LMFDB - Embedded newform 3072.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.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:	$24.5300435009$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 48)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.68554$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.79793 q^{5} +2.15894 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.79793 q^{5} +2.15894 q^{7} +1.00000 q^{9} -2.54266 q^{11} +1.95687 q^{13} +3.79793 q^{15} +0.224777 q^{17} +0.224777 q^{19} -2.15894 q^{21} -2.82843 q^{23} +9.42429 q^{25} -1.00000 q^{27} +2.62636 q^{29} +1.84106 q^{31} +2.54266 q^{33} -8.19951 q^{35} -5.18944 q^{37} -1.95687 q^{39} +5.88163 q^{41} +10.9670 q^{43} -3.79793 q^{45} +2.82843 q^{47} -2.33897 q^{49} -0.224777 q^{51} -10.6264 q^{53} +9.65685 q^{55} -0.224777 q^{57} +5.65685 q^{59} -8.46742 q^{61} +2.15894 q^{63} -7.43208 q^{65} +14.7422 q^{67} +2.82843 q^{69} -4.31788 q^{71} +5.97474 q^{73} -9.42429 q^{75} -5.48946 q^{77} -15.0075 q^{79} +1.00000 q^{81} -14.3059 q^{83} -0.853690 q^{85} -2.62636 q^{87} -1.42847 q^{89} +4.22478 q^{91} -1.84106 q^{93} -0.853690 q^{95} -16.3990 q^{97} -2.54266 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9} - 8 q^{13} + 4 q^{15} - 4 q^{21} + 4 q^{25} - 4 q^{27} - 12 q^{29} + 12 q^{31} - 16 q^{37} + 8 q^{39} - 4 q^{45} + 4 q^{49} - 20 q^{53} + 16 q^{55} - 16 q^{61} + 4 q^{63} - 8 q^{65} + 16 q^{67} - 8 q^{71} - 8 q^{73} - 4 q^{75} - 24 q^{77} + 12 q^{79} + 4 q^{81} - 24 q^{85} + 12 q^{87} - 8 q^{89} + 16 q^{91} - 12 q^{93} - 24 q^{95}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9 - 8 * q^13 + 4 * q^15 - 4 * q^21 + 4 * q^25 - 4 * q^27 - 12 * q^29 + 12 * q^31 - 16 * q^37 + 8 * q^39 - 4 * q^45 + 4 * q^49 - 20 * q^53 + 16 * q^55 - 16 * q^61 + 4 * q^63 - 8 * q^65 + 16 * q^67 - 8 * q^71 - 8 * q^73 - 4 * q^75 - 24 * q^77 + 12 * q^79 + 4 * q^81 - 24 * q^85 + 12 * q^87 - 8 * q^89 + 16 * q^91 - 12 * q^93 - 24 * q^95

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
$5$	−3.79793	−1.69849	−0.849244	−	0.528001i	$-0.822942\pi$
$5$	−3.79793	−1.69849	−0.849244	+	0.528001i	$0.822942\pi$
$6$	0	0
$7$	2.15894	0.816003	0.408002	−	0.912981i	$-0.366226\pi$
$7$	2.15894	0.816003	0.408002	+	0.912981i	$0.366226\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.54266	−0.766641	−0.383321	−	0.923615i	$-0.625220\pi$
$11$	−2.54266	−0.766641	−0.383321	+	0.923615i	$0.625220\pi$
$12$	0	0
$13$	1.95687	0.542739	0.271370	−	0.962475i	$-0.412523\pi$
$13$	1.95687	0.542739	0.271370	+	0.962475i	$0.412523\pi$
$14$	0	0
$15$	3.79793	0.980622
$16$	0	0
$17$	0.224777	0.0545165	0.0272583	−	0.999628i	$-0.491322\pi$
$17$	0.224777	0.0545165	0.0272583	+	0.999628i	$0.491322\pi$
$18$	0	0
$19$	0.224777	0.0515675	0.0257837	−	0.999668i	$-0.491792\pi$
$19$	0.224777	0.0515675	0.0257837	+	0.999668i	$0.491792\pi$
$20$	0	0
$21$	−2.15894	−0.471120
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	9.42429	1.88486
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	2.62636	0.487703	0.243851	−	0.969813i	$-0.421589\pi$
$29$	2.62636	0.487703	0.243851	+	0.969813i	$0.421589\pi$
$30$	0	0
$31$	1.84106	0.330664	0.165332	−	0.986238i	$-0.447130\pi$
$31$	1.84106	0.330664	0.165332	+	0.986238i	$0.447130\pi$
$32$	0	0
$33$	2.54266	0.442620
$34$	0	0
$35$	−8.19951	−1.38597
$36$	0	0
$37$	−5.18944	−0.853138	−0.426569	−	0.904455i	$-0.640278\pi$
$37$	−5.18944	−0.853138	−0.426569	+	0.904455i	$0.640278\pi$
$38$	0	0
$39$	−1.95687	−0.313351
$40$	0	0
$41$	5.88163	0.918557	0.459278	−	0.888292i	$-0.348108\pi$
$41$	5.88163	0.918557	0.459278	+	0.888292i	$0.348108\pi$
$42$	0	0
$43$	10.9670	1.67244	0.836222	−	0.548391i	$-0.184759\pi$
$43$	10.9670	1.67244	0.836222	+	0.548391i	$0.184759\pi$
$44$	0	0
$45$	−3.79793	−0.566162
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−2.33897	−0.334139
$50$	0	0
$51$	−0.224777	−0.0314751
$52$	0	0
$53$	−10.6264	−1.45964	−0.729821	−	0.683638i	$-0.760397\pi$
$53$	−10.6264	−1.45964	−0.729821	+	0.683638i	$0.760397\pi$
$54$	0	0
$55$	9.65685	1.30213
$56$	0	0
$57$	−0.224777	−0.0297725
$58$	0	0
$59$	5.65685	0.736460	0.368230	−	0.929735i	$-0.379964\pi$
$59$	5.65685	0.736460	0.368230	+	0.929735i	$0.379964\pi$
$60$	0	0
$61$	−8.46742	−1.08414	−0.542071	−	0.840333i	$-0.682360\pi$
$61$	−8.46742	−1.08414	−0.542071	+	0.840333i	$0.682360\pi$
$62$	0	0
$63$	2.15894	0.272001
$64$	0	0
$65$	−7.43208	−0.921836
$66$	0	0
$67$	14.7422	1.80104	0.900522	−	0.434811i	$-0.143185\pi$
$67$	14.7422	1.80104	0.900522	+	0.434811i	$0.143185\pi$
$68$	0	0
$69$	2.82843	0.340503
$70$	0	0
$71$	−4.31788	−0.512438	−0.256219	−	0.966619i	$-0.582477\pi$
$71$	−4.31788	−0.512438	−0.256219	+	0.966619i	$0.582477\pi$
$72$	0	0
$73$	5.97474	0.699290	0.349645	−	0.936882i	$-0.386302\pi$
$73$	5.97474	0.699290	0.349645	+	0.936882i	$0.386302\pi$
$74$	0	0
$75$	−9.42429	−1.08822
$76$	0	0
$77$	−5.48946	−0.625582
$78$	0	0
$79$	−15.0075	−1.68848	−0.844239	−	0.535966i	$-0.819947\pi$
$79$	−15.0075	−1.68848	−0.844239	+	0.535966i	$0.819947\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.3059	−1.57028	−0.785140	−	0.619319i	$-0.787409\pi$
$83$	−14.3059	−1.57028	−0.785140	+	0.619319i	$0.787409\pi$
$84$	0	0
$85$	−0.853690	−0.0925956
$86$	0	0
$87$	−2.62636	−0.281575
$88$	0	0
$89$	−1.42847	−0.151417	−0.0757086	−	0.997130i	$-0.524122\pi$
$89$	−1.42847	−0.151417	−0.0757086	+	0.997130i	$0.524122\pi$
$90$	0	0
$91$	4.22478	0.442877
$92$	0	0
$93$	−1.84106	−0.190909
$94$	0	0
$95$	−0.853690	−0.0875867
$96$	0	0
$97$	−16.3990	−1.66507	−0.832535	−	0.553973i	$-0.813111\pi$
$97$	−16.3990	−1.66507	−0.832535	+	0.553973i	$0.813111\pi$
$98$	0	0
$99$	−2.54266	−0.255547
$100$	0	0
$101$	0.115816	0.0115241	0.00576206	−	0.999983i	$-0.498166\pi$
$101$	0.115816	0.0115241	0.00576206	+	0.999983i	$0.498166\pi$
$102$	0	0
$103$	13.3507	1.31548	0.657740	−	0.753245i	$-0.271512\pi$
$103$	13.3507	1.31548	0.657740	+	0.753245i	$0.271512\pi$
$104$	0	0
$105$	8.19951	0.800191
$106$	0	0
$107$	10.2926	0.995025	0.497513	−	0.867457i	$-0.334247\pi$
$107$	10.2926	0.995025	0.497513	+	0.867457i	$0.334247\pi$
$108$	0	0
$109$	−9.95687	−0.953696	−0.476848	−	0.878986i	$-0.658221\pi$
$109$	−9.95687	−0.953696	−0.476848	+	0.878986i	$0.658221\pi$
$110$	0	0
$111$	5.18944	0.492559
$112$	0	0
$113$	−18.8486	−1.77313	−0.886563	−	0.462608i	$-0.846914\pi$
$113$	−18.8486	−1.77313	−0.886563	+	0.462608i	$0.846914\pi$
$114$	0	0
$115$	10.7422	1.00171
$116$	0	0
$117$	1.95687	0.180913
$118$	0	0
$119$	0.485281	0.0444857
$120$	0	0
$121$	−4.53488	−0.412261
$122$	0	0
$123$	−5.88163	−0.530329
$124$	0	0
$125$	−16.8032	−1.50292
$126$	0	0
$127$	−3.81580	−0.338597	−0.169299	−	0.985565i	$-0.554150\pi$
$127$	−3.81580	−0.338597	−0.169299	+	0.985565i	$0.554150\pi$
$128$	0	0
$129$	−10.9670	−0.965586
$130$	0	0
$131$	1.08532	0.0948250	0.0474125	−	0.998875i	$-0.484902\pi$
$131$	1.08532	0.0948250	0.0474125	+	0.998875i	$0.484902\pi$
$132$	0	0
$133$	0.485281	0.0420792
$134$	0	0
$135$	3.79793	0.326874
$136$	0	0
$137$	−5.31010	−0.453672	−0.226836	−	0.973933i	$-0.572838\pi$
$137$	−5.31010	−0.453672	−0.226836	+	0.973933i	$0.572838\pi$
$138$	0	0
$139$	−12.3990	−1.05167	−0.525836	−	0.850586i	$-0.676247\pi$
$139$	−12.3990	−1.05167	−0.525836	+	0.850586i	$0.676247\pi$
$140$	0	0
$141$	−2.82843	−0.238197
$142$	0	0
$143$	−4.97567	−0.416086
$144$	0	0
$145$	−9.97474	−0.828357
$146$	0	0
$147$	2.33897	0.192915
$148$	0	0
$149$	1.45479	0.119181	0.0595904	−	0.998223i	$-0.481021\pi$
$149$	1.45479	0.119181	0.0595904	+	0.998223i	$0.481021\pi$
$150$	0	0
$151$	2.03696	0.165766	0.0828829	−	0.996559i	$-0.473587\pi$
$151$	2.03696	0.165766	0.0828829	+	0.996559i	$0.473587\pi$
$152$	0	0
$153$	0.224777	0.0181722
$154$	0	0
$155$	−6.99222	−0.561628
$156$	0	0
$157$	−8.61790	−0.687784	−0.343892	−	0.939009i	$-0.611745\pi$
$157$	−8.61790	−0.687784	−0.343892	+	0.939009i	$0.611745\pi$
$158$	0	0
$159$	10.6264	0.842725
$160$	0	0
$161$	−6.10641	−0.481252
$162$	0	0
$163$	−4.86054	−0.380707	−0.190354	−	0.981716i	$-0.560963\pi$
$163$	−4.86054	−0.380707	−0.190354	+	0.981716i	$0.560963\pi$
$164$	0	0
$165$	−9.65685	−0.751785
$166$	0	0
$167$	−21.7023	−1.67937	−0.839686	−	0.543072i	$-0.817261\pi$
$167$	−21.7023	−1.67937	−0.839686	+	0.543072i	$0.817261\pi$
$168$	0	0
$169$	−9.17064	−0.705434
$170$	0	0
$171$	0.224777	0.0171892
$172$	0	0
$173$	−12.3695	−0.940433	−0.470217	−	0.882551i	$-0.655824\pi$
$173$	−12.3695	−0.940433	−0.470217	+	0.882551i	$0.655824\pi$
$174$	0	0
$175$	20.3465	1.53805
$176$	0	0
$177$	−5.65685	−0.425195
$178$	0	0
$179$	11.6413	0.870111	0.435055	−	0.900404i	$-0.356729\pi$
$179$	11.6413	0.870111	0.435055	+	0.900404i	$0.356729\pi$
$180$	0	0
$181$	−9.50732	−0.706673	−0.353337	−	0.935496i	$-0.614953\pi$
$181$	−9.50732	−0.706673	−0.353337	+	0.935496i	$0.614953\pi$
$182$	0	0
$183$	8.46742	0.625930
$184$	0	0
$185$	19.7091	1.44904
$186$	0	0
$187$	−0.571533	−0.0417946
$188$	0	0
$189$	−2.15894	−0.157040
$190$	0	0
$191$	−20.8032	−1.50526	−0.752632	−	0.658441i	$-0.771216\pi$
$191$	−20.8032	−1.50526	−0.752632	+	0.658441i	$0.771216\pi$
$192$	0	0
$193$	14.1454	1.01821	0.509103	−	0.860705i	$-0.329977\pi$
$193$	14.1454	1.01821	0.509103	+	0.860705i	$0.329977\pi$
$194$	0	0
$195$	7.43208	0.532222
$196$	0	0
$197$	−3.43463	−0.244707	−0.122354	−	0.992487i	$-0.539044\pi$
$197$	−3.43463	−0.244707	−0.122354	+	0.992487i	$0.539044\pi$
$198$	0	0
$199$	0.306182	0.0217047	0.0108523	−	0.999941i	$-0.496546\pi$
$199$	0.306182	0.0217047	0.0108523	+	0.999941i	$0.496546\pi$
$200$	0	0
$201$	−14.7422	−1.03983
$202$	0	0
$203$	5.67016	0.397967
$204$	0	0
$205$	−22.3380	−1.56016
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	−0.571533	−0.0395337
$210$	0	0
$211$	10.2284	0.704151	0.352076	−	0.935972i	$-0.385476\pi$
$211$	10.2284	0.704151	0.352076	+	0.935972i	$0.385476\pi$
$212$	0	0
$213$	4.31788	0.295856
$214$	0	0
$215$	−41.6517	−2.84063
$216$	0	0
$217$	3.97474	0.269823
$218$	0	0
$219$	−5.97474	−0.403735
$220$	0	0
$221$	0.439861	0.0295883
$222$	0	0
$223$	−1.71908	−0.115118	−0.0575591	−	0.998342i	$-0.518332\pi$
$223$	−1.71908	−0.115118	−0.0575591	+	0.998342i	$0.518332\pi$
$224$	0	0
$225$	9.42429	0.628286
$226$	0	0
$227$	−14.3059	−0.949518	−0.474759	−	0.880116i	$-0.657465\pi$
$227$	−14.3059	−0.949518	−0.474759	+	0.880116i	$0.657465\pi$
$228$	0	0
$229$	−16.9981	−1.12327	−0.561634	−	0.827386i	$-0.689827\pi$
$229$	−16.9981	−1.12327	−0.561634	+	0.827386i	$0.689827\pi$
$230$	0	0
$231$	5.48946	0.361180
$232$	0	0
$233$	13.3779	0.876418	0.438209	−	0.898873i	$-0.355613\pi$
$233$	13.3779	0.876418	0.438209	+	0.898873i	$0.355613\pi$
$234$	0	0
$235$	−10.7422	−0.700742
$236$	0	0
$237$	15.0075	0.974844
$238$	0	0
$239$	−13.3675	−0.864670	−0.432335	−	0.901713i	$-0.642310\pi$
$239$	−13.3675	−0.864670	−0.432335	+	0.901713i	$0.642310\pi$
$240$	0	0
$241$	−0.211474	−0.0136222	−0.00681112	−	0.999977i	$-0.502168\pi$
$241$	−0.211474	−0.0136222	−0.00681112	+	0.999977i	$0.502168\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	8.88325	0.567530
$246$	0	0
$247$	0.439861	0.0279877
$248$	0	0
$249$	14.3059	0.906601
$250$	0	0
$251$	14.7555	0.931358	0.465679	−	0.884954i	$-0.345810\pi$
$251$	14.7555	0.931358	0.465679	+	0.884954i	$0.345810\pi$
$252$	0	0
$253$	7.19173	0.452140
$254$	0	0
$255$	0.853690	0.0534601
$256$	0	0
$257$	−0.742176	−0.0462957	−0.0231478	−	0.999732i	$-0.507369\pi$
$257$	−0.742176	−0.0462957	−0.0231478	+	0.999732i	$0.507369\pi$
$258$	0	0
$259$	−11.2037	−0.696163
$260$	0	0
$261$	2.62636	0.162568
$262$	0	0
$263$	5.48435	0.338180	0.169090	−	0.985601i	$-0.445917\pi$
$263$	5.48435	0.338180	0.169090	+	0.985601i	$0.445917\pi$
$264$	0	0
$265$	40.3582	2.47918
$266$	0	0
$267$	1.42847	0.0874208
$268$	0	0
$269$	20.4694	1.24804	0.624021	−	0.781407i	$-0.285498\pi$
$269$	20.4694	1.24804	0.624021	+	0.781407i	$0.285498\pi$
$270$	0	0
$271$	14.0370	0.852685	0.426342	−	0.904562i	$-0.359802\pi$
$271$	14.0370	0.852685	0.426342	+	0.904562i	$0.359802\pi$
$272$	0	0
$273$	−4.22478	−0.255695
$274$	0	0
$275$	−23.9628	−1.44501
$276$	0	0
$277$	−13.4211	−0.806394	−0.403197	−	0.915113i	$-0.632101\pi$
$277$	−13.4211	−0.806394	−0.403197	+	0.915113i	$0.632101\pi$
$278$	0	0
$279$	1.84106	0.110221
$280$	0	0
$281$	3.89359	0.232272	0.116136	−	0.993233i	$-0.462949\pi$
$281$	3.89359	0.232272	0.116136	+	0.993233i	$0.462949\pi$
$282$	0	0
$283$	−17.6569	−1.04959	−0.524796	−	0.851228i	$-0.675858\pi$
$283$	−17.6569	−1.04959	−0.524796	+	0.851228i	$0.675858\pi$
$284$	0	0
$285$	0.853690	0.0505682
$286$	0	0
$287$	12.6981	0.749545
$288$	0	0
$289$	−16.9495	−0.997028
$290$	0	0
$291$	16.3990	0.961328
$292$	0	0
$293$	−15.7759	−0.921639	−0.460819	−	0.887494i	$-0.652444\pi$
$293$	−15.7759	−0.921639	−0.460819	+	0.887494i	$0.652444\pi$
$294$	0	0
$295$	−21.4844	−1.25087
$296$	0	0
$297$	2.54266	0.147540
$298$	0	0
$299$	−5.53488	−0.320090
$300$	0	0
$301$	23.6770	1.36472
$302$	0	0
$303$	−0.115816	−0.00665345
$304$	0	0
$305$	32.1587	1.84140
$306$	0	0
$307$	−7.64129	−0.436111	−0.218056	−	0.975936i	$-0.569971\pi$
$307$	−7.64129	−0.436111	−0.218056	+	0.975936i	$0.569971\pi$
$308$	0	0
$309$	−13.3507	−0.759493
$310$	0	0
$311$	24.1623	1.37012	0.685059	−	0.728488i	$-0.259776\pi$
$311$	24.1623	1.37012	0.685059	+	0.728488i	$0.259776\pi$
$312$	0	0
$313$	−16.6105	−0.938881	−0.469441	−	0.882964i	$-0.655544\pi$
$313$	−16.6105	−0.938881	−0.469441	+	0.882964i	$0.655544\pi$
$314$	0	0
$315$	−8.19951	−0.461990
$316$	0	0
$317$	−2.56213	−0.143903	−0.0719517	−	0.997408i	$-0.522923\pi$
$317$	−2.56213	−0.143903	−0.0719517	+	0.997408i	$0.522923\pi$
$318$	0	0
$319$	−6.67794	−0.373893
$320$	0	0
$321$	−10.2926	−0.574478
$322$	0	0
$323$	0.0505249	0.00281128
$324$	0	0
$325$	18.4422	1.02299
$326$	0	0
$327$	9.95687	0.550616
$328$	0	0
$329$	6.10641	0.336657
$330$	0	0
$331$	−19.1275	−1.05134	−0.525671	−	0.850688i	$-0.676186\pi$
$331$	−19.1275	−1.05134	−0.525671	+	0.850688i	$0.676186\pi$
$332$	0	0
$333$	−5.18944	−0.284379
$334$	0	0
$335$	−55.9898	−3.05905
$336$	0	0
$337$	1.12615	0.0613454	0.0306727	−	0.999529i	$-0.490235\pi$
$337$	1.12615	0.0613454	0.0306727	+	0.999529i	$0.490235\pi$
$338$	0	0
$339$	18.8486	1.02371
$340$	0	0
$341$	−4.68119	−0.253500
$342$	0	0
$343$	−20.1623	−1.08866
$344$	0	0
$345$	−10.7422	−0.578339
$346$	0	0
$347$	29.4068	1.57864	0.789320	−	0.613982i	$-0.210433\pi$
$347$	29.4068	1.57864	0.789320	+	0.613982i	$0.210433\pi$
$348$	0	0
$349$	27.2738	1.45993	0.729967	−	0.683482i	$-0.239535\pi$
$349$	27.2738	1.45993	0.729967	+	0.683482i	$0.239535\pi$
$350$	0	0
$351$	−1.95687	−0.104450
$352$	0	0
$353$	25.5908	1.36206	0.681029	−	0.732256i	$-0.261533\pi$
$353$	25.5908	1.36206	0.681029	+	0.732256i	$0.261533\pi$
$354$	0	0
$355$	16.3990	0.870370
$356$	0	0
$357$	−0.485281	−0.0256838
$358$	0	0
$359$	3.77296	0.199129	0.0995645	−	0.995031i	$-0.468255\pi$
$359$	3.77296	0.199129	0.0995645	+	0.995031i	$0.468255\pi$
$360$	0	0
$361$	−18.9495	−0.997341
$362$	0	0
$363$	4.53488	0.238019
$364$	0	0
$365$	−22.6917	−1.18774
$366$	0	0
$367$	27.4474	1.43274	0.716371	−	0.697720i	$-0.245802\pi$
$367$	27.4474	1.43274	0.716371	+	0.697720i	$0.245802\pi$
$368$	0	0
$369$	5.88163	0.306186
$370$	0	0
$371$	−22.9417	−1.19107
$372$	0	0
$373$	−17.8518	−0.924332	−0.462166	−	0.886794i	$-0.652928\pi$
$373$	−17.8518	−0.924332	−0.462166	+	0.886794i	$0.652928\pi$
$374$	0	0
$375$	16.8032	0.867712
$376$	0	0
$377$	5.13946	0.264695
$378$	0	0
$379$	−16.5018	−0.847642	−0.423821	−	0.905746i	$-0.639311\pi$
$379$	−16.5018	−0.847642	−0.423821	+	0.905746i	$0.639311\pi$
$380$	0	0
$381$	3.81580	0.195489
$382$	0	0
$383$	−17.1885	−0.878291	−0.439145	−	0.898416i	$-0.644719\pi$
$383$	−17.1885	−0.878291	−0.439145	+	0.898416i	$0.644719\pi$
$384$	0	0
$385$	20.8486	1.06254
$386$	0	0
$387$	10.9670	0.557482
$388$	0	0
$389$	−2.66209	−0.134973	−0.0674866	−	0.997720i	$-0.521498\pi$
$389$	−2.66209	−0.134973	−0.0674866	+	0.997720i	$0.521498\pi$
$390$	0	0
$391$	−0.635767	−0.0321521
$392$	0	0
$393$	−1.08532	−0.0547472
$394$	0	0
$395$	56.9976	2.86786
$396$	0	0
$397$	11.8959	0.597037	0.298519	−	0.954404i	$-0.403507\pi$
$397$	11.8959	0.597037	0.298519	+	0.954404i	$0.403507\pi$
$398$	0	0
$399$	−0.485281	−0.0242945
$400$	0	0
$401$	−1.12389	−0.0561242	−0.0280621	−	0.999606i	$-0.508934\pi$
$401$	−1.12389	−0.0561242	−0.0280621	+	0.999606i	$0.508934\pi$
$402$	0	0
$403$	3.60272	0.179464
$404$	0	0
$405$	−3.79793	−0.188721
$406$	0	0
$407$	13.1950	0.654051
$408$	0	0
$409$	13.7211	0.678464	0.339232	−	0.940703i	$-0.389833\pi$
$409$	13.7211	0.678464	0.339232	+	0.940703i	$0.389833\pi$
$410$	0	0
$411$	5.31010	0.261928
$412$	0	0
$413$	12.2128	0.600953
$414$	0	0
$415$	54.3329	2.66710
$416$	0	0
$417$	12.3990	0.607183
$418$	0	0
$419$	13.1629	0.643048	0.321524	−	0.946901i	$-0.395805\pi$
$419$	13.1629	0.643048	0.321524	+	0.946901i	$0.395805\pi$
$420$	0	0
$421$	11.9413	0.581984	0.290992	−	0.956726i	$-0.406015\pi$
$421$	11.9413	0.581984	0.290992	+	0.956726i	$0.406015\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	2.11837	0.102756
$426$	0	0
$427$	−18.2807	−0.884663
$428$	0	0
$429$	4.97567	0.240227
$430$	0	0
$431$	30.6054	1.47421	0.737105	−	0.675778i	$-0.236192\pi$
$431$	30.6054	1.47421	0.737105	+	0.675778i	$0.236192\pi$
$432$	0	0
$433$	15.3137	0.735930	0.367965	−	0.929840i	$-0.380055\pi$
$433$	15.3137	0.735930	0.367965	+	0.929840i	$0.380055\pi$
$434$	0	0
$435$	9.97474	0.478252
$436$	0	0
$437$	−0.635767	−0.0304128
$438$	0	0
$439$	33.3676	1.59255	0.796274	−	0.604936i	$-0.206801\pi$
$439$	33.3676	1.59255	0.796274	+	0.604936i	$0.206801\pi$
$440$	0	0
$441$	−2.33897	−0.111380
$442$	0	0
$443$	3.23617	0.153755	0.0768776	−	0.997041i	$-0.475505\pi$
$443$	3.23617	0.153755	0.0768776	+	0.997041i	$0.475505\pi$
$444$	0	0
$445$	5.42522	0.257180
$446$	0	0
$447$	−1.45479	−0.0688091
$448$	0	0
$449$	−27.4165	−1.29387	−0.646933	−	0.762547i	$-0.723948\pi$
$449$	−27.4165	−1.29387	−0.646933	+	0.762547i	$0.723948\pi$
$450$	0	0
$451$	−14.9550	−0.704203
$452$	0	0
$453$	−2.03696	−0.0957049
$454$	0	0
$455$	−16.0454	−0.752221
$456$	0	0
$457$	−10.9147	−0.510567	−0.255284	−	0.966866i	$-0.582169\pi$
$457$	−10.9147	−0.510567	−0.255284	+	0.966866i	$0.582169\pi$
$458$	0	0
$459$	−0.224777	−0.0104917
$460$	0	0
$461$	25.2181	1.17452	0.587261	−	0.809398i	$-0.300206\pi$
$461$	25.2181	1.17452	0.587261	+	0.809398i	$0.300206\pi$
$462$	0	0
$463$	−22.4937	−1.04537	−0.522686	−	0.852525i	$-0.675070\pi$
$463$	−22.4937	−1.04537	−0.522686	+	0.852525i	$0.675070\pi$
$464$	0	0
$465$	6.99222	0.324256
$466$	0	0
$467$	34.2482	1.58482	0.792408	−	0.609991i	$-0.208827\pi$
$467$	34.2482	1.58482	0.792408	+	0.609991i	$0.208827\pi$
$468$	0	0
$469$	31.8275	1.46966
$470$	0	0
$471$	8.61790	0.397092
$472$	0	0
$473$	−27.8852	−1.28216
$474$	0	0
$475$	2.11837	0.0971974
$476$	0	0
$477$	−10.6264	−0.486548
$478$	0	0
$479$	−36.2362	−1.65568	−0.827838	−	0.560968i	$-0.810429\pi$
$479$	−36.2362	−1.65568	−0.827838	+	0.560968i	$0.810429\pi$
$480$	0	0
$481$	−10.1551	−0.463032
$482$	0	0
$483$	6.10641	0.277851
$484$	0	0
$485$	62.2824	2.82810
$486$	0	0
$487$	16.8200	0.762186	0.381093	−	0.924537i	$-0.375548\pi$
$487$	16.8200	0.762186	0.381093	+	0.924537i	$0.375548\pi$
$488$	0	0
$489$	4.86054	0.219801
$490$	0	0
$491$	−8.63577	−0.389727	−0.194863	−	0.980830i	$-0.562426\pi$
$491$	−8.63577	−0.389727	−0.194863	+	0.980830i	$0.562426\pi$
$492$	0	0
$493$	0.590346	0.0265879
$494$	0	0
$495$	9.65685	0.434043
$496$	0	0
$497$	−9.32206	−0.418151
$498$	0	0
$499$	−27.8275	−1.24573	−0.622865	−	0.782329i	$-0.714031\pi$
$499$	−27.8275	−1.24573	−0.622865	+	0.782329i	$0.714031\pi$
$500$	0	0
$501$	21.7023	0.969586
$502$	0	0
$503$	−25.7308	−1.14728	−0.573639	−	0.819108i	$-0.694469\pi$
$503$	−25.7308	−1.14728	−0.573639	+	0.819108i	$0.694469\pi$
$504$	0	0
$505$	−0.439861	−0.0195736
$506$	0	0
$507$	9.17064	0.407283
$508$	0	0
$509$	−2.45386	−0.108765	−0.0543826	−	0.998520i	$-0.517319\pi$
$509$	−2.45386	−0.108765	−0.0543826	+	0.998520i	$0.517319\pi$
$510$	0	0
$511$	12.8991	0.570623
$512$	0	0
$513$	−0.224777	−0.00992417
$514$	0	0
$515$	−50.7050	−2.23433
$516$	0	0
$517$	−7.19173	−0.316292
$518$	0	0
$519$	12.3695	0.542959
$520$	0	0
$521$	−33.5944	−1.47180	−0.735898	−	0.677092i	$-0.763240\pi$
$521$	−33.5944	−1.47180	−0.735898	+	0.677092i	$0.763240\pi$
$522$	0	0
$523$	30.8522	1.34907	0.674536	−	0.738242i	$-0.264344\pi$
$523$	30.8522	1.34907	0.674536	+	0.738242i	$0.264344\pi$
$524$	0	0
$525$	−20.3465	−0.887994
$526$	0	0
$527$	0.413828	0.0180266
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	5.65685	0.245487
$532$	0	0
$533$	11.5096	0.498537
$534$	0	0
$535$	−39.0907	−1.69004
$536$	0	0
$537$	−11.6413	−0.502359
$538$	0	0
$539$	5.94721	0.256164
$540$	0	0
$541$	−38.4825	−1.65449	−0.827245	−	0.561841i	$-0.810093\pi$
$541$	−38.4825	−1.65449	−0.827245	+	0.561841i	$0.810093\pi$
$542$	0	0
$543$	9.50732	0.407998
$544$	0	0
$545$	37.8155	1.61984
$546$	0	0
$547$	9.61829	0.411248	0.205624	−	0.978631i	$-0.434078\pi$
$547$	9.61829	0.411248	0.205624	+	0.978631i	$0.434078\pi$
$548$	0	0
$549$	−8.46742	−0.361381
$550$	0	0
$551$	0.590346	0.0251496
$552$	0	0
$553$	−32.4004	−1.37780
$554$	0	0
$555$	−19.7091	−0.836606
$556$	0	0
$557$	−6.07174	−0.257268	−0.128634	−	0.991692i	$-0.541059\pi$
$557$	−6.07174	−0.257268	−0.128634	+	0.991692i	$0.541059\pi$
$558$	0	0
$559$	21.4609	0.907701
$560$	0	0
$561$	0.571533	0.0241301
$562$	0	0
$563$	−14.2554	−0.600793	−0.300397	−	0.953814i	$-0.597119\pi$
$563$	−14.2554	−0.600793	−0.300397	+	0.953814i	$0.597119\pi$
$564$	0	0
$565$	71.5857	3.01163
$566$	0	0
$567$	2.15894	0.0906670
$568$	0	0
$569$	−32.5018	−1.36255	−0.681274	−	0.732029i	$-0.738574\pi$
$569$	−32.5018	−1.36255	−0.681274	+	0.732029i	$0.738574\pi$
$570$	0	0
$571$	−12.9706	−0.542801	−0.271401	−	0.962466i	$-0.587487\pi$
$571$	−12.9706	−0.542801	−0.271401	+	0.962466i	$0.587487\pi$
$572$	0	0
$573$	20.8032	0.869065
$574$	0	0
$575$	−26.6559	−1.11163
$576$	0	0
$577$	11.7536	0.489308	0.244654	−	0.969611i	$-0.421326\pi$
$577$	11.7536	0.489308	0.244654	+	0.969611i	$0.421326\pi$
$578$	0	0
$579$	−14.1454	−0.587862
$580$	0	0
$581$	−30.8857	−1.28135
$582$	0	0
$583$	27.0192	1.11902
$584$	0	0
$585$	−7.43208	−0.307279
$586$	0	0
$587$	−9.13585	−0.377077	−0.188538	−	0.982066i	$-0.560375\pi$
$587$	−9.13585	−0.377077	−0.188538	+	0.982066i	$0.560375\pi$
$588$	0	0
$589$	0.413828	0.0170515
$590$	0	0
$591$	3.43463	0.141282
$592$	0	0
$593$	−5.49270	−0.225558	−0.112779	−	0.993620i	$-0.535975\pi$
$593$	−5.49270	−0.225558	−0.112779	+	0.993620i	$0.535975\pi$
$594$	0	0
$595$	−1.84307	−0.0755583
$596$	0	0
$597$	−0.306182	−0.0125312
$598$	0	0
$599$	−36.4348	−1.48868	−0.744342	−	0.667799i	$-0.767237\pi$
$599$	−36.4348	−1.48868	−0.744342	+	0.667799i	$0.767237\pi$
$600$	0	0
$601$	−9.97474	−0.406878	−0.203439	−	0.979088i	$-0.565212\pi$
$601$	−9.97474	−0.406878	−0.203439	+	0.979088i	$0.565212\pi$
$602$	0	0
$603$	14.7422	0.600348
$604$	0	0
$605$	17.2232	0.700221
$606$	0	0
$607$	4.51900	0.183421	0.0917103	−	0.995786i	$-0.470767\pi$
$607$	4.51900	0.183421	0.0917103	+	0.995786i	$0.470767\pi$
$608$	0	0
$609$	−5.67016	−0.229766
$610$	0	0
$611$	5.53488	0.223917
$612$	0	0
$613$	−11.9316	−0.481913	−0.240957	−	0.970536i	$-0.577461\pi$
$613$	−11.9316	−0.481913	−0.240957	+	0.970536i	$0.577461\pi$
$614$	0	0
$615$	22.3380	0.900757
$616$	0	0
$617$	−32.1201	−1.29311	−0.646554	−	0.762869i	$-0.723790\pi$
$617$	−32.1201	−1.29311	−0.646554	+	0.762869i	$0.723790\pi$
$618$	0	0
$619$	−21.2715	−0.854975	−0.427488	−	0.904021i	$-0.640601\pi$
$619$	−21.2715	−0.854975	−0.427488	+	0.904021i	$0.640601\pi$
$620$	0	0
$621$	2.82843	0.113501
$622$	0	0
$623$	−3.08398	−0.123557
$624$	0	0
$625$	16.6958	0.667833
$626$	0	0
$627$	0.571533	0.0228248
$628$	0	0
$629$	−1.16647	−0.0465101
$630$	0	0
$631$	36.4685	1.45179	0.725894	−	0.687807i	$-0.241426\pi$
$631$	36.4685	1.45179	0.725894	+	0.687807i	$0.241426\pi$
$632$	0	0
$633$	−10.2284	−0.406542
$634$	0	0
$635$	14.4921	0.575103
$636$	0	0
$637$	−4.57707	−0.181350
$638$	0	0
$639$	−4.31788	−0.170813
$640$	0	0
$641$	14.0036	0.553109	0.276555	−	0.960998i	$-0.410807\pi$
$641$	14.0036	0.553109	0.276555	+	0.960998i	$0.410807\pi$
$642$	0	0
$643$	−23.4807	−0.925990	−0.462995	−	0.886361i	$-0.653225\pi$
$643$	−23.4807	−0.925990	−0.462995	+	0.886361i	$0.653225\pi$
$644$	0	0
$645$	41.6517	1.64004
$646$	0	0
$647$	−12.1908	−0.479270	−0.239635	−	0.970863i	$-0.577028\pi$
$647$	−12.1908	−0.479270	−0.239635	+	0.970863i	$0.577028\pi$
$648$	0	0
$649$	−14.3835	−0.564600
$650$	0	0
$651$	−3.97474	−0.155782
$652$	0	0
$653$	1.39055	0.0544165	0.0272083	−	0.999630i	$-0.491338\pi$
$653$	1.39055	0.0544165	0.0272083	+	0.999630i	$0.491338\pi$
$654$	0	0
$655$	−4.12198	−0.161059
$656$	0	0
$657$	5.97474	0.233097
$658$	0	0
$659$	−25.5349	−0.994698	−0.497349	−	0.867551i	$-0.665693\pi$
$659$	−25.5349	−0.994698	−0.497349	+	0.867551i	$0.665693\pi$
$660$	0	0
$661$	6.44726	0.250769	0.125385	−	0.992108i	$-0.459983\pi$
$661$	6.44726	0.250769	0.125385	+	0.992108i	$0.459983\pi$
$662$	0	0
$663$	−0.439861	−0.0170828
$664$	0	0
$665$	−1.84307	−0.0714710
$666$	0	0
$667$	−7.42847	−0.287631
$668$	0	0
$669$	1.71908	0.0664635
$670$	0	0
$671$	21.5298	0.831148
$672$	0	0
$673$	−10.8569	−0.418504	−0.209252	−	0.977862i	$-0.567103\pi$
$673$	−10.8569	−0.418504	−0.209252	+	0.977862i	$0.567103\pi$
$674$	0	0
$675$	−9.42429	−0.362741
$676$	0	0
$677$	33.5262	1.28852	0.644259	−	0.764807i	$-0.277166\pi$
$677$	33.5262	1.28852	0.644259	+	0.764807i	$0.277166\pi$
$678$	0	0
$679$	−35.4045	−1.35870
$680$	0	0
$681$	14.3059	0.548204
$682$	0	0
$683$	25.2206	0.965040	0.482520	−	0.875885i	$-0.339722\pi$
$683$	25.2206	0.965040	0.482520	+	0.875885i	$0.339722\pi$
$684$	0	0
$685$	20.1674	0.770557
$686$	0	0
$687$	16.9981	0.648519
$688$	0	0
$689$	−20.7945	−0.792205
$690$	0	0
$691$	15.3523	0.584028	0.292014	−	0.956414i	$-0.405675\pi$
$691$	15.3523	0.584028	0.292014	+	0.956414i	$0.405675\pi$
$692$	0	0
$693$	−5.48946	−0.208527
$694$	0	0
$695$	47.0907	1.78625
$696$	0	0
$697$	1.32206	0.0500765
$698$	0	0
$699$	−13.3779	−0.506000
$700$	0	0
$701$	8.61079	0.325225	0.162613	−	0.986690i	$-0.448008\pi$
$701$	8.61079	0.325225	0.162613	+	0.986690i	$0.448008\pi$
$702$	0	0
$703$	−1.16647	−0.0439942
$704$	0	0
$705$	10.7422	0.404574
$706$	0	0
$707$	0.250040	0.00940372
$708$	0	0
$709$	32.3624	1.21539	0.607697	−	0.794169i	$-0.292094\pi$
$709$	32.3624	1.21539	0.607697	+	0.794169i	$0.292094\pi$
$710$	0	0
$711$	−15.0075	−0.562826
$712$	0	0
$713$	−5.20730	−0.195015
$714$	0	0
$715$	18.8972	0.706717
$716$	0	0
$717$	13.3675	0.499218
$718$	0	0
$719$	1.46744	0.0547262	0.0273631	−	0.999626i	$-0.491289\pi$
$719$	1.46744	0.0547262	0.0273631	+	0.999626i	$0.491289\pi$
$720$	0	0
$721$	28.8233	1.07344
$722$	0	0
$723$	0.211474	0.00786481
$724$	0	0
$725$	24.7516	0.919251
$726$	0	0
$727$	−15.3928	−0.570889	−0.285445	−	0.958395i	$-0.592141\pi$
$727$	−15.3928	−0.570889	−0.285445	+	0.958395i	$0.592141\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.46512	0.0911759
$732$	0	0
$733$	17.5624	0.648683	0.324342	−	0.945940i	$-0.394857\pi$
$733$	17.5624	0.648683	0.324342	+	0.945940i	$0.394857\pi$
$734$	0	0
$735$	−8.88325	−0.327664
$736$	0	0
$737$	−37.4844	−1.38075
$738$	0	0
$739$	20.7266	0.762441	0.381220	−	0.924484i	$-0.375504\pi$
$739$	20.7266	0.762441	0.381220	+	0.924484i	$0.375504\pi$
$740$	0	0
$741$	−0.439861	−0.0161587
$742$	0	0
$743$	−31.7821	−1.16597	−0.582986	−	0.812482i	$-0.698116\pi$
$743$	−31.7821	−1.16597	−0.582986	+	0.812482i	$0.698116\pi$
$744$	0	0
$745$	−5.52518	−0.202427
$746$	0	0
$747$	−14.3059	−0.523426
$748$	0	0
$749$	22.2212	0.811944
$750$	0	0
$751$	−29.7594	−1.08594	−0.542968	−	0.839753i	$-0.682699\pi$
$751$	−29.7594	−1.08594	−0.542968	+	0.839753i	$0.682699\pi$
$752$	0	0
$753$	−14.7555	−0.537720
$754$	0	0
$755$	−7.73625	−0.281551
$756$	0	0
$757$	−22.1119	−0.803672	−0.401836	−	0.915712i	$-0.631628\pi$
$757$	−22.1119	−0.803672	−0.401836	+	0.915712i	$0.631628\pi$
$758$	0	0
$759$	−7.19173	−0.261043
$760$	0	0
$761$	−4.55957	−0.165284	−0.0826422	−	0.996579i	$-0.526336\pi$
$761$	−4.55957	−0.165284	−0.0826422	+	0.996579i	$0.526336\pi$
$762$	0	0
$763$	−21.4963	−0.778219
$764$	0	0
$765$	−0.853690	−0.0308652
$766$	0	0
$767$	11.0698	0.399706
$768$	0	0
$769$	36.5794	1.31909	0.659543	−	0.751667i	$-0.270750\pi$
$769$	36.5794	1.31909	0.659543	+	0.751667i	$0.270750\pi$
$770$	0	0
$771$	0.742176	0.0267288
$772$	0	0
$773$	−26.4611	−0.951739	−0.475869	−	0.879516i	$-0.657867\pi$
$773$	−26.4611	−0.951739	−0.475869	+	0.879516i	$0.657867\pi$
$774$	0	0
$775$	17.3507	0.623255
$776$	0	0
$777$	11.2037	0.401930
$778$	0	0
$779$	1.32206	0.0473676
$780$	0	0
$781$	10.9789	0.392856
$782$	0	0
$783$	−2.62636	−0.0938584
$784$	0	0
$785$	32.7302	1.16819
$786$	0	0
$787$	18.9164	0.674298	0.337149	−	0.941451i	$-0.390537\pi$
$787$	18.9164	0.674298	0.337149	+	0.941451i	$0.390537\pi$
$788$	0	0
$789$	−5.48435	−0.195248
$790$	0	0
$791$	−40.6930	−1.44688
$792$	0	0
$793$	−16.5697	−0.588406
$794$	0	0
$795$	−40.3582	−1.43136
$796$	0	0
$797$	47.8065	1.69339	0.846697	−	0.532075i	$-0.178588\pi$
$797$	47.8065	1.69339	0.846697	+	0.532075i	$0.178588\pi$
$798$	0	0
$799$	0.635767	0.0224918
$800$	0	0
$801$	−1.42847	−0.0504724
$802$	0	0
$803$	−15.1917	−0.536105
$804$	0	0
$805$	23.1917	0.817401
$806$	0	0
$807$	−20.4694	−0.720558
$808$	0	0
$809$	29.9862	1.05426	0.527129	−	0.849785i	$-0.323268\pi$
$809$	29.9862	1.05426	0.527129	+	0.849785i	$0.323268\pi$
$810$	0	0
$811$	11.3899	0.399954	0.199977	−	0.979801i	$-0.435913\pi$
$811$	11.3899	0.399954	0.199977	+	0.979801i	$0.435913\pi$
$812$	0	0
$813$	−14.0370	−0.492298
$814$	0	0
$815$	18.4600	0.646626
$816$	0	0
$817$	2.46512	0.0862438
$818$	0	0
$819$	4.22478	0.147626
$820$	0	0
$821$	−19.6929	−0.687286	−0.343643	−	0.939100i	$-0.611661\pi$
$821$	−19.6929	−0.687286	−0.343643	+	0.939100i	$0.611661\pi$
$822$	0	0
$823$	22.4666	0.783137	0.391568	−	0.920149i	$-0.371933\pi$
$823$	22.4666	0.783137	0.391568	+	0.920149i	$0.371933\pi$
$824$	0	0
$825$	23.9628	0.834277
$826$	0	0
$827$	10.7927	0.375299	0.187649	−	0.982236i	$-0.439913\pi$
$827$	10.7927	0.375299	0.187649	+	0.982236i	$0.439913\pi$
$828$	0	0
$829$	45.9421	1.59563	0.797817	−	0.602900i	$-0.205988\pi$
$829$	45.9421	1.59563	0.797817	+	0.602900i	$0.205988\pi$
$830$	0	0
$831$	13.4211	0.465572
$832$	0	0
$833$	−0.525748	−0.0182161
$834$	0	0
$835$	82.4238	2.85239
$836$	0	0
$837$	−1.84106	−0.0636363
$838$	0	0
$839$	22.9142	0.791085	0.395542	−	0.918448i	$-0.370557\pi$
$839$	22.9142	0.791085	0.395542	+	0.918448i	$0.370557\pi$
$840$	0	0
$841$	−22.1022	−0.762146
$842$	0	0
$843$	−3.89359	−0.134102
$844$	0	0
$845$	34.8295	1.19817
$846$	0	0
$847$	−9.79053	−0.336407
$848$	0	0
$849$	17.6569	0.605982
$850$	0	0
$851$	14.6779	0.503153
$852$	0	0
$853$	−55.0728	−1.88566	−0.942829	−	0.333278i	$-0.891845\pi$
$853$	−55.0728	−1.88566	−0.942829	+	0.333278i	$0.891845\pi$
$854$	0	0
$855$	−0.853690	−0.0291956
$856$	0	0
$857$	−1.79079	−0.0611723	−0.0305861	−	0.999532i	$-0.509737\pi$
$857$	−1.79079	−0.0611723	−0.0305861	+	0.999532i	$0.509737\pi$
$858$	0	0
$859$	25.5468	0.871647	0.435823	−	0.900032i	$-0.356457\pi$
$859$	25.5468	0.871647	0.435823	+	0.900032i	$0.356457\pi$
$860$	0	0
$861$	−12.6981	−0.432750
$862$	0	0
$863$	42.1150	1.43361	0.716806	−	0.697273i	$-0.245603\pi$
$863$	42.1150	1.43361	0.716806	+	0.697273i	$0.245603\pi$
$864$	0	0
$865$	46.9784	1.59731
$866$	0	0
$867$	16.9495	0.575634
$868$	0	0
$869$	38.1590	1.29446
$870$	0	0
$871$	28.8486	0.977497
$872$	0	0
$873$	−16.3990	−0.555023
$874$	0	0
$875$	−36.2771	−1.22639
$876$	0	0
$877$	1.01044	0.0341203	0.0170601	−	0.999854i	$-0.494569\pi$
$877$	1.01044	0.0341203	0.0170601	+	0.999854i	$0.494569\pi$
$878$	0	0
$879$	15.7759	0.532108
$880$	0	0
$881$	44.3972	1.49578	0.747889	−	0.663823i	$-0.231067\pi$
$881$	44.3972	1.49578	0.747889	+	0.663823i	$0.231067\pi$
$882$	0	0
$883$	1.82389	0.0613787	0.0306894	−	0.999529i	$-0.490230\pi$
$883$	1.82389	0.0613787	0.0306894	+	0.999529i	$0.490230\pi$
$884$	0	0
$885$	21.4844	0.722189
$886$	0	0
$887$	4.38532	0.147245	0.0736223	−	0.997286i	$-0.476544\pi$
$887$	4.38532	0.147245	0.0736223	+	0.997286i	$0.476544\pi$
$888$	0	0
$889$	−8.23808	−0.276296
$890$	0	0
$891$	−2.54266	−0.0851823
$892$	0	0
$893$	0.635767	0.0212751
$894$	0	0
$895$	−44.2128	−1.47787
$896$	0	0
$897$	5.53488	0.184804
$898$	0	0
$899$	4.83528	0.161266
$900$	0	0
$901$	−2.38857	−0.0795747
$902$	0	0
$903$	−23.6770	−0.787922
$904$	0	0
$905$	36.1082	1.20028
$906$	0	0
$907$	4.06248	0.134893	0.0674463	−	0.997723i	$-0.478515\pi$
$907$	4.06248	0.134893	0.0674463	+	0.997723i	$0.478515\pi$
$908$	0	0
$909$	0.115816	0.00384137
$910$	0	0
$911$	−42.3784	−1.40406	−0.702029	−	0.712149i	$-0.747722\pi$
$911$	−42.3784	−1.40406	−0.702029	+	0.712149i	$0.747722\pi$
$912$	0	0
$913$	36.3751	1.20384
$914$	0	0
$915$	−32.1587	−1.06313
$916$	0	0
$917$	2.34315	0.0773775
$918$	0	0
$919$	44.8603	1.47980	0.739902	−	0.672715i	$-0.234872\pi$
$919$	44.8603	1.47980	0.739902	+	0.672715i	$0.234872\pi$
$920$	0	0
$921$	7.64129	0.251789
$922$	0	0
$923$	−8.44955	−0.278120
$924$	0	0
$925$	−48.9068	−1.60804
$926$	0	0
$927$	13.3507	0.438494
$928$	0	0
$929$	−2.96695	−0.0973426	−0.0486713	−	0.998815i	$-0.515499\pi$
$929$	−2.96695	−0.0973426	−0.0486713	+	0.998815i	$0.515499\pi$
$930$	0	0
$931$	−0.525748	−0.0172307
$932$	0	0
$933$	−24.1623	−0.791038
$934$	0	0
$935$	2.17064	0.0709876
$936$	0	0
$937$	54.7669	1.78916	0.894579	−	0.446910i	$-0.147476\pi$
$937$	54.7669	1.78916	0.894579	+	0.446910i	$0.147476\pi$
$938$	0	0
$939$	16.6105	0.542063
$940$	0	0
$941$	−8.89558	−0.289988	−0.144994	−	0.989433i	$-0.546316\pi$
$941$	−8.89558	−0.289988	−0.144994	+	0.989433i	$0.546316\pi$
$942$	0	0
$943$	−16.6358	−0.541735
$944$	0	0
$945$	8.19951	0.266730
$946$	0	0
$947$	−15.8541	−0.515189	−0.257595	−	0.966253i	$-0.582930\pi$
$947$	−15.8541	−0.515189	−0.257595	+	0.966253i	$0.582930\pi$
$948$	0	0
$949$	11.6918	0.379532
$950$	0	0
$951$	2.56213	0.0830826
$952$	0	0
$953$	−30.2807	−0.980887	−0.490443	−	0.871473i	$-0.663165\pi$
$953$	−30.2807	−0.980887	−0.490443	+	0.871473i	$0.663165\pi$
$954$	0	0
$955$	79.0090	2.55667
$956$	0	0
$957$	6.67794	0.215867
$958$	0	0
$959$	−11.4642	−0.370198
$960$	0	0
$961$	−27.6105	−0.890661
$962$	0	0
$963$	10.2926	0.331675
$964$	0	0
$965$	−53.7232	−1.72941
$966$	0	0
$967$	−10.5273	−0.338537	−0.169268	−	0.985570i	$-0.554140\pi$
$967$	−10.5273	−0.338537	−0.169268	+	0.985570i	$0.554140\pi$
$968$	0	0
$969$	−0.0505249	−0.00162309
$970$	0	0
$971$	−38.7050	−1.24210	−0.621051	−	0.783771i	$-0.713294\pi$
$971$	−38.7050	−1.24210	−0.621051	+	0.783771i	$0.713294\pi$
$972$	0	0
$973$	−26.7688	−0.858168
$974$	0	0
$975$	−18.4422	−0.590622
$976$	0	0
$977$	16.9009	0.540706	0.270353	−	0.962761i	$-0.412860\pi$
$977$	16.9009	0.540706	0.270353	+	0.962761i	$0.412860\pi$
$978$	0	0
$979$	3.63211	0.116083
$980$	0	0
$981$	−9.95687	−0.317899
$982$	0	0
$983$	−10.0798	−0.321496	−0.160748	−	0.986995i	$-0.551391\pi$
$983$	−10.0798	−0.321496	−0.160748	+	0.986995i	$0.551391\pi$
$984$	0	0
$985$	13.0445	0.415632
$986$	0	0
$987$	−6.10641	−0.194369
$988$	0	0
$989$	−31.0192	−0.986354
$990$	0	0
$991$	−42.3446	−1.34512	−0.672561	−	0.740042i	$-0.734806\pi$
$991$	−42.3446	−1.34512	−0.672561	+	0.740042i	$0.734806\pi$
$992$	0	0
$993$	19.1275	0.606993
$994$	0	0
$995$	−1.16286	−0.0368651
$996$	0	0
$997$	−47.6132	−1.50793	−0.753963	−	0.656917i	$-0.771860\pi$
$997$	−47.6132	−1.50793	−0.753963	+	0.656917i	$0.771860\pi$
$998$	0	0
$999$	5.18944	0.164186

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.i.1.1		4
3.2	odd	2		9216.2.a.bo.1.4		4
4.3	odd	2		3072.2.a.o.1.1		4
8.3	odd	2		3072.2.a.n.1.4		4
8.5	even	2		3072.2.a.t.1.4		4
12.11	even	2		9216.2.a.bn.1.4		4
16.3	odd	4		3072.2.d.i.1537.8		8
16.5	even	4		3072.2.d.f.1537.5		8
16.11	odd	4		3072.2.d.i.1537.1		8
16.13	even	4		3072.2.d.f.1537.4		8
24.5	odd	2		9216.2.a.y.1.1		4
24.11	even	2		9216.2.a.x.1.1		4
32.3	odd	8		384.2.j.a.289.4		8
32.5	even	8		48.2.j.a.37.3	yes	8
32.11	odd	8		384.2.j.a.97.4		8
32.13	even	8		48.2.j.a.13.3	✓	8
32.19	odd	8		192.2.j.a.145.1		8
32.21	even	8		384.2.j.b.97.2		8
32.27	odd	8		192.2.j.a.49.1		8
32.29	even	8		384.2.j.b.289.2		8
96.5	odd	8		144.2.k.b.37.2		8
96.11	even	8		1152.2.k.f.865.1		8
96.29	odd	8		1152.2.k.c.289.1		8
96.35	even	8		1152.2.k.f.289.1		8
96.53	odd	8		1152.2.k.c.865.1		8
96.59	even	8		576.2.k.b.433.4		8
96.77	odd	8		144.2.k.b.109.2		8
96.83	even	8		576.2.k.b.145.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.2.j.a.13.3	✓	8	32.13	even	8
48.2.j.a.37.3	yes	8	32.5	even	8
144.2.k.b.37.2		8	96.5	odd	8
144.2.k.b.109.2		8	96.77	odd	8
192.2.j.a.49.1		8	32.27	odd	8
192.2.j.a.145.1		8	32.19	odd	8
384.2.j.a.97.4		8	32.11	odd	8
384.2.j.a.289.4		8	32.3	odd	8
384.2.j.b.97.2		8	32.21	even	8
384.2.j.b.289.2		8	32.29	even	8
576.2.k.b.145.4		8	96.83	even	8
576.2.k.b.433.4		8	96.59	even	8
1152.2.k.c.289.1		8	96.29	odd	8
1152.2.k.c.865.1		8	96.53	odd	8
1152.2.k.f.289.1		8	96.35	even	8
1152.2.k.f.865.1		8	96.11	even	8
3072.2.a.i.1.1		4	1.1	even	1	trivial
3072.2.a.n.1.4		4	8.3	odd	2
3072.2.a.o.1.1		4	4.3	odd	2
3072.2.a.t.1.4		4	8.5	even	2
3072.2.d.f.1537.4		8	16.13	even	4
3072.2.d.f.1537.5		8	16.5	even	4
3072.2.d.i.1537.1		8	16.11	odd	4
3072.2.d.i.1537.8		8	16.3	odd	4
9216.2.a.x.1.1		4	24.11	even	2
9216.2.a.y.1.1		4	24.5	odd	2
9216.2.a.bn.1.4		4	12.11	even	2
9216.2.a.bo.1.4		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.79793 q^{5} +2.15894 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.79793 q^{5} +2.15894 q^{7} +1.00000 q^{9} -2.54266 q^{11} +1.95687 q^{13} +3.79793 q^{15} +0.224777 q^{17} +0.224777 q^{19} -2.15894 q^{21} -2.82843 q^{23} +9.42429 q^{25} -1.00000 q^{27} +2.62636 q^{29} +1.84106 q^{31} +2.54266 q^{33} -8.19951 q^{35} -5.18944 q^{37} -1.95687 q^{39} +5.88163 q^{41} +10.9670 q^{43} -3.79793 q^{45} +2.82843 q^{47} -2.33897 q^{49} -0.224777 q^{51} -10.6264 q^{53} +9.65685 q^{55} -0.224777 q^{57} +5.65685 q^{59} -8.46742 q^{61} +2.15894 q^{63} -7.43208 q^{65} +14.7422 q^{67} +2.82843 q^{69} -4.31788 q^{71} +5.97474 q^{73} -9.42429 q^{75} -5.48946 q^{77} -15.0075 q^{79} +1.00000 q^{81} -14.3059 q^{83} -0.853690 q^{85} -2.62636 q^{87} -1.42847 q^{89} +4.22478 q^{91} -1.84106 q^{93} -0.853690 q^{95} -16.3990 q^{97} -2.54266 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9} - 8 q^{13} + 4 q^{15} - 4 q^{21} + 4 q^{25} - 4 q^{27} - 12 q^{29} + 12 q^{31} - 16 q^{37} + 8 q^{39} - 4 q^{45} + 4 q^{49} - 20 q^{53} + 16 q^{55} - 16 q^{61} + 4 q^{63} - 8 q^{65} + 16 q^{67} - 8 q^{71} - 8 q^{73} - 4 q^{75} - 24 q^{77} + 12 q^{79} + 4 q^{81} - 24 q^{85} + 12 q^{87} - 8 q^{89} + 16 q^{91} - 12 q^{93} - 24 q^{95}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9 - 8 * q^13 + 4 * q^15 - 4 * q^21 + 4 * q^25 - 4 * q^27 - 12 * q^29 + 12 * q^31 - 16 * q^37 + 8 * q^39 - 4 * q^45 + 4 * q^49 - 20 * q^53 + 16 * q^55 - 16 * q^61 + 4 * q^63 - 8 * q^65 + 16 * q^67 - 8 * q^71 - 8 * q^73 - 4 * q^75 - 24 * q^77 + 12 * q^79 + 4 * q^81 - 24 * q^85 + 12 * q^87 - 8 * q^89 + 16 * q^91 - 12 * q^93 - 24 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more