LMFDB - Embedded newform 6864.2.a.cf.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2172244.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 16x^{2} + 5x - 8 $ x^5 - 2x^4 - 8x^3 + 16x^2 + 5x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.19338$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.47314 q^{5} +3.67591 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.47314 q^{5} +3.67591 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -3.47314 q^{15} +2.76229 q^{17} -6.53152 q^{19} +3.67591 q^{21} +2.05410 q^{23} +7.06267 q^{25} +1.00000 q^{27} +5.71513 q^{29} -4.73430 q^{31} +1.00000 q^{33} -12.7670 q^{35} -5.04350 q^{37} +1.00000 q^{39} +9.96544 q^{41} -0.157873 q^{43} -3.47314 q^{45} +12.5681 q^{47} +6.51235 q^{49} +2.76229 q^{51} -9.05447 q^{53} -3.47314 q^{55} -6.53152 q^{57} -0.0906653 q^{59} +10.0719 q^{61} +3.67591 q^{63} -3.47314 q^{65} +0.535436 q^{67} +2.05410 q^{69} +13.9113 q^{71} -12.6222 q^{73} +7.06267 q^{75} +3.67591 q^{77} -6.77087 q^{79} +1.00000 q^{81} -10.4063 q^{83} -9.59382 q^{85} +5.71513 q^{87} -7.25547 q^{89} +3.67591 q^{91} -4.73430 q^{93} +22.6849 q^{95} -0.269986 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 4 q^{19} + 5 q^{21} + 5 q^{23} + 4 q^{25} + 5 q^{27} + 11 q^{29} + 8 q^{31} + 5 q^{33} + 5 q^{35} - 8 q^{37} + 5 q^{39} - q^{41} - q^{43} + q^{45} + 18 q^{47} + 10 q^{49} + 2 q^{53} + q^{55} + 4 q^{57} + 13 q^{59} + 9 q^{61} + 5 q^{63} + q^{65} - 5 q^{67} + 5 q^{69} + 24 q^{71} - 13 q^{73} + 4 q^{75} + 5 q^{77} + 6 q^{79} + 5 q^{81} + 22 q^{83} - 22 q^{85} + 11 q^{87} - 14 q^{89} + 5 q^{91} + 8 q^{93} + 32 q^{95} - 20 q^{97} + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 4 * q^19 + 5 * q^21 + 5 * q^23 + 4 * q^25 + 5 * q^27 + 11 * q^29 + 8 * q^31 + 5 * q^33 + 5 * q^35 - 8 * q^37 + 5 * q^39 - q^41 - q^43 + q^45 + 18 * q^47 + 10 * q^49 + 2 * q^53 + q^55 + 4 * q^57 + 13 * q^59 + 9 * q^61 + 5 * q^63 + q^65 - 5 * q^67 + 5 * q^69 + 24 * q^71 - 13 * q^73 + 4 * q^75 + 5 * q^77 + 6 * q^79 + 5 * q^81 + 22 * q^83 - 22 * q^85 + 11 * q^87 - 14 * q^89 + 5 * q^91 + 8 * q^93 + 32 * q^95 - 20 * q^97 + 5 * 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.47314	−1.55323	−0.776617	−	0.629973i	$-0.783066\pi$
$5$	−3.47314	−1.55323	−0.776617	+	0.629973i	$0.783066\pi$
$6$	0	0
$7$	3.67591	1.38937	0.694683	−	0.719316i	$-0.255545\pi$
$7$	3.67591	1.38937	0.694683	+	0.719316i	$0.255545\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−3.47314	−0.896760
$16$	0	0
$17$	2.76229	0.669955	0.334977	−	0.942226i	$-0.391271\pi$
$17$	2.76229	0.669955	0.334977	+	0.942226i	$0.391271\pi$
$18$	0	0
$19$	−6.53152	−1.49843	−0.749217	−	0.662325i	$-0.769570\pi$
$19$	−6.53152	−1.49843	−0.749217	+	0.662325i	$0.769570\pi$
$20$	0	0
$21$	3.67591	0.802150
$22$	0	0
$23$	2.05410	0.428309	0.214155	−	0.976800i	$-0.431300\pi$
$23$	2.05410	0.428309	0.214155	+	0.976800i	$0.431300\pi$
$24$	0	0
$25$	7.06267	1.41253
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	5.71513	1.06127	0.530636	−	0.847600i	$-0.321953\pi$
$29$	5.71513	1.06127	0.530636	+	0.847600i	$0.321953\pi$
$30$	0	0
$31$	−4.73430	−0.850305	−0.425153	−	0.905122i	$-0.639780\pi$
$31$	−4.73430	−0.850305	−0.425153	+	0.905122i	$0.639780\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−12.7670	−2.15801
$36$	0	0
$37$	−5.04350	−0.829146	−0.414573	−	0.910016i	$-0.636069\pi$
$37$	−5.04350	−0.829146	−0.414573	+	0.910016i	$0.636069\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	9.96544	1.55634	0.778170	−	0.628053i	$-0.216148\pi$
$41$	9.96544	1.55634	0.778170	+	0.628053i	$0.216148\pi$
$42$	0	0
$43$	−0.157873	−0.0240753	−0.0120377	−	0.999928i	$-0.503832\pi$
$43$	−0.157873	−0.0240753	−0.0120377	+	0.999928i	$0.503832\pi$
$44$	0	0
$45$	−3.47314	−0.517744
$46$	0	0
$47$	12.5681	1.83324	0.916622	−	0.399755i	$-0.130905\pi$
$47$	12.5681	1.83324	0.916622	+	0.399755i	$0.130905\pi$
$48$	0	0
$49$	6.51235	0.930336
$50$	0	0
$51$	2.76229	0.386799
$52$	0	0
$53$	−9.05447	−1.24373	−0.621863	−	0.783126i	$-0.713624\pi$
$53$	−9.05447	−1.24373	−0.621863	+	0.783126i	$0.713624\pi$
$54$	0	0
$55$	−3.47314	−0.468318
$56$	0	0
$57$	−6.53152	−0.865121
$58$	0	0
$59$	−0.0906653	−0.0118036	−0.00590181	−	0.999983i	$-0.501879\pi$
$59$	−0.0906653	−0.0118036	−0.00590181	+	0.999983i	$0.501879\pi$
$60$	0	0
$61$	10.0719	1.28957	0.644785	−	0.764364i	$-0.276947\pi$
$61$	10.0719	1.28957	0.644785	+	0.764364i	$0.276947\pi$
$62$	0	0
$63$	3.67591	0.463122
$64$	0	0
$65$	−3.47314	−0.430789
$66$	0	0
$67$	0.535436	0.0654140	0.0327070	−	0.999465i	$-0.489587\pi$
$67$	0.535436	0.0654140	0.0327070	+	0.999465i	$0.489587\pi$
$68$	0	0
$69$	2.05410	0.247284
$70$	0	0
$71$	13.9113	1.65097	0.825486	−	0.564422i	$-0.190901\pi$
$71$	13.9113	1.65097	0.825486	+	0.564422i	$0.190901\pi$
$72$	0	0
$73$	−12.6222	−1.47732	−0.738658	−	0.674081i	$-0.764540\pi$
$73$	−12.6222	−1.47732	−0.738658	+	0.674081i	$0.764540\pi$
$74$	0	0
$75$	7.06267	0.815527
$76$	0	0
$77$	3.67591	0.418909
$78$	0	0
$79$	−6.77087	−0.761782	−0.380891	−	0.924620i	$-0.624383\pi$
$79$	−6.77087	−0.761782	−0.380891	+	0.924620i	$0.624383\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.4063	−1.14224	−0.571120	−	0.820867i	$-0.693491\pi$
$83$	−10.4063	−1.14224	−0.571120	+	0.820867i	$0.693491\pi$
$84$	0	0
$85$	−9.59382	−1.04060
$86$	0	0
$87$	5.71513	0.612726
$88$	0	0
$89$	−7.25547	−0.769078	−0.384539	−	0.923109i	$-0.625640\pi$
$89$	−7.25547	−0.769078	−0.384539	+	0.923109i	$0.625640\pi$
$90$	0	0
$91$	3.67591	0.385341
$92$	0	0
$93$	−4.73430	−0.490924
$94$	0	0
$95$	22.6849	2.32742
$96$	0	0
$97$	−0.269986	−0.0274130	−0.0137065	−	0.999906i	$-0.504363\pi$
$97$	−0.269986	−0.0274130	−0.0137065	+	0.999906i	$0.504363\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	1.34590	0.133922	0.0669612	−	0.997756i	$-0.478670\pi$
$101$	1.34590	0.133922	0.0669612	+	0.997756i	$0.478670\pi$
$102$	0	0
$103$	6.85561	0.675503	0.337751	−	0.941235i	$-0.390334\pi$
$103$	6.85561	0.675503	0.337751	+	0.941235i	$0.390334\pi$
$104$	0	0
$105$	−12.7670	−1.24593
$106$	0	0
$107$	4.84703	0.468580	0.234290	−	0.972167i	$-0.424723\pi$
$107$	4.84703	0.468580	0.234290	+	0.972167i	$0.424723\pi$
$108$	0	0
$109$	7.83643	0.750594	0.375297	−	0.926905i	$-0.377541\pi$
$109$	7.83643	0.750594	0.375297	+	0.926905i	$0.377541\pi$
$110$	0	0
$111$	−5.04350	−0.478708
$112$	0	0
$113$	15.6377	1.47107	0.735535	−	0.677487i	$-0.236931\pi$
$113$	15.6377	1.47107	0.735535	+	0.677487i	$0.236931\pi$
$114$	0	0
$115$	−7.13416	−0.665264
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	10.1540	0.930812
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	9.96544	0.898554
$124$	0	0
$125$	−7.16394	−0.640762
$126$	0	0
$127$	−0.562162	−0.0498838	−0.0249419	−	0.999689i	$-0.507940\pi$
$127$	−0.562162	−0.0498838	−0.0249419	+	0.999689i	$0.507940\pi$
$128$	0	0
$129$	−0.157873	−0.0138999
$130$	0	0
$131$	6.74741	0.589524	0.294762	−	0.955571i	$-0.404760\pi$
$131$	6.74741	0.589524	0.294762	+	0.955571i	$0.404760\pi$
$132$	0	0
$133$	−24.0093	−2.08187
$134$	0	0
$135$	−3.47314	−0.298920
$136$	0	0
$137$	−16.4643	−1.40664	−0.703321	−	0.710873i	$-0.748300\pi$
$137$	−16.4643	−1.40664	−0.703321	+	0.710873i	$0.748300\pi$
$138$	0	0
$139$	10.3894	0.881218	0.440609	−	0.897699i	$-0.354763\pi$
$139$	10.3894	0.881218	0.440609	+	0.897699i	$0.354763\pi$
$140$	0	0
$141$	12.5681	1.05842
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−19.8494	−1.64840
$146$	0	0
$147$	6.51235	0.537129
$148$	0	0
$149$	−3.75801	−0.307868	−0.153934	−	0.988081i	$-0.549194\pi$
$149$	−3.75801	−0.307868	−0.153934	+	0.988081i	$0.549194\pi$
$150$	0	0
$151$	9.10984	0.741348	0.370674	−	0.928763i	$-0.379127\pi$
$151$	9.10984	0.741348	0.370674	+	0.928763i	$0.379127\pi$
$152$	0	0
$153$	2.76229	0.223318
$154$	0	0
$155$	16.4429	1.32072
$156$	0	0
$157$	−14.2318	−1.13582	−0.567909	−	0.823091i	$-0.692248\pi$
$157$	−14.2318	−1.13582	−0.567909	+	0.823091i	$0.692248\pi$
$158$	0	0
$159$	−9.05447	−0.718066
$160$	0	0
$161$	7.55069	0.595078
$162$	0	0
$163$	−16.5636	−1.29736	−0.648679	−	0.761062i	$-0.724678\pi$
$163$	−16.5636	−1.29736	−0.648679	+	0.761062i	$0.724678\pi$
$164$	0	0
$165$	−3.47314	−0.270383
$166$	0	0
$167$	2.56608	0.198569	0.0992845	−	0.995059i	$-0.468345\pi$
$167$	2.56608	0.198569	0.0992845	+	0.995059i	$0.468345\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.53152	−0.499478
$172$	0	0
$173$	6.52583	0.496149	0.248075	−	0.968741i	$-0.420202\pi$
$173$	6.52583	0.496149	0.248075	+	0.968741i	$0.420202\pi$
$174$	0	0
$175$	25.9618	1.96253
$176$	0	0
$177$	−0.0906653	−0.00681482
$178$	0	0
$179$	18.9183	1.41402	0.707009	−	0.707204i	$-0.250044\pi$
$179$	18.9183	1.41402	0.707009	+	0.707204i	$0.250044\pi$
$180$	0	0
$181$	21.2809	1.58180	0.790900	−	0.611946i	$-0.209613\pi$
$181$	21.2809	1.58180	0.790900	+	0.611946i	$0.209613\pi$
$182$	0	0
$183$	10.0719	0.744534
$184$	0	0
$185$	17.5168	1.28786
$186$	0	0
$187$	2.76229	0.201999
$188$	0	0
$189$	3.67591	0.267383
$190$	0	0
$191$	0.0735212	0.00531981	0.00265990	−	0.999996i	$-0.499153\pi$
$191$	0.0735212	0.00531981	0.00265990	+	0.999996i	$0.499153\pi$
$192$	0	0
$193$	15.4778	1.11412	0.557058	−	0.830474i	$-0.311930\pi$
$193$	15.4778	1.11412	0.557058	+	0.830474i	$0.311930\pi$
$194$	0	0
$195$	−3.47314	−0.248716
$196$	0	0
$197$	10.5229	0.749729	0.374865	−	0.927080i	$-0.377689\pi$
$197$	10.5229	0.749729	0.374865	+	0.927080i	$0.377689\pi$
$198$	0	0
$199$	−11.1267	−0.788754	−0.394377	−	0.918949i	$-0.629040\pi$
$199$	−11.1267	−0.788754	−0.394377	+	0.918949i	$0.629040\pi$
$200$	0	0
$201$	0.535436	0.0377668
$202$	0	0
$203$	21.0083	1.47450
$204$	0	0
$205$	−34.6113	−2.41736
$206$	0	0
$207$	2.05410	0.142770
$208$	0	0
$209$	−6.53152	−0.451795
$210$	0	0
$211$	−26.3391	−1.81326	−0.906629	−	0.421929i	$-0.861353\pi$
$211$	−26.3391	−1.81326	−0.906629	+	0.421929i	$0.861353\pi$
$212$	0	0
$213$	13.9113	0.953190
$214$	0	0
$215$	0.548313	0.0373946
$216$	0	0
$217$	−17.4029	−1.18138
$218$	0	0
$219$	−12.6222	−0.852928
$220$	0	0
$221$	2.76229	0.185812
$222$	0	0
$223$	19.4380	1.30166	0.650831	−	0.759223i	$-0.274421\pi$
$223$	19.4380	1.30166	0.650831	+	0.759223i	$0.274421\pi$
$224$	0	0
$225$	7.06267	0.470845
$226$	0	0
$227$	25.3994	1.68582	0.842908	−	0.538058i	$-0.180842\pi$
$227$	25.3994	1.68582	0.842908	+	0.538058i	$0.180842\pi$
$228$	0	0
$229$	15.9460	1.05374	0.526871	−	0.849945i	$-0.323365\pi$
$229$	15.9460	1.05374	0.526871	+	0.849945i	$0.323365\pi$
$230$	0	0
$231$	3.67591	0.241857
$232$	0	0
$233$	−13.9924	−0.916674	−0.458337	−	0.888779i	$-0.651555\pi$
$233$	−13.9924	−0.916674	−0.458337	+	0.888779i	$0.651555\pi$
$234$	0	0
$235$	−43.6507	−2.84746
$236$	0	0
$237$	−6.77087	−0.439815
$238$	0	0
$239$	−12.8121	−0.828745	−0.414373	−	0.910107i	$-0.635999\pi$
$239$	−12.8121	−0.828745	−0.414373	+	0.910107i	$0.635999\pi$
$240$	0	0
$241$	−5.64919	−0.363896	−0.181948	−	0.983308i	$-0.558240\pi$
$241$	−5.64919	−0.363896	−0.181948	+	0.983308i	$0.558240\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−22.6183	−1.44503
$246$	0	0
$247$	−6.53152	−0.415591
$248$	0	0
$249$	−10.4063	−0.659472
$250$	0	0
$251$	26.7690	1.68964	0.844821	−	0.535049i	$-0.179707\pi$
$251$	26.7690	1.68964	0.844821	+	0.535049i	$0.179707\pi$
$252$	0	0
$253$	2.05410	0.129140
$254$	0	0
$255$	−9.59382	−0.600788
$256$	0	0
$257$	25.3436	1.58089	0.790445	−	0.612532i	$-0.209849\pi$
$257$	25.3436	1.58089	0.790445	+	0.612532i	$0.209849\pi$
$258$	0	0
$259$	−18.5395	−1.15199
$260$	0	0
$261$	5.71513	0.353758
$262$	0	0
$263$	7.11046	0.438450	0.219225	−	0.975674i	$-0.429647\pi$
$263$	7.11046	0.438450	0.219225	+	0.975674i	$0.429647\pi$
$264$	0	0
$265$	31.4474	1.93180
$266$	0	0
$267$	−7.25547	−0.444028
$268$	0	0
$269$	−3.90037	−0.237810	−0.118905	−	0.992906i	$-0.537938\pi$
$269$	−3.90037	−0.237810	−0.118905	+	0.992906i	$0.537938\pi$
$270$	0	0
$271$	23.1368	1.40546	0.702730	−	0.711457i	$-0.251964\pi$
$271$	23.1368	1.40546	0.702730	+	0.711457i	$0.251964\pi$
$272$	0	0
$273$	3.67591	0.222476
$274$	0	0
$275$	7.06267	0.425895
$276$	0	0
$277$	14.2773	0.857839	0.428920	−	0.903343i	$-0.358894\pi$
$277$	14.2773	0.857839	0.428920	+	0.903343i	$0.358894\pi$
$278$	0	0
$279$	−4.73430	−0.283435
$280$	0	0
$281$	−30.6045	−1.82571	−0.912857	−	0.408280i	$-0.866129\pi$
$281$	−30.6045	−1.82571	−0.912857	+	0.408280i	$0.866129\pi$
$282$	0	0
$283$	−5.13681	−0.305352	−0.152676	−	0.988276i	$-0.548789\pi$
$283$	−5.13681	−0.305352	−0.152676	+	0.988276i	$0.548789\pi$
$284$	0	0
$285$	22.6849	1.34373
$286$	0	0
$287$	36.6321	2.16233
$288$	0	0
$289$	−9.36973	−0.551161
$290$	0	0
$291$	−0.269986	−0.0158269
$292$	0	0
$293$	−13.6584	−0.797931	−0.398966	−	0.916966i	$-0.630631\pi$
$293$	−13.6584	−0.797931	−0.398966	+	0.916966i	$0.630631\pi$
$294$	0	0
$295$	0.314893	0.0183338
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	2.05410	0.118792
$300$	0	0
$301$	−0.580326	−0.0334494
$302$	0	0
$303$	1.34590	0.0773202
$304$	0	0
$305$	−34.9809	−2.00300
$306$	0	0
$307$	−9.02002	−0.514800	−0.257400	−	0.966305i	$-0.582866\pi$
$307$	−9.02002	−0.514800	−0.257400	+	0.966305i	$0.582866\pi$
$308$	0	0
$309$	6.85561	0.390002
$310$	0	0
$311$	30.5299	1.73119	0.865595	−	0.500745i	$-0.166941\pi$
$311$	30.5299	1.73119	0.865595	+	0.500745i	$0.166941\pi$
$312$	0	0
$313$	−6.37842	−0.360529	−0.180265	−	0.983618i	$-0.557695\pi$
$313$	−6.37842	−0.360529	−0.180265	+	0.983618i	$0.557695\pi$
$314$	0	0
$315$	−12.7670	−0.719336
$316$	0	0
$317$	8.71917	0.489717	0.244859	−	0.969559i	$-0.421258\pi$
$317$	8.71917	0.489717	0.244859	+	0.969559i	$0.421258\pi$
$318$	0	0
$319$	5.71513	0.319986
$320$	0	0
$321$	4.84703	0.270535
$322$	0	0
$323$	−18.0420	−1.00388
$324$	0	0
$325$	7.06267	0.391767
$326$	0	0
$327$	7.83643	0.433356
$328$	0	0
$329$	46.1992	2.54705
$330$	0	0
$331$	10.7678	0.591853	0.295926	−	0.955211i	$-0.404372\pi$
$331$	10.7678	0.591853	0.295926	+	0.955211i	$0.404372\pi$
$332$	0	0
$333$	−5.04350	−0.276382
$334$	0	0
$335$	−1.85964	−0.101603
$336$	0	0
$337$	7.19746	0.392070	0.196035	−	0.980597i	$-0.437193\pi$
$337$	7.19746	0.392070	0.196035	+	0.980597i	$0.437193\pi$
$338$	0	0
$339$	15.6377	0.849323
$340$	0	0
$341$	−4.73430	−0.256377
$342$	0	0
$343$	−1.79256	−0.0967894
$344$	0	0
$345$	−7.13416	−0.384090
$346$	0	0
$347$	−8.45206	−0.453730	−0.226865	−	0.973926i	$-0.572848\pi$
$347$	−8.45206	−0.453730	−0.226865	+	0.973926i	$0.572848\pi$
$348$	0	0
$349$	−28.6958	−1.53605	−0.768026	−	0.640418i	$-0.778761\pi$
$349$	−28.6958	−1.53605	−0.768026	+	0.640418i	$0.778761\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	10.8859	0.579396	0.289698	−	0.957118i	$-0.406445\pi$
$353$	10.8859	0.579396	0.289698	+	0.957118i	$0.406445\pi$
$354$	0	0
$355$	−48.3160	−2.56435
$356$	0	0
$357$	10.1540	0.537404
$358$	0	0
$359$	0.215887	0.0113941	0.00569705	−	0.999984i	$-0.498187\pi$
$359$	0.215887	0.0113941	0.00569705	+	0.999984i	$0.498187\pi$
$360$	0	0
$361$	23.6608	1.24530
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	43.8386	2.29462
$366$	0	0
$367$	32.4070	1.69163	0.845817	−	0.533473i	$-0.179114\pi$
$367$	32.4070	1.69163	0.845817	+	0.533473i	$0.179114\pi$
$368$	0	0
$369$	9.96544	0.518780
$370$	0	0
$371$	−33.2835	−1.72799
$372$	0	0
$373$	36.5925	1.89469	0.947345	−	0.320215i	$-0.103755\pi$
$373$	36.5925	1.89469	0.947345	+	0.320215i	$0.103755\pi$
$374$	0	0
$375$	−7.16394	−0.369944
$376$	0	0
$377$	5.71513	0.294344
$378$	0	0
$379$	16.8592	0.866000	0.433000	−	0.901394i	$-0.357455\pi$
$379$	16.8592	0.866000	0.433000	+	0.901394i	$0.357455\pi$
$380$	0	0
$381$	−0.562162	−0.0288004
$382$	0	0
$383$	22.6567	1.15771	0.578853	−	0.815432i	$-0.303501\pi$
$383$	22.6567	1.15771	0.578853	+	0.815432i	$0.303501\pi$
$384$	0	0
$385$	−12.7670	−0.650664
$386$	0	0
$387$	−0.157873	−0.00802511
$388$	0	0
$389$	−3.25220	−0.164893	−0.0824466	−	0.996595i	$-0.526273\pi$
$389$	−3.25220	−0.164893	−0.0824466	+	0.996595i	$0.526273\pi$
$390$	0	0
$391$	5.67403	0.286948
$392$	0	0
$393$	6.74741	0.340362
$394$	0	0
$395$	23.5161	1.18323
$396$	0	0
$397$	8.08044	0.405545	0.202773	−	0.979226i	$-0.435005\pi$
$397$	8.08044	0.405545	0.202773	+	0.979226i	$0.435005\pi$
$398$	0	0
$399$	−24.0093	−1.20197
$400$	0	0
$401$	34.0056	1.69816	0.849080	−	0.528264i	$-0.177157\pi$
$401$	34.0056	1.69816	0.849080	+	0.528264i	$0.177157\pi$
$402$	0	0
$403$	−4.73430	−0.235832
$404$	0	0
$405$	−3.47314	−0.172581
$406$	0	0
$407$	−5.04350	−0.249997
$408$	0	0
$409$	−26.6961	−1.32004	−0.660018	−	0.751250i	$-0.729451\pi$
$409$	−26.6961	−1.32004	−0.660018	+	0.751250i	$0.729451\pi$
$410$	0	0
$411$	−16.4643	−0.812125
$412$	0	0
$413$	−0.333278	−0.0163995
$414$	0	0
$415$	36.1425	1.77416
$416$	0	0
$417$	10.3894	0.508771
$418$	0	0
$419$	6.16191	0.301029	0.150514	−	0.988608i	$-0.451907\pi$
$419$	6.16191	0.301029	0.150514	+	0.988608i	$0.451907\pi$
$420$	0	0
$421$	−32.3136	−1.57487	−0.787433	−	0.616400i	$-0.788590\pi$
$421$	−32.3136	−1.57487	−0.787433	+	0.616400i	$0.788590\pi$
$422$	0	0
$423$	12.5681	0.611081
$424$	0	0
$425$	19.5092	0.946334
$426$	0	0
$427$	37.0233	1.79168
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	2.21967	0.106918	0.0534590	−	0.998570i	$-0.482975\pi$
$431$	2.21967	0.106918	0.0534590	+	0.998570i	$0.482975\pi$
$432$	0	0
$433$	24.6612	1.18514	0.592570	−	0.805519i	$-0.298114\pi$
$433$	24.6612	1.18514	0.592570	+	0.805519i	$0.298114\pi$
$434$	0	0
$435$	−19.8494	−0.951707
$436$	0	0
$437$	−13.4164	−0.641793
$438$	0	0
$439$	33.9454	1.62012	0.810062	−	0.586344i	$-0.199433\pi$
$439$	33.9454	1.62012	0.810062	+	0.586344i	$0.199433\pi$
$440$	0	0
$441$	6.51235	0.310112
$442$	0	0
$443$	13.6590	0.648959	0.324479	−	0.945893i	$-0.394811\pi$
$443$	13.6590	0.648959	0.324479	+	0.945893i	$0.394811\pi$
$444$	0	0
$445$	25.1992	1.19456
$446$	0	0
$447$	−3.75801	−0.177748
$448$	0	0
$449$	−4.75021	−0.224176	−0.112088	−	0.993698i	$-0.535754\pi$
$449$	−4.75021	−0.224176	−0.112088	+	0.993698i	$0.535754\pi$
$450$	0	0
$451$	9.96544	0.469254
$452$	0	0
$453$	9.10984	0.428017
$454$	0	0
$455$	−12.7670	−0.598524
$456$	0	0
$457$	19.4730	0.910909	0.455454	−	0.890259i	$-0.349477\pi$
$457$	19.4730	0.910909	0.455454	+	0.890259i	$0.349477\pi$
$458$	0	0
$459$	2.76229	0.128933
$460$	0	0
$461$	4.64226	0.216211	0.108106	−	0.994139i	$-0.465521\pi$
$461$	4.64226	0.216211	0.108106	+	0.994139i	$0.465521\pi$
$462$	0	0
$463$	6.24009	0.290001	0.145001	−	0.989432i	$-0.453682\pi$
$463$	6.24009	0.290001	0.145001	+	0.989432i	$0.453682\pi$
$464$	0	0
$465$	16.4429	0.762520
$466$	0	0
$467$	−23.0698	−1.06754	−0.533772	−	0.845628i	$-0.679226\pi$
$467$	−23.0698	−1.06754	−0.533772	+	0.845628i	$0.679226\pi$
$468$	0	0
$469$	1.96822	0.0908839
$470$	0	0
$471$	−14.2318	−0.655765
$472$	0	0
$473$	−0.157873	−0.00725899
$474$	0	0
$475$	−46.1300	−2.11659
$476$	0	0
$477$	−9.05447	−0.414576
$478$	0	0
$479$	28.1806	1.28760	0.643802	−	0.765192i	$-0.277356\pi$
$479$	28.1806	1.28760	0.643802	+	0.765192i	$0.277356\pi$
$480$	0	0
$481$	−5.04350	−0.229964
$482$	0	0
$483$	7.55069	0.343568
$484$	0	0
$485$	0.937699	0.0425787
$486$	0	0
$487$	12.1584	0.550949	0.275474	−	0.961308i	$-0.411165\pi$
$487$	12.1584	0.550949	0.275474	+	0.961308i	$0.411165\pi$
$488$	0	0
$489$	−16.5636	−0.749030
$490$	0	0
$491$	33.2716	1.50153	0.750764	−	0.660571i	$-0.229686\pi$
$491$	33.2716	1.50153	0.750764	+	0.660571i	$0.229686\pi$
$492$	0	0
$493$	15.7869	0.711005
$494$	0	0
$495$	−3.47314	−0.156106
$496$	0	0
$497$	51.1369	2.29380
$498$	0	0
$499$	25.2022	1.12821	0.564103	−	0.825704i	$-0.309222\pi$
$499$	25.2022	1.12821	0.564103	+	0.825704i	$0.309222\pi$
$500$	0	0
$501$	2.56608	0.114644
$502$	0	0
$503$	−1.27414	−0.0568113	−0.0284056	−	0.999596i	$-0.509043\pi$
$503$	−1.27414	−0.0568113	−0.0284056	+	0.999596i	$0.509043\pi$
$504$	0	0
$505$	−4.67451	−0.208013
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	2.16092	0.0957811	0.0478905	−	0.998853i	$-0.484750\pi$
$509$	2.16092	0.0957811	0.0478905	+	0.998853i	$0.484750\pi$
$510$	0	0
$511$	−46.3981	−2.05253
$512$	0	0
$513$	−6.53152	−0.288374
$514$	0	0
$515$	−23.8104	−1.04921
$516$	0	0
$517$	12.5681	0.552744
$518$	0	0
$519$	6.52583	0.286452
$520$	0	0
$521$	−36.1007	−1.58160	−0.790801	−	0.612074i	$-0.790336\pi$
$521$	−36.1007	−1.58160	−0.790801	+	0.612074i	$0.790336\pi$
$522$	0	0
$523$	−2.28286	−0.0998226	−0.0499113	−	0.998754i	$-0.515894\pi$
$523$	−2.28286	−0.0998226	−0.0499113	+	0.998754i	$0.515894\pi$
$524$	0	0
$525$	25.9618	1.13306
$526$	0	0
$527$	−13.0775	−0.569666
$528$	0	0
$529$	−18.7807	−0.816551
$530$	0	0
$531$	−0.0906653	−0.00393454
$532$	0	0
$533$	9.96544	0.431651
$534$	0	0
$535$	−16.8344	−0.727815
$536$	0	0
$537$	18.9183	0.816384
$538$	0	0
$539$	6.51235	0.280507
$540$	0	0
$541$	9.86029	0.423927	0.211964	−	0.977278i	$-0.432014\pi$
$541$	9.86029	0.423927	0.211964	+	0.977278i	$0.432014\pi$
$542$	0	0
$543$	21.2809	0.913252
$544$	0	0
$545$	−27.2170	−1.16585
$546$	0	0
$547$	7.92623	0.338901	0.169451	−	0.985539i	$-0.445801\pi$
$547$	7.92623	0.338901	0.169451	+	0.985539i	$0.445801\pi$
$548$	0	0
$549$	10.0719	0.429857
$550$	0	0
$551$	−37.3285	−1.59025
$552$	0	0
$553$	−24.8891	−1.05839
$554$	0	0
$555$	17.5168	0.743545
$556$	0	0
$557$	−36.9855	−1.56712	−0.783562	−	0.621313i	$-0.786600\pi$
$557$	−36.9855	−1.56712	−0.783562	+	0.621313i	$0.786600\pi$
$558$	0	0
$559$	−0.157873	−0.00667730
$560$	0	0
$561$	2.76229	0.116624
$562$	0	0
$563$	−4.05599	−0.170940	−0.0854698	−	0.996341i	$-0.527239\pi$
$563$	−4.05599	−0.170940	−0.0854698	+	0.996341i	$0.527239\pi$
$564$	0	0
$565$	−54.3118	−2.28492
$566$	0	0
$567$	3.67591	0.154374
$568$	0	0
$569$	−13.2846	−0.556920	−0.278460	−	0.960448i	$-0.589824\pi$
$569$	−13.2846	−0.556920	−0.278460	+	0.960448i	$0.589824\pi$
$570$	0	0
$571$	−38.0764	−1.59345	−0.796724	−	0.604343i	$-0.793436\pi$
$571$	−38.0764	−1.59345	−0.796724	+	0.604343i	$0.793436\pi$
$572$	0	0
$573$	0.0735212	0.00307139
$574$	0	0
$575$	14.5074	0.605001
$576$	0	0
$577$	−25.9104	−1.07867	−0.539333	−	0.842093i	$-0.681323\pi$
$577$	−25.9104	−1.07867	−0.539333	+	0.842093i	$0.681323\pi$
$578$	0	0
$579$	15.4778	0.643235
$580$	0	0
$581$	−38.2527	−1.58699
$582$	0	0
$583$	−9.05447	−0.374998
$584$	0	0
$585$	−3.47314	−0.143596
$586$	0	0
$587$	18.4856	0.762981	0.381491	−	0.924373i	$-0.375411\pi$
$587$	18.4856	0.762981	0.381491	+	0.924373i	$0.375411\pi$
$588$	0	0
$589$	30.9222	1.27413
$590$	0	0
$591$	10.5229	0.432856
$592$	0	0
$593$	−41.8715	−1.71945	−0.859727	−	0.510753i	$-0.829367\pi$
$593$	−41.8715	−1.71945	−0.859727	+	0.510753i	$0.829367\pi$
$594$	0	0
$595$	−35.2661	−1.44577
$596$	0	0
$597$	−11.1267	−0.455388
$598$	0	0
$599$	−41.8712	−1.71081	−0.855406	−	0.517958i	$-0.826692\pi$
$599$	−41.8712	−1.71081	−0.855406	+	0.517958i	$0.826692\pi$
$600$	0	0
$601$	−15.7284	−0.641573	−0.320787	−	0.947152i	$-0.603947\pi$
$601$	−15.7284	−0.641573	−0.320787	+	0.947152i	$0.603947\pi$
$602$	0	0
$603$	0.535436	0.0218047
$604$	0	0
$605$	−3.47314	−0.141203
$606$	0	0
$607$	−0.562162	−0.0228174	−0.0114087	−	0.999935i	$-0.503632\pi$
$607$	−0.562162	−0.0228174	−0.0114087	+	0.999935i	$0.503632\pi$
$608$	0	0
$609$	21.0083	0.851300
$610$	0	0
$611$	12.5681	0.508450
$612$	0	0
$613$	19.6031	0.791763	0.395882	−	0.918302i	$-0.370439\pi$
$613$	19.6031	0.791763	0.395882	+	0.918302i	$0.370439\pi$
$614$	0	0
$615$	−34.6113	−1.39566
$616$	0	0
$617$	−1.04930	−0.0422434	−0.0211217	−	0.999777i	$-0.506724\pi$
$617$	−1.04930	−0.0422434	−0.0211217	+	0.999777i	$0.506724\pi$
$618$	0	0
$619$	−43.7446	−1.75824	−0.879122	−	0.476597i	$-0.841870\pi$
$619$	−43.7446	−1.75824	−0.879122	+	0.476597i	$0.841870\pi$
$620$	0	0
$621$	2.05410	0.0824282
$622$	0	0
$623$	−26.6705	−1.06853
$624$	0	0
$625$	−10.4320	−0.417281
$626$	0	0
$627$	−6.53152	−0.260844
$628$	0	0
$629$	−13.9316	−0.555490
$630$	0	0
$631$	−11.6353	−0.463194	−0.231597	−	0.972812i	$-0.574395\pi$
$631$	−11.6353	−0.463194	−0.231597	+	0.972812i	$0.574395\pi$
$632$	0	0
$633$	−26.3391	−1.04688
$634$	0	0
$635$	1.95246	0.0774812
$636$	0	0
$637$	6.51235	0.258029
$638$	0	0
$639$	13.9113	0.550324
$640$	0	0
$641$	−45.8876	−1.81245	−0.906226	−	0.422794i	$-0.861049\pi$
$641$	−45.8876	−1.81245	−0.906226	+	0.422794i	$0.861049\pi$
$642$	0	0
$643$	−7.24732	−0.285806	−0.142903	−	0.989737i	$-0.545644\pi$
$643$	−7.24732	−0.285806	−0.142903	+	0.989737i	$0.545644\pi$
$644$	0	0
$645$	0.548313	0.0215898
$646$	0	0
$647$	−23.3987	−0.919900	−0.459950	−	0.887945i	$-0.652133\pi$
$647$	−23.3987	−0.919900	−0.459950	+	0.887945i	$0.652133\pi$
$648$	0	0
$649$	−0.0906653	−0.00355892
$650$	0	0
$651$	−17.4029	−0.682073
$652$	0	0
$653$	6.82950	0.267259	0.133630	−	0.991031i	$-0.457337\pi$
$653$	6.82950	0.267259	0.133630	+	0.991031i	$0.457337\pi$
$654$	0	0
$655$	−23.4347	−0.915668
$656$	0	0
$657$	−12.6222	−0.492438
$658$	0	0
$659$	−28.3348	−1.10377	−0.551883	−	0.833921i	$-0.686091\pi$
$659$	−28.3348	−1.10377	−0.551883	+	0.833921i	$0.686091\pi$
$660$	0	0
$661$	9.10922	0.354307	0.177154	−	0.984183i	$-0.443311\pi$
$661$	9.10922	0.354307	0.177154	+	0.984183i	$0.443311\pi$
$662$	0	0
$663$	2.76229	0.107279
$664$	0	0
$665$	83.3876	3.23363
$666$	0	0
$667$	11.7394	0.454553
$668$	0	0
$669$	19.4380	0.751515
$670$	0	0
$671$	10.0719	0.388820
$672$	0	0
$673$	−17.2284	−0.664105	−0.332053	−	0.943261i	$-0.607741\pi$
$673$	−17.2284	−0.664105	−0.332053	+	0.943261i	$0.607741\pi$
$674$	0	0
$675$	7.06267	0.271842
$676$	0	0
$677$	−39.0073	−1.49917	−0.749586	−	0.661907i	$-0.769747\pi$
$677$	−39.0073	−1.49917	−0.749586	+	0.661907i	$0.769747\pi$
$678$	0	0
$679$	−0.992447	−0.0380866
$680$	0	0
$681$	25.3994	0.973306
$682$	0	0
$683$	3.03126	0.115988	0.0579940	−	0.998317i	$-0.481530\pi$
$683$	3.03126	0.115988	0.0579940	+	0.998317i	$0.481530\pi$
$684$	0	0
$685$	57.1828	2.18484
$686$	0	0
$687$	15.9460	0.608379
$688$	0	0
$689$	−9.05447	−0.344948
$690$	0	0
$691$	50.0554	1.90420	0.952099	−	0.305789i	$-0.0989202\pi$
$691$	50.0554	1.90420	0.952099	+	0.305789i	$0.0989202\pi$
$692$	0	0
$693$	3.67591	0.139636
$694$	0	0
$695$	−36.0838	−1.36874
$696$	0	0
$697$	27.5275	1.04268
$698$	0	0
$699$	−13.9924	−0.529242
$700$	0	0
$701$	22.6512	0.855523	0.427761	−	0.903892i	$-0.359302\pi$
$701$	22.6512	0.855523	0.427761	+	0.903892i	$0.359302\pi$
$702$	0	0
$703$	32.9417	1.24242
$704$	0	0
$705$	−43.6507	−1.64398
$706$	0	0
$707$	4.94743	0.186067
$708$	0	0
$709$	−13.6217	−0.511574	−0.255787	−	0.966733i	$-0.582335\pi$
$709$	−13.6217	−0.511574	−0.255787	+	0.966733i	$0.582335\pi$
$710$	0	0
$711$	−6.77087	−0.253927
$712$	0	0
$713$	−9.72472	−0.364194
$714$	0	0
$715$	−3.47314	−0.129888
$716$	0	0
$717$	−12.8121	−0.478476
$718$	0	0
$719$	10.2561	0.382488	0.191244	−	0.981543i	$-0.438748\pi$
$719$	10.2561	0.382488	0.191244	+	0.981543i	$0.438748\pi$
$720$	0	0
$721$	25.2006	0.938520
$722$	0	0
$723$	−5.64919	−0.210096
$724$	0	0
$725$	40.3641	1.49908
$726$	0	0
$727$	30.5316	1.13236	0.566178	−	0.824283i	$-0.308422\pi$
$727$	30.5316	1.13236	0.566178	+	0.824283i	$0.308422\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−0.436090	−0.0161294
$732$	0	0
$733$	12.2705	0.453222	0.226611	−	0.973985i	$-0.427235\pi$
$733$	12.2705	0.453222	0.226611	+	0.973985i	$0.427235\pi$
$734$	0	0
$735$	−22.6183	−0.834287
$736$	0	0
$737$	0.535436	0.0197231
$738$	0	0
$739$	−27.8695	−1.02520	−0.512598	−	0.858629i	$-0.671317\pi$
$739$	−27.8695	−1.02520	−0.512598	+	0.858629i	$0.671317\pi$
$740$	0	0
$741$	−6.53152	−0.239941
$742$	0	0
$743$	−15.8705	−0.582232	−0.291116	−	0.956688i	$-0.594027\pi$
$743$	−15.8705	−0.582232	−0.291116	+	0.956688i	$0.594027\pi$
$744$	0	0
$745$	13.0521	0.478191
$746$	0	0
$747$	−10.4063	−0.380747
$748$	0	0
$749$	17.8173	0.651029
$750$	0	0
$751$	−44.3597	−1.61871	−0.809355	−	0.587320i	$-0.800183\pi$
$751$	−44.3597	−1.61871	−0.809355	+	0.587320i	$0.800183\pi$
$752$	0	0
$753$	26.7690	0.975515
$754$	0	0
$755$	−31.6397	−1.15149
$756$	0	0
$757$	12.6678	0.460420	0.230210	−	0.973141i	$-0.426059\pi$
$757$	12.6678	0.460420	0.230210	+	0.973141i	$0.426059\pi$
$758$	0	0
$759$	2.05410	0.0745591
$760$	0	0
$761$	−42.5016	−1.54068	−0.770341	−	0.637632i	$-0.779914\pi$
$761$	−42.5016	−1.54068	−0.770341	+	0.637632i	$0.779914\pi$
$762$	0	0
$763$	28.8061	1.04285
$764$	0	0
$765$	−9.59382	−0.346865
$766$	0	0
$767$	−0.0906653	−0.00327373
$768$	0	0
$769$	−2.60442	−0.0939178	−0.0469589	−	0.998897i	$-0.514953\pi$
$769$	−2.60442	−0.0939178	−0.0469589	+	0.998897i	$0.514953\pi$
$770$	0	0
$771$	25.3436	0.912728
$772$	0	0
$773$	−52.5354	−1.88957	−0.944783	−	0.327697i	$-0.893728\pi$
$773$	−52.5354	−1.88957	−0.944783	+	0.327697i	$0.893728\pi$
$774$	0	0
$775$	−33.4368	−1.20109
$776$	0	0
$777$	−18.5395	−0.665100
$778$	0	0
$779$	−65.0895	−2.33207
$780$	0	0
$781$	13.9113	0.497787
$782$	0	0
$783$	5.71513	0.204242
$784$	0	0
$785$	49.4288	1.76419
$786$	0	0
$787$	21.8081	0.777376	0.388688	−	0.921369i	$-0.372928\pi$
$787$	21.8081	0.777376	0.388688	+	0.921369i	$0.372928\pi$
$788$	0	0
$789$	7.11046	0.253139
$790$	0	0
$791$	57.4828	2.04385
$792$	0	0
$793$	10.0719	0.357662
$794$	0	0
$795$	31.4474	1.11532
$796$	0	0
$797$	6.70416	0.237474	0.118737	−	0.992926i	$-0.462116\pi$
$797$	6.70416	0.237474	0.118737	+	0.992926i	$0.462116\pi$
$798$	0	0
$799$	34.7168	1.22819
$800$	0	0
$801$	−7.25547	−0.256359
$802$	0	0
$803$	−12.6222	−0.445427
$804$	0	0
$805$	−26.2246	−0.924295
$806$	0	0
$807$	−3.90037	−0.137300
$808$	0	0
$809$	24.5335	0.862554	0.431277	−	0.902220i	$-0.358063\pi$
$809$	24.5335	0.862554	0.431277	+	0.902220i	$0.358063\pi$
$810$	0	0
$811$	−7.17969	−0.252113	−0.126057	−	0.992023i	$-0.540232\pi$
$811$	−7.17969	−0.252113	−0.126057	+	0.992023i	$0.540232\pi$
$812$	0	0
$813$	23.1368	0.811443
$814$	0	0
$815$	57.5275	2.01510
$816$	0	0
$817$	1.03115	0.0360753
$818$	0	0
$819$	3.67591	0.128447
$820$	0	0
$821$	30.9938	1.08169	0.540846	−	0.841122i	$-0.318104\pi$
$821$	30.9938	1.08169	0.540846	+	0.841122i	$0.318104\pi$
$822$	0	0
$823$	6.28586	0.219111	0.109556	−	0.993981i	$-0.465057\pi$
$823$	6.28586	0.219111	0.109556	+	0.993981i	$0.465057\pi$
$824$	0	0
$825$	7.06267	0.245891
$826$	0	0
$827$	−27.4508	−0.954559	−0.477280	−	0.878752i	$-0.658377\pi$
$827$	−27.4508	−0.954559	−0.477280	+	0.878752i	$0.658377\pi$
$828$	0	0
$829$	11.7463	0.407965	0.203983	−	0.978975i	$-0.434611\pi$
$829$	11.7463	0.407965	0.203983	+	0.978975i	$0.434611\pi$
$830$	0	0
$831$	14.2773	0.495274
$832$	0	0
$833$	17.9890	0.623283
$834$	0	0
$835$	−8.91233	−0.308424
$836$	0	0
$837$	−4.73430	−0.163641
$838$	0	0
$839$	−7.69255	−0.265576	−0.132788	−	0.991144i	$-0.542393\pi$
$839$	−7.69255	−0.265576	−0.132788	+	0.991144i	$0.542393\pi$
$840$	0	0
$841$	3.66268	0.126299
$842$	0	0
$843$	−30.6045	−1.05408
$844$	0	0
$845$	−3.47314	−0.119479
$846$	0	0
$847$	3.67591	0.126306
$848$	0	0
$849$	−5.13681	−0.176295
$850$	0	0
$851$	−10.3598	−0.355131
$852$	0	0
$853$	38.1816	1.30731	0.653656	−	0.756792i	$-0.273234\pi$
$853$	38.1816	1.30731	0.653656	+	0.756792i	$0.273234\pi$
$854$	0	0
$855$	22.6849	0.775806
$856$	0	0
$857$	39.0580	1.33420	0.667098	−	0.744970i	$-0.267536\pi$
$857$	39.0580	1.33420	0.667098	+	0.744970i	$0.267536\pi$
$858$	0	0
$859$	−35.3952	−1.20767	−0.603834	−	0.797110i	$-0.706361\pi$
$859$	−35.3952	−1.20767	−0.603834	+	0.797110i	$0.706361\pi$
$860$	0	0
$861$	36.6321	1.24842
$862$	0	0
$863$	−26.4550	−0.900537	−0.450269	−	0.892893i	$-0.648672\pi$
$863$	−26.4550	−0.900537	−0.450269	+	0.892893i	$0.648672\pi$
$864$	0	0
$865$	−22.6651	−0.770636
$866$	0	0
$867$	−9.36973	−0.318213
$868$	0	0
$869$	−6.77087	−0.229686
$870$	0	0
$871$	0.535436	0.0181426
$872$	0	0
$873$	−0.269986	−0.00913765
$874$	0	0
$875$	−26.3340	−0.890252
$876$	0	0
$877$	−43.7998	−1.47901	−0.739506	−	0.673150i	$-0.764941\pi$
$877$	−43.7998	−1.47901	−0.739506	+	0.673150i	$0.764941\pi$
$878$	0	0
$879$	−13.6584	−0.460686
$880$	0	0
$881$	1.26646	0.0426680	0.0213340	−	0.999772i	$-0.493209\pi$
$881$	1.26646	0.0426680	0.0213340	+	0.999772i	$0.493209\pi$
$882$	0	0
$883$	27.5866	0.928363	0.464182	−	0.885740i	$-0.346348\pi$
$883$	27.5866	0.928363	0.464182	+	0.885740i	$0.346348\pi$
$884$	0	0
$885$	0.314893	0.0105850
$886$	0	0
$887$	−51.9630	−1.74475	−0.872374	−	0.488840i	$-0.837420\pi$
$887$	−51.9630	−1.74475	−0.872374	+	0.488840i	$0.837420\pi$
$888$	0	0
$889$	−2.06646	−0.0693068
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−82.0887	−2.74699
$894$	0	0
$895$	−65.7057	−2.19630
$896$	0	0
$897$	2.05410	0.0685844
$898$	0	0
$899$	−27.0571	−0.902406
$900$	0	0
$901$	−25.0111	−0.833241
$902$	0	0
$903$	−0.580326	−0.0193120
$904$	0	0
$905$	−73.9116	−2.45690
$906$	0	0
$907$	−45.9092	−1.52439	−0.762195	−	0.647348i	$-0.775878\pi$
$907$	−45.9092	−1.52439	−0.762195	+	0.647348i	$0.775878\pi$
$908$	0	0
$909$	1.34590	0.0446408
$910$	0	0
$911$	−0.500631	−0.0165866	−0.00829332	−	0.999966i	$-0.502640\pi$
$911$	−0.500631	−0.0165866	−0.00829332	+	0.999966i	$0.502640\pi$
$912$	0	0
$913$	−10.4063	−0.344398
$914$	0	0
$915$	−34.9809	−1.15643
$916$	0	0
$917$	24.8029	0.819064
$918$	0	0
$919$	25.8572	0.852951	0.426476	−	0.904499i	$-0.359755\pi$
$919$	25.8572	0.852951	0.426476	+	0.904499i	$0.359755\pi$
$920$	0	0
$921$	−9.02002	−0.297220
$922$	0	0
$923$	13.9113	0.457897
$924$	0	0
$925$	−35.6206	−1.17120
$926$	0	0
$927$	6.85561	0.225168
$928$	0	0
$929$	−16.0106	−0.525291	−0.262645	−	0.964892i	$-0.584595\pi$
$929$	−16.0106	−0.525291	−0.262645	+	0.964892i	$0.584595\pi$
$930$	0	0
$931$	−42.5355	−1.39405
$932$	0	0
$933$	30.5299	0.999503
$934$	0	0
$935$	−9.59382	−0.313752
$936$	0	0
$937$	−9.62509	−0.314438	−0.157219	−	0.987564i	$-0.550253\pi$
$937$	−9.62509	−0.314438	−0.157219	+	0.987564i	$0.550253\pi$
$938$	0	0
$939$	−6.37842	−0.208152
$940$	0	0
$941$	23.6425	0.770723	0.385361	−	0.922766i	$-0.374077\pi$
$941$	23.6425	0.770723	0.385361	+	0.922766i	$0.374077\pi$
$942$	0	0
$943$	20.4700	0.666595
$944$	0	0
$945$	−12.7670	−0.415309
$946$	0	0
$947$	42.0705	1.36711	0.683553	−	0.729900i	$-0.260434\pi$
$947$	42.0705	1.36711	0.683553	+	0.729900i	$0.260434\pi$
$948$	0	0
$949$	−12.6222	−0.409733
$950$	0	0
$951$	8.71917	0.282738
$952$	0	0
$953$	32.1039	1.03995	0.519974	−	0.854182i	$-0.325942\pi$
$953$	32.1039	1.03995	0.519974	+	0.854182i	$0.325942\pi$
$954$	0	0
$955$	−0.255349	−0.00826290
$956$	0	0
$957$	5.71513	0.184744
$958$	0	0
$959$	−60.5214	−1.95434
$960$	0	0
$961$	−8.58641	−0.276981
$962$	0	0
$963$	4.84703	0.156193
$964$	0	0
$965$	−53.7565	−1.73048
$966$	0	0
$967$	16.8199	0.540892	0.270446	−	0.962735i	$-0.412829\pi$
$967$	16.8199	0.540892	0.270446	+	0.962735i	$0.412829\pi$
$968$	0	0
$969$	−18.0420	−0.579592
$970$	0	0
$971$	19.2716	0.618456	0.309228	−	0.950988i	$-0.399929\pi$
$971$	19.2716	0.618456	0.309228	+	0.950988i	$0.399929\pi$
$972$	0	0
$973$	38.1906	1.22433
$974$	0	0
$975$	7.06267	0.226186
$976$	0	0
$977$	−25.9084	−0.828884	−0.414442	−	0.910076i	$-0.636023\pi$
$977$	−25.9084	−0.828884	−0.414442	+	0.910076i	$0.636023\pi$
$978$	0	0
$979$	−7.25547	−0.231886
$980$	0	0
$981$	7.83643	0.250198
$982$	0	0
$983$	−1.20529	−0.0384427	−0.0192213	−	0.999815i	$-0.506119\pi$
$983$	−1.20529	−0.0384427	−0.0192213	+	0.999815i	$0.506119\pi$
$984$	0	0
$985$	−36.5476	−1.16450
$986$	0	0
$987$	46.1992	1.47054
$988$	0	0
$989$	−0.324286	−0.0103117
$990$	0	0
$991$	1.20970	0.0384272	0.0192136	−	0.999815i	$-0.493884\pi$
$991$	1.20970	0.0384272	0.0192136	+	0.999815i	$0.493884\pi$
$992$	0	0
$993$	10.7678	0.341706
$994$	0	0
$995$	38.6447	1.22512
$996$	0	0
$997$	5.22232	0.165393	0.0826963	−	0.996575i	$-0.473647\pi$
$997$	5.22232	0.165393	0.0826963	+	0.996575i	$0.473647\pi$
$998$	0	0
$999$	−5.04350	−0.159569

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cf.1.1		5
4.3	odd	2		3432.2.a.w.1.1	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.w.1.1	✓	5	4.3	odd	2
6864.2.a.cf.1.1		5	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.47314 q^{5} +3.67591 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.47314 q^{5} +3.67591 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -3.47314 q^{15} +2.76229 q^{17} -6.53152 q^{19} +3.67591 q^{21} +2.05410 q^{23} +7.06267 q^{25} +1.00000 q^{27} +5.71513 q^{29} -4.73430 q^{31} +1.00000 q^{33} -12.7670 q^{35} -5.04350 q^{37} +1.00000 q^{39} +9.96544 q^{41} -0.157873 q^{43} -3.47314 q^{45} +12.5681 q^{47} +6.51235 q^{49} +2.76229 q^{51} -9.05447 q^{53} -3.47314 q^{55} -6.53152 q^{57} -0.0906653 q^{59} +10.0719 q^{61} +3.67591 q^{63} -3.47314 q^{65} +0.535436 q^{67} +2.05410 q^{69} +13.9113 q^{71} -12.6222 q^{73} +7.06267 q^{75} +3.67591 q^{77} -6.77087 q^{79} +1.00000 q^{81} -10.4063 q^{83} -9.59382 q^{85} +5.71513 q^{87} -7.25547 q^{89} +3.67591 q^{91} -4.73430 q^{93} +22.6849 q^{95} -0.269986 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + q^{5} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 5 q^{13} + q^{15} + 4 q^{19} + 5 q^{21} + 5 q^{23} + 4 q^{25} + 5 q^{27} + 11 q^{29} + 8 q^{31} + 5 q^{33} + 5 q^{35} - 8 q^{37} + 5 q^{39} - q^{41} - q^{43} + q^{45} + 18 q^{47} + 10 q^{49} + 2 q^{53} + q^{55} + 4 q^{57} + 13 q^{59} + 9 q^{61} + 5 q^{63} + q^{65} - 5 q^{67} + 5 q^{69} + 24 q^{71} - 13 q^{73} + 4 q^{75} + 5 q^{77} + 6 q^{79} + 5 q^{81} + 22 q^{83} - 22 q^{85} + 11 q^{87} - 14 q^{89} + 5 q^{91} + 8 q^{93} + 32 q^{95} - 20 q^{97} + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + q^5 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 5 * q^13 + q^15 + 4 * q^19 + 5 * q^21 + 5 * q^23 + 4 * q^25 + 5 * q^27 + 11 * q^29 + 8 * q^31 + 5 * q^33 + 5 * q^35 - 8 * q^37 + 5 * q^39 - q^41 - q^43 + q^45 + 18 * q^47 + 10 * q^49 + 2 * q^53 + q^55 + 4 * q^57 + 13 * q^59 + 9 * q^61 + 5 * q^63 + q^65 - 5 * q^67 + 5 * q^69 + 24 * q^71 - 13 * q^73 + 4 * q^75 + 5 * q^77 + 6 * q^79 + 5 * q^81 + 22 * q^83 - 22 * q^85 + 11 * q^87 - 14 * q^89 + 5 * q^91 + 8 * q^93 + 32 * q^95 - 20 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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