LMFDB - Embedded newform 816.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [816,2,Mod(1,816)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(816, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("816.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 816 = 2^{4} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	816.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:	$6.51579280494$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	816.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.56155 q^{5} +1.00000 q^{9} -1.56155 q^{11} +0.438447 q^{13} +3.56155 q^{15} +1.00000 q^{17} +4.68466 q^{19} +2.43845 q^{23} +7.68466 q^{25} +1.00000 q^{27} -8.24621 q^{29} -3.12311 q^{31} -1.56155 q^{33} -5.12311 q^{37} +0.438447 q^{39} -3.56155 q^{41} -4.68466 q^{43} +3.56155 q^{45} +11.1231 q^{47} -7.00000 q^{49} +1.00000 q^{51} +12.2462 q^{53} -5.56155 q^{55} +4.68466 q^{57} -7.12311 q^{59} +9.12311 q^{61} +1.56155 q^{65} -4.00000 q^{67} +2.43845 q^{69} +6.24621 q^{71} -12.2462 q^{73} +7.68466 q^{75} +9.36932 q^{79} +1.00000 q^{81} +0.876894 q^{83} +3.56155 q^{85} -8.24621 q^{87} -1.12311 q^{89} -3.12311 q^{93} +16.6847 q^{95} -2.87689 q^{97} -1.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 3 q^{5} + 2 q^{9} + q^{11} + 5 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} + 9 q^{23} + 3 q^{25} + 2 q^{27} + 2 q^{31} + q^{33} - 2 q^{37} + 5 q^{39} - 3 q^{41} + 3 q^{43} + 3 q^{45} + 14 q^{47}+ \cdots + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 3 * q^5 + 2 * q^9 + q^11 + 5 * q^13 + 3 * q^15 + 2 * q^17 - 3 * q^19 + 9 * q^23 + 3 * q^25 + 2 * q^27 + 2 * q^31 + q^33 - 2 * q^37 + 5 * q^39 - 3 * q^41 + 3 * q^43 + 3 * q^45 + 14 * q^47 - 14 * q^49 + 2 * q^51 + 8 * q^53 - 7 * q^55 - 3 * q^57 - 6 * q^59 + 10 * q^61 - q^65 - 8 * q^67 + 9 * q^69 - 4 * q^71 - 8 * q^73 + 3 * q^75 - 6 * q^79 + 2 * q^81 + 10 * q^83 + 3 * q^85 + 6 * q^89 + 2 * q^93 + 21 * q^95 - 14 * q^97 + q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.56155	1.59277	0.796387	−	0.604787i	$-0.206742\pi$
$5$	3.56155	1.59277	0.796387	+	0.604787i	$0.206742\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.56155	−0.470826	−0.235413	−	0.971895i	$-0.575644\pi$
$11$	−1.56155	−0.470826	−0.235413	+	0.971895i	$0.575644\pi$
$12$	0	0
$13$	0.438447	0.121603	0.0608017	−	0.998150i	$-0.480634\pi$
$13$	0.438447	0.121603	0.0608017	+	0.998150i	$0.480634\pi$
$14$	0	0
$15$	3.56155	0.919589
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	4.68466	1.07473	0.537367	−	0.843348i	$-0.319419\pi$
$19$	4.68466	1.07473	0.537367	+	0.843348i	$0.319419\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.43845	0.508451	0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845	0.508451	0.254226	+	0.967145i	$0.418179\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.24621	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	−8.24621	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	−3.12311	−0.560926	−0.280463	−	0.959865i	$-0.590488\pi$
$31$	−3.12311	−0.560926	−0.280463	+	0.959865i	$0.590488\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−5.12311	−0.842233	−0.421117	−	0.907006i	$-0.638362\pi$
$37$	−5.12311	−0.842233	−0.421117	+	0.907006i	$0.638362\pi$
$38$	0	0
$39$	0.438447	0.0702077
$40$	0	0
$41$	−3.56155	−0.556221	−0.278111	−	0.960549i	$-0.589708\pi$
$41$	−3.56155	−0.556221	−0.278111	+	0.960549i	$0.589708\pi$
$42$	0	0
$43$	−4.68466	−0.714404	−0.357202	−	0.934027i	$-0.616269\pi$
$43$	−4.68466	−0.714404	−0.357202	+	0.934027i	$0.616269\pi$
$44$	0	0
$45$	3.56155	0.530925
$46$	0	0
$47$	11.1231	1.62247	0.811236	−	0.584719i	$-0.198795\pi$
$47$	11.1231	1.62247	0.811236	+	0.584719i	$0.198795\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	0	0
$55$	−5.56155	−0.749920
$56$	0	0
$57$	4.68466	0.620498
$58$	0	0
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	0	0
$61$	9.12311	1.16809	0.584047	−	0.811720i	$-0.301468\pi$
$61$	9.12311	1.16809	0.584047	+	0.811720i	$0.301468\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.56155	0.193687
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	2.43845	0.293555
$70$	0	0
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	0	0
$73$	−12.2462	−1.43331	−0.716655	−	0.697428i	$-0.754328\pi$
$73$	−12.2462	−1.43331	−0.716655	+	0.697428i	$0.754328\pi$
$74$	0	0
$75$	7.68466	0.887348
$76$	0	0
$77$	0	0
$78$	0	0
$79$	9.36932	1.05413	0.527065	−	0.849825i	$-0.323292\pi$
$79$	9.36932	1.05413	0.527065	+	0.849825i	$0.323292\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.876894	0.0962517	0.0481258	−	0.998841i	$-0.484675\pi$
$83$	0.876894	0.0962517	0.0481258	+	0.998841i	$0.484675\pi$
$84$	0	0
$85$	3.56155	0.386305
$86$	0	0
$87$	−8.24621	−0.884087
$88$	0	0
$89$	−1.12311	−0.119049	−0.0595245	−	0.998227i	$-0.518958\pi$
$89$	−1.12311	−0.119049	−0.0595245	+	0.998227i	$0.518958\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.12311	−0.323851
$94$	0	0
$95$	16.6847	1.71181
$96$	0	0
$97$	−2.87689	−0.292104	−0.146052	−	0.989277i	$-0.546657\pi$
$97$	−2.87689	−0.292104	−0.146052	+	0.989277i	$0.546657\pi$
$98$	0	0
$99$	−1.56155	−0.156942
$100$	0	0
$101$	10.8769	1.08229	0.541146	−	0.840929i	$-0.317991\pi$
$101$	10.8769	1.08229	0.541146	+	0.840929i	$0.317991\pi$
$102$	0	0
$103$	−16.6847	−1.64399	−0.821994	−	0.569496i	$-0.807138\pi$
$103$	−16.6847	−1.64399	−0.821994	+	0.569496i	$0.807138\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.68466	0.452883	0.226442	−	0.974025i	$-0.427291\pi$
$107$	4.68466	0.452883	0.226442	+	0.974025i	$0.427291\pi$
$108$	0	0
$109$	−6.87689	−0.658687	−0.329344	−	0.944210i	$-0.606827\pi$
$109$	−6.87689	−0.658687	−0.329344	+	0.944210i	$0.606827\pi$
$110$	0	0
$111$	−5.12311	−0.486264
$112$	0	0
$113$	−0.438447	−0.0412456	−0.0206228	−	0.999787i	$-0.506565\pi$
$113$	−0.438447	−0.0412456	−0.0206228	+	0.999787i	$0.506565\pi$
$114$	0	0
$115$	8.68466	0.809849
$116$	0	0
$117$	0.438447	0.0405345
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−8.56155	−0.778323
$122$	0	0
$123$	−3.56155	−0.321134
$124$	0	0
$125$	9.56155	0.855211
$126$	0	0
$127$	−19.8078	−1.75765	−0.878827	−	0.477140i	$-0.841674\pi$
$127$	−19.8078	−1.75765	−0.878827	+	0.477140i	$0.841674\pi$
$128$	0	0
$129$	−4.68466	−0.412461
$130$	0	0
$131$	−14.4384	−1.26149	−0.630746	−	0.775989i	$-0.717251\pi$
$131$	−14.4384	−1.26149	−0.630746	+	0.775989i	$0.717251\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.56155	0.306530
$136$	0	0
$137$	0.246211	0.0210352	0.0105176	−	0.999945i	$-0.496652\pi$
$137$	0.246211	0.0210352	0.0105176	+	0.999945i	$0.496652\pi$
$138$	0	0
$139$	0.876894	0.0743772	0.0371886	−	0.999308i	$-0.488160\pi$
$139$	0.876894	0.0743772	0.0371886	+	0.999308i	$0.488160\pi$
$140$	0	0
$141$	11.1231	0.936734
$142$	0	0
$143$	−0.684658	−0.0572540
$144$	0	0
$145$	−29.3693	−2.43899
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	12.2462	1.00325	0.501624	−	0.865086i	$-0.332736\pi$
$149$	12.2462	1.00325	0.501624	+	0.865086i	$0.332736\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	1.00000	0.0808452
$154$	0	0
$155$	−11.1231	−0.893429
$156$	0	0
$157$	6.68466	0.533494	0.266747	−	0.963767i	$-0.414051\pi$
$157$	6.68466	0.533494	0.266747	+	0.963767i	$0.414051\pi$
$158$	0	0
$159$	12.2462	0.971188
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.1231	1.18453	0.592267	−	0.805742i	$-0.298233\pi$
$163$	15.1231	1.18453	0.592267	+	0.805742i	$0.298233\pi$
$164$	0	0
$165$	−5.56155	−0.432966
$166$	0	0
$167$	−19.8078	−1.53277	−0.766385	−	0.642381i	$-0.777947\pi$
$167$	−19.8078	−1.53277	−0.766385	+	0.642381i	$0.777947\pi$
$168$	0	0
$169$	−12.8078	−0.985213
$170$	0	0
$171$	4.68466	0.358245
$172$	0	0
$173$	1.80776	0.137442	0.0687209	−	0.997636i	$-0.478108\pi$
$173$	1.80776	0.137442	0.0687209	+	0.997636i	$0.478108\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−7.12311	−0.535405
$178$	0	0
$179$	0.876894	0.0655422	0.0327711	−	0.999463i	$-0.489567\pi$
$179$	0.876894	0.0655422	0.0327711	+	0.999463i	$0.489567\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	9.12311	0.674399
$184$	0	0
$185$	−18.2462	−1.34149
$186$	0	0
$187$	−1.56155	−0.114192
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.87689	0.352880	0.176440	−	0.984311i	$-0.443542\pi$
$191$	4.87689	0.352880	0.176440	+	0.984311i	$0.443542\pi$
$192$	0	0
$193$	−7.75379	−0.558130	−0.279065	−	0.960272i	$-0.590024\pi$
$193$	−7.75379	−0.558130	−0.279065	+	0.960272i	$0.590024\pi$
$194$	0	0
$195$	1.56155	0.111825
$196$	0	0
$197$	−8.93087	−0.636298	−0.318149	−	0.948041i	$-0.603061\pi$
$197$	−8.93087	−0.636298	−0.318149	+	0.948041i	$0.603061\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	−4.00000	−0.282138
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.6847	−0.885935
$206$	0	0
$207$	2.43845	0.169484
$208$	0	0
$209$	−7.31534	−0.506013
$210$	0	0
$211$	−13.3693	−0.920382	−0.460191	−	0.887820i	$-0.652219\pi$
$211$	−13.3693	−0.920382	−0.460191	+	0.887820i	$0.652219\pi$
$212$	0	0
$213$	6.24621	0.427983
$214$	0	0
$215$	−16.6847	−1.13788
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−12.2462	−0.827522
$220$	0	0
$221$	0.438447	0.0294931
$222$	0	0
$223$	14.9309	0.999845	0.499922	−	0.866070i	$-0.333362\pi$
$223$	14.9309	0.999845	0.499922	+	0.866070i	$0.333362\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	0	0
$227$	14.0540	0.932795	0.466398	−	0.884575i	$-0.345552\pi$
$227$	14.0540	0.932795	0.466398	+	0.884575i	$0.345552\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.56155	−0.233325	−0.116663	−	0.993172i	$-0.537220\pi$
$233$	−3.56155	−0.233325	−0.116663	+	0.993172i	$0.537220\pi$
$234$	0	0
$235$	39.6155	2.58423
$236$	0	0
$237$	9.36932	0.608603
$238$	0	0
$239$	6.24621	0.404034	0.202017	−	0.979382i	$-0.435250\pi$
$239$	6.24621	0.404034	0.202017	+	0.979382i	$0.435250\pi$
$240$	0	0
$241$	3.36932	0.217037	0.108518	−	0.994094i	$-0.465389\pi$
$241$	3.36932	0.217037	0.108518	+	0.994094i	$0.465389\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−24.9309	−1.59277
$246$	0	0
$247$	2.05398	0.130691
$248$	0	0
$249$	0.876894	0.0555709
$250$	0	0
$251$	−8.49242	−0.536037	−0.268018	−	0.963414i	$-0.586369\pi$
$251$	−8.49242	−0.536037	−0.268018	+	0.963414i	$0.586369\pi$
$252$	0	0
$253$	−3.80776	−0.239392
$254$	0	0
$255$	3.56155	0.223033
$256$	0	0
$257$	−15.3693	−0.958712	−0.479356	−	0.877621i	$-0.659130\pi$
$257$	−15.3693	−0.958712	−0.479356	+	0.877621i	$0.659130\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−8.24621	−0.510428
$262$	0	0
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	0	0
$265$	43.6155	2.67928
$266$	0	0
$267$	−1.12311	−0.0687329
$268$	0	0
$269$	16.4384	1.00227	0.501135	−	0.865369i	$-0.332916\pi$
$269$	16.4384	1.00227	0.501135	+	0.865369i	$0.332916\pi$
$270$	0	0
$271$	−19.8078	−1.20324	−0.601618	−	0.798784i	$-0.705477\pi$
$271$	−19.8078	−1.20324	−0.601618	+	0.798784i	$0.705477\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	−10.8769	−0.648861	−0.324431	−	0.945910i	$-0.605173\pi$
$281$	−10.8769	−0.648861	−0.324431	+	0.945910i	$0.605173\pi$
$282$	0	0
$283$	−21.3693	−1.27027	−0.635137	−	0.772399i	$-0.719056\pi$
$283$	−21.3693	−1.27027	−0.635137	+	0.772399i	$0.719056\pi$
$284$	0	0
$285$	16.6847	0.988314
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.87689	−0.168647
$292$	0	0
$293$	1.12311	0.0656125	0.0328063	−	0.999462i	$-0.489556\pi$
$293$	1.12311	0.0656125	0.0328063	+	0.999462i	$0.489556\pi$
$294$	0	0
$295$	−25.3693	−1.47706
$296$	0	0
$297$	−1.56155	−0.0906105
$298$	0	0
$299$	1.06913	0.0618294
$300$	0	0
$301$	0	0
$302$	0	0
$303$	10.8769	0.624861
$304$	0	0
$305$	32.4924	1.86051
$306$	0	0
$307$	−32.4924	−1.85444	−0.927220	−	0.374516i	$-0.877809\pi$
$307$	−32.4924	−1.85444	−0.927220	+	0.374516i	$0.877809\pi$
$308$	0	0
$309$	−16.6847	−0.949157
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	33.6155	1.90006	0.950031	−	0.312156i	$-0.101051\pi$
$313$	33.6155	1.90006	0.950031	+	0.312156i	$0.101051\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	12.8769	0.720968
$320$	0	0
$321$	4.68466	0.261472
$322$	0	0
$323$	4.68466	0.260661
$324$	0	0
$325$	3.36932	0.186896
$326$	0	0
$327$	−6.87689	−0.380293
$328$	0	0
$329$	0	0
$330$	0	0
$331$	34.9309	1.91997	0.959987	−	0.280044i	$-0.0903491\pi$
$331$	34.9309	1.91997	0.959987	+	0.280044i	$0.0903491\pi$
$332$	0	0
$333$	−5.12311	−0.280744
$334$	0	0
$335$	−14.2462	−0.778354
$336$	0	0
$337$	−16.7386	−0.911811	−0.455906	−	0.890028i	$-0.650685\pi$
$337$	−16.7386	−0.911811	−0.455906	+	0.890028i	$0.650685\pi$
$338$	0	0
$339$	−0.438447	−0.0238132
$340$	0	0
$341$	4.87689	0.264099
$342$	0	0
$343$	0	0
$344$	0	0
$345$	8.68466	0.467566
$346$	0	0
$347$	8.49242	0.455897	0.227949	−	0.973673i	$-0.426798\pi$
$347$	8.49242	0.455897	0.227949	+	0.973673i	$0.426798\pi$
$348$	0	0
$349$	11.5616	0.618876	0.309438	−	0.950920i	$-0.399859\pi$
$349$	11.5616	0.618876	0.309438	+	0.950920i	$0.399859\pi$
$350$	0	0
$351$	0.438447	0.0234026
$352$	0	0
$353$	−10.4924	−0.558455	−0.279228	−	0.960225i	$-0.590078\pi$
$353$	−10.4924	−0.558455	−0.279228	+	0.960225i	$0.590078\pi$
$354$	0	0
$355$	22.2462	1.18071
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.2462	−0.751886	−0.375943	−	0.926643i	$-0.622681\pi$
$359$	−14.2462	−0.751886	−0.375943	+	0.926643i	$0.622681\pi$
$360$	0	0
$361$	2.94602	0.155054
$362$	0	0
$363$	−8.56155	−0.449365
$364$	0	0
$365$	−43.6155	−2.28294
$366$	0	0
$367$	1.75379	0.0915470	0.0457735	−	0.998952i	$-0.485425\pi$
$367$	1.75379	0.0915470	0.0457735	+	0.998952i	$0.485425\pi$
$368$	0	0
$369$	−3.56155	−0.185407
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−0.246211	−0.0127483	−0.00637417	−	0.999980i	$-0.502029\pi$
$373$	−0.246211	−0.0127483	−0.00637417	+	0.999980i	$0.502029\pi$
$374$	0	0
$375$	9.56155	0.493756
$376$	0	0
$377$	−3.61553	−0.186209
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	−19.8078	−1.01478
$382$	0	0
$383$	−6.24621	−0.319166	−0.159583	−	0.987184i	$-0.551015\pi$
$383$	−6.24621	−0.319166	−0.159583	+	0.987184i	$0.551015\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4.68466	−0.238135
$388$	0	0
$389$	35.8617	1.81826	0.909131	−	0.416510i	$-0.136747\pi$
$389$	35.8617	1.81826	0.909131	+	0.416510i	$0.136747\pi$
$390$	0	0
$391$	2.43845	0.123318
$392$	0	0
$393$	−14.4384	−0.728323
$394$	0	0
$395$	33.3693	1.67899
$396$	0	0
$397$	−19.3693	−0.972118	−0.486059	−	0.873926i	$-0.661566\pi$
$397$	−19.3693	−0.972118	−0.486059	+	0.873926i	$0.661566\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	39.1771	1.95641	0.978205	−	0.207641i	$-0.0665787\pi$
$401$	39.1771	1.95641	0.978205	+	0.207641i	$0.0665787\pi$
$402$	0	0
$403$	−1.36932	−0.0682105
$404$	0	0
$405$	3.56155	0.176975
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	0	0
$411$	0.246211	0.0121447
$412$	0	0
$413$	0	0
$414$	0	0
$415$	3.12311	0.153307
$416$	0	0
$417$	0.876894	0.0429417
$418$	0	0
$419$	0.492423	0.0240564	0.0120282	−	0.999928i	$-0.496171\pi$
$419$	0.492423	0.0240564	0.0120282	+	0.999928i	$0.496171\pi$
$420$	0	0
$421$	24.4384	1.19106	0.595529	−	0.803334i	$-0.296943\pi$
$421$	24.4384	1.19106	0.595529	+	0.803334i	$0.296943\pi$
$422$	0	0
$423$	11.1231	0.540824
$424$	0	0
$425$	7.68466	0.372761
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−0.684658	−0.0330556
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	26.6847	1.28238	0.641191	−	0.767381i	$-0.278440\pi$
$433$	26.6847	1.28238	0.641191	+	0.767381i	$0.278440\pi$
$434$	0	0
$435$	−29.3693	−1.40815
$436$	0	0
$437$	11.4233	0.546450
$438$	0	0
$439$	22.2462	1.06175	0.530877	−	0.847449i	$-0.321863\pi$
$439$	22.2462	1.06175	0.530877	+	0.847449i	$0.321863\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	31.1231	1.47870	0.739352	−	0.673319i	$-0.235132\pi$
$443$	31.1231	1.47870	0.739352	+	0.673319i	$0.235132\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	0	0
$447$	12.2462	0.579226
$448$	0	0
$449$	36.7386	1.73380	0.866902	−	0.498479i	$-0.166108\pi$
$449$	36.7386	1.73380	0.866902	+	0.498479i	$0.166108\pi$
$450$	0	0
$451$	5.56155	0.261883
$452$	0	0
$453$	−8.00000	−0.375873
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.8078	0.645900	0.322950	−	0.946416i	$-0.395325\pi$
$457$	13.8078	0.645900	0.322950	+	0.946416i	$0.395325\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−8.24621	−0.384064	−0.192032	−	0.981389i	$-0.561508\pi$
$461$	−8.24621	−0.384064	−0.192032	+	0.981389i	$0.561508\pi$
$462$	0	0
$463$	40.9848	1.90473	0.952364	−	0.304965i	$-0.0986447\pi$
$463$	40.9848	1.90473	0.952364	+	0.304965i	$0.0986447\pi$
$464$	0	0
$465$	−11.1231	−0.515822
$466$	0	0
$467$	−21.3693	−0.988854	−0.494427	−	0.869219i	$-0.664622\pi$
$467$	−21.3693	−0.988854	−0.494427	+	0.869219i	$0.664622\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	6.68466	0.308013
$472$	0	0
$473$	7.31534	0.336360
$474$	0	0
$475$	36.0000	1.65179
$476$	0	0
$477$	12.2462	0.560715
$478$	0	0
$479$	−24.3002	−1.11030	−0.555152	−	0.831749i	$-0.687340\pi$
$479$	−24.3002	−1.11030	−0.555152	+	0.831749i	$0.687340\pi$
$480$	0	0
$481$	−2.24621	−0.102418
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−10.2462	−0.465256
$486$	0	0
$487$	−17.3693	−0.787079	−0.393539	−	0.919308i	$-0.628750\pi$
$487$	−17.3693	−0.787079	−0.393539	+	0.919308i	$0.628750\pi$
$488$	0	0
$489$	15.1231	0.683890
$490$	0	0
$491$	21.3693	0.964384	0.482192	−	0.876066i	$-0.339841\pi$
$491$	21.3693	0.964384	0.482192	+	0.876066i	$0.339841\pi$
$492$	0	0
$493$	−8.24621	−0.371391
$494$	0	0
$495$	−5.56155	−0.249973
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−13.3693	−0.598493	−0.299246	−	0.954176i	$-0.596735\pi$
$499$	−13.3693	−0.598493	−0.299246	+	0.954176i	$0.596735\pi$
$500$	0	0
$501$	−19.8078	−0.884946
$502$	0	0
$503$	29.5616	1.31808	0.659042	−	0.752106i	$-0.270962\pi$
$503$	29.5616	1.31808	0.659042	+	0.752106i	$0.270962\pi$
$504$	0	0
$505$	38.7386	1.72385
$506$	0	0
$507$	−12.8078	−0.568813
$508$	0	0
$509$	25.1231	1.11356	0.556781	−	0.830659i	$-0.312036\pi$
$509$	25.1231	1.11356	0.556781	+	0.830659i	$0.312036\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.68466	0.206833
$514$	0	0
$515$	−59.4233	−2.61850
$516$	0	0
$517$	−17.3693	−0.763902
$518$	0	0
$519$	1.80776	0.0793520
$520$	0	0
$521$	−35.5616	−1.55798	−0.778990	−	0.627036i	$-0.784268\pi$
$521$	−35.5616	−1.55798	−0.778990	+	0.627036i	$0.784268\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.12311	−0.136045
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	−7.12311	−0.309116
$532$	0	0
$533$	−1.56155	−0.0676384
$534$	0	0
$535$	16.6847	0.721341
$536$	0	0
$537$	0.876894	0.0378408
$538$	0	0
$539$	10.9309	0.470826
$540$	0	0
$541$	34.1080	1.46642	0.733208	−	0.680005i	$-0.238022\pi$
$541$	34.1080	1.46642	0.733208	+	0.680005i	$0.238022\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	−24.4924	−1.04914
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	9.12311	0.389365
$550$	0	0
$551$	−38.6307	−1.64572
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−18.2462	−0.774509
$556$	0	0
$557$	26.4924	1.12252	0.561260	−	0.827640i	$-0.310317\pi$
$557$	26.4924	1.12252	0.561260	+	0.827640i	$0.310317\pi$
$558$	0	0
$559$	−2.05398	−0.0868739
$560$	0	0
$561$	−1.56155	−0.0659288
$562$	0	0
$563$	−31.1231	−1.31168	−0.655841	−	0.754899i	$-0.727686\pi$
$563$	−31.1231	−1.31168	−0.655841	+	0.754899i	$0.727686\pi$
$564$	0	0
$565$	−1.56155	−0.0656950
$566$	0	0
$567$	0	0
$568$	0	0
$569$	21.1231	0.885527	0.442763	−	0.896639i	$-0.353998\pi$
$569$	21.1231	0.885527	0.442763	+	0.896639i	$0.353998\pi$
$570$	0	0
$571$	30.7386	1.28637	0.643186	−	0.765710i	$-0.277612\pi$
$571$	30.7386	1.28637	0.643186	+	0.765710i	$0.277612\pi$
$572$	0	0
$573$	4.87689	0.203735
$574$	0	0
$575$	18.7386	0.781455
$576$	0	0
$577$	−3.94602	−0.164275	−0.0821376	−	0.996621i	$-0.526175\pi$
$577$	−3.94602	−0.164275	−0.0821376	+	0.996621i	$0.526175\pi$
$578$	0	0
$579$	−7.75379	−0.322236
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−19.1231	−0.791998
$584$	0	0
$585$	1.56155	0.0645623
$586$	0	0
$587$	28.9848	1.19633	0.598166	−	0.801372i	$-0.295896\pi$
$587$	28.9848	1.19633	0.598166	+	0.801372i	$0.295896\pi$
$588$	0	0
$589$	−14.6307	−0.602847
$590$	0	0
$591$	−8.93087	−0.367367
$592$	0	0
$593$	27.7538	1.13971	0.569856	−	0.821745i	$-0.306999\pi$
$593$	27.7538	1.13971	0.569856	+	0.821745i	$0.306999\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−16.0000	−0.654836
$598$	0	0
$599$	0.384472	0.0157091	0.00785455	−	0.999969i	$-0.497500\pi$
$599$	0.384472	0.0157091	0.00785455	+	0.999969i	$0.497500\pi$
$600$	0	0
$601$	−30.9848	−1.26390	−0.631949	−	0.775010i	$-0.717745\pi$
$601$	−30.9848	−1.26390	−0.631949	+	0.775010i	$0.717745\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	0	0
$605$	−30.4924	−1.23969
$606$	0	0
$607$	9.36932	0.380289	0.190144	−	0.981756i	$-0.439104\pi$
$607$	9.36932	0.380289	0.190144	+	0.981756i	$0.439104\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.87689	0.197298
$612$	0	0
$613$	14.6847	0.593108	0.296554	−	0.955016i	$-0.404163\pi$
$613$	14.6847	0.593108	0.296554	+	0.955016i	$0.404163\pi$
$614$	0	0
$615$	−12.6847	−0.511495
$616$	0	0
$617$	−44.2462	−1.78129	−0.890643	−	0.454704i	$-0.849745\pi$
$617$	−44.2462	−1.78129	−0.890643	+	0.454704i	$0.849745\pi$
$618$	0	0
$619$	−5.36932	−0.215811	−0.107906	−	0.994161i	$-0.534414\pi$
$619$	−5.36932	−0.215811	−0.107906	+	0.994161i	$0.534414\pi$
$620$	0	0
$621$	2.43845	0.0978515
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	−7.31534	−0.292147
$628$	0	0
$629$	−5.12311	−0.204272
$630$	0	0
$631$	0.684658	0.0272558	0.0136279	−	0.999907i	$-0.495662\pi$
$631$	0.684658	0.0272558	0.0136279	+	0.999907i	$0.495662\pi$
$632$	0	0
$633$	−13.3693	−0.531383
$634$	0	0
$635$	−70.5464	−2.79955
$636$	0	0
$637$	−3.06913	−0.121603
$638$	0	0
$639$	6.24621	0.247096
$640$	0	0
$641$	−28.9309	−1.14270	−0.571350	−	0.820706i	$-0.693580\pi$
$641$	−28.9309	−1.14270	−0.571350	+	0.820706i	$0.693580\pi$
$642$	0	0
$643$	13.7538	0.542396	0.271198	−	0.962524i	$-0.412580\pi$
$643$	13.7538	0.542396	0.271198	+	0.962524i	$0.412580\pi$
$644$	0	0
$645$	−16.6847	−0.656958
$646$	0	0
$647$	9.36932	0.368346	0.184173	−	0.982894i	$-0.441039\pi$
$647$	9.36932	0.368346	0.184173	+	0.982894i	$0.441039\pi$
$648$	0	0
$649$	11.1231	0.436620
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.9309	−1.28868	−0.644342	−	0.764737i	$-0.722869\pi$
$653$	−32.9309	−1.28868	−0.644342	+	0.764737i	$0.722869\pi$
$654$	0	0
$655$	−51.4233	−2.00927
$656$	0	0
$657$	−12.2462	−0.477770
$658$	0	0
$659$	9.86174	0.384159	0.192079	−	0.981379i	$-0.438477\pi$
$659$	9.86174	0.384159	0.192079	+	0.981379i	$0.438477\pi$
$660$	0	0
$661$	13.3153	0.517907	0.258953	−	0.965890i	$-0.416622\pi$
$661$	13.3153	0.517907	0.258953	+	0.965890i	$0.416622\pi$
$662$	0	0
$663$	0.438447	0.0170279
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.1080	−0.778583
$668$	0	0
$669$	14.9309	0.577261
$670$	0	0
$671$	−14.2462	−0.549969
$672$	0	0
$673$	−0.738634	−0.0284722	−0.0142361	−	0.999899i	$-0.504532\pi$
$673$	−0.738634	−0.0284722	−0.0142361	+	0.999899i	$0.504532\pi$
$674$	0	0
$675$	7.68466	0.295783
$676$	0	0
$677$	−1.31534	−0.0505527	−0.0252763	−	0.999681i	$-0.508047\pi$
$677$	−1.31534	−0.0505527	−0.0252763	+	0.999681i	$0.508047\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	14.0540	0.538550
$682$	0	0
$683$	9.56155	0.365863	0.182931	−	0.983126i	$-0.441441\pi$
$683$	9.56155	0.365863	0.182931	+	0.983126i	$0.441441\pi$
$684$	0	0
$685$	0.876894	0.0335044
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	5.36932	0.204555
$690$	0	0
$691$	28.9848	1.10264	0.551318	−	0.834295i	$-0.314125\pi$
$691$	28.9848	1.10264	0.551318	+	0.834295i	$0.314125\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.12311	0.118466
$696$	0	0
$697$	−3.56155	−0.134903
$698$	0	0
$699$	−3.56155	−0.134710
$700$	0	0
$701$	15.3693	0.580491	0.290246	−	0.956952i	$-0.406263\pi$
$701$	15.3693	0.580491	0.290246	+	0.956952i	$0.406263\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	39.6155	1.49201
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−44.7386	−1.68019	−0.840097	−	0.542436i	$-0.817502\pi$
$709$	−44.7386	−1.68019	−0.840097	+	0.542436i	$0.817502\pi$
$710$	0	0
$711$	9.36932	0.351377
$712$	0	0
$713$	−7.61553	−0.285204
$714$	0	0
$715$	−2.43845	−0.0911928
$716$	0	0
$717$	6.24621	0.233269
$718$	0	0
$719$	11.8078	0.440355	0.220178	−	0.975460i	$-0.429336\pi$
$719$	11.8078	0.440355	0.220178	+	0.975460i	$0.429336\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	3.36932	0.125306
$724$	0	0
$725$	−63.3693	−2.35348
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−4.68466	−0.173268
$732$	0	0
$733$	−11.7538	−0.434136	−0.217068	−	0.976156i	$-0.569649\pi$
$733$	−11.7538	−0.434136	−0.217068	+	0.976156i	$0.569649\pi$
$734$	0	0
$735$	−24.9309	−0.919589
$736$	0	0
$737$	6.24621	0.230082
$738$	0	0
$739$	20.6847	0.760897	0.380449	−	0.924802i	$-0.375770\pi$
$739$	20.6847	0.760897	0.380449	+	0.924802i	$0.375770\pi$
$740$	0	0
$741$	2.05398	0.0754547
$742$	0	0
$743$	−28.4924	−1.04529	−0.522643	−	0.852552i	$-0.675054\pi$
$743$	−28.4924	−1.04529	−0.522643	+	0.852552i	$0.675054\pi$
$744$	0	0
$745$	43.6155	1.59795
$746$	0	0
$747$	0.876894	0.0320839
$748$	0	0
$749$	0	0
$750$	0	0
$751$	25.3693	0.925740	0.462870	−	0.886426i	$-0.346820\pi$
$751$	25.3693	0.925740	0.462870	+	0.886426i	$0.346820\pi$
$752$	0	0
$753$	−8.49242	−0.309481
$754$	0	0
$755$	−28.4924	−1.03695
$756$	0	0
$757$	16.0540	0.583492	0.291746	−	0.956496i	$-0.405764\pi$
$757$	16.0540	0.583492	0.291746	+	0.956496i	$0.405764\pi$
$758$	0	0
$759$	−3.80776	−0.138213
$760$	0	0
$761$	−15.7538	−0.571074	−0.285537	−	0.958368i	$-0.592172\pi$
$761$	−15.7538	−0.571074	−0.285537	+	0.958368i	$0.592172\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	3.56155	0.128768
$766$	0	0
$767$	−3.12311	−0.112769
$768$	0	0
$769$	40.5464	1.46214	0.731070	−	0.682302i	$-0.239021\pi$
$769$	40.5464	1.46214	0.731070	+	0.682302i	$0.239021\pi$
$770$	0	0
$771$	−15.3693	−0.553512
$772$	0	0
$773$	−8.63068	−0.310424	−0.155212	−	0.987881i	$-0.549606\pi$
$773$	−8.63068	−0.310424	−0.155212	+	0.987881i	$0.549606\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.6847	−0.597790
$780$	0	0
$781$	−9.75379	−0.349018
$782$	0	0
$783$	−8.24621	−0.294696
$784$	0	0
$785$	23.8078	0.849736
$786$	0	0
$787$	10.2462	0.365238	0.182619	−	0.983184i	$-0.441543\pi$
$787$	10.2462	0.365238	0.182619	+	0.983184i	$0.441543\pi$
$788$	0	0
$789$	20.4924	0.729550
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	43.6155	1.54688
$796$	0	0
$797$	−9.61553	−0.340599	−0.170300	−	0.985392i	$-0.554474\pi$
$797$	−9.61553	−0.340599	−0.170300	+	0.985392i	$0.554474\pi$
$798$	0	0
$799$	11.1231	0.393507
$800$	0	0
$801$	−1.12311	−0.0396830
$802$	0	0
$803$	19.1231	0.674840
$804$	0	0
$805$	0	0
$806$	0	0
$807$	16.4384	0.578661
$808$	0	0
$809$	15.9460	0.560632	0.280316	−	0.959908i	$-0.409561\pi$
$809$	15.9460	0.560632	0.280316	+	0.959908i	$0.409561\pi$
$810$	0	0
$811$	45.3693	1.59313	0.796566	−	0.604551i	$-0.206648\pi$
$811$	45.3693	1.59313	0.796566	+	0.604551i	$0.206648\pi$
$812$	0	0
$813$	−19.8078	−0.694689
$814$	0	0
$815$	53.8617	1.88669
$816$	0	0
$817$	−21.9460	−0.767794
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−12.4384	−0.434105	−0.217052	−	0.976160i	$-0.569644\pi$
$821$	−12.4384	−0.434105	−0.217052	+	0.976160i	$0.569644\pi$
$822$	0	0
$823$	−3.50758	−0.122266	−0.0611332	−	0.998130i	$-0.519471\pi$
$823$	−3.50758	−0.122266	−0.0611332	+	0.998130i	$0.519471\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	−47.4233	−1.64907	−0.824535	−	0.565811i	$-0.808563\pi$
$827$	−47.4233	−1.64907	−0.824535	+	0.565811i	$0.808563\pi$
$828$	0	0
$829$	17.5076	0.608063	0.304032	−	0.952662i	$-0.401667\pi$
$829$	17.5076	0.608063	0.304032	+	0.952662i	$0.401667\pi$
$830$	0	0
$831$	6.00000	0.208138
$832$	0	0
$833$	−7.00000	−0.242536
$834$	0	0
$835$	−70.5464	−2.44136
$836$	0	0
$837$	−3.12311	−0.107950
$838$	0	0
$839$	26.0540	0.899483	0.449742	−	0.893159i	$-0.351516\pi$
$839$	26.0540	0.899483	0.449742	+	0.893159i	$0.351516\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	−10.8769	−0.374620
$844$	0	0
$845$	−45.6155	−1.56922
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−21.3693	−0.733393
$850$	0	0
$851$	−12.4924	−0.428235
$852$	0	0
$853$	−28.7386	−0.983992	−0.491996	−	0.870597i	$-0.663733\pi$
$853$	−28.7386	−0.983992	−0.491996	+	0.870597i	$0.663733\pi$
$854$	0	0
$855$	16.6847	0.570603
$856$	0	0
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.75379	0.332023	0.166011	−	0.986124i	$-0.446911\pi$
$863$	9.75379	0.332023	0.166011	+	0.986124i	$0.446911\pi$
$864$	0	0
$865$	6.43845	0.218914
$866$	0	0
$867$	1.00000	0.0339618
$868$	0	0
$869$	−14.6307	−0.496312
$870$	0	0
$871$	−1.75379	−0.0594249
$872$	0	0
$873$	−2.87689	−0.0973681
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	1.12311	0.0378814
$880$	0	0
$881$	40.2462	1.35593	0.677965	−	0.735095i	$-0.262862\pi$
$881$	40.2462	1.35593	0.677965	+	0.735095i	$0.262862\pi$
$882$	0	0
$883$	23.4233	0.788257	0.394128	−	0.919055i	$-0.371047\pi$
$883$	23.4233	0.788257	0.394128	+	0.919055i	$0.371047\pi$
$884$	0	0
$885$	−25.3693	−0.852780
$886$	0	0
$887$	−18.4384	−0.619102	−0.309551	−	0.950883i	$-0.600179\pi$
$887$	−18.4384	−0.619102	−0.309551	+	0.950883i	$0.600179\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−1.56155	−0.0523140
$892$	0	0
$893$	52.1080	1.74373
$894$	0	0
$895$	3.12311	0.104394
$896$	0	0
$897$	1.06913	0.0356972
$898$	0	0
$899$	25.7538	0.858937
$900$	0	0
$901$	12.2462	0.407980
$902$	0	0
$903$	0	0
$904$	0	0
$905$	21.3693	0.710340
$906$	0	0
$907$	9.86174	0.327454	0.163727	−	0.986506i	$-0.447648\pi$
$907$	9.86174	0.327454	0.163727	+	0.986506i	$0.447648\pi$
$908$	0	0
$909$	10.8769	0.360764
$910$	0	0
$911$	−24.3002	−0.805101	−0.402551	−	0.915398i	$-0.631876\pi$
$911$	−24.3002	−0.805101	−0.402551	+	0.915398i	$0.631876\pi$
$912$	0	0
$913$	−1.36932	−0.0453178
$914$	0	0
$915$	32.4924	1.07417
$916$	0	0
$917$	0	0
$918$	0	0
$919$	16.6847	0.550376	0.275188	−	0.961390i	$-0.411260\pi$
$919$	16.6847	0.550376	0.275188	+	0.961390i	$0.411260\pi$
$920$	0	0
$921$	−32.4924	−1.07066
$922$	0	0
$923$	2.73863	0.0901432
$924$	0	0
$925$	−39.3693	−1.29446
$926$	0	0
$927$	−16.6847	−0.547996
$928$	0	0
$929$	3.06913	0.100695	0.0503474	−	0.998732i	$-0.483967\pi$
$929$	3.06913	0.100695	0.0503474	+	0.998732i	$0.483967\pi$
$930$	0	0
$931$	−32.7926	−1.07473
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.56155	−0.181882
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	33.6155	1.09700
$940$	0	0
$941$	30.0000	0.977972	0.488986	−	0.872292i	$-0.337367\pi$
$941$	30.0000	0.977972	0.488986	+	0.872292i	$0.337367\pi$
$942$	0	0
$943$	−8.68466	−0.282811
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	0	0
$949$	−5.36932	−0.174295
$950$	0	0
$951$	−18.0000	−0.583690
$952$	0	0
$953$	36.3542	1.17763	0.588813	−	0.808269i	$-0.299595\pi$
$953$	36.3542	1.17763	0.588813	+	0.808269i	$0.299595\pi$
$954$	0	0
$955$	17.3693	0.562058
$956$	0	0
$957$	12.8769	0.416251
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−21.2462	−0.685362
$962$	0	0
$963$	4.68466	0.150961
$964$	0	0
$965$	−27.6155	−0.888975
$966$	0	0
$967$	−42.4384	−1.36473	−0.682364	−	0.731012i	$-0.739048\pi$
$967$	−42.4384	−1.36473	−0.682364	+	0.731012i	$0.739048\pi$
$968$	0	0
$969$	4.68466	0.150493
$970$	0	0
$971$	43.6155	1.39969	0.699844	−	0.714295i	$-0.253253\pi$
$971$	43.6155	1.39969	0.699844	+	0.714295i	$0.253253\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	3.36932	0.107904
$976$	0	0
$977$	8.24621	0.263820	0.131910	−	0.991262i	$-0.457889\pi$
$977$	8.24621	0.263820	0.131910	+	0.991262i	$0.457889\pi$
$978$	0	0
$979$	1.75379	0.0560513
$980$	0	0
$981$	−6.87689	−0.219562
$982$	0	0
$983$	−30.9309	−0.986542	−0.493271	−	0.869876i	$-0.664199\pi$
$983$	−30.9309	−0.986542	−0.493271	+	0.869876i	$0.664199\pi$
$984$	0	0
$985$	−31.8078	−1.01348
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.4233	−0.363240
$990$	0	0
$991$	−42.7386	−1.35764	−0.678819	−	0.734306i	$-0.737508\pi$
$991$	−42.7386	−1.35764	−0.678819	+	0.734306i	$0.737508\pi$
$992$	0	0
$993$	34.9309	1.10850
$994$	0	0
$995$	−56.9848	−1.80654
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	−5.12311	−0.162088

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	816.2.a.m.1.2		2
3.2	odd	2		2448.2.a.v.1.1		2
4.3	odd	2		51.2.a.b.1.1	✓	2
8.3	odd	2		3264.2.a.bl.1.1		2
8.5	even	2		3264.2.a.bg.1.1		2
12.11	even	2		153.2.a.e.1.2		2
20.3	even	4		1275.2.b.d.1174.4		4
20.7	even	4		1275.2.b.d.1174.1		4
20.19	odd	2		1275.2.a.n.1.2		2
24.5	odd	2		9792.2.a.cz.1.2		2
24.11	even	2		9792.2.a.cy.1.2		2
28.27	even	2		2499.2.a.o.1.1		2
44.43	even	2		6171.2.a.p.1.2		2
52.51	odd	2		8619.2.a.q.1.2		2
60.59	even	2		3825.2.a.s.1.1		2
68.3	even	16		867.2.h.j.757.1		16
68.7	even	16		867.2.h.j.712.4		16
68.11	even	16		867.2.h.j.733.2		16
68.15	odd	8		867.2.e.f.616.2		8
68.19	odd	8		867.2.e.f.616.1		8
68.23	even	16		867.2.h.j.733.1		16
68.27	even	16		867.2.h.j.712.3		16
68.31	even	16		867.2.h.j.757.2		16
68.39	even	16		867.2.h.j.688.4		16
68.43	odd	8		867.2.e.f.829.3		8
68.47	odd	4		867.2.d.c.577.3		4
68.55	odd	4		867.2.d.c.577.4		4
68.59	odd	8		867.2.e.f.829.4		8
68.63	even	16		867.2.h.j.688.3		16
68.67	odd	2		867.2.a.f.1.1		2
84.83	odd	2		7497.2.a.v.1.2		2
204.203	even	2		2601.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.b.1.1	✓	2	4.3	odd	2
153.2.a.e.1.2		2	12.11	even	2
816.2.a.m.1.2		2	1.1	even	1	trivial
867.2.a.f.1.1		2	68.67	odd	2
867.2.d.c.577.3		4	68.47	odd	4
867.2.d.c.577.4		4	68.55	odd	4
867.2.e.f.616.1		8	68.19	odd	8
867.2.e.f.616.2		8	68.15	odd	8
867.2.e.f.829.3		8	68.43	odd	8
867.2.e.f.829.4		8	68.59	odd	8
867.2.h.j.688.3		16	68.63	even	16
867.2.h.j.688.4		16	68.39	even	16
867.2.h.j.712.3		16	68.27	even	16
867.2.h.j.712.4		16	68.7	even	16
867.2.h.j.733.1		16	68.23	even	16
867.2.h.j.733.2		16	68.11	even	16
867.2.h.j.757.1		16	68.3	even	16
867.2.h.j.757.2		16	68.31	even	16
1275.2.a.n.1.2		2	20.19	odd	2
1275.2.b.d.1174.1		4	20.7	even	4
1275.2.b.d.1174.4		4	20.3	even	4
2448.2.a.v.1.1		2	3.2	odd	2
2499.2.a.o.1.1		2	28.27	even	2
2601.2.a.t.1.2		2	204.203	even	2
3264.2.a.bg.1.1		2	8.5	even	2
3264.2.a.bl.1.1		2	8.3	odd	2
3825.2.a.s.1.1		2	60.59	even	2
6171.2.a.p.1.2		2	44.43	even	2
7497.2.a.v.1.2		2	84.83	odd	2
8619.2.a.q.1.2		2	52.51	odd	2
9792.2.a.cy.1.2		2	24.11	even	2
9792.2.a.cz.1.2		2	24.5	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:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.56155 q^{5} +1.00000 q^{9} -1.56155 q^{11} +0.438447 q^{13} +3.56155 q^{15} +1.00000 q^{17} +4.68466 q^{19} +2.43845 q^{23} +7.68466 q^{25} +1.00000 q^{27} -8.24621 q^{29} -3.12311 q^{31} -1.56155 q^{33} -5.12311 q^{37} +0.438447 q^{39} -3.56155 q^{41} -4.68466 q^{43} +3.56155 q^{45} +11.1231 q^{47} -7.00000 q^{49} +1.00000 q^{51} +12.2462 q^{53} -5.56155 q^{55} +4.68466 q^{57} -7.12311 q^{59} +9.12311 q^{61} +1.56155 q^{65} -4.00000 q^{67} +2.43845 q^{69} +6.24621 q^{71} -12.2462 q^{73} +7.68466 q^{75} +9.36932 q^{79} +1.00000 q^{81} +0.876894 q^{83} +3.56155 q^{85} -8.24621 q^{87} -1.12311 q^{89} -3.12311 q^{93} +16.6847 q^{95} -2.87689 q^{97} -1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 3 q^{5} + 2 q^{9} + q^{11} + 5 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} + 9 q^{23} + 3 q^{25} + 2 q^{27} + 2 q^{31} + q^{33} - 2 q^{37} + 5 q^{39} - 3 q^{41} + 3 q^{43} + 3 q^{45} + 14 q^{47}+ \cdots + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 3 * q^5 + 2 * q^9 + q^11 + 5 * q^13 + 3 * q^15 + 2 * q^17 - 3 * q^19 + 9 * q^23 + 3 * q^25 + 2 * q^27 + 2 * q^31 + q^33 - 2 * q^37 + 5 * q^39 - 3 * q^41 + 3 * q^43 + 3 * q^45 + 14 * q^47 - 14 * q^49 + 2 * q^51 + 8 * q^53 - 7 * q^55 - 3 * q^57 - 6 * q^59 + 10 * q^61 - q^65 - 8 * q^67 + 9 * q^69 - 4 * q^71 - 8 * q^73 + 3 * q^75 - 6 * q^79 + 2 * q^81 + 10 * q^83 + 3 * q^85 + 6 * q^89 + 2 * q^93 + 21 * q^95 - 14 * q^97 + q^99

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