LMFDB - Embedded newform 2898.2.a.bc.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(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.70156$ of defining polynomial
Character	$\chi$	$=$	2898.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.70156 q^{10} +5.70156 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.40312 q^{19} +3.70156 q^{20} +1.00000 q^{23} +8.70156 q^{25} +5.70156 q^{26} -1.00000 q^{28} -0.298438 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.70156 q^{35} +3.70156 q^{37} -5.40312 q^{38} +3.70156 q^{40} -7.70156 q^{41} +5.70156 q^{43} +1.00000 q^{46} -3.70156 q^{47} +1.00000 q^{49} +8.70156 q^{50} +5.70156 q^{52} -9.40312 q^{53} -1.00000 q^{56} -0.298438 q^{58} +0.596876 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +21.1047 q^{65} +4.00000 q^{67} -3.70156 q^{70} -7.40312 q^{71} -5.40312 q^{73} +3.70156 q^{74} -5.40312 q^{76} +3.70156 q^{80} -7.70156 q^{82} +2.59688 q^{83} +5.70156 q^{86} +15.4031 q^{89} -5.70156 q^{91} +1.00000 q^{92} -3.70156 q^{94} -20.0000 q^{95} -1.10469 q^{97} +1.00000 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} + 5 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{19} + q^{20} + 2 q^{23} + 11 q^{25} + 5 q^{26} - 2 q^{28} - 7 q^{29} + 12 q^{31} + 2 q^{32} - q^{35} + q^{37} + 2 q^{38} + q^{40} - 9 q^{41} + 5 q^{43} + 2 q^{46} - q^{47} + 2 q^{49} + 11 q^{50} + 5 q^{52} - 6 q^{53} - 2 q^{56} - 7 q^{58} + 14 q^{59} + 20 q^{61} + 12 q^{62} + 2 q^{64} + 23 q^{65} + 8 q^{67} - q^{70} - 2 q^{71} + 2 q^{73} + q^{74} + 2 q^{76} + q^{80} - 9 q^{82} + 18 q^{83} + 5 q^{86} + 18 q^{89} - 5 q^{91} + 2 q^{92} - q^{94} - 40 q^{95} + 17 q^{97} + 2 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8 + q^10 + 5 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^19 + q^20 + 2 * q^23 + 11 * q^25 + 5 * q^26 - 2 * q^28 - 7 * q^29 + 12 * q^31 + 2 * q^32 - q^35 + q^37 + 2 * q^38 + q^40 - 9 * q^41 + 5 * q^43 + 2 * q^46 - q^47 + 2 * q^49 + 11 * q^50 + 5 * q^52 - 6 * q^53 - 2 * q^56 - 7 * q^58 + 14 * q^59 + 20 * q^61 + 12 * q^62 + 2 * q^64 + 23 * q^65 + 8 * q^67 - q^70 - 2 * q^71 + 2 * q^73 + q^74 + 2 * q^76 + q^80 - 9 * q^82 + 18 * q^83 + 5 * q^86 + 18 * q^89 - 5 * q^91 + 2 * q^92 - q^94 - 40 * q^95 + 17 * q^97 + 2 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.70156	1.65539	0.827694	−	0.561179i	$-0.189652\pi$
$5$	3.70156	1.65539	0.827694	+	0.561179i	$0.189652\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	3.70156	1.17054
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	5.70156	1.58133	0.790664	−	0.612250i	$-0.209735\pi$
$13$	5.70156	1.58133	0.790664	+	0.612250i	$0.209735\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−5.40312	−1.23956	−0.619781	−	0.784775i	$-0.712779\pi$
$19$	−5.40312	−1.23956	−0.619781	+	0.784775i	$0.712779\pi$
$20$	3.70156	0.827694
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	8.70156	1.74031
$26$	5.70156	1.11817
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−0.298438	−0.0554185	−0.0277093	−	0.999616i	$-0.508821\pi$
$29$	−0.298438	−0.0554185	−0.0277093	+	0.999616i	$0.508821\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	−3.70156	−0.625678
$36$	0	0
$37$	3.70156	0.608533	0.304267	−	0.952587i	$-0.401589\pi$
$37$	3.70156	0.608533	0.304267	+	0.952587i	$0.401589\pi$
$38$	−5.40312	−0.876502
$39$	0	0
$40$	3.70156	0.585268
$41$	−7.70156	−1.20278	−0.601391	−	0.798955i	$-0.705387\pi$
$41$	−7.70156	−1.20278	−0.601391	+	0.798955i	$0.705387\pi$
$42$	0	0
$43$	5.70156	0.869480	0.434740	−	0.900556i	$-0.356840\pi$
$43$	5.70156	0.869480	0.434740	+	0.900556i	$0.356840\pi$
$44$	0	0
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.70156	−0.539928	−0.269964	−	0.962870i	$-0.587012\pi$
$47$	−3.70156	−0.539928	−0.269964	+	0.962870i	$0.587012\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	8.70156	1.23059
$51$	0	0
$52$	5.70156	0.790664
$53$	−9.40312	−1.29162	−0.645809	−	0.763499i	$-0.723480\pi$
$53$	−9.40312	−1.29162	−0.645809	+	0.763499i	$0.723480\pi$
$54$	0	0
$55$	0	0
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−0.298438	−0.0391868
$59$	0.596876	0.0777066	0.0388533	−	0.999245i	$-0.487629\pi$
$59$	0.596876	0.0777066	0.0388533	+	0.999245i	$0.487629\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	21.1047	2.61771
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	−3.70156	−0.442421
$71$	−7.40312	−0.878589	−0.439295	−	0.898343i	$-0.644772\pi$
$71$	−7.40312	−0.878589	−0.439295	+	0.898343i	$0.644772\pi$
$72$	0	0
$73$	−5.40312	−0.632388	−0.316194	−	0.948695i	$-0.602405\pi$
$73$	−5.40312	−0.632388	−0.316194	+	0.948695i	$0.602405\pi$
$74$	3.70156	0.430298
$75$	0	0
$76$	−5.40312	−0.619781
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	3.70156	0.413847
$81$	0	0
$82$	−7.70156	−0.850495
$83$	2.59688	0.285044	0.142522	−	0.989792i	$-0.454479\pi$
$83$	2.59688	0.285044	0.142522	+	0.989792i	$0.454479\pi$
$84$	0	0
$85$	0	0
$86$	5.70156	0.614815
$87$	0	0
$88$	0	0
$89$	15.4031	1.63273	0.816364	−	0.577538i	$-0.195986\pi$
$89$	15.4031	1.63273	0.816364	+	0.577538i	$0.195986\pi$
$90$	0	0
$91$	−5.70156	−0.597686
$92$	1.00000	0.104257
$93$	0	0
$94$	−3.70156	−0.381787
$95$	−20.0000	−2.05196
$96$	0	0
$97$	−1.10469	−0.112164	−0.0560820	−	0.998426i	$-0.517861\pi$
$97$	−1.10469	−0.112164	−0.0560820	+	0.998426i	$0.517861\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	8.70156	0.870156
$101$	−4.59688	−0.457406	−0.228703	−	0.973496i	$-0.573449\pi$
$101$	−4.59688	−0.457406	−0.228703	+	0.973496i	$0.573449\pi$
$102$	0	0
$103$	−17.1047	−1.68537	−0.842687	−	0.538403i	$-0.819028\pi$
$103$	−17.1047	−1.68537	−0.842687	+	0.538403i	$0.819028\pi$
$104$	5.70156	0.559084
$105$	0	0
$106$	−9.40312	−0.913312
$107$	−14.8062	−1.43137	−0.715687	−	0.698421i	$-0.753886\pi$
$107$	−14.8062	−1.43137	−0.715687	+	0.698421i	$0.753886\pi$
$108$	0	0
$109$	15.7016	1.50394	0.751968	−	0.659199i	$-0.229105\pi$
$109$	15.7016	1.50394	0.751968	+	0.659199i	$0.229105\pi$
$110$	0	0
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	4.29844	0.404363	0.202182	−	0.979348i	$-0.435197\pi$
$113$	4.29844	0.404363	0.202182	+	0.979348i	$0.435197\pi$
$114$	0	0
$115$	3.70156	0.345172
$116$	−0.298438	−0.0277093
$117$	0	0
$118$	0.596876	0.0549469
$119$	0	0
$120$	0	0
$121$	−11.0000	−1.00000
$122$	10.0000	0.905357
$123$	0	0
$124$	6.00000	0.538816
$125$	13.7016	1.22550
$126$	0	0
$127$	17.1047	1.51780	0.758898	−	0.651210i	$-0.225738\pi$
$127$	17.1047	1.51780	0.758898	+	0.651210i	$0.225738\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	21.1047	1.85100
$131$	−15.4031	−1.34578	−0.672889	−	0.739744i	$-0.734947\pi$
$131$	−15.4031	−1.34578	−0.672889	+	0.739744i	$0.734947\pi$
$132$	0	0
$133$	5.40312	0.468510
$134$	4.00000	0.345547
$135$	0	0
$136$	0	0
$137$	−0.895314	−0.0764918	−0.0382459	−	0.999268i	$-0.512177\pi$
$137$	−0.895314	−0.0764918	−0.0382459	+	0.999268i	$0.512177\pi$
$138$	0	0
$139$	−21.1047	−1.79008	−0.895038	−	0.445990i	$-0.852852\pi$
$139$	−21.1047	−1.79008	−0.895038	+	0.445990i	$0.852852\pi$
$140$	−3.70156	−0.312839
$141$	0	0
$142$	−7.40312	−0.621256
$143$	0	0
$144$	0	0
$145$	−1.10469	−0.0917392
$146$	−5.40312	−0.447166
$147$	0	0
$148$	3.70156	0.304267
$149$	17.4031	1.42572	0.712860	−	0.701307i	$-0.247400\pi$
$149$	17.4031	1.42572	0.712860	+	0.701307i	$0.247400\pi$
$150$	0	0
$151$	9.10469	0.740929	0.370464	−	0.928847i	$-0.379199\pi$
$151$	9.10469	0.740929	0.370464	+	0.928847i	$0.379199\pi$
$152$	−5.40312	−0.438251
$153$	0	0
$154$	0	0
$155$	22.2094	1.78390
$156$	0	0
$157$	−13.4031	−1.06969	−0.534843	−	0.844952i	$-0.679629\pi$
$157$	−13.4031	−1.06969	−0.534843	+	0.844952i	$0.679629\pi$
$158$	0	0
$159$	0	0
$160$	3.70156	0.292634
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−12.0000	−0.939913	−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000	−0.939913	−0.469956	+	0.882690i	$0.655730\pi$
$164$	−7.70156	−0.601391
$165$	0	0
$166$	2.59688	0.201557
$167$	21.4031	1.65622	0.828112	−	0.560563i	$-0.189415\pi$
$167$	21.4031	1.65622	0.828112	+	0.560563i	$0.189415\pi$
$168$	0	0
$169$	19.5078	1.50060
$170$	0	0
$171$	0	0
$172$	5.70156	0.434740
$173$	3.40312	0.258735	0.129367	−	0.991597i	$-0.458705\pi$
$173$	3.40312	0.258735	0.129367	+	0.991597i	$0.458705\pi$
$174$	0	0
$175$	−8.70156	−0.657776
$176$	0	0
$177$	0	0
$178$	15.4031	1.15451
$179$	−5.70156	−0.426155	−0.213077	−	0.977035i	$-0.568349\pi$
$179$	−5.70156	−0.426155	−0.213077	+	0.977035i	$0.568349\pi$
$180$	0	0
$181$	16.8062	1.24920	0.624599	−	0.780945i	$-0.285262\pi$
$181$	16.8062	1.24920	0.624599	+	0.780945i	$0.285262\pi$
$182$	−5.70156	−0.422628
$183$	0	0
$184$	1.00000	0.0737210
$185$	13.7016	1.00736
$186$	0	0
$187$	0	0
$188$	−3.70156	−0.269964
$189$	0	0
$190$	−20.0000	−1.45095
$191$	−22.8062	−1.65020	−0.825101	−	0.564985i	$-0.808882\pi$
$191$	−22.8062	−1.65020	−0.825101	+	0.564985i	$0.808882\pi$
$192$	0	0
$193$	−7.10469	−0.511407	−0.255703	−	0.966755i	$-0.582307\pi$
$193$	−7.10469	−0.511407	−0.255703	+	0.966755i	$0.582307\pi$
$194$	−1.10469	−0.0793119
$195$	0	0
$196$	1.00000	0.0714286
$197$	−6.50781	−0.463662	−0.231831	−	0.972756i	$-0.574472\pi$
$197$	−6.50781	−0.463662	−0.231831	+	0.972756i	$0.574472\pi$
$198$	0	0
$199$	1.10469	0.0783091	0.0391546	−	0.999233i	$-0.487534\pi$
$199$	1.10469	0.0783091	0.0391546	+	0.999233i	$0.487534\pi$
$200$	8.70156	0.615293
$201$	0	0
$202$	−4.59688	−0.323435
$203$	0.298438	0.0209462
$204$	0	0
$205$	−28.5078	−1.99107
$206$	−17.1047	−1.19174
$207$	0	0
$208$	5.70156	0.395332
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	−9.40312	−0.645809
$213$	0	0
$214$	−14.8062	−1.01213
$215$	21.1047	1.43933
$216$	0	0
$217$	−6.00000	−0.407307
$218$	15.7016	1.06344
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−20.8062	−1.39329	−0.696645	−	0.717416i	$-0.745325\pi$
$223$	−20.8062	−1.39329	−0.696645	+	0.717416i	$0.745325\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	4.29844	0.285928
$227$	26.5078	1.75939	0.879693	−	0.475543i	$-0.157748\pi$
$227$	26.5078	1.75939	0.879693	+	0.475543i	$0.157748\pi$
$228$	0	0
$229$	28.2094	1.86413	0.932064	−	0.362294i	$-0.118006\pi$
$229$	28.2094	1.86413	0.932064	+	0.362294i	$0.118006\pi$
$230$	3.70156	0.244074
$231$	0	0
$232$	−0.298438	−0.0195934
$233$	−28.8062	−1.88716	−0.943580	−	0.331145i	$-0.892565\pi$
$233$	−28.8062	−1.88716	−0.943580	+	0.331145i	$0.892565\pi$
$234$	0	0
$235$	−13.7016	−0.893791
$236$	0.596876	0.0388533
$237$	0	0
$238$	0	0
$239$	18.8062	1.21648	0.608238	−	0.793755i	$-0.291877\pi$
$239$	18.8062	1.21648	0.608238	+	0.793755i	$0.291877\pi$
$240$	0	0
$241$	5.10469	0.328822	0.164411	−	0.986392i	$-0.447428\pi$
$241$	5.10469	0.328822	0.164411	+	0.986392i	$0.447428\pi$
$242$	−11.0000	−0.707107
$243$	0	0
$244$	10.0000	0.640184
$245$	3.70156	0.236484
$246$	0	0
$247$	−30.8062	−1.96015
$248$	6.00000	0.381000
$249$	0	0
$250$	13.7016	0.866563
$251$	−13.9109	−0.878050	−0.439025	−	0.898475i	$-0.644676\pi$
$251$	−13.9109	−0.878050	−0.439025	+	0.898475i	$0.644676\pi$
$252$	0	0
$253$	0	0
$254$	17.1047	1.07324
$255$	0	0
$256$	1.00000	0.0625000
$257$	−23.6125	−1.47291	−0.736454	−	0.676488i	$-0.763501\pi$
$257$	−23.6125	−1.47291	−0.736454	+	0.676488i	$0.763501\pi$
$258$	0	0
$259$	−3.70156	−0.230004
$260$	21.1047	1.30886
$261$	0	0
$262$	−15.4031	−0.951608
$263$	16.5078	1.01792	0.508958	−	0.860792i	$-0.330031\pi$
$263$	16.5078	1.01792	0.508958	+	0.860792i	$0.330031\pi$
$264$	0	0
$265$	−34.8062	−2.13813
$266$	5.40312	0.331287
$267$	0	0
$268$	4.00000	0.244339
$269$	3.40312	0.207492	0.103746	−	0.994604i	$-0.466917\pi$
$269$	3.40312	0.207492	0.103746	+	0.994604i	$0.466917\pi$
$270$	0	0
$271$	17.4031	1.05716	0.528582	−	0.848882i	$-0.322724\pi$
$271$	17.4031	1.05716	0.528582	+	0.848882i	$0.322724\pi$
$272$	0	0
$273$	0	0
$274$	−0.895314	−0.0540879
$275$	0	0
$276$	0	0
$277$	−13.4031	−0.805316	−0.402658	−	0.915351i	$-0.631914\pi$
$277$	−13.4031	−0.805316	−0.402658	+	0.915351i	$0.631914\pi$
$278$	−21.1047	−1.26577
$279$	0	0
$280$	−3.70156	−0.221211
$281$	−4.29844	−0.256423	−0.128212	−	0.991747i	$-0.540924\pi$
$281$	−4.29844	−0.256423	−0.128212	+	0.991747i	$0.540924\pi$
$282$	0	0
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	−7.40312	−0.439295
$285$	0	0
$286$	0	0
$287$	7.70156	0.454609
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−1.10469	−0.0648694
$291$	0	0
$292$	−5.40312	−0.316194
$293$	3.19375	0.186581	0.0932905	−	0.995639i	$-0.470261\pi$
$293$	3.19375	0.186581	0.0932905	+	0.995639i	$0.470261\pi$
$294$	0	0
$295$	2.20937	0.128635
$296$	3.70156	0.215149
$297$	0	0
$298$	17.4031	1.00814
$299$	5.70156	0.329730
$300$	0	0
$301$	−5.70156	−0.328633
$302$	9.10469	0.523916
$303$	0	0
$304$	−5.40312	−0.309890
$305$	37.0156	2.11951
$306$	0	0
$307$	5.10469	0.291340	0.145670	−	0.989333i	$-0.453466\pi$
$307$	5.10469	0.291340	0.145670	+	0.989333i	$0.453466\pi$
$308$	0	0
$309$	0	0
$310$	22.2094	1.26141
$311$	16.2094	0.919149	0.459575	−	0.888139i	$-0.348002\pi$
$311$	16.2094	0.919149	0.459575	+	0.888139i	$0.348002\pi$
$312$	0	0
$313$	−4.59688	−0.259831	−0.129915	−	0.991525i	$-0.541471\pi$
$313$	−4.59688	−0.259831	−0.129915	+	0.991525i	$0.541471\pi$
$314$	−13.4031	−0.756382
$315$	0	0
$316$	0	0
$317$	−7.70156	−0.432563	−0.216281	−	0.976331i	$-0.569393\pi$
$317$	−7.70156	−0.432563	−0.216281	+	0.976331i	$0.569393\pi$
$318$	0	0
$319$	0	0
$320$	3.70156	0.206924
$321$	0	0
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	0	0
$325$	49.6125	2.75201
$326$	−12.0000	−0.664619
$327$	0	0
$328$	−7.70156	−0.425248
$329$	3.70156	0.204074
$330$	0	0
$331$	−17.6125	−0.968070	−0.484035	−	0.875049i	$-0.660829\pi$
$331$	−17.6125	−0.968070	−0.484035	+	0.875049i	$0.660829\pi$
$332$	2.59688	0.142522
$333$	0	0
$334$	21.4031	1.17113
$335$	14.8062	0.808952
$336$	0	0
$337$	17.4031	0.948009	0.474004	−	0.880523i	$-0.342808\pi$
$337$	17.4031	0.948009	0.474004	+	0.880523i	$0.342808\pi$
$338$	19.5078	1.06109
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.70156	0.307408
$345$	0	0
$346$	3.40312	0.182953
$347$	−20.5078	−1.10092	−0.550458	−	0.834863i	$-0.685547\pi$
$347$	−20.5078	−1.10092	−0.550458	+	0.834863i	$0.685547\pi$
$348$	0	0
$349$	34.2094	1.83119	0.915593	−	0.402107i	$-0.131722\pi$
$349$	34.2094	1.83119	0.915593	+	0.402107i	$0.131722\pi$
$350$	−8.70156	−0.465118
$351$	0	0
$352$	0	0
$353$	33.3141	1.77313	0.886564	−	0.462606i	$-0.153085\pi$
$353$	33.3141	1.77313	0.886564	+	0.462606i	$0.153085\pi$
$354$	0	0
$355$	−27.4031	−1.45441
$356$	15.4031	0.816364
$357$	0	0
$358$	−5.70156	−0.301337
$359$	−9.70156	−0.512029	−0.256014	−	0.966673i	$-0.582409\pi$
$359$	−9.70156	−0.512029	−0.256014	+	0.966673i	$0.582409\pi$
$360$	0	0
$361$	10.1938	0.536513
$362$	16.8062	0.883317
$363$	0	0
$364$	−5.70156	−0.298843
$365$	−20.0000	−1.04685
$366$	0	0
$367$	15.9109	0.830544	0.415272	−	0.909697i	$-0.363686\pi$
$367$	15.9109	0.830544	0.415272	+	0.909697i	$0.363686\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	13.7016	0.712310
$371$	9.40312	0.488186
$372$	0	0
$373$	−4.80625	−0.248858	−0.124429	−	0.992229i	$-0.539710\pi$
$373$	−4.80625	−0.248858	−0.124429	+	0.992229i	$0.539710\pi$
$374$	0	0
$375$	0	0
$376$	−3.70156	−0.190893
$377$	−1.70156	−0.0876349
$378$	0	0
$379$	−17.7016	−0.909268	−0.454634	−	0.890678i	$-0.650230\pi$
$379$	−17.7016	−0.909268	−0.454634	+	0.890678i	$0.650230\pi$
$380$	−20.0000	−1.02598
$381$	0	0
$382$	−22.8062	−1.16687
$383$	−6.80625	−0.347783	−0.173892	−	0.984765i	$-0.555634\pi$
$383$	−6.80625	−0.347783	−0.173892	+	0.984765i	$0.555634\pi$
$384$	0	0
$385$	0	0
$386$	−7.10469	−0.361619
$387$	0	0
$388$	−1.10469	−0.0560820
$389$	−36.8062	−1.86615	−0.933075	−	0.359681i	$-0.882886\pi$
$389$	−36.8062	−1.86615	−0.933075	+	0.359681i	$0.882886\pi$
$390$	0	0
$391$	0	0
$392$	1.00000	0.0505076
$393$	0	0
$394$	−6.50781	−0.327859
$395$	0	0
$396$	0	0
$397$	6.20937	0.311639	0.155820	−	0.987786i	$-0.450198\pi$
$397$	6.20937	0.311639	0.155820	+	0.987786i	$0.450198\pi$
$398$	1.10469	0.0553729
$399$	0	0
$400$	8.70156	0.435078
$401$	−28.8062	−1.43852	−0.719258	−	0.694743i	$-0.755518\pi$
$401$	−28.8062	−1.43852	−0.719258	+	0.694743i	$0.755518\pi$
$402$	0	0
$403$	34.2094	1.70409
$404$	−4.59688	−0.228703
$405$	0	0
$406$	0.298438	0.0148112
$407$	0	0
$408$	0	0
$409$	−32.2094	−1.59265	−0.796325	−	0.604868i	$-0.793226\pi$
$409$	−32.2094	−1.59265	−0.796325	+	0.604868i	$0.793226\pi$
$410$	−28.5078	−1.40790
$411$	0	0
$412$	−17.1047	−0.842687
$413$	−0.596876	−0.0293703
$414$	0	0
$415$	9.61250	0.471859
$416$	5.70156	0.279542
$417$	0	0
$418$	0	0
$419$	2.59688	0.126866	0.0634328	−	0.997986i	$-0.479795\pi$
$419$	2.59688	0.126866	0.0634328	+	0.997986i	$0.479795\pi$
$420$	0	0
$421$	15.7016	0.765247	0.382624	−	0.923904i	$-0.375021\pi$
$421$	15.7016	0.765247	0.382624	+	0.923904i	$0.375021\pi$
$422$	−12.0000	−0.584151
$423$	0	0
$424$	−9.40312	−0.456656
$425$	0	0
$426$	0	0
$427$	−10.0000	−0.483934
$428$	−14.8062	−0.715687
$429$	0	0
$430$	21.1047	1.01776
$431$	−5.10469	−0.245884	−0.122942	−	0.992414i	$-0.539233\pi$
$431$	−5.10469	−0.245884	−0.122942	+	0.992414i	$0.539233\pi$
$432$	0	0
$433$	17.1047	0.821999	0.410999	−	0.911636i	$-0.365180\pi$
$433$	17.1047	0.821999	0.410999	+	0.911636i	$0.365180\pi$
$434$	−6.00000	−0.288009
$435$	0	0
$436$	15.7016	0.751968
$437$	−5.40312	−0.258466
$438$	0	0
$439$	3.19375	0.152429	0.0762147	−	0.997091i	$-0.475717\pi$
$439$	3.19375	0.152429	0.0762147	+	0.997091i	$0.475717\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−17.1047	−0.812668	−0.406334	−	0.913725i	$-0.633193\pi$
$443$	−17.1047	−0.812668	−0.406334	+	0.913725i	$0.633193\pi$
$444$	0	0
$445$	57.0156	2.70280
$446$	−20.8062	−0.985204
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−27.6125	−1.30311	−0.651557	−	0.758600i	$-0.725884\pi$
$449$	−27.6125	−1.30311	−0.651557	+	0.758600i	$0.725884\pi$
$450$	0	0
$451$	0	0
$452$	4.29844	0.202182
$453$	0	0
$454$	26.5078	1.24407
$455$	−21.1047	−0.989403
$456$	0	0
$457$	−16.2094	−0.758242	−0.379121	−	0.925347i	$-0.623774\pi$
$457$	−16.2094	−0.758242	−0.379121	+	0.925347i	$0.623774\pi$
$458$	28.2094	1.31814
$459$	0	0
$460$	3.70156	0.172586
$461$	−27.4031	−1.27629	−0.638145	−	0.769916i	$-0.720298\pi$
$461$	−27.4031	−1.27629	−0.638145	+	0.769916i	$0.720298\pi$
$462$	0	0
$463$	−13.1047	−0.609026	−0.304513	−	0.952508i	$-0.598494\pi$
$463$	−13.1047	−0.609026	−0.304513	+	0.952508i	$0.598494\pi$
$464$	−0.298438	−0.0138546
$465$	0	0
$466$	−28.8062	−1.33442
$467$	8.29844	0.384006	0.192003	−	0.981394i	$-0.438502\pi$
$467$	8.29844	0.384006	0.192003	+	0.981394i	$0.438502\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	−13.7016	−0.632006
$471$	0	0
$472$	0.596876	0.0274734
$473$	0	0
$474$	0	0
$475$	−47.0156	−2.15722
$476$	0	0
$477$	0	0
$478$	18.8062	0.860178
$479$	−7.40312	−0.338257	−0.169129	−	0.985594i	$-0.554095\pi$
$479$	−7.40312	−0.338257	−0.169129	+	0.985594i	$0.554095\pi$
$480$	0	0
$481$	21.1047	0.962291
$482$	5.10469	0.232512
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−4.08907	−0.185675
$486$	0	0
$487$	−13.1047	−0.593830	−0.296915	−	0.954904i	$-0.595958\pi$
$487$	−13.1047	−0.593830	−0.296915	+	0.954904i	$0.595958\pi$
$488$	10.0000	0.452679
$489$	0	0
$490$	3.70156	0.167220
$491$	17.6125	0.794841	0.397420	−	0.917637i	$-0.369905\pi$
$491$	17.6125	0.794841	0.397420	+	0.917637i	$0.369905\pi$
$492$	0	0
$493$	0	0
$494$	−30.8062	−1.38604
$495$	0	0
$496$	6.00000	0.269408
$497$	7.40312	0.332076
$498$	0	0
$499$	37.0156	1.65705	0.828523	−	0.559954i	$-0.189181\pi$
$499$	37.0156	1.65705	0.828523	+	0.559954i	$0.189181\pi$
$500$	13.7016	0.612752
$501$	0	0
$502$	−13.9109	−0.620875
$503$	11.4031	0.508440	0.254220	−	0.967146i	$-0.418181\pi$
$503$	11.4031	0.508440	0.254220	+	0.967146i	$0.418181\pi$
$504$	0	0
$505$	−17.0156	−0.757185
$506$	0	0
$507$	0	0
$508$	17.1047	0.758898
$509$	14.8062	0.656275	0.328138	−	0.944630i	$-0.393579\pi$
$509$	14.8062	0.656275	0.328138	+	0.944630i	$0.393579\pi$
$510$	0	0
$511$	5.40312	0.239020
$512$	1.00000	0.0441942
$513$	0	0
$514$	−23.6125	−1.04150
$515$	−63.3141	−2.78995
$516$	0	0
$517$	0	0
$518$	−3.70156	−0.162637
$519$	0	0
$520$	21.1047	0.925502
$521$	−15.4031	−0.674823	−0.337412	−	0.941357i	$-0.609551\pi$
$521$	−15.4031	−0.674823	−0.337412	+	0.941357i	$0.609551\pi$
$522$	0	0
$523$	−38.4187	−1.67993	−0.839967	−	0.542637i	$-0.817426\pi$
$523$	−38.4187	−1.67993	−0.839967	+	0.542637i	$0.817426\pi$
$524$	−15.4031	−0.672889
$525$	0	0
$526$	16.5078	0.719775
$527$	0	0
$528$	0	0
$529$	1.00000	0.0434783
$530$	−34.8062	−1.51189
$531$	0	0
$532$	5.40312	0.234255
$533$	−43.9109	−1.90199
$534$	0	0
$535$	−54.8062	−2.36948
$536$	4.00000	0.172774
$537$	0	0
$538$	3.40312	0.146719
$539$	0	0
$540$	0	0
$541$	−20.2094	−0.868869	−0.434434	−	0.900703i	$-0.643052\pi$
$541$	−20.2094	−0.868869	−0.434434	+	0.900703i	$0.643052\pi$
$542$	17.4031	0.747528
$543$	0	0
$544$	0	0
$545$	58.1203	2.48960
$546$	0	0
$547$	−14.2094	−0.607549	−0.303774	−	0.952744i	$-0.598247\pi$
$547$	−14.2094	−0.607549	−0.303774	+	0.952744i	$0.598247\pi$
$548$	−0.895314	−0.0382459
$549$	0	0
$550$	0	0
$551$	1.61250	0.0686947
$552$	0	0
$553$	0	0
$554$	−13.4031	−0.569444
$555$	0	0
$556$	−21.1047	−0.895038
$557$	−21.4031	−0.906879	−0.453440	−	0.891287i	$-0.649803\pi$
$557$	−21.4031	−0.906879	−0.453440	+	0.891287i	$0.649803\pi$
$558$	0	0
$559$	32.5078	1.37493
$560$	−3.70156	−0.156420
$561$	0	0
$562$	−4.29844	−0.181319
$563$	−44.7172	−1.88460	−0.942302	−	0.334763i	$-0.891344\pi$
$563$	−44.7172	−1.88460	−0.942302	+	0.334763i	$0.891344\pi$
$564$	0	0
$565$	15.9109	0.669378
$566$	−10.0000	−0.420331
$567$	0	0
$568$	−7.40312	−0.310628
$569$	−11.1047	−0.465533	−0.232766	−	0.972533i	$-0.574778\pi$
$569$	−11.1047	−0.465533	−0.232766	+	0.972533i	$0.574778\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	7.70156	0.321457
$575$	8.70156	0.362880
$576$	0	0
$577$	−23.6125	−0.983001	−0.491501	−	0.870877i	$-0.663551\pi$
$577$	−23.6125	−0.983001	−0.491501	+	0.870877i	$0.663551\pi$
$578$	−17.0000	−0.707107
$579$	0	0
$580$	−1.10469	−0.0458696
$581$	−2.59688	−0.107737
$582$	0	0
$583$	0	0
$584$	−5.40312	−0.223583
$585$	0	0
$586$	3.19375	0.131933
$587$	1.19375	0.0492714	0.0246357	−	0.999696i	$-0.492157\pi$
$587$	1.19375	0.0492714	0.0246357	+	0.999696i	$0.492157\pi$
$588$	0	0
$589$	−32.4187	−1.33579
$590$	2.20937	0.0909584
$591$	0	0
$592$	3.70156	0.152133
$593$	41.9109	1.72108	0.860538	−	0.509386i	$-0.170128\pi$
$593$	41.9109	1.72108	0.860538	+	0.509386i	$0.170128\pi$
$594$	0	0
$595$	0	0
$596$	17.4031	0.712860
$597$	0	0
$598$	5.70156	0.233154
$599$	29.6125	1.20993	0.604967	−	0.796251i	$-0.293186\pi$
$599$	29.6125	1.20993	0.604967	+	0.796251i	$0.293186\pi$
$600$	0	0
$601$	25.4031	1.03622	0.518108	−	0.855315i	$-0.326637\pi$
$601$	25.4031	1.03622	0.518108	+	0.855315i	$0.326637\pi$
$602$	−5.70156	−0.232378
$603$	0	0
$604$	9.10469	0.370464
$605$	−40.7172	−1.65539
$606$	0	0
$607$	−2.00000	−0.0811775	−0.0405887	−	0.999176i	$-0.512923\pi$
$607$	−2.00000	−0.0811775	−0.0405887	+	0.999176i	$0.512923\pi$
$608$	−5.40312	−0.219126
$609$	0	0
$610$	37.0156	1.49872
$611$	−21.1047	−0.853804
$612$	0	0
$613$	−25.3141	−1.02243	−0.511213	−	0.859454i	$-0.670804\pi$
$613$	−25.3141	−1.02243	−0.511213	+	0.859454i	$0.670804\pi$
$614$	5.10469	0.206008
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	0	0
$619$	−30.0000	−1.20580	−0.602901	−	0.797816i	$-0.705989\pi$
$619$	−30.0000	−1.20580	−0.602901	+	0.797816i	$0.705989\pi$
$620$	22.2094	0.891950
$621$	0	0
$622$	16.2094	0.649937
$623$	−15.4031	−0.617113
$624$	0	0
$625$	7.20937	0.288375
$626$	−4.59688	−0.183728
$627$	0	0
$628$	−13.4031	−0.534843
$629$	0	0
$630$	0	0
$631$	9.19375	0.365997	0.182999	−	0.983113i	$-0.441420\pi$
$631$	9.19375	0.365997	0.182999	+	0.983113i	$0.441420\pi$
$632$	0	0
$633$	0	0
$634$	−7.70156	−0.305868
$635$	63.3141	2.51254
$636$	0	0
$637$	5.70156	0.225904
$638$	0	0
$639$	0	0
$640$	3.70156	0.146317
$641$	11.7016	0.462184	0.231092	−	0.972932i	$-0.425770\pi$
$641$	11.7016	0.462184	0.231092	+	0.972932i	$0.425770\pi$
$642$	0	0
$643$	3.19375	0.125949	0.0629746	−	0.998015i	$-0.479941\pi$
$643$	3.19375	0.125949	0.0629746	+	0.998015i	$0.479941\pi$
$644$	−1.00000	−0.0394055
$645$	0	0
$646$	0	0
$647$	4.20937	0.165488	0.0827438	−	0.996571i	$-0.473632\pi$
$647$	4.20937	0.165488	0.0827438	+	0.996571i	$0.473632\pi$
$648$	0	0
$649$	0	0
$650$	49.6125	1.94596
$651$	0	0
$652$	−12.0000	−0.469956
$653$	−47.1047	−1.84335	−0.921674	−	0.387964i	$-0.873178\pi$
$653$	−47.1047	−1.84335	−0.921674	+	0.387964i	$0.873178\pi$
$654$	0	0
$655$	−57.0156	−2.22778
$656$	−7.70156	−0.300695
$657$	0	0
$658$	3.70156	0.144302
$659$	−25.6125	−0.997721	−0.498861	−	0.866682i	$-0.666248\pi$
$659$	−25.6125	−0.997721	−0.498861	+	0.866682i	$0.666248\pi$
$660$	0	0
$661$	−20.8062	−0.809269	−0.404635	−	0.914478i	$-0.632601\pi$
$661$	−20.8062	−0.809269	−0.404635	+	0.914478i	$0.632601\pi$
$662$	−17.6125	−0.684529
$663$	0	0
$664$	2.59688	0.100778
$665$	20.0000	0.775567
$666$	0	0
$667$	−0.298438	−0.0115556
$668$	21.4031	0.828112
$669$	0	0
$670$	14.8062	0.572015
$671$	0	0
$672$	0	0
$673$	−8.29844	−0.319881	−0.159941	−	0.987127i	$-0.551130\pi$
$673$	−8.29844	−0.319881	−0.159941	+	0.987127i	$0.551130\pi$
$674$	17.4031	0.670343
$675$	0	0
$676$	19.5078	0.750300
$677$	3.19375	0.122746	0.0613729	−	0.998115i	$-0.480452\pi$
$677$	3.19375	0.122746	0.0613729	+	0.998115i	$0.480452\pi$
$678$	0	0
$679$	1.10469	0.0423940
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	−3.31406	−0.126624
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	5.70156	0.217370
$689$	−53.6125	−2.04247
$690$	0	0
$691$	−41.7016	−1.58640	−0.793201	−	0.608960i	$-0.791587\pi$
$691$	−41.7016	−1.58640	−0.793201	+	0.608960i	$0.791587\pi$
$692$	3.40312	0.129367
$693$	0	0
$694$	−20.5078	−0.778466
$695$	−78.1203	−2.96327
$696$	0	0
$697$	0	0
$698$	34.2094	1.29484
$699$	0	0
$700$	−8.70156	−0.328888
$701$	−14.5969	−0.551316	−0.275658	−	0.961256i	$-0.588896\pi$
$701$	−14.5969	−0.551316	−0.275658	+	0.961256i	$0.588896\pi$
$702$	0	0
$703$	−20.0000	−0.754314
$704$	0	0
$705$	0	0
$706$	33.3141	1.25379
$707$	4.59688	0.172883
$708$	0	0
$709$	−15.1938	−0.570613	−0.285307	−	0.958436i	$-0.592095\pi$
$709$	−15.1938	−0.570613	−0.285307	+	0.958436i	$0.592095\pi$
$710$	−27.4031	−1.02842
$711$	0	0
$712$	15.4031	0.577256
$713$	6.00000	0.224702
$714$	0	0
$715$	0	0
$716$	−5.70156	−0.213077
$717$	0	0
$718$	−9.70156	−0.362059
$719$	−14.5078	−0.541050	−0.270525	−	0.962713i	$-0.587197\pi$
$719$	−14.5078	−0.541050	−0.270525	+	0.962713i	$0.587197\pi$
$720$	0	0
$721$	17.1047	0.637012
$722$	10.1938	0.379372
$723$	0	0
$724$	16.8062	0.624599
$725$	−2.59688	−0.0964455
$726$	0	0
$727$	−44.0000	−1.63187	−0.815935	−	0.578144i	$-0.803777\pi$
$727$	−44.0000	−1.63187	−0.815935	+	0.578144i	$0.803777\pi$
$728$	−5.70156	−0.211314
$729$	0	0
$730$	−20.0000	−0.740233
$731$	0	0
$732$	0	0
$733$	31.0156	1.14559	0.572794	−	0.819699i	$-0.305859\pi$
$733$	31.0156	1.14559	0.572794	+	0.819699i	$0.305859\pi$
$734$	15.9109	0.587283
$735$	0	0
$736$	1.00000	0.0368605
$737$	0	0
$738$	0	0
$739$	38.2094	1.40555	0.702777	−	0.711410i	$-0.251943\pi$
$739$	38.2094	1.40555	0.702777	+	0.711410i	$0.251943\pi$
$740$	13.7016	0.503679
$741$	0	0
$742$	9.40312	0.345200
$743$	21.6125	0.792886	0.396443	−	0.918059i	$-0.370245\pi$
$743$	21.6125	0.792886	0.396443	+	0.918059i	$0.370245\pi$
$744$	0	0
$745$	64.4187	2.36012
$746$	−4.80625	−0.175969
$747$	0	0
$748$	0	0
$749$	14.8062	0.541009
$750$	0	0
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−3.70156	−0.134982
$753$	0	0
$754$	−1.70156	−0.0619672
$755$	33.7016	1.22653
$756$	0	0
$757$	7.19375	0.261461	0.130731	−	0.991418i	$-0.458268\pi$
$757$	7.19375	0.261461	0.130731	+	0.991418i	$0.458268\pi$
$758$	−17.7016	−0.642950
$759$	0	0
$760$	−20.0000	−0.725476
$761$	−15.6125	−0.565953	−0.282976	−	0.959127i	$-0.591322\pi$
$761$	−15.6125	−0.565953	−0.282976	+	0.959127i	$0.591322\pi$
$762$	0	0
$763$	−15.7016	−0.568435
$764$	−22.8062	−0.825101
$765$	0	0
$766$	−6.80625	−0.245920
$767$	3.40312	0.122880
$768$	0	0
$769$	−29.1047	−1.04954	−0.524771	−	0.851243i	$-0.675849\pi$
$769$	−29.1047	−1.04954	−0.524771	+	0.851243i	$0.675849\pi$
$770$	0	0
$771$	0	0
$772$	−7.10469	−0.255703
$773$	−44.7172	−1.60837	−0.804183	−	0.594382i	$-0.797397\pi$
$773$	−44.7172	−1.60837	−0.804183	+	0.594382i	$0.797397\pi$
$774$	0	0
$775$	52.2094	1.87542
$776$	−1.10469	−0.0396559
$777$	0	0
$778$	−36.8062	−1.31957
$779$	41.6125	1.49092
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	−49.6125	−1.77075
$786$	0	0
$787$	49.8219	1.77596	0.887979	−	0.459884i	$-0.152109\pi$
$787$	49.8219	1.77596	0.887979	+	0.459884i	$0.152109\pi$
$788$	−6.50781	−0.231831
$789$	0	0
$790$	0	0
$791$	−4.29844	−0.152835
$792$	0	0
$793$	57.0156	2.02468
$794$	6.20937	0.220362
$795$	0	0
$796$	1.10469	0.0391546
$797$	31.7016	1.12293	0.561463	−	0.827502i	$-0.310239\pi$
$797$	31.7016	1.12293	0.561463	+	0.827502i	$0.310239\pi$
$798$	0	0
$799$	0	0
$800$	8.70156	0.307647
$801$	0	0
$802$	−28.8062	−1.01718
$803$	0	0
$804$	0	0
$805$	−3.70156	−0.130463
$806$	34.2094	1.20497
$807$	0	0
$808$	−4.59688	−0.161718
$809$	21.4031	0.752494	0.376247	−	0.926519i	$-0.377214\pi$
$809$	21.4031	0.752494	0.376247	+	0.926519i	$0.377214\pi$
$810$	0	0
$811$	11.3141	0.397290	0.198645	−	0.980071i	$-0.436346\pi$
$811$	11.3141	0.397290	0.198645	+	0.980071i	$0.436346\pi$
$812$	0.298438	0.0104731
$813$	0	0
$814$	0	0
$815$	−44.4187	−1.55592
$816$	0	0
$817$	−30.8062	−1.07777
$818$	−32.2094	−1.12617
$819$	0	0
$820$	−28.5078	−0.995536
$821$	−6.00000	−0.209401	−0.104701	−	0.994504i	$-0.533388\pi$
$821$	−6.00000	−0.209401	−0.104701	+	0.994504i	$0.533388\pi$
$822$	0	0
$823$	46.7172	1.62846	0.814229	−	0.580543i	$-0.197160\pi$
$823$	46.7172	1.62846	0.814229	+	0.580543i	$0.197160\pi$
$824$	−17.1047	−0.595870
$825$	0	0
$826$	−0.596876	−0.0207680
$827$	33.0156	1.14807	0.574033	−	0.818832i	$-0.305378\pi$
$827$	33.0156	1.14807	0.574033	+	0.818832i	$0.305378\pi$
$828$	0	0
$829$	−30.2094	−1.04921	−0.524607	−	0.851344i	$-0.675788\pi$
$829$	−30.2094	−1.04921	−0.524607	+	0.851344i	$0.675788\pi$
$830$	9.61250	0.333655
$831$	0	0
$832$	5.70156	0.197666
$833$	0	0
$834$	0	0
$835$	79.2250	2.74169
$836$	0	0
$837$	0	0
$838$	2.59688	0.0897076
$839$	19.4031	0.669870	0.334935	−	0.942241i	$-0.391286\pi$
$839$	19.4031	0.669870	0.334935	+	0.942241i	$0.391286\pi$
$840$	0	0
$841$	−28.9109	−0.996929
$842$	15.7016	0.541112
$843$	0	0
$844$	−12.0000	−0.413057
$845$	72.2094	2.48408
$846$	0	0
$847$	11.0000	0.377964
$848$	−9.40312	−0.322905
$849$	0	0
$850$	0	0
$851$	3.70156	0.126888
$852$	0	0
$853$	41.1047	1.40740	0.703699	−	0.710498i	$-0.251530\pi$
$853$	41.1047	1.40740	0.703699	+	0.710498i	$0.251530\pi$
$854$	−10.0000	−0.342193
$855$	0	0
$856$	−14.8062	−0.506067
$857$	−2.08907	−0.0713611	−0.0356806	−	0.999363i	$-0.511360\pi$
$857$	−2.08907	−0.0713611	−0.0356806	+	0.999363i	$0.511360\pi$
$858$	0	0
$859$	−3.91093	−0.133439	−0.0667197	−	0.997772i	$-0.521253\pi$
$859$	−3.91093	−0.133439	−0.0667197	+	0.997772i	$0.521253\pi$
$860$	21.1047	0.719664
$861$	0	0
$862$	−5.10469	−0.173866
$863$	10.8062	0.367849	0.183924	−	0.982940i	$-0.441120\pi$
$863$	10.8062	0.367849	0.183924	+	0.982940i	$0.441120\pi$
$864$	0	0
$865$	12.5969	0.428307
$866$	17.1047	0.581241
$867$	0	0
$868$	−6.00000	−0.203653
$869$	0	0
$870$	0	0
$871$	22.8062	0.772760
$872$	15.7016	0.531722
$873$	0	0
$874$	−5.40312	−0.182763
$875$	−13.7016	−0.463197
$876$	0	0
$877$	−48.2094	−1.62791	−0.813957	−	0.580925i	$-0.802691\pi$
$877$	−48.2094	−1.62791	−0.813957	+	0.580925i	$0.802691\pi$
$878$	3.19375	0.107784
$879$	0	0
$880$	0	0
$881$	10.2094	0.343963	0.171981	−	0.985100i	$-0.444983\pi$
$881$	10.2094	0.343963	0.171981	+	0.985100i	$0.444983\pi$
$882$	0	0
$883$	16.5969	0.558529	0.279265	−	0.960214i	$-0.409909\pi$
$883$	16.5969	0.558529	0.279265	+	0.960214i	$0.409909\pi$
$884$	0	0
$885$	0	0
$886$	−17.1047	−0.574643
$887$	52.2094	1.75302	0.876510	−	0.481384i	$-0.159866\pi$
$887$	52.2094	1.75302	0.876510	+	0.481384i	$0.159866\pi$
$888$	0	0
$889$	−17.1047	−0.573673
$890$	57.0156	1.91117
$891$	0	0
$892$	−20.8062	−0.696645
$893$	20.0000	0.669274
$894$	0	0
$895$	−21.1047	−0.705452
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−27.6125	−0.921441
$899$	−1.79063	−0.0597208
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	4.29844	0.142964
$905$	62.2094	2.06791
$906$	0	0
$907$	−2.89531	−0.0961373	−0.0480687	−	0.998844i	$-0.515307\pi$
$907$	−2.89531	−0.0961373	−0.0480687	+	0.998844i	$0.515307\pi$
$908$	26.5078	0.879693
$909$	0	0
$910$	−21.1047	−0.699614
$911$	−1.70156	−0.0563753	−0.0281876	−	0.999603i	$-0.508974\pi$
$911$	−1.70156	−0.0563753	−0.0281876	+	0.999603i	$0.508974\pi$
$912$	0	0
$913$	0	0
$914$	−16.2094	−0.536158
$915$	0	0
$916$	28.2094	0.932064
$917$	15.4031	0.508656
$918$	0	0
$919$	1.19375	0.0393782	0.0196891	−	0.999806i	$-0.493732\pi$
$919$	1.19375	0.0393782	0.0196891	+	0.999806i	$0.493732\pi$
$920$	3.70156	0.122037
$921$	0	0
$922$	−27.4031	−0.902474
$923$	−42.2094	−1.38934
$924$	0	0
$925$	32.2094	1.05904
$926$	−13.1047	−0.430647
$927$	0	0
$928$	−0.298438	−0.00979670
$929$	−23.7016	−0.777623	−0.388812	−	0.921317i	$-0.627114\pi$
$929$	−23.7016	−0.777623	−0.388812	+	0.921317i	$0.627114\pi$
$930$	0	0
$931$	−5.40312	−0.177080
$932$	−28.8062	−0.943580
$933$	0	0
$934$	8.29844	0.271533
$935$	0	0
$936$	0	0
$937$	47.3141	1.54568	0.772841	−	0.634599i	$-0.218835\pi$
$937$	47.3141	1.54568	0.772841	+	0.634599i	$0.218835\pi$
$938$	−4.00000	−0.130605
$939$	0	0
$940$	−13.7016	−0.446896
$941$	9.31406	0.303630	0.151815	−	0.988409i	$-0.451488\pi$
$941$	9.31406	0.303630	0.151815	+	0.988409i	$0.451488\pi$
$942$	0	0
$943$	−7.70156	−0.250797
$944$	0.596876	0.0194267
$945$	0	0
$946$	0	0
$947$	−3.49219	−0.113481	−0.0567405	−	0.998389i	$-0.518071\pi$
$947$	−3.49219	−0.113481	−0.0567405	+	0.998389i	$0.518071\pi$
$948$	0	0
$949$	−30.8062	−1.00001
$950$	−47.0156	−1.52539
$951$	0	0
$952$	0	0
$953$	20.8062	0.673980	0.336990	−	0.941508i	$-0.390591\pi$
$953$	20.8062	0.673980	0.336990	+	0.941508i	$0.390591\pi$
$954$	0	0
$955$	−84.4187	−2.73173
$956$	18.8062	0.608238
$957$	0	0
$958$	−7.40312	−0.239184
$959$	0.895314	0.0289112
$960$	0	0
$961$	5.00000	0.161290
$962$	21.1047	0.680442
$963$	0	0
$964$	5.10469	0.164411
$965$	−26.2984	−0.846577
$966$	0	0
$967$	20.4187	0.656623	0.328311	−	0.944570i	$-0.393521\pi$
$967$	20.4187	0.656623	0.328311	+	0.944570i	$0.393521\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−4.08907	−0.131292
$971$	33.4031	1.07196	0.535979	−	0.844232i	$-0.319943\pi$
$971$	33.4031	1.07196	0.535979	+	0.844232i	$0.319943\pi$
$972$	0	0
$973$	21.1047	0.676585
$974$	−13.1047	−0.419901
$975$	0	0
$976$	10.0000	0.320092
$977$	40.2984	1.28926	0.644631	−	0.764494i	$-0.277011\pi$
$977$	40.2984	1.28926	0.644631	+	0.764494i	$0.277011\pi$
$978$	0	0
$979$	0	0
$980$	3.70156	0.118242
$981$	0	0
$982$	17.6125	0.562037
$983$	55.8219	1.78044	0.890221	−	0.455530i	$-0.150550\pi$
$983$	55.8219	1.78044	0.890221	+	0.455530i	$0.150550\pi$
$984$	0	0
$985$	−24.0891	−0.767541
$986$	0	0
$987$	0	0
$988$	−30.8062	−0.980077
$989$	5.70156	0.181299
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	7.40312	0.234813
$995$	4.08907	0.129632
$996$	0	0
$997$	48.5969	1.53908	0.769539	−	0.638600i	$-0.220486\pi$
$997$	48.5969	1.53908	0.769539	+	0.638600i	$0.220486\pi$
$998$	37.0156	1.17171
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.m.1.1	✓	2	3.2	odd	2
2898.2.a.bc.1.2		2	1.1	even	1	trivial
6762.2.a.bq.1.2		2	21.20	even	2
7728.2.a.z.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.70156 q^{5} -1.00000 q^{7} +1.00000 q^{8} +3.70156 q^{10} +5.70156 q^{13} -1.00000 q^{14} +1.00000 q^{16} -5.40312 q^{19} +3.70156 q^{20} +1.00000 q^{23} +8.70156 q^{25} +5.70156 q^{26} -1.00000 q^{28} -0.298438 q^{29} +6.00000 q^{31} +1.00000 q^{32} -3.70156 q^{35} +3.70156 q^{37} -5.40312 q^{38} +3.70156 q^{40} -7.70156 q^{41} +5.70156 q^{43} +1.00000 q^{46} -3.70156 q^{47} +1.00000 q^{49} +8.70156 q^{50} +5.70156 q^{52} -9.40312 q^{53} -1.00000 q^{56} -0.298438 q^{58} +0.596876 q^{59} +10.0000 q^{61} +6.00000 q^{62} +1.00000 q^{64} +21.1047 q^{65} +4.00000 q^{67} -3.70156 q^{70} -7.40312 q^{71} -5.40312 q^{73} +3.70156 q^{74} -5.40312 q^{76} +3.70156 q^{80} -7.70156 q^{82} +2.59688 q^{83} +5.70156 q^{86} +15.4031 q^{89} -5.70156 q^{91} +1.00000 q^{92} -3.70156 q^{94} -20.0000 q^{95} -1.10469 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + q^{5} - 2 q^{7} + 2 q^{8} + q^{10} + 5 q^{13} - 2 q^{14} + 2 q^{16} + 2 q^{19} + q^{20} + 2 q^{23} + 11 q^{25} + 5 q^{26} - 2 q^{28} - 7 q^{29} + 12 q^{31} + 2 q^{32} - q^{35} + q^{37} + 2 q^{38} + q^{40} - 9 q^{41} + 5 q^{43} + 2 q^{46} - q^{47} + 2 q^{49} + 11 q^{50} + 5 q^{52} - 6 q^{53} - 2 q^{56} - 7 q^{58} + 14 q^{59} + 20 q^{61} + 12 q^{62} + 2 q^{64} + 23 q^{65} + 8 q^{67} - q^{70} - 2 q^{71} + 2 q^{73} + q^{74} + 2 q^{76} + q^{80} - 9 q^{82} + 18 q^{83} + 5 q^{86} + 18 q^{89} - 5 q^{91} + 2 q^{92} - q^{94} - 40 q^{95} + 17 q^{97} + 2 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + q^5 - 2 * q^7 + 2 * q^8 + q^10 + 5 * q^13 - 2 * q^14 + 2 * q^16 + 2 * q^19 + q^20 + 2 * q^23 + 11 * q^25 + 5 * q^26 - 2 * q^28 - 7 * q^29 + 12 * q^31 + 2 * q^32 - q^35 + q^37 + 2 * q^38 + q^40 - 9 * q^41 + 5 * q^43 + 2 * q^46 - q^47 + 2 * q^49 + 11 * q^50 + 5 * q^52 - 6 * q^53 - 2 * q^56 - 7 * q^58 + 14 * q^59 + 20 * q^61 + 12 * q^62 + 2 * q^64 + 23 * q^65 + 8 * q^67 - q^70 - 2 * q^71 + 2 * q^73 + q^74 + 2 * q^76 + q^80 - 9 * q^82 + 18 * q^83 + 5 * q^86 + 18 * q^89 - 5 * q^91 + 2 * q^92 - q^94 - 40 * q^95 + 17 * q^97 + 2 * q^98

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