LMFDB - Embedded newform 855.2.a.k.1.1

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			1.1
Root			$-1.48119$ 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)

$f(q)$	$=$	$q-1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} -3.28726 q^{7} +2.67513 q^{8} +O(q^{10})$	\(q-1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} -3.28726 q^{7} +2.67513 q^{8} +1.48119 q^{10} +2.86907 q^{11} -7.02539 q^{13} +4.86907 q^{14} -4.35026 q^{16} +1.84367 q^{17} +1.00000 q^{19} -0.193937 q^{20} -4.24965 q^{22} -0.156325 q^{23} +1.00000 q^{25} +10.4060 q^{26} -0.637519 q^{28} -9.18172 q^{29} +9.50659 q^{31} +1.09332 q^{32} -2.73084 q^{34} +3.28726 q^{35} -1.36248 q^{37} -1.48119 q^{38} -2.67513 q^{40} +9.05571 q^{41} +5.59991 q^{43} +0.556417 q^{44} +0.231548 q^{46} +4.93207 q^{47} +3.80606 q^{49} -1.48119 q^{50} -1.36248 q^{52} +13.6932 q^{53} -2.86907 q^{55} -8.79384 q^{56} +13.5999 q^{58} +1.35026 q^{59} -1.76845 q^{61} -14.0811 q^{62} +7.08110 q^{64} +7.02539 q^{65} -11.2750 q^{67} +0.357556 q^{68} -4.86907 q^{70} +6.96239 q^{71} +2.57452 q^{73} +2.01810 q^{74} +0.193937 q^{76} -9.43136 q^{77} +8.18664 q^{79} +4.35026 q^{80} -13.4133 q^{82} -8.46898 q^{83} -1.84367 q^{85} -8.29455 q^{86} +7.67513 q^{88} +8.16854 q^{89} +23.0943 q^{91} -0.0303172 q^{92} -7.30536 q^{94} -1.00000 q^{95} +7.98778 q^{97} -5.63752 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.48119	−1.04736	−0.523681	−	0.851914i	$-0.675442\pi$
$2$	−1.48119	−1.04736	−0.523681	+	0.851914i	$0.675442\pi$
$3$	0	0
$3$	0	0
$4$	0.193937	0.0969683
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.28726	−1.24247	−0.621233	−	0.783626i	$-0.713368\pi$
$7$	−3.28726	−1.24247	−0.621233	+	0.783626i	$0.713368\pi$
$8$	2.67513	0.945802
$9$	0	0
$10$	1.48119	0.468395
$11$	2.86907	0.865056	0.432528	−	0.901620i	$-0.357622\pi$
$11$	2.86907	0.865056	0.432528	+	0.901620i	$0.357622\pi$
$12$	0	0
$13$	−7.02539	−1.94849	−0.974247	−	0.225485i	$-0.927603\pi$
$13$	−7.02539	−1.94849	−0.974247	+	0.225485i	$0.927603\pi$
$14$	4.86907	1.30131
$15$	0	0
$16$	−4.35026	−1.08757
$17$	1.84367	0.447157	0.223578	−	0.974686i	$-0.428226\pi$
$17$	1.84367	0.447157	0.223578	+	0.974686i	$0.428226\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.193937	−0.0433655
$21$	0	0
$22$	−4.24965	−0.906028
$23$	−0.156325	−0.0325961	−0.0162980	−	0.999867i	$-0.505188\pi$
$23$	−0.156325	−0.0325961	−0.0162980	+	0.999867i	$0.505188\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	10.4060	2.04078
$27$	0	0
$28$	−0.637519	−0.120480
$29$	−9.18172	−1.70500	−0.852501	−	0.522725i	$-0.824915\pi$
$29$	−9.18172	−1.70500	−0.852501	+	0.522725i	$0.824915\pi$
$30$	0	0
$31$	9.50659	1.70743	0.853717	−	0.520738i	$-0.174343\pi$
$31$	9.50659	1.70743	0.853717	+	0.520738i	$0.174343\pi$
$32$	1.09332	0.193274
$33$	0	0
$34$	−2.73084	−0.468335
$35$	3.28726	0.555648
$36$	0	0
$37$	−1.36248	−0.223990	−0.111995	−	0.993709i	$-0.535724\pi$
$37$	−1.36248	−0.223990	−0.111995	+	0.993709i	$0.535724\pi$
$38$	−1.48119	−0.240281
$39$	0	0
$40$	−2.67513	−0.422975
$41$	9.05571	1.41426	0.707132	−	0.707081i	$-0.249989\pi$
$41$	9.05571	1.41426	0.707132	+	0.707081i	$0.249989\pi$
$42$	0	0
$43$	5.59991	0.853978	0.426989	−	0.904257i	$-0.359574\pi$
$43$	5.59991	0.853978	0.426989	+	0.904257i	$0.359574\pi$
$44$	0.556417	0.0838830
$45$	0	0
$46$	0.231548	0.0341399
$47$	4.93207	0.719417	0.359708	−	0.933065i	$-0.382876\pi$
$47$	4.93207	0.719417	0.359708	+	0.933065i	$0.382876\pi$
$48$	0	0
$49$	3.80606	0.543723
$50$	−1.48119	−0.209473
$51$	0	0
$52$	−1.36248	−0.188942
$53$	13.6932	1.88091	0.940455	−	0.339919i	$-0.110400\pi$
$53$	13.6932	1.88091	0.940455	+	0.339919i	$0.110400\pi$
$54$	0	0
$55$	−2.86907	−0.386865
$56$	−8.79384	−1.17513
$57$	0	0
$58$	13.5999	1.78576
$59$	1.35026	0.175789	0.0878946	−	0.996130i	$-0.471986\pi$
$59$	1.35026	0.175789	0.0878946	+	0.996130i	$0.471986\pi$
$60$	0	0
$61$	−1.76845	−0.226427	−0.113214	−	0.993571i	$-0.536114\pi$
$61$	−1.76845	−0.226427	−0.113214	+	0.993571i	$0.536114\pi$
$62$	−14.0811	−1.78830
$63$	0	0
$64$	7.08110	0.885138
$65$	7.02539	0.871393
$66$	0	0
$67$	−11.2750	−1.37747	−0.688733	−	0.725015i	$-0.741833\pi$
$67$	−11.2750	−1.37747	−0.688733	+	0.725015i	$0.741833\pi$
$68$	0.357556	0.0433600
$69$	0	0
$70$	−4.86907	−0.581965
$71$	6.96239	0.826284	0.413142	−	0.910667i	$-0.364431\pi$
$71$	6.96239	0.826284	0.413142	+	0.910667i	$0.364431\pi$
$72$	0	0
$73$	2.57452	0.301324	0.150662	−	0.988585i	$-0.451859\pi$
$73$	2.57452	0.301324	0.150662	+	0.988585i	$0.451859\pi$
$74$	2.01810	0.234599
$75$	0	0
$76$	0.193937	0.0222460
$77$	−9.43136	−1.07480
$78$	0	0
$79$	8.18664	0.921069	0.460535	−	0.887642i	$-0.347658\pi$
$79$	8.18664	0.921069	0.460535	+	0.887642i	$0.347658\pi$
$80$	4.35026	0.486374
$81$	0	0
$82$	−13.4133	−1.48125
$83$	−8.46898	−0.929591	−0.464795	−	0.885418i	$-0.653872\pi$
$83$	−8.46898	−0.929591	−0.464795	+	0.885418i	$0.653872\pi$
$84$	0	0
$85$	−1.84367	−0.199975
$86$	−8.29455	−0.894425
$87$	0	0
$88$	7.67513	0.818172
$89$	8.16854	0.865864	0.432932	−	0.901427i	$-0.357479\pi$
$89$	8.16854	0.865864	0.432932	+	0.901427i	$0.357479\pi$
$90$	0	0
$91$	23.0943	2.42094
$92$	−0.0303172	−0.00316078
$93$	0	0
$94$	−7.30536	−0.753490
$95$	−1.00000	−0.102598
$96$	0	0
$97$	7.98778	0.811036	0.405518	−	0.914087i	$-0.367091\pi$
$97$	7.98778	0.811036	0.405518	+	0.914087i	$0.367091\pi$
$98$	−5.63752	−0.569475
$99$	0	0
$100$	0.193937	0.0193937
$101$	15.5877	1.55103	0.775517	−	0.631327i	$-0.217490\pi$
$101$	15.5877	1.55103	0.775517	+	0.631327i	$0.217490\pi$
$102$	0	0
$103$	−1.92478	−0.189654	−0.0948270	−	0.995494i	$-0.530230\pi$
$103$	−1.92478	−0.189654	−0.0948270	+	0.995494i	$0.530230\pi$
$104$	−18.7938	−1.84289
$105$	0	0
$106$	−20.2823	−1.96999
$107$	−5.73813	−0.554726	−0.277363	−	0.960765i	$-0.589461\pi$
$107$	−5.73813	−0.554726	−0.277363	+	0.960765i	$0.589461\pi$
$108$	0	0
$109$	12.1114	1.16006	0.580032	−	0.814594i	$-0.303040\pi$
$109$	12.1114	1.16006	0.580032	+	0.814594i	$0.303040\pi$
$110$	4.24965	0.405188
$111$	0	0
$112$	14.3004	1.35126
$113$	10.9927	1.03411	0.517053	−	0.855953i	$-0.327029\pi$
$113$	10.9927	1.03411	0.517053	+	0.855953i	$0.327029\pi$
$114$	0	0
$115$	0.156325	0.0145774
$116$	−1.78067	−0.165331
$117$	0	0
$118$	−2.00000	−0.184115
$119$	−6.06063	−0.555577
$120$	0	0
$121$	−2.76845	−0.251677
$122$	2.61942	0.237151
$123$	0	0
$124$	1.84367	0.165567
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	4.88717	0.433666	0.216833	−	0.976209i	$-0.430427\pi$
$127$	4.88717	0.433666	0.216833	+	0.976209i	$0.430427\pi$
$128$	−12.6751	−1.12033
$129$	0	0
$130$	−10.4060	−0.912664
$131$	14.0181	1.22477	0.612383	−	0.790561i	$-0.290211\pi$
$131$	14.0181	1.22477	0.612383	+	0.790561i	$0.290211\pi$
$132$	0	0
$133$	−3.28726	−0.285041
$134$	16.7005	1.44271
$135$	0	0
$136$	4.93207	0.422922
$137$	18.4387	1.57532	0.787660	−	0.616110i	$-0.211292\pi$
$137$	18.4387	1.57532	0.787660	+	0.616110i	$0.211292\pi$
$138$	0	0
$139$	0.337088	0.0285914	0.0142957	−	0.999898i	$-0.495449\pi$
$139$	0.337088	0.0285914	0.0142957	+	0.999898i	$0.495449\pi$
$140$	0.637519	0.0538802
$141$	0	0
$142$	−10.3127	−0.865418
$143$	−20.1563	−1.68556
$144$	0	0
$145$	9.18172	0.762500
$146$	−3.81336	−0.315596
$147$	0	0
$148$	−0.264235	−0.0217200
$149$	1.29948	0.106457	0.0532286	−	0.998582i	$-0.483049\pi$
$149$	1.29948	0.106457	0.0532286	+	0.998582i	$0.483049\pi$
$150$	0	0
$151$	−21.4821	−1.74819	−0.874096	−	0.485753i	$-0.838546\pi$
$151$	−21.4821	−1.74819	−0.874096	+	0.485753i	$0.838546\pi$
$152$	2.67513	0.216982
$153$	0	0
$154$	13.9697	1.12571
$155$	−9.50659	−0.763587
$156$	0	0
$157$	−8.77575	−0.700381	−0.350190	−	0.936679i	$-0.613883\pi$
$157$	−8.77575	−0.700381	−0.350190	+	0.936679i	$0.613883\pi$
$158$	−12.1260	−0.964693
$159$	0	0
$160$	−1.09332	−0.0864346
$161$	0.513881	0.0404995
$162$	0	0
$163$	14.1138	1.10548	0.552739	−	0.833355i	$-0.313583\pi$
$163$	14.1138	1.10548	0.552739	+	0.833355i	$0.313583\pi$
$164$	1.75623	0.137139
$165$	0	0
$166$	12.5442	0.973619
$167$	−22.2071	−1.71844	−0.859219	−	0.511608i	$-0.829050\pi$
$167$	−22.2071	−1.71844	−0.859219	+	0.511608i	$0.829050\pi$
$168$	0	0
$169$	36.3561	2.79663
$170$	2.73084	0.209446
$171$	0	0
$172$	1.08603	0.0828088
$173$	−19.3054	−1.46776	−0.733880	−	0.679280i	$-0.762292\pi$
$173$	−19.3054	−1.46776	−0.733880	+	0.679280i	$0.762292\pi$
$174$	0	0
$175$	−3.28726	−0.248493
$176$	−12.4812	−0.940805
$177$	0	0
$178$	−12.0992	−0.906873
$179$	−19.5877	−1.46405	−0.732026	−	0.681276i	$-0.761425\pi$
$179$	−19.5877	−1.46405	−0.732026	+	0.681276i	$0.761425\pi$
$180$	0	0
$181$	−19.2750	−1.43270	−0.716351	−	0.697740i	$-0.754189\pi$
$181$	−19.2750	−1.43270	−0.716351	+	0.697740i	$0.754189\pi$
$182$	−34.2071	−2.53560
$183$	0	0
$184$	−0.418190	−0.0308294
$185$	1.36248	0.100172
$186$	0	0
$187$	5.28963	0.386816
$188$	0.956509	0.0697606
$189$	0	0
$190$	1.48119	0.107457
$191$	0.607202	0.0439356	0.0219678	−	0.999759i	$-0.493007\pi$
$191$	0.607202	0.0439356	0.0219678	+	0.999759i	$0.493007\pi$
$192$	0	0
$193$	−26.2252	−1.88773	−0.943866	−	0.330329i	$-0.892840\pi$
$193$	−26.2252	−1.88773	−0.943866	+	0.330329i	$0.892840\pi$
$194$	−11.8315	−0.849449
$195$	0	0
$196$	0.738135	0.0527239
$197$	−5.31994	−0.379030	−0.189515	−	0.981878i	$-0.560692\pi$
$197$	−5.31994	−0.379030	−0.189515	+	0.981878i	$0.560692\pi$
$198$	0	0
$199$	−10.8872	−0.771771	−0.385885	−	0.922547i	$-0.626104\pi$
$199$	−10.8872	−0.771771	−0.385885	+	0.922547i	$0.626104\pi$
$200$	2.67513	0.189160
$201$	0	0
$202$	−23.0884	−1.62449
$203$	30.1827	2.11841
$204$	0	0
$205$	−9.05571	−0.632478
$206$	2.85097	0.198636
$207$	0	0
$208$	30.5623	2.11911
$209$	2.86907	0.198458
$210$	0	0
$211$	−6.36344	−0.438077	−0.219038	−	0.975716i	$-0.570292\pi$
$211$	−6.36344	−0.438077	−0.219038	+	0.975716i	$0.570292\pi$
$212$	2.65562	0.182389
$213$	0	0
$214$	8.49929	0.581000
$215$	−5.59991	−0.381911
$216$	0	0
$217$	−31.2506	−2.12143
$218$	−17.9394	−1.21501
$219$	0	0
$220$	−0.556417	−0.0375136
$221$	−12.9525	−0.871282
$222$	0	0
$223$	−4.00000	−0.267860	−0.133930	−	0.990991i	$-0.542760\pi$
$223$	−4.00000	−0.267860	−0.133930	+	0.990991i	$0.542760\pi$
$224$	−3.59403	−0.240136
$225$	0	0
$226$	−16.2823	−1.08308
$227$	−5.58181	−0.370478	−0.185239	−	0.982694i	$-0.559306\pi$
$227$	−5.58181	−0.370478	−0.185239	+	0.982694i	$0.559306\pi$
$228$	0	0
$229$	10.9829	0.725768	0.362884	−	0.931834i	$-0.381792\pi$
$229$	10.9829	0.725768	0.362884	+	0.931834i	$0.381792\pi$
$230$	−0.231548	−0.0152678
$231$	0	0
$232$	−24.5623	−1.61259
$233$	17.7889	1.16539	0.582695	−	0.812691i	$-0.301998\pi$
$233$	17.7889	1.16539	0.582695	+	0.812691i	$0.301998\pi$
$234$	0	0
$235$	−4.93207	−0.321733
$236$	0.261865	0.0170460
$237$	0	0
$238$	8.97698	0.581891
$239$	−0.0933212	−0.00603644	−0.00301822	−	0.999995i	$-0.500961\pi$
$239$	−0.0933212	−0.00603644	−0.00301822	+	0.999995i	$0.500961\pi$
$240$	0	0
$241$	12.7005	0.818113	0.409056	−	0.912509i	$-0.365858\pi$
$241$	12.7005	0.818113	0.409056	+	0.912509i	$0.365858\pi$
$242$	4.10062	0.263598
$243$	0	0
$244$	−0.342968	−0.0219562
$245$	−3.80606	−0.243160
$246$	0	0
$247$	−7.02539	−0.447015
$248$	25.4314	1.61489
$249$	0	0
$250$	1.48119	0.0936790
$251$	4.74306	0.299379	0.149690	−	0.988733i	$-0.452173\pi$
$251$	4.74306	0.299379	0.149690	+	0.988733i	$0.452173\pi$
$252$	0	0
$253$	−0.448507	−0.0281974
$254$	−7.23884	−0.454205
$255$	0	0
$256$	4.61213	0.288258
$257$	25.2955	1.57789	0.788945	−	0.614464i	$-0.210628\pi$
$257$	25.2955	1.57789	0.788945	+	0.614464i	$0.210628\pi$
$258$	0	0
$259$	4.47882	0.278301
$260$	1.36248	0.0844975
$261$	0	0
$262$	−20.7635	−1.28278
$263$	1.64244	0.101277	0.0506387	−	0.998717i	$-0.483874\pi$
$263$	1.64244	0.101277	0.0506387	+	0.998717i	$0.483874\pi$
$264$	0	0
$265$	−13.6932	−0.841168
$266$	4.86907	0.298542
$267$	0	0
$268$	−2.18664	−0.133570
$269$	14.3453	0.874651	0.437325	−	0.899303i	$-0.355926\pi$
$269$	14.3453	0.874651	0.437325	+	0.899303i	$0.355926\pi$
$270$	0	0
$271$	25.6483	1.55802	0.779012	−	0.627009i	$-0.215721\pi$
$271$	25.6483	1.55802	0.779012	+	0.627009i	$0.215721\pi$
$272$	−8.02047	−0.486312
$273$	0	0
$274$	−27.3112	−1.64993
$275$	2.86907	0.173011
$276$	0	0
$277$	14.5647	0.875106	0.437553	−	0.899193i	$-0.355845\pi$
$277$	14.5647	0.875106	0.437553	+	0.899193i	$0.355845\pi$
$278$	−0.499293	−0.0299456
$279$	0	0
$280$	8.79384	0.525533
$281$	−11.7054	−0.698288	−0.349144	−	0.937069i	$-0.613528\pi$
$281$	−11.7054	−0.698288	−0.349144	+	0.937069i	$0.613528\pi$
$282$	0	0
$283$	−4.89938	−0.291238	−0.145619	−	0.989341i	$-0.546517\pi$
$283$	−4.89938	−0.291238	−0.145619	+	0.989341i	$0.546517\pi$
$284$	1.35026	0.0801233
$285$	0	0
$286$	29.8554	1.76539
$287$	−29.7685	−1.75718
$288$	0	0
$289$	−13.6009	−0.800051
$290$	−13.5999	−0.798614
$291$	0	0
$292$	0.499293	0.0292189
$293$	28.2071	1.64788	0.823938	−	0.566679i	$-0.191772\pi$
$293$	28.2071	1.64788	0.823938	+	0.566679i	$0.191772\pi$
$294$	0	0
$295$	−1.35026	−0.0786153
$296$	−3.64481	−0.211850
$297$	0	0
$298$	−1.92478	−0.111499
$299$	1.09825	0.0635132
$300$	0	0
$301$	−18.4083	−1.06104
$302$	31.8192	1.83099
$303$	0	0
$304$	−4.35026	−0.249505
$305$	1.76845	0.101261
$306$	0	0
$307$	−7.41090	−0.422962	−0.211481	−	0.977382i	$-0.567829\pi$
$307$	−7.41090	−0.422962	−0.211481	+	0.977382i	$0.567829\pi$
$308$	−1.82909	−0.104222
$309$	0	0
$310$	14.0811	0.799753
$311$	−8.59261	−0.487242	−0.243621	−	0.969870i	$-0.578335\pi$
$311$	−8.59261	−0.487242	−0.243621	+	0.969870i	$0.578335\pi$
$312$	0	0
$313$	6.23743	0.352560	0.176280	−	0.984340i	$-0.443594\pi$
$313$	6.23743	0.352560	0.176280	+	0.984340i	$0.443594\pi$
$314$	12.9986	0.733553
$315$	0	0
$316$	1.58769	0.0893145
$317$	−8.26774	−0.464363	−0.232181	−	0.972673i	$-0.574586\pi$
$317$	−8.26774	−0.464363	−0.232181	+	0.972673i	$0.574586\pi$
$318$	0	0
$319$	−26.3430	−1.47492
$320$	−7.08110	−0.395846
$321$	0	0
$322$	−0.761158	−0.0424177
$323$	1.84367	0.102585
$324$	0	0
$325$	−7.02539	−0.389699
$326$	−20.9053	−1.15784
$327$	0	0
$328$	24.2252	1.33761
$329$	−16.2130	−0.893851
$330$	0	0
$331$	20.5442	1.12921	0.564606	−	0.825361i	$-0.309028\pi$
$331$	20.5442	1.12921	0.564606	+	0.825361i	$0.309028\pi$
$332$	−1.64244	−0.0901408
$333$	0	0
$334$	32.8930	1.79983
$335$	11.2750	0.616021
$336$	0	0
$337$	0.0630040	0.00343205	0.00171602	−	0.999999i	$-0.499454\pi$
$337$	0.0630040	0.00343205	0.00171602	+	0.999999i	$0.499454\pi$
$338$	−53.8505	−2.92908
$339$	0	0
$340$	−0.357556	−0.0193912
$341$	27.2750	1.47703
$342$	0	0
$343$	10.4993	0.566909
$344$	14.9805	0.807694
$345$	0	0
$346$	28.5950	1.53728
$347$	24.3185	1.30549	0.652744	−	0.757579i	$-0.273618\pi$
$347$	24.3185	1.30549	0.652744	+	0.757579i	$0.273618\pi$
$348$	0	0
$349$	4.72496	0.252921	0.126461	−	0.991972i	$-0.459638\pi$
$349$	4.72496	0.252921	0.126461	+	0.991972i	$0.459638\pi$
$350$	4.86907	0.260263
$351$	0	0
$352$	3.13681	0.167193
$353$	−2.77575	−0.147738	−0.0738690	−	0.997268i	$-0.523535\pi$
$353$	−2.77575	−0.147738	−0.0738690	+	0.997268i	$0.523535\pi$
$354$	0	0
$355$	−6.96239	−0.369525
$356$	1.58418	0.0839613
$357$	0	0
$358$	29.0132	1.53339
$359$	−21.9429	−1.15810	−0.579050	−	0.815292i	$-0.696577\pi$
$359$	−21.9429	−1.15810	−0.579050	+	0.815292i	$0.696577\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	28.5501	1.50056
$363$	0	0
$364$	4.47882	0.234754
$365$	−2.57452	−0.134756
$366$	0	0
$367$	−11.8011	−0.616014	−0.308007	−	0.951384i	$-0.599662\pi$
$367$	−11.8011	−0.616014	−0.308007	+	0.951384i	$0.599662\pi$
$368$	0.680055	0.0354503
$369$	0	0
$370$	−2.01810	−0.104916
$371$	−45.0132	−2.33697
$372$	0	0
$373$	12.3611	0.640032	0.320016	−	0.947412i	$-0.396312\pi$
$373$	12.3611	0.640032	0.320016	+	0.947412i	$0.396312\pi$
$374$	−7.83497	−0.405136
$375$	0	0
$376$	13.1939	0.680425
$377$	64.5052	3.32219
$378$	0	0
$379$	−24.6048	−1.26387	−0.631933	−	0.775023i	$-0.717738\pi$
$379$	−24.6048	−1.26387	−0.631933	+	0.775023i	$0.717738\pi$
$380$	−0.193937	−0.00994874
$381$	0	0
$382$	−0.899385	−0.0460165
$383$	24.1622	1.23463	0.617315	−	0.786716i	$-0.288220\pi$
$383$	24.1622	1.23463	0.617315	+	0.786716i	$0.288220\pi$
$384$	0	0
$385$	9.43136	0.480667
$386$	38.8446	1.97714
$387$	0	0
$388$	1.54912	0.0786448
$389$	−29.8496	−1.51343	−0.756716	−	0.653743i	$-0.773198\pi$
$389$	−29.8496	−1.51343	−0.756716	+	0.653743i	$0.773198\pi$
$390$	0	0
$391$	−0.288213	−0.0145755
$392$	10.1817	0.514254
$393$	0	0
$394$	7.87987	0.396982
$395$	−8.18664	−0.411915
$396$	0	0
$397$	28.0870	1.40965	0.704823	−	0.709384i	$-0.251027\pi$
$397$	28.0870	1.40965	0.704823	+	0.709384i	$0.251027\pi$
$398$	16.1260	0.808324
$399$	0	0
$400$	−4.35026	−0.217513
$401$	−23.2931	−1.16320	−0.581602	−	0.813474i	$-0.697574\pi$
$401$	−23.2931	−1.16320	−0.581602	+	0.813474i	$0.697574\pi$
$402$	0	0
$403$	−66.7875	−3.32692
$404$	3.02302	0.150401
$405$	0	0
$406$	−44.7064	−2.21874
$407$	−3.90905	−0.193764
$408$	0	0
$409$	8.32724	0.411755	0.205878	−	0.978578i	$-0.433995\pi$
$409$	8.32724	0.411755	0.205878	+	0.978578i	$0.433995\pi$
$410$	13.4133	0.662434
$411$	0	0
$412$	−0.373285	−0.0183904
$413$	−4.43866	−0.218412
$414$	0	0
$415$	8.46898	0.415726
$416$	−7.68101	−0.376593
$417$	0	0
$418$	−4.24965	−0.207857
$419$	22.4060	1.09460	0.547302	−	0.836935i	$-0.315655\pi$
$419$	22.4060	1.09460	0.547302	+	0.836935i	$0.315655\pi$
$420$	0	0
$421$	−4.11142	−0.200378	−0.100189	−	0.994968i	$-0.531945\pi$
$421$	−4.11142	−0.200378	−0.100189	+	0.994968i	$0.531945\pi$
$422$	9.42548	0.458825
$423$	0	0
$424$	36.6312	1.77897
$425$	1.84367	0.0894314
$426$	0	0
$427$	5.81336	0.281328
$428$	−1.11283	−0.0537909
$429$	0	0
$430$	8.29455	0.399999
$431$	−16.7513	−0.806882	−0.403441	−	0.915006i	$-0.632186\pi$
$431$	−16.7513	−0.806882	−0.403441	+	0.915006i	$0.632186\pi$
$432$	0	0
$433$	27.0860	1.30167	0.650836	−	0.759219i	$-0.274419\pi$
$433$	27.0860	1.30167	0.650836	+	0.759219i	$0.274419\pi$
$434$	46.2882	2.22191
$435$	0	0
$436$	2.34885	0.112489
$437$	−0.156325	−0.00747805
$438$	0	0
$439$	−2.85097	−0.136069	−0.0680347	−	0.997683i	$-0.521673\pi$
$439$	−2.85097	−0.136069	−0.0680347	+	0.997683i	$0.521673\pi$
$440$	−7.67513	−0.365898
$441$	0	0
$442$	19.1852	0.912548
$443$	29.2809	1.39118	0.695589	−	0.718440i	$-0.255143\pi$
$443$	29.2809	1.39118	0.695589	+	0.718440i	$0.255143\pi$
$444$	0	0
$445$	−8.16854	−0.387226
$446$	5.92478	0.280546
$447$	0	0
$448$	−23.2774	−1.09975
$449$	19.5042	0.920461	0.460230	−	0.887799i	$-0.347767\pi$
$449$	19.5042	0.920461	0.460230	+	0.887799i	$0.347767\pi$
$450$	0	0
$451$	25.9814	1.22342
$452$	2.13189	0.100276
$453$	0	0
$454$	8.26774	0.388024
$455$	−23.0943	−1.08268
$456$	0	0
$457$	−4.71511	−0.220564	−0.110282	−	0.993900i	$-0.535175\pi$
$457$	−4.71511	−0.220564	−0.110282	+	0.993900i	$0.535175\pi$
$458$	−16.2677	−0.760142
$459$	0	0
$460$	0.0303172	0.00141355
$461$	−3.47627	−0.161906	−0.0809530	−	0.996718i	$-0.525796\pi$
$461$	−3.47627	−0.161906	−0.0809530	+	0.996718i	$0.525796\pi$
$462$	0	0
$463$	18.5379	0.861527	0.430764	−	0.902465i	$-0.358244\pi$
$463$	18.5379	0.861527	0.430764	+	0.902465i	$0.358244\pi$
$464$	39.9429	1.85430
$465$	0	0
$466$	−26.3488	−1.22059
$467$	12.8667	0.595400	0.297700	−	0.954660i	$-0.403781\pi$
$467$	12.8667	0.595400	0.297700	+	0.954660i	$0.403781\pi$
$468$	0	0
$469$	37.0640	1.71145
$470$	7.30536	0.336971
$471$	0	0
$472$	3.61213	0.166262
$473$	16.0665	0.738739
$474$	0	0
$475$	1.00000	0.0458831
$476$	−1.17538	−0.0538734
$477$	0	0
$478$	0.138227	0.00632235
$479$	28.8808	1.31960	0.659799	−	0.751442i	$-0.270641\pi$
$479$	28.8808	1.31960	0.659799	+	0.751442i	$0.270641\pi$
$480$	0	0
$481$	9.57196	0.436444
$482$	−18.8119	−0.856861
$483$	0	0
$484$	−0.536904	−0.0244047
$485$	−7.98778	−0.362706
$486$	0	0
$487$	−5.59754	−0.253649	−0.126824	−	0.991925i	$-0.540478\pi$
$487$	−5.59754	−0.253649	−0.126824	+	0.991925i	$0.540478\pi$
$488$	−4.73084	−0.214155
$489$	0	0
$490$	5.63752	0.254677
$491$	26.2800	1.18600	0.592999	−	0.805203i	$-0.297944\pi$
$491$	26.2800	1.18600	0.592999	+	0.805203i	$0.297944\pi$
$492$	0	0
$493$	−16.9281	−0.762403
$494$	10.4060	0.468187
$495$	0	0
$496$	−41.3561	−1.85695
$497$	−22.8872	−1.02663
$498$	0	0
$499$	13.2144	0.591558	0.295779	−	0.955256i	$-0.404421\pi$
$499$	13.2144	0.591558	0.295779	+	0.955256i	$0.404421\pi$
$500$	−0.193937	−0.00867311
$501$	0	0
$502$	−7.02539	−0.313559
$503$	−0.432779	−0.0192967	−0.00964833	−	0.999953i	$-0.503071\pi$
$503$	−0.432779	−0.0192967	−0.00964833	+	0.999953i	$0.503071\pi$
$504$	0	0
$505$	−15.5877	−0.693643
$506$	0.664327	0.0295329
$507$	0	0
$508$	0.947800	0.0420518
$509$	−4.43041	−0.196374	−0.0981872	−	0.995168i	$-0.531304\pi$
$509$	−4.43041	−0.196374	−0.0981872	+	0.995168i	$0.531304\pi$
$510$	0	0
$511$	−8.46310	−0.374385
$512$	18.5188	0.818423
$513$	0	0
$514$	−37.4676	−1.65262
$515$	1.92478	0.0848158
$516$	0	0
$517$	14.1504	0.622336
$518$	−6.63401	−0.291482
$519$	0	0
$520$	18.7938	0.824165
$521$	7.98190	0.349693	0.174847	−	0.984596i	$-0.444057\pi$
$521$	7.98190	0.349693	0.174847	+	0.984596i	$0.444057\pi$
$522$	0	0
$523$	−21.1754	−0.925935	−0.462967	−	0.886375i	$-0.653215\pi$
$523$	−21.1754	−0.925935	−0.462967	+	0.886375i	$0.653215\pi$
$524$	2.71862	0.118764
$525$	0	0
$526$	−2.43278	−0.106074
$527$	17.5271	0.763491
$528$	0	0
$529$	−22.9756	−0.998937
$530$	20.2823	0.881008
$531$	0	0
$532$	−0.637519	−0.0276400
$533$	−63.6199	−2.75568
$534$	0	0
$535$	5.73813	0.248081
$536$	−30.1622	−1.30281
$537$	0	0
$538$	−21.2482	−0.916076
$539$	10.9199	0.470351
$540$	0	0
$541$	6.15045	0.264428	0.132214	−	0.991221i	$-0.457791\pi$
$541$	6.15045	0.264428	0.132214	+	0.991221i	$0.457791\pi$
$542$	−37.9902	−1.63182
$543$	0	0
$544$	2.01573	0.0864237
$545$	−12.1114	−0.518796
$546$	0	0
$547$	42.2736	1.80749	0.903745	−	0.428072i	$-0.140807\pi$
$547$	42.2736	1.80749	0.903745	+	0.428072i	$0.140807\pi$
$548$	3.57593	0.152756
$549$	0	0
$550$	−4.24965	−0.181206
$551$	−9.18172	−0.391154
$552$	0	0
$553$	−26.9116	−1.14440
$554$	−21.5731	−0.916553
$555$	0	0
$556$	0.0653737	0.00277246
$557$	5.97556	0.253193	0.126596	−	0.991954i	$-0.459595\pi$
$557$	5.97556	0.253193	0.126596	+	0.991954i	$0.459595\pi$
$558$	0	0
$559$	−39.3416	−1.66397
$560$	−14.3004	−0.604304
$561$	0	0
$562$	17.3380	0.731361
$563$	21.3707	0.900669	0.450334	−	0.892860i	$-0.351305\pi$
$563$	21.3707	0.900669	0.450334	+	0.892860i	$0.351305\pi$
$564$	0	0
$565$	−10.9927	−0.462467
$566$	7.25694	0.305032
$567$	0	0
$568$	18.6253	0.781500
$569$	26.6072	1.11543	0.557716	−	0.830032i	$-0.311678\pi$
$569$	26.6072	1.11543	0.557716	+	0.830032i	$0.311678\pi$
$570$	0	0
$571$	29.0884	1.21731	0.608656	−	0.793434i	$-0.291709\pi$
$571$	29.0884	1.21731	0.608656	+	0.793434i	$0.291709\pi$
$572$	−3.90905	−0.163446
$573$	0	0
$574$	44.0929	1.84040
$575$	−0.156325	−0.00651921
$576$	0	0
$577$	−20.1016	−0.836839	−0.418420	−	0.908254i	$-0.637416\pi$
$577$	−20.1016	−0.836839	−0.418420	+	0.908254i	$0.637416\pi$
$578$	20.1455	0.837943
$579$	0	0
$580$	1.78067	0.0739383
$581$	27.8397	1.15499
$582$	0	0
$583$	39.2868	1.62709
$584$	6.88717	0.284993
$585$	0	0
$586$	−41.7802	−1.72592
$587$	−9.17935	−0.378872	−0.189436	−	0.981893i	$-0.560666\pi$
$587$	−9.17935	−0.378872	−0.189436	+	0.981893i	$0.560666\pi$
$588$	0	0
$589$	9.50659	0.391712
$590$	2.00000	0.0823387
$591$	0	0
$592$	5.92715	0.243604
$593$	6.64974	0.273072	0.136536	−	0.990635i	$-0.456403\pi$
$593$	6.64974	0.273072	0.136536	+	0.990635i	$0.456403\pi$
$594$	0	0
$595$	6.06063	0.248462
$596$	0.252016	0.0103230
$597$	0	0
$598$	−1.62672	−0.0665213
$599$	−35.5223	−1.45140	−0.725701	−	0.688010i	$-0.758485\pi$
$599$	−35.5223	−1.45140	−0.725701	+	0.688010i	$0.758485\pi$
$600$	0	0
$601$	−1.87399	−0.0764417	−0.0382209	−	0.999269i	$-0.512169\pi$
$601$	−1.87399	−0.0764417	−0.0382209	+	0.999269i	$0.512169\pi$
$602$	27.2663	1.11129
$603$	0	0
$604$	−4.16617	−0.169519
$605$	2.76845	0.112554
$606$	0	0
$607$	−1.07381	−0.0435845	−0.0217923	−	0.999763i	$-0.506937\pi$
$607$	−1.07381	−0.0435845	−0.0217923	+	0.999763i	$0.506937\pi$
$608$	1.09332	0.0443400
$609$	0	0
$610$	−2.61942	−0.106057
$611$	−34.6497	−1.40178
$612$	0	0
$613$	−13.6121	−0.549789	−0.274894	−	0.961474i	$-0.588643\pi$
$613$	−13.6121	−0.549789	−0.274894	+	0.961474i	$0.588643\pi$
$614$	10.9770	0.442995
$615$	0	0
$616$	−25.2301	−1.01655
$617$	6.84226	0.275459	0.137730	−	0.990470i	$-0.456020\pi$
$617$	6.84226	0.275459	0.137730	+	0.990470i	$0.456020\pi$
$618$	0	0
$619$	−30.4894	−1.22547	−0.612737	−	0.790287i	$-0.709932\pi$
$619$	−30.4894	−1.22547	−0.612737	+	0.790287i	$0.709932\pi$
$620$	−1.84367	−0.0740438
$621$	0	0
$622$	12.7273	0.510319
$623$	−26.8521	−1.07581
$624$	0	0
$625$	1.00000	0.0400000
$626$	−9.23884	−0.369258
$627$	0	0
$628$	−1.70194	−0.0679147
$629$	−2.51197	−0.100159
$630$	0	0
$631$	−12.0362	−0.479153	−0.239577	−	0.970877i	$-0.577009\pi$
$631$	−12.0362	−0.479153	−0.239577	+	0.970877i	$0.577009\pi$
$632$	21.9003	0.871149
$633$	0	0
$634$	12.2461	0.486356
$635$	−4.88717	−0.193941
$636$	0	0
$637$	−26.7391	−1.05944
$638$	39.0191	1.54478
$639$	0	0
$640$	12.6751	0.501029
$641$	−2.84463	−0.112356	−0.0561781	−	0.998421i	$-0.517891\pi$
$641$	−2.84463	−0.112356	−0.0561781	+	0.998421i	$0.517891\pi$
$642$	0	0
$643$	−35.3136	−1.39263	−0.696316	−	0.717735i	$-0.745179\pi$
$643$	−35.3136	−1.39263	−0.696316	+	0.717735i	$0.745179\pi$
$644$	0.0996603	0.00392717
$645$	0	0
$646$	−2.73084	−0.107443
$647$	29.2203	1.14877	0.574384	−	0.818586i	$-0.305242\pi$
$647$	29.2203	1.14877	0.574384	+	0.818586i	$0.305242\pi$
$648$	0	0
$649$	3.87399	0.152067
$650$	10.4060	0.408156
$651$	0	0
$652$	2.73718	0.107196
$653$	12.6458	0.494867	0.247434	−	0.968905i	$-0.420413\pi$
$653$	12.6458	0.494867	0.247434	+	0.968905i	$0.420413\pi$
$654$	0	0
$655$	−14.0181	−0.547732
$656$	−39.3947	−1.53810
$657$	0	0
$658$	24.0146	0.936186
$659$	−24.1768	−0.941794	−0.470897	−	0.882188i	$-0.656070\pi$
$659$	−24.1768	−0.941794	−0.470897	+	0.882188i	$0.656070\pi$
$660$	0	0
$661$	−10.9380	−0.425437	−0.212718	−	0.977114i	$-0.568232\pi$
$661$	−10.9380	−0.425437	−0.212718	+	0.977114i	$0.568232\pi$
$662$	−30.4299	−1.18269
$663$	0	0
$664$	−22.6556	−0.879208
$665$	3.28726	0.127474
$666$	0	0
$667$	1.43533	0.0555763
$668$	−4.30677	−0.166634
$669$	0	0
$670$	−16.7005	−0.645198
$671$	−5.07381	−0.195872
$672$	0	0
$673$	12.0630	0.464995	0.232497	−	0.972597i	$-0.425310\pi$
$673$	12.0630	0.464995	0.232497	+	0.972597i	$0.425310\pi$
$674$	−0.0933212	−0.00359460
$675$	0	0
$676$	7.05079	0.271184
$677$	22.3430	0.858710	0.429355	−	0.903136i	$-0.358741\pi$
$677$	22.3430	0.858710	0.429355	+	0.903136i	$0.358741\pi$
$678$	0	0
$679$	−26.2579	−1.00769
$680$	−4.93207	−0.189136
$681$	0	0
$682$	−40.3996	−1.54698
$683$	−35.6629	−1.36460	−0.682302	−	0.731071i	$-0.739021\pi$
$683$	−35.6629	−1.36460	−0.682302	+	0.731071i	$0.739021\pi$
$684$	0	0
$685$	−18.4387	−0.704505
$686$	−15.5515	−0.593759
$687$	0	0
$688$	−24.3611	−0.928757
$689$	−96.2003	−3.66494
$690$	0	0
$691$	25.5633	0.972472	0.486236	−	0.873828i	$-0.338370\pi$
$691$	25.5633	0.972472	0.486236	+	0.873828i	$0.338370\pi$
$692$	−3.74401	−0.142326
$693$	0	0
$694$	−36.0205	−1.36732
$695$	−0.337088	−0.0127865
$696$	0	0
$697$	16.6958	0.632398
$698$	−6.99859	−0.264900
$699$	0	0
$700$	−0.637519	−0.0240960
$701$	−21.4617	−0.810597	−0.405298	−	0.914185i	$-0.632832\pi$
$701$	−21.4617	−0.810597	−0.405298	+	0.914185i	$0.632832\pi$
$702$	0	0
$703$	−1.36248	−0.0513869
$704$	20.3162	0.765694
$705$	0	0
$706$	4.11142	0.154735
$707$	−51.2408	−1.92711
$708$	0	0
$709$	−2.17091	−0.0815304	−0.0407652	−	0.999169i	$-0.512980\pi$
$709$	−2.17091	−0.0815304	−0.0407652	+	0.999169i	$0.512980\pi$
$710$	10.3127	0.387027
$711$	0	0
$712$	21.8519	0.818935
$713$	−1.48612	−0.0556556
$714$	0	0
$715$	20.1563	0.753804
$716$	−3.79877	−0.141967
$717$	0	0
$718$	32.5017	1.21295
$719$	15.1309	0.564289	0.282144	−	0.959372i	$-0.408954\pi$
$719$	15.1309	0.564289	0.282144	+	0.959372i	$0.408954\pi$
$720$	0	0
$721$	6.32724	0.235639
$722$	−1.48119	−0.0551243
$723$	0	0
$724$	−3.73813	−0.138927
$725$	−9.18172	−0.341000
$726$	0	0
$727$	42.7635	1.58601	0.793006	−	0.609214i	$-0.208515\pi$
$727$	42.7635	1.58601	0.793006	+	0.609214i	$0.208515\pi$
$728$	61.7802	2.28973
$729$	0	0
$730$	3.81336	0.141139
$731$	10.3244	0.381862
$732$	0	0
$733$	−26.5745	−0.981552	−0.490776	−	0.871286i	$-0.663287\pi$
$733$	−26.5745	−0.981552	−0.490776	+	0.871286i	$0.663287\pi$
$734$	17.4798	0.645190
$735$	0	0
$736$	−0.170914	−0.00629996
$737$	−32.3488	−1.19159
$738$	0	0
$739$	7.51247	0.276351	0.138175	−	0.990408i	$-0.455876\pi$
$739$	7.51247	0.276351	0.138175	+	0.990408i	$0.455876\pi$
$740$	0.264235	0.00971346
$741$	0	0
$742$	66.6733	2.44765
$743$	−44.5647	−1.63492	−0.817460	−	0.575986i	$-0.804618\pi$
$743$	−44.5647	−1.63492	−0.817460	+	0.575986i	$0.804618\pi$
$744$	0	0
$745$	−1.29948	−0.0476091
$746$	−18.3091	−0.670345
$747$	0	0
$748$	1.02585	0.0375089
$749$	18.8627	0.689229
$750$	0	0
$751$	14.0059	0.511082	0.255541	−	0.966798i	$-0.417746\pi$
$751$	14.0059	0.511082	0.255541	+	0.966798i	$0.417746\pi$
$752$	−21.4558	−0.782413
$753$	0	0
$754$	−95.5447	−3.47953
$755$	21.4821	0.781815
$756$	0	0
$757$	3.52705	0.128193	0.0640965	−	0.997944i	$-0.479583\pi$
$757$	3.52705	0.128193	0.0640965	+	0.997944i	$0.479583\pi$
$758$	36.4445	1.32373
$759$	0	0
$760$	−2.67513	−0.0970372
$761$	16.6107	0.602138	0.301069	−	0.953602i	$-0.402657\pi$
$761$	16.6107	0.602138	0.301069	+	0.953602i	$0.402657\pi$
$762$	0	0
$763$	−39.8134	−1.44134
$764$	0.117759	0.00426036
$765$	0	0
$766$	−35.7889	−1.29311
$767$	−9.48612	−0.342524
$768$	0	0
$769$	−24.0752	−0.868175	−0.434087	−	0.900871i	$-0.642929\pi$
$769$	−24.0752	−0.868175	−0.434087	+	0.900871i	$0.642929\pi$
$770$	−13.9697	−0.503432
$771$	0	0
$772$	−5.08603	−0.183050
$773$	25.7539	0.926302	0.463151	−	0.886279i	$-0.346719\pi$
$773$	25.7539	0.926302	0.463151	+	0.886279i	$0.346719\pi$
$774$	0	0
$775$	9.50659	0.341487
$776$	21.3684	0.767079
$777$	0	0
$778$	44.2130	1.58511
$779$	9.05571	0.324454
$780$	0	0
$781$	19.9756	0.714782
$782$	0.426899	0.0152659
$783$	0	0
$784$	−16.5574	−0.591335
$785$	8.77575	0.313220
$786$	0	0
$787$	33.8496	1.20661	0.603303	−	0.797512i	$-0.293851\pi$
$787$	33.8496	1.20661	0.603303	+	0.797512i	$0.293851\pi$
$788$	−1.03173	−0.0367539
$789$	0	0
$790$	12.1260	0.431424
$791$	−36.1359	−1.28484
$792$	0	0
$793$	12.4241	0.441192
$794$	−41.6023	−1.47641
$795$	0	0
$796$	−2.11142	−0.0748373
$797$	25.6688	0.909235	0.454618	−	0.890687i	$-0.349776\pi$
$797$	25.6688	0.909235	0.454618	+	0.890687i	$0.349776\pi$
$798$	0	0
$799$	9.09314	0.321692
$800$	1.09332	0.0386547
$801$	0	0
$802$	34.5017	1.21830
$803$	7.38646	0.260662
$804$	0	0
$805$	−0.513881	−0.0181119
$806$	98.9253	3.48449
$807$	0	0
$808$	41.6991	1.46697
$809$	−14.0263	−0.493140	−0.246570	−	0.969125i	$-0.579304\pi$
$809$	−14.0263	−0.493140	−0.246570	+	0.969125i	$0.579304\pi$
$810$	0	0
$811$	−39.8007	−1.39759	−0.698795	−	0.715322i	$-0.746280\pi$
$811$	−39.8007	−1.39759	−0.698795	+	0.715322i	$0.746280\pi$
$812$	5.85352	0.205418
$813$	0	0
$814$	5.79006	0.202942
$815$	−14.1138	−0.494384
$816$	0	0
$817$	5.59991	0.195916
$818$	−12.3343	−0.431257
$819$	0	0
$820$	−1.75623	−0.0613303
$821$	−17.5125	−0.611189	−0.305595	−	0.952162i	$-0.598855\pi$
$821$	−17.5125	−0.611189	−0.305595	+	0.952162i	$0.598855\pi$
$822$	0	0
$823$	−10.7734	−0.375536	−0.187768	−	0.982213i	$-0.560125\pi$
$823$	−10.7734	−0.375536	−0.187768	+	0.982213i	$0.560125\pi$
$824$	−5.14903	−0.179375
$825$	0	0
$826$	6.57452	0.228757
$827$	−35.9405	−1.24977	−0.624887	−	0.780715i	$-0.714855\pi$
$827$	−35.9405	−1.24977	−0.624887	+	0.780715i	$0.714855\pi$
$828$	0	0
$829$	2.93795	0.102039	0.0510196	−	0.998698i	$-0.483753\pi$
$829$	2.93795	0.102039	0.0510196	+	0.998698i	$0.483753\pi$
$830$	−12.5442	−0.435415
$831$	0	0
$832$	−49.7475	−1.72469
$833$	7.01714	0.243130
$834$	0	0
$835$	22.2071	0.768509
$836$	0.556417	0.0192441
$837$	0	0
$838$	−33.1876	−1.14645
$839$	−50.5402	−1.74484	−0.872421	−	0.488755i	$-0.837451\pi$
$839$	−50.5402	−1.74484	−0.872421	+	0.488755i	$0.837451\pi$
$840$	0	0
$841$	55.3039	1.90703
$842$	6.08981	0.209869
$843$	0	0
$844$	−1.23410	−0.0424796
$845$	−36.3561	−1.25069
$846$	0	0
$847$	9.10062	0.312701
$848$	−59.5691	−2.04561
$849$	0	0
$850$	−2.73084	−0.0936671
$851$	0.212990	0.00730120
$852$	0	0
$853$	7.52705	0.257721	0.128861	−	0.991663i	$-0.458868\pi$
$853$	7.52705	0.257721	0.128861	+	0.991663i	$0.458868\pi$
$854$	−8.61071	−0.294653
$855$	0	0
$856$	−15.3503	−0.524661
$857$	−11.5164	−0.393394	−0.196697	−	0.980464i	$-0.563022\pi$
$857$	−11.5164	−0.393394	−0.196697	+	0.980464i	$0.563022\pi$
$858$	0	0
$859$	23.5731	0.804304	0.402152	−	0.915573i	$-0.368262\pi$
$859$	23.5731	0.804304	0.402152	+	0.915573i	$0.368262\pi$
$860$	−1.08603	−0.0370332
$861$	0	0
$862$	24.8119	0.845098
$863$	−41.7381	−1.42078	−0.710391	−	0.703807i	$-0.751482\pi$
$863$	−41.7381	−1.42078	−0.710391	+	0.703807i	$0.751482\pi$
$864$	0	0
$865$	19.3054	0.656402
$866$	−40.1197	−1.36332
$867$	0	0
$868$	−6.06063	−0.205711
$869$	23.4880	0.796777
$870$	0	0
$871$	79.2116	2.68398
$872$	32.3996	1.09719
$873$	0	0
$874$	0.231548	0.00783223
$875$	3.28726	0.111130
$876$	0	0
$877$	21.6145	0.729870	0.364935	−	0.931033i	$-0.381091\pi$
$877$	21.6145	0.729870	0.364935	+	0.931033i	$0.381091\pi$
$878$	4.22284	0.142514
$879$	0	0
$880$	12.4812	0.420741
$881$	45.2750	1.52535	0.762677	−	0.646779i	$-0.223884\pi$
$881$	45.2750	1.52535	0.762677	+	0.646779i	$0.223884\pi$
$882$	0	0
$883$	−24.3757	−0.820306	−0.410153	−	0.912017i	$-0.634525\pi$
$883$	−24.3757	−0.820306	−0.410153	+	0.912017i	$0.634525\pi$
$884$	−2.51197	−0.0844867
$885$	0	0
$886$	−43.3707	−1.45707
$887$	24.2520	0.814303	0.407152	−	0.913361i	$-0.366522\pi$
$887$	24.2520	0.814303	0.407152	+	0.913361i	$0.366522\pi$
$888$	0	0
$889$	−16.0654	−0.538815
$890$	12.0992	0.405566
$891$	0	0
$892$	−0.775746	−0.0259739
$893$	4.93207	0.165045
$894$	0	0
$895$	19.5877	0.654744
$896$	41.6664	1.39198
$897$	0	0
$898$	−28.8895	−0.964056
$899$	−87.2868	−2.91118
$900$	0	0
$901$	25.2459	0.841062
$902$	−38.4836	−1.28136
$903$	0	0
$904$	29.4069	0.978060
$905$	19.2750	0.640724
$906$	0	0
$907$	28.7005	0.952985	0.476493	−	0.879178i	$-0.341908\pi$
$907$	28.7005	0.952985	0.476493	+	0.879178i	$0.341908\pi$
$908$	−1.08252	−0.0359246
$909$	0	0
$910$	34.2071	1.13395
$911$	18.3272	0.607209	0.303604	−	0.952798i	$-0.401810\pi$
$911$	18.3272	0.607209	0.303604	+	0.952798i	$0.401810\pi$
$912$	0	0
$913$	−24.2981	−0.804148
$914$	6.98400	0.231010
$915$	0	0
$916$	2.12998	0.0703764
$917$	−46.0811	−1.52173
$918$	0	0
$919$	−40.4993	−1.33595	−0.667974	−	0.744184i	$-0.732838\pi$
$919$	−40.4993	−1.33595	−0.667974	+	0.744184i	$0.732838\pi$
$920$	0.418190	0.0137873
$921$	0	0
$922$	5.14903	0.169574
$923$	−48.9135	−1.61001
$924$	0	0
$925$	−1.36248	−0.0447981
$926$	−27.4582	−0.902331
$927$	0	0
$928$	−10.0386	−0.329532
$929$	49.3503	1.61913	0.809565	−	0.587031i	$-0.199703\pi$
$929$	49.3503	1.61913	0.809565	+	0.587031i	$0.199703\pi$
$930$	0	0
$931$	3.80606	0.124739
$932$	3.44992	0.113006
$933$	0	0
$934$	−19.0581	−0.623599
$935$	−5.28963	−0.172989
$936$	0	0
$937$	−23.8886	−0.780406	−0.390203	−	0.920729i	$-0.627595\pi$
$937$	−23.8886	−0.780406	−0.390203	+	0.920729i	$0.627595\pi$
$938$	−54.8989	−1.79251
$939$	0	0
$940$	−0.956509	−0.0311979
$941$	12.2095	0.398018	0.199009	−	0.979998i	$-0.436228\pi$
$941$	12.2095	0.398018	0.199009	+	0.979998i	$0.436228\pi$
$942$	0	0
$943$	−1.41564	−0.0460994
$944$	−5.87399	−0.191182
$945$	0	0
$946$	−23.7976	−0.773728
$947$	−8.78163	−0.285364	−0.142682	−	0.989769i	$-0.545573\pi$
$947$	−8.78163	−0.285364	−0.142682	+	0.989769i	$0.545573\pi$
$948$	0	0
$949$	−18.0870	−0.587128
$950$	−1.48119	−0.0480563
$951$	0	0
$952$	−16.2130	−0.525466
$953$	−0.381994	−0.0123740	−0.00618699	−	0.999981i	$-0.501969\pi$
$953$	−0.381994	−0.0123740	−0.00618699	+	0.999981i	$0.501969\pi$
$954$	0	0
$955$	−0.607202	−0.0196486
$956$	−0.0180984	−0.000585344 0
$957$	0	0
$958$	−42.7781	−1.38210
$959$	−60.6126	−1.95728
$960$	0	0
$961$	59.3752	1.91533
$962$	−14.1779	−0.457115
$963$	0	0
$964$	2.46310	0.0793310
$965$	26.2252	0.844219
$966$	0	0
$967$	1.79640	0.0577683	0.0288842	−	0.999583i	$-0.490805\pi$
$967$	1.79640	0.0577683	0.0288842	+	0.999583i	$0.490805\pi$
$968$	−7.40597	−0.238037
$969$	0	0
$970$	11.8315	0.379885
$971$	−46.8481	−1.50343	−0.751714	−	0.659489i	$-0.770773\pi$
$971$	−46.8481	−1.50343	−0.751714	+	0.659489i	$0.770773\pi$
$972$	0	0
$973$	−1.10809	−0.0355239
$974$	8.29104	0.265662
$975$	0	0
$976$	7.69323	0.246254
$977$	4.41819	0.141350	0.0706752	−	0.997499i	$-0.477485\pi$
$977$	4.41819	0.141350	0.0706752	+	0.997499i	$0.477485\pi$
$978$	0	0
$979$	23.4361	0.749021
$980$	−0.738135	−0.0235789
$981$	0	0
$982$	−38.9257	−1.24217
$983$	−15.1451	−0.483052	−0.241526	−	0.970394i	$-0.577648\pi$
$983$	−15.1451	−0.483052	−0.241526	+	0.970394i	$0.577648\pi$
$984$	0	0
$985$	5.31994	0.169508
$986$	25.0738	0.798513
$987$	0	0
$988$	−1.36248	−0.0433463
$989$	−0.875407	−0.0278363
$990$	0	0
$991$	−50.0263	−1.58914	−0.794570	−	0.607173i	$-0.792303\pi$
$991$	−50.0263	−1.58914	−0.794570	+	0.607173i	$0.792303\pi$
$992$	10.3938	0.330002
$993$	0	0
$994$	33.9003	1.07525
$995$	10.8872	0.345146
$996$	0	0
$997$	1.36011	0.0430751	0.0215376	−	0.999768i	$-0.493144\pi$
$997$	1.36011	0.0430751	0.0215376	+	0.999768i	$0.493144\pi$
$998$	−19.5731	−0.619576
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} -3.28726 q^{7} +2.67513 q^{8} +O(q^{10})\)	\(q-1.48119 q^{2} +0.193937 q^{4} -1.00000 q^{5} -3.28726 q^{7} +2.67513 q^{8} +1.48119 q^{10} +2.86907 q^{11} -7.02539 q^{13} +4.86907 q^{14} -4.35026 q^{16} +1.84367 q^{17} +1.00000 q^{19} -0.193937 q^{20} -4.24965 q^{22} -0.156325 q^{23} +1.00000 q^{25} +10.4060 q^{26} -0.637519 q^{28} -9.18172 q^{29} +9.50659 q^{31} +1.09332 q^{32} -2.73084 q^{34} +3.28726 q^{35} -1.36248 q^{37} -1.48119 q^{38} -2.67513 q^{40} +9.05571 q^{41} +5.59991 q^{43} +0.556417 q^{44} +0.231548 q^{46} +4.93207 q^{47} +3.80606 q^{49} -1.48119 q^{50} -1.36248 q^{52} +13.6932 q^{53} -2.86907 q^{55} -8.79384 q^{56} +13.5999 q^{58} +1.35026 q^{59} -1.76845 q^{61} -14.0811 q^{62} +7.08110 q^{64} +7.02539 q^{65} -11.2750 q^{67} +0.357556 q^{68} -4.86907 q^{70} +6.96239 q^{71} +2.57452 q^{73} +2.01810 q^{74} +0.193937 q^{76} -9.43136 q^{77} +8.18664 q^{79} +4.35026 q^{80} -13.4133 q^{82} -8.46898 q^{83} -1.84367 q^{85} -8.29455 q^{86} +7.67513 q^{88} +8.16854 q^{89} +23.0943 q^{91} -0.0303172 q^{92} -7.30536 q^{94} -1.00000 q^{95} +7.98778 q^{97} -5.63752 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + q^{4} - 3 q^{5} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) 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

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