LMFDB - Embedded newform 2898.2.a.bg.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:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.76156$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -2.62620 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -2.62620 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.62620 q^{10} -3.52311 q^{11} +3.76156 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.38776 q^{17} +2.86464 q^{19} -2.62620 q^{20} -3.52311 q^{22} -1.00000 q^{23} +1.89692 q^{25} +3.76156 q^{26} -1.00000 q^{28} +4.89692 q^{29} +6.38776 q^{31} +1.00000 q^{32} -4.38776 q^{34} +2.62620 q^{35} +2.89692 q^{37} +2.86464 q^{38} -2.62620 q^{40} +10.1493 q^{41} +6.35548 q^{43} -3.52311 q^{44} -1.00000 q^{46} +7.49084 q^{47} +1.00000 q^{49} +1.89692 q^{50} +3.76156 q^{52} +1.79383 q^{53} +9.25240 q^{55} -1.00000 q^{56} +4.89692 q^{58} -6.77551 q^{61} +6.38776 q^{62} +1.00000 q^{64} -9.87859 q^{65} -7.04623 q^{67} -4.38776 q^{68} +2.62620 q^{70} -10.9817 q^{71} +7.72928 q^{73} +2.89692 q^{74} +2.86464 q^{76} +3.52311 q^{77} +5.52311 q^{79} -2.62620 q^{80} +10.1493 q^{82} +8.11704 q^{83} +11.5231 q^{85} +6.35548 q^{86} -3.52311 q^{88} +4.32320 q^{89} -3.76156 q^{91} -1.00000 q^{92} +7.49084 q^{94} -7.52311 q^{95} +0.238443 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{7} + 3 q^{8} + q^{10} + 2 q^{11} + 5 q^{13} - 3 q^{14} + 3 q^{16} + 2 q^{17} + 6 q^{19} + q^{20} + 2 q^{22} - 3 q^{23} + 2 q^{25} + 5 q^{26} - 3 q^{28} + 11 q^{29} + 4 q^{31} + 3 q^{32} + 2 q^{34} - q^{35} + 5 q^{37} + 6 q^{38} + q^{40} + 9 q^{41} + 5 q^{43} + 2 q^{44} - 3 q^{46} + 11 q^{47} + 3 q^{49} + 2 q^{50} + 5 q^{52} - 2 q^{53} + 10 q^{55} - 3 q^{56} + 11 q^{58} + 10 q^{61} + 4 q^{62} + 3 q^{64} - 3 q^{65} + 4 q^{67} + 2 q^{68} - q^{70} - 10 q^{71} + 18 q^{73} + 5 q^{74} + 6 q^{76} - 2 q^{77} + 4 q^{79} + q^{80} + 9 q^{82} + 4 q^{83} + 22 q^{85} + 5 q^{86} + 2 q^{88} - 5 q^{91} - 3 q^{92} + 11 q^{94} - 10 q^{95} + 7 q^{97} + 3 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 3 * q^7 + 3 * q^8 + q^10 + 2 * q^11 + 5 * q^13 - 3 * q^14 + 3 * q^16 + 2 * q^17 + 6 * q^19 + q^20 + 2 * q^22 - 3 * q^23 + 2 * q^25 + 5 * q^26 - 3 * q^28 + 11 * q^29 + 4 * q^31 + 3 * q^32 + 2 * q^34 - q^35 + 5 * q^37 + 6 * q^38 + q^40 + 9 * q^41 + 5 * q^43 + 2 * q^44 - 3 * q^46 + 11 * q^47 + 3 * q^49 + 2 * q^50 + 5 * q^52 - 2 * q^53 + 10 * q^55 - 3 * q^56 + 11 * q^58 + 10 * q^61 + 4 * q^62 + 3 * q^64 - 3 * q^65 + 4 * q^67 + 2 * q^68 - q^70 - 10 * q^71 + 18 * q^73 + 5 * q^74 + 6 * q^76 - 2 * q^77 + 4 * q^79 + q^80 + 9 * q^82 + 4 * q^83 + 22 * q^85 + 5 * q^86 + 2 * q^88 - 5 * q^91 - 3 * q^92 + 11 * q^94 - 10 * q^95 + 7 * q^97 + 3 * 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
$5$	−2.62620	−1.17447	−0.587236	−	0.809416i	$-0.699784\pi$
$5$	−2.62620	−1.17447	−0.587236	+	0.809416i	$0.699784\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−2.62620	−0.830477
$11$	−3.52311	−1.06226	−0.531129	−	0.847291i	$-0.678232\pi$
$11$	−3.52311	−1.06226	−0.531129	+	0.847291i	$0.678232\pi$
$12$	0	0
$13$	3.76156	1.04327	0.521634	−	0.853169i	$-0.325323\pi$
$13$	3.76156	1.04327	0.521634	+	0.853169i	$0.325323\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−4.38776	−1.06419	−0.532093	−	0.846686i	$-0.678595\pi$
$17$	−4.38776	−1.06419	−0.532093	+	0.846686i	$0.678595\pi$
$18$	0	0
$19$	2.86464	0.657194	0.328597	−	0.944470i	$-0.393424\pi$
$19$	2.86464	0.657194	0.328597	+	0.944470i	$0.393424\pi$
$20$	−2.62620	−0.587236
$21$	0	0
$22$	−3.52311	−0.751131
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.89692	0.379383
$26$	3.76156	0.737702
$27$	0	0
$28$	−1.00000	−0.188982
$29$	4.89692	0.909335	0.454667	−	0.890661i	$-0.349758\pi$
$29$	4.89692	0.909335	0.454667	+	0.890661i	$0.349758\pi$
$30$	0	0
$31$	6.38776	1.14727	0.573637	−	0.819109i	$-0.305532\pi$
$31$	6.38776	1.14727	0.573637	+	0.819109i	$0.305532\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.38776	−0.752494
$35$	2.62620	0.443908
$36$	0	0
$37$	2.89692	0.476250	0.238125	−	0.971235i	$-0.423467\pi$
$37$	2.89692	0.476250	0.238125	+	0.971235i	$0.423467\pi$
$38$	2.86464	0.464706
$39$	0	0
$40$	−2.62620	−0.415238
$41$	10.1493	1.58506	0.792528	−	0.609835i	$-0.208764\pi$
$41$	10.1493	1.58506	0.792528	+	0.609835i	$0.208764\pi$
$42$	0	0
$43$	6.35548	0.969202	0.484601	−	0.874735i	$-0.338965\pi$
$43$	6.35548	0.969202	0.484601	+	0.874735i	$0.338965\pi$
$44$	−3.52311	−0.531129
$45$	0	0
$46$	−1.00000	−0.147442
$47$	7.49084	1.09265	0.546326	−	0.837573i	$-0.316026\pi$
$47$	7.49084	1.09265	0.546326	+	0.837573i	$0.316026\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.89692	0.268264
$51$	0	0
$52$	3.76156	0.521634
$53$	1.79383	0.246402	0.123201	−	0.992382i	$-0.460684\pi$
$53$	1.79383	0.246402	0.123201	+	0.992382i	$0.460684\pi$
$54$	0	0
$55$	9.25240	1.24759
$56$	−1.00000	−0.133631
$57$	0	0
$58$	4.89692	0.642997
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−6.77551	−0.867515	−0.433758	−	0.901030i	$-0.642813\pi$
$61$	−6.77551	−0.867515	−0.433758	+	0.901030i	$0.642813\pi$
$62$	6.38776	0.811246
$63$	0	0
$64$	1.00000	0.125000
$65$	−9.87859	−1.22529
$66$	0	0
$67$	−7.04623	−0.860834	−0.430417	−	0.902630i	$-0.641633\pi$
$67$	−7.04623	−0.860834	−0.430417	+	0.902630i	$0.641633\pi$
$68$	−4.38776	−0.532093
$69$	0	0
$70$	2.62620	0.313891
$71$	−10.9817	−1.30329	−0.651643	−	0.758526i	$-0.725920\pi$
$71$	−10.9817	−1.30329	−0.651643	+	0.758526i	$0.725920\pi$
$72$	0	0
$73$	7.72928	0.904644	0.452322	−	0.891855i	$-0.350596\pi$
$73$	7.72928	0.904644	0.452322	+	0.891855i	$0.350596\pi$
$74$	2.89692	0.336760
$75$	0	0
$76$	2.86464	0.328597
$77$	3.52311	0.401496
$78$	0	0
$79$	5.52311	0.621399	0.310699	−	0.950508i	$-0.399437\pi$
$79$	5.52311	0.621399	0.310699	+	0.950508i	$0.399437\pi$
$80$	−2.62620	−0.293618
$81$	0	0
$82$	10.1493	1.12080
$83$	8.11704	0.890961	0.445480	−	0.895292i	$-0.353033\pi$
$83$	8.11704	0.890961	0.445480	+	0.895292i	$0.353033\pi$
$84$	0	0
$85$	11.5231	1.24986
$86$	6.35548	0.685329
$87$	0	0
$88$	−3.52311	−0.375565
$89$	4.32320	0.458259	0.229129	−	0.973396i	$-0.426412\pi$
$89$	4.32320	0.458259	0.229129	+	0.973396i	$0.426412\pi$
$90$	0	0
$91$	−3.76156	−0.394318
$92$	−1.00000	−0.104257
$93$	0	0
$94$	7.49084	0.772621
$95$	−7.52311	−0.771855
$96$	0	0
$97$	0.238443	0.0242102	0.0121051	−	0.999927i	$-0.496147\pi$
$97$	0.238443	0.0242102	0.0121051	+	0.999927i	$0.496147\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	1.89692	0.189692
$101$	3.07081	0.305557	0.152778	−	0.988260i	$-0.451178\pi$
$101$	3.07081	0.305557	0.152778	+	0.988260i	$0.451178\pi$
$102$	0	0
$103$	2.12141	0.209028	0.104514	−	0.994523i	$-0.466671\pi$
$103$	2.12141	0.209028	0.104514	+	0.994523i	$0.466671\pi$
$104$	3.76156	0.368851
$105$	0	0
$106$	1.79383	0.174232
$107$	−11.5231	−1.11398	−0.556991	−	0.830519i	$-0.688044\pi$
$107$	−11.5231	−1.11398	−0.556991	+	0.830519i	$0.688044\pi$
$108$	0	0
$109$	6.14931	0.588997	0.294499	−	0.955652i	$-0.404847\pi$
$109$	6.14931	0.588997	0.294499	+	0.955652i	$0.404847\pi$
$110$	9.25240	0.882181
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	2.62620	0.247052	0.123526	−	0.992341i	$-0.460580\pi$
$113$	2.62620	0.247052	0.123526	+	0.992341i	$0.460580\pi$
$114$	0	0
$115$	2.62620	0.244894
$116$	4.89692	0.454667
$117$	0	0
$118$	0	0
$119$	4.38776	0.402225
$120$	0	0
$121$	1.41233	0.128394
$122$	−6.77551	−0.613426
$123$	0	0
$124$	6.38776	0.573637
$125$	8.14931	0.728897
$126$	0	0
$127$	8.62620	0.765451	0.382726	−	0.923862i	$-0.374985\pi$
$127$	8.62620	0.765451	0.382726	+	0.923862i	$0.374985\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−9.87859	−0.866410
$131$	−6.20617	−0.542235	−0.271118	−	0.962546i	$-0.587393\pi$
$131$	−6.20617	−0.542235	−0.271118	+	0.962546i	$0.587393\pi$
$132$	0	0
$133$	−2.86464	−0.248396
$134$	−7.04623	−0.608701
$135$	0	0
$136$	−4.38776	−0.376247
$137$	12.3555	1.05560	0.527800	−	0.849369i	$-0.323017\pi$
$137$	12.3555	1.05560	0.527800	+	0.849369i	$0.323017\pi$
$138$	0	0
$139$	17.4663	1.48147	0.740734	−	0.671798i	$-0.234478\pi$
$139$	17.4663	1.48147	0.740734	+	0.671798i	$0.234478\pi$
$140$	2.62620	0.221954
$141$	0	0
$142$	−10.9817	−0.921562
$143$	−13.2524	−1.10822
$144$	0	0
$145$	−12.8603	−1.06799
$146$	7.72928	0.639680
$147$	0	0
$148$	2.89692	0.238125
$149$	13.7572	1.12703	0.563516	−	0.826105i	$-0.309448\pi$
$149$	13.7572	1.12703	0.563516	+	0.826105i	$0.309448\pi$
$150$	0	0
$151$	−13.8786	−1.12942	−0.564712	−	0.825288i	$-0.691013\pi$
$151$	−13.8786	−1.12942	−0.564712	+	0.825288i	$0.691013\pi$
$152$	2.86464	0.232353
$153$	0	0
$154$	3.52311	0.283901
$155$	−16.7755	−1.34744
$156$	0	0
$157$	−8.50479	−0.678756	−0.339378	−	0.940650i	$-0.610217\pi$
$157$	−8.50479	−0.678756	−0.339378	+	0.940650i	$0.610217\pi$
$158$	5.52311	0.439395
$159$	0	0
$160$	−2.62620	−0.207619
$161$	1.00000	0.0788110
$162$	0	0
$163$	18.2341	1.42820	0.714101	−	0.700042i	$-0.246836\pi$
$163$	18.2341	1.42820	0.714101	+	0.700042i	$0.246836\pi$
$164$	10.1493	0.792528
$165$	0	0
$166$	8.11704	0.630004
$167$	10.3878	0.803829	0.401914	−	0.915677i	$-0.368345\pi$
$167$	10.3878	0.803829	0.401914	+	0.915677i	$0.368345\pi$
$168$	0	0
$169$	1.14931	0.0884086
$170$	11.5231	0.883783
$171$	0	0
$172$	6.35548	0.484601
$173$	7.43398	0.565195	0.282598	−	0.959239i	$-0.408804\pi$
$173$	7.43398	0.565195	0.282598	+	0.959239i	$0.408804\pi$
$174$	0	0
$175$	−1.89692	−0.143393
$176$	−3.52311	−0.265565
$177$	0	0
$178$	4.32320	0.324038
$179$	−1.10308	−0.0824483	−0.0412242	−	0.999150i	$-0.513126\pi$
$179$	−1.10308	−0.0824483	−0.0412242	+	0.999150i	$0.513126\pi$
$180$	0	0
$181$	2.77551	0.206302	0.103151	−	0.994666i	$-0.467107\pi$
$181$	2.77551	0.206302	0.103151	+	0.994666i	$0.467107\pi$
$182$	−3.76156	−0.278825
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−7.60788	−0.559342
$186$	0	0
$187$	15.4586	1.13044
$188$	7.49084	0.546326
$189$	0	0
$190$	−7.52311	−0.545784
$191$	4.02791	0.291449	0.145725	−	0.989325i	$-0.453449\pi$
$191$	4.02791	0.291449	0.145725	+	0.989325i	$0.453449\pi$
$192$	0	0
$193$	−9.81404	−0.706430	−0.353215	−	0.935542i	$-0.614912\pi$
$193$	−9.81404	−0.706430	−0.353215	+	0.935542i	$0.614912\pi$
$194$	0.238443	0.0171192
$195$	0	0
$196$	1.00000	0.0714286
$197$	−17.1310	−1.22053	−0.610266	−	0.792196i	$-0.708938\pi$
$197$	−17.1310	−1.22053	−0.610266	+	0.792196i	$0.708938\pi$
$198$	0	0
$199$	−19.1310	−1.35616	−0.678080	−	0.734988i	$-0.737188\pi$
$199$	−19.1310	−1.35616	−0.678080	+	0.734988i	$0.737188\pi$
$200$	1.89692	0.134132
$201$	0	0
$202$	3.07081	0.216061
$203$	−4.89692	−0.343696
$204$	0	0
$205$	−26.6541	−1.86160
$206$	2.12141	0.147805
$207$	0	0
$208$	3.76156	0.260817
$209$	−10.0925	−0.698110
$210$	0	0
$211$	−16.0279	−1.10341	−0.551703	−	0.834040i	$-0.686022\pi$
$211$	−16.0279	−1.10341	−0.551703	+	0.834040i	$0.686022\pi$
$212$	1.79383	0.123201
$213$	0	0
$214$	−11.5231	−0.787704
$215$	−16.6907	−1.13830
$216$	0	0
$217$	−6.38776	−0.433629
$218$	6.14931	0.416484
$219$	0	0
$220$	9.25240	0.623796
$221$	−16.5048	−1.11023
$222$	0	0
$223$	17.8463	1.19508	0.597539	−	0.801840i	$-0.296145\pi$
$223$	17.8463	1.19508	0.597539	+	0.801840i	$0.296145\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	2.62620	0.174692
$227$	−6.23844	−0.414060	−0.207030	−	0.978335i	$-0.566380\pi$
$227$	−6.23844	−0.414060	−0.207030	+	0.978335i	$0.566380\pi$
$228$	0	0
$229$	26.2341	1.73360	0.866798	−	0.498659i	$-0.166174\pi$
$229$	26.2341	1.73360	0.866798	+	0.498659i	$0.166174\pi$
$230$	2.62620	0.173166
$231$	0	0
$232$	4.89692	0.321498
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−19.6724	−1.28329
$236$	0	0
$237$	0	0
$238$	4.38776	0.284416
$239$	−10.9817	−0.710346	−0.355173	−	0.934801i	$-0.615578\pi$
$239$	−10.9817	−0.710346	−0.355173	+	0.934801i	$0.615578\pi$
$240$	0	0
$241$	−1.55539	−0.100191	−0.0500957	−	0.998744i	$-0.515953\pi$
$241$	−1.55539	−0.100191	−0.0500957	+	0.998744i	$0.515953\pi$
$242$	1.41233	0.0907883
$243$	0	0
$244$	−6.77551	−0.433758
$245$	−2.62620	−0.167782
$246$	0	0
$247$	10.7755	0.685629
$248$	6.38776	0.405623
$249$	0	0
$250$	8.14931	0.515408
$251$	7.61994	0.480966	0.240483	−	0.970653i	$-0.422694\pi$
$251$	7.61994	0.480966	0.240483	+	0.970653i	$0.422694\pi$
$252$	0	0
$253$	3.52311	0.221496
$254$	8.62620	0.541256
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.7755	0.672158	0.336079	−	0.941834i	$-0.390899\pi$
$257$	10.7755	0.672158	0.336079	+	0.941834i	$0.390899\pi$
$258$	0	0
$259$	−2.89692	−0.180006
$260$	−9.87859	−0.612644
$261$	0	0
$262$	−6.20617	−0.383418
$263$	−1.64452	−0.101405	−0.0507027	−	0.998714i	$-0.516146\pi$
$263$	−1.64452	−0.101405	−0.0507027	+	0.998714i	$0.516146\pi$
$264$	0	0
$265$	−4.71096	−0.289392
$266$	−2.86464	−0.175642
$267$	0	0
$268$	−7.04623	−0.430417
$269$	−25.1633	−1.53423	−0.767116	−	0.641509i	$-0.778309\pi$
$269$	−25.1633	−1.53423	−0.767116	+	0.641509i	$0.778309\pi$
$270$	0	0
$271$	24.9571	1.51604	0.758018	−	0.652234i	$-0.226168\pi$
$271$	24.9571	1.51604	0.758018	+	0.652234i	$0.226168\pi$
$272$	−4.38776	−0.266047
$273$	0	0
$274$	12.3555	0.746422
$275$	−6.68305	−0.403003
$276$	0	0
$277$	31.8496	1.91366	0.956830	−	0.290649i	$-0.0938710\pi$
$277$	31.8496	1.91366	0.956830	+	0.290649i	$0.0938710\pi$
$278$	17.4663	1.04756
$279$	0	0
$280$	2.62620	0.156945
$281$	6.14931	0.366837	0.183419	−	0.983035i	$-0.441284\pi$
$281$	6.14931	0.366837	0.183419	+	0.983035i	$0.441284\pi$
$282$	0	0
$283$	25.0216	1.48738	0.743691	−	0.668523i	$-0.233073\pi$
$283$	25.0216	1.48738	0.743691	+	0.668523i	$0.233073\pi$
$284$	−10.9817	−0.651643
$285$	0	0
$286$	−13.2524	−0.783631
$287$	−10.1493	−0.599095
$288$	0	0
$289$	2.25240	0.132494
$290$	−12.8603	−0.755181
$291$	0	0
$292$	7.72928	0.452322
$293$	−11.3169	−0.661143	−0.330572	−	0.943781i	$-0.607241\pi$
$293$	−11.3169	−0.661143	−0.330572	+	0.943781i	$0.607241\pi$
$294$	0	0
$295$	0	0
$296$	2.89692	0.168380
$297$	0	0
$298$	13.7572	0.796933
$299$	−3.76156	−0.217536
$300$	0	0
$301$	−6.35548	−0.366324
$302$	−13.8786	−0.798623
$303$	0	0
$304$	2.86464	0.164298
$305$	17.7938	1.01887
$306$	0	0
$307$	−18.0077	−1.02775	−0.513877	−	0.857864i	$-0.671791\pi$
$307$	−18.0077	−1.02775	−0.513877	+	0.857864i	$0.671791\pi$
$308$	3.52311	0.200748
$309$	0	0
$310$	−16.7755	−0.952785
$311$	−9.07081	−0.514358	−0.257179	−	0.966364i	$-0.582793\pi$
$311$	−9.07081	−0.514358	−0.257179	+	0.966364i	$0.582793\pi$
$312$	0	0
$313$	7.61224	0.430270	0.215135	−	0.976584i	$-0.430981\pi$
$313$	7.61224	0.430270	0.215135	+	0.976584i	$0.430981\pi$
$314$	−8.50479	−0.479953
$315$	0	0
$316$	5.52311	0.310699
$317$	−24.7187	−1.38834	−0.694169	−	0.719812i	$-0.744228\pi$
$317$	−24.7187	−1.38834	−0.694169	+	0.719812i	$0.744228\pi$
$318$	0	0
$319$	−17.2524	−0.965949
$320$	−2.62620	−0.146809
$321$	0	0
$322$	1.00000	0.0557278
$323$	−12.5693	−0.699377
$324$	0	0
$325$	7.13536	0.395799
$326$	18.2341	1.00989
$327$	0	0
$328$	10.1493	0.560402
$329$	−7.49084	−0.412983
$330$	0	0
$331$	−17.7938	−0.978037	−0.489019	−	0.872273i	$-0.662645\pi$
$331$	−17.7938	−0.978037	−0.489019	+	0.872273i	$0.662645\pi$
$332$	8.11704	0.445480
$333$	0	0
$334$	10.3878	0.568393
$335$	18.5048	1.01102
$336$	0	0
$337$	−20.8680	−1.13675	−0.568375	−	0.822769i	$-0.692428\pi$
$337$	−20.8680	−1.13675	−0.568375	+	0.822769i	$0.692428\pi$
$338$	1.14931	0.0625144
$339$	0	0
$340$	11.5231	0.624929
$341$	−22.5048	−1.21870
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	6.35548	0.342665
$345$	0	0
$346$	7.43398	0.399653
$347$	−28.7187	−1.54170	−0.770849	−	0.637018i	$-0.780168\pi$
$347$	−28.7187	−1.54170	−0.770849	+	0.637018i	$0.780168\pi$
$348$	0	0
$349$	−19.3049	−1.03337	−0.516683	−	0.856177i	$-0.672834\pi$
$349$	−19.3049	−1.03337	−0.516683	+	0.856177i	$0.672834\pi$
$350$	−1.89692	−0.101394
$351$	0	0
$352$	−3.52311	−0.187783
$353$	28.9527	1.54100	0.770499	−	0.637441i	$-0.220007\pi$
$353$	28.9527	1.54100	0.770499	+	0.637441i	$0.220007\pi$
$354$	0	0
$355$	28.8401	1.53067
$356$	4.32320	0.229129
$357$	0	0
$358$	−1.10308	−0.0582998
$359$	−5.37380	−0.283618	−0.141809	−	0.989894i	$-0.545292\pi$
$359$	−5.37380	−0.283618	−0.141809	+	0.989894i	$0.545292\pi$
$360$	0	0
$361$	−10.7938	−0.568096
$362$	2.77551	0.145878
$363$	0	0
$364$	−3.76156	−0.197159
$365$	−20.2986	−1.06248
$366$	0	0
$367$	6.42003	0.335123	0.167561	−	0.985862i	$-0.446411\pi$
$367$	6.42003	0.335123	0.167561	+	0.985862i	$0.446411\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−7.60788	−0.395515
$371$	−1.79383	−0.0931311
$372$	0	0
$373$	−0.710960	−0.0368121	−0.0184060	−	0.999831i	$-0.505859\pi$
$373$	−0.710960	−0.0368121	−0.0184060	+	0.999831i	$0.505859\pi$
$374$	15.4586	0.799343
$375$	0	0
$376$	7.49084	0.386311
$377$	18.4200	0.948680
$378$	0	0
$379$	−8.08476	−0.415286	−0.207643	−	0.978205i	$-0.566579\pi$
$379$	−8.08476	−0.415286	−0.207643	+	0.978205i	$0.566579\pi$
$380$	−7.52311	−0.385928
$381$	0	0
$382$	4.02791	0.206086
$383$	−17.2524	−0.881556	−0.440778	−	0.897616i	$-0.645297\pi$
$383$	−17.2524	−0.881556	−0.440778	+	0.897616i	$0.645297\pi$
$384$	0	0
$385$	−9.25240	−0.471546
$386$	−9.81404	−0.499522
$387$	0	0
$388$	0.238443	0.0121051
$389$	29.5510	1.49830	0.749148	−	0.662403i	$-0.230463\pi$
$389$	29.5510	1.49830	0.749148	+	0.662403i	$0.230463\pi$
$390$	0	0
$391$	4.38776	0.221898
$392$	1.00000	0.0505076
$393$	0	0
$394$	−17.1310	−0.863047
$395$	−14.5048	−0.729815
$396$	0	0
$397$	9.34153	0.468838	0.234419	−	0.972136i	$-0.424681\pi$
$397$	9.34153	0.468838	0.234419	+	0.972136i	$0.424681\pi$
$398$	−19.1310	−0.958950
$399$	0	0
$400$	1.89692	0.0948458
$401$	0.504792	0.0252081	0.0126041	−	0.999921i	$-0.495988\pi$
$401$	0.504792	0.0252081	0.0126041	+	0.999921i	$0.495988\pi$
$402$	0	0
$403$	24.0279	1.19692
$404$	3.07081	0.152778
$405$	0	0
$406$	−4.89692	−0.243030
$407$	−10.2062	−0.505901
$408$	0	0
$409$	−15.5510	−0.768948	−0.384474	−	0.923136i	$-0.625617\pi$
$409$	−15.5510	−0.768948	−0.384474	+	0.923136i	$0.625617\pi$
$410$	−26.6541	−1.31635
$411$	0	0
$412$	2.12141	0.104514
$413$	0	0
$414$	0	0
$415$	−21.3169	−1.04641
$416$	3.76156	0.184426
$417$	0	0
$418$	−10.0925	−0.493638
$419$	−11.7047	−0.571812	−0.285906	−	0.958258i	$-0.592295\pi$
$419$	−11.7047	−0.571812	−0.285906	+	0.958258i	$0.592295\pi$
$420$	0	0
$421$	24.1127	1.17518	0.587590	−	0.809159i	$-0.300077\pi$
$421$	24.1127	1.17518	0.587590	+	0.809159i	$0.300077\pi$
$422$	−16.0279	−0.780226
$423$	0	0
$424$	1.79383	0.0871162
$425$	−8.32320	−0.403735
$426$	0	0
$427$	6.77551	0.327890
$428$	−11.5231	−0.556991
$429$	0	0
$430$	−16.6907	−0.804899
$431$	1.94315	0.0935980	0.0467990	−	0.998904i	$-0.485098\pi$
$431$	1.94315	0.0935980	0.0467990	+	0.998904i	$0.485098\pi$
$432$	0	0
$433$	−10.8723	−0.522491	−0.261246	−	0.965272i	$-0.584133\pi$
$433$	−10.8723	−0.522491	−0.261246	+	0.965272i	$0.584133\pi$
$434$	−6.38776	−0.306622
$435$	0	0
$436$	6.14931	0.294499
$437$	−2.86464	−0.137034
$438$	0	0
$439$	−37.9667	−1.81205	−0.906025	−	0.423223i	$-0.860899\pi$
$439$	−37.9667	−1.81205	−0.906025	+	0.423223i	$0.860899\pi$
$440$	9.25240	0.441091
$441$	0	0
$442$	−16.5048	−0.785053
$443$	14.0848	0.669187	0.334594	−	0.942363i	$-0.391401\pi$
$443$	14.0848	0.669187	0.334594	+	0.942363i	$0.391401\pi$
$444$	0	0
$445$	−11.3536	−0.538212
$446$	17.8463	0.845048
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−24.3911	−1.15109	−0.575543	−	0.817771i	$-0.695209\pi$
$449$	−24.3911	−1.15109	−0.575543	+	0.817771i	$0.695209\pi$
$450$	0	0
$451$	−35.7572	−1.68374
$452$	2.62620	0.123526
$453$	0	0
$454$	−6.23844	−0.292785
$455$	9.87859	0.463116
$456$	0	0
$457$	18.0558	0.844615	0.422308	−	0.906453i	$-0.361220\pi$
$457$	18.0558	0.844615	0.422308	+	0.906453i	$0.361220\pi$
$458$	26.2341	1.22584
$459$	0	0
$460$	2.62620	0.122447
$461$	21.4061	0.996980	0.498490	−	0.866895i	$-0.333888\pi$
$461$	21.4061	0.996980	0.498490	+	0.866895i	$0.333888\pi$
$462$	0	0
$463$	6.59829	0.306649	0.153324	−	0.988176i	$-0.451002\pi$
$463$	6.59829	0.306649	0.153324	+	0.988176i	$0.451002\pi$
$464$	4.89692	0.227334
$465$	0	0
$466$	−10.0000	−0.463241
$467$	−4.98605	−0.230727	−0.115363	−	0.993323i	$-0.536803\pi$
$467$	−4.98605	−0.230727	−0.115363	+	0.993323i	$0.536803\pi$
$468$	0	0
$469$	7.04623	0.325365
$470$	−19.6724	−0.907421
$471$	0	0
$472$	0	0
$473$	−22.3911	−1.02954
$474$	0	0
$475$	5.43398	0.249328
$476$	4.38776	0.201112
$477$	0	0
$478$	−10.9817	−0.502290
$479$	23.4094	1.06960	0.534801	−	0.844978i	$-0.320386\pi$
$479$	23.4094	1.06960	0.534801	+	0.844978i	$0.320386\pi$
$480$	0	0
$481$	10.8969	0.496857
$482$	−1.55539	−0.0708461
$483$	0	0
$484$	1.41233	0.0641970
$485$	−0.626198	−0.0284342
$486$	0	0
$487$	−33.4017	−1.51358	−0.756788	−	0.653660i	$-0.773232\pi$
$487$	−33.4017	−1.51358	−0.756788	+	0.653660i	$0.773232\pi$
$488$	−6.77551	−0.306713
$489$	0	0
$490$	−2.62620	−0.118640
$491$	37.0375	1.67148	0.835739	−	0.549126i	$-0.185039\pi$
$491$	37.0375	1.67148	0.835739	+	0.549126i	$0.185039\pi$
$492$	0	0
$493$	−21.4865	−0.967702
$494$	10.7755	0.484813
$495$	0	0
$496$	6.38776	0.286819
$497$	10.9817	0.492596
$498$	0	0
$499$	40.0837	1.79439	0.897197	−	0.441631i	$-0.145600\pi$
$499$	40.0837	1.79439	0.897197	+	0.441631i	$0.145600\pi$
$500$	8.14931	0.364448
$501$	0	0
$502$	7.61994	0.340095
$503$	−9.79383	−0.436685	−0.218343	−	0.975872i	$-0.570065\pi$
$503$	−9.79383	−0.436685	−0.218343	+	0.975872i	$0.570065\pi$
$504$	0	0
$505$	−8.06455	−0.358868
$506$	3.52311	0.156622
$507$	0	0
$508$	8.62620	0.382726
$509$	26.4157	1.17085	0.585427	−	0.810725i	$-0.300927\pi$
$509$	26.4157	1.17085	0.585427	+	0.810725i	$0.300927\pi$
$510$	0	0
$511$	−7.72928	−0.341923
$512$	1.00000	0.0441942
$513$	0	0
$514$	10.7755	0.475287
$515$	−5.57123	−0.245498
$516$	0	0
$517$	−26.3911	−1.16068
$518$	−2.89692	−0.127283
$519$	0	0
$520$	−9.87859	−0.433205
$521$	39.2557	1.71982	0.859912	−	0.510442i	$-0.170518\pi$
$521$	39.2557	1.71982	0.859912	+	0.510442i	$0.170518\pi$
$522$	0	0
$523$	6.68638	0.292375	0.146187	−	0.989257i	$-0.453300\pi$
$523$	6.68638	0.292375	0.146187	+	0.989257i	$0.453300\pi$
$524$	−6.20617	−0.271118
$525$	0	0
$526$	−1.64452	−0.0717045
$527$	−28.0279	−1.22091
$528$	0	0
$529$	1.00000	0.0434783
$530$	−4.71096	−0.204631
$531$	0	0
$532$	−2.86464	−0.124198
$533$	38.1772	1.65364
$534$	0	0
$535$	30.2620	1.30834
$536$	−7.04623	−0.304351
$537$	0	0
$538$	−25.1633	−1.08487
$539$	−3.52311	−0.151751
$540$	0	0
$541$	−29.4586	−1.26652	−0.633261	−	0.773938i	$-0.718284\pi$
$541$	−29.4586	−1.26652	−0.633261	+	0.773938i	$0.718284\pi$
$542$	24.9571	1.07200
$543$	0	0
$544$	−4.38776	−0.188123
$545$	−16.1493	−0.691761
$546$	0	0
$547$	−5.25240	−0.224576	−0.112288	−	0.993676i	$-0.535818\pi$
$547$	−5.25240	−0.224576	−0.112288	+	0.993676i	$0.535818\pi$
$548$	12.3555	0.527800
$549$	0	0
$550$	−6.68305	−0.284966
$551$	14.0279	0.597609
$552$	0	0
$553$	−5.52311	−0.234867
$554$	31.8496	1.35316
$555$	0	0
$556$	17.4663	0.740734
$557$	31.4219	1.33139	0.665695	−	0.746224i	$-0.268135\pi$
$557$	31.4219	1.33139	0.665695	+	0.746224i	$0.268135\pi$
$558$	0	0
$559$	23.9065	1.01114
$560$	2.62620	0.110977
$561$	0	0
$562$	6.14931	0.259393
$563$	−22.2384	−0.937239	−0.468619	−	0.883400i	$-0.655248\pi$
$563$	−22.2384	−0.937239	−0.468619	+	0.883400i	$0.655248\pi$
$564$	0	0
$565$	−6.89692	−0.290155
$566$	25.0216	1.05174
$567$	0	0
$568$	−10.9817	−0.460781
$569$	−22.7312	−0.952940	−0.476470	−	0.879191i	$-0.658084\pi$
$569$	−22.7312	−0.952940	−0.476470	+	0.879191i	$0.658084\pi$
$570$	0	0
$571$	−19.5877	−0.819718	−0.409859	−	0.912149i	$-0.634422\pi$
$571$	−19.5877	−0.819718	−0.409859	+	0.912149i	$0.634422\pi$
$572$	−13.2524	−0.554111
$573$	0	0
$574$	−10.1493	−0.423624
$575$	−1.89692	−0.0791069
$576$	0	0
$577$	−28.5693	−1.18936	−0.594679	−	0.803963i	$-0.702721\pi$
$577$	−28.5693	−1.18936	−0.594679	+	0.803963i	$0.702721\pi$
$578$	2.25240	0.0936873
$579$	0	0
$580$	−12.8603	−0.533994
$581$	−8.11704	−0.336751
$582$	0	0
$583$	−6.31988	−0.261743
$584$	7.72928	0.319840
$585$	0	0
$586$	−11.3169	−0.467499
$587$	25.7293	1.06196	0.530981	−	0.847384i	$-0.321824\pi$
$587$	25.7293	1.06196	0.530981	+	0.847384i	$0.321824\pi$
$588$	0	0
$589$	18.2986	0.753982
$590$	0	0
$591$	0	0
$592$	2.89692	0.119063
$593$	−19.1676	−0.787120	−0.393560	−	0.919299i	$-0.628757\pi$
$593$	−19.1676	−0.787120	−0.393560	+	0.919299i	$0.628757\pi$
$594$	0	0
$595$	−11.5231	−0.472402
$596$	13.7572	0.563516
$597$	0	0
$598$	−3.76156	−0.153822
$599$	44.0558	1.80007	0.900036	−	0.435816i	$-0.143540\pi$
$599$	44.0558	1.80007	0.900036	+	0.435816i	$0.143540\pi$
$600$	0	0
$601$	43.9142	1.79130	0.895649	−	0.444762i	$-0.146712\pi$
$601$	43.9142	1.79130	0.895649	+	0.444762i	$0.146712\pi$
$602$	−6.35548	−0.259030
$603$	0	0
$604$	−13.8786	−0.564712
$605$	−3.70907	−0.150795
$606$	0	0
$607$	5.13536	0.208438	0.104219	−	0.994554i	$-0.466766\pi$
$607$	5.13536	0.208438	0.104219	+	0.994554i	$0.466766\pi$
$608$	2.86464	0.116177
$609$	0	0
$610$	17.7938	0.720451
$611$	28.1772	1.13993
$612$	0	0
$613$	−30.5616	−1.23437	−0.617187	−	0.786817i	$-0.711728\pi$
$613$	−30.5616	−1.23437	−0.617187	+	0.786817i	$0.711728\pi$
$614$	−18.0077	−0.726731
$615$	0	0
$616$	3.52311	0.141950
$617$	11.9634	0.481627	0.240813	−	0.970571i	$-0.422586\pi$
$617$	11.9634	0.481627	0.240813	+	0.970571i	$0.422586\pi$
$618$	0	0
$619$	−30.4523	−1.22398	−0.611991	−	0.790865i	$-0.709631\pi$
$619$	−30.4523	−1.22398	−0.611991	+	0.790865i	$0.709631\pi$
$620$	−16.7755	−0.673721
$621$	0	0
$622$	−9.07081	−0.363706
$623$	−4.32320	−0.173206
$624$	0	0
$625$	−30.8863	−1.23545
$626$	7.61224	0.304246
$627$	0	0
$628$	−8.50479	−0.339378
$629$	−12.7110	−0.506819
$630$	0	0
$631$	−35.7851	−1.42458	−0.712291	−	0.701884i	$-0.752342\pi$
$631$	−35.7851	−1.42458	−0.712291	+	0.701884i	$0.752342\pi$
$632$	5.52311	0.219698
$633$	0	0
$634$	−24.7187	−0.981703
$635$	−22.6541	−0.899001
$636$	0	0
$637$	3.76156	0.149038
$638$	−17.2524	−0.683029
$639$	0	0
$640$	−2.62620	−0.103810
$641$	38.1493	1.50681	0.753404	−	0.657558i	$-0.228411\pi$
$641$	38.1493	1.50681	0.753404	+	0.657558i	$0.228411\pi$
$642$	0	0
$643$	26.1449	1.03106	0.515528	−	0.856873i	$-0.327596\pi$
$643$	26.1449	1.03106	0.515528	+	0.856873i	$0.327596\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	−12.5693	−0.494534
$647$	−5.74135	−0.225716	−0.112858	−	0.993611i	$-0.536000\pi$
$647$	−5.74135	−0.225716	−0.112858	+	0.993611i	$0.536000\pi$
$648$	0	0
$649$	0	0
$650$	7.13536	0.279872
$651$	0	0
$652$	18.2341	0.714101
$653$	3.94315	0.154307	0.0771536	−	0.997019i	$-0.475417\pi$
$653$	3.94315	0.154307	0.0771536	+	0.997019i	$0.475417\pi$
$654$	0	0
$655$	16.2986	0.636840
$656$	10.1493	0.396264
$657$	0	0
$658$	−7.49084	−0.292023
$659$	29.0096	1.13005	0.565026	−	0.825073i	$-0.308866\pi$
$659$	29.0096	1.13005	0.565026	+	0.825073i	$0.308866\pi$
$660$	0	0
$661$	−37.5231	−1.45948	−0.729740	−	0.683725i	$-0.760359\pi$
$661$	−37.5231	−1.45948	−0.729740	+	0.683725i	$0.760359\pi$
$662$	−17.7938	−0.691577
$663$	0	0
$664$	8.11704	0.315002
$665$	7.52311	0.291734
$666$	0	0
$667$	−4.89692	−0.189609
$668$	10.3878	0.401914
$669$	0	0
$670$	18.5048	0.714902
$671$	23.8709	0.921526
$672$	0	0
$673$	51.6637	1.99149	0.995744	−	0.0921576i	$-0.0293764\pi$
$673$	51.6637	1.99149	0.995744	+	0.0921576i	$0.0293764\pi$
$674$	−20.8680	−0.803804
$675$	0	0
$676$	1.14931	0.0442043
$677$	10.9046	0.419098	0.209549	−	0.977798i	$-0.432800\pi$
$677$	10.9046	0.419098	0.209549	+	0.977798i	$0.432800\pi$
$678$	0	0
$679$	−0.238443	−0.00915060
$680$	11.5231	0.441891
$681$	0	0
$682$	−22.5048	−0.861753
$683$	−9.52311	−0.364392	−0.182196	−	0.983262i	$-0.558321\pi$
$683$	−9.52311	−0.364392	−0.182196	+	0.983262i	$0.558321\pi$
$684$	0	0
$685$	−32.4479	−1.23977
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	6.35548	0.242300
$689$	6.74760	0.257063
$690$	0	0
$691$	34.5896	1.31585	0.657924	−	0.753084i	$-0.271435\pi$
$691$	34.5896	1.31585	0.657924	+	0.753084i	$0.271435\pi$
$692$	7.43398	0.282598
$693$	0	0
$694$	−28.7187	−1.09015
$695$	−45.8699	−1.73994
$696$	0	0
$697$	−44.5327	−1.68680
$698$	−19.3049	−0.730701
$699$	0	0
$700$	−1.89692	−0.0716967
$701$	0.917127	0.0346394	0.0173197	−	0.999850i	$-0.494487\pi$
$701$	0.917127	0.0346394	0.0173197	+	0.999850i	$0.494487\pi$
$702$	0	0
$703$	8.29862	0.312989
$704$	−3.52311	−0.132782
$705$	0	0
$706$	28.9527	1.08965
$707$	−3.07081	−0.115490
$708$	0	0
$709$	39.8130	1.49521	0.747604	−	0.664144i	$-0.231204\pi$
$709$	39.8130	1.49521	0.747604	+	0.664144i	$0.231204\pi$
$710$	28.8401	1.08235
$711$	0	0
$712$	4.32320	0.162019
$713$	−6.38776	−0.239223
$714$	0	0
$715$	34.8034	1.30157
$716$	−1.10308	−0.0412242
$717$	0	0
$718$	−5.37380	−0.200549
$719$	−22.8357	−0.851628	−0.425814	−	0.904811i	$-0.640012\pi$
$719$	−22.8357	−0.851628	−0.425814	+	0.904811i	$0.640012\pi$
$720$	0	0
$721$	−2.12141	−0.0790053
$722$	−10.7938	−0.401705
$723$	0	0
$724$	2.77551	0.103151
$725$	9.28904	0.344986
$726$	0	0
$727$	−25.7851	−0.956316	−0.478158	−	0.878274i	$-0.658695\pi$
$727$	−25.7851	−0.956316	−0.478158	+	0.878274i	$0.658695\pi$
$728$	−3.76156	−0.139413
$729$	0	0
$730$	−20.2986	−0.751286
$731$	−27.8863	−1.03141
$732$	0	0
$733$	−9.34485	−0.345160	−0.172580	−	0.984996i	$-0.555210\pi$
$733$	−9.34485	−0.345160	−0.172580	+	0.984996i	$0.555210\pi$
$734$	6.42003	0.236968
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	24.8247	0.914428
$738$	0	0
$739$	−43.7851	−1.61066	−0.805330	−	0.592826i	$-0.798012\pi$
$739$	−43.7851	−1.61066	−0.805330	+	0.592826i	$0.798012\pi$
$740$	−7.60788	−0.279671
$741$	0	0
$742$	−1.79383	−0.0658537
$743$	−44.0279	−1.61523	−0.807614	−	0.589712i	$-0.799241\pi$
$743$	−44.0279	−1.61523	−0.807614	+	0.589712i	$0.799241\pi$
$744$	0	0
$745$	−36.1291	−1.32367
$746$	−0.710960	−0.0260301
$747$	0	0
$748$	15.4586	0.565221
$749$	11.5231	0.421045
$750$	0	0
$751$	−37.9634	−1.38530	−0.692651	−	0.721273i	$-0.743557\pi$
$751$	−37.9634	−1.38530	−0.692651	+	0.721273i	$0.743557\pi$
$752$	7.49084	0.273163
$753$	0	0
$754$	18.4200	0.670818
$755$	36.4479	1.32648
$756$	0	0
$757$	49.2311	1.78934	0.894668	−	0.446731i	$-0.147412\pi$
$757$	49.2311	1.78934	0.894668	+	0.446731i	$0.147412\pi$
$758$	−8.08476	−0.293652
$759$	0	0
$760$	−7.52311	−0.272892
$761$	−41.8776	−1.51806	−0.759030	−	0.651056i	$-0.774326\pi$
$761$	−41.8776	−1.51806	−0.759030	+	0.651056i	$0.774326\pi$
$762$	0	0
$763$	−6.14931	−0.222620
$764$	4.02791	0.145725
$765$	0	0
$766$	−17.2524	−0.623354
$767$	0	0
$768$	0	0
$769$	29.0140	1.04627	0.523135	−	0.852250i	$-0.324762\pi$
$769$	29.0140	1.04627	0.523135	+	0.852250i	$0.324762\pi$
$770$	−9.25240	−0.333433
$771$	0	0
$772$	−9.81404	−0.353215
$773$	−4.35548	−0.156656	−0.0783279	−	0.996928i	$-0.524958\pi$
$773$	−4.35548	−0.156656	−0.0783279	+	0.996928i	$0.524958\pi$
$774$	0	0
$775$	12.1170	0.435257
$776$	0.238443	0.00855960
$777$	0	0
$778$	29.5510	1.05946
$779$	29.0741	1.04169
$780$	0	0
$781$	38.6897	1.38443
$782$	4.38776	0.156906
$783$	0	0
$784$	1.00000	0.0357143
$785$	22.3353	0.797180
$786$	0	0
$787$	6.15368	0.219355	0.109678	−	0.993967i	$-0.465018\pi$
$787$	6.15368	0.219355	0.109678	+	0.993967i	$0.465018\pi$
$788$	−17.1310	−0.610266
$789$	0	0
$790$	−14.5048	−0.516057
$791$	−2.62620	−0.0933769
$792$	0	0
$793$	−25.4865	−0.905051
$794$	9.34153	0.331518
$795$	0	0
$796$	−19.1310	−0.678080
$797$	9.13973	0.323746	0.161873	−	0.986812i	$-0.448247\pi$
$797$	9.13973	0.323746	0.161873	+	0.986812i	$0.448247\pi$
$798$	0	0
$799$	−32.8680	−1.16279
$800$	1.89692	0.0670661
$801$	0	0
$802$	0.504792	0.0178248
$803$	−27.2311	−0.960966
$804$	0	0
$805$	−2.62620	−0.0925613
$806$	24.0279	0.846347
$807$	0	0
$808$	3.07081	0.108031
$809$	32.6339	1.14735	0.573673	−	0.819084i	$-0.305518\pi$
$809$	32.6339	1.14735	0.573673	+	0.819084i	$0.305518\pi$
$810$	0	0
$811$	−3.72159	−0.130683	−0.0653413	−	0.997863i	$-0.520814\pi$
$811$	−3.72159	−0.130683	−0.0653413	+	0.997863i	$0.520814\pi$
$812$	−4.89692	−0.171848
$813$	0	0
$814$	−10.2062	−0.357726
$815$	−47.8863	−1.67738
$816$	0	0
$817$	18.2062	0.636953
$818$	−15.5510	−0.543729
$819$	0	0
$820$	−26.6541	−0.930802
$821$	−21.4094	−0.747193	−0.373597	−	0.927591i	$-0.621876\pi$
$821$	−21.4094	−0.747193	−0.373597	+	0.927591i	$0.621876\pi$
$822$	0	0
$823$	−39.0173	−1.36006	−0.680028	−	0.733186i	$-0.738033\pi$
$823$	−39.0173	−1.36006	−0.680028	+	0.733186i	$0.738033\pi$
$824$	2.12141	0.0739027
$825$	0	0
$826$	0	0
$827$	14.1570	0.492287	0.246144	−	0.969233i	$-0.420836\pi$
$827$	14.1570	0.492287	0.246144	+	0.969233i	$0.420836\pi$
$828$	0	0
$829$	−4.15368	−0.144263	−0.0721317	−	0.997395i	$-0.522980\pi$
$829$	−4.15368	−0.144263	−0.0721317	+	0.997395i	$0.522980\pi$
$830$	−21.3169	−0.739922
$831$	0	0
$832$	3.76156	0.130409
$833$	−4.38776	−0.152027
$834$	0	0
$835$	−27.2803	−0.944074
$836$	−10.0925	−0.349055
$837$	0	0
$838$	−11.7047	−0.404332
$839$	−35.5144	−1.22609	−0.613046	−	0.790047i	$-0.710056\pi$
$839$	−35.5144	−1.22609	−0.613046	+	0.790047i	$0.710056\pi$
$840$	0	0
$841$	−5.02021	−0.173111
$842$	24.1127	0.830977
$843$	0	0
$844$	−16.0279	−0.551703
$845$	−3.01832	−0.103833
$846$	0	0
$847$	−1.41233	−0.0485284
$848$	1.79383	0.0616005
$849$	0	0
$850$	−8.32320	−0.285484
$851$	−2.89692	−0.0993050
$852$	0	0
$853$	1.60455	0.0549387	0.0274694	−	0.999623i	$-0.491255\pi$
$853$	1.60455	0.0549387	0.0274694	+	0.999623i	$0.491255\pi$
$854$	6.77551	0.231853
$855$	0	0
$856$	−11.5231	−0.393852
$857$	5.55206	0.189655	0.0948274	−	0.995494i	$-0.469770\pi$
$857$	5.55206	0.189655	0.0948274	+	0.995494i	$0.469770\pi$
$858$	0	0
$859$	35.5587	1.21325	0.606624	−	0.794989i	$-0.292523\pi$
$859$	35.5587	1.21325	0.606624	+	0.794989i	$0.292523\pi$
$860$	−16.6907	−0.569150
$861$	0	0
$862$	1.94315	0.0661838
$863$	8.36318	0.284686	0.142343	−	0.989817i	$-0.454536\pi$
$863$	8.36318	0.284686	0.142343	+	0.989817i	$0.454536\pi$
$864$	0	0
$865$	−19.5231	−0.663806
$866$	−10.8723	−0.369457
$867$	0	0
$868$	−6.38776	−0.216815
$869$	−19.4586	−0.660087
$870$	0	0
$871$	−26.5048	−0.898081
$872$	6.14931	0.208242
$873$	0	0
$874$	−2.86464	−0.0968979
$875$	−8.14931	−0.275497
$876$	0	0
$877$	−0.569343	−0.0192254	−0.00961268	−	0.999954i	$-0.503060\pi$
$877$	−0.569343	−0.0192254	−0.00961268	+	0.999954i	$0.503060\pi$
$878$	−37.9667	−1.28131
$879$	0	0
$880$	9.25240	0.311898
$881$	36.9850	1.24606	0.623028	−	0.782199i	$-0.285902\pi$
$881$	36.9850	1.24606	0.623028	+	0.782199i	$0.285902\pi$
$882$	0	0
$883$	6.06455	0.204088	0.102044	−	0.994780i	$-0.467462\pi$
$883$	6.06455	0.204088	0.102044	+	0.994780i	$0.467462\pi$
$884$	−16.5048	−0.555116
$885$	0	0
$886$	14.0848	0.473187
$887$	29.0216	0.974452	0.487226	−	0.873276i	$-0.338009\pi$
$887$	29.0216	0.974452	0.487226	+	0.873276i	$0.338009\pi$
$888$	0	0
$889$	−8.62620	−0.289313
$890$	−11.3536	−0.380573
$891$	0	0
$892$	17.8463	0.597539
$893$	21.4586	0.718083
$894$	0	0
$895$	2.89692	0.0968332
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−24.3911	−0.813941
$899$	31.2803	1.04326
$900$	0	0
$901$	−7.87090	−0.262218
$902$	−35.7572	−1.19058
$903$	0	0
$904$	2.62620	0.0873460
$905$	−7.28904	−0.242296
$906$	0	0
$907$	9.28134	0.308182	0.154091	−	0.988057i	$-0.450755\pi$
$907$	9.28134	0.308182	0.154091	+	0.988057i	$0.450755\pi$
$908$	−6.23844	−0.207030
$909$	0	0
$910$	9.87859	0.327472
$911$	15.5800	0.516187	0.258094	−	0.966120i	$-0.416906\pi$
$911$	15.5800	0.516187	0.258094	+	0.966120i	$0.416906\pi$
$912$	0	0
$913$	−28.5972	−0.946431
$914$	18.0558	0.597233
$915$	0	0
$916$	26.2341	0.866798
$917$	6.20617	0.204946
$918$	0	0
$919$	14.6464	0.483140	0.241570	−	0.970383i	$-0.422338\pi$
$919$	14.6464	0.483140	0.241570	+	0.970383i	$0.422338\pi$
$920$	2.62620	0.0865832
$921$	0	0
$922$	21.4061	0.704972
$923$	−41.3082	−1.35968
$924$	0	0
$925$	5.49521	0.180681
$926$	6.59829	0.216833
$927$	0	0
$928$	4.89692	0.160749
$929$	−2.44794	−0.0803142	−0.0401571	−	0.999193i	$-0.512786\pi$
$929$	−2.44794	−0.0803142	−0.0401571	+	0.999193i	$0.512786\pi$
$930$	0	0
$931$	2.86464	0.0938848
$932$	−10.0000	−0.327561
$933$	0	0
$934$	−4.98605	−0.163148
$935$	−40.5972	−1.32767
$936$	0	0
$937$	33.1989	1.08456	0.542280	−	0.840198i	$-0.317561\pi$
$937$	33.1989	1.08456	0.542280	+	0.840198i	$0.317561\pi$
$938$	7.04623	0.230068
$939$	0	0
$940$	−19.6724	−0.641644
$941$	−35.2234	−1.14825	−0.574126	−	0.818767i	$-0.694658\pi$
$941$	−35.2234	−1.14825	−0.574126	+	0.818767i	$0.694658\pi$
$942$	0	0
$943$	−10.1493	−0.330507
$944$	0	0
$945$	0	0
$946$	−22.3911	−0.727997
$947$	−13.9711	−0.453998	−0.226999	−	0.973895i	$-0.572891\pi$
$947$	−13.9711	−0.453998	−0.226999	+	0.973895i	$0.572891\pi$
$948$	0	0
$949$	29.0741	0.943786
$950$	5.43398	0.176302
$951$	0	0
$952$	4.38776	0.142208
$953$	50.4190	1.63323	0.816616	−	0.577182i	$-0.195848\pi$
$953$	50.4190	1.63323	0.816616	+	0.577182i	$0.195848\pi$
$954$	0	0
$955$	−10.5781	−0.342299
$956$	−10.9817	−0.355173
$957$	0	0
$958$	23.4094	0.756324
$959$	−12.3555	−0.398979
$960$	0	0
$961$	9.80342	0.316239
$962$	10.8969	0.351331
$963$	0	0
$964$	−1.55539	−0.0500957
$965$	25.7736	0.829682
$966$	0	0
$967$	−11.1020	−0.357018	−0.178509	−	0.983938i	$-0.557127\pi$
$967$	−11.1020	−0.357018	−0.178509	+	0.983938i	$0.557127\pi$
$968$	1.41233	0.0453942
$969$	0	0
$970$	−0.626198	−0.0201060
$971$	15.2924	0.490755	0.245378	−	0.969428i	$-0.421088\pi$
$971$	15.2924	0.490755	0.245378	+	0.969428i	$0.421088\pi$
$972$	0	0
$973$	−17.4663	−0.559943
$974$	−33.4017	−1.07026
$975$	0	0
$976$	−6.77551	−0.216879
$977$	30.5770	0.978246	0.489123	−	0.872215i	$-0.337317\pi$
$977$	30.5770	0.978246	0.489123	+	0.872215i	$0.337317\pi$
$978$	0	0
$979$	−15.2311	−0.486789
$980$	−2.62620	−0.0838908
$981$	0	0
$982$	37.0375	1.18191
$983$	−58.5048	−1.86601	−0.933007	−	0.359859i	$-0.882825\pi$
$983$	−58.5048	−1.86601	−0.933007	+	0.359859i	$0.882825\pi$
$984$	0	0
$985$	44.9894	1.43348
$986$	−21.4865	−0.684269
$987$	0	0
$988$	10.7755	0.342815
$989$	−6.35548	−0.202093
$990$	0	0
$991$	−30.6098	−0.972351	−0.486176	−	0.873861i	$-0.661608\pi$
$991$	−30.6098	−0.972351	−0.486176	+	0.873861i	$0.661608\pi$
$992$	6.38776	0.202811
$993$	0	0
$994$	10.9817	0.348318
$995$	50.2418	1.59277
$996$	0	0
$997$	16.0804	0.509271	0.254636	−	0.967037i	$-0.418044\pi$
$997$	16.0804	0.509271	0.254636	+	0.967037i	$0.418044\pi$
$998$	40.0837	1.26883
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.bg.1.1	yes	3
3.2	odd	2		2898.2.a.bf.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.a.bf.1.3	✓	3	3.2	odd	2
2898.2.a.bg.1.1	yes	3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.62620 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -2.62620 q^{5} -1.00000 q^{7} +1.00000 q^{8} -2.62620 q^{10} -3.52311 q^{11} +3.76156 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.38776 q^{17} +2.86464 q^{19} -2.62620 q^{20} -3.52311 q^{22} -1.00000 q^{23} +1.89692 q^{25} +3.76156 q^{26} -1.00000 q^{28} +4.89692 q^{29} +6.38776 q^{31} +1.00000 q^{32} -4.38776 q^{34} +2.62620 q^{35} +2.89692 q^{37} +2.86464 q^{38} -2.62620 q^{40} +10.1493 q^{41} +6.35548 q^{43} -3.52311 q^{44} -1.00000 q^{46} +7.49084 q^{47} +1.00000 q^{49} +1.89692 q^{50} +3.76156 q^{52} +1.79383 q^{53} +9.25240 q^{55} -1.00000 q^{56} +4.89692 q^{58} -6.77551 q^{61} +6.38776 q^{62} +1.00000 q^{64} -9.87859 q^{65} -7.04623 q^{67} -4.38776 q^{68} +2.62620 q^{70} -10.9817 q^{71} +7.72928 q^{73} +2.89692 q^{74} +2.86464 q^{76} +3.52311 q^{77} +5.52311 q^{79} -2.62620 q^{80} +10.1493 q^{82} +8.11704 q^{83} +11.5231 q^{85} +6.35548 q^{86} -3.52311 q^{88} +4.32320 q^{89} -3.76156 q^{91} -1.00000 q^{92} +7.49084 q^{94} -7.52311 q^{95} +0.238443 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + q^{5} - 3 q^{7} + 3 q^{8} + q^{10} + 2 q^{11} + 5 q^{13} - 3 q^{14} + 3 q^{16} + 2 q^{17} + 6 q^{19} + q^{20} + 2 q^{22} - 3 q^{23} + 2 q^{25} + 5 q^{26} - 3 q^{28} + 11 q^{29} + 4 q^{31} + 3 q^{32} + 2 q^{34} - q^{35} + 5 q^{37} + 6 q^{38} + q^{40} + 9 q^{41} + 5 q^{43} + 2 q^{44} - 3 q^{46} + 11 q^{47} + 3 q^{49} + 2 q^{50} + 5 q^{52} - 2 q^{53} + 10 q^{55} - 3 q^{56} + 11 q^{58} + 10 q^{61} + 4 q^{62} + 3 q^{64} - 3 q^{65} + 4 q^{67} + 2 q^{68} - q^{70} - 10 q^{71} + 18 q^{73} + 5 q^{74} + 6 q^{76} - 2 q^{77} + 4 q^{79} + q^{80} + 9 q^{82} + 4 q^{83} + 22 q^{85} + 5 q^{86} + 2 q^{88} - 5 q^{91} - 3 q^{92} + 11 q^{94} - 10 q^{95} + 7 q^{97} + 3 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + q^5 - 3 * q^7 + 3 * q^8 + q^10 + 2 * q^11 + 5 * q^13 - 3 * q^14 + 3 * q^16 + 2 * q^17 + 6 * q^19 + q^20 + 2 * q^22 - 3 * q^23 + 2 * q^25 + 5 * q^26 - 3 * q^28 + 11 * q^29 + 4 * q^31 + 3 * q^32 + 2 * q^34 - q^35 + 5 * q^37 + 6 * q^38 + q^40 + 9 * q^41 + 5 * q^43 + 2 * q^44 - 3 * q^46 + 11 * q^47 + 3 * q^49 + 2 * q^50 + 5 * q^52 - 2 * q^53 + 10 * q^55 - 3 * q^56 + 11 * q^58 + 10 * q^61 + 4 * q^62 + 3 * q^64 - 3 * q^65 + 4 * q^67 + 2 * q^68 - q^70 - 10 * q^71 + 18 * q^73 + 5 * q^74 + 6 * q^76 - 2 * q^77 + 4 * q^79 + q^80 + 9 * q^82 + 4 * q^83 + 22 * q^85 + 5 * q^86 + 2 * q^88 - 5 * q^91 - 3 * q^92 + 11 * q^94 - 10 * q^95 + 7 * q^97 + 3 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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