LMFDB - Embedded newform 8281.2.a.cv.1.16

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:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
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)

$f(q)$	$=$	$q+1.02884 q^{2} +2.66326 q^{3} -0.941498 q^{4} -1.49113 q^{5} +2.74006 q^{6} -3.02632 q^{8} +4.09295 q^{9} +O(q^{10})$	\(q+1.02884 q^{2} +2.66326 q^{3} -0.941498 q^{4} -1.49113 q^{5} +2.74006 q^{6} -3.02632 q^{8} +4.09295 q^{9} -1.53412 q^{10} -4.04132 q^{11} -2.50745 q^{12} -3.97125 q^{15} -1.23059 q^{16} +3.68875 q^{17} +4.21097 q^{18} +5.28437 q^{19} +1.40389 q^{20} -4.15786 q^{22} +7.27170 q^{23} -8.05987 q^{24} -2.77654 q^{25} +2.91081 q^{27} -7.56677 q^{29} -4.08577 q^{30} +1.67723 q^{31} +4.78656 q^{32} -10.7631 q^{33} +3.79512 q^{34} -3.85350 q^{36} -7.02128 q^{37} +5.43675 q^{38} +4.51262 q^{40} -0.409255 q^{41} -6.43516 q^{43} +3.80490 q^{44} -6.10310 q^{45} +7.48138 q^{46} -5.13884 q^{47} -3.27737 q^{48} -2.85661 q^{50} +9.82410 q^{51} -1.58189 q^{53} +2.99474 q^{54} +6.02612 q^{55} +14.0736 q^{57} -7.78497 q^{58} -7.64640 q^{59} +3.73893 q^{60} +11.2893 q^{61} +1.72560 q^{62} +7.38576 q^{64} -11.0734 q^{66} -13.1577 q^{67} -3.47295 q^{68} +19.3664 q^{69} -8.71275 q^{71} -12.3866 q^{72} -15.1914 q^{73} -7.22374 q^{74} -7.39466 q^{75} -4.97522 q^{76} -0.753837 q^{79} +1.83496 q^{80} -4.52661 q^{81} -0.421056 q^{82} +3.96146 q^{83} -5.50039 q^{85} -6.62072 q^{86} -20.1523 q^{87} +12.2303 q^{88} +3.69313 q^{89} -6.27909 q^{90} -6.84629 q^{92} +4.46690 q^{93} -5.28702 q^{94} -7.87966 q^{95} +12.7479 q^{96} -8.43912 q^{97} -16.5409 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.02884	0.727497	0.363748	−	0.931497i	$-0.381497\pi$
$2$	1.02884	0.727497	0.363748	+	0.931497i	$0.381497\pi$
$3$	2.66326	1.53763	0.768817	−	0.639469i	$-0.220846\pi$
$3$	2.66326	1.53763	0.768817	+	0.639469i	$0.220846\pi$
$4$	−0.941498	−0.470749
$5$	−1.49113	−0.666852	−0.333426	−	0.942776i	$-0.608205\pi$
$5$	−1.49113	−0.666852	−0.333426	+	0.942776i	$0.608205\pi$
$6$	2.74006	1.11862
$7$	0	0
$7$	0	0
$8$	−3.02632	−1.06996
$9$	4.09295	1.36432
$10$	−1.53412	−0.485132
$11$	−4.04132	−1.21850	−0.609252	−	0.792976i	$-0.708530\pi$
$11$	−4.04132	−1.21850	−0.609252	+	0.792976i	$0.708530\pi$
$12$	−2.50745	−0.723839
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.97125	−1.02537
$16$	−1.23059	−0.307647
$17$	3.68875	0.894654	0.447327	−	0.894371i	$-0.352376\pi$
$17$	3.68875	0.894654	0.447327	+	0.894371i	$0.352376\pi$
$18$	4.21097	0.992535
$19$	5.28437	1.21232	0.606159	−	0.795344i	$-0.292710\pi$
$19$	5.28437	1.21232	0.606159	+	0.795344i	$0.292710\pi$
$20$	1.40389	0.313920
$21$	0	0
$22$	−4.15786	−0.886458
$23$	7.27170	1.51625	0.758127	−	0.652107i	$-0.226115\pi$
$23$	7.27170	1.51625	0.758127	+	0.652107i	$0.226115\pi$
$24$	−8.05987	−1.64521
$25$	−2.77654	−0.555309
$26$	0	0
$27$	2.91081	0.560185
$28$	0	0
$29$	−7.56677	−1.40511	−0.702557	−	0.711627i	$-0.747959\pi$
$29$	−7.56677	−1.40511	−0.702557	+	0.711627i	$0.747959\pi$
$30$	−4.08577	−0.745956
$31$	1.67723	0.301240	0.150620	−	0.988592i	$-0.451873\pi$
$31$	1.67723	0.301240	0.150620	+	0.988592i	$0.451873\pi$
$32$	4.78656	0.846153
$33$	−10.7631	−1.87361
$34$	3.79512	0.650858
$35$	0	0
$36$	−3.85350	−0.642250
$37$	−7.02128	−1.15429	−0.577145	−	0.816641i	$-0.695833\pi$
$37$	−7.02128	−1.15429	−0.577145	+	0.816641i	$0.695833\pi$
$38$	5.43675	0.881957
$39$	0	0
$40$	4.51262	0.713508
$41$	−0.409255	−0.0639148	−0.0319574	−	0.999489i	$-0.510174\pi$
$41$	−0.409255	−0.0639148	−0.0319574	+	0.999489i	$0.510174\pi$
$42$	0	0
$43$	−6.43516	−0.981353	−0.490677	−	0.871342i	$-0.663250\pi$
$43$	−6.43516	−0.981353	−0.490677	+	0.871342i	$0.663250\pi$
$44$	3.80490	0.573610
$45$	−6.10310	−0.909797
$46$	7.48138	1.10307
$47$	−5.13884	−0.749576	−0.374788	−	0.927110i	$-0.622285\pi$
$47$	−5.13884	−0.749576	−0.374788	+	0.927110i	$0.622285\pi$
$48$	−3.27737	−0.473048
$49$	0	0
$50$	−2.85661	−0.403985
$51$	9.82410	1.37565
$52$	0	0
$53$	−1.58189	−0.217289	−0.108645	−	0.994081i	$-0.534651\pi$
$53$	−1.58189	−0.217289	−0.108645	+	0.994081i	$0.534651\pi$
$54$	2.99474	0.407533
$55$	6.02612	0.812562
$56$	0	0
$57$	14.0736	1.86410
$58$	−7.78497	−1.02222
$59$	−7.64640	−0.995476	−0.497738	−	0.867327i	$-0.665836\pi$
$59$	−7.64640	−0.995476	−0.497738	+	0.867327i	$0.665836\pi$
$60$	3.73893	0.482693
$61$	11.2893	1.44545	0.722726	−	0.691135i	$-0.242889\pi$
$61$	11.2893	1.44545	0.722726	+	0.691135i	$0.242889\pi$
$62$	1.72560	0.219151
$63$	0	0
$64$	7.38576	0.923220
$65$	0	0
$66$	−11.0734	−1.36305
$67$	−13.1577	−1.60746	−0.803732	−	0.594991i	$-0.797155\pi$
$67$	−13.1577	−1.60746	−0.803732	+	0.594991i	$0.797155\pi$
$68$	−3.47295	−0.421157
$69$	19.3664	2.33144
$70$	0	0
$71$	−8.71275	−1.03401	−0.517007	−	0.855981i	$-0.672954\pi$
$71$	−8.71275	−1.03401	−0.517007	+	0.855981i	$0.672954\pi$
$72$	−12.3866	−1.45977
$73$	−15.1914	−1.77802	−0.889008	−	0.457891i	$-0.848605\pi$
$73$	−15.1914	−1.77802	−0.889008	+	0.457891i	$0.848605\pi$
$74$	−7.22374	−0.839743
$75$	−7.39466	−0.853862
$76$	−4.97522	−0.570697
$77$	0	0
$78$	0	0
$79$	−0.753837	−0.0848133	−0.0424066	−	0.999100i	$-0.513503\pi$
$79$	−0.753837	−0.0848133	−0.0424066	+	0.999100i	$0.513503\pi$
$80$	1.83496	0.205155
$81$	−4.52661	−0.502957
$82$	−0.421056	−0.0464978
$83$	3.96146	0.434827	0.217413	−	0.976080i	$-0.430238\pi$
$83$	3.96146	0.434827	0.217413	+	0.976080i	$0.430238\pi$
$84$	0	0
$85$	−5.50039	−0.596601
$86$	−6.62072	−0.713931
$87$	−20.1523	−2.16055
$88$	12.2303	1.30376
$89$	3.69313	0.391471	0.195736	−	0.980657i	$-0.437291\pi$
$89$	3.69313	0.391471	0.195736	+	0.980657i	$0.437291\pi$
$90$	−6.27909	−0.661874
$91$	0	0
$92$	−6.84629	−0.713775
$93$	4.46690	0.463196
$94$	−5.28702	−0.545314
$95$	−7.87966	−0.808436
$96$	12.7479	1.30107
$97$	−8.43912	−0.856863	−0.428432	−	0.903574i	$-0.640934\pi$
$97$	−8.43912	−0.856863	−0.428432	+	0.903574i	$0.640934\pi$
$98$	0	0
$99$	−16.5409	−1.66243
$100$	2.61411	0.261411
$101$	8.47591	0.843385	0.421692	−	0.906739i	$-0.361436\pi$
$101$	8.47591	0.843385	0.421692	+	0.906739i	$0.361436\pi$
$102$	10.1074	1.00078
$103$	−15.9093	−1.56759	−0.783796	−	0.621019i	$-0.786719\pi$
$103$	−15.9093	−1.56759	−0.783796	+	0.621019i	$0.786719\pi$
$104$	0	0
$105$	0	0
$106$	−1.62750	−0.158077
$107$	7.14711	0.690937	0.345469	−	0.938430i	$-0.387720\pi$
$107$	7.14711	0.690937	0.345469	+	0.938430i	$0.387720\pi$
$108$	−2.74052	−0.263706
$109$	1.11245	0.106553	0.0532767	−	0.998580i	$-0.483033\pi$
$109$	1.11245	0.106553	0.0532767	+	0.998580i	$0.483033\pi$
$110$	6.19988	0.591136
$111$	−18.6995	−1.77488
$112$	0	0
$113$	−13.0433	−1.22701	−0.613504	−	0.789692i	$-0.710240\pi$
$113$	−13.0433	−1.22701	−0.613504	+	0.789692i	$0.710240\pi$
$114$	14.4795	1.35613
$115$	−10.8430	−1.01112
$116$	7.12410	0.661456
$117$	0	0
$118$	−7.86688	−0.724205
$119$	0	0
$120$	12.0183	1.09711
$121$	5.33229	0.484753
$122$	11.6149	1.05156
$123$	−1.08995	−0.0982776
$124$	−1.57911	−0.141808
$125$	11.5958	1.03716
$126$	0	0
$127$	11.4482	1.01587	0.507933	−	0.861396i	$-0.330410\pi$
$127$	11.4482	1.01587	0.507933	+	0.861396i	$0.330410\pi$
$128$	−1.97439	−0.174513
$129$	−17.1385	−1.50896
$130$	0	0
$131$	1.78846	0.156258	0.0781290	−	0.996943i	$-0.475105\pi$
$131$	1.78846	0.156258	0.0781290	+	0.996943i	$0.475105\pi$
$132$	10.1334	0.882001
$133$	0	0
$134$	−13.5371	−1.16942
$135$	−4.34038	−0.373560
$136$	−11.1633	−0.957248
$137$	−6.35470	−0.542919	−0.271459	−	0.962450i	$-0.587506\pi$
$137$	−6.35470	−0.542919	−0.271459	+	0.962450i	$0.587506\pi$
$138$	19.9249	1.69612
$139$	10.6138	0.900250	0.450125	−	0.892966i	$-0.351380\pi$
$139$	10.6138	0.900250	0.450125	+	0.892966i	$0.351380\pi$
$140$	0	0
$141$	−13.6861	−1.15257
$142$	−8.96399	−0.752241
$143$	0	0
$144$	−5.03673	−0.419727
$145$	11.2830	0.937003
$146$	−15.6294	−1.29350
$147$	0	0
$148$	6.61052	0.543381
$149$	−4.80540	−0.393674	−0.196837	−	0.980436i	$-0.563067\pi$
$149$	−4.80540	−0.393674	−0.196837	+	0.980436i	$0.563067\pi$
$150$	−7.60789	−0.621181
$151$	−4.13077	−0.336158	−0.168079	−	0.985774i	$-0.553756\pi$
$151$	−4.13077	−0.336158	−0.168079	+	0.985774i	$0.553756\pi$
$152$	−15.9922	−1.29714
$153$	15.0979	1.22059
$154$	0	0
$155$	−2.50096	−0.200882
$156$	0	0
$157$	5.95919	0.475595	0.237797	−	0.971315i	$-0.423575\pi$
$157$	5.95919	0.475595	0.237797	+	0.971315i	$0.423575\pi$
$158$	−0.775574	−0.0617014
$159$	−4.21298	−0.334111
$160$	−7.13737	−0.564258
$161$	0	0
$162$	−4.65714	−0.365900
$163$	8.86820	0.694611	0.347305	−	0.937752i	$-0.387097\pi$
$163$	8.86820	0.694611	0.347305	+	0.937752i	$0.387097\pi$
$164$	0.385312	0.0300878
$165$	16.0491	1.24942
$166$	4.07569	0.316335
$167$	11.9219	0.922547	0.461274	−	0.887258i	$-0.347393\pi$
$167$	11.9219	0.922547	0.461274	+	0.887258i	$0.347393\pi$
$168$	0	0
$169$	0	0
$170$	−5.65900	−0.434025
$171$	21.6287	1.65398
$172$	6.05869	0.461971
$173$	−8.83119	−0.671423	−0.335711	−	0.941965i	$-0.608977\pi$
$173$	−8.83119	−0.671423	−0.335711	+	0.941965i	$0.608977\pi$
$174$	−20.7334	−1.57179
$175$	0	0
$176$	4.97320	0.374869
$177$	−20.3643	−1.53068
$178$	3.79963	0.284794
$179$	−18.1054	−1.35326	−0.676632	−	0.736322i	$-0.736561\pi$
$179$	−18.1054	−1.35326	−0.676632	+	0.736322i	$0.736561\pi$
$180$	5.74606	0.428286
$181$	−21.3354	−1.58585	−0.792923	−	0.609322i	$-0.791442\pi$
$181$	−21.3354	−1.58585	−0.792923	+	0.609322i	$0.791442\pi$
$182$	0	0
$183$	30.0664	2.22257
$184$	−22.0065	−1.62234
$185$	10.4696	0.769741
$186$	4.59571	0.336974
$187$	−14.9074	−1.09014
$188$	4.83820	0.352862
$189$	0	0
$190$	−8.10687	−0.588134
$191$	−16.2289	−1.17428	−0.587140	−	0.809485i	$-0.699746\pi$
$191$	−16.2289	−1.17428	−0.587140	+	0.809485i	$0.699746\pi$
$192$	19.6702	1.41957
$193$	−2.05173	−0.147687	−0.0738434	−	0.997270i	$-0.523527\pi$
$193$	−2.05173	−0.147687	−0.0738434	+	0.997270i	$0.523527\pi$
$194$	−8.68247	−0.623365
$195$	0	0
$196$	0	0
$197$	−2.49922	−0.178062	−0.0890309	−	0.996029i	$-0.528377\pi$
$197$	−2.49922	−0.178062	−0.0890309	+	0.996029i	$0.528377\pi$
$198$	−17.0179	−1.20941
$199$	−15.7466	−1.11625	−0.558123	−	0.829758i	$-0.688478\pi$
$199$	−15.7466	−1.11625	−0.558123	+	0.829758i	$0.688478\pi$
$200$	8.40270	0.594161
$201$	−35.0423	−2.47169
$202$	8.72032	0.613559
$203$	0	0
$204$	−9.24937	−0.647585
$205$	0.610250	0.0426217
$206$	−16.3681	−1.14042
$207$	29.7627	2.06865
$208$	0	0
$209$	−21.3558	−1.47721
$210$	0	0
$211$	7.75205	0.533673	0.266837	−	0.963742i	$-0.414022\pi$
$211$	7.75205	0.533673	0.266837	+	0.963742i	$0.414022\pi$
$212$	1.48935	0.102289
$213$	−23.2043	−1.58993
$214$	7.35320	0.502654
$215$	9.59564	0.654417
$216$	−8.80902	−0.599378
$217$	0	0
$218$	1.14453	0.0775173
$219$	−40.4586	−2.73394
$220$	−5.67358	−0.382512
$221$	0	0
$222$	−19.2387	−1.29122
$223$	4.75374	0.318334	0.159167	−	0.987252i	$-0.449119\pi$
$223$	4.75374	0.318334	0.159167	+	0.987252i	$0.449119\pi$
$224$	0	0
$225$	−11.3643	−0.757617
$226$	−13.4194	−0.892644
$227$	−12.5900	−0.835625	−0.417813	−	0.908533i	$-0.637203\pi$
$227$	−12.5900	−0.835625	−0.417813	+	0.908533i	$0.637203\pi$
$228$	−13.2503	−0.877523
$229$	5.25748	0.347424	0.173712	−	0.984796i	$-0.444424\pi$
$229$	5.25748	0.347424	0.173712	+	0.984796i	$0.444424\pi$
$230$	−11.1557	−0.735583
$231$	0	0
$232$	22.8995	1.50342
$233$	−8.45633	−0.553993	−0.276996	−	0.960871i	$-0.589339\pi$
$233$	−8.45633	−0.553993	−0.276996	+	0.960871i	$0.589339\pi$
$234$	0	0
$235$	7.66265	0.499856
$236$	7.19907	0.468619
$237$	−2.00766	−0.130412
$238$	0	0
$239$	6.55677	0.424122	0.212061	−	0.977256i	$-0.431982\pi$
$239$	6.55677	0.424122	0.212061	+	0.977256i	$0.431982\pi$
$240$	4.88697	0.315453
$241$	−11.5930	−0.746773	−0.373387	−	0.927676i	$-0.621803\pi$
$241$	−11.5930	−0.746773	−0.373387	+	0.927676i	$0.621803\pi$
$242$	5.48605	0.352656
$243$	−20.7880	−1.33355
$244$	−10.6289	−0.680445
$245$	0	0
$246$	−1.12138	−0.0714966
$247$	0	0
$248$	−5.07584	−0.322316
$249$	10.5504	0.668604
$250$	11.9302	0.754530
$251$	−19.3214	−1.21956	−0.609778	−	0.792572i	$-0.708741\pi$
$251$	−19.3214	−1.21956	−0.609778	+	0.792572i	$0.708741\pi$
$252$	0	0
$253$	−29.3873	−1.84756
$254$	11.7784	0.739040
$255$	−14.6490	−0.917354
$256$	−16.8028	−1.05018
$257$	−17.3130	−1.07996	−0.539979	−	0.841678i	$-0.681568\pi$
$257$	−17.3130	−1.07996	−0.539979	+	0.841678i	$0.681568\pi$
$258$	−17.6327	−1.09776
$259$	0	0
$260$	0	0
$261$	−30.9704	−1.91702
$262$	1.84003	0.113677
$263$	25.3937	1.56584	0.782921	−	0.622121i	$-0.213729\pi$
$263$	25.3937	1.56584	0.782921	+	0.622121i	$0.213729\pi$
$264$	32.5725	2.00470
$265$	2.35880	0.144900
$266$	0	0
$267$	9.83577	0.601939
$268$	12.3879	0.756712
$269$	11.6884	0.712655	0.356327	−	0.934361i	$-0.384029\pi$
$269$	11.6884	0.712655	0.356327	+	0.934361i	$0.384029\pi$
$270$	−4.46554	−0.271764
$271$	−3.41391	−0.207380	−0.103690	−	0.994610i	$-0.533065\pi$
$271$	−3.41391	−0.207380	−0.103690	+	0.994610i	$0.533065\pi$
$272$	−4.53933	−0.275237
$273$	0	0
$274$	−6.53794	−0.394971
$275$	11.2209	0.676646
$276$	−18.2334	−1.09752
$277$	1.91564	0.115100	0.0575499	−	0.998343i	$-0.481671\pi$
$277$	1.91564	0.115100	0.0575499	+	0.998343i	$0.481671\pi$
$278$	10.9198	0.654928
$279$	6.86483	0.410986
$280$	0	0
$281$	14.8090	0.883432	0.441716	−	0.897155i	$-0.354370\pi$
$281$	14.8090	0.883432	0.441716	+	0.897155i	$0.354370\pi$
$282$	−14.0807	−0.838493
$283$	33.5594	1.99490	0.997449	−	0.0713875i	$-0.0227427\pi$
$283$	33.5594	1.99490	0.997449	+	0.0713875i	$0.0227427\pi$
$284$	8.20303	0.486761
$285$	−20.9856	−1.24308
$286$	0	0
$287$	0	0
$288$	19.5912	1.15442
$289$	−3.39311	−0.199594
$290$	11.6084	0.681666
$291$	−22.4756	−1.31754
$292$	14.3026	0.836999
$293$	−5.03064	−0.293893	−0.146946	−	0.989144i	$-0.546944\pi$
$293$	−5.03064	−0.293893	−0.146946	+	0.989144i	$0.546944\pi$
$294$	0	0
$295$	11.4017	0.663835
$296$	21.2486	1.23505
$297$	−11.7635	−0.682588
$298$	−4.94397	−0.286396
$299$	0	0
$300$	6.96205	0.401954
$301$	0	0
$302$	−4.24989	−0.244553
$303$	22.5735	1.29682
$304$	−6.50288	−0.372966
$305$	−16.8338	−0.963902
$306$	15.5332	0.887976
$307$	−14.1518	−0.807688	−0.403844	−	0.914828i	$-0.632326\pi$
$307$	−14.1518	−0.807688	−0.403844	+	0.914828i	$0.632326\pi$
$308$	0	0
$309$	−42.3706	−2.41038
$310$	−2.57308	−0.146141
$311$	−8.69534	−0.493067	−0.246534	−	0.969134i	$-0.579292\pi$
$311$	−8.69534	−0.493067	−0.246534	+	0.969134i	$0.579292\pi$
$312$	0	0
$313$	20.8777	1.18008	0.590038	−	0.807375i	$-0.299113\pi$
$313$	20.8777	1.18008	0.590038	+	0.807375i	$0.299113\pi$
$314$	6.13102	0.345994
$315$	0	0
$316$	0.709736	0.0399258
$317$	−17.4902	−0.982346	−0.491173	−	0.871062i	$-0.663432\pi$
$317$	−17.4902	−0.982346	−0.491173	+	0.871062i	$0.663432\pi$
$318$	−4.33447	−0.243065
$319$	30.5798	1.71214
$320$	−11.0131	−0.615651
$321$	19.0346	1.06241
$322$	0	0
$323$	19.4927	1.08460
$324$	4.26180	0.236767
$325$	0	0
$326$	9.12392	0.505327
$327$	2.96274	0.163840
$328$	1.23853	0.0683866
$329$	0	0
$330$	16.5119	0.908950
$331$	−4.57887	−0.251678	−0.125839	−	0.992051i	$-0.540162\pi$
$331$	−4.57887	−0.251678	−0.125839	+	0.992051i	$0.540162\pi$
$332$	−3.72971	−0.204694
$333$	−28.7377	−1.57482
$334$	12.2657	0.671150
$335$	19.6197	1.07194
$336$	0	0
$337$	23.5671	1.28378	0.641891	−	0.766796i	$-0.278150\pi$
$337$	23.5671	1.28378	0.641891	+	0.766796i	$0.278150\pi$
$338$	0	0
$339$	−34.7376	−1.88669
$340$	5.17861	0.280849
$341$	−6.77824	−0.367062
$342$	22.2523	1.20327
$343$	0	0
$344$	19.4748	1.05001
$345$	−28.8778	−1.55473
$346$	−9.08584	−0.488458
$347$	26.9884	1.44882	0.724408	−	0.689372i	$-0.242113\pi$
$347$	26.9884	1.44882	0.724408	+	0.689372i	$0.242113\pi$
$348$	18.9733	1.01708
$349$	−24.7255	−1.32352	−0.661762	−	0.749714i	$-0.730191\pi$
$349$	−24.7255	−1.32352	−0.661762	+	0.749714i	$0.730191\pi$
$350$	0	0
$351$	0	0
$352$	−19.3440	−1.03104
$353$	−1.44590	−0.0769576	−0.0384788	−	0.999259i	$-0.512251\pi$
$353$	−1.44590	−0.0769576	−0.0384788	+	0.999259i	$0.512251\pi$
$354$	−20.9516	−1.11356
$355$	12.9918	0.689533
$356$	−3.47708	−0.184285
$357$	0	0
$358$	−18.6275	−0.984494
$359$	16.5493	0.873439	0.436719	−	0.899598i	$-0.356140\pi$
$359$	16.5493	0.873439	0.436719	+	0.899598i	$0.356140\pi$
$360$	18.4699	0.973450
$361$	8.92456	0.469714
$362$	−21.9506	−1.15370
$363$	14.2013	0.745373
$364$	0	0
$365$	22.6523	1.18567
$366$	30.9334	1.61692
$367$	−32.7598	−1.71005	−0.855023	−	0.518590i	$-0.826457\pi$
$367$	−32.7598	−1.71005	−0.855023	+	0.518590i	$0.826457\pi$
$368$	−8.94845	−0.466470
$369$	−1.67506	−0.0872000
$370$	10.7715	0.559984
$371$	0	0
$372$	−4.20558	−0.218049
$373$	16.0257	0.829778	0.414889	−	0.909872i	$-0.363820\pi$
$373$	16.0257	0.829778	0.414889	+	0.909872i	$0.363820\pi$
$374$	−15.3373	−0.793073
$375$	30.8826	1.59477
$376$	15.5518	0.802020
$377$	0	0
$378$	0	0
$379$	−4.97761	−0.255682	−0.127841	−	0.991795i	$-0.540805\pi$
$379$	−4.97761	−0.255682	−0.127841	+	0.991795i	$0.540805\pi$
$380$	7.41868	0.380570
$381$	30.4896	1.56203
$382$	−16.6968	−0.854285
$383$	−22.2516	−1.13700	−0.568501	−	0.822683i	$-0.692476\pi$
$383$	−22.2516	−1.13700	−0.568501	+	0.822683i	$0.692476\pi$
$384$	−5.25832	−0.268338
$385$	0	0
$386$	−2.11089	−0.107442
$387$	−26.3388	−1.33888
$388$	7.94541	0.403367
$389$	1.01854	0.0516423	0.0258211	−	0.999667i	$-0.491780\pi$
$389$	1.01854	0.0516423	0.0258211	+	0.999667i	$0.491780\pi$
$390$	0	0
$391$	26.8235	1.35652
$392$	0	0
$393$	4.76312	0.240268
$394$	−2.57128	−0.129539
$395$	1.12407	0.0565579
$396$	15.5732	0.782585
$397$	−21.4075	−1.07441	−0.537205	−	0.843452i	$-0.680520\pi$
$397$	−21.4075	−1.07441	−0.537205	+	0.843452i	$0.680520\pi$
$398$	−16.2006	−0.812065
$399$	0	0
$400$	3.41678	0.170839
$401$	−1.60522	−0.0801609	−0.0400804	−	0.999196i	$-0.512761\pi$
$401$	−1.60522	−0.0801609	−0.0400804	+	0.999196i	$0.512761\pi$
$402$	−36.0527	−1.79815
$403$	0	0
$404$	−7.98005	−0.397022
$405$	6.74975	0.335398
$406$	0	0
$407$	28.3752	1.40651
$408$	−29.7309	−1.47190
$409$	35.7170	1.76609	0.883047	−	0.469284i	$-0.155488\pi$
$409$	35.7170	1.76609	0.883047	+	0.469284i	$0.155488\pi$
$410$	0.627847	0.0310071
$411$	−16.9242	−0.834810
$412$	14.9786	0.737942
$413$	0	0
$414$	30.6209	1.50494
$415$	−5.90703	−0.289965
$416$	0	0
$417$	28.2673	1.38425
$418$	−21.9716	−1.07467
$419$	12.3317	0.602443	0.301221	−	0.953554i	$-0.402606\pi$
$419$	12.3317	0.602443	0.301221	+	0.953554i	$0.402606\pi$
$420$	0	0
$421$	−12.8739	−0.627436	−0.313718	−	0.949516i	$-0.601575\pi$
$421$	−12.8739	−0.627436	−0.313718	+	0.949516i	$0.601575\pi$
$422$	7.97559	0.388245
$423$	−21.0330	−1.02266
$424$	4.78730	0.232492
$425$	−10.2420	−0.496809
$426$	−23.8734	−1.15667
$427$	0	0
$428$	−6.72899	−0.325258
$429$	0	0
$430$	9.87233	0.476086
$431$	16.6218	0.800642	0.400321	−	0.916375i	$-0.368899\pi$
$431$	16.6218	0.800642	0.400321	+	0.916375i	$0.368899\pi$
$432$	−3.58200	−0.172339
$433$	7.34167	0.352818	0.176409	−	0.984317i	$-0.443552\pi$
$433$	7.34167	0.352818	0.176409	+	0.984317i	$0.443552\pi$
$434$	0	0
$435$	30.0496	1.44077
$436$	−1.04737	−0.0501599
$437$	38.4263	1.83818
$438$	−41.6252	−1.98893
$439$	−1.71182	−0.0817005	−0.0408503	−	0.999165i	$-0.513007\pi$
$439$	−1.71182	−0.0817005	−0.0408503	+	0.999165i	$0.513007\pi$
$440$	−18.2369	−0.869412
$441$	0	0
$442$	0	0
$443$	7.20501	0.342320	0.171160	−	0.985243i	$-0.445248\pi$
$443$	7.20501	0.342320	0.171160	+	0.985243i	$0.445248\pi$
$444$	17.6055	0.835521
$445$	−5.50693	−0.261053
$446$	4.89081	0.231587
$447$	−12.7980	−0.605326
$448$	0	0
$449$	7.17254	0.338493	0.169247	−	0.985574i	$-0.445867\pi$
$449$	7.17254	0.338493	0.169247	+	0.985574i	$0.445867\pi$
$450$	−11.6919	−0.551164
$451$	1.65393	0.0778805
$452$	12.2802	0.577612
$453$	−11.0013	−0.516887
$454$	−12.9530	−0.607915
$455$	0	0
$456$	−42.5913	−1.99452
$457$	36.0974	1.68856	0.844282	−	0.535899i	$-0.180027\pi$
$457$	36.0974	1.68856	0.844282	+	0.535899i	$0.180027\pi$
$458$	5.40908	0.252750
$459$	10.7372	0.501172
$460$	10.2087	0.475982
$461$	−21.0255	−0.979257	−0.489628	−	0.871931i	$-0.662868\pi$
$461$	−21.0255	−0.979257	−0.489628	+	0.871931i	$0.662868\pi$
$462$	0	0
$463$	25.7348	1.19600	0.597999	−	0.801497i	$-0.295963\pi$
$463$	25.7348	1.19600	0.597999	+	0.801497i	$0.295963\pi$
$464$	9.31157	0.432279
$465$	−6.66072	−0.308883
$466$	−8.70017	−0.403028
$467$	26.7403	1.23739	0.618697	−	0.785629i	$-0.287661\pi$
$467$	26.7403	1.23739	0.618697	+	0.785629i	$0.287661\pi$
$468$	0	0
$469$	0	0
$470$	7.88361	0.363644
$471$	15.8709	0.731290
$472$	23.1404	1.06512
$473$	26.0066	1.19578
$474$	−2.06555	−0.0948741
$475$	−14.6723	−0.673211
$476$	0	0
$477$	−6.47460	−0.296451
$478$	6.74584	0.308547
$479$	−31.0007	−1.41646	−0.708228	−	0.705984i	$-0.750505\pi$
$479$	−31.0007	−1.41646	−0.708228	+	0.705984i	$0.750505\pi$
$480$	−19.0087	−0.867622
$481$	0	0
$482$	−11.9273	−0.543275
$483$	0	0
$484$	−5.02034	−0.228197
$485$	12.5838	0.571401
$486$	−21.3874	−0.970152
$487$	−8.61392	−0.390334	−0.195167	−	0.980770i	$-0.562525\pi$
$487$	−8.61392	−0.390334	−0.195167	+	0.980770i	$0.562525\pi$
$488$	−34.1651	−1.54658
$489$	23.6183	1.06806
$490$	0	0
$491$	−0.762083	−0.0343923	−0.0171962	−	0.999852i	$-0.505474\pi$
$491$	−0.762083	−0.0343923	−0.0171962	+	0.999852i	$0.505474\pi$
$492$	1.02619	0.0462641
$493$	−27.9120	−1.25709
$494$	0	0
$495$	24.6646	1.10859
$496$	−2.06398	−0.0926754
$497$	0	0
$498$	10.8546	0.486407
$499$	31.4152	1.40634	0.703169	−	0.711022i	$-0.251768\pi$
$499$	31.4152	1.40634	0.703169	+	0.711022i	$0.251768\pi$
$500$	−10.9174	−0.488242
$501$	31.7512	1.41854
$502$	−19.8785	−0.887223
$503$	−32.0350	−1.42837	−0.714185	−	0.699957i	$-0.753202\pi$
$503$	−32.0350	−1.42837	−0.714185	+	0.699957i	$0.753202\pi$
$504$	0	0
$505$	−12.6386	−0.562412
$506$	−30.2347	−1.34409
$507$	0	0
$508$	−10.7785	−0.478218
$509$	−10.5666	−0.468355	−0.234178	−	0.972194i	$-0.575240\pi$
$509$	−10.5666	−0.468355	−0.234178	+	0.972194i	$0.575240\pi$
$510$	−15.0714	−0.667372
$511$	0	0
$512$	−13.3386	−0.589487
$513$	15.3818	0.679122
$514$	−17.8123	−0.785666
$515$	23.7228	1.04535
$516$	16.1359	0.710342
$517$	20.7677	0.913362
$518$	0	0
$519$	−23.5197	−1.03240
$520$	0	0
$521$	20.7893	0.910796	0.455398	−	0.890288i	$-0.349497\pi$
$521$	20.7893	0.910796	0.455398	+	0.890288i	$0.349497\pi$
$522$	−31.8635	−1.39463
$523$	−6.06193	−0.265070	−0.132535	−	0.991178i	$-0.542312\pi$
$523$	−6.06193	−0.265070	−0.132535	+	0.991178i	$0.542312\pi$
$524$	−1.68383	−0.0735583
$525$	0	0
$526$	26.1259	1.13915
$527$	6.18690	0.269505
$528$	13.2449	0.576411
$529$	29.8776	1.29903
$530$	2.42681	0.105414
$531$	−31.2963	−1.35814
$532$	0	0
$533$	0	0
$534$	10.1194	0.437909
$535$	−10.6572	−0.460753
$536$	39.8193	1.71993
$537$	−48.2194	−2.08082
$538$	12.0254	0.518454
$539$	0	0
$540$	4.08646	0.175853
$541$	33.1378	1.42471	0.712353	−	0.701821i	$-0.247629\pi$
$541$	33.1378	1.42471	0.712353	+	0.701821i	$0.247629\pi$
$542$	−3.51235	−0.150868
$543$	−56.8216	−2.43845
$544$	17.6564	0.757014
$545$	−1.65880	−0.0710554
$546$	0	0
$547$	23.8568	1.02004	0.510021	−	0.860162i	$-0.329637\pi$
$547$	23.8568	1.02004	0.510021	+	0.860162i	$0.329637\pi$
$548$	5.98293	0.255578
$549$	46.2067	1.97205
$550$	11.5445	0.492258
$551$	−39.9856	−1.70345
$552$	−58.6089	−2.49456
$553$	0	0
$554$	1.97088	0.0837347
$555$	27.8833	1.18358
$556$	−9.99285	−0.423791
$557$	−33.6023	−1.42378	−0.711888	−	0.702293i	$-0.752160\pi$
$557$	−33.6023	−1.42378	−0.711888	+	0.702293i	$0.752160\pi$
$558$	7.06278	0.298991
$559$	0	0
$560$	0	0
$561$	−39.7024	−1.67624
$562$	15.2360	0.642694
$563$	31.6250	1.33284	0.666418	−	0.745579i	$-0.267827\pi$
$563$	31.6250	1.33284	0.666418	+	0.745579i	$0.267827\pi$
$564$	12.8854	0.542573
$565$	19.4491	0.818232
$566$	34.5271	1.45128
$567$	0	0
$568$	26.3675	1.10636
$569$	16.8054	0.704518	0.352259	−	0.935903i	$-0.385414\pi$
$569$	16.8054	0.704518	0.352259	+	0.935903i	$0.385414\pi$
$570$	−21.5907	−0.904335
$571$	14.0607	0.588420	0.294210	−	0.955741i	$-0.404943\pi$
$571$	14.0607	0.588420	0.294210	+	0.955741i	$0.404943\pi$
$572$	0	0
$573$	−43.2217	−1.80561
$574$	0	0
$575$	−20.1902	−0.841989
$576$	30.2295	1.25956
$577$	14.2128	0.591686	0.295843	−	0.955237i	$-0.404400\pi$
$577$	14.2128	0.591686	0.295843	+	0.955237i	$0.404400\pi$
$578$	−3.49095	−0.145204
$579$	−5.46429	−0.227088
$580$	−10.6229	−0.441093
$581$	0	0
$582$	−23.1237	−0.958507
$583$	6.39293	0.264768
$584$	45.9739	1.90241
$585$	0	0
$586$	−5.17570	−0.213806
$587$	17.9188	0.739590	0.369795	−	0.929113i	$-0.379428\pi$
$587$	17.9188	0.739590	0.369795	+	0.929113i	$0.379428\pi$
$588$	0	0
$589$	8.86312	0.365198
$590$	11.7305	0.482938
$591$	−6.65606	−0.273794
$592$	8.64029	0.355114
$593$	5.30406	0.217812	0.108906	−	0.994052i	$-0.465265\pi$
$593$	5.30406	0.217812	0.108906	+	0.994052i	$0.465265\pi$
$594$	−12.1027	−0.496580
$595$	0	0
$596$	4.52427	0.185321
$597$	−41.9372	−1.71638
$598$	0	0
$599$	−19.9244	−0.814088	−0.407044	−	0.913409i	$-0.633440\pi$
$599$	−19.9244	−0.814088	−0.407044	+	0.913409i	$0.633440\pi$
$600$	22.3786	0.913602
$601$	3.44445	0.140502	0.0702509	−	0.997529i	$-0.477620\pi$
$601$	3.44445	0.140502	0.0702509	+	0.997529i	$0.477620\pi$
$602$	0	0
$603$	−53.8536	−2.19309
$604$	3.88911	0.158246
$605$	−7.95111	−0.323259
$606$	23.2245	0.943429
$607$	23.2451	0.943491	0.471745	−	0.881735i	$-0.343624\pi$
$607$	23.2451	0.943491	0.471745	+	0.881735i	$0.343624\pi$
$608$	25.2940	1.02581
$609$	0	0
$610$	−17.3192	−0.701235
$611$	0	0
$612$	−14.2146	−0.574592
$613$	−8.46236	−0.341791	−0.170896	−	0.985289i	$-0.554666\pi$
$613$	−8.46236	−0.341791	−0.170896	+	0.985289i	$0.554666\pi$
$614$	−14.5599	−0.587590
$615$	1.62525	0.0655366
$616$	0	0
$617$	14.9847	0.603262	0.301631	−	0.953425i	$-0.402469\pi$
$617$	14.9847	0.603262	0.301631	+	0.953425i	$0.402469\pi$
$618$	−43.5924	−1.75354
$619$	31.6261	1.27116	0.635580	−	0.772035i	$-0.280761\pi$
$619$	31.6261	1.27116	0.635580	+	0.772035i	$0.280761\pi$
$620$	2.35465	0.0945651
$621$	21.1665	0.849383
$622$	−8.94607	−0.358705
$623$	0	0
$624$	0	0
$625$	−3.40808	−0.136323
$626$	21.4797	0.858501
$627$	−56.8761	−2.27141
$628$	−5.61056	−0.223886
$629$	−25.8998	−1.03269
$630$	0	0
$631$	26.9817	1.07412	0.537062	−	0.843543i	$-0.319534\pi$
$631$	26.9817	1.07412	0.537062	+	0.843543i	$0.319534\pi$
$632$	2.28135	0.0907472
$633$	20.6457	0.820594
$634$	−17.9945	−0.714653
$635$	−17.0708	−0.677433
$636$	3.96651	0.157283
$637$	0	0
$638$	31.4616	1.24557
$639$	−35.6608	−1.41072
$640$	2.94407	0.116375
$641$	−35.0330	−1.38372	−0.691861	−	0.722031i	$-0.743209\pi$
$641$	−35.0330	−1.38372	−0.691861	+	0.722031i	$0.743209\pi$
$642$	19.5835	0.772898
$643$	38.2944	1.51019	0.755093	−	0.655618i	$-0.227592\pi$
$643$	38.2944	1.51019	0.755093	+	0.655618i	$0.227592\pi$
$644$	0	0
$645$	25.5557	1.00625
$646$	20.0548	0.789046
$647$	−14.8643	−0.584377	−0.292188	−	0.956361i	$-0.594383\pi$
$647$	−14.8643	−0.584377	−0.292188	+	0.956361i	$0.594383\pi$
$648$	13.6990	0.538146
$649$	30.9016	1.21299
$650$	0	0
$651$	0	0
$652$	−8.34939	−0.326987
$653$	−26.3386	−1.03071	−0.515354	−	0.856977i	$-0.672340\pi$
$653$	−26.3386	−1.03071	−0.515354	+	0.856977i	$0.672340\pi$
$654$	3.04818	0.119193
$655$	−2.66681	−0.104201
$656$	0.503623	0.0196632
$657$	−62.1775	−2.42578
$658$	0	0
$659$	3.18074	0.123904	0.0619521	−	0.998079i	$-0.480267\pi$
$659$	3.18074	0.123904	0.0619521	+	0.998079i	$0.480267\pi$
$660$	−15.1102	−0.588164
$661$	−1.09086	−0.0424295	−0.0212147	−	0.999775i	$-0.506753\pi$
$661$	−1.09086	−0.0424295	−0.0212147	+	0.999775i	$0.506753\pi$
$662$	−4.71091	−0.183095
$663$	0	0
$664$	−11.9886	−0.465249
$665$	0	0
$666$	−29.5664	−1.14567
$667$	−55.0233	−2.13051
$668$	−11.2245	−0.434288
$669$	12.6604	0.489481
$670$	20.1855	0.779833
$671$	−45.6239	−1.76129
$672$	0	0
$673$	−3.63442	−0.140096	−0.0700482	−	0.997544i	$-0.522315\pi$
$673$	−3.63442	−0.140096	−0.0700482	+	0.997544i	$0.522315\pi$
$674$	24.2467	0.933947
$675$	−8.08198	−0.311076
$676$	0	0
$677$	−4.18936	−0.161010	−0.0805052	−	0.996754i	$-0.525653\pi$
$677$	−4.18936	−0.161010	−0.0805052	+	0.996754i	$0.525653\pi$
$678$	−35.7393	−1.37256
$679$	0	0
$680$	16.6459	0.638342
$681$	−33.5303	−1.28489
$682$	−6.97369	−0.267036
$683$	−37.5157	−1.43550	−0.717750	−	0.696301i	$-0.754828\pi$
$683$	−37.5157	−1.43550	−0.717750	+	0.696301i	$0.754828\pi$
$684$	−20.3633	−0.778611
$685$	9.47565	0.362046
$686$	0	0
$687$	14.0020	0.534211
$688$	7.91903	0.301910
$689$	0	0
$690$	−29.7105	−1.13106
$691$	−15.1255	−0.575402	−0.287701	−	0.957720i	$-0.592891\pi$
$691$	−15.1255	−0.575402	−0.287701	+	0.957720i	$0.592891\pi$
$692$	8.31454	0.316071
$693$	0	0
$694$	27.7667	1.05401
$695$	−15.8265	−0.600333
$696$	60.9872	2.31171
$697$	−1.50964	−0.0571816
$698$	−25.4384	−0.962859
$699$	−22.5214	−0.851837
$700$	0	0
$701$	−38.7255	−1.46264	−0.731320	−	0.682034i	$-0.761096\pi$
$701$	−38.7255	−1.46264	−0.731320	+	0.682034i	$0.761096\pi$
$702$	0	0
$703$	−37.1030	−1.39937
$704$	−29.8482	−1.12495
$705$	20.4076	0.768596
$706$	−1.48760	−0.0559864
$707$	0	0
$708$	19.1730	0.720565
$709$	−4.43025	−0.166382	−0.0831908	−	0.996534i	$-0.526511\pi$
$709$	−4.43025	−0.166382	−0.0831908	+	0.996534i	$0.526511\pi$
$710$	13.3664	0.501633
$711$	−3.08542	−0.115712
$712$	−11.1766	−0.418861
$713$	12.1963	0.456756
$714$	0	0
$715$	0	0
$716$	17.0462	0.637047
$717$	17.4624	0.652145
$718$	17.0265	0.635424
$719$	−34.4659	−1.28536	−0.642681	−	0.766134i	$-0.722178\pi$
$719$	−34.4659	−1.28536	−0.642681	+	0.766134i	$0.722178\pi$
$720$	7.51040	0.279896
$721$	0	0
$722$	9.18191	0.341715
$723$	−30.8753	−1.14826
$724$	20.0872	0.746535
$725$	21.0095	0.780273
$726$	14.6108	0.542256
$727$	0.889602	0.0329935	0.0164968	−	0.999864i	$-0.494749\pi$
$727$	0.889602	0.0329935	0.0164968	+	0.999864i	$0.494749\pi$
$728$	0	0
$729$	−41.7839	−1.54755
$730$	23.3054	0.862573
$731$	−23.7377	−0.877971
$732$	−28.3075	−1.04627
$733$	−5.58187	−0.206171	−0.103086	−	0.994672i	$-0.532872\pi$
$733$	−5.58187	−0.206171	−0.103086	+	0.994672i	$0.532872\pi$
$734$	−33.7044	−1.24405
$735$	0	0
$736$	34.8064	1.28298
$737$	53.1743	1.95870
$738$	−1.72336	−0.0634377
$739$	25.2877	0.930225	0.465112	−	0.885252i	$-0.346014\pi$
$739$	25.2877	0.930225	0.465112	+	0.885252i	$0.346014\pi$
$740$	−9.85711	−0.362355
$741$	0	0
$742$	0	0
$743$	−12.7921	−0.469298	−0.234649	−	0.972080i	$-0.575394\pi$
$743$	−12.7921	−0.469298	−0.234649	+	0.972080i	$0.575394\pi$
$744$	−13.5183	−0.495604
$745$	7.16545	0.262522
$746$	16.4878	0.603661
$747$	16.2141	0.593241
$748$	14.0353	0.513182
$749$	0	0
$750$	31.7731	1.16019
$751$	36.1678	1.31978	0.659890	−	0.751362i	$-0.270603\pi$
$751$	36.1678	1.31978	0.659890	+	0.751362i	$0.270603\pi$
$752$	6.32379	0.230605
$753$	−51.4579	−1.87523
$754$	0	0
$755$	6.15950	0.224167
$756$	0	0
$757$	27.7793	1.00966	0.504828	−	0.863220i	$-0.331556\pi$
$757$	27.7793	1.00966	0.504828	+	0.863220i	$0.331556\pi$
$758$	−5.12114	−0.186008
$759$	−78.2659	−2.84087
$760$	23.8463	0.864998
$761$	15.7076	0.569402	0.284701	−	0.958616i	$-0.408106\pi$
$761$	15.7076	0.569402	0.284701	+	0.958616i	$0.408106\pi$
$762$	31.3688	1.13637
$763$	0	0
$764$	15.2795	0.552791
$765$	−22.5128	−0.813953
$766$	−22.8932	−0.827164
$767$	0	0
$768$	−44.7503	−1.61479
$769$	23.0847	0.832455	0.416228	−	0.909260i	$-0.363352\pi$
$769$	23.0847	0.832455	0.416228	+	0.909260i	$0.363352\pi$
$770$	0	0
$771$	−46.1091	−1.66058
$772$	1.93170	0.0695234
$773$	25.7386	0.925752	0.462876	−	0.886423i	$-0.346818\pi$
$773$	25.7386	0.925752	0.462876	+	0.886423i	$0.346818\pi$
$774$	−27.0983	−0.974028
$775$	−4.65691	−0.167281
$776$	25.5395	0.916813
$777$	0	0
$778$	1.04791	0.0375696
$779$	−2.16265	−0.0774851
$780$	0	0
$781$	35.2110	1.25995
$782$	27.5970	0.986865
$783$	−22.0254	−0.787124
$784$	0	0
$785$	−8.88589	−0.317151
$786$	4.90047	0.174794
$787$	−4.47918	−0.159666	−0.0798328	−	0.996808i	$-0.525439\pi$
$787$	−4.47918	−0.159666	−0.0798328	+	0.996808i	$0.525439\pi$
$788$	2.35301	0.0838224
$789$	67.6300	2.40769
$790$	1.15648	0.0411457
$791$	0	0
$792$	50.0581	1.77874
$793$	0	0
$794$	−22.0248	−0.781630
$795$	6.28209	0.222803
$796$	14.8254	0.525471
$797$	−20.6676	−0.732083	−0.366042	−	0.930598i	$-0.619287\pi$
$797$	−20.6676	−0.732083	−0.366042	+	0.930598i	$0.619287\pi$
$798$	0	0
$799$	−18.9559	−0.670611
$800$	−13.2901	−0.469876
$801$	15.1158	0.534091
$802$	−1.65151	−0.0583168
$803$	61.3933	2.16652
$804$	32.9922	1.16355
$805$	0	0
$806$	0	0
$807$	31.1293	1.09580
$808$	−25.6508	−0.902392
$809$	−47.4881	−1.66959	−0.834797	−	0.550558i	$-0.814415\pi$
$809$	−47.4881	−1.66959	−0.834797	+	0.550558i	$0.814415\pi$
$810$	6.94438	0.244001
$811$	26.3338	0.924704	0.462352	−	0.886697i	$-0.347006\pi$
$811$	26.3338	0.924704	0.462352	+	0.886697i	$0.347006\pi$
$812$	0	0
$813$	−9.09212	−0.318875
$814$	29.1935	1.02323
$815$	−13.2236	−0.463202
$816$	−12.0894	−0.423214
$817$	−34.0058	−1.18971
$818$	36.7470	1.28483
$819$	0	0
$820$	−0.574549	−0.0200641
$821$	−10.5083	−0.366741	−0.183371	−	0.983044i	$-0.558701\pi$
$821$	−10.5083	−0.366741	−0.183371	+	0.983044i	$0.558701\pi$
$822$	−17.4122	−0.607321
$823$	−12.3350	−0.429973	−0.214986	−	0.976617i	$-0.568971\pi$
$823$	−12.3350	−0.429973	−0.214986	+	0.976617i	$0.568971\pi$
$824$	48.1466	1.67727
$825$	29.8842	1.04043
$826$	0	0
$827$	29.2074	1.01564	0.507820	−	0.861463i	$-0.330452\pi$
$827$	29.2074	1.01564	0.507820	+	0.861463i	$0.330452\pi$
$828$	−28.0215	−0.973814
$829$	8.47220	0.294252	0.147126	−	0.989118i	$-0.452998\pi$
$829$	8.47220	0.294252	0.147126	+	0.989118i	$0.452998\pi$
$830$	−6.07737	−0.210948
$831$	5.10185	0.176981
$832$	0	0
$833$	0	0
$834$	29.0824	1.00704
$835$	−17.7771	−0.615202
$836$	20.1065	0.695397
$837$	4.88210	0.168750
$838$	12.6873	0.438275
$839$	32.3795	1.11786	0.558932	−	0.829213i	$-0.311211\pi$
$839$	32.3795	1.11786	0.558932	+	0.829213i	$0.311211\pi$
$840$	0	0
$841$	28.2561	0.974347
$842$	−13.2451	−0.456458
$843$	39.4403	1.35839
$844$	−7.29854	−0.251226
$845$	0	0
$846$	−21.6395	−0.743981
$847$	0	0
$848$	1.94665	0.0668483
$849$	89.3773	3.06742
$850$	−10.5373	−0.361427
$851$	−51.0566	−1.75020
$852$	21.8468	0.748459
$853$	39.8315	1.36381	0.681903	−	0.731443i	$-0.261153\pi$
$853$	39.8315	1.36381	0.681903	+	0.731443i	$0.261153\pi$
$854$	0	0
$855$	−32.2510	−1.10296
$856$	−21.6294	−0.739278
$857$	2.60560	0.0890056	0.0445028	−	0.999009i	$-0.485830\pi$
$857$	2.60560	0.0890056	0.0445028	+	0.999009i	$0.485830\pi$
$858$	0	0
$859$	−46.5165	−1.58712	−0.793562	−	0.608489i	$-0.791776\pi$
$859$	−46.5165	−1.58712	−0.793562	+	0.608489i	$0.791776\pi$
$860$	−9.03427	−0.308066
$861$	0	0
$862$	17.1011	0.582464
$863$	−14.1020	−0.480039	−0.240019	−	0.970768i	$-0.577154\pi$
$863$	−14.1020	−0.480039	−0.240019	+	0.970768i	$0.577154\pi$
$864$	13.9328	0.474002
$865$	13.1684	0.447739
$866$	7.55337	0.256674
$867$	−9.03672	−0.306903
$868$	0	0
$869$	3.04650	0.103345
$870$	30.9161	1.04815
$871$	0	0
$872$	−3.36663	−0.114008
$873$	−34.5409	−1.16903
$874$	39.5344	1.33727
$875$	0	0
$876$	38.0917	1.28700
$877$	10.6105	0.358290	0.179145	−	0.983823i	$-0.442667\pi$
$877$	10.6105	0.358290	0.179145	+	0.983823i	$0.442667\pi$
$878$	−1.76118	−0.0594368
$879$	−13.3979	−0.451900
$880$	−7.41566	−0.249982
$881$	33.4123	1.12569	0.562844	−	0.826563i	$-0.309707\pi$
$881$	33.4123	1.12569	0.562844	+	0.826563i	$0.309707\pi$
$882$	0	0
$883$	12.9066	0.434341	0.217170	−	0.976134i	$-0.430317\pi$
$883$	12.9066	0.434341	0.217170	+	0.976134i	$0.430317\pi$
$884$	0	0
$885$	30.3658	1.02073
$886$	7.41277	0.249037
$887$	−17.5198	−0.588256	−0.294128	−	0.955766i	$-0.595029\pi$
$887$	−17.5198	−0.588256	−0.294128	+	0.955766i	$0.595029\pi$
$888$	56.5906	1.89905
$889$	0	0
$890$	−5.66572	−0.189915
$891$	18.2935	0.612856
$892$	−4.47563	−0.149855
$893$	−27.1555	−0.908725
$894$	−13.1671	−0.440372
$895$	26.9975	0.902426
$896$	0	0
$897$	0	0
$898$	7.37937	0.246253
$899$	−12.6912	−0.423276
$900$	10.6994	0.356647
$901$	−5.83520	−0.194399
$902$	1.70162	0.0566578
$903$	0	0
$904$	39.4731	1.31285
$905$	31.8137	1.05752
$906$	−11.3185	−0.376034
$907$	19.2077	0.637782	0.318891	−	0.947791i	$-0.396690\pi$
$907$	19.2077	0.637782	0.318891	+	0.947791i	$0.396690\pi$
$908$	11.8534	0.393370
$909$	34.6915	1.15064
$910$	0	0
$911$	5.98108	0.198162	0.0990811	−	0.995079i	$-0.468410\pi$
$911$	5.98108	0.198162	0.0990811	+	0.995079i	$0.468410\pi$
$912$	−17.3188	−0.573484
$913$	−16.0095	−0.529838
$914$	37.1383	1.22842
$915$	−44.8328	−1.48213
$916$	−4.94991	−0.163549
$917$	0	0
$918$	11.0469	0.364601
$919$	−46.0843	−1.52018	−0.760090	−	0.649818i	$-0.774845\pi$
$919$	−46.0843	−1.52018	−0.760090	+	0.649818i	$0.774845\pi$
$920$	32.8144	1.08186
$921$	−37.6900	−1.24193
$922$	−21.6318	−0.712406
$923$	0	0
$924$	0	0
$925$	19.4949	0.640988
$926$	26.4769	0.870084
$927$	−65.1160	−2.13869
$928$	−36.2188	−1.18894
$929$	4.22690	0.138680	0.0693401	−	0.997593i	$-0.477911\pi$
$929$	4.22690	0.138680	0.0693401	+	0.997593i	$0.477911\pi$
$930$	−6.85278	−0.224712
$931$	0	0
$932$	7.96161	0.260791
$933$	−23.1579	−0.758157
$934$	27.5114	0.900200
$935$	22.2289	0.726962
$936$	0	0
$937$	45.0155	1.47059	0.735296	−	0.677746i	$-0.237043\pi$
$937$	45.0155	1.47059	0.735296	+	0.677746i	$0.237043\pi$
$938$	0	0
$939$	55.6027	1.81452
$940$	−7.21437	−0.235307
$941$	−20.9767	−0.683822	−0.341911	−	0.939732i	$-0.611074\pi$
$941$	−20.9767	−0.683822	−0.341911	+	0.939732i	$0.611074\pi$
$942$	16.3285	0.532011
$943$	−2.97598	−0.0969111
$944$	9.40956	0.306255
$945$	0	0
$946$	26.7565	0.869928
$947$	56.4128	1.83317	0.916584	−	0.399842i	$-0.130935\pi$
$947$	56.4128	1.83317	0.916584	+	0.399842i	$0.130935\pi$
$948$	1.89021	0.0613912
$949$	0	0
$950$	−15.0954	−0.489758
$951$	−46.5809	−1.51049
$952$	0	0
$953$	−46.8932	−1.51902	−0.759510	−	0.650495i	$-0.774561\pi$
$953$	−46.8932	−1.51902	−0.759510	+	0.650495i	$0.774561\pi$
$954$	−6.66129	−0.215667
$955$	24.1993	0.783071
$956$	−6.17318	−0.199655
$957$	81.4419	2.63264
$958$	−31.8946	−1.03047
$959$	0	0
$960$	−29.3307	−0.946645
$961$	−28.1869	−0.909255
$962$	0	0
$963$	29.2528	0.942657
$964$	10.9148	0.351543
$965$	3.05939	0.0984852
$966$	0	0
$967$	−44.8315	−1.44168	−0.720842	−	0.693100i	$-0.756245\pi$
$967$	−44.8315	−1.44168	−0.720842	+	0.693100i	$0.756245\pi$
$968$	−16.1372	−0.518669
$969$	51.9142	1.66772
$970$	12.9467	0.415692
$971$	−12.1544	−0.390053	−0.195027	−	0.980798i	$-0.562479\pi$
$971$	−12.1544	−0.390053	−0.195027	+	0.980798i	$0.562479\pi$
$972$	19.5718	0.627767
$973$	0	0
$974$	−8.86231	−0.283967
$975$	0	0
$976$	−13.8925	−0.444688
$977$	27.9617	0.894573	0.447286	−	0.894391i	$-0.352391\pi$
$977$	27.9617	0.894573	0.447286	+	0.894391i	$0.352391\pi$
$978$	24.2994	0.777008
$979$	−14.9251	−0.477010
$980$	0	0
$981$	4.55320	0.145373
$982$	−0.784058	−0.0250203
$983$	−50.0256	−1.59557	−0.797784	−	0.602943i	$-0.793995\pi$
$983$	−50.0256	−1.59557	−0.797784	+	0.602943i	$0.793995\pi$
$984$	3.29854	0.105154
$985$	3.72665	0.118741
$986$	−28.7168	−0.914530
$987$	0	0
$988$	0	0
$989$	−46.7946	−1.48798
$990$	25.3758	0.806496
$991$	−14.0583	−0.446576	−0.223288	−	0.974752i	$-0.571679\pi$
$991$	−14.0583	−0.446576	−0.223288	+	0.974752i	$0.571679\pi$
$992$	8.02818	0.254895
$993$	−12.1947	−0.386988
$994$	0	0
$995$	23.4801	0.744370
$996$	−9.93317	−0.314745
$997$	−7.47365	−0.236693	−0.118346	−	0.992972i	$-0.537759\pi$
$997$	−7.47365	−0.236693	−0.118346	+	0.992972i	$0.537759\pi$
$998$	32.3211	1.02311
$999$	−20.4376	−0.646617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cv.1.16		24
7.2	even	3		1183.2.e.k.508.9	yes	48
7.4	even	3		1183.2.e.k.170.9	✓	48
7.6	odd	2		8281.2.a.cw.1.16		24
13.12	even	2		8281.2.a.cu.1.9		24
91.25	even	6		1183.2.e.l.170.16	yes	48
91.51	even	6		1183.2.e.l.508.16	yes	48
91.90	odd	2		8281.2.a.ct.1.9		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.9	✓	48	7.4	even	3
1183.2.e.k.508.9	yes	48	7.2	even	3
1183.2.e.l.170.16	yes	48	91.25	even	6
1183.2.e.l.508.16	yes	48	91.51	even	6
8281.2.a.ct.1.9		24	91.90	odd	2
8281.2.a.cu.1.9		24	13.12	even	2
8281.2.a.cv.1.16		24	1.1	even	1	trivial
8281.2.a.cw.1.16		24	7.6	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.02884 q^{2} +2.66326 q^{3} -0.941498 q^{4} -1.49113 q^{5} +2.74006 q^{6} -3.02632 q^{8} +4.09295 q^{9} +O(q^{10})\)	\(q+1.02884 q^{2} +2.66326 q^{3} -0.941498 q^{4} -1.49113 q^{5} +2.74006 q^{6} -3.02632 q^{8} +4.09295 q^{9} -1.53412 q^{10} -4.04132 q^{11} -2.50745 q^{12} -3.97125 q^{15} -1.23059 q^{16} +3.68875 q^{17} +4.21097 q^{18} +5.28437 q^{19} +1.40389 q^{20} -4.15786 q^{22} +7.27170 q^{23} -8.05987 q^{24} -2.77654 q^{25} +2.91081 q^{27} -7.56677 q^{29} -4.08577 q^{30} +1.67723 q^{31} +4.78656 q^{32} -10.7631 q^{33} +3.79512 q^{34} -3.85350 q^{36} -7.02128 q^{37} +5.43675 q^{38} +4.51262 q^{40} -0.409255 q^{41} -6.43516 q^{43} +3.80490 q^{44} -6.10310 q^{45} +7.48138 q^{46} -5.13884 q^{47} -3.27737 q^{48} -2.85661 q^{50} +9.82410 q^{51} -1.58189 q^{53} +2.99474 q^{54} +6.02612 q^{55} +14.0736 q^{57} -7.78497 q^{58} -7.64640 q^{59} +3.73893 q^{60} +11.2893 q^{61} +1.72560 q^{62} +7.38576 q^{64} -11.0734 q^{66} -13.1577 q^{67} -3.47295 q^{68} +19.3664 q^{69} -8.71275 q^{71} -12.3866 q^{72} -15.1914 q^{73} -7.22374 q^{74} -7.39466 q^{75} -4.97522 q^{76} -0.753837 q^{79} +1.83496 q^{80} -4.52661 q^{81} -0.421056 q^{82} +3.96146 q^{83} -5.50039 q^{85} -6.62072 q^{86} -20.1523 q^{87} +12.2303 q^{88} +3.69313 q^{89} -6.27909 q^{90} -6.84629 q^{92} +4.46690 q^{93} -5.28702 q^{94} -7.87966 q^{95} +12.7479 q^{96} -8.43912 q^{97} -16.5409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 23 q^{4} - 13 q^{5} - 14 q^{6} + 26 q^{9}+O(q^{10}) \) 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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