LMFDB - Embedded newform 1680.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(1,1680)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1680, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1680.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 840)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1680.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.65685 q^{11} +2.00000 q^{13} -1.00000 q^{15} +7.65685 q^{17} -5.65685 q^{19} +1.00000 q^{21} -5.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.65685 q^{29} -4.00000 q^{31} -5.65685 q^{33} -1.00000 q^{35} +0.343146 q^{37} -2.00000 q^{39} +2.00000 q^{41} +9.65685 q^{43} +1.00000 q^{45} +13.6569 q^{47} +1.00000 q^{49} -7.65685 q^{51} +7.65685 q^{53} +5.65685 q^{55} +5.65685 q^{57} +4.00000 q^{59} +11.6569 q^{61} -1.00000 q^{63} +2.00000 q^{65} -1.65685 q^{67} +5.65685 q^{69} -15.3137 q^{71} +6.00000 q^{73} -1.00000 q^{75} -5.65685 q^{77} +11.3137 q^{79} +1.00000 q^{81} +4.00000 q^{83} +7.65685 q^{85} +7.65685 q^{87} -14.0000 q^{89} -2.00000 q^{91} +4.00000 q^{93} -5.65685 q^{95} +6.00000 q^{97} +5.65685 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 2 q^{35} + 12 q^{37} - 4 q^{39} + 4 q^{41} + 8 q^{43} + 2 q^{45} + 16 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{59} + 12 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} - 8 q^{71} + 12 q^{73} - 2 q^{75} + 2 q^{81} + 8 q^{83} + 4 q^{85} + 4 q^{87} - 28 q^{89} - 4 q^{91} + 8 q^{93} + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^31 - 2 * q^35 + 12 * q^37 - 4 * q^39 + 4 * q^41 + 8 * q^43 + 2 * q^45 + 16 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^59 + 12 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 - 8 * q^71 + 12 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 4 * q^85 + 4 * q^87 - 28 * q^89 - 4 * q^91 + 8 * q^93 + 12 * q^97

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.65685	1.70561	0.852803	−	0.522233i	$-0.174901\pi$
$11$	5.65685	1.70561	0.852803	+	0.522233i	$0.174901\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	7.65685	1.85706	0.928530	−	0.371257i	$-0.121073\pi$
$17$	7.65685	1.85706	0.928530	+	0.371257i	$0.121073\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−5.65685	−0.984732
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	0.343146	0.0564128	0.0282064	−	0.999602i	$-0.491020\pi$
$37$	0.343146	0.0564128	0.0282064	+	0.999602i	$0.491020\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	9.65685	1.47266	0.736328	−	0.676625i	$-0.236558\pi$
$43$	9.65685	1.47266	0.736328	+	0.676625i	$0.236558\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	13.6569	1.99206	0.996028	−	0.0890354i	$-0.0283784\pi$
$47$	13.6569	1.99206	0.996028	+	0.0890354i	$0.0283784\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−7.65685	−1.07217
$52$	0	0
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	0	0
$55$	5.65685	0.762770
$56$	0	0
$57$	5.65685	0.749269
$58$	0	0
$59$	4.00000	0.520756	0.260378	−	0.965507i	$-0.416153\pi$
$59$	4.00000	0.520756	0.260378	+	0.965507i	$0.416153\pi$
$60$	0	0
$61$	11.6569	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$61$	11.6569	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	−1.65685	−0.202417	−0.101208	−	0.994865i	$-0.532271\pi$
$67$	−1.65685	−0.202417	−0.101208	+	0.994865i	$0.532271\pi$
$68$	0	0
$69$	5.65685	0.681005
$70$	0	0
$71$	−15.3137	−1.81740	−0.908701	−	0.417447i	$-0.862925\pi$
$71$	−15.3137	−1.81740	−0.908701	+	0.417447i	$0.862925\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.65685	−0.644658
$78$	0	0
$79$	11.3137	1.27289	0.636446	−	0.771321i	$-0.280404\pi$
$79$	11.3137	1.27289	0.636446	+	0.771321i	$0.280404\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	7.65685	0.830502
$86$	0	0
$87$	7.65685	0.820901
$88$	0	0
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	−2.00000	−0.209657
$92$	0	0
$93$	4.00000	0.414781
$94$	0	0
$95$	−5.65685	−0.580381
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	5.65685	0.568535
$100$	0	0
$101$	−5.31371	−0.528734	−0.264367	−	0.964422i	$-0.585163\pi$
$101$	−5.31371	−0.528734	−0.264367	+	0.964422i	$0.585163\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	17.6569	1.70695	0.853476	−	0.521132i	$-0.174490\pi$
$107$	17.6569	1.70695	0.853476	+	0.521132i	$0.174490\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	−0.343146	−0.0325700
$112$	0	0
$113$	−15.6569	−1.47287	−0.736436	−	0.676507i	$-0.763493\pi$
$113$	−15.6569	−1.47287	−0.736436	+	0.676507i	$0.763493\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	−7.65685	−0.701903
$120$	0	0
$121$	21.0000	1.90909
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.31371	−0.294044	−0.147022	−	0.989133i	$-0.546969\pi$
$127$	−3.31371	−0.294044	−0.147022	+	0.989133i	$0.546969\pi$
$128$	0	0
$129$	−9.65685	−0.850239
$130$	0	0
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	5.65685	0.490511
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−15.6569	−1.33766	−0.668828	−	0.743417i	$-0.733204\pi$
$137$	−15.6569	−1.33766	−0.668828	+	0.743417i	$0.733204\pi$
$138$	0	0
$139$	2.34315	0.198743	0.0993715	−	0.995050i	$-0.468317\pi$
$139$	2.34315	0.198743	0.0993715	+	0.995050i	$0.468317\pi$
$140$	0	0
$141$	−13.6569	−1.15011
$142$	0	0
$143$	11.3137	0.946100
$144$	0	0
$145$	−7.65685	−0.635867
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	14.9706	1.22644	0.613218	−	0.789914i	$-0.289875\pi$
$149$	14.9706	1.22644	0.613218	+	0.789914i	$0.289875\pi$
$150$	0	0
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	7.65685	0.619020
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	−7.65685	−0.607228
$160$	0	0
$161$	5.65685	0.445823
$162$	0	0
$163$	6.34315	0.496834	0.248417	−	0.968653i	$-0.420090\pi$
$163$	6.34315	0.496834	0.248417	+	0.968653i	$0.420090\pi$
$164$	0	0
$165$	−5.65685	−0.440386
$166$	0	0
$167$	2.34315	0.181318	0.0906590	−	0.995882i	$-0.471103\pi$
$167$	2.34315	0.181318	0.0906590	+	0.995882i	$0.471103\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−5.65685	−0.432590
$172$	0	0
$173$	1.31371	0.0998794	0.0499397	−	0.998752i	$-0.484097\pi$
$173$	1.31371	0.0998794	0.0499397	+	0.998752i	$0.484097\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−2.34315	−0.175135	−0.0875675	−	0.996159i	$-0.527909\pi$
$179$	−2.34315	−0.175135	−0.0875675	+	0.996159i	$0.527909\pi$
$180$	0	0
$181$	0.343146	0.0255058	0.0127529	−	0.999919i	$-0.495941\pi$
$181$	0.343146	0.0255058	0.0127529	+	0.999919i	$0.495941\pi$
$182$	0	0
$183$	−11.6569	−0.861699
$184$	0	0
$185$	0.343146	0.0252286
$186$	0	0
$187$	43.3137	3.16741
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	7.31371	0.529201	0.264601	−	0.964358i	$-0.414760\pi$
$191$	7.31371	0.529201	0.264601	+	0.964358i	$0.414760\pi$
$192$	0	0
$193$	−9.31371	−0.670415	−0.335208	−	0.942144i	$-0.608806\pi$
$193$	−9.31371	−0.670415	−0.335208	+	0.942144i	$0.608806\pi$
$194$	0	0
$195$	−2.00000	−0.143223
$196$	0	0
$197$	−0.343146	−0.0244481	−0.0122241	−	0.999925i	$-0.503891\pi$
$197$	−0.343146	−0.0244481	−0.0122241	+	0.999925i	$0.503891\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	1.65685	0.116865
$202$	0	0
$203$	7.65685	0.537406
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	−32.0000	−2.21349
$210$	0	0
$211$	18.6274	1.28236	0.641182	−	0.767389i	$-0.278444\pi$
$211$	18.6274	1.28236	0.641182	+	0.767389i	$0.278444\pi$
$212$	0	0
$213$	15.3137	1.04928
$214$	0	0
$215$	9.65685	0.658592
$216$	0	0
$217$	4.00000	0.271538
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	15.3137	1.03011
$222$	0	0
$223$	−3.31371	−0.221902	−0.110951	−	0.993826i	$-0.535390\pi$
$223$	−3.31371	−0.221902	−0.110951	+	0.993826i	$0.535390\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−18.6274	−1.23635	−0.618173	−	0.786042i	$-0.712127\pi$
$227$	−18.6274	−1.23635	−0.618173	+	0.786042i	$0.712127\pi$
$228$	0	0
$229$	−18.9706	−1.25361	−0.626805	−	0.779176i	$-0.715638\pi$
$229$	−18.9706	−1.25361	−0.626805	+	0.779176i	$0.715638\pi$
$230$	0	0
$231$	5.65685	0.372194
$232$	0	0
$233$	−18.9706	−1.24280	−0.621401	−	0.783492i	$-0.713436\pi$
$233$	−18.9706	−1.24280	−0.621401	+	0.783492i	$0.713436\pi$
$234$	0	0
$235$	13.6569	0.890875
$236$	0	0
$237$	−11.3137	−0.734904
$238$	0	0
$239$	−0.686292	−0.0443925	−0.0221963	−	0.999754i	$-0.507066\pi$
$239$	−0.686292	−0.0443925	−0.0221963	+	0.999754i	$0.507066\pi$
$240$	0	0
$241$	21.3137	1.37294	0.686468	−	0.727160i	$-0.259160\pi$
$241$	21.3137	1.37294	0.686468	+	0.727160i	$0.259160\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−11.3137	−0.719874
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−7.31371	−0.461637	−0.230819	−	0.972997i	$-0.574140\pi$
$251$	−7.31371	−0.461637	−0.230819	+	0.972997i	$0.574140\pi$
$252$	0	0
$253$	−32.0000	−2.01182
$254$	0	0
$255$	−7.65685	−0.479491
$256$	0	0
$257$	−16.3431	−1.01946	−0.509729	−	0.860335i	$-0.670254\pi$
$257$	−16.3431	−1.01946	−0.509729	+	0.860335i	$0.670254\pi$
$258$	0	0
$259$	−0.343146	−0.0213220
$260$	0	0
$261$	−7.65685	−0.473947
$262$	0	0
$263$	−18.3431	−1.13109	−0.565543	−	0.824719i	$-0.691334\pi$
$263$	−18.3431	−1.13109	−0.565543	+	0.824719i	$0.691334\pi$
$264$	0	0
$265$	7.65685	0.470357
$266$	0	0
$267$	14.0000	0.856786
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	0	0
$275$	5.65685	0.341121
$276$	0	0
$277$	−4.34315	−0.260954	−0.130477	−	0.991451i	$-0.541651\pi$
$277$	−4.34315	−0.260954	−0.130477	+	0.991451i	$0.541651\pi$
$278$	0	0
$279$	−4.00000	−0.239474
$280$	0	0
$281$	13.3137	0.794229	0.397115	−	0.917769i	$-0.370012\pi$
$281$	13.3137	0.794229	0.397115	+	0.917769i	$0.370012\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	0	0
$285$	5.65685	0.335083
$286$	0	0
$287$	−2.00000	−0.118056
$288$	0	0
$289$	41.6274	2.44867
$290$	0	0
$291$	−6.00000	−0.351726
$292$	0	0
$293$	−24.6274	−1.43875	−0.719375	−	0.694622i	$-0.755571\pi$
$293$	−24.6274	−1.43875	−0.719375	+	0.694622i	$0.755571\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	0	0
$297$	−5.65685	−0.328244
$298$	0	0
$299$	−11.3137	−0.654289
$300$	0	0
$301$	−9.65685	−0.556612
$302$	0	0
$303$	5.31371	0.305265
$304$	0	0
$305$	11.6569	0.667470
$306$	0	0
$307$	26.6274	1.51971	0.759853	−	0.650094i	$-0.225271\pi$
$307$	26.6274	1.51971	0.759853	+	0.650094i	$0.225271\pi$
$308$	0	0
$309$	−8.00000	−0.455104
$310$	0	0
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	−6.68629	−0.377932	−0.188966	−	0.981984i	$-0.560514\pi$
$313$	−6.68629	−0.377932	−0.188966	+	0.981984i	$0.560514\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−0.343146	−0.0192730	−0.00963649	−	0.999954i	$-0.503067\pi$
$317$	−0.343146	−0.0192730	−0.00963649	+	0.999954i	$0.503067\pi$
$318$	0	0
$319$	−43.3137	−2.42510
$320$	0	0
$321$	−17.6569	−0.985510
$322$	0	0
$323$	−43.3137	−2.41004
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	2.00000	0.110600
$328$	0	0
$329$	−13.6569	−0.752927
$330$	0	0
$331$	−18.6274	−1.02386	−0.511928	−	0.859029i	$-0.671068\pi$
$331$	−18.6274	−1.02386	−0.511928	+	0.859029i	$0.671068\pi$
$332$	0	0
$333$	0.343146	0.0188043
$334$	0	0
$335$	−1.65685	−0.0905236
$336$	0	0
$337$	−9.31371	−0.507350	−0.253675	−	0.967290i	$-0.581639\pi$
$337$	−9.31371	−0.507350	−0.253675	+	0.967290i	$0.581639\pi$
$338$	0	0
$339$	15.6569	0.850364
$340$	0	0
$341$	−22.6274	−1.22534
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	5.65685	0.304555
$346$	0	0
$347$	9.65685	0.518407	0.259204	−	0.965823i	$-0.416540\pi$
$347$	9.65685	0.518407	0.259204	+	0.965823i	$0.416540\pi$
$348$	0	0
$349$	−31.6569	−1.69455	−0.847276	−	0.531152i	$-0.821759\pi$
$349$	−31.6569	−1.69455	−0.847276	+	0.531152i	$0.821759\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	34.9706	1.86130	0.930648	−	0.365917i	$-0.119244\pi$
$353$	34.9706	1.86130	0.930648	+	0.365917i	$0.119244\pi$
$354$	0	0
$355$	−15.3137	−0.812767
$356$	0	0
$357$	7.65685	0.405244
$358$	0	0
$359$	−28.0000	−1.47778	−0.738892	−	0.673824i	$-0.764651\pi$
$359$	−28.0000	−1.47778	−0.738892	+	0.673824i	$0.764651\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−21.0000	−1.10221
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	30.6274	1.59874	0.799369	−	0.600840i	$-0.205167\pi$
$367$	30.6274	1.59874	0.799369	+	0.600840i	$0.205167\pi$
$368$	0	0
$369$	2.00000	0.104116
$370$	0	0
$371$	−7.65685	−0.397524
$372$	0	0
$373$	8.34315	0.431992	0.215996	−	0.976394i	$-0.430700\pi$
$373$	8.34315	0.431992	0.215996	+	0.976394i	$0.430700\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−15.3137	−0.788696
$378$	0	0
$379$	−10.6274	−0.545894	−0.272947	−	0.962029i	$-0.587998\pi$
$379$	−10.6274	−0.545894	−0.272947	+	0.962029i	$0.587998\pi$
$380$	0	0
$381$	3.31371	0.169766
$382$	0	0
$383$	−5.65685	−0.289052	−0.144526	−	0.989501i	$-0.546166\pi$
$383$	−5.65685	−0.289052	−0.144526	+	0.989501i	$0.546166\pi$
$384$	0	0
$385$	−5.65685	−0.288300
$386$	0	0
$387$	9.65685	0.490885
$388$	0	0
$389$	−2.97056	−0.150614	−0.0753068	−	0.997160i	$-0.523994\pi$
$389$	−2.97056	−0.150614	−0.0753068	+	0.997160i	$0.523994\pi$
$390$	0	0
$391$	−43.3137	−2.19047
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	11.3137	0.569254
$396$	0	0
$397$	13.3137	0.668196	0.334098	−	0.942538i	$-0.391568\pi$
$397$	13.3137	0.668196	0.334098	+	0.942538i	$0.391568\pi$
$398$	0	0
$399$	−5.65685	−0.283197
$400$	0	0
$401$	−9.31371	−0.465104	−0.232552	−	0.972584i	$-0.574708\pi$
$401$	−9.31371	−0.465104	−0.232552	+	0.972584i	$0.574708\pi$
$402$	0	0
$403$	−8.00000	−0.398508
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	1.94113	0.0962180
$408$	0	0
$409$	−1.31371	−0.0649587	−0.0324794	−	0.999472i	$-0.510340\pi$
$409$	−1.31371	−0.0649587	−0.0324794	+	0.999472i	$0.510340\pi$
$410$	0	0
$411$	15.6569	0.772296
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	−2.34315	−0.114744
$418$	0	0
$419$	26.6274	1.30083	0.650417	−	0.759577i	$-0.274594\pi$
$419$	26.6274	1.30083	0.650417	+	0.759577i	$0.274594\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	0	0
$423$	13.6569	0.664019
$424$	0	0
$425$	7.65685	0.371412
$426$	0	0
$427$	−11.6569	−0.564115
$428$	0	0
$429$	−11.3137	−0.546231
$430$	0	0
$431$	−39.3137	−1.89367	−0.946837	−	0.321713i	$-0.895741\pi$
$431$	−39.3137	−1.89367	−0.946837	+	0.321713i	$0.895741\pi$
$432$	0	0
$433$	1.31371	0.0631328	0.0315664	−	0.999502i	$-0.489950\pi$
$433$	1.31371	0.0631328	0.0315664	+	0.999502i	$0.489950\pi$
$434$	0	0
$435$	7.65685	0.367118
$436$	0	0
$437$	32.0000	1.53077
$438$	0	0
$439$	−0.686292	−0.0327549	−0.0163775	−	0.999866i	$-0.505213\pi$
$439$	−0.686292	−0.0327549	−0.0163775	+	0.999866i	$0.505213\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−28.9706	−1.37643	−0.688216	−	0.725505i	$-0.741606\pi$
$443$	−28.9706	−1.37643	−0.688216	+	0.725505i	$0.741606\pi$
$444$	0	0
$445$	−14.0000	−0.663664
$446$	0	0
$447$	−14.9706	−0.708083
$448$	0	0
$449$	13.3137	0.628313	0.314156	−	0.949371i	$-0.398278\pi$
$449$	13.3137	0.628313	0.314156	+	0.949371i	$0.398278\pi$
$450$	0	0
$451$	11.3137	0.532742
$452$	0	0
$453$	16.0000	0.751746
$454$	0	0
$455$	−2.00000	−0.0937614
$456$	0	0
$457$	34.0000	1.59045	0.795226	−	0.606313i	$-0.207352\pi$
$457$	34.0000	1.59045	0.795226	+	0.606313i	$0.207352\pi$
$458$	0	0
$459$	−7.65685	−0.357391
$460$	0	0
$461$	31.9411	1.48765	0.743823	−	0.668376i	$-0.233010\pi$
$461$	31.9411	1.48765	0.743823	+	0.668376i	$0.233010\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	0	0
$467$	8.68629	0.401954	0.200977	−	0.979596i	$-0.435588\pi$
$467$	8.68629	0.401954	0.200977	+	0.979596i	$0.435588\pi$
$468$	0	0
$469$	1.65685	0.0765064
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	54.6274	2.51177
$474$	0	0
$475$	−5.65685	−0.259554
$476$	0	0
$477$	7.65685	0.350583
$478$	0	0
$479$	−22.6274	−1.03387	−0.516937	−	0.856024i	$-0.672928\pi$
$479$	−22.6274	−1.03387	−0.516937	+	0.856024i	$0.672928\pi$
$480$	0	0
$481$	0.686292	0.0312922
$482$	0	0
$483$	−5.65685	−0.257396
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	22.6274	1.02535	0.512673	−	0.858584i	$-0.328655\pi$
$487$	22.6274	1.02535	0.512673	+	0.858584i	$0.328655\pi$
$488$	0	0
$489$	−6.34315	−0.286847
$490$	0	0
$491$	5.65685	0.255290	0.127645	−	0.991820i	$-0.459258\pi$
$491$	5.65685	0.255290	0.127645	+	0.991820i	$0.459258\pi$
$492$	0	0
$493$	−58.6274	−2.64045
$494$	0	0
$495$	5.65685	0.254257
$496$	0	0
$497$	15.3137	0.686914
$498$	0	0
$499$	10.6274	0.475749	0.237874	−	0.971296i	$-0.423549\pi$
$499$	10.6274	0.475749	0.237874	+	0.971296i	$0.423549\pi$
$500$	0	0
$501$	−2.34315	−0.104684
$502$	0	0
$503$	−2.34315	−0.104476	−0.0522379	−	0.998635i	$-0.516635\pi$
$503$	−2.34315	−0.104476	−0.0522379	+	0.998635i	$0.516635\pi$
$504$	0	0
$505$	−5.31371	−0.236457
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	14.0000	0.620539	0.310270	−	0.950649i	$-0.399581\pi$
$509$	14.0000	0.620539	0.310270	+	0.950649i	$0.399581\pi$
$510$	0	0
$511$	−6.00000	−0.265424
$512$	0	0
$513$	5.65685	0.249756
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	77.2548	3.39766
$518$	0	0
$519$	−1.31371	−0.0576654
$520$	0	0
$521$	6.68629	0.292932	0.146466	−	0.989216i	$-0.453210\pi$
$521$	6.68629	0.292932	0.146466	+	0.989216i	$0.453210\pi$
$522$	0	0
$523$	18.6274	0.814520	0.407260	−	0.913312i	$-0.366484\pi$
$523$	18.6274	0.814520	0.407260	+	0.913312i	$0.366484\pi$
$524$	0	0
$525$	1.00000	0.0436436
$526$	0	0
$527$	−30.6274	−1.33415
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	4.00000	0.173585
$532$	0	0
$533$	4.00000	0.173259
$534$	0	0
$535$	17.6569	0.763372
$536$	0	0
$537$	2.34315	0.101114
$538$	0	0
$539$	5.65685	0.243658
$540$	0	0
$541$	−29.3137	−1.26029	−0.630147	−	0.776476i	$-0.717006\pi$
$541$	−29.3137	−1.26029	−0.630147	+	0.776476i	$0.717006\pi$
$542$	0	0
$543$	−0.343146	−0.0147258
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−24.2843	−1.03832	−0.519160	−	0.854677i	$-0.673755\pi$
$547$	−24.2843	−1.03832	−0.519160	+	0.854677i	$0.673755\pi$
$548$	0	0
$549$	11.6569	0.497502
$550$	0	0
$551$	43.3137	1.84523
$552$	0	0
$553$	−11.3137	−0.481108
$554$	0	0
$555$	−0.343146	−0.0145657
$556$	0	0
$557$	−38.9706	−1.65124	−0.825618	−	0.564230i	$-0.809173\pi$
$557$	−38.9706	−1.65124	−0.825618	+	0.564230i	$0.809173\pi$
$558$	0	0
$559$	19.3137	0.816883
$560$	0	0
$561$	−43.3137	−1.82871
$562$	0	0
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	0	0
$565$	−15.6569	−0.658689
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	−39.9411	−1.67442	−0.837210	−	0.546882i	$-0.815815\pi$
$569$	−39.9411	−1.67442	−0.837210	+	0.546882i	$0.815815\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	0	0
$573$	−7.31371	−0.305535
$574$	0	0
$575$	−5.65685	−0.235907
$576$	0	0
$577$	10.6863	0.444876	0.222438	−	0.974947i	$-0.428598\pi$
$577$	10.6863	0.444876	0.222438	+	0.974947i	$0.428598\pi$
$578$	0	0
$579$	9.31371	0.387065
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	43.3137	1.79387
$584$	0	0
$585$	2.00000	0.0826898
$586$	0	0
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	0	0
$589$	22.6274	0.932346
$590$	0	0
$591$	0.343146	0.0141151
$592$	0	0
$593$	−30.9706	−1.27181	−0.635904	−	0.771768i	$-0.719373\pi$
$593$	−30.9706	−1.27181	−0.635904	+	0.771768i	$0.719373\pi$
$594$	0	0
$595$	−7.65685	−0.313900
$596$	0	0
$597$	−4.00000	−0.163709
$598$	0	0
$599$	−23.3137	−0.952572	−0.476286	−	0.879290i	$-0.658017\pi$
$599$	−23.3137	−0.952572	−0.476286	+	0.879290i	$0.658017\pi$
$600$	0	0
$601$	−20.6274	−0.841410	−0.420705	−	0.907198i	$-0.638217\pi$
$601$	−20.6274	−0.841410	−0.420705	+	0.907198i	$0.638217\pi$
$602$	0	0
$603$	−1.65685	−0.0674723
$604$	0	0
$605$	21.0000	0.853771
$606$	0	0
$607$	−41.9411	−1.70234	−0.851169	−	0.524892i	$-0.824106\pi$
$607$	−41.9411	−1.70234	−0.851169	+	0.524892i	$0.824106\pi$
$608$	0	0
$609$	−7.65685	−0.310271
$610$	0	0
$611$	27.3137	1.10499
$612$	0	0
$613$	6.97056	0.281538	0.140769	−	0.990042i	$-0.455042\pi$
$613$	6.97056	0.281538	0.140769	+	0.990042i	$0.455042\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−2.97056	−0.119590	−0.0597952	−	0.998211i	$-0.519045\pi$
$617$	−2.97056	−0.119590	−0.0597952	+	0.998211i	$0.519045\pi$
$618$	0	0
$619$	7.02944	0.282537	0.141268	−	0.989971i	$-0.454882\pi$
$619$	7.02944	0.282537	0.141268	+	0.989971i	$0.454882\pi$
$620$	0	0
$621$	5.65685	0.227002
$622$	0	0
$623$	14.0000	0.560898
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	32.0000	1.27796
$628$	0	0
$629$	2.62742	0.104762
$630$	0	0
$631$	−30.6274	−1.21926	−0.609629	−	0.792687i	$-0.708682\pi$
$631$	−30.6274	−1.21926	−0.609629	+	0.792687i	$0.708682\pi$
$632$	0	0
$633$	−18.6274	−0.740373
$634$	0	0
$635$	−3.31371	−0.131501
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	−15.3137	−0.605801
$640$	0	0
$641$	−12.6274	−0.498753	−0.249376	−	0.968407i	$-0.580226\pi$
$641$	−12.6274	−0.498753	−0.249376	+	0.968407i	$0.580226\pi$
$642$	0	0
$643$	15.3137	0.603914	0.301957	−	0.953322i	$-0.402360\pi$
$643$	15.3137	0.603914	0.301957	+	0.953322i	$0.402360\pi$
$644$	0	0
$645$	−9.65685	−0.380238
$646$	0	0
$647$	−2.34315	−0.0921186	−0.0460593	−	0.998939i	$-0.514666\pi$
$647$	−2.34315	−0.0921186	−0.0460593	+	0.998939i	$0.514666\pi$
$648$	0	0
$649$	22.6274	0.888204
$650$	0	0
$651$	−4.00000	−0.156772
$652$	0	0
$653$	18.9706	0.742375	0.371188	−	0.928558i	$-0.378951\pi$
$653$	18.9706	0.742375	0.371188	+	0.928558i	$0.378951\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	−16.9706	−0.661079	−0.330540	−	0.943792i	$-0.607231\pi$
$659$	−16.9706	−0.661079	−0.330540	+	0.943792i	$0.607231\pi$
$660$	0	0
$661$	10.2843	0.400012	0.200006	−	0.979795i	$-0.435904\pi$
$661$	10.2843	0.400012	0.200006	+	0.979795i	$0.435904\pi$
$662$	0	0
$663$	−15.3137	−0.594735
$664$	0	0
$665$	5.65685	0.219363
$666$	0	0
$667$	43.3137	1.67711
$668$	0	0
$669$	3.31371	0.128115
$670$	0	0
$671$	65.9411	2.54563
$672$	0	0
$673$	−26.6863	−1.02868	−0.514340	−	0.857586i	$-0.671963\pi$
$673$	−26.6863	−1.02868	−0.514340	+	0.857586i	$0.671963\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	18.6863	0.718173	0.359086	−	0.933304i	$-0.383088\pi$
$677$	18.6863	0.718173	0.359086	+	0.933304i	$0.383088\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	18.6274	0.713804
$682$	0	0
$683$	−33.6569	−1.28784	−0.643922	−	0.765091i	$-0.722694\pi$
$683$	−33.6569	−1.28784	−0.643922	+	0.765091i	$0.722694\pi$
$684$	0	0
$685$	−15.6569	−0.598218
$686$	0	0
$687$	18.9706	0.723772
$688$	0	0
$689$	15.3137	0.583406
$690$	0	0
$691$	−24.9706	−0.949925	−0.474962	−	0.880006i	$-0.657538\pi$
$691$	−24.9706	−0.949925	−0.474962	+	0.880006i	$0.657538\pi$
$692$	0	0
$693$	−5.65685	−0.214886
$694$	0	0
$695$	2.34315	0.0888806
$696$	0	0
$697$	15.3137	0.580048
$698$	0	0
$699$	18.9706	0.717533
$700$	0	0
$701$	−7.65685	−0.289195	−0.144598	−	0.989491i	$-0.546189\pi$
$701$	−7.65685	−0.289195	−0.144598	+	0.989491i	$0.546189\pi$
$702$	0	0
$703$	−1.94113	−0.0732109
$704$	0	0
$705$	−13.6569	−0.514347
$706$	0	0
$707$	5.31371	0.199843
$708$	0	0
$709$	−5.31371	−0.199561	−0.0997803	−	0.995009i	$-0.531814\pi$
$709$	−5.31371	−0.199561	−0.0997803	+	0.995009i	$0.531814\pi$
$710$	0	0
$711$	11.3137	0.424297
$712$	0	0
$713$	22.6274	0.847403
$714$	0	0
$715$	11.3137	0.423109
$716$	0	0
$717$	0.686292	0.0256300
$718$	0	0
$719$	−12.6863	−0.473119	−0.236559	−	0.971617i	$-0.576020\pi$
$719$	−12.6863	−0.473119	−0.236559	+	0.971617i	$0.576020\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	−21.3137	−0.792665
$724$	0	0
$725$	−7.65685	−0.284368
$726$	0	0
$727$	−16.0000	−0.593407	−0.296704	−	0.954970i	$-0.595887\pi$
$727$	−16.0000	−0.593407	−0.296704	+	0.954970i	$0.595887\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	73.9411	2.73481
$732$	0	0
$733$	2.00000	0.0738717	0.0369358	−	0.999318i	$-0.488240\pi$
$733$	2.00000	0.0738717	0.0369358	+	0.999318i	$0.488240\pi$
$734$	0	0
$735$	−1.00000	−0.0368856
$736$	0	0
$737$	−9.37258	−0.345244
$738$	0	0
$739$	−0.686292	−0.0252456	−0.0126228	−	0.999920i	$-0.504018\pi$
$739$	−0.686292	−0.0252456	−0.0126228	+	0.999920i	$0.504018\pi$
$740$	0	0
$741$	11.3137	0.415619
$742$	0	0
$743$	−2.34315	−0.0859617	−0.0429808	−	0.999076i	$-0.513685\pi$
$743$	−2.34315	−0.0859617	−0.0429808	+	0.999076i	$0.513685\pi$
$744$	0	0
$745$	14.9706	0.548479
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	−17.6569	−0.645167
$750$	0	0
$751$	11.3137	0.412843	0.206422	−	0.978463i	$-0.433818\pi$
$751$	11.3137	0.412843	0.206422	+	0.978463i	$0.433818\pi$
$752$	0	0
$753$	7.31371	0.266526
$754$	0	0
$755$	−16.0000	−0.582300
$756$	0	0
$757$	−52.9117	−1.92311	−0.961554	−	0.274616i	$-0.911449\pi$
$757$	−52.9117	−1.92311	−0.961554	+	0.274616i	$0.911449\pi$
$758$	0	0
$759$	32.0000	1.16153
$760$	0	0
$761$	35.9411	1.30286	0.651432	−	0.758707i	$-0.274168\pi$
$761$	35.9411	1.30286	0.651432	+	0.758707i	$0.274168\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	7.65685	0.276834
$766$	0	0
$767$	8.00000	0.288863
$768$	0	0
$769$	−39.9411	−1.44031	−0.720157	−	0.693811i	$-0.755930\pi$
$769$	−39.9411	−1.44031	−0.720157	+	0.693811i	$0.755930\pi$
$770$	0	0
$771$	16.3431	0.588584
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−4.00000	−0.143684
$776$	0	0
$777$	0.343146	0.0123103
$778$	0	0
$779$	−11.3137	−0.405356
$780$	0	0
$781$	−86.6274	−3.09977
$782$	0	0
$783$	7.65685	0.273634
$784$	0	0
$785$	10.0000	0.356915
$786$	0	0
$787$	−39.3137	−1.40138	−0.700691	−	0.713465i	$-0.747125\pi$
$787$	−39.3137	−1.40138	−0.700691	+	0.713465i	$0.747125\pi$
$788$	0	0
$789$	18.3431	0.653033
$790$	0	0
$791$	15.6569	0.556694
$792$	0	0
$793$	23.3137	0.827894
$794$	0	0
$795$	−7.65685	−0.271561
$796$	0	0
$797$	−5.31371	−0.188221	−0.0941106	−	0.995562i	$-0.530001\pi$
$797$	−5.31371	−0.188221	−0.0941106	+	0.995562i	$0.530001\pi$
$798$	0	0
$799$	104.569	3.69937
$800$	0	0
$801$	−14.0000	−0.494666
$802$	0	0
$803$	33.9411	1.19776
$804$	0	0
$805$	5.65685	0.199378
$806$	0	0
$807$	2.00000	0.0704033
$808$	0	0
$809$	40.6274	1.42838	0.714192	−	0.699950i	$-0.246794\pi$
$809$	40.6274	1.42838	0.714192	+	0.699950i	$0.246794\pi$
$810$	0	0
$811$	2.34315	0.0822790	0.0411395	−	0.999153i	$-0.486901\pi$
$811$	2.34315	0.0822790	0.0411395	+	0.999153i	$0.486901\pi$
$812$	0	0
$813$	12.0000	0.420858
$814$	0	0
$815$	6.34315	0.222191
$816$	0	0
$817$	−54.6274	−1.91117
$818$	0	0
$819$	−2.00000	−0.0698857
$820$	0	0
$821$	18.2843	0.638125	0.319063	−	0.947734i	$-0.396632\pi$
$821$	18.2843	0.638125	0.319063	+	0.947734i	$0.396632\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	−5.65685	−0.196946
$826$	0	0
$827$	−41.6569	−1.44855	−0.724275	−	0.689511i	$-0.757826\pi$
$827$	−41.6569	−1.44855	−0.724275	+	0.689511i	$0.757826\pi$
$828$	0	0
$829$	−12.3431	−0.428695	−0.214348	−	0.976757i	$-0.568763\pi$
$829$	−12.3431	−0.428695	−0.214348	+	0.976757i	$0.568763\pi$
$830$	0	0
$831$	4.34315	0.150662
$832$	0	0
$833$	7.65685	0.265294
$834$	0	0
$835$	2.34315	0.0810879
$836$	0	0
$837$	4.00000	0.138260
$838$	0	0
$839$	4.68629	0.161789	0.0808944	−	0.996723i	$-0.474222\pi$
$839$	4.68629	0.161789	0.0808944	+	0.996723i	$0.474222\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	−13.3137	−0.458548
$844$	0	0
$845$	−9.00000	−0.309609
$846$	0	0
$847$	−21.0000	−0.721569
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−1.94113	−0.0665409
$852$	0	0
$853$	−39.9411	−1.36756	−0.683779	−	0.729689i	$-0.739665\pi$
$853$	−39.9411	−1.36756	−0.683779	+	0.729689i	$0.739665\pi$
$854$	0	0
$855$	−5.65685	−0.193460
$856$	0	0
$857$	55.6569	1.90120	0.950601	−	0.310416i	$-0.100468\pi$
$857$	55.6569	1.90120	0.950601	+	0.310416i	$0.100468\pi$
$858$	0	0
$859$	−15.0294	−0.512798	−0.256399	−	0.966571i	$-0.582536\pi$
$859$	−15.0294	−0.512798	−0.256399	+	0.966571i	$0.582536\pi$
$860$	0	0
$861$	2.00000	0.0681598
$862$	0	0
$863$	29.6569	1.00953	0.504766	−	0.863256i	$-0.331579\pi$
$863$	29.6569	1.00953	0.504766	+	0.863256i	$0.331579\pi$
$864$	0	0
$865$	1.31371	0.0446674
$866$	0	0
$867$	−41.6274	−1.41374
$868$	0	0
$869$	64.0000	2.17105
$870$	0	0
$871$	−3.31371	−0.112281
$872$	0	0
$873$	6.00000	0.203069
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	13.0294	0.439973	0.219986	−	0.975503i	$-0.429399\pi$
$877$	13.0294	0.439973	0.219986	+	0.975503i	$0.429399\pi$
$878$	0	0
$879$	24.6274	0.830662
$880$	0	0
$881$	−44.6274	−1.50354	−0.751768	−	0.659428i	$-0.770799\pi$
$881$	−44.6274	−1.50354	−0.751768	+	0.659428i	$0.770799\pi$
$882$	0	0
$883$	−22.3431	−0.751907	−0.375953	−	0.926639i	$-0.622685\pi$
$883$	−22.3431	−0.751907	−0.375953	+	0.926639i	$0.622685\pi$
$884$	0	0
$885$	−4.00000	−0.134459
$886$	0	0
$887$	−12.2843	−0.412465	−0.206233	−	0.978503i	$-0.566120\pi$
$887$	−12.2843	−0.412465	−0.206233	+	0.978503i	$0.566120\pi$
$888$	0	0
$889$	3.31371	0.111138
$890$	0	0
$891$	5.65685	0.189512
$892$	0	0
$893$	−77.2548	−2.58523
$894$	0	0
$895$	−2.34315	−0.0783227
$896$	0	0
$897$	11.3137	0.377754
$898$	0	0
$899$	30.6274	1.02148
$900$	0	0
$901$	58.6274	1.95316
$902$	0	0
$903$	9.65685	0.321360
$904$	0	0
$905$	0.343146	0.0114066
$906$	0	0
$907$	−4.97056	−0.165045	−0.0825224	−	0.996589i	$-0.526298\pi$
$907$	−4.97056	−0.165045	−0.0825224	+	0.996589i	$0.526298\pi$
$908$	0	0
$909$	−5.31371	−0.176245
$910$	0	0
$911$	−37.9411	−1.25705	−0.628523	−	0.777791i	$-0.716340\pi$
$911$	−37.9411	−1.25705	−0.628523	+	0.777791i	$0.716340\pi$
$912$	0	0
$913$	22.6274	0.748858
$914$	0	0
$915$	−11.6569	−0.385364
$916$	0	0
$917$	−20.0000	−0.660458
$918$	0	0
$919$	−25.9411	−0.855719	−0.427859	−	0.903845i	$-0.640732\pi$
$919$	−25.9411	−0.855719	−0.427859	+	0.903845i	$0.640732\pi$
$920$	0	0
$921$	−26.6274	−0.877403
$922$	0	0
$923$	−30.6274	−1.00811
$924$	0	0
$925$	0.343146	0.0112826
$926$	0	0
$927$	8.00000	0.262754
$928$	0	0
$929$	−33.3137	−1.09299	−0.546494	−	0.837463i	$-0.684038\pi$
$929$	−33.3137	−1.09299	−0.546494	+	0.837463i	$0.684038\pi$
$930$	0	0
$931$	−5.65685	−0.185396
$932$	0	0
$933$	8.00000	0.261908
$934$	0	0
$935$	43.3137	1.41651
$936$	0	0
$937$	49.3137	1.61101	0.805504	−	0.592590i	$-0.201895\pi$
$937$	49.3137	1.61101	0.805504	+	0.592590i	$0.201895\pi$
$938$	0	0
$939$	6.68629	0.218199
$940$	0	0
$941$	14.0000	0.456387	0.228193	−	0.973616i	$-0.426718\pi$
$941$	14.0000	0.456387	0.228193	+	0.973616i	$0.426718\pi$
$942$	0	0
$943$	−11.3137	−0.368425
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	40.2843	1.30906	0.654531	−	0.756035i	$-0.272866\pi$
$947$	40.2843	1.30906	0.654531	+	0.756035i	$0.272866\pi$
$948$	0	0
$949$	12.0000	0.389536
$950$	0	0
$951$	0.343146	0.0111273
$952$	0	0
$953$	−7.65685	−0.248030	−0.124015	−	0.992280i	$-0.539577\pi$
$953$	−7.65685	−0.248030	−0.124015	+	0.992280i	$0.539577\pi$
$954$	0	0
$955$	7.31371	0.236666
$956$	0	0
$957$	43.3137	1.40013
$958$	0	0
$959$	15.6569	0.505586
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	17.6569	0.568984
$964$	0	0
$965$	−9.31371	−0.299819
$966$	0	0
$967$	25.9411	0.834210	0.417105	−	0.908858i	$-0.363045\pi$
$967$	25.9411	0.834210	0.417105	+	0.908858i	$0.363045\pi$
$968$	0	0
$969$	43.3137	1.39144
$970$	0	0
$971$	13.3726	0.429147	0.214573	−	0.976708i	$-0.431164\pi$
$971$	13.3726	0.429147	0.214573	+	0.976708i	$0.431164\pi$
$972$	0	0
$973$	−2.34315	−0.0751178
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	−36.3431	−1.16272	−0.581360	−	0.813646i	$-0.697479\pi$
$977$	−36.3431	−1.16272	−0.581360	+	0.813646i	$0.697479\pi$
$978$	0	0
$979$	−79.1960	−2.53111
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	10.3431	0.329895	0.164948	−	0.986302i	$-0.447255\pi$
$983$	10.3431	0.329895	0.164948	+	0.986302i	$0.447255\pi$
$984$	0	0
$985$	−0.343146	−0.0109335
$986$	0	0
$987$	13.6569	0.434702
$988$	0	0
$989$	−54.6274	−1.73705
$990$	0	0
$991$	11.3137	0.359392	0.179696	−	0.983722i	$-0.442489\pi$
$991$	11.3137	0.359392	0.179696	+	0.983722i	$0.442489\pi$
$992$	0	0
$993$	18.6274	0.591123
$994$	0	0
$995$	4.00000	0.126809
$996$	0	0
$997$	5.31371	0.168287	0.0841434	−	0.996454i	$-0.473185\pi$
$997$	5.31371	0.168287	0.0841434	+	0.996454i	$0.473185\pi$
$998$	0	0
$999$	−0.343146	−0.0108567

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.a.u.1.2		2
3.2	odd	2		5040.2.a.br.1.1		2
4.3	odd	2		840.2.a.k.1.1	✓	2
5.4	even	2		8400.2.a.de.1.2		2
8.3	odd	2		6720.2.a.co.1.2		2
8.5	even	2		6720.2.a.cu.1.1		2
12.11	even	2		2520.2.a.v.1.2		2
20.3	even	4		4200.2.t.u.1849.3		4
20.7	even	4		4200.2.t.u.1849.1		4
20.19	odd	2		4200.2.a.bg.1.1		2
28.27	even	2		5880.2.a.bl.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.a.k.1.1	✓	2	4.3	odd	2
1680.2.a.u.1.2		2	1.1	even	1	trivial
2520.2.a.v.1.2		2	12.11	even	2
4200.2.a.bg.1.1		2	20.19	odd	2
4200.2.t.u.1849.1		4	20.7	even	4
4200.2.t.u.1849.3		4	20.3	even	4
5040.2.a.br.1.1		2	3.2	odd	2
5880.2.a.bl.1.1		2	28.27	even	2
6720.2.a.co.1.2		2	8.3	odd	2
6720.2.a.cu.1.1		2	8.5	even	2
8400.2.a.de.1.2		2	5.4	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.65685 q^{11} +2.00000 q^{13} -1.00000 q^{15} +7.65685 q^{17} -5.65685 q^{19} +1.00000 q^{21} -5.65685 q^{23} +1.00000 q^{25} -1.00000 q^{27} -7.65685 q^{29} -4.00000 q^{31} -5.65685 q^{33} -1.00000 q^{35} +0.343146 q^{37} -2.00000 q^{39} +2.00000 q^{41} +9.65685 q^{43} +1.00000 q^{45} +13.6569 q^{47} +1.00000 q^{49} -7.65685 q^{51} +7.65685 q^{53} +5.65685 q^{55} +5.65685 q^{57} +4.00000 q^{59} +11.6569 q^{61} -1.00000 q^{63} +2.00000 q^{65} -1.65685 q^{67} +5.65685 q^{69} -15.3137 q^{71} +6.00000 q^{73} -1.00000 q^{75} -5.65685 q^{77} +11.3137 q^{79} +1.00000 q^{81} +4.00000 q^{83} +7.65685 q^{85} +7.65685 q^{87} -14.0000 q^{89} -2.00000 q^{91} +4.00000 q^{93} -5.65685 q^{95} +6.00000 q^{97} +5.65685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} + 4 q^{13} - 2 q^{15} + 4 q^{17} + 2 q^{21} + 2 q^{25} - 2 q^{27} - 4 q^{29} - 8 q^{31} - 2 q^{35} + 12 q^{37} - 4 q^{39} + 4 q^{41} + 8 q^{43} + 2 q^{45} + 16 q^{47} + 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{59} + 12 q^{61} - 2 q^{63} + 4 q^{65} + 8 q^{67} - 8 q^{71} + 12 q^{73} - 2 q^{75} + 2 q^{81} + 8 q^{83} + 4 q^{85} + 4 q^{87} - 28 q^{89} - 4 q^{91} + 8 q^{93} + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 + 4 * q^13 - 2 * q^15 + 4 * q^17 + 2 * q^21 + 2 * q^25 - 2 * q^27 - 4 * q^29 - 8 * q^31 - 2 * q^35 + 12 * q^37 - 4 * q^39 + 4 * q^41 + 8 * q^43 + 2 * q^45 + 16 * q^47 + 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^59 + 12 * q^61 - 2 * q^63 + 4 * q^65 + 8 * q^67 - 8 * q^71 + 12 * q^73 - 2 * q^75 + 2 * q^81 + 8 * q^83 + 4 * q^85 + 4 * q^87 - 28 * q^89 - 4 * q^91 + 8 * q^93 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more