LMFDB - Embedded newform 2898.2.a.bi.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:	$4$
Coefficient field:	4.4.271296.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 18x^{2} - 8x + 24 $ x^4 - 18x^2 - 8x + 24
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.974808$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.974808 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.974808 q^{5} +1.00000 q^{7} +1.00000 q^{8} -0.974808 q^{10} +1.02519 q^{11} -0.655375 q^{13} +1.00000 q^{14} +1.00000 q^{16} +0.655375 q^{17} +0.319433 q^{19} -0.974808 q^{20} +1.02519 q^{22} +1.00000 q^{23} -4.04975 q^{25} -0.655375 q^{26} +1.00000 q^{28} +2.00000 q^{29} +9.70512 q^{31} +1.00000 q^{32} +0.655375 q^{34} -0.974808 q^{35} +11.3353 q^{37} +0.319433 q^{38} -0.974808 q^{40} -2.00000 q^{41} +4.28556 q^{43} +1.02519 q^{44} +1.00000 q^{46} +5.70512 q^{47} +1.00000 q^{49} -4.04975 q^{50} -0.655375 q^{52} -7.33531 q^{53} -0.999365 q^{55} +1.00000 q^{56} +2.00000 q^{58} -2.05038 q^{59} +2.92442 q^{61} +9.70512 q^{62} +1.00000 q^{64} +0.638865 q^{65} +7.61367 q^{67} +0.655375 q^{68} -0.974808 q^{70} +10.3101 q^{71} -6.36050 q^{73} +11.3353 q^{74} +0.319433 q^{76} +1.02519 q^{77} -2.36050 q^{79} -0.974808 q^{80} -2.00000 q^{82} -15.7297 q^{83} -0.638865 q^{85} +4.28556 q^{86} +1.02519 q^{88} -18.0152 q^{89} -0.655375 q^{91} +1.00000 q^{92} +5.70512 q^{94} -0.311385 q^{95} +16.0656 q^{97} +1.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 8 q^{22} + 4 q^{23} + 16 q^{25} + 4 q^{28} + 8 q^{29} + 4 q^{31} + 4 q^{32} + 4 q^{37} - 8 q^{41} + 8 q^{43} + 8 q^{44} + 4 q^{46} - 12 q^{47} + 4 q^{49} + 16 q^{50} + 12 q^{53} + 36 q^{55} + 4 q^{56} + 8 q^{58} - 16 q^{59} + 4 q^{62} + 4 q^{64} + 24 q^{67} - 4 q^{71} + 12 q^{73} + 4 q^{74} + 8 q^{77} + 28 q^{79} - 8 q^{82} + 8 q^{83} + 8 q^{86} + 8 q^{88} + 8 q^{89} + 4 q^{92} - 12 q^{94} - 36 q^{95} - 8 q^{97} + 4 q^{98}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 8 * q^22 + 4 * q^23 + 16 * q^25 + 4 * q^28 + 8 * q^29 + 4 * q^31 + 4 * q^32 + 4 * q^37 - 8 * q^41 + 8 * q^43 + 8 * q^44 + 4 * q^46 - 12 * q^47 + 4 * q^49 + 16 * q^50 + 12 * q^53 + 36 * q^55 + 4 * q^56 + 8 * q^58 - 16 * q^59 + 4 * q^62 + 4 * q^64 + 24 * q^67 - 4 * q^71 + 12 * q^73 + 4 * q^74 + 8 * q^77 + 28 * q^79 - 8 * q^82 + 8 * q^83 + 8 * q^86 + 8 * q^88 + 8 * q^89 + 4 * q^92 - 12 * q^94 - 36 * q^95 - 8 * q^97 + 4 * 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
$5$	−0.974808	−0.435947	−0.217974	−	0.975955i	$-0.569945\pi$
$5$	−0.974808	−0.435947	−0.217974	+	0.975955i	$0.569945\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.974808	−0.308261
$11$	1.02519	0.309107	0.154554	−	0.987984i	$-0.450606\pi$
$11$	1.02519	0.309107	0.154554	+	0.987984i	$0.450606\pi$
$12$	0	0
$13$	−0.655375	−0.181768	−0.0908842	−	0.995861i	$-0.528969\pi$
$13$	−0.655375	−0.181768	−0.0908842	+	0.995861i	$0.528969\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	0.655375	0.158952	0.0794759	−	0.996837i	$-0.474675\pi$
$17$	0.655375	0.158952	0.0794759	+	0.996837i	$0.474675\pi$
$18$	0	0
$19$	0.319433	0.0732829	0.0366414	−	0.999328i	$-0.488334\pi$
$19$	0.319433	0.0732829	0.0366414	+	0.999328i	$0.488334\pi$
$20$	−0.974808	−0.217974
$21$	0	0
$22$	1.02519	0.218572
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.04975	−0.809950
$26$	−0.655375	−0.128530
$27$	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
$31$	9.70512	1.74309	0.871546	−	0.490314i	$-0.163118\pi$
$31$	9.70512	1.74309	0.871546	+	0.490314i	$0.163118\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0.655375	0.112396
$35$	−0.974808	−0.164773
$36$	0	0
$37$	11.3353	1.86351	0.931757	−	0.363084i	$-0.118276\pi$
$37$	11.3353	1.86351	0.931757	+	0.363084i	$0.118276\pi$
$38$	0.319433	0.0518188
$39$	0	0
$40$	−0.974808	−0.154131
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.28556	0.653542	0.326771	−	0.945104i	$-0.394039\pi$
$43$	4.28556	0.653542	0.326771	+	0.945104i	$0.394039\pi$
$44$	1.02519	0.154554
$45$	0	0
$46$	1.00000	0.147442
$47$	5.70512	0.832178	0.416089	−	0.909324i	$-0.363400\pi$
$47$	5.70512	0.832178	0.416089	+	0.909324i	$0.363400\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−4.04975	−0.572721
$51$	0	0
$52$	−0.655375	−0.0908842
$53$	−7.33531	−1.00758	−0.503791	−	0.863826i	$-0.668062\pi$
$53$	−7.33531	−1.00758	−0.503791	+	0.863826i	$0.668062\pi$
$54$	0	0
$55$	−0.999365	−0.134754
$56$	1.00000	0.133631
$57$	0	0
$58$	2.00000	0.262613
$59$	−2.05038	−0.266937	−0.133469	−	0.991053i	$-0.542612\pi$
$59$	−2.05038	−0.266937	−0.133469	+	0.991053i	$0.542612\pi$
$60$	0	0
$61$	2.92442	0.374434	0.187217	−	0.982319i	$-0.440053\pi$
$61$	2.92442	0.374434	0.187217	+	0.982319i	$0.440053\pi$
$62$	9.70512	1.23255
$63$	0	0
$64$	1.00000	0.125000
$65$	0.638865	0.0792415
$66$	0	0
$67$	7.61367	0.930158	0.465079	−	0.885269i	$-0.346026\pi$
$67$	7.61367	0.930158	0.465079	+	0.885269i	$0.346026\pi$
$68$	0.655375	0.0794759
$69$	0	0
$70$	−0.974808	−0.116512
$71$	10.3101	1.22359	0.611793	−	0.791018i	$-0.290449\pi$
$71$	10.3101	1.22359	0.611793	+	0.791018i	$0.290449\pi$
$72$	0	0
$73$	−6.36050	−0.744440	−0.372220	−	0.928144i	$-0.621403\pi$
$73$	−6.36050	−0.744440	−0.372220	+	0.928144i	$0.621403\pi$
$74$	11.3353	1.31770
$75$	0	0
$76$	0.319433	0.0366414
$77$	1.02519	0.116831
$78$	0	0
$79$	−2.36050	−0.265577	−0.132789	−	0.991144i	$-0.542393\pi$
$79$	−2.36050	−0.265577	−0.132789	+	0.991144i	$0.542393\pi$
$80$	−0.974808	−0.108987
$81$	0	0
$82$	−2.00000	−0.220863
$83$	−15.7297	−1.72656	−0.863278	−	0.504728i	$-0.831593\pi$
$83$	−15.7297	−1.72656	−0.863278	+	0.504728i	$0.831593\pi$
$84$	0	0
$85$	−0.638865	−0.0692947
$86$	4.28556	0.462124
$87$	0	0
$88$	1.02519	0.109286
$89$	−18.0152	−1.90961	−0.954806	−	0.297230i	$-0.903937\pi$
$89$	−18.0152	−1.90961	−0.954806	+	0.297230i	$0.903937\pi$
$90$	0	0
$91$	−0.655375	−0.0687020
$92$	1.00000	0.104257
$93$	0	0
$94$	5.70512	0.588439
$95$	−0.311385	−0.0319475
$96$	0	0
$97$	16.0656	1.63122	0.815609	−	0.578604i	$-0.196402\pi$
$97$	16.0656	1.63122	0.815609	+	0.578604i	$0.196402\pi$
$98$	1.00000	0.101015
$99$	0	0
$100$	−4.04975	−0.404975
$101$	7.91574	0.787646	0.393823	−	0.919186i	$-0.371152\pi$
$101$	7.91574	0.787646	0.393823	+	0.919186i	$0.371152\pi$
$102$	0	0
$103$	−3.36113	−0.331182	−0.165591	−	0.986194i	$-0.552953\pi$
$103$	−3.36113	−0.331182	−0.165591	+	0.986194i	$0.552953\pi$
$104$	−0.655375	−0.0642648
$105$	0	0
$106$	−7.33531	−0.712468
$107$	1.02519	0.0991090	0.0495545	−	0.998771i	$-0.484220\pi$
$107$	1.02519	0.0991090	0.0495545	+	0.998771i	$0.484220\pi$
$108$	0	0
$109$	20.3851	1.95253	0.976267	−	0.216570i	$-0.0694868\pi$
$109$	20.3851	1.95253	0.976267	+	0.216570i	$0.0694868\pi$
$110$	−0.999365	−0.0952857
$111$	0	0
$112$	1.00000	0.0944911
$113$	17.3095	1.62834	0.814170	−	0.580627i	$-0.197193\pi$
$113$	17.3095	1.62834	0.814170	+	0.580627i	$0.197193\pi$
$114$	0	0
$115$	−0.974808	−0.0909013
$116$	2.00000	0.185695
$117$	0	0
$118$	−2.05038	−0.188753
$119$	0.655375	0.0600782
$120$	0	0
$121$	−9.94898	−0.904453
$122$	2.92442	0.264765
$123$	0	0
$124$	9.70512	0.871546
$125$	8.82177	0.789043
$126$	0	0
$127$	2.68925	0.238632	0.119316	−	0.992856i	$-0.461930\pi$
$127$	2.68925	0.238632	0.119316	+	0.992856i	$0.461930\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.638865	0.0560322
$131$	1.94962	0.170339	0.0851694	−	0.996366i	$-0.472857\pi$
$131$	1.94962	0.170339	0.0851694	+	0.996366i	$0.472857\pi$
$132$	0	0
$133$	0.319433	0.0276983
$134$	7.61367	0.657721
$135$	0	0
$136$	0.655375	0.0561980
$137$	3.94962	0.337438	0.168719	−	0.985664i	$-0.446037\pi$
$137$	3.94962	0.337438	0.168719	+	0.985664i	$0.446037\pi$
$138$	0	0
$139$	3.36113	0.285088	0.142544	−	0.989788i	$-0.454472\pi$
$139$	3.36113	0.285088	0.142544	+	0.989788i	$0.454472\pi$
$140$	−0.974808	−0.0823863
$141$	0	0
$142$	10.3101	0.865206
$143$	−0.671886	−0.0561859
$144$	0	0
$145$	−1.94962	−0.161907
$146$	−6.36050	−0.526399
$147$	0	0
$148$	11.3353	0.931757
$149$	−2.28556	−0.187240	−0.0936201	−	0.995608i	$-0.529844\pi$
$149$	−2.28556	−0.187240	−0.0936201	+	0.995608i	$0.529844\pi$
$150$	0	0
$151$	14.4109	1.17274	0.586371	−	0.810043i	$-0.300556\pi$
$151$	14.4109	1.17274	0.586371	+	0.810043i	$0.300556\pi$
$152$	0.319433	0.0259094
$153$	0	0
$154$	1.02519	0.0826123
$155$	−9.46063	−0.759896
$156$	0	0
$157$	−14.3347	−1.14403	−0.572016	−	0.820243i	$-0.693838\pi$
$157$	−14.3347	−1.14403	−0.572016	+	0.820243i	$0.693838\pi$
$158$	−2.36050	−0.187791
$159$	0	0
$160$	−0.974808	−0.0770653
$161$	1.00000	0.0788110
$162$	0	0
$163$	8.31012	0.650899	0.325449	−	0.945560i	$-0.394484\pi$
$163$	8.31012	0.650899	0.325449	+	0.945560i	$0.394484\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	−15.7297	−1.22086
$167$	−20.2936	−1.57037	−0.785183	−	0.619264i	$-0.787431\pi$
$167$	−20.2936	−1.57037	−0.785183	+	0.619264i	$0.787431\pi$
$168$	0	0
$169$	−12.5705	−0.966960
$170$	−0.638865	−0.0489987
$171$	0	0
$172$	4.28556	0.326771
$173$	14.7549	1.12179	0.560896	−	0.827886i	$-0.310457\pi$
$173$	14.7549	1.12179	0.560896	+	0.827886i	$0.310457\pi$
$174$	0	0
$175$	−4.04975	−0.306132
$176$	1.02519	0.0772768
$177$	0	0
$178$	−18.0152	−1.35030
$179$	0.671886	0.0502191	0.0251095	−	0.999685i	$-0.492007\pi$
$179$	0.671886	0.0502191	0.0251095	+	0.999685i	$0.492007\pi$
$180$	0	0
$181$	−15.6454	−1.16292	−0.581458	−	0.813577i	$-0.697517\pi$
$181$	−15.6454	−1.16292	−0.581458	+	0.813577i	$0.697517\pi$
$182$	−0.655375	−0.0485797
$183$	0	0
$184$	1.00000	0.0737210
$185$	−11.0497	−0.812394
$186$	0	0
$187$	0.671886	0.0491331
$188$	5.70512	0.416089
$189$	0	0
$190$	−0.311385	−0.0225903
$191$	16.3101	1.18016	0.590079	−	0.807345i	$-0.299096\pi$
$191$	16.3101	1.18016	0.590079	+	0.807345i	$0.299096\pi$
$192$	0	0
$193$	19.0497	1.37123	0.685615	−	0.727964i	$-0.259533\pi$
$193$	19.0497	1.37123	0.685615	+	0.727964i	$0.259533\pi$
$194$	16.0656	1.15344
$195$	0	0
$196$	1.00000	0.0714286
$197$	−20.0318	−1.42720	−0.713602	−	0.700552i	$-0.752937\pi$
$197$	−20.0318	−1.42720	−0.713602	+	0.700552i	$0.752937\pi$
$198$	0	0
$199$	9.20998	0.652878	0.326439	−	0.945218i	$-0.394151\pi$
$199$	9.20998	0.652878	0.326439	+	0.945218i	$0.394151\pi$
$200$	−4.04975	−0.286361
$201$	0	0
$202$	7.91574	0.556950
$203$	2.00000	0.140372
$204$	0	0
$205$	1.94962	0.136167
$206$	−3.36113	−0.234181
$207$	0	0
$208$	−0.655375	−0.0454421
$209$	0.327480	0.0226522
$210$	0	0
$211$	10.2597	0.706309	0.353155	−	0.935565i	$-0.385109\pi$
$211$	10.2597	0.706309	0.353155	+	0.935565i	$0.385109\pi$
$212$	−7.33531	−0.503791
$213$	0	0
$214$	1.02519	0.0700807
$215$	−4.17760	−0.284910
$216$	0	0
$217$	9.70512	0.658827
$218$	20.3851	1.38065
$219$	0	0
$220$	−0.999365	−0.0673772
$221$	−0.429517	−0.0288924
$222$	0	0
$223$	−0.916377	−0.0613651	−0.0306826	−	0.999529i	$-0.509768\pi$
$223$	−0.916377	−0.0613651	−0.0306826	+	0.999529i	$0.509768\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	17.3095	1.15141
$227$	10.9900	0.729435	0.364718	−	0.931118i	$-0.381166\pi$
$227$	10.9900	0.729435	0.364718	+	0.931118i	$0.381166\pi$
$228$	0	0
$229$	−10.9244	−0.721906	−0.360953	−	0.932584i	$-0.617549\pi$
$229$	−10.9244	−0.721906	−0.360953	+	0.932584i	$0.617549\pi$
$230$	−0.974808	−0.0642769
$231$	0	0
$232$	2.00000	0.131306
$233$	−16.9490	−1.11036	−0.555182	−	0.831729i	$-0.687352\pi$
$233$	−16.9490	−1.11036	−0.555182	+	0.831729i	$0.687352\pi$
$234$	0	0
$235$	−5.56140	−0.362786
$236$	−2.05038	−0.133469
$237$	0	0
$238$	0.655375	0.0424817
$239$	−23.4102	−1.51428	−0.757142	−	0.653251i	$-0.773405\pi$
$239$	−23.4102	−1.51428	−0.757142	+	0.653251i	$0.773405\pi$
$240$	0	0
$241$	15.9649	1.02839	0.514193	−	0.857674i	$-0.328091\pi$
$241$	15.9649	1.02839	0.514193	+	0.857674i	$0.328091\pi$
$242$	−9.94898	−0.639545
$243$	0	0
$244$	2.92442	0.187217
$245$	−0.974808	−0.0622782
$246$	0	0
$247$	−0.209348	−0.0133205
$248$	9.70512	0.616276
$249$	0	0
$250$	8.82177	0.557938
$251$	−26.4189	−1.66755	−0.833774	−	0.552106i	$-0.813824\pi$
$251$	−26.4189	−1.66755	−0.833774	+	0.552106i	$0.813824\pi$
$252$	0	0
$253$	1.02519	0.0644533
$254$	2.68925	0.168739
$255$	0	0
$256$	1.00000	0.0625000
$257$	−17.3095	−1.07974	−0.539868	−	0.841750i	$-0.681526\pi$
$257$	−17.3095	−1.07974	−0.539868	+	0.841750i	$0.681526\pi$
$258$	0	0
$259$	11.3353	0.704342
$260$	0.638865	0.0396207
$261$	0	0
$262$	1.94962	0.120448
$263$	−24.7210	−1.52436	−0.762181	−	0.647364i	$-0.775871\pi$
$263$	−24.7210	−1.52436	−0.762181	+	0.647364i	$0.775871\pi$
$264$	0	0
$265$	7.15052	0.439253
$266$	0.319433	0.0195857
$267$	0	0
$268$	7.61367	0.465079
$269$	8.65538	0.527728	0.263864	−	0.964560i	$-0.415003\pi$
$269$	8.65538	0.527728	0.263864	+	0.964560i	$0.415003\pi$
$270$	0	0
$271$	−16.7058	−1.01480	−0.507401	−	0.861710i	$-0.669394\pi$
$271$	−16.7058	−1.01480	−0.507401	+	0.861710i	$0.669394\pi$
$272$	0.655375	0.0397380
$273$	0	0
$274$	3.94962	0.238605
$275$	−4.15177	−0.250361
$276$	0	0
$277$	−6.04911	−0.363456	−0.181728	−	0.983349i	$-0.558169\pi$
$277$	−6.04911	−0.363456	−0.181728	+	0.983349i	$0.558169\pi$
$278$	3.36113	0.201587
$279$	0	0
$280$	−0.974808	−0.0582559
$281$	8.52073	0.508304	0.254152	−	0.967164i	$-0.418204\pi$
$281$	8.52073	0.508304	0.254152	+	0.967164i	$0.418204\pi$
$282$	0	0
$283$	11.5798	0.688348	0.344174	−	0.938906i	$-0.388159\pi$
$283$	11.5798	0.688348	0.344174	+	0.938906i	$0.388159\pi$
$284$	10.3101	0.611793
$285$	0	0
$286$	−0.671886	−0.0397294
$287$	−2.00000	−0.118056
$288$	0	0
$289$	−16.5705	−0.974734
$290$	−1.94962	−0.114485
$291$	0	0
$292$	−6.36050	−0.372220
$293$	3.07431	0.179603	0.0898015	−	0.995960i	$-0.471377\pi$
$293$	3.07431	0.179603	0.0898015	+	0.995960i	$0.471377\pi$
$294$	0	0
$295$	1.99873	0.116371
$296$	11.3353	0.658851
$297$	0	0
$298$	−2.28556	−0.132399
$299$	−0.655375	−0.0379013
$300$	0	0
$301$	4.28556	0.247016
$302$	14.4109	0.829253
$303$	0	0
$304$	0.319433	0.0183207
$305$	−2.85075	−0.163234
$306$	0	0
$307$	1.41152	0.0805596	0.0402798	−	0.999188i	$-0.487175\pi$
$307$	1.41152	0.0805596	0.0402798	+	0.999188i	$0.487175\pi$
$308$	1.02519	0.0584157
$309$	0	0
$310$	−9.46063	−0.537328
$311$	−30.6367	−1.73725	−0.868625	−	0.495470i	$-0.834996\pi$
$311$	−30.6367	−1.73725	−0.868625	+	0.495470i	$0.834996\pi$
$312$	0	0
$313$	0.554607	0.0313483	0.0156741	−	0.999877i	$-0.495011\pi$
$313$	0.554607	0.0313483	0.0156741	+	0.999877i	$0.495011\pi$
$314$	−14.3347	−0.808952
$315$	0	0
$316$	−2.36050	−0.132789
$317$	19.3108	1.08460	0.542300	−	0.840185i	$-0.317554\pi$
$317$	19.3108	1.08460	0.542300	+	0.840185i	$0.317554\pi$
$318$	0	0
$319$	2.05038	0.114799
$320$	−0.974808	−0.0544934
$321$	0	0
$322$	1.00000	0.0557278
$323$	0.209348	0.0116484
$324$	0	0
$325$	2.65411	0.147223
$326$	8.31012	0.460255
$327$	0	0
$328$	−2.00000	−0.110432
$329$	5.70512	0.314534
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−15.7297	−0.863278
$333$	0	0
$334$	−20.2936	−1.11042
$335$	−7.42187	−0.405500
$336$	0	0
$337$	11.3108	0.616136	0.308068	−	0.951364i	$-0.400318\pi$
$337$	11.3108	0.616136	0.308068	+	0.951364i	$0.400318\pi$
$338$	−12.5705	−0.683744
$339$	0	0
$340$	−0.638865	−0.0346473
$341$	9.94962	0.538802
$342$	0	0
$343$	1.00000	0.0539949
$344$	4.28556	0.231062
$345$	0	0
$346$	14.7549	0.793227
$347$	−16.4109	−0.880982	−0.440491	−	0.897757i	$-0.645196\pi$
$347$	−16.4109	−0.880982	−0.440491	+	0.897757i	$0.645196\pi$
$348$	0	0
$349$	−12.8053	−0.685450	−0.342725	−	0.939436i	$-0.611350\pi$
$349$	−12.8053	−0.685450	−0.342725	+	0.939436i	$0.611350\pi$
$350$	−4.04975	−0.216468
$351$	0	0
$352$	1.02519	0.0546429
$353$	−0.948981	−0.0505092	−0.0252546	−	0.999681i	$-0.508040\pi$
$353$	−0.948981	−0.0505092	−0.0252546	+	0.999681i	$0.508040\pi$
$354$	0	0
$355$	−10.0504	−0.533419
$356$	−18.0152	−0.954806
$357$	0	0
$358$	0.671886	0.0355103
$359$	18.4096	0.971622	0.485811	−	0.874064i	$-0.338524\pi$
$359$	18.4096	0.971622	0.485811	+	0.874064i	$0.338524\pi$
$360$	0	0
$361$	−18.8980	−0.994630
$362$	−15.6454	−0.822305
$363$	0	0
$364$	−0.655375	−0.0343510
$365$	6.20027	0.324537
$366$	0	0
$367$	−15.3095	−0.799148	−0.399574	−	0.916701i	$-0.630842\pi$
$367$	−15.3095	−0.799148	−0.399574	+	0.916701i	$0.630842\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	−11.0497	−0.574449
$371$	−7.33531	−0.380830
$372$	0	0
$373$	6.76419	0.350237	0.175118	−	0.984547i	$-0.443969\pi$
$373$	6.76419	0.350237	0.175118	+	0.984547i	$0.443969\pi$
$374$	0.671886	0.0347424
$375$	0	0
$376$	5.70512	0.294219
$377$	−1.31075	−0.0675071
$378$	0	0
$379$	−23.7966	−1.22235	−0.611174	−	0.791496i	$-0.709302\pi$
$379$	−23.7966	−1.22235	−0.611174	+	0.791496i	$0.709302\pi$
$380$	−0.311385	−0.0159737
$381$	0	0
$382$	16.3101	0.834498
$383$	−32.7210	−1.67197	−0.835983	−	0.548756i	$-0.815102\pi$
$383$	−32.7210	−1.67197	−0.835983	+	0.548756i	$0.815102\pi$
$384$	0	0
$385$	−0.999365	−0.0509324
$386$	19.0497	0.969607
$387$	0	0
$388$	16.0656	0.815609
$389$	−18.8133	−0.953872	−0.476936	−	0.878938i	$-0.658253\pi$
$389$	−18.8133	−0.953872	−0.476936	+	0.878938i	$0.658253\pi$
$390$	0	0
$391$	0.655375	0.0331438
$392$	1.00000	0.0505076
$393$	0	0
$394$	−20.0318	−1.00919
$395$	2.30103	0.115778
$396$	0	0
$397$	−23.3446	−1.17163	−0.585817	−	0.810444i	$-0.699226\pi$
$397$	−23.3446	−1.17163	−0.585817	+	0.810444i	$0.699226\pi$
$398$	9.20998	0.461655
$399$	0	0
$400$	−4.04975	−0.202487
$401$	−12.5698	−0.627708	−0.313854	−	0.949471i	$-0.601620\pi$
$401$	−12.5698	−0.627708	−0.313854	+	0.949471i	$0.601620\pi$
$402$	0	0
$403$	−6.36050	−0.316839
$404$	7.91574	0.393823
$405$	0	0
$406$	2.00000	0.0992583
$407$	11.6209	0.576025
$408$	0	0
$409$	16.0995	0.796069	0.398034	−	0.917370i	$-0.369692\pi$
$409$	16.0995	0.796069	0.398034	+	0.917370i	$0.369692\pi$
$410$	1.94962	0.0962847
$411$	0	0
$412$	−3.36113	−0.165591
$413$	−2.05038	−0.100893
$414$	0	0
$415$	15.3334	0.752688
$416$	−0.655375	−0.0321324
$417$	0	0
$418$	0.327480	0.0160176
$419$	−3.59843	−0.175795	−0.0878975	−	0.996130i	$-0.528015\pi$
$419$	−3.59843	−0.175795	−0.0878975	+	0.996130i	$0.528015\pi$
$420$	0	0
$421$	−8.90706	−0.434104	−0.217052	−	0.976160i	$-0.569644\pi$
$421$	−8.90706	−0.434104	−0.217052	+	0.976160i	$0.569644\pi$
$422$	10.2597	0.499436
$423$	0	0
$424$	−7.33531	−0.356234
$425$	−2.65411	−0.128743
$426$	0	0
$427$	2.92442	0.141523
$428$	1.02519	0.0495545
$429$	0	0
$430$	−4.17760	−0.201462
$431$	8.14988	0.392566	0.196283	−	0.980547i	$-0.437113\pi$
$431$	8.14988	0.392566	0.196283	+	0.980547i	$0.437113\pi$
$432$	0	0
$433$	−3.24386	−0.155890	−0.0779449	−	0.996958i	$-0.524836\pi$
$433$	−3.24386	−0.155890	−0.0779449	+	0.996958i	$0.524836\pi$
$434$	9.70512	0.465861
$435$	0	0
$436$	20.3851	0.976267
$437$	0.319433	0.0152805
$438$	0	0
$439$	−23.2942	−1.11177	−0.555887	−	0.831258i	$-0.687621\pi$
$439$	−23.2942	−1.11177	−0.555887	+	0.831258i	$0.687621\pi$
$440$	−0.999365	−0.0476429
$441$	0	0
$442$	−0.429517	−0.0204300
$443$	−4.51165	−0.214355	−0.107178	−	0.994240i	$-0.534181\pi$
$443$	−4.51165	−0.214355	−0.107178	+	0.994240i	$0.534181\pi$
$444$	0	0
$445$	17.5614	0.832490
$446$	−0.916377	−0.0433917
$447$	0	0
$448$	1.00000	0.0472456
$449$	−19.9987	−0.943798	−0.471899	−	0.881652i	$-0.656431\pi$
$449$	−19.9987	−0.943798	−0.471899	+	0.881652i	$0.656431\pi$
$450$	0	0
$451$	−2.05038	−0.0965488
$452$	17.3095	0.814170
$453$	0	0
$454$	10.9900	0.515789
$455$	0.638865	0.0299505
$456$	0	0
$457$	26.0491	1.21853	0.609263	−	0.792968i	$-0.291465\pi$
$457$	26.0491	1.21853	0.609263	+	0.792968i	$0.291465\pi$
$458$	−10.9244	−0.510465
$459$	0	0
$460$	−0.974808	−0.0454507
$461$	−20.1334	−0.937705	−0.468852	−	0.883277i	$-0.655332\pi$
$461$	−20.1334	−0.937705	−0.468852	+	0.883277i	$0.655332\pi$
$462$	0	0
$463$	−26.0995	−1.21295	−0.606473	−	0.795104i	$-0.707416\pi$
$463$	−26.0995	−1.21295	−0.606473	+	0.795104i	$0.707416\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−16.9490	−0.785146
$467$	33.1412	1.53359	0.766796	−	0.641891i	$-0.221850\pi$
$467$	33.1412	1.53359	0.766796	+	0.641891i	$0.221850\pi$
$468$	0	0
$469$	7.61367	0.351567
$470$	−5.56140	−0.256528
$471$	0	0
$472$	−2.05038	−0.0943766
$473$	4.39352	0.202014
$474$	0	0
$475$	−1.29362	−0.0593554
$476$	0.655375	0.0300391
$477$	0	0
$478$	−23.4102	−1.07076
$479$	10.6892	0.488404	0.244202	−	0.969724i	$-0.421474\pi$
$479$	10.6892	0.488404	0.244202	+	0.969724i	$0.421474\pi$
$480$	0	0
$481$	−7.42888	−0.338728
$482$	15.9649	0.727179
$483$	0	0
$484$	−9.94898	−0.452226
$485$	−15.6609	−0.711125
$486$	0	0
$487$	33.8980	1.53606	0.768032	−	0.640412i	$-0.221236\pi$
$487$	33.8980	1.53606	0.768032	+	0.640412i	$0.221236\pi$
$488$	2.92442	0.132382
$489$	0	0
$490$	−0.974808	−0.0440373
$491$	−11.0324	−0.497885	−0.248942	−	0.968518i	$-0.580083\pi$
$491$	−11.0324	−0.497885	−0.248942	+	0.968518i	$0.580083\pi$
$492$	0	0
$493$	1.31075	0.0590332
$494$	−0.209348	−0.00941902
$495$	0	0
$496$	9.70512	0.435773
$497$	10.3101	0.462472
$498$	0	0
$499$	1.68861	0.0755928	0.0377964	−	0.999285i	$-0.487966\pi$
$499$	1.68861	0.0755928	0.0377964	+	0.999285i	$0.487966\pi$
$500$	8.82177	0.394521
$501$	0	0
$502$	−26.4189	−1.17913
$503$	−36.1499	−1.61184	−0.805922	−	0.592022i	$-0.798330\pi$
$503$	−36.1499	−1.61184	−0.805922	+	0.592022i	$0.798330\pi$
$504$	0	0
$505$	−7.71633	−0.343372
$506$	1.02519	0.0455753
$507$	0	0
$508$	2.68925	0.119316
$509$	31.8641	1.41235	0.706175	−	0.708037i	$-0.250419\pi$
$509$	31.8641	1.41235	0.706175	+	0.708037i	$0.250419\pi$
$510$	0	0
$511$	−6.36050	−0.281372
$512$	1.00000	0.0441942
$513$	0	0
$514$	−17.3095	−0.763489
$515$	3.27646	0.144378
$516$	0	0
$517$	5.84885	0.257232
$518$	11.3353	0.498045
$519$	0	0
$520$	0.638865	0.0280161
$521$	29.1748	1.27817	0.639086	−	0.769135i	$-0.279313\pi$
$521$	29.1748	1.27817	0.639086	+	0.769135i	$0.279313\pi$
$522$	0	0
$523$	−6.26905	−0.274126	−0.137063	−	0.990562i	$-0.543766\pi$
$523$	−6.26905	−0.274126	−0.137063	+	0.990562i	$0.543766\pi$
$524$	1.94962	0.0851694
$525$	0	0
$526$	−24.7210	−1.07789
$527$	6.36050	0.277068
$528$	0	0
$529$	1.00000	0.0434783
$530$	7.15052	0.310599
$531$	0	0
$532$	0.319433	0.0138492
$533$	1.31075	0.0567749
$534$	0	0
$535$	−0.999365	−0.0432063
$536$	7.61367	0.328861
$537$	0	0
$538$	8.65538	0.373160
$539$	1.02519	0.0441581
$540$	0	0
$541$	−30.0491	−1.29191	−0.645956	−	0.763374i	$-0.723541\pi$
$541$	−30.0491	−1.29191	−0.645956	+	0.763374i	$0.723541\pi$
$542$	−16.7058	−0.717574
$543$	0	0
$544$	0.655375	0.0280990
$545$	−19.8715	−0.851202
$546$	0	0
$547$	−15.4780	−0.661791	−0.330896	−	0.943667i	$-0.607351\pi$
$547$	−15.4780	−0.661791	−0.330896	+	0.943667i	$0.607351\pi$
$548$	3.94962	0.168719
$549$	0	0
$550$	−4.15177	−0.177032
$551$	0.638865	0.0272166
$552$	0	0
$553$	−2.36050	−0.100379
$554$	−6.04911	−0.257002
$555$	0	0
$556$	3.36113	0.142544
$557$	−27.6784	−1.17277	−0.586387	−	0.810031i	$-0.699450\pi$
$557$	−27.6784	−1.17277	−0.586387	+	0.810031i	$0.699450\pi$
$558$	0	0
$559$	−2.80865	−0.118793
$560$	−0.974808	−0.0411932
$561$	0	0
$562$	8.52073	0.359425
$563$	19.5612	0.824405	0.412202	−	0.911092i	$-0.364760\pi$
$563$	19.5612	0.824405	0.412202	+	0.911092i	$0.364760\pi$
$564$	0	0
$565$	−16.8734	−0.709870
$566$	11.5798	0.486735
$567$	0	0
$568$	10.3101	0.432603
$569$	25.2273	1.05759	0.528793	−	0.848751i	$-0.322645\pi$
$569$	25.2273	1.05759	0.528793	+	0.848751i	$0.322645\pi$
$570$	0	0
$571$	5.76483	0.241250	0.120625	−	0.992698i	$-0.461510\pi$
$571$	5.76483	0.241250	0.120625	+	0.992698i	$0.461510\pi$
$572$	−0.671886	−0.0280929
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	−4.04975	−0.168886
$576$	0	0
$577$	−29.7030	−1.23655	−0.618276	−	0.785961i	$-0.712169\pi$
$577$	−29.7030	−1.23655	−0.618276	+	0.785961i	$0.712169\pi$
$578$	−16.5705	−0.689241
$579$	0	0
$580$	−1.94962	−0.0809534
$581$	−15.7297	−0.652577
$582$	0	0
$583$	−7.52010	−0.311451
$584$	−6.36050	−0.263199
$585$	0	0
$586$	3.07431	0.126998
$587$	27.8806	1.15076	0.575378	−	0.817888i	$-0.304855\pi$
$587$	27.8806	1.15076	0.575378	+	0.817888i	$0.304855\pi$
$588$	0	0
$589$	3.10013	0.127739
$590$	1.99873	0.0822864
$591$	0	0
$592$	11.3353	0.465878
$593$	41.1079	1.68810	0.844051	−	0.536264i	$-0.180165\pi$
$593$	41.1079	1.68810	0.844051	+	0.536264i	$0.180165\pi$
$594$	0	0
$595$	−0.638865	−0.0261909
$596$	−2.28556	−0.0936201
$597$	0	0
$598$	−0.655375	−0.0268003
$599$	−46.8205	−1.91303	−0.956517	−	0.291677i	$-0.905787\pi$
$599$	−46.8205	−1.91303	−0.956517	+	0.291677i	$0.905787\pi$
$600$	0	0
$601$	−30.3605	−1.23843	−0.619215	−	0.785221i	$-0.712549\pi$
$601$	−30.3605	−1.23843	−0.619215	+	0.785221i	$0.712549\pi$
$602$	4.28556	0.174666
$603$	0	0
$604$	14.4109	0.586371
$605$	9.69835	0.394294
$606$	0	0
$607$	−18.9151	−0.767741	−0.383870	−	0.923387i	$-0.625409\pi$
$607$	−18.9151	−0.767741	−0.383870	+	0.923387i	$0.625409\pi$
$608$	0.319433	0.0129547
$609$	0	0
$610$	−2.85075	−0.115424
$611$	−3.73900	−0.151264
$612$	0	0
$613$	6.85668	0.276939	0.138469	−	0.990367i	$-0.455782\pi$
$613$	6.85668	0.276939	0.138469	+	0.990367i	$0.455782\pi$
$614$	1.41152	0.0569643
$615$	0	0
$616$	1.02519	0.0413062
$617$	26.8218	1.07980	0.539902	−	0.841728i	$-0.318461\pi$
$617$	26.8218	1.07980	0.539902	+	0.841728i	$0.318461\pi$
$618$	0	0
$619$	6.90791	0.277653	0.138826	−	0.990317i	$-0.455667\pi$
$619$	6.90791	0.277653	0.138826	+	0.990317i	$0.455667\pi$
$620$	−9.46063	−0.379948
$621$	0	0
$622$	−30.6367	−1.22842
$623$	−18.0152	−0.721765
$624$	0	0
$625$	11.6492	0.465969
$626$	0.554607	0.0221666
$627$	0	0
$628$	−14.3347	−0.572016
$629$	7.42888	0.296209
$630$	0	0
$631$	−17.8902	−0.712196	−0.356098	−	0.934449i	$-0.615893\pi$
$631$	−17.8902	−0.712196	−0.356098	+	0.934449i	$0.615893\pi$
$632$	−2.36050	−0.0938957
$633$	0	0
$634$	19.3108	0.766928
$635$	−2.62150	−0.104031
$636$	0	0
$637$	−0.655375	−0.0259669
$638$	2.05038	0.0811755
$639$	0	0
$640$	−0.974808	−0.0385327
$641$	6.25065	0.246886	0.123443	−	0.992352i	$-0.460606\pi$
$641$	6.25065	0.246886	0.123443	+	0.992352i	$0.460606\pi$
$642$	0	0
$643$	−24.8906	−0.981587	−0.490794	−	0.871276i	$-0.663293\pi$
$643$	−24.8906	−0.981587	−0.490794	+	0.871276i	$0.663293\pi$
$644$	1.00000	0.0394055
$645$	0	0
$646$	0.209348	0.00823670
$647$	−17.3937	−0.683818	−0.341909	−	0.939733i	$-0.611074\pi$
$647$	−17.3937	−0.683818	−0.341909	+	0.939733i	$0.611074\pi$
$648$	0	0
$649$	−2.10204	−0.0825122
$650$	2.65411	0.104103
$651$	0	0
$652$	8.31012	0.325449
$653$	37.4924	1.46719	0.733595	−	0.679587i	$-0.237841\pi$
$653$	37.4924	1.46719	0.733595	+	0.679587i	$0.237841\pi$
$654$	0	0
$655$	−1.90050	−0.0742587
$656$	−2.00000	−0.0780869
$657$	0	0
$658$	5.70512	0.222409
$659$	21.8631	0.851664	0.425832	−	0.904802i	$-0.359982\pi$
$659$	21.8631	0.851664	0.425832	+	0.904802i	$0.359982\pi$
$660$	0	0
$661$	36.2165	1.40866	0.704330	−	0.709873i	$-0.251247\pi$
$661$	36.2165	1.40866	0.704330	+	0.709873i	$0.251247\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−15.7297	−0.610430
$665$	−0.311385	−0.0120750
$666$	0	0
$667$	2.00000	0.0774403
$668$	−20.2936	−0.785183
$669$	0	0
$670$	−7.42187	−0.286732
$671$	2.99810	0.115740
$672$	0	0
$673$	−13.1770	−0.507935	−0.253967	−	0.967213i	$-0.581736\pi$
$673$	−13.1770	−0.507935	−0.253967	+	0.967213i	$0.581736\pi$
$674$	11.3108	0.435674
$675$	0	0
$676$	−12.5705	−0.483480
$677$	37.6958	1.44877	0.724384	−	0.689397i	$-0.242124\pi$
$677$	37.6958	1.44877	0.724384	+	0.689397i	$0.242124\pi$
$678$	0	0
$679$	16.0656	0.616542
$680$	−0.638865	−0.0244994
$681$	0	0
$682$	9.94962	0.380990
$683$	14.2597	0.545633	0.272817	−	0.962066i	$-0.412045\pi$
$683$	14.2597	0.545633	0.272817	+	0.962066i	$0.412045\pi$
$684$	0	0
$685$	−3.85012	−0.147105
$686$	1.00000	0.0381802
$687$	0	0
$688$	4.28556	0.163385
$689$	4.80738	0.183147
$690$	0	0
$691$	−17.8819	−0.680258	−0.340129	−	0.940379i	$-0.610471\pi$
$691$	−17.8819	−0.680258	−0.340129	+	0.940379i	$0.610471\pi$
$692$	14.7549	0.560896
$693$	0	0
$694$	−16.4109	−0.622949
$695$	−3.27646	−0.124283
$696$	0	0
$697$	−1.31075	−0.0496482
$698$	−12.8053	−0.484686
$699$	0	0
$700$	−4.04975	−0.153066
$701$	42.5362	1.60657	0.803285	−	0.595595i	$-0.203083\pi$
$701$	42.5362	1.60657	0.803285	+	0.595595i	$0.203083\pi$
$702$	0	0
$703$	3.62087	0.136564
$704$	1.02519	0.0386384
$705$	0	0
$706$	−0.948981	−0.0357154
$707$	7.91574	0.297702
$708$	0	0
$709$	5.80693	0.218084	0.109042	−	0.994037i	$-0.465222\pi$
$709$	5.80693	0.218084	0.109042	+	0.994037i	$0.465222\pi$
$710$	−10.0504	−0.377184
$711$	0	0
$712$	−18.0152	−0.675150
$713$	9.70512	0.363460
$714$	0	0
$715$	0.654959	0.0244941
$716$	0.671886	0.0251095
$717$	0	0
$718$	18.4096	0.687041
$719$	−34.8053	−1.29802	−0.649009	−	0.760781i	$-0.724816\pi$
$719$	−34.8053	−1.29802	−0.649009	+	0.760781i	$0.724816\pi$
$720$	0	0
$721$	−3.36113	−0.125175
$722$	−18.8980	−0.703309
$723$	0	0
$724$	−15.6454	−0.581458
$725$	−8.09950	−0.300808
$726$	0	0
$727$	−13.1092	−0.486194	−0.243097	−	0.970002i	$-0.578163\pi$
$727$	−13.1092	−0.486194	−0.243097	+	0.970002i	$0.578163\pi$
$728$	−0.655375	−0.0242898
$729$	0	0
$730$	6.20027	0.229482
$731$	2.80865	0.103882
$732$	0	0
$733$	−16.2843	−0.601474	−0.300737	−	0.953707i	$-0.597233\pi$
$733$	−16.2843	−0.601474	−0.300737	+	0.953707i	$0.597233\pi$
$734$	−15.3095	−0.565083
$735$	0	0
$736$	1.00000	0.0368605
$737$	7.80548	0.287518
$738$	0	0
$739$	0.209348	0.00770100	0.00385050	−	0.999993i	$-0.498774\pi$
$739$	0.209348	0.00770100	0.00385050	+	0.999993i	$0.498774\pi$
$740$	−11.0497	−0.406197
$741$	0	0
$742$	−7.33531	−0.269288
$743$	−34.8800	−1.27962	−0.639811	−	0.768532i	$-0.720987\pi$
$743$	−34.8800	−1.27962	−0.639811	+	0.768532i	$0.720987\pi$
$744$	0	0
$745$	2.22798	0.0816269
$746$	6.76419	0.247655
$747$	0	0
$748$	0.671886	0.0245666
$749$	1.02519	0.0374597
$750$	0	0
$751$	10.0504	0.366744	0.183372	−	0.983044i	$-0.441299\pi$
$751$	10.0504	0.366744	0.183372	+	0.983044i	$0.441299\pi$
$752$	5.70512	0.208044
$753$	0	0
$754$	−1.31075	−0.0477347
$755$	−14.0478	−0.511253
$756$	0	0
$757$	−22.8567	−0.830740	−0.415370	−	0.909653i	$-0.636348\pi$
$757$	−22.8567	−0.830740	−0.415370	+	0.909653i	$0.636348\pi$
$758$	−23.7966	−0.864331
$759$	0	0
$760$	−0.311385	−0.0112951
$761$	31.5027	1.14197	0.570987	−	0.820959i	$-0.306561\pi$
$761$	31.5027	1.14197	0.570987	+	0.820959i	$0.306561\pi$
$762$	0	0
$763$	20.3851	0.737989
$764$	16.3101	0.590079
$765$	0	0
$766$	−32.7210	−1.18226
$767$	1.34377	0.0485208
$768$	0	0
$769$	−17.6457	−0.636319	−0.318159	−	0.948037i	$-0.603065\pi$
$769$	−17.6457	−0.636319	−0.318159	+	0.948037i	$0.603065\pi$
$770$	−0.999365	−0.0360146
$771$	0	0
$772$	19.0497	0.685615
$773$	−28.8554	−1.03786	−0.518928	−	0.854818i	$-0.673669\pi$
$773$	−28.8554	−1.03786	−0.518928	+	0.854818i	$0.673669\pi$
$774$	0	0
$775$	−39.3033	−1.41182
$776$	16.0656	0.576722
$777$	0	0
$778$	−18.8133	−0.674490
$779$	−0.638865	−0.0228897
$780$	0	0
$781$	10.5698	0.378219
$782$	0.655375	0.0234362
$783$	0	0
$784$	1.00000	0.0357143
$785$	13.9736	0.498738
$786$	0	0
$787$	28.7898	1.02624	0.513122	−	0.858315i	$-0.328489\pi$
$787$	28.7898	1.02624	0.513122	+	0.858315i	$0.328489\pi$
$788$	−20.0318	−0.713602
$789$	0	0
$790$	2.30103	0.0818671
$791$	17.3095	0.615454
$792$	0	0
$793$	−1.91660	−0.0680603
$794$	−23.3446	−0.828470
$795$	0	0
$796$	9.20998	0.326439
$797$	−33.1061	−1.17268	−0.586338	−	0.810066i	$-0.699431\pi$
$797$	−33.1061	−1.17268	−0.586338	+	0.810066i	$0.699431\pi$
$798$	0	0
$799$	3.73900	0.132276
$800$	−4.04975	−0.143180
$801$	0	0
$802$	−12.5698	−0.443857
$803$	−6.52073	−0.230112
$804$	0	0
$805$	−0.974808	−0.0343575
$806$	−6.36050	−0.224039
$807$	0	0
$808$	7.91574	0.278475
$809$	13.8715	0.487697	0.243848	−	0.969813i	$-0.421590\pi$
$809$	13.8715	0.487697	0.243848	+	0.969813i	$0.421590\pi$
$810$	0	0
$811$	−29.9496	−1.05167	−0.525837	−	0.850586i	$-0.676248\pi$
$811$	−29.9496	−1.05167	−0.525837	+	0.850586i	$0.676248\pi$
$812$	2.00000	0.0701862
$813$	0	0
$814$	11.6209	0.407311
$815$	−8.10077	−0.283758
$816$	0	0
$817$	1.36895	0.0478934
$818$	16.0995	0.562906
$819$	0	0
$820$	1.94962	0.0680835
$821$	44.6376	1.55786	0.778931	−	0.627109i	$-0.215762\pi$
$821$	44.6376	1.55786	0.778931	+	0.627109i	$0.215762\pi$
$822$	0	0
$823$	−21.7534	−0.758275	−0.379138	−	0.925340i	$-0.623779\pi$
$823$	−21.7534	−0.758275	−0.379138	+	0.925340i	$0.623779\pi$
$824$	−3.36113	−0.117091
$825$	0	0
$826$	−2.05038	−0.0713420
$827$	−10.8740	−0.378127	−0.189064	−	0.981965i	$-0.560545\pi$
$827$	−10.8740	−0.378127	−0.189064	+	0.981965i	$0.560545\pi$
$828$	0	0
$829$	−42.1160	−1.46275	−0.731375	−	0.681976i	$-0.761121\pi$
$829$	−42.1160	−1.46275	−0.731375	+	0.681976i	$0.761121\pi$
$830$	15.3334	0.532231
$831$	0	0
$832$	−0.655375	−0.0227211
$833$	0.655375	0.0227074
$834$	0	0
$835$	19.7824	0.684597
$836$	0.327480	0.0113261
$837$	0	0
$838$	−3.59843	−0.124306
$839$	19.7985	0.683519	0.341759	−	0.939787i	$-0.388977\pi$
$839$	19.7985	0.683519	0.341759	+	0.939787i	$0.388977\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−8.90706	−0.306958
$843$	0	0
$844$	10.2597	0.353155
$845$	12.2538	0.421544
$846$	0	0
$847$	−9.94898	−0.341851
$848$	−7.33531	−0.251896
$849$	0	0
$850$	−2.65411	−0.0910351
$851$	11.3353	0.388569
$852$	0	0
$853$	17.1748	0.588055	0.294028	−	0.955797i	$-0.405004\pi$
$853$	17.1748	0.588055	0.294028	+	0.955797i	$0.405004\pi$
$854$	2.92442	0.100072
$855$	0	0
$856$	1.02519	0.0350403
$857$	3.77202	0.128850	0.0644249	−	0.997923i	$-0.479479\pi$
$857$	3.77202	0.128850	0.0644249	+	0.997923i	$0.479479\pi$
$858$	0	0
$859$	−24.5872	−0.838905	−0.419452	−	0.907777i	$-0.637778\pi$
$859$	−24.5872	−0.838905	−0.419452	+	0.907777i	$0.637778\pi$
$860$	−4.17760	−0.142455
$861$	0	0
$862$	8.14988	0.277586
$863$	−2.89079	−0.0984035	−0.0492017	−	0.998789i	$-0.515668\pi$
$863$	−2.89079	−0.0984035	−0.0492017	+	0.998789i	$0.515668\pi$
$864$	0	0
$865$	−14.3832	−0.489042
$866$	−3.24386	−0.110231
$867$	0	0
$868$	9.70512	0.329413
$869$	−2.41997	−0.0820917
$870$	0	0
$871$	−4.98981	−0.169073
$872$	20.3851	0.690325
$873$	0	0
$874$	0.319433	0.0108050
$875$	8.82177	0.298230
$876$	0	0
$877$	23.6622	0.799015	0.399507	−	0.916730i	$-0.369181\pi$
$877$	23.6622	0.799015	0.399507	+	0.916730i	$0.369181\pi$
$878$	−23.2942	−0.786143
$879$	0	0
$880$	−0.999365	−0.0336886
$881$	28.0165	0.943900	0.471950	−	0.881625i	$-0.343550\pi$
$881$	28.0165	0.943900	0.471950	+	0.881625i	$0.343550\pi$
$882$	0	0
$883$	41.5015	1.39664	0.698318	−	0.715788i	$-0.253932\pi$
$883$	41.5015	1.39664	0.698318	+	0.715788i	$0.253932\pi$
$884$	−0.429517	−0.0144462
$885$	0	0
$886$	−4.51165	−0.151572
$887$	−19.8724	−0.667249	−0.333624	−	0.942706i	$-0.608272\pi$
$887$	−19.8724	−0.667249	−0.333624	+	0.942706i	$0.608272\pi$
$888$	0	0
$889$	2.68925	0.0901945
$890$	17.5614	0.588660
$891$	0	0
$892$	−0.916377	−0.0306826
$893$	1.82240	0.0609844
$894$	0	0
$895$	−0.654959	−0.0218929
$896$	1.00000	0.0334077
$897$	0	0
$898$	−19.9987	−0.667366
$899$	19.4102	0.647368
$900$	0	0
$901$	−4.80738	−0.160157
$902$	−2.05038	−0.0682703
$903$	0	0
$904$	17.3095	0.575705
$905$	15.2513	0.506970
$906$	0	0
$907$	26.4672	0.878829	0.439414	−	0.898284i	$-0.355186\pi$
$907$	26.4672	0.878829	0.439414	+	0.898284i	$0.355186\pi$
$908$	10.9900	0.364718
$909$	0	0
$910$	0.638865	0.0211782
$911$	−36.1086	−1.19633	−0.598165	−	0.801373i	$-0.704103\pi$
$911$	−36.1086	−1.19633	−0.598165	+	0.801373i	$0.704103\pi$
$912$	0	0
$913$	−16.1259	−0.533691
$914$	26.0491	0.861628
$915$	0	0
$916$	−10.9244	−0.360953
$917$	1.94962	0.0643820
$918$	0	0
$919$	−46.3580	−1.52921	−0.764604	−	0.644500i	$-0.777065\pi$
$919$	−46.3580	−1.52921	−0.764604	+	0.644500i	$0.777065\pi$
$920$	−0.974808	−0.0321385
$921$	0	0
$922$	−20.1334	−0.663057
$923$	−6.75700	−0.222409
$924$	0	0
$925$	−45.9052	−1.50935
$926$	−26.0995	−0.857683
$927$	0	0
$928$	2.00000	0.0656532
$929$	−31.4420	−1.03158	−0.515789	−	0.856715i	$-0.672501\pi$
$929$	−31.4420	−1.03158	−0.515789	+	0.856715i	$0.672501\pi$
$930$	0	0
$931$	0.319433	0.0104690
$932$	−16.9490	−0.555182
$933$	0	0
$934$	33.1412	1.08441
$935$	−0.654959	−0.0214195
$936$	0	0
$937$	−10.0152	−0.327184	−0.163592	−	0.986528i	$-0.552308\pi$
$937$	−10.0152	−0.327184	−0.163592	+	0.986528i	$0.552308\pi$
$938$	7.61367	0.248595
$939$	0	0
$940$	−5.56140	−0.181393
$941$	40.2513	1.31215	0.656077	−	0.754694i	$-0.272215\pi$
$941$	40.2513	1.31215	0.656077	+	0.754694i	$0.272215\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	−2.05038	−0.0667343
$945$	0	0
$946$	4.39352	0.142846
$947$	−39.2314	−1.27485	−0.637424	−	0.770513i	$-0.720000\pi$
$947$	−39.2314	−1.27485	−0.637424	+	0.770513i	$0.720000\pi$
$948$	0	0
$949$	4.16852	0.135316
$950$	−1.29362	−0.0419706
$951$	0	0
$952$	0.655375	0.0212408
$953$	−21.5601	−0.698401	−0.349201	−	0.937048i	$-0.613547\pi$
$953$	−21.5601	−0.698401	−0.349201	+	0.937048i	$0.613547\pi$
$954$	0	0
$955$	−15.8992	−0.514487
$956$	−23.4102	−0.757142
$957$	0	0
$958$	10.6892	0.345354
$959$	3.94962	0.127540
$960$	0	0
$961$	63.1894	2.03837
$962$	−7.42888	−0.239517
$963$	0	0
$964$	15.9649	0.514193
$965$	−18.5698	−0.597785
$966$	0	0
$967$	7.92317	0.254792	0.127396	−	0.991852i	$-0.459338\pi$
$967$	7.92317	0.254792	0.127396	+	0.991852i	$0.459338\pi$
$968$	−9.94898	−0.319772
$969$	0	0
$970$	−15.6609	−0.502841
$971$	−12.9727	−0.416313	−0.208157	−	0.978096i	$-0.566746\pi$
$971$	−12.9727	−0.416313	−0.208157	+	0.978096i	$0.566746\pi$
$972$	0	0
$973$	3.36113	0.107753
$974$	33.8980	1.08616
$975$	0	0
$976$	2.92442	0.0936085
$977$	25.2273	0.807094	0.403547	−	0.914959i	$-0.367777\pi$
$977$	25.2273	0.807094	0.403547	+	0.914959i	$0.367777\pi$
$978$	0	0
$979$	−18.4691	−0.590274
$980$	−0.974808	−0.0311391
$981$	0	0
$982$	−11.0324	−0.352058
$983$	33.8488	1.07961	0.539805	−	0.841790i	$-0.318498\pi$
$983$	33.8488	1.07961	0.539805	+	0.841790i	$0.318498\pi$
$984$	0	0
$985$	19.5271	0.622186
$986$	1.31075	0.0417428
$987$	0	0
$988$	−0.209348	−0.00666026
$989$	4.28556	0.136273
$990$	0	0
$991$	2.51165	0.0797853	0.0398926	−	0.999204i	$-0.487298\pi$
$991$	2.51165	0.0797853	0.0398926	+	0.999204i	$0.487298\pi$
$992$	9.70512	0.308138
$993$	0	0
$994$	10.3101	0.327017
$995$	−8.97797	−0.284621
$996$	0	0
$997$	54.0152	1.71068	0.855340	−	0.518067i	$-0.173348\pi$
$997$	54.0152	1.71068	0.855340	+	0.518067i	$0.173348\pi$
$998$	1.68861	0.0534522
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2898.2.a.bi.1.2	yes	4
3.2	odd	2		2898.2.a.bh.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2898.2.a.bh.1.3	✓	4	3.2	odd	2
2898.2.a.bi.1.2	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.974808 q^{5} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.974808 q^{5} +1.00000 q^{7} +1.00000 q^{8} -0.974808 q^{10} +1.02519 q^{11} -0.655375 q^{13} +1.00000 q^{14} +1.00000 q^{16} +0.655375 q^{17} +0.319433 q^{19} -0.974808 q^{20} +1.02519 q^{22} +1.00000 q^{23} -4.04975 q^{25} -0.655375 q^{26} +1.00000 q^{28} +2.00000 q^{29} +9.70512 q^{31} +1.00000 q^{32} +0.655375 q^{34} -0.974808 q^{35} +11.3353 q^{37} +0.319433 q^{38} -0.974808 q^{40} -2.00000 q^{41} +4.28556 q^{43} +1.02519 q^{44} +1.00000 q^{46} +5.70512 q^{47} +1.00000 q^{49} -4.04975 q^{50} -0.655375 q^{52} -7.33531 q^{53} -0.999365 q^{55} +1.00000 q^{56} +2.00000 q^{58} -2.05038 q^{59} +2.92442 q^{61} +9.70512 q^{62} +1.00000 q^{64} +0.638865 q^{65} +7.61367 q^{67} +0.655375 q^{68} -0.974808 q^{70} +10.3101 q^{71} -6.36050 q^{73} +11.3353 q^{74} +0.319433 q^{76} +1.02519 q^{77} -2.36050 q^{79} -0.974808 q^{80} -2.00000 q^{82} -15.7297 q^{83} -0.638865 q^{85} +4.28556 q^{86} +1.02519 q^{88} -18.0152 q^{89} -0.655375 q^{91} +1.00000 q^{92} +5.70512 q^{94} -0.311385 q^{95} +16.0656 q^{97} +1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{7} + 4 q^{8} + 8 q^{11} + 4 q^{14} + 4 q^{16} + 8 q^{22} + 4 q^{23} + 16 q^{25} + 4 q^{28} + 8 q^{29} + 4 q^{31} + 4 q^{32} + 4 q^{37} - 8 q^{41} + 8 q^{43} + 8 q^{44} + 4 q^{46} - 12 q^{47} + 4 q^{49} + 16 q^{50} + 12 q^{53} + 36 q^{55} + 4 q^{56} + 8 q^{58} - 16 q^{59} + 4 q^{62} + 4 q^{64} + 24 q^{67} - 4 q^{71} + 12 q^{73} + 4 q^{74} + 8 q^{77} + 28 q^{79} - 8 q^{82} + 8 q^{83} + 8 q^{86} + 8 q^{88} + 8 q^{89} + 4 q^{92} - 12 q^{94} - 36 q^{95} - 8 q^{97} + 4 q^{98}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^7 + 4 * q^8 + 8 * q^11 + 4 * q^14 + 4 * q^16 + 8 * q^22 + 4 * q^23 + 16 * q^25 + 4 * q^28 + 8 * q^29 + 4 * q^31 + 4 * q^32 + 4 * q^37 - 8 * q^41 + 8 * q^43 + 8 * q^44 + 4 * q^46 - 12 * q^47 + 4 * q^49 + 16 * q^50 + 12 * q^53 + 36 * q^55 + 4 * q^56 + 8 * q^58 - 16 * q^59 + 4 * q^62 + 4 * q^64 + 24 * q^67 - 4 * q^71 + 12 * q^73 + 4 * q^74 + 8 * q^77 + 28 * q^79 - 8 * q^82 + 8 * q^83 + 8 * q^86 + 8 * q^88 + 8 * q^89 + 4 * q^92 - 12 * q^94 - 36 * q^95 - 8 * q^97 + 4 * q^98

Introduction

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