LMFDB - Embedded newform 2352.4.a.bv.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2352,4,Mod(1,2352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2352, 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("2352.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2352 = 2^{4} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2352.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:	$138.772492334$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{177}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 44 $ x^2 - x - 44
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.15207$ of defining polynomial
Character	$\chi$	$=$	2352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +6.30413 q^{5} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +6.30413 q^{5} +9.00000 q^{9} -48.9124 q^{11} +2.60827 q^{13} +18.9124 q^{15} -136.737 q^{17} +45.2165 q^{19} +38.1289 q^{23} -85.2579 q^{25} +27.0000 q^{27} +52.7835 q^{29} -14.7835 q^{31} -146.737 q^{33} +333.908 q^{37} +7.82481 q^{39} -227.263 q^{41} +398.433 q^{43} +56.7372 q^{45} -184.608 q^{47} -410.212 q^{51} +359.825 q^{53} -308.350 q^{55} +135.650 q^{57} +99.9075 q^{59} +674.516 q^{61} +16.4429 q^{65} +376.959 q^{67} +114.387 q^{69} +1187.60 q^{71} +735.825 q^{73} -255.774 q^{75} +836.774 q^{79} +81.0000 q^{81} +293.732 q^{83} -862.010 q^{85} +158.350 q^{87} -1298.89 q^{89} -44.3504 q^{93} +285.051 q^{95} +201.041 q^{97} -440.212 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{3} - 14 q^{5} + 18 q^{9}+O(q^{10}) $ 2 * q + 6 * q^3 - 14 * q^5 + 18 * q^9	$ 2 q + 6 q^{3} - 14 q^{5} + 18 q^{9} - 18 q^{11} - 48 q^{13} - 42 q^{15} - 34 q^{17} - 16 q^{19} - 110 q^{23} + 202 q^{25} + 54 q^{27} + 212 q^{29} - 136 q^{31} - 54 q^{33} - 24 q^{37} - 144 q^{39} - 694 q^{41} + 584 q^{43} - 126 q^{45} - 316 q^{47} - 102 q^{51} + 560 q^{53} - 936 q^{55} - 48 q^{57} - 492 q^{59} + 604 q^{61} + 1044 q^{65} + 1020 q^{67} - 330 q^{69} + 1710 q^{71} + 1312 q^{73} + 606 q^{75} + 556 q^{79} + 162 q^{81} - 264 q^{83} - 2948 q^{85} + 636 q^{87} - 70 q^{89} - 408 q^{93} + 1528 q^{95} + 136 q^{97} - 162 q^{99}+O(q^{100}) $ 2 * q + 6 * q^3 - 14 * q^5 + 18 * q^9 - 18 * q^11 - 48 * q^13 - 42 * q^15 - 34 * q^17 - 16 * q^19 - 110 * q^23 + 202 * q^25 + 54 * q^27 + 212 * q^29 - 136 * q^31 - 54 * q^33 - 24 * q^37 - 144 * q^39 - 694 * q^41 + 584 * q^43 - 126 * q^45 - 316 * q^47 - 102 * q^51 + 560 * q^53 - 936 * q^55 - 48 * q^57 - 492 * q^59 + 604 * q^61 + 1044 * q^65 + 1020 * q^67 - 330 * q^69 + 1710 * q^71 + 1312 * q^73 + 606 * q^75 + 556 * q^79 + 162 * q^81 - 264 * q^83 - 2948 * q^85 + 636 * q^87 - 70 * q^89 - 408 * q^93 + 1528 * q^95 + 136 * q^97 - 162 * 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
$5$	6.30413	0.563859	0.281929	−	0.959435i	$-0.409026\pi$
$5$	6.30413	0.563859	0.281929	+	0.959435i	$0.409026\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−48.9124	−1.34069	−0.670347	−	0.742047i	$-0.733855\pi$
$11$	−48.9124	−1.34069	−0.670347	+	0.742047i	$0.733855\pi$
$12$	0	0
$13$	2.60827	0.0556464	0.0278232	−	0.999613i	$-0.491142\pi$
$13$	2.60827	0.0556464	0.0278232	+	0.999613i	$0.491142\pi$
$14$	0	0
$15$	18.9124	0.325544
$16$	0	0
$17$	−136.737	−1.95080	−0.975401	−	0.220436i	$-0.929252\pi$
$17$	−136.737	−1.95080	−0.975401	+	0.220436i	$0.929252\pi$
$18$	0	0
$19$	45.2165	0.545968	0.272984	−	0.962019i	$-0.411989\pi$
$19$	45.2165	0.545968	0.272984	+	0.962019i	$0.411989\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	38.1289	0.345671	0.172836	−	0.984951i	$-0.444707\pi$
$23$	38.1289	0.345671	0.172836	+	0.984951i	$0.444707\pi$
$24$	0	0
$25$	−85.2579	−0.682063
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	52.7835	0.337988	0.168994	−	0.985617i	$-0.445948\pi$
$29$	52.7835	0.337988	0.168994	+	0.985617i	$0.445948\pi$
$30$	0	0
$31$	−14.7835	−0.0856512	−0.0428256	−	0.999083i	$-0.513636\pi$
$31$	−14.7835	−0.0856512	−0.0428256	+	0.999083i	$0.513636\pi$
$32$	0	0
$33$	−146.737	−0.774051
$34$	0	0
$35$	0	0
$36$	0	0
$37$	333.908	1.48362	0.741812	−	0.670608i	$-0.233967\pi$
$37$	333.908	1.48362	0.741812	+	0.670608i	$0.233967\pi$
$38$	0	0
$39$	7.82481	0.0321275
$40$	0	0
$41$	−227.263	−0.865670	−0.432835	−	0.901473i	$-0.642487\pi$
$41$	−227.263	−0.865670	−0.432835	+	0.901473i	$0.642487\pi$
$42$	0	0
$43$	398.433	1.41303	0.706517	−	0.707696i	$-0.250265\pi$
$43$	398.433	1.41303	0.706517	+	0.707696i	$0.250265\pi$
$44$	0	0
$45$	56.7372	0.187953
$46$	0	0
$47$	−184.608	−0.572934	−0.286467	−	0.958090i	$-0.592481\pi$
$47$	−184.608	−0.572934	−0.286467	+	0.958090i	$0.592481\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−410.212	−1.12630
$52$	0	0
$53$	359.825	0.932561	0.466281	−	0.884637i	$-0.345594\pi$
$53$	359.825	0.932561	0.466281	+	0.884637i	$0.345594\pi$
$54$	0	0
$55$	−308.350	−0.755963
$56$	0	0
$57$	135.650	0.315215
$58$	0	0
$59$	99.9075	0.220455	0.110228	−	0.993906i	$-0.464842\pi$
$59$	99.9075	0.220455	0.110228	+	0.993906i	$0.464842\pi$
$60$	0	0
$61$	674.516	1.41579	0.707893	−	0.706320i	$-0.249646\pi$
$61$	674.516	1.41579	0.707893	+	0.706320i	$0.249646\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.4429	0.0313767
$66$	0	0
$67$	376.959	0.687356	0.343678	−	0.939088i	$-0.388327\pi$
$67$	376.959	0.687356	0.343678	+	0.939088i	$0.388327\pi$
$68$	0	0
$69$	114.387	0.199573
$70$	0	0
$71$	1187.60	1.98511	0.992553	−	0.121810i	$-0.0388698\pi$
$71$	1187.60	1.98511	0.992553	+	0.121810i	$0.0388698\pi$
$72$	0	0
$73$	735.825	1.17975	0.589875	−	0.807494i	$-0.299177\pi$
$73$	735.825	1.17975	0.589875	+	0.807494i	$0.299177\pi$
$74$	0	0
$75$	−255.774	−0.393789
$76$	0	0
$77$	0	0
$78$	0	0
$79$	836.774	1.19170	0.595851	−	0.803095i	$-0.296815\pi$
$79$	836.774	1.19170	0.595851	+	0.803095i	$0.296815\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	293.732	0.388450	0.194225	−	0.980957i	$-0.437781\pi$
$83$	293.732	0.388450	0.194225	+	0.980957i	$0.437781\pi$
$84$	0	0
$85$	−862.010	−1.09998
$86$	0	0
$87$	158.350	0.195137
$88$	0	0
$89$	−1298.89	−1.54699	−0.773496	−	0.633801i	$-0.781494\pi$
$89$	−1298.89	−1.54699	−0.773496	+	0.633801i	$0.781494\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−44.3504	−0.0494508
$94$	0	0
$95$	285.051	0.307849
$96$	0	0
$97$	201.041	0.210440	0.105220	−	0.994449i	$-0.466445\pi$
$97$	201.041	0.210440	0.105220	+	0.994449i	$0.466445\pi$
$98$	0	0
$99$	−440.212	−0.446898
$100$	0	0
$101$	1053.51	1.03790	0.518952	−	0.854804i	$-0.326322\pi$
$101$	1053.51	1.03790	0.518952	+	0.854804i	$0.326322\pi$
$102$	0	0
$103$	1025.73	0.981247	0.490623	−	0.871372i	$-0.336769\pi$
$103$	1025.73	0.981247	0.490623	+	0.871372i	$0.336769\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	103.418	0.0934377	0.0467188	−	0.998908i	$-0.485124\pi$
$107$	103.418	0.0934377	0.0467188	+	0.998908i	$0.485124\pi$
$108$	0	0
$109$	−677.134	−0.595024	−0.297512	−	0.954718i	$-0.596157\pi$
$109$	−677.134	−0.595024	−0.297512	+	0.954718i	$0.596157\pi$
$110$	0	0
$111$	1001.72	0.856570
$112$	0	0
$113$	−452.083	−0.376357	−0.188179	−	0.982135i	$-0.560258\pi$
$113$	−452.083	−0.376357	−0.188179	+	0.982135i	$0.560258\pi$
$114$	0	0
$115$	240.370	0.194910
$116$	0	0
$117$	23.4744	0.0185488
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1061.42	0.797463
$122$	0	0
$123$	−681.788	−0.499795
$124$	0	0
$125$	−1325.49	−0.948446
$126$	0	0
$127$	1182.88	0.826482	0.413241	−	0.910622i	$-0.364397\pi$
$127$	1182.88	0.826482	0.413241	+	0.910622i	$0.364397\pi$
$128$	0	0
$129$	1195.30	0.815816
$130$	0	0
$131$	1257.75	0.838857	0.419429	−	0.907788i	$-0.362230\pi$
$131$	1257.75	0.838857	0.419429	+	0.907788i	$0.362230\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	170.212	0.108515
$136$	0	0
$137$	−56.5256	−0.0352504	−0.0176252	−	0.999845i	$-0.505611\pi$
$137$	−56.5256	−0.0352504	−0.0176252	+	0.999845i	$0.505611\pi$
$138$	0	0
$139$	2113.57	1.28972	0.644858	−	0.764303i	$-0.276917\pi$
$139$	2113.57	1.28972	0.644858	+	0.764303i	$0.276917\pi$
$140$	0	0
$141$	−553.825	−0.330783
$142$	0	0
$143$	−127.577	−0.0746049
$144$	0	0
$145$	332.754	0.190577
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1794.96	0.986904	0.493452	−	0.869773i	$-0.335735\pi$
$149$	1794.96	0.986904	0.493452	+	0.869773i	$0.335735\pi$
$150$	0	0
$151$	−377.032	−0.203195	−0.101597	−	0.994826i	$-0.532395\pi$
$151$	−377.032	−0.203195	−0.101597	+	0.994826i	$0.532395\pi$
$152$	0	0
$153$	−1230.63	−0.650268
$154$	0	0
$155$	−93.1969	−0.0482952
$156$	0	0
$157$	898.701	0.456842	0.228421	−	0.973562i	$-0.426644\pi$
$157$	898.701	0.456842	0.228421	+	0.973562i	$0.426644\pi$
$158$	0	0
$159$	1079.47	0.538414
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3863.52	−1.85653	−0.928264	−	0.371922i	$-0.878699\pi$
$163$	−3863.52	−1.85653	−0.928264	+	0.371922i	$0.878699\pi$
$164$	0	0
$165$	−925.051	−0.436455
$166$	0	0
$167$	−2861.44	−1.32590	−0.662948	−	0.748666i	$-0.730695\pi$
$167$	−2861.44	−1.32590	−0.662948	+	0.748666i	$0.730695\pi$
$168$	0	0
$169$	−2190.20	−0.996903
$170$	0	0
$171$	406.949	0.181989
$172$	0	0
$173$	−979.005	−0.430245	−0.215122	−	0.976587i	$-0.569015\pi$
$173$	−979.005	−0.430245	−0.215122	+	0.976587i	$0.569015\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	299.723	0.127280
$178$	0	0
$179$	1146.27	0.478640	0.239320	−	0.970941i	$-0.423075\pi$
$179$	1146.27	0.478640	0.239320	+	0.970941i	$0.423075\pi$
$180$	0	0
$181$	−3929.33	−1.61362	−0.806809	−	0.590812i	$-0.798808\pi$
$181$	−3929.33	−1.61362	−0.806809	+	0.590812i	$0.798808\pi$
$182$	0	0
$183$	2023.55	0.817404
$184$	0	0
$185$	2105.00	0.836554
$186$	0	0
$187$	6688.15	2.61543
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2937.36	1.11277	0.556386	−	0.830924i	$-0.312187\pi$
$191$	2937.36	1.11277	0.556386	+	0.830924i	$0.312187\pi$
$192$	0	0
$193$	−3533.17	−1.31774	−0.658868	−	0.752259i	$-0.728964\pi$
$193$	−3533.17	−1.31774	−0.658868	+	0.752259i	$0.728964\pi$
$194$	0	0
$195$	49.3286	0.0181154
$196$	0	0
$197$	584.856	0.211519	0.105760	−	0.994392i	$-0.466273\pi$
$197$	584.856	0.211519	0.105760	+	0.994392i	$0.466273\pi$
$198$	0	0
$199$	2158.08	0.768756	0.384378	−	0.923176i	$-0.374416\pi$
$199$	2158.08	0.768756	0.384378	+	0.923176i	$0.374416\pi$
$200$	0	0
$201$	1130.88	0.396845
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−1432.70	−0.488116
$206$	0	0
$207$	343.160	0.115224
$208$	0	0
$209$	−2211.65	−0.731976
$210$	0	0
$211$	4290.04	1.39971	0.699855	−	0.714285i	$-0.253248\pi$
$211$	4290.04	1.39971	0.699855	+	0.714285i	$0.253248\pi$
$212$	0	0
$213$	3562.81	1.14610
$214$	0	0
$215$	2511.78	0.796752
$216$	0	0
$217$	0	0
$218$	0	0
$219$	2207.47	0.681129
$220$	0	0
$221$	−356.647	−0.108555
$222$	0	0
$223$	−2743.78	−0.823932	−0.411966	−	0.911199i	$-0.635158\pi$
$223$	−2743.78	−0.823932	−0.411966	+	0.911199i	$0.635158\pi$
$224$	0	0
$225$	−767.321	−0.227354
$226$	0	0
$227$	−1724.79	−0.504311	−0.252155	−	0.967687i	$-0.581139\pi$
$227$	−1724.79	−0.504311	−0.252155	+	0.967687i	$0.581139\pi$
$228$	0	0
$229$	−4201.70	−1.21247	−0.606237	−	0.795284i	$-0.707322\pi$
$229$	−4201.70	−1.21247	−0.606237	+	0.795284i	$0.707322\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1274.94	0.358472	0.179236	−	0.983806i	$-0.442637\pi$
$233$	1274.94	0.358472	0.179236	+	0.983806i	$0.442637\pi$
$234$	0	0
$235$	−1163.80	−0.323054
$236$	0	0
$237$	2510.32	0.688029
$238$	0	0
$239$	5967.16	1.61499	0.807497	−	0.589872i	$-0.200822\pi$
$239$	5967.16	1.61499	0.807497	+	0.589872i	$0.200822\pi$
$240$	0	0
$241$	−4881.64	−1.30479	−0.652395	−	0.757879i	$-0.726236\pi$
$241$	−4881.64	−1.30479	−0.652395	+	0.757879i	$0.726236\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	117.937	0.0303812
$248$	0	0
$249$	881.197	0.224271
$250$	0	0
$251$	2262.63	0.568988	0.284494	−	0.958678i	$-0.408174\pi$
$251$	2262.63	0.568988	0.284494	+	0.958678i	$0.408174\pi$
$252$	0	0
$253$	−1864.98	−0.463439
$254$	0	0
$255$	−2586.03	−0.635072
$256$	0	0
$257$	6210.60	1.50742	0.753709	−	0.657209i	$-0.228263\pi$
$257$	6210.60	1.50742	0.753709	+	0.657209i	$0.228263\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	475.051	0.112663
$262$	0	0
$263$	2972.69	0.696973	0.348486	−	0.937314i	$-0.386696\pi$
$263$	2972.69	0.696973	0.348486	+	0.937314i	$0.386696\pi$
$264$	0	0
$265$	2268.38	0.525833
$266$	0	0
$267$	−3896.68	−0.893157
$268$	0	0
$269$	4443.42	1.00714	0.503569	−	0.863955i	$-0.332020\pi$
$269$	4443.42	1.00714	0.503569	+	0.863955i	$0.332020\pi$
$270$	0	0
$271$	−6840.25	−1.53327	−0.766634	−	0.642084i	$-0.778070\pi$
$271$	−6840.25	−1.53327	−0.766634	+	0.642084i	$0.778070\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4170.17	0.914439
$276$	0	0
$277$	−3228.67	−0.700332	−0.350166	−	0.936688i	$-0.613875\pi$
$277$	−3228.67	−0.700332	−0.350166	+	0.936688i	$0.613875\pi$
$278$	0	0
$279$	−133.051	−0.0285504
$280$	0	0
$281$	−6453.83	−1.37012	−0.685059	−	0.728488i	$-0.740224\pi$
$281$	−6453.83	−1.37012	−0.685059	+	0.728488i	$0.740224\pi$
$282$	0	0
$283$	3840.72	0.806739	0.403369	−	0.915037i	$-0.367839\pi$
$283$	3840.72	0.806739	0.403369	+	0.915037i	$0.367839\pi$
$284$	0	0
$285$	855.153	0.177737
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13784.1	2.80563
$290$	0	0
$291$	603.124	0.121497
$292$	0	0
$293$	−8801.01	−1.75481	−0.877407	−	0.479747i	$-0.840728\pi$
$293$	−8801.01	−1.75481	−0.877407	+	0.479747i	$0.840728\pi$
$294$	0	0
$295$	629.830	0.124306
$296$	0	0
$297$	−1320.63	−0.258017
$298$	0	0
$299$	99.4506	0.0192354
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3160.53	0.599234
$304$	0	0
$305$	4252.24	0.798303
$306$	0	0
$307$	−3926.72	−0.730000	−0.365000	−	0.931008i	$-0.618931\pi$
$307$	−3926.72	−0.730000	−0.365000	+	0.931008i	$0.618931\pi$
$308$	0	0
$309$	3077.20	0.566523
$310$	0	0
$311$	−6143.13	−1.12008	−0.560040	−	0.828466i	$-0.689214\pi$
$311$	−6143.13	−1.12008	−0.560040	+	0.828466i	$0.689214\pi$
$312$	0	0
$313$	−3824.19	−0.690594	−0.345297	−	0.938493i	$-0.612222\pi$
$313$	−3824.19	−0.690594	−0.345297	+	0.938493i	$0.612222\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7949.68	1.40851	0.704257	−	0.709946i	$-0.251280\pi$
$317$	7949.68	1.40851	0.704257	+	0.709946i	$0.251280\pi$
$318$	0	0
$319$	−2581.77	−0.453138
$320$	0	0
$321$	310.255	0.0539463
$322$	0	0
$323$	−6182.78	−1.06508
$324$	0	0
$325$	−222.376	−0.0379544
$326$	0	0
$327$	−2031.40	−0.343537
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11236.4	1.86588	0.932940	−	0.360032i	$-0.117234\pi$
$331$	11236.4	1.86588	0.932940	+	0.360032i	$0.117234\pi$
$332$	0	0
$333$	3005.17	0.494541
$334$	0	0
$335$	2376.40	0.387572
$336$	0	0
$337$	8425.41	1.36190	0.680951	−	0.732329i	$-0.261567\pi$
$337$	8425.41	1.36190	0.680951	+	0.732329i	$0.261567\pi$
$338$	0	0
$339$	−1356.25	−0.217290
$340$	0	0
$341$	723.095	0.114832
$342$	0	0
$343$	0	0
$344$	0	0
$345$	721.110	0.112531
$346$	0	0
$347$	−7177.15	−1.11034	−0.555172	−	0.831735i	$-0.687348\pi$
$347$	−7177.15	−1.11034	−0.555172	+	0.831735i	$0.687348\pi$
$348$	0	0
$349$	−1549.41	−0.237644	−0.118822	−	0.992916i	$-0.537912\pi$
$349$	−1549.41	−0.237644	−0.118822	+	0.992916i	$0.537912\pi$
$350$	0	0
$351$	70.4233	0.0107092
$352$	0	0
$353$	566.231	0.0853752	0.0426876	−	0.999088i	$-0.486408\pi$
$353$	566.231	0.0853752	0.0426876	+	0.999088i	$0.486408\pi$
$354$	0	0
$355$	7486.81	1.11932
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3848.19	−0.565738	−0.282869	−	0.959159i	$-0.591286\pi$
$359$	−3848.19	−0.565738	−0.282869	+	0.959159i	$0.591286\pi$
$360$	0	0
$361$	−4814.46	−0.701919
$362$	0	0
$363$	3184.27	0.460415
$364$	0	0
$365$	4638.74	0.665213
$366$	0	0
$367$	12542.2	1.78392	0.891960	−	0.452113i	$-0.149330\pi$
$367$	12542.2	1.78392	0.891960	+	0.452113i	$0.149330\pi$
$368$	0	0
$369$	−2045.37	−0.288557
$370$	0	0
$371$	0	0
$372$	0	0
$373$	6345.87	0.880903	0.440452	−	0.897776i	$-0.354818\pi$
$373$	6345.87	0.880903	0.440452	+	0.897776i	$0.354818\pi$
$374$	0	0
$375$	−3976.48	−0.547586
$376$	0	0
$377$	137.673	0.0188078
$378$	0	0
$379$	12200.2	1.65351	0.826757	−	0.562559i	$-0.190183\pi$
$379$	12200.2	1.65351	0.826757	+	0.562559i	$0.190183\pi$
$380$	0	0
$381$	3548.63	0.477170
$382$	0	0
$383$	−2770.91	−0.369679	−0.184839	−	0.982769i	$-0.559176\pi$
$383$	−2770.91	−0.369679	−0.184839	+	0.982769i	$0.559176\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3585.90	0.471011
$388$	0	0
$389$	1581.89	0.206183	0.103091	−	0.994672i	$-0.467127\pi$
$389$	1581.89	0.206183	0.103091	+	0.994672i	$0.467127\pi$
$390$	0	0
$391$	−5213.65	−0.674336
$392$	0	0
$393$	3773.26	0.484314
$394$	0	0
$395$	5275.13	0.671951
$396$	0	0
$397$	14235.9	1.79970	0.899848	−	0.436203i	$-0.143677\pi$
$397$	14235.9	1.79970	0.899848	+	0.436203i	$0.143677\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9556.18	−1.19006	−0.595028	−	0.803705i	$-0.702859\pi$
$401$	−9556.18	−1.19006	−0.595028	+	0.803705i	$0.702859\pi$
$402$	0	0
$403$	−38.5592	−0.00476619
$404$	0	0
$405$	510.635	0.0626510
$406$	0	0
$407$	−16332.2	−1.98909
$408$	0	0
$409$	−2858.17	−0.345544	−0.172772	−	0.984962i	$-0.555272\pi$
$409$	−2858.17	−0.345544	−0.172772	+	0.984962i	$0.555272\pi$
$410$	0	0
$411$	−169.577	−0.0203518
$412$	0	0
$413$	0	0
$414$	0	0
$415$	1851.73	0.219031
$416$	0	0
$417$	6340.70	0.744617
$418$	0	0
$419$	−13333.3	−1.55460	−0.777299	−	0.629132i	$-0.783411\pi$
$419$	−13333.3	−1.55460	−0.777299	+	0.629132i	$0.783411\pi$
$420$	0	0
$421$	−13567.4	−1.57063	−0.785314	−	0.619098i	$-0.787499\pi$
$421$	−13567.4	−1.57063	−0.785314	+	0.619098i	$0.787499\pi$
$422$	0	0
$423$	−1661.47	−0.190978
$424$	0	0
$425$	11657.9	1.33057
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−382.730	−0.0430732
$430$	0	0
$431$	−14207.3	−1.58780	−0.793898	−	0.608051i	$-0.791952\pi$
$431$	−14207.3	−1.58780	−0.793898	+	0.608051i	$0.791952\pi$
$432$	0	0
$433$	10530.6	1.16875	0.584375	−	0.811484i	$-0.301340\pi$
$433$	10530.6	1.16875	0.584375	+	0.811484i	$0.301340\pi$
$434$	0	0
$435$	998.262	0.110030
$436$	0	0
$437$	1724.06	0.188725
$438$	0	0
$439$	−4038.23	−0.439030	−0.219515	−	0.975609i	$-0.570447\pi$
$439$	−4038.23	−0.439030	−0.219515	+	0.975609i	$0.570447\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4574.15	0.490574	0.245287	−	0.969450i	$-0.421118\pi$
$443$	4574.15	0.490574	0.245287	+	0.969450i	$0.421118\pi$
$444$	0	0
$445$	−8188.40	−0.872286
$446$	0	0
$447$	5384.88	0.569789
$448$	0	0
$449$	14957.1	1.57209	0.786044	−	0.618171i	$-0.212126\pi$
$449$	14957.1	1.57209	0.786044	+	0.618171i	$0.212126\pi$
$450$	0	0
$451$	11116.0	1.16060
$452$	0	0
$453$	−1131.09	−0.117314
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3027.65	0.309907	0.154954	−	0.987922i	$-0.450477\pi$
$457$	3027.65	0.309907	0.154954	+	0.987922i	$0.450477\pi$
$458$	0	0
$459$	−3691.90	−0.375432
$460$	0	0
$461$	10877.7	1.09897	0.549484	−	0.835504i	$-0.314824\pi$
$461$	10877.7	1.09897	0.549484	+	0.835504i	$0.314824\pi$
$462$	0	0
$463$	4038.28	0.405345	0.202673	−	0.979247i	$-0.435037\pi$
$463$	4038.28	0.405345	0.202673	+	0.979247i	$0.435037\pi$
$464$	0	0
$465$	−279.591	−0.0278833
$466$	0	0
$467$	8411.80	0.833515	0.416758	−	0.909018i	$-0.363166\pi$
$467$	8411.80	0.833515	0.416758	+	0.909018i	$0.363166\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2696.10	0.263758
$472$	0	0
$473$	−19488.3	−1.89445
$474$	0	0
$475$	−3855.07	−0.372384
$476$	0	0
$477$	3238.42	0.310854
$478$	0	0
$479$	7172.70	0.684194	0.342097	−	0.939665i	$-0.388863\pi$
$479$	7172.70	0.684194	0.342097	+	0.939665i	$0.388863\pi$
$480$	0	0
$481$	870.921	0.0825584
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1267.39	0.118658
$486$	0	0
$487$	5580.52	0.519256	0.259628	−	0.965709i	$-0.416400\pi$
$487$	5580.52	0.519256	0.259628	+	0.965709i	$0.416400\pi$
$488$	0	0
$489$	−11590.6	−1.07187
$490$	0	0
$491$	−12489.4	−1.14794	−0.573972	−	0.818875i	$-0.694598\pi$
$491$	−12489.4	−1.14794	−0.573972	+	0.818875i	$0.694598\pi$
$492$	0	0
$493$	−7217.46	−0.659347
$494$	0	0
$495$	−2775.15	−0.251988
$496$	0	0
$497$	0	0
$498$	0	0
$499$	15216.6	1.36511	0.682556	−	0.730834i	$-0.260869\pi$
$499$	15216.6	1.36511	0.682556	+	0.730834i	$0.260869\pi$
$500$	0	0
$501$	−8584.31	−0.765506
$502$	0	0
$503$	−1814.89	−0.160879	−0.0804393	−	0.996760i	$-0.525632\pi$
$503$	−1814.89	−0.160879	−0.0804393	+	0.996760i	$0.525632\pi$
$504$	0	0
$505$	6641.47	0.585231
$506$	0	0
$507$	−6570.59	−0.575562
$508$	0	0
$509$	4853.68	0.422663	0.211332	−	0.977414i	$-0.432220\pi$
$509$	4853.68	0.422663	0.211332	+	0.977414i	$0.432220\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	1220.85	0.105072
$514$	0	0
$515$	6466.35	0.553285
$516$	0	0
$517$	9029.63	0.768129
$518$	0	0
$519$	−2937.01	−0.248402
$520$	0	0
$521$	9913.18	0.833598	0.416799	−	0.908999i	$-0.363152\pi$
$521$	9913.18	0.833598	0.416799	+	0.908999i	$0.363152\pi$
$522$	0	0
$523$	−4524.29	−0.378267	−0.189133	−	0.981951i	$-0.560568\pi$
$523$	−4524.29	−0.378267	−0.189133	+	0.981951i	$0.560568\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2021.45	0.167089
$528$	0	0
$529$	−10713.2	−0.880512
$530$	0	0
$531$	899.168	0.0734850
$532$	0	0
$533$	−592.763	−0.0481715
$534$	0	0
$535$	651.963	0.0526857
$536$	0	0
$537$	3438.82	0.276343
$538$	0	0
$539$	0	0
$540$	0	0
$541$	3724.94	0.296022	0.148011	−	0.988986i	$-0.452713\pi$
$541$	3724.94	0.296022	0.148011	+	0.988986i	$0.452713\pi$
$542$	0	0
$543$	−11788.0	−0.931623
$544$	0	0
$545$	−4268.74	−0.335510
$546$	0	0
$547$	−16121.6	−1.26016	−0.630082	−	0.776528i	$-0.716979\pi$
$547$	−16121.6	−1.26016	−0.630082	+	0.776528i	$0.716979\pi$
$548$	0	0
$549$	6070.64	0.471928
$550$	0	0
$551$	2386.69	0.184530
$552$	0	0
$553$	0	0
$554$	0	0
$555$	6314.99	0.482985
$556$	0	0
$557$	20451.7	1.55577	0.777887	−	0.628405i	$-0.216292\pi$
$557$	20451.7	1.55577	0.777887	+	0.628405i	$0.216292\pi$
$558$	0	0
$559$	1039.22	0.0786303
$560$	0	0
$561$	20064.4	1.51002
$562$	0	0
$563$	−10046.7	−0.752078	−0.376039	−	0.926604i	$-0.622714\pi$
$563$	−10046.7	−0.752078	−0.376039	+	0.926604i	$0.622714\pi$
$564$	0	0
$565$	−2849.99	−0.212212
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−15356.4	−1.13141	−0.565705	−	0.824608i	$-0.691396\pi$
$569$	−15356.4	−1.13141	−0.565705	+	0.824608i	$0.691396\pi$
$570$	0	0
$571$	19333.6	1.41696	0.708480	−	0.705731i	$-0.249381\pi$
$571$	19333.6	1.41696	0.708480	+	0.705731i	$0.249381\pi$
$572$	0	0
$573$	8812.07	0.642460
$574$	0	0
$575$	−3250.79	−0.235769
$576$	0	0
$577$	−26258.8	−1.89458	−0.947288	−	0.320384i	$-0.896188\pi$
$577$	−26258.8	−1.89458	−0.947288	+	0.320384i	$0.896188\pi$
$578$	0	0
$579$	−10599.5	−0.760795
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−17599.9	−1.25028
$584$	0	0
$585$	147.986	0.0104589
$586$	0	0
$587$	−4868.98	−0.342358	−0.171179	−	0.985240i	$-0.554758\pi$
$587$	−4868.98	−0.342358	−0.171179	+	0.985240i	$0.554758\pi$
$588$	0	0
$589$	−668.457	−0.0467628
$590$	0	0
$591$	1754.57	0.122121
$592$	0	0
$593$	13647.1	0.945055	0.472528	−	0.881316i	$-0.343342\pi$
$593$	13647.1	0.945055	0.472528	+	0.881316i	$0.343342\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6474.25	0.443841
$598$	0	0
$599$	−7543.11	−0.514529	−0.257265	−	0.966341i	$-0.582821\pi$
$599$	−7543.11	−0.514529	−0.257265	+	0.966341i	$0.582821\pi$
$600$	0	0
$601$	19522.4	1.32501	0.662507	−	0.749056i	$-0.269493\pi$
$601$	19522.4	1.32501	0.662507	+	0.749056i	$0.269493\pi$
$602$	0	0
$603$	3392.63	0.229119
$604$	0	0
$605$	6691.36	0.449657
$606$	0	0
$607$	13804.5	0.923079	0.461539	−	0.887120i	$-0.347297\pi$
$607$	13804.5	0.923079	0.461539	+	0.887120i	$0.347297\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−481.508	−0.0318817
$612$	0	0
$613$	21718.8	1.43102	0.715508	−	0.698605i	$-0.246195\pi$
$613$	21718.8	1.43102	0.715508	+	0.698605i	$0.246195\pi$
$614$	0	0
$615$	−4298.09	−0.281814
$616$	0	0
$617$	−5183.85	−0.338240	−0.169120	−	0.985595i	$-0.554093\pi$
$617$	−5183.85	−0.338240	−0.169120	+	0.985595i	$0.554093\pi$
$618$	0	0
$619$	22003.7	1.42876	0.714382	−	0.699756i	$-0.246708\pi$
$619$	22003.7	1.42876	0.714382	+	0.699756i	$0.246708\pi$
$620$	0	0
$621$	1029.48	0.0665244
$622$	0	0
$623$	0	0
$624$	0	0
$625$	2301.14	0.147273
$626$	0	0
$627$	−6634.95	−0.422607
$628$	0	0
$629$	−45657.6	−2.89426
$630$	0	0
$631$	985.836	0.0621957	0.0310979	−	0.999516i	$-0.490100\pi$
$631$	985.836	0.0621957	0.0310979	+	0.999516i	$0.490100\pi$
$632$	0	0
$633$	12870.1	0.808123
$634$	0	0
$635$	7457.01	0.466020
$636$	0	0
$637$	0	0
$638$	0	0
$639$	10688.4	0.661702
$640$	0	0
$641$	24282.6	1.49626	0.748131	−	0.663551i	$-0.230951\pi$
$641$	24282.6	1.49626	0.748131	+	0.663551i	$0.230951\pi$
$642$	0	0
$643$	−4743.12	−0.290903	−0.145451	−	0.989365i	$-0.546463\pi$
$643$	−4743.12	−0.290903	−0.145451	+	0.989365i	$0.546463\pi$
$644$	0	0
$645$	7535.33	0.460005
$646$	0	0
$647$	29641.1	1.80110	0.900549	−	0.434754i	$-0.143165\pi$
$647$	29641.1	1.80110	0.900549	+	0.434754i	$0.143165\pi$
$648$	0	0
$649$	−4886.72	−0.295563
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23046.9	−1.38116	−0.690578	−	0.723258i	$-0.742644\pi$
$653$	−23046.9	−1.38116	−0.690578	+	0.723258i	$0.742644\pi$
$654$	0	0
$655$	7929.04	0.472997
$656$	0	0
$657$	6622.42	0.393250
$658$	0	0
$659$	−5795.12	−0.342558	−0.171279	−	0.985223i	$-0.554790\pi$
$659$	−5795.12	−0.342558	−0.171279	+	0.985223i	$0.554790\pi$
$660$	0	0
$661$	−2592.59	−0.152557	−0.0762784	−	0.997087i	$-0.524304\pi$
$661$	−2592.59	−0.152557	−0.0762784	+	0.997087i	$0.524304\pi$
$662$	0	0
$663$	−1069.94	−0.0626744
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2012.58	0.116833
$668$	0	0
$669$	−8231.33	−0.475697
$670$	0	0
$671$	−32992.2	−1.89814
$672$	0	0
$673$	28156.0	1.61268	0.806341	−	0.591451i	$-0.201445\pi$
$673$	28156.0	1.61268	0.806341	+	0.591451i	$0.201445\pi$
$674$	0	0
$675$	−2301.96	−0.131263
$676$	0	0
$677$	−20271.4	−1.15080	−0.575402	−	0.817871i	$-0.695154\pi$
$677$	−20271.4	−1.15080	−0.575402	+	0.817871i	$0.695154\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−5174.38	−0.291164
$682$	0	0
$683$	13267.0	0.743264	0.371632	−	0.928380i	$-0.378798\pi$
$683$	13267.0	0.743264	0.371632	+	0.928380i	$0.378798\pi$
$684$	0	0
$685$	−356.345	−0.0198763
$686$	0	0
$687$	−12605.1	−0.700022
$688$	0	0
$689$	938.520	0.0518937
$690$	0	0
$691$	−15966.3	−0.878995	−0.439497	−	0.898244i	$-0.644843\pi$
$691$	−15966.3	−0.878995	−0.439497	+	0.898244i	$0.644843\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	13324.2	0.727217
$696$	0	0
$697$	31075.3	1.68875
$698$	0	0
$699$	3824.82	0.206964
$700$	0	0
$701$	7525.29	0.405458	0.202729	−	0.979235i	$-0.435019\pi$
$701$	7525.29	0.405458	0.202729	+	0.979235i	$0.435019\pi$
$702$	0	0
$703$	15098.1	0.810010
$704$	0	0
$705$	−3491.39	−0.186515
$706$	0	0
$707$	0	0
$708$	0	0
$709$	20033.6	1.06118	0.530591	−	0.847628i	$-0.321970\pi$
$709$	20033.6	1.06118	0.530591	+	0.847628i	$0.321970\pi$
$710$	0	0
$711$	7530.96	0.397234
$712$	0	0
$713$	−563.678	−0.0296071
$714$	0	0
$715$	−804.261	−0.0420666
$716$	0	0
$717$	17901.5	0.932417
$718$	0	0
$719$	−8081.69	−0.419188	−0.209594	−	0.977788i	$-0.567214\pi$
$719$	−8081.69	−0.419188	−0.209594	+	0.977788i	$0.567214\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−14644.9	−0.753321
$724$	0	0
$725$	−4500.21	−0.230529
$726$	0	0
$727$	34117.8	1.74052	0.870262	−	0.492590i	$-0.163950\pi$
$727$	34117.8	1.74052	0.870262	+	0.492590i	$0.163950\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−54480.6	−2.75655
$732$	0	0
$733$	−20048.0	−1.01022	−0.505110	−	0.863055i	$-0.668548\pi$
$733$	−20048.0	−1.01022	−0.505110	+	0.863055i	$0.668548\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18438.0	−0.921534
$738$	0	0
$739$	407.607	0.0202897	0.0101448	−	0.999949i	$-0.496771\pi$
$739$	407.607	0.0202897	0.0101448	+	0.999949i	$0.496771\pi$
$740$	0	0
$741$	353.811	0.0175406
$742$	0	0
$743$	128.374	0.00633861	0.00316930	−	0.999995i	$-0.498991\pi$
$743$	128.374	0.00633861	0.00316930	+	0.999995i	$0.498991\pi$
$744$	0	0
$745$	11315.7	0.556475
$746$	0	0
$747$	2643.59	0.129483
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−25076.0	−1.21842	−0.609212	−	0.793007i	$-0.708514\pi$
$751$	−25076.0	−1.21842	−0.609212	+	0.793007i	$0.708514\pi$
$752$	0	0
$753$	6787.90	0.328506
$754$	0	0
$755$	−2376.86	−0.114573
$756$	0	0
$757$	−21824.1	−1.04783	−0.523916	−	0.851770i	$-0.675530\pi$
$757$	−21824.1	−1.04783	−0.523916	+	0.851770i	$0.675530\pi$
$758$	0	0
$759$	−5594.93	−0.267567
$760$	0	0
$761$	38939.2	1.85485	0.927427	−	0.374004i	$-0.122015\pi$
$761$	38939.2	1.85485	0.927427	+	0.374004i	$0.122015\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−7758.09	−0.366659
$766$	0	0
$767$	260.586	0.0122675
$768$	0	0
$769$	−3079.42	−0.144404	−0.0722021	−	0.997390i	$-0.523003\pi$
$769$	−3079.42	−0.144404	−0.0722021	+	0.997390i	$0.523003\pi$
$770$	0	0
$771$	18631.8	0.870308
$772$	0	0
$773$	−31770.9	−1.47829	−0.739146	−	0.673545i	$-0.764771\pi$
$773$	−31770.9	−1.47829	−0.739146	+	0.673545i	$0.764771\pi$
$774$	0	0
$775$	1260.41	0.0584195
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−10276.0	−0.472628
$780$	0	0
$781$	−58088.5	−2.66142
$782$	0	0
$783$	1425.15	0.0650458
$784$	0	0
$785$	5665.53	0.257594
$786$	0	0
$787$	6736.02	0.305099	0.152550	−	0.988296i	$-0.451252\pi$
$787$	6736.02	0.305099	0.152550	+	0.988296i	$0.451252\pi$
$788$	0	0
$789$	8918.06	0.402397
$790$	0	0
$791$	0	0
$792$	0	0
$793$	1759.32	0.0787834
$794$	0	0
$795$	6805.15	0.303590
$796$	0	0
$797$	−13445.1	−0.597552	−0.298776	−	0.954323i	$-0.596578\pi$
$797$	−13445.1	−0.597552	−0.298776	+	0.954323i	$0.596578\pi$
$798$	0	0
$799$	25242.8	1.11768
$800$	0	0
$801$	−11690.0	−0.515664
$802$	0	0
$803$	−35991.0	−1.58169
$804$	0	0
$805$	0	0
$806$	0	0
$807$	13330.3	0.581472
$808$	0	0
$809$	20337.7	0.883853	0.441927	−	0.897051i	$-0.354295\pi$
$809$	20337.7	0.883853	0.441927	+	0.897051i	$0.354295\pi$
$810$	0	0
$811$	−23069.9	−0.998884	−0.499442	−	0.866347i	$-0.666462\pi$
$811$	−23069.9	−0.998884	−0.499442	+	0.866347i	$0.666462\pi$
$812$	0	0
$813$	−20520.8	−0.885233
$814$	0	0
$815$	−24356.1	−1.04682
$816$	0	0
$817$	18015.8	0.771471
$818$	0	0
$819$	0	0
$820$	0	0
$821$	8230.74	0.349884	0.174942	−	0.984579i	$-0.444026\pi$
$821$	8230.74	0.349884	0.174942	+	0.984579i	$0.444026\pi$
$822$	0	0
$823$	−17577.9	−0.744506	−0.372253	−	0.928131i	$-0.621415\pi$
$823$	−17577.9	−0.744506	−0.372253	+	0.928131i	$0.621415\pi$
$824$	0	0
$825$	12510.5	0.527951
$826$	0	0
$827$	6440.14	0.270793	0.135396	−	0.990792i	$-0.456769\pi$
$827$	6440.14	0.270793	0.135396	+	0.990792i	$0.456769\pi$
$828$	0	0
$829$	−5084.23	−0.213007	−0.106503	−	0.994312i	$-0.533966\pi$
$829$	−5084.23	−0.213007	−0.106503	+	0.994312i	$0.533966\pi$
$830$	0	0
$831$	−9686.01	−0.404337
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−18038.9	−0.747618
$836$	0	0
$837$	−399.153	−0.0164836
$838$	0	0
$839$	−25150.5	−1.03491	−0.517456	−	0.855710i	$-0.673121\pi$
$839$	−25150.5	−1.03491	−0.517456	+	0.855710i	$0.673121\pi$
$840$	0	0
$841$	−21602.9	−0.885764
$842$	0	0
$843$	−19361.5	−0.791038
$844$	0	0
$845$	−13807.3	−0.562113
$846$	0	0
$847$	0	0
$848$	0	0
$849$	11522.2	0.465771
$850$	0	0
$851$	12731.5	0.512846
$852$	0	0
$853$	6408.37	0.257232	0.128616	−	0.991694i	$-0.458947\pi$
$853$	6408.37	0.257232	0.128616	+	0.991694i	$0.458947\pi$
$854$	0	0
$855$	2565.46	0.102616
$856$	0	0
$857$	−17248.6	−0.687515	−0.343758	−	0.939058i	$-0.611700\pi$
$857$	−17248.6	−0.687515	−0.343758	+	0.939058i	$0.611700\pi$
$858$	0	0
$859$	3159.07	0.125479	0.0627393	−	0.998030i	$-0.480016\pi$
$859$	3159.07	0.125479	0.0627393	+	0.998030i	$0.480016\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41071.9	−1.62005	−0.810025	−	0.586395i	$-0.800547\pi$
$863$	−41071.9	−1.62005	−0.810025	+	0.586395i	$0.800547\pi$
$864$	0	0
$865$	−6171.78	−0.242597
$866$	0	0
$867$	41352.2	1.61983
$868$	0	0
$869$	−40928.6	−1.59771
$870$	0	0
$871$	983.210	0.0382489
$872$	0	0
$873$	1809.37	0.0701466
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1034.90	−0.0398472	−0.0199236	−	0.999802i	$-0.506342\pi$
$877$	−1034.90	−0.0398472	−0.0199236	+	0.999802i	$0.506342\pi$
$878$	0	0
$879$	−26403.0	−1.01314
$880$	0	0
$881$	3109.73	0.118921	0.0594606	−	0.998231i	$-0.481062\pi$
$881$	3109.73	0.118921	0.0594606	+	0.998231i	$0.481062\pi$
$882$	0	0
$883$	19782.8	0.753955	0.376978	−	0.926222i	$-0.376963\pi$
$883$	19782.8	0.753955	0.376978	+	0.926222i	$0.376963\pi$
$884$	0	0
$885$	1889.49	0.0717678
$886$	0	0
$887$	9355.56	0.354148	0.177074	−	0.984198i	$-0.443337\pi$
$887$	9355.56	0.354148	0.177074	+	0.984198i	$0.443337\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−3961.90	−0.148966
$892$	0	0
$893$	−8347.35	−0.312803
$894$	0	0
$895$	7226.27	0.269886
$896$	0	0
$897$	298.352	0.0111055
$898$	0	0
$899$	−780.322	−0.0289491
$900$	0	0
$901$	−49201.4	−1.81924
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24771.0	−0.909853
$906$	0	0
$907$	−16211.9	−0.593505	−0.296752	−	0.954954i	$-0.595904\pi$
$907$	−16211.9	−0.593505	−0.296752	+	0.954954i	$0.595904\pi$
$908$	0	0
$909$	9481.60	0.345968
$910$	0	0
$911$	−14540.6	−0.528816	−0.264408	−	0.964411i	$-0.585177\pi$
$911$	−14540.6	−0.528816	−0.264408	+	0.964411i	$0.585177\pi$
$912$	0	0
$913$	−14367.2	−0.520792
$914$	0	0
$915$	12756.7	0.460901
$916$	0	0
$917$	0	0
$918$	0	0
$919$	37239.2	1.33668	0.668339	−	0.743857i	$-0.267005\pi$
$919$	37239.2	1.33668	0.668339	+	0.743857i	$0.267005\pi$
$920$	0	0
$921$	−11780.2	−0.421466
$922$	0	0
$923$	3097.59	0.110464
$924$	0	0
$925$	−28468.2	−1.01192
$926$	0	0
$927$	9231.59	0.327082
$928$	0	0
$929$	−22062.5	−0.779167	−0.389583	−	0.920991i	$-0.627381\pi$
$929$	−22062.5	−0.779167	−0.389583	+	0.920991i	$0.627381\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−18429.4	−0.646678
$934$	0	0
$935$	42163.0	1.47473
$936$	0	0
$937$	15597.3	0.543800	0.271900	−	0.962326i	$-0.412348\pi$
$937$	15597.3	0.543800	0.271900	+	0.962326i	$0.412348\pi$
$938$	0	0
$939$	−11472.6	−0.398715
$940$	0	0
$941$	15887.1	0.550377	0.275188	−	0.961390i	$-0.411260\pi$
$941$	15887.1	0.550377	0.275188	+	0.961390i	$0.411260\pi$
$942$	0	0
$943$	−8665.29	−0.299237
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−54754.6	−1.87887	−0.939433	−	0.342733i	$-0.888648\pi$
$947$	−54754.6	−1.87887	−0.939433	+	0.342733i	$0.888648\pi$
$948$	0	0
$949$	1919.23	0.0656489
$950$	0	0
$951$	23849.0	0.813205
$952$	0	0
$953$	12091.3	0.410992	0.205496	−	0.978658i	$-0.434119\pi$
$953$	12091.3	0.410992	0.205496	+	0.978658i	$0.434119\pi$
$954$	0	0
$955$	18517.5	0.627447
$956$	0	0
$957$	−7745.30	−0.261620
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29572.4	−0.992664
$962$	0	0
$963$	930.765	0.0311459
$964$	0	0
$965$	−22273.6	−0.743017
$966$	0	0
$967$	27415.1	0.911695	0.455848	−	0.890058i	$-0.349336\pi$
$967$	27415.1	0.911695	0.455848	+	0.890058i	$0.349336\pi$
$968$	0	0
$969$	−18548.4	−0.614921
$970$	0	0
$971$	55397.0	1.83087	0.915435	−	0.402466i	$-0.131847\pi$
$971$	55397.0	1.83087	0.915435	+	0.402466i	$0.131847\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−667.127	−0.0219130
$976$	0	0
$977$	18294.3	0.599065	0.299532	−	0.954086i	$-0.403169\pi$
$977$	18294.3	0.599065	0.299532	+	0.954086i	$0.403169\pi$
$978$	0	0
$979$	63532.0	2.07405
$980$	0	0
$981$	−6094.20	−0.198341
$982$	0	0
$983$	−23803.2	−0.772334	−0.386167	−	0.922429i	$-0.626201\pi$
$983$	−23803.2	−0.772334	−0.386167	+	0.922429i	$0.626201\pi$
$984$	0	0
$985$	3687.01	0.119267
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15191.8	0.488445
$990$	0	0
$991$	13624.5	0.436726	0.218363	−	0.975868i	$-0.429928\pi$
$991$	13624.5	0.436726	0.218363	+	0.975868i	$0.429928\pi$
$992$	0	0
$993$	33709.1	1.07727
$994$	0	0
$995$	13604.8	0.433470
$996$	0	0
$997$	46834.4	1.48772	0.743861	−	0.668334i	$-0.232992\pi$
$997$	46834.4	1.48772	0.743861	+	0.668334i	$0.232992\pi$
$998$	0	0
$999$	9015.50	0.285523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2352.4.a.bv.1.2		2
4.3	odd	2		1176.4.a.p.1.2		2
7.6	odd	2		336.4.a.n.1.1		2
21.20	even	2		1008.4.a.y.1.2		2
28.27	even	2		168.4.a.h.1.1	✓	2
56.13	odd	2		1344.4.a.bl.1.2		2
56.27	even	2		1344.4.a.bd.1.2		2
84.83	odd	2		504.4.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.h.1.1	✓	2	28.27	even	2
336.4.a.n.1.1		2	7.6	odd	2
504.4.a.j.1.2		2	84.83	odd	2
1008.4.a.y.1.2		2	21.20	even	2
1176.4.a.p.1.2		2	4.3	odd	2
1344.4.a.bd.1.2		2	56.27	even	2
1344.4.a.bl.1.2		2	56.13	odd	2
2352.4.a.bv.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2352.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +6.30413 q^{5} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +6.30413 q^{5} +9.00000 q^{9} -48.9124 q^{11} +2.60827 q^{13} +18.9124 q^{15} -136.737 q^{17} +45.2165 q^{19} +38.1289 q^{23} -85.2579 q^{25} +27.0000 q^{27} +52.7835 q^{29} -14.7835 q^{31} -146.737 q^{33} +333.908 q^{37} +7.82481 q^{39} -227.263 q^{41} +398.433 q^{43} +56.7372 q^{45} -184.608 q^{47} -410.212 q^{51} +359.825 q^{53} -308.350 q^{55} +135.650 q^{57} +99.9075 q^{59} +674.516 q^{61} +16.4429 q^{65} +376.959 q^{67} +114.387 q^{69} +1187.60 q^{71} +735.825 q^{73} -255.774 q^{75} +836.774 q^{79} +81.0000 q^{81} +293.732 q^{83} -862.010 q^{85} +158.350 q^{87} -1298.89 q^{89} -44.3504 q^{93} +285.051 q^{95} +201.041 q^{97} -440.212 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{3} - 14 q^{5} + 18 q^{9}+O(q^{10}) \) 2 * q + 6 * q^3 - 14 * q^5 + 18 * q^9	\( 2 q + 6 q^{3} - 14 q^{5} + 18 q^{9} - 18 q^{11} - 48 q^{13} - 42 q^{15} - 34 q^{17} - 16 q^{19} - 110 q^{23} + 202 q^{25} + 54 q^{27} + 212 q^{29} - 136 q^{31} - 54 q^{33} - 24 q^{37} - 144 q^{39} - 694 q^{41} + 584 q^{43} - 126 q^{45} - 316 q^{47} - 102 q^{51} + 560 q^{53} - 936 q^{55} - 48 q^{57} - 492 q^{59} + 604 q^{61} + 1044 q^{65} + 1020 q^{67} - 330 q^{69} + 1710 q^{71} + 1312 q^{73} + 606 q^{75} + 556 q^{79} + 162 q^{81} - 264 q^{83} - 2948 q^{85} + 636 q^{87} - 70 q^{89} - 408 q^{93} + 1528 q^{95} + 136 q^{97} - 162 q^{99}+O(q^{100}) \) 2 * q + 6 * q^3 - 14 * q^5 + 18 * q^9 - 18 * q^11 - 48 * q^13 - 42 * q^15 - 34 * q^17 - 16 * q^19 - 110 * q^23 + 202 * q^25 + 54 * q^27 + 212 * q^29 - 136 * q^31 - 54 * q^33 - 24 * q^37 - 144 * q^39 - 694 * q^41 + 584 * q^43 - 126 * q^45 - 316 * q^47 - 102 * q^51 + 560 * q^53 - 936 * q^55 - 48 * q^57 - 492 * q^59 + 604 * q^61 + 1044 * q^65 + 1020 * q^67 - 330 * q^69 + 1710 * q^71 + 1312 * q^73 + 606 * q^75 + 556 * q^79 + 162 * q^81 - 264 * q^83 - 2948 * q^85 + 636 * q^87 - 70 * q^89 - 408 * q^93 + 1528 * q^95 + 136 * q^97 - 162 * q^99

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