LMFDB - Embedded newform 1416.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1416,2,Mod(1,1416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1416, 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("1416.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1416 = 2^{3} \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1416.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:	$11.3068169262$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	1416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +0.454904 q^{5} -3.12398 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +0.454904 q^{5} -3.12398 q^{7} +1.00000 q^{9} -3.00000 q^{11} +6.33816 q^{13} -0.454904 q^{15} +2.66908 q^{17} +1.21417 q^{19} +3.12398 q^{21} -4.12398 q^{23} -4.79306 q^{25} -1.00000 q^{27} -0.330921 q^{29} +1.66908 q^{31} +3.00000 q^{33} -1.42111 q^{35} -10.0072 q^{37} -6.33816 q^{39} -5.55233 q^{41} -3.00000 q^{43} +0.454904 q^{45} -1.42111 q^{47} +2.75927 q^{49} -2.66908 q^{51} -4.45490 q^{53} -1.36471 q^{55} -1.21417 q^{57} +1.00000 q^{59} -11.0072 q^{61} -3.12398 q^{63} +2.88325 q^{65} +4.70287 q^{67} +4.12398 q^{69} -14.3382 q^{71} -8.09743 q^{73} +4.79306 q^{75} +9.37195 q^{77} +5.24797 q^{79} +1.00000 q^{81} -0.876017 q^{83} +1.21417 q^{85} +0.330921 q^{87} -3.24073 q^{89} -19.8003 q^{91} -1.66908 q^{93} +0.552333 q^{95} +17.5596 q^{97} -3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 3 * q^9	$ 3 q - 3 q^{3} + 3 q^{9} - 9 q^{11} + 3 q^{13} - 3 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 9 q^{29} - 3 q^{31} + 9 q^{33} - 15 q^{35} - 6 q^{37} - 3 q^{39} + 6 q^{41} - 9 q^{43} - 15 q^{47} + 3 q^{49} - 12 q^{53} + 3 q^{57} + 3 q^{59} - 9 q^{61} - 6 q^{65} - 6 q^{67} + 3 q^{69} - 27 q^{71} - 3 q^{73} - 3 q^{75} - 3 q^{79} + 3 q^{81} - 12 q^{83} - 3 q^{85} + 9 q^{87} - 15 q^{89} - 18 q^{91} + 3 q^{93} - 21 q^{95} + 6 q^{97} - 9 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^9 - 9 * q^11 + 3 * q^13 - 3 * q^19 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 9 * q^29 - 3 * q^31 + 9 * q^33 - 15 * q^35 - 6 * q^37 - 3 * q^39 + 6 * q^41 - 9 * q^43 - 15 * q^47 + 3 * q^49 - 12 * q^53 + 3 * q^57 + 3 * q^59 - 9 * q^61 - 6 * q^65 - 6 * q^67 + 3 * q^69 - 27 * q^71 - 3 * q^73 - 3 * q^75 - 3 * q^79 + 3 * q^81 - 12 * q^83 - 3 * q^85 + 9 * q^87 - 15 * q^89 - 18 * q^91 + 3 * q^93 - 21 * q^95 + 6 * q^97 - 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0.454904	0.203439	0.101720	−	0.994813i	$-0.467566\pi$
$5$	0.454904	0.203439	0.101720	+	0.994813i	$0.467566\pi$
$6$	0	0
$7$	−3.12398	−1.18075	−0.590377	−	0.807127i	$-0.701021\pi$
$7$	−3.12398	−1.18075	−0.590377	+	0.807127i	$0.701021\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	6.33816	1.75789	0.878944	−	0.476924i	$-0.158248\pi$
$13$	6.33816	1.75789	0.878944	+	0.476924i	$0.158248\pi$
$14$	0	0
$15$	−0.454904	−0.117456
$16$	0	0
$17$	2.66908	0.647347	0.323673	−	0.946169i	$-0.395082\pi$
$17$	2.66908	0.647347	0.323673	+	0.946169i	$0.395082\pi$
$18$	0	0
$19$	1.21417	0.278551	0.139275	−	0.990254i	$-0.455523\pi$
$19$	1.21417	0.278551	0.139275	+	0.990254i	$0.455523\pi$
$20$	0	0
$21$	3.12398	0.681709
$22$	0	0
$23$	−4.12398	−0.859910	−0.429955	−	0.902850i	$-0.641471\pi$
$23$	−4.12398	−0.859910	−0.429955	+	0.902850i	$0.641471\pi$
$24$	0	0
$25$	−4.79306	−0.958612
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.330921	−0.0614505	−0.0307252	−	0.999528i	$-0.509782\pi$
$29$	−0.330921	−0.0614505	−0.0307252	+	0.999528i	$0.509782\pi$
$30$	0	0
$31$	1.66908	0.299775	0.149888	−	0.988703i	$-0.452109\pi$
$31$	1.66908	0.299775	0.149888	+	0.988703i	$0.452109\pi$
$32$	0	0
$33$	3.00000	0.522233
$34$	0	0
$35$	−1.42111	−0.240212
$36$	0	0
$37$	−10.0072	−1.64518	−0.822590	−	0.568635i	$-0.807472\pi$
$37$	−10.0072	−1.64518	−0.822590	+	0.568635i	$0.807472\pi$
$38$	0	0
$39$	−6.33816	−1.01492
$40$	0	0
$41$	−5.55233	−0.867129	−0.433564	−	0.901123i	$-0.642744\pi$
$41$	−5.55233	−0.867129	−0.433564	+	0.901123i	$0.642744\pi$
$42$	0	0
$43$	−3.00000	−0.457496	−0.228748	−	0.973486i	$-0.573463\pi$
$43$	−3.00000	−0.457496	−0.228748	+	0.973486i	$0.573463\pi$
$44$	0	0
$45$	0.454904	0.0678131
$46$	0	0
$47$	−1.42111	−0.207291	−0.103645	−	0.994614i	$-0.533051\pi$
$47$	−1.42111	−0.207291	−0.103645	+	0.994614i	$0.533051\pi$
$48$	0	0
$49$	2.75927	0.394182
$50$	0	0
$51$	−2.66908	−0.373746
$52$	0	0
$53$	−4.45490	−0.611928	−0.305964	−	0.952043i	$-0.598979\pi$
$53$	−4.45490	−0.611928	−0.305964	+	0.952043i	$0.598979\pi$
$54$	0	0
$55$	−1.36471	−0.184018
$56$	0	0
$57$	−1.21417	−0.160821
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−11.0072	−1.40933	−0.704666	−	0.709539i	$-0.748903\pi$
$61$	−11.0072	−1.40933	−0.704666	+	0.709539i	$0.748903\pi$
$62$	0	0
$63$	−3.12398	−0.393585
$64$	0	0
$65$	2.88325	0.357624
$66$	0	0
$67$	4.70287	0.574547	0.287274	−	0.957849i	$-0.407251\pi$
$67$	4.70287	0.574547	0.287274	+	0.957849i	$0.407251\pi$
$68$	0	0
$69$	4.12398	0.496469
$70$	0	0
$71$	−14.3382	−1.70163	−0.850813	−	0.525468i	$-0.823890\pi$
$71$	−14.3382	−1.70163	−0.850813	+	0.525468i	$0.823890\pi$
$72$	0	0
$73$	−8.09743	−0.947732	−0.473866	−	0.880597i	$-0.657142\pi$
$73$	−8.09743	−0.947732	−0.473866	+	0.880597i	$0.657142\pi$
$74$	0	0
$75$	4.79306	0.553455
$76$	0	0
$77$	9.37195	1.06803
$78$	0	0
$79$	5.24797	0.590442	0.295221	−	0.955429i	$-0.404607\pi$
$79$	5.24797	0.590442	0.295221	+	0.955429i	$0.404607\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−0.876017	−0.0961553	−0.0480777	−	0.998844i	$-0.515309\pi$
$83$	−0.876017	−0.0961553	−0.0480777	+	0.998844i	$0.515309\pi$
$84$	0	0
$85$	1.21417	0.131696
$86$	0	0
$87$	0.330921	0.0354784
$88$	0	0
$89$	−3.24073	−0.343517	−0.171758	−	0.985139i	$-0.554945\pi$
$89$	−3.24073	−0.343517	−0.171758	+	0.985139i	$0.554945\pi$
$90$	0	0
$91$	−19.8003	−2.07564
$92$	0	0
$93$	−1.66908	−0.173075
$94$	0	0
$95$	0.552333	0.0566682
$96$	0	0
$97$	17.5596	1.78290	0.891452	−	0.453115i	$-0.149687\pi$
$97$	17.5596	1.78290	0.891452	+	0.453115i	$0.149687\pi$
$98$	0	0
$99$	−3.00000	−0.301511
$100$	0	0
$101$	12.3116	1.22505	0.612525	−	0.790451i	$-0.290154\pi$
$101$	12.3116	1.22505	0.612525	+	0.790451i	$0.290154\pi$
$102$	0	0
$103$	15.8341	1.56018	0.780090	−	0.625668i	$-0.215173\pi$
$103$	15.8341	1.56018	0.780090	+	0.625668i	$0.215173\pi$
$104$	0	0
$105$	1.42111	0.138686
$106$	0	0
$107$	−18.5523	−1.79352	−0.896761	−	0.442515i	$-0.854086\pi$
$107$	−18.5523	−1.79352	−0.896761	+	0.442515i	$0.854086\pi$
$108$	0	0
$109$	−2.30437	−0.220718	−0.110359	−	0.993892i	$-0.535200\pi$
$109$	−2.30437	−0.220718	−0.110359	+	0.993892i	$0.535200\pi$
$110$	0	0
$111$	10.0072	0.949845
$112$	0	0
$113$	−16.4694	−1.54931	−0.774654	−	0.632385i	$-0.782076\pi$
$113$	−16.4694	−1.54931	−0.774654	+	0.632385i	$0.782076\pi$
$114$	0	0
$115$	−1.87602	−0.174939
$116$	0	0
$117$	6.33816	0.585963
$118$	0	0
$119$	−8.33816	−0.764358
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	5.55233	0.500637
$124$	0	0
$125$	−4.45490	−0.398459
$126$	0	0
$127$	3.54510	0.314576	0.157288	−	0.987553i	$-0.449725\pi$
$127$	3.54510	0.314576	0.157288	+	0.987553i	$0.449725\pi$
$128$	0	0
$129$	3.00000	0.264135
$130$	0	0
$131$	−14.4283	−1.26061	−0.630305	−	0.776347i	$-0.717070\pi$
$131$	−14.4283	−1.26061	−0.630305	+	0.776347i	$0.717070\pi$
$132$	0	0
$133$	−3.79306	−0.328900
$134$	0	0
$135$	−0.454904	−0.0391519
$136$	0	0
$137$	−0.338158	−0.0288908	−0.0144454	−	0.999896i	$-0.504598\pi$
$137$	−0.338158	−0.0288908	−0.0144454	+	0.999896i	$0.504598\pi$
$138$	0	0
$139$	−5.97345	−0.506661	−0.253331	−	0.967380i	$-0.581526\pi$
$139$	−5.97345	−0.506661	−0.253331	+	0.967380i	$0.581526\pi$
$140$	0	0
$141$	1.42111	0.119679
$142$	0	0
$143$	−19.0145	−1.59007
$144$	0	0
$145$	−0.150537	−0.0125014
$146$	0	0
$147$	−2.75927	−0.227581
$148$	0	0
$149$	−16.7367	−1.37112	−0.685560	−	0.728016i	$-0.740443\pi$
$149$	−16.7367	−1.37112	−0.685560	+	0.728016i	$0.740443\pi$
$150$	0	0
$151$	−3.25520	−0.264905	−0.132452	−	0.991189i	$-0.542285\pi$
$151$	−3.25520	−0.264905	−0.132452	+	0.991189i	$0.542285\pi$
$152$	0	0
$153$	2.66908	0.215782
$154$	0	0
$155$	0.759271	0.0609861
$156$	0	0
$157$	6.70287	0.534947	0.267474	−	0.963565i	$-0.413811\pi$
$157$	6.70287	0.534947	0.267474	+	0.963565i	$0.413811\pi$
$158$	0	0
$159$	4.45490	0.353297
$160$	0	0
$161$	12.8833	1.01534
$162$	0	0
$163$	−0.578887	−0.0453420	−0.0226710	−	0.999743i	$-0.507217\pi$
$163$	−0.578887	−0.0453420	−0.0226710	+	0.999743i	$0.507217\pi$
$164$	0	0
$165$	1.36471	0.106243
$166$	0	0
$167$	12.5934	0.974504	0.487252	−	0.873262i	$-0.337999\pi$
$167$	12.5934	0.974504	0.487252	+	0.873262i	$0.337999\pi$
$168$	0	0
$169$	27.1722	2.09017
$170$	0	0
$171$	1.21417	0.0928503
$172$	0	0
$173$	−20.0483	−1.52424	−0.762121	−	0.647435i	$-0.775842\pi$
$173$	−20.0483	−1.52424	−0.762121	+	0.647435i	$0.775842\pi$
$174$	0	0
$175$	14.9734	1.13189
$176$	0	0
$177$	−1.00000	−0.0751646
$178$	0	0
$179$	−3.48146	−0.260216	−0.130108	−	0.991500i	$-0.541532\pi$
$179$	−3.48146	−0.260216	−0.130108	+	0.991500i	$0.541532\pi$
$180$	0	0
$181$	10.0974	0.750536	0.375268	−	0.926916i	$-0.377551\pi$
$181$	10.0974	0.750536	0.375268	+	0.926916i	$0.377551\pi$
$182$	0	0
$183$	11.0072	0.813678
$184$	0	0
$185$	−4.55233	−0.334694
$186$	0	0
$187$	−8.00724	−0.585547
$188$	0	0
$189$	3.12398	0.227236
$190$	0	0
$191$	−3.18433	−0.230410	−0.115205	−	0.993342i	$-0.536752\pi$
$191$	−3.18433	−0.230410	−0.115205	+	0.993342i	$0.536752\pi$
$192$	0	0
$193$	12.7931	0.920865	0.460432	−	0.887695i	$-0.347694\pi$
$193$	12.7931	0.920865	0.460432	+	0.887695i	$0.347694\pi$
$194$	0	0
$195$	−2.88325	−0.206474
$196$	0	0
$197$	−6.06758	−0.432297	−0.216149	−	0.976360i	$-0.569350\pi$
$197$	−6.06758	−0.432297	−0.216149	+	0.976360i	$0.569350\pi$
$198$	0	0
$199$	−3.43559	−0.243542	−0.121771	−	0.992558i	$-0.538857\pi$
$199$	−3.43559	−0.243542	−0.121771	+	0.992558i	$0.538857\pi$
$200$	0	0
$201$	−4.70287	−0.331715
$202$	0	0
$203$	1.03379	0.0725579
$204$	0	0
$205$	−2.52578	−0.176408
$206$	0	0
$207$	−4.12398	−0.286637
$208$	0	0
$209$	−3.64252	−0.251959
$210$	0	0
$211$	−4.43559	−0.305358	−0.152679	−	0.988276i	$-0.548790\pi$
$211$	−4.43559	−0.305358	−0.152679	+	0.988276i	$0.548790\pi$
$212$	0	0
$213$	14.3382	0.982434
$214$	0	0
$215$	−1.36471	−0.0930726
$216$	0	0
$217$	−5.21417	−0.353961
$218$	0	0
$219$	8.09743	0.547174
$220$	0	0
$221$	16.9170	1.13796
$222$	0	0
$223$	25.8833	1.73327	0.866635	−	0.498942i	$-0.166278\pi$
$223$	25.8833	1.73327	0.866635	+	0.498942i	$0.166278\pi$
$224$	0	0
$225$	−4.79306	−0.319537
$226$	0	0
$227$	−6.24073	−0.414212	−0.207106	−	0.978319i	$-0.566404\pi$
$227$	−6.24073	−0.414212	−0.207106	+	0.978319i	$0.566404\pi$
$228$	0	0
$229$	8.28176	0.547274	0.273637	−	0.961833i	$-0.411773\pi$
$229$	8.28176	0.547274	0.273637	+	0.961833i	$0.411773\pi$
$230$	0	0
$231$	−9.37195	−0.616629
$232$	0	0
$233$	−4.60873	−0.301928	−0.150964	−	0.988539i	$-0.548238\pi$
$233$	−4.60873	−0.301928	−0.150964	+	0.988539i	$0.548238\pi$
$234$	0	0
$235$	−0.646470	−0.0421711
$236$	0	0
$237$	−5.24797	−0.340892
$238$	0	0
$239$	5.36471	0.347014	0.173507	−	0.984833i	$-0.444490\pi$
$239$	5.36471	0.347014	0.173507	+	0.984833i	$0.444490\pi$
$240$	0	0
$241$	−23.4694	−1.51180	−0.755898	−	0.654690i	$-0.772799\pi$
$241$	−23.4694	−1.51180	−0.755898	+	0.654690i	$0.772799\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.25520	0.0801920
$246$	0	0
$247$	7.69563	0.489661
$248$	0	0
$249$	0.876017	0.0555153
$250$	0	0
$251$	5.61268	0.354269	0.177135	−	0.984187i	$-0.443317\pi$
$251$	5.61268	0.354269	0.177135	+	0.984187i	$0.443317\pi$
$252$	0	0
$253$	12.3719	0.777818
$254$	0	0
$255$	−1.21417	−0.0760346
$256$	0	0
$257$	13.5225	0.843510	0.421755	−	0.906710i	$-0.361414\pi$
$257$	13.5225	0.843510	0.421755	+	0.906710i	$0.361414\pi$
$258$	0	0
$259$	31.2624	1.94255
$260$	0	0
$261$	−0.330921	−0.0204835
$262$	0	0
$263$	−10.9581	−0.675704	−0.337852	−	0.941199i	$-0.609700\pi$
$263$	−10.9581	−0.675704	−0.337852	+	0.941199i	$0.609700\pi$
$264$	0	0
$265$	−2.02655	−0.124490
$266$	0	0
$267$	3.24073	0.198329
$268$	0	0
$269$	17.0821	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$269$	17.0821	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$270$	0	0
$271$	21.0821	1.28064	0.640322	−	0.768107i	$-0.278801\pi$
$271$	21.0821	1.28064	0.640322	+	0.768107i	$0.278801\pi$
$272$	0	0
$273$	19.8003	1.19837
$274$	0	0
$275$	14.3792	0.867098
$276$	0	0
$277$	0.440430	0.0264628	0.0132314	−	0.999912i	$-0.495788\pi$
$277$	0.440430	0.0264628	0.0132314	+	0.999912i	$0.495788\pi$
$278$	0	0
$279$	1.66908	0.0999252
$280$	0	0
$281$	26.8075	1.59920	0.799602	−	0.600531i	$-0.205044\pi$
$281$	26.8075	1.59920	0.799602	+	0.600531i	$0.205044\pi$
$282$	0	0
$283$	17.3454	1.03108	0.515538	−	0.856866i	$-0.327592\pi$
$283$	17.3454	1.03108	0.515538	+	0.856866i	$0.327592\pi$
$284$	0	0
$285$	−0.552333	−0.0327174
$286$	0	0
$287$	17.3454	1.02387
$288$	0	0
$289$	−9.87602	−0.580942
$290$	0	0
$291$	−17.5596	−1.02936
$292$	0	0
$293$	16.5523	0.966998	0.483499	−	0.875345i	$-0.339366\pi$
$293$	16.5523	0.966998	0.483499	+	0.875345i	$0.339366\pi$
$294$	0	0
$295$	0.454904	0.0264855
$296$	0	0
$297$	3.00000	0.174078
$298$	0	0
$299$	−26.1385	−1.51163
$300$	0	0
$301$	9.37195	0.540190
$302$	0	0
$303$	−12.3116	−0.707283
$304$	0	0
$305$	−5.00724	−0.286714
$306$	0	0
$307$	−9.58612	−0.547109	−0.273555	−	0.961857i	$-0.588199\pi$
$307$	−9.58612	−0.547109	−0.273555	+	0.961857i	$0.588199\pi$
$308$	0	0
$309$	−15.8341	−0.900770
$310$	0	0
$311$	14.4887	0.821579	0.410789	−	0.911730i	$-0.365253\pi$
$311$	14.4887	0.821579	0.410789	+	0.911730i	$0.365253\pi$
$312$	0	0
$313$	16.6199	0.939413	0.469707	−	0.882823i	$-0.344360\pi$
$313$	16.6199	0.939413	0.469707	+	0.882823i	$0.344360\pi$
$314$	0	0
$315$	−1.42111	−0.0800706
$316$	0	0
$317$	−8.95084	−0.502729	−0.251365	−	0.967892i	$-0.580879\pi$
$317$	−8.95084	−0.502729	−0.251365	+	0.967892i	$0.580879\pi$
$318$	0	0
$319$	0.992763	0.0555840
$320$	0	0
$321$	18.5523	1.03549
$322$	0	0
$323$	3.24073	0.180319
$324$	0	0
$325$	−30.3792	−1.68513
$326$	0	0
$327$	2.30437	0.127432
$328$	0	0
$329$	4.43953	0.244759
$330$	0	0
$331$	−34.3300	−1.88695	−0.943474	−	0.331445i	$-0.892464\pi$
$331$	−34.3300	−1.88695	−0.943474	+	0.331445i	$0.892464\pi$
$332$	0	0
$333$	−10.0072	−0.548393
$334$	0	0
$335$	2.13936	0.116885
$336$	0	0
$337$	−15.0483	−0.819731	−0.409866	−	0.912146i	$-0.634424\pi$
$337$	−15.0483	−0.819731	−0.409866	+	0.912146i	$0.634424\pi$
$338$	0	0
$339$	16.4694	0.894494
$340$	0	0
$341$	−5.00724	−0.271157
$342$	0	0
$343$	13.2480	0.715323
$344$	0	0
$345$	1.87602	0.101001
$346$	0	0
$347$	29.7511	1.59712	0.798562	−	0.601912i	$-0.205594\pi$
$347$	29.7511	1.59712	0.798562	+	0.601912i	$0.205594\pi$
$348$	0	0
$349$	6.03708	0.323158	0.161579	−	0.986860i	$-0.448341\pi$
$349$	6.03708	0.323158	0.161579	+	0.986860i	$0.448341\pi$
$350$	0	0
$351$	−6.33816	−0.338306
$352$	0	0
$353$	31.4766	1.67533	0.837666	−	0.546183i	$-0.183920\pi$
$353$	31.4766	1.67533	0.837666	+	0.546183i	$0.183920\pi$
$354$	0	0
$355$	−6.52249	−0.346178
$356$	0	0
$357$	8.33816	0.441302
$358$	0	0
$359$	12.5596	0.662869	0.331434	−	0.943478i	$-0.392467\pi$
$359$	12.5596	0.662869	0.331434	+	0.943478i	$0.392467\pi$
$360$	0	0
$361$	−17.5258	−0.922409
$362$	0	0
$363$	2.00000	0.104973
$364$	0	0
$365$	−3.68355	−0.192806
$366$	0	0
$367$	33.4468	1.74591	0.872954	−	0.487803i	$-0.162202\pi$
$367$	33.4468	1.74591	0.872954	+	0.487803i	$0.162202\pi$
$368$	0	0
$369$	−5.55233	−0.289043
$370$	0	0
$371$	13.9170	0.722537
$372$	0	0
$373$	−3.28176	−0.169923	−0.0849615	−	0.996384i	$-0.527077\pi$
$373$	−3.28176	−0.169923	−0.0849615	+	0.996384i	$0.527077\pi$
$374$	0	0
$375$	4.45490	0.230050
$376$	0	0
$377$	−2.09743	−0.108023
$378$	0	0
$379$	−27.8606	−1.43111	−0.715553	−	0.698559i	$-0.753825\pi$
$379$	−27.8606	−1.43111	−0.715553	+	0.698559i	$0.753825\pi$
$380$	0	0
$381$	−3.54510	−0.181621
$382$	0	0
$383$	−31.2624	−1.59744	−0.798718	−	0.601705i	$-0.794488\pi$
$383$	−31.2624	−1.59744	−0.798718	+	0.601705i	$0.794488\pi$
$384$	0	0
$385$	4.26334	0.217280
$386$	0	0
$387$	−3.00000	−0.152499
$388$	0	0
$389$	−22.8567	−1.15888	−0.579440	−	0.815015i	$-0.696729\pi$
$389$	−22.8567	−1.15888	−0.579440	+	0.815015i	$0.696729\pi$
$390$	0	0
$391$	−11.0072	−0.556660
$392$	0	0
$393$	14.4283	0.727814
$394$	0	0
$395$	2.38732	0.120119
$396$	0	0
$397$	−4.52249	−0.226977	−0.113489	−	0.993539i	$-0.536203\pi$
$397$	−4.52249	−0.226977	−0.113489	+	0.993539i	$0.536203\pi$
$398$	0	0
$399$	3.79306	0.189891
$400$	0	0
$401$	19.8905	0.993284	0.496642	−	0.867956i	$-0.334566\pi$
$401$	19.8905	0.993284	0.496642	+	0.867956i	$0.334566\pi$
$402$	0	0
$403$	10.5789	0.526972
$404$	0	0
$405$	0.454904	0.0226044
$406$	0	0
$407$	30.0217	1.48812
$408$	0	0
$409$	−3.32368	−0.164346	−0.0821728	−	0.996618i	$-0.526186\pi$
$409$	−3.32368	−0.164346	−0.0821728	+	0.996618i	$0.526186\pi$
$410$	0	0
$411$	0.338158	0.0166801
$412$	0	0
$413$	−3.12398	−0.153721
$414$	0	0
$415$	−0.398504	−0.0195618
$416$	0	0
$417$	5.97345	0.292521
$418$	0	0
$419$	−30.1988	−1.47531	−0.737654	−	0.675179i	$-0.764066\pi$
$419$	−30.1988	−1.47531	−0.737654	+	0.675179i	$0.764066\pi$
$420$	0	0
$421$	37.2093	1.81347	0.906736	−	0.421699i	$-0.138566\pi$
$421$	37.2093	1.81347	0.906736	+	0.421699i	$0.138566\pi$
$422$	0	0
$423$	−1.42111	−0.0690969
$424$	0	0
$425$	−12.7931	−0.620555
$426$	0	0
$427$	34.3864	1.66408
$428$	0	0
$429$	19.0145	0.918027
$430$	0	0
$431$	−14.7062	−0.708371	−0.354185	−	0.935175i	$-0.615242\pi$
$431$	−14.7062	−0.708371	−0.354185	+	0.935175i	$0.615242\pi$
$432$	0	0
$433$	−28.5032	−1.36977	−0.684887	−	0.728649i	$-0.740149\pi$
$433$	−28.5032	−1.36977	−0.684887	+	0.728649i	$0.740149\pi$
$434$	0	0
$435$	0.150537	0.00721771
$436$	0	0
$437$	−5.00724	−0.239529
$438$	0	0
$439$	2.17315	0.103719	0.0518593	−	0.998654i	$-0.483485\pi$
$439$	2.17315	0.103719	0.0518593	+	0.998654i	$0.483485\pi$
$440$	0	0
$441$	2.75927	0.131394
$442$	0	0
$443$	−17.1505	−0.814847	−0.407423	−	0.913239i	$-0.633573\pi$
$443$	−17.1505	−0.814847	−0.407423	+	0.913239i	$0.633573\pi$
$444$	0	0
$445$	−1.47422	−0.0698848
$446$	0	0
$447$	16.7367	0.791617
$448$	0	0
$449$	−3.54904	−0.167490	−0.0837448	−	0.996487i	$-0.526688\pi$
$449$	−3.54904	−0.167490	−0.0837448	+	0.996487i	$0.526688\pi$
$450$	0	0
$451$	16.6570	0.784347
$452$	0	0
$453$	3.25520	0.152943
$454$	0	0
$455$	−9.00724	−0.422266
$456$	0	0
$457$	−22.3599	−1.04595	−0.522975	−	0.852348i	$-0.675178\pi$
$457$	−22.3599	−1.04595	−0.522975	+	0.852348i	$0.675178\pi$
$458$	0	0
$459$	−2.66908	−0.124582
$460$	0	0
$461$	−39.9725	−1.86171	−0.930854	−	0.365392i	$-0.880935\pi$
$461$	−39.9725	−1.86171	−0.930854	+	0.365392i	$0.880935\pi$
$462$	0	0
$463$	−13.1611	−0.611647	−0.305823	−	0.952088i	$-0.598932\pi$
$463$	−13.1611	−0.611647	−0.305823	+	0.952088i	$0.598932\pi$
$464$	0	0
$465$	−0.759271	−0.0352103
$466$	0	0
$467$	0.559570	0.0258938	0.0129469	−	0.999916i	$-0.495879\pi$
$467$	0.559570	0.0258938	0.0129469	+	0.999916i	$0.495879\pi$
$468$	0	0
$469$	−14.6917	−0.678399
$470$	0	0
$471$	−6.70287	−0.308852
$472$	0	0
$473$	9.00000	0.413820
$474$	0	0
$475$	−5.81962	−0.267022
$476$	0	0
$477$	−4.45490	−0.203976
$478$	0	0
$479$	−6.48475	−0.296296	−0.148148	−	0.988965i	$-0.547331\pi$
$479$	−6.48475	−0.296296	−0.148148	+	0.988965i	$0.547331\pi$
$480$	0	0
$481$	−63.4275	−2.89204
$482$	0	0
$483$	−12.8833	−0.586208
$484$	0	0
$485$	7.98792	0.362713
$486$	0	0
$487$	18.4211	0.834740	0.417370	−	0.908737i	$-0.362952\pi$
$487$	18.4211	0.834740	0.417370	+	0.908737i	$0.362952\pi$
$488$	0	0
$489$	0.578887	0.0261782
$490$	0	0
$491$	−24.4130	−1.10174	−0.550871	−	0.834590i	$-0.685704\pi$
$491$	−24.4130	−1.10174	−0.550871	+	0.834590i	$0.685704\pi$
$492$	0	0
$493$	−0.883254	−0.0397798
$494$	0	0
$495$	−1.36471	−0.0613393
$496$	0	0
$497$	44.7922	2.00920
$498$	0	0
$499$	25.4468	1.13915	0.569577	−	0.821938i	$-0.307107\pi$
$499$	25.4468	1.13915	0.569577	+	0.821938i	$0.307107\pi$
$500$	0	0
$501$	−12.5934	−0.562630
$502$	0	0
$503$	16.3309	0.728160	0.364080	−	0.931368i	$-0.381384\pi$
$503$	16.3309	0.728160	0.364080	+	0.931368i	$0.381384\pi$
$504$	0	0
$505$	5.60060	0.249223
$506$	0	0
$507$	−27.1722	−1.20676
$508$	0	0
$509$	−44.7206	−1.98221	−0.991104	−	0.133092i	$-0.957509\pi$
$509$	−44.7206	−1.98221	−0.991104	+	0.133092i	$0.957509\pi$
$510$	0	0
$511$	25.2962	1.11904
$512$	0	0
$513$	−1.21417	−0.0536071
$514$	0	0
$515$	7.20299	0.317402
$516$	0	0
$517$	4.26334	0.187501
$518$	0	0
$519$	20.0483	0.880021
$520$	0	0
$521$	−5.48870	−0.240464	−0.120232	−	0.992746i	$-0.538364\pi$
$521$	−5.48870	−0.240464	−0.120232	+	0.992746i	$0.538364\pi$
$522$	0	0
$523$	−9.77859	−0.427588	−0.213794	−	0.976879i	$-0.568582\pi$
$523$	−9.77859	−0.427588	−0.213794	+	0.976879i	$0.568582\pi$
$524$	0	0
$525$	−14.9734	−0.653495
$526$	0	0
$527$	4.45490	0.194059
$528$	0	0
$529$	−5.99276	−0.260555
$530$	0	0
$531$	1.00000	0.0433963
$532$	0	0
$533$	−35.1916	−1.52432
$534$	0	0
$535$	−8.43953	−0.364873
$536$	0	0
$537$	3.48146	0.150236
$538$	0	0
$539$	−8.27781	−0.356551
$540$	0	0
$541$	31.2769	1.34470	0.672350	−	0.740234i	$-0.265285\pi$
$541$	31.2769	1.34470	0.672350	+	0.740234i	$0.265285\pi$
$542$	0	0
$543$	−10.0974	−0.433322
$544$	0	0
$545$	−1.04827	−0.0449028
$546$	0	0
$547$	16.4251	0.702285	0.351142	−	0.936322i	$-0.385793\pi$
$547$	16.4251	0.702285	0.351142	+	0.936322i	$0.385793\pi$
$548$	0	0
$549$	−11.0072	−0.469777
$550$	0	0
$551$	−0.401796	−0.0171171
$552$	0	0
$553$	−16.3946	−0.697168
$554$	0	0
$555$	4.55233	0.193236
$556$	0	0
$557$	20.9098	0.885977	0.442989	−	0.896527i	$-0.353918\pi$
$557$	20.9098	0.885977	0.442989	+	0.896527i	$0.353918\pi$
$558$	0	0
$559$	−19.0145	−0.804227
$560$	0	0
$561$	8.00724	0.338066
$562$	0	0
$563$	−21.9734	−0.926070	−0.463035	−	0.886340i	$-0.653240\pi$
$563$	−21.9734	−0.926070	−0.463035	+	0.886340i	$0.653240\pi$
$564$	0	0
$565$	−7.49199	−0.315190
$566$	0	0
$567$	−3.12398	−0.131195
$568$	0	0
$569$	−18.8124	−0.788656	−0.394328	−	0.918970i	$-0.629023\pi$
$569$	−18.8124	−0.788656	−0.394328	+	0.918970i	$0.629023\pi$
$570$	0	0
$571$	−23.6272	−0.988766	−0.494383	−	0.869244i	$-0.664606\pi$
$571$	−23.6272	−0.988766	−0.494383	+	0.869244i	$0.664606\pi$
$572$	0	0
$573$	3.18433	0.133027
$574$	0	0
$575$	19.7665	0.824320
$576$	0	0
$577$	17.8036	0.741173	0.370587	−	0.928798i	$-0.379157\pi$
$577$	17.8036	0.741173	0.370587	+	0.928798i	$0.379157\pi$
$578$	0	0
$579$	−12.7931	−0.531662
$580$	0	0
$581$	2.73666	0.113536
$582$	0	0
$583$	13.3647	0.553510
$584$	0	0
$585$	2.88325	0.119208
$586$	0	0
$587$	6.33816	0.261604	0.130802	−	0.991409i	$-0.458245\pi$
$587$	6.33816	0.261604	0.130802	+	0.991409i	$0.458245\pi$
$588$	0	0
$589$	2.02655	0.0835027
$590$	0	0
$591$	6.06758	0.249587
$592$	0	0
$593$	−10.3454	−0.424834	−0.212417	−	0.977179i	$-0.568134\pi$
$593$	−10.3454	−0.424834	−0.212417	+	0.977179i	$0.568134\pi$
$594$	0	0
$595$	−3.79306	−0.155500
$596$	0	0
$597$	3.43559	0.140609
$598$	0	0
$599$	18.6803	0.763255	0.381627	−	0.924316i	$-0.375364\pi$
$599$	18.6803	0.763255	0.381627	+	0.924316i	$0.375364\pi$
$600$	0	0
$601$	−6.12398	−0.249802	−0.124901	−	0.992169i	$-0.539861\pi$
$601$	−6.12398	−0.249802	−0.124901	+	0.992169i	$0.539861\pi$
$602$	0	0
$603$	4.70287	0.191516
$604$	0	0
$605$	−0.909808	−0.0369890
$606$	0	0
$607$	15.3687	0.623795	0.311897	−	0.950116i	$-0.399036\pi$
$607$	15.3687	0.623795	0.311897	+	0.950116i	$0.399036\pi$
$608$	0	0
$609$	−1.03379	−0.0418913
$610$	0	0
$611$	−9.00724	−0.364394
$612$	0	0
$613$	11.5370	0.465973	0.232987	−	0.972480i	$-0.425150\pi$
$613$	11.5370	0.465973	0.232987	+	0.972480i	$0.425150\pi$
$614$	0	0
$615$	2.52578	0.101849
$616$	0	0
$617$	21.3599	0.859916	0.429958	−	0.902849i	$-0.358528\pi$
$617$	21.3599	0.859916	0.429958	+	0.902849i	$0.358528\pi$
$618$	0	0
$619$	−21.0105	−0.844484	−0.422242	−	0.906483i	$-0.638757\pi$
$619$	−21.0105	−0.844484	−0.422242	+	0.906483i	$0.638757\pi$
$620$	0	0
$621$	4.12398	0.165490
$622$	0	0
$623$	10.1240	0.405609
$624$	0	0
$625$	21.9388	0.877550
$626$	0	0
$627$	3.64252	0.145468
$628$	0	0
$629$	−26.7101	−1.06500
$630$	0	0
$631$	36.8341	1.46634	0.733171	−	0.680044i	$-0.238039\pi$
$631$	36.8341	1.46634	0.733171	+	0.680044i	$0.238039\pi$
$632$	0	0
$633$	4.43559	0.176299
$634$	0	0
$635$	1.61268	0.0639972
$636$	0	0
$637$	17.4887	0.692927
$638$	0	0
$639$	−14.3382	−0.567209
$640$	0	0
$641$	24.1617	0.954331	0.477165	−	0.878814i	$-0.341664\pi$
$641$	24.1617	0.954331	0.477165	+	0.878814i	$0.341664\pi$
$642$	0	0
$643$	20.3333	0.801868	0.400934	−	0.916107i	$-0.368686\pi$
$643$	20.3333	0.801868	0.400934	+	0.916107i	$0.368686\pi$
$644$	0	0
$645$	1.36471	0.0537355
$646$	0	0
$647$	2.64737	0.104079	0.0520394	−	0.998645i	$-0.483428\pi$
$647$	2.64737	0.104079	0.0520394	+	0.998645i	$0.483428\pi$
$648$	0	0
$649$	−3.00000	−0.117760
$650$	0	0
$651$	5.21417	0.204360
$652$	0	0
$653$	37.8108	1.47965	0.739826	−	0.672798i	$-0.234908\pi$
$653$	37.8108	1.47965	0.739826	+	0.672798i	$0.234908\pi$
$654$	0	0
$655$	−6.56352	−0.256458
$656$	0	0
$657$	−8.09743	−0.315911
$658$	0	0
$659$	−28.0257	−1.09172	−0.545862	−	0.837875i	$-0.683798\pi$
$659$	−28.0257	−1.09172	−0.545862	+	0.837875i	$0.683798\pi$
$660$	0	0
$661$	−3.66908	−0.142711	−0.0713553	−	0.997451i	$-0.522732\pi$
$661$	−3.66908	−0.142711	−0.0713553	+	0.997451i	$0.522732\pi$
$662$	0	0
$663$	−16.9170	−0.657004
$664$	0	0
$665$	−1.72548	−0.0669112
$666$	0	0
$667$	1.36471	0.0528419
$668$	0	0
$669$	−25.8833	−1.00070
$670$	0	0
$671$	33.0217	1.27479
$672$	0	0
$673$	18.4959	0.712966	0.356483	−	0.934302i	$-0.383976\pi$
$673$	18.4959	0.712966	0.356483	+	0.934302i	$0.383976\pi$
$674$	0	0
$675$	4.79306	0.184485
$676$	0	0
$677$	9.15777	0.351962	0.175981	−	0.984394i	$-0.443690\pi$
$677$	9.15777	0.351962	0.175981	+	0.984394i	$0.443690\pi$
$678$	0	0
$679$	−54.8558	−2.10517
$680$	0	0
$681$	6.24073	0.239145
$682$	0	0
$683$	29.4573	1.12715	0.563576	−	0.826064i	$-0.309425\pi$
$683$	29.4573	1.12715	0.563576	+	0.826064i	$0.309425\pi$
$684$	0	0
$685$	−0.153830	−0.00587752
$686$	0	0
$687$	−8.28176	−0.315969
$688$	0	0
$689$	−28.2359	−1.07570
$690$	0	0
$691$	24.8075	0.943723	0.471862	−	0.881673i	$-0.343582\pi$
$691$	24.8075	0.943723	0.471862	+	0.881673i	$0.343582\pi$
$692$	0	0
$693$	9.37195	0.356011
$694$	0	0
$695$	−2.71734	−0.103075
$696$	0	0
$697$	−14.8196	−0.561333
$698$	0	0
$699$	4.60873	0.174318
$700$	0	0
$701$	42.6425	1.61059	0.805293	−	0.592877i	$-0.202008\pi$
$701$	42.6425	1.61059	0.805293	+	0.592877i	$0.202008\pi$
$702$	0	0
$703$	−12.1505	−0.458266
$704$	0	0
$705$	0.646470	0.0243475
$706$	0	0
$707$	−38.4612	−1.44648
$708$	0	0
$709$	−41.9693	−1.57619	−0.788094	−	0.615555i	$-0.788932\pi$
$709$	−41.9693	−1.57619	−0.788094	+	0.615555i	$0.788932\pi$
$710$	0	0
$711$	5.24797	0.196814
$712$	0	0
$713$	−6.88325	−0.257780
$714$	0	0
$715$	−8.64976	−0.323483
$716$	0	0
$717$	−5.36471	−0.200349
$718$	0	0
$719$	−5.15448	−0.192230	−0.0961149	−	0.995370i	$-0.530642\pi$
$719$	−5.15448	−0.192230	−0.0961149	+	0.995370i	$0.530642\pi$
$720$	0	0
$721$	−49.4654	−1.84219
$722$	0	0
$723$	23.4694	0.872836
$724$	0	0
$725$	1.58612	0.0589072
$726$	0	0
$727$	0.116746	0.00432987	0.00216493	−	0.999998i	$-0.499311\pi$
$727$	0.116746	0.00432987	0.00216493	+	0.999998i	$0.499311\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.00724	−0.296158
$732$	0	0
$733$	−41.3261	−1.52641	−0.763207	−	0.646154i	$-0.776376\pi$
$733$	−41.3261	−1.52641	−0.763207	+	0.646154i	$0.776376\pi$
$734$	0	0
$735$	−1.25520	−0.0462989
$736$	0	0
$737$	−14.1086	−0.519697
$738$	0	0
$739$	26.7367	0.983524	0.491762	−	0.870730i	$-0.336353\pi$
$739$	26.7367	0.983524	0.491762	+	0.870730i	$0.336353\pi$
$740$	0	0
$741$	−7.69563	−0.282706
$742$	0	0
$743$	7.03379	0.258045	0.129022	−	0.991642i	$-0.458816\pi$
$743$	7.03379	0.258045	0.129022	+	0.991642i	$0.458816\pi$
$744$	0	0
$745$	−7.61358	−0.278940
$746$	0	0
$747$	−0.876017	−0.0320518
$748$	0	0
$749$	57.9572	2.11771
$750$	0	0
$751$	−33.3382	−1.21653	−0.608263	−	0.793735i	$-0.708134\pi$
$751$	−33.3382	−1.21653	−0.608263	+	0.793735i	$0.708134\pi$
$752$	0	0
$753$	−5.61268	−0.204537
$754$	0	0
$755$	−1.48081	−0.0538920
$756$	0	0
$757$	−10.6730	−0.387918	−0.193959	−	0.981010i	$-0.562133\pi$
$757$	−10.6730	−0.387918	−0.193959	+	0.981010i	$0.562133\pi$
$758$	0	0
$759$	−12.3719	−0.449073
$760$	0	0
$761$	−25.6272	−0.928984	−0.464492	−	0.885577i	$-0.653763\pi$
$761$	−25.6272	−0.928984	−0.464492	+	0.885577i	$0.653763\pi$
$762$	0	0
$763$	7.19880	0.260614
$764$	0	0
$765$	1.21417	0.0438986
$766$	0	0
$767$	6.33816	0.228858
$768$	0	0
$769$	−50.6875	−1.82784	−0.913919	−	0.405897i	$-0.866959\pi$
$769$	−50.6875	−1.82784	−0.913919	+	0.405897i	$0.866959\pi$
$770$	0	0
$771$	−13.5225	−0.487000
$772$	0	0
$773$	−12.4815	−0.448927	−0.224463	−	0.974483i	$-0.572063\pi$
$773$	−12.4815	−0.448927	−0.224463	+	0.974483i	$0.572063\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	−31.2624	−1.12153
$778$	0	0
$779$	−6.74150	−0.241539
$780$	0	0
$781$	43.0145	1.53918
$782$	0	0
$783$	0.330921	0.0118261
$784$	0	0
$785$	3.04916	0.108829
$786$	0	0
$787$	−43.0516	−1.53462	−0.767311	−	0.641275i	$-0.778406\pi$
$787$	−43.0516	−1.53462	−0.767311	+	0.641275i	$0.778406\pi$
$788$	0	0
$789$	10.9581	0.390118
$790$	0	0
$791$	51.4501	1.82935
$792$	0	0
$793$	−69.7656	−2.47745
$794$	0	0
$795$	2.02655	0.0718745
$796$	0	0
$797$	14.4920	0.513333	0.256666	−	0.966500i	$-0.417376\pi$
$797$	14.4920	0.513333	0.256666	+	0.966500i	$0.417376\pi$
$798$	0	0
$799$	−3.79306	−0.134189
$800$	0	0
$801$	−3.24073	−0.114506
$802$	0	0
$803$	24.2923	0.857256
$804$	0	0
$805$	5.86064	0.206561
$806$	0	0
$807$	−17.0821	−0.601317
$808$	0	0
$809$	−16.3044	−0.573231	−0.286616	−	0.958046i	$-0.592530\pi$
$809$	−16.3044	−0.573231	−0.286616	+	0.958046i	$0.592530\pi$
$810$	0	0
$811$	−34.1843	−1.20037	−0.600187	−	0.799860i	$-0.704907\pi$
$811$	−34.1843	−1.20037	−0.600187	+	0.799860i	$0.704907\pi$
$812$	0	0
$813$	−21.0821	−0.739380
$814$	0	0
$815$	−0.263338	−0.00922433
$816$	0	0
$817$	−3.64252	−0.127436
$818$	0	0
$819$	−19.8003	−0.691878
$820$	0	0
$821$	−44.9050	−1.56719	−0.783597	−	0.621269i	$-0.786617\pi$
$821$	−44.9050	−1.56719	−0.783597	+	0.621269i	$0.786617\pi$
$822$	0	0
$823$	44.2359	1.54197	0.770983	−	0.636856i	$-0.219765\pi$
$823$	44.2359	1.54197	0.770983	+	0.636856i	$0.219765\pi$
$824$	0	0
$825$	−14.3792	−0.500619
$826$	0	0
$827$	6.75598	0.234928	0.117464	−	0.993077i	$-0.462523\pi$
$827$	6.75598	0.234928	0.117464	+	0.993077i	$0.462523\pi$
$828$	0	0
$829$	−40.5403	−1.40802	−0.704011	−	0.710189i	$-0.748609\pi$
$829$	−40.5403	−1.40802	−0.704011	+	0.710189i	$0.748609\pi$
$830$	0	0
$831$	−0.440430	−0.0152783
$832$	0	0
$833$	7.36471	0.255172
$834$	0	0
$835$	5.72877	0.198252
$836$	0	0
$837$	−1.66908	−0.0576918
$838$	0	0
$839$	−42.2359	−1.45814	−0.729072	−	0.684437i	$-0.760048\pi$
$839$	−42.2359	−1.45814	−0.729072	+	0.684437i	$0.760048\pi$
$840$	0	0
$841$	−28.8905	−0.996224
$842$	0	0
$843$	−26.8075	−0.923301
$844$	0	0
$845$	12.3608	0.425223
$846$	0	0
$847$	6.24797	0.214683
$848$	0	0
$849$	−17.3454	−0.595292
$850$	0	0
$851$	41.2697	1.41471
$852$	0	0
$853$	14.7327	0.504439	0.252219	−	0.967670i	$-0.418840\pi$
$853$	14.7327	0.504439	0.252219	+	0.967670i	$0.418840\pi$
$854$	0	0
$855$	0.552333	0.0188894
$856$	0	0
$857$	−12.1086	−0.413622	−0.206811	−	0.978381i	$-0.566309\pi$
$857$	−12.1086	−0.413622	−0.206811	+	0.978381i	$0.566309\pi$
$858$	0	0
$859$	−9.76651	−0.333229	−0.166614	−	0.986022i	$-0.553284\pi$
$859$	−9.76651	−0.333229	−0.166614	+	0.986022i	$0.553284\pi$
$860$	0	0
$861$	−17.3454	−0.591129
$862$	0	0
$863$	29.6570	1.00954	0.504768	−	0.863255i	$-0.331578\pi$
$863$	29.6570	1.00954	0.504768	+	0.863255i	$0.331578\pi$
$864$	0	0
$865$	−9.12004	−0.310091
$866$	0	0
$867$	9.87602	0.335407
$868$	0	0
$869$	−15.7439	−0.534075
$870$	0	0
$871$	29.8075	1.00999
$872$	0	0
$873$	17.5596	0.594301
$874$	0	0
$875$	13.9170	0.470482
$876$	0	0
$877$	23.2254	0.784265	0.392132	−	0.919909i	$-0.371738\pi$
$877$	23.2254	0.784265	0.392132	+	0.919909i	$0.371738\pi$
$878$	0	0
$879$	−16.5523	−0.558296
$880$	0	0
$881$	5.20299	0.175293	0.0876466	−	0.996152i	$-0.472065\pi$
$881$	5.20299	0.175293	0.0876466	+	0.996152i	$0.472065\pi$
$882$	0	0
$883$	−45.3816	−1.52721	−0.763606	−	0.645683i	$-0.776573\pi$
$883$	−45.3816	−1.52721	−0.763606	+	0.645683i	$0.776573\pi$
$884$	0	0
$885$	−0.454904	−0.0152914
$886$	0	0
$887$	36.6642	1.23106	0.615532	−	0.788112i	$-0.288941\pi$
$887$	36.6642	1.23106	0.615532	+	0.788112i	$0.288941\pi$
$888$	0	0
$889$	−11.0748	−0.371438
$890$	0	0
$891$	−3.00000	−0.100504
$892$	0	0
$893$	−1.72548	−0.0577410
$894$	0	0
$895$	−1.58373	−0.0529382
$896$	0	0
$897$	26.1385	0.872738
$898$	0	0
$899$	−0.552333	−0.0184213
$900$	0	0
$901$	−11.8905	−0.396130
$902$	0	0
$903$	−9.37195	−0.311879
$904$	0	0
$905$	4.59336	0.152689
$906$	0	0
$907$	21.6724	0.719619	0.359810	−	0.933026i	$-0.382842\pi$
$907$	21.6724	0.719619	0.359810	+	0.933026i	$0.382842\pi$
$908$	0	0
$909$	12.3116	0.408350
$910$	0	0
$911$	57.3864	1.90130	0.950649	−	0.310270i	$-0.100419\pi$
$911$	57.3864	1.90130	0.950649	+	0.310270i	$0.100419\pi$
$912$	0	0
$913$	2.62805	0.0869758
$914$	0	0
$915$	5.00724	0.165534
$916$	0	0
$917$	45.0739	1.48847
$918$	0	0
$919$	0.375242	0.0123781	0.00618904	−	0.999981i	$-0.498030\pi$
$919$	0.375242	0.0123781	0.00618904	+	0.999981i	$0.498030\pi$
$920$	0	0
$921$	9.58612	0.315874
$922$	0	0
$923$	−90.8775	−2.99127
$924$	0	0
$925$	47.9653	1.57709
$926$	0	0
$927$	15.8341	0.520060
$928$	0	0
$929$	5.25850	0.172526	0.0862628	−	0.996272i	$-0.472508\pi$
$929$	5.25850	0.172526	0.0862628	+	0.996272i	$0.472508\pi$
$930$	0	0
$931$	3.35024	0.109800
$932$	0	0
$933$	−14.4887	−0.474339
$934$	0	0
$935$	−3.64252	−0.119123
$936$	0	0
$937$	−5.73995	−0.187516	−0.0937581	−	0.995595i	$-0.529888\pi$
$937$	−5.73995	−0.187516	−0.0937581	+	0.995595i	$0.529888\pi$
$938$	0	0
$939$	−16.6199	−0.542370
$940$	0	0
$941$	24.5225	0.799410	0.399705	−	0.916644i	$-0.369113\pi$
$941$	24.5225	0.799410	0.399705	+	0.916644i	$0.369113\pi$
$942$	0	0
$943$	22.8977	0.745653
$944$	0	0
$945$	1.42111	0.0462288
$946$	0	0
$947$	−22.1650	−0.720266	−0.360133	−	0.932901i	$-0.617269\pi$
$947$	−22.1650	−0.720266	−0.360133	+	0.932901i	$0.617269\pi$
$948$	0	0
$949$	−51.3228	−1.66601
$950$	0	0
$951$	8.95084	0.290251
$952$	0	0
$953$	−53.9122	−1.74639	−0.873194	−	0.487373i	$-0.837955\pi$
$953$	−53.9122	−1.74639	−0.873194	+	0.487373i	$0.837955\pi$
$954$	0	0
$955$	−1.44856	−0.0468744
$956$	0	0
$957$	−0.992763	−0.0320915
$958$	0	0
$959$	1.05640	0.0341129
$960$	0	0
$961$	−28.2142	−0.910135
$962$	0	0
$963$	−18.5523	−0.597841
$964$	0	0
$965$	5.81962	0.187340
$966$	0	0
$967$	56.6941	1.82316	0.911579	−	0.411124i	$-0.134864\pi$
$967$	56.6941	1.82316	0.911579	+	0.411124i	$0.134864\pi$
$968$	0	0
$969$	−3.24073	−0.104107
$970$	0	0
$971$	−16.7327	−0.536978	−0.268489	−	0.963283i	$-0.586524\pi$
$971$	−16.7327	−0.536978	−0.268489	+	0.963283i	$0.586524\pi$
$972$	0	0
$973$	18.6609	0.598242
$974$	0	0
$975$	30.3792	0.972913
$976$	0	0
$977$	−19.4612	−0.622620	−0.311310	−	0.950308i	$-0.600768\pi$
$977$	−19.4612	−0.622620	−0.311310	+	0.950308i	$0.600768\pi$
$978$	0	0
$979$	9.72219	0.310722
$980$	0	0
$981$	−2.30437	−0.0735728
$982$	0	0
$983$	8.07966	0.257701	0.128851	−	0.991664i	$-0.458871\pi$
$983$	8.07966	0.257701	0.128851	+	0.991664i	$0.458871\pi$
$984$	0	0
$985$	−2.76017	−0.0879463
$986$	0	0
$987$	−4.43953	−0.141312
$988$	0	0
$989$	12.3719	0.393405
$990$	0	0
$991$	−46.8148	−1.48712	−0.743560	−	0.668669i	$-0.766864\pi$
$991$	−46.8148	−1.48712	−0.743560	+	0.668669i	$0.766864\pi$
$992$	0	0
$993$	34.3300	1.08943
$994$	0	0
$995$	−1.56286	−0.0495461
$996$	0	0
$997$	−0.759271	−0.0240464	−0.0120232	−	0.999928i	$-0.503827\pi$
$997$	−0.759271	−0.0240464	−0.0120232	+	0.999928i	$0.503827\pi$
$998$	0	0
$999$	10.0072	0.316615

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1416.2.a.d.1.2	✓	3
3.2	odd	2		4248.2.a.m.1.2		3
4.3	odd	2		2832.2.a.u.1.2		3
12.11	even	2		8496.2.a.bk.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1416.2.a.d.1.2	✓	3	1.1	even	1	trivial
2832.2.a.u.1.2		3	4.3	odd	2
4248.2.a.m.1.2		3	3.2	odd	2
8496.2.a.bk.1.2		3	12.11	even	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:	\( N \)	\(=\)	\( 1416 = 2^{3} \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1416.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +0.454904 q^{5} -3.12398 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +0.454904 q^{5} -3.12398 q^{7} +1.00000 q^{9} -3.00000 q^{11} +6.33816 q^{13} -0.454904 q^{15} +2.66908 q^{17} +1.21417 q^{19} +3.12398 q^{21} -4.12398 q^{23} -4.79306 q^{25} -1.00000 q^{27} -0.330921 q^{29} +1.66908 q^{31} +3.00000 q^{33} -1.42111 q^{35} -10.0072 q^{37} -6.33816 q^{39} -5.55233 q^{41} -3.00000 q^{43} +0.454904 q^{45} -1.42111 q^{47} +2.75927 q^{49} -2.66908 q^{51} -4.45490 q^{53} -1.36471 q^{55} -1.21417 q^{57} +1.00000 q^{59} -11.0072 q^{61} -3.12398 q^{63} +2.88325 q^{65} +4.70287 q^{67} +4.12398 q^{69} -14.3382 q^{71} -8.09743 q^{73} +4.79306 q^{75} +9.37195 q^{77} +5.24797 q^{79} +1.00000 q^{81} -0.876017 q^{83} +1.21417 q^{85} +0.330921 q^{87} -3.24073 q^{89} -19.8003 q^{91} -1.66908 q^{93} +0.552333 q^{95} +17.5596 q^{97} -3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{9} - 9 q^{11} + 3 q^{13} - 3 q^{19} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 9 q^{29} - 3 q^{31} + 9 q^{33} - 15 q^{35} - 6 q^{37} - 3 q^{39} + 6 q^{41} - 9 q^{43} - 15 q^{47} + 3 q^{49} - 12 q^{53} + 3 q^{57} + 3 q^{59} - 9 q^{61} - 6 q^{65} - 6 q^{67} + 3 q^{69} - 27 q^{71} - 3 q^{73} - 3 q^{75} - 3 q^{79} + 3 q^{81} - 12 q^{83} - 3 q^{85} + 9 q^{87} - 15 q^{89} - 18 q^{91} + 3 q^{93} - 21 q^{95} + 6 q^{97} - 9 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^9 - 9 * q^11 + 3 * q^13 - 3 * q^19 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 9 * q^29 - 3 * q^31 + 9 * q^33 - 15 * q^35 - 6 * q^37 - 3 * q^39 + 6 * q^41 - 9 * q^43 - 15 * q^47 + 3 * q^49 - 12 * q^53 + 3 * q^57 + 3 * q^59 - 9 * q^61 - 6 * q^65 - 6 * q^67 + 3 * q^69 - 27 * q^71 - 3 * q^73 - 3 * q^75 - 3 * q^79 + 3 * q^81 - 12 * q^83 - 3 * q^85 + 9 * q^87 - 15 * q^89 - 18 * q^91 + 3 * q^93 - 21 * q^95 + 6 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more