LMFDB - Embedded newform 4840.2.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.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:	$38.6475945783$
Analytic rank:	$1$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{3} +1.00000 q^{5} +0.414214 q^{7} -2.82843 q^{9} -2.00000 q^{13} -0.414214 q^{15} +5.65685 q^{17} +0.828427 q^{19} -0.171573 q^{21} -7.65685 q^{23} +1.00000 q^{25} +2.41421 q^{27} -2.00000 q^{29} +5.65685 q^{31} +0.414214 q^{35} -4.00000 q^{37} +0.828427 q^{39} -7.48528 q^{41} -6.41421 q^{43} -2.82843 q^{45} +10.4142 q^{47} -6.82843 q^{49} -2.34315 q^{51} +3.65685 q^{53} -0.343146 q^{57} +11.3137 q^{59} -1.00000 q^{61} -1.17157 q^{63} -2.00000 q^{65} -3.58579 q^{67} +3.17157 q^{69} -4.00000 q^{71} -4.00000 q^{73} -0.414214 q^{75} -2.48528 q^{79} +7.48528 q^{81} -9.31371 q^{83} +5.65685 q^{85} +0.828427 q^{87} -8.65685 q^{89} -0.828427 q^{91} -2.34315 q^{93} +0.828427 q^{95} +5.31371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{13} + 2 q^{15} - 4 q^{19} - 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 2 q^{35} - 8 q^{37} - 4 q^{39} + 2 q^{41} - 10 q^{43} + 18 q^{47} - 8 q^{49} - 16 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^13 + 2 * q^15 - 4 * q^19 - 6 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 2 * q^35 - 8 * q^37 - 4 * q^39 + 2 * q^41 - 10 * q^43 + 18 * q^47 - 8 * q^49 - 16 * q^51 - 4 * q^53 - 12 * q^57 - 2 * q^61 - 8 * q^63 - 4 * q^65 - 10 * q^67 + 12 * q^69 - 8 * q^71 - 8 * q^73 + 2 * q^75 + 12 * q^79 - 2 * q^81 + 4 * q^83 - 4 * q^87 - 6 * q^89 + 4 * q^91 - 16 * q^93 - 4 * q^95 - 12 * q^97

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
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.414214	0.156558	0.0782790	−	0.996931i	$-0.475058\pi$
$7$	0.414214	0.156558	0.0782790	+	0.996931i	$0.475058\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−0.414214	−0.106949
$16$	0	0
$17$	5.65685	1.37199	0.685994	−	0.727607i	$-0.259367\pi$
$17$	5.65685	1.37199	0.685994	+	0.727607i	$0.259367\pi$
$18$	0	0
$19$	0.828427	0.190054	0.0950271	−	0.995475i	$-0.469706\pi$
$19$	0.828427	0.190054	0.0950271	+	0.995475i	$0.469706\pi$
$20$	0	0
$21$	−0.171573	−0.0374403
$22$	0	0
$23$	−7.65685	−1.59656	−0.798282	−	0.602284i	$-0.794258\pi$
$23$	−7.65685	−1.59656	−0.798282	+	0.602284i	$0.794258\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.414214	0.0700149
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	0	0
$39$	0.828427	0.132655
$40$	0	0
$41$	−7.48528	−1.16900	−0.584502	−	0.811392i	$-0.698710\pi$
$41$	−7.48528	−1.16900	−0.584502	+	0.811392i	$0.698710\pi$
$42$	0	0
$43$	−6.41421	−0.978158	−0.489079	−	0.872239i	$-0.662667\pi$
$43$	−6.41421	−0.978158	−0.489079	+	0.872239i	$0.662667\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	10.4142	1.51907	0.759535	−	0.650467i	$-0.225427\pi$
$47$	10.4142	1.51907	0.759535	+	0.650467i	$0.225427\pi$
$48$	0	0
$49$	−6.82843	−0.975490
$50$	0	0
$51$	−2.34315	−0.328106
$52$	0	0
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.343146	−0.0454508
$58$	0	0
$59$	11.3137	1.47292	0.736460	−	0.676481i	$-0.236496\pi$
$59$	11.3137	1.47292	0.736460	+	0.676481i	$0.236496\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	0	0
$63$	−1.17157	−0.147604
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−3.58579	−0.438074	−0.219037	−	0.975717i	$-0.570291\pi$
$67$	−3.58579	−0.438074	−0.219037	+	0.975717i	$0.570291\pi$
$68$	0	0
$69$	3.17157	0.381813
$70$	0	0
$71$	−4.00000	−0.474713	−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000	−0.474713	−0.237356	+	0.971423i	$0.576281\pi$
$72$	0	0
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	0	0
$75$	−0.414214	−0.0478293
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	−9.31371	−1.02231	−0.511156	−	0.859488i	$-0.670783\pi$
$83$	−9.31371	−1.02231	−0.511156	+	0.859488i	$0.670783\pi$
$84$	0	0
$85$	5.65685	0.613572
$86$	0	0
$87$	0.828427	0.0888167
$88$	0	0
$89$	−8.65685	−0.917625	−0.458812	−	0.888533i	$-0.651725\pi$
$89$	−8.65685	−0.917625	−0.458812	+	0.888533i	$0.651725\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	0	0
$93$	−2.34315	−0.242973
$94$	0	0
$95$	0.828427	0.0849948
$96$	0	0
$97$	5.31371	0.539525	0.269763	−	0.962927i	$-0.413055\pi$
$97$	5.31371	0.539525	0.269763	+	0.962927i	$0.413055\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.8284	−1.57499	−0.787494	−	0.616323i	$-0.788622\pi$
$101$	−15.8284	−1.57499	−0.787494	+	0.616323i	$0.788622\pi$
$102$	0	0
$103$	−9.31371	−0.917707	−0.458853	−	0.888512i	$-0.651740\pi$
$103$	−9.31371	−0.917707	−0.458853	+	0.888512i	$0.651740\pi$
$104$	0	0
$105$	−0.171573	−0.0167438
$106$	0	0
$107$	−6.75736	−0.653259	−0.326629	−	0.945153i	$-0.605913\pi$
$107$	−6.75736	−0.653259	−0.326629	+	0.945153i	$0.605913\pi$
$108$	0	0
$109$	−5.34315	−0.511781	−0.255890	−	0.966706i	$-0.582369\pi$
$109$	−5.34315	−0.511781	−0.255890	+	0.966706i	$0.582369\pi$
$110$	0	0
$111$	1.65685	0.157262
$112$	0	0
$113$	−5.31371	−0.499872	−0.249936	−	0.968262i	$-0.580410\pi$
$113$	−5.31371	−0.499872	−0.249936	+	0.968262i	$0.580410\pi$
$114$	0	0
$115$	−7.65685	−0.714005
$116$	0	0
$117$	5.65685	0.522976
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	0	0
$122$	0	0
$123$	3.10051	0.279563
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.58579	0.140716	0.0703579	−	0.997522i	$-0.477586\pi$
$127$	1.58579	0.140716	0.0703579	+	0.997522i	$0.477586\pi$
$128$	0	0
$129$	2.65685	0.233923
$130$	0	0
$131$	−14.4853	−1.26558	−0.632792	−	0.774321i	$-0.718091\pi$
$131$	−14.4853	−1.26558	−0.632792	+	0.774321i	$0.718091\pi$
$132$	0	0
$133$	0.343146	0.0297545
$134$	0	0
$135$	2.41421	0.207782
$136$	0	0
$137$	−9.31371	−0.795724	−0.397862	−	0.917445i	$-0.630248\pi$
$137$	−9.31371	−0.795724	−0.397862	+	0.917445i	$0.630248\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	−4.31371	−0.363280
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	2.82843	0.233285
$148$	0	0
$149$	17.0000	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$149$	17.0000	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$150$	0	0
$151$	5.65685	0.460348	0.230174	−	0.973149i	$-0.426070\pi$
$151$	5.65685	0.460348	0.230174	+	0.973149i	$0.426070\pi$
$152$	0	0
$153$	−16.0000	−1.29352
$154$	0	0
$155$	5.65685	0.454369
$156$	0	0
$157$	6.34315	0.506238	0.253119	−	0.967435i	$-0.418544\pi$
$157$	6.34315	0.506238	0.253119	+	0.967435i	$0.418544\pi$
$158$	0	0
$159$	−1.51472	−0.120125
$160$	0	0
$161$	−3.17157	−0.249955
$162$	0	0
$163$	10.4142	0.815704	0.407852	−	0.913048i	$-0.366278\pi$
$163$	10.4142	0.815704	0.407852	+	0.913048i	$0.366278\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3.58579	−0.277476	−0.138738	−	0.990329i	$-0.544305\pi$
$167$	−3.58579	−0.277476	−0.138738	+	0.990329i	$0.544305\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.34315	−0.179185
$172$	0	0
$173$	−7.65685	−0.582140	−0.291070	−	0.956702i	$-0.594011\pi$
$173$	−7.65685	−0.582140	−0.291070	+	0.956702i	$0.594011\pi$
$174$	0	0
$175$	0.414214	0.0313116
$176$	0	0
$177$	−4.68629	−0.352243
$178$	0	0
$179$	21.7990	1.62933	0.814667	−	0.579930i	$-0.196920\pi$
$179$	21.7990	1.62933	0.814667	+	0.579930i	$0.196920\pi$
$180$	0	0
$181$	−20.1716	−1.49934	−0.749671	−	0.661811i	$-0.769788\pi$
$181$	−20.1716	−1.49934	−0.749671	+	0.661811i	$0.769788\pi$
$182$	0	0
$183$	0.414214	0.0306195
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	0	0
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	11.1716	0.808347	0.404173	−	0.914682i	$-0.367559\pi$
$191$	11.1716	0.808347	0.404173	+	0.914682i	$0.367559\pi$
$192$	0	0
$193$	21.3137	1.53419	0.767097	−	0.641531i	$-0.221700\pi$
$193$	21.3137	1.53419	0.767097	+	0.641531i	$0.221700\pi$
$194$	0	0
$195$	0.828427	0.0593249
$196$	0	0
$197$	−26.9706	−1.92157	−0.960787	−	0.277289i	$-0.910564\pi$
$197$	−26.9706	−1.92157	−0.960787	+	0.277289i	$0.910564\pi$
$198$	0	0
$199$	−12.8284	−0.909383	−0.454692	−	0.890649i	$-0.650250\pi$
$199$	−12.8284	−0.909383	−0.454692	+	0.890649i	$0.650250\pi$
$200$	0	0
$201$	1.48528	0.104764
$202$	0	0
$203$	−0.828427	−0.0581442
$204$	0	0
$205$	−7.48528	−0.522795
$206$	0	0
$207$	21.6569	1.50526
$208$	0	0
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	1.65685	0.113526
$214$	0	0
$215$	−6.41421	−0.437446
$216$	0	0
$217$	2.34315	0.159063
$218$	0	0
$219$	1.65685	0.111960
$220$	0	0
$221$	−11.3137	−0.761042
$222$	0	0
$223$	22.0711	1.47799	0.738994	−	0.673712i	$-0.235301\pi$
$223$	22.0711	1.47799	0.738994	+	0.673712i	$0.235301\pi$
$224$	0	0
$225$	−2.82843	−0.188562
$226$	0	0
$227$	−11.9289	−0.791751	−0.395876	−	0.918304i	$-0.629559\pi$
$227$	−11.9289	−0.791751	−0.395876	+	0.918304i	$0.629559\pi$
$228$	0	0
$229$	−5.48528	−0.362478	−0.181239	−	0.983439i	$-0.558011\pi$
$229$	−5.48528	−0.362478	−0.181239	+	0.983439i	$0.558011\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−15.6569	−1.02571	−0.512857	−	0.858474i	$-0.671413\pi$
$233$	−15.6569	−1.02571	−0.512857	+	0.858474i	$0.671413\pi$
$234$	0	0
$235$	10.4142	0.679348
$236$	0	0
$237$	1.02944	0.0668691
$238$	0	0
$239$	−4.14214	−0.267932	−0.133966	−	0.990986i	$-0.542771\pi$
$239$	−4.14214	−0.267932	−0.133966	+	0.990986i	$0.542771\pi$
$240$	0	0
$241$	−21.8284	−1.40609	−0.703046	−	0.711144i	$-0.748177\pi$
$241$	−21.8284	−1.40609	−0.703046	+	0.711144i	$0.748177\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	−6.82843	−0.436252
$246$	0	0
$247$	−1.65685	−0.105423
$248$	0	0
$249$	3.85786	0.244482
$250$	0	0
$251$	23.3137	1.47155	0.735774	−	0.677227i	$-0.236819\pi$
$251$	23.3137	1.47155	0.735774	+	0.677227i	$0.236819\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−2.34315	−0.146733
$256$	0	0
$257$	−20.9706	−1.30811	−0.654054	−	0.756448i	$-0.726933\pi$
$257$	−20.9706	−1.30811	−0.654054	+	0.756448i	$0.726933\pi$
$258$	0	0
$259$	−1.65685	−0.102952
$260$	0	0
$261$	5.65685	0.350150
$262$	0	0
$263$	20.6274	1.27194	0.635971	−	0.771713i	$-0.280600\pi$
$263$	20.6274	1.27194	0.635971	+	0.771713i	$0.280600\pi$
$264$	0	0
$265$	3.65685	0.224639
$266$	0	0
$267$	3.58579	0.219447
$268$	0	0
$269$	6.31371	0.384954	0.192477	−	0.981302i	$-0.438348\pi$
$269$	6.31371	0.384954	0.192477	+	0.981302i	$0.438348\pi$
$270$	0	0
$271$	−11.3137	−0.687259	−0.343629	−	0.939105i	$-0.611656\pi$
$271$	−11.3137	−0.687259	−0.343629	+	0.939105i	$0.611656\pi$
$272$	0	0
$273$	0.343146	0.0207681
$274$	0	0
$275$	0	0
$276$	0	0
$277$	16.6274	0.999045	0.499522	−	0.866301i	$-0.333509\pi$
$277$	16.6274	0.999045	0.499522	+	0.866301i	$0.333509\pi$
$278$	0	0
$279$	−16.0000	−0.957895
$280$	0	0
$281$	−17.3137	−1.03285	−0.516425	−	0.856333i	$-0.672737\pi$
$281$	−17.3137	−1.03285	−0.516425	+	0.856333i	$0.672737\pi$
$282$	0	0
$283$	−22.2132	−1.32044	−0.660219	−	0.751073i	$-0.729537\pi$
$283$	−22.2132	−1.32044	−0.660219	+	0.751073i	$0.729537\pi$
$284$	0	0
$285$	−0.343146	−0.0203262
$286$	0	0
$287$	−3.10051	−0.183017
$288$	0	0
$289$	15.0000	0.882353
$290$	0	0
$291$	−2.20101	−0.129025
$292$	0	0
$293$	−3.31371	−0.193589	−0.0967945	−	0.995304i	$-0.530859\pi$
$293$	−3.31371	−0.193589	−0.0967945	+	0.995304i	$0.530859\pi$
$294$	0	0
$295$	11.3137	0.658710
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.3137	0.885615
$300$	0	0
$301$	−2.65685	−0.153139
$302$	0	0
$303$	6.55635	0.376652
$304$	0	0
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	2.68629	0.153315	0.0766574	−	0.997057i	$-0.475575\pi$
$307$	2.68629	0.153315	0.0766574	+	0.997057i	$0.475575\pi$
$308$	0	0
$309$	3.85786	0.219466
$310$	0	0
$311$	7.17157	0.406663	0.203331	−	0.979110i	$-0.434823\pi$
$311$	7.17157	0.406663	0.203331	+	0.979110i	$0.434823\pi$
$312$	0	0
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	0	0
$315$	−1.17157	−0.0660107
$316$	0	0
$317$	−25.3137	−1.42176	−0.710880	−	0.703314i	$-0.751703\pi$
$317$	−25.3137	−1.42176	−0.710880	+	0.703314i	$0.751703\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	2.79899	0.156224
$322$	0	0
$323$	4.68629	0.260752
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	2.21320	0.122390
$328$	0	0
$329$	4.31371	0.237822
$330$	0	0
$331$	−0.828427	−0.0455345	−0.0227672	−	0.999741i	$-0.507248\pi$
$331$	−0.828427	−0.0455345	−0.0227672	+	0.999741i	$0.507248\pi$
$332$	0	0
$333$	11.3137	0.619987
$334$	0	0
$335$	−3.58579	−0.195912
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	0	0
$339$	2.20101	0.119542
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−5.72792	−0.309279
$344$	0	0
$345$	3.17157	0.170752
$346$	0	0
$347$	35.3848	1.89955	0.949777	−	0.312927i	$-0.101310\pi$
$347$	35.3848	1.89955	0.949777	+	0.312927i	$0.101310\pi$
$348$	0	0
$349$	−17.3137	−0.926782	−0.463391	−	0.886154i	$-0.653367\pi$
$349$	−17.3137	−0.926782	−0.463391	+	0.886154i	$0.653367\pi$
$350$	0	0
$351$	−4.82843	−0.257722
$352$	0	0
$353$	10.6274	0.565640	0.282820	−	0.959173i	$-0.408730\pi$
$353$	10.6274	0.565640	0.282820	+	0.959173i	$0.408730\pi$
$354$	0	0
$355$	−4.00000	−0.212298
$356$	0	0
$357$	−0.970563	−0.0513676
$358$	0	0
$359$	8.82843	0.465947	0.232973	−	0.972483i	$-0.425155\pi$
$359$	8.82843	0.465947	0.232973	+	0.972483i	$0.425155\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.00000	−0.209370
$366$	0	0
$367$	−3.24264	−0.169264	−0.0846322	−	0.996412i	$-0.526972\pi$
$367$	−3.24264	−0.169264	−0.0846322	+	0.996412i	$0.526972\pi$
$368$	0	0
$369$	21.1716	1.10215
$370$	0	0
$371$	1.51472	0.0786403
$372$	0	0
$373$	−12.9706	−0.671590	−0.335795	−	0.941935i	$-0.609005\pi$
$373$	−12.9706	−0.671590	−0.335795	+	0.941935i	$0.609005\pi$
$374$	0	0
$375$	−0.414214	−0.0213899
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	8.14214	0.418233	0.209117	−	0.977891i	$-0.432941\pi$
$379$	8.14214	0.418233	0.209117	+	0.977891i	$0.432941\pi$
$380$	0	0
$381$	−0.656854	−0.0336517
$382$	0	0
$383$	−2.97056	−0.151789	−0.0758943	−	0.997116i	$-0.524181\pi$
$383$	−2.97056	−0.151789	−0.0758943	+	0.997116i	$0.524181\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	18.1421	0.922217
$388$	0	0
$389$	3.68629	0.186902	0.0934512	−	0.995624i	$-0.470210\pi$
$389$	3.68629	0.186902	0.0934512	+	0.995624i	$0.470210\pi$
$390$	0	0
$391$	−43.3137	−2.19047
$392$	0	0
$393$	6.00000	0.302660
$394$	0	0
$395$	−2.48528	−0.125048
$396$	0	0
$397$	−24.0000	−1.20453	−0.602263	−	0.798298i	$-0.705734\pi$
$397$	−24.0000	−1.20453	−0.602263	+	0.798298i	$0.705734\pi$
$398$	0	0
$399$	−0.142136	−0.00711568
$400$	0	0
$401$	−16.3137	−0.814668	−0.407334	−	0.913279i	$-0.633541\pi$
$401$	−16.3137	−0.814668	−0.407334	+	0.913279i	$0.633541\pi$
$402$	0	0
$403$	−11.3137	−0.563576
$404$	0	0
$405$	7.48528	0.371947
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−33.7696	−1.66980	−0.834898	−	0.550404i	$-0.814474\pi$
$409$	−33.7696	−1.66980	−0.834898	+	0.550404i	$0.814474\pi$
$410$	0	0
$411$	3.85786	0.190294
$412$	0	0
$413$	4.68629	0.230597
$414$	0	0
$415$	−9.31371	−0.457192
$416$	0	0
$417$	4.97056	0.243410
$418$	0	0
$419$	−34.4853	−1.68472	−0.842358	−	0.538918i	$-0.818833\pi$
$419$	−34.4853	−1.68472	−0.842358	+	0.538918i	$0.818833\pi$
$420$	0	0
$421$	−30.3137	−1.47740	−0.738700	−	0.674034i	$-0.764560\pi$
$421$	−30.3137	−1.47740	−0.738700	+	0.674034i	$0.764560\pi$
$422$	0	0
$423$	−29.4558	−1.43219
$424$	0	0
$425$	5.65685	0.274398
$426$	0	0
$427$	−0.414214	−0.0200452
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.4853	−1.66110	−0.830549	−	0.556946i	$-0.811973\pi$
$431$	−34.4853	−1.66110	−0.830549	+	0.556946i	$0.811973\pi$
$432$	0	0
$433$	4.97056	0.238870	0.119435	−	0.992842i	$-0.461892\pi$
$433$	4.97056	0.238870	0.119435	+	0.992842i	$0.461892\pi$
$434$	0	0
$435$	0.828427	0.0397200
$436$	0	0
$437$	−6.34315	−0.303434
$438$	0	0
$439$	−32.2843	−1.54084	−0.770422	−	0.637534i	$-0.779954\pi$
$439$	−32.2843	−1.54084	−0.770422	+	0.637534i	$0.779954\pi$
$440$	0	0
$441$	19.3137	0.919700
$442$	0	0
$443$	0.757359	0.0359832	0.0179916	−	0.999838i	$-0.494273\pi$
$443$	0.757359	0.0359832	0.0179916	+	0.999838i	$0.494273\pi$
$444$	0	0
$445$	−8.65685	−0.410374
$446$	0	0
$447$	−7.04163	−0.333058
$448$	0	0
$449$	10.4558	0.493442	0.246721	−	0.969087i	$-0.420647\pi$
$449$	10.4558	0.493442	0.246721	+	0.969087i	$0.420647\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−2.34315	−0.110091
$454$	0	0
$455$	−0.828427	−0.0388373
$456$	0	0
$457$	19.3137	0.903457	0.451729	−	0.892155i	$-0.350808\pi$
$457$	19.3137	0.903457	0.451729	+	0.892155i	$0.350808\pi$
$458$	0	0
$459$	13.6569	0.637447
$460$	0	0
$461$	−19.1421	−0.891538	−0.445769	−	0.895148i	$-0.647070\pi$
$461$	−19.1421	−0.891538	−0.445769	+	0.895148i	$0.647070\pi$
$462$	0	0
$463$	6.75736	0.314041	0.157021	−	0.987595i	$-0.449811\pi$
$463$	6.75736	0.314041	0.157021	+	0.987595i	$0.449811\pi$
$464$	0	0
$465$	−2.34315	−0.108661
$466$	0	0
$467$	11.4437	0.529549	0.264775	−	0.964310i	$-0.414702\pi$
$467$	11.4437	0.529549	0.264775	+	0.964310i	$0.414702\pi$
$468$	0	0
$469$	−1.48528	−0.0685839
$470$	0	0
$471$	−2.62742	−0.121065
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0.828427	0.0380108
$476$	0	0
$477$	−10.3431	−0.473580
$478$	0	0
$479$	−37.7990	−1.72708	−0.863540	−	0.504280i	$-0.831758\pi$
$479$	−37.7990	−1.72708	−0.863540	+	0.504280i	$0.831758\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	1.31371	0.0597758
$484$	0	0
$485$	5.31371	0.241283
$486$	0	0
$487$	41.3137	1.87210	0.936051	−	0.351863i	$-0.114452\pi$
$487$	41.3137	1.87210	0.936051	+	0.351863i	$0.114452\pi$
$488$	0	0
$489$	−4.31371	−0.195073
$490$	0	0
$491$	−34.4853	−1.55630	−0.778149	−	0.628079i	$-0.783841\pi$
$491$	−34.4853	−1.55630	−0.778149	+	0.628079i	$0.783841\pi$
$492$	0	0
$493$	−11.3137	−0.509544
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.65685	−0.0743201
$498$	0	0
$499$	1.79899	0.0805338	0.0402669	−	0.999189i	$-0.487179\pi$
$499$	1.79899	0.0805338	0.0402669	+	0.999189i	$0.487179\pi$
$500$	0	0
$501$	1.48528	0.0663575
$502$	0	0
$503$	27.3848	1.22103	0.610513	−	0.792006i	$-0.290963\pi$
$503$	27.3848	1.22103	0.610513	+	0.792006i	$0.290963\pi$
$504$	0	0
$505$	−15.8284	−0.704356
$506$	0	0
$507$	3.72792	0.165563
$508$	0	0
$509$	−5.48528	−0.243131	−0.121565	−	0.992583i	$-0.538791\pi$
$509$	−5.48528	−0.243131	−0.121565	+	0.992583i	$0.538791\pi$
$510$	0	0
$511$	−1.65685	−0.0732949
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	−9.31371	−0.410411
$516$	0	0
$517$	0	0
$518$	0	0
$519$	3.17157	0.139217
$520$	0	0
$521$	39.9706	1.75114	0.875571	−	0.483089i	$-0.160485\pi$
$521$	39.9706	1.75114	0.875571	+	0.483089i	$0.160485\pi$
$522$	0	0
$523$	−9.31371	−0.407260	−0.203630	−	0.979048i	$-0.565274\pi$
$523$	−9.31371	−0.407260	−0.203630	+	0.979048i	$0.565274\pi$
$524$	0	0
$525$	−0.171573	−0.00748805
$526$	0	0
$527$	32.0000	1.39394
$528$	0	0
$529$	35.6274	1.54902
$530$	0	0
$531$	−32.0000	−1.38868
$532$	0	0
$533$	14.9706	0.648447
$534$	0	0
$535$	−6.75736	−0.292146
$536$	0	0
$537$	−9.02944	−0.389649
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−23.1421	−0.994958	−0.497479	−	0.867476i	$-0.665741\pi$
$541$	−23.1421	−0.994958	−0.497479	+	0.867476i	$0.665741\pi$
$542$	0	0
$543$	8.35534	0.358562
$544$	0	0
$545$	−5.34315	−0.228875
$546$	0	0
$547$	−26.0000	−1.11168	−0.555840	−	0.831289i	$-0.687603\pi$
$547$	−26.0000	−1.11168	−0.555840	+	0.831289i	$0.687603\pi$
$548$	0	0
$549$	2.82843	0.120714
$550$	0	0
$551$	−1.65685	−0.0705844
$552$	0	0
$553$	−1.02944	−0.0437761
$554$	0	0
$555$	1.65685	0.0703295
$556$	0	0
$557$	22.6274	0.958754	0.479377	−	0.877609i	$-0.340863\pi$
$557$	22.6274	0.958754	0.479377	+	0.877609i	$0.340863\pi$
$558$	0	0
$559$	12.8284	0.542585
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.8995	0.712229	0.356114	−	0.934442i	$-0.384101\pi$
$563$	16.8995	0.712229	0.356114	+	0.934442i	$0.384101\pi$
$564$	0	0
$565$	−5.31371	−0.223549
$566$	0	0
$567$	3.10051	0.130209
$568$	0	0
$569$	−34.4558	−1.44446	−0.722232	−	0.691651i	$-0.756884\pi$
$569$	−34.4558	−1.44446	−0.722232	+	0.691651i	$0.756884\pi$
$570$	0	0
$571$	27.4558	1.14899	0.574496	−	0.818508i	$-0.305198\pi$
$571$	27.4558	1.14899	0.574496	+	0.818508i	$0.305198\pi$
$572$	0	0
$573$	−4.62742	−0.193313
$574$	0	0
$575$	−7.65685	−0.319313
$576$	0	0
$577$	33.9411	1.41299	0.706494	−	0.707719i	$-0.250276\pi$
$577$	33.9411	1.41299	0.706494	+	0.707719i	$0.250276\pi$
$578$	0	0
$579$	−8.82843	−0.366897
$580$	0	0
$581$	−3.85786	−0.160051
$582$	0	0
$583$	0	0
$584$	0	0
$585$	5.65685	0.233882
$586$	0	0
$587$	−22.6985	−0.936867	−0.468433	−	0.883499i	$-0.655181\pi$
$587$	−22.6985	−0.936867	−0.468433	+	0.883499i	$0.655181\pi$
$588$	0	0
$589$	4.68629	0.193095
$590$	0	0
$591$	11.1716	0.459537
$592$	0	0
$593$	12.3431	0.506872	0.253436	−	0.967352i	$-0.418439\pi$
$593$	12.3431	0.506872	0.253436	+	0.967352i	$0.418439\pi$
$594$	0	0
$595$	2.34315	0.0960596
$596$	0	0
$597$	5.31371	0.217476
$598$	0	0
$599$	44.1421	1.80360	0.901799	−	0.432155i	$-0.142247\pi$
$599$	44.1421	1.80360	0.901799	+	0.432155i	$0.142247\pi$
$600$	0	0
$601$	27.9411	1.13974	0.569871	−	0.821734i	$-0.306993\pi$
$601$	27.9411	1.13974	0.569871	+	0.821734i	$0.306993\pi$
$602$	0	0
$603$	10.1421	0.413020
$604$	0	0
$605$	0	0
$606$	0	0
$607$	45.5980	1.85076	0.925382	−	0.379035i	$-0.123744\pi$
$607$	45.5980	1.85076	0.925382	+	0.379035i	$0.123744\pi$
$608$	0	0
$609$	0.343146	0.0139050
$610$	0	0
$611$	−20.8284	−0.842628
$612$	0	0
$613$	18.9706	0.766214	0.383107	−	0.923704i	$-0.374854\pi$
$613$	18.9706	0.766214	0.383107	+	0.923704i	$0.374854\pi$
$614$	0	0
$615$	3.10051	0.125024
$616$	0	0
$617$	−40.0000	−1.61034	−0.805170	−	0.593045i	$-0.797926\pi$
$617$	−40.0000	−1.61034	−0.805170	+	0.593045i	$0.797926\pi$
$618$	0	0
$619$	−15.3137	−0.615510	−0.307755	−	0.951466i	$-0.599578\pi$
$619$	−15.3137	−0.615510	−0.307755	+	0.951466i	$0.599578\pi$
$620$	0	0
$621$	−18.4853	−0.741789
$622$	0	0
$623$	−3.58579	−0.143662
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−22.6274	−0.902214
$630$	0	0
$631$	45.6569	1.81757	0.908785	−	0.417264i	$-0.137011\pi$
$631$	45.6569	1.81757	0.908785	+	0.417264i	$0.137011\pi$
$632$	0	0
$633$	−3.31371	−0.131708
$634$	0	0
$635$	1.58579	0.0629300
$636$	0	0
$637$	13.6569	0.541104
$638$	0	0
$639$	11.3137	0.447563
$640$	0	0
$641$	−3.37258	−0.133209	−0.0666045	−	0.997779i	$-0.521217\pi$
$641$	−3.37258	−0.133209	−0.0666045	+	0.997779i	$0.521217\pi$
$642$	0	0
$643$	25.7279	1.01461	0.507305	−	0.861767i	$-0.330642\pi$
$643$	25.7279	1.01461	0.507305	+	0.861767i	$0.330642\pi$
$644$	0	0
$645$	2.65685	0.104614
$646$	0	0
$647$	23.8701	0.938429	0.469214	−	0.883084i	$-0.344537\pi$
$647$	23.8701	0.938429	0.469214	+	0.883084i	$0.344537\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−0.970563	−0.0380394
$652$	0	0
$653$	3.65685	0.143104	0.0715519	−	0.997437i	$-0.477205\pi$
$653$	3.65685	0.143104	0.0715519	+	0.997437i	$0.477205\pi$
$654$	0	0
$655$	−14.4853	−0.565987
$656$	0	0
$657$	11.3137	0.441390
$658$	0	0
$659$	28.8284	1.12300	0.561498	−	0.827478i	$-0.310225\pi$
$659$	28.8284	1.12300	0.561498	+	0.827478i	$0.310225\pi$
$660$	0	0
$661$	46.6569	1.81474	0.907371	−	0.420331i	$-0.138086\pi$
$661$	46.6569	1.81474	0.907371	+	0.420331i	$0.138086\pi$
$662$	0	0
$663$	4.68629	0.182000
$664$	0	0
$665$	0.343146	0.0133066
$666$	0	0
$667$	15.3137	0.592949
$668$	0	0
$669$	−9.14214	−0.353455
$670$	0	0
$671$	0	0
$672$	0	0
$673$	16.9706	0.654167	0.327084	−	0.944995i	$-0.393934\pi$
$673$	16.9706	0.654167	0.327084	+	0.944995i	$0.393934\pi$
$674$	0	0
$675$	2.41421	0.0929231
$676$	0	0
$677$	31.3137	1.20348	0.601742	−	0.798691i	$-0.294474\pi$
$677$	31.3137	1.20348	0.601742	+	0.798691i	$0.294474\pi$
$678$	0	0
$679$	2.20101	0.0844670
$680$	0	0
$681$	4.94113	0.189344
$682$	0	0
$683$	−33.8701	−1.29600	−0.648001	−	0.761640i	$-0.724395\pi$
$683$	−33.8701	−1.29600	−0.648001	+	0.761640i	$0.724395\pi$
$684$	0	0
$685$	−9.31371	−0.355859
$686$	0	0
$687$	2.27208	0.0866852
$688$	0	0
$689$	−7.31371	−0.278630
$690$	0	0
$691$	2.48528	0.0945446	0.0472723	−	0.998882i	$-0.484947\pi$
$691$	2.48528	0.0945446	0.0472723	+	0.998882i	$0.484947\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.0000	−0.455186
$696$	0	0
$697$	−42.3431	−1.60386
$698$	0	0
$699$	6.48528	0.245296
$700$	0	0
$701$	27.9411	1.05532	0.527661	−	0.849455i	$-0.323069\pi$
$701$	27.9411	1.05532	0.527661	+	0.849455i	$0.323069\pi$
$702$	0	0
$703$	−3.31371	−0.124979
$704$	0	0
$705$	−4.31371	−0.162464
$706$	0	0
$707$	−6.55635	−0.246577
$708$	0	0
$709$	−6.85786	−0.257553	−0.128776	−	0.991674i	$-0.541105\pi$
$709$	−6.85786	−0.257553	−0.128776	+	0.991674i	$0.541105\pi$
$710$	0	0
$711$	7.02944	0.263624
$712$	0	0
$713$	−43.3137	−1.62211
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.71573	0.0640751
$718$	0	0
$719$	−36.8284	−1.37347	−0.686734	−	0.726909i	$-0.740956\pi$
$719$	−36.8284	−1.37347	−0.686734	+	0.726909i	$0.740956\pi$
$720$	0	0
$721$	−3.85786	−0.143674
$722$	0	0
$723$	9.04163	0.336262
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	37.9289	1.40671	0.703353	−	0.710841i	$-0.251685\pi$
$727$	37.9289	1.40671	0.703353	+	0.710841i	$0.251685\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−36.2843	−1.34202
$732$	0	0
$733$	16.2843	0.601473	0.300737	−	0.953707i	$-0.402767\pi$
$733$	16.2843	0.601473	0.300737	+	0.953707i	$0.402767\pi$
$734$	0	0
$735$	2.82843	0.104328
$736$	0	0
$737$	0	0
$738$	0	0
$739$	50.4853	1.85713	0.928566	−	0.371168i	$-0.121043\pi$
$739$	50.4853	1.85713	0.928566	+	0.371168i	$0.121043\pi$
$740$	0	0
$741$	0.686292	0.0252115
$742$	0	0
$743$	−29.5858	−1.08540	−0.542699	−	0.839928i	$-0.682597\pi$
$743$	−29.5858	−1.08540	−0.542699	+	0.839928i	$0.682597\pi$
$744$	0	0
$745$	17.0000	0.622832
$746$	0	0
$747$	26.3431	0.963845
$748$	0	0
$749$	−2.79899	−0.102273
$750$	0	0
$751$	−0.828427	−0.0302297	−0.0151149	−	0.999886i	$-0.504811\pi$
$751$	−0.828427	−0.0302297	−0.0151149	+	0.999886i	$0.504811\pi$
$752$	0	0
$753$	−9.65685	−0.351915
$754$	0	0
$755$	5.65685	0.205874
$756$	0	0
$757$	2.97056	0.107967	0.0539835	−	0.998542i	$-0.482808\pi$
$757$	2.97056	0.107967	0.0539835	+	0.998542i	$0.482808\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.00000	−0.217500	−0.108750	−	0.994069i	$-0.534685\pi$
$761$	−6.00000	−0.217500	−0.108750	+	0.994069i	$0.534685\pi$
$762$	0	0
$763$	−2.21320	−0.0801233
$764$	0	0
$765$	−16.0000	−0.578481
$766$	0	0
$767$	−22.6274	−0.817029
$768$	0	0
$769$	18.0000	0.649097	0.324548	−	0.945869i	$-0.394788\pi$
$769$	18.0000	0.649097	0.324548	+	0.945869i	$0.394788\pi$
$770$	0	0
$771$	8.68629	0.312829
$772$	0	0
$773$	4.97056	0.178779	0.0893894	−	0.995997i	$-0.471508\pi$
$773$	4.97056	0.178779	0.0893894	+	0.995997i	$0.471508\pi$
$774$	0	0
$775$	5.65685	0.203200
$776$	0	0
$777$	0.686292	0.0246206
$778$	0	0
$779$	−6.20101	−0.222174
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−4.82843	−0.172554
$784$	0	0
$785$	6.34315	0.226397
$786$	0	0
$787$	42.2132	1.50474	0.752369	−	0.658742i	$-0.228911\pi$
$787$	42.2132	1.50474	0.752369	+	0.658742i	$0.228911\pi$
$788$	0	0
$789$	−8.54416	−0.304180
$790$	0	0
$791$	−2.20101	−0.0782589
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	−1.51472	−0.0537215
$796$	0	0
$797$	45.6569	1.61725	0.808624	−	0.588325i	$-0.200213\pi$
$797$	45.6569	1.61725	0.808624	+	0.588325i	$0.200213\pi$
$798$	0	0
$799$	58.9117	2.08415
$800$	0	0
$801$	24.4853	0.865145
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−3.17157	−0.111783
$806$	0	0
$807$	−2.61522	−0.0920602
$808$	0	0
$809$	20.6274	0.725221	0.362611	−	0.931941i	$-0.381885\pi$
$809$	20.6274	0.725221	0.362611	+	0.931941i	$0.381885\pi$
$810$	0	0
$811$	−5.65685	−0.198639	−0.0993195	−	0.995056i	$-0.531667\pi$
$811$	−5.65685	−0.198639	−0.0993195	+	0.995056i	$0.531667\pi$
$812$	0	0
$813$	4.68629	0.164355
$814$	0	0
$815$	10.4142	0.364794
$816$	0	0
$817$	−5.31371	−0.185903
$818$	0	0
$819$	2.34315	0.0818761
$820$	0	0
$821$	−29.9706	−1.04598	−0.522990	−	0.852339i	$-0.675183\pi$
$821$	−29.9706	−1.04598	−0.522990	+	0.852339i	$0.675183\pi$
$822$	0	0
$823$	−16.0711	−0.560202	−0.280101	−	0.959971i	$-0.590368\pi$
$823$	−16.0711	−0.560202	−0.280101	+	0.959971i	$0.590368\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38.3553	−1.33375	−0.666873	−	0.745171i	$-0.732368\pi$
$827$	−38.3553	−1.33375	−0.666873	+	0.745171i	$0.732368\pi$
$828$	0	0
$829$	−34.9411	−1.21356	−0.606778	−	0.794872i	$-0.707538\pi$
$829$	−34.9411	−1.21356	−0.606778	+	0.794872i	$0.707538\pi$
$830$	0	0
$831$	−6.88730	−0.238918
$832$	0	0
$833$	−38.6274	−1.33836
$834$	0	0
$835$	−3.58579	−0.124091
$836$	0	0
$837$	13.6569	0.472050
$838$	0	0
$839$	2.20101	0.0759873	0.0379937	−	0.999278i	$-0.487903\pi$
$839$	2.20101	0.0759873	0.0379937	+	0.999278i	$0.487903\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	7.17157	0.247002
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	0	0
$848$	0	0
$849$	9.20101	0.315778
$850$	0	0
$851$	30.6274	1.04989
$852$	0	0
$853$	18.3431	0.628057	0.314029	−	0.949413i	$-0.398321\pi$
$853$	18.3431	0.628057	0.314029	+	0.949413i	$0.398321\pi$
$854$	0	0
$855$	−2.34315	−0.0801339
$856$	0	0
$857$	−22.9706	−0.784659	−0.392330	−	0.919825i	$-0.628331\pi$
$857$	−22.9706	−0.784659	−0.392330	+	0.919825i	$0.628331\pi$
$858$	0	0
$859$	−36.1421	−1.23315	−0.616577	−	0.787295i	$-0.711481\pi$
$859$	−36.1421	−1.23315	−0.616577	+	0.787295i	$0.711481\pi$
$860$	0	0
$861$	1.28427	0.0437678
$862$	0	0
$863$	45.8701	1.56143	0.780717	−	0.624884i	$-0.214854\pi$
$863$	45.8701	1.56143	0.780717	+	0.624884i	$0.214854\pi$
$864$	0	0
$865$	−7.65685	−0.260341
$866$	0	0
$867$	−6.21320	−0.211011
$868$	0	0
$869$	0	0
$870$	0	0
$871$	7.17157	0.242999
$872$	0	0
$873$	−15.0294	−0.508669
$874$	0	0
$875$	0.414214	0.0140030
$876$	0	0
$877$	24.6863	0.833597	0.416798	−	0.908999i	$-0.363152\pi$
$877$	24.6863	0.833597	0.416798	+	0.908999i	$0.363152\pi$
$878$	0	0
$879$	1.37258	0.0462961
$880$	0	0
$881$	19.4853	0.656476	0.328238	−	0.944595i	$-0.393545\pi$
$881$	19.4853	0.656476	0.328238	+	0.944595i	$0.393545\pi$
$882$	0	0
$883$	−43.6569	−1.46917	−0.734585	−	0.678517i	$-0.762623\pi$
$883$	−43.6569	−1.46917	−0.734585	+	0.678517i	$0.762623\pi$
$884$	0	0
$885$	−4.68629	−0.157528
$886$	0	0
$887$	37.3848	1.25526	0.627629	−	0.778513i	$-0.284026\pi$
$887$	37.3848	1.25526	0.627629	+	0.778513i	$0.284026\pi$
$888$	0	0
$889$	0.656854	0.0220302
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.62742	0.288706
$894$	0	0
$895$	21.7990	0.728660
$896$	0	0
$897$	−6.34315	−0.211791
$898$	0	0
$899$	−11.3137	−0.377333
$900$	0	0
$901$	20.6863	0.689160
$902$	0	0
$903$	1.10051	0.0366225
$904$	0	0
$905$	−20.1716	−0.670526
$906$	0	0
$907$	−18.6985	−0.620873	−0.310436	−	0.950594i	$-0.600475\pi$
$907$	−18.6985	−0.620873	−0.310436	+	0.950594i	$0.600475\pi$
$908$	0	0
$909$	44.7696	1.48491
$910$	0	0
$911$	−32.8284	−1.08765	−0.543827	−	0.839197i	$-0.683025\pi$
$911$	−32.8284	−1.08765	−0.543827	+	0.839197i	$0.683025\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0.414214	0.0136935
$916$	0	0
$917$	−6.00000	−0.198137
$918$	0	0
$919$	52.4264	1.72939	0.864694	−	0.502299i	$-0.167512\pi$
$919$	52.4264	1.72939	0.864694	+	0.502299i	$0.167512\pi$
$920$	0	0
$921$	−1.11270	−0.0366647
$922$	0	0
$923$	8.00000	0.263323
$924$	0	0
$925$	−4.00000	−0.131519
$926$	0	0
$927$	26.3431	0.865222
$928$	0	0
$929$	34.0000	1.11550	0.557752	−	0.830008i	$-0.311664\pi$
$929$	34.0000	1.11550	0.557752	+	0.830008i	$0.311664\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	0	0
$933$	−2.97056	−0.0972519
$934$	0	0
$935$	0	0
$936$	0	0
$937$	30.0000	0.980057	0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000	0.980057	0.490029	+	0.871706i	$0.336986\pi$
$938$	0	0
$939$	−0.544156	−0.0177579
$940$	0	0
$941$	−46.1716	−1.50515	−0.752575	−	0.658506i	$-0.771189\pi$
$941$	−46.1716	−1.50515	−0.752575	+	0.658506i	$0.771189\pi$
$942$	0	0
$943$	57.3137	1.86639
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	10.0000	0.324956	0.162478	−	0.986712i	$-0.448051\pi$
$947$	10.0000	0.324956	0.162478	+	0.986712i	$0.448051\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	0	0
$951$	10.4853	0.340009
$952$	0	0
$953$	32.2843	1.04579	0.522895	−	0.852397i	$-0.324852\pi$
$953$	32.2843	1.04579	0.522895	+	0.852397i	$0.324852\pi$
$954$	0	0
$955$	11.1716	0.361504
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.85786	−0.124577
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	19.1127	0.615898
$964$	0	0
$965$	21.3137	0.686113
$966$	0	0
$967$	37.5980	1.20907	0.604535	−	0.796579i	$-0.293359\pi$
$967$	37.5980	1.20907	0.604535	+	0.796579i	$0.293359\pi$
$968$	0	0
$969$	−1.94113	−0.0623579
$970$	0	0
$971$	−60.2843	−1.93461	−0.967307	−	0.253608i	$-0.918383\pi$
$971$	−60.2843	−1.93461	−0.967307	+	0.253608i	$0.918383\pi$
$972$	0	0
$973$	−4.97056	−0.159349
$974$	0	0
$975$	0.828427	0.0265309
$976$	0	0
$977$	−42.6274	−1.36377	−0.681886	−	0.731459i	$-0.738840\pi$
$977$	−42.6274	−1.36377	−0.681886	+	0.731459i	$0.738840\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	15.1127	0.482511
$982$	0	0
$983$	−20.0711	−0.640168	−0.320084	−	0.947389i	$-0.603711\pi$
$983$	−20.0711	−0.640168	−0.320084	+	0.947389i	$0.603711\pi$
$984$	0	0
$985$	−26.9706	−0.859354
$986$	0	0
$987$	−1.78680	−0.0568744
$988$	0	0
$989$	49.1127	1.56169
$990$	0	0
$991$	41.5147	1.31876	0.659379	−	0.751810i	$-0.270819\pi$
$991$	41.5147	1.31876	0.659379	+	0.751810i	$0.270819\pi$
$992$	0	0
$993$	0.343146	0.0108894
$994$	0	0
$995$	−12.8284	−0.406688
$996$	0	0
$997$	6.68629	0.211757	0.105878	−	0.994379i	$-0.466235\pi$
$997$	6.68629	0.211757	0.105878	+	0.994379i	$0.466235\pi$
$998$	0	0
$999$	−9.65685	−0.305529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.o.1.1	✓	2
4.3	odd	2		9680.2.a.bk.1.2		2
11.10	odd	2		4840.2.a.p.1.1	yes	2
44.43	even	2		9680.2.a.bj.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4840.2.a.o.1.1	✓	2	1.1	even	1	trivial
4840.2.a.p.1.1	yes	2	11.10	odd	2
9680.2.a.bj.1.2		2	44.43	even	2
9680.2.a.bk.1.2		2	4.3	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:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} +1.00000 q^{5} +0.414214 q^{7} -2.82843 q^{9} -2.00000 q^{13} -0.414214 q^{15} +5.65685 q^{17} +0.828427 q^{19} -0.171573 q^{21} -7.65685 q^{23} +1.00000 q^{25} +2.41421 q^{27} -2.00000 q^{29} +5.65685 q^{31} +0.414214 q^{35} -4.00000 q^{37} +0.828427 q^{39} -7.48528 q^{41} -6.41421 q^{43} -2.82843 q^{45} +10.4142 q^{47} -6.82843 q^{49} -2.34315 q^{51} +3.65685 q^{53} -0.343146 q^{57} +11.3137 q^{59} -1.00000 q^{61} -1.17157 q^{63} -2.00000 q^{65} -3.58579 q^{67} +3.17157 q^{69} -4.00000 q^{71} -4.00000 q^{73} -0.414214 q^{75} -2.48528 q^{79} +7.48528 q^{81} -9.31371 q^{83} +5.65685 q^{85} +0.828427 q^{87} -8.65685 q^{89} -0.828427 q^{91} -2.34315 q^{93} +0.828427 q^{95} +5.31371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} - 4 q^{13} + 2 q^{15} - 4 q^{19} - 6 q^{21} - 4 q^{23} + 2 q^{25} + 2 q^{27} - 4 q^{29} - 2 q^{35} - 8 q^{37} - 4 q^{39} + 2 q^{41} - 10 q^{43} + 18 q^{47} - 8 q^{49} - 16 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 - 4 * q^13 + 2 * q^15 - 4 * q^19 - 6 * q^21 - 4 * q^23 + 2 * q^25 + 2 * q^27 - 4 * q^29 - 2 * q^35 - 8 * q^37 - 4 * q^39 + 2 * q^41 - 10 * q^43 + 18 * q^47 - 8 * q^49 - 16 * q^51 - 4 * q^53 - 12 * q^57 - 2 * q^61 - 8 * q^63 - 4 * q^65 - 10 * q^67 + 12 * q^69 - 8 * q^71 - 8 * q^73 + 2 * q^75 + 12 * q^79 - 2 * q^81 + 4 * q^83 - 4 * q^87 - 6 * q^89 + 4 * q^91 - 16 * q^93 - 4 * q^95 - 12 * q^97

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