LMFDB - Embedded newform 2898.2.a.bd.1.2

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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.23607 q^{10} -6.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.76393 q^{17} +4.47214 q^{19} +3.23607 q^{20} +1.00000 q^{23} +5.47214 q^{25} -6.47214 q^{26} -1.00000 q^{28} +2.00000 q^{29} +3.23607 q^{31} +1.00000 q^{32} +2.76393 q^{34} -3.23607 q^{35} +6.00000 q^{37} +4.47214 q^{38} +3.23607 q^{40} +10.0000 q^{41} +2.47214 q^{43} +1.00000 q^{46} +11.2361 q^{47} +1.00000 q^{49} +5.47214 q^{50} -6.47214 q^{52} +6.00000 q^{53} -1.00000 q^{56} +2.00000 q^{58} -6.76393 q^{59} -1.70820 q^{61} +3.23607 q^{62} +1.00000 q^{64} -20.9443 q^{65} +4.00000 q^{67} +2.76393 q^{68} -3.23607 q^{70} -6.47214 q^{71} +13.4164 q^{73} +6.00000 q^{74} +4.47214 q^{76} -8.94427 q^{79} +3.23607 q^{80} +10.0000 q^{82} -10.9443 q^{83} +8.94427 q^{85} +2.47214 q^{86} -6.18034 q^{89} +6.47214 q^{91} +1.00000 q^{92} +11.2361 q^{94} +14.4721 q^{95} -14.1803 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	3.23607	1.02333
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	−6.47214	−1.79505	−0.897524	−	0.440966i	$-0.854636\pi$
$13$	−6.47214	−1.79505	−0.897524	+	0.440966i	$0.854636\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	2.76393	0.670352	0.335176	−	0.942156i	$-0.391204\pi$
$17$	2.76393	0.670352	0.335176	+	0.942156i	$0.391204\pi$
$18$	0	0
$19$	4.47214	1.02598	0.512989	−	0.858395i	$-0.328538\pi$
$19$	4.47214	1.02598	0.512989	+	0.858395i	$0.328538\pi$
$20$	3.23607	0.723607
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	−6.47214	−1.26929
$27$	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
$31$	3.23607	0.581215	0.290607	−	0.956842i	$-0.406143\pi$
$31$	3.23607	0.581215	0.290607	+	0.956842i	$0.406143\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	2.76393	0.474010
$35$	−3.23607	−0.546995
$36$	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
$40$	3.23607	0.511667
$41$	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
$43$	2.47214	0.376997	0.188499	−	0.982073i	$-0.439638\pi$
$43$	2.47214	0.376997	0.188499	+	0.982073i	$0.439638\pi$
$44$	0	0
$45$	0	0
$46$	1.00000	0.147442
$47$	11.2361	1.63895	0.819474	−	0.573116i	$-0.194265\pi$
$47$	11.2361	1.63895	0.819474	+	0.573116i	$0.194265\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	5.47214	0.773877
$51$	0	0
$52$	−6.47214	−0.897524
$53$	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
$59$	−6.76393	−0.880589	−0.440294	−	0.897853i	$-0.645126\pi$
$59$	−6.76393	−0.880589	−0.440294	+	0.897853i	$0.645126\pi$
$60$	0	0
$61$	−1.70820	−0.218713	−0.109357	−	0.994003i	$-0.534879\pi$
$61$	−1.70820	−0.218713	−0.109357	+	0.994003i	$0.534879\pi$
$62$	3.23607	0.410981
$63$	0	0
$64$	1.00000	0.125000
$65$	−20.9443	−2.59782
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	2.76393	0.335176
$69$	0	0
$70$	−3.23607	−0.386784
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	13.4164	1.57027	0.785136	−	0.619324i	$-0.212593\pi$
$73$	13.4164	1.57027	0.785136	+	0.619324i	$0.212593\pi$
$74$	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	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	3.23607	0.361803
$81$	0	0
$82$	10.0000	1.10432
$83$	−10.9443	−1.20129	−0.600645	−	0.799516i	$-0.705089\pi$
$83$	−10.9443	−1.20129	−0.600645	+	0.799516i	$0.705089\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	2.47214	0.266577
$87$	0	0
$88$	0	0
$89$	−6.18034	−0.655115	−0.327557	−	0.944831i	$-0.606225\pi$
$89$	−6.18034	−0.655115	−0.327557	+	0.944831i	$0.606225\pi$
$90$	0	0
$91$	6.47214	0.678464
$92$	1.00000	0.104257
$93$	0	0
$94$	11.2361	1.15891
$95$	14.4721	1.48481
$96$	0	0
$97$	−14.1803	−1.43980	−0.719898	−	0.694080i	$-0.755811\pi$
$97$	−14.1803	−1.43980	−0.719898	+	0.694080i	$0.755811\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	5.47214	0.547214
$101$	5.52786	0.550043	0.275022	−	0.961438i	$-0.411315\pi$
$101$	5.52786	0.550043	0.275022	+	0.961438i	$0.411315\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−6.47214	−0.634645
$105$	0	0
$106$	6.00000	0.582772
$107$	1.52786	0.147704	0.0738521	−	0.997269i	$-0.476471\pi$
$107$	1.52786	0.147704	0.0738521	+	0.997269i	$0.476471\pi$
$108$	0	0
$109$	12.4721	1.19461	0.597307	−	0.802013i	$-0.296237\pi$
$109$	12.4721	1.19461	0.597307	+	0.802013i	$0.296237\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−12.4721	−1.17328	−0.586640	−	0.809848i	$-0.699550\pi$
$113$	−12.4721	−1.17328	−0.586640	+	0.809848i	$0.699550\pi$
$114$	0	0
$115$	3.23607	0.301765
$116$	2.00000	0.185695
$117$	0	0
$118$	−6.76393	−0.622670
$119$	−2.76393	−0.253369
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−1.70820	−0.154654
$123$	0	0
$124$	3.23607	0.290607
$125$	1.52786	0.136656
$126$	0	0
$127$	−19.4164	−1.72293	−0.861464	−	0.507819i	$-0.830452\pi$
$127$	−19.4164	−1.72293	−0.861464	+	0.507819i	$0.830452\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−20.9443	−1.83693
$131$	17.2361	1.50592	0.752961	−	0.658065i	$-0.228625\pi$
$131$	17.2361	1.50592	0.752961	+	0.658065i	$0.228625\pi$
$132$	0	0
$133$	−4.47214	−0.387783
$134$	4.00000	0.345547
$135$	0	0
$136$	2.76393	0.237005
$137$	9.41641	0.804498	0.402249	−	0.915530i	$-0.368229\pi$
$137$	9.41641	0.804498	0.402249	+	0.915530i	$0.368229\pi$
$138$	0	0
$139$	−23.1246	−1.96140	−0.980702	−	0.195509i	$-0.937364\pi$
$139$	−23.1246	−1.96140	−0.980702	+	0.195509i	$0.937364\pi$
$140$	−3.23607	−0.273498
$141$	0	0
$142$	−6.47214	−0.543130
$143$	0	0
$144$	0	0
$145$	6.47214	0.537482
$146$	13.4164	1.11035
$147$	0	0
$148$	6.00000	0.493197
$149$	−21.4164	−1.75450	−0.877250	−	0.480033i	$-0.840625\pi$
$149$	−21.4164	−1.75450	−0.877250	+	0.480033i	$0.840625\pi$
$150$	0	0
$151$	−9.52786	−0.775367	−0.387683	−	0.921793i	$-0.626725\pi$
$151$	−9.52786	−0.775367	−0.387683	+	0.921793i	$0.626725\pi$
$152$	4.47214	0.362738
$153$	0	0
$154$	0	0
$155$	10.4721	0.841142
$156$	0	0
$157$	−18.6525	−1.48863	−0.744315	−	0.667829i	$-0.767224\pi$
$157$	−18.6525	−1.48863	−0.744315	+	0.667829i	$0.767224\pi$
$158$	−8.94427	−0.711568
$159$	0	0
$160$	3.23607	0.255834
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−6.47214	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$163$	−6.47214	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$164$	10.0000	0.780869
$165$	0	0
$166$	−10.9443	−0.849440
$167$	6.65248	0.514784	0.257392	−	0.966307i	$-0.417137\pi$
$167$	6.65248	0.514784	0.257392	+	0.966307i	$0.417137\pi$
$168$	0	0
$169$	28.8885	2.22220
$170$	8.94427	0.685994
$171$	0	0
$172$	2.47214	0.188499
$173$	−3.05573	−0.232323	−0.116161	−	0.993230i	$-0.537059\pi$
$173$	−3.05573	−0.232323	−0.116161	+	0.993230i	$0.537059\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	0	0
$177$	0	0
$178$	−6.18034	−0.463236
$179$	−16.9443	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$179$	−16.9443	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$180$	0	0
$181$	8.76393	0.651418	0.325709	−	0.945470i	$-0.394397\pi$
$181$	8.76393	0.651418	0.325709	+	0.945470i	$0.394397\pi$
$182$	6.47214	0.479747
$183$	0	0
$184$	1.00000	0.0737210
$185$	19.4164	1.42752
$186$	0	0
$187$	0	0
$188$	11.2361	0.819474
$189$	0	0
$190$	14.4721	1.04992
$191$	−3.05573	−0.221105	−0.110552	−	0.993870i	$-0.535262\pi$
$191$	−3.05573	−0.221105	−0.110552	+	0.993870i	$0.535262\pi$
$192$	0	0
$193$	−5.05573	−0.363919	−0.181960	−	0.983306i	$-0.558244\pi$
$193$	−5.05573	−0.363919	−0.181960	+	0.983306i	$0.558244\pi$
$194$	−14.1803	−1.01809
$195$	0	0
$196$	1.00000	0.0714286
$197$	19.8885	1.41700	0.708500	−	0.705711i	$-0.249372\pi$
$197$	19.8885	1.41700	0.708500	+	0.705711i	$0.249372\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	5.47214	0.386938
$201$	0	0
$202$	5.52786	0.388939
$203$	−2.00000	−0.140372
$204$	0	0
$205$	32.3607	2.26017
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−6.47214	−0.448762
$209$	0	0
$210$	0	0
$211$	16.9443	1.16649	0.583246	−	0.812296i	$-0.301782\pi$
$211$	16.9443	1.16649	0.583246	+	0.812296i	$0.301782\pi$
$212$	6.00000	0.412082
$213$	0	0
$214$	1.52786	0.104443
$215$	8.00000	0.545595
$216$	0	0
$217$	−3.23607	−0.219679
$218$	12.4721	0.844720
$219$	0	0
$220$	0	0
$221$	−17.8885	−1.20331
$222$	0	0
$223$	−10.6525	−0.713343	−0.356671	−	0.934230i	$-0.616088\pi$
$223$	−10.6525	−0.713343	−0.356671	+	0.934230i	$0.616088\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−12.4721	−0.829634
$227$	21.4164	1.42146	0.710728	−	0.703466i	$-0.248365\pi$
$227$	21.4164	1.42146	0.710728	+	0.703466i	$0.248365\pi$
$228$	0	0
$229$	−15.2361	−1.00683	−0.503414	−	0.864045i	$-0.667923\pi$
$229$	−15.2361	−1.00683	−0.503414	+	0.864045i	$0.667923\pi$
$230$	3.23607	0.213380
$231$	0	0
$232$	2.00000	0.131306
$233$	16.4721	1.07913	0.539563	−	0.841945i	$-0.318590\pi$
$233$	16.4721	1.07913	0.539563	+	0.841945i	$0.318590\pi$
$234$	0	0
$235$	36.3607	2.37191
$236$	−6.76393	−0.440294
$237$	0	0
$238$	−2.76393	−0.179159
$239$	−6.47214	−0.418648	−0.209324	−	0.977846i	$-0.567126\pi$
$239$	−6.47214	−0.418648	−0.209324	+	0.977846i	$0.567126\pi$
$240$	0	0
$241$	0.291796	0.0187962	0.00939812	−	0.999956i	$-0.497008\pi$
$241$	0.291796	0.0187962	0.00939812	+	0.999956i	$0.497008\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	−1.70820	−0.109357
$245$	3.23607	0.206745
$246$	0	0
$247$	−28.9443	−1.84168
$248$	3.23607	0.205491
$249$	0	0
$250$	1.52786	0.0966306
$251$	4.47214	0.282279	0.141139	−	0.989990i	$-0.454923\pi$
$251$	4.47214	0.282279	0.141139	+	0.989990i	$0.454923\pi$
$252$	0	0
$253$	0	0
$254$	−19.4164	−1.21829
$255$	0	0
$256$	1.00000	0.0625000
$257$	26.9443	1.68074	0.840369	−	0.542015i	$-0.182338\pi$
$257$	26.9443	1.68074	0.840369	+	0.542015i	$0.182338\pi$
$258$	0	0
$259$	−6.00000	−0.372822
$260$	−20.9443	−1.29891
$261$	0	0
$262$	17.2361	1.06485
$263$	−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
$265$	19.4164	1.19274
$266$	−4.47214	−0.274204
$267$	0	0
$268$	4.00000	0.244339
$269$	−3.05573	−0.186311	−0.0931555	−	0.995652i	$-0.529695\pi$
$269$	−3.05573	−0.186311	−0.0931555	+	0.995652i	$0.529695\pi$
$270$	0	0
$271$	−15.2361	−0.925525	−0.462763	−	0.886482i	$-0.653142\pi$
$271$	−15.2361	−0.925525	−0.462763	+	0.886482i	$0.653142\pi$
$272$	2.76393	0.167588
$273$	0	0
$274$	9.41641	0.568866
$275$	0	0
$276$	0	0
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	−23.1246	−1.38692
$279$	0	0
$280$	−3.23607	−0.193392
$281$	−5.41641	−0.323116	−0.161558	−	0.986863i	$-0.551652\pi$
$281$	−5.41641	−0.323116	−0.161558	+	0.986863i	$0.551652\pi$
$282$	0	0
$283$	4.47214	0.265841	0.132920	−	0.991127i	$-0.457565\pi$
$283$	4.47214	0.265841	0.132920	+	0.991127i	$0.457565\pi$
$284$	−6.47214	−0.384051
$285$	0	0
$286$	0	0
$287$	−10.0000	−0.590281
$288$	0	0
$289$	−9.36068	−0.550628
$290$	6.47214	0.380057
$291$	0	0
$292$	13.4164	0.785136
$293$	29.1246	1.70148	0.850739	−	0.525588i	$-0.176155\pi$
$293$	29.1246	1.70148	0.850739	+	0.525588i	$0.176155\pi$
$294$	0	0
$295$	−21.8885	−1.27440
$296$	6.00000	0.348743
$297$	0	0
$298$	−21.4164	−1.24062
$299$	−6.47214	−0.374293
$300$	0	0
$301$	−2.47214	−0.142492
$302$	−9.52786	−0.548267
$303$	0	0
$304$	4.47214	0.256495
$305$	−5.52786	−0.316525
$306$	0	0
$307$	29.2361	1.66859	0.834295	−	0.551318i	$-0.185875\pi$
$307$	29.2361	1.66859	0.834295	+	0.551318i	$0.185875\pi$
$308$	0	0
$309$	0	0
$310$	10.4721	0.594777
$311$	−25.1246	−1.42469	−0.712343	−	0.701831i	$-0.752366\pi$
$311$	−25.1246	−1.42469	−0.712343	+	0.701831i	$0.752366\pi$
$312$	0	0
$313$	26.1803	1.47980	0.739900	−	0.672717i	$-0.234873\pi$
$313$	26.1803	1.47980	0.739900	+	0.672717i	$0.234873\pi$
$314$	−18.6525	−1.05262
$315$	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
$320$	3.23607	0.180902
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	12.3607	0.687767
$324$	0	0
$325$	−35.4164	−1.96455
$326$	−6.47214	−0.358458
$327$	0	0
$328$	10.0000	0.552158
$329$	−11.2361	−0.619464
$330$	0	0
$331$	−30.4721	−1.67490	−0.837450	−	0.546514i	$-0.815955\pi$
$331$	−30.4721	−1.67490	−0.837450	+	0.546514i	$0.815955\pi$
$332$	−10.9443	−0.600645
$333$	0	0
$334$	6.65248	0.364007
$335$	12.9443	0.707221
$336$	0	0
$337$	−35.8885	−1.95497	−0.977487	−	0.210997i	$-0.932329\pi$
$337$	−35.8885	−1.95497	−0.977487	+	0.210997i	$0.932329\pi$
$338$	28.8885	1.57133
$339$	0	0
$340$	8.94427	0.485071
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	2.47214	0.133289
$345$	0	0
$346$	−3.05573	−0.164277
$347$	−24.3607	−1.30775	−0.653875	−	0.756603i	$-0.726858\pi$
$347$	−24.3607	−1.30775	−0.653875	+	0.756603i	$0.726858\pi$
$348$	0	0
$349$	−3.05573	−0.163569	−0.0817847	−	0.996650i	$-0.526062\pi$
$349$	−3.05573	−0.163569	−0.0817847	+	0.996650i	$0.526062\pi$
$350$	−5.47214	−0.292498
$351$	0	0
$352$	0	0
$353$	−0.472136	−0.0251293	−0.0125646	−	0.999921i	$-0.504000\pi$
$353$	−0.472136	−0.0251293	−0.0125646	+	0.999921i	$0.504000\pi$
$354$	0	0
$355$	−20.9443	−1.11161
$356$	−6.18034	−0.327557
$357$	0	0
$358$	−16.9443	−0.895533
$359$	−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
$362$	8.76393	0.460622
$363$	0	0
$364$	6.47214	0.339232
$365$	43.4164	2.27252
$366$	0	0
$367$	24.3607	1.27162	0.635809	−	0.771847i	$-0.280667\pi$
$367$	24.3607	1.27162	0.635809	+	0.771847i	$0.280667\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	19.4164	1.00941
$371$	−6.00000	−0.311504
$372$	0	0
$373$	−10.5836	−0.547998	−0.273999	−	0.961730i	$-0.588346\pi$
$373$	−10.5836	−0.547998	−0.273999	+	0.961730i	$0.588346\pi$
$374$	0	0
$375$	0	0
$376$	11.2361	0.579456
$377$	−12.9443	−0.666664
$378$	0	0
$379$	−17.8885	−0.918873	−0.459436	−	0.888211i	$-0.651949\pi$
$379$	−17.8885	−0.918873	−0.459436	+	0.888211i	$0.651949\pi$
$380$	14.4721	0.742405
$381$	0	0
$382$	−3.05573	−0.156345
$383$	0.583592	0.0298202	0.0149101	−	0.999889i	$-0.495254\pi$
$383$	0.583592	0.0298202	0.0149101	+	0.999889i	$0.495254\pi$
$384$	0	0
$385$	0	0
$386$	−5.05573	−0.257330
$387$	0	0
$388$	−14.1803	−0.719898
$389$	−9.41641	−0.477431	−0.238715	−	0.971090i	$-0.576726\pi$
$389$	−9.41641	−0.477431	−0.238715	+	0.971090i	$0.576726\pi$
$390$	0	0
$391$	2.76393	0.139778
$392$	1.00000	0.0505076
$393$	0	0
$394$	19.8885	1.00197
$395$	−28.9443	−1.45634
$396$	0	0
$397$	14.4721	0.726336	0.363168	−	0.931724i	$-0.381695\pi$
$397$	14.4721	0.726336	0.363168	+	0.931724i	$0.381695\pi$
$398$	−12.0000	−0.601506
$399$	0	0
$400$	5.47214	0.273607
$401$	37.7771	1.88650	0.943249	−	0.332087i	$-0.107753\pi$
$401$	37.7771	1.88650	0.943249	+	0.332087i	$0.107753\pi$
$402$	0	0
$403$	−20.9443	−1.04331
$404$	5.52786	0.275022
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	0	0
$408$	0	0
$409$	28.4721	1.40786	0.703928	−	0.710271i	$-0.251428\pi$
$409$	28.4721	1.40786	0.703928	+	0.710271i	$0.251428\pi$
$410$	32.3607	1.59818
$411$	0	0
$412$	−4.00000	−0.197066
$413$	6.76393	0.332831
$414$	0	0
$415$	−35.4164	−1.73852
$416$	−6.47214	−0.317323
$417$	0	0
$418$	0	0
$419$	−34.3607	−1.67863	−0.839315	−	0.543646i	$-0.817043\pi$
$419$	−34.3607	−1.67863	−0.839315	+	0.543646i	$0.817043\pi$
$420$	0	0
$421$	35.8885	1.74910	0.874550	−	0.484935i	$-0.161157\pi$
$421$	35.8885	1.74910	0.874550	+	0.484935i	$0.161157\pi$
$422$	16.9443	0.824834
$423$	0	0
$424$	6.00000	0.291386
$425$	15.1246	0.733651
$426$	0	0
$427$	1.70820	0.0826658
$428$	1.52786	0.0738521
$429$	0	0
$430$	8.00000	0.385794
$431$	−20.9443	−1.00885	−0.504425	−	0.863455i	$-0.668296\pi$
$431$	−20.9443	−1.00885	−0.504425	+	0.863455i	$0.668296\pi$
$432$	0	0
$433$	6.76393	0.325054	0.162527	−	0.986704i	$-0.448036\pi$
$433$	6.76393	0.325054	0.162527	+	0.986704i	$0.448036\pi$
$434$	−3.23607	−0.155336
$435$	0	0
$436$	12.4721	0.597307
$437$	4.47214	0.213931
$438$	0	0
$439$	−8.76393	−0.418280	−0.209140	−	0.977886i	$-0.567066\pi$
$439$	−8.76393	−0.418280	−0.209140	+	0.977886i	$0.567066\pi$
$440$	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
$446$	−10.6525	−0.504409
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−9.41641	−0.444388	−0.222194	−	0.975003i	$-0.571322\pi$
$449$	−9.41641	−0.444388	−0.222194	+	0.975003i	$0.571322\pi$
$450$	0	0
$451$	0	0
$452$	−12.4721	−0.586640
$453$	0	0
$454$	21.4164	1.00512
$455$	20.9443	0.981883
$456$	0	0
$457$	22.3607	1.04599	0.522994	−	0.852336i	$-0.324815\pi$
$457$	22.3607	1.04599	0.522994	+	0.852336i	$0.324815\pi$
$458$	−15.2361	−0.711935
$459$	0	0
$460$	3.23607	0.150882
$461$	−3.05573	−0.142319	−0.0711597	−	0.997465i	$-0.522670\pi$
$461$	−3.05573	−0.142319	−0.0711597	+	0.997465i	$0.522670\pi$
$462$	0	0
$463$	−11.0557	−0.513803	−0.256902	−	0.966438i	$-0.582702\pi$
$463$	−11.0557	−0.513803	−0.256902	+	0.966438i	$0.582702\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	16.4721	0.763057
$467$	11.5279	0.533446	0.266723	−	0.963773i	$-0.414059\pi$
$467$	11.5279	0.533446	0.266723	+	0.963773i	$0.414059\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	36.3607	1.67719
$471$	0	0
$472$	−6.76393	−0.311335
$473$	0	0
$474$	0	0
$475$	24.4721	1.12286
$476$	−2.76393	−0.126685
$477$	0	0
$478$	−6.47214	−0.296029
$479$	34.8328	1.59155	0.795776	−	0.605591i	$-0.207063\pi$
$479$	34.8328	1.59155	0.795776	+	0.605591i	$0.207063\pi$
$480$	0	0
$481$	−38.8328	−1.77062
$482$	0.291796	0.0132909
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−45.8885	−2.08369
$486$	0	0
$487$	0	0		−	1.00000i	$-0.5\pi$
$487$	0	0			1.00000i	$0.5\pi$
$488$	−1.70820	−0.0773268
$489$	0	0
$490$	3.23607	0.146191
$491$	7.05573	0.318421	0.159210	−	0.987245i	$-0.449105\pi$
$491$	7.05573	0.318421	0.159210	+	0.987245i	$0.449105\pi$
$492$	0	0
$493$	5.52786	0.248962
$494$	−28.9443	−1.30226
$495$	0	0
$496$	3.23607	0.145304
$497$	6.47214	0.290315
$498$	0	0
$499$	−8.94427	−0.400401	−0.200200	−	0.979755i	$-0.564159\pi$
$499$	−8.94427	−0.400401	−0.200200	+	0.979755i	$0.564159\pi$
$500$	1.52786	0.0683282
$501$	0	0
$502$	4.47214	0.199601
$503$	4.94427	0.220454	0.110227	−	0.993906i	$-0.464842\pi$
$503$	4.94427	0.220454	0.110227	+	0.993906i	$0.464842\pi$
$504$	0	0
$505$	17.8885	0.796030
$506$	0	0
$507$	0	0
$508$	−19.4164	−0.861464
$509$	−33.8885	−1.50208	−0.751042	−	0.660255i	$-0.770448\pi$
$509$	−33.8885	−1.50208	−0.751042	+	0.660255i	$0.770448\pi$
$510$	0	0
$511$	−13.4164	−0.593507
$512$	1.00000	0.0441942
$513$	0	0
$514$	26.9443	1.18846
$515$	−12.9443	−0.570393
$516$	0	0
$517$	0	0
$518$	−6.00000	−0.263625
$519$	0	0
$520$	−20.9443	−0.918467
$521$	−11.7082	−0.512946	−0.256473	−	0.966551i	$-0.582560\pi$
$521$	−11.7082	−0.512946	−0.256473	+	0.966551i	$0.582560\pi$
$522$	0	0
$523$	10.5836	0.462788	0.231394	−	0.972860i	$-0.425671\pi$
$523$	10.5836	0.462788	0.231394	+	0.972860i	$0.425671\pi$
$524$	17.2361	0.752961
$525$	0	0
$526$	−12.0000	−0.523225
$527$	8.94427	0.389619
$528$	0	0
$529$	1.00000	0.0434783
$530$	19.4164	0.843395
$531$	0	0
$532$	−4.47214	−0.193892
$533$	−64.7214	−2.80339
$534$	0	0
$535$	4.94427	0.213760
$536$	4.00000	0.172774
$537$	0	0
$538$	−3.05573	−0.131742
$539$	0	0
$540$	0	0
$541$	3.88854	0.167182	0.0835908	−	0.996500i	$-0.473361\pi$
$541$	3.88854	0.167182	0.0835908	+	0.996500i	$0.473361\pi$
$542$	−15.2361	−0.654445
$543$	0	0
$544$	2.76393	0.118503
$545$	40.3607	1.72886
$546$	0	0
$547$	−22.4721	−0.960839	−0.480420	−	0.877039i	$-0.659516\pi$
$547$	−22.4721	−0.960839	−0.480420	+	0.877039i	$0.659516\pi$
$548$	9.41641	0.402249
$549$	0	0
$550$	0	0
$551$	8.94427	0.381039
$552$	0	0
$553$	8.94427	0.380349
$554$	19.8885	0.844983
$555$	0	0
$556$	−23.1246	−0.980702
$557$	17.4164	0.737957	0.368978	−	0.929438i	$-0.379708\pi$
$557$	17.4164	0.737957	0.368978	+	0.929438i	$0.379708\pi$
$558$	0	0
$559$	−16.0000	−0.676728
$560$	−3.23607	−0.136749
$561$	0	0
$562$	−5.41641	−0.228477
$563$	−6.58359	−0.277465	−0.138733	−	0.990330i	$-0.544303\pi$
$563$	−6.58359	−0.277465	−0.138733	+	0.990330i	$0.544303\pi$
$564$	0	0
$565$	−40.3607	−1.69799
$566$	4.47214	0.187978
$567$	0	0
$568$	−6.47214	−0.271565
$569$	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
$571$	−33.3050	−1.39377	−0.696884	−	0.717183i	$-0.745431\pi$
$571$	−33.3050	−1.39377	−0.696884	+	0.717183i	$0.745431\pi$
$572$	0	0
$573$	0	0
$574$	−10.0000	−0.417392
$575$	5.47214	0.228204
$576$	0	0
$577$	−25.4164	−1.05810	−0.529049	−	0.848591i	$-0.677451\pi$
$577$	−25.4164	−1.05810	−0.529049	+	0.848591i	$0.677451\pi$
$578$	−9.36068	−0.389353
$579$	0	0
$580$	6.47214	0.268741
$581$	10.9443	0.454045
$582$	0	0
$583$	0	0
$584$	13.4164	0.555175
$585$	0	0
$586$	29.1246	1.20313
$587$	−39.7082	−1.63893	−0.819466	−	0.573127i	$-0.805730\pi$
$587$	−39.7082	−1.63893	−0.819466	+	0.573127i	$0.805730\pi$
$588$	0	0
$589$	14.4721	0.596314
$590$	−21.8885	−0.901137
$591$	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
$596$	−21.4164	−0.877250
$597$	0	0
$598$	−6.47214	−0.264665
$599$	−38.8328	−1.58667	−0.793333	−	0.608788i	$-0.791656\pi$
$599$	−38.8328	−1.58667	−0.793333	+	0.608788i	$0.791656\pi$
$600$	0	0
$601$	−4.47214	−0.182422	−0.0912111	−	0.995832i	$-0.529074\pi$
$601$	−4.47214	−0.182422	−0.0912111	+	0.995832i	$0.529074\pi$
$602$	−2.47214	−0.100757
$603$	0	0
$604$	−9.52786	−0.387683
$605$	−35.5967	−1.44721
$606$	0	0
$607$	0.763932	0.0310070	0.0155035	−	0.999880i	$-0.495065\pi$
$607$	0.763932	0.0310070	0.0155035	+	0.999880i	$0.495065\pi$
$608$	4.47214	0.181369
$609$	0	0
$610$	−5.52786	−0.223817
$611$	−72.7214	−2.94199
$612$	0	0
$613$	−36.2492	−1.46409	−0.732046	−	0.681255i	$-0.761434\pi$
$613$	−36.2492	−1.46409	−0.732046	+	0.681255i	$0.761434\pi$
$614$	29.2361	1.17987
$615$	0	0
$616$	0	0
$617$	−9.05573	−0.364570	−0.182285	−	0.983246i	$-0.558349\pi$
$617$	−9.05573	−0.364570	−0.182285	+	0.983246i	$0.558349\pi$
$618$	0	0
$619$	1.05573	0.0424333	0.0212166	−	0.999775i	$-0.493246\pi$
$619$	1.05573	0.0424333	0.0212166	+	0.999775i	$0.493246\pi$
$620$	10.4721	0.420571
$621$	0	0
$622$	−25.1246	−1.00741
$623$	6.18034	0.247610
$624$	0	0
$625$	−22.4164	−0.896656
$626$	26.1803	1.04638
$627$	0	0
$628$	−18.6525	−0.744315
$629$	16.5836	0.661231
$630$	0	0
$631$	−2.11146	−0.0840557	−0.0420279	−	0.999116i	$-0.513382\pi$
$631$	−2.11146	−0.0840557	−0.0420279	+	0.999116i	$0.513382\pi$
$632$	−8.94427	−0.355784
$633$	0	0
$634$	−10.0000	−0.397151
$635$	−62.8328	−2.49344
$636$	0	0
$637$	−6.47214	−0.256435
$638$	0	0
$639$	0	0
$640$	3.23607	0.127917
$641$	−34.9443	−1.38022	−0.690108	−	0.723707i	$-0.742437\pi$
$641$	−34.9443	−1.38022	−0.690108	+	0.723707i	$0.742437\pi$
$642$	0	0
$643$	44.2492	1.74502	0.872510	−	0.488597i	$-0.162491\pi$
$643$	44.2492	1.74502	0.872510	+	0.488597i	$0.162491\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	12.3607	0.486324
$647$	−1.34752	−0.0529766	−0.0264883	−	0.999649i	$-0.508432\pi$
$647$	−1.34752	−0.0529766	−0.0264883	+	0.999649i	$0.508432\pi$
$648$	0	0
$649$	0	0
$650$	−35.4164	−1.38915
$651$	0	0
$652$	−6.47214	−0.253468
$653$	−29.7771	−1.16527	−0.582634	−	0.812735i	$-0.697978\pi$
$653$	−29.7771	−1.16527	−0.582634	+	0.812735i	$0.697978\pi$
$654$	0	0
$655$	55.7771	2.17939
$656$	10.0000	0.390434
$657$	0	0
$658$	−11.2361	−0.438028
$659$	46.2492	1.80161	0.900807	−	0.434220i	$-0.142976\pi$
$659$	46.2492	1.80161	0.900807	+	0.434220i	$0.142976\pi$
$660$	0	0
$661$	5.12461	0.199324	0.0996621	−	0.995021i	$-0.468224\pi$
$661$	5.12461	0.199324	0.0996621	+	0.995021i	$0.468224\pi$
$662$	−30.4721	−1.18433
$663$	0	0
$664$	−10.9443	−0.424720
$665$	−14.4721	−0.561205
$666$	0	0
$667$	2.00000	0.0774403
$668$	6.65248	0.257392
$669$	0	0
$670$	12.9443	0.500081
$671$	0	0
$672$	0	0
$673$	−29.4164	−1.13392	−0.566960	−	0.823746i	$-0.691880\pi$
$673$	−29.4164	−1.13392	−0.566960	+	0.823746i	$0.691880\pi$
$674$	−35.8885	−1.38238
$675$	0	0
$676$	28.8885	1.11110
$677$	−23.2361	−0.893035	−0.446517	−	0.894775i	$-0.647336\pi$
$677$	−23.2361	−0.893035	−0.446517	+	0.894775i	$0.647336\pi$
$678$	0	0
$679$	14.1803	0.544191
$680$	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
$685$	30.4721	1.16428
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	2.47214	0.0942493
$689$	−38.8328	−1.47941
$690$	0	0
$691$	49.0132	1.86455	0.932274	−	0.361753i	$-0.117821\pi$
$691$	49.0132	1.86455	0.932274	+	0.361753i	$0.117821\pi$
$692$	−3.05573	−0.116161
$693$	0	0
$694$	−24.3607	−0.924719
$695$	−74.8328	−2.83857
$696$	0	0
$697$	27.6393	1.04691
$698$	−3.05573	−0.115661
$699$	0	0
$700$	−5.47214	−0.206827
$701$	21.0557	0.795264	0.397632	−	0.917545i	$-0.369832\pi$
$701$	21.0557	0.795264	0.397632	+	0.917545i	$0.369832\pi$
$702$	0	0
$703$	26.8328	1.01202
$704$	0	0
$705$	0	0
$706$	−0.472136	−0.0177691
$707$	−5.52786	−0.207897
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	−20.9443	−0.786025
$711$	0	0
$712$	−6.18034	−0.231618
$713$	3.23607	0.121192
$714$	0	0
$715$	0	0
$716$	−16.9443	−0.633237
$717$	0	0
$718$	−12.0000	−0.447836
$719$	32.5410	1.21358	0.606788	−	0.794864i	$-0.292458\pi$
$719$	32.5410	1.21358	0.606788	+	0.794864i	$0.292458\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	1.00000	0.0372161
$723$	0	0
$724$	8.76393	0.325709
$725$	10.9443	0.406460
$726$	0	0
$727$	7.05573	0.261682	0.130841	−	0.991403i	$-0.458232\pi$
$727$	7.05573	0.261682	0.130841	+	0.991403i	$0.458232\pi$
$728$	6.47214	0.239873
$729$	0	0
$730$	43.4164	1.60691
$731$	6.83282	0.252721
$732$	0	0
$733$	3.59675	0.132849	0.0664245	−	0.997791i	$-0.478841\pi$
$733$	3.59675	0.132849	0.0664245	+	0.997791i	$0.478841\pi$
$734$	24.3607	0.899169
$735$	0	0
$736$	1.00000	0.0368605
$737$	0	0
$738$	0	0
$739$	24.3607	0.896122	0.448061	−	0.894003i	$-0.352115\pi$
$739$	24.3607	0.896122	0.448061	+	0.894003i	$0.352115\pi$
$740$	19.4164	0.713761
$741$	0	0
$742$	−6.00000	−0.220267
$743$	48.9443	1.79559	0.897796	−	0.440412i	$-0.145168\pi$
$743$	48.9443	1.79559	0.897796	+	0.440412i	$0.145168\pi$
$744$	0	0
$745$	−69.3050	−2.53914
$746$	−10.5836	−0.387493
$747$	0	0
$748$	0	0
$749$	−1.52786	−0.0558269
$750$	0	0
$751$	−40.9443	−1.49408	−0.747039	−	0.664780i	$-0.768525\pi$
$751$	−40.9443	−1.49408	−0.747039	+	0.664780i	$0.768525\pi$
$752$	11.2361	0.409737
$753$	0	0
$754$	−12.9443	−0.471403
$755$	−30.8328	−1.12212
$756$	0	0
$757$	23.5279	0.855135	0.427567	−	0.903983i	$-0.359371\pi$
$757$	23.5279	0.855135	0.427567	+	0.903983i	$0.359371\pi$
$758$	−17.8885	−0.649741
$759$	0	0
$760$	14.4721	0.524960
$761$	0.472136	0.0171149	0.00855746	−	0.999963i	$-0.497276\pi$
$761$	0.472136	0.0171149	0.00855746	+	0.999963i	$0.497276\pi$
$762$	0	0
$763$	−12.4721	−0.451522
$764$	−3.05573	−0.110552
$765$	0	0
$766$	0.583592	0.0210860
$767$	43.7771	1.58070
$768$	0	0
$769$	21.2361	0.765792	0.382896	−	0.923791i	$-0.374927\pi$
$769$	21.2361	0.765792	0.382896	+	0.923791i	$0.374927\pi$
$770$	0	0
$771$	0	0
$772$	−5.05573	−0.181960
$773$	−16.1803	−0.581966	−0.290983	−	0.956728i	$-0.593982\pi$
$773$	−16.1803	−0.581966	−0.290983	+	0.956728i	$0.593982\pi$
$774$	0	0
$775$	17.7082	0.636097
$776$	−14.1803	−0.509045
$777$	0	0
$778$	−9.41641	−0.337595
$779$	44.7214	1.60231
$780$	0	0
$781$	0	0
$782$	2.76393	0.0988380
$783$	0	0
$784$	1.00000	0.0357143
$785$	−60.3607	−2.15437
$786$	0	0
$787$	−35.8885	−1.27929	−0.639644	−	0.768671i	$-0.720918\pi$
$787$	−35.8885	−1.27929	−0.639644	+	0.768671i	$0.720918\pi$
$788$	19.8885	0.708500
$789$	0	0
$790$	−28.9443	−1.02979
$791$	12.4721	0.443458
$792$	0	0
$793$	11.0557	0.392600
$794$	14.4721	0.513597
$795$	0	0
$796$	−12.0000	−0.425329
$797$	13.3475	0.472794	0.236397	−	0.971657i	$-0.424033\pi$
$797$	13.3475	0.472794	0.236397	+	0.971657i	$0.424033\pi$
$798$	0	0
$799$	31.0557	1.09867
$800$	5.47214	0.193469
$801$	0	0
$802$	37.7771	1.33396
$803$	0	0
$804$	0	0
$805$	−3.23607	−0.114056
$806$	−20.9443	−0.737731
$807$	0	0
$808$	5.52786	0.194470
$809$	32.8328	1.15434	0.577170	−	0.816624i	$-0.304157\pi$
$809$	32.8328	1.15434	0.577170	+	0.816624i	$0.304157\pi$
$810$	0	0
$811$	21.5967	0.758364	0.379182	−	0.925322i	$-0.376205\pi$
$811$	21.5967	0.758364	0.379182	+	0.925322i	$0.376205\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	0	0
$815$	−20.9443	−0.733646
$816$	0	0
$817$	11.0557	0.386791
$818$	28.4721	0.995505
$819$	0	0
$820$	32.3607	1.13008
$821$	2.94427	0.102756	0.0513779	−	0.998679i	$-0.483639\pi$
$821$	2.94427	0.102756	0.0513779	+	0.998679i	$0.483639\pi$
$822$	0	0
$823$	−11.4164	−0.397951	−0.198975	−	0.980004i	$-0.563761\pi$
$823$	−11.4164	−0.397951	−0.198975	+	0.980004i	$0.563761\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	6.76393	0.235347
$827$	12.5836	0.437574	0.218787	−	0.975773i	$-0.429790\pi$
$827$	12.5836	0.437574	0.218787	+	0.975773i	$0.429790\pi$
$828$	0	0
$829$	13.8885	0.482369	0.241185	−	0.970479i	$-0.422464\pi$
$829$	13.8885	0.482369	0.241185	+	0.970479i	$0.422464\pi$
$830$	−35.4164	−1.22932
$831$	0	0
$832$	−6.47214	−0.224381
$833$	2.76393	0.0957646
$834$	0	0
$835$	21.5279	0.745002
$836$	0	0
$837$	0	0
$838$	−34.3607	−1.18697
$839$	−28.3607	−0.979119	−0.489560	−	0.871970i	$-0.662842\pi$
$839$	−28.3607	−0.979119	−0.489560	+	0.871970i	$0.662842\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	35.8885	1.23680
$843$	0	0
$844$	16.9443	0.583246
$845$	93.4853	3.21599
$846$	0	0
$847$	11.0000	0.377964
$848$	6.00000	0.206041
$849$	0	0
$850$	15.1246	0.518770
$851$	6.00000	0.205677
$852$	0	0
$853$	11.4164	0.390890	0.195445	−	0.980715i	$-0.437385\pi$
$853$	11.4164	0.390890	0.195445	+	0.980715i	$0.437385\pi$
$854$	1.70820	0.0584535
$855$	0	0
$856$	1.52786	0.0522213
$857$	11.8885	0.406105	0.203052	−	0.979168i	$-0.434914\pi$
$857$	11.8885	0.406105	0.203052	+	0.979168i	$0.434914\pi$
$858$	0	0
$859$	2.76393	0.0943041	0.0471521	−	0.998888i	$-0.484985\pi$
$859$	2.76393	0.0943041	0.0471521	+	0.998888i	$0.484985\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−20.9443	−0.713365
$863$	3.41641	0.116296	0.0581479	−	0.998308i	$-0.481480\pi$
$863$	3.41641	0.116296	0.0581479	+	0.998308i	$0.481480\pi$
$864$	0	0
$865$	−9.88854	−0.336221
$866$	6.76393	0.229848
$867$	0	0
$868$	−3.23607	−0.109839
$869$	0	0
$870$	0	0
$871$	−25.8885	−0.877200
$872$	12.4721	0.422360
$873$	0	0
$874$	4.47214	0.151272
$875$	−1.52786	−0.0516512
$876$	0	0
$877$	26.9443	0.909843	0.454922	−	0.890531i	$-0.349667\pi$
$877$	26.9443	0.909843	0.454922	+	0.890531i	$0.349667\pi$
$878$	−8.76393	−0.295768
$879$	0	0
$880$	0	0
$881$	21.2361	0.715461	0.357731	−	0.933825i	$-0.383551\pi$
$881$	21.2361	0.715461	0.357731	+	0.933825i	$0.383551\pi$
$882$	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
$887$	35.5967	1.19522	0.597611	−	0.801786i	$-0.296117\pi$
$887$	35.5967	1.19522	0.597611	+	0.801786i	$0.296117\pi$
$888$	0	0
$889$	19.4164	0.651205
$890$	−20.0000	−0.670402
$891$	0	0
$892$	−10.6525	−0.356671
$893$	50.2492	1.68153
$894$	0	0
$895$	−54.8328	−1.83286
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−9.41641	−0.314230
$899$	6.47214	0.215858
$900$	0	0
$901$	16.5836	0.552480
$902$	0	0
$903$	0	0
$904$	−12.4721	−0.414817
$905$	28.3607	0.942741
$906$	0	0
$907$	−1.52786	−0.0507319	−0.0253659	−	0.999678i	$-0.508075\pi$
$907$	−1.52786	−0.0507319	−0.0253659	+	0.999678i	$0.508075\pi$
$908$	21.4164	0.710728
$909$	0	0
$910$	20.9443	0.694296
$911$	−21.8885	−0.725200	−0.362600	−	0.931945i	$-0.618111\pi$
$911$	−21.8885	−0.725200	−0.362600	+	0.931945i	$0.618111\pi$
$912$	0	0
$913$	0	0
$914$	22.3607	0.739626
$915$	0	0
$916$	−15.2361	−0.503414
$917$	−17.2361	−0.569185
$918$	0	0
$919$	−10.1115	−0.333546	−0.166773	−	0.985995i	$-0.553335\pi$
$919$	−10.1115	−0.333546	−0.166773	+	0.985995i	$0.553335\pi$
$920$	3.23607	0.106690
$921$	0	0
$922$	−3.05573	−0.100635
$923$	41.8885	1.37878
$924$	0	0
$925$	32.8328	1.07954
$926$	−11.0557	−0.363314
$927$	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
$932$	16.4721	0.539563
$933$	0	0
$934$	11.5279	0.377203
$935$	0	0
$936$	0	0
$937$	50.7639	1.65839	0.829193	−	0.558963i	$-0.188801\pi$
$937$	50.7639	1.65839	0.829193	+	0.558963i	$0.188801\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	36.3607	1.18595
$941$	−19.5967	−0.638836	−0.319418	−	0.947614i	$-0.603487\pi$
$941$	−19.5967	−0.638836	−0.319418	+	0.947614i	$0.603487\pi$
$942$	0	0
$943$	10.0000	0.325645
$944$	−6.76393	−0.220147
$945$	0	0
$946$	0	0
$947$	−42.2492	−1.37292	−0.686458	−	0.727170i	$-0.740835\pi$
$947$	−42.2492	−1.37292	−0.686458	+	0.727170i	$0.740835\pi$
$948$	0	0
$949$	−86.8328	−2.81871
$950$	24.4721	0.793981
$951$	0	0
$952$	−2.76393	−0.0895796
$953$	−45.7771	−1.48287	−0.741433	−	0.671027i	$-0.765853\pi$
$953$	−45.7771	−1.48287	−0.741433	+	0.671027i	$0.765853\pi$
$954$	0	0
$955$	−9.88854	−0.319986
$956$	−6.47214	−0.209324
$957$	0	0
$958$	34.8328	1.12540
$959$	−9.41641	−0.304072
$960$	0	0
$961$	−20.5279	−0.662189
$962$	−38.8328	−1.25202
$963$	0	0
$964$	0.291796	0.00939812
$965$	−16.3607	−0.526669
$966$	0	0
$967$	−9.88854	−0.317994	−0.158997	−	0.987279i	$-0.550826\pi$
$967$	−9.88854	−0.317994	−0.158997	+	0.987279i	$0.550826\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−45.8885	−1.47339
$971$	35.8885	1.15172	0.575859	−	0.817549i	$-0.304668\pi$
$971$	35.8885	1.15172	0.575859	+	0.817549i	$0.304668\pi$
$972$	0	0
$973$	23.1246	0.741341
$974$	0	0
$975$	0	0
$976$	−1.70820	−0.0546783
$977$	12.4721	0.399019	0.199509	−	0.979896i	$-0.436065\pi$
$977$	12.4721	0.399019	0.199509	+	0.979896i	$0.436065\pi$
$978$	0	0
$979$	0	0
$980$	3.23607	0.103372
$981$	0	0
$982$	7.05573	0.225157
$983$	13.5279	0.431472	0.215736	−	0.976452i	$-0.430785\pi$
$983$	13.5279	0.431472	0.215736	+	0.976452i	$0.430785\pi$
$984$	0	0
$985$	64.3607	2.05070
$986$	5.52786	0.176043
$987$	0	0
$988$	−28.9443	−0.920840
$989$	2.47214	0.0786094
$990$	0	0
$991$	21.3050	0.676774	0.338387	−	0.941007i	$-0.390119\pi$
$991$	21.3050	0.676774	0.338387	+	0.941007i	$0.390119\pi$
$992$	3.23607	0.102745
$993$	0	0
$994$	6.47214	0.205284
$995$	−38.8328	−1.23108
$996$	0	0
$997$	10.8328	0.343079	0.171539	−	0.985177i	$-0.445126\pi$
$997$	10.8328	0.343079	0.171539	+	0.985177i	$0.445126\pi$
$998$	−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.1.2		2
3.2	odd	2		322.2.a.e.1.2	✓	2
12.11	even	2		2576.2.a.t.1.1		2
15.14	odd	2		8050.2.a.bf.1.1		2
21.20	even	2		2254.2.a.k.1.1		2
69.68	even	2		7406.2.a.j.1.2		2

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.23607 q^{10} -6.47214 q^{13} -1.00000 q^{14} +1.00000 q^{16} +2.76393 q^{17} +4.47214 q^{19} +3.23607 q^{20} +1.00000 q^{23} +5.47214 q^{25} -6.47214 q^{26} -1.00000 q^{28} +2.00000 q^{29} +3.23607 q^{31} +1.00000 q^{32} +2.76393 q^{34} -3.23607 q^{35} +6.00000 q^{37} +4.47214 q^{38} +3.23607 q^{40} +10.0000 q^{41} +2.47214 q^{43} +1.00000 q^{46} +11.2361 q^{47} +1.00000 q^{49} +5.47214 q^{50} -6.47214 q^{52} +6.00000 q^{53} -1.00000 q^{56} +2.00000 q^{58} -6.76393 q^{59} -1.70820 q^{61} +3.23607 q^{62} +1.00000 q^{64} -20.9443 q^{65} +4.00000 q^{67} +2.76393 q^{68} -3.23607 q^{70} -6.47214 q^{71} +13.4164 q^{73} +6.00000 q^{74} +4.47214 q^{76} -8.94427 q^{79} +3.23607 q^{80} +10.0000 q^{82} -10.9443 q^{83} +8.94427 q^{85} +2.47214 q^{86} -6.18034 q^{89} +6.47214 q^{91} +1.00000 q^{92} +11.2361 q^{94} +14.4721 q^{95} -14.1803 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 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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