LMFDB - Embedded newform 5520.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} -1.41421 q^{11} +3.41421 q^{13} +1.00000 q^{15} +0.414214 q^{17} +4.58579 q^{19} +3.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.41421 q^{29} +1.00000 q^{31} -1.41421 q^{33} +3.82843 q^{35} +9.48528 q^{37} +3.41421 q^{39} -1.24264 q^{41} -3.65685 q^{43} +1.00000 q^{45} -2.24264 q^{47} +7.65685 q^{49} +0.414214 q^{51} -3.24264 q^{53} -1.41421 q^{55} +4.58579 q^{57} +11.7279 q^{59} -1.41421 q^{61} +3.82843 q^{63} +3.41421 q^{65} -1.48528 q^{67} +1.00000 q^{69} -0.0710678 q^{71} -6.24264 q^{73} +1.00000 q^{75} -5.41421 q^{77} +16.9706 q^{79} +1.00000 q^{81} -11.2426 q^{83} +0.414214 q^{85} -6.41421 q^{87} -14.8284 q^{89} +13.0711 q^{91} +1.00000 q^{93} +4.58579 q^{95} -10.1421 q^{97} -1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} + 12 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 4 q^{43} + 2 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} + 12 q^{57} - 2 q^{59} + 2 q^{63} + 4 q^{65} + 14 q^{67} + 2 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 8 q^{77} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 24 q^{89} + 12 q^{91} + 2 q^{93} + 12 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 2 * q^17 + 12 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 2 * q^31 + 2 * q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 + 4 * q^43 + 2 * q^45 + 4 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^53 + 12 * q^57 - 2 * q^59 + 2 * q^63 + 4 * q^65 + 14 * q^67 + 2 * q^69 + 14 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 2 * q^81 - 14 * q^83 - 2 * q^85 - 10 * q^87 - 24 * q^89 + 12 * q^91 + 2 * q^93 + 12 * q^95 + 8 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.82843	1.44701	0.723505	−	0.690319i	$-0.242530\pi$
$7$	3.82843	1.44701	0.723505	+	0.690319i	$0.242530\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.41421	−0.426401	−0.213201	−	0.977008i	$-0.568389\pi$
$11$	−1.41421	−0.426401	−0.213201	+	0.977008i	$0.568389\pi$
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.414214	0.100462	0.0502308	−	0.998738i	$-0.484004\pi$
$17$	0.414214	0.100462	0.0502308	+	0.998738i	$0.484004\pi$
$18$	0	0
$19$	4.58579	1.05205	0.526026	−	0.850469i	$-0.323682\pi$
$19$	4.58579	1.05205	0.526026	+	0.850469i	$0.323682\pi$
$20$	0	0
$21$	3.82843	0.835431
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.41421	−1.19109	−0.595545	−	0.803322i	$-0.703064\pi$
$29$	−6.41421	−1.19109	−0.595545	+	0.803322i	$0.703064\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	−1.41421	−0.246183
$34$	0	0
$35$	3.82843	0.647122
$36$	0	0
$37$	9.48528	1.55937	0.779685	−	0.626172i	$-0.215379\pi$
$37$	9.48528	1.55937	0.779685	+	0.626172i	$0.215379\pi$
$38$	0	0
$39$	3.41421	0.546712
$40$	0	0
$41$	−1.24264	−0.194068	−0.0970339	−	0.995281i	$-0.530936\pi$
$41$	−1.24264	−0.194068	−0.0970339	+	0.995281i	$0.530936\pi$
$42$	0	0
$43$	−3.65685	−0.557665	−0.278833	−	0.960340i	$-0.589947\pi$
$43$	−3.65685	−0.557665	−0.278833	+	0.960340i	$0.589947\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−2.24264	−0.327123	−0.163561	−	0.986533i	$-0.552298\pi$
$47$	−2.24264	−0.327123	−0.163561	+	0.986533i	$0.552298\pi$
$48$	0	0
$49$	7.65685	1.09384
$50$	0	0
$51$	0.414214	0.0580015
$52$	0	0
$53$	−3.24264	−0.445411	−0.222705	−	0.974886i	$-0.571489\pi$
$53$	−3.24264	−0.445411	−0.222705	+	0.974886i	$0.571489\pi$
$54$	0	0
$55$	−1.41421	−0.190693
$56$	0	0
$57$	4.58579	0.607402
$58$	0	0
$59$	11.7279	1.52685	0.763423	−	0.645899i	$-0.223517\pi$
$59$	11.7279	1.52685	0.763423	+	0.645899i	$0.223517\pi$
$60$	0	0
$61$	−1.41421	−0.181071	−0.0905357	−	0.995893i	$-0.528858\pi$
$61$	−1.41421	−0.181071	−0.0905357	+	0.995893i	$0.528858\pi$
$62$	0	0
$63$	3.82843	0.482336
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	−1.48528	−0.181456	−0.0907280	−	0.995876i	$-0.528919\pi$
$67$	−1.48528	−0.181456	−0.0907280	+	0.995876i	$0.528919\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−0.0710678	−0.00843420	−0.00421710	−	0.999991i	$-0.501342\pi$
$71$	−0.0710678	−0.00843420	−0.00421710	+	0.999991i	$0.501342\pi$
$72$	0	0
$73$	−6.24264	−0.730646	−0.365323	−	0.930881i	$-0.619041\pi$
$73$	−6.24264	−0.730646	−0.365323	+	0.930881i	$0.619041\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−5.41421	−0.617007
$78$	0	0
$79$	16.9706	1.90934	0.954669	−	0.297670i	$-0.0962096\pi$
$79$	16.9706	1.90934	0.954669	+	0.297670i	$0.0962096\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.2426	−1.23404	−0.617020	−	0.786947i	$-0.711660\pi$
$83$	−11.2426	−1.23404	−0.617020	+	0.786947i	$0.711660\pi$
$84$	0	0
$85$	0.414214	0.0449278
$86$	0	0
$87$	−6.41421	−0.687676
$88$	0	0
$89$	−14.8284	−1.57181	−0.785905	−	0.618347i	$-0.787803\pi$
$89$	−14.8284	−1.57181	−0.785905	+	0.618347i	$0.787803\pi$
$90$	0	0
$91$	13.0711	1.37022
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	4.58579	0.470492
$96$	0	0
$97$	−10.1421	−1.02978	−0.514889	−	0.857257i	$-0.672167\pi$
$97$	−10.1421	−1.02978	−0.514889	+	0.857257i	$0.672167\pi$
$98$	0	0
$99$	−1.41421	−0.142134
$100$	0	0
$101$	4.07107	0.405086	0.202543	−	0.979273i	$-0.435079\pi$
$101$	4.07107	0.405086	0.202543	+	0.979273i	$0.435079\pi$
$102$	0	0
$103$	18.4853	1.82141	0.910704	−	0.413059i	$-0.135540\pi$
$103$	18.4853	1.82141	0.910704	+	0.413059i	$0.135540\pi$
$104$	0	0
$105$	3.82843	0.373616
$106$	0	0
$107$	−6.41421	−0.620085	−0.310043	−	0.950723i	$-0.600343\pi$
$107$	−6.41421	−0.620085	−0.310043	+	0.950723i	$0.600343\pi$
$108$	0	0
$109$	−4.58579	−0.439239	−0.219619	−	0.975586i	$-0.570482\pi$
$109$	−4.58579	−0.439239	−0.219619	+	0.975586i	$0.570482\pi$
$110$	0	0
$111$	9.48528	0.900303
$112$	0	0
$113$	−0.899495	−0.0846174	−0.0423087	−	0.999105i	$-0.513471\pi$
$113$	−0.899495	−0.0846174	−0.0423087	+	0.999105i	$0.513471\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	3.41421	0.315644
$118$	0	0
$119$	1.58579	0.145369
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	−1.24264	−0.112045
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.5563	−1.02546	−0.512730	−	0.858550i	$-0.671366\pi$
$127$	−11.5563	−1.02546	−0.512730	+	0.858550i	$0.671366\pi$
$128$	0	0
$129$	−3.65685	−0.321968
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	17.5563	1.52233
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	0	0
$139$	0.514719	0.0436579	0.0218289	−	0.999762i	$-0.493051\pi$
$139$	0.514719	0.0436579	0.0218289	+	0.999762i	$0.493051\pi$
$140$	0	0
$141$	−2.24264	−0.188864
$142$	0	0
$143$	−4.82843	−0.403773
$144$	0	0
$145$	−6.41421	−0.532671
$146$	0	0
$147$	7.65685	0.631527
$148$	0	0
$149$	6.24264	0.511417	0.255709	−	0.966754i	$-0.417691\pi$
$149$	6.24264	0.511417	0.255709	+	0.966754i	$0.417691\pi$
$150$	0	0
$151$	−1.31371	−0.106908	−0.0534540	−	0.998570i	$-0.517023\pi$
$151$	−1.31371	−0.106908	−0.0534540	+	0.998570i	$0.517023\pi$
$152$	0	0
$153$	0.414214	0.0334872
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	0	0
$159$	−3.24264	−0.257158
$160$	0	0
$161$	3.82843	0.301722
$162$	0	0
$163$	14.9706	1.17258	0.586292	−	0.810099i	$-0.300587\pi$
$163$	14.9706	1.17258	0.586292	+	0.810099i	$0.300587\pi$
$164$	0	0
$165$	−1.41421	−0.110096
$166$	0	0
$167$	10.2426	0.792599	0.396300	−	0.918121i	$-0.370294\pi$
$167$	10.2426	0.792599	0.396300	+	0.918121i	$0.370294\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	4.58579	0.350684
$172$	0	0
$173$	10.3431	0.786375	0.393187	−	0.919458i	$-0.371372\pi$
$173$	10.3431	0.786375	0.393187	+	0.919458i	$0.371372\pi$
$174$	0	0
$175$	3.82843	0.289402
$176$	0	0
$177$	11.7279	0.881525
$178$	0	0
$179$	8.34315	0.623596	0.311798	−	0.950148i	$-0.399069\pi$
$179$	8.34315	0.623596	0.311798	+	0.950148i	$0.399069\pi$
$180$	0	0
$181$	−20.4853	−1.52266	−0.761329	−	0.648365i	$-0.775453\pi$
$181$	−20.4853	−1.52266	−0.761329	+	0.648365i	$0.775453\pi$
$182$	0	0
$183$	−1.41421	−0.104542
$184$	0	0
$185$	9.48528	0.697372
$186$	0	0
$187$	−0.585786	−0.0428369
$188$	0	0
$189$	3.82843	0.278477
$190$	0	0
$191$	−14.2426	−1.03056	−0.515281	−	0.857021i	$-0.672312\pi$
$191$	−14.2426	−1.03056	−0.515281	+	0.857021i	$0.672312\pi$
$192$	0	0
$193$	−10.1421	−0.730047	−0.365023	−	0.930998i	$-0.618939\pi$
$193$	−10.1421	−0.730047	−0.365023	+	0.930998i	$0.618939\pi$
$194$	0	0
$195$	3.41421	0.244497
$196$	0	0
$197$	−12.8284	−0.913988	−0.456994	−	0.889470i	$-0.651074\pi$
$197$	−12.8284	−0.913988	−0.456994	+	0.889470i	$0.651074\pi$
$198$	0	0
$199$	−2.48528	−0.176177	−0.0880885	−	0.996113i	$-0.528076\pi$
$199$	−2.48528	−0.176177	−0.0880885	+	0.996113i	$0.528076\pi$
$200$	0	0
$201$	−1.48528	−0.104764
$202$	0	0
$203$	−24.5563	−1.72352
$204$	0	0
$205$	−1.24264	−0.0867898
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−6.48528	−0.448596
$210$	0	0
$211$	18.7990	1.29418	0.647088	−	0.762415i	$-0.275987\pi$
$211$	18.7990	1.29418	0.647088	+	0.762415i	$0.275987\pi$
$212$	0	0
$213$	−0.0710678	−0.00486949
$214$	0	0
$215$	−3.65685	−0.249395
$216$	0	0
$217$	3.82843	0.259891
$218$	0	0
$219$	−6.24264	−0.421839
$220$	0	0
$221$	1.41421	0.0951303
$222$	0	0
$223$	−14.9706	−1.00250	−0.501252	−	0.865302i	$-0.667127\pi$
$223$	−14.9706	−1.00250	−0.501252	+	0.865302i	$0.667127\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	23.3137	1.54739	0.773693	−	0.633561i	$-0.218408\pi$
$227$	23.3137	1.54739	0.773693	+	0.633561i	$0.218408\pi$
$228$	0	0
$229$	0.485281	0.0320683	0.0160341	−	0.999871i	$-0.494896\pi$
$229$	0.485281	0.0320683	0.0160341	+	0.999871i	$0.494896\pi$
$230$	0	0
$231$	−5.41421	−0.356229
$232$	0	0
$233$	5.65685	0.370593	0.185296	−	0.982683i	$-0.440675\pi$
$233$	5.65685	0.370593	0.185296	+	0.982683i	$0.440675\pi$
$234$	0	0
$235$	−2.24264	−0.146294
$236$	0	0
$237$	16.9706	1.10236
$238$	0	0
$239$	−1.58579	−0.102576	−0.0512880	−	0.998684i	$-0.516333\pi$
$239$	−1.58579	−0.102576	−0.0512880	+	0.998684i	$0.516333\pi$
$240$	0	0
$241$	27.2132	1.75296	0.876478	−	0.481441i	$-0.159887\pi$
$241$	27.2132	1.75296	0.876478	+	0.481441i	$0.159887\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	7.65685	0.489178
$246$	0	0
$247$	15.6569	0.996222
$248$	0	0
$249$	−11.2426	−0.712473
$250$	0	0
$251$	−18.9706	−1.19741	−0.598706	−	0.800969i	$-0.704318\pi$
$251$	−18.9706	−1.19741	−0.598706	+	0.800969i	$0.704318\pi$
$252$	0	0
$253$	−1.41421	−0.0889108
$254$	0	0
$255$	0.414214	0.0259391
$256$	0	0
$257$	−8.58579	−0.535567	−0.267783	−	0.963479i	$-0.586291\pi$
$257$	−8.58579	−0.535567	−0.267783	+	0.963479i	$0.586291\pi$
$258$	0	0
$259$	36.3137	2.25642
$260$	0	0
$261$	−6.41421	−0.397030
$262$	0	0
$263$	−9.58579	−0.591085	−0.295542	−	0.955330i	$-0.595500\pi$
$263$	−9.58579	−0.591085	−0.295542	+	0.955330i	$0.595500\pi$
$264$	0	0
$265$	−3.24264	−0.199194
$266$	0	0
$267$	−14.8284	−0.907485
$268$	0	0
$269$	−2.07107	−0.126275	−0.0631376	−	0.998005i	$-0.520111\pi$
$269$	−2.07107	−0.126275	−0.0631376	+	0.998005i	$0.520111\pi$
$270$	0	0
$271$	4.31371	0.262039	0.131020	−	0.991380i	$-0.458175\pi$
$271$	4.31371	0.262039	0.131020	+	0.991380i	$0.458175\pi$
$272$	0	0
$273$	13.0711	0.791097
$274$	0	0
$275$	−1.41421	−0.0852803
$276$	0	0
$277$	−14.9706	−0.899494	−0.449747	−	0.893156i	$-0.648486\pi$
$277$	−14.9706	−0.899494	−0.449747	+	0.893156i	$0.648486\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	12.2426	0.730335	0.365167	−	0.930942i	$-0.381012\pi$
$281$	12.2426	0.730335	0.365167	+	0.930942i	$0.381012\pi$
$282$	0	0
$283$	−19.6274	−1.16673	−0.583364	−	0.812211i	$-0.698264\pi$
$283$	−19.6274	−1.16673	−0.583364	+	0.812211i	$0.698264\pi$
$284$	0	0
$285$	4.58579	0.271639
$286$	0	0
$287$	−4.75736	−0.280818
$288$	0	0
$289$	−16.8284	−0.989907
$290$	0	0
$291$	−10.1421	−0.594543
$292$	0	0
$293$	−8.89949	−0.519914	−0.259957	−	0.965620i	$-0.583708\pi$
$293$	−8.89949	−0.519914	−0.259957	+	0.965620i	$0.583708\pi$
$294$	0	0
$295$	11.7279	0.682826
$296$	0	0
$297$	−1.41421	−0.0820610
$298$	0	0
$299$	3.41421	0.197449
$300$	0	0
$301$	−14.0000	−0.806947
$302$	0	0
$303$	4.07107	0.233877
$304$	0	0
$305$	−1.41421	−0.0809776
$306$	0	0
$307$	15.8995	0.907432	0.453716	−	0.891146i	$-0.350098\pi$
$307$	15.8995	0.907432	0.453716	+	0.891146i	$0.350098\pi$
$308$	0	0
$309$	18.4853	1.05159
$310$	0	0
$311$	−8.68629	−0.492554	−0.246277	−	0.969199i	$-0.579207\pi$
$311$	−8.68629	−0.492554	−0.246277	+	0.969199i	$0.579207\pi$
$312$	0	0
$313$	−7.14214	−0.403697	−0.201849	−	0.979417i	$-0.564695\pi$
$313$	−7.14214	−0.403697	−0.201849	+	0.979417i	$0.564695\pi$
$314$	0	0
$315$	3.82843	0.215707
$316$	0	0
$317$	−4.58579	−0.257563	−0.128782	−	0.991673i	$-0.541107\pi$
$317$	−4.58579	−0.257563	−0.128782	+	0.991673i	$0.541107\pi$
$318$	0	0
$319$	9.07107	0.507882
$320$	0	0
$321$	−6.41421	−0.358006
$322$	0	0
$323$	1.89949	0.105691
$324$	0	0
$325$	3.41421	0.189386
$326$	0	0
$327$	−4.58579	−0.253595
$328$	0	0
$329$	−8.58579	−0.473350
$330$	0	0
$331$	18.1716	0.998800	0.499400	−	0.866372i	$-0.333554\pi$
$331$	18.1716	0.998800	0.499400	+	0.866372i	$0.333554\pi$
$332$	0	0
$333$	9.48528	0.519790
$334$	0	0
$335$	−1.48528	−0.0811496
$336$	0	0
$337$	−19.6569	−1.07078	−0.535389	−	0.844606i	$-0.679835\pi$
$337$	−19.6569	−1.07078	−0.535389	+	0.844606i	$0.679835\pi$
$338$	0	0
$339$	−0.899495	−0.0488539
$340$	0	0
$341$	−1.41421	−0.0765840
$342$	0	0
$343$	2.51472	0.135782
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	12.4853	0.670245	0.335123	−	0.942175i	$-0.391222\pi$
$347$	12.4853	0.670245	0.335123	+	0.942175i	$0.391222\pi$
$348$	0	0
$349$	−10.1716	−0.544472	−0.272236	−	0.962231i	$-0.587763\pi$
$349$	−10.1716	−0.544472	−0.272236	+	0.962231i	$0.587763\pi$
$350$	0	0
$351$	3.41421	0.182237
$352$	0	0
$353$	26.0416	1.38606	0.693028	−	0.720911i	$-0.256276\pi$
$353$	26.0416	1.38606	0.693028	+	0.720911i	$0.256276\pi$
$354$	0	0
$355$	−0.0710678	−0.00377189
$356$	0	0
$357$	1.58579	0.0839287
$358$	0	0
$359$	−9.07107	−0.478753	−0.239376	−	0.970927i	$-0.576943\pi$
$359$	−9.07107	−0.478753	−0.239376	+	0.970927i	$0.576943\pi$
$360$	0	0
$361$	2.02944	0.106812
$362$	0	0
$363$	−9.00000	−0.472377
$364$	0	0
$365$	−6.24264	−0.326755
$366$	0	0
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	0	0
$369$	−1.24264	−0.0646893
$370$	0	0
$371$	−12.4142	−0.644514
$372$	0	0
$373$	6.34315	0.328436	0.164218	−	0.986424i	$-0.447490\pi$
$373$	6.34315	0.328436	0.164218	+	0.986424i	$0.447490\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−21.8995	−1.12788
$378$	0	0
$379$	25.6569	1.31790	0.658952	−	0.752185i	$-0.271000\pi$
$379$	25.6569	1.31790	0.658952	+	0.752185i	$0.271000\pi$
$380$	0	0
$381$	−11.5563	−0.592050
$382$	0	0
$383$	11.7279	0.599269	0.299634	−	0.954054i	$-0.403135\pi$
$383$	11.7279	0.599269	0.299634	+	0.954054i	$0.403135\pi$
$384$	0	0
$385$	−5.41421	−0.275934
$386$	0	0
$387$	−3.65685	−0.185888
$388$	0	0
$389$	31.4558	1.59487	0.797437	−	0.603402i	$-0.206188\pi$
$389$	31.4558	1.59487	0.797437	+	0.603402i	$0.206188\pi$
$390$	0	0
$391$	0.414214	0.0209477
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	16.9706	0.853882
$396$	0	0
$397$	−10.3431	−0.519108	−0.259554	−	0.965729i	$-0.583575\pi$
$397$	−10.3431	−0.519108	−0.259554	+	0.965729i	$0.583575\pi$
$398$	0	0
$399$	17.5563	0.878917
$400$	0	0
$401$	21.5147	1.07439	0.537197	−	0.843457i	$-0.319483\pi$
$401$	21.5147	1.07439	0.537197	+	0.843457i	$0.319483\pi$
$402$	0	0
$403$	3.41421	0.170074
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−13.4142	−0.664918
$408$	0	0
$409$	14.1716	0.700739	0.350370	−	0.936612i	$-0.386056\pi$
$409$	14.1716	0.700739	0.350370	+	0.936612i	$0.386056\pi$
$410$	0	0
$411$	−2.82843	−0.139516
$412$	0	0
$413$	44.8995	2.20936
$414$	0	0
$415$	−11.2426	−0.551880
$416$	0	0
$417$	0.514719	0.0252059
$418$	0	0
$419$	35.3553	1.72722	0.863611	−	0.504159i	$-0.168198\pi$
$419$	35.3553	1.72722	0.863611	+	0.504159i	$0.168198\pi$
$420$	0	0
$421$	−20.8701	−1.01714	−0.508572	−	0.861019i	$-0.669827\pi$
$421$	−20.8701	−1.01714	−0.508572	+	0.861019i	$0.669827\pi$
$422$	0	0
$423$	−2.24264	−0.109041
$424$	0	0
$425$	0.414214	0.0200923
$426$	0	0
$427$	−5.41421	−0.262012
$428$	0	0
$429$	−4.82843	−0.233119
$430$	0	0
$431$	−11.5147	−0.554644	−0.277322	−	0.960777i	$-0.589447\pi$
$431$	−11.5147	−0.554644	−0.277322	+	0.960777i	$0.589447\pi$
$432$	0	0
$433$	26.3137	1.26456	0.632278	−	0.774742i	$-0.282120\pi$
$433$	26.3137	1.26456	0.632278	+	0.774742i	$0.282120\pi$
$434$	0	0
$435$	−6.41421	−0.307538
$436$	0	0
$437$	4.58579	0.219368
$438$	0	0
$439$	−5.79899	−0.276771	−0.138385	−	0.990378i	$-0.544191\pi$
$439$	−5.79899	−0.276771	−0.138385	+	0.990378i	$0.544191\pi$
$440$	0	0
$441$	7.65685	0.364612
$442$	0	0
$443$	3.89949	0.185271	0.0926353	−	0.995700i	$-0.470471\pi$
$443$	3.89949	0.185271	0.0926353	+	0.995700i	$0.470471\pi$
$444$	0	0
$445$	−14.8284	−0.702935
$446$	0	0
$447$	6.24264	0.295267
$448$	0	0
$449$	20.8995	0.986308	0.493154	−	0.869942i	$-0.335844\pi$
$449$	20.8995	0.986308	0.493154	+	0.869942i	$0.335844\pi$
$450$	0	0
$451$	1.75736	0.0827508
$452$	0	0
$453$	−1.31371	−0.0617234
$454$	0	0
$455$	13.0711	0.612781
$456$	0	0
$457$	40.4558	1.89244	0.946222	−	0.323517i	$-0.104865\pi$
$457$	40.4558	1.89244	0.946222	+	0.323517i	$0.104865\pi$
$458$	0	0
$459$	0.414214	0.0193338
$460$	0	0
$461$	23.4558	1.09245	0.546224	−	0.837639i	$-0.316065\pi$
$461$	23.4558	1.09245	0.546224	+	0.837639i	$0.316065\pi$
$462$	0	0
$463$	26.0416	1.21026	0.605129	−	0.796128i	$-0.293122\pi$
$463$	26.0416	1.21026	0.605129	+	0.796128i	$0.293122\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−14.4142	−0.667010	−0.333505	−	0.942748i	$-0.608231\pi$
$467$	−14.4142	−0.667010	−0.333505	+	0.942748i	$0.608231\pi$
$468$	0	0
$469$	−5.68629	−0.262569
$470$	0	0
$471$	−3.00000	−0.138233
$472$	0	0
$473$	5.17157	0.237789
$474$	0	0
$475$	4.58579	0.210410
$476$	0	0
$477$	−3.24264	−0.148470
$478$	0	0
$479$	−39.8995	−1.82305	−0.911527	−	0.411240i	$-0.865096\pi$
$479$	−39.8995	−1.82305	−0.911527	+	0.411240i	$0.865096\pi$
$480$	0	0
$481$	32.3848	1.47662
$482$	0	0
$483$	3.82843	0.174199
$484$	0	0
$485$	−10.1421	−0.460531
$486$	0	0
$487$	−16.9289	−0.767123	−0.383562	−	0.923515i	$-0.625303\pi$
$487$	−16.9289	−0.767123	−0.383562	+	0.923515i	$0.625303\pi$
$488$	0	0
$489$	14.9706	0.676992
$490$	0	0
$491$	−42.8995	−1.93603	−0.968014	−	0.250898i	$-0.919274\pi$
$491$	−42.8995	−1.93603	−0.968014	+	0.250898i	$0.919274\pi$
$492$	0	0
$493$	−2.65685	−0.119659
$494$	0	0
$495$	−1.41421	−0.0635642
$496$	0	0
$497$	−0.272078	−0.0122044
$498$	0	0
$499$	−26.4558	−1.18433	−0.592163	−	0.805818i	$-0.701726\pi$
$499$	−26.4558	−1.18433	−0.592163	+	0.805818i	$0.701726\pi$
$500$	0	0
$501$	10.2426	0.457607
$502$	0	0
$503$	−29.2426	−1.30386	−0.651932	−	0.758277i	$-0.726041\pi$
$503$	−29.2426	−1.30386	−0.651932	+	0.758277i	$0.726041\pi$
$504$	0	0
$505$	4.07107	0.181160
$506$	0	0
$507$	−1.34315	−0.0596512
$508$	0	0
$509$	−3.51472	−0.155787	−0.0778936	−	0.996962i	$-0.524819\pi$
$509$	−3.51472	−0.155787	−0.0778936	+	0.996962i	$0.524819\pi$
$510$	0	0
$511$	−23.8995	−1.05725
$512$	0	0
$513$	4.58579	0.202467
$514$	0	0
$515$	18.4853	0.814559
$516$	0	0
$517$	3.17157	0.139486
$518$	0	0
$519$	10.3431	0.454014
$520$	0	0
$521$	0.727922	0.0318908	0.0159454	−	0.999873i	$-0.494924\pi$
$521$	0.727922	0.0318908	0.0159454	+	0.999873i	$0.494924\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	3.82843	0.167086
$526$	0	0
$527$	0.414214	0.0180434
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	11.7279	0.508948
$532$	0	0
$533$	−4.24264	−0.183769
$534$	0	0
$535$	−6.41421	−0.277311
$536$	0	0
$537$	8.34315	0.360033
$538$	0	0
$539$	−10.8284	−0.466413
$540$	0	0
$541$	19.1127	0.821719	0.410860	−	0.911699i	$-0.365229\pi$
$541$	19.1127	0.821719	0.410860	+	0.911699i	$0.365229\pi$
$542$	0	0
$543$	−20.4853	−0.879108
$544$	0	0
$545$	−4.58579	−0.196434
$546$	0	0
$547$	−17.6569	−0.754953	−0.377476	−	0.926019i	$-0.623208\pi$
$547$	−17.6569	−0.754953	−0.377476	+	0.926019i	$0.623208\pi$
$548$	0	0
$549$	−1.41421	−0.0603572
$550$	0	0
$551$	−29.4142	−1.25309
$552$	0	0
$553$	64.9706	2.76283
$554$	0	0
$555$	9.48528	0.402628
$556$	0	0
$557$	−31.5269	−1.33584	−0.667919	−	0.744234i	$-0.732815\pi$
$557$	−31.5269	−1.33584	−0.667919	+	0.744234i	$0.732815\pi$
$558$	0	0
$559$	−12.4853	−0.528071
$560$	0	0
$561$	−0.585786	−0.0247319
$562$	0	0
$563$	46.4142	1.95613	0.978063	−	0.208310i	$-0.0667962\pi$
$563$	46.4142	1.95613	0.978063	+	0.208310i	$0.0667962\pi$
$564$	0	0
$565$	−0.899495	−0.0378420
$566$	0	0
$567$	3.82843	0.160779
$568$	0	0
$569$	1.51472	0.0635003	0.0317502	−	0.999496i	$-0.489892\pi$
$569$	1.51472	0.0635003	0.0317502	+	0.999496i	$0.489892\pi$
$570$	0	0
$571$	−17.8995	−0.749071	−0.374535	−	0.927213i	$-0.622198\pi$
$571$	−17.8995	−0.749071	−0.374535	+	0.927213i	$0.622198\pi$
$572$	0	0
$573$	−14.2426	−0.594995
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−32.2843	−1.34401	−0.672006	−	0.740546i	$-0.734567\pi$
$577$	−32.2843	−1.34401	−0.672006	+	0.740546i	$0.734567\pi$
$578$	0	0
$579$	−10.1421	−0.421493
$580$	0	0
$581$	−43.0416	−1.78567
$582$	0	0
$583$	4.58579	0.189924
$584$	0	0
$585$	3.41421	0.141160
$586$	0	0
$587$	8.62742	0.356092	0.178046	−	0.984022i	$-0.443022\pi$
$587$	8.62742	0.356092	0.178046	+	0.984022i	$0.443022\pi$
$588$	0	0
$589$	4.58579	0.188954
$590$	0	0
$591$	−12.8284	−0.527691
$592$	0	0
$593$	−21.0711	−0.865285	−0.432643	−	0.901566i	$-0.642419\pi$
$593$	−21.0711	−0.865285	−0.432643	+	0.901566i	$0.642419\pi$
$594$	0	0
$595$	1.58579	0.0650109
$596$	0	0
$597$	−2.48528	−0.101716
$598$	0	0
$599$	−10.1421	−0.414396	−0.207198	−	0.978299i	$-0.566435\pi$
$599$	−10.1421	−0.414396	−0.207198	+	0.978299i	$0.566435\pi$
$600$	0	0
$601$	11.3431	0.462697	0.231348	−	0.972871i	$-0.425686\pi$
$601$	11.3431	0.462697	0.231348	+	0.972871i	$0.425686\pi$
$602$	0	0
$603$	−1.48528	−0.0604853
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	−12.7279	−0.516610	−0.258305	−	0.966063i	$-0.583164\pi$
$607$	−12.7279	−0.516610	−0.258305	+	0.966063i	$0.583164\pi$
$608$	0	0
$609$	−24.5563	−0.995073
$610$	0	0
$611$	−7.65685	−0.309763
$612$	0	0
$613$	40.6274	1.64093	0.820463	−	0.571700i	$-0.193716\pi$
$613$	40.6274	1.64093	0.820463	+	0.571700i	$0.193716\pi$
$614$	0	0
$615$	−1.24264	−0.0501081
$616$	0	0
$617$	8.75736	0.352558	0.176279	−	0.984340i	$-0.443594\pi$
$617$	8.75736	0.352558	0.176279	+	0.984340i	$0.443594\pi$
$618$	0	0
$619$	40.2843	1.61916	0.809581	−	0.587008i	$-0.199694\pi$
$619$	40.2843	1.61916	0.809581	+	0.587008i	$0.199694\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−56.7696	−2.27442
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−6.48528	−0.258997
$628$	0	0
$629$	3.92893	0.156657
$630$	0	0
$631$	1.41421	0.0562990	0.0281495	−	0.999604i	$-0.491039\pi$
$631$	1.41421	0.0562990	0.0281495	+	0.999604i	$0.491039\pi$
$632$	0	0
$633$	18.7990	0.747193
$634$	0	0
$635$	−11.5563	−0.458600
$636$	0	0
$637$	26.1421	1.03579
$638$	0	0
$639$	−0.0710678	−0.00281140
$640$	0	0
$641$	−18.5858	−0.734094	−0.367047	−	0.930202i	$-0.619631\pi$
$641$	−18.5858	−0.734094	−0.367047	+	0.930202i	$0.619631\pi$
$642$	0	0
$643$	−42.6569	−1.68222	−0.841111	−	0.540862i	$-0.818098\pi$
$643$	−42.6569	−1.68222	−0.841111	+	0.540862i	$0.818098\pi$
$644$	0	0
$645$	−3.65685	−0.143988
$646$	0	0
$647$	−34.3848	−1.35181	−0.675903	−	0.736991i	$-0.736246\pi$
$647$	−34.3848	−1.35181	−0.675903	+	0.736991i	$0.736246\pi$
$648$	0	0
$649$	−16.5858	−0.651049
$650$	0	0
$651$	3.82843	0.150048
$652$	0	0
$653$	8.72792	0.341550	0.170775	−	0.985310i	$-0.445373\pi$
$653$	8.72792	0.341550	0.170775	+	0.985310i	$0.445373\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	−6.24264	−0.243549
$658$	0	0
$659$	−30.3848	−1.18362	−0.591811	−	0.806076i	$-0.701587\pi$
$659$	−30.3848	−1.18362	−0.591811	+	0.806076i	$0.701587\pi$
$660$	0	0
$661$	−49.1127	−1.91026	−0.955131	−	0.296183i	$-0.904286\pi$
$661$	−49.1127	−1.91026	−0.955131	+	0.296183i	$0.904286\pi$
$662$	0	0
$663$	1.41421	0.0549235
$664$	0	0
$665$	17.5563	0.680806
$666$	0	0
$667$	−6.41421	−0.248359
$668$	0	0
$669$	−14.9706	−0.578795
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	32.7279	1.26157	0.630784	−	0.775958i	$-0.282733\pi$
$673$	32.7279	1.26157	0.630784	+	0.775958i	$0.282733\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	33.1838	1.27536	0.637678	−	0.770303i	$-0.279895\pi$
$677$	33.1838	1.27536	0.637678	+	0.770303i	$0.279895\pi$
$678$	0	0
$679$	−38.8284	−1.49010
$680$	0	0
$681$	23.3137	0.893383
$682$	0	0
$683$	39.0122	1.49276	0.746380	−	0.665520i	$-0.231790\pi$
$683$	39.0122	1.49276	0.746380	+	0.665520i	$0.231790\pi$
$684$	0	0
$685$	−2.82843	−0.108069
$686$	0	0
$687$	0.485281	0.0185146
$688$	0	0
$689$	−11.0711	−0.421774
$690$	0	0
$691$	−18.9706	−0.721674	−0.360837	−	0.932629i	$-0.617509\pi$
$691$	−18.9706	−0.721674	−0.360837	+	0.932629i	$0.617509\pi$
$692$	0	0
$693$	−5.41421	−0.205669
$694$	0	0
$695$	0.514719	0.0195244
$696$	0	0
$697$	−0.514719	−0.0194964
$698$	0	0
$699$	5.65685	0.213962
$700$	0	0
$701$	18.4437	0.696607	0.348304	−	0.937382i	$-0.386758\pi$
$701$	18.4437	0.696607	0.348304	+	0.937382i	$0.386758\pi$
$702$	0	0
$703$	43.4975	1.64054
$704$	0	0
$705$	−2.24264	−0.0844627
$706$	0	0
$707$	15.5858	0.586164
$708$	0	0
$709$	−41.6985	−1.56602	−0.783010	−	0.622009i	$-0.786317\pi$
$709$	−41.6985	−1.56602	−0.783010	+	0.622009i	$0.786317\pi$
$710$	0	0
$711$	16.9706	0.636446
$712$	0	0
$713$	1.00000	0.0374503
$714$	0	0
$715$	−4.82843	−0.180573
$716$	0	0
$717$	−1.58579	−0.0592223
$718$	0	0
$719$	−1.44365	−0.0538391	−0.0269195	−	0.999638i	$-0.508570\pi$
$719$	−1.44365	−0.0538391	−0.0269195	+	0.999638i	$0.508570\pi$
$720$	0	0
$721$	70.7696	2.63560
$722$	0	0
$723$	27.2132	1.01207
$724$	0	0
$725$	−6.41421	−0.238218
$726$	0	0
$727$	−9.00000	−0.333792	−0.166896	−	0.985975i	$-0.553374\pi$
$727$	−9.00000	−0.333792	−0.166896	+	0.985975i	$0.553374\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.51472	−0.0560239
$732$	0	0
$733$	−6.31371	−0.233202	−0.116601	−	0.993179i	$-0.537200\pi$
$733$	−6.31371	−0.233202	−0.116601	+	0.993179i	$0.537200\pi$
$734$	0	0
$735$	7.65685	0.282427
$736$	0	0
$737$	2.10051	0.0773731
$738$	0	0
$739$	11.1421	0.409870	0.204935	−	0.978776i	$-0.434302\pi$
$739$	11.1421	0.409870	0.204935	+	0.978776i	$0.434302\pi$
$740$	0	0
$741$	15.6569	0.575169
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	6.24264	0.228713
$746$	0	0
$747$	−11.2426	−0.411347
$748$	0	0
$749$	−24.5563	−0.897269
$750$	0	0
$751$	30.9289	1.12861	0.564306	−	0.825565i	$-0.309144\pi$
$751$	30.9289	1.12861	0.564306	+	0.825565i	$0.309144\pi$
$752$	0	0
$753$	−18.9706	−0.691326
$754$	0	0
$755$	−1.31371	−0.0478107
$756$	0	0
$757$	3.82843	0.139147	0.0695733	−	0.997577i	$-0.477836\pi$
$757$	3.82843	0.139147	0.0695733	+	0.997577i	$0.477836\pi$
$758$	0	0
$759$	−1.41421	−0.0513327
$760$	0	0
$761$	−38.0711	−1.38008	−0.690038	−	0.723774i	$-0.742406\pi$
$761$	−38.0711	−1.38008	−0.690038	+	0.723774i	$0.742406\pi$
$762$	0	0
$763$	−17.5563	−0.635583
$764$	0	0
$765$	0.414214	0.0149759
$766$	0	0
$767$	40.0416	1.44582
$768$	0	0
$769$	−40.7279	−1.46869	−0.734343	−	0.678778i	$-0.762510\pi$
$769$	−40.7279	−1.46869	−0.734343	+	0.678778i	$0.762510\pi$
$770$	0	0
$771$	−8.58579	−0.309210
$772$	0	0
$773$	25.7990	0.927925	0.463963	−	0.885855i	$-0.346427\pi$
$773$	25.7990	0.927925	0.463963	+	0.885855i	$0.346427\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	36.3137	1.30275
$778$	0	0
$779$	−5.69848	−0.204169
$780$	0	0
$781$	0.100505	0.00359635
$782$	0	0
$783$	−6.41421	−0.229225
$784$	0	0
$785$	−3.00000	−0.107075
$786$	0	0
$787$	−37.2843	−1.32904	−0.664520	−	0.747270i	$-0.731364\pi$
$787$	−37.2843	−1.32904	−0.664520	+	0.747270i	$0.731364\pi$
$788$	0	0
$789$	−9.58579	−0.341263
$790$	0	0
$791$	−3.44365	−0.122442
$792$	0	0
$793$	−4.82843	−0.171462
$794$	0	0
$795$	−3.24264	−0.115005
$796$	0	0
$797$	−15.1005	−0.534887	−0.267444	−	0.963573i	$-0.586179\pi$
$797$	−15.1005	−0.534887	−0.267444	+	0.963573i	$0.586179\pi$
$798$	0	0
$799$	−0.928932	−0.0328633
$800$	0	0
$801$	−14.8284	−0.523937
$802$	0	0
$803$	8.82843	0.311548
$804$	0	0
$805$	3.82843	0.134934
$806$	0	0
$807$	−2.07107	−0.0729050
$808$	0	0
$809$	4.27208	0.150198	0.0750991	−	0.997176i	$-0.476073\pi$
$809$	4.27208	0.150198	0.0750991	+	0.997176i	$0.476073\pi$
$810$	0	0
$811$	−23.1421	−0.812630	−0.406315	−	0.913733i	$-0.633186\pi$
$811$	−23.1421	−0.812630	−0.406315	+	0.913733i	$0.633186\pi$
$812$	0	0
$813$	4.31371	0.151288
$814$	0	0
$815$	14.9706	0.524396
$816$	0	0
$817$	−16.7696	−0.586692
$818$	0	0
$819$	13.0711	0.456740
$820$	0	0
$821$	−17.4558	−0.609213	−0.304607	−	0.952478i	$-0.598525\pi$
$821$	−17.4558	−0.609213	−0.304607	+	0.952478i	$0.598525\pi$
$822$	0	0
$823$	42.3431	1.47599	0.737995	−	0.674807i	$-0.235773\pi$
$823$	42.3431	1.47599	0.737995	+	0.674807i	$0.235773\pi$
$824$	0	0
$825$	−1.41421	−0.0492366
$826$	0	0
$827$	−27.7279	−0.964194	−0.482097	−	0.876118i	$-0.660125\pi$
$827$	−27.7279	−0.964194	−0.482097	+	0.876118i	$0.660125\pi$
$828$	0	0
$829$	−2.17157	−0.0754218	−0.0377109	−	0.999289i	$-0.512007\pi$
$829$	−2.17157	−0.0754218	−0.0377109	+	0.999289i	$0.512007\pi$
$830$	0	0
$831$	−14.9706	−0.519323
$832$	0	0
$833$	3.17157	0.109888
$834$	0	0
$835$	10.2426	0.354461
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−39.6569	−1.36911	−0.684553	−	0.728963i	$-0.740003\pi$
$839$	−39.6569	−1.36911	−0.684553	+	0.728963i	$0.740003\pi$
$840$	0	0
$841$	12.1421	0.418694
$842$	0	0
$843$	12.2426	0.421659
$844$	0	0
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	−34.4558	−1.18392
$848$	0	0
$849$	−19.6274	−0.673611
$850$	0	0
$851$	9.48528	0.325151
$852$	0	0
$853$	−10.4853	−0.359009	−0.179505	−	0.983757i	$-0.557449\pi$
$853$	−10.4853	−0.359009	−0.179505	+	0.983757i	$0.557449\pi$
$854$	0	0
$855$	4.58579	0.156831
$856$	0	0
$857$	−53.3137	−1.82116	−0.910581	−	0.413331i	$-0.864365\pi$
$857$	−53.3137	−1.82116	−0.910581	+	0.413331i	$0.864365\pi$
$858$	0	0
$859$	48.1716	1.64359	0.821796	−	0.569781i	$-0.192972\pi$
$859$	48.1716	1.64359	0.821796	+	0.569781i	$0.192972\pi$
$860$	0	0
$861$	−4.75736	−0.162130
$862$	0	0
$863$	−29.5147	−1.00469	−0.502346	−	0.864666i	$-0.667530\pi$
$863$	−29.5147	−1.00469	−0.502346	+	0.864666i	$0.667530\pi$
$864$	0	0
$865$	10.3431	0.351678
$866$	0	0
$867$	−16.8284	−0.571523
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−5.07107	−0.171827
$872$	0	0
$873$	−10.1421	−0.343259
$874$	0	0
$875$	3.82843	0.129424
$876$	0	0
$877$	−12.8284	−0.433185	−0.216593	−	0.976262i	$-0.569494\pi$
$877$	−12.8284	−0.433185	−0.216593	+	0.976262i	$0.569494\pi$
$878$	0	0
$879$	−8.89949	−0.300173
$880$	0	0
$881$	−25.0711	−0.844666	−0.422333	−	0.906441i	$-0.638789\pi$
$881$	−25.0711	−0.844666	−0.422333	+	0.906441i	$0.638789\pi$
$882$	0	0
$883$	47.0711	1.58407	0.792034	−	0.610477i	$-0.209022\pi$
$883$	47.0711	1.58407	0.792034	+	0.610477i	$0.209022\pi$
$884$	0	0
$885$	11.7279	0.394230
$886$	0	0
$887$	−0.828427	−0.0278159	−0.0139079	−	0.999903i	$-0.504427\pi$
$887$	−0.828427	−0.0278159	−0.0139079	+	0.999903i	$0.504427\pi$
$888$	0	0
$889$	−44.2426	−1.48385
$890$	0	0
$891$	−1.41421	−0.0473779
$892$	0	0
$893$	−10.2843	−0.344150
$894$	0	0
$895$	8.34315	0.278881
$896$	0	0
$897$	3.41421	0.113997
$898$	0	0
$899$	−6.41421	−0.213926
$900$	0	0
$901$	−1.34315	−0.0447467
$902$	0	0
$903$	−14.0000	−0.465891
$904$	0	0
$905$	−20.4853	−0.680954
$906$	0	0
$907$	−23.8284	−0.791210	−0.395605	−	0.918421i	$-0.629465\pi$
$907$	−23.8284	−0.791210	−0.395605	+	0.918421i	$0.629465\pi$
$908$	0	0
$909$	4.07107	0.135029
$910$	0	0
$911$	4.97056	0.164682	0.0823410	−	0.996604i	$-0.473760\pi$
$911$	4.97056	0.164682	0.0823410	+	0.996604i	$0.473760\pi$
$912$	0	0
$913$	15.8995	0.526196
$914$	0	0
$915$	−1.41421	−0.0467525
$916$	0	0
$917$	61.2548	2.02281
$918$	0	0
$919$	32.2843	1.06496	0.532480	−	0.846443i	$-0.321260\pi$
$919$	32.2843	1.06496	0.532480	+	0.846443i	$0.321260\pi$
$920$	0	0
$921$	15.8995	0.523906
$922$	0	0
$923$	−0.242641	−0.00798662
$924$	0	0
$925$	9.48528	0.311874
$926$	0	0
$927$	18.4853	0.607136
$928$	0	0
$929$	−41.2426	−1.35313	−0.676564	−	0.736384i	$-0.736532\pi$
$929$	−41.2426	−1.35313	−0.676564	+	0.736384i	$0.736532\pi$
$930$	0	0
$931$	35.1127	1.15077
$932$	0	0
$933$	−8.68629	−0.284376
$934$	0	0
$935$	−0.585786	−0.0191573
$936$	0	0
$937$	−0.485281	−0.0158535	−0.00792673	−	0.999969i	$-0.502523\pi$
$937$	−0.485281	−0.0158535	−0.00792673	+	0.999969i	$0.502523\pi$
$938$	0	0
$939$	−7.14214	−0.233075
$940$	0	0
$941$	−58.5269	−1.90792	−0.953961	−	0.299929i	$-0.903037\pi$
$941$	−58.5269	−1.90792	−0.953961	+	0.299929i	$0.903037\pi$
$942$	0	0
$943$	−1.24264	−0.0404659
$944$	0	0
$945$	3.82843	0.124539
$946$	0	0
$947$	11.5147	0.374178	0.187089	−	0.982343i	$-0.440095\pi$
$947$	11.5147	0.374178	0.187089	+	0.982343i	$0.440095\pi$
$948$	0	0
$949$	−21.3137	−0.691872
$950$	0	0
$951$	−4.58579	−0.148704
$952$	0	0
$953$	40.6274	1.31605	0.658026	−	0.752996i	$-0.271392\pi$
$953$	40.6274	1.31605	0.658026	+	0.752996i	$0.271392\pi$
$954$	0	0
$955$	−14.2426	−0.460881
$956$	0	0
$957$	9.07107	0.293226
$958$	0	0
$959$	−10.8284	−0.349668
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−6.41421	−0.206695
$964$	0	0
$965$	−10.1421	−0.326487
$966$	0	0
$967$	−5.61522	−0.180573	−0.0902867	−	0.995916i	$-0.528778\pi$
$967$	−5.61522	−0.180573	−0.0902867	+	0.995916i	$0.528778\pi$
$968$	0	0
$969$	1.89949	0.0610206
$970$	0	0
$971$	−50.4264	−1.61826	−0.809130	−	0.587629i	$-0.800061\pi$
$971$	−50.4264	−1.61826	−0.809130	+	0.587629i	$0.800061\pi$
$972$	0	0
$973$	1.97056	0.0631733
$974$	0	0
$975$	3.41421	0.109342
$976$	0	0
$977$	3.87006	0.123814	0.0619071	−	0.998082i	$-0.480282\pi$
$977$	3.87006	0.123814	0.0619071	+	0.998082i	$0.480282\pi$
$978$	0	0
$979$	20.9706	0.670222
$980$	0	0
$981$	−4.58579	−0.146413
$982$	0	0
$983$	−27.5269	−0.877972	−0.438986	−	0.898494i	$-0.644662\pi$
$983$	−27.5269	−0.877972	−0.438986	+	0.898494i	$0.644662\pi$
$984$	0	0
$985$	−12.8284	−0.408748
$986$	0	0
$987$	−8.58579	−0.273289
$988$	0	0
$989$	−3.65685	−0.116281
$990$	0	0
$991$	0.455844	0.0144804	0.00724018	−	0.999974i	$-0.497695\pi$
$991$	0.455844	0.0144804	0.00724018	+	0.999974i	$0.497695\pi$
$992$	0	0
$993$	18.1716	0.576657
$994$	0	0
$995$	−2.48528	−0.0787887
$996$	0	0
$997$	36.4853	1.15550	0.577750	−	0.816214i	$-0.303931\pi$
$997$	36.4853	1.15550	0.577750	+	0.816214i	$0.303931\pi$
$998$	0	0
$999$	9.48528	0.300101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bt.1.2		2
4.3	odd	2		2760.2.a.n.1.1	✓	2
12.11	even	2		8280.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.n.1.1	✓	2	4.3	odd	2
5520.2.a.bt.1.2		2	1.1	even	1	trivial
8280.2.a.x.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +3.82843 q^{7} +1.00000 q^{9} -1.41421 q^{11} +3.41421 q^{13} +1.00000 q^{15} +0.414214 q^{17} +4.58579 q^{19} +3.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -6.41421 q^{29} +1.00000 q^{31} -1.41421 q^{33} +3.82843 q^{35} +9.48528 q^{37} +3.41421 q^{39} -1.24264 q^{41} -3.65685 q^{43} +1.00000 q^{45} -2.24264 q^{47} +7.65685 q^{49} +0.414214 q^{51} -3.24264 q^{53} -1.41421 q^{55} +4.58579 q^{57} +11.7279 q^{59} -1.41421 q^{61} +3.82843 q^{63} +3.41421 q^{65} -1.48528 q^{67} +1.00000 q^{69} -0.0710678 q^{71} -6.24264 q^{73} +1.00000 q^{75} -5.41421 q^{77} +16.9706 q^{79} +1.00000 q^{81} -11.2426 q^{83} +0.414214 q^{85} -6.41421 q^{87} -14.8284 q^{89} +13.0711 q^{91} +1.00000 q^{93} +4.58579 q^{95} -10.1421 q^{97} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} + 12 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 4 q^{43} + 2 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} + 12 q^{57} - 2 q^{59} + 2 q^{63} + 4 q^{65} + 14 q^{67} + 2 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 8 q^{77} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 24 q^{89} + 12 q^{91} + 2 q^{93} + 12 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 2 * q^17 + 12 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 2 * q^31 + 2 * q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 + 4 * q^43 + 2 * q^45 + 4 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^53 + 12 * q^57 - 2 * q^59 + 2 * q^63 + 4 * q^65 + 14 * q^67 + 2 * q^69 + 14 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 2 * q^81 - 14 * q^83 - 2 * q^85 - 10 * q^87 - 24 * q^89 + 12 * q^91 + 2 * q^93 + 12 * q^95 + 8 * q^97

Introduction

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