LMFDB - Embedded newform 855.2.a.e.1.2

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	$q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.585786 q^{7} -1.58579 q^{8} +O(q^{10})$	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.585786 q^{7} -1.58579 q^{8} +0.414214 q^{10} -1.41421 q^{11} +5.41421 q^{13} +0.242641 q^{14} +3.00000 q^{16} +1.17157 q^{17} +1.00000 q^{19} -1.82843 q^{20} -0.585786 q^{22} -7.65685 q^{23} +1.00000 q^{25} +2.24264 q^{26} -1.07107 q^{28} +9.07107 q^{29} +6.48528 q^{31} +4.41421 q^{32} +0.485281 q^{34} +0.585786 q^{35} +11.0711 q^{37} +0.414214 q^{38} -1.58579 q^{40} +7.41421 q^{41} +0.585786 q^{43} +2.58579 q^{44} -3.17157 q^{46} -0.343146 q^{47} -6.65685 q^{49} +0.414214 q^{50} -9.89949 q^{52} -4.00000 q^{53} -1.41421 q^{55} -0.928932 q^{56} +3.75736 q^{58} -8.48528 q^{59} +5.65685 q^{61} +2.68629 q^{62} -4.17157 q^{64} +5.41421 q^{65} +12.0000 q^{67} -2.14214 q^{68} +0.242641 q^{70} +4.48528 q^{71} -2.00000 q^{73} +4.58579 q^{74} -1.82843 q^{76} -0.828427 q^{77} -11.3137 q^{79} +3.00000 q^{80} +3.07107 q^{82} +10.4853 q^{83} +1.17157 q^{85} +0.242641 q^{86} +2.24264 q^{88} -10.7279 q^{89} +3.17157 q^{91} +14.0000 q^{92} -0.142136 q^{94} +1.00000 q^{95} -4.24264 q^{97} -2.75736 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8} - 2 q^{10} + 8 q^{13} - 8 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} + 4 q^{29} - 4 q^{31} + 6 q^{32} - 16 q^{34} + 4 q^{35} + 8 q^{37} - 2 q^{38} - 6 q^{40} + 12 q^{41} + 4 q^{43} + 8 q^{44} - 12 q^{46} - 12 q^{47} - 2 q^{49} - 2 q^{50} - 8 q^{53} - 16 q^{56} + 16 q^{58} + 28 q^{62} - 14 q^{64} + 8 q^{65} + 24 q^{67} + 24 q^{68} - 8 q^{70} - 8 q^{71} - 4 q^{73} + 12 q^{74} + 2 q^{76} + 4 q^{77} + 6 q^{80} - 8 q^{82} + 4 q^{83} + 8 q^{85} - 8 q^{86} - 4 q^{88} + 4 q^{89} + 12 q^{91} + 28 q^{92} + 28 q^{94} + 2 q^{95} - 14 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8 - 2 * q^10 + 8 * q^13 - 8 * q^14 + 6 * q^16 + 8 * q^17 + 2 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 - 4 * q^26 + 12 * q^28 + 4 * q^29 - 4 * q^31 + 6 * q^32 - 16 * q^34 + 4 * q^35 + 8 * q^37 - 2 * q^38 - 6 * q^40 + 12 * q^41 + 4 * q^43 + 8 * q^44 - 12 * q^46 - 12 * q^47 - 2 * q^49 - 2 * q^50 - 8 * q^53 - 16 * q^56 + 16 * q^58 + 28 * q^62 - 14 * q^64 + 8 * q^65 + 24 * q^67 + 24 * q^68 - 8 * q^70 - 8 * q^71 - 4 * q^73 + 12 * q^74 + 2 * q^76 + 4 * q^77 + 6 * q^80 - 8 * q^82 + 4 * q^83 + 8 * q^85 - 8 * q^86 - 4 * q^88 + 4 * q^89 + 12 * q^91 + 28 * q^92 + 28 * q^94 + 2 * q^95 - 14 * 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 $

$2$	0.414214	0.292893	0.146447	−	0.989219i	$-0.453216\pi$
$2$	0.414214	0.292893	0.146447	+	0.989219i	$0.453216\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.585786	0.221406	0.110703	−	0.993854i	$-0.464690\pi$
$7$	0.585786	0.221406	0.110703	+	0.993854i	$0.464690\pi$
$8$	−1.58579	−0.560660
$9$	0	0
$10$	0.414214	0.130986
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	5.41421	1.50163	0.750816	−	0.660511i	$-0.229660\pi$
$13$	5.41421	1.50163	0.750816	+	0.660511i	$0.229660\pi$
$14$	0.242641	0.0648485
$15$	0	0
$16$	3.00000	0.750000
$17$	1.17157	0.284148	0.142074	−	0.989856i	$-0.454623\pi$
$17$	1.17157	0.284148	0.142074	+	0.989856i	$0.454623\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.82843	−0.408849
$21$	0	0
$22$	−0.585786	−0.124890
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.24264	0.439818
$27$	0	0
$28$	−1.07107	−0.202413
$29$	9.07107	1.68446	0.842228	−	0.539122i	$-0.181244\pi$
$29$	9.07107	1.68446	0.842228	+	0.539122i	$0.181244\pi$
$30$	0	0
$31$	6.48528	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$31$	6.48528	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$32$	4.41421	0.780330
$33$	0	0
$34$	0.485281	0.0832251
$35$	0.585786	0.0990160
$36$	0	0
$37$	11.0711	1.82007	0.910036	−	0.414529i	$-0.136054\pi$
$37$	11.0711	1.82007	0.910036	+	0.414529i	$0.136054\pi$
$38$	0.414214	0.0671943
$39$	0	0
$40$	−1.58579	−0.250735
$41$	7.41421	1.15791	0.578953	−	0.815361i	$-0.303462\pi$
$41$	7.41421	1.15791	0.578953	+	0.815361i	$0.303462\pi$
$42$	0	0
$43$	0.585786	0.0893316	0.0446658	−	0.999002i	$-0.485778\pi$
$43$	0.585786	0.0893316	0.0446658	+	0.999002i	$0.485778\pi$
$44$	2.58579	0.389822
$45$	0	0
$46$	−3.17157	−0.467623
$47$	−0.343146	−0.0500530	−0.0250265	−	0.999687i	$-0.507967\pi$
$47$	−0.343146	−0.0500530	−0.0250265	+	0.999687i	$0.507967\pi$
$48$	0	0
$49$	−6.65685	−0.950979
$50$	0.414214	0.0585786
$51$	0	0
$52$	−9.89949	−1.37281
$53$	−4.00000	−0.549442	−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000	−0.549442	−0.274721	+	0.961524i	$0.588586\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	−0.928932	−0.124134
$57$	0	0
$58$	3.75736	0.493365
$59$	−8.48528	−1.10469	−0.552345	−	0.833616i	$-0.686267\pi$
$59$	−8.48528	−1.10469	−0.552345	+	0.833616i	$0.686267\pi$
$60$	0	0
$61$	5.65685	0.724286	0.362143	−	0.932123i	$-0.382045\pi$
$61$	5.65685	0.724286	0.362143	+	0.932123i	$0.382045\pi$
$62$	2.68629	0.341159
$63$	0	0
$64$	−4.17157	−0.521447
$65$	5.41421	0.671551
$66$	0	0
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	−2.14214	−0.259772
$69$	0	0
$70$	0.242641	0.0290011
$71$	4.48528	0.532305	0.266152	−	0.963931i	$-0.414248\pi$
$71$	4.48528	0.532305	0.266152	+	0.963931i	$0.414248\pi$
$72$	0	0
$73$	−2.00000	−0.234082	−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000	−0.234082	−0.117041	+	0.993127i	$0.537341\pi$
$74$	4.58579	0.533087
$75$	0	0
$76$	−1.82843	−0.209735
$77$	−0.828427	−0.0944080
$78$	0	0
$79$	−11.3137	−1.27289	−0.636446	−	0.771321i	$-0.719596\pi$
$79$	−11.3137	−1.27289	−0.636446	+	0.771321i	$0.719596\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	3.07107	0.339143
$83$	10.4853	1.15091	0.575455	−	0.817834i	$-0.304825\pi$
$83$	10.4853	1.15091	0.575455	+	0.817834i	$0.304825\pi$
$84$	0	0
$85$	1.17157	0.127075
$86$	0.242641	0.0261646
$87$	0	0
$88$	2.24264	0.239066
$89$	−10.7279	−1.13716	−0.568579	−	0.822629i	$-0.692507\pi$
$89$	−10.7279	−1.13716	−0.568579	+	0.822629i	$0.692507\pi$
$90$	0	0
$91$	3.17157	0.332471
$92$	14.0000	1.45960
$93$	0	0
$94$	−0.142136	−0.0146602
$95$	1.00000	0.102598
$96$	0	0
$97$	−4.24264	−0.430775	−0.215387	−	0.976529i	$-0.569101\pi$
$97$	−4.24264	−0.430775	−0.215387	+	0.976529i	$0.569101\pi$
$98$	−2.75736	−0.278535
$99$	0	0
$100$	−1.82843	−0.182843
$101$	4.82843	0.480446	0.240223	−	0.970718i	$-0.422779\pi$
$101$	4.82843	0.480446	0.240223	+	0.970718i	$0.422779\pi$
$102$	0	0
$103$	−1.65685	−0.163255	−0.0816274	−	0.996663i	$-0.526012\pi$
$103$	−1.65685	−0.163255	−0.0816274	+	0.996663i	$0.526012\pi$
$104$	−8.58579	−0.841906
$105$	0	0
$106$	−1.65685	−0.160928
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	0	0
$109$	3.17157	0.303782	0.151891	−	0.988397i	$-0.451464\pi$
$109$	3.17157	0.303782	0.151891	+	0.988397i	$0.451464\pi$
$110$	−0.585786	−0.0558525
$111$	0	0
$112$	1.75736	0.166055
$113$	−12.4853	−1.17452	−0.587258	−	0.809400i	$-0.699793\pi$
$113$	−12.4853	−1.17452	−0.587258	+	0.809400i	$0.699793\pi$
$114$	0	0
$115$	−7.65685	−0.714005
$116$	−16.5858	−1.53995
$117$	0	0
$118$	−3.51472	−0.323556
$119$	0.686292	0.0629122
$120$	0	0
$121$	−9.00000	−0.818182
$122$	2.34315	0.212138
$123$	0	0
$124$	−11.8579	−1.06487
$125$	1.00000	0.0894427
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−10.5563	−0.933058
$129$	0	0
$130$	2.24264	0.196693
$131$	−16.7279	−1.46153	−0.730763	−	0.682632i	$-0.760835\pi$
$131$	−16.7279	−1.46153	−0.730763	+	0.682632i	$0.760835\pi$
$132$	0	0
$133$	0.585786	0.0507941
$134$	4.97056	0.429391
$135$	0	0
$136$	−1.85786	−0.159311
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	1.17157	0.0993715	0.0496858	−	0.998765i	$-0.484178\pi$
$139$	1.17157	0.0993715	0.0496858	+	0.998765i	$0.484178\pi$
$140$	−1.07107	−0.0905218
$141$	0	0
$142$	1.85786	0.155909
$143$	−7.65685	−0.640298
$144$	0	0
$145$	9.07107	0.753311
$146$	−0.828427	−0.0685611
$147$	0	0
$148$	−20.2426	−1.66393
$149$	3.65685	0.299581	0.149791	−	0.988718i	$-0.452140\pi$
$149$	3.65685	0.299581	0.149791	+	0.988718i	$0.452140\pi$
$150$	0	0
$151$	17.7990	1.44846	0.724231	−	0.689558i	$-0.242195\pi$
$151$	17.7990	1.44846	0.724231	+	0.689558i	$0.242195\pi$
$152$	−1.58579	−0.128624
$153$	0	0
$154$	−0.343146	−0.0276515
$155$	6.48528	0.520910
$156$	0	0
$157$	−21.7990	−1.73975	−0.869874	−	0.493273i	$-0.835800\pi$
$157$	−21.7990	−1.73975	−0.869874	+	0.493273i	$0.835800\pi$
$158$	−4.68629	−0.372821
$159$	0	0
$160$	4.41421	0.348974
$161$	−4.48528	−0.353490
$162$	0	0
$163$	7.89949	0.618736	0.309368	−	0.950942i	$-0.399882\pi$
$163$	7.89949	0.618736	0.309368	+	0.950942i	$0.399882\pi$
$164$	−13.5563	−1.05857
$165$	0	0
$166$	4.34315	0.337093
$167$	−10.0000	−0.773823	−0.386912	−	0.922117i	$-0.626458\pi$
$167$	−10.0000	−0.773823	−0.386912	+	0.922117i	$0.626458\pi$
$168$	0	0
$169$	16.3137	1.25490
$170$	0.485281	0.0372194
$171$	0	0
$172$	−1.07107	−0.0816682
$173$	6.14214	0.466978	0.233489	−	0.972359i	$-0.424986\pi$
$173$	6.14214	0.466978	0.233489	+	0.972359i	$0.424986\pi$
$174$	0	0
$175$	0.585786	0.0442813
$176$	−4.24264	−0.319801
$177$	0	0
$178$	−4.44365	−0.333066
$179$	−17.1716	−1.28346	−0.641732	−	0.766929i	$-0.721784\pi$
$179$	−17.1716	−1.28346	−0.641732	+	0.766929i	$0.721784\pi$
$180$	0	0
$181$	−19.1716	−1.42501	−0.712506	−	0.701666i	$-0.752440\pi$
$181$	−19.1716	−1.42501	−0.712506	+	0.701666i	$0.752440\pi$
$182$	1.31371	0.0973786
$183$	0	0
$184$	12.1421	0.895130
$185$	11.0711	0.813961
$186$	0	0
$187$	−1.65685	−0.121161
$188$	0.627417	0.0457591
$189$	0	0
$190$	0.414214	0.0300502
$191$	−1.89949	−0.137443	−0.0687213	−	0.997636i	$-0.521892\pi$
$191$	−1.89949	−0.137443	−0.0687213	+	0.997636i	$0.521892\pi$
$192$	0	0
$193$	−15.0711	−1.08484	−0.542420	−	0.840108i	$-0.682492\pi$
$193$	−15.0711	−1.08484	−0.542420	+	0.840108i	$0.682492\pi$
$194$	−1.75736	−0.126171
$195$	0	0
$196$	12.1716	0.869398
$197$	14.8284	1.05648	0.528241	−	0.849095i	$-0.322852\pi$
$197$	14.8284	1.05648	0.528241	+	0.849095i	$0.322852\pi$
$198$	0	0
$199$	−16.4853	−1.16861	−0.584305	−	0.811534i	$-0.698633\pi$
$199$	−16.4853	−1.16861	−0.584305	+	0.811534i	$0.698633\pi$
$200$	−1.58579	−0.112132
$201$	0	0
$202$	2.00000	0.140720
$203$	5.31371	0.372949
$204$	0	0
$205$	7.41421	0.517831
$206$	−0.686292	−0.0478162
$207$	0	0
$208$	16.2426	1.12622
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	−15.3137	−1.05424	−0.527120	−	0.849791i	$-0.676728\pi$
$211$	−15.3137	−1.05424	−0.527120	+	0.849791i	$0.676728\pi$
$212$	7.31371	0.502308
$213$	0	0
$214$	−3.31371	−0.226520
$215$	0.585786	0.0399503
$216$	0	0
$217$	3.79899	0.257892
$218$	1.31371	0.0889756
$219$	0	0
$220$	2.58579	0.174334
$221$	6.34315	0.426686
$222$	0	0
$223$	6.34315	0.424768	0.212384	−	0.977186i	$-0.431877\pi$
$223$	6.34315	0.424768	0.212384	+	0.977186i	$0.431877\pi$
$224$	2.58579	0.172770
$225$	0	0
$226$	−5.17157	−0.344008
$227$	18.9706	1.25912	0.629560	−	0.776952i	$-0.283235\pi$
$227$	18.9706	1.25912	0.629560	+	0.776952i	$0.283235\pi$
$228$	0	0
$229$	−1.65685	−0.109488	−0.0547440	−	0.998500i	$-0.517434\pi$
$229$	−1.65685	−0.109488	−0.0547440	+	0.998500i	$0.517434\pi$
$230$	−3.17157	−0.209127
$231$	0	0
$232$	−14.3848	−0.944407
$233$	8.34315	0.546578	0.273289	−	0.961932i	$-0.411889\pi$
$233$	8.34315	0.546578	0.273289	+	0.961932i	$0.411889\pi$
$234$	0	0
$235$	−0.343146	−0.0223844
$236$	15.5147	1.00992
$237$	0	0
$238$	0.284271	0.0184266
$239$	−2.58579	−0.167261	−0.0836303	−	0.996497i	$-0.526651\pi$
$239$	−2.58579	−0.167261	−0.0836303	+	0.996497i	$0.526651\pi$
$240$	0	0
$241$	−14.9706	−0.964339	−0.482169	−	0.876078i	$-0.660151\pi$
$241$	−14.9706	−0.964339	−0.482169	+	0.876078i	$0.660151\pi$
$242$	−3.72792	−0.239640
$243$	0	0
$244$	−10.3431	−0.662152
$245$	−6.65685	−0.425291
$246$	0	0
$247$	5.41421	0.344498
$248$	−10.2843	−0.653052
$249$	0	0
$250$	0.414214	0.0261972
$251$	−12.9289	−0.816067	−0.408033	−	0.912967i	$-0.633785\pi$
$251$	−12.9289	−0.816067	−0.408033	+	0.912967i	$0.633785\pi$
$252$	0	0
$253$	10.8284	0.680777
$254$	3.31371	0.207921
$255$	0	0
$256$	3.97056	0.248160
$257$	−1.17157	−0.0730807	−0.0365404	−	0.999332i	$-0.511634\pi$
$257$	−1.17157	−0.0730807	−0.0365404	+	0.999332i	$0.511634\pi$
$258$	0	0
$259$	6.48528	0.402976
$260$	−9.89949	−0.613941
$261$	0	0
$262$	−6.92893	−0.428071
$263$	32.1421	1.98197	0.990984	−	0.133977i	$-0.0427747\pi$
$263$	32.1421	1.98197	0.990984	+	0.133977i	$0.0427747\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0.242641	0.0148773
$267$	0	0
$268$	−21.9411	−1.34027
$269$	−8.38478	−0.511229	−0.255614	−	0.966779i	$-0.582278\pi$
$269$	−8.38478	−0.511229	−0.255614	+	0.966779i	$0.582278\pi$
$270$	0	0
$271$	−30.8284	−1.87269	−0.936347	−	0.351076i	$-0.885816\pi$
$271$	−30.8284	−1.87269	−0.936347	+	0.351076i	$0.885816\pi$
$272$	3.51472	0.213111
$273$	0	0
$274$	5.79899	0.350330
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	10.9706	0.659157	0.329579	−	0.944128i	$-0.393093\pi$
$277$	10.9706	0.659157	0.329579	+	0.944128i	$0.393093\pi$
$278$	0.485281	0.0291052
$279$	0	0
$280$	−0.928932	−0.0555143
$281$	14.7279	0.878594	0.439297	−	0.898342i	$-0.355228\pi$
$281$	14.7279	0.878594	0.439297	+	0.898342i	$0.355228\pi$
$282$	0	0
$283$	6.24264	0.371086	0.185543	−	0.982636i	$-0.440596\pi$
$283$	6.24264	0.371086	0.185543	+	0.982636i	$0.440596\pi$
$284$	−8.20101	−0.486640
$285$	0	0
$286$	−3.17157	−0.187539
$287$	4.34315	0.256368
$288$	0	0
$289$	−15.6274	−0.919260
$290$	3.75736	0.220640
$291$	0	0
$292$	3.65685	0.214001
$293$	31.7990	1.85772	0.928858	−	0.370435i	$-0.120791\pi$
$293$	31.7990	1.85772	0.928858	+	0.370435i	$0.120791\pi$
$294$	0	0
$295$	−8.48528	−0.494032
$296$	−17.5563	−1.02044
$297$	0	0
$298$	1.51472	0.0877453
$299$	−41.4558	−2.39745
$300$	0	0
$301$	0.343146	0.0197786
$302$	7.37258	0.424244
$303$	0	0
$304$	3.00000	0.172062
$305$	5.65685	0.323911
$306$	0	0
$307$	7.79899	0.445112	0.222556	−	0.974920i	$-0.428560\pi$
$307$	7.79899	0.445112	0.222556	+	0.974920i	$0.428560\pi$
$308$	1.51472	0.0863091
$309$	0	0
$310$	2.68629	0.152571
$311$	32.2426	1.82831	0.914156	−	0.405362i	$-0.132855\pi$
$311$	32.2426	1.82831	0.914156	+	0.405362i	$0.132855\pi$
$312$	0	0
$313$	−9.51472	−0.537804	−0.268902	−	0.963168i	$-0.586661\pi$
$313$	−9.51472	−0.537804	−0.268902	+	0.963168i	$0.586661\pi$
$314$	−9.02944	−0.509561
$315$	0	0
$316$	20.6863	1.16369
$317$	−11.3137	−0.635441	−0.317721	−	0.948184i	$-0.602917\pi$
$317$	−11.3137	−0.635441	−0.317721	+	0.948184i	$0.602917\pi$
$318$	0	0
$319$	−12.8284	−0.718254
$320$	−4.17157	−0.233198
$321$	0	0
$322$	−1.85786	−0.103535
$323$	1.17157	0.0651881
$324$	0	0
$325$	5.41421	0.300327
$326$	3.27208	0.181224
$327$	0	0
$328$	−11.7574	−0.649192
$329$	−0.201010	−0.0110820
$330$	0	0
$331$	7.17157	0.394185	0.197093	−	0.980385i	$-0.436850\pi$
$331$	7.17157	0.394185	0.197093	+	0.980385i	$0.436850\pi$
$332$	−19.1716	−1.05218
$333$	0	0
$334$	−4.14214	−0.226648
$335$	12.0000	0.655630
$336$	0	0
$337$	28.2426	1.53847	0.769237	−	0.638963i	$-0.220636\pi$
$337$	28.2426	1.53847	0.769237	+	0.638963i	$0.220636\pi$
$338$	6.75736	0.367552
$339$	0	0
$340$	−2.14214	−0.116174
$341$	−9.17157	−0.496669
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−0.928932	−0.0500847
$345$	0	0
$346$	2.54416	0.136775
$347$	−10.4853	−0.562879	−0.281440	−	0.959579i	$-0.590812\pi$
$347$	−10.4853	−0.562879	−0.281440	+	0.959579i	$0.590812\pi$
$348$	0	0
$349$	29.3137	1.56913	0.784563	−	0.620049i	$-0.212887\pi$
$349$	29.3137	1.56913	0.784563	+	0.620049i	$0.212887\pi$
$350$	0.242641	0.0129697
$351$	0	0
$352$	−6.24264	−0.332734
$353$	−3.65685	−0.194635	−0.0973174	−	0.995253i	$-0.531026\pi$
$353$	−3.65685	−0.194635	−0.0973174	+	0.995253i	$0.531026\pi$
$354$	0	0
$355$	4.48528	0.238054
$356$	19.6152	1.03960
$357$	0	0
$358$	−7.11270	−0.375918
$359$	−9.89949	−0.522475	−0.261238	−	0.965275i	$-0.584131\pi$
$359$	−9.89949	−0.522475	−0.261238	+	0.965275i	$0.584131\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−7.94113	−0.417376
$363$	0	0
$364$	−5.79899	−0.303950
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−19.4142	−1.01341	−0.506707	−	0.862118i	$-0.669137\pi$
$367$	−19.4142	−1.01341	−0.506707	+	0.862118i	$0.669137\pi$
$368$	−22.9706	−1.19742
$369$	0	0
$370$	4.58579	0.238404
$371$	−2.34315	−0.121650
$372$	0	0
$373$	−9.89949	−0.512576	−0.256288	−	0.966600i	$-0.582500\pi$
$373$	−9.89949	−0.512576	−0.256288	+	0.966600i	$0.582500\pi$
$374$	−0.686292	−0.0354873
$375$	0	0
$376$	0.544156	0.0280627
$377$	49.1127	2.52943
$378$	0	0
$379$	24.1421	1.24010	0.620049	−	0.784563i	$-0.287113\pi$
$379$	24.1421	1.24010	0.620049	+	0.784563i	$0.287113\pi$
$380$	−1.82843	−0.0937963
$381$	0	0
$382$	−0.786797	−0.0402560
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	−0.828427	−0.0422206
$386$	−6.24264	−0.317742
$387$	0	0
$388$	7.75736	0.393820
$389$	−14.9706	−0.759038	−0.379519	−	0.925184i	$-0.623910\pi$
$389$	−14.9706	−0.759038	−0.379519	+	0.925184i	$0.623910\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	10.5563	0.533176
$393$	0	0
$394$	6.14214	0.309436
$395$	−11.3137	−0.569254
$396$	0	0
$397$	35.6569	1.78957	0.894783	−	0.446501i	$-0.147330\pi$
$397$	35.6569	1.78957	0.894783	+	0.446501i	$0.147330\pi$
$398$	−6.82843	−0.342278
$399$	0	0
$400$	3.00000	0.150000
$401$	−30.0416	−1.50021	−0.750104	−	0.661320i	$-0.769996\pi$
$401$	−30.0416	−1.50021	−0.750104	+	0.661320i	$0.769996\pi$
$402$	0	0
$403$	35.1127	1.74909
$404$	−8.82843	−0.439231
$405$	0	0
$406$	2.20101	0.109234
$407$	−15.6569	−0.776081
$408$	0	0
$409$	9.51472	0.470473	0.235236	−	0.971938i	$-0.424414\pi$
$409$	9.51472	0.470473	0.235236	+	0.971938i	$0.424414\pi$
$410$	3.07107	0.151669
$411$	0	0
$412$	3.02944	0.149250
$413$	−4.97056	−0.244585
$414$	0	0
$415$	10.4853	0.514702
$416$	23.8995	1.17177
$417$	0	0
$418$	−0.585786	−0.0286518
$419$	34.8701	1.70351	0.851757	−	0.523937i	$-0.175537\pi$
$419$	34.8701	1.70351	0.851757	+	0.523937i	$0.175537\pi$
$420$	0	0
$421$	−14.6863	−0.715766	−0.357883	−	0.933766i	$-0.616501\pi$
$421$	−14.6863	−0.715766	−0.357883	+	0.933766i	$0.616501\pi$
$422$	−6.34315	−0.308780
$423$	0	0
$424$	6.34315	0.308050
$425$	1.17157	0.0568296
$426$	0	0
$427$	3.31371	0.160362
$428$	14.6274	0.707043
$429$	0	0
$430$	0.242641	0.0117012
$431$	−3.51472	−0.169298	−0.0846490	−	0.996411i	$-0.526977\pi$
$431$	−3.51472	−0.169298	−0.0846490	+	0.996411i	$0.526977\pi$
$432$	0	0
$433$	0.928932	0.0446416	0.0223208	−	0.999751i	$-0.492894\pi$
$433$	0.928932	0.0446416	0.0223208	+	0.999751i	$0.492894\pi$
$434$	1.57359	0.0755349
$435$	0	0
$436$	−5.79899	−0.277721
$437$	−7.65685	−0.366277
$438$	0	0
$439$	0.970563	0.0463224	0.0231612	−	0.999732i	$-0.492627\pi$
$439$	0.970563	0.0463224	0.0231612	+	0.999732i	$0.492627\pi$
$440$	2.24264	0.106914
$441$	0	0
$442$	2.62742	0.124973
$443$	1.31371	0.0624162	0.0312081	−	0.999513i	$-0.490065\pi$
$443$	1.31371	0.0624162	0.0312081	+	0.999513i	$0.490065\pi$
$444$	0	0
$445$	−10.7279	−0.508552
$446$	2.62742	0.124412
$447$	0	0
$448$	−2.44365	−0.115452
$449$	−7.89949	−0.372800	−0.186400	−	0.982474i	$-0.559682\pi$
$449$	−7.89949	−0.372800	−0.186400	+	0.982474i	$0.559682\pi$
$450$	0	0
$451$	−10.4853	−0.493733
$452$	22.8284	1.07376
$453$	0	0
$454$	7.85786	0.368788
$455$	3.17157	0.148686
$456$	0	0
$457$	28.8284	1.34854	0.674268	−	0.738486i	$-0.264459\pi$
$457$	28.8284	1.34854	0.674268	+	0.738486i	$0.264459\pi$
$458$	−0.686292	−0.0320683
$459$	0	0
$460$	14.0000	0.652753
$461$	−10.6863	−0.497710	−0.248855	−	0.968541i	$-0.580054\pi$
$461$	−10.6863	−0.497710	−0.248855	+	0.968541i	$0.580054\pi$
$462$	0	0
$463$	−8.10051	−0.376462	−0.188231	−	0.982125i	$-0.560275\pi$
$463$	−8.10051	−0.376462	−0.188231	+	0.982125i	$0.560275\pi$
$464$	27.2132	1.26334
$465$	0	0
$466$	3.45584	0.160089
$467$	−16.3431	−0.756271	−0.378135	−	0.925750i	$-0.623435\pi$
$467$	−16.3431	−0.756271	−0.378135	+	0.925750i	$0.623435\pi$
$468$	0	0
$469$	7.02944	0.324589
$470$	−0.142136	−0.00655623
$471$	0	0
$472$	13.4558	0.619355
$473$	−0.828427	−0.0380911
$474$	0	0
$475$	1.00000	0.0458831
$476$	−1.25483	−0.0575152
$477$	0	0
$478$	−1.07107	−0.0489895
$479$	38.1838	1.74466	0.872330	−	0.488917i	$-0.162608\pi$
$479$	38.1838	1.74466	0.872330	+	0.488917i	$0.162608\pi$
$480$	0	0
$481$	59.9411	2.73308
$482$	−6.20101	−0.282448
$483$	0	0
$484$	16.4558	0.747993
$485$	−4.24264	−0.192648
$486$	0	0
$487$	−2.82843	−0.128168	−0.0640841	−	0.997944i	$-0.520413\pi$
$487$	−2.82843	−0.128168	−0.0640841	+	0.997944i	$0.520413\pi$
$488$	−8.97056	−0.406078
$489$	0	0
$490$	−2.75736	−0.124565
$491$	−21.8995	−0.988310	−0.494155	−	0.869374i	$-0.664523\pi$
$491$	−21.8995	−0.988310	−0.494155	+	0.869374i	$0.664523\pi$
$492$	0	0
$493$	10.6274	0.478635
$494$	2.24264	0.100901
$495$	0	0
$496$	19.4558	0.873593
$497$	2.62742	0.117856
$498$	0	0
$499$	−23.7990	−1.06539	−0.532695	−	0.846308i	$-0.678821\pi$
$499$	−23.7990	−1.06539	−0.532695	+	0.846308i	$0.678821\pi$
$500$	−1.82843	−0.0817697
$501$	0	0
$502$	−5.35534	−0.239020
$503$	−0.828427	−0.0369377	−0.0184689	−	0.999829i	$-0.505879\pi$
$503$	−0.828427	−0.0369377	−0.0184689	+	0.999829i	$0.505879\pi$
$504$	0	0
$505$	4.82843	0.214862
$506$	4.48528	0.199395
$507$	0	0
$508$	−14.6274	−0.648987
$509$	−32.3848	−1.43543	−0.717715	−	0.696337i	$-0.754812\pi$
$509$	−32.3848	−1.43543	−0.717715	+	0.696337i	$0.754812\pi$
$510$	0	0
$511$	−1.17157	−0.0518273
$512$	22.7574	1.00574
$513$	0	0
$514$	−0.485281	−0.0214048
$515$	−1.65685	−0.0730097
$516$	0	0
$517$	0.485281	0.0213427
$518$	2.68629	0.118029
$519$	0	0
$520$	−8.58579	−0.376512
$521$	24.3848	1.06832	0.534158	−	0.845385i	$-0.320629\pi$
$521$	24.3848	1.06832	0.534158	+	0.845385i	$0.320629\pi$
$522$	0	0
$523$	−15.7990	−0.690842	−0.345421	−	0.938448i	$-0.612264\pi$
$523$	−15.7990	−0.690842	−0.345421	+	0.938448i	$0.612264\pi$
$524$	30.5858	1.33615
$525$	0	0
$526$	13.3137	0.580505
$527$	7.59798	0.330973
$528$	0	0
$529$	35.6274	1.54902
$530$	−1.65685	−0.0719691
$531$	0	0
$532$	−1.07107	−0.0464367
$533$	40.1421	1.73875
$534$	0	0
$535$	−8.00000	−0.345870
$536$	−19.0294	−0.821946
$537$	0	0
$538$	−3.47309	−0.149735
$539$	9.41421	0.405499
$540$	0	0
$541$	−32.6274	−1.40276	−0.701381	−	0.712786i	$-0.747433\pi$
$541$	−32.6274	−1.40276	−0.701381	+	0.712786i	$0.747433\pi$
$542$	−12.7696	−0.548499
$543$	0	0
$544$	5.17157	0.221729
$545$	3.17157	0.135855
$546$	0	0
$547$	−34.1421	−1.45981	−0.729906	−	0.683547i	$-0.760436\pi$
$547$	−34.1421	−1.45981	−0.729906	+	0.683547i	$0.760436\pi$
$548$	−25.5980	−1.09349
$549$	0	0
$550$	−0.585786	−0.0249780
$551$	9.07107	0.386440
$552$	0	0
$553$	−6.62742	−0.281826
$554$	4.54416	0.193063
$555$	0	0
$556$	−2.14214	−0.0908468
$557$	−10.0000	−0.423714	−0.211857	−	0.977301i	$-0.567951\pi$
$557$	−10.0000	−0.423714	−0.211857	+	0.977301i	$0.567951\pi$
$558$	0	0
$559$	3.17157	0.134143
$560$	1.75736	0.0742620
$561$	0	0
$562$	6.10051	0.257334
$563$	−42.2843	−1.78207	−0.891035	−	0.453935i	$-0.850020\pi$
$563$	−42.2843	−1.78207	−0.891035	+	0.453935i	$0.850020\pi$
$564$	0	0
$565$	−12.4853	−0.525260
$566$	2.58579	0.108689
$567$	0	0
$568$	−7.11270	−0.298442
$569$	6.72792	0.282049	0.141025	−	0.990006i	$-0.454960\pi$
$569$	6.72792	0.282049	0.141025	+	0.990006i	$0.454960\pi$
$570$	0	0
$571$	19.7990	0.828562	0.414281	−	0.910149i	$-0.364033\pi$
$571$	19.7990	0.828562	0.414281	+	0.910149i	$0.364033\pi$
$572$	14.0000	0.585369
$573$	0	0
$574$	1.79899	0.0750884
$575$	−7.65685	−0.319313
$576$	0	0
$577$	−37.7990	−1.57359	−0.786796	−	0.617213i	$-0.788262\pi$
$577$	−37.7990	−1.57359	−0.786796	+	0.617213i	$0.788262\pi$
$578$	−6.47309	−0.269245
$579$	0	0
$580$	−16.5858	−0.688687
$581$	6.14214	0.254819
$582$	0	0
$583$	5.65685	0.234283
$584$	3.17157	0.131241
$585$	0	0
$586$	13.1716	0.544113
$587$	−11.6569	−0.481130	−0.240565	−	0.970633i	$-0.577333\pi$
$587$	−11.6569	−0.481130	−0.240565	+	0.970633i	$0.577333\pi$
$588$	0	0
$589$	6.48528	0.267221
$590$	−3.51472	−0.144699
$591$	0	0
$592$	33.2132	1.36505
$593$	29.3137	1.20377	0.601885	−	0.798583i	$-0.294417\pi$
$593$	29.3137	1.20377	0.601885	+	0.798583i	$0.294417\pi$
$594$	0	0
$595$	0.686292	0.0281352
$596$	−6.68629	−0.273881
$597$	0	0
$598$	−17.1716	−0.702198
$599$	21.9411	0.896490	0.448245	−	0.893911i	$-0.352049\pi$
$599$	21.9411	0.896490	0.448245	+	0.893911i	$0.352049\pi$
$600$	0	0
$601$	−28.8284	−1.17594	−0.587968	−	0.808884i	$-0.700072\pi$
$601$	−28.8284	−1.17594	−0.587968	+	0.808884i	$0.700072\pi$
$602$	0.142136	0.00579302
$603$	0	0
$604$	−32.5442	−1.32420
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−2.14214	−0.0869466	−0.0434733	−	0.999055i	$-0.513842\pi$
$607$	−2.14214	−0.0869466	−0.0434733	+	0.999055i	$0.513842\pi$
$608$	4.41421	0.179020
$609$	0	0
$610$	2.34315	0.0948712
$611$	−1.85786	−0.0751611
$612$	0	0
$613$	25.1127	1.01429	0.507146	−	0.861860i	$-0.330700\pi$
$613$	25.1127	1.01429	0.507146	+	0.861860i	$0.330700\pi$
$614$	3.23045	0.130370
$615$	0	0
$616$	1.31371	0.0529308
$617$	−10.1421	−0.408307	−0.204154	−	0.978939i	$-0.565444\pi$
$617$	−10.1421	−0.408307	−0.204154	+	0.978939i	$0.565444\pi$
$618$	0	0
$619$	−43.7990	−1.76043	−0.880215	−	0.474575i	$-0.842602\pi$
$619$	−43.7990	−1.76043	−0.880215	+	0.474575i	$0.842602\pi$
$620$	−11.8579	−0.476223
$621$	0	0
$622$	13.3553	0.535500
$623$	−6.28427	−0.251774
$624$	0	0
$625$	1.00000	0.0400000
$626$	−3.94113	−0.157519
$627$	0	0
$628$	39.8579	1.59050
$629$	12.9706	0.517170
$630$	0	0
$631$	22.6274	0.900783	0.450392	−	0.892831i	$-0.351284\pi$
$631$	22.6274	0.900783	0.450392	+	0.892831i	$0.351284\pi$
$632$	17.9411	0.713660
$633$	0	0
$634$	−4.68629	−0.186116
$635$	8.00000	0.317470
$636$	0	0
$637$	−36.0416	−1.42802
$638$	−5.31371	−0.210372
$639$	0	0
$640$	−10.5563	−0.417276
$641$	8.58579	0.339118	0.169559	−	0.985520i	$-0.445766\pi$
$641$	8.58579	0.339118	0.169559	+	0.985520i	$0.445766\pi$
$642$	0	0
$643$	6.04163	0.238259	0.119129	−	0.992879i	$-0.461990\pi$
$643$	6.04163	0.238259	0.119129	+	0.992879i	$0.461990\pi$
$644$	8.20101	0.323165
$645$	0	0
$646$	0.485281	0.0190931
$647$	−45.1127	−1.77356	−0.886782	−	0.462189i	$-0.847064\pi$
$647$	−45.1127	−1.77356	−0.886782	+	0.462189i	$0.847064\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	2.24264	0.0879636
$651$	0	0
$652$	−14.4437	−0.565657
$653$	−42.4264	−1.66027	−0.830137	−	0.557560i	$-0.811738\pi$
$653$	−42.4264	−1.66027	−0.830137	+	0.557560i	$0.811738\pi$
$654$	0	0
$655$	−16.7279	−0.653614
$656$	22.2426	0.868429
$657$	0	0
$658$	−0.0832611	−0.00324586
$659$	20.0000	0.779089	0.389545	−	0.921008i	$-0.372632\pi$
$659$	20.0000	0.779089	0.389545	+	0.921008i	$0.372632\pi$
$660$	0	0
$661$	18.4853	0.718994	0.359497	−	0.933146i	$-0.382948\pi$
$661$	18.4853	0.718994	0.359497	+	0.933146i	$0.382948\pi$
$662$	2.97056	0.115454
$663$	0	0
$664$	−16.6274	−0.645269
$665$	0.585786	0.0227158
$666$	0	0
$667$	−69.4558	−2.68934
$668$	18.2843	0.707440
$669$	0	0
$670$	4.97056	0.192030
$671$	−8.00000	−0.308837
$672$	0	0
$673$	21.8995	0.844163	0.422082	−	0.906558i	$-0.361300\pi$
$673$	21.8995	0.844163	0.422082	+	0.906558i	$0.361300\pi$
$674$	11.6985	0.450609
$675$	0	0
$676$	−29.8284	−1.14725
$677$	44.9706	1.72836	0.864180	−	0.503184i	$-0.167838\pi$
$677$	44.9706	1.72836	0.864180	+	0.503184i	$0.167838\pi$
$678$	0	0
$679$	−2.48528	−0.0953763
$680$	−1.85786	−0.0712458
$681$	0	0
$682$	−3.79899	−0.145471
$683$	−5.65685	−0.216454	−0.108227	−	0.994126i	$-0.534517\pi$
$683$	−5.65685	−0.216454	−0.108227	+	0.994126i	$0.534517\pi$
$684$	0	0
$685$	14.0000	0.534913
$686$	−3.31371	−0.126518
$687$	0	0
$688$	1.75736	0.0669987
$689$	−21.6569	−0.825060
$690$	0	0
$691$	−23.1127	−0.879248	−0.439624	−	0.898182i	$-0.644888\pi$
$691$	−23.1127	−0.879248	−0.439624	+	0.898182i	$0.644888\pi$
$692$	−11.2304	−0.426918
$693$	0	0
$694$	−4.34315	−0.164864
$695$	1.17157	0.0444403
$696$	0	0
$697$	8.68629	0.329017
$698$	12.1421	0.459587
$699$	0	0
$700$	−1.07107	−0.0404826
$701$	0.343146	0.0129604	0.00648022	−	0.999979i	$-0.497937\pi$
$701$	0.343146	0.0129604	0.00648022	+	0.999979i	$0.497937\pi$
$702$	0	0
$703$	11.0711	0.417553
$704$	5.89949	0.222346
$705$	0	0
$706$	−1.51472	−0.0570072
$707$	2.82843	0.106374
$708$	0	0
$709$	−35.3137	−1.32623	−0.663117	−	0.748516i	$-0.730767\pi$
$709$	−35.3137	−1.32623	−0.663117	+	0.748516i	$0.730767\pi$
$710$	1.85786	0.0697244
$711$	0	0
$712$	17.0122	0.637559
$713$	−49.6569	−1.85966
$714$	0	0
$715$	−7.65685	−0.286350
$716$	31.3970	1.17336
$717$	0	0
$718$	−4.10051	−0.153029
$719$	16.4437	0.613245	0.306622	−	0.951831i	$-0.400801\pi$
$719$	16.4437	0.613245	0.306622	+	0.951831i	$0.400801\pi$
$720$	0	0
$721$	−0.970563	−0.0361456
$722$	0.414214	0.0154154
$723$	0	0
$724$	35.0538	1.30277
$725$	9.07107	0.336891
$726$	0	0
$727$	−4.58579	−0.170077	−0.0850387	−	0.996378i	$-0.527101\pi$
$727$	−4.58579	−0.170077	−0.0850387	+	0.996378i	$0.527101\pi$
$728$	−5.02944	−0.186403
$729$	0	0
$730$	−0.828427	−0.0306615
$731$	0.686292	0.0253834
$732$	0	0
$733$	−1.31371	−0.0485229	−0.0242615	−	0.999706i	$-0.507723\pi$
$733$	−1.31371	−0.0485229	−0.0242615	+	0.999706i	$0.507723\pi$
$734$	−8.04163	−0.296822
$735$	0	0
$736$	−33.7990	−1.24585
$737$	−16.9706	−0.625119
$738$	0	0
$739$	−9.65685	−0.355233	−0.177617	−	0.984100i	$-0.556839\pi$
$739$	−9.65685	−0.355233	−0.177617	+	0.984100i	$0.556839\pi$
$740$	−20.2426	−0.744134
$741$	0	0
$742$	−0.970563	−0.0356305
$743$	15.3137	0.561805	0.280903	−	0.959736i	$-0.409366\pi$
$743$	15.3137	0.561805	0.280903	+	0.959736i	$0.409366\pi$
$744$	0	0
$745$	3.65685	0.133977
$746$	−4.10051	−0.150130
$747$	0	0
$748$	3.02944	0.110767
$749$	−4.68629	−0.171233
$750$	0	0
$751$	−37.1127	−1.35426	−0.677131	−	0.735863i	$-0.736777\pi$
$751$	−37.1127	−1.35426	−0.677131	+	0.735863i	$0.736777\pi$
$752$	−1.02944	−0.0375397
$753$	0	0
$754$	20.3431	0.740854
$755$	17.7990	0.647772
$756$	0	0
$757$	16.8284	0.611640	0.305820	−	0.952089i	$-0.401069\pi$
$757$	16.8284	0.611640	0.305820	+	0.952089i	$0.401069\pi$
$758$	10.0000	0.363216
$759$	0	0
$760$	−1.58579	−0.0575225
$761$	10.2843	0.372805	0.186402	−	0.982474i	$-0.440317\pi$
$761$	10.2843	0.372805	0.186402	+	0.982474i	$0.440317\pi$
$762$	0	0
$763$	1.85786	0.0672592
$764$	3.47309	0.125652
$765$	0	0
$766$	−4.97056	−0.179594
$767$	−45.9411	−1.65884
$768$	0	0
$769$	−35.6569	−1.28582	−0.642910	−	0.765942i	$-0.722273\pi$
$769$	−35.6569	−1.28582	−0.642910	+	0.765942i	$0.722273\pi$
$770$	−0.343146	−0.0123661
$771$	0	0
$772$	27.5563	0.991775
$773$	32.9706	1.18587	0.592934	−	0.805251i	$-0.297969\pi$
$773$	32.9706	1.18587	0.592934	+	0.805251i	$0.297969\pi$
$774$	0	0
$775$	6.48528	0.232958
$776$	6.72792	0.241518
$777$	0	0
$778$	−6.20101	−0.222317
$779$	7.41421	0.265642
$780$	0	0
$781$	−6.34315	−0.226976
$782$	−3.71573	−0.132874
$783$	0	0
$784$	−19.9706	−0.713234
$785$	−21.7990	−0.778039
$786$	0	0
$787$	13.4558	0.479649	0.239825	−	0.970816i	$-0.422910\pi$
$787$	13.4558	0.479649	0.239825	+	0.970816i	$0.422910\pi$
$788$	−27.1127	−0.965850
$789$	0	0
$790$	−4.68629	−0.166731
$791$	−7.31371	−0.260046
$792$	0	0
$793$	30.6274	1.08761
$794$	14.7696	0.524152
$795$	0	0
$796$	30.1421	1.06836
$797$	−10.8284	−0.383563	−0.191781	−	0.981438i	$-0.561426\pi$
$797$	−10.8284	−0.383563	−0.191781	+	0.981438i	$0.561426\pi$
$798$	0	0
$799$	−0.402020	−0.0142225
$800$	4.41421	0.156066
$801$	0	0
$802$	−12.4437	−0.439401
$803$	2.82843	0.0998130
$804$	0	0
$805$	−4.48528	−0.158085
$806$	14.5442	0.512296
$807$	0	0
$808$	−7.65685	−0.269367
$809$	−49.3137	−1.73378	−0.866889	−	0.498502i	$-0.833884\pi$
$809$	−49.3137	−1.73378	−0.866889	+	0.498502i	$0.833884\pi$
$810$	0	0
$811$	−15.3137	−0.537737	−0.268869	−	0.963177i	$-0.586650\pi$
$811$	−15.3137	−0.537737	−0.268869	+	0.963177i	$0.586650\pi$
$812$	−9.71573	−0.340955
$813$	0	0
$814$	−6.48528	−0.227309
$815$	7.89949	0.276707
$816$	0	0
$817$	0.585786	0.0204941
$818$	3.94113	0.137798
$819$	0	0
$820$	−13.5563	−0.473408
$821$	−51.4558	−1.79582	−0.897911	−	0.440178i	$-0.854915\pi$
$821$	−51.4558	−1.79582	−0.897911	+	0.440178i	$0.854915\pi$
$822$	0	0
$823$	−2.72792	−0.0950894	−0.0475447	−	0.998869i	$-0.515140\pi$
$823$	−2.72792	−0.0950894	−0.0475447	+	0.998869i	$0.515140\pi$
$824$	2.62742	0.0915304
$825$	0	0
$826$	−2.05887	−0.0716374
$827$	−48.6274	−1.69094	−0.845470	−	0.534022i	$-0.820680\pi$
$827$	−48.6274	−1.69094	−0.845470	+	0.534022i	$0.820680\pi$
$828$	0	0
$829$	−38.4853	−1.33665	−0.668325	−	0.743870i	$-0.732988\pi$
$829$	−38.4853	−1.33665	−0.668325	+	0.743870i	$0.732988\pi$
$830$	4.34315	0.150753
$831$	0	0
$832$	−22.5858	−0.783021
$833$	−7.79899	−0.270219
$834$	0	0
$835$	−10.0000	−0.346064
$836$	2.58579	0.0894313
$837$	0	0
$838$	14.4437	0.498948
$839$	−27.1127	−0.936034	−0.468017	−	0.883719i	$-0.655031\pi$
$839$	−27.1127	−0.936034	−0.468017	+	0.883719i	$0.655031\pi$
$840$	0	0
$841$	53.2843	1.83739
$842$	−6.08326	−0.209643
$843$	0	0
$844$	28.0000	0.963800
$845$	16.3137	0.561209
$846$	0	0
$847$	−5.27208	−0.181151
$848$	−12.0000	−0.412082
$849$	0	0
$850$	0.485281	0.0166450
$851$	−84.7696	−2.90586
$852$	0	0
$853$	−9.51472	−0.325778	−0.162889	−	0.986644i	$-0.552081\pi$
$853$	−9.51472	−0.325778	−0.162889	+	0.986644i	$0.552081\pi$
$854$	1.37258	0.0469688
$855$	0	0
$856$	12.6863	0.433609
$857$	−37.9411	−1.29604	−0.648022	−	0.761622i	$-0.724404\pi$
$857$	−37.9411	−1.29604	−0.648022	+	0.761622i	$0.724404\pi$
$858$	0	0
$859$	25.9411	0.885100	0.442550	−	0.896744i	$-0.354074\pi$
$859$	25.9411	0.885100	0.442550	+	0.896744i	$0.354074\pi$
$860$	−1.07107	−0.0365231
$861$	0	0
$862$	−1.45584	−0.0495862
$863$	31.3137	1.06593	0.532966	−	0.846137i	$-0.321078\pi$
$863$	31.3137	1.06593	0.532966	+	0.846137i	$0.321078\pi$
$864$	0	0
$865$	6.14214	0.208839
$866$	0.384776	0.0130752
$867$	0	0
$868$	−6.94618	−0.235769
$869$	16.0000	0.542763
$870$	0	0
$871$	64.9706	2.20144
$872$	−5.02944	−0.170318
$873$	0	0
$874$	−3.17157	−0.107280
$875$	0.585786	0.0198032
$876$	0	0
$877$	17.8995	0.604423	0.302211	−	0.953241i	$-0.402275\pi$
$877$	17.8995	0.604423	0.302211	+	0.953241i	$0.402275\pi$
$878$	0.402020	0.0135675
$879$	0	0
$880$	−4.24264	−0.143019
$881$	47.4558	1.59883	0.799414	−	0.600781i	$-0.205143\pi$
$881$	47.4558	1.59883	0.799414	+	0.600781i	$0.205143\pi$
$882$	0	0
$883$	46.5269	1.56576	0.782878	−	0.622176i	$-0.213751\pi$
$883$	46.5269	1.56576	0.782878	+	0.622176i	$0.213751\pi$
$884$	−11.5980	−0.390082
$885$	0	0
$886$	0.544156	0.0182813
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	0	0
$889$	4.68629	0.157173
$890$	−4.44365	−0.148952
$891$	0	0
$892$	−11.5980	−0.388329
$893$	−0.343146	−0.0114829
$894$	0	0
$895$	−17.1716	−0.573982
$896$	−6.18377	−0.206585
$897$	0	0
$898$	−3.27208	−0.109191
$899$	58.8284	1.96204
$900$	0	0
$901$	−4.68629	−0.156123
$902$	−4.34315	−0.144611
$903$	0	0
$904$	19.7990	0.658505
$905$	−19.1716	−0.637285
$906$	0	0
$907$	38.1421	1.26649	0.633244	−	0.773952i	$-0.281723\pi$
$907$	38.1421	1.26649	0.633244	+	0.773952i	$0.281723\pi$
$908$	−34.6863	−1.15111
$909$	0	0
$910$	1.31371	0.0435490
$911$	23.3137	0.772418	0.386209	−	0.922411i	$-0.373784\pi$
$911$	23.3137	0.772418	0.386209	+	0.922411i	$0.373784\pi$
$912$	0	0
$913$	−14.8284	−0.490749
$914$	11.9411	0.394977
$915$	0	0
$916$	3.02944	0.100095
$917$	−9.79899	−0.323591
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	12.1421	0.400314
$921$	0	0
$922$	−4.42641	−0.145776
$923$	24.2843	0.799327
$924$	0	0
$925$	11.0711	0.364014
$926$	−3.35534	−0.110263
$927$	0	0
$928$	40.0416	1.31443
$929$	51.4558	1.68821	0.844106	−	0.536177i	$-0.180132\pi$
$929$	51.4558	1.68821	0.844106	+	0.536177i	$0.180132\pi$
$930$	0	0
$931$	−6.65685	−0.218170
$932$	−15.2548	−0.499689
$933$	0	0
$934$	−6.76955	−0.221507
$935$	−1.65685	−0.0541849
$936$	0	0
$937$	−18.7696	−0.613175	−0.306587	−	0.951843i	$-0.599187\pi$
$937$	−18.7696	−0.613175	−0.306587	+	0.951843i	$0.599187\pi$
$938$	2.91169	0.0950700
$939$	0	0
$940$	0.627417	0.0204641
$941$	17.5563	0.572321	0.286160	−	0.958182i	$-0.407621\pi$
$941$	17.5563	0.572321	0.286160	+	0.958182i	$0.407621\pi$
$942$	0	0
$943$	−56.7696	−1.84867
$944$	−25.4558	−0.828517
$945$	0	0
$946$	−0.343146	−0.0111566
$947$	−12.8284	−0.416868	−0.208434	−	0.978036i	$-0.566837\pi$
$947$	−12.8284	−0.416868	−0.208434	+	0.978036i	$0.566837\pi$
$948$	0	0
$949$	−10.8284	−0.351506
$950$	0.414214	0.0134389
$951$	0	0
$952$	−1.08831	−0.0352724
$953$	5.85786	0.189755	0.0948774	−	0.995489i	$-0.469754\pi$
$953$	5.85786	0.189755	0.0948774	+	0.995489i	$0.469754\pi$
$954$	0	0
$955$	−1.89949	−0.0614662
$956$	4.72792	0.152912
$957$	0	0
$958$	15.8162	0.510999
$959$	8.20101	0.264824
$960$	0	0
$961$	11.0589	0.356738
$962$	24.8284	0.800501
$963$	0	0
$964$	27.3726	0.881612
$965$	−15.0711	−0.485155
$966$	0	0
$967$	−27.8995	−0.897187	−0.448594	−	0.893736i	$-0.648075\pi$
$967$	−27.8995	−0.897187	−0.448594	+	0.893736i	$0.648075\pi$
$968$	14.2721	0.458722
$969$	0	0
$970$	−1.75736	−0.0564254
$971$	−17.6569	−0.566635	−0.283318	−	0.959026i	$-0.591435\pi$
$971$	−17.6569	−0.566635	−0.283318	+	0.959026i	$0.591435\pi$
$972$	0	0
$973$	0.686292	0.0220015
$974$	−1.17157	−0.0375396
$975$	0	0
$976$	16.9706	0.543214
$977$	−39.5980	−1.26685	−0.633426	−	0.773803i	$-0.718352\pi$
$977$	−39.5980	−1.26685	−0.633426	+	0.773803i	$0.718352\pi$
$978$	0	0
$979$	15.1716	0.484886
$980$	12.1716	0.388807
$981$	0	0
$982$	−9.07107	−0.289469
$983$	−31.9411	−1.01876	−0.509382	−	0.860541i	$-0.670126\pi$
$983$	−31.9411	−1.01876	−0.509382	+	0.860541i	$0.670126\pi$
$984$	0	0
$985$	14.8284	0.472473
$986$	4.40202	0.140189
$987$	0	0
$988$	−9.89949	−0.314945
$989$	−4.48528	−0.142624
$990$	0	0
$991$	−61.6569	−1.95859	−0.979297	−	0.202427i	$-0.935117\pi$
$991$	−61.6569	−1.95859	−0.979297	+	0.202427i	$0.935117\pi$
$992$	28.6274	0.908921
$993$	0	0
$994$	1.08831	0.0345192
$995$	−16.4853	−0.522619
$996$	0	0
$997$	−50.4853	−1.59888	−0.799442	−	0.600743i	$-0.794872\pi$
$997$	−50.4853	−1.59888	−0.799442	+	0.600743i	$0.794872\pi$
$998$	−9.85786	−0.312045
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.2.a.e.1.2		2
3.2	odd	2		285.2.a.f.1.1	✓	2
5.4	even	2		4275.2.a.x.1.1		2
12.11	even	2		4560.2.a.bj.1.2		2
15.2	even	4		1425.2.c.j.799.2		4
15.8	even	4		1425.2.c.j.799.3		4
15.14	odd	2		1425.2.a.l.1.2		2
57.56	even	2		5415.2.a.p.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.f.1.1	✓	2	3.2	odd	2
855.2.a.e.1.2		2	1.1	even	1	trivial
1425.2.a.l.1.2		2	15.14	odd	2
1425.2.c.j.799.2		4	15.2	even	4
1425.2.c.j.799.3		4	15.8	even	4
4275.2.a.x.1.1		2	5.4	even	2
4560.2.a.bj.1.2		2	12.11	even	2
5415.2.a.p.1.2		2	57.56	even	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)

\(f(q)\)	\(=\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.585786 q^{7} -1.58579 q^{8} +O(q^{10})\)	\(q+0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +0.585786 q^{7} -1.58579 q^{8} +0.414214 q^{10} -1.41421 q^{11} +5.41421 q^{13} +0.242641 q^{14} +3.00000 q^{16} +1.17157 q^{17} +1.00000 q^{19} -1.82843 q^{20} -0.585786 q^{22} -7.65685 q^{23} +1.00000 q^{25} +2.24264 q^{26} -1.07107 q^{28} +9.07107 q^{29} +6.48528 q^{31} +4.41421 q^{32} +0.485281 q^{34} +0.585786 q^{35} +11.0711 q^{37} +0.414214 q^{38} -1.58579 q^{40} +7.41421 q^{41} +0.585786 q^{43} +2.58579 q^{44} -3.17157 q^{46} -0.343146 q^{47} -6.65685 q^{49} +0.414214 q^{50} -9.89949 q^{52} -4.00000 q^{53} -1.41421 q^{55} -0.928932 q^{56} +3.75736 q^{58} -8.48528 q^{59} +5.65685 q^{61} +2.68629 q^{62} -4.17157 q^{64} +5.41421 q^{65} +12.0000 q^{67} -2.14214 q^{68} +0.242641 q^{70} +4.48528 q^{71} -2.00000 q^{73} +4.58579 q^{74} -1.82843 q^{76} -0.828427 q^{77} -11.3137 q^{79} +3.00000 q^{80} +3.07107 q^{82} +10.4853 q^{83} +1.17157 q^{85} +0.242641 q^{86} +2.24264 q^{88} -10.7279 q^{89} +3.17157 q^{91} +14.0000 q^{92} -0.142136 q^{94} +1.00000 q^{95} -4.24264 q^{97} -2.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} + 4 q^{7} - 6 q^{8} - 2 q^{10} + 8 q^{13} - 8 q^{14} + 6 q^{16} + 8 q^{17} + 2 q^{19} + 2 q^{20} - 4 q^{22} - 4 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} + 4 q^{29} - 4 q^{31} + 6 q^{32} - 16 q^{34} + 4 q^{35} + 8 q^{37} - 2 q^{38} - 6 q^{40} + 12 q^{41} + 4 q^{43} + 8 q^{44} - 12 q^{46} - 12 q^{47} - 2 q^{49} - 2 q^{50} - 8 q^{53} - 16 q^{56} + 16 q^{58} + 28 q^{62} - 14 q^{64} + 8 q^{65} + 24 q^{67} + 24 q^{68} - 8 q^{70} - 8 q^{71} - 4 q^{73} + 12 q^{74} + 2 q^{76} + 4 q^{77} + 6 q^{80} - 8 q^{82} + 4 q^{83} + 8 q^{85} - 8 q^{86} - 4 q^{88} + 4 q^{89} + 12 q^{91} + 28 q^{92} + 28 q^{94} + 2 q^{95} - 14 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 + 4 * q^7 - 6 * q^8 - 2 * q^10 + 8 * q^13 - 8 * q^14 + 6 * q^16 + 8 * q^17 + 2 * q^19 + 2 * q^20 - 4 * q^22 - 4 * q^23 + 2 * q^25 - 4 * q^26 + 12 * q^28 + 4 * q^29 - 4 * q^31 + 6 * q^32 - 16 * q^34 + 4 * q^35 + 8 * q^37 - 2 * q^38 - 6 * q^40 + 12 * q^41 + 4 * q^43 + 8 * q^44 - 12 * q^46 - 12 * q^47 - 2 * q^49 - 2 * q^50 - 8 * q^53 - 16 * q^56 + 16 * q^58 + 28 * q^62 - 14 * q^64 + 8 * q^65 + 24 * q^67 + 24 * q^68 - 8 * q^70 - 8 * q^71 - 4 * q^73 + 12 * q^74 + 2 * q^76 + 4 * q^77 + 6 * q^80 - 8 * q^82 + 4 * q^83 + 8 * q^85 - 8 * q^86 - 4 * q^88 + 4 * q^89 + 12 * q^91 + 28 * q^92 + 28 * q^94 + 2 * q^95 - 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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