LMFDB - Embedded newform 2898.2.a.bd.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:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 322)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.23607 q^{10} +2.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.23607 q^{17} -4.47214 q^{19} -1.23607 q^{20} +1.00000 q^{23} -3.47214 q^{25} +2.47214 q^{26} -1.00000 q^{28} +2.00000 q^{29} -1.23607 q^{31} +1.00000 q^{32} +7.23607 q^{34} +1.23607 q^{35} +6.00000 q^{37} -4.47214 q^{38} -1.23607 q^{40} +10.0000 q^{41} -6.47214 q^{43} +1.00000 q^{46} +6.76393 q^{47} +1.00000 q^{49} -3.47214 q^{50} +2.47214 q^{52} +6.00000 q^{53} -1.00000 q^{56} +2.00000 q^{58} -11.2361 q^{59} +11.7082 q^{61} -1.23607 q^{62} +1.00000 q^{64} -3.05573 q^{65} +4.00000 q^{67} +7.23607 q^{68} +1.23607 q^{70} +2.47214 q^{71} -13.4164 q^{73} +6.00000 q^{74} -4.47214 q^{76} +8.94427 q^{79} -1.23607 q^{80} +10.0000 q^{82} +6.94427 q^{83} -8.94427 q^{85} -6.47214 q^{86} +16.1803 q^{89} -2.47214 q^{91} +1.00000 q^{92} +6.76393 q^{94} +5.52786 q^{95} +8.18034 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} + 2 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} + 2 q^{20} + 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} + 4 q^{29} + 2 q^{31} + 2 q^{32} + 10 q^{34} - 2 q^{35} + 12 q^{37} + 2 q^{40} + 20 q^{41} - 4 q^{43} + 2 q^{46} + 18 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 2 q^{56} + 4 q^{58} - 18 q^{59} + 10 q^{61} + 2 q^{62} + 2 q^{64} - 24 q^{65} + 8 q^{67} + 10 q^{68} - 2 q^{70} - 4 q^{71} + 12 q^{74} + 2 q^{80} + 20 q^{82} - 4 q^{83} - 4 q^{86} + 10 q^{89} + 4 q^{91} + 2 q^{92} + 18 q^{94} + 20 q^{95} - 6 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8 + 2 * q^10 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 + 2 * q^20 + 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 + 4 * q^29 + 2 * q^31 + 2 * q^32 + 10 * q^34 - 2 * q^35 + 12 * q^37 + 2 * q^40 + 20 * q^41 - 4 * q^43 + 2 * q^46 + 18 * q^47 + 2 * q^49 + 2 * q^50 - 4 * q^52 + 12 * q^53 - 2 * q^56 + 4 * q^58 - 18 * q^59 + 10 * q^61 + 2 * q^62 + 2 * q^64 - 24 * q^65 + 8 * q^67 + 10 * q^68 - 2 * q^70 - 4 * q^71 + 12 * q^74 + 2 * q^80 + 20 * q^82 - 4 * q^83 - 4 * q^86 + 10 * q^89 + 4 * q^91 + 2 * q^92 + 18 * q^94 + 20 * q^95 - 6 * 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
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.23607	−0.390879
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.47214	0.685647	0.342824	−	0.939400i	$-0.388617\pi$
$13$	2.47214	0.685647	0.342824	+	0.939400i	$0.388617\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	7.23607	1.75500	0.877502	−	0.479573i	$-0.159208\pi$
$17$	7.23607	1.75500	0.877502	+	0.479573i	$0.159208\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	−1.23607	−0.276393
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−3.47214	−0.694427
$26$	2.47214	0.484826
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	7.23607	1.24098
$35$	1.23607	0.208934
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	−4.47214	−0.725476
$39$	0	0
$40$	−1.23607	−0.195440
$41$	10.0000	1.56174	0.780869	−	0.624695i	$-0.214777\pi$
$41$	10.0000	1.56174	0.780869	+	0.624695i	$0.214777\pi$
$42$	0	0
$43$	−6.47214	−0.986991	−0.493496	−	0.869748i	$-0.664281\pi$
$43$	−6.47214	−0.986991	−0.493496	+	0.869748i	$0.664281\pi$
$44$	0	0
$45$	0	0
$46$	1.00000	0.147442
$47$	6.76393	0.986621	0.493310	−	0.869853i	$-0.335787\pi$
$47$	6.76393	0.986621	0.493310	+	0.869853i	$0.335787\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−3.47214	−0.491034
$51$	0	0
$52$	2.47214	0.342824
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	−11.2361	−1.46281	−0.731406	−	0.681943i	$-0.761135\pi$
$59$	−11.2361	−1.46281	−0.731406	+	0.681943i	$0.761135\pi$
$60$	0	0
$61$	11.7082	1.49908	0.749541	−	0.661958i	$-0.230274\pi$
$61$	11.7082	1.49908	0.749541	+	0.661958i	$0.230274\pi$
$62$	−1.23607	−0.156981
$63$	0	0
$64$	1.00000	0.125000
$65$	−3.05573	−0.379016
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	7.23607	0.877502
$69$	0	0
$70$	1.23607	0.147738
$71$	2.47214	0.293389	0.146694	−	0.989182i	$-0.453137\pi$
$71$	2.47214	0.293389	0.146694	+	0.989182i	$0.453137\pi$
$72$	0	0
$73$	−13.4164	−1.57027	−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164	−1.57027	−0.785136	+	0.619324i	$0.787407\pi$
$74$	6.00000	0.697486
$75$	0	0
$76$	−4.47214	−0.512989
$77$	0	0
$78$	0	0
$79$	8.94427	1.00631	0.503155	−	0.864196i	$-0.332173\pi$
$79$	8.94427	1.00631	0.503155	+	0.864196i	$0.332173\pi$
$80$	−1.23607	−0.138197
$81$	0	0
$82$	10.0000	1.10432
$83$	6.94427	0.762233	0.381116	−	0.924527i	$-0.375540\pi$
$83$	6.94427	0.762233	0.381116	+	0.924527i	$0.375540\pi$
$84$	0	0
$85$	−8.94427	−0.970143
$86$	−6.47214	−0.697908
$87$	0	0
$88$	0	0
$89$	16.1803	1.71511	0.857556	−	0.514390i	$-0.171982\pi$
$89$	16.1803	1.71511	0.857556	+	0.514390i	$0.171982\pi$
$90$	0	0
$91$	−2.47214	−0.259150
$92$	1.00000	0.104257
$93$	0	0
$94$	6.76393	0.697646
$95$	5.52786	0.567147
$96$	0	0
$97$	8.18034	0.830588	0.415294	−	0.909687i	$-0.363679\pi$
$97$	8.18034	0.830588	0.415294	+	0.909687i	$0.363679\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−3.47214	−0.347214
$101$	14.4721	1.44003	0.720016	−	0.693958i	$-0.244135\pi$
$101$	14.4721	1.44003	0.720016	+	0.693958i	$0.244135\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	2.47214	0.242413
$105$	0	0
$106$	6.00000	0.582772
$107$	10.4721	1.01238	0.506190	−	0.862422i	$-0.331054\pi$
$107$	10.4721	1.01238	0.506190	+	0.862422i	$0.331054\pi$
$108$	0	0
$109$	3.52786	0.337908	0.168954	−	0.985624i	$-0.445961\pi$
$109$	3.52786	0.337908	0.168954	+	0.985624i	$0.445961\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−3.52786	−0.331874	−0.165937	−	0.986136i	$-0.553065\pi$
$113$	−3.52786	−0.331874	−0.165937	+	0.986136i	$0.553065\pi$
$114$	0	0
$115$	−1.23607	−0.115264
$116$	2.00000	0.185695
$117$	0	0
$118$	−11.2361	−1.03436
$119$	−7.23607	−0.663329
$120$	0	0
$121$	−11.0000	−1.00000
$122$	11.7082	1.06001
$123$	0	0
$124$	−1.23607	−0.111002
$125$	10.4721	0.936656
$126$	0	0
$127$	7.41641	0.658100	0.329050	−	0.944313i	$-0.393272\pi$
$127$	7.41641	0.658100	0.329050	+	0.944313i	$0.393272\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−3.05573	−0.268005
$131$	12.7639	1.11519	0.557595	−	0.830113i	$-0.311724\pi$
$131$	12.7639	1.11519	0.557595	+	0.830113i	$0.311724\pi$
$132$	0	0
$133$	4.47214	0.387783
$134$	4.00000	0.345547
$135$	0	0
$136$	7.23607	0.620488
$137$	−17.4164	−1.48798	−0.743992	−	0.668188i	$-0.767070\pi$
$137$	−17.4164	−1.48798	−0.743992	+	0.668188i	$0.767070\pi$
$138$	0	0
$139$	17.1246	1.45249	0.726245	−	0.687436i	$-0.241264\pi$
$139$	17.1246	1.45249	0.726245	+	0.687436i	$0.241264\pi$
$140$	1.23607	0.104467
$141$	0	0
$142$	2.47214	0.207457
$143$	0	0
$144$	0	0
$145$	−2.47214	−0.205300
$146$	−13.4164	−1.11035
$147$	0	0
$148$	6.00000	0.493197
$149$	5.41641	0.443729	0.221865	−	0.975077i	$-0.428786\pi$
$149$	5.41641	0.443729	0.221865	+	0.975077i	$0.428786\pi$
$150$	0	0
$151$	−18.4721	−1.50324	−0.751621	−	0.659596i	$-0.770728\pi$
$151$	−18.4721	−1.50324	−0.751621	+	0.659596i	$0.770728\pi$
$152$	−4.47214	−0.362738
$153$	0	0
$154$	0	0
$155$	1.52786	0.122721
$156$	0	0
$157$	12.6525	1.00978	0.504889	−	0.863184i	$-0.331534\pi$
$157$	12.6525	1.00978	0.504889	+	0.863184i	$0.331534\pi$
$158$	8.94427	0.711568
$159$	0	0
$160$	−1.23607	−0.0977198
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	2.47214	0.193633	0.0968163	−	0.995302i	$-0.469134\pi$
$163$	2.47214	0.193633	0.0968163	+	0.995302i	$0.469134\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	6.94427	0.538980
$167$	−24.6525	−1.90767	−0.953833	−	0.300338i	$-0.902901\pi$
$167$	−24.6525	−1.90767	−0.953833	+	0.300338i	$0.902901\pi$
$168$	0	0
$169$	−6.88854	−0.529888
$170$	−8.94427	−0.685994
$171$	0	0
$172$	−6.47214	−0.493496
$173$	−20.9443	−1.59236	−0.796182	−	0.605058i	$-0.793150\pi$
$173$	−20.9443	−1.59236	−0.796182	+	0.605058i	$0.793150\pi$
$174$	0	0
$175$	3.47214	0.262469
$176$	0	0
$177$	0	0
$178$	16.1803	1.21277
$179$	0.944272	0.0705782	0.0352891	−	0.999377i	$-0.488765\pi$
$179$	0.944272	0.0705782	0.0352891	+	0.999377i	$0.488765\pi$
$180$	0	0
$181$	13.2361	0.983829	0.491915	−	0.870643i	$-0.336297\pi$
$181$	13.2361	0.983829	0.491915	+	0.870643i	$0.336297\pi$
$182$	−2.47214	−0.183247
$183$	0	0
$184$	1.00000	0.0737210
$185$	−7.41641	−0.545265
$186$	0	0
$187$	0	0
$188$	6.76393	0.493310
$189$	0	0
$190$	5.52786	0.401033
$191$	−20.9443	−1.51547	−0.757737	−	0.652560i	$-0.773695\pi$
$191$	−20.9443	−1.51547	−0.757737	+	0.652560i	$0.773695\pi$
$192$	0	0
$193$	−22.9443	−1.65156	−0.825782	−	0.563989i	$-0.809266\pi$
$193$	−22.9443	−1.65156	−0.825782	+	0.563989i	$0.809266\pi$
$194$	8.18034	0.587314
$195$	0	0
$196$	1.00000	0.0714286
$197$	−15.8885	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$197$	−15.8885	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	−3.47214	−0.245517
$201$	0	0
$202$	14.4721	1.01826
$203$	−2.00000	−0.140372
$204$	0	0
$205$	−12.3607	−0.863307
$206$	−4.00000	−0.278693
$207$	0	0
$208$	2.47214	0.171412
$209$	0	0
$210$	0	0
$211$	−0.944272	−0.0650064	−0.0325032	−	0.999472i	$-0.510348\pi$
$211$	−0.944272	−0.0650064	−0.0325032	+	0.999472i	$0.510348\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	10.4721	0.715860
$215$	8.00000	0.545595
$216$	0	0
$217$	1.23607	0.0839098
$218$	3.52786	0.238937
$219$	0	0
$220$	0	0
$221$	17.8885	1.20331
$222$	0	0
$223$	20.6525	1.38299	0.691496	−	0.722380i	$-0.256952\pi$
$223$	20.6525	1.38299	0.691496	+	0.722380i	$0.256952\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−3.52786	−0.234670
$227$	−5.41641	−0.359500	−0.179750	−	0.983712i	$-0.557529\pi$
$227$	−5.41641	−0.359500	−0.179750	+	0.983712i	$0.557529\pi$
$228$	0	0
$229$	−10.7639	−0.711301	−0.355650	−	0.934619i	$-0.615741\pi$
$229$	−10.7639	−0.711301	−0.355650	+	0.934619i	$0.615741\pi$
$230$	−1.23607	−0.0815039
$231$	0	0
$232$	2.00000	0.131306
$233$	7.52786	0.493167	0.246583	−	0.969122i	$-0.420692\pi$
$233$	7.52786	0.493167	0.246583	+	0.969122i	$0.420692\pi$
$234$	0	0
$235$	−8.36068	−0.545391
$236$	−11.2361	−0.731406
$237$	0	0
$238$	−7.23607	−0.469045
$239$	2.47214	0.159909	0.0799546	−	0.996799i	$-0.474522\pi$
$239$	2.47214	0.159909	0.0799546	+	0.996799i	$0.474522\pi$
$240$	0	0
$241$	13.7082	0.883023	0.441512	−	0.897256i	$-0.354442\pi$
$241$	13.7082	0.883023	0.441512	+	0.897256i	$0.354442\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	11.7082	0.749541
$245$	−1.23607	−0.0789695
$246$	0	0
$247$	−11.0557	−0.703459
$248$	−1.23607	−0.0784904
$249$	0	0
$250$	10.4721	0.662316
$251$	−4.47214	−0.282279	−0.141139	−	0.989990i	$-0.545077\pi$
$251$	−4.47214	−0.282279	−0.141139	+	0.989990i	$0.545077\pi$
$252$	0	0
$253$	0	0
$254$	7.41641	0.465347
$255$	0	0
$256$	1.00000	0.0625000
$257$	9.05573	0.564881	0.282440	−	0.959285i	$-0.408856\pi$
$257$	9.05573	0.564881	0.282440	+	0.959285i	$0.408856\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	−3.05573	−0.189508
$261$	0	0
$262$	12.7639	0.788558
$263$	−12.0000	−0.739952	−0.369976	−	0.929041i	$-0.620634\pi$
$263$	−12.0000	−0.739952	−0.369976	+	0.929041i	$0.620634\pi$
$264$	0	0
$265$	−7.41641	−0.455586
$266$	4.47214	0.274204
$267$	0	0
$268$	4.00000	0.244339
$269$	−20.9443	−1.27699	−0.638497	−	0.769624i	$-0.720444\pi$
$269$	−20.9443	−1.27699	−0.638497	+	0.769624i	$0.720444\pi$
$270$	0	0
$271$	−10.7639	−0.653862	−0.326931	−	0.945048i	$-0.606015\pi$
$271$	−10.7639	−0.653862	−0.326931	+	0.945048i	$0.606015\pi$
$272$	7.23607	0.438751
$273$	0	0
$274$	−17.4164	−1.05216
$275$	0	0
$276$	0	0
$277$	−15.8885	−0.954650	−0.477325	−	0.878727i	$-0.658394\pi$
$277$	−15.8885	−0.954650	−0.477325	+	0.878727i	$0.658394\pi$
$278$	17.1246	1.02707
$279$	0	0
$280$	1.23607	0.0738692
$281$	21.4164	1.27760	0.638798	−	0.769375i	$-0.279432\pi$
$281$	21.4164	1.27760	0.638798	+	0.769375i	$0.279432\pi$
$282$	0	0
$283$	−4.47214	−0.265841	−0.132920	−	0.991127i	$-0.542435\pi$
$283$	−4.47214	−0.265841	−0.132920	+	0.991127i	$0.542435\pi$
$284$	2.47214	0.146694
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	35.3607	2.08004
$290$	−2.47214	−0.145169
$291$	0	0
$292$	−13.4164	−0.785136
$293$	−11.1246	−0.649907	−0.324953	−	0.945730i	$-0.605349\pi$
$293$	−11.1246	−0.649907	−0.324953	+	0.945730i	$0.605349\pi$
$294$	0	0
$295$	13.8885	0.808622
$296$	6.00000	0.348743
$297$	0	0
$298$	5.41641	0.313764
$299$	2.47214	0.142967
$300$	0	0
$301$	6.47214	0.373048
$302$	−18.4721	−1.06295
$303$	0	0
$304$	−4.47214	−0.256495
$305$	−14.4721	−0.828672
$306$	0	0
$307$	24.7639	1.41335	0.706676	−	0.707537i	$-0.250194\pi$
$307$	24.7639	1.41335	0.706676	+	0.707537i	$0.250194\pi$
$308$	0	0
$309$	0	0
$310$	1.52786	0.0867768
$311$	15.1246	0.857638	0.428819	−	0.903390i	$-0.358930\pi$
$311$	15.1246	0.857638	0.428819	+	0.903390i	$0.358930\pi$
$312$	0	0
$313$	3.81966	0.215900	0.107950	−	0.994156i	$-0.465571\pi$
$313$	3.81966	0.215900	0.107950	+	0.994156i	$0.465571\pi$
$314$	12.6525	0.714021
$315$	0	0
$316$	8.94427	0.503155
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	0	0
$319$	0	0
$320$	−1.23607	−0.0690983
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	−32.3607	−1.80060
$324$	0	0
$325$	−8.58359	−0.476132
$326$	2.47214	0.136919
$327$	0	0
$328$	10.0000	0.552158
$329$	−6.76393	−0.372908
$330$	0	0
$331$	−21.5279	−1.18328	−0.591639	−	0.806203i	$-0.701519\pi$
$331$	−21.5279	−1.18328	−0.591639	+	0.806203i	$0.701519\pi$
$332$	6.94427	0.381116
$333$	0	0
$334$	−24.6525	−1.34892
$335$	−4.94427	−0.270134
$336$	0	0
$337$	−0.111456	−0.00607140	−0.00303570	−	0.999995i	$-0.500966\pi$
$337$	−0.111456	−0.00607140	−0.00303570	+	0.999995i	$0.500966\pi$
$338$	−6.88854	−0.374687
$339$	0	0
$340$	−8.94427	−0.485071
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−6.47214	−0.348954
$345$	0	0
$346$	−20.9443	−1.12597
$347$	20.3607	1.09302	0.546509	−	0.837453i	$-0.315956\pi$
$347$	20.3607	1.09302	0.546509	+	0.837453i	$0.315956\pi$
$348$	0	0
$349$	−20.9443	−1.12112	−0.560561	−	0.828113i	$-0.689414\pi$
$349$	−20.9443	−1.12112	−0.560561	+	0.828113i	$0.689414\pi$
$350$	3.47214	0.185593
$351$	0	0
$352$	0	0
$353$	8.47214	0.450926	0.225463	−	0.974252i	$-0.427610\pi$
$353$	8.47214	0.450926	0.225463	+	0.974252i	$0.427610\pi$
$354$	0	0
$355$	−3.05573	−0.162181
$356$	16.1803	0.857556
$357$	0	0
$358$	0.944272	0.0499063
$359$	−12.0000	−0.633336	−0.316668	−	0.948536i	$-0.602564\pi$
$359$	−12.0000	−0.633336	−0.316668	+	0.948536i	$0.602564\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	13.2361	0.695672
$363$	0	0
$364$	−2.47214	−0.129575
$365$	16.5836	0.868025
$366$	0	0
$367$	−20.3607	−1.06282	−0.531409	−	0.847115i	$-0.678337\pi$
$367$	−20.3607	−1.06282	−0.531409	+	0.847115i	$0.678337\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−7.41641	−0.385561
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−37.4164	−1.93735	−0.968674	−	0.248336i	$-0.920116\pi$
$373$	−37.4164	−1.93735	−0.968674	+	0.248336i	$0.920116\pi$
$374$	0	0
$375$	0	0
$376$	6.76393	0.348823
$377$	4.94427	0.254643
$378$	0	0
$379$	17.8885	0.918873	0.459436	−	0.888211i	$-0.348051\pi$
$379$	17.8885	0.918873	0.459436	+	0.888211i	$0.348051\pi$
$380$	5.52786	0.283573
$381$	0	0
$382$	−20.9443	−1.07160
$383$	27.4164	1.40091	0.700456	−	0.713695i	$-0.252980\pi$
$383$	27.4164	1.40091	0.700456	+	0.713695i	$0.252980\pi$
$384$	0	0
$385$	0	0
$386$	−22.9443	−1.16783
$387$	0	0
$388$	8.18034	0.415294
$389$	17.4164	0.883047	0.441523	−	0.897250i	$-0.354438\pi$
$389$	17.4164	0.883047	0.441523	+	0.897250i	$0.354438\pi$
$390$	0	0
$391$	7.23607	0.365944
$392$	1.00000	0.0505076
$393$	0	0
$394$	−15.8885	−0.800453
$395$	−11.0557	−0.556274
$396$	0	0
$397$	5.52786	0.277436	0.138718	−	0.990332i	$-0.455702\pi$
$397$	5.52786	0.277436	0.138718	+	0.990332i	$0.455702\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	−3.47214	−0.173607
$401$	−33.7771	−1.68675	−0.843374	−	0.537328i	$-0.819434\pi$
$401$	−33.7771	−1.68675	−0.843374	+	0.537328i	$0.819434\pi$
$402$	0	0
$403$	−3.05573	−0.152217
$404$	14.4721	0.720016
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	0	0
$409$	19.5279	0.965591	0.482795	−	0.875733i	$-0.339621\pi$
$409$	19.5279	0.965591	0.482795	+	0.875733i	$0.339621\pi$
$410$	−12.3607	−0.610450
$411$	0	0
$412$	−4.00000	−0.197066
$413$	11.2361	0.552891
$414$	0	0
$415$	−8.58359	−0.421352
$416$	2.47214	0.121206
$417$	0	0
$418$	0	0
$419$	10.3607	0.506152	0.253076	−	0.967446i	$-0.418558\pi$
$419$	10.3607	0.506152	0.253076	+	0.967446i	$0.418558\pi$
$420$	0	0
$421$	0.111456	0.00543204	0.00271602	−	0.999996i	$-0.499135\pi$
$421$	0.111456	0.00543204	0.00271602	+	0.999996i	$0.499135\pi$
$422$	−0.944272	−0.0459664
$423$	0	0
$424$	6.00000	0.291386
$425$	−25.1246	−1.21872
$426$	0	0
$427$	−11.7082	−0.566600
$428$	10.4721	0.506190
$429$	0	0
$430$	8.00000	0.385794
$431$	−3.05573	−0.147189	−0.0735946	−	0.997288i	$-0.523447\pi$
$431$	−3.05573	−0.147189	−0.0735946	+	0.997288i	$0.523447\pi$
$432$	0	0
$433$	11.2361	0.539971	0.269985	−	0.962864i	$-0.412981\pi$
$433$	11.2361	0.539971	0.269985	+	0.962864i	$0.412981\pi$
$434$	1.23607	0.0593332
$435$	0	0
$436$	3.52786	0.168954
$437$	−4.47214	−0.213931
$438$	0	0
$439$	−13.2361	−0.631723	−0.315862	−	0.948805i	$-0.602294\pi$
$439$	−13.2361	−0.631723	−0.315862	+	0.948805i	$0.602294\pi$
$440$	0	0
$441$	0	0
$442$	17.8885	0.850871
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	20.6525	0.977923
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	17.4164	0.821931	0.410966	−	0.911651i	$-0.365192\pi$
$449$	17.4164	0.821931	0.410966	+	0.911651i	$0.365192\pi$
$450$	0	0
$451$	0	0
$452$	−3.52786	−0.165937
$453$	0	0
$454$	−5.41641	−0.254205
$455$	3.05573	0.143255
$456$	0	0
$457$	−22.3607	−1.04599	−0.522994	−	0.852336i	$-0.675185\pi$
$457$	−22.3607	−1.04599	−0.522994	+	0.852336i	$0.675185\pi$
$458$	−10.7639	−0.502966
$459$	0	0
$460$	−1.23607	−0.0576320
$461$	−20.9443	−0.975472	−0.487736	−	0.872991i	$-0.662177\pi$
$461$	−20.9443	−0.975472	−0.487736	+	0.872991i	$0.662177\pi$
$462$	0	0
$463$	−28.9443	−1.34515	−0.672577	−	0.740027i	$-0.734813\pi$
$463$	−28.9443	−1.34515	−0.672577	+	0.740027i	$0.734813\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	7.52786	0.348722
$467$	20.4721	0.947337	0.473669	−	0.880703i	$-0.342929\pi$
$467$	20.4721	0.947337	0.473669	+	0.880703i	$0.342929\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−8.36068	−0.385649
$471$	0	0
$472$	−11.2361	−0.517182
$473$	0	0
$474$	0	0
$475$	15.5279	0.712467
$476$	−7.23607	−0.331665
$477$	0	0
$478$	2.47214	0.113073
$479$	−18.8328	−0.860493	−0.430247	−	0.902711i	$-0.641573\pi$
$479$	−18.8328	−0.860493	−0.430247	+	0.902711i	$0.641573\pi$
$480$	0	0
$481$	14.8328	0.676318
$482$	13.7082	0.624392
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−10.1115	−0.459138
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	11.7082	0.530005
$489$	0	0
$490$	−1.23607	−0.0558399
$491$	24.9443	1.12572	0.562860	−	0.826553i	$-0.309701\pi$
$491$	24.9443	1.12572	0.562860	+	0.826553i	$0.309701\pi$
$492$	0	0
$493$	14.4721	0.651792
$494$	−11.0557	−0.497421
$495$	0	0
$496$	−1.23607	−0.0555011
$497$	−2.47214	−0.110890
$498$	0	0
$499$	8.94427	0.400401	0.200200	−	0.979755i	$-0.435841\pi$
$499$	8.94427	0.400401	0.200200	+	0.979755i	$0.435841\pi$
$500$	10.4721	0.468328
$501$	0	0
$502$	−4.47214	−0.199601
$503$	−12.9443	−0.577157	−0.288578	−	0.957456i	$-0.593183\pi$
$503$	−12.9443	−0.577157	−0.288578	+	0.957456i	$0.593183\pi$
$504$	0	0
$505$	−17.8885	−0.796030
$506$	0	0
$507$	0	0
$508$	7.41641	0.329050
$509$	1.88854	0.0837082	0.0418541	−	0.999124i	$-0.486674\pi$
$509$	1.88854	0.0837082	0.0418541	+	0.999124i	$0.486674\pi$
$510$	0	0
$511$	13.4164	0.593507
$512$	1.00000	0.0441942
$513$	0	0
$514$	9.05573	0.399431
$515$	4.94427	0.217871
$516$	0	0
$517$	0	0
$518$	−6.00000	−0.263625
$519$	0	0
$520$	−3.05573	−0.134003
$521$	1.70820	0.0748378	0.0374189	−	0.999300i	$-0.488086\pi$
$521$	1.70820	0.0748378	0.0374189	+	0.999300i	$0.488086\pi$
$522$	0	0
$523$	37.4164	1.63611	0.818053	−	0.575143i	$-0.195054\pi$
$523$	37.4164	1.63611	0.818053	+	0.575143i	$0.195054\pi$
$524$	12.7639	0.557595
$525$	0	0
$526$	−12.0000	−0.523225
$527$	−8.94427	−0.389619
$528$	0	0
$529$	1.00000	0.0434783
$530$	−7.41641	−0.322148
$531$	0	0
$532$	4.47214	0.193892
$533$	24.7214	1.07080
$534$	0	0
$535$	−12.9443	−0.559630
$536$	4.00000	0.172774
$537$	0	0
$538$	−20.9443	−0.902972
$539$	0	0
$540$	0	0
$541$	−31.8885	−1.37100	−0.685498	−	0.728075i	$-0.740415\pi$
$541$	−31.8885	−1.37100	−0.685498	+	0.728075i	$0.740415\pi$
$542$	−10.7639	−0.462350
$543$	0	0
$544$	7.23607	0.310244
$545$	−4.36068	−0.186791
$546$	0	0
$547$	−13.5279	−0.578410	−0.289205	−	0.957267i	$-0.593391\pi$
$547$	−13.5279	−0.578410	−0.289205	+	0.957267i	$0.593391\pi$
$548$	−17.4164	−0.743992
$549$	0	0
$550$	0	0
$551$	−8.94427	−0.381039
$552$	0	0
$553$	−8.94427	−0.380349
$554$	−15.8885	−0.675040
$555$	0	0
$556$	17.1246	0.726245
$557$	−9.41641	−0.398986	−0.199493	−	0.979899i	$-0.563930\pi$
$557$	−9.41641	−0.398986	−0.199493	+	0.979899i	$0.563930\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	1.23607	0.0522334
$561$	0	0
$562$	21.4164	0.903397
$563$	−33.4164	−1.40833	−0.704167	−	0.710035i	$-0.748679\pi$
$563$	−33.4164	−1.40833	−0.704167	+	0.710035i	$0.748679\pi$
$564$	0	0
$565$	4.36068	0.183455
$566$	−4.47214	−0.187978
$567$	0	0
$568$	2.47214	0.103729
$569$	2.00000	0.0838444	0.0419222	−	0.999121i	$-0.486652\pi$
$569$	2.00000	0.0838444	0.0419222	+	0.999121i	$0.486652\pi$
$570$	0	0
$571$	29.3050	1.22637	0.613187	−	0.789938i	$-0.289887\pi$
$571$	29.3050	1.22637	0.613187	+	0.789938i	$0.289887\pi$
$572$	0	0
$573$	0	0
$574$	−10.0000	−0.417392
$575$	−3.47214	−0.144798
$576$	0	0
$577$	1.41641	0.0589658	0.0294829	−	0.999565i	$-0.490614\pi$
$577$	1.41641	0.0589658	0.0294829	+	0.999565i	$0.490614\pi$
$578$	35.3607	1.47081
$579$	0	0
$580$	−2.47214	−0.102650
$581$	−6.94427	−0.288097
$582$	0	0
$583$	0	0
$584$	−13.4164	−0.555175
$585$	0	0
$586$	−11.1246	−0.459553
$587$	−26.2918	−1.08518	−0.542589	−	0.839998i	$-0.682556\pi$
$587$	−26.2918	−1.08518	−0.542589	+	0.839998i	$0.682556\pi$
$588$	0	0
$589$	5.52786	0.227772
$590$	13.8885	0.571782
$591$	0	0
$592$	6.00000	0.246598
$593$	38.0000	1.56047	0.780236	−	0.625485i	$-0.215099\pi$
$593$	38.0000	1.56047	0.780236	+	0.625485i	$0.215099\pi$
$594$	0	0
$595$	8.94427	0.366679
$596$	5.41641	0.221865
$597$	0	0
$598$	2.47214	0.101093
$599$	14.8328	0.606052	0.303026	−	0.952982i	$-0.402003\pi$
$599$	14.8328	0.606052	0.303026	+	0.952982i	$0.402003\pi$
$600$	0	0
$601$	4.47214	0.182422	0.0912111	−	0.995832i	$-0.470926\pi$
$601$	4.47214	0.182422	0.0912111	+	0.995832i	$0.470926\pi$
$602$	6.47214	0.263785
$603$	0	0
$604$	−18.4721	−0.751621
$605$	13.5967	0.552786
$606$	0	0
$607$	5.23607	0.212525	0.106263	−	0.994338i	$-0.466112\pi$
$607$	5.23607	0.212525	0.106263	+	0.994338i	$0.466112\pi$
$608$	−4.47214	−0.181369
$609$	0	0
$610$	−14.4721	−0.585960
$611$	16.7214	0.676474
$612$	0	0
$613$	44.2492	1.78721	0.893605	−	0.448855i	$-0.148168\pi$
$613$	44.2492	1.78721	0.893605	+	0.448855i	$0.148168\pi$
$614$	24.7639	0.999391
$615$	0	0
$616$	0	0
$617$	−26.9443	−1.08474	−0.542368	−	0.840141i	$-0.682472\pi$
$617$	−26.9443	−1.08474	−0.542368	+	0.840141i	$0.682472\pi$
$618$	0	0
$619$	18.9443	0.761435	0.380717	−	0.924691i	$-0.375677\pi$
$619$	18.9443	0.761435	0.380717	+	0.924691i	$0.375677\pi$
$620$	1.52786	0.0613605
$621$	0	0
$622$	15.1246	0.606442
$623$	−16.1803	−0.648252
$624$	0	0
$625$	4.41641	0.176656
$626$	3.81966	0.152664
$627$	0	0
$628$	12.6525	0.504889
$629$	43.4164	1.73113
$630$	0	0
$631$	−37.8885	−1.50832	−0.754160	−	0.656691i	$-0.771955\pi$
$631$	−37.8885	−1.50832	−0.754160	+	0.656691i	$0.771955\pi$
$632$	8.94427	0.355784
$633$	0	0
$634$	−10.0000	−0.397151
$635$	−9.16718	−0.363789
$636$	0	0
$637$	2.47214	0.0979496
$638$	0	0
$639$	0	0
$640$	−1.23607	−0.0488599
$641$	−17.0557	−0.673661	−0.336830	−	0.941565i	$-0.609355\pi$
$641$	−17.0557	−0.673661	−0.336830	+	0.941565i	$0.609355\pi$
$642$	0	0
$643$	−36.2492	−1.42953	−0.714765	−	0.699365i	$-0.753466\pi$
$643$	−36.2492	−1.42953	−0.714765	+	0.699365i	$0.753466\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	−32.3607	−1.27321
$647$	−32.6525	−1.28370	−0.641851	−	0.766830i	$-0.721833\pi$
$647$	−32.6525	−1.28370	−0.641851	+	0.766830i	$0.721833\pi$
$648$	0	0
$649$	0	0
$650$	−8.58359	−0.336676
$651$	0	0
$652$	2.47214	0.0968163
$653$	41.7771	1.63486	0.817432	−	0.576025i	$-0.195397\pi$
$653$	41.7771	1.63486	0.817432	+	0.576025i	$0.195397\pi$
$654$	0	0
$655$	−15.7771	−0.616462
$656$	10.0000	0.390434
$657$	0	0
$658$	−6.76393	−0.263686
$659$	−34.2492	−1.33416	−0.667080	−	0.744986i	$-0.732456\pi$
$659$	−34.2492	−1.33416	−0.667080	+	0.744986i	$0.732456\pi$
$660$	0	0
$661$	−35.1246	−1.36619	−0.683095	−	0.730330i	$-0.739366\pi$
$661$	−35.1246	−1.36619	−0.683095	+	0.730330i	$0.739366\pi$
$662$	−21.5279	−0.836704
$663$	0	0
$664$	6.94427	0.269490
$665$	−5.52786	−0.214361
$666$	0	0
$667$	2.00000	0.0774403
$668$	−24.6525	−0.953833
$669$	0	0
$670$	−4.94427	−0.191014
$671$	0	0
$672$	0	0
$673$	−2.58359	−0.0995902	−0.0497951	−	0.998759i	$-0.515857\pi$
$673$	−2.58359	−0.0995902	−0.0497951	+	0.998759i	$0.515857\pi$
$674$	−0.111456	−0.00429313
$675$	0	0
$676$	−6.88854	−0.264944
$677$	−18.7639	−0.721156	−0.360578	−	0.932729i	$-0.617421\pi$
$677$	−18.7639	−0.721156	−0.360578	+	0.932729i	$0.617421\pi$
$678$	0	0
$679$	−8.18034	−0.313933
$680$	−8.94427	−0.342997
$681$	0	0
$682$	0	0
$683$	4.00000	0.153056	0.0765279	−	0.997067i	$-0.475617\pi$
$683$	4.00000	0.153056	0.0765279	+	0.997067i	$0.475617\pi$
$684$	0	0
$685$	21.5279	0.822537
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	−6.47214	−0.246748
$689$	14.8328	0.565085
$690$	0	0
$691$	−27.0132	−1.02763	−0.513814	−	0.857901i	$-0.671768\pi$
$691$	−27.0132	−1.02763	−0.513814	+	0.857901i	$0.671768\pi$
$692$	−20.9443	−0.796182
$693$	0	0
$694$	20.3607	0.772881
$695$	−21.1672	−0.802917
$696$	0	0
$697$	72.3607	2.74086
$698$	−20.9443	−0.792752
$699$	0	0
$700$	3.47214	0.131234
$701$	38.9443	1.47090	0.735452	−	0.677576i	$-0.236970\pi$
$701$	38.9443	1.47090	0.735452	+	0.677576i	$0.236970\pi$
$702$	0	0
$703$	−26.8328	−1.01202
$704$	0	0
$705$	0	0
$706$	8.47214	0.318853
$707$	−14.4721	−0.544281
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	−3.05573	−0.114679
$711$	0	0
$712$	16.1803	0.606384
$713$	−1.23607	−0.0462911
$714$	0	0
$715$	0	0
$716$	0.944272	0.0352891
$717$	0	0
$718$	−12.0000	−0.447836
$719$	−34.5410	−1.28816	−0.644081	−	0.764957i	$-0.722760\pi$
$719$	−34.5410	−1.28816	−0.644081	+	0.764957i	$0.722760\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	1.00000	0.0372161
$723$	0	0
$724$	13.2361	0.491915
$725$	−6.94427	−0.257904
$726$	0	0
$727$	24.9443	0.925132	0.462566	−	0.886585i	$-0.346929\pi$
$727$	24.9443	0.925132	0.462566	+	0.886585i	$0.346929\pi$
$728$	−2.47214	−0.0916235
$729$	0	0
$730$	16.5836	0.613786
$731$	−46.8328	−1.73217
$732$	0	0
$733$	−45.5967	−1.68415	−0.842077	−	0.539357i	$-0.818667\pi$
$733$	−45.5967	−1.68415	−0.842077	+	0.539357i	$0.818667\pi$
$734$	−20.3607	−0.751526
$735$	0	0
$736$	1.00000	0.0368605
$737$	0	0
$738$	0	0
$739$	−20.3607	−0.748980	−0.374490	−	0.927231i	$-0.622182\pi$
$739$	−20.3607	−0.748980	−0.374490	+	0.927231i	$0.622182\pi$
$740$	−7.41641	−0.272633
$741$	0	0
$742$	−6.00000	−0.220267
$743$	31.0557	1.13932	0.569662	−	0.821879i	$-0.307074\pi$
$743$	31.0557	1.13932	0.569662	+	0.821879i	$0.307074\pi$
$744$	0	0
$745$	−6.69505	−0.245288
$746$	−37.4164	−1.36991
$747$	0	0
$748$	0	0
$749$	−10.4721	−0.382644
$750$	0	0
$751$	−23.0557	−0.841315	−0.420658	−	0.907219i	$-0.638201\pi$
$751$	−23.0557	−0.841315	−0.420658	+	0.907219i	$0.638201\pi$
$752$	6.76393	0.246655
$753$	0	0
$754$	4.94427	0.180060
$755$	22.8328	0.830971
$756$	0	0
$757$	32.4721	1.18022	0.590110	−	0.807323i	$-0.299084\pi$
$757$	32.4721	1.18022	0.590110	+	0.807323i	$0.299084\pi$
$758$	17.8885	0.649741
$759$	0	0
$760$	5.52786	0.200517
$761$	−8.47214	−0.307115	−0.153557	−	0.988140i	$-0.549073\pi$
$761$	−8.47214	−0.307115	−0.153557	+	0.988140i	$0.549073\pi$
$762$	0	0
$763$	−3.52786	−0.127717
$764$	−20.9443	−0.757737
$765$	0	0
$766$	27.4164	0.990595
$767$	−27.7771	−1.00297
$768$	0	0
$769$	16.7639	0.604523	0.302261	−	0.953225i	$-0.402258\pi$
$769$	16.7639	0.604523	0.302261	+	0.953225i	$0.402258\pi$
$770$	0	0
$771$	0	0
$772$	−22.9443	−0.825782
$773$	6.18034	0.222291	0.111146	−	0.993804i	$-0.464548\pi$
$773$	6.18034	0.222291	0.111146	+	0.993804i	$0.464548\pi$
$774$	0	0
$775$	4.29180	0.154166
$776$	8.18034	0.293657
$777$	0	0
$778$	17.4164	0.624408
$779$	−44.7214	−1.60231
$780$	0	0
$781$	0	0
$782$	7.23607	0.258761
$783$	0	0
$784$	1.00000	0.0357143
$785$	−15.6393	−0.558191
$786$	0	0
$787$	−0.111456	−0.00397298	−0.00198649	−	0.999998i	$-0.500632\pi$
$787$	−0.111456	−0.00397298	−0.00198649	+	0.999998i	$0.500632\pi$
$788$	−15.8885	−0.566006
$789$	0	0
$790$	−11.0557	−0.393345
$791$	3.52786	0.125436
$792$	0	0
$793$	28.9443	1.02784
$794$	5.52786	0.196177
$795$	0	0
$796$	−12.0000	−0.425329
$797$	44.6525	1.58167	0.790836	−	0.612028i	$-0.209646\pi$
$797$	44.6525	1.58167	0.790836	+	0.612028i	$0.209646\pi$
$798$	0	0
$799$	48.9443	1.73152
$800$	−3.47214	−0.122759
$801$	0	0
$802$	−33.7771	−1.19271
$803$	0	0
$804$	0	0
$805$	1.23607	0.0435657
$806$	−3.05573	−0.107633
$807$	0	0
$808$	14.4721	0.509128
$809$	−20.8328	−0.732443	−0.366221	−	0.930528i	$-0.619349\pi$
$809$	−20.8328	−0.732443	−0.366221	+	0.930528i	$0.619349\pi$
$810$	0	0
$811$	−27.5967	−0.969053	−0.484526	−	0.874777i	$-0.661008\pi$
$811$	−27.5967	−0.969053	−0.484526	+	0.874777i	$0.661008\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	0	0
$815$	−3.05573	−0.107037
$816$	0	0
$817$	28.9443	1.01263
$818$	19.5279	0.682776
$819$	0	0
$820$	−12.3607	−0.431654
$821$	−14.9443	−0.521559	−0.260779	−	0.965398i	$-0.583980\pi$
$821$	−14.9443	−0.521559	−0.260779	+	0.965398i	$0.583980\pi$
$822$	0	0
$823$	15.4164	0.537382	0.268691	−	0.963226i	$-0.413409\pi$
$823$	15.4164	0.537382	0.268691	+	0.963226i	$0.413409\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	11.2361	0.390953
$827$	39.4164	1.37064	0.685321	−	0.728241i	$-0.259662\pi$
$827$	39.4164	1.37064	0.685321	+	0.728241i	$0.259662\pi$
$828$	0	0
$829$	−21.8885	−0.760221	−0.380110	−	0.924941i	$-0.624114\pi$
$829$	−21.8885	−0.760221	−0.380110	+	0.924941i	$0.624114\pi$
$830$	−8.58359	−0.297941
$831$	0	0
$832$	2.47214	0.0857059
$833$	7.23607	0.250715
$834$	0	0
$835$	30.4721	1.05453
$836$	0	0
$837$	0	0
$838$	10.3607	0.357904
$839$	16.3607	0.564833	0.282417	−	0.959292i	$-0.408864\pi$
$839$	16.3607	0.564833	0.282417	+	0.959292i	$0.408864\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0.111456	0.00384103
$843$	0	0
$844$	−0.944272	−0.0325032
$845$	8.51471	0.292915
$846$	0	0
$847$	11.0000	0.377964
$848$	6.00000	0.206041
$849$	0	0
$850$	−25.1246	−0.861767
$851$	6.00000	0.205677
$852$	0	0
$853$	−15.4164	−0.527848	−0.263924	−	0.964544i	$-0.585017\pi$
$853$	−15.4164	−0.527848	−0.263924	+	0.964544i	$0.585017\pi$
$854$	−11.7082	−0.400646
$855$	0	0
$856$	10.4721	0.357930
$857$	−23.8885	−0.816017	−0.408009	−	0.912978i	$-0.633777\pi$
$857$	−23.8885	−0.816017	−0.408009	+	0.912978i	$0.633777\pi$
$858$	0	0
$859$	7.23607	0.246891	0.123446	−	0.992351i	$-0.460606\pi$
$859$	7.23607	0.246891	0.123446	+	0.992351i	$0.460606\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−3.05573	−0.104079
$863$	−23.4164	−0.797104	−0.398552	−	0.917146i	$-0.630487\pi$
$863$	−23.4164	−0.797104	−0.398552	+	0.917146i	$0.630487\pi$
$864$	0	0
$865$	25.8885	0.880237
$866$	11.2361	0.381817
$867$	0	0
$868$	1.23607	0.0419549
$869$	0	0
$870$	0	0
$871$	9.88854	0.335061
$872$	3.52786	0.119469
$873$	0	0
$874$	−4.47214	−0.151272
$875$	−10.4721	−0.354023
$876$	0	0
$877$	9.05573	0.305790	0.152895	−	0.988242i	$-0.451140\pi$
$877$	9.05573	0.305790	0.152895	+	0.988242i	$0.451140\pi$
$878$	−13.2361	−0.446696
$879$	0	0
$880$	0	0
$881$	16.7639	0.564791	0.282396	−	0.959298i	$-0.408871\pi$
$881$	16.7639	0.564791	0.282396	+	0.959298i	$0.408871\pi$
$882$	0	0
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	17.8885	0.601657
$885$	0	0
$886$	−4.00000	−0.134383
$887$	−13.5967	−0.456534	−0.228267	−	0.973599i	$-0.573306\pi$
$887$	−13.5967	−0.456534	−0.228267	+	0.973599i	$0.573306\pi$
$888$	0	0
$889$	−7.41641	−0.248738
$890$	−20.0000	−0.670402
$891$	0	0
$892$	20.6525	0.691496
$893$	−30.2492	−1.01225
$894$	0	0
$895$	−1.16718	−0.0390147
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	17.4164	0.581193
$899$	−2.47214	−0.0824504
$900$	0	0
$901$	43.4164	1.44641
$902$	0	0
$903$	0	0
$904$	−3.52786	−0.117335
$905$	−16.3607	−0.543847
$906$	0	0
$907$	−10.4721	−0.347722	−0.173861	−	0.984770i	$-0.555624\pi$
$907$	−10.4721	−0.347722	−0.173861	+	0.984770i	$0.555624\pi$
$908$	−5.41641	−0.179750
$909$	0	0
$910$	3.05573	0.101296
$911$	13.8885	0.460148	0.230074	−	0.973173i	$-0.426103\pi$
$911$	13.8885	0.460148	0.230074	+	0.973173i	$0.426103\pi$
$912$	0	0
$913$	0	0
$914$	−22.3607	−0.739626
$915$	0	0
$916$	−10.7639	−0.355650
$917$	−12.7639	−0.421502
$918$	0	0
$919$	−45.8885	−1.51372	−0.756862	−	0.653575i	$-0.773268\pi$
$919$	−45.8885	−1.51372	−0.756862	+	0.653575i	$0.773268\pi$
$920$	−1.23607	−0.0407520
$921$	0	0
$922$	−20.9443	−0.689763
$923$	6.11146	0.201161
$924$	0	0
$925$	−20.8328	−0.684979
$926$	−28.9443	−0.951168
$927$	0	0
$928$	2.00000	0.0656532
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	−4.47214	−0.146568
$932$	7.52786	0.246583
$933$	0	0
$934$	20.4721	0.669869
$935$	0	0
$936$	0	0
$937$	55.2361	1.80448	0.902242	−	0.431230i	$-0.141920\pi$
$937$	55.2361	1.80448	0.902242	+	0.431230i	$0.141920\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−8.36068	−0.272695
$941$	29.5967	0.964826	0.482413	−	0.875944i	$-0.339760\pi$
$941$	29.5967	0.964826	0.482413	+	0.875944i	$0.339760\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	−11.2361	−0.365703
$945$	0	0
$946$	0	0
$947$	38.2492	1.24293	0.621466	−	0.783441i	$-0.286537\pi$
$947$	38.2492	1.24293	0.621466	+	0.783441i	$0.286537\pi$
$948$	0	0
$949$	−33.1672	−1.07665
$950$	15.5279	0.503790
$951$	0	0
$952$	−7.23607	−0.234522
$953$	25.7771	0.835002	0.417501	−	0.908677i	$-0.362906\pi$
$953$	25.7771	0.835002	0.417501	+	0.908677i	$0.362906\pi$
$954$	0	0
$955$	25.8885	0.837734
$956$	2.47214	0.0799546
$957$	0	0
$958$	−18.8328	−0.608461
$959$	17.4164	0.562405
$960$	0	0
$961$	−29.4721	−0.950714
$962$	14.8328	0.478229
$963$	0	0
$964$	13.7082	0.441512
$965$	28.3607	0.912963
$966$	0	0
$967$	25.8885	0.832519	0.416260	−	0.909246i	$-0.363341\pi$
$967$	25.8885	0.832519	0.416260	+	0.909246i	$0.363341\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−10.1115	−0.324659
$971$	0.111456	0.00357680	0.00178840	−	0.999998i	$-0.499431\pi$
$971$	0.111456	0.00357680	0.00178840	+	0.999998i	$0.499431\pi$
$972$	0	0
$973$	−17.1246	−0.548990
$974$	0	0
$975$	0	0
$976$	11.7082	0.374770
$977$	3.52786	0.112866	0.0564332	−	0.998406i	$-0.482027\pi$
$977$	3.52786	0.112866	0.0564332	+	0.998406i	$0.482027\pi$
$978$	0	0
$979$	0	0
$980$	−1.23607	−0.0394847
$981$	0	0
$982$	24.9443	0.796004
$983$	22.4721	0.716750	0.358375	−	0.933578i	$-0.383331\pi$
$983$	22.4721	0.716750	0.358375	+	0.933578i	$0.383331\pi$
$984$	0	0
$985$	19.6393	0.625761
$986$	14.4721	0.460887
$987$	0	0
$988$	−11.0557	−0.351730
$989$	−6.47214	−0.205802
$990$	0	0
$991$	−41.3050	−1.31210	−0.656048	−	0.754720i	$-0.727773\pi$
$991$	−41.3050	−1.31210	−0.656048	+	0.754720i	$0.727773\pi$
$992$	−1.23607	−0.0392452
$993$	0	0
$994$	−2.47214	−0.0784114
$995$	14.8328	0.470232
$996$	0	0
$997$	−42.8328	−1.35653	−0.678264	−	0.734818i	$-0.737268\pi$
$997$	−42.8328	−1.35653	−0.678264	+	0.734818i	$0.737268\pi$
$998$	8.94427	0.283126
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.bd.1.1		2
3.2	odd	2		322.2.a.e.1.1	✓	2
12.11	even	2		2576.2.a.t.1.2		2
15.14	odd	2		8050.2.a.bf.1.2		2
21.20	even	2		2254.2.a.k.1.2		2
69.68	even	2		7406.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
322.2.a.e.1.1	✓	2	3.2	odd	2
2254.2.a.k.1.2		2	21.20	even	2
2576.2.a.t.1.2		2	12.11	even	2
2898.2.a.bd.1.1		2	1.1	even	1	trivial
7406.2.a.j.1.1		2	69.68	even	2
8050.2.a.bf.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2898 = 2 \cdot 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2898.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} -1.23607 q^{10} +2.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} +7.23607 q^{17} -4.47214 q^{19} -1.23607 q^{20} +1.00000 q^{23} -3.47214 q^{25} +2.47214 q^{26} -1.00000 q^{28} +2.00000 q^{29} -1.23607 q^{31} +1.00000 q^{32} +7.23607 q^{34} +1.23607 q^{35} +6.00000 q^{37} -4.47214 q^{38} -1.23607 q^{40} +10.0000 q^{41} -6.47214 q^{43} +1.00000 q^{46} +6.76393 q^{47} +1.00000 q^{49} -3.47214 q^{50} +2.47214 q^{52} +6.00000 q^{53} -1.00000 q^{56} +2.00000 q^{58} -11.2361 q^{59} +11.7082 q^{61} -1.23607 q^{62} +1.00000 q^{64} -3.05573 q^{65} +4.00000 q^{67} +7.23607 q^{68} +1.23607 q^{70} +2.47214 q^{71} -13.4164 q^{73} +6.00000 q^{74} -4.47214 q^{76} +8.94427 q^{79} -1.23607 q^{80} +10.0000 q^{82} +6.94427 q^{83} -8.94427 q^{85} -6.47214 q^{86} +16.1803 q^{89} -2.47214 q^{91} +1.00000 q^{92} +6.76393 q^{94} +5.52786 q^{95} +8.18034 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8} + 2 q^{10} - 4 q^{13} - 2 q^{14} + 2 q^{16} + 10 q^{17} + 2 q^{20} + 2 q^{23} + 2 q^{25} - 4 q^{26} - 2 q^{28} + 4 q^{29} + 2 q^{31} + 2 q^{32} + 10 q^{34} - 2 q^{35} + 12 q^{37} + 2 q^{40} + 20 q^{41} - 4 q^{43} + 2 q^{46} + 18 q^{47} + 2 q^{49} + 2 q^{50} - 4 q^{52} + 12 q^{53} - 2 q^{56} + 4 q^{58} - 18 q^{59} + 10 q^{61} + 2 q^{62} + 2 q^{64} - 24 q^{65} + 8 q^{67} + 10 q^{68} - 2 q^{70} - 4 q^{71} + 12 q^{74} + 2 q^{80} + 20 q^{82} - 4 q^{83} - 4 q^{86} + 10 q^{89} + 4 q^{91} + 2 q^{92} + 18 q^{94} + 20 q^{95} - 6 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^7 + 2 * q^8 + 2 * q^10 - 4 * q^13 - 2 * q^14 + 2 * q^16 + 10 * q^17 + 2 * q^20 + 2 * q^23 + 2 * q^25 - 4 * q^26 - 2 * q^28 + 4 * q^29 + 2 * q^31 + 2 * q^32 + 10 * q^34 - 2 * q^35 + 12 * q^37 + 2 * q^40 + 20 * q^41 - 4 * q^43 + 2 * q^46 + 18 * q^47 + 2 * q^49 + 2 * q^50 - 4 * q^52 + 12 * q^53 - 2 * q^56 + 4 * q^58 - 18 * q^59 + 10 * q^61 + 2 * q^62 + 2 * q^64 - 24 * q^65 + 8 * q^67 + 10 * q^68 - 2 * q^70 - 4 * q^71 + 12 * q^74 + 2 * q^80 + 20 * q^82 - 4 * q^83 - 4 * q^86 + 10 * q^89 + 4 * q^91 + 2 * q^92 + 18 * q^94 + 20 * q^95 - 6 * q^97 + 2 * q^98

Introduction

L-functions

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