LMFDB - Embedded newform 4410.2.a.bz.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4410,2,Mod(1,4410)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4410, 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("4410.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4410.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:	$35.2140272914$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1470)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4410.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +6.24264 q^{11} +5.65685 q^{13} +1.00000 q^{16} +5.41421 q^{17} +1.17157 q^{19} +1.00000 q^{20} +6.24264 q^{22} -8.82843 q^{23} +1.00000 q^{25} +5.65685 q^{26} -5.41421 q^{29} +1.75736 q^{31} +1.00000 q^{32} +5.41421 q^{34} +8.24264 q^{37} +1.17157 q^{38} +1.00000 q^{40} -6.48528 q^{41} -5.07107 q^{43} +6.24264 q^{44} -8.82843 q^{46} +6.24264 q^{47} +1.00000 q^{50} +5.65685 q^{52} -11.6569 q^{53} +6.24264 q^{55} -5.41421 q^{58} -8.82843 q^{59} +0.343146 q^{61} +1.75736 q^{62} +1.00000 q^{64} +5.65685 q^{65} -5.07107 q^{67} +5.41421 q^{68} -12.4853 q^{71} +2.00000 q^{73} +8.24264 q^{74} +1.17157 q^{76} +8.00000 q^{79} +1.00000 q^{80} -6.48528 q^{82} +10.8284 q^{83} +5.41421 q^{85} -5.07107 q^{86} +6.24264 q^{88} +4.82843 q^{89} -8.82843 q^{92} +6.24264 q^{94} +1.17157 q^{95} -10.4853 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 4 q^{11} + 2 q^{16} + 8 q^{17} + 8 q^{19} + 2 q^{20} + 4 q^{22} - 12 q^{23} + 2 q^{25} - 8 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 8 q^{37} + 8 q^{38} + 2 q^{40} + 4 q^{41} + 4 q^{43} + 4 q^{44} - 12 q^{46} + 4 q^{47} + 2 q^{50} - 12 q^{53} + 4 q^{55} - 8 q^{58} - 12 q^{59} + 12 q^{61} + 12 q^{62} + 2 q^{64} + 4 q^{67} + 8 q^{68} - 8 q^{71} + 4 q^{73} + 8 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} + 8 q^{85} + 4 q^{86} + 4 q^{88} + 4 q^{89} - 12 q^{92} + 4 q^{94} + 8 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 4 * q^11 + 2 * q^16 + 8 * q^17 + 8 * q^19 + 2 * q^20 + 4 * q^22 - 12 * q^23 + 2 * q^25 - 8 * q^29 + 12 * q^31 + 2 * q^32 + 8 * q^34 + 8 * q^37 + 8 * q^38 + 2 * q^40 + 4 * q^41 + 4 * q^43 + 4 * q^44 - 12 * q^46 + 4 * q^47 + 2 * q^50 - 12 * q^53 + 4 * q^55 - 8 * q^58 - 12 * q^59 + 12 * q^61 + 12 * q^62 + 2 * q^64 + 4 * q^67 + 8 * q^68 - 8 * q^71 + 4 * q^73 + 8 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 + 8 * q^85 + 4 * q^86 + 4 * q^88 + 4 * q^89 - 12 * q^92 + 4 * q^94 + 8 * q^95 - 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	6.24264	1.88223	0.941113	−	0.338091i	$-0.109781\pi$
$11$	6.24264	1.88223	0.941113	+	0.338091i	$0.109781\pi$
$12$	0	0
$13$	5.65685	1.56893	0.784465	−	0.620174i	$-0.212938\pi$
$13$	5.65685	1.56893	0.784465	+	0.620174i	$0.212938\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	5.41421	1.31314	0.656570	−	0.754265i	$-0.272007\pi$
$17$	5.41421	1.31314	0.656570	+	0.754265i	$0.272007\pi$
$18$	0	0
$19$	1.17157	0.268777	0.134389	−	0.990929i	$-0.457093\pi$
$19$	1.17157	0.268777	0.134389	+	0.990929i	$0.457093\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	6.24264	1.33094
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	5.65685	1.10940
$27$	0	0
$28$	0	0
$29$	−5.41421	−1.00539	−0.502697	−	0.864463i	$-0.667659\pi$
$29$	−5.41421	−1.00539	−0.502697	+	0.864463i	$0.667659\pi$
$30$	0	0
$31$	1.75736	0.315631	0.157816	−	0.987469i	$-0.449555\pi$
$31$	1.75736	0.315631	0.157816	+	0.987469i	$0.449555\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.41421	0.928530
$35$	0	0
$36$	0	0
$37$	8.24264	1.35508	0.677541	−	0.735485i	$-0.263046\pi$
$37$	8.24264	1.35508	0.677541	+	0.735485i	$0.263046\pi$
$38$	1.17157	0.190054
$39$	0	0
$40$	1.00000	0.158114
$41$	−6.48528	−1.01283	−0.506415	−	0.862290i	$-0.669030\pi$
$41$	−6.48528	−1.01283	−0.506415	+	0.862290i	$0.669030\pi$
$42$	0	0
$43$	−5.07107	−0.773331	−0.386665	−	0.922220i	$-0.626373\pi$
$43$	−5.07107	−0.773331	−0.386665	+	0.922220i	$0.626373\pi$
$44$	6.24264	0.941113
$45$	0	0
$46$	−8.82843	−1.30168
$47$	6.24264	0.910583	0.455291	−	0.890343i	$-0.349535\pi$
$47$	6.24264	0.910583	0.455291	+	0.890343i	$0.349535\pi$
$48$	0	0
$49$	0	0
$50$	1.00000	0.141421
$51$	0	0
$52$	5.65685	0.784465
$53$	−11.6569	−1.60119	−0.800596	−	0.599204i	$-0.795484\pi$
$53$	−11.6569	−1.60119	−0.800596	+	0.599204i	$0.795484\pi$
$54$	0	0
$55$	6.24264	0.841757
$56$	0	0
$57$	0	0
$58$	−5.41421	−0.710921
$59$	−8.82843	−1.14936	−0.574682	−	0.818377i	$-0.694874\pi$
$59$	−8.82843	−1.14936	−0.574682	+	0.818377i	$0.694874\pi$
$60$	0	0
$61$	0.343146	0.0439353	0.0219677	−	0.999759i	$-0.493007\pi$
$61$	0.343146	0.0439353	0.0219677	+	0.999759i	$0.493007\pi$
$62$	1.75736	0.223185
$63$	0	0
$64$	1.00000	0.125000
$65$	5.65685	0.701646
$66$	0	0
$67$	−5.07107	−0.619530	−0.309765	−	0.950813i	$-0.600250\pi$
$67$	−5.07107	−0.619530	−0.309765	+	0.950813i	$0.600250\pi$
$68$	5.41421	0.656570
$69$	0	0
$70$	0	0
$71$	−12.4853	−1.48173	−0.740865	−	0.671654i	$-0.765584\pi$
$71$	−12.4853	−1.48173	−0.740865	+	0.671654i	$0.765584\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	8.24264	0.958188
$75$	0	0
$76$	1.17157	0.134389
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−6.48528	−0.716180
$83$	10.8284	1.18857	0.594287	−	0.804253i	$-0.297434\pi$
$83$	10.8284	1.18857	0.594287	+	0.804253i	$0.297434\pi$
$84$	0	0
$85$	5.41421	0.587254
$86$	−5.07107	−0.546827
$87$	0	0
$88$	6.24264	0.665468
$89$	4.82843	0.511812	0.255906	−	0.966702i	$-0.417626\pi$
$89$	4.82843	0.511812	0.255906	+	0.966702i	$0.417626\pi$
$90$	0	0
$91$	0	0
$92$	−8.82843	−0.920427
$93$	0	0
$94$	6.24264	0.643879
$95$	1.17157	0.120201
$96$	0	0
$97$	−10.4853	−1.06462	−0.532310	−	0.846550i	$-0.678676\pi$
$97$	−10.4853	−1.06462	−0.532310	+	0.846550i	$0.678676\pi$
$98$	0	0
$99$	0	0
$100$	1.00000	0.100000
$101$	6.34315	0.631167	0.315583	−	0.948898i	$-0.397800\pi$
$101$	6.34315	0.631167	0.315583	+	0.948898i	$0.397800\pi$
$102$	0	0
$103$	−9.17157	−0.903702	−0.451851	−	0.892093i	$-0.649236\pi$
$103$	−9.17157	−0.903702	−0.451851	+	0.892093i	$0.649236\pi$
$104$	5.65685	0.554700
$105$	0	0
$106$	−11.6569	−1.13221
$107$	1.65685	0.160174	0.0800871	−	0.996788i	$-0.474480\pi$
$107$	1.65685	0.160174	0.0800871	+	0.996788i	$0.474480\pi$
$108$	0	0
$109$	7.65685	0.733394	0.366697	−	0.930341i	$-0.380489\pi$
$109$	7.65685	0.733394	0.366697	+	0.930341i	$0.380489\pi$
$110$	6.24264	0.595212
$111$	0	0
$112$	0	0
$113$	−10.0000	−0.940721	−0.470360	−	0.882474i	$-0.655876\pi$
$113$	−10.0000	−0.940721	−0.470360	+	0.882474i	$0.655876\pi$
$114$	0	0
$115$	−8.82843	−0.823255
$116$	−5.41421	−0.502697
$117$	0	0
$118$	−8.82843	−0.812723
$119$	0	0
$120$	0	0
$121$	27.9706	2.54278
$122$	0.343146	0.0310670
$123$	0	0
$124$	1.75736	0.157816
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.82843	−0.250982	−0.125491	−	0.992095i	$-0.540051\pi$
$127$	−2.82843	−0.250982	−0.125491	+	0.992095i	$0.540051\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	5.65685	0.496139
$131$	−8.82843	−0.771343	−0.385672	−	0.922636i	$-0.626030\pi$
$131$	−8.82843	−0.771343	−0.385672	+	0.922636i	$0.626030\pi$
$132$	0	0
$133$	0	0
$134$	−5.07107	−0.438074
$135$	0	0
$136$	5.41421	0.464265
$137$	−2.34315	−0.200188	−0.100094	−	0.994978i	$-0.531914\pi$
$137$	−2.34315	−0.200188	−0.100094	+	0.994978i	$0.531914\pi$
$138$	0	0
$139$	13.6569	1.15836	0.579180	−	0.815200i	$-0.303373\pi$
$139$	13.6569	1.15836	0.579180	+	0.815200i	$0.303373\pi$
$140$	0	0
$141$	0	0
$142$	−12.4853	−1.04774
$143$	35.3137	2.95308
$144$	0	0
$145$	−5.41421	−0.449626
$146$	2.00000	0.165521
$147$	0	0
$148$	8.24264	0.677541
$149$	14.3848	1.17845	0.589223	−	0.807970i	$-0.299434\pi$
$149$	14.3848	1.17845	0.589223	+	0.807970i	$0.299434\pi$
$150$	0	0
$151$	−8.14214	−0.662598	−0.331299	−	0.943526i	$-0.607487\pi$
$151$	−8.14214	−0.662598	−0.331299	+	0.943526i	$0.607487\pi$
$152$	1.17157	0.0950271
$153$	0	0
$154$	0	0
$155$	1.75736	0.141154
$156$	0	0
$157$	19.6569	1.56879	0.784394	−	0.620263i	$-0.212974\pi$
$157$	19.6569	1.56879	0.784394	+	0.620263i	$0.212974\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	1.00000	0.0790569
$161$	0	0
$162$	0	0
$163$	−2.92893	−0.229412	−0.114706	−	0.993400i	$-0.536593\pi$
$163$	−2.92893	−0.229412	−0.114706	+	0.993400i	$0.536593\pi$
$164$	−6.48528	−0.506415
$165$	0	0
$166$	10.8284	0.840449
$167$	−6.24264	−0.483070	−0.241535	−	0.970392i	$-0.577651\pi$
$167$	−6.24264	−0.483070	−0.241535	+	0.970392i	$0.577651\pi$
$168$	0	0
$169$	19.0000	1.46154
$170$	5.41421	0.415251
$171$	0	0
$172$	−5.07107	−0.386665
$173$	15.1716	1.15347	0.576737	−	0.816930i	$-0.304326\pi$
$173$	15.1716	1.15347	0.576737	+	0.816930i	$0.304326\pi$
$174$	0	0
$175$	0	0
$176$	6.24264	0.470557
$177$	0	0
$178$	4.82843	0.361906
$179$	−17.7574	−1.32725	−0.663624	−	0.748067i	$-0.730982\pi$
$179$	−17.7574	−1.32725	−0.663624	+	0.748067i	$0.730982\pi$
$180$	0	0
$181$	−1.51472	−0.112588	−0.0562941	−	0.998414i	$-0.517928\pi$
$181$	−1.51472	−0.112588	−0.0562941	+	0.998414i	$0.517928\pi$
$182$	0	0
$183$	0	0
$184$	−8.82843	−0.650840
$185$	8.24264	0.606011
$186$	0	0
$187$	33.7990	2.47163
$188$	6.24264	0.455291
$189$	0	0
$190$	1.17157	0.0849948
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	0.828427	0.0596315	0.0298157	−	0.999555i	$-0.490508\pi$
$193$	0.828427	0.0596315	0.0298157	+	0.999555i	$0.490508\pi$
$194$	−10.4853	−0.752799
$195$	0	0
$196$	0	0
$197$	−13.7990	−0.983137	−0.491569	−	0.870839i	$-0.663576\pi$
$197$	−13.7990	−0.983137	−0.491569	+	0.870839i	$0.663576\pi$
$198$	0	0
$199$	14.2426	1.00963	0.504817	−	0.863226i	$-0.331560\pi$
$199$	14.2426	1.00963	0.504817	+	0.863226i	$0.331560\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.34315	0.446302
$203$	0	0
$204$	0	0
$205$	−6.48528	−0.452952
$206$	−9.17157	−0.639014
$207$	0	0
$208$	5.65685	0.392232
$209$	7.31371	0.505900
$210$	0	0
$211$	−0.686292	−0.0472463	−0.0236231	−	0.999721i	$-0.507520\pi$
$211$	−0.686292	−0.0472463	−0.0236231	+	0.999721i	$0.507520\pi$
$212$	−11.6569	−0.800596
$213$	0	0
$214$	1.65685	0.113260
$215$	−5.07107	−0.345844
$216$	0	0
$217$	0	0
$218$	7.65685	0.518588
$219$	0	0
$220$	6.24264	0.420879
$221$	30.6274	2.06022
$222$	0	0
$223$	−15.3137	−1.02548	−0.512741	−	0.858543i	$-0.671370\pi$
$223$	−15.3137	−1.02548	−0.512741	+	0.858543i	$0.671370\pi$
$224$	0	0
$225$	0	0
$226$	−10.0000	−0.665190
$227$	8.97056	0.595397	0.297699	−	0.954660i	$-0.403781\pi$
$227$	8.97056	0.595397	0.297699	+	0.954660i	$0.403781\pi$
$228$	0	0
$229$	−12.1421	−0.802375	−0.401187	−	0.915996i	$-0.631402\pi$
$229$	−12.1421	−0.802375	−0.401187	+	0.915996i	$0.631402\pi$
$230$	−8.82843	−0.582129
$231$	0	0
$232$	−5.41421	−0.355461
$233$	14.9706	0.980754	0.490377	−	0.871510i	$-0.336859\pi$
$233$	14.9706	0.980754	0.490377	+	0.871510i	$0.336859\pi$
$234$	0	0
$235$	6.24264	0.407225
$236$	−8.82843	−0.574682
$237$	0	0
$238$	0	0
$239$	−3.31371	−0.214346	−0.107173	−	0.994240i	$-0.534180\pi$
$239$	−3.31371	−0.214346	−0.107173	+	0.994240i	$0.534180\pi$
$240$	0	0
$241$	−11.5563	−0.744410	−0.372205	−	0.928151i	$-0.621398\pi$
$241$	−11.5563	−0.744410	−0.372205	+	0.928151i	$0.621398\pi$
$242$	27.9706	1.79802
$243$	0	0
$244$	0.343146	0.0219677
$245$	0	0
$246$	0	0
$247$	6.62742	0.421692
$248$	1.75736	0.111592
$249$	0	0
$250$	1.00000	0.0632456
$251$	20.9706	1.32365	0.661825	−	0.749658i	$-0.269782\pi$
$251$	20.9706	1.32365	0.661825	+	0.749658i	$0.269782\pi$
$252$	0	0
$253$	−55.1127	−3.46491
$254$	−2.82843	−0.177471
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.4142	−1.33578	−0.667891	−	0.744259i	$-0.732803\pi$
$257$	−21.4142	−1.33578	−0.667891	+	0.744259i	$0.732803\pi$
$258$	0	0
$259$	0	0
$260$	5.65685	0.350823
$261$	0	0
$262$	−8.82843	−0.545422
$263$	−16.1421	−0.995367	−0.497683	−	0.867359i	$-0.665816\pi$
$263$	−16.1421	−0.995367	−0.497683	+	0.867359i	$0.665816\pi$
$264$	0	0
$265$	−11.6569	−0.716075
$266$	0	0
$267$	0	0
$268$	−5.07107	−0.309765
$269$	23.6569	1.44238	0.721192	−	0.692735i	$-0.243595\pi$
$269$	23.6569	1.44238	0.721192	+	0.692735i	$0.243595\pi$
$270$	0	0
$271$	27.6985	1.68256	0.841282	−	0.540597i	$-0.181802\pi$
$271$	27.6985	1.68256	0.841282	+	0.540597i	$0.181802\pi$
$272$	5.41421	0.328285
$273$	0	0
$274$	−2.34315	−0.141555
$275$	6.24264	0.376445
$276$	0	0
$277$	15.5563	0.934690	0.467345	−	0.884075i	$-0.345211\pi$
$277$	15.5563	0.934690	0.467345	+	0.884075i	$0.345211\pi$
$278$	13.6569	0.819084
$279$	0	0
$280$	0	0
$281$	−27.4558	−1.63788	−0.818939	−	0.573880i	$-0.805437\pi$
$281$	−27.4558	−1.63788	−0.818939	+	0.573880i	$0.805437\pi$
$282$	0	0
$283$	−26.4853	−1.57439	−0.787193	−	0.616706i	$-0.788467\pi$
$283$	−26.4853	−1.57439	−0.787193	+	0.616706i	$0.788467\pi$
$284$	−12.4853	−0.740865
$285$	0	0
$286$	35.3137	2.08814
$287$	0	0
$288$	0	0
$289$	12.3137	0.724336
$290$	−5.41421	−0.317934
$291$	0	0
$292$	2.00000	0.117041
$293$	−1.31371	−0.0767477	−0.0383738	−	0.999263i	$-0.512218\pi$
$293$	−1.31371	−0.0767477	−0.0383738	+	0.999263i	$0.512218\pi$
$294$	0	0
$295$	−8.82843	−0.514011
$296$	8.24264	0.479094
$297$	0	0
$298$	14.3848	0.833288
$299$	−49.9411	−2.88817
$300$	0	0
$301$	0	0
$302$	−8.14214	−0.468527
$303$	0	0
$304$	1.17157	0.0671943
$305$	0.343146	0.0196485
$306$	0	0
$307$	−15.3137	−0.874000	−0.437000	−	0.899462i	$-0.643959\pi$
$307$	−15.3137	−0.874000	−0.437000	+	0.899462i	$0.643959\pi$
$308$	0	0
$309$	0	0
$310$	1.75736	0.0998113
$311$	−5.17157	−0.293253	−0.146626	−	0.989192i	$-0.546842\pi$
$311$	−5.17157	−0.293253	−0.146626	+	0.989192i	$0.546842\pi$
$312$	0	0
$313$	−13.5147	−0.763897	−0.381949	−	0.924184i	$-0.624747\pi$
$313$	−13.5147	−0.763897	−0.381949	+	0.924184i	$0.624747\pi$
$314$	19.6569	1.10930
$315$	0	0
$316$	8.00000	0.450035
$317$	2.68629	0.150877	0.0754386	−	0.997150i	$-0.475964\pi$
$317$	2.68629	0.150877	0.0754386	+	0.997150i	$0.475964\pi$
$318$	0	0
$319$	−33.7990	−1.89238
$320$	1.00000	0.0559017
$321$	0	0
$322$	0	0
$323$	6.34315	0.352942
$324$	0	0
$325$	5.65685	0.313786
$326$	−2.92893	−0.162219
$327$	0	0
$328$	−6.48528	−0.358090
$329$	0	0
$330$	0	0
$331$	−17.6569	−0.970508	−0.485254	−	0.874373i	$-0.661273\pi$
$331$	−17.6569	−0.970508	−0.485254	+	0.874373i	$0.661273\pi$
$332$	10.8284	0.594287
$333$	0	0
$334$	−6.24264	−0.341582
$335$	−5.07107	−0.277062
$336$	0	0
$337$	−29.1127	−1.58587	−0.792935	−	0.609306i	$-0.791448\pi$
$337$	−29.1127	−1.58587	−0.792935	+	0.609306i	$0.791448\pi$
$338$	19.0000	1.03346
$339$	0	0
$340$	5.41421	0.293627
$341$	10.9706	0.594089
$342$	0	0
$343$	0	0
$344$	−5.07107	−0.273414
$345$	0	0
$346$	15.1716	0.815629
$347$	33.6569	1.80679	0.903397	−	0.428805i	$-0.141065\pi$
$347$	33.6569	1.80679	0.903397	+	0.428805i	$0.141065\pi$
$348$	0	0
$349$	27.4558	1.46968	0.734839	−	0.678242i	$-0.237258\pi$
$349$	27.4558	1.46968	0.734839	+	0.678242i	$0.237258\pi$
$350$	0	0
$351$	0	0
$352$	6.24264	0.332734
$353$	−32.0416	−1.70540	−0.852702	−	0.522398i	$-0.825038\pi$
$353$	−32.0416	−1.70540	−0.852702	+	0.522398i	$0.825038\pi$
$354$	0	0
$355$	−12.4853	−0.662650
$356$	4.82843	0.255906
$357$	0	0
$358$	−17.7574	−0.938506
$359$	11.5147	0.607724	0.303862	−	0.952716i	$-0.401724\pi$
$359$	11.5147	0.607724	0.303862	+	0.952716i	$0.401724\pi$
$360$	0	0
$361$	−17.6274	−0.927759
$362$	−1.51472	−0.0796118
$363$	0	0
$364$	0	0
$365$	2.00000	0.104685
$366$	0	0
$367$	5.17157	0.269954	0.134977	−	0.990849i	$-0.456904\pi$
$367$	5.17157	0.269954	0.134977	+	0.990849i	$0.456904\pi$
$368$	−8.82843	−0.460214
$369$	0	0
$370$	8.24264	0.428514
$371$	0	0
$372$	0	0
$373$	9.41421	0.487450	0.243725	−	0.969844i	$-0.421631\pi$
$373$	9.41421	0.487450	0.243725	+	0.969844i	$0.421631\pi$
$374$	33.7990	1.74770
$375$	0	0
$376$	6.24264	0.321940
$377$	−30.6274	−1.57739
$378$	0	0
$379$	−16.4853	−0.846792	−0.423396	−	0.905945i	$-0.639162\pi$
$379$	−16.4853	−0.846792	−0.423396	+	0.905945i	$0.639162\pi$
$380$	1.17157	0.0601004
$381$	0	0
$382$	−4.00000	−0.204658
$383$	16.5858	0.847494	0.423747	−	0.905781i	$-0.360715\pi$
$383$	16.5858	0.847494	0.423747	+	0.905781i	$0.360715\pi$
$384$	0	0
$385$	0	0
$386$	0.828427	0.0421658
$387$	0	0
$388$	−10.4853	−0.532310
$389$	10.1005	0.512116	0.256058	−	0.966661i	$-0.417576\pi$
$389$	10.1005	0.512116	0.256058	+	0.966661i	$0.417576\pi$
$390$	0	0
$391$	−47.7990	−2.41730
$392$	0	0
$393$	0	0
$394$	−13.7990	−0.695183
$395$	8.00000	0.402524
$396$	0	0
$397$	−29.3137	−1.47121	−0.735606	−	0.677409i	$-0.763103\pi$
$397$	−29.3137	−1.47121	−0.735606	+	0.677409i	$0.763103\pi$
$398$	14.2426	0.713919
$399$	0	0
$400$	1.00000	0.0500000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	0	0
$403$	9.94113	0.495203
$404$	6.34315	0.315583
$405$	0	0
$406$	0	0
$407$	51.4558	2.55057
$408$	0	0
$409$	4.24264	0.209785	0.104893	−	0.994484i	$-0.466550\pi$
$409$	4.24264	0.209785	0.104893	+	0.994484i	$0.466550\pi$
$410$	−6.48528	−0.320285
$411$	0	0
$412$	−9.17157	−0.451851
$413$	0	0
$414$	0	0
$415$	10.8284	0.531547
$416$	5.65685	0.277350
$417$	0	0
$418$	7.31371	0.357725
$419$	−3.02944	−0.147998	−0.0739988	−	0.997258i	$-0.523576\pi$
$419$	−3.02944	−0.147998	−0.0739988	+	0.997258i	$0.523576\pi$
$420$	0	0
$421$	7.65685	0.373172	0.186586	−	0.982439i	$-0.440258\pi$
$421$	7.65685	0.373172	0.186586	+	0.982439i	$0.440258\pi$
$422$	−0.686292	−0.0334081
$423$	0	0
$424$	−11.6569	−0.566107
$425$	5.41421	0.262628
$426$	0	0
$427$	0	0
$428$	1.65685	0.0800871
$429$	0	0
$430$	−5.07107	−0.244549
$431$	37.4558	1.80418	0.902092	−	0.431543i	$-0.142031\pi$
$431$	37.4558	1.80418	0.902092	+	0.431543i	$0.142031\pi$
$432$	0	0
$433$	−21.7990	−1.04759	−0.523796	−	0.851844i	$-0.675485\pi$
$433$	−21.7990	−1.04759	−0.523796	+	0.851844i	$0.675485\pi$
$434$	0	0
$435$	0	0
$436$	7.65685	0.366697
$437$	−10.3431	−0.494780
$438$	0	0
$439$	−35.6985	−1.70380	−0.851898	−	0.523708i	$-0.824548\pi$
$439$	−35.6985	−1.70380	−0.851898	+	0.523708i	$0.824548\pi$
$440$	6.24264	0.297606
$441$	0	0
$442$	30.6274	1.45680
$443$	20.2843	0.963735	0.481867	−	0.876244i	$-0.339959\pi$
$443$	20.2843	0.963735	0.481867	+	0.876244i	$0.339959\pi$
$444$	0	0
$445$	4.82843	0.228889
$446$	−15.3137	−0.725125
$447$	0	0
$448$	0	0
$449$	−6.48528	−0.306059	−0.153030	−	0.988222i	$-0.548903\pi$
$449$	−6.48528	−0.306059	−0.153030	+	0.988222i	$0.548903\pi$
$450$	0	0
$451$	−40.4853	−1.90638
$452$	−10.0000	−0.470360
$453$	0	0
$454$	8.97056	0.421009
$455$	0	0
$456$	0	0
$457$	−14.4853	−0.677593	−0.338796	−	0.940860i	$-0.610020\pi$
$457$	−14.4853	−0.677593	−0.338796	+	0.940860i	$0.610020\pi$
$458$	−12.1421	−0.567365
$459$	0	0
$460$	−8.82843	−0.411628
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	−14.3431	−0.666583	−0.333291	−	0.942824i	$-0.608159\pi$
$463$	−14.3431	−0.666583	−0.333291	+	0.942824i	$0.608159\pi$
$464$	−5.41421	−0.251349
$465$	0	0
$466$	14.9706	0.693498
$467$	−4.48528	−0.207554	−0.103777	−	0.994601i	$-0.533093\pi$
$467$	−4.48528	−0.207554	−0.103777	+	0.994601i	$0.533093\pi$
$468$	0	0
$469$	0	0
$470$	6.24264	0.287952
$471$	0	0
$472$	−8.82843	−0.406361
$473$	−31.6569	−1.45558
$474$	0	0
$475$	1.17157	0.0537555
$476$	0	0
$477$	0	0
$478$	−3.31371	−0.151565
$479$	8.48528	0.387702	0.193851	−	0.981031i	$-0.437902\pi$
$479$	8.48528	0.387702	0.193851	+	0.981031i	$0.437902\pi$
$480$	0	0
$481$	46.6274	2.12603
$482$	−11.5563	−0.526377
$483$	0	0
$484$	27.9706	1.27139
$485$	−10.4853	−0.476112
$486$	0	0
$487$	5.17157	0.234346	0.117173	−	0.993111i	$-0.462617\pi$
$487$	5.17157	0.234346	0.117173	+	0.993111i	$0.462617\pi$
$488$	0.343146	0.0155335
$489$	0	0
$490$	0	0
$491$	2.92893	0.132181	0.0660904	−	0.997814i	$-0.478947\pi$
$491$	2.92893	0.132181	0.0660904	+	0.997814i	$0.478947\pi$
$492$	0	0
$493$	−29.3137	−1.32022
$494$	6.62742	0.298182
$495$	0	0
$496$	1.75736	0.0789078
$497$	0	0
$498$	0	0
$499$	1.17157	0.0524468	0.0262234	−	0.999656i	$-0.491652\pi$
$499$	1.17157	0.0524468	0.0262234	+	0.999656i	$0.491652\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	20.9706	0.935962
$503$	−7.41421	−0.330583	−0.165292	−	0.986245i	$-0.552857\pi$
$503$	−7.41421	−0.330583	−0.165292	+	0.986245i	$0.552857\pi$
$504$	0	0
$505$	6.34315	0.282266
$506$	−55.1127	−2.45006
$507$	0	0
$508$	−2.82843	−0.125491
$509$	16.9706	0.752207	0.376103	−	0.926578i	$-0.377264\pi$
$509$	16.9706	0.752207	0.376103	+	0.926578i	$0.377264\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	0	0
$514$	−21.4142	−0.944540
$515$	−9.17157	−0.404148
$516$	0	0
$517$	38.9706	1.71392
$518$	0	0
$519$	0	0
$520$	5.65685	0.248069
$521$	−34.7696	−1.52328	−0.761641	−	0.647999i	$-0.775606\pi$
$521$	−34.7696	−1.52328	−0.761641	+	0.647999i	$0.775606\pi$
$522$	0	0
$523$	−2.20101	−0.0962435	−0.0481217	−	0.998841i	$-0.515324\pi$
$523$	−2.20101	−0.0962435	−0.0481217	+	0.998841i	$0.515324\pi$
$524$	−8.82843	−0.385672
$525$	0	0
$526$	−16.1421	−0.703831
$527$	9.51472	0.414468
$528$	0	0
$529$	54.9411	2.38874
$530$	−11.6569	−0.506341
$531$	0	0
$532$	0	0
$533$	−36.6863	−1.58906
$534$	0	0
$535$	1.65685	0.0716321
$536$	−5.07107	−0.219037
$537$	0	0
$538$	23.6569	1.01992
$539$	0	0
$540$	0	0
$541$	9.79899	0.421291	0.210646	−	0.977562i	$-0.432443\pi$
$541$	9.79899	0.421291	0.210646	+	0.977562i	$0.432443\pi$
$542$	27.6985	1.18975
$543$	0	0
$544$	5.41421	0.232132
$545$	7.65685	0.327984
$546$	0	0
$547$	8.38478	0.358507	0.179254	−	0.983803i	$-0.442632\pi$
$547$	8.38478	0.358507	0.179254	+	0.983803i	$0.442632\pi$
$548$	−2.34315	−0.100094
$549$	0	0
$550$	6.24264	0.266187
$551$	−6.34315	−0.270227
$552$	0	0
$553$	0	0
$554$	15.5563	0.660926
$555$	0	0
$556$	13.6569	0.579180
$557$	38.0000	1.61011	0.805056	−	0.593199i	$-0.202135\pi$
$557$	38.0000	1.61011	0.805056	+	0.593199i	$0.202135\pi$
$558$	0	0
$559$	−28.6863	−1.21330
$560$	0	0
$561$	0	0
$562$	−27.4558	−1.15815
$563$	−3.51472	−0.148128	−0.0740639	−	0.997254i	$-0.523597\pi$
$563$	−3.51472	−0.148128	−0.0740639	+	0.997254i	$0.523597\pi$
$564$	0	0
$565$	−10.0000	−0.420703
$566$	−26.4853	−1.11326
$567$	0	0
$568$	−12.4853	−0.523871
$569$	−42.9706	−1.80142	−0.900710	−	0.434421i	$-0.856953\pi$
$569$	−42.9706	−1.80142	−0.900710	+	0.434421i	$0.856953\pi$
$570$	0	0
$571$	27.3137	1.14304	0.571522	−	0.820587i	$-0.306353\pi$
$571$	27.3137	1.14304	0.571522	+	0.820587i	$0.306353\pi$
$572$	35.3137	1.47654
$573$	0	0
$574$	0	0
$575$	−8.82843	−0.368171
$576$	0	0
$577$	32.1421	1.33809	0.669047	−	0.743220i	$-0.266702\pi$
$577$	32.1421	1.33809	0.669047	+	0.743220i	$0.266702\pi$
$578$	12.3137	0.512183
$579$	0	0
$580$	−5.41421	−0.224813
$581$	0	0
$582$	0	0
$583$	−72.7696	−3.01381
$584$	2.00000	0.0827606
$585$	0	0
$586$	−1.31371	−0.0542688
$587$	−10.8284	−0.446937	−0.223469	−	0.974711i	$-0.571738\pi$
$587$	−10.8284	−0.446937	−0.223469	+	0.974711i	$0.571738\pi$
$588$	0	0
$589$	2.05887	0.0848344
$590$	−8.82843	−0.363461
$591$	0	0
$592$	8.24264	0.338770
$593$	1.21320	0.0498203	0.0249101	−	0.999690i	$-0.492070\pi$
$593$	1.21320	0.0498203	0.0249101	+	0.999690i	$0.492070\pi$
$594$	0	0
$595$	0	0
$596$	14.3848	0.589223
$597$	0	0
$598$	−49.9411	−2.04224
$599$	−4.00000	−0.163436	−0.0817178	−	0.996656i	$-0.526041\pi$
$599$	−4.00000	−0.163436	−0.0817178	+	0.996656i	$0.526041\pi$
$600$	0	0
$601$	40.2426	1.64153	0.820766	−	0.571265i	$-0.193547\pi$
$601$	40.2426	1.64153	0.820766	+	0.571265i	$0.193547\pi$
$602$	0	0
$603$	0	0
$604$	−8.14214	−0.331299
$605$	27.9706	1.13717
$606$	0	0
$607$	−4.00000	−0.162355	−0.0811775	−	0.996700i	$-0.525868\pi$
$607$	−4.00000	−0.162355	−0.0811775	+	0.996700i	$0.525868\pi$
$608$	1.17157	0.0475136
$609$	0	0
$610$	0.343146	0.0138936
$611$	35.3137	1.42864
$612$	0	0
$613$	3.07107	0.124039	0.0620196	−	0.998075i	$-0.480246\pi$
$613$	3.07107	0.124039	0.0620196	+	0.998075i	$0.480246\pi$
$614$	−15.3137	−0.618011
$615$	0	0
$616$	0	0
$617$	33.9411	1.36642	0.683209	−	0.730223i	$-0.260584\pi$
$617$	33.9411	1.36642	0.683209	+	0.730223i	$0.260584\pi$
$618$	0	0
$619$	43.1127	1.73285	0.866423	−	0.499311i	$-0.166413\pi$
$619$	43.1127	1.73285	0.866423	+	0.499311i	$0.166413\pi$
$620$	1.75736	0.0705772
$621$	0	0
$622$	−5.17157	−0.207361
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.5147	−0.540157
$627$	0	0
$628$	19.6569	0.784394
$629$	44.6274	1.77941
$630$	0	0
$631$	20.1421	0.801846	0.400923	−	0.916112i	$-0.368690\pi$
$631$	20.1421	0.801846	0.400923	+	0.916112i	$0.368690\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	2.68629	0.106686
$635$	−2.82843	−0.112243
$636$	0	0
$637$	0	0
$638$	−33.7990	−1.33811
$639$	0	0
$640$	1.00000	0.0395285
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	0	0
$643$	−23.1716	−0.913798	−0.456899	−	0.889519i	$-0.651040\pi$
$643$	−23.1716	−0.913798	−0.456899	+	0.889519i	$0.651040\pi$
$644$	0	0
$645$	0	0
$646$	6.34315	0.249568
$647$	−44.8701	−1.76402	−0.882012	−	0.471227i	$-0.843811\pi$
$647$	−44.8701	−1.76402	−0.882012	+	0.471227i	$0.843811\pi$
$648$	0	0
$649$	−55.1127	−2.16336
$650$	5.65685	0.221880
$651$	0	0
$652$	−2.92893	−0.114706
$653$	−37.7990	−1.47919	−0.739594	−	0.673053i	$-0.764983\pi$
$653$	−37.7990	−1.47919	−0.739594	+	0.673053i	$0.764983\pi$
$654$	0	0
$655$	−8.82843	−0.344955
$656$	−6.48528	−0.253208
$657$	0	0
$658$	0	0
$659$	16.3848	0.638260	0.319130	−	0.947711i	$-0.396609\pi$
$659$	16.3848	0.638260	0.319130	+	0.947711i	$0.396609\pi$
$660$	0	0
$661$	38.9706	1.51578	0.757890	−	0.652383i	$-0.226231\pi$
$661$	38.9706	1.51578	0.757890	+	0.652383i	$0.226231\pi$
$662$	−17.6569	−0.686253
$663$	0	0
$664$	10.8284	0.420224
$665$	0	0
$666$	0	0
$667$	47.7990	1.85078
$668$	−6.24264	−0.241535
$669$	0	0
$670$	−5.07107	−0.195912
$671$	2.14214	0.0826962
$672$	0	0
$673$	30.2843	1.16737	0.583686	−	0.811979i	$-0.301610\pi$
$673$	30.2843	1.16737	0.583686	+	0.811979i	$0.301610\pi$
$674$	−29.1127	−1.12138
$675$	0	0
$676$	19.0000	0.730769
$677$	40.4264	1.55371	0.776857	−	0.629678i	$-0.216813\pi$
$677$	40.4264	1.55371	0.776857	+	0.629678i	$0.216813\pi$
$678$	0	0
$679$	0	0
$680$	5.41421	0.207626
$681$	0	0
$682$	10.9706	0.420085
$683$	−7.79899	−0.298420	−0.149210	−	0.988806i	$-0.547673\pi$
$683$	−7.79899	−0.298420	−0.149210	+	0.988806i	$0.547673\pi$
$684$	0	0
$685$	−2.34315	−0.0895270
$686$	0	0
$687$	0	0
$688$	−5.07107	−0.193333
$689$	−65.9411	−2.51216
$690$	0	0
$691$	−36.9706	−1.40643	−0.703213	−	0.710979i	$-0.748252\pi$
$691$	−36.9706	−1.40643	−0.703213	+	0.710979i	$0.748252\pi$
$692$	15.1716	0.576737
$693$	0	0
$694$	33.6569	1.27760
$695$	13.6569	0.518034
$696$	0	0
$697$	−35.1127	−1.32999
$698$	27.4558	1.03922
$699$	0	0
$700$	0	0
$701$	−25.2132	−0.952290	−0.476145	−	0.879367i	$-0.657966\pi$
$701$	−25.2132	−0.952290	−0.476145	+	0.879367i	$0.657966\pi$
$702$	0	0
$703$	9.65685	0.364215
$704$	6.24264	0.235278
$705$	0	0
$706$	−32.0416	−1.20590
$707$	0	0
$708$	0	0
$709$	40.1421	1.50757	0.753785	−	0.657121i	$-0.228226\pi$
$709$	40.1421	1.50757	0.753785	+	0.657121i	$0.228226\pi$
$710$	−12.4853	−0.468564
$711$	0	0
$712$	4.82843	0.180953
$713$	−15.5147	−0.581031
$714$	0	0
$715$	35.3137	1.32066
$716$	−17.7574	−0.663624
$717$	0	0
$718$	11.5147	0.429725
$719$	−49.9411	−1.86249	−0.931245	−	0.364394i	$-0.881276\pi$
$719$	−49.9411	−1.86249	−0.931245	+	0.364394i	$0.881276\pi$
$720$	0	0
$721$	0	0
$722$	−17.6274	−0.656025
$723$	0	0
$724$	−1.51472	−0.0562941
$725$	−5.41421	−0.201079
$726$	0	0
$727$	−14.6274	−0.542501	−0.271250	−	0.962509i	$-0.587437\pi$
$727$	−14.6274	−0.542501	−0.271250	+	0.962509i	$0.587437\pi$
$728$	0	0
$729$	0	0
$730$	2.00000	0.0740233
$731$	−27.4558	−1.01549
$732$	0	0
$733$	34.0000	1.25582	0.627909	−	0.778287i	$-0.283911\pi$
$733$	34.0000	1.25582	0.627909	+	0.778287i	$0.283911\pi$
$734$	5.17157	0.190886
$735$	0	0
$736$	−8.82843	−0.325420
$737$	−31.6569	−1.16610
$738$	0	0
$739$	−51.1127	−1.88021	−0.940106	−	0.340884i	$-0.889274\pi$
$739$	−51.1127	−1.88021	−0.940106	+	0.340884i	$0.889274\pi$
$740$	8.24264	0.303005
$741$	0	0
$742$	0	0
$743$	28.2843	1.03765	0.518825	−	0.854881i	$-0.326370\pi$
$743$	28.2843	1.03765	0.518825	+	0.854881i	$0.326370\pi$
$744$	0	0
$745$	14.3848	0.527017
$746$	9.41421	0.344679
$747$	0	0
$748$	33.7990	1.23581
$749$	0	0
$750$	0	0
$751$	−17.7990	−0.649494	−0.324747	−	0.945801i	$-0.605279\pi$
$751$	−17.7990	−0.649494	−0.324747	+	0.945801i	$0.605279\pi$
$752$	6.24264	0.227646
$753$	0	0
$754$	−30.6274	−1.11538
$755$	−8.14214	−0.296323
$756$	0	0
$757$	6.10051	0.221727	0.110863	−	0.993836i	$-0.464638\pi$
$757$	6.10051	0.221727	0.110863	+	0.993836i	$0.464638\pi$
$758$	−16.4853	−0.598772
$759$	0	0
$760$	1.17157	0.0424974
$761$	6.97056	0.252683	0.126341	−	0.991987i	$-0.459677\pi$
$761$	6.97056	0.252683	0.126341	+	0.991987i	$0.459677\pi$
$762$	0	0
$763$	0	0
$764$	−4.00000	−0.144715
$765$	0	0
$766$	16.5858	0.599269
$767$	−49.9411	−1.80327
$768$	0	0
$769$	20.7279	0.747468	0.373734	−	0.927536i	$-0.378077\pi$
$769$	20.7279	0.747468	0.373734	+	0.927536i	$0.378077\pi$
$770$	0	0
$771$	0	0
$772$	0.828427	0.0298157
$773$	10.9706	0.394584	0.197292	−	0.980345i	$-0.436785\pi$
$773$	10.9706	0.394584	0.197292	+	0.980345i	$0.436785\pi$
$774$	0	0
$775$	1.75736	0.0631262
$776$	−10.4853	−0.376400
$777$	0	0
$778$	10.1005	0.362121
$779$	−7.59798	−0.272226
$780$	0	0
$781$	−77.9411	−2.78895
$782$	−47.7990	−1.70929
$783$	0	0
$784$	0	0
$785$	19.6569	0.701583
$786$	0	0
$787$	−49.6569	−1.77008	−0.885038	−	0.465519i	$-0.845868\pi$
$787$	−49.6569	−1.77008	−0.885038	+	0.465519i	$0.845868\pi$
$788$	−13.7990	−0.491569
$789$	0	0
$790$	8.00000	0.284627
$791$	0	0
$792$	0	0
$793$	1.94113	0.0689314
$794$	−29.3137	−1.04030
$795$	0	0
$796$	14.2426	0.504817
$797$	−28.1421	−0.996846	−0.498423	−	0.866934i	$-0.666087\pi$
$797$	−28.1421	−0.996846	−0.498423	+	0.866934i	$0.666087\pi$
$798$	0	0
$799$	33.7990	1.19572
$800$	1.00000	0.0353553
$801$	0	0
$802$	6.00000	0.211867
$803$	12.4853	0.440596
$804$	0	0
$805$	0	0
$806$	9.94113	0.350161
$807$	0	0
$808$	6.34315	0.223151
$809$	−17.5147	−0.615785	−0.307892	−	0.951421i	$-0.599624\pi$
$809$	−17.5147	−0.615785	−0.307892	+	0.951421i	$0.599624\pi$
$810$	0	0
$811$	−21.6569	−0.760475	−0.380238	−	0.924889i	$-0.624158\pi$
$811$	−21.6569	−0.760475	−0.380238	+	0.924889i	$0.624158\pi$
$812$	0	0
$813$	0	0
$814$	51.4558	1.80353
$815$	−2.92893	−0.102596
$816$	0	0
$817$	−5.94113	−0.207854
$818$	4.24264	0.148340
$819$	0	0
$820$	−6.48528	−0.226476
$821$	41.2132	1.43835	0.719175	−	0.694829i	$-0.244520\pi$
$821$	41.2132	1.43835	0.719175	+	0.694829i	$0.244520\pi$
$822$	0	0
$823$	−4.97056	−0.173263	−0.0866315	−	0.996240i	$-0.527610\pi$
$823$	−4.97056	−0.173263	−0.0866315	+	0.996240i	$0.527610\pi$
$824$	−9.17157	−0.319507
$825$	0	0
$826$	0	0
$827$	−52.4853	−1.82509	−0.912546	−	0.408974i	$-0.865887\pi$
$827$	−52.4853	−1.82509	−0.912546	+	0.408974i	$0.865887\pi$
$828$	0	0
$829$	−10.6863	−0.371150	−0.185575	−	0.982630i	$-0.559415\pi$
$829$	−10.6863	−0.371150	−0.185575	+	0.982630i	$0.559415\pi$
$830$	10.8284	0.375860
$831$	0	0
$832$	5.65685	0.196116
$833$	0	0
$834$	0	0
$835$	−6.24264	−0.216035
$836$	7.31371	0.252950
$837$	0	0
$838$	−3.02944	−0.104650
$839$	−8.48528	−0.292944	−0.146472	−	0.989215i	$-0.546792\pi$
$839$	−8.48528	−0.292944	−0.146472	+	0.989215i	$0.546792\pi$
$840$	0	0
$841$	0.313708	0.0108175
$842$	7.65685	0.263873
$843$	0	0
$844$	−0.686292	−0.0236231
$845$	19.0000	0.653620
$846$	0	0
$847$	0	0
$848$	−11.6569	−0.400298
$849$	0	0
$850$	5.41421	0.185706
$851$	−72.7696	−2.49451
$852$	0	0
$853$	−4.62742	−0.158440	−0.0792199	−	0.996857i	$-0.525243\pi$
$853$	−4.62742	−0.158440	−0.0792199	+	0.996857i	$0.525243\pi$
$854$	0	0
$855$	0	0
$856$	1.65685	0.0566301
$857$	−0.727922	−0.0248653	−0.0124327	−	0.999923i	$-0.503958\pi$
$857$	−0.727922	−0.0248653	−0.0124327	+	0.999923i	$0.503958\pi$
$858$	0	0
$859$	−43.5980	−1.48754	−0.743772	−	0.668433i	$-0.766965\pi$
$859$	−43.5980	−1.48754	−0.743772	+	0.668433i	$0.766965\pi$
$860$	−5.07107	−0.172922
$861$	0	0
$862$	37.4558	1.27575
$863$	−34.3431	−1.16905	−0.584527	−	0.811374i	$-0.698720\pi$
$863$	−34.3431	−1.16905	−0.584527	+	0.811374i	$0.698720\pi$
$864$	0	0
$865$	15.1716	0.515849
$866$	−21.7990	−0.740760
$867$	0	0
$868$	0	0
$869$	49.9411	1.69414
$870$	0	0
$871$	−28.6863	−0.971998
$872$	7.65685	0.259294
$873$	0	0
$874$	−10.3431	−0.349862
$875$	0	0
$876$	0	0
$877$	−4.24264	−0.143264	−0.0716319	−	0.997431i	$-0.522821\pi$
$877$	−4.24264	−0.143264	−0.0716319	+	0.997431i	$0.522821\pi$
$878$	−35.6985	−1.20477
$879$	0	0
$880$	6.24264	0.210439
$881$	26.0000	0.875962	0.437981	−	0.898984i	$-0.355694\pi$
$881$	26.0000	0.875962	0.437981	+	0.898984i	$0.355694\pi$
$882$	0	0
$883$	−25.7574	−0.866804	−0.433402	−	0.901201i	$-0.642687\pi$
$883$	−25.7574	−0.866804	−0.433402	+	0.901201i	$0.642687\pi$
$884$	30.6274	1.03011
$885$	0	0
$886$	20.2843	0.681463
$887$	20.8701	0.700748	0.350374	−	0.936610i	$-0.386054\pi$
$887$	20.8701	0.700748	0.350374	+	0.936610i	$0.386054\pi$
$888$	0	0
$889$	0	0
$890$	4.82843	0.161849
$891$	0	0
$892$	−15.3137	−0.512741
$893$	7.31371	0.244744
$894$	0	0
$895$	−17.7574	−0.593563
$896$	0	0
$897$	0	0
$898$	−6.48528	−0.216417
$899$	−9.51472	−0.317334
$900$	0	0
$901$	−63.1127	−2.10259
$902$	−40.4853	−1.34801
$903$	0	0
$904$	−10.0000	−0.332595
$905$	−1.51472	−0.0503510
$906$	0	0
$907$	36.8701	1.22425	0.612125	−	0.790761i	$-0.290315\pi$
$907$	36.8701	1.22425	0.612125	+	0.790761i	$0.290315\pi$
$908$	8.97056	0.297699
$909$	0	0
$910$	0	0
$911$	−11.0294	−0.365422	−0.182711	−	0.983167i	$-0.558487\pi$
$911$	−11.0294	−0.365422	−0.182711	+	0.983167i	$0.558487\pi$
$912$	0	0
$913$	67.5980	2.23717
$914$	−14.4853	−0.479131
$915$	0	0
$916$	−12.1421	−0.401187
$917$	0	0
$918$	0	0
$919$	42.7696	1.41084	0.705419	−	0.708791i	$-0.250759\pi$
$919$	42.7696	1.41084	0.705419	+	0.708791i	$0.250759\pi$
$920$	−8.82843	−0.291065
$921$	0	0
$922$	2.00000	0.0658665
$923$	−70.6274	−2.32473
$924$	0	0
$925$	8.24264	0.271016
$926$	−14.3431	−0.471345
$927$	0	0
$928$	−5.41421	−0.177730
$929$	44.3431	1.45485	0.727426	−	0.686186i	$-0.240717\pi$
$929$	44.3431	1.45485	0.727426	+	0.686186i	$0.240717\pi$
$930$	0	0
$931$	0	0
$932$	14.9706	0.490377
$933$	0	0
$934$	−4.48528	−0.146763
$935$	33.7990	1.10535
$936$	0	0
$937$	−47.6569	−1.55688	−0.778441	−	0.627718i	$-0.783989\pi$
$937$	−47.6569	−1.55688	−0.778441	+	0.627718i	$0.783989\pi$
$938$	0	0
$939$	0	0
$940$	6.24264	0.203612
$941$	28.2843	0.922041	0.461020	−	0.887390i	$-0.347483\pi$
$941$	28.2843	0.922041	0.461020	+	0.887390i	$0.347483\pi$
$942$	0	0
$943$	57.2548	1.86447
$944$	−8.82843	−0.287341
$945$	0	0
$946$	−31.6569	−1.02925
$947$	14.8284	0.481859	0.240930	−	0.970543i	$-0.422548\pi$
$947$	14.8284	0.481859	0.240930	+	0.970543i	$0.422548\pi$
$948$	0	0
$949$	11.3137	0.367259
$950$	1.17157	0.0380108
$951$	0	0
$952$	0	0
$953$	−27.3137	−0.884778	−0.442389	−	0.896823i	$-0.645869\pi$
$953$	−27.3137	−0.884778	−0.442389	+	0.896823i	$0.645869\pi$
$954$	0	0
$955$	−4.00000	−0.129437
$956$	−3.31371	−0.107173
$957$	0	0
$958$	8.48528	0.274147
$959$	0	0
$960$	0	0
$961$	−27.9117	−0.900377
$962$	46.6274	1.50333
$963$	0	0
$964$	−11.5563	−0.372205
$965$	0.828427	0.0266680
$966$	0	0
$967$	−21.6569	−0.696437	−0.348219	−	0.937413i	$-0.613213\pi$
$967$	−21.6569	−0.696437	−0.348219	+	0.937413i	$0.613213\pi$
$968$	27.9706	0.899008
$969$	0	0
$970$	−10.4853	−0.336662
$971$	28.9706	0.929710	0.464855	−	0.885387i	$-0.346107\pi$
$971$	28.9706	0.929710	0.464855	+	0.885387i	$0.346107\pi$
$972$	0	0
$973$	0	0
$974$	5.17157	0.165708
$975$	0	0
$976$	0.343146	0.0109838
$977$	6.68629	0.213913	0.106957	−	0.994264i	$-0.465889\pi$
$977$	6.68629	0.213913	0.106957	+	0.994264i	$0.465889\pi$
$978$	0	0
$979$	30.1421	0.963347
$980$	0	0
$981$	0	0
$982$	2.92893	0.0934660
$983$	−2.92893	−0.0934184	−0.0467092	−	0.998909i	$-0.514873\pi$
$983$	−2.92893	−0.0934184	−0.0467092	+	0.998909i	$0.514873\pi$
$984$	0	0
$985$	−13.7990	−0.439672
$986$	−29.3137	−0.933539
$987$	0	0
$988$	6.62742	0.210846
$989$	44.7696	1.42359
$990$	0	0
$991$	−49.7990	−1.58192	−0.790959	−	0.611870i	$-0.790418\pi$
$991$	−49.7990	−1.58192	−0.790959	+	0.611870i	$0.790418\pi$
$992$	1.75736	0.0557962
$993$	0	0
$994$	0	0
$995$	14.2426	0.451522
$996$	0	0
$997$	11.3726	0.360173	0.180087	−	0.983651i	$-0.442362\pi$
$997$	11.3726	0.360173	0.180087	+	0.983651i	$0.442362\pi$
$998$	1.17157	0.0370855
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4410.2.a.bz.1.2		2
3.2	odd	2		1470.2.a.t.1.1	yes	2
7.6	odd	2		4410.2.a.bw.1.2		2
15.14	odd	2		7350.2.a.dh.1.1		2
21.2	odd	6		1470.2.i.w.361.2		4
21.5	even	6		1470.2.i.x.361.2		4
21.11	odd	6		1470.2.i.w.961.2		4
21.17	even	6		1470.2.i.x.961.2		4
21.20	even	2		1470.2.a.s.1.1	✓	2
105.104	even	2		7350.2.a.dl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1470.2.a.s.1.1	✓	2	21.20	even	2
1470.2.a.t.1.1	yes	2	3.2	odd	2
1470.2.i.w.361.2		4	21.2	odd	6
1470.2.i.w.961.2		4	21.11	odd	6
1470.2.i.x.361.2		4	21.5	even	6
1470.2.i.x.961.2		4	21.17	even	6
4410.2.a.bw.1.2		2	7.6	odd	2
4410.2.a.bz.1.2		2	1.1	even	1	trivial
7350.2.a.dh.1.1		2	15.14	odd	2
7350.2.a.dl.1.1		2	105.104	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:	\( N \)	\(=\)	\( 4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4410.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} +6.24264 q^{11} +5.65685 q^{13} +1.00000 q^{16} +5.41421 q^{17} +1.17157 q^{19} +1.00000 q^{20} +6.24264 q^{22} -8.82843 q^{23} +1.00000 q^{25} +5.65685 q^{26} -5.41421 q^{29} +1.75736 q^{31} +1.00000 q^{32} +5.41421 q^{34} +8.24264 q^{37} +1.17157 q^{38} +1.00000 q^{40} -6.48528 q^{41} -5.07107 q^{43} +6.24264 q^{44} -8.82843 q^{46} +6.24264 q^{47} +1.00000 q^{50} +5.65685 q^{52} -11.6569 q^{53} +6.24264 q^{55} -5.41421 q^{58} -8.82843 q^{59} +0.343146 q^{61} +1.75736 q^{62} +1.00000 q^{64} +5.65685 q^{65} -5.07107 q^{67} +5.41421 q^{68} -12.4853 q^{71} +2.00000 q^{73} +8.24264 q^{74} +1.17157 q^{76} +8.00000 q^{79} +1.00000 q^{80} -6.48528 q^{82} +10.8284 q^{83} +5.41421 q^{85} -5.07107 q^{86} +6.24264 q^{88} +4.82843 q^{89} -8.82843 q^{92} +6.24264 q^{94} +1.17157 q^{95} -10.4853 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 2 q^{8} + 2 q^{10} + 4 q^{11} + 2 q^{16} + 8 q^{17} + 8 q^{19} + 2 q^{20} + 4 q^{22} - 12 q^{23} + 2 q^{25} - 8 q^{29} + 12 q^{31} + 2 q^{32} + 8 q^{34} + 8 q^{37} + 8 q^{38} + 2 q^{40} + 4 q^{41} + 4 q^{43} + 4 q^{44} - 12 q^{46} + 4 q^{47} + 2 q^{50} - 12 q^{53} + 4 q^{55} - 8 q^{58} - 12 q^{59} + 12 q^{61} + 12 q^{62} + 2 q^{64} + 4 q^{67} + 8 q^{68} - 8 q^{71} + 4 q^{73} + 8 q^{74} + 8 q^{76} + 16 q^{79} + 2 q^{80} + 4 q^{82} + 16 q^{83} + 8 q^{85} + 4 q^{86} + 4 q^{88} + 4 q^{89} - 12 q^{92} + 4 q^{94} + 8 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 2 * q^8 + 2 * q^10 + 4 * q^11 + 2 * q^16 + 8 * q^17 + 8 * q^19 + 2 * q^20 + 4 * q^22 - 12 * q^23 + 2 * q^25 - 8 * q^29 + 12 * q^31 + 2 * q^32 + 8 * q^34 + 8 * q^37 + 8 * q^38 + 2 * q^40 + 4 * q^41 + 4 * q^43 + 4 * q^44 - 12 * q^46 + 4 * q^47 + 2 * q^50 - 12 * q^53 + 4 * q^55 - 8 * q^58 - 12 * q^59 + 12 * q^61 + 12 * q^62 + 2 * q^64 + 4 * q^67 + 8 * q^68 - 8 * q^71 + 4 * q^73 + 8 * q^74 + 8 * q^76 + 16 * q^79 + 2 * q^80 + 4 * q^82 + 16 * q^83 + 8 * q^85 + 4 * q^86 + 4 * q^88 + 4 * q^89 - 12 * q^92 + 4 * q^94 + 8 * q^95 - 4 * q^97

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