LMFDB - Embedded newform 2898.2.a.bb.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2898,2,Mod(1,2898)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2898, 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("2898.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2898.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:	$23.1406465058$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	2898.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.70156 q^{10} +4.00000 q^{11} +5.70156 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} -5.40312 q^{19} -3.70156 q^{20} +4.00000 q^{22} -1.00000 q^{23} +8.70156 q^{25} +5.70156 q^{26} +1.00000 q^{28} -0.298438 q^{29} -2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -3.70156 q^{35} +4.29844 q^{37} -5.40312 q^{38} -3.70156 q^{40} +0.298438 q^{41} -1.70156 q^{43} +4.00000 q^{44} -1.00000 q^{46} +11.1047 q^{47} +1.00000 q^{49} +8.70156 q^{50} +5.70156 q^{52} +9.40312 q^{53} -14.8062 q^{55} +1.00000 q^{56} -0.298438 q^{58} +7.40312 q^{59} -2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -21.1047 q^{65} +14.8062 q^{67} -4.00000 q^{68} -3.70156 q^{70} +7.40312 q^{71} +1.40312 q^{73} +4.29844 q^{74} -5.40312 q^{76} +4.00000 q^{77} +8.00000 q^{79} -3.70156 q^{80} +0.298438 q^{82} -13.4031 q^{83} +14.8062 q^{85} -1.70156 q^{86} +4.00000 q^{88} +11.4031 q^{89} +5.70156 q^{91} -1.00000 q^{92} +11.1047 q^{94} +20.0000 q^{95} +6.29844 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8} - q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} - q^{20} + 8 q^{22} - 2 q^{23} + 11 q^{25} + 5 q^{26} + 2 q^{28} - 7 q^{29} - 4 q^{31} + 2 q^{32} - 8 q^{34} - q^{35} + 15 q^{37} + 2 q^{38} - q^{40} + 7 q^{41} + 3 q^{43} + 8 q^{44} - 2 q^{46} + 3 q^{47} + 2 q^{49} + 11 q^{50} + 5 q^{52} + 6 q^{53} - 4 q^{55} + 2 q^{56} - 7 q^{58} + 2 q^{59} - 4 q^{61} - 4 q^{62} + 2 q^{64} - 23 q^{65} + 4 q^{67} - 8 q^{68} - q^{70} + 2 q^{71} - 10 q^{73} + 15 q^{74} + 2 q^{76} + 8 q^{77} + 16 q^{79} - q^{80} + 7 q^{82} - 14 q^{83} + 4 q^{85} + 3 q^{86} + 8 q^{88} + 10 q^{89} + 5 q^{91} - 2 q^{92} + 3 q^{94} + 40 q^{95} + 19 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8 - q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^19 - q^20 + 8 * q^22 - 2 * q^23 + 11 * q^25 + 5 * q^26 + 2 * q^28 - 7 * q^29 - 4 * q^31 + 2 * q^32 - 8 * q^34 - q^35 + 15 * q^37 + 2 * q^38 - q^40 + 7 * q^41 + 3 * q^43 + 8 * q^44 - 2 * q^46 + 3 * q^47 + 2 * q^49 + 11 * q^50 + 5 * q^52 + 6 * q^53 - 4 * q^55 + 2 * q^56 - 7 * q^58 + 2 * q^59 - 4 * q^61 - 4 * q^62 + 2 * q^64 - 23 * q^65 + 4 * q^67 - 8 * q^68 - q^70 + 2 * q^71 - 10 * q^73 + 15 * q^74 + 2 * q^76 + 8 * q^77 + 16 * q^79 - q^80 + 7 * q^82 - 14 * q^83 + 4 * q^85 + 3 * q^86 + 8 * q^88 + 10 * q^89 + 5 * q^91 - 2 * q^92 + 3 * q^94 + 40 * q^95 + 19 * q^97 + 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−3.70156	−1.65539	−0.827694	−	0.561179i	$-0.810348\pi$
$5$	−3.70156	−1.65539	−0.827694	+	0.561179i	$0.810348\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.70156	−1.17054
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	5.70156	1.58133	0.790664	−	0.612250i	$-0.209735\pi$
$13$	5.70156	1.58133	0.790664	+	0.612250i	$0.209735\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	−5.40312	−1.23956	−0.619781	−	0.784775i	$-0.712779\pi$
$19$	−5.40312	−1.23956	−0.619781	+	0.784775i	$0.712779\pi$
$20$	−3.70156	−0.827694
$21$	0	0
$22$	4.00000	0.852803
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	8.70156	1.74031
$26$	5.70156	1.11817
$27$	0	0
$28$	1.00000	0.188982
$29$	−0.298438	−0.0554185	−0.0277093	−	0.999616i	$-0.508821\pi$
$29$	−0.298438	−0.0554185	−0.0277093	+	0.999616i	$0.508821\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.00000	−0.685994
$35$	−3.70156	−0.625678
$36$	0	0
$37$	4.29844	0.706659	0.353329	−	0.935499i	$-0.385049\pi$
$37$	4.29844	0.706659	0.353329	+	0.935499i	$0.385049\pi$
$38$	−5.40312	−0.876502
$39$	0	0
$40$	−3.70156	−0.585268
$41$	0.298438	0.0466082	0.0233041	−	0.999728i	$-0.492581\pi$
$41$	0.298438	0.0466082	0.0233041	+	0.999728i	$0.492581\pi$
$42$	0	0
$43$	−1.70156	−0.259486	−0.129743	−	0.991548i	$-0.541415\pi$
$43$	−1.70156	−0.259486	−0.129743	+	0.991548i	$0.541415\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−1.00000	−0.147442
$47$	11.1047	1.61978	0.809892	−	0.586578i	$-0.199525\pi$
$47$	11.1047	1.61978	0.809892	+	0.586578i	$0.199525\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.70156	1.23059
$51$	0	0
$52$	5.70156	0.790664
$53$	9.40312	1.29162	0.645809	−	0.763499i	$-0.276520\pi$
$53$	9.40312	1.29162	0.645809	+	0.763499i	$0.276520\pi$
$54$	0	0
$55$	−14.8062	−1.99647
$56$	1.00000	0.133631
$57$	0	0
$58$	−0.298438	−0.0391868
$59$	7.40312	0.963805	0.481902	−	0.876225i	$-0.339946\pi$
$59$	7.40312	0.963805	0.481902	+	0.876225i	$0.339946\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−21.1047	−2.61771
$66$	0	0
$67$	14.8062	1.80887	0.904436	−	0.426610i	$-0.140292\pi$
$67$	14.8062	1.80887	0.904436	+	0.426610i	$0.140292\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	−3.70156	−0.442421
$71$	7.40312	0.878589	0.439295	−	0.898343i	$-0.355228\pi$
$71$	7.40312	0.878589	0.439295	+	0.898343i	$0.355228\pi$
$72$	0	0
$73$	1.40312	0.164223	0.0821116	−	0.996623i	$-0.473834\pi$
$73$	1.40312	0.164223	0.0821116	+	0.996623i	$0.473834\pi$
$74$	4.29844	0.499683
$75$	0	0
$76$	−5.40312	−0.619781
$77$	4.00000	0.455842
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−3.70156	−0.413847
$81$	0	0
$82$	0.298438	0.0329570
$83$	−13.4031	−1.47118	−0.735592	−	0.677425i	$-0.763096\pi$
$83$	−13.4031	−1.47118	−0.735592	+	0.677425i	$0.763096\pi$
$84$	0	0
$85$	14.8062	1.60596
$86$	−1.70156	−0.183484
$87$	0	0
$88$	4.00000	0.426401
$89$	11.4031	1.20873	0.604364	−	0.796708i	$-0.293427\pi$
$89$	11.4031	1.20873	0.604364	+	0.796708i	$0.293427\pi$
$90$	0	0
$91$	5.70156	0.597686
$92$	−1.00000	−0.104257
$93$	0	0
$94$	11.1047	1.14536
$95$	20.0000	2.05196
$96$	0	0
$97$	6.29844	0.639509	0.319755	−	0.947500i	$-0.396399\pi$
$97$	6.29844	0.639509	0.319755	+	0.947500i	$0.396399\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	8.70156	0.870156
$101$	3.40312	0.338624	0.169312	−	0.985563i	$-0.445846\pi$
$101$	3.40312	0.338624	0.169312	+	0.985563i	$0.445846\pi$
$102$	0	0
$103$	2.29844	0.226472	0.113236	−	0.993568i	$-0.463878\pi$
$103$	2.29844	0.226472	0.113236	+	0.993568i	$0.463878\pi$
$104$	5.70156	0.559084
$105$	0	0
$106$	9.40312	0.913312
$107$	−18.8062	−1.81807	−0.909034	−	0.416721i	$-0.863179\pi$
$107$	−18.8062	−1.81807	−0.909034	+	0.416721i	$0.863179\pi$
$108$	0	0
$109$	−7.70156	−0.737676	−0.368838	−	0.929494i	$-0.620244\pi$
$109$	−7.70156	−0.737676	−0.368838	+	0.929494i	$0.620244\pi$
$110$	−14.8062	−1.41172
$111$	0	0
$112$	1.00000	0.0944911
$113$	−7.10469	−0.668353	−0.334176	−	0.942511i	$-0.608458\pi$
$113$	−7.10469	−0.668353	−0.334176	+	0.942511i	$0.608458\pi$
$114$	0	0
$115$	3.70156	0.345172
$116$	−0.298438	−0.0277093
$117$	0	0
$118$	7.40312	0.681513
$119$	−4.00000	−0.366679
$120$	0	0
$121$	5.00000	0.454545
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−2.00000	−0.179605
$125$	−13.7016	−1.22550
$126$	0	0
$127$	−5.70156	−0.505932	−0.252966	−	0.967475i	$-0.581406\pi$
$127$	−5.70156	−0.505932	−0.252966	+	0.967475i	$0.581406\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−21.1047	−1.85100
$131$	7.40312	0.646814	0.323407	−	0.946260i	$-0.395172\pi$
$131$	7.40312	0.646814	0.323407	+	0.946260i	$0.395172\pi$
$132$	0	0
$133$	−5.40312	−0.468510
$134$	14.8062	1.27907
$135$	0	0
$136$	−4.00000	−0.342997
$137$	19.7016	1.68322	0.841609	−	0.540087i	$-0.181609\pi$
$137$	19.7016	1.68322	0.841609	+	0.540087i	$0.181609\pi$
$138$	0	0
$139$	17.7016	1.50143	0.750713	−	0.660628i	$-0.229710\pi$
$139$	17.7016	1.50143	0.750713	+	0.660628i	$0.229710\pi$
$140$	−3.70156	−0.312839
$141$	0	0
$142$	7.40312	0.621256
$143$	22.8062	1.90715
$144$	0	0
$145$	1.10469	0.0917392
$146$	1.40312	0.116123
$147$	0	0
$148$	4.29844	0.353329
$149$	20.2094	1.65562	0.827808	−	0.561011i	$-0.189588\pi$
$149$	20.2094	1.65562	0.827808	+	0.561011i	$0.189588\pi$
$150$	0	0
$151$	23.9109	1.94584	0.972922	−	0.231133i	$-0.0742433\pi$
$151$	23.9109	1.94584	0.972922	+	0.231133i	$0.0742433\pi$
$152$	−5.40312	−0.438251
$153$	0	0
$154$	4.00000	0.322329
$155$	7.40312	0.594633
$156$	0	0
$157$	−1.40312	−0.111982	−0.0559908	−	0.998431i	$-0.517832\pi$
$157$	−1.40312	−0.111982	−0.0559908	+	0.998431i	$0.517832\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	−3.70156	−0.292634
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−10.8062	−0.846411	−0.423205	−	0.906034i	$-0.639095\pi$
$163$	−10.8062	−0.846411	−0.423205	+	0.906034i	$0.639095\pi$
$164$	0.298438	0.0233041
$165$	0	0
$166$	−13.4031	−1.04028
$167$	13.4031	1.03716	0.518582	−	0.855028i	$-0.326460\pi$
$167$	13.4031	1.03716	0.518582	+	0.855028i	$0.326460\pi$
$168$	0	0
$169$	19.5078	1.50060
$170$	14.8062	1.13559
$171$	0	0
$172$	−1.70156	−0.129743
$173$	−19.4031	−1.47519	−0.737596	−	0.675242i	$-0.764039\pi$
$173$	−19.4031	−1.47519	−0.737596	+	0.675242i	$0.764039\pi$
$174$	0	0
$175$	8.70156	0.657776
$176$	4.00000	0.301511
$177$	0	0
$178$	11.4031	0.854700
$179$	25.1047	1.87641	0.938206	−	0.346077i	$-0.112486\pi$
$179$	25.1047	1.87641	0.938206	+	0.346077i	$0.112486\pi$
$180$	0	0
$181$	−16.8062	−1.24920	−0.624599	−	0.780945i	$-0.714738\pi$
$181$	−16.8062	−1.24920	−0.624599	+	0.780945i	$0.714738\pi$
$182$	5.70156	0.422628
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−15.9109	−1.16980
$186$	0	0
$187$	−16.0000	−1.17004
$188$	11.1047	0.809892
$189$	0	0
$190$	20.0000	1.45095
$191$	−22.8062	−1.65020	−0.825101	−	0.564985i	$-0.808882\pi$
$191$	−22.8062	−1.65020	−0.825101	+	0.564985i	$0.808882\pi$
$192$	0	0
$193$	−23.1047	−1.66311	−0.831556	−	0.555441i	$-0.812549\pi$
$193$	−23.1047	−1.66311	−0.831556	+	0.555441i	$0.812549\pi$
$194$	6.29844	0.452201
$195$	0	0
$196$	1.00000	0.0714286
$197$	−22.5078	−1.60362	−0.801808	−	0.597582i	$-0.796128\pi$
$197$	−22.5078	−1.60362	−0.801808	+	0.597582i	$0.796128\pi$
$198$	0	0
$199$	−2.29844	−0.162932	−0.0814660	−	0.996676i	$-0.525960\pi$
$199$	−2.29844	−0.162932	−0.0814660	+	0.996676i	$0.525960\pi$
$200$	8.70156	0.615293
$201$	0	0
$202$	3.40312	0.239443
$203$	−0.298438	−0.0209462
$204$	0	0
$205$	−1.10469	−0.0771546
$206$	2.29844	0.160140
$207$	0	0
$208$	5.70156	0.395332
$209$	−21.6125	−1.49497
$210$	0	0
$211$	10.8062	0.743933	0.371966	−	0.928246i	$-0.378684\pi$
$211$	10.8062	0.743933	0.371966	+	0.928246i	$0.378684\pi$
$212$	9.40312	0.645809
$213$	0	0
$214$	−18.8062	−1.28557
$215$	6.29844	0.429550
$216$	0	0
$217$	−2.00000	−0.135769
$218$	−7.70156	−0.521616
$219$	0	0
$220$	−14.8062	−0.998237
$221$	−22.8062	−1.53411
$222$	0	0
$223$	−22.0000	−1.47323	−0.736614	−	0.676313i	$-0.763577\pi$
$223$	−22.0000	−1.47323	−0.736614	+	0.676313i	$0.763577\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−7.10469	−0.472597
$227$	23.1047	1.53351	0.766756	−	0.641939i	$-0.221870\pi$
$227$	23.1047	1.53351	0.766756	+	0.641939i	$0.221870\pi$
$228$	0	0
$229$	−14.5969	−0.964589	−0.482294	−	0.876009i	$-0.660196\pi$
$229$	−14.5969	−0.964589	−0.482294	+	0.876009i	$0.660196\pi$
$230$	3.70156	0.244074
$231$	0	0
$232$	−0.298438	−0.0195934
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−41.1047	−2.68137
$236$	7.40312	0.481902
$237$	0	0
$238$	−4.00000	−0.259281
$239$	−10.8062	−0.698998	−0.349499	−	0.936937i	$-0.613648\pi$
$239$	−10.8062	−0.698998	−0.349499	+	0.936937i	$0.613648\pi$
$240$	0	0
$241$	−15.9109	−1.02491	−0.512457	−	0.858713i	$-0.671264\pi$
$241$	−15.9109	−1.02491	−0.512457	+	0.858713i	$0.671264\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	−2.00000	−0.128037
$245$	−3.70156	−0.236484
$246$	0	0
$247$	−30.8062	−1.96015
$248$	−2.00000	−0.127000
$249$	0	0
$250$	−13.7016	−0.866563
$251$	−19.7016	−1.24355	−0.621776	−	0.783195i	$-0.713588\pi$
$251$	−19.7016	−1.24355	−0.621776	+	0.783195i	$0.713588\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	−5.70156	−0.357748
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.00000	−0.124757	−0.0623783	−	0.998053i	$-0.519869\pi$
$257$	−2.00000	−0.124757	−0.0623783	+	0.998053i	$0.519869\pi$
$258$	0	0
$259$	4.29844	0.267092
$260$	−21.1047	−1.30886
$261$	0	0
$262$	7.40312	0.457367
$263$	−9.70156	−0.598224	−0.299112	−	0.954218i	$-0.596690\pi$
$263$	−9.70156	−0.598224	−0.299112	+	0.954218i	$0.596690\pi$
$264$	0	0
$265$	−34.8062	−2.13813
$266$	−5.40312	−0.331287
$267$	0	0
$268$	14.8062	0.904436
$269$	10.2094	0.622476	0.311238	−	0.950332i	$-0.399256\pi$
$269$	10.2094	0.622476	0.311238	+	0.950332i	$0.399256\pi$
$270$	0	0
$271$	1.40312	0.0852337	0.0426169	−	0.999091i	$-0.486431\pi$
$271$	1.40312	0.0852337	0.0426169	+	0.999091i	$0.486431\pi$
$272$	−4.00000	−0.242536
$273$	0	0
$274$	19.7016	1.19021
$275$	34.8062	2.09890
$276$	0	0
$277$	−21.4031	−1.28599	−0.642995	−	0.765871i	$-0.722308\pi$
$277$	−21.4031	−1.28599	−0.642995	+	0.765871i	$0.722308\pi$
$278$	17.7016	1.06167
$279$	0	0
$280$	−3.70156	−0.221211
$281$	−14.5078	−0.865463	−0.432732	−	0.901523i	$-0.642450\pi$
$281$	−14.5078	−0.865463	−0.432732	+	0.901523i	$0.642450\pi$
$282$	0	0
$283$	4.80625	0.285702	0.142851	−	0.989744i	$-0.454373\pi$
$283$	4.80625	0.285702	0.142851	+	0.989744i	$0.454373\pi$
$284$	7.40312	0.439295
$285$	0	0
$286$	22.8062	1.34856
$287$	0.298438	0.0176162
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	1.10469	0.0648694
$291$	0	0
$292$	1.40312	0.0821116
$293$	20.8062	1.21551	0.607757	−	0.794123i	$-0.292069\pi$
$293$	20.8062	1.21551	0.607757	+	0.794123i	$0.292069\pi$
$294$	0	0
$295$	−27.4031	−1.59547
$296$	4.29844	0.249842
$297$	0	0
$298$	20.2094	1.17070
$299$	−5.70156	−0.329730
$300$	0	0
$301$	−1.70156	−0.0980764
$302$	23.9109	1.37592
$303$	0	0
$304$	−5.40312	−0.309890
$305$	7.40312	0.423902
$306$	0	0
$307$	11.9109	0.679793	0.339896	−	0.940463i	$-0.389608\pi$
$307$	11.9109	0.679793	0.339896	+	0.940463i	$0.389608\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	7.40312	0.420469
$311$	−13.4031	−0.760021	−0.380011	−	0.924982i	$-0.624080\pi$
$311$	−13.4031	−0.760021	−0.380011	+	0.924982i	$0.624080\pi$
$312$	0	0
$313$	−6.20937	−0.350974	−0.175487	−	0.984482i	$-0.556150\pi$
$313$	−6.20937	−0.350974	−0.175487	+	0.984482i	$0.556150\pi$
$314$	−1.40312	−0.0791829
$315$	0	0
$316$	8.00000	0.450035
$317$	8.29844	0.466087	0.233043	−	0.972466i	$-0.425132\pi$
$317$	8.29844	0.466087	0.233043	+	0.972466i	$0.425132\pi$
$318$	0	0
$319$	−1.19375	−0.0668373
$320$	−3.70156	−0.206924
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	21.6125	1.20255
$324$	0	0
$325$	49.6125	2.75201
$326$	−10.8062	−0.598503
$327$	0	0
$328$	0.298438	0.0164785
$329$	11.1047	0.612221
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−13.4031	−0.735592
$333$	0	0
$334$	13.4031	0.733386
$335$	−54.8062	−2.99439
$336$	0	0
$337$	−4.20937	−0.229299	−0.114650	−	0.993406i	$-0.536575\pi$
$337$	−4.20937	−0.229299	−0.114650	+	0.993406i	$0.536575\pi$
$338$	19.5078	1.06109
$339$	0	0
$340$	14.8062	0.802982
$341$	−8.00000	−0.433224
$342$	0	0
$343$	1.00000	0.0539949
$344$	−1.70156	−0.0917421
$345$	0	0
$346$	−19.4031	−1.04312
$347$	10.2984	0.552849	0.276425	−	0.961036i	$-0.410850\pi$
$347$	10.2984	0.552849	0.276425	+	0.961036i	$0.410850\pi$
$348$	0	0
$349$	12.5969	0.674295	0.337148	−	0.941452i	$-0.390538\pi$
$349$	12.5969	0.674295	0.337148	+	0.941452i	$0.390538\pi$
$350$	8.70156	0.465118
$351$	0	0
$352$	4.00000	0.213201
$353$	−4.29844	−0.228783	−0.114391	−	0.993436i	$-0.536492\pi$
$353$	−4.29844	−0.228783	−0.114391	+	0.993436i	$0.536492\pi$
$354$	0	0
$355$	−27.4031	−1.45441
$356$	11.4031	0.604364
$357$	0	0
$358$	25.1047	1.32682
$359$	2.89531	0.152809	0.0764044	−	0.997077i	$-0.475656\pi$
$359$	2.89531	0.152809	0.0764044	+	0.997077i	$0.475656\pi$
$360$	0	0
$361$	10.1938	0.536513
$362$	−16.8062	−0.883317
$363$	0	0
$364$	5.70156	0.298843
$365$	−5.19375	−0.271853
$366$	0	0
$367$	−17.1047	−0.892857	−0.446429	−	0.894819i	$-0.647304\pi$
$367$	−17.1047	−0.892857	−0.446429	+	0.894819i	$0.647304\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−15.9109	−0.827170
$371$	9.40312	0.488186
$372$	0	0
$373$	−0.806248	−0.0417460	−0.0208730	−	0.999782i	$-0.506645\pi$
$373$	−0.806248	−0.0417460	−0.0208730	+	0.999782i	$0.506645\pi$
$374$	−16.0000	−0.827340
$375$	0	0
$376$	11.1047	0.572680
$377$	−1.70156	−0.0876349
$378$	0	0
$379$	13.7016	0.703802	0.351901	−	0.936037i	$-0.385535\pi$
$379$	13.7016	0.703802	0.351901	+	0.936037i	$0.385535\pi$
$380$	20.0000	1.02598
$381$	0	0
$382$	−22.8062	−1.16687
$383$	−29.6125	−1.51313	−0.756564	−	0.653920i	$-0.773123\pi$
$383$	−29.6125	−1.51313	−0.756564	+	0.653920i	$0.773123\pi$
$384$	0	0
$385$	−14.8062	−0.754596
$386$	−23.1047	−1.17600
$387$	0	0
$388$	6.29844	0.319755
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	1.00000	0.0505076
$393$	0	0
$394$	−22.5078	−1.13393
$395$	−29.6125	−1.48997
$396$	0	0
$397$	8.59688	0.431465	0.215732	−	0.976453i	$-0.430786\pi$
$397$	8.59688	0.431465	0.215732	+	0.976453i	$0.430786\pi$
$398$	−2.29844	−0.115210
$399$	0	0
$400$	8.70156	0.435078
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	−11.4031	−0.568030
$404$	3.40312	0.169312
$405$	0	0
$406$	−0.298438	−0.0148112
$407$	17.1938	0.852263
$408$	0	0
$409$	5.40312	0.267167	0.133584	−	0.991038i	$-0.457352\pi$
$409$	5.40312	0.267167	0.133584	+	0.991038i	$0.457352\pi$
$410$	−1.10469	−0.0545566
$411$	0	0
$412$	2.29844	0.113236
$413$	7.40312	0.364284
$414$	0	0
$415$	49.6125	2.43538
$416$	5.70156	0.279542
$417$	0	0
$418$	−21.6125	−1.05710
$419$	0.209373	0.0102285	0.00511426	−	0.999987i	$-0.498372\pi$
$419$	0.209373	0.0102285	0.00511426	+	0.999987i	$0.498372\pi$
$420$	0	0
$421$	−15.7016	−0.765247	−0.382624	−	0.923904i	$-0.624979\pi$
$421$	−15.7016	−0.765247	−0.382624	+	0.923904i	$0.624979\pi$
$422$	10.8062	0.526040
$423$	0	0
$424$	9.40312	0.456656
$425$	−34.8062	−1.68835
$426$	0	0
$427$	−2.00000	−0.0967868
$428$	−18.8062	−0.909034
$429$	0	0
$430$	6.29844	0.303738
$431$	3.91093	0.188383	0.0941916	−	0.995554i	$-0.469973\pi$
$431$	3.91093	0.188383	0.0941916	+	0.995554i	$0.469973\pi$
$432$	0	0
$433$	31.3141	1.50486	0.752429	−	0.658674i	$-0.228882\pi$
$433$	31.3141	1.50486	0.752429	+	0.658674i	$0.228882\pi$
$434$	−2.00000	−0.0960031
$435$	0	0
$436$	−7.70156	−0.368838
$437$	5.40312	0.258466
$438$	0	0
$439$	−22.0000	−1.05000	−0.525001	−	0.851101i	$-0.675935\pi$
$439$	−22.0000	−1.05000	−0.525001	+	0.851101i	$0.675935\pi$
$440$	−14.8062	−0.705860
$441$	0	0
$442$	−22.8062	−1.08478
$443$	13.7016	0.650981	0.325490	−	0.945545i	$-0.394471\pi$
$443$	13.7016	0.650981	0.325490	+	0.945545i	$0.394471\pi$
$444$	0	0
$445$	−42.2094	−2.00092
$446$	−22.0000	−1.04173
$447$	0	0
$448$	1.00000	0.0472456
$449$	38.4187	1.81309	0.906546	−	0.422106i	$-0.138709\pi$
$449$	38.4187	1.81309	0.906546	+	0.422106i	$0.138709\pi$
$450$	0	0
$451$	1.19375	0.0562116
$452$	−7.10469	−0.334176
$453$	0	0
$454$	23.1047	1.08436
$455$	−21.1047	−0.989403
$456$	0	0
$457$	28.2094	1.31958	0.659789	−	0.751451i	$-0.270645\pi$
$457$	28.2094	1.31958	0.659789	+	0.751451i	$0.270645\pi$
$458$	−14.5969	−0.682067
$459$	0	0
$460$	3.70156	0.172586
$461$	2.20937	0.102901	0.0514504	−	0.998676i	$-0.483616\pi$
$461$	2.20937	0.102901	0.0514504	+	0.998676i	$0.483616\pi$
$462$	0	0
$463$	−22.2984	−1.03630	−0.518148	−	0.855291i	$-0.673378\pi$
$463$	−22.2984	−1.03630	−0.518148	+	0.855291i	$0.673378\pi$
$464$	−0.298438	−0.0138546
$465$	0	0
$466$	26.0000	1.20443
$467$	3.70156	0.171288	0.0856439	−	0.996326i	$-0.472705\pi$
$467$	3.70156	0.171288	0.0856439	+	0.996326i	$0.472705\pi$
$468$	0	0
$469$	14.8062	0.683689
$470$	−41.1047	−1.89602
$471$	0	0
$472$	7.40312	0.340756
$473$	−6.80625	−0.312952
$474$	0	0
$475$	−47.0156	−2.15722
$476$	−4.00000	−0.183340
$477$	0	0
$478$	−10.8062	−0.494266
$479$	−16.5969	−0.758331	−0.379165	−	0.925329i	$-0.623789\pi$
$479$	−16.5969	−0.758331	−0.379165	+	0.925329i	$0.623789\pi$
$480$	0	0
$481$	24.5078	1.11746
$482$	−15.9109	−0.724723
$483$	0	0
$484$	5.00000	0.227273
$485$	−23.3141	−1.05864
$486$	0	0
$487$	−6.29844	−0.285409	−0.142705	−	0.989765i	$-0.545580\pi$
$487$	−6.29844	−0.285409	−0.142705	+	0.989765i	$0.545580\pi$
$488$	−2.00000	−0.0905357
$489$	0	0
$490$	−3.70156	−0.167220
$491$	9.61250	0.433806	0.216903	−	0.976193i	$-0.430404\pi$
$491$	9.61250	0.433806	0.216903	+	0.976193i	$0.430404\pi$
$492$	0	0
$493$	1.19375	0.0537639
$494$	−30.8062	−1.38604
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	7.40312	0.332076
$498$	0	0
$499$	−23.4031	−1.04767	−0.523834	−	0.851820i	$-0.675499\pi$
$499$	−23.4031	−1.04767	−0.523834	+	0.851820i	$0.675499\pi$
$500$	−13.7016	−0.612752
$501$	0	0
$502$	−19.7016	−0.879324
$503$	−4.59688	−0.204965	−0.102482	−	0.994735i	$-0.532678\pi$
$503$	−4.59688	−0.204965	−0.102482	+	0.994735i	$0.532678\pi$
$504$	0	0
$505$	−12.5969	−0.560554
$506$	−4.00000	−0.177822
$507$	0	0
$508$	−5.70156	−0.252966
$509$	−37.6125	−1.66714	−0.833572	−	0.552410i	$-0.813708\pi$
$509$	−37.6125	−1.66714	−0.833572	+	0.552410i	$0.813708\pi$
$510$	0	0
$511$	1.40312	0.0620706
$512$	1.00000	0.0441942
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	−8.50781	−0.374899
$516$	0	0
$517$	44.4187	1.95353
$518$	4.29844	0.188863
$519$	0	0
$520$	−21.1047	−0.925502
$521$	20.5969	0.902366	0.451183	−	0.892432i	$-0.351002\pi$
$521$	20.5969	0.902366	0.451183	+	0.892432i	$0.351002\pi$
$522$	0	0
$523$	20.8062	0.909794	0.454897	−	0.890544i	$-0.349676\pi$
$523$	20.8062	0.909794	0.454897	+	0.890544i	$0.349676\pi$
$524$	7.40312	0.323407
$525$	0	0
$526$	−9.70156	−0.423008
$527$	8.00000	0.348485
$528$	0	0
$529$	1.00000	0.0434783
$530$	−34.8062	−1.51189
$531$	0	0
$532$	−5.40312	−0.234255
$533$	1.70156	0.0737028
$534$	0	0
$535$	69.6125	3.00961
$536$	14.8062	0.639533
$537$	0	0
$538$	10.2094	0.440157
$539$	4.00000	0.172292
$540$	0	0
$541$	33.4031	1.43611	0.718056	−	0.695985i	$-0.245032\pi$
$541$	33.4031	1.43611	0.718056	+	0.695985i	$0.245032\pi$
$542$	1.40312	0.0602693
$543$	0	0
$544$	−4.00000	−0.171499
$545$	28.5078	1.22114
$546$	0	0
$547$	29.0156	1.24062	0.620309	−	0.784357i	$-0.287007\pi$
$547$	29.0156	1.24062	0.620309	+	0.784357i	$0.287007\pi$
$548$	19.7016	0.841609
$549$	0	0
$550$	34.8062	1.48414
$551$	1.61250	0.0686947
$552$	0	0
$553$	8.00000	0.340195
$554$	−21.4031	−0.909332
$555$	0	0
$556$	17.7016	0.750713
$557$	37.4031	1.58482	0.792411	−	0.609988i	$-0.208826\pi$
$557$	37.4031	1.58482	0.792411	+	0.609988i	$0.208826\pi$
$558$	0	0
$559$	−9.70156	−0.410332
$560$	−3.70156	−0.156420
$561$	0	0
$562$	−14.5078	−0.611975
$563$	3.10469	0.130847	0.0654235	−	0.997858i	$-0.479160\pi$
$563$	3.10469	0.130847	0.0654235	+	0.997858i	$0.479160\pi$
$564$	0	0
$565$	26.2984	1.10638
$566$	4.80625	0.202022
$567$	0	0
$568$	7.40312	0.310628
$569$	−31.7016	−1.32900	−0.664499	−	0.747289i	$-0.731355\pi$
$569$	−31.7016	−1.32900	−0.664499	+	0.747289i	$0.731355\pi$
$570$	0	0
$571$	−14.8062	−0.619622	−0.309811	−	0.950798i	$-0.600266\pi$
$571$	−14.8062	−0.619622	−0.309811	+	0.950798i	$0.600266\pi$
$572$	22.8062	0.953577
$573$	0	0
$574$	0.298438	0.0124566
$575$	−8.70156	−0.362880
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	−1.00000	−0.0415945
$579$	0	0
$580$	1.10469	0.0458696
$581$	−13.4031	−0.556055
$582$	0	0
$583$	37.6125	1.55775
$584$	1.40312	0.0580617
$585$	0	0
$586$	20.8062	0.859498
$587$	−14.8062	−0.611119	−0.305560	−	0.952173i	$-0.598844\pi$
$587$	−14.8062	−0.611119	−0.305560	+	0.952173i	$0.598844\pi$
$588$	0	0
$589$	10.8062	0.445264
$590$	−27.4031	−1.12817
$591$	0	0
$592$	4.29844	0.176665
$593$	−43.7016	−1.79461	−0.897304	−	0.441413i	$-0.854477\pi$
$593$	−43.7016	−1.79461	−0.897304	+	0.441413i	$0.854477\pi$
$594$	0	0
$595$	14.8062	0.606997
$596$	20.2094	0.827808
$597$	0	0
$598$	−5.70156	−0.233154
$599$	−13.6125	−0.556192	−0.278096	−	0.960553i	$-0.589703\pi$
$599$	−13.6125	−0.556192	−0.278096	+	0.960553i	$0.589703\pi$
$600$	0	0
$601$	−13.4031	−0.546725	−0.273362	−	0.961911i	$-0.588136\pi$
$601$	−13.4031	−0.546725	−0.273362	+	0.961911i	$0.588136\pi$
$602$	−1.70156	−0.0693505
$603$	0	0
$604$	23.9109	0.972922
$605$	−18.5078	−0.752450
$606$	0	0
$607$	−39.6125	−1.60782	−0.803911	−	0.594750i	$-0.797251\pi$
$607$	−39.6125	−1.60782	−0.803911	+	0.594750i	$0.797251\pi$
$608$	−5.40312	−0.219126
$609$	0	0
$610$	7.40312	0.299744
$611$	63.3141	2.56141
$612$	0	0
$613$	3.70156	0.149505	0.0747523	−	0.997202i	$-0.476183\pi$
$613$	3.70156	0.149505	0.0747523	+	0.997202i	$0.476183\pi$
$614$	11.9109	0.480686
$615$	0	0
$616$	4.00000	0.161165
$617$	19.1938	0.772711	0.386356	−	0.922350i	$-0.373734\pi$
$617$	19.1938	0.772711	0.386356	+	0.922350i	$0.373734\pi$
$618$	0	0
$619$	−7.19375	−0.289141	−0.144571	−	0.989494i	$-0.546180\pi$
$619$	−7.19375	−0.289141	−0.144571	+	0.989494i	$0.546180\pi$
$620$	7.40312	0.297317
$621$	0	0
$622$	−13.4031	−0.537416
$623$	11.4031	0.456857
$624$	0	0
$625$	7.20937	0.288375
$626$	−6.20937	−0.248176
$627$	0	0
$628$	−1.40312	−0.0559908
$629$	−17.1938	−0.685560
$630$	0	0
$631$	−29.6125	−1.17885	−0.589427	−	0.807821i	$-0.700647\pi$
$631$	−29.6125	−1.17885	−0.589427	+	0.807821i	$0.700647\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	8.29844	0.329573
$635$	21.1047	0.837514
$636$	0	0
$637$	5.70156	0.225904
$638$	−1.19375	−0.0472611
$639$	0	0
$640$	−3.70156	−0.146317
$641$	20.7172	0.818280	0.409140	−	0.912472i	$-0.365829\pi$
$641$	20.7172	0.818280	0.409140	+	0.912472i	$0.365829\pi$
$642$	0	0
$643$	0.806248	0.0317953	0.0158977	−	0.999874i	$-0.494939\pi$
$643$	0.806248	0.0317953	0.0158977	+	0.999874i	$0.494939\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	21.6125	0.850332
$647$	6.59688	0.259350	0.129675	−	0.991557i	$-0.458607\pi$
$647$	6.59688	0.259350	0.129675	+	0.991557i	$0.458607\pi$
$648$	0	0
$649$	29.6125	1.16239
$650$	49.6125	1.94596
$651$	0	0
$652$	−10.8062	−0.423205
$653$	−7.10469	−0.278028	−0.139014	−	0.990290i	$-0.544393\pi$
$653$	−7.10469	−0.278028	−0.139014	+	0.990290i	$0.544393\pi$
$654$	0	0
$655$	−27.4031	−1.07073
$656$	0.298438	0.0116520
$657$	0	0
$658$	11.1047	0.432906
$659$	−8.00000	−0.311636	−0.155818	−	0.987786i	$-0.549801\pi$
$659$	−8.00000	−0.311636	−0.155818	+	0.987786i	$0.549801\pi$
$660$	0	0
$661$	42.4187	1.64990	0.824949	−	0.565207i	$-0.191204\pi$
$661$	42.4187	1.64990	0.824949	+	0.565207i	$0.191204\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−13.4031	−0.520142
$665$	20.0000	0.775567
$666$	0	0
$667$	0.298438	0.0115556
$668$	13.4031	0.518582
$669$	0	0
$670$	−54.8062	−2.11735
$671$	−8.00000	−0.308837
$672$	0	0
$673$	7.70156	0.296873	0.148437	−	0.988922i	$-0.452576\pi$
$673$	7.70156	0.296873	0.148437	+	0.988922i	$0.452576\pi$
$674$	−4.20937	−0.162139
$675$	0	0
$676$	19.5078	0.750300
$677$	50.4187	1.93775	0.968875	−	0.247551i	$-0.0796257\pi$
$677$	50.4187	1.93775	0.968875	+	0.247551i	$0.0796257\pi$
$678$	0	0
$679$	6.29844	0.241712
$680$	14.8062	0.567794
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−33.6125	−1.28615	−0.643073	−	0.765805i	$-0.722341\pi$
$683$	−33.6125	−1.28615	−0.643073	+	0.765805i	$0.722341\pi$
$684$	0	0
$685$	−72.9266	−2.78638
$686$	1.00000	0.0381802
$687$	0	0
$688$	−1.70156	−0.0648714
$689$	53.6125	2.04247
$690$	0	0
$691$	−0.507811	−0.0193180	−0.00965901	−	0.999953i	$-0.503075\pi$
$691$	−0.507811	−0.0193180	−0.00965901	+	0.999953i	$0.503075\pi$
$692$	−19.4031	−0.737596
$693$	0	0
$694$	10.2984	0.390923
$695$	−65.5234	−2.48545
$696$	0	0
$697$	−1.19375	−0.0452166
$698$	12.5969	0.476799
$699$	0	0
$700$	8.70156	0.328888
$701$	−47.0156	−1.77576	−0.887878	−	0.460079i	$-0.847821\pi$
$701$	−47.0156	−1.77576	−0.887878	+	0.460079i	$0.847821\pi$
$702$	0	0
$703$	−23.2250	−0.875947
$704$	4.00000	0.150756
$705$	0	0
$706$	−4.29844	−0.161774
$707$	3.40312	0.127988
$708$	0	0
$709$	23.1938	0.871060	0.435530	−	0.900174i	$-0.356561\pi$
$709$	23.1938	0.871060	0.435530	+	0.900174i	$0.356561\pi$
$710$	−27.4031	−1.02842
$711$	0	0
$712$	11.4031	0.427350
$713$	2.00000	0.0749006
$714$	0	0
$715$	−84.4187	−3.15708
$716$	25.1047	0.938206
$717$	0	0
$718$	2.89531	0.108052
$719$	5.91093	0.220441	0.110220	−	0.993907i	$-0.464844\pi$
$719$	5.91093	0.220441	0.110220	+	0.993907i	$0.464844\pi$
$720$	0	0
$721$	2.29844	0.0855983
$722$	10.1938	0.379372
$723$	0	0
$724$	−16.8062	−0.624599
$725$	−2.59688	−0.0964455
$726$	0	0
$727$	−12.0000	−0.445055	−0.222528	−	0.974926i	$-0.571431\pi$
$727$	−12.0000	−0.445055	−0.222528	+	0.974926i	$0.571431\pi$
$728$	5.70156	0.211314
$729$	0	0
$730$	−5.19375	−0.192229
$731$	6.80625	0.251738
$732$	0	0
$733$	52.2094	1.92840	0.964199	−	0.265181i	$-0.0854318\pi$
$733$	52.2094	1.92840	0.964199	+	0.265181i	$0.0854318\pi$
$734$	−17.1047	−0.631345
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	59.2250	2.18158
$738$	0	0
$739$	−29.0156	−1.06736	−0.533678	−	0.845687i	$-0.679191\pi$
$739$	−29.0156	−1.06736	−0.533678	+	0.845687i	$0.679191\pi$
$740$	−15.9109	−0.584898
$741$	0	0
$742$	9.40312	0.345200
$743$	−32.0000	−1.17397	−0.586983	−	0.809599i	$-0.699684\pi$
$743$	−32.0000	−1.17397	−0.586983	+	0.809599i	$0.699684\pi$
$744$	0	0
$745$	−74.8062	−2.74069
$746$	−0.806248	−0.0295189
$747$	0	0
$748$	−16.0000	−0.585018
$749$	−18.8062	−0.687165
$750$	0	0
$751$	29.6125	1.08058	0.540288	−	0.841480i	$-0.318315\pi$
$751$	29.6125	1.08058	0.540288	+	0.841480i	$0.318315\pi$
$752$	11.1047	0.404946
$753$	0	0
$754$	−1.70156	−0.0619672
$755$	−88.5078	−3.22113
$756$	0	0
$757$	−26.4187	−0.960206	−0.480103	−	0.877212i	$-0.659401\pi$
$757$	−26.4187	−0.960206	−0.480103	+	0.877212i	$0.659401\pi$
$758$	13.7016	0.497663
$759$	0	0
$760$	20.0000	0.725476
$761$	35.6125	1.29095	0.645476	−	0.763781i	$-0.276659\pi$
$761$	35.6125	1.29095	0.645476	+	0.763781i	$0.276659\pi$
$762$	0	0
$763$	−7.70156	−0.278815
$764$	−22.8062	−0.825101
$765$	0	0
$766$	−29.6125	−1.06994
$767$	42.2094	1.52409
$768$	0	0
$769$	15.9109	0.573763	0.286881	−	0.957966i	$-0.407381\pi$
$769$	15.9109	0.573763	0.286881	+	0.957966i	$0.407381\pi$
$770$	−14.8062	−0.533580
$771$	0	0
$772$	−23.1047	−0.831556
$773$	7.10469	0.255538	0.127769	−	0.991804i	$-0.459218\pi$
$773$	7.10469	0.255538	0.127769	+	0.991804i	$0.459218\pi$
$774$	0	0
$775$	−17.4031	−0.625139
$776$	6.29844	0.226101
$777$	0	0
$778$	6.00000	0.215110
$779$	−1.61250	−0.0577737
$780$	0	0
$781$	29.6125	1.05962
$782$	4.00000	0.143040
$783$	0	0
$784$	1.00000	0.0357143
$785$	5.19375	0.185373
$786$	0	0
$787$	−18.5969	−0.662907	−0.331454	−	0.943472i	$-0.607539\pi$
$787$	−18.5969	−0.662907	−0.331454	+	0.943472i	$0.607539\pi$
$788$	−22.5078	−0.801808
$789$	0	0
$790$	−29.6125	−1.05357
$791$	−7.10469	−0.252614
$792$	0	0
$793$	−11.4031	−0.404937
$794$	8.59688	0.305092
$795$	0	0
$796$	−2.29844	−0.0814660
$797$	−15.7016	−0.556178	−0.278089	−	0.960555i	$-0.589701\pi$
$797$	−15.7016	−0.556178	−0.278089	+	0.960555i	$0.589701\pi$
$798$	0	0
$799$	−44.4187	−1.57142
$800$	8.70156	0.307647
$801$	0	0
$802$	−30.0000	−1.05934
$803$	5.61250	0.198061
$804$	0	0
$805$	3.70156	0.130463
$806$	−11.4031	−0.401658
$807$	0	0
$808$	3.40312	0.119721
$809$	−31.0156	−1.09045	−0.545226	−	0.838289i	$-0.683556\pi$
$809$	−31.0156	−1.09045	−0.545226	+	0.838289i	$0.683556\pi$
$810$	0	0
$811$	52.5078	1.84380	0.921899	−	0.387430i	$-0.126637\pi$
$811$	52.5078	1.84380	0.921899	+	0.387430i	$0.126637\pi$
$812$	−0.298438	−0.0104731
$813$	0	0
$814$	17.1938	0.602641
$815$	40.0000	1.40114
$816$	0	0
$817$	9.19375	0.321649
$818$	5.40312	0.188916
$819$	0	0
$820$	−1.10469	−0.0385773
$821$	−27.6125	−0.963683	−0.481841	−	0.876258i	$-0.660032\pi$
$821$	−27.6125	−0.963683	−0.481841	+	0.876258i	$0.660032\pi$
$822$	0	0
$823$	−13.7016	−0.477606	−0.238803	−	0.971068i	$-0.576755\pi$
$823$	−13.7016	−0.477606	−0.238803	+	0.971068i	$0.576755\pi$
$824$	2.29844	0.0800699
$825$	0	0
$826$	7.40312	0.257588
$827$	47.4031	1.64837	0.824184	−	0.566322i	$-0.191634\pi$
$827$	47.4031	1.64837	0.824184	+	0.566322i	$0.191634\pi$
$828$	0	0
$829$	55.4031	1.92423	0.962115	−	0.272644i	$-0.0878981\pi$
$829$	55.4031	1.92423	0.962115	+	0.272644i	$0.0878981\pi$
$830$	49.6125	1.72207
$831$	0	0
$832$	5.70156	0.197666
$833$	−4.00000	−0.138592
$834$	0	0
$835$	−49.6125	−1.71691
$836$	−21.6125	−0.747484
$837$	0	0
$838$	0.209373	0.00723266
$839$	−42.2094	−1.45723	−0.728615	−	0.684924i	$-0.759835\pi$
$839$	−42.2094	−1.45723	−0.728615	+	0.684924i	$0.759835\pi$
$840$	0	0
$841$	−28.9109	−0.996929
$842$	−15.7016	−0.541112
$843$	0	0
$844$	10.8062	0.371966
$845$	−72.2094	−2.48408
$846$	0	0
$847$	5.00000	0.171802
$848$	9.40312	0.322905
$849$	0	0
$850$	−34.8062	−1.19384
$851$	−4.29844	−0.147349
$852$	0	0
$853$	9.10469	0.311739	0.155869	−	0.987778i	$-0.450182\pi$
$853$	9.10469	0.311739	0.155869	+	0.987778i	$0.450182\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	−18.8062	−0.642784
$857$	0.298438	0.0101944	0.00509722	−	0.999987i	$-0.498377\pi$
$857$	0.298438	0.0101944	0.00509722	+	0.999987i	$0.498377\pi$
$858$	0	0
$859$	−15.4922	−0.528587	−0.264293	−	0.964442i	$-0.585139\pi$
$859$	−15.4922	−0.528587	−0.264293	+	0.964442i	$0.585139\pi$
$860$	6.29844	0.214775
$861$	0	0
$862$	3.91093	0.133207
$863$	6.38750	0.217433	0.108717	−	0.994073i	$-0.465326\pi$
$863$	6.38750	0.217433	0.108717	+	0.994073i	$0.465326\pi$
$864$	0	0
$865$	71.8219	2.44202
$866$	31.3141	1.06410
$867$	0	0
$868$	−2.00000	−0.0678844
$869$	32.0000	1.08553
$870$	0	0
$871$	84.4187	2.86042
$872$	−7.70156	−0.260808
$873$	0	0
$874$	5.40312	0.182763
$875$	−13.7016	−0.463197
$876$	0	0
$877$	−17.4031	−0.587662	−0.293831	−	0.955857i	$-0.594930\pi$
$877$	−17.4031	−0.587662	−0.293831	+	0.955857i	$0.594930\pi$
$878$	−22.0000	−0.742464
$879$	0	0
$880$	−14.8062	−0.499119
$881$	−6.20937	−0.209199	−0.104600	−	0.994514i	$-0.533356\pi$
$881$	−6.20937	−0.209199	−0.104600	+	0.994514i	$0.533356\pi$
$882$	0	0
$883$	−24.5969	−0.827751	−0.413875	−	0.910334i	$-0.635825\pi$
$883$	−24.5969	−0.827751	−0.413875	+	0.910334i	$0.635825\pi$
$884$	−22.8062	−0.767057
$885$	0	0
$886$	13.7016	0.460313
$887$	28.2094	0.947178	0.473589	−	0.880746i	$-0.342958\pi$
$887$	28.2094	0.947178	0.473589	+	0.880746i	$0.342958\pi$
$888$	0	0
$889$	−5.70156	−0.191224
$890$	−42.2094	−1.41486
$891$	0	0
$892$	−22.0000	−0.736614
$893$	−60.0000	−2.00782
$894$	0	0
$895$	−92.9266	−3.10619
$896$	1.00000	0.0334077
$897$	0	0
$898$	38.4187	1.28205
$899$	0.596876	0.0199069
$900$	0	0
$901$	−37.6125	−1.25305
$902$	1.19375	0.0397476
$903$	0	0
$904$	−7.10469	−0.236298
$905$	62.2094	2.06791
$906$	0	0
$907$	20.5078	0.680951	0.340475	−	0.940253i	$-0.389412\pi$
$907$	20.5078	0.680951	0.340475	+	0.940253i	$0.389412\pi$
$908$	23.1047	0.766756
$909$	0	0
$910$	−21.1047	−0.699614
$911$	−53.1047	−1.75944	−0.879718	−	0.475495i	$-0.842269\pi$
$911$	−53.1047	−1.75944	−0.879718	+	0.475495i	$0.842269\pi$
$912$	0	0
$913$	−53.6125	−1.77431
$914$	28.2094	0.933083
$915$	0	0
$916$	−14.5969	−0.482294
$917$	7.40312	0.244473
$918$	0	0
$919$	−52.4187	−1.72913	−0.864567	−	0.502517i	$-0.832407\pi$
$919$	−52.4187	−1.72913	−0.864567	+	0.502517i	$0.832407\pi$
$920$	3.70156	0.122037
$921$	0	0
$922$	2.20937	0.0727618
$923$	42.2094	1.38934
$924$	0	0
$925$	37.4031	1.22981
$926$	−22.2984	−0.732772
$927$	0	0
$928$	−0.298438	−0.00979670
$929$	45.9109	1.50629	0.753144	−	0.657855i	$-0.228536\pi$
$929$	45.9109	1.50629	0.753144	+	0.657855i	$0.228536\pi$
$930$	0	0
$931$	−5.40312	−0.177080
$932$	26.0000	0.851658
$933$	0	0
$934$	3.70156	0.121119
$935$	59.2250	1.93686
$936$	0	0
$937$	25.1047	0.820134	0.410067	−	0.912055i	$-0.365505\pi$
$937$	25.1047	0.820134	0.410067	+	0.912055i	$0.365505\pi$
$938$	14.8062	0.483441
$939$	0	0
$940$	−41.1047	−1.34069
$941$	−25.3141	−0.825215	−0.412607	−	0.910909i	$-0.635382\pi$
$941$	−25.3141	−0.825215	−0.412607	+	0.910909i	$0.635382\pi$
$942$	0	0
$943$	−0.298438	−0.00971847
$944$	7.40312	0.240951
$945$	0	0
$946$	−6.80625	−0.221290
$947$	−15.9109	−0.517036	−0.258518	−	0.966006i	$-0.583234\pi$
$947$	−15.9109	−0.517036	−0.258518	+	0.966006i	$0.583234\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−47.0156	−1.52539
$951$	0	0
$952$	−4.00000	−0.129641
$953$	−61.2250	−1.98327	−0.991636	−	0.129066i	$-0.958802\pi$
$953$	−61.2250	−1.98327	−0.991636	+	0.129066i	$0.958802\pi$
$954$	0	0
$955$	84.4187	2.73173
$956$	−10.8062	−0.349499
$957$	0	0
$958$	−16.5969	−0.536221
$959$	19.7016	0.636197
$960$	0	0
$961$	−27.0000	−0.870968
$962$	24.5078	0.790164
$963$	0	0
$964$	−15.9109	−0.512457
$965$	85.5234	2.75310
$966$	0	0
$967$	−14.8062	−0.476137	−0.238068	−	0.971248i	$-0.576514\pi$
$967$	−14.8062	−0.476137	−0.238068	+	0.971248i	$0.576514\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	−23.3141	−0.748569
$971$	−49.8219	−1.59886	−0.799430	−	0.600759i	$-0.794865\pi$
$971$	−49.8219	−1.59886	−0.799430	+	0.600759i	$0.794865\pi$
$972$	0	0
$973$	17.7016	0.567486
$974$	−6.29844	−0.201815
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	−16.7172	−0.534830	−0.267415	−	0.963581i	$-0.586169\pi$
$977$	−16.7172	−0.534830	−0.267415	+	0.963581i	$0.586169\pi$
$978$	0	0
$979$	45.6125	1.45778
$980$	−3.70156	−0.118242
$981$	0	0
$982$	9.61250	0.306747
$983$	−42.2094	−1.34627	−0.673135	−	0.739520i	$-0.735053\pi$
$983$	−42.2094	−1.34627	−0.673135	+	0.739520i	$0.735053\pi$
$984$	0	0
$985$	83.3141	2.65461
$986$	1.19375	0.0380168
$987$	0	0
$988$	−30.8062	−0.980077
$989$	1.70156	0.0541065
$990$	0	0
$991$	28.0000	0.889449	0.444725	−	0.895667i	$-0.353302\pi$
$991$	28.0000	0.889449	0.444725	+	0.895667i	$0.353302\pi$
$992$	−2.00000	−0.0635001
$993$	0	0
$994$	7.40312	0.234813
$995$	8.50781	0.269716
$996$	0	0
$997$	−23.4031	−0.741184	−0.370592	−	0.928796i	$-0.620845\pi$
$997$	−23.4031	−0.741184	−0.370592	+	0.928796i	$0.620845\pi$
$998$	−23.4031	−0.740813
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.bb.1.1		2
3.2	odd	2		966.2.a.n.1.2	✓	2
12.11	even	2		7728.2.a.bc.1.2		2
21.20	even	2		6762.2.a.bo.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.n.1.2	✓	2	3.2	odd	2
2898.2.a.bb.1.1		2	1.1	even	1	trivial
6762.2.a.bo.1.1		2	21.20	even	2
7728.2.a.bc.1.2		2	12.11	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:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.70156 q^{5} +1.00000 q^{7} +1.00000 q^{8} -3.70156 q^{10} +4.00000 q^{11} +5.70156 q^{13} +1.00000 q^{14} +1.00000 q^{16} -4.00000 q^{17} -5.40312 q^{19} -3.70156 q^{20} +4.00000 q^{22} -1.00000 q^{23} +8.70156 q^{25} +5.70156 q^{26} +1.00000 q^{28} -0.298438 q^{29} -2.00000 q^{31} +1.00000 q^{32} -4.00000 q^{34} -3.70156 q^{35} +4.29844 q^{37} -5.40312 q^{38} -3.70156 q^{40} +0.298438 q^{41} -1.70156 q^{43} +4.00000 q^{44} -1.00000 q^{46} +11.1047 q^{47} +1.00000 q^{49} +8.70156 q^{50} +5.70156 q^{52} +9.40312 q^{53} -14.8062 q^{55} +1.00000 q^{56} -0.298438 q^{58} +7.40312 q^{59} -2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} -21.1047 q^{65} +14.8062 q^{67} -4.00000 q^{68} -3.70156 q^{70} +7.40312 q^{71} +1.40312 q^{73} +4.29844 q^{74} -5.40312 q^{76} +4.00000 q^{77} +8.00000 q^{79} -3.70156 q^{80} +0.298438 q^{82} -13.4031 q^{83} +14.8062 q^{85} -1.70156 q^{86} +4.00000 q^{88} +11.4031 q^{89} +5.70156 q^{91} -1.00000 q^{92} +11.1047 q^{94} +20.0000 q^{95} +6.29844 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} - q^{5} + 2 q^{7} + 2 q^{8} - q^{10} + 8 q^{11} + 5 q^{13} + 2 q^{14} + 2 q^{16} - 8 q^{17} + 2 q^{19} - q^{20} + 8 q^{22} - 2 q^{23} + 11 q^{25} + 5 q^{26} + 2 q^{28} - 7 q^{29} - 4 q^{31} + 2 q^{32} - 8 q^{34} - q^{35} + 15 q^{37} + 2 q^{38} - q^{40} + 7 q^{41} + 3 q^{43} + 8 q^{44} - 2 q^{46} + 3 q^{47} + 2 q^{49} + 11 q^{50} + 5 q^{52} + 6 q^{53} - 4 q^{55} + 2 q^{56} - 7 q^{58} + 2 q^{59} - 4 q^{61} - 4 q^{62} + 2 q^{64} - 23 q^{65} + 4 q^{67} - 8 q^{68} - q^{70} + 2 q^{71} - 10 q^{73} + 15 q^{74} + 2 q^{76} + 8 q^{77} + 16 q^{79} - q^{80} + 7 q^{82} - 14 q^{83} + 4 q^{85} + 3 q^{86} + 8 q^{88} + 10 q^{89} + 5 q^{91} - 2 q^{92} + 3 q^{94} + 40 q^{95} + 19 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - q^5 + 2 * q^7 + 2 * q^8 - q^10 + 8 * q^11 + 5 * q^13 + 2 * q^14 + 2 * q^16 - 8 * q^17 + 2 * q^19 - q^20 + 8 * q^22 - 2 * q^23 + 11 * q^25 + 5 * q^26 + 2 * q^28 - 7 * q^29 - 4 * q^31 + 2 * q^32 - 8 * q^34 - q^35 + 15 * q^37 + 2 * q^38 - q^40 + 7 * q^41 + 3 * q^43 + 8 * q^44 - 2 * q^46 + 3 * q^47 + 2 * q^49 + 11 * q^50 + 5 * q^52 + 6 * q^53 - 4 * q^55 + 2 * q^56 - 7 * q^58 + 2 * q^59 - 4 * q^61 - 4 * q^62 + 2 * q^64 - 23 * q^65 + 4 * q^67 - 8 * q^68 - q^70 + 2 * q^71 - 10 * q^73 + 15 * q^74 + 2 * q^76 + 8 * q^77 + 16 * q^79 - q^80 + 7 * q^82 - 14 * q^83 + 4 * q^85 + 3 * q^86 + 8 * q^88 + 10 * q^89 + 5 * q^91 - 2 * q^92 + 3 * q^94 + 40 * q^95 + 19 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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