LMFDB - Embedded newform 3380.2.a.r.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3380,2,Mod(1,3380)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3380, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3380.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3380 = 2^{2} \cdot 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3380.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:	$26.9894358832$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 19x^{7} + 16x^{6} + 106x^{5} - 87x^{4} - 153x^{3} + 149x^{2} - 26x + 1 $ x^9 - x^8 - 19x^7 + 16x^6 + 106x^5 - 87x^4 - 153x^3 + 149x^2 - 26*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.81781$ of defining polynomial
Character	$\chi$	$=$	3380.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.81781 q^{3} -1.00000 q^{5} +0.974186 q^{7} +4.94004 q^{9} +O(q^{10})$	$q+2.81781 q^{3} -1.00000 q^{5} +0.974186 q^{7} +4.94004 q^{9} +3.42555 q^{11} -2.81781 q^{15} +0.283829 q^{17} -6.03105 q^{19} +2.74507 q^{21} +8.91511 q^{23} +1.00000 q^{25} +5.46666 q^{27} +0.461096 q^{29} -7.96959 q^{31} +9.65254 q^{33} -0.974186 q^{35} +2.95339 q^{37} +8.76466 q^{41} +8.81154 q^{43} -4.94004 q^{45} +10.3775 q^{47} -6.05096 q^{49} +0.799776 q^{51} +4.96945 q^{53} -3.42555 q^{55} -16.9943 q^{57} -7.88601 q^{59} +11.4869 q^{61} +4.81252 q^{63} +3.96340 q^{67} +25.1211 q^{69} +6.22524 q^{71} -9.94784 q^{73} +2.81781 q^{75} +3.33712 q^{77} +6.63735 q^{79} +0.583871 q^{81} -7.05049 q^{83} -0.283829 q^{85} +1.29928 q^{87} -8.66990 q^{89} -22.4568 q^{93} +6.03105 q^{95} +0.332881 q^{97} +16.9224 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	$ 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) $ 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * 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
$3$	2.81781	1.62686	0.813431	−	0.581661i	$-0.197597\pi$
$3$	2.81781	1.62686	0.813431	+	0.581661i	$0.197597\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.974186	0.368208	0.184104	−	0.982907i	$-0.441062\pi$
$7$	0.974186	0.368208	0.184104	+	0.982907i	$0.441062\pi$
$8$	0	0
$9$	4.94004	1.64668
$10$	0	0
$11$	3.42555	1.03284	0.516421	−	0.856335i	$-0.327264\pi$
$11$	3.42555	1.03284	0.516421	+	0.856335i	$0.327264\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.81781	−0.727555
$16$	0	0
$17$	0.283829	0.0688387	0.0344193	−	0.999407i	$-0.489042\pi$
$17$	0.283829	0.0688387	0.0344193	+	0.999407i	$0.489042\pi$
$18$	0	0
$19$	−6.03105	−1.38362	−0.691809	−	0.722081i	$-0.743186\pi$
$19$	−6.03105	−1.38362	−0.691809	+	0.722081i	$0.743186\pi$
$20$	0	0
$21$	2.74507	0.599023
$22$	0	0
$23$	8.91511	1.85893	0.929465	−	0.368911i	$-0.120269\pi$
$23$	8.91511	1.85893	0.929465	+	0.368911i	$0.120269\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.46666	1.05206
$28$	0	0
$29$	0.461096	0.0856233	0.0428117	−	0.999083i	$-0.486368\pi$
$29$	0.461096	0.0856233	0.0428117	+	0.999083i	$0.486368\pi$
$30$	0	0
$31$	−7.96959	−1.43138	−0.715690	−	0.698418i	$-0.753888\pi$
$31$	−7.96959	−1.43138	−0.715690	+	0.698418i	$0.753888\pi$
$32$	0	0
$33$	9.65254	1.68029
$34$	0	0
$35$	−0.974186	−0.164668
$36$	0	0
$37$	2.95339	0.485535	0.242767	−	0.970085i	$-0.421945\pi$
$37$	2.95339	0.485535	0.242767	+	0.970085i	$0.421945\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.76466	1.36881	0.684405	−	0.729102i	$-0.260062\pi$
$41$	8.76466	1.36881	0.684405	+	0.729102i	$0.260062\pi$
$42$	0	0
$43$	8.81154	1.34375	0.671873	−	0.740666i	$-0.265490\pi$
$43$	8.81154	1.34375	0.671873	+	0.740666i	$0.265490\pi$
$44$	0	0
$45$	−4.94004	−0.736418
$46$	0	0
$47$	10.3775	1.51372	0.756859	−	0.653578i	$-0.226733\pi$
$47$	10.3775	1.51372	0.756859	+	0.653578i	$0.226733\pi$
$48$	0	0
$49$	−6.05096	−0.864423
$50$	0	0
$51$	0.799776	0.111991
$52$	0	0
$53$	4.96945	0.682606	0.341303	−	0.939953i	$-0.389132\pi$
$53$	4.96945	0.682606	0.341303	+	0.939953i	$0.389132\pi$
$54$	0	0
$55$	−3.42555	−0.461901
$56$	0	0
$57$	−16.9943	−2.25096
$58$	0	0
$59$	−7.88601	−1.02667	−0.513336	−	0.858188i	$-0.671590\pi$
$59$	−7.88601	−1.02667	−0.513336	+	0.858188i	$0.671590\pi$
$60$	0	0
$61$	11.4869	1.47075	0.735375	−	0.677661i	$-0.237006\pi$
$61$	11.4869	1.47075	0.735375	+	0.677661i	$0.237006\pi$
$62$	0	0
$63$	4.81252	0.606320
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.96340	0.484206	0.242103	−	0.970251i	$-0.422163\pi$
$67$	3.96340	0.484206	0.242103	+	0.970251i	$0.422163\pi$
$68$	0	0
$69$	25.1211	3.02422
$70$	0	0
$71$	6.22524	0.738801	0.369400	−	0.929270i	$-0.379563\pi$
$71$	6.22524	0.738801	0.369400	+	0.929270i	$0.379563\pi$
$72$	0	0
$73$	−9.94784	−1.16431	−0.582153	−	0.813079i	$-0.697790\pi$
$73$	−9.94784	−1.16431	−0.582153	+	0.813079i	$0.697790\pi$
$74$	0	0
$75$	2.81781	0.325372
$76$	0	0
$77$	3.33712	0.380301
$78$	0	0
$79$	6.63735	0.746760	0.373380	−	0.927679i	$-0.378199\pi$
$79$	6.63735	0.746760	0.373380	+	0.927679i	$0.378199\pi$
$80$	0	0
$81$	0.583871	0.0648745
$82$	0	0
$83$	−7.05049	−0.773891	−0.386946	−	0.922103i	$-0.626470\pi$
$83$	−7.05049	−0.773891	−0.386946	+	0.922103i	$0.626470\pi$
$84$	0	0
$85$	−0.283829	−0.0307856
$86$	0	0
$87$	1.29928	0.139297
$88$	0	0
$89$	−8.66990	−0.919007	−0.459504	−	0.888176i	$-0.651973\pi$
$89$	−8.66990	−0.919007	−0.459504	+	0.888176i	$0.651973\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−22.4568	−2.32866
$94$	0	0
$95$	6.03105	0.618773
$96$	0	0
$97$	0.332881	0.0337990	0.0168995	−	0.999857i	$-0.494620\pi$
$97$	0.332881	0.0337990	0.0168995	+	0.999857i	$0.494620\pi$
$98$	0	0
$99$	16.9224	1.70076
$100$	0	0
$101$	−14.0197	−1.39501	−0.697506	−	0.716579i	$-0.745707\pi$
$101$	−14.0197	−1.39501	−0.697506	+	0.716579i	$0.745707\pi$
$102$	0	0
$103$	−18.6039	−1.83310	−0.916549	−	0.399923i	$-0.869037\pi$
$103$	−18.6039	−1.83310	−0.916549	+	0.399923i	$0.869037\pi$
$104$	0	0
$105$	−2.74507	−0.267891
$106$	0	0
$107$	−10.8007	−1.04414	−0.522071	−	0.852902i	$-0.674840\pi$
$107$	−10.8007	−1.04414	−0.522071	+	0.852902i	$0.674840\pi$
$108$	0	0
$109$	−2.87817	−0.275679	−0.137839	−	0.990455i	$-0.544016\pi$
$109$	−2.87817	−0.275679	−0.137839	+	0.990455i	$0.544016\pi$
$110$	0	0
$111$	8.32209	0.789898
$112$	0	0
$113$	9.77628	0.919675	0.459838	−	0.888003i	$-0.347908\pi$
$113$	9.77628	0.919675	0.459838	+	0.888003i	$0.347908\pi$
$114$	0	0
$115$	−8.91511	−0.831338
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.276502	0.0253469
$120$	0	0
$121$	0.734394	0.0667631
$122$	0	0
$123$	24.6971	2.22686
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	14.3824	1.27623	0.638114	−	0.769942i	$-0.279715\pi$
$127$	14.3824	1.27623	0.638114	+	0.769942i	$0.279715\pi$
$128$	0	0
$129$	24.8292	2.18609
$130$	0	0
$131$	16.9106	1.47749	0.738745	−	0.673985i	$-0.235419\pi$
$131$	16.9106	1.47749	0.738745	+	0.673985i	$0.235419\pi$
$132$	0	0
$133$	−5.87537	−0.509459
$134$	0	0
$135$	−5.46666	−0.470495
$136$	0	0
$137$	−7.71889	−0.659469	−0.329735	−	0.944074i	$-0.606959\pi$
$137$	−7.71889	−0.659469	−0.329735	+	0.944074i	$0.606959\pi$
$138$	0	0
$139$	−18.7958	−1.59424	−0.797120	−	0.603821i	$-0.793644\pi$
$139$	−18.7958	−1.59424	−0.797120	+	0.603821i	$0.793644\pi$
$140$	0	0
$141$	29.2419	2.46261
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−0.461096	−0.0382919
$146$	0	0
$147$	−17.0504	−1.40630
$148$	0	0
$149$	−3.52113	−0.288462	−0.144231	−	0.989544i	$-0.546071\pi$
$149$	−3.52113	−0.288462	−0.144231	+	0.989544i	$0.546071\pi$
$150$	0	0
$151$	11.0498	0.899217	0.449608	−	0.893226i	$-0.351564\pi$
$151$	11.0498	0.899217	0.449608	+	0.893226i	$0.351564\pi$
$152$	0	0
$153$	1.40213	0.113355
$154$	0	0
$155$	7.96959	0.640133
$156$	0	0
$157$	4.47826	0.357404	0.178702	−	0.983903i	$-0.442810\pi$
$157$	4.47826	0.357404	0.178702	+	0.983903i	$0.442810\pi$
$158$	0	0
$159$	14.0029	1.11051
$160$	0	0
$161$	8.68498	0.684472
$162$	0	0
$163$	11.3490	0.888923	0.444462	−	0.895798i	$-0.353395\pi$
$163$	11.3490	0.888923	0.444462	+	0.895798i	$0.353395\pi$
$164$	0	0
$165$	−9.65254	−0.751449
$166$	0	0
$167$	8.82022	0.682529	0.341265	−	0.939967i	$-0.389145\pi$
$167$	8.82022	0.682529	0.341265	+	0.939967i	$0.389145\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−29.7936	−2.27838
$172$	0	0
$173$	−4.64718	−0.353319	−0.176659	−	0.984272i	$-0.556529\pi$
$173$	−4.64718	−0.353319	−0.176659	+	0.984272i	$0.556529\pi$
$174$	0	0
$175$	0.974186	0.0736416
$176$	0	0
$177$	−22.2213	−1.67025
$178$	0	0
$179$	−25.3186	−1.89240	−0.946200	−	0.323581i	$-0.895113\pi$
$179$	−25.3186	−1.89240	−0.946200	+	0.323581i	$0.895113\pi$
$180$	0	0
$181$	4.32715	0.321635	0.160817	−	0.986984i	$-0.448587\pi$
$181$	4.32715	0.321635	0.160817	+	0.986984i	$0.448587\pi$
$182$	0	0
$183$	32.3679	2.39271
$184$	0	0
$185$	−2.95339	−0.217138
$186$	0	0
$187$	0.972271	0.0710995
$188$	0	0
$189$	5.32554	0.387376
$190$	0	0
$191$	−2.62340	−0.189823	−0.0949114	−	0.995486i	$-0.530257\pi$
$191$	−2.62340	−0.189823	−0.0949114	+	0.995486i	$0.530257\pi$
$192$	0	0
$193$	21.6639	1.55940	0.779700	−	0.626153i	$-0.215371\pi$
$193$	21.6639	1.55940	0.779700	+	0.626153i	$0.215371\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.8601	−1.48622	−0.743110	−	0.669169i	$-0.766650\pi$
$197$	−20.8601	−1.48622	−0.743110	+	0.669169i	$0.766650\pi$
$198$	0	0
$199$	22.7819	1.61497	0.807485	−	0.589888i	$-0.200828\pi$
$199$	22.7819	1.61497	0.807485	+	0.589888i	$0.200828\pi$
$200$	0	0
$201$	11.1681	0.787736
$202$	0	0
$203$	0.449193	0.0315272
$204$	0	0
$205$	−8.76466	−0.612150
$206$	0	0
$207$	44.0410	3.06106
$208$	0	0
$209$	−20.6597	−1.42906
$210$	0	0
$211$	15.9891	1.10073	0.550366	−	0.834923i	$-0.314488\pi$
$211$	15.9891	1.10073	0.550366	+	0.834923i	$0.314488\pi$
$212$	0	0
$213$	17.5415	1.20193
$214$	0	0
$215$	−8.81154	−0.600942
$216$	0	0
$217$	−7.76387	−0.527046
$218$	0	0
$219$	−28.0311	−1.89417
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.54613	0.505326	0.252663	−	0.967554i	$-0.418694\pi$
$223$	7.54613	0.505326	0.252663	+	0.967554i	$0.418694\pi$
$224$	0	0
$225$	4.94004	0.329336
$226$	0	0
$227$	28.4988	1.89153	0.945765	−	0.324853i	$-0.105315\pi$
$227$	28.4988	1.89153	0.945765	+	0.324853i	$0.105315\pi$
$228$	0	0
$229$	19.3902	1.28134	0.640671	−	0.767815i	$-0.278656\pi$
$229$	19.3902	1.28134	0.640671	+	0.767815i	$0.278656\pi$
$230$	0	0
$231$	9.40337	0.618696
$232$	0	0
$233$	−12.7622	−0.836080	−0.418040	−	0.908429i	$-0.637283\pi$
$233$	−12.7622	−0.836080	−0.418040	+	0.908429i	$0.637283\pi$
$234$	0	0
$235$	−10.3775	−0.676955
$236$	0	0
$237$	18.7028	1.21488
$238$	0	0
$239$	11.1424	0.720741	0.360371	−	0.932809i	$-0.382650\pi$
$239$	11.1424	0.720741	0.360371	+	0.932809i	$0.382650\pi$
$240$	0	0
$241$	5.48277	0.353176	0.176588	−	0.984285i	$-0.443494\pi$
$241$	5.48277	0.353176	0.176588	+	0.984285i	$0.443494\pi$
$242$	0	0
$243$	−14.7547	−0.946517
$244$	0	0
$245$	6.05096	0.386582
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−19.8669	−1.25901
$250$	0	0
$251$	−24.4904	−1.54582	−0.772910	−	0.634516i	$-0.781199\pi$
$251$	−24.4904	−1.54582	−0.772910	+	0.634516i	$0.781199\pi$
$252$	0	0
$253$	30.5392	1.91998
$254$	0	0
$255$	−0.799776	−0.0500839
$256$	0	0
$257$	−14.7828	−0.922128	−0.461064	−	0.887367i	$-0.652532\pi$
$257$	−14.7828	−0.922128	−0.461064	+	0.887367i	$0.652532\pi$
$258$	0	0
$259$	2.87715	0.178778
$260$	0	0
$261$	2.27783	0.140994
$262$	0	0
$263$	3.01732	0.186056	0.0930279	−	0.995664i	$-0.470345\pi$
$263$	3.01732	0.186056	0.0930279	+	0.995664i	$0.470345\pi$
$264$	0	0
$265$	−4.96945	−0.305271
$266$	0	0
$267$	−24.4301	−1.49510
$268$	0	0
$269$	−8.46709	−0.516248	−0.258124	−	0.966112i	$-0.583104\pi$
$269$	−8.46709	−0.516248	−0.258124	+	0.966112i	$0.583104\pi$
$270$	0	0
$271$	−25.7542	−1.56445	−0.782227	−	0.622994i	$-0.785916\pi$
$271$	−25.7542	−1.56445	−0.782227	+	0.622994i	$0.785916\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.42555	0.206568
$276$	0	0
$277$	14.6848	0.882322	0.441161	−	0.897428i	$-0.354567\pi$
$277$	14.6848	0.882322	0.441161	+	0.897428i	$0.354567\pi$
$278$	0	0
$279$	−39.3701	−2.35703
$280$	0	0
$281$	−25.1933	−1.50290	−0.751452	−	0.659788i	$-0.770646\pi$
$281$	−25.1933	−1.50290	−0.751452	+	0.659788i	$0.770646\pi$
$282$	0	0
$283$	−19.3058	−1.14761	−0.573807	−	0.818991i	$-0.694534\pi$
$283$	−19.3058	−1.14761	−0.573807	+	0.818991i	$0.694534\pi$
$284$	0	0
$285$	16.9943	1.00666
$286$	0	0
$287$	8.53841	0.504006
$288$	0	0
$289$	−16.9194	−0.995261
$290$	0	0
$291$	0.937995	0.0549862
$292$	0	0
$293$	−1.23340	−0.0720561	−0.0360281	−	0.999351i	$-0.511471\pi$
$293$	−1.23340	−0.0720561	−0.0360281	+	0.999351i	$0.511471\pi$
$294$	0	0
$295$	7.88601	0.459141
$296$	0	0
$297$	18.7263	1.08661
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.58408	0.494778
$302$	0	0
$303$	−39.5048	−2.26949
$304$	0	0
$305$	−11.4869	−0.657739
$306$	0	0
$307$	−7.36503	−0.420344	−0.210172	−	0.977664i	$-0.567402\pi$
$307$	−7.36503	−0.420344	−0.210172	+	0.977664i	$0.567402\pi$
$308$	0	0
$309$	−52.4222	−2.98220
$310$	0	0
$311$	−7.77672	−0.440977	−0.220489	−	0.975390i	$-0.570765\pi$
$311$	−7.77672	−0.440977	−0.220489	+	0.975390i	$0.570765\pi$
$312$	0	0
$313$	0.104459	0.00590438	0.00295219	−	0.999996i	$-0.499060\pi$
$313$	0.104459	0.00590438	0.00295219	+	0.999996i	$0.499060\pi$
$314$	0	0
$315$	−4.81252	−0.271155
$316$	0	0
$317$	−32.3972	−1.81961	−0.909803	−	0.415040i	$-0.863768\pi$
$317$	−32.3972	−1.81961	−0.909803	+	0.415040i	$0.863768\pi$
$318$	0	0
$319$	1.57951	0.0884354
$320$	0	0
$321$	−30.4342	−1.69867
$322$	0	0
$323$	−1.71179	−0.0952464
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−8.11013	−0.448491
$328$	0	0
$329$	10.1096	0.557363
$330$	0	0
$331$	19.1973	1.05518	0.527589	−	0.849500i	$-0.323096\pi$
$331$	19.1973	1.05518	0.527589	+	0.849500i	$0.323096\pi$
$332$	0	0
$333$	14.5899	0.799520
$334$	0	0
$335$	−3.96340	−0.216543
$336$	0	0
$337$	−21.6042	−1.17686	−0.588429	−	0.808549i	$-0.700253\pi$
$337$	−21.6042	−1.17686	−0.588429	+	0.808549i	$0.700253\pi$
$338$	0	0
$339$	27.5477	1.49618
$340$	0	0
$341$	−27.3002	−1.47839
$342$	0	0
$343$	−12.7141	−0.686495
$344$	0	0
$345$	−25.1211	−1.35247
$346$	0	0
$347$	−12.4830	−0.670120	−0.335060	−	0.942197i	$-0.608757\pi$
$347$	−12.4830	−0.670120	−0.335060	+	0.942197i	$0.608757\pi$
$348$	0	0
$349$	−26.1232	−1.39834	−0.699171	−	0.714954i	$-0.746448\pi$
$349$	−26.1232	−1.39834	−0.699171	+	0.714954i	$0.746448\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.7057	1.04883	0.524413	−	0.851464i	$-0.324285\pi$
$353$	19.7057	1.04883	0.524413	+	0.851464i	$0.324285\pi$
$354$	0	0
$355$	−6.22524	−0.330402
$356$	0	0
$357$	0.779130	0.0412360
$358$	0	0
$359$	6.56325	0.346395	0.173198	−	0.984887i	$-0.444590\pi$
$359$	6.56325	0.346395	0.173198	+	0.984887i	$0.444590\pi$
$360$	0	0
$361$	17.3736	0.914399
$362$	0	0
$363$	2.06938	0.108614
$364$	0	0
$365$	9.94784	0.520694
$366$	0	0
$367$	−33.1532	−1.73058	−0.865291	−	0.501270i	$-0.832866\pi$
$367$	−33.1532	−1.73058	−0.865291	+	0.501270i	$0.832866\pi$
$368$	0	0
$369$	43.2978	2.25399
$370$	0	0
$371$	4.84117	0.251341
$372$	0	0
$373$	30.0243	1.55460	0.777300	−	0.629130i	$-0.216589\pi$
$373$	30.0243	1.55460	0.777300	+	0.629130i	$0.216589\pi$
$374$	0	0
$375$	−2.81781	−0.145511
$376$	0	0
$377$	0	0
$378$	0	0
$379$	11.1570	0.573097	0.286549	−	0.958066i	$-0.407492\pi$
$379$	11.1570	0.573097	0.286549	+	0.958066i	$0.407492\pi$
$380$	0	0
$381$	40.5267	2.07625
$382$	0	0
$383$	11.5707	0.591233	0.295616	−	0.955307i	$-0.404475\pi$
$383$	11.5707	0.591233	0.295616	+	0.955307i	$0.404475\pi$
$384$	0	0
$385$	−3.33712	−0.170076
$386$	0	0
$387$	43.5293	2.21272
$388$	0	0
$389$	−18.6724	−0.946727	−0.473363	−	0.880867i	$-0.656960\pi$
$389$	−18.6724	−0.946727	−0.473363	+	0.880867i	$0.656960\pi$
$390$	0	0
$391$	2.53037	0.127966
$392$	0	0
$393$	47.6509	2.40367
$394$	0	0
$395$	−6.63735	−0.333961
$396$	0	0
$397$	−14.7838	−0.741977	−0.370988	−	0.928638i	$-0.620981\pi$
$397$	−14.7838	−0.741977	−0.370988	+	0.928638i	$0.620981\pi$
$398$	0	0
$399$	−16.5557	−0.828819
$400$	0	0
$401$	13.5374	0.676025	0.338012	−	0.941142i	$-0.390245\pi$
$401$	13.5374	0.676025	0.338012	+	0.941142i	$0.390245\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−0.583871	−0.0290128
$406$	0	0
$407$	10.1170	0.501481
$408$	0	0
$409$	−9.79211	−0.484189	−0.242094	−	0.970253i	$-0.577834\pi$
$409$	−9.79211	−0.484189	−0.242094	+	0.970253i	$0.577834\pi$
$410$	0	0
$411$	−21.7504	−1.07287
$412$	0	0
$413$	−7.68244	−0.378028
$414$	0	0
$415$	7.05049	0.346095
$416$	0	0
$417$	−52.9630	−2.59361
$418$	0	0
$419$	−19.3450	−0.945065	−0.472533	−	0.881313i	$-0.656660\pi$
$419$	−19.3450	−0.945065	−0.472533	+	0.881313i	$0.656660\pi$
$420$	0	0
$421$	−10.8582	−0.529199	−0.264599	−	0.964358i	$-0.585240\pi$
$421$	−10.8582	−0.529199	−0.264599	+	0.964358i	$0.585240\pi$
$422$	0	0
$423$	51.2654	2.49261
$424$	0	0
$425$	0.283829	0.0137677
$426$	0	0
$427$	11.1904	0.541541
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8.47538	−0.408245	−0.204122	−	0.978945i	$-0.565434\pi$
$431$	−8.47538	−0.408245	−0.204122	+	0.978945i	$0.565434\pi$
$432$	0	0
$433$	−14.6325	−0.703193	−0.351596	−	0.936152i	$-0.614361\pi$
$433$	−14.6325	−0.703193	−0.351596	+	0.936152i	$0.614361\pi$
$434$	0	0
$435$	−1.29928	−0.0622957
$436$	0	0
$437$	−53.7675	−2.57205
$438$	0	0
$439$	10.2299	0.488245	0.244123	−	0.969744i	$-0.421500\pi$
$439$	10.2299	0.488245	0.244123	+	0.969744i	$0.421500\pi$
$440$	0	0
$441$	−29.8920	−1.42343
$442$	0	0
$443$	1.89017	0.0898045	0.0449023	−	0.998991i	$-0.485702\pi$
$443$	1.89017	0.0898045	0.0449023	+	0.998991i	$0.485702\pi$
$444$	0	0
$445$	8.66990	0.410993
$446$	0	0
$447$	−9.92185	−0.469288
$448$	0	0
$449$	−10.6031	−0.500393	−0.250196	−	0.968195i	$-0.580495\pi$
$449$	−10.6031	−0.500393	−0.250196	+	0.968195i	$0.580495\pi$
$450$	0	0
$451$	30.0238	1.41376
$452$	0	0
$453$	31.1361	1.46290
$454$	0	0
$455$	0	0
$456$	0	0
$457$	28.2158	1.31988	0.659939	−	0.751319i	$-0.270582\pi$
$457$	28.2158	1.31988	0.659939	+	0.751319i	$0.270582\pi$
$458$	0	0
$459$	1.55160	0.0724223
$460$	0	0
$461$	14.1918	0.660976	0.330488	−	0.943810i	$-0.392787\pi$
$461$	14.1918	0.660976	0.330488	+	0.943810i	$0.392787\pi$
$462$	0	0
$463$	18.9826	0.882195	0.441098	−	0.897459i	$-0.354589\pi$
$463$	18.9826	0.882195	0.441098	+	0.897459i	$0.354589\pi$
$464$	0	0
$465$	22.4568	1.04141
$466$	0	0
$467$	−9.88823	−0.457573	−0.228786	−	0.973477i	$-0.573476\pi$
$467$	−9.88823	−0.457573	−0.228786	+	0.973477i	$0.573476\pi$
$468$	0	0
$469$	3.86109	0.178288
$470$	0	0
$471$	12.6189	0.581447
$472$	0	0
$473$	30.1844	1.38788
$474$	0	0
$475$	−6.03105	−0.276724
$476$	0	0
$477$	24.5493	1.12403
$478$	0	0
$479$	−26.4914	−1.21042	−0.605211	−	0.796065i	$-0.706911\pi$
$479$	−26.4914	−1.21042	−0.605211	+	0.796065i	$0.706911\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	24.4726	1.11354
$484$	0	0
$485$	−0.332881	−0.0151154
$486$	0	0
$487$	−34.8349	−1.57852	−0.789261	−	0.614058i	$-0.789536\pi$
$487$	−34.8349	−1.57852	−0.789261	+	0.614058i	$0.789536\pi$
$488$	0	0
$489$	31.9793	1.44616
$490$	0	0
$491$	−0.113245	−0.00511068	−0.00255534	−	0.999997i	$-0.500813\pi$
$491$	−0.113245	−0.00511068	−0.00255534	+	0.999997i	$0.500813\pi$
$492$	0	0
$493$	0.130872	0.00589420
$494$	0	0
$495$	−16.9224	−0.760603
$496$	0	0
$497$	6.06455	0.272032
$498$	0	0
$499$	−7.69924	−0.344665	−0.172333	−	0.985039i	$-0.555130\pi$
$499$	−7.69924	−0.344665	−0.172333	+	0.985039i	$0.555130\pi$
$500$	0	0
$501$	24.8537	1.11038
$502$	0	0
$503$	−1.09165	−0.0486742	−0.0243371	−	0.999704i	$-0.507748\pi$
$503$	−1.09165	−0.0486742	−0.0243371	+	0.999704i	$0.507748\pi$
$504$	0	0
$505$	14.0197	0.623869
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−3.78054	−0.167569	−0.0837847	−	0.996484i	$-0.526701\pi$
$509$	−3.78054	−0.167569	−0.0837847	+	0.996484i	$0.526701\pi$
$510$	0	0
$511$	−9.69105	−0.428707
$512$	0	0
$513$	−32.9697	−1.45565
$514$	0	0
$515$	18.6039	0.819786
$516$	0	0
$517$	35.5487	1.56343
$518$	0	0
$519$	−13.0949	−0.574800
$520$	0	0
$521$	−8.70631	−0.381430	−0.190715	−	0.981645i	$-0.561081\pi$
$521$	−8.70631	−0.381430	−0.190715	+	0.981645i	$0.561081\pi$
$522$	0	0
$523$	−11.8023	−0.516077	−0.258038	−	0.966135i	$-0.583076\pi$
$523$	−11.8023	−0.516077	−0.258038	+	0.966135i	$0.583076\pi$
$524$	0	0
$525$	2.74507	0.119805
$526$	0	0
$527$	−2.26200	−0.0985343
$528$	0	0
$529$	56.4792	2.45562
$530$	0	0
$531$	−38.9572	−1.69060
$532$	0	0
$533$	0	0
$534$	0	0
$535$	10.8007	0.466954
$536$	0	0
$537$	−71.3429	−3.07867
$538$	0	0
$539$	−20.7279	−0.892813
$540$	0	0
$541$	21.0010	0.902902	0.451451	−	0.892296i	$-0.350907\pi$
$541$	21.0010	0.902902	0.451451	+	0.892296i	$0.350907\pi$
$542$	0	0
$543$	12.1931	0.523255
$544$	0	0
$545$	2.87817	0.123287
$546$	0	0
$547$	−19.6759	−0.841282	−0.420641	−	0.907227i	$-0.638195\pi$
$547$	−19.6759	−0.841282	−0.420641	+	0.907227i	$0.638195\pi$
$548$	0	0
$549$	56.7458	2.42185
$550$	0	0
$551$	−2.78089	−0.118470
$552$	0	0
$553$	6.46601	0.274963
$554$	0	0
$555$	−8.32209	−0.353253
$556$	0	0
$557$	−21.5516	−0.913172	−0.456586	−	0.889679i	$-0.650928\pi$
$557$	−21.5516	−0.913172	−0.456586	+	0.889679i	$0.650928\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	2.73967	0.115669
$562$	0	0
$563$	10.6586	0.449205	0.224602	−	0.974450i	$-0.427892\pi$
$563$	10.6586	0.449205	0.224602	+	0.974450i	$0.427892\pi$
$564$	0	0
$565$	−9.77628	−0.411291
$566$	0	0
$567$	0.568799	0.0238873
$568$	0	0
$569$	27.5445	1.15473	0.577363	−	0.816488i	$-0.304082\pi$
$569$	27.5445	1.15473	0.577363	+	0.816488i	$0.304082\pi$
$570$	0	0
$571$	−42.1038	−1.76199	−0.880994	−	0.473127i	$-0.843125\pi$
$571$	−42.1038	−1.76199	−0.880994	+	0.473127i	$0.843125\pi$
$572$	0	0
$573$	−7.39224	−0.308815
$574$	0	0
$575$	8.91511	0.371786
$576$	0	0
$577$	−33.7205	−1.40380	−0.701901	−	0.712274i	$-0.747665\pi$
$577$	−33.7205	−1.40380	−0.701901	+	0.712274i	$0.747665\pi$
$578$	0	0
$579$	61.0447	2.53693
$580$	0	0
$581$	−6.86849	−0.284953
$582$	0	0
$583$	17.0231	0.705024
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.3282	−0.550114	−0.275057	−	0.961428i	$-0.588697\pi$
$587$	−13.3282	−0.550114	−0.275057	+	0.961428i	$0.588697\pi$
$588$	0	0
$589$	48.0650	1.98048
$590$	0	0
$591$	−58.7797	−2.41788
$592$	0	0
$593$	1.07378	0.0440949	0.0220474	−	0.999757i	$-0.492982\pi$
$593$	1.07378	0.0440949	0.0220474	+	0.999757i	$0.492982\pi$
$594$	0	0
$595$	−0.276502	−0.0113355
$596$	0	0
$597$	64.1951	2.62733
$598$	0	0
$599$	−18.3111	−0.748169	−0.374085	−	0.927395i	$-0.622043\pi$
$599$	−18.3111	−0.748169	−0.374085	+	0.927395i	$0.622043\pi$
$600$	0	0
$601$	−1.14497	−0.0467043	−0.0233521	−	0.999727i	$-0.507434\pi$
$601$	−1.14497	−0.0467043	−0.0233521	+	0.999727i	$0.507434\pi$
$602$	0	0
$603$	19.5793	0.797332
$604$	0	0
$605$	−0.734394	−0.0298573
$606$	0	0
$607$	−11.3920	−0.462385	−0.231193	−	0.972908i	$-0.574263\pi$
$607$	−11.3920	−0.462385	−0.231193	+	0.972908i	$0.574263\pi$
$608$	0	0
$609$	1.26574	0.0512904
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−6.51518	−0.263146	−0.131573	−	0.991306i	$-0.542003\pi$
$613$	−6.51518	−0.263146	−0.131573	+	0.991306i	$0.542003\pi$
$614$	0	0
$615$	−24.6971	−0.995884
$616$	0	0
$617$	4.43338	0.178481	0.0892405	−	0.996010i	$-0.471556\pi$
$617$	4.43338	0.178481	0.0892405	+	0.996010i	$0.471556\pi$
$618$	0	0
$619$	−2.32341	−0.0933857	−0.0466929	−	0.998909i	$-0.514868\pi$
$619$	−2.32341	−0.0933857	−0.0466929	+	0.998909i	$0.514868\pi$
$620$	0	0
$621$	48.7359	1.95570
$622$	0	0
$623$	−8.44610	−0.338386
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−58.2150	−2.32488
$628$	0	0
$629$	0.838259	0.0334236
$630$	0	0
$631$	−25.4463	−1.01300	−0.506501	−	0.862239i	$-0.669061\pi$
$631$	−25.4463	−1.01300	−0.506501	+	0.862239i	$0.669061\pi$
$632$	0	0
$633$	45.0541	1.79074
$634$	0	0
$635$	−14.3824	−0.570746
$636$	0	0
$637$	0	0
$638$	0	0
$639$	30.7530	1.21657
$640$	0	0
$641$	9.30285	0.367440	0.183720	−	0.982979i	$-0.441186\pi$
$641$	9.30285	0.367440	0.183720	+	0.982979i	$0.441186\pi$
$642$	0	0
$643$	−49.3771	−1.94724	−0.973621	−	0.228172i	$-0.926725\pi$
$643$	−49.3771	−1.94724	−0.973621	+	0.228172i	$0.926725\pi$
$644$	0	0
$645$	−24.8292	−0.977649
$646$	0	0
$647$	26.1654	1.02867	0.514334	−	0.857590i	$-0.328039\pi$
$647$	26.1654	1.02867	0.514334	+	0.857590i	$0.328039\pi$
$648$	0	0
$649$	−27.0139	−1.06039
$650$	0	0
$651$	−21.8771	−0.857430
$652$	0	0
$653$	−12.5177	−0.489857	−0.244929	−	0.969541i	$-0.578765\pi$
$653$	−12.5177	−0.489857	−0.244929	+	0.969541i	$0.578765\pi$
$654$	0	0
$655$	−16.9106	−0.660753
$656$	0	0
$657$	−49.1427	−1.91724
$658$	0	0
$659$	−9.30544	−0.362488	−0.181244	−	0.983438i	$-0.558012\pi$
$659$	−9.30544	−0.362488	−0.181244	+	0.983438i	$0.558012\pi$
$660$	0	0
$661$	0.778487	0.0302796	0.0151398	−	0.999885i	$-0.495181\pi$
$661$	0.778487	0.0302796	0.0151398	+	0.999885i	$0.495181\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.87537	0.227837
$666$	0	0
$667$	4.11072	0.159168
$668$	0	0
$669$	21.2635	0.822096
$670$	0	0
$671$	39.3490	1.51905
$672$	0	0
$673$	18.6069	0.717243	0.358621	−	0.933483i	$-0.383247\pi$
$673$	18.6069	0.717243	0.358621	+	0.933483i	$0.383247\pi$
$674$	0	0
$675$	5.46666	0.210412
$676$	0	0
$677$	0.451238	0.0173425	0.00867125	−	0.999962i	$-0.497240\pi$
$677$	0.451238	0.0173425	0.00867125	+	0.999962i	$0.497240\pi$
$678$	0	0
$679$	0.324288	0.0124450
$680$	0	0
$681$	80.3040	3.07726
$682$	0	0
$683$	−35.3929	−1.35427	−0.677136	−	0.735858i	$-0.736779\pi$
$683$	−35.3929	−1.35427	−0.677136	+	0.735858i	$0.736779\pi$
$684$	0	0
$685$	7.71889	0.294924
$686$	0	0
$687$	54.6379	2.08457
$688$	0	0
$689$	0	0
$690$	0	0
$691$	39.0369	1.48503	0.742516	−	0.669828i	$-0.233632\pi$
$691$	39.0369	1.48503	0.742516	+	0.669828i	$0.233632\pi$
$692$	0	0
$693$	16.4855	0.626233
$694$	0	0
$695$	18.7958	0.712966
$696$	0	0
$697$	2.48767	0.0942270
$698$	0	0
$699$	−35.9614	−1.36019
$700$	0	0
$701$	18.0395	0.681344	0.340672	−	0.940182i	$-0.389345\pi$
$701$	18.0395	0.681344	0.340672	+	0.940182i	$0.389345\pi$
$702$	0	0
$703$	−17.8121	−0.671795
$704$	0	0
$705$	−29.2419	−1.10131
$706$	0	0
$707$	−13.6578	−0.513654
$708$	0	0
$709$	19.7805	0.742871	0.371436	−	0.928459i	$-0.378866\pi$
$709$	19.7805	0.742871	0.371436	+	0.928459i	$0.378866\pi$
$710$	0	0
$711$	32.7888	1.22967
$712$	0	0
$713$	−71.0498	−2.66084
$714$	0	0
$715$	0	0
$716$	0	0
$717$	31.3971	1.17255
$718$	0	0
$719$	−10.2849	−0.383562	−0.191781	−	0.981438i	$-0.561426\pi$
$719$	−10.2849	−0.383562	−0.191781	+	0.981438i	$0.561426\pi$
$720$	0	0
$721$	−18.1237	−0.674961
$722$	0	0
$723$	15.4494	0.574569
$724$	0	0
$725$	0.461096	0.0171247
$726$	0	0
$727$	−7.08016	−0.262589	−0.131294	−	0.991343i	$-0.541913\pi$
$727$	−7.08016	−0.262589	−0.131294	+	0.991343i	$0.541913\pi$
$728$	0	0
$729$	−43.3276	−1.60473
$730$	0	0
$731$	2.50097	0.0925017
$732$	0	0
$733$	−6.29532	−0.232523	−0.116261	−	0.993219i	$-0.537091\pi$
$733$	−6.29532	−0.232523	−0.116261	+	0.993219i	$0.537091\pi$
$734$	0	0
$735$	17.0504	0.628915
$736$	0	0
$737$	13.5768	0.500108
$738$	0	0
$739$	33.4301	1.22975	0.614873	−	0.788626i	$-0.289207\pi$
$739$	33.4301	1.22975	0.614873	+	0.788626i	$0.289207\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.84274	0.140976	0.0704882	−	0.997513i	$-0.477544\pi$
$743$	3.84274	0.140976	0.0704882	+	0.997513i	$0.477544\pi$
$744$	0	0
$745$	3.52113	0.129004
$746$	0	0
$747$	−34.8297	−1.27435
$748$	0	0
$749$	−10.5219	−0.384461
$750$	0	0
$751$	−34.7285	−1.26726	−0.633630	−	0.773636i	$-0.718436\pi$
$751$	−34.7285	−1.26726	−0.633630	+	0.773636i	$0.718436\pi$
$752$	0	0
$753$	−69.0092	−2.51483
$754$	0	0
$755$	−11.0498	−0.402142
$756$	0	0
$757$	8.10935	0.294739	0.147370	−	0.989081i	$-0.452919\pi$
$757$	8.10935	0.294739	0.147370	+	0.989081i	$0.452919\pi$
$758$	0	0
$759$	86.0535	3.12354
$760$	0	0
$761$	−10.4717	−0.379598	−0.189799	−	0.981823i	$-0.560784\pi$
$761$	−10.4717	−0.379598	−0.189799	+	0.981823i	$0.560784\pi$
$762$	0	0
$763$	−2.80388	−0.101507
$764$	0	0
$765$	−1.40213	−0.0506940
$766$	0	0
$767$	0	0
$768$	0	0
$769$	44.2939	1.59728	0.798639	−	0.601810i	$-0.205554\pi$
$769$	44.2939	1.59728	0.798639	+	0.601810i	$0.205554\pi$
$770$	0	0
$771$	−41.6552	−1.50017
$772$	0	0
$773$	12.2760	0.441537	0.220769	−	0.975326i	$-0.429143\pi$
$773$	12.2760	0.441537	0.220769	+	0.975326i	$0.429143\pi$
$774$	0	0
$775$	−7.96959	−0.286276
$776$	0	0
$777$	8.10727	0.290847
$778$	0	0
$779$	−52.8601	−1.89391
$780$	0	0
$781$	21.3249	0.763064
$782$	0	0
$783$	2.52065	0.0900808
$784$	0	0
$785$	−4.47826	−0.159836
$786$	0	0
$787$	7.08241	0.252461	0.126230	−	0.992001i	$-0.459712\pi$
$787$	7.08241	0.252461	0.126230	+	0.992001i	$0.459712\pi$
$788$	0	0
$789$	8.50222	0.302687
$790$	0	0
$791$	9.52392	0.338632
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−14.0029	−0.496633
$796$	0	0
$797$	−19.6799	−0.697099	−0.348550	−	0.937290i	$-0.613326\pi$
$797$	−19.6799	−0.697099	−0.348550	+	0.937290i	$0.613326\pi$
$798$	0	0
$799$	2.94544	0.104202
$800$	0	0
$801$	−42.8296	−1.51331
$802$	0	0
$803$	−34.0768	−1.20254
$804$	0	0
$805$	−8.68498	−0.306105
$806$	0	0
$807$	−23.8586	−0.839864
$808$	0	0
$809$	26.3066	0.924890	0.462445	−	0.886648i	$-0.346972\pi$
$809$	26.3066	0.924890	0.462445	+	0.886648i	$0.346972\pi$
$810$	0	0
$811$	−52.2096	−1.83333	−0.916663	−	0.399662i	$-0.869128\pi$
$811$	−52.2096	−1.83333	−0.916663	+	0.399662i	$0.869128\pi$
$812$	0	0
$813$	−72.5702	−2.54515
$814$	0	0
$815$	−11.3490	−0.397539
$816$	0	0
$817$	−53.1428	−1.85923
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.9551	1.70855	0.854273	−	0.519825i	$-0.174003\pi$
$821$	48.9551	1.70855	0.854273	+	0.519825i	$0.174003\pi$
$822$	0	0
$823$	28.3073	0.986731	0.493366	−	0.869822i	$-0.335767\pi$
$823$	28.3073	0.986731	0.493366	+	0.869822i	$0.335767\pi$
$824$	0	0
$825$	9.65254	0.336058
$826$	0	0
$827$	−11.1223	−0.386759	−0.193380	−	0.981124i	$-0.561945\pi$
$827$	−11.1223	−0.386759	−0.193380	+	0.981124i	$0.561945\pi$
$828$	0	0
$829$	−1.14566	−0.0397905	−0.0198952	−	0.999802i	$-0.506333\pi$
$829$	−1.14566	−0.0397905	−0.0198952	+	0.999802i	$0.506333\pi$
$830$	0	0
$831$	41.3788	1.43542
$832$	0	0
$833$	−1.71744	−0.0595057
$834$	0	0
$835$	−8.82022	−0.305236
$836$	0	0
$837$	−43.5670	−1.50590
$838$	0	0
$839$	28.3167	0.977602	0.488801	−	0.872395i	$-0.337434\pi$
$839$	28.3167	0.977602	0.488801	+	0.872395i	$0.337434\pi$
$840$	0	0
$841$	−28.7874	−0.992669
$842$	0	0
$843$	−70.9898	−2.44502
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.715436	0.0245827
$848$	0	0
$849$	−54.4002	−1.86701
$850$	0	0
$851$	26.3298	0.902574
$852$	0	0
$853$	−29.6180	−1.01410	−0.507051	−	0.861916i	$-0.669264\pi$
$853$	−29.6180	−1.01410	−0.507051	+	0.861916i	$0.669264\pi$
$854$	0	0
$855$	29.7936	1.01892
$856$	0	0
$857$	14.3690	0.490835	0.245417	−	0.969418i	$-0.421075\pi$
$857$	14.3690	0.490835	0.245417	+	0.969418i	$0.421075\pi$
$858$	0	0
$859$	−9.17850	−0.313166	−0.156583	−	0.987665i	$-0.550048\pi$
$859$	−9.17850	−0.313166	−0.156583	+	0.987665i	$0.550048\pi$
$860$	0	0
$861$	24.0596	0.819949
$862$	0	0
$863$	32.1458	1.09426	0.547128	−	0.837049i	$-0.315721\pi$
$863$	32.1458	1.09426	0.547128	+	0.837049i	$0.315721\pi$
$864$	0	0
$865$	4.64718	0.158009
$866$	0	0
$867$	−47.6757	−1.61915
$868$	0	0
$869$	22.7366	0.771285
$870$	0	0
$871$	0	0
$872$	0	0
$873$	1.64445	0.0556561
$874$	0	0
$875$	−0.974186	−0.0329335
$876$	0	0
$877$	25.9633	0.876720	0.438360	−	0.898800i	$-0.355560\pi$
$877$	25.9633	0.876720	0.438360	+	0.898800i	$0.355560\pi$
$878$	0	0
$879$	−3.47549	−0.117225
$880$	0	0
$881$	−17.1165	−0.576671	−0.288335	−	0.957529i	$-0.593102\pi$
$881$	−17.1165	−0.576671	−0.288335	+	0.957529i	$0.593102\pi$
$882$	0	0
$883$	−18.3273	−0.616762	−0.308381	−	0.951263i	$-0.599787\pi$
$883$	−18.3273	−0.616762	−0.308381	+	0.951263i	$0.599787\pi$
$884$	0	0
$885$	22.2213	0.746960
$886$	0	0
$887$	−36.2866	−1.21838	−0.609192	−	0.793023i	$-0.708506\pi$
$887$	−36.2866	−1.21838	−0.609192	+	0.793023i	$0.708506\pi$
$888$	0	0
$889$	14.0111	0.469917
$890$	0	0
$891$	2.00008	0.0670051
$892$	0	0
$893$	−62.5874	−2.09441
$894$	0	0
$895$	25.3186	0.846307
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.67475	−0.122560
$900$	0	0
$901$	1.41047	0.0469897
$902$	0	0
$903$	24.1883	0.804935
$904$	0	0
$905$	−4.32715	−0.143839
$906$	0	0
$907$	10.7409	0.356645	0.178323	−	0.983972i	$-0.442933\pi$
$907$	10.7409	0.356645	0.178323	+	0.983972i	$0.442933\pi$
$908$	0	0
$909$	−69.2579	−2.29714
$910$	0	0
$911$	−31.8649	−1.05573	−0.527866	−	0.849327i	$-0.677008\pi$
$911$	−31.8649	−1.05573	−0.527866	+	0.849327i	$0.677008\pi$
$912$	0	0
$913$	−24.1518	−0.799308
$914$	0	0
$915$	−32.3679	−1.07005
$916$	0	0
$917$	16.4741	0.544023
$918$	0	0
$919$	14.6467	0.483149	0.241574	−	0.970382i	$-0.422336\pi$
$919$	14.6467	0.483149	0.241574	+	0.970382i	$0.422336\pi$
$920$	0	0
$921$	−20.7532	−0.683842
$922$	0	0
$923$	0	0
$924$	0	0
$925$	2.95339	0.0971069
$926$	0	0
$927$	−91.9040	−3.01852
$928$	0	0
$929$	56.9334	1.86792	0.933962	−	0.357373i	$-0.116328\pi$
$929$	56.9334	1.86792	0.933962	+	0.357373i	$0.116328\pi$
$930$	0	0
$931$	36.4937	1.19603
$932$	0	0
$933$	−21.9133	−0.717409
$934$	0	0
$935$	−0.972271	−0.0317967
$936$	0	0
$937$	3.76516	0.123002	0.0615012	−	0.998107i	$-0.480411\pi$
$937$	3.76516	0.123002	0.0615012	+	0.998107i	$0.480411\pi$
$938$	0	0
$939$	0.294346	0.00960562
$940$	0	0
$941$	28.9802	0.944728	0.472364	−	0.881404i	$-0.343401\pi$
$941$	28.9802	0.944728	0.472364	+	0.881404i	$0.343401\pi$
$942$	0	0
$943$	78.1379	2.54452
$944$	0	0
$945$	−5.32554	−0.173240
$946$	0	0
$947$	9.77047	0.317498	0.158749	−	0.987319i	$-0.449254\pi$
$947$	9.77047	0.317498	0.158749	+	0.987319i	$0.449254\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−91.2890	−2.96025
$952$	0	0
$953$	42.3749	1.37266	0.686330	−	0.727291i	$-0.259221\pi$
$953$	42.3749	1.37266	0.686330	+	0.727291i	$0.259221\pi$
$954$	0	0
$955$	2.62340	0.0848913
$956$	0	0
$957$	4.45075	0.143872
$958$	0	0
$959$	−7.51964	−0.242822
$960$	0	0
$961$	32.5144	1.04885
$962$	0	0
$963$	−53.3558	−1.71937
$964$	0	0
$965$	−21.6639	−0.697385
$966$	0	0
$967$	5.65301	0.181788	0.0908942	−	0.995861i	$-0.471027\pi$
$967$	5.65301	0.181788	0.0908942	+	0.995861i	$0.471027\pi$
$968$	0	0
$969$	−4.82349	−0.154953
$970$	0	0
$971$	40.0547	1.28542	0.642709	−	0.766111i	$-0.277811\pi$
$971$	40.0547	1.28542	0.642709	+	0.766111i	$0.277811\pi$
$972$	0	0
$973$	−18.3106	−0.587012
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−15.7119	−0.502668	−0.251334	−	0.967900i	$-0.580869\pi$
$977$	−15.7119	−0.502668	−0.251334	+	0.967900i	$0.580869\pi$
$978$	0	0
$979$	−29.6992	−0.949190
$980$	0	0
$981$	−14.2183	−0.453955
$982$	0	0
$983$	−46.6671	−1.48845	−0.744225	−	0.667929i	$-0.767181\pi$
$983$	−46.6671	−1.48845	−0.744225	+	0.667929i	$0.767181\pi$
$984$	0	0
$985$	20.8601	0.664658
$986$	0	0
$987$	28.4870	0.906752
$988$	0	0
$989$	78.5558	2.49793
$990$	0	0
$991$	1.23358	0.0391859	0.0195930	−	0.999808i	$-0.493763\pi$
$991$	1.23358	0.0391859	0.0195930	+	0.999808i	$0.493763\pi$
$992$	0	0
$993$	54.0942	1.71663
$994$	0	0
$995$	−22.7819	−0.722236
$996$	0	0
$997$	32.9285	1.04286	0.521428	−	0.853295i	$-0.325400\pi$
$997$	32.9285	1.04286	0.521428	+	0.853295i	$0.325400\pi$
$998$	0	0
$999$	16.1452	0.510811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3380.2.a.r.1.9	✓	9
13.5	odd	4		3380.2.f.j.3041.18		18
13.8	odd	4		3380.2.f.j.3041.17		18
13.12	even	2		3380.2.a.s.1.9	yes	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3380.2.a.r.1.9	✓	9	1.1	even	1	trivial
3380.2.a.s.1.9	yes	9	13.12	even	2
3380.2.f.j.3041.17		18	13.8	odd	4
3380.2.f.j.3041.18		18	13.5	odd	4

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:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)

\(f(q)\)	\(=\)	\(q+2.81781 q^{3} -1.00000 q^{5} +0.974186 q^{7} +4.94004 q^{9} +O(q^{10})\)	\(q+2.81781 q^{3} -1.00000 q^{5} +0.974186 q^{7} +4.94004 q^{9} +3.42555 q^{11} -2.81781 q^{15} +0.283829 q^{17} -6.03105 q^{19} +2.74507 q^{21} +8.91511 q^{23} +1.00000 q^{25} +5.46666 q^{27} +0.461096 q^{29} -7.96959 q^{31} +9.65254 q^{33} -0.974186 q^{35} +2.95339 q^{37} +8.76466 q^{41} +8.81154 q^{43} -4.94004 q^{45} +10.3775 q^{47} -6.05096 q^{49} +0.799776 q^{51} +4.96945 q^{53} -3.42555 q^{55} -16.9943 q^{57} -7.88601 q^{59} +11.4869 q^{61} +4.81252 q^{63} +3.96340 q^{67} +25.1211 q^{69} +6.22524 q^{71} -9.94784 q^{73} +2.81781 q^{75} +3.33712 q^{77} +6.63735 q^{79} +0.583871 q^{81} -7.05049 q^{83} -0.283829 q^{85} +1.29928 q^{87} -8.66990 q^{89} -22.4568 q^{93} +6.03105 q^{95} +0.332881 q^{97} +16.9224 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9	\( 9 q - q^{3} - 9 q^{5} - q^{7} + 12 q^{9} + 7 q^{11} + q^{15} + 13 q^{17} + 4 q^{19} + 3 q^{21} + 12 q^{23} + 9 q^{25} - 4 q^{27} + 16 q^{29} - 13 q^{31} - 34 q^{33} + q^{35} - q^{37} + 6 q^{41} + q^{43} - 12 q^{45} + 2 q^{47} + 20 q^{49} + 11 q^{51} + 30 q^{53} - 7 q^{55} - 38 q^{57} - 15 q^{59} + 21 q^{61} + 17 q^{63} + 7 q^{67} + 15 q^{69} + 7 q^{71} + 28 q^{73} - q^{75} + 46 q^{77} + 31 q^{79} + 41 q^{81} - 45 q^{83} - 13 q^{85} + 28 q^{87} + 41 q^{89} + 11 q^{93} - 4 q^{95} - 8 q^{97} + 81 q^{99}+O(q^{100}) \) 9 * q - q^3 - 9 * q^5 - q^7 + 12 * q^9 + 7 * q^11 + q^15 + 13 * q^17 + 4 * q^19 + 3 * q^21 + 12 * q^23 + 9 * q^25 - 4 * q^27 + 16 * q^29 - 13 * q^31 - 34 * q^33 + q^35 - q^37 + 6 * q^41 + q^43 - 12 * q^45 + 2 * q^47 + 20 * q^49 + 11 * q^51 + 30 * q^53 - 7 * q^55 - 38 * q^57 - 15 * q^59 + 21 * q^61 + 17 * q^63 + 7 * q^67 + 15 * q^69 + 7 * q^71 + 28 * q^73 - q^75 + 46 * q^77 + 31 * q^79 + 41 * q^81 - 45 * q^83 - 13 * q^85 + 28 * q^87 + 41 * q^89 + 11 * q^93 - 4 * q^95 - 8 * q^97 + 81 * q^99

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