LMFDB - Embedded newform 8281.2.a.cw.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.09197 q^{2} -1.39541 q^{3} -0.807612 q^{4} -1.62650 q^{5} +1.52374 q^{6} +3.06581 q^{8} -1.05283 q^{9} +O(q^{10})$	\(q-1.09197 q^{2} -1.39541 q^{3} -0.807612 q^{4} -1.62650 q^{5} +1.52374 q^{6} +3.06581 q^{8} -1.05283 q^{9} +1.77608 q^{10} +1.29380 q^{11} +1.12695 q^{12} +2.26964 q^{15} -1.73254 q^{16} +5.41114 q^{17} +1.14965 q^{18} -1.51098 q^{19} +1.31358 q^{20} -1.41279 q^{22} -2.64242 q^{23} -4.27807 q^{24} -2.35449 q^{25} +5.65536 q^{27} +5.81805 q^{29} -2.47837 q^{30} +7.28103 q^{31} -4.23976 q^{32} -1.80538 q^{33} -5.90878 q^{34} +0.850278 q^{36} +6.95363 q^{37} +1.64994 q^{38} -4.98656 q^{40} -8.09856 q^{41} +11.1034 q^{43} -1.04489 q^{44} +1.71243 q^{45} +2.88543 q^{46} +7.17743 q^{47} +2.41760 q^{48} +2.57102 q^{50} -7.55076 q^{51} +4.66907 q^{53} -6.17546 q^{54} -2.10437 q^{55} +2.10844 q^{57} -6.35311 q^{58} +0.773673 q^{59} -1.83299 q^{60} -8.74201 q^{61} -7.95063 q^{62} +8.09475 q^{64} +1.97142 q^{66} -6.37515 q^{67} -4.37010 q^{68} +3.68726 q^{69} -11.9090 q^{71} -3.22778 q^{72} +7.21294 q^{73} -7.59312 q^{74} +3.28548 q^{75} +1.22029 q^{76} -11.7793 q^{79} +2.81798 q^{80} -4.73306 q^{81} +8.84334 q^{82} +8.42105 q^{83} -8.80124 q^{85} -12.1246 q^{86} -8.11857 q^{87} +3.96656 q^{88} -1.66642 q^{89} -1.86992 q^{90} +2.13405 q^{92} -10.1600 q^{93} -7.83751 q^{94} +2.45762 q^{95} +5.91620 q^{96} +12.4755 q^{97} -1.36215 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9}+O(q^{10}) $ 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9	$ 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27} + 4 q^{29} - 22 q^{30} + 40 q^{31} + 6 q^{32} + 24 q^{33} + 36 q^{34} - 15 q^{36} + 4 q^{37} + 29 q^{38} + 4 q^{40} + 49 q^{41} + 13 q^{43} - 10 q^{44} + 58 q^{45} + 10 q^{46} + 62 q^{47} - 89 q^{48} + 23 q^{50} - 21 q^{51} - 18 q^{53} + 12 q^{54} + 14 q^{55} + 13 q^{57} - 56 q^{58} + 79 q^{59} - 22 q^{60} - 13 q^{61} - 12 q^{62} + 18 q^{64} + 38 q^{66} + 2 q^{67} + 12 q^{68} + 28 q^{69} + 19 q^{71} - 81 q^{72} + 17 q^{73} + 17 q^{74} - 24 q^{75} + 58 q^{76} - 9 q^{79} + 63 q^{80} + 16 q^{81} + 22 q^{82} + 81 q^{83} + 34 q^{85} - 22 q^{86} - 70 q^{87} - 33 q^{88} + 72 q^{89} - q^{90} - 4 q^{92} - 19 q^{93} + 30 q^{94} + 13 q^{95} + 11 q^{96} + 45 q^{97} + 39 q^{99}+O(q^{100}) $ 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.09197	−0.772136	−0.386068	−	0.922470i	$-0.626167\pi$
$2$	−1.09197	−0.772136	−0.386068	+	0.922470i	$0.626167\pi$
$3$	−1.39541	−0.805640	−0.402820	−	0.915279i	$-0.631970\pi$
$3$	−1.39541	−0.805640	−0.402820	+	0.915279i	$0.631970\pi$
$4$	−0.807612	−0.403806
$5$	−1.62650	−0.727394	−0.363697	−	0.931517i	$-0.618486\pi$
$5$	−1.62650	−0.727394	−0.363697	+	0.931517i	$0.618486\pi$
$6$	1.52374	0.622064
$7$	0	0
$7$	0	0
$8$	3.06581	1.08393
$9$	−1.05283	−0.350943
$10$	1.77608	0.561647
$11$	1.29380	0.390096	0.195048	−	0.980794i	$-0.437514\pi$
$11$	1.29380	0.390096	0.195048	+	0.980794i	$0.437514\pi$
$12$	1.12695	0.325322
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.26964	0.586018
$16$	−1.73254	−0.433135
$17$	5.41114	1.31239	0.656197	−	0.754589i	$-0.272164\pi$
$17$	5.41114	1.31239	0.656197	+	0.754589i	$0.272164\pi$
$18$	1.14965	0.270976
$19$	−1.51098	−0.346643	−0.173322	−	0.984865i	$-0.555450\pi$
$19$	−1.51098	−0.346643	−0.173322	+	0.984865i	$0.555450\pi$
$20$	1.31358	0.293726
$21$	0	0
$22$	−1.41279	−0.301207
$23$	−2.64242	−0.550982	−0.275491	−	0.961304i	$-0.588840\pi$
$23$	−2.64242	−0.550982	−0.275491	+	0.961304i	$0.588840\pi$
$24$	−4.27807	−0.873257
$25$	−2.35449	−0.470898
$26$	0	0
$27$	5.65536	1.08837
$28$	0	0
$29$	5.81805	1.08038	0.540192	−	0.841542i	$-0.318351\pi$
$29$	5.81805	1.08038	0.540192	+	0.841542i	$0.318351\pi$
$30$	−2.47837	−0.452486
$31$	7.28103	1.30771	0.653856	−	0.756619i	$-0.273150\pi$
$31$	7.28103	1.30771	0.653856	+	0.756619i	$0.273150\pi$
$32$	−4.23976	−0.749490
$33$	−1.80538	−0.314277
$34$	−5.90878	−1.01335
$35$	0	0
$36$	0.850278	0.141713
$37$	6.95363	1.14317	0.571585	−	0.820543i	$-0.306329\pi$
$37$	6.95363	1.14317	0.571585	+	0.820543i	$0.306329\pi$
$38$	1.64994	0.267656
$39$	0	0
$40$	−4.98656	−0.788444
$41$	−8.09856	−1.26478	−0.632391	−	0.774649i	$-0.717926\pi$
$41$	−8.09856	−1.26478	−0.632391	+	0.774649i	$0.717926\pi$
$42$	0	0
$43$	11.1034	1.69326	0.846629	−	0.532183i	$-0.178628\pi$
$43$	11.1034	1.69326	0.846629	+	0.532183i	$0.178628\pi$
$44$	−1.04489	−0.157523
$45$	1.71243	0.255274
$46$	2.88543	0.425433
$47$	7.17743	1.04694	0.523468	−	0.852045i	$-0.324638\pi$
$47$	7.17743	1.04694	0.523468	+	0.852045i	$0.324638\pi$
$48$	2.41760	0.348951
$49$	0	0
$50$	2.57102	0.363597
$51$	−7.55076	−1.05732
$52$	0	0
$53$	4.66907	0.641346	0.320673	−	0.947190i	$-0.396091\pi$
$53$	4.66907	0.641346	0.320673	+	0.947190i	$0.396091\pi$
$54$	−6.17546	−0.840373
$55$	−2.10437	−0.283754
$56$	0	0
$57$	2.10844	0.279270
$58$	−6.35311	−0.834204
$59$	0.773673	0.100724	0.0503618	−	0.998731i	$-0.483963\pi$
$59$	0.773673	0.100724	0.0503618	+	0.998731i	$0.483963\pi$
$60$	−1.83299	−0.236638
$61$	−8.74201	−1.11930	−0.559650	−	0.828729i	$-0.689064\pi$
$61$	−8.74201	−1.11930	−0.559650	+	0.828729i	$0.689064\pi$
$62$	−7.95063	−1.00973
$63$	0	0
$64$	8.09475	1.01184
$65$	0	0
$66$	1.97142	0.242665
$67$	−6.37515	−0.778849	−0.389425	−	0.921058i	$-0.627326\pi$
$67$	−6.37515	−0.778849	−0.389425	+	0.921058i	$0.627326\pi$
$68$	−4.37010	−0.529953
$69$	3.68726	0.443893
$70$	0	0
$71$	−11.9090	−1.41334	−0.706672	−	0.707542i	$-0.749804\pi$
$71$	−11.9090	−1.41334	−0.706672	+	0.707542i	$0.749804\pi$
$72$	−3.22778	−0.380398
$73$	7.21294	0.844211	0.422105	−	0.906547i	$-0.361291\pi$
$73$	7.21294	0.844211	0.422105	+	0.906547i	$0.361291\pi$
$74$	−7.59312	−0.882682
$75$	3.28548	0.379374
$76$	1.22029	0.139977
$77$	0	0
$78$	0	0
$79$	−11.7793	−1.32528	−0.662638	−	0.748940i	$-0.730563\pi$
$79$	−11.7793	−1.32528	−0.662638	+	0.748940i	$0.730563\pi$
$80$	2.81798	0.315060
$81$	−4.73306	−0.525895
$82$	8.84334	0.976584
$83$	8.42105	0.924330	0.462165	−	0.886794i	$-0.347073\pi$
$83$	8.42105	0.924330	0.462165	+	0.886794i	$0.347073\pi$
$84$	0	0
$85$	−8.80124	−0.954629
$86$	−12.1246	−1.30743
$87$	−8.11857	−0.870402
$88$	3.96656	0.422837
$89$	−1.66642	−0.176640	−0.0883202	−	0.996092i	$-0.528150\pi$
$89$	−1.66642	−0.176640	−0.0883202	+	0.996092i	$0.528150\pi$
$90$	−1.86992	−0.197106
$91$	0	0
$92$	2.13405	0.222490
$93$	−10.1600	−1.05354
$94$	−7.83751	−0.808377
$95$	2.45762	0.252146
$96$	5.91620	0.603819
$97$	12.4755	1.26670	0.633350	−	0.773866i	$-0.281679\pi$
$97$	12.4755	1.26670	0.633350	+	0.773866i	$0.281679\pi$
$98$	0	0
$99$	−1.36215	−0.136902
$100$	1.90151	0.190151
$101$	13.7324	1.36642	0.683211	−	0.730221i	$-0.260583\pi$
$101$	13.7324	1.36642	0.683211	+	0.730221i	$0.260583\pi$
$102$	8.24517	0.816394
$103$	7.96767	0.785078	0.392539	−	0.919735i	$-0.371597\pi$
$103$	7.96767	0.785078	0.392539	+	0.919735i	$0.371597\pi$
$104$	0	0
$105$	0	0
$106$	−5.09846	−0.495207
$107$	5.04414	0.487635	0.243818	−	0.969821i	$-0.421600\pi$
$107$	5.04414	0.487635	0.243818	+	0.969821i	$0.421600\pi$
$108$	−4.56734	−0.439492
$109$	13.2177	1.26603	0.633013	−	0.774141i	$-0.281818\pi$
$109$	13.2177	1.26603	0.633013	+	0.774141i	$0.281818\pi$
$110$	2.29790	0.219096
$111$	−9.70316	−0.920983
$112$	0	0
$113$	3.36605	0.316651	0.158325	−	0.987387i	$-0.449390\pi$
$113$	3.36605	0.316651	0.158325	+	0.987387i	$0.449390\pi$
$114$	−2.30235	−0.215634
$115$	4.29790	0.400781
$116$	−4.69872	−0.436266
$117$	0	0
$118$	−0.844824	−0.0777724
$119$	0	0
$120$	6.95829	0.635202
$121$	−9.32608	−0.847825
$122$	9.54597	0.864252
$123$	11.3008	1.01896
$124$	−5.88024	−0.528061
$125$	11.9621	1.06992
$126$	0	0
$127$	−18.5546	−1.64646	−0.823229	−	0.567710i	$-0.807830\pi$
$127$	−18.5546	−1.64646	−0.823229	+	0.567710i	$0.807830\pi$
$128$	−0.359671	−0.0317908
$129$	−15.4938	−1.36416
$130$	0	0
$131$	−6.30544	−0.550909	−0.275454	−	0.961314i	$-0.588828\pi$
$131$	−6.30544	−0.550909	−0.275454	+	0.961314i	$0.588828\pi$
$132$	1.45805	0.126907
$133$	0	0
$134$	6.96145	0.601377
$135$	−9.19846	−0.791678
$136$	16.5896	1.42254
$137$	−15.7226	−1.34328	−0.671638	−	0.740880i	$-0.734409\pi$
$137$	−15.7226	−1.34328	−0.671638	+	0.740880i	$0.734409\pi$
$138$	−4.02636	−0.342746
$139$	−4.15899	−0.352761	−0.176380	−	0.984322i	$-0.556439\pi$
$139$	−4.15899	−0.352761	−0.176380	+	0.984322i	$0.556439\pi$
$140$	0	0
$141$	−10.0155	−0.843454
$142$	13.0043	1.09129
$143$	0	0
$144$	1.82407	0.152006
$145$	−9.46308	−0.785866
$146$	−7.87628	−0.651845
$147$	0	0
$148$	−5.61583	−0.461618
$149$	−16.2402	−1.33045	−0.665225	−	0.746643i	$-0.731665\pi$
$149$	−16.2402	−1.33045	−0.665225	+	0.746643i	$0.731665\pi$
$150$	−3.58763	−0.292928
$151$	13.9110	1.13206	0.566032	−	0.824383i	$-0.308478\pi$
$151$	13.9110	1.13206	0.566032	+	0.824383i	$0.308478\pi$
$152$	−4.63240	−0.375737
$153$	−5.69702	−0.460576
$154$	0	0
$155$	−11.8426	−0.951221
$156$	0	0
$157$	−16.2744	−1.29884	−0.649419	−	0.760431i	$-0.724988\pi$
$157$	−16.2744	−1.29884	−0.649419	+	0.760431i	$0.724988\pi$
$158$	12.8626	1.02329
$159$	−6.51527	−0.516695
$160$	6.89598	0.545175
$161$	0	0
$162$	5.16833	0.406063
$163$	1.59294	0.124769	0.0623844	−	0.998052i	$-0.480130\pi$
$163$	1.59294	0.124769	0.0623844	+	0.998052i	$0.480130\pi$
$164$	6.54049	0.510726
$165$	2.93646	0.228603
$166$	−9.19549	−0.713709
$167$	−16.0351	−1.24083	−0.620415	−	0.784274i	$-0.713036\pi$
$167$	−16.0351	−1.24083	−0.620415	+	0.784274i	$0.713036\pi$
$168$	0	0
$169$	0	0
$170$	9.61065	0.737103
$171$	1.59081	0.121652
$172$	−8.96727	−0.683748
$173$	−21.2070	−1.61234	−0.806170	−	0.591684i	$-0.798463\pi$
$173$	−21.2070	−1.61234	−0.806170	+	0.591684i	$0.798463\pi$
$174$	8.86519	0.672069
$175$	0	0
$176$	−2.24156	−0.168964
$177$	−1.07959	−0.0811470
$178$	1.81968	0.136390
$179$	−9.24292	−0.690848	−0.345424	−	0.938447i	$-0.612265\pi$
$179$	−9.24292	−0.690848	−0.345424	+	0.938447i	$0.612265\pi$
$180$	−1.38298	−0.103081
$181$	3.45063	0.256484	0.128242	−	0.991743i	$-0.459067\pi$
$181$	3.45063	0.256484	0.128242	+	0.991743i	$0.459067\pi$
$182$	0	0
$183$	12.1987	0.901753
$184$	−8.10116	−0.597226
$185$	−11.3101	−0.831535
$186$	11.0944	0.813480
$187$	7.00095	0.511960
$188$	−5.79658	−0.422759
$189$	0	0
$190$	−2.68363	−0.194691
$191$	−21.1307	−1.52897	−0.764483	−	0.644644i	$-0.777005\pi$
$191$	−21.1307	−1.52897	−0.764483	+	0.644644i	$0.777005\pi$
$192$	−11.2955	−0.815182
$193$	−11.1138	−0.799988	−0.399994	−	0.916518i	$-0.630988\pi$
$193$	−11.1138	−0.799988	−0.399994	+	0.916518i	$0.630988\pi$
$194$	−13.6229	−0.978064
$195$	0	0
$196$	0	0
$197$	17.6129	1.25487	0.627435	−	0.778669i	$-0.284105\pi$
$197$	17.6129	1.25487	0.627435	+	0.778669i	$0.284105\pi$
$198$	1.48743	0.105707
$199$	12.8366	0.909966	0.454983	−	0.890500i	$-0.349645\pi$
$199$	12.8366	0.909966	0.454983	+	0.890500i	$0.349645\pi$
$200$	−7.21842	−0.510420
$201$	8.89595	0.627472
$202$	−14.9953	−1.05506
$203$	0	0
$204$	6.09809	0.426951
$205$	13.1723	0.919995
$206$	−8.70042	−0.606187
$207$	2.78202	0.193364
$208$	0	0
$209$	−1.95491	−0.135224
$210$	0	0
$211$	−15.2632	−1.05076	−0.525382	−	0.850866i	$-0.676078\pi$
$211$	−15.2632	−1.05076	−0.525382	+	0.850866i	$0.676078\pi$
$212$	−3.77080	−0.258979
$213$	16.6180	1.13865
$214$	−5.50802	−0.376521
$215$	−18.0598	−1.23167
$216$	17.3383	1.17972
$217$	0	0
$218$	−14.4333	−0.977545
$219$	−10.0650	−0.680130
$220$	1.69952	0.114581
$221$	0	0
$222$	10.5955	0.711125
$223$	11.0502	0.739973	0.369986	−	0.929037i	$-0.379362\pi$
$223$	11.0502	0.739973	0.369986	+	0.929037i	$0.379362\pi$
$224$	0	0
$225$	2.47888	0.165258
$226$	−3.67561	−0.244498
$227$	25.4692	1.69045	0.845225	−	0.534410i	$-0.179466\pi$
$227$	25.4692	1.69045	0.845225	+	0.534410i	$0.179466\pi$
$228$	−1.70280	−0.112771
$229$	11.4928	0.759462	0.379731	−	0.925097i	$-0.376016\pi$
$229$	11.4928	0.759462	0.379731	+	0.925097i	$0.376016\pi$
$230$	−4.69316	−0.309458
$231$	0	0
$232$	17.8371	1.17106
$233$	27.0906	1.77477	0.887383	−	0.461034i	$-0.152521\pi$
$233$	27.0906	1.77477	0.887383	+	0.461034i	$0.152521\pi$
$234$	0	0
$235$	−11.6741	−0.761536
$236$	−0.624827	−0.0406728
$237$	16.4370	1.06770
$238$	0	0
$239$	2.02896	0.131243	0.0656213	−	0.997845i	$-0.479097\pi$
$239$	2.02896	0.131243	0.0656213	+	0.997845i	$0.479097\pi$
$240$	−3.93224	−0.253825
$241$	−22.8571	−1.47235	−0.736177	−	0.676789i	$-0.763371\pi$
$241$	−22.8571	−1.47235	−0.736177	+	0.676789i	$0.763371\pi$
$242$	10.1838	0.654636
$243$	−10.3615	−0.664692
$244$	7.06015	0.451980
$245$	0	0
$246$	−12.3401	−0.786776
$247$	0	0
$248$	22.3223	1.41747
$249$	−11.7508	−0.744678
$250$	−13.0622	−0.826126
$251$	15.5048	0.978653	0.489326	−	0.872101i	$-0.337243\pi$
$251$	15.5048	0.978653	0.489326	+	0.872101i	$0.337243\pi$
$252$	0	0
$253$	−3.41877	−0.214936
$254$	20.2610	1.27129
$255$	12.2813	0.769087
$256$	−15.7967	−0.987297
$257$	−0.164954	−0.0102896	−0.00514478	−	0.999987i	$-0.501638\pi$
$257$	−0.164954	−0.0102896	−0.00514478	+	0.999987i	$0.501638\pi$
$258$	16.9187	1.05332
$259$	0	0
$260$	0	0
$261$	−6.12542	−0.379154
$262$	6.88532	0.425377
$263$	2.39179	0.147484	0.0737420	−	0.997277i	$-0.476506\pi$
$263$	2.39179	0.147484	0.0737420	+	0.997277i	$0.476506\pi$
$264$	−5.53498	−0.340654
$265$	−7.59426	−0.466512
$266$	0	0
$267$	2.32534	0.142309
$268$	5.14865	0.314504
$269$	−4.96352	−0.302631	−0.151316	−	0.988486i	$-0.548351\pi$
$269$	−4.96352	−0.302631	−0.151316	+	0.988486i	$0.548351\pi$
$270$	10.0444	0.611283
$271$	24.3486	1.47907	0.739535	−	0.673118i	$-0.235046\pi$
$271$	24.3486	1.47907	0.739535	+	0.673118i	$0.235046\pi$
$272$	−9.37502	−0.568444
$273$	0	0
$274$	17.1686	1.03719
$275$	−3.04624	−0.183695
$276$	−2.97787	−0.179247
$277$	−16.5246	−0.992864	−0.496432	−	0.868076i	$-0.665357\pi$
$277$	−16.5246	−0.992864	−0.496432	+	0.868076i	$0.665357\pi$
$278$	4.54147	0.272379
$279$	−7.66569	−0.458933
$280$	0	0
$281$	−3.36281	−0.200608	−0.100304	−	0.994957i	$-0.531982\pi$
$281$	−3.36281	−0.200608	−0.100304	+	0.994957i	$0.531982\pi$
$282$	10.9365	0.651262
$283$	−26.8550	−1.59636	−0.798181	−	0.602418i	$-0.794204\pi$
$283$	−26.8550	−1.59636	−0.798181	+	0.602418i	$0.794204\pi$
$284$	9.61788	0.570716
$285$	−3.42939	−0.203139
$286$	0	0
$287$	0	0
$288$	4.46374	0.263029
$289$	12.2805	0.722380
$290$	10.3334	0.606795
$291$	−17.4085	−1.02050
$292$	−5.82525	−0.340897
$293$	−1.36620	−0.0798144	−0.0399072	−	0.999203i	$-0.512706\pi$
$293$	−1.36620	−0.0798144	−0.0399072	+	0.999203i	$0.512706\pi$
$294$	0	0
$295$	−1.25838	−0.0732658
$296$	21.3185	1.23911
$297$	7.31692	0.424571
$298$	17.7338	1.02729
$299$	0	0
$300$	−2.65339	−0.153193
$301$	0	0
$302$	−15.1904	−0.874107
$303$	−19.1623	−1.10084
$304$	2.61784	0.150143
$305$	14.2189	0.814172
$306$	6.22094	0.355628
$307$	20.3099	1.15915	0.579574	−	0.814919i	$-0.303219\pi$
$307$	20.3099	1.15915	0.579574	+	0.814919i	$0.303219\pi$
$308$	0	0
$309$	−11.1182	−0.632491
$310$	12.9317	0.734472
$311$	−15.5524	−0.881897	−0.440949	−	0.897532i	$-0.645358\pi$
$311$	−15.5524	−0.881897	−0.440949	+	0.897532i	$0.645358\pi$
$312$	0	0
$313$	14.9628	0.845749	0.422874	−	0.906188i	$-0.361021\pi$
$313$	14.9628	0.845749	0.422874	+	0.906188i	$0.361021\pi$
$314$	17.7711	1.00288
$315$	0	0
$316$	9.51311	0.535154
$317$	−16.9460	−0.951784	−0.475892	−	0.879504i	$-0.657875\pi$
$317$	−16.9460	−0.951784	−0.475892	+	0.879504i	$0.657875\pi$
$318$	7.11445	0.398959
$319$	7.52741	0.421454
$320$	−13.1661	−0.736009
$321$	−7.03864	−0.392858
$322$	0	0
$323$	−8.17615	−0.454933
$324$	3.82247	0.212360
$325$	0	0
$326$	−1.73944	−0.0963385
$327$	−18.4441	−1.01996
$328$	−24.8287	−1.37093
$329$	0	0
$330$	−3.20652	−0.176513
$331$	−0.841094	−0.0462307	−0.0231154	−	0.999733i	$-0.507359\pi$
$331$	−0.841094	−0.0462307	−0.0231154	+	0.999733i	$0.507359\pi$
$332$	−6.80094	−0.373250
$333$	−7.32099	−0.401188
$334$	17.5097	0.958090
$335$	10.3692	0.566530
$336$	0	0
$337$	−2.11527	−0.115226	−0.0576130	−	0.998339i	$-0.518349\pi$
$337$	−2.11527	−0.115226	−0.0576130	+	0.998339i	$0.518349\pi$
$338$	0	0
$339$	−4.69701	−0.255107
$340$	7.10798	0.385485
$341$	9.42021	0.510133
$342$	−1.73711	−0.0939321
$343$	0	0
$344$	34.0411	1.83537
$345$	−5.99733	−0.322886
$346$	23.1573	1.24495
$347$	13.3398	0.716120	0.358060	−	0.933699i	$-0.383438\pi$
$347$	13.3398	0.716120	0.358060	+	0.933699i	$0.383438\pi$
$348$	6.55665	0.351473
$349$	0.969523	0.0518974	0.0259487	−	0.999663i	$-0.491739\pi$
$349$	0.969523	0.0518974	0.0259487	+	0.999663i	$0.491739\pi$
$350$	0	0
$351$	0	0
$352$	−5.48541	−0.292373
$353$	2.63810	0.140412	0.0702060	−	0.997533i	$-0.477634\pi$
$353$	2.63810	0.140412	0.0702060	+	0.997533i	$0.477634\pi$
$354$	1.17888	0.0626566
$355$	19.3701	1.02806
$356$	1.34582	0.0713284
$357$	0	0
$358$	10.0929	0.533429
$359$	1.85497	0.0979013	0.0489507	−	0.998801i	$-0.484412\pi$
$359$	1.85497	0.0979013	0.0489507	+	0.998801i	$0.484412\pi$
$360$	5.25000	0.276699
$361$	−16.7169	−0.879838
$362$	−3.76797	−0.198040
$363$	13.0137	0.683042
$364$	0	0
$365$	−11.7319	−0.614074
$366$	−13.3205	−0.696276
$367$	−21.1956	−1.10640	−0.553200	−	0.833048i	$-0.686594\pi$
$367$	−21.1956	−1.10640	−0.553200	+	0.833048i	$0.686594\pi$
$368$	4.57809	0.238650
$369$	8.52641	0.443867
$370$	12.3502	0.642058
$371$	0	0
$372$	8.20535	0.425428
$373$	28.1321	1.45662	0.728312	−	0.685246i	$-0.240305\pi$
$373$	28.1321	1.45662	0.728312	+	0.685246i	$0.240305\pi$
$374$	−7.64479	−0.395303
$375$	−16.6920	−0.861973
$376$	22.0047	1.13480
$377$	0	0
$378$	0	0
$379$	2.95326	0.151699	0.0758493	−	0.997119i	$-0.475833\pi$
$379$	2.95326	0.151699	0.0758493	+	0.997119i	$0.475833\pi$
$380$	−1.98480	−0.101818
$381$	25.8913	1.32645
$382$	23.0740	1.18057
$383$	30.9138	1.57962	0.789811	−	0.613351i	$-0.210179\pi$
$383$	30.9138	1.57962	0.789811	+	0.613351i	$0.210179\pi$
$384$	0.501889	0.0256119
$385$	0	0
$386$	12.1359	0.617700
$387$	−11.6900	−0.594238
$388$	−10.0754	−0.511501
$389$	−6.06090	−0.307300	−0.153650	−	0.988125i	$-0.549103\pi$
$389$	−6.06090	−0.307300	−0.153650	+	0.988125i	$0.549103\pi$
$390$	0	0
$391$	−14.2985	−0.723106
$392$	0	0
$393$	8.79868	0.443834
$394$	−19.2327	−0.968931
$395$	19.1591	0.963999
$396$	1.10009	0.0552817
$397$	25.8241	1.29607	0.648037	−	0.761609i	$-0.275590\pi$
$397$	25.8241	1.29607	0.648037	+	0.761609i	$0.275590\pi$
$398$	−14.0172	−0.702617
$399$	0	0
$400$	4.07924	0.203962
$401$	0.285935	0.0142789	0.00713945	−	0.999975i	$-0.497727\pi$
$401$	0.285935	0.0142789	0.00713945	+	0.999975i	$0.497727\pi$
$402$	−9.71407	−0.484494
$403$	0	0
$404$	−11.0904	−0.551769
$405$	7.69833	0.382533
$406$	0	0
$407$	8.99662	0.445946
$408$	−23.1492	−1.14606
$409$	12.3168	0.609026	0.304513	−	0.952508i	$-0.401506\pi$
$409$	12.3168	0.609026	0.304513	+	0.952508i	$0.401506\pi$
$410$	−14.3837	−0.710362
$411$	21.9395	1.08220
$412$	−6.43478	−0.317019
$413$	0	0
$414$	−3.03787	−0.149303
$415$	−13.6969	−0.672352
$416$	0	0
$417$	5.80350	0.284199
$418$	2.13470	0.104411
$419$	−2.22674	−0.108783	−0.0543916	−	0.998520i	$-0.517322\pi$
$419$	−2.22674	−0.108783	−0.0543916	+	0.998520i	$0.517322\pi$
$420$	0	0
$421$	19.3019	0.940715	0.470358	−	0.882476i	$-0.344125\pi$
$421$	19.3019	0.940715	0.470358	+	0.882476i	$0.344125\pi$
$422$	16.6669	0.811334
$423$	−7.55662	−0.367416
$424$	14.3145	0.695174
$425$	−12.7405	−0.618004
$426$	−18.1463	−0.879190
$427$	0	0
$428$	−4.07370	−0.196910
$429$	0	0
$430$	19.7206	0.951014
$431$	−11.1891	−0.538959	−0.269479	−	0.963006i	$-0.586852\pi$
$431$	−11.1891	−0.538959	−0.269479	+	0.963006i	$0.586852\pi$
$432$	−9.79814	−0.471413
$433$	32.7754	1.57509	0.787543	−	0.616260i	$-0.211353\pi$
$433$	32.7754	1.57509	0.787543	+	0.616260i	$0.211353\pi$
$434$	0	0
$435$	13.2049	0.633125
$436$	−10.6748	−0.511229
$437$	3.99265	0.190994
$438$	10.9906	0.525153
$439$	15.2789	0.729221	0.364610	−	0.931160i	$-0.381202\pi$
$439$	15.2789	0.729221	0.364610	+	0.931160i	$0.381202\pi$
$440$	−6.45162	−0.307569
$441$	0	0
$442$	0	0
$443$	−8.66065	−0.411480	−0.205740	−	0.978607i	$-0.565960\pi$
$443$	−8.66065	−0.411480	−0.205740	+	0.978607i	$0.565960\pi$
$444$	7.83639	0.371898
$445$	2.71044	0.128487
$446$	−12.0664	−0.571360
$447$	22.6618	1.07186
$448$	0	0
$449$	−1.31595	−0.0621035	−0.0310518	−	0.999518i	$-0.509886\pi$
$449$	−1.31595	−0.0621035	−0.0310518	+	0.999518i	$0.509886\pi$
$450$	−2.70685	−0.127602
$451$	−10.4779	−0.493387
$452$	−2.71846	−0.127865
$453$	−19.4116	−0.912037
$454$	−27.8115	−1.30526
$455$	0	0
$456$	6.46409	0.302709
$457$	−1.05378	−0.0492936	−0.0246468	−	0.999696i	$-0.507846\pi$
$457$	−1.05378	−0.0492936	−0.0246468	+	0.999696i	$0.507846\pi$
$458$	−12.5497	−0.586408
$459$	30.6020	1.42838
$460$	−3.47103	−0.161838
$461$	17.1485	0.798686	0.399343	−	0.916802i	$-0.369238\pi$
$461$	17.1485	0.798686	0.399343	+	0.916802i	$0.369238\pi$
$462$	0	0
$463$	−0.0166360	−0.000773140 0	−0.000386570	−	1.00000i	$-0.500123\pi$
$463$	−0.0166360	−0.000773140 0	−0.000386570		1.00000i	$0.500123\pi$
$464$	−10.0800	−0.467952
$465$	16.5253	0.766342
$466$	−29.5820	−1.37036
$467$	−32.5683	−1.50708	−0.753541	−	0.657401i	$-0.771656\pi$
$467$	−32.5683	−1.50708	−0.753541	+	0.657401i	$0.771656\pi$
$468$	0	0
$469$	0	0
$470$	12.7477	0.588009
$471$	22.7095	1.04640
$472$	2.37194	0.109177
$473$	14.3657	0.660533
$474$	−17.9486	−0.824407
$475$	3.55759	0.163234
$476$	0	0
$477$	−4.91574	−0.225076
$478$	−2.21555	−0.101337
$479$	−1.66330	−0.0759980	−0.0379990	−	0.999278i	$-0.512098\pi$
$479$	−1.66330	−0.0759980	−0.0379990	+	0.999278i	$0.512098\pi$
$480$	−9.62271	−0.439215
$481$	0	0
$482$	24.9591	1.13686
$483$	0	0
$484$	7.53185	0.342357
$485$	−20.2915	−0.921390
$486$	11.3144	0.513233
$487$	34.3094	1.55471	0.777354	−	0.629063i	$-0.216561\pi$
$487$	34.3094	1.55471	0.777354	+	0.629063i	$0.216561\pi$
$488$	−26.8014	−1.21324
$489$	−2.22281	−0.100519
$490$	0	0
$491$	26.5547	1.19840	0.599198	−	0.800601i	$-0.295486\pi$
$491$	26.5547	1.19840	0.599198	+	0.800601i	$0.295486\pi$
$492$	−9.12667	−0.411462
$493$	31.4823	1.41789
$494$	0	0
$495$	2.21555	0.0995815
$496$	−12.6147	−0.566415
$497$	0	0
$498$	12.8315	0.574992
$499$	11.4889	0.514314	0.257157	−	0.966370i	$-0.417214\pi$
$499$	11.4889	0.514314	0.257157	+	0.966370i	$0.417214\pi$
$500$	−9.66073	−0.432041
$501$	22.3755	0.999663
$502$	−16.9307	−0.755653
$503$	−5.62545	−0.250826	−0.125413	−	0.992105i	$-0.540026\pi$
$503$	−5.62545	−0.250826	−0.125413	+	0.992105i	$0.540026\pi$
$504$	0	0
$505$	−22.3357	−0.993927
$506$	3.73317	0.165960
$507$	0	0
$508$	14.9849	0.664849
$509$	31.7054	1.40532	0.702659	−	0.711526i	$-0.251996\pi$
$509$	31.7054	1.40532	0.702659	+	0.711526i	$0.251996\pi$
$510$	−13.4108	−0.593840
$511$	0	0
$512$	17.9688	0.794118
$513$	−8.54516	−0.377278
$514$	0.180124	0.00794495
$515$	−12.9594	−0.571061
$516$	12.5130	0.550855
$517$	9.28618	0.408406
$518$	0	0
$519$	29.5925	1.29897
$520$	0	0
$521$	37.0513	1.62324	0.811622	−	0.584182i	$-0.198585\pi$
$521$	37.0513	1.62324	0.811622	+	0.584182i	$0.198585\pi$
$522$	6.68875	0.292758
$523$	24.5882	1.07517	0.537584	−	0.843210i	$-0.319337\pi$
$523$	24.5882	1.07517	0.537584	+	0.843210i	$0.319337\pi$
$524$	5.09235	0.222460
$525$	0	0
$526$	−2.61175	−0.113878
$527$	39.3987	1.71623
$528$	3.12790	0.136124
$529$	−16.0176	−0.696419
$530$	8.29267	0.360211
$531$	−0.814547	−0.0353483
$532$	0	0
$533$	0	0
$534$	−2.53919	−0.109882
$535$	−8.20430	−0.354703
$536$	−19.5450	−0.844217
$537$	12.8977	0.556575
$538$	5.41999	0.233672
$539$	0	0
$540$	7.42878	0.319684
$541$	−36.4451	−1.56690	−0.783449	−	0.621456i	$-0.786541\pi$
$541$	−36.4451	−1.56690	−0.783449	+	0.621456i	$0.786541\pi$
$542$	−26.5878	−1.14204
$543$	−4.81505	−0.206634
$544$	−22.9419	−0.983627
$545$	−21.4986	−0.920901
$546$	0	0
$547$	−33.2861	−1.42321	−0.711606	−	0.702578i	$-0.752032\pi$
$547$	−33.2861	−1.42321	−0.711606	+	0.702578i	$0.752032\pi$
$548$	12.6978	0.542422
$549$	9.20386	0.392811
$550$	3.32639	0.141838
$551$	−8.79098	−0.374508
$552$	11.3044	0.481149
$553$	0	0
$554$	18.0442	0.766626
$555$	15.7822	0.669918
$556$	3.35885	0.142447
$557$	28.0776	1.18969	0.594844	−	0.803841i	$-0.297214\pi$
$557$	28.0776	1.18969	0.594844	+	0.803841i	$0.297214\pi$
$558$	8.37066	0.354358
$559$	0	0
$560$	0	0
$561$	−9.76920	−0.412456
$562$	3.67207	0.154897
$563$	25.9343	1.09300	0.546500	−	0.837459i	$-0.315960\pi$
$563$	25.9343	1.09300	0.546500	+	0.837459i	$0.315960\pi$
$564$	8.08861	0.340592
$565$	−5.47488	−0.230330
$566$	29.3247	1.23261
$567$	0	0
$568$	−36.5109	−1.53196
$569$	23.1477	0.970400	0.485200	−	0.874403i	$-0.338747\pi$
$569$	23.1477	0.970400	0.485200	+	0.874403i	$0.338747\pi$
$570$	3.74477	0.156851
$571$	−11.6082	−0.485786	−0.242893	−	0.970053i	$-0.578096\pi$
$571$	−11.6082	−0.485786	−0.242893	+	0.970053i	$0.578096\pi$
$572$	0	0
$573$	29.4860	1.23180
$574$	0	0
$575$	6.22154	0.259456
$576$	−8.52240	−0.355100
$577$	−27.4823	−1.14410	−0.572051	−	0.820218i	$-0.693852\pi$
$577$	−27.4823	−1.14410	−0.572051	+	0.820218i	$0.693852\pi$
$578$	−13.4098	−0.557776
$579$	15.5083	0.644503
$580$	7.64249	0.317337
$581$	0	0
$582$	19.0095	0.787968
$583$	6.04086	0.250187
$584$	22.1135	0.915064
$585$	0	0
$586$	1.49185	0.0616276
$587$	2.96435	0.122352	0.0611758	−	0.998127i	$-0.480515\pi$
$587$	2.96435	0.122352	0.0611758	+	0.998127i	$0.480515\pi$
$588$	0	0
$589$	−11.0015	−0.453309
$590$	1.37411	0.0565712
$591$	−24.5773	−1.01097
$592$	−12.0474	−0.495147
$593$	37.3843	1.53519	0.767594	−	0.640936i	$-0.221454\pi$
$593$	37.3843	1.53519	0.767594	+	0.640936i	$0.221454\pi$
$594$	−7.98982	−0.327826
$595$	0	0
$596$	13.1158	0.537244
$597$	−17.9124	−0.733105
$598$	0	0
$599$	−3.13658	−0.128157	−0.0640787	−	0.997945i	$-0.520411\pi$
$599$	−3.13658	−0.128157	−0.0640787	+	0.997945i	$0.520411\pi$
$600$	10.0727	0.411215
$601$	−34.6248	−1.41237	−0.706187	−	0.708026i	$-0.749586\pi$
$601$	−34.6248	−1.41237	−0.706187	+	0.708026i	$0.749586\pi$
$602$	0	0
$603$	6.71196	0.273332
$604$	−11.2347	−0.457134
$605$	15.1689	0.616703
$606$	20.9245	0.850002
$607$	5.91569	0.240110	0.120055	−	0.992767i	$-0.461693\pi$
$607$	5.91569	0.240110	0.120055	+	0.992767i	$0.461693\pi$
$608$	6.40620	0.259806
$609$	0	0
$610$	−15.5266	−0.628652
$611$	0	0
$612$	4.60098	0.185983
$613$	1.78229	0.0719860	0.0359930	−	0.999352i	$-0.488541\pi$
$613$	1.78229	0.0719860	0.0359930	+	0.999352i	$0.488541\pi$
$614$	−22.1777	−0.895021
$615$	−18.3808	−0.741185
$616$	0	0
$617$	−17.1213	−0.689276	−0.344638	−	0.938736i	$-0.611998\pi$
$617$	−17.1213	−0.689276	−0.344638	+	0.938736i	$0.611998\pi$
$618$	12.1407	0.488369
$619$	16.6280	0.668334	0.334167	−	0.942514i	$-0.391545\pi$
$619$	16.6280	0.668334	0.334167	+	0.942514i	$0.391545\pi$
$620$	9.56423	0.384109
$621$	−14.9438	−0.599675
$622$	16.9827	0.680945
$623$	0	0
$624$	0	0
$625$	−7.68395	−0.307358
$626$	−16.3389	−0.653033
$627$	2.72791	0.108942
$628$	13.1434	0.524479
$629$	37.6271	1.50029
$630$	0	0
$631$	−18.0007	−0.716595	−0.358297	−	0.933608i	$-0.616643\pi$
$631$	−18.0007	−0.716595	−0.358297	+	0.933608i	$0.616643\pi$
$632$	−36.1132	−1.43651
$633$	21.2985	0.846539
$634$	18.5045	0.734907
$635$	30.1792	1.19762
$636$	5.26181	0.208644
$637$	0	0
$638$	−8.21967	−0.325420
$639$	12.5382	0.496004
$640$	0.585007	0.0231244
$641$	−0.791851	−0.0312762	−0.0156381	−	0.999878i	$-0.504978\pi$
$641$	−0.791851	−0.0312762	−0.0156381	+	0.999878i	$0.504978\pi$
$642$	7.68595	0.303340
$643$	−5.45711	−0.215207	−0.107604	−	0.994194i	$-0.534318\pi$
$643$	−5.45711	−0.215207	−0.107604	+	0.994194i	$0.534318\pi$
$644$	0	0
$645$	25.2008	0.992280
$646$	8.92807	0.351270
$647$	−26.9726	−1.06040	−0.530202	−	0.847871i	$-0.677884\pi$
$647$	−26.9726	−1.06040	−0.530202	+	0.847871i	$0.677884\pi$
$648$	−14.5107	−0.570033
$649$	1.00098	0.0392919
$650$	0	0
$651$	0	0
$652$	−1.28648	−0.0503824
$653$	−13.7576	−0.538378	−0.269189	−	0.963087i	$-0.586756\pi$
$653$	−13.7576	−0.538378	−0.269189	+	0.963087i	$0.586756\pi$
$654$	20.1403	0.787550
$655$	10.2558	0.400728
$656$	14.0311	0.547821
$657$	−7.59400	−0.296270
$658$	0	0
$659$	9.56828	0.372727	0.186364	−	0.982481i	$-0.440330\pi$
$659$	9.56828	0.372727	0.186364	+	0.982481i	$0.440330\pi$
$660$	−2.37152	−0.0923114
$661$	−0.0106544	−0.000414408 0	−0.000207204	−	1.00000i	$-0.500066\pi$
$661$	−0.0106544	−0.000414408 0	−0.000207204		1.00000i	$0.500066\pi$
$662$	0.918446	0.0356964
$663$	0	0
$664$	25.8174	1.00191
$665$	0	0
$666$	7.99427	0.309772
$667$	−15.3737	−0.595273
$668$	12.9501	0.501055
$669$	−15.4195	−0.596152
$670$	−11.3228	−0.437439
$671$	−11.3104	−0.436634
$672$	0	0
$673$	43.9527	1.69425	0.847127	−	0.531391i	$-0.178330\pi$
$673$	43.9527	1.69425	0.847127	+	0.531391i	$0.178330\pi$
$674$	2.30980	0.0889702
$675$	−13.3155	−0.512513
$676$	0	0
$677$	−16.3300	−0.627615	−0.313807	−	0.949487i	$-0.601605\pi$
$677$	−16.3300	−0.627615	−0.313807	+	0.949487i	$0.601605\pi$
$678$	5.12898	0.196977
$679$	0	0
$680$	−26.9830	−1.03475
$681$	−35.5400	−1.36190
$682$	−10.2865	−0.393892
$683$	−23.9545	−0.916592	−0.458296	−	0.888800i	$-0.651540\pi$
$683$	−23.9545	−0.916592	−0.458296	+	0.888800i	$0.651540\pi$
$684$	−1.28476	−0.0491239
$685$	25.5729	0.977091
$686$	0	0
$687$	−16.0371	−0.611854
$688$	−19.2371	−0.733409
$689$	0	0
$690$	6.54888	0.249312
$691$	34.1717	1.29995	0.649976	−	0.759955i	$-0.274779\pi$
$691$	34.1717	1.29995	0.649976	+	0.759955i	$0.274779\pi$
$692$	17.1270	0.651072
$693$	0	0
$694$	−14.5666	−0.552942
$695$	6.76461	0.256596
$696$	−24.8900	−0.943454
$697$	−43.8225	−1.65989
$698$	−1.05869	−0.0400718
$699$	−37.8025	−1.42982
$700$	0	0
$701$	−0.381723	−0.0144175	−0.00720874	−	0.999974i	$-0.502295\pi$
$701$	−0.381723	−0.0144175	−0.00720874	+	0.999974i	$0.502295\pi$
$702$	0	0
$703$	−10.5068	−0.396272
$704$	10.4730	0.394716
$705$	16.2902	0.613524
$706$	−2.88072	−0.108417
$707$	0	0
$708$	0.871890	0.0327676
$709$	−11.3399	−0.425878	−0.212939	−	0.977065i	$-0.568304\pi$
$709$	−11.3399	−0.425878	−0.212939	+	0.977065i	$0.568304\pi$
$710$	−21.1515	−0.793800
$711$	12.4016	0.465097
$712$	−5.10894	−0.191466
$713$	−19.2395	−0.720525
$714$	0	0
$715$	0	0
$716$	7.46469	0.278969
$717$	−2.83123	−0.105734
$718$	−2.02556	−0.0755932
$719$	19.2554	0.718107	0.359054	−	0.933317i	$-0.383100\pi$
$719$	19.2554	0.718107	0.359054	+	0.933317i	$0.383100\pi$
$720$	−2.96686	−0.110568
$721$	0	0
$722$	18.2543	0.679355
$723$	31.8950	1.18619
$724$	−2.78677	−0.103570
$725$	−13.6985	−0.508751
$726$	−14.2105	−0.527402
$727$	18.7802	0.696519	0.348259	−	0.937398i	$-0.386773\pi$
$727$	18.7802	0.696519	0.348259	+	0.937398i	$0.386773\pi$
$728$	0	0
$729$	28.6578	1.06140
$730$	12.8108	0.474149
$731$	60.0823	2.22222
$732$	−9.85181	−0.364133
$733$	−36.4640	−1.34683	−0.673414	−	0.739265i	$-0.735173\pi$
$733$	−36.4640	−1.34683	−0.673414	+	0.739265i	$0.735173\pi$
$734$	23.1448	0.854292
$735$	0	0
$736$	11.2032	0.412956
$737$	−8.24819	−0.303826
$738$	−9.31054	−0.342726
$739$	33.7745	1.24241	0.621207	−	0.783646i	$-0.286642\pi$
$739$	33.7745	1.24241	0.621207	+	0.783646i	$0.286642\pi$
$740$	9.13417	0.335779
$741$	0	0
$742$	0	0
$743$	31.9131	1.17078	0.585389	−	0.810752i	$-0.300942\pi$
$743$	31.9131	1.17078	0.585389	+	0.810752i	$0.300942\pi$
$744$	−31.1487	−1.14197
$745$	26.4148	0.967762
$746$	−30.7192	−1.12471
$747$	−8.86593	−0.324388
$748$	−5.65405	−0.206732
$749$	0	0
$750$	18.2271	0.665560
$751$	3.06193	0.111731	0.0558656	−	0.998438i	$-0.482208\pi$
$751$	3.06193	0.111731	0.0558656	+	0.998438i	$0.482208\pi$
$752$	−12.4352	−0.453465
$753$	−21.6355	−0.788442
$754$	0	0
$755$	−22.6263	−0.823457
$756$	0	0
$757$	−43.7489	−1.59008	−0.795041	−	0.606555i	$-0.792551\pi$
$757$	−43.7489	−1.59008	−0.795041	+	0.606555i	$0.792551\pi$
$758$	−3.22485	−0.117132
$759$	4.77058	0.173161
$760$	7.53460	0.273309
$761$	43.4455	1.57490	0.787449	−	0.616380i	$-0.211402\pi$
$761$	43.4455	1.57490	0.787449	+	0.616380i	$0.211402\pi$
$762$	−28.2724	−1.02420
$763$	0	0
$764$	17.0654	0.617405
$765$	9.26621	0.335021
$766$	−33.7568	−1.21968
$767$	0	0
$768$	22.0429	0.795406
$769$	−9.02061	−0.325291	−0.162646	−	0.986685i	$-0.552003\pi$
$769$	−9.02061	−0.325291	−0.162646	+	0.986685i	$0.552003\pi$
$770$	0	0
$771$	0.230179	0.00828969
$772$	8.97563	0.323040
$773$	18.5721	0.667991	0.333996	−	0.942575i	$-0.391603\pi$
$773$	18.5721	0.667991	0.333996	+	0.942575i	$0.391603\pi$
$774$	12.7651	0.458833
$775$	−17.1431	−0.615798
$776$	38.2477	1.37301
$777$	0	0
$778$	6.61829	0.237277
$779$	12.2368	0.438428
$780$	0	0
$781$	−15.4079	−0.551340
$782$	15.6135	0.558336
$783$	32.9032	1.17586
$784$	0	0
$785$	26.4704	0.944768
$786$	−9.60785	−0.342701
$787$	0.132844	0.00473538	0.00236769	−	0.999997i	$-0.499246\pi$
$787$	0.132844	0.00473538	0.00236769	+	0.999997i	$0.499246\pi$
$788$	−14.2244	−0.506724
$789$	−3.33753	−0.118819
$790$	−20.9211	−0.744338
$791$	0	0
$792$	−4.17611	−0.148392
$793$	0	0
$794$	−28.1990	−1.00075
$795$	10.5971	0.375841
$796$	−10.3670	−0.367449
$797$	2.38876	0.0846143	0.0423071	−	0.999105i	$-0.486529\pi$
$797$	2.38876	0.0846143	0.0423071	+	0.999105i	$0.486529\pi$
$798$	0	0
$799$	38.8381	1.37399
$800$	9.98245	0.352933
$801$	1.75446	0.0619908
$802$	−0.312231	−0.0110253
$803$	9.33212	0.329323
$804$	−7.18448	−0.253377
$805$	0	0
$806$	0	0
$807$	6.92615	0.243812
$808$	42.1009	1.48110
$809$	50.2919	1.76817	0.884085	−	0.467326i	$-0.154782\pi$
$809$	50.2919	1.76817	0.884085	+	0.467326i	$0.154782\pi$
$810$	−8.40631	−0.295368
$811$	−49.0464	−1.72225	−0.861126	−	0.508392i	$-0.830240\pi$
$811$	−49.0464	−1.72225	−0.861126	+	0.508392i	$0.830240\pi$
$812$	0	0
$813$	−33.9762	−1.19160
$814$	−9.82400	−0.344331
$815$	−2.59092	−0.0907561
$816$	13.0820	0.457962
$817$	−16.7771	−0.586957
$818$	−13.4495	−0.470251
$819$	0	0
$820$	−10.6381	−0.371499
$821$	−4.96434	−0.173257	−0.0866283	−	0.996241i	$-0.527609\pi$
$821$	−4.96434	−0.173257	−0.0866283	+	0.996241i	$0.527609\pi$
$822$	−23.9572	−0.835603
$823$	21.7553	0.758343	0.379172	−	0.925326i	$-0.376209\pi$
$823$	21.7553	0.758343	0.379172	+	0.925326i	$0.376209\pi$
$824$	24.4274	0.850969
$825$	4.25076	0.147992
$826$	0	0
$827$	4.61342	0.160424	0.0802122	−	0.996778i	$-0.474440\pi$
$827$	4.61342	0.160424	0.0802122	+	0.996778i	$0.474440\pi$
$828$	−2.24679	−0.0780813
$829$	−41.6942	−1.44810	−0.724050	−	0.689748i	$-0.757721\pi$
$829$	−41.6942	−1.44810	−0.724050	+	0.689748i	$0.757721\pi$
$830$	14.9565	0.519147
$831$	23.0585	0.799892
$832$	0	0
$833$	0	0
$834$	−6.33722	−0.219440
$835$	26.0811	0.902573
$836$	1.57881	0.0546043
$837$	41.1768	1.42328
$838$	2.43152	0.0839954
$839$	6.16946	0.212993	0.106497	−	0.994313i	$-0.466037\pi$
$839$	6.16946	0.212993	0.106497	+	0.994313i	$0.466037\pi$
$840$	0	0
$841$	4.84970	0.167231
$842$	−21.0770	−0.726360
$843$	4.69250	0.161618
$844$	12.3268	0.424305
$845$	0	0
$846$	8.25157	0.283695
$847$	0	0
$848$	−8.08935	−0.277790
$849$	37.4737	1.28609
$850$	13.9122	0.477183
$851$	−18.3744	−0.629866
$852$	−13.4209	−0.459792
$853$	37.8265	1.29516	0.647578	−	0.761999i	$-0.275782\pi$
$853$	37.8265	1.29516	0.647578	+	0.761999i	$0.275782\pi$
$854$	0	0
$855$	−2.58746	−0.0884891
$856$	15.4644	0.528562
$857$	−5.74818	−0.196354	−0.0981770	−	0.995169i	$-0.531301\pi$
$857$	−5.74818	−0.196354	−0.0981770	+	0.995169i	$0.531301\pi$
$858$	0	0
$859$	−6.01434	−0.205206	−0.102603	−	0.994722i	$-0.532717\pi$
$859$	−6.01434	−0.205206	−0.102603	+	0.994722i	$0.532717\pi$
$860$	14.5853	0.497354
$861$	0	0
$862$	12.2181	0.416149
$863$	48.6042	1.65451	0.827254	−	0.561828i	$-0.189902\pi$
$863$	48.6042	1.65451	0.827254	+	0.561828i	$0.189902\pi$
$864$	−23.9773	−0.815726
$865$	34.4933	1.17281
$866$	−35.7896	−1.21618
$867$	−17.1363	−0.581979
$868$	0	0
$869$	−15.2401	−0.516985
$870$	−14.4193	−0.488859
$871$	0	0
$872$	40.5230	1.37228
$873$	−13.1346	−0.444540
$874$	−4.35983	−0.147474
$875$	0	0
$876$	8.12862	0.274641
$877$	−3.48848	−0.117798	−0.0588989	−	0.998264i	$-0.518759\pi$
$877$	−3.48848	−0.117798	−0.0588989	+	0.998264i	$0.518759\pi$
$878$	−16.6840	−0.563058
$879$	1.90641	0.0643017
$880$	3.64591	0.122904
$881$	51.2874	1.72792	0.863958	−	0.503564i	$-0.167978\pi$
$881$	51.2874	1.72792	0.863958	+	0.503564i	$0.167978\pi$
$882$	0	0
$883$	15.1882	0.511123	0.255562	−	0.966793i	$-0.417740\pi$
$883$	15.1882	0.511123	0.255562	+	0.966793i	$0.417740\pi$
$884$	0	0
$885$	1.75596	0.0590259
$886$	9.45713	0.317718
$887$	45.2051	1.51784	0.758920	−	0.651184i	$-0.225727\pi$
$887$	45.2051	1.51784	0.758920	+	0.651184i	$0.225727\pi$
$888$	−29.7481	−0.998281
$889$	0	0
$890$	−2.95971	−0.0992096
$891$	−6.12364	−0.205150
$892$	−8.92423	−0.298805
$893$	−10.8450	−0.362914
$894$	−24.7459	−0.827625
$895$	15.0336	0.502519
$896$	0	0
$897$	0	0
$898$	1.43697	0.0479524
$899$	42.3614	1.41283
$900$	−2.00197	−0.0667323
$901$	25.2650	0.841700
$902$	11.4415	0.380962
$903$	0	0
$904$	10.3197	0.343227
$905$	−5.61247	−0.186565
$906$	21.1968	0.704216
$907$	6.11178	0.202938	0.101469	−	0.994839i	$-0.467646\pi$
$907$	6.11178	0.202938	0.101469	+	0.994839i	$0.467646\pi$
$908$	−20.5692	−0.682614
$909$	−14.4579	−0.479537
$910$	0	0
$911$	7.28480	0.241356	0.120678	−	0.992692i	$-0.461493\pi$
$911$	7.28480	0.241356	0.120678	+	0.992692i	$0.461493\pi$
$912$	−3.65296	−0.120962
$913$	10.8952	0.360577
$914$	1.15069	0.0380613
$915$	−19.8412	−0.655930
$916$	−9.28168	−0.306675
$917$	0	0
$918$	−33.4163	−1.10290
$919$	−39.4004	−1.29970	−0.649849	−	0.760063i	$-0.725168\pi$
$919$	−39.4004	−1.29970	−0.649849	+	0.760063i	$0.725168\pi$
$920$	13.1766	0.434418
$921$	−28.3407	−0.933857
$922$	−18.7256	−0.616694
$923$	0	0
$924$	0	0
$925$	−16.3722	−0.538316
$926$	0.0181659	0.000596969 0
$927$	−8.38861	−0.275518
$928$	−24.6671	−0.809738
$929$	−42.7417	−1.40231	−0.701155	−	0.713009i	$-0.747332\pi$
$929$	−42.7417	−1.40231	−0.701155	+	0.713009i	$0.747332\pi$
$930$	−18.0451	−0.591721
$931$	0	0
$932$	−21.8787	−0.716661
$933$	21.7020	0.710492
$934$	35.5635	1.16367
$935$	−11.3871	−0.372397
$936$	0	0
$937$	−31.3277	−1.02343	−0.511716	−	0.859155i	$-0.670990\pi$
$937$	−31.3277	−1.02343	−0.511716	+	0.859155i	$0.670990\pi$
$938$	0	0
$939$	−20.8793	−0.681369
$940$	9.42815	0.307512
$941$	8.97428	0.292553	0.146277	−	0.989244i	$-0.453271\pi$
$941$	8.97428	0.292553	0.146277	+	0.989244i	$0.453271\pi$
$942$	−24.7979	−0.807961
$943$	21.3998	0.696872
$944$	−1.34042	−0.0436269
$945$	0	0
$946$	−15.6868	−0.510022
$947$	−32.6800	−1.06196	−0.530978	−	0.847386i	$-0.678175\pi$
$947$	−32.6800	−1.06196	−0.530978	+	0.847386i	$0.678175\pi$
$948$	−13.2747	−0.431142
$949$	0	0
$950$	−3.88477	−0.126038
$951$	23.6467	0.766796
$952$	0	0
$953$	−20.8795	−0.676355	−0.338177	−	0.941082i	$-0.609810\pi$
$953$	−20.8795	−0.676355	−0.338177	+	0.941082i	$0.609810\pi$
$954$	5.36782	0.173790
$955$	34.3692	1.11216
$956$	−1.63861	−0.0529965
$957$	−10.5038	−0.339540
$958$	1.81626	0.0586808
$959$	0	0
$960$	18.3722	0.592959
$961$	22.0133	0.710108
$962$	0	0
$963$	−5.31062	−0.171132
$964$	18.4596	0.594545
$965$	18.0766	0.581907
$966$	0	0
$967$	9.98931	0.321235	0.160617	−	0.987017i	$-0.448652\pi$
$967$	9.98931	0.321235	0.160617	+	0.987017i	$0.448652\pi$
$968$	−28.5920	−0.918982
$969$	11.4091	0.366512
$970$	22.1576	0.711438
$971$	46.0240	1.47698	0.738490	−	0.674265i	$-0.235539\pi$
$971$	46.0240	1.47698	0.738490	+	0.674265i	$0.235539\pi$
$972$	8.36809	0.268407
$973$	0	0
$974$	−37.4647	−1.20045
$975$	0	0
$976$	15.1459	0.484808
$977$	10.6475	0.340644	0.170322	−	0.985388i	$-0.445519\pi$
$977$	10.6475	0.340644	0.170322	+	0.985388i	$0.445519\pi$
$978$	2.42723	0.0776142
$979$	−2.15602	−0.0689067
$980$	0	0
$981$	−13.9160	−0.444304
$982$	−28.9968	−0.925325
$983$	17.0501	0.543815	0.271908	−	0.962323i	$-0.412346\pi$
$983$	17.0501	0.543815	0.271908	+	0.962323i	$0.412346\pi$
$984$	34.6462	1.10448
$985$	−28.6475	−0.912785
$986$	−34.3776	−1.09481
$987$	0	0
$988$	0	0
$989$	−29.3399	−0.932955
$990$	−2.41930	−0.0768905
$991$	−57.2242	−1.81779	−0.908894	−	0.417027i	$-0.863072\pi$
$991$	−57.2242	−1.81779	−0.908894	+	0.417027i	$0.863072\pi$
$992$	−30.8698	−0.980116
$993$	1.17367	0.0372453
$994$	0	0
$995$	−20.8788	−0.661904
$996$	9.49009	0.300705
$997$	1.01279	0.0320753	0.0160376	−	0.999871i	$-0.494895\pi$
$997$	1.01279	0.0320753	0.0160376	+	0.999871i	$0.494895\pi$
$998$	−12.5455	−0.397121
$999$	39.3253	1.24420

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cw.1.9		24
7.3	odd	6		1183.2.e.k.170.16	✓	48
7.5	odd	6		1183.2.e.k.508.16	yes	48
7.6	odd	2		8281.2.a.cv.1.9		24
13.12	even	2		8281.2.a.ct.1.16		24
91.12	odd	6		1183.2.e.l.508.9	yes	48
91.38	odd	6		1183.2.e.l.170.9	yes	48
91.90	odd	2		8281.2.a.cu.1.16		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.16	✓	48	7.3	odd	6
1183.2.e.k.508.16	yes	48	7.5	odd	6
1183.2.e.l.170.9	yes	48	91.38	odd	6
1183.2.e.l.508.9	yes	48	91.12	odd	6
8281.2.a.ct.1.16		24	13.12	even	2
8281.2.a.cu.1.16		24	91.90	odd	2
8281.2.a.cv.1.9		24	7.6	odd	2
8281.2.a.cw.1.9		24	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-1.09197 q^{2} -1.39541 q^{3} -0.807612 q^{4} -1.62650 q^{5} +1.52374 q^{6} +3.06581 q^{8} -1.05283 q^{9} +O(q^{10})\)	\(q-1.09197 q^{2} -1.39541 q^{3} -0.807612 q^{4} -1.62650 q^{5} +1.52374 q^{6} +3.06581 q^{8} -1.05283 q^{9} +1.77608 q^{10} +1.29380 q^{11} +1.12695 q^{12} +2.26964 q^{15} -1.73254 q^{16} +5.41114 q^{17} +1.14965 q^{18} -1.51098 q^{19} +1.31358 q^{20} -1.41279 q^{22} -2.64242 q^{23} -4.27807 q^{24} -2.35449 q^{25} +5.65536 q^{27} +5.81805 q^{29} -2.47837 q^{30} +7.28103 q^{31} -4.23976 q^{32} -1.80538 q^{33} -5.90878 q^{34} +0.850278 q^{36} +6.95363 q^{37} +1.64994 q^{38} -4.98656 q^{40} -8.09856 q^{41} +11.1034 q^{43} -1.04489 q^{44} +1.71243 q^{45} +2.88543 q^{46} +7.17743 q^{47} +2.41760 q^{48} +2.57102 q^{50} -7.55076 q^{51} +4.66907 q^{53} -6.17546 q^{54} -2.10437 q^{55} +2.10844 q^{57} -6.35311 q^{58} +0.773673 q^{59} -1.83299 q^{60} -8.74201 q^{61} -7.95063 q^{62} +8.09475 q^{64} +1.97142 q^{66} -6.37515 q^{67} -4.37010 q^{68} +3.68726 q^{69} -11.9090 q^{71} -3.22778 q^{72} +7.21294 q^{73} -7.59312 q^{74} +3.28548 q^{75} +1.22029 q^{76} -11.7793 q^{79} +2.81798 q^{80} -4.73306 q^{81} +8.84334 q^{82} +8.42105 q^{83} -8.80124 q^{85} -12.1246 q^{86} -8.11857 q^{87} +3.96656 q^{88} -1.66642 q^{89} -1.86992 q^{90} +2.13405 q^{92} -10.1600 q^{93} -7.83751 q^{94} +2.45762 q^{95} +5.91620 q^{96} +12.4755 q^{97} -1.36215 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9}+O(q^{10}) \) 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27} + 4 q^{29} - 22 q^{30} + 40 q^{31} + 6 q^{32} + 24 q^{33} + 36 q^{34} - 15 q^{36} + 4 q^{37} + 29 q^{38} + 4 q^{40} + 49 q^{41} + 13 q^{43} - 10 q^{44} + 58 q^{45} + 10 q^{46} + 62 q^{47} - 89 q^{48} + 23 q^{50} - 21 q^{51} - 18 q^{53} + 12 q^{54} + 14 q^{55} + 13 q^{57} - 56 q^{58} + 79 q^{59} - 22 q^{60} - 13 q^{61} - 12 q^{62} + 18 q^{64} + 38 q^{66} + 2 q^{67} + 12 q^{68} + 28 q^{69} + 19 q^{71} - 81 q^{72} + 17 q^{73} + 17 q^{74} - 24 q^{75} + 58 q^{76} - 9 q^{79} + 63 q^{80} + 16 q^{81} + 22 q^{82} + 81 q^{83} + 34 q^{85} - 22 q^{86} - 70 q^{87} - 33 q^{88} + 72 q^{89} - q^{90} - 4 q^{92} - 19 q^{93} + 30 q^{94} + 13 q^{95} + 11 q^{96} + 45 q^{97} + 39 q^{99}+O(q^{100}) \) 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

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