LMFDB - Embedded newform 9680.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9680,2,Mod(1,9680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9680, 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("9680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9680 = 2^{4} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9680.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:	$77.2951891566$
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:	no (minimal twist has level 4840)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9680.1

$q$-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	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.414214	−0.156558	−0.0782790	−	0.996931i	$-0.524942\pi$
$7$	−0.414214	−0.156558	−0.0782790	+	0.996931i	$0.524942\pi$
$8$	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	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	0	0
$21$	−0.171573	−0.0374403
$22$	0	0
$23$	7.65685	1.59656	0.798282	−	0.602284i	$-0.205742\pi$
$23$	7.65685	1.59656	0.798282	+	0.602284i	$0.205742\pi$
$24$	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	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	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	$-0.337333\pi$
$43$	6.41421	0.978158	0.489079	+	0.872239i	$0.337333\pi$
$44$	0	0
$45$	−2.82843	−0.421637
$46$	0	0
$47$	−10.4142	−1.51907	−0.759535	−	0.650467i	$-0.774573\pi$
$47$	−10.4142	−1.51907	−0.759535	+	0.650467i	$0.774573\pi$
$48$	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	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	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	$-0.429709\pi$
$67$	3.58579	0.438074	0.219037	+	0.975717i	$0.429709\pi$
$68$	0	0
$69$	3.17157	0.381813
$70$	0	0
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	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	$-0.455351\pi$
$79$	2.48528	0.279616	0.139808	+	0.990179i	$0.455351\pi$
$80$	0	0
$81$	7.48528	0.831698
$82$	0	0
$83$	9.31371	1.02231	0.511156	−	0.859488i	$-0.329217\pi$
$83$	9.31371	1.02231	0.511156	+	0.859488i	$0.329217\pi$
$84$	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	$-0.348260\pi$
$103$	9.31371	0.917707	0.458853	+	0.888512i	$0.348260\pi$
$104$	0	0
$105$	−0.171573	−0.0167438
$106$	0	0
$107$	6.75736	0.653259	0.326629	−	0.945153i	$-0.394087\pi$
$107$	6.75736	0.653259	0.326629	+	0.945153i	$0.394087\pi$
$108$	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	$-0.522414\pi$
$127$	−1.58579	−0.140716	−0.0703579	+	0.997522i	$0.522414\pi$
$128$	0	0
$129$	2.65685	0.233923
$130$	0	0
$131$	14.4853	1.26558	0.632792	−	0.774321i	$-0.281909\pi$
$131$	14.4853	1.26558	0.632792	+	0.774321i	$0.281909\pi$
$132$	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	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	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	$-0.573930\pi$
$151$	−5.65685	−0.460348	−0.230174	+	0.973149i	$0.573930\pi$
$152$	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	$-0.633722\pi$
$163$	−10.4142	−0.815704	−0.407852	+	0.913048i	$0.633722\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.58579	0.277476	0.138738	−	0.990329i	$-0.455695\pi$
$167$	3.58579	0.277476	0.138738	+	0.990329i	$0.455695\pi$
$168$	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	$-0.803080\pi$
$179$	−21.7990	−1.62933	−0.814667	+	0.579930i	$0.803080\pi$
$180$	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	$-0.632441\pi$
$191$	−11.1716	−0.808347	−0.404173	+	0.914682i	$0.632441\pi$
$192$	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	$-0.349750\pi$
$199$	12.8284	0.909383	0.454692	+	0.890649i	$0.349750\pi$
$200$	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	$-0.588801\pi$
$211$	−8.00000	−0.550743	−0.275371	+	0.961338i	$0.588801\pi$
$212$	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	$-0.764699\pi$
$223$	−22.0711	−1.47799	−0.738994	+	0.673712i	$0.764699\pi$
$224$	0	0
$225$	−2.82843	−0.188562
$226$	0	0
$227$	11.9289	0.791751	0.395876	−	0.918304i	$-0.370441\pi$
$227$	11.9289	0.791751	0.395876	+	0.918304i	$0.370441\pi$
$228$	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	$-0.457229\pi$
$239$	4.14214	0.267932	0.133966	+	0.990986i	$0.457229\pi$
$240$	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	$-0.763181\pi$
$251$	−23.3137	−1.47155	−0.735774	+	0.677227i	$0.763181\pi$
$252$	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	$-0.719400\pi$
$263$	−20.6274	−1.27194	−0.635971	+	0.771713i	$0.719400\pi$
$264$	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	$-0.388344\pi$
$271$	11.3137	0.687259	0.343629	+	0.939105i	$0.388344\pi$
$272$	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	$-0.270463\pi$
$283$	22.2132	1.32044	0.660219	+	0.751073i	$0.270463\pi$
$284$	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	$-0.524425\pi$
$307$	−2.68629	−0.153315	−0.0766574	+	0.997057i	$0.524425\pi$
$308$	0	0
$309$	3.85786	0.219466
$310$	0	0
$311$	−7.17157	−0.406663	−0.203331	−	0.979110i	$-0.565177\pi$
$311$	−7.17157	−0.406663	−0.203331	+	0.979110i	$0.565177\pi$
$312$	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	$-0.492752\pi$
$331$	0.828427	0.0455345	0.0227672	+	0.999741i	$0.492752\pi$
$332$	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	$-0.898690\pi$
$347$	−35.3848	−1.89955	−0.949777	+	0.312927i	$0.898690\pi$
$348$	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	$-0.574845\pi$
$359$	−8.82843	−0.465947	−0.232973	+	0.972483i	$0.574845\pi$
$360$	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	$-0.473028\pi$
$367$	3.24264	0.169264	0.0846322	+	0.996412i	$0.473028\pi$
$368$	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	$-0.567059\pi$
$379$	−8.14214	−0.418233	−0.209117	+	0.977891i	$0.567059\pi$
$380$	0	0
$381$	−0.656854	−0.0336517
$382$	0	0
$383$	2.97056	0.151789	0.0758943	−	0.997116i	$-0.475819\pi$
$383$	2.97056	0.151789	0.0758943	+	0.997116i	$0.475819\pi$
$384$	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	$-0.181167\pi$
$419$	34.4853	1.68472	0.842358	+	0.538918i	$0.181167\pi$
$420$	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	$-0.188027\pi$
$431$	34.4853	1.66110	0.830549	+	0.556946i	$0.188027\pi$
$432$	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	$-0.220046\pi$
$439$	32.2843	1.54084	0.770422	+	0.637534i	$0.220046\pi$
$440$	0	0
$441$	19.3137	0.919700
$442$	0	0
$443$	−0.757359	−0.0359832	−0.0179916	−	0.999838i	$-0.505727\pi$
$443$	−0.757359	−0.0359832	−0.0179916	+	0.999838i	$0.505727\pi$
$444$	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	$-0.550189\pi$
$463$	−6.75736	−0.314041	−0.157021	+	0.987595i	$0.550189\pi$
$464$	0	0
$465$	−2.34315	−0.108661
$466$	0	0
$467$	−11.4437	−0.529549	−0.264775	−	0.964310i	$-0.585298\pi$
$467$	−11.4437	−0.529549	−0.264775	+	0.964310i	$0.585298\pi$
$468$	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	$-0.168242\pi$
$479$	37.7990	1.72708	0.863540	+	0.504280i	$0.168242\pi$
$480$	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	$-0.885548\pi$
$487$	−41.3137	−1.87210	−0.936051	+	0.351863i	$0.885548\pi$
$488$	0	0
$489$	−4.31371	−0.195073
$490$	0	0
$491$	34.4853	1.55630	0.778149	−	0.628079i	$-0.216159\pi$
$491$	34.4853	1.55630	0.778149	+	0.628079i	$0.216159\pi$
$492$	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	$-0.512821\pi$
$499$	−1.79899	−0.0805338	−0.0402669	+	0.999189i	$0.512821\pi$
$500$	0	0
$501$	1.48528	0.0663575
$502$	0	0
$503$	−27.3848	−1.22103	−0.610513	−	0.792006i	$-0.709037\pi$
$503$	−27.3848	−1.22103	−0.610513	+	0.792006i	$0.709037\pi$
$504$	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	$-0.434726\pi$
$523$	9.31371	0.407260	0.203630	+	0.979048i	$0.434726\pi$
$524$	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	$-0.312397\pi$
$547$	26.0000	1.11168	0.555840	+	0.831289i	$0.312397\pi$
$548$	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	$-0.615899\pi$
$563$	−16.8995	−0.712229	−0.356114	+	0.934442i	$0.615899\pi$
$564$	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	$-0.694802\pi$
$571$	−27.4558	−1.14899	−0.574496	+	0.818508i	$0.694802\pi$
$572$	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	$-0.344819\pi$
$587$	22.6985	0.936867	0.468433	+	0.883499i	$0.344819\pi$
$588$	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	$-0.857753\pi$
$599$	−44.1421	−1.80360	−0.901799	+	0.432155i	$0.857753\pi$
$600$	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	$-0.876256\pi$
$607$	−45.5980	−1.85076	−0.925382	+	0.379035i	$0.876256\pi$
$608$	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	$-0.400422\pi$
$619$	15.3137	0.615510	0.307755	+	0.951466i	$0.400422\pi$
$620$	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	$-0.862989\pi$
$631$	−45.6569	−1.81757	−0.908785	+	0.417264i	$0.862989\pi$
$632$	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	$-0.669358\pi$
$643$	−25.7279	−1.01461	−0.507305	+	0.861767i	$0.669358\pi$
$644$	0	0
$645$	2.65685	0.104614
$646$	0	0
$647$	−23.8701	−0.938429	−0.469214	−	0.883084i	$-0.655463\pi$
$647$	−23.8701	−0.938429	−0.469214	+	0.883084i	$0.655463\pi$
$648$	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	$-0.689775\pi$
$659$	−28.8284	−1.12300	−0.561498	+	0.827478i	$0.689775\pi$
$660$	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	$-0.275605\pi$
$683$	33.8701	1.29600	0.648001	+	0.761640i	$0.275605\pi$
$684$	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	$-0.515053\pi$
$691$	−2.48528	−0.0945446	−0.0472723	+	0.998882i	$0.515053\pi$
$692$	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	$-0.259044\pi$
$719$	36.8284	1.37347	0.686734	+	0.726909i	$0.259044\pi$
$720$	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	$-0.748315\pi$
$727$	−37.9289	−1.40671	−0.703353	+	0.710841i	$0.748315\pi$
$728$	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	$-0.878957\pi$
$739$	−50.4853	−1.85713	−0.928566	+	0.371168i	$0.878957\pi$
$740$	0	0
$741$	0.686292	0.0252115
$742$	0	0
$743$	29.5858	1.08540	0.542699	−	0.839928i	$-0.317403\pi$
$743$	29.5858	1.08540	0.542699	+	0.839928i	$0.317403\pi$
$744$	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	$-0.495189\pi$
$751$	0.828427	0.0302297	0.0151149	+	0.999886i	$0.495189\pi$
$752$	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	$-0.771089\pi$
$787$	−42.2132	−1.50474	−0.752369	+	0.658742i	$0.771089\pi$
$788$	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	$-0.468333\pi$
$811$	5.65685	0.198639	0.0993195	+	0.995056i	$0.468333\pi$
$812$	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	$-0.409632\pi$
$823$	16.0711	0.560202	0.280101	+	0.959971i	$0.409632\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.3553	1.33375	0.666873	−	0.745171i	$-0.267632\pi$
$827$	38.3553	1.33375	0.666873	+	0.745171i	$0.267632\pi$
$828$	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	$-0.512097\pi$
$839$	−2.20101	−0.0759873	−0.0379937	+	0.999278i	$0.512097\pi$
$840$	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	$-0.288519\pi$
$859$	36.1421	1.23315	0.616577	+	0.787295i	$0.288519\pi$
$860$	0	0
$861$	1.28427	0.0437678
$862$	0	0
$863$	−45.8701	−1.56143	−0.780717	−	0.624884i	$-0.785146\pi$
$863$	−45.8701	−1.56143	−0.780717	+	0.624884i	$0.785146\pi$
$864$	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	$-0.237377\pi$
$883$	43.6569	1.46917	0.734585	+	0.678517i	$0.237377\pi$
$884$	0	0
$885$	−4.68629	−0.157528
$886$	0	0
$887$	−37.3848	−1.25526	−0.627629	−	0.778513i	$-0.715974\pi$
$887$	−37.3848	−1.25526	−0.627629	+	0.778513i	$0.715974\pi$
$888$	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	$-0.399525\pi$
$907$	18.6985	0.620873	0.310436	+	0.950594i	$0.399525\pi$
$908$	0	0
$909$	44.7696	1.48491
$910$	0	0
$911$	32.8284	1.08765	0.543827	−	0.839197i	$-0.316975\pi$
$911$	32.8284	1.08765	0.543827	+	0.839197i	$0.316975\pi$
$912$	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	$-0.832488\pi$
$919$	−52.4264	−1.72939	−0.864694	+	0.502299i	$0.832488\pi$
$920$	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	$-0.551949\pi$
$947$	−10.0000	−0.324956	−0.162478	+	0.986712i	$0.551949\pi$
$948$	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	$-0.706641\pi$
$967$	−37.5980	−1.20907	−0.604535	+	0.796579i	$0.706641\pi$
$968$	0	0
$969$	−1.94113	−0.0623579
$970$	0	0
$971$	60.2843	1.93461	0.967307	−	0.253608i	$-0.0816172\pi$
$971$	60.2843	1.93461	0.967307	+	0.253608i	$0.0816172\pi$
$972$	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	$-0.396289\pi$
$983$	20.0711	0.640168	0.320084	+	0.947389i	$0.396289\pi$
$984$	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	$-0.729181\pi$
$991$	−41.5147	−1.31876	−0.659379	+	0.751810i	$0.729181\pi$
$992$	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	9680.2.a.bk.1.2		2
4.3	odd	2		4840.2.a.o.1.1	✓	2
11.10	odd	2		9680.2.a.bj.1.2		2
44.43	even	2		4840.2.a.p.1.1	yes	2

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

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:	\( N \)	\(=\)	\( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9680.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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