LMFDB - Embedded newform 5445.2.a.bh.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5445,2,Mod(1,5445)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5445, 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("5445.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5445 = 3^{2} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5445.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:	$43.4785439006$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.48704.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 6x^{2} + 4x + 6 $ x^4 - 2x^3 - 6x^2 + 4*x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 495)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.26270$ of defining polynomial
Character	$\chi$	$=$	5445.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.262696 q^{2} -1.93099 q^{4} +1.00000 q^{5} -0.704647 q^{7} -1.03266 q^{8} +O(q^{10})$	$q+0.262696 q^{2} -1.93099 q^{4} +1.00000 q^{5} -0.704647 q^{7} -1.03266 q^{8} +0.262696 q^{10} -3.82075 q^{13} -0.185108 q^{14} +3.59071 q^{16} +7.15733 q^{17} -7.33659 q^{19} -1.93099 q^{20} +5.86198 q^{23} +1.00000 q^{25} -1.00370 q^{26} +1.36067 q^{28} -8.97808 q^{29} -0.590706 q^{31} +3.00858 q^{32} +1.88021 q^{34} -0.704647 q^{35} +10.4527 q^{37} -1.92729 q^{38} -1.03266 q^{40} +7.92729 q^{41} -3.29535 q^{43} +1.53992 q^{46} +3.64149 q^{47} -6.50347 q^{49} +0.262696 q^{50} +7.37782 q^{52} +11.8620 q^{53} +0.727658 q^{56} -2.35851 q^{58} -10.8112 q^{59} -5.64149 q^{61} -0.155176 q^{62} -6.39107 q^{64} -3.82075 q^{65} -10.9128 q^{67} -13.8207 q^{68} -0.185108 q^{70} -11.8620 q^{71} -3.82075 q^{73} +2.74588 q^{74} +14.1669 q^{76} +3.33659 q^{79} +3.59071 q^{80} +2.08247 q^{82} +4.04124 q^{83} +7.15733 q^{85} -0.865677 q^{86} -7.05079 q^{89} +2.69228 q^{91} -11.3194 q^{92} +0.956606 q^{94} -7.33659 q^{95} +6.81120 q^{97} -1.70844 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 8 q^{4} + 4 q^{5} - 4 q^{7} - 6 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 8 * q^4 + 4 * q^5 - 4 * q^7 - 6 * q^8	$ 4 q - 2 q^{2} + 8 q^{4} + 4 q^{5} - 4 q^{7} - 6 q^{8} - 2 q^{10} - 8 q^{13} - 8 q^{14} + 12 q^{16} - 4 q^{17} - 4 q^{19} + 8 q^{20} - 8 q^{23} + 4 q^{25} - 16 q^{26} + 8 q^{28} + 4 q^{29} - 14 q^{32} + 4 q^{34} - 4 q^{35} + 8 q^{37} + 20 q^{38} - 6 q^{40} + 4 q^{41} - 12 q^{43} + 16 q^{46} + 20 q^{49} - 2 q^{50} - 20 q^{52} + 16 q^{53} - 48 q^{56} - 24 q^{58} - 24 q^{59} - 8 q^{61} + 20 q^{62} - 8 q^{65} - 48 q^{68} - 8 q^{70} - 16 q^{71} - 8 q^{73} - 12 q^{74} + 36 q^{76} - 12 q^{79} + 12 q^{80} - 40 q^{82} - 8 q^{83} - 4 q^{85} + 16 q^{86} - 16 q^{89} - 16 q^{91} - 72 q^{92} + 40 q^{94} - 4 q^{95} + 8 q^{97} - 62 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 8 * q^4 + 4 * q^5 - 4 * q^7 - 6 * q^8 - 2 * q^10 - 8 * q^13 - 8 * q^14 + 12 * q^16 - 4 * q^17 - 4 * q^19 + 8 * q^20 - 8 * q^23 + 4 * q^25 - 16 * q^26 + 8 * q^28 + 4 * q^29 - 14 * q^32 + 4 * q^34 - 4 * q^35 + 8 * q^37 + 20 * q^38 - 6 * q^40 + 4 * q^41 - 12 * q^43 + 16 * q^46 + 20 * q^49 - 2 * q^50 - 20 * q^52 + 16 * q^53 - 48 * q^56 - 24 * q^58 - 24 * q^59 - 8 * q^61 + 20 * q^62 - 8 * q^65 - 48 * q^68 - 8 * q^70 - 16 * q^71 - 8 * q^73 - 12 * q^74 + 36 * q^76 - 12 * q^79 + 12 * q^80 - 40 * q^82 - 8 * q^83 - 4 * q^85 + 16 * q^86 - 16 * q^89 - 16 * q^91 - 72 * q^92 + 40 * q^94 - 4 * q^95 + 8 * q^97 - 62 * 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 $

$2$	0.262696	0.185754	0.0928772	−	0.995678i	$-0.470394\pi$
$2$	0.262696	0.185754	0.0928772	+	0.995678i	$0.470394\pi$
$3$	0	0
$3$	0	0
$4$	−1.93099	−0.965495
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.704647	−0.266332	−0.133166	−	0.991094i	$-0.542514\pi$
$7$	−0.704647	−0.266332	−0.133166	+	0.991094i	$0.542514\pi$
$8$	−1.03266	−0.365099
$9$	0	0
$10$	0.262696	0.0830719
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−3.82075	−1.05968	−0.529842	−	0.848096i	$-0.677749\pi$
$13$	−3.82075	−1.05968	−0.529842	+	0.848096i	$0.677749\pi$
$14$	−0.185108	−0.0494722
$15$	0	0
$16$	3.59071	0.897677
$17$	7.15733	1.73591	0.867954	−	0.496644i	$-0.165435\pi$
$17$	7.15733	1.73591	0.867954	+	0.496644i	$0.165435\pi$
$18$	0	0
$19$	−7.33659	−1.68313	−0.841564	−	0.540157i	$-0.818365\pi$
$19$	−7.33659	−1.68313	−0.841564	+	0.540157i	$0.818365\pi$
$20$	−1.93099	−0.431783
$21$	0	0
$22$	0	0
$23$	5.86198	1.22231	0.611154	−	0.791512i	$-0.290706\pi$
$23$	5.86198	1.22231	0.611154	+	0.791512i	$0.290706\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.00370	−0.196841
$27$	0	0
$28$	1.36067	0.257142
$29$	−8.97808	−1.66719	−0.833594	−	0.552378i	$-0.813721\pi$
$29$	−8.97808	−1.66719	−0.833594	+	0.552378i	$0.813721\pi$
$30$	0	0
$31$	−0.590706	−0.106094	−0.0530470	−	0.998592i	$-0.516893\pi$
$31$	−0.590706	−0.106094	−0.0530470	+	0.998592i	$0.516893\pi$
$32$	3.00858	0.531847
$33$	0	0
$34$	1.88021	0.322453
$35$	−0.704647	−0.119107
$36$	0	0
$37$	10.4527	1.71841	0.859206	−	0.511630i	$-0.170958\pi$
$37$	10.4527	1.71841	0.859206	+	0.511630i	$0.170958\pi$
$38$	−1.92729	−0.312649
$39$	0	0
$40$	−1.03266	−0.163277
$41$	7.92729	1.23804	0.619018	−	0.785377i	$-0.287531\pi$
$41$	7.92729	1.23804	0.619018	+	0.785377i	$0.287531\pi$
$42$	0	0
$43$	−3.29535	−0.502537	−0.251268	−	0.967917i	$-0.580848\pi$
$43$	−3.29535	−0.502537	−0.251268	+	0.967917i	$0.580848\pi$
$44$	0	0
$45$	0	0
$46$	1.53992	0.227049
$47$	3.64149	0.531166	0.265583	−	0.964088i	$-0.414436\pi$
$47$	3.64149	0.531166	0.265583	+	0.964088i	$0.414436\pi$
$48$	0	0
$49$	−6.50347	−0.929068
$50$	0.262696	0.0371509
$51$	0	0
$52$	7.37782	1.02312
$53$	11.8620	1.62937	0.814684	−	0.579905i	$-0.196910\pi$
$53$	11.8620	1.62937	0.814684	+	0.579905i	$0.196910\pi$
$54$	0	0
$55$	0	0
$56$	0.727658	0.0972374
$57$	0	0
$58$	−2.35851	−0.309687
$59$	−10.8112	−1.40750	−0.703749	−	0.710449i	$-0.748492\pi$
$59$	−10.8112	−1.40750	−0.703749	+	0.710449i	$0.748492\pi$
$60$	0	0
$61$	−5.64149	−0.722319	−0.361159	−	0.932504i	$-0.617619\pi$
$61$	−5.64149	−0.722319	−0.361159	+	0.932504i	$0.617619\pi$
$62$	−0.155176	−0.0197074
$63$	0	0
$64$	−6.39107	−0.798884
$65$	−3.82075	−0.473905
$66$	0	0
$67$	−10.9128	−1.33321	−0.666603	−	0.745413i	$-0.732252\pi$
$67$	−10.9128	−1.33321	−0.666603	+	0.745413i	$0.732252\pi$
$68$	−13.8207	−1.67601
$69$	0	0
$70$	−0.185108	−0.0221247
$71$	−11.8620	−1.40776	−0.703879	−	0.710320i	$-0.748550\pi$
$71$	−11.8620	−1.40776	−0.703879	+	0.710320i	$0.748550\pi$
$72$	0	0
$73$	−3.82075	−0.447184	−0.223592	−	0.974683i	$-0.571778\pi$
$73$	−3.82075	−0.447184	−0.223592	+	0.974683i	$0.571778\pi$
$74$	2.74588	0.319202
$75$	0	0
$76$	14.1669	1.62505
$77$	0	0
$78$	0	0
$79$	3.33659	0.375396	0.187698	−	0.982227i	$-0.439897\pi$
$79$	3.33659	0.375396	0.187698	+	0.982227i	$0.439897\pi$
$80$	3.59071	0.401453
$81$	0	0
$82$	2.08247	0.229970
$83$	4.04124	0.443583	0.221792	−	0.975094i	$-0.428810\pi$
$83$	4.04124	0.443583	0.221792	+	0.975094i	$0.428810\pi$
$84$	0	0
$85$	7.15733	0.776322
$86$	−0.865677	−0.0933484
$87$	0	0
$88$	0	0
$89$	−7.05079	−0.747382	−0.373691	−	0.927553i	$-0.621908\pi$
$89$	−7.05079	−0.747382	−0.373691	+	0.927553i	$0.621908\pi$
$90$	0	0
$91$	2.69228	0.282227
$92$	−11.3194	−1.18013
$93$	0	0
$94$	0.956606	0.0986664
$95$	−7.33659	−0.752718
$96$	0	0
$97$	6.81120	0.691572	0.345786	−	0.938313i	$-0.387612\pi$
$97$	6.81120	0.691572	0.345786	+	0.938313i	$0.387612\pi$
$98$	−1.70844	−0.172578
$99$	0	0
$100$	−1.93099	−0.193099
$101$	−6.38737	−0.635567	−0.317784	−	0.948163i	$-0.602939\pi$
$101$	−6.38737	−0.635567	−0.317784	+	0.948163i	$0.602939\pi$
$102$	0	0
$103$	−11.4019	−1.12346	−0.561731	−	0.827320i	$-0.689865\pi$
$103$	−11.4019	−1.12346	−0.561731	+	0.827320i	$0.689865\pi$
$104$	3.94552	0.386890
$105$	0	0
$106$	3.11610	0.302662
$107$	−5.58116	−0.539551	−0.269775	−	0.962923i	$-0.586949\pi$
$107$	−5.58116	−0.539551	−0.269775	+	0.962923i	$0.586949\pi$
$108$	0	0
$109$	−0.949215	−0.0909183	−0.0454591	−	0.998966i	$-0.514475\pi$
$109$	−0.949215	−0.0909183	−0.0454591	+	0.998966i	$0.514475\pi$
$110$	0	0
$111$	0	0
$112$	−2.53018	−0.239080
$113$	9.27128	0.872168	0.436084	−	0.899906i	$-0.356365\pi$
$113$	9.27128	0.872168	0.436084	+	0.899906i	$0.356365\pi$
$114$	0	0
$115$	5.86198	0.546633
$116$	17.3366	1.60966
$117$	0	0
$118$	−2.84006	−0.261449
$119$	−5.04339	−0.462327
$120$	0	0
$121$	0	0
$122$	−1.48200	−0.134174
$123$	0	0
$124$	1.14065	0.102433
$125$	1.00000	0.0894427
$126$	0	0
$127$	11.0193	0.977806	0.488903	−	0.872338i	$-0.337397\pi$
$127$	11.0193	0.977806	0.488903	+	0.872338i	$0.337397\pi$
$128$	−7.69607	−0.680243
$129$	0	0
$130$	−1.00370	−0.0880299
$131$	3.64149	0.318159	0.159079	−	0.987266i	$-0.449147\pi$
$131$	3.64149	0.318159	0.159079	+	0.987266i	$0.449147\pi$
$132$	0	0
$133$	5.16970	0.448270
$134$	−2.86674	−0.247649
$135$	0	0
$136$	−7.39107	−0.633779
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	2.07271	0.175805	0.0879023	−	0.996129i	$-0.471984\pi$
$139$	2.07271	0.175805	0.0879023	+	0.996129i	$0.471984\pi$
$140$	1.36067	0.114997
$141$	0	0
$142$	−3.11610	−0.261497
$143$	0	0
$144$	0	0
$145$	−8.97808	−0.745589
$146$	−1.00370	−0.0830665
$147$	0	0
$148$	−20.1840	−1.65912
$149$	19.4382	1.59244	0.796218	−	0.605010i	$-0.206831\pi$
$149$	19.4382	1.59244	0.796218	+	0.605010i	$0.206831\pi$
$150$	0	0
$151$	−7.33659	−0.597043	−0.298522	−	0.954403i	$-0.596493\pi$
$151$	−7.33659	−0.597043	−0.298522	+	0.954403i	$0.596493\pi$
$152$	7.57618	0.614509
$153$	0	0
$154$	0	0
$155$	−0.590706	−0.0474467
$156$	0	0
$157$	−11.2639	−0.898956	−0.449478	−	0.893291i	$-0.648390\pi$
$157$	−11.2639	−0.898956	−0.449478	+	0.893291i	$0.648390\pi$
$158$	0.876510	0.0697314
$159$	0	0
$160$	3.00858	0.237849
$161$	−4.13063	−0.325539
$162$	0	0
$163$	−13.5035	−1.05767	−0.528837	−	0.848724i	$-0.677372\pi$
$163$	−13.5035	−1.05767	−0.528837	+	0.848724i	$0.677372\pi$
$164$	−15.3075	−1.19532
$165$	0	0
$166$	1.06162	0.0823975
$167$	−16.0412	−1.24131	−0.620654	−	0.784085i	$-0.713133\pi$
$167$	−16.0412	−1.24131	−0.620654	+	0.784085i	$0.713133\pi$
$168$	0	0
$169$	1.59810	0.122931
$170$	1.88021	0.144205
$171$	0	0
$172$	6.36330	0.485197
$173$	−4.84267	−0.368181	−0.184091	−	0.982909i	$-0.558934\pi$
$173$	−4.84267	−0.368181	−0.184091	+	0.982909i	$0.558934\pi$
$174$	0	0
$175$	−0.704647	−0.0532663
$176$	0	0
$177$	0	0
$178$	−1.85222	−0.138829
$179$	−1.05079	−0.0785394	−0.0392697	−	0.999229i	$-0.512503\pi$
$179$	−1.05079	−0.0785394	−0.0392697	+	0.999229i	$0.512503\pi$
$180$	0	0
$181$	0.598098	0.0444563	0.0222281	−	0.999753i	$-0.492924\pi$
$181$	0.598098	0.0444563	0.0222281	+	0.999753i	$0.492924\pi$
$182$	0.707251	0.0524249
$183$	0	0
$184$	−6.05341	−0.446264
$185$	10.4527	0.768497
$186$	0	0
$187$	0	0
$188$	−7.03169	−0.512838
$189$	0	0
$190$	−1.92729	−0.139821
$191$	−24.7747	−1.79264	−0.896319	−	0.443410i	$-0.853769\pi$
$191$	−24.7747	−1.79264	−0.896319	+	0.443410i	$0.853769\pi$
$192$	0	0
$193$	−23.9032	−1.72059	−0.860296	−	0.509796i	$-0.829721\pi$
$193$	−23.9032	−1.72059	−0.860296	+	0.509796i	$0.829721\pi$
$194$	1.78928	0.128463
$195$	0	0
$196$	12.5581	0.897010
$197$	−16.5666	−1.18032	−0.590162	−	0.807285i	$-0.700936\pi$
$197$	−16.5666	−1.18032	−0.590162	+	0.807285i	$0.700936\pi$
$198$	0	0
$199$	−5.05079	−0.358041	−0.179020	−	0.983845i	$-0.557293\pi$
$199$	−5.05079	−0.358041	−0.179020	+	0.983845i	$0.557293\pi$
$200$	−1.03266	−0.0730199
$201$	0	0
$202$	−1.67794	−0.118059
$203$	6.32638	0.444025
$204$	0	0
$205$	7.92729	0.553666
$206$	−2.99524	−0.208688
$207$	0	0
$208$	−13.7192	−0.951254
$209$	0	0
$210$	0	0
$211$	10.1552	0.699111	0.349556	−	0.936916i	$-0.386333\pi$
$211$	10.1552	0.699111	0.349556	+	0.936916i	$0.386333\pi$
$212$	−22.9054	−1.57315
$213$	0	0
$214$	−1.46615	−0.100224
$215$	−3.29535	−0.224741
$216$	0	0
$217$	0.416239	0.0282562
$218$	−0.249355	−0.0168885
$219$	0	0
$220$	0	0
$221$	−27.3464	−1.83951
$222$	0	0
$223$	10.3147	0.690721	0.345361	−	0.938470i	$-0.387757\pi$
$223$	10.3147	0.690721	0.345361	+	0.938470i	$0.387757\pi$
$224$	−2.11999	−0.141648
$225$	0	0
$226$	2.43553	0.162009
$227$	−9.00955	−0.597985	−0.298992	−	0.954255i	$-0.596651\pi$
$227$	−9.00955	−0.597985	−0.298992	+	0.954255i	$0.596651\pi$
$228$	0	0
$229$	9.28298	0.613437	0.306718	−	0.951800i	$-0.400769\pi$
$229$	9.28298	0.613437	0.306718	+	0.951800i	$0.400769\pi$
$230$	1.53992	0.101539
$231$	0	0
$232$	9.27128	0.608689
$233$	−3.51584	−0.230331	−0.115165	−	0.993346i	$-0.536740\pi$
$233$	−3.51584	−0.230331	−0.115165	+	0.993346i	$0.536740\pi$
$234$	0	0
$235$	3.64149	0.237545
$236$	20.8763	1.35893
$237$	0	0
$238$	−1.32488	−0.0858793
$239$	−15.3655	−0.993909	−0.496954	−	0.867777i	$-0.665548\pi$
$239$	−15.3655	−0.993909	−0.496954	+	0.867777i	$0.665548\pi$
$240$	0	0
$241$	0.590706	0.0380507	0.0190254	−	0.999819i	$-0.493944\pi$
$241$	0.590706	0.0380507	0.0190254	+	0.999819i	$0.493944\pi$
$242$	0	0
$243$	0	0
$244$	10.8937	0.697396
$245$	−6.50347	−0.415492
$246$	0	0
$247$	28.0312	1.78359
$248$	0.609997	0.0387348
$249$	0	0
$250$	0.262696	0.0166144
$251$	23.5859	1.48873	0.744366	−	0.667772i	$-0.232752\pi$
$251$	23.5859	1.48873	0.744366	+	0.667772i	$0.232752\pi$
$252$	0	0
$253$	0	0
$254$	2.89473	0.181632
$255$	0	0
$256$	10.7604	0.672526
$257$	−20.3147	−1.26719	−0.633597	−	0.773663i	$-0.718422\pi$
$257$	−20.3147	−1.26719	−0.633597	+	0.773663i	$0.718422\pi$
$258$	0	0
$259$	−7.36545	−0.457667
$260$	7.37782	0.457553
$261$	0	0
$262$	0.956606	0.0590993
$263$	−19.8958	−1.22683	−0.613415	−	0.789761i	$-0.710205\pi$
$263$	−19.8958	−1.22683	−0.613415	+	0.789761i	$0.710205\pi$
$264$	0	0
$265$	11.8620	0.728676
$266$	1.35806	0.0832681
$267$	0	0
$268$	21.0725	1.28720
$269$	−19.2830	−1.17570	−0.587852	−	0.808968i	$-0.700026\pi$
$269$	−19.2830	−1.17570	−0.587852	+	0.808968i	$0.700026\pi$
$270$	0	0
$271$	−9.43816	−0.573327	−0.286664	−	0.958031i	$-0.592546\pi$
$271$	−9.43816	−0.573327	−0.286664	+	0.958031i	$0.592546\pi$
$272$	25.6999	1.55828
$273$	0	0
$274$	−1.57618	−0.0952204
$275$	0	0
$276$	0	0
$277$	−22.5764	−1.35648	−0.678242	−	0.734839i	$-0.737258\pi$
$277$	−22.5764	−1.35648	−0.678242	+	0.734839i	$0.737258\pi$
$278$	0.544492	0.0326565
$279$	0	0
$280$	0.727658	0.0434859
$281$	−7.65126	−0.456436	−0.228218	−	0.973610i	$-0.573290\pi$
$281$	−7.65126	−0.456436	−0.228218	+	0.973610i	$0.573290\pi$
$282$	0	0
$283$	30.8887	1.83614	0.918071	−	0.396416i	$-0.129746\pi$
$283$	30.8887	1.83614	0.918071	+	0.396416i	$0.129746\pi$
$284$	22.9054	1.35918
$285$	0	0
$286$	0	0
$287$	−5.58594	−0.329728
$288$	0	0
$289$	34.2274	2.01338
$290$	−2.35851	−0.138496
$291$	0	0
$292$	7.37782	0.431755
$293$	−10.3344	−0.603744	−0.301872	−	0.953348i	$-0.597612\pi$
$293$	−10.3344	−0.603744	−0.301872	+	0.953348i	$0.597612\pi$
$294$	0	0
$295$	−10.8112	−0.629452
$296$	−10.7940	−0.627391
$297$	0	0
$298$	5.10633	0.295802
$299$	−22.3971	−1.29526
$300$	0	0
$301$	2.32206	0.133841
$302$	−1.92729	−0.110903
$303$	0	0
$304$	−26.3435	−1.51091
$305$	−5.64149	−0.323031
$306$	0	0
$307$	13.3969	0.764603	0.382301	−	0.924038i	$-0.375132\pi$
$307$	13.3969	0.764603	0.382301	+	0.924038i	$0.375132\pi$
$308$	0	0
$309$	0	0
$310$	−0.155176	−0.00881342
$311$	6.37022	0.361222	0.180611	−	0.983555i	$-0.442193\pi$
$311$	6.37022	0.361222	0.180611	+	0.983555i	$0.442193\pi$
$312$	0	0
$313$	−1.27128	−0.0718567	−0.0359284	−	0.999354i	$-0.511439\pi$
$313$	−1.27128	−0.0718567	−0.0359284	+	0.999354i	$0.511439\pi$
$314$	−2.95898	−0.166985
$315$	0	0
$316$	−6.44292	−0.362443
$317$	−4.55426	−0.255793	−0.127896	−	0.991788i	$-0.540822\pi$
$317$	−4.55426	−0.255793	−0.127896	+	0.991788i	$0.540822\pi$
$318$	0	0
$319$	0	0
$320$	−6.39107	−0.357272
$321$	0	0
$322$	−1.08510	−0.0604703
$323$	−52.5104	−2.92176
$324$	0	0
$325$	−3.82075	−0.211937
$326$	−3.54731	−0.196467
$327$	0	0
$328$	−8.18617	−0.452006
$329$	−2.56597	−0.141466
$330$	0	0
$331$	−30.5469	−1.67901	−0.839504	−	0.543354i	$-0.817154\pi$
$331$	−30.5469	−1.67901	−0.839504	+	0.543354i	$0.817154\pi$
$332$	−7.80359	−0.428278
$333$	0	0
$334$	−4.21397	−0.230578
$335$	−10.9128	−0.596228
$336$	0	0
$337$	22.2179	1.21029	0.605143	−	0.796117i	$-0.293116\pi$
$337$	22.2179	1.21029	0.605143	+	0.796117i	$0.293116\pi$
$338$	0.419814	0.0228349
$339$	0	0
$340$	−13.8207	−0.749535
$341$	0	0
$342$	0	0
$343$	9.51518	0.513771
$344$	3.40297	0.183476
$345$	0	0
$346$	−1.27215	−0.0683912
$347$	−0.186646	−0.0100197	−0.00500984	−	0.999987i	$-0.501595\pi$
$347$	−0.186646	−0.0100197	−0.00500984	+	0.999987i	$0.501595\pi$
$348$	0	0
$349$	−8.44530	−0.452066	−0.226033	−	0.974120i	$-0.572576\pi$
$349$	−8.44530	−0.452066	−0.226033	+	0.974120i	$0.572576\pi$
$350$	−0.185108	−0.00989445
$351$	0	0
$352$	0	0
$353$	−17.0434	−0.907128	−0.453564	−	0.891224i	$-0.649848\pi$
$353$	−17.0434	−0.907128	−0.453564	+	0.891224i	$0.649848\pi$
$354$	0	0
$355$	−11.8620	−0.629569
$356$	13.6150	0.721594
$357$	0	0
$358$	−0.276037	−0.0145890
$359$	19.4961	1.02896	0.514482	−	0.857501i	$-0.327984\pi$
$359$	19.4961	1.02896	0.514482	+	0.857501i	$0.327984\pi$
$360$	0	0
$361$	34.8255	1.83292
$362$	0.157118	0.00825794
$363$	0	0
$364$	−5.19876	−0.272489
$365$	−3.82075	−0.199987
$366$	0	0
$367$	−18.5907	−0.970427	−0.485213	−	0.874396i	$-0.661258\pi$
$367$	−18.5907	−0.970427	−0.485213	+	0.874396i	$0.661258\pi$
$368$	21.0487	1.09724
$369$	0	0
$370$	2.74588	0.142752
$371$	−8.35851	−0.433952
$372$	0	0
$373$	−10.6393	−0.550884	−0.275442	−	0.961318i	$-0.588824\pi$
$373$	−10.6393	−0.550884	−0.275442	+	0.961318i	$0.588824\pi$
$374$	0	0
$375$	0	0
$376$	−3.76041	−0.193928
$377$	34.3030	1.76669
$378$	0	0
$379$	−32.6293	−1.67606	−0.838028	−	0.545627i	$-0.816292\pi$
$379$	−32.6293	−1.67606	−0.838028	+	0.545627i	$0.816292\pi$
$380$	14.1669	0.726746
$381$	0	0
$382$	−6.50824	−0.332990
$383$	−19.4019	−0.991391	−0.495695	−	0.868496i	$-0.665087\pi$
$383$	−19.4019	−0.991391	−0.495695	+	0.868496i	$0.665087\pi$
$384$	0	0
$385$	0	0
$386$	−6.27929	−0.319607
$387$	0	0
$388$	−13.1524	−0.667710
$389$	5.95616	0.301989	0.150995	−	0.988535i	$-0.451752\pi$
$389$	5.95616	0.301989	0.150995	+	0.988535i	$0.451752\pi$
$390$	0	0
$391$	41.9562	2.12181
$392$	6.71586	0.339202
$393$	0	0
$394$	−4.35199	−0.219250
$395$	3.33659	0.167882
$396$	0	0
$397$	−0.00739165	−0.000370976 0	−0.000185488	−	1.00000i	$-0.500059\pi$
$397$	−0.00739165	−0.000370976 0	−0.000185488		1.00000i	$0.500059\pi$
$398$	−1.32682	−0.0665076
$399$	0	0
$400$	3.59071	0.179535
$401$	−1.55902	−0.0778538	−0.0389269	−	0.999242i	$-0.512394\pi$
$401$	−1.55902	−0.0778538	−0.0389269	+	0.999242i	$0.512394\pi$
$402$	0	0
$403$	2.25694	0.112426
$404$	12.3340	0.613637
$405$	0	0
$406$	1.66192	0.0824795
$407$	0	0
$408$	0	0
$409$	27.6801	1.36869	0.684347	−	0.729156i	$-0.260087\pi$
$409$	27.6801	1.36869	0.684347	+	0.729156i	$0.260087\pi$
$410$	2.08247	0.102846
$411$	0	0
$412$	22.0170	1.08470
$413$	7.61808	0.374861
$414$	0	0
$415$	4.04124	0.198376
$416$	−11.4950	−0.563589
$417$	0	0
$418$	0	0
$419$	−12.7747	−0.624087	−0.312044	−	0.950068i	$-0.601014\pi$
$419$	−12.7747	−0.624087	−0.312044	+	0.950068i	$0.601014\pi$
$420$	0	0
$421$	−20.3221	−0.990437	−0.495218	−	0.868769i	$-0.664912\pi$
$421$	−20.3221	−0.990437	−0.495218	+	0.868769i	$0.664912\pi$
$422$	2.66773	0.129863
$423$	0	0
$424$	−12.2494	−0.594881
$425$	7.15733	0.347182
$426$	0	0
$427$	3.97526	0.192376
$428$	10.7772	0.520934
$429$	0	0
$430$	−0.865677	−0.0417467
$431$	−7.28298	−0.350809	−0.175404	−	0.984496i	$-0.556123\pi$
$431$	−7.28298	−0.350809	−0.175404	+	0.984496i	$0.556123\pi$
$432$	0	0
$433$	2.37022	0.113905	0.0569527	−	0.998377i	$-0.481862\pi$
$433$	2.37022	0.113905	0.0569527	+	0.998377i	$0.481862\pi$
$434$	0.109345	0.00524871
$435$	0	0
$436$	1.83292	0.0877812
$437$	−43.0069	−2.05730
$438$	0	0
$439$	17.9273	0.855623	0.427812	−	0.903868i	$-0.359285\pi$
$439$	17.9273	0.855623	0.427812	+	0.903868i	$0.359285\pi$
$440$	0	0
$441$	0	0
$442$	−7.18379	−0.341698
$443$	−14.2205	−0.675636	−0.337818	−	0.941211i	$-0.609689\pi$
$443$	−14.2205	−0.675636	−0.337818	+	0.941211i	$0.609689\pi$
$444$	0	0
$445$	−7.05079	−0.334239
$446$	2.70963	0.128304
$447$	0	0
$448$	4.50345	0.212768
$449$	17.1376	0.808772	0.404386	−	0.914588i	$-0.367485\pi$
$449$	17.1376	0.808772	0.404386	+	0.914588i	$0.367485\pi$
$450$	0	0
$451$	0	0
$452$	−17.9027	−0.842074
$453$	0	0
$454$	−2.36678	−0.111078
$455$	2.69228	0.126216
$456$	0	0
$457$	16.7261	0.782415	0.391207	−	0.920303i	$-0.372057\pi$
$457$	16.7261	0.782415	0.391207	+	0.920303i	$0.372057\pi$
$458$	2.43861	0.113949
$459$	0	0
$460$	−11.3194	−0.527771
$461$	−9.77757	−0.455387	−0.227693	−	0.973733i	$-0.573118\pi$
$461$	−9.77757	−0.455387	−0.227693	+	0.973733i	$0.573118\pi$
$462$	0	0
$463$	27.2200	1.26502	0.632511	−	0.774551i	$-0.282024\pi$
$463$	27.2200	1.26502	0.632511	+	0.774551i	$0.282024\pi$
$464$	−32.2376	−1.49660
$465$	0	0
$466$	−0.923599	−0.0427849
$467$	24.8689	1.15080	0.575398	−	0.817873i	$-0.304847\pi$
$467$	24.8689	1.15080	0.575398	+	0.817873i	$0.304847\pi$
$468$	0	0
$469$	7.68965	0.355075
$470$	0.956606	0.0441250
$471$	0	0
$472$	11.1643	0.513876
$473$	0	0
$474$	0	0
$475$	−7.33659	−0.336626
$476$	9.73875	0.446375
$477$	0	0
$478$	−4.03645	−0.184623
$479$	−4.22788	−0.193177	−0.0965884	−	0.995324i	$-0.530793\pi$
$479$	−4.22788	−0.193177	−0.0965884	+	0.995324i	$0.530793\pi$
$480$	0	0
$481$	−39.9371	−1.82097
$482$	0.155176	0.00706809
$483$	0	0
$484$	0	0
$485$	6.81120	0.309280
$486$	0	0
$487$	−21.2756	−0.964089	−0.482045	−	0.876147i	$-0.660106\pi$
$487$	−21.2756	−0.964089	−0.482045	+	0.876147i	$0.660106\pi$
$488$	5.82572	0.263718
$489$	0	0
$490$	−1.70844	−0.0771794
$491$	−35.7240	−1.61220	−0.806100	−	0.591779i	$-0.798426\pi$
$491$	−35.7240	−1.61220	−0.806100	+	0.591779i	$0.798426\pi$
$492$	0	0
$493$	−64.2591	−2.89409
$494$	7.36370	0.331309
$495$	0	0
$496$	−2.12105	−0.0952381
$497$	8.35851	0.374930
$498$	0	0
$499$	−9.76780	−0.437267	−0.218633	−	0.975807i	$-0.570160\pi$
$499$	−9.76780	−0.437267	−0.218633	+	0.975807i	$0.570160\pi$
$500$	−1.93099	−0.0863565
$501$	0	0
$502$	6.19594	0.276538
$503$	23.9106	1.06612	0.533061	−	0.846077i	$-0.321042\pi$
$503$	23.9106	1.06612	0.533061	+	0.846077i	$0.321042\pi$
$504$	0	0
$505$	−6.38737	−0.284234
$506$	0	0
$507$	0	0
$508$	−21.2782	−0.944067
$509$	−16.9054	−0.749318	−0.374659	−	0.927163i	$-0.622240\pi$
$509$	−16.9054	−0.749318	−0.374659	+	0.927163i	$0.622240\pi$
$510$	0	0
$511$	2.69228	0.119099
$512$	18.2189	0.805167
$513$	0	0
$514$	−5.33659	−0.235387
$515$	−11.4019	−0.502428
$516$	0	0
$517$	0	0
$518$	−1.93488	−0.0850136
$519$	0	0
$520$	3.94552	0.173022
$521$	−20.8038	−0.911431	−0.455716	−	0.890125i	$-0.650617\pi$
$521$	−20.8038	−0.911431	−0.455716	+	0.890125i	$0.650617\pi$
$522$	0	0
$523$	−31.9247	−1.39597	−0.697985	−	0.716113i	$-0.745920\pi$
$523$	−31.9247	−1.39597	−0.697985	+	0.716113i	$0.745920\pi$
$524$	−7.03169	−0.307181
$525$	0	0
$526$	−5.22656	−0.227889
$527$	−4.22788	−0.184169
$528$	0	0
$529$	11.3628	0.494036
$530$	3.11610	0.135355
$531$	0	0
$532$	−9.98265	−0.432803
$533$	−30.2882	−1.31193
$534$	0	0
$535$	−5.58116	−0.241294
$536$	11.2691	0.486753
$537$	0	0
$538$	−5.06557	−0.218392
$539$	0	0
$540$	0	0
$541$	35.8255	1.54026	0.770130	−	0.637887i	$-0.220191\pi$
$541$	35.8255	1.54026	0.770130	+	0.637887i	$0.220191\pi$
$542$	−2.47937	−0.106498
$543$	0	0
$544$	21.5334	0.923237
$545$	−0.949215	−0.0406599
$546$	0	0
$547$	20.8930	0.893320	0.446660	−	0.894704i	$-0.352613\pi$
$547$	20.8930	0.893320	0.446660	+	0.894704i	$0.352613\pi$
$548$	11.5859	0.494927
$549$	0	0
$550$	0	0
$551$	65.8685	2.80609
$552$	0	0
$553$	−2.35112	−0.0999797
$554$	−5.93074	−0.251973
$555$	0	0
$556$	−4.00237	−0.169738
$557$	−21.2589	−0.900769	−0.450384	−	0.892835i	$-0.648713\pi$
$557$	−21.2589	−0.900769	−0.450384	+	0.892835i	$0.648713\pi$
$558$	0	0
$559$	12.5907	0.532530
$560$	−2.53018	−0.106920
$561$	0	0
$562$	−2.00996	−0.0847849
$563$	28.2543	1.19078	0.595389	−	0.803438i	$-0.296998\pi$
$563$	28.2543	1.19078	0.595389	+	0.803438i	$0.296998\pi$
$564$	0	0
$565$	9.27128	0.390045
$566$	8.11434	0.341071
$567$	0	0
$568$	12.2494	0.513972
$569$	−1.75804	−0.0737007	−0.0368504	−	0.999321i	$-0.511732\pi$
$569$	−1.75804	−0.0737007	−0.0368504	+	0.999321i	$0.511732\pi$
$570$	0	0
$571$	12.7459	0.533399	0.266699	−	0.963780i	$-0.414067\pi$
$571$	12.7459	0.533399	0.266699	+	0.963780i	$0.414067\pi$
$572$	0	0
$573$	0	0
$574$	−1.46741	−0.0612484
$575$	5.86198	0.244462
$576$	0	0
$577$	12.9245	0.538053	0.269026	−	0.963133i	$-0.413298\pi$
$577$	12.9245	0.538053	0.269026	+	0.963133i	$0.413298\pi$
$578$	8.99142	0.373994
$579$	0	0
$580$	17.3366	0.719863
$581$	−2.84764	−0.118140
$582$	0	0
$583$	0	0
$584$	3.94552	0.163267
$585$	0	0
$586$	−2.71482	−0.112148
$587$	25.8472	1.06683	0.533414	−	0.845854i	$-0.320909\pi$
$587$	25.8472	1.06683	0.533414	+	0.845854i	$0.320909\pi$
$588$	0	0
$589$	4.33377	0.178570
$590$	−2.84006	−0.116923
$591$	0	0
$592$	37.5325	1.54258
$593$	40.4789	1.66227	0.831136	−	0.556070i	$-0.187691\pi$
$593$	40.4789	1.66227	0.831136	+	0.556070i	$0.187691\pi$
$594$	0	0
$595$	−5.04339	−0.206759
$596$	−37.5349	−1.53749
$597$	0	0
$598$	−5.88365	−0.240600
$599$	−31.2830	−1.27819	−0.639094	−	0.769129i	$-0.720691\pi$
$599$	−31.2830	−1.27819	−0.639094	+	0.769129i	$0.720691\pi$
$600$	0	0
$601$	−12.9245	−0.527200	−0.263600	−	0.964632i	$-0.584910\pi$
$601$	−12.9245	−0.527200	−0.263600	+	0.964632i	$0.584910\pi$
$602$	0.609997	0.0248616
$603$	0	0
$604$	14.1669	0.576442
$605$	0	0
$606$	0	0
$607$	11.0193	0.447260	0.223630	−	0.974674i	$-0.428209\pi$
$607$	11.0193	0.447260	0.223630	+	0.974674i	$0.428209\pi$
$608$	−22.0727	−0.895166
$609$	0	0
$610$	−1.48200	−0.0600044
$611$	−13.9132	−0.562868
$612$	0	0
$613$	18.0529	0.729152	0.364576	−	0.931174i	$-0.381214\pi$
$613$	18.0529	0.729152	0.364576	+	0.931174i	$0.381214\pi$
$614$	3.51932	0.142028
$615$	0	0
$616$	0	0
$617$	31.0820	1.25132	0.625658	−	0.780098i	$-0.284831\pi$
$617$	31.0820	1.25132	0.625658	+	0.780098i	$0.284831\pi$
$618$	0	0
$619$	−30.5469	−1.22778	−0.613891	−	0.789391i	$-0.710397\pi$
$619$	−30.5469	−1.22778	−0.613891	+	0.789391i	$0.710397\pi$
$620$	1.14065	0.0458095
$621$	0	0
$622$	1.67343	0.0670985
$623$	4.96831	0.199051
$624$	0	0
$625$	1.00000	0.0400000
$626$	−0.333959	−0.0133477
$627$	0	0
$628$	21.7505	0.867938
$629$	74.8134	2.98300
$630$	0	0
$631$	4.33377	0.172525	0.0862623	−	0.996272i	$-0.472508\pi$
$631$	4.33377	0.172525	0.0862623	+	0.996272i	$0.472508\pi$
$632$	−3.44555	−0.137057
$633$	0	0
$634$	−1.19639	−0.0475146
$635$	11.0193	0.437288
$636$	0	0
$637$	24.8481	0.984518
$638$	0	0
$639$	0	0
$640$	−7.69607	−0.304214
$641$	−9.85459	−0.389233	−0.194616	−	0.980879i	$-0.562346\pi$
$641$	−9.85459	−0.389233	−0.194616	+	0.980879i	$0.562346\pi$
$642$	0	0
$643$	−40.0312	−1.57868	−0.789339	−	0.613958i	$-0.789577\pi$
$643$	−40.0312	−1.57868	−0.789339	+	0.613958i	$0.789577\pi$
$644$	7.97620	0.314306
$645$	0	0
$646$	−13.7943	−0.542729
$647$	31.1259	1.22368	0.611842	−	0.790980i	$-0.290429\pi$
$647$	31.1259	1.22368	0.611842	+	0.790980i	$0.290429\pi$
$648$	0	0
$649$	0	0
$650$	−1.00370	−0.0393682
$651$	0	0
$652$	26.0751	1.02118
$653$	18.2131	0.712734	0.356367	−	0.934346i	$-0.384015\pi$
$653$	18.2131	0.712734	0.356367	+	0.934346i	$0.384015\pi$
$654$	0	0
$655$	3.64149	0.142285
$656$	28.4646	1.11136
$657$	0	0
$658$	−0.674070	−0.0262780
$659$	−15.1524	−0.590252	−0.295126	−	0.955458i	$-0.595362\pi$
$659$	−15.1524	−0.590252	−0.295126	+	0.955458i	$0.595362\pi$
$660$	0	0
$661$	−26.6293	−1.03576	−0.517881	−	0.855453i	$-0.673279\pi$
$661$	−26.6293	−1.03576	−0.517881	+	0.855453i	$0.673279\pi$
$662$	−8.02455	−0.311883
$663$	0	0
$664$	−4.17321	−0.161952
$665$	5.16970	0.200473
$666$	0	0
$667$	−52.6293	−2.03782
$668$	30.9755	1.19848
$669$	0	0
$670$	−2.86674	−0.110752
$671$	0	0
$672$	0	0
$673$	−24.3924	−0.940256	−0.470128	−	0.882598i	$-0.655792\pi$
$673$	−24.3924	−0.940256	−0.470128	+	0.882598i	$0.655792\pi$
$674$	5.83656	0.224816
$675$	0	0
$676$	−3.08591	−0.118689
$677$	−3.20549	−0.123197	−0.0615985	−	0.998101i	$-0.519620\pi$
$677$	−3.20549	−0.123197	−0.0615985	+	0.998101i	$0.519620\pi$
$678$	0	0
$679$	−4.79949	−0.184187
$680$	−7.39107	−0.283435
$681$	0	0
$682$	0	0
$683$	3.36545	0.128776	0.0643878	−	0.997925i	$-0.479491\pi$
$683$	3.36545	0.128776	0.0643878	+	0.997925i	$0.479491\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	2.49960	0.0954353
$687$	0	0
$688$	−11.8326	−0.451115
$689$	−45.3216	−1.72662
$690$	0	0
$691$	47.8889	1.82178	0.910890	−	0.412649i	$-0.135397\pi$
$691$	47.8889	1.82178	0.910890	+	0.412649i	$0.135397\pi$
$692$	9.35114	0.355477
$693$	0	0
$694$	−0.0490312	−0.00186120
$695$	2.07271	0.0786222
$696$	0	0
$697$	56.7383	2.14912
$698$	−2.21855	−0.0839733
$699$	0	0
$700$	1.36067	0.0514284
$701$	−23.3175	−0.880689	−0.440345	−	0.897829i	$-0.645144\pi$
$701$	−23.3175	−0.880689	−0.440345	+	0.897829i	$0.645144\pi$
$702$	0	0
$703$	−76.6871	−2.89231
$704$	0	0
$705$	0	0
$706$	−4.47724	−0.168503
$707$	4.50084	0.169272
$708$	0	0
$709$	17.2274	0.646990	0.323495	−	0.946230i	$-0.395142\pi$
$709$	17.2274	0.646990	0.323495	+	0.946230i	$0.395142\pi$
$710$	−3.11610	−0.116945
$711$	0	0
$712$	7.28104	0.272869
$713$	−3.46271	−0.129679
$714$	0	0
$715$	0	0
$716$	2.02906	0.0758294
$717$	0	0
$718$	5.12155	0.191134
$719$	−0.586390	−0.0218687	−0.0109343	−	0.999940i	$-0.503481\pi$
$719$	−0.586390	−0.0218687	−0.0109343	+	0.999940i	$0.503481\pi$
$720$	0	0
$721$	8.03432	0.299214
$722$	9.14854	0.340473
$723$	0	0
$724$	−1.15492	−0.0429223
$725$	−8.97808	−0.333438
$726$	0	0
$727$	−32.1649	−1.19293	−0.596466	−	0.802638i	$-0.703429\pi$
$727$	−32.1649	−1.19293	−0.596466	+	0.802638i	$0.703429\pi$
$728$	−2.78020	−0.103041
$729$	0	0
$730$	−1.00370	−0.0371484
$731$	−23.5859	−0.872358
$732$	0	0
$733$	−31.3993	−1.15976	−0.579880	−	0.814702i	$-0.696900\pi$
$733$	−31.3993	−1.15976	−0.579880	+	0.814702i	$0.696900\pi$
$734$	−4.88371	−0.180261
$735$	0	0
$736$	17.6362	0.650080
$737$	0	0
$738$	0	0
$739$	−38.0922	−1.40125	−0.700623	−	0.713532i	$-0.747094\pi$
$739$	−38.0922	−1.40125	−0.700623	+	0.713532i	$0.747094\pi$
$740$	−20.1840	−0.741980
$741$	0	0
$742$	−2.19575	−0.0806085
$743$	−1.23743	−0.0453969	−0.0226985	−	0.999742i	$-0.507226\pi$
$743$	−1.23743	−0.0453969	−0.0226985	+	0.999742i	$0.507226\pi$
$744$	0	0
$745$	19.4382	0.712159
$746$	−2.79491	−0.102329
$747$	0	0
$748$	0	0
$749$	3.93274	0.143699
$750$	0	0
$751$	35.8546	1.30835	0.654176	−	0.756342i	$-0.273015\pi$
$751$	35.8546	1.30835	0.654176	+	0.756342i	$0.273015\pi$
$752$	13.0755	0.476815
$753$	0	0
$754$	9.01126	0.328171
$755$	−7.33659	−0.267006
$756$	0	0
$757$	14.5235	0.527864	0.263932	−	0.964541i	$-0.414981\pi$
$757$	14.5235	0.527864	0.263932	+	0.964541i	$0.414981\pi$
$758$	−8.57161	−0.311335
$759$	0	0
$760$	7.57618	0.274817
$761$	23.8448	0.864374	0.432187	−	0.901784i	$-0.357742\pi$
$761$	23.8448	0.864374	0.432187	+	0.901784i	$0.357742\pi$
$762$	0	0
$763$	0.668861	0.0242144
$764$	47.8398	1.73078
$765$	0	0
$766$	−5.09681	−0.184155
$767$	41.3068	1.49150
$768$	0	0
$769$	−32.4453	−1.17001	−0.585004	−	0.811031i	$-0.698907\pi$
$769$	−32.4453	−1.17001	−0.585004	+	0.811031i	$0.698907\pi$
$770$	0	0
$771$	0	0
$772$	46.1569	1.66122
$773$	51.3216	1.84591	0.922955	−	0.384908i	$-0.125767\pi$
$773$	51.3216	1.84591	0.922955	+	0.384908i	$0.125767\pi$
$774$	0	0
$775$	−0.590706	−0.0212188
$776$	−7.03363	−0.252493
$777$	0	0
$778$	1.56466	0.0560958
$779$	−58.1593	−2.08377
$780$	0	0
$781$	0	0
$782$	11.0217	0.394136
$783$	0	0
$784$	−23.3521	−0.834002
$785$	−11.2639	−0.402025
$786$	0	0
$787$	−17.1209	−0.610294	−0.305147	−	0.952305i	$-0.598706\pi$
$787$	−17.1209	−0.610294	−0.305147	+	0.952305i	$0.598706\pi$
$788$	31.9900	1.13960
$789$	0	0
$790$	0.876510	0.0311848
$791$	−6.53298	−0.232286
$792$	0	0
$793$	21.5547	0.765430
$794$	−0.00194176	−6.89105e−5 0
$795$	0	0
$796$	9.75302	0.345687
$797$	−17.2565	−0.611256	−0.305628	−	0.952151i	$-0.598866\pi$
$797$	−17.2565	−0.611256	−0.305628	+	0.952151i	$0.598866\pi$
$798$	0	0
$799$	26.0634	0.922056
$800$	3.00858	0.106369
$801$	0	0
$802$	−0.409549	−0.0144617
$803$	0	0
$804$	0	0
$805$	−4.13063	−0.145585
$806$	0.592889	0.0208836
$807$	0	0
$808$	6.59596	0.232045
$809$	14.9342	0.525060	0.262530	−	0.964924i	$-0.415443\pi$
$809$	14.9342	0.525060	0.262530	+	0.964924i	$0.415443\pi$
$810$	0	0
$811$	−21.9273	−0.769971	−0.384986	−	0.922923i	$-0.625794\pi$
$811$	−21.9273	−0.769971	−0.384986	+	0.922923i	$0.625794\pi$
$812$	−12.2162	−0.428704
$813$	0	0
$814$	0	0
$815$	−13.5035	−0.473006
$816$	0	0
$817$	24.1767	0.845834
$818$	7.27147	0.254241
$819$	0	0
$820$	−15.3075	−0.534562
$821$	34.8036	1.21465	0.607327	−	0.794452i	$-0.292242\pi$
$821$	34.8036	1.21465	0.607327	+	0.794452i	$0.292242\pi$
$822$	0	0
$823$	−27.8429	−0.970542	−0.485271	−	0.874364i	$-0.661279\pi$
$823$	−27.8429	−0.970542	−0.485271	+	0.874364i	$0.661279\pi$
$824$	11.7743	0.410175
$825$	0	0
$826$	2.00124	0.0696321
$827$	34.5590	1.20174	0.600868	−	0.799348i	$-0.294822\pi$
$827$	34.5590	1.20174	0.600868	+	0.799348i	$0.294822\pi$
$828$	0	0
$829$	−21.0243	−0.730204	−0.365102	−	0.930968i	$-0.618966\pi$
$829$	−21.0243	−0.730204	−0.365102	+	0.930968i	$0.618966\pi$
$830$	1.06162	0.0368493
$831$	0	0
$832$	24.4187	0.846564
$833$	−46.5475	−1.61278
$834$	0	0
$835$	−16.0412	−0.555130
$836$	0	0
$837$	0	0
$838$	−3.35588	−0.115927
$839$	37.3390	1.28908	0.644542	−	0.764569i	$-0.277048\pi$
$839$	37.3390	1.28908	0.644542	+	0.764569i	$0.277048\pi$
$840$	0	0
$841$	51.6059	1.77951
$842$	−5.33853	−0.183978
$843$	0	0
$844$	−19.6096	−0.674989
$845$	1.59810	0.0549762
$846$	0	0
$847$	0	0
$848$	42.5929	1.46265
$849$	0	0
$850$	1.88021	0.0644905
$851$	61.2735	2.10043
$852$	0	0
$853$	22.9831	0.786925	0.393462	−	0.919341i	$-0.371277\pi$
$853$	22.9831	0.786925	0.393462	+	0.919341i	$0.371277\pi$
$854$	1.04429	0.0357347
$855$	0	0
$856$	5.76342	0.196990
$857$	46.0584	1.57332	0.786662	−	0.617383i	$-0.211807\pi$
$857$	46.0584	1.57332	0.786662	+	0.617383i	$0.211807\pi$
$858$	0	0
$859$	29.3655	1.00194	0.500968	−	0.865466i	$-0.332977\pi$
$859$	29.3655	1.00194	0.500968	+	0.865466i	$0.332977\pi$
$860$	6.36330	0.216987
$861$	0	0
$862$	−1.91321	−0.0651643
$863$	−33.7795	−1.14987	−0.574934	−	0.818200i	$-0.694972\pi$
$863$	−33.7795	−1.14987	−0.574934	+	0.818200i	$0.694972\pi$
$864$	0	0
$865$	−4.84267	−0.164656
$866$	0.622647	0.0211584
$867$	0	0
$868$	−0.803754	−0.0272812
$869$	0	0
$870$	0	0
$871$	41.6949	1.41278
$872$	0.980213	0.0331942
$873$	0	0
$874$	−11.2978	−0.382153
$875$	−0.704647	−0.0238214
$876$	0	0
$877$	−43.9609	−1.48446	−0.742228	−	0.670148i	$-0.766231\pi$
$877$	−43.9609	−1.48446	−0.742228	+	0.670148i	$0.766231\pi$
$878$	4.70943	0.158936
$879$	0	0
$880$	0	0
$881$	−7.86937	−0.265126	−0.132563	−	0.991175i	$-0.542321\pi$
$881$	−7.86937	−0.265126	−0.132563	+	0.991175i	$0.542321\pi$
$882$	0	0
$883$	−8.62934	−0.290400	−0.145200	−	0.989402i	$-0.546383\pi$
$883$	−8.62934	−0.290400	−0.145200	+	0.989402i	$0.546383\pi$
$884$	52.8056	1.77604
$885$	0	0
$886$	−3.73567	−0.125502
$887$	−14.5013	−0.486906	−0.243453	−	0.969913i	$-0.578280\pi$
$887$	−14.5013	−0.486906	−0.243453	+	0.969913i	$0.578280\pi$
$888$	0	0
$889$	−7.76473	−0.260421
$890$	−1.85222	−0.0620864
$891$	0	0
$892$	−19.9175	−0.666888
$893$	−26.7161	−0.894021
$894$	0	0
$895$	−1.05079	−0.0351239
$896$	5.42301	0.181170
$897$	0	0
$898$	4.50198	0.150233
$899$	5.30341	0.176879
$900$	0	0
$901$	84.9002	2.82843
$902$	0	0
$903$	0	0
$904$	−9.57404	−0.318428
$905$	0.598098	0.0198814
$906$	0	0
$907$	−13.5977	−0.451503	−0.225751	−	0.974185i	$-0.572484\pi$
$907$	−13.5977	−0.451503	−0.225751	+	0.974185i	$0.572484\pi$
$908$	17.3974	0.577352
$909$	0	0
$910$	0.707251	0.0234451
$911$	−25.0126	−0.828704	−0.414352	−	0.910117i	$-0.635992\pi$
$911$	−25.0126	−0.828704	−0.414352	+	0.910117i	$0.635992\pi$
$912$	0	0
$913$	0	0
$914$	4.39389	0.145337
$915$	0	0
$916$	−17.9254	−0.592270
$917$	−2.56597	−0.0847357
$918$	0	0
$919$	−0.580940	−0.0191634	−0.00958172	−	0.999954i	$-0.503050\pi$
$919$	−0.580940	−0.0191634	−0.00958172	+	0.999954i	$0.503050\pi$
$920$	−6.05341	−0.199575
$921$	0	0
$922$	−2.56853	−0.0845901
$923$	45.3216	1.49178
$924$	0	0
$925$	10.4527	0.343682
$926$	7.15061	0.234983
$927$	0	0
$928$	−27.0113	−0.886688
$929$	27.5786	0.904823	0.452411	−	0.891809i	$-0.350564\pi$
$929$	27.5786	0.904823	0.452411	+	0.891809i	$0.350564\pi$
$930$	0	0
$931$	47.7133	1.56374
$932$	6.78906	0.222383
$933$	0	0
$934$	6.53298	0.213765
$935$	0	0
$936$	0	0
$937$	17.9900	0.587708	0.293854	−	0.955850i	$-0.405062\pi$
$937$	17.9900	0.587708	0.293854	+	0.955850i	$0.405062\pi$
$938$	2.02004	0.0659567
$939$	0	0
$940$	−7.03169	−0.229348
$941$	0.431214	0.0140572	0.00702858	−	0.999975i	$-0.497763\pi$
$941$	0.431214	0.0140572	0.00702858	+	0.999975i	$0.497763\pi$
$942$	0	0
$943$	46.4697	1.51326
$944$	−38.8198	−1.26348
$945$	0	0
$946$	0	0
$947$	−17.4670	−0.567602	−0.283801	−	0.958883i	$-0.591596\pi$
$947$	−17.4670	−0.567602	−0.283801	+	0.958883i	$0.591596\pi$
$948$	0	0
$949$	14.5981	0.473874
$950$	−1.92729	−0.0625297
$951$	0	0
$952$	5.20809	0.168795
$953$	24.8622	0.805366	0.402683	−	0.915340i	$-0.368078\pi$
$953$	24.8622	0.805366	0.402683	+	0.915340i	$0.368078\pi$
$954$	0	0
$955$	−24.7747	−0.801692
$956$	29.6705	0.959614
$957$	0	0
$958$	−1.11065	−0.0358834
$959$	4.22788	0.136525
$960$	0	0
$961$	−30.6511	−0.988744
$962$	−10.4913	−0.338254
$963$	0	0
$964$	−1.14065	−0.0367378
$965$	−23.9032	−0.769472
$966$	0	0
$967$	18.1139	0.582505	0.291253	−	0.956646i	$-0.405928\pi$
$967$	18.1139	0.582505	0.291253	+	0.956646i	$0.405928\pi$
$968$	0	0
$969$	0	0
$970$	1.78928	0.0574502
$971$	18.5426	0.595059	0.297529	−	0.954713i	$-0.403837\pi$
$971$	18.5426	0.595059	0.297529	+	0.954713i	$0.403837\pi$
$972$	0	0
$973$	−1.46053	−0.0468223
$974$	−5.58902	−0.179084
$975$	0	0
$976$	−20.2569	−0.648409
$977$	−24.2665	−0.776355	−0.388177	−	0.921585i	$-0.626895\pi$
$977$	−24.2665	−0.776355	−0.388177	+	0.921585i	$0.626895\pi$
$978$	0	0
$979$	0	0
$980$	12.5581	0.401155
$981$	0	0
$982$	−9.38455	−0.299473
$983$	44.9514	1.43373	0.716863	−	0.697214i	$-0.245577\pi$
$983$	44.9514	1.43373	0.716863	+	0.697214i	$0.245577\pi$
$984$	0	0
$985$	−16.5666	−0.527857
$986$	−16.8806	−0.537589
$987$	0	0
$988$	−54.1281	−1.72204
$989$	−19.3173	−0.614254
$990$	0	0
$991$	43.1185	1.36970	0.684852	−	0.728682i	$-0.259867\pi$
$991$	43.1185	1.36970	0.684852	+	0.728682i	$0.259867\pi$
$992$	−1.77719	−0.0564257
$993$	0	0
$994$	2.19575	0.0696449
$995$	−5.05079	−0.160121
$996$	0	0
$997$	1.38541	0.0438763	0.0219381	−	0.999759i	$-0.493016\pi$
$997$	1.38541	0.0438763	0.0219381	+	0.999759i	$0.493016\pi$
$998$	−2.56597	−0.0812242
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5445.2.a.bh.1.3		4
3.2	odd	2		5445.2.a.bs.1.2		4
11.10	odd	2		495.2.a.g.1.2	yes	4
33.32	even	2		495.2.a.f.1.3	✓	4
44.43	even	2		7920.2.a.cn.1.3		4
55.32	even	4		2475.2.c.s.199.4		8
55.43	even	4		2475.2.c.s.199.5		8
55.54	odd	2		2475.2.a.bf.1.3		4
132.131	odd	2		7920.2.a.cm.1.3		4
165.32	odd	4		2475.2.c.t.199.5		8
165.98	odd	4		2475.2.c.t.199.4		8
165.164	even	2		2475.2.a.bj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
495.2.a.f.1.3	✓	4	33.32	even	2
495.2.a.g.1.2	yes	4	11.10	odd	2
2475.2.a.bf.1.3		4	55.54	odd	2
2475.2.a.bj.1.2		4	165.164	even	2
2475.2.c.s.199.4		8	55.32	even	4
2475.2.c.s.199.5		8	55.43	even	4
2475.2.c.t.199.4		8	165.98	odd	4
2475.2.c.t.199.5		8	165.32	odd	4
5445.2.a.bh.1.3		4	1.1	even	1	trivial
5445.2.a.bs.1.2		4	3.2	odd	2
7920.2.a.cm.1.3		4	132.131	odd	2
7920.2.a.cn.1.3		4	44.43	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:	\( N \)	\(=\)	\( 5445 = 3^{2} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5445.a (trivial)

\(f(q)\)	\(=\)	\(q+0.262696 q^{2} -1.93099 q^{4} +1.00000 q^{5} -0.704647 q^{7} -1.03266 q^{8} +O(q^{10})\)	\(q+0.262696 q^{2} -1.93099 q^{4} +1.00000 q^{5} -0.704647 q^{7} -1.03266 q^{8} +0.262696 q^{10} -3.82075 q^{13} -0.185108 q^{14} +3.59071 q^{16} +7.15733 q^{17} -7.33659 q^{19} -1.93099 q^{20} +5.86198 q^{23} +1.00000 q^{25} -1.00370 q^{26} +1.36067 q^{28} -8.97808 q^{29} -0.590706 q^{31} +3.00858 q^{32} +1.88021 q^{34} -0.704647 q^{35} +10.4527 q^{37} -1.92729 q^{38} -1.03266 q^{40} +7.92729 q^{41} -3.29535 q^{43} +1.53992 q^{46} +3.64149 q^{47} -6.50347 q^{49} +0.262696 q^{50} +7.37782 q^{52} +11.8620 q^{53} +0.727658 q^{56} -2.35851 q^{58} -10.8112 q^{59} -5.64149 q^{61} -0.155176 q^{62} -6.39107 q^{64} -3.82075 q^{65} -10.9128 q^{67} -13.8207 q^{68} -0.185108 q^{70} -11.8620 q^{71} -3.82075 q^{73} +2.74588 q^{74} +14.1669 q^{76} +3.33659 q^{79} +3.59071 q^{80} +2.08247 q^{82} +4.04124 q^{83} +7.15733 q^{85} -0.865677 q^{86} -7.05079 q^{89} +2.69228 q^{91} -11.3194 q^{92} +0.956606 q^{94} -7.33659 q^{95} +6.81120 q^{97} -1.70844 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{4} + 4 q^{5} - 4 q^{7} - 6 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 8 * q^4 + 4 * q^5 - 4 * q^7 - 6 * q^8	\( 4 q - 2 q^{2} + 8 q^{4} + 4 q^{5} - 4 q^{7} - 6 q^{8} - 2 q^{10} - 8 q^{13} - 8 q^{14} + 12 q^{16} - 4 q^{17} - 4 q^{19} + 8 q^{20} - 8 q^{23} + 4 q^{25} - 16 q^{26} + 8 q^{28} + 4 q^{29} - 14 q^{32} + 4 q^{34} - 4 q^{35} + 8 q^{37} + 20 q^{38} - 6 q^{40} + 4 q^{41} - 12 q^{43} + 16 q^{46} + 20 q^{49} - 2 q^{50} - 20 q^{52} + 16 q^{53} - 48 q^{56} - 24 q^{58} - 24 q^{59} - 8 q^{61} + 20 q^{62} - 8 q^{65} - 48 q^{68} - 8 q^{70} - 16 q^{71} - 8 q^{73} - 12 q^{74} + 36 q^{76} - 12 q^{79} + 12 q^{80} - 40 q^{82} - 8 q^{83} - 4 q^{85} + 16 q^{86} - 16 q^{89} - 16 q^{91} - 72 q^{92} + 40 q^{94} - 4 q^{95} + 8 q^{97} - 62 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 8 * q^4 + 4 * q^5 - 4 * q^7 - 6 * q^8 - 2 * q^10 - 8 * q^13 - 8 * q^14 + 12 * q^16 - 4 * q^17 - 4 * q^19 + 8 * q^20 - 8 * q^23 + 4 * q^25 - 16 * q^26 + 8 * q^28 + 4 * q^29 - 14 * q^32 + 4 * q^34 - 4 * q^35 + 8 * q^37 + 20 * q^38 - 6 * q^40 + 4 * q^41 - 12 * q^43 + 16 * q^46 + 20 * q^49 - 2 * q^50 - 20 * q^52 + 16 * q^53 - 48 * q^56 - 24 * q^58 - 24 * q^59 - 8 * q^61 + 20 * q^62 - 8 * q^65 - 48 * q^68 - 8 * q^70 - 16 * q^71 - 8 * q^73 - 12 * q^74 + 36 * q^76 - 12 * q^79 + 12 * q^80 - 40 * q^82 - 8 * q^83 - 4 * q^85 + 16 * q^86 - 16 * q^89 - 16 * q^91 - 72 * q^92 + 40 * q^94 - 4 * q^95 + 8 * q^97 - 62 * q^98

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