LMFDB - Embedded newform 9280.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9280,2,Mod(1,9280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9280, 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("9280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9280 = 2^{6} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9280.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:	$74.1011730757$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	9280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{3} -1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} -4.82843 q^{11} +2.00000 q^{13} +2.00000 q^{15} -2.82843 q^{17} +0.828427 q^{19} +1.65685 q^{21} +8.82843 q^{23} +1.00000 q^{25} +4.00000 q^{27} -1.00000 q^{29} +10.4853 q^{31} +9.65685 q^{33} +0.828427 q^{35} -8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} -1.00000 q^{45} +0.343146 q^{47} -6.31371 q^{49} +5.65685 q^{51} -7.65685 q^{53} +4.82843 q^{55} -1.65685 q^{57} -7.65685 q^{61} -0.828427 q^{63} -2.00000 q^{65} -10.4853 q^{67} -17.6569 q^{69} -7.31371 q^{71} -8.48528 q^{73} -2.00000 q^{75} +4.00000 q^{77} -14.4853 q^{79} -11.0000 q^{81} +12.8284 q^{83} +2.82843 q^{85} +2.00000 q^{87} +3.65685 q^{89} -1.65685 q^{91} -20.9706 q^{93} -0.828427 q^{95} +4.48528 q^{97} -4.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{19} - 8 q^{21} + 12 q^{23} + 2 q^{25} + 8 q^{27} - 2 q^{29} + 4 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 12 q^{41} - 12 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 - 4 * q^19 - 8 * q^21 + 12 * q^23 + 2 * q^25 + 8 * q^27 - 2 * q^29 + 4 * q^31 + 8 * q^33 - 4 * q^35 - 8 * q^39 - 12 * q^41 - 12 * q^43 - 2 * q^45 + 12 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 + 8 * q^57 - 4 * q^61 + 4 * q^63 - 4 * q^65 - 4 * q^67 - 24 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^77 - 12 * q^79 - 22 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 + 8 * q^91 - 8 * q^93 + 4 * q^95 - 8 * q^97 - 4 * q^99

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
$3$	−2.00000	−1.15470	−0.577350	−	0.816497i	$-0.695913\pi$
$3$	−2.00000	−1.15470	−0.577350	+	0.816497i	$0.695913\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	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
$21$	1.65685	0.361555
$22$	0	0
$23$	8.82843	1.84085	0.920427	−	0.390914i	$-0.127841\pi$
$23$	8.82843	1.84085	0.920427	+	0.390914i	$0.127841\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	10.4853	1.88321	0.941606	−	0.336717i	$-0.109316\pi$
$31$	10.4853	1.88321	0.941606	+	0.336717i	$0.109316\pi$
$32$	0	0
$33$	9.65685	1.68104
$34$	0	0
$35$	0.828427	0.140030
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	−4.00000	−0.640513
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	0	0
$51$	5.65685	0.792118
$52$	0	0
$53$	−7.65685	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$53$	−7.65685	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$54$	0	0
$55$	4.82843	0.651065
$56$	0	0
$57$	−1.65685	−0.219456
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	0	0
$63$	−0.828427	−0.104372
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	−10.4853	−1.28098	−0.640490	−	0.767966i	$-0.721269\pi$
$67$	−10.4853	−1.28098	−0.640490	+	0.767966i	$0.721269\pi$
$68$	0	0
$69$	−17.6569	−2.12564
$70$	0	0
$71$	−7.31371	−0.867978	−0.433989	−	0.900918i	$-0.642894\pi$
$71$	−7.31371	−0.867978	−0.433989	+	0.900918i	$0.642894\pi$
$72$	0	0
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	0	0
$75$	−2.00000	−0.230940
$76$	0	0
$77$	4.00000	0.455842
$78$	0	0
$79$	−14.4853	−1.62972	−0.814861	−	0.579657i	$-0.803187\pi$
$79$	−14.4853	−1.62972	−0.814861	+	0.579657i	$0.803187\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	12.8284	1.40810	0.704051	−	0.710149i	$-0.251372\pi$
$83$	12.8284	1.40810	0.704051	+	0.710149i	$0.251372\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	2.00000	0.214423
$88$	0	0
$89$	3.65685	0.387626	0.193813	−	0.981039i	$-0.437915\pi$
$89$	3.65685	0.387626	0.193813	+	0.981039i	$0.437915\pi$
$90$	0	0
$91$	−1.65685	−0.173686
$92$	0	0
$93$	−20.9706	−2.17455
$94$	0	0
$95$	−0.828427	−0.0849948
$96$	0	0
$97$	4.48528	0.455411	0.227706	−	0.973730i	$-0.426878\pi$
$97$	4.48528	0.455411	0.227706	+	0.973730i	$0.426878\pi$
$98$	0	0
$99$	−4.82843	−0.485275
$100$	0	0
$101$	−4.34315	−0.432159	−0.216080	−	0.976376i	$-0.569327\pi$
$101$	−4.34315	−0.432159	−0.216080	+	0.976376i	$0.569327\pi$
$102$	0	0
$103$	12.1421	1.19640	0.598200	−	0.801347i	$-0.295883\pi$
$103$	12.1421	1.19640	0.598200	+	0.801347i	$0.295883\pi$
$104$	0	0
$105$	−1.65685	−0.161692
$106$	0	0
$107$	−8.14214	−0.787130	−0.393565	−	0.919297i	$-0.628758\pi$
$107$	−8.14214	−0.787130	−0.393565	+	0.919297i	$0.628758\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	16.9706	1.61077
$112$	0	0
$113$	2.82843	0.266076	0.133038	−	0.991111i	$-0.457527\pi$
$113$	2.82843	0.266076	0.133038	+	0.991111i	$0.457527\pi$
$114$	0	0
$115$	−8.82843	−0.823255
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	12.3137	1.11943
$122$	0	0
$123$	12.0000	1.08200
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	12.0000	1.05654
$130$	0	0
$131$	16.1421	1.41034	0.705172	−	0.709036i	$-0.250870\pi$
$131$	16.1421	1.41034	0.705172	+	0.709036i	$0.250870\pi$
$132$	0	0
$133$	−0.686292	−0.0595090
$134$	0	0
$135$	−4.00000	−0.344265
$136$	0	0
$137$	−10.8284	−0.925135	−0.462567	−	0.886584i	$-0.653072\pi$
$137$	−10.8284	−0.925135	−0.462567	+	0.886584i	$0.653072\pi$
$138$	0	0
$139$	10.3431	0.877294	0.438647	−	0.898659i	$-0.355458\pi$
$139$	10.3431	0.877294	0.438647	+	0.898659i	$0.355458\pi$
$140$	0	0
$141$	−0.686292	−0.0577962
$142$	0	0
$143$	−9.65685	−0.807547
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	12.6274	1.04149
$148$	0	0
$149$	13.3137	1.09070	0.545351	−	0.838208i	$-0.316396\pi$
$149$	13.3137	1.09070	0.545351	+	0.838208i	$0.316396\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	−10.4853	−0.842198
$156$	0	0
$157$	16.4853	1.31567	0.657834	−	0.753163i	$-0.271473\pi$
$157$	16.4853	1.31567	0.657834	+	0.753163i	$0.271473\pi$
$158$	0	0
$159$	15.3137	1.21446
$160$	0	0
$161$	−7.31371	−0.576401
$162$	0	0
$163$	−19.6569	−1.53964	−0.769822	−	0.638259i	$-0.779655\pi$
$163$	−19.6569	−1.53964	−0.769822	+	0.638259i	$0.779655\pi$
$164$	0	0
$165$	−9.65685	−0.751785
$166$	0	0
$167$	−14.4853	−1.12090	−0.560452	−	0.828187i	$-0.689373\pi$
$167$	−14.4853	−1.12090	−0.560452	+	0.828187i	$0.689373\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0.828427	0.0633514
$172$	0	0
$173$	5.31371	0.403994	0.201997	−	0.979386i	$-0.435257\pi$
$173$	5.31371	0.403994	0.201997	+	0.979386i	$0.435257\pi$
$174$	0	0
$175$	−0.828427	−0.0626232
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	15.3137	1.13202
$184$	0	0
$185$	8.48528	0.623850
$186$	0	0
$187$	13.6569	0.998688
$188$	0	0
$189$	−3.31371	−0.241037
$190$	0	0
$191$	15.1716	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$191$	15.1716	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$192$	0	0
$193$	−12.4853	−0.898710	−0.449355	−	0.893353i	$-0.648346\pi$
$193$	−12.4853	−0.898710	−0.449355	+	0.893353i	$0.648346\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	8.34315	0.594425	0.297212	−	0.954811i	$-0.403943\pi$
$197$	8.34315	0.594425	0.297212	+	0.954811i	$0.403943\pi$
$198$	0	0
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	0	0
$201$	20.9706	1.47915
$202$	0	0
$203$	0.828427	0.0581442
$204$	0	0
$205$	6.00000	0.419058
$206$	0	0
$207$	8.82843	0.613618
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	4.82843	0.332403	0.166201	−	0.986092i	$-0.446850\pi$
$211$	4.82843	0.332403	0.166201	+	0.986092i	$0.446850\pi$
$212$	0	0
$213$	14.6274	1.00225
$214$	0	0
$215$	6.00000	0.409197
$216$	0	0
$217$	−8.68629	−0.589664
$218$	0	0
$219$	16.9706	1.14676
$220$	0	0
$221$	−5.65685	−0.380521
$222$	0	0
$223$	−21.7990	−1.45977	−0.729884	−	0.683571i	$-0.760426\pi$
$223$	−21.7990	−1.45977	−0.729884	+	0.683571i	$0.760426\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−8.14214	−0.540413	−0.270206	−	0.962802i	$-0.587092\pi$
$227$	−8.14214	−0.540413	−0.270206	+	0.962802i	$0.587092\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	−8.00000	−0.526361
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	0	0
$235$	−0.343146	−0.0223844
$236$	0	0
$237$	28.9706	1.88184
$238$	0	0
$239$	23.3137	1.50804	0.754019	−	0.656852i	$-0.228113\pi$
$239$	23.3137	1.50804	0.754019	+	0.656852i	$0.228113\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	0	0
$243$	10.0000	0.641500
$244$	0	0
$245$	6.31371	0.403368
$246$	0	0
$247$	1.65685	0.105423
$248$	0	0
$249$	−25.6569	−1.62594
$250$	0	0
$251$	3.17157	0.200188	0.100094	−	0.994978i	$-0.468086\pi$
$251$	3.17157	0.200188	0.100094	+	0.994978i	$0.468086\pi$
$252$	0	0
$253$	−42.6274	−2.67996
$254$	0	0
$255$	−5.65685	−0.354246
$256$	0	0
$257$	29.3137	1.82854	0.914269	−	0.405107i	$-0.132766\pi$
$257$	29.3137	1.82854	0.914269	+	0.405107i	$0.132766\pi$
$258$	0	0
$259$	7.02944	0.436788
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	0	0
$263$	8.34315	0.514460	0.257230	−	0.966350i	$-0.417190\pi$
$263$	8.34315	0.514460	0.257230	+	0.966350i	$0.417190\pi$
$264$	0	0
$265$	7.65685	0.470357
$266$	0	0
$267$	−7.31371	−0.447592
$268$	0	0
$269$	−1.31371	−0.0800982	−0.0400491	−	0.999198i	$-0.512751\pi$
$269$	−1.31371	−0.0800982	−0.0400491	+	0.999198i	$0.512751\pi$
$270$	0	0
$271$	−29.7990	−1.81016	−0.905080	−	0.425242i	$-0.860189\pi$
$271$	−29.7990	−1.81016	−0.905080	+	0.425242i	$0.860189\pi$
$272$	0	0
$273$	3.31371	0.200555
$274$	0	0
$275$	−4.82843	−0.291165
$276$	0	0
$277$	−7.65685	−0.460056	−0.230028	−	0.973184i	$-0.573882\pi$
$277$	−7.65685	−0.460056	−0.230028	+	0.973184i	$0.573882\pi$
$278$	0	0
$279$	10.4853	0.627737
$280$	0	0
$281$	−6.68629	−0.398871	−0.199435	−	0.979911i	$-0.563911\pi$
$281$	−6.68629	−0.398871	−0.199435	+	0.979911i	$0.563911\pi$
$282$	0	0
$283$	−0.828427	−0.0492449	−0.0246224	−	0.999697i	$-0.507838\pi$
$283$	−0.828427	−0.0492449	−0.0246224	+	0.999697i	$0.507838\pi$
$284$	0	0
$285$	1.65685	0.0981436
$286$	0	0
$287$	4.97056	0.293403
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	−8.97056	−0.525864
$292$	0	0
$293$	8.48528	0.495715	0.247858	−	0.968796i	$-0.420273\pi$
$293$	8.48528	0.495715	0.247858	+	0.968796i	$0.420273\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−19.3137	−1.12070
$298$	0	0
$299$	17.6569	1.02112
$300$	0	0
$301$	4.97056	0.286498
$302$	0	0
$303$	8.68629	0.499014
$304$	0	0
$305$	7.65685	0.438430
$306$	0	0
$307$	10.9706	0.626123	0.313062	−	0.949733i	$-0.398645\pi$
$307$	10.9706	0.626123	0.313062	+	0.949733i	$0.398645\pi$
$308$	0	0
$309$	−24.2843	−1.38148
$310$	0	0
$311$	2.48528	0.140927	0.0704637	−	0.997514i	$-0.477552\pi$
$311$	2.48528	0.140927	0.0704637	+	0.997514i	$0.477552\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0.828427	0.0466766
$316$	0	0
$317$	2.82843	0.158860	0.0794301	−	0.996840i	$-0.474690\pi$
$317$	2.82843	0.158860	0.0794301	+	0.996840i	$0.474690\pi$
$318$	0	0
$319$	4.82843	0.270340
$320$	0	0
$321$	16.2843	0.908899
$322$	0	0
$323$	−2.34315	−0.130376
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	4.00000	0.221201
$328$	0	0
$329$	−0.284271	−0.0156724
$330$	0	0
$331$	−17.7990	−0.978321	−0.489160	−	0.872194i	$-0.662697\pi$
$331$	−17.7990	−0.978321	−0.489160	+	0.872194i	$0.662697\pi$
$332$	0	0
$333$	−8.48528	−0.464991
$334$	0	0
$335$	10.4853	0.572872
$336$	0	0
$337$	−6.82843	−0.371968	−0.185984	−	0.982553i	$-0.559547\pi$
$337$	−6.82843	−0.371968	−0.185984	+	0.982553i	$0.559547\pi$
$338$	0	0
$339$	−5.65685	−0.307238
$340$	0	0
$341$	−50.6274	−2.74163
$342$	0	0
$343$	11.0294	0.595534
$344$	0	0
$345$	17.6569	0.950613
$346$	0	0
$347$	20.1421	1.08129	0.540643	−	0.841252i	$-0.318181\pi$
$347$	20.1421	1.08129	0.540643	+	0.841252i	$0.318181\pi$
$348$	0	0
$349$	24.6274	1.31828	0.659138	−	0.752022i	$-0.270921\pi$
$349$	24.6274	1.31828	0.659138	+	0.752022i	$0.270921\pi$
$350$	0	0
$351$	8.00000	0.427008
$352$	0	0
$353$	−15.6569	−0.833330	−0.416665	−	0.909060i	$-0.636801\pi$
$353$	−15.6569	−0.833330	−0.416665	+	0.909060i	$0.636801\pi$
$354$	0	0
$355$	7.31371	0.388171
$356$	0	0
$357$	−4.68629	−0.248025
$358$	0	0
$359$	32.1421	1.69640	0.848199	−	0.529678i	$-0.177687\pi$
$359$	32.1421	1.69640	0.848199	+	0.529678i	$0.177687\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	0	0
$363$	−24.6274	−1.29260
$364$	0	0
$365$	8.48528	0.444140
$366$	0	0
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	6.34315	0.329320
$372$	0	0
$373$	−26.9706	−1.39648	−0.698241	−	0.715862i	$-0.746034\pi$
$373$	−26.9706	−1.39648	−0.698241	+	0.715862i	$0.746034\pi$
$374$	0	0
$375$	2.00000	0.103280
$376$	0	0
$377$	−2.00000	−0.103005
$378$	0	0
$379$	5.51472	0.283272	0.141636	−	0.989919i	$-0.454764\pi$
$379$	5.51472	0.283272	0.141636	+	0.989919i	$0.454764\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	0	0
$383$	14.4853	0.740163	0.370082	−	0.928999i	$-0.379330\pi$
$383$	14.4853	0.740163	0.370082	+	0.928999i	$0.379330\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−6.00000	−0.304997
$388$	0	0
$389$	6.68629	0.339008	0.169504	−	0.985529i	$-0.445783\pi$
$389$	6.68629	0.339008	0.169504	+	0.985529i	$0.445783\pi$
$390$	0	0
$391$	−24.9706	−1.26282
$392$	0	0
$393$	−32.2843	−1.62853
$394$	0	0
$395$	14.4853	0.728834
$396$	0	0
$397$	8.34315	0.418730	0.209365	−	0.977838i	$-0.432860\pi$
$397$	8.34315	0.418730	0.209365	+	0.977838i	$0.432860\pi$
$398$	0	0
$399$	1.37258	0.0687151
$400$	0	0
$401$	−29.3137	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$401$	−29.3137	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$402$	0	0
$403$	20.9706	1.04462
$404$	0	0
$405$	11.0000	0.546594
$406$	0	0
$407$	40.9706	2.03084
$408$	0	0
$409$	30.9706	1.53140	0.765698	−	0.643200i	$-0.222394\pi$
$409$	30.9706	1.53140	0.765698	+	0.643200i	$0.222394\pi$
$410$	0	0
$411$	21.6569	1.06825
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−12.8284	−0.629723
$416$	0	0
$417$	−20.6863	−1.01301
$418$	0	0
$419$	−4.97056	−0.242828	−0.121414	−	0.992602i	$-0.538743\pi$
$419$	−4.97056	−0.242828	−0.121414	+	0.992602i	$0.538743\pi$
$420$	0	0
$421$	14.9706	0.729621	0.364810	−	0.931082i	$-0.381134\pi$
$421$	14.9706	0.729621	0.364810	+	0.931082i	$0.381134\pi$
$422$	0	0
$423$	0.343146	0.0166843
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	6.34315	0.306966
$428$	0	0
$429$	19.3137	0.932475
$430$	0	0
$431$	19.3137	0.930309	0.465154	−	0.885230i	$-0.345999\pi$
$431$	19.3137	0.930309	0.465154	+	0.885230i	$0.345999\pi$
$432$	0	0
$433$	−34.8284	−1.67375	−0.836874	−	0.547396i	$-0.815619\pi$
$433$	−34.8284	−1.67375	−0.836874	+	0.547396i	$0.815619\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	0	0
$437$	7.31371	0.349862
$438$	0	0
$439$	21.6569	1.03363	0.516813	−	0.856099i	$-0.327118\pi$
$439$	21.6569	1.03363	0.516813	+	0.856099i	$0.327118\pi$
$440$	0	0
$441$	−6.31371	−0.300653
$442$	0	0
$443$	3.65685	0.173742	0.0868712	−	0.996220i	$-0.472313\pi$
$443$	3.65685	0.173742	0.0868712	+	0.996220i	$0.472313\pi$
$444$	0	0
$445$	−3.65685	−0.173352
$446$	0	0
$447$	−26.6274	−1.25943
$448$	0	0
$449$	0.343146	0.0161940	0.00809702	−	0.999967i	$-0.497423\pi$
$449$	0.343146	0.0161940	0.00809702	+	0.999967i	$0.497423\pi$
$450$	0	0
$451$	28.9706	1.36417
$452$	0	0
$453$	−24.0000	−1.12762
$454$	0	0
$455$	1.65685	0.0776745
$456$	0	0
$457$	8.34315	0.390276	0.195138	−	0.980776i	$-0.437485\pi$
$457$	8.34315	0.390276	0.195138	+	0.980776i	$0.437485\pi$
$458$	0	0
$459$	−11.3137	−0.528079
$460$	0	0
$461$	24.3431	1.13377	0.566887	−	0.823796i	$-0.308148\pi$
$461$	24.3431	1.13377	0.566887	+	0.823796i	$0.308148\pi$
$462$	0	0
$463$	17.7990	0.827189	0.413595	−	0.910461i	$-0.364273\pi$
$463$	17.7990	0.827189	0.413595	+	0.910461i	$0.364273\pi$
$464$	0	0
$465$	20.9706	0.972487
$466$	0	0
$467$	−22.9706	−1.06295	−0.531475	−	0.847074i	$-0.678362\pi$
$467$	−22.9706	−1.06295	−0.531475	+	0.847074i	$0.678362\pi$
$468$	0	0
$469$	8.68629	0.401096
$470$	0	0
$471$	−32.9706	−1.51920
$472$	0	0
$473$	28.9706	1.33207
$474$	0	0
$475$	0.828427	0.0380108
$476$	0	0
$477$	−7.65685	−0.350583
$478$	0	0
$479$	−12.8284	−0.586146	−0.293073	−	0.956090i	$-0.594678\pi$
$479$	−12.8284	−0.586146	−0.293073	+	0.956090i	$0.594678\pi$
$480$	0	0
$481$	−16.9706	−0.773791
$482$	0	0
$483$	14.6274	0.665571
$484$	0	0
$485$	−4.48528	−0.203666
$486$	0	0
$487$	29.7990	1.35032	0.675161	−	0.737671i	$-0.264074\pi$
$487$	29.7990	1.35032	0.675161	+	0.737671i	$0.264074\pi$
$488$	0	0
$489$	39.3137	1.77783
$490$	0	0
$491$	43.4558	1.96113	0.980567	−	0.196183i	$-0.0628545\pi$
$491$	43.4558	1.96113	0.980567	+	0.196183i	$0.0628545\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	0	0
$495$	4.82843	0.217022
$496$	0	0
$497$	6.05887	0.271778
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	0	0
$501$	28.9706	1.29431
$502$	0	0
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	4.34315	0.193267
$506$	0	0
$507$	18.0000	0.799408
$508$	0	0
$509$	44.6274	1.97808	0.989038	−	0.147663i	$-0.0471751\pi$
$509$	44.6274	1.97808	0.989038	+	0.147663i	$0.0471751\pi$
$510$	0	0
$511$	7.02944	0.310964
$512$	0	0
$513$	3.31371	0.146304
$514$	0	0
$515$	−12.1421	−0.535046
$516$	0	0
$517$	−1.65685	−0.0728684
$518$	0	0
$519$	−10.6274	−0.466492
$520$	0	0
$521$	1.31371	0.0575546	0.0287773	−	0.999586i	$-0.490839\pi$
$521$	1.31371	0.0575546	0.0287773	+	0.999586i	$0.490839\pi$
$522$	0	0
$523$	−14.4853	−0.633397	−0.316699	−	0.948526i	$-0.602574\pi$
$523$	−14.4853	−0.633397	−0.316699	+	0.948526i	$0.602574\pi$
$524$	0	0
$525$	1.65685	0.0723110
$526$	0	0
$527$	−29.6569	−1.29187
$528$	0	0
$529$	54.9411	2.38874
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.0000	−0.519778
$534$	0	0
$535$	8.14214	0.352015
$536$	0	0
$537$	1.37258	0.0592313
$538$	0	0
$539$	30.4853	1.31309
$540$	0	0
$541$	38.9706	1.67548	0.837738	−	0.546073i	$-0.183878\pi$
$541$	38.9706	1.67548	0.837738	+	0.546073i	$0.183878\pi$
$542$	0	0
$543$	−12.0000	−0.514969
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	14.4853	0.619346	0.309673	−	0.950843i	$-0.399780\pi$
$547$	14.4853	0.619346	0.309673	+	0.950843i	$0.399780\pi$
$548$	0	0
$549$	−7.65685	−0.326787
$550$	0	0
$551$	−0.828427	−0.0352922
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	−16.9706	−0.720360
$556$	0	0
$557$	−39.9411	−1.69236	−0.846180	−	0.532897i	$-0.821103\pi$
$557$	−39.9411	−1.69236	−0.846180	+	0.532897i	$0.821103\pi$
$558$	0	0
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−27.3137	−1.15319
$562$	0	0
$563$	−3.65685	−0.154118	−0.0770590	−	0.997027i	$-0.524553\pi$
$563$	−3.65685	−0.154118	−0.0770590	+	0.997027i	$0.524553\pi$
$564$	0	0
$565$	−2.82843	−0.118993
$566$	0	0
$567$	9.11270	0.382697
$568$	0	0
$569$	16.3431	0.685140	0.342570	−	0.939492i	$-0.388703\pi$
$569$	16.3431	0.685140	0.342570	+	0.939492i	$0.388703\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	−30.3431	−1.26760
$574$	0	0
$575$	8.82843	0.368171
$576$	0	0
$577$	−15.7990	−0.657721	−0.328860	−	0.944379i	$-0.606665\pi$
$577$	−15.7990	−0.657721	−0.328860	+	0.944379i	$0.606665\pi$
$578$	0	0
$579$	24.9706	1.03774
$580$	0	0
$581$	−10.6274	−0.440900
$582$	0	0
$583$	36.9706	1.53116
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	9.79899	0.404448	0.202224	−	0.979339i	$-0.435183\pi$
$587$	9.79899	0.404448	0.202224	+	0.979339i	$0.435183\pi$
$588$	0	0
$589$	8.68629	0.357912
$590$	0	0
$591$	−16.6863	−0.686382
$592$	0	0
$593$	3.65685	0.150169	0.0750845	−	0.997177i	$-0.476077\pi$
$593$	3.65685	0.150169	0.0750845	+	0.997177i	$0.476077\pi$
$594$	0	0
$595$	−2.34315	−0.0960596
$596$	0	0
$597$	24.0000	0.982255
$598$	0	0
$599$	−1.79899	−0.0735047	−0.0367524	−	0.999324i	$-0.511701\pi$
$599$	−1.79899	−0.0735047	−0.0367524	+	0.999324i	$0.511701\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	0	0
$603$	−10.4853	−0.426994
$604$	0	0
$605$	−12.3137	−0.500623
$606$	0	0
$607$	−42.9706	−1.74412	−0.872061	−	0.489398i	$-0.837217\pi$
$607$	−42.9706	−1.74412	−0.872061	+	0.489398i	$0.837217\pi$
$608$	0	0
$609$	−1.65685	−0.0671391
$610$	0	0
$611$	0.686292	0.0277644
$612$	0	0
$613$	2.00000	0.0807792	0.0403896	−	0.999184i	$-0.487140\pi$
$613$	2.00000	0.0807792	0.0403896	+	0.999184i	$0.487140\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−14.8284	−0.596970	−0.298485	−	0.954414i	$-0.596481\pi$
$617$	−14.8284	−0.596970	−0.298485	+	0.954414i	$0.596481\pi$
$618$	0	0
$619$	29.7990	1.19772	0.598861	−	0.800853i	$-0.295620\pi$
$619$	29.7990	1.19772	0.598861	+	0.800853i	$0.295620\pi$
$620$	0	0
$621$	35.3137	1.41709
$622$	0	0
$623$	−3.02944	−0.121372
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	8.00000	0.319489
$628$	0	0
$629$	24.0000	0.956943
$630$	0	0
$631$	−3.02944	−0.120600	−0.0603000	−	0.998180i	$-0.519206\pi$
$631$	−3.02944	−0.120600	−0.0603000	+	0.998180i	$0.519206\pi$
$632$	0	0
$633$	−9.65685	−0.383825
$634$	0	0
$635$	6.00000	0.238103
$636$	0	0
$637$	−12.6274	−0.500316
$638$	0	0
$639$	−7.31371	−0.289326
$640$	0	0
$641$	−44.6274	−1.76268	−0.881338	−	0.472485i	$-0.843357\pi$
$641$	−44.6274	−1.76268	−0.881338	+	0.472485i	$0.843357\pi$
$642$	0	0
$643$	−31.4558	−1.24050	−0.620249	−	0.784405i	$-0.712968\pi$
$643$	−31.4558	−1.24050	−0.620249	+	0.784405i	$0.712968\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	−21.1127	−0.830026	−0.415013	−	0.909816i	$-0.636223\pi$
$647$	−21.1127	−0.830026	−0.415013	+	0.909816i	$0.636223\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	17.3726	0.680885
$652$	0	0
$653$	22.8284	0.893345	0.446673	−	0.894697i	$-0.352609\pi$
$653$	22.8284	0.893345	0.446673	+	0.894697i	$0.352609\pi$
$654$	0	0
$655$	−16.1421	−0.630725
$656$	0	0
$657$	−8.48528	−0.331042
$658$	0	0
$659$	−37.7990	−1.47244	−0.736220	−	0.676743i	$-0.763391\pi$
$659$	−37.7990	−1.47244	−0.736220	+	0.676743i	$0.763391\pi$
$660$	0	0
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	0	0
$663$	11.3137	0.439388
$664$	0	0
$665$	0.686292	0.0266132
$666$	0	0
$667$	−8.82843	−0.341838
$668$	0	0
$669$	43.5980	1.68560
$670$	0	0
$671$	36.9706	1.42723
$672$	0	0
$673$	−10.9706	−0.422884	−0.211442	−	0.977391i	$-0.567816\pi$
$673$	−10.9706	−0.422884	−0.211442	+	0.977391i	$0.567816\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−36.7696	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$677$	−36.7696	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$678$	0	0
$679$	−3.71573	−0.142597
$680$	0	0
$681$	16.2843	0.624015
$682$	0	0
$683$	40.1421	1.53600	0.767998	−	0.640452i	$-0.221253\pi$
$683$	40.1421	1.53600	0.767998	+	0.640452i	$0.221253\pi$
$684$	0	0
$685$	10.8284	0.413733
$686$	0	0
$687$	−4.00000	−0.152610
$688$	0	0
$689$	−15.3137	−0.583406
$690$	0	0
$691$	−11.0294	−0.419580	−0.209790	−	0.977747i	$-0.567278\pi$
$691$	−11.0294	−0.419580	−0.209790	+	0.977747i	$0.567278\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	−10.3431	−0.392338
$696$	0	0
$697$	16.9706	0.642806
$698$	0	0
$699$	−36.0000	−1.36165
$700$	0	0
$701$	−29.3137	−1.10716	−0.553582	−	0.832795i	$-0.686739\pi$
$701$	−29.3137	−1.10716	−0.553582	+	0.832795i	$0.686739\pi$
$702$	0	0
$703$	−7.02944	−0.265120
$704$	0	0
$705$	0.686292	0.0258472
$706$	0	0
$707$	3.59798	0.135316
$708$	0	0
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	−14.4853	−0.543240
$712$	0	0
$713$	92.5685	3.46672
$714$	0	0
$715$	9.65685	0.361146
$716$	0	0
$717$	−46.6274	−1.74133
$718$	0	0
$719$	−10.6274	−0.396336	−0.198168	−	0.980168i	$-0.563499\pi$
$719$	−10.6274	−0.396336	−0.198168	+	0.980168i	$0.563499\pi$
$720$	0	0
$721$	−10.0589	−0.374612
$722$	0	0
$723$	−20.0000	−0.743808
$724$	0	0
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−43.9411	−1.62969	−0.814843	−	0.579682i	$-0.803177\pi$
$727$	−43.9411	−1.62969	−0.814843	+	0.579682i	$0.803177\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	16.9706	0.627679
$732$	0	0
$733$	17.1716	0.634247	0.317123	−	0.948384i	$-0.397283\pi$
$733$	17.1716	0.634247	0.317123	+	0.948384i	$0.397283\pi$
$734$	0	0
$735$	−12.6274	−0.465769
$736$	0	0
$737$	50.6274	1.86488
$738$	0	0
$739$	−2.48528	−0.0914226	−0.0457113	−	0.998955i	$-0.514555\pi$
$739$	−2.48528	−0.0914226	−0.0457113	+	0.998955i	$0.514555\pi$
$740$	0	0
$741$	−3.31371	−0.121732
$742$	0	0
$743$	7.37258	0.270474	0.135237	−	0.990813i	$-0.456820\pi$
$743$	7.37258	0.270474	0.135237	+	0.990813i	$0.456820\pi$
$744$	0	0
$745$	−13.3137	−0.487777
$746$	0	0
$747$	12.8284	0.469368
$748$	0	0
$749$	6.74517	0.246463
$750$	0	0
$751$	12.1421	0.443073	0.221536	−	0.975152i	$-0.428893\pi$
$751$	12.1421	0.443073	0.221536	+	0.975152i	$0.428893\pi$
$752$	0	0
$753$	−6.34315	−0.231157
$754$	0	0
$755$	−12.0000	−0.436725
$756$	0	0
$757$	−36.4853	−1.32608	−0.663040	−	0.748584i	$-0.730734\pi$
$757$	−36.4853	−1.32608	−0.663040	+	0.748584i	$0.730734\pi$
$758$	0	0
$759$	85.2548	3.09455
$760$	0	0
$761$	−36.6274	−1.32774	−0.663871	−	0.747847i	$-0.731088\pi$
$761$	−36.6274	−1.32774	−0.663871	+	0.747847i	$0.731088\pi$
$762$	0	0
$763$	1.65685	0.0599822
$764$	0	0
$765$	2.82843	0.102262
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−4.34315	−0.156618	−0.0783089	−	0.996929i	$-0.524952\pi$
$769$	−4.34315	−0.156618	−0.0783089	+	0.996929i	$0.524952\pi$
$770$	0	0
$771$	−58.6274	−2.11141
$772$	0	0
$773$	−8.48528	−0.305194	−0.152597	−	0.988288i	$-0.548764\pi$
$773$	−8.48528	−0.305194	−0.152597	+	0.988288i	$0.548764\pi$
$774$	0	0
$775$	10.4853	0.376642
$776$	0	0
$777$	−14.0589	−0.504359
$778$	0	0
$779$	−4.97056	−0.178089
$780$	0	0
$781$	35.3137	1.26362
$782$	0	0
$783$	−4.00000	−0.142948
$784$	0	0
$785$	−16.4853	−0.588385
$786$	0	0
$787$	−21.7990	−0.777050	−0.388525	−	0.921438i	$-0.627015\pi$
$787$	−21.7990	−0.777050	−0.388525	+	0.921438i	$0.627015\pi$
$788$	0	0
$789$	−16.6863	−0.594048
$790$	0	0
$791$	−2.34315	−0.0833127
$792$	0	0
$793$	−15.3137	−0.543806
$794$	0	0
$795$	−15.3137	−0.543121
$796$	0	0
$797$	34.1421	1.20938	0.604688	−	0.796462i	$-0.293298\pi$
$797$	34.1421	1.20938	0.604688	+	0.796462i	$0.293298\pi$
$798$	0	0
$799$	−0.970563	−0.0343360
$800$	0	0
$801$	3.65685	0.129209
$802$	0	0
$803$	40.9706	1.44582
$804$	0	0
$805$	7.31371	0.257774
$806$	0	0
$807$	2.62742	0.0924895
$808$	0	0
$809$	−14.2843	−0.502208	−0.251104	−	0.967960i	$-0.580794\pi$
$809$	−14.2843	−0.502208	−0.251104	+	0.967960i	$0.580794\pi$
$810$	0	0
$811$	26.3431	0.925033	0.462516	−	0.886611i	$-0.346947\pi$
$811$	26.3431	0.925033	0.462516	+	0.886611i	$0.346947\pi$
$812$	0	0
$813$	59.5980	2.09019
$814$	0	0
$815$	19.6569	0.688550
$816$	0	0
$817$	−4.97056	−0.173898
$818$	0	0
$819$	−1.65685	−0.0578952
$820$	0	0
$821$	45.3137	1.58146	0.790730	−	0.612165i	$-0.209701\pi$
$821$	45.3137	1.58146	0.790730	+	0.612165i	$0.209701\pi$
$822$	0	0
$823$	−2.97056	−0.103547	−0.0517737	−	0.998659i	$-0.516487\pi$
$823$	−2.97056	−0.103547	−0.0517737	+	0.998659i	$0.516487\pi$
$824$	0	0
$825$	9.65685	0.336209
$826$	0	0
$827$	−5.31371	−0.184776	−0.0923879	−	0.995723i	$-0.529450\pi$
$827$	−5.31371	−0.184776	−0.0923879	+	0.995723i	$0.529450\pi$
$828$	0	0
$829$	24.6274	0.855346	0.427673	−	0.903934i	$-0.359334\pi$
$829$	24.6274	0.855346	0.427673	+	0.903934i	$0.359334\pi$
$830$	0	0
$831$	15.3137	0.531227
$832$	0	0
$833$	17.8579	0.618738
$834$	0	0
$835$	14.4853	0.501284
$836$	0	0
$837$	41.9411	1.44970
$838$	0	0
$839$	14.4853	0.500087	0.250044	−	0.968235i	$-0.419555\pi$
$839$	14.4853	0.500087	0.250044	+	0.968235i	$0.419555\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	13.3726	0.460576
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	−10.2010	−0.350511
$848$	0	0
$849$	1.65685	0.0568631
$850$	0	0
$851$	−74.9117	−2.56794
$852$	0	0
$853$	11.1127	0.380492	0.190246	−	0.981736i	$-0.439072\pi$
$853$	11.1127	0.380492	0.190246	+	0.981736i	$0.439072\pi$
$854$	0	0
$855$	−0.828427	−0.0283316
$856$	0	0
$857$	48.6274	1.66108	0.830540	−	0.556958i	$-0.188032\pi$
$857$	48.6274	1.66108	0.830540	+	0.556958i	$0.188032\pi$
$858$	0	0
$859$	28.4264	0.969896	0.484948	−	0.874543i	$-0.338838\pi$
$859$	28.4264	0.969896	0.484948	+	0.874543i	$0.338838\pi$
$860$	0	0
$861$	−9.94113	−0.338793
$862$	0	0
$863$	7.85786	0.267485	0.133742	−	0.991016i	$-0.457301\pi$
$863$	7.85786	0.267485	0.133742	+	0.991016i	$0.457301\pi$
$864$	0	0
$865$	−5.31371	−0.180672
$866$	0	0
$867$	18.0000	0.611312
$868$	0	0
$869$	69.9411	2.37259
$870$	0	0
$871$	−20.9706	−0.710560
$872$	0	0
$873$	4.48528	0.151804
$874$	0	0
$875$	0.828427	0.0280059
$876$	0	0
$877$	18.2843	0.617416	0.308708	−	0.951157i	$-0.400103\pi$
$877$	18.2843	0.617416	0.308708	+	0.951157i	$0.400103\pi$
$878$	0	0
$879$	−16.9706	−0.572403
$880$	0	0
$881$	6.68629	0.225267	0.112633	−	0.993637i	$-0.464071\pi$
$881$	6.68629	0.225267	0.112633	+	0.993637i	$0.464071\pi$
$882$	0	0
$883$	2.48528	0.0836364	0.0418182	−	0.999125i	$-0.486685\pi$
$883$	2.48528	0.0836364	0.0418182	+	0.999125i	$0.486685\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.3137	0.984258	0.492129	−	0.870522i	$-0.336219\pi$
$887$	29.3137	0.984258	0.492129	+	0.870522i	$0.336219\pi$
$888$	0	0
$889$	4.97056	0.166707
$890$	0	0
$891$	53.1127	1.77934
$892$	0	0
$893$	0.284271	0.00951277
$894$	0	0
$895$	0.686292	0.0229402
$896$	0	0
$897$	−35.3137	−1.17909
$898$	0	0
$899$	−10.4853	−0.349704
$900$	0	0
$901$	21.6569	0.721494
$902$	0	0
$903$	−9.94113	−0.330820
$904$	0	0
$905$	−6.00000	−0.199447
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	0	0
$909$	−4.34315	−0.144053
$910$	0	0
$911$	−3.85786	−0.127817	−0.0639084	−	0.997956i	$-0.520357\pi$
$911$	−3.85786	−0.127817	−0.0639084	+	0.997956i	$0.520357\pi$
$912$	0	0
$913$	−61.9411	−2.04995
$914$	0	0
$915$	−15.3137	−0.506256
$916$	0	0
$917$	−13.3726	−0.441602
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	−21.9411	−0.722985
$922$	0	0
$923$	−14.6274	−0.481467
$924$	0	0
$925$	−8.48528	−0.278994
$926$	0	0
$927$	12.1421	0.398800
$928$	0	0
$929$	40.6274	1.33294	0.666471	−	0.745531i	$-0.267804\pi$
$929$	40.6274	1.33294	0.666471	+	0.745531i	$0.267804\pi$
$930$	0	0
$931$	−5.23045	−0.171421
$932$	0	0
$933$	−4.97056	−0.162729
$934$	0	0
$935$	−13.6569	−0.446627
$936$	0	0
$937$	8.34315	0.272559	0.136279	−	0.990670i	$-0.456486\pi$
$937$	8.34315	0.272559	0.136279	+	0.990670i	$0.456486\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	0	0
$941$	−39.9411	−1.30204	−0.651022	−	0.759059i	$-0.725659\pi$
$941$	−39.9411	−1.30204	−0.651022	+	0.759059i	$0.725659\pi$
$942$	0	0
$943$	−52.9706	−1.72496
$944$	0	0
$945$	3.31371	0.107795
$946$	0	0
$947$	−56.9117	−1.84938	−0.924691	−	0.380719i	$-0.875676\pi$
$947$	−56.9117	−1.84938	−0.924691	+	0.380719i	$0.875676\pi$
$948$	0	0
$949$	−16.9706	−0.550888
$950$	0	0
$951$	−5.65685	−0.183436
$952$	0	0
$953$	6.68629	0.216590	0.108295	−	0.994119i	$-0.465461\pi$
$953$	6.68629	0.216590	0.108295	+	0.994119i	$0.465461\pi$
$954$	0	0
$955$	−15.1716	−0.490941
$956$	0	0
$957$	−9.65685	−0.312162
$958$	0	0
$959$	8.97056	0.289675
$960$	0	0
$961$	78.9411	2.54649
$962$	0	0
$963$	−8.14214	−0.262377
$964$	0	0
$965$	12.4853	0.401915
$966$	0	0
$967$	18.9706	0.610052	0.305026	−	0.952344i	$-0.401335\pi$
$967$	18.9706	0.610052	0.305026	+	0.952344i	$0.401335\pi$
$968$	0	0
$969$	4.68629	0.150545
$970$	0	0
$971$	−0.142136	−0.00456135	−0.00228067	−	0.999997i	$-0.500726\pi$
$971$	−0.142136	−0.00456135	−0.00228067	+	0.999997i	$0.500726\pi$
$972$	0	0
$973$	−8.56854	−0.274695
$974$	0	0
$975$	−4.00000	−0.128103
$976$	0	0
$977$	−25.3137	−0.809857	−0.404929	−	0.914348i	$-0.632704\pi$
$977$	−25.3137	−0.809857	−0.404929	+	0.914348i	$0.632704\pi$
$978$	0	0
$979$	−17.6569	−0.564316
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−13.3137	−0.424641	−0.212321	−	0.977200i	$-0.568102\pi$
$983$	−13.3137	−0.424641	−0.212321	+	0.977200i	$0.568102\pi$
$984$	0	0
$985$	−8.34315	−0.265835
$986$	0	0
$987$	0.568542	0.0180969
$988$	0	0
$989$	−52.9706	−1.68437
$990$	0	0
$991$	−52.0000	−1.65183	−0.825917	−	0.563791i	$-0.809342\pi$
$991$	−52.0000	−1.65183	−0.825917	+	0.563791i	$0.809342\pi$
$992$	0	0
$993$	35.5980	1.12967
$994$	0	0
$995$	12.0000	0.380426
$996$	0	0
$997$	−1.17157	−0.0371041	−0.0185520	−	0.999828i	$-0.505906\pi$
$997$	−1.17157	−0.0371041	−0.0185520	+	0.999828i	$0.505906\pi$
$998$	0	0
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9280.2.a.w.1.1		2
4.3	odd	2		9280.2.a.be.1.2		2
8.3	odd	2		145.2.a.b.1.1	✓	2
8.5	even	2		2320.2.a.k.1.1		2
24.11	even	2		1305.2.a.n.1.2		2
40.3	even	4		725.2.b.c.349.4		4
40.19	odd	2		725.2.a.c.1.2		2
40.27	even	4		725.2.b.c.349.1		4
56.27	even	2		7105.2.a.e.1.1		2
120.59	even	2		6525.2.a.p.1.1		2
232.115	odd	2		4205.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.1	✓	2	8.3	odd	2
725.2.a.c.1.2		2	40.19	odd	2
725.2.b.c.349.1		4	40.27	even	4
725.2.b.c.349.4		4	40.3	even	4
1305.2.a.n.1.2		2	24.11	even	2
2320.2.a.k.1.1		2	8.5	even	2
4205.2.a.d.1.2		2	232.115	odd	2
6525.2.a.p.1.1		2	120.59	even	2
7105.2.a.e.1.1		2	56.27	even	2
9280.2.a.w.1.1		2	1.1	even	1	trivial
9280.2.a.be.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:	\( N \)	\(=\)	\( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9280.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{3} -1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} -4.82843 q^{11} +2.00000 q^{13} +2.00000 q^{15} -2.82843 q^{17} +0.828427 q^{19} +1.65685 q^{21} +8.82843 q^{23} +1.00000 q^{25} +4.00000 q^{27} -1.00000 q^{29} +10.4853 q^{31} +9.65685 q^{33} +0.828427 q^{35} -8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} -6.00000 q^{43} -1.00000 q^{45} +0.343146 q^{47} -6.31371 q^{49} +5.65685 q^{51} -7.65685 q^{53} +4.82843 q^{55} -1.65685 q^{57} -7.65685 q^{61} -0.828427 q^{63} -2.00000 q^{65} -10.4853 q^{67} -17.6569 q^{69} -7.31371 q^{71} -8.48528 q^{73} -2.00000 q^{75} +4.00000 q^{77} -14.4853 q^{79} -11.0000 q^{81} +12.8284 q^{83} +2.82843 q^{85} +2.00000 q^{87} +3.65685 q^{89} -1.65685 q^{91} -20.9706 q^{93} -0.828427 q^{95} +4.48528 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} - 4 q^{19} - 8 q^{21} + 12 q^{23} + 2 q^{25} + 8 q^{27} - 2 q^{29} + 4 q^{31} + 8 q^{33} - 4 q^{35} - 8 q^{39} - 12 q^{41} - 12 q^{43}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 - 4 * q^19 - 8 * q^21 + 12 * q^23 + 2 * q^25 + 8 * q^27 - 2 * q^29 + 4 * q^31 + 8 * q^33 - 4 * q^35 - 8 * q^39 - 12 * q^41 - 12 * q^43 - 2 * q^45 + 12 * q^47 + 10 * q^49 - 4 * q^53 + 4 * q^55 + 8 * q^57 - 4 * q^61 + 4 * q^63 - 4 * q^65 - 4 * q^67 - 24 * q^69 + 8 * q^71 - 4 * q^75 + 8 * q^77 - 12 * q^79 - 22 * q^81 + 20 * q^83 + 4 * q^87 - 4 * q^89 + 8 * q^91 - 8 * q^93 + 4 * q^95 - 8 * q^97 - 4 * q^99

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