LMFDB - Embedded newform 6036.2.a.i.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6036,2,Mod(1,6036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6036, 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("6036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6036 = 2^{2} \cdot 3 \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6036.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:	$48.1977026600$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	6036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +1.90366 q^{5} -3.61918 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.90366 q^{5} -3.61918 q^{7} +1.00000 q^{9} +5.57755 q^{11} +5.61913 q^{13} -1.90366 q^{15} +6.17169 q^{17} -0.419270 q^{19} +3.61918 q^{21} +3.21783 q^{23} -1.37606 q^{25} -1.00000 q^{27} +3.62615 q^{29} +5.39266 q^{31} -5.57755 q^{33} -6.88970 q^{35} -2.73595 q^{37} -5.61913 q^{39} -2.39155 q^{41} +2.76555 q^{43} +1.90366 q^{45} +0.129160 q^{47} +6.09845 q^{49} -6.17169 q^{51} +8.82207 q^{53} +10.6178 q^{55} +0.419270 q^{57} -0.0305566 q^{59} -3.48688 q^{61} -3.61918 q^{63} +10.6969 q^{65} -11.2996 q^{67} -3.21783 q^{69} +3.48404 q^{71} -7.80164 q^{73} +1.37606 q^{75} -20.1862 q^{77} -14.5184 q^{79} +1.00000 q^{81} +5.10465 q^{83} +11.7488 q^{85} -3.62615 q^{87} +11.8290 q^{89} -20.3366 q^{91} -5.39266 q^{93} -0.798150 q^{95} -11.7288 q^{97} +5.57755 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	$ 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) $ 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	1.90366	0.851345	0.425672	−	0.904877i	$-0.360038\pi$
$5$	1.90366	0.851345	0.425672	+	0.904877i	$0.360038\pi$
$6$	0	0
$7$	−3.61918	−1.36792	−0.683960	−	0.729519i	$-0.739744\pi$
$7$	−3.61918	−1.36792	−0.683960	+	0.729519i	$0.739744\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.57755	1.68170	0.840848	−	0.541272i	$-0.182057\pi$
$11$	5.57755	1.68170	0.840848	+	0.541272i	$0.182057\pi$
$12$	0	0
$13$	5.61913	1.55847	0.779234	−	0.626734i	$-0.215609\pi$
$13$	5.61913	1.55847	0.779234	+	0.626734i	$0.215609\pi$
$14$	0	0
$15$	−1.90366	−0.491524
$16$	0	0
$17$	6.17169	1.49685	0.748427	−	0.663217i	$-0.230809\pi$
$17$	6.17169	1.49685	0.748427	+	0.663217i	$0.230809\pi$
$18$	0	0
$19$	−0.419270	−0.0961872	−0.0480936	−	0.998843i	$-0.515315\pi$
$19$	−0.419270	−0.0961872	−0.0480936	+	0.998843i	$0.515315\pi$
$20$	0	0
$21$	3.61918	0.789770
$22$	0	0
$23$	3.21783	0.670964	0.335482	−	0.942047i	$-0.391101\pi$
$23$	3.21783	0.670964	0.335482	+	0.942047i	$0.391101\pi$
$24$	0	0
$25$	−1.37606	−0.275212
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.62615	0.673360	0.336680	−	0.941619i	$-0.390696\pi$
$29$	3.62615	0.673360	0.336680	+	0.941619i	$0.390696\pi$
$30$	0	0
$31$	5.39266	0.968550	0.484275	−	0.874916i	$-0.339083\pi$
$31$	5.39266	0.968550	0.484275	+	0.874916i	$0.339083\pi$
$32$	0	0
$33$	−5.57755	−0.970927
$34$	0	0
$35$	−6.88970	−1.16457
$36$	0	0
$37$	−2.73595	−0.449787	−0.224894	−	0.974383i	$-0.572203\pi$
$37$	−2.73595	−0.449787	−0.224894	+	0.974383i	$0.572203\pi$
$38$	0	0
$39$	−5.61913	−0.899781
$40$	0	0
$41$	−2.39155	−0.373497	−0.186749	−	0.982408i	$-0.559795\pi$
$41$	−2.39155	−0.373497	−0.186749	+	0.982408i	$0.559795\pi$
$42$	0	0
$43$	2.76555	0.421743	0.210872	−	0.977514i	$-0.432370\pi$
$43$	2.76555	0.421743	0.210872	+	0.977514i	$0.432370\pi$
$44$	0	0
$45$	1.90366	0.283782
$46$	0	0
$47$	0.129160	0.0188399	0.00941994	−	0.999956i	$-0.497001\pi$
$47$	0.129160	0.0188399	0.00941994	+	0.999956i	$0.497001\pi$
$48$	0	0
$49$	6.09845	0.871208
$50$	0	0
$51$	−6.17169	−0.864209
$52$	0	0
$53$	8.82207	1.21180	0.605902	−	0.795539i	$-0.292812\pi$
$53$	8.82207	1.21180	0.605902	+	0.795539i	$0.292812\pi$
$54$	0	0
$55$	10.6178	1.43170
$56$	0	0
$57$	0.419270	0.0555337
$58$	0	0
$59$	−0.0305566	−0.00397813	−0.00198906	−	0.999998i	$-0.500633\pi$
$59$	−0.0305566	−0.00397813	−0.00198906	+	0.999998i	$0.500633\pi$
$60$	0	0
$61$	−3.48688	−0.446450	−0.223225	−	0.974767i	$-0.571658\pi$
$61$	−3.48688	−0.446450	−0.223225	+	0.974767i	$0.571658\pi$
$62$	0	0
$63$	−3.61918	−0.455974
$64$	0	0
$65$	10.6969	1.32679
$66$	0	0
$67$	−11.2996	−1.38046	−0.690232	−	0.723588i	$-0.742492\pi$
$67$	−11.2996	−1.38046	−0.690232	+	0.723588i	$0.742492\pi$
$68$	0	0
$69$	−3.21783	−0.387381
$70$	0	0
$71$	3.48404	0.413480	0.206740	−	0.978396i	$-0.433715\pi$
$71$	3.48404	0.413480	0.206740	+	0.978396i	$0.433715\pi$
$72$	0	0
$73$	−7.80164	−0.913113	−0.456556	−	0.889694i	$-0.650917\pi$
$73$	−7.80164	−0.913113	−0.456556	+	0.889694i	$0.650917\pi$
$74$	0	0
$75$	1.37606	0.158894
$76$	0	0
$77$	−20.1862	−2.30043
$78$	0	0
$79$	−14.5184	−1.63344	−0.816722	−	0.577032i	$-0.804211\pi$
$79$	−14.5184	−1.63344	−0.816722	+	0.577032i	$0.804211\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.10465	0.560308	0.280154	−	0.959955i	$-0.409614\pi$
$83$	5.10465	0.560308	0.280154	+	0.959955i	$0.409614\pi$
$84$	0	0
$85$	11.7488	1.27434
$86$	0	0
$87$	−3.62615	−0.388764
$88$	0	0
$89$	11.8290	1.25387	0.626936	−	0.779070i	$-0.284309\pi$
$89$	11.8290	1.25387	0.626936	+	0.779070i	$0.284309\pi$
$90$	0	0
$91$	−20.3366	−2.13186
$92$	0	0
$93$	−5.39266	−0.559193
$94$	0	0
$95$	−0.798150	−0.0818884
$96$	0	0
$97$	−11.7288	−1.19088	−0.595441	−	0.803399i	$-0.703023\pi$
$97$	−11.7288	−1.19088	−0.595441	+	0.803399i	$0.703023\pi$
$98$	0	0
$99$	5.57755	0.560565
$100$	0	0
$101$	18.1101	1.80202	0.901012	−	0.433794i	$-0.142825\pi$
$101$	18.1101	1.80202	0.901012	+	0.433794i	$0.142825\pi$
$102$	0	0
$103$	−13.1885	−1.29950	−0.649750	−	0.760148i	$-0.725126\pi$
$103$	−13.1885	−1.29950	−0.649750	+	0.760148i	$0.725126\pi$
$104$	0	0
$105$	6.88970	0.672366
$106$	0	0
$107$	0.332326	0.0321272	0.0160636	−	0.999871i	$-0.494887\pi$
$107$	0.332326	0.0321272	0.0160636	+	0.999871i	$0.494887\pi$
$108$	0	0
$109$	−17.5340	−1.67946	−0.839728	−	0.543007i	$-0.817286\pi$
$109$	−17.5340	−1.67946	−0.839728	+	0.543007i	$0.817286\pi$
$110$	0	0
$111$	2.73595	0.259685
$112$	0	0
$113$	4.03768	0.379833	0.189916	−	0.981800i	$-0.439178\pi$
$113$	4.03768	0.379833	0.189916	+	0.981800i	$0.439178\pi$
$114$	0	0
$115$	6.12567	0.571222
$116$	0	0
$117$	5.61913	0.519489
$118$	0	0
$119$	−22.3364	−2.04758
$120$	0	0
$121$	20.1091	1.82810
$122$	0	0
$123$	2.39155	0.215639
$124$	0	0
$125$	−12.1379	−1.08565
$126$	0	0
$127$	−6.02421	−0.534562	−0.267281	−	0.963619i	$-0.586125\pi$
$127$	−6.02421	−0.534562	−0.267281	+	0.963619i	$0.586125\pi$
$128$	0	0
$129$	−2.76555	−0.243493
$130$	0	0
$131$	−9.22093	−0.805636	−0.402818	−	0.915280i	$-0.631969\pi$
$131$	−9.22093	−0.805636	−0.402818	+	0.915280i	$0.631969\pi$
$132$	0	0
$133$	1.51741	0.131576
$134$	0	0
$135$	−1.90366	−0.163841
$136$	0	0
$137$	14.9935	1.28098	0.640491	−	0.767966i	$-0.278731\pi$
$137$	14.9935	1.28098	0.640491	+	0.767966i	$0.278731\pi$
$138$	0	0
$139$	−11.1669	−0.947161	−0.473580	−	0.880751i	$-0.657039\pi$
$139$	−11.1669	−0.947161	−0.473580	+	0.880751i	$0.657039\pi$
$140$	0	0
$141$	−0.129160	−0.0108772
$142$	0	0
$143$	31.3410	2.62087
$144$	0	0
$145$	6.90298	0.573261
$146$	0	0
$147$	−6.09845	−0.502992
$148$	0	0
$149$	19.6013	1.60580	0.802900	−	0.596114i	$-0.203289\pi$
$149$	19.6013	1.60580	0.802900	+	0.596114i	$0.203289\pi$
$150$	0	0
$151$	−3.73694	−0.304108	−0.152054	−	0.988372i	$-0.548589\pi$
$151$	−3.73694	−0.304108	−0.152054	+	0.988372i	$0.548589\pi$
$152$	0	0
$153$	6.17169	0.498951
$154$	0	0
$155$	10.2658	0.824570
$156$	0	0
$157$	24.8232	1.98111	0.990553	−	0.137131i	$-0.0437882\pi$
$157$	24.8232	1.98111	0.990553	+	0.137131i	$0.0437882\pi$
$158$	0	0
$159$	−8.82207	−0.699635
$160$	0	0
$161$	−11.6459	−0.917826
$162$	0	0
$163$	19.1568	1.50048	0.750239	−	0.661167i	$-0.229939\pi$
$163$	19.1568	1.50048	0.750239	+	0.661167i	$0.229939\pi$
$164$	0	0
$165$	−10.6178	−0.826594
$166$	0	0
$167$	−16.2557	−1.25790	−0.628951	−	0.777445i	$-0.716516\pi$
$167$	−16.2557	−1.25790	−0.628951	+	0.777445i	$0.716516\pi$
$168$	0	0
$169$	18.5747	1.42882
$170$	0	0
$171$	−0.419270	−0.0320624
$172$	0	0
$173$	13.3360	1.01392	0.506958	−	0.861971i	$-0.330770\pi$
$173$	13.3360	1.01392	0.506958	+	0.861971i	$0.330770\pi$
$174$	0	0
$175$	4.98021	0.376469
$176$	0	0
$177$	0.0305566	0.00229677
$178$	0	0
$179$	24.0711	1.79916	0.899579	−	0.436757i	$-0.143873\pi$
$179$	24.0711	1.79916	0.899579	+	0.436757i	$0.143873\pi$
$180$	0	0
$181$	6.94634	0.516318	0.258159	−	0.966102i	$-0.416884\pi$
$181$	6.94634	0.516318	0.258159	+	0.966102i	$0.416884\pi$
$182$	0	0
$183$	3.48688	0.257758
$184$	0	0
$185$	−5.20833	−0.382924
$186$	0	0
$187$	34.4229	2.51725
$188$	0	0
$189$	3.61918	0.263257
$190$	0	0
$191$	−12.5873	−0.910784	−0.455392	−	0.890291i	$-0.650501\pi$
$191$	−12.5873	−0.910784	−0.455392	+	0.890291i	$0.650501\pi$
$192$	0	0
$193$	−5.10938	−0.367781	−0.183890	−	0.982947i	$-0.558869\pi$
$193$	−5.10938	−0.367781	−0.183890	+	0.982947i	$0.558869\pi$
$194$	0	0
$195$	−10.6969	−0.766024
$196$	0	0
$197$	−2.47196	−0.176120	−0.0880599	−	0.996115i	$-0.528067\pi$
$197$	−2.47196	−0.176120	−0.0880599	+	0.996115i	$0.528067\pi$
$198$	0	0
$199$	25.0861	1.77830	0.889152	−	0.457612i	$-0.151295\pi$
$199$	25.0861	1.77830	0.889152	+	0.457612i	$0.151295\pi$
$200$	0	0
$201$	11.2996	0.797011
$202$	0	0
$203$	−13.1237	−0.921103
$204$	0	0
$205$	−4.55271	−0.317975
$206$	0	0
$207$	3.21783	0.223655
$208$	0	0
$209$	−2.33850	−0.161758
$210$	0	0
$211$	15.0915	1.03894	0.519471	−	0.854488i	$-0.326129\pi$
$211$	15.0915	1.03894	0.519471	+	0.854488i	$0.326129\pi$
$212$	0	0
$213$	−3.48404	−0.238723
$214$	0	0
$215$	5.26469	0.359049
$216$	0	0
$217$	−19.5170	−1.32490
$218$	0	0
$219$	7.80164	0.527186
$220$	0	0
$221$	34.6795	2.33280
$222$	0	0
$223$	−27.7017	−1.85504	−0.927522	−	0.373768i	$-0.878066\pi$
$223$	−27.7017	−1.85504	−0.927522	+	0.373768i	$0.878066\pi$
$224$	0	0
$225$	−1.37606	−0.0917374
$226$	0	0
$227$	−14.0651	−0.933536	−0.466768	−	0.884380i	$-0.654582\pi$
$227$	−14.0651	−0.933536	−0.466768	+	0.884380i	$0.654582\pi$
$228$	0	0
$229$	−10.9817	−0.725690	−0.362845	−	0.931850i	$-0.618195\pi$
$229$	−10.9817	−0.725690	−0.362845	+	0.931850i	$0.618195\pi$
$230$	0	0
$231$	20.1862	1.32815
$232$	0	0
$233$	−16.3683	−1.07232	−0.536162	−	0.844115i	$-0.680127\pi$
$233$	−16.3683	−1.07232	−0.536162	+	0.844115i	$0.680127\pi$
$234$	0	0
$235$	0.245877	0.0160392
$236$	0	0
$237$	14.5184	0.943069
$238$	0	0
$239$	−27.8170	−1.79933	−0.899665	−	0.436581i	$-0.856189\pi$
$239$	−27.8170	−1.79933	−0.899665	+	0.436581i	$0.856189\pi$
$240$	0	0
$241$	10.8561	0.699302	0.349651	−	0.936880i	$-0.386300\pi$
$241$	10.8561	0.699302	0.349651	+	0.936880i	$0.386300\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	11.6094	0.741698
$246$	0	0
$247$	−2.35594	−0.149905
$248$	0	0
$249$	−5.10465	−0.323494
$250$	0	0
$251$	−4.31403	−0.272299	−0.136150	−	0.990688i	$-0.543473\pi$
$251$	−4.31403	−0.272299	−0.136150	+	0.990688i	$0.543473\pi$
$252$	0	0
$253$	17.9476	1.12836
$254$	0	0
$255$	−11.7488	−0.735740
$256$	0	0
$257$	−26.7299	−1.66737	−0.833683	−	0.552244i	$-0.813772\pi$
$257$	−26.7299	−1.66737	−0.833683	+	0.552244i	$0.813772\pi$
$258$	0	0
$259$	9.90189	0.615273
$260$	0	0
$261$	3.62615	0.224453
$262$	0	0
$263$	−7.27645	−0.448685	−0.224342	−	0.974510i	$-0.572023\pi$
$263$	−7.27645	−0.448685	−0.224342	+	0.974510i	$0.572023\pi$
$264$	0	0
$265$	16.7943	1.03166
$266$	0	0
$267$	−11.8290	−0.723924
$268$	0	0
$269$	6.84338	0.417248	0.208624	−	0.977996i	$-0.433101\pi$
$269$	6.84338	0.417248	0.208624	+	0.977996i	$0.433101\pi$
$270$	0	0
$271$	−24.3794	−1.48094	−0.740471	−	0.672089i	$-0.765397\pi$
$271$	−24.3794	−1.48094	−0.740471	+	0.672089i	$0.765397\pi$
$272$	0	0
$273$	20.3366	1.23083
$274$	0	0
$275$	−7.67505	−0.462823
$276$	0	0
$277$	−6.73872	−0.404891	−0.202445	−	0.979294i	$-0.564889\pi$
$277$	−6.73872	−0.404891	−0.202445	+	0.979294i	$0.564889\pi$
$278$	0	0
$279$	5.39266	0.322850
$280$	0	0
$281$	14.6602	0.874552	0.437276	−	0.899327i	$-0.355943\pi$
$281$	14.6602	0.874552	0.437276	+	0.899327i	$0.355943\pi$
$282$	0	0
$283$	10.3854	0.617346	0.308673	−	0.951168i	$-0.400115\pi$
$283$	10.3854	0.617346	0.308673	+	0.951168i	$0.400115\pi$
$284$	0	0
$285$	0.798150	0.0472783
$286$	0	0
$287$	8.65545	0.510915
$288$	0	0
$289$	21.0897	1.24057
$290$	0	0
$291$	11.7288	0.687556
$292$	0	0
$293$	16.0050	0.935020	0.467510	−	0.883988i	$-0.345151\pi$
$293$	16.0050	0.935020	0.467510	+	0.883988i	$0.345151\pi$
$294$	0	0
$295$	−0.0581695	−0.00338676
$296$	0	0
$297$	−5.57755	−0.323642
$298$	0	0
$299$	18.0814	1.04568
$300$	0	0
$301$	−10.0090	−0.576911
$302$	0	0
$303$	−18.1101	−1.04040
$304$	0	0
$305$	−6.63786	−0.380082
$306$	0	0
$307$	15.5437	0.887126	0.443563	−	0.896243i	$-0.353714\pi$
$307$	15.5437	0.887126	0.443563	+	0.896243i	$0.353714\pi$
$308$	0	0
$309$	13.1885	0.750266
$310$	0	0
$311$	5.59429	0.317223	0.158611	−	0.987341i	$-0.449298\pi$
$311$	5.59429	0.317223	0.158611	+	0.987341i	$0.449298\pi$
$312$	0	0
$313$	24.5636	1.38841	0.694207	−	0.719775i	$-0.255755\pi$
$313$	24.5636	1.38841	0.694207	+	0.719775i	$0.255755\pi$
$314$	0	0
$315$	−6.88970	−0.388191
$316$	0	0
$317$	−3.67825	−0.206591	−0.103295	−	0.994651i	$-0.532939\pi$
$317$	−3.67825	−0.206591	−0.103295	+	0.994651i	$0.532939\pi$
$318$	0	0
$319$	20.2251	1.13239
$320$	0	0
$321$	−0.332326	−0.0185486
$322$	0	0
$323$	−2.58760	−0.143978
$324$	0	0
$325$	−7.73227	−0.428909
$326$	0	0
$327$	17.5340	0.969634
$328$	0	0
$329$	−0.467452	−0.0257715
$330$	0	0
$331$	25.4670	1.39979	0.699896	−	0.714245i	$-0.253230\pi$
$331$	25.4670	1.39979	0.699896	+	0.714245i	$0.253230\pi$
$332$	0	0
$333$	−2.73595	−0.149929
$334$	0	0
$335$	−21.5106	−1.17525
$336$	0	0
$337$	−27.7981	−1.51426	−0.757130	−	0.653264i	$-0.773399\pi$
$337$	−27.7981	−1.51426	−0.757130	+	0.653264i	$0.773399\pi$
$338$	0	0
$339$	−4.03768	−0.219296
$340$	0	0
$341$	30.0778	1.62881
$342$	0	0
$343$	3.26286	0.176178
$344$	0	0
$345$	−6.12567	−0.329795
$346$	0	0
$347$	12.8933	0.692147	0.346073	−	0.938207i	$-0.387515\pi$
$347$	12.8933	0.692147	0.346073	+	0.938207i	$0.387515\pi$
$348$	0	0
$349$	25.7314	1.37737	0.688684	−	0.725061i	$-0.258189\pi$
$349$	25.7314	1.37737	0.688684	+	0.725061i	$0.258189\pi$
$350$	0	0
$351$	−5.61913	−0.299927
$352$	0	0
$353$	−9.39146	−0.499857	−0.249929	−	0.968264i	$-0.580407\pi$
$353$	−9.39146	−0.499857	−0.249929	+	0.968264i	$0.580407\pi$
$354$	0	0
$355$	6.63245	0.352014
$356$	0	0
$357$	22.3364	1.18217
$358$	0	0
$359$	32.7278	1.72731	0.863653	−	0.504086i	$-0.168171\pi$
$359$	32.7278	1.72731	0.863653	+	0.504086i	$0.168171\pi$
$360$	0	0
$361$	−18.8242	−0.990748
$362$	0	0
$363$	−20.1091	−1.05545
$364$	0	0
$365$	−14.8517	−0.777374
$366$	0	0
$367$	10.5432	0.550353	0.275176	−	0.961394i	$-0.411264\pi$
$367$	10.5432	0.550353	0.275176	+	0.961394i	$0.411264\pi$
$368$	0	0
$369$	−2.39155	−0.124499
$370$	0	0
$371$	−31.9286	−1.65765
$372$	0	0
$373$	−15.5545	−0.805382	−0.402691	−	0.915336i	$-0.631925\pi$
$373$	−15.5545	−0.805382	−0.402691	+	0.915336i	$0.631925\pi$
$374$	0	0
$375$	12.1379	0.626798
$376$	0	0
$377$	20.3758	1.04941
$378$	0	0
$379$	−2.27475	−0.116846	−0.0584231	−	0.998292i	$-0.518607\pi$
$379$	−2.27475	−0.116846	−0.0584231	+	0.998292i	$0.518607\pi$
$380$	0	0
$381$	6.02421	0.308630
$382$	0	0
$383$	−3.50779	−0.179240	−0.0896199	−	0.995976i	$-0.528565\pi$
$383$	−3.50779	−0.179240	−0.0896199	+	0.995976i	$0.528565\pi$
$384$	0	0
$385$	−38.4277	−1.95846
$386$	0	0
$387$	2.76555	0.140581
$388$	0	0
$389$	31.3820	1.59113	0.795565	−	0.605868i	$-0.207174\pi$
$389$	31.3820	1.59113	0.795565	+	0.605868i	$0.207174\pi$
$390$	0	0
$391$	19.8594	1.00434
$392$	0	0
$393$	9.22093	0.465134
$394$	0	0
$395$	−27.6381	−1.39062
$396$	0	0
$397$	−19.6125	−0.984325	−0.492163	−	0.870503i	$-0.663793\pi$
$397$	−19.6125	−0.984325	−0.492163	+	0.870503i	$0.663793\pi$
$398$	0	0
$399$	−1.51741	−0.0759657
$400$	0	0
$401$	7.08248	0.353682	0.176841	−	0.984239i	$-0.443412\pi$
$401$	7.08248	0.353682	0.176841	+	0.984239i	$0.443412\pi$
$402$	0	0
$403$	30.3021	1.50945
$404$	0	0
$405$	1.90366	0.0945939
$406$	0	0
$407$	−15.2599	−0.756405
$408$	0	0
$409$	−16.5777	−0.819714	−0.409857	−	0.912150i	$-0.634421\pi$
$409$	−16.5777	−0.819714	−0.409857	+	0.912150i	$0.634421\pi$
$410$	0	0
$411$	−14.9935	−0.739575
$412$	0	0
$413$	0.110590	0.00544176
$414$	0	0
$415$	9.71755	0.477016
$416$	0	0
$417$	11.1669	0.546844
$418$	0	0
$419$	−36.5142	−1.78384	−0.891918	−	0.452196i	$-0.850641\pi$
$419$	−36.5142	−1.78384	−0.891918	+	0.452196i	$0.850641\pi$
$420$	0	0
$421$	−19.7421	−0.962172	−0.481086	−	0.876673i	$-0.659758\pi$
$421$	−19.7421	−0.962172	−0.481086	+	0.876673i	$0.659758\pi$
$422$	0	0
$423$	0.129160	0.00627996
$424$	0	0
$425$	−8.49262	−0.411952
$426$	0	0
$427$	12.6197	0.610708
$428$	0	0
$429$	−31.3410	−1.51316
$430$	0	0
$431$	3.77006	0.181597	0.0907986	−	0.995869i	$-0.471058\pi$
$431$	3.77006	0.181597	0.0907986	+	0.995869i	$0.471058\pi$
$432$	0	0
$433$	11.6330	0.559045	0.279522	−	0.960139i	$-0.409824\pi$
$433$	11.6330	0.559045	0.279522	+	0.960139i	$0.409824\pi$
$434$	0	0
$435$	−6.90298	−0.330972
$436$	0	0
$437$	−1.34914	−0.0645382
$438$	0	0
$439$	36.0749	1.72176	0.860880	−	0.508809i	$-0.169914\pi$
$439$	36.0749	1.72176	0.860880	+	0.508809i	$0.169914\pi$
$440$	0	0
$441$	6.09845	0.290403
$442$	0	0
$443$	28.7802	1.36739	0.683694	−	0.729769i	$-0.260373\pi$
$443$	28.7802	1.36739	0.683694	+	0.729769i	$0.260373\pi$
$444$	0	0
$445$	22.5185	1.06748
$446$	0	0
$447$	−19.6013	−0.927109
$448$	0	0
$449$	−13.3412	−0.629609	−0.314805	−	0.949157i	$-0.601939\pi$
$449$	−13.3412	−0.629609	−0.314805	+	0.949157i	$0.601939\pi$
$450$	0	0
$451$	−13.3390	−0.628109
$452$	0	0
$453$	3.73694	0.175577
$454$	0	0
$455$	−38.7142	−1.81495
$456$	0	0
$457$	−30.3061	−1.41766	−0.708830	−	0.705379i	$-0.750777\pi$
$457$	−30.3061	−1.41766	−0.708830	+	0.705379i	$0.750777\pi$
$458$	0	0
$459$	−6.17169	−0.288070
$460$	0	0
$461$	9.62387	0.448228	0.224114	−	0.974563i	$-0.428051\pi$
$461$	9.62387	0.448228	0.224114	+	0.974563i	$0.428051\pi$
$462$	0	0
$463$	1.99653	0.0927867	0.0463934	−	0.998923i	$-0.485227\pi$
$463$	1.99653	0.0927867	0.0463934	+	0.998923i	$0.485227\pi$
$464$	0	0
$465$	−10.2658	−0.476066
$466$	0	0
$467$	0.964828	0.0446469	0.0223235	−	0.999751i	$-0.492894\pi$
$467$	0.964828	0.0446469	0.0223235	+	0.999751i	$0.492894\pi$
$468$	0	0
$469$	40.8952	1.88837
$470$	0	0
$471$	−24.8232	−1.14379
$472$	0	0
$473$	15.4250	0.709243
$474$	0	0
$475$	0.576941	0.0264719
$476$	0	0
$477$	8.82207	0.403935
$478$	0	0
$479$	−20.4762	−0.935581	−0.467790	−	0.883839i	$-0.654950\pi$
$479$	−20.4762	−0.935581	−0.467790	+	0.883839i	$0.654950\pi$
$480$	0	0
$481$	−15.3737	−0.700979
$482$	0	0
$483$	11.6459	0.529907
$484$	0	0
$485$	−22.3277	−1.01385
$486$	0	0
$487$	−13.3594	−0.605370	−0.302685	−	0.953091i	$-0.597883\pi$
$487$	−13.3594	−0.605370	−0.302685	+	0.953091i	$0.597883\pi$
$488$	0	0
$489$	−19.1568	−0.866301
$490$	0	0
$491$	1.02733	0.0463629	0.0231815	−	0.999731i	$-0.492620\pi$
$491$	1.02733	0.0463629	0.0231815	+	0.999731i	$0.492620\pi$
$492$	0	0
$493$	22.3795	1.00792
$494$	0	0
$495$	10.6178	0.477234
$496$	0	0
$497$	−12.6094	−0.565608
$498$	0	0
$499$	−38.8368	−1.73858	−0.869288	−	0.494307i	$-0.835422\pi$
$499$	−38.8368	−1.73858	−0.869288	+	0.494307i	$0.835422\pi$
$500$	0	0
$501$	16.2557	0.726250
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	34.4756	1.53414
$506$	0	0
$507$	−18.5747	−0.824930
$508$	0	0
$509$	−12.4332	−0.551093	−0.275547	−	0.961288i	$-0.588859\pi$
$509$	−12.4332	−0.551093	−0.275547	+	0.961288i	$0.588859\pi$
$510$	0	0
$511$	28.2355	1.24907
$512$	0	0
$513$	0.419270	0.0185112
$514$	0	0
$515$	−25.1064	−1.10632
$516$	0	0
$517$	0.720395	0.0316830
$518$	0	0
$519$	−13.3360	−0.585385
$520$	0	0
$521$	0.694905	0.0304444	0.0152222	−	0.999884i	$-0.495154\pi$
$521$	0.694905	0.0304444	0.0152222	+	0.999884i	$0.495154\pi$
$522$	0	0
$523$	−11.4278	−0.499703	−0.249852	−	0.968284i	$-0.580382\pi$
$523$	−11.4278	−0.499703	−0.249852	+	0.968284i	$0.580382\pi$
$524$	0	0
$525$	−4.98021	−0.217354
$526$	0	0
$527$	33.2818	1.44978
$528$	0	0
$529$	−12.6456	−0.549807
$530$	0	0
$531$	−0.0305566	−0.00132604
$532$	0	0
$533$	−13.4384	−0.582084
$534$	0	0
$535$	0.632638	0.0273513
$536$	0	0
$537$	−24.0711	−1.03874
$538$	0	0
$539$	34.0144	1.46511
$540$	0	0
$541$	26.3541	1.13305	0.566526	−	0.824044i	$-0.308287\pi$
$541$	26.3541	1.13305	0.566526	+	0.824044i	$0.308287\pi$
$542$	0	0
$543$	−6.94634	−0.298096
$544$	0	0
$545$	−33.3789	−1.42980
$546$	0	0
$547$	6.56641	0.280759	0.140380	−	0.990098i	$-0.455168\pi$
$547$	6.56641	0.280759	0.140380	+	0.990098i	$0.455168\pi$
$548$	0	0
$549$	−3.48688	−0.148817
$550$	0	0
$551$	−1.52034	−0.0647686
$552$	0	0
$553$	52.5446	2.23442
$554$	0	0
$555$	5.20833	0.221081
$556$	0	0
$557$	33.8341	1.43360	0.716799	−	0.697280i	$-0.245607\pi$
$557$	33.8341	1.43360	0.716799	+	0.697280i	$0.245607\pi$
$558$	0	0
$559$	15.5400	0.657273
$560$	0	0
$561$	−34.4229	−1.45334
$562$	0	0
$563$	−0.556859	−0.0234688	−0.0117344	−	0.999931i	$-0.503735\pi$
$563$	−0.556859	−0.0234688	−0.0117344	+	0.999931i	$0.503735\pi$
$564$	0	0
$565$	7.68638	0.323368
$566$	0	0
$567$	−3.61918	−0.151991
$568$	0	0
$569$	21.0750	0.883509	0.441754	−	0.897136i	$-0.354356\pi$
$569$	21.0750	0.883509	0.441754	+	0.897136i	$0.354356\pi$
$570$	0	0
$571$	3.38512	0.141663	0.0708314	−	0.997488i	$-0.477435\pi$
$571$	3.38512	0.141663	0.0708314	+	0.997488i	$0.477435\pi$
$572$	0	0
$573$	12.5873	0.525841
$574$	0	0
$575$	−4.42793	−0.184658
$576$	0	0
$577$	−12.4257	−0.517289	−0.258645	−	0.965973i	$-0.583276\pi$
$577$	−12.4257	−0.517289	−0.258645	+	0.965973i	$0.583276\pi$
$578$	0	0
$579$	5.10938	0.212338
$580$	0	0
$581$	−18.4746	−0.766458
$582$	0	0
$583$	49.2055	2.03789
$584$	0	0
$585$	10.6969	0.442264
$586$	0	0
$587$	20.9155	0.863274	0.431637	−	0.902047i	$-0.357936\pi$
$587$	20.9155	0.863274	0.431637	+	0.902047i	$0.357936\pi$
$588$	0	0
$589$	−2.26098	−0.0931621
$590$	0	0
$591$	2.47196	0.101683
$592$	0	0
$593$	8.03087	0.329788	0.164894	−	0.986311i	$-0.447272\pi$
$593$	8.03087	0.329788	0.164894	+	0.986311i	$0.447272\pi$
$594$	0	0
$595$	−42.5211	−1.74319
$596$	0	0
$597$	−25.0861	−1.02670
$598$	0	0
$599$	13.4615	0.550024	0.275012	−	0.961441i	$-0.411318\pi$
$599$	13.4615	0.550024	0.275012	+	0.961441i	$0.411318\pi$
$600$	0	0
$601$	−25.1994	−1.02790	−0.513952	−	0.857819i	$-0.671819\pi$
$601$	−25.1994	−1.02790	−0.513952	+	0.857819i	$0.671819\pi$
$602$	0	0
$603$	−11.2996	−0.460155
$604$	0	0
$605$	38.2810	1.55634
$606$	0	0
$607$	9.94928	0.403829	0.201914	−	0.979403i	$-0.435284\pi$
$607$	9.94928	0.403829	0.201914	+	0.979403i	$0.435284\pi$
$608$	0	0
$609$	13.1237	0.531799
$610$	0	0
$611$	0.725766	0.0293613
$612$	0	0
$613$	42.0466	1.69825	0.849123	−	0.528195i	$-0.177131\pi$
$613$	42.0466	1.69825	0.849123	+	0.528195i	$0.177131\pi$
$614$	0	0
$615$	4.55271	0.183583
$616$	0	0
$617$	−2.69200	−0.108376	−0.0541880	−	0.998531i	$-0.517257\pi$
$617$	−2.69200	−0.108376	−0.0541880	+	0.998531i	$0.517257\pi$
$618$	0	0
$619$	−7.98069	−0.320771	−0.160386	−	0.987054i	$-0.551274\pi$
$619$	−7.98069	−0.320771	−0.160386	+	0.987054i	$0.551274\pi$
$620$	0	0
$621$	−3.21783	−0.129127
$622$	0	0
$623$	−42.8113	−1.71520
$624$	0	0
$625$	−16.2262	−0.649046
$626$	0	0
$627$	2.33850	0.0933908
$628$	0	0
$629$	−16.8854	−0.673266
$630$	0	0
$631$	9.56812	0.380901	0.190450	−	0.981697i	$-0.439005\pi$
$631$	9.56812	0.380901	0.190450	+	0.981697i	$0.439005\pi$
$632$	0	0
$633$	−15.0915	−0.599834
$634$	0	0
$635$	−11.4681	−0.455097
$636$	0	0
$637$	34.2680	1.35775
$638$	0	0
$639$	3.48404	0.137827
$640$	0	0
$641$	22.5848	0.892045	0.446023	−	0.895022i	$-0.352840\pi$
$641$	22.5848	0.892045	0.446023	+	0.895022i	$0.352840\pi$
$642$	0	0
$643$	19.5119	0.769473	0.384736	−	0.923026i	$-0.374292\pi$
$643$	19.5119	0.769473	0.384736	+	0.923026i	$0.374292\pi$
$644$	0	0
$645$	−5.26469	−0.207297
$646$	0	0
$647$	10.9375	0.429996	0.214998	−	0.976614i	$-0.431025\pi$
$647$	10.9375	0.429996	0.214998	+	0.976614i	$0.431025\pi$
$648$	0	0
$649$	−0.170431	−0.00669000
$650$	0	0
$651$	19.5170	0.764932
$652$	0	0
$653$	−10.6523	−0.416858	−0.208429	−	0.978037i	$-0.566835\pi$
$653$	−10.6523	−0.416858	−0.208429	+	0.978037i	$0.566835\pi$
$654$	0	0
$655$	−17.5535	−0.685874
$656$	0	0
$657$	−7.80164	−0.304371
$658$	0	0
$659$	−17.2770	−0.673017	−0.336509	−	0.941680i	$-0.609246\pi$
$659$	−17.2770	−0.673017	−0.336509	+	0.941680i	$0.609246\pi$
$660$	0	0
$661$	−42.7456	−1.66261	−0.831306	−	0.555815i	$-0.812406\pi$
$661$	−42.7456	−1.66261	−0.831306	+	0.555815i	$0.812406\pi$
$662$	0	0
$663$	−34.6795	−1.34684
$664$	0	0
$665$	2.88865	0.112017
$666$	0	0
$667$	11.6683	0.451800
$668$	0	0
$669$	27.7017	1.07101
$670$	0	0
$671$	−19.4483	−0.750792
$672$	0	0
$673$	2.48147	0.0956535	0.0478268	−	0.998856i	$-0.484770\pi$
$673$	2.48147	0.0956535	0.0478268	+	0.998856i	$0.484770\pi$
$674$	0	0
$675$	1.37606	0.0529646
$676$	0	0
$677$	19.1210	0.734882	0.367441	−	0.930047i	$-0.380234\pi$
$677$	19.1210	0.734882	0.367441	+	0.930047i	$0.380234\pi$
$678$	0	0
$679$	42.4487	1.62903
$680$	0	0
$681$	14.0651	0.538977
$682$	0	0
$683$	−44.5689	−1.70538	−0.852690	−	0.522417i	$-0.825030\pi$
$683$	−44.5689	−1.70538	−0.852690	+	0.522417i	$0.825030\pi$
$684$	0	0
$685$	28.5426	1.09056
$686$	0	0
$687$	10.9817	0.418977
$688$	0	0
$689$	49.5724	1.88856
$690$	0	0
$691$	25.0227	0.951908	0.475954	−	0.879470i	$-0.342103\pi$
$691$	25.0227	0.951908	0.475954	+	0.879470i	$0.342103\pi$
$692$	0	0
$693$	−20.1862	−0.766809
$694$	0	0
$695$	−21.2580	−0.806360
$696$	0	0
$697$	−14.7599	−0.559071
$698$	0	0
$699$	16.3683	0.619107
$700$	0	0
$701$	25.2592	0.954028	0.477014	−	0.878896i	$-0.341719\pi$
$701$	25.2592	0.954028	0.477014	+	0.878896i	$0.341719\pi$
$702$	0	0
$703$	1.14710	0.0432638
$704$	0	0
$705$	−0.245877	−0.00926026
$706$	0	0
$707$	−65.5438	−2.46503
$708$	0	0
$709$	9.60807	0.360839	0.180419	−	0.983590i	$-0.442255\pi$
$709$	9.60807	0.360839	0.180419	+	0.983590i	$0.442255\pi$
$710$	0	0
$711$	−14.5184	−0.544481
$712$	0	0
$713$	17.3527	0.649863
$714$	0	0
$715$	59.6628	2.23126
$716$	0	0
$717$	27.8170	1.03884
$718$	0	0
$719$	−18.8898	−0.704472	−0.352236	−	0.935911i	$-0.614579\pi$
$719$	−18.8898	−0.704472	−0.352236	+	0.935911i	$0.614579\pi$
$720$	0	0
$721$	47.7315	1.77761
$722$	0	0
$723$	−10.8561	−0.403742
$724$	0	0
$725$	−4.98981	−0.185317
$726$	0	0
$727$	1.60379	0.0594811	0.0297406	−	0.999558i	$-0.490532\pi$
$727$	1.60379	0.0594811	0.0297406	+	0.999558i	$0.490532\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.0681	0.631288
$732$	0	0
$733$	38.2841	1.41405	0.707027	−	0.707186i	$-0.250036\pi$
$733$	38.2841	1.41405	0.707027	+	0.707186i	$0.250036\pi$
$734$	0	0
$735$	−11.6094	−0.428220
$736$	0	0
$737$	−63.0240	−2.32152
$738$	0	0
$739$	−11.0649	−0.407028	−0.203514	−	0.979072i	$-0.565236\pi$
$739$	−11.0649	−0.407028	−0.203514	+	0.979072i	$0.565236\pi$
$740$	0	0
$741$	2.35594	0.0865474
$742$	0	0
$743$	−12.2036	−0.447706	−0.223853	−	0.974623i	$-0.571864\pi$
$743$	−12.2036	−0.447706	−0.223853	+	0.974623i	$0.571864\pi$
$744$	0	0
$745$	37.3143	1.36709
$746$	0	0
$747$	5.10465	0.186769
$748$	0	0
$749$	−1.20275	−0.0439475
$750$	0	0
$751$	20.0871	0.732987	0.366493	−	0.930421i	$-0.380558\pi$
$751$	20.0871	0.732987	0.366493	+	0.930421i	$0.380558\pi$
$752$	0	0
$753$	4.31403	0.157212
$754$	0	0
$755$	−7.11388	−0.258901
$756$	0	0
$757$	−0.813516	−0.0295677	−0.0147839	−	0.999891i	$-0.504706\pi$
$757$	−0.813516	−0.0295677	−0.0147839	+	0.999891i	$0.504706\pi$
$758$	0	0
$759$	−17.9476	−0.651457
$760$	0	0
$761$	−40.9585	−1.48474	−0.742372	−	0.669988i	$-0.766299\pi$
$761$	−40.9585	−1.48474	−0.742372	+	0.669988i	$0.766299\pi$
$762$	0	0
$763$	63.4588	2.29736
$764$	0	0
$765$	11.7488	0.424780
$766$	0	0
$767$	−0.171701	−0.00619978
$768$	0	0
$769$	23.5116	0.847852	0.423926	−	0.905697i	$-0.360652\pi$
$769$	23.5116	0.847852	0.423926	+	0.905697i	$0.360652\pi$
$770$	0	0
$771$	26.7299	0.962654
$772$	0	0
$773$	−7.38215	−0.265517	−0.132759	−	0.991148i	$-0.542384\pi$
$773$	−7.38215	−0.265517	−0.132759	+	0.991148i	$0.542384\pi$
$774$	0	0
$775$	−7.42063	−0.266557
$776$	0	0
$777$	−9.90189	−0.355228
$778$	0	0
$779$	1.00271	0.0359257
$780$	0	0
$781$	19.4324	0.695347
$782$	0	0
$783$	−3.62615	−0.129588
$784$	0	0
$785$	47.2550	1.68660
$786$	0	0
$787$	−42.1085	−1.50101	−0.750503	−	0.660867i	$-0.770189\pi$
$787$	−42.1085	−1.50101	−0.750503	+	0.660867i	$0.770189\pi$
$788$	0	0
$789$	7.27645	0.259048
$790$	0	0
$791$	−14.6131	−0.519581
$792$	0	0
$793$	−19.5933	−0.695777
$794$	0	0
$795$	−16.7943	−0.595631
$796$	0	0
$797$	−43.0432	−1.52467	−0.762335	−	0.647183i	$-0.775947\pi$
$797$	−43.0432	−1.52467	−0.762335	+	0.647183i	$0.775947\pi$
$798$	0	0
$799$	0.797134	0.0282006
$800$	0	0
$801$	11.8290	0.417958
$802$	0	0
$803$	−43.5141	−1.53558
$804$	0	0
$805$	−22.1699	−0.781386
$806$	0	0
$807$	−6.84338	−0.240898
$808$	0	0
$809$	−14.2585	−0.501301	−0.250650	−	0.968078i	$-0.580644\pi$
$809$	−14.2585	−0.501301	−0.250650	+	0.968078i	$0.580644\pi$
$810$	0	0
$811$	12.1148	0.425407	0.212704	−	0.977117i	$-0.431773\pi$
$811$	12.1148	0.425407	0.212704	+	0.977117i	$0.431773\pi$
$812$	0	0
$813$	24.3794	0.855022
$814$	0	0
$815$	36.4681	1.27742
$816$	0	0
$817$	−1.15951	−0.0405663
$818$	0	0
$819$	−20.3366	−0.710620
$820$	0	0
$821$	−50.6102	−1.76631	−0.883155	−	0.469082i	$-0.844585\pi$
$821$	−50.6102	−1.76631	−0.883155	+	0.469082i	$0.844585\pi$
$822$	0	0
$823$	32.3566	1.12788	0.563939	−	0.825816i	$-0.309285\pi$
$823$	32.3566	1.12788	0.563939	+	0.825816i	$0.309285\pi$
$824$	0	0
$825$	7.67505	0.267211
$826$	0	0
$827$	11.8726	0.412849	0.206425	−	0.978462i	$-0.433817\pi$
$827$	11.8726	0.412849	0.206425	+	0.978462i	$0.433817\pi$
$828$	0	0
$829$	16.3237	0.566947	0.283473	−	0.958980i	$-0.408513\pi$
$829$	16.3237	0.566947	0.283473	+	0.958980i	$0.408513\pi$
$830$	0	0
$831$	6.73872	0.233764
$832$	0	0
$833$	37.6377	1.30407
$834$	0	0
$835$	−30.9454	−1.07091
$836$	0	0
$837$	−5.39266	−0.186398
$838$	0	0
$839$	48.5929	1.67761	0.838807	−	0.544428i	$-0.183253\pi$
$839$	48.5929	1.67761	0.838807	+	0.544428i	$0.183253\pi$
$840$	0	0
$841$	−15.8510	−0.546587
$842$	0	0
$843$	−14.6602	−0.504923
$844$	0	0
$845$	35.3599	1.21642
$846$	0	0
$847$	−72.7784	−2.50070
$848$	0	0
$849$	−10.3854	−0.356425
$850$	0	0
$851$	−8.80382	−0.301791
$852$	0	0
$853$	−29.7272	−1.01784	−0.508920	−	0.860814i	$-0.669955\pi$
$853$	−29.7272	−1.01784	−0.508920	+	0.860814i	$0.669955\pi$
$854$	0	0
$855$	−0.798150	−0.0272961
$856$	0	0
$857$	−6.14166	−0.209795	−0.104897	−	0.994483i	$-0.533451\pi$
$857$	−6.14166	−0.209795	−0.104897	+	0.994483i	$0.533451\pi$
$858$	0	0
$859$	−19.2930	−0.658269	−0.329135	−	0.944283i	$-0.606757\pi$
$859$	−19.2930	−0.658269	−0.329135	+	0.944283i	$0.606757\pi$
$860$	0	0
$861$	−8.65545	−0.294977
$862$	0	0
$863$	21.5629	0.734009	0.367004	−	0.930219i	$-0.380383\pi$
$863$	21.5629	0.734009	0.367004	+	0.930219i	$0.380383\pi$
$864$	0	0
$865$	25.3873	0.863193
$866$	0	0
$867$	−21.0897	−0.716244
$868$	0	0
$869$	−80.9769	−2.74695
$870$	0	0
$871$	−63.4939	−2.15141
$872$	0	0
$873$	−11.7288	−0.396961
$874$	0	0
$875$	43.9292	1.48508
$876$	0	0
$877$	−43.7559	−1.47753	−0.738766	−	0.673962i	$-0.764591\pi$
$877$	−43.7559	−1.47753	−0.738766	+	0.673962i	$0.764591\pi$
$878$	0	0
$879$	−16.0050	−0.539834
$880$	0	0
$881$	37.3214	1.25739	0.628696	−	0.777651i	$-0.283589\pi$
$881$	37.3214	1.25739	0.628696	+	0.777651i	$0.283589\pi$
$882$	0	0
$883$	−3.04779	−0.102566	−0.0512831	−	0.998684i	$-0.516331\pi$
$883$	−3.04779	−0.102566	−0.0512831	+	0.998684i	$0.516331\pi$
$884$	0	0
$885$	0.0581695	0.00195535
$886$	0	0
$887$	10.4385	0.350490	0.175245	−	0.984525i	$-0.443928\pi$
$887$	10.4385	0.350490	0.175245	+	0.984525i	$0.443928\pi$
$888$	0	0
$889$	21.8027	0.731239
$890$	0	0
$891$	5.57755	0.186855
$892$	0	0
$893$	−0.0541528	−0.00181216
$894$	0	0
$895$	45.8233	1.53170
$896$	0	0
$897$	−18.0814	−0.603721
$898$	0	0
$899$	19.5546	0.652183
$900$	0	0
$901$	54.4470	1.81389
$902$	0	0
$903$	10.0090	0.333080
$904$	0	0
$905$	13.2235	0.439564
$906$	0	0
$907$	−45.4241	−1.50828	−0.754141	−	0.656713i	$-0.771946\pi$
$907$	−45.4241	−1.50828	−0.754141	+	0.656713i	$0.771946\pi$
$908$	0	0
$909$	18.1101	0.600675
$910$	0	0
$911$	23.1594	0.767305	0.383652	−	0.923478i	$-0.374666\pi$
$911$	23.1594	0.767305	0.383652	+	0.923478i	$0.374666\pi$
$912$	0	0
$913$	28.4715	0.942268
$914$	0	0
$915$	6.63786	0.219441
$916$	0	0
$917$	33.3722	1.10205
$918$	0	0
$919$	1.92326	0.0634425	0.0317213	−	0.999497i	$-0.489901\pi$
$919$	1.92326	0.0634425	0.0317213	+	0.999497i	$0.489901\pi$
$920$	0	0
$921$	−15.5437	−0.512182
$922$	0	0
$923$	19.5773	0.644395
$924$	0	0
$925$	3.76483	0.123787
$926$	0	0
$927$	−13.1885	−0.433167
$928$	0	0
$929$	−32.2206	−1.05712	−0.528561	−	0.848895i	$-0.677268\pi$
$929$	−32.2206	−1.05712	−0.528561	+	0.848895i	$0.677268\pi$
$930$	0	0
$931$	−2.55690	−0.0837990
$932$	0	0
$933$	−5.59429	−0.183149
$934$	0	0
$935$	65.5297	2.14305
$936$	0	0
$937$	−6.72064	−0.219554	−0.109777	−	0.993956i	$-0.535014\pi$
$937$	−6.72064	−0.219554	−0.109777	+	0.993956i	$0.535014\pi$
$938$	0	0
$939$	−24.5636	−0.801602
$940$	0	0
$941$	52.5431	1.71286	0.856428	−	0.516266i	$-0.172678\pi$
$941$	52.5431	1.71286	0.856428	+	0.516266i	$0.172678\pi$
$942$	0	0
$943$	−7.69561	−0.250603
$944$	0	0
$945$	6.88970	0.224122
$946$	0	0
$947$	−9.59653	−0.311845	−0.155923	−	0.987769i	$-0.549835\pi$
$947$	−9.59653	−0.311845	−0.155923	+	0.987769i	$0.549835\pi$
$948$	0	0
$949$	−43.8385	−1.42306
$950$	0	0
$951$	3.67825	0.119275
$952$	0	0
$953$	21.5868	0.699264	0.349632	−	0.936887i	$-0.386307\pi$
$953$	21.5868	0.699264	0.349632	+	0.936887i	$0.386307\pi$
$954$	0	0
$955$	−23.9620	−0.775391
$956$	0	0
$957$	−20.2251	−0.653783
$958$	0	0
$959$	−54.2642	−1.75228
$960$	0	0
$961$	−1.91921	−0.0619100
$962$	0	0
$963$	0.332326	0.0107091
$964$	0	0
$965$	−9.72654	−0.313108
$966$	0	0
$967$	34.0890	1.09623	0.548114	−	0.836404i	$-0.315346\pi$
$967$	34.0890	1.09623	0.548114	+	0.836404i	$0.315346\pi$
$968$	0	0
$969$	2.58760	0.0831258
$970$	0	0
$971$	−10.3510	−0.332179	−0.166089	−	0.986111i	$-0.553114\pi$
$971$	−10.3510	−0.332179	−0.166089	+	0.986111i	$0.553114\pi$
$972$	0	0
$973$	40.4149	1.29564
$974$	0	0
$975$	7.73227	0.247631
$976$	0	0
$977$	−57.8074	−1.84942	−0.924711	−	0.380670i	$-0.875693\pi$
$977$	−57.8074	−1.84942	−0.924711	+	0.380670i	$0.875693\pi$
$978$	0	0
$979$	65.9769	2.10863
$980$	0	0
$981$	−17.5340	−0.559819
$982$	0	0
$983$	−61.7179	−1.96849	−0.984247	−	0.176797i	$-0.943426\pi$
$983$	−61.7179	−1.96849	−0.984247	+	0.176797i	$0.943426\pi$
$984$	0	0
$985$	−4.70578	−0.149939
$986$	0	0
$987$	0.467452	0.0148792
$988$	0	0
$989$	8.89909	0.282974
$990$	0	0
$991$	25.5848	0.812727	0.406364	−	0.913711i	$-0.366797\pi$
$991$	25.5848	0.812727	0.406364	+	0.913711i	$0.366797\pi$
$992$	0	0
$993$	−25.4670	−0.808170
$994$	0	0
$995$	47.7554	1.51395
$996$	0	0
$997$	−21.9572	−0.695391	−0.347695	−	0.937608i	$-0.613036\pi$
$997$	−21.9572	−0.695391	−0.347695	+	0.937608i	$0.613036\pi$
$998$	0	0
$999$	2.73595	0.0865616

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6036.2.a.i.1.18	✓	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6036.2.a.i.1.18	✓	26	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6036 = 2^{2} \cdot 3 \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.90366 q^{5} -3.61918 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.90366 q^{5} -3.61918 q^{7} +1.00000 q^{9} +5.57755 q^{11} +5.61913 q^{13} -1.90366 q^{15} +6.17169 q^{17} -0.419270 q^{19} +3.61918 q^{21} +3.21783 q^{23} -1.37606 q^{25} -1.00000 q^{27} +3.62615 q^{29} +5.39266 q^{31} -5.57755 q^{33} -6.88970 q^{35} -2.73595 q^{37} -5.61913 q^{39} -2.39155 q^{41} +2.76555 q^{43} +1.90366 q^{45} +0.129160 q^{47} +6.09845 q^{49} -6.17169 q^{51} +8.82207 q^{53} +10.6178 q^{55} +0.419270 q^{57} -0.0305566 q^{59} -3.48688 q^{61} -3.61918 q^{63} +10.6969 q^{65} -11.2996 q^{67} -3.21783 q^{69} +3.48404 q^{71} -7.80164 q^{73} +1.37606 q^{75} -20.1862 q^{77} -14.5184 q^{79} +1.00000 q^{81} +5.10465 q^{83} +11.7488 q^{85} -3.62615 q^{87} +11.8290 q^{89} -20.3366 q^{91} -5.39266 q^{93} -0.798150 q^{95} -11.7288 q^{97} +5.57755 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9}+O(q^{10}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9	\( 26 q - 26 q^{3} + 6 q^{5} + 5 q^{7} + 26 q^{9} - 11 q^{11} + 13 q^{13} - 6 q^{15} + 12 q^{17} - q^{19} - 5 q^{21} - 22 q^{23} + 48 q^{25} - 26 q^{27} + 6 q^{29} + 19 q^{31} + 11 q^{33} - 21 q^{35} + 20 q^{37} - 13 q^{39} + 25 q^{41} + 4 q^{43} + 6 q^{45} + 8 q^{47} + 67 q^{49} - 12 q^{51} - 5 q^{53} + 20 q^{55} + q^{57} - 18 q^{59} + 43 q^{61} + 5 q^{63} + 41 q^{65} + 5 q^{67} + 22 q^{69} - q^{71} + 22 q^{73} - 48 q^{75} + 23 q^{77} + 16 q^{79} + 26 q^{81} - 19 q^{83} + 29 q^{85} - 6 q^{87} + 49 q^{89} - 13 q^{91} - 19 q^{93} - 26 q^{95} + 25 q^{97} - 11 q^{99}+O(q^{100}) \) 26 * q - 26 * q^3 + 6 * q^5 + 5 * q^7 + 26 * q^9 - 11 * q^11 + 13 * q^13 - 6 * q^15 + 12 * q^17 - q^19 - 5 * q^21 - 22 * q^23 + 48 * q^25 - 26 * q^27 + 6 * q^29 + 19 * q^31 + 11 * q^33 - 21 * q^35 + 20 * q^37 - 13 * q^39 + 25 * q^41 + 4 * q^43 + 6 * q^45 + 8 * q^47 + 67 * q^49 - 12 * q^51 - 5 * q^53 + 20 * q^55 + q^57 - 18 * q^59 + 43 * q^61 + 5 * q^63 + 41 * q^65 + 5 * q^67 + 22 * q^69 - q^71 + 22 * q^73 - 48 * q^75 + 23 * q^77 + 16 * q^79 + 26 * q^81 - 19 * q^83 + 29 * q^85 - 6 * q^87 + 49 * q^89 - 13 * q^91 - 19 * q^93 - 26 * q^95 + 25 * q^97 - 11 * q^99

Introduction

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