LMFDB - Embedded newform 4012.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.92101 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.92101 q^{5} +1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} +4.92101 q^{13} +3.92101 q^{15} +1.00000 q^{17} +1.92101 q^{19} -1.00000 q^{21} -6.10817 q^{23} +10.3743 q^{25} +5.00000 q^{27} -5.92101 q^{29} +3.73385 q^{31} +2.00000 q^{33} -3.92101 q^{35} +8.65486 q^{37} -4.92101 q^{39} +1.92101 q^{41} -0.266149 q^{43} +7.84202 q^{45} +2.10817 q^{47} -6.00000 q^{49} -1.00000 q^{51} +9.37432 q^{53} +7.84202 q^{55} -1.92101 q^{57} -1.00000 q^{59} -6.92101 q^{61} -2.00000 q^{63} -19.2953 q^{65} +10.2163 q^{67} +6.10817 q^{69} +2.37432 q^{71} +2.00000 q^{73} -10.3743 q^{75} -2.00000 q^{77} -5.00000 q^{79} +1.00000 q^{81} +4.65486 q^{83} -3.92101 q^{85} +5.92101 q^{87} +5.73385 q^{89} +4.92101 q^{91} -3.73385 q^{93} -7.53230 q^{95} -3.45331 q^{97} +4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9	$ 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 3 q^{21} + 20 q^{25} + 15 q^{27} - 5 q^{29} + 4 q^{31} + 6 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 7 q^{41} - 8 q^{43} - 2 q^{45} - 12 q^{47} - 18 q^{49} - 3 q^{51} + 17 q^{53} - 2 q^{55} + 7 q^{57} - 3 q^{59} - 8 q^{61} - 6 q^{63} - 34 q^{65} - 6 q^{67} - 4 q^{71} + 6 q^{73} - 20 q^{75} - 6 q^{77} - 15 q^{79} + 3 q^{81} - 6 q^{83} + q^{85} + 5 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} - 37 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9 - 6 * q^11 + 2 * q^13 - q^15 + 3 * q^17 - 7 * q^19 - 3 * q^21 + 20 * q^25 + 15 * q^27 - 5 * q^29 + 4 * q^31 + 6 * q^33 + q^35 + 6 * q^37 - 2 * q^39 - 7 * q^41 - 8 * q^43 - 2 * q^45 - 12 * q^47 - 18 * q^49 - 3 * q^51 + 17 * q^53 - 2 * q^55 + 7 * q^57 - 3 * q^59 - 8 * q^61 - 6 * q^63 - 34 * q^65 - 6 * q^67 - 4 * q^71 + 6 * q^73 - 20 * q^75 - 6 * q^77 - 15 * q^79 + 3 * q^81 - 6 * q^83 + q^85 + 5 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 - 37 * q^95 - 12 * q^97 + 12 * 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	−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000	−0.577350	−0.288675	+	0.957427i	$0.593215\pi$
$4$	0	0
$5$	−3.92101	−1.75353	−0.876764	−	0.480920i	$-0.840303\pi$
$5$	−3.92101	−1.75353	−0.876764	+	0.480920i	$0.840303\pi$
$6$	0	0
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.92101	1.36484	0.682421	−	0.730959i	$-0.260927\pi$
$13$	4.92101	1.36484	0.682421	+	0.730959i	$0.260927\pi$
$14$	0	0
$15$	3.92101	1.01240
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	1.92101	0.440710	0.220355	−	0.975420i	$-0.429278\pi$
$19$	1.92101	0.440710	0.220355	+	0.975420i	$0.429278\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−6.10817	−1.27364	−0.636821	−	0.771012i	$-0.719751\pi$
$23$	−6.10817	−1.27364	−0.636821	+	0.771012i	$0.719751\pi$
$24$	0	0
$25$	10.3743	2.07486
$26$	0	0
$27$	5.00000	0.962250
$28$	0	0
$29$	−5.92101	−1.09950	−0.549752	−	0.835328i	$-0.685278\pi$
$29$	−5.92101	−1.09950	−0.549752	+	0.835328i	$0.685278\pi$
$30$	0	0
$31$	3.73385	0.670619	0.335310	−	0.942108i	$-0.391159\pi$
$31$	3.73385	0.670619	0.335310	+	0.942108i	$0.391159\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	−3.92101	−0.662772
$36$	0	0
$37$	8.65486	1.42285	0.711425	−	0.702762i	$-0.248050\pi$
$37$	8.65486	1.42285	0.711425	+	0.702762i	$0.248050\pi$
$38$	0	0
$39$	−4.92101	−0.787992
$40$	0	0
$41$	1.92101	0.300011	0.150006	−	0.988685i	$-0.452071\pi$
$41$	1.92101	0.300011	0.150006	+	0.988685i	$0.452071\pi$
$42$	0	0
$43$	−0.266149	−0.0405873	−0.0202937	−	0.999794i	$-0.506460\pi$
$43$	−0.266149	−0.0405873	−0.0202937	+	0.999794i	$0.506460\pi$
$44$	0	0
$45$	7.84202	1.16902
$46$	0	0
$47$	2.10817	0.307508	0.153754	−	0.988109i	$-0.450864\pi$
$47$	2.10817	0.307508	0.153754	+	0.988109i	$0.450864\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−1.00000	−0.140028
$52$	0	0
$53$	9.37432	1.28766	0.643831	−	0.765168i	$-0.277344\pi$
$53$	9.37432	1.28766	0.643831	+	0.765168i	$0.277344\pi$
$54$	0	0
$55$	7.84202	1.05742
$56$	0	0
$57$	−1.92101	−0.254444
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−6.92101	−0.886144	−0.443072	−	0.896486i	$-0.646111\pi$
$61$	−6.92101	−0.886144	−0.443072	+	0.896486i	$0.646111\pi$
$62$	0	0
$63$	−2.00000	−0.251976
$64$	0	0
$65$	−19.2953	−2.39329
$66$	0	0
$67$	10.2163	1.24812	0.624062	−	0.781375i	$-0.285481\pi$
$67$	10.2163	1.24812	0.624062	+	0.781375i	$0.285481\pi$
$68$	0	0
$69$	6.10817	0.735337
$70$	0	0
$71$	2.37432	0.281780	0.140890	−	0.990025i	$-0.455004\pi$
$71$	2.37432	0.281780	0.140890	+	0.990025i	$0.455004\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−10.3743	−1.19792
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	−5.00000	−0.562544	−0.281272	−	0.959628i	$-0.590756\pi$
$79$	−5.00000	−0.562544	−0.281272	+	0.959628i	$0.590756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.65486	0.510937	0.255469	−	0.966817i	$-0.417770\pi$
$83$	4.65486	0.510937	0.255469	+	0.966817i	$0.417770\pi$
$84$	0	0
$85$	−3.92101	−0.425293
$86$	0	0
$87$	5.92101	0.634799
$88$	0	0
$89$	5.73385	0.607787	0.303893	−	0.952706i	$-0.401713\pi$
$89$	5.73385	0.607787	0.303893	+	0.952706i	$0.401713\pi$
$90$	0	0
$91$	4.92101	0.515862
$92$	0	0
$93$	−3.73385	−0.387182
$94$	0	0
$95$	−7.53230	−0.772797
$96$	0	0
$97$	−3.45331	−0.350630	−0.175315	−	0.984512i	$-0.556094\pi$
$97$	−3.45331	−0.350630	−0.175315	+	0.984512i	$0.556094\pi$
$98$	0	0
$99$	4.00000	0.402015
$100$	0	0
$101$	−17.0292	−1.69447	−0.847233	−	0.531221i	$-0.821733\pi$
$101$	−17.0292	−1.69447	−0.847233	+	0.531221i	$0.821733\pi$
$102$	0	0
$103$	9.13735	0.900330	0.450165	−	0.892946i	$-0.351365\pi$
$103$	9.13735	0.900330	0.450165	+	0.892946i	$0.351365\pi$
$104$	0	0
$105$	3.92101	0.382651
$106$	0	0
$107$	−6.84202	−0.661443	−0.330721	−	0.943728i	$-0.607292\pi$
$107$	−6.84202	−0.661443	−0.330721	+	0.943728i	$0.607292\pi$
$108$	0	0
$109$	−6.21634	−0.595417	−0.297709	−	0.954657i	$-0.596222\pi$
$109$	−6.21634	−0.595417	−0.297709	+	0.954657i	$0.596222\pi$
$110$	0	0
$111$	−8.65486	−0.821483
$112$	0	0
$113$	−17.1373	−1.61215	−0.806073	−	0.591816i	$-0.798411\pi$
$113$	−17.1373	−1.61215	−0.806073	+	0.591816i	$0.798411\pi$
$114$	0	0
$115$	23.9502	2.23337
$116$	0	0
$117$	−9.84202	−0.909895
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−1.92101	−0.173212
$124$	0	0
$125$	−21.0728	−1.88480
$126$	0	0
$127$	−5.76303	−0.511386	−0.255693	−	0.966758i	$-0.582304\pi$
$127$	−5.76303	−0.511386	−0.255693	+	0.966758i	$0.582304\pi$
$128$	0	0
$129$	0.266149	0.0234331
$130$	0	0
$131$	−15.4677	−1.35142	−0.675710	−	0.737168i	$-0.736163\pi$
$131$	−15.4677	−1.35142	−0.675710	+	0.737168i	$0.736163\pi$
$132$	0	0
$133$	1.92101	0.166573
$134$	0	0
$135$	−19.6050	−1.68733
$136$	0	0
$137$	−2.84202	−0.242810	−0.121405	−	0.992603i	$-0.538740\pi$
$137$	−2.84202	−0.242810	−0.121405	+	0.992603i	$0.538740\pi$
$138$	0	0
$139$	2.37432	0.201387	0.100693	−	0.994917i	$-0.467894\pi$
$139$	2.37432	0.201387	0.100693	+	0.994917i	$0.467894\pi$
$140$	0	0
$141$	−2.10817	−0.177540
$142$	0	0
$143$	−9.84202	−0.823031
$144$	0	0
$145$	23.2163	1.92801
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	−1.07899	−0.0883943	−0.0441972	−	0.999023i	$-0.514073\pi$
$149$	−1.07899	−0.0883943	−0.0441972	+	0.999023i	$0.514073\pi$
$150$	0	0
$151$	−8.81284	−0.717179	−0.358589	−	0.933495i	$-0.616742\pi$
$151$	−8.81284	−0.717179	−0.358589	+	0.933495i	$0.616742\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	−14.6405	−1.17595
$156$	0	0
$157$	−1.45331	−0.115987	−0.0579933	−	0.998317i	$-0.518470\pi$
$157$	−1.45331	−0.115987	−0.0579933	+	0.998317i	$0.518470\pi$
$158$	0	0
$159$	−9.37432	−0.743432
$160$	0	0
$161$	−6.10817	−0.481391
$162$	0	0
$163$	−18.0584	−1.41444	−0.707220	−	0.706994i	$-0.750051\pi$
$163$	−18.0584	−1.41444	−0.707220	+	0.706994i	$0.750051\pi$
$164$	0	0
$165$	−7.84202	−0.610500
$166$	0	0
$167$	9.21634	0.713182	0.356591	−	0.934261i	$-0.383939\pi$
$167$	9.21634	0.713182	0.356591	+	0.934261i	$0.383939\pi$
$168$	0	0
$169$	11.2163	0.862795
$170$	0	0
$171$	−3.84202	−0.293807
$172$	0	0
$173$	17.2953	1.31494	0.657470	−	0.753481i	$-0.271627\pi$
$173$	17.2953	1.31494	0.657470	+	0.753481i	$0.271627\pi$
$174$	0	0
$175$	10.3743	0.784225
$176$	0	0
$177$	1.00000	0.0751646
$178$	0	0
$179$	−12.9210	−0.965762	−0.482881	−	0.875686i	$-0.660410\pi$
$179$	−12.9210	−0.965762	−0.482881	+	0.875686i	$0.660410\pi$
$180$	0	0
$181$	2.82763	0.210176	0.105088	−	0.994463i	$-0.466488\pi$
$181$	2.82763	0.210176	0.105088	+	0.994463i	$0.466488\pi$
$182$	0	0
$183$	6.92101	0.511616
$184$	0	0
$185$	−33.9358	−2.49501
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	5.00000	0.363696
$190$	0	0
$191$	−19.5759	−1.41646	−0.708230	−	0.705982i	$-0.750506\pi$
$191$	−19.5759	−1.41646	−0.708230	+	0.705982i	$0.750506\pi$
$192$	0	0
$193$	−13.7630	−0.990685	−0.495342	−	0.868698i	$-0.664957\pi$
$193$	−13.7630	−0.990685	−0.495342	+	0.868698i	$0.664957\pi$
$194$	0	0
$195$	19.2953	1.38177
$196$	0	0
$197$	16.5907	1.18204	0.591018	−	0.806659i	$-0.298726\pi$
$197$	16.5907	1.18204	0.591018	+	0.806659i	$0.298726\pi$
$198$	0	0
$199$	−13.3743	−0.948080	−0.474040	−	0.880503i	$-0.657205\pi$
$199$	−13.3743	−0.948080	−0.474040	+	0.880503i	$0.657205\pi$
$200$	0	0
$201$	−10.2163	−0.720605
$202$	0	0
$203$	−5.92101	−0.415573
$204$	0	0
$205$	−7.53230	−0.526079
$206$	0	0
$207$	12.2163	0.849094
$208$	0	0
$209$	−3.84202	−0.265758
$210$	0	0
$211$	13.9502	0.960371	0.480185	−	0.877167i	$-0.340569\pi$
$211$	13.9502	0.960371	0.480185	+	0.877167i	$0.340569\pi$
$212$	0	0
$213$	−2.37432	−0.162686
$214$	0	0
$215$	1.04357	0.0711711
$216$	0	0
$217$	3.73385	0.253470
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	4.92101	0.331023
$222$	0	0
$223$	−19.4677	−1.30365	−0.651827	−	0.758368i	$-0.725997\pi$
$223$	−19.4677	−1.30365	−0.651827	+	0.758368i	$0.725997\pi$
$224$	0	0
$225$	−20.7486	−1.38324
$226$	0	0
$227$	6.97082	0.462670	0.231335	−	0.972874i	$-0.425691\pi$
$227$	6.97082	0.462670	0.231335	+	0.972874i	$0.425691\pi$
$228$	0	0
$229$	−13.5615	−0.896168	−0.448084	−	0.893992i	$-0.647893\pi$
$229$	−13.5615	−0.896168	−0.448084	+	0.893992i	$0.647893\pi$
$230$	0	0
$231$	2.00000	0.131590
$232$	0	0
$233$	−22.4969	−1.47382	−0.736910	−	0.675991i	$-0.763716\pi$
$233$	−22.4969	−1.47382	−0.736910	+	0.675991i	$0.763716\pi$
$234$	0	0
$235$	−8.26615	−0.539224
$236$	0	0
$237$	5.00000	0.324785
$238$	0	0
$239$	−0.0789903	−0.00510946	−0.00255473	−	0.999997i	$-0.500813\pi$
$239$	−0.0789903	−0.00510946	−0.00255473	+	0.999997i	$0.500813\pi$
$240$	0	0
$241$	−2.23697	−0.144096	−0.0720480	−	0.997401i	$-0.522953\pi$
$241$	−2.23697	−0.144096	−0.0720480	+	0.997401i	$0.522953\pi$
$242$	0	0
$243$	−16.0000	−1.02640
$244$	0	0
$245$	23.5261	1.50302
$246$	0	0
$247$	9.45331	0.601500
$248$	0	0
$249$	−4.65486	−0.294990
$250$	0	0
$251$	11.3887	0.718849	0.359425	−	0.933174i	$-0.382973\pi$
$251$	11.3887	0.718849	0.359425	+	0.933174i	$0.382973\pi$
$252$	0	0
$253$	12.2163	0.768035
$254$	0	0
$255$	3.92101	0.245543
$256$	0	0
$257$	12.4677	0.777714	0.388857	−	0.921298i	$-0.372870\pi$
$257$	12.4677	0.777714	0.388857	+	0.921298i	$0.372870\pi$
$258$	0	0
$259$	8.65486	0.537787
$260$	0	0
$261$	11.8420	0.733003
$262$	0	0
$263$	3.92101	0.241780	0.120890	−	0.992666i	$-0.461425\pi$
$263$	3.92101	0.241780	0.120890	+	0.992666i	$0.461425\pi$
$264$	0	0
$265$	−36.7568	−2.25795
$266$	0	0
$267$	−5.73385	−0.350906
$268$	0	0
$269$	21.4035	1.30499	0.652497	−	0.757791i	$-0.273721\pi$
$269$	21.4035	1.30499	0.652497	+	0.757791i	$0.273721\pi$
$270$	0	0
$271$	−30.5117	−1.85345	−0.926726	−	0.375738i	$-0.877389\pi$
$271$	−30.5117	−1.85345	−0.926726	+	0.375738i	$0.877389\pi$
$272$	0	0
$273$	−4.92101	−0.297833
$274$	0	0
$275$	−20.7486	−1.25119
$276$	0	0
$277$	13.9210	0.836432	0.418216	−	0.908348i	$-0.362655\pi$
$277$	13.9210	0.836432	0.418216	+	0.908348i	$0.362655\pi$
$278$	0	0
$279$	−7.46770	−0.447080
$280$	0	0
$281$	18.1230	1.08112	0.540562	−	0.841304i	$-0.318212\pi$
$281$	18.1230	1.08112	0.540562	+	0.841304i	$0.318212\pi$
$282$	0	0
$283$	−13.6696	−0.812576	−0.406288	−	0.913745i	$-0.633177\pi$
$283$	−13.6696	−0.812576	−0.406288	+	0.913745i	$0.633177\pi$
$284$	0	0
$285$	7.53230	0.446175
$286$	0	0
$287$	1.92101	0.113394
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	3.45331	0.202436
$292$	0	0
$293$	16.9004	0.987330	0.493665	−	0.869652i	$-0.335657\pi$
$293$	16.9004	0.987330	0.493665	+	0.869652i	$0.335657\pi$
$294$	0	0
$295$	3.92101	0.228290
$296$	0	0
$297$	−10.0000	−0.580259
$298$	0	0
$299$	−30.0584	−1.73832
$300$	0	0
$301$	−0.266149	−0.0153406
$302$	0	0
$303$	17.0292	0.978301
$304$	0	0
$305$	27.1373	1.55388
$306$	0	0
$307$	17.5467	1.00144	0.500721	−	0.865609i	$-0.333068\pi$
$307$	17.5467	1.00144	0.500721	+	0.865609i	$0.333068\pi$
$308$	0	0
$309$	−9.13735	−0.519805
$310$	0	0
$311$	−15.0000	−0.850572	−0.425286	−	0.905059i	$-0.639826\pi$
$311$	−15.0000	−0.850572	−0.425286	+	0.905059i	$0.639826\pi$
$312$	0	0
$313$	14.9210	0.843385	0.421693	−	0.906739i	$-0.361436\pi$
$313$	14.9210	0.843385	0.421693	+	0.906739i	$0.361436\pi$
$314$	0	0
$315$	7.84202	0.441848
$316$	0	0
$317$	−16.0584	−0.901927	−0.450964	−	0.892542i	$-0.648920\pi$
$317$	−16.0584	−0.901927	−0.450964	+	0.892542i	$0.648920\pi$
$318$	0	0
$319$	11.8420	0.663026
$320$	0	0
$321$	6.84202	0.381884
$322$	0	0
$323$	1.92101	0.106888
$324$	0	0
$325$	51.0521	2.83186
$326$	0	0
$327$	6.21634	0.343764
$328$	0	0
$329$	2.10817	0.116227
$330$	0	0
$331$	−18.0790	−0.993711	−0.496856	−	0.867833i	$-0.665512\pi$
$331$	−18.0790	−0.993711	−0.496856	+	0.867833i	$0.665512\pi$
$332$	0	0
$333$	−17.3097	−0.948567
$334$	0	0
$335$	−40.0584	−2.18862
$336$	0	0
$337$	−13.3097	−0.725027	−0.362513	−	0.931979i	$-0.618081\pi$
$337$	−13.3097	−0.725027	−0.362513	+	0.931979i	$0.618081\pi$
$338$	0	0
$339$	17.1373	0.930773
$340$	0	0
$341$	−7.46770	−0.404399
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	−23.9502	−1.28943
$346$	0	0
$347$	1.93579	0.103919	0.0519594	−	0.998649i	$-0.483453\pi$
$347$	1.93579	0.103919	0.0519594	+	0.998649i	$0.483453\pi$
$348$	0	0
$349$	11.0292	0.590378	0.295189	−	0.955439i	$-0.404617\pi$
$349$	11.0292	0.590378	0.295189	+	0.955439i	$0.404617\pi$
$350$	0	0
$351$	24.6050	1.31332
$352$	0	0
$353$	18.4969	0.984490	0.492245	−	0.870457i	$-0.336176\pi$
$353$	18.4969	0.984490	0.492245	+	0.870457i	$0.336176\pi$
$354$	0	0
$355$	−9.30972	−0.494109
$356$	0	0
$357$	−1.00000	−0.0529256
$358$	0	0
$359$	4.29533	0.226699	0.113349	−	0.993555i	$-0.463842\pi$
$359$	4.29533	0.226699	0.113349	+	0.993555i	$0.463842\pi$
$360$	0	0
$361$	−15.3097	−0.805775
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−7.84202	−0.410470
$366$	0	0
$367$	−21.9358	−1.14504	−0.572520	−	0.819891i	$-0.694034\pi$
$367$	−21.9358	−1.14504	−0.572520	+	0.819891i	$0.694034\pi$
$368$	0	0
$369$	−3.84202	−0.200008
$370$	0	0
$371$	9.37432	0.486690
$372$	0	0
$373$	−8.37432	−0.433606	−0.216803	−	0.976215i	$-0.569563\pi$
$373$	−8.37432	−0.433606	−0.216803	+	0.976215i	$0.569563\pi$
$374$	0	0
$375$	21.0728	1.08819
$376$	0	0
$377$	−29.1373	−1.50065
$378$	0	0
$379$	4.68404	0.240603	0.120301	−	0.992737i	$-0.461614\pi$
$379$	4.68404	0.240603	0.120301	+	0.992737i	$0.461614\pi$
$380$	0	0
$381$	5.76303	0.295249
$382$	0	0
$383$	−22.5907	−1.15433	−0.577164	−	0.816628i	$-0.695841\pi$
$383$	−22.5907	−1.15433	−0.577164	+	0.816628i	$0.695841\pi$
$384$	0	0
$385$	7.84202	0.399666
$386$	0	0
$387$	0.532298	0.0270582
$388$	0	0
$389$	−9.46770	−0.480032	−0.240016	−	0.970769i	$-0.577153\pi$
$389$	−9.46770	−0.480032	−0.240016	+	0.970769i	$0.577153\pi$
$390$	0	0
$391$	−6.10817	−0.308903
$392$	0	0
$393$	15.4677	0.780242
$394$	0	0
$395$	19.6050	0.986437
$396$	0	0
$397$	0.330355	0.0165801	0.00829003	−	0.999966i	$-0.497361\pi$
$397$	0.330355	0.0165801	0.00829003	+	0.999966i	$0.497361\pi$
$398$	0	0
$399$	−1.92101	−0.0961708
$400$	0	0
$401$	−0.704673	−0.0351897	−0.0175948	−	0.999845i	$-0.505601\pi$
$401$	−0.704673	−0.0351897	−0.0175948	+	0.999845i	$0.505601\pi$
$402$	0	0
$403$	18.3743	0.915290
$404$	0	0
$405$	−3.92101	−0.194837
$406$	0	0
$407$	−17.3097	−0.858011
$408$	0	0
$409$	4.70467	0.232631	0.116316	−	0.993212i	$-0.462892\pi$
$409$	4.70467	0.232631	0.116316	+	0.993212i	$0.462892\pi$
$410$	0	0
$411$	2.84202	0.140186
$412$	0	0
$413$	−1.00000	−0.0492068
$414$	0	0
$415$	−18.2518	−0.895943
$416$	0	0
$417$	−2.37432	−0.116271
$418$	0	0
$419$	−22.5467	−1.10148	−0.550739	−	0.834678i	$-0.685654\pi$
$419$	−22.5467	−1.10148	−0.550739	+	0.834678i	$0.685654\pi$
$420$	0	0
$421$	−10.8568	−0.529128	−0.264564	−	0.964368i	$-0.585228\pi$
$421$	−10.8568	−0.529128	−0.264564	+	0.964368i	$0.585228\pi$
$422$	0	0
$423$	−4.21634	−0.205005
$424$	0	0
$425$	10.3743	0.503228
$426$	0	0
$427$	−6.92101	−0.334931
$428$	0	0
$429$	9.84202	0.475177
$430$	0	0
$431$	36.2163	1.74448	0.872240	−	0.489078i	$-0.162667\pi$
$431$	36.2163	1.74448	0.872240	+	0.489078i	$0.162667\pi$
$432$	0	0
$433$	−20.4677	−0.983615	−0.491807	−	0.870704i	$-0.663664\pi$
$433$	−20.4677	−0.983615	−0.491807	+	0.870704i	$0.663664\pi$
$434$	0	0
$435$	−23.2163	−1.11314
$436$	0	0
$437$	−11.7339	−0.561306
$438$	0	0
$439$	−10.3743	−0.495139	−0.247570	−	0.968870i	$-0.579632\pi$
$439$	−10.3743	−0.495139	−0.247570	+	0.968870i	$0.579632\pi$
$440$	0	0
$441$	12.0000	0.571429
$442$	0	0
$443$	35.4179	1.68275	0.841377	−	0.540448i	$-0.181745\pi$
$443$	35.4179	1.68275	0.841377	+	0.540448i	$0.181745\pi$
$444$	0	0
$445$	−22.4825	−1.06577
$446$	0	0
$447$	1.07899	0.0510345
$448$	0	0
$449$	20.6696	0.975461	0.487730	−	0.872994i	$-0.337825\pi$
$449$	20.6696	0.975461	0.487730	+	0.872994i	$0.337825\pi$
$450$	0	0
$451$	−3.84202	−0.180914
$452$	0	0
$453$	8.81284	0.414063
$454$	0	0
$455$	−19.2953	−0.904579
$456$	0	0
$457$	−26.1809	−1.22469	−0.612346	−	0.790590i	$-0.709774\pi$
$457$	−26.1809	−1.22469	−0.612346	+	0.790590i	$0.709774\pi$
$458$	0	0
$459$	5.00000	0.233380
$460$	0	0
$461$	29.4677	1.37245	0.686224	−	0.727390i	$-0.259267\pi$
$461$	29.4677	1.37245	0.686224	+	0.727390i	$0.259267\pi$
$462$	0	0
$463$	37.4762	1.74167	0.870834	−	0.491576i	$-0.163579\pi$
$463$	37.4762	1.74167	0.870834	+	0.491576i	$0.163579\pi$
$464$	0	0
$465$	14.6405	0.678935
$466$	0	0
$467$	0.157981	0.00731047	0.00365523	−	0.999993i	$-0.498837\pi$
$467$	0.157981	0.00731047	0.00365523	+	0.999993i	$0.498837\pi$
$468$	0	0
$469$	10.2163	0.471747
$470$	0	0
$471$	1.45331	0.0669649
$472$	0	0
$473$	0.532298	0.0244751
$474$	0	0
$475$	19.9292	0.914413
$476$	0	0
$477$	−18.7486	−0.858441
$478$	0	0
$479$	−35.1517	−1.60612	−0.803062	−	0.595895i	$-0.796797\pi$
$479$	−35.1517	−1.60612	−0.803062	+	0.595895i	$0.796797\pi$
$480$	0	0
$481$	42.5907	1.94197
$482$	0	0
$483$	6.10817	0.277931
$484$	0	0
$485$	13.5405	0.614840
$486$	0	0
$487$	17.9650	0.814071	0.407035	−	0.913412i	$-0.366563\pi$
$487$	17.9650	0.814071	0.407035	+	0.913412i	$0.366563\pi$
$488$	0	0
$489$	18.0584	0.816627
$490$	0	0
$491$	−21.0728	−0.951000	−0.475500	−	0.879716i	$-0.657733\pi$
$491$	−21.0728	−0.951000	−0.475500	+	0.879716i	$0.657733\pi$
$492$	0	0
$493$	−5.92101	−0.266669
$494$	0	0
$495$	−15.6840	−0.704945
$496$	0	0
$497$	2.37432	0.106503
$498$	0	0
$499$	−9.21634	−0.412580	−0.206290	−	0.978491i	$-0.566139\pi$
$499$	−9.21634	−0.412580	−0.206290	+	0.978491i	$0.566139\pi$
$500$	0	0
$501$	−9.21634	−0.411756
$502$	0	0
$503$	−38.1226	−1.69980	−0.849901	−	0.526943i	$-0.823338\pi$
$503$	−38.1226	−1.69980	−0.849901	+	0.526943i	$0.823338\pi$
$504$	0	0
$505$	66.7716	2.97130
$506$	0	0
$507$	−11.2163	−0.498135
$508$	0	0
$509$	−16.6050	−0.736006	−0.368003	−	0.929825i	$-0.619958\pi$
$509$	−16.6050	−0.736006	−0.368003	+	0.929825i	$0.619958\pi$
$510$	0	0
$511$	2.00000	0.0884748
$512$	0	0
$513$	9.60505	0.424073
$514$	0	0
$515$	−35.8276	−1.57875
$516$	0	0
$517$	−4.21634	−0.185434
$518$	0	0
$519$	−17.2953	−0.759181
$520$	0	0
$521$	13.4677	0.590031	0.295015	−	0.955493i	$-0.404675\pi$
$521$	13.4677	0.590031	0.295015	+	0.955493i	$0.404675\pi$
$522$	0	0
$523$	12.3537	0.540189	0.270094	−	0.962834i	$-0.412945\pi$
$523$	12.3537	0.540189	0.270094	+	0.962834i	$0.412945\pi$
$524$	0	0
$525$	−10.3743	−0.452772
$526$	0	0
$527$	3.73385	0.162649
$528$	0	0
$529$	14.3097	0.622162
$530$	0	0
$531$	2.00000	0.0867926
$532$	0	0
$533$	9.45331	0.409468
$534$	0	0
$535$	26.8276	1.15986
$536$	0	0
$537$	12.9210	0.557583
$538$	0	0
$539$	12.0000	0.516877
$540$	0	0
$541$	−7.38910	−0.317682	−0.158841	−	0.987304i	$-0.550776\pi$
$541$	−7.38910	−0.317682	−0.158841	+	0.987304i	$0.550776\pi$
$542$	0	0
$543$	−2.82763	−0.121345
$544$	0	0
$545$	24.3743	1.04408
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	13.8420	0.590763
$550$	0	0
$551$	−11.3743	−0.484562
$552$	0	0
$553$	−5.00000	−0.212622
$554$	0	0
$555$	33.9358	1.44049
$556$	0	0
$557$	40.5261	1.71714	0.858572	−	0.512693i	$-0.171352\pi$
$557$	40.5261	1.71714	0.858572	+	0.512693i	$0.171352\pi$
$558$	0	0
$559$	−1.30972	−0.0553953
$560$	0	0
$561$	2.00000	0.0844401
$562$	0	0
$563$	−7.46770	−0.314726	−0.157363	−	0.987541i	$-0.550299\pi$
$563$	−7.46770	−0.314726	−0.157363	+	0.987541i	$0.550299\pi$
$564$	0	0
$565$	67.1957	2.82694
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−1.93579	−0.0811527	−0.0405763	−	0.999176i	$-0.512919\pi$
$569$	−1.93579	−0.0811527	−0.0405763	+	0.999176i	$0.512919\pi$
$570$	0	0
$571$	42.5264	1.77968	0.889838	−	0.456276i	$-0.150817\pi$
$571$	42.5264	1.77968	0.889838	+	0.456276i	$0.150817\pi$
$572$	0	0
$573$	19.5759	0.817794
$574$	0	0
$575$	−63.3681	−2.64263
$576$	0	0
$577$	25.0000	1.04076	0.520382	−	0.853934i	$-0.325790\pi$
$577$	25.0000	1.04076	0.520382	+	0.853934i	$0.325790\pi$
$578$	0	0
$579$	13.7630	0.571972
$580$	0	0
$581$	4.65486	0.193116
$582$	0	0
$583$	−18.7486	−0.776489
$584$	0	0
$585$	38.5907	1.59553
$586$	0	0
$587$	−14.1665	−0.584715	−0.292358	−	0.956309i	$-0.594440\pi$
$587$	−14.1665	−0.584715	−0.292358	+	0.956309i	$0.594440\pi$
$588$	0	0
$589$	7.17276	0.295549
$590$	0	0
$591$	−16.5907	−0.682448
$592$	0	0
$593$	27.8070	1.14190	0.570948	−	0.820986i	$-0.306576\pi$
$593$	27.8070	1.14190	0.570948	+	0.820986i	$0.306576\pi$
$594$	0	0
$595$	−3.92101	−0.160746
$596$	0	0
$597$	13.3743	0.547374
$598$	0	0
$599$	10.6696	0.435950	0.217975	−	0.975954i	$-0.430055\pi$
$599$	10.6696	0.435950	0.217975	+	0.975954i	$0.430055\pi$
$600$	0	0
$601$	−41.9587	−1.71153	−0.855766	−	0.517363i	$-0.826914\pi$
$601$	−41.9587	−1.71153	−0.855766	+	0.517363i	$0.826914\pi$
$602$	0	0
$603$	−20.4327	−0.832083
$604$	0	0
$605$	27.4471	1.11588
$606$	0	0
$607$	−27.1167	−1.10063	−0.550317	−	0.834956i	$-0.685493\pi$
$607$	−27.1167	−1.10063	−0.550317	+	0.834956i	$0.685493\pi$
$608$	0	0
$609$	5.92101	0.239931
$610$	0	0
$611$	10.3743	0.419700
$612$	0	0
$613$	5.41789	0.218827	0.109413	−	0.993996i	$-0.465103\pi$
$613$	5.41789	0.218827	0.109413	+	0.993996i	$0.465103\pi$
$614$	0	0
$615$	7.53230	0.303732
$616$	0	0
$617$	−26.6696	−1.07368	−0.536840	−	0.843684i	$-0.680382\pi$
$617$	−26.6696	−1.07368	−0.536840	+	0.843684i	$0.680382\pi$
$618$	0	0
$619$	−9.05836	−0.364086	−0.182043	−	0.983291i	$-0.558271\pi$
$619$	−9.05836	−0.364086	−0.182043	+	0.983291i	$0.558271\pi$
$620$	0	0
$621$	−30.5408	−1.22556
$622$	0	0
$623$	5.73385	0.229722
$624$	0	0
$625$	30.7549	1.23019
$626$	0	0
$627$	3.84202	0.153435
$628$	0	0
$629$	8.65486	0.345092
$630$	0	0
$631$	−28.9650	−1.15308	−0.576539	−	0.817070i	$-0.695597\pi$
$631$	−28.9650	−1.15308	−0.576539	+	0.817070i	$0.695597\pi$
$632$	0	0
$633$	−13.9502	−0.554470
$634$	0	0
$635$	22.5969	0.896730
$636$	0	0
$637$	−29.5261	−1.16987
$638$	0	0
$639$	−4.74863	−0.187853
$640$	0	0
$641$	19.0934	0.754143	0.377072	−	0.926184i	$-0.376931\pi$
$641$	19.0934	0.754143	0.377072	+	0.926184i	$0.376931\pi$
$642$	0	0
$643$	−27.4031	−1.08067	−0.540337	−	0.841449i	$-0.681703\pi$
$643$	−27.4031	−1.08067	−0.540337	+	0.841449i	$0.681703\pi$
$644$	0	0
$645$	−1.04357	−0.0410906
$646$	0	0
$647$	15.8214	0.622003	0.311001	−	0.950409i	$-0.399336\pi$
$647$	15.8214	0.622003	0.311001	+	0.950409i	$0.399336\pi$
$648$	0	0
$649$	2.00000	0.0785069
$650$	0	0
$651$	−3.73385	−0.146341
$652$	0	0
$653$	33.2891	1.30270	0.651351	−	0.758776i	$-0.274202\pi$
$653$	33.2891	1.30270	0.651351	+	0.758776i	$0.274202\pi$
$654$	0	0
$655$	60.6490	2.36975
$656$	0	0
$657$	−4.00000	−0.156055
$658$	0	0
$659$	−28.9296	−1.12694	−0.563468	−	0.826138i	$-0.690533\pi$
$659$	−28.9296	−1.12694	−0.563468	+	0.826138i	$0.690533\pi$
$660$	0	0
$661$	22.1517	0.861603	0.430801	−	0.902447i	$-0.358231\pi$
$661$	22.1517	0.861603	0.430801	+	0.902447i	$0.358231\pi$
$662$	0	0
$663$	−4.92101	−0.191116
$664$	0	0
$665$	−7.53230	−0.292090
$666$	0	0
$667$	36.1665	1.40037
$668$	0	0
$669$	19.4677	0.752665
$670$	0	0
$671$	13.8420	0.534365
$672$	0	0
$673$	−34.9152	−1.34588	−0.672940	−	0.739697i	$-0.734969\pi$
$673$	−34.9152	−1.34588	−0.672940	+	0.739697i	$0.734969\pi$
$674$	0	0
$675$	51.8716	1.99654
$676$	0	0
$677$	0.590654	0.0227007	0.0113503	−	0.999936i	$-0.496387\pi$
$677$	0.590654	0.0227007	0.0113503	+	0.999936i	$0.496387\pi$
$678$	0	0
$679$	−3.45331	−0.132526
$680$	0	0
$681$	−6.97082	−0.267122
$682$	0	0
$683$	−20.2019	−0.773006	−0.386503	−	0.922288i	$-0.626317\pi$
$683$	−20.2019	−0.773006	−0.386503	+	0.922288i	$0.626317\pi$
$684$	0	0
$685$	11.1436	0.425775
$686$	0	0
$687$	13.5615	0.517403
$688$	0	0
$689$	46.1311	1.75746
$690$	0	0
$691$	−0.827236	−0.0314695	−0.0157348	−	0.999876i	$-0.505009\pi$
$691$	−0.827236	−0.0314695	−0.0157348	+	0.999876i	$0.505009\pi$
$692$	0	0
$693$	4.00000	0.151947
$694$	0	0
$695$	−9.30972	−0.353138
$696$	0	0
$697$	1.92101	0.0727634
$698$	0	0
$699$	22.4969	0.850910
$700$	0	0
$701$	2.98522	0.112750	0.0563750	−	0.998410i	$-0.482046\pi$
$701$	2.98522	0.112750	0.0563750	+	0.998410i	$0.482046\pi$
$702$	0	0
$703$	16.6261	0.627064
$704$	0	0
$705$	8.26615	0.311321
$706$	0	0
$707$	−17.0292	−0.640448
$708$	0	0
$709$	34.2953	1.28799	0.643994	−	0.765031i	$-0.277276\pi$
$709$	34.2953	1.28799	0.643994	+	0.765031i	$0.277276\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	0	0
$713$	−22.8070	−0.854129
$714$	0	0
$715$	38.5907	1.44321
$716$	0	0
$717$	0.0789903	0.00294995
$718$	0	0
$719$	−44.1311	−1.64581	−0.822906	−	0.568177i	$-0.807649\pi$
$719$	−44.1311	−1.64581	−0.822906	+	0.568177i	$0.807649\pi$
$720$	0	0
$721$	9.13735	0.340293
$722$	0	0
$723$	2.23697	0.0831938
$724$	0	0
$725$	−61.4264	−2.28132
$726$	0	0
$727$	−18.5907	−0.689489	−0.344745	−	0.938697i	$-0.612034\pi$
$727$	−18.5907	−0.689489	−0.344745	+	0.938697i	$0.612034\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	−0.266149	−0.00984387
$732$	0	0
$733$	5.12295	0.189221	0.0946103	−	0.995514i	$-0.469840\pi$
$733$	5.12295	0.189221	0.0946103	+	0.995514i	$0.469840\pi$
$734$	0	0
$735$	−23.5261	−0.867772
$736$	0	0
$737$	−20.4327	−0.752647
$738$	0	0
$739$	−51.4120	−1.89122	−0.945611	−	0.325299i	$-0.894535\pi$
$739$	−51.4120	−1.89122	−0.945611	+	0.325299i	$0.894535\pi$
$740$	0	0
$741$	−9.45331	−0.347276
$742$	0	0
$743$	−9.06460	−0.332548	−0.166274	−	0.986080i	$-0.553174\pi$
$743$	−9.06460	−0.332548	−0.166274	+	0.986080i	$0.553174\pi$
$744$	0	0
$745$	4.23073	0.155002
$746$	0	0
$747$	−9.30972	−0.340625
$748$	0	0
$749$	−6.84202	−0.250002
$750$	0	0
$751$	−33.4677	−1.22125	−0.610627	−	0.791918i	$-0.709082\pi$
$751$	−33.4677	−1.22125	−0.610627	+	0.791918i	$0.709082\pi$
$752$	0	0
$753$	−11.3887	−0.415028
$754$	0	0
$755$	34.5552	1.25759
$756$	0	0
$757$	−12.9416	−0.470372	−0.235186	−	0.971950i	$-0.575570\pi$
$757$	−12.9416	−0.470372	−0.235186	+	0.971950i	$0.575570\pi$
$758$	0	0
$759$	−12.2163	−0.443425
$760$	0	0
$761$	−27.9650	−1.01373	−0.506865	−	0.862026i	$-0.669196\pi$
$761$	−27.9650	−1.01373	−0.506865	+	0.862026i	$0.669196\pi$
$762$	0	0
$763$	−6.21634	−0.225047
$764$	0	0
$765$	7.84202	0.283529
$766$	0	0
$767$	−4.92101	−0.177687
$768$	0	0
$769$	5.07899	0.183153	0.0915765	−	0.995798i	$-0.470809\pi$
$769$	5.07899	0.183153	0.0915765	+	0.995798i	$0.470809\pi$
$770$	0	0
$771$	−12.4677	−0.449013
$772$	0	0
$773$	−36.1521	−1.30030	−0.650151	−	0.759805i	$-0.725294\pi$
$773$	−36.1521	−1.30030	−0.650151	+	0.759805i	$0.725294\pi$
$774$	0	0
$775$	38.7362	1.39144
$776$	0	0
$777$	−8.65486	−0.310491
$778$	0	0
$779$	3.69028	0.132218
$780$	0	0
$781$	−4.74863	−0.169920
$782$	0	0
$783$	−29.6050	−1.05800
$784$	0	0
$785$	5.69843	0.203386
$786$	0	0
$787$	−34.5907	−1.23302	−0.616512	−	0.787346i	$-0.711455\pi$
$787$	−34.5907	−1.23302	−0.616512	+	0.787346i	$0.711455\pi$
$788$	0	0
$789$	−3.92101	−0.139592
$790$	0	0
$791$	−17.1373	−0.609334
$792$	0	0
$793$	−34.0584	−1.20945
$794$	0	0
$795$	36.7568	1.30363
$796$	0	0
$797$	−26.0728	−0.923544	−0.461772	−	0.886999i	$-0.652786\pi$
$797$	−26.0728	−0.923544	−0.461772	+	0.886999i	$0.652786\pi$
$798$	0	0
$799$	2.10817	0.0745816
$800$	0	0
$801$	−11.4677	−0.405191
$802$	0	0
$803$	−4.00000	−0.141157
$804$	0	0
$805$	23.9502	0.844133
$806$	0	0
$807$	−21.4035	−0.753439
$808$	0	0
$809$	−11.7778	−0.414086	−0.207043	−	0.978332i	$-0.566384\pi$
$809$	−11.7778	−0.414086	−0.207043	+	0.978332i	$0.566384\pi$
$810$	0	0
$811$	15.5903	0.547448	0.273724	−	0.961808i	$-0.411744\pi$
$811$	15.5903	0.547448	0.273724	+	0.961808i	$0.411744\pi$
$812$	0	0
$813$	30.5117	1.07009
$814$	0	0
$815$	70.8070	2.48026
$816$	0	0
$817$	−0.511275	−0.0178872
$818$	0	0
$819$	−9.84202	−0.343908
$820$	0	0
$821$	3.71946	0.129810	0.0649050	−	0.997891i	$-0.479326\pi$
$821$	3.71946	0.129810	0.0649050	+	0.997891i	$0.479326\pi$
$822$	0	0
$823$	28.5467	0.995075	0.497538	−	0.867442i	$-0.334238\pi$
$823$	28.5467	0.995075	0.497538	+	0.867442i	$0.334238\pi$
$824$	0	0
$825$	20.7486	0.722375
$826$	0	0
$827$	10.0584	0.349763	0.174882	−	0.984589i	$-0.444046\pi$
$827$	10.0584	0.349763	0.174882	+	0.984589i	$0.444046\pi$
$828$	0	0
$829$	4.43891	0.154170	0.0770849	−	0.997025i	$-0.475439\pi$
$829$	4.43891	0.154170	0.0770849	+	0.997025i	$0.475439\pi$
$830$	0	0
$831$	−13.9210	−0.482914
$832$	0	0
$833$	−6.00000	−0.207888
$834$	0	0
$835$	−36.1373	−1.25058
$836$	0	0
$837$	18.6693	0.645304
$838$	0	0
$839$	−38.8712	−1.34198	−0.670991	−	0.741465i	$-0.734131\pi$
$839$	−38.8712	−1.34198	−0.670991	+	0.741465i	$0.734131\pi$
$840$	0	0
$841$	6.05836	0.208909
$842$	0	0
$843$	−18.1230	−0.624188
$844$	0	0
$845$	−43.9794	−1.51294
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	13.6696	0.469141
$850$	0	0
$851$	−52.8653	−1.81220
$852$	0	0
$853$	48.9148	1.67481	0.837405	−	0.546583i	$-0.184072\pi$
$853$	48.9148	1.67481	0.837405	+	0.546583i	$0.184072\pi$
$854$	0	0
$855$	15.0646	0.515198
$856$	0	0
$857$	15.3595	0.524672	0.262336	−	0.964977i	$-0.415507\pi$
$857$	15.3595	0.524672	0.262336	+	0.964977i	$0.415507\pi$
$858$	0	0
$859$	−36.3829	−1.24137	−0.620684	−	0.784061i	$-0.713145\pi$
$859$	−36.3829	−1.24137	−0.620684	+	0.784061i	$0.713145\pi$
$860$	0	0
$861$	−1.92101	−0.0654678
$862$	0	0
$863$	40.5907	1.38172	0.690861	−	0.722988i	$-0.257232\pi$
$863$	40.5907	1.38172	0.690861	+	0.722988i	$0.257232\pi$
$864$	0	0
$865$	−67.8151	−2.30578
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	10.0000	0.339227
$870$	0	0
$871$	50.2747	1.70349
$872$	0	0
$873$	6.90662	0.233754
$874$	0	0
$875$	−21.0728	−0.712389
$876$	0	0
$877$	47.8214	1.61481	0.807407	−	0.589995i	$-0.200870\pi$
$877$	47.8214	1.61481	0.807407	+	0.589995i	$0.200870\pi$
$878$	0	0
$879$	−16.9004	−0.570036
$880$	0	0
$881$	26.9296	0.907280	0.453640	−	0.891185i	$-0.350125\pi$
$881$	26.9296	0.907280	0.453640	+	0.891185i	$0.350125\pi$
$882$	0	0
$883$	−27.2019	−0.915418	−0.457709	−	0.889102i	$-0.651330\pi$
$883$	−27.2019	−0.915418	−0.457709	+	0.889102i	$0.651330\pi$
$884$	0	0
$885$	−3.92101	−0.131803
$886$	0	0
$887$	43.7424	1.46873	0.734363	−	0.678757i	$-0.237481\pi$
$887$	43.7424	1.46873	0.734363	+	0.678757i	$0.237481\pi$
$888$	0	0
$889$	−5.76303	−0.193286
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	4.04981	0.135522
$894$	0	0
$895$	50.6634	1.69349
$896$	0	0
$897$	30.0584	1.00362
$898$	0	0
$899$	−22.1082	−0.737349
$900$	0	0
$901$	9.37432	0.312304
$902$	0	0
$903$	0.266149	0.00885688
$904$	0	0
$905$	−11.0871	−0.368549
$906$	0	0
$907$	42.6840	1.41730	0.708650	−	0.705560i	$-0.249304\pi$
$907$	42.6840	1.41730	0.708650	+	0.705560i	$0.249304\pi$
$908$	0	0
$909$	34.0584	1.12964
$910$	0	0
$911$	−57.1751	−1.89429	−0.947147	−	0.320799i	$-0.896049\pi$
$911$	−57.1751	−1.89429	−0.947147	+	0.320799i	$0.896049\pi$
$912$	0	0
$913$	−9.30972	−0.308107
$914$	0	0
$915$	−27.1373	−0.897133
$916$	0	0
$917$	−15.4677	−0.510789
$918$	0	0
$919$	37.9358	1.25139	0.625693	−	0.780069i	$-0.284816\pi$
$919$	37.9358	1.25139	0.625693	+	0.780069i	$0.284816\pi$
$920$	0	0
$921$	−17.5467	−0.578183
$922$	0	0
$923$	11.6840	0.384585
$924$	0	0
$925$	89.7883	2.95222
$926$	0	0
$927$	−18.2747	−0.600220
$928$	0	0
$929$	−12.6903	−0.416355	−0.208177	−	0.978091i	$-0.566753\pi$
$929$	−12.6903	−0.416355	−0.208177	+	0.978091i	$0.566753\pi$
$930$	0	0
$931$	−11.5261	−0.377751
$932$	0	0
$933$	15.0000	0.491078
$934$	0	0
$935$	7.84202	0.256461
$936$	0	0
$937$	−51.0521	−1.66780	−0.833900	−	0.551916i	$-0.813897\pi$
$937$	−51.0521	−1.66780	−0.833900	+	0.551916i	$0.813897\pi$
$938$	0	0
$939$	−14.9210	−0.486929
$940$	0	0
$941$	−11.3241	−0.369156	−0.184578	−	0.982818i	$-0.559092\pi$
$941$	−11.3241	−0.369156	−0.184578	+	0.982818i	$0.559092\pi$
$942$	0	0
$943$	−11.7339	−0.382107
$944$	0	0
$945$	−19.6050	−0.637752
$946$	0	0
$947$	16.5261	0.537025	0.268512	−	0.963276i	$-0.413468\pi$
$947$	16.5261	0.537025	0.268512	+	0.963276i	$0.413468\pi$
$948$	0	0
$949$	9.84202	0.319485
$950$	0	0
$951$	16.0584	0.520728
$952$	0	0
$953$	−1.03503	−0.0335279	−0.0167639	−	0.999859i	$-0.505336\pi$
$953$	−1.03503	−0.0335279	−0.0167639	+	0.999859i	$0.505336\pi$
$954$	0	0
$955$	76.7572	2.48380
$956$	0	0
$957$	−11.8420	−0.382798
$958$	0	0
$959$	−2.84202	−0.0917736
$960$	0	0
$961$	−17.0584	−0.550270
$962$	0	0
$963$	13.6840	0.440962
$964$	0	0
$965$	53.9650	1.73719
$966$	0	0
$967$	−7.24552	−0.233000	−0.116500	−	0.993191i	$-0.537168\pi$
$967$	−7.24552	−0.233000	−0.116500	+	0.993191i	$0.537168\pi$
$968$	0	0
$969$	−1.92101	−0.0617117
$970$	0	0
$971$	28.6696	0.920053	0.460026	−	0.887905i	$-0.347840\pi$
$971$	28.6696	0.920053	0.460026	+	0.887905i	$0.347840\pi$
$972$	0	0
$973$	2.37432	0.0761171
$974$	0	0
$975$	−51.0521	−1.63498
$976$	0	0
$977$	17.8506	0.571090	0.285545	−	0.958365i	$-0.407825\pi$
$977$	17.8506	0.571090	0.285545	+	0.958365i	$0.407825\pi$
$978$	0	0
$979$	−11.4677	−0.366509
$980$	0	0
$981$	12.4327	0.396945
$982$	0	0
$983$	−31.7424	−1.01243	−0.506213	−	0.862409i	$-0.668955\pi$
$983$	−31.7424	−1.01243	−0.506213	+	0.862409i	$0.668955\pi$
$984$	0	0
$985$	−65.0521	−2.07273
$986$	0	0
$987$	−2.10817	−0.0671037
$988$	0	0
$989$	1.62568	0.0516937
$990$	0	0
$991$	1.12880	0.0358576	0.0179288	−	0.999839i	$-0.494293\pi$
$991$	1.12880	0.0358576	0.0179288	+	0.999839i	$0.494293\pi$
$992$	0	0
$993$	18.0790	0.573719
$994$	0	0
$995$	52.4408	1.66249
$996$	0	0
$997$	−60.9731	−1.93104	−0.965519	−	0.260332i	$-0.916168\pi$
$997$	−60.9731	−1.93104	−0.965519	+	0.260332i	$0.916168\pi$
$998$	0	0
$999$	43.2743	1.36914

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.f.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.f.1.1	✓	3	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.92101 q^{5} +1.00000 q^{7} -2.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.92101 q^{5} +1.00000 q^{7} -2.00000 q^{9} -2.00000 q^{11} +4.92101 q^{13} +3.92101 q^{15} +1.00000 q^{17} +1.92101 q^{19} -1.00000 q^{21} -6.10817 q^{23} +10.3743 q^{25} +5.00000 q^{27} -5.92101 q^{29} +3.73385 q^{31} +2.00000 q^{33} -3.92101 q^{35} +8.65486 q^{37} -4.92101 q^{39} +1.92101 q^{41} -0.266149 q^{43} +7.84202 q^{45} +2.10817 q^{47} -6.00000 q^{49} -1.00000 q^{51} +9.37432 q^{53} +7.84202 q^{55} -1.92101 q^{57} -1.00000 q^{59} -6.92101 q^{61} -2.00000 q^{63} -19.2953 q^{65} +10.2163 q^{67} +6.10817 q^{69} +2.37432 q^{71} +2.00000 q^{73} -10.3743 q^{75} -2.00000 q^{77} -5.00000 q^{79} +1.00000 q^{81} +4.65486 q^{83} -3.92101 q^{85} +5.92101 q^{87} +5.73385 q^{89} +4.92101 q^{91} -3.73385 q^{93} -7.53230 q^{95} -3.45331 q^{97} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9	\( 3 q - 3 q^{3} + q^{5} + 3 q^{7} - 6 q^{9} - 6 q^{11} + 2 q^{13} - q^{15} + 3 q^{17} - 7 q^{19} - 3 q^{21} + 20 q^{25} + 15 q^{27} - 5 q^{29} + 4 q^{31} + 6 q^{33} + q^{35} + 6 q^{37} - 2 q^{39} - 7 q^{41} - 8 q^{43} - 2 q^{45} - 12 q^{47} - 18 q^{49} - 3 q^{51} + 17 q^{53} - 2 q^{55} + 7 q^{57} - 3 q^{59} - 8 q^{61} - 6 q^{63} - 34 q^{65} - 6 q^{67} - 4 q^{71} + 6 q^{73} - 20 q^{75} - 6 q^{77} - 15 q^{79} + 3 q^{81} - 6 q^{83} + q^{85} + 5 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{93} - 37 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + q^5 + 3 * q^7 - 6 * q^9 - 6 * q^11 + 2 * q^13 - q^15 + 3 * q^17 - 7 * q^19 - 3 * q^21 + 20 * q^25 + 15 * q^27 - 5 * q^29 + 4 * q^31 + 6 * q^33 + q^35 + 6 * q^37 - 2 * q^39 - 7 * q^41 - 8 * q^43 - 2 * q^45 - 12 * q^47 - 18 * q^49 - 3 * q^51 + 17 * q^53 - 2 * q^55 + 7 * q^57 - 3 * q^59 - 8 * q^61 - 6 * q^63 - 34 * q^65 - 6 * q^67 - 4 * q^71 + 6 * q^73 - 20 * q^75 - 6 * q^77 - 15 * q^79 + 3 * q^81 - 6 * q^83 + q^85 + 5 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^93 - 37 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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