LMFDB - Embedded newform 5776.2.a.br.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5776,2,Mod(1,5776)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5776, 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("5776.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5776 = 2^{4} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5776.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:	$46.1215922075$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ of defining polynomial
Character	$\chi$	$=$	5776.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.652704 q^{3} -1.34730 q^{5} -1.53209 q^{7} -2.57398 q^{9} +O(q^{10})$	$q+0.652704 q^{3} -1.34730 q^{5} -1.53209 q^{7} -2.57398 q^{9} +1.18479 q^{11} +2.71688 q^{13} -0.879385 q^{15} +3.87939 q^{17} -1.00000 q^{21} +5.06418 q^{23} -3.18479 q^{25} -3.63816 q^{27} -4.65270 q^{29} +3.83750 q^{31} +0.773318 q^{33} +2.06418 q^{35} -4.10607 q^{37} +1.77332 q^{39} -9.98545 q^{41} +8.70233 q^{43} +3.46791 q^{45} -0.573978 q^{47} -4.65270 q^{49} +2.53209 q^{51} +2.94356 q^{53} -1.59627 q^{55} +3.93582 q^{59} -4.51754 q^{61} +3.94356 q^{63} -3.66044 q^{65} +3.88713 q^{67} +3.30541 q^{69} +6.93582 q^{71} +6.12836 q^{73} -2.07873 q^{75} -1.81521 q^{77} -9.80840 q^{79} +5.34730 q^{81} -12.3182 q^{83} -5.22668 q^{85} -3.03684 q^{87} -2.42602 q^{89} -4.16250 q^{91} +2.50475 q^{93} +7.36959 q^{97} -3.04963 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5	$ 3 q + 3 q^{3} - 3 q^{5} + 3 q^{15} + 6 q^{17} - 3 q^{21} + 6 q^{23} - 6 q^{25} + 6 q^{27} - 15 q^{29} + 9 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{39} - 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} + 3 q^{51} - 6 q^{53} + 9 q^{55} + 21 q^{59} + 9 q^{61} - 3 q^{63} + 12 q^{65} - 18 q^{67} + 12 q^{69} + 30 q^{71} - 15 q^{75} - 9 q^{77} + 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} - 15 q^{89} - 15 q^{91} + 24 q^{93} + 15 q^{97} + 18 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^15 + 6 * q^17 - 3 * q^21 + 6 * q^23 - 6 * q^25 + 6 * q^27 - 15 * q^29 + 9 * q^31 + 9 * q^33 - 3 * q^35 + 12 * q^39 - 12 * q^41 + 15 * q^45 + 6 * q^47 - 15 * q^49 + 3 * q^51 - 6 * q^53 + 9 * q^55 + 21 * q^59 + 9 * q^61 - 3 * q^63 + 12 * q^65 - 18 * q^67 + 12 * q^69 + 30 * q^71 - 15 * q^75 - 9 * q^77 + 9 * q^79 + 15 * q^81 - 9 * q^85 - 21 * q^87 - 15 * q^89 - 15 * q^91 + 24 * q^93 + 15 * q^97 + 18 * q^99

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
$3$	0.652704	0.376839	0.188419	−	0.982089i	$-0.439664\pi$
$3$	0.652704	0.376839	0.188419	+	0.982089i	$0.439664\pi$
$4$	0	0
$5$	−1.34730	−0.602529	−0.301265	−	0.953541i	$-0.597409\pi$
$5$	−1.34730	−0.602529	−0.301265	+	0.953541i	$0.597409\pi$
$6$	0	0
$7$	−1.53209	−0.579075	−0.289538	−	0.957167i	$-0.593502\pi$
$7$	−1.53209	−0.579075	−0.289538	+	0.957167i	$0.593502\pi$
$8$	0	0
$9$	−2.57398	−0.857993
$10$	0	0
$11$	1.18479	0.357228	0.178614	−	0.983919i	$-0.442839\pi$
$11$	1.18479	0.357228	0.178614	+	0.983919i	$0.442839\pi$
$12$	0	0
$13$	2.71688	0.753527	0.376764	−	0.926309i	$-0.377037\pi$
$13$	2.71688	0.753527	0.376764	+	0.926309i	$0.377037\pi$
$14$	0	0
$15$	−0.879385	−0.227056
$16$	0	0
$17$	3.87939	0.940889	0.470445	−	0.882430i	$-0.344094\pi$
$17$	3.87939	0.940889	0.470445	+	0.882430i	$0.344094\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	5.06418	1.05595	0.527977	−	0.849259i	$-0.322951\pi$
$23$	5.06418	1.05595	0.527977	+	0.849259i	$0.322951\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	0	0
$27$	−3.63816	−0.700163
$28$	0	0
$29$	−4.65270	−0.863985	−0.431993	−	0.901877i	$-0.642189\pi$
$29$	−4.65270	−0.863985	−0.431993	+	0.901877i	$0.642189\pi$
$30$	0	0
$31$	3.83750	0.689235	0.344617	−	0.938743i	$-0.388009\pi$
$31$	3.83750	0.689235	0.344617	+	0.938743i	$0.388009\pi$
$32$	0	0
$33$	0.773318	0.134617
$34$	0	0
$35$	2.06418	0.348910
$36$	0	0
$37$	−4.10607	−0.675033	−0.337517	−	0.941320i	$-0.609587\pi$
$37$	−4.10607	−0.675033	−0.337517	+	0.941320i	$0.609587\pi$
$38$	0	0
$39$	1.77332	0.283958
$40$	0	0
$41$	−9.98545	−1.55947	−0.779733	−	0.626112i	$-0.784645\pi$
$41$	−9.98545	−1.55947	−0.779733	+	0.626112i	$0.784645\pi$
$42$	0	0
$43$	8.70233	1.32709	0.663547	−	0.748135i	$-0.269050\pi$
$43$	8.70233	1.32709	0.663547	+	0.748135i	$0.269050\pi$
$44$	0	0
$45$	3.46791	0.516966
$46$	0	0
$47$	−0.573978	−0.0837233	−0.0418616	−	0.999123i	$-0.513329\pi$
$47$	−0.573978	−0.0837233	−0.0418616	+	0.999123i	$0.513329\pi$
$48$	0	0
$49$	−4.65270	−0.664672
$50$	0	0
$51$	2.53209	0.354563
$52$	0	0
$53$	2.94356	0.404329	0.202165	−	0.979352i	$-0.435202\pi$
$53$	2.94356	0.404329	0.202165	+	0.979352i	$0.435202\pi$
$54$	0	0
$55$	−1.59627	−0.215241
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.93582	0.512400	0.256200	−	0.966624i	$-0.417529\pi$
$59$	3.93582	0.512400	0.256200	+	0.966624i	$0.417529\pi$
$60$	0	0
$61$	−4.51754	−0.578412	−0.289206	−	0.957267i	$-0.593391\pi$
$61$	−4.51754	−0.578412	−0.289206	+	0.957267i	$0.593391\pi$
$62$	0	0
$63$	3.94356	0.496842
$64$	0	0
$65$	−3.66044	−0.454022
$66$	0	0
$67$	3.88713	0.474888	0.237444	−	0.971401i	$-0.423690\pi$
$67$	3.88713	0.474888	0.237444	+	0.971401i	$0.423690\pi$
$68$	0	0
$69$	3.30541	0.397924
$70$	0	0
$71$	6.93582	0.823131	0.411565	−	0.911380i	$-0.364982\pi$
$71$	6.93582	0.823131	0.411565	+	0.911380i	$0.364982\pi$
$72$	0	0
$73$	6.12836	0.717270	0.358635	−	0.933478i	$-0.383242\pi$
$73$	6.12836	0.717270	0.358635	+	0.933478i	$0.383242\pi$
$74$	0	0
$75$	−2.07873	−0.240031
$76$	0	0
$77$	−1.81521	−0.206862
$78$	0	0
$79$	−9.80840	−1.10353	−0.551766	−	0.833999i	$-0.686046\pi$
$79$	−9.80840	−1.10353	−0.551766	+	0.833999i	$0.686046\pi$
$80$	0	0
$81$	5.34730	0.594144
$82$	0	0
$83$	−12.3182	−1.35210	−0.676049	−	0.736857i	$-0.736309\pi$
$83$	−12.3182	−1.35210	−0.676049	+	0.736857i	$0.736309\pi$
$84$	0	0
$85$	−5.22668	−0.566913
$86$	0	0
$87$	−3.03684	−0.325583
$88$	0	0
$89$	−2.42602	−0.257158	−0.128579	−	0.991699i	$-0.541042\pi$
$89$	−2.42602	−0.257158	−0.128579	+	0.991699i	$0.541042\pi$
$90$	0	0
$91$	−4.16250	−0.436349
$92$	0	0
$93$	2.50475	0.259730
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.36959	0.748268	0.374134	−	0.927375i	$-0.377940\pi$
$97$	7.36959	0.748268	0.374134	+	0.927375i	$0.377940\pi$
$98$	0	0
$99$	−3.04963	−0.306499
$100$	0	0
$101$	2.17024	0.215947	0.107974	−	0.994154i	$-0.465564\pi$
$101$	2.17024	0.215947	0.107974	+	0.994154i	$0.465564\pi$
$102$	0	0
$103$	12.4757	1.22926	0.614631	−	0.788815i	$-0.289305\pi$
$103$	12.4757	1.22926	0.614631	+	0.788815i	$0.289305\pi$
$104$	0	0
$105$	1.34730	0.131483
$106$	0	0
$107$	6.68004	0.645784	0.322892	−	0.946436i	$-0.395345\pi$
$107$	6.68004	0.645784	0.322892	+	0.946436i	$0.395345\pi$
$108$	0	0
$109$	9.45336	0.905468	0.452734	−	0.891646i	$-0.350449\pi$
$109$	9.45336	0.905468	0.452734	+	0.891646i	$0.350449\pi$
$110$	0	0
$111$	−2.68004	−0.254379
$112$	0	0
$113$	1.31046	0.123278	0.0616388	−	0.998099i	$-0.480367\pi$
$113$	1.31046	0.123278	0.0616388	+	0.998099i	$0.480367\pi$
$114$	0	0
$115$	−6.82295	−0.636243
$116$	0	0
$117$	−6.99319	−0.646521
$118$	0	0
$119$	−5.94356	−0.544846
$120$	0	0
$121$	−9.59627	−0.872388
$122$	0	0
$123$	−6.51754	−0.587667
$124$	0	0
$125$	11.0273	0.986315
$126$	0	0
$127$	14.5030	1.28693	0.643466	−	0.765474i	$-0.277496\pi$
$127$	14.5030	1.28693	0.643466	+	0.765474i	$0.277496\pi$
$128$	0	0
$129$	5.68004	0.500100
$130$	0	0
$131$	19.8084	1.73067	0.865334	−	0.501196i	$-0.167106\pi$
$131$	19.8084	1.73067	0.865334	+	0.501196i	$0.167106\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.90167	0.421869
$136$	0	0
$137$	10.2044	0.871820	0.435910	−	0.899990i	$-0.356427\pi$
$137$	10.2044	0.871820	0.435910	+	0.899990i	$0.356427\pi$
$138$	0	0
$139$	1.66044	0.140837	0.0704185	−	0.997518i	$-0.477567\pi$
$139$	1.66044	0.140837	0.0704185	+	0.997518i	$0.477567\pi$
$140$	0	0
$141$	−0.374638	−0.0315502
$142$	0	0
$143$	3.21894	0.269181
$144$	0	0
$145$	6.26857	0.520576
$146$	0	0
$147$	−3.03684	−0.250474
$148$	0	0
$149$	11.2071	0.918120	0.459060	−	0.888405i	$-0.348186\pi$
$149$	11.2071	0.918120	0.459060	+	0.888405i	$0.348186\pi$
$150$	0	0
$151$	11.0419	0.898576	0.449288	−	0.893387i	$-0.351678\pi$
$151$	11.0419	0.898576	0.449288	+	0.893387i	$0.351678\pi$
$152$	0	0
$153$	−9.98545	−0.807276
$154$	0	0
$155$	−5.17024	−0.415284
$156$	0	0
$157$	10.9932	0.877352	0.438676	−	0.898645i	$-0.355448\pi$
$157$	10.9932	0.877352	0.438676	+	0.898645i	$0.355448\pi$
$158$	0	0
$159$	1.92127	0.152367
$160$	0	0
$161$	−7.75877	−0.611477
$162$	0	0
$163$	6.33275	0.496019	0.248010	−	0.968758i	$-0.420224\pi$
$163$	6.33275	0.496019	0.248010	+	0.968758i	$0.420224\pi$
$164$	0	0
$165$	−1.04189	−0.0811110
$166$	0	0
$167$	13.7784	1.06620	0.533101	−	0.846051i	$-0.321027\pi$
$167$	13.7784	1.06620	0.533101	+	0.846051i	$0.321027\pi$
$168$	0	0
$169$	−5.61856	−0.432197
$170$	0	0
$171$	0	0
$172$	0	0
$173$	25.2472	1.91951	0.959755	−	0.280838i	$-0.0906124\pi$
$173$	25.2472	1.91951	0.959755	+	0.280838i	$0.0906124\pi$
$174$	0	0
$175$	4.87939	0.368847
$176$	0	0
$177$	2.56893	0.193092
$178$	0	0
$179$	−5.83069	−0.435806	−0.217903	−	0.975970i	$-0.569922\pi$
$179$	−5.83069	−0.435806	−0.217903	+	0.975970i	$0.569922\pi$
$180$	0	0
$181$	13.5621	1.00806	0.504032	−	0.863685i	$-0.331849\pi$
$181$	13.5621	1.00806	0.504032	+	0.863685i	$0.331849\pi$
$182$	0	0
$183$	−2.94862	−0.217968
$184$	0	0
$185$	5.53209	0.406727
$186$	0	0
$187$	4.59627	0.336112
$188$	0	0
$189$	5.57398	0.405447
$190$	0	0
$191$	10.2841	0.744128	0.372064	−	0.928207i	$-0.378650\pi$
$191$	10.2841	0.744128	0.372064	+	0.928207i	$0.378650\pi$
$192$	0	0
$193$	13.8007	0.993393	0.496697	−	0.867924i	$-0.334546\pi$
$193$	13.8007	0.993393	0.496697	+	0.867924i	$0.334546\pi$
$194$	0	0
$195$	−2.38919	−0.171093
$196$	0	0
$197$	−7.94087	−0.565764	−0.282882	−	0.959155i	$-0.591290\pi$
$197$	−7.94087	−0.565764	−0.282882	+	0.959155i	$0.591290\pi$
$198$	0	0
$199$	−27.0351	−1.91647	−0.958233	−	0.285988i	$-0.907678\pi$
$199$	−27.0351	−1.91647	−0.958233	+	0.285988i	$0.907678\pi$
$200$	0	0
$201$	2.53714	0.178956
$202$	0	0
$203$	7.12836	0.500312
$204$	0	0
$205$	13.4534	0.939624
$206$	0	0
$207$	−13.0351	−0.906001
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.07192	0.555694	0.277847	−	0.960625i	$-0.410379\pi$
$211$	8.07192	0.555694	0.277847	+	0.960625i	$0.410379\pi$
$212$	0	0
$213$	4.52704	0.310187
$214$	0	0
$215$	−11.7246	−0.799613
$216$	0	0
$217$	−5.87939	−0.399119
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	10.5398	0.708986
$222$	0	0
$223$	15.4757	1.03633	0.518163	−	0.855282i	$-0.326616\pi$
$223$	15.4757	1.03633	0.518163	+	0.855282i	$0.326616\pi$
$224$	0	0
$225$	8.19759	0.546506
$226$	0	0
$227$	−9.87258	−0.655266	−0.327633	−	0.944805i	$-0.606251\pi$
$227$	−9.87258	−0.655266	−0.327633	+	0.944805i	$0.606251\pi$
$228$	0	0
$229$	20.1189	1.32949	0.664746	−	0.747070i	$-0.268540\pi$
$229$	20.1189	1.32949	0.664746	+	0.747070i	$0.268540\pi$
$230$	0	0
$231$	−1.18479	−0.0779536
$232$	0	0
$233$	3.53478	0.231571	0.115785	−	0.993274i	$-0.463061\pi$
$233$	3.53478	0.231571	0.115785	+	0.993274i	$0.463061\pi$
$234$	0	0
$235$	0.773318	0.0504457
$236$	0	0
$237$	−6.40198	−0.415853
$238$	0	0
$239$	−11.9736	−0.774507	−0.387254	−	0.921973i	$-0.626576\pi$
$239$	−11.9736	−0.774507	−0.387254	+	0.921973i	$0.626576\pi$
$240$	0	0
$241$	12.9017	0.831070	0.415535	−	0.909577i	$-0.363594\pi$
$241$	12.9017	0.831070	0.415535	+	0.909577i	$0.363594\pi$
$242$	0	0
$243$	14.4047	0.924060
$244$	0	0
$245$	6.26857	0.400484
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−8.04013	−0.509523
$250$	0	0
$251$	−14.3628	−0.906571	−0.453285	−	0.891366i	$-0.649748\pi$
$251$	−14.3628	−0.906571	−0.453285	+	0.891366i	$0.649748\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	0	0
$255$	−3.41147	−0.213635
$256$	0	0
$257$	4.97771	0.310501	0.155251	−	0.987875i	$-0.450382\pi$
$257$	4.97771	0.310501	0.155251	+	0.987875i	$0.450382\pi$
$258$	0	0
$259$	6.29086	0.390895
$260$	0	0
$261$	11.9760	0.741293
$262$	0	0
$263$	24.0428	1.48254	0.741272	−	0.671205i	$-0.234223\pi$
$263$	24.0428	1.48254	0.741272	+	0.671205i	$0.234223\pi$
$264$	0	0
$265$	−3.96585	−0.243620
$266$	0	0
$267$	−1.58347	−0.0969070
$268$	0	0
$269$	−13.1111	−0.799399	−0.399700	−	0.916646i	$-0.630886\pi$
$269$	−13.1111	−0.799399	−0.399700	+	0.916646i	$0.630886\pi$
$270$	0	0
$271$	26.5699	1.61400	0.807002	−	0.590549i	$-0.201089\pi$
$271$	26.5699	1.61400	0.807002	+	0.590549i	$0.201089\pi$
$272$	0	0
$273$	−2.71688	−0.164433
$274$	0	0
$275$	−3.77332	−0.227540
$276$	0	0
$277$	−16.5107	−0.992034	−0.496017	−	0.868313i	$-0.665205\pi$
$277$	−16.5107	−0.992034	−0.496017	+	0.868313i	$0.665205\pi$
$278$	0	0
$279$	−9.87763	−0.591358
$280$	0	0
$281$	−19.3901	−1.15672	−0.578359	−	0.815783i	$-0.696307\pi$
$281$	−19.3901	−1.15672	−0.578359	+	0.815783i	$0.696307\pi$
$282$	0	0
$283$	11.3105	0.672337	0.336169	−	0.941802i	$-0.390869\pi$
$283$	11.3105	0.672337	0.336169	+	0.941802i	$0.390869\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.2986	0.903048
$288$	0	0
$289$	−1.95037	−0.114728
$290$	0	0
$291$	4.81016	0.281976
$292$	0	0
$293$	3.89899	0.227781	0.113891	−	0.993493i	$-0.463669\pi$
$293$	3.89899	0.227781	0.113891	+	0.993493i	$0.463669\pi$
$294$	0	0
$295$	−5.30272	−0.308736
$296$	0	0
$297$	−4.31046	−0.250118
$298$	0	0
$299$	13.7588	0.795690
$300$	0	0
$301$	−13.3327	−0.768487
$302$	0	0
$303$	1.41653	0.0813773
$304$	0	0
$305$	6.08647	0.348510
$306$	0	0
$307$	−23.1753	−1.32268	−0.661342	−	0.750084i	$-0.730013\pi$
$307$	−23.1753	−1.32268	−0.661342	+	0.750084i	$0.730013\pi$
$308$	0	0
$309$	8.14290	0.463234
$310$	0	0
$311$	3.46110	0.196261	0.0981306	−	0.995174i	$-0.468714\pi$
$311$	3.46110	0.196261	0.0981306	+	0.995174i	$0.468714\pi$
$312$	0	0
$313$	−22.8898	−1.29381	−0.646904	−	0.762571i	$-0.723937\pi$
$313$	−22.8898	−1.29381	−0.646904	+	0.762571i	$0.723937\pi$
$314$	0	0
$315$	−5.31315	−0.299362
$316$	0	0
$317$	26.1206	1.46708	0.733540	−	0.679646i	$-0.237867\pi$
$317$	26.1206	1.46708	0.733540	+	0.679646i	$0.237867\pi$
$318$	0	0
$319$	−5.51249	−0.308640
$320$	0	0
$321$	4.36009	0.243356
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−8.65270	−0.479966
$326$	0	0
$327$	6.17024	0.341215
$328$	0	0
$329$	0.879385	0.0484821
$330$	0	0
$331$	−19.0446	−1.04678	−0.523392	−	0.852092i	$-0.675334\pi$
$331$	−19.0446	−1.04678	−0.523392	+	0.852092i	$0.675334\pi$
$332$	0	0
$333$	10.5689	0.579174
$334$	0	0
$335$	−5.23711	−0.286134
$336$	0	0
$337$	1.70140	0.0926812	0.0463406	−	0.998926i	$-0.485244\pi$
$337$	1.70140	0.0926812	0.0463406	+	0.998926i	$0.485244\pi$
$338$	0	0
$339$	0.855342	0.0464558
$340$	0	0
$341$	4.54664	0.246214
$342$	0	0
$343$	17.8530	0.963970
$344$	0	0
$345$	−4.45336	−0.239761
$346$	0	0
$347$	4.90167	0.263136	0.131568	−	0.991307i	$-0.457999\pi$
$347$	4.90167	0.263136	0.131568	+	0.991307i	$0.457999\pi$
$348$	0	0
$349$	−28.1293	−1.50573	−0.752863	−	0.658177i	$-0.771328\pi$
$349$	−28.1293	−1.50573	−0.752863	+	0.658177i	$0.771328\pi$
$350$	0	0
$351$	−9.88444	−0.527592
$352$	0	0
$353$	−8.31996	−0.442827	−0.221413	−	0.975180i	$-0.571067\pi$
$353$	−8.31996	−0.442827	−0.221413	+	0.975180i	$0.571067\pi$
$354$	0	0
$355$	−9.34461	−0.495960
$356$	0	0
$357$	−3.87939	−0.205319
$358$	0	0
$359$	−24.9290	−1.31570	−0.657852	−	0.753148i	$-0.728535\pi$
$359$	−24.9290	−1.31570	−0.657852	+	0.753148i	$0.728535\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−6.26352	−0.328749
$364$	0	0
$365$	−8.25671	−0.432176
$366$	0	0
$367$	2.58584	0.134980	0.0674898	−	0.997720i	$-0.478501\pi$
$367$	2.58584	0.134980	0.0674898	+	0.997720i	$0.478501\pi$
$368$	0	0
$369$	25.7023	1.33801
$370$	0	0
$371$	−4.50980	−0.234137
$372$	0	0
$373$	−23.3833	−1.21074	−0.605371	−	0.795943i	$-0.706975\pi$
$373$	−23.3833	−1.21074	−0.605371	+	0.795943i	$0.706975\pi$
$374$	0	0
$375$	7.19759	0.371682
$376$	0	0
$377$	−12.6408	−0.651037
$378$	0	0
$379$	−25.4388	−1.30670	−0.653352	−	0.757054i	$-0.726638\pi$
$379$	−25.4388	−1.30670	−0.653352	+	0.757054i	$0.726638\pi$
$380$	0	0
$381$	9.46616	0.484966
$382$	0	0
$383$	27.4807	1.40420	0.702099	−	0.712079i	$-0.252246\pi$
$383$	27.4807	1.40420	0.702099	+	0.712079i	$0.252246\pi$
$384$	0	0
$385$	2.44562	0.124640
$386$	0	0
$387$	−22.3996	−1.13864
$388$	0	0
$389$	−3.34224	−0.169458	−0.0847292	−	0.996404i	$-0.527003\pi$
$389$	−3.34224	−0.169458	−0.0847292	+	0.996404i	$0.527003\pi$
$390$	0	0
$391$	19.6459	0.993536
$392$	0	0
$393$	12.9290	0.652183
$394$	0	0
$395$	13.2148	0.664910
$396$	0	0
$397$	−13.1233	−0.658640	−0.329320	−	0.944218i	$-0.606819\pi$
$397$	−13.1233	−0.658640	−0.329320	+	0.944218i	$0.606819\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	17.1138	0.854623	0.427311	−	0.904105i	$-0.359461\pi$
$401$	17.1138	0.854623	0.427311	+	0.904105i	$0.359461\pi$
$402$	0	0
$403$	10.4260	0.519357
$404$	0	0
$405$	−7.20439	−0.357989
$406$	0	0
$407$	−4.86484	−0.241141
$408$	0	0
$409$	−8.79797	−0.435032	−0.217516	−	0.976057i	$-0.569795\pi$
$409$	−8.79797	−0.435032	−0.217516	+	0.976057i	$0.569795\pi$
$410$	0	0
$411$	6.66044	0.328535
$412$	0	0
$413$	−6.03003	−0.296718
$414$	0	0
$415$	16.5963	0.814679
$416$	0	0
$417$	1.08378	0.0530728
$418$	0	0
$419$	−6.84018	−0.334165	−0.167082	−	0.985943i	$-0.553435\pi$
$419$	−6.84018	−0.334165	−0.167082	+	0.985943i	$0.553435\pi$
$420$	0	0
$421$	−4.82295	−0.235056	−0.117528	−	0.993070i	$-0.537497\pi$
$421$	−4.82295	−0.235056	−0.117528	+	0.993070i	$0.537497\pi$
$422$	0	0
$423$	1.47741	0.0718340
$424$	0	0
$425$	−12.3550	−0.599307
$426$	0	0
$427$	6.92127	0.334944
$428$	0	0
$429$	2.10101	0.101438
$430$	0	0
$431$	−1.30365	−0.0627947	−0.0313974	−	0.999507i	$-0.509996\pi$
$431$	−1.30365	−0.0627947	−0.0313974	+	0.999507i	$0.509996\pi$
$432$	0	0
$433$	19.8239	0.952675	0.476337	−	0.879263i	$-0.341964\pi$
$433$	19.8239	0.952675	0.476337	+	0.879263i	$0.341964\pi$
$434$	0	0
$435$	4.09152	0.196173
$436$	0	0
$437$	0	0
$438$	0	0
$439$	34.5672	1.64980	0.824901	−	0.565278i	$-0.191231\pi$
$439$	34.5672	1.64980	0.824901	+	0.565278i	$0.191231\pi$
$440$	0	0
$441$	11.9760	0.570284
$442$	0	0
$443$	−17.0101	−0.808174	−0.404087	−	0.914720i	$-0.632411\pi$
$443$	−17.0101	−0.808174	−0.404087	+	0.914720i	$0.632411\pi$
$444$	0	0
$445$	3.26857	0.154945
$446$	0	0
$447$	7.31490	0.345983
$448$	0	0
$449$	37.4097	1.76547	0.882737	−	0.469868i	$-0.155698\pi$
$449$	37.4097	1.76547	0.882737	+	0.469868i	$0.155698\pi$
$450$	0	0
$451$	−11.8307	−0.557085
$452$	0	0
$453$	7.20708	0.338618
$454$	0	0
$455$	5.60813	0.262913
$456$	0	0
$457$	9.11112	0.426200	0.213100	−	0.977030i	$-0.431644\pi$
$457$	9.11112	0.426200	0.213100	+	0.977030i	$0.431644\pi$
$458$	0	0
$459$	−14.1138	−0.658776
$460$	0	0
$461$	−24.4483	−1.13867	−0.569336	−	0.822105i	$-0.692800\pi$
$461$	−24.4483	−1.13867	−0.569336	+	0.822105i	$0.692800\pi$
$462$	0	0
$463$	0.250725	0.0116522	0.00582609	−	0.999983i	$-0.498145\pi$
$463$	0.250725	0.0116522	0.00582609	+	0.999983i	$0.498145\pi$
$464$	0	0
$465$	−3.37464	−0.156495
$466$	0	0
$467$	−15.3618	−0.710861	−0.355431	−	0.934703i	$-0.615666\pi$
$467$	−15.3618	−0.710861	−0.355431	+	0.934703i	$0.615666\pi$
$468$	0	0
$469$	−5.95542	−0.274996
$470$	0	0
$471$	7.17530	0.330620
$472$	0	0
$473$	10.3105	0.474075
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.57667	−0.346912
$478$	0	0
$479$	0.719246	0.0328632	0.0164316	−	0.999865i	$-0.494769\pi$
$479$	0.719246	0.0328632	0.0164316	+	0.999865i	$0.494769\pi$
$480$	0	0
$481$	−11.1557	−0.508656
$482$	0	0
$483$	−5.06418	−0.230428
$484$	0	0
$485$	−9.92902	−0.450853
$486$	0	0
$487$	−11.7469	−0.532303	−0.266152	−	0.963931i	$-0.585752\pi$
$487$	−11.7469	−0.532303	−0.266152	+	0.963931i	$0.585752\pi$
$488$	0	0
$489$	4.13341	0.186919
$490$	0	0
$491$	−0.0888306	−0.00400887	−0.00200443	−	0.999998i	$-0.500638\pi$
$491$	−0.0888306	−0.00400887	−0.00200443	+	0.999998i	$0.500638\pi$
$492$	0	0
$493$	−18.0496	−0.812914
$494$	0	0
$495$	4.10876	0.184675
$496$	0	0
$497$	−10.6263	−0.476655
$498$	0	0
$499$	−14.6905	−0.657636	−0.328818	−	0.944393i	$-0.606650\pi$
$499$	−14.6905	−0.657636	−0.328818	+	0.944393i	$0.606650\pi$
$500$	0	0
$501$	8.99319	0.401786
$502$	0	0
$503$	4.90404	0.218660	0.109330	−	0.994005i	$-0.465129\pi$
$503$	4.90404	0.218660	0.109330	+	0.994005i	$0.465129\pi$
$504$	0	0
$505$	−2.92396	−0.130115
$506$	0	0
$507$	−3.66725	−0.162868
$508$	0	0
$509$	6.41384	0.284288	0.142144	−	0.989846i	$-0.454600\pi$
$509$	6.41384	0.284288	0.142144	+	0.989846i	$0.454600\pi$
$510$	0	0
$511$	−9.38919	−0.415353
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−16.8084	−0.740667
$516$	0	0
$517$	−0.680045	−0.0299083
$518$	0	0
$519$	16.4789	0.723346
$520$	0	0
$521$	−35.8135	−1.56902	−0.784508	−	0.620119i	$-0.787084\pi$
$521$	−35.8135	−1.56902	−0.784508	+	0.620119i	$0.787084\pi$
$522$	0	0
$523$	−38.7725	−1.69540	−0.847701	−	0.530474i	$-0.822014\pi$
$523$	−38.7725	−1.69540	−0.847701	+	0.530474i	$0.822014\pi$
$524$	0	0
$525$	3.18479	0.138996
$526$	0	0
$527$	14.8871	0.648493
$528$	0	0
$529$	2.64590	0.115039
$530$	0	0
$531$	−10.1307	−0.439636
$532$	0	0
$533$	−27.1293	−1.17510
$534$	0	0
$535$	−9.00000	−0.389104
$536$	0	0
$537$	−3.80571	−0.164229
$538$	0	0
$539$	−5.51249	−0.237440
$540$	0	0
$541$	9.49020	0.408016	0.204008	−	0.978969i	$-0.434603\pi$
$541$	9.49020	0.408016	0.204008	+	0.978969i	$0.434603\pi$
$542$	0	0
$543$	8.85204	0.379878
$544$	0	0
$545$	−12.7365	−0.545571
$546$	0	0
$547$	−14.2121	−0.607667	−0.303833	−	0.952725i	$-0.598267\pi$
$547$	−14.2121	−0.607667	−0.303833	+	0.952725i	$0.598267\pi$
$548$	0	0
$549$	11.6281	0.496273
$550$	0	0
$551$	0	0
$552$	0	0
$553$	15.0273	0.639028
$554$	0	0
$555$	3.61081	0.153271
$556$	0	0
$557$	22.5398	0.955043	0.477522	−	0.878620i	$-0.341535\pi$
$557$	22.5398	0.955043	0.477522	+	0.878620i	$0.341535\pi$
$558$	0	0
$559$	23.6432	1.00000
$560$	0	0
$561$	3.00000	0.126660
$562$	0	0
$563$	42.9718	1.81105	0.905524	−	0.424296i	$-0.139478\pi$
$563$	42.9718	1.81105	0.905524	+	0.424296i	$0.139478\pi$
$564$	0	0
$565$	−1.76558	−0.0742784
$566$	0	0
$567$	−8.19253	−0.344054
$568$	0	0
$569$	7.42696	0.311354	0.155677	−	0.987808i	$-0.450244\pi$
$569$	7.42696	0.311354	0.155677	+	0.987808i	$0.450244\pi$
$570$	0	0
$571$	−4.04458	−0.169260	−0.0846301	−	0.996412i	$-0.526971\pi$
$571$	−4.04458	−0.169260	−0.0846301	+	0.996412i	$0.526971\pi$
$572$	0	0
$573$	6.71244	0.280416
$574$	0	0
$575$	−16.1284	−0.672599
$576$	0	0
$577$	3.23442	0.134651	0.0673254	−	0.997731i	$-0.478553\pi$
$577$	3.23442	0.134651	0.0673254	+	0.997731i	$0.478553\pi$
$578$	0	0
$579$	9.00774	0.374349
$580$	0	0
$581$	18.8726	0.782966
$582$	0	0
$583$	3.48751	0.144438
$584$	0	0
$585$	9.42190	0.389548
$586$	0	0
$587$	40.8084	1.68434	0.842171	−	0.539210i	$-0.181277\pi$
$587$	40.8084	1.68434	0.842171	+	0.539210i	$0.181277\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−5.18304	−0.213202
$592$	0	0
$593$	−11.0642	−0.454351	−0.227176	−	0.973854i	$-0.572949\pi$
$593$	−11.0642	−0.454351	−0.227176	+	0.973854i	$0.572949\pi$
$594$	0	0
$595$	8.00774	0.328285
$596$	0	0
$597$	−17.6459	−0.722198
$598$	0	0
$599$	44.5577	1.82058	0.910289	−	0.413974i	$-0.135860\pi$
$599$	44.5577	1.82058	0.910289	+	0.413974i	$0.135860\pi$
$600$	0	0
$601$	−4.99907	−0.203916	−0.101958	−	0.994789i	$-0.532511\pi$
$601$	−4.99907	−0.203916	−0.101958	+	0.994789i	$0.532511\pi$
$602$	0	0
$603$	−10.0054	−0.407450
$604$	0	0
$605$	12.9290	0.525639
$606$	0	0
$607$	31.1881	1.26589	0.632943	−	0.774199i	$-0.281847\pi$
$607$	31.1881	1.26589	0.632943	+	0.774199i	$0.281847\pi$
$608$	0	0
$609$	4.65270	0.188537
$610$	0	0
$611$	−1.55943	−0.0630878
$612$	0	0
$613$	16.3696	0.661161	0.330581	−	0.943778i	$-0.392755\pi$
$613$	16.3696	0.661161	0.330581	+	0.943778i	$0.392755\pi$
$614$	0	0
$615$	8.78106	0.354086
$616$	0	0
$617$	16.0583	0.646483	0.323241	−	0.946317i	$-0.395227\pi$
$617$	16.0583	0.646483	0.323241	+	0.946317i	$0.395227\pi$
$618$	0	0
$619$	−23.8425	−0.958313	−0.479156	−	0.877730i	$-0.659057\pi$
$619$	−23.8425	−0.958313	−0.479156	+	0.877730i	$0.659057\pi$
$620$	0	0
$621$	−18.4243	−0.739340
$622$	0	0
$623$	3.71688	0.148914
$624$	0	0
$625$	1.06687	0.0426746
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−15.9290	−0.635131
$630$	0	0
$631$	−21.4730	−0.854825	−0.427413	−	0.904057i	$-0.640575\pi$
$631$	−21.4730	−0.854825	−0.427413	+	0.904057i	$0.640575\pi$
$632$	0	0
$633$	5.26857	0.209407
$634$	0	0
$635$	−19.5398	−0.775414
$636$	0	0
$637$	−12.6408	−0.500848
$638$	0	0
$639$	−17.8527	−0.706240
$640$	0	0
$641$	12.7537	0.503742	0.251871	−	0.967761i	$-0.418954\pi$
$641$	12.7537	0.503742	0.251871	+	0.967761i	$0.418954\pi$
$642$	0	0
$643$	−28.6081	−1.12819	−0.564097	−	0.825708i	$-0.690776\pi$
$643$	−28.6081	−1.12819	−0.564097	+	0.825708i	$0.690776\pi$
$644$	0	0
$645$	−7.65270	−0.301325
$646$	0	0
$647$	−16.7128	−0.657046	−0.328523	−	0.944496i	$-0.606551\pi$
$647$	−16.7128	−0.657046	−0.328523	+	0.944496i	$0.606551\pi$
$648$	0	0
$649$	4.66313	0.183044
$650$	0	0
$651$	−3.83750	−0.150403
$652$	0	0
$653$	27.0000	1.05659	0.528296	−	0.849060i	$-0.322831\pi$
$653$	27.0000	1.05659	0.528296	+	0.849060i	$0.322831\pi$
$654$	0	0
$655$	−26.6878	−1.04278
$656$	0	0
$657$	−15.7743	−0.615412
$658$	0	0
$659$	43.9009	1.71013	0.855067	−	0.518517i	$-0.173516\pi$
$659$	43.9009	1.71013	0.855067	+	0.518517i	$0.173516\pi$
$660$	0	0
$661$	−10.7561	−0.418363	−0.209182	−	0.977877i	$-0.567080\pi$
$661$	−10.7561	−0.418363	−0.209182	+	0.977877i	$0.567080\pi$
$662$	0	0
$663$	6.87939	0.267173
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.5621	−0.912329
$668$	0	0
$669$	10.1010	0.390528
$670$	0	0
$671$	−5.35235	−0.206625
$672$	0	0
$673$	4.65776	0.179543	0.0897717	−	0.995962i	$-0.471386\pi$
$673$	4.65776	0.179543	0.0897717	+	0.995962i	$0.471386\pi$
$674$	0	0
$675$	11.5868	0.445975
$676$	0	0
$677$	3.26857	0.125621	0.0628107	−	0.998025i	$-0.479994\pi$
$677$	3.26857	0.125621	0.0628107	+	0.998025i	$0.479994\pi$
$678$	0	0
$679$	−11.2909	−0.433303
$680$	0	0
$681$	−6.44387	−0.246930
$682$	0	0
$683$	−6.21894	−0.237961	−0.118981	−	0.992897i	$-0.537963\pi$
$683$	−6.21894	−0.237961	−0.118981	+	0.992897i	$0.537963\pi$
$684$	0	0
$685$	−13.7483	−0.525297
$686$	0	0
$687$	13.1317	0.501004
$688$	0	0
$689$	7.99731	0.304673
$690$	0	0
$691$	−22.2175	−0.845194	−0.422597	−	0.906318i	$-0.638881\pi$
$691$	−22.2175	−0.845194	−0.422597	+	0.906318i	$0.638881\pi$
$692$	0	0
$693$	4.67230	0.177486
$694$	0	0
$695$	−2.23711	−0.0848584
$696$	0	0
$697$	−38.7374	−1.46728
$698$	0	0
$699$	2.30716	0.0872649
$700$	0	0
$701$	−27.7725	−1.04895	−0.524476	−	0.851425i	$-0.675739\pi$
$701$	−27.7725	−1.04895	−0.524476	+	0.851425i	$0.675739\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0.504748	0.0190099
$706$	0	0
$707$	−3.32501	−0.125050
$708$	0	0
$709$	−6.10782	−0.229384	−0.114692	−	0.993401i	$-0.536588\pi$
$709$	−6.10782	−0.229384	−0.114692	+	0.993401i	$0.536588\pi$
$710$	0	0
$711$	25.2466	0.946822
$712$	0	0
$713$	19.4338	0.727800
$714$	0	0
$715$	−4.33687	−0.162190
$716$	0	0
$717$	−7.81521	−0.291864
$718$	0	0
$719$	−38.7238	−1.44415	−0.722077	−	0.691813i	$-0.756812\pi$
$719$	−38.7238	−1.44415	−0.722077	+	0.691813i	$0.756812\pi$
$720$	0	0
$721$	−19.1138	−0.711835
$722$	0	0
$723$	8.42097	0.313179
$724$	0	0
$725$	14.8179	0.550323
$726$	0	0
$727$	11.0779	0.410857	0.205428	−	0.978672i	$-0.434141\pi$
$727$	11.0779	0.410857	0.205428	+	0.978672i	$0.434141\pi$
$728$	0	0
$729$	−6.63991	−0.245923
$730$	0	0
$731$	33.7597	1.24865
$732$	0	0
$733$	−15.8075	−0.583862	−0.291931	−	0.956439i	$-0.594298\pi$
$733$	−15.8075	−0.583862	−0.291931	+	0.956439i	$0.594298\pi$
$734$	0	0
$735$	4.09152	0.150918
$736$	0	0
$737$	4.60544	0.169643
$738$	0	0
$739$	−1.54933	−0.0569928	−0.0284964	−	0.999594i	$-0.509072\pi$
$739$	−1.54933	−0.0569928	−0.0284964	+	0.999594i	$0.509072\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−38.1634	−1.40008	−0.700040	−	0.714103i	$-0.746835\pi$
$743$	−38.1634	−1.40008	−0.700040	+	0.714103i	$0.746835\pi$
$744$	0	0
$745$	−15.0993	−0.553194
$746$	0	0
$747$	31.7068	1.16009
$748$	0	0
$749$	−10.2344	−0.373958
$750$	0	0
$751$	25.3482	0.924970	0.462485	−	0.886627i	$-0.346958\pi$
$751$	25.3482	0.924970	0.462485	+	0.886627i	$0.346958\pi$
$752$	0	0
$753$	−9.37464	−0.341631
$754$	0	0
$755$	−14.8767	−0.541418
$756$	0	0
$757$	42.3705	1.53998	0.769991	−	0.638054i	$-0.220260\pi$
$757$	42.3705	1.53998	0.769991	+	0.638054i	$0.220260\pi$
$758$	0	0
$759$	3.91622	0.142150
$760$	0	0
$761$	−2.85710	−0.103570	−0.0517848	−	0.998658i	$-0.516491\pi$
$761$	−2.85710	−0.103570	−0.0517848	+	0.998658i	$0.516491\pi$
$762$	0	0
$763$	−14.4834	−0.524334
$764$	0	0
$765$	13.4534	0.486407
$766$	0	0
$767$	10.6932	0.386108
$768$	0	0
$769$	19.1206	0.689507	0.344754	−	0.938693i	$-0.387963\pi$
$769$	19.1206	0.689507	0.344754	+	0.938693i	$0.387963\pi$
$770$	0	0
$771$	3.24897	0.117009
$772$	0	0
$773$	2.51485	0.0904530	0.0452265	−	0.998977i	$-0.485599\pi$
$773$	2.51485	0.0904530	0.0452265	+	0.998977i	$0.485599\pi$
$774$	0	0
$775$	−12.2216	−0.439014
$776$	0	0
$777$	4.10607	0.147304
$778$	0	0
$779$	0	0
$780$	0	0
$781$	8.21751	0.294046
$782$	0	0
$783$	16.9273	0.604931
$784$	0	0
$785$	−14.8111	−0.528630
$786$	0	0
$787$	−2.72605	−0.0971733	−0.0485866	−	0.998819i	$-0.515472\pi$
$787$	−2.72605	−0.0971733	−0.0485866	+	0.998819i	$0.515472\pi$
$788$	0	0
$789$	15.6928	0.558680
$790$	0	0
$791$	−2.00774	−0.0713870
$792$	0	0
$793$	−12.2736	−0.435849
$794$	0	0
$795$	−2.58853	−0.0918056
$796$	0	0
$797$	22.0327	0.780439	0.390219	−	0.920722i	$-0.372399\pi$
$797$	22.0327	0.780439	0.390219	+	0.920722i	$0.372399\pi$
$798$	0	0
$799$	−2.22668	−0.0787743
$800$	0	0
$801$	6.24453	0.220640
$802$	0	0
$803$	7.26083	0.256229
$804$	0	0
$805$	10.4534	0.368433
$806$	0	0
$807$	−8.55768	−0.301244
$808$	0	0
$809$	54.7205	1.92387	0.961935	−	0.273278i	$-0.0881077\pi$
$809$	54.7205	1.92387	0.961935	+	0.273278i	$0.0881077\pi$
$810$	0	0
$811$	2.31046	0.0811312	0.0405656	−	0.999177i	$-0.487084\pi$
$811$	2.31046	0.0811312	0.0405656	+	0.999177i	$0.487084\pi$
$812$	0	0
$813$	17.3422	0.608219
$814$	0	0
$815$	−8.53209	−0.298866
$816$	0	0
$817$	0	0
$818$	0	0
$819$	10.7142	0.374384
$820$	0	0
$821$	1.11112	0.0387783	0.0193892	−	0.999812i	$-0.493828\pi$
$821$	1.11112	0.0387783	0.0193892	+	0.999812i	$0.493828\pi$
$822$	0	0
$823$	−20.6477	−0.719732	−0.359866	−	0.933004i	$-0.617178\pi$
$823$	−20.6477	−0.719732	−0.359866	+	0.933004i	$0.617178\pi$
$824$	0	0
$825$	−2.46286	−0.0857457
$826$	0	0
$827$	−36.3054	−1.26246	−0.631231	−	0.775595i	$-0.717450\pi$
$827$	−36.3054	−1.26246	−0.631231	+	0.775595i	$0.717450\pi$
$828$	0	0
$829$	−7.14971	−0.248320	−0.124160	−	0.992262i	$-0.539624\pi$
$829$	−7.14971	−0.248320	−0.124160	+	0.992262i	$0.539624\pi$
$830$	0	0
$831$	−10.7766	−0.373837
$832$	0	0
$833$	−18.0496	−0.625383
$834$	0	0
$835$	−18.5635	−0.642418
$836$	0	0
$837$	−13.9614	−0.482577
$838$	0	0
$839$	34.6067	1.19476	0.597378	−	0.801960i	$-0.296209\pi$
$839$	34.6067	1.19476	0.597378	+	0.801960i	$0.296209\pi$
$840$	0	0
$841$	−7.35235	−0.253529
$842$	0	0
$843$	−12.6560	−0.435896
$844$	0	0
$845$	7.56986	0.260411
$846$	0	0
$847$	14.7023	0.505178
$848$	0	0
$849$	7.38238	0.253363
$850$	0	0
$851$	−20.7939	−0.712804
$852$	0	0
$853$	33.2508	1.13849	0.569243	−	0.822169i	$-0.307236\pi$
$853$	33.2508	1.13849	0.569243	+	0.822169i	$0.307236\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.88619	−0.132750	−0.0663749	−	0.997795i	$-0.521143\pi$
$857$	−3.88619	−0.132750	−0.0663749	+	0.997795i	$0.521143\pi$
$858$	0	0
$859$	−1.65776	−0.0565619	−0.0282810	−	0.999600i	$-0.509003\pi$
$859$	−1.65776	−0.0565619	−0.0282810	+	0.999600i	$0.509003\pi$
$860$	0	0
$861$	9.98545	0.340303
$862$	0	0
$863$	52.7187	1.79457	0.897284	−	0.441455i	$-0.145537\pi$
$863$	52.7187	1.79457	0.897284	+	0.441455i	$0.145537\pi$
$864$	0	0
$865$	−34.0155	−1.15656
$866$	0	0
$867$	−1.27301	−0.0432338
$868$	0	0
$869$	−11.6209	−0.394213
$870$	0	0
$871$	10.5609	0.357841
$872$	0	0
$873$	−18.9691	−0.642008
$874$	0	0
$875$	−16.8949	−0.571151
$876$	0	0
$877$	−21.1898	−0.715530	−0.357765	−	0.933812i	$-0.616461\pi$
$877$	−21.1898	−0.715530	−0.357765	+	0.933812i	$0.616461\pi$
$878$	0	0
$879$	2.54488	0.0858367
$880$	0	0
$881$	32.1010	1.08151	0.540755	−	0.841180i	$-0.318138\pi$
$881$	32.1010	1.08151	0.540755	+	0.841180i	$0.318138\pi$
$882$	0	0
$883$	47.2968	1.59167	0.795833	−	0.605516i	$-0.207033\pi$
$883$	47.2968	1.59167	0.795833	+	0.605516i	$0.207033\pi$
$884$	0	0
$885$	−3.46110	−0.116344
$886$	0	0
$887$	−10.5631	−0.354673	−0.177336	−	0.984150i	$-0.556748\pi$
$887$	−10.5631	−0.354673	−0.177336	+	0.984150i	$0.556748\pi$
$888$	0	0
$889$	−22.2199	−0.745231
$890$	0	0
$891$	6.33544	0.212245
$892$	0	0
$893$	0	0
$894$	0	0
$895$	7.85567	0.262586
$896$	0	0
$897$	8.98040	0.299847
$898$	0	0
$899$	−17.8547	−0.595489
$900$	0	0
$901$	11.4192	0.380429
$902$	0	0
$903$	−8.70233	−0.289596
$904$	0	0
$905$	−18.2722	−0.607388
$906$	0	0
$907$	−42.9205	−1.42515	−0.712575	−	0.701596i	$-0.752471\pi$
$907$	−42.9205	−1.42515	−0.712575	+	0.701596i	$0.752471\pi$
$908$	0	0
$909$	−5.58616	−0.185281
$910$	0	0
$911$	55.1411	1.82691	0.913454	−	0.406942i	$-0.133405\pi$
$911$	55.1411	1.82691	0.913454	+	0.406942i	$0.133405\pi$
$912$	0	0
$913$	−14.5945	−0.483008
$914$	0	0
$915$	3.97266	0.131332
$916$	0	0
$917$	−30.3482	−1.00219
$918$	0	0
$919$	24.5577	0.810083	0.405041	−	0.914298i	$-0.367257\pi$
$919$	24.5577	0.810083	0.405041	+	0.914298i	$0.367257\pi$
$920$	0	0
$921$	−15.1266	−0.498438
$922$	0	0
$923$	18.8438	0.620251
$924$	0	0
$925$	13.0770	0.429968
$926$	0	0
$927$	−32.1121	−1.05470
$928$	0	0
$929$	−22.2772	−0.730893	−0.365446	−	0.930832i	$-0.619084\pi$
$929$	−22.2772	−0.730893	−0.365446	+	0.930832i	$0.619084\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	2.25908	0.0739588
$934$	0	0
$935$	−6.19253	−0.202517
$936$	0	0
$937$	−9.55169	−0.312040	−0.156020	−	0.987754i	$-0.549866\pi$
$937$	−9.55169	−0.312040	−0.156020	+	0.987754i	$0.549866\pi$
$938$	0	0
$939$	−14.9403	−0.487557
$940$	0	0
$941$	−55.7256	−1.81660	−0.908301	−	0.418318i	$-0.862620\pi$
$941$	−55.7256	−1.81660	−0.908301	+	0.418318i	$0.862620\pi$
$942$	0	0
$943$	−50.5681	−1.64672
$944$	0	0
$945$	−7.50980	−0.244294
$946$	0	0
$947$	27.0428	0.878774	0.439387	−	0.898298i	$-0.355196\pi$
$947$	27.0428	0.878774	0.439387	+	0.898298i	$0.355196\pi$
$948$	0	0
$949$	16.6500	0.540482
$950$	0	0
$951$	17.0490	0.552852
$952$	0	0
$953$	23.1310	0.749288	0.374644	−	0.927169i	$-0.377765\pi$
$953$	23.1310	0.749288	0.374644	+	0.927169i	$0.377765\pi$
$954$	0	0
$955$	−13.8557	−0.448359
$956$	0	0
$957$	−3.59802	−0.116308
$958$	0	0
$959$	−15.6340	−0.504849
$960$	0	0
$961$	−16.2736	−0.524956
$962$	0	0
$963$	−17.1943	−0.554078
$964$	0	0
$965$	−18.5936	−0.598548
$966$	0	0
$967$	−39.0351	−1.25528	−0.627642	−	0.778502i	$-0.715980\pi$
$967$	−39.0351	−1.25528	−0.627642	+	0.778502i	$0.715980\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.2026	1.32226	0.661128	−	0.750273i	$-0.270078\pi$
$971$	41.2026	1.32226	0.661128	+	0.750273i	$0.270078\pi$
$972$	0	0
$973$	−2.54395	−0.0815552
$974$	0	0
$975$	−5.64765	−0.180870
$976$	0	0
$977$	22.4938	0.719641	0.359821	−	0.933022i	$-0.382838\pi$
$977$	22.4938	0.719641	0.359821	+	0.933022i	$0.382838\pi$
$978$	0	0
$979$	−2.87433	−0.0918641
$980$	0	0
$981$	−24.3327	−0.776885
$982$	0	0
$983$	44.5461	1.42080	0.710401	−	0.703798i	$-0.248514\pi$
$983$	44.5461	1.42080	0.710401	+	0.703798i	$0.248514\pi$
$984$	0	0
$985$	10.6987	0.340889
$986$	0	0
$987$	0.573978	0.0182699
$988$	0	0
$989$	44.0702	1.40135
$990$	0	0
$991$	45.3296	1.43994	0.719971	−	0.694005i	$-0.244155\pi$
$991$	45.3296	1.43994	0.719971	+	0.694005i	$0.244155\pi$
$992$	0	0
$993$	−12.4305	−0.394469
$994$	0	0
$995$	36.4243	1.15473
$996$	0	0
$997$	−10.4911	−0.332258	−0.166129	−	0.986104i	$-0.553127\pi$
$997$	−10.4911	−0.332258	−0.166129	+	0.986104i	$0.553127\pi$
$998$	0	0
$999$	14.9385	0.472634

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5776.2.a.br.1.2		3
4.3	odd	2		361.2.a.g.1.1		3
12.11	even	2		3249.2.a.z.1.3		3
19.4	even	9		304.2.u.b.225.1		6
19.5	even	9		304.2.u.b.177.1		6
19.18	odd	2		5776.2.a.bi.1.2		3
20.19	odd	2		9025.2.a.bd.1.3		3
76.3	even	18		361.2.e.b.28.1		6
76.7	odd	6		361.2.c.i.68.3		6
76.11	odd	6		361.2.c.i.292.3		6
76.15	even	18		361.2.e.h.54.1		6
76.23	odd	18		19.2.e.a.16.1	yes	6
76.27	even	6		361.2.c.h.292.1		6
76.31	even	6		361.2.c.h.68.1		6
76.35	odd	18		361.2.e.f.28.1		6
76.43	odd	18		19.2.e.a.6.1	✓	6
76.47	odd	18		361.2.e.g.62.1		6
76.51	even	18		361.2.e.b.245.1		6
76.55	odd	18		361.2.e.g.99.1		6
76.59	even	18		361.2.e.a.99.1		6
76.63	odd	18		361.2.e.f.245.1		6
76.67	even	18		361.2.e.a.62.1		6
76.71	even	18		361.2.e.h.234.1		6
76.75	even	2		361.2.a.h.1.3		3
228.23	even	18		171.2.u.c.73.1		6
228.119	even	18		171.2.u.c.82.1		6
228.227	odd	2		3249.2.a.s.1.1		3
380.23	even	36		475.2.u.a.149.1		12
380.43	even	36		475.2.u.a.424.2		12
380.99	odd	18		475.2.l.a.301.1		6
380.119	odd	18		475.2.l.a.101.1		6
380.327	even	36		475.2.u.a.149.2		12
380.347	even	36		475.2.u.a.424.1		12
380.379	even	2		9025.2.a.x.1.1		3
532.23	odd	18		931.2.v.b.263.1		6
532.195	even	18		931.2.w.a.785.1		6
532.251	even	18		931.2.w.a.491.1		6
532.271	even	18		931.2.x.b.557.1		6
532.327	even	18		931.2.v.a.263.1		6
532.347	odd	18		931.2.v.b.177.1		6
532.403	odd	18		931.2.x.a.814.1		6
532.423	even	18		931.2.v.a.177.1		6
532.479	even	18		931.2.x.b.814.1		6
532.499	odd	18		931.2.x.a.557.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.e.a.6.1	✓	6	76.43	odd	18
19.2.e.a.16.1	yes	6	76.23	odd	18
171.2.u.c.73.1		6	228.23	even	18
171.2.u.c.82.1		6	228.119	even	18
304.2.u.b.177.1		6	19.5	even	9
304.2.u.b.225.1		6	19.4	even	9
361.2.a.g.1.1		3	4.3	odd	2
361.2.a.h.1.3		3	76.75	even	2
361.2.c.h.68.1		6	76.31	even	6
361.2.c.h.292.1		6	76.27	even	6
361.2.c.i.68.3		6	76.7	odd	6
361.2.c.i.292.3		6	76.11	odd	6
361.2.e.a.62.1		6	76.67	even	18
361.2.e.a.99.1		6	76.59	even	18
361.2.e.b.28.1		6	76.3	even	18
361.2.e.b.245.1		6	76.51	even	18
361.2.e.f.28.1		6	76.35	odd	18
361.2.e.f.245.1		6	76.63	odd	18
361.2.e.g.62.1		6	76.47	odd	18
361.2.e.g.99.1		6	76.55	odd	18
361.2.e.h.54.1		6	76.15	even	18
361.2.e.h.234.1		6	76.71	even	18
475.2.l.a.101.1		6	380.119	odd	18
475.2.l.a.301.1		6	380.99	odd	18
475.2.u.a.149.1		12	380.23	even	36
475.2.u.a.149.2		12	380.327	even	36
475.2.u.a.424.1		12	380.347	even	36
475.2.u.a.424.2		12	380.43	even	36
931.2.v.a.177.1		6	532.423	even	18
931.2.v.a.263.1		6	532.327	even	18
931.2.v.b.177.1		6	532.347	odd	18
931.2.v.b.263.1		6	532.23	odd	18
931.2.w.a.491.1		6	532.251	even	18
931.2.w.a.785.1		6	532.195	even	18
931.2.x.a.557.1		6	532.499	odd	18
931.2.x.a.814.1		6	532.403	odd	18
931.2.x.b.557.1		6	532.271	even	18
931.2.x.b.814.1		6	532.479	even	18
3249.2.a.s.1.1		3	228.227	odd	2
3249.2.a.z.1.3		3	12.11	even	2
5776.2.a.bi.1.2		3	19.18	odd	2
5776.2.a.br.1.2		3	1.1	even	1	trivial
9025.2.a.x.1.1		3	380.379	even	2
9025.2.a.bd.1.3		3	20.19	odd	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:	\( N \)	\(=\)	\( 5776 = 2^{4} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5776.a (trivial)

\(f(q)\)	\(=\)	\(q+0.652704 q^{3} -1.34730 q^{5} -1.53209 q^{7} -2.57398 q^{9} +O(q^{10})\)	\(q+0.652704 q^{3} -1.34730 q^{5} -1.53209 q^{7} -2.57398 q^{9} +1.18479 q^{11} +2.71688 q^{13} -0.879385 q^{15} +3.87939 q^{17} -1.00000 q^{21} +5.06418 q^{23} -3.18479 q^{25} -3.63816 q^{27} -4.65270 q^{29} +3.83750 q^{31} +0.773318 q^{33} +2.06418 q^{35} -4.10607 q^{37} +1.77332 q^{39} -9.98545 q^{41} +8.70233 q^{43} +3.46791 q^{45} -0.573978 q^{47} -4.65270 q^{49} +2.53209 q^{51} +2.94356 q^{53} -1.59627 q^{55} +3.93582 q^{59} -4.51754 q^{61} +3.94356 q^{63} -3.66044 q^{65} +3.88713 q^{67} +3.30541 q^{69} +6.93582 q^{71} +6.12836 q^{73} -2.07873 q^{75} -1.81521 q^{77} -9.80840 q^{79} +5.34730 q^{81} -12.3182 q^{83} -5.22668 q^{85} -3.03684 q^{87} -2.42602 q^{89} -4.16250 q^{91} +2.50475 q^{93} +7.36959 q^{97} -3.04963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5	\( 3 q + 3 q^{3} - 3 q^{5} + 3 q^{15} + 6 q^{17} - 3 q^{21} + 6 q^{23} - 6 q^{25} + 6 q^{27} - 15 q^{29} + 9 q^{31} + 9 q^{33} - 3 q^{35} + 12 q^{39} - 12 q^{41} + 15 q^{45} + 6 q^{47} - 15 q^{49} + 3 q^{51} - 6 q^{53} + 9 q^{55} + 21 q^{59} + 9 q^{61} - 3 q^{63} + 12 q^{65} - 18 q^{67} + 12 q^{69} + 30 q^{71} - 15 q^{75} - 9 q^{77} + 9 q^{79} + 15 q^{81} - 9 q^{85} - 21 q^{87} - 15 q^{89} - 15 q^{91} + 24 q^{93} + 15 q^{97} + 18 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 + 3 * q^15 + 6 * q^17 - 3 * q^21 + 6 * q^23 - 6 * q^25 + 6 * q^27 - 15 * q^29 + 9 * q^31 + 9 * q^33 - 3 * q^35 + 12 * q^39 - 12 * q^41 + 15 * q^45 + 6 * q^47 - 15 * q^49 + 3 * q^51 - 6 * q^53 + 9 * q^55 + 21 * q^59 + 9 * q^61 - 3 * q^63 + 12 * q^65 - 18 * q^67 + 12 * q^69 + 30 * q^71 - 15 * q^75 - 9 * q^77 + 9 * q^79 + 15 * q^81 - 9 * q^85 - 21 * q^87 - 15 * q^89 - 15 * q^91 + 24 * q^93 + 15 * q^97 + 18 * q^99

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