LMFDB - Embedded newform 1392.4.a.k.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1392,4,Mod(1,1392)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1392, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1392.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1392 = 2^{4} \cdot 3 \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1392.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:	$82.1306587280$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{41}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 10 $ x^2 - x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.70156$ of defining polynomial
Character	$\chi$	$=$	1392.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +9.10469 q^{5} +5.59688 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +9.10469 q^{5} +5.59688 q^{7} +9.00000 q^{9} -9.29844 q^{11} -77.1203 q^{13} +27.3141 q^{15} -61.8062 q^{17} +39.3141 q^{19} +16.7906 q^{21} +37.4031 q^{23} -42.1047 q^{25} +27.0000 q^{27} +29.0000 q^{29} -26.5969 q^{31} -27.8953 q^{33} +50.9578 q^{35} -279.105 q^{37} -231.361 q^{39} -194.570 q^{41} -309.889 q^{43} +81.9422 q^{45} -109.325 q^{47} -311.675 q^{49} -185.419 q^{51} -560.303 q^{53} -84.6594 q^{55} +117.942 q^{57} -147.461 q^{59} +278.125 q^{61} +50.3719 q^{63} -702.156 q^{65} +970.795 q^{67} +112.209 q^{69} +519.684 q^{71} -264.744 q^{73} -126.314 q^{75} -52.0422 q^{77} -875.066 q^{79} +81.0000 q^{81} -141.297 q^{83} -562.727 q^{85} +87.0000 q^{87} +970.342 q^{89} -431.633 q^{91} -79.7906 q^{93} +357.942 q^{95} -2.93122 q^{97} -83.6859 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - q^{5} + 24 q^{7} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - q^5 + 24 * q^7 + 18 * q^9	$ 2 q + 6 q^{3} - q^{5} + 24 q^{7} + 18 q^{9} - 25 q^{11} - 71 q^{13} - 3 q^{15} - 98 q^{17} + 21 q^{19} + 72 q^{21} + 62 q^{23} - 65 q^{25} + 54 q^{27} + 58 q^{29} - 66 q^{31} - 75 q^{33} - 135 q^{35} - 539 q^{37} - 213 q^{39} - 101 q^{41} + 155 q^{43} - 9 q^{45} - 526 q^{47} - 316 q^{49} - 294 q^{51} - 698 q^{53} + 74 q^{55} + 63 q^{57} - 455 q^{59} + 44 q^{61} + 216 q^{63} - 764 q^{65} + 1551 q^{67} + 186 q^{69} - 126 q^{71} - 760 q^{73} - 195 q^{75} - 341 q^{77} + 158 q^{79} + 162 q^{81} + 934 q^{83} - 197 q^{85} + 174 q^{87} - 691 q^{89} - 319 q^{91} - 198 q^{93} + 543 q^{95} + 532 q^{97} - 225 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - q^5 + 24 * q^7 + 18 * q^9 - 25 * q^11 - 71 * q^13 - 3 * q^15 - 98 * q^17 + 21 * q^19 + 72 * q^21 + 62 * q^23 - 65 * q^25 + 54 * q^27 + 58 * q^29 - 66 * q^31 - 75 * q^33 - 135 * q^35 - 539 * q^37 - 213 * q^39 - 101 * q^41 + 155 * q^43 - 9 * q^45 - 526 * q^47 - 316 * q^49 - 294 * q^51 - 698 * q^53 + 74 * q^55 + 63 * q^57 - 455 * q^59 + 44 * q^61 + 216 * q^63 - 764 * q^65 + 1551 * q^67 + 186 * q^69 - 126 * q^71 - 760 * q^73 - 195 * q^75 - 341 * q^77 + 158 * q^79 + 162 * q^81 + 934 * q^83 - 197 * q^85 + 174 * q^87 - 691 * q^89 - 319 * q^91 - 198 * q^93 + 543 * q^95 + 532 * q^97 - 225 * 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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	9.10469	0.814348	0.407174	−	0.913351i	$-0.366514\pi$
$5$	9.10469	0.814348	0.407174	+	0.913351i	$0.366514\pi$
$6$	0	0
$7$	5.59688	0.302203	0.151101	−	0.988518i	$-0.451718\pi$
$7$	5.59688	0.302203	0.151101	+	0.988518i	$0.451718\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−9.29844	−0.254871	−0.127436	−	0.991847i	$-0.540675\pi$
$11$	−9.29844	−0.254871	−0.127436	+	0.991847i	$0.540675\pi$
$12$	0	0
$13$	−77.1203	−1.64533	−0.822666	−	0.568524i	$-0.807514\pi$
$13$	−77.1203	−1.64533	−0.822666	+	0.568524i	$0.807514\pi$
$14$	0	0
$15$	27.3141	0.470164
$16$	0	0
$17$	−61.8062	−0.881777	−0.440889	−	0.897562i	$-0.645337\pi$
$17$	−61.8062	−0.881777	−0.440889	+	0.897562i	$0.645337\pi$
$18$	0	0
$19$	39.3141	0.474698	0.237349	−	0.971424i	$-0.423721\pi$
$19$	39.3141	0.474698	0.237349	+	0.971424i	$0.423721\pi$
$20$	0	0
$21$	16.7906	0.174477
$22$	0	0
$23$	37.4031	0.339091	0.169545	−	0.985522i	$-0.445770\pi$
$23$	37.4031	0.339091	0.169545	+	0.985522i	$0.445770\pi$
$24$	0	0
$25$	−42.1047	−0.336837
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−26.5969	−0.154095	−0.0770474	−	0.997027i	$-0.524549\pi$
$31$	−26.5969	−0.154095	−0.0770474	+	0.997027i	$0.524549\pi$
$32$	0	0
$33$	−27.8953	−0.147150
$34$	0	0
$35$	50.9578	0.246098
$36$	0	0
$37$	−279.105	−1.24012	−0.620061	−	0.784553i	$-0.712892\pi$
$37$	−279.105	−1.24012	−0.620061	+	0.784553i	$0.712892\pi$
$38$	0	0
$39$	−231.361	−0.949933
$40$	0	0
$41$	−194.570	−0.741141	−0.370570	−	0.928804i	$-0.620838\pi$
$41$	−194.570	−0.741141	−0.370570	+	0.928804i	$0.620838\pi$
$42$	0	0
$43$	−309.889	−1.09901	−0.549507	−	0.835489i	$-0.685185\pi$
$43$	−309.889	−1.09901	−0.549507	+	0.835489i	$0.685185\pi$
$44$	0	0
$45$	81.9422	0.271449
$46$	0	0
$47$	−109.325	−0.339291	−0.169646	−	0.985505i	$-0.554262\pi$
$47$	−109.325	−0.339291	−0.169646	+	0.985505i	$0.554262\pi$
$48$	0	0
$49$	−311.675	−0.908673
$50$	0	0
$51$	−185.419	−0.509094
$52$	0	0
$53$	−560.303	−1.45214	−0.726071	−	0.687620i	$-0.758656\pi$
$53$	−560.303	−1.45214	−0.726071	+	0.687620i	$0.758656\pi$
$54$	0	0
$55$	−84.6594	−0.207554
$56$	0	0
$57$	117.942	0.274067
$58$	0	0
$59$	−147.461	−0.325386	−0.162693	−	0.986677i	$-0.552018\pi$
$59$	−147.461	−0.325386	−0.162693	+	0.986677i	$0.552018\pi$
$60$	0	0
$61$	278.125	0.583775	0.291887	−	0.956453i	$-0.405717\pi$
$61$	278.125	0.583775	0.291887	+	0.956453i	$0.405717\pi$
$62$	0	0
$63$	50.3719	0.100734
$64$	0	0
$65$	−702.156	−1.33987
$66$	0	0
$67$	970.795	1.77017	0.885086	−	0.465428i	$-0.154099\pi$
$67$	970.795	1.77017	0.885086	+	0.465428i	$0.154099\pi$
$68$	0	0
$69$	112.209	0.195774
$70$	0	0
$71$	519.684	0.868665	0.434332	−	0.900753i	$-0.356984\pi$
$71$	519.684	0.868665	0.434332	+	0.900753i	$0.356984\pi$
$72$	0	0
$73$	−264.744	−0.424465	−0.212232	−	0.977219i	$-0.568073\pi$
$73$	−264.744	−0.424465	−0.212232	+	0.977219i	$0.568073\pi$
$74$	0	0
$75$	−126.314	−0.194473
$76$	0	0
$77$	−52.0422	−0.0770228
$78$	0	0
$79$	−875.066	−1.24623	−0.623117	−	0.782128i	$-0.714134\pi$
$79$	−875.066	−1.24623	−0.623117	+	0.782128i	$0.714134\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−141.297	−0.186860	−0.0934298	−	0.995626i	$-0.529783\pi$
$83$	−141.297	−0.186860	−0.0934298	+	0.995626i	$0.529783\pi$
$84$	0	0
$85$	−562.727	−0.718074
$86$	0	0
$87$	87.0000	0.107211
$88$	0	0
$89$	970.342	1.15569	0.577843	−	0.816148i	$-0.303895\pi$
$89$	970.342	1.15569	0.577843	+	0.816148i	$0.303895\pi$
$90$	0	0
$91$	−431.633	−0.497224
$92$	0	0
$93$	−79.7906	−0.0889667
$94$	0	0
$95$	357.942	0.386569
$96$	0	0
$97$	−2.93122	−0.00306825	−0.00153412	−	0.999999i	$-0.500488\pi$
$97$	−2.93122	−0.00306825	−0.00153412	+	0.999999i	$0.500488\pi$
$98$	0	0
$99$	−83.6859	−0.0849571
$100$	0	0
$101$	1773.40	1.74713	0.873565	−	0.486708i	$-0.161803\pi$
$101$	1773.40	1.74713	0.873565	+	0.486708i	$0.161803\pi$
$102$	0	0
$103$	−1357.23	−1.29837	−0.649184	−	0.760632i	$-0.724889\pi$
$103$	−1357.23	−1.29837	−0.649184	+	0.760632i	$0.724889\pi$
$104$	0	0
$105$	152.873	0.142085
$106$	0	0
$107$	−118.814	−0.107348	−0.0536738	−	0.998559i	$-0.517093\pi$
$107$	−118.814	−0.107348	−0.0536738	+	0.998559i	$0.517093\pi$
$108$	0	0
$109$	1033.18	0.907899	0.453949	−	0.891027i	$-0.350015\pi$
$109$	1033.18	0.907899	0.453949	+	0.891027i	$0.350015\pi$
$110$	0	0
$111$	−837.314	−0.715985
$112$	0	0
$113$	431.459	0.359188	0.179594	−	0.983741i	$-0.442522\pi$
$113$	431.459	0.359188	0.179594	+	0.983741i	$0.442522\pi$
$114$	0	0
$115$	340.544	0.276138
$116$	0	0
$117$	−694.083	−0.548444
$118$	0	0
$119$	−345.922	−0.266476
$120$	0	0
$121$	−1244.54	−0.935041
$122$	0	0
$123$	−583.711	−0.427898
$124$	0	0
$125$	−1521.44	−1.08865
$126$	0	0
$127$	1897.09	1.32551	0.662755	−	0.748836i	$-0.269387\pi$
$127$	1897.09	1.32551	0.662755	+	0.748836i	$0.269387\pi$
$128$	0	0
$129$	−929.667	−0.634516
$130$	0	0
$131$	−2027.28	−1.35209	−0.676046	−	0.736860i	$-0.736308\pi$
$131$	−2027.28	−1.35209	−0.676046	+	0.736860i	$0.736308\pi$
$132$	0	0
$133$	220.036	0.143455
$134$	0	0
$135$	245.827	0.156721
$136$	0	0
$137$	−1010.04	−0.629882	−0.314941	−	0.949111i	$-0.601985\pi$
$137$	−1010.04	−0.629882	−0.314941	+	0.949111i	$0.601985\pi$
$138$	0	0
$139$	2087.35	1.27371	0.636857	−	0.770982i	$-0.280234\pi$
$139$	2087.35	1.27371	0.636857	+	0.770982i	$0.280234\pi$
$140$	0	0
$141$	−327.975	−0.195890
$142$	0	0
$143$	717.098	0.419348
$144$	0	0
$145$	264.036	0.151221
$146$	0	0
$147$	−935.025	−0.524623
$148$	0	0
$149$	728.820	0.400720	0.200360	−	0.979722i	$-0.435789\pi$
$149$	728.820	0.400720	0.200360	+	0.979722i	$0.435789\pi$
$150$	0	0
$151$	−1045.01	−0.563189	−0.281594	−	0.959534i	$-0.590863\pi$
$151$	−1045.01	−0.563189	−0.281594	+	0.959534i	$0.590863\pi$
$152$	0	0
$153$	−556.256	−0.293926
$154$	0	0
$155$	−242.156	−0.125487
$156$	0	0
$157$	−1838.74	−0.934698	−0.467349	−	0.884073i	$-0.654791\pi$
$157$	−1838.74	−0.934698	−0.467349	+	0.884073i	$0.654791\pi$
$158$	0	0
$159$	−1680.91	−0.838395
$160$	0	0
$161$	209.341	0.102474
$162$	0	0
$163$	−1986.07	−0.954364	−0.477182	−	0.878805i	$-0.658342\pi$
$163$	−1986.07	−0.954364	−0.477182	+	0.878805i	$0.658342\pi$
$164$	0	0
$165$	−253.978	−0.119831
$166$	0	0
$167$	−184.097	−0.0853045	−0.0426523	−	0.999090i	$-0.513581\pi$
$167$	−184.097	−0.0853045	−0.0426523	+	0.999090i	$0.513581\pi$
$168$	0	0
$169$	3750.54	1.70712
$170$	0	0
$171$	353.827	0.158233
$172$	0	0
$173$	−1060.75	−0.466169	−0.233084	−	0.972457i	$-0.574882\pi$
$173$	−1060.75	−0.466169	−0.233084	+	0.972457i	$0.574882\pi$
$174$	0	0
$175$	−235.655	−0.101793
$176$	0	0
$177$	−442.383	−0.187862
$178$	0	0
$179$	−3090.07	−1.29030	−0.645148	−	0.764058i	$-0.723204\pi$
$179$	−3090.07	−1.29030	−0.645148	+	0.764058i	$0.723204\pi$
$180$	0	0
$181$	−3324.07	−1.36506	−0.682531	−	0.730857i	$-0.739121\pi$
$181$	−3324.07	−1.36506	−0.682531	+	0.730857i	$0.739121\pi$
$182$	0	0
$183$	834.375	0.337042
$184$	0	0
$185$	−2541.16	−1.00989
$186$	0	0
$187$	574.702	0.224740
$188$	0	0
$189$	151.116	0.0581590
$190$	0	0
$191$	4448.15	1.68512	0.842558	−	0.538606i	$-0.181049\pi$
$191$	4448.15	1.68512	0.842558	+	0.538606i	$0.181049\pi$
$192$	0	0
$193$	354.653	0.132272	0.0661359	−	0.997811i	$-0.478933\pi$
$193$	354.653	0.132272	0.0661359	+	0.997811i	$0.478933\pi$
$194$	0	0
$195$	−2106.47	−0.773576
$196$	0	0
$197$	−2386.43	−0.863078	−0.431539	−	0.902094i	$-0.642029\pi$
$197$	−2386.43	−0.863078	−0.431539	+	0.902094i	$0.642029\pi$
$198$	0	0
$199$	−2957.89	−1.05366	−0.526832	−	0.849970i	$-0.676620\pi$
$199$	−2957.89	−1.05366	−0.526832	+	0.849970i	$0.676620\pi$
$200$	0	0
$201$	2912.39	1.02201
$202$	0	0
$203$	162.309	0.0561177
$204$	0	0
$205$	−1771.50	−0.603547
$206$	0	0
$207$	336.628	0.113030
$208$	0	0
$209$	−365.559	−0.120987
$210$	0	0
$211$	5067.01	1.65321	0.826605	−	0.562782i	$-0.190269\pi$
$211$	5067.01	1.65321	0.826605	+	0.562782i	$0.190269\pi$
$212$	0	0
$213$	1559.05	0.501524
$214$	0	0
$215$	−2821.44	−0.894980
$216$	0	0
$217$	−148.859	−0.0465679
$218$	0	0
$219$	−794.231	−0.245065
$220$	0	0
$221$	4766.52	1.45082
$222$	0	0
$223$	3734.21	1.12135	0.560675	−	0.828036i	$-0.310542\pi$
$223$	3734.21	1.12135	0.560675	+	0.828036i	$0.310542\pi$
$224$	0	0
$225$	−378.942	−0.112279
$226$	0	0
$227$	−5393.99	−1.57714	−0.788571	−	0.614943i	$-0.789179\pi$
$227$	−5393.99	−1.57714	−0.788571	+	0.614943i	$0.789179\pi$
$228$	0	0
$229$	−908.554	−0.262179	−0.131089	−	0.991371i	$-0.541848\pi$
$229$	−908.554	−0.262179	−0.131089	+	0.991371i	$0.541848\pi$
$230$	0	0
$231$	−156.127	−0.0444692
$232$	0	0
$233$	5044.02	1.41822	0.709108	−	0.705100i	$-0.249098\pi$
$233$	5044.02	1.41822	0.709108	+	0.705100i	$0.249098\pi$
$234$	0	0
$235$	−995.370	−0.276301
$236$	0	0
$237$	−2625.20	−0.719514
$238$	0	0
$239$	−822.622	−0.222640	−0.111320	−	0.993785i	$-0.535508\pi$
$239$	−822.622	−0.222640	−0.111320	+	0.993785i	$0.535508\pi$
$240$	0	0
$241$	−957.597	−0.255951	−0.127976	−	0.991777i	$-0.540848\pi$
$241$	−957.597	−0.255951	−0.127976	+	0.991777i	$0.540848\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−2837.70	−0.739976
$246$	0	0
$247$	−3031.91	−0.781036
$248$	0	0
$249$	−423.890	−0.107883
$250$	0	0
$251$	−3408.72	−0.857197	−0.428599	−	0.903495i	$-0.640993\pi$
$251$	−3408.72	−0.857197	−0.428599	+	0.903495i	$0.640993\pi$
$252$	0	0
$253$	−347.791	−0.0864245
$254$	0	0
$255$	−1688.18	−0.414580
$256$	0	0
$257$	4903.44	1.19015	0.595075	−	0.803670i	$-0.297122\pi$
$257$	4903.44	1.19015	0.595075	+	0.803670i	$0.297122\pi$
$258$	0	0
$259$	−1562.11	−0.374769
$260$	0	0
$261$	261.000	0.0618984
$262$	0	0
$263$	−8416.48	−1.97332	−0.986658	−	0.162804i	$-0.947946\pi$
$263$	−8416.48	−1.97332	−0.986658	+	0.162804i	$0.947946\pi$
$264$	0	0
$265$	−5101.38	−1.18255
$266$	0	0
$267$	2911.03	0.667236
$268$	0	0
$269$	5282.51	1.19732	0.598662	−	0.801002i	$-0.295699\pi$
$269$	5282.51	1.19732	0.598662	+	0.801002i	$0.295699\pi$
$270$	0	0
$271$	7049.65	1.58020	0.790102	−	0.612975i	$-0.210027\pi$
$271$	7049.65	1.58020	0.790102	+	0.612975i	$0.210027\pi$
$272$	0	0
$273$	−1294.90	−0.287073
$274$	0	0
$275$	391.508	0.0858502
$276$	0	0
$277$	507.867	0.110162	0.0550808	−	0.998482i	$-0.482458\pi$
$277$	507.867	0.110162	0.0550808	+	0.998482i	$0.482458\pi$
$278$	0	0
$279$	−239.372	−0.0513649
$280$	0	0
$281$	−8776.99	−1.86331	−0.931657	−	0.363339i	$-0.881637\pi$
$281$	−8776.99	−1.86331	−0.931657	+	0.363339i	$0.881637\pi$
$282$	0	0
$283$	1061.07	0.222877	0.111438	−	0.993771i	$-0.464454\pi$
$283$	1061.07	0.222877	0.111438	+	0.993771i	$0.464454\pi$
$284$	0	0
$285$	1073.83	0.223186
$286$	0	0
$287$	−1088.99	−0.223975
$288$	0	0
$289$	−1092.99	−0.222468
$290$	0	0
$291$	−8.79365	−0.00177145
$292$	0	0
$293$	−6565.00	−1.30898	−0.654491	−	0.756070i	$-0.727117\pi$
$293$	−6565.00	−1.30898	−0.654491	+	0.756070i	$0.727117\pi$
$294$	0	0
$295$	−1342.59	−0.264977
$296$	0	0
$297$	−251.058	−0.0490500
$298$	0	0
$299$	−2884.54	−0.557917
$300$	0	0
$301$	−1734.41	−0.332125
$302$	0	0
$303$	5320.20	1.00871
$304$	0	0
$305$	2532.24	0.475396
$306$	0	0
$307$	−6787.60	−1.26185	−0.630926	−	0.775843i	$-0.717325\pi$
$307$	−6787.60	−1.26185	−0.630926	+	0.775843i	$0.717325\pi$
$308$	0	0
$309$	−4071.69	−0.749613
$310$	0	0
$311$	10607.3	1.93404	0.967019	−	0.254704i	$-0.0819781\pi$
$311$	10607.3	1.93404	0.967019	+	0.254704i	$0.0819781\pi$
$312$	0	0
$313$	2519.55	0.454994	0.227497	−	0.973779i	$-0.426946\pi$
$313$	2519.55	0.454994	0.227497	+	0.973779i	$0.426946\pi$
$314$	0	0
$315$	458.620	0.0820328
$316$	0	0
$317$	−5471.65	−0.969459	−0.484729	−	0.874664i	$-0.661082\pi$
$317$	−5471.65	−0.969459	−0.484729	+	0.874664i	$0.661082\pi$
$318$	0	0
$319$	−269.655	−0.0473284
$320$	0	0
$321$	−356.443	−0.0619772
$322$	0	0
$323$	−2429.85	−0.418578
$324$	0	0
$325$	3247.13	0.554210
$326$	0	0
$327$	3099.55	0.524176
$328$	0	0
$329$	−611.879	−0.102535
$330$	0	0
$331$	−1867.25	−0.310070	−0.155035	−	0.987909i	$-0.549549\pi$
$331$	−1867.25	−0.310070	−0.155035	+	0.987909i	$0.549549\pi$
$332$	0	0
$333$	−2511.94	−0.413374
$334$	0	0
$335$	8838.79	1.44154
$336$	0	0
$337$	−10112.3	−1.63457	−0.817285	−	0.576233i	$-0.804522\pi$
$337$	−10112.3	−1.63457	−0.817285	+	0.576233i	$0.804522\pi$
$338$	0	0
$339$	1294.38	0.207377
$340$	0	0
$341$	247.309	0.0392744
$342$	0	0
$343$	−3664.13	−0.576807
$344$	0	0
$345$	1021.63	0.159428
$346$	0	0
$347$	6032.03	0.933188	0.466594	−	0.884472i	$-0.345481\pi$
$347$	6032.03	0.933188	0.466594	+	0.884472i	$0.345481\pi$
$348$	0	0
$349$	8001.09	1.22719	0.613594	−	0.789622i	$-0.289723\pi$
$349$	8001.09	1.22719	0.613594	+	0.789622i	$0.289723\pi$
$350$	0	0
$351$	−2082.25	−0.316644
$352$	0	0
$353$	−11369.0	−1.71419	−0.857095	−	0.515159i	$-0.827733\pi$
$353$	−11369.0	−1.71419	−0.857095	+	0.515159i	$0.827733\pi$
$354$	0	0
$355$	4731.56	0.707395
$356$	0	0
$357$	−1037.77	−0.153850
$358$	0	0
$359$	−4888.10	−0.718618	−0.359309	−	0.933219i	$-0.616988\pi$
$359$	−4888.10	−0.718618	−0.359309	+	0.933219i	$0.616988\pi$
$360$	0	0
$361$	−5313.40	−0.774662
$362$	0	0
$363$	−3733.62	−0.539846
$364$	0	0
$365$	−2410.41	−0.345662
$366$	0	0
$367$	−1649.48	−0.234611	−0.117306	−	0.993096i	$-0.537426\pi$
$367$	−1649.48	−0.234611	−0.117306	+	0.993096i	$0.537426\pi$
$368$	0	0
$369$	−1751.13	−0.247047
$370$	0	0
$371$	−3135.95	−0.438842
$372$	0	0
$373$	12769.2	1.77256	0.886280	−	0.463151i	$-0.153281\pi$
$373$	12769.2	1.77256	0.886280	+	0.463151i	$0.153281\pi$
$374$	0	0
$375$	−4564.31	−0.628533
$376$	0	0
$377$	−2236.49	−0.305531
$378$	0	0
$379$	11762.9	1.59424	0.797122	−	0.603818i	$-0.206355\pi$
$379$	11762.9	1.59424	0.797122	+	0.603818i	$0.206355\pi$
$380$	0	0
$381$	5691.28	0.765284
$382$	0	0
$383$	−2282.30	−0.304491	−0.152245	−	0.988343i	$-0.548650\pi$
$383$	−2282.30	−0.304491	−0.152245	+	0.988343i	$0.548650\pi$
$384$	0	0
$385$	−473.828	−0.0627234
$386$	0	0
$387$	−2789.00	−0.366338
$388$	0	0
$389$	2444.93	0.318670	0.159335	−	0.987225i	$-0.449065\pi$
$389$	2444.93	0.318670	0.159335	+	0.987225i	$0.449065\pi$
$390$	0	0
$391$	−2311.75	−0.299003
$392$	0	0
$393$	−6081.83	−0.780630
$394$	0	0
$395$	−7967.20	−1.01487
$396$	0	0
$397$	8906.54	1.12596	0.562980	−	0.826470i	$-0.309655\pi$
$397$	8906.54	1.12596	0.562980	+	0.826470i	$0.309655\pi$
$398$	0	0
$399$	660.108	0.0828239
$400$	0	0
$401$	−9730.86	−1.21181	−0.605905	−	0.795537i	$-0.707189\pi$
$401$	−9730.86	−1.21181	−0.605905	+	0.795537i	$0.707189\pi$
$402$	0	0
$403$	2051.16	0.253537
$404$	0	0
$405$	737.480	0.0904831
$406$	0	0
$407$	2595.24	0.316072
$408$	0	0
$409$	−75.4913	−0.00912666	−0.00456333	−	0.999990i	$-0.501453\pi$
$409$	−75.4913	−0.00912666	−0.00456333	+	0.999990i	$0.501453\pi$
$410$	0	0
$411$	−3030.13	−0.363663
$412$	0	0
$413$	−825.321	−0.0983326
$414$	0	0
$415$	−1286.46	−0.152169
$416$	0	0
$417$	6262.04	0.735379
$418$	0	0
$419$	5408.69	0.630625	0.315313	−	0.948988i	$-0.397891\pi$
$419$	5408.69	0.630625	0.315313	+	0.948988i	$0.397891\pi$
$420$	0	0
$421$	14091.3	1.63127	0.815637	−	0.578564i	$-0.196387\pi$
$421$	14091.3	1.63127	0.815637	+	0.578564i	$0.196387\pi$
$422$	0	0
$423$	−983.925	−0.113097
$424$	0	0
$425$	2602.33	0.297016
$426$	0	0
$427$	1556.63	0.176418
$428$	0	0
$429$	2151.30	0.242111
$430$	0	0
$431$	14739.4	1.64727	0.823635	−	0.567121i	$-0.191943\pi$
$431$	14739.4	1.64727	0.823635	+	0.567121i	$0.191943\pi$
$432$	0	0
$433$	−3554.21	−0.394467	−0.197234	−	0.980357i	$-0.563196\pi$
$433$	−3554.21	−0.394467	−0.197234	+	0.980357i	$0.563196\pi$
$434$	0	0
$435$	792.108	0.0873073
$436$	0	0
$437$	1470.47	0.160966
$438$	0	0
$439$	13317.5	1.44786	0.723929	−	0.689875i	$-0.242334\pi$
$439$	13317.5	1.44786	0.723929	+	0.689875i	$0.242334\pi$
$440$	0	0
$441$	−2805.07	−0.302891
$442$	0	0
$443$	602.583	0.0646266	0.0323133	−	0.999478i	$-0.489713\pi$
$443$	602.583	0.0646266	0.0323133	+	0.999478i	$0.489713\pi$
$444$	0	0
$445$	8834.66	0.941130
$446$	0	0
$447$	2186.46	0.231356
$448$	0	0
$449$	−16961.3	−1.78275	−0.891376	−	0.453266i	$-0.850259\pi$
$449$	−16961.3	−1.78275	−0.891376	+	0.453266i	$0.850259\pi$
$450$	0	0
$451$	1809.20	0.188896
$452$	0	0
$453$	−3135.02	−0.325157
$454$	0	0
$455$	−3929.88	−0.404914
$456$	0	0
$457$	13470.4	1.37882	0.689410	−	0.724371i	$-0.257870\pi$
$457$	13470.4	1.37882	0.689410	+	0.724371i	$0.257870\pi$
$458$	0	0
$459$	−1668.77	−0.169698
$460$	0	0
$461$	5394.71	0.545025	0.272513	−	0.962152i	$-0.412145\pi$
$461$	5394.71	0.545025	0.272513	+	0.962152i	$0.412145\pi$
$462$	0	0
$463$	−13083.9	−1.31331	−0.656654	−	0.754192i	$-0.728029\pi$
$463$	−13083.9	−1.31331	−0.656654	+	0.754192i	$0.728029\pi$
$464$	0	0
$465$	−726.469	−0.0724498
$466$	0	0
$467$	−5391.09	−0.534197	−0.267099	−	0.963669i	$-0.586065\pi$
$467$	−5391.09	−0.534197	−0.267099	+	0.963669i	$0.586065\pi$
$468$	0	0
$469$	5433.42	0.534951
$470$	0	0
$471$	−5516.23	−0.539648
$472$	0	0
$473$	2881.48	0.280107
$474$	0	0
$475$	−1655.31	−0.159896
$476$	0	0
$477$	−5042.73	−0.484047
$478$	0	0
$479$	−15518.9	−1.48033	−0.740163	−	0.672428i	$-0.765252\pi$
$479$	−15518.9	−1.48033	−0.740163	+	0.672428i	$0.765252\pi$
$480$	0	0
$481$	21524.6	2.04041
$482$	0	0
$483$	628.022	0.0591635
$484$	0	0
$485$	−26.6878	−0.00249862
$486$	0	0
$487$	9407.95	0.875390	0.437695	−	0.899123i	$-0.355795\pi$
$487$	9407.95	0.875390	0.437695	+	0.899123i	$0.355795\pi$
$488$	0	0
$489$	−5958.22	−0.551002
$490$	0	0
$491$	705.615	0.0648553	0.0324277	−	0.999474i	$-0.489676\pi$
$491$	705.615	0.0648553	0.0324277	+	0.999474i	$0.489676\pi$
$492$	0	0
$493$	−1792.38	−0.163742
$494$	0	0
$495$	−761.934	−0.0691846
$496$	0	0
$497$	2908.61	0.262513
$498$	0	0
$499$	−9363.34	−0.840001	−0.420000	−	0.907524i	$-0.637970\pi$
$499$	−9363.34	−0.840001	−0.420000	+	0.907524i	$0.637970\pi$
$500$	0	0
$501$	−552.291	−0.0492506
$502$	0	0
$503$	−2275.13	−0.201676	−0.100838	−	0.994903i	$-0.532152\pi$
$503$	−2275.13	−0.201676	−0.100838	+	0.994903i	$0.532152\pi$
$504$	0	0
$505$	16146.3	1.42277
$506$	0	0
$507$	11251.6	0.985606
$508$	0	0
$509$	2648.37	0.230623	0.115311	−	0.993329i	$-0.463213\pi$
$509$	2648.37	0.230623	0.115311	+	0.993329i	$0.463213\pi$
$510$	0	0
$511$	−1481.74	−0.128274
$512$	0	0
$513$	1061.48	0.0913557
$514$	0	0
$515$	−12357.2	−1.05732
$516$	0	0
$517$	1016.55	0.0864756
$518$	0	0
$519$	−3182.25	−0.269143
$520$	0	0
$521$	−10097.1	−0.849061	−0.424530	−	0.905414i	$-0.639561\pi$
$521$	−10097.1	−0.849061	−0.424530	+	0.905414i	$0.639561\pi$
$522$	0	0
$523$	8221.06	0.687346	0.343673	−	0.939089i	$-0.388329\pi$
$523$	8221.06	0.687346	0.343673	+	0.939089i	$0.388329\pi$
$524$	0	0
$525$	−706.964	−0.0587704
$526$	0	0
$527$	1643.85	0.135877
$528$	0	0
$529$	−10768.0	−0.885017
$530$	0	0
$531$	−1327.15	−0.108462
$532$	0	0
$533$	15005.3	1.21942
$534$	0	0
$535$	−1081.77	−0.0874183
$536$	0	0
$537$	−9270.22	−0.744952
$538$	0	0
$539$	2898.09	0.231595
$540$	0	0
$541$	22669.4	1.80154	0.900770	−	0.434297i	$-0.143003\pi$
$541$	22669.4	1.80154	0.900770	+	0.434297i	$0.143003\pi$
$542$	0	0
$543$	−9972.21	−0.788119
$544$	0	0
$545$	9406.81	0.739345
$546$	0	0
$547$	9705.61	0.758650	0.379325	−	0.925263i	$-0.376156\pi$
$547$	9705.61	0.758650	0.379325	+	0.925263i	$0.376156\pi$
$548$	0	0
$549$	2503.12	0.194592
$550$	0	0
$551$	1140.11	0.0881492
$552$	0	0
$553$	−4897.63	−0.376616
$554$	0	0
$555$	−7623.48	−0.583061
$556$	0	0
$557$	4989.90	0.379585	0.189792	−	0.981824i	$-0.439219\pi$
$557$	4989.90	0.379585	0.189792	+	0.981824i	$0.439219\pi$
$558$	0	0
$559$	23898.7	1.80824
$560$	0	0
$561$	1724.10	0.129754
$562$	0	0
$563$	2332.02	0.174570	0.0872852	−	0.996183i	$-0.472181\pi$
$563$	2332.02	0.174570	0.0872852	+	0.996183i	$0.472181\pi$
$564$	0	0
$565$	3928.30	0.292504
$566$	0	0
$567$	453.347	0.0335781
$568$	0	0
$569$	−23481.5	−1.73004	−0.865022	−	0.501733i	$-0.832696\pi$
$569$	−23481.5	−1.73004	−0.865022	+	0.501733i	$0.832696\pi$
$570$	0	0
$571$	−7973.60	−0.584387	−0.292193	−	0.956359i	$-0.594385\pi$
$571$	−7973.60	−0.584387	−0.292193	+	0.956359i	$0.594385\pi$
$572$	0	0
$573$	13344.5	0.972902
$574$	0	0
$575$	−1574.85	−0.114219
$576$	0	0
$577$	−15243.9	−1.09985	−0.549924	−	0.835215i	$-0.685343\pi$
$577$	−15243.9	−1.09985	−0.549924	+	0.835215i	$0.685343\pi$
$578$	0	0
$579$	1063.96	0.0763672
$580$	0	0
$581$	−790.821	−0.0564695
$582$	0	0
$583$	5209.94	0.370109
$584$	0	0
$585$	−6319.41	−0.446624
$586$	0	0
$587$	−5044.41	−0.354693	−0.177347	−	0.984148i	$-0.556751\pi$
$587$	−5044.41	−0.354693	−0.177347	+	0.984148i	$0.556751\pi$
$588$	0	0
$589$	−1045.63	−0.0731485
$590$	0	0
$591$	−7159.30	−0.498298
$592$	0	0
$593$	−89.3318	−0.00618620	−0.00309310	−	0.999995i	$-0.500985\pi$
$593$	−89.3318	−0.00618620	−0.00309310	+	0.999995i	$0.500985\pi$
$594$	0	0
$595$	−3149.51	−0.217004
$596$	0	0
$597$	−8873.66	−0.608333
$598$	0	0
$599$	−7097.18	−0.484112	−0.242056	−	0.970262i	$-0.577822\pi$
$599$	−7097.18	−0.484112	−0.242056	+	0.970262i	$0.577822\pi$
$600$	0	0
$601$	6290.97	0.426978	0.213489	−	0.976945i	$-0.431517\pi$
$601$	6290.97	0.426978	0.213489	+	0.976945i	$0.431517\pi$
$602$	0	0
$603$	8737.16	0.590057
$604$	0	0
$605$	−11331.1	−0.761448
$606$	0	0
$607$	18590.2	1.24309	0.621544	−	0.783379i	$-0.286506\pi$
$607$	18590.2	1.24309	0.621544	+	0.783379i	$0.286506\pi$
$608$	0	0
$609$	486.928	0.0323996
$610$	0	0
$611$	8431.18	0.558247
$612$	0	0
$613$	−17971.9	−1.18414	−0.592069	−	0.805887i	$-0.701689\pi$
$613$	−17971.9	−1.18414	−0.592069	+	0.805887i	$0.701689\pi$
$614$	0	0
$615$	−5314.50	−0.348458
$616$	0	0
$617$	−1586.93	−0.103545	−0.0517726	−	0.998659i	$-0.516487\pi$
$617$	−1586.93	−0.103545	−0.0517726	+	0.998659i	$0.516487\pi$
$618$	0	0
$619$	15260.6	0.990911	0.495455	−	0.868633i	$-0.335001\pi$
$619$	15260.6	0.990911	0.495455	+	0.868633i	$0.335001\pi$
$620$	0	0
$621$	1009.88	0.0652581
$622$	0	0
$623$	5430.88	0.349252
$624$	0	0
$625$	−8589.11	−0.549703
$626$	0	0
$627$	−1096.68	−0.0698518
$628$	0	0
$629$	17250.4	1.09351
$630$	0	0
$631$	8338.07	0.526043	0.263022	−	0.964790i	$-0.415281\pi$
$631$	8338.07	0.526043	0.263022	+	0.964790i	$0.415281\pi$
$632$	0	0
$633$	15201.0	0.954481
$634$	0	0
$635$	17272.4	1.07943
$636$	0	0
$637$	24036.5	1.49507
$638$	0	0
$639$	4677.16	0.289555
$640$	0	0
$641$	−1971.38	−0.121474	−0.0607370	−	0.998154i	$-0.519345\pi$
$641$	−1971.38	−0.121474	−0.0607370	+	0.998154i	$0.519345\pi$
$642$	0	0
$643$	−22321.5	−1.36901	−0.684507	−	0.729007i	$-0.739982\pi$
$643$	−22321.5	−1.36901	−0.684507	+	0.729007i	$0.739982\pi$
$644$	0	0
$645$	−8464.33	−0.516717
$646$	0	0
$647$	28115.8	1.70842	0.854210	−	0.519928i	$-0.174041\pi$
$647$	28115.8	1.70842	0.854210	+	0.519928i	$0.174041\pi$
$648$	0	0
$649$	1371.16	0.0829316
$650$	0	0
$651$	−446.578	−0.0268860
$652$	0	0
$653$	14472.2	0.867289	0.433644	−	0.901084i	$-0.357227\pi$
$653$	14472.2	0.867289	0.433644	+	0.901084i	$0.357227\pi$
$654$	0	0
$655$	−18457.7	−1.10107
$656$	0	0
$657$	−2382.69	−0.141488
$658$	0	0
$659$	6799.54	0.401931	0.200965	−	0.979598i	$-0.435592\pi$
$659$	6799.54	0.401931	0.200965	+	0.979598i	$0.435592\pi$
$660$	0	0
$661$	5028.70	0.295906	0.147953	−	0.988994i	$-0.452732\pi$
$661$	5028.70	0.295906	0.147953	+	0.988994i	$0.452732\pi$
$662$	0	0
$663$	14299.6	0.837630
$664$	0	0
$665$	2003.36	0.116822
$666$	0	0
$667$	1084.69	0.0629676
$668$	0	0
$669$	11202.6	0.647412
$670$	0	0
$671$	−2586.13	−0.148787
$672$	0	0
$673$	6053.04	0.346698	0.173349	−	0.984860i	$-0.444541\pi$
$673$	6053.04	0.346698	0.173349	+	0.984860i	$0.444541\pi$
$674$	0	0
$675$	−1136.83	−0.0648244
$676$	0	0
$677$	8724.62	0.495294	0.247647	−	0.968850i	$-0.420343\pi$
$677$	8724.62	0.495294	0.247647	+	0.968850i	$0.420343\pi$
$678$	0	0
$679$	−16.4057	−0.000927234 0
$680$	0	0
$681$	−16182.0	−0.910564
$682$	0	0
$683$	4392.89	0.246104	0.123052	−	0.992400i	$-0.460732\pi$
$683$	4392.89	0.246104	0.123052	+	0.992400i	$0.460732\pi$
$684$	0	0
$685$	−9196.13	−0.512943
$686$	0	0
$687$	−2725.66	−0.151369
$688$	0	0
$689$	43210.7	2.38926
$690$	0	0
$691$	22695.9	1.24948	0.624741	−	0.780832i	$-0.285205\pi$
$691$	22695.9	1.24948	0.624741	+	0.780832i	$0.285205\pi$
$692$	0	0
$693$	−468.380	−0.0256743
$694$	0	0
$695$	19004.6	1.03725
$696$	0	0
$697$	12025.7	0.653521
$698$	0	0
$699$	15132.0	0.818808
$700$	0	0
$701$	−7055.27	−0.380134	−0.190067	−	0.981771i	$-0.560871\pi$
$701$	−7055.27	−0.380134	−0.190067	+	0.981771i	$0.560871\pi$
$702$	0	0
$703$	−10972.7	−0.588684
$704$	0	0
$705$	−2986.11	−0.159523
$706$	0	0
$707$	9925.51	0.527987
$708$	0	0
$709$	12058.3	0.638727	0.319363	−	0.947632i	$-0.396531\pi$
$709$	12058.3	0.638727	0.319363	+	0.947632i	$0.396531\pi$
$710$	0	0
$711$	−7875.59	−0.415412
$712$	0	0
$713$	−994.806	−0.0522522
$714$	0	0
$715$	6528.96	0.341495
$716$	0	0
$717$	−2467.86	−0.128541
$718$	0	0
$719$	−25400.6	−1.31750	−0.658750	−	0.752362i	$-0.728914\pi$
$719$	−25400.6	−1.31750	−0.658750	+	0.752362i	$0.728914\pi$
$720$	0	0
$721$	−7596.25	−0.392370
$722$	0	0
$723$	−2872.79	−0.147773
$724$	0	0
$725$	−1221.04	−0.0625492
$726$	0	0
$727$	797.068	0.0406625	0.0203312	−	0.999793i	$-0.493528\pi$
$727$	797.068	0.0406625	0.0203312	+	0.999793i	$0.493528\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	19153.1	0.969086
$732$	0	0
$733$	7975.10	0.401865	0.200933	−	0.979605i	$-0.435603\pi$
$733$	7975.10	0.401865	0.200933	+	0.979605i	$0.435603\pi$
$734$	0	0
$735$	−8513.11	−0.427226
$736$	0	0
$737$	−9026.88	−0.451166
$738$	0	0
$739$	17739.6	0.883035	0.441517	−	0.897253i	$-0.354440\pi$
$739$	17739.6	0.883035	0.441517	+	0.897253i	$0.354440\pi$
$740$	0	0
$741$	−9095.74	−0.450932
$742$	0	0
$743$	−2085.57	−0.102977	−0.0514887	−	0.998674i	$-0.516397\pi$
$743$	−2085.57	−0.102977	−0.0514887	+	0.998674i	$0.516397\pi$
$744$	0	0
$745$	6635.68	0.326325
$746$	0	0
$747$	−1271.67	−0.0622865
$748$	0	0
$749$	−664.988	−0.0324408
$750$	0	0
$751$	34644.3	1.68334	0.841669	−	0.539994i	$-0.181573\pi$
$751$	34644.3	1.68334	0.841669	+	0.539994i	$0.181573\pi$
$752$	0	0
$753$	−10226.2	−0.494903
$754$	0	0
$755$	−9514.47	−0.458632
$756$	0	0
$757$	−11557.8	−0.554923	−0.277461	−	0.960737i	$-0.589493\pi$
$757$	−11557.8	−0.554923	−0.277461	+	0.960737i	$0.589493\pi$
$758$	0	0
$759$	−1043.37	−0.0498972
$760$	0	0
$761$	29295.6	1.39548	0.697742	−	0.716349i	$-0.254188\pi$
$761$	29295.6	1.39548	0.697742	+	0.716349i	$0.254188\pi$
$762$	0	0
$763$	5782.60	0.274370
$764$	0	0
$765$	−5064.54	−0.239358
$766$	0	0
$767$	11372.2	0.535368
$768$	0	0
$769$	−5777.96	−0.270948	−0.135474	−	0.990781i	$-0.543256\pi$
$769$	−5777.96	−0.270948	−0.135474	+	0.990781i	$0.543256\pi$
$770$	0	0
$771$	14710.3	0.687133
$772$	0	0
$773$	10733.4	0.499424	0.249712	−	0.968320i	$-0.419664\pi$
$773$	10733.4	0.499424	0.249712	+	0.968320i	$0.419664\pi$
$774$	0	0
$775$	1119.85	0.0519049
$776$	0	0
$777$	−4686.34	−0.216373
$778$	0	0
$779$	−7649.35	−0.351818
$780$	0	0
$781$	−4832.25	−0.221398
$782$	0	0
$783$	783.000	0.0357371
$784$	0	0
$785$	−16741.2	−0.761169
$786$	0	0
$787$	7268.94	0.329237	0.164619	−	0.986357i	$-0.447361\pi$
$787$	7268.94	0.329237	0.164619	+	0.986357i	$0.447361\pi$
$788$	0	0
$789$	−25249.4	−1.13929
$790$	0	0
$791$	2414.82	0.108548
$792$	0	0
$793$	−21449.1	−0.960504
$794$	0	0
$795$	−15304.2	−0.682745
$796$	0	0
$797$	7518.00	0.334130	0.167065	−	0.985946i	$-0.446571\pi$
$797$	7518.00	0.334130	0.167065	+	0.985946i	$0.446571\pi$
$798$	0	0
$799$	6756.97	0.299179
$800$	0	0
$801$	8733.08	0.385229
$802$	0	0
$803$	2461.70	0.108184
$804$	0	0
$805$	1905.98	0.0834497
$806$	0	0
$807$	15847.5	0.691275
$808$	0	0
$809$	−41676.1	−1.81119	−0.905595	−	0.424143i	$-0.860576\pi$
$809$	−41676.1	−1.81119	−0.905595	+	0.424143i	$0.860576\pi$
$810$	0	0
$811$	15200.3	0.658144	0.329072	−	0.944305i	$-0.393264\pi$
$811$	15200.3	0.658144	0.329072	+	0.944305i	$0.393264\pi$
$812$	0	0
$813$	21148.9	0.912332
$814$	0	0
$815$	−18082.6	−0.777184
$816$	0	0
$817$	−12183.0	−0.521700
$818$	0	0
$819$	−3884.70	−0.165741
$820$	0	0
$821$	−10265.1	−0.436362	−0.218181	−	0.975908i	$-0.570012\pi$
$821$	−10265.1	−0.436362	−0.218181	+	0.975908i	$0.570012\pi$
$822$	0	0
$823$	6168.86	0.261280	0.130640	−	0.991430i	$-0.458297\pi$
$823$	6168.86	0.261280	0.130640	+	0.991430i	$0.458297\pi$
$824$	0	0
$825$	1174.52	0.0495656
$826$	0	0
$827$	−9610.39	−0.404094	−0.202047	−	0.979376i	$-0.564759\pi$
$827$	−9610.39	−0.404094	−0.202047	+	0.979376i	$0.564759\pi$
$828$	0	0
$829$	12791.4	0.535902	0.267951	−	0.963433i	$-0.413653\pi$
$829$	12791.4	0.535902	0.267951	+	0.963433i	$0.413653\pi$
$830$	0	0
$831$	1523.60	0.0636018
$832$	0	0
$833$	19263.5	0.801248
$834$	0	0
$835$	−1676.15	−0.0694676
$836$	0	0
$837$	−718.116	−0.0296556
$838$	0	0
$839$	−37025.1	−1.52354	−0.761769	−	0.647848i	$-0.775669\pi$
$839$	−37025.1	−1.52354	−0.761769	+	0.647848i	$0.775669\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	−26331.0	−1.07578
$844$	0	0
$845$	34147.5	1.39019
$846$	0	0
$847$	−6965.53	−0.282572
$848$	0	0
$849$	3183.21	0.128678
$850$	0	0
$851$	−10439.4	−0.420514
$852$	0	0
$853$	37893.8	1.52105	0.760527	−	0.649306i	$-0.224941\pi$
$853$	37893.8	1.52105	0.760527	+	0.649306i	$0.224941\pi$
$854$	0	0
$855$	3221.48	0.128856
$856$	0	0
$857$	2766.91	0.110287	0.0551434	−	0.998478i	$-0.482438\pi$
$857$	2766.91	0.110287	0.0551434	+	0.998478i	$0.482438\pi$
$858$	0	0
$859$	34378.3	1.36551	0.682754	−	0.730648i	$-0.260782\pi$
$859$	34378.3	1.36551	0.682754	+	0.730648i	$0.260782\pi$
$860$	0	0
$861$	−3266.96	−0.129312
$862$	0	0
$863$	17326.9	0.683448	0.341724	−	0.939800i	$-0.388989\pi$
$863$	17326.9	0.683448	0.341724	+	0.939800i	$0.388989\pi$
$864$	0	0
$865$	−9657.78	−0.379624
$866$	0	0
$867$	−3278.96	−0.128442
$868$	0	0
$869$	8136.74	0.317630
$870$	0	0
$871$	−74868.0	−2.91252
$872$	0	0
$873$	−26.3810	−0.00102275
$874$	0	0
$875$	−8515.29	−0.328993
$876$	0	0
$877$	−14753.1	−0.568046	−0.284023	−	0.958817i	$-0.591669\pi$
$877$	−14753.1	−0.568046	−0.284023	+	0.958817i	$0.591669\pi$
$878$	0	0
$879$	−19695.0	−0.755741
$880$	0	0
$881$	−6902.00	−0.263943	−0.131972	−	0.991253i	$-0.542131\pi$
$881$	−6902.00	−0.263943	−0.131972	+	0.991253i	$0.542131\pi$
$882$	0	0
$883$	25055.7	0.954915	0.477458	−	0.878655i	$-0.341558\pi$
$883$	25055.7	0.954915	0.477458	+	0.878655i	$0.341558\pi$
$884$	0	0
$885$	−4027.76	−0.152985
$886$	0	0
$887$	−1065.02	−0.0403157	−0.0201579	−	0.999797i	$-0.506417\pi$
$887$	−1065.02	−0.0403157	−0.0201579	+	0.999797i	$0.506417\pi$
$888$	0	0
$889$	10617.8	0.400573
$890$	0	0
$891$	−753.173	−0.0283190
$892$	0	0
$893$	−4298.01	−0.161061
$894$	0	0
$895$	−28134.1	−1.05075
$896$	0	0
$897$	−8653.62	−0.322114
$898$	0	0
$899$	−771.309	−0.0286147
$900$	0	0
$901$	34630.2	1.28047
$902$	0	0
$903$	−5203.23	−0.191753
$904$	0	0
$905$	−30264.6	−1.11164
$906$	0	0
$907$	30648.4	1.12201	0.561005	−	0.827812i	$-0.310415\pi$
$907$	30648.4	1.12201	0.561005	+	0.827812i	$0.310415\pi$
$908$	0	0
$909$	15960.6	0.582376
$910$	0	0
$911$	30520.2	1.10997	0.554983	−	0.831861i	$-0.312725\pi$
$911$	30520.2	1.10997	0.554983	+	0.831861i	$0.312725\pi$
$912$	0	0
$913$	1313.84	0.0476251
$914$	0	0
$915$	7596.72	0.274470
$916$	0	0
$917$	−11346.4	−0.408606
$918$	0	0
$919$	44926.8	1.61262	0.806310	−	0.591493i	$-0.201461\pi$
$919$	44926.8	1.61262	0.806310	+	0.591493i	$0.201461\pi$
$920$	0	0
$921$	−20362.8	−0.728531
$922$	0	0
$923$	−40078.2	−1.42924
$924$	0	0
$925$	11751.6	0.417720
$926$	0	0
$927$	−12215.1	−0.432789
$928$	0	0
$929$	5122.97	0.180925	0.0904624	−	0.995900i	$-0.471166\pi$
$929$	5122.97	0.180925	0.0904624	+	0.995900i	$0.471166\pi$
$930$	0	0
$931$	−12253.2	−0.431346
$932$	0	0
$933$	31822.0	1.11662
$934$	0	0
$935$	5232.48	0.183016
$936$	0	0
$937$	30750.7	1.07212	0.536062	−	0.844178i	$-0.319911\pi$
$937$	30750.7	1.07212	0.536062	+	0.844178i	$0.319911\pi$
$938$	0	0
$939$	7558.64	0.262691
$940$	0	0
$941$	21790.9	0.754902	0.377451	−	0.926030i	$-0.376801\pi$
$941$	21790.9	0.754902	0.377451	+	0.926030i	$0.376801\pi$
$942$	0	0
$943$	−7277.54	−0.251314
$944$	0	0
$945$	1375.86	0.0473616
$946$	0	0
$947$	−30011.6	−1.02983	−0.514914	−	0.857242i	$-0.672176\pi$
$947$	−30011.6	−1.02983	−0.514914	+	0.857242i	$0.672176\pi$
$948$	0	0
$949$	20417.1	0.698385
$950$	0	0
$951$	−16414.9	−0.559717
$952$	0	0
$953$	−9552.71	−0.324704	−0.162352	−	0.986733i	$-0.551908\pi$
$953$	−9552.71	−0.324704	−0.162352	+	0.986733i	$0.551908\pi$
$954$	0	0
$955$	40499.1	1.37227
$956$	0	0
$957$	−808.964	−0.0273251
$958$	0	0
$959$	−5653.09	−0.190352
$960$	0	0
$961$	−29083.6	−0.976255
$962$	0	0
$963$	−1069.33	−0.0357826
$964$	0	0
$965$	3229.00	0.107715
$966$	0	0
$967$	−310.364	−0.0103212	−0.00516062	−	0.999987i	$-0.501643\pi$
$967$	−310.364	−0.0103212	−0.00516062	+	0.999987i	$0.501643\pi$
$968$	0	0
$969$	−7289.56	−0.241666
$970$	0	0
$971$	−42733.1	−1.41233	−0.706164	−	0.708048i	$-0.749576\pi$
$971$	−42733.1	−1.41233	−0.706164	+	0.708048i	$0.749576\pi$
$972$	0	0
$973$	11682.6	0.384920
$974$	0	0
$975$	9741.38	0.319973
$976$	0	0
$977$	−46297.0	−1.51604	−0.758021	−	0.652230i	$-0.773834\pi$
$977$	−46297.0	−1.51604	−0.758021	+	0.652230i	$0.773834\pi$
$978$	0	0
$979$	−9022.67	−0.294551
$980$	0	0
$981$	9298.65	0.302633
$982$	0	0
$983$	50156.9	1.62742	0.813711	−	0.581270i	$-0.197444\pi$
$983$	50156.9	1.62742	0.813711	+	0.581270i	$0.197444\pi$
$984$	0	0
$985$	−21727.7	−0.702845
$986$	0	0
$987$	−1835.64	−0.0591985
$988$	0	0
$989$	−11590.8	−0.372666
$990$	0	0
$991$	−48947.5	−1.56899	−0.784495	−	0.620135i	$-0.787078\pi$
$991$	−48947.5	−1.56899	−0.784495	+	0.620135i	$0.787078\pi$
$992$	0	0
$993$	−5601.74	−0.179019
$994$	0	0
$995$	−26930.6	−0.858048
$996$	0	0
$997$	−27637.1	−0.877911	−0.438956	−	0.898509i	$-0.644651\pi$
$997$	−27637.1	−0.877911	−0.438956	+	0.898509i	$0.644651\pi$
$998$	0	0
$999$	−7535.83	−0.238662

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1392.4.a.k.1.2		2
4.3	odd	2		87.4.a.b.1.1	✓	2
12.11	even	2		261.4.a.a.1.2		2
20.19	odd	2		2175.4.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.4.a.b.1.1	✓	2	4.3	odd	2
261.4.a.a.1.2		2	12.11	even	2
1392.4.a.k.1.2		2	1.1	even	1	trivial
2175.4.a.f.1.2		2	20.19	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:	\( N \)	\(=\)	\( 1392 = 2^{4} \cdot 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1392.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +9.10469 q^{5} +5.59688 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +9.10469 q^{5} +5.59688 q^{7} +9.00000 q^{9} -9.29844 q^{11} -77.1203 q^{13} +27.3141 q^{15} -61.8062 q^{17} +39.3141 q^{19} +16.7906 q^{21} +37.4031 q^{23} -42.1047 q^{25} +27.0000 q^{27} +29.0000 q^{29} -26.5969 q^{31} -27.8953 q^{33} +50.9578 q^{35} -279.105 q^{37} -231.361 q^{39} -194.570 q^{41} -309.889 q^{43} +81.9422 q^{45} -109.325 q^{47} -311.675 q^{49} -185.419 q^{51} -560.303 q^{53} -84.6594 q^{55} +117.942 q^{57} -147.461 q^{59} +278.125 q^{61} +50.3719 q^{63} -702.156 q^{65} +970.795 q^{67} +112.209 q^{69} +519.684 q^{71} -264.744 q^{73} -126.314 q^{75} -52.0422 q^{77} -875.066 q^{79} +81.0000 q^{81} -141.297 q^{83} -562.727 q^{85} +87.0000 q^{87} +970.342 q^{89} -431.633 q^{91} -79.7906 q^{93} +357.942 q^{95} -2.93122 q^{97} -83.6859 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - q^{5} + 24 q^{7} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - q^5 + 24 * q^7 + 18 * q^9	\( 2 q + 6 q^{3} - q^{5} + 24 q^{7} + 18 q^{9} - 25 q^{11} - 71 q^{13} - 3 q^{15} - 98 q^{17} + 21 q^{19} + 72 q^{21} + 62 q^{23} - 65 q^{25} + 54 q^{27} + 58 q^{29} - 66 q^{31} - 75 q^{33} - 135 q^{35} - 539 q^{37} - 213 q^{39} - 101 q^{41} + 155 q^{43} - 9 q^{45} - 526 q^{47} - 316 q^{49} - 294 q^{51} - 698 q^{53} + 74 q^{55} + 63 q^{57} - 455 q^{59} + 44 q^{61} + 216 q^{63} - 764 q^{65} + 1551 q^{67} + 186 q^{69} - 126 q^{71} - 760 q^{73} - 195 q^{75} - 341 q^{77} + 158 q^{79} + 162 q^{81} + 934 q^{83} - 197 q^{85} + 174 q^{87} - 691 q^{89} - 319 q^{91} - 198 q^{93} + 543 q^{95} + 532 q^{97} - 225 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - q^5 + 24 * q^7 + 18 * q^9 - 25 * q^11 - 71 * q^13 - 3 * q^15 - 98 * q^17 + 21 * q^19 + 72 * q^21 + 62 * q^23 - 65 * q^25 + 54 * q^27 + 58 * q^29 - 66 * q^31 - 75 * q^33 - 135 * q^35 - 539 * q^37 - 213 * q^39 - 101 * q^41 + 155 * q^43 - 9 * q^45 - 526 * q^47 - 316 * q^49 - 294 * q^51 - 698 * q^53 + 74 * q^55 + 63 * q^57 - 455 * q^59 + 44 * q^61 + 216 * q^63 - 764 * q^65 + 1551 * q^67 + 186 * q^69 - 126 * q^71 - 760 * q^73 - 195 * q^75 - 341 * q^77 + 158 * q^79 + 162 * q^81 + 934 * q^83 - 197 * q^85 + 174 * q^87 - 691 * q^89 - 319 * q^91 - 198 * q^93 + 543 * q^95 + 532 * q^97 - 225 * q^99

Introduction

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