.974666i	$-0.571803\pi$
$919$	−13.5609	−0.447334	−0.223667	+	0.974666i	$0.571803\pi$
$920$	−7.12783	−0.234997
$921$	0	0
$922$	−19.5174	−0.642773
$923$	1.23060	0.0405056
$924$	0	0
$925$	−9.80098	−0.322254
$926$	−56.2628	−1.84891
$927$	0	0
$928$	7.48747	0.245788
$929$	47.0784	1.54459	0.772296	−	0.635263i	$-0.219108\pi$
$929$	47.0784	1.54459	0.772296	+	0.635263i	$0.219108\pi$
$930$	0	0
$931$	1.29072	0.0423018
$932$	37.2039	1.21866
$933$	0	0
$934$	−32.1217	−1.05105
$935$	−28.1711	−0.921295
$936$	0	0
$937$	−55.8576	−1.82479	−0.912394	−	0.409312i	$-0.865769\pi$
$937$	−55.8576	−1.82479	−0.912394	+	0.409312i	$0.865769\pi$
$938$	35.0037	1.14291
$939$	0	0
$940$	−27.6514	−0.901890
$941$	29.6547	0.966717	0.483358	−	0.875423i	$-0.339417\pi$
$941$	29.6547	0.966717	0.483358	+	0.875423i	$0.339417\pi$
$942$	0	0
$943$	−32.0866	−1.04488
$944$	1.91548	0.0623435
$945$	0	0
$946$	−93.4968	−3.03984
$947$	15.1545	0.492455	0.246227	−	0.969212i	$-0.420809\pi$
$947$	15.1545	0.492455	0.246227	+	0.969212i	$0.420809\pi$
$948$	0	0
$949$	35.3028	1.14598
$950$	2.17009	0.0704069
$951$	0	0
$952$	29.3874	0.952450
$953$	−39.4101	−1.27662	−0.638310	−	0.769780i	$-0.720366\pi$
$953$	−39.4101	−1.27662	−0.638310	+	0.769780i	$0.720366\pi$
$954$	0	0
$955$	−4.74539	−0.153557
$956$	23.2690	0.752573
$957$	0	0
$958$	−81.8876	−2.64567
$959$	47.9520	1.54845
$960$	0	0
$961$	−25.0098	−0.806769
$962$	76.9420	2.48071
$963$	0	0
$964$	22.0989	0.711758
$965$	−8.66475	−0.278928
$966$	0	0
$967$	−49.6463	−1.59652	−0.798259	−	0.602314i	$-0.794245\pi$
$967$	−49.6463	−1.59652	−0.798259	+	0.602314i	$0.794245\pi$
$968$	10.8504	0.348746
$969$	0	0
$970$	5.90829	0.189704
$971$	36.2388	1.16296	0.581480	−	0.813561i	$-0.302474\pi$
$971$	36.2388	1.16296	0.581480	+	0.813561i	$0.302474\pi$
$972$	0	0
$973$	35.0805	1.12463
$974$	85.0493	2.72515
$975$	0	0
$976$	−16.7298	−0.535508
$977$	−3.12783	−0.100068	−0.0500340	−	0.998748i	$-0.515933\pi$
$977$	−3.12783	−0.100068	−0.0500340	+	0.998748i	$0.515933\pi$
$978$	0	0
$979$	59.8681	1.91339
$980$	−3.49693	−0.111705
$981$	0	0
$982$	0.507992	0.0162107
$983$	25.1812	0.803154	0.401577	−	0.915825i	$-0.368462\pi$
$983$	25.1812	0.803154	0.401577	+	0.915825i	$0.368462\pi$
$984$	0	0
$985$	15.6248	0.497846
$986$	−14.1978	−0.452150
$987$	0	0
$988$	−9.80098	−0.311811
$989$	−46.9627	−1.49333
$990$	0	0
$991$	−21.7899	−0.692180	−0.346090	−	0.938201i	$-0.612491\pi$
$991$	−21.7899	−0.692180	−0.346090	+	0.938201i	$0.612491\pi$
$992$	−18.5730	−0.589695
$993$	0	0
$994$	−2.12556	−0.0674186
$995$	−11.0205	−0.349374
$996$	0	0
$997$	−19.2495	−0.609638	−0.304819	−	0.952410i	$-0.598596\pi$
$997$	−19.2495	−0.609638	−0.304819	+	0.952410i	$0.598596\pi$
$998$	46.6369	1.47626
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.k.1.3	yes	3
3.2	odd	2		855.2.a.j.1.1	✓	3
5.4	even	2		4275.2.a.bc.1.1		3
15.14	odd	2		4275.2.a.bl.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
855.2.a.j.1.1	✓	3	3.2	odd	2
855.2.a.k.1.3	yes	3	1.1	even	1	trivial
4275.2.a.bc.1.1		3	5.4	even	2
4275.2.a.bl.1.3		3	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	855.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +2.87936 q^{7} +1.53919 q^{8} +O(q^{10})\)	\(q+2.17009 q^{2} +2.70928 q^{4} -1.00000 q^{5} +2.87936 q^{7} +1.53919 q^{8} -2.17009 q^{10} +4.24846 q^{11} -3.61757 q^{13} +6.24846 q^{14} -2.07838 q^{16} +6.63090 q^{17} +1.00000 q^{19} -2.70928 q^{20} +9.21953 q^{22} +4.63090 q^{23} +1.00000 q^{25} -7.85043 q^{26} +7.80098 q^{28} -0.986669 q^{29} +2.44748 q^{31} -7.58864 q^{32} +14.3896 q^{34} -2.87936 q^{35} -9.80098 q^{37} +2.17009 q^{38} -1.53919 q^{40} -6.92881 q^{41} -10.1412 q^{43} +11.5103 q^{44} +10.0494 q^{46} +10.2062 q^{47} +1.29072 q^{49} +2.17009 q^{50} -9.80098 q^{52} -10.7298 q^{53} -4.24846 q^{55} +4.43188 q^{56} -2.14116 q^{58} -0.921622 q^{59} +8.04945 q^{61} +5.31124 q^{62} -12.3112 q^{64} +3.61757 q^{65} +5.60197 q^{67} +17.9649 q^{68} -6.24846 q^{70} -0.340173 q^{71} -9.75872 q^{73} -21.2690 q^{74} +2.70928 q^{76} +12.2329 q^{77} -9.17727 q^{79} +2.07838 q^{80} -15.0361 q^{82} +5.89269 q^{83} -6.63090 q^{85} -22.0072 q^{86} +6.53919 q^{88} +14.0917 q^{89} -10.4163 q^{91} +12.5464 q^{92} +22.1483 q^{94} -1.00000 q^{95} -2.72261 q^{97} +2.80098 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8	\( 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8} - q^{10} + 4 q^{11} - 6 q^{13} + 10 q^{14} - 3 q^{16} + 16 q^{17} + 3 q^{19} - q^{20} + 4 q^{22} + 10 q^{23} + 3 q^{25} + 4 q^{26} + 14 q^{28} - 2 q^{29} + 8 q^{31} - 3 q^{32} + 14 q^{34} + 4 q^{35} - 20 q^{37} + q^{38} - 3 q^{40} + 10 q^{41} - 10 q^{43} + 18 q^{44} + 12 q^{46} + 6 q^{47} + 11 q^{49} + q^{50} - 20 q^{52} + 8 q^{53} - 4 q^{55} + 14 q^{58} - 6 q^{59} + 6 q^{61} - 10 q^{62} - 11 q^{64} + 6 q^{65} - 2 q^{67} + 4 q^{68} - 10 q^{70} + 10 q^{71} - 4 q^{73} - 22 q^{74} + q^{76} + 14 q^{77} + 12 q^{79} + 3 q^{80} - 26 q^{82} + 6 q^{83} - 16 q^{85} - 32 q^{86} + 18 q^{88} + 40 q^{89} - 4 q^{91} + 2 q^{92} + 12 q^{94} - 3 q^{95} - 2 q^{97} - q^{98}+O(q^{100}) \) 3 * q + q^2 + q^4 - 3 * q^5 - 4 * q^7 + 3 * q^8 - q^10 + 4 * q^11 - 6 * q^13 + 10 * q^14 - 3 * q^16 + 16 * q^17 + 3 * q^19 - q^20 + 4 * q^22 + 10 * q^23 + 3 * q^25 + 4 * q^26 + 14 * q^28 - 2 * q^29 + 8 * q^31 - 3 * q^32 + 14 * q^34 + 4 * q^35 - 20 * q^37 + q^38 - 3 * q^40 + 10 * q^41 - 10 * q^43 + 18 * q^44 + 12 * q^46 + 6 * q^47 + 11 * q^49 + q^50 - 20 * q^52 + 8 * q^53 - 4 * q^55 + 14 * q^58 - 6 * q^59 + 6 * q^61 - 10 * q^62 - 11 * q^64 + 6 * q^65 - 2 * q^67 + 4 * q^68 - 10 * q^70 + 10 * q^71 - 4 * q^73 - 22 * q^74 + q^76 + 14 * q^77 + 12 * q^79 + 3 * q^80 - 26 * q^82 + 6 * q^83 - 16 * q^85 - 32 * q^86 + 18 * q^88 + 40 * q^89 - 4 * q^91 + 2 * q^92 + 12 * q^94 - 3 * q^95 - 2 * q^97 - q^98

Introduction

L-functions

Modular forms

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