LMFDB - Embedded newform 4020.2.a.b.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(1,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.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:	$32.0998616126$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.9301.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 3 $ x^4 - x^3 - 5*x^2 + x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.869986$ of defining polynomial
Character	$\chi$	$=$	4020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -2.24312 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} -2.24312 q^{7} +1.00000 q^{9} +1.86999 q^{11} +4.82147 q^{13} +1.00000 q^{15} -6.69146 q^{17} -5.44833 q^{19} +2.24312 q^{21} +5.48625 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.91357 q^{29} +4.93458 q^{31} -1.86999 q^{33} +2.24312 q^{35} -6.15669 q^{37} -4.82147 q^{39} -9.11878 q^{41} +2.66478 q^{43} -1.00000 q^{45} +7.75195 q^{47} -1.96839 q^{49} +6.69146 q^{51} +13.4145 q^{53} -1.86999 q^{55} +5.44833 q^{57} -12.5276 q^{59} -5.58739 q^{61} -2.24312 q^{63} -4.82147 q^{65} +1.00000 q^{67} -5.48625 q^{69} +11.8770 q^{71} +9.24093 q^{73} -1.00000 q^{75} -4.19461 q^{77} -5.01197 q^{79} +1.00000 q^{81} +0.956416 q^{83} +6.69146 q^{85} -3.91357 q^{87} -2.14692 q^{89} -10.8152 q^{91} -4.93458 q^{93} +5.44833 q^{95} -2.38374 q^{97} +1.86999 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^5 - q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9} + 5 q^{11} - q^{13} + 4 q^{15} - 4 q^{17} - 7 q^{19} + q^{21} + 6 q^{23} + 4 q^{25} - 4 q^{27} - q^{29} - 11 q^{31} - 5 q^{33} + q^{35} + q^{39} - 13 q^{41} + 15 q^{43} - 4 q^{45} + 7 q^{47} - 3 q^{49} + 4 q^{51} + 13 q^{53} - 5 q^{55} + 7 q^{57} + q^{59} - 15 q^{61} - q^{63} + q^{65} + 4 q^{67} - 6 q^{69} - 6 q^{71} + 8 q^{73} - 4 q^{75} + 9 q^{77} - q^{79} + 4 q^{81} + 18 q^{83} + 4 q^{85} + q^{87} - 24 q^{89} - 29 q^{91} + 11 q^{93} + 7 q^{95} - 23 q^{97} + 5 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 - q^7 + 4 * q^9 + 5 * q^11 - q^13 + 4 * q^15 - 4 * q^17 - 7 * q^19 + q^21 + 6 * q^23 + 4 * q^25 - 4 * q^27 - q^29 - 11 * q^31 - 5 * q^33 + q^35 + q^39 - 13 * q^41 + 15 * q^43 - 4 * q^45 + 7 * q^47 - 3 * q^49 + 4 * q^51 + 13 * q^53 - 5 * q^55 + 7 * q^57 + q^59 - 15 * q^61 - q^63 + q^65 + 4 * q^67 - 6 * q^69 - 6 * q^71 + 8 * q^73 - 4 * q^75 + 9 * q^77 - q^79 + 4 * q^81 + 18 * q^83 + 4 * q^85 + q^87 - 24 * q^89 - 29 * q^91 + 11 * q^93 + 7 * q^95 - 23 * q^97 + 5 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.24312	−0.847822	−0.423911	−	0.905704i	$-0.639343\pi$
$7$	−2.24312	−0.847822	−0.423911	+	0.905704i	$0.639343\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.86999	0.563822	0.281911	−	0.959441i	$-0.409032\pi$
$11$	1.86999	0.563822	0.281911	+	0.959441i	$0.409032\pi$
$12$	0	0
$13$	4.82147	1.33724	0.668618	−	0.743606i	$-0.266886\pi$
$13$	4.82147	1.33724	0.668618	+	0.743606i	$0.266886\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.69146	−1.62292	−0.811458	−	0.584410i	$-0.801326\pi$
$17$	−6.69146	−1.62292	−0.811458	+	0.584410i	$0.801326\pi$
$18$	0	0
$19$	−5.44833	−1.24993	−0.624967	−	0.780651i	$-0.714887\pi$
$19$	−5.44833	−1.24993	−0.624967	+	0.780651i	$0.714887\pi$
$20$	0	0
$21$	2.24312	0.489490
$22$	0	0
$23$	5.48625	1.14396	0.571981	−	0.820267i	$-0.306175\pi$
$23$	5.48625	1.14396	0.571981	+	0.820267i	$0.306175\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.91357	0.726732	0.363366	−	0.931647i	$-0.381628\pi$
$29$	3.91357	0.726732	0.363366	+	0.931647i	$0.381628\pi$
$30$	0	0
$31$	4.93458	0.886277	0.443139	−	0.896453i	$-0.353865\pi$
$31$	4.93458	0.886277	0.443139	+	0.896453i	$0.353865\pi$
$32$	0	0
$33$	−1.86999	−0.325523
$34$	0	0
$35$	2.24312	0.379157
$36$	0	0
$37$	−6.15669	−1.01215	−0.506077	−	0.862488i	$-0.668905\pi$
$37$	−6.15669	−1.01215	−0.506077	+	0.862488i	$0.668905\pi$
$38$	0	0
$39$	−4.82147	−0.772053
$40$	0	0
$41$	−9.11878	−1.42411	−0.712057	−	0.702122i	$-0.752236\pi$
$41$	−9.11878	−1.42411	−0.712057	+	0.702122i	$0.752236\pi$
$42$	0	0
$43$	2.66478	0.406375	0.203187	−	0.979140i	$-0.434870\pi$
$43$	2.66478	0.406375	0.203187	+	0.979140i	$0.434870\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	7.75195	1.13074	0.565369	−	0.824838i	$-0.308734\pi$
$47$	7.75195	1.13074	0.565369	+	0.824838i	$0.308734\pi$
$48$	0	0
$49$	−1.96839	−0.281199
$50$	0	0
$51$	6.69146	0.936992
$52$	0	0
$53$	13.4145	1.84263	0.921313	−	0.388821i	$-0.127117\pi$
$53$	13.4145	1.84263	0.921313	+	0.388821i	$0.127117\pi$
$54$	0	0
$55$	−1.86999	−0.252149
$56$	0	0
$57$	5.44833	0.721649
$58$	0	0
$59$	−12.5276	−1.63096	−0.815480	−	0.578786i	$-0.803527\pi$
$59$	−12.5276	−1.63096	−0.815480	+	0.578786i	$0.803527\pi$
$60$	0	0
$61$	−5.58739	−0.715392	−0.357696	−	0.933838i	$-0.616438\pi$
$61$	−5.58739	−0.715392	−0.357696	+	0.933838i	$0.616438\pi$
$62$	0	0
$63$	−2.24312	−0.282607
$64$	0	0
$65$	−4.82147	−0.598030
$66$	0	0
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	0	0
$69$	−5.48625	−0.660467
$70$	0	0
$71$	11.8770	1.40954	0.704772	−	0.709434i	$-0.251049\pi$
$71$	11.8770	1.40954	0.704772	+	0.709434i	$0.251049\pi$
$72$	0	0
$73$	9.24093	1.08157	0.540784	−	0.841161i	$-0.318128\pi$
$73$	9.24093	1.08157	0.540784	+	0.841161i	$0.318128\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−4.19461	−0.478020
$78$	0	0
$79$	−5.01197	−0.563891	−0.281946	−	0.959430i	$-0.590980\pi$
$79$	−5.01197	−0.563891	−0.281946	+	0.959430i	$0.590980\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.956416	0.104980	0.0524902	−	0.998621i	$-0.483284\pi$
$83$	0.956416	0.104980	0.0524902	+	0.998621i	$0.483284\pi$
$84$	0	0
$85$	6.69146	0.725791
$86$	0	0
$87$	−3.91357	−0.419579
$88$	0	0
$89$	−2.14692	−0.227573	−0.113786	−	0.993505i	$-0.536298\pi$
$89$	−2.14692	−0.227573	−0.113786	+	0.993505i	$0.536298\pi$
$90$	0	0
$91$	−10.8152	−1.13374
$92$	0	0
$93$	−4.93458	−0.511692
$94$	0	0
$95$	5.44833	0.558987
$96$	0	0
$97$	−2.38374	−0.242032	−0.121016	−	0.992651i	$-0.538615\pi$
$97$	−2.38374	−0.242032	−0.121016	+	0.992651i	$0.538615\pi$
$98$	0	0
$99$	1.86999	0.187941
$100$	0	0
$101$	−13.3472	−1.32810	−0.664048	−	0.747690i	$-0.731163\pi$
$101$	−13.3472	−1.32810	−0.664048	+	0.747690i	$0.731163\pi$
$102$	0	0
$103$	−2.66478	−0.262568	−0.131284	−	0.991345i	$-0.541910\pi$
$103$	−2.66478	−0.262568	−0.131284	+	0.991345i	$0.541910\pi$
$104$	0	0
$105$	−2.24312	−0.218907
$106$	0	0
$107$	12.4236	1.20103	0.600516	−	0.799613i	$-0.294962\pi$
$107$	12.4236	1.20103	0.600516	+	0.799613i	$0.294962\pi$
$108$	0	0
$109$	−15.8193	−1.51521	−0.757606	−	0.652712i	$-0.773631\pi$
$109$	−15.8193	−1.51521	−0.757606	+	0.652712i	$0.773631\pi$
$110$	0	0
$111$	6.15669	0.584368
$112$	0	0
$113$	−11.5051	−1.08231	−0.541153	−	0.840924i	$-0.682012\pi$
$113$	−11.5051	−1.08231	−0.541153	+	0.840924i	$0.682012\pi$
$114$	0	0
$115$	−5.48625	−0.511595
$116$	0	0
$117$	4.82147	0.445745
$118$	0	0
$119$	15.0098	1.37594
$120$	0	0
$121$	−7.50315	−0.682105
$122$	0	0
$123$	9.11878	0.822212
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.995070	0.0882982	0.0441491	−	0.999025i	$-0.485942\pi$
$127$	0.995070	0.0882982	0.0441491	+	0.999025i	$0.485942\pi$
$128$	0	0
$129$	−2.66478	−0.234621
$130$	0	0
$131$	−8.88827	−0.776571	−0.388286	−	0.921539i	$-0.626933\pi$
$131$	−8.88827	−0.776571	−0.388286	+	0.921539i	$0.626933\pi$
$132$	0	0
$133$	12.2213	1.05972
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−22.0981	−1.88797	−0.943985	−	0.329989i	$-0.892955\pi$
$137$	−22.0981	−1.88797	−0.943985	+	0.329989i	$0.892955\pi$
$138$	0	0
$139$	2.65354	0.225070	0.112535	−	0.993648i	$-0.464103\pi$
$139$	2.65354	0.225070	0.112535	+	0.993648i	$0.464103\pi$
$140$	0	0
$141$	−7.75195	−0.652832
$142$	0	0
$143$	9.01608	0.753963
$144$	0	0
$145$	−3.91357	−0.325004
$146$	0	0
$147$	1.96839	0.162350
$148$	0	0
$149$	7.85875	0.643814	0.321907	−	0.946771i	$-0.395676\pi$
$149$	7.85875	0.643814	0.321907	+	0.946771i	$0.395676\pi$
$150$	0	0
$151$	13.1327	1.06872	0.534360	−	0.845257i	$-0.320552\pi$
$151$	13.1327	1.06872	0.534360	+	0.845257i	$0.320552\pi$
$152$	0	0
$153$	−6.69146	−0.540972
$154$	0	0
$155$	−4.93458	−0.396355
$156$	0	0
$157$	−18.8102	−1.50122	−0.750610	−	0.660745i	$-0.770240\pi$
$157$	−18.8102	−1.50122	−0.750610	+	0.660745i	$0.770240\pi$
$158$	0	0
$159$	−13.4145	−1.06384
$160$	0	0
$161$	−12.3063	−0.969876
$162$	0	0
$163$	−19.6949	−1.54263	−0.771313	−	0.636456i	$-0.780400\pi$
$163$	−19.6949	−1.54263	−0.771313	+	0.636456i	$0.780400\pi$
$164$	0	0
$165$	1.86999	0.145578
$166$	0	0
$167$	−6.92554	−0.535915	−0.267957	−	0.963431i	$-0.586349\pi$
$167$	−6.92554	−0.535915	−0.267957	+	0.963431i	$0.586349\pi$
$168$	0	0
$169$	10.2466	0.788200
$170$	0	0
$171$	−5.44833	−0.416644
$172$	0	0
$173$	−0.430692	−0.0327449	−0.0163725	−	0.999866i	$-0.505212\pi$
$173$	−0.430692	−0.0327449	−0.0163725	+	0.999866i	$0.505212\pi$
$174$	0	0
$175$	−2.24312	−0.169564
$176$	0	0
$177$	12.5276	0.941635
$178$	0	0
$179$	−5.26350	−0.393412	−0.196706	−	0.980462i	$-0.563024\pi$
$179$	−5.26350	−0.393412	−0.196706	+	0.980462i	$0.563024\pi$
$180$	0	0
$181$	−4.39160	−0.326425	−0.163213	−	0.986591i	$-0.552186\pi$
$181$	−4.39160	−0.326425	−0.163213	+	0.986591i	$0.552186\pi$
$182$	0	0
$183$	5.58739	0.413032
$184$	0	0
$185$	6.15669	0.452649
$186$	0	0
$187$	−12.5129	−0.915036
$188$	0	0
$189$	2.24312	0.163163
$190$	0	0
$191$	−13.7091	−0.991956	−0.495978	−	0.868335i	$-0.665190\pi$
$191$	−13.7091	−0.991956	−0.495978	+	0.868335i	$0.665190\pi$
$192$	0	0
$193$	7.76665	0.559056	0.279528	−	0.960138i	$-0.409822\pi$
$193$	7.76665	0.559056	0.279528	+	0.960138i	$0.409822\pi$
$194$	0	0
$195$	4.82147	0.345273
$196$	0	0
$197$	2.86368	0.204029	0.102014	−	0.994783i	$-0.467471\pi$
$197$	2.86368	0.204029	0.102014	+	0.994783i	$0.467471\pi$
$198$	0	0
$199$	−13.2368	−0.938333	−0.469167	−	0.883110i	$-0.655446\pi$
$199$	−13.2368	−0.938333	−0.469167	+	0.883110i	$0.655446\pi$
$200$	0	0
$201$	−1.00000	−0.0705346
$202$	0	0
$203$	−8.77863	−0.616139
$204$	0	0
$205$	9.11878	0.636883
$206$	0	0
$207$	5.48625	0.381321
$208$	0	0
$209$	−10.1883	−0.704740
$210$	0	0
$211$	−17.6351	−1.21405	−0.607024	−	0.794683i	$-0.707637\pi$
$211$	−17.6351	−1.21405	−0.607024	+	0.794683i	$0.707637\pi$
$212$	0	0
$213$	−11.8770	−0.813801
$214$	0	0
$215$	−2.66478	−0.181736
$216$	0	0
$217$	−11.0689	−0.751405
$218$	0	0
$219$	−9.24093	−0.624444
$220$	0	0
$221$	−32.2627	−2.17022
$222$	0	0
$223$	−28.6407	−1.91793	−0.958963	−	0.283531i	$-0.908494\pi$
$223$	−28.6407	−1.91793	−0.958963	+	0.283531i	$0.908494\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	8.65354	0.574356	0.287178	−	0.957877i	$-0.407283\pi$
$227$	8.65354	0.574356	0.287178	+	0.957877i	$0.407283\pi$
$228$	0	0
$229$	−21.3224	−1.40903	−0.704513	−	0.709691i	$-0.748834\pi$
$229$	−21.3224	−1.40903	−0.704513	+	0.709691i	$0.748834\pi$
$230$	0	0
$231$	4.19461	0.275985
$232$	0	0
$233$	−19.4463	−1.27397	−0.636986	−	0.770875i	$-0.719819\pi$
$233$	−19.4463	−1.27397	−0.636986	+	0.770875i	$0.719819\pi$
$234$	0	0
$235$	−7.75195	−0.505681
$236$	0	0
$237$	5.01197	0.325563
$238$	0	0
$239$	15.3694	0.994163	0.497081	−	0.867704i	$-0.334405\pi$
$239$	15.3694	0.994163	0.497081	+	0.867704i	$0.334405\pi$
$240$	0	0
$241$	−0.718221	−0.0462647	−0.0231323	−	0.999732i	$-0.507364\pi$
$241$	−0.718221	−0.0462647	−0.0231323	+	0.999732i	$0.507364\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	1.96839	0.125756
$246$	0	0
$247$	−26.2690	−1.67146
$248$	0	0
$249$	−0.956416	−0.0606104
$250$	0	0
$251$	4.56027	0.287842	0.143921	−	0.989589i	$-0.454029\pi$
$251$	4.56027	0.287842	0.143921	+	0.989589i	$0.454029\pi$
$252$	0	0
$253$	10.2592	0.644991
$254$	0	0
$255$	−6.69146	−0.419035
$256$	0	0
$257$	2.12289	0.132422	0.0662110	−	0.997806i	$-0.478909\pi$
$257$	2.12289	0.132422	0.0662110	+	0.997806i	$0.478909\pi$
$258$	0	0
$259$	13.8102	0.858126
$260$	0	0
$261$	3.91357	0.242244
$262$	0	0
$263$	14.7617	0.910247	0.455123	−	0.890428i	$-0.349595\pi$
$263$	14.7617	0.910247	0.455123	+	0.890428i	$0.349595\pi$
$264$	0	0
$265$	−13.4145	−0.824048
$266$	0	0
$267$	2.14692	0.131389
$268$	0	0
$269$	−5.70205	−0.347660	−0.173830	−	0.984776i	$-0.555614\pi$
$269$	−5.70205	−0.347660	−0.173830	+	0.984776i	$0.555614\pi$
$270$	0	0
$271$	−13.2840	−0.806944	−0.403472	−	0.914992i	$-0.632197\pi$
$271$	−13.2840	−0.806944	−0.403472	+	0.914992i	$0.632197\pi$
$272$	0	0
$273$	10.8152	0.654564
$274$	0	0
$275$	1.86999	0.112764
$276$	0	0
$277$	27.4200	1.64751	0.823754	−	0.566947i	$-0.191875\pi$
$277$	27.4200	1.64751	0.823754	+	0.566947i	$0.191875\pi$
$278$	0	0
$279$	4.93458	0.295426
$280$	0	0
$281$	−27.1371	−1.61886	−0.809431	−	0.587216i	$-0.800224\pi$
$281$	−27.1371	−1.61886	−0.809431	+	0.587216i	$0.800224\pi$
$282$	0	0
$283$	−25.0841	−1.49110	−0.745549	−	0.666451i	$-0.767813\pi$
$283$	−25.0841	−1.49110	−0.745549	+	0.666451i	$0.767813\pi$
$284$	0	0
$285$	−5.44833	−0.322731
$286$	0	0
$287$	20.4546	1.20739
$288$	0	0
$289$	27.7756	1.63386
$290$	0	0
$291$	2.38374	0.139737
$292$	0	0
$293$	−4.02530	−0.235161	−0.117580	−	0.993063i	$-0.537514\pi$
$293$	−4.02530	−0.235161	−0.117580	+	0.993063i	$0.537514\pi$
$294$	0	0
$295$	12.5276	0.729387
$296$	0	0
$297$	−1.86999	−0.108508
$298$	0	0
$299$	26.4518	1.52975
$300$	0	0
$301$	−5.97743	−0.344533
$302$	0	0
$303$	13.3472	0.766776
$304$	0	0
$305$	5.58739	0.319933
$306$	0	0
$307$	22.5747	1.28841	0.644203	−	0.764855i	$-0.277189\pi$
$307$	22.5747	1.28841	0.644203	+	0.764855i	$0.277189\pi$
$308$	0	0
$309$	2.66478	0.151594
$310$	0	0
$311$	15.1962	0.861696	0.430848	−	0.902425i	$-0.358215\pi$
$311$	15.1962	0.861696	0.430848	+	0.902425i	$0.358215\pi$
$312$	0	0
$313$	−12.0850	−0.683083	−0.341541	−	0.939867i	$-0.610949\pi$
$313$	−12.0850	−0.683083	−0.341541	+	0.939867i	$0.610949\pi$
$314$	0	0
$315$	2.24312	0.126386
$316$	0	0
$317$	35.2451	1.97956	0.989782	−	0.142589i	$-0.0455428\pi$
$317$	35.2451	1.97956	0.989782	+	0.142589i	$0.0455428\pi$
$318$	0	0
$319$	7.31832	0.409747
$320$	0	0
$321$	−12.4236	−0.693416
$322$	0	0
$323$	36.4573	2.02854
$324$	0	0
$325$	4.82147	0.267447
$326$	0	0
$327$	15.8193	0.874808
$328$	0	0
$329$	−17.3886	−0.958664
$330$	0	0
$331$	3.22622	0.177329	0.0886646	−	0.996062i	$-0.471740\pi$
$331$	3.22622	0.177329	0.0886646	+	0.996062i	$0.471740\pi$
$332$	0	0
$333$	−6.15669	−0.337385
$334$	0	0
$335$	−1.00000	−0.0546358
$336$	0	0
$337$	16.5922	0.903837	0.451918	−	0.892059i	$-0.350740\pi$
$337$	16.5922	0.903837	0.451918	+	0.892059i	$0.350740\pi$
$338$	0	0
$339$	11.5051	0.624869
$340$	0	0
$341$	9.22760	0.499702
$342$	0	0
$343$	20.1172	1.08623
$344$	0	0
$345$	5.48625	0.295370
$346$	0	0
$347$	−19.9921	−1.07323	−0.536615	−	0.843827i	$-0.680297\pi$
$347$	−19.9921	−1.07323	−0.536615	+	0.843827i	$0.680297\pi$
$348$	0	0
$349$	11.2992	0.604833	0.302417	−	0.953176i	$-0.402207\pi$
$349$	11.2992	0.604833	0.302417	+	0.953176i	$0.402207\pi$
$350$	0	0
$351$	−4.82147	−0.257351
$352$	0	0
$353$	31.2766	1.66468	0.832342	−	0.554263i	$-0.187000\pi$
$353$	31.2766	1.66468	0.832342	+	0.554263i	$0.187000\pi$
$354$	0	0
$355$	−11.8770	−0.630367
$356$	0	0
$357$	−15.0098	−0.794402
$358$	0	0
$359$	−11.3507	−0.599065	−0.299533	−	0.954086i	$-0.596831\pi$
$359$	−11.3507	−0.599065	−0.299533	+	0.954086i	$0.596831\pi$
$360$	0	0
$361$	10.6843	0.562333
$362$	0	0
$363$	7.50315	0.393813
$364$	0	0
$365$	−9.24093	−0.483692
$366$	0	0
$367$	−4.80163	−0.250643	−0.125322	−	0.992116i	$-0.539996\pi$
$367$	−4.80163	−0.250643	−0.125322	+	0.992116i	$0.539996\pi$
$368$	0	0
$369$	−9.11878	−0.474705
$370$	0	0
$371$	−30.0905	−1.56222
$372$	0	0
$373$	−31.4074	−1.62621	−0.813107	−	0.582115i	$-0.802225\pi$
$373$	−31.4074	−1.62621	−0.813107	+	0.582115i	$0.802225\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	18.8692	0.971811
$378$	0	0
$379$	−7.54298	−0.387457	−0.193729	−	0.981055i	$-0.562058\pi$
$379$	−7.54298	−0.387457	−0.193729	+	0.981055i	$0.562058\pi$
$380$	0	0
$381$	−0.995070	−0.0509790
$382$	0	0
$383$	−0.553579	−0.0282866	−0.0141433	−	0.999900i	$-0.504502\pi$
$383$	−0.553579	−0.0282866	−0.0141433	+	0.999900i	$0.504502\pi$
$384$	0	0
$385$	4.19461	0.213777
$386$	0	0
$387$	2.66478	0.135458
$388$	0	0
$389$	−18.1790	−0.921711	−0.460855	−	0.887475i	$-0.652457\pi$
$389$	−18.1790	−0.921711	−0.460855	+	0.887475i	$0.652457\pi$
$390$	0	0
$391$	−36.7110	−1.85656
$392$	0	0
$393$	8.88827	0.448354
$394$	0	0
$395$	5.01197	0.252180
$396$	0	0
$397$	−25.9213	−1.30095	−0.650475	−	0.759527i	$-0.725430\pi$
$397$	−25.9213	−1.30095	−0.650475	+	0.759527i	$0.725430\pi$
$398$	0	0
$399$	−12.2213	−0.611830
$400$	0	0
$401$	9.37725	0.468277	0.234139	−	0.972203i	$-0.424773\pi$
$401$	9.37725	0.468277	0.234139	+	0.972203i	$0.424773\pi$
$402$	0	0
$403$	23.7920	1.18516
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−11.5129	−0.570675
$408$	0	0
$409$	−28.6837	−1.41832	−0.709159	−	0.705049i	$-0.750925\pi$
$409$	−28.6837	−1.41832	−0.709159	+	0.705049i	$0.750925\pi$
$410$	0	0
$411$	22.0981	1.09002
$412$	0	0
$413$	28.1011	1.38276
$414$	0	0
$415$	−0.956416	−0.0469486
$416$	0	0
$417$	−2.65354	−0.129944
$418$	0	0
$419$	17.5298	0.856388	0.428194	−	0.903687i	$-0.359150\pi$
$419$	17.5298	0.856388	0.428194	+	0.903687i	$0.359150\pi$
$420$	0	0
$421$	−34.3261	−1.67295	−0.836476	−	0.548004i	$-0.815388\pi$
$421$	−34.3261	−1.67295	−0.836476	+	0.548004i	$0.815388\pi$
$422$	0	0
$423$	7.75195	0.376912
$424$	0	0
$425$	−6.69146	−0.324583
$426$	0	0
$427$	12.5332	0.606524
$428$	0	0
$429$	−9.01608	−0.435301
$430$	0	0
$431$	−30.0620	−1.44803	−0.724017	−	0.689782i	$-0.757706\pi$
$431$	−30.0620	−1.44803	−0.724017	+	0.689782i	$0.757706\pi$
$432$	0	0
$433$	−16.1570	−0.776458	−0.388229	−	0.921563i	$-0.626913\pi$
$433$	−16.1570	−0.776458	−0.388229	+	0.921563i	$0.626913\pi$
$434$	0	0
$435$	3.91357	0.187641
$436$	0	0
$437$	−29.8909	−1.42988
$438$	0	0
$439$	−13.6779	−0.652812	−0.326406	−	0.945230i	$-0.605838\pi$
$439$	−13.6779	−0.652812	−0.326406	+	0.945230i	$0.605838\pi$
$440$	0	0
$441$	−1.96839	−0.0937329
$442$	0	0
$443$	40.9485	1.94552	0.972760	−	0.231814i	$-0.0744659\pi$
$443$	40.9485	1.94552	0.972760	+	0.231814i	$0.0744659\pi$
$444$	0	0
$445$	2.14692	0.101774
$446$	0	0
$447$	−7.85875	−0.371706
$448$	0	0
$449$	19.7174	0.930522	0.465261	−	0.885174i	$-0.345961\pi$
$449$	19.7174	0.930522	0.465261	+	0.885174i	$0.345961\pi$
$450$	0	0
$451$	−17.0520	−0.802946
$452$	0	0
$453$	−13.1327	−0.617026
$454$	0	0
$455$	10.8152	0.507023
$456$	0	0
$457$	−27.6286	−1.29241	−0.646206	−	0.763163i	$-0.723645\pi$
$457$	−27.6286	−1.29241	−0.646206	+	0.763163i	$0.723645\pi$
$458$	0	0
$459$	6.69146	0.312331
$460$	0	0
$461$	24.8915	1.15931	0.579655	−	0.814862i	$-0.303187\pi$
$461$	24.8915	1.15931	0.579655	+	0.814862i	$0.303187\pi$
$462$	0	0
$463$	6.95769	0.323351	0.161676	−	0.986844i	$-0.448310\pi$
$463$	6.95769	0.323351	0.161676	+	0.986844i	$0.448310\pi$
$464$	0	0
$465$	4.93458	0.228836
$466$	0	0
$467$	−3.51493	−0.162651	−0.0813257	−	0.996688i	$-0.525915\pi$
$467$	−3.51493	−0.162651	−0.0813257	+	0.996688i	$0.525915\pi$
$468$	0	0
$469$	−2.24312	−0.103578
$470$	0	0
$471$	18.8102	0.866730
$472$	0	0
$473$	4.98310	0.229123
$474$	0	0
$475$	−5.44833	−0.249987
$476$	0	0
$477$	13.4145	0.614209
$478$	0	0
$479$	1.27409	0.0582148	0.0291074	−	0.999576i	$-0.490734\pi$
$479$	1.27409	0.0582148	0.0291074	+	0.999576i	$0.490734\pi$
$480$	0	0
$481$	−29.6843	−1.35349
$482$	0	0
$483$	12.3063	0.559958
$484$	0	0
$485$	2.38374	0.108240
$486$	0	0
$487$	−12.4427	−0.563831	−0.281916	−	0.959439i	$-0.590970\pi$
$487$	−12.4427	−0.563831	−0.281916	+	0.959439i	$0.590970\pi$
$488$	0	0
$489$	19.6949	0.890635
$490$	0	0
$491$	25.7758	1.16324	0.581622	−	0.813459i	$-0.302418\pi$
$491$	25.7758	1.16324	0.581622	+	0.813459i	$0.302418\pi$
$492$	0	0
$493$	−26.1875	−1.17942
$494$	0	0
$495$	−1.86999	−0.0840496
$496$	0	0
$497$	−26.6417	−1.19504
$498$	0	0
$499$	20.4614	0.915978	0.457989	−	0.888958i	$-0.348570\pi$
$499$	20.4614	0.915978	0.457989	+	0.888958i	$0.348570\pi$
$500$	0	0
$501$	6.92554	0.309410
$502$	0	0
$503$	−25.6772	−1.14489	−0.572445	−	0.819943i	$-0.694005\pi$
$503$	−25.6772	−1.14489	−0.572445	+	0.819943i	$0.694005\pi$
$504$	0	0
$505$	13.3472	0.593942
$506$	0	0
$507$	−10.2466	−0.455067
$508$	0	0
$509$	42.6786	1.89169	0.945847	−	0.324612i	$-0.105234\pi$
$509$	42.6786	1.89169	0.945847	+	0.324612i	$0.105234\pi$
$510$	0	0
$511$	−20.7286	−0.916977
$512$	0	0
$513$	5.44833	0.240550
$514$	0	0
$515$	2.66478	0.117424
$516$	0	0
$517$	14.4960	0.637535
$518$	0	0
$519$	0.430692	0.0189053
$520$	0	0
$521$	30.8917	1.35339	0.676694	−	0.736264i	$-0.263412\pi$
$521$	30.8917	1.35339	0.676694	+	0.736264i	$0.263412\pi$
$522$	0	0
$523$	2.55596	0.111764	0.0558821	−	0.998437i	$-0.482203\pi$
$523$	2.55596	0.111764	0.0558821	+	0.998437i	$0.482203\pi$
$524$	0	0
$525$	2.24312	0.0978980
$526$	0	0
$527$	−33.0196	−1.43835
$528$	0	0
$529$	7.09894	0.308650
$530$	0	0
$531$	−12.5276	−0.543653
$532$	0	0
$533$	−43.9659	−1.90438
$534$	0	0
$535$	−12.4236	−0.537118
$536$	0	0
$537$	5.26350	0.227137
$538$	0	0
$539$	−3.68086	−0.158546
$540$	0	0
$541$	−14.5548	−0.625758	−0.312879	−	0.949793i	$-0.601293\pi$
$541$	−14.5548	−0.625758	−0.312879	+	0.949793i	$0.601293\pi$
$542$	0	0
$543$	4.39160	0.188462
$544$	0	0
$545$	15.8193	0.677623
$546$	0	0
$547$	24.2141	1.03532	0.517659	−	0.855587i	$-0.326803\pi$
$547$	24.2141	1.03532	0.517659	+	0.855587i	$0.326803\pi$
$548$	0	0
$549$	−5.58739	−0.238464
$550$	0	0
$551$	−21.3224	−0.908366
$552$	0	0
$553$	11.2425	0.478079
$554$	0	0
$555$	−6.15669	−0.261337
$556$	0	0
$557$	−21.4607	−0.909317	−0.454659	−	0.890666i	$-0.650239\pi$
$557$	−21.4607	−0.909317	−0.454659	+	0.890666i	$0.650239\pi$
$558$	0	0
$559$	12.8482	0.543419
$560$	0	0
$561$	12.5129	0.528296
$562$	0	0
$563$	34.7349	1.46390	0.731950	−	0.681358i	$-0.238610\pi$
$563$	34.7349	1.46390	0.731950	+	0.681358i	$0.238610\pi$
$564$	0	0
$565$	11.5051	0.484022
$566$	0	0
$567$	−2.24312	−0.0942024
$568$	0	0
$569$	−1.11520	−0.0467517	−0.0233758	−	0.999727i	$-0.507441\pi$
$569$	−1.11520	−0.0467517	−0.0233758	+	0.999727i	$0.507441\pi$
$570$	0	0
$571$	−18.0582	−0.755712	−0.377856	−	0.925864i	$-0.623339\pi$
$571$	−18.0582	−0.755712	−0.377856	+	0.925864i	$0.623339\pi$
$572$	0	0
$573$	13.7091	0.572706
$574$	0	0
$575$	5.48625	0.228792
$576$	0	0
$577$	−16.2886	−0.678104	−0.339052	−	0.940768i	$-0.610106\pi$
$577$	−16.2886	−0.678104	−0.339052	+	0.940768i	$0.610106\pi$
$578$	0	0
$579$	−7.76665	−0.322771
$580$	0	0
$581$	−2.14536	−0.0890046
$582$	0	0
$583$	25.0850	1.03891
$584$	0	0
$585$	−4.82147	−0.199343
$586$	0	0
$587$	−9.51293	−0.392641	−0.196320	−	0.980540i	$-0.562899\pi$
$587$	−9.51293	−0.392641	−0.196320	+	0.980540i	$0.562899\pi$
$588$	0	0
$589$	−26.8852	−1.10779
$590$	0	0
$591$	−2.86368	−0.117796
$592$	0	0
$593$	−26.8704	−1.10344	−0.551718	−	0.834031i	$-0.686028\pi$
$593$	−26.8704	−1.10344	−0.551718	+	0.834031i	$0.686028\pi$
$594$	0	0
$595$	−15.0098	−0.615341
$596$	0	0
$597$	13.2368	0.541747
$598$	0	0
$599$	−12.7200	−0.519727	−0.259864	−	0.965645i	$-0.583678\pi$
$599$	−12.7200	−0.519727	−0.259864	+	0.965645i	$0.583678\pi$
$600$	0	0
$601$	−40.8202	−1.66509	−0.832546	−	0.553956i	$-0.813117\pi$
$601$	−40.8202	−1.66509	−0.832546	+	0.553956i	$0.813117\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	0	0
$605$	7.50315	0.305047
$606$	0	0
$607$	−35.2979	−1.43270	−0.716348	−	0.697743i	$-0.754188\pi$
$607$	−35.2979	−1.43270	−0.716348	+	0.697743i	$0.754188\pi$
$608$	0	0
$609$	8.77863	0.355728
$610$	0	0
$611$	37.3758	1.51206
$612$	0	0
$613$	−1.78302	−0.0720155	−0.0360078	−	0.999352i	$-0.511464\pi$
$613$	−1.78302	−0.0720155	−0.0360078	+	0.999352i	$0.511464\pi$
$614$	0	0
$615$	−9.11878	−0.367705
$616$	0	0
$617$	−10.3464	−0.416529	−0.208265	−	0.978072i	$-0.566782\pi$
$617$	−10.3464	−0.416529	−0.208265	+	0.978072i	$0.566782\pi$
$618$	0	0
$619$	15.8066	0.635319	0.317660	−	0.948205i	$-0.397103\pi$
$619$	15.8066	0.635319	0.317660	+	0.948205i	$0.397103\pi$
$620$	0	0
$621$	−5.48625	−0.220156
$622$	0	0
$623$	4.81581	0.192941
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.1883	0.406882
$628$	0	0
$629$	41.1973	1.64264
$630$	0	0
$631$	43.0680	1.71451	0.857255	−	0.514893i	$-0.172168\pi$
$631$	43.0680	1.71451	0.857255	+	0.514893i	$0.172168\pi$
$632$	0	0
$633$	17.6351	0.700932
$634$	0	0
$635$	−0.995070	−0.0394882
$636$	0	0
$637$	−9.49054	−0.376029
$638$	0	0
$639$	11.8770	0.469848
$640$	0	0
$641$	−3.47510	−0.137258	−0.0686290	−	0.997642i	$-0.521862\pi$
$641$	−3.47510	−0.137258	−0.0686290	+	0.997642i	$0.521862\pi$
$642$	0	0
$643$	−6.36874	−0.251159	−0.125579	−	0.992084i	$-0.540079\pi$
$643$	−6.36874	−0.251159	−0.125579	+	0.992084i	$0.540079\pi$
$644$	0	0
$645$	2.66478	0.104926
$646$	0	0
$647$	30.0038	1.17957	0.589786	−	0.807560i	$-0.299212\pi$
$647$	30.0038	1.17957	0.589786	+	0.807560i	$0.299212\pi$
$648$	0	0
$649$	−23.4265	−0.919571
$650$	0	0
$651$	11.0689	0.433824
$652$	0	0
$653$	3.66953	0.143600	0.0717999	−	0.997419i	$-0.477126\pi$
$653$	3.66953	0.143600	0.0717999	+	0.997419i	$0.477126\pi$
$654$	0	0
$655$	8.88827	0.347293
$656$	0	0
$657$	9.24093	0.360523
$658$	0	0
$659$	−19.4832	−0.758959	−0.379479	−	0.925200i	$-0.623897\pi$
$659$	−19.4832	−0.758959	−0.379479	+	0.925200i	$0.623897\pi$
$660$	0	0
$661$	43.7764	1.70271	0.851353	−	0.524593i	$-0.175783\pi$
$661$	43.7764	1.70271	0.851353	+	0.524593i	$0.175783\pi$
$662$	0	0
$663$	32.2627	1.25298
$664$	0	0
$665$	−12.2213	−0.473921
$666$	0	0
$667$	21.4708	0.831353
$668$	0	0
$669$	28.6407	1.10732
$670$	0	0
$671$	−10.4483	−0.403353
$672$	0	0
$673$	48.6374	1.87483	0.937416	−	0.348210i	$-0.113210\pi$
$673$	48.6374	1.87483	0.937416	+	0.348210i	$0.113210\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−47.9022	−1.84103	−0.920515	−	0.390708i	$-0.872230\pi$
$677$	−47.9022	−1.84103	−0.920515	+	0.390708i	$0.872230\pi$
$678$	0	0
$679$	5.34702	0.205200
$680$	0	0
$681$	−8.65354	−0.331604
$682$	0	0
$683$	−15.0785	−0.576962	−0.288481	−	0.957486i	$-0.593150\pi$
$683$	−15.0785	−0.576962	−0.288481	+	0.957486i	$0.593150\pi$
$684$	0	0
$685$	22.0981	0.844326
$686$	0	0
$687$	21.3224	0.813501
$688$	0	0
$689$	64.6778	2.46403
$690$	0	0
$691$	10.5255	0.400411	0.200205	−	0.979754i	$-0.435839\pi$
$691$	10.5255	0.400411	0.200205	+	0.979754i	$0.435839\pi$
$692$	0	0
$693$	−4.19461	−0.159340
$694$	0	0
$695$	−2.65354	−0.100655
$696$	0	0
$697$	61.0179	2.31122
$698$	0	0
$699$	19.4463	0.735528
$700$	0	0
$701$	27.5710	1.04134	0.520672	−	0.853757i	$-0.325681\pi$
$701$	27.5710	1.04134	0.520672	+	0.853757i	$0.325681\pi$
$702$	0	0
$703$	33.5437	1.26513
$704$	0	0
$705$	7.75195	0.291955
$706$	0	0
$707$	29.9394	1.12599
$708$	0	0
$709$	−40.4313	−1.51843	−0.759216	−	0.650839i	$-0.774417\pi$
$709$	−40.4313	−1.51843	−0.759216	+	0.650839i	$0.774417\pi$
$710$	0	0
$711$	−5.01197	−0.187964
$712$	0	0
$713$	27.0724	1.01387
$714$	0	0
$715$	−9.01608	−0.337182
$716$	0	0
$717$	−15.3694	−0.573980
$718$	0	0
$719$	18.8194	0.701844	0.350922	−	0.936405i	$-0.385868\pi$
$719$	18.8194	0.701844	0.350922	+	0.936405i	$0.385868\pi$
$720$	0	0
$721$	5.97743	0.222611
$722$	0	0
$723$	0.718221	0.0267109
$724$	0	0
$725$	3.91357	0.145346
$726$	0	0
$727$	17.6814	0.655767	0.327883	−	0.944718i	$-0.393665\pi$
$727$	17.6814	0.655767	0.327883	+	0.944718i	$0.393665\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−17.8312	−0.659513
$732$	0	0
$733$	11.0554	0.408339	0.204169	−	0.978936i	$-0.434551\pi$
$733$	11.0554	0.408339	0.204169	+	0.978936i	$0.434551\pi$
$734$	0	0
$735$	−1.96839	−0.0726052
$736$	0	0
$737$	1.86999	0.0688818
$738$	0	0
$739$	−49.2681	−1.81235	−0.906177	−	0.422898i	$-0.861013\pi$
$739$	−49.2681	−1.81235	−0.906177	+	0.422898i	$0.861013\pi$
$740$	0	0
$741$	26.2690	0.965015
$742$	0	0
$743$	−11.2430	−0.412467	−0.206234	−	0.978503i	$-0.566121\pi$
$743$	−11.2430	−0.412467	−0.206234	+	0.978503i	$0.566121\pi$
$744$	0	0
$745$	−7.85875	−0.287922
$746$	0	0
$747$	0.956416	0.0349934
$748$	0	0
$749$	−27.8676	−1.01826
$750$	0	0
$751$	31.5515	1.15133	0.575665	−	0.817685i	$-0.304743\pi$
$751$	31.5515	1.15133	0.575665	+	0.817685i	$0.304743\pi$
$752$	0	0
$753$	−4.56027	−0.166185
$754$	0	0
$755$	−13.1327	−0.477947
$756$	0	0
$757$	14.4556	0.525397	0.262698	−	0.964878i	$-0.415388\pi$
$757$	14.4556	0.525397	0.262698	+	0.964878i	$0.415388\pi$
$758$	0	0
$759$	−10.2592	−0.372386
$760$	0	0
$761$	−14.6316	−0.530395	−0.265198	−	0.964194i	$-0.585437\pi$
$761$	−14.6316	−0.530395	−0.265198	+	0.964194i	$0.585437\pi$
$762$	0	0
$763$	35.4846	1.28463
$764$	0	0
$765$	6.69146	0.241930
$766$	0	0
$767$	−60.4016	−2.18098
$768$	0	0
$769$	25.9277	0.934975	0.467488	−	0.884000i	$-0.345159\pi$
$769$	25.9277	0.934975	0.467488	+	0.884000i	$0.345159\pi$
$770$	0	0
$771$	−2.12289	−0.0764539
$772$	0	0
$773$	2.87821	0.103522	0.0517609	−	0.998660i	$-0.483517\pi$
$773$	2.87821	0.103522	0.0517609	+	0.998660i	$0.483517\pi$
$774$	0	0
$775$	4.93458	0.177255
$776$	0	0
$777$	−13.8102	−0.495439
$778$	0	0
$779$	49.6821	1.78005
$780$	0	0
$781$	22.2099	0.794732
$782$	0	0
$783$	−3.91357	−0.139860
$784$	0	0
$785$	18.8102	0.671366
$786$	0	0
$787$	10.8613	0.387164	0.193582	−	0.981084i	$-0.437990\pi$
$787$	10.8613	0.387164	0.193582	+	0.981084i	$0.437990\pi$
$788$	0	0
$789$	−14.7617	−0.525531
$790$	0	0
$791$	25.8073	0.917602
$792$	0	0
$793$	−26.9394	−0.956647
$794$	0	0
$795$	13.4145	0.475764
$796$	0	0
$797$	−45.1469	−1.59919	−0.799593	−	0.600542i	$-0.794951\pi$
$797$	−45.1469	−1.59919	−0.799593	+	0.600542i	$0.794951\pi$
$798$	0	0
$799$	−51.8718	−1.83509
$800$	0	0
$801$	−2.14692	−0.0758576
$802$	0	0
$803$	17.2804	0.609812
$804$	0	0
$805$	12.3063	0.433742
$806$	0	0
$807$	5.70205	0.200722
$808$	0	0
$809$	−14.0337	−0.493399	−0.246700	−	0.969092i	$-0.579346\pi$
$809$	−14.0337	−0.493399	−0.246700	+	0.969092i	$0.579346\pi$
$810$	0	0
$811$	47.9240	1.68284	0.841420	−	0.540382i	$-0.181720\pi$
$811$	47.9240	1.68284	0.841420	+	0.540382i	$0.181720\pi$
$812$	0	0
$813$	13.2840	0.465889
$814$	0	0
$815$	19.6949	0.689883
$816$	0	0
$817$	−14.5186	−0.507941
$818$	0	0
$819$	−10.8152	−0.377912
$820$	0	0
$821$	−23.8534	−0.832491	−0.416245	−	0.909252i	$-0.636654\pi$
$821$	−23.8534	−0.832491	−0.416245	+	0.909252i	$0.636654\pi$
$822$	0	0
$823$	23.3817	0.815036	0.407518	−	0.913197i	$-0.366394\pi$
$823$	23.3817	0.815036	0.407518	+	0.913197i	$0.366394\pi$
$824$	0	0
$825$	−1.86999	−0.0651045
$826$	0	0
$827$	48.4243	1.68388	0.841939	−	0.539573i	$-0.181414\pi$
$827$	48.4243	1.68388	0.841939	+	0.539573i	$0.181414\pi$
$828$	0	0
$829$	−28.5883	−0.992914	−0.496457	−	0.868061i	$-0.665366\pi$
$829$	−28.5883	−0.992914	−0.496457	+	0.868061i	$0.665366\pi$
$830$	0	0
$831$	−27.4200	−0.951190
$832$	0	0
$833$	13.1714	0.456362
$834$	0	0
$835$	6.92554	0.239668
$836$	0	0
$837$	−4.93458	−0.170564
$838$	0	0
$839$	18.8953	0.652338	0.326169	−	0.945312i	$-0.394242\pi$
$839$	18.8953	0.652338	0.326169	+	0.945312i	$0.394242\pi$
$840$	0	0
$841$	−13.6840	−0.471861
$842$	0	0
$843$	27.1371	0.934650
$844$	0	0
$845$	−10.2466	−0.352494
$846$	0	0
$847$	16.8305	0.578303
$848$	0	0
$849$	25.0841	0.860886
$850$	0	0
$851$	−33.7772	−1.15787
$852$	0	0
$853$	23.5278	0.805578	0.402789	−	0.915293i	$-0.368041\pi$
$853$	23.5278	0.805578	0.402789	+	0.915293i	$0.368041\pi$
$854$	0	0
$855$	5.44833	0.186329
$856$	0	0
$857$	−11.2009	−0.382616	−0.191308	−	0.981530i	$-0.561273\pi$
$857$	−11.2009	−0.382616	−0.191308	+	0.981530i	$0.561273\pi$
$858$	0	0
$859$	−5.26945	−0.179791	−0.0898957	−	0.995951i	$-0.528653\pi$
$859$	−5.26945	−0.179791	−0.0898957	+	0.995951i	$0.528653\pi$
$860$	0	0
$861$	−20.4546	−0.697089
$862$	0	0
$863$	−31.5506	−1.07399	−0.536997	−	0.843584i	$-0.680441\pi$
$863$	−31.5506	−1.07399	−0.536997	+	0.843584i	$0.680441\pi$
$864$	0	0
$865$	0.430692	0.0146440
$866$	0	0
$867$	−27.7756	−0.943309
$868$	0	0
$869$	−9.37232	−0.317934
$870$	0	0
$871$	4.82147	0.163369
$872$	0	0
$873$	−2.38374	−0.0806772
$874$	0	0
$875$	2.24312	0.0758315
$876$	0	0
$877$	7.73533	0.261203	0.130602	−	0.991435i	$-0.458309\pi$
$877$	7.73533	0.261203	0.130602	+	0.991435i	$0.458309\pi$
$878$	0	0
$879$	4.02530	0.135770
$880$	0	0
$881$	−32.9267	−1.10933	−0.554665	−	0.832074i	$-0.687153\pi$
$881$	−32.9267	−1.10933	−0.554665	+	0.832074i	$0.687153\pi$
$882$	0	0
$883$	32.5202	1.09439	0.547196	−	0.837005i	$-0.315695\pi$
$883$	32.5202	1.09439	0.547196	+	0.837005i	$0.315695\pi$
$884$	0	0
$885$	−12.5276	−0.421112
$886$	0	0
$887$	25.9593	0.871627	0.435813	−	0.900037i	$-0.356461\pi$
$887$	25.9593	0.871627	0.435813	+	0.900037i	$0.356461\pi$
$888$	0	0
$889$	−2.23207	−0.0748611
$890$	0	0
$891$	1.86999	0.0626469
$892$	0	0
$893$	−42.2352	−1.41335
$894$	0	0
$895$	5.26350	0.175939
$896$	0	0
$897$	−26.4518	−0.883200
$898$	0	0
$899$	19.3118	0.644086
$900$	0	0
$901$	−89.7627	−2.99043
$902$	0	0
$903$	5.97743	0.198916
$904$	0	0
$905$	4.39160	0.145982
$906$	0	0
$907$	−7.71203	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$907$	−7.71203	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$908$	0	0
$909$	−13.3472	−0.442699
$910$	0	0
$911$	−38.0215	−1.25971	−0.629854	−	0.776714i	$-0.716885\pi$
$911$	−38.0215	−1.25971	−0.629854	+	0.776714i	$0.716885\pi$
$912$	0	0
$913$	1.78848	0.0591902
$914$	0	0
$915$	−5.58739	−0.184713
$916$	0	0
$917$	19.9375	0.658394
$918$	0	0
$919$	−32.5946	−1.07520	−0.537599	−	0.843201i	$-0.680669\pi$
$919$	−32.5946	−1.07520	−0.537599	+	0.843201i	$0.680669\pi$
$920$	0	0
$921$	−22.5747	−0.743861
$922$	0	0
$923$	57.2648	1.88489
$924$	0	0
$925$	−6.15669	−0.202431
$926$	0	0
$927$	−2.66478	−0.0875228
$928$	0	0
$929$	48.9258	1.60520	0.802602	−	0.596515i	$-0.203449\pi$
$929$	48.9258	1.60520	0.802602	+	0.596515i	$0.203449\pi$
$930$	0	0
$931$	10.7244	0.351479
$932$	0	0
$933$	−15.1962	−0.497500
$934$	0	0
$935$	12.5129	0.409217
$936$	0	0
$937$	19.9189	0.650721	0.325360	−	0.945590i	$-0.394514\pi$
$937$	19.9189	0.650721	0.325360	+	0.945590i	$0.394514\pi$
$938$	0	0
$939$	12.0850	0.394378
$940$	0	0
$941$	59.2687	1.93210	0.966052	−	0.258347i	$-0.0831778\pi$
$941$	59.2687	1.93210	0.966052	+	0.258347i	$0.0831778\pi$
$942$	0	0
$943$	−50.0279	−1.62913
$944$	0	0
$945$	−2.24312	−0.0729689
$946$	0	0
$947$	35.7863	1.16290	0.581449	−	0.813583i	$-0.302486\pi$
$947$	35.7863	1.16290	0.581449	+	0.813583i	$0.302486\pi$
$948$	0	0
$949$	44.5549	1.44631
$950$	0	0
$951$	−35.2451	−1.14290
$952$	0	0
$953$	−4.00593	−0.129765	−0.0648823	−	0.997893i	$-0.520667\pi$
$953$	−4.00593	−0.129765	−0.0648823	+	0.997893i	$0.520667\pi$
$954$	0	0
$955$	13.7091	0.443616
$956$	0	0
$957$	−7.31832	−0.236568
$958$	0	0
$959$	49.5688	1.60066
$960$	0	0
$961$	−6.64989	−0.214513
$962$	0	0
$963$	12.4236	0.400344
$964$	0	0
$965$	−7.76665	−0.250017
$966$	0	0
$967$	39.8831	1.28256	0.641278	−	0.767309i	$-0.278405\pi$
$967$	39.8831	1.28256	0.641278	+	0.767309i	$0.278405\pi$
$968$	0	0
$969$	−36.4573	−1.17118
$970$	0	0
$971$	12.7307	0.408549	0.204274	−	0.978914i	$-0.434517\pi$
$971$	12.7307	0.408549	0.204274	+	0.978914i	$0.434517\pi$
$972$	0	0
$973$	−5.95222	−0.190820
$974$	0	0
$975$	−4.82147	−0.154411
$976$	0	0
$977$	8.93203	0.285761	0.142880	−	0.989740i	$-0.454364\pi$
$977$	8.93203	0.285761	0.142880	+	0.989740i	$0.454364\pi$
$978$	0	0
$979$	−4.01471	−0.128311
$980$	0	0
$981$	−15.8193	−0.505071
$982$	0	0
$983$	8.51502	0.271587	0.135793	−	0.990737i	$-0.456642\pi$
$983$	8.51502	0.271587	0.135793	+	0.990737i	$0.456642\pi$
$984$	0	0
$985$	−2.86368	−0.0912444
$986$	0	0
$987$	17.3886	0.553485
$988$	0	0
$989$	14.6196	0.464877
$990$	0	0
$991$	−26.1181	−0.829668	−0.414834	−	0.909897i	$-0.636160\pi$
$991$	−26.1181	−0.829668	−0.414834	+	0.909897i	$0.636160\pi$
$992$	0	0
$993$	−3.22622	−0.102381
$994$	0	0
$995$	13.2368	0.419635
$996$	0	0
$997$	22.5736	0.714913	0.357456	−	0.933930i	$-0.383644\pi$
$997$	22.5736	0.714913	0.357456	+	0.933930i	$0.383644\pi$
$998$	0	0
$999$	6.15669	0.194789

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.a.b.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.a.b.1.2	✓	4	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.24312 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} -2.24312 q^{7} +1.00000 q^{9} +1.86999 q^{11} +4.82147 q^{13} +1.00000 q^{15} -6.69146 q^{17} -5.44833 q^{19} +2.24312 q^{21} +5.48625 q^{23} +1.00000 q^{25} -1.00000 q^{27} +3.91357 q^{29} +4.93458 q^{31} -1.86999 q^{33} +2.24312 q^{35} -6.15669 q^{37} -4.82147 q^{39} -9.11878 q^{41} +2.66478 q^{43} -1.00000 q^{45} +7.75195 q^{47} -1.96839 q^{49} +6.69146 q^{51} +13.4145 q^{53} -1.86999 q^{55} +5.44833 q^{57} -12.5276 q^{59} -5.58739 q^{61} -2.24312 q^{63} -4.82147 q^{65} +1.00000 q^{67} -5.48625 q^{69} +11.8770 q^{71} +9.24093 q^{73} -1.00000 q^{75} -4.19461 q^{77} -5.01197 q^{79} +1.00000 q^{81} +0.956416 q^{83} +6.69146 q^{85} -3.91357 q^{87} -2.14692 q^{89} -10.8152 q^{91} -4.93458 q^{93} +5.44833 q^{95} -2.38374 q^{97} +1.86999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^5 - q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{5} - q^{7} + 4 q^{9} + 5 q^{11} - q^{13} + 4 q^{15} - 4 q^{17} - 7 q^{19} + q^{21} + 6 q^{23} + 4 q^{25} - 4 q^{27} - q^{29} - 11 q^{31} - 5 q^{33} + q^{35} + q^{39} - 13 q^{41} + 15 q^{43} - 4 q^{45} + 7 q^{47} - 3 q^{49} + 4 q^{51} + 13 q^{53} - 5 q^{55} + 7 q^{57} + q^{59} - 15 q^{61} - q^{63} + q^{65} + 4 q^{67} - 6 q^{69} - 6 q^{71} + 8 q^{73} - 4 q^{75} + 9 q^{77} - q^{79} + 4 q^{81} + 18 q^{83} + 4 q^{85} + q^{87} - 24 q^{89} - 29 q^{91} + 11 q^{93} + 7 q^{95} - 23 q^{97} + 5 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 - q^7 + 4 * q^9 + 5 * q^11 - q^13 + 4 * q^15 - 4 * q^17 - 7 * q^19 + q^21 + 6 * q^23 + 4 * q^25 - 4 * q^27 - q^29 - 11 * q^31 - 5 * q^33 + q^35 + q^39 - 13 * q^41 + 15 * q^43 - 4 * q^45 + 7 * q^47 - 3 * q^49 + 4 * q^51 + 13 * q^53 - 5 * q^55 + 7 * q^57 + q^59 - 15 * q^61 - q^63 + q^65 + 4 * q^67 - 6 * q^69 - 6 * q^71 + 8 * q^73 - 4 * q^75 + 9 * q^77 - q^79 + 4 * q^81 + 18 * q^83 + 4 * q^85 + q^87 - 24 * q^89 - 29 * q^91 + 11 * q^93 + 7 * q^95 - 23 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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